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<?xml version="1.0" encoding="ISO-8859-1" standalone="yes"?>
<archimedes xmlns:xlink="http://www.w3.org/1999/xlink"> <info>
<author>Cardano, Girolamo</author>
<title>Opus novum de proportionibus</title>
<date>1570</date>
<place>Basel</place>
<translator/>
<lang>la</lang>
<cvs_file>carda_propo_015_la_1570.xml</cvs_file>
<cvs_version/>
<locator>015.xml</locator>
</info> <text> <front> <section> <pb xlink:href="015/01/001.jpg"/><pb xlink:href="015/01/002.jpg"/><pb xlink:href="015/01/003.jpg"/><pb xlink:href="015/01/004.jpg"/><p type="head">
<s id="id2648798">HIERONYMI <lb/>CARDANI MEDIO <lb/>LANENSIS, CIVISQV'E BONO<lb/>NIENSIS, PHILOSOPHI, MEDICI ET <lb/>Mathematici clarisimi,</s></p><p type="head">
<s id="id2644031">OPVS NOVVM DE <lb/>PROPORTIONIBVS NVMERORVM, MO<lb/>TVVM, PONDERVM, SONORVM, ALIARVMQV'E RERVM <lb/>menurandarum, non olm Geometrico more tabilitum, ed etiam <lb/>uarijs experimentis & oberuationibus rerum in natura, olerti <lb/>demontratione illutratum, ad multiplices uus ac<lb/>commodatum, & in V libros digetum.</s></p><p type="head">
<s id="id2648832">PRAETEREA.</s></p><p type="head">
<s id="id2648840">ARTIS MAGN, SIVE DE REGVLIS <lb/>ALGEBRAICIS, LIBER VNVS, ABSTRVSISSIMVS <lb/>& inexhautus plane totius Arithmetic theaurus, ab <lb/>authore recens multis in locis recogni<lb/>tus & auctus.</s></p><p type="head">
<s id="id2648848">ITEM.</s></p><p type="head">
<s id="id2648856">DE ALIZA REGVLA LIBER, HOC EST, ALGEBRAICAE <lb/>logitic u, numeros recondita numerandi ubtilitate, ecundum Geo<lb/>metricas quantitates inquirentis, necearia Coronis, <lb/>nunc demum in lucem edita.</s></p><p type="head">
<s id="id2643963">O<emph type="italics"/>pus<emph.end type="italics"/> P<emph type="italics"/>hyicis &<emph.end type="italics"/> M<emph type="italics"/>athematicis imprimis <lb/>utile & necearium.<emph.end type="italics"/></s></p><p type="head">
<s id="id2644053">Cum C. </s>
<s id="id2644061">Maiet. </s>
<s id="id2644068">Gratia & Priuilegio.</s></p><p type="head">
<s id="id2644073">BASILE.</s></p></section><section><pb xlink:href="015/01/005.jpg"/><pb xlink:href="015/01/006.jpg"/><p type="head">
<s id="id2644094">IN LIBRVM DE <lb/>PROPORTIONIBVS HIERONYMI <lb/>CARDANI MEDIOLANENSIS, CIVISQV'E <lb/>Bononienis, Medici, Prfatio ad M. A. </s>
<s id="id2644116">Amulium <lb/>Venetum Card. </s>
<s id="id2644123">Illutrisimum.</s></p><p type="main">
<s id="id2643768">Bene Dictum et meo iudicio Platone M. <lb/>A. </s>
<s id="id2643781">Amuli optime, beatas fore Repub. </s>
<s id="id2643788">i uel <lb/>illarum domini apienti amatores eent, <lb/>aut qui apienti eent amatores domina<lb/>rentur, hoc ipum clar intelligens, tudio a <lb/>pienti nihil ee utilius humano generi: <lb/>quo imul & pietas, & iutitia, & mutuus <lb/>amor hominum inter e & eorum commo<lb/>da continerentur. </s>
<s id="id2643860">Nempe hice quatuor tota notra felicitas com<lb/>prehenditur. </s>
<s id="id2643874">Si quidem pietate in Deos nihil nii anctum, & pu<lb/>rum, & illutre apimus: hoc ipo primum quod upra nos et, intel<lb/>ligimus, Deos ueneramur, gratias agimus, timor cum ueneratione <lb/>notros animos ubit, & de futura uita cogitamus, hc ipa morta<lb/>lia i non negligentes altem paruifacientes. </s>
<s id="id2645038">Iutitiam autem ade <lb/>neceariam humano generi ee cimus, ut ine illa neque ee, nedum <lb/>ben ee posmus, ut neque latronum ctus abque ea diu tare po<lb/>int. </s>
<s id="id2644555">Porr quid dicam de concordia, & mutua hominum beneuo<lb/>lentia, in quibus omnis uit human dulcedo repoita et: nec quis <lb/>utineat uiuere, qui e omnibus odioum ee entiat. </s>
<s id="id2644464">His ipis fi<lb/>lios in pem alimus, parentes fouemus, fratres tuemur, & adiuua<lb/>mus, amicis opitulamur, cum hominibus hilarem & iucundam ui<lb/>tam ducimus. </s>
<s id="id2644490">Si quis erpentem in lecto haberet, nunquam om<lb/>num caperet: ita nihil moletius et in hac uita, quam ee cum quo <lb/>nolis, & priuari conuetudine eorum cum quibus maxim uiuere <lb/>cupias. </s>
<s id="id2644352">Quid enim habent Principes prcipuum cum tota illa po<lb/>tentia quam habent, nii hoc unum, quod uis quos amant bene fa<lb/>cere posint: nam reliqua omnia exerceri, uenari, edere, bibere, dor<lb/>mire, iter agere, loca amna inuiere multis alijs conceum et, ma<lb/>ioreque commodo qui in uita priuata degunt. </s>
<s id="id2644400">Si ergo principatum <lb/>cum tot laboribus, curis, periculis, & merit omnes appetunt: nec <lb/>et in eo quicquam prcipuum prter hoc, cui dubium et quin <lb/>hoc non it ummum huius uit hominibus bonum? </s>
<s id="id2645043">propter cu<lb/>ius uel dubiam pem eorum, qu habent obliti mortales pericli<lb/>tantur. </s>
<s id="id2645062">Succedunt inde tot commoda, non olum utilia, ed pleraque<pb xlink:href="015/01/007.jpg"/>etiam necearia, qu nos apientia docet: huiumodi ergo omnia <lb/>cm libris contineantur, merit optimus quique librorum bono<lb/>rum perpetuitati atque in columitati fauere debet. </s>
<s id="id2645109">C. </s>
<s id="id2645115">Caligulam exe<lb/>cramur olum ob id quod Vergilij, & T. </s>
<s id="id2645125">Liuij cripta delere cogi<lb/>tauerit. </s>
<s id="id2645136">Quid facturi eemus, i feciet quod cogitauerat? </s>
<s id="id2645152">Et in a<lb/>pientum monumentis bonum ine malo, mens ine corporea labe: <lb/>Virtutes abque uitijs, grati & iucunditas ine orde, & immundi<lb/>tia, uoluptas ine dolore, conueratio abque tdio, deliti abque mie <lb/>ria nuda, omnia bona prtant, atque laudabilia ab omnibus morta<lb/>litatis exuuijs libera, tantum commodi afferunt libri. </s>
<s id="id2644200">Sed & in eo<lb/>rum electione ac tudijs modus, ac medio critas qudam eruanda <lb/>et, qu i quis neglexerit non leui incommodo afficietur: eam an<lb/>tiqui rationem alij proportionem appellarunt, non equidem etiam <lb/>in pertritis tam <expan abbr="facillimã">facillimam</expan>, ut rentur homines: nam in alijs rebus per<lb/>obcuram ee fatentur, ego difficillimam puto undique, & magis for <lb/>an ubi non exitimamus. </s>
<s id="id2644267">Vnde plures decidere uidemus magnis <lb/>cum auxilijs, & euidenti pe: quid aliud et in caua qum ignota <lb/>menura rerum? </s>
<s id="id2644292">quam tamen plerique tenere e putant. </s>
<s id="id2644299">Ergo, cm <lb/>ummum bonum in hac menura itum ee cernerem, ut clar oten <lb/>dunt muic uoces, qu non nii indiuiduo (ut ita dicam) pacio <lb/>eu loco tare pount, ita & in figuris picturarum & tatuarum, & <lb/>diebus decretorijs, & negocijs ciuilibus operprecium me factu<lb/>rum exitimaui, i omnia hc qu lat patebant breuiter in unum <lb/>redegiem, <expan abbr="nõ">non</expan> tantum ne lectorem tdio afficerem, qum ut qud <lb/>alis do cui, breuibus tractationibus, & plura continerentur, & faci <lb/>lius docerentur. </s>
<s id="id2644675">Cum uer bona fortuna qudam effeciet, ut tibi <lb/>libellum dedicaem de Prouidentia ex contitutione temporum, <lb/>longe meliore occaione nominis tui typographi obliti int, indi<lb/>gnum fore putaui, ut non rea (quemadmodum cum Glauco Dio<lb/>medes) cum aureis commutarem. </s>
<s id="id2644719">Itaque infinitis licet circumuentus <lb/>negocijs totus huic oper in cubui, atque ade ut prter pem unius <lb/>anni pen pacio liber abolueretur. </s>
<s id="id2644750">Qui cum tibi (ut dixi) iam iur <lb/>deberetur, e tamen magis dedicandum putaui, quod non ego o<lb/>lum quanquam id maxim, ed communis conenus ho<lb/>minum exitimet, te ingulari uirtute omnibus <lb/>tudiois plurimum fauere, <lb/>Vale.</s></p></section><section><pb xlink:href="015/01/008.jpg"/><p type="head">
<s id="id2644817">TABVLA PRO<lb/>POSITIONVM DE <lb/>PROPORTIONIBVS.<lb/><arrow.to.target n="table1"/></s></p><table><table.target id="table1"/><row><cell>I.</cell><cell>Proportionem <emph type="italics"/>in proportionem duci, et uperiores numeros atque inferiores inuicem ducere.<emph.end type="italics"/></cell><cell><emph type="italics"/>pagina<emph.end type="italics"/> 6</cell></row><row><cell>II.</cell><cell>P<emph type="italics"/>roportio extremorum producitur ex intermedijs.<emph.end type="italics"/></cell><cell>7</cell></row><row><cell>III.</cell><cell>S<emph type="italics"/>i proportio ex duabus proportionibus in quatuor terminis producatur, ipa uer proportio inter duas alias quantitates fuerit contituta: conurgent trecen-ti exaginta modi productionis proportionis.<emph.end type="italics"/></cell><cell>7</cell></row><row><cell>IIII.</cell><cell>S<emph type="italics"/>i fuerit proportio primi ad ecundum, producta ex proportionibus tertij ad quartum, & quinti ad extum, producetur etiam ex proportione tertij ad extum, & quinti ad quartum.<emph.end type="italics"/></cell><cell>8</cell></row><row><cell>V.</cell><cell>S<emph type="italics"/>i fuerit proportio primi ad ecundum, producta ex proportione tertij ad quartum, & quinti ad extum: erit proportio tertij ad extum, producta ex proportionibus primi ad ecur dum, & quarti ad quintum.<emph.end type="italics"/></cell><cell>8</cell></row><row><cell>VI.</cell><cell>E<emph type="italics"/>x trecentis exaginta modis producendarum proportionum triginta ex tantum ee necearios.<emph.end type="italics"/></cell><cell>9</cell></row><row><cell>VII.</cell><cell>I<emph type="italics"/>n modis qui neceari producuntur ex duabus proportionibus, cum du quantitates ex illis qu modos conficiunt, quales fuerint: proportio producta ad quatuor quanti-tates omiologas reducetur.<emph.end type="italics"/></cell><cell>10</cell></row><row><cell>VIII.</cell><cell>S<emph type="italics"/>i duarum proportionum uperiores numeri alternatim cum inferioribus multiplicen-tur atque coniungantur, erit proportio aggregati ad productum ex inferioribus in-uicem proportio, ex primis proportionibus compoita.<emph.end type="italics"/></cell><cell>11</cell></row><row><cell>IX.</cell><cell>S<emph type="italics"/>i duarum proportionum uperiores numeri alternatim cum inferioribus multiplicen-tur, minusque productum ex maiore detrahatur, erit reidui ad productum ex ine-rioribus proportio uelut illa, qu relinquitur detracta minore proportione ex ma-iore.<emph.end type="italics"/></cell><cell>11</cell></row><row><cell>X.</cell><cell>S<emph type="italics"/>i fuerit alicuius quantitatis ad unam partem proportio, uelut alterius partis ad ecun-dam quantitatem, erit proportio cuiuuis quantitatis eiudem generis ad ecundam compoita proportio, ex proportionibus eiudem quantitatis, aumpt ad utranque partem prim quantitatis eorum.<emph.end type="italics"/></cell><cell>11</cell></row><row><cell>XI.</cell><cell>P<emph type="italics"/>roportio aggregati quarumlibet duarum quantitatum ad aggregatum duarum qua-lium <expan abbr="quantitatũ">quantitatum</expan> et, compoita ex proportionibus primis, & diuia per duplam.<emph.end type="italics"/></cell><cell>12</cell></row><row><cell>XII.</cell><cell>P<emph type="italics"/>ropoitis duabus proportionibus unam alteri iungere abque multiplicatione.<emph.end type="italics"/></cell><cell>12</cell></row><row><cell>XIII.</cell><cell>P<emph type="italics"/>roportio confua aggregata prim & terti quatuor quantitatum omiologarum ad aggregatum ecund & quart, et uelut compoita ex eidem diuia per du-plam.<emph.end type="italics"/></cell><cell>13</cell></row><row><cell>XIIII.</cell><cell>P<emph type="italics"/>roportiones confu & coniunct in tribus quantitatibus inuicem commutantur.<emph.end type="italics"/></cell><cell>13</cell></row><row><cell>XV.</cell><cell>S<emph type="italics"/>i fuerint quatuor quantitates proportio confua, aggregati prim & terti, ad aggre-gatum ecund & quart, erit ut monadis addito prouentu, qui fit diuia differentia, differentiarum prim & ecund, atque quart & terti, per aggregatum terti & quart ad ipam monadem.<emph.end type="italics"/></cell><cell>14</cell></row><row><cell>XVI.</cell><cell>O<emph type="italics"/>mnium quatuor quantitatum propoita prima, qu non minorem habet proportio-nem ad uam correpondentem qum alia ad aliam, erit proportio confua illarum,<emph.end type="italics"/></cell><cell/></row><pb xlink:href="015/01/009.jpg"/><row><cell/><cell><emph type="italics"/>ut producti ex aggregato prim & terti, in tertiam ad productum ex iggre gato terti & omiotat ad ecundam in ipam quartam.<emph.end type="italics"/></cell><cell>14</cell></row><row><cell>XVII.</cell><cell>O<emph type="italics"/>mnes du proportiones conuer producunt qualem proportionem.<emph.end type="italics"/></cell><cell>15</cell></row><row><cell>XVIII.</cell><cell>S<emph type="italics"/>i fuerint quotlibet quantitates in continua proportione multiplici prter, <expan abbr="ultimã">ultimam</expan> proportio uer penultim ad ultimam, qualis reidui prim ad ecundam, erit prim ad aggregatum reliquarum, uelut penultim ad ultimam.<emph.end type="italics"/></cell><cell>15</cell></row><row><cell>XIX.</cell><cell>S<emph type="italics"/>i fuerint aliquot quantitates arithmetic omiolog, quarum exceus it qualis minim, omnibus autem deficientibus upplementa ad qualitatem maxim adiungantur, erunt quadrata omnium quantitatum qualium, adiecto rurus quadrato prim cum eo quod fit ex minima primi ordinis in aggregatum o-mnium quantitatum eiudem, tripla aggregato quadratorum omnium quanti tatum primi ordinis pariter acceptis.<emph.end type="italics"/></cell><cell>17</cell></row><row><cell>XX.</cell><cell>C<emph type="italics"/>um fuerint quatuor quantitates, fueritque <expan abbr="&longs;ecũda">ecunda</expan> qualis terti, aut prima qualis quart, erit proportio prim ad quartam, aut terti ad ecundam, producta ex proportionibus prim ad ecundam & terti ad quartam.<emph.end type="italics"/></cell><cell>21</cell></row><row><cell>XXI.</cell><cell>C<emph type="italics"/>um decuatim ducta fuerit prima in quartam, & ecunda in tertiam, produ-ctumque prim in quartam, diuium fuerit per productum ecund in tertiam, erit proportio prim ad ecundam, diuia per proportonem terti ad quar-tam.<emph.end type="italics"/> E<emph type="italics"/>t imiliter interpoita omiologa.<emph.end type="italics"/></cell><cell>22</cell></row><row><cell>XXII.</cell><cell>C<emph type="italics"/>um fuerit proportio prim ad ecundam maior qum terti ad quartam, erit confua ex his maior qum terti ad quartam, minor autem qum prim ad ecundam.<emph.end type="italics"/></cell><cell>23</cell></row><row><cell>XXIII.</cell><cell>O<emph type="italics"/>mnis motus naturalis ad locum uum et: ide per rectam lineam fit.<emph.end type="italics"/></cell><cell>23</cell></row><row><cell>XXIIII.</cell><cell>O<emph type="italics"/>mnis motus circularis uoluntarius et.<emph.end type="italics"/></cell><cell>23</cell></row><row><cell>XXV.</cell><cell>T<emph type="italics"/>res unt motus omnino implices naturalis, uoluntarius, & uiolentus.<emph.end type="italics"/></cell><cell>24</cell></row><row><cell>XXVI.</cell><cell>M<emph type="italics"/>otus ergo compoiti quatuor neceari unt pecies.<emph.end type="italics"/></cell><cell>24</cell></row><row><cell>XXVII.</cell><cell>M<emph type="italics"/>otus uoluntarius et in loco: naturalis ad locum: uiolentus ex loco.<emph.end type="italics"/></cell><cell>25</cell></row><row><cell>XXVIII.</cell><cell>M<emph type="italics"/>otus quilibet uoluntarius aut uiolentus in aliquo medio fit.<emph.end type="italics"/></cell><cell>25</cell></row><row><cell>XXIX.</cell><cell>O<emph type="italics"/>mnis motus uoluntarius qualis et emper: impliciter etiam quilibet alius mo-tus.<emph.end type="italics"/></cell><cell>25</cell></row><row><cell>XXX.</cell><cell>I<emph type="italics"/>n omni corpore mobili in medio partes medij reitunt obui, ali impel-lunt.<emph.end type="italics"/></cell><cell>26</cell></row><row><cell>XXXI.</cell><cell>O<emph type="italics"/>mnis motus naturalis in quali medio ualidior et in fine qum in principio.<emph.end type="italics"/>V<emph type="italics"/>iolentus contr.<emph.end type="italics"/></cell><cell>26</cell></row><row><cell>XXXII.</cell><cell>O<emph type="italics"/>mne mobile naturaliter motum eu uiolenter uelocius mouetur in medio rariore qum deniore.<emph.end type="italics"/> M<emph type="italics"/>aior quoque et proportio finis motus in corpore rariore ad finem motus in corpore deniore qum principij.<emph.end type="italics"/> I<emph type="italics"/>n uiolento autem celerius perueniret ad finem motus in corpore deniore.<emph.end type="italics"/></cell><cell>27</cell></row><row><cell>XXXIII.</cell><cell>O<emph type="italics"/>mnia duo mobilia qualis undique magnitudinis qu quali in tempore qualia pacia pertraneunt in diueris ubtantia medijs necee et, ut it ponderis ad pondus, quem ad modum medij ad medium proportio duplicata.<emph.end type="italics"/></cell><cell>27</cell></row><row><cell>XXXIIII.</cell><cell>P<emph type="italics"/>roportio corporis cubi ad uam uperficiem quadratam, et uelut eiudem uperfi ciei, ad latus eiudem uer ad monadem.<emph.end type="italics"/></cell><cell>28</cell></row><row><cell>XXXV.</cell><cell>V<emph type="italics"/>ocum magnitudines excrecunt in acumine, non in grauitate, finis autem et in utroque extremo.<emph.end type="italics"/> P<emph type="italics"/>ropter hoc minima facta uariatione in hypate acut uix ferunt.<emph.end type="italics"/></cell><cell>29</cell></row><row><cell>XXXVI.</cell><cell>S<emph type="italics"/>i proportio per proportionem minorem quali ducatur, proportio minor pro-<emph.end type="italics"/></cell><cell/></row><pb xlink:href="015/01/010.jpg"/><row><cell/><cell><emph type="italics"/>ducetur.<emph.end type="italics"/> V<emph type="italics"/>nde manifetum et duas proportiones minores qualitate <expan abbr="inuic&etilde;">inuicem</expan> du ctas proportionem minorem unaquaque illarum producere.<emph.end type="italics"/></cell><cell>30</cell></row><row><cell>XXXVII.</cell><cell>S<emph type="italics"/>i plures homines, quorum per e nauim mouere point, aut pondus ferre imul iuncti eam moueant, aut pondus ferant, erunt ill proportiones coniunct non product.<emph.end type="italics"/></cell><cell>30</cell></row><row><cell>XXXVIII.</cell><cell>O<emph type="italics"/>mne corpus tantum reitit motui contrario uo natrali, quantum mouetur oc-culto motu quiecendo.<emph.end type="italics"/></cell><cell>31</cell></row><row><cell>XXXIX.</cell><cell>A<emph type="italics"/>b quali aut minore ui qum it impedimentum non fit motus.<emph.end type="italics"/></cell><cell>31</cell></row><row><cell>XL.</cell><cell>O<emph type="italics"/>mne corpus pb ricum tangens planum in puncto mouetur ad latus per quam-cunque uim, qu medium diuidere potet.<emph.end type="italics"/></cell><cell>31</cell></row><row><cell>XLI.</cell><cell>S<emph type="italics"/>i fuerint du quantitates umaturque toties <expan abbr="aggregatũ">aggregatum</expan> maioris & minoris, quo-ties aggregatum minoris & maioris, erit proportio confua maioris aggregati ad minus, minor quam multiplicis maioris ad multiplex minoris.<emph.end type="italics"/></cell><cell>32</cell></row><row><cell>XLII.</cell><cell>T<emph type="italics"/>rahentium nauim, aut ferentium pondera proportiones in e inuicem, quomodo ducere oporteat coniderare.<emph.end type="italics"/></cell><cell>32</cell></row><row><cell>XLIII.</cell><cell>P<emph type="italics"/>roductionem ad additionem retrabere.<emph.end type="italics"/></cell><cell>33</cell></row><row><cell>XLIIII.</cell><cell>S<emph type="italics"/>i fuerit proportio motoris ad id quod et maximum non mouens, & pacium & tempus, nota erit etiam reliquorum nota.<emph.end type="italics"/></cell><cell>33</cell></row><row><cell>XLV.</cell><cell>R<emph type="italics"/>ationem tater otendere.<emph.end type="italics"/></cell><cell>34</cell></row><row><cell>XLVI.</cell><cell>A<emph type="italics"/>n it aliqua proportio & qualis inter animam & uitas, & ua corpora conide-rare.<emph.end type="italics"/></cell><cell>35</cell></row><row><cell>XLVII.</cell><cell>S<emph type="italics"/>i duo mobilia qualister in eodem circulo iuxta proprios motus moueantur, pro-ductum temporis circuituum inuicem, erit quale producto differenti tempo rum circuitus duct in tempus coniunctionis prim.<emph.end type="italics"/></cell><cell>36</cell></row><row><cell>XLVIII.</cell><cell>S<emph type="italics"/>i tria mobilia ex eodem puncto dicedant, fuerintque duorum ac duorum coniun-ctiones in temporibus commenis, illa tria mobilia denuo coniungentur in tem pore producto ex denominatore diuiionis temporis maioris per minus in mi-nus aut numeratore in maius.<emph.end type="italics"/></cell><cell>37</cell></row><row><cell>XLIX.</cell><cell>P<emph type="italics"/>ropofitio mobilis in circulo circuitus tempore dataque ratione ditanti ab illo mo bilis circuitum inuenire, quod ex <expan abbr="eod&etilde;">eodem</expan> puncto dicedens <expan abbr="cũalio">cunalio</expan> mobili in dato puncto <expan abbr="cõueniat">conueniat</expan> ub <expan abbr="quocũque">quocunque</expan> numero <expan abbr="circuituũ">circuituum</expan> <expan abbr="t&etilde;pus">tempus</expan> quoque <expan abbr="cõiunctionis">coniunctionis</expan>.<emph.end type="italics"/></cell><cell>39</cell></row><row><cell>L.</cell><cell>O<emph type="italics"/>mnes circuituum portiones in eidem temporibus repetuntur.<emph.end type="italics"/></cell><cell>40</cell></row><row><cell>LI.</cell><cell>O<emph type="italics"/>perationes dictas exemplo declarare.<emph.end type="italics"/></cell><cell>41</cell></row><row><cell>LII.</cell><cell>T<emph type="italics"/>ria mobilia coniuncta in <expan abbr="eod&etilde;">eodem</expan> puncto, quorum duo & duo conueniant in partib. incommenis inter e, in perpetuum in nullo unquam puncto conuenient.<emph.end type="italics"/></cell><cell>42</cell></row><row><cell>LIII.</cell><cell>C<emph type="italics"/>irculorum e in aduerum mouentium proportionem declarare.<emph.end type="italics"/></cell><cell>43</cell></row><row><cell>LIIII.</cell><cell>P<emph type="italics"/>roportio circuli ad uum diametrum per imilitudinem et quarta pars periphe-ri.<emph.end type="italics"/> R<emph type="italics"/>urusque eiudem circuli ad peripheriam diametri quarta pars.<emph.end type="italics"/></cell><cell>44</cell></row><row><cell>LV.</cell><cell>P<emph type="italics"/>roportionem medicamentorum per ordines up poita quali proportione in or-dinibus per quantitates & proportiones demontrare.<emph.end type="italics"/></cell><cell>44</cell></row><row><cell>LVI.</cell><cell>P<emph type="italics"/>roportio cuiuuis binomij ad uum recium, uel ei commenum et duplicata ei qu ad numeri latus.<emph.end type="italics"/></cell><cell>49</cell></row><row><cell>LVII.</cell><cell>M<emph type="italics"/>otus rationem ad pondus inuenire.<emph.end type="italics"/></cell><cell>49</cell></row><row><cell>LVIII.</cell><cell>Q<emph type="italics"/>u ex alto decendunt, cur non eandem pro ditantia motus rationem in libero are eruent coniderare.<emph.end type="italics"/></cell><cell>49</cell></row><row><cell>LIX.</cell><cell>O<emph type="italics"/>mne mobile motum duobus motibus non ad idem tendentibus utroque eorum tar dius mouetur imili motu.<emph.end type="italics"/></cell><cell>50</cell></row><row><cell>LX.</cell><cell>O<emph type="italics"/>mne mobile motu naturali decendentis parte, decendit grauiore ecundum gra-<emph.end type="italics"/></cell><cell/></row><pb xlink:href="015/01/011.jpg"/><row><cell/><cell><emph type="italics"/>uitatis centrum.<emph.end type="italics"/></cell><cell>51</cell></row><row><cell>LXI.</cell><cell>P<emph type="italics"/>roportionum ictus ad pondus rei & ditantiam generaliter coniderare.<emph.end type="italics"/></cell><cell>52</cell></row><row><cell>LXII.</cell><cell>P<emph type="italics"/>roportionem motoris in plano ad motorem, qui eleuat pondus iuxta id quod mouet, inuenire.<emph.end type="italics"/></cell><cell>53</cell></row><row><cell>LXIII.</cell><cell>O<emph type="italics"/>mne graue quanto proximius alligatum plano, tant facilius trabitur.<emph.end type="italics"/></cell><cell>53</cell></row><row><cell>LXIIII.</cell><cell>O<emph type="italics"/>mne mobile quant latius tanto tardius moustur in plano.<emph.end type="italics"/></cell><cell>54</cell></row><row><cell>LXV.</cell><cell>P<emph type="italics"/>roportionem duorum mobilium inter e cum auxilio medij inuenire.<emph.end type="italics"/></cell><cell>54</cell></row><row><cell>LXVI.</cell><cell>P<emph type="italics"/>roportionem laterum eptagoni, & ubtenarum coniderare, & qu reflexa proportione pendent.<emph.end type="italics"/></cell><cell>55</cell></row><row><cell>LXVII.</cell><cell>S<emph type="italics"/>i fuerint aliquot quantitates ab una quantitate alique totidem ab eadem analo-g, erit proportio terti unius ordinis ad tertiam alterius, ut ecund ad e-cundum duplicata, & quart ad quartam triplicata, quint ad quintam quadruplicata, atque ic de alijs.<emph.end type="italics"/></cell><cell>57</cell></row><row><cell>LXVIII.</cell><cell>P<emph type="italics"/>ropoitio collectorum ab<emph.end type="italics"/> E<emph type="italics"/>uclide &<emph.end type="italics"/> A<emph type="italics"/>rchimede.<emph.end type="italics"/></cell><cell>57</cell></row><row><cell>LXIX.</cell><cell>P<emph type="italics"/>ropoitio collectorum ex quatuor libris<emph.end type="italics"/> A<emph type="italics"/>pollonij<emph.end type="italics"/> P<emph type="italics"/>ergei &<emph.end type="italics"/> <expan abbr="q.">que</expan> S<emph type="italics"/>ereni.<emph.end type="italics"/></cell><cell>59</cell></row><row><cell>LXX.</cell><cell>S<emph type="italics"/>Si fuerint tres quantitates in continua proportione, alique totidem in continua proportione poterunt contituere tres quantitates in quali differentia per-uerim copulat.<emph.end type="italics"/></cell><cell>62</cell></row><row><cell>LXXI.</cell><cell>P<emph type="italics"/>roportionem leuitatis ponderis per uirgam torcularem attracti ad rectam u-penionem inuenire.<emph.end type="italics"/></cell><cell>63</cell></row><row><cell>LXXII.</cell><cell>P<emph type="italics"/>roportionem ponderis phr pendentis ad acendentem per accliue planum inuenire.<emph.end type="italics"/></cell><cell>63</cell></row><row><cell>LXXIII.</cell><cell>P<emph type="italics"/>roportionem ponderum attractorum penes figuram in plano inuenire.<emph.end type="italics"/></cell><cell>64</cell></row><row><cell>LXXIIII.</cell><cell>P<emph type="italics"/>roportionem concutientis ad concuum intabili inuenire.<emph.end type="italics"/></cell><cell>64</cell></row><row><cell>LXXV.</cell><cell>P<emph type="italics"/><expan abbr="roportion&etilde;">roportionem</expan> immoti in aqua, ad <expan abbr="immotũ">immotum</expan> in terra in excipiendo <expan abbr="ictũ">ictum</expan> inuenire.<emph.end type="italics"/></cell><cell>65</cell></row><row><cell>LXXVI.</cell><cell>P<emph type="italics"/>roportionem <expan abbr="duorũ">duorum</expan> mobilium ibi <expan abbr="inuic&etilde;">inuicem</expan> <expan abbr="concurrentiũ">concurrentium</expan> per <expan abbr="rectã">rectam</expan> inuenire.<emph.end type="italics"/></cell><cell>66</cell></row><row><cell>LXXVII.</cell><cell>P<emph type="italics"/>roportionem motus obliqui ad motum rectum in nauibus inuenire.<emph.end type="italics"/></cell><cell>66</cell></row><row><cell>LXXVIII.</cell><cell>P<emph type="italics"/>roportionem nauis ad triremes quotuis concurrentes demontrare.<emph.end type="italics"/></cell><cell>67</cell></row><row><cell>LXXIX.</cell><cell>P<emph type="italics"/>roportionem medicamentorum purgantium inuicem declarare<emph.end type="italics"/></cell><cell>68</cell></row><row><cell>LXXX.</cell><cell>P<emph type="italics"/>roportionem motus ecundum obliquum ad rectum in pacio declarare.<emph.end type="italics"/></cell><cell>69</cell></row><row><cell>LXXXI.</cell><cell>Q<emph type="italics"/>uualis it angulus, per quem potet moueri nauis ad rectum explorare.<emph.end type="italics"/></cell><cell>70</cell></row><row><cell>LXXXII.</cell><cell>P<emph type="italics"/>roportionem uelorum indagare.<emph.end type="italics"/></cell><cell>70</cell></row><row><cell>LXXXIII.</cell><cell>P<emph type="italics"/>roportionem receus recta uia ad obliquitatem inuetigare.<emph.end type="italics"/></cell><cell>72</cell></row><row><cell>LXXXIIII.</cell><cell>D<emph type="italics"/><expan abbr="i&longs;tantiã">itantiam</expan> centri terr centro mundi per motum lapidis<emph.end type="italics"/> H<emph type="italics"/>erculei declarare.<emph.end type="italics"/></cell><cell>73</cell></row><row><cell>LXXXV.</cell><cell>P<emph type="italics"/>roportio ponderis unius grauis ad aliud ub eadem menura et ueluti eiudem ad differentiam ponderis uais repleti ex altero graui, & ex ambobus de-tracto priore.<emph.end type="italics"/></cell><cell>74</cell></row><row><cell>LXXXVI.</cell><cell>S<emph type="italics"/>i circuli in quales eu in phra eu in plano e ecuerint, nunqum oppoitos angulos quales habent.<emph.end type="italics"/></cell><cell>77</cell></row><row><cell>LXXXVII.</cell><cell>P<emph type="italics"/>roportiones craitiei aqu ad <expan abbr="a&etilde;r&etilde;">aerrem</expan> in <expan abbr="cõparatione">comparatione</expan> ad radios demontrare.<emph.end type="italics"/></cell><cell>78</cell></row><row><cell>LXXXVIII.</cell><cell>I<emph type="italics"/><expan abbr="n&longs;trumentũ">ntrumentum</expan><emph.end type="italics"/> A<emph type="italics"/>colingen, quo momenta temporum <expan abbr="deprehendãtur">deprehendantur</expan> fabricare.<emph.end type="italics"/></cell><cell>79</cell></row><row><cell>LXXXIX.</cell><cell>P<emph type="italics"/>roportionem denitatis aqu ad arem per pondera inuenire.<emph.end type="italics"/></cell><cell>82</cell></row><row><cell>XC.</cell><cell>R<emph type="italics"/>ationem impetus uiolenti extra mii ponderis ad qualitatem reducere.<emph.end type="italics"/></cell><cell>82</cell></row><row><cell>XCI.</cell><cell>P<emph type="italics"/>roportionem grauis cubi, & phrici qualium in accliui, & decenus eorum demontrare.<emph.end type="italics"/></cell><cell>83</cell></row><row><cell>XCII.</cell><cell>P<emph type="italics"/><expan abbr="roportion&etilde;">roportionem</expan> ponderis qualis iuxta longitudinis <expan abbr="cõparation&etilde;">comparationem</expan> demontrare.<emph.end type="italics"/></cell><cell>85</cell></row><row><cell>XCIII.</cell><cell>P<emph type="italics"/>ropter qd in <expan abbr="cõcußione">concuione</expan> <expan abbr="etiã">etiam</expan> leui nauis loco moueatar <expan abbr="o&longs;t&etilde;dere">otendere</expan>.<emph.end type="italics"/> V<emph type="italics"/>nde manifi <expan abbr="&longs;iũ">ium</expan> et duas naues ibi <expan abbr="inuic&etilde;">inuicem</expan> occurantes retrocedere, & <expan abbr="quãtũ">quantum</expan> <expan abbr="retrocedãt">retrocedant</expan> amb.<emph.end type="italics"/></cell><cell>86</cell></row><pb xlink:href="015/01/012.jpg"/><row><cell>XCIIII.</cell><cell>S<emph type="italics"/>i <expan abbr="quãtitas">quantitas</expan> aliqua nota atque proportio erit producta, <expan abbr="quãtitas">quantitas</expan> nota imiliter.<emph.end type="italics"/> E<emph type="italics"/>t i du proportiones not fuerint, erit producta ex his atque diuia coniunctaque atque detra-cta nota.<emph.end type="italics"/> E<emph type="italics"/>t i fuerit totius ad partem proportio nota, erit et ad aliam partem nota: & alterius partis ad <expan abbr="alterã">alteram</expan> uno minor.<emph.end type="italics"/> E<emph type="italics"/>t i fuerit partis ad partem, erit ad totum monade minor atque nota.<emph.end type="italics"/> E<emph type="italics"/>t i fuerit unius <expan abbr="quãtitatis">quantitatis</expan> ad duas <expan abbr="quãtitates">quantitates</expan> proportio nota, erit & <expan abbr="cõfu&longs;a">confua</expan> ex eis nota.<emph.end type="italics"/> E<emph type="italics"/>t i fuerint trium quantitatum omiologarum, aut quatuor analogarum omnes prter unam cognit, erunt & illa alia cognita.<emph.end type="italics"/></cell><cell>87</cell></row><row><cell>XCV.</cell><cell>C<emph type="italics"/>uiuuis trigoni rectanguli, aut cuius duo auguli int in dupla proportione, aut qui circulo incriptus it cognita quantitate unius lateris in comparatione ad dimetien <expan abbr="t&etilde;">tem</expan>, i proportio duorum laterum cognita fuerit, <expan abbr="erũt">erunt</expan> omnia eius latera cognita.<emph.end type="italics"/></cell><cell>88</cell></row><row><cell>XCVI.</cell><cell>C<emph type="italics"/>um in <expan abbr="per&longs;picuũ">perpicuum</expan> denum radij luminoi inciderint, quatuor fiunt luminis genera.<emph.end type="italics"/></cell><cell>89</cell></row><row><cell>XCVII.</cell><cell>M<emph type="italics"/><expan abbr="otũ">otum</expan> inuerionis in figuris in <expan abbr="cõparatione">comparatione</expan> ad <expan abbr="motũ">motum</expan> phr in plano inuetigare.<emph.end type="italics"/></cell><cell>91</cell></row><row><cell>XCVIII.</cell><cell>P<emph type="italics"/>roportionem ponderum qualium per differentiam angulorum inuenire.<emph.end type="italics"/></cell><cell>92</cell></row><row><cell>XCIX.</cell><cell>P<emph type="italics"/>roportionem grauitatum per multitudinem uppoitorum orbium otendere.<emph.end type="italics"/></cell><cell>93</cell></row><row><cell>C.</cell><cell>P<emph type="italics"/><expan abbr="roportion&etilde;">roportionem</expan> grauitatis <expan abbr="ponderũ">ponderum</expan> attractorum per <expan abbr="trochlearũ">trochlearum</expan> <expan abbr="numerũ">numerum</expan> inuetigare.<emph.end type="italics"/></cell><cell>93</cell></row><row><cell>CI.</cell><cell>P<emph type="italics"/>roportionem precij gemmarum ex tribus in eodem genere cognitis inuenire.<emph.end type="italics"/></cell><cell>94</cell></row><row><cell>CII.</cell><cell>P<emph type="italics"/>roportionem motuum inuerionis, & attractionis in plano inuenire.<emph.end type="italics"/></cell><cell>95</cell></row><row><cell>CIII.</cell><cell>P<emph type="italics"/>roportionem eorundem in accliui demontrare.<emph.end type="italics"/></cell><cell>95</cell></row><row><cell>CIIII.</cell><cell>P<emph type="italics"/>roportionem motus attractionis in decliui ad motum in plano determinare.<emph.end type="italics"/></cell><cell>95</cell></row><row><cell>CV.</cell><cell>P<emph type="italics"/>roportionem ferentium pondus in pertica inuenire.<emph.end type="italics"/></cell><cell>96</cell></row><row><cell>CVI.</cell><cell>Q<emph type="italics"/>uales proportiones angulorum doceant laterum proportiones.<emph.end type="italics"/> A<emph type="italics"/>tque uiciim deter-minare.<emph.end type="italics"/></cell><cell>97</cell></row><row><cell>CVII.</cell><cell>S<emph type="italics"/>i in circulo du diametri ad rectum angulum e ecauerint: ali uer ad perpendicu-lum ex diametro exicrint ad circum ferentiam, ingul upra diametrum erunt ma iores portionibus reliquis diametri uperioribus, infra autem minores.<emph.end type="italics"/> D<emph type="italics"/>imidium autem portionis uperioris reiduum ad centrum maius agitta habebit.<emph.end type="italics"/> I<emph type="italics"/>n aliqua prterea portionis uperioris parte, qu uerus diametrum tranuerum poita et, maior et differentia partis diametri ei <expan abbr="corre&longs;põdentis">correpondentis</expan>, <expan abbr="&qtilde;">quae</expan> line tranuer.<emph.end type="italics"/></cell><cell>100</cell></row><row><cell>CVIII.</cell><cell>P<emph type="italics"/>unctum qualitatis differenti decenus & remotionis centro inuenire.<emph.end type="italics"/></cell><cell>100</cell></row><row><cell>CIX.</cell><cell>R<emph type="italics"/>ationem libr expendere.<emph.end type="italics"/></cell><cell>101</cell></row><row><cell>CX.</cell><cell>S<emph type="italics"/>i du phr ex eadem materia decendant in are, eodem temporis momento ad planum ueniunt.<emph.end type="italics"/></cell><cell>104</cell></row><row><cell>CXI.</cell><cell>C<emph type="italics"/>ur ex medio tela ualidiorem ictum, & naues in calmo remo ac malo recipiant in-de ex puppi explorare.<emph.end type="italics"/></cell><cell>105</cell></row><row><cell>CXII.</cell><cell>C<emph type="italics"/>ur ex imo leuia longis ferantur declarare,<emph.end type="italics"/></cell><cell>106</cell></row><row><cell>CXIII.</cell><cell>C<emph type="italics"/>ur uirga longius mittatur puero quam uiro inueftigare.<emph.end type="italics"/></cell><cell>107</cell></row><row><cell>CXIIII.</cell><cell>C<emph type="italics"/>ircularis motus differentias quatuor ee, earumque rationem contemplari.<emph.end type="italics"/></cell><cell>108</cell></row><row><cell>CXV.</cell><cell>P<emph type="italics"/>roportionem motuum impulionis, & attractionis inter e, ab eadem ui decla-rare.<emph.end type="italics"/></cell><cell>110</cell></row><row><cell>CXVI.</cell><cell>C<emph type="italics"/>ur machin oblong igne longius emittant phram explorare.<emph.end type="italics"/></cell><cell>111</cell></row><row><cell>CXVII.</cell><cell>I<emph type="italics"/>n curriculis maior et uis pulueris copioioris ampliore in pacio, qum paucioris in minore iuxta proportionem eandem.<emph.end type="italics"/></cell><cell>112</cell></row><row><cell>CXVIII.</cell><cell>Q<emph type="italics"/>uanta proportione decedat ictus in obliquum parietem ab eo qui et ad perpendi-culum declarare.<emph.end type="italics"/></cell><cell>114</cell></row><row><cell>CXIX.</cell><cell>Q<emph type="italics"/>uantum ictus machin procliuis ad angulum minuatur explorare.<emph.end type="italics"/></cell><cell>115</cell></row><row><cell>CXX</cell><cell>P<emph type="italics"/>roportionem partium nauis ad eundem obliquum uentum explorare.<emph.end type="italics"/></cell><cell>118</cell></row><row><cell>CXXI.</cell><cell>F<emph type="italics"/>labelli uires atque naturam declarare.<emph.end type="italics"/></cell><cell>219</cell></row><row><cell>CXXII.</cell><cell>C<emph type="italics"/>ontemptus circa<emph.end type="italics"/> S<emph type="italics"/>olis rationem in umbris declarare.<emph.end type="italics"/></cell><cell>120</cell></row><pb xlink:href="015/01/013.jpg"/><row><cell>CXXIII.</cell><cell>C<emph type="italics"/>ognita ratione umbr ad gnomonem inum, & arcum altitudinis ab horizon-te, quouis tempore dignocere.<emph.end type="italics"/></cell><cell>121</cell></row><row><cell>CXXIIII.</cell><cell>P<emph type="italics"/>roportionem umbr uer ee ad gnomonem, uelut gnomonis ad umbram ueram.<emph.end type="italics"/></cell><cell>122</cell></row><row><cell>CXXV.</cell><cell>P<emph type="italics"/>roportionem dimetientis, & peripheri cuiuslibet circuli paralleli quino-ctiali per cognitam partem magni circuli demontrare.<emph.end type="italics"/></cell><cell>123</cell></row><row><cell>CXXVI.</cell><cell>C<emph type="italics"/>irculi horarij naturam declarare.<emph.end type="italics"/></cell><cell>123</cell></row><row><cell>CXXVII.</cell><cell>D<emph type="italics"/>ata poli altitudine ortus amplitudinem demonftrare.<emph.end type="italics"/></cell><cell>124</cell></row><row><cell>CXXVIII.</cell><cell>N<emph type="italics"/>ota amplitudine ortus, cuiuque puncti arcum emidiurnum inuenire.<emph.end type="italics"/></cell><cell>124</cell></row><row><cell>CXXIX.</cell><cell>D<emph type="italics"/>ata altitudine<emph.end type="italics"/> S<emph type="italics"/>olis in quacunque regione, quacunque die ditantiam<emph.end type="italics"/> S<emph type="italics"/>olis meri-diano cognocere.<emph.end type="italics"/></cell><cell>124</cell></row><row><cell>CXXX.</cell><cell>D<emph type="italics"/>ata regionis altitudine, & loco<emph.end type="italics"/> S<emph type="italics"/>olis proportionem gnomonis, tam ad um-bram rectam qum ueram, uel etiam in cylindro determinare.<emph.end type="italics"/></cell><cell>125</cell></row><row><cell>CXXXI.</cell><cell>S<emph type="italics"/>i line alicui duplum alterius adiungatur, erit proportio duarum ad primam maior qum dupli cum prima ad primam cum una adiecta.<emph.end type="italics"/></cell><cell>126</cell></row><row><cell>CXXXII.</cell><cell>S<emph type="italics"/>i ad duas lineas quarum una alteri dupla it eadem linea addatur, erit aggrega-ti ex minore, & adiecta ad ipam minorem, minor proportio qum aggre-gati ex maiore, & adiecta ad ipam maiorem duplicata.<emph.end type="italics"/></cell><cell>126</cell></row><row><cell>CXXXIII.</cell><cell>S<emph type="italics"/>i fuerint du quantitates, <expan abbr="quarũ">quarum</expan> una alteri dupla it: minuatur minore qu-dam quantitas, <expan abbr="ead&etilde;que">eadenque</expan> maiori addatur, erit minoris ad reiduum maior pro-portio, qum aggregati ad maiorem duplicata.<emph.end type="italics"/> S<emph type="italics"/>i uer minori addatur, & maiore detrabatur, erit aggregati ad minorem minor proportio qum maioris ad reiduum duplicata.<emph.end type="italics"/></cell><cell>127</cell></row><row><cell>CXXXIIII.</cell><cell>S<emph type="italics"/>i rectangula uperficies it, cuius pars tertia quadrata it corpus, quod ex la-tere quadrat in reiduum uperficiei contat, maius et quouis corpore ex eadem uperficies, aliter diuia contituto.<emph.end type="italics"/></cell><cell>127</cell></row><row><cell>CXXXV.</cell><cell>S<emph type="italics"/>i linea in duas partes, quarum una fit alteri dupla diuidatur, erit quod fit ex tertia parte in quadratum reidui parallelipedum maius omni pararalleli-pedo, quod ex diuiione eiudem line creari poit.<emph.end type="italics"/></cell><cell>128</cell></row><row><cell>CXXXVI.</cell><cell>D<emph type="italics"/>enominationes in infinitum extendere.<emph.end type="italics"/></cell><cell>129</cell></row><row><cell>CXXXVII.</cell><cell>R<emph type="italics"/>ationem numerorum ex progreione declarare.<emph.end type="italics"/></cell><cell>131</cell></row><row><cell>CXXXVIII.</cell><cell>M<emph type="italics"/>odos uus horum numerorum declarare.<emph.end type="italics"/></cell><cell>131</cell></row><row><cell>CXXXIX.</cell><cell>R<emph type="italics"/>adices omnes propoitis numeris extrahere.<emph.end type="italics"/></cell><cell>132</cell></row><row><cell>CXL.</cell><cell>R<emph type="italics"/>adices per numeros fractos determinare.<emph.end type="italics"/></cell><cell>133</cell></row><row><cell>CXLI.</cell><cell>N<emph type="italics"/>umeros fractos ad minores in ea <expan abbr="i&etilde;">iem</expan> proportione ualde propinqud deducere<emph.end type="italics"/></cell><cell>136</cell></row><row><cell>CXLII.</cell><cell>D<emph type="italics"/><expan abbr="enominationũ">enominationum</expan> in <expan abbr="crem&etilde;ta">crementa</expan> ex extrema cognita inuenire.<emph.end type="italics"/> E<emph type="italics"/>t <expan abbr="cõuer&longs;o">conuero</expan> modo.<emph.end type="italics"/></cell><cell>137</cell></row><row><cell>CXLIII.</cell><cell>S<emph type="italics"/>i linea in duas partes diuidatur, corpora qu fiunt ex una parte in alterius quadratum mutuo qualia unt corpori, quod fit ex tota linea in uperfi-ciem unius partis in alteram.<emph.end type="italics"/></cell><cell>138</cell></row><row><cell>CXLIIII.</cell><cell>D<emph type="italics"/>uplum cubi medietatis maius et aggregato corporum mutuorum, cuiuslibet diuiionis quantum et, quod fit ex tota in quadratum differenti.<emph.end type="italics"/></cell><cell>139</cell></row><row><cell>CXLV.</cell><cell>S<emph type="italics"/>i linea in duas partes diuidatur quadrata ambarum partium detracto eo, quod fit ex una parte in alteram, qualia unt producto unius in alteram cum quadrato differenti.<emph.end type="italics"/></cell><cell>139</cell></row><row><cell>CXLVI.</cell><cell>C<emph type="italics"/>orpus quod fit ex linea diuia in uperficiem qualem quadratis ambarum par tium detracta uperficie unius partis in alteram, et quale aggregato cubo-rum ambarum partium.<emph.end type="italics"/></cell><cell>139</cell></row><row><cell>CXLVII.</cell><cell>P<emph type="italics"/>ropoita linea diuia duas ei line as adijcere, ut proportio <expan abbr="additarũ">additarum</expan> ingularium<emph.end type="italics"/></cell><cell/></row><pb xlink:href="015/01/014.jpg"/><row><cell/><cell><emph type="italics"/>& partium imul iunctarum ad additas it mutua.<emph.end type="italics"/></cell><cell>148</cell></row><row><cell>CXLVIII.</cell><cell>P<emph type="italics"/>ropoitis tribus lineis primam ic diuidere, ut adiectis duabus alijs lineis, ecun-dum <expan abbr="ration&etilde;">rationem</expan> mutuam ingularum ingulis, <expan abbr="aggregatũ">aggregatum</expan> ex una <expan abbr="adiectarũ">adiectarum</expan>, & par te ad <expan abbr="aggregatũ">aggregatum</expan> ex alia parte, & adiecta e habeat, ut ecunda ad <expan abbr="tertiã">tertiam</expan>.<emph.end type="italics"/></cell><cell>140</cell></row><row><cell>CXLIX.</cell><cell>D<emph type="italics"/>atam lineam ic diuidere, ut proportio quadratorum ad dupium unius partis in alteram it, ut line dat ad lineam datam.<emph.end type="italics"/></cell><cell>141</cell></row><row><cell>CL.</cell><cell>P<emph type="italics"/>ropoitis duabus lineis, lineam communem utrique adiungere, ut it maioris ad ad-ditam proportio, uelut quadratorum minoris, & adiect ad duplum unius in alteram.<emph.end type="italics"/></cell><cell>141</cell></row><row><cell>CLI.</cell><cell>P<emph type="italics"/>roportio differenti quadratorum partium cuiuuis line, ad quadratum diffe-renti illarum et, uelut totius line ad differentiam.<emph.end type="italics"/></cell><cell>142</cell></row><row><cell>CLII.</cell><cell>S<emph type="italics"/>i linea in duas partes quales, duasque inquales diuidatur, fueritque proportio ag-gregati ex maiore, & dimidio ad ipam maiorem, uelut ex minore, & ali-qua linea ad ipam minorem, & rurus aggregati ex minore, & dimidio ad ipam minorem, uelut aggregati ex maiore, & alia addita ad ipam maiorem, erit proportio dimidij ad partem unam inqualem, uelut alterius partis in-qualis ad uam additam mutu, & etiam proportio additarum inuicem, uelut proportio <expan abbr="partiũ">partium</expan> <expan abbr="inæqualiũ">inqualium</expan> duplicata, & rurus ipum <expan abbr="dimidiũ">dimidium</expan> line aum-pt <expan abbr="mediũ">medium</expan>, erit proportione inter additas.<emph.end type="italics"/> D<emph type="italics"/><expan abbr="emũ">emum</expan> proportio dimidij <expan abbr="cũ">cum</expan> addita maiore ad <expan abbr="dimidiũ">dimidium</expan>, cum addita minore, uelut maioris partis ad <expan abbr="minor&etilde;">minorem</expan>.<emph.end type="italics"/></cell><cell>142</cell></row><row><cell>CLIII.</cell><cell>V<emph type="italics"/>im quamcunque manus multiplicare.<emph.end type="italics"/></cell><cell>144</cell></row><row><cell>CLIIII.</cell><cell>S<emph type="italics"/>i line dat alia linea adiungatur, ab extremitatibus autem prioris line du rect in unum punctum concurrant proportionem habentes, quam mediam inter tota m & adiectam, & adiectam erit punctus, concurus puncto extre-mo line adiect ditans per lineam mediam.<emph.end type="italics"/> Q<emph type="italics"/>uod i ab extremo alicuius li-ne qua'is medi, eu peripheria circuli, cuius emidiameter it media linea du line ad prdicta puncta producantur, ip erunt in proportione medi ad adiectam.<emph.end type="italics"/></cell><cell>145</cell></row><row><cell>CLV.</cell><cell>Q<emph type="italics"/>uadr atorum numerum proportionem & inuentionem coniderare.<emph.end type="italics"/></cell><cell>147</cell></row><row><cell>CLVI.</cell><cell>H<emph type="italics"/>orologiorum tempus multiplicare.<emph.end type="italics"/></cell><cell>152</cell></row><row><cell>CLVII.</cell><cell>H<emph type="italics"/>orologiorum molarium rationem otendere.<emph.end type="italics"/></cell><cell>154</cell></row><row><cell>CLVIII.</cell><cell>R<emph type="italics"/>ationem indicis mobilis cum rota, qua horarum numerus per ictus indicatur ex-plicare.<emph.end type="italics"/></cell><cell>156</cell></row><row><cell>CLIX.</cell><cell>N<emph type="italics"/>ullus angulus rectilineus qualis ee potet alicui angulo contento recta, & cir culi portione.<emph.end type="italics"/></cell><cell>158</cell></row><row><cell>CLX.</cell><cell>P<emph type="italics"/>ropoita linea tribusque in ea ignis punctum inuenire, ex quo duct tres line ad igna int in proportionibus datis.<emph.end type="italics"/></cell><cell>162</cell></row><row><cell>CLXI.</cell><cell>S<emph type="italics"/>i fuerint duo trianguli, quorum baes in eadem linea int contituti, & quales ad unum punctum terminati, & latus unum commune inter reliqua quantita-te medium necee et angulum maioribus lineis <expan abbr="contentũ">contentum</expan> minorem ee.<emph.end type="italics"/></cell><cell>162</cell></row><row><cell>CLXII.</cell><cell>P<emph type="italics"/>roportionem duorum orbium, quorum diametrorum conuex partis, & conca-u proportiones dat int inuetigare.<emph.end type="italics"/></cell><cell>164</cell></row><row><cell>CLXIII.</cell><cell>P<emph type="italics"/>roportionem uirium tellarum per motus uos indagare.<emph.end type="italics"/></cell><cell>165</cell></row><row><cell>CLXIIII.</cell><cell>S<emph type="italics"/>yderum proportionem in magnitudine otendere.<emph.end type="italics"/></cell><cell>166</cell></row><row><cell>CLXV.</cell><cell>P<emph type="italics"/>roportionem motuum omnium tellarum ad<emph.end type="italics"/> S<emph type="italics"/>olem coniderare.<emph.end type="italics"/></cell><cell>167</cell></row><row><cell>CLXVI.</cell><cell>P<emph type="italics"/>roportiones muicas uperpartientes in eas, qu particul una tantum abundant reducere.<emph.end type="italics"/></cell><cell>168</cell></row><pb xlink:href="015/01/015.jpg"/><row><cell>CLXVII.</cell><cell>P<emph type="italics"/>roportionem muicam ad apores & odores coaptare.<emph.end type="italics"/></cell><cell>176</cell></row><row><cell>CLXVIII.</cell><cell>P<emph type="italics"/>icturarum proportiones explicare.<emph.end type="italics"/></cell><cell>179</cell></row><row><cell>CLXIX.</cell><cell>P<emph type="italics"/>roportionem muicam in intrumentis declarare iuxta compoitionis ra-tionem.<emph.end type="italics"/></cell><cell>182</cell></row><row><cell>CLXX.</cell><cell>C<emph type="italics"/>oniugationes cuiuuis numeri breuiter inuenire.<emph.end type="italics"/></cell><cell>185</cell></row><row><cell>CLXXI.</cell><cell>P<emph type="italics"/>ropoitis duobus quibuslibet numeris, quotuis alios eu in continuum eu medios in continua proportione arithmetica, geometrica & muica in-uenire.<emph.end type="italics"/></cell><cell>187</cell></row><row><cell>CLXXII.</cell><cell>P<emph type="italics"/>roportiones<emph.end type="italics"/> S<emph type="italics"/>tiphelij decribere.<emph.end type="italics"/></cell><cell>191</cell></row><row><cell>CLXXIII.</cell><cell>C<emph type="italics"/>irculum uper centro uo mouere qualiter, ita quod omnia illius puncta per rectam lineam moueantur ultro citroque.<emph.end type="italics"/></cell><cell>192</cell></row><row><cell>CLXXIIII.</cell><cell>P<emph type="italics"/>rogreus & regreus, tam ine latitudine qum cum latitudine in planetis per olos concentricos circulos qualiter motos demontrare.<emph.end type="italics"/></cell><cell>194</cell></row><row><cell>CLXXV.</cell><cell>C<emph type="italics"/>auam uarietatis diametrorum ex uppoitis concentricis demontra-re.<emph.end type="italics"/></cell><cell>195</cell></row><row><cell>CLXXVI.</cell><cell>R<emph type="italics"/>ationem centri grauitatis declarare.<emph.end type="italics"/></cell><cell>197</cell></row><row><cell>CLXXVII.</cell><cell>S<emph type="italics"/>i proportio aliqua ex duabus proportionibus eiudem quantitatis ad alias duas componatur, erit proportio illarum duarum eadem proportioni producti ex proportione in primam duarum quantitatum, detracta prio-re illa quantitate, qu ad duas comparatur, ad eandem priorem quanti-tatem.<emph.end type="italics"/></cell><cell>198</cell></row><row><cell>CLXXVIII.</cell><cell>P<emph type="italics"/>roportionem mitionis metallorum, maxim auri & argenti declara-re.<emph.end type="italics"/></cell><cell>199</cell></row><row><cell>CLXXIX.</cell><cell>S<emph type="italics"/>i duobus totis du portiones imiles abcindantur ab eidem denu, & ab-ciis portionibus partes edem auferantur, denuoque ac denu quoties libuerit portionibus, & reiduis iparum quantitatum partes edem auferantur, erit reidu ad reiduum, ueluti totius ad totum.<emph.end type="italics"/></cell><cell>200</cell></row><row><cell>CLXXX.</cell><cell>S<emph type="italics"/>i aliqua quantitas in duas partes diuidatur, fueritque alicuius quantitatis ad partes illas compoita proportio, non poterit eiudem quantitatis ad par-tes alias quantitatis diuia, aliter proportio eadem componi.<emph.end type="italics"/></cell><cell>202</cell></row><row><cell>CLXXXI.</cell><cell>C<emph type="italics"/>um fuerit aliqua proportio, compoita ex proportionibus prim ad ecun-dam & tertiam, & rurus quart ad quintam & extam: ita e habebit proportio ecund ad tertiam, ad proportionem quint ad extam, uelut producti ex proportione in ecundam detracta prima ad primam ad pro-ductum ex proportione in quintam, detracta quarta ad quartam.<emph.end type="italics"/></cell><cell>203</cell></row><row><cell>CLXXXII.</cell><cell>P<emph type="italics"/>ropoita differentia proportionum partium imilium ad partes aumptas, propoitaque proportione totius ad reidua eadem, differentiam propor-tionum totius ad reliquum reidui inuenire.<emph.end type="italics"/></cell><cell>203</cell></row><row><cell>CLXXXIII.</cell><cell>S<emph type="italics"/>pacium uit naturalis per pacium uit fortuitum declarare.<emph.end type="italics"/></cell><cell>204</cell></row><row><cell>CLXXXIIII.</cell><cell>Q<emph type="italics"/>ucunque grauia in uorticibus aquarum, merguntur, in medio uorticis, pri-mum uera mergantur.<emph.end type="italics"/></cell><cell>211</cell></row><row><cell>CLXXXV.</cell><cell>C<emph type="italics"/>ur homo edens quanto altius edet, & quanto magis crura ad fmora, & fmora ad pectus reclinata habet, facilius conurgat, cum tamen hc op-poito modo inuicem e habeant, declarare.<emph.end type="italics"/></cell><cell>213</cell></row><row><cell>CLXXXVI.</cell><cell>S<emph type="italics"/>i fuerit proportio prim & ecund quantitatis ad tertiam, ut prim & quart ad quintam, fueritque quarta ecunda maior, erit proportio quar-t ad quintam maior qum ecund ad tertiam.<emph.end type="italics"/> Q<emph type="italics"/>uod i fuerit maior<emph.end type="italics"/></cell><cell/></row><pb xlink:href="015/01/016.jpg"/><row><cell/><cell><emph type="italics"/>quart ad quintam qum ecund ad tertiam, necee et quartam ecunda ee maiorem.<emph.end type="italics"/></cell><cell>214</cell></row><row><cell>CLXXXVII.</cell><cell>S<emph type="italics"/>i eidem uiribus & eadem proportione cum auxilio ponderis tertij quar-tum pondus moueatur quibus ecundum, auxilio primi necee et <expan abbr="quartũ">quartum</expan> pon dus tardius & maiore cum difficultate moueri qum ecundum.<emph.end type="italics"/></cell><cell>214</cell></row><row><cell>CLXXXVIII.</cell><cell>S<emph type="italics"/>i uires aliqu moueant cum ponderibus aliqua pondera, ut compoita pro-portio it eadem proportioni uirium & duorum ponderum mouentium ag-gregatum quale duorum ponderum, ubi maior fuerit partium in qualitas, ibi erit maior difficultas.<emph.end type="italics"/></cell><cell>214</cell></row><row><cell>CLXXXIX.</cell><cell>S<emph type="italics"/>i pondus minus ad longitudinem minorem ub quali proportione coapte-tar, facilius deorum trahetur qum quod maius et & propius.<emph.end type="italics"/></cell><cell>215</cell></row><row><cell>CXC.</cell><cell>S<emph type="italics"/>i fuerit primum graue minus ecundo, & ecundum minus tertio, proportio autem primi ad ecundum multo maior qum ecundi ad tertium, poibile erit propoitis uiribus eidem addere pondus <expan abbr="&longs;ecũdo">ecundo</expan>, ut ipum & tertium mouea-tur facilis ab eidem uiribus, & primo uel ecundo qum antea.<emph.end type="italics"/></cell><cell>215</cell></row><row><cell>CXCL.</cell><cell>C<emph type="italics"/>um fuerint duo pondera & uires, duxerisque aggregatum ex uiribus & mi-nore pondere in maius, addiderisque inuper quantum et productum dimidij ui rium in e latus aggregati detracto dimidio uirium, dicetur pondus auxiliare qualis proportionis.<emph.end type="italics"/></cell><cell>215</cell></row><row><cell>CXCII.</cell><cell>S<emph type="italics"/>i ex medio diametri linea ad perpendiculum erigatur ad circuli peripheri-am, ex eo puncto autem quotlibet line ducantur eu intus ad circun ferentiam uque, eu extra ad diametrum, erit proportio totius line ad totam uelut mu-tuo partis ad partem.<emph.end type="italics"/></cell><cell>217</cell></row><row><cell>CXCIII.</cell><cell>R<emph type="italics"/>ationem ponderis triplicem explicare.<emph.end type="italics"/></cell><cell>218</cell></row><row><cell>CXCIIII.</cell><cell>P<emph type="italics"/>roportionem ponderis longioris in medio upeni, ad breuius illi quale & in medio upenum declarare.<emph.end type="italics"/></cell><cell>219</cell></row><row><cell>CXCV.</cell><cell>S<emph type="italics"/>i lectus fiat dupla longitudine ad latitudinem, melius uffulcietur retibus ex medio ad angulos & eius quiditantibus qum ecundum longitudinem & latitudinem.<emph.end type="italics"/></cell><cell>220</cell></row><row><cell>CXCVI.</cell><cell>S<emph type="italics"/>i duo circuli uper eodem centro eodem motu trans feruntur, quale pacium uperant.<emph.end type="italics"/></cell><cell>221</cell></row><row><cell>CXCVII.</cell><cell>C<emph type="italics"/>ur lances ad locum uum upeni redeant, impendentes <expan abbr="nõ">non</expan>, <expan abbr="demõ&longs;trare">demontrare</expan>.<emph.end type="italics"/></cell><cell>224</cell></row><row><cell>CXCVIII.</cell><cell>C<emph type="italics"/>ur olidum quod cubus uocatur<emph.end type="italics"/> P<emph type="italics"/>yramide tabilius it otendere.<emph.end type="italics"/></cell><cell>225</cell></row><row><cell>CXCIX.</cell><cell>R<emph type="italics"/>ationem remorum nauim impellentium inuenire.<emph.end type="italics"/></cell><cell>227</cell></row><row><cell>CC.</cell><cell>C<emph type="italics"/>ur temo cum paruus it, magnam nauim agere potet, & cur cm uarietas it in prora, ipe contituatur in puppi.<emph.end type="italics"/> E<emph type="italics"/>t cum transuerim ab aqua prematur rect nauim dirigat.<emph.end type="italics"/></cell><cell>228</cell></row><row><cell>CCI.</cell><cell>S<emph type="italics"/>i du line non ecantes circuli peripheriam in unum punctum ex ea coe-ant exterius, necee et illas peripheria contenta ee maiores.<emph.end type="italics"/></cell><cell>229</cell></row><row><cell>CCII.</cell><cell>R<emph type="italics"/>ationem trepitus otendere.<emph.end type="italics"/></cell><cell>232</cell></row><row><cell>CCIII.</cell><cell>C<emph type="italics"/>ur cytalis onera portentur facilis, explorare.<emph.end type="italics"/></cell><cell>233</cell></row><row><cell>CCIIII.</cell><cell>C<emph type="italics"/>ur pluribus trochleis, pondera facilius eleuentur otendere.<emph.end type="italics"/></cell><cell>233</cell></row><row><cell>CCV.</cell><cell>S<emph type="italics"/>uper uerbis<emph.end type="italics"/> P<emph type="italics"/>latonis de fine<emph.end type="italics"/> R<emph type="italics"/>eipublic.<emph.end type="italics"/></cell><cell>234</cell></row><row><cell>CCVI.</cell><cell>R<emph type="italics"/>hombi paiones quadam declarare.<emph.end type="italics"/></cell><cell>235</cell></row><row><cell>CCVII.</cell><cell>P<emph type="italics"/>roportionem agentium naturalium in tranmutatione coniderare.<emph.end type="italics"/></cell><cell>238</cell></row><row><cell>CCVIII.</cell><cell>M<emph type="italics"/>ota res centro grauitatis per <expan abbr="prior&etilde;">priorem</expan> motum, in reditu uelocius mouetur quam i quieuerit.<emph.end type="italics"/></cell><cell>238</cell></row><pb xlink:href="015/01/017.jpg"/><row><cell>CCIX.</cell><cell>S<emph type="italics"/>i uperficies rectangula in duas partes quales diuia intelligatur, qu am-b quadrat int, itemque in duas inquales, erit parallelipedum ex latere medi partis in totam uperficiem maius aggregato parallelipedorum ex partibus inqualibus in latera alterius partis mutuo, in eo, quod fit ex dif ferentia lateris minoris partis medi latere in differentiam maioris par-tis uperficiei media uperficie bis, & ex differentia amborum laterum inqualium iunctorum ad ambo latera, qualia iuncta in minorem par-tem uperficiei.<emph.end type="italics"/></cell><cell>241</cell></row><row><cell>CCX.</cell><cell>S<emph type="italics"/>i du line ad quales angulos ab eodem puncto peripheri circuli refle-ctantur, necee et angulos cum dimetiente factos quales ee.<emph.end type="italics"/> V<emph type="italics"/>nde ma-nifetum et, protractam diametrum angulum uppoitum per qualia di-uidere.<emph.end type="italics"/></cell><cell>242</cell></row><row><cell>CCXI.</cell><cell>S<emph type="italics"/>i du line ex duobus punctis peripheriam contingentes, in eandem par-tem protrahantur, emper magis ditabunt inuicem ea ex parte, & nun-quam concurrent.<emph.end type="italics"/></cell><cell>243</cell></row><row><cell>CCXII.</cell><cell>S<emph type="italics"/>i ab eodem puncto ad circuli peripheriam line quotuis ducantur, tres inue-nire lineas, qu non in alium punctum reflectentur.<emph.end type="italics"/></cell><cell>244</cell></row><row><cell>CCXIII.</cell><cell>P<emph type="italics"/>ropoito circulo, atque in eius peripheria puncto ignato, lineas contingentes ultra ctraque, & eam ab ipomet deducere.<emph.end type="italics"/></cell><cell>245</cell></row><row><cell>CCXIIII.</cell><cell>S<emph type="italics"/>i extra circulum duo puncta qualiter centro ditantia ignentur, erit pun-ctum reflexionis qualis in medio arcus intercepti inter lineas, qu cen tro ducuntur ad illa puncta.<emph.end type="italics"/> S<emph type="italics"/>i uer unum centro proximius fuerit altero, punctum qualitatis in peripheria tant longius, uerus breuiorem line-am, quant punctum aliud centro magis diteterit.<emph.end type="italics"/></cell><cell>245</cell></row><row><cell>CCXV.</cell><cell>P<emph type="italics"/>unctum reflexionis punctorum inqualiter ditantium centro, qualiter ditat lineis, ductis centro ad puncta qualiter ditantia alterutrin-que.<emph.end type="italics"/></cell><cell>246</cell></row><row><cell>CCXVI.</cell><cell>S<emph type="italics"/>i fuerint circuli duo inquales, & extra utrunqe punctum ad illud ex mi-nore reflex per magnam partem minoris maiore perueuire pote-runt.<emph.end type="italics"/></cell><cell>247</cell></row><row><cell>CCXVII.</cell><cell>O<emph type="italics"/>culus uidet partem uperficiei<emph.end type="italics"/> L<emph type="italics"/>un illuminatam <emph.end type="italics"/> S<emph type="italics"/>ole per radios reflexos <emph.end type="italics"/> S<emph type="italics"/>olis corpore: nec tamen potet uidere imaginem ipius in<emph.end type="italics"/> L<emph type="italics"/>una tan quam in peculo.<emph.end type="italics"/></cell><cell>248</cell></row><row><cell>CCXVIII.</cell><cell>R<emph type="italics"/>ationem macul<emph.end type="italics"/> L<emph type="italics"/>un indagare.<emph.end type="italics"/></cell><cell>248</cell></row><row><cell>CCXIX.</cell><cell>R<emph type="italics"/>ationem eorum qu apparent circa<emph.end type="italics"/> S<emph type="italics"/>olem peculo in aqua poito decla-rare.<emph.end type="italics"/></cell><cell>150</cell></row><row><cell>CCXX.</cell><cell>C<emph type="italics"/>auam cur<emph.end type="italics"/> S<emph type="italics"/>ol tiuis diebus exoriens umbram ad meridiem, cum in meridie ad boream mittat, explorare.<emph.end type="italics"/></cell><cell>252</cell></row><row><cell>CCXXI.</cell><cell>M<emph type="italics"/>agnitudo<emph.end type="italics"/> L<emph type="italics"/>un & cterorum atrorum dignocitur ex proportione alio-rum ad eam iuxta ditantiam: ipius uer iuxta rationem pupill ad<emph.end type="italics"/> L<emph type="italics"/>u-nam ditanti ratione.<emph.end type="italics"/></cell><cell>354</cell></row><row><cell>CCXXII.</cell><cell>Q<emph type="italics"/>uantitates qu quales ee non pount in eodem genere, maius tamen & minus recipiunt, unt in proportione potetatis.<emph.end type="italics"/></cell><cell>255</cell></row><row><cell>CCXXIII.</cell><cell>Q<emph type="italics"/>uantitates qu actu quales ee non pount, in nulla proportione actu ee pount.<emph.end type="italics"/></cell><cell>256</cell></row><row><cell>CCXXIIII.</cell><cell>N<emph type="italics"/>eque temporis totius, ut imaginamur, ipum ee infinitum, neque ui ui-tarum proportio ulla et ad tempus, quod potetate et, utpot diem<emph.end type="italics"/></cell><cell/></row><pb xlink:href="015/01/018.jpg"/><row><cell/><cell><emph type="italics"/>uel menem.<emph.end type="italics"/></cell><cell>256</cell></row><row><cell>CCXXV.</cell><cell>P<emph type="italics"/>roportio media non et ex ratione agentis, ed patientis.<emph.end type="italics"/></cell><cell>256</cell></row><row><cell>CCXXVI.</cell><cell>P<emph type="italics"/>roportio ublimis non conitit in magnitudine, ed ordine, iuxta quem diffe-rentia et eius quod et ante & pot.<emph.end type="italics"/></cell><cell>257</cell></row><row><cell>CCXXVII.</cell><cell>V<emph type="italics"/>it iuxta numerum perfectionum in comparatione ad cogitationem no-tram proportionem quand am habent.<emph.end type="italics"/></cell><cell>259</cell></row><row><cell>CCXXVIII.</cell><cell>P<emph type="italics"/>roportionem cienti futurorum & cterorum occultorum conidera-re.<emph.end type="italics"/></cell><cell>260</cell></row><row><cell>CCXXIX.</cell><cell>I<emph type="italics"/>ncorporea omnia unum unt, neque numerus et eorum.<emph.end type="italics"/></cell><cell>261</cell></row><row><cell>CCXXX.</cell><cell>P<emph type="italics"/>roportio incorporeorum acendentium emper maior et.<emph.end type="italics"/></cell><cell>262</cell></row><row><cell>CCXXXI.</cell><cell>T<emph type="italics"/>res ee mundos atque inter ipos nullam ee proportionem: nec numero cos definiri.<emph.end type="italics"/></cell><cell>263</cell></row><row><cell>CCXXXII.</cell><cell>O<emph type="italics"/>mnis motus naturalis quanto uelocior et tanto propior et & magis imil limus quieti.<emph.end type="italics"/></cell><cell>264</cell></row><row><cell>CCXXXIII.</cell><cell>Q<emph type="italics"/>uod et in mundo incorporeo ternum et, beatum, ecurum, immutabile ecundum locum, olum iuxta eentiam fit: iuxta quod uelut leui u-urro aqu & aura tiua demulcetur.<emph.end type="italics"/></cell><cell>270</cell></row></table><p type="head">
<s id="id2657316">FINIS.</s></p><pb xlink:href="015/01/019.jpg"/></section></front> <body> <chap>
<pb xlink:href="015/01/020.jpg" pagenum="1"/><p type="head">
<s id="id2657343">HIERONYMI CAR <lb/>DANI MEDIOLANENSIS, CI<lb/>VI'SQVE BONONIENSIS, MEDICI<lb/>de Proportionibus, eu Ope<lb/>ris Perfecti <lb/>LIBER QVINTVS.</s></p>
<p type="main">
<s id="id2657375">Prima diffinitio.</s></p><p type="main">
<s id="id2657384">Proportio ab Euclide ic decribitur, Qud <lb/>it duarum quantitatum eiudem generis, <lb/>quod ad magnitudinem attinet, compara<lb/>tio certa.</s></p><p type="main">
<s id="id2657416">Secunda diffinitio.</s></p><p type="main">
<s id="id2657424">Proportiones per imilitudinem <expan abbr="dicũtur">dicuntur</expan>, <lb/>cm quantitas quantitati <expan abbr="compara&ttilde;">comparatur</expan> alterius <lb/>generis, cui fingitur qualis ee potetate.</s></p><p type="main">
<s id="id2657473">Velut i a b fingatur monas in comparatione <lb/>ad b c erit rectangulum a c quale line b c.</s></p><figure id="id.015.01.020.1.jpg" xlink:href="015/01/020/1.jpg"/><p type="main">
<s id="id2657505">Tertia diffinitio.</s></p><p type="main">
<s id="id2657513">Proportio qualis proportioni et, cm eodem modo termini <lb/>e habent inuicem in utraque</s></p><p type="main">
<s id="id2657534">Quarta diffinitio.</s></p><p type="main">
<s id="id2657542">Proportiones ecundum genus not dicuntur, cm nouimus, <lb/>qud int maiores, aut minores. </s>
<s id="id2657564">Nam cm quales unt, imul ne<lb/>ceffe et, ut cognocamus genus, & peciem.</s></p><p type="main">
<s id="id2657596">Quinta diffinitio.</s></p><p type="main">
<s id="id2657605">Datum poitione et: quod neceari ex poitis certam habet <lb/>quantitatem.</s></p><p type="main">
<s id="id2657632">Sexta diffinitio.</s></p><p type="main">
<s id="id2657641">Datum impliciter dicitur, quod ex propoitis cognoci potet, <lb/>quantum it.</s></p><p type="main">
<s id="id2657665">Septima diffinitio.</s></p><p type="main">
<s id="id2657674">Proportiones potetate <expan abbr="dicun&ttilde;">dicuntur</expan>, quub comparatione aliarum <lb/><expan abbr="quantitatũ">quantitatum</expan> neceariam habentium <expan abbr="cõnexionem">connexionem</expan> <expan abbr="&longs;olũ">olum</expan> <expan abbr="cogno&longs;cun&ttilde;">cognocuntur</expan>.</s></p><p type="main">
<s id="id2657747">H autem unt aliquando eiudem generis, cum primis ut nu<lb/>meri: aliquand alterius, ut linearum & uperficierum, angulorum, <lb/>& arcuum: aliquando eiudem generis, & diuenarum pecierum, <lb/>ut arcuum per inus, qua utuntur Atronomi.</s></p><p type="main">
<s id="id2657796">Octaua diffinitio.</s></p><p type="main">
<s id="id2657805">Proportio homonyma dicitur duarum quantitatum diueri ge</s></p><p type="main">
<s id="id2657818"><arrow.to.target n="marg1"/><lb/>neris, ed alterius a b altero dependentium, uelut motus ad tem
<pb xlink:href="015/01/021.jpg" pagenum="2"/>pus. </s>
<s id="id2657842">Dicimus enim motum tardum, uel uelocem in comparatione <lb/>ad tempus.</s></p><p type="margin">
<s id="id2657854"><margin.target id="marg1"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2657881">Nona diffinitio.</s></p><p type="main">
<s id="id2657889">Proportionum ali dicuntur rhete, ali alog, rhet qu unt <lb/>ut numeri ad numerum, alog qu non unt numeri ad numerum.</s></p><p type="main">
<s id="id2657926">Decima diffinitio</s></p><p type="main">
<s id="id2657934">Proportio rhete alia qualis, alia multiplex, uel ubmultiplex: <lb/>alia unius partis exceus, aut defectus, alia plurium, quam uper<lb/>partientem, aut upartientem uocant.</s></p><p type="main">
<s id="id2657967">Vndecima diffinitio.</s></p><p type="main">
<s id="id2657975">Cum diuio denominatore per numeratorem exit quantitas alo <lb/>ga, proportio dicitur aloga: i autem numerus integer, aut pars nu<lb/>meri nota dicitur rhete.</s></p><p type="main">
<s id="id2657999">Duodecima diffinitio.</s></p><p type="main">
<s id="id2658007">Proportionem in proportionem duci et, quoties recto ordine <lb/>tres quantitates in eidem collo <expan abbr="can&ttilde;">cantur</expan>: ut int tres quan <lb/><figure id="id.015.01.021.1.jpg" xlink:href="015/01/021/1.jpg"/><lb/>titates a b c dicetur proportio a ad c producta ex pro <lb/>portione a ad b & b ad c, & imiliter proportio c ad <lb/>a producitur ex proportione b ad a, & c ad b.</s></p><p type="main">
<s id="id2658060">Tertiadecima diffinitio.</s></p><p type="main">
<s id="id2658068">Proportionem per proportionem diuidi et, quoties ad eandem <lb/>quantitatem du quantitates comparantur, tunc illarum propor<lb/>tio et, qu prodit una per alteram diuia.</s></p><p type="main">
<s id="id2658099">Sint proportiones a & b ad c & interponatur b inter a & c, dico <lb/>proportionem a ad c diuiam per proportionem a ad b, & prodire <lb/>proportionem b ad c, contat ex conuera prcedentis.</s></p><p type="main">
<s id="id2658128">Quartadecima diffinitio.</s></p><p type="main">
<s id="id2658136">Additio proportionum intelligitur quotiens duarum quanti<lb/>tatum ad unam tertiam, proportiones per aggregatum iparum <lb/>quantitatum ad eandem coniunguntur.</s></p><p type="main">
<s id="id2658158">Velut i comparentur a b & b c ad d, inde tota <lb/><figure id="id.015.01.021.2.jpg" xlink:href="015/01/021/2.jpg"/><lb/>a c ad d dicemus proportionem, ac ad d ee con <lb/><expan abbr="iunctã">iunctam</expan> ex duabus proportionibus a b ad d & b c <lb/>ad <expan abbr="eand&etilde;">eandem</expan> d. </s>
<s id="id2658207">Hoc & duo equentes icut & du <expan abbr="anteced&etilde;tes">antecedentes</expan> demon<lb/>trabitur ee. </s>
<s id="id2658238">nunc olum quomodo <expan abbr="intelligendũ">intelligendum</expan> it proponimus.</s></p><p type="main">
<s id="id2658261">Quintadecima diffinitio.</s></p><p type="main">
<s id="id2658270">Detractionem proportionis proportione intelligimus fieri <lb/>per <expan abbr="detraction&etilde;">detractionem</expan> minoris quantitatis maiore, comparatam ad ean<lb/>dem quantitatem.</s></p><p type="main">
<s id="id2658301">Velut in exemplo uperiore detracta proportione b c ad d ex
<pb xlink:href="015/01/022.jpg" pagenum="3"/>proportione a c ad d, relinquetur proportio a b ad d. </s>
<s id="id2658319">& probatur <lb/>ex conuerione prcedentis.</s></p><p type="main">
<s id="id2658335">Sextadecima diffinitio.</s></p><p type="main">
<s id="id2658344">Extractio radicum alicuius proportionis fit per extractionem <lb/>radicum quantitatum illius iuxta unam, & eandem rationem.</s></p><p type="main">
<s id="id2658358">Velut quadrat, uel cub, uel pronic, uel unineralis, uel alte<lb/>rius modi.</s></p><p type="main">
<s id="id2658382">Decimaeptima diffinitio.</s></p><p type="main">
<s id="id2658393">Cm fuerint du proportiones imiles in tribus terminis con<lb/>tinuat, dicetur proportio prim quantitatis ad tertiam ueluti <lb/>prim ad ecundam duplicata. </s>
<s id="id2658425">Et i int tres proportiones imiles <lb/>in quatuor terminis, dicetur proportio prim quantitatis ad quar<lb/>tam triplicat ei, qu et prim ad ecundam,</s></p><p type="main">
<s id="id2658467">Decimaoctaua diffinitio.</s></p><p type="main">
<s id="id2658475">Confua proportio dicitur implicis, aut compoit quantitatis <lb/>ad compoitam in comparatione ad proportiones ad partes.</s></p><p type="main">
<s id="id2658502">Decimanona diffinitio.</s></p><p type="main">
<s id="id2658510">Quantitates qu in continua unt proportione Analog <expan abbr="uocan&ttilde;">uocantur</expan>.</s></p><p type="main">
<s id="id2658536">Dictum et hoc ad fugiendum nomen barbarum, etiam ut bre<lb/>uiter tamen poemus ententiam explicare.</s></p><p type="main">
<s id="id2658559">Vigeima diffinitio.</s></p><p type="main">
<s id="id2658570">Reflexa proportio dicitur cum trium quantitatum aggregatum <lb/>prim, & terti e habet ad ecundam uelut ecunda ad tertiam,</s></p><p type="main">
<s id="id2658597">Vigeima prima diffinitio.</s></p><p type="main">
<s id="id2658608">Trium quantitatum analogarum ali quidem Geometric, <lb/>cm proportio imilis et: Ali Arithmetic, cum fuerit qualis <lb/>exceus hucind: Ali muic cum fuerit proportio prim ad ter <lb/>tiam multiplex, aut implex, aut compoita exceus qu implici <lb/>iuncta it ad multiplicis perfectionem: eadem autem it proportio <lb/>exceus prim, & ecund ad exceum ecund upra tertiam.</s></p><p type="main">
<s id="id2658719">Velut proportio 6. 4. 3. dupla et utrinque, & 6. 3. 2 tripla. </s>
<s id="id2658728">& 28. 24. <lb/>21. & 45. 40. 36. Geometrica uer & arithmetica facilius continuan<lb/>tur in quotquot quantitatibus, ed & muica uelut 12. 8. 6. 4. 3. & <lb/>proportio 8 ad 5 muica et: quia proportio 5 ad 4 muica et, & <lb/>bene onans, igitur contitutis 8. 5. 4. cum 8 ad 4 ben onet, & 5 <lb/>ad 4, & 4 it extrema non media inde 8. & 5 ben <expan abbr="&longs;onãt">onant</expan>. </s>
<s id="id2658802">nam in me<lb/>dijs <expan abbr="nõ">non</expan> et <expan abbr="uerũ">uerum</expan>, ut in 9. 6. 4 bis diapente, & 16. 12. 9 bis diatearon.</s></p><p type="main">
<s id="id2658839">Vigeima ecunda diffinitio.</s></p><p type="main">
<s id="id2658853">Quantitates qu imilem habent proportionem non continua<lb/>tam, omiolog appellantur.</s></p><p type="main">
<s id="id2658875">Vigeima tertia diffinitio.</s></p><p type="main">
<s id="id2658886">Prima operatione conitere dicuntur proportiones, cm inter <lb/>primo conflatas quantitates contiterint.</s></p>
<pb xlink:href="015/01/023.jpg" pagenum="4"/><p type="main">
<s id="id2658918">PRIMA Animi communis ententia.</s></p><p type="main">
<s id="id2658929">Omnis Proportio et, aut qualitatis, aut maior inqualis, <lb/>aut minor.</s></p><p type="main">
<s id="id2658949">Secunda animi communis ententia.</s></p><p type="main">
<s id="id2658961">Quilibet numerus tantus dicitur, quanta et illius proportio ad <lb/>monadem.</s></p><p type="main">
<s id="id2658975">Dicimus enim quatuor, quod monadem quater contineat. </s>
<s id="id2658980">Et <lb/>duo cum dimidio cm monadem bis & emis contineat.</s></p><p type="main">
<s id="id2658998">Tertia animi communis ententia.</s></p><p type="main">
<s id="id2659009">Proportionem defectus, eu detract quantitatis ad defectum <lb/>ee poe, ut quantitatis ad quantitatem dicuntur communes ani<lb/>mi entcnti, qu ex intellectu olo terminorum, quod uer int, <lb/>cognocuntur. </s>
<s id="id2659058">Si ergo defectus et quantitas, & quantitas eiudem <lb/>peciei, quia detrahitur, & defectus non et implicitur, ed detra<lb/>cto ergo per quartam petitionem: uel primam diffinitionem erit <lb/>proportio interillas. </s>
<s id="id2659090">Sunt enim amb detract.</s></p><p type="main">
<s id="id2659104">Quarta animi communis ententia.</s></p><p type="main">
<s id="id2659116">Inter quantitatem, & defectum minorem quantitate, cuius et de <lb/>fectus, et proportio, quatenus et quantitas. </s>
<s id="id2659134">Sit a b linea, & detra<lb/>cta quantitas b c, non maior a b & d it alia quuis quantitas eiu<lb/><figure id="id.015.01.023.1.jpg" xlink:href="015/01/023/1.jpg"/><lb/><expan abbr="d&etilde;">dem</expan> generis, dico qud inter d & b c et propor<lb/>tio quatenus b c et quantitas, quia unt eiu<lb/>dem generis ideo unt in aliqua proportione <lb/>per primam diffinitionem. </s>
<s id="id2659200">Sed ut b c et defectus, nulla et propor<lb/>tio: quia quanto b c augetur, tanto augetur proportio d ad b c, & <lb/>hoc et contra demontrata ab Euclide.</s></p><p type="main">
<s id="id2659231">Quinta animi communis ententia.</s></p><p type="main">
<s id="id2659242">Cum proportio producitur ex proportionibus qulibet illa<lb/>rum dicetur producta diuia per alteram.</s></p><p type="main">
<s id="id2659262">Sexta animi communis ententia.</s></p><p type="main">
<s id="id2659273">qualium quantitatum eu proportionum ad tertiam compa<lb/>rabilium eadem et proportio atque uicisim. </s>
<s id="id2659292">Hc eti demontre<lb/>tur ab Euclide, et tamen hic generalior: & atis per e nota. </s>
<s id="id2659318">Vt it <lb/>propior animi communi ententi, qum rei demontrand.</s></p><p type="main">
<s id="id2659346">Septima animi communis ententia.</s></p><p type="main">
<s id="id2659357">Ad quod quantitas proportionem habet infinitam, id in genere <lb/>illius quantitatis non comprehenditur.</s></p><p type="main">
<s id="id2659371">Nam proportio et duarum quantitatum eiudem generis com<lb/>paratio certa: at hc comparatio certa non et: non igitur quantita<lb/>tes amb unt, aut non eiudem generis.</s></p>
<pb xlink:href="015/01/024.jpg" pagenum="5"/><p type="main">
<s id="id2659416">PRIMA Petitio.</s></p><p type="main">
<s id="id2659425">Si fuerit primi ad ecundum, ut tertij ad quartum, & ex primo in <lb/>ecundum producatur quale, aut maius, aut minus primo, uel <lb/>ecundo, producetur eodem modo ex tertio in quartum quale aut <lb/>maius, aut minus tertio, uel quarto eadem ratione & ordine.</s></p><p type="main">
<s id="id2659458">Secunda petitio.</s></p><p type="main">
<s id="id2659466">Proportiones pount duci, diuidi, iungi, & auferri, & umi radix <lb/>in eis cuiucunque generis, atque earum quantitatis, ut libet, poe <lb/>tranponere.</s></p><p type="main">
<s id="id2659498">Tertia petitio.</s></p><p type="main">
<s id="id2659506">Proportionis cuiuuis nomen denominatore upr cripto, & <lb/>numeratore infr cripto umitur.</s></p><p type="main">
<s id="id2659540">Quarta petitio.</s></p><p type="main">
<s id="id2659549">Diuia quauis quantitate per aliam eiudem generis, quod exit <lb/>proportio dicitur.</s></p><p type="main">
<s id="id2659567">Quinta petitio.</s></p><p type="main">
<s id="id2659575">Qulibet proportio et uel inter duas quantitates, uel per unam <lb/>ignificatur.</s></p><p type="main">
<s id="id2659593">Nam per tertiam petitionem i int du quantitates, qu non h <lb/>beant unius rationem, nomen umit proportio duobus numeris, <lb/>in autem it altera monas, erit per ecundam animi communem en <lb/>tentiam, proportio numerus ipe Ide patet, quod dicitur.</s></p><p type="main">
<s id="id2659647">Sexta petitio.</s></p><p type="main">
<s id="id2659656">Propoita proportione quacunque, & monade quantitatem inue <lb/>nire, qu e habeat ad monadem in proportione propoita.</s></p><p type="main">
<s id="id2659679">Nam cm per quartam petitionem diuia quantitate per quan<lb/>titatem exeat proportio, & numerus ad <expan abbr="monad&etilde;">monadem</expan> e habeat, ut pro<lb/>portio, ideo umpta monade ecundum illum numerum, ille nume <lb/>rus et quantitas quita.</s></p><p type="main">
<s id="id2659730">Septima petitio.</s></p><p type="main">
<s id="id2659739">Quamlibet quantitatem per aliam eiudem generis diuidere <lb/>poe.</s></p><p type="main">
<s id="id2659758">Octaua petitio.</s></p><p type="main">
<s id="id2659766">Proportionem in proportionem ducere poe: quamuis int in<lb/>ter quantitates diueri generis.</s></p><p type="main">
<s id="id2659789">Quod dicitur de multiplicatione intelligendum et de alijs ope<lb/>rationibus upr enumeratis.</s></p><p type="main">
<s id="id2659811">Nona petitio.</s></p><p type="main">
<s id="id2659820">Monadem emper umere in quo cunque genere poe propoi<lb/>ta proportione.</s></p>
<pb xlink:href="015/01/025.jpg" pagenum="6"/><p type="main">
<s id="id2659853">Nam licet diuidere per eptimam petitionem quantitatem per <lb/>quantitatem proportionis: & quod exit, et proportio per quar<lb/>tam petitionem, & per ecundam animi communem ententiam <lb/>illa proportio et numero qualis: ergo diuia proportione, per i<lb/>milem numerum tatuetur monas.</s></p><p type="main">
<s id="id2659904">Decima petitio.</s></p><p type="main">
<s id="id2659912">In quouis genere quantitatum umere poe quantitatem, qu <lb/><arrow.to.target n="marg2"/><lb/>e habeat ad monadem in proportione data. </s>
<s id="id2659937">Similem huic propo<lb/>nit Euclides in lineis generaliter: nos autem contr generaliter in <lb/>omnibus quantitatibus, ed de monade tantum.</s></p><p type="margin">
<s id="id2659960"><margin.target id="marg2"/>D<emph type="italics"/>uodecima <lb/>exti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/>Vndecima petitio.</s></p><p type="main">
<s id="id2660001">Monadem in quancunque quantitatem ductam quale ipi pro<lb/>ducere. </s>
<s id="id2660015">Similiter & proportionem qualem.</s></p><p type="main">
<s id="id2660027">Nam cum aliqua quantitas augeat ducta aliqua minuat, necee <lb/>et aliquam ee, qu nec augeat, nec minuat, & hc et monas. <lb/></s>
<s id="id2660057">Idem dico de diuiione. </s>
<s id="id2660064">Aequalitas etiam ducta, uel diuidens non <lb/><arrow.to.target n="marg3"/><lb/>mutat proportionem: nec quantitatem ipam, igitur monas qua<lb/>litatem refert. </s>
<s id="id2660087">Quod etiam et perpicuum ex upradictis.</s></p><p type="margin">
<s id="id2660104"><margin.target id="marg3"/>S<emph type="italics"/>ecunda ani <lb/>mi <expan abbr="cõmunis">communis</expan> <lb/>ententia.<emph.end type="italics"/></s></p><p type="main">
<s id="id2660144">Duodecima petitio.</s></p><p type="main">
<s id="id2660153">Cum fuerint quatuor quantitates & ad primam, & tertiam qu <lb/>multiplicibus aumptis, item que ad ecundam & quartam, & i mul<lb/>tiplex prim maius et multiplici ecund, multiplex terti it ma<lb/>ius multiplici quart, & i minus minus, & i quale quale, idque<lb/>emper quouis modo aumptis his proportionibus ad primam & <lb/>tertiam, & ad ecundam & quartam erit proportio prim ad ecun<lb/>dam, ut terti ad quartam. </s>
<s id="id2660245">Hc etiam aumitur ab Euclide. </s>
<s id="id2660258">Et per <lb/><arrow.to.target n="marg4"/><lb/>hanc intelligimus etiam conueram.</s></p><p type="margin">
<s id="id2660277"><margin.target id="marg4"/>Q<emph type="italics"/>uinto<emph.end type="italics"/> E<emph type="italics"/>le. <lb/>diff.<emph.end type="italics"/> 6.</s>
</p><p type="main">
<s id="id2660320">Tertiadecima petitio.</s></p><p type="main">
<s id="id2660328">Quantitates quales, atque proportiones in quauis quanti<lb/>tates duct eandem eruant rationem. </s>
<s id="id2660348">Euclides hanc demontrat, <lb/>nos autem ad uitandum tdium petimus concedi, ub qua in<lb/><arrow.to.target n="marg5"/><lb/>cluduntur diuiio etiam additio, detractio, laterum omnium in<lb/>uentio.</s></p><p type="margin">
<s id="id2660385"><margin.target id="marg5"/>Q<emph type="italics"/>uarta quin <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="main">
<s id="id2660425">Quartadecima petitio.</s></p><p type="main">
<s id="id2660433">Cm termini alicuius quantitatis eandem eruant rationem in <lb/>omnibus, & firmi unt ac tabiles eiudem rationis comparatione <lb/>content partes qualem eruant exceum, eu proportionem.</s></p><p type="main">
<s id="id2660476">PROPOSITIO prima.</s></p><p type="main">
<s id="id2660485">Proportionem in proportionem duci et uperiores nume<lb/>ros atque inferiores inuicem ducere.</s></p>
<pb xlink:href="015/01/026.jpg" pagenum="7"/><p type="main">
<s id="id2660512">Sit proportio line a ad lineam b, ut anguli cad angulum d, ta<lb/><arrow.to.target n="marg6"/><lb/>tuatur e monas in genere a <lb/><figure id="id.015.01.026.1.jpg" xlink:href="015/01/026/1.jpg"/><lb/>b, & fiat fad e, ut cad d, & du <lb/><arrow.to.target n="marg7"/><lb/>catur a in f & b in e, & pro<lb/>ducantur g & h. </s>
<s id="id2660560">Quia ergo <lb/><arrow.to.target n="marg8"/><lb/>fet proportio ipa, erit g ad <lb/><arrow.to.target n="marg9"/><lb/>a ut c ad d, ed h et qualis <lb/>b, igitur a ad h ut ad b. </s>
<s id="id2660598">Du<lb/>cta ergo dicetur proportio a <lb/><arrow.to.target n="marg10"/><lb/>ad b in proportionem c ad d <lb/>ducendo terminos proportionis, eu quantitatis recta cilicet u<lb/>periores cum uperioribus, & inferiores cum inferioribus. </s>
<s id="id2660636">Nam i <lb/><arrow.to.target n="marg11"/><lb/>rurum contituantur fad e ut a ad b cm f it proportio, & k ad f ut <lb/><arrow.to.target n="marg12"/><lb/>c ad d, erit k ad e, ut g ad h, k autem fit ex ductu proportionis a ad b, <lb/>qu et fin proportionem c ad d, liquet igitur propoitum.</s></p><p type="margin">
<s id="id2660689"><margin.target id="marg6"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s id="id2660715"><margin.target id="marg7"/>P<emph type="italics"/>er<emph.end type="italics"/> 9. P<emph type="italics"/>etit.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2660751"><margin.target id="marg8"/>P<emph type="italics"/>er<emph.end type="italics"/> 10. P<emph type="italics"/>et.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2660785"><margin.target id="marg9"/>P<emph type="italics"/>er<emph.end type="italics"/> 8. P<emph type="italics"/>etit.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2660820"><margin.target id="marg10"/>P<emph type="italics"/>er<emph.end type="italics"/> 2. A<emph type="italics"/>ni<lb/>mi entent.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2660860"><margin.target id="marg11"/>P<emph type="italics"/>er<emph.end type="italics"/> 11. P<emph type="italics"/>et.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2660895"><margin.target id="marg12"/>P<emph type="italics"/>er<emph.end type="italics"/> 8. P<emph type="italics"/>etit.<emph.end type="italics"/></s></p><p type="main">
<s id="id2660930">Propoitio <expan abbr="&longs;ecũnda">ecunnda</expan>.</s></p><p type="main">
<s id="id2660953">Proportio extremorum producitur ex intermedijs.<lb/><arrow.to.target n="marg13"/></s></p><p type="margin">
<s id="id2660968"><margin.target id="marg13"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2660993">Sint a b c quantitates dico proportio<lb/><figure id="id.015.01.026.2.jpg" xlink:href="015/01/026/2.jpg"/><lb/>nem a ad c, produci ex proportione a ad b </s></p><p type="main">
<s id="id2661017"><arrow.to.target n="marg14"/><lb/>& b ad c, tatuantur totidem monade d e <lb/>f, erntque ex demontrantis ab Euclide in <lb/>quinto <expan abbr="Elem&etilde;torum">Elementorum</expan> in eadem proportio<lb/>ne, ftatuatur ergo d prima quantitas e e<lb/>cunda & tertia f quarta. </s>
<s id="id2661066">eritqe per prce<lb/><arrow.to.target n="marg15"/><lb/>dentem proportio productorum ex d in e <lb/>& it g, & in f & it h, producta ex propor<lb/>tionibus d ad e & e ad f, quare ex propor<lb/>tionibus a ad b & b ad e, ed ex dictis cum <lb/>e it eadem, erit proportio d ad f, ut g ad h & proportio, d ad f per <lb/>quam proportionem ab Euclide demontratam, ut a ad c, igitur <lb/><arrow.to.target n="marg16"/><lb/>proportio a ad c producitur ex proportionibus a ad b & b ad c, & <lb/>et proportio ipa a ad c d numerus, ut otenum et.</s></p><p type="margin">
<s id="id2661155"><margin.target id="marg14"/>P<emph type="italics"/>er<emph.end type="italics"/> 6. <emph type="italics"/>&<emph.end type="italics"/> 9. <lb/>P<emph type="italics"/>etit.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2661204"><margin.target id="marg15"/>P<emph type="italics"/>er<emph.end type="italics"/> 13. P<emph type="italics"/>et.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2661239"><margin.target id="marg16"/>P<emph type="italics"/>er<emph.end type="italics"/> 13. P<emph type="italics"/>et.<emph.end type="italics"/></s></p><p type="main">
<s id="id2661274">Ex hoc equitur, qud cm fuerit quantitas tertia monas ex pro<lb/><arrow.to.target n="marg17"/><lb/>portionibus inuicem ductis producetur prima quantitas.<lb/><arrow.to.target n="marg18"/></s></p><p type="margin">
<s id="id2661307"><margin.target id="marg17"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 2.</s></p><p type="margin">
<s id="id2661332"><margin.target id="marg18"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 3</s></p><p type="main">
<s id="id2661356">Ex hoc equitur, qud conuera proportio producitur ex con<lb/>ueris proportionibus.</s></p><p type="main">
<s id="id2661380">Propoitio tertia.</s></p><p type="main">
<s id="id2661391">Si proportio ex duabus proportionibus in quatuor terminis <lb/>producatur, ipa uer proportio inter duas alias quantitates fue
<pb xlink:href="015/01/027.jpg" pagenum="8"/>rit contituta: conurgent trecenti exaginta modi productionis <lb/>proportionis.</s></p><p type="main">
<s id="id2661433"><arrow.to.target n="marg19"/></s></p><p type="margin">
<s id="id2661444"><margin.target id="marg19"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2661469">Hc propoitio ut prcedens & <expan abbr="&longs;equ&etilde;tes">equentes</expan> tres ab Alchindo um<lb/>pt unt, & ab eo demontrantur. </s>
<s id="id2661509">Sit ergo proportio a ad b, pro<lb/><arrow.to.target n="table2"/><lb/><figure id="id.015.01.027.1.jpg" xlink:href="015/01/027/1.jpg"/>ducta ex proportione c ad d & e ad f, contat <lb/>qud cum int ex quantitates, qud fieri pote<lb/>runt quindecim coniugationes, quas poui la<lb/>tere facilitatis gratia, quibus repondent totidem <lb/><arrow.to.target n="table3"/><lb/>conuer: erunt ergo triginta. </s>
<s id="id2661580">Singul autem ha <lb/>rum produci pount duodecim modis: ductis <lb/><figure id="id.015.01.027.2.jpg" xlink:href="015/01/027/2.jpg"/>duodecim in triginta, fiunt trecenti exaginta mo <lb/>di. </s>
<s id="id2661612">Et hoc et clarum pere, modo <expan abbr="demõ&longs;tremus">demontremus</expan>, <lb/>quod inguli horum modorum posint produ<lb/>ci duodecim modis, & capiamus ab primam qu <lb/>potet produci ex c d & e f: Item ambabus con<lb/>ueris d c & fe: & rurus altera recta altera con<lb/>uera: & hoc bifariam c d & f e, & d c & e f, unt er<lb/>go iam quatuor modi. </s>
