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<?xml version="1.0"?>
<!DOCTYPE archimedes SYSTEM "../dtd/archimedes.dtd" ><archimedes> <info>
<author>Cardano, Girolamo</author>
<title>Opus novum de proportionibus</title>
<date>1570</date>
<place>Basel</place>
<translator></translator>
<lang>la</lang>
<cvs_file>carda_propo_01_la_1570</cvs_file>
<cvs_version></cvs_version>
<locator>0000000015.xml</locator>
</info> <text> <front> </front> <body> <chap> <pb/><pb/><pb/><p type="head">
<s>HIERONYMI <lb/>CARDANI MEDIO <lb/>LANENSIS, CIVISQV'E BONO­<lb/>NIENSIS, PHILOSOPHI, MEDICI ET <lb/>Mathematici clari&longs;simi,</s></p><p type="head">
<s>OPVS NOVVM DE <lb/>PROPORTIONIBVS NVMERORVM, MO <lb/>TVVM, PONDERVM, SONORVM, ALIARVMQV'E RERVM <lb/>men&longs;urandarum, non &longs;olùm Geometrico more &longs;tabilitum, &longs;ed etiam <lb/>uarijs experimentis & ob&longs;eruationibus rerum in natura, &longs;olerti <lb/>demon&longs;tratione illu&longs;tratum, ad multiplices u&longs;us ac­<lb/>commodatum, & in Vlibros dige&longs;tum.</s></p><p type="head">
<s>PRAETEREA.</s></p><p type="head">
<s>ARTIS MAGNÆ, SIVE DE REGVLIS <lb/>ALGEBRAICIS, LIBER VNVS, ABSTRVSISSIMVS <lb/>& inexhau&longs;tus plane totius Arithmeticæ the&longs;aurus, ab <lb/>authore recens multis in locis recogni­<lb/>tus & auctus.</s></p><p type="head">
<s>ITEM.</s></p><p type="head">
<s>DE ALIZA REGVLA LIBER, HOC EST, ALGEBRAICAE <lb/>logi&longs;ticæ &longs;uæ, numeros recondita numerandi &longs;ubtilitate, &longs;ecundum Geo­<lb/>metricas quantitates inquirentis, nece&longs;&longs;aria Coronis, <lb/>nunc demum in lucem edita.</s></p><p type="head">
<s>O<emph type="italics"/>pus<emph.end type="italics"/> P<emph type="italics"/>hy&longs;icis &<emph.end type="italics"/> M<emph type="italics"/>athematicis imprimis <lb/>utile & nece&longs;&longs;arium.<emph.end type="italics"/></s></p><figure></figure><p type="head">
<s>Cum Cæ&longs;. </s>
<s>Maie&longs;t. </s>
<s>Gratia & Priuilegio.</s></p><p type="head">
<s>BASILEÆ.</s></p><pb/><p type="head">
<s>IN LIBRVM DE <lb/>PROPORTIONIBVS HIERONYMI <lb/>CARDANI MEDIOLANENSIS, CIVISQV'E <lb/>Bononien&longs;is, Medici, Præfatio ad M. A. </s>
<s>Amulium <lb/>Venetum Card. </s>
<s>Illu&longs;tri&longs;simum.</s></p><p type="main">
<s>Bene Dictum e&longs;t meo iudicio à Platone M. <lb/>A. </s>
<s>Amuli optime, beatas fore Re&longs;pub. </s>
<s>&longs;i uel <lb/>illarum domini &longs;apientiæ amatores e&longs;&longs;ent, <lb/>aut qui &longs;apientiæ e&longs;&longs;ent amatores domina­<lb/>rentur, hoc ip&longs;um clarè intelligens, &longs;tudio &longs;a <lb/>pientiæ nihil e&longs;&longs;e utilius humano generi: <lb/>quo &longs;imul & pietas, & iu&longs;titia, & mutuus <lb/>amor hominum inter &longs;e & eorum commo­<lb/>da continerentur. </s>
<s>Nempe hi&longs;ce quatuor tota no&longs;tra felicitas com­<lb/>prehenditur. </s>
<s>Si quidem pietate in Deos nihil ni&longs;i &longs;anctum, & pu­<lb/>rum, & illu&longs;tre &longs;apimus: hoc ip&longs;o primum quod &longs;upra nos e&longs;t, intel­<lb/>ligimus, Deos ueneramur, gratias agimus, timor cum ueneratione <lb/>no&longs;tros animos &longs;ubit, & de futura uita cogitamus, hæc ip&longs;a morta­<lb/>lia &longs;i non negligentes &longs;altem paruifacientes. </s>
<s>Iu&longs;titiam autem adeò <lb/>nece&longs;&longs;ariam humano generi e&longs;&longs;e &longs;cimus, ut &longs;ine illa neque e&longs;&longs;e, nedum <lb/>benè e&longs;&longs;e po&longs;símus, ut neque latronum cœtus ab&longs;que ea diu &longs;tare po&longs;­<lb/>&longs;int. </s>
<s>Porrò quid dicam de concordia, & mutua hominum beneuo­<lb/>lentia, in quibus omnis uit&ecedil; human&ecedil; dulcedo repo&longs;ita e&longs;t: nec quis <lb/>&longs;u&longs;tineat uiuere, qui &longs;e omnibus odio&longs;um e&longs;&longs;e &longs;entiat. </s>
<s>His ip&longs;is fi­<lb/>lios in &longs;pem alimus, parentes fouemus, fratres tuemur, & adiuua­<lb/>mus, amicis opitulamur, cum hominibus hilarem & iucundam ui­<lb/>tam ducimus. </s>
<s>Si quis &longs;erpentem in lecto haberet, nunquam &longs;om­<lb/>num caperet: ita nihil mole&longs;tius e&longs;t in hac uita, quam e&longs;&longs;e cum quo <lb/>nolis, & priuari con&longs;uetudine eorum cum quibus maximè uiuere <lb/>cupias. </s>
<s>Quid enim habent Principes præcipuum cum tota illa po­<lb/>tentia quam habent, ni&longs;i hoc unum, quod &longs;uis quos amant bene fa­<lb/>cere po&longs;sint: nam reliqua omnia exerceri, uenari, edere, bibere, dor­<lb/>mire, iter agere, loca amæna inui&longs;ere multis alijs conce&longs;&longs;um e&longs;t, ma­<lb/>ioreque commodo qui in uita priuata degunt. </s>
<s>Si ergo principatum <lb/>cum tot laboribus, curis, periculis, & meritò omnes appetunt: nec <lb/>e&longs;t in eo quicquam præcipuum præter hoc, cui dubium e&longs;t quin <lb/>hoc non &longs;it &longs;ummum huius uitæ hominibus bonum? </s>
<s>propter cu­<lb/>ius uel dubiam &longs;pem eorum, quæ habent obliti mortales pericli­<lb/>tantur. </s>
<s>Succedunt inde tot commoda, non &longs;olum utilia, &longs;ed pleraque<pb/>etiam nece&longs;&longs;aria, quæ nos &longs;apientia docet: huiu&longs;modi ergo omnia <lb/>cùm libris contineantur, meritò optimus qui&longs;que librorum bono­<lb/>rum perpetuitati atque in columitati fauere debet. </s>
<s>C. </s>
<s>Caligulam exe­<lb/>cramur &longs;olum ob id quod Vergilij, & T. </s>
<s>Liuij &longs;cripta delere cogi­<lb/>tauerit. </s>
<s>Quid facturi e&longs;&longs;emus, &longs;i feci&longs;&longs;et quod cogitauerat? </s>
<s>E&longs;t in &longs;a­<lb/>pientum monumentis bonum &longs;ine malo, mens &longs;ine corporea labe: <lb/>Virtutes ab&longs;que uitijs, gratiæ & iucunditas &longs;ine &longs;orde, & immundi­<lb/>tia, uoluptas &longs;ine dolore, conuer&longs;atio ab&longs;que tædio, delitiæ ab&longs;que mi&longs;e <lb/>ria nuda, omnia bona præ&longs;tant, atque laudabilia ab omnibus morta­<lb/>litatis exuuijs libera, tantum commodi afferunt libri. </s>
<s>Sed & in eo­<lb/>rum electione ac &longs;tudijs modus, ac medio critas quædam &longs;eruanda <lb/>e&longs;t, quæ &longs;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/>ob&longs;curam e&longs;&longs;e fatentur, ego difficillimam puto undique, & magis for <lb/>&longs;an ubi non exi&longs;timamus. </s>
<s>Vnde plures decidere uidemus magnis <lb/>cum auxilijs, & euidenti &longs;pe: quid aliud e&longs;t in cau&longs;a quàm ignota <lb/>men&longs;ura rerum? </s>
<s>quam tamen plerique tenere &longs;e putant. </s>
<s>Ergo, cùm <lb/>&longs;ummum bonum in hac men&longs;ura &longs;itum e&longs;&longs;e cernerem, ut clarè o&longs;ten <lb/>dunt mu&longs;icæ uoces, quæ non ni&longs;i indiuiduo (ut ita dicam) &longs;pacio <lb/>&longs;eu loco &longs;tare po&longs;&longs;unt, ita & in figuris picturarum & &longs;tatuarum, & <lb/>diebus decretorijs, & negocijs ciuilibus oper&ecedil;precium me factu­<lb/>rum exi&longs;timaui, &longs;i omnia hæc quæ latè patebant breuiter in unum <lb/>redegi&longs;&longs;em, <expan abbr="nõ">non</expan> tantum ne lectorem tædio afficerem, quàm ut quòd <lb/>aliàs do cui, breuibus tractationibus, & plura continerentur, & faci <lb/>lius docerentur. </s>
<s>Cum uerò bona fortuna quædam effeci&longs;&longs;et, ut tibi <lb/>libellum dedica&longs;&longs;em de Prouidentia ex con&longs;titutione temporum, <lb/>longe meliore occa&longs;ione nominis tui typographi obliti &longs;int, indi­<lb/>gnum fore putaui, ut non ærea (quemadmodum cum Glauco Dio <lb/>medes) cum aureis commutarem. </s>
<s>Itaque infinitis licet circumuentus <lb/>negocijs totus huic operæ in cubui, atque adeò ut præter &longs;pem unius <lb/>anni penè &longs;pacio liber ab&longs;olueretur. </s>
<s>Qui cum tibi (ut dixi) iam iurè <lb/>deberetur, eò tamen magis dedicandum putaui, quod non ego &longs;o­<lb/>lum quanquam id maximè, &longs;ed communis con&longs;en&longs;us ho­<lb/>minum exi&longs;timet, te &longs;ingulari uirtute omnibus <lb/>&longs;tudio&longs;is plurimum fauere, <lb/>Vale.</s></p><pb/><p type="head">
<s>TABVLA PRO­<lb/>POSITIONVM DE <lb/>PROPORTIONIBVS.<lb/><arrow.to.target n="table1"></arrow.to.target></s></p><table><table.target id="table1"></table.target><row><cell>I.</cell><cell>Proportionem <emph type="italics"/>in proportionem duci, e&longs;t &longs;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, ip&longs;a uerò proportio inter duas alias quantitates fuerit con&longs;tituta: con&longs;urgent trecen-ti &longs;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 &longs;ecundum, producta ex proportionibus tertij ad quartum, & quinti ad &longs;extum, producetur etiam ex proportione tertij ad &longs;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 &longs;ecundum, producta ex proportione tertij ad quartum, & quinti ad &longs;extum: erit proportio tertij ad &longs;extum, producta ex proportionibus primi ad &longs;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 &longs;exaginta modis producendarum proportionum triginta &longs;ex tantum e&longs;&longs;e nece&longs;&longs;arios.<emph.end type="italics"/></cell><cell>9</cell></row><row><cell>VII.</cell><cell>I<emph type="italics"/>n modis qui nece&longs;&longs;ariò producuntur ex duabus proportionibus, cum duæ quantitates ex illis quæ modos conficiunt, æquales 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 &longs;uperiores numeri alternatim cum inferioribus multiplicen-tur atque coniungantur, erit proportio aggregati ad productum ex inferioribus in-uicem proportio, ex primis proportionibus compo&longs;ita.<emph.end type="italics"/></cell><cell>11</cell></row><row><cell>IX.</cell><cell>S<emph type="italics"/>i duarum proportionum &longs;uperiores numeri alternatim cum inferioribus multiplicen-tur, minusque productum ex maiore detrahatur, erit re&longs;idui ad productum ex in&longs;e-rioribus proportio uelut illa, quæ 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 &longs;ecun-dam quantitatem, erit proportio cuiu&longs;uis quantitatis eiu&longs;dem generis ad &longs;ecundam compo&longs;ita proportio, ex proportionibus eiu&longs;dem quantitatis, a&longs;&longs;umptæ ad utranque partem primæ quantitatis &longs;eor&longs;um.<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 æqua-lium <expan abbr="quantitatũ">quantitatum</expan> e&longs;t, compo&longs;ita ex proportionibus primis, & diui&longs;a per duplam.<emph.end type="italics"/></cell><cell>12</cell></row><row><cell>XII.</cell><cell>P<emph type="italics"/>ropo&longs;itis duabus proportionibus unam alteri iungere ab&longs;que multiplicatione.<emph.end type="italics"/></cell><cell>12</cell></row><row><cell>XIII.</cell><cell>P<emph type="italics"/>roportio confu&longs;a aggregata primæ & tertiæ quatuor quantitatum omiologarum ad aggregatum &longs;ecundæ & quartæ, e&longs;t uelut compo&longs;ita ex ei&longs;dem diui&longs;a per du-plam.<emph.end type="italics"/></cell><cell>13</cell></row><row><cell>XIIII.</cell><cell>P<emph type="italics"/>roportiones confu&longs;æ & coniunctæ 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 confu&longs;a, aggregati primæ & tertiæ, ad aggre-gatum &longs;ecundæ & quartæ, erit ut monadis addito prouentu, qui fit diui&longs;a differentia, differentiarum primæ & &longs;ecundæ, atque quartæ & tertiæ, per aggregatum tertiæ & quartæ ad ip&longs;am monadem.<emph.end type="italics"/></cell><cell>14</cell></row><row><cell>XVI.</cell><cell>O<emph type="italics"/>mnium quatuor quantitatum propo&longs;ita prima, quæ non minorem habet proportio-nem ad &longs;uam corre&longs;pondentem quàm alia ad aliam, erit proportio confu&longs;a illarum,<emph.end type="italics"/></cell><cell></cell></row><pb/><row><cell></cell><cell><emph type="italics"/>ut producti ex aggregato primæ & tertiæ, in tertiam ad productum ex iggre gato tertiæ & omiotatæ ad &longs;ecundam in ip&longs;am quartam.<emph.end type="italics"/></cell><cell>14</cell></row><row><cell>XVII.</cell><cell>O<emph type="italics"/>mnes duæ proportiones conuer&longs;æ producunt æqualem 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 præter, <expan abbr="ultimã">ultimam</expan> proportio uerò penultimæ ad ultimam, qualis re&longs;idui primæ ad &longs;ecundam, erit primæ ad aggregatum reliquarum, uelut penultimæ ad ultimam.<emph.end type="italics"/></cell><cell>15</cell></row><row><cell>XIX.</cell><cell>S<emph type="italics"/>i fuerint aliquot quantitates arithmeticæ omiologæ, quarum exce&longs;&longs;us &longs;it æqualis minimè, omnibus autem deficientibus &longs;upplementa ad æqualitatem maximè adiungantur, erunt quadrata omnium quantitatum æqualium, adiecto rur&longs;us quadrato primæ cum eo quod fit ex minima primi ordinis in aggregatum o-mnium quantitatum eiu&longs;dem, 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">&longs;ecunda</expan> æqualis tertiæ, aut prima æqualis quartæ, erit proportio primæ ad quartam, aut tertiæ ad &longs;ecundam, producta ex proportionibus primæ ad &longs;ecundam & tertiæ ad quartam.<emph.end type="italics"/></cell><cell>21</cell></row><row><cell>XXI.</cell><cell>C<emph type="italics"/>um decu&longs;&longs;atim ducta fuerit prima in quartam, & &longs;ecunda in tertiam, produ-ctumque primæ in quartam, diui&longs;um fu<gap/>rit per productum &longs;ecundæ in tertiam, erit proportio primæ ad &longs;ecundam, diui&longs;a per proportíonem tertiæ ad quar-tam.<emph.end type="italics"/> E<emph type="italics"/>t &longs;imiliter interpo&longs;ita omiologa.<emph.end type="italics"/></cell><cell>22</cell></row><row><cell>XXII.</cell><cell>C<emph type="italics"/>um fuerit proportio primæ ad &longs;ecundam maior quàm tertiæ ad quartam, erit confu&longs;a ex his maior quàm tertiæ ad quartam, minor autem quàm primæ ad &longs;ecundam.<emph.end type="italics"/></cell><cell>23</cell></row><row><cell>XXIII.</cell><cell>O<emph type="italics"/>mnis motus naturalis ad locum &longs;uum e&longs;t: ideò 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 e&longs;t.<emph.end type="italics"/></cell><cell>23</cell></row><row><cell>XXV.</cell><cell>T<emph type="italics"/>res &longs;unt motus omnino &longs;implices naturalis, uoluntarius, & uiolentus.<emph.end type="italics"/></cell><cell>24</cell></row><row><cell>XXVI.</cell><cell>M<emph type="italics"/>otus ergo compo&longs;iti quatuor nece&longs;&longs;ariò &longs;unt &longs;pecies.<emph.end type="italics"/></cell><cell>24</cell></row><row><cell>XXVII.</cell><cell>M<emph type="italics"/>otus uoluntarius e&longs;t 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 æqualis e&longs;t &longs;emper: &longs;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 re&longs;i&longs;tunt obuiæ, aliæ impel-lunt.<emph.end type="italics"/></cell><cell>26</cell></row><row><cell>XXXI.</cell><cell>O<emph type="italics"/>mnis motus naturalis in æquali medio ualidior e&longs;t in fine quàm in principio.<emph.end type="italics"/>V<emph type="italics"/>iolentus contrà.<emph.end type="italics"/></cell><cell>26</cell></row><row><cell>XXXII.</cell><cell>O<emph type="italics"/>mne mobile naturaliter motum &longs;eu uiolenter uelocius mouetur in medio rariore quàm den&longs;iore.<emph.end type="italics"/> M<emph type="italics"/>aior quoque e&longs;t proportio finis motus in corpore rariore ad finem motus in corpore den&longs;iore quàm principij.<emph.end type="italics"/> I<emph type="italics"/>n uiolento autem celerius perueniret ad finem motus in corpore den&longs;iore.<emph.end type="italics"/></cell><cell>27</cell></row><row><cell>XXXIII.</cell><cell>O<emph type="italics"/>mnia duo mobilia æqualis undique magnitudinis quæ æquali in tempore æqualia &longs;pacia pertran&longs;eunt in diuer&longs;is &longs;ub&longs;tantia medijs nece&longs;&longs;e e&longs;t, ut &longs;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 &longs;uam &longs;uperficiem quadratam, e&longs;t uelut eiu&longs;dem &longs;uperfi ciei, ad latus eiu&longs;dem uerò ad monadem.<emph.end type="italics"/></cell><cell>28</cell></row><row><cell>XXXV.</cell><cell>V<emph type="italics"/>ocum magnitudines excre&longs;cunt in acumine, non in grauitate, finis autem e&longs;t in utroque extremo.<emph.end type="italics"/> P<emph type="italics"/>ropter hoc minima facta uariatione in hypate acutæ 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 æquali ducatur, proportio minor pro-<emph.end type="italics"/></cell><cell></cell></row><pb/><row><cell></cell><cell><emph type="italics"/>ducetur.<emph.end type="italics"/> V<emph type="italics"/>nde manife&longs;tum e&longs;t duas proportiones minores æqualitate inuice<gap/> 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 &longs;e nauim mouere poßint, aut pondus ferre &longs;imul iuncti eam moueant, aut pondus ferant, erunt illæ proportiones coniunctæ non productæ.<emph.end type="italics"/></cell><cell>30</cell></row><row><cell>XXXVIII.</cell><cell>O<emph type="italics"/>mne corpus tantum re&longs;i&longs;tit motui contrario &longs;uo natúrali, quantum mouetur oc-culto motu quie&longs;cendo.<emph.end type="italics"/></cell><cell>31</cell></row><row><cell>XXXIX.</cell><cell>A<emph type="italics"/>b æquali aut minore ui quàm &longs;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 &longs;pb æricum tangens planum in puncto mouetur ad latus per quam-cunque uim, quæ medium diuidere pote&longs;t.<emph.end type="italics"/></cell><cell>31</cell></row><row><cell>XLI.</cell><cell>S<emph type="italics"/>i fuerint duæ quantitates &longs;umaturque toties <expan abbr="aggregatũ">aggregatum</expan> maioris & minoris, quo-ties aggregatum minoris & maioris, erit proportio confu&longs;a 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 &longs;e inuicem, quomodo ducere oporteat con&longs;iderare.<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 e&longs;t maximum non mouens, & &longs;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 &longs;tateræ o&longs;tendere.<emph.end type="italics"/></cell><cell>34</cell></row><row><cell>XLVI.</cell><cell>A<emph type="italics"/>n &longs;it aliqua proportio & qualis inter animam & uitas, & &longs;ua corpora con&longs;ide-rare.<emph.end type="italics"/></cell><cell>35</cell></row><row><cell>XLVII.</cell><cell>S<emph type="italics"/>i duo mobilia æqualister in eodem circulo iuxta proprios motus moueantur, pro-ductum temporis circuituum inuicem, erit æquale producto differentiæ tempo rum circuitus ductæ in tempus coniunctionis primæ.<emph.end type="italics"/></cell><cell>36</cell></row><row><cell>XLVIII.</cell><cell>S<emph type="italics"/>i tria mobilia ex eodem puncto di&longs;cedant, fuerintque duorum ac duorum coniun-ctiones in temporibus commen&longs;is, illa tria mobilia denuo coniungentur in tem pore producto ex denominatore diui&longs;ionis 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 di&longs;tantiæ ab illo mo bilis circuitum inuenire, quod ex <expan abbr="eod&etilde;">eodem</expan> puncto di&longs;cedens <expan abbr="cũalio">cunalio</expan> mobili in dato puncto <expan abbr="cõueniat">conueniat</expan> &longs;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 ei&longs;dem 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. incommen&longs;is inter &longs;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 &longs;e in aduer&longs;um mouentium proportionem declarare.<emph.end type="italics"/></cell><cell>43</cell></row><row><cell>LIIII.</cell><cell>P<emph type="italics"/>roportio circuli ad &longs;uum diametrum per &longs;imilitudinem e&longs;t quarta pars periphe-riæ.<emph.end type="italics"/> R<emph type="italics"/>ur&longs;usque eiu&longs;dem 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 &longs;up po&longs;ita æquali proportione in or-dinibus per quantitates & proportiones demon&longs;trare.<emph.end type="italics"/></cell><cell>44</cell></row><row><cell>LVI.</cell><cell>P<emph type="italics"/>roportio cuiu&longs;uis binomij ad &longs;uum reci&longs;um, uel ei commen&longs;um e&longs;t duplicata ei quæ 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"/>uæ ex alto de&longs;cendunt, cur non eandem pro di&longs;tantia motus rationem in libero aëre &longs;eruent con&longs;iderare.<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 &longs;eor&longs;um tar dius mouetur &longs;imili motu.<emph.end type="italics"/></cell><cell>50</cell></row><row><cell>LX.</cell><cell>O<emph type="italics"/>mne mobile motu naturali de&longs;cendentis parte, de&longs;cendit grauiore &longs;ecundum gra-<emph.end type="italics"/></cell><cell></cell></row><pb/><row><cell></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 & di&longs;tantiam generaliter con&longs;iderare.<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, tantò facilius trabitur.<emph.end type="italics"/></cell><cell>53</cell></row><row><cell>LXIIII.</cell><cell>O<emph type="italics"/>mne mobile quantò 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 &longs;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, & &longs;ubten&longs;arum con&longs;iderare, & quæ à 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 aliæque totidem ab eadem analo-gæ, erit proportio tertiæ unius ordinis ad tertiam alterius, ut &longs;ecundæ ad &longs;e-cundum duplicata, & quartæ ad quartam triplicata, quintæ ad quintam quadruplicata, atque &longs;ic de alijs.<emph.end type="italics"/></cell><cell>57</cell></row><row><cell>LXVIII.</cell><cell>P<emph type="italics"/>ropo&longs;itio 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"/>ropo&longs;itio 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"/>i fuerint tres quantitates <gap/> ontinua proportione, aliæque totidem in continua proportione poterunt con&longs;tituere tres quantitates in æquali differentia per-uer&longs;im copulatæ.<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 &longs;u-&longs;pen&longs;ionem inuenire.<emph.end type="italics"/></cell><cell>63</cell></row><row><cell>LXXII.</cell><cell>P<emph type="italics"/>roportionem ponderis &longs;phæræ pendentis ad a&longs;cendentem 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 concu&longs;&longs;um in&longs;tabili 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 &longs;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 demon&longs;trare.<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 &longs;ecundum obliquum ad rectum in &longs;pacio declarare.<emph.end type="italics"/></cell><cell>69</cell></row><row><cell>LXXXI.</cell><cell>Q<emph type="italics"/>uualis &longs;it angulus, per quem pote&longs;t 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 rece&longs;&longs;us à recta uia ad obliquitatem inue&longs;tigare.<emph.end type="italics"/></cell><cell>72</cell></row><row><cell>LXXXIIII.</cell><cell>D<emph type="italics"/><expan abbr="i&longs;tantiã">i&longs;tantiam</expan> centri terræ à 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 &longs;ub eadem men&longs;ura e&longs;t ueluti eiu&longs;dem ad differentiam ponderis ua&longs;is 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 æ quales &longs;eu in &longs;phæra &longs;eu in plano &longs;e &longs;ecuerint, nunquàm oppo&longs;itos angulos æquales habent.<emph.end type="italics"/></cell><cell>77</cell></row><row><cell>LXXXVII.</cell><cell>P<emph type="italics"/>roportiones craßitiei aquæ ad <expan abbr="a&etilde;r&etilde;">aerrem</expan> in <expan abbr="cõparatione">comparatione</expan> ad radios demon&longs;trare.<emph.end type="italics"/></cell><cell>78</cell></row><row><cell>LXXXVIII.</cell><cell>I<emph type="italics"/><expan abbr="n&longs;trumentũ">n&longs;trumentum</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 den&longs;itatis aquæ ad aërem 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 mißi ponderis ad æqualitatem reducere.<emph.end type="italics"/></cell><cell>82</cell></row><row><cell>XCI.</cell><cell>P<emph type="italics"/>roportionem grauis cubi, & &longs;phærici æqualium in accliui, & de&longs;cen&longs;us eorum demon&longs;trare.<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 æqualis iuxta longitudinis <expan abbr="cõparation&etilde;">comparationem</expan> demon&longs;trare.<emph.end type="italics"/></cell><cell>85</cell></row><row><cell>XCIII.</cell><cell>P<emph type="italics"/>ropter qd in <expan abbr="cõcußione">concußione</expan> <expan abbr="etiã">etiam</expan> leui nauis loco moueatar <expan abbr="o&longs;t&etilde;dere">o&longs;tendere</expan>.<emph.end type="italics"/> V<emph type="italics"/>nde manifi <expan abbr="&longs;iũ">&longs;ium</expan> e&longs;t duas naues &longs;ibi <expan abbr="inuic&etilde;">inuicem</expan> occur&longs;antes retrocedere, & <expan abbr="quãtũ">quantum</expan> <expan abbr="retrocedãt">retrocedant</expan> ambæ.<emph.end type="italics"/></cell><cell>86</cell></row><pb/><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 &longs;imiliter.<emph.end type="italics"/> E<emph type="italics"/>t &longs;i duæ proportiones notæ fuerint, erit producta ex his atque diui&longs;a coniunctaque atque detra-cta nota.<emph.end type="italics"/> E<emph type="italics"/>t &longs;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 &longs;i fuerit partis ad partem, erit ad to<gap/>um monade minor atque nota.<emph.end type="italics"/> E<emph type="italics"/>t &longs;i fuerit unius <expan abbr="quãtitatis">quantitatis</expan> ad duas <expan abbr="quãtitates">quantitates</expan> proportio nota, erit & <expan abbr="cõfu&longs;a">confu&longs;a</expan> ex eis nota.<emph.end type="italics"/> E<emph type="italics"/>t &longs;i fuerint trium quantitatum omiologarum, aut quatuor analogarum omnes præter unam cognitæ, erunt | & illa alia cognita.<emph.end type="italics"/></cell><cell>87</cell></row><row><cell>XCV.</cell><cell>C<emph type="italics"/>uiu&longs;uis trigoni rectanguli, aut cuius duo auguli &longs;int in dupla proportione, aut qui circulo in&longs;criptus &longs;it cognita quantitate unius lateris in comparatione ad dimetien <expan abbr="t&etilde;">tem</expan>, &longs;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">per&longs;picuum</expan> den&longs;um radij lumino&longs;i 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> inuer&longs;ionis in figuris in <expan abbr="cõparatione">comparatione</expan> ad <expan abbr="motũ">motum</expan> &longs;phæræ in plano inue&longs;tigare.<emph.end type="italics"/></cell><cell>91</cell></row><row><cell>XCVIII.</cell><cell>P<emph type="italics"/>roportionem ponderum æqualium 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 multitudin<gap/> &longs;uppo&longs;itorum orbium o&longs;tendere.<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> inue&longs;tigare.<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 inuer&longs;ionis, & 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 demon&longs;trare.<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 uicißim deter-minare.<emph.end type="italics"/></cell><cell>97</cell></row><row><cell>CVII.</cell><cell>S<emph type="italics"/>i in circulo duæ diametri ad rectum angulum &longs;e &longs;ecauerint: aliæ uerò ad perpendicu-lum ex diametro exicrint ad circum ferentiam, &longs;ingulæ &longs;upra diametrum erunt ma iores portionibus reliquis diametri &longs;uperioribus, infra autem minores.<emph.end type="italics"/> D<emph type="italics"/>imidium autem portionis &longs;uperioris re&longs;iduum ad centrum maius &longs;agitta habebit.<emph.end type="italics"/> I<emph type="italics"/>n aliqua præterea portionis &longs;uperioris parte, quæ uer&longs;us diametrum tran&longs;uer&longs;um po&longs;ita e&longs;t, maior e&longs;t differentia partis diametri ei <expan abbr="corre&longs;põdentis">corre&longs;pondentis</expan>, <expan abbr="&qtilde;">quae</expan> line æ tran&longs;uer&longs;æ.<emph.end type="italics"/></cell><cell>100</cell></row><row><cell>CVIII.</cell><cell>P<emph type="italics"/>unctum æqualitatis differentiæ de&longs;cen&longs;us & remotionis à centro inuenire.<emph.end type="italics"/></cell><cell>100</cell></row><row><cell>CIX.</cell><cell>R<emph type="italics"/>ationem libræ expendere.<emph.end type="italics"/></cell><cell>101</cell></row><row><cell>CX.</cell><cell>S<emph type="italics"/>i duæ &longs;phæræ ex eadem materia de&longs;cendant in aëre, 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 &longs;calmo à 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 longiùs ferantur declarare,<emph.end type="italics"/></cell><cell>106</cell></row><row><cell>CXIII.</cell><cell>C<emph type="italics"/>ur uirga longius mittatur à puero quam à uiro inueftigare.<emph.end type="italics"/></cell><cell>107</cell></row><row><cell>CXIIII.</cell><cell>C<emph type="italics"/>ircularis motus differentias quatuor e&longs;&longs;e, earumque rationem contemplari.<emph.end type="italics"/></cell><cell>108</cell></row><row><cell>CXV.</cell><cell>P<emph type="italics"/>roportionem motuum impul&longs;ionis, & attractionis inter &longs;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 machinæ oblongæ igneæ longius emittant &longs;phæram explorare.<emph.end type="italics"/></cell><cell>111</cell></row><row><cell>CXVII.</cell><cell>I<emph type="italics"/>n curriculis maior e&longs;t uis pulueris copio&longs;ioris ampliore in &longs;pacio, quàm 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 e&longs;t ad perpendi-culum declarare.<emph.end type="italics"/></cell><cell>114</cell></row><row><cell>CXIX.</cell><cell>Q<emph type="italics"/>uantum ictus machinæ 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/><row><cell>CXXIII.</cell><cell>C<emph type="italics"/>ognita ratione umbræ ad gnomonem &longs;inum, & arcum altitudinis ab horizon-te, quouis tempore digno&longs;cere.<emph.end type="italics"/></cell><cell>121</cell></row><row><cell>CXXIIII.</cell><cell>P<emph type="italics"/>roportionem umbræ uer&longs;æ e&longs;&longs;e ad gnomonem, uelut gnomonis ad umbram uer&longs;am.<emph.end type="italics"/></cell><cell>122</cell></row><row><cell>CXXV.</cell><cell>P<emph type="italics"/>roportionem dimetientis, & peripheriæ cuiuslibet circuli paralleli æquino-ctiali per cognitam partem magni circuli demon&longs;trare.<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, cuiu&longs;que puncti arcum &longs;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 di&longs;tantiam<emph.end type="italics"/> S<emph type="italics"/>olis à meri-diano cogno&longs;cere.<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 quàm uer&longs;am, uel etiam in cylindro determinare.<emph.end type="italics"/></cell><cell>125</cell></row><row><cell>CXXXI.</cell><cell>S<emph type="italics"/>i lineæ alicui duplum alterius adiungatur, erit proportio d<gap/>arum ad primam maior quàm 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 &longs;it eadem linea addatur, erit aggrega-ti ex minore, & adiecta ad ip&longs;am minorem, minor proportio quàm aggre-gati ex maiore, & adiecta ad ip&longs;am maiorem duplicata.<emph.end type="italics"/></cell><cell>126</cell></row><row><cell>CXXXIII.</cell><cell>S<emph type="italics"/>i fuerint duæ quantitates, <expan abbr="quarũ">quarum</expan> una alteri dupla &longs;it: minuatur à minore quæ-dam quantitas, <expan abbr="ead&etilde;que">eadenque</expan> maiori addatur, erit minoris ad re&longs;iduum maior pro-portio, quàm aggregati ad maiorem duplicata.<emph.end type="italics"/> S<emph type="italics"/>i uerò minori addatur, & à maiore detrabatur, erit aggregati ad minorem minor proportio quàm maioris ad re&longs;iduum duplicata.<emph.end type="italics"/></cell><cell>127</cell></row><row><cell>CXXXIIII.</cell><cell>S<emph type="italics"/>i rectangula &longs;uperficies &longs;it, cuius pars tertia quadrata &longs;it corpus, quod ex la-tere quadratæ in re&longs;iduum &longs;uperficiei con&longs;tat, maius e&longs;t quouis corpore ex eadem &longs;uperficies, aliter diui&longs;a con&longs;tituto.<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 re&longs;idui parallelipedum maius omni pararalleli-pedo, quod ex diui&longs;ione eiu&longs;dem lineæ creari poßit.<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 progreßione declarare.<emph.end type="italics"/></cell><cell>131</cell></row><row><cell>CXXXVIII.</cell><cell>M<emph type="italics"/>odos u&longs;us horum numerorum declarare.<emph.end type="italics"/></cell><cell>131</cell></row><row><cell>CXXXIX.</cell><cell>R<emph type="italics"/>adices omnes à propo&longs;itis 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">conuer&longs;o</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 quæ fiunt ex una parte in alterius quadratum mutuo æqualia &longs;unt corpori, quod fit ex tota linea in &longs;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 e&longs;t aggregato corporum mutuorum, cuiuslibet diui&longs;ionis quantum e&longs;t, quod fit ex tota in quadratum differentiæ.<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, æqualia &longs;unt producto unius in alteram cum quadrato differentiæ.<emph.end type="italics"/></cell><cell>139</cell></row><row><cell>CXLVI.</cell><cell>C<emph type="italics"/>orpus quod fit ex linea diui&longs;a in &longs;uperficiem æqualem quadratis ambarum par tium detracta &longs;uperficie unius partis in alteram, e&longs;t æquale aggregato cubo-rum ambarum partium.<emph.end type="italics"/></cell><cell>139</cell></row><row><cell>CXLVII.</cell><cell>P<emph type="italics"/>ropo&longs;ita linea diui&longs;a duas ei line as adijcere, ut proportio <expan abbr="additarũ">additarum</expan> &longs;ingularium<emph.end type="italics"/></cell><cell></cell></row><pb/><row><cell></cell><cell><emph type="italics"/>& partium &longs;imul iunctarum ad additas &longs;it mutua.<emph.end type="italics"/></cell><cell>148</cell></row><row><cell>CXLVIII.</cell><cell>P<emph type="italics"/>ropo&longs;itis tribus lineis primam &longs;ic diuidere, ut adiectis duabus alijs lineis, &longs;ecun-dum <expan abbr="ration&etilde;">rationem</expan> mutuam &longs;ingularum &longs;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 &longs;e habeat, ut &longs;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 &longs;ic diuidere, ut proportio quadratorum ad dupium unius partis in alteram &longs;it, ut lineæ datæ ad lineam datam.<emph.end type="italics"/></cell><cell>141</cell></row><row><cell>CL.</cell><cell>P<emph type="italics"/>ropo&longs;itis duabus lineis, lineam communem utrique adiungere, ut &longs;it maioris ad ad-ditam proportio, uelut quadratorum minoris, & adiectæ ad duplum unius in alteram.<emph.end type="italics"/></cell><cell>141</cell></row><row><cell>CLI.</cell><cell>P<emph type="italics"/>roportio differentiæ quadratorum partium cuiu&longs;uis lineæ, ad quadratum diffe-rentiæ illarum e&longs;t, uelut totius lineæ 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 æquales, duasque inæquales diuidatur, fueritque proportio ag-gregati ex maiore, & dimidio ad ip&longs;am maiorem, uelut ex minore, & ali-qua linea ad ip&longs;am minorem, & rur&longs;us aggregati ex minore, & dimidio ad ip&longs;am minorem, uelut aggregati ex maiore, & alia addita ad ip&longs;am maiorem, erit proportio dimidij ad partem unam inæqualem, uelut alterius partis inæ-qualis ad &longs;uam additam mutuò, & etiam proportio additarum inuicem, uelut proportio <expan abbr="partiũ">partium</expan> <expan abbr="inæqualiũ">inæqualium</expan> duplicata, & rur&longs;us ip&longs;um <expan abbr="dimidiũ">dimidium</expan> lineæ a&longs;&longs;um-ptæ <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 lineæ datæ alia linea adiungatur, ab extremitatibus autem prioris lineæ duæ rectæ in unum punctum concurrant proportionem habentes, quam mediam inter tota m & adiectam, & adiectam erit punctus, concur&longs;us à puncto extre-mo lineæ adiectæ di&longs;tans per lineam mediam.<emph.end type="italics"/> Q<emph type="italics"/>uod &longs;i ab extremo alicuius li-neæ æqua'is mediæ, &longs;eu peripheria circuli, cuius &longs;emidiameter &longs;it media linea duæ lineæ ad prædicta puncta producantur, ip&longs;æ erunt in proportione mediæ 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 con&longs;iderare.<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 o&longs;tendere.<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 æqualis e&longs;&longs;e pote&longs;t 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"/>ropo&longs;ita linea tribusque in ea &longs;ignis punctum inuenire, ex quo ductæ tres lineæ ad &longs;igna &longs;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 ba&longs;es in eadem linea &longs;int con&longs;tituti, & æquales ad unum punctum terminati, & latus unum commune inter reliqua quantita-te medium nece&longs;&longs;e e&longs;t angulum à maioribus lineis <expan abbr="contentũ">contentum</expan> minorem e&longs;&longs;e.<emph.end type="italics"/></cell><cell>162</cell></row><row><cell>CLXII.</cell><cell>P<emph type="italics"/>roportionem duorum orbium, quorum diametrorum conuexæ partis, & conca-uæ proportiones datæ &longs;int inue&longs;tigare.<emph.end type="italics"/></cell><cell>164</cell></row><row><cell>CLXIII.</cell><cell>P<emph type="italics"/>roportionem uirium &longs;tellarum per motus &longs;uos indagare.<emph.end type="italics"/></cell><cell>165</cell></row><row><cell>CLXIIII.</cell><cell>S<emph type="italics"/>yderum proportionem in magnitudine o&longs;tendere.<emph.end type="italics"/></cell><cell>166</cell></row><row><cell>CLXV.</cell><cell>P<emph type="italics"/>roportionem motuum omnium &longs;tellarum ad<emph.end type="italics"/> S<emph type="italics"/>olem con&longs;iderare.<emph.end type="italics"/></cell><cell>167</cell></row><row><cell>CLXVI.</cell><cell>P<emph type="italics"/>roportiones mu&longs;icas &longs;uperpartientes in eas, quæ particulá una tantum abundant reducere.<emph.end type="italics"/></cell><cell>168</cell></row><pb/><row><cell>CLXVII.</cell><cell>P<emph type="italics"/>roportionem mu&longs;icam ad &longs;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 mu&longs;icam in in&longs;trumentis declarare iuxta compo&longs;itionis ra-tionem.<emph.end type="italics"/></cell><cell>182</cell></row><row><cell>CLXX.</cell><cell>C<emph type="italics"/>oniugationes cuiu&longs;uis numeri breuiter inuenire.<emph.end type="italics"/></cell><cell>185</cell></row><row><cell>CLXXI.</cell><cell>P<emph type="italics"/>ropo&longs;itis duobus quibuslibet numeris, quotuis alios &longs;eu in continuum &longs;eu medios in continua proportione arithmetica, geometrica & mu&longs;ica 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 de&longs;cribere.<emph.end type="italics"/></cell><cell>191</cell></row><row><cell>CLXXIII.</cell><cell>C<emph type="italics"/>irculum &longs;uper centro &longs;uo mouere æqualiter, 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"/>rogre&longs;&longs;us & regre&longs;&longs;us, tam &longs;ine latitudine quàm cum latitudine in planetis per &longs;olos concentricos circulos æqualiter motos demon&longs;trare.<emph.end type="italics"/></cell><cell>194</cell></row><row><cell>CLXXV.</cell><cell>C<emph type="italics"/>au&longs;am uarietatis diametrorum ex &longs;uppo&longs;itis concentricis demon&longs;tra-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 eiu&longs;dem quantitatis ad alias duas componatur, erit proportio illarum duarum eadem proportioni producti ex proportione in primam duarum quantitatum, detracta prio-re illa quantitate, quæ 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 mi&longs;tionis metallorum, maximè 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 duæ portiones &longs;imiles ab&longs;cindantur ab ei&longs;dem denuò, & ab-&longs;cißis portionibus partes eædem auferantur, denuoque ac denuò quoties libuerit à portionibus, & ù re&longs;iduis ip&longs;arum quantitatum partes eædem auferantur, erit re&longs;iduí ad re&longs;iduum, 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 compo&longs;ita proportio, non poterit eiu&longs;dem quantitatis ad par-tes alias quantitatis diui&longs;a, 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, compo&longs;ita ex proportionibus primæ ad &longs;ecun-dam & tertiam, & rur&longs;us quartæ ad quintam & &longs;extam: ita &longs;e habebit proportio &longs;ecundæ ad tertiam, ad proportionem quintæ ad &longs;extam, uelut producti ex proportione in &longs;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"/>ropo&longs;ita differentia proportionum partium &longs;imilium ad partes a&longs;&longs;umptas, propo&longs;itaque proportione totius ad re&longs;idua eadem, differentiam propor-tionum totius ad reliquum re&longs;idui inuenire.<emph.end type="italics"/></cell><cell>203</cell></row><row><cell>CLXXXIII.</cell><cell>S<emph type="italics"/>pacium uitæ naturalis per &longs;pacium uitæ fortuitum declarare.<emph.end type="italics"/></cell><cell>204</cell></row><row><cell>CLXXXIIII.</cell><cell>Q<emph type="italics"/>uæcunque grauia in uorticibus aquarum, merguntur, in medio uorticis, pri-mum uer&longs;a mergantur.<emph.end type="italics"/></cell><cell>211</cell></row><row><cell>CLXXXV.</cell><cell>C<emph type="italics"/>ur homo &longs;edens quanto altius &longs;edet, & quanto magis crura ad fœmora, & fœmora ad pectus reclinata habet, facilius con&longs;urgat, cum tamen hæc op-po&longs;ito modo inuicem &longs;e habeant, declarare.<emph.end type="italics"/></cell><cell>213</cell></row><row><cell>CLXXXVI.</cell><cell>S<emph type="italics"/>i fuerit proportio primæ & &longs;ecundæ quantitatis ad tertiam, ut primæ & quartæ ad quintam, fueritque quarta &longs;ecunda maior, erit proportio quar-tæ ad quintam maior quàm &longs;ecundæ ad tertiam.<emph.end type="italics"/> Q<emph type="italics"/>uod &longs;i fuerit maior<emph.end type="italics"/></cell><cell></cell></row><pb/><row><cell></cell><cell><emph type="italics"/>quartæ ad quintam quàm &longs;ecundæ ad tertiam, nece&longs;&longs;e e&longs;t quartam &longs;ecunda e&longs;&longs;e maiorem.<emph.end type="italics"/></cell><cell>214</cell></row><row><cell>CLXXXVII.</cell><cell>S<emph type="italics"/>i ei&longs;dem uiribus & ‘eadem’ proportione cum auxilio ponderis tertij quar-tum pondus moueatur quibus &longs;ecundum, auxilio primi nece&longs;&longs;e e&longs;t <expan abbr="quartũ">quartum</expan> pon dus tardius & maiore cum difficultate moueri quàm &longs;ecundum.<emph.end type="italics"/></cell><cell>214</cell></row><row><cell>CLXXXVIII.</cell><cell>S<emph type="italics"/>i uires aliquæ moueant cum ponderibus aliqua pondera, ut compo&longs;ita pro-portio &longs;it eadem proportioni uirium & duorum ponderum mouentium ag-gregatum æquale duorum ponderum, ubi maior fuerit partium in æqual<gap/>as, 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 &longs;ub æquali proportione coapte-tar, facilius deor&longs;um trahetur quàm quod maius e&longs;t & propius.<emph.end type="italics"/></cell><cell>215</cell></row><row><cell>CXC.</cell><cell>S<emph type="italics"/>i fuerit primum graue minus &longs;ecundo, & &longs;ecundum minus tertio, proportio autem primi ad &longs;ecundum multo maior quàm &longs;ecundi ad tertium, po&longs;ibile erit propo&longs;itis uiribus ei&longs;dem addere pondus <expan abbr="&longs;ecũdo">&longs;ecundo</expan>, ut ip&longs;um & tertium mouea-tur faciliùs ab ei&longs;dem uiribus, & primo uel &longs;ecundo quàm 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 in&longs;uper quantum e&longs;t productum dimidij ui rium in &longs;e latus aggregati detracto dimidio uirium, dice<gap/> pondus auxiliare æqualis 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 lineæ ducantur &longs;eu intus ad circun ferentia<gap/>u&longs;que, &longs;eu extra ad diametrum, erit proportio totius lineæ 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 &longs;u&longs;pen&longs;i, ad breuius illi æquale & in medio &longs;u&longs;pen&longs;um 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 &longs;uffulcietur re&longs;tibus ex medio ad angulos & eius æquidi&longs;tantibus quàm &longs;ecundum longitudinem & latitudinem.<emph.end type="italics"/></cell><cell>220</cell></row><row><cell>CXCVI.</cell><cell>S<emph type="italics"/>i duo circuli &longs;uper eodem centro eodem motu trans feruntur, æquale &longs;pacium &longs;uperant.<emph.end type="italics"/></cell><cell>221</cell></row><row><cell>CXCVII.</cell><cell>C<emph type="italics"/>ur lances ad locum &longs;uum &longs;u&longs;pen&longs;i redeant, impendentes <expan abbr="nõ">non</expan>, <expan abbr="demõ&longs;trare">demon&longs;trare</expan>.<emph.end type="italics"/></cell><cell>224</cell></row><row><cell>CXCVIII.</cell><cell>C<emph type="italics"/>ur &longs;olidum quod cubus uocatur<emph.end type="italics"/> P<emph type="italics"/>yramide &longs;tabilius &longs;it o&longs;tendere.<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 &longs;it, magnam nauim agere pote&longs;t, & cur cùm uarietas &longs;it in prora, ip&longs;e con&longs;tituatur in puppi.<emph.end type="italics"/> E<emph type="italics"/>t cum transuer&longs;im ab aqua prematur rectà nauim dirigat.<emph.end type="italics"/></cell><cell>228</cell></row><row><cell>CCI.</cell><cell>S<emph type="italics"/>i duæ lineæ non &longs;ecantes circuli peripheriam in unum punctum ex ea coe-ant exterius, nece&longs;&longs;e e&longs;t illas peripheria contenta e&longs;&longs;e maiores.<emph.end type="italics"/></cell><cell>229</cell></row><row><cell>CCII.</cell><cell>R<emph type="italics"/>ationem &longs;trepitus o&longs;tendere.<emph.end type="italics"/></cell><cell>232</cell></row><row><cell>CCIII.</cell><cell>C<emph type="italics"/>ur &longs;cytalis onera portentur faciliùs, 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 o&longs;tendere.<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"/>eipublicæ.<emph.end type="italics"/></cell><cell>234</cell></row><row><cell>CCVI.</cell><cell>R<emph type="italics"/>hombi paßiones qua&longs;dam declarare.<emph.end type="italics"/></cell><cell>235</cell></row><row><cell>CCVII.</cell><cell>P<emph type="italics"/>roportionem agentium naturalium in tran&longs;mutatione con&longs;iderare.<emph.end type="italics"/></cell><cell>238</cell></row><row><cell>CCVIII.</cell><cell>M<emph type="italics"/>ota res à centro grauitatis per <expan abbr="prior&etilde;">priorem</expan> motum, in reditu uelocius mouetur quam &longs;i quieuerit.<emph.end type="italics"/></cell><cell>238</cell></row><pb/><row><cell>CCIX.</cell><cell>S<emph type="italics"/>i &longs;uperficies rectangula in duas partes æquales diui&longs;a intelligatur, quæ am-bæ quadratæ &longs;int, itemque in duas inæquales, erit parallelipedum ex latere mediæ partis in totam &longs;uperficiem maius aggregato parallelipedorum ex partibus inæqualibus in latera alterius partis mutuo, in eo, quod fit ex dif ferentia lateris minoris partis à mediæ latere in differentiam maioris par-tis &longs;uperficiei à media &longs;uperficie bis, & ex differentia amborum laterum inæqualium iunctorum ad ambo latera, æqualia iuncta in minorem par-tem &longs;uperficiei.<emph.end type="italics"/></cell><cell>241</cell></row><row><cell>CCX.</cell><cell>S<emph type="italics"/>i duæ lineæ ad æquales angulos ab eodem puncto peripheriæ circuli refle-ctantur, nece&longs;&longs;e e&longs;t angulos cum dimetiente factos æquales e&longs;&longs;e.<emph.end type="italics"/> V<emph type="italics"/>nde ma-nife&longs;tum e&longs;t, protractam diametrum angulum &longs;uppo&longs;itum per æqualia di-uidere.<emph.end type="italics"/></cell><cell>242</cell></row><row><cell>CCXI.</cell><cell>S<emph type="italics"/>i duæ lineæ ex duobus punctis peripheriam contingentes, in eandem par-tem protrahantur, &longs;emper magis di&longs;tabunt 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 lineæ quotuis ducantur, tres inue-nire lineas, quæ non in alium punctum reflectentur.<emph.end type="italics"/></cell><cell>244</cell></row><row><cell>CCXIII.</cell><cell>P<emph type="italics"/>ropo&longs;ito circulo, atque in eius peripheria puncto &longs;ignato, lineas contingentes ultra cítraque, & eam ab ip&longs;omet deducere.<emph.end type="italics"/></cell><cell>245</cell></row><row><cell>CCXIIII.</cell><cell>S<emph type="italics"/>i extra circulum duo puncta æqualiter à centro di&longs;tantia &longs;ignentur, erit pun-ctum reflexionis æqualis in medio arcus intercepti inter lineas, quæ à cen tro ducuntur ad illa puncta.<emph.end type="italics"/> S<emph type="italics"/>i uerò unum centro proximius fuerit altero, punctum æqualitatis in peripheria tantò longius, uer&longs;us breuiorem line-am, quantò punctum aliud à centro magis di&longs;teterit.<emph.end type="italics"/></cell><cell>245</cell></row><row><cell>CCXV.</cell><cell>P<emph type="italics"/>unctum reflexionis punctorum inæqualiter di&longs;tantium à centro, æqualiter di&longs;tat à lineis, ductis à centro ad puncta æqualiter di&longs;tantia alterutrin-que.<emph.end type="italics"/></cell><cell>246</cell></row><row><cell>CCXVI.</cell><cell>S<emph type="italics"/>i fuerint circuli duo inæquales, & extra utrunqúe punctum ad illud ex mi-nore reflexè per magnam partem minoris à 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 &longs;uperficiei<emph.end type="italics"/> L<emph type="italics"/>unæ illuminatam à<emph.end type="italics"/> S<emph type="italics"/>ole per radios reflexos à<emph.end type="italics"/> S<emph type="italics"/>olis corpore: nec tamen pote&longs;t uidere imaginem ip&longs;ius in<emph.end type="italics"/> L<emph type="italics"/>una tan quam in &longs;peculo.<emph.end type="italics"/></cell><cell>248</cell></row><row><cell>CCXVIII.</cell><cell>R<emph type="italics"/>ationem maculæ<emph.end type="italics"/> L<emph type="italics"/>unæ indagare.<emph.end type="italics"/></cell><cell>248</cell></row><row><cell>CCXIX.</cell><cell>R<emph type="italics"/>ationem eorum quæ apparent circa<emph.end type="italics"/> S<emph type="italics"/>olem &longs;peculo in aqua po&longs;ito decla-rare.<emph.end type="italics"/></cell><cell>150</cell></row><row><cell>CCXX.</cell><cell>C<emph type="italics"/>au&longs;am cur<emph.end type="italics"/> S<emph type="italics"/>ol æ&longs;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"/>unæ & cæterorum a&longs;trorum digno&longs;citur ex proportione alio-rum ad eam iuxta di&longs;tantiam: ip&longs;ius uerò iuxta rationem pupillæ ad<emph.end type="italics"/> L<emph type="italics"/>u-nam di&longs;tantiæ ratione.<emph.end type="italics"/></cell><cell>354</cell></row><row><cell>CCXXII.</cell><cell>Q<emph type="italics"/>uantitates quæ æquales e&longs;&longs;e non po&longs;&longs;unt in eodem genere, maius tamen & minus recipiunt, &longs;unt in proportione pote&longs;tatis.<emph.end type="italics"/></cell><cell>255</cell></row><row><cell>CCXXIII.</cell><cell>Q<emph type="italics"/>uantitates quæ actu æquales e&longs;&longs;e non po&longs;&longs;unt, in nulla proportione actu e&longs;&longs;e po&longs;&longs;unt.<emph.end type="italics"/></cell><cell>256</cell></row><row><cell>CCXXIIII.</cell><cell>N<emph type="italics"/>eque temporis totius, ut imaginamur, ip&longs;um e&longs;&longs;e infinitum, neque æui ui-tarum proportio ulla e&longs;t ad tempus, quod pote&longs;tate e&longs;t, utpotè diem<emph.end type="italics"/></cell><cell></cell></row><pb/><row><cell></cell><cell><emph type="italics"/>uel men&longs;em.<emph.end type="italics"/></cell><cell>256</cell></row><row><cell>CCXXV.</cell><cell>P<emph type="italics"/>roportio media non e&longs;t ex ratione agentis, &longs;ed patientis.<emph.end type="italics"/></cell><cell>256</cell></row><row><cell>CCXXVI.</cell><cell>P<emph type="italics"/>roportio &longs;ublimis non con&longs;i&longs;tit in magnitudine, &longs;ed ordine, iuxta quem diffe-rentia e&longs;t eius quod e&longs;t ante & po&longs;t.<emph.end type="italics"/></cell><cell>257</cell></row><row><cell>CCXXVII.</cell><cell>V<emph type="italics"/>itæ iuxta numerum perfectionum in comparatione ad cogitationem no-&longs;tram proportionem quand am habent.<emph.end type="italics"/></cell><cell>259</cell></row><row><cell>CCXXVIII.</cell><cell>P<emph type="italics"/>roportionem &longs;cientiæ futurorum & cæterorum occultorum con&longs;idera-re.<emph.end type="italics"/></cell><cell>260</cell></row><row><cell>CCXXIX.</cell><cell>I<emph type="italics"/>ncorporea omnia unum &longs;unt, neque numerus e&longs;t eorum.<emph.end type="italics"/></cell><cell>261</cell></row><row><cell>CCXXX.</cell><cell>P<emph type="italics"/>roportio incorporeorum a&longs;cendentium &longs;emper maior e&longs;t.<emph.end type="italics"/></cell><cell>262</cell></row><row><cell>CCXXXI.</cell><cell>T<emph type="italics"/>res e&longs;&longs;e mundos atque inter ip&longs;os nullam e&longs;&longs;e 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 e&longs;t tanto propior e&longs;t & magis &longs;imil limus quieti.<emph.end type="italics"/></cell><cell>264</cell></row><row><cell>CCXXXIII.</cell><cell>Q<emph type="italics"/>uod e&longs;t in mundo incorporeo æternum e&longs;t, beatum, &longs;ecurum, immutabile &longs;ecundum locum, &longs;olum iuxta e&longs;&longs;entiam fit: iuxta quod uelut à leui &longs;u-&longs;urro aquæ & aura æ&longs;tiua demulcetur.<emph.end type="italics"/></cell><cell>270</cell></row></table><p type="head">
<s>FINIS.</s></p><pb pagenum="1"/><p type="head">
<s>HIERONYMI CAR <lb/>DANI MEDIOLANENSIS, CI­<lb/>VI'SQVE BONONIENSIS, MEDICI­<lb/>de Proportionibus, &longs;eu Ope­<lb/>ris Perfecti <lb/>LIBER QVINTVS.</s></p><p type="main">
<s>Prima diffinitio.</s></p><p type="main">
<s>Proportio ab Euclide &longs;ic de&longs;cribitur, Quòd <lb/>&longs;it duarum quantitatum eiu&longs;dem generis, <lb/>quod ad magnitudinem attinet, compara­<lb/>tio certa.</s></p><p type="main">
<s>Secunda diffinitio.</s></p><p type="main">
<s>Proportiones per &longs;imilitudinem <expan abbr="dicũtur">dicuntur</expan>, <lb/>cùm quantitas quantitati <expan abbr="compara&ttilde;">comparatur</expan> alterius <lb/>generis, cui fingitur æqualis e&longs;&longs;e pote&longs;tate.<gap/></s></p><p type="main">
<s>Velut &longs;i a b fingatur monas in comparatione <lb/>ad b c erit rectangulum a c æquale lineæ b c.</s></p><figure></figure><p type="main">
<s>Tertia diffinitio.</s></p><p type="main">
<s>Proportio æqualis proportioni e&longs;t, cùm eodem modo termini <lb/>&longs;e habent inuicem in utraque</s></p><p type="main">
<s>Quarta diffinitio.</s></p><p type="main">
<s>Proportiones &longs;ecundum genus notæ dicuntur, cùm nouimus, <lb/>quòd &longs;int maiores, aut minores. </s>
<s>Nam cùm æquales &longs;unt, &longs;imul ne­<lb/>ceffe e&longs;t, ut cogno&longs;camus genus, & &longs;peciem.</s></p><p type="main">
<s>Quinta diffinitio.</s></p><p type="main">
<s>Datum po&longs;itione e&longs;t: quod nece&longs;&longs;ariò ex po&longs;itis certam habet <lb/>quantitatem.</s></p><p type="main">
<s>Sexta diffinitio.</s></p><p type="main">
<s>Datum &longs;impliciter dicitur, quod ex propo&longs;itis cogno&longs;ci pote&longs;t, <lb/>quantum &longs;it.</s></p><p type="main">
<s>Septima diffinitio.</s></p><p type="main">
<s>Proportiones pote&longs;tate <expan abbr="dicun&ttilde;">dicuntur</expan>, quæ&longs;ub comparatione aliarum <lb/><expan abbr="quantitatũ">quantitatum</expan> nece&longs;&longs;ariam habentium <expan abbr="cõnexionem">connexionem</expan> <expan abbr="&longs;olũ">&longs;olum</expan> <expan abbr="cogno&longs;cun&ttilde;">cogno&longs;cuntur</expan>.</s></p><p type="main">
<s>Hæ autem &longs;unt aliquando eiu&longs;dem generis, cum primis ut nu­<lb/>meri: aliquandò alterius, ut linearum & &longs;uperficierum, angulorum, <lb/>& arcuum: aliquando eiu&longs;dem generis, & diuen&longs;arum &longs;pecierum, <lb/>ut arcuum per &longs;inus, qua utuntur A&longs;tronomi.</s></p><p type="main">
<s>Octaua diffinitio.</s></p><p type="main">
<s>Proportio homonyma dicitur duarum quantitatum diuer&longs;i ge­</s></p><p type="main">
<s><arrow.to.target n="marg1"></arrow.to.target><lb/>neris, &longs;ed alterius a b altero dependentium, uelut motus ad tem­<pb pagenum="2"/>pus. </s>
<s>Dicimus enim motum tardum, uel uelocem in comparatione <lb/>ad tempus.</s></p><p type="margin">
<s><margin.target id="marg1"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Nona diffinitio.</s></p><p type="main">
<s>Proportionum aliæ dicuntur rhete, aliæ alogæ, rhetæ quæ &longs;unt <lb/>ut numeri ad numerum, alogæ quæ non &longs;unt numeri ad numerum.</s></p><p type="main">
<s>Decima diffinitio</s></p><p type="main">
<s>Proportio rhete alia æqualis, alia multiplex, uel &longs;ubmultiplex: <lb/>alia unius partis exce&longs;&longs;us, aut defectus, alia plurium, quam &longs;uper­<lb/>partientem, aut &longs;upartientem uocant.</s></p><p type="main">
<s>Vndecima diffinitio.</s></p><p type="main">
<s>Cum diui&longs;o denominatore per numeratorem exit quantitas alo <lb/>ga, proportio dicitur aloga: &longs;i autem numerus integer, aut pars nu­<lb/>meri nota dicitur rhete.</s></p><p type="main">
<s>Duodecima diffinitio.</s></p><p type="main">
<s>Proportionem in proportionem duci e&longs;t, quoties recto ordine <lb/>tres quantitates in ei&longs;dem collo <expan abbr="can&ttilde;">cantur</expan>: ut &longs;int tres quan <lb/><figure id="fig1"></figure><lb/>titates a b c dicetur proportio a ad c producta ex pro <lb/>portione a ad b & b ad c, & &longs;imiliter proportio c ad <lb/>a producitur ex proportione b ad a, & c ad b.</s></p><p type="main">
<s>Tertiadecima diffinitio.</s></p><p type="main">
<s>Proportionem per proportionem diuidi e&longs;t, quoties ad eandem <lb/>quantitatem duæ quantitates comparantur, tunc illarum propor­<lb/>tio e&longs;t, quæ prodit una per alteram diui&longs;a.</s></p><p type="main">
<s>Sint proportiones a & b ad c & interponatur b inter a & c, dico <lb/>proportionem a ad c diui&longs;am per proportionem a ad b, & prodire <lb/>proportionem b ad c, con&longs;tat ex conuer&longs;a præcedentis.</s></p><p type="main">
<s>Quartadecima diffinitio.</s></p><p type="main">
<s>Additio proportionum intelligitur quotiens duarum quanti­<lb/>tatum ad unam tertiam, proportiones per aggregatum ip&longs;arum <lb/>quantitatum ad eandem coniunguntur.</s></p><p type="main">
<s>Velut &longs;i comparentur a b & b c ad d, inde tota <lb/><figure id="fig2"></figure><lb/>a c ad d dicemus proportionem, ac ad d e&longs;&longs;e 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>Hoc & duo &longs;equentes &longs;icut & du&ecedil; <expan abbr="anteced&etilde;tes">antecedentes</expan> demon­<lb/>&longs;trabitur e&longs;&longs;e. </s>
<s>nunc &longs;olum quomodo <expan abbr="intelligendũ">intelligendum</expan> &longs;it proponimus.</s></p><p type="main">
<s>Quintadecima diffinitio.</s></p><p type="main">
<s>Detractionem proportionis à proportione intelligimus fieri <lb/>per <expan abbr="detraction&etilde;">detractionem</expan> minoris quantitatis à maiore, comparatam ad ean­<lb/>dem quantitatem.</s></p><p type="main">
<s>Velut in exemplo &longs;uperiore detracta proportione b c ad d ex <pb pagenum="3"/>proportione a c ad d, relinquetur proportio a b ad d. </s>
<s>& probatur <lb/>ex conuer&longs;ione præcedentis.</s></p><p type="main">
<s>Sextadecima diffinitio.</s></p><p type="main">
<s>Extractio radicum alicuius proportionis fit per extractionem <lb/>radicum quantitatum illius iuxta unam, & eandem rationem.</s></p><p type="main">
<s>Velut quadratæ, uel cubæ, uel pronicæ, uel uniner&longs;alis, uel alte­<lb/>rius modi.</s></p><p type="main">
<s>Decima&longs;eptima diffinitio.</s></p><p type="main">
<s>Cùm fuerint duæ proportiones &longs;imiles in tribus terminis con­<lb/>tinuatæ, dicetur proportio primæ quantitatis ad tertiam ueluti <lb/>primæ ad &longs;ecundam duplicata. </s>
<s>Et &longs;i &longs;int tres proportiones &longs;imiles <lb/>in quatuor terminis, dicetur proportio primæ quantitatis ad quar­<lb/>tam triplicatà ei, quæ e&longs;t primæ ad &longs;ecundam,</s></p><p type="main">
<s>Decimaoctaua diffinitio.</s></p><p type="main">
<s>Confu&longs;a proportio dicitur &longs;implicis, aut compo&longs;itæ quantitatis <lb/>ad compo&longs;itam in comparatione ad proportiones ad partes.</s></p><p type="main">
<s>Decimanona diffinitio.</s></p><p type="main">
<s>Quantitates qu&ecedil; in continua &longs;unt proportione Analogæ <expan abbr="uocan&ttilde;">uocantur</expan>.</s></p><p type="main">
<s>Dictum e&longs;t hoc ad fugiendum nomen barbarum, etiam ut bre­<lb/>uiter tamen po&longs;&longs;emus &longs;ententiam explicare.</s></p><p type="main">
<s>Vige&longs;ima diffinitio.</s></p><p type="main">
<s>Reflexa proportio dicitur cum trium quantitatum aggregatum <lb/>primæ, & tertiæ &longs;e habet ad &longs;ecundam uelut &longs;ecunda ad tertiam,</s></p><p type="main">
<s>Vige&longs;ima prima diffinitio.</s></p><p type="main">
<s>Trium quantitatum analogarum aliæ quidem Geometricæ, <lb/>cùm proportio &longs;imilis e&longs;t: Aliæ Arithmeticæ, cum fuerit æqualis <lb/>exce&longs;&longs;us hucindè: Aliæ mu&longs;icæ cum fuerit proportio primæ ad ter <lb/>tiam multiplex, aut &longs;implex, aut compo&longs;ita exce&longs;&longs;us quæ &longs;implici <lb/>iuncta &longs;it ad multiplicis perfectionem: eadem autem &longs;it proportio <lb/>exce&longs;&longs;us primæ, & &longs;ecundæ ad exce&longs;&longs;um &longs;ecundæ &longs;upra tertiam.</s></p><p type="main">
<s>Velut proportio 6. 4. 3. dupla e&longs;t utrinque, & 6. 3. 2 tripla. </s>
<s>& 28. 24. <lb/>21. & 45. 40. 36. Geometrica uerò & arithmetica facilius continuan­<lb/>tur in quotquot quantitatibus, &longs;ed & mu&longs;ica uelut 12. 8. 6. 4. 3. & <lb/>proportio 8 ad 5 mu&longs;ica e&longs;t: quia proportio 5 ad 4 mu&longs;ica e&longs;t, & <lb/>bene &longs;onans, igitur con&longs;titutis 8. 5. 4. cum 8 ad 4 benè &longs;onet, & 5 <lb/>ad 4, & 4 &longs;it extrema non media inde 8. & 5 benè <expan abbr="&longs;onãt">&longs;onant</expan>. </s>
<s>nam in me­<lb/>dijs <expan abbr="nõ">non</expan> e&longs;t <expan abbr="uerũ">uerum</expan>, ut in 9. 6. 4 bis diapente, & 16. 12. 9 bis diate&longs;&longs;aron.</s></p><p type="main">
<s>Vige&longs;ima &longs;ecunda diffinitio.</s></p><p type="main">
<s>Quantitates quæ &longs;imilem habent proportionem non continua­<lb/>tam, omiologæ appellantur.</s></p><p type="main">
<s>Vige&longs;ima tertia diffinitio.</s></p><p type="main">
<s>Prima operatione con&longs;i&longs;tere dicuntur proportiones, cùm inter <lb/>primo conflatas quantitates con&longs;titerint.</s></p><pb pagenum="4"/><p type="main">
<s>PRIMA Animi communis &longs;ententia.</s></p><p type="main">
<s>Omnis Proportio e&longs;t, aut æqualitatis, aut maior inæqualis, <lb/>aut minor.</s></p><p type="main">
<s>Secunda animi communis &longs;ententia.</s></p><p type="main">
<s>Quilibet numerus tantus dicitur, quanta e&longs;t illius proportio ad <lb/>monadem.</s></p><p type="main">
<s>Dicimus enim quatuor, quod monadem quater contineat. </s>
<s>Et <lb/>duo cum dimidio cùm monadem bis & &longs;emis contineat.</s></p><p type="main">
<s>Tertia animi communis &longs;ententia.</s></p><p type="main">
<s>Proportionem defectus, &longs;eu detractæ quantitatis ad defectum <lb/>e&longs;&longs;e po&longs;&longs;e, ut quantitatis ad quantitatem dicuntur communes ani­<lb/>mi &longs;entcntiæ, quæ ex intellectu &longs;olo terminorum, quod ueræ &longs;int, <lb/>cogno&longs;cuntur. </s>
<s>Si ergo defectus e&longs;t quantitas, & quantitas eiu&longs;dem <lb/>&longs;peciei, quia detrahitur, & defectus non e&longs;t &longs;implicitur, &longs;ed detra­<lb/>cto ergo per quartam petitionem: uel primam diffinitionem erit <lb/>proportio interillas. </s>
<s>Sunt enim ambæ detractæ.</s></p><p type="main">
<s>Quarta animi communis &longs;ententia.</s></p><p type="main">
<s>Inter quantitatem, & defectum minorem quantitate, cuius e&longs;t de <lb/>fectus, e&longs;t proportio, quatenus e&longs;t quantitas. </s>
<s>Sit a b linea, & detra­<lb/>cta quantitas b c, non maior a b & d &longs;it alia quæuis quantitas eiu&longs;­<lb/><figure id="fig3"></figure><lb/><expan abbr="d&etilde;">dem</expan> generis, dico quòd inter d & b c e&longs;t propor­<lb/>tio quatenus b c e&longs;t quantitas, quia &longs;unt eiu&longs;­<lb/>dem generis ideo &longs;unt in aliqua proportione <lb/>per primam diffinitionem. </s>
<s>Sed ut b c e&longs;t defectus, nulla e&longs;t propor­<lb/>tio: quia quanto b c augetur, tanto augetur proportio d ad b c, & <lb/>hoc e&longs;t contra demon&longs;trata ab Euclide.</s></p><p type="main">
<s>Quinta animi communis &longs;ententia.</s></p><p type="main">
<s>Cum proportio producitur ex proportionibus quælibet illa­<lb/>rum dicetur producta diui&longs;a per alteram.</s></p><p type="main">
<s>Sexta animi communis &longs;ententia.</s></p><p type="main">
<s>Æqualium quantitatum &longs;eu proportionum ad tertiam compa­<lb/>rabilium eadem e&longs;t proportio atque uici&longs;sim. </s>
<s>Hæc et&longs;i demon&longs;tre­<lb/>tur ab Euclide, e&longs;t tamen hic generalior: & &longs;atis per &longs;e nota. </s>
<s>Vt &longs;it <lb/>propior animi communi &longs;ententiæ, quàm rei demon&longs;trandæ.</s></p><p type="main">
<s>Septima animi communis &longs;ententia.</s></p><p type="main">
<s>Ad quod quantitas proportionem habet infinitam, id in genere <lb/>illius quantitatis non comprehenditur.</s></p><p type="main">
<s>Nam proportio e&longs;t duarum quantitatum eiu&longs;dem generis com­<lb/>paratio certa: at hæc comparatio certa non e&longs;t: non igitur quantita­<lb/>tes ambæ &longs;unt, aut non eiu&longs;dem generis.</s></p><pb pagenum="5"/><p type="main">
<s>PRIMA Petitio.</s></p><p type="main">
<s>Si fuerit primi ad &longs;ecundum, ut tertij ad quartum, & ex primo in <lb/>&longs;ecundum producatur æquale, aut maius, aut minus primo, uel <lb/>&longs;ecundo, producetur eodem modo ex tertio in quartum &ecedil;quale aut <lb/>maius, aut minus tertio, uel quarto eadem ratione & ordine.</s></p><p type="main">
<s>Secunda petitio.</s></p><p type="main">
<s>Proportiones po&longs;&longs;unt duci, diuidi, iungi, & auferri, & &longs;umi radix <lb/>in eis cuiu&longs;cunque generis, atque earum quantitatis, ut libet, po&longs;&longs;e <lb/>tran&longs;ponere.</s></p><p type="main">
<s>Tertia petitio.</s></p><p type="main">
<s>Proportionis cuiu&longs;uis nomen à denominatore &longs;uprà &longs;cripto, & <lb/>numeratore infrà &longs;cripto &longs;umitur.</s></p><p type="main">
<s>Quarta petitio.</s></p><p type="main">
<s>Diui&longs;a quauis quantitate per aliam eiu&longs;dem generis, quod exit <lb/>proportio dicitur.</s></p><p type="main">
<s>Quinta petitio.</s></p><p type="main">
<s>Qu&ecedil;libet proportio e&longs;t uel inter duas quantitates, uel per unam <lb/>&longs;ignificatur.</s></p><p type="main">
<s>Nam per tertiam petitionem &longs;i &longs;int duæ quantitates, quæ non hæ <lb/>beant unius rationem, nomen &longs;umit proportio à duobus numeris, <lb/>&longs;in autem &longs;it altera monas, erit per &longs;ecundam animi communem &longs;en <lb/>tentiam, proportio numerus ip&longs;e Ideò patet, quod dicitur.</s></p><p type="main">
<s>Sexta petitio.</s></p><p type="main">
<s>Propo&longs;ita proportione quacunque, & monade quantitatem inue <lb/>nire, quæ &longs;e habeat ad monadem in proportione propo&longs;ita.</s></p><p type="main">
<s>Nam cùm per quartam petitionem diui&longs;a quantitate per quan­<lb/>titatem exeat proportio, & numerus ad <expan abbr="monad&etilde;">monadem</expan> &longs;e habeat, ut pro­<lb/>portio, ideo &longs;umpta monade &longs;ecundum illum numerum, ille nume <lb/>rus e&longs;t quantitas quæ&longs;ita.</s></p><p type="main">
<s>Septima petitio.</s></p><p type="main">
<s>Quamlibet quantitatem per aliam eiu&longs;dem generis diuidere <lb/>po&longs;&longs;e.</s></p><p type="main">
<s>Octaua petitio.</s></p><p type="main">
<s>Proportionem in proportionem ducere po&longs;&longs;e: quamuis &longs;int in­<lb/>ter quantitates diuer&longs;i generis.</s></p><p type="main">
<s>Quod dicitur de multiplicatione intelligendum e&longs;t de alijs ope­<lb/>rationibus &longs;uprà enumeratis.</s></p><p type="main">
<s>Nona petitio.</s></p><p type="main">
<s>Monadem &longs;emper &longs;umere in quo cunque genere po&longs;&longs;e propo&longs;i­<lb/>ta proportione.</s></p><pb pagenum="6"/><p type="main">
<s>Nam licet diuidere per &longs;eptimam petitionem quantitatem per <lb/>quantitatem proportionis: & quod exit, e&longs;t proportio per quar­<lb/>tam petitionem, & per &longs;ecundam animi communem &longs;ententiam <lb/>illa proportio e&longs;t numero æqualis: ergo diui&longs;a proportione, per &longs;i­<lb/>milem numerum &longs;tatuetur monas.</s></p><p type="main">
<s>Decima petitio.</s></p><p type="main">
<s>In quouis genere quantitatum &longs;umere po&longs;&longs;e quantitatem, quæ <lb/><arrow.to.target n="marg2"></arrow.to.target><lb/>&longs;e habeat ad monadem in proportione data. </s>
<s>Similem huic propo­<lb/>nit Euclides in lineis generaliter: nos autem contrà generaliter in <lb/>omnibus quantitatibus, &longs;ed de monade tantum.</s></p><p type="margin">
<s><margin.target id="marg2"></margin.target>D<emph type="italics"/>uodecima <lb/>&longs;exti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="main">
<s>Vndecima petitio.</s></p><p type="main">
<s>Monadem in quancunque quantitatem ductam æquale ip&longs;i pro­<lb/>ducere. </s>
<s>Similiter & proportionem æqualem.</s></p><p type="main">
<s>Nam cum aliqua quantitas augeat ducta aliqua minuat, nece&longs;&longs;e <lb/>e&longs;t aliquam e&longs;&longs;e, quæ nec augeat, nec minuat, & hæc e&longs;t monas. <lb/></s>
<s>Idem dico de diui&longs;ione. </s>
<s>Aequalitas etiam ducta, uel diuidens non <lb/><arrow.to.target n="marg3"></arrow.to.target><lb/>mutat proportionem: nec quantitatem ip&longs;am, igitur monas æqua­<lb/>litatem refert. </s>
<s>Quod etiam e&longs;t per&longs;picuum ex &longs;upradictis.</s></p><p type="margin">
<s><margin.target id="marg3"></margin.target>S<emph type="italics"/>ecunda ani <lb/>mi <expan abbr="cõmunis">communis</expan> <lb/>&longs;ententia.<emph.end type="italics"/></s></p><p type="main">
<s>Duodecima petitio.</s></p><p type="main">
<s>Cum fuerint quatuor quantitates & ad primam, & tertiam æquè <lb/>multiplicibus a&longs;&longs;umptis, item que ad &longs;ecundam & quartam, & &longs;i mul­<lb/>tiplex primæ maius e&longs;t multiplici &longs;ecundæ, multiplex tertiæ &longs;it ma­<lb/>ius multiplici quartæ, & &longs;i minus minus, & &longs;i æquale æquale, idque<lb/>&longs;emper quouis modo a&longs;&longs;umptis his proportionibus ad primam & <lb/>tertiam, & ad &longs;ecundam & quartam erit proportio primæ ad &longs;ecun <lb/>dam, ut tertiæ ad quartam. </s>
<s>Hæc etiam a&longs;&longs;umitur ab Euclide. </s>
<s>Et per <lb/><arrow.to.target n="marg4"></arrow.to.target><lb/>hanc intelligimus etiam conuer&longs;am.</s></p><p type="margin">
<s><margin.target id="marg4"></margin.target>Q<emph type="italics"/>uinto<emph.end type="italics"/> E<emph type="italics"/>le. <lb/></s>
<s>diff.<emph.end type="italics"/> 6.</s></p><p type="main">
<s>Tertiadecima petitio.</s></p><p type="main">
<s>Quantitates æquales, atque proportiones in qua&longs;uis quanti­<lb/>tates ductæ eandem &longs;eruant rationem. </s>
<s>Euclides hanc demon&longs;trat, <lb/>nos autem ad uitandum tædium petimus concedi, &longs;ub qua in­<lb/><arrow.to.target n="marg5"></arrow.to.target><lb/>cluduntur diui&longs;io etiam additio, detractio, laterum omnium in­<lb/>uentio.</s></p><p type="margin">
<s><margin.target id="marg5"></margin.target>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>Quartadecima petitio.</s></p><p type="main">
<s>Cùm termini alicuius quantitatis eandem &longs;eruant rationem in <lb/>omnibus, & firmi &longs;unt ac &longs;tabiles eiu&longs;dem rationis comparatione <lb/>contentæ partes æqualem &longs;eruant exce&longs;&longs;um, &longs;eu proportionem.</s></p><p type="main">
<s>PROPOSITIO prima.</s></p><p type="main">
<s>Proportionem in proportionem duci e&longs;t &longs;uperiores nume­<lb/>ros atque inferiores inuicem ducere.</s></p><pb pagenum="7"/><p type="main">
<s>Sit proportio lineæ a ad lineam b, ut anguli cad angulum d, &longs;ta­<lb/><arrow.to.target n="marg6"></arrow.to.target><lb/>tuatur e monas in genere a <lb/><figure id="fig4"></figure><lb/>b, & fiat fad e, ut cad d, & du <lb/><arrow.to.target n="marg7"></arrow.to.target><lb/>catur<gap/>a in f & b in e, & pro­<lb/>ducantur g & h. </s>
<s>Quia ergo <lb/><arrow.to.target n="marg8"></arrow.to.target><lb/>fe&longs;t proportio ip&longs;a, erit g ad <lb/><arrow.to.target n="marg9"></arrow.to.target><lb/>a ut c ad d, &longs;ed h e&longs;t æqualis <lb/>b, igitur a ad h ut ad b. </s>
<s>Du­<lb/>cta ergo dicetur proportio a <lb/><arrow.to.target n="marg10"></arrow.to.target><lb/>ad b in proportionem c ad d <lb/>ducendo terminos proportionis, &longs;eu quantitatis recta &longs;cilicet &longs;u­<lb/>periores cum &longs;uperioribus, & inferiores cum inferioribus. </s>
<s>Nam &longs;i <lb/><arrow.to.target n="marg11"></arrow.to.target><lb/>rur&longs;um con&longs;tituantur fad e ut a ad b cùm f &longs;it proportio, & k ad f ut <lb/><arrow.to.target n="marg12"></arrow.to.target><lb/>c ad d, erit k ad e, ut g ad h, k autem fit ex ductu proportionis a ad b, <lb/>quæ e&longs;t fin proportionem c ad d, liquet igitur propo&longs;itum.</s></p><p type="margin">
<s><margin.target id="marg6"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s><margin.target id="marg7"></margin.target>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><margin.target id="marg8"></margin.target>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><margin.target id="marg9"></margin.target>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><margin.target id="marg10"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 2. A<emph type="italics"/>ni­<lb/><gap/>i &longs;entent.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg11"></margin.target>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><margin.target id="marg12"></margin.target>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>Propo&longs;itio <expan abbr="&longs;ecũnda">&longs;ecunnda</expan>.</s></p><p type="main">
<s>Proportio extremorum producitur ex intermedijs.<lb/><arrow.to.target n="marg13"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg13"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Sint a b c quantitates dico proportio­<lb/><figure id="fig5"></figure><lb/>nem a ad c, produci ex proportione a ad b </s></p><p type="main">
<s><arrow.to.target n="marg14"></arrow.to.target><lb/>& b ad c, &longs;tatuantur totidem à monade d e <lb/>f, erúntque ex demon&longs;trantis ab Euclide in <lb/>quinto <expan abbr="Elem&etilde;torum">Elementorum</expan> in eadem proportio­<lb/>ne, ftatuatur ergo d prima quantitas e &longs;e­<lb/>cunda & tertia f quarta. </s>
<s>eritqúe per præce­<lb/><arrow.to.target n="marg15"></arrow.to.target><lb/>dentem proportio productorum ex d in e <lb/>& &longs;it g, & in f & &longs;it h, producta ex propor­<lb/>tionibus d ad e & e ad f, quare ex propor­<lb/>tionibus a ad b & b ad e, &longs;ed ex dictis cum <lb/>e &longs;it eadem, erit proportio d ad f, ut g ad h & proportio, d ad f per <lb/>æquam proportionem ab Euclide demon&longs;tratam, ut a ad c, igitur <lb/><arrow.to.target n="marg16"></arrow.to.target><lb/>proportio a ad c producitur ex proportionibus a ad b & b ad c, & <lb/>e&longs;t proportio ip&longs;a a ad c d numerus, ut o&longs;ten&longs;um e&longs;t.</s></p><p type="margin">
<s><margin.target id="marg14"></margin.target>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><margin.target id="marg15"></margin.target>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><margin.target id="marg16"></margin.target>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>Ex hoc &longs;equitur, quòd cùm fuerit quantitas tertia monas ex pro­<lb/><arrow.to.target n="marg17"></arrow.to.target><lb/>portionibus inuicem ductis producetur prima quantitas.<lb/><arrow.to.target n="marg18"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg17"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 2.</s></p><p type="margin">
<s><margin.target id="marg18"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 3</s></p><p type="main">
<s>Ex hoc &longs;equitur, quòd conuer&longs;a proportio producitur ex con­<lb/>uer&longs;is proportionibus.</s></p><p type="main">
<s>Propo&longs;itio tertia.</s></p><p type="main">
<s>Si proportio ex duabus proportionibus in quatuor terminis <lb/>producatur, ip&longs;a uerò proportio inter duas alias quantitates fue­<pb pagenum="8"/>rit con&longs;tituta: con&longs;urgent trecenti &longs;exaginta modi productionis <lb/>proportionis.</s></p><p type="main">
<s><arrow.to.target n="marg19"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg19"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>H&ecedil;c propo&longs;itio ut præcedens & <expan abbr="&longs;equ&etilde;tes">&longs;equentes</expan> tres ab Alchindo &longs;um­<lb/>ptæ &longs;unt, & ab eo demon&longs;trantur. </s>
<s>Sit ergo proportio a ad b, pro­<lb/><arrow.to.target n="table2"></arrow.to.target><lb/>ducta ex proportione c ad d & e ad f, con&longs;tat <lb/>quòd cum &longs;int &longs;ex quantitates, quòd fieri pote­<lb/>runt quindecim coniugationes, quas po&longs;ui à la­<lb/>tere facilitatis gratia, quibus re&longs;pondent totidem <lb/><arrow.to.target n="table3"></arrow.to.target><lb/>conuer&longs;æ: erunt ergo triginta. </s>
<s>Singulæ autem ha <lb/>rum produci po&longs;&longs;unt duodecim modis: ductis <lb/>duodecim in triginta, fiunt trecenti &longs;exaginta mo <lb/>di. </s>
<s>Et hoc e&longs;t clarum per&longs;e, modo <expan abbr="demõ&longs;tremus">demon&longs;tremus</expan>, <lb/>quod &longs;inguli horum modorum po&longs;sint produ­<lb/>ci duodecim modis, & capiamus ab primam qu&ecedil; <lb/>pote&longs;t produci ex c d & e f: Item ambabus con­<lb/>uer&longs;is d c & fe: & rur&longs;us altera recta altera con­<lb/>uer&longs;a: & hoc bifariam c d & f e, & d c & e f, &longs;unt er­<lb/>go iam quatuor modi. </s>
<s>Totidem ex c e & d f, toti­<lb/>demque ex c f & d e, igitur erunt duodecim mo­<lb/>di, quibus produci po&longs;&longs;e intelligitur propor­<lb/>tio a ad b.</s></p><table><table.target id="table2"></table.target><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"></table.target><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>Propo&longs;itio quarta.</s></p><p type="main">
<s>Si fuerit proportio primi ad &longs;ecundum produ­<lb/>cta ex proportionibus tertij ad quartum, & quin <lb/>ti ad &longs;extum, producetur etiam ex proportione <lb/>tertij ad &longs;extum, & quinti ad quartum.</s></p><p type="main">
<s>Sit proportio a b producta ex proportioni­<lb/><arrow.to.target n="table4"></arrow.to.target><lb/>bus c ad d, & e ad f, dico quod etiam erit produ­</s></p><table><table.target id="table4"></table.target><row><cell>a</cell><cell>b</cell><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><arrow.to.target n="marg20"></arrow.to.target><lb/>cta ex proportionibus c ad f, & e ad d, di&longs;ponan­<lb/>tur ut in figura & fiat ex c in e g, & ex d in fh, ergo <lb/><arrow.to.target n="marg21"></arrow.to.target><lb/>per primam harum g ad h ut a ad b, &longs;ed per præ­<lb/>&longs;uppo&longs;ita in &longs;ecunda productione etiam prode­<lb/>unt g & h, igitur per primam propo&longs;itionem ha­<lb/>rum a ad b proportio producitur ex proportionibus c ad f tertiæ <lb/>&longs;cilicet ad &longs;extam, & e ad d quint&ecedil; ad quartam, quod fuit <expan abbr="propo&longs;itũ">propo&longs;itum</expan>.</s></p><p type="margin">
<s><margin.target id="marg20"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 8. <emph type="italics"/>petit.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg21"></margin.target>I<emph type="italics"/>n<emph.end type="italics"/> 13. <emph type="italics"/>petit.<emph.end type="italics"/></s></p><p type="main">
<s>Propo&longs;itio quinta.</s></p><p type="main">
<s>Si fuerit proportio primi ad &longs;ecundum producta ex proportio­<lb/>ne tertij ad quartum, & quinta ad &longs;extum: erit proportio tertij ad <lb/>&longs;extum producta ex proportionibus primi ad &longs;ecundum, & quar­<lb/>ti ad quintum.</s></p><pb pagenum="9"/><p type="main">
<s>Sit proportio a ad b producta ex proportio­<lb/><arrow.to.target n="marg22"></arrow.to.target><lb/><arrow.to.target n="table5"></arrow.to.target><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>In­<lb/>terponam d e inter c & f, eritque ex &longs;ecunda pro­<lb/>po&longs;itione repetita proportio c ad f producta ex <lb/>tribus proportionibus c ad d, d ad e, e ad f, &longs;ed <lb/>proportiones c ad d, & e ad f producunt pro­<lb/>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"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg22"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><table><table.target id="table5"></table.target><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"></table.target><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>Propo&longs;itio &longs;exta.</s></p><p type="main">
<s>Ex trecentis &longs;exaginta modis producenda­<lb/>rum proportionum triginta &longs;ex tantum e&longs;&longs;e ne­<lb/>ce&longs;&longs;arios.<lb/><arrow.to.target n="table7"></arrow.to.target></s></p><table><table.target id="table7"></table.target><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">
<s>Per quartam enim proportio a ad b produ­<lb/><arrow.to.target n="marg23"></arrow.to.target><lb/>citur bifariam, & ex c ad d, & e ad f, & ex c ad f, & <lb/>e ad d. </s>
<s>& perpræ cedentem c ad f producitur ex <lb/>a ad b, & d ad e, & per quartam rur&longs;us ex a ad e, <lb/>& d ad b. </s>
<s>Et per præcedentem rut&longs;us 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>Quare per præceden­<lb/>tem c ad f ex a ad e, & d ad b, & ita di&longs;ponemus <lb/>hos modos in tabula. </s>
<s>Vides etiam <lb/><arrow.to.target n="table8"></arrow.to.target><lb/>aliquos modos non produci, ut pri­<lb/>mi ad quartum nec ad &longs;extum, & li­<lb/>quet, quòd cùm &longs;int quindecim o­<lb/>mnes modi qui produci po&longs;&longs;e intelli­<lb/>guntur, & nouem tantum producan­<lb/>tur &longs;ex e&longs;&longs;e, qui non producuntur, quos <lb/>&longs;eor&longs;um in tabula coniunxi. </s>
<s>Et con­<lb/>&longs;tat etiam, quod totidem conuer&longs;i &longs;ci­<lb/>licet decem octo <expan abbr="producũtur">producuntur</expan>, de qui­<lb/>bus diximus, ut &longs;int omnes triginta <lb/>&longs;ex, qui con&longs;tat ex duabus propo&longs;i­<lb/>tionibus præmi&longs;sis, & hac tertia, <expan abbr="quã">quam</expan> <lb/>adiungemus &longs;cilicet, quòd propor­<lb/>tio primi ad tertium producatur ex <lb/>proportionibus <expan abbr="&longs;ecũdi">&longs;ecundi</expan> ad quartum, <lb/>& quinti ad <expan abbr="&longs;extũ">&longs;extum</expan>. </s>
<s>Hoc enim ex præ­<lb/>cedentibus non liquet: benè liquet <lb/>permutatis ordinibus, quod &longs;i pro­<lb/>portio primi ad tertium producitur, <pb pagenum="10"/>quod etiam propor­<lb/><arrow.to.target n="marg24"></arrow.to.target><lb/>tio primi ad <expan abbr="quintũ">quintum</expan>. <lb/></s>
<s>Nam tertium, & quin <lb/>tum, item que quartum, <lb/>& &longs;extum non <expan abbr="diffe-rũt">diffe­<lb/>runt</expan> ni&longs;i ordine uolun <lb/>tario. </s>
<s>Ergo interpo&longs;i­<lb/>to e inter a, & c per &longs;e­<lb/>cundam propo&longs;itio­<lb/>nem proportio a ad c <lb/>producitur ex proportionibus a ad <lb/>e, & e ad c, ut ex demon&longs;tratis in præ­<lb/>&longs;enti proportio a ad c producitur ex <lb/>c ad f & b ad d. </s>
<s>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, quæ e&longs;t e ad f per <lb/><expan abbr="&longs;ecũdam">&longs;ecundam</expan> propo&longs;itionem. </s>
<s>Igitur pro­<lb/>portio a ad c producitur ex propor­<lb/>tionibus b ad d &longs;ecundi ad quartum, <lb/>& e ad f quinti ad &longs;extum. </s>
<s>Hæc Al­<lb/>chindus in &longs;uo libello: &longs;ed licet inge­<lb/>nio &longs;a ualde: parum <expan abbr="tam&etilde;">tamen</expan> utilia olim <lb/><expan abbr="erãt">erant</expan> nece&longs;&longs;aria ad intelligendum ma­<lb/>gnam <expan abbr="cõpo&longs;itionem">compo&longs;itionem</expan> Ptolem&ecedil;i, nunc <lb/>po&longs;tquam Heber has &longs;ex quantita­<lb/>tes traduxit ad quatuor, pror&longs;us hæc <lb/>&longs;cientia ulli u&longs;ui e&longs;&longs;e de&longs;ijt.<lb/><arrow.to.target n="table9"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg23"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s><margin.target id="marg24"></margin.target>Modi qui <expan abbr="nõ">non</expan> <lb/>producuntur <lb/>pri. </s>
<s>ad quartu <lb/>pri. </s>
<s>ad &longs;extum <lb/>&longs;ec. </s>
<s>ad <expan abbr="tertiũ">tertium</expan> <lb/>&longs;ec. </s>
<s>ad <expan abbr="quintũ">quintum</expan> <lb/>tert. </s>
<s>ad quint. <lb/></s>
<s>quart. </s>
<s>ad &longs;ext.</s></p><table><table.target id="table8"></table.target><row><cell></cell><cell>Primi ad &longs;ecundum.</cell></row><row><cell>1</cell><cell>tertij ad <expan abbr="quartũ">quartum</expan>, & quin</cell></row><row><cell></cell><cell>ti ad &longs;extum.</cell></row><row><cell>2</cell><cell>tertij ad &longs;extum, & qu<gap/>n</cell></row><row><cell></cell><cell>ti ad quartum.</cell></row><row><cell></cell><cell>Primi ad tertium.</cell></row><row><cell>3</cell><cell>&longs;ecundi ad quartum, &</cell></row><row><cell></cell><cell>quinti ad &longs;extum.</cell></row><row><cell>4</cell><cell>&longs;ecundi ad &longs;extum, &</cell></row><row><cell></cell><cell>quinti ad quartum.</cell></row><row><cell></cell><cell>Primi ad quintum.</cell></row><row><cell>5</cell><cell>&longs;ecundi ad <expan abbr="&longs;extũ">&longs;extum</expan>, & ter-</cell></row><row><cell></cell><cell>tij ad quartum.</cell></row><row><cell>6</cell><cell>&longs;ecundi ad quartum, &</cell></row><row><cell></cell><cell>tertij ad &longs;extum.</cell></row><row><cell></cell><cell>Secundi ad quartum.</cell></row><row><cell>7</cell><cell>primi ad tertium, & &longs;ex</cell></row><row><cell></cell><cell>ti ad quintum.</cell></row><row><cell>8</cell><cell>primi ad quintum, et &longs;ex</cell></row><row><cell></cell><cell>ti ad tertium.</cell></row><row><cell></cell><cell>Secundi ad &longs;extum.</cell></row><row><cell>9</cell><cell>primi ad <expan abbr="quintũ">quintum</expan>, & quar</cell></row><row><cell></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><cell>ti ad quintum.</cell></row><row><cell></cell><cell>Tertij ad quartum.</cell></row><row><cell>11</cell><cell>primi ad &longs;ecundum, &</cell></row><row><cell></cell><cell>&longs;exti ad quintum.</cell></row><row><cell>12</cell><cell>primi ad quintum, & &longs;ex</cell></row><row><cell></cell><cell>ti ad &longs;ecundum.</cell></row><row><cell></cell><cell>Tertij ad &longs;extum.</cell></row><row><cell>13</cell><cell>primi ad &longs;ecundum, &</cell></row><row><cell></cell><cell>quarti ad quintum.</cell></row><row><cell>14</cell><cell>primi ad quintum, &</cell></row><row><cell></cell><cell>quarti ad &longs;ecundum.</cell></row><row><cell></cell><cell>Quarti ad quintum.</cell></row><row><cell>15</cell><cell>&longs;ecundi ad primum, &</cell></row><row><cell></cell><cell>tertij ad &longs;extum.</cell></row><row><cell>16</cell><cell>&longs;ecundi ad &longs;extum, & ter</cell></row><row><cell></cell><cell>tij ad primum.</cell></row><row><cell></cell><cell>Quinti ad &longs;extum.</cell></row><row><cell>17</cell><cell>primi ad &longs;ecundum, &</cell></row><row><cell></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><cell>ti ad &longs;ecundum.</cell></row></table><table><table.target id="table9"></table.target><row><cell>a</cell><cell>e c</cell><cell>a e</cell><cell>e c</cell></row><row><cell></cell><cell></cell><cell>c b</cell><cell>e</cell></row><row><cell></cell><cell></cell><cell>f d</cell><cell>c</cell></row><row><cell></cell><cell></cell><cell></cell><cell>f</cell></row></table><p type="main">
<s>Propo&longs;itio &longs;eptima.</s></p><p type="main">
<s>In modis qui nece&longs;&longs;ariò produ­<lb/>cuntur ex duabus proportionibus, <lb/>cum du&ecedil; quantitates ex illis, qu&ecedil; mo <lb/>dos conficiunt, æquales fuerint: pro­<lb/><arrow.to.target n="table10"></arrow.to.target><lb/>portio producta ad quatuor quanti­<lb/>tates omiologas reducetur.<lb/><arrow.to.target n="marg25"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg25"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><table><table.target id="table10"></table.target><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>Sint &longs;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 &longs;cis, quòd <lb/>modi recepti &longs;unt prima cum &longs;ecunda, tertia uel quinta, & &longs;ecunda <lb/>cum quarta, & &longs;exta, & tertia &longs;imiliter cum ei&longs;dem, & quinta eodem <lb/>modo cum ei&longs;dem: &longs;i igitur du&ecedil; quantitates ex his, qu&ecedil; faciunt pro­<pb pagenum="11"/>portionem productam inter &longs;e fuerint æquales reducetur hæc pro­<lb/>portio ad quatuor quantitates omologas, &longs;cilicer abiectis amba­<lb/>bus æqualibus. </s>
<s>Sit gratia exempli prima æqualis quintæ: & quia <lb/>in octauo modo proportio <expan abbr="&longs;ecũdi">&longs;ecundi</expan> ad quartum producitur ex pro­<lb/>portione primi ad quintum, & &longs;exti ad tertium, ergo per expo&longs;ita <lb/>proportio &longs;ecundi ad quartum, ut &longs;exti ad tertium, & ita permutan­<lb/>do, & conuertendo &longs;ecundi ad &longs;extum, ut quarti ad tertium, & tertij </s></p><p type="main">
<s><arrow.to.target n="marg26"></arrow.to.target><lb/>ad quartum, ut &longs;exti ad &longs;ecundum.</s></p><p type="margin">
<s><margin.target id="marg26"></margin.target>V<emph type="italics"/>ndecima <lb/>petitione.<emph.end type="italics"/></s></p><p type="main">
<s>Propo&longs;itio octaua.</s></p><p type="main">
<s>Si duarum <expan abbr="proportionũ">proportionum</expan> &longs;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 compo&longs;ita.</s></p><figure></figure><p type="main">
<s>Sit proportio una a ad b, alia c ad d, ducatur b in <lb/><arrow.to.target n="marg27"></arrow.to.target><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/>po&longs;itam e&longs;&longs;e ex proportione a ad b, & c ad d. </s>
<s>Quia <lb/><arrow.to.target n="marg28"></arrow.to.target><lb/>enim ex b in c fit e, & ex b in d fit g, erit proportio e <lb/>ad g, ut c ad d, & &longs;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>Sed e & f componunt h, igitur proportio h ad g e&longs;t com <lb/>po&longs;ita ex proportionibus e & f ad g, igitur per communem animi <lb/>&longs;ententiam, & diffinitionem compo&longs;itæ proportionis, proportio h <lb/><arrow.to.target n="marg29"></arrow.to.target><lb/>ad g compo&longs;ita e&longs;t ex proportionibus a ad b, & c ad d, quod e&longs;t <lb/>propo&longs;itum.</s></p><p type="margin">
<s><margin.target id="marg27"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s><margin.target id="marg28"></margin.target>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><margin.target id="marg29"></margin.target>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>Propo&longs;itio nona.</s></p><p type="main">
<s>Si duarum proportionum &longs;uperiores numeri alternatim cum <lb/>inferioribus multiplicentur, minusque productum ex maiore detra­<lb/>hatur, erit re&longs;idui ad productum ex inferioribus proportio uelut <lb/>illa, quæ relinquitur detracta minore proportione ex maiore.</s></p><p type="main">
<s>Hæc eodem modo probatur, ut præcedens, ni&longs;i quod h fit de­<lb/><arrow.to.target n="marg30"></arrow.to.target><lb/>tracto è minore: gratia exempli ex f, & ita ex diffinitione patet pro­<lb/>po&longs;itum.</s></p><p type="margin">
<s><margin.target id="marg30"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>_{m}. <lb/>152.</s></p><p type="main">
<s>Propo&longs;itio decima.</s></p><p type="main">
<s>Si fuerit alicuius quantitatis ad unam partem proportio uelut al <lb/>terius partis ad <expan abbr="&longs;ecũdam">&longs;ecundam</expan> quantitatem erit proportio cuiu&longs;uis quan <lb/>titatis eiu&longs;dem generis ad &longs;ecundam compo&longs;ita proportio ex pro­<lb/>portionibus eiu&longs;dem quantitatis a&longs;&longs;umptæ ad utran que partem pri­<lb/>mæ quantitatis &longs;eor&longs;um.<lb/><arrow.to.target n="marg31"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg31"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><figure></figure><p type="main">
<s>Sit a b quantitas diui&longs;a in c, & &longs;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, & &longs;umatur quædam quantitas e eiu&longs;­<pb pagenum="12"/>dem tamen generis, cum illis dico quòd proportio e ad d e&longs;t com­<lb/>po&longs;ita ex proportionibus e ad a c, & e ad b c. </s>
<s>Po&longs;ita ergo e tan<08> &longs;u­<lb/>periore numero, & a c & c b inferioribus, erit ex octaua propo&longs;itio­<lb/>ne huius proportio productorum ex e in a c, & coniunctorum, & <lb/>ex con&longs;equenti per primam &longs;ecundi Elementorum producti ex e in <lb/>a b ad productum ex a c in c b compo&longs;ita ex proportionibus e ad <lb/>a c, & e ad c b: at quod fit ex a c in c b, e&longs;t æquale ei quod fit ex a b in <lb/>d, eo quòd a b, a c, c b & d &longs;unt omiologæ per decimam&longs;extam &longs;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 e&longs;t compo&longs;ita 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, e&longs;t uelut e <lb/><arrow.to.target n="marg32"></arrow.to.target><lb/>ad d. </s>
<s>per &longs;uppo&longs;ita igitur proportio e ad d e&longs;t compo&longs;ita ex propor<lb/>tionibus e ad a c, & e ad b c, quod fuit demon&longs;trandum.</s></p><p type="margin">
<s><margin.target id="marg32"></margin.target>13. P<emph type="italics"/>etit.<emph.end type="italics"/></s></p><p type="main">
<s>Propo&longs;itio undecima.</s></p><p type="main">
<s>Proportio aggregati quarumlibet duarum quantitatum ad ag­<lb/>gregatum duarum æqualium quantitatum e&longs;t compo&longs;ita ex pro­<lb/>portionibus primis, & diui&longs;a per duplam.<lb/><arrow.to.target n="marg33"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg33"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Sit proportio a ad c, & b ad d, & &longs;int c & d <lb/><figure id="fig6"></figure><lb/>æquales, dico quòd proportio a b ad c d e&longs;t <lb/>compo&longs;ita ex proportionibus a ad c, & b ad <lb/>d diui&longs;o compo&longs;ito per duplam. </s>
<s>Quia enim </s></p><p type="main">
<s><arrow.to.target n="marg34"></arrow.to.target><lb/>c & d &longs;unt æquales, erit b ad c, ut b ad d, qua­<lb/>re ex diffinitione cùm proportio a b ad c d <lb/><arrow.to.target n="marg35"></arrow.to.target><lb/>&longs;it compo&longs;ita ex proportionibus a ad c, & b <lb/>ad c, erit etiam compo&longs;ita ex dictis ex propo&longs;itione a ad c, & b ad d, <lb/><arrow.to.target n="marg36"></arrow.to.target><lb/>&longs;tatuatur ergo e æqualis c d media inter a b & c. </s>
<s>Et erit per &longs;ecun­<lb/>dam propo&longs;itionem proportio aggregati a b ad c producta ex <lb/><arrow.to.target n="marg37"></arrow.to.target><lb/>proportione aggregati a b ad c, & e ad c, igitur proportio a b ad e <lb/>erit proportio a b ad c, diui&longs;a per proportionem e ad c, &longs;ed e ad c e&longs;t <lb/><arrow.to.target n="marg38"></arrow.to.target><lb/>dupla: igitur proportio a b ad c d e&longs;t proportio a b ad c diui&longs;a per <lb/>duplam.</s></p><p type="margin">
<s><margin.target id="marg34"></margin.target>E<emph type="italics"/>x &longs;exta<emph.end type="italics"/> A<emph type="italics"/>nim. <lb/></s>
<s>com. </s>
<s>&longs;ententia.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg35"></margin.target>D<emph type="italics"/>ecimaquarta<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg36"></margin.target>13. P<emph type="italics"/>etit.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg37"></margin.target>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><margin.target id="marg38"></margin.target>P<emph type="italics"/>er quintam<emph.end type="italics"/><lb/>A<emph type="italics"/>nim. </s>
<s>com. </s>
<s>&longs;en <lb/>tentiam.<emph.end type="italics"/></s></p><p type="main">
<s>Propo&longs;itio duodecima.</s></p><p type="main">
<s>Propo&longs;itis duabus proportionibus unam alteri iungere ab&longs;que <lb/>multiplicatione.<lb/><arrow.to.target n="marg39"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg39"></margin.target>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>Sint propo&longs;itæ proportiones a ad c & <lb/><figure id="fig7"></figure><lb/>b ad d, & a&longs;&longs;umo e ad c, iuxta ea quæ Eu­<lb/>clides demon&longs;trauit, ut b ad d, erit igitur </s></p><p type="main">
<s><arrow.to.target n="marg40"></arrow.to.target><lb/>proportio a e ad c, compo&longs;ita ex proportionibus a ad c, & e ad c, <lb/>&longs;ed proportio e ad c e&longs;t, ut b ad d, igitur proportio a e ad c compo­<lb/>&longs;ita e&longs;t ex proportionibus a ad c, & b ad d.</s></p><p type="margin">
<s><margin.target id="marg40"></margin.target>E<emph type="italics"/>x generali <lb/>com.<emph.end type="italics"/> A<emph type="italics"/>nim. </s>
<s>&longs;en <lb/>tentia.<emph.end type="italics"/></s></p><p type="main">
<s>Aliter ex b in c fiat fex a in d, g ex c in d h coniunctum ex f g, k.</s></p><pb pagenum="13"/><figure></figure><p type="main">
<s>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, &longs;ed a ad c, ut g ad h igi <lb/><arrow.to.target n="marg41"></arrow.to.target><lb/>tur a e ad c, ut k ad h, &longs;ed k ad h cómponitur ex <lb/>proportionibus a ad c, & b ad d. </s>
<s>Ex octaua ha <lb/>rum igitur proportio a c ad c compo&longs;ita e&longs;t ex <lb/>ei&longs;dem. </s>
<s>For&longs;an quis dicat hanc eandem e&longs;&longs;e <lb/>octauæ &longs;ed <expan abbr="nõ">non</expan> e&longs;t, in illa enim proportio com­<lb/>paratur ad productum, in hac ad unam ex <lb/>quantitatibus.</s></p><p type="margin">
<s><margin.target id="marg41"></margin.target>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>Ex hoc &longs;equitur quòd: Quælibet duæ quantitates quarum ag­<lb/><arrow.to.target n="marg42"></arrow.to.target><lb/>gregatum e&longs;tidem ad eam quantitatem, componunt eandem pro­<lb/>portionem.</s></p><p type="margin">
<s><margin.target id="marg42"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Propo&longs;itio tertiadecima.</s></p><p type="main">
<s>Proportio confu&longs;a aggregati primæ & tertiæ quatuor quantita­<lb/>tum omiologarum ad <expan abbr="aggregatũ">aggregatum</expan> &longs;ecundæ & quartæ, e&longs;t uelut com <lb/>po&longs;ita ex ei&longs;dem diui&longs;a per duplam.<lb/><arrow.to.target n="marg43"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg43"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Sint a ad b, ut c ad d, dico, quòd erit confu&longs;a <lb/><arrow.to.target n="table11"></arrow.to.target><lb/>proportio a c aggregati ad <expan abbr="aggregatũ">aggregatum</expan> b d, com <lb/>po&longs;itæ ex his proportionibus diui&longs;æ per du­<lb/>plam æqualis. </s>
<s>Erit enim aggregati ex a c ad aggregatum ex b d, ue­<lb/>lut a ad b per 18 quinti Elementorum. </s>
<s>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 uerò ex a in d <lb/>e&longs;t æquale producto ex b in c per decimam&longs;extam &longs;exti Elemento­<lb/>rum, & proportio producti ex b in c ad productum ex b in d e&longs;t ue <lb/>lut c ad d, quare ut aggregati a c ad aggregatum b d, igitur propor­<lb/>tio compo&longs;ita ex a ad b, & c ad d, e&longs;t uelut confu&longs;a bis &longs;umpta. </s>
<s>Igi­<lb/>tur confu&longs;a e&longs;t uelut compo&longs;ita diui&longs;a per duplam per modum un­<lb/>decimæ huius.</s></p><table><table.target id="table11"></table.target><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>Propo&longs;itio quartadecima.</s></p><p type="main">
<s>Proportiones confu&longs;æ, & coniunctæ in tribus quantitatibus in­<lb/>uicem commutantur.</s></p><figure></figure><p type="main">
<s>Sint tres quantitates, dico, quod proportio c </s></p><p type="main">
<s><arrow.to.target n="marg44"></arrow.to.target><lb/>ad a b confu&longs;a e&longs;t, conuer&longs;a coniunctæ a & b ad <lb/><arrow.to.target n="marg45"></arrow.to.target><lb/>c. </s>
<s>Nam per dicta proportio a b ad c efficit con­<lb/>iunctam ex a b ad c: &longs;ed c ad a b conuer&longs;a e&longs;t eius, quæ e&longs;t a b ad c, & <lb/>proportio c ad a b e&longs;t confu&longs;a eius, quæ e&longs;t c ad a & b. </s>
<s>Igitur pro­<lb/>portio confu&longs;a in tribus quantitatibus e&longs;t contraria coniunctæ in <lb/>ei&longs;dem.</s></p><p type="margin">
<s><margin.target id="marg44"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s><margin.target id="marg45"></margin.target>14. <emph type="italics"/>diff.<emph.end type="italics"/></s></p><p type="main">
<s>Ex quauis ergo illarum data, data erit & reliqua.<lb/><arrow.to.target n="marg46"></arrow.to.target></s></p><pb pagenum="14"/><p type="margin">
<s><margin.target id="marg46"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 18. <emph type="italics"/>diff.<emph.end type="italics"/></s></p><p type="main">
<s>Propo&longs;itio quintadecima.</s></p><p type="main">
<s>Si fuerint quatuor quantitas-proportio confu&longs;a aggregati pri­<lb/>mæ & tertiæ ad aggregatum &longs;ecundæ, & quartæ erit ut monadis <lb/>addito prouentu, qui fit diui&longs;a differentia differentiarum primæ & <lb/>&longs;ecundæ, atque quartæ & tertiæ per aggregatum tertiæ, & quartæ ad <lb/>ip&longs;am monadem.</s></p><figure></figure><p type="main">
<s>Sint quatuor quantitates a b, c, d, e f, & <lb/><arrow.to.target n="marg47"></arrow.to.target><lb/>&longs;it a b maior cin a h, & e fmaior d in f g, & <lb/>differentia f g & a h &longs;it a k: dico proportio­<lb/>nem a b, & d confu&longs;am ad c & e f, e&longs;&longs;e ut mo <lb/>nadis addito prouentu, uel detracto a k diui&longs;æ per aggregatum c. <lb/></s>
<s>& e f ad ip&longs;am monadem, & manife&longs;tum e&longs;t, quòd pote&longs;t continge­<lb/>re pluribus modis: Primus ut a b &longs;it maior c & e f minor d, & tunc <lb/>differentiæ coniungentur, & prouentus, addetur monadi. </s>
<s>Idem fa­<lb/>ciendum erit &longs;i a b &longs;it maior c, & e f &longs;it minor d, &longs;ed exce&longs;&longs;us &longs;uperet <lb/>defectum. </s>
<s>At &longs;i uel a b &longs;it minor c, & e f maior d, uel ita minor, ut c <lb/>exce&longs;&longs;us &longs;upra b a &longs;it maior defectu, detrahemus prouentum à mo­<lb/>nade. </s>
<s>Alia cautio e&longs;t quòd &longs;i fuerint utrinque exce&longs;&longs;us, aut defectus, <lb/>minuemus minorem de maiore: &longs;i autem unus &longs;it exce&longs;&longs;us alter de­<lb/>fectus, iungemus illos, & po&longs;t diuidemus. </s>
<s>uno ergo demon&longs;trato <lb/>ut pote primo intelligentur reliqui. </s>
<s>Quia ergo b h e&longs;t æqualis c & <lb/>e g æqualis d & h k æqualis g f, erit ex communi animi &longs;ententia ag <lb/>gregatum ex d & k b æquale aggregato ex c & e f, igitur per dicta <lb/>proportio aggregati ad aggregatum e&longs;t unum. </s>
<s>at uerò diui&longs;a k a <lb/>per c & e f fit quantum diui&longs;a eadem per b k, & d, &longs;ed diui&longs;a k a per b <lb/>k, & d iunctas, exit proportio a k ad aggregatum b k & d: igitur di­<lb/>ui&longs;a a k per aggregatum e f & c, exibit eadem proportio, igitur a b <lb/>& d ad aggregatum c & e f e&longs;t coninncta ex monade & proportio­<lb/>ne a k ad aggregatum c & e f, quod erat demon&longs;trandum.</s></p><p type="margin">
<s><margin.target id="marg47"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><figure></figure><p type="main">
<s>Ex hoc patet quod proportionum confu&longs;io <lb/><arrow.to.target n="marg48"></arrow.to.target><lb/>fit iunctis denominatoribus numeratoris: mul­<lb/>tiplicatio multiplicatis: additio multiplicatis <lb/>decu&longs;&longs;atim in numeratores ad productum ex <lb/>denominatoribus, ut in exemplis.</s></p><p type="margin">
<s><margin.target id="marg48"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Propo&longs;itio &longs;extadecima.</s></p><p type="main">
<s>Omnium quatuor quantitatum propo&longs;ita <lb/>prima, quæ non minorem habet proportionem <lb/>ad &longs;uam corre&longs;pondentem, quàm alia ad aliam <lb/><figure id="fig8"></figure><lb/>erit proportio confu&longs;a illarum, ut pro­<lb/>ducti ex aggregato primæ & tertiæ in <pb pagenum="15"/>tertiam, ad productum ex aggregato tertiæ & omiotatæ ad &longs;ecun­<lb/>dam in ip&longs;am quartam.</s></p><p type="main">
<s>Hæc magis reducit confu&longs;am proportionem ad notitiam, quàm, <lb/>præcedens, quia reducit ad proportionem <expan abbr="productã">productam</expan>, qu&ecedil; operatio <lb/>e&longs;t &longs;implici&longs;sima, &longs;iue per multiplicationem quantitatum fiat, duæ <lb/>&longs;unt tantum multiplicationes, &longs;iue per eundem terminum &longs;ufficit <lb/>alium addere. </s>
<s>Summatur ergo a b, c, d & e, & non &longs;it maior propor­<lb/>tio d ad e, quàm a b ad c, & &longs;tatuatur tunc prima a b, &longs;ecunda c, ter­<lb/>tia d, quarta e, & po&longs;tquam non e&longs;t minor ratio a b ad c, quàm d ad <lb/>e, &longs;umatur a f ad c, ut d ad e. </s>
<s>licet enim hoc facere. </s>
<s>Dico quod pro­<lb/>portio confufa a b & d ad c & e e&longs;t uelut producti ex aggregato a b <lb/>& d in d ad productum ex aggregato a f & d in e. </s>
<s>Statuatur aggre­<lb/><arrow.to.target n="marg49"></arrow.to.target><lb/>gatum a b & d linea a d prima quantitas, & aggregatum a f & d, <lb/><figure id="fig9"></figure><lb/>a d &longs;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 propo&longs;itionem g ad h pro­<lb/><arrow.to.target n="marg50"></arrow.to.target><lb/>ducta ex proportionibus a b d ad <lb/>a f d, & d ad c. </s>
<s>Sed proportio a f d <lb/>ad aggregatum c e, e&longs;t uelut d ad <lb/>e. </s>
<s>Proportio uerò 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 &longs;ecundam propo&longs;itionem, harum igitur con­<lb/>&longs;u&longs;a a b ad c, & d ad e, & e&longs;t proportio a b d ad c & e, producuntur <lb/>ex proportionibus a b d ad a f d, & d ad e. </s>
<s>Ergo proportio g ad h <lb/>e&longs;t confu&longs;a ex a b ad e, & d ad e, quod erat demon&longs;trandum.</s></p><p type="margin">
<s><margin.target id="marg49"></margin.target>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><margin.target id="marg50"></margin.target>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>Propo&longs;itio decima&longs;eptima.</s></p><p type="main">
<s>Omnes du&ecedil; proportiones conuer&longs;æ producunt æqualem pro­<lb/>portionem.<lb/><arrow.to.target n="table12"></arrow.to.target></s></p><table><table.target id="table12"></table.target><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>Sint duæ proportiones a ad b & b ad a conuer&longs;a, <lb/><arrow.to.target n="marg51"></arrow.to.target><lb/>dico, quòd producunt proportionem æqualem. </s>
<s>fiat <lb/>enim b ad c, ut b ad a, erit igitur a æqualis c & b c con <lb/><arrow.to.target n="marg52"></arrow.to.target><lb/>uer&longs;a eius quæ e&longs;t a ad b, &longs;ed per &longs;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><margin.target id="marg51"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s><margin.target id="marg52"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 6. A<emph type="italics"/>ni­<lb/>mi <expan abbr="commun&etilde;">communem</expan> <lb/>&longs;ententiam.<emph.end type="italics"/></s></p><p type="main">
<s>Propo&longs;itio decimaoctaua.</s></p><p type="main">
<s>Si fuerint quotlibet quantitates in continua proportione multi­<lb/>plici præter ultimam: proportio uerò penultimæ ad ultimam qua­<lb/>lis re&longs;idui primæ ad &longs;ecundam, erit primæ ad aggregatum reliqua­<lb/>rum uelut penultimæ ad ultimam.<pb pagenum="16"/><arrow.to.target n="marg53"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg53"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Sint quantitates a b c d in continua proportione multiplici, &longs;ed <lb/>d ad e &longs;it uelut re&longs;idui a & b ad b, dico proportionem a ad b c d e <lb/>e&longs;&longs;e ut d ad e. </s>
<s>Quia enim e&longs;t gnomonis e ad quadratum d, ut d ad e <lb/>ex &longs;uppo&longs;ito erit per coniunctam proportionem c & d ad d & e, u<gap/></s></p><p type="main">
<s><arrow.to.target n="marg54"></arrow.to.target><lb/>d ad e, &longs;ed e gnomo cum quadrato d efficit qua­<lb/><figure id="fig10"></figure><lb/>dratum e, igitur ut c quadrati ad d & eiuncta, ita <lb/>d ad e. </s>
<s>Rur&longs;us, quia b quadrati ad c quadratum, <lb/><arrow.to.target n="marg55"></arrow.to.target><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"></arrow.to.target><lb/>gnomonum b c cum quadrato d ad aggrega­<lb/>tum c d e quadratorum, ut d ad e, &longs;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>Et ita <lb/>repetendo de quotuis quantitatibus in infi <lb/>nitum u&longs;que. </s>
<s>Hæc proponitur ab Archimede in libro de quadrato <lb/>æquali parabolæ, & minus generaliter & pluribus demon&longs;tratur. <lb/></s>
<s>Ego tamen quia e&longs;t generalis, de&longs;cribam illam per corrolarium: ad­<lb/>damque aliud quod ex hoc &longs;equitur.<lb/><arrow.to.target n="marg57"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg54"></margin.target>13. P<emph type="italics"/>ropo&longs;. <lb/></s>
<s>quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg55"></margin.target>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><margin.target id="marg56"></margin.target>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><margin.target id="marg57"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 1.</s></p><p type="main">
<s>Si fuerint quotlibet <expan abbr="quãtitates">quantitates</expan> omnes analogæ præter ultimam, <lb/>&longs;it autem penultima ad ultimam qualis re&longs;idui primæ & &longs;ecundæ <lb/>ad &longs;ecundam, erit proportio primæ ad aggregatum omnium alia­<lb/>rum ueluti penultimæ ad ultimam.</s></p><p type="main">
<s><arrow.to.target n="marg58"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg58"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Hæc enim e&longs;t euidens, quia conuenit ei demon&longs;tratio propo&longs;ita. <lb/><figure id="fig11"></figure><lb/>exemplo autem in numeris à latere <lb/>po&longs;ito uides declarationem. </s>
<s>nam <lb/>proportio 16 ad 32 e&longs;t uelut 27 re&longs;i <lb/>dui primæ & &longs;ecundæ ad ip&longs;am &longs;e­<lb/>cundam &longs;cilicet ad 54.</s></p><p type="main">
<s><arrow.to.target n="marg59"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg59"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 2.</s></p><p type="main">
<s>Ex hoc patet etiam quòd a&longs;&longs;umptis omnibus, &longs;ub multiplicibus <lb/>analogiæ u&longs;que in infinitum prima quantitas e&longs;t multiplex aggre­<lb/>gati omnium reliquarum numero 1 m: quo prima e&longs;t multiplex <lb/>&longs;ecundæ.</s></p><p type="main">
<s><arrow.to.target n="marg60"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg60"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 3.</s></p><p type="main">
<s>Si fuerint quotlibet quantitates in &longs;uper particulari proportio­<lb/>ne analogæ, erit proportio primæ ad aggregatum omnium in infi­<lb/>nitum iuxta proportionem multiplicem conuer&longs;am illius partis.</s></p><p type="main">
<s><arrow.to.target n="marg61"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg61"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Velut collectæ in &longs;e&longs;quialtera duplæ in &longs;exquitertia triplæ in <lb/>&longs;exqui&longs;eptima &longs;eptuplæ. </s>
<s>Vt capio 512 448 392 343, & ita deinceps <lb/>u&longs;que in infinitum aggregatum omnium earum erit 3584. Septu­<pb pagenum="17"/>plum 512, & aggregatum 18. 12. 8. 5 2/3, & ita deinceps in &longs;<gap/>xquialtera <lb/>erit 54 duplum 27 primæ in eo ordine.</s></p><p type="head">
<s>SCHOLIVM.</s></p><p type="main">
<s>Ex quo patet genus demon&longs;trandi nouun & pulchrum: nam <lb/>&longs;upponatur 54, aggregatum duplum 27, primæ igitur addito 27 <lb/>ad 54, cum &longs;it dimidium, & addito 13 1/2, dimidio 27 ad 27, nam ex <lb/>&longs;uppo&longs;ito quantitas &longs;equens e&longs;t &longs;exquialtera ad 27, igitur 81 e&longs;t du­</s></p><p type="main">
<s><arrow.to.target n="marg62"></arrow.to.target><lb/>plum ad 40 1/2. Igitur conuertendo e&longs;t proportio aggregati prioris <lb/>ad 27 e&longs;t dupla, ergo aggregatum e&longs;t 54.<lb/><arrow.to.target n="marg63"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg62"></margin.target>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><margin.target id="marg63"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 4.</s></p><p type="main">
<s>Ex hoc patet eandem generaliter quod proportio maioris quan <lb/>titatis ad aggregatum reliquarum analogarum e&longs;t, uelut eius quod <lb/>prouenit diui&longs;o quadrato maioris termini per differentiam eius, & <lb/>&longs;equentis maioris in eadem proportione ad ip&longs;um maiorem.</s></p><p type="main">
<s><arrow.to.target n="marg64"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg64"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Exemplum &longs;it proportio augens 25 & 35 duarum quintarum, uo <lb/>lo &longs;cire quantum &longs;it aggregatum omnium citra 25, maximam acci­<lb/>pio 35, ulteriorem ad 25, cuius differentia a 25 e&longs;t 10, cum quo diui­<lb/>do 625 quadratum, exit 62 1/2 aggregatum quantitatum. </s>
<s>Et facile po­</s></p><p type="main">
<s><arrow.to.target n="marg65"></arrow.to.target><lb/>re&longs;t demon&longs;trari. </s>
<s>Si quis dicat in qua proportione &longs;unt infinitæ <lb/>quantitates analogæ cum 12, quæiunctæ 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> illæ quantitates, in qua &longs;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, & demon&longs;tratur ex prioribus.</s></p><p type="margin">
<s><margin.target id="marg65"></margin.target>Q<emph type="italics"/>uæftio.<emph.end type="italics"/></s></p><p type="main">
<s>Propo&longs;itio decimanona.</s></p><p type="main">
<s>Si fu erint aliquot quantitates arithmeticæ omiologæ, quarum <lb/>exce&longs;&longs;us &longs;it æqualis minimè, omnibus autem deficientibus &longs;upple­<lb/>menta ad &ecedil;qualitatem maximè adiungantur, erunt quadrata omni­<lb/>um quantitatum æqualium adiecto rur&longs;us quadrato primæ cum <lb/>eo quod fit ex minima primi ordinis in <expan abbr="aggregatũ">aggregatum</expan> omnium quan­<lb/>titatum eiu&longs;dem tripla aggregato quadra­<lb/><figure id="fig12"></figure><lb/>torum omnium quantitatum primi ordinis <lb/><arrow.to.target n="marg66"></arrow.to.target><lb/>pariter acceptis.</s></p><p type="margin">
<s><margin.target id="marg66"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Sint aliquot quantitates a b c d e f g h in <lb/>continua proportione. </s>
<s>Arithmetica di&longs;po&longs;it&ecedil; <lb/>ita ut minima <expan abbr="earũ">earum</expan> qu&ecedil; &longs;it h, &longs;it &ecedil;qualis diffe­<lb/>renti&ecedil; quantitatum <expan abbr="&longs;ecundũ">&longs;ecundum</expan> ordinem di&longs;po <lb/><expan abbr="&longs;itarũ">&longs;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">&longs;upplementa</expan> &longs;in <lb/>gulis harum, quæ &longs;int i k l m n o p, ita ut <expan abbr="o&etilde;s">oens</expan> <lb/>fiant &ecedil;quales <expan abbr="cũ">cum</expan> &longs;uis &longs;upplementis ip&longs;i line&ecedil; <lb/>à maiori. </s>
<s>E&longs;tque <expan abbr="id&etilde;">idem</expan> ac &longs;i e&longs;&longs;ent aliquot quanti <pb pagenum="18"/>tates, & <expan abbr="diuideren&ttilde;">diuiderentur</expan> &longs;ingul&ecedil; <expan abbr="&longs;ecundũ">&longs;ecundum</expan> numerum <expan abbr="illarũ">illarum</expan>, &longs;i quatuor in <lb/>quatuor partes æquales, &longs;i quinque in quinque, &longs;i decem in decem, eara<lb/>tione ut ultima <expan abbr="diuidere&ttilde;">diuideretur</expan>, ubi e&longs;t finis primæ partis, penultima ubi <lb/>e&longs;t finis &longs;ecundæ partis, antepenultima ubi e&longs;t finis tertiæ, & &longs;ic de <lb/>alijs. </s>
<s>Vocabo ergo primas <expan abbr="quãtitates">quantitates</expan> propo&longs;itas a b c d e f g h quan­<lb/>titates primi ordinis, &longs;ed quantitates æquales quæ <expan abbr="con&longs;tãt">con&longs;tant</expan> ex quan <lb/>titatib. </s>
<s>primi ordinis, & fupplementis, appellabo quantitates &longs;ecun <lb/>di ordinis: ex quo patet quòd prima <expan abbr="quãtitas">quantitas</expan> erit ex utro que ordine, <lb/>quia non e&longs;t diui&longs;a, reliquæ omnes differunt, quantitates uerò quas <lb/>adiunxi nominabo <expan abbr="&longs;upplem&etilde;ta">&longs;upplementa</expan>, & &longs;unt una minus <expan abbr="quã">quam</expan> quantitates <lb/>ordinum: ut &longs;i <expan abbr="quãtitates">quantitates</expan> ordinum &longs;int octo, erunt &longs;upplementa &longs;e­<lb/>ptem, & &longs;i quantitates <expan abbr="ordinũ">ordinum</expan>, e&longs;&longs;ent &longs;eptem e&longs;&longs;ent <expan abbr="&longs;upplem&etilde;ta">&longs;upplementa</expan> &longs;ex, <lb/>quia inter &longs;upplementa <expan abbr="nõ">non</expan> <expan abbr="adnumera&ttilde;">adnumeratur</expan> quantitas indiui&longs;a. </s>
<s>Erunt er <lb/>go &longs;upplementa i k l m n o p, quætanto erunt maiora quanto quan <lb/>titates primi ordinis &longs;unt minores, & contrà tanto maiora, quanto <lb/><expan abbr="quãtitates">quantitates</expan> primi ordinis &longs;unt maiores. </s>
<s>quantitates <expan abbr="aũt">aunt</expan> &longs;ecundi ordi <lb/>nis <expan abbr="appellabun&ttilde;">appellabuntur</expan> a, b i, ck, dl, em, fn, go, & hp. </s>
<s>Hæcuolui pluribus <lb/>agere, ut dilucidior e&longs;&longs;et propo&longs;itio. </s>
<s>quæ licet <expan abbr="nõ">non</expan> &longs;it difficilis, e&longs;t <expan abbr="tam&etilde;">tamen</expan> <lb/>confu&longs;a ualde propter multitudinem <expan abbr="quantitatũ">quantitatum</expan> & ordinum. </s>
<s>Dico <lb/>ergo &qring;d aggregatum <expan abbr="quadratorũ">quadratorum</expan> quantitatum &longs;ecundi ordinis pri <lb/>mo quadrato bis repetito, &longs;eu uno addito <expan abbr="cũ">cum</expan> eo quod fit ex minima <lb/>in aggregatum quantitatum primi ordinis e&longs;t <expan abbr="triplũ">triplum</expan> aggregato ex <lb/>quadratis omnibus <expan abbr="quantitatũ">quantitatum</expan> <expan abbr="eiu&longs;d&etilde;">eiu&longs;dem</expan> primi ordinis, & utres exem <lb/>plo facilius innote&longs;cat, &longs;int <expan abbr="quãtitates">quantitates</expan> primi ordinis 8. 7. 6. 5. 4. 3. 2. 1. <lb/>quorum quadrata &longs;int 64. 49. 36. 25. 16. & 9.4 & 1. quæ iuncta <expan abbr="faciũt">faciunt</expan> <lb/>204, dico quod &longs;i &longs;umamus quadrata omnium <expan abbr="quãtitatum">quantitatum</expan> &longs;ecundi <lb/>ordinis, quæ &longs;unt octies 64, & eis addiderimus unum <expan abbr="quadratũ">quadratum</expan> ex <lb/>his, ut fiant nouies 64, & erunt 556, &longs;imul iuncta & eis addamus, &qring;d <lb/>fit ex 1 quantitate minima primi ordinis in 36 aggregatum quanti­<lb/>tatum omnium primi ordinis, & e&longs;t tale <expan abbr="productũ">productum</expan> 36, ut fiat totum <lb/>612, quod tale 612 e&longs;t triplum 204, aggregati <expan abbr="quadratorũ">quadratorum</expan> primi or­<lb/>dinis unius demon&longs;tratio h&ecedil;c e&longs;t. </s>
<s>Quia ex quarta &longs;ecundi Element. <lb/></s>
<s>Euclidis &longs;ingula quadrata <expan abbr="quantitatũ">quantitatum</expan> <expan abbr="diui&longs;arũ">diui&longs;arum</expan> &longs;ecundi ordinis con <lb/>&longs;tant ex quatuor partibus quarum du&ecedil; &longs;unt quadrata partium, reli­<lb/>quæ duæ &longs;unt producta ex partibus <expan abbr="inuic&etilde;">inuicem</expan> bis, & quia h fuit æqua­<lb/>lis 1, & p &ecedil;qualis b, quia &longs;upplementa <expan abbr="fuerũt&ecedil;qualia">fuerunt&ecedil;qualia</expan> mutuò quanti <lb/>tatibus, & ita c æqualis o & k æqualis g & d, æqualis n & l, æqualis <lb/>f, e <expan abbr="aũt">aunt</expan> &ecedil;qualis m. </s>
<s><expan abbr="Sequi&ttilde;">Sequitur</expan> ergo quod &longs;umptis duabus quantitatibus <lb/>&longs;ecundi ordinis hab entibus <expan abbr="&longs;upplem&etilde;ta">&longs;upplementa</expan> mutuò æqualia ip&longs;is quan <lb/>titatibus quod quadrata partium <expan abbr="erũt">erunt</expan> dupla quadratis primarum <lb/>quantitatum: ueluti capio b i &longs;ecundam & h p ultimam, <expan abbr="quarũ">quarum</expan> qua­<pb pagenum="19"/>drata partium &longs;unt quadrata b & i, & h & p, &longs;ed b e&longs;t æqualis p, & h <lb/>æqualis i. </s>
<s>Ergo quatuor quadrata b i & h p &longs;unt dupla quadratis b <lb/>& h, & ita <expan abbr="concludã">concludam</expan> de omnibus ubi duæ quantitates duabus com <lb/>parantur: &longs;ed in e m quia e&longs;t &longs;ola una quantitas, i&longs;tud e&longs;t etiam cla­<lb/>rius, quia quadrata e & m &longs;unt dupla quadrato e &longs;oli eo, quod & m <lb/><arrow.to.target n="marg67"></arrow.to.target><lb/>&longs;unt æquales. </s>
<s>Igitur per demon&longs;trata 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>atuerò 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 qu<gap/>drati a &longs;cilicet &longs;emel, <lb/>quia a e&longs;t ex &longs;ecundo ordine quantitatum, & &longs;emel, quia hoc fuit a&longs;­<lb/>&longs;umptum in Problemate. </s>
<s>Sequitur ut quadrata omnia <expan abbr="quãtitatum">quantitatum</expan> <lb/>&longs;ecundi ordinis, prout &longs;unt diui&longs;a in partes addito quadrato a, &longs;int <lb/>dupla quadratis primarum quantítatum, &longs;imul pariter acceptis. </s>
<s>Re <lb/>liquum e&longs;t modo ut o&longs;tendamus dupla <expan abbr="illorũ">illorum</expan> productorum, cum <lb/>eo quod fit ex minima quantitate, &longs;cilicet h in aggregatum ip&longs;arum <lb/>quantitatum primi ordinis e&longs;&longs;e æquale quadratis, <expan abbr="quantitatũ">quantitatum</expan> eiu&longs;­<lb/>dem primi ordinis pariter acceptis. </s>
<s>Con&longs;tatigitur, quod duplum <gap/><lb/>in b e&longs;t æquale duplo h in ip&longs;um b, quia h & i &longs;unt æquales, & du­<lb/>plum k in ip&longs;um c, e&longs;t æquale quadruplo h in idem c, quia k e&longs;t du­<lb/>pla h, & &longs;imiliter duplum l in ip&longs;um d e&longs;t æquale &longs;excuplo, h in d, <lb/>quia l e&longs;t tripla h, & ita procedendo erunt illa dupla producta æ­<lb/>qualia productis ex h in ip&longs;as quantitates toties &longs;umptis quantus <lb/>e&longs;t numerus, qui prouenit duplicato numero, &longs;ecundum <expan abbr="qu&etilde;">quem</expan> h con <lb/>tinetur in illo &longs;upplemento, exemplum uolo duplum producti lin <lb/>d bis, &longs;cio quòd &longs;upplementum l continet h ter, duplicabo tria & fi­<lb/>ent &longs;ex, <expan abbr="igi&ttilde;">igitur</expan> <expan abbr="duplũ">duplum</expan> lin d æquale e&longs;t &longs;excuplo h in ip&longs;um d. </s>
<s>Quo con­<lb/>&longs;tituto, cum &longs;uppo&longs;itum &longs;it producta illa duplicata cum producto h <lb/>in aggregatum primarum <expan abbr="quãtitatum">quantitatum</expan> e&longs;&longs;e æqualia quadratis ip&longs;a­<lb/>rum quantitatum, igitur addemus <expan abbr="productũ">productum</expan> ex h in &longs;ingulas quan­<lb/>titates productis illis prioribus, & fiet productum h in a &longs;emel, in b <lb/>ter, in c quinquies, in d &longs;epties, in e nouies, in f undecies, in g trede­<lb/>cies, & in h quindecies æquale duplo producti uniu&longs;cuiu&longs;que quan­<lb/>titatis in &longs;uum &longs;upplementum cum producto h in <expan abbr="aggregatũ">aggregatum</expan> ip&longs;a­<lb/>rum quantitarum, at quadratum a e&longs;t &ecedil;quale producto ex h in eam, <lb/>qu&ecedil; talem habet proportionem ad ip&longs;um a, <expan abbr="qual&etilde;">qualem</expan> habet a ad ip&longs;um <lb/><arrow.to.target n="marg68"></arrow.to.target><lb/>h per demon&longs;trata ab Euclide, & pariter de quadrato b, quod e&longs;t &ecedil;­<lb/>quale ei quod fit ex h in eam quæ toties continet b, quotiens b con <lb/>tinet h, & ita quadratum c æquale e&longs;t ei, quod continetur &longs;ub h, & <lb/>habente proportionem ad b eandem, quam b ad h, & &longs;imiliter de <lb/>quadrato c & omnibus reliquis, u&longs;que ad h ip&longs;um. </s>
<s>Gratia ergo exem <pb pagenum="20"/>pli quadratum a, erit æquale producto ex h in omnes quatitates &longs;e­<lb/>cundas, quia quotus e&longs;t numerus quantitatum, totus e&longs;t numerus <lb/>&longs;ecundum quem a continet h, & &longs;imiliter quotus e&longs;t numerus quan <lb/>títatum incipiendo à b, & quotus e&longs;t numerus quantitatum incipi­<lb/>endo à c, toties b uel c <expan abbr="contin&etilde;t">continent</expan> h, & ita de alijs, quadrata ergo om­<lb/>nium quantitatum &longs;imul iuncta &longs;unt æqualia productis ex h in &longs;in­<lb/>gulas illarum toties &longs;umptis, quoties illæ <expan abbr="cõtinent">continent</expan> h, &longs;eu quotus e&longs;t <lb/>numerus illius quantitatis, incipiendo ab h, & <expan abbr="numerãdo">numerando</expan> uer&longs;us a. <lb/></s>
<s>Rur&longs;us dico, quod productum multiplicis cuiuslibet <expan abbr="quãtitatis">quantitatis</expan> in <lb/>minimam, &longs;eu quadratum eiu&longs;dem quantitatis &ecedil;quale e&longs;t producto <lb/>eiu&longs;dem quantitatis, & dupli omnium &longs;equentium primi ordinis in <lb/>ip&longs;am minimam quantitatem, uelut quadratum a e&longs;t æquale produ <lb/>cto ex h in a, & in duplum b c d e f g h, hoc <expan abbr="aut&etilde;">autem</expan> facile e&longs;t probare in <lb/>his quantitatibus, quia &longs;i quadratum a e&longs;t æquale producto h in o­<lb/>mnes quantitates &longs;ecundi ordinis, & omnes quantitates &longs;ecundi or <lb/>dinis &longs;imul &longs;umptæ &longs;unt &ecedil;quales ip&longs;i a, & duplo <expan abbr="reliquarũ">reliquarum</expan> primi or <lb/>dinis, quia tales quantitates &longs;unt æquales &longs;uis &longs;upplementis uici&longs;­<lb/>&longs;im, ut h cum i, k cum g, f cum l, e <expan abbr="cũ">cum</expan> m, ergo tam &longs;upplementa, quàm <lb/>quantitates primi ordinis &longs;unt dimidium quantitatum &longs;ecundi or­<lb/>dinis, ergo duplum quantitatum primi ordinis e&longs;t dimidium quan <lb/>titatum &longs;ecundi ordinis, uerùm de b dico idem accidere, quia qua­<lb/>dratum b e&longs;t &ecedil;quale producto ex h in b, & in duplum reliquarum à <lb/>b, &longs;cilicet duplum c d e f g h, & hoc e&longs;t o&longs;tendere, quod i&longs;t&ecedil; quantita <lb/>tes &longs;unt dimidium totidem quantitatum æqualium b, nam c e&longs;t mi­<lb/>nor b in h, & &longs;upplementum p quod e&longs;t æquale ip&longs;i b, &longs;i tota h p fiat <lb/>æqualis ip&longs;i b, ut pote h q erit ip&longs;a q dempta h æqualis ip&longs;i c, ergo <lb/>quantitates primi ordinis &longs;emper &longs;unt æquales &longs;upplementis non <lb/>ueris, &longs;ed prioris quantitatis a&longs;&longs;umptæ, &longs;eu in comparatione ad il­<lb/>lam, quadratum igitur b e&longs;t æquale producto ex h in b, & in duplum <lb/>c d e f g h, & &longs;imiliter per eadem, quadratum c e&longs;t æquale producto <lb/>ex h in c, & in duplum d e f g h, & &longs;ic de alijs. </s>
<s>Habemus ergo, quod <lb/>quadrata a b c d e f g h &longs;imul iuncta &longs;unt æqualia producto ex h in <lb/>a, & in duplum reliquarum, & ex h in b, & in duplum reliquarum <lb/>&longs;equentium, & producto ex h in c &longs;emel, & in duplum &longs;equentium <lb/>u&longs;que ad h, & ita de reliquis. </s>
<s>hoc enim e&longs;t, quod nuper demon&longs;traui­<lb/>mus. </s>
<s>Antea quo que <expan abbr="demõ&longs;tratum">demon&longs;tratum</expan> e&longs;t, 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 &ecedil;quale productis ex h in a &longs;emel, & in b ter, & in c quinquies, in <lb/>d &longs;epties, in e nouies, in fundecies, in g tredecies, in &longs;eip&longs;am h quin­<lb/>decies, detractis ergo p <expan abbr="ordin&etilde;">ordinem</expan>, &qring;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 &longs;emel <pb pagenum="21"/>cum &longs;uis duplicatis &longs;equentibus, & in c, & in d, & in reliquis pa­<lb/>riter conduplicatis &longs;uis &longs;equentibus ex altera, quod fit ex h in b &longs;e­<lb/>mel, in c ter, in d quinquies, in e &longs;epties, in f nouies, in g undecies, <lb/>in h tredecies, detractis ergo rur&longs;us quod fit ex h in b &longs;emel, & ex <lb/>h in c d e f g h bis relinquetur, quod fit ex h in c, & duplo &longs;equen­<lb/>tium, & d & duplo &longs;equentium, & e & aliarum pariter: & ex alia <lb/>parte, quod fit ex h in c &longs;emel, & in d ter, & in e quinquies, in f &longs;e­<lb/>pties, in g nouies, in h undecies. </s>
<s>Ab his rur&longs;us detractis, quòd fit <lb/>ex h in c &longs;emel, & in &longs;equentes bis, relinquetur h in d &longs;emel cum &longs;uis <lb/>&longs;equentibus bis, & in e &longs;emel cum &longs;uis &longs;equentibus & in f, & in g & <lb/>in h pariter, & ex alia parte, quod fit ex h in d &longs;emel, in e ter, f quin­<lb/>quies, g &longs;epties, h nouies, ab his rur&longs;us detraho, quod fit ex h in d <lb/>&longs;emel, & in &longs;equentes bis, relinquetur ex una parte, quod fit ex h <lb/>in e f g h cum duplo &longs;equentium ex alia, quod fit ex h in e &longs;e­<lb/>mel, f ter, g quinquies, h &longs;epties, & &longs;imiliter ab his detractis, quod <lb/>fit ex h in e &longs;emel, & bis in &longs;equentes, relinquetur ex una par­<lb/>te; quod fit ex h in f &longs;emel, & in g h bis, & in g &longs;emel, & in h bis, <lb/>& in h &longs;emel, & ex alia, quod fit ex h in f &longs;emel, in g ter, in h quin­<lb/>quies. </s>
<s>Iterum detractis, quod fit ex h in f &longs;emel, & in g h bis com­<lb/>muniter relin quetur, quod fit ex h in g &longs;emel, & in h bis, & in h &longs;e­<lb/>mel, & ex alia parte quod fit ex h in g &longs;emel, & ex h in h ter. </s>
<s>Sed <lb/>i&longs;ta, quæ relicta &longs;unt iam, &longs;unt manife&longs;tè æqualia, ergo etiam pri­<lb/>ma aggregata ab initio fuere æqualia, ergo & æqualia illis qua­<lb/>drata a b c d e f g h his, quæ fiunt, ex h in ea&longs;dem 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, &longs;ed iam his quadratis a b c d e f g h demon&longs;trata &longs;unt e&longs;&longs;e 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 &longs;ecundi ordinis cum <lb/>quadrato a rur&longs;us repetito, & producto h in aggregatum quanti­<lb/>tatum primi ordinis &longs;unt tripla quadratis quantitatum primi ordi­<lb/>nis pariter acceptis, quod fuit propo&longs;itum, & fuit Archimedis in li <lb/>bro de lineis &longs;piralibus, & ego adieci hic propter modum demon <lb/>&longs;trandi, qui e&longs;t eleganti&longs;simus, & procedit ex principijs arithmeti­<lb/>cis, & diuer&longs;is à communibus, & ideo non reuoluitur, ut &longs;olentre­<lb/>liquæ quæ&longs;tiones.</s></p><p type="margin">
<s><margin.target id="marg67"></margin.target>I<emph type="italics"/>n<emph.end type="italics"/> 5. E<emph type="italics"/>l<gap/><emph.end type="italics"/><lb/>P<emph type="italics"/>rop.<emph.end type="italics"/> 12.</s></p><p type="margin">
<s><margin.target id="marg68"></margin.target>L<emph type="italics"/>ib.<emph.end type="italics"/> 6. E<emph type="italics"/>l<gap/>.<emph.end type="italics"/><lb/>P<emph type="italics"/>rop.<emph.end type="italics"/> 17.</s></p><p type="main">
<s>Propo&longs;itio uige&longs;ima.</s></p><p type="main">
<s>Cùm fuerint quatuor quantitates, fueritque &longs;ecunda æqualis ter­<lb/>tiæ, aut primæ æqualis quartæ, erit proportio primæ ad quartam, <lb/>aut tertiæ ad &longs;ecundam producta ex proportionibus primæ ad &longs;e­<lb/>cundam, & tertiæ ad quartam.<lb/><arrow.to.target n="marg69"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg69"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Cùm enim quantitates hæ non fuerint &ecedil;quales, <expan abbr="cõ&longs;tat">con&longs;tat</expan> per &longs;ecun­<pb pagenum="22"/>dam harum, quod proportio primæ ad <expan abbr="quartã">quartam</expan> producitur ex pro­<lb/>portione primæ ad &longs;ecundam, &longs;ecund&ecedil; ad tertiam, & terti&ecedil; ad quar <lb/>tam: ergo non ex &longs;olis proportionibus primæ ad &longs;ecundam, & ter­<lb/>tiæ ad quartam, & &longs;imiliter ex prima harum proportio prim&ecedil; ad &longs;e­<lb/>cundam, & tertiæ ad quartam producunt proportionem producti <lb/>primæ in &longs;ecundam ad productum tertiæ in quartam. </s>
<s>Et in multi­<lb/>plicatione proportio, quæ &longs;olet e&longs;&longs;e inter producta illa, & e&longs;t qua&longs;i <lb/>duplicata e&longs;t inter ip&longs;as quantitates. </s>
<s>Sint igitur quantitates a b c d, <lb/>& &longs;it b æqualis c, ponantur ergo recto ordine a b c d, eritque propor <lb/><figure id="fig13"></figure><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>proportio c ad f, erit igitur pro­<lb/>portio e ad f, &longs;i multiplicetur per pro­<lb/>portionem b ad c eadem quæ prius, & </s></p><p type="main">
<s><arrow.to.target n="marg70"></arrow.to.target><lb/>producta iam e&longs;t eadem ei, quæ e&longs;t a <lb/>ad d, ergo proportio a ad d erit producta ex proportionibus a ad <lb/>b, c ad d per primam propo&longs;itionem. </s>
<s>Quod uerò diximus de pri­<lb/>ma & quarta &longs;i &longs;int æquales, manife&longs;tum e&longs;t, quòd res redit ad idem <lb/>&longs;olum tran&longs;mutato ordine, ut tertia, & quarta præmittantur prim&ecedil;, <lb/>& &longs;ecundæ. </s>
<s>Hæcigitur propo&longs;itio nihil aliud innuit, quàm quod <lb/>in hoc ca&longs;u productio, quæ&longs;olet fieri ex tribus proportionibus fiat <lb/>ex duabus tantum.</s></p><p type="margin">
<s><margin.target id="marg70"></margin.target>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>Propo&longs;itio uige&longs;imaprima.</s></p><p type="main">
<s>Cùm decu&longs;&longs;atim ducta fuerit prima in quartam, & &longs;ecunda in ter <lb/>tiam; productumque primæ in quartam diui&longs;um fuerit per produ­<lb/>ctum &longs;ecundæ in tertiam erit proportio primæ ad &longs;ecundam diui­<lb/>&longs;a per proportionem tertiæ ad quartam. </s>
<s>Et &longs;imiliter interpo&longs;ita <lb/>omiologa.<lb/><arrow.to.target n="marg71"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg71"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><figure></figure><p type="main">
<s>Primum exponamus &longs;ecundam partem, &longs;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"></arrow.to.target><lb/>ergo per <expan abbr="&longs;ecũdam">&longs;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/>diui&longs;a proportione a ad b per proportionem c ad d exit proportio <lb/>a ad e, & hic e&longs;t &longs;ecundus modus. </s>
<s>Primus autem modus ducatur a <lb/>in d & fiat f, & b in c & fiat g, dico proportione f ad g e&longs;&longs;e 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, & &longs;imiliter quia k producitur ex d in g, & g producitur ex b in <pb 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>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 &longs;it eadem, quæ a ad b, ergo <lb/>proportio a ad b producitur ex c ad d, & f ad g, ergo diui&longs;a propor­<lb/>tione a ad b prodibit proportio f ad g, quod fuit propo&longs;itum.</s></p><p type="margin">
<s><margin.target id="marg72"></margin.target>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>Propo&longs;itio uige&longs;ima&longs;ecunda.</s></p><p type="main">
<s>Cùm fuerit proportio primæ ad &longs;ecundam maior, quàm tertiæ <lb/>ad quartam, erit confu&longs;a ex his maior quàm tertiæ ad quartam, mi­<lb/>nor autem quàm primæ ad &longs;ecundam.</s></p><figure></figure><p type="main">
<s>Sit proportio a ad b maior quàm c <lb/><arrow.to.target n="marg73"></arrow.to.target><lb/>ad d, dico, quod confu&longs;a ex a c ad b d <lb/>e&longs;t maior, quàm c ad d, et minor quàm <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"></arrow.to.target><lb/>rum e c ad b d confu&longs;a minor quàm a c ad b d, nam e e&longs;t minor a, <lb/>quia proportionem habent minorem ad b quam a eo quòd e ha­<lb/>bet proportionem ad b, quam c ad d, quæ <expan abbr="aut&etilde;">autem</expan> c ad d minor, quám <lb/>a ad b, ut &longs;uppo&longs;itum e&longs;t, igitur e c ad b d minor, quàm a b ad c d, e b <lb/>autem ad c d e&longs;t, ut demon&longs;tratum e&longs;t qualis c ad d, ergo c ad d mi­<lb/>nor, quàm confu&longs;a a b ad c d, quod e&longs;t &longs;ecundum per idem proba­<lb/>bitur, & primum po&longs;ita f ad d, ut a ad b, eritque a maior c, igitur ma­<lb/>ior proportio a f ad b d, quàm a c ad b d, &longs;ed a f ad b d, ut a ad b per <lb/>candem tertiamdecimam huius ergo proportio confu&longs;a a b ad c d <lb/>e&longs;t minor, quàm a ad b.</s></p><p type="margin">
<s><margin.target id="marg73"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s><margin.target id="marg74"></margin.target>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>Propo&longs;itio uige&longs;imatertia.</s></p><p type="main">
<s>Omnis motus naturalis ad locum &longs;uum e&longs;t: ideo per rectam li­<lb/>neam fit.<lb/><arrow.to.target n="marg75"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg75"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Motus naturalis e&longs;t, ut con&longs;eruetur corpus, & conueniat locus <lb/>corpori, igitur fit ad &longs;uum locum. </s>
<s>Locus autem dicitur in compara<lb/>tione ad uniuer&longs;um. </s>
<s>ideo omnis motus naturalis e&longs;t à centro mun­<lb/>di &longs;ur&longs;um, uel ad centrum deor&longs;um. </s>
<s>Et quia quanto natura celerius <lb/>&longs;uum finem pote&longs;t a&longs;&longs;equi (quia finis bonus e&longs;t aliter non illum ap­<lb/>peteret) eum quærit, cùm &longs;it &longs;apienti&longs;simæ uitæ mini&longs;tra: at linea re­</s></p><p type="main">
<s><arrow.to.target n="marg76"></arrow.to.target><lb/>cta breui&longs;sima e&longs;t Euclide te&longs;te à puncto ad punctum, igitur omnis <lb/>motus naturalis e&longs;t &longs;ur&longs;um aut deor&longs;um per rectam lineam.</s></p><p type="margin">
<s><margin.target id="marg76"></margin.target>D<emph type="italics"/>i&longs;t. </s>
<s>tertia <lb/>primi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="main">
<s>Propo&longs;itio uige&longs;imaquarta.</s></p><p type="main">
<s>Omnis motus circularis uoluntarius e&longs;t.</s></p><p type="main">
<s>Sit motus in circulo &longs;eu per circulum in orbe cuius &longs;it centrum, <lb/>&longs;it c mundi centrum: igitur ex diffinitione circuli tantum di&longs;tabit a, <lb/>quantum b ab ip&longs;o c: &longs;ed in motu naturali per pr&ecedil;cedentem nece&longs;&longs;e <lb/>e&longs;t, ut recta feratur ad c, uel recedat, igitur motus a e&longs;t uoluntarius, <pb pagenum="24"/><figure id="fig14"></figure><lb/>non naturalis. </s>
<s>nam &longs;i uiolentus e&longs;&longs;et, non <lb/>e&longs;&longs;et perpetuus. </s>
<s>Omnia ergo a&longs;tra feruntur <lb/>circa centrum mundi. </s>
<s>Sit modo rota e f g, di <lb/>co enon moueri motu circulari nam linea <lb/>e <expan abbr="clõgior">clongior</expan> e&longs;t g c, ergo recta mouetur ad cen <lb/>trum non circa centrum. </s>
<s>Indicio etiamid <lb/>e&longs;t: quòd &longs;i in e ponatur fru&longs;tum aliquod <lb/>in&longs;igne plumbi in motu ad g per f de&longs;cen­<lb/>det raptim: at dum ex g in e magna cum dif­<lb/>ficultate, igitur motus hic non e&longs;t naturalis, <lb/>nec circularis. </s>
<s>nihil etiam hoc modo &longs;ponte mouetur. </s>
<s>Sed cum non <lb/>moueatur per rectam naturaliter, nec æquidi&longs;tans à centro per cir­<lb/>culum relinquitur, ut moueatur motu uiolento, aut mi&longs;to, &longs;ed non <lb/>ex uoluntario, cum nullo modo moueatur æquidi&longs;tans à centro, <lb/>&longs;ed &longs;emper ab e lineæ ad centrum fiant breuiores, liquet e&longs;&longs;e mo­<lb/>tum uiolentum: aut mi&longs;tum ex naturali, & uiolento.</s></p><p type="main">
<s>Propo&longs;itio uige&longs;imaquinta.</s></p><p type="main">
<s>Tres &longs;unt motus omnino &longs;implices naturalis, uoluntarius & <lb/>uiolentus.<lb/><arrow.to.target n="marg77"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg77"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Tres &longs;unt modi, quibus po&longs;&longs;unt moueri in comparatione ad cen <lb/>trum &longs;cilicet uel recta cum centro, uel æquidi&longs;tando à centro, uel <lb/>neutro modo, igitur tres motus. </s>
<s>Rur&longs;us uel à principio interiore <lb/>non intelligente, & e&longs;t naturalis, uel intelligente & e&longs;t uoluntarius: <lb/>uel exteriore & e&longs;t uiolentus. </s>
<s>Hæc autem diui&longs;io e&longs;t &longs;olum propria <lb/>non prima. </s>
<s>Nam e&longs;t uiolentus in recta ad centrum: ideo omnis, qui <lb/>non e&longs;t in recta ad centrum, nec æquidi&longs;tat, uiolentus e&longs;t: non ta­<lb/>men omnis uiolentus e&longs;t extra rectam. </s>
<s>Attractio autem, quæ fit ob <lb/>raritatem corporum, &longs;eu, ut dicunt, à uacuo, uiolenta e&longs;t non natu­<lb/>ralis ni&longs;i ratione finis, non agentis. </s>
<s>Sunt enim quatuor genera mo­</s></p><p type="main">
<s><arrow.to.target n="marg78"></arrow.to.target><lb/>tus uiolenti ab Ari&longs;totele po&longs;ita, uectio, tractio, pul&longs;io, & uolutio: <lb/>quanquam his non opus &longs;it in demon&longs;tratiua &longs;cientia. </s>
<s><expan abbr="cõ&longs;tat">con&longs;tat</expan> enim <lb/>uolutionem ex tractione, & pul&longs;ione apud illum con&longs;i&longs;tere.</s></p><p type="margin">
<s><margin.target id="marg78"></margin.target>7. P<emph type="italics"/>hy&longs;. <lb/></s>
<s>cap.<emph.end type="italics"/> 2.</s></p><p type="main">
<s>Propo&longs;itio uige&longs;ima.</s></p><p type="main">
<s>Motus ergo compo&longs;iti quatuor nece&longs;&longs;ariò &longs;unt &longs;pecies.</s></p><p type="main">
<s>Si tantum &longs;unt tres &longs;pecies &longs;implicium, con&longs;tat ratione arithme­<lb/>tica quatuor e&longs;&longs;e compo&longs;itorum. </s>
<s>Di&longs;quiramus ergo an &longs;int natura­<lb/>liter tot &longs;pecies, for&longs;an enim repugnabit aliquis alicui. </s>
<s>Porrò uidea­<lb/>mus primò, quot &longs;int uiolentorum &longs;pecies: Prima erit cum non &longs;e­<lb/>cundum rectam lineam fuerit: nec à centro æquidi&longs;tantem. </s>
<s>Secun­<lb/>da cum fuerit &longs;ecundum rectam, &longs;ed non ad centrum. </s>
<s>Tertia cum <lb/>fuerit in recta ad centrum, &longs;ed contrario modo, uelut terræ &longs;ur&longs;um. <pb pagenum="25"/>Quarta cùm in recta ad centrum, &longs;ecundum naturam, &longs;ed <expan abbr="nõ">non</expan> à prin <lb/>cipio naturali. </s>
<s>Velut cum quis proij cit lapidem rectà in terram è <lb/>turri uiolentius, quàm ille &longs;ua grauitate de&longs;cen&longs;urus e&longs;&longs;et. </s>
<s>Hic igi­<lb/>tur motus e&longs;t compo&longs;itus ex naturali, & uiolento. </s>
<s>Animalium au­<lb/>tem motus uoluntarius e&longs;t, cum &longs;it à principio interiore cogno&longs;cen <lb/>te: & &longs;it quatenus à principio in linea circulari æqualiter di&longs;tante à <lb/>centro: &longs;ed quia ob&longs;tat grauitas, ideò mi&longs;tus e&longs;t ex naturali, & uo­<lb/>luntario. </s>
<s>Sed circularis, & uiolentus &longs;oli e&longs;&longs;e non po&longs;&longs;unt: nam uio <lb/>lentus e&longs;t nece&longs;&longs;ariò in corpore graui aut leui: &longs;ed omne corpus gra<lb/>ue aut leue, cùm mouetur, naturaliter mouetur &longs;altem in fine: & per <lb/>totum motum, motu ócculto, qui maximè in hoc libro dignus e&longs;t <lb/>con&longs;ideratione, igitur motus uoluntarius, & uiolentus non po&longs;­<lb/>&longs;unt e&longs;&longs;e &longs;imul &longs;oli. </s>
<s>Eruntergo &longs;ecundum naturam tantùm tres &longs;pe­<lb/>cies. </s>
<s>Velut cùm quis &longs;candit, aut&longs;alit: E&longs;t enim motus naturalis &longs;al­<lb/>tem in fine, & uoluntarius, & uiolentus. </s>
<s>Si quis autem uelit uiolen­<lb/>tum cum uoluntario copulare dicemus con&longs;tare eam compo&longs;itio­<lb/>nem in initio &longs;aliendi. </s>
<s>Motum autem occultum uocamus grauita­<lb/>tem aut leuitatem.</s></p><p type="main">
<s>Propo&longs;itio uige&longs;ima&longs;eptima.</s></p><p type="main">
<s>Motus uoluntarius e&longs;t in loco: naturalis ad locum: uiolentus <lb/>exloco.</s></p><p type="main">
<s>Hæc e&longs;t tertia differentia primarum &longs;pecierum motuum uolun­<lb/>tarius fit manente corpore toto in eodem loco, ideo proprius e&longs;t <lb/>cœlo, corpora autem animalium in eodem loco feruntur: quia in <lb/>eodem orbe nata redire ad proprium locum. </s>
<s>Et ideò, ut dixi, e&longs;t mo <lb/>tus mi&longs;tus ex naturali, & uoluntario, qui &longs;i per &longs;e fieret, non fatiga­<lb/>ret mobile, cùm ex utroque principio ab interiore ui procedat. </s>
<s>Sed <lb/>quia fit per mu&longs;culos, qui trahuntur: hic autem motus e&longs;t uiolen­<lb/>tus, ideò per con&longs;equentiam fatigat. </s>
<s>Qui uerò naturalis, e&longs;t ut re­<lb/>deat corpus ad &longs;uum locum, igitur naturalis e&longs;t ad locum. </s>
<s>Sed <lb/>uiolenti finis e&longs;t, ut protrudatur ex loco in quo e&longs;t, non habens cer­<lb/>tum finem. </s>
<s>licet enim qui trahit, ad &longs;uum locum trabat, non tamen <lb/>ad locum mobilis.</s></p><p type="main">
<s>Propo&longs;itio uige&longs;imaoctaua.</s></p><p type="main">
<s>Motus quilibet naturalis aut uiolentus in aliquo medio fit.<lb/><arrow.to.target n="marg79"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg79"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Cùm uacuum non detur, & omnis motus naturalis &longs;it ad locum, <lb/>et uiolentus ex loco per præcedentem, igitur cùm non &longs;it in medio, <lb/>uacuum erit in aliquo corpore, uelut aere, aqua, igne, ligno.</s></p><p type="main">
<s>Propo&longs;itio uige&longs;imanona.</s></p><p type="main">
<s>Omnis motus uoluntarius æqualis e&longs;t &longs;emper: &longs;impliciter etiam <lb/>quilibet alius motus.</s></p><pb pagenum="26"/><p type="main">
<s><arrow.to.target n="marg80"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg80"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>m.</s></p><p type="main">
<s>Motus uoluntarius non habet, quòd fatiget, & &longs;umma perfectio <lb/>e&longs;t æqualitas, & natura quæ mouet non debilitatur, igitur perpe­<lb/>tuo per&longs;euerat æqualis. </s>
<s>neque enim e&longs;t, ut dixi, per medium corpus. <lb/></s>
<s>Naturalis quoque, & uiolentus cum ratione proportionis mouentis <lb/>&longs;upra mobile per&longs;e non uarientur, & ab &ecedil;quali proportione &ecedil;qua­<lb/>lis uelo citas proueniat, igitur natura tales motus &longs;unt &ecedil;quales, nam <lb/>in utroque mouens, mouet &longs;ecundum ultimam &longs;uam uim.</s></p><p type="main">
<s>Propo&longs;itio trige&longs;ima.</s></p><p type="main">
<s>In omni corpore mobili in medio, partes medij re&longs;i&longs;tunt obuiæ, <lb/>aliæ impellunt.</s></p><p type="main">
<s><arrow.to.target n="marg81"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg81"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Sit mobile a cui partes &longs;ubiaceant directæ b, & &longs;it graue. </s>
<s>Et pa­<lb/>tet ne diuidatur b re&longs;i&longs;tere, cum autem &longs;uperauerit, partes b de&longs;cen­<lb/>dunt ante a, & trahunt partes c & d adh&ecedil;rentes &longs;ecum, atque ita e c d f <lb/><figure id="fig15"></figure><lb/>adiuuant ad de&longs;cen&longs;um partes etiam laterales <lb/>g & h cum a tran&longs;it in b, ne detur uacuum, tran­<lb/>&longs;eunt in k uelo ci motu, ergo propellunt a maio<lb/>reimpetu inferius.</s></p><p type="main">
<s><arrow.to.target n="marg82"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg82"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Ex quo patet, quod in omni motu naturali, <lb/>uel uiolento fit augumentum uelocitatis ab initio &longs;altem u&longs;que <lb/>ad aliquid.</s></p><p type="main">
<s><arrow.to.target n="marg83"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg83"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Et ideò etiam bellicæ machinæ cuiu&longs;cunque generis certam exi­<lb/>gunt di&longs;tantiam, ut uiolentius feriant.</s></p><p type="main">
<s>Propo&longs;itio trige&longs;imaprima.</s></p><p type="main">
<s>Omnis motus naturalis in æquali medio ualidior e&longs;t in fine, <lb/>quàm in principio: uiolentus contrà.</s></p><p type="main">
<s><arrow.to.target n="marg84"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg84"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Cùm enim ex præcedenti augeantur &longs;emper ob medium, & cau­<lb/>fa, quæ mouet, &longs;it perpetua, & à principio æterno, quod per dictæ <lb/>æqualiter mouet, igitur motus ille fiet uelo cior in fine quàm in alia <lb/>parte temporis. </s>
<s>In uiolento autem, cùm perueniat ad finem de&longs;init </s></p><p type="main">
<s><arrow.to.target n="marg85"></arrow.to.target><lb/>uis illa nece&longs;&longs;ariò, quæ mouet, & &longs;uperatur à ui naturali, quæ mo­<lb/>uet in contrarium, igitur antequam ce&longs;&longs;et motus fiet tardi&longs;simus <lb/>in fine.</s></p><p type="margin">
<s><margin.target id="marg85"></margin.target><gap/> 29. P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/></s></p><p type="main">
<s>Ex quo patet, quòd motus quadrifariam mi&longs;ti dicuntur, aut &longs;pe­<lb/><arrow.to.target n="marg86"></arrow.to.target><lb/>cie, ut cùm quis iacit lapidem è turri: uel ex occulto naturali, & uio­<lb/>lento manife&longs;to: uelut cùm quis iacit lapidem, & de&longs;cendit po&longs;tmo <lb/><figure id="fig16"></figure><lb/>dum ex b in c motu utroque manife&longs;to, &longs;ed ex a <lb/>in b motu uiolento manife&longs;to, & naturali oc­<lb/>culto: uel ratione medij, & hoc modo omnis <lb/>motus naturalis etiam non &longs;olum uiolentus e&longs;t <lb/>mi&longs;tus ex proportione uirtutis mouentis, cum motu medij, ad me­<lb/>dium ip&longs;um, uel &longs;i uiolentus &longs;it ex proportione uirtutis mouentis, <pb pagenum="27"/>& medij ad mobile, ac medium, quod re&longs;i&longs;tit. </s>
<s>Quarto ex motibus <lb/>imperfectis natura &longs;ua, & non e&longs;t uera mi&longs;tio, & hoc apparet in mo­<lb/>tibus uoluntarijs animalium, qui non &longs;unt neque æquales, neque perfe <lb/>ctè circa medium: &longs;ed &longs;unt potius &longs;imiles uoluntarijs. </s>
<s>Etideo de­<lb/>mon&longs;trationes illæ Ari&longs;totelis quoad u&longs;um nihil iuuant nos.</s></p><p type="margin">
<s><margin.target id="marg86"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Propo&longs;itio trige&longs;ima&longs;ecunda.</s></p><p type="main">
<s>Omne mobile naturaliter motum, &longs;eu uiolenter uelo cius moue­<lb/>tur in medio rariore, quàm den&longs;iore. </s>
<s>Maior quoque e&longs;t proportio fi­<lb/>nis motus in corpore rariore ad finem motus in corpore den&longs;iore, <lb/>quàm principij. </s>
<s>In uiolento autem celeriùs perueniet ad finem mo <lb/>tus in corpore den&longs;iore.</s></p><figure></figure><p type="main">
<s>A mobile moueatur in b medio rariore, & in c den&longs;io­<lb/><arrow.to.target n="marg87"></arrow.to.target><lb/>re, igitur b minus re&longs;i&longs;tit, quàm c & magis adiuuat, quia <lb/>uelociùs mouetur: igitur duplici de cau&longs;a a mouebitur <lb/>uelociùs in b quàm in c: & quia per corrolarium trige&longs;i­<lb/>mæ, & præcedentis proportio finis (ubi æqualiter moueantur) ad <lb/>&longs;ua principia maior erit in d, quàm in e: ergo per <expan abbr="demõ&longs;trata">demon&longs;trata</expan> à Cam <lb/>pano po&longs;ita d prima, b &longs;ecunda, e tertia, c quarta, maior erit propor­<lb/>tio d ad e, quàm b ad c quod fuit propo&longs;itum in naturali.</s></p><p type="margin">
<s><margin.target id="marg87"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Propo&longs;itio trige&longs;imatertia.</s></p><p type="main">
<s>Omnia duo mobilia æqualis undique magnitudinis, quæ æquali <lb/>in tempore æqualia &longs;patia pertran&longs;eunt in diuer&longs;is &longs;ub&longs;tantia me­<lb/>dijs, nece&longs;&longs;e e&longs;t, ut &longs;it ponderis ad pondus, quemadmodum medij <lb/>ad medium, proportio duplicata.<lb/><arrow.to.target n="marg88"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg88"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Sint duo mobilia a & b magnitudine, & forma omnino paria, <lb/>& &longs;int media c & d, exempli gratia: & pertran&longs;eant æquale &longs;patium <lb/>in utroque in eodem tempore, e dico proportionem ponderis b ad <lb/>pondus a e&longs;&longs;e duplicatam ei quæ e&longs;t raritatis c ad raritatem d. </s>
<s>Quia <lb/>enim feruntur æqualiter, nam in æquali tem­<lb/><figure id="fig17"></figure><lb/>pore, &longs;eu eodem æqualia &longs;patia pertran&longs;e­<lb/>unt, erit proportio potentiæ a cum &longs;uo auxi­<lb/>lio ad id, quod re&longs;i&longs;tit ex c ut b cum &longs;uo au­<lb/>xilio ad id, quod re&longs;i&longs;tit ex d, permutando igi <lb/>tur d ad c, ut b ad a, &longs;ed c ad d proportio rari­<lb/>tatis duplicat actionem, tum minus re&longs;i&longs;ten­<lb/>do, tum adiuuando motum a, igitur proportio differentiæ motus <lb/>e&longs;t duplicata proportioni raritatis: &longs;ed proportio motus e&longs;t æqua­<lb/>lis proportioni ponderis uici&longs;sim per uige&longs;imam&longs;extam &longs;exti Ele­<lb/>mentorum b ad a, igitur proportio b ad a ponderis e&longs;t duplicata ei, <lb/>quæ e&longs;t raritatis c ad raritatem d.</s></p><pb pagenum="28"/><p type="head">
<s>SCHOLIVM PRIMVM.</s></p><p type="main">
<s>Ne tamen &longs;ine exemplo intelligas hanc duplicatam rationem, <lb/>proponatur craritas quatuor, d unum, a pondus duodecim libra­<lb/><figure id="fig18"></figure><lb/>rum, tunc c re&longs;i&longs;tit &longs;olum ex quarta parte, & effi­<lb/>cit a quadruplo maioris actionis, &longs;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 e&longs;t unum, exibit <expan abbr="põdus">pondus</expan> b centum nonaginta duo. </s>