<s id="id2661687">Totidem ex c e & d f, toti<lb/>demque ex c f & d e, igitur erunt duodecim mo<lb/>di, quibus produci poe intelligitur propor<lb/>tio a ad b.</s></p><table><table.target id="table2"/><row><cell>a</cell><cell>b</cell></row><row><cell>---</cell><cell>---</cell></row><row><cell>c</cell><cell>d</cell></row><row><cell>---</cell><cell>---</cell></row><row><cell>e</cell><cell>f</cell></row><row><cell>---</cell><cell>---</cell></row></table><table><table.target id="table3"/><row><cell>a b</cell><cell>b a</cell></row><row><cell>a c</cell><cell>c a</cell></row><row><cell>a d</cell><cell>d a</cell></row><row><cell>a e</cell><cell>e a</cell></row><row><cell>a f</cell><cell>f a</cell></row><row><cell>b c</cell><cell>c b</cell></row><row><cell>b d</cell><cell>d b</cell></row><row><cell>b e</cell><cell>e b</cell></row><row><cell>b f</cell><cell>f b</cell></row><row><cell>c d</cell><cell>d c</cell></row><row><cell>c e</cell><cell>e c</cell></row><row><cell>c f</cell><cell>f c</cell></row><row><cell>d e</cell><cell>e d</cell></row><row><cell>d f</cell><cell>f d</cell></row><row><cell>e f</cell><cell>f e</cell></row><row><cell>direc.</cell><cell>conuer.</cell></row></table><p type="main">
<s id="id2661881">Propoitio quarta.</s></p><p type="main">
<s id="id2661892">Si fuerit proportio primi ad ecundum produ<lb/>cta ex proportionibus tertij ad quartum, & quin <lb/>ti ad extum, producetur etiam ex proportione <lb/>tertij ad extum, & quinti ad quartum.</s></p><p type="main">
<s id="id2661922">Sit proportio a b producta ex proportioni<lb/><arrow.to.target n="table4"/><lb/><figure id="id.015.01.027.3.jpg" xlink:href="015/01/027/3.jpg"/>bus c ad d, & e ad f, dico quod etiam erit produ</s></p><table><table.target id="table4"/><row><cell>a</cell><cell>b</cell><cell/></row><row><cell>c</cell><cell>e</cell><cell>g</cell></row><row><cell>d</cell><cell>f</cell><cell>h</cell></row><row><cell>---</cell><cell>---</cell><cell>---</cell></row><row><cell>c</cell><cell>e</cell><cell>g</cell></row><row><cell>f</cell><cell>d</cell><cell>h</cell></row></table><p type="main">
<s id="id2662017"><arrow.to.target n="marg20"/><lb/>cta ex proportionibus c ad f, & e ad d, diponan<lb/>tur ut in figura & fiat ex c in e g, & ex d in fh, ergo <lb/><arrow.to.target n="marg21"/><lb/>per primam harum g ad h ut a ad b, ed per pr<lb/>uppoita in ecunda productione etiam prode<lb/>unt g & h, igitur per primam propoitionem ha<lb/>rum a ad b proportio producitur ex proportionibus c ad f terti <lb/>cilicet ad extam, & e ad d quint ad quartam, quod fuit <expan abbr="propo&longs;itũ">propoitum</expan>.</s></p><p type="margin">
<s id="id2662106"><margin.target id="marg20"/>P<emph type="italics"/>er<emph.end type="italics"/> 8. <emph type="italics"/>petit.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2662139"><margin.target id="marg21"/>I<emph type="italics"/>n<emph.end type="italics"/> 13. <emph type="italics"/>petit.<emph.end type="italics"/></s></p><p type="main">
<s id="id2662173">Propoitio quinta.</s></p><p type="main">
<s id="id2662184">Si fuerit proportio primi ad ecundum producta ex proportio<lb/>ne tertij ad quartum, & quinta ad extum: erit proportio tertij ad <lb/>extum producta ex proportionibus primi ad ecundum, & quar<lb/>ti ad quintum.</s></p>
<pb xlink:href="015/01/028.jpg" pagenum="9"/><p type="main"><figure id="id.015.01.028.1.jpg" xlink:href="015/01/028/1.jpg"/>
<s id="id2662233">Sit proportio a ad b producta ex proportio<lb/><arrow.to.target n="marg22"/><lb/><arrow.to.target n="table5"/><lb/>nibus c ad d, & e ad f, dico quod proportio c ad <lb/>f producitur ex proportione a ad b & d ad e. </s>
<s id="id2662258">In<lb/>terponam d e inter c & f, eritque ex ecunda pro<lb/>poitione repetita proportio c ad f producta ex <lb/>tribus proportionibus c ad d, d ad e, e ad f, ed <lb/>proportiones c ad d, & e ad f producunt pro<lb/><figure id="id.015.01.028.2.jpg" xlink:href="015/01/028/2.jpg"/>portionem a ad b, igitur proportio c ad f produ <lb/>citur ex proportionibus a ad b, & e ad f.<lb/><arrow.to.target n="table6"/></s></p><p type="margin">
<s id="id2662315"><margin.target id="marg22"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><table><table.target id="table5"/><row><cell>a</cell><cell>b</cell></row><row><cell>---</cell><cell>---</cell></row><row><cell>c</cell><cell>e</cell></row><row><cell>---</cell><cell>---</cell></row><row><cell>d</cell><cell>f</cell></row><row><cell>---</cell><cell>---</cell></row></table><table><table.target id="table6"/><row><cell>c</cell></row><row><cell>-----</cell></row><row><cell>d</cell></row><row><cell>-----</cell></row><row><cell>e</cell></row><row><cell>-----</cell></row><row><cell>f</cell></row><row><cell>-----</cell></row></table><p type="main">
<s id="id2662429">Propoitio exta.</s></p><p type="main">
<s id="id2662442">Ex trecentis exaginta modis producenda<lb/>rum proportionum triginta ex tantum ee ne<lb/>cearios.<lb/><arrow.to.target n="table7"/></s></p><table><table.target id="table7"/><row><cell>c</cell><cell>p</cell></row><row><cell>---</cell><cell>---</cell></row><row><cell>a</cell><cell>d</cell></row><row><cell>---</cell><cell>---</cell></row><row><cell>b</cell><cell>e</cell></row><row><cell>---</cell><cell>---</cell></row></table><p type="main"><figure id="id.015.01.028.3.jpg" xlink:href="015/01/028/3.jpg"/>
<s id="id2662537">Per quartam enim proportio a ad b produ<lb/><arrow.to.target n="marg23"/><lb/>citur bifariam, & ex c ad d, & e ad f, & ex c ad f, & <lb/>e ad d. </s>
<s id="id2662553">& perpr cedentem c ad f producitur ex <lb/>a ad b, & d ad e, & per quartam rurus ex a ad e, <lb/>& d ad b. </s>
<s id="id2662571">Et per prcedentem rutus a ad e ex c <lb/>ad f & b ad d, igitur per quartam eadem produ<lb/>cetur ex c ad d & b ad f. </s>
<s id="id2662590">Quare per prceden<lb/>tem c ad f ex a ad e, & d ad b, & ita diponemus <lb/>hos modos in tabula. </s>
<s id="id2662608">Vides etiam <lb/><arrow.to.target n="table8"/><lb/><figure id="id.015.01.028.4.jpg" xlink:href="015/01/028/4.jpg"/>aliquos modos non produci, ut pri<lb/>mi ad quartum nec ad extum, & li<lb/>quet, qud cm int quindecim o<lb/>mnes modi qui produci poe intelli<lb/>guntur, & nouem tantum producan<lb/>tur ex ee, qui non producuntur, quos <lb/>eorum in tabula coniunxi. </s>
<s id="id2662682">Et con<lb/>tat etiam, quod totidem conueri ci<lb/>licet decem octo <expan abbr="producũtur">producuntur</expan>, de qui<lb/>bus diximus, ut int omnes triginta <lb/>ex, qui contat ex duabus propoi<lb/>tionibus prmisis, & hac tertia, <expan abbr="quã">quam</expan> <lb/>adiungemus cilicet, qud propor<lb/>tio primi ad tertium producatur ex <lb/>proportionibus <expan abbr="&longs;ecũdi">ecundi</expan> ad quartum, <lb/>& quinti ad <expan abbr="&longs;extũ">extum</expan>. </s>
<s id="id2662790">Hoc enim ex pr<lb/>cedentibus non liquet: ben liquet <lb/>permutatis ordinibus, quod i pro<lb/>portio primi ad tertium producitur,
<pb xlink:href="015/01/029.jpg" pagenum="9 [=10]"/>quod etiam propor<lb/><figure id="id.015.01.029.1.jpg" xlink:href="015/01/029/1.jpg"/><arrow.to.target n="marg24"/><lb/>tio primi ad <expan abbr="quintũ">quintum</expan>. <lb/></s>
<s id="id2662851">Nam tertium, & quin <lb/>tum, item que quartum, <lb/>& extum non <expan abbr="diffe-rũt">diffe<lb/>runt</expan> nii ordine uolun <lb/>tario. </s>
<s id="id2662884">Ergo interpoi<lb/>to e inter a, & c per e<lb/>cundam propoitio<lb/>nem proportio a ad c <lb/>producitur ex proportionibus a ad <lb/>e, & e ad c, ut ex demontratis in pr<lb/>enti proportio a ad c producitur ex <lb/>c ad f & b ad d. </s>
<s id="id2662932">Proportio ergo a ad <lb/>c producitur ex proportionibus e <lb/>ad c & c ad f & b ad d, at e ad c & c ad <lb/>f producunt eam, qu et e ad f per <lb/><expan abbr="&longs;ecũdam">ecundam</expan> propoitionem. </s>
<s id="id2662968">Igitur pro<lb/>portio a ad c producitur ex propor<lb/>tionibus b ad d ecundi ad quartum, <lb/>& e ad f quinti ad extum. </s>
<s id="id2662990">Hc Al<lb/>chindus in uo libello: ed licet inge<lb/>nio a ualde: parum <expan abbr="tam&etilde;">tamen</expan> utilia olim <lb/><expan abbr="erãt">erant</expan> necearia ad intelligendum ma<lb/>gnam <expan abbr="cõpo&longs;itionem">compoitionem</expan> Ptolemi, nunc <lb/>potquam Heber has ex quantita<lb/>tes traduxit ad quatuor, prorus hc <lb/>cientia ulli uui ee deijt.<lb/><arrow.to.target n="table9"/></s></p><p type="margin">
<s id="id2663103"><margin.target id="marg23"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s id="id2663129"><margin.target id="marg24"/>Modi qui <expan abbr="nõ">non</expan> <lb/>producuntur <lb/>pri. ad quartu <lb/>pri. ad extum <lb/>ec. ad <expan abbr="tertiũ">tertium</expan> <lb/>ec. ad <expan abbr="quintũ">quintum</expan> <lb/>tert. </s>
<s id="id2663195">ad quint. <lb/></s>
<s id="id2663201">quart. </s>
<s id="id2663205">ad ext.</s></p><table><table.target id="table8"/><row><cell/><cell>Primi ad ecundum.</cell></row><row><cell>1</cell><cell>tertij ad <expan abbr="quartũ">quartum</expan>, & quin</cell></row><row><cell/><cell>ti ad extum.</cell></row><row><cell>2</cell><cell>tertij ad extum, & quin</cell></row><row><cell/><cell>ti ad quartum.</cell></row><row><cell/><cell>Primi ad tertium.</cell></row><row><cell>3</cell><cell>ecundi ad quartum, &</cell></row><row><cell/><cell>quinti ad extum.</cell></row><row><cell>4</cell><cell>ecundi ad extum, &</cell></row><row><cell/><cell>quinti ad quartum.</cell></row><row><cell/><cell>Primi ad quintum.</cell></row><row><cell>5</cell><cell>ecundi ad <expan abbr="&longs;extũ">extum</expan>, & ter-</cell></row><row><cell/><cell>tij ad quartum.</cell></row><row><cell>6</cell><cell>ecundi ad quartum, &</cell></row><row><cell/><cell>tertij ad extum.</cell></row><row><cell/><cell>Secundi ad quartum.</cell></row><row><cell>7</cell><cell>primi ad tertium, & ex</cell></row><row><cell/><cell>ti ad quintum.</cell></row><row><cell>8</cell><cell>primi ad quintum, et ex</cell></row><row><cell/><cell>ti ad tertium.</cell></row><row><cell/><cell>Secundi ad extum.</cell></row><row><cell>9</cell><cell>primi ad <expan abbr="quintũ">quintum</expan>, & quar</cell></row><row><cell/><cell>ti ad tertium.</cell></row><row><cell>10</cell><cell>primi ad <expan abbr="tertiũ">tertium</expan>, & quar-</cell></row><row><cell/><cell>ti ad quintum.</cell></row><row><cell/><cell>Tertij ad quartum.</cell></row><row><cell>11</cell><cell>primi ad ecundum, &</cell></row><row><cell/><cell>exti ad quintum.</cell></row><row><cell>12</cell><cell>primi ad quintum, & ex</cell></row><row><cell/><cell>ti ad ecundum.</cell></row><row><cell/><cell>Tertij ad extum.</cell></row><row><cell>13</cell><cell>primi ad ecundum, &</cell></row><row><cell/><cell>quarti ad quintum.</cell></row><row><cell>14</cell><cell>primi ad quintum, &</cell></row><row><cell/><cell>quarti ad ecundum.</cell></row><row><cell/><cell>Quarti ad quintum.</cell></row><row><cell>15</cell><cell>ecundi ad primum, &</cell></row><row><cell/><cell>tertij ad extum.</cell></row><row><cell>16</cell><cell>ecundi ad extum, & ter</cell></row><row><cell/><cell>tij ad primum.</cell></row><row><cell/><cell>Quinti ad extum.</cell></row><row><cell>17</cell><cell>primi ad ecundum, &</cell></row><row><cell/><cell>quarti ad tertium.</cell></row><row><cell>18</cell><cell>primi ad <expan abbr="tertiũ">tertium</expan>, & quar-</cell></row><row><cell/><cell>ti ad ecundum.</cell></row></table><table><table.target id="table9"/><row><cell>a</cell><cell>e c</cell><cell>a e</cell><cell>e c</cell></row><row><cell/><cell/><cell>c b</cell><cell>e</cell></row><row><cell/><cell/><cell>f d</cell><cell>c</cell></row><row><cell/><cell/><cell/><cell>f</cell></row></table><p type="main">
<s id="id2663663">Propoitio eptima.</s></p><figure id="id.015.01.029.2.jpg" xlink:href="015/01/029/2.jpg"/><p type="main">
<s id="id2663685">In modis qui neceari produ<lb/>cuntur ex duabus proportionibus, <lb/>cum du quantitates ex illis, qu mo <lb/><figure id="id.015.01.029.3.jpg" xlink:href="015/01/029/3.jpg"/>dos conficiunt, quales fuerint: pro<lb/><arrow.to.target n="table10"/><lb/>portio producta ad quatuor quanti<lb/>tates omiologas reducetur.<lb/><arrow.to.target n="marg25"/></s></p><p type="margin">
<s id="id2663748"><margin.target id="marg25"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><table><table.target id="table10"/><row><cell>a</cell><cell>b</cell></row><row><cell>---</cell><cell>---</cell></row><row><cell>c</cell><cell>e</cell></row><row><cell>---</cell><cell>---</cell></row><row><cell>d</cell><cell>f</cell></row><row><cell>---</cell><cell>---</cell></row></table><p type="main">
<s id="id2663822">Sint ex quantitates a b c d e f, & <lb/>producatur proportio a ad b ex pro<lb/>portione c ad d, & e ad f, tu cis, qud <lb/>modi recepti unt prima cum ecunda, tertia uel quinta, & ecunda <lb/>cum quarta, & exta, & tertia imiliter cum eidem, & quinta eodem <lb/>modo cum eidem: i igitur du quantitates ex his, qu faciunt pro
<pb xlink:href="015/01/030.jpg" pagenum="11"/>portionem productam inter e fuerint quales reducetur hc pro<lb/>portio ad quatuor quantitates omologas, cilicer abiectis amba<lb/>bus qualibus. </s>
<s id="id2663916">Sit gratia exempli prima qualis quint: & quia <lb/>in octauo modo proportio <expan abbr="&longs;ecũdi">ecundi</expan> ad quartum producitur ex pro<lb/>portione primi ad quintum, & exti ad tertium, ergo per expoita <lb/>proportio ecundi ad quartum, ut exti ad tertium, & ita permutan<lb/>do, & conuertendo ecundi ad extum, ut quarti ad tertium, & tertij </s></p><p type="main">
<s id="id2663978"><arrow.to.target n="marg26"/><lb/>ad quartum, ut exti ad ecundum.</s></p><p type="margin">
<s id="id2663998"><margin.target id="marg26"/>V<emph type="italics"/>ndecima <lb/>petitione.<emph.end type="italics"/></s></p><p type="main">
<s id="id2664025">Propoitio octaua.</s></p><p type="main">
<s id="id2664036">Si duarum <expan abbr="proportionũ">proportionum</expan> uperiores numeri alternatim cum infe <lb/>rioribus multiplicentur, atque coniungantur: erit proportio aggre<lb/>gati ad productum ex inferioribus inuicem proportio ex primis <lb/>proportionibus compoita.</s></p><figure id="id.015.01.030.1.jpg" xlink:href="015/01/030/1.jpg"/><p type="main">
<s id="id2664079">Sit proportio una a ad b, alia c ad d, ducatur b in <lb/><arrow.to.target n="marg27"/><lb/>c, fiatque e & a in d, & fiat f, iunganturque e & f & fiat h, <lb/>& ducatur b in d et fiat g: dico <expan abbr="proportion&etilde;">proportionem</expan> h g com<lb/>poitam ee ex proportione a ad b, & c ad d. </s>
<s id="id2664119">Quia <lb/><arrow.to.target n="marg28"/><lb/>enim ex b in c fit e, & ex b in d fit g, erit proportio e <lb/>ad g, ut c ad d, & imiliter, quia ex d in a fit f, & ex d in b fit g, erit f ad <lb/>g ut a ad b. </s>
<s id="id2664145">Sed e & f componunt h, igitur proportio h ad g et com <lb/>poita ex proportionibus e & f ad g, igitur per communem animi <lb/>ententiam, & diffinitionem compoit proportionis, proportio h <lb/><arrow.to.target n="marg29"/><lb/>ad g compoita et ex proportionibus a ad b, & c ad d, quod et <lb/>propoitum.</s></p><p type="margin">
<s id="id2664201"><margin.target id="marg27"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s id="id2664227"><margin.target id="marg28"/>E<emph type="italics"/>x<emph.end type="italics"/> 13 <emph type="italics"/>peti<lb/>tione.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2664265"><margin.target id="marg29"/>P<emph type="italics"/>er<emph.end type="italics"/> 14 <emph type="italics"/>diffi <lb/>nitionem.<emph.end type="italics"/></s></p><p type="main">
<s id="id2664301">Propoitio nona.</s></p><p type="main">
<s id="id2664312">Si duarum proportionum uperiores numeri alternatim cum <lb/>inferioribus multiplicentur, minusque productum ex maiore detra<lb/>hatur, erit reidui ad productum ex inferioribus proportio uelut <lb/>illa, qu relinquitur detracta minore proportione ex maiore.</s></p><p type="main">
<s id="id2664341">Hc eodem modo probatur, ut prcedens, nii quod h fit de<lb/><arrow.to.target n="marg30"/><lb/>tracto minore: gratia exempli ex f, & ita ex diffinitione patet pro<lb/>poitum.</s></p><p type="margin">
<s id="id2664379"><margin.target id="marg30"/>C<emph type="italics"/>or<emph.end type="italics"/>_{m}. <lb/>152.</s></p><p type="main">
<s id="id2664407">Propoitio decima.</s></p><p type="main">
<s id="id2664418">Si fuerit alicuius quantitatis ad unam partem proportio uelut al <lb/>terius partis ad <expan abbr="&longs;ecũdam">ecundam</expan> quantitatem erit proportio cuiuuis quan <lb/>titatis eiudem generis ad ecundam compoita proportio ex pro<lb/>portionibus eiudem quantitatis aumpt ad utran que partem pri<lb/>m quantitatis eorum.<lb/><arrow.to.target n="marg31"/></s></p><p type="margin">
<s id="id2664490"><margin.target id="marg31"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><figure id="id.015.01.030.2.jpg" xlink:href="015/01/030/2.jpg"/><p type="main">
<s id="id2664524">Sit a b quantitas diuia in c, & i cut a b ad a c, <lb/>ita b c ad d: eritque iterum permutando a b ad b c, <lb/>ut a c ad d, & umatur qudam quantitas e eiu
<pb xlink:href="015/01/031.jpg" pagenum="12"/>dem tamen generis, cum illis dico qud proportio e ad d et com<lb/>poita ex proportionibus e ad a c, & e ad b c. </s>
<s id="id2664574">Poita ergo e tan<08> u<lb/>periore numero, & a c & c b inferioribus, erit ex octaua propoitio<lb/>ne huius proportio productorum ex e in a c, & coniunctorum, & <lb/>ex conequenti per primam ecundi Elementorum producti ex e in <lb/>a b ad productum ex a c in c b compoita ex proportionibus e ad <lb/>a c, & e ad c b: at quod fit ex a c in c b, et quale ei quod fit ex a b in <lb/>d, eo qud a b, a c, c b & d unt omiolog per decimamextam exti <lb/><expan abbr="Elem&etilde;torum">Elementorum</expan>: Proportio igitur producti ex e in a b ad productum <lb/>ex d in a b et compoita ex proportionibus e ad a c, & e ad e b: At <lb/>proportio producti ex e in a b ad productum ex d in a b, et uelut e <lb/><arrow.to.target n="marg32"/><lb/>ad d. </s>
<s id="id2664681">per uppoita igitur proportio e ad d et compoita ex propor<lb/>tionibus e ad a c, & e ad b c, quod fuit demontrandum.</s></p><p type="margin">
<s id="id2664709"><margin.target id="marg32"/>13. P<emph type="italics"/>etit.<emph.end type="italics"/></s></p><p type="main">
<s id="id2664734">Propoitio undecima.</s></p><p type="main">
<s id="id2664744">Proportio aggregati quarumlibet duarum quantitatum ad ag<lb/>gregatum duarum qualium quantitatum et compoita ex pro<lb/>portionibus primis, & diuia per duplam.<lb/><arrow.to.target n="marg33"/></s></p><p type="margin">
<s id="id2664780"><margin.target id="marg33"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2664806">Sit proportio a ad c, & b ad d, & int c & d <lb/><figure id="id.015.01.031.1.jpg" xlink:href="015/01/031/1.jpg"/><lb/>quales, dico qud proportio a b ad c d et <lb/>compoita ex proportionibus a ad c, & b ad <lb/>d diuio compoito per duplam. </s>
<s id="id2664848">Quia enim </s></p><p type="main">
<s id="id2664856"><arrow.to.target n="marg34"/><lb/>c & d unt quales, erit b ad c, ut b ad d, qua<lb/>re ex diffinitione cm proportio a b ad c d <lb/><arrow.to.target n="marg35"/><lb/>it compoita ex proportionibus a ad c, & b <lb/>ad c, erit etiam compoita ex dictis ex propoitione a ad c, & b ad d, <lb/><arrow.to.target n="marg36"/><lb/>tatuatur ergo e qualis c d media inter a b & c. </s>
<s id="id2664918">Et erit per ecun<lb/>dam propoitionem proportio aggregati a b ad c producta ex <lb/><arrow.to.target n="marg37"/><lb/>proportione aggregati a b ad c, & e ad c, igitur proportio a b ad e <lb/>erit proportio a b ad c, diuia per proportionem e ad c, ed e ad c et <lb/><arrow.to.target n="marg38"/><lb/>dupla: igitur proportio a b ad c d et proportio a b ad c diuia per <lb/>duplam.</s></p><p type="margin">
<s id="id2664978"><margin.target id="marg34"/>E<emph type="italics"/>x exta<emph.end type="italics"/> A<emph type="italics"/>nim. <lb/>com. </s>
<s id="id2665011">ententia.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2665024"><margin.target id="marg35"/>D<emph type="italics"/>ecimaquarta<emph.end type="italics"/></s></p><p type="margin">
<s id="id2665047"><margin.target id="marg36"/>13. P<emph type="italics"/>etit.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2665072"><margin.target id="marg37"/>P<emph type="italics"/>er<emph.end type="italics"/> 2. P<emph type="italics"/>etit.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2665106"><margin.target id="marg38"/>P<emph type="italics"/>er quintam<emph.end type="italics"/><lb/>A<emph type="italics"/>nim. </s>
<s id="id2665133">com. </s>
<s id="id2665136">en <lb/>tentiam.<emph.end type="italics"/></s></p><p type="main">
<s id="id2665153">Propoitio duodecima.</s></p><p type="main">
<s id="id2665164">Propoitis duabus proportionibus unam alteri iungere abque <lb/>multiplicatione.<lb/><arrow.to.target n="marg39"/></s></p><p type="margin">
<s id="id2665188"><margin.target id="marg39"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. <lb/>10. P<emph type="italics"/>etit.<emph.end type="italics"/></s></p><p type="main">
<s id="id2665226">Sint propoit proportiones a ad c & <lb/><figure id="id.015.01.031.2.jpg" xlink:href="015/01/031/2.jpg"/><lb/>b ad d, & aumo e ad c, iuxta ea qu Eu<lb/>clides demontrauit, ut b ad d, erit igitur </s></p><p type="main">
<s id="id2665266"><arrow.to.target n="marg40"/><lb/>proportio a e ad c, compoita ex proportionibus a ad c, & e ad c, <lb/>ed proportio e ad c et, ut b ad d, igitur proportio a e ad c compo<lb/>ita et ex proportionibus a ad c, & b ad d.</s></p><p type="margin">
<s id="id2665303"><margin.target id="marg40"/>E<emph type="italics"/>x generali <lb/>com.<emph.end type="italics"/> A<emph type="italics"/>nim. en <lb/>tentia.<emph.end type="italics"/></s>
</p><p type="main">
<s id="id2665348">Aliter ex b in c fiat fex a in d, g ex c in d h coniunctum ex f g, k.</s></p>
<pb xlink:href="015/01/032.jpg" pagenum="13"/><figure id="id.015.01.032.1.jpg" xlink:href="015/01/032/1.jpg"/><p type="main">
<s id="id2665374">Quia ergo ex c in b fit f, ex c in d h, erit f ad h, <lb/>ut b ad d, igitur ut e ad c, ed a ad c, ut g ad h igi <lb/><arrow.to.target n="marg41"/><lb/>tur a e ad c, ut k ad h, ed k ad h cmponitur ex <lb/>proportionibus a ad c, & b ad d. </s>
<s id="id2665401">Ex octaua ha <lb/>rum igitur proportio a c ad c compoita et ex <lb/>eidem. </s>
<s id="id2665424">Foran quis dicat hanc eandem ee <lb/>octau ed <expan abbr="nõ">non</expan> et, in illa enim proportio com<lb/>paratur ad productum, in hac ad unam ex <lb/>quantitatibus.</s></p><p type="margin">
<s id="id2665468"><margin.target id="marg41"/>P<emph type="italics"/>er<emph.end type="italics"/> 13. P<emph type="italics"/>et.<emph.end type="italics"/></s></p><p type="main">
<s id="id2665503">Ex hoc equitur qud: Qulibet du quantitates quarum ag<lb/><arrow.to.target n="marg42"/><lb/>gregatum etidem ad eam quantitatem, componunt eandem pro<lb/>portionem.</s></p><p type="margin">
<s id="id2665539"><margin.target id="marg42"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2665565">Propoitio tertiadecima.</s></p><p type="main">
<s id="id2665576">Proportio confua aggregati prim & terti quatuor quantita<lb/>tum omiologarum ad <expan abbr="aggregatũ">aggregatum</expan> ecund & quart, et uelut com <lb/>poita ex eidem diuia per duplam.<lb/><arrow.to.target n="marg43"/></s></p><p type="margin">
<s id="id2665635"><margin.target id="marg43"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2665660">Sint a ad b, ut c ad d, dico, qud erit confua <lb/><figure id="id.015.01.032.2.jpg" xlink:href="015/01/032/2.jpg"/><arrow.to.target n="table11"/><lb/>proportio a c aggregati ad <expan abbr="aggregatũ">aggregatum</expan> b d, com <lb/>poit ex his proportionibus diui per du<lb/>plam qualis. </s>
<s id="id2665715">Erit enim aggregati ex a c ad aggregatum ex b d, ue<lb/>lut a ad b per 18 quinti Elementorum. </s>
<s id="id2665726">Sed proportiones a ad b, <lb/>& c ad d componunt proportionem producti a in d, & c in b per <lb/>octauam harum, ad <expan abbr="productũ">productum</expan> ex b in d, productum uer ex a in d <lb/>et quale producto ex b in c per decimamextam exti Elemento<lb/>rum, & proportio producti ex b in c ad productum ex b in d et ue <lb/>lut c ad d, quare ut aggregati a c ad aggregatum b d, igitur propor<lb/>tio compoita ex a ad b, & c ad d, et uelut confua bis umpta. </s>
<s id="id2665793">Igi<lb/>tur confua et uelut compoita diuia per duplam per modum un<lb/>decim huius.</s></p><table><table.target id="table11"/><row><cell>a</cell><cell>c</cell></row><row><cell>-----</cell><cell>-----</cell></row><row><cell>b</cell><cell>d</cell></row><row><cell>---</cell><cell>---</cell></row></table><p type="main">
<s id="id2665860">Propoitio quartadecima.</s></p><p type="main">
<s id="id2665871">Proportiones confu, & coniunct in tribus quantitatibus in<lb/>uicem commutantur.</s></p><figure id="id.015.01.032.3.jpg" xlink:href="015/01/032/3.jpg"/><p type="main">
<s id="id2665899">Sint tres quantitates, dico, quod proportio c </s></p><p type="main">
<s id="id2665908"><arrow.to.target n="marg44"/><lb/>ad a b confua et, conuera coniunct a & b ad <lb/><arrow.to.target n="marg45"/><lb/>c. </s>
<s id="id2665939">Nam per dicta proportio a b ad c efficit con<lb/>iunctam ex a b ad c: ed c ad a b conuera et eius, qu et a b ad c, & <lb/>proportio c ad a b et confua eius, qu et c ad a & b. </s>
<s id="id2665977">Igitur pro<lb/>portio confua in tribus quantitatibus et contraria coniunct in <lb/>eidem.</s></p><p type="margin">
<s id="id2666004"><margin.target id="marg44"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s id="id2666030"><margin.target id="marg45"/>14. <emph type="italics"/>diff.<emph.end type="italics"/></s></p><p type="main">
<s id="id2666053">Ex quauis ergo illarum data, data erit & reliqua.<lb/><arrow.to.target n="marg46"/></s></p>
<pb xlink:href="015/01/033.jpg" pagenum="14"/><p type="margin">
<s id="id2666076"><margin.target id="marg46"/>P<emph type="italics"/>er<emph.end type="italics"/> 18. <emph type="italics"/>diff.<emph.end type="italics"/></s></p><p type="main">
<s id="id2666110">Propoitio quintadecima.</s></p><p type="main">
<s id="id2666121">Si fuerint quatuor quantitas-proportio confua aggregati pri<lb/>m & terti ad aggregatum ecund, & quart erit ut monadis <lb/>addito prouentu, qui fit diuia differentia differentiarum prim & <lb/>ecund, atque quart & terti per aggregatum terti, & quart ad <lb/>ipam monadem.</s></p><figure id="id.015.01.033.1.jpg" xlink:href="015/01/033/1.jpg"/><p type="main">
<s id="id2666195">Sint quatuor quantitates a b, c, d, e f, & <lb/><arrow.to.target n="marg47"/><lb/>it a b maior cin a h, & e fmaior d in f g, & <lb/>differentia f g & a h it a k: dico proportio<lb/>nem a b, & d confuam ad c & e f, ee ut mo <lb/>nadis addito prouentu, uel detracto a k diui per aggregatum c. <lb/>& e f ad ipam monadem, & manifetum et, qud potet continge<lb/>re pluribus modis: Primus ut a b it maior c & e f minor d, & tunc <lb/>differenti coniungentur, & prouentus, addetur monadi. </s>
<s id="id2666271">Idem fa<lb/>ciendum erit i a b it maior c, & e f it minor d, ed exceus uperet <lb/>defectum. </s>
<s id="id2666300">At i uel a b it minor c, & e f maior d, uel ita minor, ut c <lb/>exceus upra b a it maior defectu, detrahemus prouentum mo<lb/>nade. </s>
<s id="id2666332">Alia cautio et qud i fuerint utrinque exceus, aut defectus, <lb/>minuemus minorem de maiore: i autem unus it exceus alter de<lb/>fectus, iungemus illos, & pot diuidemus. </s>
<s id="id2666370">uno ergo demontrato <lb/>ut pote primo intelligentur reliqui. </s>
<s id="id2666381">Quia ergo b h et qualis c & <lb/>e g qualis d & h k qualis g f, erit ex communi animi ententia ag <lb/>gregatum ex d & k b quale aggregato ex c & e f, igitur per dicta <lb/>proportio aggregati ad aggregatum et unum. </s>
<s id="id2666416">at uer diuia k a <lb/>per c & e f fit quantum diuia eadem per b k, & d, ed diuia k a per b <lb/>k, & d iunctas, exit proportio a k ad aggregatum b k & d: igitur di<lb/>uia a k per aggregatum e f & c, exibit eadem proportio, igitur a b <lb/>& d ad aggregatum c & e f et coninncta ex monade & proportio<lb/>ne a k ad aggregatum c & e f, quod erat demontrandum.</s></p><p type="margin">
<s id="id2666472"><margin.target id="marg47"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><figure id="id.015.01.033.2.jpg" xlink:href="015/01/033/2.jpg"/><p type="main">
<s id="id2666507">Ex hoc patet quod proportionum confuio <lb/><arrow.to.target n="marg48"/><lb/>fit iunctis denominatoribus numeratoris: mul<lb/>tiplicatio multiplicatis: additio multiplicatis <lb/>decuatim in numeratores ad productum ex <lb/>denominatoribus, ut in exemplis.</s></p><p type="margin">
<s id="id2666543"><margin.target id="marg48"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2666569">Propoitio extadecima.</s></p><p type="main">
<s id="id2666582">Omnium quatuor quantitatum propoita <lb/>prima, qu non minorem habet proportionem <lb/>ad uam correpondentem, qum alia ad aliam <lb/><figure id="id.015.01.033.3.jpg" xlink:href="015/01/033/3.jpg"/><lb/>erit proportio confua illarum, ut pro<lb/>ducti ex aggregato prim & terti in
<pb xlink:href="015/01/034.jpg" pagenum="15"/>tertiam, ad productum ex aggregato terti & omiotat ad ecun<lb/>dam in ipam quartam.</s></p><p type="main">
<s id="id2666661">Hc magis reducit confuam proportionem ad notitiam, qum, <lb/>prcedens, quia reducit ad proportionem <expan abbr="productã">productam</expan>, qu operatio <lb/>et implicisima, iue per multiplicationem quantitatum fiat, du <lb/>unt tantum multiplicationes, iue per eundem terminum ufficit <lb/>alium addere. </s>
<s id="id2666722">Summatur ergo a b, c, d & e, & non it maior propor<lb/>tio d ad e, qum a b ad c, & tatuatur tunc prima a b, ecunda c, ter<lb/>tia d, quarta e, & potquam non et minor ratio a b ad c, qum d ad <lb/>e, umatur a f ad c, ut d ad e. </s>
<s id="id2666763">licet enim hoc facere. </s>
<s id="id2666767">Dico quod pro<lb/>portio confufa a b & d ad c & e et uelut producti ex aggregato a b <lb/>& d in d ad productum ex aggregato a f & d in e. </s>
<s id="id2666784">Statuatur aggre<lb/><arrow.to.target n="marg49"/><lb/>gatum a b & d linea a d prima quantitas, & aggregatum a f & d, <lb/><figure id="id.015.01.034.1.jpg" xlink:href="015/01/034/1.jpg"/><lb/>a d ecunda quantitas, & d tertia, <lb/>& c quarta, & ex a b in d fiat g, ex <lb/>a d in e fiat h, erit ergo per pri<lb/>mam propoitionem g ad h pro<lb/><arrow.to.target n="marg50"/><lb/>ducta ex proportionibus a b d ad <lb/>a f d, & d ad c. </s>
<s id="id2666845">Sed proportio a f d <lb/>ad aggregatum c e, et uelut d ad <lb/>e. </s>
<s id="id2666858">Proportio uer a b d ad a f d, & <lb/>a f d ad e c producunt proportio<lb/>nem a b d ad c & e per ecundam propoitionem, harum igitur con<lb/>ua a b ad c, & d ad e, & et proportio a b d ad c & e, producuntur <lb/>ex proportionibus a b d ad a f d, & d ad e. </s>
<s id="id2666896">Ergo proportio g ad h <lb/>et confua ex a b ad e, & d ad e, quod erat demontrandum.</s></p><p type="margin">
<s id="id2666917"><margin.target id="marg49"/>P<emph type="italics"/>er<emph.end type="italics"/> 10. P<emph type="italics"/>et.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2666953"><margin.target id="marg50"/>P<emph type="italics"/>er<emph.end type="italics"/> 13. P<emph type="italics"/>et.<emph.end type="italics"/></s></p><p type="main">
<s id="id2666989">Propoitio decimaeptima.</s></p><p type="main">
<s id="id2667001">Omnes du proportiones conuer producunt qualem pro<lb/>portionem.<lb/><arrow.to.target n="table12"/></s></p><table><table.target id="table12"/><row><cell>a</cell></row><row><cell>-----</cell></row><row><cell>b</cell></row><row><cell>---</cell></row><row><cell>c</cell></row><row><cell>----</cell></row></table><p type="main">
<s id="id2667062">Sint du proportiones a ad b & b ad a conuera, <lb/><figure id="id.015.01.034.2.jpg" xlink:href="015/01/034/2.jpg"/><arrow.to.target n="marg51"/><lb/>dico, qud producunt proportionem qualem. </s>
<s id="id2667095">fiat <lb/>enim b ad c, ut b ad a, erit igitur a qualis c & b c con <lb/><arrow.to.target n="marg52"/><lb/>uera eius qu et a ad b, ed per ecundam harum <lb/>proportiones a ad b, & b ad c producunt propor<lb/>tionem a ad c, igitur proportiones etiam a ad b & b ad a produ<lb/>cunt eandem.</s></p><p type="margin">
<s id="id2667148"><margin.target id="marg51"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s id="id2667174"><margin.target id="marg52"/>P<emph type="italics"/>er<emph.end type="italics"/> 6. A<emph type="italics"/>ni<lb/>mi <expan abbr="commun&etilde;">communem</expan> <lb/>ententiam.<emph.end type="italics"/></s></p><p type="main">
<s id="id2667225">Propoitio decimaoctaua.</s></p><p type="main">
<s id="id2667236">Si fuerint quotlibet quantitates in continua proportione multi<lb/>plici prter ultimam: proportio uer penultim ad ultimam qua<lb/>lis reidui prim ad ecundam, erit prim ad aggregatum reliqua<lb/>rum uelut penultim ad ultimam.
<pb xlink:href="015/01/035.jpg" pagenum="16"/><arrow.to.target n="marg53"/></s></p><p type="margin">
<s id="id2667291"><margin.target id="marg53"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2667318">Sint quantitates a b c d in continua proportione multiplici, ed <lb/>d ad e it uelut reidui a & b ad b, dico proportionem a ad b c d e <lb/>ee ut d ad e. </s>
<s id="id2667343">Quia enim et gnomonis e ad quadratum d, ut d ad e <lb/>ex uppoito erit per coniunctam proportionem c & d ad d & e, ut</s></p><p type="main">
<s id="id2667364"><arrow.to.target n="marg54"/><lb/>d ad e, ed e gnomo cum quadrato d efficit qua<lb/><figure id="id.015.01.035.1.jpg" xlink:href="015/01/035/1.jpg"/><lb/>dratum e, igitur ut c quadrati ad d & eiuncta, ita <lb/>d ad e. </s>
<s id="id2667395">Rurus, quia b quadrati ad c quadratum, <lb/><arrow.to.target n="marg55"/><lb/>ut c ad d erit gnomonis b ad quadratum c, ut <lb/>gnomonis c ad quadratum d, & ita d ad e, igitur <lb/><arrow.to.target n="marg56"/><lb/>gnomonum b c cum quadrato d ad aggrega<lb/>tum c d e quadratorum, ut d ad e, ed c gno<lb/>mo cum d quadrato perficit c quadratum, <lb/>& c quadratum cum gnomone b perficit <lb/>quadratum b, igitur proportio quadrati b <lb/>ad quadrata c d e, ut d quadrati a d e. </s>
<s id="id2667447">Et ita <lb/>repetendo de quotuis quantitatibus in infi <lb/>nitum uque. </s>
<s id="id2667460">Hc proponitur ab Archimede in libro de quadrato <lb/>quali parabol, & minus generaliter & pluribus demontratur. <lb/></s>
<s id="id2667480">Ego tamen quia et generalis, decribam illam per corrolarium: ad<lb/>damque aliud quod ex hoc equitur.<lb/><arrow.to.target n="marg57"/></s></p><p type="margin">
<s id="id2667508"><margin.target id="marg54"/>13. P<emph type="italics"/>ropo. <lb/>quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
</p><p type="margin">
<s id="id2667552"><margin.target id="marg55"/>P<emph type="italics"/>er<emph.end type="italics"/> 19. <emph type="italics"/>quin <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2667602"><margin.target id="marg56"/>P<emph type="italics"/>er<emph.end type="italics"/> 12. <emph type="italics"/>quin <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2667650"><margin.target id="marg57"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 1.</s></p><p type="main">
<s id="id2667676">Si fuerint quotlibet <expan abbr="quãtitates">quantitates</expan> omnes analog prter ultimam, <lb/>it autem penultima ad ultimam qualis reidui prim & ecund <lb/>ad ecundam, erit proportio prim ad aggregatum omnium alia<lb/>rum ueluti penultim ad ultimam.</s></p><p type="main">
<s id="id2667731"><arrow.to.target n="marg58"/></s></p><p type="margin">
<s id="id2667743"><margin.target id="marg58"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2667770">Hc enim et euidens, quia conuenit ei demontratio propoita. <lb/><figure id="id.015.01.035.2.jpg" xlink:href="015/01/035/2.jpg"/><lb/>exemplo autem in numeris latere <lb/>poito uides declarationem. </s>
<s id="id2667806">nam <lb/>proportio 16 ad 32 et uelut 27 rei <lb/>dui prim & ecund ad ipam e<lb/>cundam cilicet ad 54.</s></p><p type="main">
<s id="id2667847"><arrow.to.target n="marg59"/></s></p><p type="margin">
<s id="id2667858"><margin.target id="marg59"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 2.</s></p><p type="main">
<s id="id2667885">Ex hoc patet etiam qud aumptis omnibus, ub multiplicibus <lb/>analogi uque in infinitum prima quantitas et multiplex aggre<lb/>gati omnium reliquarum numero 1 m: quo prima et multiplex <lb/>ecund.</s></p><p type="main">
<s id="id2667929"><arrow.to.target n="marg60"/></s></p><p type="margin">
<s id="id2667940"><margin.target id="marg60"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 3.</s></p><p type="main">
<s id="id2667967">Si fuerint quotlibet quantitates in uper particulari proportio<lb/>ne analog, erit proportio prim ad aggregatum omnium in infi<lb/>nitum iuxta proportionem multiplicem conueram illius partis.</s></p><p type="main">
<s id="id2667996"><arrow.to.target n="marg61"/></s></p><p type="margin">
<s id="id2668008"><margin.target id="marg61"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2668034">Velut collect in equialtera dupl in exquitertia tripl in <lb/>exquieptima eptupl. </s>
<s id="id2668067">Vt capio 512 448 392 343, & ita deinceps <lb/>uque in infinitum aggregatum omnium earum erit 3584. Septu
<pb xlink:href="015/01/036.jpg" pagenum="17"/>plum 512, & aggregatum 18. 12. 8. 5 2/3, & ita deinceps in exquialtera <lb/>erit 54 duplum 27 prim in eo ordine.</s></p><p type="head">
<s id="id2668105">SCHOLIVM.</s></p><p type="main">
<s id="id2668115">Ex quo patet genus demontrandi nouun & pulchrum: nam <lb/>upponatur 54, aggregatum duplum 27, prim igitur addito 27 <lb/>ad 54, cum it dimidium, & addito 13 1/2, dimidio 27 ad 27, nam ex <lb/>uppoito quantitas equens et exquialtera ad 27, igitur 81 et du</s></p><p type="main">
<s id="id2668162"><arrow.to.target n="marg62"/><lb/>plum ad 40 1/2. Igitur conuertendo et proportio aggregati prioris <lb/>ad 27 et dupla, ergo aggregatum et 54.<lb/><arrow.to.target n="marg63"/></s></p><p type="margin">
<s id="id2668195"><margin.target id="marg62"/>P<emph type="italics"/>er<emph.end type="italics"/> 18. <emph type="italics"/>quin <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2668245"><margin.target id="marg63"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 4.</s></p><p type="main">
<s id="id2668270">Ex hoc patet eandem generaliter quod proportio maioris quan <lb/>titatis ad aggregatum reliquarum analogarum et, uelut eius quod <lb/>prouenit diuio quadrato maioris termini per differentiam eius, & <lb/>equentis maioris in eadem proportione ad ipum maiorem.</s></p><p type="main">
<s id="id2668300"><arrow.to.target n="marg64"/></s></p><p type="margin">
<s id="id2668312"><margin.target id="marg64"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2668338">Exemplum it proportio augens 25 & 35 duarum quintarum, uo <lb/>lo cire quantum it aggregatum omnium citra 25, maximam acci<lb/>pio 35, ulteriorem ad 25, cuius differentia a 25 et 10, cum quo diui<lb/>do 625 quadratum, exit 62 1/2 aggregatum quantitatum. </s>
<s id="id2668367">Et facile po</s></p><p type="main">
<s id="id2668377"><arrow.to.target n="marg65"/><lb/>ret demontrari. </s>
<s id="id2668392">Si quis dicat in qua proportione unt infinit <lb/>quantitates analog cum 12, quiunct efficiunt 10, iunge 10 cum <lb/>12 fit 22, duc 22 in 12 fit 264, diuide 264 per 10, exit 26 2/3, & in ea pro<lb/>portione <expan abbr="erũt">erunt</expan> ill quantitates, in qua unt 26 2/3 ad 12: duc per 5 fiunt <lb/>60, & 132 diuide per 12, exeunt 11 & 5, & ita eruntin proportione 11 <lb/>ad 5 experiaris, & inuenies, & demontratur ex prioribus.</s></p><p type="margin">
<s id="id2668454"><margin.target id="marg65"/>Q<emph type="italics"/>uftio.<emph.end type="italics"/></s></p><p type="main">
<s id="id2668481">Propoitio decimanona.</s></p><p type="main">
<s id="id2668492">Si fu erint aliquot quantitates arithmetic omiolog, quarum <lb/>exceus it qualis minim, omnibus autem deficientibus upple<lb/>menta ad qualitatem maxim adiungantur, erunt quadrata omni<lb/>um quantitatum qualium adiecto rurus quadrato prim cum <lb/>eo quod fit ex minima primi ordinis in <expan abbr="aggregatũ">aggregatum</expan> omnium quan<lb/>titatum eiudem tripla aggregato quadra<lb/><figure id="id.015.01.036.1.jpg" xlink:href="015/01/036/1.jpg"/><lb/>torum omnium quantitatum primi ordinis <lb/><arrow.to.target n="marg66"/><lb/>pariter acceptis.</s></p><p type="margin">
<s id="id2668591"><margin.target id="marg66"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2668617">Sint aliquot quantitates a b c d e f g h in <lb/>continua proportione. </s>
<s id="id2668624">Arithmetica dipoit <lb/>ita ut minima <expan abbr="earũ">earum</expan> qu it h, it qualis diffe<lb/>renti quantitatum <expan abbr="&longs;ecundũ">ecundum</expan> ordinem dipo <lb/><expan abbr="&longs;itarũ">itarum</expan>, uelut differentia a & b, & b & c, & c & <lb/>d, et ita de alijs, addantur <expan abbr="aũt">aunt</expan> <expan abbr="&longs;upplem&etilde;ta">upplementa</expan> in <lb/>gulis harum, qu int i k l m n o p, ita ut <expan abbr="o&etilde;s">oens</expan> <lb/>fiant quales <expan abbr="cũ">cum</expan> uis upplementis ipi line <lb/> maiori. </s>
<s id="id2668769">Etque <expan abbr="id&etilde;">idem</expan> ac i eent aliquot quanti
<pb xlink:href="015/01/037.jpg" pagenum="18"/>tates, & <expan abbr="diuideren&ttilde;">diuiderentur</expan> ingul <expan abbr="&longs;ecundũ">ecundum</expan> numerum <expan abbr="illarũ">illarum</expan>, i quatuor in <lb/>quatuor partes quales, i quinque in quinque, i decem in decem, eara<lb/>tione ut ultima <expan abbr="diuidere&ttilde;">diuideretur</expan>, ubi et finis prim partis, penultima ubi <lb/>et finis ecund partis, antepenultima ubi et finis terti, & ic de <lb/>alijs. </s>
<s id="id2668887">Vocabo ergo primas <expan abbr="quãtitates">quantitates</expan> propoitas a b c d e f g h quan<lb/>titates primi ordinis, ed quantitates quales qu <expan abbr="con&longs;tãt">contant</expan> ex quan <lb/>titatib. </s>
<s id="id2668935">primi ordinis, & fupplementis, appellabo quantitates ecun<lb/>di ordinis: ex quo patet qud prima <expan abbr="quãtitas">quantitas</expan> erit ex utro que ordine, <lb/>quia non et diuia, reliqu omnes differunt, quantitates uer quas <lb/>adiunxi nominabo <expan abbr="&longs;upplem&etilde;ta">upplementa</expan>, & unt una minus <expan abbr="quã">quam</expan> quantitates <lb/>ordinum: ut i <expan abbr="quãtitates">quantitates</expan> ordinum int octo, erunt upplementa e<lb/>ptem, & i quantitates <expan abbr="ordinũ">ordinum</expan>, eent eptem eent <expan abbr="&longs;upplem&etilde;ta">upplementa</expan> ex, <lb/>quia inter upplementa <expan abbr="nõ">non</expan> <expan abbr="adnumera&ttilde;">adnumeratur</expan> quantitas indiuia. </s>
<s id="id2669092">Erunt er <lb/>go upplementa i k l m n o p, qutanto erunt maiora quanto quan <lb/>titates primi ordinis unt minores, & contr tanto maiora, quanto <lb/><expan abbr="quãtitates">quantitates</expan> primi ordinis unt maiores. </s>
<s id="id2669129">quantitates <expan abbr="aũt">aunt</expan> ecundi ordi <lb/>nis <expan abbr="appellabun&ttilde;">appellabuntur</expan> a, b i, ck, dl, em, fn, go, & hp. </s>
<s id="id2669159">Hcuolui pluribus <lb/>agere, ut dilucidior eet propoitio. </s>
<s id="id2669175">qu licet <expan abbr="nõ">non</expan> it difficilis, et <expan abbr="tam&etilde;">tamen</expan> <lb/>confua ualde propter multitudinem <expan abbr="quantitatũ">quantitatum</expan> & ordinum. </s>
<s id="id2669219">Dico <lb/>ergo d aggregatum <expan abbr="quadratorũ">quadratorum</expan> quantitatum ecundi ordinis pri <lb/>mo quadrato bis repetito, eu uno addito <expan abbr="cũ">cum</expan> eo quod fit ex minima <lb/>in aggregatum quantitatum primi ordinis et <expan abbr="triplũ">triplum</expan> aggregato ex <lb/>quadratis omnibus <expan abbr="quantitatũ">quantitatum</expan> <expan abbr="eiu&longs;d&etilde;">eiudem</expan> primi ordinis, & utres exem <lb/>plo facilius innotecat, int <expan abbr="quãtitates">quantitates</expan> primi ordinis 8. 7. 6. 5. 4. 3. 2. 1. <lb/>quorum quadrata int 64. 49. 36. 25. 16. & 9.4 & 1. qu iuncta <expan abbr="faciũt">faciunt</expan> <lb/>204, dico quod i umamus quadrata omnium <expan abbr="quãtitatum">quantitatum</expan> ecundi <lb/>ordinis, qu unt octies 64, & eis addiderimus unum <expan abbr="quadratũ">quadratum</expan> ex <lb/>his, ut fiant nouies 64, & erunt 556, imul iuncta & eis addamus, d <lb/>fit ex 1 quantitate minima primi ordinis in 36 aggregatum quanti<lb/>tatum omnium primi ordinis, & et tale <expan abbr="productũ">productum</expan> 36, ut fiat totum <lb/>612, quod tale 612 et triplum 204, aggregati <expan abbr="quadratorũ">quadratorum</expan> primi or<lb/>dinis unius demontratio hc et. </s>
<s id="id2669430">Quia ex quarta ecundi Element. <lb/>Euclidis ingula quadrata <expan abbr="quantitatũ">quantitatum</expan> <expan abbr="diui&longs;arũ">diuiarum</expan> ecundi ordinis con <lb/>tant ex quatuor partibus quarum du unt quadrata partium, reli<lb/>qu du unt producta ex partibus <expan abbr="inuic&etilde;">inuicem</expan> bis, & quia h fuit qua<lb/>lis 1, & p qualis b, quia upplementa <expan abbr="fuerũt&ecedil;qualia">fueruntqualia</expan> mutu quanti <lb/>tatibus, & ita c qualis o & k qualis g & d, qualis n & l, qualis <lb/>f, e <expan abbr="aũt">aunt</expan> qualis m. </s>
<s id="id2669563"><expan abbr="Sequi&ttilde;">Sequitur</expan> ergo quod umptis duabus quantitatibus <lb/>ecundi ordinis hab entibus <expan abbr="&longs;upplem&etilde;ta">upplementa</expan> mutu qualia ipis quan <lb/>titatibus quod quadrata partium <expan abbr="erũt">erunt</expan> dupla quadratis primarum <lb/>quantitatum: ueluti capio b i ecundam & h p ultimam, <expan abbr="quarũ">quarum</expan> qua
<pb xlink:href="015/01/038.jpg" pagenum="19"/>drata partium unt quadrata b & i, & h & p, ed b et qualis p, & h <lb/>qualis i. </s>
<s id="id2669658">Ergo quatuor quadrata b i & h p unt dupla quadratis b <lb/>& h, & ita <expan abbr="concludã">concludam</expan> de omnibus ubi du quantitates duabus com <lb/>parantur: ed in e m quia et ola una quantitas, itud et etiam cla<lb/>rius, quia quadrata e & m unt dupla quadrato e oli eo, quod & m <lb/><arrow.to.target n="marg67"/><lb/>unt quales. </s>
<s id="id2669721">Igitur per demontrata ab Euclide erit proportio o<lb/>mnium quadratorum b i, c k, d l, e m, f n, g o, h p, ad quadrata b c d e <lb/>f g h, pariter accepta proportio dupla. </s>
<s id="id2669737">atuer addito quadrato a <lb/>quadratis b c d e f g h, & erunt quadrata omnium quantitatum, & <lb/>quadratis b i, c k, d l, e m, f n, g o, h p, duplo quadrati a cilicet emel, <lb/>quia a et ex ecundo ordine quantitatum, & emel, quia hoc fuit a<lb/>umptum in Problemate. </s>
<s id="id2669779">Sequitur ut quadrata omnia <expan abbr="quãtitatum">quantitatum</expan> <lb/>ecundi ordinis, prout unt diuia in partes addito quadrato a, int <lb/>dupla quadratis primarum quanttatum, imul pariter acceptis. </s>
<s id="id2669815">Re <lb/>liquum et modo ut otendamus dupla <expan abbr="illorũ">illorum</expan> productorum, cum <lb/>eo quod fit ex minima quantitate, cilicet h in aggregatum iparum <lb/>quantitatum primi ordinis ee quale quadratis, <expan abbr="quantitatũ">quantitatum</expan> eiu<lb/>dem primi ordinis pariter acceptis. </s>
<s id="id2669870">Contatigitur, quod duplum i<lb/>in b et quale duplo h in ipum b, quia h & i unt quales, & du<lb/>plum k in ipum c, et quale quadruplo h in idem c, quia k et du<lb/>pla h, & imiliter duplum l in ipum d et quale excuplo, h in d, <lb/>quia l et tripla h, & ita procedendo erunt illa dupla producta <lb/>qualia productis ex h in ipas quantitates toties umptis quantus <lb/>et numerus, qui prouenit duplicato numero, ecundum <expan abbr="qu&etilde;">quem</expan> h con <lb/>tinetur in illo upplemento, exemplum uolo duplum producti lin <lb/>d bis, cio qud upplementum l continet h ter, duplicabo tria & fi<lb/>ent ex, <expan abbr="igi&ttilde;">igitur</expan> <expan abbr="duplũ">duplum</expan> lin d quale et excuplo h in ipum d. </s>
<s id="id2670022">Quo con<lb/>tituto, cum uppoitum it producta illa duplicata cum producto h <lb/>in aggregatum primarum <expan abbr="quãtitatum">quantitatum</expan> ee qualia quadratis ipa<lb/>rum quantitatum, igitur addemus <expan abbr="productũ">productum</expan> ex h in ingulas quan<lb/>titates productis illis prioribus, & fiet productum h in a emel, in b <lb/>ter, in c quinquies, in d epties, in e nouies, in f undecies, in g trede<lb/>cies, & in h quindecies quale duplo producti uniucuiuque quan<lb/>titatis in uum upplementum cum producto h in <expan abbr="aggregatũ">aggregatum</expan> ipa<lb/>rum quantitarum, at quadratum a et quale producto ex h in eam, <lb/>qu talem habet proportionem ad ipum a, <expan abbr="qual&etilde;">qualem</expan> habet a ad ipum <lb/><arrow.to.target n="marg68"/><lb/>h per demontrata ab Euclide, & pariter de quadrato b, quod et <lb/>quale ei quod fit ex h in eam qu toties continet b, quotiens b con <lb/>tinet h, & ita quadratum c quale et ei, quod continetur ub h, & <lb/>habente proportionem ad b eandem, quam b ad h, & imiliter de <lb/>quadrato c & omnibus reliquis, uque ad h ipum. </s>
<s id="id2670214">Gratia ergo exem
<pb xlink:href="015/01/039.jpg" pagenum="20"/>pli quadratum a, erit quale producto ex h in omnes quatitates e<lb/>cundas, quia quotus et numerus quantitatum, totus et numerus <lb/>ecundum quem a continet h, & imiliter quotus et numerus quan <lb/>ttatum incipiendo b, & quotus et numerus quantitatum incipi<lb/>endo c, toties b uel c <expan abbr="contin&etilde;t">continent</expan> h, & ita de alijs, quadrata ergo om<lb/>nium quantitatum imul iuncta unt qualia productis ex h in in<lb/>gulas illarum toties umptis, quoties ill <expan abbr="cõtinent">continent</expan> h, eu quotus et <lb/>numerus illius quantitatis, incipiendo ab h, & <expan abbr="numerãdo">numerando</expan> uerus a. <lb/></s>
<s id="id2670345">Rurus dico, quod productum multiplicis cuiuslibet <expan abbr="quãtitatis">quantitatis</expan> in <lb/>minimam, eu quadratum eiudem quantitatis quale et producto <lb/>eiudem quantitatis, & dupli omnium equentium primi ordinis in <lb/>ipam minimam quantitatem, uelut quadratum a et quale produ <lb/>cto ex h in a, & in duplum b c d e f g h, hoc <expan abbr="aut&etilde;">autem</expan> facile et probare in <lb/>his quantitatibus, quia i quadratum a et quale producto h in o<lb/>mnes quantitates ecundi ordinis, & omnes quantitates ecundi or <lb/>dinis imul umpt unt quales ipi a, & duplo <expan abbr="reliquarũ">reliquarum</expan> primi or <lb/>dinis, quia tales quantitates unt quales uis upplementis uici<lb/>im, ut h cum i, k cum g, f cum l, e <expan abbr="cũ">cum</expan> m, ergo tam upplementa, qum <lb/>quantitates primi ordinis unt dimidium quantitatum ecundi or<lb/>dinis, ergo duplum quantitatum primi ordinis et dimidium quan <lb/>titatum ecundi ordinis, uerm de b dico idem accidere, quia qua<lb/>dratum b et quale producto ex h in b, & in duplum reliquarum <lb/>b, cilicet duplum c d e f g h, & hoc et otendere, quod it quantita <lb/>tes unt dimidium totidem quantitatum qualium b, nam c et mi<lb/>nor b in h, & upplementum p quod et quale ipi b, i tota h p fiat <lb/>qualis ipi b, ut pote h q erit ipa q dempta h qualis ipi c, ergo <lb/>quantitates primi ordinis emper unt quales upplementis non <lb/>ueris, ed prioris quantitatis aumpt, eu in comparatione ad il<lb/>lam, quadratum igitur b et quale producto ex h in b, & in duplum <lb/>c d e f g h, & imiliter per eadem, quadratum c et quale producto <lb/>ex h in c, & in duplum d e f g h, & ic de alijs. </s>
<s id="id2670665">Habemus ergo, quod <lb/>quadrata a b c d e f g h imul iuncta unt qualia producto ex h in <lb/>a, & in duplum reliquarum, & ex h in b, & in duplum reliquarum <lb/>equentium, & producto ex h in c emel, & in duplum equentium <lb/>uque ad h, & ita de reliquis. </s>
<s id="id2670704">hoc enim et, quod nuper demontraui<lb/>mus. </s>
<s id="id2670718">Antea quo que <expan abbr="demõ&longs;tratum">demontratum</expan> et, quod duplum b in i, c in k, d in <lb/>l, e in m, f in n, g in o, h in p, <expan abbr="cũ">cum</expan> producto h in <expan abbr="aggregatũ">aggregatum</expan> a b c d e f g h <lb/>erat quale productis ex h in a emel, & in b ter, & in c quinquies, in <lb/>d epties, in e nouies, in fundecies, in g tredecies, in eipam h quin<lb/>decies, detractis ergo p <expan abbr="ordin&etilde;">ordinem</expan>, d fit ex h in a ab utro que aggregato, <lb/>& ex h in b c d e f g h bis <expan abbr="relinque&ttilde;">relinquetur</expan> ex una parte, quae fit ex h in b emel
<pb xlink:href="015/01/040.jpg" pagenum="21"/>cum uis duplicatis equentibus, & in c, & in d, & in reliquis pa<lb/>riter conduplicatis uis equentibus ex altera, quod fit ex h in b e<lb/>mel, in c ter, in d quinquies, in e epties, in f nouies, in g undecies, <lb/>in h tredecies, detractis ergo rurus quod fit ex h in b emel, & ex <lb/>h in c d e f g h bis relinquetur, quod fit ex h in c, & duplo equen<lb/>tium, & d & duplo equentium, & e & aliarum pariter: & ex alia <lb/>parte, quod fit ex h in c emel, & in d ter, & in e quinquies, in f e<lb/>pties, in g nouies, in h undecies. </s>
<s id="id2670889">Ab his rurus detractis, qud fit <lb/>ex h in c emel, & in equentes bis, relinquetur h in d emel cum uis <lb/>equentibus bis, & in e emel cum uis equentibus & in f, & in g & <lb/>in h pariter, & ex alia parte, quod fit ex h in d emel, in e ter, f quin<lb/>quies, g epties, h nouies, ab his rurus detraho, quod fit ex h in d <lb/>emel, & in equentes bis, relinquetur ex una parte, quod fit ex h <lb/>in e f g h cum duplo equentium ex alia, quod fit ex h in e e<lb/>mel, f ter, g quinquies, h epties, & imiliter ab his detractis, quod <lb/>fit ex h in e emel, & bis in equentes, relinquetur ex una par<lb/>te; quod fit ex h in f emel, & in g h bis, & in g emel, & in h bis, <lb/>& in h emel, & ex alia, quod fit ex h in f emel, in g ter, in h quin<lb/>quies. </s>
<s id="id2671009">Iterum detractis, quod fit ex h in f emel, & in g h bis com<lb/>muniter relin quetur, quod fit ex h in g emel, & in h bis, & in h e<lb/>mel, & ex alia parte quod fit ex h in g emel, & ex h in h ter. </s>
<s id="id2671039">Sed <lb/>ita, qu relicta unt iam, unt manifet qualia, ergo etiam pri<lb/>ma aggregata ab initio fuere qualia, ergo & qualia illis qua<lb/>drata a b c d e f g h his, qu fiunt, ex h in eadem quantita<lb/>tes cum duplo producti b in i, cin k, d in l, e in m, f in n, g in o, <lb/>h in p, ed iam his quadratis a b c d e f g h demontrata unt ee du<lb/>pla quadrata h p, g o, f n, e m, d l, c k, b i, cum duplo quadra<lb/>ti a, ergo quadrata omnium quantitatum ecundi ordinis cum <lb/>quadrato a rurus repetito, & producto h in aggregatum quanti<lb/>tatum primi ordinis unt tripla quadratis quantitatum primi ordi<lb/>nis pariter acceptis, quod fuit propoitum, & fuit Archimedis in li <lb/>bro de lineis piralibus, & ego adieci hic propter modum demon <lb/>trandi, qui et elegantisimus, & procedit ex principijs arithmeti<lb/>cis, & diueris communibus, & ideo non reuoluitur, ut olentre<lb/>liqu qutiones.</s></p><p type="margin">
<s id="id2671195"><margin.target id="marg67"/>I<emph type="italics"/>n<emph.end type="italics"/> 5. E<emph type="italics"/>lem.<emph.end type="italics"/><lb/>P<emph type="italics"/>rop.<emph.end type="italics"/> 12.</s></p><p type="margin">
<s id="id2671243"><margin.target id="marg68"/>L<emph type="italics"/>ib.<emph.end type="italics"/> 6. E<emph type="italics"/>le.<emph.end type="italics"/><lb/>P<emph type="italics"/>rop.<emph.end type="italics"/> 17.</s></p><p type="main">
<s id="id2671291">Propoitio uigeima.</s></p><p type="main">
<s id="id2671304">Cm fuerint quatuor quantitates, fueritque ecunda qualis ter<lb/>ti, aut prim qualis quart, erit proportio prim ad quartam, <lb/>aut terti ad ecundam producta ex proportionibus prim ad e<lb/>cundam, & terti ad quartam.<lb/><arrow.to.target n="marg69"/></s></p><p type="margin">
<s id="id2671368"><margin.target id="marg69"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2671394">Cm enim quantitates h non fuerint quales, <expan abbr="cõ&longs;tat">contat</expan> per ecun
<pb xlink:href="015/01/041.jpg" pagenum="22"/>dam harum, quod proportio prim ad <expan abbr="quartã">quartam</expan> producitur ex pro<lb/>portione prim ad ecundam, ecund ad tertiam, & terti ad quar <lb/>tam: ergo non ex olis proportionibus prim ad ecundam, & ter<lb/>ti ad quartam, & imiliter ex prima harum proportio prim ad e<lb/>cundam, & terti ad quartam producunt proportionem producti <lb/>prim in ecundam ad productum terti in quartam. </s>
<s id="id2671511">Et in multi<lb/>plicatione proportio, qu olet ee inter producta illa, & et quai <lb/>duplicata et inter ipas quantitates. </s>
<s id="id2671544">Sint igitur quantitates a b c d, <lb/>& it b qualis c, ponantur ergo recto ordine a b c d, eritque propor <lb/><figure id="id.015.01.041.1.jpg" xlink:href="015/01/041/1.jpg"/><lb/>tio a ad d producta ex proportioni<lb/>bus a ad b, b ad c, & c ad d, producan<lb/>tur igitur ex proportionibus a ad b, c <lb/>ad d. </s>
<s id="id2671586">proportio c ad f, erit igitur pro<lb/>portio e ad f, i multiplicetur per pro<lb/>portionem b ad c eadem qu prius, & </s></p><p type="main">
<s id="id2671609"><arrow.to.target n="marg70"/><lb/>producta iam et eadem ei, qu et a <lb/>ad d, ergo proportio a ad d erit producta ex proportionibus a ad <lb/>b, c ad d per primam propoitionem. </s>
<s id="id2671637">Quod uer diximus de pri<lb/>ma & quarta i int quales, manifetum et, qud res redit ad idem <lb/>olum tranmutato ordine, ut tertia, & quarta prmittantur prim, <lb/>& ecund. </s>
<s id="id2671687">Hcigitur propoitio nihil aliud innuit, qum quod <lb/>in hoc cau productio, quolet fieri ex tribus proportionibus fiat <lb/>ex duabus tantum.</s></p><p type="margin">
<s id="id2671719"><margin.target id="marg70"/>P<emph type="italics"/>er<emph.end type="italics"/> 16. P<emph type="italics"/>et.<emph.end type="italics"/></s></p><p type="main">
<s id="id2671754">Propoitio uigeimaprima.</s></p><p type="main">
<s id="id2671766">Cm decuatim ducta fuerit prima in quartam, & ecunda in ter <lb/>tiam; productumque prim in quartam diuium fuerit per produ<lb/>ctum ecund in tertiam erit proportio prim ad ecundam diui<lb/>a per proportionem terti ad quartam. </s>
<s id="id2671815">Et imiliter interpoita <lb/>omiologa.<lb/><arrow.to.target n="marg71"/></s></p><p type="margin">
<s id="id2671838"><margin.target id="marg71"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><figure id="id.015.01.041.2.jpg" xlink:href="015/01/041/2.jpg"/><p type="main">
<s id="id2671873">Primum exponamus ecundam partem, it <lb/>proportio a ad b, quam uolo diuidere per <lb/>proportionem c ad d, facio e ad b, ut c ad d, erit <lb/><arrow.to.target n="marg72"/><lb/>ergo per <expan abbr="&longs;ecũdam">ecundam</expan> harum proportio ad b pro<lb/>ducta ex proportione a ad e, & e ad b, quare ex a ad e, & c ad d, ergo <lb/>diuia proportione a ad b per proportionem c ad d exit proportio <lb/>a ad e, & hic et ecundus modus. </s>
<s id="id2671932">Primus autem modus ducatur a <lb/>in d & fiat f, & b in c & fiat g, dico proportione f ad g ee prouen<lb/>tum proportionis a ad b, diuide per proportionem c ad d, ducatur <lb/>igitur c in f & fiat h, & d in g & fiat k, quia igitur h producitur ex c <lb/>in f, & f producitur ex a in d, ergo h producetur ex producto c in d, <lb/>in a, & imiliter quia k producitur ex d in g, & g producitur ex b in
<pb xlink:href="015/01/042.jpg" pagenum="23"/>c, ergo k producetur ex c d in b, ergo ex c d in a fit h, ex c d in b fit k. <lb/></s>
<s id="id2671980">erit a ad b ut h ad k, igitur ex prima harum cum ex c in f producatur <lb/>h, & ex d in g k, & dicatur produci proportio h ad k ex proportio<lb/>ne c ad d, & f ad g, & proportio h ad k it eadem, qu a ad b, ergo <lb/>proportio a ad b producitur ex c ad d, & f ad g, ergo diuia propor<lb/>tione a ad b prodibit proportio f ad g, quod fuit propoitum.</s></p><p type="margin">
<s id="id2672022"><margin.target id="marg72"/>P<emph type="italics"/>er<emph.end type="italics"/> 10. P<emph type="italics"/>et.<emph.end type="italics"/></s></p><p type="main">
<s id="id2672058">Propoitio uigeimaecunda.</s></p><p type="main">
<s id="id2672073">Cm fuerit proportio prim ad ecundam maior, qum terti <lb/>ad quartam, erit confua ex his maior qum terti ad quartam, mi<lb/>nor autem qum prim ad ecundam.</s></p><figure id="id.015.01.042.1.jpg" xlink:href="015/01/042/1.jpg"/><p type="main">
<s id="id2672129">Sit proportio a ad b maior qum c <lb/><arrow.to.target n="marg73"/><lb/>ad d, dico, quod confua ex a c ad b d <lb/>et maior, qum c ad d, et minor qum <lb/>a ad b, ut enim c ad d ita fiat e ad b, erit que per tertiamdecimam ha<lb/><arrow.to.target n="marg74"/><lb/>rum e c ad b d confua minor qum a c ad b d, nam e et minor a, <lb/>quia proportionem habent minorem ad b quam a eo qud e ha<lb/>bet proportionem ad b, quam c ad d, qu <expan abbr="aut&etilde;">autem</expan> c ad d minor, qum <lb/>a ad b, ut uppoitum et, igitur e c ad b d minor, qum a b ad c d, e b <lb/>autem ad c d et, ut demontratum et qualis c ad d, ergo c ad d mi<lb/>nor, qum confua a b ad c d, quod et ecundum per idem proba<lb/>bitur, & primum poita f ad d, ut a ad b, eritque a maior c, igitur ma<lb/>ior proportio a f ad b d, qum a c ad b d, ed a f ad b d, ut a ad b per <lb/>candem tertiamdecimam huius ergo proportio confua a b ad c d <lb/>et minor, qum a ad b.</s></p><p type="margin">
<s id="id2672287"><margin.target id="marg73"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s id="id2672313"><margin.target id="marg74"/>P<emph type="italics"/>er<emph.end type="italics"/> 10. P<emph type="italics"/>et.<emph.end type="italics"/></s></p><p type="main">
<s id="id2672348">Propoitio uigeimatertia.</s></p><p type="main">
<s id="id2672361">Omnis motus naturalis ad locum uum et: ideo per rectam li<lb/>neam fit.<lb/><arrow.to.target n="marg75"/></s></p><p type="margin">
<s id="id2672386"><margin.target id="marg75"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2672412">Motus naturalis et, ut coneruetur corpus, & conueniat locus <lb/>corpori, igitur fit ad uum locum. </s>
<s id="id2672427">Locus autem dicitur in compara<lb/>tione ad uniuerum. </s>
<s id="id2672437">ideo omnis motus naturalis et centro mun<lb/>di urum, uel ad centrum deorum. </s>
<s id="id2672460">Et quia quanto natura celerius <lb/>uum finem potet aequi (quia finis bonus et aliter non illum ap<lb/>peteret) eum qurit, cm it apientisim uit minitra: at linea re</s></p><p type="main">
<s id="id2672512"><arrow.to.target n="marg76"/><lb/>cta breuisima et Euclide tete puncto ad punctum, igitur omnis <lb/>motus naturalis et urum aut deorum per rectam lineam.</s></p><p type="margin">
<s id="id2672551"><margin.target id="marg76"/>D<emph type="italics"/>it. tertia <lb/>primi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
</p><p type="main">
<s id="id2672597">Propoitio uigeimaquarta.</s></p><p type="main">
<s id="id2672610">Omnis motus circularis uoluntarius et.</s></p><p type="main">
<s id="id2672622">Sit motus in circulo eu per circulum in orbe cuius it centrum, <lb/>it c mundi centrum: igitur ex diffinitione circuli tantum ditabit a, <lb/>quantum b ab ipo c: ed in motu naturali per prcedentem necee <lb/>et, ut recta feratur ad c, uel recedat, igitur motus a et uoluntarius,
<pb xlink:href="015/01/043.jpg" pagenum="24"/><figure id="id.015.01.043.1.jpg" xlink:href="015/01/043/1.jpg"/><lb/>non naturalis. </s>
<s id="id2672685">nam i uiolentus eet, non <lb/>eet perpetuus. </s>
<s id="id2672703">Omnia ergo atra feruntur <lb/>circa centrum mundi. </s>
<s id="id2672713">Sit modo rota e f g, di<lb/>co e non moueri motu circulari nam linea <lb/>e c <expan abbr="lõgior">longior</expan> et g c, ergo recta mouetur ad cen<lb/>trum non circa centrum. </s>
<s id="id2672740">Indicio etiam id <lb/>et: qud i in e ponatur frutum aliquod <lb/>inigne plumbi in motu ad g per f decen<lb/>det raptim: at dum ex g in e magna cum dif<lb/>ficultate, igitur motus hic non et naturalis, <lb/>nec circularis. </s>
<s id="id2672782">nihil etiam hoc modo ponte mouetur. </s>
<s id="id2672790">Sed cum non <lb/>moueatur per rectam naturaliter, nec quiditans centro per cir<lb/>culum relinquitur, ut moueatur motu uiolento, aut mito, ed non <lb/>ex uoluntario, cum nullo modo moueatur quiditans centro, <lb/>ed emper ab e line ad centrum fiant breuiores, liquet ee mo<lb/>tum uiolentum: aut mitum ex naturali, & uiolento.</s></p><p type="main">
<s id="id2672855">Propoitio uigeimaquinta.</s></p><p type="main">
<s id="id2672869">Tres unt motus omnino implices naturalis, uoluntarius & <lb/>uiolentus.<lb/><arrow.to.target n="marg77"/></s></p><p type="margin">
<s id="id2672893"><margin.target id="marg77"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2672918">Tres unt modi, quibus pount moueri in comparatione ad cen <lb/>trum cilicet uel recta cum centro, uel quiditando centro, uel <lb/>neutro modo, igitur tres motus. </s>
<s id="id2672947">Rurus uel principio interiore <lb/>non intelligente, & et naturalis, uel intelligente & et uoluntarius: <lb/>uel exteriore & et uiolentus. </s>
<s id="id2672973">Hc autem diuiio et olum propria <lb/>non prima. </s>
<s id="id2672992">Nam et uiolentus in recta ad centrum: ideo omnis, qui <lb/>non et in recta ad centrum, nec quiditat, uiolentus et: non ta<lb/>men omnis uiolentus et extra rectam. </s>
<s id="id2673021">Attractio autem, qu fit ob <lb/>raritatem corporum, eu, ut dicunt, uacuo, uiolenta et non natu<lb/>ralis nii ratione finis, non agentis. </s>
<s id="id2673047">Sunt enim quatuor genera mo</s></p><p type="main">
<s id="id2673057"><arrow.to.target n="marg78"/><lb/>tus uiolenti ab Aritotele poita, uectio, tractio, pulio, & uolutio: <lb/>quanquam his non opus it in demontratiua cientia. </s>
<s id="id2673088"><expan abbr="cõ&longs;tat">contat</expan> enim <lb/>uolutionem ex tractione, & pulione apud illum conitere.</s></p><p type="margin">
<s id="id2673120"><margin.target id="marg78"/>7. P<emph type="italics"/>hy. <lb/>cap.<emph.end type="italics"/> 2.</s>
</p><p type="main">
<s id="id2673155">Propoitio uigeima.</s></p><p type="main">
<s id="id2673168">Motus ergo compoiti quatuor neceari unt pecies.</s></p><p type="main">
<s id="id2673192">Si tantum unt tres pecies implicium, contat ratione arithme<lb/>tica quatuor ee compoitorum. </s>
<s id="id2673219">Diquiramus ergo an int natura<lb/>liter tot pecies, foran enim repugnabit aliquis alicui. </s>
<s id="id2673239">Porr uidea<lb/>mus prim, quot int uiolentorum pecies: Prima erit cum non e<lb/>cundum rectam lineam fuerit: nec centro quiditantem. </s>
<s id="id2673274">Secun<lb/>da cum fuerit ecundum rectam, ed non ad centrum. </s>
<s id="id2673288">Tertia cum <lb/>fuerit in recta ad centrum, ed contrario modo, uelut terr urum.
<pb xlink:href="015/01/044.jpg" pagenum="25"/>Quarta cm in recta ad centrum, ecundum naturam, ed <expan abbr="nõ">non</expan> prin <lb/>cipio naturali. </s>
<s id="id2673338">Velut cum quis proij cit lapidem rect in terram <lb/>turri uiolentius, qum ille ua grauitate decenurus eet. </s>
<s id="id2673367">Hic igi<lb/>tur motus et compoitus ex naturali, & uiolento. </s>
<s id="id2673382">Animalium au<lb/>tem motus uoluntarius et, cum it principio interiore cognocen <lb/>te: & it quatenus principio in linea circulari qualiter ditante <lb/>centro: ed quia obtat grauitas, ide mitus et ex naturali, & uo<lb/>luntario. </s>
<s id="id2673440">Sed circularis, & uiolentus oli ee non pount: nam uio <lb/>lentus et neceari in corpore graui aut leui: ed omne corpus gra<lb/>ue aut leue, cm mouetur, naturaliter mouetur altem in fine: & per <lb/>totum motum, motu cculto, qui maxim in hoc libro dignus et <lb/>conideratione, igitur motus uoluntarius, & uiolentus non po<lb/>unt ee imul oli. </s>
<s id="id2673517">Eruntergo ecundum naturam tantm tres pe<lb/>cies. </s>
<s id="id2673534">Velut cm quis candit, autalit: Et enim motus naturalis al<lb/>tem in fine, & uoluntarius, & uiolentus. </s>
<s id="id2673557">Si quis autem uelit uiolen<lb/>tum cum uoluntario copulare dicemus contare eam compoitio<lb/>nem in initio aliendi. </s>
<s id="id2673579">Motum autem occultum uocamus grauita<lb/>tem aut leuitatem.</s></p><p type="main">
<s id="id2673593">Propoitio uigeimaeptima.</s></p><p type="main">
<s id="id2673609">Motus uoluntarius et in loco: naturalis ad locum: uiolentus <lb/>exloco.</s></p><p type="main">
<s id="id2673624">Hc et tertia differentia primarum pecierum motuum uolun<lb/>tarius fit manente corpore toto in eodem loco, ideo proprius et <lb/>clo, corpora autem animalium in eodem loco feruntur: quia in <lb/>eodem orbe nata redire ad proprium locum. </s>
<s id="id2673655">Et ide, ut dixi, et mo<lb/>tus mitus ex naturali, & uoluntario, qui i per e fieret, non fatiga<lb/>ret mobile, cm ex utroque principio ab interiore ui procedat. </s>
<s id="id2673685">Sed <lb/>quia fit per muculos, qui trahuntur: hic autem motus et uiolen<lb/>tus, ide per conequentiam fatigat. </s>
<s id="id2673708">Qui uer naturalis, et ut re<lb/>deat corpus ad uum locum, igitur naturalis et ad locum. </s>
<s id="id2673728">Sed <lb/>uiolenti finis et, ut protrudatur ex loco in quo et, non habens cer<lb/>tum finem. </s>
<s id="id2673745">licet enim qui trahit, ad uum locum trabat, non tamen <lb/>ad locum mobilis.</s></p><p type="main">
<s id="id2673761">Propoitio uigeimaoctaua.</s></p><p type="main">
<s id="id2673774">Motus quilibet naturalis aut uiolentus in aliquo medio fit.<lb/><arrow.to.target n="marg79"/></s></p><p type="margin">
<s id="id2673790"><margin.target id="marg79"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2673816">Cm uacuum non detur, & omnis motus naturalis it ad locum, <lb/>et uiolentus ex loco per prcedentem, igitur cm non it in medio, <lb/>uacuum erit in aliquo corpore, uelut aere, aqua, igne, ligno.</s></p><p type="main">
<s id="id2673848">Propoitio uigeimanona.</s></p><p type="main">
<s id="id2673860">Omnis motus uoluntarius qualis et emper: impliciter etiam <lb/>quilibet alius motus.</s></p>
<pb xlink:href="015/01/045.jpg" pagenum="26"/><p type="main">
<s id="id2673892"><arrow.to.target n="marg80"/></s></p><p type="margin">
<s id="id2673903"><margin.target id="marg80"/>C<emph type="italics"/>o<emph.end type="italics"/>m.</s></p><p type="main">
<s id="id2673930">Motus uoluntarius non habet, qud fatiget, & umma perfectio <lb/>et qualitas, & natura qu mouet non debilitatur, igitur perpe<lb/>tuo pereuerat qualis. </s>
<s id="id2673961">neque enim et, ut dixi, per medium corpus. <lb/></s>
<s id="id2673970">Naturalis quoque, & uiolentus cum ratione proportionis mouentis <lb/>upra mobile pere non uarientur, & ab quali proportione qua<lb/>lis uelo citas proueniat, igitur natura tales motus unt quales, nam <lb/>in utroque mouens, mouet ecundum ultimam uam uim.</s></p><p type="main">
<s id="id2674015">Propoitio trigeima.</s></p><p type="main">
<s id="id2674028">In omni corpore mobili in medio, partes medij reitunt obui, <lb/>ali impellunt.</s></p><p type="main">
<s id="id2674051"><arrow.to.target n="marg81"/></s></p><p type="margin">
<s id="id2674063"><margin.target id="marg81"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2674090">Sit mobile a cui partes ubiaceant direct b, & it graue. </s>
<s id="id2674102">Et pa<lb/>tet ne diuidatur b reitere, cum autem uperauerit, partes b decen<lb/>dunt ante a, & trahunt partes c & d adhrentes ecum, atque ita e c d f <lb/><figure id="id.015.01.045.1.jpg" xlink:href="015/01/045/1.jpg"/><lb/>adiuuant ad decenum partes etiam laterales <lb/>g & h cum a tranit in b, ne detur uacuum, tran<lb/>eunt in k uelo ci motu, ergo propellunt a maio<lb/>reimpetu inferius.</s></p><p type="main">
<s id="id2674173"><arrow.to.target n="marg82"/></s></p><p type="margin">
<s id="id2674184"><margin.target id="marg82"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2674210">Ex quo patet, quod in omni motu naturali, <lb/>uel uiolento fit augumentum uelocitatis ab initio altem uque <lb/>ad aliquid.</s></p><p type="main">
<s id="id2674230"><arrow.to.target n="marg83"/></s></p><p type="margin">
<s id="id2674241"><margin.target id="marg83"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2674268">Et ide etiam bellic machin cuiucunque generis certam exi<lb/>gunt ditantiam, ut uiolentius feriant.</s></p><p type="main">
<s id="id2674294">Propoitio trigeimaprima.</s></p><p type="main">
<s id="id2674308">Omnis motus naturalis in quali medio ualidior et in fine, <lb/>qum in principio: uiolentus contr.</s></p><p type="main">
<s id="id2674331"><arrow.to.target n="marg84"/></s></p><p type="margin">
<s id="id2674342"><margin.target id="marg84"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2674369">Cm enim ex prcedenti augeantur emper ob medium, & cau<lb/>fa, qu mouet, it perpetua, & principio terno, quod per dict <lb/>qualiter mouet, igitur motus ille fiet uelo cior in fine qum in alia <lb/>parte temporis. </s>
<s id="id2674411">In uiolento autem, cm perueniat ad finem deinit </s></p><p type="main">
<s id="id2674426"><arrow.to.target n="marg85"/><lb/>uis illa neceari, qu mouet, & uperatur ui naturali, qu mo<lb/>uet in contrarium, igitur antequam ceet motus fiet tardisimus <lb/>in fine.</s></p><p type="margin">
<s id="id2674473"><margin.target id="marg85"/> 29. P<emph type="italics"/>ropo.<emph.end type="italics"/></s></p><p type="main">
<s id="id2674501">Ex quo patet, qud motus quadrifariam miti dicuntur, aut pe<lb/><arrow.to.target n="marg86"/><lb/>cie, ut cm quis iacit lapidem turri: uel ex occulto naturali, & uio<lb/>lento manifeto: uelut cm quis iacit lapidem, & decendit potmo <lb/><figure id="id.015.01.045.2.jpg" xlink:href="015/01/045/2.jpg"/><lb/>dum ex b in c motu utroque manifeto, ed ex a <lb/>in b motu uiolento manifeto, & naturali oc<lb/>culto: uel ratione medij, & hoc modo omnis <lb/>motus naturalis etiam non olum uiolentus et <lb/>mitus ex proportione uirtutis mouentis, cum motu medij, ad me<lb/>dium ipum, uel i uiolentus it ex proportione uirtutis mouentis,
<pb xlink:href="015/01/046.jpg" pagenum="27"/>& medij ad mobile, ac medium, quod reitit. </s>
<s id="id2674620">Quarto ex motibus <lb/>imperfectis natura ua, & non et uera mitio, & hoc apparet in mo<lb/>tibus uoluntarijs animalium, qui non unt neque quales, neque perfe <lb/>ct circa medium: ed unt potius imiles uoluntarijs. </s>
<s id="id2674661">Etideo de<lb/>montrationes ill Aritotelis quoad uum nihil iuuant nos.</s></p><p type="margin">
<s id="id2674685"><margin.target id="marg86"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2674712">Propoitio trigeimaecunda.</s></p><p type="main">
<s id="id2674728">Omne mobile naturaliter motum, eu uiolenter uelo cius moue<lb/>tur in medio rariore, qum deniore. </s>
<s id="id2674746">Maior quoque et proportio fi<lb/>nis motus in corpore rariore ad finem motus in corpore deniore, <lb/>qum principij. </s>
<s id="id2674766">In uiolento autem celeris perueniet ad finem mo<lb/>tus in corpore deniore.</s></p><figure id="id.015.01.046.1.jpg" xlink:href="015/01/046/1.jpg"/><p type="main">
<s id="id2674793">A mobile moueatur in b medio rariore, & in c denio<lb/><arrow.to.target n="marg87"/><lb/>re, igitur b minus reitit, qum c & magis adiuuat, quia <lb/>uelocis mouetur: igitur duplici de caua a mouebitur <lb/>uelocis in b qum in c: & quia per corrolarium trigei<lb/>m, & prcedentis proportio finis (ubi qualiter moueantur) ad <lb/>ua principia maior erit in d, qum in e: ergo per <expan abbr="demõ&longs;trata">demontrata</expan> Cam <lb/>pano poita d prima, b ecunda, e tertia, c quarta, maior erit propor<lb/>tio d ad e, qum b ad c quod fuit propoitum in naturali.</s></p><p type="margin">
<s id="id2674902"><margin.target id="marg87"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2674929">Propoitio trigeimatertia.</s></p><p type="main">
<s id="id2674942">Omnia duo mobilia qualis undique magnitudinis, qu quali <lb/>in tempore qualia patia pertraneunt in diueris ubtantia me<lb/>dijs, necee et, ut it ponderis ad pondus, quemadmodum medij <lb/>ad medium, proportio duplicata.<lb/><arrow.to.target n="marg88"/></s></p><p type="margin">
<s id="id2675003"><margin.target id="marg88"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2675028">Sint duo mobilia a & b magnitudine, & forma omnino paria, <lb/>& int media c & d, exempli gratia: & pertraneant quale patium <lb/>in utroque in eodem tempore, e dico proportionem ponderis b ad <lb/>pondus a ee duplicatam ei qu et raritatis c ad raritatem d. </s>
<s id="id2675068">Quia <lb/>enim feruntur qualiter, nam in quali tem<lb/><figure id="id.015.01.046.2.jpg" xlink:href="015/01/046/2.jpg"/><lb/>pore, eu eodem qualia patia pertrane<lb/>unt, erit proportio potenti a cum uo auxi<lb/>lio ad id, quod reitit ex c ut b cum uo au<lb/>xilio ad id, quod reitit ex d, permutando igi <lb/>tur d ad c, ut b ad a, ed c ad d proportio rari<lb/>tatis duplicat actionem, tum minus reiten<lb/>do, tum adiuuando motum a, igitur proportio differenti motus <lb/>et duplicata proportioni raritatis: ed proportio motus et qua<lb/>lis proportioni ponderis uicisim per uigeimamextam exti Ele<lb/>mentorum b ad a, igitur proportio b ad a ponderis et duplicata ei, <lb/>qu et raritatis c ad raritatem d.</s></p>
<pb xlink:href="015/01/047.jpg" pagenum="28"/><p type="head">
<s id="id2675224">SCHOLIVM PRIMVM.</s></p><p type="main">
<s id="id2675234">Ne tamen ine exemplo intelligas hanc duplicatam rationem, <lb/>proponatur craritas quatuor, d unum, a pondus duodecim libra<lb/><figure id="id.015.01.047.1.jpg" xlink:href="015/01/047/1.jpg"/><lb/>rum, tunc c reitit olum ex quarta parte, & effi<lb/>cit a quadruplo maioris actionis, cilicet ut qua<lb/>draginta octo, tota igitur proportio, qua mo<lb/>uebitur a in c, erit centum nonaginta duorum, & hoc diuidemus <lb/>per d, quod et unum, exibit <expan abbr="põdus">pondus</expan> b centum nonaginta duo. </s>
<s id="id2675303">Pro<lb/>portio igitur b ad a et exde cupla, & hc et duplicata quadrupl <lb/>raritatis c ad raritatem d.</s></p><p type="main">
<s id="id2675335">Qud i quis neget tantundem augere c actionem a, quanto mi<lb/>nus reitit, ed aut magis aut minus, & it proportio b ad a dupli<lb/>cata ipi f, dico fee proportionem c ad d, nam proportio b ad a <lb/>et uelut actionis c ad d per decimamextam exti Elementorum, <lb/>ergo ex auxilio c in proportionem a ad c fit proportio b ad a, ed ex <lb/>fin e fit proportio b ad a ex diffinitione proportionis duplicat. <lb/></s>
<s id="id2675403">Sed ex duabus proportionibus a ad c, & actionis ex c ad a produ<lb/>citur proportio b ad a, igitur per <expan abbr="decimam&longs;eptimã">decimameptimam</expan> exti Elemento<lb/>rum proportio c ad d et media inter proportiones a ad c, & actio<lb/>nis a in c, quare qualis f, igitur proportio b ad a duplicata ei, qu <lb/>et c ad d quod erat demontrandum.</s></p><p type="head">
<s id="id2675462">SCHOLIVM SECVNDVM.</s></p><p type="main">
<s id="id2675472">Si autem media fuerint diuerarum rationum, ut aqua, & ar non <lb/>demontrat argumentum, quia pondera inter e non eruant ratio<lb/>nem. </s>
<s id="id2675499">Nam lignum centum librarum ex alicis arbore, non magis <lb/>decendit, qum lignum libr unius. </s>
<s id="id2675518">Ide nec in comparatione ad <lb/>medium aris.</s></p><p type="main">
<s id="id2675534">Propoitio trigeimaquarta.</s></p><p type="main">
<s id="id2675548">Proportio corporis cubi ad uam uperficiem quadratam, et ue<lb/>lut eiudem uperficiei ad latus, eiudem uer ad monadem.</s></p><p type="main">
<s id="id2675581"><arrow.to.target n="marg89"/></s></p><p type="margin">
<s id="id2675592"><margin.target id="marg89"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2675619">Sit cubus a b c eius quadrata, uperficies a <lb/><figure id="id.015.01.047.2.jpg" xlink:href="015/01/047/2.jpg"/><lb/>c, latus a b, monas d, dico eas ee inuicem <lb/>analogas. </s>
<s id="id2675645">Quia enim proportio a b c ad a c <lb/>et, ut quoties aumitur a c in a b c, & toties <lb/>ctiam aumitur a b in a c ex diffinitione Eucli </s></p><p type="main">
<s id="id2675672"><arrow.to.target n="marg90"/><lb/>dis ecundo Elementorum, i ergo monas et <lb/>in continua proportione, habeo intentum: i <lb/>non ponatur e media inter a e & d, erit ergo <lb/>per decimam noni Elementorum elatus a c, <lb/>ergo qualis a b, igitur cum a c, e & d int analog, erunt & a b c, <lb/>a b, & d analog, quod fuit demontrandum.</s></p>
<pb xlink:href="015/01/048.jpg" pagenum="39 [=29]"/><p type="margin">
<s id="id2675738"><margin.target id="marg90"/>P<emph type="italics"/>rima ex<emph.end type="italics"/><lb/>C<emph type="italics"/>ampano.<emph.end type="italics"/></s></p><p type="main">
<s id="id2675774">Propoitio trigeimaquinta.</s></p><p type="main">
<s id="id2675787">Vocum magnitudines excrecunt in acumine non in grauitate, <lb/>finis autem et in utroque extremo, propter hoc minima facta uaria<lb/>tione in hypate acut uix ferunt.<lb/><arrow.to.target n="marg91"/></s></p><p type="margin">
<s id="id2675820"><margin.target id="marg91"/>C<emph type="italics"/>o<emph.end type="italics"/>m.</s></p><p type="main">
<s id="id2675846">Quoniam facta uariatione in hypate, qu et <lb/>in Diapaon, uel bis Dapaon maiore interual<lb/><figure id="id.015.01.048.1.jpg" xlink:href="015/01/048/1.jpg"/><lb/>lo ditat, uelut ex a in b in grauiore, maius et in<lb/>teruallum ex c in d, igitur maior et b d, qum a c <lb/>ergo ingul uoces inter b & d magis ditant, <lb/>qum inter a & c, & quanto magis appropin<lb/>quant ad d, igitur d maius et qum b. </s>
<s id="id2675926">Ergo magnitudo et ratione <lb/>acuitatis, non grauitatis, cum uppouerimus d ee acutiorem b & <lb/>cipo a. </s>
<s id="id2675952">Otenditur etiam idem quia uox grauis fit ex priuatione <lb/>motus icut acuta ex uehementia. </s>
<s id="id2675966">Motus autem et res, quies, <lb/>priuatio.</s></p><p type="main">
<s id="id2675980">Secundum ic: nam remisio mota non feriet aurem, ide onum <lb/>non pariet ob nimiam tarditatem. </s>
<s id="id2675999">At in uelo cisimo motu oportet <lb/>uel fidem uel arteriam contrahi, & non contrahitur nii per mucu<lb/>los, igitur contentio illa finem habet. </s>
<s id="id2676021">Si autem non it necearium <lb/>habere, uel ualde procul posit extendi contentio, ut in machinis <lb/>igneis trepitus fit maximus, nam motus, ut motus et etiam in are <lb/>nullum finem per e habet nii ratione intrumenti, ergo trepitus <lb/>tantus ee potet, ut ferm oburdecant, qui audierint, ut ferunt de <lb/>Nili cataractis.</s></p><figure id="id.015.01.048.2.jpg" xlink:href="015/01/048/2.jpg"/><p type="main">
<s id="id2676100">Tertium ic it a b humi<lb/>lior uox, qu excrecat e<lb/>mitonio minore olum in <lb/>c, & it d e dupla ad ab e<lb/>cundum naturam, ut in uo<lb/>cibus medijs fiet, ut i e debeat excrecere emitonio minore per de<lb/>cimamnonam quinti <expan abbr="Elem&etilde;torum">Elementorum</expan> fe dupla c b, & in acutis ubi ex<lb/>creuerit ad diapaon quadrupla: pueri autem uox, qu iam diapa<lb/>on altior et d e, erit bis diapaon, & ide quadrupla b c, ed in acu<lb/>tioribus erit dupla, nullus enim puer et adeo fract uocis, quiu<lb/>pra humillimam non acendat per diapaon, igitur interuallum uo<lb/>cum erit octuplum a d, b c, ed communiter acen dunt ad bis diapa <lb/>on, igitur interuallum unius uocis etiam cum emitonio propor<lb/>tionem habentis et quale ferm toti a b, cum autem in diapaon <lb/>int duodecim emitonia, & duo comata, manifetum et, quod ex<lb/>tenio illa erit maxima in <expan abbr="cõparatíone">comparatone</expan> grauioris uo cis a b. </s>
<s id="id2676291">Etide <lb/>minimum in crementum in humilioribus uocibus, ubi quis coga
<pb xlink:href="015/01/049.jpg" pagenum="40 [=30]"/>tur acendere, maximum ee uidetur, ade ut gr pluribus fera<lb/>tur, quibudam non omnino feratur.</s></p><p type="head">
<s id="id2676345">SCHOLIVM.</s></p><p type="main">
<s id="id2676355">Ob hoc natura fecit, ut non quemadmodum in fidibus uoces ex <lb/>breuitate intenderentur, ed ex contrictione ligul, ut dicunt, u<lb/>per aperam arteriam uox ad diapaon acueretur addito impetu <lb/>proportione, ut ex contrictione, & impetu <expan abbr="cõ&longs;urgeret">conurgeret</expan> dupla pro<lb/>portio. </s>
<s id="id2676409">Hoc autem manifet experimur in elymis in quibus null <lb/>prorus facta mutatione intrumenti contantibus digitis omni<lb/>bus prter pollicem initr uocem exacuimus ad diapaon, inde <lb/>etiam ad bis diapaon: icut declarauimus in commentarijs Epi<lb/>demiorum.</s></p><p type="main">
<s id="id2676470">Propoitio trigeimaexta.</s></p><p type="main">
<s id="id2676487">Si proportio per proportionem minorem quali ducatur, pro<lb/>portio minor producetur. </s>
<s id="id2676499">Vnde manifetum et duas proportio<lb/>nes minores qualitate inuicem ductas proportionem minorem <lb/>unaquaque illarum producere.</s></p><p type="main">
<s id="id2676525"><arrow.to.target n="marg92"/></s></p><p type="margin">
<s id="id2676536"><margin.target id="marg92"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><figure id="id.015.01.049.1.jpg" xlink:href="015/01/049/1.jpg"/><p type="main">
<s id="id2676570">Proportio a b ad c, qualicunque it, duca<lb/>tur in proportionem minorem qualitate <lb/>fad g, dico quod producta proportio erit <lb/>minor ea, qu et a b ad c fiat d ad a b, ut f <lb/>ad g, et erit per ecundam huius d ad c pro<lb/>ducta ex proportionibus a b ad c, & f g. </s>
<s id="id2676612">Itemque per decimamquar<lb/><arrow.to.target n="marg93"/><lb/>tam quinti <expan abbr="Elementorũ">Elementorum</expan> erit d minor a b, igitur maior a b ad c, qum <lb/>d ad c. igitur qum proportio a b ad c in proportionem f ad g. </s>
<s id="id2676648">Sit <lb/>autem utraque minor qualitate ea, qu a b ad c, & ea qu f ad g, di<lb/>co productam unaquaque earum ee minorem. </s>
<s id="id2676673">Quod enim (manen<lb/>tibus his, qu dicta unt) minor it d ad c, quam a b ad c ex prima <lb/>parte otenum et. </s>
<s id="id2676700">Qud uer etiam minor it d ad c, qum d ad <lb/>a b, & ex conequenti qum f ad g demontratur ic. </s>
<s id="id2676729">Quia enim mi<lb/>nor et a b ad c, qualitate erit a b minor c, fiat ergo h qualis a b, <lb/>erit ergo d ad h, ut d ad a b per eptimam quinti Elementorum, at d <lb/>ad c minor qum d ad h per octauam eiudem, igitur minor d ad c, <lb/>qum d ad a b, igitur patet propoitum.</s></p><p type="margin">
<s id="id2676776"><margin.target id="marg93"/>P<emph type="italics"/>er<emph.end type="italics"/> 10. P<emph type="italics"/>et.<emph.end type="italics"/></s></p><p type="main">
<s id="id2676812">Propoitio trigeimaeptima.</s></p><p type="main">
<s id="id2676828">Si plures homines, quorum nulli per e nauim mouere posint, <lb/>aut pondus ferre imul iuncti eam moueant, aut pondus ferant, <lb/>erunt ill proportiones coniunct non product.<lb/><arrow.to.target n="marg94"/></s></p><p type="margin">
<s id="id2676867"><margin.target id="marg94"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2676892">Cm enim primus non posit mouere nec ecundus, erunt pro<lb/>portiones minores qualitate, Ide per ecundam partem prce<lb/>dentis multo minus mouerent duo, qum unus. </s>
<s id="id2676927">Et i quatuor mo
<pb xlink:href="015/01/050.jpg" pagenum="41 [=31]"/>uerent unusque per e mouere non poet, adderetur i proportio <lb/>produceretur, fieret minor, ergo minus mouerent quinque qum <lb/>quatuor ex ijdem, quod et aburdum.</s></p><p type="main">
<s id="id2676978">Propoitio trigeimao ctaua.</s></p><p type="main">
<s id="id2676991">Omne corpus tantm reitit motui contrario uo naturali quan <lb/>cum mouetur occulto motu quiecendo.</s></p><p type="main">
<s id="id2677017"><arrow.to.target n="marg95"/></s></p><p type="margin">
<s id="id2677029"><margin.target id="marg95"/>C<emph type="italics"/>o<emph.end type="italics"/>m.</s></p><p type="main">
<s id="id2677055">Sit a corpus quiecens in pauimento b, & mouetur in eo occul</s></p><p type="main">
<s id="id2677068"><arrow.to.target n="marg96"/><lb/>to motu uerus centrum, ut upr uium et, contra<lb/><figure id="id.015.01.050.1.jpg" xlink:href="015/01/050/1.jpg"/><lb/>rius illi it motus ad c, i ergo a quieceret in c moue<lb/>retur ad b occulto motu certa ui, ergo eadem retitit, <lb/>ne traheretur ad c. </s>
<s id="id2677126">Manifetum et autem, quod hic <lb/><arrow.to.target n="marg97"/><lb/>motus occultus et minor manifeto.<lb/><arrow.to.target n="marg98"/></s></p><p type="margin">
<s id="id2677161"><margin.target id="marg96"/>I<emph type="italics"/>n commen.<emph.end type="italics"/><lb/>26. P<emph type="italics"/>ropo.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2677200"><margin.target id="marg97"/>P<emph type="italics"/>er<emph.end type="italics"/> 30. P<emph type="italics"/>ro <lb/>po.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2677239"><margin.target id="marg98"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2677265">Ex hoc patet cur naues & currus ab initio tard & difficulter mo<lb/>ueantur, ubi moueri cperint motus augetur: quoniam reitunt </s></p><p type="main">
<s id="id2677288"><arrow.to.target n="marg99"/><lb/>per motum occultum naturalem qui maximus et dum quiecunt, <lb/>ut etiam do cebat philoophus in mechanicis, nam motus ille natu<lb/>ralis et, & ide contrarius uiolento: Ergo cum iam mouetur uio<lb/>lenter minus, mouetur naturaliter, igitur minus reitit. </s>
<s id="id2677331">Declarabi<lb/>tur enim infr qud omne quod mouetur duobus motibus tanto <lb/><arrow.to.target n="marg100"/><lb/>minus uno mouetur quanto magis altero.</s></p><p type="margin">
<s id="id2677360"><margin.target id="marg99"/>Q<emph type="italics"/>uet.<emph.end type="italics"/> 31.</s></p><p type="margin">
<s id="id2677387"><margin.target id="marg100"/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 59.</s></p><p type="main">
<s id="id2677414">Propoitio trigeimanona.</s></p><p type="main">
<s id="id2677427">Ab quali aut minore ui, qum it <expan abbr="impedimentũ">impedimentum</expan>, non fit motus.</s></p><p type="main">
<s id="id2677453">Sit a quod reitat, ne urum trahatur per decem, dico, quod <expan abbr="nõ">non</expan> <lb/><arrow.to.target n="marg101"/><lb/>urum trahetur neque decem, neque minore: nam i impedimen<lb/>tum non eet, moueretur infra ut decem, ergo i traheretur urum <lb/>per decem tantum moueretur urum, <expan abbr="quantũ">quantum</expan> deorum, ergo quie<lb/>ceret. </s>
<s id="id2677537">Si uer minore moueretur maiore ui deorum, quam ur<lb/>um, ergo deorum impliciter non urum.</s></p><p type="margin">
<s id="id2677577"><margin.target id="marg101"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2677604">Propoitio quadrageima.</s></p><p type="main">
<s id="id2677617">Omne corpus phricum tangens planum in puncto mouetur <lb/>ad latus per quancunque uim, qu medium diuidere potet.</s></p><figure id="id.015.01.050.2.jpg" xlink:href="015/01/050/2.jpg"/><p type="main">
<s id="id2677650">Sit corpus ad unguem phricum a tan<lb/><arrow.to.target n="marg102"/><lb/>gens planum b in puncto c (et enim hoc <lb/>necearium ex demontratis ab Euclide in <lb/>decimaexta Propoitione tertij Elemento<lb/>rum) dico, quod mouebitur ui, qu potet <lb/>cindere arem. </s>
<s id="id2677711">Nam cum non acendat, nec <lb/>decendat, ed quai in circulo ad centrum <lb/>mundi moueatur, pondus non affert. </s>
<s id="id2677732">Neque<lb/>ratione magnitudinis contactus, cum it in <lb/>puncto olo, igitur remanet olum aris impedimentum.