<s>Pro­<lb/>portio igitur b ad a e&longs;t &longs;exde cupla, & hæc e&longs;t duplicata quadruplæ <lb/>raritatis c ad raritatem d.</s></p><p type="main">
<s>Quòd &longs;i quis neget tantundem augere c actionem a, quanto mi­<lb/>nus re&longs;i&longs;tit, &longs;ed aut magis aut minus, & &longs;it proportio b ad a dupli­<lb/>cata ip&longs;i f, dico fe&longs;&longs;e proportionem c ad d, nam proportio b ad a <lb/>e&longs;t uelut actionis c ad d per decimam&longs;extam &longs;exti Elementorum, <lb/>ergo ex auxilio c in proportionem a ad c fit proportio b ad a, &longs;ed ex <lb/>fin &longs;e fit proportio b ad a ex diffinitione proportionis duplicatæ. <lb/></s>
<s>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">decimam&longs;eptimam</expan> &longs;exti Elemento­<lb/>rum proportio c ad d e&longs;t media inter proportiones a ad c, & actio­<lb/>nis a in c, quare æqualis f, igitur proportio b ad a duplicata ei, quæ <lb/>e&longs;t c ad d quod erat demon&longs;trandum.</s></p><p type="head">
<s>SCHOLIVM SECVNDVM.</s></p><p type="main">
<s>Si autem media fuerint diuer&longs;arum rationum, ut aqua, & aër non <lb/>demon&longs;trat argumentum, quia pondera inter &longs;e non &longs;eruant ratio­<lb/>nem. </s>
<s>Nam lignum centum librarum ex &longs;alicis arbore, non magis <lb/>de&longs;cendit, quàm lignum libræ unius. </s>
<s>Ideò nec in comparatione ad <lb/>medium aëris.</s></p><p type="main">
<s>Propo&longs;itio trige&longs;imaquarta.</s></p><p type="main">
<s>Proportio corporis cubi ad &longs;uam &longs;uperficiem quadratam, e&longs;t ue­<lb/>lut eiu&longs;dem &longs;uperficiei ad latus, eiu&longs;dem uerò ad monadem.</s></p><p type="main">
<s><arrow.to.target n="marg89"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg89"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Sit cubus a b c eius quadrata, &longs;uperficies a <lb/><figure id="fig19"></figure><lb/>c, latus a b, monas d, dico eas e&longs;&longs;e inuicem <lb/>analogas. </s>
<s>Quia enim proportio a b c ad a c <lb/>e&longs;t, ut quoties a&longs;&longs;umitur a c in a b c, & toties <lb/>ctiam a&longs;&longs;umitur a b in a c ex diffinitione Eucli </s></p><p type="main">
<s><arrow.to.target n="marg90"></arrow.to.target><lb/>dis &longs;ecundo Elementorum, &longs;i ergo monas e&longs;t <lb/>in continua proportione, habeo intentum: &longs;i <lb/>non ponatur e media inter a e & d, erit ergo <lb/>per decimam noni Elementorum elatus a c, <lb/>ergo æqualis a b, igitur cum a c, e & d &longs;int analogæ, erunt & a b c, <lb/>a b, & d analogæ, quod fuit demon&longs;trandum.</s></p><pb pagenum="29"/><p type="margin">
<s><margin.target id="marg90"></margin.target>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>Propo&longs;itio trige&longs;imaquinta.</s></p><p type="main">
<s>Vocum magnitudines excre&longs;cunt in acumine non in grauitate, <lb/>finis autem e&longs;t in utroque extremo, propter hoc minima facta uaria­<lb/>tione in hypate acutæ uix ferunt.<lb/><arrow.to.target n="marg91"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg91"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>m.</s></p><p type="main">
<s>Quoniam facta uariatione in hypate, quæ e&longs;t <lb/>in Diapa&longs;on, uel bis Díapa&longs;on maiore interual­<lb/><figure id="fig20"></figure><lb/>lo di&longs;tat, uelut ex a in b in grauiore, maius e&longs;t in­<lb/>teruallum ex c in d, igitur maior e&longs;t b d, quàm a c <lb/>ergo &longs;ingulæ uoces inter b & d magis di&longs;tant, <lb/><figure id="fig21"></figure><lb/>quàm inter a & c, & quanto magis appropin­<lb/>quant ad d, igitur d maius e&longs;t quàm b. </s>
<s>Ergo magnitudo e&longs;t ratione <lb/>acuitatis, non grauitatis, cum &longs;uppo&longs;uerimus d e&longs;&longs;e acutiorem b & <lb/>cip&longs;o a. </s>
<s>O&longs;tenditur etiam idem quia uox grauis fit ex priuatione <lb/>motus &longs;icut acuta ex uehementia. </s>
<s>Motus autem e&longs;t res, quies, <lb/>priuatio.</s></p><p type="main">
<s>Secundum &longs;ic: nam remi&longs;sio mota non feriet aurem, ideò &longs;onum <lb/>non pariet ob nimiam tarditatem. </s>
<s>At in uelo ci&longs;simo motu oportet <lb/>uel fidem uel arteriam contrahi, & non contrahitur ni&longs;i per mu&longs;cu­<lb/>los, igitur contentio illa finem habet. </s>
<s>Si autem non &longs;it nece&longs;&longs;arium <lb/>habere, uel ualde procul po&longs;sit extendi contentio, ut in machinis <lb/>igneis &longs;trepitus fit maximus, nam motus, ut motus e&longs;t etiam in aëre <lb/>nullum finem per &longs;e habet ni&longs;i ratione in&longs;trumenti, ergo &longs;trepitus <lb/>tantus e&longs;&longs;e pote&longs;t, ut fermè ob&longs;urde&longs;cant, qui audierint, ut ferunt de <lb/>Nili cataractis.</s></p><figure></figure><p type="main">
<s>Tertium &longs;ic &longs;it a b humi­<lb/>lior uox, quæ excre&longs;cat &longs;e­<lb/>mitonio minore &longs;olum in <lb/>c, & &longs;it d e dupla ad ab &longs;e­<lb/><figure id="fig22"></figure><lb/>cundum naturam, ut in uo­<lb/>cibus medijs fiet, ut &longs;i e debeat excre&longs;cere &longs;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 diapa&longs;on quadrupla: pueri autem uox, quæ iam diapa­<lb/>&longs;on altior e&longs;t d e, erit bis diapa&longs;on, & ideò quadrupla b c, &longs;ed in acu­<lb/>tioribus erit dupla, nullus enim puer e&longs;t adeo fractæ uocis, qui&longs;u­<lb/>pra humillimam non a&longs;cendat per diapa&longs;on, igitur interuallum uo­<lb/>cum erit octuplum a d, b c, &longs;ed communiter a&longs;cen dunt ad bis diapa <lb/>&longs;on, igitur interuallum unius uocis etiam cum &longs;emitonio propor­<lb/>tionem habentis e&longs;t æquale fermè toti a b, cum autem in diapa&longs;on <lb/>&longs;int duodecim &longs;emitonia, & duo comata, manife&longs;tum e&longs;t, quod ex­<lb/>ten&longs;io illa erit maxima in <expan abbr="cõparatíone">comparatíone</expan> grauioris uo cis a b. </s>
<s>Etideò <lb/>minimum in crementum in humilioribus uocibus, ubi quis coga­<pb pagenum="30"/>tur a&longs;cendere, maximum e&longs;&longs;e uidetur, adeò ut ægrè à pluribus fera­<lb/>tur, à quibu&longs;dam non omnino feratur.</s></p><p type="head">
<s>SCHOLIVM.</s></p><p type="main">
<s>Ob hoc natura fecit, ut non quemadmodum in fidibus uoces ex <lb/>breuitate intenderentur, &longs;ed ex con&longs;trictione ligulæ, ut dicunt, &longs;u­<lb/>per a&longs;peram arteriam uox ad diapa&longs;on acueretur addito impetu <lb/>proportione, ut ex con&longs;trictione, & impetu <expan abbr="cõ&longs;urgeret">con&longs;urgeret</expan> dupla pro­<lb/>portio. </s>
<s>Hoc autem manife&longs;tè experimur in elymis in quibus nullæ <lb/>pror&longs;us facta mutatione in&longs;trumenti con&longs;tantibus digitis omni­<lb/>bus præter pollicem &longs;ini&longs;træ uocem exacuimus ad diapa&longs;on, inde <lb/>etiam ad bis diapa&longs;on: &longs;icut declarauimus in commentarijs Epi­<lb/>demiorum.</s></p><p type="main">
<s>Propo&longs;itio trige&longs;ima&longs;exta.</s></p><p type="main">
<s>Si proportio per proportionem minorem æquali ducatur, pro­<lb/>portio minor producetur. </s>
<s>Vnde manife&longs;tum e&longs;t duas proportio­<lb/>nes minores æqualitate inuicem ductas proportionem minorem <lb/>unaquaque illarum producere.</s></p><p type="main">
<s><arrow.to.target n="marg92"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg92"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><figure></figure><p type="main">
<s>Proportio a b ad c, quali&longs;cunque &longs;it, duca­<lb/>tur in proportionem minorem æqualitate <lb/>fad g, dico quod producta proportio erit <lb/>minor ea, quæ e&longs;t a b ad c fiat d ad a b, ut f <lb/>ad g, et erit per &longs;ecundam huius d ad c pro­<lb/>ducta ex proportionibus a b ad c, & f g. </s>
<s>Itemque per decimamquar­<lb/><arrow.to.target n="marg93"></arrow.to.target><lb/>tam quinti <expan abbr="Elementorũ">Elementorum</expan> erit d minor a b, igitur maior a b ad c, quàm <lb/>d ad c. </s>
<s>igitur quàm proportio a b ad c in proportionem f ad g. </s>
<s>Sit <lb/>autem utraque minor æqualitate ea, quæ a b ad c, & ea quæ f ad g, di­<lb/>co productam unaquaque earum e&longs;&longs;e minorem. </s>
<s>Quod enim (manen<lb/>tibus his, quæ dicta &longs;unt) minor &longs;it d ad c, quam a b ad c ex prima <lb/>parte o&longs;ten&longs;um e&longs;t. </s>
<s>Quòd uerò etiam minor &longs;it d ad c, quàm d ad <lb/>a b, & ex con&longs;equenti quàm f ad g demon&longs;tratur &longs;ic. </s>
<s>Quia enim mi­<lb/>nor e&longs;t a b ad c, æqualitate erit a b minor c, fiat ergo h æqualis a b, <lb/>erit ergo d ad h, ut d ad a b per &longs;eptimam quinti Elementorum, at d <lb/>ad c minor quàm d ad h per octauam eiu&longs;dem, igitur minor d ad c, <lb/>quàm d ad a b, igitur patet propo&longs;itum.</s></p><p type="margin">
<s><margin.target id="marg93"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 1 <gap/>. </s>
<s>P<emph type="italics"/>et.<emph.end type="italics"/></s></p><p type="main">
<s>Propo&longs;itio trige&longs;ima&longs;eptima.</s></p><p type="main">
<s>Si plures homines, quorum nulli per &longs;e nauim mouere po&longs;sint, <lb/>aut pondus ferre &longs;imul iuncti eam moueant, aut pondus ferant, <lb/>erunt illæ proportiones coniunctæ non productæ.<lb/><arrow.to.target n="marg94"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg94"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Cùm enim primus non po&longs;sit mouere nec &longs;ecundus, erunt pro­<lb/>portiones minores æqualitate, Ideò per &longs;ecundam partem præce­<lb/>dentis multo minus mouerent duo, quàm unus. </s>
<s>Et &longs;i quatuor mo­<pb pagenum="31"/>uerent unusque per &longs;e mouere non po&longs;&longs;et, adderetur &longs;i proportio <lb/>produceretur, fieret minor, ergo minus mouerent quinque quàm <lb/>quatuor ex ij&longs;dem, quod e&longs;t ab&longs;urdum.</s></p><p type="main">
<s>Propo&longs;itio trige&longs;imao ctaua.</s></p><p type="main">
<s>Omne corpus tantùm re&longs;i&longs;tit motui contrario &longs;uo naturali quan <lb/>cum mouetur occulto motu quie&longs;cendo.</s></p><p type="main">
<s><arrow.to.target n="marg95"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg95"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>m.</s></p><p type="main">
<s>Sit a corpus quie&longs;cens in pauimento b, & mouetur in eo occul­</s></p><p type="main">
<s><arrow.to.target n="marg96"></arrow.to.target><lb/>to motu uer&longs;us centrum, ut &longs;uprà ui&longs;um e&longs;t, contra­<lb/><figure id="fig23"></figure><lb/>rius illi &longs;it motus ad c, &longs;i ergo a quie&longs;ceret in c moue­<lb/>retur ad b occulto motu certa ui, ergo eadem re&longs;titit, <lb/>ne traheretur ad c. </s>
<s>Manife&longs;tum e&longs;t autem, quod hic <lb/><arrow.to.target n="marg97"></arrow.to.target><lb/>motus occultus e&longs;t minor manife&longs;to.<lb/><arrow.to.target n="marg98"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg96"></margin.target>I<emph type="italics"/>n commen.<emph.end type="italics"/><lb/>26. P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg97"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 30. P<emph type="italics"/>ro <lb/>po&longs;.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg98"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Ex hoc patet cur naues & currus ab initio tardè & difficulter mo<lb/>ueantur, ubi moueri cœperint motus augetur: quoniam re&longs;i&longs;tunt </s></p><p type="main">
<s><arrow.to.target n="marg99"></arrow.to.target><lb/>per motum occultum naturalem qui maximus e&longs;t dum quie&longs;cunt, <lb/>ut etiam do cebat philo&longs;ophus in mechanicis, nam motus ille natu­<lb/>ralis e&longs;t, & ideò contrarius uiolento: Ergo cum iam mouetur uio­<lb/>lenter minus, mouetur naturaliter, igitur minus re&longs;i&longs;tit. </s>
<s>Declarabi­<lb/>tur enim infrà quòd omne quod mouetur duobus motibus tanto <lb/><arrow.to.target n="marg100"></arrow.to.target><lb/>minus uno mouetur quanto magis altero.</s></p><p type="margin">
<s><margin.target id="marg99"></margin.target>Q<emph type="italics"/>ue&longs;t.<emph.end type="italics"/> 31.</s></p><p type="margin">
<s><margin.target id="marg100"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 59.</s></p><p type="main">
<s>Propo&longs;itio trige&longs;imanona.</s></p><p type="main">
<s>Ab æquali aut minore ui, quàm &longs;it <expan abbr="impedimentũ">impedimentum</expan>, non fit motus.</s></p><p type="main">
<s>Sit a quod re&longs;i&longs;tat, ne &longs;ur&longs;um trahatur per decem, dico, quod <expan abbr="nõ">non</expan> <lb/><arrow.to.target n="marg101"></arrow.to.target><lb/>&longs;ur&longs;um trahetur neque à decem, neque minore: nam &longs;i impedimen­<lb/>tum non e&longs;&longs;et, moueretur infra ut decem, ergo &longs;i traheretur &longs;ur&longs;um <lb/>per decem tantum moueretur &longs;ur&longs;um, <expan abbr="quantũ">quantum</expan> deor&longs;um, ergo quie­<lb/>&longs;ceret. </s>
<s>Si uerò à minore moueretur à maiore ui deor&longs;um, quam &longs;ur­<lb/>&longs;um, ergo deor&longs;um &longs;impliciter non &longs;ur&longs;um.</s></p><p type="margin">
<s><margin.target id="marg101"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Propo&longs;itio quadrage&longs;ima.</s></p><p type="main">
<s>Omne corpus &longs;phæricum tangens planum in puncto mouetur <lb/>ad latus per quancunque uim, quæ medium diuidere pote&longs;t.</s></p><figure></figure><p type="main">
<s>Sit corpus ad unguem &longs;phæricum a tan­<lb/><arrow.to.target n="marg102"></arrow.to.target><lb/>gens planum b in puncto c (e&longs;t enim hoc <lb/>nece&longs;&longs;arium ex demon&longs;tratis ab Euclide in <lb/>decima&longs;exta Propo&longs;itione tertij Elemento­<lb/>rum) dico, quod mouebitur à ui, quæ pote&longs;t <lb/>&longs;cindere aërem. </s>
<s>Nam cum non a&longs;cendat, nec <lb/>de&longs;cendat, &longs;ed qua&longs;i in circulo ad centrum <lb/>mundi moueatur, pondus non affert. </s>
<s>Neque<lb/>ratione magnitudinis contactus, cum &longs;it in <lb/>puncto &longs;olo, igitur remanet &longs;olum aëris impedimentum.<pb pagenum="32"/><arrow.to.target n="marg103"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg102"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s><margin.target id="marg103"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 1.</s></p><p type="main">
<s>Ex hoc liquet, quod oportet b planum e&longs;&longs;e ex duri&longs;sima mate­<lb/>ria, quæ nullo modo cedat, aliter tanget plu&longs;quàm in puncto.</s></p><p type="main">
<s><arrow.to.target n="marg104"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg104"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 2.</s></p><p type="main">
<s>Vix fieri pote&longs;t, utin elementaribus &longs;phæra tangat planum in <lb/>puncto. </s>
<s>Vel quia planum non erit exactè rectum, uel non durum, <lb/>ut pror&longs;us non cedat, uel non ad æquilibrium po&longs;itum, uel &longs;phæra <lb/>non erit exactè rotunda.</s></p><p type="main">
<s>Propo&longs;itio quadrage&longs;imaprima.</s></p><p type="main">
<s>Si fuerint duæ quantitates &longs;umaturque totius aggregatum maio­<lb/>ris & minoris, quoties aggregatum minoris, & maioris, erit pro­<lb/>portio confu&longs;a maioris aggregati ad minus, minor quàm multipli­<lb/>cis maioris ad multiplex minoris.</s></p><p type="main">
<s><arrow.to.target n="marg105"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg105"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Sint duæ magnitudines a & b, & &longs;it a maior <lb/><figure id="fig24"></figure><lb/>b, & &longs;umatur exempli gratia a quater cum b &longs;e­<lb/>mel, & b quater cum a &longs;emel, dico, quod propor<lb/>tio (quam confu&longs;am e&longs;&longs;e liquet) aggregati primi ad &longs;ecundum, e&longs;t </s></p><p type="main">
<s><arrow.to.target n="marg106"></arrow.to.target><lb/>minor quàm quadrupla. </s>
<s>Con&longs;tat enim quod proportio quadru­<lb/>pli a ad a e&longs;t maior, quam b ad quadruplum b, cum una &longs;it quadru­<lb/>pla, alia &longs;ub quadrupla, igitur per uige&longs;imam&longs;ecundam huius ag­<lb/>gregati quadrupli a cum b &longs;emel, ad quadruplum b cum a &longs;emel mi <lb/><arrow.to.target n="marg107"></arrow.to.target><lb/>nor, quàm quadrupli a ad a, & maior quàm b ad quadruplum b, & <lb/>e&longs;t pro intellectu Archimedis.</s></p><p type="margin">
<s><margin.target id="marg106"></margin.target>E<emph type="italics"/>x<emph.end type="italics"/> 18. <emph type="italics"/>diff.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg107"></margin.target>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&longs;.<emph.end type="italics"/> 10.</s></p><p type="main">
<s>Propo&longs;itio quadrage&longs;ima&longs;ecunda.</s></p><p type="main">
<s>Trahentium nauim, ut ferentium pondera proportiones in &longs;e in­<lb/>uicem, quomodo ducere oporteat con&longs;iderare.<lb/><arrow.to.target n="marg108"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg108"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Hoc quomodo non po&longs;sit fieri &longs;uprà docuimus, nunc etiam ge­</s></p><p type="main">
<s><arrow.to.target n="marg109"></arrow.to.target><lb/>neraliter dicam, cum con&longs;i&longs;tant hæc in duobus terminis, productio <lb/>uerò præ&longs;upponit quatuor terminos, ut in prima propo&longs;itione, aut <lb/>&longs;altem tres, atque in his medius habet rationem mouentis, & moti, <lb/>ergo cum in huiu&longs;modi <expan abbr="nõ">non</expan> &longs;int quatuor termini, nec tres, è quibus <lb/>unus &longs;it mouens, & motum proportio non poterit produci. </s>
<s>Illud <lb/>etiam patet exemplo, nam &longs;i e&longs;&longs;et lapis, aut nauis ob&longs;i&longs;tens ut &longs;ex, & <lb/>e&longs;&longs;ent homines uiribus &longs;inguli, ut quatuor cum dimidio, tres mo­<lb/>uerent in proportione dupla &longs;exquiquarta perdicta &longs;uperius eo­<lb/>dem loco, at &longs;i proportio duci po&longs;&longs;et aliquorum hominum nume­<lb/>rus po&longs;&longs;et mouere in duplicata proportione ad unguem &longs;cilicet <lb/>5 1/16 ut e&longs;&longs;et uix hominum collectorum 30 3/8 at nullus e&longs;t numerus ho <lb/>minum qui collectus faciat hunc numerum, nam &longs;ex homines ex­<lb/>plentnumerum 27, & &longs;eptem 31 1/2, & ideò non pote&longs;t duci propor­<lb/>tio. </s>
<s>Et ideò maximus e&longs;t error dicendo decem homines mouent na <lb/>uim proportione tripla, ergo triginta alij additis illis &longs;imiles robo­<lb/>re mouebunt à proportione uiginti &longs;eptupla &longs;cilicet ducta nonu­<pb pagenum="33"/>pla in triplam. </s>
<s>Sed &longs;umpta proportione alio modo producitur. </s>
<s>Ve <lb/>lut &longs;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 &longs;cilicet quadrupla in triplam ducta. </s>
<s>Cum ergo <lb/>addo triginta homines, qui mouent in proportione nonupla, non <lb/>oportet ducere nonuplam in triplam, &longs;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>Vnde &longs;i duo moueant in proportione &longs;ex­<lb/>quialtera, & &longs;ex in proportione quadrupla cum dimidia, & iungan <lb/>tur, ut fiant octo, non oportebit ducere &longs;exquialteram, in quadru­<lb/>plam &longs;exquialteram, &longs;ed cum octo ad duo &longs;it in proportione qua­<lb/>drupla, &longs;umemus quadruplam ad &longs;exquialteram, qu&ecedil; erit &longs;excupla, <lb/>& octo mouebunt, aut pondus gerentin proportione &longs;excupla.</s></p><p type="margin">
<s><margin.target id="marg109"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 37.</s></p><p type="main">
<s>Propo&longs;itio quadrage&longs;imatertia.</s></p><p type="main">
<s>Productionem ad additionem retrahere.<lb/><arrow.to.target n="marg110"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg110"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><figure></figure><p type="main">
<s>Sit proportio a ad b dupla pote&longs;tate li­<lb/>cet &longs;int quinque homines, & &longs;int quindecim <lb/>homines c, & habebunt ad b &longs;excuplam <lb/>proportionem per præcedentem. </s>
<s>Iuncta <lb/>ergo a, & c per octauam huius <expan abbr="mouebũt">mouebunt</expan> <lb/>b proportione octupla, dico, quod &longs;i du­<lb/>xeris <expan abbr="proportion&etilde;">proportionem</expan> c ad a plus uno. </s>
<s>i. </s>
<s>qua­<lb/>druplam in proportionem a ad b, quæ e&longs;t dupla, proueniet eadem <lb/>octupla. </s>
<s>Nam quia in coniunctione &longs;ufficit iungere c cum a, & &longs;u­<lb/>mitur &longs;ecundum proportionem a ad b, igitur cum proportio a ad <lb/>b co mparata ad proportionem c & a ad b &longs;it, &longs;icut proportio c & a <lb/>ad a, & proportio c & a ad a &longs;it, &longs;icut proportio c ad a, & a ad a, & <lb/>proportio a ad a habet rationem unius, igitur proportio aggregati <lb/>c a ad b e&longs;t producta ex proportione c ad a plus monade in propor<lb/>tionem a ad b, quod erat demon&longs;trandum.</s></p><p type="main">
<s>Propo&longs;itio quadrage&longs;imaquarta.</s></p><p type="main">
<s>Si fuerit proportio motoris ad id, quod e&longs;t maximum non mo­<lb/>uens & &longs;patium, & tempus, nota erit etiam reliquorum nota.</s></p><p type="main">
<s>Sæpe contingit, ut quinque homines moueant nauim, & &longs;patium <lb/>ad tempus notum, & etiam cognitum maximum, quod mouere <lb/>non pote&longs;t. </s>
<s>Sit ergo a numerus hominum, b na­<lb/><figure id="fig25"></figure><lb/>uis, c maximum, quod non mouere pote&longs;t, d <lb/>tempus, e &longs;patium, f motor alius &longs;iue numerus <lb/>hominum notus, & g tempus, dico, quod h &longs;patium notum erit, &longs;eu <lb/><expan abbr="notũ">notum</expan> g tempus, & h &longs;patium, dico, quod erit f motor, &longs;eu numerus <pb pagenum="34"/>hominum notus. </s>
<s>Quoniam ergo notum e&longs;t a & c, quia e&longs;t æquale <lb/>b, igitur proportio a ad b nota e&longs;t: &longs;ed iuxta illam a mouet b in d <lb/>tempore per e &longs;patium, igitur per præcedentem, ut f ad a ita &longs;patij <lb/>ad e in d tempore. </s>
<s>Sed per eadem ut temporis d ad &longs;patium illud, <lb/>ita g ad h, ergo cum nota &longs;int d e f g erit etiam h, & ita conuertendo.</s></p><p type="main">
<s>Propo&longs;itio quadrage&longs;imaquinta.</s></p><p type="main">
<s>Rationem &longs;tateræ o&longs;tendere.<lb/><arrow.to.target n="marg111"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg111"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Archimedes nititur huic fundamento, quod pondera, quæ pro­<lb/>portionem mutuam habent, ut di&longs;tantiæ à libella a, quæ &longs;u&longs;pen­<lb/>duntur, æqualiter ponderant, &longs;it ergo libella a b, & &longs;u&longs;pen&longs;a in a cen <lb/>trum mundi c, ad quod dirigitur pondus, & liquet, quod ip&longs;um <lb/>non &longs;e inclin abit ex uige&longs;imatertia propo&longs;itione. </s>
<s>Si ergo ponantur <lb/>lo co lineæ b d in e & f, & &longs;it proportio e b <lb/><figure id="fig26"></figure><lb/>ad b f, ut g ad h, dico, quòd erit æquili­<lb/>brium, per eandem enim h mouebitur in k, <lb/>&longs;cilicet ut perueniat in rectam a d, &longs;i enim <lb/>non e&longs;&longs;et |&longs;u&longs;pen&longs;um h, moueretur in re­<lb/>cta e h per eandem, quia ergo retinetur, mo­<lb/>uetur per obliquam h k, & &longs;umatur in pro­<lb/>pin quum punctum in b e, & n in æquali di­<lb/>&longs;tantia in e f, quia ergo e b totum mouetur <lb/>eadem ui in &longs;ingulis partibus, quia a pon­<lb/>dere h, & in h mouetur per h k in m per m <lb/>p, ergo qualis e&longs;t proportio magnitudinis h k ad m p, talis e&longs;t uis <lb/>in m p ad uim in h k, & ita in b erit penè infinita: quia quanta ui ex­<lb/>tenditur ex h in k tanta puncta b, &longs;e circumuertit ergo propor­<lb/>tio hypomochlij ad &longs;patium, uelut roboris ad robur, at eadem n o <lb/>ad h k, e&longs;t enim n o æqualis m p, & n b, & b m æquales, ut uerò 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>nam idem pondus &longs;cilicet g mouet totam b f, igitur ut g &longs;e habet </s></p><p type="main">
<s><arrow.to.target n="marg112"></arrow.to.target><lb/>ad n o, ita h ad m p, &longs;ed m p & n o &longs;unt æquales, ergo tanta e&longs;t uis g <lb/>in f, quanta h in e.<lb/><arrow.to.target n="marg113"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg112"></margin.target>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><margin.target id="marg113"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 1.</s></p><p type="main">
<s>Ex quo patet, quod hypomo chlion moueretur infinita ui, &longs;i po&longs;­<lb/>&longs;et e&longs;&longs;e punctus: &longs;ed quia in extrema &longs;uperficie cylindri, ideò pote&longs;t <lb/>aliqua ui retineri.</s></p><p type="main">
<s><arrow.to.target n="marg114"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg114"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 2.</s></p><p type="main">
<s>Et &longs;i quis po&longs;&longs;et capere ha&longs;tam in extremo puncto, non po&longs;&longs;et <lb/>eam mouere, etiam quod haberet robur infinitum, quia ab æquali <lb/>non fit motus per trige&longs;imamnonam propo&longs;itionem.</s></p><p type="main">
<s><arrow.to.target n="marg115"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg115"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 3.</s></p><p type="main">
<s>Et libella nihil retinet ni&longs;i quantum e&longs;t pondus eius quod cu­<pb pagenum="35"/>pit ad centrum peruenire, & pondus ei appen&longs;um non prohi­<lb/>bet motum, etiam &longs;i e&longs;&longs;et infinitum, ni&longs;i quatenus non uult recede­<lb/>re ex directo centri mundi: & ut grauat hypomochlion faciens im­<lb/>pre&longs;sionem.</s></p><p type="main">
<s><arrow.to.target n="marg116"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg116"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 4.</s></p><p type="main">
<s>Et &longs;i terra tota e&longs;&longs;et appen&longs;a polo, moueretur magna ui: quoni­<lb/>am uis eadem e&longs;t in polo, quæ in circulo toto æquinoctij.</s></p><p type="main">
<s><arrow.to.target n="marg117"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg117"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 5.</s></p><p type="main">
<s>Etrota, quanto uelocius mouetur in ambitu, tanto mi<lb/>norem habet uim: &longs;ed propter aërem, qui &longs;ecum circum­<lb/><figure id="fig27"></figure><lb/>fertur, mouetur magno impetu, & magnas facit læ&longs;iones. <lb/></s>
<s>Ideò hoc in cono non accidit.</s></p><p type="main">
<s><arrow.to.target n="marg118"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg118"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 6.</s></p><p type="main">
<s>Ex quo patet ratio eleuandi pondera magna per tra­<lb/>bem, ut à latere uides.</s></p><p type="main">
<s>Propo&longs;itio quadrage&longs;ima&longs;exta.</s></p><p type="main">
<s>An &longs;it aliqua proportio, & qualis inter animam, & ui­<lb/>tas, & &longs;ua corpora con&longs;iderare.</s></p><p type="main">
<s><arrow.to.target n="marg119"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg119"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Declarauimus motum cœli e&longs;&longs;e uoluntarium, ob&longs;equente cœ­<lb/>lo per uirtutem in eo infu&longs;am. </s>
<s>In animalibus autem, & præcipuè <lb/>in homine notius e&longs;t hoc experientibus nobis in ip&longs;is: &longs;ed motus <lb/>hic, ut dixi &longs;upra, mi&longs;tus e&longs;t, ille uerò cœle&longs;tis ignotior e&longs;t. </s>
<s>Certum </s></p><p type="main">
<s><arrow.to.target n="marg120"></arrow.to.target><lb/>tamen e&longs;t plenè ob&longs;equi cœlum uitæ, nec pror&longs;us repugnare. </s>
<s>So­<lb/>let Ari&longs;toteli imponi, quòd &longs;i adderetur a&longs;trum cœlo, quòd cœlum <lb/>aut quie&longs;ceret, aut tardius moueretur: quod e&longs;t, ac &longs;i diceremus, <lb/>quòd homo paruus &longs;i fieret maior, non e&longs;&longs;et adeò agilis, tanquam <lb/>motus ille e&longs;&longs;et ab externa cau&longs;a. </s>
<s>Imò perinde e&longs;&longs;et, ac&longs;i quis dice­<lb/>ret, quod lapides magni minus uelociter de&longs;cenderent, quam par­<lb/>ui. </s>
<s>Quin potius ut lapis magnus uelociùs mouetur: quàm par­<lb/>uus naturali motu, & tardius præternaturali, ita cœlum motu uo­<lb/>luntario, &longs;i ita dici po&longs;&longs;et æqualius & maiore cum efficacia, quan­<lb/>to den&longs;ius. </s>
<s>Et ita &longs;i Ari&longs;toteles illud dixi&longs;&longs;et, o&longs;tendi&longs;&longs;et magnam <lb/>imperitiam. </s>
<s>Ideò quale iudicium debemus facere de Alexandro, & <lb/><arrow.to.target n="marg121"></arrow.to.target><lb/>Aueroe, qui hoc ei tribuunt. </s>
<s><expan abbr="legi&ttilde;">legitur</expan> enim in textu Arabico tale quip­<lb/>piam. </s>
<s>De Animalibus for&longs;an po&longs;&longs;et hoc dici, <expan abbr="quoniã">quoniam</expan>, ut &longs;uprà dixi­<lb/>mus, motus ille mi&longs;tus e&longs;t. </s>
<s>Remanet ergo difficultas, <expan abbr="quoniã">quoniam</expan> &longs;i mo­<lb/>tus i&longs;te non à proportione fit, quare non e&longs;t infinitus? </s>
<s>& dico quae in <lb/>animalibus tres &longs;unt cau&longs;æ, una, quia e&longs;t mi&longs;tus, & habet repugnan<lb/>tiam: &longs;ecunda, quia e&longs;t de loco ad locum, motus autem cœli e&longs;t in lo <lb/>co: tertia e&longs;t communis etiam cœlo, et e&longs;t, <expan abbr="quoniã">quoniam</expan> non e&longs;t ratio finis. <lb/></s>
<s>Natura enim diuina non appetit mouere <expan abbr="tã">tam</expan> celeriter. </s>
<s>Quid e&longs;t ergo <lb/>proportio, <expan abbr="cũ">cum</expan> &longs;it <expan abbr="ultimũ">ultimum</expan> uoluntatis uit&ecedil;, ut obtemperet primæ cau&longs;æ, <lb/>ideo illud e&longs;t <expan abbr="ultimũ">ultimum</expan>, &qring; mouet. </s>
<s>E&longs;t <expan abbr="aũt">aunt</expan> idem uelle, & po&longs;&longs;e. </s>
<s>In natura <pb pagenum="36"/>enim cœli e&longs;t ille appetitus, cuius prin cipium e&longs;t uita: & eíus uolun <lb/>tatis bonum ip&longs;um. </s>
<s>Et ideo hæc proportio <expan abbr="nõ">non</expan> diuiditur. </s>
<s>In anima­<lb/>libus autem non e&longs;t uis illa ni&longs;i, cum proportione, quia primum in­<lb/>&longs;trumentum, quod recipit, & e&longs;t &longs;piritus uim habet determinatam, <lb/>cum &longs;it uirtus in materia: ideo <expan abbr="nõ">non</expan> mouet ni&longs;i cum certa proportio­<lb/>ne, uelut lumen in medio in &longs;e non habet proportionem ni&longs;i ad lu­<lb/>cem, &longs;ed ut e&longs;t in illo, pote&longs;t e&longs;&longs;e remi&longs;&longs;um, <expan abbr="ob&longs;curũ">ob&longs;curum</expan> & hebes. </s>
<s>Quæ­<lb/>ritur ergo quantitas illius? </s>
<s>&longs;i dicas, quòd e&longs;t à luce: quæro quanti­<lb/>tas lucis, unde &longs;it? </s>
<s>for&longs;an dicendum, quòd uelutin motibus, quanto <lb/>den&longs;iora &longs;unt corpora tanto <expan abbr="mouen&ttilde;">mouentur</expan> maiore nixu, & robore. </s>
<s>Nam <lb/>calor in materia augetur iuxta illius quantitatem: idem in luce, & <lb/>reliquis. </s>
<s>Dico ergo proportionem e&longs;&longs;e infinitam: nam &longs;i corpus e&longs;­<lb/>&longs;et infinitum & optimè di&longs;po&longs;itum infinita ui moueretur & agili­<lb/>tate, ut enim maius e&longs;t eo maiores uires habet.</s></p><p type="margin">
<s><margin.target id="marg120"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 27.</s></p><p type="margin">
<s><margin.target id="marg121"></margin.target>T<emph type="italics"/>ex.<emph.end type="italics"/> 71. <lb/>2. <emph type="italics"/>de<emph.end type="italics"/> C<emph type="italics"/><gap/>.<emph.end type="italics"/></s></p><p type="main">
<s>Propo&longs;itio quadrage&longs;ima&longs;eptima.</s></p><p type="main">
<s>Si duo mobilia æqualiter in eodem circulo iuxta proprios mo­<lb/>tus moueantur, productum temporis circuituum inuicem erit æ­<lb/>quale producto differentiæ temporum circuitus ductæ in tempus <lb/>coniunctionis primæ.<lb/><arrow.to.target n="marg122"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg122"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Sint duo mobilia a & b in eodem pun­<lb/><figure id="fig28"></figure><lb/>cto, quæ æqualiter uer&longs;us eandem partem <lb/>moueantur æqualibus in temporibus, inui <lb/>cem tamen in æqualiter, ita quod a in f & b <lb/>in g temporibus ab&longs;oluant circulum, & ho <lb/>rum differentia &longs;it h. </s>
<s>Dum ita que a perficit <lb/>circulum b perueniat in c, igitur c d b e&longs;t dif <lb/>ferentia, quæ &longs;uperanda e&longs;t, & proportio <lb/>circuli ad b c ut g ad f, quare reliqui ad reli­<lb/>quum, ut re&longs;idui ad re&longs;iduum, &longs;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/>diui&longs;o per certam quantitatem & cum circulo & b c & c d b diffe­<lb/>rentia, & &longs;it &longs;productum exfin g, dico quod diui&longs;a &longs; per h exibit k <lb/>tempus coniunctionis primæ, &longs;it itaque d locus coniunctionis, dico <lb/>igitur quod differentia &longs;patij pertran&longs;iti a b, a & a, b in reditu ex con <lb/>iunctione prima ad d e&longs;t unus circulus completus, non enim po&longs;­<lb/>&longs;unt e&longs;&longs;e plures, nam &longs;equeretur, quòd a aliquando pertran&longs;i&longs;&longs;et b, <lb/>et &longs;ic non e&longs;&longs;et prima coniunctio, nec pote&longs;t e&longs;&longs;e minus, nam &longs;ic cum <lb/>a & b &longs;int in d ultra perfectas circulationes uterque eorum pertran <lb/>&longs;iuit arcum b c, igitur nullo modo differentia pote&longs;t e&longs;&longs;e minor cir­<lb/>culo, neque maior, ut declaratum e&longs;t, igitur e&longs;t unus circulus ad un­<pb pagenum="37"/>guem. </s>
<s>Hoc declarato ponatur m &longs;patium compofitum ex circulis <lb/>pertran&longs;itis a b a cum &longs;patio b d, etenim &longs;patium, quod pertran&longs;it <lb/>b a coniunctione in a, ad coniunctionem primam in d, & erit ex de­<lb/>mon&longs;tratis horum differentia circulus qui uocetur o, & &longs;it p &longs;pa­<lb/>tium, quod pertran&longs;it b in tempore eodem, in quo a pertran&longs;it o, & <lb/>&longs;it q differentia o, & p qu&ecedil; in circulo e&longs;t c d l b, quia igitur in eodem <lb/>tempore a pertran&longs;it 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 &longs;it differentia m & n, & q, differentia o & p erit ex <lb/>decimanona quinti Euclidis, m ad o, ut o ad q, & ita circulus e&longs;t ana <lb/>logus inter &longs;patium pertran&longs;itum à motore uelociori, & inter diffe­<lb/>rentiam &longs;patij quæ accidit, dum uelocior motor pertran&longs;it circu­<lb/>lum, id e&longs;t quòd circulus a c d e&longs;t analogus inter c d l b, & circulos <lb/>pertran&longs;itos a b a cum portione b d. </s>
<s>Reuertor igitur ad propo&longs;i­<lb/>tum, cum &longs;it m ad o, ut o ad q, & m ad o, ut n ad p, ex &longs;extadecima <lb/>quinti Euclidis, erit ex undecima eiu&longs;dem n ad p, ut o ad q, quare ex <lb/>&longs;extadecima &longs;exti Elementorum ducto o, id e&longs;t circulo, &longs;eu maiore <lb/>numero in p &longs;patium pertran&longs;itum a b, &longs;eu ducto fin g, & diui&longs;o per <lb/>q differentiam &longs;patiorum, &longs;eu per h exibit n, &longs;eu &longs;patium quod <lb/>pertran&longs;it b ab una coniunctione ad aliam quod erat demon­<lb/>&longs;trandum.</s></p><p type="main">
<s><arrow.to.target n="marg123"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg123"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Ex hoc patet, quod proportio temporis coniunctionis ad tem­<lb/>pus tardioris motus circuitionis e&longs;t 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>Propo&longs;itio quadrage&longs;imao ctaua.</s></p><p type="main">
<s>Si tria mobilia ex eodem puncto di&longs;cedant, fuerintque duorum, ac <lb/>duorum coniunctiones in temporibus commen&longs;is illa tria mobi­<lb/>lia denuò coniungentur in tempore producto ex denominatore di <lb/>ui&longs;ionis temporis maioris per minus in minus, aut numeratore <lb/>in maius.</s></p><p type="main">
<s><arrow.to.target n="marg124"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg124"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Sint tria mobilia a, quod circuat in duobus annis b in quinque, <lb/>c in &longs;eptem. </s>
<s>Dico quod primum redibunt in numero producto ex <lb/>&longs;eptem quinque & duobus, qui &longs;unt numeri primi, & erit ille nume­<lb/>rus &longs;eptuaginta annorum. </s>
<s>Nam in &longs;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>O&longs;tendo modo quod <lb/>non ante: nam &longs;i &longs;ic: &longs;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 fui&longs;&longs;et &longs;eptuaginta minimus numeratus ab a b c, cum <pb pagenum="38"/>ergo iam &longs;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>Quod &longs;i dicas a b c coniungi in decem <lb/>&longs;eptem annis numero non numerato ab ali <lb/><figure id="fig29"></figure><lb/>quo illorum temporum, auferantur perfe­<lb/>ctæ circulationes, & <expan abbr="remanebũt">remanebunt</expan> dimidium <lb/>ex a, duæ quintæ ex b, tres &longs;eptimæ ex c, igi­<lb/>tur oportebit ut hæ portiones &longs;int æqua­<lb/>les, ut po&longs;t perfectas circulationes in idem <lb/>punctum, <expan abbr="cõueniant">conueniant</expan>, ergo 1/2 & 2/5 & 3/7 æqui­<lb/>ualebunt, quare proportio 7 ad 3 & 5 ad 2 <lb/>& 2 ad 1, e&longs;t una, quare permutando 3 ad 2 <lb/>ut 7 ad 5, &longs;ed 7 & 5 &longs;unt contra &longs;e primi, ergo in &longs;ua proportione mi <lb/>nimi per dicta in &longs;eptimo Elementorum: ergo tria, & duo non &longs;unt <lb/>in eadem proportione. </s>
<s>Rur&longs;us dicantur conuenire in annis qua­</s></p><p type="main">
<s><arrow.to.target n="marg125"></arrow.to.target><lb/>tuordecim cum dimidio, ergo in uiginti nouem conuenient ite­<lb/>rum: ergo per &longs;ecundam partem erit &longs;eptem ad unum, ut duo ad <lb/>unum, igitur permutando unius ad unum, ut &longs;eptem ad duo, &longs;ed <lb/>unum e&longs;t æquale uni, ergo duo erunt æqualia &longs;eptem. </s>
<s>Rur&longs;us dica­<lb/>mus, quod in tempore annorum <02> quadrata decem &longs;imiliter aufe­<lb/>ram integras reuolutiones, quas potero, & erunt <02> 2 1/2 m: 1, & <02> 2/5 & <lb/><02> 10/49 æqualia. </s>
<s>Hic uides infinita &longs;equi in conuenientia, quæ longum <lb/>e&longs;&longs;et numerare, nam &longs;eptem e&longs;&longs;et æquale quinque, & proportio reci&longs;i <lb/>ad potentia rethe, ut numeri ad numerum. </s>
<s>Igitur non conueniunt <lb/>ante &longs;eptuaginta annos.<lb/><arrow.to.target n="marg126"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg125"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 23</s></p><p type="margin">
<s><margin.target id="marg126"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 1.</s></p><p type="main">
<s>Ex hoc &longs;equitur, quòd nullibi conuenient præterquàm in eo­<lb/>dem puncto, &longs;cilicet in quo ab initio coniuncti fuerunt.</s></p><p type="main">
<s><arrow.to.target n="marg127"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg127"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>m. </s>
<s>2.</s></p><p type="main">
<s>Sequitur denuo ex propo&longs;itione ip&longs;a repetita, & primo corrola­<lb/>rio, quod nullibi alibi conuenient quàm in dato primo puncto, in <lb/>quo coniuncti fuerant ab initio etiam u&longs;que in æternum.</s></p><p type="main">
<s>Sit rur&longs;us ut a circuat in annis duobus cum dimidio, b in tribus <lb/>cum tertia parte, cin quatuor cum quarta parte ducam per &longs;uos <lb/>denominatores, & erit ut a in quinque annis. </s>
<s>b in decem, c in decem­<lb/>&longs;eptem circuant, & redeant ad idem punctum, & quia quin que nu­<lb/>merat decem, & decem, & decem&longs;eptem &longs;unt numeri inuicem pri­<lb/>mi, ducam decem in decem&longs;eptem fiunt centum &longs;eptuaginta. </s>
<s>Con­<lb/>&longs;tat igitur c quadragíes, b quinquagies &longs;emel, a &longs;exagies octies cir­<lb/>cumuerti, & redire ad idem punctum: ergo rur&longs;us coibunt po&longs;t tot <lb/>annos in eo, dico modo, quod non ante: nam &longs;i non &longs;it, ut in trigin­<lb/>ta tribus annis. </s>
<s>gratia exempli, aufero <expan abbr="decem&longs;ept&etilde;">decem&longs;eptem</expan>, decem, & quin­<lb/>que, & relinquentur &longs;exdecim tria & tria, & rur&longs;us ex &longs;exde cim tres <pb pagenum="39"/>cir cuitus c, & relinquentur 3 3/4 &longs;equetur igitur, ut &longs;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 &longs;i iam &longs;uppo<gap/>­<lb/>mus 17 & 10 e&longs;&longs;e primos inuicem, ut in &longs;ecunda demon&longs;tratione<gap/><lb/>Igitur &longs;equuntur eadem corrolaria, quæ dicta &longs;unt.</s></p><p type="main">
<s>Propo&longs;itio quadrage&longs;imanona.</s></p><p type="main">
<s>Propo&longs;ito mobilis in circulo circuitus tempore, dataque ratione <lb/>di&longs;tantiæ ab illo mobilis circuitum inuenire, quod ex eodem pun­<lb/>cto di&longs;cedens cum alio mobili in dato puncto conueniat &longs;ub quo­<lb/>cunque numero circuituum tempus quoque coniunctionis.</s></p><p type="main">
<s><arrow.to.target n="marg128"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg128"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><figure></figure><p type="main">
<s>Sit in circuli peripheria a <expan abbr="pũctus">punctus</expan>, qui cir <lb/>cuat æquali motu (hoc enim &longs;emper intel­<lb/>ligitur) in b tempore: & &longs;it datus punctus c <lb/>in quo di&longs;cedens e mobile ex coniunctio­<lb/>ne cum a po&longs;t certos circuitus proprios, <lb/>aut etiam. </s>
<s>&longs;ine ulla circuitione perfecta de­<lb/>beat conuenire. </s>
<s>Volo &longs;cire tempus circui­<lb/>tionis e: & etiam tempus coniunctionis. <lb/></s>
<s>Sit ergo primum ut ab&longs;que circuitione ulla e, a debeat comprehen­<lb/>dere e in c po&longs;t numerum circuitionum ip&longs;ius a, qui &longs;it f. </s>
<s>nam &longs;i a o c <lb/>currit e in prima circuitione ip&longs;ius e, igitur a mouetur uelocius <lb/>quàm e, cum ergo debeat attingere ip&longs;um e, nece&longs;&longs;e e&longs;t ut a pertran­<lb/>&longs;eat prius per punctum ex quo di&longs;ce&longs;sit antequam redeat ad con­<lb/>iunctionem e: ergo perficiet &longs;altem unam circuitionem. </s>
<s>Ducemus <lb/>ergo f in b, & fiet g tempus circuitus aut circuituum a, & quia &longs;pa­<lb/>tium a c datum e&longs;t, &longs;it b temporis circuitus a ad h, uelut circuli to­<lb/><arrow.to.target n="marg129"></arrow.to.target><lb/>tius ad a c, & iungatur g cum h & fiat k. </s>
<s>Fiat quoque, ut monadis <lb/>ad h, ita l ad monadem, & ducatur l in k, & fiat m: dico m e&longs;&longs;e tem­<lb/>pus circuitus e. </s>
<s>Con&longs;tat enim ex &longs;uppo&longs;ito, quod k e&longs;t tempus to­<lb/>tum in quo a peruenit po&longs;t b circuitiones in c, &longs;i ergo e moueretur <lb/>per m tempus totum ex &longs;uppo&longs;ito 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 &longs;i in m tran&longs;it to <lb/>tum circuitum in monade tran&longs;it a c: &longs;ed monas ducta in k facit k, <lb/>igitur e in tempore k perueniet in c, quod erat demon&longs;trandum. <lb/></s>
<s>Proponatur modo tempus reuolutionum e ip&longs;um d: eodem mo­<lb/><arrow.to.target n="marg130"></arrow.to.target><lb/>do agemus ducendo fin b fit g, addatur h & fiat k, diuidatur k per <lb/>aggregatum d & a e, & exeat m, (idem enim e&longs;t diuidere per aggre­<lb/>gatum d & h, & multiplicare per l) dico ergo ut in demon&longs;tratione <lb/>priore, quod m e&longs;t tempus circuitus e. </s>
<s>Nam cum k &longs;it tempus, in <lb/>quo a po&longs;t circuitus f peruenit ad c, ergo diui&longs;o ip&longs;o toto tempore <pb pagenum="40"/>per numerum reuolutionum d, & partem reuolutionis exibit tem­<lb/>pus unius reuolutionis.</s></p><p type="margin">
<s><margin.target id="marg129"></margin.target>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><margin.target id="marg130"></margin.target>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>Exemplum primi in repaulò ob&longs;curiore: &longs;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 e&longs;t 2 fit 12, diuide per 4/5 &longs;eu mul­<lb/>tiplica per 5/4 quod idem e&longs;t, fit 15 circuitus e, in quatuor ergo circui­<lb/>tibus, & 4/5 qui &longs;unt duo decim anni perueniet a ad c, & in duodecim <lb/>annis e perueniet ad c, nam 12 &longs;unt 4/5 ip&longs;ius 15. Similiter in &longs;ecundo <lb/>ca&longs;u &longs;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 e&longs;t totius circuitus, id e&longs;t 1/7, e&longs;t autem 1/7 2 1/3, 1/3 <lb/>fient 9 1/3, &longs;imiliter ponatur d 5, & quia a c e&longs;t 1/7 erunt 36/7, diuide ergo <lb/>9 2/3 id e&longs;t 29/3 per 36/7 exeunt 203/108 tempus reuolutionis e. </s>
<s>Quin que ergo <lb/>reuolutiones e erunt 1015/108 addita &longs;eptima parte, quæ e&longs;t 29/108 fient 2044/108 <lb/>&longs;eu 261/27, & &longs;unt anni 9 18/27 &longs;eu 9 2/3, ergo in tanto tempore a faciet qua­<lb/>tuor circuitus, & &longs;eptimam partem, & e quinque circuitus, & &longs;e­<lb/>ptimam.<lb/><arrow.to.target n="marg131"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg131"></margin.target>C<emph type="italics"/><gap/><emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Ex hoc patet, quod non coniungentur in alio loco, neque alio tem <lb/>pore ante prædictum tempus.</s></p><p type="main">
<s>Propo&longs;itio quinquage&longs;ima.</s></p><p type="main">
<s>Omnes circuituum portiones in eiu&longs;dem temporibus <expan abbr="repetun&ttilde;">repetuntur</expan>.</s></p><p type="main">
<s>Sint in circulo a b c d e f g: a & b iuncta, & in primo congre&longs;&longs;u <lb/>iungantur in c, in &longs;ecundo in d, in tertio in e, in quarto in f, in quinto <lb/>in g, in &longs;exto in h, in &longs;eptimo in k, in octauo in l. </s>
<s>Et &longs;ic deinceps <expan abbr="cũquetempora">cunque<lb/>tempora</expan> &longs;int æqualia, erunt & circuitus totidem numero, & exce&longs;­<lb/>&longs;us æquales etiam a c, c d, d e, e f, f g, g h, h k, <lb/><figure id="fig30"></figure><lb/>k l. </s>
<s>Et &longs;i aggregatum a &longs;cilicet circulorum, <lb/>& portionis fuerit commen&longs;um circulo, & <lb/>ita de b erunt omnia <expan abbr="cõmen&longs;a">commen&longs;a</expan> ad circulum, </s></p><p type="main">
<s><arrow.to.target n="marg132"></arrow.to.target><lb/>& etiam inter &longs;e. </s>
<s>Et &longs;i inter &longs;e aggregata, uel <lb/>portiones erunt, & eodem modo reliqua. <lb/></s>
<s>Et quoniam circuli circulis commen&longs;i &longs;unt: <lb/>&longs;i portiones erunt inuicem commen&longs;æ <expan abbr="erũt">erunt</expan>, <lb/>& toti circuitus cum partibus commen&longs;i, & <lb/>&longs;i non commen&longs;i, neque erunt inter &longs;e, neque ad circulum. </s>
<s>Et &longs;i totum <lb/>&longs;patium cum circuitibus erit unius generis, erunt duplicata, & tri­<lb/>plicata, & quadruplicata eiu&longs;dem generis: quare cum &longs;patia ip&longs;a <lb/>detractis circuitibus uelut rhete habeant naturam reci&longs;i, & &longs;patia <lb/>ip&longs;a tota &longs;int eiu&longs;dem generis, erunt &longs;patia, quæ relinquuntur eiu&longs;­<lb/>dem generis. </s>
<s>Erunt tamen incommen&longs;a nece&longs;&longs;ariò, &longs;i partes fuerint <lb/>incommen&longs;æ toti. </s>
<s>Ponatur a c incommen&longs;a toti circulo dico, quod <lb/>a k <expan abbr="etiã">etiam</expan> e&longs;t incommen&longs;a toti circulo: & <expan abbr="etiã">etiam</expan> a k, & k c. </s>
<s>Quia enim a c <lb/>e&longs;t incommen&longs;a circulo, & k a cum toto circulo &longs;emel e&longs;t commen­<pb pagenum="41"/>&longs;a a c, quia multiplex ei. </s>
<s>igitur cum circulus, & a k diuidantur in cir­<lb/><arrow.to.target n="marg133"></arrow.to.target><lb/>culum et a k, & circulus &longs;it incommen&longs;us circulo, cum a k erit aggre. <lb/></s>
<s>gatum ex circulo, & a k incommen&longs;um ip&longs;i a k, & a k pariter incom <lb/><arrow.to.target n="marg134"></arrow.to.target><lb/>men&longs;a circulo. </s>
<s>Rur&longs;us quia a k e&longs;t incommen&longs;a circulo cum a k, & <lb/>circulus cum a k &longs;it multiplex ad a c, erit a k incommen&longs;a a c, quare <lb/><arrow.to.target n="marg135"></arrow.to.target><lb/>erit c k incommen&longs;a a k & a c, & circulo ad dita a k. </s>
<s>Si ergo a c &longs;it <lb/>commen&longs;a circulo, erunt omnes portiones e genere numeri, & &longs;i <lb/><arrow.to.target n="marg136"></arrow.to.target><lb/>potentia rhete erunt omnes, uel potentia rhete, uel circulis detra­<lb/>ctis, ut a k & a l reci&longs;a: & a c &longs;it potentia &longs;ecunda rhete, id e&longs;t radix cu <lb/>bica erunt omnes c d, d e, e f, potentia &longs;ecunda rhete, et radices cubi­<lb/>cæ numeri, &longs;eu latera corporum rhete, a k uero & a l, & huiu&longs;modi <lb/>in infinitum reci&longs;a potentia rhete.<lb/><arrow.to.target n="marg137"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg132"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> C<emph type="italics"/>or<emph.end type="italics"/>^{m}. <lb/><emph type="italics"/>præcedentis.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg133"></margin.target>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><margin.target id="marg134"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 17. <lb/><emph type="italics"/>eiu&longs;dem.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg135"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 14. <lb/><emph type="italics"/>rur&longs;us.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg136"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 17. <lb/><emph type="italics"/>rur&longs;us.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg137"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Ex hoc patet, quod cum circulus po&longs;sit diuidi in infinita gene­</s></p><p type="main">
<s><arrow.to.target n="marg138"></arrow.to.target><lb/>ra quantitatum, quæ non &longs;unt inuicem commen&longs;æ cumque coniun­<lb/>ctiones hæ &longs;emper in eodem genere maneant, quod infinita pun­<lb/>cta, & infinitis in &longs;peciebus quantitatum remanebunt in quibus a <lb/>& b in perpetuum nunquam conuenient. </s>
<s>Velut &longs;i coniunctio pri­<lb/>ma fiat in <02> cu. </s>
<s>1/2 alicuius circuli, nunquam conuenient, neque in me­<lb/>dietate, neque in quarta parte, nec octaua, nec tertia, nec &longs;exta, nec no­<lb/>na, nec quinta, nec decima, & &longs;ic de &longs;ingulis in genere commen&longs;a­<lb/>rum toti circulo. </s>
<s>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>3 nec 2 m: <02> cub. </s>
<s>4, & &longs;ic <lb/>de alijs.</s></p><p type="margin">
<s><margin.target id="marg138"></margin.target>P<emph type="italics"/>er penulti­<lb/>mam uige&longs;i­<lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s></p><p type="main">
<s>Propo&longs;itio quinquage&longs;imaprima.</s></p><p type="main">
<s>Operationes dictas exemplo declarare.<lb/><arrow.to.target n="marg139"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg139"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Supponamus in circulo prædicto a c <02> 7 con&longs;tat, quod e&longs;&longs;e non <lb/>pote&longs;t, quia <02> 7 e&longs;t maior monade, ideo toto circulo, quare non po<lb/>terit e&longs;&longs;e pars circuli, &longs;ed referetur ad <expan abbr="quantitat&etilde;">quantitatem</expan> certam, uelut quod <lb/>circulus &longs;it 10. &longs;emper ergo diuidemus <02> 7, &longs;eu eam portionem per <lb/>10 quantitatem circuli & exibit <02> 7/100, & hæc erit portio circuli, & ita <lb/>&longs;i portio &longs;it <02> cub. </s>
<s>16, diuidemus <02> cub. </s>
<s>16 per 10 exibit <02> cu 2/125, & <lb/>ita de alijs.</s></p><p type="main">
<s>Sed cum ex repetitione cre&longs;cat portio illa, donec exuperet mo­<lb/>nadem, aut aliquem quemuis numerum detracta monade aut nu­<lb/>mero circuituum habebit rationem reci&longs;i. </s>
<s>Velut <02> 7/100 quater &longs;um­<lb/>pta efficit <02> 112/100. Et hoc e&longs;t potentia rhete, &longs;ed &longs;i quis auferat mona­<lb/>dem fiet <02> 112/100 m: 1, & hoc e&longs;t reci&longs;um 1, &longs;cilicet 1 p: <02> v: 23/25 m: <02> 28/25, &longs;ed ta <lb/>men uerè e&longs;t linea media.</s></p><p type="main">
<s>Quod uerò non contingat coniungi in alio loco, neque tem­<lb/>pore &longs;it, ut a b iungantur in c, & &longs;it reuolutio a triplex integra, & b <pb pagenum="42"/>&longs;excuplex, & tempus totum decem annorum: ita ut a c &longs;it tertia <lb/>pars circuitus, & a circuitus tres anni, & quia circuitus b &longs;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="fig31"></figure><lb/>puncto. </s>
<s>Si enim primùm 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 e&longs;t nece&longs;&longs;arium, &longs;i debet perueni­<lb/>re ad c, & erunt anni tres, & 23/19, non ergo <lb/>anni quatuor. </s>
<s>Cum enim tempora di­<lb/>uer&longs;a diuiduntur per numeros haben­<lb/>tes proportionem erunt, qui prodeunt <lb/><arrow.to.target n="table13"></arrow.to.target><lb/>numeri in eadem ratione. </s>
<s>Diui&longs;o ergo <lb/>10 per 1 11/19 exit 6 2/3, & diui&longs;o 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 pote&longs;t e&longs;&longs;e æquale 1/3. Si enim per <lb/>præcedentem repetuntur, ergo non po&longs;­<lb/>&longs;unt redire, doneciterum coniung antur in ip&longs;o a. </s>
<s>Si enim aliter &longs;it <lb/>ut ex e, igitur e c e&longs;t æqualis a c pars toti, quod contingere non po­<lb/>te&longs;t. </s>
<s>Sin uerò coniunctio fiat in d, igitur per præcedentem d e e&longs;t <lb/>pars a c &longs;ubmultiplex quomodolibet, quare non fuerunt a&longs;&longs;um­<lb/>pti primi numeri. </s>
<s>Veluti in exemplo con&longs;tituimus, quod a, & b <lb/>conueniunt in c in decem annis, & a c e&longs;t tertia pars circuitus: er­<lb/>go in triginta annis conueniunt in a, & in quadraginta rur&longs;us in c. <lb/></s>
<s>&longs;i ergo quis a&longs;&longs;ump&longs;i&longs;&longs;et quadraginta annos ab initio pro con­<lb/>gre&longs;&longs;u, & diui&longs;i&longs;&longs;et per 1 12/19 exiret 25 1/3, & &longs;i per 3 exiret 13 1/3, & mani­<lb/>fe&longs;tum e&longs;t, quod uterque numerus pote&longs;t diuidi per eundem nu­<lb/>merum, utpote 4 & exit numerus cum eadem parte &longs;cilicet 6 1/3 & <lb/>3 1/3 ergo conuenient ante, non ergo a&longs;&longs;ump&longs;i&longs;ti minimos in ea pro­<lb/>portione. </s>
<s>Illi autem nequaquam amplius diuidi non po&longs;&longs;unt eo­<lb/>dem modo.</s></p><table><table.target id="table13"></table.target><row><cell>Decem</cell><cell></cell><cell>Quatuor</cell><cell></cell></row><row><cell>3</cell><cell>3 1/3</cell><cell>1 11/19</cell><cell>2/(<gap/>/2<gap/>)</cell></row><row><cell>1 11/19</cell><cell>6 1/3</cell><cell></cell><cell></cell></row></table><p type="main">
<s>Propo&longs;itio quinquage&longs;ima&longs;ecunda.</s></p><p type="main">
<s>Tria mobilia coniuncta in eodem puncto, quorum duo, & duo <lb/>conueniant in partibus in commen&longs;is inter &longs;e, in perpetuum in nul­<lb/>lo unquam puncto conuenient.</s></p><p type="main">
<s><arrow.to.target n="marg140"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg140"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Sint a b c iuncta, & primo iungantur a & b, iterum in d & b, & <lb/>c in e, & &longs;int a d, a e inconimen&longs;æ, dico quòd a b c nunquam con­<lb/>uenient in aliquo puncto, &longs;eu primo, &longs;eu alio à prim o: &longs;i non con­<pb pagenum="43"/><figure id="fig32"></figure><lb/>ueniant in f, erunt ergo in g tempore re­<lb/>uolutiones integræ, & portio a f in&longs;uper. <lb/></s>
<s>Et quia hæ con&longs;tituuntur per congre&longs;&longs;us <lb/>b cum a, & &longs;unt &longs;patia a d, & b cum c, & <lb/>&longs;unt &longs;patia e f, igitur &longs;patium a f erit ex ge­<lb/>nere quantitatis a d, & a e per quinqua­<lb/>ge&longs;imam, harum ergo erunt commen&longs;æ: <lb/>quod e&longs;t contra &longs;uppo&longs;itum. </s>
<s>Et harum <lb/>propo&longs;itionum principium e&longs;t traditum <lb/>à Campano Nouarien&longs;i Euclidis expo&longs;itore, in quodam libello <lb/>non edito qui diligentia patris mei Facij ad me peruenit.</s></p><p type="main">
<s>Propo&longs;itio quinquage&longs;imatertia.</s></p><p type="main">
<s><expan abbr="Circulorũ">Circulorum</expan> &longs;e in aduer&longs;um mouentium proportionem declarare.</s></p><p type="main">
<s><arrow.to.target n="marg141"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg141"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Sit orbis a b cuius cen­<lb/><figure id="fig33"></figure><lb/>centrum c, manubrium c <lb/>d f e, &longs;eu uero tangat circu <lb/>lum g, &longs;eu more gemmas <lb/>&longs;culpentium aligetur al­<lb/>teri orbi funiculo a l b, & <lb/>&longs;it in uertice axis k m or­<lb/>biculus &longs;olidus aut &longs;emi­<lb/>circulari forma m, dico <lb/>quod proportio motus a <lb/>b ad motum m e&longs;t produ <lb/>cta ex duabus proportio­<lb/>nibus c n <expan abbr="&longs;emidimeti&etilde;tis">&longs;emidimetientis</expan>, <lb/>& &longs;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 e&longs;t <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, &longs;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 &longs;i fiat comparatio &longs;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 &longs;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> e&longs;t, ut rectanguli &longs;ub c n, <lb/>& &longs;emidimetiente m ad quadratum k o, quod erat <expan abbr="demon&longs;trandũ">demon&longs;trandum</expan>.</s></p><p type="main">
<s>Manife&longs;tum e&longs;t autem ex ip&longs;a &longs;ola con&longs;titutione, quod &longs;i a b mo­</s></p><p type="main">
<s><arrow.to.target n="marg142"></arrow.to.target><pb pagenum="44"/>uetur &longs;ur&longs;um à dextro in &longs;ini&longs;trum in inferiore parte, mouebitur à <lb/>&longs;ini&longs;tro in dextrum, & uterque circulorum g & k in &longs;uperiore parte, <lb/>& in inferiore mouebitur contrario motu, &longs;cilicet in &longs;uperiore à &longs;ini <lb/>&longs;tro in dextrum, & inferiore à dextro in &longs;ini&longs;trum, illi uerò duo or­<lb/>bes &longs;imili motu mouebuntur tam in parte &longs;uperiore, quàm inferio­<lb/>re, & proportio motuum eorum inter &longs;e erit uelut dimetientium <lb/>corundem.<lb/><arrow.to.target n="marg143"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg142"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 1.</s></p><p type="margin">
<s><margin.target id="marg143"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 2.</s></p><p type="main">
<s>Rur&longs;us 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 e&longs;t, ergo in eodem tempo­<lb/>re, in quo axis circumuertitur in eodem orbis: ergo tanto tardius <lb/>uidebitur moueri axis ip&longs;o orbe, quanta e&longs;t proportio minoris in <lb/>æqualitatis ip&longs;ius axis, &longs;eu ambitus, &longs;eu &longs;emidimetientis ad ambi­<lb/>tum, &longs;eu &longs;emidimetientem orbis.</s></p><p type="main">
<s>Propo&longs;itio quinquage&longs;imaquarta.</s></p><p type="main">
<s>Proportio circuli ad &longs;uum diametrum per <expan abbr="&longs;imilitudin&etilde;">&longs;imilitudinem</expan> e&longs;t quar­<lb/>ta pars peripheriæ. </s>
<s>Rur&longs;usque eiu&longs;dem circuli ad peripheriam diame<lb/>tri quarta pars.</s></p><p type="main">
<s><arrow.to.target n="marg144"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg144"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Quoniam enim &longs;uperficies circuli, ut ab <lb/><figure id="fig34"></figure><lb/>Archimede demon&longs;tratum e&longs;t, fit ex dimi­</s></p><p type="main">
<s><arrow.to.target n="marg145"></arrow.to.target><lb/>dio diametri in <expan abbr="dimidiũ">dimidium</expan> peripheriæ 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&ecedil;. </s>
<s>ergo proportio are&ecedil; <lb/>circuli ad diametrum per &longs;imilitudinem <lb/><arrow.to.target n="marg146"></arrow.to.target><lb/>e&longs;t quarta pars peripheri&ecedil;, & proportio are&ecedil; <lb/>ad <expan abbr="peripheriã">peripheriam</expan> e&longs;t quarta pars dimetientis, quod erat probandum.</s></p><p type="margin">
<s><margin.target id="marg145"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 16. <emph type="italics"/>&longs;ex <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg146"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 2. <emph type="italics"/>diff.<emph.end type="italics"/></s></p><p type="main">
<s>Propo&longs;itio quinquage&longs;imaquinta.</s></p><p type="main">
<s>Proportionem medicamentorum per ordines &longs;uppo&longs;ita æquali <lb/>proportione in ordinibus per quantitates, & proportiones de­<lb/>mon&longs;trare.<lb/><arrow.to.target n="marg147"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg147"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Galenus libro quinto de Simplicibus medicamentis, quem &longs;e­</s></p><p type="main">
<s><arrow.to.target n="marg148"></arrow.to.target><lb/>quuti &longs;unt alij medici, ponit quatuor ordines <expan abbr="medicamentorũ">medicamentorum</expan> iux­<lb/>ta qualitates calidi, frigidi, &longs;icci, & humidi, & primus e&longs;t cum <expan abbr="medi-camentũ">medi­<lb/>camentum</expan> non &longs;entitur quale &longs;it licet operetur, uelut cam&ecedil;melon, ab­<lb/>&longs;ynthium, & oriza: &longs;ecundus e&longs;t, cum &longs;entitur, &longs;ed non lædit, ut nux <lb/>myri&longs;tica, &longs;aluia, ozimum: tertius e&longs;t cum &longs;entitur, & lædit, &longs;ed <lb/>non de&longs;truit, neque corrumpit corpus, uelut a&longs;&longs;arum apium &longs;ta­<lb/>phi&longs;agria, cappares, myrrha, ruta: quartus e&longs;t, cum de&longs;truit ue­<lb/>lut pyretrum, piper, euphorbium cæpe aggre&longs;te, & &longs;inapis, cina­<pb pagenum="45"/>momum autem, & gingiber numerantur inter medicinas calídas <lb/>tertij gradus, & hoc opus comparatur ad corpus &longs;icut dicit Gale­<lb/>nus, & Serapio non ad linguam, ut medici no&longs;tri temporis interpre <lb/>tantur. </s>
<s>Ex quo patet, quod aliqua medicina poterit e&longs;&longs;e quarti ordl <lb/>nis, & non lædere linguam in gu&longs;tu, & alia tertij ordinis, quæ non <lb/>&longs;olum lædet linguam, &longs;ed &longs;en&longs;um eius corrumpet, et de&longs;truet, quod <lb/>contingit propter &longs;ub&longs;tantiam tenuem cra&longs;&longs;æ mi&longs;tam cum &longs;iccitate <lb/>pari ip&longs;i calori. </s>
<s>Sed non oportet h&ecedil;c nunc tractar, enon &longs;olum quia <lb/>non &longs;it locus, &longs;ed etiam quòd con&longs;u&longs;a &longs;it per &longs;eip&longs;a materia ab&longs;que <lb/>eo, quod difficultatem difficultati addamus, &longs;olum ergo eas dubita<lb/>tiones adiungemus, quas <expan abbr="uol&etilde;tes">uolentes</expan> declarare propo&longs;itionem præ&longs;en <lb/>tem, neque &longs;uperfugere, neque declinare po&longs;&longs;umus. </s>
<s>Nam de &longs;icco, <lb/>& humido, cum &longs;int longè minoris actionis, quàm calidum, & fri­<lb/>gidum, & præcipuè humidum, non uideo quomodo po&longs;sit Gale­<lb/>nus &longs;tatuere medicinam humidam tertij gradus, nedum quarti, <lb/>cum non po&longs;sit inueniri medicina, quæ de&longs;truat corpus no&longs;trum <lb/>propter humidam qualitatem. </s>
<s>Et licet Serapio po&longs;uerit gingiber <lb/><arrow.to.target n="marg149"></arrow.to.target><lb/>& enulam & zelim in tertio ordine calidorum & humidorum: & <lb/>inter frigidas, & humidas in tertio portulacam, aizoum, & uirgam <lb/>pa&longs;toris, & fungos. </s>
<s>Primum non au&longs;us e&longs;t ponere medicinas ullas <lb/>calidas, aut frigidas in quarto ordine, qu&ecedil; &longs;int humidæ. </s>
<s>&longs;ecundum, <lb/>quando dicit medicinas calídas, aut frigidas, atque humídas in ter­<lb/>tio ordine, intelligit &longs;olum de qualitate actiua &longs;cilicet caliditate, uel <lb/>frigiditate, & non de humida qualitate, quod o&longs;tendit de gingibe­<lb/>re, & enula, dicens, quod &longs;unt calidæ in tertio ordine, & humidæ <lb/>humido crudo, non au&longs;us addere ordinem, quia non uídit ratio­<lb/>nem, qua po&longs;&longs;ent dici humidæ in tertio. </s>
<s>Et clarius in capite de zei­<lb/>len, quem &longs;tatuerat inter medicinas calidas, & humidas in tertio, di <lb/>cit quod e&longs;t calida in tertio, & humida in primo, ergo non intelligit <lb/>per medicinas calidas & humidas in tertio ordine, quod &longs;int humi­<lb/>dæ in tertio ordine. </s>
<s>Clarius etiam de frigidis & humidis, nam por­<lb/>tula cam dicit e&longs;&longs;e frigidam in tertio, humidam in &longs;ecundo, & quod <lb/>maius, e&longs;t cum collo ca&longs;&longs;et aizoum inter medicinas frigidas, & hu­<lb/>midas in tertio ordine, dicit, quod e&longs;t frigidum in tertio ordine, ad­<lb/>ijcit, quod e&longs;t &longs;iccum parum, & de uirga pa&longs;toris nihil dicit de hu­<lb/>mido, &longs;ed dicit, quod a&longs;tringit, ex quo concludo, quod &longs;ecun­<lb/>dum mentem Serapionis nulla e&longs;t medicina humidior portulaca, <lb/>etiam uidetur innuere de fungis, &longs;atis e&longs;t quod non excedunt &longs;ecun <lb/>dum ordinem in humido neque calida neque frigida, &longs;ed frigida &longs;unt <lb/>humidiora, ut fungi, & portulaca, quia frigiditas in generatione <lb/>humidum magis admittit, quàm caliditas, & calida magis hu­<pb pagenum="46"/>mectant, quia magis penetrat uis medicamenti, & hæc regula de <lb/>humido, & &longs;icco e&longs;t generalis apud Serapionem, quod non intelli­<lb/>gitur ordo in pa&longs;siuis, ni&longs;i &longs;pecialiter exprimatur, nam de &longs;iccitate <lb/>non nego, quin inueniantur medicinæ &longs;iccæ in tertio, & for&longs;an in <lb/>quarto ordine, &longs;ed de hac Galeni o&longs;citantia, quæ in illo peculiaris <lb/>e&longs;t dum uult &longs;equi &longs;uas methodos &longs;ine alio di&longs;crimine, medicis con <lb/>&longs;i derandum relinquo.</s></p><p type="margin">
<s><margin.target id="marg148"></margin.target>C<emph type="italics"/>ap. </s>
<s>ult.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg149"></margin.target>C<emph type="italics"/>ap.<emph.end type="italics"/> 336. <lb/>337. & <lb/>338.</s></p><p type="main">
<s>Secunda difficultas e&longs;t maior, & magis pertinet ad nos, & e&longs;t, <lb/>quòd non declarauit an i&longs;ti ordines inter &longs;e <expan abbr="aliquã">aliquam</expan> proportionem <lb/>&longs;eruarent, an omnino nullam, &longs;i enim nulla proportio &longs;eruatur, fieri <lb/>nullo modo pote&longs;t, ut per cognitionem temperaturæ &longs;implicium <lb/>medicamentorum cogno &longs;camus temperaturam compo&longs;itorum ex <lb/>illis ratione ulla, &longs;ed oportebit &longs;olum experiri. </s>
<s>Sed &longs;i ordines &longs;er­<lb/>uant proportionem, adhuc relinquitur dubium, an illa proportio <lb/>&longs;it Arithmetica, uel Geometrica, uel Mu&longs;ica, & nihil mirum e&longs;&longs;et, <lb/>quod e&longs;&longs;et Mu&longs;ica, ut aliâs docuimus, ubitractauimus de differen­<lb/>tia inter &longs;en&longs;um auditus, et ui&longs;us. </s>
<s>Sed quia de hac nullus medicus ui <lb/>detur intellexi&longs;&longs;e, omittam hanc tractationem. </s>
<s>Et quanquàm Gale­<lb/>nus po&longs;sit uideri non exi&longs;tima&longs;&longs;e, quòd hi ordines non &longs;eruent <lb/>proportionem ullam, quia non au&longs;us e&longs;t tractare de temperamen­<lb/>to medicamentorum compo&longs;itorum per rationem temperamen­<lb/>ti &longs;implicium, nihilominus &longs;uppo&longs;ito quod ita e&longs;&longs;et, quod &longs;eruetur <lb/>altera proportionum, uolo o&longs;tendere rationem componendi in <lb/>utraque proportione & Arithmetica, & Geometrica. </s>
<s>Ex quo &longs;e­<lb/>quitur, quod Aueroes quàm o&longs;citanter tractauerit in quinto &longs;uo­<lb/>rum collectaneorum de hoc, & non di&longs;tinguit, neque docet pri­<lb/>mum an &longs;it aliqua proportio, deinde &longs;i qua &longs;it, cuius generis &longs;it, & <lb/>cum in re tam clara pugnet pror&longs;us, ut cœcus ictus maximos eden­<lb/>do, &longs;ed in ca&longs;&longs;um plero&longs;que, quàm malè agant qui ei in arduis tan­<lb/>tum tribuunt fidei, & authoritatis, &longs;ed hæc e&longs;t infelicitas no&longs;tra, & <lb/>ira Deorum. </s>
<s>Suppo&longs;ito ergo quod primò ordines di&longs;tinguantur <lb/>per proportionem arithmeticam, &longs;it &longs;uperficies a b pro quantitate, <lb/><figure id="fig35"></figure><lb/>& a &longs;it calida in primo gradu, & b in ter­<lb/>tio, erit ergo perinde ac &longs;i duo corpora <lb/>e&longs;&longs;ent unum altitudinis unius cum ba&longs;i <lb/>quadrilatera rectangula a, aliud altitu­<lb/>dinis trium, ba&longs;i autem quadrilatera &longs;u­<lb/>perficie rectangula b, hoc igitur erit to­<lb/>tum mi&longs;tum, & quia quantitas medicamenti non mutatur quæ e&longs;t <lb/>a, b, ergo talia corpora æquantur uni corpori, cuius ba&longs;is e&longs;t a b, <lb/>cum ergo talia corpora producantur ex a in unum, & b in tria, ergo <pb pagenum="47"/>diui&longs;o aggregato per a b prodibit altitudo, &longs;eu ordo qualitatis to­<lb/>tius medicamenti, iuxta quod con&longs;tituitur regula prima libri artis <lb/>medendi paruæ huiu&longs;modi, & reliquæ, traduxi autem illas ad hunc <lb/>locuin, “quia pendent ex demon&longs;tratione hac: “duc numerum ordi­<lb/>nis &longs;ingulorum medicamentorum in numerum quantitatis, &longs;imilia <lb/>iunge, di&longs;similia detrahe, quod fit, diuide per aggregatum, quanti­<lb/>tatum, exibit numerus ordinis compo&longs;iti. </s>
<s>Sic mi&longs;cendo calidum in <lb/>&longs;ecundo ordine cum duplo pondere temperati conflabit calidum <lb/>in be&longs;&longs;e. </s>
<s>Secunda &longs;i ex pluribus diuer&longs;arum, qualitatum, & ordi­<lb/>num temperatum efficere uelis, duc quæ &longs;unt eiu&longs;dem qualitatis in <lb/>&longs;uas quantitates, & iunge, quod fit, diuide per numerum or dinis <lb/>medicamenti contrarij, exibit quantitas illius, &longs;ub qua &longs;i iungatur, <lb/>fiet medicamentum temperatum. </s>
<s>Tertia cum nolueris ex tempera­<lb/>to, & alio cuiu&longs;cunque ordinis medicamen conficere ordinis re­<lb/>mi&longs;sionis, detrahe numerum ordinis eius, quod conficere uis ex nu <lb/>mero ordinis eius, quod habes, & cum re&longs;iduo diuide numerum <lb/>medicaminis, quod conficere uis, quod exit e&longs;t numerus quantita­<lb/>tis medicamenti non temperati in comparatione ad temperatum.” <lb/>Ex his potes propo&longs;itis quibu&longs;cunque medicamentis conficere <lb/>antidotum &longs;ub quo cunque ordine remi&longs;siore potenti&longs;simo ex il­<lb/>lis. </s>
<s>Quarta in compo&longs;itione, quæ non fermente&longs;cit calida, calidis <lb/>iuncta &longs;emper opus augent, ut mel cum pipere. </s>
<s>Quæ autem &longs;ub mi<lb/>nore quantitate exhibentur non &longs;ub remi&longs;siore ordine agant, &longs;ed <lb/>uel facilius impediuntur, uel minorem corporis partem, uel leuius <lb/>immutant.</s></p><p type="main">
<s>Quod &longs;i &longs;tatuamus proportionem e&longs;&longs;e Geometricam, modus <lb/>erit idem in omnibus, & quo ad numerum etiam in primo, & &longs;ecun <lb/>do ordine, quia in proportione dupla Geometrica &longs;ecundus ordo <lb/>tantundem di&longs;tat à primo, quantum primus ab æqualitate, quia <lb/>unum & duo &longs;eruant proportionem, & æqualem di&longs;tantiam, &longs;ed in <lb/>cæteris ordinibus non ita erit, quia qui e&longs;&longs;et trium in Arithmetica, <lb/>&longs;cilicet totius ordo e&longs;t, quatuor in Geometrica, & quartus ordo, <lb/>qui e&longs;&longs;et quatuor in Arithmetica, e&longs;&longs;et octo in Geometrica, ideo <lb/><figure id="fig36"></figure><lb/>&longs;cribemus ordines hoc modo, & operabimur cum <lb/>numeris loco ordinum, exemplum ergo primum <lb/>&longs;it medicina calida in tertio ordine quatuor uncia­<lb/>rum, & medicina frigida in <expan abbr="&longs;ecũdo">&longs;ecundo</expan> ordine duarum <lb/>unciarum, duco quatuor in tria, &longs;i proportio &longs;it Arithmetica, fit <lb/>duodecim, duco duo in duo fit quatuor, detraho quatuor in duo­<lb/>decim, quia omnis medicina tantum retondit de contrario, &longs;eu mi­<lb/>nuit relin quuntur octo &longs;cilicet caliditatis, diuido per &longs;ex ag­<pb pagenum="48"/>gregatum unciarum exit unum, & tertia, ergo erit calida in princi­<lb/>pio &longs;ecundi ordinis. </s>
<s>Secundum exemplum &longs;int eædem medicinæ, <lb/>& &longs;it proportio Geometrica, ducemus ergo quatuor in quatuor, & <lb/>fiunt &longs;exdecim, & duo in duo fiunt quatuor, detrahe quatuor ex &longs;ex <lb/>decim, & remanent duodecim, diuide per &longs;ex, ut prius, exeunt duo, <lb/>ergo erit calida in fine &longs;ecund i gradus uides ergo di&longs;crimen. </s>
<s>rur&longs;us <lb/>&longs;int ambæ medicinæ calidæ, & ducemus, ut prius in tertio exem­<lb/>plo, ubi proportio &longs;it Arithmetica iungendo duodecim cum qua­<lb/>tuor, & fient &longs;exdecim, diuide per &longs;ex, exeunt duo, & duæ tertiæ, er­<lb/>go erit calida in medio tertij gradus, rur&longs;us in quarto exemplo iun <lb/>gemus &longs;edecim cum quatuor, & fient uiginti, diuide per &longs;ex exi­<lb/>bunt tria & tertia, & ita erit in medio tertij gradus, ut prius, &longs;ed &longs;i <lb/>ille quatuor unciæ e&longs;&longs;ent calidæ in quarto gradu, & illæ duæ unciæ <lb/>in &longs;ecundo gradu, ut prius ducendo quatuor in quatuor fiunt &longs;ex­<lb/>decim, & duo in duo fiunt quatuor, iunge, & fient uiginti, diuide <lb/>per &longs;ex exeunt tria cum tertia, ergo erit calida in principio quarti <lb/>gradus &longs;ecundum proportionem Arithmeticam, &longs;ed &longs;ecundum <lb/>Geometricam duc quatuor in octo, fiunt triginta duo, adde qua­<lb/>tuor ut prius, &longs;cilicet productum duorum in duo fiunt triginta &longs;ex, <lb/>diuide per &longs;ex, exeunt &longs;ex, & quia &longs;ex ad quatuor maiorem habent <lb/>proportionem, quàm octo ad &longs;ex ideo hæc medicina erit calida ul­<lb/>tra medium quarti gradus, iam ergo uides rationem, & differen­<lb/>tiam horum.</s></p><p type="main">
<s>Quod &longs;i quis dicat, an debeat attendi Geometrica proportio in <lb/>medicamentis, an Arithmetica, re&longs;pondeo, quòd ueri&longs;imilius e&longs;t de <lb/>Arithmetica, quia illa proportio etiam quod &longs;it minor quatuor ad <lb/>trium, quàm trium ad duo, & multò minor quàm duo ad unum ni­<lb/>hilominus longè plus operatur, quia tertius ordo iam incipit e&longs;&longs;e <lb/>præter naturam, & uidemus, quod læ&longs;io facta in uulnerato, etiam <lb/>quòd &longs;it quadruplo minor, plus nocet longè, quàm in &longs;ano qua­<lb/>druplo maior: quia termini præter naturam &longs;unt ualdè angu&longs;ti in <lb/>comparatione ad latitudinem naturalem, &longs;icut etiam uidemus in­<lb/>tendendis chordis &longs;corpionum, quod ultima pars e&longs;t breuis & ta­<lb/>men homini tantam difficultatem adijcit. </s>
<s>Notandum e&longs;t etiam, <lb/>quòd ob hoc diui&longs;erunt ordines in tres partes, uelut gingiber e&longs;t <lb/>calidum in fine tertij ordinis, origanum in medio, cinamomum in <lb/>principio, & ita euphorbium e&longs;t calidum in principio quarti gra­<lb/>dus, &longs;ed in fine principij piper, in prin cipio principij aqua &longs;epara­<lb/>tionis in medio quarti ordinis, &longs;ed oleum chalcanthi factum ea ar­<lb/>te, ut exurat paleas, &longs;icut ignis e&longs;t calidum in fine quarti ordinis, & <lb/>ita &longs;ufficiet diuidere propter eandem cau&longs;am primum, & &longs;ecun­<pb pagenum="49"/>dum ordinem in duas tantum partes non ratione latitudinis, quæ <lb/>e&longs;t æqualis, uel etiam for&longs;an maior, &longs;ed ratione uarietatis operatio­<lb/>nis quæ minus &longs;entitur, & maximè in primo ordine.</s></p><p type="main">
<s>Propo&longs;itio quinquage&longs;ima&longs;exta.</s></p><p type="main">
<s>Proportio cuiu&longs;uis binomij ad &longs;uum reci&longs;um, uel ei commen­<lb/>&longs;um e&longs;t duplicata ei, quæ ad numeri latus.<lb/><arrow.to.target n="marg150"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg150"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>m.</s></p><p type="main">
<s>Cum enim proportionis medium &longs;itlatus numeri eo quod ex bi <lb/>nomio in reci&longs;um &longs;uum fit numerus ex his, quæ demon&longs;trata &longs;unt <lb/>generaliter in tertio Arithmeticæ de omnibus binomijs cum &longs;uis </s></p><p type="main">
<s><arrow.to.target n="marg151"></arrow.to.target><lb/>reci&longs;is, uel in quadratis lateribus erit <02> numeri media proportione <lb/>inter binomium, & &longs;uum reci&longs;um, igitur cum proportio producto­<lb/>rum ex binomio in commen&longs;a reci&longs;o &longs;it, ut commen&longs;orum ad reci­<lb/><arrow.to.target n="marg152"></arrow.to.target><lb/>&longs;a crunt omnia producta ex binomio in commen&longs;a reci&longs;o &longs;uo <02> nu <lb/><arrow.to.target n="marg153"></arrow.to.target><lb/>meri, igitur proportio binomij ad reci&longs;um &longs;uum, & omnia com­<lb/>men&longs;a illi, e&longs;t duplicata ei quæ ad <02> numeri.<lb/><arrow.to.target n="marg154"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg151"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 6. P<emph type="italics"/>ro­<lb/>po&longs;. </s>
<s>lib.