<pb xlink:href="015/01/051.jpg" pagenum="42 [=32]"/><arrow.to.target n="marg103"/></s></p><p type="margin">
<s id="id2677770"><margin.target id="marg102"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s id="id2677797"><margin.target id="marg103"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 1.</s></p><p type="main">
<s id="id2677823">Ex hoc liquet, quod oportet b planum ee ex durisima mate<lb/>ria, qu nullo modo cedat, aliter tanget pluqum in puncto.</s></p><p type="main">
<s id="id2677851"><arrow.to.target n="marg104"/></s></p><p type="margin">
<s id="id2677862"><margin.target id="marg104"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 2.</s></p><p type="main">
<s id="id2677889">Vix fieri potet, utin elementaribus phra tangat planum in <lb/>puncto. </s>
<s id="id2677904">Vel quia planum non erit exact rectum, uel non durum, <lb/>ut prorus non cedat, uel non ad quilibrium poitum, uel phra <lb/>non erit exact rotunda.</s></p><p type="main">
<s id="id2677939">Propoitio quadrageimaprima.</s></p><p type="main">
<s id="id2677953">Si fuerint du quantitates umaturque totius aggregatum maio<lb/>ris & minoris, quoties aggregatum minoris, & maioris, erit pro<lb/>portio confua maioris aggregati ad minus, minor qum multipli<lb/>cis maioris ad multiplex minoris.</s></p><p type="main">
<s id="id2677989"><arrow.to.target n="marg105"/></s></p><p type="margin">
<s id="id2678000"><margin.target id="marg105"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2678026">Sint du magnitudines a & b, & it a maior <lb/><figure id="id.015.01.051.1.jpg" xlink:href="015/01/051/1.jpg"/><lb/>b, & umatur exempli gratia a quater cum b e<lb/>mel, & b quater cum a emel, dico, quod propor<lb/>tio (quam confuam ee liquet) aggregati primi ad ecundum, et </s></p><p type="main">
<s id="id2678083"><arrow.to.target n="marg106"/><lb/>minor qum quadrupla. </s>
<s id="id2678096">Contat enim quod proportio quadru<lb/>pli a ad a et maior, quam b ad quadruplum b, cum una it quadru<lb/>pla, alia ub quadrupla, igitur per uigeimamecundam huius ag<lb/>gregati quadrupli a cum b emel, ad quadruplum b cum a emel mi <lb/><arrow.to.target n="marg107"/><lb/>nor, qum quadrupli a ad a, & maior qum b ad quadruplum b, & <lb/>et pro intellectu Archimedis.</s></p><p type="margin">
<s id="id2678164"><margin.target id="marg106"/>E<emph type="italics"/>x<emph.end type="italics"/> 18. <emph type="italics"/>diff.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2678198"><margin.target id="marg107"/>I<emph type="italics"/>n<emph.end type="italics"/> 2. <emph type="italics"/>lib.
de<emph.end type="italics"/><lb/>A<emph type="italics"/>tqui pon<lb/>deran.<emph.end type="italics"/><lb/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 10.</s></p><p type="main">
<s id="id2678265">Propoitio quadrageimaecunda.</s></p><p type="main">
<s id="id2678281">Trahentium nauim, ut ferentium pondera proportiones in e in<lb/>uicem, quomodo ducere oporteat coniderare.<lb/><arrow.to.target n="marg108"/></s></p><p type="margin">
<s id="id2678307"><margin.target id="marg108"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2678333">Hoc quomodo non posit fieri upr docuimus, nunc etiam ge</s></p><p type="main">
<s id="id2678351"><arrow.to.target n="marg109"/><lb/>neraliter dicam, cum conitant hc in duobus terminis, productio <lb/>uer prupponit quatuor terminos, ut in prima propoitione, aut <lb/>altem tres, atque in his medius habet rationem mouentis, & moti, <lb/>ergo cum in huiumodi <expan abbr="nõ">non</expan> int quatuor termini, nec tres, quibus <lb/>unus it mouens, & motum proportio non poterit produci. </s>
<s id="id2678417">Illud <lb/>etiam patet exemplo, nam i eet lapis, aut nauis obitens ut ex, & <lb/>eent homines uiribus inguli, ut quatuor cum dimidio, tres mo<lb/>uerent in proportione dupla exquiquarta perdicta uperius eo<lb/>dem loco, at i proportio duci poet aliquorum hominum nume<lb/>rus poet mouere in duplicata proportione ad unguem cilicet <lb/>5 1/16 ut eet uix hominum collectorum 30 3/8 at nullus et numerus ho <lb/>minum qui collectus faciat hunc numerum, nam ex homines ex<lb/>plentnumerum 27, & eptem 31 1/2, & ide non potet duci propor<lb/>tio. </s>
<s id="id2678519">Et ide maximus et error dicendo decem homines mouent na <lb/>uim proportione tripla, ergo triginta alij additis illis imiles robo<lb/>re mouebunt proportione uiginti eptupla cilicet ducta nonu
<pb xlink:href="015/01/052.jpg" pagenum="33"/>pla in triplam. </s>
<s id="id2678560">Sed umpta proportione alio modo producitur. </s>
<s id="id2678567">Ve <lb/>lut i dicam, homines decem mouent nauim, aut <expan abbr="ferũt">ferunt</expan> pondus pro<lb/>portione tripla, igitur quadraginta homines idem facient propor<lb/>tione duodecupla cilicet quadrupla in triplam ducta. </s>
<s id="id2678600">Cum ergo <lb/>addo triginta homines, qui mouent in proportione nonupla, non <lb/>oportet ducere nonuplam in triplam, ed totum numerum accipe<lb/>re, & quam proportionem habet ad partem, tandem habet uis mo<lb/>uens ad uim <expan abbr="mou&etilde;tem">mouentem</expan>. </s>
<s id="id2678635">Vnde i duo moueant in proportione ex<lb/>quialtera, & ex in proportione quadrupla cum dimidia, & iungan <lb/>tur, ut fiant octo, non oportebit ducere exquialteram, in quadru<lb/>plam exquialteram, ed cum octo ad duo it in proportione qua<lb/>drupla, umemus quadruplam ad exquialteram, qu erit excupla, <lb/>& octo mouebunt, aut pondus gerentin proportione excupla.</s></p><p type="margin">
<s id="id2678701"><margin.target id="marg109"/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 37.</s></p><p type="main">
<s id="id2678729">Propoitio quadrageimatertia.</s></p><p type="main">
<s id="id2678742">Productionem ad additionem retrahere.<lb/><arrow.to.target n="marg110"/></s></p><p type="margin">
<s id="id2678758"><margin.target id="marg110"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><figure id="id.015.01.052.1.jpg" xlink:href="015/01/052/1.jpg"/><p type="main">
<s id="id2678792">Sit proportio a ad b dupla potetate li<lb/>cet int quinque homines, & int quindecim <lb/>homines c, & habebunt ad b excuplam <lb/>proportionem per prcedentem. </s>
<s id="id2678822">Iuncta <lb/>ergo a, & c per octauam huius <expan abbr="mouebũt">mouebunt</expan> <lb/>b proportione octupla, dico, quod i du<lb/>xeris <expan abbr="proportion&etilde;">proportionem</expan> c ad a plus uno. </s>
<s id="id2678859">i. </s>
<s id="id2678863">qua<lb/>druplam in proportionem a ad b, qu et dupla, proueniet eadem <lb/>octupla. </s>
<s id="id2678880">Nam quia in coniunctione ufficit iungere c cum a, & u<lb/>mitur ecundum proportionem a ad b, igitur cum proportio a ad <lb/>b co mparata ad proportionem c & a ad b it, icut proportio c & a <lb/>ad a, & proportio c & a ad a it, icut proportio c ad a, & a ad a, & <lb/>proportio a ad a habet rationem unius, igitur proportio aggregati <lb/>c a ad b et producta ex proportione c ad a plus monade in propor<lb/>tionem a ad b, quod erat demontrandum.</s></p><p type="main">
<s id="id2678940">Propoitio quadrageimaquarta.</s></p><p type="main">
<s id="id2678954">Si fuerit proportio motoris ad id, quod et maximum non mo<lb/>uens & patium, & tempus, nota erit etiam reliquorum nota.</s></p><p type="main">
<s id="id2678975">Spe contingit, ut quinque homines moueant nauim, & patium <lb/>ad tempus notum, & etiam cognitum maximum, quod mouere <lb/>non potet. </s>
<s id="id2678997">Sit ergo a numerus hominum, b na<lb/><figure id="id.015.01.052.2.jpg" xlink:href="015/01/052/2.jpg"/><lb/>uis, c maximum, quod non mouere potet, d <lb/>tempus, e patium, f motor alius iue numerus <lb/>hominum notus, & g tempus, dico, quod h patium notum erit, eu <lb/><expan abbr="notũ">notum</expan> g tempus, & h patium, dico, quod erit f motor, eu numerus
<pb xlink:href="015/01/053.jpg" pagenum="34"/>hominum notus. </s>
<s id="id2679063">Quoniam ergo notum et a & c, quia et quale <lb/>b, igitur proportio a ad b nota et: ed iuxta illam a mouet b in d <lb/>tempore per e patium, igitur per prcedentem, ut f ad a ita patij <lb/>ad e in d tempore. </s>
<s id="id2679100">Sed per eadem ut temporis d ad patium illud, <lb/>ita g ad h, ergo cum nota int d e f g erit etiam h, & ita conuertendo.</s></p><p type="main">
<s id="id2679118">Propoitio quadrageimaquinta.</s></p><p type="main">
<s id="id2679132">Rationem tater otendere.<lb/><arrow.to.target n="marg111"/></s></p><p type="margin">
<s id="id2679156"><margin.target id="marg111"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2679182">Archimedes nititur huic fundamento, quod pondera, qu pro<lb/>portionem mutuam habent, ut ditanti libella a, qu upen<lb/>duntur, qualiter ponderant, it ergo libella a b, & upena in a cen <lb/>trum mundi c, ad quod dirigitur pondus, & liquet, quod ipum <lb/>non e inclin abit ex uigeimatertia propoitione. </s>
<s id="id2679248">Si ergo ponantur <lb/>lo co line b d in e & f, & it proportio e b <lb/><figure id="id.015.01.053.1.jpg" xlink:href="015/01/053/1.jpg"/><lb/>ad b f, ut g ad h, dico, qud erit quili<lb/>brium, per eandem enim h mouebitur in k, <lb/>cilicet ut perueniat in rectam a d, i enim <lb/>non eet upenum h, moueretur in re<lb/>cta e h per eandem, quia ergo retinetur, mo<lb/>uetur per obliquam h k, & umatur in pro<lb/>pin quum punctum in b e, & n in quali di<lb/>tantia in e f, quia ergo e b totum mouetur <lb/>eadem ui in ingulis partibus, quia a pon<lb/>dere h, & in h mouetur per h k in m per m <lb/>p, ergo qualis et proportio magnitudinis h k ad m p, talis et uis <lb/>in m p ad uim in h k, & ita in b erit pen infinita: quia quanta ui ex<lb/>tenditur ex h in k tanta puncta b, e circumuertit ergo propor<lb/>tio hypomochlij ad patium, uelut roboris ad robur, at eadem n o <lb/>ad h k, et enim n o qualis m p, & n b, & b m quales, ut uer g ad <lb/>h, ita e b ad b f: ergo ut e b ad b f, ita uirium n o ad h k, ut igitur g ad <lb/>h, ita uirium m p ad h k: ut etiam g l ad n o, ita uirium f b ad n b. <lb/></s>
<s id="id2679404">nam idem pondus cilicet g mouet totam b f, igitur ut g e habet </s></p><p type="main">
<s id="id2679418"><arrow.to.target n="marg112"/><lb/>ad n o, ita h ad m p, ed m p & n o unt quales, ergo tanta et uis g <lb/>in f, quanta h in e.<lb/><arrow.to.target n="marg113"/></s></p><p type="margin">
<s id="id2679454"><margin.target id="marg112"/>P<emph type="italics"/>er<emph.end type="italics"/> 9. <emph type="italics"/>quin<lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2679504"><margin.target id="marg113"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 1.</s></p><p type="main">
<s id="id2679530">Ex quo patet, quod hypomo chlion moueretur infinita ui, i po<lb/>et ee punctus: ed quia in extrema uperficie cylindri, ide potet <lb/>aliqua ui retineri.</s></p><p type="main">
<s id="id2679568"><arrow.to.target n="marg114"/></s></p><p type="margin">
<s id="id2679578"><margin.target id="marg114"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 2.</s></p><p type="main">
<s id="id2679605">Et i quis poet capere hatam in extremo puncto, non poet <lb/>eam mouere, etiam quod haberet robur infinitum, quia ab quali <lb/>non fit motus per trigeimamnonam propoitionem.</s></p><p type="main">
<s id="id2679642"><arrow.to.target n="marg115"/></s></p><p type="margin">
<s id="id2679653"><margin.target id="marg115"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 3.</s></p><p type="main">
<s id="id2679679">Et libella nihil retinet nii quantum et pondus eius quod cu
<pb xlink:href="015/01/054.jpg" pagenum="35"/>pit ad centrum peruenire, & pondus ei appenum non prohi<lb/>bet motum, etiam i eet infinitum, nii quatenus non uult recede<lb/>re ex directo centri mundi: & ut grauat hypomochlion faciens im<lb/>presionem.</s></p><p type="main">
<s id="id2679736"><arrow.to.target n="marg116"/></s></p><p type="margin">
<s id="id2679748"><margin.target id="marg116"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 4.</s></p><p type="main">
<s id="id2679775">Et i terra tota eet appena polo, moueretur magna ui: quoni<lb/>am uis eadem et in polo, qu in circulo toto quinoctij.</s></p><p type="main">
<s id="id2679805"><arrow.to.target n="marg117"/></s></p><p type="margin">
<s id="id2679816"><margin.target id="marg117"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 5.</s></p><p type="main">
<s id="id2679843">Etrota, quanto uelocius mouetur in ambitu, tanto mi<lb/>norem habet uim: ed propter arem, qui ecum circum<lb/><figure id="id.015.01.054.1.jpg" xlink:href="015/01/054/1.jpg"/><lb/>fertur, mouetur magno impetu, & magnas facit liones. <lb/></s>
<s id="id2679879">Ide hoc in cono non accidit.</s></p><p type="main">
<s id="id2679891"><arrow.to.target n="marg118"/></s></p><p type="margin">
<s id="id2679902"><margin.target id="marg118"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 6.</s></p><p type="main">
<s id="id2679929">Ex quo patet ratio eleuandi pondera magna per tra<lb/>bem, ut latere uides.</s></p><p type="main">
<s id="id2679944">Propoitio quadrageimaexta.</s></p><p type="main">
<s id="id2679961">An it aliqua proportio, & qualis inter animam, & ui<lb/>tas, & ua corpora coniderare.</s></p><p type="main">
<s id="id2679984"><arrow.to.target n="marg119"/></s></p><p type="margin">
<s id="id2679995"><margin.target id="marg119"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2680021">Declarauimus motum cli ee uoluntarium, obequente c<lb/>lo per uirtutem in eo infuam. </s>
<s id="id2680043">In animalibus autem, & prcipu <lb/>in homine notius et hoc experientibus nobis in ipis: ed motus <lb/>hic, ut dixi upra, mitus et, ille uer cletis ignotior et. </s>
<s id="id2680088">Certum </s></p><p type="main">
<s id="id2680096"><arrow.to.target n="marg120"/><lb/>tamen et plen obequi clum uit, nec prorus repugnare. </s>
<s id="id2680122">So<lb/>let Aritoteli imponi, qud i adderetur atrum clo, qud clum <lb/>aut quieceret, aut tardius moueretur: quod et, ac i diceremus, <lb/>qud homo paruus i fieret maior, non eet ade agilis, tanquam <lb/>motus ille eet ab externa caua. </s>
<s id="id2680188">Im perinde eet, aci quis dice<lb/>ret, quod lapides magni minus uelociter decenderent, quam par<lb/>ui. </s>
<s id="id2680214">Quin potius ut lapis magnus uelocis mouetur: qum par<lb/>uus naturali motu, & tardius prternaturali, ita clum motu uo<lb/>luntario, i ita dici poet qualius & maiore cum efficacia, quan<lb/>to denius. </s>
<s id="id2680257">Et ita i Aritoteles illud dixiet, otendiet magnam <lb/>imperitiam. </s>
<s id="id2680281">Ide quale iudicium debemus facere de Alexandro, & <lb/><arrow.to.target n="marg121"/><lb/>Aueroe, qui hoc ei tribuunt. </s>
<s id="id2680298"><expan abbr="legi&ttilde;">legitur</expan> enim in textu Arabico tale quip<lb/>piam. </s>
<s id="id2680314">De Animalibus foran poet hoc dici, <expan abbr="quoniã">quoniam</expan>, ut upr dixi<lb/>mus, motus ille mitus et. </s>
<s id="id2680349">Remanet ergo difficultas, <expan abbr="quoniã">quoniam</expan> i mo<lb/>tus ite non proportione fit, quare non et infinitus? </s>
<s id="id2680377">& dico quae in <lb/>animalibus tres unt cau, una, quia et mitus, & habet repugnan<lb/>tiam: ecunda, quia et de loco ad locum, motus autem cli et in lo <lb/>co: tertia et communis etiam clo, et et, <expan abbr="quoniã">quoniam</expan> non et ratio finis. <lb/></s>
<s id="id2680437">Natura enim diuina non appetit mouere <expan abbr="tã">tam</expan> celeriter. </s>
<s id="id2680450">Quid et ergo <lb/>proportio, <expan abbr="cũ">cum</expan> it <expan abbr="ultimũ">ultimum</expan> uoluntatis uit, ut obtemperet prim cau, <lb/>ideo illud et <expan abbr="ultimũ">ultimum</expan>, mouet. </s>
<s id="id2680505">Et <expan abbr="aũt">aunt</expan> idem uelle, & poe. </s>
<s id="id2680526">In natura
<pb xlink:href="015/01/055.jpg" pagenum="46 [=36]"/>enim cli et ille appetitus, cuius prin cipium et uita: & eus uolun <lb/>tatis bonum ipum. </s>
<s id="id2680557">Et ideo hc proportio <expan abbr="nõ">non</expan> diuiditur. </s>
<s id="id2680572">In anima<lb/>libus autem non et uis illa nii, cum proportione, quia primum in<lb/>trumentum, quod recipit, & et piritus uim habet determinatam, <lb/>cum it uirtus in materia: ideo <expan abbr="nõ">non</expan> mouet nii cum certa proportio<lb/>ne, uelut lumen in medio in e non habet proportionem nii ad lu<lb/>cem, ed ut et in illo, potet ee remium, <expan abbr="ob&longs;curũ">obcurum</expan> & hebes. </s>
<s id="id2680662">Qu<lb/>ritur ergo quantitas illius? </s>
<s id="id2680672">i dicas, qud et luce: quro quanti<lb/>tas lucis, unde it? </s>
<s id="id2680696">foran dicendum, qud uelutin motibus, quanto <lb/>deniora unt corpora tanto <expan abbr="mouen&ttilde;">mouentur</expan> maiore nixu, & robore. </s>
<s id="id2680723">Nam <lb/>calor in materia augetur iuxta illius quantitatem: idem in luce, & <lb/>reliquis. </s>
<s id="id2680735">Dico ergo proportionem ee infinitam: nam i corpus e<lb/>et infinitum & optim dipoitum infinita ui moueretur & agili<lb/>tate, ut enim maius et eo maiores uires habet.</s></p><p type="margin">
<s id="id2680775"><margin.target id="marg120"/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 27.</s></p><p type="margin">
<s id="id2680802"><margin.target id="marg121"/>T<emph type="italics"/>ex.<emph.end type="italics"/> 71. <lb/>2. <emph type="italics"/>de<emph.end type="italics"/> C<emph type="italics"/>lo.<emph.end type="italics"/></s></p><p type="main">
<s id="id2680853">Propoitio quadrageimaeptima.</s></p><p type="main">
<s id="id2680869">Si duo mobilia qualiter in eodem circulo iuxta proprios mo<lb/>tus moueantur, productum temporis circuituum inuicem erit <lb/>quale producto differenti temporum circuitus duct in tempus <lb/>coniunctionis prim.<lb/><arrow.to.target n="marg122"/></s></p><p type="margin">
<s id="id2680910"><margin.target id="marg122"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2680936">Sint duo mobilia a & b in eodem pun<lb/><figure id="id.015.01.055.1.jpg" xlink:href="015/01/055/1.jpg"/><lb/>cto, qu qualiter uerus eandem partem <lb/>moueantur qualibus in temporibus, inui <lb/>cem tamen in qualiter, ita quod a in f & b <lb/>in g temporibus aboluant circulum, & ho <lb/>rum differentia it h. </s>
<s id="id2680987">Dum itaque a perficit <lb/>circulum b perueniat in c, igitur c d b et dif <lb/>ferentia, qu uperanda et, & proportio <lb/>circuli ad b c ut g ad f, quare reliqui ad reli<lb/>quum, ut reidui ad reiduum, cilicet circu<lb/>li ad c d b, ut g ad h, & b c ad c d b ut f ad h, coniungantur igitur in k <lb/>tempore, eruntque k f g h omiologa, ut productum ex circulo in b c <lb/>diuio per certam quantitatem & cum circulo & b c & c d b diffe<lb/>rentia, & it productum exfin g, dico quod diuia per h exibit k <lb/>tempus coniunctionis prim, it itaque d locus coniunctionis, dico <lb/>igitur quod differentia patij pertraniti a b, a & a, b in reditu ex con <lb/>iunctione prima ad d et unus circulus completus, non enim po<lb/>unt ee plures, nam equeretur, qud a aliquando pertraniet b, <lb/>et ic non eet prima coniunctio, nec potet ee minus, nam ic cum <lb/>a & b int in d ultra perfectas circulationes uterque eorum pertran <lb/>iuit arcum b c, igitur nullo modo differentia potet ee minor cir<lb/>culo, neque maior, ut declaratum et, igitur et unus circulus ad un
<pb xlink:href="015/01/056.jpg" pagenum="37"/>guem. </s>
<s id="id2681172">Hoc declarato ponatur m patium compofitum ex circulis <lb/>pertranitis a b a cum patio b d, etenim patium, quod pertranit <lb/>b a coniunctione in a, ad coniunctionem primam in d, & erit ex de<lb/>montratis horum differentia circulus qui uocetur o, & it p pa<lb/>tium, quod pertranit b in tempore eodem, in quo a pertranit o, & <lb/>it q differentia o, & p qu in circulo et c d l b, quia igitur in eodem <lb/>tempore a pertranit m & b, n, erit m ad n, ut a ad b, & eadem ratio<lb/>ne a ad b, ut o ad p, igitur ex undecima quinti Euclidis m ad n, ut o <lb/>ad p, quare cum o it differentia m & n, & q, differentia o & p erit ex <lb/>decimanona quinti Euclidis, m ad o, ut o ad q, & ita circulus et ana <lb/>logus inter patium pertranitum motore uelociori, & inter diffe<lb/>rentiam patij qu accidit, dum uelocior motor pertranit circu<lb/>lum, id et qud circulus a c d et analogus inter c d l b, & circulos <lb/>pertranitos a b a cum portione b d. </s>
<s id="id2681308">Reuertor igitur ad propoi<lb/>tum, cum it m ad o, ut o ad q, & m ad o, ut n ad p, ex extadecima <lb/>quinti Euclidis, erit ex undecima eiudem n ad p, ut o ad q, quare ex <lb/>extadecima exti Elementorum ducto o, id et circulo, eu maiore <lb/>numero in p patium pertranitum a b, eu ducto fin g, & diuio per <lb/>q differentiam patiorum, eu per h exibit n, eu patium quod <lb/>pertranit b ab una coniunctione ad aliam quod erat demon<lb/>trandum.</s></p><p type="main">
<s id="id2681390"><arrow.to.target n="marg123"/></s></p><p type="margin">
<s id="id2681402"><margin.target id="marg123"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2681428">Ex hoc patet, quod proportio temporis coniunctionis ad tem<lb/>pus tardioris motus circuitionis et ueluti temporis circuitus uelo <lb/>cioris motoris ad differentiam temporis motus tardioris, & uelo<lb/>cioris motoris in uno circuitu.</s></p><p type="main">
<s id="id2681455">Propoitio quadrageimao ctaua.</s></p><p type="main">
<s id="id2681468">Si tria mobilia ex eodem puncto dicedant, fuerintque duorum, ac <lb/>duorum coniunctiones in temporibus commenis illa tria mobi<lb/>lia denu coniungentur in tempore producto ex denominatore di <lb/>uiionis temporis maioris per minus in minus, aut numeratore <lb/>in maius.</s></p><p type="main">
<s id="id2681503"><arrow.to.target n="marg124"/></s></p><p type="margin">
<s id="id2681515"><margin.target id="marg124"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2681542">Sint tria mobilia a, quod circuat in duobus annis b in quinque, <lb/>c in eptem. </s>
<s id="id2681551">Dico quod primum redibunt in numero producto ex <lb/>eptem quinque & duobus, qui unt numeri primi, & erit ille nume<lb/>rus eptuaginta annorum. </s>
<s id="id2681572">Nam in eptuaginta annis a perficiet tri<lb/>gintaquinque reuolutiones b quatuordecim, c decem: ergo <expan abbr="redibũt">redibunt</expan> <lb/>per perfectos circuitus ad idem punctum. </s>
<s id="id2681597">Otendo modo quod <lb/>non ante: nam i ic: it, ut in trigintaquinque annis igitur b & c per<lb/>ficient perfectos circuitus, ergo <expan abbr="redibũt">redibunt</expan> ad idem punctum, a autem <lb/>non redibit, quoniam eius circuitus non numerat trigintaquinque<lb/>aliter non fuiet eptuaginta minimus numeratus ab a b c, cum
<pb xlink:href="015/01/057.jpg" pagenum="38"/>ergo iam upponatur numerari a b & c non numerabitur a b a, er<lb/>go a non perficiet circuitus, ergo non redibit ad primum <expan abbr="locũ">locum</expan>, ergo <lb/>non erit iunctus cum b & c. </s>
<s id="id2681679">Quod i dicas a b c coniungi in decem <lb/>eptem annis numero non numerato ab ali <lb/><figure id="id.015.01.057.1.jpg" xlink:href="015/01/057/1.jpg"/><lb/>quo illorum temporum, auferantur perfe<lb/>ct circulationes, & <expan abbr="remanebũt">remanebunt</expan> dimidium <lb/>ex a, du quint ex b, tres eptim ex c, igi<lb/>tur oportebit ut h portiones int qua<lb/>les, ut pot perfectas circulationes in idem <lb/>punctum, <expan abbr="cõueniant">conueniant</expan>, ergo 1/2 & 2/5 & 3/7 qui<lb/>ualebunt, quare proportio 7 ad 3 & 5 ad 2 <lb/>& 2 ad 1, et una, quare permutando 3 ad 2 <lb/>ut 7 ad 5, ed 7 & 5 unt contra e primi, ergo in ua proportione mi <lb/>nimi per dicta in eptimo Elementorum: ergo tria, & duo non unt <lb/>in eadem proportione. </s>
<s id="id2681809">Rurus dicantur conuenire in annis qua</s></p><p type="main">
<s id="id2681822"><arrow.to.target n="marg125"/><lb/>tuordecim cum dimidio, ergo in uiginti nouem conuenient ite<lb/>rum: ergo per ecundam partem erit eptem ad unum, ut duo ad <lb/>unum, igitur permutando unius ad unum, ut eptem ad duo, ed <lb/>unum et quale uni, ergo duo erunt qualia eptem. </s>
<s id="id2681866">Rurus dica<lb/>mus, quod in tempore annorum <02> quadrata decem imiliter aufe<lb/>ram integras reuolutiones, quas potero, & erunt <02> 2 1/2 m: 1, & <02> 2/5 & <lb/><02> 10/49 qualia. </s>
<s id="id2681892">Hic uides infinita equi in conuenientia, qu longum <lb/>eet numerare, nam eptem eet quale quinque, & proportio recii <lb/>ad potentia rethe, ut numeri ad numerum. </s>
<s id="id2681926">Igitur non conueniunt <lb/>ante eptuaginta annos.<lb/><arrow.to.target n="marg126"/></s></p><p type="margin">
<s id="id2681947"><margin.target id="marg125"/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 23</s></p><p type="margin">
<s id="id2681974"><margin.target id="marg126"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 1.</s></p><p type="main">
<s id="id2681999">Ex hoc equitur, qud nullibi conuenient prterqum in eo<lb/>dem puncto, cilicet in quo ab initio coniuncti fuerunt.</s></p><p type="main">
<s id="id2682026"><arrow.to.target n="marg127"/></s></p><p type="margin">
<s id="id2682037"><margin.target id="marg127"/>C<emph type="italics"/>or<emph.end type="italics"/>m. </s>
<s id="id2682061">2.</s></p><p type="main">
<s id="id2682069">Sequitur denuo ex propoitione ipa repetita, & primo corrola<lb/>rio, quod nullibi alibi conuenient qum in dato primo puncto, in <lb/>quo coniuncti fuerant ab initio etiam uque in ternum.</s></p><p type="main">
<s id="id2682100">Sit rurus ut a circuat in annis duobus cum dimidio, b in tribus <lb/>cum tertia parte, cin quatuor cum quarta parte ducam per uos <lb/>denominatores, & erit ut a in quinque annis. </s>
<s id="id2682119">b in decem, c in decem<lb/>eptem circuant, & redeant ad idem punctum, & quia quin que nu<lb/>merat decem, & decem, & decemeptem unt numeri inuicem pri<lb/>mi, ducam decem in decemeptem fiunt centum eptuaginta. </s>
<s id="id2682152">Con<lb/>tat igitur c quadrages, b quinquagies emel, a exagies octies cir<lb/>cumuerti, & redire ad idem punctum: ergo rurus coibunt pot tot <lb/>annos in eo, dico modo, quod non ante: nam i non it, ut in trigin<lb/>ta tribus annis. </s>
<s id="id2682196">gratia exempli, aufero <expan abbr="decem&longs;ept&etilde;">decemeptem</expan>, decem, & quin<lb/>que, & relinquentur exdecim tria & tria, & rurus ex exde cim tres
<pb xlink:href="015/01/058.jpg" pagenum="39"/>cir cuitus c, & relinquentur 3 3/4 equetur igitur, ut it proportio 17 ad <lb/>13, & 2 1/2 ad 1/2 & 3 1/3 ad 3 eadem, & ita 17/13, 5/2 & 10/9 eadem i iam upponi/><lb/>mus 17 & 10 ee primos inuicem, ut in ecunda demontratione./><lb/></s>
<s id="id2682271">Igitur equuntur eadem corrolaria, qu dicta unt.</s></p><p type="main">
<s id="id2682288">Propoitio quadrageimanona.</s></p><p type="main">
<s id="id2682302">Propoito mobilis in circulo circuitus tempore, dataque ratione <lb/>ditanti ab illo mobilis circuitum inuenire, quod ex eodem pun<lb/>cto dicedens cum alio mobili in dato puncto conueniat ub quo<lb/>cunque numero circuituum tempus quoque coniunctionis.</s></p><p type="main">
<s id="id2682340"><arrow.to.target n="marg128"/></s></p><p type="margin">
<s id="id2682351"><margin.target id="marg128"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><figure id="id.015.01.058.1.jpg" xlink:href="015/01/058/1.jpg"/><p type="main">
<s id="id2682386">Sit in circuli peripheria a <expan abbr="pũctus">punctus</expan>, qui cir <lb/>cuat quali motu (hoc enim emper intel<lb/>ligitur) in b tempore: & it datus punctus c <lb/>in quo dicedens e mobile ex coniunctio<lb/>ne cum a pot certos circuitus proprios, <lb/>aut etiam. </s>
<s id="id2682432">ine ulla circuitione perfecta de<lb/>beat conuenire. </s>
<s id="id2682442">Volo cire tempus circui<lb/>tionis e: & etiam tempus coniunctionis. <lb/></s>
<s id="id2682457">Sit ergo primum ut abque circuitione ulla e, a debeat comprehen<lb/>dere e in c pot numerum circuitionum ipius a, qui it f. </s>
<s id="id2682477">nam i a o c <lb/>currit e in prima circuitione ipius e, igitur a mouetur uelocius <lb/>qum e, cum ergo debeat attingere ipum e, necee et ut a pertran<lb/>eat prius per punctum ex quo dicesit antequam redeat ad con<lb/>iunctionem e: ergo perficiet altem unam circuitionem. </s>
<s id="id2682526">Ducemus <lb/>ergo f in b, & fiet g tempus circuitus aut circuituum a, & quia pa<lb/>tium a c datum et, it b temporis circuitus a ad h, uelut circuli to<lb/><arrow.to.target n="marg129"/><lb/>tius ad a c, & iungatur g cum h & fiat k. </s>
<s id="id2682560">Fiat quoque, ut monadis <lb/>ad h, ita l ad monadem, & ducatur l in k, & fiat m: dico m ee tem<lb/>pus circuitus e. </s>
<s id="id2682579">Contat enim ex uppoito, quod k et tempus to<lb/>tum in quo a peruenit pot b circuitiones in c, i ergo e moueretur <lb/>per m tempus totum ex uppoito perficeret circuitum, at quia cir<lb/>cuitus ad a c, ut monadis ad h, igitur etiam ut l ad monadem, ergo <lb/>proportio circuitus ad a c, ut m ad monadem: ergo i in m tranit to <lb/>tum circuitum in monade tranit a c: ed monas ducta in k facit k, <lb/>igitur e in tempore k perueniet in c, quod erat demontrandum. <lb/></s>
<s id="id2682645">Proponatur modo tempus reuolutionum e ipum d: eodem mo<lb/><arrow.to.target n="marg130"/><lb/>do agemus ducendo fin b fit g, addatur h & fiat k, diuidatur k per <lb/>aggregatum d & a e, & exeat m, (idem enim et diuidere per aggre<lb/>gatum d & h, & multiplicare per l) dico ergo ut in demontratione <lb/>priore, quod m et tempus circuitus e. </s>
<s id="id2682687">Nam cum k it tempus, in <lb/>quo a pot circuitus f peruenit ad c, ergo diuio ipo toto tempore
<pb xlink:href="015/01/059.jpg" pagenum="40"/>per numerum reuolutionum d, & partem reuolutionis exibit tem<lb/>pus unius reuolutionis.</s></p><p type="margin">
<s id="id2682727"><margin.target id="marg129"/>P<emph type="italics"/>er<emph.end type="italics"/> 10. P<emph type="italics"/>et.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2682763"><margin.target id="marg130"/>P<emph type="italics"/>er<emph.end type="italics"/> 11. P<emph type="italics"/>et.<emph.end type="italics"/></s></p><p type="main">
<s id="id2682798">Exemplum primi in repaul obcuriore: it f 4 & b 2 1/2 & a c 4/5, du <lb/>cemus 4 in 2 1/2 fit 10, adde 4/5 6 quod et 2 fit 12, diuide per 4/5 eu mul<lb/>tiplica per 5/4 quod idem et, fit 15 circuitus e, in quatuor ergo circui<lb/>tibus, & 4/5 qui unt duo decim anni perueniet a ad c, & in duodecim <lb/>annis e perueniet ad c, nam 12 unt 4/5 ipius 15. Similiter in ecundo <lb/>cau it f 4 ut prius b 2 1/3 a c 1/7, ducemus 4 in 2 1/3 fit 9 1/3, addemusque h <lb/>portionem b qualis a c et totius circuitus, id et 1/7, et autem 1/7 2 1/3, 1/3 <lb/>fient 9 1/3, imiliter ponatur d 5, & quia a c et 1/7 erunt 36/7, diuide ergo <lb/>9 2/3 id et 29/3 per 36/7 exeunt 203/108 tempus reuolutionis e. </s>
<s id="id2682885">Quin que ergo <lb/>reuolutiones e erunt 1015/108 addita eptima parte, qu et 29/108 fient 2044/108 <lb/>eu 261/27, & unt anni 9 18/27 eu 9 2/3, ergo in tanto tempore a faciet qua<lb/>tuor circuitus, & eptimam partem, & e quinque circuitus, & e<lb/>ptimam.<lb/><arrow.to.target n="marg131"/></s></p><p type="margin">
<s id="id2682939"><margin.target id="marg131"/>C<emph type="italics"/>om./><emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2682966">Ex hoc patet, quod non coniungentur in alio loco, neque alio tem <lb/>pore ante prdictum tempus.</s></p><p type="main">
<s id="id2682981">Propoitio quinquageima.</s></p><p type="main">
<s id="id2682994">Omnes circuituum portiones in eiudem temporibus <expan abbr="repetun&ttilde;">repetuntur</expan>.</s></p><p type="main">
<s id="id2683014">Sint in circulo a b c d e f g: a & b iuncta, & in primo congreu <lb/>iungantur in c, in ecundo in d, in tertio in e, in quarto in f, in quinto <lb/>in g, in exto in h, in eptimo in k, in octauo in l. </s>
<s id="id2683040">Et ic deinceps <expan abbr="cũquetempora">cunque<lb/>tempora</expan> int qualia, erunt & circuitus totidem numero, & exce<lb/>us quales etiam a c, c d, d e, e f, f g, g h, h k, <lb/><figure id="id.015.01.059.1.jpg" xlink:href="015/01/059/1.jpg"/><lb/>k l. </s>
<s id="id2683092">Et i aggregatum a cilicet circulorum, <lb/>& portionis fuerit commenum circulo, & <lb/>ita de b erunt omnia <expan abbr="cõmen&longs;a">commena</expan> ad circulum, </s></p><p type="main">
<s id="id2683132"><arrow.to.target n="marg132"/><lb/>& etiam inter e. </s>
<s id="id2683144">Et i inter e aggregata, uel <lb/>portiones erunt, & eodem modo reliqua. <lb/></s>
<s id="id2683161">Et quoniam circuli circulis commeni unt: <lb/>i portiones erunt inuicem commen <expan abbr="erũt">erunt</expan>, <lb/>& toti circuitus cum partibus commeni, & <lb/>i non commeni, neque erunt inter e, neque ad circulum. </s>
<s id="id2683206">Et i totum <lb/>patium cum circuitibus erit unius generis, erunt duplicata, & tri<lb/>plicata, & quadruplicata eiudem generis: quare cum patia ipa <lb/>detractis circuitibus uelut rhete habeant naturam recii, & patia <lb/>ipa tota int eiudem generis, erunt patia, qu relinquuntur eiu<lb/>dem generis. </s>
<s id="id2683266">Erunt tamen incommena neceari, i partes fuerint <lb/>incommen toti. </s>
<s id="id2683290">Ponatur a c incommena toti circulo dico, quod <lb/>a k <expan abbr="etiã">etiam</expan> et incommena toti circulo: & <expan abbr="etiã">etiam</expan> a k, & k c. </s>
<s id="id2683325">Quia enim a c <lb/>et incommena circulo, & k a cum toto circulo emel et commen
<pb xlink:href="015/01/060.jpg" pagenum="41"/>a a c, quia multiplex ei. </s>
<s id="id2683357">igitur cum circulus, & a k diuidantur in cir<lb/><arrow.to.target n="marg133"/><lb/>culum et a k, & circulus it incommenus circulo, cum a k erit aggre. <lb/></s>
<s id="id2683379">gatum ex circulo, & a k incommenum ipi a k, & a k pariter incom <lb/><arrow.to.target n="marg134"/><lb/>mena circulo. </s>
<s id="id2683402">Rurus quia a k et incommena circulo cum a k, & <lb/>circulus cum a k it multiplex ad a c, erit a k incommena a c, quare <lb/><arrow.to.target n="marg135"/><lb/>erit c k incommena a k & a c, & circulo ad dita a k. </s>
<s id="id2683437">Si ergo a c it <lb/>commena circulo, erunt omnes portiones e genere numeri, & i <lb/><arrow.to.target n="marg136"/><lb/>potentia rhete erunt omnes, uel potentia rhete, uel circulis detra<lb/>ctis, ut a k & a l recia: & a c it potentia ecunda rhete, id et radix cu <lb/>bica erunt omnes c d, d e, e f, potentia ecunda rhete, et radices cubi<lb/>c numeri, eu latera corporum rhete, a k uero & a l, & huiumodi <lb/>in infinitum recia potentia rhete.<lb/><arrow.to.target n="marg137"/></s></p><p type="margin">
<s id="id2683516"><margin.target id="marg132"/>P<emph type="italics"/>er<emph.end type="italics"/> C<emph type="italics"/>or<emph.end type="italics"/>^{m}. <lb/><emph type="italics"/>prcedentis.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2683565"><margin.target id="marg133"/>P<emph type="italics"/>er<emph.end type="italics"/> 14. <emph type="italics"/>deci <lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2683614"><margin.target id="marg134"/>P<emph type="italics"/>er<emph.end type="italics"/> 17. <lb/><emph type="italics"/>eiudem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2683652"><margin.target id="marg135"/>P<emph type="italics"/>er<emph.end type="italics"/> 14. <lb/><emph type="italics"/>rurus.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2683690"><margin.target id="marg136"/>P<emph type="italics"/>er<emph.end type="italics"/> 17. <lb/><emph type="italics"/>rurus.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2683728"><margin.target id="marg137"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2683754">Ex hoc patet, quod cum circulus posit diuidi in infinita gene</s></p><p type="main">
<s id="id2683766"><arrow.to.target n="marg138"/><lb/>ra quantitatum, qu non unt inuicem commen cumque coniun<lb/>ctiones h emper in eodem genere maneant, quod infinita pun<lb/>cta, & infinitis in peciebus quantitatum remanebunt in quibus a <lb/>& b in perpetuum nunquam conuenient. </s>
<s id="id2683809">Velut i coniunctio pri<lb/>ma fiat in <02> cu. </s>
<s id="id2683821">1/2 alicuius circuli, nunquam conuenient, neque in me<lb/>dietate, neque in quarta parte, nec octaua, nec tertia, nec exta, nec no<lb/>na, nec quinta, nec decima, & ic de ingulis in genere commena<lb/>rum toti circulo. </s>
<s id="id2683852">Neque in <02> quadrata 1/2 uel 1/3 uel 1/5 neque <02> 1/6 uel 1/20, <lb/>neque in <02> 3 m: 1, nec 2 m: <02> 3 nec in <02> <02> 2 aut 3 aut 7 nec in <02> rela<lb/>ta alicuius numeri, nec in 2 m: <02> <02> cub. </s>
<s id="id2683869">3 nec 2 m: <02> cub. </s>
<s id="id2683874">4, & ic <lb/>de alijs.</s></p><p type="margin">
<s id="id2683887"><margin.target id="marg138"/>P<emph type="italics"/>er penulti<lb/>mam uigei<lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s></p><p type="main">
<s id="id2683935">Propoitio quinquageimaprima.</s></p><p type="main">
<s id="id2683948">Operationes dictas exemplo declarare.<lb/><arrow.to.target n="marg139"/></s></p><p type="margin">
<s id="id2683964"><margin.target id="marg139"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2683990">Supponamus in circulo prdicto a c <02> 7 contat, quod ee non <lb/>potet, quia <02> 7 et maior monade, ideo toto circulo, quare non po<lb/>terit ee pars circuli, ed referetur ad <expan abbr="quantitat&etilde;">quantitatem</expan> certam, uelut quod <lb/>circulus it 10. emper ergo diuidemus <02> 7, eu eam portionem per <lb/>10 quantitatem circuli & exibit <02> 7/100, & hc erit portio circuli, & ita <lb/>i portio it <02> cub. </s>
<s id="id2684058">16, diuidemus <02> cub. </s>
<s id="id2684062">16 per 10 exibit <02> cu 2/125, & <lb/>ita de alijs.</s></p><p type="main">
<s id="id2684075">Sed cum ex repetitione crecat portio illa, donec exuperet mo<lb/>nadem, aut aliquem quemuis numerum detracta monade aut nu<lb/>mero circuituum habebit rationem recii. </s>
<s id="id2684094">Velut <02> 7/100 quater um<lb/>pta efficit <02> 112/100. Et hoc et potentia rhete, ed i quis auferat mona<lb/>dem fiet <02> 112/100 m: 1, & hoc et recium 1, cilicet 1 p: <02> v: 23/25 m: <02> 28/25, ed ta <lb/>men uer et linea media.</s></p><p type="main">
<s id="id2684147">Quod uer non contingat coniungi in alio loco, neque tem<lb/>pore it, ut a b iungantur in c, & it reuolutio a triplex integra, & b
<pb xlink:href="015/01/061.jpg" pagenum="42"/>excuplex, & tempus totum decem annorum: ita ut a c it tertia <lb/>pars circuitus, & a circuitus tres anni, & quia circuitus b unt fex <lb/>cum tertia, diuidemus decem per 6 1/3 exit <lb/>1 11/29, dico quod non prius, neque in alio <lb/><figure id="id.015.01.061.1.jpg" xlink:href="015/01/061/1.jpg"/><lb/>puncto. </s>
<s id="id2684209">Si enim primm in eodem pun<lb/>cto, &, gratia exempli, in quatuor annis <lb/>congruit enim, & b dicamus quod per<lb/>egerit duas reuolutiones cum tertia, hoc <lb/>enim et necearium, i debet perueni<lb/>re ad c, & erunt anni tres, & 23/19, non ergo <lb/>anni quatuor. </s>
<s id="id2684253">Cum enim tempora di<lb/>uera diuiduntur per numeros haben<lb/>tes proportionem erunt, qui prodeunt <lb/><arrow.to.target n="table13"/><lb/>numeri in eadem ratione. </s>
<s id="id2684278">Diuio ergo <lb/>10 per 1 11/19 exit 6 2/3, & diuio 4 per 1 11/19 exit <lb/>2 8/15, igitur 6 1/3 ad 2 8/15, ut 10 ad 4, igitur 8/25 <lb/>non potet ee quale 1/3. Si enim per <lb/>prcedentem repetuntur, ergo non po<lb/>unt redire, doneciterum coniung antur in ipo a. </s>
<s id="id2684324">Si enim aliter it <lb/>ut ex e, igitur e c et qualis a c pars toti, quod contingere non po<lb/>tet. </s>
<s id="id2684348">Sin uer coniunctio fiat in d, igitur per prcedentem d e et <lb/>pars a c ubmultiplex quomodolibet, quare non fuerunt aum<lb/>pti primi numeri. </s>
<s id="id2684376">Veluti in exemplo contituimus, quod a, & b <lb/>conueniunt in c in decem annis, & a c et tertia pars circuitus: er<lb/>go in triginta annis conueniunt in a, & in quadraginta rurus in c. <lb/>i ergo quis aumpiet quadraginta annos ab initio pro con<lb/>greu, & diuiiet per 1 12/19 exiret 25 1/3, & i per 3 exiret 13 1/3, & mani<lb/>fetum et, quod uterque numerus potet diuidi per eundem nu<lb/>merum, utpote 4 & exit numerus cum eadem parte cilicet 6 1/3 & <lb/>3 1/3 ergo conuenient ante, non ergo aumpiti minimos in ea pro<lb/>portione. </s>
<s id="id2684476">Illi autem nequaquam amplius diuidi non pount eo<lb/>dem modo.</s></p><table><table.target id="table13"/><row><cell>Decem</cell><cell/><cell>Quatuor</cell><cell/></row><row><cell>3</cell><cell>3 1/3</cell><cell>1 11/19</cell><cell>2 8/15)</cell></row><row><cell>1 11/19</cell><cell>6 1/3</cell><cell/><cell/></row></table><p type="main">
<s id="id2684533">Propoitio quinquageimaecunda.</s></p><p type="main">
<s id="id2684549">Tria mobilia coniuncta in eodem puncto, quorum duo, & duo <lb/>conueniant in partibus in commenis inter e, in perpetuum in nul<lb/>lo unquam puncto conuenient.</s></p><p type="main">
<s id="id2684574"><arrow.to.target n="marg140"/></s></p><p type="margin">
<s id="id2684585"><margin.target id="marg140"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2684610">Sint a b c iuncta, & primo iungantur a & b, iterum in d & b, & <lb/>c in e, & int a d, a e inconimen, dico qud a b c nunquam con<lb/>uenient in aliquo puncto, eu primo, eu alio prim o: i non con
<pb xlink:href="015/01/062.jpg" pagenum="43"/><figure id="id.015.01.062.1.jpg" xlink:href="015/01/062/1.jpg"/><lb/>ueniant in f, erunt ergo in g tempore re<lb/>uolutiones integr, & portio a f inuper. <lb/></s>
<s id="id2684678">Et quia h contituuntur per congreus <lb/>b cum a, & unt patia a d, & b cum c, & <lb/>unt patia e f, igitur patium a f erit ex ge<lb/>nere quantitatis a d, & a e per quinqua<lb/>geimam, harum ergo erunt commen: <lb/>quod et contra uppoitum. </s>
<s id="id2684738">Et harum <lb/>propoitionum principium et traditum <lb/> Campano Nouarieni Euclidis expoitore, in quodam libello <lb/>non edito qui diligentia patris mei Facij ad me peruenit.</s></p><p type="main">
<s id="id2684770">Propoitio quinquageimatertia.</s></p><p type="main">
<s id="id2684783"><expan abbr="Circulorũ">Circulorum</expan> e in aduerum mouentium proportionem declarare.</s></p><p type="main">
<s id="id2684804"><arrow.to.target n="marg141"/></s></p><p type="margin">
<s id="id2684816"><margin.target id="marg141"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2684842">Sit orbis a b cuius cen<lb/><figure id="id.015.01.062.2.jpg" xlink:href="015/01/062/2.jpg"/><lb/>centrum c, manubrium c <lb/>d f e, eu uero tangat circu <lb/>lum g, eu more gemmas <lb/>culpentium aligetur al<lb/>teri orbi funiculo a l b, & <lb/>it in uertice axis k m or<lb/>biculus olidus aut emi<lb/>circulari forma m, dico <lb/>quod proportio motus a <lb/>b ad motum m et produ <lb/>cta ex duabus proportio<lb/>nibus c n <expan abbr="&longs;emidimeti&etilde;tis">emidimetientis</expan>, <lb/>& emidimetientis m ad k <lb/>o, quare ut rectanguli c n <lb/>in dimidium dimetientis <lb/>m ad quadratum o, ut enim a b ad ol orbem, id et <expan abbr="peripheriarũ">peripheriarum</expan> ita <lb/>c n ad o k, quoniam o l mouetur toties in una circuitione a b, quo<lb/>ties <expan abbr="peripheriã">peripheriam</expan> o l <expan abbr="contine&ttilde;">continetur</expan> in peripheria a b, ergo quoties o k con<lb/>tinetur in c n toties in una circuitione a b o l circumuertitur, ed <lb/>quoties circumuertitur ol, toties etiam m, quia uterque mouetur eo<lb/>dem circuitu k m axis, ergo quoties m circumducitur in circuitu a <lb/>b toties o k continetur in c n, ergo i fiat comparatio emidiametri <lb/>m ad c n, erit product a proportio circuitus a b ad circuitum m ex <lb/>proportione c n ad o k, et emidimetientis m ad <expan abbr="id&etilde;">idem</expan> o k, ergo per 26 <lb/>proportio numeri circuitus unius p <expan abbr="alterũ">alterum</expan> et, ut rectanguli ub c n, <lb/>& emidimetiente m ad quadratum k o, quod erat <expan abbr="demon&longs;trandũ">demontrandum</expan>.</s></p><p type="main">
<s id="id2685068">Manifetum et autem ex ipa ola contitutione, quod i a b mo</s></p><p type="main">
<s id="id2685093"><arrow.to.target n="marg142"/>
<pb xlink:href="015/01/063.jpg" pagenum="44"/>uetur urum dextro in initrum in inferiore parte, mouebitur <lb/>initro in dextrum, & uterque circulorum g & k in uperiore parte, <lb/>& in inferiore mouebitur contrario motu, cilicet in uperiore ini <lb/>tro in dextrum, & inferiore dextro in initrum, illi uer duo or<lb/>bes imili motu mouebuntur tam in parte uperiore, qum inferio<lb/>re, & proportio motuum eorum inter e erit uelut dimetientium <lb/>corundem.<lb/><arrow.to.target n="marg143"/></s></p><p type="margin">
<s id="id2685205"><margin.target id="marg142"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 1.</s></p><p type="margin">
<s id="id2685231"><margin.target id="marg143"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 2.</s></p><p type="main">
<s id="id2685257">Rurus cum a b circumuertatur cum manubrio c d f e, tanto uelo <lb/>cius circumuertetur, & in ea proportione, qua d f continetur in c n, <lb/>& in eodem tempore, in quo manubrium circumuertitur in eodem <lb/>axis circumuertitur, & orbis, ut dictum et, ergo in eodem tempo<lb/>re, in quo axis circumuertitur in eodem orbis: ergo tanto tardius <lb/>uidebitur moueri axis ipo orbe, quanta et proportio minoris in <lb/>qualitatis ipius axis, eu ambitus, eu emidimetientis ad ambi<lb/>tum, eu emidimetientem orbis.</s></p><p type="main">
<s id="id2685326">Propoitio quinquageimaquarta.</s></p><p type="main">
<s id="id2685340">Proportio circuli ad uum diametrum per <expan abbr="&longs;imilitudin&etilde;">imilitudinem</expan> et quar<lb/>ta pars peripheri. </s>
<s id="id2685368">Rurusque eiudem circuli ad peripheriam diame<lb/>tri quarta pars.</s></p><p type="main">
<s id="id2685386"><arrow.to.target n="marg144"/></s></p><p type="margin">
<s id="id2685397"><margin.target id="marg144"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2685423">Quoniam enim uperficies circuli, ut ab <lb/><figure id="id.015.01.063.1.jpg" xlink:href="015/01/063/1.jpg"/><lb/>Archimede demontratum et, fit ex dimi</s></p><p type="main">
<s id="id2685454"><arrow.to.target n="marg145"/><lb/>dio diametri in <expan abbr="dimidiũ">dimidium</expan> peripheri erit, ut <lb/>eadem fiat ex tota peripheria in <expan abbr="quartã">quartam</expan> par <lb/>tem diametri, & ex tota diametro in quar<lb/>tam <expan abbr="part&etilde;">partem</expan> peripheri. </s>
<s id="id2685506">ergo proportio are <lb/>circuli ad diametrum per imilitudinem <lb/><arrow.to.target n="marg146"/><lb/>et quarta pars peripheri, & proportio are <lb/>ad <expan abbr="peripheriã">peripheriam</expan> et quarta pars dimetientis, quod erat probandum.</s></p><p type="margin">
<s id="id2685555"><margin.target id="marg145"/>P<emph type="italics"/>er<emph.end type="italics"/> 16. <emph type="italics"/>ex <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2685606"><margin.target id="marg146"/>P<emph type="italics"/>er<emph.end type="italics"/> 2. <emph type="italics"/>diff.<emph.end type="italics"/></s></p><p type="main">
<s id="id2685639">Propoitio quinquageimaquinta.</s></p><p type="main">
<s id="id2685652">Proportionem medicamentorum per ordines uppoita quali <lb/>proportione in ordinibus per quantitates, & proportiones de<lb/>montrare.<lb/><arrow.to.target n="marg147"/></s></p><p type="margin">
<s id="id2685687"><margin.target id="marg147"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2685714">Galenus libro quinto de Simplicibus medicamentis, quem e</s></p><p type="main">
<s id="id2685726"><arrow.to.target n="marg148"/><lb/>quuti unt alij medici, ponit quatuor ordines <expan abbr="medicamentorũ">medicamentorum</expan> iux<lb/>ta qualitates calidi, frigidi, icci, & humidi, & primus et cum <expan abbr="medi-camentũ">medi<lb/>camentum</expan> non entitur quale it licet operetur, uelut cammelon, ab<lb/>ynthium, & oriza: ecundus et, cum entitur, ed non ldit, ut nux <lb/>myritica, aluia, ozimum: tertius et cum entitur, & ldit, ed <lb/>non detruit, neque corrumpit corpus, uelut aarum apium ta<lb/>phiagria, cappares, myrrha, ruta: quartus et, cum detruit ue<lb/>lut pyretrum, piper, euphorbium cpe aggrete, & inapis, cina
<pb xlink:href="015/01/064.jpg" pagenum="45"/>momum autem, & gingiber numerantur inter medicinas caldas <lb/>tertij gradus, & hoc opus comparatur ad corpus icut dicit Gale<lb/>nus, & Serapio non ad linguam, ut medici notri temporis interpre <lb/>tantur. </s>
<s id="id2685895">Ex quo patet, quod aliqua medicina poterit ee quarti ordl <lb/>nis, & non ldere linguam in gutu, & alia tertij ordinis, qu non <lb/>olum ldet linguam, ed enum eius corrumpet, et detruet, quod <lb/>contingit propter ubtantiam tenuem cra mitam cum iccitate <lb/>pari ipi calori. </s>
<s id="id2685961">Sed non oportet hc nunc tractar, enon olum quia <lb/>non it locus, ed etiam qud conua it per eipa materia abque <lb/>eo, quod difficultatem difficultati addamus, olum ergo eas dubita<lb/>tiones adiungemus, quas <expan abbr="uol&etilde;tes">uolentes</expan> declarare propoitionem pren <lb/>tem, neque uperfugere, neque declinare poumus. </s>
<s id="id2686035">Nam de icco, <lb/>& humido, cum int long minoris actionis, qum calidum, & fri<lb/>gidum, & prcipu humidum, non uideo quomodo posit Gale<lb/>nus tatuere medicinam humidam tertij gradus, nedum quarti, <lb/>cum non posit inueniri medicina, qu detruat corpus notrum <lb/>propter humidam qualitatem. </s>
<s id="id2686093">Et licet Serapio pouerit gingiber <lb/><arrow.to.target n="marg149"/><lb/>& enulam & zelim in tertio ordine calidorum & humidorum: & <lb/>inter frigidas, & humidas in tertio portulacam, aizoum, & uirgam <lb/>patoris, & fungos. </s>
<s id="id2686121">Primum non auus et ponere medicinas ullas <lb/>calidas, aut frigidas in quarto ordine, qu int humid. </s>
<s id="id2686143">ecundum, <lb/>quando dicit medicinas caldas, aut frigidas, atque humdas in ter<lb/>tio ordine, intelligit olum de qualitate actiua cilicet caliditate, uel <lb/>frigiditate, & non de humida qualitate, quod otendit de gingibe<lb/>re, & enula, dicens, quod unt calid in tertio ordine, & humid <lb/>humido crudo, non auus addere ordinem, quia non udit ratio<lb/>nem, qua poent dici humid in tertio. </s>
<s id="id2686211">Et clarius in capite de zei<lb/>len, quem tatuerat inter medicinas calidas, & humidas in tertio, di <lb/>cit quod et calida in tertio, & humida in primo, ergo non intelligit <lb/>per medicinas calidas & humidas in tertio ordine, quod int humi<lb/>d in tertio ordine. </s>
<s id="id2686248">Clarius etiam de frigidis & humidis, nam por<lb/>tula cam dicit ee frigidam in tertio, humidam in ecundo, & quod <lb/>maius, et cum collo caet aizoum inter medicinas frigidas, & hu<lb/>midas in tertio ordine, dicit, quod et frigidum in tertio ordine, ad<lb/>ijcit, quod et iccum parum, & de uirga patoris nihil dicit de hu<lb/>mido, ed dicit, quod atringit, ex quo concludo, quod ecun<lb/>dum mentem Serapionis nulla et medicina humidior portulaca, <lb/>etiam uidetur innuere de fungis, atis et quod non excedunt ecun<lb/>dum ordinem in humido neque calida neque frigida, ed frigida unt <lb/>humidiora, ut fungi, & portulaca, quia frigiditas in generatione <lb/>humidum magis admittit, qum caliditas, & calida magis hu
<pb xlink:href="015/01/065.jpg" pagenum="46"/>mectant, quia magis penetrat uis medicamenti, & hc regula de <lb/>humido, & icco et generalis apud Serapionem, quod non intelli<lb/>gitur ordo in pasiuis, nii pecialiter exprimatur, nam de iccitate <lb/>non nego, quin inueniantur medicin icc in tertio, & foran in <lb/>quarto ordine, ed de hac Galeni ocitantia, qu in illo peculiaris <lb/>et dum uult equi uas methodos ine alio dicrimine, medicis con <lb/>i derandum relinquo.</s></p><p type="margin">
<s id="id2686445"><margin.target id="marg148"/>C<emph type="italics"/>ap. </s>
<s id="id2686462">ult.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2686473"><margin.target id="marg149"/>C<emph type="italics"/>ap.<emph.end type="italics"/> 336. <lb/>337. & <lb/>338.</s></p><p type="main">
<s id="id2686503">Secunda difficultas et maior, & magis pertinet ad nos, & et, <lb/>qud non declarauit an iti ordines inter e <expan abbr="aliquã">aliquam</expan> proportionem <lb/>eruarent, an omnino nullam, i enim nulla proportio eruatur, fieri <lb/>nullo modo potet, ut per cognitionem temperatur implicium <lb/>medicamentorum cognocamus temperaturam compoitorum ex <lb/>illis ratione ulla, ed oportebit olum experiri. </s>
<s id="id2686574">Sed i ordines er<lb/>uant proportionem, adhuc relinquitur dubium, an illa proportio <lb/>it Arithmetica, uel Geometrica, uel Muica, & nihil mirum eet, <lb/>quod eet Muica, ut alis docuimus, ubitractauimus de differen<lb/>tia inter enum auditus, et uius. </s>
<s id="id2686629">Sed quia de hac nullus medicus ui <lb/>detur intellexie, omittam hanc tractationem. </s>
<s id="id2686641">Et quanqum Gale<lb/>nus posit uideri non exitimae, qud hi ordines non eruent <lb/>proportionem ullam, quia non auus et tractare de temperamen<lb/>to medicamentorum compoitorum per rationem temperamen<lb/>ti implicium, nihilominus uppoito quod ita eet, quod eruetur <lb/>altera proportionum, uolo otendere rationem componendi in <lb/>utraque proportione & Arithmetica, & Geometrica. </s>
<s id="id2686715">Ex quo e<lb/>quitur, quod Aueroes qum ocitanter tractauerit in quinto uo<lb/>rum collectaneorum de hoc, & non ditinguit, neque docet pri<lb/>mum an it aliqua proportio, deinde i qua it, cuius generis it, & <lb/>cum in re tam clara pugnet prorus, ut ccus ictus maximos eden<lb/>do, ed in caum pleroque, qum mal agant qui ei in arduis tan<lb/>tum tribuunt fidei, & authoritatis, ed hc et infelicitas notra, & <lb/>ira Deorum. </s>
<s id="id2686807">Suppoito ergo quod prim ordines ditinguantur <lb/>per proportionem arithmeticam, it uperficies a b pro quantitate, <lb/><figure id="id.015.01.065.1.jpg" xlink:href="015/01/065/1.jpg"/><lb/>& a it calida in primo gradu, & b in ter<lb/>tio, erit ergo perinde ac i duo corpora <lb/>eent unum altitudinis unius cum bai <lb/>quadrilatera rectangula a, aliud altitu<lb/>dinis trium, bai autem quadrilatera u<lb/>perficie rectangula b, hoc igitur erit to<lb/>tum mitum, & quia quantitas medicamenti non mutatur qu et <lb/>a, b, ergo talia corpora quantur uni corpori, cuius bais et a b, <lb/>cum ergo talia corpora producantur ex a in unum, & b in tria, ergo
<pb xlink:href="015/01/066.jpg" pagenum="47"/>diuio aggregato per a b prodibit altitudo, eu ordo qualitatis to<lb/>tius medicamenti, iuxta quod contituitur regula prima libri artis <lb/>medendi paru huiumodi, & reliqu, traduxi autem illas ad hunc <lb/>locuin, quia pendent ex demontratione hac: duc numerum ordi<lb/>nis ingulorum medicamentorum in numerum quantitatis, imilia <lb/>iunge, disimilia detrahe, quod fit, diuide per aggregatum, quanti<lb/>tatum, exibit numerus ordinis compoiti. </s>
<s id="id2686984">Sic micendo calidum in <lb/>ecundo ordine cum duplo pondere temperati conflabit calidum <lb/>in bee. </s>
<s id="id2687003">Secunda i ex pluribus diuerarum, qualitatum, & ordi<lb/>num temperatum efficere uelis, duc qu unt eiudem qualitatis in <lb/>uas quantitates, & iunge, quod fit, diuide per numerum or dinis <lb/>medicamenti contrarij, exibit quantitas illius, ub qua i iungatur, <lb/>fiet medicamentum temperatum. </s>
<s id="id2687046">Tertia cum nolueris ex tempera<lb/>to, & alio cuiucunque ordinis medicamen conficere ordinis re<lb/>misionis, detrahe numerum ordinis eius, quod conficere uis ex nu<lb/>mero ordinis eius, quod habes, & cum reiduo diuide numerum <lb/>medicaminis, quod conficere uis, quod exit et numerus quantita<lb/>tis medicamenti non temperati in comparatione ad temperatum. <lb/>Ex his potes propoitis quibucunque medicamentis conficere <lb/>antidotum ub quo cunque ordine remisiore potentisimo ex il<lb/>lis. </s>
<s id="id2687115">Quarta in compoitione, qu non fermentecit calida, calidis <lb/>iuncta emper opus augent, ut mel cum pipere. </s>
<s id="id2687134">Qu autem ub mi<lb/>nore quantitate exhibentur non ub remisiore ordine agant, ed <lb/>uel facilius impediuntur, uel minorem corporis partem, uel leuius <lb/>immutant.</s></p><p type="main">
<s id="id2687166">Quod i tatuamus proportionem ee Geometricam, modus <lb/>erit idem in omnibus, & quo ad numerum etiam in primo, & ecun<lb/>do ordine, quia in proportione dupla Geometrica ecundus ordo <lb/>tantundem ditat primo, quantum primus ab qualitate, quia <lb/>unum & duo eruant proportionem, & qualem ditantiam, ed in <lb/>cteris ordinibus non ita erit, quia qui eet trium in Arithmetica, <lb/>cilicet totius ordo et, quatuor in Geometrica, & quartus ordo, <lb/>qui eet quatuor in Arithmetica, eet octo in Geometrica, ideo <lb/><figure id="id.015.01.066.1.jpg" xlink:href="015/01/066/1.jpg"/><lb/>cribemus ordines hoc modo, & operabimur cum <lb/>numeris loco ordinum, exemplum ergo primum <lb/>it medicina calida in tertio ordine quatuor uncia<lb/>rum, & medicina frigida in <expan abbr="&longs;ecũdo">ecundo</expan> ordine duarum <lb/>unciarum, duco quatuor in tria, i proportio it Arithmetica, fit <lb/>duodecim, duco duo in duo fit quatuor, detraho quatuor in duo<lb/>decim, quia omnis medicina tantum retondit de contrario, eu mi<lb/>nuit relin quuntur octo cilicet caliditatis, diuido per ex ag
<pb xlink:href="015/01/067.jpg" pagenum="48"/>gregatum unciarum exit unum, & tertia, ergo erit calida in princi<lb/>pio ecundi ordinis. </s>
<s id="id2687348">Secundum exemplum int edem medicin, <lb/>& it proportio Geometrica, ducemus ergo quatuor in quatuor, & <lb/>fiunt exdecim, & duo in duo fiunt quatuor, detrahe quatuor ex ex <lb/>decim, & remanent duodecim, diuide per ex, ut prius, exeunt duo, <lb/>ergo erit calida in fine ecund i gradus uides ergo dicrimen. </s>
<s id="id2687395">rurus <lb/>int amb medicin calid, & ducemus, ut prius in tertio exem<lb/>plo, ubi proportio it Arithmetica iungendo duodecim cum qua<lb/>tuor, & fient exdecim, diuide per ex, exeunt duo, & du terti, er<lb/>go erit calida in medio tertij gradus, rurus in quarto exemplo iun <lb/>gemus edecim cum quatuor, & fient uiginti, diuide per ex exi<lb/>bunt tria & tertia, & ita erit in medio tertij gradus, ut prius, ed i <lb/>ille quatuor unci eent calid in quarto gradu, & ill du unci <lb/>in ecundo gradu, ut prius ducendo quatuor in quatuor fiunt ex<lb/>decim, & duo in duo fiunt quatuor, iunge, & fient uiginti, diuide <lb/>per ex exeunt tria cum tertia, ergo erit calida in principio quarti <lb/>gradus ecundum proportionem Arithmeticam, ed ecundum <lb/>Geometricam duc quatuor in octo, fiunt triginta duo, adde qua<lb/>tuor ut prius, cilicet productum duorum in duo fiunt triginta ex, <lb/>diuide per ex, exeunt ex, & quia ex ad quatuor maiorem habent <lb/>proportionem, qum octo ad ex ideo hc medicina erit calida ul<lb/>tra medium quarti gradus, iam ergo uides rationem, & differen<lb/>tiam horum.</s></p><p type="main">
<s id="id2687578">Quod i quis dicat, an debeat attendi Geometrica proportio in <lb/>medicamentis, an Arithmetica, repondeo, qud ueriimilius et de <lb/>Arithmetica, quia illa proportio etiam quod it minor quatuor ad <lb/>trium, qum trium ad duo, & mult minor qum duo ad unum ni<lb/>hilominus long plus operatur, quia tertius ordo iam incipit ee <lb/>prter naturam, & uidemus, quod lio facta in uulnerato, etiam <lb/>qud it quadruplo minor, plus nocet long, qum in ano qua<lb/>druplo maior: quia termini prter naturam unt uald anguti in <lb/>comparatione ad latitudinem naturalem, icut etiam uidemus in<lb/>tendendis chordis corpionum, quod ultima pars et breuis & ta<lb/>men homini tantam difficultatem adijcit. </s>
<s id="id2687697">Notandum et etiam, <lb/>qud ob hoc diuierunt ordines in tres partes, uelut gingiber et <lb/>calidum in fine tertij ordinis, origanum in medio, cinamomum in <lb/>principio, & ita euphorbium et calidum in principio quarti gra<lb/>dus, ed in fine principij piper, in principio principij aqua epara<lb/>tionis in medio quarti ordinis, ed oleum chalcanthi factum ea ar<lb/>te, ut exurat paleas, icut ignis et calidum in fine quarti ordinis, & <lb/>ita ufficiet diuidere propter eandem cauam primum, & ecun
<pb xlink:href="015/01/068.jpg" pagenum="49"/>dum ordinem in duas tantum partes non ratione latitudinis, qu <lb/>et qualis, uel etiam foran maior, ed ratione uarietatis operatio<lb/>nis qu minus entitur, & maxim in primo ordine.</s></p><p type="main">
<s id="id2687815">Propoitio quinquageimaexta.</s></p><p type="main">
<s id="id2687832">Proportio cuiuuis binomij ad uum recium, uel ei commen<lb/>um et duplicata ei, qu ad numeri latus.<lb/><arrow.to.target n="marg150"/></s></p><p type="margin">
<s id="id2687867"><margin.target id="marg150"/>C<emph type="italics"/>o<emph.end type="italics"/>m.</s></p><p type="main">
<s id="id2687893">Cum enim proportionis medium it latus numeri eo quod ex bi <lb/>nomio in recium uum fit numerus ex his, qu demontrata unt <lb/>generaliter in tertio Arithmetic de omnibus binomijs cum uis </s></p><p type="main">
<s id="id2687930"><arrow.to.target n="marg151"/><lb/>reciis, uel in quadratis lateribus erit <02> numeri media proportione <lb/>inter binomium, & uum recium, igitur cum proportio producto<lb/>rum ex binomio in commena recio it, ut commenorum ad reci<lb/><arrow.to.target n="marg152"/><lb/>a crunt omnia producta ex binomio in commena recio uo <02> nu <lb/><arrow.to.target n="marg153"/><lb/>meri, igitur proportio binomij ad recium uum, & omnia com<lb/>mena illi, et duplicata ei qu ad <02> numeri.<lb/><arrow.to.target n="marg154"/></s></p><p type="margin">
<s id="id2688028"><margin.target id="marg151"/>P<emph type="italics"/>er<emph.end type="italics"/> 6. P<emph type="italics"/>ro<lb/>po. </s>
<s id="id2688060">lib.
de<emph.end type="italics"/><lb/>A<emph type="italics"/>liza.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2688085"><margin.target id="marg152"/>P<emph type="italics"/>er<emph.end type="italics"/> 17. <emph type="italics"/>ex <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2688136"><margin.target id="marg153"/>P<emph type="italics"/>er<emph.end type="italics"/> 17. <lb/><emph type="italics"/>eptimi <lb/>eiudem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2688177"><margin.target id="marg154"/>P<emph type="italics"/>er<emph.end type="italics"/> 6. <emph type="italics"/>deci<lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lement:<emph.end type="italics"/></s></p><p type="main">
<s id="id2688226">Propoitio quinquageimaeptima.</s></p><p type="main">
<s id="id2688242">Motus rationem ad pondus inuenire.</s></p><p type="main">
<s id="id2688250"><arrow.to.target n="marg155"/></s></p><p type="margin">
<s id="id2688262"><margin.target id="marg155"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2688288">Otenum et antea, quod motus naturalis uelocior fit in fine, ac <lb/>magis augetur ob aris motum, ubi uer hret et ac i quiecat. <lb/></s>
<s id="id2688322">Eadem autem et ratio in motis uiolenter, & naturaliter dum qua<lb/>li impetu feruntur. </s>
<s id="id2688337">Sed ubit pot etiam, quod motus qualiter <lb/>augerentur minus tamen crecit proportio uiolenti cilicet ob im<lb/><figure id="id.015.01.068.1.jpg" xlink:href="015/01/068/1.jpg"/><lb/>pedimentum naturale. </s>
<s id="id2688376">Sed i uis mouens fuerit <lb/>ade ualida ut proportio incrementi ex are it <lb/>maior, qum impedimentum, & in crementum al <lb/>terius mobilis naturaliter moti, motus ille uelo<lb/>cior fiet naturali, ut in phris ferreis ex machina <lb/>igne excusis, quod ergo attinet ad prentem <lb/>motum ratio et eadem. </s>
<s id="id2688430">Quicun que ergo motus <lb/>minoris grauis cogit decendere lancem ex ad<lb/>uero proportionem habet eandem ad uum mo <lb/>bile quam habet graue quiponderans. </s>
<s id="id2688457">Sit ergo <lb/>ut a ex b, c, d, e, eleuet eodem ordine pondera e, f, <lb/>g, h, erit ergo ponderum h, g, f, e, ad e inuicem, & ad a qualis mo<lb/>tuum ob ditantiam intentorum. </s>
<s id="id2688480">Experimentum ergo docet, qud <lb/>dimidium ponderis quilibrium facit ex palmo minoris dimidio <lb/>motum manifetum, & ex palmo quarta pars ponderis, ergo e ha<lb/>bent prope portionem.</s></p><p type="main">
<s id="id2688513">Propoitio quinquageimaoctaua.</s></p><p type="main">
<s id="id2688526">Qu ex alto decendunt cur non eandem pro ditantia motus ra<lb/>tionem in libero are eruent coniderare.</s></p>
<pb xlink:href="015/01/069.jpg" pagenum="50"/><p type="main">
<s id="id2688563">Ar in ublimiore eius regione emper naturali motu fertur ex <lb/>Oriente in Occidentem, ed & infra uerum minus manifet. </s>
<s id="id2688588">At ca<lb/>u plerun que contingit, ut moueatur long uehementius, eu ad ean<lb/>dem partem, eu aliam. </s>
<s id="id2688612">Qui uer naturalis et, debilis <lb/><figure id="id.015.01.069.1.jpg" xlink:href="015/01/069/1.jpg"/><lb/>et, quoniam in tenui ualde ubtantia et: nec <expan abbr="cõtinuus">continuus</expan> <lb/>ed intar motus aqu maris fluit ac refluit: aliter ne<lb/>cee eet, ut ingulis horis per mille milliaria procede<lb/>ret, ut ic ne que latere poet, quarndoquidem fortuiti mo<lb/>tus, qui unt multo tardiores non latentnos. </s>
<s id="id2688699">Nam tardiores illos <lb/>ee <expan abbr="cõ&longs;tat">contat</expan>, cum in hora int pulus arteriarum, quatuor millia <expan abbr="ictuũ">ictuum</expan> <lb/>in homine prope temperamentum: i igitur motus naturalis aris <lb/>eet continuus, in hora ar procederet ob ambitum terr millies <lb/>mille paus, <expan abbr="igi&ttilde;">igitur</expan> in ictu pulus uperaret paus 250. At experimur <lb/>nullum uentum aut procellam uperare quinquaginta paus, cum <lb/>etiam continuus ee nunquam oleat, im ne posit quidem, ita que<lb/>cum hic multo tardior etiam in ublimi, dum et, nos latere non <lb/>queat, multo minus poet naturalis latere, i ade uelox & in ea<lb/>dem parte <expan abbr="a&etilde;ris">aerris</expan> eet at que continuus. </s>
<s id="id2688853">Prterea tantus impetus nun<lb/>quam minore motu, aut caua uperaretur, ade ut emper flatum <lb/>aris orientalem entiremus. </s>
<s id="id2688888">Quotidie etiam aduenire ad nos a<lb/>rem ex Illyrico, Macedonia, Myia, Ponto, Bythnia, Capado cia, Sy <lb/>ria, Babylonia, Hyrcanomar, Bactrianis, Sacs, Scythis, ac Seris, to<lb/>to prterea Oceano orientali tam uato, & Gallica noua, terra que flo <lb/>rida non olum res et admirabilis', & incredibilis, ed etiam aliena <lb/> enu, & ab his, qu eueniunt. </s>
<s id="id2688950">A'enu quidem, quoniam nebul, <lb/>qu in are mouentur, primm non in eandem partem emper mo<lb/>uentur: nun quam autem ade celeriter: at i ar ic circumuoluere<lb/>tur, mouerentur & illa, qu in eo continentur, quotidieque arem ex<lb/>periremur & nubiloum, & madidum propter mare. </s>
<s id="id2689010">Nechis, qu <lb/>eueniunt hoc atis repondet, nec nobis id contingeret, ut i peti<lb/>aliqua in regione notra directa uiret, ut ar ingulis diebus la<lb/>be ea infectus ad nos deferretur. </s>
<s id="id2689054">Moueri uer arem emper mani<lb/>fetisimum et tum experimento, tum ratione: ratione iquidem, <lb/>quod aqua & clum naturaliter perpetu mouentur, quare etiam <lb/>ar. </s>
<s id="id2689097">Experimento, qud ubi hiant otia, & ianu, ibi perpetuus en<lb/>titur flatus. </s>
<s id="id2689117">Ergo i a pondus decendat in c, ex alto fertur rect, ed <lb/>i ex ublimi transferetur in b, & indirecta, & ad latus, unde ex <lb/>hoc equitur.</s></p><p type="main">
<s id="id2689152">Propoitio quin quageimanona.</s></p><p type="main">
<s id="id2689166"><arrow.to.target n="marg156"/></s></p><p type="margin">
<s id="id2689177"><margin.target id="marg156"/>C<emph type="italics"/>o<emph.end type="italics"/>m.</s></p><p type="main">
<s id="id2689204">Omne mobile motum duobus motibus non ad idem tendenti<lb/>bus, utro que eorum tardius mouetur imili motu.</s></p>
<pb xlink:href="015/01/070.jpg" pagenum="51"/><p type="main">
<s id="id2689233">Sit a mobile, quod moueatur per a b c impulu uenti aut uiolen</s></p><p type="main">
<s id="id2689246"><arrow.to.target n="marg157"/><lb/><figure id="id.015.01.070.1.jpg" xlink:href="015/01/070/1.jpg"/><lb/>to cum naturali coniuncto: & it terminus naturalis e, <lb/><arrow.to.target n="marg158"/><lb/>& uiolenti d: uter que in directo c, dico, quod tardius per<lb/>ueniet ad c quam d, uel e. </s>
<s id="id2689285">De e manifetum et, quoniam <lb/>motus aris, qui intendit motum a, diuditur in partem, <lb/>qu iuuat motum ad d, & partem, qu mouetur ad e, <lb/>igitur fit minor adiectio. </s>
<s id="id2689315">Et etiam quia a c et longior <lb/>a e ex diffinitione rect: quare tardius perueniet ad c qum ad e du <lb/>plici ratione. </s>
<s id="id2689334">Dico etiam, quod tardius ad c qum d. </s>
<s id="id2689344">Quia enim <lb/>uis, qu fert ad d repugnat ei, qu fert ad e, & uis, qu fert ad e, re<lb/>pugnat ei qu fert ad d, igitur tardius perueniet ad c, qum d. </s>
<s id="id2689372">Nec <lb/>potes dicere, qud uis, qu fert ad c adiuuet ad motum regione <lb/>d, nam cum unus motus non posit perfici ine altero, igitur quan<lb/>tum motus ad eretar dabit motum ad d, tanto motus a c erit tard<lb/>or abolut motu ad d. </s>
<s id="id2689414">Verum etiam et, quod c e breuior erit a d, <lb/>quia motus ad e emper contrahit motum ad d naturalis uiolen<lb/>rum ob cauam dictam. </s>
<s id="id2689435">Vtrm uer motus ad c abolut it tardi<lb/>or, qum ad d, non uppoito, quod c e it qualis a d, ed minor, <lb/>nunc non et locus determinandi.</s></p><p type="margin">
<s id="id2689484"><margin.target id="marg157"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s id="id2689511"><margin.target id="marg158"/>P<emph type="italics"/>er<emph.end type="italics"/> 20. <emph type="italics"/>bu-ius.<emph.end type="italics"/></s></p><p type="main">
<s id="id2689545">Ex hoc patet, quod motus quiditantis mobilis, finis et mini<lb/><arrow.to.target n="marg159"/><lb/>mus omnium: quoniam mobile quai quiecit in illo. </s>
<s id="id2689573">Velut i a mo<lb/>ueatur ad b, inde deflectat ad c minimus motus erit in b, ubi incipit <lb/>naturalis: nam cum incipiat, erit debilisimus, quia non <lb/><figure id="id.015.01.070.2.jpg" xlink:href="015/01/070/2.jpg"/><lb/>et motus actu: uiolentus autem qualis et naturali, <lb/>dum minimus et: ergo cum ex ditantia medij palmi <lb/>duplicetur, naturalis erit motus in b minimus, nii b c <lb/><arrow.to.target n="marg160"/><lb/>eet minor dimidio palmi. </s>
<s id="id2689640">Et etiam qud eet minor, quia ut di<lb/>ctum et, uter que imul iunctus et qualis uni eorum non impedito <lb/>uel minor.</s></p><p type="margin">
<s id="id2689674"><margin.target id="marg159"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s id="id2689700"><margin.target id="marg160"/>P<emph type="italics"/>er<emph.end type="italics"/> 57. <emph type="italics"/>bu-ius.<emph.end type="italics"/></s></p><p type="main">
<s id="id2689734">Propoitio exageima.</s></p><p type="main">
<s id="id2689750">Omne mobile motu naturali decendens parte, decendit gra<lb/>uiore ecundum grauitatis centrum.</s></p><p type="main">
<s id="id2689771">Sit a mobile, grauitatis centrum b, cuius pars ei pro<lb/><arrow.to.target n="marg161"/><lb/><figure id="id.015.01.070.3.jpg" xlink:href="015/01/070/3.jpg"/><lb/>ximior it c a, dico quod decendat motu naturali c a, <lb/>parte tangendo terram, quia enim totum a non potet <lb/>decendere ad centrum decendit b, quia eadem et na<lb/>tura partis, & totius: totius autem terr natura et ut <lb/>centrum, totius it centrum grauitatis, quare b breuiore uia fertur <lb/><arrow.to.target n="marg162"/><lb/>ad centrum, ergo per c d proximiorem partem ipi b. </s>
<s id="id2689851">Sed pars pro<lb/>ximior neceari et grauior, quia centrum et in medio grauita
<pb xlink:href="015/01/071.jpg" pagenum="52"/>tis, ergo omne mobile decendit motu naturali per ui grauio<lb/>rem partem.<lb/><arrow.to.target n="marg163"/></s></p><p type="margin">
<s id="id2689904"><margin.target id="marg161"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s id="id2689929"><margin.target id="marg162"/>P<emph type="italics"/>er<emph.end type="italics"/> 23. <emph type="italics"/>bu-ius.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2689963"><margin.target id="marg163"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2689988">Ex hoc equitur, qud graue habens partes inquales, eu ub<lb/>tantia, cu forma, i ita excutiatur, ut pars grauior <expan abbr="nõ">non</expan> it, infr opor<lb/>tet, ut circumuoluatur.</s></p><p type="main">
<s id="id2690040">Propoitio exageimaprima.</s></p><p type="main">
<s id="id2690056">Proportionem ictus ad pondus rei, & ditantiam generaliter <lb/>coniderare.</s></p><p type="main">
<s id="id2690074"><arrow.to.target n="marg164"/></s></p><p type="margin">
<s id="id2690086"><margin.target id="marg164"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2690112">Dictum et uperius de proportione decenus ad grauitatem: </s></p><p type="main">
<s id="id2690131"><arrow.to.target n="marg165"/><lb/>& qud i graue decendat ex alto impeditur motu aris: & qud <lb/><arrow.to.target n="marg166"/><lb/>res, qu mouetur duobus motibus non ad idem tendentibus tar<lb/><arrow.to.target n="marg167"/><lb/>dius mouetur, quam motus it unuquique. </s>
<s id="id2690188">Demm qud graue <lb/><arrow.to.target n="marg168"/><lb/>decendens circumuoluitur, i pars grauior non it, deorum: & an<lb/>tea ubi egimus de proportione motus ad grauitatem, quod hcin<lb/>telligenda unt prout pount intelligi de motu etiam uiolento. <lb/></s>
<s id="id2690238">Cum ergo uideamus duo hc, quod res acuta frangit caput, i ex <lb/>alto incidat, ed non concutit, lata concutit, ed non diuidit, premit <lb/>tamen carnem ubiectam: nec hoc accidit merito ponderis: nam ut <lb/>uium et emilibra lapidis, uel ferri cadens ex alto contundit caput, <lb/>& uulnerat, & non eleuat in quilibrio, ut pot ex alto cadens loco <lb/>per patium octo palmorum pondus exdecim librarum, & a pon<lb/>dere exdecim librarum homo non lditur, nec uulneratur, ergo id <lb/>accidit ex alia caua, & et, quod ar interceptus inter graue, & cor<lb/>pus notrum non potet dilabi tam cit, ergo ne corpus penetret, <lb/>cogitur ingredi locum, cui et obuius, at que ita concutere, & diuide<lb/>re. </s>
<s id="id2690343">Ex quibus equuntur omnia hc.<lb/><arrow.to.target n="marg169"/></s></p><p type="margin">
<s id="id2690364"><margin.target id="marg165"/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 57.</s></p><p type="margin">
<s id="id2690390"><margin.target id="marg166"/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 58.</s></p><p type="margin">
<s id="id2690418"><margin.target id="marg167"/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 59.</s></p><p type="margin">
<s id="id2690445"><margin.target id="marg168"/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 60.</s></p><p type="margin">
<s id="id2690472"><margin.target id="marg169"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2690497">Primm i quod incidit, molle fuerit, non uulneratur caput, uel <lb/>pars ubiecta, quia reilit in corpus molle: nec molli, quia retundi<lb/>tur, potet uulnerari: ergo nullo modo. </s>
<s id="id2690526">Sed neque ade concutit, <lb/>quia ar rediens, & receptus in molli corpore pro parte, non uer<lb/>berat locum.</s></p><p type="main">
<s id="id2690551"><arrow.to.target n="marg170"/></s></p><p type="margin">
<s id="id2690562"><margin.target id="marg170"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2690588">Secundum in omni colliione eu duri, eu mollis, ed magis du<lb/>ri, dilabuntur partes aris ad latera, ideo quod partes medi pre<lb/>muntur. </s>
<s id="id2690617">Et quanto motus et tardior.</s></p><p type="main">
<s id="id2690628"><arrow.to.target n="marg171"/></s></p><p type="margin">
<s id="id2690640"><margin.target id="marg171"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2690667">Tertium in motu uelo ci fit maior ictus & lio, & maiora omnia <lb/>quam proproportione motus: quoniam ob uelo <expan abbr="citat&etilde;">citatem</expan> minus diffu <lb/>git aris. </s>
<s id="id2690693">Et ide fiunt grauia uulnera ex modico incremento uelo<lb/>citatis motus.</s></p><p type="main">
<s id="id2690710"><arrow.to.target n="marg172"/></s></p><p type="margin">
<s id="id2690721"><margin.target id="marg172"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2690747">Quartum res lat, dur concutiunt, & non uulnerant nii int <lb/>cum magno impetu, aut ualde graues: acut autem uulnerant, ed <lb/>non concutiunt, nii parti acut lata uccedat.</s></p>
<pb xlink:href="015/01/072.jpg" pagenum="53"/><p type="main">
<s id="id2690797">Quintum, corpora dura magis lduntur latis, quia cindun</s></p><p type="main">
<s id="id2690815"><arrow.to.target n="marg173"/><lb/>tur, mollia autem tenuibus, quia diuiduntur: nam mollitie excipi<lb/>unt arem, & ita latis non ade patiuntur, & etiam, quoniam nec <lb/>franguntur, nec ponte cinduntur.</s></p><p type="margin">
<s id="id2690856"><margin.target id="marg173"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2690882">Sextum, etiam in duris penetrat aliquid aris, aliter tota frange<lb/><arrow.to.target n="marg174"/><lb/>rentur. </s>
<s id="id2690899">Contat etiam omnem lapidem marmoreum, aut iliceum <lb/>ee poroum, ut dicunt. </s>
<s id="id2690919">Et etiam quia recipitur in mollioribus, er<lb/>go etiam in durioribus & in durisimis: quod i non recipiant ut ui <lb/>trum, & gemm tota franguntur. </s>
<s id="id2690940">Hoc etiam uidetur enie Philo <lb/>ophus, qui uult, qud res franguntur ob poros.</s></p><p type="margin">
<s id="id2690966"><margin.target id="marg174"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2690992">Propoitio exageimaecunda.</s></p><p type="main">
<s id="id2691011">Proportionem motoris in plano ad motorem, qui eleuat pon<lb/>dus iuxta id, quod mouet inuenire.</s></p><p type="main">
<s id="id2691025">Contitutum et inuenire proportionem uirium, qu eleuant <lb/><arrow.to.target n="marg175"/><lb/>pondus ad uires, qu ipum in plano leui trahere po<lb/><figure id="id.015.01.072.1.jpg" xlink:href="015/01/072/1.jpg"/><lb/>unt. </s>
<s id="id2691069">Vires enim, qu eleuant pondus a unt edem <lb/>puta b, qu uero trahunt c, ed h pount uariari, nam <lb/>quanto uinculum altius, aut decliuis locus magis, aut <lb/>apera uperficies eu ponderis eu plani, tanto difficilius trahitur, <lb/>& maiores expocit uires: hoc enim experimento deprehenditur. <lb/></s>
<s id="id2691124">Du uer potrem cau etiam per e perpicu unt, nec demon <lb/>tratione indigent: nii quod i planum it durisimum, ac leuisi<lb/>mum, quod et aperum facilius trahitur, quia minore ui parte pla<lb/>num tangit. </s>
<s id="id2691190">Nos prterea upponimus planum quale undique <lb/>leue durum, & corpus undique ibi imile, id et cubi formam refe<lb/>rens, & uinculum in imo: Demontrare igitur expedit primum, <lb/>qud in hoc cau b et duplum ad c. </s>
<s id="id2691237">Quia enim cum a eleuatur b ui <lb/>res uperant motum obcurum eu occultum, eu pondus a, & i <lb/>permitteretur ine eo, quod utineret, decenderet iuxta pondus <lb/>uum, quod it d: nititur ergo per pondus d, at quia trahendo duci<lb/>tur circa medium, nam plana uperficies parum differt rotunda <lb/>terr ob terr magnitudinem, media erit repugnantia: in eo enim <lb/>quod mouetur, grauitatem habet d in eo, quod <expan abbr="nõ">non</expan> remouetur nul<lb/>lam habet grauitatem, mediam ergo retinet grauitatem, quare ut b <lb/>ad d, ita c ad dimidium, grauitatis a, at b et primum, quod potet <lb/>mouere d, igitur c et primum, quod potet mouere dimidium a, ut <lb/>ergo dimidium a ad d, ita c ad b, et igitur c dimidium b.</s></p><p type="margin">
<s id="id2691347"><margin.target id="marg175"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2691373">Propoitio exageimatertia.</s></p><p type="main">
<s id="id2691389">Omne graue quanto proximius alligatum plano, tanto faci<lb/>lius trahitur.
<pb xlink:href="015/01/073.jpg" pagenum="54"/><arrow.to.target n="marg176"/></s></p><p type="margin">
<s id="id2691414"><margin.target id="marg176"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2691440">Sit graue a b c alligatum funibus in d ef, dico, <lb/><figure id="id.015.01.073.1.jpg" xlink:href="015/01/073/1.jpg"/><lb/>qud facilius trahetur per fe qum c b & e b, qum <lb/>d a, quia i debet trahi ex a uel b, aut cadet, aut uis ex <lb/>a & b communicabitur c, igitur erit minor qum in <lb/>c, & hoc naturaliter. </s>
<s id="id2691483">Mathematica autem ratione quoniam ex a tra<lb/>hetur c, quai per lineam d c: at attractio recta et ualidior obliqua<lb/>igitur attractio c per d et debilior, qum per f. </s>
<s id="id2691509">Rurus i e trahitur <lb/>per d cm a peruenerit in d, erit perinde ac, i attractum eet per li<lb/>neam c d, ed linea c d mouet duobus motibus, uno ad uperiora, al </s></p><p type="main">
<s id="id2691546"><arrow.to.target n="marg177"/><lb/>tero ad latus, ergo lentius ad f per d c qum f c, quod erat demon<lb/>trandum.</s></p><p type="margin">
<s id="id2691569"><margin.target id="marg177"/>P<emph type="italics"/>er<emph.end type="italics"/> 59. <emph type="italics"/>bu-ius.<emph.end type="italics"/></s></p><p type="main">
<s id="id2691604">Propoitio exageimaquarta.</s></p><p type="main">
<s id="id2691619">Omne mobile quanto latius tanto tardius mouetur in plano.<lb/><arrow.to.target n="marg178"/></s></p><p type="margin">
<s id="id2691635"><margin.target id="marg178"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2691661">Demontratum et uperius qud i mobile it phricum, & tan </s></p><p type="main">
<s id="id2691690"><arrow.to.target n="marg179"/><lb/>gat planum in puncto, qud mouetur per quancunque uim aptam <lb/>diuidere medium. </s>
<s id="id2691706">Quia ergo i tangat in puncto facillime moue<lb/>tur, i in linea paul difficilius, i per uperficiem adhuc difficilius, <lb/>igitur cum fiat attritio in motu quanto latius et mobile eo diffici<lb/>lius mouetur. </s>
<s id="id2691741">Sit ergo mobile a b, quod moueatur uerus c, & quia <lb/>pars b eu dimidium mouetur iuxta rationem me<lb/><figure id="id.015.01.073.2.jpg" xlink:href="015/01/073/2.jpg"/><lb/>dietatis, & pars a eodem modo, ergo conduplicata <lb/>difficultate, quia medietas b impedit medietatem, a <lb/>quanto latius et, & longius a b, tanto difficilius <lb/><arrow.to.target n="marg180"/><lb/>mouetur. </s>
<s id="id2691790">Et hoc intelligitur de corporibus ualde <lb/>latis propter dicta uperius.</s></p><p type="margin">
<s id="id2691805"><margin.target id="marg179"/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 40.</s></p><p type="margin">
<s id="id2691833"><margin.target id="marg180"/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 62</s></p><p type="main">
<s id="id2691861">Propoitio exageimaquinta.</s></p><p type="main">
<s id="id2691876">Proportionem duorum mobilium inter e cum auxilio medij <lb/>inuenire.<lb/><arrow.to.target n="marg181"/></s></p><p type="margin">
<s id="id2691898"><margin.target id="marg181"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2691924">Graue decendit naturaliter quatuor cauis: prima et ponderis <lb/>magnitudo, unde quod grauius et celerius decendit. </s>
<s id="id2691945">Secund ob <lb/>paruam medij repugnantiam, ideo quanto medium et rarius & <lb/>mobile tenuius, tanto celerius decendit: contr uer tardius. </s>
<s id="id2691970">Ter<lb/>ti ob impetum aris ub equentis: & ideo mobile qud ex eadem </s></p><p type="main">
<s id="id2691998"><arrow.to.target n="marg182"/><lb/>materia contat, emper decendit parte acutiore uprapoita, ne ar <lb/>cogatur celerius ferri: & quanto diutius decendit, tanto magis in<lb/>tenditur motus, at que augetur, ut upr de claratum et. </s>
<s id="id2692044">Quarta caua <lb/>et, quod non impediatur ab are tranuerfim moto, et latere: ideo <lb/>leuia mobilia & magna non olum lentius decendunt, quoniam <lb/><arrow.to.target n="marg183"/><lb/>paruam uim habeant, & magnam repugnantiam, ed quia tranuer <lb/><arrow.to.target n="marg184"/><lb/>im impula minus mouentur motu recto, ut upra uium et. </s>
<s id="id2692110">Por
<pb xlink:href="015/01/074.jpg" pagenum="55"/>r proportio ratione decenus aucta, declarata et paulo ant, <lb/>quare cum medium upponatur eiudem generis, & figura non <lb/>eiumodi, nec leuitas, ut prorus non impellat, nedum ut moueat la <lb/>tus: figura quo que eadem ambobus relinquetur proportio motus <lb/>ad motum producta ex proportionibus incrementi in proportio<lb/><arrow.to.target n="marg185"/><lb/>nem ponderum, & iam habuimus proportionem incrementi ex <lb/><arrow.to.target n="marg186"/><lb/>motu aris, ergo proportio unius motus producti ad alteram no<lb/>ta erit.</s></p><p type="margin">
<s id="id2692198"><margin.target id="marg182"/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 30.</s></p><p type="margin">
<s id="id2692226"><margin.target id="marg183"/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 59.</s></p><p type="margin">
<s id="id2692253"><margin.target id="marg184"/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 62.</s></p><p type="margin">
<s id="id2692280"><margin.target id="marg185"/>P<emph type="italics"/>er<emph.end type="italics"/> 42. <emph type="italics"/>ha-rum.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2692314"><margin.target id="marg186"/>I<emph type="italics"/>n<emph.end type="italics"/> 61. <emph type="italics"/>ha-rum.<emph.end type="italics"/></s></p><p type="main">
<s id="id2692347">Propoitio exageimaexta.</s></p><p type="main">
<s id="id2692366">Proportionem laterum eptagoni, & ubtenarum coniderare, <lb/>& qu reflexa proportione pendent.<lb/><arrow.to.target n="marg187"/></s></p><p type="margin">
<s id="id2692398"><margin.target id="marg187"/>C<emph type="italics"/>o<emph.end type="italics"/>m.</s></p><p type="main">
<s id="id2692424">Sit eptagonus a b d f g e c, & ubten b <lb/><figure id="id.015.01.074.1.jpg" xlink:href="015/01/074/1.jpg"/><lb/>c, & f e duobus lateribus, tribus autem d c <lb/>d e, & erunt (quia intelligitur eptagono <lb/>quilatero, & quiangulo) b c & e finuicem <lb/>quales: & item d c, & d e quales: & i du<lb/>cerentur b e & c f inuicem quales: & ad a c <lb/>& d g: quare cum angulus cb d conitatin </s></p><p type="main">
<s id="id2692495"><arrow.to.target n="marg188"/><lb/>arcu c e g f d, & angulus b d c in arcu b a c, <lb/>& angulus b c d in arcu b d; & it arcus c e g <lb/>f d duplus arcus b a c, quia c e g f d ubtendit quatuor latera epta<lb/>goni, & arcus b a c duo, & ita arcus etiam b a c duplus arcui b d <lb/>erit angulus d b e duplus angulo c d b, & angulus c d b duplus an<lb/><arrow.to.target n="marg189"/><lb/>gulo b c d, quare per demontrata nobis proportio laterum b d, <lb/>b c, c d, et reflexa, igitur proportio d b & b c, ad d c, ut d e ad b c, & <lb/><arrow.to.target n="marg190"/><lb/>rurus proportio b d & d e ad b e, ut b e ad b d. </s>
<s id="id2692567">Quare uppoita <lb/>d b 1, b c 1 poitione, erit d c latus 1 quad. </s>
<s id="id2692583">p: 1 poitione. </s>
<s id="id2692590">Proportio <lb/><arrow.to.target n="marg191"/><lb/>uer, ut dictum et b d & d c ad b c, id et p: <02> 1 quad. </s>
<s id="id2692612">p: 1 pos, ad 1 <lb/>pos et, ut b c ad b d, id et 1 pos ad 1, igitur 1 p: <02> v: 1 quad. </s>
<s id="id2692626">p: 1 pos <lb/>quatur quadrato b c, quod et 1 quad. </s>
<s id="id2692638">igitur 1 quad. </s>
<s id="id2692642">m: 1 quatur <lb/><02> v: 1 quad. </s>
<s id="id2692652">p: 1 pos quare 1 quad. </s>
<s id="id2692657">quad. </s>
<s id="id2692661">m: 2, quad. </s>
<s id="id2692665">p: 1 quatur 1 <lb/>quad. </s>
<s id="id2692675">p: 1 pos. </s>
<s id="id2692679">Additis igitur communiter quatuor quadratis fient <lb/>1 quad. </s>
<s id="id2692687">quad. </s>
<s id="id2692691">p: 2 quad. </s>
<s id="id2692695">p: 1 qualia 5 quad. </s>
<s id="id2692702">p: 1 pos. </s>
<s id="id2692706">Et reducitur ad <lb/>1 cu. </s>
<s id="id2692713">qualem 1 3/4 pos p: 7/8.</s></p><p type="margin">
<s id="id2692723"><margin.target id="marg188"/>P<emph type="italics"/>er<emph.end type="italics"/> 28. & 29. <emph type="italics"/>tertij<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2692770"><margin.target id="marg189"/>P<emph type="italics"/>er ult. </s>
<s id="id2692786">exti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2692812"><margin.target id="marg190"/>D<emph type="italics"/>e<emph.end type="italics"/> S<emph type="italics"/>uh. lib.<emph.end type="italics"/> 16.</s>
</p><p type="margin">
<s id="id2692851"><margin.target id="marg191"/>P<emph type="italics"/>er<emph.end type="italics"/> 20. <emph type="italics"/>diff.<emph.end type="italics"/></s></p><p type="main">
<s id="id2692885">Aliter tante uppoitione ut Ludouicus Ferrarius ex demon<lb/>tratis Ptolemo quadratum b c, & et 1 quad et quale produ<lb/>cto ex b d in c e, quod et 1, & a b in d c, igitur detracto 1, produ<lb/>cto b d in c e ex 1 quad. </s>
<s id="id2692931">quadrato c b, relinquitur productum ex <lb/>a b in c d 1 quad. </s>
<s id="id2692939">m: 1, ergo diuio co per a b, qu et 1, relinquitur <lb/>c d 1 quad. </s>
<s id="id2692955">m: 1 huius uer quadratum per <expan abbr="ead&etilde;">eadem</expan> demontrata Pto
<pb xlink:href="015/01/075.jpg" pagenum="56"/>lemo, quale et rectangulis ex b c in de, & b d in c e, igitur 1 quad. <lb/></s>
<s id="id2692997">quad. </s>
<s id="id2693001">m: 2 quad. </s>
<s id="id2693006">p: 1 et quale 1 producto b d in c e, & producto b <lb/>cin d e detracto 1 communi, relin quetur productum ex b c in d e 1 <lb/>quad. </s>
<s id="id2693023">quad. </s>
<s id="id2693027">m: 2 quad. </s>
<s id="id2693031">igitur diuio 1 quad. </s>
<s id="id2693038">quad. </s>
<s id="id2693042">m: 2 quad. </s>
<s id="id2693046">per 1 <lb/>pos, exit 1 cu. </s>
<s id="id2693054">m: 2 pos qualia d e, & d e et qualis d c, ut ab initio <lb/>demontrauimus, & d c fuit 1 quad. </s>
<s id="id2693074">m: 1, igitur 1 cu. </s>
<s id="id2693078">m: 2 quantur 1 <lb/>quad. </s>
<s id="id2693088">m: 1, igitur 1 cu. </s>
<s id="id2693092">p: 1 quantur 1 quad. </s>
<s id="id2693099">p: 2 pos.</s></p><p type="main">
<s id="id2693107">Aliter ut Pacciolus, concurrant latera eptagoni b d, c e in a, & du <lb/>cantur perpendiculares a f, d g & c d, & it c e i ca 1 pos, & quia ut <lb/><arrow.to.target n="marg192"/><lb/>a e ad a c, ita d e ad b c, erit ergo b c (1 posp: 1)/(1 pos) quare b f (1/2 pos 1/2,)/(2 pos) & <lb/>quia d h et dimidium d e, erit d h, & g f <lb/><figure id="id.015.01.075.1.jpg" xlink:href="015/01/075/1.jpg"/><lb/>1/2, cum ergo b f it (1/2 pos p: 1/2)/pos erit ergo di<lb/>uia 1/2 pos per 1 pos, & exit 1/2, b f 1/2p: 1/2/pos <lb/>igitur detracta g f relinquetur g b 1/2/(1 pos). <lb/>& eius quadratum 1/4/(1 quad). igitur cum qua<lb/>dratum b d it 1, erit quadratum g d 1 m: <lb/>2/4/(2 quad)g c autem et compoita ex e f, qu <lb/>et 1/2p: 1/2/(1 pos) & f g qu et 1/2, erit igitur c <lb/>g 1 p: 1/2/(1 pos), & <expan abbr="quadratũ">quadratum</expan> eius 1 p: 1/pos et 1/4/(1 quad.) quare <expan abbr="&qtilde;dratũ">quadratum</expan> e d d et <lb/><arrow.to.target n="marg193"/><lb/>compoitum ex quadratis c g & g d erit 2 p: 1/pos c a uer et qua<lb/>lis c d, quia, ut demontratum et angulus d c e et eptima pars <lb/>duorum rectorum, & angulus b c e ei duplus, quare cum c f a it re<lb/>ctus erit ex trigeimaecunda primi Elementorum f a c tres epti<lb/>m unius recti, ergo d a c 6/7 unius recti, d c a uer 2/7 unius recti, quia <lb/><arrow.to.target n="marg194"/><lb/>et eptima pars duorum rectorum, gitur a d c et 6/7 unius recti: igi<lb/>tur c d et qualis c a, ergo quadratum quadrato: igitur 1 quad. </s>
<s id="id2693336">p: 2 <lb/>pos p: 1, quatur 2 p: 1/(1 pos) igitur 1 quad. </s>
<s id="id2693346">p: 2 pos, quantur 1 p: 1/(1 pos). <lb/>Quare 1 cub. </s>
<s id="id2693356">p: 2 quad. </s>
<s id="id2693360">quatur 1 pos p: 1. <lb/><figure id="id.015.01.075.2.jpg" xlink:href="015/01/075/2.jpg"/><lb/>Sit etiam angulus a duplus b, & b c dupla <lb/>b a: & erit per eadem proportio a c, & a b <lb/>ad c b, ut c b ad c a. </s>
<s id="id2693387">Ponamus ergo ab 1, erit <lb/>b c 2, & a c 1 pos, & a c, a b 1 pos p: 1, & du<lb/>cta in a c fit 1 quad. </s>
<s id="id2693400">p: 1 pos, & hoc et quale 4 quadrato b c per re<lb/>flex proportionis diffinitionem. </s>
<s id="id2693418">Igitur a c et <02> 4 1/4 m: 1/2, & ita <lb/>de alijs.</s></p><p type="margin">
<s id="id2693432"><margin.target id="marg192"/>P<emph type="italics"/>er<emph.end type="italics"/> 42. <emph type="italics"/>pri mi<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2693479"><margin.target id="marg193"/>P<emph type="italics"/>er<emph.end type="italics"/> 32. <emph type="italics"/>pri mi<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2693525"><margin.target id="marg194"/>P<emph type="italics"/>er extam eiudem.<emph.end type="italics"/></s></p><p type="main">
<s id="id2693553">Propoitio exageimaeptima.</s></p><p type="main">
<s id="id2693571">Si fuerint aliquot quantitates ab una quantitate, alique totidem
<pb xlink:href="015/01/076.jpg" pagenum="57"/>ab eadem analo g, erit proportio terti unius ordinis ad tertiam <lb/>alterius, ut ecund ad ecundam duplicata, & quart ad quartam <lb/>triplicata, quint ad quintam quadruplicata, at que ic de alijs.<lb/><arrow.to.target n="marg195"/></s></p><p type="margin">
<s id="id2693628"><margin.target id="marg195"/>C<emph type="italics"/>o<emph.end type="italics"/>_{m}.</s></p><p type="main">
<s id="id2693654">Sint quantitates b c d e f, ab a in continua proportio<lb/><figure id="id.015.01.076.1.jpg" xlink:href="015/01/076/1.jpg"/><arrow.to.target n="table14"/><lb/>ne, & ali totidem g h k l m, dico quod proportio h c et <lb/>duplicata ei, qu et g ad b, & k ad d triplicata, & l ad e <lb/>quadruplicata, & ic deinceps, umatur enim unum, & ab </s></p><table><table.target id="table14"/><row><cell/><cell>a</cell><cell/></row><row><cell>b</cell><cell/><cell>g</cell></row><row><cell>c</cell><cell/><cell>h</cell></row><row><cell>d</cell><cell/><cell>k</cell></row><row><cell>e</cell><cell/><cell>l</cell></row><row><cell>f</cell><cell/><cell>m</cell></row><row><cell/><cell>n</cell><cell/></row><row><cell>o</cell><cell/><cell>t</cell></row><row><cell>p</cell><cell><foreign lang="greek">a</foreign></cell><cell>u</cell></row><row><cell>q</cell><cell><foreign lang="greek">b g</foreign></cell><cell>x</cell></row><row><cell>z</cell><cell/><cell>y</cell></row><row><cell>s</cell><cell/><cell>z</cell></row></table><p type="main">
<s id="id2693822"><arrow.to.target n="marg196"/><lb/>co o p q r s in proportione b ad a, & tuxyz in propor<lb/>tione g ad a, erit igitur p quadratum o, & u quadratum t, <lb/>& q cubus o, & x cubus t, & ita de alijs: ergo proportio <lb/><arrow.to.target n="marg197"/><lb/>n ad p duplicata ei, qu t ad o, & x ad q triplicata ei, qut <lb/>ad o, & potet etiam demontrari generaliter ultra qua<lb/><arrow.to.target n="marg198"/><lb/>dratum, & cubum: nam i ducatur t in o, fiat que <foreign lang="greek">a</foreign> erit, pro<lb/>portio enim ad <foreign lang="greek">a</foreign> eadem qu t ad o, & proportio a ad p, <lb/>ut t ad o, igitur per diffinitionem proportionis duplicat <lb/><arrow.to.target n="marg199"/><lb/>poitam in quinto libro ab Euclide u ad p duplicata ei, <lb/>qu t ad o, & imiliter ex t in p fit <foreign lang="greek">b</foreign> ex o in u, <foreign lang="greek">g</foreign> eruntque<lb/><arrow.to.target n="marg200"/><lb/>q <foreign lang="greek">b g</foreign> x in continua proportione per eandem. </s>
<s id="id2693961">Quia ergo propor<lb/>tio q ad <foreign lang="greek">b</foreign> et ut o ad t, patet, quod x ad q et triplicata ei, qu et t ad <lb/>o, & ita de reliquis, cum ergo proportio p ad o it, ut e ad b, & o ad <lb/><arrow.to.target n="marg201"/><lb/>n, ut b ad a, & n ad t, ut a ad g, & t ad u, ut g ad h, equitur ut it t ad a, <lb/>ut g ad b, & u ad p, ut h ad c, igitur cum it ut u ad p duplicata ei, qu <lb/>et t ad o erit h ad e, duplicata ei qu et g ad b, & ita de reliquis, & <lb/>no refert, eu dicas u ad p duplicatam ei, qu et t ad o, eu dicas p <lb/><arrow.to.target n="marg202"/><lb/>ad u duplicatam ei, qu et o ad t. </s>
<s id="id2694068">Aliter & euidentius in duabus <lb/>oleo demontrare: cum enim it e & h duplicata ei qu et b & g <lb/>ad a, ut upra, & quadrati b ad quadratum a, & quadrati g ad qua<lb/><arrow.to.target n="marg203"/><lb/>dratum a duplicata his qu b & g ad a erunt b & g quadratorum <lb/>ad quadratum a, uelut c & h ad a. </s>
<s id="id2694114">Et conuertendo qua<lb/><arrow.to.target n="table15"/><lb/>drati a ad quadratum g, ut a ad h, contituantur ergo <lb/><figure id="id.015.01.076.2.jpg" xlink:href="015/01/076/2.jpg"/>hic & erit quadrati b ad <expan abbr="quadratũ">quadratum</expan> g, ita c ad h: ed qua<lb/>drati b ad quadratum g, ut b ad g proportio duplicata <lb/>igitur e ad h, ut b ad g duplicata.</s></p><p type="margin">
<s id="id2694168"><margin.target id="marg196"/>P<emph type="italics"/>er<emph.end type="italics"/> 8. <emph type="italics"/>noni<emph.end type="italics"/> E<emph type="italics"/>le.<emph.end type="italics"/> & 22. & 23. <emph type="italics"/>octa ui.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2694223"><margin.target id="marg197"/>V<emph type="italics"/>ide per<emph.end type="italics"/> 23. P<emph type="italics"/>etit.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2694258"><margin.target id="marg198"/>P<emph type="italics"/>er<emph.end type="italics"/> 23. <emph type="italics"/>ex ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/> & 33. <emph type="italics"/>undeci-mi.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2694315"><margin.target id="marg199"/>P<emph type="italics"/>er<emph.end type="italics"/> 17. <emph type="italics"/>e-ptimi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2694363"><margin.target id="marg200"/>D<emph type="italics"/>iff.<emph.end type="italics"/> 10.</s></p><p type="margin">
<s id="id2694387"><margin.target id="marg201"/>P<emph type="italics"/>er<emph.end type="italics"/> 24. <emph type="italics"/>quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2694434"><margin.target id="marg202"/>P<emph type="italics"/>er<emph.end type="italics"/> 10 <emph type="italics"/>diff. </s>
<s id="id2694460">quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2694484"><margin.target id="marg203"/>P<emph type="italics"/>er<emph.end type="italics"/> 20. <emph type="italics"/>ex ti<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s></p><table><table.target id="table15"/><row><cell><expan abbr="&qtilde;d">quad</expan>.</cell><cell>b</cell><cell>e</cell></row><row><cell><expan abbr="&qtilde;d">quad</expan>.</cell><cell>a</cell><cell>a</cell></row><row><cell><expan abbr="&qtilde;d">quad</expan>.</cell><cell>g</cell><cell>h</cell></row></table><p type="main">
<s id="id2694588">Propoitio exageimaoctaua, collectorum ab Euclide <lb/>& Archimede.</s></p><p type="main">
<s id="id2694608">Omnis cylindrus cono habenti baim, & altitudinem eandem <lb/><arrow.to.target n="marg204"/><lb/>triplus et. </s>
<s id="id2694627">Omnis cylindrus phr habenti eundem magnum <lb/><arrow.to.target n="marg205"/><lb/>circulum, & altitudinem exquialter et. </s>
<s id="id2694655">Omnis phra dupla et <lb/><arrow.to.target n="marg206"/><lb/>cono, cuius bais et eius circulus magnus, & altitudo eadem, qu <lb/>phr ipius. </s>
<s id="id2694698">Omnis uperficies phr quadrupla et maiori <lb/><arrow.to.target n="marg207"/><lb/>uo circulo. </s>
<s id="id2694726">Superficies portionis phr et qualis circulo, cu <lb/><arrow.to.target n="marg208"/>
<pb xlink:href="015/01/077.jpg" pagenum="58"/>ius emidiameter et linea ducta uertice portionis ad finem illius.</s></p><p type="margin">
<s id="id2694773"><margin.target id="marg204"/>1</s></p><p type="margin">
<s id="id2694787"><margin.target id="marg205"/>2</s></p><p type="margin">
<s id="id2694801"><margin.target id="marg206"/>3</s></p><p type="margin">
<s id="id2694814"><margin.target id="marg207"/>4</s></p><p type="margin">
<s id="id2694828"><margin.target id="marg208"/>5</s></p><p type="main">
<s id="id2694842">Quilibet ector phr qualis et cono, cuius bais et circu<lb/>lus qualis uperficiei eiudem portionis, altitudo uer phr e<lb/>midiameter. </s>
<s id="id2694898">Proportio phr ad ectorem datum, et duplica<lb/>ta ei, qu et dimetientis ad lineam, qu uertice portionis ad lim<lb/>bum. </s>
<s id="id2694936">Cum enim phra it qualis cono, cuius bais et maior cir<lb/>culus, altitudo uer dupla dimetienti per tertiam harum, qu hic <lb/><arrow.to.target n="marg209"/><lb/>proponuntur: erit phra qualis cono baim habenti circulum, <lb/>cuius emidiameter it qualis diametro phr, altitudo uer e<lb/>midiameter phr. </s>
<s id="id2695026">At per extam harum ector phr et qua<lb/>lis cono habenti altitudinem cmidiametrum phr, baim autem <lb/><arrow.to.target n="marg210"/><lb/>ipam portionis uperficiem: igitur proportio phr ad ecto<lb/>rem, uelut circuli cuius diameter et dupla dimetienti phr ad <lb/>crculum qualem uperficiei portionis: at uperficies portionis <lb/>per quintam harum et qualis circulo, cuius emidiameter et li<lb/>nea uertice portionis ad limbum eiudem: ergo proportio ph<lb/>r ad uum ectorem et uelut circuli, cuius dimetiens et duplus di <lb/>metienti phr, aut emidimetiens et qualis dimetienti phr <lb/>ad circulum, cuius emidimetiens et linea uertice portionis ad <lb/>limbum. </s>
<s id="id2695214">Sed proportio talium circulorum et duplicata propor<lb/><arrow.to.target n="marg211"/><lb/>tioni emidimetientium, igitur proportio phr ad uum ecto<lb/>rem et ueluti dimetientis phr ad lineam, qu uertice portio<lb/><arrow.to.target n="marg212"/><lb/>nis ad limbum duplicata. </s>
<s id="id2695280">Cuicunque portioni phr conus ille <lb/>habetur qualis, qui baim hab eat eandem cum portione, altitudi<lb/>nem uer lineam rectam, qu ad altitudinem portionis eandem <lb/>habeat proportionem, quam emidiametros phr un cum alti<lb/>tudine reliqu portionis habet ad eandem reliqu portionis alti<lb/><arrow.to.target n="marg213"/><lb/>tudinem. </s>
<s id="id2695349">Earum phr portionum, qu qualibus uperfi<lb/><arrow.to.target n="marg214"/><lb/>ciebus continentur medietas phr maxima exitit. </s>
<s id="id2695391">Proportio <lb/>uperficiei phr plano diui ad reliqu portionis uperficiem, <lb/>& reidui ectoris ad ectorem, et uelut quadratorum duarum li<lb/>nearum qu uerticulis ectionum ad communem uperficiem <lb/>plani portiones ecantis decendunt: nam ectorem phr, dico <lb/><arrow.to.target n="marg215"/><lb/>corpus compoitum ex portione, & cono illo. </s>
<s id="id2695479">Ille idem etiam defi<lb/>nit Ellipim coni a cuti anguli ectionem, quam dicit etiam fieri e<lb/><arrow.to.target n="marg216"/><lb/>cto cylindro per planum non ad angulos rectos tante uper cylin<lb/>dri axem. </s>
<s id="id2695517">Ab hac igitur coni acuti anguli ectione eu ellipi cir<lb/><arrow.to.target n="marg217"/><lb/>cumacta figura phroides corpus quod baim rotundam habet, <lb/>uocat: id que duplex ob longum, quod fit diametro longiore quie<lb/>cente, & prolatum quod fit quiecente breuiore: icut reliquam ci <lb/>licet parabolen aut hyperbolen, quia inferius non et terminata,
<pb xlink:href="015/01/078.jpg" pagenum="59"/>in cono rectangulo uocat rectanguli coni ectionem: ex qua cir<lb/>cumacta fit conoidale, quia planam habet baim. </s>
<s id="id2695594">Si ergo in ea<lb/><arrow.to.target n="marg218"/><lb/>dem rectanguli coni ectione plano portiones quales habentes <lb/>diametros abcindantur, ill portiones erunt quales. </s>
<s id="id2695628">Et triangu<lb/>li in eidem portionibus incripti quales erunt. </s>
<s id="id2695645">Diametrum uo<lb/>cat in <expan abbr="quacunqũe">quacunqune</expan> portione lineam, qu omnes lineas bai quidi<lb/>tantes per qualia diuidit. </s>
<s id="id2695681">Omnis circuli cuius diameter et ma <lb/><arrow.to.target n="marg219"/><lb/>ior diameter ellipis proportio ad ellipim et uelut direct diame<lb/>tri ellipis ad diametrum tranueram. </s>
<s id="id2695721">Ex quo patet quod pro<lb/><arrow.to.target n="marg220"/><lb/>portio cuiuslibet circuli ad ellipim et uelut quadrati u diame<lb/>tri ad rectangulum recta, & tranuera diametro ellipis compre<lb/>henum. </s>
<s id="id2695767">Ex hoc rurus equitur quod ellipis ad ellipim, ut re<lb/><arrow.to.target n="marg221"/><lb/>ctanguli ex diametris unius ad rectangulum ex diametris alterius.</s></p><p type="margin">
<s id="id2695798"><margin.target id="marg209"/>P<emph type="italics"/>er<emph.end type="italics"/> 14. & 15. <emph type="italics"/>duodeci mi<emph.end type="italics"/> E<emph type="italics"/>le.<emph.end type="italics"/> E<emph type="italics"/>ucl.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2695855"><margin.target id="marg210"/>P<emph type="italics"/>er<emph.end type="italics"/> 11. <emph type="italics"/>duo decimi<emph.end type="italics"/> E<emph type="italics"/>le.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2695901"><margin.target id="marg211"/>P<emph type="italics"/>er<emph.end type="italics"/> 2. <emph type="italics"/>duode cimi<emph.end type="italics"/>, & 20. <emph type="italics"/>exti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2695959"><margin.target id="marg212"/>8</s></p><p type="margin">
<s id="id2695972"><margin.target id="marg213"/>9</s></p><p type="margin">
<s id="id2695986"><margin.target id="marg214"/>10</s></p><p type="margin">
<s id="id2696000"><margin.target id="marg215"/>P<emph type="italics"/>er<emph.end type="italics"/> 22. <emph type="italics"/>quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2696046"><margin.target id="marg216"/>P<emph type="italics"/>er<emph.end type="italics"/> 20. <emph type="italics"/>ex ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2696094"><margin.target id="marg217"/>P<emph type="italics"/>er<emph.end type="italics"/> 11. <emph type="italics"/>quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2696140"><margin.target id="marg218"/>11</s></p><p type="margin">
<s id="id2696154"><margin.target id="marg219"/>12</s></p><p type="margin">
<s id="id2696168"><margin.target id="marg220"/>13</s></p><p type="margin">
<s id="id2696181"><margin.target id="marg221"/>14</s></p><p type="main">
<s id="id2696194">Si conoides & phroides ecet plano quiditanti axi fiet e<lb/><arrow.to.target n="marg222"/><lb/>ctio conoidalis imilis ei qua conoides eu phroides decri<lb/>ptum et. </s>
<s id="id2696248">Sin autem upra axem plano ad perpendiculum erecto <lb/>ectio circulus erit. </s>
<s id="id2696260">Et i ecentur obliqu fiet ellipis, modo omnia <lb/>latera comprehendat. </s>
<s id="id2696279">Omnis portio conoidalis rectanguli, quam <lb/><arrow.to.target n="marg223"/><lb/>planum ecat, exquialtera et, cono qui baim & axem eandem ha<lb/>bet. </s>
<s id="id2696308">Ex quo patet, quod i portio conoidalis rectanguli & ph<lb/><arrow.to.target n="marg224"/><lb/>r medietas eandem baim habeant & axem eundem, medietas <lb/>phr exquitertia erit conoidali portioni. </s>
<s id="id2696351">Et i eiudem rectan <lb/><arrow.to.target n="marg225"/><lb/>guli conoidalis portiones abcin dantur erit portionum propor<lb/>tio uelut quadratorum axium. </s>
<s id="id2696377">Cuiuslibet phroidis pars pla<lb/><arrow.to.target n="marg226"/><lb/>no per centrum abcia dupla et cono baim & axem eadem ha<lb/>benti. </s>
<s id="id2696414">Si autem non uper centrum erit proportio earum ad co<lb/><arrow.to.target n="marg227"/><lb/>num baim, & axem eandem habentem uelut coniunct ex axe al<lb/>terius partis & dimidio axis phroidis ad axem alterius partis.</s></p><p type="margin">
<s id="id2696453"><margin.target id="marg222"/>15</s></p><p type="margin">
<s id="id2696468"><margin.target id="marg223"/>16</s></p><p type="margin">
<s id="id2696482"><margin.target id="marg224"/>17</s></p><p type="margin">
<s id="id2696495"><margin.target id="marg225"/>18</s></p><p type="margin">
<s id="id2696509"><margin.target id="marg226"/>19</s></p><p type="margin">
<s id="id2696523"><margin.target id="marg227"/>20</s></p><p type="main">
<s id="id2696537">Demum proportio partis conoidis obtui anguli plano abci<lb/><arrow.to.target n="marg228"/><lb/> ad conum, baim & axem eadem habentem et ueluti line, com <lb/>poit ex axe portionis & triplo adiect ad compoitum ex axe <lb/>portionis & duplo eiudem adiect. </s>
<s id="id2696593">Adiectam uocat hyperbolis <lb/>tranueram. </s>
<s id="id2696606">Omnis cylindrus cono triplus et habenti eandem <lb/><arrow.to.target n="marg229"/><lb/>baim & altitudinem. </s>
<s id="id2696625">Omnes cylindri coni phr unt in pro<lb/><arrow.to.target n="marg230"/><lb/>portione corporum imilium planis uperficiebus contentarum.</s></p><p type="margin">
<s id="id2696661"><margin.target id="marg228"/>21</s></p><p type="margin">
<s id="id2696676"><margin.target id="marg229"/>22</s></p><p type="margin">
<s id="id2696690"><margin.target id="marg230"/>23</s></p><p type="main">
<s id="id2696704">Propoitio exageimanona, collectorum ex quatuor libris <lb/>Apollonij Pergei & <expan abbr="q.">que</expan> Sereni.</s></p><p type="main">
<s id="id2696730">Si fuerit linea bifariam diuia, eique in longum alia addita, & rur<lb/><arrow.to.target n="marg231"/><lb/>us alia detracta, fueritque totius cum addita ad eam, qu addita et <lb/>ueluti reidui ad detractam erit line com<lb/><figure id="id.015.01.078.1.jpg" xlink:href="015/01/078/1.jpg"/><lb/>poit ex addita, & dimidia ad dimidiam
<pb xlink:href="015/01/079.jpg" pagenum="60"/>ipam uelut dimidi ad differentiam eius, & detract. </s>
<s id="id2696804">Rurusque li<lb/>ne compoit ex dimidio & reiduo dimidi ac detract ad li<lb/>neam compoitam ex addita & detracta ut reidui dimidi, & de<lb/>tract ad partem detractam. </s>
<s id="id2696852">Et rurus totius compoit ad com<lb/>poitam ex dimidia & addita, uelut compoit ex addita, & diffe<lb/>rentia ad ipam additam. </s>
<s id="id2696887">Velut it propoita a b per qualia diuia <lb/>in c, addita b d, & detracta b e, it proportio a d ad d b, ut a e ad e b, <lb/>dico ee, ut c d ad cb, ita ab ad c e. </s>
<s id="id2696917">Et ut a e ad e d ut c e ad e b. </s>
<s id="id2696921">Etite<lb/><arrow.to.target n="marg232"/><lb/>rum ut a d ad c d uelut e d ad d b. </s>
<s id="id2696936">In parabole proportio partium <lb/>diametri ad uerticem terminantium duplicata et proportioni li<lb/>nearum ab eidem punctis ordinatim ductarum ad ipam ectio<lb/><arrow.to.target n="marg233"/><lb/>nem. </s>
<s id="id2696970">In hyperbole autem & ellipi & circuli circumferentia erit <lb/>quadratorum linearum ordinatim ductarum inter e uelut rectan<lb/><arrow.to.target n="marg234"/><lb/>gulorum partium diametri ad eadem puncta terminantium. </s>
<s id="id2696996">Et in <lb/>eidem i puncto peripheri contingens ad diametrum ducatur, <lb/>& ab eodem ordinata, erit ut partis diametri intercept inter extre<lb/>mum, & ordinatam ad partem inter ordinatam & peripheriam, ue<lb/>lut intercept inter extremum & contingentem ad interceptam <lb/><arrow.to.target n="marg235"/><lb/>exterius inter finem contingentis & peripheriam. </s>
<s id="id2697047">Et in eidem <lb/>quadratum emidiametri quale ee rectangulo ex intercepta in<lb/>ter centrum & caum contingentis in inter ceptam inter centrum & <lb/><arrow.to.target n="marg236"/><lb/>caum ordinat loco contactus product. </s>
<s id="id2697095">Si parabolen recta <lb/>linea contingens ad diametrum perueniat, umptoque puncto alio <lb/>in ectione quiditans ab eo ducatur contingenti: & ab utroque <lb/>etiam ad diametrum ordinat, demum uertice quiditans illis, <lb/>& priore puncto diametro quiditans donec concurrant, erit <lb/>triangulus ex ordinata, & quiditante ecundo puncto, & dia<lb/>metri parte contentus rectangulo ex prima ordinata & parte dia<lb/>metri inter uerticem & ecundam ordinatam contento qualis.<lb/><arrow.to.target n="marg237"/></s></p><p type="margin">
<s id="id2697187"><margin.target id="marg231"/>1</s></p><p type="margin">
<s id="id2697201"><margin.target id="marg232"/>2</s></p><p type="margin">
<s id="id2697215"><margin.target id="marg233"/>3</s></p><p type="margin">
<s id="id2697229"><margin.target id="marg234"/>4</s></p><p type="margin">
<s id="id2697243"><margin.target id="marg235"/>5</s></p><p type="margin">
<s id="id2697257"><margin.target id="marg236"/>6</s></p><p type="margin">
<s id="id2697271"><margin.target id="marg237"/>7</s></p><p type="main">
<s id="id2697285">Si in parabole contingente ad diametrum ducta ex alio puncto <lb/>ei quiditans ducatur ex ipa ectione, ubi iterum ecat ectionem/><lb/>intercepta per qualia diuidetur linea puncto contingentis dia</s></p><p type="main">
<s id="id2697324"><arrow.to.target n="marg238"/><lb/>metro quiditanti ducta. </s>
<s id="id2697339">Idem uer ferm continget ducta li<lb/>nea centro in locum contactus, ecabit enim omnes contingenti <lb/><arrow.to.target n="marg239"/><lb/>quiditantes in hyperbole, ellipi at que circulo. </s>
<s id="id2697375">Et autem omne <lb/>centrum in medio diametri: diameter autem in circulo & ellipi il<lb/>las per qualia diuidit intus enim et: in contrapoitis inter uerti<lb/>cem, & uerticem poita et exterius utriuque contingenti ad per<lb/>pendiculum initens. </s>
<s id="id2697426">In hyperbole autem exterius etiam adiacet, <lb/>ut in contrapoitis eadem & tranuera uo catur: cuius terminus et <lb/>punctus concurus cum latere trianguli, qui conum per axem diui
<pb xlink:href="015/01/080.jpg" pagenum="61"/>dit: linea uer tangens uerticem hyperbolis ad quam ordinat <lb/><arrow.to.target n="marg240"/><lb/>pount, Recta appellabitur. </s>
<s id="id2697483">Datarecta linea poitione, aliaque ma <lb/>gnitudine data & anglo parabolen, & hyperbolen, & ellipim, <lb/>& contrapoitas circa datam poitione tanqum diametrum de<lb/>cribere tanqum cono erecto, ut angulus ad uerticem ectionis <lb/>comprehenus it, & per rectam rectangulum quale comprehen<lb/>datur quadrato dat line magnitudine. </s>
<s id="id2697548">Si linea in duas partes <lb/><arrow.to.target n="marg241"/><lb/>diuidatur, eique utrinque quales line adiun<lb/><figure id="id.015.01.080.1.jpg" xlink:href="015/01/080/1.jpg"/><lb/>gantur erit rectangulum ex partibus totius <lb/>quale rectangulis partium prioris line, & ex <lb/>priore linea cum una adiecta in eam, qu adiecta et. </s>
<s id="id2697599">Si hyperbo <lb/><arrow.to.target n="marg242"/><lb/>len recta linea in uertice contingat, & utrinque abcindatur, quan<lb/>tum et, quod potet in quartam partem rectanguli ex diametro <lb/>tranuera hyperbolis, qu exterius adiacetin eam, qu recta dici<lb/>tur, ad quam, qu ordinatim ducuntur, unt quiditantes line, <lb/>qu ectionis centro ad terminos contingentis ducuntur emper <lb/>ipi ectioni magis appropinquabunt, nec unquam conuenient: & <lb/>ob id aymptoton appellantur. </s>
<s id="id2697690">Nec ull ali intra <expan abbr="angulũ">angulum</expan> illum <lb/><arrow.to.target n="marg243"/><lb/>inueniri poterunt. </s>
<s id="id2697716">Vnde etiam intra <expan abbr="datũ">datum</expan> angulum decribere do<lb/>cemur hyperbolen cuius anguli latera int aymptota. </s>
<s id="id2697742">Aymptotis <lb/><arrow.to.target n="marg244"/><lb/>duabus propoitis uni hyperboli, in finitas alas eidem aymptotas <lb/>inuenire. </s>
<s id="id2697769">Duabus rectis aymptotis infinitas ubijci poe hyperbo <lb/>les illis rectis, & inter e aymptotas. </s>
<s id="id2697792">Cum in duabus uperficie<lb/><arrow.to.target n="marg245"/><lb/>bus quiditantibus duo circuli quales, quorum linea per cen<lb/>tra non et ad perpendiculum earum infinitis planis ecantur, fiunt <lb/>in ipis line peripheria in peripheriam rect qu corpus cylin<lb/>dricum claudunt quod calenus cylindrus appellatur: long alius <lb/>ab eo, qui fit recto cylindro per duo plana quiditantia, ed non <lb/>ad perpendiculum poita diecto. </s>
<s id="id2697878">nam eius extrem uperficies <lb/>non circuli, ed ellipes unt. </s>
<s id="id2697899">Si calenus cylindrus plano non <lb/><arrow.to.target n="marg246"/><lb/>quiditanti bai, ed ita ut angulos interiores quales faciat angu<lb/>lis bais ectio circulus erit: uo caturque hcectio ubcontraria: nec <lb/>ulla prter hanc & bai quiditantem ectio circulus ee potet: <lb/>ed unt ellipes. </s>
<s id="id2697981">Super eundem circulum, & ub eadem altitudi<lb/><arrow.to.target n="marg247"/><lb/>ne ellipes imiles in cono & cylindro ee pount, qu ab eodem <lb/>plano fiant, docetque uel bai uel cono uel cylindro, aut cono pro<lb/>poito reliqua facere, quod et ualde admirabile: cum ellipis cylin<lb/>drica emper qualis it in utraque parte diametro tranuera <lb/>utrinque qualiter ditante, conica uer minor neceari it in u<lb/>periore parte uerus coni uerticem latior in inferiore, ubi partes a <lb/>diametro tranuera qualiter diteterint: ip autem non olum i
<pb xlink:href="015/01/081.jpg" pagenum="62"/><arrow.to.target n="marg248"/><lb/>miles, ed unam perpe in utri que ee uult. </s>
<s id="id2698143">Sed & hoc Archime<lb/>des dicere uidetur: line duct uertice conicaleni ad perpendi<lb/>culum uper baes ingulas omnium triangulorum per axem/> coni <lb/>traneuntium in peripheriam unius circuli cadunt.</s></p><p type="margin">
<s id="id2698187"><margin.target id="marg238"/>8</s></p><p type="margin">
<s id="id2698202"><margin.target id="marg239"/>9</s></p><p type="margin">
<s id="id2698216"><margin.target id="marg240"/>10</s></p><p type="margin">
<s id="id2698229"><margin.target id="marg241"/>11</s></p><p type="margin">
<s id="id2698243"><margin.target id="marg242"/>12</s></p><p type="margin">
<s id="id2698257"><margin.target id="marg243"/>13</s></p><p type="margin">
<s id="id2698271"><margin.target id="marg244"/>14</s></p><p type="margin">
<s id="id2698284"><margin.target id="marg245"/>15</s></p><p type="margin">
<s id="id2698298"><margin.target id="marg246"/>16</s></p><p type="margin">
<s id="id2698312"><margin.target id="marg247"/>17</s></p><p type="margin">
<s id="id2698326"><margin.target id="marg248"/>18</s></p><p type="main">
<s id="id2698340">Propoitio eptuageima.</s></p><p type="main">
<s id="id2698356">Si fuerint tres quantitates in continua proportione, alique toti<lb/>dem in continua proportione, poterunt contituere tres quantita<lb/>tes in quali differentia peruerim copulat.<lb/><arrow.to.target n="marg249"/></s></p><p type="margin">
<s id="id2698395"><margin.target id="marg249"/>C<emph type="italics"/>o<emph.end type="italics"/>m.</s></p><p type="main">
<s id="id2698421">Velut int a b c primi ordi<lb/><figure id="id.015.01.081.1.jpg" xlink:href="015/01/081/1.jpg"/><lb/>nis, & d ef ecundi, & it 28, </s></p><p type="main">
<s id="id2698451"><arrow.to.target n="marg250"/><lb/>b 4, c 2, & d 2 1/4, e 1 1/2, f 1, tunc <lb/>iunctis a & e fit 9 1/2, & b & d b <lb/>1/4, & e cum f 3, at 3 & 6 1/4 & 9 1/2 <lb/>qualiter ditant, nam diffe<lb/>rentia et 3 1/4. At i iungatur <lb/>cum e, & b cum f, & c cum d <lb/>idem poterit contingere: ut in <lb/>figura uides, nam a e et 8 1/2, <lb/>p: <02> 1 1/4, & b f 7, & c d 5 1/2, m: <02> 1 1/4, & differentia b f ab utro que com<lb/>poito, et 1 1/2 p: <02> 1 1/4, qua excedit & exceditur. </s>
<s id="id2698516">Dico modo, quai <lb/>ex ordine coniungantur qualecun que proportiones fuerint, modo <lb/>non int amb qualitatis 1, ut b iungatur cum c, & reliqu ut li<lb/>bet, uelut a cum d, & c cum f, uel a cum f, & e cum d, nunquam fient <lb/><arrow.to.target n="marg251"/><lb/>quales exceus, nam de primo et clarum: nam i a cum diun<lb/>gatur, & amb fuerint maxim, maior et differentia a ad b, qum <lb/>b ad c, & maior etiam d ad e qum e ad f, ideo maior erit differentia <lb/>a & d ad b e qum b e ad c f, quod erat probandum. </s>
<s id="id2698601">Eodem modo <lb/>ed laborioius demontratur reliquus modus cilicet, quod con<lb/>iunctio a f ad b e et maior aut minor qum b e ad c d, ex hoce<lb/>quuntur corrolaria.</s></p><p type="margin">
<s id="id2698641"><margin.target id="marg250"/>16</s></p><p type="margin">
<s id="id2698655"><margin.target id="marg251"/>17</s></p><p type="main">
<s id="id2698669">Primum, tres quales quantitates non pount diuidi in tres, & <lb/>tres quantitates in continua proportione ordinat, ut dixi, nii u<lb/>triuque ordinis tres, ac tres inuicem int quales.</s></p><p type="main">
<s id="id2698707">Secundum, tres quantitates in quali exceu ordinate, ut dixi, <lb/>non pount diuidi in tres, & tres quantitates, qu int in eadem <lb/>proportione quantumcun que proportiones ill duorum ordinum <lb/>fint diuer .</s></p><p type="main">
<s id="id2698749">Tertium, tres quantitates, qu intin eadem proportione non <lb/>pount diuidi ordinate in tres ac tres, qu int in continua propor<lb/>tione nii int amb proportiones edem cum proportione ipa<lb/>rum quantitatum.</s></p>
<pb xlink:href="015/01/082.jpg" pagenum="63"/><p type="main">
<s id="id2698807">Propoitio eptuageimaprima.</s></p><p type="main">
<s id="id2698824">Proportionem leuitatis ponderis per uirgam torcularem attra<lb/>cti ad rectam upenfionem inuenire.</s></p><figure id="id.015.01.082.1.jpg" xlink:href="015/01/082/1.jpg"/><p type="main">
<s id="id2698852">Sit torcularis uirga, cuius pir a b per circui<lb/><arrow.to.target n="marg252"/><lb/>tum int centupl ad altitudinem a b, & axis d c <lb/><arrow.to.target n="marg253"/><lb/>emidiametro b c centupla, & quoniam per upe<lb/>rius aumpta, qualis et proportio patij ad pa<lb/>tium, talis leuitatis ad <expan abbr="leuitat&etilde;">leuitatem</expan>, <expan abbr="igi&ttilde;">igitur</expan> e pondus acen <lb/>dens per a b leuius quam per b <expan abbr="crectã">crectam</expan> centuplo, et <lb/>imiliter cum circuitus b c, & d c int in eodem tem <lb/>pore, & circuitus d c, it centuplus ad piralem b c <lb/>per demontrata ab Euclide, ergo e erit centuplo <lb/>leuius circum ductum per d qum b, ed per b circumductum cen<lb/>tuplo leuius et, qum per rectam, igitur e ponderat folum particu<lb/>lam ex decem millibus recti ponderis.</s></p><p type="margin">
<s id="id2698997"><margin.target id="marg252"/>C<emph type="italics"/>o<emph.end type="italics"/>m.</s></p><p type="margin">
<s id="id2699022"><margin.target id="marg253"/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 45.</s></p><p type="main">
<s id="id2699049">Propoitio eptuageimaecunda.</s></p><p type="main">
<s id="id2699068">Proportionem ponderis phr pendentis ad acendentem per <lb/>accliue planum inuenire</s></p><figure id="id.015.01.082.2.jpg" xlink:href="015/01/082/2.jpg"/><p type="main">
<s id="id2699100">Sit phra qualis ponderig in pun<lb/><arrow.to.target n="marg254"/><lb/>cto b, qu debeat trahi uper b c accli<lb/>ue planum b e ad perpendiculum pla<lb/><arrow.to.target n="marg255"/><lb/>ni b f. </s>
<s id="id2699142">Quia ergo in b e mouetur a, qua<lb/>uis modica ui per dicta uperius, erit per <lb/>communem animi ententiam uis, qu <lb/>mouebit a per e b nulla: per dicta uer <lb/>a mouebitur ad f emper, a contanti ui <lb/>quali g, & per b c a contanti ui qua<lb/>li k, icut per b d a contanti quali h, ergo per ultimam petitio<lb/>nem, cum termini eruent, quo ad partes eandem rationem in<lb/>guli per e, & motus per b e it a nulla ui, erit proportio g ad k, ue<lb/>lut proportio uis, qu mouet per b f ad uim, qu mouet per <lb/>b c, & uelut anguli per e b f recti ad angulum e b c, & ita uis, <lb/>qu mouet a per b f, & et, ut dictum et, g ad uim, qu mouet <lb/>per b d, & et h ex uppoito, ut c b f ad e b d, igitur proportio dif<lb/>ficultatis motus a per b d ad idem a per b c, et uelut h ad k, quod <lb/>erat demontrandum.</s></p>
<pb xlink:href="015/01/083.jpg" pagenum="64"/><p type="margin">
<s id="id2699291"><margin.target id="marg254"/>C<emph type="italics"/>o<emph.end type="italics"/>m.</s></p><p type="margin">
<s id="id2699318"><margin.target id="marg255"/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 40. 7</s></p><p type="main">
<s id="id2699345">Propoitio eptuageimatertia.</s></p><p type="main">
<s id="id2699361">Proportionem ponderum attractorum penes figuram in pla<lb/>no inuenire.<lb/><arrow.to.target n="marg256"/></s></p><p type="margin">
<s id="id2699381"><margin.target id="marg256"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2699407">Sint duo pondera qualia in plano a & b, & it <lb/><figure id="id.015.01.083.1.jpg" xlink:href="015/01/083/1.jpg"/><lb/>a uperficies qua planum tangit dupla b uperfi<lb/>ciei, qua planum tangit: dico quod i trahantur ab <lb/>imo, quod erunt qualia: upendantur, & erunt <lb/>qualia ex uppoito, ed a quiecens in plano et <lb/>dimidium a upeni, & b quiecens in plano et di <lb/>midium b upeni ex demontratis uperius, igi<lb/>tur per communem animi ententiam a & b in pla<lb/>no unt qualia.</s></p><p type="main">
<s id="id2699529"><arrow.to.target n="marg257"/></s></p><p type="margin">
<s id="id2699540"><margin.target id="marg257"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2699567">Ex hoc manifetum et, quod proportio uirium trahentium pon<lb/>dera in plano eadem et, qu iporum ponderum dum upendun<lb/>tur. </s>
<s id="id2699598">Vbiplanum quale it, & olidum.</s></p><p type="main">
<s id="id2699614"><arrow.to.target n="marg258"/></s></p><p type="margin">
<s id="id2699626"><margin.target id="marg258"/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 62.</s></p><p type="main">
<s id="id2699654">Propoitio eptuageimaquarta.</s></p><p type="main">
<s id="id2699670">Proportionem concutientis ad concuum tabili inuenire.</s></p><p type="main">
<s id="id2699685"><arrow.to.target n="marg259"/></s></p><p type="margin">
<s id="id2699697"><margin.target id="marg259"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2699724">Intelligo concutiens ee olidum, quod non frangitur, idque gra<lb/>uitate, & impetu concutere, nam de duritie upponitur, & grauitas, <lb/>ut demontrabitur in corrolario et iuxta uperficiem inferiorem <lb/>ponderi comparatam. </s>
<s id="id2699758">Cum ergo motus concusionis magnitudo <lb/>contet ex grauitate, impetu & figura, concusi autem ex pondere <lb/>& connexione: multiplicatis inuicem partibus productorum pro<lb/>portio, erit proportio concusionis: ut it grauitas decem, impetus <lb/>quadraginta: pondus icti centum connexio ut duo, ducemus qua<lb/>dragintain decem, & fient quadringenta, et duo in centum, fient du <lb/>centa, igitur concusio erit dupla.</s></p><p type="main">
<s id="id2699812"><arrow.to.target n="marg260"/></s></p><p type="margin">
<s id="id2699823"><margin.target id="marg260"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 1.</s></p><p type="main">
<s id="id2699850">Cum fuerit figura rotunda, concusio erit integra in puncto: <lb/>quia phra iacens in plano totum pondus in punctum cogit.</s></p><p type="main">
<s id="id2699870"><arrow.to.target n="marg261"/></s></p><p type="margin">
<s id="id2699882"><margin.target id="marg261"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 2.</s></p><p type="main">
<s id="id2699909">Si autem planum et, quod ijcitur, proportio totius ad totum et <lb/>minor, qum partis ad partem pro ratione quantitatis latitudinis. </s></p><p type="main">
<s id="id2699929"><arrow.to.target n="marg262"/><lb/>ed maior ratione aris compreheni, de quo infr.<lb/><arrow.to.target n="marg263"/></s></p><p type="margin">
<s id="id2699960"><margin.target id="marg262"/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 84.</s></p><p type="margin">
<s id="id2699987"><margin.target id="marg263"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 3.</s></p><p type="main">
<s id="id2700013">Cum proportio minor fuerit tabile, non poterit in olido plano <lb/>moueri: aliter fieret motus debiliore, & per prcedentem etiam <lb/>poet pari ratione eleuari.</s></p><p type="main">
<s id="id2700043"><arrow.to.target n="marg264"/></s></p><p type="margin">
<s id="id2700055"><margin.target id="marg264"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 4.</s></p><p type="main">
<s id="id2700082">Cumque tabile non mouetur, & omne agens agat aliquid necee <lb/>et, ut tabilis partes cedant, aut dioluantur. </s>
<s id="id2700107">Quanto ergo magis <lb/>cedit, tanto minus dioluitur.</s></p>
<pb xlink:href="015/01/084.jpg" pagenum="65"/><p type="main">
<s id="id2700131">Cau igitur qu alleuiant ictum, ne dioluatur, unt eptem le</s></p><p type="main">
<s id="id2700158"><arrow.to.target n="marg265"/><lb/>uitas ictus, ponderis, fractura, mollities eius, quodicitur, mollities <lb/>eius, quod excipit ictum, motus eiudem, & figura lata, & inqua<lb/>lis. </s>
<s id="id2700183">Durities ergo, quatenus fractur opponitur, aliud et, quam ut <lb/>molliciei: & utra que et caua, qu augetictum, ut reliqu <lb/> oppoit minuunt, dicemus autem de his inferius.</s></p><p type="margin">
<s id="id2700220"><margin.target id="marg265"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 9.</s></p><p type="main">
<s id="id2700246">Propoitio eptuageimaquints.</s></p><p type="main">
<s id="id2700262">Proportionem immoti in aqua ad immotum in terra in excipien <lb/>do ictum inuenire.</s></p><p type="main">
<s id="id2700275">Sit pondus a in terra quale b eiudem natur magnitudinis fi<lb/><arrow.to.target n="marg266"/><lb/>gur, & eodem in itu, quod it in aqua porr a, i eet affixum ter<lb/>r oportet, ut conuellatur, aut dioluatur aut frangatur. </s>
<s id="id2700327">Et clarum <lb/><figure id="id.015.01.084.1.jpg" xlink:href="015/01/084/1.jpg"/><lb/>et, quod totum ictum excipit. </s>
<s id="id2700347">Si uer <lb/>affixum non it, euertitur, & tanto mino<lb/>rem partem excipit ictus, quanto faci<lb/>lior et ad euerionem. </s>
<s id="id2700374">Vnde nata fabu<lb/>la de quercu, qu cum immobilis eet, <lb/>& taret uento euera et, arundo flecten<lb/>do e, cecidit quidem, ed non et eradi<lb/>cata. </s>
<s id="id2700418">Sermo igitur et de b inidenti aqu <lb/>in comparatione ad a, quando excipit <lb/>plenum ictum. </s>
<s id="id2700437">Cum ergo b tangitur, ex<lb/>cipit plenum ictum illo intanti, ed quia <lb/>non excipitur ictus cedente materia, & <lb/>antequam materia cedat b mouetur loco, quia inidet aqu, ergo <lb/>non excipit ictum. </s>
<s id="id2700468">Proponatur ergo, quod moueatur b per cpa<lb/>tium in d tempore, & it, ut idem b ab e ui trahatur per idem pa<lb/>tium in eodem tempore ex loco directo ad eandem partem: qua<lb/>lis ergo proportio e ad b, & arem, qui cum eo reitit, talis propor<lb/>tio ictus f grauis puta in a ad ictum Y in b. </s>
<s id="id2700510">Quia per demontra<lb/><arrow.to.target n="marg267"/><lb/>ta uperius proportio f ad a producitur ex proportionibus e ad b, <lb/><arrow.to.target n="marg268"/><lb/>& a ad e, ergo diuia proportione f ad a per proportionem c ad b <lb/>exibit proportio ictus Y in a ad ictum Y in b quod erat demon<lb/>trandum.</s></p><p type="margin">
<s id="id2700557"><margin.target id="marg266"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s id="id2700584"><margin.target id="marg267"/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 2.</s></p><p type="margin">
<s id="id2700612"><margin.target id="marg268"/>P<emph type="italics"/>er<emph.end type="italics"/> 42. & 43. P<emph type="italics"/>ropo.<emph.end type="italics"/></s></p><p type="main">
<s id="id2700651">Ex hoc patet, quod b quanto mollius, leuius, & trictius in imo, <lb/><arrow.to.target n="marg269"/><lb/>& in tenuiore aqua, eo minus ldetur. </s>
<s id="id2700671">Et quanto ictus lentior fue<lb/>rit etiam quod it grauius Y.</s></p>
<pb xlink:href="015/01/085.jpg" pagenum="66"/><p type="margin">
<s id="id2700698"><margin.target id="marg269"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2700724">Propoitio eptuageimaexta.</s></p><p type="main">
<s id="id2700742">Proportionem duorum mobilium ibi inuicem concurrentium <lb/>per rectam inuenire.<lb/><arrow.to.target n="marg270"/></s></p><p type="margin">
<s id="id2700764"><margin.target id="marg270"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2700789">Iam cognito, quod mobilia, qu loco mouentur per prceden<lb/>tes, ed omnino quiecunt integros excipiuntictus: alia quidem, <lb/>qu concurrunt, non omnino reiliunt, alia uero reiliunt, & qu <lb/>reiliunt minores excipiuntictus, equitur ut diuera it compara<lb/>tio: nam erunt, qu tando excipient ictus, & hc integros ut mu<lb/>ri, & qu concurrendo, nec reiliendo, ut equi curu incitati: & qu <lb/>tando, ed reiliendo, ut naues tantes: & qu concurrendo, rei<lb/>liendo qe ut naues uentis, & triremes ab impulu: bifariam ergo <lb/>contingit intelligi, quod proponitur. </s>
<s id="id2700900">Sed in utroque etiam enu <lb/>uarietas et: nam ut concurrit pars altera celerius, ita etiam magis <lb/>concutitur. </s>
<s id="id2700920">Et ideo it, ut proportio icts it in comparatione ad <lb/>grauitatem dupl, & concurrant qualiter, & int qu grauia, & <lb/>neutrum reiliat, erunt in proportione quadrupla, & eodem mo<lb/>do i utrunque reiliat. </s>
<s id="id2700968">At i diuero impetu ferantur, ut dixi, tria <lb/>erunt prcipu conideranda grauitas eu pondus, impetus, & an <lb/>reiliat. </s>
<s id="id2700998">Quanto enim grauiora fuerint, & maiore impetu agen<lb/>tur, & non reilierint eo maiorem ictum recipient: quanto leuio<lb/>ra, & minore impetu, & magis reilierint, minus ldentur. </s>
<s id="id2701022">Sed & <lb/>in debilitando ictum coniderare oportet tria, quod reiliat, quod <lb/>diffugiat, quod circumuertatur: reiliunt naues, i rotris concur<lb/>rant pleno ictu: i uer non pleno ictu concurrant, ed diffugiant <lb/>hoc experimento compertum et minimum ee ictum: i rotro <lb/>tranuerum nauis feriatur medium, et hoc.</s></p><figure id="id.015.01.085.1.jpg" xlink:href="015/01/085/1.jpg"/><p type="main">
<s id="id2701101">Sit ergo ut a b nauis tangat rotro b c ic ut <lb/>diffugiat, erit hypomochlium c, & i tangat <lb/>e f hypomochlium et in d dupla, ergo et c b <lb/>ipi d e, igitur ictus duplo minor excipitur <lb/>c b qum ef. </s>
<s id="id2701140">Et etiam tempus long maius, <lb/>quo excipit ictum ef, qum b c: tatim enim dicedit b c occurrit que<lb/>alijs partibus, in c f autem impingit, & angulus a d c et long ma<lb/>ior recto, qum a b f: ob hc igitur long maior et ictus c f qum <lb/>b c: uocant autem hoc declinationem.</s></p><p type="main">
<s id="id2701198">Propoitio eptuageimaeptima.</s></p><p type="main">
<s id="id2701217">Proportionem motus obliqui ad motum rectum in nauibus <lb/>inuenire.</s></p><p type="main">
<s id="id2701229"><arrow.to.target n="marg271"/></s></p><p type="margin">
<s id="id2701240"><margin.target id="marg271"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2701266">Cm uentus fertur ad puppim rect, nauiqe gubernaculum di
<pb xlink:href="015/01/086.jpg" pagenum="67"/>rigitur, tendunturqe uela ac expanduntur umma in parte mali, <lb/>tunc motus et uelocisimus: fingamus autem, quod omnia ad <lb/>idem tendant prter uentum, qui non directus it ad puppim, ed <lb/> latere, ut uides, & temo itin contrarium tantundem directus, & <lb/>upponamus pro nune, quod uelum it olum in anteriore parte <lb/>nauis, nam ecus eet nimis magna differentia, <lb/><figure id="id.015.01.086.1.jpg" xlink:href="015/01/086/1.jpg"/><lb/>quod nauis una ageretur tribus malis alia una: <lb/>Quritur igitur proportio motus b c ad mo<lb/>tum d e: fiat ergo c f qualis e g, ita ut f angulus <lb/>rectus it, & manifetum et, quod h c maior et <lb/>c f, cum ergo angulus f rectus it, quanto maior <lb/>erit angulus h c f, tanto maior erit proportio h c <lb/>ad c f, quod et primum a, ide noto angulo h c f <lb/>per ea, qu tradita unt ab Atrologis de inu & <lb/>arcu erit nota proportio c h ad c f, ideo ad e g <lb/>fiat ergo c k qualis c h, igitur c k erit maior e g, i ergo perambula<lb/>bit qualiter c, ut c h, erit temporis motus e g ad motum e f, ut c k <lb/>ad c f, igitur cum nota it c k, et enim qualis c h, erit temporis ad <lb/>tempus proportio nota. </s>
<s id="id2701458">Quod autem in quali tempore mouebi<lb/>tur nauis per c k & h c patet ex aumpto inferius declarando.</s></p><p type="main">
<s id="id2701479"><arrow.to.target n="marg272"/></s></p><p type="margin">
<s id="id2701490"><margin.target id="marg272"/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 99.</s></p><p type="main">
<s id="id2701518">Propoitio eptuageimaoctaua.</s></p><p type="main">
<s id="id2701534">Propoitionem nauis ad triremes quotuis concurrentes de<lb/>montrare.</s></p><p type="main">
<s id="id2701552">Sit nauis deferens pondus decuplo maius triremi, & contat, </s></p><p type="main">
<s id="id2701565"><arrow.to.target n="marg273"/><lb/>quod impulu quabitur decem triremibus, ubi flante uento e <lb/>puppi qualiter feratur in aduerum, quantum triremes ui homi<lb/>num. </s>
<s id="id2701594">Sed quoniam triremes impediuntur uento licet ine uelis <lb/>int, habent enim & ip malum, & uelum, ed exigua comparatio<lb/><arrow.to.target n="marg274"/><lb/>ne nauium, ideo ictus ille multo ualidior et ex demontratis. </s>
<s id="id2701633">Cum <lb/>uero uis illa imul it, liquet, 'qud hoc in cau nii machin obta<lb/>rent una nauis mille poet obruere triremes diiunctas per tantum <lb/>patium inter e, quantum et id, in quo nauis potetuenti impul<lb/>um recipere. </s>
<s id="id2701690">At impedimentorum maximum unt machin, qu <lb/>in nauim collimant lateribus, cum triremes quaqu uerum e a<lb/>g ant, & ob id proram olam exponunt ictibus, in quam difficile <lb/>et collimare, & i tangatur pars ea robutior et, nec periculum <lb/>euerionis ade in currit, ut lateribus: nec enim ade anguta et a <lb/>prora ad puppim nauis, quam latere ad latus: his tot cauis mi<lb/>nus et obnoxia machinis triremis, qum nauis. </s>
<s id="id2701780">Sed & alia caua <lb/>et, quoniam necee et ut ob angulum laterum ad proram
<pb xlink:href="015/01/087.jpg" pagenum="68"/>ictus dilabatur pius olum traiecta uperficie. </s>
<s id="id2701819">Secundum impe<lb/>dimentum et uento, i ualde obliquus it, nam ad rectum impul<lb/>um, multum debilitatur: aut i incontans it, uiribusque remittatur. <lb/></s>
<s id="id2701855">Tertium uer i triremes inuicem connex int, ac e tangant, in <lb/>quas nauis dirigitur. </s>
<s id="id2701876">Sed & hoc infr demontrabitur nauim, ut le<lb/><arrow.to.target n="marg275"/><lb/>uior fuerit facilius elabi, ed ut pondere magis onerata grauiores <lb/>ictus inferre: ob hoc triremem inuenerunt mediam maximi uus <lb/><foreign lang="greek">a)mfh/rhn. </foreign></s>
<s id="id2701914">Galeonum uulg uocant.</s></p><p type="margin">
<s id="id2701925"><margin.target id="marg273"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s id="id2701952"><margin.target id="marg274"/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 74.</s></p><p type="margin">
<s id="id2701979"><margin.target id="marg275"/>P<emph type="italics"/>rop.<emph.end type="italics"/> 109.</s></p><p type="main">
<s id="id2702004">Propoitio eptuageimanona.</s></p><p type="main">
<s id="id2702020">Proportionem medicamentorum purgantium inuicem de<lb/>clarare.<lb/><arrow.to.target n="marg276"/></s></p><p type="margin">
<s id="id2702039"><margin.target id="marg276"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2702065">Scio, qum multa concurrant, etiam per e ad purgationem mul <lb/>titudo humorum prparatio locus propinquus, ed nobis er<lb/>mo et pariub conditione, ut it dimidia uncia Casi nigr in tri<lb/>bus uicibus expurget libram humorum, & uelim cire ab una un<lb/>cia, quoties expurgabitur, & quantum. </s>
<s id="id2702121">Dico, quod in camonio, & <lb/>agarico hc ratio deprehendi potet: in his autem medicamentis, <lb/>qu magis leniunt, qum proprietate educant, ut et casia nigra, <lb/>ratio hc non ualet, quoniam feces quando que pro maiore par<lb/>te educuntur, ita ut etiam multiplicato medicamento deit, quod <lb/>educatur. </s>
<s id="id2702172">Et quamuis humores iuxta proportionem trahat, cum <lb/>tamen feces proportionem non eruent, equitur: ut aggregati ad </s></p><p type="main">
<s id="id2702190"><arrow.to.target n="marg277"/><lb/>aggregatum proportio non eruetur. </s>
<s id="id2702203">At non et facile potmo<lb/>dum internocere feces ab humoribus, quocirca uidetur propor<lb/>tio illa confundi. </s>
<s id="id2702224">Quod i medicamentum leniens, fiat ob quanti<lb/>tatem purgans humores, ut de multa casia nigra, tuncnon potet <lb/>asignari illa comparatio nii ut et medicamentum purgans. </s>
<s id="id2702254">Et it <lb/>gratia exempli, primum ut grana ex camonij purgent aliquem <lb/>ter, & uncias decem bilis, dico iuxta rationem uprapoitam, quod <lb/><arrow.to.target n="marg278"/><lb/>grana duodecim purgabunt iuxta proportionem duplam exqui<lb/>alteram, i duo grana nil purgant, ed commouent. </s>
<s id="id2702302">qualia enim <lb/><arrow.to.target n="marg279"/><lb/>unt: ut quatuor int dupla, & ex tripla, & mouent ter, quia exqui<lb/>alteram habent proportionem ad exceum, igitur duodecim du<lb/>plam, & exquialteram ad quatuor, nam decem ad quatuor et du<lb/>pla exquialtera, & purgabit epties cum nixu libras duas fer<lb/>me bilis. </s>
<s id="id2702363">Vt comparatio fiat exceus ad uim, qu reitit eodem <lb/>modo. </s>
<s id="id2702382">In casia ergo nigra i uncia <expan abbr="unanõ">unanon</expan> purga, ed lenit tantum, <lb/>& du unci purgant ter, & libram unam bilis, tres unci duplam
<pb xlink:href="015/01/088.jpg" pagenum="69"/>habent proportionem iuxta exceum ad unam, exceus igitur <lb/>duplum purgabunt, & duplo magis, id et prter feces libras <lb/>duas bilis in ex uicibus.</s></p><p type="margin">
<s id="id2702453"><margin.target id="marg277"/>E<emph type="italics"/>x conuera<emph.end type="italics"/> 18. <emph type="italics"/>quint.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2702490"><margin.target id="marg278"/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 37.</s></p><p type="margin">
<s id="id2702518"><margin.target id="marg279"/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 42.</s></p><p type="main">
<s id="id2702546">Propoitio octuageima.</s></p><p type="main">
<s id="id2702559">Proportionem motus ecundum obliquum ad rectum in pa<lb/>tio declarare.</s></p><p type="main">
<s id="id2702578">Hc udetur imilis uperiori cuidam propoitioni, ed tamen in <lb/><arrow.to.target n="marg280"/><lb/>hoc differt, quoniam in c a upponimus nauim moueri, ut concu<lb/>tiat, hic autem iuxta motum olum: ut proponamus b nauim ferri <lb/><figure id="id.015.01.088.1.jpg" xlink:href="015/01/088/1.jpg"/><lb/>uerus a uento recto ex b in a: it autem uentus ex <lb/>cin a mouens nauim ex b in a: nn enim moue<lb/>bit ut quidam putant in ratione c a ad b a: ut i ca <lb/>it exquiquarta ad b a, ut quali impetu ex b & <lb/>c flante uento moueretur tardius per c a, quam <lb/>per b a, quia qualiter ex uppoito: ergo tanto <lb/>tardius c fertur in a, quam b in idem quanto lon<lb/>gior et c a, b a igitur i b perueniet in a in qua<lb/>tuor diebus c perueniet in idem a in quinque <lb/>diebus. </s>
<s id="id2702698">Hoc enim et per e manifetum: ed non qurimus id, ed <lb/>ut uento c a quali per c a ei, qui et b a per b a, ubi b moueatur uen <lb/>to c a per b a, quanto tardius mouebitur. </s>
<s id="id2702732">Mouebitur. </s>
<s id="id2702736">n. </s>
<s id="id2702740">tardius ad <lb/>a per b a, quam per c a, at per c a tardius, quam ex b in a per qua<lb/>lem uim, ergo multo tardius ex b in a per c a uentum, quam per uen <lb/>tum ex b in a. </s>
<s id="id2702759">Qurimus ergo compoitionem horum, ut it c <lb/>nauis, qu debeat transferri ad a per uentum ex b, & equitur, <lb/>quod tardius, quam ex c per uentum ex c in a, & tardius ex b per <lb/>uentum ex cin a. </s>
<s id="id2702789">Ergo malus, qui in prora et conuoluto eo, qui <lb/>et in puppi, ut etiam Aritoteles docet tantundem nititur ad re<lb/><arrow.to.target n="marg281"/><lb/>ctum ex cin quiditantem locum ab a quantum c ditat ab con<lb/>tra temo, qui in puppi et dirigitur ad h, & i ualidius it uentus e<lb/>tiam adiuuante temonem, eu contra nitente, quantum licet mo<lb/>bili pondere nauis ad id latus, premitur enim nauis, quai ubmer<lb/>gi debeat, uento in aduerum premente, ut i uentus repente huic <lb/>contrarius exoriatur, <expan abbr="periculũ">periculum</expan> ubeat, ne obruatur. </s>
<s id="id2702881">Cum ergo uen<lb/>tus ex b feratur, quiditans c h, & c feratur per temonem in k, & ab <lb/>oppoitis qualis actio equatur, im tota impeditur, ex c in h fere<lb/>tur iuxta proportionem anguli, quem contituit h c cum a c ad to<lb/>um rectum, Si igitur ex c in a debuit ferri in duodecim horis ob
<pb xlink:href="015/01/089.jpg" pagenum="70"/>uim uenti, & ui longitudinem, angulus uer h c a it exta re<lb/>cti pars, feretur ex c uerus a ad quantitatem b a in quatuorde<lb/>cim horis: igitur rurus quanta et proportio c a ad b a tan<lb/>tum et temporis, in quo fertur ex c ad a ad quatuordecim horas <lb/>per uentum b a.</s></p><p type="margin">
<s id="id2702978"><margin.target id="marg280"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s id="id2703004"><margin.target id="marg281"/>Q<emph type="italics"/>ut.<emph.end type="italics"/> 7. M<emph type="italics"/>echanica.<emph.end type="italics"/></s></p><p type="main">
<s id="id2703043">Propoitio octuageimaprima.</s></p><p type="main">
<s id="id2703056">Qualis it angulus, per quem potet moueri nauis ad rectum <lb/>explorare.<lb/><arrow.to.target n="marg282"/></s></p><p type="margin">
<s id="id2703079"><margin.target id="marg282"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2703106">Cum in prcedenti propoitione otenum it angulum k c a <lb/>oportere ee qualem angulo h c a, ut feratur, c in a uento c h, nec <lb/>tamen prorus, ed temo magis inflectit uerus k quam uentus co<lb/>git uerus h: icut contra maiori ui uentus dirigit ad h, qum temo <lb/>ad k, ut necee it nauim flecti ad k pondere, ideo i uentus eet <lb/>tranuerus periclitaretur, necee et, ut per omnes uentos, qui fe<lb/>runt ab ea, qu ad perpendiculum uper c a, & unt quatuordecim: <lb/>ed quoniam, ut dixi, pondere adiuuante uis uenti minor fit, nece<lb/>e et, ut per uentos debiliores feratur magis ab extremis, qui pro<lb/>pe perpendiculum unt: ita ut numerus omnium it, cum leuisimi <lb/>fuerint, quatuordecim, cum uiolentisimi, tres tantum proprius, & <lb/>qui ditant trigeimaecunda parte totius circuli, id et partibus un <lb/>decimi, cum quarta reliqui undecim, medij unt: ut tanto plures a<lb/>umi posint Nauclero, quanto molliores unt uenti, tanto pau<lb/>ciores, quo uiolentiores. </s>
<s id="id2703282">Tutius autem fuerit in ualidis uentis diri<lb/>gere nauim per uentum proximiorem, quam per ipummet, qui re</s></p><p type="main">
<s id="id2703300"><arrow.to.target n="marg283"/><lb/>ct tendit ad locum. </s>
<s id="id2703313">Veluti tendat nauis ex a in b, uentus tendat in <lb/>cualidior, cumque magnus fuerit angulus c a b, ut pot dodrans to<lb/>tius recti, ut eet temo dirigendus ad extum uentum altrinecus di <lb/>rigemus olum ad quintum, ut feratur in d, & hoc erit tanto cele<lb/>rius, & celerius feratur per a d & d b, qum i nauis recta lata eet <lb/>ex a in b. </s>
<s id="id2703364">inuper tutius.</s></p><p type="margin">
<s id="id2703375"><margin.target id="marg283"/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 83</s></p><p type="main">
<s id="id2703403">Propoitio octuageimaecunda.</s></p><p type="main">
<s id="id2703419">Proportionem uelorum indagare.<lb/><arrow.to.target n="marg284"/></s></p><p type="margin">
<s id="id2703434"><margin.target id="marg284"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2703461">Vela tribus in locis diponi olent dolo b, quod in prora con<lb/>tituitur, & in malo, qui ponitur in medio ratione, qu inferius <lb/>otendetur, ed non ad unguem, quia cum malus in anteriorem <lb/>partem uento impellatur, i eet in medio, emper prmeretur <lb/>nauis in anteriorem partem, ex quo duo magna incommoda eque <lb/>rentur: primm ut periculum ubiret, ne inuera in anteriorem par
<pb xlink:href="015/01/090.jpg" pagenum="71"/>tem ubmergeretur. </s>
<s id="id2703540">Secundum ne prea in parte anteriore dif<lb/>ficilius aquas diecaret, & ob id longe tardiu, moueretur. </s>
<s id="id2703559">Pro<lb/>pter hc duo incommoda igitur malus etiam i unicus eet <lb/>(quod uulgatisimum maloribus notris fuit) in parte magis <lb/>pror proxima locabatur gubernatoribus, ut eet quai in trien <lb/>te rotro in bee puppi: Rarum fuit, & memorabile, quod nunc <lb/>pasim habet olim Antigoni <foreign lang="greek">triame/ou&</foreign> 1, uelorum trium: quorum <lb/>potremum Epidromus ut ipa uoce intelligamus non fuie ue<lb/>lum in malo ipo medio, ed in puppi contitutum. </s>
<s id="id2703662">Caua Dolonis <lb/>inferius exponetur: quod autem eet paruum, & omnium mini<lb/>mum, ut nauis acile ab eo inuerteretur. </s>
<s id="id2703684">Vnde etiam nunc minus <lb/>minime habent tam quantitate, quam etiam altitudine, quod uo<lb/>cant Trinehetum, olum enim utinet nauim, qu uentis, uel un<lb/>dis mergi olet: ab undis ubi humilior et, uentis lateribus, et an<lb/>teriore parte. </s>
<s id="id2703731">Vnde humile, & exiguum uelum efficit, ut nauis ante<lb/>riore parte leuis, nec mergatur prona uentis, nec aquas ea exci<lb/>piat, nec tamen impelli potet nauis in copulos, nec euerti ob cau<lb/>as dictas: ob qu in magnis tempetatibus hoc ipo duntaxat uti <lb/>olent. </s>
<s id="id2703775">Quod eti nimium uierint, etiam illud demittunt, & i <lb/>fieri potet, etiam malum ipam quamuis ine uelo it. </s>
<s id="id2703804">Sed plerun<lb/>que circumuolutam, & implicatam olet antennam annexam, at<lb/>que upenam habere. </s>
<s id="id2703828">Sed & ne nauis prorum obruatur, quo<lb/>niam ea pars omnem uentorum uim excipere olet, & ut leuisima <lb/>it ijdem Gubernatores puppim multa arena, lapillis qe onerant. <lb/></s>
<s id="id2703856">Ergo uelocitas nauis uentorum impetu, eorumqe rectitudi<lb/>ne uelorum magnitudine, & loco humiliore, aut ublimiore ha<lb/>betur: tum nauis leuitate, & forma. </s>
<s id="id2703882">Qu enim non merguntur ut <lb/><foreign lang="greek">droma/des</foreign> (ic enim uocat Aritophanes) eas, quas nunc uulgus fre<lb/>gatas appellat) quai aquas innatantes curu unt uelocisim. </s>
<s id="id2703922">Et <lb/>longiores latis. </s>
<s id="id2703929">Pot has unt, qu carinam habent tenuem, ut fa<lb/>cile aquas diuidant. </s>
<s id="id2703946">Vltimo loco, qu quai medi, ante quidem <lb/>tenues, pt latiores ad uelocem curum, & ferendum onera apt, <lb/>& humiles altis: & leui ex ligno. </s>
<s id="id2703976">Sed nos de uelorum uarieta<lb/>te loquimur, non ea', qu ad malos pertinet. </s>
<s id="id2703988">Contat enim me<lb/>dio loco plus mouere, quam in extremis, ut infr docebi<lb/>mus. </s>
<s id="id2704007">Antiquo enim tempore opus non fuit malorum mul<lb/>titudine, quoniam yderibus uias dirigebant ob id non ad <lb/>amusim, quoniam linea dirigi non poterat maxim ob mo<lb/>tus obliquitatem in circulo uius: ide mali multi confu<lb/>ionem in curu, & impedimentum in naui, maiuqe pericu<lb/>lum attulient. </s>
<s id="id2704062">At nunc inuenta pyxide, & lapidis Her
<pb xlink:href="015/01/091.jpg" pagenum="72"/>culei auxilio pluribus locis uela dipoita melius dirigunt iter, ut <lb/>quai craa minerua depictum, & potetate deformatum, ad amu<lb/>im contrahant. </s>
<s id="id2704105">Motus ergo magnitudo non impliciter contat, <lb/>ed comparatione upericiei ueli ad uelum longitudine quidem, </s></p><p type="main">
<s id="id2704130"><arrow.to.target n="marg285"/><lb/>ac latitudine conflata per multiplicationem. </s>
<s id="id2704140">Altitudinis quo que ut <lb/><arrow.to.target n="marg286"/><lb/>infr exponetur. </s>
<s id="id2704156">Ex quorum omnium ductu, quai cubica, uel tri<lb/>plicata ratione, ut uperius otenum et, ratio uelocitatis motus na <lb/>uium conflatur.</s></p><p type="margin">
<s id="id2704187"><margin.target id="marg285"/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 86.</s></p><p type="margin">
<s id="id2704215"><margin.target id="marg286"/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 42.</s></p><p type="main">
<s id="id2704242">Propoitio octuageimatertia.</s></p><p type="main">
<s id="id2704255">Proportionem receus recta uia ad obliquitatem inuetigare.<lb/><arrow.to.target n="marg287"/></s></p><p type="margin">
<s id="id2704280"><margin.target id="marg287"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2704307">Sit nauis in a itura in b (uentus rectus ad c, medius ad e) per <expan abbr="ob-liquũ">ob<lb/>liquum</expan>, cum ergo tardius moueatur per a e qum a c & per a b, quam <lb/>per a d, & int ad perpendiculum b e, b d quas contat ee breuisi<lb/>mas earum, qu ad a c & ad a d. </s>
<s id="id2704353">Queritur igitur quando uelocius <lb/><figure id="id.015.01.091.1.jpg" xlink:href="015/01/091/1.jpg"/><lb/>ferretur ad b, an cum per a c, c b, an cum per a d, d b, <lb/>an cum per a b impliciter. </s>
<s id="id2704377">Et contat quod a d & d b <lb/>longiores unt a b, itud enim demontratum et ab <lb/>Euclide in primo Elementorum, dico modo a c, & </s></p><p type="main">
<s id="id2704408"><arrow.to.target n="marg288"/><lb/>c b ee longiores a d & d b, nam quadrata a d & d b <lb/>& a c & c b unt qualia quadrato a b per dicta ibi<lb/><arrow.to.target n="marg289"/><lb/>dem, & ideo quadrata a c & c b qualia quadratis a d <lb/>& d b, ed a d et longior a c, quia ducta c d angulus <lb/>d c a et obtuus, igitur ad maiorem a c per decimam <lb/>nonam primi Elementorum: quare per communem <lb/>animi ententiam quadratum a d maius et quadrato a c, quarerur<lb/>us per communem animi ententiam quadratum c b maius et <lb/>quadrato d b. </s>
<s id="id2704491">Cum ergo quadrata a d & d b qualia int quadra<lb/>tis a c & c b, & a d it maior a c & c b maior d b, equitur per nonam <lb/>ecundi Elementorum, quod a c & c d int maiores a d & d b pari<lb/>ter acceptis. </s>
<s id="id2704527">Si ergo maior fuerit exceus qum proportio motus <lb/>per temonem cohibiti, ut upra uium et, tardius mouebitur per <lb/>a d, d b qum a b per a c, c b qum per a d, d b, ed i contr maior it <lb/>proportio motus cohibiti temone ad motum liberum qum ex<lb/><arrow.to.target n="marg290"/><lb/>ceus ad exceum uelocius mouebitur per a d d b, qum per a b, <lb/>& per a c qum per a b. </s>
<s id="id2704606">Accedit huc e incommodo longioris ui, <lb/>quod uento a c non poterit ferri nauis ex c d in b, quoniam antea <lb/>gre ferebatur: & nunc grius per c b qum a b, plus enim ditat <lb/>uentus a c ab itinere c a qum uento a b, ut uium et uperius, igi<lb/>tur multo melius et (ni quid obtet) ire per a b qum per <expan abbr="ullã">ullam</expan> aliam <lb/><arrow.to.target n="marg291"/><lb/>uiam: nii tationes int in c d, uel periculum immineat in a b. </s>
<s id="id2704686">Vbi ta <lb/>men uenti ecundarent, tantum et uirium in recto curu, & quali
<pb xlink:href="015/01/092.jpg" pagenum="73"/>uelocitate ferretur citius ex a in b per a d d b, & etiam citius per a c, <lb/>c b in b quam per ipam a b, quod fuit propoitum declarare.</s></p><p type="margin">
<s id="id2704728"><margin.target id="marg288"/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 20.</s></p><p type="margin">
<s id="id2704757"><margin.target id="marg289"/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 47.</s></p><p type="margin">
<s id="id2704784"><margin.target id="marg290"/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 80.</s></p><p type="margin">
<s id="id2704812"><margin.target id="marg291"/>P<emph type="italics"/>er<emph.end type="italics"/> 81. P<emph type="italics"/>ropo.<emph.end type="italics"/></s></p><p type="main">
<s id="id2704850">Propoitio octuageimaquarta.</s></p><p type="main">
<s id="id2704863">Ditantiam centri terr centro mundi per motum lapidis Her <lb/>culei declarare.<lb/><arrow.to.target n="marg292"/></s></p><p type="margin">
<s id="id2704890"><margin.target id="marg292"/>C<emph type="italics"/>o<emph.end type="italics"/>_{m}.</s></p><p type="main">
<s id="id2704916">Non me later Aritotelem exitimare centrum mundi ee cen<lb/>trum terr illudque probae, quod tamen ex demontratione notra <lb/>mathematica apparet nuncubijciam, & quid ad illius rationes di<lb/>cendum it, alis etiam dicendum erit: nam liber hic, ut mathemati<lb/>ca decet, ee debet ab omnibus contentionibus abolutus. </s>
<s id="id2704977">Con<lb/>tat an non ee propriam uim lapidis illius, ut qui non it circum<lb/>criptus ed frutulum quoduis id potet, neque per e, ed in ferro & <lb/>pendulo, nec fieri potet, ut it illius <expan abbr="tãquam">tanquam</expan> peciei unius lapidum, <lb/>ed quai perfect portionis cuiudam generis terr, qu abolu<lb/>ta it, cuius indicium et illius copia, neque enim ullibi non inuenitur, <lb/>& ubi ferrum effoditur, ut in Ilua Inula Tyrrheno mari, et ergo fer <lb/><figure id="id.015.01.092.1.jpg" xlink:href="015/01/092/1.jpg"/><lb/>ri uis terr marit, qu perfecta in uo ge<lb/>nere, ubi uim fcundam acceperit macu<lb/>lo cilicet Herculeo lapide, qurit primum <lb/>ut decendat, ubi hoc non posit <expan abbr="&longs;alt&etilde;">altem</expan> qu<lb/>rit, ut quiecere posit. </s>
<s id="id2705161">Vt ergo quiecat <lb/>motu cli qui et ab Oriente in Occiden<lb/>tem iuxta axis cli itum e dirigit, quod <lb/>ille olus quiecat in uo motu, uel altem <lb/>tardisim moueatur: indicio et quod i <lb/>extra itum illum acus ferrea imbuta eo lapide ponatur, tatim tre<lb/>mit uchementer, ade ut nec momento ullo conitat, ed mier & <lb/>grauiter torqueri uideatur, non ergo quod entiat polorum locum <lb/>qui tantum abet ab illa, ut nec ab homine perito mathematicarum, <lb/>ed quod uix illa cli entiatur circa centrum mundi. </s>
<s id="id2705274">Cuius indi<lb/>cio et Oceani maris, aquarum fluxus & refluxus. </s>
<s id="id2705286">Duos ergo ha<lb/>bet motus terra perfecta, eu ferrum lapide Herculeo <expan abbr="imbutũ">imbutum</expan> ub<lb/>ordinatos imperfectum perfecto: perfectus et, ut decendat ad cen <lb/>trum terr, ut ibi quiecat: imperfectum, cum perfecto prohibe<lb/>tur, ut quiecat altem extra centrum cum in clinatione ad centrum, <lb/>et hoc fiet i ecundum longitudinem acus dirigatur per axem mun <lb/>di, cum itu tamen decenui ad terr centrum proximiore, ut pi<lb/>us uperius declarauimus, dum de motu grauium & prcipu li<lb/>br, & centro grauitatis loqueremur. </s>
<s id="id2705391">Quibus demontratis tum <lb/>experimento tum ratione Fortunio Affaytato Cremoneni Me<lb/>dico, cum per hc potmodum cogeretur fateri acum ad polum
<pb xlink:href="015/01/093.jpg" pagenum="74"/>tendere, cum tamen tendat dextro latere cilicet ab Oriente no<lb/>uem partibus, eu decima parte unius recti in centro terr, qu et <lb/>quadrageima totius ambitus cli. </s>
<s id="id2705457">Statuatur centrum mundia, & <lb/>b a c axis, ecundum quam mouetur motu diurno, ital a dextra exit <lb/>oriens, k a initra occidens, & tatuatur d centrum terr, eu upr <lb/>eu infr, non tamen in linea b c, ed uel upr in dextra parte, uel in<lb/>fr in initra, ita ut ducta linea per illud punctum arcus b g it no<lb/>uem partium. </s>
<s id="id2705528">Contituta ergo acu in e puncto, ubilinea h ad g ecat <lb/>peripheriam terr dico, quod acus dirigetur per h g, & non per b c, <lb/>nam acus mouetur ad centrum per eam, & in eo itu tota dirigitur, <lb/>quia omnes partes grauis conentiunt in motu principij grauitatis <lb/>ad centrum, hoc enim demontratum: nixus ergo et ut moueatur <lb/>per c d, & in eo nixu qui et quies cuto dit lineam axis, qu et a b, <lb/>ut quiecat, ergo non quiecet, nii in linea d g, quod erat demon<lb/>trandum. </s>
<s id="id2705601">Qu autem equuntur ex his corrolaria omnia concor<lb/>dant cum experimentis. </s>
<s id="id2705616">Ergo hic ermo et demontratiuus, ut e<lb/>nim bene dixit Auerroes: Sermo demontratiuus atisfacit omni<lb/>bus problematibus qu <expan abbr="cõtingunt">contingunt</expan> circa principale quitum. </s>
<s id="id2705660">Ex <lb/>hoc ergo patet, quod angulus ditantia d ab a in latitudine et de ci<lb/>ma pars recti, et quod quanto magis ditatin longitudine centrum <lb/>terr centro mundi, tanto etiam minus ditatin latitudine. </s>
<s id="id2705693">Hc <lb/>enim unt demontrata clar in mathematicis. </s>
<s id="id2705712">Vnde fieri poet <lb/>quod hc quantitas ditanti eet res, per quam exigua etiam i <lb/>non eet maior quatuor digitis ufficeret, modo etiam per ualde <lb/>paruum patium ditaret ab eodem in longitudine. </s>
<s id="id2705758">De caua au<lb/>tem huius differenti alis dicendum erit, hiclo cus non et, ed uf<lb/>ficit cire quod ita it, quod i mobilis it punctus d, clarum et ali<lb/>quando futurum ut minus ditet g b, aliquando ut it idem. </s>
<s id="id2705815">Et <lb/>qualicunque motus it, necee et eam ditantiam uariari.</s></p><p type="main">
<s id="id2705842">Propoitio octuageimaquinta.</s></p><p type="main">
<s id="id2705856">Proportio ponderis unius grauis ad aliud ub eadem menura <lb/>et, ueluti eiudem ad differentiam ponderis uais repleti ex altero <lb/>graui, & ex ambobus detracto priore.</s></p><p type="main">
<s id="id2705887"><arrow.to.target n="marg293"/></s></p><p type="margin">
<s id="id2705898"><margin.target id="marg293"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2705924">Sit aurum a, & liquor b, qu repleant uas c, & <lb/>pondus amborum it librarum quadraginta, & <lb/><figure id="id.015.01.093.1.jpg" xlink:href="015/01/093/1.jpg"/><lb/>uas repletum liquore olo it librarum xxix, au<lb/>rum autem it ponderis librarum xij, igitur reli<lb/>quum erit ponderis xxviij, differentia ergo ua<lb/>is pleni, & non pleni liquore et libra una, pon<lb/>dus auri et librarum duodecim: dico quod au<lb/>ri pondus et duode cuplum ponderi liquoris, &
<pb xlink:href="015/01/094.jpg" pagenum="75"/>i fuiet pondus amborum libr xxxix, manentibus reliquis, eque <lb/>retur quod pondus liquoris eet xxvij, & quia plenum uas uppo<lb/>nitur ee librarum xxix, eet differentia librij, at auri pondus et <lb/>libr xij, igitur proportio ponderis auri ad liquorem eet excu<lb/>pla. </s>
<s id="id2706063">Nam i uas plenum liquore ex uppoito et librarum xxix, & <lb/>cum auro xl, gratia exempli, & auri pondus et xij, igitur liquoris <lb/>pondus et xxviij librarum: ed cum liquor it corpus imilium par<lb/>tium, igitur loci ad lo cum, ut ponderis ad pondus, ergo dum adet <lb/>aurum, liquor occupat xxviij partes cxxxix, totius uais igitur au<lb/>rum continet unam partem tantum, & cum aurum pondus habeat <lb/>librarum xij, & liquor unius: quia totum uas cxxxix librarum dum <lb/>et plenum, & et diuium in xxix partes, igitur pondus unius par<lb/>tis liquoris et una libra, igitur pondus auri et duode cuplum ad <lb/>pondus liquoris quod fuit propoitum.</s></p><p type="main">
<s id="id2706159"><arrow.to.target n="marg294"/></s></p><p type="margin">
<s id="id2706170"><margin.target id="marg294"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 1.</s></p><p type="main">
<s id="id2706197">Ex quo equitur qud i ducatur pondus illud partis per pon<lb/>dus repleti uais ex alio graui, & productum diuidatur per differen <lb/>tiam illam, prodibit pondus uais repleti liquore graui.</s></p><p type="main">
<s id="id2706229"><arrow.to.target n="marg295"/></s></p><p type="margin">
<s id="id2706240"><margin.target id="marg295"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2706267">Exemplum, i pondus auri fuerit librarum xij, pondus uais re<lb/>pleti liquore xxix librarum, pondus auri & liquoris replentium <lb/>uas xxxix librarum, ducemus xij in xxix fit cccxlviij, diuido perij <lb/>differentiam xxvij ponderis uais, repleti ex ambobus detracto au<lb/>ri pondere, & xxix ponderis uais repleti liquore exit clxxiiij, & tan <lb/>tum auri uas illud continebit, nam cum du partes quas occupa<lb/>bat aurum eent ponderis librarum xij, totum quod erat partium <lb/>xxix, continebit decies & quater cum dimidio illud aurum xij, aut <lb/>ductum in xiiij cum dimidio, efficit cclxxiiij ut prius.</s></p><p type="head">
<s id="id2706333">EXEMPLVM.</s></p><p type="main">
<s id="id2706341">Quia ergo in uperiore propoitione docui, quod ferrum et ue<lb/>ra terra: uolui cire qualis eet proportio ferri ad aquam. </s>
<s id="id2706365">Accepi ur <lb/>ceum cuius aqua dum plenus eet ponderis, fuit unciarum ex, & <lb/>eptuncis unci, & eptuncis duodecim partis unci & pondus <lb/>ferri unci eptem, & triens unci & triens duodecim partis un<lb/>ci: & uais aqu & ferro eodem repleti unci tredecim, & duode<lb/>cima & eptunx duode cim partis unci. </s>
<s id="id2706441">Detrahemus ergo vij & <lb/>trientem & trientem duodecim. </s>
<s id="id2706452">i. </s>
<s id="id2706456">7 & 64/144 pondus ferri ex 13 19/144, & <lb/>relinquentur 5 99/144, detrahe ex 6 81/144, pondere aqu totius uais relin <lb/>quuntur 17/18, diuide 7 64/144 per 17/18 exit proportio ponderis ferri ad pon <lb/>dus aqu 7 15/17. Ethoc et proximum ei quod dixit Philoophus de <lb/>proportione ponderis terr & aqu.</s></p><p type="main">
<s id="id2706500"><arrow.to.target n="marg296"/></s></p><p type="margin">
<s id="id2706511"><margin.target id="marg296"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 2.</s></p><p type="main">
<s id="id2706538">Ex hoc patet olutio problematis cuiudam propoiti aliasque mi <lb/>nus bene oluti cm cauam habeat manifetisimam, cilicet quod
<pb xlink:href="015/01/095.jpg" pagenum="76"/>uae aqua pleno impoitis enim centum aureis coronatis nihil ef<lb/>funditur, non quod quicquam abumatur in metallo, ed caua et <lb/>quod cum aurum it duplum pondere ferro, erit ex demontratis <lb/>ex decuplum ad pondus aqu. </s>
<s id="id2706622">Igitur cum it proportio ponderis <lb/>auri ad differentiam patij eadem, i it uas aqu ponderis libr <lb/>unius & medi, erit pondus totum xxiij unciarum, igitur aqua de<lb/>ficiet olum ex decimaoctaua parte eu crecet ex impoitione auri, <lb/>ed illa pars in tumore aqu abumitur, <expan abbr="nõ">non</expan> olum, quia <lb/><figure id="id.015.01.095.1.jpg" xlink:href="015/01/095/1.jpg"/><lb/>dum aureos imponimus plana olum it, ed quia non ex <lb/>quauis rotunditate defluit, aliter in urceo tam exiguo <lb/>non poet apparere rotunda: quod enim rotunditas to<lb/>tius terr, qu etiam planam otendit totam unam re<lb/>gionem ad rotun ditatem qu apparet in exiguo urceo <lb/>aqu. </s>
<s id="id2706749">Et igitur rotunditas illa potius ob lentorem aqu qui auge<lb/>tur lentore argenti, & etiam magis auri, cum enu digitorum per<lb/>cipiatur.</s></p><p type="main">
<s id="id2706782"><arrow.to.target n="marg297"/></s></p><p type="margin">
<s id="id2706794"><margin.target id="marg297"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 3.</s></p><p type="main">
<s id="id2706820">Ex hoc apparet ratio quomodo Archimedes potuerit deprehen <lb/>dere coronam Hierone propoitam quantum auri & argenti con <lb/>tineret. </s>
<s id="id2706836">Sit ergo uas a b aqua <expan abbr="plenũ">plenum</expan> ponderis un ciarum triginta, & <lb/>cum libra auri it ponderis unciarum quadraginta unius, & cum li<lb/>bra argenti ponderis unciarum quadraginta cum dimidio, igitur <lb/>erit auri pondus ad aqu pondus duodecuplum, argenti autem <lb/>ad idem octuplum, quare auri ad <expan abbr="arg&etilde;tum">argentum</expan> pondus exquialterum. <lb/></s>
<s id="id2706885">Ponamus ergo quod corona impoita ex auro & argento olo fa<lb/>bricata (hoc enim upponere oportet) fuerit un ciarum exaginta, <lb/>pondus autem aqu content cum corona in uae unciarum uigin <lb/>tiquatuor cum dimidio, cilicet totum octuaginta quatuor cum di<lb/>midia, erit ergo proportio ponderis coron ad pondus aqu, ut <lb/>cxx ad xi, aurum igitur et proportione duodecuplum, argentum <lb/>autem octuplum, corona ut cxx ad xi. </s>
<s id="id2706944">Contituantur ub eidem ra<lb/>tionibus ducen do lxxxviij. </s>
<s id="id2706962">cxx. </s>
<s id="id2706966">cxxxij. </s>
<s id="id2706970">hoc et ac i dicamus, accipe <lb/>partes ex cxxxij & lxxxviij, tot ut faciant integrum & componant <lb/>cxx. </s>
<s id="id2706988">Et ide reduces ad minores numeros, cilicet xxxiij. </s>
<s id="id2706998">xxij. </s>
<s id="id2707002">et xxx. </s></p><p type="main">
<s id="id2707011"><arrow.to.target n="marg298"/><lb/>& operaberis per regulam de conolatione monetarum, quas po<lb/>nemus infr, & fient auri partes octo & argen <lb/><figure id="id.015.01.095.2.jpg" xlink:href="015/01/095/2.jpg"/><lb/>ti partes iij, nam cum duxeris iij in octo pon<lb/>dus argenti fiet xxiiij, & cum duxeris viij in <lb/>xij, pondus auri fiet xcvi, igitur totum pon<lb/>dus erit cxx, diuidendum per xi, aggregatum <lb/>partium auri & argenti, ita uero uncia ad unciam, ut tota corona mi <lb/>ta ad coronam puram auri & argenti.</s></p>
<pb xlink:href="015/01/096.jpg" pagenum="77"/><p type="margin">
<s id="id2707083"><margin.target id="marg298"/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 178.</s></p><p type="main">
<s id="id2707112">Ex hoc etiam patet modus <expan abbr="cogno&longs;c&etilde;di">cognocendi</expan> proportionem grauium <lb/><arrow.to.target n="marg299"/><lb/>inuicem per olam aquam, uelut auri ad plumbum, ad lapides uel <lb/>s, aut ris ad lapidem & imilia, ut in prcedenti operatione de<lb/>prehenditi: nam cum it nota proportio auri ad aquam & ris uel <lb/>lapidis ad eandem, erit auri ad s uel lapidem nota.</s></p><p type="margin">
<s id="id2707180"><margin.target id="marg299"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 4.</s></p><p type="main">
<s id="id2707207">Et imiliter ciemus per hoc accipere partes diuerorum, qu iun <lb/><arrow.to.target n="marg300"/><lb/>ct faciant contitutum pondus. </s>
<s id="id2707236">Velut uolo facere maam ex mel<lb/><figure id="id.015.01.096.1.jpg" xlink:href="015/01/096/1.jpg"/><lb/>le & aqua, qu impleat uas, quod mellis contineat <lb/>quindecim, aqu duodecim, uolo ut contentum it <lb/>ponderis quatuorde cim, operabor, ut in <expan abbr="cõ&longs;olatio-nibus">conolatio<lb/>nibus</expan>, ponam duas partes mellis & unam aqu, ut <lb/>uides in operatione latere.</s></p><p type="margin">
<s id="id2707307"><margin.target id="marg300"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 5.</s></p><p type="main">
<s id="id2707334">Propoitio octuageimaexta.</s></p><p type="main">
<s id="id2707349">Si circuli in quales, eu in phra, eu in plano e ecuerint nun<lb/>quam oppoitos angulos quales habent.</s></p><p type="main">
<s id="id2707388">Capiantur tres quart cir culorum magnorum a b, a c, b c, & alia <lb/><arrow.to.target n="marg301"/><lb/>b d ad rectos angulos <expan abbr="erũtque">eruntque</expan> uicisim poli, & ducatur per medium <lb/>parallelus, erit ergo e f qualis e g, & f e qualis f g, ed bais c g et <lb/><figure id="id.015.01.096.2.jpg" xlink:href="015/01/096/2.jpg"/><lb/>quarta circuli, & bais c b dimidium quart <lb/>circuli eo quod tota b a et quarta circuli, igi<lb/>tur per modum 25 primi Elementorum qu <lb/>tenet, erit angulus c f g maior oppoito c f b. <lb/></s>
<s id="id2707477">Hoc autem tenet in eiudem rationis uperfi<lb/>ciebus, quales unt h, qu unt uperficies eiudem phr. </s>
<s id="id2707516">poet <lb/>etiam demontrari per modum quart primi Elementorum. </s>
<s id="id2707535">Et eti<lb/>am contituta phra e f g, cuius hic circulus eet maior circulus, & <lb/>non tangeret nii in illa linea phra maiorem, & utrin que ecaret eo<lb/>dem circulo. </s>
<s id="id2707576">Et etiam per cordas & trigonos rectilineos, auxilio <lb/><expan abbr="tam&etilde;">tamen</expan> regul dialectic. </s>
<s id="id2707598">Ex hoc equitur auxilio regul dialectic, <lb/><figure id="id.015.01.096.3.jpg" xlink:href="015/01/096/3.jpg"/><lb/>quod in omnibus parallelis a c d & e f g cum b c circulo <lb/>maiore, & per aliam regulam dialecticam in omnibus cira <lb/>culis inqualibus inter e ad quales angulos ecanti<lb/>bus & ex tertia demum regula dialectica, equitur in o<lb/>mnibus circulis in qualibus e ecantibus ad quemuis <lb/>angulum in phr uperficie. </s>
<s id="id2707680">Sunt autem h regul medi inter <lb/>axiomata & demontrata. </s>
<s id="id2707698">Et ex logica propria illi arti. </s>
<s id="id2707703">In plano au<lb/><arrow.to.target n="marg302"/><lb/>tem patium d b c minus et a b c, ed patium c b d et unum, ergo <lb/>per communem animi ententiam patium a b d, maius et patio <lb/>c b c, quod fuit probandum.</s></p>
<pb xlink:href="015/01/097.jpg" pagenum="78"/><p type="margin">
<s id="id2707761"><margin.target id="marg301"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s id="id2707788"><margin.target id="marg302"/>P<emph type="italics"/>er<emph.end type="italics"/> 13. <emph type="italics"/>terd <lb/>tij<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s></p><p type="main">
<s id="id2707837">Propoitio octuageimaeptima.</s></p><p type="main">
<s id="id2707853">Proportionem crasitiei aqu ad arem in comparatione ad ra<lb/>dios demontrare.<lb/><arrow.to.target n="marg303"/></s></p><p type="margin">
<s id="id2707883"><margin.target id="marg303"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2707909">Sit in aheno a b c d in imo e dena <lb/><figure id="id.015.01.097.1.jpg" xlink:href="015/01/097/1.jpg"/><lb/>rius argenteus cera affixus uel cla<lb/>uo, quem uideat ex h impoita aqua <lb/>clara uque ad f, uideat ex k, igitur per <lb/>aquam deflectitur perpendiculo <lb/>per angulum k f n, & in l, per angu<lb/>lum l g o crecente aqua demum in <lb/>labro m a p, & it e annexus, & tabu <lb/>la h k l m it affixa olo uel pondere <lb/>firma foraminibus obliquis infr <lb/>pectantibus, & per a apicientibus extremitatem e. </s>
<s id="id2707985">Poumus ergo <lb/>imaginari primum, qud omnes inclinationes int perpendicu<lb/>lari, dum exit aqua, & ita denarius uideretur, uel in uperficie aqu <lb/>in directo e, uel in recta ex oculo in imo, quorum neutrum uerum <lb/>et. </s>
<s id="id2708027">Secundus modus et, ut radius delatus e a flectatur ad k uell, & <lb/>hoc non quia in a non et mutatio medij. </s>
<s id="id2708042">Tertius et, ut linea ex ocu <lb/>lo ducta perueniat per punctum a ad uperficiem aqu, & ex ea <lb/>per directum ad denarium, & tunc quia oculus iudicat e uidere <lb/>per rectam, ideo iudicabit e uidere per l a g in q, eo quod emper in <lb/>directo loci in quo et e. </s>