de<emph.end type="italics"/><lb/>A<emph type="italics"/>liza.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg152"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 17. <emph type="italics"/>&longs;ex <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg153"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 17. <lb/><emph type="italics"/>&longs;eptimi <lb/>eiu&longs;dem.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg154"></margin.target>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>Propo&longs;itio quinquage&longs;ima&longs;eptima.</s></p><p type="main">
<s>Motus rationem ad pondus inuenire.</s></p><p type="main">
<s><arrow.to.target n="marg155"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg155"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>O&longs;ten&longs;um e&longs;t antea, quod motus naturalis uelocior fit in fine, ac <lb/>magis augetur ob aëris motum, ubi uerò hæret e&longs;t ac &longs;i quie&longs;cat. <lb/></s>
<s>Eadem autem e&longs;t ratio in motis uiolenter, & naturaliter dum &ecedil;qua­<lb/>li impetu feruntur. </s>
<s>Sed &longs;ubitò po&longs;t etiam, quod motus æqualiter <lb/>augerentur minus tamen cre&longs;cit proportio uiolenti &longs;cilicet ob im­<lb/><figure id="fig37"></figure><lb/>pedimentum naturale. </s>
<s>Sed &longs;i uis mouens fuerit <lb/>adeò ualida ut proportio incrementi ex aëre &longs;it <lb/>maior, quàm impedimentum, & in crementum al <lb/>terius mobilis naturaliter moti, motus ille uelo­<lb/>cior fiet naturali, ut in &longs;phæris ferreis ex machina <lb/>igne excu&longs;sis, quod ergo attinet ad præ&longs;entem <lb/>motum ratio e&longs;t eadem. </s>
<s>Quicun que ergo motus <lb/>minoris grauis cogit de&longs;cendere lancem ex ad­<lb/>uer&longs;o proportionem habet eandem ad &longs;uum mo <lb/>bile quam habet graue æquiponderans. </s>
<s>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 &longs;e inuicem, & ad a qualis mo­<lb/>tuum ob di&longs;tantiam intentorum. </s>
<s>Experimentum ergo docet, quòd <lb/>dimidium ponderis æquilibrium facit ex palmo minoris dimidio <lb/>motum manife&longs;tum, & ex palmo quarta pars ponderis, ergo &longs;e ha­<lb/>bent prope portionem.</s></p><p type="main">
<s>Propo&longs;itio quinquage&longs;imaoctaua.</s></p><p type="main">
<s>Qu&ecedil; ex alto de&longs;cendunt cur non eandem pro di&longs;tantia motus ra<lb/>tionem in libero aëre &longs;eruent con&longs;iderare.</s></p><pb pagenum="50"/><p type="main">
<s>Aër in &longs;ublimiore eius regione &longs;emper naturali motu fertur ex <lb/>Oriente in Occidentem, &longs;ed & infra uerum minus manife&longs;tè. </s>
<s>At ca­<lb/>&longs;u plerun que contingit, ut moueatur longè uehementius, &longs;eu ad ean­<lb/>dem partem, &longs;eu aliam. </s>
<s>Qui uerò naturalis e&longs;t, debilis <lb/><figure id="fig38"></figure><lb/>e&longs;t, quoniam in tenui ualde &longs;ub&longs;tantia e&longs;t: nec <expan abbr="cõtinuus">continuus</expan> <lb/>&longs;ed in&longs;tar motus aquæ maris fluit ac refluit: aliter ne­<lb/>ce&longs;&longs;e e&longs;&longs;et, ut &longs;ingulis horis per mille milliaria procede­<lb/>ret, ut &longs;ic ne que latere po&longs;&longs;et, quarndoquidem fortuiti mo <lb/>tus, qui &longs;unt multo tardiores non latentnos. </s>
<s>Nam tardiores illos <lb/>e&longs;&longs;e <expan abbr="cõ&longs;tat">con&longs;tat</expan>, cum in hora &longs;int pul&longs;us arteriarum, quatuor millia <expan abbr="ictuũ">ictuum</expan> <lb/>in homine prope temperamentum: &longs;i igitur motus naturalis aëris <lb/>e&longs;&longs;et continuus, in hora aër procederet ob ambitum terræ millies <lb/>mille pa&longs;&longs;us, <expan abbr="igi&ttilde;">igitur</expan> in ictu pul&longs;us &longs;uperaret pa&longs;&longs;us 250. At experimur <lb/>nullum uentum aut procellam &longs;uperare quinquaginta pa&longs;&longs;us, cum <lb/>etiam continuus e&longs;&longs;e nunquam &longs;oleat, imò ne po&longs;sit quidem, ita que<lb/>cum hic multo tardior etiam in &longs;ublimi, dum e&longs;t, nos latere non <lb/>queat, multo minus po&longs;&longs;et naturalis latere, &longs;i adeò uelox & in ea­<lb/>dem parte <expan abbr="a&etilde;ris">aerris</expan> e&longs;&longs;et at que continuus. </s>
<s>Præterea tantus impetus nun­<lb/>quam à minore motu, aut cau&longs;a &longs;uperaretur, adeò ut &longs;emper flatum <lb/>aëris orientalem &longs;entiremus. </s>
<s>Quotidie etiam aduenire ad nos aë­<lb/>rem ex Illyrico, Macedonia, My&longs;ia, Ponto, Bythínia, Capado cia, Sy <lb/>ria, Babylonia, Hyrcanomarí, Bactrianis, Sacís, Scythis, ac Seris, to­<lb/>to præterea Oceano orientali tam ua&longs;to, & Gallica noua, terra que flo <lb/>rida non &longs;olum res e&longs;t admirabilis', & incredibilis, &longs;ed etiam aliena <lb/>à &longs;en&longs;u, & ab his, quæ eueniunt. </s>
<s>A'&longs;en&longs;u quidem, quoniam nebul&ecedil;, <lb/>quæ in aëre mouentur, primùm non in eandem partem &longs;emper mo<lb/>uentur: nun quam autem adeò celeriter: at &longs;i aër &longs;ic circumuoluere­<lb/>tur, mouerentur & illa, qu&ecedil; in eo continentur, quotidieque aërem ex­<lb/>periremur & nubilo&longs;um, & madidum propter mare. </s>
<s>Nechis, quæ <lb/>eueniunt hoc &longs;atis re&longs;pondet, nec nobis id contingeret, ut &longs;i pe&longs;ti­<lb/>aliqua in regione no&longs;tra directa &longs;æuiret, ut aër &longs;ingulis diebus la­<lb/>be ea infectus ad nos deferretur. </s>
<s>Moueri uerò aërem &longs;emper mani­<lb/>fe&longs;ti&longs;simum e&longs;t tum experimento, tum ratione: ratione &longs;iquidem, <lb/>quod aqua & cœlum naturaliter perpetuò mouentur, quare etiam <lb/>aër. </s>
<s>Experimento, quòd ubi hiant o&longs;tia, & ianuæ, ibi perpetuus &longs;en­<lb/>titur flatus. </s>
<s>Ergo &longs;i a pondus de&longs;cendat in c, ex alto fertur rectà, &longs;ed <lb/>&longs;i ex &longs;ublimi transferetur in b, & indirecta, & ad latus, unde ex <lb/>hoc &longs;equitur.</s></p><p type="main">
<s>Propo&longs;itio quin quage&longs;imanona.</s></p><p type="main">
<s><arrow.to.target n="marg156"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg156"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>m.</s></p><p type="main">
<s>Omne mobile motum duobus motibus non ad idem tendenti­<lb/>bus, utro que &longs;eor&longs;um tardius mouetur &longs;imili motu.</s></p><pb pagenum="51"/><p type="main">
<s>Sit a mobile, quod moueatur per a b c impul&longs;u uenti aut uiolen­</s></p><p type="main">
<s><arrow.to.target n="marg157"></arrow.to.target><lb/><figure id="fig39"></figure><lb/>to cum naturali coniuncto: & &longs;it terminus naturalis e, <lb/><arrow.to.target n="marg158"></arrow.to.target><lb/>& uiolenti d: uter que in directo c, dico, quod tardius per­<lb/>ueniet ad c quam d, uel e. </s>
<s>De e manife&longs;tum e&longs;t, quoniam <lb/>motus aëris, qui intendit motum a, diuíditur in partem, <lb/>quæ iuuat motum ad d, & partem, quæ mouetur ad e, <lb/>igitur fit minor adiectio. </s>
<s>Et etiam quia a c e&longs;t longior <lb/>a e ex diffinitione rectæ: quare tardius perueniet ad c quàm ad e du <lb/>plici ratione. </s>
<s>Dico etiam, quod tardius ad c quàm d. </s>
<s>Quia enim <lb/>uis, quæ fert ad d repugnat ei, quæ fert ad e, & uis, quæ fert ad e, re­<lb/>pugnat ei quæ fert ad d, igitur tardius perueniet ad c, quàm d. </s>
<s>Nec <lb/>potes dicere, quòd uis, quæ fert ad c adiuuet ad motum è regione <lb/>d, nam cum unus motus non po&longs;sit perfici &longs;ine altero, igitur quan­<lb/>tum motus ad eretar dabit motum ad d, tanto motus a c erit tardí­<lb/>or ab&longs;olutè motu ad d. </s>
<s>Verum etiam e&longs;t, quod c e breuior erit a d, <lb/>quia motus ad e &longs;emper contrahit motum ad d naturalis uiolen­<lb/>rum ob cau&longs;am dictam. </s>
<s>Vtrùm uerò motus ad c ab&longs;olutè &longs;it tardi­<lb/>or, quàm ad d, non &longs;uppo&longs;ito, quod c e &longs;it æqualis a d, &longs;ed minor, <lb/>nunc non e&longs;t locus determinandi.</s></p><p type="margin">
<s><margin.target id="marg157"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s><margin.target id="marg158"></margin.target>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>Ex hoc patet, quod motus æquidi&longs;tantis mobilis, finis e&longs;t mini­<lb/><arrow.to.target n="marg159"></arrow.to.target><lb/>mus omnium: quoniam mobile qua&longs;i quie&longs;cit in illo. </s>
<s>Velut &longs;i a mo<lb/>ueatur ad b, inde deflectat ad c minimus motus erit in b, ubi incipit <lb/>naturalis: nam cum incipiat, erit debili&longs;simus, quia non <lb/><figure id="fig40"></figure><lb/>e&longs;t motus actu: uiolentus autem æqualis e&longs;t naturali, <lb/>dum minimus e&longs;t: ergo cum ex di&longs;tantia medij palmi <lb/>duplicetur, naturalis erit motus in b minimus, ni&longs;i b c <lb/><arrow.to.target n="marg160"></arrow.to.target><lb/>e&longs;&longs;et minor dimidio palmi. </s>
<s>Et etiam quòd e&longs;&longs;et minor, quia ut di­<lb/>ctum e&longs;t, uter que &longs;imul iunctus e&longs;t æqualis uni eorum non impedito <lb/>uel minor.</s></p><p type="margin">
<s><margin.target id="marg159"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s><margin.target id="marg160"></margin.target>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>Propo&longs;itio &longs;exage&longs;ima.</s></p><p type="main">
<s>Omne mobile motu naturali de&longs;cendens parte, de&longs;cendit gra­<lb/>uiore &longs;ecundum grauitatis centrum.</s></p><p type="main">
<s>Sit a mobile, grauitatis centrum b, cuius pars ei pro­<lb/><arrow.to.target n="marg161"></arrow.to.target><lb/><figure id="fig41"></figure><lb/>ximior &longs;it c a, dico quod de&longs;cendat motu naturali c a, <lb/>parte tangendo terram, quia enim totum a non pote&longs;t <lb/>de&longs;cendere ad centrum de&longs;cendit b, quia eadem e&longs;t na­<lb/>tura partis, & totius: totius autem terræ natura e&longs;t ut <lb/>centrum, totius &longs;it centrum grauitatis, quare b breuiore uia fertur <lb/><arrow.to.target n="marg162"></arrow.to.target><lb/>ad centrum, ergo per c d proximiorem partem ip&longs;i b. </s>
<s>Sed pars pro­<lb/>ximior nece&longs;&longs;ariò e&longs;t grauior, quia centrum e&longs;t in medio grauita­<pb pagenum="52"/>tis, ergo omne mobile de&longs;cendit motu naturali per &longs;ui grauio­<lb/>rem partem.<lb/><arrow.to.target n="marg163"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg161"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s><margin.target id="marg162"></margin.target>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><margin.target id="marg163"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Ex hoc &longs;equitur, quòd graue habens partes inæquales, &longs;eu &longs;ub­<lb/>&longs;tantia, &longs;cu forma, &longs;i ita excutiatur, ut pars grauior <expan abbr="nõ">non</expan> &longs;it, infrà opor­<lb/>tet, ut circumuoluatur.</s></p><p type="main">
<s>Propo&longs;itio &longs;exage&longs;imaprima.</s></p><p type="main">
<s>Proportionem ictus ad pondus rei, & di&longs;tantiam generaliter <lb/>con&longs;iderare.</s></p><p type="main">
<s><arrow.to.target n="marg164"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg164"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Dictum e&longs;t &longs;uperius de proportione de&longs;cenfus ad grauitatem: </s></p><p type="main">
<s><arrow.to.target n="marg165"></arrow.to.target><lb/>& quòd &longs;i graue de&longs;cendat ex alto impeditur à motu aëris: & quòd <lb/><arrow.to.target n="marg166"></arrow.to.target><lb/>res, quæ mouetur duobus motibus non ad idem tendentibus tar­<lb/><arrow.to.target n="marg167"></arrow.to.target><lb/>dius mouetur, quam motus &longs;it unu&longs;qui&longs;que. </s>
<s>Demùm quòd graue <lb/><arrow.to.target n="marg168"></arrow.to.target><lb/>de&longs;cendens circumuoluitur, &longs;i pars grauior non &longs;it, deor&longs;um: & an­<lb/>tea ubi egimus de proportione motus ad grauitatem, quod h&ecedil;cin­<lb/>telligenda &longs;unt prout po&longs;&longs;unt intelligi de motu etiam uiolento. <lb/></s>
<s>Cum ergo uideamus duo hæc, quodres acuta frangit caput, &longs;i ex <lb/>alto incidat, &longs;ed non concutit, lata concutit, &longs;ed non diuidit, premit <lb/>tamen carnem &longs;ubiectam: nec hoc accidit merito ponderis: nam ut <lb/>ui&longs;um e&longs;t &longs;emilibra lapidis, uel ferri cadens ex alto contundit caput, <lb/>& uulnerat, & non eleuat in æquilibrio, ut potè ex alto cadens loco <lb/>per &longs;patium octo palmorum pondus &longs;exdecim librarum, & a pon­<lb/>dere &longs;exdecim librarum homo non læditur, nec uulneratur, ergo id <lb/>accidit ex alia cau&longs;a, & e&longs;t, quod aër interceptus inter graue, & cor­<lb/>pus no&longs;trum non pote&longs;t dilabi tam citò, ergo ne corpus penetret, <lb/>cogitur ingredi locum, cui e&longs;t obuius, at que ita concutere, & diuide­<lb/>re. </s>
<s>Ex quibus &longs;equuntur omnia hæc.<lb/><arrow.to.target n="marg169"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg165"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 57.</s></p><p type="margin">
<s><margin.target id="marg166"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 58.</s></p><p type="margin">
<s><margin.target id="marg167"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 59.</s></p><p type="margin">
<s><margin.target id="marg168"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 60.</s></p><p type="margin">
<s><margin.target id="marg169"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Primùm &longs;i quod incidit, molle fuerit, non uulneratur caput, uel <lb/>pars &longs;ubiecta, quia re&longs;ilit in corpus molle: nec à molli, quia retundi­<lb/>tur, pote&longs;t uulnerari: ergo nullo modo. </s>
<s>Sed neque adeò concutit, <lb/>quia aër rediens, & receptus in molli corpore pro parte, non uer­<lb/>berat locum.</s></p><p type="main">
<s><arrow.to.target n="marg170"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg170"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Secundum in omni colli&longs;ione &longs;eu duri, &longs;eu mollis, &longs;ed magis du­<lb/>ri, dilabuntur partes aëris ad latera, ideo quod partes mediæ pre­<lb/>muntur. </s>
<s>Et quanto motus e&longs;t tardior.</s></p><p type="main">
<s><arrow.to.target n="marg171"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg171"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Tertium in motu uelo ci fit maior ictus & læ&longs;io, & maiora omnia <lb/>quam proproportione motus: quoniam ob uelo <expan abbr="citat&etilde;">citatem</expan> minus diffu <lb/>git aëris. </s>
<s>Et ideò fiunt grauia uulnera ex modico incremento uelo­<lb/>citatis motus.</s></p><p type="main">
<s><arrow.to.target n="marg172"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg172"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Quartum res latæ, duræ concutiunt, & non uulnerant ni&longs;i &longs;int <lb/>cum magno impetu, aut ualde graues: acutæ autem uulnerant, &longs;ed <lb/>non concutiunt, ni&longs;i parti acutæ lata &longs;uccedat.</s></p><pb pagenum="53"/><p type="main">
<s>Quintum, corpora dura magis læduntur à latis, quia &longs;cindun­</s></p><p type="main">
<s><arrow.to.target n="marg173"></arrow.to.target><lb/>tur, mollia autem à tenuibus, quia diuiduntur: nam mollitie excipi­<lb/>unt aërem, & ita à latis non adeò patiuntur, & etiam, quoniam nec <lb/>franguntur, nec &longs;ponte &longs;cinduntur.</s></p><p type="margin">
<s><margin.target id="marg173"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Sextum, etiam in duris penetrat aliquid aëris, aliter tota frange­<lb/><arrow.to.target n="marg174"></arrow.to.target><lb/>rentur. </s>
<s>Con&longs;tat etiam omnem lapidem marmoreum, aut &longs;iliceum <lb/>e&longs;&longs;e poro&longs;um, ut dicunt. </s>
<s>Et etiam quia recipitur in mollioribus, er­<lb/>go etiam in durioribus & in duri&longs;simis: quod &longs;i non recipiant ut ui <lb/>trum, & gemmæ tota franguntur. </s>
<s>Hoc etiam uidetur &longs;en&longs;i&longs;&longs;e Philo <lb/>&longs;ophus, qui uult, quòd res franguntur ob poros.</s></p><p type="margin">
<s><margin.target id="marg174"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Propo&longs;itio &longs;exage&longs;ima&longs;ecunda.</s></p><p type="main">
<s>Proportionem motoris in plano ad motorem, qui eleuat pon­<lb/>dus iuxta id, quod mouet inuenire.</s></p><p type="main">
<s>Con&longs;titutum e&longs;t inuenire proportionem uirium, quæ eleuant <lb/><arrow.to.target n="marg175"></arrow.to.target><lb/>pondus ad uires, quæ ip&longs;um in plano leui trahere po&longs;­<lb/><figure id="fig42"></figure><lb/>&longs;unt. </s>
<s>Vires enim, quæ eleuant pondus a &longs;unt eædem <lb/>puta b, quæ uero trahunt c, &longs;ed hæ po&longs;&longs;unt uariari, nam <lb/>quanto uinculum altius, aut decliuis locus magis, aut <lb/>a&longs;pera &longs;uperficies &longs;eu ponderis &longs;eu plani, tanto difficilius trahitur, <lb/>& maiores expo&longs;cit uires: hoc enim experimento deprehenditur. <lb/></s>
<s>Duæ uerò po&longs;tremæ cau&longs;æ etiam per &longs;e per&longs;picuæ &longs;unt, nec demon <lb/>&longs;tratione indigent: ni&longs;i quod &longs;i planum &longs;it duri&longs;simum, ac leui&longs;si­<lb/>mum, quod e&longs;t a&longs;perum facilius trahitur, quia minore &longs;ui parte pla­<lb/>num tangit. </s>
<s>Nos præterea &longs;upponimus planum æquale undique <lb/>leue durum, & corpus undique &longs;ibi &longs;imile, id e&longs;t cubi formam refe­<lb/>rens, & uinculum in imo: Demon&longs;trare igitur expedit primum, <lb/>quòd in hoc ca&longs;u b e&longs;t duplum ad c. </s>
<s>Quia enim cum a eleuatur b ui <lb/>res &longs;uperant motum ob&longs;curum &longs;eu occultum, &longs;eu pondus a, & &longs;i <lb/>permitteretur &longs;ine eo, quod &longs;u&longs;tineret, de&longs;cenderet iuxta pondus <lb/>&longs;uum, quod &longs;it d: nititur ergo per pondus d, at quia trahendo duci­<lb/>tur circa medium, nam plana &longs;uperficies parum differt à rotunda <lb/>terræ ob terræ 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 e&longs;t primum, quod pote&longs;t <lb/>mouere d, igitur c e&longs;t primum, quod pote&longs;t mouere dimidium a, ut <lb/>ergo dimidium a ad d, ita c ad b, e&longs;t igitur c dimidium b.</s></p><p type="margin">
<s><margin.target id="marg175"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Propo&longs;itio &longs;exage&longs;imatertia.</s></p><p type="main">
<s>Omne graue quanto proximius alligatum plano, tanto faci­<lb/>lius trahitur.<pb pagenum="54"/><arrow.to.target n="marg176"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg176"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Sit graue a b c alligatum funibus in d ef, dico, <lb/><figure id="fig43"></figure><lb/>quòd facilius trahetur per fe quàm c b & e b, quàm <lb/>d a, quia &longs;i debet trahi ex a uel b, aut cadet, aut uis ex <lb/>a & b communicabitur c, igitur erit minor quàm in <lb/>c, & hoc naturaliter. </s>
<s>Mathematica autem ratione quoniam ex a tra­<lb/>hetur c, qua&longs;i per lineam d c: at attractio recta e&longs;t ualidior obliqua­<lb/>igitur attractio c per d e&longs;t debilior, quàm per f. </s>
<s>Rur&longs;us &longs;i e trahitur <lb/>per d cùm a peruenerit in d, erit perinde ac, &longs;i attractum e&longs;&longs;et per li­<lb/>neam c d, &longs;ed linea c d mouet duobus motibus, uno ad &longs;uperiora, al </s></p><p type="main">
<s><arrow.to.target n="marg177"></arrow.to.target><lb/>tero ad latus, ergo lentius ad f per d c quàm f c, quod erat demon­<lb/>&longs;trandum.</s></p><p type="margin">
<s><margin.target id="marg177"></margin.target>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>Propo&longs;itio &longs;exage&longs;imaquarta.</s></p><p type="main">
<s>Omne mobile quanto latius tanto tardius mouetur in plano.<lb/><arrow.to.target n="marg178"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg178"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Demon&longs;tratum e&longs;t &longs;uperius quòd &longs;i mobile &longs;it &longs;ph&ecedil;ricum, & tan </s></p><p type="main">
<s><arrow.to.target n="marg179"></arrow.to.target><lb/>gat planum in puncto, quòd mouetur per quancunque uim aptam <lb/>diuidere medium. </s>
<s>Quia ergo &longs;i tangat in puncto facillime moue­<lb/>tur, &longs;i in linea paulò difficilius, &longs;i per &longs;uperficiem adhuc difficilius, <lb/>igitur cum fiat attritio in motu quanto latius e&longs;t mobile eo diffici­<lb/>lius mouetur. </s>
<s>Sit ergo mobile a b, quod moueatur uer&longs;us c, & quia <lb/>pars b &longs;eu dimidium mouetur iuxta rationem me­<lb/><figure id="fig44"></figure><lb/>dietatis, & pars a eodem modo, ergo conduplicata <lb/>difficultate, quia medietas b impedit medietatem, a <lb/>quanto latius e&longs;t, & longius a b, tanto difficilius <lb/><arrow.to.target n="marg180"></arrow.to.target><lb/>mouetur. </s>
<s>Et hoc intelligitur de corporibus ualde <lb/>latis propter dicta &longs;uperius.</s></p><p type="margin">
<s><margin.target id="marg179"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 40.</s></p><p type="margin">
<s><margin.target id="marg180"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 62</s></p><p type="main">
<s>Propo&longs;itio &longs;exage&longs;imaquinta.</s></p><p type="main">
<s>Proportionem duorum mobilium inter &longs;e cum auxilio medij <lb/>inuenire.<lb/><arrow.to.target n="marg181"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg181"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Graue de&longs;cendit naturaliter quatuor cau&longs;is: prima e&longs;t ponderis <lb/>magnitudo, unde quod grauius e&longs;t celerius de&longs;cendit. </s>
<s>Secundò ob <lb/>paruam medij repugnantiam, ideo quanto medium e&longs;t rarius & <lb/>mobile tenuius, tanto celerius de&longs;cendit: contrà uerò tardius. </s>
<s>Ter­<lb/>tiò ob impetum aëris &longs;ub &longs;equentis: & ideo mobile quòd ex eadem </s></p><p type="main">
<s><arrow.to.target n="marg182"></arrow.to.target><lb/>materia con&longs;tat, &longs;emper de&longs;cendit parte acutiore &longs;uprapo&longs;ita, ne aër <lb/>cogatur celerius ferri: & quanto diutius de&longs;cendit, tanto magis in­<lb/>tenditur motus, at que augetur, ut &longs;uprà de claratum e&longs;t. </s>
<s>Quarta cau&longs;a <lb/>e&longs;t, quod non impediatur ab aëre tran&longs;uerfim moto, et à latere: ideo <lb/>leuia mobilia & magna non &longs;olum lentius de&longs;cendunt, quoniam <lb/><arrow.to.target n="marg183"></arrow.to.target><lb/>paruam uim habeant, & magnam repugnantiam, &longs;ed quia tran&longs;uer <lb/><arrow.to.target n="marg184"></arrow.to.target><lb/>&longs;im impul&longs;a minus mouentur motu recto, ut &longs;upra ui&longs;um e&longs;t. </s>
<s>Por­<pb pagenum="55"/>rò proportio ratione de&longs;cen&longs;us aucta, declarata e&longs;t paulo antè, <lb/>quare cum medium &longs;upponatur eiu&longs;dem generis, & figura non <lb/>eiu&longs;modi, nec leuitas, ut pror&longs;us 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"></arrow.to.target><lb/>nem ponderum, & iam habuimus proportionem incrementi ex <lb/><arrow.to.target n="marg186"></arrow.to.target><lb/>motu aëris, ergo proportio unius motus producti ad alteram no­<lb/>ta erit.</s></p><p type="margin">
<s><margin.target id="marg182"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 30.</s></p><p type="margin">
<s><margin.target id="marg183"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 59.</s></p><p type="margin">
<s><margin.target id="marg184"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 62.</s></p><p type="margin">
<s><margin.target id="marg185"></margin.target>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><margin.target id="marg186"></margin.target>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>Propo&longs;itio &longs;exage&longs;ima&longs;exta.</s></p><p type="main">
<s>Proportionem laterum eptagoni, & &longs;ubten&longs;arum con&longs;iderare, <lb/>& quæ à reflexa proportione pendent.<lb/><arrow.to.target n="marg187"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg187"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>m.</s></p><p type="main">
<s>Sit eptagonus a b d f g e c, & &longs;ubten&longs;æ b <lb/><figure id="fig45"></figure><lb/>c, & f e duobus lateribus, tribus autem d c <lb/>d e, & erunt (quia intelligitur eptagono æ­<lb/>quilatero, & æquiangulo) b c & e finuicem <lb/>æquales: & item d c, & d e æquales: & &longs;i du­<lb/>cerentur b e & c f inuicem æquales: & ad a c <lb/>& d g: quare cum angulus cb d con&longs;i&longs;tatin </s></p><p type="main">
<s><arrow.to.target n="marg188"></arrow.to.target><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; & &longs;it arcus c e g <lb/>f d duplus arcus b a c, quia c e g f d &longs;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"></arrow.to.target><lb/>gulo b c d, quare per demon&longs;trata à nobis proportio laterum b d, <lb/>b c, c d, e&longs;t reflexa, igitur proportio d b & b c, ad d c, ut d e ad b c, & <lb/><arrow.to.target n="marg190"></arrow.to.target><lb/>rur&longs;us proportio b d & d e ad b e, ut b e ad b d. </s>
<s>Quare &longs;uppo&longs;ita <lb/>d b 1, b c 1 po&longs;itione, erit d c latus 1 quad. </s>
<s>p: 1 po&longs;itione. </s>
<s>Proportio <lb/><arrow.to.target n="marg191"></arrow.to.target><lb/>uerò, ut dictum e&longs;t b d & d c ad b c, id e&longs;t p: <02> 1 quad. </s>
<s>p: 1 pos, ad 1 <lb/>pos e&longs;t, ut b c ad b d, id e&longs;t 1 pos ad 1, igitur 1 p: <02> v: 1 quad. </s>
<s>p: 1 pos <lb/>æquatur quadrato b c, quod e&longs;t 1 quad. </s>
<s>igitur 1 quad. </s>
<s>m: 1 æquatur <lb/><02> v: 1 quad. </s>
<s>p: 1 pos quare 1 quad. </s>
<s>quad. </s>
<s>m: 2, quad. </s>
<s>p: 1 æquatur 1 <lb/>quad. </s>
<s>p: 1 pos. </s>
<s>Additis igitur communiter quatuor quadratis fient <lb/>1 quad. </s>
<s>quad. </s>
<s>p: 2 quad. </s>
<s>p: 1 æqualia 5 quad. </s>
<s>p: 1 pos. </s>
<s>Et reducitur ad <lb/>1 cu. </s>
<s>æqualem 1 3/4 pos p: 7/8.</s></p><p type="margin">
<s><margin.target id="marg188"></margin.target>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><margin.target id="marg189"></margin.target>P<emph type="italics"/>er ult. </s>
<s>&longs;exti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg190"></margin.target>D<emph type="italics"/>e<emph.end type="italics"/> S<emph type="italics"/>uh. </s>
<s>lib.<emph.end type="italics"/> 16.</s></p><p type="margin">
<s><margin.target id="marg191"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 20. <emph type="italics"/>diff.<emph.end type="italics"/></s></p><p type="main">
<s>Aliter &longs;tante &longs;uppo&longs;itione ut Ludouicus Ferrarius ex demon­<lb/>&longs;tratis à Ptolemæo quadratum b c, & e&longs;t 1 quad e&longs;t æquale produ­<lb/>cto ex b d in c e, quod e&longs;t 1, & a b in d c, igitur detracto 1, produ­<lb/>cto b d in c e ex 1 quad. </s>
<s>quadrato c b, relinquitur productum ex <lb/>a b in c d 1 quad. </s>
<s>m: 1, ergo diui&longs;o co per a b, quæ e&longs;t 1, relinquitur <lb/>c d 1 quad. </s>
<s>m: 1 huius uerò quadratum per <expan abbr="ead&etilde;">eadem</expan> demon&longs;trata à Pto­<pb pagenum="56"/>lemæo, &ecedil;quale e&longs;t rectangulis ex b c in de, & b d in c e, igitur 1 quad. <lb/></s>
<s>quad. </s>
<s>m: 2 quad. </s>
<s>p: 1 e&longs;t æquale 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>quad. </s>
<s>m: 2 quad. </s>
<s>igitur diui&longs;o 1 quad. </s>
<s>quad. </s>
<s>m: 2 quad. </s>
<s>per 1 <lb/>pos, exit 1 cu. </s>
<s>m: 2 pos æqualia d e, & d e e&longs;t æqualis d c, ut ab initio <lb/>demon&longs;trauimus, & d c fuit 1 quad. </s>
<s>m: 1, igitur 1 cu. </s>
<s>m: 2 æquantur 1 <lb/>quad. </s>
<s>m: 1, igitur 1 cu. </s>
<s>p: 1 æquantur 1 quad. </s>
<s>p: 2 pos.</s></p><p type="main">
<s>Aliter ut Pacciolus, concurrant latera eptagoni b d, c e in a, & du <lb/>cantur perpendiculares a f, d g & c d, & &longs;it c e i ca 1 pos, & quia ut <lb/><arrow.to.target n="marg192"></arrow.to.target><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 e&longs;t dimidium d e, erit d h, & g f <lb/><figure id="fig46"></figure><lb/>1/2, cum ergo b f &longs;it (1/2 pos p: 1/2)/pos erit ergo di­<lb/>ui&longs;a 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 &longs;it 1, erit quadratum g d 1 m: <lb/>2/4/(2 quad)g c autem e&longs;t compo&longs;ita ex e f, quæ <lb/>e&longs;t 1/2p: 1/2/(1 pos) & f g quæ e&longs;t 1/2, erit igitur c <lb/>g 1 p: 1/2/(1 pos), & <expan abbr="quadratũ">quadratum</expan> eius 1 p: 1/pos e&longs;t 1/4/(1 quad.) quare <expan abbr="&qtilde;dratũ">quadratum</expan> e d &qring;d e&longs;t <lb/><arrow.to.target n="marg193"></arrow.to.target><lb/>compo&longs;itum ex quadratis c g & g d erit 2 p: 1/pos c a uerò e&longs;t æqua­<lb/>lis c d, quia, ut demon&longs;tratum e&longs;t angulus d c e e&longs;t &longs;eptima pars <lb/>duorum rectorum, & angulus b c e ei duplus, quare cum c f a &longs;it re­<lb/>ctus erit ex trige&longs;ima&longs;ecunda primi Elementorum f a c tres &longs;epti­<lb/>mæ unius recti, ergo d a c 6/7 unius recti, d c a uerò 2/7 unius recti, quia <lb/><arrow.to.target n="marg194"></arrow.to.target><lb/>e&longs;t &longs;eptima pars duorum rectorum, ígitur a d c e&longs;t 6/7 unius recti: igi­<lb/>tur c d e&longs;t æqualis c a, ergo quadratum quadrato: igitur 1 quad. </s>
<s>p: 2 <lb/>pos p: 1, æquatur 2 p: 1/(1 pos) igitur 1 quad. </s>
<s>p: 2 pos, æquantur 1 p: 1/(1 pos). <lb/>Quare 1 cub. </s>
<s>p: 2 quad. </s>
<s>æquatur 1 pos p: 1. <lb/><figure id="fig47"></figure><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>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>p: 1 pos, & hoc e&longs;t æquale 4 quadrato b c per re­<lb/>flexæ proportionis diffinitionem. </s>
<s>Igitur a c e&longs;t <02> 4 1/4 m: 1/2, & ita <lb/>de alijs.</s></p><p type="margin">
<s><margin.target id="marg192"></margin.target>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><margin.target id="marg193"></margin.target>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><margin.target id="marg194"></margin.target>P<emph type="italics"/>er &longs;extam eiu&longs;dem.<emph.end type="italics"/></s></p><p type="main">
<s>Propo&longs;itio &longs;exage&longs;ima&longs;eptima.</s></p><p type="main">
<s>Si fuerint aliquot quantitates ab una quantitate, aliæque totidem <pb pagenum="57"/>ab eadem analo gæ, erit proportio tertiæ unius ordinis ad tertiam <lb/>alterius, ut &longs;ecundæ ad &longs;ecundam duplicata, & quartæ ad quartam <lb/>triplicata, quintæ ad quintam quadruplicata, at que &longs;ic de alijs.<lb/><arrow.to.target n="marg195"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg195"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>_{m}.</s></p><p type="main">
<s>Sint quantitates b c d e f, ab a in continua proportio­<lb/><arrow.to.target n="table14"></arrow.to.target><lb/>ne, & aliæ totidem g h k l m, dico quod proportio h c e&longs;t <lb/>duplicata ei, quæ e&longs;t g ad b, & k ad d triplicata, & l ad e <lb/>quadruplicata, & &longs;ic deinceps, &longs;umatur enim unum, & ab </s></p><table><table.target id="table14"></table.target><row><cell></cell><cell>a</cell><cell></cell></row><row><cell>b</cell><cell></cell><cell>g</cell></row><row><cell>c</cell><cell></cell><cell>h</cell></row><row><cell>d</cell><cell></cell><cell>k</cell></row><row><cell>e</cell><cell></cell><cell>l</cell></row><row><cell>f</cell><cell></cell><cell>m</cell></row><row><cell></cell><cell>n</cell><cell></cell></row><row><cell>o</cell><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><cell>y</cell></row><row><cell>s</cell><cell></cell><cell>z</cell></row></table><p type="main">
<s><arrow.to.target n="marg196"></arrow.to.target><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"></arrow.to.target><lb/>n ad p duplicata ei, quæ t ad o, & x ad q triplicata ei, quæt <lb/>ad o, & pote&longs;t etiam demon&longs;trari generaliter ultra qua­<lb/><arrow.to.target n="marg198"></arrow.to.target><lb/>dratum, & cubum: nam &longs;i ducatur t in o, fiat que <foreign lang="greek">a</foreign> erit, pro­<lb/>portio enim ad <foreign lang="greek">a</foreign> eadem quæ t ad o, & proportio a ad p, <lb/>ut t ad o, igitur per diffinitionem proportionis duplicatæ <lb/><arrow.to.target n="marg199"></arrow.to.target><lb/>po&longs;itam in quinto libro ab Euclide u ad p duplicata ei, <lb/>quæ t ad o, & &longs;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"></arrow.to.target><lb/>q <foreign lang="greek">b g</foreign> x in continua proportione per eandem. </s>
<s>Quia ergo propor­<lb/>tio q ad <foreign lang="greek">b</foreign> e&longs;t ut o ad t, patet, quod x ad q e&longs;t triplicata ei, quæ e&longs;t t ad <lb/>o, & ita de reliquis, cum ergo proportio p ad o &longs;it, ut e ad b, & o ad <lb/><arrow.to.target n="marg201"></arrow.to.target><lb/>n, ut b ad a, & n ad t, ut a ad g, & t ad u, ut g ad h, &longs;equitur ut &longs;it t ad a, <lb/>ut g ad b, & u ad p, ut h ad c, igitur cum &longs;it ut u ad p duplicata ei, qu&ecedil; <lb/>e&longs;t t ad o erit h ad e, duplicata ei quæ e&longs;t g ad b, & ita de reliquis, & <lb/>no&ngrave; refert, &longs;eu dicas u ad p duplicatam ei, quæ e&longs;t t ad o, &longs;eu dicas p <lb/><arrow.to.target n="marg202"></arrow.to.target><lb/>ad u duplicatam ei, quæ e&longs;t o ad t. </s>
<s>Aliter & euidentius in duabus <lb/>&longs;oleo demon&longs;trare: cum enim &longs;it e & h duplicata ei quæ e&longs;t b & g <lb/>ad a, ut &longs;upra, & quadrati b ad quadratum a, & quadrati g ad qua­<lb/><arrow.to.target n="marg203"></arrow.to.target><lb/>dratum a duplicata his quæ b & g ad a erunt b & g quadratorum <lb/>ad quadratum a, uelut c & h ad a. </s>
<s>Et conuertendo qua­<lb/><arrow.to.target n="table15"></arrow.to.target><lb/>drati a ad quadratum g, ut a ad h, con&longs;tituantur ergo <lb/>hic & erit quadrati b ad <expan abbr="quadratũ">quadratum</expan> g, ita c ad h: &longs;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><margin.target id="marg196"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 8. <emph type="italics"/>non<gap/><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><margin.target id="marg197"></margin.target>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><margin.target id="marg198"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 23. <emph type="italics"/>&longs;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><margin.target id="marg199"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 17. <emph type="italics"/>&longs;e-ptimi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg200"></margin.target>D<emph type="italics"/>iff.<emph.end type="italics"/> 10.</s></p><p type="margin">
<s><margin.target id="marg201"></margin.target>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><margin.target id="marg202"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 10 <emph type="italics"/>diff. </s>
<s>quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg203"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 20. <emph type="italics"/>&longs;ex ti<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s></p><table><table.target id="table15"></table.target><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>Propo&longs;itio &longs;exage&longs;imaoctaua, collectorum ab Euclide <lb/>& Archimede.</s></p><p type="main">
<s>Omnis cylindrus cono habenti ba&longs;im, & altitudinem eandem <lb/><arrow.to.target n="marg204"></arrow.to.target><lb/>triplus e&longs;t. </s>
<s>Omnis cylindrus &longs;phæræ habenti eundem magnum <lb/><arrow.to.target n="marg205"></arrow.to.target><lb/>circulum, & altitudinem &longs;exquialter e&longs;t. </s>
<s>Omnis &longs;phæra dupla e&longs;t <lb/><arrow.to.target n="marg206"></arrow.to.target><lb/>cono, cuius ba&longs;is e&longs;t eius circulus magnus, & altitudo eadem, quæ <lb/>&longs;phæræ ip&longs;ius. </s>
<s>Omnis &longs;uperficies &longs;phæræ quadrupla e&longs;t maiori <lb/><arrow.to.target n="marg207"></arrow.to.target><lb/>&longs;uo circulo. </s>
<s>Superficies portionis &longs;phæræ e&longs;t æqualis circulo, cu <lb/><arrow.to.target n="marg208"></arrow.to.target><pb pagenum="58"/>ius &longs;emidiameter e&longs;t linea ducta à uertice portionis ad finem illius.</s></p><p type="margin">
<s><margin.target id="marg204"></margin.target>1</s></p><p type="margin">
<s><margin.target id="marg205"></margin.target>2</s></p><p type="margin">
<s><margin.target id="marg206"></margin.target>3</s></p><p type="margin">
<s><margin.target id="marg207"></margin.target>4</s></p><p type="margin">
<s><margin.target id="marg208"></margin.target>5</s></p><p type="main">
<s>Quilibet &longs;ector &longs;phæræ æqualis e&longs;t cono, cuius ba&longs;is e&longs;t circu­<lb/>lus æqualis &longs;uperficiei eiu&longs;dem portionis, altitudo uerò &longs;phæræ &longs;e­<lb/>midiameter. </s>
<s>Proportio &longs;phæræ ad &longs;ectorem datum, e&longs;t duplica­<lb/>ta ei, qu&ecedil; e&longs;t dimetientis ad lineam, quæ à uertice portionis ad lim­<lb/>bum. </s>
<s>Cum enim &longs;phæra &longs;it æqualis cono, cuius ba&longs;is e&longs;t maior cir­<lb/>culus, altitudo uerò dupla dimetienti per tertiam harum, quæ hic <lb/><arrow.to.target n="marg209"></arrow.to.target><lb/>proponuntur: erit &longs;phæra æqualis cono ba&longs;im habenti circulum, <lb/>cuius &longs;emidiameter &longs;it æqualis diametro &longs;phæræ, altitudo uerò &longs;e­<lb/>midiameter &longs;phæræ. </s>
<s>At per &longs;extam harum &longs;ector &longs;phæræ e&longs;t æqua­<lb/>lis cono habenti altitudinem &longs;cmidiametrum &longs;phær&ecedil;, ba&longs;im autem <lb/><arrow.to.target n="marg210"></arrow.to.target><lb/>ip&longs;am portionis &longs;uperficiem: igitur proportio &longs;phæræ ad &longs;ecto­<lb/>rem, uelut circuli cuius diameter e&longs;t dupla dimetienti &longs;phæræ ad <lb/>círculum æqualem &longs;uperficiei portionis: at &longs;uperficies portionis <lb/>per quintam harum e&longs;t æqualis circulo, cuius &longs;emidiameter e&longs;t li­<lb/>nea à uertice portionis ad limbum eiu&longs;dem: ergo proportio &longs;phæ­<lb/>ræ ad &longs;uum &longs;ectorem e&longs;t uelut circuli, cuius dimetiens e&longs;t duplus di <lb/>metienti &longs;phæræ, aut &longs;emidimetiens e&longs;t æqualis dimetienti &longs;phæræ <lb/>ad circulum, cuius &longs;emidimetiens e&longs;t linea à uertice portionis ad <lb/>limbum. </s>
<s>Sed proportio talium circulorum e&longs;t duplicata propor­<lb/><arrow.to.target n="marg211"></arrow.to.target><lb/>tioni &longs;emidimetientium, igitur proportio &longs;phæræ ad &longs;uum &longs;ecto­<lb/>rem e&longs;t ueluti dimetientis &longs;phæræ ad lineam, quæ á uertice portio­<lb/><arrow.to.target n="marg212"></arrow.to.target><lb/>nis ad limbum duplicata. </s>
<s>Cuicunque portioni &longs;phæræ conus ille <lb/>habetur æqualis, qui ba&longs;im hab eat eandem cum portione, altitudi­<lb/>nem uerò lineam rectam, quæ ad altitudinem portionis eandem <lb/>habeat proportionem, quam &longs;emidiametros &longs;phæræ unà cum alti­<lb/>tudine reliquæ portionis habet ad eandem reliquæ portionis alti­<lb/><arrow.to.target n="marg213"></arrow.to.target><lb/>tudinem. </s>
<s>Earum &longs;phæræ portionum, quæ æqualibus &longs;uperfi­<lb/><arrow.to.target n="marg214"></arrow.to.target><lb/>ciebus continentur medietas &longs;phæræ maxima exi&longs;tit. </s>
<s>Proportio <lb/>&longs;uperficiei &longs;phæræ plano diui&longs;æ ad reliquæ portionis &longs;uperficiem, <lb/>& re&longs;idui &longs;ectoris ad &longs;ectorem, e&longs;t uelut quadratorum duarum li­<lb/>nearum quæ à uerticulis &longs;ectionum ad communem &longs;uperficiem <lb/>plani portiones &longs;ecantis de&longs;cendunt: nam &longs;ectorem &longs;phæræ, dico <lb/><arrow.to.target n="marg215"></arrow.to.target><lb/>corpus compo&longs;itum ex portione, & cono illo. </s>
<s>Ille idem etiam defi­<lb/>nit Ellip&longs;im coni a cuti anguli &longs;ectionem, quam dicit etiam fieri &longs;e­<lb/><arrow.to.target n="marg216"></arrow.to.target><lb/>cto cylindro per planum non ad angulos rectos &longs;tante &longs;uper cylin­<lb/>dri axem. </s>
<s>Ab hac igitur coni acuti anguli &longs;ectione &longs;eu ellip&longs;i cir­<lb/><arrow.to.target n="marg217"></arrow.to.target><lb/>cumacta figura &longs;phæroides corpus quod ba&longs;im rotundam habet, <lb/>uocat: id que duplex ob longum, quod fit diametro longiore quie­<lb/>&longs;cente, & prolatum quod fit quie&longs;cente breuiore: &longs;icut reliquam &longs;ci <lb/>licet parabolen aut hyperbolen, quia inferius non e&longs;t terminata, <pb pagenum="59"/>in cono rectangulo uocat rectanguli coni &longs;ectionem: ex qua cir­<lb/>cumacta fit conoidale, quia planam habet ba&longs;im. </s>
<s>Si ergo in ea­<lb/><arrow.to.target n="marg218"></arrow.to.target><lb/>dem rectanguli coni &longs;ectione à plano portiones æquales habentes <lb/>diametros ab&longs;cindantur, illæ portiones erunt æquales. </s>
<s>Et triangu­<lb/>li in ei&longs;dem portionibus in&longs;cripti æquales erunt. </s>
<s>Diametrum uo­<lb/>cat in <expan abbr="quacunqũe">quacunqune</expan> portione lineam, quæ omnes lineas ba&longs;i æquidi­<lb/>&longs;tantes per æqualia diuidit. </s>
<s>Omnis circuli cuius diameter e&longs;t ma <lb/><arrow.to.target n="marg219"></arrow.to.target><lb/>ior diameter ellip&longs;is proportio ad ellip&longs;im e&longs;t uelut directè diame­<lb/>tri ellip&longs;is ad diametrum tran&longs;uer&longs;am. </s>
<s>Ex quo patet quod pro­<lb/><arrow.to.target n="marg220"></arrow.to.target><lb/>portio cuiuslibet circuli ad ellip&longs;im e&longs;t uelut quadrati &longs;uæ diame­<lb/>tri ad rectangulum recta, & tran&longs;uer&longs;a diametro ellip&longs;is compre­<lb/>hen&longs;um. </s>
<s>Ex hoc rur&longs;us &longs;equitur quod ellip&longs;is ad ellip&longs;im, ut re­<lb/><arrow.to.target n="marg221"></arrow.to.target><lb/>ctanguli ex diametris unius ad rectangulum ex diametris alterius.</s></p><p type="margin">
<s><margin.target id="marg209"></margin.target>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><margin.target id="marg210"></margin.target>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><margin.target id="marg211"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 2. <emph type="italics"/>duode cimi<emph.end type="italics"/>, & 20. <emph type="italics"/>&longs;exti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg212"></margin.target>8</s></p><p type="margin">
<s><margin.target id="marg213"></margin.target>9</s></p><p type="margin">
<s><margin.target id="marg214"></margin.target>10</s></p><p type="margin">
<s><margin.target id="marg215"></margin.target>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><margin.target id="marg216"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 20. <emph type="italics"/>&longs;ex ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg217"></margin.target>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><margin.target id="marg218"></margin.target>11</s></p><p type="margin">
<s><margin.target id="marg219"></margin.target>12</s></p><p type="margin">
<s><margin.target id="marg220"></margin.target>13</s></p><p type="margin">
<s><margin.target id="marg221"></margin.target>14</s></p><p type="main">
<s>Si conoides & &longs;phæroides &longs;ecet plano æquidi&longs;tanti axi fiet &longs;e­<lb/><arrow.to.target n="marg222"></arrow.to.target><lb/>ctio conoidalis &longs;imilis ei à qua conoides &longs;eu &longs;phæroides de&longs;cri­<lb/>ptum e&longs;t. </s>
<s>Sin autem &longs;upra axem plano ad perpendiculum erecto <lb/>&longs;ectio circulus erit. </s>
<s>Et &longs;i &longs;ecentur obliquè fiet ellip&longs;is, modo omnia <lb/>latera comprehendat. </s>
<s>Omnis portio conoidalis rectanguli, quam <lb/><arrow.to.target n="marg223"></arrow.to.target><lb/>planum &longs;ecat, &longs;exquialtera e&longs;t, cono qui ba&longs;im & axem eandem ha­<lb/>bet. </s>
<s>Ex quo patet, quod &longs;i portio conoidalis rectanguli & &longs;phæ­<lb/><arrow.to.target n="marg224"></arrow.to.target><lb/>ræ medietas eandem ba&longs;im habeant & axem eundem, medietas <lb/>&longs;phæræ &longs;exquitertia erit conoidali portioni. </s>
<s>Et &longs;i eiu&longs;dem rectan <lb/><arrow.to.target n="marg225"></arrow.to.target><lb/>guli conoidalis portiones ab&longs;cin dantur erit portionum propor­<lb/>tio uelut quadratorum axium. </s>
<s>Cuiuslibet &longs;phæroidis pars pla­<lb/><arrow.to.target n="marg226"></arrow.to.target><lb/>no per centrum ab&longs;ci&longs;&longs;a dupla e&longs;t cono ba&longs;im & axem eadem ha­<lb/>benti. </s>
<s>Si autem non &longs;uper centrum erit proportio earum ad co­<lb/><arrow.to.target n="marg227"></arrow.to.target><lb/>num ba&longs;im, & axem eandem habentem uelut coniunctæ ex axe al­<lb/>terius partis & dimidio axis &longs;phæroidis ad axem alterius partis.</s></p><p type="margin">
<s><margin.target id="marg222"></margin.target>15</s></p><p type="margin">
<s><margin.target id="marg223"></margin.target>16</s></p><p type="margin">
<s><margin.target id="marg224"></margin.target>17</s></p><p type="margin">
<s><margin.target id="marg225"></margin.target>18</s></p><p type="margin">
<s><margin.target id="marg226"></margin.target>19</s></p><p type="margin">
<s><margin.target id="marg227"></margin.target>20</s></p><p type="main">
<s>Demum proportio partis conoidis obtu&longs;i anguli plano ab&longs;ci&longs;­<lb/><arrow.to.target n="marg228"></arrow.to.target><lb/>&longs;æ ad conum, ba&longs;im & axem eadem habentem e&longs;t ueluti lineæ, com <lb/>po&longs;itæ ex axe portionis & triplo adiectæ ad compo&longs;itum ex axe <lb/>portionis & duplo eiu&longs;dem adiectæ. </s>
<s>Adiectam uocat hyperbolis <lb/>tran&longs;uer&longs;am. </s>
<s>Omnis cylindrus cono triplus e&longs;t habenti eandem <lb/><arrow.to.target n="marg229"></arrow.to.target><lb/>ba&longs;im & altitudinem. </s>
<s>Omnes cylindri coni &longs;phæræ &longs;unt in pro­<lb/><arrow.to.target n="marg230"></arrow.to.target><lb/>portione corporum &longs;imilium planis &longs;uperficiebus contentarum.</s></p><p type="margin">
<s><margin.target id="marg228"></margin.target>21</s></p><p type="margin">
<s><margin.target id="marg229"></margin.target>22</s></p><p type="margin">
<s><margin.target id="marg230"></margin.target>23</s></p><p type="main">
<s>Propo&longs;itio &longs;exage&longs;imanona, collectorum ex quatuor libris <lb/>Apollonij Pergei & <expan abbr="q.">que</expan> Sereni.</s></p><p type="main">
<s>Si fuerit linea bifariam diui&longs;a, eique in longum alia addita, & rur­<lb/><arrow.to.target n="marg231"></arrow.to.target><lb/>&longs;us alia detracta, fueritque totius cum addita ad eam, quæ addita e&longs;t <lb/>ueluti re&longs;idui ad detractam erit lineæ com­<lb/><figure id="fig48"></figure><lb/>po&longs;itæ ex addita, & dimidia ad dimidiam <pb pagenum="60"/>ip&longs;am uelut dimidiæ ad differentiam eius, & detractæ. </s>
<s>Rur&longs;usque li­<lb/>neæ compo&longs;itæ ex dimidio & re&longs;iduo dimidiæ ac detractæ ad li­<lb/>neam compo&longs;itam ex addita & detracta ut re&longs;idui dimidiæ, & de­<lb/>tractæ ad partem detractam. </s>
<s>Et rur&longs;us totius compo&longs;itæ ad com­<lb/>po&longs;itam ex dimidia & addita, uelut compo&longs;itæ ex addita, & diffe­<lb/>rentia ad ip&longs;am additam. </s>
<s>Velut &longs;it propo&longs;ita a b per æqualia diui&longs;a <lb/>in c, addita b d, & detracta b e, &longs;it proportio a d ad d b, ut a e ad e b, <lb/>dico e&longs;&longs;e, ut c d ad cb, ita ab ad c e. </s>
<s>Et ut a e ad e d ut c e ad e b. </s>
<s>Etite­<lb/><arrow.to.target n="marg232"></arrow.to.target><lb/>rum ut a d ad c d uelut e d ad d b. </s>
<s>In parabole proportio partium <lb/>diametri ad uerticem terminantium duplicata e&longs;t proportioni li­<lb/>nearum ab ei&longs;dem punctis ordinatim ductarum ad ip&longs;am &longs;ectio­<lb/><arrow.to.target n="marg233"></arrow.to.target><lb/>nem. </s>
<s>In hyperbole autem & ellip&longs;i & circuli circumferentia erit <lb/>quadratorum linearum ordinatim ductarum inter &longs;e uelut rectan­<lb/><arrow.to.target n="marg234"></arrow.to.target><lb/>gulorum partium diametri ad eadem puncta terminantium. </s>
<s>Et in <lb/>ei&longs;dem &longs;i à puncto peripheriæ contingens ad diametrum ducatur, <lb/>& ab eodem ordinata, erit ut partis diametri intercept&ecedil; inter extre­<lb/>mum, & ordinatam ad partem inter ordinatam & peripheriam, ue­<lb/>lut interceptæ inter extremum & contingentem ad interceptam <lb/><arrow.to.target n="marg235"></arrow.to.target><lb/>exterius inter finem contingentis & peripheriam. </s>
<s>Et in ei&longs;dem <lb/>quadratum &longs;emidiametri æquale e&longs;&longs;e rectangulo ex intercepta in­<lb/>ter centrum & ca&longs;um contingentis in inter ceptam inter centrum & <lb/><arrow.to.target n="marg236"></arrow.to.target><lb/>ca&longs;um ordinatæ à loco contactus productæ. </s>
<s>Si parabolen recta <lb/>linea contingens ad diametrum perueniat, &longs;umptoque puncto alio <lb/>in &longs;ectione æquidi&longs;tans ab eo ducatur contingenti: & ab utroque <lb/>etiam ad diametrum ordinatæ, demum à uertice æquidi&longs;tans illis, <lb/>& à priore puncto diametro æquidi&longs;tans donec concurrant, erit <lb/>triangulus ex ordinata, & æquidi&longs;tante à &longs;ecundo puncto, & dia­<lb/>metri parte contentus rectangulo ex prima ordinata & parte dia­<lb/>metri inter uerticem & &longs;ecundam ordinatam contento æqualis.<lb/><arrow.to.target n="marg237"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg231"></margin.target>1</s></p><p type="margin">
<s><margin.target id="marg232"></margin.target>2</s></p><p type="margin">
<s><margin.target id="marg233"></margin.target>3</s></p><p type="margin">
<s><margin.target id="marg234"></margin.target>4</s></p><p type="margin">
<s><margin.target id="marg235"></margin.target>5</s></p><p type="margin">
<s><margin.target id="marg236"></margin.target>6</s></p><p type="margin">
<s><margin.target id="marg237"></margin.target>7</s></p><p type="main">
<s>Si in parabole contingente ad diametrum ducta ex alio puncto <lb/>ei æquidi&longs;tans ducatur ex ip&longs;a &longs;ectione, ubi iterum &longs;ecat &longs;ectione<gap/><lb/>intercepta per æqualia diuidetur linea à puncto contingentis dia­</s></p><p type="main">
<s><arrow.to.target n="marg238"></arrow.to.target><lb/>metro æquidi&longs;tanti ducta. </s>
<s>Idem uerò fermè continget ducta li­<lb/>nea à centro in locum contactus, &longs;ecabit enim omnes contingenti <lb/><arrow.to.target n="marg239"></arrow.to.target><lb/>æquidi&longs;tantes in hyperbole, ellip&longs;i at que circulo. </s>
<s>E&longs;t autem omne <lb/>centrum in medio diametri: diameter autem in circulo & ellip&longs;i il­<lb/>las per æqualia diuidit intus enim e&longs;t: in contrapo&longs;itis inter uerti­<lb/>cem, & uerticem po&longs;ita e&longs;t exterius utriu&longs;que contingenti ad per­<lb/>pendiculum in&longs;i&longs;tens. </s>
<s>In hyperbole autem exterius etiam adiacet, <lb/>ut in contrapo&longs;itis eadem & tran&longs;uer&longs;a uo catur: cuius terminus e&longs;t <lb/>punctus concur&longs;us cum latere trianguli, qui conum per axem diui­<pb pagenum="61"/>dit: linea uerò tangens uerticem hyperbolis ad quam ordinatæ <lb/><arrow.to.target n="marg240"></arrow.to.target><lb/>po&longs;&longs;unt, Recta appellabitur. </s>
<s>Datarecta linea po&longs;itione, aliaque ma <lb/>gnitudine data & angülo parabolen, & hyperbolen, & ellip&longs;im, <lb/>& contrapo&longs;itas circa datam po&longs;itione tanquàm diametrum de­<lb/>&longs;cribere tanquàm cono erecto, ut angulus ad uerticem &longs;ectionis <lb/>comprehen&longs;us &longs;it, & per rectam rectangulum æquale comprehen­<lb/>datur quadrato datæ lineæ magnitudine. </s>
<s>Si linea in duas partes <lb/><arrow.to.target n="marg241"></arrow.to.target><lb/>diuidatur, eique utrinque æquales lineæ adiun­<lb/><figure id="fig49"></figure><lb/>gantur erit rectangulum ex partibus totius æ­<lb/>quale rectangulis partium prioris lineæ, & ex <lb/>priore linea cum una adiecta in eam, quæ adiecta e&longs;t. </s>
<s>Si hyperbo <lb/><arrow.to.target n="marg242"></arrow.to.target><lb/>len recta linea in uertice contingat, & utrinque ab&longs;cindatur, quan­<lb/>tum e&longs;t, quod pote&longs;t in quartam partem rectanguli ex diametro <lb/>tran&longs;uer&longs;a hyperbolis, quæ exterius adiacetin eam, quæ recta dici­<lb/>tur, ad quam, quæ ordinatim ducuntur, &longs;unt æquidi&longs;tantes lineæ, <lb/>quæ à &longs;ectionis centro ad terminos contingentis ducuntur &longs;emper <lb/>ip&longs;i &longs;ectioni magis appropinquabunt, nec unquam conuenient: & <lb/>ob id a&longs;ymptoton appellantur. </s>
<s>Nec ullæ aliæ intra <expan abbr="angulũ">angulum</expan> illum <lb/><arrow.to.target n="marg243"></arrow.to.target><lb/>inueniri poterunt. </s>
<s>Vnde etiam intra <expan abbr="datũ">datum</expan> angulum de&longs;cribere do­<lb/>cemur hyperbolen cuius anguli latera &longs;int a&longs;ymptota. </s>
<s>A&longs;ymptotis <lb/><arrow.to.target n="marg244"></arrow.to.target><lb/>duabus propo&longs;itis uni hyperboli, in finitas alías eidem a&longs;ymptotas <lb/>inuenire. </s>
<s>Duabus rectis a&longs;ymptotis infinitas &longs;ubijci po&longs;&longs;e hyperbo <lb/>les illis rectis, & inter &longs;e a&longs;ymptotas. </s>
<s>Cum in duabus &longs;uperficie­<lb/><arrow.to.target n="marg245"></arrow.to.target><lb/>bus æquidi&longs;tantibus duo circuli æquales, quorum linea per cen­<lb/>tra non e&longs;t ad perpendiculum earum infinitis planis &longs;ecantur, fiunt <lb/>in ip&longs;is lineæ à peripheria in peripheriam rectæ quæ corpus cylin­<lb/>dricum claudunt quod &longs;calenus cylindrus appellatur: longè alius <lb/>ab eo, qui fit recto cylindro per duo plana æquidi&longs;tantia, &longs;ed non <lb/>ad perpendiculum po&longs;ita di&longs;&longs;ecto. </s>
<s>nam eius extremæ &longs;uperficies <lb/>non circuli, &longs;ed ellip&longs;es &longs;unt. </s>
<s>Si &longs;calenus cylindrus plano non æ­<lb/><arrow.to.target n="marg246"></arrow.to.target><lb/>quidi&longs;tanti ba&longs;i, &longs;ed ita ut angulos interiores æquales faciat angu­<lb/>lis ba&longs;is &longs;ectio circulus erit: uo caturque hæc&longs;ectio &longs;ubcontraria: nec <lb/>ulla præter hanc & ba&longs;i æquidi&longs;tantem &longs;ectio circulus e&longs;&longs;e pote&longs;t: <lb/>&longs;ed &longs;unt ellip&longs;es. </s>
<s>Super eundem circulum, & &longs;ub eadem altitudi­<lb/><arrow.to.target n="marg247"></arrow.to.target><lb/>ne ellip&longs;es &longs;imiles in cono & cylindro e&longs;&longs;e po&longs;&longs;unt, quæ ab eodem <lb/>plano fiant, docetque uel ba&longs;i uel cono uel cylindro, aut cono pro­<lb/>po&longs;ito reliqua facere, quod e&longs;t ualde admirabile: cum ellip&longs;is cylin­<lb/>drica &longs;emper æqualis &longs;it in utraque parte à diametro tran&longs;uer&longs;a <lb/>utrinque æqualiter di&longs;tante, conica uerò minor nece&longs;&longs;ariò &longs;it in &longs;u­<lb/>periore parte uer&longs;us coni uerticem latior in inferiore, ubi partes a <lb/>diametro tran&longs;uer&longs;a æqualiter di&longs;teterint: ip&longs;&ecedil; autem non &longs;olum &longs;i­<pb pagenum="62"/><arrow.to.target n="marg248"></arrow.to.target><lb/>miles, &longs;ed unam per&longs;æpe in utri&longs; que e&longs;&longs;e uult. </s>
<s>Sed & hoc Archime­<lb/>des dicere uidetur: lineæ ductæ à uertice coni&longs;caleni ad perpendi­<lb/>culum &longs;uper ba&longs;es &longs;ingulas omnium triangulorum per axe<gap/> coni <lb/>tran&longs;euntium in peripheriam unius circuli cadunt.</s></p><p type="margin">
<s><margin.target id="marg238"></margin.target>8</s></p><p type="margin">
<s><margin.target id="marg239"></margin.target>9</s></p><p type="margin">
<s><margin.target id="marg240"></margin.target>10</s></p><p type="margin">
<s><margin.target id="marg241"></margin.target>11</s></p><p type="margin">
<s><margin.target id="marg242"></margin.target>12</s></p><p type="margin">
<s><margin.target id="marg243"></margin.target>13</s></p><p type="margin">
<s><margin.target id="marg244"></margin.target>14</s></p><p type="margin">
<s><margin.target id="marg245"></margin.target>15</s></p><p type="margin">
<s><margin.target id="marg246"></margin.target>16</s></p><p type="margin">
<s><margin.target id="marg247"></margin.target>17</s></p><p type="margin">
<s><margin.target id="marg248"></margin.target>18</s></p><p type="main">
<s>Propo&longs;itio &longs;eptuage&longs;ima.</s></p><p type="main">
<s>Si fuerint tres quantitates in continua proportione, aliæque toti­<lb/>dem in continua proportione, poterunt con&longs;tituere tres quantita­<lb/>tes in æquali differentia peruer&longs;im copulatæ.<lb/><arrow.to.target n="marg249"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg249"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>m.</s></p><p type="main">
<s>Velut &longs;int a b c primi ordi­<lb/><figure id="fig50"></figure><lb/>nis, & d ef &longs;ecundi, & &longs;it 28, </s></p><p type="main">
<s><arrow.to.target n="marg250"></arrow.to.target><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/>æqualiter di&longs;tant, nam diffe­<lb/>rentia e&longs;t 3 1/4. At &longs;i iungatur <lb/>cum e, & b cum f, & c cum d <lb/>idem poterit contingere: ut in <lb/>figura uides, nam a e e&longs;t 8 1/2, <lb/>p: <02> 1 1/<gap/>4, & b f 7, & c d 5 1/2, m: <02> 1 1/4, & differentia b f ab utro que com­<lb/>po&longs;ito, e&longs;t 1 1/2 p: <02> 1 1/4, qua excedit & exceditur. </s>
<s>Dico modo, qua&longs;i <lb/>ex ordine coniungantur quale&longs;cun que proportiones fuerint, modo <lb/>non &longs;int ambæ æqualitatis 1, ut b iungatur cum c, & reliquæ 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"></arrow.to.target><lb/>æquales exce&longs;&longs;us, nam de primo e&longs;t clarum: nam &longs;i a cum diun­<lb/>gatur, & ambæ fuerint maximæ, maior e&longs;t differentia a ad b, quàm <lb/>b ad c, & maior etiam d ad e quàm e ad f, ideo maior erit differentia <lb/>a & d ad b e quàm b e ad c f, quod erat probandum. </s>
<s>Eodem modo <lb/>&longs;ed laborio&longs;ius demon&longs;tratur reliquus modus &longs;cilicet, quod con­<lb/>iunctio a f ad b e e&longs;t maior aut minor quàm b e ad c d, ex hoc&longs;e­<lb/>quuntur corrolaria.</s></p><p type="margin">
<s><margin.target id="marg250"></margin.target>16</s></p><p type="margin">
<s><margin.target id="marg251"></margin.target>17</s></p><p type="main">
<s>Primum, tres æquales quantitates non po&longs;&longs;unt diuidi in tres, & <lb/>tres quantitates in continua proportione ordinatè, ut dixi, ni&longs;i u­<lb/>triu&longs;que ordinis tres, ac tres inuicem &longs;int æquales.</s></p><p type="main">
<s>Secundum, tres quantitates in æquali exce&longs;&longs;u ordinate, ut dixi, <lb/>non po&longs;&longs;unt diuidi in tres, & tres quantitates, quæ &longs;int in eadem <lb/>proportione quantumcun que proportiones illæ duorum ordinum <lb/>fint diuer &longs;æ.</s></p><p type="main">
<s>Tertium, tres quantitates, quæ &longs;intin eadem proportione non <lb/>po&longs;&longs;unt diuidi ordinate in tres ac tres, quæ &longs;int in continua propor<lb/>tione ni&longs;i &longs;int ambæ proportiones eædem cum proportione ip&longs;a­<lb/>rum quantitatum.</s></p><pb pagenum="63"/><p type="main">
<s>Propo&longs;itio &longs;eptuage&longs;imaprima.</s></p><p type="main">
<s>Proportionem leuitatis ponderis per uirgam torcularem attra­<lb/>cti ad rectam &longs;u&longs;penfionem inuenire.</s></p><figure></figure><p type="main">
<s>Sit torcularis uirga, cuius &longs;piræ a b per circui­<lb/><arrow.to.target n="marg252"></arrow.to.target><lb/>tum &longs;int centuplæ ad altitudinem a b, & axis d c <lb/><arrow.to.target n="marg253"></arrow.to.target><lb/>&longs;emidiametro b c centupla, & quoniam per &longs;upe­<lb/>rius a&longs;&longs;umpta, qualis e&longs;t proportio &longs;patij ad &longs;pa­<lb/>tium, talis leuitatis ad <expan abbr="leuitat&etilde;">leuitatem</expan>, <expan abbr="igi&ttilde;">igitur</expan> e pondus a&longs;cen <lb/>dens per a b leuius quam per b <expan abbr="crectã">crectam</expan> centuplo, et <lb/>&longs;imiliter cum circuitus b c, & d c &longs;int in eodem tem <lb/>pore, & circuitus d c, &longs;it centuplus ad &longs;piralem b c <lb/>per demon&longs;trata ab Euclide, ergo e erit centuplo <lb/>leuius circum ductum per d quàm b, &longs;ed per b circumductum cen­<lb/>tuplo leuius e&longs;t, quàm per rectam, igitur e ponderat folum particu­<lb/>lam ex decem millibus recti ponderis.</s></p><p type="margin">
<s><margin.target id="marg252"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>m.</s></p><p type="margin">
<s><margin.target id="marg253"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 45.</s></p><p type="main">
<s>Propo&longs;itio &longs;eptuage&longs;ima&longs;ecunda.</s></p><p type="main">
<s>Proportionem ponde<gap/>is &longs;ph&ecedil;ræ pendentis ad a&longs;cendentem per <lb/>accliue planum inueni<gap/></s></p><figure></figure><p type="main">
<s>Sit &longs;phæra æqualis ponderig in pun­<lb/><arrow.to.target n="marg254"></arrow.to.target><lb/>cto b, quæ debeat trahi &longs;uper b c accli­<lb/>ue planum b e ad perpendiculum pla­<lb/><arrow.to.target n="marg255"></arrow.to.target><lb/>ni b f. </s>
<s>Quia ergo in b e mouetur a, qua­<lb/>uis modica ui per dicta &longs;uperius, erit per <lb/>communem animi &longs;ententiam uis, quæ <lb/>mouebit a per e b nulla: per dicta uerò <lb/>a mouebitur ad f &longs;emper, a con&longs;tanti ui <lb/>æquali g, & per b c a con&longs;tanti ui æqua­<lb/>li k, &longs;icut per b d a con&longs;tanti æquali h, ergo per ultimam petitio­<lb/>nem, cum termini &longs;eruent, quo ad partes eandem rationem &longs;in­<lb/>guli per &longs;e, & motus per b e &longs;it a nulla ui, erit proportio g ad k, ue­<lb/>lut proportio uis, quæ mouet per b f ad uim, quæ mouet per <lb/>b c, & uelut anguli per e b f recti ad angulum e b c, & ita uis, <lb/>quæ mouet a per b f, & e&longs;t, ut dictum e&longs;t, g ad uim, quæ mouet <lb/>per b d, & e&longs;t h ex &longs;uppo&longs;ito, ut c b f ad e b d, igitur proportio dif­<lb/>ficultatis motus a per b d ad idem a per b c, e&longs;t uelut h ad k, quod <lb/>erat demon&longs;trandum.</s></p><pb pagenum="64"/><p type="margin">
<s><margin.target id="marg254"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>m.</s></p><p type="margin">
<s><margin.target id="marg255"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 40. 7</s></p><p type="main">
<s>Propo&longs;itio &longs;eptuage&longs;imatertia.</s></p><p type="main">
<s>Proportionem ponderum attractorum penes figuram in pla­<lb/>no inuenire.<lb/><arrow.to.target n="marg256"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg256"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Sint duo pondera æqualia in plano a & b, & &longs;it <lb/><figure id="fig51"></figure><lb/>a &longs;uperficies qua planum tangit dupla b &longs;uperfi­<lb/>ciei, qua planum tangit: dico quod &longs;i trahantur ab <lb/>imo, quod erunt æqualia: &longs;u&longs;pendantur, & erunt <lb/>æqualia ex &longs;uppo&longs;ito, &longs;ed a quie&longs;cens in plano e&longs;t <lb/>dimidium a &longs;u&longs;pen&longs;i, & b quie&longs;cens in plano e&longs;t di <lb/>midium b &longs;u&longs;pen&longs;i ex demon&longs;tratis &longs;uperius, igi­<lb/>tur per communem animi &longs;ententiam a & b in pla­<lb/>no &longs;unt æqualia.</s></p><p type="main">
<s><arrow.to.target n="marg257"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg257"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Ex hoc manife&longs;tum e&longs;t, quod proportio uirium trahentium pon <lb/>dera in plano eadem e&longs;t, quæ ip&longs;orum ponderum dum &longs;u&longs;pendun­<lb/>tur. </s>
<s>Vbiplanum æquale &longs;it, & &longs;olidum.</s></p><p type="main">
<s><arrow.to.target n="marg258"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg258"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 62.</s></p><p type="main">
<s>Propo&longs;itio &longs;eptuage&longs;imaquarta.</s></p><p type="main">
<s>Proportionem concutientis ad concu&longs;&longs;um &longs;tabili inuenire.</s></p><p type="main">
<s><arrow.to.target n="marg259"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg259"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Intelligo concutiens e&longs;&longs;e &longs;olidum, quod non frangitur, idque gra­<lb/>uitate, & impetu concutere, nam de duritie &longs;upponitur, & grauitas, <lb/>ut demon&longs;trabitur in corrolario e&longs;t iuxta &longs;uperficiem inferiorem <lb/>ponderi comparatam. </s>
<s>Cum ergo motus concu&longs;sionis magnitudo <lb/>con&longs;tet ex grauitate, impetu & figura, concu&longs;si autem ex pondere <lb/>& connexione: multiplicatis inuicem partibus productorum pro­<lb/>portio, erit proportio concu&longs;sionis: ut &longs;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 concu&longs;sio erit dupla.</s></p><p type="main">
<s><arrow.to.target n="marg260"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg260"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 1.</s></p><p type="main">
<s>Cum fuerit figura rotunda, concu&longs;sio erit integra in puncto: <lb/>quia &longs;phæra iacens in plano totum pondus in punctum cogit.</s></p><p type="main">
<s><arrow.to.target n="marg261"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg261"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 2.</s></p><p type="main">
<s>Si autem planum e&longs;t, quod ijcitur, proportio totius ad totum e&longs;t <lb/>minor, quàm partis ad partem pro ratione quantitatis latitudinis. </s></p><p type="main">
<s><arrow.to.target n="marg262"></arrow.to.target><lb/>&longs;ed maior ratione aëris comprehen&longs;i, de quo infrà.<lb/><arrow.to.target n="marg263"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg262"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 84.</s></p><p type="margin">
<s><margin.target id="marg263"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 3.</s></p><p type="main">
<s>Cum proportio minor fuerit &longs;tabile, non poterit in &longs;olido plano <lb/>moueri: aliter fieret motus à debiliore, & per præcedentem etiam <lb/>po&longs;&longs;et pari ratione eleuari.</s></p><p type="main">
<s><arrow.to.target n="marg264"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg264"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 4.</s></p><p type="main">
<s>Cumque &longs;tabile non mouetur, & omne agens agat aliquid nece&longs;&longs;e <lb/>e&longs;t, ut &longs;tabilis partes cedant, aut di&longs;&longs;oluantur. </s>
<s>Quanto ergo magis <lb/>cedit, tanto minus di&longs;&longs;oluitur.</s></p><pb pagenum="65"/><p type="main">
<s>Cau&longs;æ igitur quæ alleuiant ictum, ne di&longs;&longs;oluatur, &longs;unt &longs;eptem le­</s></p><p type="main">
<s><arrow.to.target n="marg265"></arrow.to.target><lb/>uitas ictus, ponderis, fractura, mollities eius, quodicitur, mollities <lb/>eius, quod excipit ictum, motus eiu&longs;dem, & figura lata, & inæqua­<lb/>lis. </s>
<s>Durities ergo, quatenus fracturæ opponitur, aliud e&longs;t, quam ut <lb/>molliciei: & utra que e&longs;t cau&longs;a, quæ augetictum, ut reliquæ <lb/> oppo&longs;itæ minuunt, dicemus autem de his inferius.</s></p><p type="margin">
<s><margin.target id="marg265"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 9.</s></p><p type="main">
<s>Propo&longs;itio &longs;eptuage&longs;imaquints.</s></p><p type="main">
<s>Proportionem immoti in aqua ad immotum in terra in excipien <lb/>do ictum inuenire.</s></p><p type="main">
<s>Sit pondus a in terra æquale b eiu&longs;dem naturæ magnitudinis fi­<lb/><arrow.to.target n="marg266"></arrow.to.target><lb/>guræ, & eodem in &longs;itu, quod &longs;it in aqua porrò a, &longs;i e&longs;&longs;et affixum ter­<lb/>ræ oportet, ut conuellatur, aut di&longs;&longs;oluatur aut frangatur. </s>
<s>Et clarum <lb/><figure id="fig52"></figure><lb/>e&longs;t, quod totum ictum excipit. </s>
<s>Si uerò <lb/>affixum non &longs;it, euertitur, & tanto mino­<lb/>rem partem excipit ictus, quanto faci­<lb/>lior e&longs;t ad euer&longs;ionem. </s>
<s>Vnde nata fabu­<lb/>la de quercu, quæ cum immobilis e&longs;&longs;et, <lb/>& &longs;taret uento euer&longs;a e&longs;t, arundo flecten­<lb/>do &longs;e, cecidit quidem, &longs;ed non e&longs;t eradi­<lb/>cata. </s>
<s>Sermo igitur e&longs;t de b in&longs;identi aqu&ecedil; <lb/>in comparatione ad a, quando excipit <lb/>plenum ictum. </s>
<s>Cum ergo b tangitur, ex­<lb/>cipit plenum ictum illo in&longs;tanti, &longs;ed quia <lb/>non excipitur ictus cedente materia, & <lb/>antequam materia cedat b mouetur loco, quia in&longs;idet aquæ, ergo <lb/>non excipit ictum. </s>
<s>Proponatur ergo, quod moueatur b per c&longs;pa­<lb/>tium in d tempore, & &longs;it, ut idem b ab e ui trahatur per idem &longs;pa­<lb/>tium in eodem tempore ex loco directo ad eandem partem: qua­<lb/>lis ergo proportio e ad b, & aërem, qui cum eo re&longs;i&longs;tit, talis propor­<lb/>tio ictus f grauis puta in a ad ictum Y in b. </s>
<s>Quia per demon&longs;tra­<lb/><arrow.to.target n="marg267"></arrow.to.target><lb/>ta &longs;uperius proportio f ad a producitur ex proportionibus e ad b, <lb/><arrow.to.target n="marg268"></arrow.to.target><lb/>& a ad e, ergo diui&longs;a 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/>&longs;trandum.</s></p><p type="margin">
<s><margin.target id="marg266"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s><margin.target id="marg267"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 2.</s></p><p type="margin">
<s><margin.target id="marg268"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 42. & 43. P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/></s></p><p type="main">
<s>Ex hoc patet, quod b quanto mollius, leuius, & &longs;trictius in imo, <lb/><arrow.to.target n="marg269"></arrow.to.target><lb/>& in tenuiore aqua, eo minus lædetur. </s>
<s>Et quanto ictus lentior fue­<lb/>rit etiam quod &longs;it grauius Y.</s></p><pb pagenum="66"/><p type="margin">
<s><margin.target id="marg269"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Propo&longs;itio &longs;eptuage&longs;ima&longs;exta.</s></p><p type="main">
<s>Proportionem duorum mobilium &longs;ibi inuicem concurrentium <lb/>per rectam inuenire.<lb/><arrow.to.target n="marg270"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg270"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Iam cognito, quod mobilia, quæ loco mouentur per præceden­<lb/>tes, &longs;ed omnino quie&longs;cunt integros excipiuntictus: alia quidem, <lb/>quæ concurrunt, non omnino re&longs;iliunt, alia uero re&longs;iliunt, & quæ <lb/>re&longs;iliunt minores excipiuntictus, &longs;equitur ut diuer&longs;a &longs;it compara­<lb/>tio: nam erunt, quæ &longs;tando excipient ictus, & hæc integros ut mu­<lb/>ri, & quæ concurrendo, nec re&longs;iliendo, ut equi cur&longs;u incitati: & quæ <lb/>&longs;tando, &longs;ed re&longs;iliendo, ut naues &longs;tantes: & quæ concurrendo, re&longs;i­<lb/>liendo qúe ut naues uentis, & triremes ab impul&longs;u: bifariam ergo <lb/>contingit intelligi, quod proponitur. </s>
<s>Sed in utroque etiam &longs;en&longs;u <lb/>uarietas e&longs;t: nam ut concurrit pars altera celerius, ita etiam magis <lb/>concutitur. </s>
<s>Et ideo &longs;it, ut proportio ictùs &longs;it in comparatione ad <lb/>grauitatem duplá, & concurrant æqualiter, & &longs;int æquè grauia, & <lb/>neutrum re&longs;iliat, erunt in proportione quadrupla, & eodem mo­<lb/>do &longs;i utrunque re&longs;iliat. </s>
<s>At &longs;i diuer&longs;o impetu ferantur, ut dixi, tria <lb/>erunt præcipuè con&longs;ideranda grauitas &longs;eu pondus, impetus, & an <lb/>re&longs;iliat. </s>
<s>Quanto enim grauiora fuerint, & maiore impetu agen­<lb/>tur, & non re&longs;ilierint eo maiorem ictum recipient: quanto leuio­<lb/>ra, & minore impetu, & magis re&longs;ilierint, minus lædentur. </s>
<s>Sed & <lb/>in debilitando ictum con&longs;iderare oportet tria, quod re&longs;iliat, quod <lb/>diffugiat, quod circumuertatur: re&longs;iliunt naues, &longs;i ro&longs;tris concur­<lb/>rant pleno ictu: &longs;i uerò non pleno ictu concurrant, &longs;ed diffugiant <lb/>hoc experimento compertum e&longs;t minimum e&longs;&longs;e ictum: &longs;i ro&longs;tro <lb/>tran&longs;uer&longs;um nauis feriatur medium, e&longs;t hoc.</s></p><figure></figure><p type="main">
<s>Sit ergo ut a b nauis tangat ro&longs;tro b c &longs;ic ut <lb/>diffugiat, erit hypomochlium c, & &longs;i tangat <lb/>e f hypomochlium e&longs;t in d dupla, ergo e&longs;t c b <lb/>ip&longs;i d e, igitur ictus duplo minor excipitur à <lb/>c b quàm ef. </s>
<s>E&longs;t etiam tempus longè maius, <lb/>quo excipit ictum ef, quàm b c: &longs;tatim enim di&longs;cedit b c occurrit que<lb/>alijs partibus, in c f autem impingit, & angulus a d c e&longs;t longè ma­<lb/>ior recto, quàm a b f: ob hæc igitur longè maior e&longs;t ictus c f quàm <lb/>b c: uocant autem hoc declinationem.</s></p><p type="main">
<s>Propo&longs;itio &longs;eptuage&longs;ima&longs;eptima.</s></p><p type="main">
<s>Proportionem motus obliqui ad motum rectum in nauibus <lb/>inuenire.</s></p><p type="main">
<s><arrow.to.target n="marg271"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg271"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Cùm uentus fertur ad puppim rectà, naui&longs;qúe gubernaculum di <pb pagenum="67"/>rigitur, tendunturqúe uela ac expanduntur &longs;umma in parte mali, <lb/>tunc motus e&longs;t ueloci&longs;simus: fingamus autem, quod omnia ad <lb/>idem tendant præter uentum, qui non directus &longs;it ad puppim, &longs;ed <lb/>à latere, ut uides, & temo &longs;itin contrarium tantundem directus, & <lb/>&longs;upponamus pro nune, quod uelum &longs;it &longs;olum in anteriore parte <lb/>nauis, nam &longs;ecus e&longs;&longs;et nimis magna differentia, <lb/><figure id="fig53"></figure><lb/>quod nauis una ageretur tribus malis alia una: <lb/>Quæritur igitur proportio motus b c ad mo­<lb/>tum d e: fiat ergo c f æqualis e g, ita ut f angulus <lb/>rectus &longs;it, & manife&longs;tum e&longs;t, quod h c maior e&longs;t <lb/>c f, cum ergo angulus f rectus &longs;it, quanto maior <lb/>erit angulus h c f, tanto maior erit proportio h c <lb/>ad c f, quod e&longs;t primum a, ińde noto angulo h c f <lb/>per ea, quæ tradita &longs;unt ab A&longs;trologis de &longs;inu & <lb/>arcu erit nota proportio c h ad c f, ideo ad e g <lb/>fiat ergo c k æqualis c h, igitur c k erit maior e g, &longs;i ergo perambula­<lb/>bit æqualiter c, ut c h, erit temporis motus e g ad motum e f, ut c k <lb/>ad c f, igitur cum nota &longs;it c k, e&longs;t enim æqualis c h, erit temporis ad <lb/>tempus proportio nota. </s>
<s>Quod autem in æquali tempore mouebi­<lb/>tur nauis per c k & h c patet ex a&longs;&longs;umpto inferius declarando.</s></p><p type="main">
<s><arrow.to.target n="marg272"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg272"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 99.</s></p><p type="main">
<s>Propo&longs;itio &longs;eptuage&longs;imaoctaua.</s></p><p type="main">
<s>Propo&longs;itionem nauis ad triremes quotuis concurrentes de­<lb/>mon&longs;trare.</s></p><p type="main">
<s>Sit nauis deferens pondus decuplo maius triremi, & con&longs;tat, </s></p><p type="main">
<s><arrow.to.target n="marg273"></arrow.to.target><lb/>quod impul&longs;u æquabitur decem triremibus, ubi flante uento e <lb/>puppi æqualiter feratur in aduer&longs;um, quantum triremes ui homi­<lb/>num. </s>
<s>Sed quoniam triremes impediuntur à uento licet &longs;ine uelis <lb/>&longs;int, habent enim & ip&longs;&ecedil; malum, & uelum, &longs;ed exigua comparatio­<lb/><arrow.to.target n="marg274"></arrow.to.target><lb/>ne nauium, ideo ictus ille multo ualidior e&longs;t ex demon&longs;tratis. </s>
<s>Cum <lb/>uero uis illa &longs;imul &longs;it, liquet, 'quòd hoc in ca&longs;u ni&longs;i machinæ ob&longs;ta­<lb/>rent una nauis mille po&longs;&longs;et obruere triremes di&longs;iunctas per tantum <lb/>&longs;patium inter &longs;e, quantum e&longs;t id, in quo nauis pote&longs;tuenti impul­<lb/>&longs;um recipere. </s>
<s>At impedimentorum maximum &longs;unt machinæ, quæ <lb/>in nauim collimant à lateribus, cum triremes quaquâ uer&longs;um &longs;e a­<lb/>g ant, & ob id proram &longs;olam exponunt ictibus, in quam difficile <lb/>e&longs;t collimare, & &longs;i tangatur pars ea robu&longs;tior e&longs;t, nec periculum <lb/>euer&longs;ionis adeò in currit, ut à lateribus: nec enim adeò angu&longs;ta e&longs;t a <lb/>prora ad puppim nauis, quam à latere ad latus: his tot cau&longs;is mi­<lb/>nus e&longs;t obnoxia machinis triremis, quám nauis. </s>
<s>Sed & alia cau&longs;a <lb/>e&longs;t, quoniam nece&longs;&longs;e e&longs;t ut ob angulum laterum ad proram <pb pagenum="68"/>ictus dilabatur &longs;&ecedil;pius &longs;olum traiecta &longs;uperficie. </s>
<s>Secundum impe­<lb/>dimentum e&longs;t à uento, &longs;i ualde obliquus &longs;it, nam ad rectum impul­<lb/>&longs;um, multum debilitatur: aut &longs;i incon&longs;tans &longs;it, uiribusque remittatur. <lb/></s>
<s>Tertium uerò &longs;i triremes inuicem connexæ &longs;int, ac &longs;e tangant, in <lb/>quas nauis dirigitur. </s>
<s>Sed & hoc infrà demon&longs;trabitur nauim, ut le­<lb/><arrow.to.target n="marg275"></arrow.to.target><lb/>uior fuerit facilius elabi, &longs;ed ut pondere magis onerata grauiores <lb/>ictus inferre: ob hoc triremem inuenerunt mediam maximi u&longs;us <lb/><foreign lang="greek">a)mfh/rhn. </foreign></s>
<s>Galeonum uulgò uocant.</s></p><p type="margin">
<s><margin.target id="marg273"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s><margin.target id="marg274"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 74.</s></p><p type="margin">
<s><margin.target id="marg275"></margin.target>P<emph type="italics"/>rop.<emph.end type="italics"/> 109.</s></p><p type="main">
<s>Propo&longs;itio &longs;eptuage&longs;imanona.</s></p><p type="main">
<s>Proportionem medicamentorum purgantium inuicem de­<lb/>clarare.<lb/><arrow.to.target n="marg276"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg276"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Scio, quàm multa concurrant, etiam per &longs;e ad purgationem mul <lb/>titudo humorum præparatio locus propinquus, &longs;ed nobis &longs;er­<lb/>mo e&longs;t pari&longs;ub conditione, ut &longs;it dimidia uncia Ca&longs;siæ nigræ in tri­<lb/>bus uicibus expurget libram humorum, & uelim &longs;cire ab una un­<lb/>cia, quoties expurgabitur, & quantum. </s>
<s>Dico, quod in &longs;camonio, & <lb/>agarico hæc ratio deprehendi pote&longs;t: in his autem medicamentis, <lb/>quæ magis leniunt, quàm à proprietate educant, ut e&longs;t ca&longs;sia nigra, <lb/>ratio hæc non ualet, quoniam feces quando que pro maiore par­<lb/>te educuntur, ita ut etiam multiplicato medicamento de&longs;it, quod <lb/>educatur. </s>
<s>Et quamuis humores iuxta proportionem trahat, cum <lb/>tamen feces proportionem non &longs;eruent, &longs;equitur: ut aggregati ad </s></p><p type="main">
<s><arrow.to.target n="marg277"></arrow.to.target><lb/>aggregatum proportio non &longs;eruetur. </s>
<s>At non e&longs;t facile po&longs;tmo­<lb/>dum interno&longs;cere feces ab humoribus, quocirca uidetur propor­<lb/>tio illa confundi. </s>
<s>Quod &longs;i medicamentum leniens, fiat ob quanti­<lb/>tatem purgans humores, ut de multa ca&longs;sia nigra, tuncnon pote&longs;t <lb/>a&longs;signari illa comparatio ni&longs;i ut e&longs;t medicamentum purgans. </s>
<s>Et &longs;it <lb/>gratia exempli, primum ut grana &longs;ex &longs;camonij purgent aliquem <lb/>ter, & uncias decem bilis, dico iuxta rationem &longs;uprapo&longs;itam, quod <lb/><arrow.to.target n="marg278"></arrow.to.target><lb/>grana duodecim purgabunt iuxta proportionem duplam &longs;exqui­<lb/>alteram, &longs;i duo grana nil purgant, &longs;ed commouent. </s>
<s>æqualia enim <lb/><arrow.to.target n="marg279"></arrow.to.target><lb/>&longs;unt: ut quatuor &longs;int dupla, & &longs;ex tripla, & mouent ter, quia &longs;exqui­<lb/>alteram habent proportionem ad exce&longs;&longs;um, igitur duodecim du­<lb/>plam, & &longs;exquialteram ad quatuor, nam decem ad quatuor e&longs;t du­<lb/>pla &longs;exquialtera, & purgabit &longs;epties cum nixu libras duas fer­<lb/>me bilis. </s>
<s>Vt comparatio fiat exce&longs;&longs;us ad uim, quæ re&longs;i&longs;tit eodem <lb/>modo. </s>
<s>In ca&longs;sia ergo nigra &longs;i uncia <expan abbr="unanõ">unanon</expan> purga, &longs;ed lenit tantum, <lb/>& duæ unciæ purgant ter, & libram unam bilis, tres unciæ duplam <pb pagenum="69"/>habent proportionem iuxta exce&longs;&longs;um ad unam, exce&longs;&longs;us igitur <lb/>duplum purgabunt, & duplo magis, id e&longs;t præter feces libras <lb/>duas bilis in &longs;ex uicibus.</s></p><p type="margin">
<s><margin.target id="marg277"></margin.target>E<emph type="italics"/>x conuer&longs;a<emph.end type="italics"/> 18. <emph type="italics"/>quint.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg278"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 37.</s></p><p type="margin">
<s><margin.target id="marg279"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 42.</s></p><p type="main">
<s>Propo&longs;itio octuage&longs;ima.</s></p><p type="main">
<s>Proportionem motus &longs;ecundum obliquum ad rectum in &longs;pa­<lb/>tio declarare.</s></p><p type="main">