<s id="id2708080">At quoniam non ex qua cunque ditantia ui<lb/>detur e, ed ex longinquiore loco, ubi uas fuerit humilius quod li<lb/>ne ad a ex oculo, quanto a fuerit humilius, tanto propius ipi e <lb/>procedunt. </s>
<s id="id2708109">Et uera uice line ex e ad a, quanto e et humilius ad <lb/>quencunque locum inflectuntur, tanto inferius <expan abbr="cadũt">cadunt</expan>. </s>
<s id="id2708135">Ergo cum fue <lb/>rint ad quilibrium h, magis ditabunt ab e, & ita e magis procul <lb/>uidebitur. </s>
<s id="id2708152">Caua ergo triplex et humilitas, uel altitudo uais: humi <lb/>litas uel altitudo aqu: & labri uais altitudo. </s>
<s id="id2708175">Sed han crelinquere <lb/>poumus. </s>
<s id="id2708186">Difficultas ergo experimenti etiam rect facti et, quo<lb/>niam poito uae n c d olum, ut altitudo it tantum n e, procul ma<lb/>gis uidebitur e, qum i uas it a b c d, & totum plenum. </s>
<s id="id2708226">Vbi autem <lb/>uas fit a b c d, magis procul uidebitur e cum uerit totum plenum, <lb/>quam cum fuerit plena ola pars n c d. </s>
<s id="id2708245">Sic difficile et coniderare <lb/>an altitudo aqu faciat ad uiionem procul, cum in humiliore, ed <lb/>disipari uae longius uideatur in pauca, quia labrum non obtat: <lb/>in eodem autem longius in pluri aqua, quia labrum etiam non ob<lb/>tat, ed alia ratione. </s>
<s id="id2708290">Vt ergo uideamus hoc experimentum, capie
<pb xlink:href="015/01/098.jpg" pagenum="79"/>mus duo uaa a b c d duplum h k l m ub eadem proportione alti<lb/>tudinis & latitudinis, & collo cabimus ita ut p n radius quiditet <lb/>f e, & collo cabimus tabulas cum foraminibus, ut prius, & g f p q <lb/><figure id="id.015.01.098.1.jpg" xlink:href="015/01/098/1.jpg"/><lb/>in quilibrio, in de uidebimus, an q p it qualis aut breuior, nam <lb/>longior ee non potet, quoniam inflectitur a minore aqua, ideo <lb/>angulus p h q non potet ee maior f a g, uppoita p h quali a f: <lb/>quod i non eet, ufficeret, ut q & p eent in quilibrio uno, & f g <lb/>alio. </s>
<s id="id2708402">Sed ueritas et quod maiore aqua maior fit reflexio: tum <lb/>quia in his, qu unt ecundum naturam corpoream, & ubtan<lb/>tiam denam, aut tenuem uarietas quantitatis uariat uires: tum <lb/>quia uidemus, quod in altiore aqua denarius uidetur magis cum <lb/>fundo elatus. </s>
<s id="id2708445">Igitur his cognitis experimentum fiat cum uae ple<lb/>no. </s>
<s id="id2708457">Et (ut dixi) coniderabimus proportionem anguli f a g ad far, <lb/>eu f e c qu an et no tabilis: ade ut it maior proportio aqu ad <lb/>arem comparatione grauium qum lucis.</s></p><p type="main">
<s id="id2708501"><arrow.to.target n="marg304"/></s></p><p type="margin">
<s id="id2708512"><margin.target id="marg304"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 1.</s></p><p type="main">
<s id="id2708539">Ex his cognocemus comparatione eiudem aqu tenuitatem <lb/>aris unius regionis in comparatione ad arem alterius: nam ubi <lb/>remotius uidebitur denarius, ibi ar erit tenuior.</s></p><p type="main">
<s id="id2708571"><arrow.to.target n="marg305"/></s></p><p type="margin">
<s id="id2708582"><margin.target id="marg305"/>C<emph type="italics"/>or<emph.end type="italics"/>_{m}. 2.</s></p><p type="main">
<s id="id2708609">Et per idem in eadem regione comparationem aquarum. </s>
<s id="id2708614">Nam <lb/>cum it idem ar, & uas, ac reliqua paria, ubi magis procul uidebi<lb/>tur denarius, aqua erit crasior ide deterior.</s></p><p type="main">
<s id="id2708642"><arrow.to.target n="marg306"/></s></p><p type="margin">
<s id="id2708653"><margin.target id="marg306"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 3.</s></p><p type="main">
<s id="id2708680">Se quitur etiam qud omnes res propiores in aqua uidentur, <lb/>quam int, & ide maiores: & ob id etiam omnis aqua profundior <lb/>et, quam uideatur. </s>
<s id="id2708704">Vtingredi perp it periculoum.</s></p><p type="main">
<s id="id2708724">Propoitio octuageimaoctaua. </s>
<s id="id2708734">De intrumento <lb/>momentorum.</s></p><p type="main">
<s id="id2708748">Intrumentum Acolingen, quo momenta temporum deprehen <lb/>dantur fabricare.</s></p>
<pb xlink:href="015/01/099.jpg" pagenum="80"/><p type="main">
<s id="id2708771"><arrow.to.target n="marg307"/></s></p><p type="margin">
<s id="id2708782"><margin.target id="marg307"/>C<emph type="italics"/>om.<emph.end type="italics"/></s></p><p type="main">
<s id="id2708808">Et quoniam motus naturales fiunt in tempore: & dicuntur ue<lb/>lociores, uel ob patium loci magnum, quod uperatur, uel ob tem <lb/>poris breuitatem in uelo cisimis motibus, quod ad patia attinet, <lb/>facilius dignocuntur uelociores, quoniam patium maius & ma<lb/>net, ut menurari commod posit: ed qud ad tempus, quanto tar<lb/>diores, quoniam in uelo cibus quantitas temporis et exigua: & e<lb/>tiam tempus ipum perpetu diffluit: ide difficillim deprehendi <lb/>potet. </s>
<s id="id2708888">Huius caua exco gitauimus intrumentum, quod uo caui<lb/>mus Acolingen: quod contat tribus rotis: prima et pedum duo<lb/>decim diametri, in ambitu autem habet denticulos ccclx qua<lb/>les, & qualiter inter e ditantes, huius peripheri funis cum pon<lb/>deribus ineritur, ita ut cum alijs duabus rotis renitentibus in una <lb/>hora circumagatur qualiter. </s>
<s id="id2708946">Duodecim ex his denticulis curru<lb/>lis duode cim denticulorum axis ecund rot ineritur: ic ut cum <lb/>rota magna duode cim conuera fuerit partibus, ecunda rota cu<lb/>ius axis it pedum duorum, cilicet excuplo maior circumuerta<lb/>tur. </s>
<s id="id2708995">Huius minoris ambitus diuius it in cxx partes quales, & <lb/>unicuique parti denticulus inertus it: ita hc rota tricies in una <lb/>hora conuertetur. </s>
<s id="id2709023">Singulis uer denticulis currulis axis rot ha<lb/>bentis denticulos quatuor ineratur, ita ut dum ecunda rota uer<lb/>titur emel minima circumuertatur tricies: nam pro ingulis qua<lb/>tuor denticulis, quibus media rota cir cumagetur, minima tota cir<lb/>cumuertetur, ideoqe nongenties in una hora. </s>
<s id="id2709067">Hc minima ro<lb/>tula beem pedis in dimetiente habebit, ut it exta pars illius, in <lb/>ambitu autem diuia erit in xl partes, ut cum circumuera fue<lb/>rit nongenties in una hora pertranierit partes xxxvi. </s>
<s id="id2709105">Et cum <lb/>pulus hominis communis int in hora <23>, uel circa nouem partes <lb/>ex his rot minoris perficient circiter unam pulationem ex diato<lb/>le & itole, eu ex ditentione & contractione perfectam: ut partis <lb/>unius conuerio fiat in nona parte, uel circa unius pulationis pul<lb/>us humani: & hoc et minimum ferm, quod ab humano en<lb/>u percipi posit. </s>
<s id="id2709181">Erit etiam proportio rotarum eadem tam in dia<lb/>metris, qum circuitibus cilicet excupla, neque motus diffor<lb/>mis, quoniam maior tanto tardius mouebitur, quanto quod ue<lb/>locius mouetur etiam minus erit, tamen proportio uelo citatis ma<lb/>ioris ad minorem in qualibus patijs uigintiquin cupla, ut ma<lb/>ioris ad mediam quintupla, nam cum it excupla in ambitu, <lb/>& tricies moueatur uelocius comparatione totius, equitur, ut <lb/>proportio patij, quod uperabit media ad patium, quod u<lb/>perabit maior in eidem temporibus, erit quintupla, emper ad un<lb/>guem. </s>
<s id="id2709266">Et ita medi ad minorem quintupla, & ide maioris ad
<pb xlink:href="015/01/100.jpg" pagenum="81"/>minorem uelo citas uiginti quincupla, ut non it difformis, neque <lb/>pcriculoa, ut in rotis moletrinis, & it diuia per medium iuxta <lb/>proportionem, cum it tanto uelo cior minor media, quanto media <lb/>maiore. </s>
<s id="id2709310">Rurus proportio partium maioris ad medi partes tripla <lb/>et cilicet ccclx ad cxx, & medi ad <expan abbr="minor&etilde;">minorem</expan> tripla cxx ad xl, & pro<lb/>portio et excupla, iterum igitur partes maioris ad mediam, & me<lb/>di ad minorem erunt in dupla proportione, utrobique, & et pul<lb/>chrum. </s>
<s id="id2709368">Ide partes etiam minim rot erunt atis magn: nam <lb/>cum diameter it bes pedis, ambitus peripheri erit duorum pe<lb/>dum. </s>
<s id="id2709400">1. unciarum uigintiquatuor: igitur diuia peripheria in xl par<lb/>ter, unaqu que pars erit maior dimidia uncia.</s></p><p type="head">
<s id="id2709419">SCHOLIVM.</s></p><p type="main">
<s id="id2709429">Et cum defuerit intrumentum, utemur menura expulu homi<lb/>nis deumpta, ed non et ade exacta. </s>
<s id="id2709457">Accedit aliud commodum, <lb/>qud cum in una hora circumuertantur partes xxxvi, id et triginta <lb/>ex mille: & octauus orbis circumuertatur in totidem annis, tot <lb/>erunt momenta ex his in una hora, quot anni in uno circuitu tella<lb/>rum fixarum. </s>
<s id="id2709489">Vtintelligamus, qum breui tranit una hora apud <lb/>nos, ita apud Deum, utita dicam (nam nulla in infinito proportio) <lb/>unus annus magnus, & reditus rerum omnium. </s>
<s id="id2709508">Comparata etiam <lb/>rota minima ad rotam moletrini ic e habet, qud cm modica ad<lb/>et, ueratur rota in una pulatione: cum atis abundans quinquies, <lb/>aut exies cum immodica duo decies.</s></p><figure id="id.015.01.100.1.jpg" xlink:href="015/01/100/1.jpg"/>
<pb xlink:href="015/01/101.jpg" pagenum="82"/><p type="main">
<s id="id2709570"><arrow.to.target n="marg308"/></s></p><p type="margin">
<s id="id2709581"><margin.target id="marg308"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2709608">Ex hoc equitur, quod homo i moueretur uelo citate motus ro<lb/>t moletrin in ex eb domadibus perueniret ad ydus Lun, nam <lb/>rotarum earum, quibus ferrum acuitur emidimetiens communi<lb/>ter et bes unius paus, ide dimetiens paus cum triente: ambi<lb/>tus ergo quatuor paus, & xxi pars, colligamus nunc integra, in <lb/>uno ictu pulus circumagitur decies, id et paus xl, in hora unt <lb/><23> pulationes: in hora igitur patium pertranitum et cxl pauum <lb/>in M. horis, ergo erunt clx M. pauum addita parte xxi, erunt clxviij <lb/>M. pauum, & tantum ditat luna terra: & M. hor unt dies pen <lb/>xlij, eb domad cilicet ex.</s></p><p type="main">
<s id="id2709751">Propoitio octuageimanona.</s></p><p type="main">
<s id="id2709765">Proportionem denitatis aqu ad arem per pondera inuenire.</s></p><p type="main">
<s id="id2709782"><arrow.to.target n="marg309"/></s></p><p type="margin">
<s id="id2709793"><margin.target id="marg309"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2709820">Contingit hoc multis modis: primum acceptis duabus phru<lb/>lis qualibus ex crytaliubtantia unaque demia ab altisima turri, <lb/>& menurato ictu per intrumentum prcedens, & ub totidem <lb/>momentis alia demia in aquam, in de ub eodem tempore dimen<lb/>a altitudine, erit proportio patij ad patium, ut denitatis aqu, ad <lb/>denitatem aris. </s>
<s id="id2709902">Item emia phrula per intrumentum in arem, <lb/>in de in aquam: & fumpta proportione. </s>
<s id="id2709927">Et uidimus corpionem, <lb/>qui <expan abbr="&longs;phærulã">phrulam</expan> creteam emittebat pedibus lxx, & in aqua per unum <lb/>& dimidium ade, ut proportio fuerit, ut quinquaginta ad unum: <lb/>ide et fallax experimentum in uiolento motu: nam cum emitte<lb/>batur in aquam erat prop, & ob id in ummo robore: cm in a<lb/>rem, emittitur enim uis. </s>
<s id="id2709995">De hoc ergo loquar.</s></p><p type="main">
<s id="id2710003"><arrow.to.target n="marg310"/></s></p><p type="margin">
<s id="id2710014"><margin.target id="marg310"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2710041">Et erumpentia ob id magis qum terra, et minus qum ex are: <lb/>diuiditur enim aqua cum graue petit fundum, & aqua feruet: & et <lb/>mirabilius, qum utile.</s></p><p type="main">
<s id="id2710073">Propoitio nonageima.</s></p><p type="main">
<s id="id2710087">Rationem impetus uiolenti extra misi ponderis ad qualita<lb/>tem reducere.</s></p><p type="main">
<s id="id2710106"><arrow.to.target n="marg311"/></s></p><p type="margin">
<s id="id2710117"><margin.target id="marg311"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2710143">Sit uiolentum a quod moueatur per b c d e, e patium, & quia <lb/>uiolentum contr nititur naturali, cadat ergo in planum in e: unt <lb/>ergo tria conideran da, primum quod, ut dixi alis, motus uiolen<lb/>tus pro certa ditantia augetur, & cauam ibireddidi, ut pot uque <lb/>ad c, ed hoc eet difficile cognitu. </s>
<s id="id2710194">Secundum, quod ubi in cipit de<lb/>crecere, emper magis ac magis decrecit propter naturalem ni<lb/>xum contra operantem. </s>
<s id="id2710216">Tertium quod ubi decendere in cipit, ibi <lb/>et qualis uis uiolentum motum agens cum naturali. </s>
<s id="id2710233">Certum et <lb/>etiam quod motus qualis intelligitur erecta ad perpendiculum <lb/>e f, donec occurrat a d: & diuia tota b f per tempus, locus ergo, in <lb/>quo mouetur per tantum patium, dicitur locus motus qualis:
<pb xlink:href="015/01/102.jpg" pagenum="83"/>qui it gratia exempli g h, cuius medium proportione it k, di<lb/>co k conitere propiorem f, qum b, etiami qualiter mouere<lb/>tur. </s>
<s id="id2710300">Primum qud in tota g f declinat, & totus motus et lentior, <lb/>qum in tota b g, & tamen tardatur tantundem, ergo per commu<lb/>nem animi ententiam, k et propior f, qum b. </s>
<s id="id2710331">Secund, quia per <lb/>ecundum uppo itum motus a uerus f, continu fit lentior, igitur <lb/>per communem animi ententiam mult longius et tempus mo<lb/>tus a k, quam f, & tanto maius patium. </s>
<s id="id2710373">Terti, quia motus ex b uer <lb/>us caugetur, & i eet qualis adhuc mult eet breuior k f quam <lb/>a k, igitur mult magis hoc modo, & triplicata ratione. </s>
<s id="id2710410">Si ergo b k <lb/><figure id="id.015.01.102.1.jpg" xlink:href="015/01/102/1.jpg"/><lb/>eet exquiquarta olum ipi k f, <lb/>erit b k dupla: ferm ex triplicata <lb/>ratione ipi k f, & iuxta eundem <lb/>modum ponemus mediam uim <lb/>xlvi pasibus corpione a quam <lb/>& hoc modo erit propid quod et.</s></p><p type="head">
<s id="id2710480">SCHOLIVM.</s></p><p type="main">
<s id="id2710489">Dubitat autem Philoophus in mechanicis qu nam uis it, qu <lb/>moueat lapidem iam excuum? </s>
<s id="id2710512">& dubium non et quin ex parte it <lb/>ar motus tum ratione, quia mouetur ergo mouet, tum experimen <lb/>to, ut in fulminibus, & his qu uento impelluntur, ut hypophyis, <lb/>ed in corpionibus & arcubus & pilis id non ufficere uidetur. </s>
<s id="id2710550">Ita<lb/>que uelut & caliditas & frigiditas in corporibus natura contrarijs <lb/>aliquandiu manent, & agunt ita & uiolentos motus, idque Alexan<lb/>der & Simplicius uolunt. </s>
<s id="id2710571">Inditio unt qud mota & emia ex lon<lb/>gioribus machinis quan quam non arem continentibus, nec in<lb/>anibus tamen, longius eijciunt agittas & misilia, quoniam uis <lb/>illa firmius imprimitur, uelut etiam de lapidibus & ferro, quod di<lb/>utius in igne moram traxit, aut continu follibus ignitum et, nam <lb/>etiam tanto tardius refrigeratur unum quod que horum, & alia urit <lb/>& accendit calore illo externo, quanquam natura frigidum it: di<lb/>cemus autem & de hoc uo loco.</s></p><p type="main">
<s id="id2710642">Propoitio nonageimaprima.</s></p><p type="main">
<s id="id2710656">Proportionem grauis cubi, & phrici qualium in accliui, & <lb/>decenus eorum demontrare.</s></p><p type="main">
<s id="id2710686">Hic non pauca unt <expan abbr="cõ&longs;ideranda">conideranda</expan>: Primum <lb/><figure id="id.015.01.102.2.jpg" xlink:href="015/01/102/2.jpg"/><lb/>qud hoc intelligi potet, uel de motibus at<lb/>tractionis, uel impulionis, uel inuerionis. <lb/></s>
<s id="id2710737">Secundum quod omne, quod impellitur uperis, tantundem gra<lb/>uat attractum, quantum ad decenum, i it rotundum, nam qua<lb/>drata, <expan abbr="etiã">etiam</expan> alia non decendunt ponte in decliui, & i it locus uald
<pb xlink:href="015/01/103.jpg" pagenum="84"/>decliuis, tanto minus decendunt, quanto unt latiora. </s>
<s id="id2710805">Quia tamen <lb/>omnia difficilis decendunt phricis, & facilius qum in plano, <lb/>ubi ponderant nii per dimidium grauitatis, ide proportio hc <lb/>contat ex proportione anguli decenus ad totum rectum, & ma<lb/>gnitudine uperficiei, qua incumbit ad pondus comparata. </s>
<s id="id2710859">Omne <lb/>enim graue, quanto grauius tam ad quietem, qum ad motum na<lb/>turalem potentius et: hoc enim perpicuum et, quia quieti natu<lb/>rali motus uiolentus, & motui naturali quies uiolenta opponitur: <lb/>quia ergo maiore ui opus et ad motum prter naturam, ergo e<lb/>cundum naturam etiam maiore ui quiecit. </s>
<s id="id2710909">Aumpimus ergo cu<lb/>bum, ut magis notum. </s>
<s id="id2710925">Sphra igitur in omni decliui decendit, <lb/>quia ut dictum et, nil habet quod reitat ad motum: & ipa gra<lb/>uior et in decliui, qum in plano, quia c pun<lb/>ctus cadit ultra e, ergo punctus contactus, & <lb/><figure id="id.015.01.103.1.jpg" xlink:href="015/01/103/1.jpg"/><lb/>centrum grauitatis, & centrum mundi, non unt <lb/>in una linea. </s>
<s id="id2710984">Si enim b c contangeretur, eet b c <lb/>plana. </s>
<s id="id2710996">Si uer tangit, angulus et maior angulo <lb/>contactus, ergo cum necearium it, quidita<lb/>re aliter non eet phricum, oportet, ut eleue<lb/>tur ex parte c, & decendat uerus b, & ide ut <lb/>continuetur motus. </s>
<s id="id2711052">Si uer it in linea conta<lb/>ctus b c f, & quiditet non erit, ut dixi punctus <lb/>contactus in linea centrorum, ed in a c, cum uppoitum it lineam <lb/>a d ee lineam centrorum: maior et ergo portio g c e, qum rei<lb/>duum, ergo decendet in b. </s>
<s id="id2711109">Cubus uer non decendet, nii cum di<lb/>midium d addito, quod inter cipitur inter lineam mediam, & qu <lb/>centro mundi ad punctum medium contactus uque qu perueniat <lb/>ad oppoitam partem, eam habuerit proportionem ad idem me<lb/>dium eadem portione detracta, quem iuncta proportioni an guli <lb/>declinationis ad reiduum recti dimidiam proportionem efficiat. <lb/></s>
<s id="id2711162">Eademque ratio aliorum planorum. </s>
<s id="id2711167">Dico prterea qud motus <lb/>phr, & etiam corporum rectarum uperficierum in decenu <lb/>alius et qualis, & alius inqualis, & quai latere, uelut i angu<lb/>lus unus prolabatur, ac fiat circumuolutio: cum ergo facilius fiat <lb/>hoc, & maxim i non retineatur qualiter, & difficile it in medio <lb/>retinere, propterea prolapus hi melius <expan abbr="retin&etilde;tur">retinentur</expan> duobus uinculis, <lb/>qum in medio, non olum ob hanc qualitatem, & complexum <lb/>meliorem, ed <expan abbr="etiã">etiam</expan>, quod omnes motus, omnes ponderum nixus fa <lb/>cilis cohibentur, & <expan abbr="deducun&ttilde;">deducuntur</expan> diuii in partes, <08> i toti contin <expan abbr="ean&ttilde;">eantur</expan>, <lb/>aut ui <expan abbr="trahãtur">trahantur</expan>. </s>
<s id="id2711318">Et ideo uin cula in rami cibus duplicia dextra, & ini <lb/>tra cilicet in <expan abbr="ead&etilde;">eadem</expan> parte tam longe unt meliora etiam ferreis, qu <lb/>olum in medio nectantur.</s></p>
<pb xlink:href="015/01/104.jpg" pagenum="85"/><p type="main">
<s id="id2711368"><arrow.to.target n="marg312"/></s></p><p type="margin">
<s id="id2711379"><margin.target id="marg312"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 1.</s></p><p type="main">
<s id="id2711406">Ex hoc etiam equitur, <lb/><figure id="id.015.01.104.1.jpg" xlink:href="015/01/104/1.jpg"/><lb/>quod cm omne graue <lb/>pont emper appropin<lb/>quet centro mundi, & a i <lb/>moueretur per planum e, <lb/>magis remoueretur cen<lb/>tro mundi, ut per e c per ea <lb/>qu diximus, & quoniam <lb/>linea ex centro mundi ad <lb/>c longior et, qum ad e, <lb/>mult potet enim ee, ut <lb/>in proportione diametri <lb/>quadrati ad latus eius, & <lb/>ctiam maior. </s>
<s id="id2711498">ergo poterit <lb/>ee ade parum decliuis <lb/>linea c d, ut c punctus ma<lb/>gis diter centro mundi, <lb/>qum d, & tamen feretur <lb/>ex d in c motu naturali, ut demontratum et, ergo per purum mo<lb/>tum naturalem poterit a remoueri centro mundi. </s>
<s id="id2711548">Hoc uolui pro<lb/>ponere, ut intelligeres in plano uero c e non moueri a ponte, quia <lb/>c neceari altior et d: i ergo mouebitur, non erit c e recta, ed <lb/>pars proportionis circuli uperficiei terr, qu enu recta ditin<lb/>gui non poterit. </s>
<s id="id2711606">Hoc ergo et primum, ex quo equitur.</s></p><p type="main">
<s id="id2711620"><arrow.to.target n="marg313"/></s></p><p type="margin">
<s id="id2711631"><margin.target id="marg313"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 2.</s></p><p type="main">
<s id="id2711658">Quod aliquid poterit uideri decliue, in quo non decendet im <lb/>erit, ut pot i aliqua linea obliqua eet inter c e, & f e, illa eet decli<lb/>uis pecie, & re, & tamen graue in illa non decenderet, quia cen<lb/>tro mundi magis remoueretur: hoc tamen et perdifficile factu, & <lb/>maxim in parua ditantia, uel etiam unius miliaris. </s>
<s id="id2711717">Atque hc <lb/>in leuigatis.</s></p><p type="main">
<s id="id2711731">Propoitio nonageimaecunda.</s></p><p type="main">
<s id="id2711747">Propprtionem ponderis qualis iuxta longitu dinis compara<lb/>tionem demontrare.</s></p><figure id="id.015.01.104.2.jpg" xlink:href="015/01/104/2.jpg"/><p type="main">
<s id="id2711776">Hoc et, quod Archimedes reliquit </s></p><p type="main">
<s id="id2711787"><arrow.to.target n="marg314"/><lb/>intactum, cum eet maxim necea<lb/>rium, & otendit magis abtrua, ed <lb/>pace illius dixerim minus utilia. </s>
<s id="id2711826">Cum <lb/>ergo umpiem uirgam b f ponderis <lb/>unciarum xxiij, fuiet b a uigeimaquarta pars, b f fuit pondus <lb/>quilibrij in b appenum librarum uigintiex cum dimidia: fuit igi<lb/>tur proportio ponderis e f ad pondus f b, ut tredecim ferme ad
<pb xlink:href="015/01/105.jpg" pagenum="86"/>unum. </s>
<s id="id2711879">Et rurus feci a b quintam partem a f, & fuit a b unciarum <lb/>quatuor, & pondus quod quauit librarum quatuor, ide du<lb/>plum ad pondus b f, icut c f ad c b: contat enim qud pondus ap<lb/>penum et quale ponderi cf. </s>
<s id="id2711922">Et rurus poui b a quartam partem <lb/>b f, & fuit pondus, quod quauit in b du libr: ex quo manife<lb/>tum et, qud proportio c f ad c b et emper uelut ponderis c f ad <lb/>totam b f. </s>
<s id="id2711965">Et hoc et, ac i dicamus, qud proportio ponderis c f ad <lb/>totam et confua ex proportione e f ad c b, & c f, quod et 1 p. </s>
<s id="id2711990">Id <lb/><arrow.to.target n="marg315"/><lb/>etiam declaratum et in primo de Subtilitate. </s>
<s id="id2712008">Proponatur ergo <lb/>lemma, iam ic proportio ponderis cf ad pondus b c, et primum <lb/>ut longitu dinis cf, i eet upena in medio ad longitudinem b c, <lb/>quia upponuntur proportione imiles uis longitudinibus ma<lb/>gnitudines, & pondera. </s>
<s id="id2712055">At c f upena in c, tanto et grauior pon<lb/>dere proprio, quanto proportionis longitudinis cf ad cb quadra<lb/>tum, quia in e ducitur proportio: igitur proportio ponderis c f in <lb/>loco uo ad b c pondus et confua ex proportione longitudinis <lb/>cf ad c b, & quadratis eiudem proportionis longitudinis cf ad c <lb/>b. </s>
<s id="id2712106">Sed quadratum proportionis longitudinis cf ad cb et quale <lb/>producto proportionis longitudinis c f in ipam c f, propterea <lb/>qud ex proportione longitudinis cf ad cb in ipam c b fit c f, igi<lb/>tur proportio ponderis c f ad pondus c b et confua ex propor<lb/>tione ponderis c f ad pondus c b, & proportione ponderis cf alicu <lb/>ius e habentis ad pondus cf, ut cf longitudo ad longitudinem <lb/>c b, igitur proportio ponderis cf ad pondus b f, ut cf ad c b in lon<lb/>gitudine, quod erat probandum.</s></p><p type="margin">
<s id="id2712168"><margin.target id="marg314"/>C<emph type="italics"/>om.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2712192"><margin.target id="marg315"/>E<emph type="italics"/>x<emph.end type="italics"/> 18. <emph type="italics"/>diff.<emph.end type="italics"/></s></p><p type="main">
<s id="id2712226">Propoitio nonageimatertia.</s></p><p type="main">
<s id="id2712239">Propter quid in concusione etiam leui nauis loco moueatur <lb/>otendere. </s>
<s id="id2712252">Vnde manifetum et, duas naues ibi inuicem occuran <lb/>tes retrocedere, & quantum retrocedant amb.<lb/><arrow.to.target n="marg316"/></s></p><p type="margin">
<s id="id2712286"><margin.target id="marg316"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2712312">Proponatur, quod proportio motus grauis in a d graue in aqua <lb/>it, uelut proportio ponderis attracti in terra ad denitatem aqu <lb/>cum profunditate, nam ubi pondus upernataret aqu, quia aqua <lb/>et rotunda, et ac i tangeret in puncto. </s>
<s id="id2712348">Quare per demontrata u<lb/>peris mouebitur quacunque ui, ergo nixus contrarius aduenit ob </s></p><p type="main">
<s id="id2712373"><arrow.to.target n="marg317"/><lb/>profunditatem, & aqu denitatem, ed quanto aqua denior et, <lb/>tanto minus nauis decendit, & quanto minus dena, tanto magis: <lb/>ergo pari modo ferm redduntur mobiles, & in aqua dulci & ala, <lb/>ubi naues int imiles forma, pondere, magnitudine. </s>
<s id="id2712428">Quia crgo ne<lb/>cee et tabulam nauis ee duriorem, quam aqua ad reitendum, <lb/>ergo pars maior ictus mouebit primo nauim, quam tabulam pe<lb/>netret, cum ergo quod facilius et, prcedat, difficilius ergo naues
<pb xlink:href="015/01/106.jpg" pagenum="87"/>utrinque mouebuntur, & quia inter duos quocunque motus contra<lb/>rios <expan abbr="nõ">non</expan> eeos, ut utar uocabulo Auerrois quinto Phyicorum, ne<lb/>cee et, ut intercedat quies media, & in quiete ab ictu, ut uium et <lb/>uperius, oportet, ut quod excipit ictum uelloco moueatur, uel ce<lb/><arrow.to.target n="marg318"/><lb/>dat, & ictus penetret, uel ar non condenetur ob tarditatem ultra <lb/>metam, nec retro cedere potet ex uppoito, & ictus et magnus, <lb/>clarum et, quod oportet, ut cedat, & i durum it confringatur. <lb/></s>
<s id="id2712568">Proportio ergo receus ad ictum et ut temporis, & magnitudinis <lb/>partis, qu cedit, & retro ceus poito ictu tanquam monade.</s></p><p type="margin">
<s id="id2712596"><margin.target id="marg317"/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 40.</s></p><p type="margin">
<s id="id2712625"><margin.target id="marg318"/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 74.</s></p><p type="main">
<s id="id2712652">Propoitio nonageimaquarta.</s></p><p type="main">
<s id="id2712665">Si quantitas aliqua nota atque proportio erit producta quantitas <lb/>nota imiliter. </s>
<s id="id2712676">Et i du proportiones not fuerint, erit producta <lb/>ex his atque diuia, coniunctaque, atque detracta nota. </s>
<s id="id2712695">Et i fuerit totius <lb/>ad partem proportio nota erit, & ad aliam partem nota, & alterius <lb/>partis ad alteram uno minor. </s>
<s id="id2712711">Et i fuerit partis ad partem, erit ad to <lb/>tum monade minor atque nota. </s>
<s id="id2712722">Et i fuerit unius quantitatis ad duas <lb/>quantitates proportio nota, erit & confua ex eis nota. </s>
<s id="id2712736">Et i fuerint <lb/>trium quantitatum omiologarum, aut quatuor analogarum, o<lb/>mnes prter unam cognit erunt, & illa alia cognita.</s></p><figure id="id.015.01.106.1.jpg" xlink:href="015/01/106/1.jpg"/><p type="main">
<s id="id2712772">Sit quantitas a b & ducta in d proportionem, <lb/><arrow.to.target n="marg319"/><lb/>producat b c: dico quod duobus quibuslibet ex <lb/>his cognitis, erit cognitum tertium: nam cogni<lb/>tum quodlibet dicitur in comparatione ad impliciter cognitum, <lb/>quod et unum per e omnibus cognitum. </s>
<s id="id2712806">Ob id Arithmetica et <lb/>prima omnium diciplinarum, quia habet principium cognitum, <lb/>& id, quod et, ad principium comparatum cognitum in illius com <lb/>paratione: neque aliter cognitum dici potet. </s>
<s id="id2712833">Quia ergo d cognita <lb/>et, erunt monades, & partes cognit in ea: aliter non eet cognita <lb/>b a, igitur cum cognita it, erit cognita per ingulas monades, quan <lb/>ta it. </s>
<s id="id2712865">Et i diceres qud b a non et cognita per partem monadis: <lb/>dico quod pars monadis non et incognita, quia cum monades <lb/>unt cognit, eet d incognita. </s>
<s id="id2712896">Omnes enim, quod componitur ex <lb/>cognito & incognito, et incognitum, quia cognitum olum ratio<lb/>ne partis cognit. </s>
<s id="id2712918">Si ergo pars monadis et cognita, erit pars a b <lb/>qulibet prout ex monade componitur impliciter cognita. </s>
<s id="id2712934">Su<lb/><arrow.to.target n="marg320"/><lb/>peret, ut olum pars partis: & dico quod illa etiam et cognita: <lb/>quia i pars ab eet, monas eet cognita: eet enim pars ipa.</s></p><p type="margin">
<s id="id2712984"><margin.target id="marg319"/>C<emph type="italics"/>om.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2713010"><margin.target id="marg320"/>E<emph type="italics"/>x ecunda <lb/>animi com<lb/>muni enter <lb/>tia.<emph.end type="italics"/></s></p><p type="main">
<s id="id2713047">Sed i it pars, erit umpta ecundum partem monadis ipius, <lb/>ide erit cognita iuxta nomen, uelut dimidium et dimidium mo<lb/>nadis, dimi dium terti partis monadis et cognitum, quia tertia <lb/>pars et cognita, & cimus, quanta pars aumatur illius. </s>
<s id="id2713098">Ergo i a b,
<pb xlink:href="015/01/107.jpg" pagenum="88"/>& d cognit unt erit & b c, quod et primum. </s>
<s id="id2713122">Per hc eadem pro<lb/>bantur quatuor equentes partes eodem modo. </s>
<s id="id2713137">Sexta ic: it pro<lb/>portio a c ad c b, nota igitur in comparatione ad monadem, ed pro <lb/>portio a c ad c b b a et monas, igitur proportio a c ad a b nota et, <lb/>quoniam aliter non poet dici proportio a c ad b c nota. </s>
<s id="id2713171">Aliter, it <lb/>proportio a c ad c b e nota, ex uppoito igitur conuera nota qu <lb/>it f ex f, igitur in a c fit b c ex g in a c, fiat a b ergo ex a c in f g fit a, cigi <lb/>tur f g et monas, f autem nota et, igitur in comparatione ad mona<lb/><arrow.to.target n="marg321"/><lb/>dem, ergo reiduum g notum. </s>
<s id="id2713220">Cum uer proportio a c ad c b com<lb/>ponatur ex proportione a b b c ad b c, & proportio b c ad b c it <lb/>monas, & proportio a c ad b c nota, erit proportio a b ad b c cogni <lb/><arrow.to.target n="marg322"/><lb/>ta, & monade minor proportione a c ad b c. </s>
<s id="id2713253">Per idem octaua pars <lb/><figure id="id.015.01.107.1.jpg" xlink:href="015/01/107/1.jpg"/><lb/>demontrabitur. </s>
<s id="id2713273">Inde it proportio a ad b, & ad c no<lb/>ta, erit ergo b, & c ad a nota, quare b c ad a nota, ed <lb/><arrow.to.target n="marg323"/><lb/>hc et conuera ad b c confua, igitur proportio a <lb/>ad b confua nota et. </s>
<s id="id2713318">Vltimum it, int a b c omiolog, & int a & b <lb/><arrow.to.target n="marg324"/><lb/>not duo, quod c nota et, nam a b, i not unt, nota et proportio <lb/>earum. </s>
<s id="id2713362">Ergo & proportio b ad c ergo per primam partem huius <lb/><arrow.to.target n="marg325"/><lb/>cum it b nota, exit & c. </s>
<s id="id2713382">Et i ponantur a c not, dico, qud b nota <lb/>erit: nam proportio a c ad c nota et, qu it d, igitur d ad monadem <lb/>ut a ad c, ergo latus notum erit, quod ductum in c producit b, b igi<lb/><arrow.to.target n="marg326"/><lb/>tur nota. </s>
<s id="id2713420">Et imiliter in analogis int a b c not: & ide erit propor<lb/>tio a ad b nota ergo c ad d. </s>
<s id="id2713442">cumque c nota it, ergo per primam par<lb/>tem huius erit d nota, quod fuit demontrandum.</s></p><p type="margin">
<s id="id2713460"><margin.target id="marg321"/>P<emph type="italics"/>er demon<lb/>trat.<emph.end type="italics"/> 12. <lb/>P<emph type="italics"/>ropo.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2713508"><margin.target id="marg322"/>P<emph type="italics"/>er<emph.end type="italics"/> 11. P<emph type="italics"/>et.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2713544"><margin.target id="marg323"/>E<emph type="italics"/>x demont.<emph.end type="italics"/><lb/>12. P<emph type="italics"/>ropo.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2713586"><margin.target id="marg324"/>P<emph type="italics"/>er<emph.end type="italics"/> 14. <lb/>P<emph type="italics"/>ropo.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2713627"><margin.target id="marg325"/>P<emph type="italics"/>er<emph.end type="italics"/> 3. P<emph type="italics"/>etit.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2713663"><margin.target id="marg326"/>E<emph type="italics"/>x<emph.end type="italics"/> 2. A<emph type="italics"/>nimi <lb/>ententia.<emph.end type="italics"/></s></p><p type="main">
<s id="id2713701">Propoitio nonageimaquinta.</s></p><p type="main">
<s id="id2713714">Cuiuuis trigoni rectanguli, aut cuius duo anguli int in dupla <lb/>proportione, aut qui circulo incriptus it cognita quantitate uni<lb/>us lateris in comparatione ad dimetientem i proportio <expan abbr="duorũ">duorum</expan> la<lb/>terum cognita fuerit, erunt omnia eius latera cognita.<lb/><arrow.to.target n="marg327"/></s></p><p type="margin">
<s id="id2713765"><margin.target id="marg327"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2713791">Non de cognitione propinqua <expan abbr="a&longs;tronomorũ">atronomorum</expan>, de qua abund ab <lb/>Heber tractatum et, ed de exacta, de qua uperius egi nunc ermo </s></p><p type="main">
<s id="id2713830"><arrow.to.target n="marg328"/><lb/>et: it igitur primum a b c trigonus orthogonius: & it a rectus, & <lb/>proportio <expan abbr="duorũ">duorum</expan> laterum cognita, dico, quod omnia latera cognita <lb/><arrow.to.target n="marg329"/><lb/><figure id="id.015.01.107.2.jpg" xlink:href="015/01/107/2.jpg"/><lb/>erunt: nam it proportio, gratia exempli, <lb/>a b ad b c, erit ergo quadrati a b ad qua<lb/>dratum b c cognita, quia duplicata: at <lb/>quadrata a b, & a c perficiunt quadratum <lb/>b c, igitur proportio quadrati a b ad a c et <lb/>et 1 p: cognita erit, quare & a b ad a c, & <expan abbr="eod&etilde;">eodem</expan> modo a c ad b c: quod <lb/>et primum. </s>
<s id="id2713920">Exemplum, ponatur b c dupla a b, erit a b quadratum <lb/>ub quadruplum quadrato a b quare ubtriplum quadrato a cigi
<pb xlink:href="015/01/108.jpg" pagenum="89"/>tur i a b ponatur 1 b c erit 2, & a c <02> 3. Rurus ponatur angulus b <lb/>duplus angulo c qualicunque it, erit per demontrata uperius pro<lb/>portio a b b c ad a c, ut a c ad a b, i igitur nota it proportio a c ad <lb/>a b, erit nota proportio a b b c ad a b per prcedentem. </s>
<s id="id2713982">Ergo per <lb/>eandem omnia nota cilicet b c ad b a, & b c ad c a. </s>
<s id="id2713993">Et i eet nota <lb/>proportio a b ad b c, dico, quod eent nota omnia, nam nota eet <lb/>a b, & b c, & quod fit ex a b in ipum aggregatum. </s>
<s id="id2714022">Sed hoc et <lb/><arrow.to.target n="marg330"/><lb/>quale quadrato a c, igitur notum et quadratum a c ergo a c: igitur <lb/>proportio a b b c ad a c, & a c ad a b. </s>
<s id="id2714048">Vt i a b eet 4 b c 5, eet a b b c <lb/>9 ducta in a b, qu et, fit 36, cuius latus et b a c cilicet. </s>
<s id="id2714078">Et i eet <lb/>trigonus aliquis in cir culo, cuius proportio duorum laterum it co <lb/>gnita ad dimetientem relata, equitur per demontrata upe<lb/>rius, quod etiam tertium latus erit cognitum in comparatione ad <lb/>eadem, & ideo etiam proportio illorum laterum ad unguem co<lb/>gnita erit.</s></p><p type="margin">
<s id="id2714126"><margin.target id="marg328"/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 97.</s></p><p type="margin">
<s id="id2714154"><margin.target id="marg329"/>P<emph type="italics"/>er<emph.end type="italics"/> 47. <emph type="italics"/>pri <lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2714203"><margin.target id="marg330"/>P<emph type="italics"/>er<emph.end type="italics"/> 17. <emph type="italics"/>ex <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/><lb/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 17.</s></p><p type="main">
<s id="id2714268">Multa prterea cognita eent in hoc genere, qu nunc prter<lb/><arrow.to.target n="marg331"/><lb/>mitto, quia non unt ad finem necearia. </s>
<s id="id2714301">Alia prterea per diligen<lb/>tem inquiitionem maioris artis qum alias edidimus. </s>
<s id="id2714319">tum uer <lb/>etiam per nouas demontrationes.</s></p><p type="margin">
<s id="id2714336"><margin.target id="marg331"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2714362">Propoitio nonageimaexta.</s></p><p type="main">
<s id="id2714378">Cum in perpicuum denum radij luminoi in ciderint, quatuor <lb/>fiunt luminis genera.<lb/><arrow.to.target n="marg332"/></s></p><p type="margin">
<s id="id2714405"><margin.target id="marg332"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2714431">Sit ol a, & perpicuum denum, exempli gratia, ut ampula <lb/>magna aqua plena b c d, & i it rotunda accendit ignem ex ad<lb/>uero ut in e. </s>
<s id="id2714460">Dico ergo in b c d ee quatuor genera luminis. </s>
<s id="id2714468">Pri<lb/>mum quod et ualidius, & rect tranit, ualidius enim et, quod <lb/>tranit qum quod tranire non potet, & etiam quia, ut dixi, <lb/>ignem accen dit. </s>
<s id="id2714506">Secundum et quod colligitur in ampula, & dein<lb/>de pargitur <expan abbr="circũcircà">circuncirc</expan>, nam id ualidius et, quia penetrat, & reilit <lb/>qum quod non penetrat, aut i penetrat, non pargitur, & hoc dif<lb/>funditur circa uas, necreflectitur rect, ed quai intro colligitur, & <lb/>diuera ratione diffunditur, et tamen imbecillius primo, ut dictum <lb/>et. </s>
<s id="id2714580">Tertium genus et, quod illuminat intus ingrediendo, ed non <lb/>pargitur, & hoc et debilius ecundo, quia <expan abbr="nõ">non</expan> potet pargi. </s>
<s id="id2714614">Quar<lb/><figure id="id.015.01.108.1.jpg" xlink:href="015/01/108/1.jpg"/><lb/>tum et, quod non ingreditur omnino, ed refle<lb/>ctitur, itud et abque dubio imbecillimum, quo<lb/>niam penetrare non potet. </s>
<s id="id2714659">Et licet in peculis <lb/>concauis radius reflexus uideatur ee ualidior, <lb/>tatim enim accendit ignem, hoc non contin<lb/>git, nii quia in peculo cauo radij omnes col
<pb xlink:href="015/01/109.jpg" pagenum="90"/><expan abbr="ligun&ttilde;">liguntur</expan> ob <expan abbr="opacũ">opacum</expan>, quod tergo et, neque <expan abbr="&longs;pargun&ttilde;">parguntur</expan>, neque <expan abbr="tran&longs;eũt">traneunt</expan>, neque<lb/>combibuntur, ut ita dicam ed omnes <expan abbr="reflectũtur">reflectuntur</expan>. </s>
<s id="id2714762">Ex quo colligitur <lb/>quin cuplex ordo radiorum iuxta rationem uirium, primus et refle <lb/><expan abbr="xorũ">xorum</expan> peculo <expan abbr="cõcauo">concauo</expan>, & hi unt <expan abbr="pot&etilde;ti&longs;simi">potentisimi</expan> ob <expan abbr="ration&etilde;">rationem</expan> <expan abbr="dictã">dictam</expan>, pot <lb/>quos unt radij, qui traneunt per perpicuum maxim rotundum, <lb/>qui & ipi generant ignem, & debiliorem primo, deinde reliqui <lb/>tres equentes upradicti. </s>
<s id="id2714866">Sextus et radiorum, qui reflectuntur <lb/>rebus non nitidis, ut muris, & tabulis, nam omnia dura reflectunt <lb/>& etiam mollium pleraque, & hc reflexio et ferm infinita, & ob id <lb/>cubicula etiam in angulis illuminantur.</s></p><p type="main">
<s id="id2714904"><arrow.to.target n="marg333"/></s></p><p type="margin">
<s id="id2714915"><margin.target id="marg333"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 1.</s></p><p type="main">
<s id="id2714941">Ex hoc equitur, qud Luna remittit lumen, non reflectit, nam <lb/>ecus non illuminaret to tum orbem, ed olum portionem oppo<lb/>itam Soli, & hoc etiam rar, ergo combibitur, & illutrat circun<lb/>circa ubique.</s></p><p type="main">
<s id="id2714983"><arrow.to.target n="marg334"/></s></p><p type="margin">
<s id="id2714994"><margin.target id="marg334"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 2.</s></p><p type="main">
<s id="id2715019">In tellis lumen Solis pertranit aliter, i reflecteretur, non illumi<lb/>naret nos, aut apparerent, uelut comet, quia pars una eet clarior <lb/>reliqua, & i conbiberent lumen, non uiderentur qu clar, cum <lb/>Sol eet propinquus, aut remotus.</s></p><p type="main">
<s id="id2715070"><arrow.to.target n="marg335"/></s></p><p type="margin">
<s id="id2715081"><margin.target id="marg335"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 3.</s></p><p type="main">
<s id="id2715108">Luna tota intus illuminatur Sole, quoniam i ante coniun<lb/>ctionem illuminatur initra parte, & combibit lumen per cor<lb/>rolarium primum, & pot coniunctionem illuminatur dex<lb/>tra, & combibit pariter lumen, ergo et tota natur perpicu, ed <lb/>uidetur obcura ex aduero, propterea qud radij ualidiores refle<lb/>xi illutrant illam ex parte Solis, diffugiunt contraria, quod ma<lb/>nifet apparet in ampula expoita Soli. </s>
<s id="id2715199">Pars enim clarior uerus <lb/>Solem uidetur, quam ex aduero, hoc autem long magis in Luna <lb/>ob ditantiam.</s></p><p type="main">
<s id="id2715225"><arrow.to.target n="marg336"/></s></p><p type="margin">
<s id="id2715236"><margin.target id="marg336"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 4.</s></p><p type="main">
<s id="id2715263">In omni Solis eclipi fit colectio radiorum ad apectum, & <lb/>ideo in regione illa, in qua centrum Solis integitur centro Lun, <lb/>& ubicunque fit, fit in cendium per tertium corrolarium. </s>
<s id="id2715286">Hoc autem <lb/>fit emper in quauis coniunctione, & dum Luna ilet in regione ae<lb/>ris, ed terris non e cundm centrum, uerm ad latitudinem, & ad <lb/>Orientem ante coniunctionem cum Sole, & ad Occidentem pot: <lb/>ed centra non unt in linea uius.</s></p><p type="main">
<s id="id2715337"><arrow.to.target n="marg337"/></s></p><p type="margin">
<s id="id2715349"><margin.target id="marg337"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 5.</s></p><p type="main">
<s id="id2715376">Ex hoc equitur, quod oportet ubtantiam Lun ee ualde cla<lb/>ram, cum uideamus ab ampula tam paruum lumen diffundi, & ra<lb/>rum, Luna uer in uniuerum orbem, & tam copioum, ut nece<lb/>arium it ubtantiam Lun ee denam, & lucidam ualde.</s></p><p type="head">
<s id="id2715446">SCHOLIVM.</s></p><p type="main">
<s id="id2715456">Et i quis dicat, qud i in cendium illud fieri poet in hora ecli<lb/>pis, equeretur, qud ut in ampula in medio Lun uideretur ma
<pb xlink:href="015/01/110.jpg" pagenum="91"/>gnus plendor, referens corpus Solis. </s>
<s id="id2715505">Propterea dico, qud uel ac<lb/>eidit, quia homo non potet ea hora intueri Solem, & etiam et im<lb/>peditus radijs circumtantibus, cuius indicio et, quod in pe<lb/>culo poito in aqua, imile uidetur tellul in centro Lun: & hic et <lb/>plen dor Solis collectus in centro Lun. </s>
<s id="id2715569">poet etiam dici, qud <lb/>Luna circa medium propter maculam non admitteret lumen, & ita <lb/>eet inqualium partium.</s></p><p type="main">
<s id="id2715599">Propoitio nonageimaeptima.</s></p><p type="main">
<s id="id2715615">Motum inuerionis in figuris in comparatione ad motum ph <lb/>r in plano inuetigare.</s></p><p type="main">
<s id="id2715641"><arrow.to.target n="marg338"/></s></p><p type="margin">
<s id="id2715652"><margin.target id="marg338"/>C<emph type="italics"/>om.<emph.end type="italics"/></s></p><p type="main">
<s id="id2715678">Voco motum inuerionis, qui imilis et motui phr, cili<lb/>cet circumuertendo graue uertice, & manifetum et, qud in <lb/>quacunque figura, qua graue inidet plano per punctum ue</s></p><p type="main">
<s id="id2715728"><arrow.to.target n="marg339"/><lb/>lut ouata ipum mouetur quauis ui, ed i inideat per uperfi<lb/>ciem, quanto maior et, & humilior, tanto difficilius mouetur, <lb/>ide in corpore uiginti baium, qud inter regularia uocata, plu<lb/>res habet, uperficies pro ratione qualis ponderis, motus erit <lb/>longe facilior. </s>
<s id="id2715789">Alia caua et inqualitas partium, unde qu ro<lb/>tunda unt, quia prominent, facile mouentur, & cum partes me<lb/>di initant plano, quanto minores erunt tanto facilius moue<lb/>buntur ratione ponderis. </s>
<s id="id2715831">Vnde patet, qud corpora ouata faci<lb/>lius mouentur, etiam qum phrica, habent enim partem me<lb/>diam minorem, & paria unt ratione inceus plani, ed aris mul<lb/>titudine tardius, quoniam enim phra ub quali ambitu plus <lb/>continet corporis, ergo ouatum quale phr habet maio<lb/>rem ambitum ipa phra. </s>
<s id="id2715911">Hc autem Theone partim de<lb/>montrata unt, partim ab Archimede, & partim nobis, ergo <lb/>motus ouati et ferm qualis motui phr, & tardior et con<lb/><figure id="id.015.01.110.1.jpg" xlink:href="015/01/110/1.jpg"/><lb/>citatus, qum phr, quia ma<lb/>iore excipitur are, & partes exte<lb/>riores non ita incumbunt in me<lb/>dium ecundum longitudinem. </s>
<s id="id2716006">Cu<lb/>bus uero tardior et propter qua<lb/>litatem, & latitudinem uperficiei in<lb/>ferioris, omnium <expan abbr="aut&etilde;">autem</expan> minime pro<lb/>pter has cauas conus ambligonius, <lb/>& quanto magis fuerit, ratio uero <lb/>eleuationis et, ut it cubus b c, cuius <lb/>medium grauitatis it b uper pla
<pb xlink:href="015/01/111.jpg" pagenum="92"/>no de, & eleuetur ex a, & manifetum et, quod inidebit per totam <lb/>lineam c f ipi plano, & proportio grauitatis totius upeni in com <lb/>paratione ad grauitatem eius, qui inuertit, et, uelut proportio par<lb/>tis terminat ad lineam c f uerus eum, qui eleuat ad partem, qu <lb/>ultra et, cum uer h partes not int iuxta perpendiculum ex <lb/>centro grauitatis, manifetum et, quod ciemus pondus corporis <lb/>a b cf, dum inuertitur in quo cunque itu ad pondus eius, dum u<lb/>penditur, & clarum et, qud cm centrum, & medium grauitatis <lb/>fuerint in una linea per c f, tunc nulla erit grauitas.</s></p><p type="margin">
<s id="id2716188"><margin.target id="marg339"/>P<emph type="italics"/>er<emph.end type="italics"/> 40.</s></p><p type="main">
<s id="id2716212">Propoitio nonageimaoctaua.</s></p><p type="main">
<s id="id2716225">Proportionem ponderum qualium per differentiam angulo<lb/>rum inuenire.<lb/><arrow.to.target n="marg340"/></s></p><p type="margin">
<s id="id2716248"><margin.target id="marg340"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2716274">Sit a b, qu i appena eet ad quidi<lb/><figure id="id.015.01.111.1.jpg" xlink:href="015/01/111/1.jpg"/><lb/>tantem terr uperficiei, nulla ui poet ele </s></p><p type="main">
<s id="id2716322"><arrow.to.target n="marg341"/><lb/>uari, inflectatur ergo ad c punctum, omia <lb/>c g, & manifetum et, quod i b c initeret <lb/><arrow.to.target n="marg342"/><lb/>ad perpendiculum, ponderaret a c i eet in <lb/>quilibrio, ponatur ergo accliuis in c d per <lb/>notum angulum. </s>
<s id="id2716377">Quia igitur b c ad c a no<lb/>ta et, erit dicta uperis notum pondus <lb/>b h, poita h c quali c a, quare totius a b, <lb/>& iam fuit e k notum, & punctus d notus: <lb/>hoc enim infr demontrabitur, qualis igitur proportio line <lb/><arrow.to.target n="marg343"/><lb/>tranuer dl ad lineam decendentem d m, talis differenti pon<lb/>derum c m, & c e, id et partis ad partem. </s>
<s id="id2716448">hc autem inferis de<lb/>montrabuntur. </s>
<s id="id2716464">Neque enim aburdum et in materijs mitis, ali<lb/><arrow.to.target n="marg344"/><lb/>quando uti nondum demontratis cum fuerint mathematica, quia <lb/>obtinent principij rationem, quod etiam facit Archimedes. </s>
<s id="id2716494">Ma<lb/>nifetum et autem, quod in angulo m c d recti dimidio, propor<lb/>tio media erit. </s>
<s id="id2716513">Sed hoc bifariam contingere potet cilicet, ut it <lb/>media, per quantitatem, & per proportionem, et autem media, ut <lb/><arrow.to.target n="marg345"/><lb/>demontrabitur infr ecundum proportionem l d ad l e, propo<lb/>natur ergo c e b, erit latus quadrati <02> 72, igitur latus octogoni et <lb/><02> v: 72 m: <02> 2592, & latus reidui <02> v: 72 p: <02> 2592. quadrata er<lb/>go partium bais differunt in <02> 10368. Quare partes bais unt <lb/>6 p: <02> 18, & 6 m: <02> 18 cilicet l e, l d autem et <02> 18, igitur differen<lb/>tia, & proportio et, qualis <02> 18 ad 6 m: <02> 18 ferm, ut 17 ad 7, & ta<lb/>lis et proportio ponderis c d ad pondus c e ratione in crementi, <lb/>eu differenti. </s>
<s id="id2716619">Vt i pondus in c e eet decem librarum in c in
<pb xlink:href="015/01/112.jpg" pagenum="93"/>quadraginta erit in c d triginta unius cum quarta, ed proportionis <lb/>ratione eet uiginti octo cum tertia.</s></p><p type="margin">
<s id="id2716654"><margin.target id="marg341"/>P<emph type="italics"/>er<emph.end type="italics"/> C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 2. <lb/>45. P<emph type="italics"/>ropo.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2716707"><margin.target id="marg342"/>P<emph type="italics"/>er<emph.end type="italics"/> 86. <lb/>P<emph type="italics"/>ropo.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2716748"><margin.target id="marg343"/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 99.</s></p><p type="margin">
<s id="id2716775"><margin.target id="marg344"/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 97.</s></p><p type="margin">
<s id="id2716802"><margin.target id="marg345"/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 98.</s></p><p type="main">
<s id="id2716830">Propoitio nonageimanona.</s></p><p type="main">
<s id="id2716843">Proportionem grauitatum per multitudinem uppoitorum or <lb/>bium otendere.<lb/><arrow.to.target n="marg346"/></s></p><p type="margin">
<s id="id2716869"><margin.target id="marg346"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2716895">Omne, quod mouetur, mouetur ecundum naturam ponderis, <lb/>qu in attractione, ut demontratum et, qualis et dimidio u<lb/>peni, cum ergo diuidatur in multiplices partes motus uniucuiu<lb/>que, et ecundum dimidium illius partis, ut, i int ex rot in cur<lb/>ru det, quod uehitur, it pondus exaginta librarum, unaqu que </s></p><p type="main">
<s id="id2716974"><arrow.to.target n="marg347"/><lb/>rota habet pondus quinque librarum, cilicet diuio triginta per <lb/>ex, & quia quod cunque mouetur phric non habet pondus, <lb/>nii quantum premitur axis, ide pondus exaginta librarum in <lb/>uehendo red ditur lus, quanto proportio producta minor et <lb/>additione. </s>
<s id="id2717029">Exemplum, it deductum pondus exaginta librarum <lb/>per ex rotas ad uigintiquatuor, quia i rot poent circumduci, <lb/>ut in inuerione dictum et, & eent quales, & in olido quali, <lb/>ac duro, nulla ui mouerentur, ed quai per e, ergo uppoito pon<lb/>dere uiginti quatuor librarum aumemus unamquamque partem, <lb/>& ducemus eam in eipam, cilicet detraham quintam partem ex <lb/>toto 30, fit 24, duc 30 in e, fit 900, duc 24 in e, fit 576, proportio ut <lb/>25 ad 16, at diuio 30 in ex partes, fit 5, detrahe quintam partem, fit <lb/>4, duc in e, fit 16, duc in ex, fit 96, igitur proportio 900 ad 96 et ut <lb/>25 ad 2 2/3, quod ergo erat 16 factum et 2 2/3, proportio ergo de<lb/>crecentis maior et diuio per plura. </s>
<s id="id2717164">Sed plerunque additis ro<lb/>tis crecit pondus nihilo ecius, redditur etiam leuius. </s>
<s id="id2717178">Sed & de <lb/>hoc in equenti.</s></p><p type="margin">
<s id="id2717192"><margin.target id="marg347"/>P<emph type="italics"/>er<emph.end type="italics"/> 40.</s></p><p type="main">
<s id="id2717218">Propoitio centeima.</s></p><p type="main">
<s id="id2717231">Proportionem grauitatis ponderum attractorum per trochlea<lb/>rum numerum inuetigare.<lb/><arrow.to.target n="marg348"/></s></p><p type="margin">
<s id="id2717253"><margin.target id="marg348"/>C<emph type="italics"/>om.<emph.end type="italics"/></s></p><p type="main">
<s id="id2717278">Aritoteles in Mechanicis cenet cauam leuitatis trochlearum </s></p><p type="main">
<s id="id2717294"><arrow.to.target n="marg349"/><lb/>ee in pondere eleuando, qud pondera auxilio uectium facilius <lb/>mouentur, qum manibus. </s>
<s id="id2717318">Rotul uer in trochleis uectes unt, <lb/>& axis mita hypomochlij, ergo facilius pondus trahitur per u<lb/>nam rotulam, qum i manu traheretur, at uer per duas tres, <lb/>unde tris paus longe facilius, & etiam facilius per quinque, unde <lb/>pentas paus, nam quinque orbiculis, quai totidem uectibus <lb/>diuium pondus manifet fit leuius, & ut dictum et, tanquam <lb/>totidem uectibus pondus eleuatur, etqe proportio produ
<pb xlink:href="015/01/113.jpg" pagenum="94"/>cta, emperque prior hypomochlij locum habet, ueruntamen minus <lb/>aumit laboris, poterior uer uectis maiorem partem ibi ponde<lb/>ris eruat, uelut in uccula etiam iugum traiectum per plures colo<lb/>pes facilius uertitur. </s>
<s id="id2717439">Et i quis dicat nnne totum pondus inidet <lb/>prim trochle per trochleam, intelligo nunc olm rotulam cum <lb/>ipo axe, eu axiculo (ut dicunt) non autem in proprio ignificato, <lb/>in quo etiam funis traiectus, & inidens rotul, eu rotulis, nam <lb/>una trochlea plures continere'potet orbiculos, & axes. </s>
<s id="id2717497">Licet ergo <lb/>pondus inideat prim trochle, eu rotul, in eo tamen, quod tra <lb/>hitur, diuiditur', licet non qualiter dico, prter id funis motum <lb/>intendi. </s>
<s id="id2717530">nam motus actionem auget, & ide quanto longior, eo fa<lb/>cilius mouet ob con cusionem, demum quia leuis et rotula circa <lb/>axem, ut plus uecte posit.</s></p><p type="margin">
<s id="id2717558"><margin.target id="marg349"/>I<emph type="italics"/>n<emph.end type="italics"/> M<emph type="italics"/>echan.<emph.end type="italics"/><lb/>Q<emph type="italics"/>ut.<emph.end type="italics"/> 18.</s></p><p type="main">
<s id="id2717611">Propoitio centeimaprima.</s></p><p type="main">
<s id="id2717624">Proportionem precij gemmarum ex tribus in eodem genere co <lb/>gnitis inuenire.<lb/><arrow.to.target n="marg350"/></s></p><p type="margin">
<s id="id2717643"><margin.target id="marg350"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2717669">Solent gemmarij uendere adamantem ponderis unius grani <lb/>uno coronato, duorum autem granorum tribus coronatis, qua<lb/>tuor autem, gratia exempli, quadraginta coronatis, quritur quan<lb/>tum ualebit adamas octo granorum, quoniam ergo proportio <lb/>non eruatur. </s>
<s id="id2717695">Et enim in pondere utraque dupla, in precio autem <lb/>ex prima habetur tripla, ex ecunda habetur proportio maior, <lb/>qum tredecim ad unum, propterea utendum et proportione <lb/>propinquiori, i atis faceret. </s>
<s id="id2717727">gratia exempli, in prima ad ditione fuit <lb/>unum granum, & acquiiuit proportionem triplam, in ecunda fue <lb/>runt duo grana, i ergo acquiiet olum excuplam proportio<lb/>nem, haberemus intentum. </s>
<s id="id2717764">Propterea in ito cau oportet demon<lb/>trare forma Geometrica, uppoito, qud it figura recta ex uno la <lb/><figure id="id.015.01.113.1.jpg" xlink:href="015/01/113/1.jpg"/><lb/>tere a b, ita ut angulus, uel minimus capiat b c qualem a b, & ex <lb/>quali b a c addito fiat b d tripla b c, & ex angulo b a e duplo b a d, <lb/>fiat b c d e quadragintupla a b, & iuxta rationem erit in infinitum. <lb/></s>
<s id="id2717819">Siue it parabole, iue hiperbole, eu it alia coincidentium.</s></p>
<pb xlink:href="015/01/114.jpg" pagenum="95"/><p type="head">
<s id="id2717847">SCHOLIVM.</s></p><p type="main">
<s id="id2717857">Et nota, qud i res hc eet naturalis, otenderet infinitum in <lb/>rebus ex regula dialectica, ed quia ex <expan abbr="uolũtaria">uoluntaria</expan>, nullas habet uires.</s></p><p type="main">
<s id="id2717898">Propoitio centeimaecunda.</s></p><p type="main">
<s id="id2717914">Proportionem motuum inuerionis, & attractionis in plano <lb/>inuenire.</s></p><p type="main">
<s id="id2717929">Et it, ut aliquid inuertatur, declaratum autem et upr, quid it </s></p><p type="main">
<s id="id2717952"><arrow.to.target n="marg351"/><lb/>inuerio, & qum diuera it rurus, & qud attractio et dimidium <lb/><arrow.to.target n="marg352"/><lb/>ponderis eleuati. </s>
<s id="id2717990">Cum ergo contet in inuerione, quanta it pro<lb/>portio ponderis upeni ad pondus inuerum, & pondus upeni <lb/><arrow.to.target n="marg353"/><lb/>it duplum ponderi attracti, equitur, ut diuifa proportione ponde <lb/>ris upeni ad pondus inuerum per medium cognocatur propor<lb/>tio attractionis ad inuerionem.</s></p><p type="margin">
<s id="id2718068"><margin.target id="marg351"/>C<emph type="italics"/>o<emph.end type="italics"/>_{m}.</s></p><p type="margin">
<s id="id2718095"><margin.target id="marg352"/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 89.</s></p><p type="margin">
<s id="id2718122"><margin.target id="marg353"/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 62.</s></p><p type="main">
<s id="id2718150">Ex hoc equitur, quod aliquod pondus trahi potet, quod non <lb/><arrow.to.target n="marg354"/><lb/>potet inuerti, hoc autem indigetlonga declaratione, quam doce<lb/>bimus inferis: & tamen attigit hocrar.</s></p><p type="margin">
<s id="id2718186"><margin.target id="marg354"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2718213">Propoitio centeimatertia.</s></p><p type="main">
<s id="id2718226">Proportionem eorundem in accliui demontrare.</s></p><p type="main">
<s id="id2718238">Dupliciter potet intelligi, uel decendendo, uel acendendo. <lb/><arrow.to.target n="marg355"/><lb/><arrow.to.target n="marg356"/><lb/>Sed ego nunc loquor de acenu, contraria ratione intelliges de <lb/>decenu, & circa inuerionem demontrata et proportio eius <lb/>iuxta angulum acenus, & imiliter declarabitur de proportione <lb/><arrow.to.target n="marg357"/><lb/>attractionis iuxta eundem angulum acenus, & nuper declarata <lb/>et proportio inuerionis in plano ad attractionem, ex quibus e<lb/>quitur per ea, qu dicam inferius, qud proportio cuiuuis mobi<lb/>lis inueri ad attractum ub quibucun que angulis nota erit.</s></p><p type="margin">
<s id="id2718357"><margin.target id="marg355"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s id="id2718384"><margin.target id="marg356"/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 72.</s></p><p type="margin">
<s id="id2718412"><margin.target id="marg357"/>I<emph type="italics"/>n equenti.<emph.end type="italics"/></s></p><p type="main">
<s id="id2718438">Propoitio centeimaquarta.</s></p><p type="main">
<s id="id2718451">Proportionem motus attractionis in decliui ad motum in pla<lb/>no determinare.</s></p><p type="main">
<s id="id2718465">Si ab accliue, eu decliue in quo d ad attra<lb/><arrow.to.target n="marg358"/><lb/><arrow.to.target n="marg359"/><lb/><figure id="id.015.01.114.1.jpg" xlink:href="015/01/114/1.jpg"/><lb/>hendum, cuius nota et ex uperioribus dif<lb/>ficultas in plano ratione figur contante, er<lb/>go ea quritur proportio acenus, & quo<lb/>niam terminus ad perpendiculum et dupla <lb/>proportio, & iam grauitas in plano et dimidium, ide quicquid <lb/>acquiritur in eleuatione et in comparatione ad illud dimidium, <lb/>cum ergo attractio ecundum eandem proportionem augeatur, er<lb/>go emper maior difficultas augebitur, ergo ab initio minimum
<pb xlink:href="015/01/115.jpg" pagenum="96"/>erit dicrimen ab attractione in plano. </s>
<s id="id2718576">Exempli gratia it, ut graue d <lb/>in plano it, ut quin que, & upenum decem, ergo in medio angulo <lb/>erit pen eptem, ed eptem minus longe <expan abbr="di&longs;tãt">ditant</expan> quin que, qum de<lb/>cem ad eptem, ergo in ecunda parte plus long augebitur difficul <lb/>tas attractionis upra difficultatem in medio angulo accliui, quam <lb/>in prima parte plano ad medium accliue, & quoniam planum in <lb/>plano decendit, tanto uehementius, quanto difficilius attrahitur, <lb/>ergo planum in decliui ublimi longe maiore impetu feretur infr <lb/>quam it proportio anguli ad angulum. </s>
<s id="id2718681">Exempli gratia, planum in <lb/>medio angulo, i incipiat decendere in dodrante multo lentius, <lb/>qum pro dimidio uirium decenus totius anguli, im initium de<lb/>cenus et medio recti ad unguem, ubi omnia plana int, & duri<lb/>ima, & caua huius et, quia omne graue tendit ad centrum, qud <lb/>maior pars ipius grauis et ultra medium grauitatis in decliui <lb/>humiliore.</s></p><p type="margin">
<s id="id2718759"><margin.target id="marg358"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s id="id2718786"><margin.target id="marg359"/>E<emph type="italics"/>x<emph.end type="italics"/> 62. & <lb/>64. P<emph type="italics"/>ropo.<emph.end type="italics"/></s></p><p type="main">
<s id="id2718827">Propoitio centeimaquinta.</s></p><p type="main">
<s id="id2718840">Proportionem ferentium pondus in pertica inuenire.</s></p><figure id="id.015.01.115.1.jpg" xlink:href="015/01/115/1.jpg"/><p type="main">
<s id="id2718859">Hc proponitur etiam Philoo<lb/><arrow.to.target n="marg360"/><lb/>pho, & ponatur ab, & i pondus it in <lb/><arrow.to.target n="marg361"/><lb/>medio d grauat qualiter utrunque, <lb/>nam in hoc conentit experimentum <lb/>cum ratione, at uer i ponatur in cita, <lb/>ut b c it tripla b a uiderentur a & b, tanquam hypomochlia, & pon <lb/><arrow.to.target n="marg362"/><lb/>dus ipum b, ut grauior eet cb, quam c a. </s>
<s id="id2718936">Aritoteles, eu author <lb/>ille hoc uidens bifariam repondet: primum qud hoc et inuer<lb/><arrow.to.target n="marg363"/><lb/>um intrumentum, cum in cteris motor it ex aduero hypomo<lb/>chlij, hic in ipo, getans enim mouet & hypomochlij intar et hu<lb/>merus. </s>
<s id="id2718999">At hoc uerum non et: quod mouet enim et pondus, & et <lb/>in c: nam a, & contingit moueri: quia i tarent, idem equeretur. </s>
<s id="id2719026">Se<lb/>cunda reponio et, quod utrun que premit cilicet ferentes & pon<lb/>dus, & qud qui longior et ab hypomochlio facilius mouet, & <lb/>redit ad idem ferm: nam in c contituitur, quod moueri debet, ca<lb/>pita uectium unt a, & b: motus autem et ipum utinere pondus. <lb/></s>
<s id="id2719087">At hoc non uidetur, quoniam ratio, qua uectis longior facilius mo<lb/>uet, et ambitus magnitudo, ob quam motus redditur tardior, & <lb/>ideo leuior: igitur non et hoc uerum de motu occulto, icut et gra<lb/>uis prementis, ed circumducente, cum in occulto uelut in tatera <lb/>contrarium accidere do cuerimus alis. </s>
<s id="id2719127">Quidam dixere b premere <lb/>c uerus a, a contr uerus b, & ide grauari magis a b, qum b ab <lb/>a, quia maiorem uim habet b e, qum a c. </s>
<s id="id2719159">Itud falum et bifariam. <lb/></s>
<s id="id2719173">Primum, quia & i a, & b int in quilibrio, ut nec unus in alterum
<pb xlink:href="015/01/116.jpg" pagenum="97"/>in cumbat, necimpellat, ed tantum utineat nihiloecius res uera <lb/>et. </s>
<s id="id2719212">Et etiam quia non et uerum, qud qui longius in cumbit, ma<lb/>iorem uim inferat. </s>
<s id="id2719226">Propterea dicendum et, qud qui ex commu<lb/>nibus propria nituntur demontrare, omnes corrumpunt dicipli<lb/>nas. </s>
<s id="id2719251">Nihil deterius et his montris. </s>
<s id="id2719261">Nam eti hc ratio uera eet: <lb/>non tamen reddit cauam, quia non et ex proprijs principijs. </s>
<s id="id2719284">Dico <lb/>ergo, quod i c decendat in e, per perpendiculum decendet, igitur <lb/>d b et longior d a, quare angulus e a b maior e b a: igitur pondus c <lb/>plus decendit comparatione a, qum b, ergo plus grauat cipum a <lb/>qum b, eu ex caua, quod magis premat, eu ex effectu, qud ma<lb/>gis deceerit. </s>
<s id="id2719346">Caua ergo erroris et, quod i ponatur angulus f b a <lb/>qualis angulo f a b, & ponatur b f qualis b c, tun c in eodem tem<lb/>pore, in quo tranit dimidium c in e, tranibit aliud dimidium c in f. <lb/></s>
<s id="id2719379">quia eparat partes grauiores unt in c b, qum c a, propter ditan<lb/>tiam ab hypomochlio, ed tunc uelo cius mouentur, & angulus fit <lb/>qualis. </s>
<s id="id2719410">Sed quando pondus et unum, & c decendit ad e, cum de<lb/>cendat inquali tempore, & peragat maiorem angulum compa<lb/>ratione a, quam b, equitur, ut uelo cius moueatur comparatione a <lb/>qum b. </s>
<s id="id2719444">Ergo i non mouetur, cum omnis potentia it imilis actui, <lb/>tum quia ab eo producitur, & effectus et imilis cau: tum quia <lb/>et initium actus, igitur etiam quod a b non in clinetur, nec decen<lb/>dat, grauius erit pondus, comparatione a qum b, quod erat de<lb/>montrandum.</s></p><p type="margin">
<s id="id2719499"><margin.target id="marg360"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s id="id2719526"><margin.target id="marg361"/>Q<emph type="italics"/>ust.<emph.end type="italics"/> 59. <lb/>M<emph type="italics"/>echanic.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2719565"><margin.target id="marg362"/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 45.</s></p><p type="margin">
<s id="id2719593"><margin.target id="marg363"/>P<emph type="italics"/>rop.<emph.end type="italics"/> 103.</s></p><p type="main">
<s id="id2719618">Ex hoc equitur, qud aliqua iuncta erunt grauiora repectu u<lb/>nius, qu erunt mutato ordine diuia leuiora. </s>
<s id="id2719640">Quoniam diuia, <lb/>qu longius ditant qualem, aut maiorem angulum faciunt, iun<lb/>cta minorem.</s></p><p type="main">
<s id="id2719668">Propoitio centeimaexta.</s></p><p type="main">
<s id="id2719683">Quales proportiones angulorum doceant laterum proportio<lb/>nes. </s>
<s id="id2719693">At que uicisim determinare.</s></p><p type="main">
<s id="id2719704">Sit circulus a b c, cuius dimetiens, nota b d it b, erit ergo latus <lb/><arrow.to.target n="marg364"/><lb/><figure id="id.015.01.116.1.jpg" xlink:href="015/01/116/1.jpg"/><lb/>exagoni a b dimidium b d, id et 3. igitur <lb/>cum angulus a it rectus, erit a d <02> 27 latus <lb/>trianguli. </s>
<s id="id2719743">Et latus quadrati per eandem <02><lb/>18. Vt latus exagoni it <02> 9. Quadrati <02> 18 <lb/>Trianguli <02> 27, & ita potetate e habent <lb/>hc ut 1. 2. 3. Et unt nota. </s>
<s id="id2719772">Et quia latus d e c <lb/>agoni et <02> 11 1/4 m, 1 1/2. & ipum erit notum. <lb/></s>
<s id="id2719788">Quare latus pentagoni et <02> v 22 1/2 m: <02><lb/>101 1/4 notum. </s>
<s id="id2719798">Et iam notum fuit latus epta<lb/>goni. </s>
<s id="id2719807">Habebimus igitur latera Trianguli
<pb xlink:href="015/01/117.jpg" pagenum="98"/>quadrati pentagoni, & eptagoni quilaterorum nota: & etiam <lb/>ubtenorum duobus ex his. </s>
<s id="id2719831">Sit, gratia exempli, a b 3 & b c <02> 11 1/4m: <lb/>1 1/2, ut prius, & ponatur b d diameter, erit ad <02> 27 & c d <02> v 22 1/2 m: <lb/><02> 101 1/4, quam ducemus in a b, & fiet <02> v 202 1/2 m: <02> 8201 1/4. Duce<lb/>mus itidem <02> 27 a d in b c <02> 11 1/4 m: 1 1/2 fiet <02> 303 3/4m: <02> 60 3/4, hoc to<lb/>tum diuide per 66, qu et b: fiet a c <02> 8 7/16 m: <02> 1 11/16 p: <02> v: 5 45/72 m: <02><lb/>6 1701/5184. Nec credas te errare, quoniam latus pentagoni eet, ac i an<lb/>gulus b rectus eet: ed quia et obtuus, ideo a c et alia linea, & <lb/>maior latere pentagoni. </s>
<s id="id2719902">Et imiliter i a b, & a c not eent, utpo<lb/><arrow.to.target n="marg365"/><lb/>te a b 3, ut prius a c 5 dico, qud b c nota et: nam a d erit <02> 27, & <lb/>quia ex b d in a c fit 30, fiet ex b c in a d pos <02> 27, et ex a b in c d <02> 324 <lb/>m: 9 quad. </s>
<s id="id2719942">igitur 30 m: pos <02> 27 quantur <02> 324 m: 9 quad. </s>
<s id="id2719951">quare <lb/>900 p: 27 quad. </s>
<s id="id2719958">m: pos <02> 97200 <expan abbr="æquãtur">quantur</expan> 324 m: 9 quad. </s>
<s id="id2719976">igitur 576 <lb/>p: 16 quad. </s>
<s id="id2719983">quantur pos <02> 97200. Quadratum igitur p: 36 quan<lb/>tur pos <02> 379 11/16, erit ergo b c <02> v: <02> 94 59/64 p: <02> 58 59/64 & imiliter i a c <lb/>it nota, puta 4 erit a b ubtena dimidio arcus a c nota. </s>
<s id="id2720015">Erit enim a e <lb/>2 ergo d e 3 p: <02> 5 et b e 3 m: <02> 5, <expan abbr="igi&ttilde;">igitur</expan> a b <02> v: 18 m, <02> 180. Igitur hoc <lb/>modo diuidendo, iungendo, & detrahendo habebimus ex quatu<lb/>or illis implicibus trianguli quadrati. </s>
<s id="id2720045">Pentagoni, & eptagoni in <lb/>numeras linearum magnitudines in circulo. </s>
<s id="id2720053">Et imiliter quouis mo <lb/>do, ut dictum et, in quauis figura quilatera, utpote uppoito <lb/><figure id="id.015.01.117.1.jpg" xlink:href="015/01/117/1.jpg"/><lb/>quod decriptum it nonangulum in <lb/>circulo quilaterum, quod etiam erit <lb/>quiangulum, & it arcus a b duplus <lb/>arcui a c, erit angulus a c b duplus an<lb/>gulo a b c, & angulus b a c in portione <lb/>b d e c excuplus a b c, & triplus a c b. <lb/></s>
<s id="id2720124">Erit ergo per demontrata proportio <lb/><arrow.to.target n="marg366"/><lb/>b a ad a c, uelut a c, & c b, ad a b: pro<lb/>portio autem a b arcus ad a c, ex up<lb/>poito maior et proportione rect a b ad a c, igitur etiam propor<lb/>tione a c & c b ad a b, ergo duo latera trianguli ad tertium minorem <lb/>habent proportionem, quam arcus ad arcum, quanto rect ad re<lb/>ctam minor et. </s>
<s id="id2720181">Sit rurus in triangulo b e d quomodolibet modo <lb/>it angulus b d e quadruplus angulo b e d, & diuidatur d per qua<lb/>lia ducta d f, erit igitur proportio f d, d e ad f e, ut e f ad f d, ed e f ad <lb/><arrow.to.target n="marg367"/><lb/>f b ut d e ad d b. </s>
<s id="id2720214">igitur proportio b d, d e ad f b <expan abbr="cõpo&longs;ita">compoita</expan> ex propor<lb/>tionibus e f ad f d, & e d ad d b. </s>
<s id="id2720238">Proportio igitur b d, d e ad f b, ut <lb/>producti ex e f in e d ad productum ex d fin d b. </s>
<s id="id2720247">Rurus ponamus, <lb/><arrow.to.target n="marg368"/><lb/>quod in quadrangulo a b c d prim figur it a b 4 b c 3 c d 5 ad 6 <lb/>dico, qud pacium contentum erit notum. </s>
<s id="id2720280">Ductis rectis a c & b d
<pb xlink:href="015/01/118.jpg" pagenum="99"/>quomodolibet, ut e ecent in e, erunt anguli d c a, & d b a quales, <lb/><arrow.to.target n="marg369"/><lb/>quia in eaem portione circuli a d, & anguli a d e quales, quia con <lb/>tra e poiti. </s>
<s id="id2720327">igitur trianguli a b e, & c d e imiles, & proportio d c ad <lb/><arrow.to.target n="marg370"/><lb/>a b, ut c e ad b e, c d autem fuit 5 a b 4, igitur i b e ponatur 4 pos c e <lb/>erit 5 pos. </s>
<s id="id2720351">Per eadem, & eodem modo a d ad b c ut d e ad e c. igitur <lb/>poita c e 5 pos erit e d 10 pos, tota igitur d b 14 pos. </s>
<s id="id2720369">Et quoniam ea<lb/><arrow.to.target n="marg371"/><lb/>dem proportio a e ad e b per eadem, & e b fuit 4 pos: igitur a e et 8 <lb/>pos, quare a e 13. pot productum igitur ex a c in d b, et 182 quad. <lb/></s>
<s id="id2720398">& hoc quatur productis a b in c d, quod et 20, & b c in a d quod <lb/>et 18, totum igitur et 38, igitur res et <02> 19/91. Quare not erunt line <lb/>b e, e d, a e, & e c, ed ufficit, ut cognita it a c, uel b d. </s>
<s id="id2720441">Per regulam <lb/>enim triangulorum erunt not are a b c, & a d e, quare tota uper<lb/>ficies a b c d. </s>
<s id="id2720463">Et et inuentum Scipionis Ferri Bononienis de quo <lb/>alis. </s>
<s id="id2720479">Potet etiam inuenta a c uel b d haberi uperficies facilius <lb/>per catheros.</s></p><p type="margin">
<s id="id2720497"><margin.target id="marg364"/>C<emph type="italics"/>o<emph.end type="italics"/>_{m}.</s></p><p type="margin">
<s id="id2720523"><margin.target id="marg365"/>P<emph type="italics"/>er<emph.end type="italics"/> 52. E<emph type="italics"/>le <lb/>ment.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2720560"><margin.target id="marg366"/>I<emph type="italics"/>n<emph.end type="italics"/> 16. <emph type="italics"/>de<emph.end type="italics"/><lb/>S<emph type="italics"/>ubtil.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2720608"><margin.target id="marg367"/>P<emph type="italics"/>er<emph.end type="italics"/> 3. <emph type="italics"/>exti<emph.end type="italics"/><lb/>E<emph type="italics"/>Elem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2720655"><margin.target id="marg368"/>P<emph type="italics"/>er<emph.end type="italics"/> 23. <emph type="italics"/>ex <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2720706"><margin.target id="marg369"/>P<emph type="italics"/>er<emph.end type="italics"/> 21. <emph type="italics"/>ter <lb/>tij<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2720755"><margin.target id="marg370"/>P<emph type="italics"/>er<emph.end type="italics"/> 15. <emph type="italics"/>pri <lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2720803"><margin.target id="marg371"/>P<emph type="italics"/>er<emph.end type="italics"/> 32. <emph type="italics"/>pri <lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="main">
<s id="id2720852">Sit modo obtui angulus a b c, & nota latera ingula, & angu<lb/>lus a b c, & producantur latera ad perpendicu<lb/><figure id="id.015.01.118.1.jpg" xlink:href="015/01/118/1.jpg"/><lb/>lum, ut int d & e recti, & quia anguli ad a unt <lb/>quales, erunt anguli e b a, & d e a emper <lb/><arrow.to.target n="marg372"/><lb/>quales. </s>
<s id="id2720907">Et hoc idem contingit in acuti angulis <lb/>triangulis intus, & et utile mechanicum: & <lb/>quia a b c notus et, & d notus, erunt anguli tri <lb/>goni d b c noti: & i fuerit angulus a notus, <expan abbr="erũt">erunt</expan> anguli d a c & e a b <lb/>noti, & ideo anguli e b a, & d c a: & emper notum, quod fit ex b a <lb/>in a d, uel c a in a e, unt enim qualia inter e: etiam not ad & a c, <lb/>quoniam duplum horum et exceus quadrati b c uper quadrata <lb/>a b, & a c. </s>
<s id="id2720982">Quod uer proponitur Monteregio de cognitione an<lb/>gulorum in triangulis non et intelligendum, ut uerba ignificant, <lb/><arrow.to.target n="marg373"/><lb/>ed olum de cognitione quoad uum tabularum.</s></p><p type="margin">
<s id="id2721022"><margin.target id="marg372"/>P<emph type="italics"/>er<emph.end type="italics"/> 32. <emph type="italics"/>pri <lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2721072"><margin.target id="marg373"/>P<emph type="italics"/>er<emph.end type="italics"/> 12. <emph type="italics"/>e<lb/>cundi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="main">
<s id="id2721124">Et iterum ponamus, qud proportio a c c b ad a b it qualis a b <lb/>ad a c, dico qud angulus c duplus et angulo b. </s>
<s id="id2721142">Si non ducatur c d <lb/><figure id="id.015.01.118.2.jpg" xlink:href="015/01/118/2.jpg"/><lb/>faciens angulum d c b duplum b, erit igitur pro<lb/>portio d c c b ad d b, ut d b ad d c. </s>
<s id="id2721166">Maior et <expan abbr="aut&etilde;">autem</expan> <lb/>d c, qum a c, aut qualis, aut minor, i qualis, <lb/>igitur maior proportio d c c b ad b d qum b a, <lb/>igitur maior proportio b d ad d c quam b a ad a c <lb/>ad a c & quales unt igitur b d maior d a pars toto, quod ee non <lb/>potet. </s>
<s id="id2721224">Si uer d c ponatur maior a c, magis ex hoc equitur b d ma<lb/>iorem ee b a. </s>
<s id="id2721242">Quod i minor it d c qum a c. </s>
<s id="id2721256">Ex demontratio<lb/>ne ipius reflex proportionis patet hoc contingere non poe. <lb/></s>
<s id="id2721279">Et imiliter patet conueras in reliquis etiam ueras ee, non olum
<pb xlink:href="015/01/119.jpg" pagenum="100"/>in proportionibus notisimis angulorum ed etiam in coniuncti<lb/>one & detractione. </s>
<s id="id2721316">Et et ex ubtilisimis operationibus, qu ho<lb/>mini in hoc genere eueniant.</s></p><p type="main">
<s id="id2721341">Propoitio centeimaeptima.</s></p><p type="main">
<s id="id2721357">Si in circulo duo diametri ad rectum angulum e ecauer int: ali <lb/>uer ad perpendiculum ex diametro exierint ad circumferentiam, <lb/>ingul upra diametrum erunt maiores portionibus reliquis dia<lb/>metri uperioribus, infra autem minores. </s>
<s id="id2721394">Dimidium autem porti<lb/>onis uperioris reiduum ad centrum maius agitta habebit. </s>
<s id="id2721412">In ali<lb/>qua prterea portionis uperioris parte, qu uerus diam etrum <lb/>tranuerum poita et, maior et differe ntia partis diametri ei cor<lb/>repondentis, quam line tranuer.</s></p><figure id="id.015.01.119.1.jpg" xlink:href="015/01/119/1.jpg"/><p type="main">
<s id="id2721480">Sint du diametri a b, c d ad perpendi <lb/>culum ecantes e in centro, & <expan abbr="ducũtur">ducuntur</expan> <lb/>upr f g k h, & infra m l ad perpendicu<lb/>lum upra a b: dico f g ee maiorem f a, <lb/>& k h k a, & contr minorem m l, qum <lb/>m a. </s>
<s id="id2721534">Per octauam enim exti, quod fit ex <lb/><arrow.to.target n="marg374"/><lb/>b f in f a quale et <expan abbr="&qtilde;drato">quadrato</expan> f g, ed b f et <lb/>maior f g, quia b f et maior c b, & ideo <lb/>e c g f, ergo f g maior et f a, m l <expan abbr="aũt">aunt</expan> minor et per <expan abbr="ead&etilde;">eadem</expan> e c, quare e a, <lb/>multo igitur minor m a, quod et primum. </s>
<s id="id2721608">Suppoito etiam, qud <lb/><arrow.to.target n="marg375"/><lb/>a g arcus it dimidium a c, dico a f <expan abbr="minor&etilde;">minorem</expan> ee f e, nam quadratum e <lb/><arrow.to.target n="marg376"/><lb/>g quale et quadratis f e, & f g, & <expan abbr="quadratũ">quadratum</expan> a g quadratis f g & f a <lb/>& e g et qualis lateri exagoni, & a g latus octogoni, igitur e g ma<lb/><arrow.to.target n="marg377"/><lb/>ior g a, & duo quadrata e f & f g maiora duobus quadratis f g & <lb/>f a, detracto igitur communi f g quadrato, patet propoitum.<lb/><arrow.to.target n="marg378"/></s></p><p type="margin">
<s id="id2721705"><margin.target id="marg374"/>P<emph type="italics"/>er<emph.end type="italics"/> 31. <emph type="italics"/>ter<lb/>tij<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2721755"><margin.target id="marg375"/>P<emph type="italics"/>er<emph.end type="italics"/> 7. <emph type="italics"/>tertij<emph.end type="italics"/><lb/>E<emph type="italics"/>lem.<emph.end type="italics"/> C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s id="id2721815"><margin.target id="marg376"/>1. <emph type="italics"/>eiudem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2721841"><margin.target id="marg377"/>P<emph type="italics"/>er<emph.end type="italics"/> 47. <emph type="italics"/>pri <lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2721890"><margin.target id="marg378"/>P<emph type="italics"/>er<emph.end type="italics"/> C<emph type="italics"/>or<emph.end type="italics"/>^{m}. <lb/>15. <emph type="italics"/>quarti<emph.end type="italics"/><lb/>E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="main">
<s id="id2721951">Cum rurus ex prima parte huius line f g & k h int maiores f a, <lb/>& k a & ea it qualis e c, necee et ut iuxta punctum c augeatur </s></p><p type="main">
<s id="id2721984"><arrow.to.target n="marg379"/><lb/>magis linea in ea, quam it differentia line tranuer ad lineam <lb/>tranueram per communem animi ententiam, quod et tertium.</s></p><p type="margin">
<s id="id2722025"><margin.target id="marg379"/>P<emph type="italics"/>er<emph.end type="italics"/> 28. <emph type="italics"/>ter<lb/>tij<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="main">
<s id="id2722076">Propoitio centeimaoctaua.</s></p><p type="main">
<s id="id2722089">Punctum qualitatis differenti decenus, & remotionis cen<lb/>tro inuenire.</s></p><p type="main">
<s id="id2722117">Per prcedentem moto puncto a uerus c emper u que ad e, c ma <lb/><arrow.to.target n="marg380"/><lb/>gis ditat <expan abbr="pũctum">punctum</expan> a linea a e, qum puncto a uerus, quia linea n h <lb/>maior et n a, & per eandem dum appropinquat ad c cum e c fiat <lb/>qualis ea, maius fit in crementum in a e, qum repectu line tran<lb/>ueralis. </s>
<s id="id2722191">Volo ergo inuenire punctum hoc in quo fit mutatio: & <lb/>diuido arcum ac per qualia in f, & dico illum ee punctum qu<lb/>itum: accepto quouis puncto in e f, puta k, duco g o h p quiditan
<pb xlink:href="015/01/120.jpg" pagenum="101"/><figure id="id.015.01.120.1.jpg" xlink:href="015/01/120/1.jpg"/><lb/>tes a b, & c d: erunt que anguli q & n recti <lb/><arrow.to.target n="marg381"/><lb/>& anguli f e a, & f e c quales, igitur uter <lb/><arrow.to.target n="marg382"/><lb/>que dimidium recti: igitur per dicta in <lb/>primo Elementorum Euclidis e n qua <lb/><arrow.to.target n="marg383"/><lb/>lis n k, igitur c q qualis e n, quare h p <lb/>qualis g o, ed quod fit ex o k in k g et <lb/><arrow.to.target n="marg384"/><lb/>quale ei, quod fit ex p k in k h, igitur <lb/><arrow.to.target n="marg385"/><lb/>k h et qualis k g ex eisdem otendi<lb/>tur f l m k quadratum ee. </s>
<s id="id2722326">Quia ergo <lb/>k h et qualis k g, & k l qualis k m, erit l g qualis m h. </s>
<s id="id2722344">Er<lb/>go decendendo ex g in f, quantum f l uperat l g, tantum decen<lb/>dendo ex f in h, f m uperat m h per communem animi ententi<lb/>am. </s>
<s id="id2722376">At f m et decenus f in linea a e, & m h ditantia, qu acqui<lb/>ritur in linea f r, n m enim et qualis f r, igitur n h excedit f r in <lb/>h m, & ita a n excedit a r in n r quali f m. </s>
<s id="id2722413">Quantum ergo in g f, <lb/>l f excedit l g, tantum in decenu ex f in h, f m, qu refert g l, ex<lb/>cedit h m, qu refert f l. </s>
<s id="id2722438">Arcus autem f g et qualis arcui f h, <lb/>quod <expan abbr="cũ">cum</expan> poem otendere pluribus modis atis contat, quia chor <lb/><arrow.to.target n="marg386"/><lb/>darum illorum quadrata unt inuicem qualia, quia line f m, & <lb/><arrow.to.target n="marg387"/><lb/>f l item que m h & l g unt quales, & anguli m, & l recti. </s>
<s id="id2722505">Igitur cum <lb/>ad quod uis punctum in linea e f emper linea decenus in parte <lb/>inferiore et maior linea ditanti tanto, quanto per qualem ar<lb/>cum in uperiore linea ditanti et maior linea, decenus equitur <lb/>per regulam Dialecticam quod punctus f, et punctus qualitatis. <lb/></s>
<s id="id2722569">Per idem diceremus in quarta parte inferiore.</s></p><p type="margin">
<s id="id2722578"><margin.target id="marg380"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s id="id2722605"><margin.target id="marg381"/>P<emph type="italics"/>er<emph.end type="italics"/> 29. <emph type="italics"/>pri <lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2722654"><margin.target id="marg382"/>P<emph type="italics"/>er<emph.end type="italics"/> 23. <emph type="italics"/>ter <lb/>tij<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2722703"><margin.target id="marg383"/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 32. <lb/>& 6.</s></p><p type="margin">
<s id="id2722734"><margin.target id="marg384"/>P<emph type="italics"/>er<emph.end type="italics"/> 34. <emph type="italics"/>pri <lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2722783"><margin.target id="marg385"/>P<emph type="italics"/>er<emph.end type="italics"/> 7. <emph type="italics"/>tertij<emph.end type="italics"/><lb/>E<emph type="italics"/>lement.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2722831"><margin.target id="marg386"/>P<emph type="italics"/>er<emph.end type="italics"/> 47. <emph type="italics"/>pri <lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2722880"><margin.target id="marg387"/>P<emph type="italics"/>er<emph.end type="italics"/> 47. <emph type="italics"/>ter<lb/>tij<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="main">
<s id="id2722930">Propoitio centeimanona.</s></p><p type="main">
<s id="id2722943">Rationem libr expendere.</s></p><p type="main">
<s id="id2722954">Cum libra moueatur, uelut rota circa axem, quia trutina manet, <lb/>ide i pondus ponatur, dum iugum fuerit in linea a b nihil mo<lb/>uebitur, quia appetitus decenus ex puncto a maximus et, & ni<lb/>hil iuuat motum extra naturam, idem dico de graui poito inuerti<lb/>ce b a. </s>
<s id="id2722994">Nam duo unt motus in rota, & in libra unus, per quem <lb/>dum fertur per arcum a f, gratia exempli decendit, quantum et <lb/><arrow.to.target n="marg388"/><lb/>a r, qu et minor dimidio e r, & ide minor e r, qu et maior di<lb/>midio, ut demontratum et, & etiam minor r f, qu qualis et r e <lb/><arrow.to.target n="marg389"/><lb/>per demontrata rurus: & hic et naturalis ut palam et: alter pr<lb/>ter <expan abbr="naturã">naturam</expan>, & et ferri ad latus, quoniam hoc et <expan abbr="propriũ">proprium</expan> immortali<lb/>bus: cun que hic it ad latus et etiam <expan abbr="cõtra">contra</expan> naturam, quia magis ditat <lb/>a centro, nam e f et longior c r, i ergo r ferretur in f, moueretur <lb/>centro, & contra naturam. </s>
<s id="id2723140">Dum ergo fertur ex a in f, multo lentius
<pb xlink:href="015/01/121.jpg" pagenum="102"/>fertur, qum ex f in c: uelo cius autem ex c uque ad medium: nam <lb/>plurimum decendit. </s>
<s id="id2723165">Ex h ad b autem celerrim, quoniam decen<lb/>dit, & appropinquat line a b, ut uter que motus it naturalis. </s>
<s id="id2723187">Non <lb/>ergo mouetur prter naturam nii quatenus longius recedit linea <lb/>a b, unde in inferiore parte mouetur ad eandem, ide de parte c b <lb/>tota perpicua et ratio, cur facillim decendat, imiliter & tota, <lb/>hoc enim et demontratum. </s>
<s id="id2723234">Similiter & quare difficillim feratur <lb/>ex b u que ad p, & ultra p u que ad directum r f: at de motu ex a in f, <lb/>quod debeat ferri, quia plus remouetur, quam decendat, nulla et <lb/>ratio: ut nec cur ex oppoito f ad a difficilem e prtet: & hoc et, <lb/>quia tertiam rationem etiam ipe Aritoteles, & qui eum equuti <lb/>unt, prtermiit. </s>
<s id="id2723299">Ea autem et, quod dum fertur ad g, uel f etiam li<lb/>cet non decendat magis, qum remoueatur, ex a <lb/><figure id="id.015.01.121.1.jpg" xlink:href="015/01/121/1.jpg"/><lb/>ad centrum terr tamen magis appropinquat. <lb/></s>
<s id="id2723334">Quia enim e a et qualis e c, quoniam prodeunt <lb/> centro circuli eiudem, & b e, & e c unt maio<lb/>res b c, ide b a erit maior b c, et autem b cen<lb/><arrow.to.target n="marg390"/><lb/>trum mundi, ergo a motum ad c, appropin qua<lb/>uit ipi b</s></p><p type="margin">
<s id="id2723386"><margin.target id="marg388"/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 98.</s></p><p type="margin">
<s id="id2723414"><margin.target id="marg389"/>I<emph type="italics"/>n prceden <lb/>ti.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2723443"><margin.target id="marg390"/>P<emph type="italics"/>er<emph.end type="italics"/> 17. <emph type="italics"/>pri <lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="main">
<s id="id2723493">Dico etiam quod libra ex chalybe tenuisimo, <lb/>& quanto <expan abbr="leuiorũ">leuiorum</expan> concharum, & longioris iugi <lb/>10 exactior, quoniam lances ill minori exceu <lb/>mouentur, quia plus ditant ab hypomochlio. <lb/></s>
<s id="id2723529">Sit ergo libra, cuius iugum a b trutin a c: lances d & e, alia libra, <lb/>cuius lances h, & k, & l m longiores, iugum f g. </s>
<s id="id2723542">Contat, quod <lb/>qualis proportio f g ad a b, talis ambitus, ad ambitum: motus er<lb/>go i it qualis utrarumque, igitur a tanto minore proportione <lb/><figure id="id.015.01.121.2.jpg" xlink:href="015/01/121/2.jpg"/>
<pb xlink:href="015/01/122.jpg" pagenum="103"/>mouebitur in h, quam in d, uelut it proportio f g ad a b dupla, ut <lb/>ergo qualiter moueantur, i it dupla exquiquarta in d cum lan<lb/>ce ad e uacuam, erit in h exquialtera, & mouebit quali tempore. <lb/></s>
<s id="id2723616">Ergo iuxta hoc fient libr, qu examinabunt decimam, & uigei<lb/>mam partem grani, quod et necearium in preciois rebus, & me<lb/>dicamentis potentibus, & long magis in mechanicis experimen<lb/>tis, & maxim qu ad demontrationem pertinent magnitudinis <lb/>uperficierum, & contat res in tribus, in longitudine, f g iungi, in le <lb/>uitate materi illius, & lancium, nam tanto maior redditur propor<lb/>tio ponderis exigui, & in firmitate iugi ac rectitudine. </s>
<s id="id2723688">ide debet <lb/>fieri ex chalybe purgato, durato ac tenuisimo, natura que leui, & ut c <lb/>it in medio, & mobilis f g.</s></p><p type="main">
<s id="id2723712">Coniderandum et demum an f l & g m int grauiores f h, & <lb/>g k. </s>
<s id="id2723728">Vt enim grauiores extiterint minus facil mouentur. </s>
<s id="id2723736">Viden<lb/>tur autem mihi, qui de his concriperunt perperam contempie <lb/>hoc, contat enim, qud dum l decendit, remouetur a b n c tru<lb/>tina, & m, qu acendit contra appropinquat. </s>
<s id="id2723779">Videtur autem hoc <lb/>bifariam contra naturam: nam ut diximus pondus applicat e ad <lb/>rectam n c, quia uerus centrum, & etiam quia facit angulum ob<lb/>tuum, cum deberet, ut ab initio altem contituere cum iugo re<lb/>ctum. </s>
<s id="id2723815">Et de m nihil mirum et, cum acutum, ut e ad lineam, qu ad <lb/>centrum retrahat. </s>
<s id="id2723831">Huiumodi prterije Aritotelem, demiror, <lb/>qu nimis fuerunt in conpicuo, ut dubitem ne non uus it ille li<lb/>ber, qui eius pen nihil apiat prter obcuritatem. </s>
<s id="id2723878">Tentan<lb/>dum et igitur horum cauas asignare. </s>
<s id="id2723894">nam qu huiumodi po<lb/>tet ee doctrina nii perfecta fuerit, in omnibus etenim necee et <lb/>aut omnia cire, aut ignorare. </s>
<s id="id2723932">In hoc igitur dico, quod h f, eu l f, <lb/>emper quiditant n c trutin, ergo cum angulus f c n in clina<lb/>to iugo fiat obtuus decendente pondere, & n c g acendente pon<lb/>dere fiat acutus, ergo angulus l f c tantundem fiet obtuior, & m g c <lb/>acutior, quanto anguli ad c tales unt. </s>
<s id="id2723979">Et caua et quia n c ratio<lb/>ne ponderis et directa ad centrum, ergo oportet, ut pondera l, uel <lb/>h, & m, uel k, i debent tendere ad centrum, ut f l, & g m quidi<lb/>tent n c, nii quantum et pro ditantia f, puncto c, & g a b eodem, <lb/>qu comparata ad <expan abbr="centrũ">centrum</expan> terr, eu mundi, et inenibilis omnino. <lb/></s>
<s id="id2724055">Circa hc <expan abbr="notandũ">notandum</expan> itud mirabile fcilicet, quod ratio motus, quan<lb/>tumuis exigua ufficit ad motus <expan abbr="modũ">modum</expan>, licet uelo citas <expan abbr="p&etilde;deat">pendeat</expan> ex gra<lb/>uitate, & alijs. </s>
<s id="id2724102">Et quae graue, quod expers et enus, debeat equi ratio <lb/>nem Geometricam uix apientibus <expan abbr="cognitã">cognitam</expan>, caua tamen una et, & <lb/>perpicua: <expan abbr="nã">nam</expan> omne graue et in linea centro <expan abbr="mũdi">mundi</expan>: i <expan abbr="aũt">aunt</expan> medium <lb/>grauis it extra <expan abbr="lineã">lineam</expan>, uertitur ad illam, qu et in eo, nam <expan abbr="centrũ">centrum</expan> em
<pb xlink:href="015/01/123.jpg" pagenum="104"/>per et in <expan abbr="ead&etilde;">eadem</expan>. </s>
<s id="id2724229">Ergo ola in clinatio ad hoc ut <expan abbr="mediũ">medium</expan> grauis it in li<lb/>nea <expan abbr="centrorũ">centrorum</expan> grauitatis & terr, ufficit. </s>
<s id="id2724265">Et ergo principium in ei<lb/>po. </s>
<s id="id2724282">In appenis imiliter. </s>
<s id="id2724292">Trutina enim, & finis iugi, & grauis <expan abbr="cen-trũ">cen<lb/>trum</expan> mundi <expan abbr="centrũ">centrum</expan> unt in <expan abbr="ead&etilde;">eadem</expan> linea, ut ee pount, cum exigua illa <lb/>& ola ditantia intercedat. </s>
<s id="id2724346">& hoc et primum. </s>
<s id="id2724353">Quia ergo <expan abbr="iugũ">iugum</expan> et <lb/>ex materia olida, mouetur ratione, qu dicta et, lances autem <lb/>oportet cum filis appeni int, ut puncta f & h, uell, & g k, uel g m <lb/>int in una linea cum centro terr. </s>
<s id="id2724398">Et quia l magis ditat a b f quam <lb/>h, & m a g magis, quam k, & oportet faciant eandem inclinatio<lb/>nem, quia anguli trutin cum iug unt ijdem, & linea cl et ma<lb/>ior c h, & c m, qum c k in quouis itu, ergo patium, quod ambitur, <lb/>et maius ergo per d e montrata uperius l et grauius h etiam <lb/>prter uinculorum additionem, & m grauius k. </s>
<s id="id2724458">Quanto igi<lb/>tur longiores unt funiculi libr extremitate eu iugi, tanto gra<lb/>uius redditur pondus, quod tamen multi putant ee falum: nec <lb/>aliquid referre, qud it longum, aut breue utentaculum.</s></p><p type="main">
<s id="id2724509">Propoitio centeimadecima.</s></p><p type="main">
<s id="id2724523">Si du phr ex eadem materia decendant in <expan abbr="a&etilde;">aem</expan> <lb/>re eodem temporis momento ad planum ueniunt.<lb/><figure id="id.015.01.123.1.jpg" xlink:href="015/01/123/1.jpg"/><lb/><arrow.to.target n="marg391"/></s></p><p type="margin">
<s id="id2724573"><margin.target id="marg391"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2724599">Supponitur quod ex eodem loco. </s>
<s id="id2724603">Sermo enim <lb/>aburda ub interpretatione nunquam nii ab inui<lb/>dioo, uel imperito intelligi debet. </s>
<s id="id2724626">Sit ergo a tripla <lb/>ad b, phrula ad phrulam ex plumbo amb fer<lb/>ro uel lapide eiudem generis, dico, qud inquali <lb/>tempore peruenient ad planum c d. </s>
<s id="id2724665">Nam a propor<lb/>tionem habet ad b, ut uigintieptem ad unum. </s>
<s id="id2724677">pro<lb/>portio autem patij a ad patium b nonupla et, & <lb/>proportio denitatis aris ad arem et tripla, propterea quod den<lb/>itas illa multiplicatur propter impetus magnitudinem. </s>
<s id="id2724715">nam i ro<lb/>bur, ut decem percutiat baculo lato, ut quatuor ictus erit maior du<lb/>plo, qum it robur, ut quinque percutiat baculo, ut duo: propter <lb/>denitatem ergo maiorem aris in a, quam in b: & quoniam i ub <lb/>maiore impetu mouetur <expan abbr="a&etilde;r">aerr</expan> ub a, quam ub b, igitur proportio <lb/>erit comparanda longitudini centro a ad longitudinem a centro <lb/>b, qu et tripla. </s>
<s id="id2724786">Si ergo ubtripla et ratio motus b ad a, quod <lb/>ad medium attinet, tripla autem propter uelo citatem diceus a<lb/>ris medio grauitatis, quod et in uperficie e regione centri graui<lb/>tatis in linea ad centrum mundi, ut dictum et in prcedenti: mani<lb/>fetum et, quod a, & b inquali tempore peruenient ad ubie<lb/>ctum planum, & quiditans centris eorum. </s>
<s id="id2724858">Similiter & in aqua:
<pb xlink:href="015/01/124.jpg" pagenum="105"/>cum uer uideatur in illa tanto celerius a decendere, qum b, <lb/>quanto et emidiameter a longior emidiametro b, liquet ex hoc, <lb/>quod quali uelo citate decendunt, ed ob uelo citatem motus in <lb/>are latet dicrimen anticipationis contactus oli a ante b, qui di<lb/>gnocitur in aqua, ex quo patet exactam ee qualitatem. </s>
<s id="id2724931">Sed rei<lb/>liunt emel in aqua amb, cum pluries in are a olo, quare etiam in <lb/>aqua perturbatur cognitio in parum accuratis, at que enu prditis, <lb/>icut etiam in cau, ne altera alteram perueniat, utra que comprehena <lb/>duobus digitis, altera alteram tangente, & uque ad centrum in <lb/>aquam demisis imul digitis dilatatis dimittend unt.</s></p><p type="main">
<s id="id2725002">Propoitio centeimaundecima.</s></p><p type="main">
<s id="id2725016">Cur ex medio tela ualidiorem ictum, & naues in calmo remo, <lb/>ac malo recipiant inde ex puppi explorare.</s></p><p type="main">
<s id="id2725035">Aritoteles uidetur in Mechanicis, & qui eum equuti unt, ui</s></p><p type="main">
<s id="id2725053"><arrow.to.target n="marg392"/><lb/>dentur rem nauticam qud ad remos attinet, referre in longitu<lb/>dinem partis, qu calmum tanqum hypomochlium interiacet <lb/>& manum: ea enim circa medium nauis cum illa ibi it latior ma<lb/>ior et. </s>
<s id="id2725094">Sed & qui lembos ducunt, & in puppe magis ditant <lb/>calmo & in prora, qum in medio nauis, nec tamen uelo cius il<lb/>lam agunt: non qud ratio illa fala it, ed quia uelo cius ferun<lb/>tur etiam ob aliam cauam, qum it hc, & magis uniueralem. <lb/></s>
<s id="id2725150">Primum igitur umamus, quod uperis demontratum et cili<lb/><arrow.to.target n="marg393"/><lb/>cet, qud ubi pondus aliquod quale undique tanquam in li<lb/>bra upenum fuerit, proportio ponderis partium inqualium <lb/>ad duas partes quales, et confua ex proportione longitudi<lb/>nis earundem, & quadrato eiudem proportionis. </s>
<s id="id2725221">Sit ergo diui<lb/>a a b in c, & fiat c e qualis c a: proportio igitur ponderis b e ad <lb/>pondus e a et compoita ex proportione b e ad e a, & quadrato <lb/><figure id="id.015.01.124.1.jpg" xlink:href="015/01/124/1.jpg"/><lb/>eius <expan abbr="&longs;ecũdum">ecundum</expan> longitudinem. </s>
<s id="id2725270">at poita agi <lb/>na d g in medio a b, proportio ponderis b e <lb/>ad pondus ea et, ueluti longitudinis b e <lb/>ad e a, igitur proportio <expan abbr="põderis">ponderis</expan> b e ad e a, <lb/>cum agina et extra medium in c, et tanto <lb/>maior proportione b c ad ea, quantum et quadratum illius pro<lb/><arrow.to.target n="marg394"/><lb/>portionis, ergo b e pondus maius et, cum agina et in c, qum in d. <lb/></s>
<s id="id2725336">igitur per <expan abbr="commun&etilde;">communem</expan> animi <expan abbr="&longs;ententiã">ententiam</expan> addito communi pondere a e, <lb/>erit pondus a b minus emper cum agina et in d, <08> in ullo alio lo<lb/>co a b. </s>
<s id="id2725374">Ergo pondus a b apprehenum in d <expan abbr="mouebi&ttilde;">mouebitur</expan> a b quali ui <lb/><arrow.to.target n="marg395"/><lb/>maiore proportione, <08> in ullo alio loco. </s>
<s id="id2725403">Hatile ergo in medio ap<lb/>prehenum maiore ui mouebitur, qum in ulla alia parte. </s>
<s id="id2725420">Et i gra
<pb xlink:href="015/01/125.jpg" pagenum="106"/>cilius it in anteriore parte propinquius comprehenum calci, & i <lb/>crasius, uel grauius propius cupidi. </s>
<s id="id2725456">Semper igitur ob hanc cau<lb/>am mota ex medio grauitatis, eu uelo, eu ramo, eu manu uelo<lb/>cius mouentur, qum ex alijs partibus. </s>
<s id="id2725482">In remo etiam potet acce<lb/>dere illud commodum, cuius meminit Aritcteles. </s>
<s id="id2725497">Propter hoc igi <lb/>tur, qui malum in naui collo carunt tantm unum, in medio ferm <lb/>eum collocarunt, ut antiqui: & qui duos aut tres, maiorem crasio<lb/><arrow.to.target n="marg396"/><lb/>rem cilicet, & altiorem in medio contituerunt.</s></p><p type="margin">
<s id="id2725538"><margin.target id="marg392"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s id="id2725565"><margin.target id="marg393"/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 86.</s></p><p type="margin">
<s id="id2725593"><margin.target id="marg394"/>P<emph type="italics"/>er<emph.end type="italics"/> 10. <lb/><emph type="italics"/>quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2725641"><margin.target id="marg395"/>P<emph type="italics"/>er<emph.end type="italics"/> 8. <emph type="italics"/>quin<lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2725691"><margin.target id="marg396"/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 82.</s></p><p type="main">
<s id="id2725718">Propoitio centeimaduodecima.</s></p><p type="main">
<s id="id2725731">Cur ex imo leuia longius ferantur declarare.</s></p><p type="main">
<s id="id2725740">Iam uer <expan abbr="cõ&longs;ideremus">conideremus</expan>, qud propoitum et, non olum in com<lb/><arrow.to.target n="marg397"/><lb/>paratione ad medium, ed extremorum inuicem, mia enim ab imo <lb/>uelo cius feruntur, qum medio non olum manu, ed corpioni<lb/>bus, & arcubus. </s>
<s id="id2725811">Videmus & hoc oberuare pueros uirgam lon<lb/>gius iacentes non ex medio, ed imo apprehenam, quoniam pars <lb/>ipa anterior, & qu manu apprehena et, uehementi impetu emit<lb/>titur: & ut recipit impetum magis qualem, longius fertur, nam <lb/>quod emittitur proportionem habet ad patium. </s>
<s id="id2725859">Cum ergo appre <lb/>hena in medio uirga olum medietate anteriore impetum recipiat <lb/>per e, ob id minus fertur: at impetus equitur proportionem, ut ui<lb/>um et, qu et circa medium ob leuitatem ponderis. </s>
<s id="id2725896">In leuibus <lb/>ergo maius patium uperabunt emia ex imo, quoniam propor<lb/>tio patij eadem et ad duplum, & ad dimidium. </s>
<s id="id2725923">igitur ex imo fer<lb/>me duplum etiam patij uperabit: non tamen omnino quia maio<lb/>rem, ut dixi proportionem habet ad id, quod ex medio comprehen <lb/>um et. </s>
<s id="id2725950">At in leuibus non et necearium, ut ex medio apprehen<lb/>dantur, quoniam etiam cum incremento illo ponderis iam leuia <lb/>unt: plus ergo facit longitudo eius, quod eiaculatur, qum impe<lb/><figure id="id.015.01.125.1.jpg" xlink:href="015/01/125/1.jpg"/><lb/>tus, cuius demontratio et hc. </s>
<s id="id2725998">Sit uirga <lb/>a b apprehena in medio ponderis unci <lb/>medi, & in a d, ut it d a palmus, & uigei<lb/>ma pars totius a b, erit ergo reiduum ad duplum, a d nonuplum, <lb/><arrow.to.target n="marg398"/><lb/>& a b tota unciarum quin que cum dimidia, i igitur grauetur, quia in <lb/>itu recto et medi unci, in quiditanti terr, quin que unciarum <lb/>cum dimidio, erit in itu dimidij recti unciarum trium. </s>
<s id="id2726071">Et igitur <lb/>proportio excupla, i apprehendatur in medio, & ad quiditan<lb/>tem, ad apprehenam in imo, & ad angulum medium: at emia ex <lb/><arrow.to.target n="marg399"/><lb/>a d habet totum arem a b circumdantem impulum ex c b olum <lb/>dimidium reliqua pars ui trahitur, ergo proportio patij a b, erit <lb/>exdecupla ferm patio b c, quoniam et triplicata corporis ad cor <lb/>pus eius, qu et longitudinis ad longitudinem, & quadruplicata
<pb xlink:href="015/01/126.jpg" pagenum="107"/>repectu aris a c, qui reitit apprehena a b in c. </s>
<s id="id2726176">Et iam minus fere<lb/>batur quinta parte, ideo longius eiaculabitur triplo ex a, qum ex <lb/>c. </s>
<s id="id2726193">Nec tamen maiore impetu, quia obliqu fertur, & qu obliqu <lb/><expan abbr="feriũt">feriunt</expan>, minore cum impetu feriunt: at que eo magis i leuia fuerint: ab <lb/>are enim circumambiente perturbantur, & in incertum trudun<lb/>tur. </s>
<s id="id2726232">Qu ergo grauia unt ex medio emia, & ad quiditantem <lb/>longius feruntur, & maiore cum impetu, quia magis direct: leuia <lb/>autem longius ex imo, ed minore cum impetu, i aliqua caua re<lb/>cto, & quiditante declinauerint. </s>
<s id="id2726285">At i uprema parte, & iuxta <lb/>cupidem, neque procul feruntur, neque cum impetu ob cauas di<lb/>ctas. </s>
<s id="id2726311">Eadem quoque ratio et omnium machinarum: ide oblon<lb/>glongius eiaculantur, quoniam proportionem eruant ad cana<lb/><arrow.to.target n="marg400"/><lb/>iem. </s>
<s id="id2726341">Sed de hoc inferius agetur.</s></p><p type="margin">
<s id="id2726350"><margin.target id="marg397"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s id="id2726377"><margin.target id="marg398"/>P<emph type="italics"/>er<emph.end type="italics"/> 86.</s></p><p type="margin">
<s id="id2726402"><margin.target id="marg399"/>P<emph type="italics"/>er<emph.end type="italics"/> 89.</s></p><p type="margin">
<s id="id2726426"><margin.target id="marg400"/>P<emph type="italics"/>rop.<emph.end type="italics"/> 107.</s></p><p type="main">
<s id="id2726451">Propoitio centeimatertia decima.</s></p><p type="main">
<s id="id2726464">Cur uirga longius mittatur puero, qum uiro inuetigare.<lb/><arrow.to.target n="marg401"/></s></p><p type="margin">
<s id="id2726490"><margin.target id="marg401"/>C<emph type="italics"/>o<emph.end type="italics"/>_{m}.</s></p><p type="main">
<s id="id2726516">Diligentia, & uus puerilis efficit, ut uirga feratur ecundum me<lb/>dium rectianguli: uir autem non contanter iacit, & ecundum re<lb/>ctum, at rectus inceus in leuibus, quia ab are in obliquum defle<lb/>ctitur uirga ob longitudinem efficit, ut inflectatur infr celerius, & <lb/>deinat citius motus, ac finiatur. </s>
<s id="id2726562">Tertia caua et, qud leuisima <lb/>non ade recipiunt impetum ut grauia: nam leuisimam & exigu<lb/>am ligni portionem maximo nixu uix excutiemus manu. </s>
<s id="id2726594">Caua <lb/>ergo et: quoniam uim, oportet, ut habeat, quod contra naturam <lb/>mouetur, ut naturaliter moueri posit, qucun que igitur naturaliter <lb/>exiguum habent motum, ut pluma, palea, fetuc nulla ratione ue<lb/>hementer contra naturam agi pount. </s>
<s id="id2726635">Qudam ergo pueris lon <lb/>gius <expan abbr="iaciũtur">iaciuntur</expan> ob olam peritiam, & exercitationem, qudam quo<lb/>niam ad angulum latiorem magis feruntur, qum it rectus, qu<lb/>dam quoniam leuisima unt. </s>
<s id="id2726686">Sed i leuiora non feruntur ualido <lb/>motu uiolento, cur tamen pueris iacta longius <expan abbr="ferũtur">feruntur</expan>? </s>
<s id="id2726709">Ratio et, <lb/>quoniam maior uis deficiente obiecto magis fatigatur, atque ide <lb/>minus mouet. </s>
<s id="id2726726">Propter hc igitur omnia non olm in pueris, ed <lb/>in machinis, qu accommodata unt, melius impelluntur, a c lon<lb/>gius feruntur, qum leuisima. </s>
<s id="id2726760">nam nec palea corpione iacta tam <lb/>procul, qum agitta fertur, cum proportio maior it, tamen ad pa<lb/>leam, qum ad agittam. </s>
<s id="id2726789">Inde fit, ut quemadmodum Turca ille lite<lb/>ras ui Prin cipis, cum timeret ad notros propius accedere, lapidi al <lb/>ligatas longius emiit. </s>
<s id="id2726810">Cauam autem huius docet Aritoteles in <lb/>Mechanicis dum qurit cur, & grauia & leuia ualde longe proijci <lb/>nequeunt: nam grauia nimis, moueri <expan abbr="nõ">non</expan> facil pount: leuia etiam <lb/>ualde ad rem mouere non ualent. </s>
<s id="id2726849">Ob hc utra que ex his paruo cum
<pb xlink:href="015/01/127.jpg" pagenum="108"/>impetu emittuntur, tameti uehementer nitaris. </s>
<s id="id2726868">Sed & leuia ferun<lb/>tur hac illac, ut non posint retinere impetum prioris uiolenti: in<lb/>natum enim et, ut duorum motuum imul in eadem re uigentium, <lb/>cum illa proprio impetu feratur, unus alterum impediat: nam i ro<lb/>ta uehatur circulariter acta, non tamen ceabit, aut iminuetur impe <lb/>tus circulationis. </s>
<s id="id2726913">Multa ergo in huiumodi anomalis motibus con <lb/>ideranda unt, ut illorum impetum robur, aclocum definiamus.</s></p><p type="main">
<s id="id2726932">Ex hoc liquet, cur plumbe phrul longius ferantur tor</s></p><p type="main">
<s id="id2726956"><arrow.to.target n="marg402"/><lb/>mento emi, qum ligne, etiam i non fran gantur.</s></p><p type="margin">
<s id="id2726984"><margin.target id="marg402"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2727010">Propoitio centeimaquartadecima.</s></p><p type="main">
<s id="id2727023">Cir cularis motus differentias quatuor ee, earum qe rationem <lb/>contemplari.</s></p><p type="main">
<s id="id2727043">In motu circulari aut axis <expan abbr="progredi&ttilde;">progreditur</expan>, aut uo loco manet. </s>
<s id="id2727058">Vtro que<lb/><arrow.to.target n="marg403"/><lb/>autem modo uel mouetur ab axe, uel circumferentia, igitur contat <lb/>quatuor ee motuum differentias: quas cum tres proponat author <lb/>libri Mechanicarum, aut Aritotelem illum ee, credendum non <lb/>et, aut illum tupidum dicere necee et, nam modum diuidendi <lb/>eum latuie quis putet. </s>
<s id="id2727115">cum rota igitur aut phra in plano cir<lb/>cumagitur, motus et ex circumferentia prgrediente axe: ut pa<lb/>lam et: motis enim loco nobis mouentur omnia, qu unt in no<lb/>bis. </s>
<s id="id2727153">Cum uer rot ub curru unt, progreditur axis earum, & rota <lb/>ob id cum quiecere nequeat, quia facilius circumuertitur, qum <lb/>trahatur, procedit, & hic et ecundus modus, quo rota ex circumfe <lb/>rentia mouetur, & ex axe initium et motus. </s>
<s id="id2727194">At uer in rota molari, <lb/>& quibus gladij exacuuntur, cum loco non moueantur, motus et <lb/>ex axe: axis enim rotam circumagit, non rota axem, quiecit tamen <lb/>in eodem loco rota, & axis cilicet, quia non progreditur, ed in lo<lb/>co mouetur: atque hic et tertius modus. </s>
<s id="id2727232">Demum uccula putei, & <lb/>ipa mouetur circulari motu, & trochle etiam, neque enim progre<lb/>diuntur: ed non ex axe mouentur, uerm uccula per coloppes cir <lb/>cumducitur, & tro chlea per funes, axis que in uccula mouetur, in tro <lb/>chleis autem quiecit prorus: dico mouetur, id et circumducitur, <lb/>non quod progrediatur: ut non olum int quatuor modi, ed po<lb/>tius quin que, nam & demontratione otenduntur, & experimento <lb/>do cente deprehenduntur. </s>
<s id="id2727307">Horum omnium liberrimus et, primus <lb/>ex cir cumferentia progrediente toto, eu attracto eu impulo & ue <lb/>locisimus, cuius cauam upr otendimus. </s>
<s id="id2727343">Proximus huic et mo<lb/><arrow.to.target n="marg404"/><lb/>tus rotarum per axem, quoniam axis premit rotam interius o<lb/>lam, & labitur: ideo que quod & axis, & rota intus int leuisima, pro<lb/>det plurimum: & aurig axungia inungunt, & nomen ab eo traxit
<pb xlink:href="015/01/128.jpg" pagenum="109"/>axungia. </s>
<s id="id2727395">Et quae rota magna it: quoniam cum <expan abbr="nõ">non</expan> rota, ed axis traha<lb/>tur in quali tempore & magna, & parua trahitur: utra que uer una <lb/>conuerione tantam <expan abbr="lineã">lineam</expan> rectam uperat, quanta et rot periphe<lb/>ria. </s>
<s id="id2727451">Quod i plures int rot celerius feruntur, quia axis minus tan<lb/>to <expan abbr="rotã">rotam</expan> premit. </s>
<s id="id2727477">Et i rectus it axis, & bene rotundus, & foramen ro <lb/>tundum, & latius, & durisimo ligno, ut non posit in clinari: & <lb/>rota ipa in ambitu qualis, omnia hc faciunt ad motus uelo cita<lb/>tem, unde Homerus.<lb/><arrow.to.target n="marg405"/></s></p><p type="margin">
<s id="id2727527"><margin.target id="marg403"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s id="id2727553"><margin.target id="marg404"/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 40.</s></p><p type="margin">
<s id="id2727581"><margin.target id="marg405"/>I<emph type="italics"/>liad.<emph.end type="italics"/> 23.</s></p><p type="main">
<s id="id2727605"><foreign lang="greek">I)/xnia tu/pte w_o/dessi w_a/r & ko/nin a)|mfi xuqu_nai</foreign>.</s></p><p type="main">
<s id="id2727620">Id et, uetigia per cusit pedibus, ante que illa puluis pedibus ex<lb/>cuus (uetigia cilicet relinquentibus) ingrederetur. </s>
<s id="id2727647">Principalis <lb/>autem caua uelo citatis et agens, uelut equi. </s>
<s id="id2727660">Sed inter <expan abbr="hũc">hunc</expan> motum <lb/>& priorem medius et Scital uocat, nam ut in primo axis proci<lb/>dit & rotundum uperficie circumagitur, licet axis etiam circum<lb/>ducatur, ut axis, & rota, aut phra duplici motu moueantur, fci<lb/>licet antrorum, & circumcirca, in rota currus duo ijdem motus <lb/>int, axis quo que antrorum moueatur, ed non circumagatur: unde <lb/>impeditior et hic motus: ita in Scytala utrun que utro que motu mo<lb/>uetur, & circumcirca, & antrorum, at que id commune et, cum pri<lb/>mo ita axis mouet rotas, non rot axem, qud ecundo motui ro<lb/>tarum in curru proprium et, ut tantum degenerent primo motu, <lb/>quanto leuius uertuntur, qum in ecundo motu. </s>
<s id="id2727777">Trahitur ergo <lb/><figure id="id.015.01.128.1.jpg" xlink:href="015/01/128/1.jpg"/><lb/>iugum in citala, uelut in rotis currus, <lb/>ed et annexum rotis non in curri<lb/>bus. </s>
<s id="id2727810">Propterea in primo motu trahi<lb/>tur, uel impellitur uperficie: in e<lb/>cundo a b axe, ed non affixo rotis, unde gr trahuntur in cyta<lb/>la ab axe affixo rot. </s>
<s id="id2727850">Quare leuius qum in curru, difficilius qum <lb/>in rota uel phra uperficie extima circumacta. </s>
<s id="id2727875">Quartus modus <lb/>et, ut dixi, circumuecta rota ab axe, quum non progreditur, ut in <lb/>moletrinis, & rotis, quibus ferrum exacuitur. </s>
<s id="id2727891">Et enim hic imilior <lb/>primo, quia contrarius, in primo enim procedit rota, & uertitur <lb/>circumferentia, hic quiecit rota, & mouetur ab axe. </s>
<s id="id2727915">Proximus huic <lb/>et, qui fit in ucculis ob firmitatem axis: nam axis et coniunctus <lb/>rot. </s>
<s id="id2727937">Vltimus et trochlearum, qui & difficillimus: it enim cir<lb/>cunferentia, & axis diiunctus et trochlea: quod ad dit difficulta<lb/>tem. </s>
<s id="id2727968">Sed & trochlea caret colloppibus. </s>
<s id="id2727973">Ergo uerum et, quod o<lb/>mnia rotunda facilius circumaguntur, ed uaria ratione: nam plus <lb/>mota uper aliquo plano, ut in plautris & cytalis: minus in uccu<lb/>lis, & rotis acuentibus ferrum, & molis: nam & i rotun ditatem iu<lb/>uet ob qualitatem ad conuerionem, non tamen in his et ad e
<pb xlink:href="015/01/129.jpg" pagenum="110"/>utilis. </s>
<s id="id2728035">Vtilitas ergo prima et, cum circumuertitur in plano, uelut <lb/>in rotis cytalis, & phris. </s>
<s id="id2728054">Secunda qu minor et, cum uperfi<lb/>cie circumuertitur, ut in trochleis. </s>
<s id="id2728074">Tertia cum coloppis, qu mi<lb/>nima et omnium, ut in ucculis. </s>
<s id="id2728094">Motus autem cli non et ex tri<lb/>plici primo genere, cum it in loco, & non ad locum, neque ut rot <lb/>molaris: nam ille et ex axe: necut in tro chlea: nam in ea axis quie<lb/>citipum autem clum circa axem non uertitur, ed cum axe, i ta<lb/>men inecabilis linea circumagi potet dici. </s>
<s id="id2728150">Relinquitur ergo, ut <lb/>Cli motus propior it motui uccul, qum alij motui. </s>
<s id="id2728171">Differt <lb/>ab eo in hoc, quod in uccula mouetur axis ab orbe: at in clo <lb/>ut non mouetur ab axe, ita nec axis ab orbe: cun que it motus im<lb/>plicisimus, in alio genere collocandus et: quando quidem in illo <lb/>nulla pars posit dici primo, quod <expan abbr="nece&longs;&longs;ariũ">necearium</expan> et in uno quo que <expan abbr="horũ">horum</expan>.</s></p><p type="main">
<s id="id2728243">Propoitio centeimaquinta decima.</s></p><p type="main">
<s id="id2728256">Proportionem motuum impulionis, & attractionis inter'e ab <lb/>eadem ui declarare.</s></p><p type="main">
<s id="id2728275">Contat, qud attractio cum fune longiore ualidior et, quam </s></p><p type="main">
<s id="id2728292"><arrow.to.target n="marg406"/><lb/>cum manibus, quoniam et cum motu quodam: motus autem au<lb/>get actionem, ideo attractio ualidior et hac de caua, ed & impul<lb/>io cum baculo ualidior et, quam cum manibus, quoniam licet col <lb/>ligere omnes uires in illo baculo, & ipum applicare loco, unde fa<lb/>cilius impelli potet. </s>
<s id="id2728342">Velut phra ex medio latere: nam ibi magis <lb/>colliguntur uires, & ad impellendum facilius et, quodcun que leui<lb/>us et. </s>
<s id="id2728367">Pars autem magis remota centro grauitatis et leuior, his <lb/>duabus cauis, phra ex medio latere facilius ac magis impellitur. <lb/></s>
<s id="id2728391">Sed nos upponimus nunc applicationem qualem ee, nam e<lb/>cus ad impellendum facilius et applicare totum corpus, qum at<lb/>tractionem. </s>
<s id="id2728423">Pectore enim magna ui impellimus, nihil et compar, <lb/>quo trahere posimus. </s>
<s id="id2728436">Sed, ut dixi, it baculus applicatus alicui la<lb/>pidi ea parte, qua facilius potet impelli & trahi, & quritur, qu <lb/>maior it uis, an attrahendi? </s>
<s id="id2728463">& dico qud homo, uel conatur trahe<lb/>re toto corpore, & impellere, at que hoc modo magis trahit, qum <lb/>impellet, quoniam corporis pondus melius adhibetur in tractione <lb/>qum impulu: uel citra corporis pondus, ed ola ui membrorum: <lb/>& tunc magis impellit, quoniam impulus fit corpore prono in <expan abbr="an-terior&etilde;">an<lb/>teriorem</expan> partem, qu in clinatio, & motus et naturalis magis, qum <lb/>in attractione in partem poteriorem. </s>
<s id="id2728533">Sed ubi nulla it diueritas <lb/>neque horum, neque figurarum qualis uis qualem efficit motum: <lb/>quia impulus impellentis comparatione et attractio repectu al<lb/>terius. </s>
<s id="id2728567">Verm non et eadem uis nec prop par impellendi, at que <lb/>attrahendi hominibus, cum attractio fiat per muculos ad origi
<pb xlink:href="015/01/130.jpg" pagenum="111"/>nem uam naturaliter e retrahentibus impului nullum intrumen <lb/>tum natura delegatum inuenio, nam ad extenionem muculi a<lb/>n ex aduero unt fabricati: cum ergo duo int tantum motus mu<lb/>culorum tenio, dum <expan abbr="retrahũtur">retrahuntur</expan> ad principium uum, & remisio, <lb/>dum membrum quiecit in naturali nullus erit locus impulioni, <lb/>nii ex conequentia non per e, quamobrem multo infirmiorem il<lb/>lum attractione in brachijs ee, necee et.</s></p><p type="margin">
<s id="id2728704"><margin.target id="marg406"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2728731">Propoitio centeimaextadecima.</s></p><p type="main">
<s id="id2728747">Cur machin ablong igne longius emittant phram ex<lb/>plorare.<lb/><arrow.to.target n="marg407"/></s></p><p type="margin">
<s id="id2728780"><margin.target id="marg407"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2728806">Quoniam ratio uperius adducta, neque in his, neque in hypophy</s></p><p type="main">
<s id="id2728819"><arrow.to.target n="marg408"/><lb/>is (uocant cerbatanas) non potet atisfacere, cum tamen idem e<lb/>quatur in his, ut in illis uidetur, quai uis ee in phrula ic emi<lb/>a, & non in are, quemadmodum dicebamus, coniuncto ee. </s>
<s id="id2728874">Ex <lb/>quo necee eet, ut quod longius ferretur, etiam ualidiores ictus <lb/><figure id="id.015.01.130.1.jpg" xlink:href="015/01/130/1.jpg"/><lb/>inferret, hoc autem <lb/>non ita e habet, ed <lb/>ictus magnitud o <lb/>ex robore machi<lb/>narum tam ignea<lb/>rum, quam corpio <lb/>num pendet, nam <lb/>it a corpio ma<lb/>gnus, ed tenuis, ex <lb/>hc palam et lon<lb/>gius mittere agit<lb/>tam, qud parua, <lb/>& breui, quantun<lb/>uis craa non lon<lb/>ge mittitur: at uer <lb/>quod b craus & paruus maiore cum impetu mittat otenditur <lb/>nam ea pondera agitt mouet, qu non potet mouere a, igitur b <lb/>ualidiore robore mouet, quam a. </s>
<s id="id2729018">Prtera illud oten dit iugum fu<lb/>nis arcus crasiora duriora, qu maioribus uiribus <expan abbr="indig&etilde;t">indigent</expan>, quam <lb/>a, qui puero tendi poterit. </s>
<s id="id2729054">Non et ergo eadem ratio mittendi <lb/>longius, & ualidiore cum robore. </s>
<s id="id2729066">Eadem ergo cum ratio it in <lb/>machinis igneis, crasiores enim, & latiores ac breuiores magis <lb/>concutiunt, quam longiores tenuiores minoris phr capaces: <lb/>non olum ob mag nitudinem phr magis ill concutiunt, ed, <lb/>ut dixi, ob maiorem impetus uim: caua ergo et manifeta in his, <lb/>ed non caua, qua longius ferantur in longiore canali. </s>
<s id="id2729132">Sed uide
<pb xlink:href="015/01/131.jpg" pagenum="112"/>tur una, eadem que ee ratio in utrique. </s>
<s id="id2729154">Contituatur can alis a b <lb/>logior, & c d breuior, ut it exqui alter a b ad c d, & it rurus <lb/><figure id="id.015.01.131.1.jpg" xlink:href="015/01/131/1.jpg"/><lb/>phrul locus e in longiore, <lb/>exqui alter in ditantia a b, qua <lb/>lis et in f a d, & erit per dicta <lb/>ab Euclide in quinto, ac exqui <lb/>altera c f. </s>
<s id="id2729223">Poemus igitur di<lb/>cere, quod uelut ab hypomo<lb/>chlio longiore patio circuma<lb/>gitur pondus: ita & a b c, & f. <lb/></s>
<s id="id2729250">Sed rurus incidimus in id, ut <lb/>maiore impetu feratur e qum f. </s>
<s id="id2729263">Ideo i concedatur maiore ferri ex <lb/>e, quam ex f non equitur, ut celerius, aut maiore impetu. </s>
<s id="id2729276">Percutit <lb/>puer pugno quanta ui potet ac celerrim, uir robutus lent, & mi<lb/>nore impetu, ed tamen ictus long maior et. </s>
<s id="id2729308">Et enim ictus robur <lb/>non uelo citate olum, ed maiore ex ponderis grauitate, qu ola <lb/>premit, urget, & frangit etiam ine motu. </s>
<s id="id2729338">Solum ergo id retat du<lb/>bium, cur i grauius et, moueatur eodem ferm impetu: nam quo <lb/>maiore impetu fertur, eo longius fertur, non tamen magis ferit, con <lb/>cutit, aut quaat, ed grauitas ad hoc plus facit impetu. </s>
<s id="id2729374">Palea maxi<lb/>mo impetu demia non ferit, non ledit, & celerius decendit, fer<lb/>rum ola grauitate actum, im etiam temperato ictu ldit graui<lb/>ter, quaat, & frangit: itaque f maiore indiget quantitate pyrij pulue<lb/>ris, qum e: iquidem tertia parte ponderis u phr: at maius <lb/>et pondus f quam e, ergo maius pondus pulueris f qum e, ergo <lb/>maior uehementia ictus, iquidem ea equitur, robur cau mouen <lb/>tis im pliciter: ut concludamus longitudinem ictus equi propor<lb/>tionem motoris ad motum, ed uehementia robur motoris: nam i <lb/>ex portione mouet quale pondus maiore cum impetu mouet, <lb/>quoniam maior et proportio: i minore igitur pondus maius et, <lb/>&, ut dixi plus facit magnitudo ponderis cum leui ictu, qum ma<lb/>gnitudo ictus cum leui pondere. </s>
<s id="id2729510">Qu ergo feruntur per longio<lb/>res canales maiore impetu feruntur, & ocietatem <expan abbr="hab&etilde;t">habent</expan> aris moti <lb/>per longius <expan abbr="&longs;patiũ">patium</expan>, ut tardius remittatur, quia longiore tempore <expan abbr="uĩs">uins</expan> <lb/>motus confirmata et, & proportio eius, qud mouet, maior et ad id, <lb/>quod <expan abbr="moue&ttilde;">mouetur</expan>, quia minus extenditur, at uer f <expan abbr="motũ">motum</expan> minore propor<lb/>tione <expan abbr="ictũ">ictum</expan> facit <expan abbr="maior&etilde;">maiorem</expan>, proa, ut dixi, <expan abbr="tãto">tanto</expan> grauius, et quod ferit. </s>
<s id="id2729632">Quod <lb/><expan abbr="aut&etilde;">autem</expan> minus <expan abbr="ext&etilde;datur">extendatur</expan> machina a b quam c d, <expan abbr="nũc">nunc</expan> <expan abbr="o&longs;t&etilde;dere">otendere</expan> oporter.</s></p><p type="margin">
<s id="id2729685"><margin.target id="marg408"/>P<emph type="italics"/>rop.<emph.end type="italics"/> 103.</s></p><p type="main">
<s id="id2729710">Propoitio centeimadecimaeptima.</s></p><p type="main">
<s id="id2729726">In cuniculis maior et uis pulueris copioioris ampliore in pa<lb/>tio, qum paucioris in minore iuxta proportionem eandem.</s></p>
<pb xlink:href="015/01/132.jpg" pagenum="113"/><p type="main">
<s id="id2729759">Sit patium f d exqui tertium b e, puluis quo que in f d patio i<lb/><arrow.to.target n="marg409"/><lb/>militer exqui tertius pulueri b e pondere, & manifetum et, quod <lb/>dum conuertitur in ignem qualicun que it proportio (modo eadem <lb/>ignis ad puluerem) erit ignis in f d pariter exqui tertius igni in b e, <lb/>dico qud i crasities f d it etiam exqui tertia crasitiei b e, quod <lb/>poterit frangi, & moueri f d quiecente b e. </s>
<s id="id2729836">Vnde idem in cuniculis <lb/>ut magnus cuniculus cum multo puluere posit mouere montem <lb/>paruus cum puluere proportione repondente priori non posit. <lb/></s>
<s id="id2729857">Nam cm qualia int omnia iuxta que rationem eandem, necee et <lb/>ut pro ratione extendantur, at in paruo patio minor fit denitas c<lb/>tera paria unt, ergo paruo patio non tantus fit impetus, quantus <lb/> magno. </s>
<s id="id2729905">Impetus etiam proportionem habet ad <expan abbr="põdus">pondus</expan>, & ad con<lb/>iunctionem, maiore igitur impetu plura, & maiora mouentur, & <lb/>conuelluntur, quam minore, ob hc igitur minores cuniculi uc<lb/>cutiunt, maiores euertunt, maximi exturbant, & proij ciunt. </s>
<s id="id2729945">Nam <lb/>qui uccutiunt, ubi pondus, aut coniunctio maior it, qum ut di<lb/>trahere posint, condenant partes proximiores, & rimas faciunt, <lb/>per quas exhalat ignis aut omnino extinguitur, aut condenatur. <lb/></s>
<s id="id2729981">At ergo in bellicis machinis, minus dilatat puluis, cum fuerit in lon <lb/>go canali, ob id ergo maiore impetu feruntur per illas, qum per <lb/>breuiores, etiam qud minor it puluis, minor it ignis. </s>
<s id="id2730004">Experimen <lb/>tum facies in canali, ubi ambuci medulla pro globulo flatu impel<lb/>lente expellitur ab que periculo: nam quanto minor fuerit canalis <lb/>ambitu ac longior eo maiore impetu pellitur. </s>
<s id="id2730026">Foran quipiam nos <lb/>merit poterit uideri <expan abbr="repreh&etilde;di&longs;&longs;e">reprehendie</expan>, qud inanis glori tudio per<lb/>nitioa humano generi do ceam. </s>
<s id="id2730075">Quibus repondeo, me nihil do cu <lb/>ie, quod n humani generis detrimentum cedat, huiumo di que pr<lb/>cepta iam obcurae, ut ne quid mali accidere poet hominibus ex <lb/>his: <expan abbr="nã">nam</expan> qud ad ea, qu declarata, unt, cauas olm retuli, effectus <lb/>ipimodi artis <expan abbr="nimiũ">nimium</expan> feruntur, ac nimio pluquam <expan abbr="uell&etilde;">uellem</expan> in telligun<lb/>tur. </s>
<s id="id2730171">Vt cum ad copiam, ad magnitudinem, ad coacta imperia mie<lb/>rorum repicio, nihil plus posit addi. </s>
<s id="id2730188">Omnia enim hucu que <expan abbr="&longs;pectãt">pectant</expan> <lb/>ad potentiorum in crementa. </s>
<s id="id2730211">An ergo uccurrere afflictis, obesis, <lb/>cinctis, quare <expan abbr="condition&etilde;">conditionem</expan>, liberare eruitute etiam rebelles <expan abbr="nõ">non</expan> li<lb/>cebit? </s>
<s id="id2730256">Ab initio fuimus omnes liberi: excogitata fuit regni ratio ad <lb/>commodum hominum, ea uera et per uim in <expan abbr="Tyrannid&etilde;">Tyrannidem</expan>. </s>
<s id="id2730278">Subtili <lb/>ergo ratione <expan abbr="occurrendũ">occurrendum</expan> et imbecillioribus: <expan abbr="nã">nam</expan> reliqua omnia ni<lb/>mis, ut dixi, qu ad cuniculos ad <expan abbr="magnitudin&etilde;">magnitudinem</expan> <expan abbr="machinarũ">machinarum</expan> ad rectos <lb/>ictus ad <expan abbr="libram&etilde;ta">libramenta</expan> ad longitudinem pacij, per quos globus ille de<lb/>fertur, nota unt improbis illis artificibus, nec notrum et pectare, <lb/>cur id licuerit, potquam Deus hanc uiolentiam ee uoluit. </s>
<s id="id2730370">Multa <lb/>damnamus, <expan abbr="&qtilde;">quae</expan> Deus ee uult: boni uiri et <expan abbr="nõ">non</expan> nii opitulari homini<lb/>bus, <expan abbr="etiã">etiam</expan> malis modo bonis futuri <expan abbr="nõ">non</expan> int <expan abbr="impedim&etilde;to">impedimento</expan>: <expan abbr="quamobr&etilde;">quamobrem</expan>
<pb xlink:href="015/01/133.jpg" pagenum="114"/>ea tradenda unt, qu oppresis int auxilio: ea unt, qu ubtilibus <lb/><expan abbr="con&longs;tãt">contant</expan> rationibus, et multiplicata <expan abbr="amittũt">amittunt</expan> uim ut quai <expan abbr="pr&ecedil;&longs;t&etilde;t">prtent</expan> pau <lb/>ca multis, & exigua magnis. </s>
<s id="id2730524">In cteris obcurare ita decet cuncta, <expan abbr="&qtilde;">quae</expan> <lb/>obee pount, aut quouis modo puerti ad malos uus <expan abbr="queãt">queant</expan>, ut di<lb/>cta <expan abbr="nõ">non</expan> dicta ee <expan abbr="put&etilde;t">putent</expan>, hoc et <expan abbr="officiũ">officium</expan> <expan abbr="nõ">non</expan> olum probi, ed <expan abbr="etiã">etiam</expan> pruden <lb/>tis uiri.</s></p><p type="margin">
<s id="id2730632"><margin.target id="marg409"/>C<emph type="italics"/>o<emph.end type="italics"/>_{m}.</s></p><p type="main">
<s id="id2730658">Propoitio centeimadecimaoctaua.</s></p><p type="main">
<s id="id2730671">Quanta proportione decedat ictus in obliquum parietem ab eo, <lb/>qui et ad perpendiculum declarare.</s></p><figure id="id.015.01.133.1.jpg" xlink:href="015/01/133/1.jpg"/><p type="main">
<s id="id2730696">Sit paries b d e, ex a <expan abbr="fera&ttilde;">feratur</expan> in dictus, qui i <lb/><arrow.to.target n="marg410"/><lb/>eet in c d <expan abbr="pariet&etilde;">parietem</expan> ee ad perpendiculum, & <lb/>ualidisimus, in uero in f g abraderet, & <expan abbr="nõ">non</expan> <lb/><expan abbr="cõqua&longs;&longs;aret">conquaaret</expan>. </s>
<s id="id2730773">Quritur ergo ex b d e muro <lb/>qualis excipietur? </s>
<s id="id2730783">erit ergo proportio anguli c d a ad <expan abbr="angulũ">angulum</expan> b d a, <lb/>ueluti ictus a d in d c ad <expan abbr="ictũ">ictum</expan> in b d, <expan abbr="manife&longs;tũ">manifetum</expan> et <expan abbr="aũt">aunt</expan> equi proportio<lb/>nem, <expan abbr="quoniã">quoniam</expan> maxima uarietate <expan abbr="cõ&longs;tat">contat</expan> dum ex angulo b d a acuto fit <lb/>acutior, <expan abbr="quoniã">quoniam</expan> i b d c it <expan abbr="&qtilde;druplus">quadruplus</expan> b d a erit reiduus ad <expan abbr="dimidiũ">dimidium</expan> b <lb/>d a nonuplus ipi dimidio, & ad <expan abbr="quartã">quartam</expan> <expan abbr="part&etilde;">partem</expan> habebit proportionem <lb/><expan abbr="decemnou&etilde;">decemnouem</expan> ad <expan abbr="unũ">unum</expan>. </s>
<s id="id2730940">Si ergo <expan abbr="etiã">etiam</expan> in <expan abbr="id&etilde;">idem</expan> tenderent, <expan abbr="nõ">non</expan> efficerent mille <lb/>ictus d tres, cuius demontratio hc et. </s>
<s id="id2730983">Supponamus <expan abbr="proportion&etilde;">proportionem</expan> <lb/>b d c ad <expan abbr="&qtilde;rtam">quartam</expan> <expan abbr="part&etilde;">partem</expan> a d b ad dito reiduo ad b d c ee <expan abbr="&longs;olũ">olum</expan> <expan abbr="decuplã">decuplam</expan>: <lb/><expan abbr="tũc">tunc</expan> ex duob. </s>
<s id="id2731052">ictibus centupla erit in d c ad <expan abbr="eã">eam</expan>, qu in b e, <expan abbr="etiã">etiam</expan> tribus <lb/>millecupla: nam <expan abbr="cõqua&longs;&longs;ata">conquaata</expan> turri in primo ictu, id d decuplo magis <lb/>ad perpendiculum <08> in b d e <expan abbr="&longs;uma&ttilde;">umatur</expan> decima pars in ambitu d, & illa <lb/>erit ergo <expan abbr="tã">tam</expan> dioluta, & infirma ex uppoito, <08> et tota b e: ed ex e <lb/>cundo ictu decuplo magis <expan abbr="cõqua&longs;&longs;abi&ttilde;">conquaabitur</expan> illa pars, <08> b e ergo tota d c <lb/>centuplo magis <expan abbr="qua&longs;&longs;abi&ttilde;">quaabitur</expan> ex duob. </s>
<s id="id2731187">ictibus c d turris, <08> b e, & ita in <lb/>tribus: ex <expan abbr="dec&etilde;">decem</expan> millibus ergo ictibus <expan abbr="etiã">etiam</expan> ad amusim directis, <expan abbr="cũ">cum</expan> ta <lb/><expan abbr="m&etilde;id">menid</expan> uix fieri posit in <expan abbr="tãta">tanta</expan> multitudine <expan abbr="nõ">non</expan> plus <expan abbr="cõminue&ttilde;">comminuetur</expan> b d e, <08><lb/>ex dec c d <expan abbr="&ptilde;ter">pnter</expan> <expan abbr="quã">quam</expan> <expan abbr="exiguũ">exiguum</expan> <expan abbr="quippiã">quippiam</expan> in uperficie. </s>
<s id="id2731308">Imo ut <expan abbr="declaratũ">declaratum</expan> <lb/>et multo minus repetita ratione multiplicis. </s>
<s id="id2731327">Ob id in arce <expan abbr="Medio-lan&etilde;&longs;i">Medio<lb/>laneni</expan> exterius lapidibus uiuis in <expan abbr="rotundũ">rotundum</expan> diducta uperficie inter<lb/><figure id="id.015.01.133.2.jpg" xlink:href="015/01/133/2.jpg"/><lb/>uallo que <expan abbr="&qtilde;">quae</expan> drato hunc in <expan abbr="modũ">modum</expan> munit unt altiores tur <lb/>res. </s>
<s id="id2731399">Fiat ergo murus cuius proportio a d c ad b d a it ex <lb/>quitertia, erit que angulus b d c <expan abbr="dodrãs">dodrans</expan> recti, & <expan abbr="parũ">parum</expan> incli <lb/>natis, <expan abbr="&longs;iquid&etilde;">iquidem</expan> b d c erit quarta pars recti, & it tant ma<lb/>gnitudinis, at que duritiei, ac ade ben coniunctus fer<lb/><arrow.to.target n="table16"/><lb/>reis cathenis, ac tolonibus, ut posit reitere <expan abbr="machinarũ">machinarum</expan> <expan abbr="fe-rentiũ">fe<lb/>rentium</expan> <expan abbr="&longs;ph&ecedil;rã">phram</expan> <expan abbr="librarũ">librarum</expan> ducentarum (qu an maxim unt) <lb/><figure id="id.015.01.133.3.jpg" xlink:href="015/01/133/3.jpg"/>quin quaginta: <expan abbr="tũc">tunc</expan> cum proportio exquitertia nouies repeti<lb/>ta, ut in numeris uides, efficiat quinquies replicatis nouem <lb/>ictibus, fiet proportio decupla quinquies producta, qu et cen <lb/><expan abbr="tũ">tum</expan> millium ad <expan abbr="unũ">unum</expan> in quadraginta quin que ictibus. </s>
<s id="id2731599"><expan abbr="Antequã">Antequam</expan> <lb/>ergo peruenit ad quinquaginta ictus rectos necee erit, ut
<pb xlink:href="015/01/134.jpg" pagenum="115"/>multo plures centum millibus ictus excipiat ante <08> euertatur, qu <lb/>recta i eet quin quaginta olm potuiet utinere. </s>
<s id="id2731656">Qu ergo hu <lb/>mana potentia ufficeret. </s>
<s id="id2731669">In arce Medio <expan abbr="lan&etilde;&longs;i">laneni</expan> uidimus uix attactas <lb/>in illis extuberationibus lapideis. </s>
<s id="id2731691">Sed quoniam hic occurritur per <lb/>inclinationem machinarum, ide de hoc <expan abbr="&longs;ermon&etilde;">ermonem</expan> um habiturus.</s></p><p type="margin">
<s id="id2731719"><margin.target id="marg410"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><table><table.target id="table16"/><row><cell>729</cell></row><row><cell>972</cell></row><row><cell>1296</cell></row><row><cell>1728</cell></row><row><cell>2304</cell></row><row><cell>3072</cell></row><row><cell>4096</cell></row><row><cell>5461 1/3</cell></row><row><cell>7281 7/9</cell></row></table><p type="main">
<s id="id2731791">Propoitio centeimadecimanona.</s></p><p type="main">
<s id="id2731804">Quantum ictus machin procliuis ad <expan abbr="angulũ">angulum</expan> <expan abbr="minua&ttilde;">minuatur</expan> explorare.</s></p><p type="main">
<s id="id2731833">Huiuce caua <expan abbr="excogitarũt">excogitarunt</expan>, ut ictus ad <expan abbr="perpendiculũ">perpendiculum</expan> <expan abbr="dirigere&ttilde;">dirigeretur</expan>, & <lb/><arrow.to.target n="marg411"/><lb/><expan abbr="quanquã">quanquam</expan> angulus d e f it quali angulo a b c, long <expan abbr="tñ">tnm</expan> maior et uis <lb/>a b <08> d e duplici caua, & <expan abbr="quoniã">quoniam</expan> a b et <expan abbr="&longs;ecundũ">ecundum</expan> nat uram impetus <lb/><figure id="id.015.01.134.1.jpg" xlink:href="015/01/134/1.jpg"/><lb/>ignis, & <expan abbr="etiã">etiam</expan> <expan abbr="eorũ">eorum</expan>, qu <expan abbr="emittun&ttilde;">emittuntur</expan> in altum: & d pars <lb/>uperior in b retineat <expan abbr="ictũ">ictum</expan>, in e non retineat. </s>
<s id="id2731990">Sed caui <lb/>tas fiat maior in inferiore parte: cuius <expan abbr="experim&etilde;tum">experimentum</expan> <lb/>quiliber facere potet <expan abbr="cũ">cum</expan> hata. </s>
<s id="id2732024">Huic ergo olerti, <expan abbr="&qtilde;">quae</expan> <lb/>tormenta iubet altius collocare obtat <expan abbr="primũ">primum</expan>, quod <lb/>ictus ex decliui itu periculoior et pro machina, & ma <lb/>xim d retro impellit, quae ex retro cea, pot <08> exone <lb/>rata et, <expan abbr="digno&longs;ci&ttilde;">dignocitur</expan>, & ad <expan abbr="collimandũ">collimandum</expan> decedit parte <expan abbr="ui-riũ">ui<lb/>rium</expan> uarum, d eti <expan abbr="paruũ">paruum</expan> it in ductu <expan abbr="tñ">tnm</expan>, & <expan abbr="ictuũ">ictuum</expan> mul <lb/>tiplicatione <expan abbr="magnũ">magnum</expan> affert dicrimen. </s>
<s id="id2732173">Habet & <expan abbr="cõmo">commo</expan> <lb/>dum itus muri accliuis <expan abbr="terrã">terram</expan> <expan abbr="&longs;uppo&longs;itã">uppoitam</expan> ad perpendiculum, <expan abbr="&qtilde;">quae</expan> ictum <lb/>utinet: ade ut omnib. </s>
<s id="id2732234"><expan abbr="inuic&etilde;">inuicem</expan> collectis, perinde it ac i ex perpen<lb/>diculo, et quiditanti ad <expan abbr="&longs;olũ">olum</expan> <expan abbr="feria&ttilde;">feriatur</expan>. </s>
<s id="id2732280">Venetus. </s>
<s id="id2732284">S. aliter Patauij cauit, <lb/>uidetur que, quae apientisimus it, & eandem equatur ubi que normam, <lb/>pot <08> in <expan abbr="rotundã">rotundam</expan> figuram <expan abbr="totũ">totum</expan> urbis ambitum formauit, & foa la <lb/>ta, ac pro fundisima aqua que perenni muniuit, & <expan abbr="&longs;ummã">ummam</expan> muri partem <lb/><expan abbr="rotundã">rotundam</expan> in hunc <expan abbr="modũ">modum</expan> effecit <expan abbr="cauã">cauam</expan> que interius undi que, ne cuniculis <lb/><figure id="id.015.01.134.2.jpg" xlink:href="015/01/134/2.jpg"/><lb/>poet euerti, lateribus uer humiles, ac crasisimas turres, ut nul <lb/>la ui poent dirui, eas que tormentis bellicis, undi que latera lutrantib. <lb/></s>
<s id="id2732416">repleet, illud diligentisime cauit, ne murus humilior eet aduera <lb/>ripa, ed ad <expan abbr="libellã">libellam</expan> tamen depreus, ut <expan abbr="etiã">etiam</expan> machinis in terram exten <lb/>is phrul non tangerent <expan abbr="murũ">murum</expan>: nam <expan abbr="cũ">cum</expan> foa it quadraginta pa<lb/>uum, excedat <expan abbr="aũt">aunt</expan> murus <expan abbr="exterior&etilde;">exteriorem</expan> aggerem uno pau, ut quicquid <lb/>in ambitu et uno ictu oculi cognoci posit, & aggeris angulus ma <lb/>ior it uno pau, <expan abbr="tũ">tum</expan> magis adiecta crasitie machin fieri non potet, <lb/>utictus in <expan abbr="murũ">murum</expan> dirigatur. </s>
<s id="id2732577">Eam ob cauam <expan abbr="etiã">etiam</expan> cauit, ne <expan abbr="&ecedil;dificiũ">dificium</expan> ul<lb/><figure id="id.015.01.134.3.jpg" xlink:href="015/01/134/3.jpg"/><lb/>lum, aut planta, uel colliculus eet cir<lb/>cum circa <expan abbr="urb&etilde;">urbem</expan> ad tria M. P. laborat hoc <lb/>periculo hc urbs, ne tota dificijs euer<lb/>is concidat. </s>
<s id="id2732650"><expan abbr="Turcarũ">Turcarum</expan> enim Princeps di<lb/>dicit, ut in Nouo catro in Melit Inul <lb/>arce S. </s>
<s id="id2732681">Elmi appellata plu <08> mille icti<lb/>bus in ingulos dies imo M D obtundere
<pb xlink:href="015/01/135.jpg" pagenum="116"/>munitiones. </s>
<s id="id2732704">Eum que impetum producere ad quindecim dies, & ui<lb/>ginti tum etiam longius, ut facil domos omnes euertat, homines <lb/>occidat: i qui uperunt tot in commodis obruuntur uigilijs, fame, <lb/>iti, puluere, ut inutiles red dantur. </s>
<s id="id2732734">Ide huic <expan abbr="incõmodo">incommodo</expan> occurrunt <lb/>aggeribus intra mnia erectis, in quos uis <expan abbr="torm&etilde;torum">tormentorum</expan> igneorum <lb/>emoritur. </s>
<s id="id2732769">Sed dices, cur ergo non pro muris erigere eos prtat, & <lb/>minore umptu atis? </s>
<s id="id2732787">quoniam ubruuntur fooribus facillim, i<lb/>ad illos peruenire posit hotis. </s>
<s id="id2732815">Ide intra m nia utilisimi unt, pro<lb/>mnijs parum prount. </s>
<s id="id2732838">Quod uer ad tetudines attinet, ub qui<lb/>bus <expan abbr="lat&etilde;t">latent</expan> foores machin laterales, & fronte & ignes, & aqua al<lb/>tior prohibent omnino iniuriam, qu ab his imminet. </s>
<s id="id2732883">Cterum hu<lb/>iumodi cum in longum <expan abbr="differun&ttilde;">differuntur</expan> morbis, illuuie, <expan abbr="incõmodis">incommodis</expan>, plu<lb/>uijs, frigoribus omnino <expan abbr="di&longs;&longs;oluũtur">dioluuntur</expan>, ut nulla multitudo huic operi <lb/>ufficere posit. </s>
<s id="id2732945">Rhodus, Alba regia, Melita, Catrum <expan abbr="nouũ">nouum</expan>, Byzan <lb/>tium, i diferri potuient tempora, non cesient uictori quantum<lb/>uis uperbo. </s>
<s id="id2732985">Vicit pertinacia, audacia que umma, <expan abbr="Corcyrã">Corcyram</expan>, Viennam <lb/>capere <expan abbr="nõ">non</expan> potuit, quoniam in <expan abbr="longũ">longum</expan> trahebatur oppugnatio. </s>
<s id="id2733021">Mul <lb/>t machin, & pauci homines prd obeorum expoit unt: <lb/>pauc, & pauci homines obidebuntur potius, quam obidebunt. <lb/></s>
<s id="id2733068">Exercitus magnus dioluitur, & emetipum conumit, i nulla fiat <lb/>accesio aut exigua quomodo tabit: i magna auxilia omnia cor<lb/>rumpuntur. </s>
<s id="id2733104">Contr obesis auxilia i ueniant lutrata, & munita, et <lb/>omnibus necearijs ornata uiri integri <expan abbr="cõtra">contra</expan> fatigatos, & feos cor <lb/>pore, armati contra inermes, alacres contra torpidos uperueniunt. <lb/></s>
<s id="id2733153">Ob id prcipuum et auxilium prter hc his, qui oppugnantur co <lb/>pia militum, qui per initia nun <08> quiecant diu noctu que, <expan abbr="uerũ">uerum</expan> noctu <lb/>duo tubicines perpe <expan abbr="exercitũ">exercitum</expan> <expan abbr="in&longs;omn&etilde;">inomnem</expan> in armis tota nocte <expan abbr="cõtine">contine</expan> <lb/><expan abbr="bũt">bunt</expan>. </s>
<s id="id2733234">Serio <expan abbr="aũt">aunt</expan> die pugnare, & noctu <expan abbr="cũ">cum</expan> minim id <expan abbr="&longs;perãt">perant</expan>, & fatigati <lb/>unt: mira euenire olent in his inperatis, ac audacibus eruptionib. <lb/></s>
<s id="id2733284">perpe <expan abbr="etiã">etiam</expan> omnino upra <expan abbr="fid&etilde;">fidem</expan>. </s>
<s id="id2733311">Ita <expan abbr="nõ">non</expan> conquiecere oportet donec, <lb/>uel omnino cepto deinat hotis, aut <expan abbr="locũ">locum</expan> occupet ibi <expan abbr="relictũ">relictum</expan> po<lb/>tius <08> <expan abbr="qu&etilde;">quem</expan> elegerit. </s>
<s id="id2733369">nam <expan abbr="experimentũ">experimentum</expan> frequens do cuit, ubi ill ma <lb/>gn uires uo arbitrio <expan abbr="locũ">locum</expan>, <expan abbr="qu&etilde;">quem</expan> <expan abbr="elegerũt">elegerunt</expan> obtinere potuerint, <expan abbr="tand&etilde;">tandem</expan> <lb/>potiri locis <expan abbr="quãtumuis">quantumuis</expan> munitis in hoc d diximus <expan abbr="cõtra">contra</expan> <expan abbr="oppona&ttilde;">opponatur</expan>. <lb/></s>
<s id="id2733463">Etenim <expan abbr="&longs;ept&etilde;">eptem</expan> modis <expan abbr="cũ">cum</expan> urbes, at que arces <expan abbr="capian&ttilde;">capiantur</expan>, <expan abbr="quorũ">quorum</expan> duo unt ex <lb/>tra <expan abbr="&ptilde;&longs;ent&etilde;">pnentem</expan> <expan abbr="con&longs;ideration&etilde;">coniderationem</expan> obidio, <expan abbr="&qtilde;">quae</expan> magnitudine ambitus loci <expan abbr="tol-li&ttilde;">tol<lb/>litur</expan>, & proditio, <expan abbr="&qtilde;">quae</expan> cuto <expan abbr="dũ">dum</expan> <expan abbr="uigilãtia">uigilantia</expan>, cuniculi, euerio uperioris muri, <lb/>euerio ab imo per machinas, cuniculi, eu uffosio, urbis euerio, eu <lb/><expan abbr="&ecedil;dificiorũ">dificiorum</expan>: & <expan abbr="&qtilde;uo">quauo</expan> cant aggresio, eu oppugnatio per calas, & crates <lb/><expan abbr="cũ">cum</expan> agittarijs: his omnib. </s>
<s id="id2733655"><expan abbr="&longs;atisfactũ">atisfactum</expan> puto, prter <08> oppugnationi pro<lb/>pter <expan abbr="humilitat&etilde;">humilitatem</expan> <expan abbr="murorũ">murorum</expan>: <expan abbr="nã">nam</expan> lignis <expan abbr="opplen&ttilde;">opplentur</expan>, at que faciculis, terra que fo <lb/>: nihil. </s>
<s id="id2733721">n. </s>
<s id="id2733725">reitit immen illi potetati, & crudelitati <expan abbr="&longs;&ecedil;ui&longs;simorũ">uisimorum</expan> ty <lb/><expan abbr="rãnorũ">rannorum</expan>. </s>
<s id="id2733774"><expan abbr="Verũ">Verum</expan>, ut dixi, terra noctu <expan abbr="effodi&ttilde;">effoditur</expan>, ligna artificiois ignib. </s>
<s id="id2733797">eru
<pb xlink:href="015/01/136.jpg" pagenum="117"/>untur. </s>
<s id="id2733809">Et longum et opus iue per paucos, iue per multos quis ef<lb/>ficere conetur: ut non minus exigat temporis, qum obidio: nam <lb/>multitudine unus alterum impedit, & mortui uiuos, ut omnino res <lb/>it non peranda nii aduerus inertisimos. </s>
<s id="id2733852">Pontes euertunt machi <lb/>n, ignes que. </s>
<s id="id2733862">Sed ubi etiam muros obtinuerint ob rotunditatem in <lb/>illis conitere non pount. </s>
<s id="id2733879">Inde defenoribus propulantur ari<lb/>is, telis, ignibus, tranueris trabibus, machinis: illudque accedit com <lb/>modi, ut quanto plures eo facilius excutiantur. </s>
<s id="id2733912">Dixi non debere <lb/>uereri maxima etiam prterid, quoniam & it ip tanto anguine <lb/>acquiit tanto deorum & hominum iniuria modica cintilla ignis <lb/>ine munitionibus, exercitibus, iue machinis, abque terr <expan abbr="cõcu&longs;sio-ne">concusio<lb/>ne</expan>, aut inundatione, uel pete euertuntur. </s>
<s id="id2733984">In illam mieram lachry<lb/>mam patris cintilla ignis inferni, cm Deo placuerit, <expan abbr="mitti&ttilde;">mittitur</expan>, ex qua, <lb/>quod <expan abbr="coalitũ">coalitum</expan> et, multis eculis imperium luxu, crudelitate, tultitia <lb/>unius filij, uix uno lutro toto dioluitur. </s>
<s id="id2734039">Hanc <expan abbr="&longs;cintillã">cintillam</expan> cum felici <lb/>etiam genio ecum ex utero detulit Alexander Magnus. </s>
<s id="id2734060">In alijs alij <lb/>genium ortiti unt, alij <expan abbr="&longs;cintillã">cintillam</expan> detulere ab Orco. </s>
<s id="id2734084">Ex imperio Ay <lb/>riorum per luxum Sardanapalus: ex Medorum per <expan abbr="&longs;cintillã">cintillam</expan> Atya<lb/>ges: ex <expan abbr="Per&longs;arũ">Perarum</expan> per tultitiam Darius: ex <expan abbr="Romanorũ">Romanorum</expan> Honorius. </s>
<s id="id2734139">Di <lb/>ces, hc quid ad proportionem? </s>
<s id="id2734149">Im uelut machina ad <expan abbr="perpendiculũ">perpendiculum</expan> <lb/>librata pauculo illo puluere Pyrio <expan abbr="urb&etilde;">urbem</expan> euertit, ita cintilla illa infer <lb/>ni ignis emini magni tyranni indita euertit at que dioluit totum re<lb/>gnum ine machinis, ut dixi, uel exercitibus ullis, & quod maius et <lb/>remedio nullo. </s>
<s id="id2734204">Sed puerulo indito luxus, ignaui, crudelitatis at que<lb/>tultiti fontibus, mirabile dictu an, & ad proportionem diuino<lb/>rum <expan abbr="in&longs;trumentorũ">intrumentorum</expan> pertinens. </s>
<s id="id2734244">Sed redeamus ad intitutum: Video <lb/>enim, quid posit obijci, cilicet muros craos, et altiores tueri <expan abbr="urb&etilde;">urbem</expan> <lb/>& dificia illius poe abque aggeris erectione, & i <expan abbr="diruan&ttilde;">diruantur</expan> manere <lb/>etiam nihilominus imo magis, quod et terram, uque <expan abbr="quoniã">quoniam</expan> eadem <lb/>ratione manet, quia concuti non posit machinis: nec hotes id cu <lb/>raturos, perantes hoc <expan abbr="&longs;olũ">olum</expan> ufficere, d mnia olo <expan abbr="æquen&ttilde;">quentur</expan>, at que id <lb/><expan abbr="factũ">factum</expan> et Mediolani, & in arce eius, <expan abbr="tũ">tum</expan> Papi & in Cremoneni arce. <lb/></s>
<s id="id2734393">Verm ni fallor, ut paruis arcibus tanta ui tormentorum nullum <lb/>et <expan abbr="præ&longs;idiũ">pridium</expan>, aut alutis pes, ita neque <expan abbr="cõuenit">conuenit</expan>, ut muris humilibus ag <lb/>geri confidant, nam & pauci homines tanto labori non ufficerent, <lb/>& agger cum foa effoa cilicet terra defenores nimis in <expan abbr="angu&longs;tũ">angutum</expan> <lb/>cogeret. </s>
<s id="id2734484">At in urbibus contra eueniet: muris enim erectis altius ma <lb/>chin lapidum frutis hominem <expan abbr="occid&etilde;t">occident</expan>: an percua uperiore par <lb/>te ob coniunctionem inferior concutitur, & in de <expan abbr="totũ">totum</expan> imul cadit, <lb/>ut uidimus Papi, quo <expan abbr="cad&etilde;te">cadente</expan>, & foa impletur, & <foreign lang="greek">tEIkole/tois</foreign> facilior <lb/>aditus ad ubruendum reliquas partes <expan abbr="pr&ecedil;be&ttilde;">prbetur</expan>: im perculi defen
<pb xlink:href="015/01/137.jpg" pagenum="118"/>ores pe muneris ui obliuicuntur, deertaque ea parte liberum <lb/>ingreum hotibus exhibent. </s>
<s id="id2734617">Tum uer magis, quod non confi<lb/>dunt animo <expan abbr="nõ">non</expan> ad id parato, poe aggerem ufficientem, & in tam <lb/>breui tempore extruere, & etiam intelligunt, antequam erigatur, <lb/>patere lateribus introitum hotibus.</s></p><p type="margin">
<s id="id2734664"><margin.target id="marg411"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2734691">Propoitio centeimauigeima.</s></p><p type="main">
<s id="id2734706">Proportionem partium nauis ad eundem obliquum uentum <lb/>explorare.<lb/><arrow.to.target n="marg412"/></s></p><p type="margin">
<s id="id2734725"><margin.target id="marg412"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2734750">Sint mali in naui a b c, ad b e, c fuentus regione g h k etiam ad <lb/>perpendiculum feratur, ut anguli g d a, h e b, k f c int quales, dico <lb/>tamen diuero modo affici: nam cum premitur a uerus l, c premi<lb/>tur uerus f: at i prematur cuerus n a, premitur uerus d, at i pre<lb/><figure id="id.015.01.137.1.jpg" xlink:href="015/01/137/1.jpg"/><lb/>matur b uerus m, & a uer<lb/>us l, ed non quantum ex <lb/>g d, & cuerus n, ed non <lb/>quantum ex k f, ab eodem <lb/>ergo uento contrarij mo<lb/>tus efficiuntur ex uelorum <lb/>diueritate, etenim per uen <lb/>tum d feretur ad meridiem <lb/>nauis, & per uelum f ad Se <lb/>ptentrionem etiam didu<lb/>cto auxilio e l a ui, quanto <lb/>magis cum illo: & i uen<lb/>tus excipiatur in f uelo, <lb/>non iuuabit clauus, & i in <lb/>d dirigetur, & temperabitur motus, & i in e medio modo. </s>
<s id="id2734884">Ergo i <lb/>uentus feratur rect iuuabit, ut dici olet omnibus, & plenis uelis <lb/>excipere, i ex obliquo demittere antennam puppis, in autem ual<lb/>de obliqu us it, olo pror uelo utemur. </s>
<s id="id2734922">Si ualidior qum oportet <lb/>humiliore. </s>
<s id="id2734932">Atque hc potmodum unt diligenter numeranda, ac <lb/>metienda: nunc ufficiat cauam reddidie, & admonuie diueri<lb/>tatis motuum, qu ex uelis contingit: nam e fertur nauis, qu <lb/>prora dirigitur. </s>
<s id="id2734981">Ergo cum puppis tanto feratur uerus meridiem <lb/>a b, quanto prora uerus meridiem a d, & quanto puppis fertur uer <lb/>us <expan abbr="meridi&etilde;">meridiem</expan>, tanto prora fertur uerus boream, igitur quanto prora <lb/>fertur uerus meridiem a d, tanto uerus boream a b f, ed itus claui <lb/>potet multo plus in comparatione ueli d, quam f cilicet, quia di<lb/>tantia a b a et o a, & ditantia e c et o c, tanto plus ergo potet cla<lb/>ui itus in comparatione ad uelum d, quam f, quanta et proportio
<pb xlink:href="015/01/138.jpg" pagenum="119"/>o a, ad o c, igitur clauus et long potentior in comparatione ueli <lb/>d, quam f, ergo uelum d minus agit nauim, quam f. </s>
<s id="id2735082">Sed ut extrema <lb/>e habent, ita medium eorum comparatione, igitur malus b e uali<lb/>dior et, multo d a, & infirmior c f. </s>
<s id="id2735099">Verm, ut dixi, ob itum impli<lb/>citer ualidius et, uelum e quam f, & etiam quia, ut dixi, altior & <lb/>erasior olet ee, ideo multo ualidior tribus his cauis, qum e f: <lb/>adde quartam qud uelum habet maius, antiquo tempore uoca<lb/>tum acatius. </s>
<s id="id2735151">At ut etiam docui c b non et in medio, nec quiditat <lb/>ab a d & c f, ed in clinatur ad proram ideoque imbecillior: cum ergo <lb/>it qualium, & paulo maiorum uirium, qum c f, & tutior, & me<lb/>lius agatur per <expan abbr="clauũ">clauum</expan> qum c f, & it a d nimis iuto imbecillis, pro<lb/>pterea b e mali, & ueli maximus et uus: ade mali nomen per an<lb/>tonomaiam de ipo impliciter intelligatur.</s></p><p type="main">
<s id="id2735234">Propoitio centeimauigeimaprima.</s></p><p type="main">
<s id="id2735250">Flabelli uires, at que naturam declarare.</s></p><p type="main">
<s id="id2735259">Sit flabellum a b c appenum, ut olet, in a, & moueatur motu </s></p><p type="main">
<s id="id2735275"><arrow.to.target n="marg413"/><lb/>quai circa axem p a q in parte inferiore, & ar comprehenus ub <lb/>b h k, & patium it 1 m figur nauicularis, qu contat ee par<lb/>tem cylindri inanis ex formatione ab Euclide cripta: nam i pro<lb/>poneretur p a q ad perpendiculum upertans plano, fieret circum<lb/>ducta a b c uperficie, qu eet lata uperius, icut etiam inferius <lb/><arrow.to.target n="marg414"/><lb/>cylindrus: at uperius a b tenuis et, & anguta, ergo fiet pars cy<lb/>lindri inanis: quia non circunuoluitur, donecredeat. </s>
<s id="id2735382">Ergo per di<lb/>cta uperius ectio illius p r q s per axem et pars cuiudam elly<lb/><arrow.to.target n="marg415"/><lb/>pis. </s>
<s id="id2735414">Et ectio quuis plan uperficiei quiditans a b cuelut tu, <lb/>item que quiditans axi p a q et uperficies rectangula, quarum <lb/>una et imilis, & qualis b h k, et in una uperficie cum axe p a q <lb/>alia uer et quiditans eidem axi maior aut minor quiditanti<lb/>um, & ipa laterum, at que rectangula ac i cylindrus tans axi plano <lb/>quiditanti ecaretur iuxta longitudinem eu altitudinem uam: <lb/>& manifetum et, quod ita duo plana, & eorum uperficies ecant <lb/>e mutu ad rectos angulos.</s></p><p type="margin">
<s id="id2735544"><margin.target id="marg413"/>C<emph type="italics"/>o<emph.end type="italics"/>m.</s></p><p type="margin">
<s id="id2735571"><margin.target id="marg414"/>L<emph type="italics"/>ib.<emph.end type="italics"/> 11. <lb/><emph type="italics"/>diff.<emph.end type="italics"/> 21.</s></p><p type="margin">
<s id="id2735608"><margin.target id="marg415"/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 69.</s></p><p type="main">
<s id="id2735635">Quibus contitutis, qui tabunt iuxta l, & m longitudines aris <lb/>moti, & loci, per quem tranit flabellum, entient magnum uentum, <lb/>quoniam cum corpus m x l ab extremis partibus it elatius a b ex<lb/>tremis, tantes, & alti tangentur uento agitato. </s>
<s id="id2735675">Si uero edeant, aer <lb/>primum non attinget illos, ut etiam quia urum pellitur non per<lb/>ueniet ad illos, im diffugiet, ergo non refrigerabuntur. </s>
<s id="id2735699">Qui uer <lb/> lateribus l x m <expan abbr="&longs;tabũt">tabunt</expan> hiccinde, uelut in f g, i teterint, <expan abbr="nõ">non</expan> refriger <lb/><expan abbr="bũtur">buntur</expan>, quia <expan abbr="quãdo">quando</expan> flabellum erit in l, uel m aer decendet, ergo fugi <lb/>et ab illis, cum autem fuerit in x, erit in loco humiliori, & mouebi
<pb xlink:href="015/01/139.jpg" pagenum="120"/>tur diuera ratione, quippe ab f in h, & non ad latera, ergo ne que <lb/><figure id="id.015.01.139.1.jpg" xlink:href="015/01/139/1.jpg"/><lb/>contactu, neque motu, qui <lb/>fiet per quiditantem f, <lb/>& g non poterunt refrige<lb/>rari. </s>
<s id="id2735812">Sed i humili loco e<lb/>deant, quoniam ar decen <lb/>dit, ex l & m uerus x, & <lb/>etiam, quia erunt proximi <lb/>h k, <expan abbr="quãdo">quando</expan> fuerit in x, <expan abbr="refri-gerabun&ttilde;">refri<lb/>gerabuntur</expan> ualde. </s>
<s id="id2735866">Qui <expan abbr="aut&etilde;">autem</expan> <lb/><expan abbr="erũt">erunt</expan> iuxta h & k minus <expan abbr="re-frigerabun&ttilde;">re<lb/>frigerabuntur</expan> utrique, ed pau <lb/>lulum in reditibus propin <lb/>quis, & neque tantes, <expan abbr="neque&longs;ed&etilde;tes">neque<lb/>edentes</expan>, ed i altius attolla<lb/>tur h k. </s>
<s id="id2735944">Rurus i b h k fue<lb/>rit grauior eodem, ut de<lb/>cendat tanto impetu, <expan abbr="quã-to">quan<lb/>to</expan> acendit attractum, ut <lb/>pote ex ligno tenui nucis, <lb/>tunc multo magis refrige<lb/>rabit, & procul, <expan abbr="nõ">non</expan> ob uim <lb/>ualidiorem, ed quoniam <lb/>celerius occurantes ibi <lb/>contrarijs motibus, ac <expan abbr="ue-hem&etilde;tibus">ue<lb/>hementibus</expan> fiet colliio par <lb/>tium aris, & ideo in ambitum impelletur, & undique cubiculum <lb/>refrigerabit, quod non faciet maius long flabellum lento motu <lb/>agitatum, aut ex materia leui. </s>
<s id="id2736050">Idem multo magis contingeret, ubi <lb/>duo eent flabella laquearibus appena, qu ad perpendiculum <lb/><expan abbr="a&etilde;rem">aerrem</expan> mouerent, eu quod uperficies eo modo e haberent: & i <lb/>flabella rotunda eent, tunc maiorem ambitum aris occuparent, <lb/>& uelocius deficientibus angulis mouebuntur.</s></p><p type="main">
<s id="id2736110">Propoitio centeimauigeimaecunda.</s></p><p type="main">
<s id="id2736128">Contemptus circa olis rationem in umbris declarare.</s></p><p type="main">
<s id="id2736140">Contat primm olem, & excentro, & toto eius ambitu illumi<lb/>nare hanc primm diueritatem, qu aliquando tota diametro <lb/>computata dimidium unius partis totius cli excedit: cioterici <lb/>negligunt, ut exiguam. </s>
<s id="id2736178">Secund etiam diueritatis illius, qua mo<lb/>d terra uerus abidem defertur, mod ad terram decendere to<lb/>tidem uariata altitudine, non parum nullam habent rationem, eu
<pb xlink:href="015/01/140.jpg" pagenum="121"/>qud tanta ne it, ut euidentem in gnomonibus faciat uarietatem, <lb/>eu qud incertum adhuc it, an id uer oli accidat. </s>
<s id="id2736248">Tertium et fi<lb/>nis umbr ipius gnomonis, qui incertus et, ut pars non contem<lb/>nenda in dubium uertatur, quoniam enim ex obcuro in illumi<lb/>natum feratur, attamen contemnitur etiam. </s>
<s id="id2736286">Quartum qud cum <lb/>ol moueatur in pira, fingitur quai in parallelo quinoctiali circu <lb/>lo circumagatur ab his, qui horologia decribunt. </s>
<s id="id2736312">Quintum qud <lb/>cum inqualiter in orbe uo moueatur quanuis exigua it hc dif<lb/>ferentia, qualiter <expan abbr="tam&etilde;">tamen</expan> moueri prupponitur. </s>
<s id="id2736352">Sextum et, qud <lb/>dies quales upponuntur, qui tamen tum ex ratione partis pera<lb/>grat, tum ratione acenus <expan abbr="eiu&longs;d&etilde;">eiudem</expan> unt inquales, & <expan abbr="tam&etilde;">tamen</expan> hc in<lb/>qualitas <expan abbr="etiã">etiam</expan> in <expan abbr="horarũ">horarum</expan> computatione prtermittitur. </s>
<s id="id2736438">Sed & hc ut <lb/>prior ratione magis, <expan abbr="quã">quam</expan> enu <expan abbr="deprehendi&ttilde;">deprehenditur</expan>. </s>
<s id="id2736470"><expan abbr="Septimũ">Septimum</expan> et dicrimen, <lb/>d oritur ex uius circulo eu horizonte, & circulo traneunte p cen <lb/><expan abbr="trũ">trum</expan> mundi, nam horizon uere <expan abbr="tãto">tanto</expan> minor et circulo magno, quan<lb/>tum et emidiameter terr, <expan abbr="cõparatus">comparatus</expan> ad <expan abbr="&longs;emidiametrũ">emidiametrum</expan> orbis cle <lb/>tis, ed et inenilis quantitatis. </s>
<s id="id2736574"><expan abbr="Octauũ">Octauum</expan> et, quod trianguli ex gno<lb/>mone umbra, & radijs olis latera non mutant lineas, qu ole ad <lb/>centrum terr deueniunt, nec qud maius et, radius olis ad uerti<lb/>cem hominis breuior habetur femidimetiente. </s>
<s id="id2736623">Hc <expan abbr="igi&ttilde;">igitur</expan> omnia <expan abbr="&longs;ci-otericorũ">ci<lb/>otericorum</expan> opifices non oberuant, ed negligunt. </s>
<s id="id2736660">Verum quatuor <lb/>tantm altitudinem poli regionis locum olis in eclyptica locum <lb/>olis in circulo quinoctialis, uel quinoctiali parallelo, ex qui<lb/>bus tribus fit altitudo olis, una in circulo cilicet uerticali ab hori<lb/>zonte, & differentia line meridian linea uerus polum, quam <lb/><arrow.to.target n="marg416"/><lb/>otendit lapis Herculeus, de qua dictum et uperius.</s></p><p type="margin">
<s id="id2736732"><margin.target id="marg416"/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 84.</s></p><p type="main">
<s id="id2736760">Propoitio centeimauigeimatertia.</s></p><p type="main">
<s id="id2736775">Cognita ratione umbr ad gno <lb/>monem inum, & arcum altitudi<lb/>nis ab horizonte quouis tempo<lb/>re dignocere.</s></p><figure id="id.015.01.140.1.jpg" xlink:href="015/01/140/1.jpg"/><p type="main">
<s id="id2736814">Sit circulus magnus, in quo ol <lb/><arrow.to.target n="marg417"/><lb/>a f g upertans ad perpendicu<lb/>lum circulo uius f e g, quos mani <lb/>fetum et tranire per idem cen<lb/>trum mundi c, quia magni unt, & <lb/>it c d erecta ad perpendiculum <lb/>uper f g, nam perinde et per e<lb/>ptimum contemptum, ac i uper<lb/><arrow.to.target n="marg418"/><lb/>ficies horizontis traneat per terr centrum, & pedes per octauum, <lb/><arrow.to.target n="marg419"/><lb/>ideo proportio e c ad c d umbr ad gnomonem, ut b e ad b a, ergo
<pb xlink:href="015/01/141.jpg" pagenum="123 [=122]"/>per demontrata b a cognita in comparatione a d e a, e a autem per <lb/>octauum contemptum et dimetiens circuli, ergo a b inus notus, <lb/>& arcus f a, quod et primum cognitum. </s>
<s id="id2736942">Et hic quidem circulus <lb/>uerticalis dicitur, quia per illum tranit, aliter non eet ad perpen<lb/>diculum horizonti.<lb/><arrow.to.target n="marg420"/></s></p><p type="margin">
<s id="id2736972"><margin.target id="marg417"/>C<emph type="italics"/>o<emph.end type="italics"/>_{m}.</s></p><p type="margin">
<s id="id2736998"><margin.target id="marg418"/>P<emph type="italics"/>rced.<emph.end type="italics"/> P<emph type="italics"/>ro <lb/>po.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2737040"><margin.target id="marg419"/>P<emph type="italics"/>rop.<emph.end type="italics"/> 113.</s></p><p type="margin">
<s id="id2737064"><margin.target id="marg420"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 1.</s></p><p type="main">
<s id="id2737090">Ex hoc equitur, quod altitudines olis quales omnes in uno <lb/>unt circulo horizonti parallelo. </s>
<s id="id2737107">Et i ol fuerit in uno circulo ho<lb/>rizonti parallelo, altitudines olis, & umbr magnitudines qua<lb/>les erunt.</s></p><p type="main">
<s id="id2737139"><arrow.to.target n="marg421"/></s></p><p type="margin">
<s id="id2737150"><margin.target id="marg421"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 2.</s></p><p type="main">
<s id="id2737177">Sol nii bis in una die potet ee in circulo horizonti parallelo, <lb/>emel ante meridiem, & emel pot, tantundem ab eodem ditans.</s></p><p type="main">
<s id="id2737208"><arrow.to.target n="marg422"/></s></p><p type="margin">
<s id="id2737219"><margin.target id="marg422"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 3.</s></p><p type="main">
<s id="id2737246">Cum ergo ita it, necee et umbras quales, & circulum hori<lb/>zonti <expan abbr="parallelũ">parallelum</expan> fieri ub in qualibus horis in diueris emper die<lb/>bus, prterquam cum in punctis fuerit qualis ab quinoctiali, & <lb/>in eandem partem declinationis, & hoc bis <expan abbr="cõtingit">contingit</expan> olum in anno <lb/>pro quolibet circulo parallelo, icut in eodem die etiam bis <expan abbr="tãtum">tantum</expan>, <lb/>ut dictum et.</s></p><p type="main">
<s id="id2737343"><arrow.to.target n="marg423"/></s></p><p type="margin">
<s id="id2737354"><margin.target id="marg423"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2737381">Nam exempli gratia, cum ol et in initio Capricorni, & in Cli <lb/>medio, minima et umbra eius diei, & totius anni. </s>
<s id="id2737400">Cum ergo fuerit <lb/>ante meridiem, uel pot, erit umbra maior ex uppoito ecudo um<lb/>bra meridiei: at ei qualis poterit ee umbra meridiei alterius diei <lb/>ex primo uppoito, ergo umbr quales diuerorum dierum fi<lb/>unt ub diuero itu olis, quo d circulum meridiei, quod erat de<lb/>montrandum.</s></p><p type="main">
<s id="id2737477"><arrow.to.target n="marg424"/></s></p><p type="margin">
<s id="id2737488"><margin.target id="marg424"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 4.</s></p><p type="main">
<s id="id2737515">Ex hoc equitur, quod horarum determinatio fit ecundum line<lb/>am in qualem obliquam, qu toti anno eruiat, ut qualium um<lb/>brarum determinatio hararum & partium eius numerum.</s></p><p type="main">
<s id="id2737550"><arrow.to.target n="marg425"/></s></p><p type="margin">
<s id="id2737561"><margin.target id="marg425"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 5.</s></p><p type="main">
<s id="id2737587">Ex quo colligitur modus faciendi gnomonem, eu per umbras <lb/>rectas, eu per ueras, qui docebit toto anno non <expan abbr="&longs;olũ">olum</expan> horas, ed mo <lb/>menta <expan abbr="pul&longs;uũ">puluum</expan>, de quibus <expan abbr="dictũ">dictum</expan> et quod MMMDC horam <expan abbr="perficiũt">perficiunt</expan>.</s></p><p type="main">
<s id="id2737659">Propoitio centeimauigeimaquarta.</s></p><p type="main">
<s id="id2737675">Proportionem umbr uer ee ad gnomonem, uelut gnomo<lb/>nis ad umbram ueram.</s></p><p type="main">
<s id="id2737702"><arrow.to.target n="marg426"/></s></p><p type="margin">
<s id="id2737713"><margin.target id="marg426"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2737739">Vmbra uera dicitur, quoties gnomo in pariete ad perpendicu<lb/>lum figitur, ic ut gnomo quiditet circulo horizontis. </s>
<s id="id2737759">Sit ergo <lb/>paries c k ad perpendiculum f g, & h k a d gnomo ad perpendicu<lb/>lum parietis & ol, ut prius in a, & it primo k h tant longitudinis </s></p><p type="main">
<s id="id2737785"><arrow.to.target n="marg427"/><lb/>ut umbr locus it <expan abbr="pũctus">punctus</expan> d, ut it radius a h d e, eritque angulus d u<lb/>trin que qualis, & propterea triangulus k h d imilis d c e. </s>
<s id="id2737825">Sit modo <lb/><arrow.to.target n="marg428"/><lb/>gnomo maior m l ipo h k & c l maior c k eu qualis, & quam an<lb/>guli k & l recti unt, & anguli l m n, & k h d qualis, quia a n, & a c
<pb xlink:href="015/01/142.jpg" pagenum="113 [=123]"/>unt quiditantes per octauum contemptum, erunt per dicta tri<lb/>anguli imiles, igitur proportio l m gnomonis ad l n umbram <lb/>ut k h gnomonis ad k d umbram, ed k h, ad k d, ut c e umbr ad c d <lb/>gnomonem: igitur proportio l m gnomonis ad l n <expan abbr="umbrã">umbram</expan>, ut um<lb/>br c e ad c d gnomonem, quod fuit demontrandum.</s></p><p type="margin">
<s id="id2737918"><margin.target id="marg427"/>P<emph type="italics"/>er<emph.end type="italics"/> 15. <emph type="italics"/>pri <lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2737967"><margin.target id="marg428"/>P<emph type="italics"/>er<emph.end type="italics"/> 4. <emph type="italics"/>exti<emph.end type="italics"/><lb/>E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="main">
<s id="id2738015">Ex hoc primm patet & prcedenti, quod cognita proportione <lb/><arrow.to.target n="marg429"/><lb/>umbr uer ad gnomonem cognocitur inus olis, & arcus altitu<lb/>dinis in circulo magno, & et altitudo ab horizontis parte, qu <lb/>proximior et loco olis, ut demontratum nobis in Geometricis.</s></p><p type="margin">
<s id="id2738079"><margin.target id="marg429"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 1.</s></p><p type="main">
<s id="id2738105">Se quitur etiam, qud cm umbra fuerit qualis gnomoni, eu <lb/><arrow.to.target n="marg430"/><lb/>recta, eu uera olis, uel Lun, uel tell, altitudo erit partium qua<lb/>draginta quin que: nam anguli d & e, uel d & h erunt quales: igitur <lb/>arcus f a medietas quart ide partium xlv. </s>
<s id="id2738162">Et i gnomo fuerit ma<lb/>ior umbra uera, uel minor recta, erit arcus f a minor xlv partibus, i <lb/>contr maior. </s>
<s id="id2738186">Et hoc ubique terrarum. </s>
<s id="id2738190">Et ubi non posit tantundem <lb/>eleuari, ut quando ol et ub circulo capricorni, nunquam nobis <lb/><arrow.to.target n="marg431"/><lb/>gnomo quabitur umbr rect ed emper erit minor, & emper <lb/><arrow.to.target n="marg432"/><lb/>maior umbra uera pari ratione.</s></p><p type="margin">
<s id="id2738251"><margin.target id="marg430"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 2.</s></p><p type="margin">
<s id="id2738277"><margin.target id="marg431"/>P<emph type="italics"/>er<emph.end type="italics"/> 5. <emph type="italics"/>primi<emph.end type="italics"/><lb/>E<emph type="italics"/>lement.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2738325"><margin.target id="marg432"/>P<emph type="italics"/>er ult. </s>
<s id="id2738340">exti<emph.end type="italics"/><lb/>E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="main">
<s id="id2738368">Propoitio centeimauigeimaquinta.</s></p><p type="main">
<s id="id2738383">Proportionem dimetientis, & peripheri cuiuslibet circuli paral <lb/>leli quinoctiali per cognitam partem magni circuli demontrare.</s></p><p type="main">
<s id="id2738405">Hc erat tam clara, ut hic locum non mereretur: tam necearia <lb/><arrow.to.target n="marg433"/><lb/>huic propoito, ut non potuerit omitti. </s>
<s id="id2738428">Sit ergo Aequinoctij circu<lb/>lus a b portio circuli magni nota, a c parallelus circulus, quinoctij <lb/>circulo c d, erit igitur inus c d notus. </s>
<s id="id2738447">Et ide <expan abbr="quadratũ">quadratum</expan> c d notum, <lb/><arrow.to.target n="marg434"/><lb/>ergo & pars utraque b d d a nota. </s>
<s id="id2738472">Quare detracta a d ex d b relin qui<lb/>tur d g qualis f c diametro paralleli asignari. </s>
<s id="id2738486">Quare proportio <lb/><arrow.to.target n="marg435"/><lb/>a b ad e f nota ex obiter upr demontratis, & pariter ambi<lb/>tus circuli a b ad ambitum circuli c d, et enim ut dimetientis ad di<lb/><arrow.to.target n="marg436"/><lb/>metientem.</s></p><p type="margin">
<s id="id2738531"><margin.target id="marg433"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s id="id2738557"><margin.target id="marg434"/>P<emph type="italics"/>er<emph.end type="italics"/> 3. <emph type="italics"/>tertij,<emph.end type="italics"/><lb/>& 8. & 17. <lb/><emph type="italics"/>exti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2738619"><margin.target id="marg435"/>P<emph type="italics"/>er<emph.end type="italics"/> 5. <emph type="italics"/>ecun<lb/>di<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2738670"><margin.target id="marg436"/>P<emph type="italics"/>er<emph.end type="italics"/> 113. <lb/>P<emph type="italics"/>ropo.<emph.end type="italics"/></s></p><p type="main">
<s id="id2738712">Propoitio centeimauigeimaexta.</s></p><p type="main">
<s id="id2738730">Circuli horarij naturam declarare.<lb/><arrow.to.target n="marg437"/></s></p><p type="margin">
<s id="id2738745"><margin.target id="marg437"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><figure id="id.015.01.142.1.jpg" xlink:href="015/01/142/1.jpg"/><p type="main">
<s id="id2738780">Circulus horarius et circulus magnus <lb/>traniens per <expan abbr="&longs;ol&etilde;">olem</expan>, aut lunam, aut quoduis <lb/>ydus, de quo agitur, & per polos mundi, <lb/>ide differt circulo priore altitudinis So<lb/>lis, quia ille tat ad perpendiculum uper <lb/>horizontem, nii cum tangitur uice meridi<lb/>ani, uterque tamen tranit per <expan abbr="centrũ">centrum</expan> mundi, <lb/>ac olis. </s>
<s id="id2738854">Hic etiam ad imiles partes qui<lb/>noctij circulum, & omnes parallelos ecat.
<pb xlink:href="015/01/143.jpg" pagenum="124"/>Et principalis et meridianus, ide ab illo Atrologi horas utrinque<lb/>ante, & pot numerant. </s>
<s id="id2738895">Ide <expan abbr="clarũ">clarum</expan> et, qud hor meridie com<lb/>putat unt <expan abbr="cõmunes">communes</expan>, habitantibus ub quauis altitudine poli, & <lb/>ubiuis it, ol mod regiones qualiter ditent fortunatis, eu int <lb/>in eadem longitudine.</s></p><p type="main">
<s id="id2738977">Propoitio centeimauigeimaeptima.</s></p><p type="main">
<s id="id2738996">Data Poli altitudine ortus amplitudinem demontrare.<lb/><arrow.to.target n="marg438"/></s></p><p type="margin">
<s id="id2739014"><margin.target id="marg438"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2739040">Sit horizon a d b quinoctij circulus <lb/><figure id="id.015.01.143.1.jpg" xlink:href="015/01/143/1.jpg"/><lb/>a k f eclyptica c g, & punctus ortus in ea g. <lb/></s>
<s id="id2739065">& c initium arietis, & g b amplitudo ortiua <lb/>& c e, c f quart circulorum, ut it e f maxi<lb/>ma olis declinatio, & polus mundi borea<lb/>lis l, quia igitur l d nota et ex uppoito, & <lb/>l k quadrans erit k h <expan abbr="re&longs;iduũ">reiduum</expan> ad dimidium <lb/>circuli notum. </s>
<s id="id2739121">Quia uer quinoctium, & <lb/>Meridianus ecant e ad angulos rectos, & <lb/>b a quiditat ab utro que polo, erit b polus <lb/>h d, quare b k, quarta circuli, & angulus k <lb/>rectus. </s>
<s id="id2739155">Igitur umus in dipoitione tabula<lb/>rum primi mobilis, ergo etiam oppoitus <lb/>triangulus, qui ei et qualis, & quiangu<lb/>lus in eadem dipoitione b m d, quare cum <lb/>data it g n declinatio <expan abbr="pũcti">puncti</expan> g dati, datus erit, & arcus g b quitus.</s></p><p type="main">
<s id="id2739221">Propoitio centeimauigeimaoctaua.</s></p><p type="main">
<s id="id2739238">Nota amplitudine ortus cuiuque <expan abbr="pũcti">puncti</expan> <expan abbr="arcũ">arcum</expan> <expan abbr="&longs;emidiurnũ">emidiurnum</expan> inuenire.</s></p><p type="main">
<s id="id2739279"><arrow.to.target n="marg439"/></s></p><p type="margin">
<s id="id2739290"><margin.target id="marg439"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2739316">Sit in eadem figura nota g b, uolo illius <expan abbr="arcũ">arcum</expan> emidiurnum. </s>
<s id="id2739331">Cum <lb/>ergo g n it declinatio, erit pars arcus Meridiani horarij per polos <lb/>traneuntis, compleatur ergo l g n o, & quia g n nota et, quia de<lb/>clinatio puncti dati, & g b nota ex uppoito, & f angulus rectus, <lb/>quia e f et portio meridiani, erit b n nota differentia acenionis a <lb/>quarta circuli k b, <expan abbr="igi&ttilde;">igitur</expan> tota k n arcus emidiurnus. </s>
<s id="id2739389"><expan abbr="Quoniã">Quoniam</expan> g p paral <lb/>lelus imilis et k n, & in eo <expan abbr="reuolui&ttilde;">reuoluitur</expan> Sol: ergo quando enim perue<lb/>niet ad p. </s>
<s id="id2739421">Poumus etiam ine inuentione arcus ortus amplitudi<lb/>nis per triangulum k m d ex notitia g n cognocere eandem n b.</s></p><p type="main">
<s id="id2739445"><arrow.to.target n="marg440"/></s></p><p type="margin">
<s id="id2739456"><margin.target id="marg440"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2739483">Ex his duabus equitur <expan abbr="cõuer&longs;a">conuera</expan> cilicet, quae data magnitudine diei <lb/><expan abbr="cuiu&longs;cũque">cuiucunque</expan> in quauis regione nota erit poli altitudo <expan abbr="eiu&longs;d&etilde;">eiudem</expan> regionis.</s></p><p type="main">
<s id="id2739544">Propoitio centeimauigeimanona.</s></p><p type="main">
<s id="id2739559">Data altitudine olis in quacunque regione quacunque die ditan<lb/>tiam olis Meridiano cognocere.</s></p><p type="main">
<s id="id2739587"><arrow.to.target n="marg441"/></s></p><p type="margin">
<s id="id2739598"><margin.target id="marg441"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2739625">Sit Horizon a b c d quinoctij circulus b e d. </s>
<s id="id2739633">Meridianus a e c <lb/>Polus mundi Borealis f uertex, g, <expan abbr="pũctus">punctus</expan> in eclyptica h ducatur ex
<pb xlink:href="015/01/144.jpg" pagenum="125"/>polo mundi circulus horarius f h k ad quinoctij circulum, & uer<lb/>ticalis circulus p h l uque ad Horizontem, & circulus parallelus <lb/>quinoctij circulo h m, it ergo h l altitudo olis nota, igitur h g nota </s></p><p type="main">
<s id="id2739686"><arrow.to.target n="marg442"/><lb/>erit reiduum quart circuli, & imiliter h k <lb/><figure id="id.015.01.144.1.jpg" xlink:href="015/01/144/1.jpg"/><lb/>nota, quia declinatio puncti dati in eclypti <lb/>ca et n nota dies, & locus olis ex uppoi<lb/>to ergo nota fh <expan abbr="re&longs;iduũ">reiduum</expan> quart circuli no<lb/>ta et <expan abbr="etiã">etiam</expan> g e, qu et qualis altitudini po<lb/>li ex uppoito, ergo reiduum quadrantis <lb/>f g, ergo triangulus f g h notorum laterum <lb/>ergo notus angulus f, ergo arcus k e ditan <lb/><arrow.to.target n="marg443"/><lb/>tia umpta in quinoctij circulo puncti h, <lb/>cui imilis et arcus h m ex parallelo h m, nam quando k perueniet <lb/><arrow.to.target n="marg444"/><lb/>in e h perueniet in m, & in quali tempore, qua diuia per quinde<lb/>cim gradus, habebimus horas <expan abbr="di&longs;tãti&ecedil;">ditanti</expan> olis Meridie ante, uel pot, <lb/>& minuta horarum dando quibuslibet gradibus quatuor minuta <lb/>hor, & quibuslibet minutis graduum quatuor ecunda hor, & <lb/>ita habebimus tempus exactisimum Meridie in quacunque regi<lb/>one, & in quacunque hora diei.</s></p><p type="margin">
<s id="id2739907"><margin.target id="marg442"/>P<emph type="italics"/>er<emph.end type="italics"/> 123. <lb/>P<emph type="italics"/>ropo.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2739948"><margin.target id="marg443"/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 34. <lb/><emph type="italics"/>lib.<emph.end type="italics"/> 4.</s></p><p type="margin">
<s id="id2739989"><margin.target id="marg444"/>D<emph type="italics"/>e<emph.end type="italics"/> T<emph type="italics"/>riang.<emph.end type="italics"/><lb/>M<emph type="italics"/>onteregij.<emph.end type="italics"/></s></p><p type="main">
<s id="id2740035">Propoitio centeimatrigeima.</s></p><p type="main">
<s id="id2740051">Data regionis altitudine, & loco olis proportionem gnomo<lb/>nis tam ad umbram rectam, quam ueram, uel etiam in cylindro de<lb/>terminare.</s></p><p type="main">
<s id="id2740075">Hc et propoitio illa pulcherrima, quam tot ambagibus tradi<lb/><arrow.to.target n="marg445"/><lb/>dere antiqui cum uis analematibus, & cioteris, nec tamen demon <lb/>trationem, nec rationem exactam intrumenortum contructio<lb/>nem, qua poemus per umbras rectas ueras, & cylindricas cire ad <lb/>unguem, qualis hora, & minutum, & ecundum diei eet quocun<lb/>que anni tempore. </s>
<s id="id2740145">Plerique autem tam laborio id conati unt de<lb/>montrare, ut tudioos deterruerint ab opere: res autem ipa facil<lb/>lima et. </s>
<s id="id2740179">Propoita ergo Poli exacta altitudine olis in Meridie <lb/>declinatione addita uel detracta, habebis reiduum eius ad qua<lb/>drantem f g, & imiliter habebis ex declinatione nota loci olis de<lb/>tracta quadrante f h & iuxta horam tuam, & minutum multi<lb/><arrow.to.target n="marg446"/><lb/>plicatum per quindecim arcum k e quare angulum f, ex quo arcum <lb/>g h, quare reiduum h l, igitur punctum umbr rect, uel uer ipi<lb/>us gnomonis ad unguem, & ita contitues horologium exactisi<lb/>mum ecundum ea, qu dixi in Corrolarijs upradictis, & quia ho<lb/><arrow.to.target n="marg447"/><lb/>rizon a b c d ecat quinoctialem in <expan abbr="c&etilde;tro">centro</expan> terr ducta g h k, erunt <lb/>anguli b h g, & k h l quales. </s>
<s id="id2740303">Igitur poito g ortu puncti eclypti<lb/>c, erit g b ortus amplitudo nota, & ide angulus b h g, & k h l
<pb xlink:href="015/01/145.jpg" pagenum="126"/><arrow.to.target n="marg448"/><lb/>notus, & ita extendemus per totum annum. </s>
<s id="id2740337">Cum uer fuerit g ele<lb/>uatus erit, ut <expan abbr="demõ&longs;tratum">demontratum</expan> et, in circulo magno uerticali, ergo an<lb/>gulus fiet in eodem circulo, quia gnomo et etiam in illius uperfi<lb/>cie. </s>
<s id="id2740380">Ergo angulus erit qualis angulo, quem faceret ol, i oriretur <lb/><arrow.to.target n="marg449"/><lb/><figure id="id.015.01.145.1.jpg" xlink:href="015/01/145/1.jpg"/><lb/>in puncto horizontis, quem ecat circulus <lb/>uerticalis ub ea altitudine: ed his et no<lb/>tus: nam in priore figura g h f et notus ea<lb/><arrow.to.target n="marg450"/><lb/><expan abbr="d&etilde;">dem</expan> ratione, qua f, & ide ei oppoitus k h n, <lb/>& k rectus, et enim f polus b d, & h k decli <lb/>natio nota ergo k n, & h n not. </s>
<s id="id2740469">At e k, & <lb/>g h fuere not. </s>
<s id="id2740479">Ergo e n, & g n, quare rei<lb/>du n l & n b not. </s>
<s id="id2740498">Et autem angulus l <lb/>rectus. </s>
<s id="id2740508">ergo ortus amplitudo punctil nota <lb/>cilicet arcus l b, ergo in prenti figura angulus m h b, ergo k h l. <lb/></s>
<s id="id2740525">igitur poterimus tatuere angulos umbrarum, & iam poumus <lb/>determinare magnitudinem: ergo punctum ad <expan abbr="ungu&etilde;">unguem</expan> umbr qua<lb/>libet hora, & parte hor ingulis diebus in quacunque regione dat <lb/>altitudinis poli uera, & rects. </s>
<s id="id2740570">In cylindrica autem eodem modo i<lb/>cut in uera, et enim pecies umbr uer, nii quod analema ob ob<lb/>liquitatem cylindri melius aptatur, rotundum cilicet cum <expan abbr="rotũdo">rotundo</expan>.</s></p><p type="margin">
<s id="id2740623"><margin.target id="marg445"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s id="id2740648"><margin.target id="marg446"/>P<emph type="italics"/>er<emph.end type="italics"/> 28. <emph type="italics"/>li.<emph.end type="italics"/> 4. <lb/><emph type="italics"/>loan. </s>
<s id="id2740687">de<emph.end type="italics"/> M<emph type="italics"/>on <lb/>teregij de<emph.end type="italics"/><lb/>T<emph type="italics"/>riang.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2740724"><margin.target id="marg447"/>P<emph type="italics"/>er<emph.end type="italics"/> 123. <lb/><emph type="italics"/>uel<emph.end type="italics"/> 124. <lb/>P<emph type="italics"/>ropo.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2740777"><margin.target id="marg448"/>P<emph type="italics"/>rop.<emph.end type="italics"/> 123. <lb/>C<emph type="italics"/>orol.<emph.end type="italics"/> 1.</s></p><p type="margin">
<s id="id2740817"><margin.target id="marg449"/>P<emph type="italics"/>er<emph.end type="italics"/> 127. <lb/>P<emph type="italics"/>ropo.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2740858"><margin.target id="marg450"/>P<emph type="italics"/>er<emph.end type="italics"/> 15. <emph type="italics"/>pri <lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="main">
<s id="id2740906">Propoitio centeimatrigeimaprima.</s></p><p type="main">
<s id="id2740922">Si line alicui dupla alterius <expan abbr="adiunga&ttilde;">adiungatur</expan>, erit proportio duarum ad <lb/><expan abbr="primã">primam</expan> maior, quam dupli, cum prima ad primam cum una adiecta.</s></p><p type="main">
<s id="id2740952">Sit a b linea, cui adiecta it b c, & rurus ad b c c d <expan abbr="æ&qacute;ualis">qualis</expan> b c <lb/>dico, quod proportio a c ad a b et maior, qum a d ad a c. </s>
<s id="id2740985">Propor <lb/><arrow.to.target n="marg451"/><lb/>tio enim c d ad c a minor et, qum ad a b per octauam quinti E<lb/>lementorum. </s>
<s id="id2741008">Ergo minor d c ad c a qum c b ad a b, quia b c & c d <lb/>unt quales, ide <expan abbr="æqual&etilde;">qualem</expan> habent <expan abbr="proportion&etilde;">proportionem</expan> <lb/>ad a b: <expan abbr="igi&ttilde;">igitur</expan> coniungendo per 28. Quinti propor <lb/><figure id="id.015.01.145.2.jpg" xlink:href="015/01/145/2.jpg"/><lb/>tio d a ad a c minor, quam c a ad a b, quod erat demontrandum.<lb/><arrow.to.target n="marg452"/></s></p><p type="margin">
<s id="id2741084"><margin.target id="marg451"/>C<emph type="italics"/>o<emph.end type="italics"/>m.</s></p><p type="margin">
<s id="id2741110"><margin.target id="marg452"/>P<emph type="italics"/>er<emph.end type="italics"/> 7. <emph type="italics"/>quin<lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="main">
<s id="id2741160">Propoitio centeimatrigeimaecunda.</s></p><p type="main">
<s id="id2741178">Si ad duas lineas, quarum una alteri dupla it eadem linea adda<lb/>tur erit aggregati ex minore, & a d adiecta ad ipam <expan abbr="minor&etilde;">minorem</expan> minor <lb/>proportio quam aggregati ex maiore, & adiecta ad ipam maio<lb/>rem duplicata.</s></p><p type="main">
<s id="id2741218"><arrow.to.target n="marg453"/></s></p><p type="margin">
<s id="id2741229"><margin.target id="marg453"/>C<emph type="italics"/>o<emph.end type="italics"/>m.</s></p><p type="main">
<s id="id2741256">Sint du line a b, & c d. </s>
<s id="id2741266">& it c d dupla ad a b, ad datur <expan abbr="cõmunis">communis</expan> <lb/><figure id="id.015.01.145.3.jpg" xlink:href="015/01/145/3.jpg"/><lb/>b e, & uo cetur iuncta c d, d f dico, <lb/>quod proportio e a ad a b, et mi<lb/>nor duplicata f c ad c d, adij cia<lb/>tur d f qualis g f, quia ergo g d <lb/>et dupla ad f d, ideo ad e b c d autem et du pla ad a b, tota igitur
<pb xlink:href="015/01/146.jpg" pagenum="127"/>g c duplatoti e a. </s>
<s id="id2741333">quare ut g c ad g d ut e a ad e b <expan abbr="permutãdo">permutando</expan>, & per <lb/>eueram ut e a ad a b, ita g c ad c d, ut g c ad c d <expan abbr="cõponitur">componitur</expan> ex g e ad <lb/>f e, & f c ad c d, igitur e a ad c b componitur ex eidem. </s>
<s id="id2741371">Proportio <lb/>autem g c ad f c et minor, quam f c ad c d, igitur minor qum du<lb/>plicata f c ad c d. </s>
<s id="id2741391">contat uer ex eidem, quod proportio c a ad a b <lb/>maior et duplicata g c ad f c.</s></p><p type="main">
<s id="id2741415">Propoitio centeimatrigeimatertia.</s></p><p type="main">
<s id="id2741432">Si fuerint du quantitates, quarum una alteri dupla it: minua<lb/>tur minore qudam <expan abbr="quãtitas">quantitas</expan> eademque maiori addatur, erit mino<lb/>ris ad <expan abbr="re&longs;iduũ">reiduum</expan> maior proportio, <expan abbr="quã">quam</expan> aggregati ad <expan abbr="maior&etilde;">maiorem</expan> duplicata. <lb/></s>
<s id="id2741498">Si uer minori addatur et maiore detrahatur, erit aggregati ad mi<lb/>nore m minor proportio qum maioris ad reiduum duplicata.</s></p><p type="main">
<s id="id2741521"><arrow.to.target n="marg454"/></s></p><p type="margin">
<s id="id2741532"><margin.target id="marg454"/>C<emph type="italics"/>o<emph.end type="italics"/>m.</s></p><figure id="id.015.01.146.1.jpg" xlink:href="015/01/146/1.jpg"/><p type="main">
<s id="id2741568">Sit a b dupla c d, & addatur qu<lb/>dam ad b a, qu it a g, eadem detraha<lb/>tur ex c d & it c h, dico, quod propor<lb/>tio e d ad d h maior et, quam duplica<lb/>ta g b ad a b, & rurus i qudam ad c & minuatur ex a b utpot <lb/>c f addatur c d, & a e minuatur ex a b, erit proportio f d ad c d mi<lb/>nor duplicata a b ad g e. </s>
<s id="id2741626"><expan abbr="Primũ">Primum</expan> ic reecentur a n & k l quales in<lb/>gul c h, igitur a l dupla et e h & a b fuit dupla a d, c d igitur ut in <lb/>priore contitution prcedentis a b ad l b, ut c d ad h d & a b ad <lb/>b l maior, quam duplicata a b ad b k ut minor qum k b ad b l. </s>
<s id="id2741681">hoc <lb/>enim demontratum et in fine, igitur c d ad h d maior, qum du<lb/>plicata a k ad k b, ed a k ad k b maior et per uigeimam tertiam, hu<lb/>ius cilicet per demontrationem illius, qum g b ad b a, igitur mul<lb/>to maior c d ad d h, qum duplicata g b ad b a, quod et primum.</s></p><p type="main">
<s id="id2741738">Secundum ic per eadem, addito enim duplo f c ipi <lb/><figure id="id.015.01.146.2.jpg" xlink:href="015/01/146/2.jpg"/><lb/>a b ut in ecunda figura, & int a m, & m n erit f d ad c d, <lb/>ut n a ad a b, quare cum n a ad a b it minor duplicata per <lb/>prcedentem in b ad a b, & a b ad e b it maior, ut demon <lb/>tratum et in uigeima tertia huius, qum m b ad a b, erit <lb/>f d ad d c multo minor duplicata a b ad b e, quod et e<lb/>cundum.</s></p><p type="main">
<s id="id2741814">Propoitio centeimatrigeimaquarta.</s></p><p type="main">
<s id="id2741831">Si rectangula uperficies it cuius pars tertia quadrata it, corpus <lb/>quod ex latere quadrat in reiduum uperficiei contat maius et <lb/>quouis corpore ex eadem uperficies aliter diuia contituto.</s></p><p type="main">
<s id="id2741877">Sit rectangulum a c cuius tertia pars c e it quadrata, dico quod <lb/><arrow.to.target n="marg455"/><lb/>corpus, quod <expan abbr="cõ&longs;tat">contat</expan> ex e d in a b et maius omni corpore, quod fue <lb/>rit ex latere partis uperficiei a b in reliquam <expan abbr="part&etilde;">partem</expan>. </s>
<s id="id2741925">Si non diuidatur <lb/>uel upra uel infra, & primo in f erit <expan abbr="aut&etilde;">autem</expan> proportio e d ad d f, ut e c ad
<pb xlink:href="015/01/147.jpg" pagenum="128"/>e k, & f a ad a e, ut uperficierum ipa<lb/><figure id="id.015.01.147.1.jpg" xlink:href="015/01/147/1.jpg"/><lb/>rum per primam exti Elementorum: at <lb/>per prcedentem maior et proportio <lb/>e d ad d f, qum a f ad a e, duplicata igi<lb/>tur maior et proportio e d ad eam, qu <lb/>potet uper f c uperficiem, quam f a ad <lb/>a e, igitur maior, qum a k ad a b ex pri<lb/>ma exti Elementorum: igitur per trige <lb/>imam quartam undecimi. </s>
<s id="id2742030">Parallelipe<lb/>dum ex e d in a b maius et parallelipedo ex ea, qu potet in f c u<lb/>perficiem in ipam uperficiem a k. </s>
<s id="id2742060">Si uer diuiio facta fuerit in g, <lb/>contat ex prcedenti, quod minor et proportio g e ad e d, qum <lb/>it duplicata e a ad a d a g, eam igitur minor proportio eius line, <lb/>qu potet in g e uperficiem ad e d quam a b ad a h, igitur paralle<lb/>lipedum ex e d in a b et maius parallelipedo ex ea, qu potet g c <lb/>in a h cum it a b ad a h, ut dictum et, uelut a e ad a g.<lb/><arrow.to.target n="marg456"/></s></p><p type="margin">
<s id="id2742139"><margin.target id="marg455"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s id="id2742165"><margin.target id="marg456"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2742191">Manifetum et autem, qud tale corpus et quale duplo cubi <lb/>lateris partis terti quadrat.</s></p><p type="main">
<s id="id2742221">Propoitio centeimatrigeimaquinta.</s></p><p type="main">