<s>Hæc uídetur &longs;imilis &longs;uperiori cuidam propo&longs;itioni, &longs;ed tamen in <lb/><arrow.to.target n="marg280"></arrow.to.target><lb/>hoc differt, quoniam in c a &longs;upponimus nauim moueri, ut concu­<lb/>tiat, hic autem iuxta motum &longs;olum: ut proponamus b nauim ferri <lb/><figure id="fig54"></figure><lb/>uer&longs;us a uento recto ex b in a: &longs;it autem uentus ex <lb/>cin a mouens nauim ex b in a: nòn enim moue­<lb/>bit ut quidam putant in ratione c a ad b a: ut &longs;i ca <lb/>&longs;it &longs;exquiquarta ad b a, ut æquali impetu ex b & <lb/>c flante uento moueretur tardius per c a, quam <lb/>per b a, quia æqualiter ex &longs;uppo&longs;ito: ergo tanto <lb/>tardius c fertur in a, quam b in idem quanto lon­<lb/>gior e&longs;t c a, b a igitur &longs;i b perueniet in a in qua­<lb/>tuor diebus c perueniet in idem a in quinque <lb/>diebus. </s>
<s>Hoc enim e&longs;t per &longs;e manife&longs;tum: &longs;ed non quærimus id, &longs;ed <lb/>ut uento c a æquali per c a ei, qui e&longs;t b a per b a, ubi b moueatur uen <lb/>to c a per b a, quanto tardius mouebitur. </s>
<s>Mouebitur. </s>
<s>n. </s>
<s>tardius ad <lb/>a per b a, quam per c a, at per c a tardius, quam ex b in a per æqua­<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>Quærimus ergo compo&longs;itionem horum, ut &longs;it c <lb/>nauis, quæ debeat transferri ad a per uentum ex b, & &longs;equitur, <lb/>quod tardius, quam ex c per uentum ex c in a, & tardius ex b per <lb/>uentum ex cin a. </s>
<s>Ergo malus, qui in prora e&longs;t conuoluto eo, qui <lb/>e&longs;t in puppi, ut etiam Ari&longs;toteles docet tantundem nititur ad re­<lb/><arrow.to.target n="marg281"></arrow.to.target><lb/>ctum ex cin æquidi&longs;tantem locum ab a quantum c di&longs;tat ab con­<lb/>tra temo, qui in puppi e&longs;t dirigitur ad h, & &longs;i ualidius &longs;it uentus e­<lb/>tiam adiuuante temonem, &longs;eu contra nitente, quantum licet mo­<lb/>bili pondere nauis ad id latus, premitur enim nauis, qua&longs;i &longs;ubmer­<lb/>gi debeat, uento in aduer&longs;um premente, ut &longs;i uentus repente huic <lb/>contrarius exoriatur, <expan abbr="periculũ">periculum</expan> &longs;ubeat, ne obruatur. </s>
<s>Cum ergo uen­<lb/>tus ex b feratur, æquidi&longs;tans c h, & c feratur per temonem in k, & ab <lb/>oppo&longs;itis æqualis actio &longs;equatur, imò tota impeditur, ex c in h fere­<lb/>tur iuxta proportionem anguli, quem con&longs;tituit h c cum a c ad to­<lb/>um rectum, Si igitur ex c in a debuit ferri in duodecim horis ob <pb pagenum="70"/>uim uenti, & uiæ longitudinem, angulus uerò h c a &longs;it &longs;exta re­<lb/>cti pars, feretur ex c uer&longs;us a ad quantitatem b a in quatuorde­<lb/>cim horis: igitur rur&longs;us quanta e&longs;t proportio c a ad b a tan­<lb/>tum e&longs;t temporis, in quo fertur ex c ad a ad quatuordecim horas <lb/>per uentum b a.</s></p><p type="margin">
<s><margin.target id="marg280"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s><margin.target id="marg281"></margin.target>Q<emph type="italics"/>uæ&longs;t.<emph.end type="italics"/> 7. M<emph type="italics"/>echanica.<emph.end type="italics"/></s></p><p type="main">
<s>Propo&longs;itio octuage&longs;imaprima.</s></p><p type="main">
<s>Qualis &longs;it angulus, per quem pote&longs;t moueri nauis ad rectum <lb/>explorare.<lb/><arrow.to.target n="marg282"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg282"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Cum in præcedenti propo&longs;itione o&longs;ten&longs;um &longs;it angulum k c a <lb/>oportere e&longs;&longs;e æqualem angulo h c a, ut feratur, c in a uento c h, nec <lb/>tamen pror&longs;us, &longs;ed temo magis inflectit uer&longs;us k quam uentus co­<lb/>git uer&longs;us h: &longs;icut contra maiori ui uentus dirigit ad h, quàm temo <lb/>ad k, ut nece&longs;&longs;e &longs;it nauim flecti ad k pondere, ideo &longs;i uentus e&longs;&longs;et <lb/>tran&longs;uer&longs;us periclitaretur, nece&longs;&longs;e e&longs;t, ut per omnes uentos, qui fe­<lb/>runt ab ea, quæ ad perpendiculum &longs;uper c a, & &longs;unt quatuordecim: <lb/>&longs;ed quoniam, ut dixi, pondere adiuuante uis uenti minor fit, nece&longs;­<lb/>&longs;e e&longs;t, ut per uentos debiliores feratur magis ab extremis, qui pro­<lb/>pe perpendiculum &longs;unt: ita ut numerus omnium &longs;it, cum leui&longs;simi <lb/>fuerint, quatuordecim, cum uiolenti&longs;simi, tres tantum proprius, & <lb/>qui di&longs;tant trige&longs;ima&longs;ecunda parte totius circuli, id e&longs;t partibus un <lb/>decimi, cum quarta reliqui undecim, medij &longs;unt: ut tanto plures a&longs;­<lb/>&longs;umi po&longs;sint à Nauclero, quanto molliores &longs;unt uenti, tanto pau­<lb/>ciores, quo uiolentiores. </s>
<s>Tutius autem fuerit in ualidis uentis diri­<lb/>gere nauim per uentum proximiorem, quam per ip&longs;ummet, qui re­</s></p><p type="main">
<s><arrow.to.target n="marg283"></arrow.to.target><lb/>ctè tendit ad locum. </s>
<s>Veluti tendat nauis ex a in b, uentus tendat in <lb/>cualidior, cumque magnus fuerit angulus c a b, ut potè dodrans to­<lb/>tius recti, ut e&longs;&longs;et temo dirigendus ad &longs;extum uentum altrin&longs;ecus di <lb/>rigemus &longs;olum ad quintum, ut feratur in d, & hoc erit tanto cele­<lb/>rius, & celerius feratur per a d & d b, quàm &longs;i nauis recta lata e&longs;&longs;et <lb/>ex a in b. </s>
<s>in&longs;uper tutius.</s></p><p type="margin">
<s><margin.target id="marg283"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 83</s></p><p type="main">
<s>Propo&longs;itio octuage&longs;ima&longs;ecunda.</s></p><p type="main">
<s>Proportionem uelorum indagare.<lb/><arrow.to.target n="marg284"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg284"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Vela tribus in locis di&longs;poni &longs;olent dolo b, quod in prora con­<lb/>&longs;tituitur, & in malo, qui ponitur in medio ratione, quæ inferius <lb/>o&longs;tendetur, &longs;ed non ad unguem, quia cum malus in anteriorem <lb/>partem à uento impellatur, &longs;i e&longs;&longs;et in medio, &longs;emper præmeretur <lb/>nauis in anteriorem partem, ex quo duo magna incommoda &longs;eque <lb/>rentur: primùm ut periculum &longs;ubiret, ne inuer&longs;a in anteriorem par­<pb pagenum="71"/>tem &longs;ubmergeretur. </s>
<s>Secundum ne pre&longs;&longs;a in parte anteriore dif­<lb/>ficilius aquas di&longs;&longs;ecaret, & ob id longe tardiu, moueretur. </s>
<s>Pro­<lb/>pter hæc duo incommoda igitur malus etiam &longs;i unicus e&longs;&longs;et <lb/>(quod uulgati&longs;simum maloribus no&longs;tris |fuit) in parte magis <lb/>proræ proxima locabatur à gubernatoribus, ut e&longs;&longs;et qua&longs;i in trien <lb/>te à ro&longs;tro in be&longs;&longs;e à puppi: Rarum fuit, & memorabile, quod nunc <lb/>pa&longs;sim habet olim Antigoni <foreign lang="greek">triame/ou&</foreign> 1, uelorum trium: quorum <lb/>po&longs;tremum Epidromus ut ip&longs;a uoce intelligamus non fui&longs;&longs;e ue­<lb/>lum in malo ip&longs;o medio, &longs;ed in puppi con&longs;titutum. </s>
<s>Cau&longs;a Dolonis <lb/>inferius exponetur: quod autem e&longs;&longs;et paruum, & omnium mini­<lb/>mum, ut nauis &longs;acile ab eo inuerteretur. </s>
<s>Vnde etiam nunc minus <lb/>minime habent tam quantitate, quam etiam altitudine, quod uo­<lb/>cant Trinehetum, &longs;olum enim &longs;u&longs;tinet nauim, quæ à uentis, uel un­<lb/>dis mergi &longs;olet: ab undis ubi humilior e&longs;t, à uentis à lateribus, et an­<lb/>teriore parte. </s>
<s>Vnde humile, & exiguum uelum efficit, ut nauis ante­<lb/>riore parte leuis, nec mergatur prona à uentis, nec aquas ea exci­<lb/>piat, nec tamen impelli pote&longs;t nauis in &longs;copulos, nec euerti ob cau­<lb/>&longs;as dictas: ob quæ in magnis tempe&longs;tatibus hoc ip&longs;o duntaxat uti <lb/>&longs;olent. </s>
<s>Quod et&longs;i nimium &longs;æuierint, etiam illud demittunt, & &longs;i <lb/>fieri pote&longs;t, etiam malum ip&longs;am quamuis &longs;ine uelo &longs;it. </s>
<s>Sed plerun­<lb/>que circumuolutam, & implicatam &longs;olet antennam annexam, at­<lb/>que &longs;u&longs;pen&longs;am habere. </s>
<s>Sed & ne nauis pror&longs;um obruatur, quo­<lb/>niam ea pars omnem uentorum uim excipere &longs;olet, & ut leui&longs;sima <lb/>&longs;it ijdem Gubernatores puppim multa arena, lapillis qúe onerant. <lb/></s>
<s>Ergo uelocitas nauis à uentorum impetu, eorumqúe rectitudi­<lb/>ne à uelorum magnitudine, & loco humiliore, aut &longs;ublimiore ha­<lb/>betur: tum nauis leuitate, & forma. </s>
<s>Quæ enim non merguntur ut <lb/><foreign lang="greek">droma/des</foreign> (&longs;ic enim uocat Ari&longs;tophanes) eas, quas nunc uulgus fre­<lb/>gatas appellat) qua&longs;i aquas innatantes cur&longs;u &longs;unt ueloci&longs;simæ. </s>
<s>Et <lb/>longiores latis. </s>
<s>Po&longs;t has &longs;unt, quæ carinam habent tenuem, ut fa­<lb/>cile aquas diuidant. </s>
<s>Vltimo loco, quæ qua&longs;i mediæ, ante quidem <lb/>tenues, pò&longs;t latiores ad uelocem cur&longs;um, & ferendum onera aptæ, <lb/>& humiles altis: & leui ex ligno. </s>
<s>Sed nos de uelorum uarieta­<lb/>te loquimur, non ea', quæ ad malos pertinet. </s>
<s>Con&longs;tat enim me­<lb/>dio loco plus mouere, quam in extremis, ut infrà docebi­<lb/>mus. </s>
<s>Antiquo enim tempore opus non fuit malorum mul­<lb/>titudine, quoniam &longs;yderibus uias dirigebant ob id non ad <lb/>amu&longs;sim, quoniam linea dirigi non poterat maximè ob mo­<lb/>tus obliquitatem in circulo ui&longs;us: ideò mali multi confu­<lb/>&longs;ionem in cur&longs;u, & impedimentum in naui, maiu&longs;qúe pericu­<lb/>lum attuli&longs;&longs;ent. </s>
<s>At nunc inuenta pyxide, & lapidis Her­<pb pagenum="72"/>culei auxilio pluribus locis uela di&longs;po&longs;ita melius dirigunt iter, ut <lb/>qua&longs;i cra&longs;&longs;a minerua depictum, & pote&longs;tate deformatum, ad amu&longs;­<lb/>&longs;im contrahant. </s>
<s>Motus ergo magnitudo non &longs;impliciter con&longs;tat, <lb/>&longs;ed comparatione &longs;uper&longs;iciei ueli ad uelum longitudine quidem, </s></p><p type="main">
<s><arrow.to.target n="marg285"></arrow.to.target><lb/>ac latitudine conflata per multiplicationem. </s>
<s>Altitudinis quo que ut <lb/><arrow.to.target n="marg286"></arrow.to.target><lb/>infrà exponetur. </s>
<s>Ex quorum omnium ductu, qua&longs;i cubica, uel tri­<lb/>plicata ratione, ut &longs;uperius o&longs;ten&longs;um e&longs;t, ratio uelocitatis motus na <lb/>uium conflatur.</s></p><p type="margin">
<s><margin.target id="marg285"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 86.</s></p><p type="margin">
<s><margin.target id="marg286"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 42.</s></p><p type="main">
<s>Propo&longs;itio octuage&longs;imatertia.</s></p><p type="main">
<s>Proportionem rece&longs;&longs;us à recta uia ad obliquitatem inue&longs;tigare.<lb/><arrow.to.target n="marg287"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg287"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>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 quàm a c & per a b, quam <lb/>per a d, & &longs;int ad perpendiculum b e, b d quas con&longs;tat e&longs;&longs;e breui&longs;si­<lb/>mas earum, quæ ad a c & ad a d. </s>
<s>Queritur igitur quando uelocius <lb/><figure id="fig55"></figure><lb/>ferretur ad b, an cum per a c, c b, an cum per a d, d b, <lb/>an cum per a b &longs;impliciter. </s>
<s>Et con&longs;tat quod a d & d b <lb/>longiores &longs;unt a b, i&longs;tud enim demon&longs;tratum e&longs;t ab <lb/>Euclide in primo Elementorum, dico modo a c, & </s></p><p type="main">
<s><arrow.to.target n="marg288"></arrow.to.target><lb/>c b e&longs;&longs;e longiores a d & d b, nam quadrata a d & d b <lb/>& a c & c b &longs;unt æqualia quadrato a b per dicta ibi­<lb/><arrow.to.target n="marg289"></arrow.to.target><lb/>dem, & ideo quadrata a c & c b &ecedil;qualia quadratis a d <lb/>& d b, &longs;ed a d e&longs;t longior a c, quia ducta c d angulus <lb/>d c a e&longs;t obtu&longs;us, igitur ad maiorem a c per decimam <lb/>nonam primi Elementorum: quare per communem <lb/>animi &longs;ententiam quadratum a d maius e&longs;t quadrato a c, quarerur­<lb/>&longs;us per communem animi &longs;ententiam quadratum c b maius e&longs;t <lb/>quadrato d b. </s>
<s>Cum ergo quadrata a d & d b æqualia &longs;int quadra­<lb/>tis a c & c b, & a d &longs;it maior a c & c b maior d b, &longs;equitur per nonam <lb/>&longs;ecundi Elementorum, quod a c & c d &longs;int maiores a d & d b pari­<lb/>ter acceptis. </s>
<s>Si ergo maior fuerit exce&longs;&longs;us quàm proportio motus <lb/>per temonem cohibiti, ut &longs;upra ui&longs;um e&longs;t, tardius mouebitur per <lb/>a d, d b quàm a b per a c, c b quàm per a d, d b, &longs;ed &longs;i contrà maior &longs;it <lb/>proportio motus cohibiti à temone ad motum liberum quàm ex­<lb/><arrow.to.target n="marg290"></arrow.to.target><lb/>ce&longs;&longs;us ad exce&longs;&longs;um uelocius mouebitur per a d d b, quàm per a b, <lb/>& per a c quàm per a b. </s>
<s>Accedit huc e incommodo longioris uiæ, <lb/>quod uento a c non poterit ferri nauis ex c d in b, quoniam antea <lb/>ægre ferebatur: & nunc ægrius per c b quàm a b, plus enim di&longs;tat <lb/>uentus a c ab itinere c a quàm à uento a b, ut ui&longs;um e&longs;t &longs;uperius, igi­<lb/>tur multo melius e&longs;t (ni quid ob&longs;tet) ire per a b quàm per <expan abbr="ullã">ullam</expan> aliam <lb/><arrow.to.target n="marg291"></arrow.to.target><lb/>uiam: ni&longs;i &longs;tationes &longs;int in c d, uel periculum immineat in a b. </s>
<s>Vbi ta <lb/>men uenti &longs;ecundarent, tantum e&longs;t uirium in recto cur&longs;u, & æquali <pb 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 ip&longs;am a b, quod fuit propo&longs;itum declarare.</s></p><p type="margin">
<s><margin.target id="marg288"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 20.</s></p><p type="margin">
<s><margin.target id="marg289"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 47.</s></p><p type="margin">
<s><margin.target id="marg290"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 80.</s></p><p type="margin">
<s><margin.target id="marg291"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 81. P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/></s></p><p type="main">
<s>Propo&longs;itio octuage&longs;imaquarta.</s></p><p type="main">
<s>Di&longs;tantiam centri terræ à centro mundi per motum lapidis Her <lb/>culei declarare.<lb/><arrow.to.target n="marg292"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg292"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>_{m}.</s></p><p type="main">
<s>Non me later Ari&longs;totelem exi&longs;timare centrum mundi e&longs;&longs;e cen­<lb/>trum terræ illudque proba&longs;&longs;e, quod tamen ex demon&longs;tratione no&longs;tra <lb/>mathematica apparet nunc&longs;ubijciam, & quid ad illius rationes di­<lb/>cendum &longs;it, aliâs etiam dicendum erit: nam liber hic, ut mathemati­<lb/>ca decet, e&longs;&longs;e debet ab omnibus contentionibus ab&longs;olutus. </s>
<s>Con­<lb/>&longs;tat &longs;anè non e&longs;&longs;e propriam uim lapidis illius, ut qui non &longs;it circum­<lb/>&longs;criptus &longs;ed fru&longs;tulum quoduis id pote&longs;t, neque per &longs;e, &longs;ed in ferro & <lb/>pendulo, nec fieri pote&longs;t, ut &longs;it illius <expan abbr="tãquam">tanquam</expan> &longs;peciei unius lapidum, <lb/>&longs;ed qua&longs;i perfectæ portionis cuiu&longs;dam generis terræ, quæ ab&longs;olu­<lb/>ta &longs;it, cuius indicium e&longs;t illius copia, neque enim ullibi non inuenitur, <lb/>& ubi ferrum effoditur, ut in Ilua In&longs;ula Tyrrheno mari, e&longs;t ergo fer <lb/><figure id="fig56"></figure><lb/>ri uis terræ maritæ, quæ perfecta in &longs;uo ge­<lb/>nere, ubi uim fœcundam acceperit à ma&longs;cu­<lb/>lo &longs;cilicet Herculeo lapide, quærit primum <lb/>ut de&longs;cendat, ubi hoc non po&longs;sit <expan abbr="&longs;alt&etilde;">&longs;altem</expan> quæ­<lb/>rit, ut quie&longs;cere po&longs;sit. </s>
<s>Vt ergo quie&longs;cat à <lb/>motu cœli qui e&longs;t ab Oriente in Occiden­<lb/>tem iuxta axis cœli &longs;itum &longs;e dirigit, quod <lb/>ille &longs;olus quie&longs;cat in &longs;uo motu, uel &longs;altem <lb/>tardi&longs;simè moueatur: indicio e&longs;t quod &longs;i <lb/>extra &longs;itum illum acus ferrea imbuta eo lapide ponatur, &longs;tatim tre­<lb/>mit uchementer, adeò ut nec momento ullo con&longs;i&longs;tat, &longs;ed mi&longs;erè & <lb/>grauiter torqueri uideatur, non ergo quod &longs;entiat polorum locum <lb/>qui tantum abe&longs;t ab illa, ut nec ab homine perito mathematicarum, <lb/>&longs;ed quod uix illa cœli &longs;entiatur circa centrum mundi. </s>
<s>Cuius indi­<lb/>cio e&longs;t Oceani maris, aquarum fluxus & refluxus. </s>
<s>Duos ergo ha­<lb/>bet motus terra perfecta, &longs;eu ferrum lapide Herculeo <expan abbr="imbutũ">imbutum</expan> &longs;ub­<lb/>ordinatos imperfectum perfecto: perfectus e&longs;t, ut de&longs;cendat ad cen <lb/>trum terræ, ut ibi quie&longs;cat: imperfectum, cum à perfecto prohibe­<lb/>tur, ut quie&longs;cat &longs;altem extra centrum cum in clinatione ad centrum, <lb/>et hoc fiet &longs;i &longs;ecundum longitudinem acus dirigatur per axem mun <lb/>di, cum &longs;itu tamen de&longs;cen&longs;ui ad terræ centrum proximiore, ut &longs;æpi­<lb/>us &longs;uperius declarauimus, dum de motu grauium & præcipuè li­<lb/>bræ, & centro grauitatis loqueremur. </s>
<s>Quibus demon&longs;tratis tum <lb/>experimento tum ratione à Fortunio Affaytato Cremonen&longs;i Me­<lb/>dico, cum per hæc po&longs;tmodum cogeretur fateri acum ad polum <pb pagenum="74"/>tendere, cum tamen tendat à dextro latere &longs;cilicet ab Oriente no­<lb/>uem partibus, &longs;eu decima parte unius recti in centro terræ, quæ e&longs;t <lb/>quadrage&longs;ima totius ambitus cœli. </s>
<s>Statuatur centrum mundia, & <lb/>b a c axis, &longs;ecundum quam mouetur motu diurno, ital a dextra exit <lb/>oriens, k a &longs;ini&longs;tra occidens, & &longs;tatuatur d centrum terræ, &longs;eu &longs;uprà <lb/>&longs;eu infrà, non tamen in linea b c, &longs;ed uel &longs;uprà in dextra parte, uel in­<lb/>frà in &longs;ini&longs;tra, ita ut ducta linea per illud punctum arcus b g &longs;it no­<lb/>uem partium. </s>
<s>Con&longs;tituta ergo acu in e puncto, ubilinea h ad g &longs;ecat <lb/>peripheriam terr&ecedil; dico, quod acus dirigetur per h g, & non per b c, <lb/>nam acus mouetur ad centrum per eam, & in eo &longs;itu tota dirigitur, <lb/>quia omnes partes grauis con&longs;entiunt in motu principij grauitatis <lb/>ad centrum, hoc enim demon&longs;tratum: nixus ergo e&longs;t ut moueatur <lb/>per c d, & in eo nixu qui e&longs;t quies cu&longs;to dit lineam axis, quæ e&longs;t a b, <lb/>ut quie&longs;cat, ergo non quie&longs;cet, ni&longs;i in linea d g, quod erat demon­<lb/>&longs;trandum. </s>
<s>Quæ autem &longs;equuntur ex his corrolaria omnia concor­<lb/>dant cum experimentis. </s>
<s>Ergo hic &longs;ermo e&longs;t demon&longs;tratiuus, ut e­<lb/>nim bene dixit Auerroes: Sermo demon&longs;tratiuus &longs;atisfacit omni­<lb/>bus problematibus quæ <expan abbr="cõtingunt">contingunt</expan> circa principale quæ&longs;itum. </s>
<s>Ex <lb/>hoc ergo patet, quod angulus di&longs;tantia d ab a in latitudine e&longs;t de ci­<lb/>ma pars recti, et quod quanto magis di&longs;tatin longitudine centrum <lb/>terræ à centro mundi, tanto etiam minus di&longs;tatin latitudine. </s>
<s>Hæc <lb/>enim &longs;unt demon&longs;trata clarè in mathematicis. </s>
<s>Vnde fieri po&longs;&longs;et <lb/>quod hæc quantitas di&longs;tantiæ e&longs;&longs;et res, per quam exigua etiam &longs;i <lb/>non e&longs;&longs;et maior quatuor digitis &longs;ufficeret, modo etiam per ualde <lb/>paruum &longs;patium di&longs;taret ab eodem in longitudine. </s>
<s>De cau&longs;a au­<lb/>tem huius differentiæ aliâs dicendum erit, hiclo cus non e&longs;t, &longs;ed &longs;uf­<lb/>ficit &longs;cire quod ita &longs;it, quod &longs;i mobilis &longs;it punctus d, clarum e&longs;t ali­<lb/>quando futurum ut minus di&longs;tet g à b, aliquando ut &longs;it idem. </s>
<s>Et <lb/>quali&longs;cunque motus &longs;it, nece&longs;&longs;e e&longs;t eam di&longs;tantiam uariari.</s></p><p type="main">
<s>Propo&longs;itio octuage&longs;imaquinta.</s></p><p type="main">
<s>Proportio ponderis unius grauis ad aliud &longs;ub eadem men&longs;ura <lb/>e&longs;t, ueluti eiu&longs;dem ad differentiam ponderis ua&longs;is repleti ex altero <lb/>graui, & ex ambobus detracto priore.</s></p><p type="main">
<s><arrow.to.target n="marg293"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg293"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Sit aurum a, & liquor b, quæ repleant uas c, & <lb/>pondus amborum &longs;it librarum quadraginta, & <lb/><figure id="fig57"></figure><lb/>uas repletum liquore &longs;olo &longs;it librarum xxix, au­<lb/>rum autem &longs;it ponderis librarum xij, igitur reli­<lb/>quum erit ponderis xxviij, differentia ergo ua­<lb/>&longs;is pleni, & non pleni liquore e&longs;t libra una, pon­<lb/>dus auri e&longs;t librarum duodecim: dico quod au­<lb/>ri pondus e&longs;t duode cuplum ponderi liquoris, & <pb pagenum="75"/>&longs;i fui&longs;&longs;et pondus amborum libræ xxxix, manentibus reliquis, &longs;eque <lb/>retur quod pondus liquoris e&longs;&longs;et xxvij, & quia plenum uas &longs;uppo­<lb/>nitur e&longs;&longs;e librarum xxix, e&longs;&longs;et differentia libræij, at auri pondus e&longs;t <lb/>libræ xij, igitur proportio ponderis auri ad liquorem e&longs;&longs;et &longs;excu­<lb/>pla. </s>
<s>Nam &longs;i uas plenum liquore ex &longs;uppo&longs;ito e&longs;t librarum xxix, & <lb/>cum auro xl, gratia exempli, & auri pondus e&longs;t xij, igitur liquoris <lb/>pondus e&longs;t xxviij librarum: &longs;ed cum liquor &longs;it corpus &longs;imilium par­<lb/>tium, igitur loci ad lo cum, ut ponderis ad pondus, ergo dum ade&longs;t <lb/>aurum, liquor occupat xxviij partes cxxxix, totius ua&longs;is igitur au­<lb/>rum continet unam partem tantum, & cum aurum pondus habeat <lb/>librarum xij, & liquor unius: quia totum uas cxxxix librarum dum <lb/>e&longs;t plenum, & e&longs;t diui&longs;um in xxix partes, igitur pondus unius par­<lb/>tis liquoris e&longs;t una libra, igitur pondus auri e&longs;t duode cuplum ad <lb/>pondus liquoris quod fuit propo&longs;itum.</s></p><p type="main">
<s><arrow.to.target n="marg294"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg294"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 1.</s></p><p type="main">
<s>Ex quo &longs;equitur quòd &longs;i ducatur pondus illud partis per pon­<lb/>dus repleti ua&longs;is ex alio graui, & productum diuidatur per differen <lb/>tiam illam, prodibit pondus ua&longs;is repleti liquore graui.</s></p><p type="main">
<s><arrow.to.target n="marg295"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg295"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Exemplum, &longs;i pondus auri fuerit librarum xij, pondus ua&longs;is 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 ua&longs;is, repleti ex ambobus detracto au­<lb/>ri pondere, & xxix ponderis ua&longs;is repleti liquore exit clxxiiij, & tan <lb/>tum auri uas illud continebit, nam cum duæ partes quas occupa­<lb/>bat aurum e&longs;&longs;ent 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>EXEMPLVM.</s></p><p type="main">
<s>Quia ergo in &longs;uperiore propo&longs;itione docui, quod ferrum e&longs;t ue­<lb/>ra terra: uolui &longs;cire qualis e&longs;&longs;et proportio ferri ad aquam. </s>
<s>Accepi ur <lb/>ceum cuius aqua dum plenus e&longs;&longs;et ponderis, fuit unciarum &longs;ex, & <lb/>&longs;eptuncis unciæ, & &longs;eptuncis duodecimæ partis unciæ & pondus <lb/>ferri unciæ &longs;eptem, & triens unciæ & triens duodecimæ partis un­<lb/>ciæ: & ua&longs;is aqu&ecedil; & ferro eodem repleti unciæ tredecim, & duode­<lb/>cima & &longs;eptunx duode cimæ partis unciæ. </s>
<s>Detrahemus ergo vij & <lb/>trientem & trientem duodecimæ. </s>
<s>i. </s>
<s>7 & 64/144 pondus ferri ex 13 19/144, & <lb/>relinquentur 5 99/144, detrahe ex 6 81/144, pondere aquæ totius ua&longs;is relin <lb/>quuntur 17/18, diuide 7 64/144 per 17/18 exit proportio ponderis ferri ad pon <lb/>dus aquæ 7 15/17. Ethoc e&longs;t proximum ei quod dixit Philo&longs;ophus de <lb/>proportione ponderis terræ & aquæ.</s></p><p type="main">
<s><arrow.to.target n="marg296"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg296"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 2.</s></p><p type="main">
<s>Ex hoc patet &longs;olutio problematis cuiu&longs;dam propo&longs;iti aliasque mi <lb/>nus bene &longs;oluti cùm cau&longs;am habeat manife&longs;ti&longs;simam, &longs;cilicet quod <pb pagenum="76"/>ua&longs;e aqua pleno impo&longs;itis &longs;en&longs;im centum aureis coronatis nihil ef­<lb/>funditur, non quod quicquam ab&longs;umatur in metallo, &longs;ed cau&longs;a e&longs;t <lb/>quod cum aurum &longs;it duplum pondere ferro, erit ex demon&longs;tratis <lb/>&longs;ex decuplum ad pondus aquæ. </s>
<s>Igitur cum &longs;it proportio ponderis <lb/>auri ad differentiam &longs;patij eadem, &longs;i &longs;it uas aquæ ponderis libræ <lb/>unius & mediæ, erit pondus totum xxiij unciarum, igitur aqua de­<lb/>ficiet &longs;olum ex decimaoctaua parte &longs;eu cre&longs;cet ex impo&longs;itione auri, <lb/>&longs;ed illa pars in tumore aquæ ab&longs;umitur, <expan abbr="nõ">non</expan> &longs;olum, quia <lb/><figure id="fig58"></figure><lb/>dum aureos imponimus plana &longs;olum &longs;it, &longs;ed quia non ex <lb/>quauis rotunditate defluit, aliter in urceo tam exiguo <lb/>non po&longs;&longs;et apparere rotunda: quod enim rotunditas to­<lb/>tius terræ, quæ etiam planam o&longs;tendit totam unam re­<lb/>gionem ad rotun ditatem quæ apparet in exiguo urceo <lb/>aquæ. </s>
<s>E&longs;t igitur rotunditas illa potius ob lentorem aqu&ecedil; qui auge­<lb/>tur à lentore argenti, & etiam magis auri, cum &longs;en&longs;u digitorum per­<lb/>cipiatur.</s></p><p type="main">
<s><arrow.to.target n="marg297"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg297"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 3.</s></p><p type="main">
<s>Ex hoc apparet ratio quomodo Archimedes potuerit deprehen <lb/>dere coronam à Hierone propo&longs;itam quantum auri & argenti con <lb/>tineret. </s>
<s>Sit ergo uas a b aqua <expan abbr="plenũ">plenum</expan> ponderis un ciarum triginta, & <lb/>cum libra auri &longs;it ponderis unciarum quadraginta unius, & cum li­<lb/>bra argenti ponderis unciarum quadraginta cum dimidio, igitur <lb/>erit auri pondus ad aquæ pondus duodecuplum, argenti autem <lb/>ad idem octuplum, quare auri ad <expan abbr="arg&etilde;tum">argentum</expan> pondus &longs;exquialterum. <lb/></s>
<s>Ponamus ergo quod corona impo&longs;ita ex auro & argento &longs;olo fa­<lb/>bricata (hoc enim &longs;upponere oportet) fuerit un ciarum &longs;exaginta, <lb/>pondus autem aquæ content&ecedil; cum corona in ua&longs;e unciarum uigin <lb/>tiquatuor cum dimidio, &longs;cilicet totum octuaginta quatuor cum di­<lb/>midia, erit ergo proportio ponderis coronæ ad pondus aquæ, ut <lb/>cxx ad xi, aurum igitur e&longs;t proportione duodecuplum, argentum <lb/>autem octuplum, corona ut cxx ad xi. </s>
<s>Con&longs;tituantur &longs;ub ei&longs;dem ra­<lb/>tionibus ducen do lxxxviij. </s>
<s>cxx. </s>
<s>cxxxij. </s>
<s>hoc e&longs;t ac &longs;i dicamus, accipe <lb/>partes ex cxxxij & lxxxviij, tot ut faciant integrum & componant <lb/>cxx. </s>
<s>Et ideò reduces ad minores numeros, &longs;cilicet xxxiij. </s>
<s>xxij. </s>
<s>et xxx. </s></p><p type="main">
<s><arrow.to.target n="marg298"></arrow.to.target><lb/>& operaberis per regulam de con&longs;olatione monetarum, quas po­<lb/>nemus infrà, & fient auri partes octo & argen <lb/><figure id="fig59"></figure><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/>&longs;ta ad coronam puram auri & argenti.</s></p><pb pagenum="77"/><p type="margin">
<s><margin.target id="marg298"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 178.</s></p><p type="main">
<s>Ex hoc etiam patet modus <expan abbr="cogno&longs;c&etilde;di">cogno&longs;cendi</expan> proportionem grauium <lb/><arrow.to.target n="marg299"></arrow.to.target><lb/>inuicem per &longs;olam aquam, uelut auri ad plumbum, ad lapides uel <lb/>æs, aut æris ad lapidem & &longs;imilia, ut in præcedenti operatione de­<lb/>prehendi&longs;ti: nam cum &longs;it nota proportio auri ad aquam & æris uel <lb/>lapidis ad eandem, erit auri ad æs uel lapidem nota.</s></p><p type="margin">
<s><margin.target id="marg299"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 4.</s></p><p type="main">
<s>Et &longs;imiliter &longs;ciemus per hoc accipere partes diuer&longs;orum, qu&ecedil; iun <lb/><arrow.to.target n="marg300"></arrow.to.target><lb/>ctæ faciant con&longs;titutum pondus. </s>
<s>Velut uolo facere ma&longs;&longs;am ex mel­<lb/><figure id="fig60"></figure><lb/>le & aqua, quæ impleat uas, quod mellis contineat <lb/>quindecim, aquæ duodecim, uolo ut contentum &longs;it <lb/>ponderis quatuorde cim, operabor, ut in <expan abbr="cõ&longs;olatio-nibus">con&longs;olatio­<lb/>nibus</expan>, ponam duas partes mellis & unam aquæ, ut <lb/>uides in operatione à latere.</s></p><p type="margin">
<s><margin.target id="marg300"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 5.</s></p><p type="main">
<s>Propo&longs;itio octuage&longs;ima&longs;exta.</s></p><p type="main">
<s>Si circuli in æquales, &longs;eu in &longs;phæra, &longs;eu in plano &longs;e &longs;ecuerint nun­<lb/>quam oppo&longs;itos angulos æquales habent.</s></p><p type="main">
<s>Capiantur tres quartæ cir culorum magnorum a b, a c, b c, & alia <lb/><arrow.to.target n="marg301"></arrow.to.target><lb/>b d ad rectos angulos <expan abbr="erũtque">eruntque</expan> uici&longs;sim poli, & ducatur per medium <lb/>parallelus, erit ergo e f æqualis e g, & f e æqualis f g, &longs;ed ba&longs;is c g e&longs;t <lb/><figure id="fig61"></figure><lb/>quarta circuli, & ba&longs;is c b dimidium quartæ <lb/>circuli eo quod tota b a e&longs;t quarta circuli, igi­<lb/>tur per modum 25 primi Elementorum quæ <lb/>tenet, erit angulus c f g maior oppo&longs;ito c f b. <lb/></s>
<s>Hoc autem tenet in eiu&longs;dem rationis &longs;uperfi­<lb/>ciebus, quales &longs;unt hæ, quæ &longs;unt &longs;uperficies eiu&longs;dem &longs;ph&ecedil;ræ. </s>
<s>po&longs;&longs;et <lb/>etiam demon&longs;trari per modum quartæ primi Elementorum. </s>
<s>Et eti­<lb/>am con&longs;tituta &longs;phæra e f g, cuius hic circulus e&longs;&longs;et maior circulus, & <lb/>non tangeret ni&longs;i in illa linea &longs;phæra maiorem, & utrin que &longs;ecaret eo­<lb/>dem circulo. </s>
<s>Et etiam per cordas & trigonos rectilineos, auxilio <lb/><expan abbr="tam&etilde;">tamen</expan> regulæ dialecticæ. </s>
<s>Ex hoc &longs;equitur auxilio regulæ dialecticæ, <lb/><figure id="fig62"></figure><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 inæqualibus inter &longs;e ad æquales angulos &longs;ecanti­<lb/>bus & ex tertia demum regula dialectica, &longs;equitur in o­<lb/>mnibus circulis in æqualibus &longs;e &longs;ecantibus ad quemuis <lb/>angulum in &longs;phæræ &longs;uperficie. </s>
<s>Sunt autem hæ regulæ mediæ inter <lb/>axiomata & demon&longs;trata. </s>
<s>Et ex logica propria illi arti. </s>
<s>In plano au­<lb/><arrow.to.target n="marg302"></arrow.to.target><lb/>tem &longs;patium d b c minus e&longs;t a b c, &longs;ed &longs;patium c b d e&longs;t unum, ergo <lb/>per communem animi &longs;ententiam &longs;patium a b d, maius e&longs;t &longs;patio <lb/>c b c, quod fuit probandum.</s></p><pb pagenum="78"/><p type="margin">
<s><margin.target id="marg301"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s><margin.target id="marg302"></margin.target>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>Propo&longs;itio octuage&longs;ima&longs;eptima.</s></p><p type="main">
<s>Proportionem cra&longs;sitiei aquæ ad aërem in comparatione ad ra­<lb/>dios demon&longs;trare.<lb/><arrow.to.target n="marg303"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg303"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Sit in aheno a b c d in imo e dena <lb/><figure id="fig63"></figure><lb/>rius argenteus cera affixus uel cla­<lb/>uo, quem uideat ex h impo&longs;ita aqua <lb/>clara u&longs;que ad f, uideat ex k, igitur per <lb/>aquam deflectitur à perpendiculo <lb/>per angulum k f n, & in l, per angu­<lb/>lum l g o cre&longs;cente aqua demum in <lb/>labro m a p, & &longs;it e annexus, & tabu <lb/>la h k l m &longs;it affixa &longs;olo uel pondere <lb/>firma foraminibus obliquis infrà <lb/>&longs;pectantibus, & per a a&longs;picientibus extremitatem e. </s>
<s>Po&longs;&longs;umus ergo <lb/>imaginari primum, quòd omnes inclinationes &longs;int à perpendicu­<lb/>lari, dum exit aqua, & ita denarius uideretur, uel in &longs;uperficie aquæ <lb/>in directo e, uel in recta ex oculo in imo, quorum neutrum uerum <lb/>e&longs;t. </s>
<s>Secundus modus e&longs;t, ut radius delatus e a flectatur ad k uell, & <lb/>hoc non quia in a non e&longs;t mutatio medij. </s>
<s>Tertius e&longs;t, ut linea ex ocu <lb/>lo ducta perueniat per punctum a ad &longs;uperficiem aquæ, & ex ea <lb/>per directum ad denarium, & tunc quia oculus iudicat &longs;e uidere <lb/>per rectam, ideo iudicabit &longs;e uidere per l a g in q, eo quod &longs;emper in <lb/>directo loci in quo e&longs;t e. </s>
<s>At quoniam non ex qua cunque di&longs;tantia ui­<lb/>detur e, &longs;ed ex longinquiore loco, ubi uas fuerit humilius quod li­<lb/>neæ ad a ex oculo, quanto a fuerit humilius, tanto propius ip&longs;i e <lb/>procedunt. </s>
<s>Et uer&longs;a uice lineæ ex e ad a, quanto e e&longs;t humilius ad <lb/>quencunque locum inflectuntur, tanto inferius <expan abbr="cadũt">cadunt</expan>. </s>
<s>Ergo cum fue <lb/>rint ad æquilibrium h, magis di&longs;tabunt ab e, & ita e magis procul <lb/>uidebitur. </s>
<s>Cau&longs;a ergo triplex e&longs;t humilitas, uel altitudo ua&longs;is: humi <lb/>litas uel altitudo aquæ: & labri ua&longs;is altitudo. </s>
<s>Sed han crelinquere <lb/>po&longs;&longs;umus. </s>
<s>Difficultas ergo experimenti etiam rectè facti e&longs;t, quo­<lb/>niam po&longs;ito ua&longs;e n c d &longs;olum, ut altitudo &longs;it tantum n e, procul ma­<lb/>gis uidebitur e, quàm &longs;i uas &longs;it a b c d, & totum plenum. </s>
<s>Vbi autem <lb/>uas fit a b c d, magis procul uidebitur e cum &longs;uerit totum plenum, <lb/>quam cum fuerit plena &longs;ola pars n c d. </s>
<s>Sic difficile e&longs;t con&longs;iderare <lb/>an altitudo aquæ faciat ad ui&longs;ionem procul, cum in humiliore, &longs;ed <lb/>di&longs;sipari ua&longs;e longius uideatur in pauca, quia labrum non ob&longs;tat: <lb/>in eodem autem longius in pluri aqua, quia labrum etiam non ob­<lb/>&longs;tat, &longs;ed alia ratione. </s>
<s>Vt ergo uideamus hoc experimentum, capie­<pb pagenum="79"/>mus duo ua&longs;a a b c d duplum h k l m &longs;ub eadem proportione alti­<lb/>tudinis & latitudinis, & collo cabimus ita ut p n radius æquidi&longs;tet <lb/>f e, & collo cabimus tabulas cum foraminibus, ut prius, & g f p q <lb/><figure id="fig64"></figure><lb/>in æquilibrio, in de uidebimus, an q p &longs;it æqualis aut breuior, nam <lb/>longior e&longs;&longs;e non pote&longs;t, quoniam inflectitur a minore aqua, ideo <lb/>angulus p h q non pote&longs;t e&longs;&longs;e maior f a g, &longs;uppo&longs;ita p h æquali a f: <lb/>quod &longs;i non e&longs;&longs;et, &longs;ufficeret, ut q & p e&longs;&longs;ent in æquilibrio uno, & f g <lb/>alio. </s>
<s>Sed ueritas e&longs;t quod à maiore aqua maior fit reflexio: tum <lb/>quia in his, quæ &longs;unt &longs;ecundum naturam corpoream, & &longs;ub&longs;tan­<lb/>tiam den&longs;am, aut tenuem uarietas quantitatis uariat uires: tum <lb/>quia uidemus, quod in altiore aqua denarius uidetur magis cum <lb/>fundo elatus. </s>
<s>Igitur his cognitis experimentum fiat cum ua&longs;e ple­<lb/>no. </s>
<s>Et (ut dixi) con&longs;iderabimus proportionem anguli f a g ad far, <lb/>&longs;eu f e c quæ &longs;anè e&longs;t no tabilis: adeò ut &longs;it maior proportio aquæ ad <lb/>aërem comparatione grauium quàm lucis.</s></p><p type="main">
<s><arrow.to.target n="marg304"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg304"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 1.</s></p><p type="main">
<s>Ex his cogno&longs;cemus comparatione eiu&longs;dem aquæ tenuitatem <lb/>aëris unius regionis in comparatione ad aërem alterius: nam ubi <lb/>remotius uidebitur denarius, ibi aër erit tenuior.</s></p><p type="main">
<s><arrow.to.target n="marg305"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg305"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>_{m}. 2.</s></p><p type="main">
<s>Et per idem in eadem regione comparationem aquarum. </s>
<s>Nam <lb/>cum &longs;it idem aër, & uas, ac reliqua paria, ubi magis procul uidebi­<lb/>tur denarius, aqua erit cra&longs;sior ideò deterior.</s></p><p type="main">
<s><arrow.to.target n="marg306"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg306"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 3.</s></p><p type="main">
<s>Se quitur etiam quòd omnes res propiores in aqua uidentur, <lb/>quam &longs;int, & ideò maiores: & ob id etiam omnis aqua profundior <lb/>e&longs;t, quam uideatur. </s>
<s>Vtingredi per&longs;æpè &longs;it periculo&longs;um.</s></p><p type="main">
<s>Propo&longs;itio octuage&longs;imaoctaua. </s>
<s>De in&longs;trumento <lb/>momentorum.</s></p><p type="main">
<s>In&longs;trumentum Acolingen, quo momenta temporum deprehen <lb/>dantur fabricare.</s></p><pb pagenum="80"/><p type="main">
<s><arrow.to.target n="marg307"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg307"></margin.target>C<emph type="italics"/>om.<emph.end type="italics"/></s></p><p type="main">
<s>Et quoniam motus naturales fiunt in tempore: & dicuntur ue­<lb/>lociores, uel ob &longs;patium loci magnum, quod &longs;uperatur, uel ob tem <lb/>poris breuitatem in uelo ci&longs;simis motibus, quod ad &longs;patia attinet, <lb/>facilius digno&longs;cuntur uelociores, quoniam &longs;patium maius & ma­<lb/>net, ut men&longs;urari commodè po&longs;sit: &longs;ed quòd ad tempus, quanto tar <lb/>diores, quoniam in uelo cibus quantitas temporis e&longs;t exigua: & e­<lb/>tiam tempus ip&longs;um perpetuò diffluit: ideò difficillimè deprehen di <lb/>pote&longs;t. </s>
<s>Huius cau&longs;a exco gitauimus in&longs;trumentum, quod uo caui­<lb/>mus Acolingen: quod con&longs;tat tribus rotis: prima e&longs;t pedum duo­<lb/>decim diametri, in ambitu autem habet denticulos ccclx æqua­<lb/>les, & æqualiter inter &longs;e di&longs;tantes, huius peripheriæ funis cum pon­<lb/>deribus in&longs;eritur, ita ut cum alijs duabus rotis renitentibus in una <lb/>hora circumagatur æqualiter. </s>
<s>Duodecim ex his denticulis curru­<lb/>lis duode cim denticulorum axis &longs;ecundæ rotæ in&longs;eritur: &longs;ic ut cum <lb/>rota magna duode cim conuer&longs;a fuerit partibus, &longs;ecunda rota cu­<lb/>ius axis &longs;it pedum duorum, &longs;cilicet &longs;excuplo maior circumuerta­<lb/>tur. </s>
<s>Huius minoris ambitus diui&longs;us &longs;it in cxx partes æquales, & <lb/>unicuique parti denticulus in&longs;ertus &longs;it: ita hæc rota tricies in una <lb/>hora conuertetur. </s>
<s>Singulis uerò denticulis currulis axis rotæ ha­<lb/>bentis denticulos quatuor in&longs;eratur, ita ut dum &longs;ecunda rota uer­<lb/>titur &longs;emel minima circumuertatur tricies: nam pro &longs;ingulis qua­<lb/>tuor denticulis, quibus media rota cir cumagetur, minima tota cir­<lb/>cumuertetur, ideoqúe nongenties in una hora. </s>
<s>Hæc minima ro­<lb/>tula be&longs;&longs;em pedis in dimetiente habebit, ut &longs;it &longs;exta pars illius, in <lb/>ambitu autem diui&longs;a erit in xl partes, ut cum circumuer&longs;a fue­<lb/>rit nongenties in una hora pertran&longs;ierit partes xxxvi. </s>
<s>Et cum <lb/>pul&longs;us hominis communis &longs;int in hora <23>, uel circa nouem partes <lb/>ex his rot&ecedil; minoris perficient circiter unam pul&longs;ationem ex dia&longs;to­<lb/>le & &longs;i&longs;tole, &longs;eu ex di&longs;tentione & contractione perfectam: ut partis <lb/>unius conuer&longs;io fiat in nona parte, uel circa unius pul&longs;ationis pul­<lb/>&longs;us humani: & hoc e&longs;t minimum fermè, quod ab humano &longs;en­<lb/>&longs;u percipi po&longs;sit. </s>
<s>Erit etiam proportio rotarum eadem tam in dia­<lb/>metris, quàm circuitibus &longs;cilicet &longs;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 æqualibus &longs;patijs uigintiquin cupla, ut ma­<lb/>ioris ad mediam quintupla, nam cum &longs;it &longs;excupla in ambitu, <lb/>& tricies moueatur uelocius comparatione totius, &longs;equitur, ut <lb/>proportio &longs;patij, quod &longs;uperabit media ad &longs;patium, quod &longs;u­<lb/>perabit maior in ei&longs;dem temporibus, erit quintupla, &longs;emper ad un­<lb/>guem. </s>
<s>Et ita mediæ ad minorem quintupla, & ideò maioris ad <pb pagenum="81"/>minorem uelo citas uiginti quincupla, ut non &longs;it difformis, neque <lb/>pcriculo&longs;a, ut in rotis moletrinis, & &longs;it diui&longs;a per medium iuxta <lb/>proportionem, cum &longs;it tanto uelo cior minor media, quanto media <lb/>maiore. </s>
<s>Rur&longs;us proportio partium maioris ad mediæ partes tripla <lb/>e&longs;t &longs;cilicet ccclx ad cxx, & mediæ ad <expan abbr="minor&etilde;">minorem</expan> tripla cxx ad xl, & pro­<lb/>portio e&longs;t &longs;excupla, iterum igitur partes maioris ad mediam, & me­<lb/>diæ ad minorem erunt in dupla proportione, utrobique, & e&longs;t pul­<lb/>chrum. </s>
<s>Ideò partes etiam minimæ rotæ erunt &longs;atis magnæ: nam <lb/>cum diameter &longs;it bes pedis, ambitus peripheriæ erit duorum pe­<lb/>dum. </s>
<s>1. unciarum uigintiquatuor: igitur diui&longs;a peripheria in xl par­<lb/>ter, unaquæ que pars erit maior dimidia uncia.</s></p><p type="head">
<s>SCHOLIVM.</s></p><p type="main">
<s>Et cum defuerit in&longs;trumentum, utemur men&longs;ura expul&longs;u homi­<lb/>nis de&longs;umpta, &longs;ed non e&longs;t adeò exacta. </s>
<s>Accedit aliud commodum, <lb/>quòd cum in una hora circumuertantur partes xxxvi, id e&longs;t triginta <lb/>&longs;ex mille: & octauus orbis circumuertatur in totidem annis, tot <lb/>erunt momenta ex his in una hora, quot anni in uno circuitu &longs;tella­<lb/>rum fixarum. </s>
<s>Vtintelligamus, quàm breui tran&longs;it una hora apud <lb/>nos, ita apud Deum, utita dicam (nam nulla in infinito proportio) <lb/>unus annus magnus, & reditus rerum omnium. </s>
<s>Comparata etiam <lb/>rota minima ad rotam moletrini &longs;ic &longs;e habet, quòd cùm modica ad­<lb/>e&longs;t, uer&longs;atur rota in una pul&longs;atione: cum &longs;atis abundans quinquies, <lb/>aut &longs;exies cum immodica duo decies.</s></p><figure></figure><pb pagenum="82"/><p type="main">
<s><arrow.to.target n="marg308"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg308"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Ex hoc &longs;equitur, quod homo &longs;i moueretur uelo citate motus ro­<lb/>tæ moletrinæ in &longs;ex eb domadibus perueniret ad &longs;ydus Lunæ, nam <lb/>rotarum earum, quibus ferrum acuitur &longs;emidimetiens communi­<lb/>ter e&longs;t bes unius pa&longs;&longs;us, ideò dimetiens pa&longs;&longs;us cum triente: ambi­<lb/>tus ergo quatuor pa&longs;&longs;us, & xxi pars, colligamus nunc integra, in <lb/>uno ictu pul&longs;us circumagitur decies, id e&longs;t pa&longs;&longs;us xl, in hora &longs;unt <lb/><23> pul&longs;ationes: in hora igitur &longs;patium pertran&longs;itum e&longs;t cxl pa&longs;&longs;uum <lb/>in M. horis, ergo erunt clx M. pa&longs;&longs;uum addita parte xxi, erunt clxviij <lb/>M. pa&longs;&longs;uum, & tantum di&longs;tat luna à terra: & M. horæ &longs;unt dies penè <lb/>xlij, eb domadæ &longs;cilicet &longs;ex.</s></p><p type="main">
<s>Propo&longs;itio octuage&longs;imanona.</s></p><p type="main">
<s>Proportionem den&longs;itatis aquæ ad aërem per pondera inuenire.</s></p><p type="main">
<s><arrow.to.target n="marg309"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg309"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Contingit hoc multis modis: primum acceptis duabus &longs;phæru­<lb/>lis æqualibus ex cry&longs;tali&longs;ub&longs;tantia unaque demi&longs;&longs;a ab alti&longs;sima turri, <lb/>& men&longs;urato ictu per in&longs;trumentum præcedens, & &longs;ub totidem <lb/>momentis alia demi&longs;&longs;a in aquam, in de &longs;ub eodem tempore dimen­<lb/>&longs;a altitudine, erit proportio &longs;patij ad &longs;patium, ut den&longs;itatis aquæ, ad <lb/>den&longs;itatem aëris. </s>
<s>Item emi&longs;&longs;a &longs;phærula per in&longs;trumentum in aërem, <lb/>in de in aquam: & fumpta proportione. </s>
<s>Et uidimus &longs;corpionem, <lb/>qui <expan abbr="&longs;phærulã">&longs;phærulam</expan> creteam emittebat pedibus lxx, & in aqua per unum <lb/>& dimidium adeò, ut proportio fuerit, ut quinquaginta ad unum: <lb/>ideò e&longs;t fallax experimentum in uiolento motu: nam cum emitte­<lb/>batur in aquam erat propè, & ob id in &longs;ummo robore: cùm in aë­<lb/>rem, emittitur &longs;en&longs;im uis. </s>
<s>De hoc ergo loquar.</s></p><p type="main">
<s><arrow.to.target n="marg310"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg310"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Et erumpentia ob id magis quàm è terra, et minus quàm ex aëre: <lb/>diuiditur enim aqua cum graue petit fundum, & aqua feruet: & e&longs;t <lb/>mirabilius, quàm utile.</s></p><p type="main">
<s>Propo&longs;itio nonage&longs;ima.</s></p><p type="main">
<s>Rationem impetus uiolenti extra mi&longs;si ponderis ad æqualita­<lb/>tem reducere.</s></p><p type="main">
<s><arrow.to.target n="marg311"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg311"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Sit uiolentum a quod moueatur per b c d e, e &longs;patium, & quia <lb/>uiolentum contrà nititur naturali, cadat ergo in planum in e: &longs;unt <lb/>ergo tria con&longs;ideran da, primum quod, ut dixi aliâs, motus uiolen­<lb/>tus pro certa di&longs;tantia augetur, & cau&longs;am ibireddidi, ut potè u&longs;que <lb/>ad c, &longs;ed hoc e&longs;&longs;et difficile cognitu. </s>
<s>Secundum, quod ubi in cipit de­<lb/>cre&longs;cere, &longs;emper magis ac magis decre&longs;cit propter naturalem ni­<lb/>xum contra operantem. </s>
<s>Tertium quod ubi de&longs;cendere in cipit, ibi <lb/>e&longs;t æqualis uis uiolentum motum agens cum naturali. </s>
<s>Certum e&longs;t <lb/>etiam quod motus æqualis intelligitur erecta ad perpendiculum <lb/>e f, donec occurrat a d: & diui&longs;a tota b f per tempus, locus ergo, in <lb/>quo mouetur per tantum &longs;patium, dicitur locus motus æqualis: <pb pagenum="83"/>qui &longs;it gratia exempli g h, cuius medium proportione &longs;it k, di­<lb/>co k con&longs;i&longs;tere propiorem f, quàm b, etiam&longs;i æqualiter mouere­<lb/>tur. </s>
<s>Primum quòd in tota g f declinat, & totus motus e&longs;t lentior, <lb/>quàm in tota b g, & tamen tardatur tantundem, ergo per commu­<lb/>nem animi &longs;ententiam, k e&longs;t propior f, quàm b. </s>
<s>Secundò, quia per <lb/>&longs;ecundum &longs;uppo &longs;itum motus a uer&longs;us f, continuè fit lentior, igitur <lb/>per communem animi &longs;ententiam multò longius e&longs;t tempus mo­<lb/>tus a k, quam f, & tanto maius &longs;patium. </s>
<s>Tertiò, quia motus ex b uer <lb/>&longs;us caugetur, & &longs;i e&longs;&longs;et æqualis adhuc multò e&longs;&longs;et breuior k f quam <lb/>a k, igitur multò magis hoc modo, & triplicata ratione. </s>
<s>Si ergo b k <lb/><figure id="fig65"></figure><lb/>e&longs;&longs;et &longs;exquiquarta &longs;olum ip&longs;i k f, <lb/>erit b k dupla: fermè ex triplicata <lb/>ratione ip&longs;i k f, & iuxta eundem <lb/>modum ponemus mediam uim <lb/>xlvi pa&longs;sibus à &longs;corpione a quam <lb/>& hoc modo erit propèid quod e&longs;t.</s></p><p type="head">
<s>SCHOLIVM.</s></p><p type="main">
<s>Dubitat autem Philo&longs;ophus in mechanicis quæ nam uis &longs;it, qu&ecedil; <lb/>moueat lapidem iam excu&longs;&longs;um? </s>
<s>& dubium non e&longs;t quin ex parte &longs;it <lb/>aër motus tum ratione, quia mouetur ergo mouet, tum experimen <lb/>to, ut in fulminibus, & his quæ uento impelluntur, ut hypophy&longs;is, <lb/>&longs;ed in &longs;corpionibus & arcubus & pilis id non &longs;ufficere uidetur. </s>
<s>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>Inditio &longs;unt quòd mota & emi&longs;&longs;a ex lon­<lb/>gioribus machinis quan quam non aërem continentibus, nec in­<lb/>anibus tamen, longius eijciunt &longs;agittas & mi&longs;silia, quoniam uis <lb/>illa firmius imprimitur, uelut etiam de lapidibus & ferro, quod di­<lb/>utius in igne moram traxit, aut continuè follibus ignitum e&longs;t, nam <lb/>etiam tanto tardius refrigeratur unum quod que horum, & alia urit <lb/>& accendit calore illo externo, quanquam natura frigidum &longs;it: di­<lb/>cemus autem & de hoc &longs;uo loco.</s></p><p type="main">
<s>Propo&longs;itio nonage&longs;imaprima.</s></p><p type="main">
<s>Proportionem grauis cubi, & &longs;phærici æqualium in accliui, & <lb/>de&longs;cen&longs;us eorum demon&longs;trare.</s></p><p type="main">
<s>Hic non pauca &longs;unt <expan abbr="cõ&longs;ideranda">con&longs;ideranda</expan>: Primum <lb/><figure id="fig66"></figure><lb/>quòd hoc intelligi pote&longs;t, uel de motibus at­<lb/>tractionis, uel impul&longs;ionis, uel inuer&longs;ionis. <lb/></s>
<s>Secundum quod omne, quod impellitur &longs;uperiùs, tantundem gra­<lb/>uat attractum, quantum ad de&longs;cen&longs;um, &longs;i &longs;it rotundum, nam qua­<lb/>drata, <expan abbr="etiã">etiam</expan> alia non de&longs;cendunt &longs;ponte in decliui, & &longs;i &longs;it locus ualdè <pb pagenum="84"/>decliuis, tanto minus de&longs;cendunt, quanto &longs;unt latiora. </s>
<s>Quia tamen <lb/>omnia difficiliùs de&longs;cendunt &longs;phæricis, & facilius quàm in plano, <lb/>ubi ponderant ni&longs;i per dimidium grauitatis, ideò proportio hæc <lb/>con&longs;tat ex proportione anguli de&longs;cen&longs;us ad totum rectum, & ma­<lb/>gnitudine &longs;uperficiei, qua incumbit ad pondus comparata. </s>
<s>Omne <lb/>enim graue, quanto grauius tam ad quietem, quàm ad motum na­<lb/>turalem potentius e&longs;t: hoc enim per&longs;picuum e&longs;t, quia quieti natu­<lb/>rali motus uiolentus, & motui naturali quies uiolenta opponitur: <lb/>quia ergo maiore ui opus e&longs;t ad motum præter naturam, ergo &longs;e­<lb/>cundum naturam etiam maiore ui quie&longs;cit. </s>
<s>A&longs;&longs;ump&longs;imus ergo cu­<lb/>bum, ut magis notum. </s>
<s>Sphæra igitur in omni decliui de&longs;cendit, <lb/>quia ut dictum e&longs;t, nil habet quod re&longs;i&longs;tat ad motum: & ip&longs;a gra­<lb/>uior e&longs;t in decliui, quàm in plano, quia c pun­<lb/>ctus cadit ultra e, ergo punctus contactus, & <lb/><figure id="fig67"></figure><lb/>centrum grauitatis, & centrum mundi, non &longs;unt <lb/>in una linea. </s>
<s>Si enim b c contangeretur, e&longs;&longs;et b c <lb/>plana. </s>
<s>Si uerò tangit, angulus e&longs;t maior angulo <lb/>contactus, ergo cum nece&longs;&longs;arium &longs;it, æquidi&longs;ta­<lb/>re aliter non e&longs;&longs;et &longs;phæricum, oportet, ut eleue­<lb/>tur ex parte c, & de&longs;cendat uer&longs;us b, & ideò ut <lb/>continuetur motus. </s>
<s>Si uerò &longs;it in linea conta­<lb/>ctus b c f, & æquidi&longs;tet non erit, ut dixi punctus <lb/>contactus in linea centrorum, &longs;ed in a c, cum &longs;uppo&longs;itum &longs;it lineam <lb/>a d e&longs;&longs;e lineam centrorum: maior e&longs;t ergo portio g c e, quàm re&longs;i­<lb/>duum, ergo de&longs;cendet in b. </s>
<s>Cubus uerò non de&longs;cendet, ni&longs;i cum di­<lb/>midium d addito, quod inter cipitur inter lineam mediam, & quæ à <lb/>centro mundi ad punctum medium contactus u&longs;que quò perueniat <lb/>ad oppo&longs;itam partem, eam habuerit proportionem ad idem me­<lb/>dium eadem portione detracta, quem iuncta proportioni an guli <lb/>declinationis ad re&longs;iduum recti dimidiam proportionem efficiat. <lb/></s>
<s>Eademque ratio aliorum planorum. </s>
<s>Dico præterea quòd motus <lb/>&longs;phæræ, & etiam corporum rectarum &longs;uperficierum in de&longs;cen&longs;u <lb/>alius e&longs;t æqualis, & alius inæqualis, & qua&longs;i à latere, uelut &longs;i angu­<lb/>lus unus prolabatur, ac fiat circumuolutio: cum ergo facilius fiat <lb/>hoc, & maximè &longs;i non retineatur æqualiter, & difficile &longs;it in medio <lb/>retinere, propterea prolap&longs;us hi melius <expan abbr="retin&etilde;tur">retinentur</expan> duobus uinculis, <lb/>quàm in medio, non &longs;olum ob hanc æqualitatem, & complexum <lb/>meliorem, &longs;ed <expan abbr="etiã">etiam</expan>, quod omnes motus, omnes ponderum nixus fa <lb/>ciliùs cohibentur, & <expan abbr="deducun&ttilde;">deducuntur</expan> diui&longs;i in partes, <08> &longs;i toti contin <expan abbr="ean&ttilde;">eantur</expan>, <lb/>aut ui <expan abbr="trahãtur">trahantur</expan>. </s>
<s>Et ideo uin cula in rami cibus duplicia dextra, & &longs;ini <lb/>&longs;tra &longs;cilicet in <expan abbr="ead&etilde;">eadem</expan> parte tamë longe &longs;unt meliora etiam ferreis, quæ <lb/>&longs;olum in medio nectantur.</s></p><pb pagenum="85"/><p type="main">
<s><arrow.to.target n="marg312"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg312"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 1.</s></p><p type="main">
<s>Ex hoc etiam &longs;equitur, <lb/><figure id="fig68"></figure><lb/>quod cùm omne graue <lb/>&longs;pontè &longs;emper appropin­<lb/>quet centro mundi, & a &longs;i <lb/>moueretur per planum e, <lb/>magis remoueretur à cen­<lb/>tro mundi, ut per e c per ea <lb/>quæ diximus, & quoniam <lb/>linea ex centro mundi ad <lb/>c longior e&longs;t, quàm ad e, <lb/>multò pote&longs;t enim e&longs;&longs;e, ut <lb/>in proportione diametri <lb/>quadrati ad latus eius, & <lb/>ctiam maior. </s>
<s>ergo poterit <lb/>e&longs;&longs;e adeò parum decliuis <lb/>linea c d, ut c punctus ma­<lb/>gis di&longs;ter à centro mundi, <lb/>quàm d, & tamen feretur <lb/>ex d in c motu naturali, ut demon&longs;tratum e&longs;t, ergo per purum mo­<lb/>tum naturalem poterit a remoueri à centro mundi. </s>
<s>Hoc uolui pro­<lb/>ponere, ut intelligeres in plano uero c e non moueri a &longs;ponte, quia <lb/>c nece&longs;&longs;ariò altior e&longs;t d: &longs;i ergo mouebitur, non erit c e recta, &longs;ed <lb/>pars proportionis circuli &longs;uperficiei terræ, quæ &longs;en&longs;u à recta di&longs;tin­<lb/>gui non poterit. </s>
<s>Hoc ergo e&longs;t primum, ex quo &longs;equitur.</s></p><p type="main">
<s><arrow.to.target n="marg313"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg313"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 2.</s></p><p type="main">
<s>Quod aliquid poterit uideri decliue, in quo non de&longs;cendet imò <lb/>erit, ut potè &longs;i aliqua linea obliqua e&longs;&longs;et inter c e, & f e, illa e&longs;&longs;et decli­<lb/>uis &longs;pecie, & re, & tamen graue in illa non de&longs;cenderet, quia à cen­<lb/>tro mundi magis remoueretur: hoc tamen e&longs;t perdifficile factu, & <lb/>maximè in parua di&longs;tantia, uel etiam unius miliaris. </s>
<s>Atque hæc <lb/>in leuigatis.</s></p><p type="main">
<s>Propo&longs;itio nonage&longs;ima&longs;ecunda.</s></p><p type="main">
<s>Propprtionem ponderis æqualis iuxta longitu dinis compara­<lb/>tionem demon&longs;trare.</s></p><figure></figure><p type="main">
<s>Hoc e&longs;t, quod Archimedes reliquit </s></p><p type="main">
<s><arrow.to.target n="marg314"></arrow.to.target><lb/>intactum, cum e&longs;&longs;et maximè nece&longs;&longs;a­<lb/>rium, & o&longs;tendit magis ab&longs;tru&longs;a, &longs;ed <lb/>pace illius dixerim minus utilia. </s>
<s>Cum <lb/>ergo &longs;ump&longs;i&longs;&longs;em uirgam b f ponderis <lb/>unciarum xxiij, fui&longs;&longs;et b a uige&longs;imaquarta pars, b f fuit pondus æ­<lb/>quilibrij in b appen&longs;um librarum uiginti&longs;ex cum dimidia: fuit igi­<lb/>tur proportio ponderis e f ad pondus f b, ut tredecim ferme ad <pb pagenum="86"/>unum. </s>
<s>Et rur&longs;us feci a b quintam partem a f, & fuit a b unciarum <lb/>quatuor, & pondus quod æquauit librarum quatuor, ideò du­<lb/>plum ad pondus b f, &longs;icut c f ad c b: con&longs;tat enim quòd pondus ap­<lb/>pen&longs;um e&longs;t æquale ponderi cf. </s>
<s>Et rur&longs;us po&longs;ui b a quartam partem <lb/>b f, & fuit pondus, quod æquauit in b duæ libræ: ex quo manife­<lb/>&longs;tum e&longs;t, quòd proportio c f ad c b e&longs;t &longs;emper uelut ponderis c f ad <lb/>totam b f. </s>
<s>Et hoc e&longs;t, ac &longs;i dicamus, quòd proportio ponderis c f ad <lb/>totam e&longs;t confu&longs;a ex proportione e f ad c b, & c f, quod e&longs;t 1 p. </s>
<s>Id <lb/><arrow.to.target n="marg315"></arrow.to.target><lb/>etiam declaratum e&longs;t in primo de Subtilitate. </s>
<s>Proponatur ergo <lb/>lemma, iam &longs;ic proportio ponderis cf ad pondus b c, e&longs;t primum <lb/>ut longitu dinis cf, &longs;i e&longs;&longs;et &longs;u&longs;pen&longs;a in medio ad longitudinem b c, <lb/>quia &longs;upponuntur proportione &longs;imiles &longs;uis longitudinibus ma­<lb/>gnitudines, & pondera. </s>
<s>At c f &longs;u&longs;pen&longs;a in c, tanto e&longs;t grauior pon­<lb/>dere proprio, quanto proportionis longitudinis cf ad cb quadra­<lb/>tum, quia in &longs;e ducitur proportio: igitur proportio ponderis c f in <lb/>loco &longs;uo ad b c pondus e&longs;t confu&longs;a ex proportione longitudinis <lb/>cf ad c b, & quadratis eiu&longs;dem proportionis longitudinis cf ad c <lb/>b. </s>
<s>Sed quadratum proportionis longitudinis cf ad cb e&longs;t æquale <lb/>producto proportionis longitudinis c f in ip&longs;am c f, propterea <lb/>quòd ex proportione longitudinis cf ad cb in ip&longs;am c b fit c f, igi­<lb/>tur proportio ponderis c f ad pondus c b e&longs;t confu&longs;a ex propor­<lb/>tione ponderis c f ad pondus c b, & proportione ponderis cf alicu <lb/>ius &longs;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><margin.target id="marg314"></margin.target>C<emph type="italics"/>om.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg315"></margin.target>E<emph type="italics"/>x<emph.end type="italics"/> 18. <emph type="italics"/>diff.<emph.end type="italics"/></s></p><p type="main">
<s>Propo&longs;itio nonage&longs;imatertia.</s></p><p type="main">
<s>Propter quid in concu&longs;sione etiam leui nauis loco moueatur <lb/>o&longs;tendere. </s>
<s>Vnde manife&longs;tum e&longs;t, duas naues &longs;ibi inuicem occur&longs;an <lb/>tes retrocedere, & quantum retrocedant ambæ.<lb/><arrow.to.target n="marg316"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg316"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Proponatur, quod proportio motus grauis in a d graue in aqua <lb/>&longs;it, uelut proportio ponderis attracti in terra ad den&longs;itatem aquæ <lb/>cum profunditate, nam ubi pondus &longs;upernataret aquæ, quia aqua <lb/>e&longs;t rotunda, e&longs;t ac &longs;i tangeret in puncto. </s>
<s>Quare per demon&longs;trata &longs;u­<lb/>periùs mouebitur à quacunque ui, ergo nixus contrarius aduenit ob </s></p><p type="main">
<s><arrow.to.target n="marg317"></arrow.to.target><lb/>profunditatem, & aquæ den&longs;itatem, &longs;ed quanto aqua den&longs;ior e&longs;t, <lb/>tanto minus nauis de&longs;cendit, & quanto minus den&longs;a, tanto magis: <lb/>ergo pari modo fermè redduntur mobiles, & in aqua dulci & &longs;al&longs;a, <lb/>ubi naues &longs;int &longs;imiles forma, pondere, magnitudine. </s>
<s>Quia crgo ne­<lb/>ce&longs;&longs;e e&longs;t tabulam nauis e&longs;&longs;e duriorem, quam aqua ad re&longs;i&longs;tendum, <lb/>ergo pars maior ictus mouebit primo nauim, quam tabulam pe­<lb/>netret, cum ergo quod facilius e&longs;t, præcedat, difficilius ergo naues <pb pagenum="87"/>utrinque mouebuntur, & quia inter duos quo&longs;cunque motus contra­<lb/>rios <expan abbr="nõ">non</expan> e&longs;&longs;eos, ut utar uocabulo Auerrois quinto Phy&longs;icorum, ne­<lb/>ce&longs;&longs;e e&longs;t, ut intercedat quies media, & in quiete ab ictu, ut ui&longs;um e&longs;t <lb/>&longs;uperius, oportet, ut quod excipit ictum uelloco moueatur, uel ce­<lb/><arrow.to.target n="marg318"></arrow.to.target><lb/>dat, & ictus penetret, uel aër non conden&longs;etur ob tarditatem ultra <lb/>metam, nec retro cedere pote&longs;t ex &longs;uppo&longs;ito, & ictus e&longs;t magnus, <lb/>clarum e&longs;t, quod oportet, ut cedat, & &longs;i durum &longs;it confringatur. <lb/></s>
<s>Proportio ergo rece&longs;&longs;us ad ictum e&longs;t ut temporis, & magnitudinis <lb/>partis, quæ cedit, & retro ce&longs;&longs;us po&longs;ito ictu tanquam monade.</s></p><p type="margin">
<s><margin.target id="marg317"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 40.</s></p><p type="margin">
<s><margin.target id="marg318"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 74.</s></p><p type="main">
<s>Propo&longs;itio nonage&longs;imaquarta.</s></p><p type="main">
<s>Si quantitas aliqua nota atque proportio erit producta quantitas <lb/>nota &longs;imiliter. </s>
<s>Et &longs;i duæ proportiones notæ fuerint, erit producta <lb/>ex his atque diui&longs;a, coniunctaque, atque detracta nota. </s>
<s>Et &longs;i fuerit totius <lb/>ad partem proportio nota erit, & ad aliam partem nota, & alterius <lb/>partis ad alteram uno minor. </s>
<s>Et &longs;i fuerit partis ad partem, erit ad to <lb/>tum monade minor atque nota. </s>
<s>Et &longs;i fuerit unius quantitatis ad duas <lb/>quantitates proportio nota, erit & confu&longs;a ex eis nota. </s>
<s>Et &longs;i fuerint <lb/>trium quantitatum omiologarum, aut quatuor analogarum, o­<lb/>mnes præter unam cognitæ erunt, & illa alia cognita.</s></p><figure></figure><p type="main">
<s>Sit quantitas a b & ducta in d proportionem, <lb/><arrow.to.target n="marg319"></arrow.to.target><lb/>producat b c: dico quod duobus quibuslibet ex <lb/>his cognitis, erit cognitum tertium: nam cogni­<lb/>tum quodlibet dicitur in comparatione ad &longs;impliciter cognitum, <lb/>quod e&longs;t unum per &longs;e omnibus cognitum. </s>
<s>Ob id Arithmetica e&longs;t <lb/>prima omnium di&longs;ciplinarum, quia habet principium cognitum, <lb/>& id, quod e&longs;t, ad principium comparatum cognitum in illius com <lb/>paratione: neque aliter cognitum dici pote&longs;t. </s>
<s>Quia ergo d cognita <lb/>e&longs;t, erunt monades, & partes cognitæ in ea: aliter non e&longs;&longs;et cognita <lb/>b a, igitur cum cognita &longs;it, erit cognita per &longs;ingulas monades, quan <lb/>ta &longs;it. </s>
<s>Et &longs;i diceres quòd b a non e&longs;t cognita per partem monadis: <lb/>dico quod pars monadis non e&longs;t incognita, quia cum monades <lb/>&longs;unt cognitæ, e&longs;&longs;et d incognita. </s>
<s>Omnes enim, quod componitur ex <lb/>cognito & incognito, e&longs;t incognitum, quia cognitum &longs;olum ratio­<lb/>ne partis cognitæ. </s>
<s>Si ergo pars monadis e&longs;t cognita, erit pars a b <lb/>quælibet prout ex monade componitur &longs;impliciter cognita. </s>
<s>Su­<lb/><arrow.to.target n="marg320"></arrow.to.target><lb/>pere&longs;t, ut &longs;olum pars partis: & dico quod illa etiam e&longs;t cognita: <lb/>quia &longs;i pars ab e&longs;&longs;et, monas e&longs;&longs;et cognita: e&longs;&longs;et enim pars ip&longs;a.</s></p><p type="margin">
<s><margin.target id="marg319"></margin.target>C<emph type="italics"/>om.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg320"></margin.target>E<emph type="italics"/>x &longs;ecunda <lb/>animi com­<lb/>muni &longs;enter <lb/>tia.<emph.end type="italics"/></s></p><p type="main">
<s>Sed &longs;i &longs;it pars, erit &longs;umpta &longs;ecundum partem monadis ip&longs;ius, <lb/>ideò erit cognita iuxta nomen, uelut dimidium e&longs;t dimidium mo­<lb/>nadis, dimi dium tertiæ partis monadis e&longs;t cognitum, quia tertia <lb/>pars e&longs;t cognita, & &longs;cimus, quanta pars a&longs;&longs;umatur illius. </s>
<s>Ergo &longs;i a b, <pb pagenum="88"/>& d cognitæ &longs;unt erit & b c, quod e&longs;t primum. </s>
<s>Per hæc eadem pro­<lb/>bantur quatuor &longs;equentes partes eodem modo. </s>
<s>Sexta &longs;ic: &longs;it pro­<lb/>portio a c ad c b, nota igitur in comparatione ad monadem, &longs;ed pro <lb/>portio a c ad c b b a e&longs;t monas, igitur proportio a c ad a b nota e&longs;t, <lb/>quoniam aliter non po&longs;&longs;et dici proportio a c ad b c nota. </s>
<s>Aliter, &longs;it <lb/>proportio a c ad c b e nota, ex &longs;uppo&longs;ito igitur conuer&longs;a nota quæ <lb/>&longs;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 e&longs;t monas, f autem nota e&longs;t, igitur in comparatione ad mona­<lb/><arrow.to.target n="marg321"></arrow.to.target><lb/>dem, ergo re&longs;iduum g notum. </s>
<s>Cum uerò proportio a c ad c b com­<lb/>ponatur ex proportione a b b c ad b c, & proportio b c ad b c &longs;it <lb/>monas, & proportio a c ad b c nota, erit proportio a b ad b c cogni <lb/><arrow.to.target n="marg322"></arrow.to.target><lb/>ta, & monade minor proportione a c ad b c. </s>
<s>Per idem octaua pars <lb/><figure id="fig69"></figure><lb/>demon&longs;trabitur. </s>
<s>Inde &longs;it proportio a ad b, & ad c no­<lb/>ta, erit ergo b, & c ad a nota, quare b c ad a nota, &longs;ed <lb/><arrow.to.target n="marg323"></arrow.to.target><lb/>hæc e&longs;t conuer&longs;a ad b c confu&longs;a, igitur proportio a <lb/>ad b confu&longs;a nota e&longs;t. </s>
<s>Vltimum &longs;it, &longs;int a b c omiologæ, & &longs;int a & b <lb/><arrow.to.target n="marg324"></arrow.to.target><lb/>notæ duo, quod c nota e&longs;t, nam a b, &longs;i notæ &longs;unt, nota e&longs;t proportio <lb/>earum. </s>
<s>Ergo & proportio b ad c ergo per primam partem huius <lb/><arrow.to.target n="marg325"></arrow.to.target><lb/>cum &longs;it b nota, exit & c. </s>
<s>Et &longs;i ponantur a c notæ, dico, quòd b nota <lb/>erit: nam proportio a c ad c nota e&longs;t, quæ &longs;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"></arrow.to.target><lb/>tur nota. </s>
<s>Et &longs;imiliter in analogis &longs;int a b c notæ: & ideò erit propor­<lb/>tio a ad b nota ergo c ad d. </s>
<s>cumque c nota &longs;it, ergo per primam par­<lb/>tem huius erit d nota, quod fuit demon&longs;trandum.</s></p><p type="margin">
<s><margin.target id="marg321"></margin.target>P<emph type="italics"/>er demon­<lb/>&longs;trat.<emph.end type="italics"/> 12. <lb/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg322"></margin.target>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><margin.target id="marg323"></margin.target>E<emph type="italics"/>x demon&longs;t.<emph.end type="italics"/><lb/>12. P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg324"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 14. <lb/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg325"></margin.target>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><margin.target id="marg326"></margin.target>E<emph type="italics"/>x<emph.end type="italics"/> 2. A<emph type="italics"/>nimi <lb/>&longs;ententia.<emph.end type="italics"/></s></p><p type="main">
<s>Propo&longs;itio nonage&longs;imaquinta.</s></p><p type="main">
<s>Cuiu&longs;uis trigoni rectanguli, aut cuius duo anguli &longs;int in dupla <lb/>proportione, aut qui circulo in&longs;criptus &longs;it cognita quantitate uni­<lb/>us lateris in comparatione ad dimetientem &longs;i proportio <expan abbr="duorũ">duorum</expan> la­<lb/>terum cognita fuerit, erunt omnia eius latera cognita.<lb/><arrow.to.target n="marg327"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg327"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Non de cognitione propinqua <expan abbr="a&longs;tronomorũ">a&longs;tronomorum</expan>, de qua abundè ab <lb/>Heber tractatum e&longs;t, &longs;ed de exacta, de qua &longs;uperius egi nunc &longs;ermo </s></p><p type="main">
<s><arrow.to.target n="marg328"></arrow.to.target><lb/>e&longs;t: &longs;it igitur primum a b c trigonus orthogonius: & &longs;it a rectus, & <lb/>proportio <expan abbr="duorũ">duorum</expan> laterum cognita, dico, quod omnia latera cognita <lb/><arrow.to.target n="marg329"></arrow.to.target><lb/><figure id="fig70"></figure><lb/>erunt: nam &longs;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/>e&longs;t 1 p: cognita erit, quare & a b ad a c, & <expan abbr="eod&etilde;">eodem</expan> modo a c ad b c: quod <lb/>e&longs;t primum. </s>
<s>Exemplum, ponatur b c dupla a b, erit a b quadratum <lb/>&longs;ub quadruplum quadrato a b quare &longs;ubtriplum quadrato a cigi­<pb pagenum="89"/>tur &longs;i a b ponatur 1 b c erit 2, & a c <02> 3. Rur&longs;us ponatur angulus b <lb/>duplus angulo c quali&longs;cunque &longs;it, erit per demon&longs;trata &longs;uperius pro­<lb/>portio a b b c ad a c, ut a c ad a b, &longs;i igitur nota &longs;it proportio a c ad <lb/>a b, erit nota proportio a b b c ad a b per præcedentem. </s>
<s>Ergo per <lb/>eandem omnia nota &longs;cilicet b c ad b a, & b c ad c a. </s>
<s>Et &longs;i e&longs;&longs;et nota <lb/>proportio a b ad b c, dico, quod e&longs;&longs;ent nota omnia, nam nota e&longs;&longs;et <lb/>a b, & b c, & quod fit ex a b in ip&longs;um aggregatum. </s>
<s>Sed hoc e&longs;t æ­<lb/><arrow.to.target n="marg330"></arrow.to.target><lb/>quale quadrato a c, igitur notum e&longs;t quadratum a c ergo a c: igitur <lb/>proportio a b b c ad a c, & a c ad a b. </s>
<s>Vt &longs;i a b e&longs;&longs;et 4 b c 5, e&longs;&longs;et a b b c <lb/>9 ducta in a b, quæ e&longs;t, fit 36, cuius latus e&longs;t b a c &longs;cilicet. </s>
<s>Et &longs;i e&longs;&longs;et <lb/>trigonus aliquis in cir culo, cuius proportio duorum laterum &longs;it co <lb/>gnita ad dimetientem relata, &longs;equitur per demon&longs;trata &longs;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><margin.target id="marg328"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 97.</s></p><p type="margin">
<s><margin.target id="marg329"></margin.target>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><margin.target id="marg330"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 17. <emph type="italics"/>&longs;ex <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/><lb/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 17.</s></p><p type="main">
<s>Multa præterea cognita e&longs;&longs;ent in hoc genere, quæ nunc præter­<lb/><arrow.to.target n="marg331"></arrow.to.target><lb/>mitto, quia non &longs;unt ad finem nece&longs;&longs;aria. </s>
<s>Alia præterea per diligen­<lb/>tem inqui&longs;itionem maioris artis quàm alias edidimus. </s>
<s>tum uerò <lb/>etiam per nouas demon&longs;trationes.</s></p><p type="margin">
<s><margin.target id="marg331"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Propo&longs;itio nonage&longs;ima&longs;exta.</s></p><p type="main">
<s>Cum in per&longs;picuum den&longs;um radij lumino&longs;i in ciderint, quatuor <lb/>fiunt luminis genera.<lb/><arrow.to.target n="marg332"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg332"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Sit &longs;ol a, & per&longs;picuum den&longs;um, exempli gratia, ut ampula <lb/>magna aqua plena b c d, & &longs;i &longs;it rotunda accendit ignem ex ad­<lb/>uer&longs;o ut in e. </s>
<s>Dico ergo in b c d e&longs;&longs;e quatuor genera luminis. </s>
<s>Pri­<lb/>mum quod e&longs;t ualidius, & rectà tran&longs;it, ualidius enim e&longs;t, quod <lb/>tran&longs;it quàm quod tran&longs;ire non pote&longs;t, & etiam quia, ut dixi, <lb/>ignem accen dit. </s>
<s>Secundum e&longs;t quod colligitur in ampula, & dein­<lb/>de &longs;pargitur <expan abbr="circũcircà">circuncircà</expan>, nam id ualidius e&longs;t, quia penetrat, & re&longs;ilit <lb/>quàm quod non penetrat, aut &longs;i penetrat, non &longs;pargitur, & hoc dif­<lb/>funditur circa uas, necreflectitur rectè, &longs;ed qua&longs;i intro colligitur, & <lb/>diuer&longs;a ratione diffunditur, e&longs;t tamen imbecillius primo, ut dictum <lb/>e&longs;t. </s>
<s>Tertium genus e&longs;t, quod illuminat intus ingrediendo, &longs;ed non <lb/>&longs;pargitur, & hoc e&longs;t debilius &longs;ecundo, quia <expan abbr="nõ">non</expan> pote&longs;t &longs;pargi. </s>
<s>Quar­<lb/><figure id="fig71"></figure><lb/>tum e&longs;t, quod non ingreditur omnino, &longs;ed refle­<lb/>ctitur, i&longs;tud e&longs;t ab&longs;que dubio imbecillimum, quo­<lb/>niam penetrare non pote&longs;t. </s>
<s>Et licet in &longs;peculis <lb/>concauis radius reflexus uideatur e&longs;&longs;e ualidior, <lb/>&longs;tatim enim accendit ignem, hoc non contin­<lb/>git, ni&longs;i quia in &longs;peculo cauo radij omnes col­<pb pagenum="90"/><expan abbr="ligun&ttilde;">liguntur</expan> ob <expan abbr="opacũ">opacum</expan>, quod à tergo e&longs;t, neque <expan abbr="&longs;pargun&ttilde;">&longs;parguntur</expan>, neque <expan abbr="tran&longs;eũt">tran&longs;eunt</expan>, neque<lb/>combibuntur, ut ita dicam &longs;ed omnes <expan abbr="reflectũtur">reflectuntur</expan>. </s>
<s>Ex quo colligitur <lb/>quin cuplex ordo radiorum iuxta rationem uirium, primus e&longs;t refle <lb/><expan abbr="xorũ">xorum</expan> à &longs;peculo <expan abbr="cõcauo">concauo</expan>, & hi &longs;unt <expan abbr="pot&etilde;ti&longs;simi">potenti&longs;simi</expan> ob <expan abbr="ration&etilde;">rationem</expan> <expan abbr="dictã">dictam</expan>, po&longs;t <lb/>quos &longs;unt radij, qui tran&longs;eunt per per&longs;picuum maximè rotundum, <lb/>qui & ip&longs;i generant ignem, & debiliorem primo, deinde reliqui <lb/>tres &longs;equentes &longs;upradicti. </s>
<s>Sextus e&longs;t radiorum, qui reflectuntur à <lb/>rebus non nitidis, ut à muris, & tabulis, nam omnia dura reflectunt <lb/>& etiam mollium pleraque, & hæc reflexio e&longs;t fermè infinita, & ob id <lb/>cubicula etiam in angulis illuminantur.</s></p><p type="main">
<s><arrow.to.target n="marg333"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg333"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 1.</s></p><p type="main">
<s>Ex hoc &longs;equitur, quòd Luna remittit lumen, non reflectit, nam <lb/>&longs;ecus non illuminaret to tum orbem, &longs;ed &longs;olum portionem oppo­<lb/>&longs;itam Soli, & hoc etiam rarò, ergo combibitur, & illu&longs;trat circun­<lb/>circa ubique.</s></p><p type="main">
<s><arrow.to.target n="marg334"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg334"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 2.</s></p><p type="main">
<s>In &longs;tellis lumen Solis pertran&longs;it aliter, &longs;i reflecteretur, non illumi­<lb/>naret nos, aut apparerent, uelut cometæ, quia pars una e&longs;&longs;et clarior <lb/>reliqua, & &longs;i conbiberent lumen, non uiderentur æquè claræ, cum <lb/>Sol e&longs;&longs;et propinquus, aut remotus.</s></p><p type="main">
<s><arrow.to.target n="marg335"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg335"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 3.</s></p><p type="main">
<s>Luna tota intus illuminatur à Sole, quoniam &longs;i ante coniun­<lb/>ctionem illuminatur à &longs;ini&longs;tra parte, & combibit lumen per cor­<lb/>rolarium primum, & po&longs;t coniunctionem illuminatur à dex­<lb/>tra, & combibit pariter lumen, ergo e&longs;t tota naturæ per&longs;picuæ, &longs;ed <lb/>uidetur ob&longs;cura ex aduer&longs;o, propterea quòd radij ualidiores refle­<lb/>xi illu&longs;trant illam ex parte Solis, diffugiunt à contraria, quod ma­<lb/>nife&longs;tè apparet in ampula expo&longs;ita Soli. </s>
<s>Pars enim clarior uer&longs;us <lb/>Solem uidetur, quam ex aduer&longs;o, hoc autem longè magis in Luna <lb/>ob di&longs;tantiam.</s></p><p type="main">
<s><arrow.to.target n="marg336"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg336"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 4.</s></p><p type="main">
<s>In omni Solis eclip&longs;i fit colectio radiorum ad a&longs;pectum, & <lb/>ideo in regione illa, in qua centrum Solis integitur à centro Lunæ, <lb/>& ubicunque fit, fit in cendium per tertium corrolarium. </s>
<s>Hoc autem <lb/>fit &longs;emper in quauis coniunctione, & dum Luna &longs;ilet in regione ae­<lb/>ris, &longs;ed terris non &longs;e cundùm centrum, uerùm ad latitudinem, & ad <lb/>Orientem ante coniunctionem cum Sole, & ad Occidentem po&longs;t: <lb/>&longs;ed centra non &longs;unt in linea ui&longs;us.</s></p><p type="main">
<s><arrow.to.target n="marg337"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg337"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 5.</s></p><p type="main">
<s>Ex hoc &longs;equitur, quod oportet &longs;ub&longs;tantiam Lunæ e&longs;&longs;e ualde cla­<lb/>ram, cum uideamus ab ampula tam paruum lumen diffundi, & ra­<lb/>rum, à Luna uerò in uniuer&longs;um orbem, & tam copio&longs;um, ut nece&longs;­<lb/>&longs;arium &longs;it &longs;ub&longs;tantiam Lunæ e&longs;&longs;e den&longs;am, & lucidam ualde.</s></p><p type="head">
<s>SCHOLIVM.</s></p><p type="main">
<s>Et &longs;i quis dicat, quòd &longs;i in cendium illud fieri po&longs;&longs;et in hora ecli­<lb/>p&longs;is, &longs;equeretur, quòd ut in ampula in medio Lunæ uideretur ma­<pb pagenum="91"/>gnus &longs;plendor, referens corpus Solis. </s>
<s>Propterea dico, quòd uel ac­<lb/>eidit, quia homo non pote&longs;t ea hora intueri Solem, & etiam e&longs;t im­<lb/>peditus à radijs circum&longs;tantibus, cuius indicio e&longs;t, quod in &longs;pe­<lb/>culo po&longs;ito in aqua, &longs;imile uidetur &longs;tellulæ in centro Lun&ecedil;: & hic e&longs;t <lb/>&longs;plen dor Solis collectus in centro Lunæ. </s>
<s>po&longs;&longs;et etiam dici, quòd <lb/>Luna circa medium propter maculam non admitteret lumen, & ita <lb/>e&longs;&longs;et inæqualium partium.</s></p><p type="main">
<s>Propo&longs;itio nonage&longs;ima&longs;eptima.</s></p><p type="main">
<s>Motum inuer&longs;ionis in figuris in comparatione ad motum &longs;phæ <lb/>ræ in plano inue&longs;tigare.</s></p><p type="main">
<s><arrow.to.target n="marg338"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg338"></margin.target>C<emph type="italics"/>om.<emph.end type="italics"/></s></p><p type="main">
<s>Voco motum inuer&longs;ionis, qui &longs;imilis e&longs;t motui &longs;phæræ, &longs;cili­<lb/>cet circumuertendo graue à uertice, & manife&longs;tum e&longs;t, quòd in <lb/>quacunque figura, qua graue in&longs;idet plano per punctum ue­</s></p><p type="main">
<s><arrow.to.target n="marg339"></arrow.to.target><lb/>lut ouata ip&longs;um mouetur à quauis ui, &longs;ed &longs;i in&longs;ideat per &longs;uperfi­<lb/>ciem, quanto maior e&longs;t, & humilior, tanto difficilius mouetur, <lb/>ideò in corpore uiginti ba&longs;ium, quòd inter regularia uocata, plu­<lb/>res habet, &longs;uperficies pro ratione æqualis ponderis, motus erit <lb/>longe facilior. </s>
<s>Alia cau&longs;a e&longs;t inæqualitas partium, unde quæ ro­<lb/>tunda &longs;unt, quia prominent, facile mouentur, & cum partes me­<lb/>diæ in&longs;i&longs;tant plano, quanto minores erunt tanto facilius moue­<lb/>buntur ratione ponderis. </s>
<s>Vnde patet, quòd corpora ouata faci­<lb/>lius mouentur, etiam quàm &longs;phærica, habent enim partem me­<lb/>diam minorem, & paria &longs;unt ratione ince&longs;&longs;us plani, &longs;ed aëris mul­<lb/>titudine tardius, quoniam enim &longs;phæra &longs;ub æquali ambitu plus <lb/>continet corporis, ergo ouatum æquale &longs;phæræ habet maio­<lb/>rem ambitum ip&longs;a &longs;phæra. </s>
<s>Hæc autem à Theone partim de­<lb/>mon&longs;trata &longs;unt, partim ab Archimede, & partim à nobis, ergo <lb/>motus ouati e&longs;t fermè æqualis motui &longs;phæræ, & tardior e&longs;t con­<lb/><figure id="fig72"></figure><lb/>citatus, quàm &longs;phæræ, quia à ma­<lb/>iore excipitur aëre, & partes exte­<lb/>riores non ita incumbunt in me­<lb/>dium &longs;ecundum longitudinem. </s>
<s>Cu­<lb/>bus uero tardior e&longs;t propter æqua­<lb/>litatem, & latitudinem &longs;uperficiei in­<lb/>ferioris, omnium <expan abbr="aut&etilde;">autem</expan> minime pro­<lb/>pter has cau&longs;as conus ambligonius, <lb/>& quanto magis fuerit, ratio uero <lb/>eleuationis e&longs;t, ut &longs;it cubus b c, cuius <lb/>medium grauitatis &longs;it b &longs;uper pla­<pb pagenum="92"/>no de, & eleuetur ex a, & manife&longs;tum e&longs;t, quod in&longs;idebit per totam <lb/>lineam c f ip&longs;i plano, & proportio grauitatis totius &longs;u&longs;pen&longs;i in com <lb/>paratione ad grauitatem eius, qui inuertit, e&longs;t, uelut proportio par­<lb/>tis terminatæ ad lineam c f uer&longs;us eum, qui eleuat ad partem, quæ <lb/>ultra e&longs;t, cum uerò hæ partes notæ &longs;int iuxta perpendiculum ex <lb/>centro grauitatis, manife&longs;tum e&longs;t, quod &longs;ciemus pondus corporis <lb/>a b cf, dum inuertitur in quo cunque &longs;itu ad pondus eius, dum &longs;u­<lb/>&longs;penditur, & clarum e&longs;t, quòd cùm centrum, & medium grauitatis <lb/>fuerint in una linea per c f, tunc nulla erit grauitas.</s></p><p type="margin">
<s><margin.target id="marg339"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 40.</s></p><p type="main">
<s>Propo&longs;itio nonage&longs;imaoctaua.</s></p><p type="main">
<s>Proportionem ponderum æqualium per differentiam angulo­<lb/>rum inuenire.<lb/><arrow.to.target n="marg340"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg340"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Sit a b, quæ &longs;i appen&longs;a e&longs;&longs;et ad æquidi­<lb/><figure id="fig73"></figure><lb/>&longs;tantem terræ &longs;uperficiei, nulla ui po&longs;&longs;et ele </s></p><p type="main">
<s><arrow.to.target n="marg341"></arrow.to.target><lb/>uari, inflectatur ergo ad c punctum, omi&longs;&longs;a <lb/>c g, & manife&longs;tum e&longs;t, quod &longs;i b c in&longs;i&longs;teret <lb/><arrow.to.target n="marg342"></arrow.to.target><lb/>ad perpendiculum, ponderaret a c &longs;i e&longs;&longs;et in <lb/>æquilibrio, ponatur ergo accliuis in c d per <lb/>notum angulum. </s>
<s>Quia igitur b c ad c a no­<lb/>ta e&longs;t, erit dicta &longs;uperiùs notum pondus <lb/>b h, po&longs;ita h c æquali c a, quare totius a b, <lb/>& iam fuit e k notum, & punctus d notus: <lb/>hoc enim infrà demon&longs;trabitur, qualis igitur proportio lineæ <lb/><arrow.to.target n="marg343"></arrow.to.target><lb/>tran&longs;uer&longs;æ dl ad lineam de&longs;cendentem d m, talis differentiæ pon­<lb/>derum c m, & c e, id e&longs;t partis ad partem. </s>
<s>hæc autem inferiùs de­<lb/>mon&longs;trabuntur. </s>
<s>Neque enim ab&longs;urdum e&longs;t in materijs mi&longs;tis, ali­<lb/><arrow.to.target n="marg344"></arrow.to.target><lb/>quando uti nondum demon&longs;tratis cum fuerint mathematica, quia <lb/>obtinent principij rationem, quod etiam facit Archimedes. </s>
<s>Ma­<lb/>nife&longs;tum e&longs;t autem, quod in angulo m c d recti dimidio, propor­<lb/>tio media erit. </s>
<s>Sed hoc bifariam contingere pote&longs;t &longs;cilicet, ut &longs;it <lb/>media, per quantitatem, & per proportionem, e&longs;t autem media, ut <lb/><arrow.to.target n="marg345"></arrow.to.target><lb/>demon&longs;trabitur infrà &longs;ecundum proportionem l d ad l e, propo­<lb/>natur ergo c e b, erit latus quadrati <02> 72, igitur latus octogoni e&longs;t <lb/><02> v: 72 m: <02> 2592, & latus re&longs;idui <02> v: 72 p: <02> 2592. quadrata er­<lb/>go partium ba&longs;is differunt in <02> 10368. Quare partes ba&longs;is &longs;unt <lb/>6 p: <02> 18, & 6 m: <02> 18 &longs;cilicet l e, l d autem e&longs;t <02> 18, igitur differen­<lb/>tia, & proportio e&longs;t, qualis <02> 18 ad 6 m: <02> 18 fermê, ut 17 ad 7, & ta­<lb/>lis e&longs;t proportio ponderis c d ad pondus c e ratione in crementi, <lb/>&longs;eu differentiæ. </s>
<s>Vt &longs;i pondus in c e e&longs;&longs;et decem librarum in c in <pb pagenum="93"/>quadraginta erit in c d triginta unius cum quarta, &longs;ed proportionis <lb/>ratione e&longs;&longs;et uiginti octo cum tertia.</s></p><p type="margin">
<s><margin.target id="marg341"></margin.target>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&longs;.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg342"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 86. <lb/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg343"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 99.</s></p><p type="margin">
<s><margin.target id="marg344"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 97.</s></p><p type="margin">
<s><margin.target id="marg345"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 98.</s></p><p type="main">
<s>Propo&longs;itio nonage&longs;imanona.</s></p><p type="main">
<s>Proportionem grauitatum per multitudinem &longs;uppo&longs;itorum or <lb/>bium o&longs;tendere.<lb/><arrow.to.target n="marg346"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg346"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Omne, quod mouetur, mouetur &longs;ecundum naturam ponderis, <lb/>quæ in attractione, ut demon&longs;tratum e&longs;t, æqualis e&longs;t dimidio &longs;u­<lb/>&longs;pen&longs;i, cum ergo diuidatur in multiplices partes motus uniu&longs;cuiu&longs;­<lb/>que, e&longs;t &longs;ecundum dimidium illius partis, ut, &longs;i &longs;int &longs;ex rotæ in cur­<lb/>ru det, quod uehitur, &longs;it pondus &longs;exaginta librarum, unaquæ que </s></p><p type="main">
<s><arrow.to.target n="marg347"></arrow.to.target><lb/>rota habet pondus quinque librarum, &longs;cilicet diui&longs;o triginta per <lb/>&longs;ex, & quia quod cunque mouetur &longs;phæricè non habet pondus, <lb/>ni&longs;i quantum premitur axis, ideò pondus &longs;exaginta librarum in <lb/>uehendo red ditur læ&longs;us, quanto proportio producta minor e&longs;t <lb/>additione. </s>
<s>Exemplum, &longs;it deductum pondus &longs;exaginta librarum <lb/>per &longs;ex rotas ad uigintiquatuor, quia &longs;i rotæ po&longs;&longs;ent circumduci, <lb/>ut in inuer&longs;ione dictum e&longs;t, & e&longs;&longs;ent æquales, & in &longs;olido æquali, <lb/>ac duro, nulla ui mouerentur, &longs;ed qua&longs;i per &longs;e, ergo &longs;uppo&longs;ito pon­<lb/>dere uiginti quatuor librarum a&longs;&longs;umemus unamquamque partem, <lb/>& ducemus eam in &longs;eip&longs;am, &longs;cilicet detraham quintam partem ex <lb/>toto 30, fit 24, duc 30 in &longs;e, fit 900, duc 24 in &longs;e, fit 576, proportio ut <lb/>25 ad 16, at diui&longs;o 30 in &longs;ex partes, fit 5, detrahe quintam partem, fit <lb/>4, duc in &longs;e, fit 16, duc in &longs;ex, fit 96, igitur proportio 900 ad 96 e&longs;t ut <lb/>25 ad 2 2/3, quod ergo erat 16 factum e&longs;t 2 2/3, proportio ergo de­<lb/>cre&longs;centis maior e&longs;t diui&longs;o per plura. </s>
<s>Sed plerunque additis ro­<lb/>tis cre&longs;cit pondus nihilo &longs;ecius, redditur etiam leuius. </s>
<s>Sed & de <lb/>hoc in &longs;equenti.</s></p><p type="margin">
<s><margin.target id="marg347"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 40.</s></p><p type="main">
<s>Propo&longs;itio cente&longs;ima.</s></p><p type="main">
<s>Proportionem grauitatis ponderum attractorum per trochlea­<lb/>rum numerum inue&longs;tigare.<lb/><arrow.to.target n="marg348"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg348"></margin.target>C<emph type="italics"/>om.<emph.end type="italics"/></s></p><p type="main">
<s>Ari&longs;toteles in Mechanicis cen&longs;et cau&longs;am leuitatis trochlearum </s></p><p type="main">
<s><arrow.to.target n="marg349"></arrow.to.target><lb/>e&longs;&longs;e in pondere eleuando, quòd pondera auxilio uectium facilius <lb/>mouentur, quàm manibus. </s>
<s>Rotulæ uerò in trochleis uectes &longs;unt, <lb/>& axis mi&longs;ta hypomochlij, ergo facilius pondus trahitur per u­<lb/>nam rotulam, quàm &longs;i manu traheretur, at uerò per duas tres, <lb/>unde tris pa&longs;&longs;us longe facilius, & etiam facilius per quinque, unde <lb/>pentas pa&longs;&longs;us, nam quinque orbiculis, qua&longs;i totidem uectibus <lb/>diui&longs;um pondus manife&longs;tè fit leuius, & ut dictum e&longs;t, tanquam <lb/>totidem uectibus pondus eleuatur, e&longs;tqúe proportio produ­<pb pagenum="94"/>cta, &longs;emperque prior hypomochlij locum habet, ueruntamen minus <lb/>a&longs;&longs;umit laboris, po&longs;terior uerò uectis maiorem partem &longs;ibi ponde­<lb/>ris &longs;eruat, uelut in &longs;uccula etiam iugum traiectum per plures colo­<lb/>pes facilius uertitur. </s>
<s>Et &longs;i quis dicat nónne totum pondus in&longs;idet <lb/>prim&ecedil; trochleæ per trochleam, intelligo nunc &longs;olùm rotulam cum <lb/>ip&longs;o axe, &longs;eu axiculo (ut dicunt) non autem in proprio &longs;ignificato, <lb/>in quo etiam funis traiectus, & in&longs;idens rotulæ, &longs;eu rotulis, nam <lb/>una trochlea plures continere'pote&longs;t orbiculos, & axes. </s>
<s>Licet ergo <lb/>pondus in&longs;ideat primæ trochleæ, &longs;eu rotulæ, in eo tamen, quod tra <lb/>hitur, diuiditur', licet non æqualiter dico, præter id funis motum <lb/>intendi. </s>
<s>nam motus actionem auget, & ideò quanto longior, eo fa­<lb/>cilius mouet ob con cu&longs;sionem, demum quia leuis e&longs;t rotula circa <lb/>axem, ut plus uecte po&longs;sit.</s></p><p type="margin">
<s><margin.target id="marg349"></margin.target>I<emph type="italics"/>n<emph.end type="italics"/> M<emph type="italics"/>echan.<emph.end type="italics"/><lb/>Q<emph type="italics"/>uæ&longs;t.<emph.end type="italics"/> 18.</s></p><p type="main">
<s>Propo&longs;itio cente&longs;imaprima.</s></p><p type="main">
<s>Proportionem precij gemmarum ex tribus in eodem genere co <lb/>gnitis inuenire.<lb/><arrow.to.target n="marg350"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg350"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Solent gemmarij uendere adamantem ponderis unius grani <lb/>uno coronato, duorum autem granorum tribus coronatis, qua­<lb/>tuor autem, gratia exempli, quadraginta coronatis, qu&ecedil;ritur quan­<lb/>tum ualebit adamas octo granorum, quoniam ergo proportio <lb/>non &longs;eruatur. </s>
<s>E&longs;t enim in pondere utraque dupla, in precio autem <lb/>ex prima habetur tripla, ex &longs;ecunda habetur proportio maior, <lb/>quàm tredecim ad unum, propterea utendum e&longs;t proportione <lb/>propinquiori, &longs;i &longs;atis faceret. </s>
<s>gratia exempli, in prima ad ditione fuit <lb/>unum granum, & acqui&longs;iuit proportionem triplam, in &longs;ecunda fue <lb/>runt duo grana, &longs;i ergo acqui&longs;i&longs;&longs;et &longs;olum &longs;excuplam proportio­<lb/>nem, haberemus intentum. </s>
<s>Propterea in i&longs;to ca&longs;u oportet demon­<lb/>&longs;trare forma Geometrica, &longs;uppo&longs;ito, quòd &longs;it figura recta ex uno la <lb/><figure id="fig74"></figure><lb/>tere a b, ita ut angulus, uel minimus capiat b c æqualem a b, & ex <lb/>æquali 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>Siue &longs;it parabole, &longs;iue hiperbole, &longs;eu &longs;it alia coincidentium.</s></p><pb pagenum="95"/><p type="head">
<s>SCHOLIVM.</s></p><p type="main">
<s>Et nota, quòd &longs;i res hæc e&longs;&longs;et naturalis, o&longs;tenderet infinitum in <lb/>rebus ex regula dialectica, &longs;ed quia ex <expan abbr="uolũtaria">uoluntaria</expan>, nullas habet uires.</s></p><p type="main">
<s>Propo&longs;itio cente&longs;ima&longs;ecunda.</s></p><p type="main">
<s>Proportionem motuum inuer&longs;ionis, & attractionis in plano <lb/>inuenire.</s></p><p type="main">
<s>Et &longs;it, ut aliquid inuertatur, declaratum autem e&longs;t &longs;uprà, quid &longs;it </s></p><p type="main">
<s><arrow.to.target n="marg351"></arrow.to.target><lb/>inuer&longs;io, & quàm diuer&longs;a &longs;it rur&longs;us, & quòd attractio e&longs;t dimidium <lb/><arrow.to.target n="marg352"></arrow.to.target><lb/>ponderis eleuati. </s>
<s>Cum ergo con&longs;tet in inuer&longs;ione, quanta &longs;it pro­<lb/>portio ponderis &longs;u&longs;pen&longs;i ad pondus inuer&longs;um, & pondus &longs;u&longs;pen&longs;i <lb/><arrow.to.target n="marg353"></arrow.to.target><lb/>&longs;it duplum ponderi attracti, &longs;equitur, ut diuifa proportione ponde <lb/>ris &longs;u&longs;pen&longs;i ad pondus inuer&longs;um per medium cogno&longs;catur propor<lb/>tio attractionis ad inuer&longs;ionem.</s></p><p type="margin">
<s><margin.target id="marg351"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>_{m}.</s></p><p type="margin">
<s><margin.target id="marg352"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 89.</s></p><p type="margin">
<s><margin.target id="marg353"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 62.</s></p><p type="main">
<s>Ex hoc &longs;equitur, quod aliquod pondus trahi pote&longs;t, quod non <lb/><arrow.to.target n="marg354"></arrow.to.target><lb/>pote&longs;t inuerti, hoc autem indigetlonga declaratione, quam doce­<lb/>bimus inferiùs: & tamen attigit hocrarò.</s></p><p type="margin">
<s><margin.target id="marg354"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Propo&longs;itio cente&longs;imatertia.</s></p><p type="main">
<s>Proportionem eorundem in accliui demon&longs;trare.</s></p><p type="main">
<s>Dupliciter pote&longs;t intelligi, uel de&longs;cendendo, uel a&longs;cendendo. <lb/><arrow.to.target n="marg355"></arrow.to.target><lb/><arrow.to.target n="marg356"></arrow.to.target><lb/>Sed ego nunc loquor de a&longs;cen&longs;u, contraria ratione intelliges de <lb/>de&longs;cen&longs;u, & circa inuer&longs;ionem demon&longs;trata e&longs;t proportio eius <lb/>iuxta angulum a&longs;cen&longs;us, & &longs;imiliter declarabitur de proportione <lb/><arrow.to.target n="marg357"></arrow.to.target><lb/>attractionis iuxta eundem angulum a&longs;cen&longs;us, & nuper declarata <lb/>e&longs;t proportio inuer&longs;ionis in plano ad attractionem, ex quibus &longs;e­<lb/>quitur per ea, quæ dicam inferius, quòd proportio cuiu&longs;uis mobi­<lb/>lis inuer&longs;i ad attractum &longs;ub quibu&longs;cun que angulis nota erit.</s></p><p type="margin">
<s><margin.target id="marg355"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s><margin.target id="marg356"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 72.</s></p><p type="margin">
<s><margin.target id="marg357"></margin.target>I<emph type="italics"/>n &longs;equenti.<emph.end type="italics"/></s></p><p type="main">
<s>Propo&longs;itio cente&longs;imaquarta.</s></p><p type="main">
<s>Proportionem motus attractionis in decliui ad motum in pla­<lb/>no determinare.</s></p><p type="main">
<s>Si ab accliue, &longs;eu decliue in quo d ad attra­<lb/><arrow.to.target n="marg358"></arrow.to.target><lb/><arrow.to.target n="marg359"></arrow.to.target><lb/><figure id="fig75"></figure><lb/>hendum, cuius nota e&longs;t ex &longs;uperioribus dif­<lb/>ficultas in plano ratione figuræ con&longs;tante, er­<lb/>go ea quæritur proportio a&longs;cen&longs;us, & quo­<lb/>niam terminus ad perpendiculum e&longs;t dupla <lb/>proportio, & iam grauitas in plano e&longs;t dimidium, ideò quicquid <lb/>acquiritur in eleuatione e&longs;t in comparatione ad illud dimidium, <lb/>cum ergo attractio &longs;ecundum eandem proportionem augeatur, er­<lb/>go &longs;emper maior difficultas augebitur, ergo ab initio minimum <pb pagenum="96"/>erit di&longs;crimen ab attractione in plano. </s>
<s>Exempli gratia &longs;it, ut graue d <lb/>in plano &longs;it, ut quin que, & &longs;u&longs;pen&longs;um decem, ergo in medio angulo <lb/>erit penè &longs;eptem, &longs;ed &longs;eptem minus longe <expan abbr="di&longs;tãt">di&longs;tant</expan> à quin que, quàm de­<lb/>cem ad &longs;eptem, ergo in &longs;ecunda parte plus longè augebitur difficul <lb/>tas attractionis &longs;upra difficultatem in medio angulo accliui, quam <lb/>in prima parte à plano ad medium accliue, & quoniam planum in <lb/>plano de&longs;cendit, tanto uehementius, quanto difficilius attrahitur, <lb/>ergo planum in decliui &longs;ublimi longe maiore impetu feretur infrà <lb/>quam &longs;it proportio anguli ad angulum. </s>
<s>Exempli gratia, planum in <lb/>medio angulo, &longs;i incipiat de&longs;cendere in dodrante multo lentius, <lb/>quàm pro dimidio uirium de&longs;cen&longs;us totius anguli, imò initium de­<lb/>&longs;cen&longs;us e&longs;t à medio recti ad unguem, ubi omnia plana &longs;int, & duri&longs;­<lb/>&longs;ima, & cau&longs;a huius e&longs;t, quia omne graue tendit ad centrum, quòd <lb/>maior pars ip&longs;ius grauis e&longs;t ultra medium grauitatis in decliui <lb/>humiliore.</s></p><p type="margin">
<s><margin.target id="marg358"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s><margin.target id="marg359"></margin.target>E<emph type="italics"/>x<emph.end type="italics"/> 62. & <lb/>64. P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/></s></p><p type="main">
<s>Propo&longs;itio cente&longs;imaquinta.</s></p><p type="main">
<s>Proportionem ferentium pondus in pertica inuenire.</s></p><figure></figure><p type="main">
<s>Hæc proponitur etiam à Philo&longs;o­<lb/><arrow.to.target n="marg360"></arrow.to.target><lb/>pho, & ponatur ab, & &longs;i pondus &longs;it in <lb/><arrow.to.target n="marg361"></arrow.to.target><lb/>medio d grauat æqualiter utrunque, <lb/>nam in hoc con&longs;entit experimentum <lb/>cum ratione, at uerò &longs;i ponatur in cita, <lb/>ut b c &longs;it tripla b a uiderentur a & b, tanquam hypomochlia, & pon <lb/><arrow.to.target n="marg362"></arrow.to.target><lb/>dus ip&longs;um b, ut grauior e&longs;&longs;et cb, quam c a. </s>
<s>Ari&longs;toteles, &longs;eu author <lb/>ille hoc uidens bifariam re&longs;pondet: primum quòd hoc e&longs;t inuer­<lb/><arrow.to.target n="marg363"></arrow.to.target><lb/>&longs;um in&longs;trumentum, cum in cæteris motor &longs;it ex aduer&longs;o hypomo­<lb/>chlij, hic in ip&longs;o, ge&longs;tans enim mouet & hypomochlij in&longs;tar e&longs;t hu­<lb/>merus. </s>
<s>At hoc uerum non e&longs;t: quod mouet enim e&longs;t pondus, & e&longs;t <lb/>in c: nam a, & contingit moueri: quia &longs;i &longs;tarent, idem &longs;equeretur. </s>
<s>Se­<lb/>cunda re&longs;pon&longs;io e&longs;t, quod utrun que premit &longs;cilicet ferentes & pon­<lb/>dus, & quòd qui longior e&longs;t ab hypomochlio facilius mouet, & <lb/>redit ad idem fermè: nam in c con&longs;tituitur, quod moueri debet, ca­<lb/>pita uectium &longs;unt a, & b: motus autem e&longs;t ip&longs;um &longs;u&longs;tinere pondus. <lb/></s>
<s>At hoc non uidetur, quoniam ratio, qua uectis longior facilius mo<lb/>uet, e&longs;t ambitus magnitudo, ob quam motus redditur tardior, & <lb/>ideo leuior: igitur non e&longs;t hoc uerum de motu occulto, &longs;icut e&longs;t gra<lb/>uis prementis, &longs;ed circumducente, cum in occulto uelut in &longs;tatera <lb/>contrarium accidere do cuerimus aliâs. </s>
<s>Quidam dixere b premere <lb/>c uer&longs;us a, a contrà uer&longs;us b, & ideò grauari magis a àb, quàm b ab <lb/>a, quia maiorem uim habet b e, quàm a c. </s>
<s>I&longs;tud fal&longs;um e&longs;t bifariam. <lb/></s>
<s>Primum, quia & &longs;i a, & b &longs;int in æquilibrio, ut nec unus in alterum <pb pagenum="97"/>in cumbat, necimpellat, &longs;ed tantum &longs;u&longs;tineat nihilo&longs;ecius res uera <lb/>e&longs;t. </s>
<s>Et etiam quia non e&longs;t uerum, quòd qui longius in cumbit, ma­<lb/>iorem uim inferat. </s>
<s>Propterea dicendum e&longs;t, quòd qui ex commu­<lb/>nibus propria nituntur demon&longs;trare, omnes corrumpunt di&longs;cipli­<lb/>nas. </s>
<s>Nihil deterius e&longs;t his mon&longs;tris. </s>
<s>Nam et&longs;i hæc ratio uera e&longs;&longs;et: <lb/>non tamen reddit cau&longs;am, quia non e&longs;t ex proprijs principijs. </s>
<s>Dico <lb/>ergo, quod &longs;i c de&longs;cendat in e, per perpendiculum de&longs;cendet, igitur <lb/>d b e&longs;t longior d a, quare angulus e a b maior e b a: igitur pondus c <lb/>plus de&longs;cendit comparatione a, quàm b, ergo plus grauat cip&longs;um a <lb/>quàm b, &longs;eu ex cau&longs;a, quod magis premat, &longs;eu ex effectu, quòd ma­<lb/>gis de&longs;ce&longs;&longs;erit. </s>
<s>Cau&longs;a ergo erroris e&longs;t, quod &longs;i ponatur angulus f b a <lb/>æqualis angulo f a b, & ponatur b f &ecedil;qualis b c, tun c in eodem tem­<lb/>pore, in quo tran&longs;it dimidium c in e, tran&longs;ibit aliud dimidium c in f. <lb/></s>
<s>quia &longs;eparat&ecedil; partes grauiores &longs;unt in c b, quàm c a, propter di&longs;tan­<lb/>tiam ab hypomochlio, &longs;ed tunc uelo cius mouentur, & angulus fit <lb/>&ecedil;qualis. </s>
<s>Sed quando pondus e&longs;t unum, & c de&longs;cendit ad e, cum de­<lb/>&longs;cendat inæquali tempore, & peragat maiorem angulum compa­<lb/>ratione a, quam b, &longs;equitur, ut uelo cius moueatur comparatione a <lb/>quàm b. </s>
<s>Ergo &longs;i non mouetur, cum omnis potentia &longs;it &longs;imilis actui, <lb/>tum quia ab eo producitur, & effectus e&longs;t &longs;imilis cau&longs;æ: tum quia <lb/>e&longs;t initium actus, igitur etiam quod a b non in clinetur, nec de&longs;cen­<lb/>dat, grauius erit pondus, comparatione a quàm b, quod erat de­<lb/>mon&longs;trandum.</s></p><p type="margin">
<s><margin.target id="marg360"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s><margin.target id="marg361"></margin.target>Q<emph type="italics"/>us&longs;t.<emph.end type="italics"/> 59. <lb/>M<emph type="italics"/>echanic.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg362"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 45.</s></p><p type="margin">
<s><margin.target id="marg363"></margin.target>P<emph type="italics"/>rop.<emph.end type="italics"/> 103.</s></p><p type="main">
<s>Ex hoc &longs;equitur, quòd aliqua iuncta erunt grauiora re&longs;pectu u­<lb/>nius, quæ erunt mutato ordine diui&longs;a leuiora. </s>
<s>Quoniam diui&longs;a, <lb/>quæ longius di&longs;tant æqualem, aut maiorem angulum faciunt, iun­<lb/>cta minorem.</s></p><p type="main">
<s>Propo&longs;itio cente&longs;ima&longs;exta.</s></p><p type="main">
<s>Quales proportiones angulorum doceant laterum proportio­<lb/>nes. </s>
<s>At que uici&longs;sim determinare.</s></p><p type="main">
<s>Sit circulus a b c, cuius dimetiens, nota b d &longs;it b, erit ergo latus <lb/><arrow.to.target n="marg364"></arrow.to.target><lb/><figure id="fig76"></figure><lb/>exagoni a b dimidium b d, id e&longs;t 3. igitur <lb/>cum angulus a &longs;it rectus, erit a d <02> 27 latus <lb/>trianguli. </s>
<s>Et latus quadrati per eandem <02><lb/>18. Vt latus exagoni &longs;it <02> 9. Quadrati <02> 18 <lb/>Trianguli <02> 27, & ita pote&longs;tate &longs;e habent <lb/>hæc ut 1. 2. 3. Et &longs;unt nota. </s>
<s>Et quia latus d e c <lb/>agoni e&longs;t <02> 11 1/4 m, 1 1/2. & ip&longs;um erit notum. <lb/></s>
<s>Quare latus pentagoni e&longs;t <02> v 22 1/2 m: <02><lb/>101 1/4 notum. </s>
<s>Et iam notum fuit latus epta­<lb/>goni. </s>
<s>Habebimus igitur latera Trianguli <pb pagenum="98"/>quadrati pentagoni, & eptagoni æquilaterorum nota: & etiam <lb/>&longs;ubten&longs;orum duobus ex his. </s>
<s>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, quæ e&longs;t 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 e&longs;&longs;et, ac &longs;i an­<lb/>gulus b rectus e&longs;&longs;et: &longs;ed quia e&longs;t obtu&longs;us, ideo a c e&longs;t alia linea, & <lb/>maior latere pentagoni. </s>
<s>Et &longs;imiliter &longs;i a b, & a c notæ e&longs;&longs;ent, utpo­<lb/><arrow.to.target n="marg365"></arrow.to.target><lb/>te a b 3, ut prius a c 5 dico, quòd b c nota e&longs;t: 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>igitur 30 m: pos <02> 27 æquantur <02> 324 m: 9 quad. </s>
<s>quare <lb/>900 p: 27 quad. </s>
<s>m: pos <02> 97200 <expan abbr="æquãtur">æquantur</expan> 324 m: 9 quad. </s>
<s>igitur 576 <lb/>p: 16 quad. </s>
<s>&ecedil;quantur pos <02> 97200. Quadratum igitur p: 36 &ecedil;quan­<lb/>tur pos <02> 379 11/16, erit ergo b c <02> v: <02> 94 59/64 p: <02> 58 59/64 & &longs;imiliter &longs;i a c <lb/>&longs;it nota, puta 4 erit a b &longs;ubten&longs;a dimidio arcus a c nota. </s>
<s>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 &longs;implicibus trianguli quadrati. </s>
<s>Pentagoni, & eptagoni in <lb/>numeras linearum magnitudines in circulo. </s>
<s>Et &longs;imiliter quouis mo <lb/>do, ut dictum e&longs;t, in quauis figura æquilatera, utpote &longs;uppo&longs;ito <lb/><figure id="fig77"></figure><lb/>quod de&longs;criptum &longs;it nonangulum in <lb/>circulo æquilaterum, quod etiam erit <lb/>æquiangulum, & &longs;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 &longs;excuplus a b c, & triplus a c b. <lb/></s>
<s>Erit ergo per demon&longs;trata proportio <lb/><arrow.to.target n="marg366"></arrow.to.target><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 &longs;up­<lb/>po&longs;ito maior e&longs;t proportione rectæ 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 rectæ ad re­<lb/>ctam minor e&longs;t. </s>
<s>Sit rur&longs;us in triangulo b e d quomodolibet modo <lb/>&longs;it angulus b d e quadruplus angulo b e d, & diuidatur d per &ecedil;qua­<lb/>lia ducta d f, erit igitur proportio f d, d e ad f e, ut e f ad f d, &longs;ed e f ad <lb/><arrow.to.target n="marg367"></arrow.to.target><lb/>f b ut d e ad d b. </s>
<s>igitur proportio b d, d e ad f b <expan abbr="cõpo&longs;ita">compo&longs;ita</expan> ex propor­<lb/>tionibus e f ad f d, & e d ad d b. </s>
<s>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>Rur&longs;us ponamus, <lb/><arrow.to.target n="marg368"></arrow.to.target><lb/>quod in quadrangulo a b c d primæ figuræ &longs;it a b 4 b c 3 c d 5 ad 6 <lb/>dico, quòd &longs;pacium contentum erit notum. </s>
<s>Ductis rectis a c & b d <pb pagenum="99"/>quomodolibet, ut &longs;e &longs;ecent in e, erunt anguli d c a, & d b a æquales, <lb/><arrow.to.target n="marg369"></arrow.to.target><lb/>quia in ea&dacute;em portione circuli a d, & anguli a d e &ecedil;quales, quia con <lb/>tra &longs;e po&longs;iti. </s>
<s>igitur trianguli a b e, & c d e &longs;imiles, & proportio d c ad <lb/><arrow.to.target n="marg370"></arrow.to.target><lb/>a b, ut c e ad b e, c d autem fuit 5 a b 4, igitur &longs;i b e ponatur 4 pos c e <lb/>erit 5 pos. </s>
<s>Per ea&longs;dem, & eodem modo a d ad b c ut d e ad e c. </s>
<s>igitur <lb/>po&longs;ita c e 5 pos erit e d 10 pos, tota igitur d b 14 pos. </s>
<s>Et quoniam ea­<lb/><arrow.to.target n="marg371"></arrow.to.target><lb/>dem proportio a e ad e b per eadem, & e b fuit 4 pos: igitur a e e&longs;t 8 <lb/>pos, quare a e 13. po&longs;t productum igitur ex a c in d b, e&longs;t 182 quad. <lb/></s>
<s>& hoc æquatur productis a b in c d, quod e&longs;t 20, & b c in a d quod <lb/>e&longs;t 18, totum igitur e&longs;t 38, igitur res e&longs;t <02> 19/91. Quare not&ecedil; erunt lineæ <lb/>b e, e d, a e, & e c, &longs;ed &longs;ufficit, ut cognita &longs;it a c, uel b d. </s>
<s>Per regulam <lb/>enim triangulorum erunt notæ areæ a b c, & a d e, quare tota &longs;uper­<lb/>ficies a b c d. </s>
<s>Et e&longs;t inuentum Scipionis Ferri Bononien&longs;is de quo <lb/>aliâs. </s>
<s>Pote&longs;t etiam inuenta a c uel b d haberi &longs;uperficies facilius <lb/>per catheros.</s></p><p type="margin">
<s><margin.target id="marg364"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>_{m}.</s></p><p type="margin">
<s><margin.target id="marg365"></margin.target>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><margin.target id="marg366"></margin.target>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><margin.target id="marg367"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 3. <emph type="italics"/>&longs;exti<emph.end type="italics"/><lb/>E<emph type="italics"/>Elem.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg368"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 23. <emph type="italics"/>&longs;ex <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg369"></margin.target>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><margin.target id="marg370"></margin.target>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><margin.target id="marg371"></margin.target>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>Sit modo obtu&longs;i angulus a b c, & nota latera &longs;ingula, & angu­<lb/>lus a b c, & producantur latera ad perpendicu­<lb/><figure id="fig78"></figure><lb/>lum, ut &longs;int d & e recti, & quia anguli ad a &longs;unt <lb/>æquales, erunt anguli e b a, & d e a &longs;emper æ­<lb/><arrow.to.target n="marg372"></arrow.to.target><lb/>quales. </s>
<s>Et hoc idem contingit in acuti angulis <lb/>triangulis intus, & e&longs;t utile mechanicum: & <lb/>quia a b c notus e&longs;t, & d notus, erunt anguli tri <lb/>goni d b c noti: & &longs;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: & &longs;emper notum, quod fit ex b a <lb/>in a d, uel c a in a e, &longs;unt enim &ecedil;qualia inter &longs;e: etiam notæ ad & a c, <lb/>quoniam duplum horum e&longs;t exce&longs;&longs;us quadrati b c &longs;uper quadrata <lb/>a b, & a c. </s>
<s>Quod uerò proponiturà Monteregio de cognitione an­<lb/>gulorum in triangulis non e&longs;t intelligendum, ut uerba &longs;ignificant, <lb/><arrow.to.target n="marg373"></arrow.to.target><lb/>&longs;ed &longs;olum de cognitione quoad u&longs;um tabularum.</s></p><p type="margin">
<s><margin.target id="marg372"></margin.target>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><margin.target id="marg373"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 12. <emph type="italics"/>&longs;e­<lb/>cundi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="main">
<s>Et iterum ponamus, quòd proportio a c c b ad a b &longs;it qualis a b <lb/>ad a c, dico quòd angulus c duplus e&longs;t angulo b. </s>
<s>Si non ducatur c d <lb/><figure id="fig79"></figure><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>Maior e&longs;t <expan abbr="aut&etilde;">autem</expan> <lb/>d c, quàm a c, aut æqualis, aut minor, &longs;i æqualis, <lb/>igitur maior proportio d c c b ad b d quàm b a, <lb/>igitur maior proportio b d ad d c quam b a ad a c <lb/>ad a c & æquales &longs;unt igitur b d maior d a pars toto, quod e&longs;&longs;e non <lb/>pote&longs;t. </s>
<s>Si uerò d c ponatur maior a c, magis ex hoc &longs;equitur b d ma­<lb/>iorem e&longs;&longs;e b a. </s>
<s>Quod &longs;i minor &longs;it d c quàm a c. </s>
<s>Ex demon&longs;tratio­<lb/>ne ip&longs;ius reflexæ proportionis patet hoc contingere non po&longs;&longs;e. <lb/></s>
<s>Et &longs;imiliter patet conuer&longs;as in reliquis etiam ueras e&longs;&longs;e, non &longs;olum <pb pagenum="100"/>in proportionibus noti&longs;simis angulorum &longs;ed etiam in coniuncti­<lb/>one & detractione. </s>
<s>Et e&longs;t ex &longs;ubtili&longs;simis operationibus, quæ ho­<lb/>mini in hoc genere eueniant.</s></p><p type="main">
<s>Propo&longs;itio cente&longs;ima&longs;eptima.</s></p><p type="main">
<s>Si in circulo duo diametri ad rectum angulum &longs;e &longs;ecauer int: ali&ecedil; <lb/>uerò ad perpendiculum ex diametro exierint ad circumferentiam, <lb/>&longs;ingulæ &longs;upra diametrum erunt maiores portionibus reliquis dia­<lb/>metri &longs;uperioribus, infra autem minores. </s>
<s>Dimidium autem porti­<lb/>onis &longs;uperioris re&longs;iduum ad centrum maius &longs;agitta habebit. </s>
<s>In ali­<lb/>qua præterea portionis &longs;uperioris parte, quæ uer&longs;us diam etrum <lb/>tran&longs;uer&longs;um po&longs;ita e&longs;t, maior e&longs;t differe ntia partis diametri ei cor­<lb/>re&longs;pondentis, quam lineæ tran&longs;uer&longs;æ.</s></p><figure></figure><p type="main">
<s>Sint du&ecedil; diametri a b, c d ad perpendi <lb/>culum &longs;ecantes &longs;e in centro, & <expan abbr="ducũtur">ducuntur</expan> <lb/>&longs;upr f g k h, & infra m l ad perpendicu­<lb/>lum &longs;upra a b: dico f g e&longs;&longs;e maiorem f a, <lb/>& k h k a, & contrà minorem m l, quàm <lb/>m a. </s>
<s>Per octauam enim &longs;exti, quod fit ex <lb/><arrow.to.target n="marg374"></arrow.to.target><lb/>b f in f a æquale e&longs;t <expan abbr="&qtilde;drato">quadrato</expan> f g, &longs;ed b f e&longs;t <lb/>maior f g, quia b f e&longs;t maior c b, & ideo <lb/>e c g f, ergo f g maior e&longs;t f a, m l <expan abbr="aũt">aunt</expan> minor e&longs;t per <expan abbr="ead&etilde;">eadem</expan> e c, quare e a, <lb/>multo igitur minor m a, quod e&longs;t primum. </s>
<s>Suppo&longs;ito etiam, quòd <lb/><arrow.to.target n="marg375"></arrow.to.target><lb/>a g arcus &longs;it dimidium a c, dico a f <expan abbr="minor&etilde;">minorem</expan> e&longs;&longs;e f e, nam quadratum e <lb/><arrow.to.target n="marg376"></arrow.to.target><lb/>g æquale e&longs;t quadratis f e, & f g, & <expan abbr="quadratũ">quadratum</expan> a g quadratis f g & f a <lb/>& e g e&longs;t &ecedil;qualis lateri exagoni, & a g latus octogoni, igitur e g ma­<lb/><arrow.to.target n="marg377"></arrow.to.target><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 propo&longs;itum.<lb/><arrow.to.target n="marg378"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg374"></margin.target>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><margin.target id="marg375"></margin.target>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><margin.target id="marg376"></margin.target>1. <emph type="italics"/>eiu&longs;dem.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg377"></margin.target>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><margin.target id="marg378"></margin.target>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>Cum rur&longs;us ex prima parte huius line&ecedil; f g & k h &longs;int maiores f a, <lb/>& k a & ea &longs;it æqualis e c, nece&longs;&longs;e e&longs;t ut iuxta punctum c augeatur </s></p><p type="main">
<s><arrow.to.target n="marg379"></arrow.to.target><lb/>magis linea in ea, quam &longs;it differentia lineæ tran&longs;uer&longs;æ ad lineam <lb/>tran&longs;uer&longs;am per communem animi &longs;ententiam, quod e&longs;t tertium.</s></p><p type="margin">
<s><margin.target id="marg379"></margin.target>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>Propo&longs;itio cente&longs;imaoctaua.</s></p><p type="main">
<s>Punctum &ecedil;qualitatis differenti&ecedil; de&longs;cen&longs;us, & remotionis à cen­<lb/>tro inuenire.</s></p><p type="main">
<s>Per præcedentem moto puncto a uer&longs;us c &longs;emper u&longs; que ad e, c ma <lb/><arrow.to.target n="marg380"></arrow.to.target><lb/>gis di&longs;tat <expan abbr="pũctum">punctum</expan> a linea a e, quàm à puncto a uer&longs;us, quia linea n h <lb/>maior e&longs;t n a, & per eandem dum appropinquat ad c cum e c fiat <lb/>&ecedil;qualis ea, maius fit in crementum in a e, quàm re&longs;pectu lineæ tran&longs;­<lb/>uer&longs;alis. </s>
<s>Volo ergo inuenire punctum hoc in quo fit mutatio: & <lb/>diuido arcum ac per æqualia in f, & dico illum e&longs;&longs;e punctum quæ­<lb/>&longs;itum: accepto quouis puncto in e f, puta k, duco g o h p &ecedil;quidi&longs;tan <pb pagenum="101"/><figure id="fig80"></figure><lb/>tes a b, & c d: erunt que anguli q & n recti <lb/><arrow.to.target n="marg381"></arrow.to.target><lb/>& anguli f e a, & f e c &ecedil;quales, igitur uter <lb/><arrow.to.target n="marg382"></arrow.to.target><lb/>que dimidium recti: igitur per dicta in <lb/>primo Elementorum Euclidis e n &ecedil;qua <lb/><arrow.to.target n="marg383"></arrow.to.target><lb/>lis n k, igitur c q æqualis e n, quare h p <lb/>æqualis g o, &longs;ed quod fit ex o k in k g e&longs;t <lb/><arrow.to.target n="marg384"></arrow.to.target><lb/>æquale ei, quod fit ex p k in k h, igitur <lb/><arrow.to.target n="marg385"></arrow.to.target><lb/>k h e&longs;t æqualis k g ex eisdem o&longs;tendi­<lb/>tur f l m k quadratum e&longs;&longs;e. </s>
<s>Quia ergo <lb/>k h e&longs;t æqualis k g, & k l æqualis k m, erit l g æqualis m h. </s>
<s>Er­<lb/>go de&longs;cendendo ex g in f, quantum f l &longs;uperat l g, tantum de&longs;cen­<lb/>dendo ex f in h, f m &longs;uperat m h per communem animi &longs;ententi­<lb/>am. </s>
<s>At f m e&longs;t de&longs;cen&longs;us f in linea a e, & m h di&longs;tantia, quæ acqui­<lb/>ritur in linea f r, n m enim e&longs;t æqualis f r, igitur n h excedit f r in <lb/>h m, & ita a n excedit a r in n r &ecedil;quali f m. </s>
<s>Quantum ergo in g f, <lb/>l f excedit l g, tantum in de&longs;cen&longs;u ex f in h, f m, quæ refert g l, ex­<lb/>cedit h m, quæ refert f l. </s>
<s>Arcus autem f g e&longs;t æqualis arcui f h, <lb/>quod <expan abbr="cũ">cum</expan> po&longs;&longs;em o&longs;tendere pluribus modis &longs;atis con&longs;tat, quia chor <lb/><arrow.to.target n="marg386"></arrow.to.target><lb/>darum illorum quadrata &longs;unt inuicem æqualia, quia lineæ f m, & <lb/><arrow.to.target n="marg387"></arrow.to.target><lb/>f l item que m h & l g &longs;unt æquales, & anguli m, & l recti. </s>
<s>Igitur cum <lb/>ad quod uis punctum in linea e f &longs;emper linea de&longs;cen&longs;us in parte <lb/>inferiore e&longs;t maior linea di&longs;tantiæ tanto, quanto per æqualem ar­<lb/>cum in &longs;uperiore linea di&longs;tantiæ e&longs;t maior linea, de&longs;cen&longs;us &longs;equitur <lb/>per regulam Dialecticam quod punctus f, e&longs;t punctus &ecedil;qualitatis. <lb/></s>
<s>Per idem diceremus in quarta parte inferiore.</s></p><p type="margin">
<s><margin.target id="marg380"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s><margin.target id="marg381"></margin.target>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><margin.target id="marg382"></margin.target>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><margin.target id="marg383"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 32. <lb/>& 6.</s></p><p type="margin">
<s><margin.target id="marg384"></margin.target>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><margin.target id="marg385"></margin.target>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><margin.target id="marg386"></margin.target>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><margin.target id="marg387"></margin.target>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>Propo&longs;itio cente&longs;imanona.</s></p><p type="main">
<s>Rationem libræ expendere.</s></p><p type="main">
<s>Cum libra moueatur, uelut rota circa axem, quia trutina manet, <lb/>ideò &longs;i pondus ponatur, dum iugum fuerit in linea a b nihil mo­<lb/>uebitur, quia appetitus de&longs;cen&longs;us ex puncto a maximus e&longs;t, & ni­<lb/>hil iuuat motum extra naturam, idem dico de graui po&longs;ito inuerti­<lb/>ce b a. </s>
<s>Nam duo &longs;unt motus in rota, & in libra unus, per quem <lb/>dum fertur per arcum a f, gratia exempli de&longs;cendit, quantum e&longs;t <lb/><arrow.to.target n="marg388"></arrow.to.target><lb/>a r, quæ e&longs;t minor dimidio e r, & ideò minor e r, quæ e&longs;t maior di­<lb/>midio, ut demon&longs;tratum e&longs;t, & etiam minor r f, quæ æqualis e&longs;t r e <lb/><arrow.to.target n="marg389"></arrow.to.target><lb/>per demon&longs;trata rur&longs;us: & hic e&longs;t naturalis ut palam e&longs;t: alter præ­<lb/>ter <expan abbr="naturã">naturam</expan>, & e&longs;t ferri ad latus, quoniam hoc e&longs;t <expan abbr="propriũ">proprium</expan> immortali­<lb/>bus: cun que hic &longs;it ad latus e&longs;t etiam <expan abbr="cõtra">contra</expan> naturam, quia magis di&longs;tat <lb/>a centro, nam e f e&longs;t longior c r, &longs;i ergo r ferretur in f, moueretur à <lb/>centro, & contra naturam. </s>
<s>Dum ergo fertur ex a in f, multo lentius <pb pagenum="102"/>fertur, quàm ex f in c: uelo cius autem ex c u&longs;que ad medium: nam <lb/>plurimum de&longs;cendit. </s>
<s>Ex h ad b autem celerrimè, quoniam de&longs;cen­<lb/>dit, & appropinquat lineæ a b, ut uter que motus &longs;it naturalis. </s>
<s>Non <lb/>ergo mouetur pr&ecedil;ter naturam ni&longs;i quatenus longius recedit à linea <lb/>a b, unde in inferiore parte mouetur ad eandem, ideò de parte c b <lb/>tota per&longs;picua e&longs;t ratio, cur facillimè de&longs;cendat, &longs;imiliter & tota, <lb/>hoc enim e&longs;t demon&longs;tratum. </s>
<s>Similiter & quare difficillimè feratur <lb/>ex b u&longs; que ad p, & ultra p u&longs; que ad directum r f: at de motu ex a in f, <lb/>quod debeat ferri, quia plus remouetur, quam de&longs;cendat, nulla e&longs;t <lb/>ratio: ut nec cur ex oppo&longs;ito f ad a difficilem &longs;e præ&longs;tet: & hoc e&longs;t, <lb/>quia tertiam rationem etiam ip&longs;e Ari&longs;toteles, & qui eum &longs;equuti <lb/>&longs;unt, prætermi&longs;it. </s>
<s>Ea autem e&longs;t, quod dum fertur ad g, uel f etiam li­<lb/>cet non de&longs;cendat magis, quàm remoueatur, ex a <lb/><figure id="fig81"></figure><lb/>ad centrum terræ tamen magis appropinquat. <lb/></s>
<s>Quia enim e a e&longs;t &ecedil;qualis e c, quoniam prodeunt <lb/>à centro circuli eiu&longs;dem, & b e, & e c &longs;unt maio­<lb/>res b c, ideò b a erit maior b c, e&longs;t autem b cen­<lb/><arrow.to.target n="marg390"></arrow.to.target><lb/>trum mundi, ergo a motum ad c, appropin qua­<lb/>uit ip&longs;i b</s></p><p type="margin">
<s><margin.target id="marg388"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 98.</s></p><p type="margin">
<s><margin.target id="marg389"></margin.target>I<emph type="italics"/>n præceden <lb/>ti.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg390"></margin.target>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>Dico etiam quod libra ex chalybe tenui&longs;simo, <lb/>& quanto <expan abbr="leuiorũ">leuiorum</expan> concharum, & longioris iugi <lb/>10 exactior, quoniam lances illæ minori exce&longs;&longs;u <lb/>mouentur, quia plus di&longs;tant ab hypomochlio. <lb/></s>
<s>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>Con&longs;tat, quod <lb/>qualis proportio f g ad a b, talis ambitus, ad ambitum: motus er­<lb/>go &longs;i &longs;it æqualis utrarumque, igitur a tanto minore proportione <lb/><figure id="fig82"></figure><pb pagenum="103"/>mouebitur in h, quam in d, uelut &longs;it proportio f g ad a b dupla, ut <lb/>ergo æqualiter moueantur, &longs;i &longs;it dupla &longs;exquiquarta in d cum lan­<lb/>ce ad e uacuam, erit in h &longs;exquialtera, & mouebit æquali tempore. <lb/></s>
<s>Ergo iuxta hoc fient libræ, quæ examinabunt decimam, & uige&longs;i­<lb/>mam partem grani, quod e&longs;t nece&longs;&longs;arium in precio&longs;is rebus, & me­<lb/>dicamentis potentibus, & longè magis in mechanicis experimen­<lb/>tis, & maximè quæ ad demon&longs;trationem pertinent magnitudinis <lb/>&longs;uperficierum, & con&longs;tat res in tribus, in longitudine, f g iungi, in le <lb/>uitate materiæ illius, & lancium, nam tanto maior redditur propor<lb/>tio ponderis exigui, & in firmitate iugi ac rectitudine. </s>
<s>ideò debet <lb/>fieri ex chalybe purgato, durato ac tenui&longs;simo, natura que leui, & ut c <lb/>&longs;it in medio, & mobilis f g.</s></p><p type="main">
<s>Con&longs;iderandum e&longs;t demum an f l & g m &longs;int grauiores f h, & <lb/>g k. </s>
<s>Vt enim grauiores extiterint minus facilè mouentur. </s>
<s>Viden­<lb/>tur autem mihi, qui de his con&longs;crip&longs;erunt perperam contemp&longs;i&longs;&longs;e <lb/>hoc, con&longs;tat enim, quòd dum l de&longs;cendit, remouetur a b n c tru­<lb/>tina, & m, quæ a&longs;cendit contra appropinquat. </s>
<s>Videtur autem hoc <lb/>bifariam contra naturam: nam ut diximus pondus applicat &longs;e ad <lb/>rectam n c, quia uer&longs;us centrum, & etiam quia facit angulum ob­<lb/>tu&longs;um, cum deberet, ut ab initio &longs;altem con&longs;tituere cum iugo re­<lb/>ctum. </s>
<s>Et de m nihil mirum e&longs;t, cum acutum, ut &longs;e ad lineam, quæ ad <lb/>centrum retrahat. </s>
<s>Huiu&longs;modi præterij&longs;&longs;e Ari&longs;totelem, demiror, <lb/>quæ nimis fuerunt in con&longs;picuo, ut dubitem ne non &longs;uus &longs;it ille li­<lb/>ber, qui eius penè nihil &longs;apiat præter ob&longs;curitatem. </s>
<s>Tentan­<lb/>dum e&longs;t igitur horum cau&longs;as a&longs;signare. </s>
<s>nam quæ huiu&longs;modi po­<lb/>te&longs;t e&longs;&longs;e doctrina ni&longs;i perfecta fuerit, in omnibus etenim nece&longs;&longs;e e&longs;t <lb/>aut omnia &longs;cire, aut ignorare. </s>
<s>In hoc igitur dico, quod h f, &longs;eu l f, <lb/>&longs;emper æquidi&longs;tant n c trutinæ, ergo cum angulus f c n in clina­<lb/>to iugo fiat obtu&longs;us de&longs;cendente pondere, & n c g a&longs;cendente pon­<lb/>dere fiat acutus, ergo angulus l f c tantundem fiet obtu&longs;ior, & m g c <lb/>acutior, quanto anguli ad c tales &longs;unt. </s>
<s>Et cau&longs;a e&longs;t quia n c ratio­<lb/>ne ponderis e&longs;t directa ad centrum, ergo oportet, ut pondera l, uel <lb/>h, & m, uel k, &longs;i debent tendere ad centrum, ut f l, & g m æquidi­<lb/>&longs;tent n c, ni&longs;i quantum e&longs;t pro di&longs;tantia f, à puncto c, & g a b eodem, <lb/>quæ comparata ad <expan abbr="centrũ">centrum</expan> terr&ecedil;, &longs;eu mundi, e&longs;t in&longs;en&longs;ibilis omnino. <lb/></s>
<s>Circa hæc <expan abbr="notandũ">notandum</expan> i&longs;tud mirabile fcilicet, quod ratio motus, quan­<lb/>tumuis exigua &longs;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>Et quae graue, quod expers e&longs;t &longs;en&longs;us, debeat &longs;equi ratio <lb/>nem Geometricam uix &longs;apientibus <expan abbr="cognitã">cognitam</expan>, cau&longs;a tamen una e&longs;t, & <lb/>per&longs;picua: <expan abbr="nã">nam</expan> omne graue e&longs;t in linea à centro <expan abbr="mũdi">mundi</expan>: &longs;i <expan abbr="aũt">aunt</expan> medium <lb/>grauis &longs;it extra <expan abbr="lineã">lineam</expan>, uertitur ad illam, qu&ecedil; e&longs;t in eo, nam <expan abbr="centrũ">centrum</expan> &longs;em <pb pagenum="104"/>per e&longs;t in <expan abbr="ead&etilde;">eadem</expan>. </s>
<s>Ergo &longs;ola in clinatio ad hoc ut <expan abbr="mediũ">medium</expan> grauis &longs;it in li­<lb/>nea <expan abbr="centrorũ">centrorum</expan> grauitatis & terræ, &longs;ufficit. </s>
<s>E&longs;t ergo principium in &longs;ei­<lb/>p&longs;o. </s>
<s>In appen&longs;is &longs;imiliter. </s>
<s>Trutina enim, & finis iugi, & grauis <expan abbr="cen-trũ">cen­<lb/>trum</expan> mundi <expan abbr="centrũ">centrum</expan> &longs;unt in <expan abbr="ead&etilde;">eadem</expan> linea, ut e&longs;&longs;e po&longs;&longs;unt, cum exigua illa <lb/>& &longs;ola di&longs;tantia intercedat. </s>
<s>& hoc e&longs;t primum. </s>
<s>Quia ergo <expan abbr="iugũ">iugum</expan> e&longs;t <lb/>ex materia &longs;olida, mouetur ratione, quæ dicta e&longs;t, lances autem <lb/>oportet cum filis appen&longs;i &longs;int, ut puncta f & h, uell, & g k, uel g m <lb/>&longs;int in una linea cum centro terræ. </s>
<s>Et quia l magis di&longs;tat a b f quam <lb/>h, & m a g magis, quam k, & oportet faciant eandem inclinatio­<lb/>nem, quia anguli trutinæ cum iugó &longs;unt ijdem, & linea cl e&longs;t ma­<lb/>ior c h, & c m, quàm c k in quouis &longs;itu, ergo &longs;patium, quod ambitur, <lb/>e&longs;t maius ergo per d e mon&longs;trata &longs;uperius l e&longs;t grauius h etiam <lb/>præter uinculorum additionem, & m grauius k. </s>
<s>Quanto igi­<lb/>tur longiores &longs;unt funiculi à libræ extremitate &longs;eu iugi, tanto gra­<lb/>uius redditur pondus, quod tamen multi putant e&longs;&longs;e fal&longs;um: nec <lb/>aliquid referre, quòd &longs;it longum, aut breue &longs;u&longs;tentaculum.</s></p><p type="main">
<s>Propo&longs;itio cente&longs;imadecima.</s></p><p type="main">
<s>Si duæ &longs;phæræ ex eadem materia de&longs;cendant in <expan abbr="a&etilde;">aem</expan> <lb/>re eodem temporis momento ad planum ueniunt.<lb/><figure id="fig83"></figure><lb/><arrow.to.target n="marg391"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg391"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Supponitur quod ex eodem loco. </s>
<s>Sermo enim <lb/>ab&longs;urda &longs;ub interpretatione nunquam ni&longs;i ab inui­<lb/>dio&longs;o, uel imperito intelligi debet. </s>
<s>Sit ergo a tripla <lb/>ad b, &longs;phærula ad &longs;phærulam ex plumbo ambæ fer­<lb/>ro uel lapide eiu&longs;dem generis, dico, quòd inæquali <lb/>tempore peruenient ad planum c d. </s>
<s>Nam a propor­<lb/>tionem habet ad b, ut uiginti&longs;eptem ad unum. </s>
<s>pro­<lb/>portio autem &longs;patij a ad &longs;patium b nonupla e&longs;t, & <lb/>proportio den&longs;itatis aëris ad aërem e&longs;t tripla, propterea quod den­<lb/>&longs;itas illa multiplicatur propter impetus magnitudinem. </s>
<s>nam &longs;i ro­<lb/>bur, ut decem percutiat baculo lato, ut quatuor ictus erit maior du­<lb/>plo, quàm &longs;it robur, ut quinque percutiat baculo, ut duo: propter <lb/>den&longs;itatem ergo maiorem aëris in a, quam in b: & quoniam &longs;i &longs;ub <lb/>maiore impetu mouetur <expan abbr="a&etilde;r">aerr</expan> &longs;ub a, quam &longs;ub b, igitur proportio <lb/>erit comparanda longitudini à centro a ad longitudinem a centro <lb/>b, quæ e&longs;t tripla. </s>
<s>Si ergo &longs;ubtripla e&longs;t ratio motus b ad a, quod <lb/>ad medium attinet, tripla autem propter uelo citatem di&longs;ce&longs;&longs;us aë­<lb/>ris à medio grauitatis, quod e&longs;t in &longs;uperficie e regione centri graui­<lb/>tatis in linea ad centrum mundi, ut dictum e&longs;t in præcedenti: mani­<lb/>fe&longs;tum e&longs;t, quod a, & b inæquali tempore peruenient ad &longs;ubie­<lb/>ctum planum, & æquidi&longs;tans centris eorum. </s>
<s>Similiter & in aqua: <pb pagenum="105"/>cum uerò uideatur in illa tanto celerius a de&longs;cendere, quàm b, <lb/>quanto e&longs;t &longs;emidiameter a longior &longs;emidiametro b, liquet ex hoc, <lb/>quod æquali uelo citate de&longs;cendunt, &longs;ed ob uelo citatem motus in <lb/>aëre latet di&longs;crimen anticipationis contactus &longs;oli a ante b, qui di­<lb/>gno&longs;citur in aqua, ex quo patet exactam e&longs;&longs;e æqualitatem. </s>
<s>Sed re&longs;i­<lb/>liunt &longs;emel in aqua ambæ, cum pluries in aëre a &longs;olo, quare etiam in <lb/>aqua perturbatur cognitio in parum accuratis, at que &longs;en&longs;u præditis, <lb/>&longs;icut etiam in ca&longs;u, ne altera alteram perueniat, utra que comprehen&longs;a <lb/>duobus digitis, altera alteram tangente, & u&longs;que ad centrum in <lb/>aquam demi&longs;sis &longs;imul digitis dilatatis dimittendæ &longs;unt.</s></p><p type="main">
<s>Propo&longs;itio cente&longs;imaundecima.</s></p><p type="main">
<s>Cur ex medio tela ualidiorem ictum, & naues in &longs;calmo à remo, <lb/>ac malo recipiant inde ex puppi explorare.</s></p><p type="main">
<s>Ari&longs;toteles uidetur in Mechanicis, & qui eum &longs;equuti &longs;unt, ui­</s></p><p type="main">
<s><arrow.to.target n="marg392"></arrow.to.target><lb/>dentur rem nauticam quòd ad remos attinet, referre in longitu­<lb/>dinem partis, quæ &longs;calmum tanquàm hypomochlium interiacet <lb/>& manum: ea enim circa medium nauis cum illa ibi &longs;it latior ma­<lb/>ior e&longs;t. </s>
<s>Sed & qui lembos ducunt, & in puppe magis di&longs;tant à <lb/>&longs;calmo & in prora, quàm in medio nauis, nec tamen uelo cius il­<lb/>lam agunt: non quòd ratio illa fal&longs;a &longs;it, &longs;ed quia uelo cius ferun­<lb/>tur etiam ob aliam cau&longs;am, quàm &longs;it hæc, & magis uniuer&longs;alem. <lb/></s>
<s>Primum igitur &longs;umamus, quod &longs;uperiùs demon&longs;tratum e&longs;t &longs;cili­<lb/><arrow.to.target n="marg393"></arrow.to.target><lb/>cet, quòd ubi pondus aliquod æquale undique tanquam in li­<lb/>bra &longs;u&longs;pen&longs;um fuerit, proportio ponderis partium inæqualium <lb/>ad duas partes æquales, e&longs;t confu&longs;a ex proportione longitudi­<lb/>nis earundem, & quadrato eiu&longs;dem proportionis. </s>
<s>Sit ergo diui­<lb/>&longs;a a b in c, & fiat c e æqualis c a: proportio igitur ponderis b e ad <lb/>pondus e a e&longs;t compo&longs;ita ex proportione b e ad e a, & quadrato <lb/><figure id="fig84"></figure><lb/>eius <expan abbr="&longs;ecũdum">&longs;ecundum</expan> longitudinem. </s>
<s>at po&longs;ita agi <lb/>na d g in medio a b, proportio ponderis b e <lb/>ad pondus ea e&longs;t, ueluti longitudinis b e <lb/>ad e a, igitur proportio <expan abbr="põderis">ponderis</expan> b e ad e a, <lb/>cum agina e&longs;t extra medium in c, e&longs;t tanto <lb/>maior proportione b c ad ea, quantum e&longs;t quadratum illius pro­<lb/><arrow.to.target n="marg394"></arrow.to.target><lb/>portionis, ergo b e pondus maius e&longs;t, cum agina e&longs;t in c, quàm in d. <lb/></s>
<s>igitur per <expan abbr="commun&etilde;">communem</expan> animi <expan abbr="&longs;ententiã">&longs;ententiam</expan> addito communi pondere a e, <lb/>erit pondus a b minus &longs;emper cum agina e&longs;t in d, <08> in ullo alio lo­<lb/>co a b. </s>
<s>Ergo pondus a b apprehen&longs;um in d <expan abbr="mouebi&ttilde;">mouebitur</expan> a b æquali ui <lb/><arrow.to.target n="marg395"></arrow.to.target><lb/>maiore proportione, <08> in ullo alio loco. </s>
<s>Ha&longs;tile ergo in medio ap­<lb/>prehen&longs;um maiore ui mouebitur, quàm in ulla alia parte. </s>
<s>Et &longs;i gra­<pb pagenum="106"/>cilius &longs;it in anteriore parte propinquius comprehen&longs;um calci, & &longs;i <lb/>cra&longs;sius, uel grauius propius cu&longs;pidi. </s>
<s>Semper igitur ob hanc cau­<lb/>&longs;am mota ex medio grauitatis, &longs;eu uelo, &longs;eu ramo, &longs;eu manu uelo­<lb/>cius mouentur, quàm ex alijs partibus. </s>
<s>In remo etiam pote&longs;t acce­<lb/>dere illud commodum, cuius meminit Ari&longs;tcteles. </s>
<s>Propter hoc igi <lb/>tur, qui malum in naui collo carunt tantùm unum, in medio fermè <lb/>eum collocarunt, ut antiqui: & qui duos aut tres, maiorem cra&longs;sio­<lb/><arrow.to.target n="marg396"></arrow.to.target><lb/>rem &longs;cilicet, & altiorem in medio con&longs;tituerunt.</s></p><p type="margin">
<s><margin.target id="marg392"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s><margin.target id="marg393"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 86.</s></p><p type="margin">
<s><margin.target id="marg394"></margin.target>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><margin.target id="marg395"></margin.target>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><margin.target id="marg396"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 82.</s></p><p type="main">
<s>Propo&longs;itio cente&longs;imaduodecima.</s></p><p type="main">
<s>Cur ex imo leuia longius ferantur declarare.</s></p><p type="main">
<s>Iam uerò <expan abbr="cõ&longs;ideremus">con&longs;ideremus</expan>, quòd propo&longs;itum e&longs;t, non &longs;olum in com­<lb/><arrow.to.target n="marg397"></arrow.to.target><lb/>paratione ad medium, &longs;ed extremorum inuicem, mi&longs;&longs;a enim ab imo <lb/>uelo cius feruntur, quàm à medio non &longs;olum manu, &longs;ed &longs;corpioni­<lb/>bus, & arcubus. </s>
<s>Videmus & hoc ob&longs;eruare pueros uirgam lon­<lb/>gius iacentes non ex medio, &longs;ed imo apprehen&longs;am, quoniam pars <lb/>ip&longs;a anterior, & quæ manu apprehen&longs;a e&longs;t, uehementi impetu emit­<lb/>titur: & ut recipit impetum magis æqualem, longius fertur, nam <lb/>quod emittitur proportionem habet ad &longs;patium. </s>
<s>Cum ergo appre <lb/>hen&longs;a in medio uirga &longs;olum medietate anteriore impetum recipiat <lb/>per &longs;e, ob id minus fertur: at impetus &longs;equitur proportionem, ut ui­<lb/>&longs;um e&longs;t, quæ e&longs;t circa medium ob leuitatem ponderis. </s>
<s>In leuibus <lb/>ergo maius &longs;patium &longs;uperabunt emi&longs;&longs;a ex imo, quoniam propor­<lb/>tio &longs;patij eadem e&longs;t ad duplum, & ad dimidium. </s>
<s>igitur ex imo fer­<lb/>me duplum etiam &longs;patij &longs;uperabit: non tamen omnino quia maio­<lb/>rem, ut dixi proportionem habet ad id, quod ex medio comprehen <lb/>&longs;um e&longs;t. </s>
<s>At in leuibus non e&longs;t nece&longs;&longs;arium, ut ex medio apprehen­<lb/>dantur, quoniam etiam cum incremento illo ponderis iam leuia <lb/>&longs;unt: plus ergo facit longitudo eius, quod eiaculatur, quàm impe­<lb/><figure id="fig85"></figure><lb/>tus, cuius demon&longs;tratio e&longs;t hæc. </s>
<s>Sit uirga <lb/>a b apprehen&longs;a in medio ponderis unciæ <lb/>mediæ, & in a d, ut &longs;it d a palmus, & uige&longs;i­<lb/>ma pars totius a b, erit ergo re&longs;iduum ad duplum, a d nonuplum, <lb/><arrow.to.target n="marg398"></arrow.to.target><lb/>& a b tota unciarum quin que cum dimidia, &longs;i igitur grauetur, quia in <lb/>&longs;itu recto e&longs;t mediæ unciæ, in æquidi&longs;tanti terræ, quin que unciarum <lb/>cum dimidio, erit in &longs;itu dimidij recti unciarum trium. </s>
<s>E&longs;t igitur <lb/>proportio &longs;excupla, &longs;i apprehendatur in medio, & ad æquidi&longs;tan­<lb/>tem, ad apprehen&longs;am in imo, & ad angulum medium: at emi&longs;&longs;a ex <lb/><arrow.to.target n="marg399"></arrow.to.target><lb/>a d habet totum aërem a b circumdantem impul&longs;um ex c b &longs;olum <lb/>dimidium reliqua pars ui trahitur, ergo proportio &longs;patij a b, erit <lb/>&longs;exdecupla fermè &longs;patio b c, quoniam e&longs;t triplicata corporis ad cor <lb/>pus eius, quæ e&longs;t longitudinis ad longitudinem, & quadruplicata <pb pagenum="107"/>re&longs;pectu aëris a c, qui re&longs;i&longs;tit apprehen&longs;a a b in c. </s>
<s>Et iam minus fere­<lb/>batur quinta parte, ideo longius eiaculabitur triplo ex a, quàm ex <lb/>c. </s>
<s>Nec tamen maiore impetu, quia obliquè fertur, & quæ obliquè <lb/><expan abbr="feriũt">feriunt</expan>, minore cum impetu feriunt: at que eo magis &longs;i leuia fuerint: ab <lb/>aëre enim circumambiente perturbantur, & in incertum trudun­<lb/>tur. </s>
<s>Quæ ergo grauia &longs;unt ex medio emi&longs;&longs;a, & ad æquidi&longs;tantem <lb/>longius feruntur, & maiore cum impetu, quia magis directè: leuia <lb/>autem longius ex imo, &longs;ed minore cum impetu, &longs;i aliqua cau&longs;a à re­<lb/>cto, & æquidi&longs;tante declinauerint. </s>
<s>At &longs;i à &longs;uprema parte, & iuxta <lb/>cu&longs;pidem, neque procul feruntur, neque cum impetu ob cau&longs;as di­<lb/>ctas. </s>
<s>Eadem quoque ratio e&longs;t omnium machinarum: ideò oblon­<lb/>g&ecedil;longius eiaculantur, quoniam proportionem &longs;eruant ad cana­<lb/><arrow.to.target n="marg400"></arrow.to.target><lb/>iem. </s>
<s>Sed de hoc inferius agetur.</s></p><p type="margin">
<s><margin.target id="marg397"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s><margin.target id="marg398"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 86.</s></p><p type="margin">
<s><margin.target id="marg399"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 89.</s></p><p type="margin">
<s><margin.target id="marg400"></margin.target>P<emph type="italics"/>rop.<emph.end type="italics"/> 107.</s></p><p type="main">
<s>Propo&longs;itio cente&longs;imatertia decima.</s></p><p type="main">
<s>Cur uirga longius mittatur à puero, quàm à uiro inue&longs;tigare.<lb/><arrow.to.target n="marg401"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg401"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>_{m}.</s></p><p type="main">
<s>Diligentia, & u&longs;us puerilis efficit, ut uirga feratur &longs;ecundum me­<lb/>dium rectianguli: uir autem non con&longs;tanter iacit, & &longs;ecundum re­<lb/>ctum, at rectus ince&longs;&longs;us in leuibus, quia ab aëre in obliquum defle­<lb/>ctitur uirga ob longitudinem efficit, ut inflectatur infrà celerius, & <lb/>de&longs;inat citius motus, ac finiatur. </s>
<s>Tertia cau&longs;a e&longs;t, quòd leui&longs;sima <lb/>non adeò recipiunt impetum ut grauia: nam leui&longs;simam & exigu­<lb/>am ligni portionem maximo nixu uix excutiemus è manu. </s>
<s>Cau&longs;a <lb/>ergo e&longs;t: quoniam uim, oportet, ut habeat, quod contra naturam <lb/>mouetur, ut naturaliter moueri po&longs;sit, quæcun que igitur naturaliter <lb/>exiguum habent motum, ut pluma, palea, fe&longs;tucæ nulla ratione ue­<lb/>hementer contra naturam agi po&longs;&longs;unt. </s>
<s>Quædam ergo à pueris lon <lb/>gius <expan abbr="iaciũtur">iaciuntur</expan> ob &longs;olam peritiam, & exercitationem, quædam quo­<lb/>niam ad angulum latiorem magis feruntur, quàm &longs;it rectus, quæ­<lb/>dam quoniam leui&longs;sima &longs;unt. </s>
<s>Sed &longs;i leuiora non feruntur ualido <lb/>motu uiolento, cur tamen à pueris iacta longius <expan abbr="ferũtur">feruntur</expan>? </s>
<s>Ratio e&longs;t, <lb/>quoniam maior uis deficiente obiecto magis fatigatur, atque ideò <lb/>minus mouet. </s>
<s>Propter hæc igitur omnia non &longs;olùm in pueris, &longs;ed <lb/>in machinis, quæ accommodata &longs;unt, melius impelluntur, aclon­<lb/>gius feruntur, quàm leui&longs;sima. </s>
<s>nam nec palea &longs;corpione iacta tam <lb/>procul, quàm &longs;agitta fertur, cum proportio maior &longs;it, tamen ad pa­<lb/>leam, quàm ad &longs;agittam. </s>
<s>Inde fit, ut quemadmodum Turca ille lite­<lb/>ras &longs;ui Prin cipis, cum timeret ad no&longs;tros propius accedere, lapidi al <lb/>ligatas longius emi&longs;it. </s>
<s>Cau&longs;am autem huius docet Ari&longs;toteles in <lb/>Mechanicis dum quærit cur, & grauia & leuia ualde longe proijci <lb/>nequeunt: nam grauia nimis, moueri <expan abbr="nõ">non</expan> facilè po&longs;&longs;unt: leuia etiam <lb/>ualde ad rem mouere non ualent. </s>
<s>Ob hæc utra que ex his paruo cum <pb pagenum="108"/>impetu emittuntur, tamet&longs;i uehementer nitaris. </s>
<s>Sed & leuia ferun­<lb/>tur hac illac, ut non po&longs;sint retinere impetum prioris uiolentiæ: in­<lb/>natum enim e&longs;t, ut duorum motuum &longs;imul in eadem re uigentium, <lb/>cum illa proprio impetu feratur, unus alterum impediat: nam &longs;i ro­<lb/>ta uehatur circulariter acta, non tamen ce&longs;&longs;abit, aut iminuetur impe <lb/>tus circulationis. </s>
<s>Multa ergo in huiu&longs;modi anomalis motibus con <lb/>&longs;ideranda &longs;unt, ut illorum impetum robur, aclocum definiamus.</s></p><p type="main">
<s>Ex hoc liquet, cur plumbeæ &longs;phærulæ longius ferantur à tor­</s></p><p type="main">
<s><arrow.to.target n="marg402"></arrow.to.target><lb/>mento emi&longs;&longs;æ, quàm ligneæ, etiam &longs;i non fran gantur.</s></p><p type="margin">
<s><margin.target id="marg402"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Propo&longs;itio cente&longs;imaquartadecima.</s></p><p type="main">
<s>Cir cularis motus differentias quatuor e&longs;&longs;e, earum qúe rationem <lb/>contemplari.</s></p><p type="main">
<s>In motu circulari aut axis <expan abbr="progredi&ttilde;">progreditur</expan>, aut &longs;uo loco manet. </s>
<s>Vtro que<lb/><arrow.to.target n="marg403"></arrow.to.target><lb/>autem modo uel mouetur ab axe, uel circumferentia, igitur con&longs;tat <lb/>quatuor e&longs;&longs;e motuum differentias: quas cum tres proponat author <lb/>libri Mechanicarum, aut Ari&longs;totelem illum e&longs;&longs;e, credendum non <lb/>e&longs;t, aut illum &longs;tupidum dicere nece&longs;&longs;e e&longs;t, nam modum diuidendi <lb/>eum latui&longs;&longs;e quis putet. </s>
<s>cum rota igitur aut &longs;phæra in plano cir­<lb/>cumagitur, motus e&longs;t ex circumferentia prægrediente axe: ut pa­<lb/>lam e&longs;t: motis enim loco nobis mouentur omnia, quæ &longs;unt in no­<lb/>bis. </s>
<s>Cum uerò rotæ &longs;ub curru &longs;unt, progreditur axis earum, & rota <lb/>ob id cum quie&longs;cere nequeat, quia facilius circumuertitur, quàm <lb/>trahatur, procedit, & hic e&longs;t &longs;ecundus modus, quo rota ex circumfe <lb/>rentia mouetur, & ex axe initium e&longs;t motus. </s>
<s>At uerò in rota molari, <lb/>& quibus gladij exacuuntur, cum loco non moueantur, motus e&longs;t <lb/>ex axe: axis enim rotam circumagit, non rota axem, quie&longs;cit tamen <lb/>in eodem loco rota, & axis &longs;cilicet, quia non progreditur, &longs;ed in lo­<lb/>co mouetur: atque hic e&longs;t tertius modus. </s>
<s>Demum &longs;uccula putei, & <lb/>ip&longs;a mouetur circulari motu, & trochleæ etiam, neque enim progre­<lb/>diuntur: &longs;ed non ex axe mouentur, uerùm &longs;uccula per coloppes cir <lb/>cumducitur, & tro chlea per funes, axis que in &longs;uccula mouetur, in tro <lb/>chleis autem quie&longs;cit pror&longs;us: dico mouetur, id e&longs;t circumducitur, <lb/>non quod progrediatur: ut non &longs;olum &longs;int quatuor modi, &longs;ed po­<lb/>tius quin que, nam & demon&longs;tratione o&longs;tenduntur, & experimento <lb/>do cente deprehenduntur. </s>
<s>Horum omnium liberrimus e&longs;t, primus <lb/>ex cir cumferentia progrediente toto, &longs;eu attracto &longs;eu impul&longs;o & ue <lb/>loci&longs;simus, cuius cau&longs;am &longs;uprà o&longs;tendimus. </s>
<s>Proximus huic e&longs;t mo­<lb/><arrow.to.target n="marg404"></arrow.to.target><lb/>tus rotarum per axem, quoniam axis premit rotam interius &longs;o­<lb/>lam, & labitur: ideo que quod & axis, & rota intus &longs;int leui&longs;sima, pro­<lb/>de&longs;t plurimum: & aurigæ axungia inungunt, & nomen ab eo traxit <pb pagenum="109"/>axungia. </s>
<s>Et quae rota magna &longs;it: quoniam cum <expan abbr="nõ">non</expan> rota, &longs;ed axis traha­<lb/>tur in æquali tempore & magna, & parua trahitur: utra que uerò una <lb/>conuer&longs;ione tantam <expan abbr="lineã">lineam</expan> rectam &longs;uperat, quanta e&longs;t rotæ periphe­<lb/>ria. </s>
<s>Quod &longs;i plures &longs;int rotæ celerius feruntur, quia axis minus tan­<lb/>to <expan abbr="rotã">rotam</expan> premit. </s>
<s>Et &longs;i rectus &longs;it axis, & bene rotundus, & foramen ro <lb/>tundum, & latius, & è duri&longs;simo ligno, ut non po&longs;sit in clinari: & <lb/>rota ip&longs;a in ambitu æqualis, omnia hæc faciunt ad motus uelo cita­<lb/>tem, unde Homerus.<lb/><arrow.to.target n="marg405"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg403"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s><margin.target id="marg404"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 40.</s></p><p type="margin">
<s><margin.target id="marg405"></margin.target>I<emph type="italics"/>liad.<emph.end type="italics"/> 23.</s></p><p type="main">
<s><foreign lang="greek"><gap/> xnia tu/pte w_o/dessi w_a/r & ko/nin a)|mfi xuqu_nai</foreign>.</s></p><p type="main">
<s>Id e&longs;t, ue&longs;tigia per cu&longs;sit pedibus, ante que illa puluis pedibus ex­<lb/>cu&longs;&longs;us (ue&longs;tigia &longs;cilicet relinquentibus) ingrederetur. </s>
<s>Principalis <lb/>autem cau&longs;a uelo citatis e&longs;t agens, uelut equi. </s>
<s>Sed inter <expan abbr="hũc">hunc</expan> motum <lb/>& priorem medius e&longs;t Scitalæ uocatæ, nam ut in primo axis proci­<lb/>dit & rotundum à &longs;uperficie circumagitur, licet axis etiam circum­<lb/>ducatur, ut axis, & rota, aut &longs;phæra duplici motu moueantur, fci­<lb/>licet antror&longs;um, & circumcirca, in rota currus duo ijdem motus <lb/>&longs;int, axis quo que antror&longs;um moueatur, &longs;ed non circumagatur: unde <lb/>impeditior e&longs;t hic motus: ita in Scytala utrun que utro que motu mo­<lb/>uetur, & circumcirca, & antror&longs;um, at que id commune e&longs;t, cum pri­<lb/>mo ita axis mouet rotas, non rotæ axem, quòd &longs;ecundo motui ro­<lb/>tarum in curru proprium e&longs;t, ut tantum degenerent à primo motu, <lb/>quanto leuius uertuntur, quàm in &longs;ecundo motu. </s>
<s>Trahitur ergo <lb/><figure id="fig86"></figure><lb/>iugum in &longs;citala, uelut in rotis currus, <lb/>&longs;ed e&longs;t annexum rotis non in curri­<lb/>bus. </s>
<s>Propterea in primo motu trahi­<lb/>tur, uel impellitur à &longs;uperficie: in &longs;e­<lb/>cundo a b axe, &longs;ed non affixo rotis, unde ægrè trahuntur in &longs;cyta­<lb/>la ab axe affixo rot&ecedil;. </s>
<s>Quare leuius quàm in curru, difficilius quàm <lb/>in rota uel &longs;phæra à &longs;uperficie extima circumacta. </s>
<s>Quartus modus <lb/>e&longs;t, ut dixi, circumuecta rota ab axe, quum non progreditur, ut in <lb/>moletrinis, & rotis, quibus ferrum exacuitur. </s>
<s>E&longs;t enim hic &longs;imilior <lb/>primo, quia contrarius, in primo enim procedit rota, & uertitur à <lb/>circumferentia, hic quie&longs;cit rota, & mouetur ab axe. </s>
<s>Proximus huic <lb/>e&longs;t, qui fit in &longs;ucculis ob firmitatem axis: nam axis e&longs;t coniunctus <lb/>rotæ. </s>
<s>Vltimus e&longs;t trochlearum, qui & difficillimus: &longs;it enim à cir­<lb/>cunferentia, & axis di&longs;iunctus e&longs;t à trochlea: quod ad dit difficulta­<lb/>tem. </s>
<s>Sed & trochlea caret colloppibus. </s>
<s>Ergo uerum e&longs;t, quod o­<lb/>mnia rotunda facilius circumaguntur, &longs;ed uaria ratione: nam plus <lb/>mota &longs;uper aliquo plano, ut in plau&longs;tris & &longs;cytalis: minus in &longs;uccu­<lb/>lis, & rotis acuentibus ferrum, & molis: nam & &longs;i rotun ditatem iu­<lb/>uet ob æqualitatem ad conuer&longs;ionem, non tamen in his e&longs;t ad eò <pb pagenum="110"/>utilis. </s>
<s>Vtilitas ergo prima e&longs;t, cum circumuertitur in plano, uelut <lb/>in rotis &longs;cytalis, & &longs;phæris. </s>
<s>Secunda quæ minor e&longs;t, cum à &longs;uperfi­<lb/>cie circumuertitur, ut in trochleis. </s>
<s>Tertia cum à coloppis, quæ mi­<lb/>nima e&longs;t omnium, ut in &longs;ucculis. </s>
<s>Motus autem cœli non e&longs;t ex tri­<lb/>plici primo genere, cum &longs;it in loco, & non ad locum, neque ut rotæ <lb/>molaris: nam ille e&longs;t ex axe: necut in tro chlea: nam in ea axis quie&longs;­<lb/>citip&longs;um autem cœlum circa axem non uertitur, &longs;ed cum axe, &longs;i ta­<lb/>men in&longs;ecabilis linea circumagi pote&longs;t dici. </s>
<s>Relinquitur ergo, ut <lb/>Cœli motus propior &longs;it motui &longs;ucculæ, quàm alij motui. </s>
<s>Differt <lb/>ab eo in hoc, quod in &longs;uccula mouetur axis ab orbe: at in cœlo <lb/>ut non mouetur ab axe, ita nec axis ab orbe: cun que &longs;it motus &longs;im­<lb/>plici&longs;simus, in alio genere collocandus e&longs;t: quando quidem in illo <lb/>nulla pars po&longs;sit dici primo, quod <expan abbr="nece&longs;&longs;ariũ">nece&longs;&longs;arium</expan> e&longs;t in uno quo que <expan abbr="horũ">horum</expan>.</s></p><p type="main">
<s>Propo&longs;itio cente&longs;imaquinta decima.</s></p><p type="main">
<s>Proportionem motuum impul&longs;ionis, & attractionis inter'&longs;e ab <lb/>eadem ui declarare.</s></p><p type="main">
<s>Con&longs;tat, quòd attractio cum fune longiore ualidior e&longs;t, quam </s></p><p type="main">
<s><arrow.to.target n="marg406"></arrow.to.target><lb/>cum manibus, quoniam e&longs;t cum motu quodam: motus autem au­<lb/>get actionem, ideo attractio ualidior e&longs;t hac de cau&longs;a, &longs;ed & impul­<lb/>&longs;io cum baculo ualidior e&longs;t, quam cum manibus, quoniam licet col <lb/>ligere omnes uires in illo baculo, & ip&longs;um applicare loco, unde fa­<lb/>cilius impelli pote&longs;t. </s>
<s>Velut &longs;phæra ex medio latere: nam ibi magis <lb/>colliguntur uires, & ad impellendum facilius e&longs;t, quodcun que leui­<lb/>us e&longs;t. </s>
<s>Pars autem magis remota à centro grauitatis e&longs;t leuior, his <lb/>duabus cau&longs;is, &longs;phæra ex medio latere facilius ac magis impellitur. <lb/></s>
<s>Sed nos &longs;upponimus nunc applicationem æqualem e&longs;&longs;e, nam &longs;e­<lb/>cus ad impellendum facilius e&longs;t applicare totum corpus, quàm at­<lb/>tractionem. </s>
<s>Pectore enim magna ui impellimus, nihil e&longs;t compar, <lb/>quo trahere po&longs;simus. </s>
<s>Sed, ut dixi, &longs;it baculus applicatus alicui la­<lb/>pidi ea parte, qua facilius pote&longs;t impelli & trahi, & quæritur, quæ <lb/>maior &longs;it uis, an attrahendi? </s>
<s>& dico quòd homo, uel conatur trahe­<lb/>re toto corpore, & impellere, at que hoc modo magis trahit, quàm <lb/>impellet, quoniam corporis pondus melius adhibetur in tractione <lb/>quàm impul&longs;u: uel citra corporis pondus, &longs;ed &longs;ola ui membrorum: <lb/>& tunc magis impellit, quoniam impul&longs;us fit corpore prono in <expan abbr="an-terior&etilde;">an­<lb/>teriorem</expan> partem, quæ in clinatio, & motus e&longs;t naturalis magis, quàm <lb/>in attractione in partem po&longs;teriorem. </s>
<s>Sed ubi nulla &longs;it diuer&longs;itas <lb/>neque horum, neque figurarum æqualis uis æqualem efficit motum: <lb/>quia impul&longs;us impellentis comparatione e&longs;t attractio re&longs;pectu al­<lb/>terius. </s>
<s>Verùm non e&longs;t eadem uis nec propè par impellendi, at que <lb/>attrahendi hominibus, cum attractio fiat per mu&longs;culos ad origi­<pb pagenum="111"/>nem &longs;uam naturaliter &longs;e retrahentibus impul&longs;ui nullum in&longs;trumen <lb/>tum à natura delegatum inuenio, nam ad exten&longs;ionem mu&longs;culi &longs;a­<lb/>nè ex aduer&longs;o &longs;unt fabricati: cum ergo duo &longs;int tantum motus mu­<lb/>&longs;culorum ten&longs;io, dum <expan abbr="retrahũtur">retrahuntur</expan> ad principium &longs;uum, & remi&longs;sio, <lb/>dum membrum quie&longs;cit in naturali nullus erit locus impul&longs;ioni, <lb/>ni&longs;i ex con&longs;equentia non per &longs;e, quamobrem multo infirmiorem il­<lb/>lum attractione in brachijs e&longs;&longs;e, nece&longs;&longs;e e&longs;t.</s></p><p type="margin">
<s><margin.target id="marg406"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Propo&longs;itio cente&longs;ima&longs;extadecima.</s></p><p type="main">
<s>Cur machinæ ablongæ igneæ longius emittant &longs;phæram ex­<lb/>plorare.<lb/><arrow.to.target n="marg407"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg407"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Quoniam ratio &longs;uperius adducta, neque in his, neque in hypophy­</s></p><p type="main">
<s><arrow.to.target n="marg408"></arrow.to.target><lb/>&longs;is (uocant cerbatanas) non pote&longs;t &longs;atisfacere, cum tamen idem &longs;e­<lb/>quatur in his, ut in illis uidetur, qua&longs;i uis e&longs;&longs;e in &longs;phærula &longs;ic emi&longs;­<lb/>&longs;a, & non in aëre, quemadmodum dicebamus, coniuncto e&longs;&longs;e. </s>
<s>Ex <lb/>quo nece&longs;&longs;e e&longs;&longs;et, ut quod longius ferretur, etiam ualidiores ictus <lb/><figure id="fig87"></figure><lb/>inferret, hoc autem <lb/>non ita &longs;e habet, &longs;ed <lb/>ictus magnitud o <lb/>ex robore machi­<lb/>narum tam ignea­<lb/>rum, quam &longs;corpio <lb/>num pendet, nam <lb/>&longs;it a &longs;corpio ma­<lb/>gnus, &longs;ed tenuis, ex <lb/>hòc palam e&longs;t lon­<lb/>gius mittere &longs;agit­<lb/>tam, quòd à parua, <lb/>& breui, quantun­<lb/>uis cra&longs;&longs;a non lon­<lb/>ge mittitur: at uerò <lb/>quod b cra&longs;&longs;us & paruus maiore cum impetu mittat o&longs;tenditur <lb/>nam ea pondera &longs;agittæ mouet, quæ non pote&longs;t mouere a, igitur b <lb/>ualidiore robore mouet, quam a. </s>
<s>Prætera illud o&longs;ten dit iugum fu­<lb/>nis arcus cra&longs;siora duriora, quæ maioribus uiribus <expan abbr="indig&etilde;t">indigent</expan>, quam <lb/>a, qui à puero tendi poterit. </s>
<s>Non e&longs;t ergo eadem ratio mittendi <lb/>longius, & ualidiore cum robore. </s>
<s>Eadem ergo cum ratio &longs;it in <lb/>machinis igneis, cra&longs;siores enim, & latiores ac breuiores magis <lb/>concutiunt, quam longiores tenuiores minoris &longs;phæræ capaces: <lb/>non &longs;olum ob mag nitudinem &longs;phæræ magis illæ concutiunt, &longs;ed, <lb/>ut dixi, ob maiorem impetus uim: cau&longs;a ergo e&longs;t manife&longs;ta in his, <lb/>&longs;ed non cau&longs;a, qua longius ferantur in longiore canali. </s>
<s>Sed uide­<pb pagenum="112"/>tur una, eadem que e&longs;&longs;e ratio in utri&longs;que. </s>
<s>Con&longs;tituatur can alis a b <lb/>lońgior, & c d breuior, ut &longs;it &longs;exqui alter a b ad c d, & &longs;it rur&longs;us <lb/><figure id="fig88"></figure><lb/>&longs;phærulæ locus e in longiore, <lb/>&longs;exqui alter in di&longs;tantia a b, qua <lb/>lis e&longs;t in f a d, & erit per dicta <lb/>ab Euclide in quinto, ac &longs;exqui <lb/>altera c f. </s>
<s>Po&longs;&longs;emus igitur di­<lb/>cere, quod uelut ab hypomo­<lb/>chlio longiore &longs;patio circuma­<lb/>gitur pondus: ita & a b c, & f. <lb/></s>
<s>Sed rur&longs;us incidimus in id, ut <lb/>maiore impetu feratur e quàm f. </s>
<s>Ideo &longs;i concedatur maiore ferri ex <lb/>e, quam ex f non &longs;equitur, ut celerius, aut maiore impetu. </s>
<s>Percutit <lb/>puer pugno quanta ui pote&longs;t ac celerrimè, uir robu&longs;tus lentè, & mi­<lb/>nore impetu, &longs;ed tamen ictus longè maior e&longs;t. </s>
<s>E&longs;t enim ictus robur <lb/>non à uelo citate &longs;olum, &longs;ed maiore ex ponderis grauitate, quæ &longs;ola <lb/>premit, urget, & frangit etiam &longs;ine motu. </s>
<s>Solum ergo id re&longs;tat du­<lb/>bium, cur &longs;i grauius e&longs;t, moueatur eodem ferm é impetu: nam quo <lb/>maiore impetu fertur, eo longius fertur, non tamen magis ferit, con <lb/>cutit, aut qua&longs;&longs;at, &longs;ed grauitas ad hoc plus facit impetu. </s>
<s>Palea maxi­<lb/>mo impetu demi&longs;&longs;a non ferit, non ledit, & celerius de&longs;cendit, fer­<lb/>rum &longs;ola grauitate actum, imò etiam temperato ictu lædit graui­<lb/>ter, qua&longs;&longs;at, & frangit: ita que f maiore indiget quantitate pyrij pulue­<lb/>ris, quàm e: &longs;iquidem tertia parte ponderis &longs;uæ &longs;phæræ: at maius <lb/>e&longs;t pondus f quam e, ergo maius pondus pulueris f quàm e, ergo <lb/>maior uehementia ictus, &longs;iquidem ea &longs;equitur, robur cau&longs;æ mouen <lb/>tis &longs;im pliciter: ut concludamus longitudinem ictus &longs;equi propor­<lb/>tionem motoris ad motum, &longs;ed uehementia robur motoris: nam &longs;i <lb/>ex portione mouet æquale pondus maiore cum impetu mouet, <lb/>quoniam maior e&longs;t proportio: &longs;i minore igitur pondus maius e&longs;t, <lb/>&, ut dixi plus facit magnitudo ponderis cum leui ictu, quàm ma­<lb/>gnitudo ictus cum leui pondere. </s>
<s>Quæ ergo feruntur per longio­<lb/>res canales maiore impetu feruntur, & &longs;ocietatem <expan abbr="hab&etilde;t">habent</expan> aëris moti <lb/>per longius <expan abbr="&longs;patiũ">&longs;patium</expan>, ut tardius remittatur, quia longiore tempore <expan abbr="uĩs">uins</expan> <lb/>motus confirmata e&longs;t, & proportio eius, quòd mouet, maior e&longs;t ad id, <lb/>quod <expan abbr="moue&ttilde;">mouetur</expan>, quia minus extenditur, at uerò 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, e&longs;t quod ferit. </s>
<s>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">o&longs;tendere</expan> oporter.</s></p><p type="margin">
<s><margin.target id="marg408"></margin.target>P<emph type="italics"/>rop.<emph.end type="italics"/> 103.</s></p><p type="main">
<s>Propo&longs;itio cente&longs;imadecima&longs;eptima.</s></p><p type="main">
<s>In cuniculis maior e&longs;t uis pulueris copio&longs;ioris ampliore in &longs;pa­<lb/>tio, quàm paucioris in minore iuxta proportionem eandem.</s></p><pb pagenum="113"/><p type="main">
<s>Sit &longs;patium f d &longs;exqui tertium b e, puluis quo que in f d &longs;patio &longs;i­<lb/><arrow.to.target n="marg409"></arrow.to.target><lb/>militer &longs;exqui tertius pulueri b e pondere, & manife&longs;tum e&longs;t, quod <lb/>dum conuertitur in ignem quali&longs;cun que &longs;it proportio (modo eadem <lb/>ignis ad puluerem) erit ignis in f d pariter &longs;exqui tertius igni in b e, <lb/>dico quòd &longs;i cra&longs;sities f d &longs;it etiam &longs;exqui tertia cra&longs;sitiei b e, quod <lb/>poterit frangi, & moueri f d quie&longs;cente b e. </s>
<s>Vnde idem in cuniculis <lb/>ut magnus cuniculus cum multo puluere po&longs;sit mouere montem <lb/>paruus cum puluere proportione re&longs;pondente priori non po&longs;sit. <lb/></s>
<s>Nam cùm æqualia &longs;int omnia iuxta que rationem eandem, nece&longs;&longs;e e&longs;t <lb/>ut pro ratione extendantur, at in paruo &longs;patio minor fit den&longs;itas c&ecedil;­<lb/>tera paria &longs;unt, ergo à paruo &longs;patio non tantus fit impetus, quantus <lb/>à magno. </s>
<s>Impetus etiam proportionem habet ad <expan abbr="põdus">pondus</expan>, & ad con­<lb/>iunctionem, à maiore igitur impetu plura, & maiora mouentur, & <lb/>conuelluntur, quam à minore, ob hæc igitur minores cuniculi &longs;uc­<lb/>cutiunt, maiores euertunt, maximi exturbant, & proij ciunt. </s>
<s>Nam <lb/>qui &longs;uccutiunt, ubi pondus, aut coniunctio maior &longs;it, quàm ut di­<lb/>&longs;trahere po&longs;sint, conden&longs;ant partes proximiores, & rimas faciunt, <lb/>per quas exhalat ignis aut omnino extinguitur, aut conden&longs;atur. <lb/></s>
<s>At ergo in bellicis machinis, minus dilatat puluis, cum fuerit in lon <lb/>go canali, ob id ergo maiore impetu feruntur per illas, quàm per <lb/>breuiores, etiam quòd minor &longs;it puluis, minor &longs;it ignis. </s>
<s>Experimen <lb/>tum facies in canali, ubi &longs;ambuci medulla pro globulo flatu impel­<lb/>lente expellitur ab&longs; que periculo: nam quanto minor fuerit canalis <lb/>ambitu ac longior eo maiore impetu pellitur. </s>
<s>For&longs;an qui&longs;piam nos <lb/>meritò poterit uideri <expan abbr="repreh&etilde;di&longs;&longs;e">reprehendi&longs;&longs;e</expan>, quòd inanis gloriæ &longs;tudio per­<lb/>nitio&longs;a humano generi do ceam. </s>
<s>Quibus re&longs;pondeo, me nihil do cu <lb/>i&longs;&longs;e, quod ín humani generis detrimentum cedat, huiu&longs;mo di que pr&ecedil;­<lb/>cepta iam ob&longs;cura&longs;&longs;e, ut ne quid mali accidere po&longs;&longs;et hominibus ex <lb/>his: <expan abbr="nã">nam</expan> quòd ad ea, quæ declarata, &longs;unt, cau&longs;as &longs;olùm retuli, effectus <lb/>ip&longs;imodi artis <expan abbr="nimiũ">nimium</expan> feruntur, ac nimio plu&longs;quam <expan abbr="uell&etilde;">uellem</expan> in telligun­<lb/>tur. </s>
<s>Vt cum ad copiam, ad magnitudinem, ad coacta imperia mi&longs;e­<lb/>rorum re&longs;picio, nihil plus po&longs;sit addi. </s>
<s>Omnia enim hucu&longs; que <expan abbr="&longs;pectãt">&longs;pectant</expan> <lb/>ad potentiorum in crementa. </s>
<s>An ergo &longs;uccurrere afflictis, ob&longs;e&longs;sis, <lb/>cinctis, æquare <expan abbr="condition&etilde;">conditionem</expan>, liberare à &longs;eruitute etiam rebelles <expan abbr="nõ">non</expan> li­<lb/>cebit? </s>
<s>Ab initio fuimus omnes liberi: excogitata fuit regni ratio ad <lb/>commodum hominum, ea uer&longs;a e&longs;t per uim in <expan abbr="Tyrannid&etilde;">Tyrannidem</expan>. </s>