<s id="id2742238">Si linea in duas partes, quarum una it alteri dupla, diuidatur <lb/>erit, quod fit ex tertia parte in quadratum reidui parallelipedum <lb/>maius omni parallelipedo, quod ex diuiione eiudem line crea<lb/>ri posit.</s></p><p type="main">
<s id="id2742275"><arrow.to.target n="marg457"/></s></p><p type="margin">
<s id="id2742286"><margin.target id="marg457"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2742313">Sit a c dupla b c, & it quadratum ad ipius a c, dico parallelipe<lb/><figure id="id.015.01.147.2.jpg" xlink:href="015/01/147/2.jpg"/><lb/>dum ex b c in a d maius ee quouis alio ex <lb/>diuiione line a b imiliter creato. </s>
<s id="id2742353">Secetur <lb/>primo in e, & fiat quadratum a f, eritque per <lb/>uigeimam quintam. </s>
<s id="id2742368">Huius proportio c b <lb/>ad b c maior duplicata a e ad a c, quare ma<lb/>ior, quam a f ad a d per uigeimam exti Ele <lb/>mentorum, igitur per trigeimam quartam <lb/>undecimi, Parallelipedum ex b c in a d maius et parallelipedo e b <lb/>in a f, quod et demontrandum. </s>
<s id="id2742407">Si uer diuiio cadat in g, fiat qua<lb/>dratum a h, et erit per uigeimamtertiam huius proportio g c ad c b <lb/>minor, quam duplicata c a ad a g: igitur minor, qum a d ad a h, igi<lb/>tur per eandem parallelipedum ex c b in a d maius et parallelipe<lb/>do ex g b in a h.</s></p><p type="main">
<s id="id2742449"><arrow.to.target n="marg458"/></s></p><p type="margin">
<s id="id2742460"><margin.target id="marg458"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2742486">Ex hoc liquet qud parallelipedum illud erit quadruplum cu<lb/>bo minoris partis, & dimidium cubi maioris.</s></p>
<pb xlink:href="015/01/148.jpg" pagenum="129"/><p type="main">
<s id="id2742512">Propoitio centeimatrigeimaexta.</s></p><p type="main">
<s id="id2742531">Denominationes in infinitum extendere.</s></p><p type="main">
<s id="id2742540">Inquit Euclides, i fuerint quotlibet quantitates ab uno in conti</s></p><p type="main">
<s id="id2742553"><arrow.to.target n="marg459"/><lb/><arrow.to.target n="marg460"/><lb/>nua proportione, erit tertius numerus quadratus, & omnes alij e<lb/>quentes uno intermio. </s>
<s id="id2742582">Tertia igitur in comparatione ad ecun<lb/>dam etiam, quod non it numerus, et quadratum: et enim tertia <lb/>ab uno quadratum ecund, qu et proportio. </s>
<s id="id2742616">Detracto igitur <lb/>uno omnes quantitates lo co pari unt quadrat: ut cias ergo cu<lb/>ius unt quadrat diuide per medium, & erit quadratum illius, er<lb/>go quadrageima erit quadratum uigeim, & uigeima decim, <lb/>& decima quint, & uigeimaexta terti decim, & ita de alijs. <lb/></s>
<s id="id2742683">Iuxta hoc dicemus, quod ecunda erit <expan abbr="quadratũ">quadratum</expan>, & quarta quadra<lb/>tum quadrati, & octaua <expan abbr="quadratũ">quadratum</expan> quadrati quadrati. </s>
<s id="id2742712">Et extadeci<lb/>ma quad quad quad quad. </s>
<s id="id2742724">& ita trigeima ecunda quad quad quad <lb/>quad quad. </s>
<s id="id2742737">Quod autem quad. </s>
<s id="id2742741">et quarta in ordine, ideo & octa<lb/>ua & duodecima & decimaexta, & ic de alijs unt quadrata qua<lb/>drati, & icut quarta et quadratum quadrati prim, ita octaua e<lb/>cund, & duodecima terti, & extadecima quart, & uigeima <lb/>quint, & ita emper diuidendo per quatuor.</s></p><p type="margin">
<s id="id2742811"><margin.target id="marg459"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s id="id2742838"><margin.target id="marg460"/>L<emph type="italics"/>ib.<emph.end type="italics"/> 9. P<emph type="italics"/>ro <lb/>po.<emph.end type="italics"/> 8.</s></p><p type="main">
<s id="id2742879">Secunda regula dicebat ibidem Euclides, i fuerint quotlibet <lb/><arrow.to.target n="marg461"/><lb/>quantitates ab uno in continua proportione quartus, ab uno erit <lb/>cubus upple ecund, & ita duobus emper intermisis, uno igi<lb/>tur ipo relicto quolibet loco ternario, ut tertia, exta, nona, duode<lb/>cima unt cubi, & cubi eius quantitatis, qu exit diuio numero per <lb/>tria, uelut tertia prim, exta ecund, nona terti, duo decima quar <lb/>t: & ita tertia erit cubus nona cubus cubi, & uigeimaeptima cu<lb/>bus cubi cubi cilicet prim. </s>
<s id="id2742978">Et trigeimanona et cubus ter<lb/>ti decim.</s></p><p type="margin">
<s id="id2743001"><margin.target id="marg461"/>L<emph type="italics"/>ib.<emph.end type="italics"/> 9. P<emph type="italics"/>ro<lb/>po.<emph.end type="italics"/> 8.</s></p><p type="main">
<s id="id2743045">Tertia regula quarta quantitas, ut uium et: et quad quad. </s>
<s id="id2743057">Et <lb/>quinta et relatum primum, quia 5 et numerus primus, & 7 et re<lb/>latum ecundum, quia et ecundus numerus primus: & undecima <lb/>tertium: & tertiadecima quartum: & decimaeptima quintum: & <lb/>decimanona extum: & uigeimatertia eptimum & uigeima quin<lb/>ta, quia et primus numerus prter quam ad quintam, ide et rela<lb/>tum quint, qu et relatum primum prim, omnes ergo numeri <lb/>primi unt relata, alij omnes unt ex natura cubi uel quadrati. </s>
<s id="id2743149">Sed <lb/>relata unt inter e omnia diuerorum generum nii <expan abbr="uige&longs;imũ">uigeimum</expan> quin<lb/>tum, quod et relatum primum primi relati, & quadrageimumno<lb/>num et relatum ecundum relati ecundi. </s>
<s id="id2743204">Et ita centeimum uigei<lb/>mum primum et relatum tertium tertij relati, reliqua, ut dixi, me<lb/>dia inter hc unt ui generis.</s></p>
<pb xlink:href="015/01/149.jpg" pagenum="130"/><p type="main">
<s id="id2743247">Quarta regula propoita quantitate ab uno in continua propor<lb/>tione, i uis cire cuius natur it detracto uno conidera, an posit <lb/>diuidi per duo, et quadratum medietatis, & ita procedes diuiden<lb/>do uque ad numerum primum, qui uel et 2, & erit ex genere quad <lb/>quad. </s>
<s id="id2743295">uel 3, & erit ex genere quadratorum cuborum, & imiliter i <lb/>it 9, erit ex genere quadratorum cubi cubi. </s>
<s id="id2743312">Et i proueniat alius nu<lb/>merus primus, ut 5. 7. 11. 13. erit quadratum relati illius ordinis. </s>
<s id="id2743323">Et i <lb/>non potet diuidi numerus quantitatum per 2 uide, i posit diuidi <lb/>per 3, tunc erit cubus illius quantitatis, & i illa quantitas, qu pro<lb/>uenit ex diuiione: fuerit 3, uel potuerit diuidi per 3, erit cubus, uel <lb/>cubus cubi, & ita deinceps. </s>
<s id="id2743364">Si uer it alius numerus primus, ut 5. <lb/>7. 11. erit cubus relati. </s>
<s id="id2743378">Et ita i <expan abbr="nõ">non</expan> posit diuidi per 2, nec per 3, erit ex <lb/>genere relati. </s>
<s id="id2743399">Et tunc i posit diuidi per alium numerum, ut 35, erit <lb/>relatum ex eo genere. </s>
<s id="id2743412">Vtpot trigeimaquinta quantitas et rela<lb/>tum ecundum relati primi, eu relatum primum relati ecundi. <lb/></s>
<s id="id2743440">Nam quoties quantitas potet diuidi per duos numeros, dicetur <lb/>ub utro que uicisim, ut duodecima potet diuidi per 4 & 3, ide di<lb/>cetur cubus quad quad. </s>
<s id="id2743465">uel quad quad. </s>
<s id="id2743469">cub. </s>
<s id="id2743474">& per 2 & 6, & dicetur <lb/>quadratum cubi quadrati, & quadratum cubicum quadrati ipius <lb/>proportionis, ad quam omnia referri debent.</s></p><p type="main">
<s id="id2743496">Quinta regula ex prcedenti pendet, & et, quod denomina<lb/>tiones, & proportiones uicisim commutantur: uelut 256 et quad <lb/>quad quad, & inter quad quad quad, & quad quad unt quatuor ter <lb/>mini ipo computato, & inter quad quad, & quod uii duo, ergo <lb/>quad quad quad continet plures proportiones, & proportiones <lb/>duplicat non contituunt quad: nam 64 continet duas duplas <lb/>ad 16, non tamen et quadratum 16, ideo oportet diligenter ani<lb/>maduertere.</s></p><p type="main">
<s id="id2743562">Sexta regula imiliter ex dictis pendet, & et, qud gratia exem<lb/>pli relatum primum comparatum ad primum terminum et exta <lb/>quantitas, cum autem comparatur ad rem, iam prupponit pro<lb/>portionem. </s>
<s id="id2743598">Exemplum relatum primum proportionis 21/20 et 4084101/3200000 <lb/>& et aliquanto maior exquiquarta, & i colligas terminos 100. <lb/>105. 110 1/4 115 61/80 121 861/1600 127 19681/32000. Tu uides qud unt ex termini in <lb/>utra que computando primum, ed in 21/20 unt duo termini, & in qua<lb/>drato tres, & in quadrato quadrati per prcedentem, adduntur <lb/>duo & ultimus cilicet extus fit ex relato ipo. </s>
<s id="id2743661">Ergo ultra propor<lb/>tionem unt tantum quatuor termini.</s></p><p type="main">
<s id="id2743677">Septima regula ad effugiendum omnes errores tu cis, qud <lb/>4096 quadratum 64 et extus a 64, ad quem habet proportionem <lb/>quadrati, & 64 et imiliter extus ab uno illo cilicet non compu
<pb xlink:href="015/01/150.jpg" pagenum="131"/>tato, & ita 64 habet rationem unius, & licet comparetur ad 2 rem, <lb/>& it extus ab eo, eo computato 4096 autem 64 it eptimus, ta<lb/>men non et eadem ratio, quia 64 non et quadratum 2.</s></p><p type="main">
<s id="id2743756">Propoitio centeimatrigeimaeptima.</s></p><p type="main">
<s id="id2743776">Rationem numerorum ex progresione declarare.</s></p><p type="main">
<s id="id2743787">Michal Stifelius rationem pulcherrimam tradidit ad inuentio<lb/><arrow.to.target n="marg462"/><lb/><arrow.to.target n="marg463"/><lb/>nem numerorum, qui uo cantur multiplicandi, & componitur hoc <lb/>modo. </s>
<s id="id2743815">Ex prima componitur 1 & 2, faciunt 3. 1. 2. 3 faciunt 6. 1. 2. 3. 4 <lb/>faciunt 10, & ita prima tabula contituit ecundam recta erie nu<lb/>merorum iunctis o<lb/>mnibus ab uno. </s>
<s id="id2743844">Ter <lb/><figure id="id.015.01.150.1.jpg" xlink:href="015/01/150/1.jpg"/><arrow.to.target n="table17"/><lb/>tia fit ex ecunda & <lb/>tertia, prim aumi <lb/>tur 10 in tertia, ut in <lb/>ecunda, & ex 10 e<lb/>cund, & 10 terti <lb/>fit 20, & ex 15 ecun<lb/>d, & 20 terti fit <lb/>35, & ex 21 ecund, <lb/>& 35 terti fit 56, & <lb/>ex 28, & 56 fit 84. Et <lb/>quanta fit ex tertia, <lb/>& ex eipa. </s>
<s id="id2743945">primum <lb/>aumendo 35 ex ter <lb/>tia, & ponitur pro <lb/>primo numero quart, & ex 35 terti, & 35 quart fit 70 numerus <lb/>ecund quart: & ita ex 56 & 70 fit 126, & ex 84, & 126. 210. & ita <lb/>quinta ex quarta & eipa, & ic in infinitum.</s></p><p type="margin">
<s id="id2743999"><margin.target id="marg462"/>C<emph type="italics"/>o<emph.end type="italics"/>_{m}.</s></p><p type="margin">
<s id="id2744026"><margin.target id="marg463"/>P<emph type="italics"/>rim u<emph.end type="italics"/><lb/>A<emph type="italics"/>rith.<emph.end type="italics"/></s></p><table><table.target id="table17"/><row><cell>1</cell><cell>2</cell><cell>3</cell><cell>4</cell><cell>5</cell><cell>6</cell><cell>7</cell><cell>8</cell></row><row><cell>1</cell><cell/><cell/><cell/><cell/><cell/><cell/><cell/></row><row><cell>2</cell><cell/><cell/><cell/><cell/><cell/><cell/><cell/></row><row><cell>3</cell><cell>3</cell><cell/><cell/><cell/><cell/><cell/><cell/></row><row><cell>4</cell><cell>6</cell><cell/><cell/><cell/><cell/><cell/><cell/></row><row><cell>5</cell><cell>10</cell><cell>10</cell><cell/><cell/><cell/><cell/><cell/></row><row><cell>6</cell><cell>15</cell><cell>20</cell><cell/><cell/><cell/><cell/><cell/></row><row><cell>7</cell><cell>21</cell><cell>35</cell><cell>35</cell><cell/><cell/><cell/><cell/></row><row><cell>8</cell><cell>28</cell><cell>56</cell><cell>70</cell><cell/><cell/><cell/><cell/></row><row><cell>9</cell><cell>36</cell><cell>84</cell><cell>126</cell><cell>126</cell><cell/><cell/><cell/></row><row><cell>10</cell><cell>45</cell><cell>120</cell><cell>210</cell><cell>252</cell><cell/><cell/><cell/></row><row><cell>11</cell><cell>55</cell><cell>165</cell><cell>330</cell><cell>462</cell><cell>462</cell><cell/><cell/></row><row><cell>12</cell><cell>66</cell><cell>220</cell><cell>495</cell><cell>792</cell><cell>924</cell><cell/><cell/></row><row><cell>13</cell><cell>78</cell><cell>286</cell><cell>715</cell><cell>1297</cell><cell>1716</cell><cell>1716</cell><cell/></row><row><cell>14</cell><cell>91</cell><cell>364</cell><cell>1001</cell><cell>2002</cell><cell>3003</cell><cell>3432</cell><cell/></row><row><cell>15</cell><cell>105</cell><cell>455</cell><cell>1365</cell><cell>3003</cell><cell>5005</cell><cell>6435</cell><cell>6435</cell></row><row><cell>16</cell><cell>120</cell><cell>560</cell><cell>1820</cell><cell>4368</cell><cell>8008</cell><cell>11440</cell><cell>12870</cell></row><row><cell>17</cell><cell>136</cell><cell>680</cell><cell>2380</cell><cell>6188</cell><cell>12376</cell><cell>19448</cell><cell>24310</cell></row></table><p type="main">
<s id="id2744422">Regula ergo et, qud binarius eruit <02> quadrat, & quia nihil <lb/>et in eius directo, olus ipe eruiet <02> quadrat. </s>
<s id="id2744456">Ternarius autem <lb/>cubic, & quia in eius directo et alter ternarius, ille etiam eruiet <lb/><02> cubic. </s>
<s id="id2744479">Quaternarius autem eruiet quadrato quadrati, & ena<lb/>rius, qui et in illius directo. </s>
<s id="id2744497">Ergo quinarius eruiet <02> relat prim, <lb/>& duo equentes numeri cilicet 10 & 10, & eo dem modo enarius <lb/>numeri duo equentes 15 & 20 eruient cubo quadrati, & ita etiam <lb/>eptenarius cum tribus equentibus numeris 21. 35 & 35 eruient <lb/>rel. </s>
<s id="id2744545">ecundi radici, & ita deinceps in infinitum.</s></p><p type="main">
<s id="id2744557">Propoitio centeimatrigeimaoctaua.</s></p><p type="main">
<s id="id2744574">Modos uus horum numerorum declarare.</s></p><p type="main">
<s id="id2744585">In quouis numero denominationis oportet tot addere o, quo<lb/><arrow.to.target n="marg464"/>
<pb xlink:href="015/01/151.jpg" pagenum="132"/>tus et ordo, & facere tot numeros equentes; quotus et ordo, & <lb/>emper minuere unam o, uelut quia quadrata <02> et prima ad 2 ad<lb/>demus o, & fiet 20, nec alium quremus numerum. </s>
<s id="id2744634">Sed quia cubi<lb/>ca et ecundo loco, habebit prima nota 00, & fiet 300, & ecundum <lb/>3 unam 0, & fiet 30, & in quadrato quadrati addemus 000 primo, <lb/>& 00 ecundo, & o tertio, & ita hab ebimus 4000. 600. 40. ed quia <lb/>in tabula non et 4 ultimum, addemus imilem primo emper. </s>
<s id="id2744678">In <lb/>relato primo, ergo habebimus 50000. 1000. 1000. 50. & in cubo <lb/>quadrati 600000. 150000. 20000. 1500. 60. Manifetum et, qud <lb/>his uice uera aumpimus 15 & 6 imiles prioribus addendo em<lb/>per ut dixi o minus, donec ad unam peruenerit. </s>
<s id="id2744722">Et ita in relato e<lb/>cundo 7000000. 2100000. 350000. 35000. 2100. 70. & ita dein ceps.</s></p><p type="margin">
<s id="id2744740"><margin.target id="marg464"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2744767">Propoitio centeimatrigeimanona.</s></p><p type="main">
<s id="id2744782">Radices omnes propoitis numeris extrahere.<lb/><arrow.to.target n="marg465"/></s></p><p type="margin">
<s id="id2744803"><margin.target id="marg465"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2744829">Propoitis quibuuis numeris utpot 916132832, uolo detrahere <lb/><02> relatam primam, primum habebo in tabula decripta relata pri<lb/>ma numerorum implicium uque ad 10 uelut in exemplo. </s>
<s id="id2744859">Dein de <lb/><figure id="id.015.01.151.1.jpg" xlink:href="015/01/151/1.jpg"/><lb/>ubcribam pun<lb/>ctum ub prima <lb/>nota dextra, & <lb/>quia et quarta in <lb/><figure id="id.015.01.151.2.jpg" xlink:href="015/01/151/2.jpg"/>ordine hoc, eu quinta denominatio ecun<lb/>dum notrum, omittam quatuor notas in<lb/>ter medias, & ubcribam punctum aliud, <lb/>& ita facerem i eent plures qum decem <lb/>not: relinquitur ergo ad <expan abbr="pũctum">punctum</expan> primum <lb/> initra 9161, cuius quro <02> relatam pri<lb/>mam in tabula, quam inuenio ee 6, nam <lb/><figure id="id.015.01.151.3.jpg" xlink:href="015/01/151/3.jpg"/>7776 eius relatum primum et <lb/>proximius ex minoribus ad 9161, <lb/>detraho igitur 7776, ex numero <lb/>propoitio relinquitur. </s>
<s id="id2745011">Dein de <lb/>pno 6 & quadratum eius, & cub. </s>
<s id="id2745022">& quadratum <lb/>quadrati, quia, ut dixi, et quarta denominatio a<lb/><figure id="id.015.01.151.4.jpg" xlink:href="015/01/151/4.jpg"/>pud illum, & regione numeros prcedentes in<lb/>uentos relati primi ex prcedenti propoitione: & duco ingulos <lb/>cum uis collateralibus, ut uides etiam in figura, et cum ultimo pro<lb/>ducto, cilicet 64800000 diuido 138532832 exit 2, huius accipio o<lb/>mnes numeros ad relatum primum uque ut uides, & pono minores <lb/> regione maiorum, utpot 2 regione 1296 & 50000, & 4 regio
<pb xlink:href="015/01/152.jpg" pagenum="133"/>ne 216 & 10000, & 8 regione 36 & 10000, & 16 regione 6, & 50, <lb/>& duco 6 in 50 fit 300, duco in 16 fit 4800, duco 36 in 1000 fit <lb/>36000, duco 36 in 8 fit 288000, duco etiam 216 in 10000 & fit <lb/>2160000, & duco hos per 4 fit 86400000, duco rurus 1296 in <lb/>50000 fit 64800000, duco in 2 fit 129600000. Demum addo 32 re<lb/>latum primum 2, & fit umma omnium 138532832, & ita habemus <lb/>radicem relatam primam dictinumeri ee 62. Et i numerus produ <lb/>ctus fuiet maior oportuiet accipere proximo minorem. </s>
<s id="id2745169">Inde per <lb/>regulam equentem addere minutias.</s></p><p type="main">
<s id="id2745183">Propoitio centeimaquadrageima.</s></p><p type="main">
<s id="id2745200">Radices per numeros fractos determinare.</s></p><p type="main">
<s id="id2745209">Duplex et modus, ut etiam docui in arithmeticis, cilicet ut pro </s></p><p type="main">
<s id="id2745223"><arrow.to.target n="marg466"/><lb/>radice quadrata addatur duo o, & pro cuba tria, & pro quadrata <lb/>quadrata quatuor, & pro relata prima quinque, & ita deinceps, & <lb/>pr decimis emel, pro centeimis bis, pro milleimis ter, pro millia<lb/>ribus eu partibus earum quater, pro centeimis milleimis quin<lb/>quies, pro milleimis milleimarum exies, & ita deinceps deinde <lb/>per prcedentem detrahere radicem, & erit ualde exacta. </s>
<s id="id2745287">Exemplo <lb/>non utar, nii qud i uelles radicem relatam 16 ad milleimas, acci<lb/>cipies radicem relatam numeri latere propoiti, & ita de alijs <lb/>1600000, 00000, 00000, & i uelles <02> cub. </s>
<s id="id2745324">5 1/5 per milleimas, pri <lb/>mo addes ter 000, & fiet 3000000000, inde ume 1/5 1000000000, <lb/>qui et 200000000, & adde ad 5000000000, fit 2500000000, <lb/>& hoc quia unum refert numerum 1000000000 ex uppoito & 1/5 <lb/>et 1/5 unius.</s></p><p type="margin">
<s id="id2745367"><margin.target id="marg466"/>C<emph type="italics"/>o<emph.end type="italics"/>m.</s></p><p type="main">
<s id="id2745394">Secundus modus et, ut accipias proxim maiorem, & multipli<lb/>ca in e, & detrahe numerum propoitum, & reiduum diuide per <lb/>duplum radicis primo inuent, i fuerit quadrata, & per triplum <lb/>quadrati eiudem i fuerit cubica, & per quadruplum cubi, i fuerit <lb/>quadrata quadrata, & per quin cuplum quadrati quadrati, & quod <lb/>exit detrahes ex priore radice, & rurus quod relinquitur, multipli<lb/>ca in e, & eodem modo agendo quod uperet numero propoi<lb/>to, diuide per duplum radicis prioris, i it radix quadrata, uel per <lb/>triplum quadrati i it cubica, & quod exit rurus detrahe, & ita a<lb/>gendo, peruenies ad exactisimam radicem, exemplum uolo radi<lb/>cem quadratam 5 proxima maior et 3, quadratum 9, differentia 4, <lb/>diuide per 6 duplum 3 exit 2/3, detrahe ex 3 fit 2 1/3, quadratum et 49/9 <lb/>quod et 5 4/9, rurus diuido 4/9 differentiam 5 4/9 & 5 per 4 2/3 duplum <lb/>radicis prim exit 2/21, detrahe ex 2 1/3, relinquitur 2 5/21, radix atis pro<lb/>pinqua, nam eius quadratum et 5 4/441, in cubica imiliter uolo <02><lb/>cu. </s>
<s id="id2745550">5, proxima maior et 2, cubus 8, differentia 3, diuide per triplum
<pb xlink:href="015/01/153.jpg" pagenum="134"/>quadrati 2 quod et 12 exit 1/4 detrahe ex 2 fit 1 3/4 cuius cubus et 5 23/64 <lb/>differentia et 23/64 diuide per triplum quadrati 1 3/4 qud et 9 3/16 exit <lb/>23/588 detrahe ex 1 3/4 <expan abbr="relinquũtur">relinquuntur</expan> 1 107/147 cuius cubus et 5 504449/3176523 Ita diuides <lb/>hunc exceum i placet per triplum quadrati 1 107/147 & et ferm 9 exit <lb/>56050/3176523 quai detrahe ex 1 107/147 relinquuntur 323159/453789.</s></p><p type="main">
<s id="id2745628">Tertius modus et ubtilior, tu cis, d duo decima denominatio <lb/>et quadrata ext, & quadrata quad, terti, & cuba quarti, quarta <lb/>autem et inter <expan abbr="tertiã">tertiam</expan> & extam ecunda quantitas in continua pro<lb/>portione: ergo inuenta <02> numeri propoiti & <02> radicis inuent <lb/><expan abbr="reducã">reducam</expan> ad unam denominationem, et inter numeratores collo cabo <lb/>duas quantitates, quod facile erit enim procedendo, & habebo <02><lb/>cu. </s>
<s id="id2745713">quitam, cilicet minorem ex duabus intermedijs. </s>
<s id="id2745725">Et imiliter <lb/>pro relata prima, capiam exaginta denominationes, & cis, qud <lb/>quintadecima et <02> <02> exageim, & decima et <02> cu. </s>
<s id="id2745762"><02> exageim, <lb/>& duodecima <02> relata prima exageim per eandem inuenta, er<lb/>go <02> numeri propoiti tanquam ille it exageima denominatio, <lb/>inueniam illius radicis inuent <02> quadratam, & cubicam, & <lb/>quia duodecima quantitas qu et <02> relata prima numeri et <lb/>ecunda, quatuor intermediarum inter ponam inter <02> quadra<lb/>tum, quadratum, & cubicam quadratam quatuor numeros in <lb/>continua proportione, & ecundus ex minoribus erit <02> relata <lb/>prima numeri propoiti. </s>
<s id="id2745848">Exemplum cubic uolo <02> cu: 5 habui <02><lb/>quadratam eius 2 5/21 ed uolo proximiorem diuidendo 4/441 per 4, <lb/>quod et ferm duplum 2 5/21 exit 1/441 detraho ex 2 5/21 relinquitur ualde <lb/>proxima <02> 5. 2 104/441 huius igitur radix quadrata, primo inuenta et 1 1/2 <lb/>ecunda proximior et 1 41/84 reduco ad eandem denominationem fi<lb/>ent 284/9261 2 416/1764 & 1 861/1764 inter 3944, & 2625, inueniemus duos nume<lb/>ros in continua proportione, ut uides, & erit ecunda quantitas <lb/><figure id="id.015.01.153.1.jpg" xlink:href="015/01/153/1.jpg"/><lb/>3006/7641, quod et 167/98 proximum ad 1 5/7, <02> cubica. </s>
<s id="id2745920">5. <lb/><expan abbr="nã">nam</expan> eius cubus et 5. 13/343 at exactisima et ergo 1 69/98. <lb/>ut liquet. </s>
<s id="id2745946">Pro relata prima ergo ponamus, ut ue<lb/>lim <02> relatam <expan abbr="primã">primam</expan> 25, accipio 5 <02> 25 cuius <02> et, ut uium et, 2 104/441 <lb/>imiliter <02> cu: 5 fuit 1 69/98 igitur reducam ad unam denominationem, <lb/>& inueniam quatuor numeros in <expan abbr="cõtinua">continua</expan> proportione inter illos, <lb/>& ecundus pot minimum ex illis erit <02> relata prima propinqui<lb/>ima 25. Quomodo uer inueniantur facillim illi termini, do<lb/>cui in exto libro operis perfecti.</s></p><p type="main">
<s id="id2746028">Quarta regula et utilior, licet minus uideatur nobilis, & et un<lb/>data in hoc, quod i a b it maior c & eis ad dantur b e, & d f qua<lb/>les dico, quod erit minor proportio a c ad c f, quam a b ad c d, & ex <lb/>conequenti per <expan abbr="uiã">uiam</expan> fracti maior pars unius erit c fipius a e, qum
<pb xlink:href="015/01/154.jpg" pagenum="135"/>c d ipius a f ex Euclide. </s>
<s id="id2746095">Dico ergo quod maior et proportio a b <lb/><figure id="id.015.01.154.1.jpg" xlink:href="015/01/154/1.jpg"/><lb/>ad c d, qum a e ad e f, fiat d g ad quam it b c ut <lb/><arrow.to.target n="marg467"/><lb/>a b ad c d, eritque a e ad c g ut a b ad c d, minor au<lb/>tem et a e ad c f, quam ad c g, igitur minor a e ad <lb/>c f qum a b ad c d quod fuit propoitum. </s>
<s id="id2746147">Simili <lb/>ter i fuerint du quantitates, a b & c d, quarum a b it maiore, c d <lb/>autem eadem e minor, dico, qud dimidium aggregati a b & c d <lb/>maiorem habebit proportionem ad e, qum c d & minor, nam iun<lb/>cta b f quali d e ad a b, ita ut f g it dimidium totius a f, qia ergo <lb/><figure id="id.015.01.154.2.jpg" xlink:href="015/01/154/2.jpg"/><lb/>f g et dimidium f a & fb et minor dimidio <lb/><arrow.to.target n="marg468"/><lb/>f a cum it minor b a, & imiliter f g et mi<lb/>nor a b, quia a b et maior dimidio a f, quia <lb/>et maior b f, ergo proportio g f ad c et ma <lb/>ior quam b f ad e, ita quam c d ad e, & mi<lb/><arrow.to.target n="marg469"/><lb/>nor qum a b ad e, quod fuit propoitum. </s>
<s id="id2746263">Quo uio uolo <02> 1000 <lb/>quadratam, & qud de quadrata dico, dico etiam de alijs radici<lb/>bus & erit ex ecunda regula harum 31 39/62 & quadratum erit 1000 <lb/>1521/3844. Iuxta ergo primam partem regul 31 38/61 erit minus, & in ueritate <lb/>in eo, quod fit ducendo, ut uides, & hoc et pro<lb/><figure id="id.015.01.154.3.jpg" xlink:href="015/01/154/3.jpg"/><lb/>ximum ad 11/160, multiplico igitur duplum 31 39/62, <lb/>quod et ferm 63 1/4 in 1/160 fient 63/160 detrahe ex <lb/>1521/3844 hoc modo, diuide 3844 per 160 exit 24 /40 <lb/>diuide 1521 per 24, exit 63 3/8, habes igitur quod <lb/>1521/3844 unt 63/160, igitur detracto 63/160 ex 63/160 nihil relinquitur, & erit <02> exa<lb/>cta ualde 1000 hoc 31 38/61 cuius quadratum 1000 41/3421 uides breuita <lb/>tem, & propinquitatem in producto differentia et 1/100 aut parum <lb/>maius quod ad radicem comparatum cum debeat diuidi per du<lb/>plum eius erit paulo maius 1/6300. Vnde facilior et, & breuior hc <lb/>uia qum per 00 ad ditus. </s>
<s id="id2746372">Rurus uolo aliquid <expan abbr="adi&mtilde;ere">adimnere</expan> & cum pro <lb/>pinquitate ita facio. </s>
<s id="id2746394">Conidero qud 31 38/61 et maius 1/6300 radice, di<lb/>uido 6300 per 62 exit 103 ferm, neque enim curo in hoc fractiones, <lb/>multiplico ergo 103 in 38/61 & habeo 3914/6283 hic denominator et proxi<lb/>mus 6300, aufero ergo 1 ex 3914, habebo ualde proximam <02> 1000, <lb/>31 3913/6283 cuius quadratum et 1000 minus 1/1048 hoc ut dixi diuium <lb/>per duplum <02> quod et 63 et omnino inenile in radice.</s></p><p type="margin">
<s id="id2746458"><margin.target id="marg467"/>8. P<emph type="italics"/>ropo. <lb/>quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/><lb/>P<emph type="italics"/>er<emph.end type="italics"/> 18. <lb/><emph type="italics"/>quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>
</p><p type="margin">
<s id="id2746537"><margin.target id="marg468"/>P<emph type="italics"/>er<emph.end type="italics"/> 11. <lb/><emph type="italics"/>quinti<emph.end type="italics"/> E<emph type="italics"/>lem. <lb/><expan abbr="amplificatã">amplificatam</expan>.<emph.end type="italics"/></s>
</p><p type="margin">
<s id="id2746596"><margin.target id="marg469"/>P<emph type="italics"/>er<emph.end type="italics"/> 8. <emph type="italics"/>quin<lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="main">
<s id="id2746647">Quinta regula et omnium pulcherrima, & et communis omni <lb/>bus & fractis & integris & omnibus generibus radicum, & it ex<lb/>emplum, uolo <02> radicis upracript cilicet 31 3913/6283 multiplico 31 <lb/>in 6283, & fit 194793, cui addo 3913, fit 198686 manifetum et igi<lb/>tur, quod 198686/6283 quiualet 31 3913/6283 hoc facto, quod et commune om
<pb xlink:href="015/01/155.jpg" pagenum="136"/>nibus radicibus extrahendis pro radice quadrata, multiplicabo n <lb/>meratorem, qui et 194686 per denominatorem, qui et 6283, & i <lb/>uoluero radicem cubicam, multiplicabo eundem numeratorem <lb/>per quadratum denominatoris, & i uoluero radicem radicis, mul<lb/>tiplicabo per cubum, multiplicabo per quadratum quadratum <lb/>6283, & ita de alijs una diminutione minore, & eius qui prouenit <lb/>numeri <02> uprapoita denominatori erit <02> eiumodi, quam uce<lb/>piti, uelut in exemplo fuit numerus 198686/6283 quia ergo uolo <02> quad. <lb/></s>
<s id="id2746778">multiplico 198686 in 6283, & fit 1248344138, huius accipio <02><lb/>quad. </s>
<s id="id2746788">qu et 35332, hc autem et diuidenda per 6283, & exeunt <lb/>5 3917/12566, ecce uides radicem exactam admodum, & facilem. </s>
<s id="id2746809">Volo rur<lb/>us <02> quadrat. </s>
<s id="id2746819">5 3917/12566, multiplico 12566 per 5 & fit 62830, cui addo <lb/>3917, & fit 66747, cui uppono 12566 denominatorem, fient ergo <lb/>66747/12566, manifetum et igitur qud hoc quiualet 5 3917/12566, i igitur mul <lb/>tiplicarem denominatorem per denominatorem & numeratorem, <lb/>quod proueniret, eet quale eidem numero, ergo <02> eius eet ea<lb/>dem cum <02> prioris, ed <02> denominatoris eet prior numerus, er<lb/>go ufficiet extrahere <02> producti ex denominatore in numerato<lb/>rem, & ita productum erit ex denominatore in numeratorem <lb/>838742802, cuius <02> et 28961, hc igitur diuia per 12566 oten<lb/>dit <02> 2 3892/12566. In hac autem quadrata et alius modus ine multiplica<lb/>tione, ed non et communis alijs, ubi tatueris denominatorem <lb/>pro denominatore <02>, utpote 12566, & numeratorem 66747, con<lb/>titues medium enim augendo.</s></p><p type="main">
<s id="id2746957">Rurus uolo <02> relatam 2 3829/12566 reduco ad denominatorem, & fit <lb/>ut prius 28961/12566, duco igitur 12566 ad quad. </s>
<s id="id2746970">quad. </s>
<s id="id2746974">ed ufficiet in hoc <lb/>cau deducere ad minores denominationes, utpot diuide 28961 <lb/>per 12566 exit 2 3829/12566 multiplico per 566 fit 1104 5862/12566, hoc detrahe <lb/>ex 28961 habebis 27856/12000, diuide igitur per 1000 habebis 12 & 27 107/125 <lb/>at 108/126 unt 6/7, igitur habes 12 pro denominatore, & 27 6/7 pro nume<lb/>ratore, quare erunt numeri 195/84, erit ergo per hanc regulam, ut ducas <lb/>84 ad quad. </s>
<s id="id2747016">quadrati, & fit 49787136, duc in 195 fit 9708491520, <lb/>cuius <02> relata prima et 99, igitur <02> relata prima 2 3829/12566 et 1 15/84 pau<lb/>lo maior, id et 1 13/70. Et nota quod i denominator haberet <02> illius <lb/>generis, quam quris, ufficeret inuenire radicem eiudem generis <lb/>abque alia numerorum multiplicatione.</s></p><p type="main">
<s id="id2747067">Propoitio centeimaquadrageimaprima. (deducere.</s></p><p type="main">
<s id="id2747084">Numeros fractos ad minores in <expan abbr="ead&etilde;">eadem</expan> proportione ualde propinqua</s></p><p type="main">
<s id="id2747102">Cum plerunque numeri fracti hab cantur per radices, ut aliquan<lb/><arrow.to.target n="marg470"/><lb/>do maiores int, aut minores eo fit, ut posint reduci ad mino<lb/>res numeros, ut melius intelligi posint & facilius tractari, &
<pb xlink:href="015/01/156.jpg" pagenum="137"/>cum hoc it exactior illa pars exemplum, ergo habeo 2 3829/12566, quem <lb/>uolo certa ratione ad minores diuiiones deducere. </s>
<s id="id2747149">Deduco pri<lb/>m totum ad fractiones ducendo 2 in 12566, & addendo 3829, & <lb/>fit 26961/12566, multiplico 12566 per 9, quia proportio unius ad alterum <lb/>et ferm, ut 9 ad 4, & fit 113094, multiplico 4 in 28961 fit 115844, <lb/>hoc igitur et maius, igitur proportio 28961 ad 12566 et maior <lb/>qum 9 ad 4, detraho igitur 12566 ex 28961, relinquitur 16395, de<lb/>traho 113094 ex 115844, relinquitur 2750, diuido 2750 per 16395 <lb/>exit 55/328 addo 2 denominatori fit 55/330, quod et 1/6, namit additiones <lb/>paru prter qud parum uariant quantitatem etiam dum ad ex<lb/>amen reducuntur, nihil impediunt, detrahe igitur 1/6 9/4, & ducendo <lb/>per 6, & detrahendo 53/23, duco igitur primos numeros cilicet 28961/12566 <lb/>mutuo in 53/23, fiunt 665998, & 666107, ita uides, quod proportio <lb/>53 ad 23 et paulo minor, qum 28961 ad 12566, & quiualent 27/23<lb/>& 2 3829/12566.</s></p><p type="margin">
<s id="id2747264"><margin.target id="marg470"/>C<emph type="italics"/>o<emph.end type="italics"/>_{m}.</s></p><p type="main">
<s id="id2747291">Propoitio centeimaquadrageimaecunda.</s></p><p type="main">
<s id="id2747309">Denominationum incrementa ex extrema cognita inuenire, & <lb/>conuero modo.</s></p><p type="main">
<s id="id2747325"><expan abbr="Quidã">Quidam</expan> per uuram <expan abbr="rediuiuã">rediuiuam</expan> fecit 40000 coronatos ex 40 in 40 <lb/><arrow.to.target n="marg471"/><lb/>annis. </s>
<s id="id2747356">Quro <expan abbr="qutãa">qutana</expan> fuerit uura, & <expan abbr="quãdo">quando</expan> habuit 1000 coronatos, <lb/><expan abbr="quidã">quidam</expan> uellent oluere per regulam trium quantitatum, in qua com<lb/>mitterentur maximi errores. </s>
<s id="id2747403">Et in ea multi unt modi, & omnes fal<lb/>i prter hanc uiam nulla et uera, adde qud uellent multi per or<lb/>tem inuentam oluere augendo per ingulos annos, quod ade <lb/>difficile eet, & pen foret imposibile. </s>
<s id="id2747453">Ide diuides 40000 per 40 <lb/>numerum ortis exit 1000, igitur in 40 annis unum fit mille, unt <lb/>ergo 40 denominationes ab uno, quarum quadrageima et 1000, <lb/>igitur uigeima et <02> 1000 cilicet 31 3913/6283, igitur decima et <02> eius <lb/><arrow.to.target n="marg472"/><lb/>5 3917/12566 huius radix, erit quinta quantitas 2 7/23, cuius <02> relata prima, <lb/><arrow.to.target n="table18"/><lb/>erit proportio 1 13/70, cuius quadratum et 1 1889/4900 eu <lb/>1 67/165 pro ecunda quantitate, duces ergo primam, <lb/><figure id="id.015.01.156.1.jpg" xlink:href="015/01/156/1.jpg"/>qu et 83/70 in quintam, qu et reducta ad mino<lb/>res fractiones facilitatis caua 53/23, & habebis ex<lb/>tam quantitatem 2 118/161, duco etiam quintam quan<lb/>titatem cilicet 53/23 in ecundam qu et 232/165, & fit e<lb/>ptimi anni quantitas, duco igitur eptem anno<lb/>rum numerum, qui et 3 14/61 in 31 38/61 fit 102 992/6283. At in <lb/>ex annis additis ad uiginti, fit tanto minus, quan<lb/>to 31 38/61 ductum in differentiam eptem, & ex an<lb/>norum qu et 60/121, fit ergo 15 35/492. Quia ergo an
<pb xlink:href="015/01/157.jpg" pagenum="138"/>nuatim olum uura adij citur orti, ufficiet diuidere 2 992/6283 per 15 35/492 <lb/>cilicet multiplicando per 12 numerum menium 2 992/6283 fit 25 5621/6283 di<lb/>uide 25 5621/6283 per 15 35/492, exit menis unus, & dies 21, detrahe ex 27 an<lb/>nis, remanent anni 26, menes 10, dies 9, in quo tempore habuit <lb/>4000 aureos coronatos. </s>
<s id="id2747678">Vura autem fuit ut uium 13/70, igitur per re<lb/>gulam trium duc 13 in 100 fit 1300, diuide 1300 per 70 exit 18 4/7 & <lb/>tanta fuit pro centum. </s>
<s id="id2747698">Et cum computaueris in tribus annis, acqui<lb/>rit modico plus bee eius, quod habet. </s>
<s id="id2747712">Et ita in 13 annis, & parua <lb/>illa parte perueniet ad decuplum eius, quod habet, cilicet 4000 au <lb/>reorum, & habebit aureos 40000, ut propoitum et.</s></p><p type="margin">
<s id="id2747737"><margin.target id="marg471"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s id="id2747764"><margin.target id="marg472"/>P<emph type="italics"/>er<emph.end type="italics"/> 136. <lb/>P<emph type="italics"/>ropo.<emph.end type="italics"/></s></p><table><table.target id="table18"/><row><cell>Anni</cell><cell>Aurei</cell></row><row><cell>1</cell><cell>1 13/70</cell></row><row><cell>2</cell><cell>1 67/165</cell></row><row><cell>5</cell><cell>2 7/23</cell></row><row><cell>6</cell><cell>2 118/161</cell></row><row><cell>7</cell><cell>3 14/61</cell></row><row><cell>10</cell><cell>5 3917/12566</cell></row><row><cell>20</cell><cell>31 38/61</cell></row><row><cell>40</cell><cell>1000</cell></row></table><p type="head">
<s id="id2747875">SCHOLIVM.</s></p><p type="main">
<s id="id2747884">In propoita proportione numero que terminorum rediuiuam u<lb/>uram inuenire.</s></p><p type="main">
<s id="id2747901">Sit gratia exempli, in ex annis uura rediuiua uigeim, erit<lb/>qe proportio 21/20, cuius numeratorem exies ducam in e primum <lb/>bis fit 441: ergo ducto 441 in e fit qe 194481 ductum in 441 <lb/>fit 85766121 exies ductum 21, quinquies autem ducam 20 deno<lb/><figure id="id.015.01.157.1.jpg" xlink:href="015/01/157/1.jpg"/><lb/>minatorem in e fit bis 400, ter 8000, <lb/>quinquies ergo 3200000, diuide nume<lb/>ratorem per denominatorem abiectis <lb/>quinque notis erit 26 2566121/3200000. Qu propor<lb/>tio et proxima 26 4/5 ad 20, & ita ut 134 ad <lb/>100. Et i pigeret tdij autlaboris poes <lb/>pro xij annis, ducere 134 in e, & fit 17956 <lb/>diuide per 100 eadem ratione, exit 179 14/25 <lb/>& ita 100 in xij annis, fit tantundem. </s>
<s id="id2748010">Et <lb/>ita pro xviij & xx annis.</s></p><p type="main">
<s id="id2748023">Propoitio centeimaquadrageimatertia.</s></p><p type="main">
<s id="id2748038">Si linea in duas partes diuidatur, corpora, qu fiunt ex una par<lb/>te in alterius quadratum mutu qualia unt corpori, quod fit ex <lb/>tota linea in uperficiem unius partis in alteram.<lb/><arrow.to.target n="marg473"/></s></p><p type="margin">
<s id="id2748076"><margin.target id="marg473"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2748102">Sit a c diuia in a b, b c quadratum a b it <lb/><figure id="id.015.01.157.2.jpg" xlink:href="015/01/157/2.jpg"/><lb/>a d, <expan abbr="quadratũ">quadratum</expan> b c, it b e <expan abbr="parallelogrammũ">parallelogrammum</expan> <lb/>ex a b in b e, a f dico qud corpora ex a b in <lb/>b e, & b c in a d qualia unt corpori ex a c <lb/>in a f. </s>
<s id="id2748163">Quia enim corpus ex a c in a f contat <lb/>ex a b in a f, & b c in a f, per primam ecun</s></p><p type="main">
<s id="id2748183"><arrow.to.target n="marg474"/><lb/>di Elementorum. </s>
<s id="id2748195">corpus autem ex a b in a f <lb/>et quale corpori ex b c in a d, & corpus <lb/>ex b c in a f et quale corpori ex a b in b c <lb/>igitur contat propoitum.</s></p>
<pb xlink:href="015/01/158.jpg" pagenum="140 [=139]"/><p type="margin">
<s id="id2748238"><margin.target id="marg474"/>I<emph type="italics"/>d et per <lb/>eius demon<lb/>trationem.<emph.end type="italics"/><lb/>P<emph type="italics"/>er<emph.end type="italics"/> 29. <emph type="italics"/>un <lb/>decimi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="main">
<s id="id2748311">Propoitio centeimaquadrageimaquarta.</s></p><p type="main">
<s id="id2748327">Duplum cubi medietatis maius et aggregato corporum mutu<lb/>orum cuiuslibet diuiionis, quantum et, quod fit ex tota in quadra <lb/>tum differenti.<lb/><arrow.to.target n="marg475"/></s></p><p type="margin">
<s id="id2748362"><margin.target id="marg475"/>C<emph type="italics"/>o<emph.end type="italics"/>m.</s></p><p type="main">
<s id="id2748386">Sit a b diuia per qualia in c, & per inqua<lb/>lia in d, dico, qud duplum cubi a c et maius ag <lb/><figure id="id.015.01.158.1.jpg" xlink:href="015/01/158/1.jpg"/><lb/>gregato corporum ex a d in quadratum b d, & b d in quadratum <lb/>a cin eo quod fit ex a b in quadratum c d, nam per <expan abbr="præcedent&etilde;">prcedentem</expan> du<lb/>plum cubi a c et quale corpori ex a b in quadratum a c: aggrega<lb/>tum quo que corporum ex a d in quadratum b d, & b d in quadra<lb/>tum a d et quale ei, quod fit ex a b in <expan abbr="rectangulũ">rectangulum</expan> ex a d in d b. </s>
<s id="id2748476"><expan abbr="qua-dratũ">qua<lb/>dratum</expan> <expan abbr="aut&etilde;">autem</expan> a c et maius rectangulo a d in d b quadrato c d differen <lb/>ti, igitur duplum cubi a c excedit aggregatum <expan abbr="corporũ">corporum</expan> <expan abbr="mutuorũ">mutuorum</expan> <lb/>in corpore ex a b in quadratum c d differenti, quod et <expan abbr="propo&longs;itũ">propoitum</expan>.</s></p><p type="main">
<s id="id2748554"><arrow.to.target n="marg476"/></s></p><p type="margin">
<s id="id2748564"><margin.target id="marg476"/>P<emph type="italics"/>er<emph.end type="italics"/> 5. <emph type="italics"/>ecun <lb/>di<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s></p><p type="main">
<s id="id2748615">Propoitio centeimaquadrageimaquinta.</s></p><p type="main">
<s id="id2748631">Si line a in duas partes diuidatur quadrata ambarum partium <lb/>detracto eo quod fit ex una partein alteram, qualia unt producto <lb/>unius in alteram cum quadrato differenti.</s></p><p type="main">
<s id="id2748655"><arrow.to.target n="marg477"/></s></p><p type="margin">
<s id="id2748667"><margin.target id="marg477"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2748694">Sit linea a c diuia in b, & it differentia a b, <lb/>b c, b d, dico quod quadrata a b & b c detracto <lb/><figure id="id.015.01.158.2.jpg" xlink:href="015/01/158/2.jpg"/><lb/>eo quod fit ex a b in b c, qualia unt producto a b in b c cum qua<lb/>drato b d. </s>
<s id="id2748732">Quoniam. </s>
<s id="id2748736">n. </s>
<s id="id2748740">quadrata a b, b c qualia quadratis a d d b <lb/>b c & productis ex a d in d b bis & quod fit ex a b in b c quale et <lb/>ei quod fit ex a d in e cum eo quod fit ex a d in d b, quia a d et qua </s></p><p type="main">
<s id="id2748774"><arrow.to.target n="marg478"/><lb/>lis b cideo quadrata a b & b c detracto eo quod fit ex a b in b c unt <lb/>qualia quadratis a d d b, & producto a d in d b emel: a c quadra<lb/><arrow.to.target n="marg479"/><lb/>tum a d cum producto a d in d b et quale producto a b in a d, & <lb/>ex conequenti in b c, igitur reiduum quadratorum a b & b c de<lb/>tracto producti a b in b c et quale a b in b c cum quadrato b d <lb/>quod fuit propoitum.</s></p><p type="margin">
<s id="id2748844"><margin.target id="marg478"/>P<emph type="italics"/>er<emph.end type="italics"/> 4. <emph type="italics"/>ecun <lb/>di<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2748895"><margin.target id="marg479"/>P<emph type="italics"/>er<emph.end type="italics"/> 1. <emph type="italics"/>ecun <lb/>di<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="main">
<s id="id2748946">Propoitio centeimaquadrageimaexta.</s></p><p type="main">
<s id="id2748964">Corpus quod fit ex linea diuia in uperficiem qual em quadra<lb/>tis ambarum partium detracta uperficie unius partis in <expan abbr="alterã">alteram</expan>, et <lb/>quale aggregato cuborum <expan abbr="ambarũ">ambarum</expan> <expan abbr="partiũ">partium</expan>.</s></p><figure id="id.015.01.158.3.jpg" xlink:href="015/01/158/3.jpg"/><p type="main">
<s id="id2749029">Sic a b diuia in e quadrata partium e f & <lb/><arrow.to.target n="marg480"/><lb/>b d detrahatur ex e f, f g qualis a d, dico cor <lb/>pus ex a b in uperficies b d, d g quale e<lb/>e cubis a c & c b pariter acceptis, quia. </s>
<s id="id2749066">n. <lb/></s>
<s id="id2749072">ex a b in b d fiunt duo corpora cubus <lb/>b d & corpus ex a d in quadratum d b hoc <lb/>autem et quale corpori ex b cin a d quia
<pb xlink:href="015/01/159.jpg" pagenum="140"/>funt ex qualibus lineis: at corpus quod fit ex a b in d g quale et <lb/>corporibus qu fiunt ex a c, c b in uperficiem d g at cubus a c con<lb/>tinet duo corpora qu fiunt & a c in d g & g f, igitur cubus a c upe<lb/>rat productum ex a b in d g in producto ex a c in f g & uperatur ab <lb/>eo in producto ex b c in d g, uperabatur etiam, ut uium et, cubus <lb/>b c producto b a in d b in producto b cin c f, igitur cubi a c c b u<lb/>perantur producto a b in ad in producto b cinc f & in d g, quare <lb/>in producto b c in f e: i quidem f e & f g unt qualia ex uppoito <lb/>uperant autem in producto ex c b in e f, igitur tantum et in in quo <lb/>uperantur quantum et id in quo uperant: ergo unt qualia.</s></p><p type="margin">
<s id="id2749214"><margin.target id="marg480"/>C<emph type="italics"/>o<emph.end type="italics"/>m.</s></p><p type="main">
<s id="id2749241">Propoitio centeimaquadrageimaeptima.</s></p><p type="main">
<s id="id2749260">Propoita linea diuia duas ei lineas adijcere, ut proportio addita<lb/>rum ingularum & partium imul iunctarum ad additas it mutua.<lb/><arrow.to.target n="marg481"/></s></p><p type="margin">
<s id="id2749294"><margin.target id="marg481"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2749320">Sit linea a b diuia in c uolo eius <lb/><figure id="id.015.01.159.1.jpg" xlink:href="015/01/159/1.jpg"/><lb/>partibus addere lineas, ut propoi</s></p><p type="main">
<s id="id2749348"><arrow.to.target n="marg482"/><lb/>tum et, tatuo mediam c d inter a e & <lb/><arrow.to.target n="marg483"/><lb/>c b qu it c d, & facio ut c d ad c a ita <lb/>c a ad a e, & ut d c ad c b ita c b ad b f, quia ergo d e media et inter <lb/><arrow.to.target n="marg484"/><lb/>a c & c b, & ut ea ad a cita d c a c b ad c f erunt omnes in continua <lb/><arrow.to.target n="marg485"/><lb/>proportione, quare proportio e c ad c a ut c f ad b f & e c ad ea ut <lb/>c f ad c b quod et propoitum.</s></p><p type="margin">
<s id="id2749423"><margin.target id="marg482"/>P<emph type="italics"/>er<emph.end type="italics"/> 13. <emph type="italics"/>ex <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2749474"><margin.target id="marg483"/>P<emph type="italics"/>er<emph.end type="italics"/> 11. <emph type="italics"/>ex <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2749524"><margin.target id="marg484"/>P<emph type="italics"/>er<emph.end type="italics"/> 11. <lb/><emph type="italics"/>quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2749572"><margin.target id="marg485"/>P<emph type="italics"/>er<emph.end type="italics"/> 18. <lb/><emph type="italics"/>quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="main">
<s id="id2749620">Propoitio cen teima quadra geimaoctaua.</s></p><p type="main">
<s id="id2749635">Propoitis tribus lineis primam ic diuidere, ut adiectis duabus <lb/>alijs lineis ecundum rationem mutuam ingularum ingulis ag<lb/>gregatum ex una adiectarum & parte ad aggregatum ex alia parte <lb/>& adiecta e habeat, ut ecunda ad tertiam.<lb/><arrow.to.target n="marg486"/></s></p><p type="margin">
<s id="id2749684"><margin.target id="marg486"/>C<emph type="italics"/>o<emph.end type="italics"/>m.</s></p><p type="main">
<s id="id2749710">Sit a, b, c, d, propoit line, <lb/><figure id="id.015.01.159.2.jpg" xlink:href="015/01/159/2.jpg"/><lb/>uolo diuidere a b ita in e ut <lb/>umpta ecundum proportio<lb/>nem alicuius quantitatis, puta <lb/>g ad a e ic b f ad e b & ut g ad <lb/>e b ic g a ad a e ut it propor<lb/>tio g e ad e f ut c ad d. </s>
<s id="id2749768">Sint ergo <lb/>omnia <expan abbr="cõ&longs;tituta">contituta</expan> & it g rectan<lb/>gulum ex a e in e b, cum ergo <lb/>g a contineat a e ut g continet e b, g autem continet e b ecundum <lb/>a e, igitur g a continet a e ecundum a c, ergo ex diffinitione qua</s></p><p type="main">
<s id="id2749814"><arrow.to.target n="marg487"/><lb/>drati a g et quadratum a e. </s>
<s id="id2749827">Pari ratione b f et quadratum b e. </s>
<s id="id2749834">pro<lb/>portio igitur g e ad e f cum it ut c ad e ex uppoito erit ut ipi pro<lb/>portioni addamus, & detrahamus ex duplo a b & dimidium rei<lb/>dui ducamus in e, & addamus aggregato quadrati a b cum ipa
<pb xlink:href="015/01/160.jpg" pagenum="141"/>a b, & latus eius detracto dimidio reidui erit b clinea, quare diui<lb/>io nota, & et ut dicamusu: olo diuidere datam lineam, ut quantita<lb/>tes adiect ub mutua proportione ad unam tertiam cum parti<lb/>bus obtineantinter e proportionem datam.</s></p><p type="margin">
<s id="id2749920"><margin.target id="marg487"/>P<emph type="italics"/>er<emph.end type="italics"/> 1. <emph type="italics"/>ecuu <lb/>di<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s></p><p type="main">
<s id="id2749971">Propoitio centeimaquadrageimanona.</s></p><p type="main">
<s id="id2749987">Datam lineam ic diuidere, ut proportio quadratorum ad du<lb/>plum unius partis in alteram it, ut line dat ad lineam datam.</s></p><p type="main">
<s id="id2750012">Sit data a b quam uolo diuidere, ut proponitur ub proportio<lb/><arrow.to.target n="marg488"/><lb/>ne c d ad e, diuido a b bifariam in f, & abcindo <lb/><figure id="id.015.01.160.1.jpg" xlink:href="015/01/160/1.jpg"/><lb/>g d qualem d e, & inter c g <expan abbr="re&longs;iduũ">reiduum</expan> & c e inter<lb/>pono proportione, & ut h ad c g ita a f medietatis a b ad fk. </s>
<s id="id2750071">Omnia <lb/>ita unt notisima ex primo & exto Elemento<lb/><figure id="id.015.01.160.2.jpg" xlink:href="015/01/160/2.jpg"/><lb/><expan abbr="rũ">rum</expan> Euclidis. </s>
<s id="id2750114">Si ergo abcindantur fk ex fa, dico <lb/>quod proportio quadratorum l k & k a ad du<lb/>plum rectanguli a k in k b et ut c d ad d e. </s>
<s id="id2750131">Quia. n. </s>
<s id="id2750138">c e ad c g dupli<lb/>cata et ei qu et h ad c g, duplicata et <expan abbr="etiã">etiam</expan> ei qu et f a ad fk, qua<lb/>re ut quadrati a f ad fk, ita c e ad c g, igitur diiungendo c g ad g e ut <lb/>reidui quadrati k f ad reiduum quadrati a f, quare c g ad g d ut <lb/>quadrati k f ad dimidium reidui quadrati a f, igitur coniunctim c d <lb/>ad d g ut quadrati k f & dimidij reidui quadrati a f ad ipum dimi<lb/>dium reidui. </s>
<s id="id2750212">At uer cum g d it qualis d e, erit c d ad d e ut qua<lb/>drati k f cum dimidio reidui pius dicti ad ipum dimidium rei<lb/>dui. </s>
<s id="id2750246">Igitur etiam ut dupli quadrati k f cum reiduo ad <expan abbr="re&longs;iduũ">reiduum</expan>, unt <lb/>enim omnia duplicata. </s>
<s id="id2750273">At <expan abbr="duplũ">duplum</expan> quadrati k f <expan abbr="cũ">cum</expan> reiduo et qua<lb/>le quadratis a f & f k, igitur quadratorum a f & f k ad differentiam <lb/>eo rum proportio et ut c d ad d e, igitur dupli quadratorum a f & <lb/>f k ad duplum differenti quadratorum a f & fk ut c d ad d e. </s>
<s id="id2750323">Ve<lb/><arrow.to.target n="marg489"/><lb/>rum duplum quadratorum a f & f k quatur quadratis b k & k a. <lb/><arrow.to.target n="marg490"/><lb/>Et duplum differenti quadratorum a f & fk et quale duplo pro <lb/>ducti b k in k a, igitur proportio quadratorum k b & k a ad <expan abbr="duplũ">duplum</expan> <lb/>producti k b in k a et ueluti c d ad d e, quod et propoitum.</s></p><p type="margin">
<s id="id2750387"><margin.target id="marg488"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s id="id2750414"><margin.target id="marg489"/>P<emph type="italics"/>er<emph.end type="italics"/> 9. <emph type="italics"/>ecun <lb/>di<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2750465"><margin.target id="marg490"/>P<emph type="italics"/>er<emph.end type="italics"/> 5. <emph type="italics"/>ecun <lb/>di<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="main">
<s id="id2750515">Propoitio centeimaquinquageima.</s></p><p type="main">
<s id="id2750531">Propoitis duabus lineis <expan abbr="lineã">lineam</expan> communem <lb/><figure id="id.015.01.160.3.jpg" xlink:href="015/01/160/3.jpg"/><lb/>utrique adiungere, ut it maioris ad additam pro<lb/>portio, uelut quadratorum minoris & adiect <lb/>ad duplum unius in alteram.</s></p><p type="main">
<s id="id2750579">Hc et quai conuera <expan abbr="præced&etilde;tis">prcedentis</expan>. </s>
<s id="id2750608">Sit a ma<lb/><arrow.to.target n="marg491"/><lb/>ior, & b c minor, & fiat b d dupla b c, uper <expan abbr="quã">quam</expan> <lb/>erigatur b f qualis a; & it rectangulum d f & <lb/>decribatur quadratum b c quod it b g reidu <lb/>uperficiei ad d f latus it h, dico h ee lineam quitam. </s>
<s id="id2750673">Superficies
<pb xlink:href="015/01/161.jpg" pagenum="142"/>enim d f cum fiat ex a in duplum b c, dupla erit uperficiei a in b c, u <lb/>perficies f d, tota quatur quadratis h & b c, igitur quadrata h & b <lb/>c dupla unt uperficiei a in b c, quod uer fit ex a in duplum b c e <lb/>habet ad id quod fit ex h in duplum b c, ut a ad h, cum per eandem <lb/>lineam ducantur, igitur quod fit ex a in duplum b c, & unt quadra<lb/>ta h & b c, e habent ad duplum h in b c, ut a ad h, quod fuit de<lb/>montrandum.</s></p><p type="margin">
<s id="id2750746"><margin.target id="marg491"/>C<emph type="italics"/>o<emph.end type="italics"/>m.</s></p><p type="main">
<s id="id2750773">Propoitio centeimaquinquageimaprima.</s></p><p type="main">
<s id="id2750789">Proportio differenti quadratorum partium, cuiuuis line ad <lb/>quadratum differenti <expan abbr="illarũ">illarum</expan> et uelut to tius line ad differentiam.<lb/><arrow.to.target n="marg492"/></s></p><p type="margin">
<s id="id2750832"><margin.target id="marg492"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2750858">Sit a b diuia in puncto c, & fiat c d qualis <lb/>c b, manifetum et quod differentia partium <lb/><figure id="id.015.01.161.1.jpg" xlink:href="015/01/161/1.jpg"/><lb/>et a d, dico proportionem differenti quadra <lb/>torum a c & c b ad quadratum a d differenti partium ee ut a b ad </s></p><p type="main">
<s id="id2750912"><arrow.to.target n="marg493"/><lb/>a d. </s>
<s id="id2750923">Quoniam differentia quadratorum a c & c b et, quod fit ex a d <lb/>in d c bis cum quadrato a d, & ide quod fit ex a d in d b cum qua<lb/>drato a d, & ide quod fit ex tota a b in a d. </s>
<s id="id2750948">Igitur differentia qua<lb/><arrow.to.target n="marg494"/><lb/>drato a c & c b et quod fit ex a b in a d, quare cum quadratum a d <lb/>fiat ex a d in a d, erit proportio a b ad a d, uelut differenti quadra<lb/><arrow.to.target n="marg495"/><lb/>torum a c & b c ad quadratum a d differenti partium. </s>
<s id="id2750986">Quod fuit <lb/>propoitum.</s></p><p type="margin">
<s id="id2751000"><margin.target id="marg493"/>P<emph type="italics"/>er<emph.end type="italics"/> 4. <emph type="italics"/>ecun <lb/>di<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2751052"><margin.target id="marg494"/>P<emph type="italics"/>er<emph.end type="italics"/> 3. <emph type="italics"/>ecun <lb/>di<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2751102"><margin.target id="marg495"/>P<emph type="italics"/>er<emph.end type="italics"/> 1. <emph type="italics"/>exti<emph.end type="italics"/><lb/>E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="main">
<s id="id2751151">Propoitio centeimaquinquageimaecunda.</s></p><p type="main">
<s id="id2751169">Si linea in duas partes quales duas que in quales diuidatur, fue<lb/>ritque proportio aggregati ex maiore & dimidio ad ipam maiorem <lb/>uelut ex minore, & aliqua linea ad ipam minorem, & rurus aggre<lb/>gati ex minore dimidio ad ipam minorem, uelut aggregati ex ma<lb/>iore & alia addita ad ipam maiorem, erit proportio dimidij'ad par <lb/>tem unam inqualem, uelut alterius partis inqualis ad uam ad<lb/>ditam mutu, & etiam proportio ad ditarum inuicem, uelut pro<lb/>portio partium inqualium duplicata, & rurus ipum dimidium <lb/>line aumpt medium erit proportione inter additas. </s>
<s id="id2751260">Demum <lb/>proportio dimidij cum ad dita maiore ad dimidium cum addita mi<lb/>nore, uelut maioris partis ad minorem.<lb/><arrow.to.target n="marg496"/></s></p><p type="margin">
<s id="id2751282"><margin.target id="marg496"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2751309">Sit propoita a b diuia per <lb/><figure id="id.015.01.161.2.jpg" xlink:href="015/01/161/2.jpg"/><lb/>qualia in c per inqualia in <lb/>d, & it ut addantur a g & b f, <lb/>ita ut proportio c a, & a d ad a d it ueluti f d ad d b, & c b & b d ad <lb/>b d, uelut g d ad d a, & hc et quarta <expan abbr="&longs;ecũdi">ecundi</expan> Archimedis de phra, <lb/>& Cylindro: quia ergo a c & a d ad a d, ut f d ad d b erit a c ad a d, <lb/>fb ad b d. </s>
<s id="id2751386">Et imiliter quia et c b & b d ad b d, uelut g d ad d a erit
<pb xlink:href="015/01/162.jpg" pagenum="143"/>c b ad b d, uelut g a ad a d, & hoc et primum. </s>
<s id="id2751410">Quia ergo c a et <lb/>qualis c b, erit c a ad b d, uelut g a ad a d, & iam fuit a d ad c a, ut b d <lb/>ad f b, per conueram igitur a d ad b d, ut g a ad a d, & ut b d ad fb, <lb/>interpoitis ergo a d & d b inter a g & b f cum compoita it pro<lb/>portio a g ad b f ex proportione a g ad a d, & ad d b, & d b <lb/>ad b f, & proportio a d ad d b, it qualis proportioni <lb/><figure id="id.015.01.162.1.jpg" xlink:href="015/01/162/1.jpg"/><lb/>a g ad a d, & d b ad b f, igitur proportio a g ad b f. </s>
<s id="id2751475">Per de<lb/>montrata ab Alchindo et duplicata proportioni a d ad <lb/>d b quod et ecundum. </s>
<s id="id2751498">Rurus quia ex primo demon<lb/>trato, uel eius conuero proportio a d ad a c et uelut b d <lb/>ad b f, & d b ad a c, ut a d ad a g, proportiones ergo <lb/><figure id="id.015.01.162.2.jpg" xlink:href="015/01/162/2.jpg"/><lb/>a d & d b ad a c componunt proportionem produ<lb/>ducti a d in d b, quod it h ad quadratum a c quod it <lb/>k, & imiliter proportio b d ad b f & a d ad a g com<lb/>ponunt proportionem producti ex b d in a d, quod <lb/>itl ad productum b f in a g, quod it m, per demontrata ab Eucli<lb/>de in exto Elementorum, igitur proportio h ad k ut l ad m, ed h & </s></p><p type="main">
<s id="id2751584"><arrow.to.target n="marg497"/><lb/>l unt quales, quia producuntur ex eidem, igitur per demontra<lb/>ta in quinto Elementorum Euclidis, k et quale m, ergo a c et me<lb/>dia pro portione inter b f & g a, quod et tertium. </s>
<s id="id2751627">Quia uer ex pri<lb/>mo demontrato et fb ad b d, ut a c ad a d, & c b ad idem b d, ut g a <lb/>ad idem a d erit coniungendo fb & b c ad b d, ut coniun<lb/><figure id="id.015.01.162.3.jpg" xlink:href="015/01/162/3.jpg"/><lb/>gendo g a & a c ad a d, ed fb & b c componunt f c & g a, <lb/>& a c componunt g c, igitur ut f c ad b d, ita g c ad a d, er<lb/>go permutando g c ad f c, ut a d ad b d, quod et quartum.</s></p><p type="margin">
<s id="id2751684"><margin.target id="marg497"/>I<emph type="italics"/>n<emph.end type="italics"/> P<emph type="italics"/>rop.<emph.end type="italics"/> 23 <lb/>P<emph type="italics"/>ropo.<emph.end type="italics"/> 9.</s></p><p type="main">
<s id="id2751737">Cum ergo punctum d fuerit datum, licet inuenire a g & b f, faci<lb/>l, ut Archimedes prup ponit proportionem g d ad d f datam & <lb/>qurit eam, qu et a d ad d b, & peruenitur ad res numero triplo <lb/>quadrati dimidij line aumpt quales cubo & numero, qui it <lb/>ex duplo cubi dimidij in 1 m: ipa proportione, & quod produci<lb/>tur diuio per 1 p: ipa proportione. </s>
<s id="id2751803">Veluti poita a b 10, & propor<lb/>tione quam uolo g d ad d f excupla, duco 5 dimidium 10 in e fit 25, <lb/>& triplico, fit 75 numerus rerum. </s>
<s id="id2751825">Inde duco 5 idem dimidium ad <lb/>cubum fit 125, duplico fit 250, duco in 5, qui et 1 m: proportione fit <lb/>1250, diuido per 7, qui et 1 p: proportione exit 178 4/7 numerus, qui <lb/>cum cubo quatur 75 rebus. </s>
<s id="id2751848">Cum ergo contituta fuerit diuiio in <lb/>c, non recipit proportionem g d ad f d quam uolueris, ed equitur <lb/>una ola ad <expan abbr="illã">illam</expan>, & et mirabile, quoniam line uidentur umi liber. <lb/></s>
<s id="id2751894">Sed non et ita. </s>
<s id="id2751901">Et <expan abbr="etiã">etiam</expan> quia Archimedes <expan abbr="uide&ttilde;">uidetur</expan> aumere <expan abbr="aliã">aliam</expan> lineam, <lb/>ed non inue tigat eam, im otendit eam ex aumptis. </s>
<s id="id2751951">At Euto ci<lb/>us oten dit ambas, <expan abbr="unã">unam</expan> ex propria inuentione, aliam ex Diocle, ed
<pb xlink:href="015/01/163.jpg" pagenum="144"/>una et uperflua, quia ut dixi, una e quitur ad aliam. </s>
<s id="id2751991">Ex hoc pa<lb/>tet cur Dio cles aumperit lineam unam, qu et a c, qu e ha<lb/>bet ad a d, & d b, ut uicisim a d, & d b ad additas, quod et pri<lb/>mum demontratum. </s>
<s id="id2752037">Sic enim omittit primum quod proponit Ar <lb/>chimedes, & aumit quod proximum et: & ide Archimedes non <lb/>pro bat, nec prupponit, quod Diocle probatur, cilicet datum <lb/>ee punctum d in linea a b, ed olum in linea g f, ide cogitur pro<lb/>bare ecundum quod demontratur ab Eutocio, & nobis demon <lb/>tratum et upr. </s>
<s id="id2752110">Archimedes <expan abbr="aũt">aunt</expan> aumit <expan abbr="lineã">lineam</expan> extra circulum, <expan abbr="quã">quam</expan> <lb/>uo cat b f, qu et qualis b c medietati: aliam aumit quam uocat <lb/>b h, cuius proportio ad b d et icut quadrati ad a d quadratum a b. <lb/></s>
<s id="id2752171">Contat ergo quod proportio g d ad d f et data. </s>
<s id="id2752181">Et imiliter f g ad <lb/>g d, & et 1 pr proportione data. </s>
<s id="id2752197">Vnde notandum quod datum <lb/>dicitur, impliciter cognitum alio modo, dicitur datum poitione, <lb/>quod et certum & tale, uelut i quis dicat, diuide 10 in duos nume<lb/>ros quadratos: hoc non et datum, potet enim diuidi pluribus mo <lb/>dis. </s>
<s id="id2752234">At i dicas ut una pars it alterius <expan abbr="quadratũ">quadratum</expan>, itud antequm ci <lb/>untur partes, dicitur datum poitione. </s>
<s id="id2752266">Ergo datum poitione et du <lb/>plex, uel ut ratio nota it, non autem quantitas, ut i dicam a b et du <lb/>pla ad b c, utra que dicitur nota poitione, quo<lb/>niam necio quanta it a b. </s>
<s id="id2752304">Vel i quantitas et <lb/><figure id="id.015.01.163.1.jpg" xlink:href="015/01/163/1.jpg"/><lb/>nota proportio ignota it, ut i a c it 10, & it, <lb/>ut b c it <02> relata, a b erit punctus b, & proportio a b ad b c data po <lb/>itione, non tamen nota. </s>
<s id="id2752350">Et i dicas igitur omnia, qu habent deter <lb/>minationem erunt data poitione? </s>
<s id="id2752366">Dico quod non, quia oportet, <lb/>ut illa determinatio comprehendatur ub una ratione, eaque altem <lb/>generaliter co gnita.</s></p><p type="main">
<s id="id2752388">Propoitio centeimaquinquageimatertia.</s></p><p type="main">
<s id="id2752404">Vim quan cun que manus multiplicare.<lb/><arrow.to.target n="marg498"/></s></p><p type="margin">
<s id="id2752419"><margin.target id="marg498"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2752445">Cum enim radimus aut trahimus manifetum et, </s></p><p type="main">
<s id="id2752459"><arrow.to.target n="marg499"/><lb/>quod ambabus manibus uis conduplicatur, & ma<lb/><figure id="id.015.01.163.2.jpg" xlink:href="015/01/163/2.jpg"/><lb/>ior redditur, quanta et proportio totius ad exce<lb/>um: uelut it a quod mouetur ab una manu uiribus <lb/>ut b, qu unt exceus b d upra a, cum ergo propor<lb/>tio c b d ad a it compoita ex proportionibus c & <lb/>b d ad a manifetum et, quod erit producta ex pro<lb/>portione c b d ad b d, & b d ad a, ed e b d et dupla <lb/>ad b d, quia e et qualis, cigitur proportio c b d ad <lb/><arrow.to.target n="marg500"/><lb/>a et maior multo qum duorum exceuum, qui mo<lb/>uerent in proportione dupla: uelut i adderemus f
<pb xlink:href="015/01/164.jpg" pagenum="145"/>ad d b qualem b, multo maior et ex communi animi ententia e f <lb/>b d <expan abbr="quã">quam</expan> f b d, quia e continet f, & quantum et d inuper: cum ergo <lb/>b cum d moueat a in proportione b d ad a & f cum d mouebit a in <lb/>proportione eadem qua b d, ergo per uiam additionis duplo ue<lb/>locius, qum dupla proportione, uerm dupla comparatione ad <lb/>proportionem b d ad a, non autem duplicata ed dupla, ut dixi, qu <lb/>erit maior qum dupla per <expan abbr="addition&etilde;">additionem</expan> exceus. </s>
<s id="id2752655">Ergo i addatur al<lb/>ter homo, erit dupla ad illam duplam, ueluti addendo qualem d b <lb/>f e, ade ut i proportio d b f e eet quintupla, mouerent illi duo in <lb/>proportione decupla. </s>
<s id="id2752687">Sed annexo baculo aut lima aut erra annu<lb/>lo h, ita ut circunuolui posit h quabit uires non olum d b f e ed <lb/>multorum hominum. </s>
<s id="id2752713">igitur multo plus aget homo ambabus ma<lb/>nibus radendo aut ecando cum g, qum quadrupla proportione <lb/>unius manus, & hocincrementum et non olum magn <lb/>utilitatis, ed ualde <expan abbr="accõmodatum">accommodatum</expan> in actionibus artificum <lb/>operum grauiorum. </s>
<s id="id2752759">Et huiumodi conduplicatio et ratio <lb/>lim quam urdam uocamus.</s></p><p type="margin">
<s id="id2752782"><margin.target id="marg499"/>P<emph type="italics"/>er<emph.end type="italics"/> 37.</s></p><p type="margin">
<s id="id2752807"><margin.target id="marg500"/>P<emph type="italics"/>er<emph.end type="italics"/> 2.</s></p><figure id="id.015.01.164.1.jpg" xlink:href="015/01/164/1.jpg"/><p type="main">
<s id="id2752842">Propoitio centeimaquadrageimaquarta.</s></p><p type="main">
<s id="id2752858">Si line dat alia linea adiungatur, ab extremitatibus autem pri<lb/>oris line du rect in unum punctum con currant proportionem <lb/>habentes quam media inter totam & adiectam, ad adiectam erit <lb/>punctus concurus puncto extremo line adiect ditans per li<lb/>neam mediam. </s>
<s id="id2752907">Qud i ab extremo alicuius line qualis medi <lb/>eu peripheria circuli cuius emidiameter it media linea du line <lb/>ad prdicta puncta producantur, ip erunt in proportione medi <lb/>ad adiectam.<lb/><arrow.to.target n="marg501"/></s></p><p type="margin">
<s id="id2752968"><margin.target id="marg501"/>C<emph type="italics"/>o<emph.end type="italics"/>m.</s></p><p type="main">
<s id="id2752994">Hc propoitio et admirabilis: & etiam decripi, ut multa ecre<lb/>ta Dialectic potius <expan abbr="aperiren&ttilde;">aperirentur</expan> quam quod huic propoito <expan abbr="multũ">multum</expan> <lb/>congrueret. </s>
<s id="id2753044">Ide potius cholij caua poita et quam ipius tracta<lb/>tionis: ut <expan abbr="modũ">modum</expan> demontrandi magis quam id, d <expan abbr="demon&longs;tra&ttilde;">demontratur</expan>, re<lb/>picere oporteat. </s>
<s id="id2753103"><expan abbr="Con&longs;titua&ttilde;">Contituatur</expan> ergo (per uiam problematis) linea a b <lb/>& proportio c ad d, & fiat d e ad c, ut c ad d, & a b ad e ut b f ad d, & <lb/>ut g ad c, eritque g media inter a f & f b, quod licet olum upponatur <lb/>ab Appollonio, <expan abbr="tam&etilde;">tamen</expan> facil demontratur & Commandino adie<lb/>cta et <expan abbr="demõ">demom</expan> tratio. </s>
<s id="id2753172">Concurrant ergo ex a & b du line in aliquod </s></p><p type="main">
<s id="id2753187"><arrow.to.target n="marg502"/><lb/>punctum, putat h ut it a h ad h b uelut c ad d, dico quod i ducat <lb/>h f quod ipa erit qualis g, ducatur b l quiditans a h, & quia <lb/><arrow.to.target n="marg503"/><lb/>ex uppoito a h ad h b, ut g ad b f, erit b h ad h a, ut b f ad g, & quia <lb/>trianguli a h f & b l f unt imiles erit proportio a h ad b l, ueluti a f <lb/><arrow.to.target n="marg504"/><lb/>ad fb, igitur per quam proportionem b e h ad b l, ut a f ad g, ed ut <lb/><arrow.to.target n="marg505"/><lb/>a f ad g ita g ad b f ex uppoito: & ut a f ad g, it a h a ad h b, ex uppo
<pb xlink:href="015/01/165.jpg" pagenum="146"/>ito igitur ut a h ad h b ita h b ad b l, ed angulus a h b et qualis <lb/>angulo h b l, ergo triangulus a h b et <lb/>imilis triangulo h b l, quare angulus <lb/>b h l et qualis angulo h a f, igitur du <lb/>orum triangulorum f a h, & fb h duo <lb/><arrow.to.target n="marg506"/><lb/>anguli unius a & f unt quales duo<lb/>bus angulis, alterius igitur propor<lb/><figure id="id.015.01.165.1.jpg" xlink:href="015/01/165/1.jpg"/><lb/>tio a f ad fh repicientium angulos <lb/><arrow.to.target n="marg507"/><lb/>quales ut a h ad h b repicientium an<lb/><arrow.to.target n="marg508"/><lb/>gulum f, ed a h ad h b ut c ad d, ex up <lb/>poito igitur a f ad f h, ut c ad d, ed ut c ad d ita a f ad g, ex uppoito <lb/>ergo h f et qualis g.<lb/><arrow.to.target n="marg509"/></s></p><p type="margin">
<s id="id2753420"><margin.target id="marg502"/>P<emph type="italics"/>er<emph.end type="italics"/> 29. <emph type="italics"/>pri <lb/>mi, &<emph.end type="italics"/> 4. <emph type="italics"/>ex <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2753484"><margin.target id="marg503"/>P<emph type="italics"/>er<emph.end type="italics"/> 22. <lb/><emph type="italics"/>quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2753532"><margin.target id="marg504"/>P<emph type="italics"/>er<emph.end type="italics"/> 11. <emph type="italics"/>quin <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2753581"><margin.target id="marg505"/>P<emph type="italics"/>er<emph.end type="italics"/> 6. <emph type="italics"/>exti<emph.end type="italics"/><lb/>E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2753630"><margin.target id="marg506"/>P<emph type="italics"/>er<emph.end type="italics"/> 32. <emph type="italics"/>pri <lb/>mi, &<emph.end type="italics"/> 4. <emph type="italics"/>ex <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2753694"><margin.target id="marg507"/>P<emph type="italics"/>er<emph.end type="italics"/> 11. <lb/><emph type="italics"/>quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2753741"><margin.target id="marg508"/>P<emph type="italics"/>er<emph.end type="italics"/> 7. <emph type="italics"/>quin<lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2753791"><margin.target id="marg509"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 1.</s></p><p type="main">
<s id="id2753817">Cum ergo hc demontratio it ex enu in uno puncto h, ide ad <lb/>qulibet puncta traduci potet, qu potero imaginari, & ita pri<lb/>ma uo cabitur enus, <expan abbr="&longs;ecũda">ecunda</expan> imaginandi: Et <expan abbr="quoniã">quoniam</expan> in demontran<lb/>do non aumimus aliquid, quod it proprium alicui puncto, nii <lb/>proportionem h a ad h b imilem ee c ad d, ideo hoc pertinet ad <lb/>intellectum, & et tertium. </s>
<s id="id2753913">Etidem dico i k eet ultra h quod po<lb/>tet contingere. </s>
<s id="id2753932">mod k a ad k b it ut c ad d & k f it qualis g idem <lb/>equetur, & comprehenditur ub tertio & pertinet ad intellectum, <lb/>& quoniam demontratur quod punctum k ubicun que umatur, et <lb/>in quali <expan abbr="di&longs;tãtia">ditantia</expan> puncto fcilicet per g lineam, erit emper in peri<lb/>pheria circuli, & hoc potet ee in infinitis locis impliciter & extra <lb/>infinitum nihil et, igitur ub hoc continetur conuerum cilicet, <lb/>quod a quolibet puncto circuli ductis lineis ad a & b ip erunt in <lb/>proportione c ad d. </s>
<s id="id2754042">Et ita abque principijs Geometricis concluditur <lb/>propoitio Geometrica & hoc et <foreign lang="greek">w_erila/mpousin</foreign> & ferm ummum in<lb/>tellectus humani. </s>
<s id="id2754075">Et potet demontrari Geometric duobus uer<lb/>bis. </s>
<s id="id2754092">Quia. n. </s>
<s id="id2754098"><expan abbr="f&longs;upponi&ttilde;">fupponitur</expan> qualis g eo qud h et in peripheria circu<lb/>li erit media inter a f & f b, quare cum angulus f it communis, erit <lb/>proportio a h ad h b, laterum repicientium angulum f in utroque </s></p><p type="main">
<s id="id2754142"><arrow.to.target n="marg510"/><lb/>triangulo, uelut h f lateris in maiori ad f b latus in minori, quare <lb/><arrow.to.target n="marg511"/><lb/>cum ex uppoito h f ad fb it ut c ad d, erit a ad b, ut c ad d. </s>
<s id="id2754172">Et uides <lb/>Apollonium, & Pappium quanta uperflua adij ciant in hac ecun<lb/><arrow.to.target n="marg512"/><lb/>da parte demontrationis, qu et prima apud illos, & ducunt <expan abbr="unã">unam</expan> <lb/>lineam non neceariam ex puncto b ad latus fh. </s>
<s id="id2754221">Vt <expan abbr="antiquorũ">antiquorum</expan> ple <lb/>rique non tantum potuerint Geometria & ingenio, qu ferunt excel <lb/>lentisima in illis, quantum nos ex Dialectica <foreign lang="greek">w_e?ila/mpousin</foreign> inducen <lb/>tes. </s>
<s id="id2754256">et enim ingulare hoc exemplum.<lb/><arrow.to.target n="marg513"/></s></p><p type="margin">
<s id="id2754277"><margin.target id="marg510"/>P<emph type="italics"/>er<emph.end type="italics"/> 6. <emph type="italics"/>exti<emph.end type="italics"/><lb/>E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2754326"><margin.target id="marg511"/>P<emph type="italics"/>er<emph.end type="italics"/> 4. <emph type="italics"/><expan abbr="eiu&longs;d&etilde;">eiudem</expan><emph.end type="italics"/></s></p><p type="margin">
<s id="id2754371"><margin.target id="marg512"/>P<emph type="italics"/>er<emph.end type="italics"/> 11. <emph type="italics"/>ex <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/><lb/>I<emph type="italics"/>n primo<emph.end type="italics"/> C<emph type="italics"/>o <lb/>nicor.<emph.end type="italics"/> A<emph type="italics"/>pol. <lb/>in<emph.end type="italics"/> P<emph type="italics"/>rfat.<emph.end type="italics"/></s>
</p><p type="margin">
<s id="id2754475"><margin.target id="marg513"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 2.</s></p><p type="main">
<s id="id2754502">Ex hoc <expan abbr="etiã">etiam</expan> patet quod i circulus duceretur ecundum f k tran<lb/>iretque per m & n eet a m ad m b & a n ad b n, ut a h ad h b.</s></p>
<pb xlink:href="015/01/166.jpg" pagenum="147"/><p type="head">
<s id="id2754544">SCHOLIVM</s></p><p type="main">
<s id="id2754552">Ex hoc pater qualiter ex uera demontratione enu otena per<lb/>uenimus ad quotquot imaginando, inde intellectu abiectis condi<lb/>tionibus non necearijs facimus infinitum & uniuerale. </s>
<s id="id2754588">Demum <lb/>ine artis pe cialis auxilio otendimus Iheorema uniuerale (quod <lb/>etiam poterat otendi Geometric, ed long pulchrius et, ac ubli<lb/>mius per <foreign lang="greek">w_erilampousin</foreign>, qa hoc ipo infinita alia do cemus generaliter <lb/>per implicem <expan abbr="compreh&etilde;&longs;ionem">comprehenionem</expan> otendere) cilicet quod quouis <lb/>puncto peripheri circuli, cuius emidiameter et media proportio<lb/>ne inter totam extenam centro uque exterius, & partem qu' et <lb/>centro ad punctum decriptum ub proportione continua <expan abbr="datarũ">datarum</expan> <lb/>linearum line duct ex eo ad punctum exterius, & punctum de<lb/>criptum unt in proportione datarum linearum.</s></p><p type="main">
<s id="id2754740">Propoitio centeimaquinquageimaquinta.</s></p><p type="main">
<s id="id2754757"><expan abbr="Quadratorũ">Quadratorum</expan> <expan abbr="numerorũ">numerorum</expan> proportionem & <expan abbr="inuention&etilde;">inuentionem</expan> <expan abbr="cõ&longs;iderare">coniderare</expan>.</s></p><figure id="id.015.01.166.1.jpg" xlink:href="015/01/166/1.jpg"/><p type="main">
<s id="id2754813">Primm oportet cire ee tres naturales <lb/>numerorum eries, primam Euclidis iuxta </s></p><p type="main">
<s id="id2754836"><arrow.to.target n="marg514"/><lb/>quamuis <expan abbr="proportion&etilde;">proportionem</expan>, in qua unum & ter<lb/>tius & quintus, & ita uno emper intermi<lb/>o unt quadrati. </s>
<s id="id2754872">Primus quo que. </s>
<s id="id2754876">1. unum & <lb/>quartus & eptimus & ita duobus intermisis unt cubi. </s>
<s id="id2754893">In ecun<lb/>do ordine et naturalis eries numerorum, ex qua colligitur alia, & <lb/>ex illa bini quilibet e equentes contituunt numerum <expan abbr="quadratũ">quadratum</expan>. <lb/></s>
<s id="id2754933">In tertia numeri impares, qui emper collati efficiunt quadratum.</s></p><p type="margin">
<s id="id2754945"><margin.target id="marg514"/>E<emph type="italics"/><expan abbr="xemplũ">xemplum</expan><emph.end type="italics"/> 1.</s></p><figure id="id.015.01.166.2.jpg" xlink:href="015/01/166/2.jpg"/><p type="main">
<s id="id2754986">Sit ergo propoitus numerus cui uelim <lb/>addere quadratum numerum, ut fiat qua<lb/><arrow.to.target n="marg515"/><lb/>dratus totus, accipe numerum quadratum <lb/>minorem illo quem uis, & detrahe propo <lb/>ito numero eu quadrato eu non reidu<lb/><arrow.to.target n="marg516"/><lb/>um, diuide per duplum <02> quadrati quod <lb/>detraxiti, d exit duc in e fiet quadratus numerus, idem que additus <lb/>numero propoito, faciet quadratum. </s>
<s id="id2755054">Velut capio 16 qui et qua<lb/>dratus, aufero 9 quadratum <expan abbr="minor&etilde;">minorem</expan> relin quitur 7, diuido per 6 du<lb/>plum <02> 9, exit 1 1/6 quadratum eius et 1 13/36 qui additus ad 16 facit 17 13/36 <lb/><expan abbr="quadratũ">quadratum</expan> cuius <02> et 4 1/6.</s></p><p type="margin">
<s id="id2755100"><margin.target id="marg515"/>E<emph type="italics"/><expan abbr="xemplũ">xemplum</expan><emph.end type="italics"/> 2.</s></p><p type="margin">
<s id="id2755131"><margin.target id="marg516"/>E<emph type="italics"/><expan abbr="xemplũ">xemplum</expan><emph.end type="italics"/> 3.</s></p><p type="main">
<s id="id2755163">Ex hoc patet propoito quouis numero <expan abbr="&qtilde;drato">quadrato</expan> modus inuenien<lb/><arrow.to.target n="marg517"/><lb/>di infinitos numeros quadratos qui <expan abbr="cũ">cum</expan> illo iuncti facient <expan abbr="quadratũ">quadratum</expan>.</s></p><p type="margin">
<s id="id2755210"><margin.target id="marg517"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 1.</s></p><p type="head">
<s id="id2755236">SCHOLIVM.</s></p><p type="main">
<s id="id2755246">Poem adducere demontrationes omnium <expan abbr="horũ">horum</expan>, ed reddere<lb/>tur res longa <expan abbr="cũ">cum</expan> int manifet ex eptimo octauo & nono Euclidis. <lb/></s>
<s id="id2755294">Exemplum ecundum capio mod 14 qui non et quadratus, aufe<lb/>ro 9, remanet 5, diuido per 6 duplum <02> 9 exit 5/6 <expan abbr="quadratũ">quadratum</expan> eius et 25/36
<pb xlink:href="015/01/167.jpg" pagenum="148"/>hic additus ad 14 contituit 14 25/36 quadratum 3 5/6. Et ita 14 et diffe<lb/>rentia duorum quadratorum, cilicet 25/36 & 14 25/36.<lb/><arrow.to.target n="marg518"/></s></p><p type="margin">
<s id="id2755356"><margin.target id="marg518"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 2.</s></p><p type="main">
<s id="id2755382">Ex hoc habebis duo quadrata in datis terminis qu different <lb/>dato numero, & et pulchrum. </s>
<s id="id2755395">Velut uolo duo quadrata qu dif<lb/>ferant in 2, & <02> minoris it inter 1 & 2, tunc capies per regulam i<lb/>pam 2, & auferes <expan abbr="numerũ">numerum</expan> quadratum ita qud reiduum diuium <lb/>per duplum radicis efficiat <expan abbr="numerũ">numerum</expan> inter 1 & 2. Veluti capio 4/9 qua<lb/>dratum, aufero ex 2, relinquitur 1 5/9 diuido per duplum 2/13 radicis 4/9 & <lb/>et 1 1/3 & exit 1 1/6, & hic et minor numerus cuius quadratum et 1 13/36 <lb/>cui i addantur 2, fient 3 13/36 numerus quadratus 1 5/6.</s></p><p type="main">
<s id="id2755479"><arrow.to.target n="marg519"/></s></p><p type="margin">
<s id="id2755490"><margin.target id="marg519"/>C<emph type="italics"/>or<emph.end type="italics"/>_{m}. 3.</s></p><p type="main">
<s id="id2755517">Cum autem uolueris duo quadrata qu differant in 100, tunc <lb/>per regulam datam i auferes 1, peruenires ad numeros magnos & <lb/>fractos, & ideo melius et quia numerus et par, ut detrahas nume<lb/>rum parem quadratum, ita quod reiduum posit diuidi per <expan abbr="duplũ">duplum</expan> <lb/>radicis, ut in hoc non detraho neque quia remanet impar, nec 16 quia <lb/>84 <expan abbr="re&longs;iduũ">reiduum</expan> non <expan abbr="põt">pont</expan> diuidi per 8 ita ut exeat integer numerus, ergo <lb/><expan abbr="detrahã">detraham</expan> 4 & <expan abbr="relinque&ttilde;">relinquetur</expan> 96, diuido per <expan abbr="duplũ">duplum</expan> radicis quod et 4 exit <lb/>24, cuius quadratum d et 576 addito 100 facit 676 <expan abbr="quadratũ">quadratum</expan> 26. <lb/>Et ita ex 433 non auferam ed 9, quia relinquetur 24 qui potet diui<lb/>di per e, duplum <02> 9 & exit 4 cuius <expan abbr="quadratũ">quadratum</expan> et 16, addito 33 fit 49.</s></p><p type="main">
<s id="id2755670">Secunda regula, cum uolueris propoito uno numero quadra<lb/>to illum diuidere infinitis modis in duos numeros quadratos, cape <lb/>quemuis numerum quadratum per primum exemplum regul pri <lb/>m, & cum eo diuide numerum propoitum, & qui proueniet erit <lb/>quadratus, <expan abbr="hũc">hunc</expan> ergo duces in partes numeri quadrati qu unt nu<lb/>meri <expan abbr="&qtilde;drati">quadrati</expan>, & fient duo quadrati numeri, & illi <expan abbr="compon&etilde;t">component</expan> <expan abbr="numerũ">numerum</expan> <lb/><expan abbr="quadratũ">quadratum</expan> <expan abbr="prior&etilde;">priorem</expan> quem diuiiti. </s>
<s id="id2755773">quia multipli catio fit per <expan abbr="eo&longs;d&etilde;">eodem</expan> nu<lb/>meros qui unt partes diuioris. </s>
<s id="id2755802">Velut uolo facere de 4 duas partes <lb/>qu int <expan abbr="&qtilde;drati">quadrati</expan> numeri, capio <expan abbr="numerũ">numerum</expan> <expan abbr="&qtilde;dratũ">quadratum</expan> qui <expan abbr="cõpona&ttilde;">componatur</expan> ex duo<lb/>bus <expan abbr="&qtilde;dratis">quadratis</expan>, uelut 25, diuido 4 per 25 exit 4/25 <expan abbr="hũc">hunc</expan> duco per 9 & 16 <expan abbr="&qtilde;dra-tos">quadra<lb/>tos</expan> numeros <expan abbr="cõponentes">componentes</expan> 25 <expan abbr="fiũt">fiunt</expan> 1 11/25 & 2 14/25 <expan abbr="&qtilde;drati">quadrati</expan> 1 2/5 & 1 3/5 Et hi <expan abbr="&qtilde;drati">quadrati</expan> <lb/><expan abbr="cõponunt">componunt</expan> 4. Et ita poes diuidere infinitis modis, puta per 17 13/36 & <lb/>per 169. Tertia regula cum unus numerus additus <lb/><figure id="id.015.01.167.1.jpg" xlink:href="015/01/167/1.jpg"/><lb/>primo & detractis <expan abbr="&longs;ecũdo">ecundo</expan> facit ambo quadrata, <expan abbr="id&etilde;">idem</expan> <lb/>numerus coniunctus cum differentia illorum nume<lb/>rorum & detractus primo & additus ecundo facit <lb/>eodem numeros quadratos, ueluti capio 10 primum <lb/>3 ecundum 6 additus ad 10 & detractus 7 efficit 6 <lb/>& 1 quadratos dico quod iunctus 16 cum 3 differen<lb/>tia 10 & 7 fit 9, qui detractus 10 & additus ad 7 effi<lb/>cit 1 & 16 numeros quadratos priores.</s></p>
<pb xlink:href="015/01/168.jpg" pagenum="149"/><p type="head">
<s id="id2756045">SCHOLIVM</s></p><p type="main">
<s id="id2756053">Sunt & alij modi plures faciendi huiumodi, ed <expan abbr="nõ">non</expan> unt ad e ge <lb/>nerales, & nihilo minus unt magis confui, & non aliquid plus.</s></p><p type="main">
<s id="id2756093">Quarta regula, <expan abbr="cũ">cum</expan> uolueris <expan abbr="numerũ">numerum</expan> aliquem non quad. </s>
<s id="id2756114">qui bifa <lb/><expan abbr="riã">riam</expan> <expan abbr="compona&ttilde;">componatur</expan> ex duob. </s>
<s id="id2756137"><expan abbr="&qtilde;d">quad</expan>. </s>
<s id="id2756148">uelut 10 ex 25, & 25 & 49 & 1, <lb/><figure id="id.015.01.168.1.jpg" xlink:href="015/01/168/1.jpg"/><lb/>& <expan abbr="&longs;uma&ttilde;">umatur</expan> a b numerus quad. </s>
<s id="id2756176">diuius in <expan abbr="&longs;upplem&etilde;ta">upplementa</expan>, ita quae c <lb/>d it portio minor eiumodi, ut adiecta illi <expan abbr="æ&qtilde;li">quali</expan> c d gnomo <lb/>cir <expan abbr="cũ&longs;criptus">cuncriptus</expan> c k l <expan abbr="cũ">cum</expan> <expan abbr="f&qtilde;drato">fquadrato</expan>, it <expan abbr="&ecedil;&qtilde;lis">qualis</expan> a b <expan abbr="&qtilde;drato">quadrato</expan>, detractis <lb/><expan abbr="igi&ttilde;">igitur</expan> c e & e d, <expan abbr="æ&qtilde;libus">qualibus</expan> erunt duo <expan abbr="&longs;upplem&etilde;ta">upplementa</expan> c k l <expan abbr="cũf">cunf</expan> qua<lb/>drato qualia duob. </s>
<s id="id2756322"><expan abbr="&longs;upplem&etilde;tis">upplementis</expan> a b <expan abbr="cũ">cum</expan> <expan abbr="&qtilde;drato">quadrato</expan> h g. </s>
<s id="id2756356">Maio<lb/>ra <expan abbr="aũt">aunt</expan> <expan abbr="&longs;upplem&etilde;ta">upplementa</expan> <expan abbr="excedũt">excedunt</expan> minora in duplo quad. </s>
<s id="id2756395">c d <expan abbr="igi&ttilde;">igitur</expan> detractis <lb/>minoribus upplementis <expan abbr="cõmunibus">communibus</expan>, erit <expan abbr="duplũ">duplum</expan> quad. </s>
<s id="id2756432">c d <expan abbr="cũ">cum</expan> f qua<lb/>drato qualia h g <expan abbr="&qtilde;drato">quadrato</expan>. </s>
<s id="id2756459">Ergo propoito numero, put 3 ducam in e <lb/>fit 9, <expan abbr="ducã">ducam</expan> 2 <expan abbr="minor&etilde;">minorem</expan> in e fit 4, duplicabo fit 8, detraho ex 9, <expan abbr="relinqui&ttilde;">relinquitur</expan> <lb/>1 numerus <expan abbr="&qtilde;dratus">quadratus</expan>, <expan abbr="igi&ttilde;">igitur</expan> <expan abbr="dicã">dicam</expan> d 3 <expan abbr="cũ">cum</expan> duplo 2, & erit <expan abbr="totũ">totum</expan> 7, et unus <lb/>numerus, alter <02> 1. 1. 1, & <expan abbr="horũ">horum</expan> <expan abbr="&qtilde;d">quad</expan>. </s>
<s id="id2756573"><expan abbr="cõponunt">componunt</expan> 50, <expan abbr="duplũ">duplum</expan> <expan abbr="&qtilde;d">quad</expan>. </s>
<s id="id2756602">5. Et imi <lb/>liter capio 6 <expan abbr="&qtilde;d">quad</expan>. </s>
<s id="id2756620">36 <expan abbr="duplũ">duplum</expan> <expan abbr="&qtilde;d">quad</expan>. </s>
<s id="id2756641">4. 32 differentia 4, numerus <expan abbr="&qtilde;d">quad</expan>. </s>
<s id="id2756654">2, ideo <lb/>6 <expan abbr="cũ">cum</expan> duplo 4, & et 14, et unus numerus, alter 2, <expan abbr="quorũ">quorum</expan> <expan abbr="&qtilde;d">quad</expan>. </s>
<s id="id2756692">unt 200, <lb/><expan abbr="dimidiũ">dimidium</expan> et 100 <expan abbr="&qtilde;d">quad</expan>. </s>
<s id="id2756718">10 <expan abbr="cõpo&longs;iti">compoiti</expan> ex 6 & 4. Et ita capio 9, <expan abbr="&qtilde;d">quad</expan>. </s>
<s id="id2756747">eius 81 du <lb/><expan abbr="plũ">plum</expan> <expan abbr="&qtilde;d">quad</expan>. </s>
<s id="id2756769">6. 72 differentia 9 numerus <expan abbr="&qtilde;d">quad</expan>. </s>
<s id="id2756782"><expan abbr="igi&ttilde;">igitur</expan> cum duplo 6, & et 21, et <lb/>unus <expan abbr="illorũ">illorum</expan>, alter 3 <expan abbr="&qtilde;d">quad</expan>. </s>
<s id="id2756818">450, <expan abbr="duplũ">duplum</expan> 225 <expan abbr="&qtilde;d">quad</expan>. </s>
<s id="id2756839">15, qui contat ex 9 & 6. Et <lb/>ita capio 11 <expan abbr="&qtilde;d">quad</expan>. </s>
<s id="id2756858">cuius et 121, <expan abbr="duplũ">duplum</expan> <expan abbr="&qtilde;d">quad</expan>. </s>
<s id="id2756881">6 et 72 differentia, 72 & 21 et <lb/>49 numerus <expan abbr="&qtilde;d">quad</expan>. </s>
<s id="id2756903">7, <expan abbr="igi&ttilde;">igitur</expan> 23 qui contat ex 11, & duplo 6 numeri mino<lb/>ris et unus numerus, alter et 7 <expan abbr="&qtilde;d">quad</expan>. </s>
<s id="id2756936"><expan abbr="quorũ">quorum</expan> unt 578. <expan abbr="duplũ">duplum</expan> 289, <expan abbr="&qtilde;d">quad</expan>. <lb/></s>
<s id="id2756968">17, qui contat ex 11 & 6. Quinta regula, per hoc inueniemus infini <lb/>tos numeros <expan abbr="&qtilde;d">quad</expan>. </s>
<s id="id2756988"><expan abbr="cõponentes">componentes</expan> 32, nam <expan abbr="cũ">cum</expan> 32 it duplus <expan abbr="&qtilde;d">quad</expan>. </s>
<s id="id2757020"><expan abbr="diuidã">diuidam</expan> per<lb/>unum <expan abbr="aggregatũ">aggregatum</expan> ex inuentis puta 578, & quia ambo ex uppoito <lb/>unt dupli ad <expan abbr="&qtilde;d">quad</expan>. </s>
<s id="id2757061">qui proueniet erit <expan abbr="&qtilde;d">quad</expan>. </s>
<s id="id2757074">cilicet 16/289, duc in numeros <expan abbr="&qtilde;-dratos">qua<lb/>dratos</expan> qui componunt 578, & unt 529 & 49, & fient 2 206/289 & 29 83/289, <lb/>& hi iuncti <expan abbr="fiũt">fiunt</expan> 32, quia unt multiplicat partes numeri, per quem <lb/>et diuius numerus. </s>
<s id="id2757124">Et ita poteris diuidere 32 in infinitos alios <expan abbr="&qtilde;d">quad</expan>.</s></p><p type="main">
<s id="id2757142">Sexta regula, ponamus mod d uelim diuidere 10, <expan abbr="cõpo&longs;itũ">compoitum</expan> ex <lb/>duob. </s>
<s id="id2757171"><expan abbr="&qtilde;d">quad</expan>. </s>
<s id="id2757182">9 & 1, & non <expan abbr="duplũ">duplum</expan> numero <expan abbr="&qtilde;d">quad</expan>. </s>
<s id="id2757203">ita d it diuius in alios <lb/>duos: <expan abbr="ducã">ducam</expan> 10 in 25 <expan abbr="cõpo&longs;itũ">compoitum</expan> ex duob. </s>
<s id="id2757243"><expan abbr="&qtilde;d">quad</expan>. </s>
<s id="id2757254">fit 250/25, at 250 <expan abbr="cõponi&ttilde;">componitur</expan> aliter <lb/>ex duob. </s>
<s id="id2757272">quad. </s>
<s id="id2757277"><08> 225/25 & 25/25, cilicet 169/25 & 81/25, id et 6 19/25 & 3 6/25, qui unt <expan abbr="&qtilde;d">quad</expan>. <lb/></s>
<s id="id2757301">2 3/5 & 1 4/5, & ita uolo diuidere 13 in duo alia <expan abbr="&qtilde;drata">quadrata</expan> <08> 9 & 4, duco 13 in <lb/>25 & fit 325/25, qui neceario <expan abbr="cõponi&ttilde;">componitur</expan> ex 225/25 & 100/25, ed ego uolo d <expan abbr="cõpo">compo</expan> <lb/><expan abbr="na&ttilde;">natur</expan> aliter, uelut ex 289/25 & 63/25, & ita ex 11 14/25 & 1 11/25, qui unt numeri <expan abbr="&qtilde;d">quad</expan>. </s>
<s id="id2757375">com <lb/>ponentes 13, & <02> unt 3 2/5 & 1 1/5, & in his opus et in dutria, cilicet ut <lb/><expan abbr="multiplice&ttilde;">multiplicetur</expan> per numeros <expan abbr="&qtilde;d">quad</expan>. </s>
<s id="id2757413">ut proueniant numeri illi <expan abbr="bifariã">bifariam</expan> compo <lb/>iti ex <expan abbr="&qtilde;dratis">quadratis</expan>. </s>
<s id="id2757438">Vt uer uideamus <expan abbr="re&longs;iduũ">reiduum</expan>, proponamus quae uelim diui <lb/>dere 6 in duos numeros <expan abbr="&qtilde;d">quad</expan>, <expan abbr="primũ">primum</expan> cire debes d non pount ee
<pb xlink:href="015/01/169.jpg" pagenum="150"/>integri exratione dicta, quia oporteret ut eent ambo impares aut <lb/>pares, & ic <expan abbr="differr&etilde;t">differrent</expan> numero pari, ergo oporteret ut eet unus me<lb/>dius numerus <expan abbr="&qtilde;d">quad</expan>. </s>
<s id="id2757539">unt & ali rationes, ed neque unus poet ee inte <lb/>ger, & alius fractus, <expan abbr="nõ">non</expan> eet. </s>
<s id="id2757574">n. </s>
<s id="id2757578">6 numerus integer: <expan abbr="relinqui&ttilde;">relinquitur</expan> ergo ut <lb/>int duo fracti: ed in numeris fractis <expan abbr="&qtilde;d">quad</expan>. </s>
<s id="id2757606">deductis ad minimas deno <lb/>minationes <expan abbr="operũ">operum</expan>, ut tam denominator <08> numerator habeat radi<lb/>ces, ergo oportet d hoc it in illis, & quia iuncti debent facere inte<lb/>gros 6, necee et ut denominator it unus, & <expan abbr="id&etilde;">idem</expan> in utroque, et d nu<lb/>meratores imul iuncti int <expan abbr="&longs;excuplũ">excuplum</expan> denominatoris, i fracti <expan abbr="deb&etilde;t">debent</expan> <lb/>quipollere 6, ergo ille denominator <expan abbr="cũ">cum</expan> it <expan abbr="&qtilde;d">quad</expan>. </s>
<s id="id2757718">& numeratores am<lb/>bo int <expan abbr="&qtilde;d">quad</expan>. </s>
<s id="id2757738">& int <expan abbr="&longs;excuplũ">excuplum</expan> denominatoris, oportebit inuenire <expan abbr="nu-merũ">nu<lb/>merum</expan> <expan abbr="&qtilde;d">quad</expan>. </s>
<s id="id2757777">qui ductus in 6, faciat <expan abbr="numerũ">numerum</expan> qui <expan abbr="cõponi&ttilde;">componitur</expan> ex duob. </s>
<s id="id2757801"><expan abbr="&qtilde;d">quad</expan>. <lb/></s>
<s id="id2757813">aut <expan abbr="cõponi&ttilde;">componitur</expan> qualiter, ergo proportio medietatis ad <expan abbr="medietat&etilde;">medietatem</expan> 6, et <lb/>ueluti totius ad 6, ed totu continet 6 in <expan abbr="&qtilde;d">quad</expan>. </s>
<s id="id2757857">quia ex 6 in <expan abbr="&qtilde;d">quad</expan>. </s>
<s id="id2757870">fit <expan abbr="totũ">totum</expan>, <lb/>ergo ex medietate in <expan abbr="&qtilde;d">quad</expan>. </s>
<s id="id2757893">idem fit medietas, ed medietas et nume<lb/>rus <expan abbr="&qtilde;d">quad</expan>. </s>
<s id="id2757916">ergo 3 eet numerus <expan abbr="&qtilde;d">quad</expan>. </s>
<s id="id2757932">d et falum, oportet <expan abbr="igi&ttilde;">igitur</expan> ut nume <lb/>ri illi int in quales, & ut 6 diuidatur in duas partes inquales, hoc <lb/><expan abbr="aũt">aunt</expan> fit diuidendo quemlibet <expan abbr="numerũ">numerum</expan> parem, qui <expan abbr="cõponi&ttilde;">componitur</expan> ex duob. <lb/></s>
<s id="id2757997">numeris <expan abbr="&qtilde;d">quad</expan>. </s>
<s id="id2758010">nam i eet impar, <expan abbr="nõ">non</expan> poet prodire numerus integer, & <lb/><expan abbr="cũ">cum</expan> prouenerit numerus <expan abbr="&qtilde;d">quad</expan>. </s>
<s id="id2758052">ille erit <expan abbr="qu&etilde;">quem</expan> qurimus, <expan abbr="nã">nam</expan> diuio 6 per to<lb/>tum <expan abbr="illũ">illum</expan> numerum, inde d prouenit multiplicato per numeros <expan abbr="&qtilde;d">quad</expan>, <lb/><expan abbr="cõponentes">componentes</expan> illum <expan abbr="numerũ">numerum</expan> productum, <expan abbr="producun&ttilde;">producuntur</expan> partes 6, qu <expan abbr="erũt">erunt</expan> <lb/>numeri <expan abbr="&qtilde;d">quad</expan>. </s>
<s id="id2758154">quia denominator utriuque partis ex uppoito et nume <lb/>rus <expan abbr="&qtilde;dratus">quadratus</expan>, qui multipli catus et per 6, & numeratores unt nume <lb/>ri <expan abbr="&qtilde;drati">quadrati</expan>, qui <expan abbr="cõponebant">componebant</expan> <expan abbr="numerũ">numerum</expan> <expan abbr="productũ">productum</expan>, et tales partes <expan abbr="&ecedil;quan&ttilde;">quantur</expan> <lb/>6, quia numerus productus <expan abbr="componi&ttilde;">componitur</expan> ex numeratoribus, & <expan abbr="produ-ci&ttilde;">produ<lb/>citur</expan> tale <expan abbr="cõpo&longs;itum">compoitum</expan> ex 6 in <expan abbr="denominator&etilde;">denominatorem</expan>, & hic et diuius per deno <lb/><expan abbr="minator&etilde;">minatorem</expan>, ergo prouenit 6, i <expan abbr="e&mtilde;">emm</expan> multiplicato 3 in 4 fit 12, diuio 12 per <lb/>4, exit neceario idem 3. Pro colligendo ergo numeros omnes, qui <lb/><expan abbr="cõponuntur">componuntur</expan> ex <expan abbr="&qtilde;dratis">quadratis</expan>, propones tibi eriem <expan abbr="&qtilde;d">quad</expan>. </s>
<s id="id2758352"><expan abbr="omniũ">omnium</expan>, & inde iun<lb/>ges, & diuides per 6, & <expan abbr="cũ">cum</expan> prodierit <expan abbr="&qtilde;dratus">quadratus</expan>, <expan abbr="inueni&ttilde;">inuenitur</expan> denominator, <lb/>& numeri <expan abbr="cõponentes">componentes</expan> ipum erunt numeratores, et uppoiti deno <lb/>minatoribus <expan abbr="cõ&longs;tituent">contituent</expan> partes. </s>
<s id="id2758432">Vt uer cognocas, ex quibus po<lb/>it componi primum ex imparibus, non oportet aumere nii 135, <lb/>quia 7 diuium per 6 relin quit 1, & 9 diuium per 6, relinquit 3, & 35 <lb/>diuium per 6 relinquit 5. ergo non potet componi numerus im<lb/>par, qui diuidatur per 6, ut up erit impar alius qum 1. 3. 5. ed 1 & 3 <lb/>& 5, & 5 componunt 4 & 1, & 1 & 3 & 5 componunt 2, cilicet abie<lb/>cto 6, ergo tales numeri <expan abbr="&qtilde;drati">quadrati</expan> i int impares, uel ambo terminan<lb/>tur in 3, ut 9 & 81, qui faciunt 90, uel in 1 & 5, ed nullus numerus <lb/>quadratus diuius per 6 terminatur in 5, quia 1 ductum in e produ<lb/>cit 1, & 3 pro ducit 3, & 5 pro ducit 1, ut 5 in 5 facit 25, & 11 in 11 produ