<s>Subtili <lb/>ergo ratione <expan abbr="occurrendũ">occurrendum</expan> e&longs;t imbecillioribus: <expan abbr="nã">nam</expan> reliqua omnia ni­<lb/>mis, ut dixi, qu&ecedil; 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 &longs;pacij, per quos globus ille de­<lb/>fertur, nota &longs;unt improbis illis artificibus, nec no&longs;trum e&longs;t &longs;pectare, <lb/>cur id licuerit, po&longs;tquam Deus hanc uiolentiam e&longs;&longs;e uoluit. </s>
<s>Multa <lb/>damnamus, <expan abbr="&qtilde;">quae</expan> Deus e&longs;&longs;e uult: boni uiri e&longs;t <expan abbr="nõ">non</expan> ni&longs;i opitulari homini­<lb/>bus, <expan abbr="etiã">etiam</expan> malis modo bonis futuri <expan abbr="nõ">non</expan> &longs;int <expan abbr="impedim&etilde;to">impedimento</expan>: <expan abbr="quamobr&etilde;">quamobrem</expan> <pb pagenum="114"/>ea tradenda &longs;unt, quæ oppre&longs;sis &longs;int auxilio: ea &longs;unt, qu&ecedil; &longs;ubtilibus <lb/><expan abbr="con&longs;tãt">con&longs;tant</expan> rationibus, et multiplicata <expan abbr="amittũt">amittunt</expan> uim ut qua&longs;i <expan abbr="pr&ecedil;&longs;t&etilde;t">pr&ecedil;&longs;tent</expan> pau <lb/>ca multis, & exigua magnis. </s>
<s>In c&ecedil;teris ob&longs;curare ita decet cuncta, <expan abbr="&qtilde;">quae</expan> <lb/>obe&longs;&longs;e po&longs;&longs;unt, aut quouis modo puerti ad malos u&longs;us <expan abbr="queãt">queant</expan>, ut di­<lb/>cta <expan abbr="nõ">non</expan> dicta e&longs;&longs;e <expan abbr="put&etilde;t">putent</expan>, hoc e&longs;t <expan abbr="officiũ">officium</expan> <expan abbr="nõ">non</expan> &longs;olum probi, &longs;ed <expan abbr="etiã">etiam</expan> pruden <lb/>tis uiri.</s></p><p type="margin">
<s><margin.target id="marg409"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>_{m}.</s></p><p type="main">
<s>Propo&longs;itio cente&longs;imadecimaoctaua.</s></p><p type="main">
<s>Quanta proportione decedat ictus in obliquum parietem ab eo, <lb/>qui e&longs;t ad perpendiculum declarare.</s></p><figure></figure><p type="main">
<s>Sit paries b d e, ex a <expan abbr="fera&ttilde;">feratur</expan> in dictus, qui &longs;i <lb/><arrow.to.target n="marg410"></arrow.to.target><lb/>e&longs;&longs;et in c d <expan abbr="pariet&etilde;">parietem</expan> e&longs;&longs;e ad perpendiculum, & <lb/>ualidi&longs;simus, &longs;in uero in f g abraderet, & <expan abbr="nõ">non</expan> <lb/><expan abbr="cõqua&longs;&longs;aret">conqua&longs;&longs;aret</expan>. </s>
<s>Quæritur ergo ex b d e muro <lb/>qualis excipietur? </s>
<s>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">manife&longs;tum</expan> e&longs;t <expan abbr="aũt">aunt</expan> &longs;equi proportio­<lb/>nem, <expan abbr="quoniã">quoniam</expan> maxima uarietate <expan abbr="cõ&longs;tat">con&longs;tat</expan> dum ex angulo b d a acuto fit <lb/>acutior, <expan abbr="quoniã">quoniam</expan> &longs;i b d c &longs;it <expan abbr="&qtilde;druplus">quadruplus</expan> b d a erit re&longs;iduus ad <expan abbr="dimidiũ">dimidium</expan> b <lb/>d a nonuplus ip&longs;i 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>Si ergo <expan abbr="etiã">etiam</expan> in <expan abbr="id&etilde;">idem</expan> tenderent, <expan abbr="nõ">non</expan> efficerent mille <lb/>ictus &qring;d tres, cuius demon&longs;tratio h&ecedil;c e&longs;t. </s>
<s>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 re&longs;iduo ad b d c e&longs;&longs;e <expan abbr="&longs;olũ">&longs;olum</expan> <expan abbr="decuplã">decuplam</expan>: <lb/><expan abbr="tũc">tunc</expan> ex duob. </s>
<s>ictibus centupla erit in d c ad <expan abbr="eã">eam</expan>, qu&ecedil; in b e, <expan abbr="etiã">etiam</expan> tribus <lb/>millecupla: nam <expan abbr="cõqua&longs;&longs;ata">conqua&longs;&longs;ata</expan> turri in primo ictu, id d decuplo magis <lb/>ad perpendiculum <08> in b d e <expan abbr="&longs;uma&ttilde;">&longs;umatur</expan> decima pars in ambitu d, & illa <lb/>erit ergo <expan abbr="tã">tam</expan> di&longs;&longs;oluta, & infirma ex &longs;uppo&longs;ito, <08> e&longs;t tota b e: &longs;ed ex &longs;e <lb/>cundo ictu decuplo magis <expan abbr="cõqua&longs;&longs;abi&ttilde;">conqua&longs;&longs;abitur</expan> illa pars, <08> b e ergo tota d c <lb/>centuplo magis <expan abbr="qua&longs;&longs;abi&ttilde;">qua&longs;&longs;abitur</expan> ex duob. </s>
<s>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 amu&longs;sim directis, <expan abbr="cũ">cum</expan> ta <lb/><expan abbr="m&etilde;id">menid</expan> uix fieri po&longs;sit 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 decë c d <expan abbr="&ptilde;ter">pnter</expan> <expan abbr="quã">quam</expan> <expan abbr="exiguũ">exiguum</expan> <expan abbr="quippiã">quippiam</expan> in &longs;uperficie. </s>
<s>Imo ut <expan abbr="declaratũ">declaratum</expan> <lb/>e&longs;t multo minus repetita ratione multiplicis. </s>
<s>Ob id in arce <expan abbr="Medio-lan&etilde;&longs;i">Medio­<lb/>lanen&longs;i</expan> exterius lapidibus uiuis in <expan abbr="rotundũ">rotundum</expan> diducta &longs;uperficie inter­<lb/><figure id="fig89"></figure><lb/>uallo que <expan abbr="&qtilde;">quae</expan> drato hunc in <expan abbr="modũ">modum</expan> munit&ecedil; &longs;unt altiores tur <lb/>res. </s>
<s>Fiat ergo murus cuius proportio a d c ad b d a &longs;it &longs;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;">&longs;iquidem</expan> b d c erit quarta pars recti, & &longs;it tant&ecedil; ma­<lb/>gnitudinis, at que duritiei, ac adeò benè coniunctus fer­<lb/><arrow.to.target n="table16"></arrow.to.target><lb/>reis cathenis, ac &longs;tolonibus, ut po&longs;sit re&longs;i&longs;tere <expan abbr="machinarũ">machinarum</expan> <expan abbr="fe-rentiũ">fe­<lb/>rentium</expan> <expan abbr="&longs;ph&ecedil;rã">&longs;ph&ecedil;ram</expan> <expan abbr="librarũ">librarum</expan> ducentarum (quæ &longs;anè maximæ &longs;unt) <lb/>quin quaginta: <expan abbr="tũc">tunc</expan> cum proportio &longs;exquitertia nouies repeti­<lb/>ta, ut in numeris uides, efficiat quinquies replicatis nouem <lb/>ictibus, fiet proportio decupla quinquies producta, qu&ecedil; e&longs;t cen <lb/><expan abbr="tũ">tum</expan> millium ad <expan abbr="unũ">unum</expan> in quadraginta quin que ictibus. </s>
<s><expan abbr="Antequã">Antequam</expan> <lb/>ergo peruenit ad quinquaginta ictus rectos nece&longs;&longs;e erit, ut <pb pagenum="115"/>multo plures centum millibus ictus excipiat ante <08> euertatur, quæ <lb/>recta &longs;i e&longs;&longs;et quin quaginta &longs;olùm potui&longs;&longs;et &longs;u&longs;tinere. </s>
<s>Quæ ergo hu <lb/>mana potentia &longs;ufficeret. </s>
<s>In arce Medio <expan abbr="lan&etilde;&longs;i">lanen&longs;i</expan> uidimus uix attactas <lb/>in illis extuberationibus lapideis. </s>
<s>Sed quoniam hic occurritur per <lb/>inclinationem machinarum, ideò de hoc <expan abbr="&longs;ermon&etilde;">&longs;ermonem</expan> &longs;um habiturus.</s></p><p type="margin">
<s><margin.target id="marg410"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><table><table.target id="table16"></table.target><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>Propo&longs;itio cente&longs;imadecimanona.</s></p><p type="main">
<s>Quantum ictus machin&ecedil; procliuis ad <expan abbr="angulũ">angulum</expan> <expan abbr="minua&ttilde;">minuatur</expan> explorare.</s></p><p type="main">
<s>Huiu&longs;ce cau&longs;a <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"></arrow.to.target><lb/><expan abbr="quanquã">quanquam</expan> angulus d e f &longs;it &ecedil;quali angulo a b c, longè <expan abbr="tñ">tnm</expan> maior e&longs;t uis <lb/>a b <08> d e duplici cau&longs;a, & <expan abbr="quoniã">quoniam</expan> a b e&longs;t <expan abbr="&longs;ecundũ">&longs;ecundum</expan> nat uram impetus <lb/><figure id="fig90"></figure><lb/>ignis, & <expan abbr="etiã">etiam</expan> <expan abbr="eorũ">eorum</expan>, qu&ecedil; <expan abbr="emittun&ttilde;">emittuntur</expan> in altum: & &qring;d pars <lb/>&longs;uperior in b retineat <expan abbr="ictũ">ictum</expan>, in e non retineat. </s>
<s>Sed caui <lb/>tas fiat maior in inferiore parte: cuius <expan abbr="experim&etilde;tum">experimentum</expan> <lb/>quiliber facere pote&longs;t <expan abbr="cũ">cum</expan> ha&longs;ta. </s>
<s>Huic ergo &longs;olertiæ, <expan abbr="&qtilde;">quae</expan> <lb/>tormenta iubet altius collocare ob&longs;tat <expan abbr="primũ">primum</expan>, quod <lb/>ictus ex decliui &longs;itu periculo&longs;ior e&longs;t pro machina, & ma <lb/>ximè &qring;d retro impellit, quae ex retro ce&longs;&longs;a, po&longs;t <08> exone <lb/>rata e&longs;t, <expan abbr="digno&longs;ci&ttilde;">digno&longs;citur</expan>, & ad <expan abbr="collimandũ">collimandum</expan> decedit parte <expan abbr="ui-riũ">ui­<lb/>rium</expan> &longs;uarum, &qring;d et&longs;i <expan abbr="paruũ">paruum</expan> &longs;it in ductu <expan abbr="tñ">tnm</expan>, & <expan abbr="ictuũ">ictuum</expan> mul <lb/>tiplicatione <expan abbr="magnũ">magnum</expan> affert di&longs;crimen. </s>
<s>Habet & <expan abbr="cõmo">commo</expan> <lb/>dum &longs;itus muri accliuis <expan abbr="terrã">terram</expan> <expan abbr="&longs;uppo&longs;itã">&longs;uppo&longs;itam</expan> ad perpendiculum, <expan abbr="&qtilde;">quae</expan> ictum <lb/>&longs;u&longs;tinet: adeò ut omnib. </s>
<s><expan abbr="inuic&etilde;">inuicem</expan> collectis, perinde &longs;it ac &longs;i ex perpen­<lb/>diculo, et &ecedil;quidi&longs;tanti ad <expan abbr="&longs;olũ">&longs;olum</expan> <expan abbr="feria&ttilde;">feriatur</expan>. </s>
<s>Venetus. </s>
<s>S. aliter Patauij cauit, <lb/>uidetur que, quae &longs;apienti&longs;simus &longs;it, & eandem &longs;equatur ubi que normam, <lb/>po&longs;t <08> in <expan abbr="rotundã">rotundam</expan> figuram <expan abbr="totũ">totum</expan> urbis ambitum formauit, & fo&longs;&longs;a la <lb/>ta, ac pro fundi&longs;sima aqua que perenni muniuit, & <expan abbr="&longs;ummã">&longs;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="fig91"></figure><lb/>po&longs;&longs;et euerti, à lateribus uerò humiles, ac cra&longs;si&longs;simas turres, ut nul <lb/>la ui po&longs;&longs;ent dirui, eas que tormentis bellicis, undi que latera lu&longs;trantib. <lb/></s>
<s>reple&longs;&longs;et, illud diligenti&longs;sime cauit, ne murus humilior e&longs;&longs;et aduer&longs;a <lb/>ripa, &longs;ed ad <expan abbr="libellã">libellam</expan> tamen depre&longs;&longs;us, ut <expan abbr="etiã">etiam</expan> machinis in terram exten <lb/>&longs;is &longs;ph&ecedil;rulæ non tangerent <expan abbr="murũ">murum</expan>: nam <expan abbr="cũ">cum</expan> fo&longs;&longs;a &longs;it quadraginta pa&longs;­<lb/>&longs;uum, excedat <expan abbr="aũt">aunt</expan> murus <expan abbr="exterior&etilde;">exteriorem</expan> aggerem uno pa&longs;&longs;u, ut quicquid <lb/>in ambitu e&longs;t uno ictu oculi cogno&longs;ci po&longs;sit, & aggeris angulus ma <lb/>ior &longs;it uno pa&longs;&longs;u, <expan abbr="tũ">tum</expan> magis adiecta cra&longs;sitie machin&ecedil; fieri non pote&longs;t, <lb/>utictus in <expan abbr="murũ">murum</expan> dirigatur. </s>
<s>Eam ob cau&longs;am <expan abbr="etiã">etiam</expan> cauit, ne <expan abbr="&ecedil;dificiũ">&ecedil;dificium</expan> ul­<lb/><figure id="fig92"></figure><lb/>lum, aut planta, uel colliculus e&longs;&longs;et cir­<lb/>cum circa <expan abbr="urb&etilde;">urbem</expan> ad tria M. P. laborat hoc <lb/>periculo h&ecedil;c urbs, ne tota &ecedil;dificijs euer­<lb/>&longs;is concidat. </s>
<s><expan abbr="Turcarũ">Turcarum</expan> enim Princeps di­<lb/>dicit, ut in Nouo ca&longs;tro in Melit&ecedil; In&longs;ul&ecedil; <lb/>arce S. </s>
<s>Elmi appellata plu&longs; <08> mille icti­<lb/>bus in &longs;ingulos dies imo M D obtundere <pb pagenum="116"/>munitiones. </s>
<s>Eum que impetum producere ad quindecim dies, & ui­<lb/>ginti tum etiam longius, ut facilè domos omnes euertat, homines <lb/>occidat: &longs;i qui &longs;uper&longs;unt tot in commodis obruuntur uigilijs, fame, <lb/>&longs;iti, puluere, ut inutiles red dantur. </s>
<s>Ideò huic <expan abbr="incõmodo">incommodo</expan> occurrunt <lb/>aggeribus intra mœnia erectis, in quos uis <expan abbr="torm&etilde;torum">tormentorum</expan> igneorum <lb/>emoritur. </s>
<s>Sed dices, cur ergo non pro muris erigere eos præ&longs;tat, & <lb/>minore &longs;umptu &longs;atis? </s>
<s>quoniam &longs;ubruuntur à fo&longs;&longs;oribus facillimè, &longs;<gap/><lb/>ad illos peruenire po&longs;sit ho&longs;tis. </s>
<s>Ideò intra m œnia utili&longs;simi &longs;unt, pro<lb/>mœnijs parum pro&longs;unt. </s>
<s>Quod uerò ad te&longs;tudines attinet, &longs;ub qui­<lb/>bus <expan abbr="lat&etilde;t">latent</expan> fo&longs;&longs;ores machinæ laterales, & à fronte & ignes, & aqua al­<lb/>tior prohibent omnino iniuriam, qu&ecedil; ab his imminet. </s>
<s>Cæterum hu­<lb/>iu&longs;modi 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">di&longs;&longs;oluuntur</expan>, ut nulla multitudo huic operi <lb/>&longs;ufficere po&longs;sit. </s>
<s>Rhodus, Alba regia, Melita, Ca&longs;trum <expan abbr="nouũ">nouum</expan>, Byzan <lb/>tium, &longs;i diferri potui&longs;&longs;ent tempora, non ce&longs;si&longs;&longs;ent uictori quantum­<lb/>uis &longs;uperbo. </s>
<s>Vicit pertinacia, audacia que &longs;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>Mul <lb/>tæ machinæ, & pauci homines prædæ ob&longs;e&longs;&longs;orum expo&longs;itæ &longs;unt: <lb/>pauc&ecedil;, & pauci homines ob&longs;idebuntur potius, quam ob&longs;idebunt. <lb/></s>
<s>Exercitus magnus di&longs;&longs;oluitur, & &longs;emetip&longs;um con&longs;umit, &longs;i nulla fiat <lb/>acce&longs;sio aut exigua quomodo &longs;tabit: &longs;i magna auxilia omnia cor­<lb/>rumpuntur. </s>
<s>Contrà ob&longs;e&longs;sis auxilia &longs;i ueniant lu&longs;trata, & munita, et <lb/>omnibus nece&longs;&longs;arijs ornata uiri integri <expan abbr="cõtra">contra</expan> fatigatos, & fe&longs;&longs;os cor <lb/>pore, armati contra inermes, alacres contra torpidos &longs;uperueniunt. <lb/></s>
<s>Ob id præcipuum e&longs;t auxilium pr&ecedil;ter h&ecedil;c his, qui oppugnantur co <lb/>pia militum, qui per initia nun <08> quie&longs;cant diu noctu que, <expan abbr="uerũ">uerum</expan> noctu <lb/>duo tubicines per&longs;æpe <expan abbr="exercitũ">exercitum</expan> <expan abbr="in&longs;omn&etilde;">in&longs;omnem</expan> in armis tota nocte <expan abbr="cõtine">contine</expan> <lb/><expan abbr="bũt">bunt</expan>. </s>
<s>Serio <expan abbr="aũt">aunt</expan> die pugnare, & noctu <expan abbr="cũ">cum</expan> minimè id <expan abbr="&longs;perãt">&longs;perant</expan>, & fatigati <lb/>&longs;unt: mira euenire &longs;olent in his in&longs;peratis, ac audacibus eruptionib. <lb/></s>
<s>per&longs;&ecedil;pe <expan abbr="etiã">etiam</expan> omnino &longs;upra <expan abbr="fid&etilde;">fidem</expan>. </s>
<s>Ita <expan abbr="nõ">non</expan> conquie&longs;cere oportet donec, <lb/>uel omnino à cepto de&longs;inat ho&longs;tis, aut <expan abbr="locũ">locum</expan> occupet &longs;ibi <expan abbr="relictũ">relictum</expan> po­<lb/>tius <08> <expan abbr="qu&etilde;">quem</expan> elegerit. </s>
<s>nam <expan abbr="experimentũ">experimentum</expan> frequens do cuit, ubi illæ ma <lb/>gn&ecedil; uires &longs;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 &qring;d diximus <expan abbr="cõtra">contra</expan> <expan abbr="oppona&ttilde;">opponatur</expan>. <lb/></s>
<s>Etenim <expan abbr="&longs;ept&etilde;">&longs;eptem</expan> modis <expan abbr="cũ">cum</expan> urbes, at que arces <expan abbr="capian&ttilde;">capiantur</expan>, <expan abbr="quorũ">quorum</expan> duo &longs;unt ex <lb/>tra <expan abbr="&ptilde;&longs;ent&etilde;">pn&longs;entem</expan> <expan abbr="con&longs;ideration&etilde;">con&longs;iderationem</expan> ob&longs;idio, <expan abbr="&qtilde;">quae</expan> magnitudine ambitus loci <expan abbr="tol-li&ttilde;">tol­<lb/>litur</expan>, & proditio, <expan abbr="&qtilde;">quae</expan> cu&longs;to <expan abbr="dũ">dum</expan> <expan abbr="uigilãtia">uigilantia</expan>, cuniculi, euer&longs;io &longs;uperioris muri, <lb/>euer&longs;io ab imo per machinas, cuniculi, &longs;eu &longs;uffo&longs;sio, urbis euer&longs;io, &longs;eu <lb/><expan abbr="&ecedil;dificiorũ">&ecedil;dificiorum</expan>: & <expan abbr="&qtilde;uo">quauo</expan> cant aggre&longs;sio, &longs;eu oppugnatio per &longs;calas, & crates <lb/><expan abbr="cũ">cum</expan> &longs;agittarijs: his omnib. </s>
<s><expan abbr="&longs;atisfactũ">&longs;atisfactum</expan> puto, pr&ecedil;ter <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 fa&longs;ciculis, terra que fo&longs; <lb/>&longs;&ecedil;: nihil. </s>
<s>n. </s>
<s>re&longs;i&longs;tit immen&longs;&ecedil; illi pote&longs;tati, & crudelitati <expan abbr="&longs;&ecedil;ui&longs;simorũ">&longs;&ecedil;ui&longs;simorum</expan> ty <lb/><expan abbr="rãnorũ">rannorum</expan>. </s>
<s><expan abbr="Verũ">Verum</expan>, ut dixi, terra noctu <expan abbr="effodi&ttilde;">effoditur</expan>, ligna artificio&longs;is ignib. </s>
<s>eru <pb pagenum="117"/>untur. </s>
<s>Et longum e&longs;t opus &longs;iue per paucos, &longs;iue per multos quis ef­<lb/>ficere conetur: ut non minus exigat temporis, quàm ob&longs;idio: nam <lb/>multitudine unus alterum impedit, & mortui uiuos, ut omnino res <lb/>&longs;it non &longs;peranda ni&longs;i aduer&longs;us inerti&longs;simos. </s>
<s>Pontes euertunt machi <lb/>næ, ignes que. </s>
<s>Sed ubi etiam muros obtinuerint ob rotunditatem in <lb/>illis con&longs;i&longs;tere non po&longs;&longs;unt. </s>
<s>Inde à defen&longs;oribus propul&longs;antur &longs;ari&longs;­<lb/>&longs;is, telis, ignibus, tran&longs;uer&longs;is trabibus, machinis: illudque accedit com <lb/>modi, ut quanto plures eo facilius excutiantur. </s>
<s>Dixi non debere <lb/>uereri maxima etiam præterid, quoniam & i&longs;t&ecedil; ip&longs;&ecedil; tanto &longs;anguine <lb/>acqui&longs;it&ecedil; tanto deorum & hominum iniuria modica &longs;cintilla ignis <lb/>&longs;ine munitionibus, exercitibus, &longs;iue machinis, ab&longs;que terræ <expan abbr="cõcu&longs;sio-ne">concu&longs;sio­<lb/>ne</expan>, aut inundatione, uel pe&longs;te euertuntur. </s>
<s>In illam mi&longs;eram lachry­<lb/>mam patris &longs;cintilla ignis inferni, cùm Deo placuerit, <expan abbr="mitti&ttilde;">mittitur</expan>, ex qua, <lb/>quod <expan abbr="coalitũ">coalitum</expan> e&longs;t, multis &longs;eculis imperium luxu, crudelitate, &longs;tultitia <lb/>unius filij, uix uno lu&longs;tro toto di&longs;&longs;oluitur. </s>
<s>Hanc <expan abbr="&longs;cintillã">&longs;cintillam</expan> cum felici <lb/>etiam genio &longs;ecum ex utero detulit Alexander Magnus. </s>
<s>In alijs alij <lb/>genium &longs;ortiti &longs;unt, alij <expan abbr="&longs;cintillã">&longs;cintillam</expan> detulere ab Orco. </s>
<s>Ex imperio A&longs;&longs;y <lb/>riorum per luxum Sardanapalus: ex Medorum per <expan abbr="&longs;cintillã">&longs;cintillam</expan> A&longs;tya­<lb/>ges: ex <expan abbr="Per&longs;arũ">Per&longs;arum</expan> per &longs;tultitiam Darius: ex <expan abbr="Romanorũ">Romanorum</expan> Honorius. </s>
<s>Di <lb/>ces, h&ecedil;c quid ad proportionem? </s>
<s>Imò uelut machina ad <expan abbr="perpendiculũ">perpendiculum</expan> <lb/>librata pauculo illo puluere Pyrio <expan abbr="urb&etilde;">urbem</expan> euertit, ita &longs;cintilla illa infer <lb/>ni ignis &longs;emini magni tyranni indita euertit at que di&longs;&longs;oluit totum re­<lb/>gnum &longs;ine machinis, ut dixi, uel exercitibus ullis, & quod maius e&longs;t <lb/>remedio nullo. </s>
<s>Sed puerulo indito luxus, ignauiæ, crudelitatis at que<lb/>&longs;tultiti&ecedil; fontibus, mirabile dictu &longs;anè, & ad proportionem diuino­<lb/>rum <expan abbr="in&longs;trumentorũ">in&longs;trumentorum</expan> pertinens. </s>
<s>Sed redeamus ad in&longs;titutum: Video <lb/>enim, quid po&longs;sit obijci, &longs;cilicet muros cra&longs;&longs;os, et altiores tueri <expan abbr="urb&etilde;">urbem</expan> <lb/>& ædificia illius po&longs;&longs;e ab&longs;que aggeris erectione, & &longs;i <expan abbr="diruan&ttilde;">diruantur</expan> manere <lb/>etiam nihilominus imo magis, quod e&longs;t terram, u&longs;que <expan abbr="quoniã">quoniam</expan> eadem <lb/>ratione manet, quia concuti non po&longs;sit à machinis: nec ho&longs;tes id cu <lb/>raturos, &longs;perantes hoc <expan abbr="&longs;olũ">&longs;olum</expan> &longs;ufficere, &qring;d mœnia &longs;olo <expan abbr="æquen&ttilde;">æquentur</expan>, at que id <lb/><expan abbr="factũ">factum</expan> e&longs;t Mediolani, & in arce eius, <expan abbr="tũ">tum</expan> Papi&ecedil; & in Cremonen&longs;i arce. <lb/></s>
<s>Verùm ni fallor, ut paruis arcibus à tanta ui tormentorum nullum <lb/>e&longs;t <expan abbr="præ&longs;idiũ">præ&longs;idium</expan>, aut &longs;alutis &longs;pes, ita neque <expan abbr="cõuenit">conuenit</expan>, ut muris humilibus ag <lb/>geri confidant, nam & pauci homines tanto labori non &longs;ufficerent, <lb/>& agger cum fo&longs;&longs;a effo&longs;&longs;a &longs;cilicet terra defen&longs;ores nimis in <expan abbr="angu&longs;tũ">angu&longs;tum</expan> <lb/>cogeret. </s>
<s>At in urbibus contra eueniet: muris enim erectis altius ma <lb/>chinæ lapidum fru&longs;tis hominem <expan abbr="occid&etilde;t">occident</expan>: an percu&longs;&longs;a &longs;uperiore par <lb/>te ob coniunctionem inferior concutitur, & in de <expan abbr="totũ">totum</expan> &longs;imul cadit, <lb/>ut uidimus Papi&ecedil;, quo <expan abbr="cad&etilde;te">cadente</expan>, & fo&longs;&longs;a impletur, & <foreign lang="greek">tEIkole/tois</foreign> facilior <lb/>aditus ad &longs;ubruendum reliquas partes <expan abbr="pr&ecedil;be&ttilde;">pr&ecedil;betur</expan>: imò percul&longs;i defen­<pb pagenum="118"/>&longs;ores &longs;æpe muneris &longs;ui obliui&longs;cuntur, de&longs;ertaque ea parte liberum <lb/>ingre&longs;&longs;um ho&longs;tibus exhibent. </s>
<s>Tum uerò magis, quod non confi­<lb/>dunt animo <expan abbr="nõ">non</expan> ad id parato, po&longs;&longs;e aggerem &longs;ufficientem, & in tam <lb/>breui tempore ex&longs;truere, & etiam intelligunt, antequam erigatur, <lb/>patere à lateribus introitum ho&longs;tibus.</s></p><p type="margin">
<s><margin.target id="marg411"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Propo&longs;itio cente&longs;imauige&longs;ima.</s></p><p type="main">
<s>Proportionem partium nauis ad eundem obliquum uentum <lb/>explorare.<lb/><arrow.to.target n="marg412"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg412"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Sint mali in naui a b c, ad b e, c fuentus è regione g h k etiam ad <lb/>perpendiculum feratur, ut anguli g d a, h e b, k f c &longs;int æquales, dico <lb/>tamen diuer&longs;o modo affici: nam cum premitur a uer&longs;us l, c premi­<lb/>tur uer&longs;us f: at &longs;i prematur cuer&longs;us n a, premitur uer&longs;us d, at &longs;i pre­<lb/><figure id="fig93"></figure><lb/>matur b uer&longs;us m, & a uer­<lb/>&longs;us l, &longs;ed non quantum ex <lb/>g d, & cuer&longs;us n, &longs;ed non <lb/>quantum ex k f, ab eodem <lb/>ergo uento contrarij mo­<lb/>tus efficiuntur ex uelorum <lb/>diuer&longs;itate, 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: & &longs;i uen­<lb/>tus excipiatur in f uelo, <lb/>non iuuabit clauus, & &longs;i in <lb/>d dirigetur, & temperabitur motus, & &longs;i in e medio modo. </s>
<s>Ergo &longs;i <lb/>uentus feratur rectè iuuabit, ut dici &longs;olet omnibus, & plenis uelis <lb/>excipere, &longs;i ex obliquo demittere antennam puppis, &longs;in autem ual­<lb/>de obliqu us &longs;it, &longs;olo proræ uelo utemur. </s>
<s>Si ualidior quàm oportet <lb/>humiliore. </s>
<s>Atque hæc po&longs;tmodum &longs;unt diligenter numeranda, ac <lb/>metienda: nunc &longs;ufficiat cau&longs;am reddidi&longs;&longs;e, & admonui&longs;&longs;e diuer&longs;i­<lb/>tatis motuum, quæ ex uelis contingit: nam eò fertur nauis, quò <lb/>prora dirigitur. </s>
<s>Ergo cum puppis tanto feratur uer&longs;us meridiem <lb/>a b, quanto prora uer&longs;us meridiem a d, & quanto puppis fertur uer <lb/>&longs;us <expan abbr="meridi&etilde;">meridiem</expan>, tanto prora fertur uer&longs;us boream, igitur quanto prora <lb/>fertur uer&longs;us meridiem a d, tanto uer&longs;us boream a b f, &longs;ed &longs;itus claui <lb/>pote&longs;t multo plus in comparatione ueli d, quam f &longs;cilicet, quia di­<lb/>&longs;tantia a b a e&longs;t o a, & di&longs;tantia e c e&longs;t o c, tanto plus ergo pote&longs;t cla­<lb/>ui &longs;itus in comparatione ad uelum d, quam f, quanta e&longs;t proportio <pb pagenum="119"/>o a, ad o c, igitur clauus e&longs;t longè potentior in comparatione ueli <lb/>d, quam f, ergo uelum d minus agit nauim, quam f. </s>
<s>Sed ut extrema <lb/>&longs;e habent, ita medium eorum comparatione, igitur malus b e uali­<lb/>dior e&longs;t, multo d a, & infirmior c f. </s>
<s>Verùm, ut dixi, ob &longs;itum &longs;impli­<lb/>citer ualidius e&longs;t, uelum e quam f, & etiam quia, ut dixi, altior & <lb/>era&longs;sior &longs;olet e&longs;&longs;e, ideo multo ualidior tribus his cau&longs;is, quàm e f: <lb/>adde quartam quòd uelum habet maius, antiquo tempore uoca­<lb/>tum acatius. </s>
<s>At ut etiam docui c b non e&longs;t in medio, nec æquidi&longs;tat <lb/>ab a d & c f, &longs;ed in clinatur ad proram ideoque imbecillior: cum ergo <lb/>&longs;it æqualium, & paulo maiorum uirium, quàm c f, & tutior, & me­<lb/>lius agatur per <expan abbr="clauũ">clauum</expan> quàm c f, & &longs;it a d nimis iu&longs;to imbecillis, pro­<lb/>pterea b e mali, & ueli maximus e&longs;t u&longs;us: adeò mali nomen per an­<lb/>tonoma&longs;iam de ip&longs;o &longs;impliciter intelligatur.</s></p><p type="main">
<s>Propo&longs;itio cente&longs;imauige&longs;imaprima.</s></p><p type="main">
<s>Flabelli uires, at que naturam declarare.</s></p><p type="main">
<s>Sit flabellum a b c appen&longs;um, ut &longs;olet, in a, & moueatur motu </s></p><p type="main">
<s><arrow.to.target n="marg413"></arrow.to.target><lb/>qua&longs;i circa axem p a q in parte inferiore, & aër comprehen&longs;us &longs;ub <lb/>b h k, & &longs;patium &longs;it 1 m figuræ nauicularis, quæ con&longs;tat e&longs;&longs;e par­<lb/>tem cylindri inanis ex formatione ab Euclide &longs;cripta: nam &longs;i pro­<lb/>poneretur p a q ad perpendiculum &longs;uper&longs;tans plano, fieret circum­<lb/>ducta a b c &longs;uperficie, quæ e&longs;&longs;et lata &longs;uperius, &longs;icut etiam inferius <lb/><arrow.to.target n="marg414"></arrow.to.target><lb/>cylindrus: at &longs;uperius a b tenuis e&longs;t, & angu&longs;ta, ergo fiet pars cy­<lb/>lindri inanis: quia non circunuoluitur, donecredeat. </s>
<s>Ergo per di­<lb/>cta &longs;uperius &longs;ectio illius p r q s per axem e&longs;t pars cuiu&longs;dam elly­<lb/><arrow.to.target n="marg415"></arrow.to.target><lb/>p&longs;is. </s>
<s>Et &longs;ectio quæuis planæ &longs;uperficiei æquidi&longs;tans a b cuelut tu, <lb/>item que æquidi&longs;tans axi p a q e&longs;t &longs;uperficies rectangula, quarum <lb/>una e&longs;t &longs;imilis, & æqualis b h k, e&longs;t in una &longs;uperficie cum axe p a q <lb/>alia uerò e&longs;t æquidi&longs;tans eidem axi maior aut minor æquidi&longs;tanti­<lb/>um, & ip&longs;a laterum, at que rectangula ac &longs;i cylindrus &longs;tans axi plano <lb/>æquidi&longs;tanti &longs;ecaretur iuxta longitudinem &longs;eu altitudinem &longs;uam: <lb/>& manife&longs;tum e&longs;t, quod i&longs;ta duo plana, & eorum &longs;uperficies &longs;ecant <lb/>&longs;e mutuò ad rectos angulos.</s></p><p type="margin">
<s><margin.target id="marg413"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>m.</s></p><p type="margin">
<s><margin.target id="marg414"></margin.target>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><margin.target id="marg415"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 6<gap/></s></p><p type="main">
<s>Quibus con&longs;titutis, qui &longs;tabunt iuxta l, & m longitudines aëris <lb/>moti, & loci, per quem tran&longs;it flabellum, &longs;entient magnum uentum, <lb/>quoniam cum corpus m x l ab extremis partibus &longs;it elatius a b ex­<lb/>tremis, &longs;tantes, & alti tangentur à uento agitato. </s>
<s>Si uero &longs;edeant, aer <lb/>primum non attinget illos, ut etiam quia &longs;ur&longs;um pellitur non per­<lb/>ueniet ad illos, imò diffugiet, ergo non refrigerabuntur. </s>
<s>Qui uerò <lb/>à lateribus l x m <expan abbr="&longs;tabũt">&longs;tabunt</expan> hiccinde, uelut in f g, &longs;i &longs;teterint, <expan abbr="nõ">non</expan> refrigeræ <lb/><expan abbr="bũtur">buntur</expan>, quia <expan abbr="quãdo">quando</expan> flabellum erit in l, uel m aer de&longs;cendet, ergo fugi <lb/>et ab illis, cum autem fuerit in x, erit in loco humiliori, & mouebi­<pb pagenum="120"/>tur diuer&longs;a ratione, quippe ab f in h, & non ad latera, ergo ne que <lb/><figure id="fig94"></figure><lb/>contactu, neque motu, qui <lb/>fiet per æquidi&longs;tantem f, <lb/>& g non poterunt refrige­<lb/>rari. </s>
<s>Sed &longs;i humili loco &longs;e­<lb/>deant, quoniam aër de&longs;cen <lb/>dit, ex l & m uer&longs;us 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>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> utri&longs;que, &longs;ed pau <lb/>lulum in reditibus propin <lb/>quis, & neque &longs;tantes, <expan abbr="neque&longs;ed&etilde;tes">neque<lb/>&longs;edentes</expan>, &longs;ed &longs;i altius attolla­<lb/>tur h k. </s>
<s>Rur&longs;us &longs;i b h k fue­<lb/>rit grauior eodem, ut de­<lb/>&longs;cendat tanto impetu, <expan abbr="quã-to">quan­<lb/>to</expan> a&longs;cendit 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, &longs;ed quoniam <lb/>celerius occur&longs;antes &longs;ibi <lb/>contrarijs motibus, ac <expan abbr="ue-hem&etilde;tibus">ue­<lb/>hementibus</expan> fiet colli&longs;io par <lb/>tium aëris, & ideo in ambitum impelletur, & undique cubiculum <lb/>refrigerabit, quod non faciet maius longè flabellum lento motu <lb/>agitatum, aut ex materia leui. </s>
<s>Idem multo magis contingeret, ubi <lb/>duo e&longs;&longs;ent flabella laquearibus appen&longs;a, quæ ad perpendiculum <lb/><expan abbr="a&etilde;rem">aerrem</expan> mouerent, &longs;eu quod &longs;uperficies eo modo &longs;e haberent: & &longs;i <lb/>flabella rotunda e&longs;&longs;ent, tunc maiorem ambitum aëris occuparent, <lb/>& uelocius deficientibus angulis mouebuntur.</s></p><p type="main">
<s>Propo&longs;itio cente&longs;imauige&longs;ima&longs;ecunda.</s></p><p type="main">
<s>Contemptus circa &longs;olis rationem in umbris declarare.</s></p><p type="main">
<s>Con&longs;tat primùm &longs;olem, & excentro, & toto eius ambitu illumi­<lb/>nare hanc primùm diuer&longs;itatem, quæ aliquando tota diametro <lb/>computata dimidium unius partis totius cœli excedit: &longs;cioterici <lb/>negligunt, ut exiguam. </s>
<s>Secundò etiam diuer&longs;itatis illius, qua mo­<lb/>dò à terra uer&longs;us ab&longs;idem defertur, modò ad terram de&longs;cendere to­<lb/>tidem uariata altitudine, non parum nullam habent rationem, &longs;eu <pb pagenum="121"/>quòd tanta ne &longs;it, ut euidentem in gnomonibus faciat uarietatem, <lb/>&longs;eu quòd incertum adhuc &longs;it, an id uerè &longs;oli accidat. </s>
<s>Tertium e&longs;t fi­<lb/>nis umbræ ip&longs;ius gnomonis, qui incertus e&longs;t, ut pars non contem­<lb/>nenda in dubium uertatur, quoniam &longs;en&longs;im ex ob&longs;curo in illumi­<lb/>natum feratur, attamen contemnitur etiam. </s>
<s>Quartum quòd cum <lb/>&longs;ol moueatur in &longs;pira, fingitur qua&longs;i in parallelo æquinoctiali circu <lb/>lo circumagatur ab his, qui horologia de&longs;cribunt. </s>
<s>Quintum quòd <lb/>cum inæqualiter in orbe &longs;uo moueatur quanuis exigua &longs;it hæc dif­<lb/>ferentia, æqualiter <expan abbr="tam&etilde;">tamen</expan> moueri præ&longs;upponitur. </s>
<s>Sextum e&longs;t, quòd <lb/>dies æquales &longs;upponuntur, qui tamen tum ex ratione partis pera­<lb/>gratæ, tum ratione a&longs;cen&longs;us <expan abbr="eiu&longs;d&etilde;">eiu&longs;dem</expan> &longs;unt inæquales, & <expan abbr="tam&etilde;">tamen</expan> hæc in­<lb/>qualitas <expan abbr="etiã">etiam</expan> in <expan abbr="horarũ">horarum</expan> computatione prætermittitur. </s>
<s>Sed & h&ecedil;c ut <lb/>prior ratione magis, <expan abbr="quã">quam</expan> &longs;en&longs;u <expan abbr="deprehendi&ttilde;">deprehenditur</expan>. </s>
<s><expan abbr="Septimũ">Septimum</expan> e&longs;t di&longs;crimen, <lb/>&qring;d oritur ex ui&longs;us circulo &longs;eu horizonte, & circulo tran&longs;eunte p cen <lb/><expan abbr="trũ">trum</expan> mundi, nam horizon uere <expan abbr="tãto">tanto</expan> minor e&longs;t circulo magno, quan­<lb/>tum e&longs;t &longs;emidiameter terr&ecedil;, <expan abbr="cõparatus">comparatus</expan> ad <expan abbr="&longs;emidiametrũ">&longs;emidiametrum</expan> orbis cœle <lb/>&longs;tis, &longs;ed e&longs;t in&longs;en&longs;ilis quantitatis. </s>
<s><expan abbr="Octauũ">Octauum</expan> e&longs;t, quod trianguli ex gno­<lb/>mone umbra, & radijs &longs;olis latera non mutant lineas, quæ à &longs;ole ad <lb/>centrum terræ deueniunt, nec quòd maius e&longs;t, radius &longs;olis ad uerti­<lb/>cem hominis breuior habetur femidimetiente. </s>
<s>Hæc <expan abbr="igi&ttilde;">igitur</expan> omnia <expan abbr="&longs;ci-otericorũ">&longs;ci­<lb/>otericorum</expan> opifices non ob&longs;eruant, &longs;ed negligunt. </s>
<s>Verum quatuor <lb/>tantùm altitudinem poli regionis locum &longs;olis in eclyptica locum <lb/>&longs;olis in circulo æquinoctialis, uel æquinoctiali parallelo, ex qui­<lb/>bus tribus fit altitudo &longs;olis, una in circulo &longs;cilicet uerticali ab hori­<lb/>zonte, & differentia lineæ meridianæ à linea uer&longs;us polum, quam <lb/><arrow.to.target n="marg416"></arrow.to.target><lb/>o&longs;tendit lapis Herculeus, de qua dictum e&longs;t &longs;uperius.</s></p><p type="margin">
<s><margin.target id="marg416"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 84.</s></p><p type="main">
<s>Propo&longs;itio cente&longs;imauige&longs;imatertia.</s></p><p type="main">
<s>Cognita ratione umbr&ecedil; ad gno <lb/>monem &longs;inum, & arcum altitudi­<lb/>nis ab horizonte quouis tempo­<lb/>re digno&longs;cere.</s></p><figure></figure><p type="main">
<s>Sit circulus magnus, in quo &longs;ol <lb/><arrow.to.target n="marg417"></arrow.to.target><lb/>a f g &longs;uper&longs;tans ad perpendicu­<lb/>lum circulo ui&longs;us f e g, quos mani <lb/>fe&longs;tum e&longs;t tran&longs;ire per idem cen­<lb/>trum mundi c, quia magni &longs;unt, & <lb/>&longs;it c d erecta ad perpendiculum <lb/>&longs;uper f g, nam perinde e&longs;t per &longs;e­<lb/>ptimum contemptum, ac &longs;i &longs;uper­<lb/><arrow.to.target n="marg418"></arrow.to.target><lb/>ficies horizontis tran&longs;eat per terr&ecedil; centrum, & pedes per octauum, <lb/><arrow.to.target n="marg419"></arrow.to.target><lb/>ideo proportio e c ad c d umbræ ad gnomonem, ut b e ad b a, ergo <pb pagenum="122"/>per demon&longs;trata b a cognita in comparatione a d e a, e a autem per <lb/>octauum contemptum e&longs;t dimetiens circuli, ergo a b &longs;inus notus, <lb/>& arcus f a, quod e&longs;t primum cognitum. </s>
<s>Et hic quidem circulus <lb/>uerticalis dicitur, quia per illum tran&longs;it, aliter non e&longs;&longs;et ad perpen­<lb/>diculum horizonti.<lb/><arrow.to.target n="marg420"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg417"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>_{m}.</s></p><p type="margin">
<s><margin.target id="marg418"></margin.target>P<emph type="italics"/>ræced.<emph.end type="italics"/> P<emph type="italics"/>ro <lb/>po&longs;.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg419"></margin.target>P<emph type="italics"/>rop.<emph.end type="italics"/> 113.</s></p><p type="margin">
<s><margin.target id="marg420"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 1.</s></p><p type="main">
<s>Ex hoc &longs;equitur, quod altitudines &longs;olis æquales omnes in uno <lb/>&longs;unt circulo horizonti parallelo. </s>
<s>Et &longs;i &longs;ol fuerit in uno circulo ho­<lb/>rizonti parallelo, altitudines &longs;olis, & umbræ magnitudines æqua­<lb/>les erunt.</s></p><p type="main">
<s><arrow.to.target n="marg421"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg421"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 2.</s></p><p type="main">
<s>Sol ni&longs;i bis in una die pote&longs;t e&longs;&longs;e in circulo horizonti parallelo, <lb/>&longs;emel ante meridiem, & &longs;emel po&longs;t, tantundem ab eodem di&longs;tans.</s></p><p type="main">
<s><arrow.to.target n="marg422"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg422"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 3.</s></p><p type="main">
<s>Cum ergo ita &longs;it, nece&longs;&longs;e e&longs;t umbras æquales, & circulum hori­<lb/>zonti <expan abbr="parallelũ">parallelum</expan> fieri &longs;ub in æqualibus horis in diuer&longs;is &longs;emper die­<lb/>bus, præterquam cum in punctis fuerit æqualis ab &ecedil;quinoctiali, & <lb/>in eandem partem declinationis, & hoc bis <expan abbr="cõtingit">contingit</expan> &longs;olum in anno <lb/>pro quolibet circulo parallelo, &longs;icut in eodem die etiam bis <expan abbr="tãtum">tantum</expan>, <lb/>ut dictum e&longs;t.</s></p><p type="main">
<s><arrow.to.target n="marg423"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg423"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Nam exempli gratia, cum &longs;ol e&longs;t in initio Capricorni, & in Cœli <lb/>medio, minima e&longs;t umbra eius diei, & totius anni. </s>
<s>Cum ergo fuerit <lb/>ante meridiem, uel po&longs;t, erit umbra maior ex &longs;uppo&longs;ito &longs;ecudo um­<lb/>bra meridiei: at ei æqualis poterit e&longs;&longs;e umbra meridiei alterius diei <lb/>ex primo &longs;uppo&longs;ito, ergo umbræ æquales diuer&longs;orum dierum fi­<lb/>unt &longs;ub diuer&longs;o &longs;itu &longs;olis, quo æd circulum meridiei, quod erat de­<lb/>mon&longs;trandum.</s></p><p type="main">
<s><arrow.to.target n="marg424"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg424"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 4.</s></p><p type="main">
<s>Ex hoc &longs;equitur, quod horarum determinatio fit &longs;ecundum line­<lb/>am in æqualem obliquam, quæ toti anno &longs;eruiat, ut æqualium um­<lb/>brarum determinatio hararum & partium eius numerum.</s></p><p type="main">
<s><arrow.to.target n="marg425"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg425"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 5.</s></p><p type="main">
<s>Ex quo colligitur modus faciendi gnomonem, &longs;eu per umbras <lb/>rectas, &longs;eu per uer&longs;as, qui docebit toto anno non <expan abbr="&longs;olũ">&longs;olum</expan> horas, &longs;ed mo <lb/>menta <expan abbr="pul&longs;uũ">pul&longs;uum</expan>, de quibus <expan abbr="dictũ">dictum</expan> e&longs;t quod MMMDC horam <expan abbr="perficiũt">perficiunt</expan>.</s></p><p type="main">
<s>Propo&longs;itio cente&longs;imauige&longs;imaquarta.</s></p><p type="main">
<s>Proportionem umbræ uer&longs;æ e&longs;&longs;e ad gnomonem, uelut gnomo­<lb/>nis ad umbram uer&longs;am.</s></p><p type="main">
<s><arrow.to.target n="marg426"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg426"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Vmbra uer&longs;a dicitur, quoties gnomo in pariete ad perpendicu­<lb/>lum figitur, &longs;ic ut gnomo æquidi&longs;tet circulo horizontis. </s>
<s>Sit ergo <lb/>paries c k ad perpendiculum f g, & h k a d gnomo ad perpendicu­<lb/>lum parietis & &longs;ol, ut prius in a, & &longs;it primo k h tantæ longitudinis </s></p><p type="main">
<s><arrow.to.target n="marg427"></arrow.to.target><lb/>ut umbræ locus &longs;it <expan abbr="pũctus">punctus</expan> d, ut &longs;it radius a h d e, eritque angulus d u­<lb/>trin que æqualis, & propterea triangulus k h d &longs;imilis d c e. </s>
<s>Sit modo <lb/><arrow.to.target n="marg428"></arrow.to.target><lb/>gnomo maior m l ip&longs;o h k & c l maior c k &longs;eu æqualis, & quam an­<lb/>guli k & l recti &longs;unt, & anguli l m n, & k h d æqualis, quia a n, & a c <pb pagenum="123"/>&longs;unt æquidi&longs;tantes per octauum contemptum, erunt per dicta tri­<lb/>anguli &longs;imiles, igitur proportio l m gnomonis ad l n umbram <lb/>ut k h gnomonis ad k d umbram, &longs;ed k h, ad k d, ut c e umbræ ad c d <lb/>gnomonem: igitur proportio l m gnomonis ad l n <expan abbr="umbrã">umbram</expan>, ut um­<lb/>bræ c e ad c d gnomonem, quod fuit demon&longs;trandum.</s></p><p type="margin">
<s><margin.target id="marg427"></margin.target>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><margin.target id="marg428"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 4. <emph type="italics"/>&longs;exti<emph.end type="italics"/><lb/>E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="main">
<s>Ex hoc primùm patet & pr&ecedil;cedenti, quod cognita proportione <lb/><arrow.to.target n="marg429"></arrow.to.target><lb/>umbr&ecedil; uer&longs;&ecedil; ad gnomonem cogno&longs;citur &longs;inus &longs;olis, & arcus altitu­<lb/>dinis in circulo magno, & e&longs;t altitudo ab horizontis parte, quæ <lb/>proximior e&longs;t loco &longs;olis, ut demon&longs;tratum à nobis in Geometricis.</s></p><p type="margin">
<s><margin.target id="marg429"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 1.</s></p><p type="main">
<s>Se quitur etiam, quòd cùm umbra fuerit æqualis gnomoni, &longs;eu <lb/><arrow.to.target n="marg430"></arrow.to.target><lb/>recta, &longs;eu uer&longs;a &longs;olis, uel Lunæ, uel &longs;tellæ, altitudo erit partium qua­<lb/>draginta quin que: nam anguli d & e, uel d & h erunt æquales: igitur <lb/>arcus f a medietas quartæ ideò partium xlv. </s>
<s>Et &longs;i gnomo fuerit ma­<lb/>ior umbra uer&longs;a, uel minor recta, erit arcus f a minor xlv partibus, &longs;i <lb/>contrà maior. </s>
<s>Et hoc ubique terrarum. </s>
<s>Et ubi non po&longs;sit tantundem <lb/>eleuari, ut quando &longs;ol e&longs;t &longs;ub circulo capricorni, nunquam nobis <lb/><arrow.to.target n="marg431"></arrow.to.target><lb/>gnomo æquabitur umbræ rectæ &longs;ed &longs;emper erit minor, & &longs;emper <lb/><arrow.to.target n="marg432"></arrow.to.target><lb/>maior umbra uer&longs;a pari ratione.</s></p><p type="margin">
<s><margin.target id="marg430"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 2.</s></p><p type="margin">
<s><margin.target id="marg431"></margin.target>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><margin.target id="marg432"></margin.target>P<emph type="italics"/>er ult. </s>
<s>&longs;exti<emph.end type="italics"/><lb/>E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="main">
<s>Propo&longs;itio cente&longs;imauige&longs;imaquinta.</s></p><p type="main">
<s>Proportionem dimetientis, & peripheri&ecedil; cuiuslibet circuli paral <lb/>leli æquinoctiali per cognitam partem magni circuli demon&longs;trare.</s></p><p type="main">
<s>Hæc erat tam clara, ut hic locum non mereretur: tam nece&longs;&longs;aria <lb/><arrow.to.target n="marg433"></arrow.to.target><lb/>huic propo&longs;ito, ut non potuerit omitti. </s>
<s>Sit ergo Aequinoctij circu­<lb/>lus a b portio circuli magni nota, a c parallelus circulus, &ecedil;quinoctij <lb/>circulo c d, erit igitur &longs;inus c d notus. </s>
<s>Et ideò <expan abbr="quadratũ">quadratum</expan> c d notum, <lb/><arrow.to.target n="marg434"></arrow.to.target><lb/>ergo & pars utraque b d d a nota. </s>
<s>Quare detracta a d ex d b relin qui­<lb/>tur d g æqualis f c diametro paralleli a&longs;signari. </s>
<s>Quare proportio <lb/><arrow.to.target n="marg435"></arrow.to.target><lb/>a b ad e f nota ex obiter &longs;uprà demon&longs;tratis, & pariter ambi­<lb/>tus circuli a b ad ambitum circuli c d, e&longs;t enim ut dimetientis ad di­<lb/><arrow.to.target n="marg436"></arrow.to.target><lb/>metientem.</s></p><p type="margin">
<s><margin.target id="marg433"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s><margin.target id="marg434"></margin.target>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"/>&longs;exti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg435"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 5. <emph type="italics"/>&longs;ecu<gap/><lb/>di<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg436"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 113. <lb/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/></s></p><p type="main">
<s>Propo&longs;itio cente&longs;imauige&longs;ima&longs;exta.</s></p><p type="main">
<s>Circuli horarij naturam declarare.<lb/><arrow.to.target n="marg437"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg437"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><figure></figure><p type="main">
<s>Circulus horarius e&longs;t circulus magnus <lb/>tran&longs;iens per <expan abbr="&longs;ol&etilde;">&longs;olem</expan>, aut lunam, aut quoduis <lb/>&longs;ydus, de quo agitur, & per polos mundi, <lb/>ideò differt à circulo priore altitudinis So­<lb/>lis, quia ille &longs;tat ad perpendiculum &longs;uper <lb/>horizontem, ni&longs;i cum tangitur uice meridi­<lb/>ani, uterque tamen tran&longs;it per <expan abbr="centrũ">centrum</expan> mundi, <lb/>ac &longs;olis. </s>
<s>Hic etiam ad &longs;imiles partes æqui­<lb/>noctij circulum, & omnes parallelos &longs;ecat. <pb pagenum="124"/>Et principalis e&longs;t meridianus, ideò ab illo A&longs;trologi horas utrinque<lb/>ante, & po&longs;t numerant. </s>
<s>Ideò <expan abbr="clarũ">clarum</expan> e&longs;t, quòd horæ à meridie com­<lb/>putatæ &longs;unt <expan abbr="cõmunes">communes</expan>, habitantibus &longs;ub quauis altitudine poli, & <lb/>ubiuis &longs;it, &longs;ol modò regiones æqualiter di&longs;tent à fortunatis, &longs;eu &longs;int <lb/>in eadem longitudine.</s></p><p type="main">
<s>Propo&longs;itio cente&longs;imauige&longs;ima&longs;eptima.</s></p><p type="main">
<s>Data Poli altitudine ortus amplitudinem demon&longs;trare.<lb/><arrow.to.target n="marg438"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg438"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Sit horizon a d b æquinoctij circulus <lb/><figure id="fig95"></figure><lb/>a k f eclyptica c g, & punctus ortus in ea g. <lb/></s>
<s>& c initium arietis, & g b amplitudo ortiua <lb/>& c e, c f quartæ circulorum, ut &longs;it e f maxi­<lb/>ma &longs;olis declinatio, & polus mundi borea­<lb/>lis l, quia igitur l d nota e&longs;t ex &longs;uppo&longs;ito, & <lb/>l k quadrans erit k h <expan abbr="re&longs;iduũ">re&longs;iduum</expan> ad dimidium <lb/>circuli notum. </s>
<s>Quia uerò æquinoctium, & <lb/>Meridianus &longs;ecant &longs;e ad angulos rectos, & <lb/>b a æquidi&longs;tat ab utro que polo, erit b polus <lb/>h d, quare b k, quarta circuli, & angulus k <lb/>rectus. </s>
<s>Igitur &longs;umus in di&longs;po&longs;itione tabula­<lb/>rum primi mobilis, ergo etiam oppo&longs;itus <lb/>triangulus, qui ei e&longs;t æqualis, & &ecedil;quiangu­<lb/>lus in eadem di&longs;po&longs;itione b m d, quare cum <lb/>data &longs;it g n declinatio <expan abbr="pũcti">puncti</expan> g dati, datus erit, & arcus g b quæ&longs;itus.</s></p><p type="main">
<s>Propo&longs;itio cente&longs;imauige&longs;imaoctaua.</s></p><p type="main">
<s>Nota amplitudine ortus cuiu&longs;que <expan abbr="pũcti">puncti</expan> <expan abbr="arcũ">arcum</expan> <expan abbr="&longs;emidiurnũ">&longs;emidiurnum</expan> inuenire.</s></p><p type="main">
<s><arrow.to.target n="marg439"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg439"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Sit in eadem figura nota g b, uolo illius <expan abbr="arcũ">arcum</expan> &longs;emidiurnum. </s>
<s>Cum <lb/>ergo g n &longs;it declinatio, erit pars arcus Meridiani horarij per polos <lb/>tran&longs;euntis, compleatur ergo l g n o, & quia g n nota e&longs;t, quia de­<lb/>clinatio puncti dati, & g b nota ex &longs;uppo&longs;ito, & f angulus rectus, <lb/>quia e f e&longs;t portio meridiani, erit b n nota differentia a&longs;cen&longs;ionis a <lb/>quarta circuli k b, <expan abbr="igi&ttilde;">igitur</expan> tota k n arcus &longs;emidiurnus. </s>
<s><expan abbr="Quoniã">Quoniam</expan> g p paral <lb/>lelus &longs;imilis e&longs;t k n, & in eo <expan abbr="reuolui&ttilde;">reuoluitur</expan> Sol: ergo quando enim perue­<lb/>niet ad p. </s>
<s>Po&longs;&longs;umus etiam &longs;ine inuentione arcus ortus amplitudi­<lb/>nis per triangulum k m d ex notitia g n cogno&longs;cere eandem n b.</s></p><p type="main">
<s><arrow.to.target n="marg440"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg440"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Ex his duabus &longs;equitur <expan abbr="cõuer&longs;a">conuer&longs;a</expan> &longs;cilicet, quae data magnitudine diei <lb/><expan abbr="cuiu&longs;cũque">cuiu&longs;cunque</expan> in quauis regione nota erit poli altitudo <expan abbr="eiu&longs;d&etilde;">eiu&longs;dem</expan> regionis.</s></p><p type="main">
<s>Propo&longs;itio cente&longs;imauige&longs;imanona.</s></p><p type="main">
<s>Data altitudine &longs;olis in quacunque regione quacunque die di&longs;tan­<lb/>tiam &longs;olis à Meridiano cogno&longs;cere.</s></p><p type="main">
<s><arrow.to.target n="marg441"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg441"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Sit Horizon a b c d æquinoctij circulus b e d. </s>
<s>Meridianus a e c <lb/>Polus mundi Borealis f uertex, g, <expan abbr="pũctus">punctus</expan> in eclyptica h ducatur ex <pb pagenum="125"/>polo mundi circulus horarius f h k ad æquinoctij circulum, & uer­<lb/>ticalis circulus p h l u&longs;que ad Horizontem, & circulus parallelus æ­<lb/>quinoctij circulo h m, &longs;it ergo h l altitudo &longs;olis nota, igitur h g nota </s></p><p type="main">
<s><arrow.to.target n="marg442"></arrow.to.target><lb/>erit re&longs;iduum quart&ecedil; circuli, & &longs;imiliter h k <lb/><figure id="fig96"></figure><lb/>nota, quia declinatio puncti dati in eclypti <lb/>ca e&longs;t n nota dies, & locus &longs;olis ex &longs;uppo&longs;i­<lb/>to ergo nota fh <expan abbr="re&longs;iduũ">re&longs;iduum</expan> quart&ecedil; circuli no­<lb/>ta e&longs;t <expan abbr="etiã">etiam</expan> g e, quæ e&longs;t &ecedil;qualis altitudini po­<lb/>li ex &longs;uppo&longs;ito, ergo re&longs;iduum quadrantis <lb/>f g, ergo triangulus f g h notorum laterum <lb/>ergo notus angulus f, ergo arcus k e di&longs;tan <lb/><arrow.to.target n="marg443"></arrow.to.target><lb/>tia &longs;umpta in æquinoctij circulo puncti h, <lb/>cui &longs;imilis e&longs;t arcus h m ex parallelo h m, nam quando k perueniet <lb/><arrow.to.target n="marg444"></arrow.to.target><lb/>in e h perueniet in m, & in æquali tempore, qua diui&longs;a per quinde­<lb/>cim gradus, habebimus horas <expan abbr="di&longs;tãti&ecedil;">di&longs;tanti&ecedil;</expan> &longs;olis à Meridie ante, uel po&longs;t, <lb/>& minuta horarum dando quibuslibet gradibus quatuor minuta <lb/>horæ, & quibuslibet minutis graduum quatuor &longs;ecunda horæ, & <lb/>ita habebimus tempus exacti&longs;simum à Meridie in quacunque regi­<lb/>one, & in quacunque hora diei.</s></p><p type="margin">
<s><margin.target id="marg442"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 123. <lb/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg443"></margin.target>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 34. <lb/><emph type="italics"/>lib.<emph.end type="italics"/> 4.</s></p><p type="margin">
<s><margin.target id="marg444"></margin.target>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>Propo&longs;itio cente&longs;imatrige&longs;ima.</s></p><p type="main">
<s>Data regionis altitudine, & loco &longs;olis proportionem gnomo­<lb/>nis tam ad umbram rectam, quam uer&longs;am, uel etiam in cylindro de­<lb/>terminare.</s></p><p type="main">
<s>H&ecedil;c e&longs;t propo&longs;itio illa pulcherrima, quam tot ambagibus tradi­<lb/><arrow.to.target n="marg445"></arrow.to.target><lb/>dere antiqui cum &longs;uis analematibus, & &longs;cioteris, nec tamen demon <lb/>&longs;trationem, nec rationem exactam in&longs;trumenortum con&longs;tructio­<lb/>nem, qua po&longs;&longs;emus per umbras rectas uer&longs;as, & cylindricas &longs;cire ad <lb/>unguem, qualis hora, & minutum, & &longs;ecundum diei e&longs;&longs;et quocun­<lb/>que anni tempore. </s>
<s>Plerique autem tam laborio&longs;è id conati &longs;unt de­<lb/>mon&longs;trare, ut &longs;tudio&longs;os deterruerint ab opere: res autem ip&longs;a facil­<lb/>lima e&longs;t. </s>
<s>Propo&longs;ita ergo Poli exacta altitudine &longs;olis in Meridie <lb/>declinatione addita uel detracta, habebis re&longs;iduum eius ad qua­<lb/>drantem f g, & &longs;imiliter habebis ex declinatione nota loci &longs;olis de­<lb/>tracta à quadrante f h & iuxta horam tuam, & minutum multi­<lb/><arrow.to.target n="marg446"></arrow.to.target><lb/>plicatum per quindecim arcum k e quare angulum f, ex quo arcum <lb/>g h, quare re&longs;iduum h l, igitur punctum umbr&ecedil; rect&ecedil;, uel uer&longs;&ecedil; ip&longs;i­<lb/>us gnomonis ad unguem, & ita con&longs;titues horologium exacti&longs;si­<lb/>mum &longs;ecundum ea, quæ dixi in Corrolarijs &longs;upradictis, & quia ho­<lb/><arrow.to.target n="marg447"></arrow.to.target><lb/>rizon a b c d &longs;ecat æquinoctialem in <expan abbr="c&etilde;tro">centro</expan> terræ ducta g h k, erunt <lb/>anguli b h g, & k h l &ecedil;quales. </s>
<s>Igitur po&longs;ito g ortu puncti eclypti­<lb/>cæ, erit g b ortus amplitudo nota, & ideò angulus b h g, & k h l <pb pagenum="126"/><arrow.to.target n="marg448"></arrow.to.target><lb/>notus, & ita extendemus per totum annum. </s>
<s>Cum uerò fuerit g ele­<lb/>uatus erit, ut <expan abbr="demõ&longs;tratum">demon&longs;tratum</expan> e&longs;t, in circulo magno uerticali, ergo an­<lb/>gulus fiet in eodem circulo, quia gnomo e&longs;t etiam in illius &longs;uperfi­<lb/>cie. </s>
<s>Ergo angulus erit æqualis angulo, quem faceret &longs;ol, &longs;i oriretur <lb/><arrow.to.target n="marg449"></arrow.to.target><lb/><figure id="fig97"></figure><lb/>in puncto horizontis, quem &longs;ecat circulus <lb/>uerticalis &longs;ub ea altitudine: &longs;ed his e&longs;t no­<lb/>tus: nam in priore figura g h f e&longs;t notus ea­<lb/><arrow.to.target n="marg450"></arrow.to.target><lb/><expan abbr="d&etilde;">dem</expan> ratione, qua f, & ideò ei oppo&longs;itus k h n, <lb/>& k rectus, e&longs;t enim f polus b d, & h k decli <lb/>natio nota ergo k n, & h n notæ. </s>
<s>At e k, & <lb/>g h fuere notæ. </s>
<s>Ergo e n, & g n, quare re&longs;i­<lb/>duæ n l & n b notæ. </s>
<s>E&longs;t autem angulus l <lb/>rectus. </s>
<s>ergo ortus amplitudo punctil nota <lb/>&longs;cilicet arcus l b, ergo in præ&longs;enti figura angulus m h b, ergo k h l. <lb/></s>
<s>igitur poterimus &longs;tatuere angulos umbrarum, & iam po&longs;&longs;umus <lb/>determinare magnitudinem: ergo punctum ad <expan abbr="ungu&etilde;">unguem</expan> umbr&ecedil; qua­<lb/>libet hora, & parte horæ &longs;ingulis diebus in quacunque regione datæ <lb/>altitudinis poli uer&longs;a, & rects. </s>
<s>In cylindrica autem eodem modo &longs;i­<lb/>cut in uer&longs;a, e&longs;t enim &longs;pecies umbr&ecedil; uer&longs;&ecedil;, ni&longs;i quod analema ob ob­<lb/>liquitatem cylindri melius aptatur, rotundum &longs;cilicet cum <expan abbr="rotũdo">rotundo</expan>.</s></p><p type="margin">
<s><margin.target id="marg445"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s><margin.target id="marg446"></margin.target>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>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><margin.target id="marg447"></margin.target>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&longs;.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg448"></margin.target>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><margin.target id="marg449"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 127. <lb/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg450"></margin.target>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>Propo&longs;itio cente&longs;imatrige&longs;imaprima.</s></p><p type="main">
<s>Si lineæ 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>Sit a b linea, cui adiecta &longs;it b c, & rur&longs;us ad b c c d <expan abbr="æ&qacute;ualis">æqualis</expan> b c <lb/>dico, quod proportio a c ad a b e&longs;t maior, quàm a d ad a c. </s>
<s>Propor <lb/><arrow.to.target n="marg451"></arrow.to.target><lb/>tio enim c d ad c a minor e&longs;t, quàm ad a b per octauam quinti E­<lb/>lementorum. </s>
<s>Ergo minor d c ad c a quàm c b ad a b, quia b c & c d <lb/>&longs;unt æquales, ideò <expan abbr="æqual&etilde;">æqualem</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="fig98"></figure><lb/>tio d a ad a c minor, quam c a ad a b, quod erat demon&longs;trandum.<lb/><arrow.to.target n="marg452"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg451"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>m.</s></p><p type="margin">
<s><margin.target id="marg452"></margin.target>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>Propo&longs;itio cente&longs;imatrige&longs;ima&longs;ecunda.</s></p><p type="main">
<s>Si ad duas lineas, quarum una alteri dupla &longs;it eadem linea adda­<lb/>tur erit aggregati ex minore, & a d adiecta ad ip&longs;am <expan abbr="minor&etilde;">minorem</expan> minor <lb/>proportio quam aggregati ex maiore, & adiecta ad ip&longs;am maio­<lb/>rem duplicata.</s></p><p type="main">
<s><arrow.to.target n="marg453"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg453"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>m.</s></p><p type="main">
<s>Sint duæ line&ecedil; a b, & c d. </s>
<s>& &longs;it c d dupla ad a b, ad datur <expan abbr="cõmunis">communis</expan> <lb/><figure id="fig99"></figure><lb/>b e, & uo cetur iuncta c d, d f dico, <lb/>quod proportio e a ad a b, e&longs;t mi­<lb/>nor duplicata f c ad c d, adij cia­<lb/>tur d f æqualis g f, quia ergo g d <lb/>e&longs;t dupla ad f d, ideo ad e b c d autem e&longs;t du pla ad a b, tota igitur <pb pagenum="127"/>g c duplatoti e a. </s>
<s>quare ut g c ad g d ut e a ad e b <expan abbr="permutãdo">permutando</expan>, & per <lb/>euer&longs;am 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 ei&longs;dem. </s>
<s>Proportio <lb/>autem g c ad f c e&longs;t minor, quam f c ad c d, igitur minor quàm du­<lb/>plicata f c ad c d. </s>
<s>con&longs;tat uerò ex ei&longs;dem, quod proportio c a ad a b <lb/>maior e&longs;t duplicata g c ad f c.</s></p><p type="main">
<s>Propo&longs;itio cente&longs;imatrige&longs;imatertia.</s></p><p type="main">
<s>Si fuerint duæ quantitates, quarum una alteri dupla &longs;it: minua­<lb/>tur à minore quædam <expan abbr="quãtitas">quantitas</expan> eademque maiori addatur, erit mino­<lb/>ris ad <expan abbr="re&longs;iduũ">re&longs;iduum</expan> maior proportio, <expan abbr="quã">quam</expan> aggregati ad <expan abbr="maior&etilde;">maiorem</expan> duplicata. <lb/></s>
<s>Si uerò minori addatur et à maiore detrahatur, erit aggregati ad mi<lb/>nore m minor proportio quàm maioris ad re&longs;iduum duplicata.</s></p><p type="main">
<s><arrow.to.target n="marg454"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg454"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>m.</s></p><figure></figure><p type="main">
<s>Sit a b dupla c d, & addatur quæ­<lb/>dam ad b a, qu&ecedil; &longs;it a g, eadem detraha­<lb/>tur ex c d & &longs;it c h, dico, quod propor­<lb/>tio e d ad d h maior e&longs;t, quam duplica­<lb/>ta g b ad a b, & rur&longs;us &longs;i quædam ad c & minuatur ex a b utpotè <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><expan abbr="Primũ">Primum</expan> &longs;ic re&longs;ecentur a n & k l æquales &longs;in­<lb/>gulæ c h, igitur a l dupla e&longs;t e h & a b fuit dupla a d, c d igitur ut in <lb/>priore con&longs;titutioné præcedentis 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 quàm k b ad b l. </s>
<s>hoc <lb/>enim demon&longs;tratum e&longs;t in fine, igitur c d ad h d maior, quàm du­<lb/>plicata a k ad k b, &longs;ed a k ad k b maior e&longs;t per uige&longs;imam tertiam, hu­<lb/>ius &longs;cilicet per demon&longs;trationem illius, quàm g b ad b a, igitur mul­<lb/>to maior c d ad d h, quàm duplicata g b ad b a, quod e&longs;t primum.</s></p><p type="main">
<s>Secundum &longs;ic per eadem, addito enim duplo f c ip&longs;i <lb/><figure id="fig100"></figure><lb/>a b ut in &longs;ecunda figura, & &longs;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 &longs;it minor duplicata per <lb/>præcedentem in b ad a b, & a b ad e b &longs;it maior, ut demon <lb/>&longs;tratum e&longs;t in uige&longs;ima tertia huius, quàm m b ad a b, erit <lb/>f d ad d c multo minor duplicata a b ad b e, quod e&longs;t &longs;e­<lb/>cundum.</s></p><p type="main">
<s>Propo&longs;itio cente&longs;imatrige&longs;imaquarta.</s></p><p type="main">
<s>Si rectangula &longs;uperficies &longs;it cuius pars tertia quadrata &longs;it, corpus <lb/>quod ex latere quadratæ in re&longs;iduum &longs;uperficiei con&longs;tat maius e&longs;t <lb/>quouis corpore ex eadem &longs;uperficies aliter diui&longs;a con&longs;tituto.</s></p><p type="main">
<s>Sit rectangulum a c cuius tertia pars c e &longs;it quadrata, dico quod <lb/><arrow.to.target n="marg455"></arrow.to.target><lb/>corpus, quod <expan abbr="cõ&longs;tat">con&longs;tat</expan> ex e d in a b e&longs;t maius omni corpore, quod fue <lb/>rit ex latere partis &longs;uperficiei a b in reliquam <expan abbr="part&etilde;">partem</expan>. </s>
<s>Si non diuidatur <lb/>uel &longs;upra uel infra, & primo in f erit <expan abbr="aut&etilde;">autem</expan> proportio e d ad d f, ut e c ad <pb pagenum="128"/>e k, & f a ad a e, ut &longs;uperficierum ip&longs;a­<lb/><figure id="fig101"></figure><lb/>rum per primam &longs;exti Elementorum: at <lb/>per præcedentem maior e&longs;t proportio <lb/>e d ad d f, quàm a f ad a e, duplicata igi­<lb/>tur maior e&longs;t proportio e d ad eam, qu&ecedil; <lb/>pote&longs;t &longs;uper f c &longs;uperficiem, quam f a ad <lb/>a e, igitur maior, quàm a k ad a b ex pri­<lb/>ma &longs;exti Elementorum: igitur per trige <lb/>&longs;imam quartam undecimi. </s>
<s>Parallelipe­<lb/>dum ex e d in a b maius e&longs;t parallelipedo ex ea, quæ pote&longs;t in f c &longs;u­<lb/>perficiem in ip&longs;am &longs;uperficiem a k. </s>
<s>Si uerò diui&longs;io facta fuerit in g, <lb/>con&longs;tat ex præcedenti, quod minor e&longs;t proportio g e ad e d, quàm <lb/>&longs;it duplicata e a ad a d a g, eam igitur minor proportio eius lineæ, <lb/>quæ pote&longs;t in g e &longs;uperficiem ad e d quam a b ad a h, igitur paralle­<lb/>lipedum ex e d in a b e&longs;t maius parallelipedo ex ea, quæ pote&longs;t g c <lb/>in a h cum &longs;it a b ad a h, ut dictum e&longs;t, uelut a e ad a g.<lb/><arrow.to.target n="marg456"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg455"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s><margin.target id="marg456"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Manife&longs;tum e&longs;t autem, quòd tale corpus e&longs;t æquale duplo cubi <lb/>lateris partis tertiæ quadratæ.</s></p><p type="main">
<s>Propo&longs;itio cente&longs;imatrige&longs;imaquinta.</s></p><p type="main">
<s>Si linea in duas partes, quarum una &longs;it alteri dupla, diuidatur <lb/>erit, quod fit ex tertia parte in quadratum re&longs;idui parallelipedum <lb/>maius omni parallelipedo, quod ex diui&longs;ione eiu&longs;dem lineæ crea­<lb/>ri po&longs;sit.</s></p><p type="main">
<s><arrow.to.target n="marg457"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg457"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Sit a c dupla b c, & &longs;it quadratum ad ip&longs;ius a c, dico parallelipe­<lb/><figure id="fig102"></figure><lb/>dum ex b c in a d maius e&longs;&longs;e quouis alio ex <lb/>diui&longs;ione lineæ a b &longs;imiliter creato. </s>
<s>Secetur <lb/>primo in e, & fiat quadratum a f, eritque per <lb/>uige&longs;imam quintam. </s>
<s>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 uige&longs;imam &longs;exti Ele <lb/>mentorum, igitur per trige&longs;imam quartam <lb/>undecimi, Parallelipedum ex b c in a d maius e&longs;t parallelipedo e b <lb/>in a f, quod e&longs;t demon&longs;trandum. </s>
<s>Si uerò diui&longs;io cadat in g, fiat qua­<lb/>dratum a h, et erit per uige&longs;imamtertiam huius proportio g c ad c b <lb/>minor, quam duplicata c a ad a g: igitur minor, quàm a d ad a h, igi­<lb/>tur per eandem parallelipedum ex c b in a d maius e&longs;t parallelipe­<lb/>do ex g b in a h.</s></p><p type="main">
<s><arrow.to.target n="marg458"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg458"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Ex hoc liquet quòd parallelipedum illud erit quadruplum cu­<lb/>bo minoris partis, & dimidium cubi maioris.</s></p><pb pagenum="129"/><p type="main">
<s>Propo&longs;itio cente&longs;imatrige&longs;ima&longs;exta.</s></p><p type="main">
<s>Denominationes in infinitum extendere.</s></p><p type="main">
<s>Inquit Euclides, &longs;i fuerint quotlibet quantitates ab uno in conti­</s></p><p type="main">
<s><arrow.to.target n="marg459"></arrow.to.target><lb/><arrow.to.target n="marg460"></arrow.to.target><lb/>nua proportione, erit tertius numerus quadratus, & omnes alij &longs;e­<lb/>quentes uno intermi&longs;&longs;o. </s>
<s>Tertia igitur in comparatione ad &longs;ecun­<lb/>dam etiam, quod non &longs;it numerus, e&longs;t quadratum: e&longs;t enim tertia <lb/>ab uno quadratum &longs;ecundæ, quæ e&longs;t proportio. </s>
<s>Detracto igitur <lb/>uno omnes quantitates lo co pari &longs;unt quadratæ: ut &longs;cias ergo cu­<lb/>ius &longs;unt quadratæ diuide per medium, & erit quadratum illius, er­<lb/>go quadrage&longs;ima erit quadratum uige&longs;imæ, & uige&longs;ima decimæ, <lb/>& decima quintæ, & uige&longs;ima&longs;exta tertiæ decimæ, & ita de alijs. <lb/></s>
<s>Iuxta hoc dicemus, quod &longs;ecunda erit <expan abbr="quadratũ">quadratum</expan>, & quarta quadra­<lb/>tum quadrati, & octaua <expan abbr="quadratũ">quadratum</expan> quadrati quadrati. </s>
<s>Et &longs;extadeci­<lb/>ma quad quad quad quad. </s>
<s>& ita trige&longs;ima &longs;ecunda quad quad quad <lb/>quad quad. </s>
<s>Quod autem quad. </s>
<s>e&longs;t quarta in ordine, ideo & octa­<lb/>ua & duodecima & decima&longs;exta, & &longs;ic de alijs &longs;unt quadrata qua­<lb/>drati, & &longs;icut quarta e&longs;t quadratum quadrati primæ, ita octaua &longs;e­<lb/>cundæ, & duodecima tertiæ, & &longs;extadecima quartæ, & uige&longs;ima <lb/>quintæ, & ita &longs;emper diuidendo per quatuor.</s></p><p type="margin">
<s><margin.target id="marg459"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s><margin.target id="marg460"></margin.target>L<emph type="italics"/>ib.<emph.end type="italics"/> 9. P<emph type="italics"/>ro <lb/>po&longs;.<emph.end type="italics"/> 8.</s></p><p type="main">
<s>Secunda regula dicebat ibidem Euclides, &longs;i fuerint quotlibet <lb/><arrow.to.target n="marg461"></arrow.to.target><lb/>quantitates ab uno in continua proportione quartus, ab uno erit <lb/>cubus &longs;upple &longs;ecundæ, & ita duobus &longs;emper intermi&longs;sis, uno igi­<lb/>tur ip&longs;o relicto quolibet loco ternario, ut tertia, &longs;exta, nona, duode­<lb/>cima &longs;unt cubi, & cubi eius quantitatis, qu&ecedil; exit diui&longs;o numero per <lb/>tria, uelut tertia primæ, &longs;exta &longs;ecundæ, nona terti&ecedil;, duo decima quar <lb/>tæ: & ita tertia erit cubus nona cubus cubi, & uige&longs;ima&longs;eptima cu­<lb/>bus cubi cubi &longs;cilicet primæ. </s>
<s>Et trige&longs;imanona e&longs;t cubus ter­<lb/>tiæ decimæ.</s></p><p type="margin">
<s><margin.target id="marg461"></margin.target>L<emph type="italics"/>ib.<emph.end type="italics"/> 9. P<emph type="italics"/>ro­<lb/>po&longs;.<emph.end type="italics"/> 8.</s></p><p type="main">
<s>Tertia regula quarta quantitas, ut ui&longs;um e&longs;t: e&longs;t quad quad. </s>
<s>Et <lb/>quinta e&longs;t relatum primum, quia 5 e&longs;t numerus primus, & 7 e&longs;t re­<lb/>latum &longs;ecundum, quia e&longs;t &longs;ecundus numerus primus: & undecima <lb/>tertium: & tertiadecima quartum: & decima&longs;eptima quintum: & <lb/>decimanona &longs;extum: & uige&longs;imatertia &longs;eptimum & uige&longs;ima quin­<lb/>ta, quia e&longs;t primus numerus præter quam ad quintam, ideò e&longs;t rela­<lb/>tum quintæ, quæ e&longs;t relatum primum primæ, omnes ergo numeri <lb/>primi &longs;unt relata, alij omnes &longs;unt ex natura cubi uel quadrati. </s>
<s>Sed <lb/>relata &longs;unt inter &longs;e omnia diuer&longs;orum generum ni&longs;i <expan abbr="uige&longs;imũ">uige&longs;imum</expan> quin­<lb/>tum, quod e&longs;t relatum primum primi relati, & quadrage&longs;imumno­<lb/>num e&longs;t relatum &longs;ecundum relati &longs;ecundi. </s>
<s>Et ita cente&longs;imum uige&longs;i­<lb/>mum primum e&longs;t relatum tertium tertij relati, reliqua, ut dixi, me­<lb/>dia inter hæc &longs;unt &longs;ui generis.</s></p><pb pagenum="130"/><p type="main">
<s>Quarta regula propo&longs;ita quantitate ab uno in continua propor<lb/>tione, &longs;i uis &longs;cire cuius naturæ &longs;it detracto uno con&longs;idera, an po&longs;sit <lb/>diuidi per duo, e&longs;t quadratum medietatis, & ita procedes diuiden­<lb/>do u&longs;que ad numerum primum, qui uel e&longs;t 2, & erit ex genere quad <lb/>quad. </s>
<s>uel 3, & erit ex genere quadratorum cuborum, & &longs;imiliter &longs;i <lb/>&longs;it 9, erit ex genere quadratorum cubi cubi. </s>
<s>Et &longs;i proueniat alius nu <lb/>merus primus, ut 5. 7. 11. 13. erit quadratum relati illius ordinis. </s>
<s>Et &longs;i <lb/>non pote&longs;t diuidi numerus quantitatum per 2 uide, &longs;i po&longs;sit diuidi <lb/>per 3, tunc erit cubus illius quantitatis, & &longs;i illa quantitas, quæ pro­<lb/>uenit ex diui&longs;ione: fuerit 3, uel potuerit diuidi per 3, erit cubus, uel <lb/>cubus cubi, & ita deinceps. </s>
<s>Si uerò &longs;it alius numerus primus, ut 5. <lb/>7. 11. erit cubus relati. </s>
<s>Et ita &longs;i <expan abbr="nõ">non</expan> po&longs;sit diuidi per 2, nec per 3, erit ex <lb/>genere relati. </s>
<s>Et tunc &longs;i po&longs;sit diuidi per alium numerum, ut 35, erit <lb/>relatum ex eo genere. </s>
<s>Vtpotè trige&longs;imaquinta quantitas e&longs;t rela­<lb/>tum &longs;ecundum relati primi, &longs;eu relatum primum relati &longs;ecundi. <lb/></s>
<s>Nam quoties quantitas pote&longs;t diuidi per duos numeros, dicetur <lb/>&longs;ub utro que uici&longs;sim, ut duodecima pote&longs;t diuidi per 4 & 3, ideò di­<lb/>cetur cubus quad quad. </s>
<s>uel quad quad. </s>
<s>cub. </s>
<s>& per 2 & 6, & dicetur <lb/>quadratum cubi quadrati, & quadratum cubicum quadrati ip&longs;ius <lb/>proportionis, ad quam omnia referri debent.</s></p><p type="main">
<s>Quinta regula ex præcedenti pendet, & e&longs;t, quod denomina­<lb/>tiones, & proportiones uici&longs;sim commutantur: uelut 256 e&longs;t quad <lb/>quad quad, & inter quad quad quad, & quad quad &longs;unt quatuor ter <lb/>mini ip&longs;o computato, & inter quad quad, & quod ui&longs;i duo, ergo <lb/>quad quad quad continet plures proportiones, & proportiones <lb/>duplicatæ non con&longs;tituunt quad: nam 64 continet duas duplas <lb/>ad 16, non tamen e&longs;t quadratum 16, ideo oportet diligenter ani­<lb/>maduertere.</s></p><p type="main">
<s>Sexta regula &longs;imiliter ex dictis pendet, & e&longs;t, quòd gratia exem­<lb/>pli relatum primum comparatum ad primum terminum e&longs;t &longs;exta <lb/>quantitas, cum autem comparatur ad rem, iam præ&longs;upponit pro­<lb/>portionem. </s>
<s>Exemplum relatum primum proportionis 21/20 e&longs;t 4084101/3200000 <lb/>& e&longs;t aliquanto maior &longs;exquiquarta, & &longs;i colligas terminos 100. <lb/>105. 110 1/4 115 61/80 121 861/1600 127 19681/32000. Tu uides quòd &longs;unt &longs;ex termini in <lb/>utra que computando primum, &longs;ed in 21/20 &longs;unt duo termini, & in qua­<lb/>drato tres, & in quadrato quadrati per præcedentem, adduntur <lb/>duo & ultimus &longs;cilicet &longs;extus fit ex relato ip&longs;o. </s>
<s>Ergo ultra propor­<lb/>tionem &longs;unt tantum quatuor termini.</s></p><p type="main">
<s>Septima regula ad effugiendum omnes errores tu &longs;cis, quòd <lb/>4096 quadratum 64 e&longs;t &longs;extus a 64, ad quem habet proportionem <lb/>quadrati, & 64 e&longs;t &longs;imiliter &longs;extus ab uno illo &longs;cilicet non compu­<pb pagenum="131"/>tato, & ita 64 habet rationem unius, & licet comparetur ad 2 rem, <lb/>& &longs;it &longs;extus ab eo, eo computato 4096 autem à 64 &longs;it &longs;eptimus, ta­<lb/>men non e&longs;t eadem ratio, quia 64 non e&longs;t quadratum 2.</s></p><p type="main">
<s>Propo&longs;itio cente&longs;imatrige&longs;ima&longs;eptima.</s></p><p type="main">
<s>Rationem numerorum ex progre&longs;sione declarare.</s></p><p type="main">
<s>Michaël Stifelius rationem pulcherrimam tradidit ad inuentio­<lb/><arrow.to.target n="marg462"></arrow.to.target><lb/><arrow.to.target n="marg463"></arrow.to.target><lb/>nem numerorum, qui uo cantur multiplicandi, & componitur hoc <lb/>modo. </s>
<s>Ex prima componitur 1 & 2, faciunt 3. 1. 2. 3 faciunt 6. 1. 2. 3. 4 <lb/>faciunt 10, & ita prima tabula con&longs;tituit &longs;ecundam recta &longs;erie nu­<lb/>merorum iunctis o­<lb/>mnibus ab uno. </s>
<s>Ter <lb/><arrow.to.target n="table17"></arrow.to.target><lb/>tia fit ex &longs;ecunda & <lb/>tertia, primò a&longs;&longs;umi <lb/>tur 10 in tertia, ut in <lb/>&longs;ecunda, & ex 10 &longs;e­<lb/>cundæ, & 10 tertiæ <lb/>fit 20, & ex 15 &longs;ecun­<lb/>dæ, & 20 tertiæ fit <lb/>35, & ex 21 &longs;ecundæ, <lb/>& 35 tertiæ fit 56, & <lb/>ex 28, & 56 fit 84. Et <lb/>quanta fit ex tertia, <lb/>& ex &longs;eip&longs;a. </s>
<s>primum <lb/>a&longs;&longs;umendo 35 ex ter <lb/>tia, & ponitur pro <lb/>primo numero quartæ, & ex 35 tertiæ, & 35 quartæ fit 70 numerus <lb/>&longs;ecundæ quartæ: & ita ex 56 & 70 fit 126, & ex 84, & 126. 210. & ita <lb/>quinta ex quarta & &longs;eip&longs;a, & &longs;ic in infinitum.</s></p><p type="margin">
<s><margin.target id="marg462"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>_{m}.</s></p><p type="margin">
<s><margin.target id="marg463"></margin.target>P<emph type="italics"/>rimæ &longs;uæ<emph.end type="italics"/><lb/>A<emph type="italics"/>rith.<emph.end type="italics"/></s></p><table><table.target id="table17"></table.target><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></cell><cell></cell><cell></cell><cell></cell></row><row><cell>2</cell><cell></cell><cell></cell><cell></cell><cell></cell><cell></cell><cell></cell><cell></cell></row><row><cell>3</cell><cell>3</cell><cell></cell><cell></cell><cell></cell><cell></cell><cell></cell><cell></cell></row><row><cell>4</cell><cell>6</cell><cell></cell><cell></cell><cell></cell><cell></cell><cell></cell><cell></cell></row><row><cell>5</cell><cell>10</cell><cell>10</cell><cell></cell><cell></cell><cell></cell><cell></cell><cell></cell></row><row><cell>6</cell><cell>15</cell><cell>20</cell><cell></cell><cell></cell><cell></cell><cell></cell><cell></cell></row><row><cell>7</cell><cell>21</cell><cell>35</cell><cell>35</cell><cell></cell><cell></cell><cell></cell><cell></cell></row><row><cell>8</cell><cell>28</cell><cell>56</cell><cell>70</cell><cell></cell><cell></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></cell><cell></cell></row><row><cell>10</cell><cell>45</cell><cell>120</cell><cell>210</cell><cell>252</cell><cell></cell><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><cell></cell></row><row><cell>12</cell><cell>66</cell><cell>220</cell><cell>495</cell><cell>792</cell><cell>924</cell><cell></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></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></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>Regula ergo e&longs;t, quòd binarius &longs;eruit <02> quadratæ, & quia nihil <lb/>e&longs;t in eius directo, &longs;olus ip&longs;e &longs;eruiet <02> quadratæ. </s>
<s>Ternarius autem <lb/>cubicæ, & quia in eius directo e&longs;t alter ternarius, ille etiam &longs;eruiet <lb/><02> cubicæ. </s>
<s>Quaternarius autem &longs;eruiet quadrato quadrati, & &longs;ena­<lb/>rius, qui e&longs;t in illius directo. </s>
<s>Ergo quinarius &longs;eruiet <02> relat&ecedil; prim&ecedil;, <lb/>& duo &longs;equentes numeri &longs;cilicet 10 & 10, & eo dem modo &longs;enarius <lb/>numeri duo &longs;equentes 15 & 20 &longs;eruient cubo quadrati, & ita etiam <lb/>&longs;eptenarius cum tribus &longs;equentibus numeris 21. 35 & 35 &longs;eruient <lb/>rel. </s>
<s>&longs;ecundi radici, & ita deinceps in infinitum.</s></p><p type="main">
<s>Propo&longs;itio cente&longs;imatrige&longs;imaoctaua.</s></p><p type="main">
<s>Modos u&longs;us horum numerorum declarare.</s></p><p type="main">
<s>In quouis numero denominationis oportet tot addere o, quo­<lb/><arrow.to.target n="marg464"></arrow.to.target><pb pagenum="132"/>tus e&longs;t ordo, & facere tot numeros &longs;equentes; quotus e&longs;t ordo, & <lb/>&longs;emper minuere unam o, uelut quia quadrata <02> e&longs;t prima ad 2 ad­<lb/>demus o, & fiet 20, nec alium qu&ecedil;remus numerum. </s>
<s>Sed quia cubi­<lb/>ca e&longs;t &longs;ecundo loco, habebit prima nota 00, & fiet 300, & &longs;ecundum <lb/>3 unam 0, & fiet 30, & in quadrato quadrati addemus 000 primo, <lb/>& 00 &longs;ecundo, & o tertio, & ita hab ebimus 4000. 600. 40. &longs;ed quia <lb/>in tabula non e&longs;t 4 ultimum, addemus &longs;imilem primo &longs;emper. </s>
<s>In <lb/>relato primo, ergo habebimus 50000. 1000. 1000. 50. & in cubo <lb/>quadrati 600000. 150000. 20000. 1500. 60. Manife&longs;tum e&longs;t, quòd <lb/>his uice uer&longs;a a&longs;&longs;ump&longs;imus 15 & 6 &longs;imiles prioribus addendo &longs;em­<lb/>per ut dixi o minus, donec ad unam peruenerit. </s>
<s>Et ita in relato &longs;e­<lb/>cundo 7000000. 2100000. 350000. 35000. 2100. 70. & ita dein ceps.</s></p><p type="margin">
<s><margin.target id="marg464"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Propo&longs;itio cente&longs;imatrige&longs;imanona.</s></p><p type="main">
<s>Radices omnes à propo&longs;itis numeris extrahere.<lb/><arrow.to.target n="marg465"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg465"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Propo&longs;itis quibu&longs;uis numeris utpotè 916132832, uolo detrahere <lb/><02> relatam primam, primum habebo in tabula de&longs;cripta relata pri­<lb/>ma numerorum &longs;implicium u&longs;que ad 10 uelut in exemplo. </s>
<s>Dein de <lb/><figure id="fig103"></figure><lb/>&longs;ub&longs;cribam pun­<lb/>ctum &longs;ub prima <lb/>nota à dextra, & <lb/>quia e&longs;t quarta in <lb/>ordine hoc, &longs;eu quinta denominatio &longs;ecun­<lb/>dum no&longs;trum, omittam quatuor notas in­<lb/>ter medias, & &longs;ub&longs;cribam punctum aliud, <lb/>& ita facerem &longs;i e&longs;&longs;ent plures quàm decem <lb/>notæ: relinquitur ergo ad <expan abbr="pũctum">punctum</expan> primum <lb/>à &longs;ini&longs;tra 9161, cuius qu&ecedil;ro <02> relatam pri­<lb/>mam in tabula, quam inuenio e&longs;&longs;e 6, nam <lb/>7776 eius relatum primum e&longs;t <lb/>proximius ex minoribus ad 9161, <lb/>detraho igitur 7776, ex numero <lb/>propo&longs;itio relinquitur. </s>
<s>Dein de <lb/>póno 6 & quadratum eius, & cub. </s>
<s>& quadratum <lb/>quadrati, quia, ut dixi, e&longs;t quarta denominatio a­<lb/>pud illum, & è regione numeros præcedentes in­<lb/>uentos relati primi ex præcedenti propo&longs;itione: & duco &longs;ingulos <lb/>cum &longs;uis collateralibus, ut uides etiam in figura, et cum ultimo pro­<lb/>ducto, &longs;cilicet 64800000 diuido 138532832 exit 2, huius accipio o­<lb/>mnes numeros ad relatum primum u&longs;que ut uides, & pono minores <lb/>è regione maiorum, utpotè 2 è regione 1296 & 50000, & 4 è regio­<pb pagenum="133"/>ne 216 & 10000, & 8 è regione 36 & 10000, & 16 è 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 rur&longs;us 1296 in <lb/>50000 fit 64800000, duco in 2 fit 129600000. Demum addo 32 re­<lb/>latum primum 2, & fit &longs;umma omnium 138532832, & ita habemus <lb/>radicem relatam primam dictinumeri e&longs;&longs;e 62. Et &longs;i numerus produ <lb/>ctus fui&longs;&longs;et maior oportui&longs;&longs;et accipere proximo minorem. </s>
<s>Inde per <lb/>regulam &longs;equentem addere minutias.</s></p><p type="main">
<s>Propo&longs;itio cente&longs;imaquadrage&longs;ima.</s></p><p type="main">
<s>Radices per numeros fractos determinare.</s></p><p type="main">
<s>Duplex e&longs;t modus, ut etiam docui in arithmeticis, &longs;cilicet ut pro </s></p><p type="main">