<pb xlink:href="015/01/170.jpg" pagenum="151"/>cit 121, quibus diuiis per 6 uperet 1. Quod etiam ic demontratur <lb/>de 5, & compoitis 5, nam diuio 5 in 3 & 2, quadratum eius <expan abbr="cõpo-nitur">compo<lb/>nitur</expan> ex duplo 3 in 2, in quo nihil uperet, i diuidatur per 6, & ex <lb/>quadrato 3, qud et 9, in quo uperet 3, & ex quadrato 2 quod et </s></p><p type="main">
<s id="id2758626"><arrow.to.target n="marg520"/><lb/>4, ed iunctis 4 & 3, & abiecto 6 uperet 1, ergo 5 in 5 <expan abbr="ductũ">ductum</expan>, & diui <lb/>o producto relin quitur 1. Et imiliter capio 17, et <expan abbr="componi&ttilde;">componitur</expan> ex 12 & <lb/>5 quadratum, ergo 17 componitur ex quadrato 12, in quo nihil u<lb/>peret, & duplo 5 in 12, in quo <expan abbr="etiã">etiam</expan> nihil uperet, i diuidatur per 6: <lb/>& ex quadrato 5, in quo uperet 1, ergo in nullo numero <expan abbr="cõpo&longs;ito">compoito</expan> <lb/>ex 5 & 6, uel compoitis ex 6, poterit produci numerus, qui diuius <lb/>per 6 relin quat 5, igitur neque talis numerus potrit <expan abbr="cõponi">componi</expan> ex duo<lb/>bus quadratis, in quib. </s>
<s id="id2758756">uperit 5 & 1, quia nullus et, in quo uper<lb/>it 5 facta diuiione per 6. Ex quo colligitur una regula: quod i quis <lb/>dicat multiplicaui 27 in e, et diuii per 13, uellem cire quid uperet, <lb/>dico quod ine multiplicatione et diuiione poteris hoc cire ex de<lb/>montratione dicta, diuide ergo 27 per 13, & relin quitur 1, duc in e <lb/>fit 1: dices ergo, quod upererit 1, & ita i ducerem 28 in e, & diuide<lb/>rem per 11, dico quod upererit 3, nam diuio 28 per 11, relin quitur <lb/>6, duc in 6 fit 36, diuide per 11, relin quitur 3, ut dictum et, & tantum <lb/><expan abbr="relinqui&ttilde;">relinquitur</expan> ducto 28 in e & fit 784, & diuio per 11. Reuertendo ergo <lb/>ad propoitum, pater quod ex duobus tantum numeris imparibus <lb/>quadratis potet conflari ille numerus, <expan abbr="quorũ">quorum</expan> radices diui per 6 <lb/>relin quunt 3. Sed de paribus uel uperet 2 uel 4 uel nihil, ed <expan abbr="&qtilde;dra-tum">quadra<lb/>tum</expan> 2 et 4, & <expan abbr="&qtilde;dratum">quadratum</expan> 4 diuium per 6 etiam relinquit 4, ergo neque<lb/>ex duobus numeris, in quibus uperint 2, neque in quibus uperint <lb/>4, neque in quibus uperint in uno 2, in altero 4 <expan abbr="poterũt">poterunt</expan> quadrata, in <lb/>quibus emper upererit 4, & iuncta faciunt 8, in uperet 2, <expan abbr="cõ">com</expan> fla<lb/>re <expan abbr="numerũ">numerum</expan> <expan abbr="dictũ">dictum</expan> eu <expan abbr="quæ&longs;itũ">quitum</expan>, qui posit diuidi per 6: neque ex <expan abbr="&qtilde;d">quad</expan>. </s>
<s id="id2759042"><expan abbr="duo-rũ">duo<lb/>rum</expan> <expan abbr="num&etilde;rorũ">numerrorum</expan>, in <expan abbr="quorũ">quorum</expan> altero nihil uperit in reliquo uperit 2 uel <lb/>4, quia in aggregato <expan abbr="&qtilde;dratorũ">quadratorum</expan> emper upererit 4. Ergo relinqui<lb/>tur quod ille numerus componetur ex duobus quadratis, uel impa <lb/>ribus, quorum latera diuia per 6 relinquunt 3, uel ex duobus pari<lb/>bus, quorum latera diuia per 6 nihil relinquant. </s>
<s id="id2759126">Oportet igitur <lb/>inuenire duos tales numeros quadratos numerorum imparium, in <lb/>quibus uperit 3, i diuidantur per 6, aut parium in quibus nihil u<lb/>perit, quorum aggregato diuio per 6 prodeat numerus <expan abbr="&qtilde;dratus'">quadratus'</expan>.</s></p><p type="margin">
<s id="id2759172"><margin.target id="marg520"/>P<emph type="italics"/>er<emph.end type="italics"/> 4. <emph type="italics"/>ecun <lb/>di<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="main">
<s id="id2759223">His uiis dico, quod contat radices talium numerorum opor<lb/>tere ee in imparibus per additionem 6 incipiendo 3, ut int <lb/>3. 9. 15. 21. 27. 33. 39. 45. 51. & ic deinceps: in paribus au<lb/>tem per additionem eiudem 6 incipiendo 6, uelut 6. 12. <lb/>18. 24. 30. 36. 42. 48. 54. 60. Dico ergo quod diui<lb/>o numero illo compoito per 6 in imparibus exibit numerus,
<pb xlink:href="015/01/171.jpg" pagenum="152"/>qui diuius per 6 upererit 3, & in paribus qui poterit diuidi per 6. <lb/>Quia <expan abbr="componun&ttilde;">componuntur</expan> ex huiumodi: uelut 3 in e facit 9, & 25 in e facit <lb/>225, qui <expan abbr="iũcti">iuncti</expan> <expan abbr="faciũt">faciunt</expan> 234, diuio 235 per 6 exit 39, qui <expan abbr="iterũ">iterum</expan> diuius per 6 <lb/>uperet 3, & imiliter capio 6 & 12, <expan abbr="quorũ">quorum</expan> <expan abbr="&qtilde;drata">quadrata</expan> unt 36 & 144, & <lb/><expan abbr="aggregatũ">aggregatum</expan> 180, qui diuius per 6 exit 30, qui <expan abbr="iterũ">iterum</expan> potet diuidi per <lb/>6. Et hoc quia <expan abbr="quilibetillorũ">quilibetillorum</expan> potet diuidi per <expan abbr="&qtilde;dratũ">quadratum</expan> 6 in paribus, <lb/>ergo aggregato diuio per 6 d prodit, <expan abbr="iterũ">iterum</expan> poterit diuidi per 6. <lb/>Et in imparibus quo dlibet <expan abbr="&qtilde;dratorũ">quadratorum</expan> exuperat upra enarios in 3, <lb/><expan abbr="igi&ttilde;">igitur</expan> <expan abbr="aggregatũ">aggregatum</expan> diuium in 2 pariet <expan abbr="numerũ">numerum</expan> qui diuius per 3, exibit <lb/>numerus impar <expan abbr="cõpo&longs;itus">compoitus</expan> ex enarijs & 3. Illud ergo <expan abbr="quadratũ">quadratum</expan>, d <lb/>prodibit, uel erit <expan abbr="cõpo&longs;itum">compoitum</expan> ex enarijs, uel upererit 3. Sed <expan abbr="cũ">cum</expan> 3 nume <lb/>ret 6, ergo tres <expan abbr="&qtilde;drati">quadrati</expan> numeri cilicet duo, qui <expan abbr="cõponunt">componunt</expan> <expan abbr="numerũ">numerum</expan>, <lb/><arrow.to.target n="marg521"/><lb/>& qui prodit per <expan abbr="diui&longs;ion&etilde;">diuiionem</expan> 6, erunt <expan abbr="cõpo&longs;iti">compoiti</expan> inter e, ergo & radices il <lb/>lorum. </s>
<s id="id2759636"><expan abbr="Igi&ttilde;">Igitur</expan> radix numeri <expan abbr="&qtilde;drati">quadrati</expan>, qui prouenit diuio aggregato <expan abbr="qua-dratorũ">qua<lb/>dratorum</expan> per 6 et ex <expan abbr="eod&etilde;">eodem</expan> ordine <expan abbr="impariũ">imparium</expan>, i impares numeri <expan abbr="&qtilde;drati">quadrati</expan> <lb/><expan abbr="fuerũt">fuerunt</expan>, aut <expan abbr="pariũ">parium</expan> i pares. </s>
<s id="id2759724">At hoc ee <expan abbr="nõ">non</expan> potet, <expan abbr="nã">nam</expan> fracti illi numeri, <lb/>qui <expan abbr="erũt">erunt</expan> radices, <expan abbr="nõ">non</expan> <expan abbr="erũt">erunt</expan> minimi, ed diuii per 3 otendent minores, <lb/>quod et contra uppoitum, quare nullo modo 6 potet diuidi in <lb/>duos numeros quadratos, neque integros, neque fractos, quod erat <lb/>demontrandum. </s>
<s id="id2759814">Habes igitur ex hoc demontrationem quando <lb/><expan abbr="nõ">non</expan> posit diuidi, & quado posit, quod posit, & quomodo imul.</s></p><p type="margin">
<s id="id2759848"><margin.target id="marg521"/>P<emph type="italics"/>er<emph.end type="italics"/> 29. <emph type="italics"/>e<lb/>ptimi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="main">
<s id="id2759901">Propoitio centeimaquinquageimaexta.</s></p><p type="main">
<s id="id2759920">Horologiorum tempus multiplicare.<lb/><arrow.to.target n="marg522"/></s></p><p type="margin">
<s id="id2759935"><margin.target id="marg522"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2759961">Contingit quando que d <expan abbr="horologiorũ">horologiorum</expan> tem <lb/><figure id="id.015.01.171.1.jpg" xlink:href="015/01/171/1.jpg"/><lb/>pus breue et, uolumus <expan abbr="aũt">aunt</expan> maius efficere: id <lb/>duob. </s>
<s id="id2760004">modis poumus, <expan abbr="quorũ">quorum</expan> unus diffici<lb/>lior et ed perpetuus, & long nobilior, nam <lb/>grauitas ponderis ueratilis efficit <expan abbr="quid&etilde;">quidem</expan> <expan abbr="tar-dior&etilde;">tar<lb/>diorem</expan>, ed di fficilius <expan abbr="mobil&etilde;">mobilem</expan>, & ob id grauio<lb/>re <expan abbr="põdere">pondere</expan> in <expan abbr="digent&etilde;">digentem</expan>. </s>
<s id="id2760095">Sit ergo rota a b uerati<lb/>lis, qu certam menuram exigit pro quacunque funis parte correperon <lb/>dentis uni denti ex centum, in quos ditincta it, curriculum <expan abbr="aũt">aunt</expan> c d <lb/>quinque <expan abbr="dentiũ">dentium</expan>, per drota exaginta dentes <expan abbr="hab&etilde;s">habens</expan> <expan abbr="circumuolua&ttilde;">circumuoluatur</expan> in <lb/><expan abbr="cõuer&longs;ione">conuerione</expan>, <expan abbr="igi&ttilde;">igitur</expan> prim rot uities <expan abbr="circumfere&ttilde;">circumferetur</expan>, <expan abbr="&longs;ecũda">ecunda</expan> <expan abbr="d&etilde;tesque">dentesque</expan> M. CC. <lb/>rurus ad <expan abbr="hãc">hanc</expan> <expan abbr="&longs;ecundã">ecundam</expan> tertia <expan abbr="necta&ttilde;">nectatur</expan> cum curriculo ex <expan abbr="dentiũ">dentium</expan>, atque in <lb/>ea <expan abbr="d&etilde;tes">dentes</expan> eptuaginta duo, ut in una <expan abbr="cõuer&longs;ione">conuerione</expan> int xiiij cccc, dentes <lb/><expan abbr="igi&ttilde;">igitur</expan> tot dentes in una <expan abbr="cõuer&longs;ione">conuerione</expan> prim rot circumuoluentur. </s>
<s id="id2760339">Iam <lb/>uer tempus illud poterit duplicari ac triplicari iuxta <expan abbr="tarditat&etilde;">tarditatem</expan> tem <lb/>poris ueratilis: <expan abbr="quãto">quanto</expan> <expan abbr="igi&ttilde;">igitur</expan> ponderoius fuerit illud <expan abbr="t&etilde;pus">tempus</expan>, tanto tar<lb/>dius <expan abbr="mouebi&ttilde;">mouebitur</expan>, pauciores que circumuolutiones neceari <expan abbr="erũt">erunt</expan> ad <expan abbr="ex-pl&etilde;dam">ex<lb/>plendam</expan> unam <expan abbr="di&etilde;">diem</expan>: id et horas 24, ed hoc in <expan abbr="cõmodi">commodi</expan> accedet, qud <lb/>reuolutio indicis tanto tardior erit, ut <expan abbr="nõ">non</expan> iut oten dat horas: pro
<pb xlink:href="015/01/172.jpg" pagenum="153"/>poitum igitur et, ut pondera tardius ferantur, index <expan abbr="aũt">aunt</expan>, & qu ad <lb/>indicem equuntur horarum demontrationes celerius aut eodem <lb/>modo ferantur. </s>
<s id="id2760526">Ponamus ergo pot<08> eadem et ratio celerioris & <lb/>qu uelocis, ponderis <expan abbr="aũt">aunt</expan> tardius decendentis, aut <expan abbr="cõtrà">contr</expan> tardio<lb/>ris, aut qualiter cir cumducti in dicis, celerioris <expan abbr="aũt">aunt</expan> decenus pon<lb/>deris, quod ad nullam <expan abbr="utilitat&etilde;">utilitatem</expan> profuturum uideo. </s>
<s id="id2760604">Sit ergo ut pon <lb/>dus uelim tardius decendere, rotam <expan abbr="aũt">aunt</expan> qualiter circumferri, dico <lb/>quod ex tempore mobili eu ueratili (& et ferrum, quod in um<lb/>mo horologij citra ultraque <expan abbr="fer&ttilde;">fertur</expan> tam in horologijs ponderum <08> mo <lb/>l) id fieri non potet: nam quantum tardabitur rota tertia ecunda <lb/>& prima, atque ob id decenus ponderum, tantum remorabitur rota <lb/>prima qu indicem otendit, ergo tantum index tardabitur quan<lb/>rum <expan abbr="põdera">pondera</expan>, & ut uno uerbo dicam, cm <expan abbr="ead&etilde;">eadem</expan> rota index circumfe<lb/>ratur, & <expan abbr="põdus">pondus</expan> decendat, <expan abbr="quantũ">quantum</expan> unum tardatur tantum & aliud.</s></p><p type="main">
<s id="id2760742">Secundus modus et, ut rota una totum tempus cum indice in ui <lb/>gintiquatuor horis circumuoluatur, & currulis in quo funis minor <lb/>fiat: necee et <expan abbr="igi&ttilde;">igitur</expan>, ut circumuoluta rota aut emel aut bis, <expan abbr="&ttilde;er">turer</expan>, qua<lb/>ter decies, & <expan abbr="circumuolua&ttilde;">circumuoluatur</expan> pleno cir cuitu index, et ine errore: quo<lb/>niam tempus & dentes menur repondent: igitur ub eidem cir<lb/>cuitibus numero eodemque tempore minus ex fune <expan abbr="de&longs;cend&etilde;t">decendent</expan> in cur <lb/>ruli paruo <08> magno: quare mutatione indiget currulis, aut ut funis <lb/>circumuoluens rotam curriculum habeat <expan abbr="annexũ">annexum</expan> rot oten denti <lb/>horas, in qua pauciores int dentes: nam in eodem tempore, & cir<lb/>cuitu paucioribus uicibus circumuoluitur rota funis qu grauita<lb/>te temporis, & multitudine <expan abbr="dentiũ">dentium</expan> certam <lb/><figure id="id.015.01.172.1.jpg" xlink:href="015/01/172/1.jpg"/><lb/>eruabit <expan abbr="men&longs;urã">menuram</expan>. </s>
<s id="id2760916">Sed in hoc necee et gra<lb/>uius efficere pondus, aut leuius <expan abbr="t&etilde;pus">tempus</expan> <expan abbr="quo-niã">quo<lb/>niam</expan> funis debilius circumuertit <expan abbr="rotã">rotam</expan>: minus <lb/><expan abbr="tñ">tnm</expan> tard <08> it pro paruitatis circuitus ratione.</s></p><p type="main">
<s id="id2760982">Tertius modus facilior et, & magis com <lb/><expan abbr="p&etilde;dio&longs;us">pendious</expan>: Sit horologium a b c, in quo rota <lb/>d qu funem <expan abbr="cõtinet">continet</expan> bais horologij e f, cui <lb/>firmiter int <expan abbr="app&etilde;&longs;&ecedil;">appen</expan> du trochle g & h, & fu <lb/>nis una parte tro chle appenus in k, <expan abbr="duca&ttilde;">ducatur</expan> <lb/>ad inferiorem aliam tro chleam lineraturque<lb/>ibi orbiculo uo, & redeat dextra uperius <lb/><expan abbr="in&longs;era&ttilde;que">ineraturque</expan> orbiculo uperioris tro chle, dedu <lb/><expan abbr="ca&ttilde;que">caturque</expan> uerus <expan abbr="&longs;ini&longs;trã">initram</expan>: atque ibi <expan abbr="de&longs;cend&etilde;s">decendens</expan> habe <lb/>at <expan abbr="põdus">pondus</expan> tractorium in m, <expan abbr="deduca&ttilde;que">deducaturque</expan> upra <lb/>ad <expan abbr="rotã">rotam</expan> horologij d, et cir cumuolutus exeat <lb/>ipum, & <expan abbr="de&longs;c&etilde;dat">decendat</expan> ad tro <expan abbr="chleãn">chleann</expan>, ub que ea circumuolutus <expan abbr="iterũ">iterum</expan> acen
<pb xlink:href="015/01/173.jpg" pagenum="154"/>dat dextra parte, et circumuoluatur h co chle rediens ad initram <lb/>ibique decendens connectatur tro chle in inferiori in o, cuius im <lb/>parti annectatur pondus remorans in imo annexum parte tro ch<lb/>lep. </s>
<s id="id2761277">Cum ergo trahitur n tro chlea, trahitur funis ade ut pon<lb/>dus m, tandem acendat cum tro chleal prope k: quia ergo in duo<lb/>decim horis pondus m decenderet per k l funem reuolutionibus <lb/>circa d rotam dicamus uiginti, ergo i debet decendere k ad l, per <lb/>funem duplicatam k l cum ipam necee it obequitantem d reuo<lb/>lutionibus quadraginta circumuolui d, nam tota o h n d m g l k lon <lb/>g maior et duplo k l, necee et m decendere tardius qum in du <lb/>plo temporis, quo decenderet per rectum funem k l, quod erat de<lb/>montrandum. </s>
<s id="id2761365">Et hanc appendicem uidi apud Carem Odonum <lb/>Apulum medicum, uirum elegantem lepidique ingenij. </s>
<s id="id2761378">Memento <lb/>uer quod ubi orbiculi non cederent funi, uel quia duriores in cir<lb/>cumuolutione, uel quia latius exciperent illum reduplicato fune <lb/>circa illos omnin o circumducuntur, ed difficilius ide egent gra<lb/>uiori pondere.</s></p><p type="main">
<s id="id2761412">Propoitio centeimaquinquageimaeptima.</s></p><p type="main">
<s id="id2761431">Horologiorum molarium rationem otendere.</s></p><p type="main">
<s id="id2761442"><arrow.to.target n="marg523"/></s></p><p type="margin">
<s id="id2761453"><margin.target id="marg523"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2761480">Sunt horum duo genera primum, & anti <lb/><figure id="id.015.01.173.1.jpg" xlink:href="015/01/173/1.jpg"/><lb/>quius licet multo poterius eo quod pon<lb/>deribus ducitur, quod funiculo ex inteti<lb/>nis ouium eu fidibus lir agitur. </s>
<s id="id2761518">Sit igitur <lb/>axis f k erectus uper plano, cui per longum <lb/>coniuncta mola multiplicis pir in fine, cu <lb/>ius cannectatur ferreo circulo, qui habeatur lo co capul b c, qu <lb/>circumuolui posit: huic <expan abbr="circũductus">circunductus</expan> funis d e multipliciter in pun <lb/>cto g, it autem e h in modum pyramidis enim in acutum, ed non <lb/>ualde per <expan abbr="&longs;pirã">piram</expan> exculptam deinentis, cui rota in uertice inerta den <lb/>iculo, & uertatur h e, colligens funiculum tractum in pira uerus <lb/>apicem: unde funiculus circumuoluet b g d, <expan abbr="cap&longs;ulã">capulam</expan> uerus c, traher <lb/>ergo molam, & contrin get uiolenter <expan abbr="quãtum">quantum</expan> fert longitudo funis <lb/>qu circumuolui potet a b e ad h: & cum trahitur in d eremittitur, <lb/>non potet mola tatim retrahere reluctantibus denticulis h l rot, <lb/>& alijs qu implicantur curriculo m, a igitur mola contructa uio<lb/>lenter mouet b g d, capulam motu contrario c in d & in g & in b, <lb/>quare funis d e trahitur, & trahit e h illum circumuoluendo contra<lb/>rio motu priori, is mouet denticulo rotam h l, illa per curriculum in <lb/>aliam <expan abbr="rotã">rotam</expan>, & ic deinceps donec tempus moueatur, & rota indicis. <lb/></s>
<s id="id2761718">Hic adet capula, & quod circumuertitur claue non et axis mol <lb/>ed extra molam, cilicet e h. </s>
<s id="id2761745">Et quoniam hac ratione quanto mola a
<pb xlink:href="015/01/174.jpg" pagenum="155"/>magis <expan abbr="explicabi&ttilde;">explicabitur</expan>, tanto lentius trahet, & uertet e h, ide hoc ex tru <lb/>ctura auxilium prtatur, ut funis in inferiore parte <expan abbr="cõplexus">complexus</expan> latio<lb/>res orbes, & regione tanto uehementius uertat e h: & ita uis qu <lb/>remittitur ob mol laxitatem, augetur tantundem ob itum & ma<lb/>gnitudinem pirarum ut ditantiorum ua extremitate ab hypomo <lb/>chlio, quod et axis coni e h, eu intar axis.</s></p><p type="main">
<s id="id2761839">Alterum genus horologiorum cum mola ine fune loco capul <lb/>habet <expan abbr="rotã">rotam</expan> plano ub tratam, plenam denticulis axis, quo circum<lb/>agitur uiolenter, non et extra molam, ed ei annexa et mola intus, <lb/>exterius <expan abbr="aũt">aunt</expan> rot; ergo circumducto axe mol uim patitur circulus <lb/>exterior, ed non <expan abbr="moue&ttilde;">mouetur</expan>, quoniam clauo <expan abbr="impedi&ttilde;">impeditur</expan>. </s>
<s id="id2761923">Vbi mola quan<lb/>tum decet contricta et ublato clauo tatim ecum trahit rotam, & <lb/>illa <expan abbr="curriculũ">curriculum</expan> rotas que alias, & tempus agitur, & index uertitur. </s>
<s id="id2761959">Sed <lb/>in hoc idem et in commodum ine remedio <lb/><figure id="id.015.01.174.1.jpg" xlink:href="015/01/174/1.jpg"/><lb/>quod fuit in priore. </s>
<s id="id2761985">Vbi enim cperit laxa<lb/>ri mola tanto tardius progrediuntur rot <lb/>atque index. </s>
<s id="id2762003">Veluti axis a b cui ecun dum lon <lb/>gitudinem mol caput interius annexum <lb/>et altero circulo rot in c d curriculum rot e, implexum rot f <lb/>clauus rotam retinens, donec circumducto a b mola contringa<lb/>tur, & latus eius trahat rotam ex c. </s>
<s id="id2762045">Inde ublato clauo circulus, eu <lb/>rota trahitur ex c in g, & in famola, qu etiam ecundum eandem <lb/>partem circumuoluta et: igitur d circumagetur rota & reliqua. <lb/></s>
<s id="id2762075">Sed ut dixi contructio hc non atisfacit.</s></p><p type="main">
<s id="id2762092">Aliam ergo oportuit excogitare qu huiumodi et. </s>
<s id="id2762104">Sub axe a b, <lb/>qui cir cumuertitur ad molam contrahendam rotam, collocant par <lb/>uam qu et, ut ita dicam, pars axis ima cui ineruntur dentes in am <lb/>bitu ea ratione, ut dum mola ten ditur, premant denticulos interio<lb/>res, atque ita elabitur, totiesque circumducitur manente g f, donec <lb/>colligatur mola, qu non ut in priore reliquo extremo ulli rot <lb/>affixa et, ed column in continenti <lb/>opercula horologij. </s>
<s id="id2762157">Cum ergo mola <lb/>tenta retrahat axem a b contrario mo<lb/><figure id="id.015.01.174.2.jpg" xlink:href="015/01/174/2.jpg"/><lb/>tu, & ille rotam mobilem, qu cum <lb/>non posit regredi propter aueros <lb/>dentes, mouet rotam f g contrario mo<lb/>tu, qu circumacta per denticulos u<lb/>os curriculum agit, & reliqua omnia <lb/>necearia. </s>
<s id="id2762216">Cur autem cum laxatur mo <lb/>la, & uertit lentius c e rotam coniun<lb/>ctam, ideoque g f, & reliqua omnia <expan abbr="nõ">non</expan> tardetur tempus, & circumuo
<pb xlink:href="015/01/175.jpg" pagenum="156"/>lutio indicis caua et alia long qum in priore, nam mola longior <lb/>fit crasior, & durior adeoque robuta, & rot leues, ac tempus dum <lb/>laxata fuerit munus uum iuto in tempore obeant: quare necee <lb/>et, ut ab initio uehementius agat, & celerius rotam cum axe qui tra<lb/>hitur mola. </s>
<s id="id2762299">Ergo excogitarunt aliud genus retinaculi forma co<lb/>chle quod ab initio moratur <expan abbr="uehem&etilde;ter">uehementer</expan> axem ne circumagatur, et <lb/>quanto magis mola explicatur eo minus retinet <expan abbr="impetũ">impetum</expan> illius, adeo <lb/>ut uehementer retineat uehementem concitationem medio criter <lb/>moderatam, egniter lentam, nullo modo iutam: ita fit, ut emper <lb/>ferm qualiter moueatur. </s>
<s id="id2762357">Difficile et tamen ad unguem eruare <lb/>moderationem, & qualitatem, & magis etiam in his horologijs, <lb/>qu uno circuitu mol tempus <expan abbr="lõgius">longius</expan> exigunt: at difficilius etiam <lb/>efficere molam, qu longo tempore duret, cum intenta ualde cele<lb/>rius moueat rotas, & ob id breui aboluat circuitum, mollior au<lb/>tem cit remittatur. </s>
<s id="id2762415">Et ob id longior & non ade <lb/>dura melior et. </s>
<s id="id2762428">Ratio autem cochle ita e habet. <lb/><figure id="id.015.01.175.1.jpg" xlink:href="015/01/175/1.jpg"/><lb/>Circa axem mol d deducitur cochlea a b c, qu <lb/>dum laxatur mola cochlea mouetur ex b in c, at que<lb/>ita pariter laxatur uis cochle retinentis axem.</s></p><p type="main">
<s id="id2762471">Propoitio centeimaquinquageimaoctaua.</s></p><p type="main">
<s id="id2762488">Rationem indicis mobilis cum rota horarum numerus per ictus <lb/>indicatur explicare.</s></p><p type="main">
<s id="id2762501"><arrow.to.target n="marg524"/></s></p><p type="margin">
<s id="id2762512"><margin.target id="marg524"/>C<emph type="italics"/>o<emph.end type="italics"/>_{m}.</s></p><p type="main">
<s id="id2762538">Hoc fieri potet in ingulo genere horologij trium <expan abbr="de&longs;criptorũ">decriptorum</expan>. <lb/></s>
<s id="id2762563">Propterea ufficiat de uno otendie. </s>
<s id="id2762577">Sed & in ingulo genere unt <lb/>multi modi, unius tamen reddidie <expan abbr="ration&etilde;">rationem</expan> ufficiat. </s>
<s id="id2762605">Hoc <expan abbr="aũt">aunt</expan> qua<lb/>tuor habet difficultates: prima ut horarum ictus conueniant cum <lb/>indice: ecunda ut conuero indice conuertatur, & rota ictuum: ter <lb/>tia ut ictuum numerus cum numero indicis conueniat. </s>
<s id="id2762637">Vnde mul<lb/>ta unt horologia, in quibus ictus unus olum auditur ingulis ho<lb/>ris, atque hic modus facilis et: quarta cur in horum pleri que i non <lb/>pulata tatim hora <expan abbr="transfera&ttilde;ur">transferaturur</expan> index, non ceat pulatio: im nec <lb/>retineri potet, donec pondus illud decenderit. </s>
<s id="id2762704">Ergo primi & ter<lb/>tij ratio hc habeatur, cum rota qu indicis rotam circumagit, per<lb/>uenerit ad hor finem, denticulo oluit aliam, eleuans obicem, illa <lb/>mouetur pondere proprio alio, cilicet ab illo quod tempus agit: <lb/>aut i it horologium mol mola alia propria, qu malleos cir<lb/>cumacta perpetu mouet, atque motura eet emper, donec pondus <lb/>ad terram decenderet: uerum dum mouetur decendit ferrum pro <lb/>quouis ictu quod in rot limbum incidit, & donec inciderit in eam <lb/>partem qu lenis et dilabitur, nec retinetur, & ita eleuatur rurus,
<pb xlink:href="015/01/176.jpg" pagenum="157"/>at uero cum in concauam partem incidit retineri necee et: atque ita <lb/>pondus non amplius decendit, rota ititur, malleus manet immo<lb/>bilis: patia ergo qu unt inter cauitates unt ecundum magnitu<lb/>dinem proportionis numerrum <expan abbr="horarũ">horarum</expan>, uel ad ex, uel ad duode<lb/>cim, uel ad uiginti<lb/><figure id="id.015.01.176.1.jpg" xlink:href="015/01/176/1.jpg"/><lb/>quatuor terminan<lb/>tium. </s>
<s id="id2762888">Ita quod, gra<lb/>tia exempli, it iam <lb/>in cauitate a duode<lb/>cim hor uncus, di <lb/>uidam circulum to<lb/>tum in duas partes <lb/>quales, quia in in <lb/>gulis medietatibus <lb/>propoitum et, duo <lb/>decim facere cauita<lb/>tes pro unco retinen<lb/>do. </s>
<s id="id2762949">Et quia in una<lb/>quaque medietate o<lb/>portet, ut pulent ho <lb/>r lxxviij, & prterea int ibi ex patia cauitatum, quarum ingul <lb/>contineant, gratia exempli, duo patia unius ictus, ut certius retinea <lb/>tur uncus, <expan abbr="erũt">erunt</expan> igitur patia omnia nonaginta: diuidemus ergo me<lb/>dietatem circuli utranque in nonaginta partes quales in cipiendo <lb/>ab a, & dabimus b prim hor quod patium et unius tantum par <lb/>tis ex nonaginta, pot decribemus c cauitatem duarum partium, <lb/>ita ubi ictum unum dederit uncus, retinebitur in c, pt accipiemus <lb/>duo patia, & int ignificata d litera, pot qu faciemus cauitatem e: <lb/>& ita uncus bis cadet in d, & pulabunt duo ictus, & pt retinebi<lb/>tur uncus in e. </s>
<s id="id2763083">Et pot accipiam patium trium partium, quod it f, <lb/>& pot decribam cauitatem g duarum partium, atque ita procedam <lb/>uque ad duodecim.</s></p><p type="main">
<s id="id2763115">Ex quo manifetum et pondus quod agit rotam uol non de</s></p><p type="main">
<s id="id2763133"><arrow.to.target n="marg525"/><lb/>cendere, nii dum hor pulant, ecus quiecere.</s></p><p type="margin">
<s id="id2763162"><margin.target id="marg525"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 1.</s></p><p type="main">
<s id="id2763189">Secundum, qud decendit illud pondus plus & minus, iuxta <lb/><arrow.to.target n="marg526"/><lb/>proportionem numeri horarum, ita quod quando pulabit una ho <lb/>ra parum ualde decendet, cum ex hor excuplo magis, cum duo<lb/>decim adhuc long magis, id et duplo plus qum cum pulant <lb/>ex hor.</s></p><p type="margin">
<s id="id2763253"><margin.target id="marg526"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 2.</s></p><p type="main">
<s id="id2763280">Secunda contructio hanc habetrationem: Cum n rota indicis <lb/>coniuncta fuerit rot, qu transfert malleum, necee et ut un fe
<pb xlink:href="015/01/177.jpg" pagenum="158"/>rantur: quinim illud magis mirum de quo illi non mirantur quia <lb/>frequens et, cilicet cur aut quomodo i diui unt ut cir <expan abbr="çũducto">unducto</expan> <lb/>indice non transferatur rota mallei, <expan abbr="põdere">pondere</expan> tamen uerata rota in<lb/>dicis in idem incidat, ut hor qu pulu declarantur ad unguem <lb/>& in eidem ectionibus <expan abbr="cõueniant">conueniant</expan> cum horis quas index oten dit.</s></p><p type="main">
<s id="id2763405">Verm quia multis modis contingit ordinem horologiorum <lb/>peruerti: in imilibus quidem i hora indicis imul & pulus un <lb/>circumferuntur, ed tardius ambo index traducitur ad locum debi<lb/>tum, inde ponderi aliquid additur. </s>
<s id="id2763442">Si uer ant proceerit quam. <lb/></s>
<s id="id2763457">Sol in dicet ablato pondere, ines tempus fluere uque ad indicis lo<lb/>cum ine motu horologij, pondus quoque ipum minues. </s>
<s id="id2763478">At i pon<lb/>dus pulus in terram deuenerit uel prop, expecta donec uper li<lb/>nea index fuerit, inde trahe, neque. </s>
<s id="id2763503">n. </s>
<s id="id2763507">excurret: nam i dum index et in <lb/>medio hor aut prop, traxeris pondus pulus, non deinet decen <lb/>dere, pulabuntqe hor donec ad terram pondus deuenerit, <lb/>qud i iam in errorem incideris pulentque hor & decendat, pon<lb/>dus, enim deducito indicem, cum. </s>
<s id="id2763573">n. </s>
<s id="id2763577">ad finem hor peruenerit ini<lb/>tiumque equentis, quoniam ferrum in interuallum deuenerit rota & <lb/>pondus firmabitur. </s>
<s id="id2763596">Inde ublato <expan abbr="põdere">pondere</expan> donec Sol ad <expan abbr="horã">horam</expan> quam <lb/>index montrat peruenerit, reddes pondus horologio. </s>
<s id="id2763627">Si ergo ho<lb/>ram pulu <expan abbr="eand&etilde;">eandem</expan> declarat quam index, bene et, i non, <expan abbr="paululũ">paululum</expan> <expan abbr="uir-gulã">uir<lb/>gulam</expan> eleua qu et iuxta fores horologij pulabitque equens hora, id <lb/>uero toties repetes immoto in dies & ublato, i uereris ne extra <expan abbr="in-teruallũ">in<lb/>teruallum</expan> ferrum feratur, & ob id excurrat rota pulus <expan abbr="horarũ">horarum</expan>, donec <lb/>hora pulet qu cum indice conuenit, tatimque pondus quo hor <lb/>pulant urum retrahes. </s>
<s id="id2763746">His quinque regulis uum dices imilium <lb/>horologiorum, unumquodque autem proprias habet: ed du pri<lb/>m omni horologi atisfaciunt. </s>
<s id="id2763780">Qud i h non atisfa ciunt iam <lb/>horologium laborat: tum uer illud dioluere oportet & deterge<lb/>re & inungere, iuuat autem uel capula uel linteo perpetuo pul<lb/>uerem ab illo arcere. </s>
<s id="id2763819">Qud i nec ic retituitur necee et diol<lb/>uere & antea coniderare impedimentum, pt denticulum qui la<lb/>borat, plerunque. </s>
<s id="id2763862">n. </s>
<s id="id2763866">aliquem inuenies huius modi, quem lima aut alia <lb/>ratione retitues, emper autm hi ferm retituuntur: at qui mola <lb/>aguntur prter rotarum & axium & indicum labores, mol etiam <lb/>inqualitati & defectibus ubiciuntur, qui i nimis uelo citer agunt <lb/>rotas cum difficultate retituuntur moderationi, i lentius rar uel <lb/>nunquam emendantur, uix etiam noua inducta mola.</s></p><p type="main">
<s id="id2763930">Propoitio centeimaquinquageimanona.</s></p><p type="main">
<s id="id2763946">Nullus angulus rectilineus qualis ee potet alicui angulo con <lb/>tento recta & circuli portione.</s></p>
<pb xlink:href="015/01/178.jpg" pagenum="159"/><p type="main">
<s id="id2763978">Sit angulus a & circulus b c, dico non poe aliquem angulum <lb/><arrow.to.target n="marg527"/><lb/>contentum recta & circuli portione ee illi <lb/><figure id="id.015.01.178.1.jpg" xlink:href="015/01/178/1.jpg"/><lb/>qualem. </s>
<s id="id2764017">i enim ee posit, it c b e. </s>
<s id="id2764032">duca<lb/>tur recta b d faciens rectilineum d b c qua <lb/><arrow.to.target n="marg528"/><lb/>lem a, erit igitur d b c qualis e b c per com<lb/>munem animi ententiam, eu ergo b d ca<lb/>dat intra circulum eu extra, erit pars qua<lb/>lis toti quod ee non potet. </s>
<s id="id2764087">Sed neque po<lb/>tet cadere recta uper b e. </s>
<s id="id2764101">namid et contra demontrata ab Eucli<lb/><arrow.to.target n="marg529"/><lb/>de. </s>
<s id="id2764121">At i it angulus c b e exterior imiliter producta b d, eu intus, <lb/>eu extr cadat, pars erit qualis toti quod ee non potet.</s></p><p type="margin">
<s id="id2764158"><margin.target id="marg527"/>C<emph type="italics"/>o<emph.end type="italics"/>_{m}.</s></p><p type="margin">
<s id="id2764185"><margin.target id="marg528"/>P<emph type="italics"/>er<emph.end type="italics"/> 23. <emph type="italics"/>pri <lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2764234"><margin.target id="marg529"/>23. E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="main">
<s id="id2764259">Ex hoc patet quod nullus angulus peripheria circuli & recta <expan abbr="cõ-">con<lb/></expan><arrow.to.target n="marg530"/><lb/>tentus potet ee qualis recto, quia rectus etiam rectilineus et.</s></p><p type="margin">
<s id="id2764299"><margin.target id="marg530"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 1.</s></p><p type="main">
<s id="id2764326">Et rurus nullus angulus peripheria & <lb/><arrow.to.target n="marg531"/><lb/><figure id="id.015.01.178.2.jpg" xlink:href="015/01/178/2.jpg"/><lb/>recta contentus recta linea per qualia <lb/>diuidi potet, patet quia una pars eet an<lb/>gulus rectilineus, alia contentus recta & pe <lb/>ripheria: iti <expan abbr="aut&etilde;">autem</expan> non pount ee quales, <lb/>quare nec prior potuit per qualia diuidi.</s></p><p type="margin">
<s id="id2764408"><margin.target id="marg531"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 2.</s></p><p type="main">
<s id="id2764435">Ex hoc etiam patet quod pacium con<lb/><arrow.to.target n="marg532"/><lb/><expan abbr="tentũ">tentum</expan> peripheria circuli nulli angulo rectilineo quale ee potet. <lb/></s>
<s id="id2764472">nam dimidium eet quale dimidio, quod et contra demontrata.</s></p><p type="margin">
<s id="id2764494"><margin.target id="marg532"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 3.</s></p><p type="head">
<s id="id2764520">LEMMA PRIMVM.</s></p><p type="main">
<s id="id2764530">Inter duos circulos qui e diuidant infinit line duci pount. <lb/></s>
<s id="id2764548">Inter circulos autem qui e tangant, rectalinea duci non potet.<lb/><arrow.to.target n="marg533"/></s></p><p type="margin">
<s id="id2764569"><margin.target id="marg533"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2764595">Sint duo circuli a b & a c, qui e diuidant </s></p><p type="main">
<s id="id2764607"><arrow.to.target n="marg534"/><lb/>in a, & ducatur ex centro inferioris d a & <lb/><figure id="id.015.01.178.3.jpg" xlink:href="015/01/178/3.jpg"/><lb/>a d, & ad d a cathetus a e, dico qud a e di<lb/>uidet angulum b a c ducatur ex centro u<lb/><arrow.to.target n="marg535"/><lb/>perioris a c b quod it f, fa cui cathetus a g, <lb/>quia ergo e a cadit infra a g, & inter a g & <lb/><arrow.to.target n="marg536"/><lb/>a b non potet duci recta, igitur e a cadit in<lb/><figure id="id.015.01.178.4.jpg" xlink:href="015/01/178/4.jpg"/><lb/>tra a c b circulum. </s>
<s id="id2764686">Rurus tangant e circuli <lb/>c d & c e, & ducatur a b per centra <expan abbr="eorũ">eorum</expan> qu <lb/>applicabit ad c, ex c ducatur cathetus c f & <lb/><expan abbr="quoniã">quoniam</expan> c f contangit <expan abbr="circulũ">circulum</expan> c e, ligitur, du<lb/>cta quauis linea infra c f, cadet intra <expan abbr="circulũ">circulum</expan> <lb/>c e. </s>
<s id="id2764749">Non ergo poterit cadere inter c d & c e.</s></p><p type="margin">
<s id="id2764759"><margin.target id="marg534"/>P<emph type="italics"/>er<emph.end type="italics"/> 11. <emph type="italics"/>pri <lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2764809"><margin.target id="marg535"/>P<emph type="italics"/>er<emph.end type="italics"/> 15. <emph type="italics"/>ter <lb/>tij<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2764858"><margin.target id="marg536"/>P<emph type="italics"/>er<emph.end type="italics"/> 11. <emph type="italics"/>ter<lb/>tij<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s></p><p type="head">
<s id="id2764908">LEMMA SECVNDVM.</s></p><p type="main">
<s id="id2764918">Dato angulo contento duabus peripherijs <expan abbr="æqualiũ">qualium</expan> circulorum <lb/>e e cantium qualem rectilineum illi fabricare.</s></p>
<pb xlink:href="015/01/179.jpg" pagenum="160"/><p type="main">
<s id="id2764955">Sit angulus a b c duabus peripherijs qualium circulorum con <lb/><arrow.to.target n="marg537"/><lb/>tentus, uolo ei qualem rectilineum fabricare, ducantur b d & b e <lb/><arrow.to.target n="marg538"/><lb/>quales, ut pote facto b centro eritque angulus d b a qualis angu<lb/>lo e b c, addito utrique communi d b e ex peri <lb/><figure id="id.015.01.179.1.jpg" xlink:href="015/01/179/1.jpg"/><lb/>pheria & recta, fiet angulus d b e ex rectis <lb/>qualis a b c ex peripherijs, quod crat de<lb/>montrandum.</s></p><p type="margin">
<s id="id2765025"><margin.target id="marg537"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s id="id2765052"><margin.target id="marg538"/>P<emph type="italics"/>er modum<emph.end type="italics"/><lb/>8. <emph type="italics"/>primi<emph.end type="italics"/> E<emph type="italics"/>l.<emph.end type="italics"/></s></p><p type="main">
<s id="id2765100">Ex hoc patet quod reliqua duo pacia <lb/><arrow.to.target n="marg539"/><lb/>non pount ee qualia rectilineo. </s>
<s id="id2765125">Nam <lb/>patium b a c demontratum et quale e<lb/>e rectilineo, & b ad non et quale rectili<lb/>neo, <expan abbr="igi&ttilde;">igitur</expan> <expan abbr="&longs;patiũ">patium</expan> c a d non potet ee quale <lb/>angulo rectilineo, nam i ic it b a c quale <lb/>f g h & c a d h g k, <expan abbr="igi&ttilde;">igitur</expan> <expan abbr="totũ">totum</expan>, b a d erit quale <lb/><arrow.to.target n="marg540"/><lb/>toti f g k d et contra <expan abbr="&longs;uppo&longs;itũ">uppoitum</expan>, ide neque<lb/>b a e quia b a c & d a e unt <expan abbr="æ&qtilde;lia">qualia</expan> rectilineis <lb/>per e, & <expan abbr="etiã">etiam</expan> pariter accepta. </s>
<s id="id2765291">Totum <expan abbr="aũt">aunt</expan> <expan abbr="&longs;patiũ">patium</expan> a et <expan abbr="&ecedil;&qtilde;le">quale</expan> quatuor, re<lb/>ctis ergo <expan abbr="re&longs;iduũ">reiduum</expan>, cilicet patia c a d & b a c pariter accepta unt <expan abbr="&ecedil;&qtilde;-lia">qua<lb/>lia</expan> rectilineis patijs, ed <expan abbr="&longs;patiũ">patium</expan> e a d non et <expan abbr="æ&qtilde;le">quale</expan> rectilineo, ergo per<lb/>demontrata hic, nec b a e, <expan abbr="nã">nam</expan> i it, it ergo b a e quale h g k & quia <lb/>ambo patia b a e & c a d unt <expan abbr="æ&qtilde;lia">qualia</expan> rectilineo ex demontratis, it <lb/>ergo qualia f g k, erit ergo ex communi animi ententia patium f <lb/>g h quale pacio c a d, quod et contra primam partem corrolarij.</s></p><p type="margin">
<s id="id2765482"><margin.target id="marg539"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 4.</s></p><p type="margin">
<s id="id2765509"><margin.target id="marg540"/>P<emph type="italics"/>er<emph.end type="italics"/> 3. C<emph type="italics"/>or<emph.end type="italics"/>^{m}. <lb/><emph type="italics"/>prentis.<emph.end type="italics"/></s></p><p type="head">
<s id="id2765559">LEMMA TERTIVM.<lb/><arrow.to.target n="marg541"/></s></p><p type="margin">
<s id="id2765575"><margin.target id="marg541"/>P<emph type="italics"/>er<emph.end type="italics"/> 11. <emph type="italics"/>pri <lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s></p><p type="main">
<s id="id2765624">Inter duas rectas lineas e tangentes circuli dati peripheriam </s></p><p type="main">
<s id="id2765636"><arrow.to.target n="marg542"/><lb/>ducere. </s>
<s id="id2765645">Sit circulus datus a b rectilineus <lb/><figure id="id.015.01.179.2.jpg" xlink:href="015/01/179/2.jpg"/><lb/>angulus c d e, uolo illum diuidere circuli <lb/>periferia data b f, duco perpendicularem <lb/>d g ex, d uper d c, & facio g d qualem a b <lb/><arrow.to.target n="marg543"/><lb/>& duco circulum per d qui it d h qui cadet <lb/>infra d c & ob id etiam upra d e, igitur di<lb/>uidet angulum c d e, quare cum circulus d h it qualis circulo b f <lb/><arrow.to.target n="marg544"/><lb/>patet propoitum.</s></p><p type="margin">
<s id="id2765720"><margin.target id="marg542"/>P<emph type="italics"/>er<emph.end type="italics"/> 3. <emph type="italics"/><expan abbr="eiu&longs;d&etilde;">eiudem</expan><emph.end type="italics"/></s></p><p type="margin">
<s id="id2765766"><margin.target id="marg543"/>P<emph type="italics"/>er<emph.end type="italics"/> 15. <emph type="italics"/>ter <lb/>tij<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2765815"><margin.target id="marg544"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 6.</s></p><p type="main">
<s id="id2765841">Ex hoc patet quod infinitis modis potet diuidi angulus c d e <lb/><arrow.to.target n="marg545"/><lb/>peripheria b f, nam diuio per rectam c d e linea d k per qualia & di <lb/><arrow.to.target n="marg546"/><lb/>uio k d e per prentem peripheria b f, patet propoitum quoniam <lb/>angulus c d e potetin infinitum recta diuidi, & ita emper per peri<lb/>pheriam, unde patet propoitum.</s></p><p type="margin">
<s id="id2765902"><margin.target id="marg545"/>P<emph type="italics"/>er<emph.end type="italics"/> 1. <emph type="italics"/>diff. <lb/></s>
<s id="id2765931">tertij <expan abbr="eiu&longs;d&etilde;">eiudem</expan>.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2765957"><margin.target id="marg546"/>P<emph type="italics"/>er<emph.end type="italics"/> 9. <emph type="italics"/>primi<emph.end type="italics"/><lb/>E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="head">
<s id="id2766005">SCHOLIVM.</s></p><p type="main">
<s id="id2766014">Atque hc omnia equuntur de mente Euclidis, qu tamen ui<lb/>dentur difficillima creditu, quoniam anguli rectilinei, et ex periphe
<pb xlink:href="015/01/180.jpg" pagenum="161"/>ria & recta unt ex genere quantitatis continu, & qud detur ma<lb/>ius & minus & nunquam detur quale, uidetur aburdum ne dum <lb/>admirabile. </s>
<s id="id2766062">Et maxim quod etiam anguli ex peripheria & recta <lb/>unt diuerorum generum inter e & infinitorum. </s>
<s id="id2766081">Prterea itud re<lb/>pugnare uidetur ipimet Euclidi, dicenti duabus magnitu dinibus <lb/><arrow.to.target n="marg547"/><lb/><arrow.to.target n="marg548"/><lb/>propoitis inqualibus, i de maiore earum plus dimidio detraha<lb/>tur, atque iterum de reiduo maius dimidio, & rurus de eo quod re<lb/>linquitur plus dimidio, necee erit ut tandem minor minore quan<lb/>titas relinquatur. </s>
<s id="id2766146">Neque illud argumentum uidetur concludere an<lb/>gulus contactus, ex recta, & circuli circumferentia non potet recta <lb/>diuidi, & rectilineus potet diuidi, ergo rectilin eus emper et ma<lb/>ior angulo contactus, quia hoc contingit in angulo contactus pro <lb/>pter modum anguli, non paruitatem: i cut etiam non ualet de figu<lb/><figure id="id.015.01.180.1.jpg" xlink:href="015/01/180/1.jpg"/><lb/>ra a lunari, & quadrangulo b. </s>
<s id="id2766201">nam potet b diuidi <lb/>ab angulo ad angulum recta & a non potet, & <lb/>tamen a maius et quam b, cum contineat ipam. <lb/></s>
<s id="id2766225">Proponantur ergo duo circuli a d e & a f g qui e contingant in a, & <lb/>corum centra int b & c & ducantur rect a f d & a g e & contat <lb/>d portiones a d & a f imiles unt, <lb/><figure id="id.015.01.180.2.jpg" xlink:href="015/01/180/2.jpg"/><lb/>itemque a e & a g, ducta enim a b c <lb/><arrow.to.target n="marg549"/><lb/>per centra circulorum ex contactu <lb/>tranibit per illa: quare anguli h a g <lb/>& h a e untijdem & imiliter h a f <lb/>& h a d ijdem, portiones ergo af & <lb/>a d itemque a g & a e imiles unt: an<lb/>gulus igitur g a e ex peripherijs & <lb/><arrow.to.target n="marg550"/><lb/>e a d ex rectis unt ijdem in puncto <lb/>a: ed quod ad basim maior et ba<lb/>is g e quam e d: hoc enim uppono <lb/>quod per e et manifetum toties <lb/><expan abbr="diuid&etilde;do">diuidendo</expan> arcum d e ut fiat minor recta g e. </s>
<s id="id2766367">Quia ergo unt du ma<lb/>gnitudines, quarum ter mini unt ijdem ex una parte, cilicet pun<lb/>ctum a, ex alia autem unus et maior altero, cilicet g e quam e f & <lb/><arrow.to.target n="marg551"/><lb/>a d e peripheria et maior recta a g e. </s>
<s id="id2766410">Ergo per regulam dialecti<lb/>cam i ub eadem proportione procederent, maius eet patium <lb/>emper inter peripherias qum rectas. </s>
<s id="id2766439">igitur angulus peripheria<lb/>rum et maior angulo rectis contento. </s>
<s id="id2766453">Cum angulus non it <lb/>nii quidam habitus propinquitatis linearum, ed angulus con<lb/>tactus ex recta & peripheria maior et contento ex peripherijs cum <lb/>habeat rationem totius ad partem, igitur angulus contactus et <lb/>maior dato angulo rectilineo.</s></p>
<pb xlink:href="015/01/181.jpg" pagenum="162"/><p type="margin">
<s id="id2766501"><margin.target id="marg547"/>1. P<emph type="italics"/>ropo.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2766530"><margin.target id="marg548"/>10. E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2766554"><margin.target id="marg549"/>P<emph type="italics"/>er<emph.end type="italics"/> 11. <emph type="italics"/>ter <lb/>tij<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2766603"><margin.target id="marg550"/>E<emph type="italics"/>x<emph.end type="italics"/> 10. <emph type="italics"/>diff. <lb/></s>
<s id="id2766630">tertij<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2766655"><margin.target id="marg551"/>P<emph type="italics"/>er<emph.end type="italics"/> 1. <emph type="italics"/>deci<lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="main">
<s id="id2766706">Propoitio centeimaexageima.</s></p><p type="main">
<s id="id2766724">Propoita linea tribus que in ea ignis punctum inuenire, ex que <lb/>duct tres line ad igna int in proportionibus datis.<lb/><arrow.to.target n="marg552"/></s></p><p type="margin">
<s id="id2766759"><margin.target id="marg552"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s id="id2766786">Sit data linea a b c in qua puncta dicta & dat tres line d e f, uo<lb/>lo inuenire punctum, puta g ex quo duct tres <lb/>line ad a b c puncta int in proportione a g ad </s></p><p type="main">
<s id="id2766816"><arrow.to.target n="marg553"/><lb/>g b, ut d ad e & g b ad g c, ut e ad f. </s>
<s id="id2766828">Per prceden <lb/><figure id="id.015.01.181.1.jpg" xlink:href="015/01/181/1.jpg"/><lb/>tia inuenio circulum ex cuius peripheria omni<lb/>bus ex punctis duct line ad a b int in pro<lb/>portione d ad e, & per idem circulum ex cuius <lb/>peripheria qulibet line duct ad b c puncta <lb/>int in proportione c ad f, i igitur iti duo circu<lb/>li e ecabunt in aliquo puncto puta g: liquet <lb/>quod line duct ex g ad a b c, erunt in propor<lb/>tione d e f.<lb/><arrow.to.target n="marg554"/></s></p><p type="margin">
<s id="id2766922"><margin.target id="marg553"/>P<emph type="italics"/>er<emph.end type="italics"/> 154.</s></p><p type="margin">
<s id="id2766947"><margin.target id="marg554"/>C<emph type="italics"/>or<emph.end type="italics"/>^{m}_{m}.</s></p><p type="main">
<s id="id2766972">Ex quo liquet quod i uoluero ducere ad tria puncta data, tres <lb/>lineas in continua proportione data d ad e, ubijciam tertiam uel in <lb/>terponam, i uoluero mediam. </s>
<s id="id2766991">Et i uellem, ut eet a g ad g b dupli<lb/>cata ei qu et g b ad b c, & uellem qud proportio d ad a d f data <lb/>eet, oporteret inuenire duas medias proportione inter d & f, in de <lb/>operari cum una earum per modum propoitum. </s>
<s id="id2767031">Differt corrola<lb/>rium hoc propoitione in hoc, quod in propoitione non quri<lb/>mus nii proportionem g a ad g b & g b ad b c, non g a ad g c, neque<lb/>comparationem proportionum: at in corrolario qurimus tres <lb/>proportiones g a g b & g c, & comparationem proportionum in<lb/>ter e, cilicet qualitatem.</s></p><p type="main">
<s id="id2767089">Propoitio centeimaexageimaprima.</s></p><p type="main">
<s id="id2767109">Si fuerint duo trianguli quorum baes in eadem linea int con<lb/>tituti & quales & ad unum punctum terminati, & latus unum <lb/>commune inter reliqua quantita<lb/><figure id="id.015.01.181.2.jpg" xlink:href="015/01/181/2.jpg"/><lb/>te medium, necee et angulum <lb/>maioribus lineis contentum mi<lb/>norem ee.</s></p><p type="main">
<s id="id2767172">Sint duo trianguli a b c, a c d, </s></p><p type="main">
<s id="id2767181"><arrow.to.target n="marg555"/><lb/>quales proponuntur, & it a d ma<lb/><arrow.to.target n="marg556"/><lb/>ior a b dico angulum d a c ee mi<lb/>norem. </s>
<s id="id2767213">Si non fiat angulus d a c <lb/>qualis ex alia parte, & oportet i non it minorut uel cadat a d u<lb/><arrow.to.target n="marg557"/><lb/>per a b & ducta a d ad qualitatem cadet infra b, ducta ergo d c erit <lb/>trigonus a d c maior a b c, quod ee non potet cum int quales.
<pb xlink:href="015/01/182.jpg" pagenum="163"/>Si autem a d cadat extra a b ducatur d e: qu i cadat upra b c uel <lb/>infra, cum totum it maius parte erit a d e, ut prius maior a b c quod <lb/><arrow.to.target n="marg558"/><lb/>et contra Euclidem. </s>
<s id="id2767299">Reliquum et ut d c cadat upra b c: hoc au<lb/><arrow.to.target n="marg559"/><lb/>tem ee non potet, nam cum uppouerimus a b ee minorem a c <lb/>erit angulus a c b minor angulo a b c, quare a c b et minor recto, & <lb/><arrow.to.target n="marg560"/><lb/>ide a c d maior recto, at a c d qualis et a c d, alteri igitur a c d et <lb/><arrow.to.target n="marg561"/><lb/>maior recto a c b minor, erit ergo pars maior toto.</s></p><p type="margin">
<s id="id2767377"><margin.target id="marg555"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s id="id2767403"><margin.target id="marg556"/>P<emph type="italics"/>er<emph.end type="italics"/> 23. <emph type="italics"/>pri <lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2767452"><margin.target id="marg557"/>P<emph type="italics"/>er<emph.end type="italics"/> 38. <emph type="italics"/>pri <lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2767501"><margin.target id="marg558"/>P<emph type="italics"/>er<emph.end type="italics"/> 18. <emph type="italics"/>pri <lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2767550"><margin.target id="marg559"/>P<emph type="italics"/>er<emph.end type="italics"/> 23. <emph type="italics"/>eiu <lb/>dem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2767589"><margin.target id="marg560"/>P<emph type="italics"/>er<emph.end type="italics"/> 13. <emph type="italics"/>eiu <lb/>dem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2767629"><margin.target id="marg561"/>P<emph type="italics"/>er<emph.end type="italics"/> 4. <emph type="italics"/>eiu<lb/>dem.<emph.end type="italics"/></s></p><p type="head">
<s id="id2767669">LEMMA.</s></p><p type="main">
<s id="id2767678">His demontratis quis dicere poet ex uperius expoitis quod <lb/><arrow.to.target n="marg562"/><lb/>angulus rectilineus emper eetmaior angulo contactus? </s>
<s id="id2767710">quia an<lb/>gulus contactus non potet diuidi nii obliqua linea, recti lineus <lb/>autem tam obliqua quam recta. </s>
<s id="id2767728">Propter hoc exponantur circuli <lb/><figure id="id.015.01.182.1.jpg" xlink:href="015/01/182/1.jpg"/><lb/>tres e tangentes a b, a c, a d hac rati<lb/>one ut a b, b c, c d int quales, erunt <lb/><arrow.to.target n="marg563"/><lb/>enim centra omnia in linea conta<lb/>ctus, & ducatur a e f g recta quomo <lb/><arrow.to.target n="marg564"/><lb/>dolibet: & erunt ductis lineis b c, <lb/><arrow.to.target n="marg565"/><lb/>c f, d g anguli e f g recti, quare om<lb/>nes trigoni a b e, a c f, a d g, imiles <lb/><arrow.to.target n="marg566"/><lb/>& ideo a e, e f, f g quales, atque por<lb/>tiones a g, a f, a e, iuxta proportio<lb/>nem circulorum, quare a g, erit ex<lb/>quialtera a f & a f dupla a e, igitur <lb/><arrow.to.target n="marg567"/><lb/>per prcedentem maior erit angu<lb/>lus e a f, quam f a g, & a d a ex recta <lb/><arrow.to.target n="marg568"/><lb/>& peripheria quam e a f, igitur augendo eadem ratione cum perue<lb/>niamus ad angulum b a g qui ferm et recto qualis cum deficiat <lb/>olo angulo contactus, liquet angulum e a g ee long maiorem <lb/>multis rectilineis. </s>
<s id="id2767885">Itud poet etiam demontrari uia Archimedis <lb/>diuidendo arcus g a in h & f a in k bifariam ducendo que lineas re<lb/>ctas g h & fk & ita diuidendo h a in 1, & k a in m bifariam, & ducen<lb/>do rectas atque ita emper appropinquando puncto a. </s>
<s id="id2767919">Concludo er<lb/>go quod angulus <expan abbr="cõtactus">contactus</expan> ex recta & peripheria et maior multis <lb/>rectilineis. </s>
<s id="id2767944">Caua autem erroris et quod multi exitimarunt corro<lb/>larium illud ee Euclidis cum non it. </s>
<s id="id2767969">Nam Euclidi ufficit hoc <lb/>qud angulus contactus <expan abbr="nõ">non</expan> posit recta diuidi, nam eo utitur pot <lb/><expan abbr="modũ">modum</expan> in demontrationibus. </s>
<s id="id2768008">Eo uer quod it minor omnibus re<lb/>ctilineis angulis non utitur, ide etiam i <expan abbr="uerũ">uerum</expan> fuiet <expan abbr="nõ">non</expan> ad didiet: <lb/>quanto minus: cum uerum non it, ide fuit <expan abbr="adiectũ">adiectum</expan> ab aliquo qui <lb/><expan abbr="id&etilde;">idem</expan> fore credidit <expan abbr="nõ">non</expan> poe diuidi rectalinea & ee minus quocunque<lb/>quod recta linea diuidi poet, quod apert ut dixi falum et.</s></p>
<pb xlink:href="015/01/183.jpg" pagenum="164"/><p type="margin">
<s id="id2768125"><margin.target id="marg562"/>L<emph type="italics"/>emmate<emph.end type="italics"/> 3. <lb/>P<emph type="italics"/>rop.<emph.end type="italics"/> 159.</s></p><p type="margin">
<s id="id2768164"><margin.target id="marg563"/>P<emph type="italics"/>er<emph.end type="italics"/> 11. <emph type="italics"/>ter <lb/>tij<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2768214"><margin.target id="marg564"/>P<emph type="italics"/>er<emph.end type="italics"/> 31. <emph type="italics"/>ter <lb/>tij<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2768263"><margin.target id="marg565"/>P<emph type="italics"/>er<emph.end type="italics"/> 32. <emph type="italics"/>pri<lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2768313"><margin.target id="marg566"/>P<emph type="italics"/>er<emph.end type="italics"/> 4. <emph type="italics"/>exti<emph.end type="italics"/><lb/>E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2768362"><margin.target id="marg567"/>P<emph type="italics"/>er<emph.end type="italics"/> 10. <emph type="italics"/>diff<lb/>tertij<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2768412"><margin.target id="marg568"/>P<emph type="italics"/>er prce<lb/>dentem.<emph.end type="italics"/></s></p><p type="head">
<s id="id2768442">SCHOLIVM.</s></p><p type="main">
<s id="id2768452">Ratio autem qud omnis angulus contactus indiuiduus it, eu <lb/>duorum circulorum, eu circuli cum recta et, quoniam cum fuerint <lb/>du rationes contrari, & una perpetu minuitur, alia manet ne<lb/>cee et, ut tandem, qu minuitur, uperetur ab ea qu manet: cum <lb/>ergo circuli curuitas maneat, & angulus tendat in punctum perpe<lb/>tua diminutione necee et, ut curuitas circuli impediat diuiio<lb/>nem rect: ed hoc habet duplicem obicem. </s>
<s id="id2768534">Primum, quia nullus <lb/>angulus ex circumferentia & recta poet diuidi: hoc autem falum <lb/>et manifet, cum olus ille qui fit ex contactu line, qu non di<lb/>uidit circulum, diuidi non posit. </s>
<s id="id2768577">Secund, quod angulus conta<lb/>ctus duorum circulorum e exterius tangentium multo minus <lb/>poet diuidi angulo contactus interioris duorum circulorum, <lb/>quod tamen falum et: & hoc animaduertit Campanus noter, uir <lb/>acutus. </s>
<s id="id2768615">Dico ergo qud in his qui e tangunt exterius, non fit diui<lb/>io nii emel: & quamuis inclinentur mutu, tamen in concuru <lb/>non aptantur, ut cum obuiat rect aut cau parti circuli quia ne<lb/>cee et, ut accedat, in alio autem dicedat: indicio et quod circu<lb/>los e exterius tangentes, in puncto facil decribes, interius uix fie<lb/>ri potet, ed uidentur coniuncti <lb/><figure id="id.015.01.183.1.jpg" xlink:href="015/01/183/1.jpg"/><lb/>per longum interuallum. </s>
<s id="id2768707">Ad aliud <lb/>dico, qud ille angulus ex recta & <lb/>peripheria conuexa circuli propter <lb/>diceum eruat maiorem inclina<lb/>tionem in quocunque puncto, qum <lb/>it acceus conuex partis exterio<lb/>ris circuli.</s></p><p type="main">
<s id="id2768761">Propoitio centeimaexageima <lb/>ecunda.</s></p><p type="main">
<s id="id2768784">Proportionem duorum orbium <lb/>quorum diametrorum <expan abbr="cõuexæ">conuex</expan> par <lb/>tis, & concau proportiones dat <lb/>int, inuetigare.</s></p><p type="main">
<s id="id2768823">Sint duo orbes a b c d & e f g h, <lb/><arrow.to.target n="marg569"/><lb/>& it proportio a d ad b c, data & e <lb/>h ad f g, data & rurus a d ad e h, di<lb/>co orbis proportionem a b c d ad <lb/><expan abbr="orb&etilde;">orbem</expan> e f g h ee <expan abbr="datã">datam</expan>. </s>
<s id="id2768873">Quia. n. </s>
<s id="id2768880">propor <lb/>tio a d phr ad b c et ueluti ad di <lb/>metientis ad b c <expan abbr="dimetient&etilde;">dimetientem</expan> triplicata, ide <expan abbr="cũ">cum</expan> nota it a d ad b c di <lb/><arrow.to.target n="marg570"/><lb/><expan abbr="metientiũ">metientium</expan>, erit nota <expan abbr="etiã">etiam</expan> a d phr ad b c <expan abbr="&longs;ph&ecedil;rã">phram</expan>. </s>
<s id="id2768972">quare orbis ad ad <lb/><expan abbr="&longs;ph&ecedil;rã">phram</expan> b c. nota et <expan abbr="etiã">etiam</expan> proportio b c <expan abbr="dimeti&etilde;tis">dimetientis</expan> ad a d & ad a d e h &
<pb xlink:href="015/01/184.jpg" pagenum="165"/>e h ad f g, igitur b c proportio dimetientis ad f g dimetientem nota. <lb/><arrow.to.target n="marg571"/><lb/>Quare phr b c ad f g phram. </s>
<s id="id2769050">atnota et proportio f g ad e h <lb/>dimetientium igitur & phrarum: igitur nota et f g phr ad or <lb/>bem e h, igitur cum nota it proportio orbis ad a d phram b c, & <lb/>b c phr ad f g phram, & f g phr ad orbem e h, erit propor<lb/>tio orbis a d ad orbem e h nota, quod et propoitum.</s></p><p type="margin">
<s id="id2769128"><margin.target id="marg569"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s id="id2769155"><margin.target id="marg570"/>P<emph type="italics"/>er<emph.end type="italics"/> 18. <emph type="italics"/>duo <lb/>decimi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s id="id2769205"><margin.target id="marg571"/>P<emph type="italics"/>er<emph.end type="italics"/> 22. <lb/><emph type="italics"/>quinti<emph.end type="italics"/> E<emph type="italics"/>lem. <lb/></s>
<s id="id2769246">&<emph.end type="italics"/> A<emph type="italics"/>lizam.<emph.end type="italics"/></s></p><p type="main">
<s id="id2769270">Propoitio centeimaexageimatertia.</s></p><p type="main">
<s id="id2769288">Proportionem uirium tellarum per motus uos indagare.</s></p><p type="main">
<s id="id2769302">Mouentur tell omnes ab Oriente in Occidentem die una, qui <lb/><arrow.to.target n="marg572"/><lb/>motus fit prima mente, qu mouet: ide quod ad hoc attinet non <lb/>et diueritas: uerm in motibus ab Occidente in Orientem <expan abbr="cũ">cum</expan> int <lb/>proprij, oportet coniderare tempus, in quo <expan abbr="circumuertũtur">circumuertuntur</expan>, & ma <lb/>gnitudinem ambitus, & inde magnitudinem orbis, qui circumagi<lb/>tur, & horum trium facta comparatione dignocitur robur uirium <lb/>tellarum & uitarum qu mouent eas. </s>
<s id="id2769391">Ponatur ergo, ut uelim pro<lb/>portionem uit Saturni ad uitam Lun: erit ergo (ut docet Alphra <lb/><arrow.to.target n="marg573"/><lb/>ganus) Luna, cum et in longitudine propiore, altitudinem habens <lb/>109000 M.P. & cum et in longitudine longiore 208500, tota igitur <lb/>dimetiens 417000 M.P. mane 218000 M.P. </s>
<s id="id2769428">Igitur proportio olida<lb/>rum phrarum et uelut 72511713 ad 10360232, remanebit ergo <lb/>proportio orbis ad phram elementorum, ut 62151481 ad <lb/>10360232, & et excuplum ferm. </s>
<s id="id2769468">Rurus proportio dimetientis al<lb/>titudinis Saturni ad contentum et uelut 2011 ad 1440, & et prop <lb/>201 ad 114, quare 67 ad 38, quare phrarum ut 300000 ad 55000 <lb/>ferme. </s>
<s id="id2769501">Igitur fer ut 60 ad 11. Rurus proportio dimetientis ph<lb/>r Saturni ad dimetientem phr Lun et prop 313, & phra<lb/>rum olidarum 306 317 10. Perinde et. </s>
<s id="id2769555">Quia ergo proportio ph<lb/>r Saturni ad phram Lun et 30631710, & orbis Lun et 5/6 <lb/>olum phr u diuidemus 30631710 per 5/6, & exibit proportio <lb/>phr Saturni ad orbem Lun 36758052, at quia proportio o<lb/>lid phr Saturni ad contentum et ut 60 ad 11, erit phr ad <lb/>orbem, ut 60 ad 49 reiduum, diuidam ergo 36758052 per 60, exe<lb/>unt 612634, & ducam per 49, id et per 100, fit 61263400, & diuiden <lb/>do per 2, exit 30631700, detraho 612634, relinquitur proportio or<lb/>bis Saturni ad orbem Lun 30019066.</s></p><p type="margin">
<s id="id2769681"><margin.target id="marg572"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s id="id2769708"><margin.target id="marg573"/>D<emph type="italics"/>iff.<emph.end type="italics"/> 21.</s></p><p type="main">
<s id="id2769733">Iam uer circuitus Saturni ad circulum Lun, proportio et 313, <lb/>ut uium et, Lun autem tempus per ex ductum et 164 dies, Sa<lb/>turni 177 anni propemodum, qui unt dies 64649 diuide, duc <lb/>ergo 313 in 164, fiunt 51332. Idem ergo peragrat Luna in <lb/>51332 diebus, quod Saturnus in 64649, & et quo ad hoc agi
<pb xlink:href="015/01/185.jpg" pagenum="166"/>lior, ut ita dicam, quarta parte: at Saturnus, ut dictum et, mouet or<lb/>bem 30019066, ed lentis quinta parte, detrahe illam fiet robur Sa <lb/>turni in comparatione ad Lunam 24015253.</s></p><p type="main">
<s id="id2769814">Et tamen Luna multo agilior ob propinquitatem, & ob uarie<lb/>tatem luminis, & magnitudinem uperficiei. </s>
<s id="id2769829">Et etiam quod maius <lb/>et ob id quod defert ad nos uires omnium yderum, nihilominus <lb/>quo ad uires uix et comparatio.</s></p><p type="head">
<s id="id2769853">SCHOLIVM.<lb/><arrow.to.target n="marg574"/></s></p><p type="margin">
<s id="id2769869"><margin.target id="marg574"/>46</s></p><p type="main">
<s id="id2769883">Multum autem differt hc propoitio uperiore, nam in illa <lb/>quiuimus uim uitarum ex proportione ad ua corpora, qu <lb/>quodammodo et quodammodo, non hic autem exponimus uim <lb/>uitarum ex earum operatione. </s>
<s id="id2769920">Propterea ubij ciemus breuiter alti<lb/>tudinem proportiones in minore longitudine & maiori<lb/><arrow.to.target n="table19"/></s></p><table><table.target id="table19"/><row><cell>Luna</cell><cell>in minore altitudine</cell><cell>51</cell><cell>in maiore</cell><cell>64</cell></row><row><cell>Mercurij</cell><cell>in minore</cell><cell>64</cell><cell>in maiore</cell><cell>167</cell></row><row><cell>Veneris</cell><cell>in minore</cell><cell>167</cell><cell>in maiore</cell><cell>1120</cell></row><row><cell>Solis</cell><cell>in minore</cell><cell>1120</cell><cell>in maiore</cell><cell>1220</cell></row><row><cell>Martis</cell><cell>in minore</cell><cell>1220</cell><cell>in maiore</cell><cell>8876</cell></row><row><cell>Iouis</cell><cell>in minore</cell><cell>8876</cell><cell>in maiore</cell><cell>14405</cell></row><row><cell>Saturni</cell><cell>in minore</cell><cell>14405</cell><cell>in maiore</cell><cell>20110</cell></row></table><p type="main">
<s id="id2770058">Stellarum fixarum propior 20110 longior non habetur. </s>
<s id="id2770062">Et h <lb/>menur unt in comparatione ad emidiametrum terr. </s>
<s id="id2770086">Et iuxta <lb/>id quod potuit e cundum rationem haberi: nam demontratio ola <lb/>et de altitudinibus Solis & Lun, & eorum magnitudinibus </s></p><p type="main">
<s id="id2770119"><arrow.to.target n="marg575"/><lb/>Ptolemo in magna compoitione.</s></p><p type="margin">
<s id="id2770139"><margin.target id="marg575"/>L<emph type="italics"/>ib.<emph.end type="italics"/> 5. <emph type="italics"/>cap.<emph.end type="italics"/><lb/>14. 15. <emph type="italics"/>&<emph.end type="italics"/><lb/>16.</s></p><p type="main">
<s id="id2770188">Propoitio centeimaexageimaquarta.</s></p><p type="main">
<s id="id2770207">Syderum proportionem in magnitudine otendere.<lb/><arrow.to.target n="table20"/></s></p><table><table.target id="table20"/><row><cell>Luna ad terram comparata</cell><cell>1/39</cell></row><row><cell>Mercurij corpus</cell><cell>1/22000</cell></row><row><cell>Veneris</cell><cell>1/29</cell></row><row><cell>Solis corpus</cell><cell>166</cell></row><row><cell>Martis</cell><cell>15/8</cell></row><row><cell>Iouis</cell><cell>95</cell></row><row><cell>Saturni</cell><cell>91</cell></row></table><p type="main">
<s id="id2770281">Stellarum autem fixarum inignium unaquque etiam minima, i <lb/><arrow.to.target n="marg576"/><lb/>credendum et Alphragano, et centies maior tota terra, unde ca<lb/>nem necee et centies mille maiorem ee, et enim in eadem altitu <lb/>dine, & dimetiens decuplus dimetienti tellarum ecund magni<lb/>tudinis, quas ille inignes uocat: aliter Saturnus non tantus ee <lb/>poet, cum it minimus apectu.</s></p>
<pb xlink:href="015/01/186.jpg" pagenum="167"/><p type="margin">
<s id="id2770376"><margin.target id="marg576"/>D<emph type="italics"/>iff.<emph.end type="italics"/> 22.</s></p><p type="main">
<s id="id2770402">Propoitio centeimaexageimaquinta.</s></p><p type="main">
<s id="id2770420">Propoitionem motuum omnium <expan abbr="&longs;tellarũ">tellarum</expan> ad olem coniderare.</s></p><p type="main">
<s id="id2770448">Videtur Sol quai Rex in Clo, nam omnes orbes cum illius <lb/><arrow.to.target n="marg577"/><lb/>motu conueniunt, & uideturres admiratione digna his, qui non <lb/>nouerunt, quanta it concordia omnium rerum, de qua infr dice<lb/>mus. </s>
<s id="id2770483">Ergo Luna primum hoc habet, ut linea qualis motu Solis <lb/>emper media it inter lineam qualis motus Lun & loci maxim <lb/>inqualitatis motus eius, ubi cilicet tardisim mouetur, Veneris <lb/>autem & Mercurij ut motus quales idem emper int cum motu <lb/>quali, & locus cumloco ipius Solis ad unguem prterid quod <lb/>infr dicemus. </s>
<s id="id2770551">Trium uer <expan abbr="&longs;uperiorũ">uperiorum</expan> ratio ic <expan abbr="cõ&longs;tat">contat</expan> ad Solem ut <lb/>Prolemo <expan abbr="ob&longs;eruatũ">oberuatum</expan> et ex Hipparcho. </s>
<s id="id2770611">In omniretitutione cuiu<lb/>libet planet uperioris numerus <expan abbr="reuolutionũ">reuolutionum</expan> Solis qualis et nu<lb/>mero <expan abbr="re&longs;titutionũ">retitutionum</expan> planet <expan abbr="&longs;ecundũ">ecundum</expan> <expan abbr="motũ">motum</expan> qualitatis & inqualita <lb/>tis pariter acceptis. </s>
<s id="id2770692">Velut Saturnus in annis quinquaginta nouem <lb/>die una & horis decem octo quinquageies epties per motum in<lb/>qualem ad <expan abbr="ungu&etilde;">unguem</expan>, per qualem autem duabus reuolutionibus par <lb/>te inuper una & quadraginta quin que minutijs, qu repondent di<lb/>ei uni, & horis decem octo ex motu Solis, & ita bis Saturnus reuol <lb/>uitur ecundum motum qualitatis & quinquageies epties per <lb/>motum inqualem & imiliter. </s>
<s id="id2770766">Iupiter in annis 70, diebus trecen<lb/>tis exaginta, horis quatuor, exaginta quinque reuolutiones inqua <lb/>les perficiet & ex quales, deficientibus ex qualibus quatuor par<lb/>tibus & dextante quod et <expan abbr="quãtum">quantum</expan> peragraret Solin quatuor die<lb/>bus, & dextante diei ad perfectionem cilicet annorum eptuaginta <lb/>atque unius. </s>
<s id="id2770828">Martis quo que tella in annis eptuaginta nouem, & die<lb/>bus tribus & horis ferm quatuor triginta nouem facit inquali<lb/>tatis reuolutiones: qualitatis autem quadraginta duas, & inuper <lb/>partes tres cum extante, quas manifetum et peragrari Sole in <lb/>diebus tribus atque horis quatuor. </s>
<s id="id2770878">Veneris quo que ydus in octo an<lb/>nis deficientibus diebus duobus & quadrante, inqualitatis quin<lb/>que perficit reuolutiones, qualitatis autem tantundem ad un <expan abbr="gu&etilde;">guem</expan> <lb/>quantum Sol deficiente eadem parte eu diebus duobus & qua<lb/>drante. </s>
<s id="id2770920">Mercurij quo que tella in quadraginta ex annis & una die <lb/>& hora una ferm quadraginta ex ferm perficit reuolutiones <lb/>qualis motus & inuper gradum unum cum portione repondenti <lb/>portioni temporis, id et, hor ferm uni: in qualitatis autem cen<lb/>um quadraginta quin que. </s>
<s id="id2770975">Atque hc unt manifetisima et ut dixi ad<lb/>miranda unt, prterea alia minus generalia, aut minus manifeta <lb/>aut non tanti momenti qu conult prtermitto, non et. </s>
<s id="id2771021">n. </s>
<s id="id2771025">locus <lb/>hic do cendi artes ingulas ed olum ea tra ctandi qu ad argumen
<pb xlink:href="015/01/187.jpg" pagenum="168"/>tum pertinent. </s>
<s id="id2771052">Igitur ut ad rem redeam. </s>
<s id="id2771056">Solis cum octauo Orbe ea <lb/>ratio et, ut linea quam ille permeat eadem it quam qu fix tell, <lb/>non. </s>
<s id="id2771084">n. </s>
<s id="id2771088">ad eandem ditantiam & mente conceptam ab quinoctijs <lb/>decendentem ac quiditantem mouetur, ed ad eam ecundum <lb/>quam tell fix in octauo orbe mouentur in comparatione ad ecli<lb/>pticam uperioris orbis. </s>
<s id="id2771135">Porr de his atque huiumodi in Paralipo<lb/>menis diximus, ubi etiam docuimus quomodo ecundum duos cir <lb/><arrow.to.target n="marg578"/><lb/>culos, qui olum circa uum centrum mouentur, punctus datus per <lb/>petu in recta linea feratur.</s></p><p type="margin">
<s id="id2771177"><margin.target id="marg577"/>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s id="id2771204"><margin.target id="marg578"/>L<emph type="italics"/>ib.<emph.end type="italics"/> 14. <lb/><emph type="italics"/>cap.<emph.end type="italics"/> 7.</s></p><p type="main">
<s id="id2771241">Propoitio centeimaexageimaexta.</s></p><p type="main">
<s id="id2771262">Proportiones muicas uperpartientes in eas qu particula una <lb/>tantum abundant reducere.<lb/><arrow.to.target n="marg579"/></s></p><p type="margin">
<s id="id2771289"><margin.target id="marg579"/>C<emph type="italics"/>o<emph.end type="italics"/>_{m}.</s></p><p type="main">
<s id="id2771315">Ptolemi hoc inuentum fuit, ut & multa alia prclara: itaque ta<lb/>tuendum et, primum uoces quales non concentum efficere, quia <lb/>diuer non unt, qu autem diuer unt, nihilominus proportio<lb/>ne contant implicisima & multiplici, tales optimam efficiunt ar<lb/>moniam. </s>
<s id="id2771376">Eiumodi unt qu in dupla unt proportione, uocatur <lb/>autem diapaon. </s>
<s id="id2771397">1. quai omnia comprehendens non numero uo<lb/>cum uelut diapente & diatearon quatuor & quin que uo cibus. </s>
<s id="id2771420">In <lb/>diapao. </s>
<s id="id2771430">n. </s>
<s id="id2771434">omnia <expan abbr="cõprehendi">comprehendi</expan> uidentur. </s>
<s id="id2771448">1. omnes uo <expan abbr="cũ">cum</expan> differenti, <lb/><expan abbr="quanquã">quanquam</expan> ex octo <expan abbr="tantũ">tantum</expan> uo cibus contet. </s>
<s id="id2771484">Pt unt qu in <expan abbr="&qtilde;drupla">quadrupla</expan>, <lb/>unde bis diapaon, pot qu in tripla, nam propior et monadi eu <lb/>qualitati: ed non ade implex ut bis diapaon. </s>
<s id="id2771540">Vocant <expan abbr="aũt">aunt</expan> hanc <lb/>diapaon diapente: inde <expan abbr="&longs;ub&longs;equi&ttilde;">ubequitur</expan> octupla qu uix in uocib. </s>
<s id="id2771579">huma<lb/>nis habetur: <expan abbr="frequ&etilde;s">frequens</expan> in intrumentis, uo <expan abbr="ca&ttilde;que">caturque</expan> tris diapaon inde ex<lb/>cupla, eu bis diapaon diapente. </s>
<s id="id2771626">Quintupla <expan abbr="aũt">aunt</expan> minus <expan abbr="cõcors">concors</expan> et: <lb/>ed de hac inferius dicemus, atque de multiplicib. </s>
<s id="id2771657">dicta unto. </s>
<s id="id2771664">Sed de <lb/><expan abbr="cõ">com</expan> centu ex particula uperaddita exquialtera exquitertia atque alijs <lb/>nunc agendum. </s>
<s id="id2771690">Clarum et. </s>
<s id="id2771696">n. </s>
<s id="id2771701">has ee implicisimas. </s>
<s id="id2771714">Cum ergo du <lb/>pla proportio non magis posit diuidi qualibus interuallis atque<lb/>implicibus proportionibus qum in exquialteram & exquiter<lb/>tiam, uelutinter 4 & 2 interpoito 3. nam proportio 3 ad 2 et ex<lb/>quialtera, & 4 ad 3 exquitertia: nec melius potet diuidi, at exqui<lb/>alteram & exquitertiam quantumuis magnis numeris diuidere <lb/>non licebat melius aut commodius quam per exquioctauas: uelu<lb/>ti umpto numero 64 cui duplus et 128, inter medius 96 qui cum <lb/>64 exquialteram facit proportionem, qu uauisima et omni<lb/>um deductis multiplicibus, uo caturque diapente. </s>
<s id="id2771815">At qu et 128 ad <lb/>96 exquitertia et minuque ben onat per e, ed in acutioribus uo<lb/>cibus olum cum alijs ben onat, uelut cum diapente, perficiens <lb/>diapaon, interuallum, ergo inter 96 & 64 diuium per exquio cta
<pb xlink:href="015/01/188.jpg" pagenum="169"/>uas producit 72 et 81, <expan abbr="nã">nam</expan> 72 ad 64 et exquio <expan abbr="ctauũ">ctauum</expan>, icut 81 ad 72. uerm <lb/>id accidebat in <expan abbr="cõmodi">commodi</expan> quae 81 ad 64 <expan abbr="nullã">nullam</expan> habet <expan abbr="proportion&etilde;">proportionem</expan> <expan abbr="commodã">commodam</expan>, <lb/>& multominus 96 ad 81, quare uium et Ptolemo ut ubtracta mona <lb/>de <expan abbr="fier&etilde;t">fierent</expan> termini 64, 72, 80, & 96, proportio <expan abbr="aũt">aunt</expan> 80 ad 64 <expan abbr="cõ&longs;tituit">contituit</expan> exqui <lb/><expan abbr="quartã">quartam</expan> atque <expan abbr="ditonũ">ditonum</expan>, proportio quo que 96 ad 72 <expan abbr="&longs;exquitertiã">exquitertiam</expan> <expan abbr="&longs;emiditonũ">emiditonum</expan> que. <lb/></s>
<s id="id2772045">Rurus proportio 128 ad 64 <expan abbr="cõponi&ttilde;">componitur</expan> ex proportionib. </s>
<s id="id2772064">80 ad 64, <expan abbr="&qtilde;">quae</expan> <expan abbr="habe&ttilde;">habetur</expan> <lb/>pro ditono ut <expan abbr="dictũ">dictum</expan> et, & et exquiquarta proportio. </s>
<s id="id2772103">At 128 cum 80 et in <lb/>proportione uperpartiente tres quintas, <expan abbr="&qtilde;">quae</expan> <expan abbr="iterũ">iterum</expan> et conona. </s>
<s id="id2772137">Regula <expan abbr="e&mtilde;">emm</expan> <lb/>et quae ubi cononantia uo <expan abbr="cũ">cum</expan> <expan abbr="diuida&ttilde;">diuidatur</expan> in duas partes, <expan abbr="quarũ">quarum</expan> una it cono <lb/>nans, <expan abbr="reliquã">reliquam</expan> <expan abbr="etiã">etiam</expan> ee <expan abbr="con&longs;onant&etilde;">cononantem</expan>, at <expan abbr="nõ">non</expan> <expan abbr="cõuerti&ttilde;">conuertitur</expan>. </s>
<s id="id2772245">Spe. </s>
<s id="id2772252">n. </s>
<s id="id2772256">fit ut ex duab. <lb/></s>
<s id="id2772262">cononantibus dionans <expan abbr="cõpo&longs;itio">compoitio</expan> <expan abbr="oria&ttilde;">oriatur</expan>, uelut ex duplici <expan abbr="diap&etilde;te">diapente</expan>, aut <lb/><expan abbr="diap&etilde;te">diapente</expan> <expan abbr="cũ">cum</expan> ditono, ed ut ad <expan abbr="propo&longs;itũ">propoitum</expan> reuertar, alia diapaon et inter 80 <lb/>& 40, at inter 48 & 40 et emiditonus ut <expan abbr="o&longs;t&etilde;&longs;um">otenum</expan> et, uelut inter 96 & <lb/>80, nam inter 45 & 40 et proportio exquioctaua, inter 48 <expan abbr="aũt">aunt</expan> & 45 ex<lb/>quiquinta decima, <expan abbr="igi&ttilde;">igitur</expan> ex regula data proportio 80 ad 48 <expan abbr="&qtilde;">quae</expan> et uperbi<lb/>partiens tertias eu olida <expan abbr="cũ">cum</expan> bee eu exta maior erit <expan abbr="cõ&longs;onans">cononans</expan>. </s>
<s id="id2772469">Iam er <lb/>go uidemus detractione aut additione exquio ctuageim, concinnas <lb/>reddi uulgatiores armonias: <expan abbr="tertiã">tertiam</expan> utran que <expan abbr="maior&etilde;">maiorem</expan> cilicet & <expan abbr="minor&etilde;">minorem</expan>, ac <lb/>rurus <expan abbr="&longs;extã">extam</expan> <expan abbr="maior&etilde;">maiorem</expan> atque minore <expan abbr="&qtilde;">quae</expan> in minoribus numeris cilicet mo<lb/>nade ad octo poit unt. </s>
<s id="id2772567">Vides prterea <expan abbr="&longs;emiditonũ">emiditonum</expan> in exquiquinta <lb/><arrow.to.target n="table21"/><lb/><expan abbr="cõ&longs;tare">contare</expan>: ade ut enario infra nihil inutile <lb/><figure id="id.015.01.188.1.jpg" xlink:href="015/01/188/1.jpg"/>reddatur. </s>
<s id="id2772628">Diatearon <expan abbr="aũt">aunt</expan> cum primum di <lb/>uidi potet, i ecus diuidatur <08> in <expan abbr="ditonũ">ditonum</expan> <lb/>& <expan abbr="&longs;emitoniũ">emitonium</expan>, aut in emiditonum & <expan abbr="tonũ">tonum</expan>, <lb/>cilicet in duo <expan abbr="tantũ">tantum</expan> interualla, non <expan abbr="cõmo-dius">commo<lb/>dius</expan> <expan abbr="quã">quam</expan> inter octo & eptem & ex diuidi <lb/>potet. </s>
<s id="id2772738">Cum ergo octo ad <expan abbr="&longs;ept&etilde;">eptem</expan> diona it, <lb/>quippe nimis remota et hc proportio en <lb/>u humano: <expan abbr="quamobr&etilde;">quamobrem</expan> ex regula data, ne<lb/>que proportio <expan abbr="&longs;ept&etilde;">eptem</expan> ad ex. </s>
<s id="id2772805">Sed dubitabis <lb/>merit, quia <expan abbr="cũ">cum</expan> diatearon diuidatur <expan abbr="bifa-riã">bifa<lb/>riam</expan>, in <expan abbr="ditonũ">ditonum</expan> & <expan abbr="&longs;emitoniũ">emitonium</expan>, ac rurus in <expan abbr="&longs;e-miditonũ">e<lb/>miditonum</expan> & <expan abbr="tonũ">tonum</expan>, quarum altera <expan abbr="cõ&longs;onans">cononans</expan> et, reliqua <expan abbr="nõ">non</expan>. </s>
<s id="id2772910"><expan abbr="Vide&ttilde;">Videtur</expan> ergo <lb/>infirmari regula illa, quae cononantia diuia i una pars <expan abbr="cõ&longs;onet">cononet</expan>, alia non <lb/>posit ee dionans, <expan abbr="nã">nam</expan> contat <expan abbr="coniũ">conium</expan> & <expan abbr="&longs;emitoniũ">emitonium</expan> tam per e quam in <lb/><expan abbr="cõpo&longs;itione">compoitione</expan> dionare: & <expan abbr="nõ">non</expan> <expan abbr="parũ">parum</expan> ed acerb. </s>
<s id="id2773036"><expan abbr="Verũ">Verum</expan> repondeo diatea <lb/>ron, ut dixi, numerari inter ambiguas coniugationes, quatenus <expan abbr="e&mtilde;">emm</expan> per <lb/>fe et, dionans et: at que ic in <expan abbr="con&longs;onant&etilde;">cononantem</expan> & dionantem diuidi potet: <lb/>quatenus <expan abbr="aũt">aunt</expan> pars et diapaon <expan abbr="cõ&longs;onans">cononans</expan> in acutis: quan <08> <expan abbr="etiã">etiam</expan> adiecta <lb/>ditono aut emiditono upr efficiat <expan abbr="&longs;extã">extam</expan> maiorem aut <expan abbr="minor&etilde;">minorem</expan> parum <lb/>ben onantes. </s>
<s id="id2773182">At quintupla proportio ut ab initio propoitum et, <expan abbr="cõ&longs;tat">contat</expan> <lb/>bis diapaon, & exquiquarta, ut plan <expan abbr="manife&longs;tũ">manifetum</expan> et: exquiquarta <expan abbr="aũt">aunt</expan>
<pb xlink:href="015/01/189.jpg" pagenum="170"/>ditonus: bis diapaon <expan abbr="aũt">aunt</expan> quindecim uo cibus. </s>
<s id="id2773268">Omnes igitur decem, & <lb/><expan abbr="&longs;ept&etilde;">eptem</expan> uoces, <expan abbr="&qtilde;">quae</expan> exdecim interuallis <expan abbr="di&longs;tinguun&ttilde;">ditinguuntur</expan>, cononantes unt: & ex <lb/>genere ditoni, & exquiquart, ed paulo minus ben <expan abbr="&longs;onãt">onant</expan> <08> ditonus <lb/>ipe. </s>
<s id="id2773348">Igitur <expan abbr="quintuplã">quintuplam</expan> multiplicem ad ex <expan abbr="quiquartã">quiquartam</expan> reduximus. </s>
<s id="id2773372">Verum <lb/>ut otenum et & decimaeptima, <expan abbr="&qtilde;">quae</expan> bis diapaon <expan abbr="cõ&longs;tat">contat</expan>, & emiditono <lb/>ben onat, hc <expan abbr="aũt">aunt</expan> inter non aginta ex & uiginti: quadrupla <expan abbr="igi&ttilde;">igitur</expan> et & <lb/>uperquadripartiens quintas. </s>
<s id="id2773456">Diapaon quo que cum exta maiore & mi<lb/>nore eandem habentrationem quam 16 ad 5, & 10 ad 3, triplam utranque, <lb/>ed altera exquiquinta, altera exquitertia: bis diapaon uer <expan abbr="cũ">cum</expan> eidem <lb/>ut uiginti ad tria, & 32 ad quin que excupla utraque: ed altera uperbipar<lb/>tiens tertias, altera quintas. </s>
<s id="id2773514"><expan abbr="Manife&longs;tũ">Manifetum</expan> et igitur hanc diuiionem <expan abbr="nõ">non</expan> o<lb/>lum concinnam magis ee & uauem ed omnem <expan abbr="tonorũ">tonorum</expan> & emitonio<lb/>rum <expan abbr="nece&longs;sitat&etilde;">necesitatem</expan> effugere. </s>
<s id="id2773590">Qud uer in caua fuit ut toni & emitonia <lb/>in uu eent, id et, quoniam in <expan abbr="di&longs;c&etilde;do">dicendo</expan> necee et eandem eruari ratio<lb/>nem in <expan abbr="crementorũ">crementorum</expan>, ne que arithmeticam ed <expan abbr="geometricã">geometricam</expan>. </s>
<s id="id2773667">Ide <expan abbr="a&longs;c&etilde;&longs;us">acenus</expan> per <lb/>tonos & emitonia <expan abbr="cõmodus">commodus</expan> fuit, nam duplicem <expan abbr="&longs;olũ">olum</expan> differentiam pue <lb/>ri uu aequi coguntur. </s>
<s id="id2773730">At uer poterat & per exquiextam diuidi dia <lb/>tearon, ut inter triginta ex & quadraginta nouem interpoitis 42, ue<lb/>rm triplex <expan abbr="&longs;equeba&ttilde;">equebatur</expan> in <expan abbr="cõueniens">conueniens</expan>: primum ut diatearon ad amusim <lb/>non eruaretur, ed incidebat in cacophoniam, addita quadrageima o<lb/>ctaua parte: deficiente <expan abbr="aũt">aunt</expan> in duabus exquieptimis numeris eu propor<lb/>tione exquitertia: ut inter 49 & 64 loco 48 & 64, uelut <expan abbr="etiã">etiam</expan> inter 48 ad <lb/>36, additaigitur monade in termino medio utrin que fit dionantia. </s>
<s id="id2773849">Se<lb/>cundum inconueniens, et quae ic diuidente non eruabatur ratio exqui<lb/>quart & exquiquint eu ditoni & emiditoni, qu uoces ben o<lb/>nant. </s>
<s id="id2773900">Tertium inconueniens erat, qud hcratio diuidendi diapentes <lb/>minim atisfaciebat, uelutinter 324 & 216. Interponere enim necee <lb/>erat 252 & 294, unde incongrua rurus erat diuiio. </s>
<s id="id2773934">His tot cauis cum <lb/>proportiones maiores non fatisfacerent ut exqui quinta qu diatea<lb/>ron nullo modo qualiter diuidere potet, & in diapente deficit exqui <lb/>uigeimaquarta, ut inter 25 & 36, coacti unt cum nec exquiexta nec <lb/>exquieptima idone eent ad exquio ctauam confugere.</s></p><table><table.target id="table21"/><row><cell>Diapaon</cell><cell>2</cell><cell>1</cell></row><row><cell>Bis diapaon</cell><cell>4</cell><cell>1</cell></row><row><cell>Diapaon diapente</cell><cell>3</cell><cell>1</cell></row><row><cell>Tris diapaon</cell><cell>8</cell><cell>1</cell></row><row><cell>Bis diapaon <expan abbr="diap&etilde;te">diapente</expan></cell><cell>6</cell><cell>1</cell></row><row><cell>Hmiolia</cell><cell>3</cell><cell>2</cell></row><row><cell>Hmitrita</cell><cell>4</cell><cell>3</cell></row><row><cell>Ditonus</cell><cell>5</cell><cell>4</cell></row><row><cell>Semiditonus</cell><cell>6</cell><cell>5</cell></row><row><cell>Sexta minor</cell><cell>8</cell><cell>5</cell></row><row><cell>Sexta maior</cell><cell>5</cell><cell>3</cell></row><row><cell>Bis diapaon ditonus</cell><cell>5</cell><cell>1</cell></row></table><p type="main">
<s id="id2774161">Et & alia diuiio toni in emitonia, <expan abbr="&qtilde;">quae</expan> et uaria <expan abbr="pon&etilde;do">ponendo</expan> <expan abbr="tonũ">tonum</expan> inter 18 <lb/>& 16, media uox et 17 emitonium maius inter 17 & 16, ed minus inter <lb/>18 & 17, <expan abbr="quorũ">quorum</expan> differentia et 1/288. Hic ubit admiratio quomodo <expan abbr="&longs;emi-toniũ">emi<lb/>tonium</expan> minus <expan abbr="apte&ttilde;">aptetur</expan> tam grat in ymphonijs, maius <expan abbr="aũt">aunt</expan> <expan abbr="nequaquã">nequaquam</expan>. </s>
<s id="id2774278">Ptole <lb/>mus hoc negaret, quia exquiquinta eu emiditonus <expan abbr="cõ&longs;tat">contat</expan> tono inte<lb/>gro, qui et inter 90 & 80, & emitonio <expan abbr="plu&longs;quã">pluquam</expan> maiore quod et inter <lb/>96 & 90, & et exquiquinta decima: <expan abbr="&qtilde;">quae</expan> maior et tono maiore 1/255. Pro<lb/>pterea dicemus cauam ee quae poito emiditono inter 81 & 96, id et, <lb/>27 & 32 ublato tono, id et, 234 & 216, remanebit 13 differentia 256 ad <lb/>243, eu qualis et 96 ad 91 & 1/8 qu et ut 768 ad 729 et redit ad <expan abbr="id&etilde;">idem</expan>, cili
<pb xlink:href="015/01/190.jpg" pagenum="171"/>cet, ut 256 ad 243, 13 autem et paulo plus decimanona, ergo multo mi<lb/>nus emitonio minore. </s>
<s id="id2774431">ecundum <expan abbr="m&etilde;tem">mentem</expan> ergo Ptolemi, poito tono <lb/>inter 135, & 120, & emitonio maiore inter 128 & 120 remanebit emito<lb/>nium minus ferm inter 19 & 18, id et, 133 & 126, qu proportio differt <lb/> 135 & 138. Si quis autem bene animaduertat, exquioctuageima illa <lb/>adimitur, ex tono & additur emitonio minori, & hc et caua qud <lb/>emitonium maius Ptolemi it concinnum, quia additur tonis imper <lb/>fectis. </s>
<s id="id2774517">Dimidium autem emitonij minoris et inter 36 & 35, & uocatur <lb/><expan abbr="cõma">comma</expan>: & et minus & maius: maius et inter 35 & 34, rurus <expan abbr="cõma">comma</expan> mi<lb/>nus diuiditur in duas diees, minorem, qu et inter 72 & 71, & maio<lb/>rem, qu et inter 71 & 70, & ide manet difficultas quomodo intenta <lb/>uoce per dieim fiat melior cononantia? </s>
<s id="id2774594">nam de remisione poemus <lb/>dicere qud accipitur loco exquio ctuageim: ed in exquioctuage<lb/>ima remittitur de tono ecundum mentem Ptolemi, in diei intendi<lb/>tur emitonium minus, icut otendit experimentum, ed foran conue <lb/>niunt quia intentio emitonij minoris deducit emiditonum ad exqui <lb/>quintam: et enim differentia emitonij minoris intenti hoc modo ad <lb/>emitonium minus, ut 136 ad 135: ed hoc et long minus exquioctua <lb/>geima, unum at et, hanc ee ultimam diuiionem toni in octo par<lb/>tes, & ut in diatonico toni dominantur, ita in chromatico emitonia in <lb/>enarmonico diees, ed diees fugitando (utita dicam) ac aures uelli<lb/>cando, mirum in modum oblectant audientes: uelut toni tando, un<lb/>de etiam nomen, emitonia medium modum obtinent.</s></p><p type="main">
<s id="id2774754">Tertium genus proportionis (omitto mod <expan abbr="diui&longs;ion&etilde;">diuiionem</expan> temporum <lb/>binarij, ternarij, quinarij, qui ultimus et eorum quos enus recipiat, <lb/>nam eptenarius propinquior et binarij diuiioni ob octonarium, & <lb/>modos illos atis notos Doricum, Lydium & Phrigium, ac eiumodi) <lb/>et Ptolemi: rurus qui cum uideret depectam futuram muic con<lb/>templationem, conatus et illius aliquod ingulare emolumentum <lb/>otendere, quemadmodum fecit & in libro de Prdictionibus, exiti<lb/>mans ni illos compouiet ueluti prmium otendentes tanti laboris <lb/>quantus necearius uideretur ad intellectum librorum Magn com<lb/>poitionis, futurum ee, ut hi negligerentur, ergo & hoc in muic li<lb/>bris otendere molitus et, cilicet, prclarum ee <expan abbr="aliqu&etilde;">aliquem</expan> huius <expan abbr="cõtem-plationis">contem<lb/>plationis</expan> finem, quod <expan abbr="utinã">utinam</expan> non feciet, ne illud uer de eo dici poet:</s></p><p type="main">
<s id="id2774963">Non omnia poumus omnes.</s></p><p type="main">
<s id="id2774978">Virum enim hunc upra omnem humani ingenij <expan abbr="metã">metam</expan> fuie <expan abbr="nõ">non</expan> nega<lb/>mus: ed hanc partem quam hic agit, ade infeliciter tractat, ut malim <lb/>credere <expan abbr="totũ">totum</expan> illum tertium <expan abbr="librũ">librum</expan> fuie ab aliquo alio <expan abbr="adiectũ">adiectum</expan>. </s>
<s id="id2775048">Etenim <lb/>quid turpius apienti homini <08> imitari uulgares illos? </s>
<s id="id2775060"><expan abbr="&longs;ept&etilde;">eptem</expan> planet, <lb/>eptem mundi miracula, <expan abbr="&longs;ept&etilde;">eptem</expan> artes liberales: quid enim imilitudo nu
<pb xlink:href="015/01/191.jpg" pagenum="172"/>meri iuuare potet, aut qum afferre utilitatem? </s>
<s id="id2775110">nimis cert in <expan abbr="dignũ">dignum</expan> et <lb/>uti <expan abbr="argum&etilde;to">argumento</expan> imilitudine umpto: tum maxim ade leui. </s>
<s id="id2775154">Sed quo<lb/>niam contat omnia qu in mundo unt ordine coniuncta ee, & ne<lb/>cesitate uinciri, ide cm finis ipe uerus it, non tam debemus Ptole<lb/>mum damnare, quae non probauerit, qum laudare, quod <expan abbr="ueritat&etilde;">ueritatem</expan> ine <lb/>ratione it aectus. </s>
<s id="id2775225">Spe enim accidit huiumodi uiris ade prtan<lb/>tibus ut ueritas detegatur, quam cm illi, ut mos et <expan abbr="hominũ">hominum</expan>, rationi<lb/>bus adornare nituntur, trangredientes metam muneris, in aburda & <lb/>ineptias <expan abbr="incidũt">incidunt</expan>. </s>
<s id="id2775283">Ergo id mod declarare aggrediar, upponens quae ue<lb/>rum et, cilicet hanc muicam <expan abbr="concinnitat&etilde;">concinnitatem</expan> cum diuinis <expan abbr="connexã">connexam</expan> ee, <lb/>& ab illis originem ducere. </s>
<s id="id2775331">Verm dubium et, an oni propter nume <lb/>ros iucundi int, an propter aliud? </s>
<s id="id2775350">& i propter aliud, cur ergo numeri <lb/>ad hoc unt necearij? </s>
<s id="id2775367">& cur oberuare eos oportet ne ab illorum ordi <lb/>ne diiungi posint? </s>
<s id="id2775383">Hoc <expan abbr="aũt">aunt</expan> perfacil <expan abbr="intelligi&ttilde;">intelligitur</expan>, & nobis alis decla<lb/>ratum et, cilicet delectare nos, qu percipiuntur quque ratione facta <lb/>uidentur, <expan abbr="quoniã">quoniam</expan> in his natur uis relucet & imago uniueri, ergo dele <lb/>ctant nos, quoniam natur ordine nos contamus. </s>
<s id="id2775455">Illud difficilius lon <lb/>g d <expan abbr="tam&etilde;">tamen</expan> diligenti oberuatione <expan abbr="dignũ">dignum</expan> uidetur, cilicet, quonam pa <lb/>cto harmonia cum rebus cletibus aut humanis <expan abbr="cõiuncta">coniuncta</expan> it. </s>
<s id="id2775511">Foran <lb/>& illud ab re non eet intelligere, cur nullum animal prter hominem <lb/>capax it harmoni? </s>
<s id="id2775537">an foran <expan abbr="quoniã">quoniam</expan> olus homo ratione participet, <lb/>& ob id olus gaudet ratione? </s>
<s id="id2775562">ordinata <expan abbr="aũt">aunt</expan> ratione <expan abbr="cõ&longs;tant">contant</expan> aut ola aut <lb/>maxim, numerus autem quid aliud et qum ordinis <expan abbr="&longs;eparatorũ">eparatorum</expan> ima<lb/>go. </s>
<s id="id2775619">Porr hc accipienda unt ex his qu enibus deprehenduntur, <lb/>qualia unt quae animus mouetur & uarios affectus in duit iuxta harmo<lb/>ni diueritatem ltiti, trititi, impetus, remisionis, timoris, pei, ira<lb/>cundi, & commierationis. </s>
<s id="id2775686">Nos enim maxim octo affectus mouent <lb/>muic modulationes. </s>
<s id="id2775702">Secundum quid autem mouent? </s>
<s id="id2775706">uel quia con<lb/>on aut dion, uel quia concitat aut tard, uel quod maius et quae<lb/>tendant in acutum ad alacritatem, uel in grauem deinant & remium <lb/>onum ad <expan abbr="cõmi&longs;erationem">commierationem</expan>, & lachrymas, aut etiam ex modo tetrachor <lb/>dorum. </s>
<s id="id2775769">Illud an non obcurum et, <expan abbr="animã">animam</expan> cum ono maxim ee con <lb/><expan abbr="iunctã">iunctam</expan>, nam neque odoribus ut odores unt, neque aporibus, aut his qu <lb/>tanguntur licet plurimum delectent, aut etiam ldant, anima mouetur <lb/>ad affectus, licet, ut dixi, magis homo delectetur, aut trititia afficiatur <lb/>quemadmodum ex onorum uaria natura, quod etiam in moris Ta <lb/>rantula (arane genus et) deprehenditur. </s>
<s id="id2775854">Quinim nec luce nec co <lb/>loribus aut pictura, nii ut hc ad memoriam <expan abbr="reuocãt">reuocant</expan> ea, propter qu <lb/>ad hilaritatem aut trititiam uel iram, uel commierationem mouemur. <lb/></s>
<s id="id2775898">Vnde <expan abbr="quo&longs;dã">quodam</expan> reges ferunt iniurias acceptas iusie depingi in aula ne <lb/>poent obliuici, at long plures <expan abbr="curarũt">curarunt</expan>, ut potius <expan abbr="eorũ">eorum</expan> facta egregia
<pb xlink:href="015/01/192.jpg" pagenum="173"/>pingerentur continuata per memoriam uoluptate, quam dum illa ge <lb/>rent, <expan abbr="cõceperant">conceperant</expan>: nihilominus, neque color ipe, nec lux aut pectaculum <lb/>uel imagines pount ade mouere animi affectus, uel onus. </s>
<s id="id2775998">Nam <lb/>duo in uniuerum ex uiu ad animi affectus mouendos habentur, tene <lb/>br ad trititiam & metum, pictura regionum <expan abbr="amœnarũ">amnarum</expan> ad iucundita <lb/>tem, ed <expan abbr="irã">iram</expan> qu moueant pictur alacritateme aut <expan abbr="cõmi&longs;erationem">commierationem</expan>, <lb/>non habemus. </s>
<s id="id2776076">Videtur ergo ob hc onus ipe magis anim intimus <lb/><08> ullum aliud enile. </s>
<s id="id2776100">Quod i odoratus et in <expan abbr="app&etilde;dicibus">appendicibus</expan> cerebri, ui <lb/>us in pupilla oculi, gutus in lingu neruis, ueriimile et magis inti<lb/>mum ee auditum, cilicet in cerebro ipo, atque ob id magis ab illo mo<lb/>ueri animam. </s>
<s id="id2776155">Neque <expan abbr="e&mtilde;">emm</expan> in <expan abbr="a&etilde;re">aerre</expan> concepto concauitatibus auris, qui no <lb/>tri pars non et: neque tympano, cm uperflua fuiet cauitas interior <lb/>omnis: neque enim inter pupillam & cerebrum pars ulla cernitur ad ui<lb/>um adiuuandum idonea: ed olus ufficit conenus pupill cum cere <lb/>bro: nam ad nos per piritus deffertur imago, non <expan abbr="e&mtilde;">emm</expan> uius eet unus, <lb/>nec in uno tempore fieret, ed ueluti <expan abbr="&longs;ecũdo">ecundo</expan> peculo & decimo imul, <lb/>& eodem tempore reflectitur imago, ut primo ita enus uius ex pu<lb/>pilla in cerebro & in corde & anima imul relucet. </s>
<s id="id2776298">At ergo non potuit <lb/>in tympano uel neruo deniore fieri auditus, ed in cerebro ipo, ob d <lb/>magis moueret affectus. </s>
<s id="id2776320">Sed & magis incorporeus et onus, ut qui <lb/>intrumentum proprium non afficiat, nii cum immoderatus fuerit, at <lb/>omnis color, omnis lux oculum afficit, ac, ut ita dicam, tingit, neque uc<lb/>cesiones illas ob id ade minutas oculus percipere potet ut auris, <lb/>ed coinquinatur, ut ita dicam, priorum obiectorum reliquijs atque ima <lb/>ginibus. </s>
<s id="id2776368">Vt in uniuerum contet puriorem ee auditus enum etiam <lb/>anim notr propiorem qum uium.</s></p><p type="main">
<s id="id2776409">Quibus contit