<s><arrow.to.target n="marg466"></arrow.to.target><lb/>radice quadrata addatur duo o, & pro cuba tria, & pro quadrata <lb/>quadrata quatuor, & pro relata prima quinque, & ita deinceps, & <lb/>pr&ecedil; decimis &longs;emel, pro cente&longs;imis bis, pro mille&longs;imis ter, pro millia­<lb/>ribus &longs;eu partibus earum quater, pro cente&longs;imis mille&longs;imis quin­<lb/>quies, pro mille&longs;imis mille&longs;imarum &longs;exies, & ita deinceps deinde <lb/>per præcedentem detrahere radicem, & erit ualde exacta. </s>
<s>Exemplo <lb/>non utar, ni&longs;i quòd &longs;i uelles radicem relatam 16 ad mille&longs;imas, acci­<lb/>cipies radicem relatam numeri à latere propo&longs;iti, & ita de alijs <lb/>1600000, 00000, 00000, & &longs;i uelles <02> cub. </s>
<s>5 1/5 per mille&longs;imas, pri <lb/>mo addes ter 000, & fiet 3000000000, inde &longs;ume 1/5 1000000000, <lb/>qui e&longs;t 200000000, & adde ad 5000000000, fit 2500000000, <lb/>& hoc quia unum refert numerum 1000000000 ex &longs;uppo&longs;ito & 1/5 <lb/>e&longs;t 1/5 unius.</s></p><p type="margin">
<s><margin.target id="marg466"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>m.</s></p><p type="main">
<s>Secundus modus e&longs;t, ut accipias proximè maiorem, & multipli­<lb/>ca in &longs;e, & detrahe numerum propo&longs;itum, & re&longs;iduum diuide per <lb/>duplum radicis primo inuentæ, &longs;i fuerit quadrata, & per triplum <lb/>quadrati eiu&longs;dem &longs;i fuerit cubica, & per quadruplum cubi, &longs;i fuerit <lb/>quadrata quadrata, & per quin cuplum quadrati quadrati, & quod <lb/>exit detrahes ex priore radice, & rur&longs;us quod relinquitur, multipli­<lb/>ca in &longs;e, & eodem modo agendo quod &longs;upere&longs;t à numero propo&longs;i­<lb/>to, diuide per duplum radicis prioris, &longs;i &longs;it radix quadrata, uel per <lb/>triplum quadrati &longs;i &longs;it cubica, & quod exit rur&longs;us detrahe, & ita a­<lb/>gendo, peruenies ad exacti&longs;simam radicem, exemplum uolo radi­<lb/>cem quadratam 5 proxima maior e&longs;t 3, quadratum 9, differentia 4, <lb/>diuide per 6 duplum 3 exit 2/3, detrahe ex 3 fit 2 1/3, quadratum e&longs;t 49/9 <lb/>quod e&longs;t 5 4/9, rur&longs;us diuido 4/9 differentiam 5 4/9 & 5 per 4 2/3 duplum <lb/>radicis primæ exit 2/21, detrahe ex 2 1/3, relinquitur 2 5/21, radix &longs;atis pro­<lb/>pinqua, nam eius quadratum e&longs;t 5 4/441, in cubica &longs;imiliter uolo <02><lb/>cu. </s>
<s>5, proxima maior e&longs;t 2, cubus 8, differentia 3, diuide per triplum <pb pagenum="134"/>quadrati 2 quod e&longs;t 12 exit 1/4 detrahe ex 2 fit 1 3/4 cuius cubus e&longs;t 5 23/64 <lb/>differentia e&longs;t 23/64 diuide per triplum quadrati 1 3/4 quòd e&longs;t 9 3/16 exit <lb/>23/588 detrahe ex 1 3/4 <expan abbr="relinquũtur">relinquuntur</expan> 1 107/147 cuius cubus e&longs;t 5 504449/3176523 Ita diuides <lb/>hunc exce&longs;&longs;um &longs;i placet per triplum quadrati 1 107/147 & e&longs;t fermè 9 exit <lb/>56050/3176523 qua&longs;i detrahe ex 1 107/147 relinquuntur 323159/453789.</s></p><p type="main">
<s>Tertius modus e&longs;t &longs;ubtilior, tu &longs;cis, &qring;d duo decima denominatio <lb/>e&longs;t quadrata &longs;ext&ecedil;, & quadrata quad, tertiæ, & cuba quarti, quarta <lb/>autem e&longs;t inter <expan abbr="tertiã">tertiam</expan> & &longs;extam &longs;ecunda quantitas in continua pro­<lb/>portione: ergo inuenta <02> numeri propo&longs;iti & <02> radicis inuentæ <lb/><expan abbr="reducã">reducam</expan> ad unam denominationem, et inter numeratores collo cabo <lb/>duas quantitates, quod facile erit &longs;en&longs;im procedendo, & habebo <02><lb/>cu. </s>
<s>quæ&longs;itam, &longs;cilicet minorem ex duabus intermedijs. </s>
<s>Et &longs;imiliter <lb/>pro relata prima, capiam &longs;exaginta denominationes, & &longs;cis, quòd <lb/>quintadecima e&longs;t <02> <02> &longs;exage&longs;im&ecedil;, & decima e&longs;t <02> cu. </s>
<s><02> &longs;exage&longs;im&ecedil;, <lb/>& duodecima <02> relata prima &longs;exage&longs;imæ per eandem inuenta, er­<lb/>go <02> numeri propo&longs;iti tanquam ille &longs;it &longs;exage&longs;ima denominatio, <lb/>inueniam illius radicis inuentæ <02> quadratam, & cubicam, & <lb/>quia duodecima quantitas quæ e&longs;t <02> relata prima numeri e&longs;t <lb/>&longs;ecunda, quatuor intermediarum inter ponam inter <02> quadra­<lb/>tum, quadratum, & cubicam quadratam quatuor numeros in <lb/>continua proportione, & &longs;ecundus ex minoribus erit <02> relata <lb/>prima numeri propo&longs;iti. </s>
<s>Exemplum cubicæ uolo <02> cu: 5 habui <02><lb/>quadratam eius 2 5/21 &longs;ed uolo proximiorem diuidendo 4/441 per 4, <lb/>quod e&longs;t fermè 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 e&longs;t 1 1/2 <lb/>&longs;ecunda proximior e&longs;t 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 &longs;ecunda quantitas <lb/><figure id="fig104"></figure><lb/>3006/7641, quod e&longs;t 167/98 proximum ad 1 5/7, <02> cubica. </s>
<s>5. <lb/><expan abbr="nã">nam</expan> eius cubus e&longs;t 5. 13/343 at exacti&longs;sima e&longs;t ergo 1 69/98. <lb/>ut liquet. </s>
<s>Pro relata prima ergo ponamus, ut ue­<lb/>lim <02> relatam <expan abbr="primã">primam</expan> 25, accipio 5 <02> 25 cuius <02> e&longs;t, ut ui&longs;um e&longs;t, 2 104/441 <lb/>&longs;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/>& &longs;ecundus po&longs;t minimum ex illis erit <02> relata prima propinqui&longs;­<lb/>&longs;ima 25. Quomodo uerò inueniantur facillimè illi termini, do­<lb/>cui in &longs;exto libro operis perfecti.</s></p><p type="main">
<s>Quarta regula e&longs;t utilior, licet minus uideatur nobilis, & e&longs;t &longs;un­<lb/>data in hoc, quod &longs;i a b &longs;it maior c & eis ad dantur b e, & d f æqua­<lb/>les dico, quod erit minor proportio a c ad c f, quam a b ad c d, & ex <lb/>con&longs;equenti per <expan abbr="uiã">uiam</expan> fracti maior pars unius erit c fip&longs;ius a e, quàm <pb pagenum="135"/>c d ip&longs;ius a f ex Euclide. </s>
<s>Dico ergo quod maior e&longs;t proportio a b <lb/><figure id="fig105"></figure><lb/>ad c d, quàm a e ad e f, fiat d g ad quam &longs;it b c ut <lb/><arrow.to.target n="marg467"></arrow.to.target><lb/>a b ad c d, eritque a e ad c g ut a b ad c d, minor au­<lb/>tem e&longs;t a e ad c f, quam ad c g, igitur minor a e ad <lb/>c f quàm a b ad c d quod fuit propo&longs;itum. </s>
<s>Simili <lb/>ter &longs;i fuerint duæ quantitates, a b & c d, quarum a b &longs;it maiore, c d <lb/>autem eadem e minor, dico, quòd dimidium aggregati a b & c d <lb/>maiorem habebit proportionem ad e, quàm c d & minor, nam iun­<lb/>cta b f æquali d e ad a b, ita ut f g &longs;it dimidium totius a f, qùia ergo <lb/><figure id="fig106"></figure><lb/>f g e&longs;t dimidium f a & fb e&longs;t minor dimidio <lb/><arrow.to.target n="marg468"></arrow.to.target><lb/>f a cum &longs;it minor b a, & &longs;imiliter f g e&longs;t mi­<lb/>nor a b, quia a b e&longs;t maior dimidio a f, quia <lb/>e&longs;t maior b f, ergo proportio g f ad c e&longs;t ma <lb/>ior quam b f ad e, ita quam c d ad e, & mi­<lb/><arrow.to.target n="marg469"></arrow.to.target><lb/>nor quàm a b ad e, quod fuit propo&longs;itum. </s>
<s>Quo ui&longs;o uolo <02> 1000 <lb/>quadratam, & quòd de quadrata dico, dico etiam de alijs radici­<lb/>bus & erit ex &longs;ecunda regula harum 31 39/62 & quadratum erit 1000 <lb/>1521/3844. Iuxta ergo primam partem regulæ 31 38/61 erit minus, & in ueritate <lb/>in eo, quod fit ducendo, ut uides, & hoc e&longs;t pro­<lb/><figure id="fig107"></figure><lb/>ximum ad 1<gap/>/160, multiplico igitur duplum 31 39/62, <lb/>quod e&longs;t fermè 63 1/4 in 1/160 fient 63/160 <figure id="fig108"></figure> detrahe ex <lb/>1521/3844 hoc modo, diuide 3844 per 160 exit 24 <gap/>/40 <lb/>diuide 1521 per 24, exit 63 3/8, habes igitur quod <lb/>1521/3844 &longs;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 e&longs;t 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 e&longs;t, & breuior hæc <lb/>uia quàm per 00 ad ditus. </s>
<s>Rur&longs;us uolo aliquid <expan abbr="adi&mtilde;ere">adimnere</expan> & cum pro <lb/>pinquitate ita facio. </s>
<s>Con&longs;idero quòd 31 38/61 e&longs;t maius 1/6300 radice, di­<lb/>uido 6300 per 62 exit 103 fermè, neque enim curo in hoc fractiones, <lb/>multiplico ergo 103 in 38/61 & habeo 3914/6283 hic denominator e&longs;t proxi­<lb/>mus 6300, aufero ergo 1 ex 3914, habebo ualde proximam <02> 1000, <lb/>31 3913/6283 cuius quadratum e&longs;t 1000 minus 1/1048 hoc ut dixi diui&longs;um <lb/>per duplum <02> quod e&longs;t 63 e&longs;t omnino in&longs;en&longs;ile in radice.</s></p><p type="margin">
<s><margin.target id="marg467"></margin.target>8. P<emph type="italics"/>ropo&longs;. <lb/></s>
<s>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><margin.target id="marg468"></margin.target>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/></s>
<s><expan abbr="amplificatã">amplificatam</expan>.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg469"></margin.target>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>Quinta regula e&longs;t omnium pulcherrima, & e&longs;t communis omni <lb/>bus & fractis & integris & omnibus generibus radicum, & &longs;it ex­<lb/>emplum, uolo <02> radicis &longs;upra&longs;criptæ &longs;cilicet 31 3913/6283 multiplico 31 <lb/>in 6283, & fit 194793, cui addo 3913, fit 198686 manife&longs;tum e&longs;t igi­<lb/>tur, quod 198686/6283 æquiualet 31 3913/6283 hoc facto, quod e&longs;t commune om­<pb pagenum="136"/>nibus radicibus extrahendis pro radice quadrata, multiplicabo nù <lb/>meratorem, qui e&longs;t 194686 per denominatorem, qui e&longs;t 6283, & &longs;i <lb/>uoluero radicem cubicam, multiplicabo eundem numeratorem <lb/>per quadratum denominatoris, & &longs;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> &longs;uprapo&longs;ita denominatori erit <02> eiu&longs;modi, quam &longs;u&longs;ce­<lb/>pi&longs;ti, uelut in exemplo fuit numerus 198686/6283 quia ergo uolo <02> quad. <lb/></s>
<s>multiplico 198686 in 6283, & fit 1248344138, huius accipio <02><lb/>quad. </s>
<s>quæ e&longs;t 35332, hæc autem e&longs;t diuidenda per 6283, & exeunt <lb/>5 3917/12566, ecce uides radicem exactam admodum, & facilem. </s>
<s>Volo rur­<lb/>&longs;us <02> quadrat. </s>
<s>5 3917/12566, multiplico 12566 per 5 & fit 62830, cui addo <lb/>3917, & fit 66747, cui &longs;uppono 12566 denominatorem, fient ergo <lb/>66747/12566, manife&longs;tum e&longs;t igitur quòd hoc æquiualet 5 3917/12566, &longs;i igitur mul <lb/>tiplicarem denominatorem per denominatorem & numeratorem, <lb/>quod proueniret, e&longs;&longs;et æquale eidem numero, ergo <02> eius e&longs;&longs;et ea­<lb/>dem cum <02> prioris, &longs;ed <02> denominatoris e&longs;&longs;et prior numerus, er­<lb/>go &longs;ufficiet extrahere <02> producti ex denominatore in numerato­<lb/>rem, & ita productum erit ex denominatore in numeratorem <lb/>838742802, cuius <02> e&longs;t 28961, hæc igitur diui&longs;a per 12566 o&longs;ten­<lb/>dit <02> 2 3892/12566. In hac autem quadrata e&longs;t alius modus &longs;ine multiplica­<lb/>tione, &longs;ed non e&longs;t communis alijs, ubi &longs;tatueris denominatorem <lb/>pro denominatore <02>, utpote 12566, & numeratorem 66747, con­<lb/>&longs;titues medium &longs;en&longs;im augendo.</s></p><p type="main">
<s>Rur&longs;us uolo <02> relatam 2 3829/12566 reduco ad denominatorem, & fit <lb/>ut prius 28961/12566, duco igitur 12566 ad quad. </s>
<s>quad. </s>
<s>&longs;ed &longs;ufficiet in hoc <lb/>ca&longs;u deducere ad minores denominationes, utpotè 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 &longs;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>quadrati, & fit 49787136, duc in 195 fit 9708491520, <lb/>cuius <02> relata prima e&longs;t 99, igitur <02> relata prima 2 3829/12566 e&longs;t 1 15/84 pau­<lb/>lo maior, id e&longs;t 1 13/70. Et nota quod &longs;i denominator haberet <02> illius <lb/>generis, quam quæris, &longs;ufficeret inuenire radicem eiu&longs;dem generis <lb/>ab&longs;que alia numerorum multiplicatione.</s></p><p type="main">
<s>Propo&longs;itio cente&longs;imaquadrage&longs;imaprima. (deducere.</s></p><p type="main">
<s>Numeros fractos ad minores in <expan abbr="ead&etilde;">eadem</expan> proportione ualde propinqua</s></p><p type="main">
<s>Cum plerunque numeri fracti hab cantur per radices, ut aliquan­<lb/><arrow.to.target n="marg470"></arrow.to.target><lb/>do maiores &longs;int, aut minores eo fit, ut po&longs;sint reduci ad mino­<lb/>res numeros, ut melius intelligi po&longs;sint & facilius tractari, & <pb pagenum="137"/>cum hoc &longs;it exactior illa pars exemplum, ergo habeo 2 3829/12566, quem <lb/>uolo certa ratione ad minores diui&longs;iones deducere. </s>
<s>Deduco pri­<lb/>mò totum ad fractiones ducendo 2 in 12566, & addendo 3829, & <lb/>fit 26961/12566, multiplico 12566 per 9, quia proportio unius ad alterum <lb/>e&longs;t fermè, ut 9 ad 4, & fit 113094, multiplico 4 in 28961 fit 115844, <lb/>hoc igitur e&longs;t maius, igitur proportio 28961 ad 12566 e&longs;t maior <lb/>quàm 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 e&longs;t 1/6, nami&longs;tæ additiones <lb/>paruæ præter quòd parum uariant quantitatem etiam dum ad ex­<lb/>amen reducuntur, nihil impediunt, detrahe igitur 1/6 à 9/4, & ducendo <lb/>per 6, & detrahendo 53/23, duco igitur primos numeros &longs;cilicet 28961/12566 <lb/>mutuo in 53/23, fiunt 665998, & 666107, ita uides, quod proportio <lb/>53 ad 23 e&longs;t paulo minor, quàm 28961 ad 12566, & æquiualent 27/2<gap/><lb/>& 2 3829/12566.</s></p><p type="margin">
<s><margin.target id="marg470"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>_{m}.</s></p><p type="main">
<s>Propo&longs;itio cente&longs;imaquadrage&longs;ima&longs;ecunda.</s></p><p type="main">
<s>Denominationum incrementa ex extrema cognita inuenire, & <lb/>conuer&longs;o modo.</s></p><p type="main">
<s><expan abbr="Quidã">Quidam</expan> per u&longs;uram <expan abbr="rediuiuã">rediuiuam</expan> fecit 40000 coronatos ex 40 in 40 <lb/><arrow.to.target n="marg471"></arrow.to.target><lb/>annis. </s>
<s>Qu&ecedil;ro <expan abbr="qutãa">qutana</expan> fuerit u&longs;ura, & <expan abbr="quãdo">quando</expan> habuit 1000 coronatos, <lb/><expan abbr="quidã">quidam</expan> uellent &longs;oluere per regulam trium quantitatum, in qua com­<lb/>mitterentur maximi errores. </s>
<s>Et in ea multi &longs;unt modi, & omnes fal­<lb/>&longs;i præter hanc uiam nulla e&longs;t uera, adde quòd uellent multi per &longs;or­<lb/>tem inuentam &longs;oluere augendo per &longs;ingulos annos, quod adeò <lb/>difficile e&longs;&longs;et, & penè foret impo&longs;sibile. </s>
<s>Ideò diuides 40000 per 40 <lb/>numerum &longs;ortis exit 1000, igitur in 40 annis unum fit mille, &longs;unt <lb/>ergo 40 denominationes ab uno, quarum quadrage&longs;ima e&longs;t 1000, <lb/>igitur uige&longs;ima e&longs;t <02> 1000 |&longs;cilicet |31 3913/6283, igitur decima e&longs;t <02> eius <lb/><arrow.to.target n="marg472"></arrow.to.target><lb/>5 3917/12566 huius radix, erit quinta quantitas 2 7/23, cuius <02> relata prima, <lb/><arrow.to.target n="table18"></arrow.to.target><lb/>erit proportio 1 13/70, cuius quadratum e&longs;t 1 1889/4900 &longs;eu <lb/>1 67/165 pro &longs;ecunda quantitate, duces ergo primam, <lb/>quæ e&longs;t 83/70 in quintam, quæ e&longs;t reducta ad mino­<lb/>res fractiones facilitatis cau&longs;a 53/23, & habebis &longs;ex­<lb/>tam quantitatem 2 118/161, duco etiam quintam quan­<lb/>titatem &longs;cilicet 53/23 in &longs;ecundam quæ e&longs;t 232/165, & fit &longs;e­<lb/>ptimi anni quantitas, duco igitur &longs;eptem anno­<lb/>rum numerum, qui e&longs;t 3 14/61 in 31 38/61 fit 102 992/6283. At in <lb/>&longs;ex annis additis ad uiginti, fit tanto minus, quan­<lb/>to 31 38/61 ductum in differentiam &longs;eptem, & &longs;ex an­<lb/>norum quæ e&longs;t 60/121, fit ergo 15 35/492. Quia ergo an­<pb pagenum="138"/>nuatim &longs;olum u&longs;ura adij citur &longs;orti, &longs;ufficiet diuidere 2 992/6283 per 15 35/492 <lb/>&longs;cilicet multiplicando per 12 numerum men&longs;ium 2 992/6283 fit 25 5621/6283 di­<lb/>uide 25 5621/6283 per 15 35/492, exit men&longs;is unus, & dies 21, detrahe ex 27 an­<lb/>nis, remanent anni 26, men&longs;es 10, dies 9, in quo tempore habuit <lb/>4000 aureos coronatos. </s>
<s>V&longs;ura autem fuit ut ui&longs;um 13/70, igitur per re­<lb/>gulam trium duc 13 in 100 fit 1300, diuide 1300 per 70 exit 18 4/7<gap/> & <lb/>tanta fuit pro centum. </s>
<s>Et cum computaueris in tribus annis, acqui­<lb/>rit modico plus be&longs;&longs;e eius, quod habet. </s>
<s>Et ita in 13 annis, & parua <lb/>illa parte perueniet ad decuplum eius, quod habet, &longs;cilicet 4000 au <lb/>reorum, & habebit aureos 40000, ut propo&longs;itum e&longs;t.</s></p><p type="margin">
<s><margin.target id="marg471"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s><margin.target id="marg472"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 136. <lb/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/></s></p><table><table.target id="table18"></table.target><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>SCHOLIVM.</s></p><p type="main">
<s>In propo&longs;ita proportione numero que terminorum rediuiuam u­<lb/>&longs;uram inuenire.</s></p><p type="main">
<s>Sit gratia exempli, in &longs;ex annis u&longs;ura rediuiua uige&longs;imæ, erit­<lb/>qúe proportio 21/20, cuius numeratorem &longs;exies ducam in &longs;e primum <lb/>bis fit 441: ergo ducto 441 in &longs;e fit qúe 194481 ductum in 441 <lb/>fit 85766121 &longs;exies ductum 21, quinquies autem ducam 20 deno­<lb/><figure id="fig109"></figure><lb/>minatorem in &longs;e fit bis 400, ter 8000, <lb/>quinquies ergo 3200000, diuide nume­<lb/>ratorem per denominatorem abiectis <lb/>quinque notis erit 26 2566121/3200000. Quæ propor<lb/>tio e&longs;t proxima 26 4/5 ad 20, & ita ut 134 ad <lb/>100. Et &longs;i pigeret tædij autlaboris po&longs;&longs;es <lb/>pro xij annis, ducere 134 in &longs;e, & fit 17956 <lb/>diuide per 100 eadem ratione, exit 179 14/25 <lb/>& ita 100 in xij annis, fit tantundem. </s>
<s>Et <lb/>ita pro xviij & xx annis.</s></p><p type="main">
<s>Propo&longs;itio cente&longs;imaquadrage&longs;imatertia.</s></p><p type="main">
<s>Si linea in duas partes diuidatur, corpora, quæ fiunt ex una par­<lb/>te in alterius quadratum mutuò æqualia &longs;unt corpori, quod fit ex <lb/>tota linea in &longs;uperficiem unius partis in alteram.<lb/><arrow.to.target n="marg473"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg473"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Sit a c diui&longs;a in a b, b c quadratum a b &longs;it <lb/><figure id="fig110"></figure><lb/>a d, <expan abbr="quadratũ">quadratum</expan> b c, &longs;it b e <expan abbr="parallelogrammũ">parallelogrammum</expan> <lb/>ex a b in b e, a f dico quòd corpora ex a b in <lb/>b e, & b c in a d æqualia &longs;unt corpori ex a c <lb/>in a f. </s>
<s>Quia enim corpus ex a c in a f con&longs;tat <lb/>ex a b in a f, & b c in a f, per primam &longs;ecun­</s></p><p type="main">
<s><arrow.to.target n="marg474"></arrow.to.target><lb/>di Elementorum. </s>
<s>corpus autem ex a b in a f <lb/>e&longs;t æquale corpori ex b c in a d, & corpus <lb/>ex b c in a f e&longs;t æquale corpori ex a b in b c <lb/>igitur con&longs;tat propo&longs;itum.</s></p><pb pagenum="139"/><p type="margin">
<s><margin.target id="marg474"></margin.target>I<emph type="italics"/>d e&longs;t per <lb/>eius demon­<lb/>&longs;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>Propo&longs;itio cente&longs;imaquadrage&longs;imaquarta.</s></p><p type="main">
<s>Duplum cubi medietatis maius e&longs;t aggregato corporum mutu­<lb/>orum cuiuslibet diui&longs;ionis, quantum e&longs;t, quod fit ex tota in quadra <lb/>tum differentiæ.<lb/><arrow.to.target n="marg475"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg475"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/><gap/></s></p><p type="main">
<s>Sit a b diui&longs;a per æqualia in c, & per inæqua­<lb/>lia in d, dico, quòd duplum cubi a c e&longs;t maius ag <lb/><figure id="fig111"></figure><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;">præcedentem</expan> du­<lb/>plum cubi a c e&longs;t æquale 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 e&longs;t &ecedil;quale ei, quod fit ex a b in <expan abbr="rectangulũ">rectangulum</expan> ex a d in d b. </s>
<s><expan abbr="qua-dratũ">qua­<lb/>dratum</expan> <expan abbr="aut&etilde;">autem</expan> a c e&longs;t maius rectangulo a d in d b quadrato c d differen <lb/>tiæ, 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&ecedil;, quod e&longs;t <expan abbr="propo&longs;itũ">propo&longs;itum</expan>.</s></p><p type="main">
<s><arrow.to.target n="marg476"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg476"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 5. <emph type="italics"/>&longs;ecun <lb/>di<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s></p><p type="main">
<s>Propo&longs;itio cente&longs;imaquadrage&longs;imaquinta.</s></p><p type="main">
<s>Si line a in duas partes diuidatur quadrata ambarum partium <lb/>detracto eo quod fit ex una partein alteram, &ecedil;qualia &longs;unt producto <lb/>unius in alteram cum quadrato differentiæ.</s></p><p type="main">
<s><arrow.to.target n="marg477"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg477"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Sit linea a c diui&longs;a in b, & &longs;it differentia a b, <lb/>b c, b d, dico quod quadrata a b & b c detracto <lb/><figure id="fig112"></figure><lb/>eo quod fit ex a b in b c, æqualia &longs;unt producto a b in b c cum qua­<lb/>drato b d. </s>
<s>Quoniam. </s>
<s>n. </s>
<s>quadrata a b, b c æqualia quadratis a d d b <lb/>b c & productis ex a d in d b bis & quod fit ex a b in b c æquale e&longs;t <lb/>ei quod fit ex a d in &longs;e cum eo quod fit ex a d in d b, quia a d e&longs;t &ecedil;qua </s></p><p type="main">
<s><arrow.to.target n="marg478"></arrow.to.target><lb/>lis b cideo quadrata a b & b c detracto eo quod fit ex a b in b c &longs;unt <lb/>æqualia quadratis a d d b, & producto a d in d b &longs;emel: a c quadra­<lb/><arrow.to.target n="marg479"></arrow.to.target><lb/>tum a d cum producto a d in d b e&longs;t æquale producto a b in a d, & <lb/>ex con&longs;equenti in b c, igitur re&longs;iduum quadratorum a b & b c de­<lb/>tracto producti a b in b c e&longs;t æquale a b in b c cum quadrato b d <lb/>quod fuit propo&longs;itum.</s></p><p type="margin">
<s><margin.target id="marg478"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 4. <emph type="italics"/>&longs;ecun <lb/>di<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg479"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 1. <emph type="italics"/>&longs;ecun <lb/>di<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="main">
<s>Propo&longs;itio cente&longs;imaquadrage&longs;ima&longs;exta.</s></p><p type="main">
<s>Corpus quod fit ex linea diui&longs;a in &longs;uperficiem &ecedil;qual em quadra­<lb/>tis ambarum partium detracta &longs;uperficie unius pa<gap/>tis in <expan abbr="alterã">alteram</expan>, e&longs;t <lb/>æquale aggregato cuborum <expan abbr="ambarũ">ambarum</expan> <expan abbr="partiũ">partium</expan>.</s></p><figure></figure><p type="main">
<s>Sic a b diui&longs;a in e quadrata partium e f & <lb/><arrow.to.target n="marg480"></arrow.to.target><lb/>b d detrahatur ex e f, f g æqualis a d, dico cor <lb/>pus ex a b in &longs;uperficies b d, d g æquale e&longs;­<lb/>&longs;e cubis a c & c b pariter acceptis, quia. </s>
<s>n. <lb/></s>
<s>ex a b in b d fiunt duo corpora cubus <lb/>b d & corpus ex a d in quadratum d b hoc <lb/>autem e&longs;t æquale corpori ex b cin a d quia <pb pagenum="140"/>fíunt ex æqualibus lineis: at corpus quod fit ex a b in d g æquale e&longs;t <lb/>corporibus quæ fiunt ex a c, c b in &longs;uperficiem d g at cubus a c con­<lb/>tinet duo corpora qu&ecedil; fiunt & a c in d g & g f, igitur cubus a c &longs;upe­<lb/>rat productum ex a b in d g in producto ex a c in f g & &longs;uperatur ab <lb/>eo in producto ex b c in d g, &longs;uperabatur etiam, ut ui&longs;um e&longs;t, cubus <lb/>b c à producto b a in d b in producto b cin c f, igitur cubi a c c b &longs;u­<lb/>perantur à producto a b in ad in producto b cinc f & in d g, quare <lb/>in producto b c in f e: &longs;i quidem f e & f g &longs;unt æqualia ex &longs;uppo&longs;ito <lb/>&longs;uperant autem in producto ex c b in e f, igitur tantum e&longs;t in in quo <lb/>&longs;uperantur quantum e&longs;t id in quo &longs;uperant: ergo &longs;unt æqualia.</s></p><p type="margin">
<s><margin.target id="marg480"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>m.</s></p><p type="main">
<s>Propo&longs;itio cente&longs;imaquadrage&longs;ima&longs;eptima.</s></p><p type="main">
<s>Propo&longs;ita linea diui&longs;a duas ei lineas adijcere, ut proportio addita­<lb/>rum &longs;ingularum & partium &longs;imul iunctarum ad additas &longs;it mutua.<lb/><arrow.to.target n="marg481"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg481"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Sit linea a b diui&longs;a in c uolo eius <lb/><figure id="fig113"></figure><lb/>partibus addere lineas, ut propo&longs;i­</s></p><p type="main">
<s><arrow.to.target n="marg482"></arrow.to.target><lb/>tum e&longs;t, &longs;tatuo mediam c d inter a e & <lb/><arrow.to.target n="marg483"></arrow.to.target><lb/>c b quæ &longs;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 e&longs;t inter <lb/><arrow.to.target n="marg484"></arrow.to.target><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"></arrow.to.target><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 e&longs;t propo&longs;itum.</s></p><p type="margin">
<s><margin.target id="marg482"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 13. <emph type="italics"/>&longs;ex <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg483"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 11. <emph type="italics"/>&longs;ex <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg484"></margin.target>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><margin.target id="marg485"></margin.target>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>Propo&longs;itio cen te&longs;ima quadra ge&longs;imaoctaua.</s></p><p type="main">
<s>Propo&longs;itis tribus lineis primam &longs;ic diuidere, ut adiectis duabus <lb/>alijs lineis &longs;ecundum rationem mutuam &longs;ingularum &longs;ingulis ag­<lb/>gregatum ex una adiectarum & parte ad aggregatum ex alia parte <lb/>& adiecta &longs;e habeat, ut &longs;ecunda ad tertiam.<lb/><arrow.to.target n="marg486"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg486"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>m.</s></p><p type="main">
<s>Sit a, b, c, d, propo&longs;itæ line&ecedil;, <lb/><figure id="fig114"></figure><lb/>uolo diuidere a b ita in e ut <lb/>&longs;umpta &longs;ecundum proportio­<lb/>nem alicuius quantitatis, puta <lb/>g ad a e &longs;ic b f ad e b & ut g ad <lb/>e b &longs;ic g a ad a e ut &longs;it propor­<lb/>tio g e ad e f ut c ad d. </s>
<s>Sint ergo <lb/>omnia <expan abbr="cõ&longs;tituta">con&longs;tituta</expan> & &longs;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 &longs;ecundum <lb/>a e, igitur g a continet a e &longs;ecundum a c, ergo ex diffinitione qua­</s></p><p type="main">
<s><arrow.to.target n="marg487"></arrow.to.target><lb/>drati a g e&longs;t quadratum a e. </s>
<s>Pari ratione b f e&longs;t quadratum b e. </s>
<s>pro­<lb/>portio igitur g e ad e f cum &longs;it ut c ad e ex &longs;uppo&longs;ito erit ut ip&longs;i pro­<lb/>portioni addamus, & detrahamus ex duplo a b & dimidium re&longs;i­<lb/>dui ducamus in &longs;e, & addamus aggregato quadrati a b cum ip&longs;a <pb pagenum="141"/>a b, & latus eius detracto dimidio re&longs;idui erit b clinea, quare diui­<lb/>&longs;io nota, & e&longs;t ut dicamusu: olo diuidere datam lineam, ut quantita­<lb/>tes adiectæ &longs;ub mutua proportione ad unam tertiam cum parti­<lb/>bus obtineantinter &longs;e proportionem datam.</s></p><p type="margin">
<s><margin.target id="marg487"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 1. <emph type="italics"/>&longs;ecuu <lb/>di<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s></p><p type="main">
<s>Propo&longs;itio cente&longs;imaquadrage&longs;imanona.</s></p><p type="main">
<s>Datam lineam &longs;ic diuidere, ut proportio quadratorum ad du­<lb/>plum unius partis in alteram &longs;it, ut line&ecedil; datæ ad lineam datam.</s></p><p type="main">
<s>Sit data a b quam uolo diuidere, ut proponitur &longs;ub proportio­<lb/><arrow.to.target n="marg488"></arrow.to.target><lb/>ne c d ad e, diuido a b bifariam in f, & ab&longs;cindo <lb/><figure id="fig115"></figure><lb/>g d æqualem d e, & inter c g <expan abbr="re&longs;iduũ">re&longs;iduum</expan> & c e inter­<lb/>pono proportione, & ut h ad c g ita a f medietatis a b ad fk. </s>
<s>Omnia <lb/>i&longs;ta &longs;unt noti&longs;sima ex primo & &longs;exto Elemento­<lb/><figure id="fig116"></figure><lb/><expan abbr="rũ">rum</expan> Euclidis. </s>
<s>Si ergo ab&longs;cindantur fk ex fa, dico <lb/>quod proportio quadratorum l k & k a ad du­<lb/>plum rectanguli a k in k b e&longs;t ut c d ad d e. </s>
<s>Quia. </s>
<s>n. </s>
<s>c e ad c g dupli­<lb/>cata e&longs;t ei qu&ecedil; e&longs;t h ad c g, duplicata e&longs;t <expan abbr="etiã">etiam</expan> ei quæ e&longs;t f a ad fk, qua­<lb/>re ut quadrati a f ad fk, ita c e ad c g, igitur di&longs;iungendo c g ad g e ut <lb/>re&longs;idui quadrati k f ad re&longs;iduum quadrati a f, quare c g ad g d ut <lb/>quadrati k f ad dimidium re&longs;idui quadrati a f, igitur coniunctim c d <lb/>ad d g ut quadrati k f & dimidij re&longs;idui quadrati a f ad ip&longs;um dimi­<lb/>dium re&longs;idui. </s>
<s>At uerò cum g d &longs;it æqualis d e, erit c d ad d e ut qua­<lb/>drati k f cum dimidio re&longs;idui &longs;æpius dicti ad ip&longs;um dimidium re&longs;i­<lb/>dui. </s>
<s>Igitur etiam ut dupli quadrati k f cum re&longs;iduo ad <expan abbr="re&longs;iduũ">re&longs;iduum</expan>, &longs;unt <lb/>enim omnia duplicata. </s>
<s>At <expan abbr="duplũ">duplum</expan> quadrati k f <expan abbr="cũ">cum</expan> re&longs;iduo e&longs;t æqua­<lb/>le quadratis a f & f k, igitur quadratorum a f & f k ad differentiam <lb/>eo rum proportio e&longs;t ut c d ad d e, igitur dupli quadratorum a f & <lb/>f k ad duplum differentiæ quadratorum a f & fk ut c d ad d e. </s>
<s>Ve­<lb/><arrow.to.target n="marg489"></arrow.to.target><lb/>rum duplum quadratorum a f & f k æquatur quadratis b k & k a. <lb/><arrow.to.target n="marg490"></arrow.to.target><lb/>Et duplum differentiæ quadratorum a f & fk e&longs;t &ecedil;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 e&longs;t ueluti c d ad d e, quod e&longs;t propo&longs;itum.</s></p><p type="margin">
<s><margin.target id="marg488"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="margin">
<s><margin.target id="marg489"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 9. <emph type="italics"/>&longs;ecun <lb/>di<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg490"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 5. <emph type="italics"/>&longs;ecun <lb/>di<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="main">
<s>Propo&longs;itio cente&longs;imaquinquage&longs;ima.</s></p><p type="main">
<s>Propo&longs;itis duabus lineis <expan abbr="lineã">lineam</expan> communem <lb/><figure id="fig117"></figure><lb/>utrique adiungere, ut &longs;it maioris ad additam pro­<lb/>portio, uelut quadratorum minoris & adiectæ <lb/>ad duplum unius in alteram.</s></p><p type="main">
<s>Hæc e&longs;t qua&longs;i conuer&longs;a <expan abbr="præced&etilde;tis">præcedentis</expan>. </s>
<s>Sit a ma­<lb/><arrow.to.target n="marg491"></arrow.to.target><lb/>ior, & b c minor, & fiat b d dupla b c, &longs;uper <expan abbr="quã">quam</expan> <lb/>erigatur b f æqualis a; & &longs;it rectangulum d f & <lb/>de&longs;cribatur quadratum b c quod &longs;it b g re&longs;idu&ecedil; <lb/>&longs;uperficiei ad d f latus &longs;it h, dico h e&longs;&longs;e lineam quæ&longs;itam. </s>
<s>Superficies <pb pagenum="142"/>enim d f cum fiat ex a in duplum b c, dupla erit &longs;uperficiei a in b c, &longs;u <lb/>perficies f d, tota æquatur quadratis h & b c, igitur quadrata h & b <lb/>c dupla &longs;unt &longs;uperficiei a in b c, quod uerò fit ex a in duplum b c &longs;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, & &longs;unt quadra­<lb/>ta h & b c, &longs;e habent ad duplum h in b c, ut a ad h, quod fuit de­<lb/>mon&longs;trandum.</s></p><p type="margin">
<s><margin.target id="marg491"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>m.</s></p><p type="main">
<s>Propo&longs;itio cente&longs;imaquinquage&longs;imaprima.</s></p><p type="main">
<s>Proportio differentiæ quadratorum partium, cuiu&longs;uis lineæ ad <lb/>quadratum differentiæ <expan abbr="illarũ">illarum</expan> e&longs;t uelut to tius line&ecedil; ad differentiam.<lb/><arrow.to.target n="marg492"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg492"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Sit a b diui&longs;a in puncto c, & fiat c d æqualis <lb/>c b, manife&longs;tum e&longs;t quod differentia partium <lb/><figure id="fig118"></figure><lb/>e&longs;t a d, dico proportionem differentiæ quadra <lb/>torum a c & c b ad quadratum a d differentiæ partium e&longs;&longs;e ut a b ad </s></p><p type="main">
<s><arrow.to.target n="marg493"></arrow.to.target><lb/>a d. </s>
<s>Quoniam differentia quadratorum a c & c b e&longs;t, quod fit ex a d <lb/>in d c bis cum quadrato a d, & ideò quod fit ex a d in d b cum qua­<lb/>drato a d, & ideò quod fit ex tota a b in a d. </s>
<s>Igitur differentia qua­<lb/><arrow.to.target n="marg494"></arrow.to.target><lb/>drato a c & c b e&longs;t 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 differentiæ quadra­<lb/><arrow.to.target n="marg495"></arrow.to.target><lb/>torum a c & b c ad quadratum a d differentiæ partium. </s>
<s>Quod fuit <lb/>propo&longs;itum.</s></p><p type="margin">
<s><margin.target id="marg493"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 4. <emph type="italics"/>&longs;ecun <lb/>di<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg494"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 3. <emph type="italics"/>&longs;ecun <lb/>di<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg495"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 1. <emph type="italics"/>&longs;exti<emph.end type="italics"/><lb/>E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="main">
<s>Propo&longs;itio cente&longs;imaquinquage&longs;ima&longs;ecunda.</s></p><p type="main">
<s>Si linea in duas partes æquales duas que in æquales diuidatur, fue­<lb/>ritque proportio aggregati ex maiore & dimidio ad ip&longs;am maiorem <lb/>uelut ex minore, & aliqua linea ad ip&longs;am minorem, & rur&longs;us aggre­<lb/>gati ex minore dimidio ad ip&longs;am minorem, uelut aggregati ex ma­<lb/>iore & alia addita ad ip&longs;am maiorem, erit proportio dimidij'ad par <lb/>tem unam inæqualem, uelut alterius partis inæqualis ad &longs;uam ad­<lb/>ditam mutuò, & etiam proportio ad ditarum inuicem, uelut pro­<lb/>portio partium inæqualium duplicata, & rur&longs;us ip&longs;um dimidium <lb/>lineæ a&longs;&longs;umptæ medium erit proportione inter additas. </s>
<s>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"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg496"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Sit propo&longs;ita a b diui&longs;a per <lb/><figure id="fig119"></figure><lb/>æqualia in c per inæqualia in <lb/>d, & &longs;it ut addantur a g & b f, <lb/>ita ut proportio c a, & a d ad a d &longs;it ueluti f d ad d b, & c b & b d ad <lb/>b d, uelut g d ad d a, & hæc e&longs;t quarta <expan abbr="&longs;ecũdi">&longs;ecundi</expan> Archimedis de &longs;ph&ecedil;ra, <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>Et &longs;imiliter quia e&longs;t c b & b d ad b d, uelut g d ad d a erit <pb pagenum="143"/>c b ad b d, uelut g a ad a d, & hoc e&longs;t primum. </s>
<s>Quia ergo c a e&longs;t æ­<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 conuer&longs;am igitur a d ad b d, ut g a ad a d, & ut b d ad fb, <lb/>interpo&longs;itis ergo a d & d b inter a g & b f cum compo&longs;ita &longs;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, &longs;it æqualis proportioni <lb/><figure id="fig120"></figure><lb/>a g ad a d, & d b ad b f, igitur proportio a g ad b f. </s>
<s>Per de­<lb/>mon&longs;trata ab Alchindo e&longs;t duplicata proportioni a d ad <lb/>d b quod e&longs;t &longs;ecundum. </s>
<s>Rur&longs;us quia ex primo demon­<lb/>&longs;trato, uel eius conuer&longs;o proportio a d ad a c e&longs;t uelut b d <lb/>ad b f, & d b ad a c, ut a d ad a g, proportiones ergo <lb/><figure id="fig121"></figure><lb/>a d & d b ad a c componunt proportionem produ­<lb/>ducti a d in d b, quod &longs;it h ad quadratum a c quod &longs;it <lb/>k, & &longs;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/>&longs;itl ad productum b f in a g, quod &longs;it m, per demon&longs;trata ab Eucli­<lb/>de in &longs;exto Elementorum, igitur proportio h ad k ut l ad m, &longs;ed h & </s></p><p type="main">
<s><arrow.to.target n="marg497"></arrow.to.target><lb/>l &longs;unt æquales, quia producuntur ex ei&longs;dem, igitur per demon&longs;tra­<lb/>ta in quinto Elementorum Euclidis, k e&longs;t æquale m, ergo a c e&longs;t me­<lb/>dia pro portione inter b f & g a, quod e&longs;t tertium. </s>
<s>Quia uerò ex pri­<lb/>mo demon&longs;trato e&longs;t 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="fig122"></figure><lb/>gendo g a & a c ad a d, &longs;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 e&longs;t quartum.</s></p><p type="margin">
<s><margin.target id="marg497"></margin.target>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&longs;.<emph.end type="italics"/> 9.</s></p><p type="main">
<s>Cum ergo punctum d fuerit datum, licet inuenire a g & b f, faci­<lb/>lè, ut Archimedes præ&longs;up ponit proportionem g d ad d f datam & <lb/>quærit eam, quæ e&longs;t a d ad d b, & peruenitur ad res numero triplo <lb/>quadrati dimidij lineæ a&longs;&longs;umptæ æquales cubo & numero, qui &longs;it <lb/>ex duplo cubi dimidij in 1 m: ip&longs;a proportione, & quod produci­<lb/>tur diui&longs;o per 1 p: ip&longs;a proportione. </s>
<s>Veluti po&longs;ita a b 10, & propor­<lb/>tione quam uolo g d ad d f &longs;excupla, duco 5 dimidium 10 in &longs;e fit 25, <lb/>& triplico, fit 75 numerus rerum. </s>
<s>Inde duco 5 idem dimidium ad <lb/>cubum fit 125, duplico fit 250, duco in 5, qui e&longs;t 1 m: proportione fit <lb/>1250, diuido per 7, qui e&longs;t 1 p: proportione exit 178 4/7 numerus, qui <lb/>cum cubo æquatur 75 rebus. </s>
<s>Cum ergo con&longs;tituta fuerit diui&longs;io in <lb/>c, non recipit proportionem g d ad f d quam uolueris, &longs;ed &longs;equitur <lb/>una &longs;ola ad <expan abbr="illã">illam</expan>, & e&longs;t mirabile, quoniam line&ecedil; uidentur &longs;umi liberè. <lb/></s>
<s>Sed non e&longs;t ita. </s>
<s>Et <expan abbr="etiã">etiam</expan> quia Archimedes <expan abbr="uide&ttilde;">uidetur</expan> a&longs;&longs;umere <expan abbr="aliã">aliam</expan> lineam, <lb/>&longs;ed non inue &longs;tigat eam, imò o&longs;tendit eam ex a&longs;&longs;umptis. </s>
<s>At Euto ci­<lb/>us o&longs;ten dit ambas, <expan abbr="unã">unam</expan> ex propria inuentione, aliam ex Diocle, &longs;ed <pb pagenum="144"/>una e&longs;t &longs;uperflua, quia ut dixi, una &longs;e quitur ad aliam. </s>
<s>Ex hoc pa­<lb/>tet cur Dio cles a&longs;&longs;ump&longs;erit lineam unam, quæ e&longs;t a c, quæ &longs;e ha­<lb/>bet ad a d, & d b, ut uici&longs;sim a d, & d b ad additas, quod e&longs;t pri­<lb/>mum demon&longs;tratum. </s>
<s>Sic enim omittit primum quod proponit Ar <lb/>chimedes, & a&longs;&longs;umit quod proximum e&longs;t: & ideò Archimedes non <lb/>pro bat, nec præ&longs;upponit, quod à Diocle probatur, &longs;cilicet datum <lb/>e&longs;&longs;e punctum d in linea a b, &longs;ed &longs;olum in linea g f, ideò cogitur pro­<lb/>bare &longs;ecundum quod demon&longs;tratur ab Eutocio, & à nobis demon <lb/>&longs;tratum e&longs;t &longs;uprà. </s>
<s>Archimedes <expan abbr="aũt">aunt</expan> a&longs;&longs;umit <expan abbr="lineã">lineam</expan> extra circulum, <expan abbr="quã">quam</expan> <lb/>uo cat b f, quæ e&longs;t æqualis b c medietati: aliam a&longs;&longs;umit quam uocat <lb/>b h, cuius proportio ad b d e&longs;t &longs;icut quadrati ad a d quadratum a b. <lb/></s>
<s>Con&longs;tat ergo quod proportio g d ad d f e&longs;t data. </s>
<s>Et &longs;imiliter f g ad <lb/>g d, & e&longs;t 1 præ proportione data. </s>
<s>Vnde notandum quod datum <lb/>dicitur, &longs;impliciter cognitum alio modo, dicitur datum po&longs;itione, <lb/>quod e&longs;t certum & tale, uelut &longs;i quis dicat, diuide 10 in duos nume­<lb/>ros quadratos: hoc non e&longs;t datum, pote&longs;t enim diuidi pluribus mo <lb/>dis. </s>
<s>At &longs;i dicas ut una pars &longs;it alterius <expan abbr="quadratũ">quadratum</expan>, i&longs;tud antequàm &longs;ci <lb/>untur partes, dicitur datum po&longs;itione. </s>
<s>Ergo datum po&longs;itione e&longs;t du <lb/>plex, uel ut ratio nota &longs;it, non autem quantitas, ut &longs;i dicam a b e&longs;t du <lb/>pla ad b c, utra que dicitur nota po&longs;itione, quo­<lb/>niam ne&longs;cio quanta &longs;it a b. </s>
<s>Vel &longs;i quantitas e&longs;t <lb/><figure id="fig123"></figure><lb/>nota proportio ignota &longs;it, ut &longs;i a c &longs;it 10, & &longs;it, <lb/>ut b c &longs;it <02> relata, a b erit punctus b, & proportio a b ad b c data po <lb/>&longs;itione, non tamen nota. </s>
<s>Et &longs;i dicas igitur omnia, quæ habent deter <lb/>minationem erunt data po&longs;itione? </s>
<s>Dico quod non, quia oportet, <lb/>ut illa determinatio comprehendatur &longs;ub una ratione, eaque &longs;altem <lb/>generaliter co gnita.</s></p><p type="main">
<s>Propo&longs;itio cente&longs;imaquinquage&longs;imatertia.</s></p><p type="main">
<s>Vim quan cun que manus multiplicare.<lb/><arrow.to.target n="marg498"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg498"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>^{m}.</s></p><p type="main">
<s>Cum enim radimus aut trahimus manife&longs;tum e&longs;t, </s></p><p type="main">
<s><arrow.to.target n="marg499"></arrow.to.target><lb/>quod ambabus manibus uis conduplicatur, & ma­<lb/><figure id="fig124"></figure><lb/>ior redditur, quanta e&longs;t proportio totius ad exce&longs;­<lb/>&longs;um: uelut &longs;it a quod mouetur ab una manu uiribus <lb/>ut b, quæ &longs;unt exce&longs;&longs;us b d &longs;upra a, cum ergo propor<lb/>tio c b d ad a &longs;it compo&longs;ita ex proportionibus c & <lb/>b d ad a manife&longs;tum e&longs;t, quod erit producta ex pro­<lb/>portione c b d ad b d, & b d ad a, &longs;ed e b d e&longs;t dupla <lb/>ad b d, quia e e&longs;t æqualis, cigitur proportio c b d ad <lb/><arrow.to.target n="marg500"></arrow.to.target><lb/>a e&longs;t maior multo quàm duorum exce&longs;&longs;uum, qui mo<lb/>uerent in proportione dupla: uelut &longs;i adderemus f <pb pagenum="145"/>ad d b æqualem b, multo maior e&longs;t ex communi animi &longs;ententia e f <lb/>b d <expan abbr="quã">quam</expan> f b d, quia e continet f, & quantum e&longs;t d in&longs;uper: 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, quàm dupla proportione, uerùm dupla comparatione ad <lb/>proportionem b d ad a, non autem duplicata &longs;ed dupla, ut dixi, qu&ecedil; <lb/>erit maior quàm dupla per <expan abbr="addition&etilde;">additionem</expan> exce&longs;&longs;us. </s>
<s>Ergo &longs;i addatur al­<lb/>ter homo, erit dupla ad illam duplam, ueluti addendo æqualem d b <lb/>f e, adeò ut &longs;i proportio d b f e e&longs;&longs;et quintupla, mouerent illi duo in <lb/>proportione decupla. </s>
<s>Sed annexo baculo aut lima aut &longs;erra annu­<lb/>lo h, ita ut circunuolui po&longs;sit h æquabit uires non &longs;olum d b f e &longs;ed <lb/>multorum hominum. </s>
<s>igitur multo plus aget homo ambabus ma­<lb/>nibus radendo aut &longs;ecando cum g, quàm quadrupla proportione <lb/>unius manus, & hocincrementum e&longs;t non &longs;olum magnæ <lb/>utilitatis, &longs;ed ualde <expan abbr="accõmodatum">accommodatum</expan> in actionibus artificum <lb/>operum grauiorum. </s>
<s>Et huiu&longs;modi conduplicatio e&longs;t ratio <lb/>limæ quam &longs;urdam uocamus.</s></p><p type="margin">
<s><margin.target id="marg499"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 37.</s></p><p type="margin">
<s><margin.target id="marg500"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 2.</s></p><figure></figure><p type="main">
<s>Propo&longs;itio cente&longs;imaquadrage&longs;imaquarta.</s></p><p type="main">
<s>Si line&ecedil; dat&ecedil; alia linea adiungatur, ab extremitatibus autem pri­<lb/>oris line&ecedil; duæ rectæ in unum punctum con currant proportionem <lb/>habentes quam media inter totam & adiectam, ad adiectam erit <lb/>punctus concur&longs;us à puncto extremo lineæ adiectæ di&longs;tans per li­<lb/>neam mediam. </s>
<s>Quòd &longs;i ab extremo alicuius lineæ æqualis mediæ <lb/>&longs;eu peripheria circuli cuius &longs;emidiameter &longs;it media linea duæ lineæ <lb/>ad prædicta puncta producantur, ip&longs;&ecedil; erunt in proportione medi&ecedil; <lb/>ad adiectam.<lb/><arrow.to.target n="marg501"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg501"></margin.target>C<emph type="italics"/>o<emph.end type="italics"/>m.</s></p><p type="main">
<s>H&ecedil;c propo&longs;itio e&longs;t admirabilis: & etiam de&longs;crip&longs;i, ut multa &longs;ecre­<lb/>ta Dialecticæ potius <expan abbr="aperiren&ttilde;">aperirentur</expan> quam quod huic propo&longs;ito <expan abbr="multũ">multum</expan> <lb/>congrueret. </s>
<s>Ideò potius &longs;cholij cau&longs;a po&longs;ita e&longs;t quam ip&longs;ius tracta­<lb/>tionis: ut <expan abbr="modũ">modum</expan> demon&longs;trandi magis quam id, &qring;d <expan abbr="demon&longs;tra&ttilde;">demon&longs;tratur</expan>, re­<lb/>&longs;picere oporteat. </s>
<s><expan abbr="Con&longs;titua&ttilde;">Con&longs;tituatur</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 &longs;olum &longs;upponatur <lb/>ab Appollonio, <expan abbr="tam&etilde;">tamen</expan> facilè demon&longs;tratur & à Commandino adie­<lb/>cta e&longs;t <expan abbr="demõ">demom</expan> &longs;tratio. </s>
<s>Concurrant ergo ex a & b du&ecedil; line&ecedil; in aliquod </s></p><p type="main">
<s><arrow.to.target n="marg502"></arrow.to.target><lb/>punctum, putat h ut &longs;it a h ad h b uelut c ad d, dico quod &longs;i ducat <lb/>h f quod ip&longs;a erit æqualis g, ducatur b l æquidi&longs;tans a h, & quia <lb/><arrow.to.target n="marg503"></arrow.to.target><lb/>ex &longs;uppo&longs;ito 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 &longs;unt &longs;imiles erit proportio a h ad b l, ueluti a f <lb/><arrow.to.target n="marg504"></arrow.to.target><lb/>ad fb, igitur per &ecedil;quam proportionem b e h ad b l, ut a f ad g, &longs;ed ut <lb/><arrow.to.target n="marg505"></arrow.to.target><lb/>a f ad g ita g ad b f ex &longs;uppo&longs;ito: & ut a f ad g, it a h a ad h b, ex &longs;uppo <pb pagenum="146"/>&longs;ito igitur ut a h ad h b ita h b ad b l, &longs;ed angulus a h b e&longs;t æqualis <lb/>angulo h b l, ergo triangulus a h b e&longs;t <lb/>&longs;imilis triangulo h b l, quare angulus <lb/>b h l e&longs;t &ecedil;qualis angulo h a f, igitur du <lb/>orum triangulorum f a h, & fb h duo <lb/><arrow.to.target n="marg506"></arrow.to.target><lb/>anguli unius a & f &longs;unt æquales duo­<lb/>bus angulis, alterius igitur propor­<lb/><figure id="fig125"></figure><lb/>tio a f ad fh re&longs;picientium angulos &ecedil;­<lb/><arrow.to.target n="marg507"></arrow.to.target><lb/>quales ut a h ad h b re&longs;picientium an­<lb/><arrow.to.target n="marg508"></arrow.to.target><lb/>gulum f, &longs;ed a h ad h b ut c ad d, ex &longs;up <lb/>po&longs;ito igitur a f ad f h, ut c ad d, &longs;ed ut c ad d ita a f ad g, ex &longs;uppo&longs;ito <lb/>ergo h f e&longs;t æqualis g.<lb/><arrow.to.target n="marg509"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg502"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 29. <emph type="italics"/>pri <lb/>mi, &<emph.end type="italics"/> 4. <emph type="italics"/>&longs;ex <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg503"></margin.target>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><margin.target id="marg504"></margin.target>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><margin.target id="marg505"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 6. <emph type="italics"/>&longs;exti<emph.end type="italics"/><lb/>E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg506"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 32. <emph type="italics"/>pri <lb/>mi, &<emph.end type="italics"/> 4. <emph type="italics"/>&longs;ex <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg507"></margin.target>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><margin.target id="marg508"></margin.target>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><margin.target id="marg509"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 1.</s></p><p type="main">
<s>Cum ergo h&ecedil;c demon&longs;tratio &longs;it ex &longs;en&longs;u in uno puncto h, ideò ad <lb/>quælibet puncta traduci pote&longs;t, quæ potero imaginari, & ita pri­<lb/>ma uo cabitur &longs;en&longs;us, <expan abbr="&longs;ecũda">&longs;ecunda</expan> imaginandi: Et <expan abbr="quoniã">quoniam</expan> in demon&longs;tran­<lb/>do non a&longs;&longs;umimus aliquid, quod &longs;it proprium alicui puncto, ni&longs;i <lb/>proportionem h a ad h b &longs;imilem e&longs;&longs;e c ad d, ideo hoc pertinet ad <lb/>intellectum, & e&longs;t tertium. </s>
<s>Etidem dico &longs;i k e&longs;&longs;et ultra h quod po­<lb/>te&longs;t contingere. </s>
<s>modò k a ad k b &longs;it ut c ad d & k f &longs;it &ecedil;qualis g idem <lb/>&longs;equetur, & comprehenditur &longs;ub tertio & pertinet ad intellectum, <lb/>& quoniam demon&longs;tratur quod punctum k ubicun que &longs;umatur, e&longs;t <lb/>in &ecedil;quali <expan abbr="di&longs;tãtia">di&longs;tantia</expan> à puncto f&longs;cilicet per g lineam, erit &longs;emper in peri­<lb/>pheria circuli, & hoc pote&longs;t e&longs;&longs;e in infinitis locis &longs;impliciter & extra <lb/>infinitum nihil e&longs;t, igitur &longs;ub hoc continetur conuer&longs;um &longs;cilicet, <lb/>quod a quolibet puncto circuli ductis lineis ad a & b ip&longs;&ecedil; erunt in <lb/>proportione c ad d. </s>
<s>Et ita ab&longs;que principijs Geometricis concluditur <lb/>propo&longs;itio Geometrica & hoc e&longs;t <foreign lang="greek">w_erila/mpousin</foreign> & fermè &longs;ummum in­<lb/>tellectus humani. </s>
<s>Et pote&longs;t demon&longs;trari Geometricè duobus uer­<lb/>bis. </s>
<s>Quia. </s>
<s>n. </s>
<s><expan abbr="f&longs;upponi&ttilde;">f&longs;upponitur</expan> æqualis g eo quòd h e&longs;t in peripheria circu­<lb/>li erit media inter a f & f b, quare cum angulus f &longs;it communis, erit <lb/>proportio a h ad h b, laterum re&longs;picientium angulum f in utroque </s></p><p type="main">
<s><arrow.to.target n="marg510"></arrow.to.target><lb/>triangulo, uelut h f lateris in maiori ad f b latus in minori, quare <lb/><arrow.to.target n="marg511"></arrow.to.target><lb/>cum ex &longs;uppo&longs;ito h f ad fb &longs;it ut c ad d, erit a ad b, ut c ad d. </s>
<s>Et uides <lb/>Apollonium, & Pappium quanta &longs;uperflua adij ciant in hac &longs;ecun­<lb/><arrow.to.target n="marg512"></arrow.to.target><lb/>da parte demon&longs;trationis, quæ e&longs;t prima apud illos, & ducunt <expan abbr="unã">unam</expan> <lb/>lineam non nece&longs;&longs;ariam ex puncto b ad latus fh. </s>
<s>Vt <expan abbr="antiquorũ">antiquorum</expan> ple <lb/>rique non tantum potuerint Geometria & ingenio, quæ ferunt excel <lb/>lenti&longs;sima in illis, quantum nos ex Dialectica <foreign lang="greek">w_e?ila/mpousin</foreign> inducen <lb/>tes. </s>
<s>e&longs;t enim &longs;ingulare hoc exemplum.<lb/><arrow.to.target n="marg513"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg510"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 6. <emph type="italics"/>&longs;exti<emph.end type="italics"/><lb/>E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg511"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 4. <emph type="italics"/><expan abbr="eiu&longs;d&etilde;">eiu&longs;dem</expan><emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg512"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 11. <emph type="italics"/>&longs;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/></s>
<s>in<emph.end type="italics"/> P<emph type="italics"/>ræfat.<emph.end type="italics"/></s></p><p type="margin">
<s><margin.target id="marg513"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 2.</s></p><p type="main">
<s>Ex hoc <expan abbr="etiã">etiam</expan> patet quod &longs;i circulus duceretur &longs;ecundum f k tran­<lb/>&longs;iretque per m & n e&longs;&longs;et a m ad m b & a n ad b n, ut a h ad h b.</s></p><pb pagenum="147"/><p type="head">
<s>SCHOLIVM</s></p><p type="main">
<s>Ex hoc pater qualiter ex uera demon&longs;tratione &longs;en&longs;u o&longs;ten&longs;a per­<lb/>uenimus ad quotquot imaginando, inde intellectu abiectis condi­<lb/>tionibus non nece&longs;&longs;arijs facimus infinitum & uniuer&longs;ale. </s>
<s>Demum <lb/>&longs;ine artis &longs;pe cialis auxilio o&longs;tendimus Iheorema uniuer&longs;ale (quod <lb/>etiam poterat o&longs;tendi Geometricè, &longs;ed longè pulchrius e&longs;t, ac &longs;ubli­<lb/>mius per <foreign lang="greek">w_erilampousin</foreign>, qa hoc ip&longs;o infinita alia do cemus generaliter <lb/>per &longs;implicem <expan abbr="compreh&etilde;&longs;ionem">comprehen&longs;ionem</expan> o&longs;tendere) &longs;cilicet quod à quouis <lb/>puncto peripheri&ecedil; circuli, cuius &longs;emidiameter e&longs;t media proportio­<lb/>ne inter totam exten&longs;am à centro u&longs;que exterius, & partem quæ' e&longs;t à <lb/>centro ad punctum de&longs;criptum &longs;ub proportione continua <expan abbr="datarũ">datarum</expan> <lb/>linearum lineæ ductæ ex eo ad punctum exterius, & punctum de­<lb/>&longs;criptum &longs;unt in proportione datarum linearum.</s></p><p type="main">
<s>Propo&longs;itio cente&longs;imaquinquage&longs;imaquinta.</s></p><p type="main">
<s><expan abbr="Quadratorũ">Quadratorum</expan> <expan abbr="numerorũ">numerorum</expan> proportionem & <expan abbr="inuention&etilde;">inuentionem</expan> <expan abbr="cõ&longs;iderare">con&longs;iderare</expan>.</s></p><figure></figure><p type="main">
<s>Primùm oportet &longs;cire e&longs;&longs;e tres naturales <lb/>numerorum &longs;eries, primam Euclidis iuxta </s></p><p type="main">
<s><arrow.to.target n="marg514"></arrow.to.target><lb/>quamuis <expan abbr="proportion&etilde;">proportionem</expan>, in qua unum & ter­<lb/>tius & quintus, & ita uno &longs;emper intermi&longs;­<lb/>&longs;o &longs;unt quadrati. </s>
<s>Primus quo que. </s>
<s>1. unum & <lb/>quartus & &longs;eptimus & ita duobus intermi&longs;sis &longs;unt cubi. </s>
<s>In &longs;ecun­<lb/>do ordine e&longs;t naturalis &longs;eries numerorum, ex qua colligitur alia, & <lb/>ex illa bini quilibet &longs;e &longs;equentes con&longs;tituunt numerum <expan abbr="quadratũ">quadratum</expan>. <lb/></s>
<s>In tertia numeri impares, qui &longs;emper collati efficiunt quadratum.</s></p><p type="margin">
<s><margin.target id="marg514"></margin.target>E<emph type="italics"/><expan abbr="xemplũ">xemplum</expan><emph.end type="italics"/> 1.</s></p><figure></figure><p type="main">
<s>Sit ergo propo&longs;itus numerus cui uelim <lb/>addere quadratum numerum, ut fiat qua­<lb/><arrow.to.target n="marg515"></arrow.to.target><lb/>dratus totus, accipe numerum quadratum <lb/>minorem illo quem uis, & detrahe à propo <lb/>&longs;ito numero &longs;eu quadrato &longs;eu non re&longs;idu­<lb/><arrow.to.target n="marg516"></arrow.to.target><lb/>um, diuide per duplum <02> quadrati quod <lb/><gap/>axi&longs;ti, &qring;d exit duc in &longs;e fiet quadratus numerus, idem que additus <lb/><gap/>umero propo&longs;ito, faciet quadratum. </s>
<s>Velut capio 16 qui e&longs;t 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 e&longs;t 1 13/36 qui additus ad 16 facit 17 13/36 <lb/><expan abbr="quadratũ">quadratum</expan> cuius <02> e&longs;t 4 1/6.</s></p><p type="margin">
<s><margin.target id="marg515"></margin.target>E<emph type="italics"/><expan abbr="xemplũ">xemplum</expan><emph.end type="italics"/> 2.</s></p><p type="margin">
<s><margin.target id="marg516"></margin.target>E<emph type="italics"/><expan abbr="xemplũ">xemplum</expan><emph.end type="italics"/> 3.</s></p><p type="main">
<s>Ex hoc patet propo&longs;ito quouis numero <expan abbr="&qtilde;drato">quadrato</expan> modus inuenien­<lb/><arrow.to.target n="marg517"></arrow.to.target><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><margin.target id="marg517"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 1.</s></p><p type="head">
<s>SCHOLIVM.</s></p><p type="main">
<s>Po&longs;&longs;em adducere demon&longs;trationes omnium <expan abbr="horũ">horum</expan>, &longs;ed reddere­<lb/>tur res longa <expan abbr="cũ">cum</expan> &longs;int manife&longs;t&ecedil; ex &longs;eptimo octauo & nono Euclidis. <lb/></s>
<s>Exemplum &longs;ecundum capio modò 14 qui non e&longs;t quadratus, aufe­<lb/>ro 9, remanet 5, diuido per 6 duplum <02> 9 exit 5/6 <expan abbr="quadratũ">quadratum</expan> eius e&longs;t 25/36 <pb pagenum="148"/>hic additus ad 14 con&longs;tituit 14 25/36 quadratum 3 5/6. Et ita 14 e&longs;t diffe­<lb/>rentia duorum quadratorum, &longs;cilicet 25/36 & 14 25/36.<lb/><arrow.to.target n="marg518"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg518"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>^{m}. 2.</s></p><p type="main">
<s>Ex hoc habebis duo quadrata in datis terminis quæ different <lb/>dato numero, & e&longs;t pulchrum. </s>
<s>Velut uolo duo quadrata quæ dif­<lb/>ferant in 2, & <02> minoris &longs;it inter 1 & 2, tunc capies per regulam i­<lb/>p&longs;am 2, & auferes <expan abbr="numerũ">numerum</expan> quadratum ita quòd re&longs;iduum diui&longs;um <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/>e&longs;t 1 1/3 & exit 1 1/6, & hic e&longs;t minor numerus cuius quadratum e&longs;t 1 13/36 <lb/>cui &longs;i addantur 2, fient 3 13/36 numerus quadratus 1 5/6.</s></p><p type="main">
<s><arrow.to.target n="marg519"></arrow.to.target></s></p><p type="margin">
<s><margin.target id="marg519"></margin.target>C<emph type="italics"/>or<emph.end type="italics"/>_{m}. 3.</s></p><p type="main">
<s>Cum autem uolueris duo quadrata quæ differant in 100, tunc <lb/>per regulam datam &longs;i auferes 1, peruenires ad numeros magnos & <lb/>fractos, & ideo melius e&longs;t quia numerus e&longs;t par, ut detrahas nume­<lb/>rum parem quadratum, ita quod re&longs;iduum po&longs;sit 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">re&longs;iduum</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 e&longs;t 4 exit <lb/>24, cuius quadratum &qring;d e&longs;t 576 addito 100 facit 676 <expan abbr="quadratũ">quadratum</expan> 26. <lb/>Et ita ex 433 non auferam &longs;ed 9, quia relinquetur 24 qui pote&longs;t diui­<lb/>di per &longs;e, duplum <02> 9 & exit 4 cuius <expan abbr="quadratũ">quadratum</expan> e&longs;t 16, addito 33 fit 49.</s></p><p type="main">
<s>Secunda regula, cum uolueris propo&longs;ito uno numero quadra­<lb/>to illum diuidere infinitis modis in duos numeros quadratos, cape <lb/>quemuis numerum quadratum per primum exemplum regul&ecedil; pri <lb/>mæ, & cum eo diuide numerum propo&longs;itum, & qui proueniet erit <lb/>quadratus, <expan abbr="hũc">hunc</expan> ergo duces in partes numeri quadrati qu&ecedil; &longs;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 diui&longs;i&longs;ti. </s>
<s>quia multipli catio fit per <expan abbr="eo&longs;d&etilde;">eo&longs;dem</expan> nu­<lb/>meros qui &longs;unt partes diui&longs;oris. </s>
<s>Velut uolo facere de 4 duas partes <lb/>qu&ecedil; &longs;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 po&longs;&longs;es diuidere infinitis modis, puta per 17 13/36 & <lb/>per 169. Tertia regula cum unus numerus additus <lb/><figure id="fig126"></figure><lb/>primo & detractis à <expan abbr="&longs;ecũdo">&longs;ecundo</expan> facit ambo quadrata, <expan abbr="id&etilde;">idem</expan> <lb/>numerus coniunctus cum differentia illorum nume­<lb/>rorum & detractus à primo & additus &longs;ecundo facit <lb/>eo&longs;dem numeros quadratos, ueluti capio 10 primum <lb/>3 &longs;ecundum 6 additus ad 10 & detractus à 7 efficit 6 <lb/>& 1 quadratos dico quod iunctus 16 cum 3 differen­<lb/>tia 10 & 7 fit 9, qui detractus à 10 & additus ad 7 effi­<lb/>cit 1 & 16 numeros quadratos priores.</s></p><pb pagenum="149"/><p type="head">
<s>SCHOLIVM</s></p><p type="main">
<s>Sunt & alij modi plures faciendi huiu&longs;modi, &longs;ed <expan abbr="nõ">non</expan> &longs;unt ad eò ge <lb/>nerales, & nihilo minus &longs;unt magis confu&longs;i, & non aliquid plus.</s></p><p type="main">
<s>Quarta regula, <expan abbr="cũ">cum</expan> uolueris <expan abbr="numerũ">numerum</expan> aliquem non quad. </s>
<s>qui bifa <lb/><expan abbr="riã">riam</expan> <expan abbr="compona&ttilde;">componatur</expan> ex duob. </s>
<s><expan abbr="&qtilde;d">quad</expan>. </s>
<s>uelut 10 ex 25, & 25 & 49 & 1, <lb/><figure id="fig127"></figure><lb/>& <expan abbr="&longs;uma&ttilde;">&longs;umatur</expan> a b numerus quad. </s>
<s>diui&longs;us in <expan abbr="&longs;upplem&etilde;ta">&longs;upplementa</expan>, ita quae c <lb/>d &longs;it portio minor eiu&longs;modi, ut adiecta illi <expan abbr="æ&qtilde;li">æquali</expan> c d gnomo <lb/>cir <expan abbr="cũ&longs;criptus">cun&longs;criptus</expan> c k l <expan abbr="cũ">cum</expan> <expan abbr="f&qtilde;drato">fquadrato</expan>, &longs;it <expan abbr="&ecedil;&qtilde;lis">&ecedil;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">æqualibus</expan> erunt duo <expan abbr="&longs;upplem&etilde;ta">&longs;upplementa</expan> c k l <expan abbr="cũf">cunf</expan> qua­<lb/>drato &ecedil;qualia duob. </s>
<s><expan abbr="&longs;upplem&etilde;tis">&longs;upplementis</expan> a b <expan abbr="cũ">cum</expan> <expan abbr="&qtilde;drato">quadrato</expan> h g. </s>
<s>Maio­<lb/>ra <expan abbr="aũt">aunt</expan> <expan abbr="&longs;upplem&etilde;ta">&longs;upplementa</expan> <expan abbr="excedũt">excedunt</expan> minora in duplo quad. </s>
<s>c d <expan abbr="igi&ttilde;">igitur</expan> detractis <lb/>minoribus &longs;upplementis <expan abbr="cõmunibus">communibus</expan>, erit <expan abbr="duplũ">duplum</expan> quad. </s>
<s>c d <expan abbr="cũ">cum</expan> f qua­<lb/>drato &ecedil;qualia h g <expan abbr="&qtilde;drato">quadrato</expan>. </s>
<s>Ergo propo&longs;ito numero, putà 3 ducam in &longs;e <lb/>fit 9, <expan abbr="ducã">ducam</expan> 2 <expan abbr="minor&etilde;">minorem</expan> in &longs;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> &qring;d 3 <expan abbr="cũ">cum</expan> duplo 2, & erit <expan abbr="totũ">totum</expan> 7, e&longs;t unus <lb/>numerus, alter <02> 1. 1. 1, & <expan abbr="horũ">horum</expan> <expan abbr="&qtilde;d">quad</expan>. </s>
<s><expan abbr="cõponunt">componunt</expan> 50, <expan abbr="duplũ">duplum</expan> <expan abbr="&qtilde;d">quad</expan>. </s>
<s>5. Et &longs;imi <lb/>liter capio 6 <expan abbr="&qtilde;d">quad</expan>. </s>
<s>36 <expan abbr="duplũ">duplum</expan> <expan abbr="&qtilde;d">quad</expan>. </s>
<s>4. 32 differentia 4, numerus <expan abbr="&qtilde;d">quad</expan>. </s>
<s>2, ideo <lb/>6 <expan abbr="cũ">cum</expan> duplo 4, & e&longs;t 14, e&longs;t unus numerus, alter 2, <expan abbr="quorũ">quorum</expan> <expan abbr="&qtilde;d">quad</expan>. </s>
<s>&longs;unt 200, <lb/><expan abbr="dimidiũ">dimidium</expan> e&longs;t 100 <expan abbr="&qtilde;d">quad</expan>. </s>
<s>10 <expan abbr="cõpo&longs;iti">compo&longs;iti</expan> ex 6 & 4. Et ita capio 9, <expan abbr="&qtilde;d">quad</expan>. </s>
<s>eius 81 du <lb/><expan abbr="plũ">plum</expan> <expan abbr="&qtilde;d">quad</expan>. </s>
<s>6. 72 differentia 9 numerus <expan abbr="&qtilde;d">quad</expan>. </s>
<s><expan abbr="igi&ttilde;">igitur</expan> cum duplo 6, & e&longs;t 21, e&longs;t <lb/>unus <expan abbr="illorũ">illorum</expan>, alter 3 <expan abbr="&qtilde;d">quad</expan>. </s>
<s>450, <expan abbr="duplũ">duplum</expan> 225 <expan abbr="&qtilde;d">quad</expan>. </s>
<s>15, qui con&longs;tat ex 9 & 6. Et <lb/>ita capio 11 <expan abbr="&qtilde;d">quad</expan>. </s>
<s>cuius e&longs;t 121, <expan abbr="duplũ">duplum</expan> <expan abbr="&qtilde;d">quad</expan>. </s>
<s>6 e&longs;t 72 differentia, 72 & 21 e&longs;t <lb/>49 numerus <expan abbr="&qtilde;d">quad</expan>. </s>
<s>7, <expan abbr="igi&ttilde;">igitur</expan> 23 qui con&longs;tat ex 11, & duplo 6 numeri mino<lb/>ris e&longs;t unus numerus, alter e&longs;t 7 <expan abbr="&qtilde;d">quad</expan>. </s>
<s><expan abbr="quorũ">quorum</expan> &longs;unt 578. <expan abbr="duplũ">duplum</expan> 289, <expan abbr="&qtilde;d">quad</expan>. <lb/></s>
<s>17, qui con&longs;tat ex 11 & 6. Quinta regula, per hoc inueniemus infini <lb/>tos numeros <expan abbr="&qtilde;d">quad</expan>. </s>
<s><expan abbr="cõponentes">componentes</expan> 32, nam <expan abbr="cũ">cum</expan> 32 &longs;it duplus <expan abbr="&qtilde;d">quad</expan>. </s>
<s><expan abbr="diuidã">diuidam</expan> per<lb/>unum <expan abbr="aggregatũ">aggregatum</expan> ex inuentis puta 578, & quia ambo ex &longs;uppo&longs;ito <lb/>&longs;unt dupli ad <expan abbr="&qtilde;d">quad</expan>. </s>
<s>qui proueniet erit <expan abbr="&qtilde;d">quad</expan>. </s>
<s>&longs;cilicet 16/289, duc in numeros <expan abbr="&qtilde;-dratos">qua­<lb/>dratos</expan> qui componunt 578, & &longs;unt 529 & 49, & fient 2 206/289 & 29 83/289, <lb/>& hi iuncti <expan abbr="fiũt">fiunt</expan> 32, quia &longs;unt multiplicatæ partes numeri, per quem <lb/>e&longs;t <gap/>iui&longs;us numerus. </s>
<s>Et ita poteris diuidere 32 in infinitos alios <expan abbr="&qtilde;d">quad</expan>.</s></p><p type="main">
<s>Sexta regula, ponamus modò &qring;d uelim diuidere 10, <expan abbr="cõpo&longs;itũ">compo&longs;itum</expan> ex <lb/>duob. </s>
<s><expan abbr="&qtilde;d">quad</expan>. </s>
<s>9 & 1, & non <expan abbr="duplũ">duplum</expan> numero <expan abbr="&qtilde;d">quad</expan>. </s>
<s>ita &qring;d &longs;it diui&longs;us in alios <lb/>duos: <expan abbr="ducã">ducam</expan> 10 in 25 <expan abbr="cõpo&longs;itũ">compo&longs;itum</expan> ex duob. </s>
<s><expan abbr="&qtilde;d">quad</expan>. </s>
<s>fit 250/25, at 250 <expan abbr="cõponi&ttilde;">componitur</expan> aliter <lb/>ex duob. </s>
<s>quad. </s>
<s><08> 225/25 & 25/25, &longs;cilicet 169/25 & 81/25, id e&longs;t 6 19/25 & 3 6/25, qui &longs;unt <expan abbr="&qtilde;d">quad</expan>. <lb/></s>
<s>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 nece&longs;&longs;ario <expan abbr="cõponi&ttilde;">componitur</expan> ex 225/25 & 100/25, &longs;ed ego uolo &qring;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 &longs;unt numeri <expan abbr="&qtilde;d">quad</expan>. </s>
<s>com <lb/>ponentes 13, & <02> &longs;unt 3 2/5 & 1 1/5, & in his opus e&longs;t in du&longs;tria, &longs;cilicet ut <lb/><expan abbr="multiplice&ttilde;">multiplicetur</expan> per numeros <expan abbr="&qtilde;d">quad</expan>. </s>
<s>ut proueniant numeri illi <expan abbr="bifariã">bifariam</expan> compo <lb/>&longs;iti ex <expan abbr="&qtilde;dratis">quadratis</expan>. </s>
<s>Vt uerò uideamus <expan abbr="re&longs;iduũ">re&longs;iduum</expan>, proponamus quae uelim diui <lb/>dere 6 in duos numeros <expan abbr="&qtilde;d">quad</expan>, <expan abbr="primũ">primum</expan> &longs;cire debes &qring;d non po&longs;&longs;unt e&longs;&longs;e <pb pagenum="150"/>integri exratione dicta, quia oporteret ut e&longs;&longs;ent ambo impares aut <lb/>pares, & &longs;ic <expan abbr="differr&etilde;t">differrent</expan> numero pari, ergo oporteret ut e&longs;&longs;et unus me­<lb/>dius numerus <expan abbr="&qtilde;d">quad</expan>. </s>
<s>&longs;unt & ali&ecedil; rationes, &longs;ed neque unus po&longs;&longs;et e&longs;&longs;e inte <lb/>ger, & alius fractus, <expan abbr="nõ">non</expan> e&longs;&longs;et. </s>
<s>n. </s>
<s>6 numerus integer: <expan abbr="relinqui&ttilde;">relinquitur</expan> ergo ut <lb/>&longs;int duo fracti: &longs;ed in numeris fractis <expan abbr="&qtilde;d">quad</expan>. </s>
<s>deductis ad minimas deno <lb/>minationes <expan abbr="operũ">operum</expan>, ut tam denominator <08> numerator habeat radi­<lb/>ces, ergo oportet &qring;d hoc &longs;it in illis, & quia iuncti debent facere inte­<lb/>gros 6, nece&longs;&longs;e e&longs;t ut denominator &longs;it unus, & <expan abbr="id&etilde;">idem</expan> in utroque, et &qring;d nu <lb/>meratores &longs;imul iuncti &longs;int <expan abbr="&longs;excuplũ">&longs;excuplum</expan> denominatoris, &longs;i fracti <expan abbr="deb&etilde;t">debent</expan> <lb/>&ecedil;quipollere 6, ergo ille denominator <expan abbr="cũ">cum</expan> &longs;it <expan abbr="&qtilde;d">quad</expan>. </s>
<s>& numeratores am­<lb/>bo &longs;int <expan abbr="&qtilde;d">quad</expan>. </s>
<s>& &longs;int <expan abbr="&longs;excuplũ">&longs;excuplum</expan> denominatoris, oportebit inuenire <expan abbr="nu-merũ">nu­<lb/>merum</expan> <expan abbr="&qtilde;d">quad</expan>. </s>
<s>qui ductus in 6, faciat <expan abbr="numerũ">numerum</expan> qui <expan abbr="cõponi&ttilde;">componitur</expan> ex duob. </s>
<s><expan abbr="&qtilde;d">quad</expan>. <lb/></s>
<s>aut <expan abbr="cõponi&ttilde;">componitur</expan> &ecedil;qualiter, ergo proportio medietatis ad <expan abbr="medietat&etilde;">medietatem</expan> 6, e&longs;t <lb/>ueluti totius ad 6, &longs;ed totu continet 6 in <expan abbr="&qtilde;d">quad</expan>. </s>
<s>quia ex 6 in <expan abbr="&qtilde;d">quad</expan>. </s>
<s>fit <expan abbr="totũ">totum</expan>, <lb/>ergo ex medietate in <expan abbr="&qtilde;d">quad</expan>. </s>
<s>idem fit medietas, &longs;ed medietas e&longs;t nume­<lb/>rus <expan abbr="&qtilde;d">quad</expan>. </s>
<s>ergo 3 e&longs;&longs;et numerus <expan abbr="&qtilde;d">quad</expan>. </s>
<s>&qring;d e&longs;t fal&longs;um, oportet <expan abbr="igi&ttilde;">igitur</expan> ut nume <lb/>ri illi &longs;int inæ quales, & ut 6 diuidatur in duas partes in&ecedil;quales, 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>numeris <expan abbr="&qtilde;d">quad</expan>. </s>
<s>nam &longs;i e&longs;&longs;et impar, <expan abbr="nõ">non</expan> po&longs;&longs;et prodire numerus integer, & <lb/><expan abbr="cũ">cum</expan> prouenerit numerus <expan abbr="&qtilde;d">quad</expan>. </s>
<s>ille erit <expan abbr="qu&etilde;">quem</expan> qu&ecedil;rimus, <expan abbr="nã">nam</expan> diui&longs;o 6 per to­<lb/>tum <expan abbr="illũ">illum</expan> numerum, inde &qring;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, quæ <expan abbr="erũt">erunt</expan> <lb/>numeri <expan abbr="&qtilde;d">quad</expan>. </s>
<s>quia denominator utriu&longs;que partis ex &longs;uppo&longs;ito e&longs;t nume <lb/>rus <expan abbr="&qtilde;dratus">quadratus</expan>, qui multipli catus e&longs;t per 6, & numeratores &longs;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;">&ecedil;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">compo&longs;itum</expan> ex 6 in <expan abbr="denominator&etilde;">denominatorem</expan>, & hic e&longs;t diui&longs;us per deno <lb/><expan abbr="minator&etilde;">minatorem</expan>, ergo prouenit 6, &longs;i <expan abbr="e&mtilde;">emm</expan> multiplicato 3 in 4 fit 12, diui&longs;o 12 per <lb/>4, exit nece&longs;&longs;ario idem 3. Pro colligendo ergo numeros omnes, qui <lb/><expan abbr="cõponuntur">componuntur</expan> ex <expan abbr="&qtilde;dratis">quadratis</expan>, propones tibi &longs;eriem <expan abbr="&qtilde;d">quad</expan>. </s>
<s><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> ip&longs;um erunt numeratores, et &longs;uppo&longs;iti deno <lb/>minatoribus <expan abbr="cõ&longs;tituent">con&longs;tituent</expan> partes. </s>
<s>Vt uerò cogno&longs;cas, ex quibus po&longs;­<lb/>&longs;it componi primum ex imparibus, non oportet a&longs;&longs;umere ni&longs;i 135, <lb/>quia 7 diui&longs;um per 6 relin quit 1, & 9 diui&longs;um per 6, relinquit 3, & 35 <lb/>diui&longs;um per 6 relinquit 5. ergo non pote&longs;t componi numerus im­<lb/>par, qui diuidatur per 6, ut &longs;up er&longs;it impar alius quàm 1. 3. 5. &longs;ed 1 & 3 <lb/>& 5, & 5 componunt 4 & 1, & 1 & 3 & 5 componunt 2, &longs;cilicet abie­<lb/>cto 6, ergo tales numeri <expan abbr="&qtilde;drati">quadrati</expan> &longs;i &longs;int impares, uel ambo terminan­<lb/>tur in 3, ut 9 & 81, qui faciunt 90, uel in 1 & 5, &longs;ed nullus numerus <lb/>quadratus diui&longs;us per 6 terminatur in 5, quia 1 ductum in &longs;e produ­<lb/>cit 1, & 3 pro ducit 3, & 5 pro ducit 1, ut 5 in 5 facit 25, & 11 in 11 produ­<pb pagenum="151"/>cit 121, quibus diui&longs;is per 6 &longs;upere&longs;t 1. Quod etiam &longs;ic demon&longs;tratur <lb/>de 5, & compo&longs;itis à 5, nam diui&longs;o 5 in 3 & 2, quadratum eius <expan abbr="cõpo-nitur">compo­<lb/>nitur</expan> ex duplo 3 in 2, in quo nihil &longs;upere&longs;t, &longs;i diuidatur per 6, & ex <lb/>quadrato 3, quòd e&longs;t 9, in quo &longs;upere&longs;t 3, & ex quadrato 2 quod e&longs;t </s></p><p type="main">
<s><arrow.to.target n="marg520"></arrow.to.target><lb/>4, &longs;ed iunctis 4 & 3, & abiecto 6 &longs;upere&longs;t 1, ergo 5 in 5 <expan abbr="ductũ">ductum</expan>, & diui <lb/>&longs;o producto relin quitur 1. Et &longs;imiliter capio 17, et <expan abbr="componi&ttilde;">componitur</expan> ex 12 & <lb/>5 quadratum, ergo 17 componitur ex quadrato 12, in quo nihil &longs;u­<lb/>pere&longs;t, & duplo 5 in 12, in quo <expan abbr="etiã">etiam</expan> nihil &longs;upere&longs;t, &longs;i diuidatur per 6: <lb/>& ex quadrato 5, in quo &longs;upere&longs;t 1, ergo in nullo numero <expan abbr="cõpo&longs;ito">compo&longs;ito</expan> <lb/>ex 5 & 6, uel compo&longs;itis ex 6, poterit produci numerus, qui diui&longs;us <lb/>per 6 relin quat 5, igitur neque talis numerus potérit <expan abbr="cõponi">componi</expan> ex duo­<lb/>bus quadratis, in quib. </s>
<s>&longs;uper&longs;it 5 & 1, quia nullus e&longs;t, in quo &longs;uper­<lb/>&longs;it 5 facta diui&longs;ione per 6. Ex quo colligitur una regula: quod &longs;i quis <lb/>dicat multiplicaui 27 in &longs;e, et diui&longs;i per 13, uellem &longs;cire quid &longs;upere&longs;t, <lb/>dico quod &longs;ine multiplicatione et diui&longs;ione poteris hoc &longs;cire ex de­<lb/>mon&longs;tratione dicta, diuide ergo 27 per 13, & relin quitur 1, duc in &longs;e <lb/>fit 1: dices ergo, quod &longs;upererit 1, & ita &longs;i ducerem 28 in &longs;e, & diuide­<lb/>rem per 11, dico quod &longs;upererit 3, nam diui&longs;o 28 per 11, relin quitur <lb/>6, duc in 6 fit 36, diuide per 11, relin quitur 3, ut dictum e&longs;t, & tantum <lb/><expan abbr="relinqui&ttilde;">relinquitur</expan> ducto 28 in &longs;e & fit 784, & diui&longs;o per 11. Reuertendo ergo <lb/>ad propo&longs;itum, pater quod ex duobus tantum numeris imparibus <lb/>quadratis pote&longs;t conflari ille numerus, <expan abbr="quorũ">quorum</expan> radices diui&longs;æ per 6 <lb/>relin quunt 3. Sed de paribus uel &longs;upere&longs;t 2 uel 4 uel nihil, &longs;ed <expan abbr="&qtilde;dra-tum">quadra­<lb/>tum</expan> 2 e&longs;t 4, & <expan abbr="&qtilde;dratum">quadratum</expan> 4 diui&longs;um per 6 etiam relinquit 4, ergo neque<lb/>ex duobus numeris, in quibus &longs;uper&longs;int 2, neque in quibus &longs;uper&longs;int <lb/>4, neque in quibus &longs;uper&longs;int in uno 2, in altero 4 <expan abbr="poterũt">poterunt</expan> quadrata, in <lb/>quibus &longs;emper &longs;upererit 4, & iuncta faciunt 8, in &qring;&longs;upere&longs;t 2, <expan abbr="cõ">com</expan> fla­<lb/>re <expan abbr="numerũ">numerum</expan> <expan abbr="dictũ">dictum</expan> &longs;eu <expan abbr="quæ&longs;itũ">quæ&longs;itum</expan>, qui po&longs;sit diuidi per 6: neque ex <expan abbr="&qtilde;d">quad</expan>. </s>
<s><expan abbr="duo-rũ">duo­<lb/>rum</expan> <expan abbr="num&etilde;rorũ">numerrorum</expan>, in <expan abbr="quorũ">quorum</expan> altero nihil &longs;uper&longs;it in reliquo &longs;uper&longs;it 2 uel <lb/>4, quia in aggregato <expan abbr="&qtilde;dratorũ">quadratorum</expan> &longs;emper &longs;upererit 4. Ergo relinqui­<lb/>tur quod ille numerus componetur ex duobus quadratis, uel impa <lb/>ribus, quorum latera diui&longs;a per 6 relinquunt 3, uel ex duobus pari­<lb/>bus, quorum latera diui&longs;a per 6 nihil relinquant. </s>
<s>Oportet igitur <lb/>inuenire duos tales numeros quadratos numerorum imparium, in <lb/>quibus &longs;uper&longs;it 3, &longs;i diuidantur per 6, aut parium in quibus nihil &longs;u­<lb/>per&longs;it, quorum aggregato diui&longs;o per 6 prodeat numerus <expan abbr="&qtilde;dratus'">quadratus'</expan>.</s></p><p type="margin">
<s><margin.target id="marg520"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 4. <emph type="italics"/>&longs;ecun <lb/>di<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s></p><p type="main">
<s>His ui&longs;is dico, quod con&longs;tat radices talium numerorum opor­<lb/>tere e&longs;&longs;e in imparibus per additionem 6 incipiendo à 3, ut &longs;int <lb/>3. 9. 15. 21. 27. 33. 39. 45. 51. & &longs;ic deinceps: in paribus au­<lb/>tem per additionem eiu&longs;dem 6 incipiendo à 6, uelut 6. 12. <lb/>18. 24. 30. 36. 42. 48. 54. 60. Dico ergo quod diui­<lb/>&longs;o numero illo compo&longs;ito per 6 in imparibus exibit numerus, <pb pagenum="152"/>qui diui&longs;us per 6 &longs;upererit 3, & in paribus qui poterit diuidi per 6. <lb/>Quia <expan abbr="componun&ttilde;">componuntur</expan> ex huiu&longs;modi: uelut 3 in &longs;e facit 9, & 25 in &longs;e facit <lb/>225, qui <expan abbr="iũcti">iuncti</expan> <expan abbr="faciũt">faciunt</expan> 234, diui&longs;o 235 per 6 exit 39, qui <expan abbr="iterũ">iterum</expan> diui&longs;us per 6 <lb/>&longs;upere&longs;t 3, & &longs;imiliter capio 6 & 12, <expan abbr="quorũ">quorum</expan> <expan abbr="&qtilde;drata">quadrata</expan> &longs;unt 36 & 144, & <lb/><expan abbr="aggregatũ">aggregatum</expan> 180, qui diui&longs;us per 6 exit 30, qui <expan abbr="iterũ">iterum</expan> pote&longs;t diuidi per <lb/>6. Et hoc quia <expan abbr="quilibetillorũ">quilibetillorum</expan> pote&longs;t diuidi per <expan abbr="&qtilde;dratũ">quadratum</expan> 6 in paribus, <lb/>ergo aggregato diui&longs;o per 6 &qring;d prodit, <expan abbr="iterũ">iterum</expan> poterit diuidi per 6. <lb/>Et in imparibus quo dlibet <expan abbr="&qtilde;dratorũ">quadratorum</expan> exuperat &longs;upra &longs;enarios in 3, <lb/><expan abbr="igi&ttilde;">igitur</expan> <expan abbr="aggregatũ">aggregatum</expan> diui&longs;um in 2 pariet <expan abbr="numerũ">numerum</expan> qui diui&longs;us per 3, exibit <lb/>numerus impar <expan abbr="cõpo&longs;itus">compo&longs;itus</expan> ex &longs;enarijs & 3. Illud ergo <expan abbr="quadratũ">quadratum</expan>, &qring;d <lb/>prodibit, uel erit <expan abbr="cõpo&longs;itum">compo&longs;itum</expan> ex &longs;enarijs, uel &longs;upererit 3. Sed <expan abbr="cũ">cum</expan> 3 nume <lb/>ret 6, ergo tres <expan abbr="&qtilde;drati">quadrati</expan> numeri &longs;cilicet duo, qui <expan abbr="cõponunt">componunt</expan> <expan abbr="numerũ">numerum</expan>, <lb/><arrow.to.target n="marg521"></arrow.to.target><lb/>& qui prodit per <expan abbr="diui&longs;ion&etilde;">diui&longs;ionem</expan> 6, erunt <expan abbr="cõpo&longs;iti">compo&longs;iti</expan> inter &longs;e, ergo & radices il <lb/>lorum. </s>
<s><expan abbr="Igi&ttilde;">Igitur</expan> radix numeri <expan abbr="&qtilde;drati">quadrati</expan>, qui prouenit diui&longs;o aggregato <expan abbr="qua-dratorũ">qua­<lb/>dratorum</expan> per 6 e&longs;t ex <expan abbr="eod&etilde;">eodem</expan> ordine <expan abbr="impariũ">imparium</expan>, &longs;i impares numeri <expan abbr="&qtilde;drati">quadrati</expan> <lb/><expan abbr="fuerũt">fuerunt</expan>, aut <expan abbr="pariũ">parium</expan> &longs;i pares. </s>
<s>At hoc e&longs;&longs;e <expan abbr="nõ">non</expan> pote&longs;t, <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, &longs;ed diui&longs;i per 3 o&longs;tendent minores, <lb/>quod e&longs;t contra &longs;uppo&longs;itum, quare nullo modo 6 pote&longs;t diuidi in <lb/>duos numeros quadratos, neque integros, neque fractos, quod erat <lb/>demon&longs;trandum. </s>
<s>Habes igitur ex hoc demon&longs;trationem quando <lb/><expan abbr="nõ">non</expan> po&longs;sit diuidi, & quado po&longs;sit, quod po&longs;sit, & quomodo &longs;imul.</s></p><p type="margin">
<s><margin.target id="marg521"></margin.target>P<emph type="italics"/>er<emph.end type="italics"/> 29. <emph type="italics"/>&longs;e­<lb/>ptimi<emph.end type="italics"