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version 1.6, 2004/01/20 14:38:02

version 1.7, 2004/01/29 11:38:50

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<?xml version="1.0"?>

<?xml version="1.0" encoding="UTF-8"?>

<!DOCTYPE archimedes SYSTEM "../dtd/archimedes.dtd">

<info>

<author>Cardano, Girolamo</author>

<title>Opus novum de proportionibus</title>

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<cvs_file>carda_propo_015_la_1570.xml</cvs_file>

<cvs_version/>

</info> <text> <front> <section> <pb xlink:href="015/01/001.jpg"/><pb xlink:href="015/01/002.jpg"/><pb xlink:href="015/01/003.jpg"/><pb xlink:href="015/01/004.jpg"/>

</info>

<text>

<s id="id000001">HIERONYMI <lb/>CARDANI MEDIO <lb/>LANENSIS, CIVISQV'E BONO<lb/>NIENSIS, PHILOSOPHI, MEDICI ET <lb/>Mathematici clari&longs;simi,</s>

<front>

<s id="id000002">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 V libros dige&longs;tum.</s>

<s id="id000003">PRAETEREA.</s>

<s id="id000004">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>

<s id="id000006">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>

<s id="id000007">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>

<s id="id000001">HIERONYMI <lb/>CARDANI MEDIO<lb/>LANENSIS, CIVISQVE BONO<lb/>NIENSIS, PHILOSOPHI, MEDICI ET <lb/>Mathematici clari&longs;simi,</s>

<s id="id000002">OPVS NOVVM DE <lb/>PROPORTIONIBVS NVMERORVM, MO<lb/>TVVM, PONDERVM, SONORVM, ALIARVMQVE 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 <var>V</var> libros dige&longs;tum.</s>

<s id="id000003">PRAETEREA.</s>

<s id="id000008">Cum Cæ&longs;. </s>

<s id="id000009">Maie&longs;t. </s>

<s id="id000010">Gratia & Priuilegio.</s>

<s id="id000010">Gratia & Priuilegio.</s>

<s id="id000011">BASILEÆ.</s></section><section><pb xlink:href="015/01/005.jpg"/><pb xlink:href="015/01/006.jpg"/>

<s id="id000012">IN LIBRVM DE <lb/>PROPORTIONIBVS HIERONYMI <lb/>CARDANI MEDIOLANENSIS, CIVISQV'E <lb/>Bononien&longs;is, Medici, Præfatio ad M. A. </s>

<s id="id000011">BASILEÆ.</s>

</section>

<s id="id000012">IN LIBRVM DE <lb/>PROPORTIONIBVS HIERONYMI <lb/>CARDANI MEDIOLANENSIS, CIVISQVE <lb/>Bononien&longs;is, Medici, Præfatio ad M. A. </s>

<s id="id000013">Amulium <lb/>Venetum Card. </s>

<s id="id000014">Illu&longs;tri&longs;simum.</s>

<s id="id000014">Illu&longs;tri&longs;simum.</s>

<s id="id000015">Bene Dictum e&longs;t meo iudicio à Platone M. <lb/>A. </s>

<s id="id000016">Amuli optime, beatas fore Re&longs;pub. </s>

<s id="id000017">&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 id="id000018">Nempe hi&longs;ce quatuor tota no&longs;tra felicitas com<lb/>prehenditur. </s>

<s id="id000019">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 id="id000020">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 c&oelig;tus ab&longs;que ea diu &longs;tare po&longs;<lb/>&longs;int. </s>

<s id="id000021">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 id="id000021">Porrò quid dicam de concordia, & mutua hominum beneuo<lb/>lentia, in quibus omnis uit&etail; human&etail; 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 id="id000022">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 id="id000023">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 id="id000024">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>

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<s id="id000033">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 id="id000034">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 id="id000035">quam tamen plerique tenere &longs;e putant. </s>

<s id="id000036">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 id="id000036">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;patio <lb/>&longs;eu loco &longs;tare po&longs;&longs;unt, ita & in figuris picturarum & &longs;tatuarum, & <lb/>diebus decretorijs, & negotijs ciuilibus oper&etail; pretium 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 id="id000037">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 id="id000038">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 id="id000038">Itaque infinitis licet circumuentus <lb/>negotijs totus huic operæ in cubui, atque adeò ut præter &longs;pem unius <lb/>anni penè &longs;patio liber ab&longs;olueretur. </s>

<s id="id000039">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></section><section><pb xlink:href="015/01/008.jpg"/>

<s id="id000040">TABVLA PRO<lb/>POSITIONVM DE <lb/>PROPORTIONIBVS.<lb/><arrow.to.target n="table1"/></s><table><table.target id="table1"/><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="quantitat&utilde;">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/></row><pb xlink:href="015/01/009.jpg"/><row><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;ec&utilde;da">&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 fuerit 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/></row><pb xlink:href="015/01/010.jpg"/><row><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 <expan abbr="inuic&etilde;">inuicem</expan> du ctas proportionem minorem unaquaque illarum producere.<emph.end type="italics"/></cell><cell>30</cell></row><row><cell>XXXVII.</cell><cell>S<emph type="italics"/>i plures homines, quorum per &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="aggregat&utilde;">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="c&utilde;alio">cunalio</expan> mobili in dato puncto <expan abbr="cõueniat">conueniat</expan> &longs;ub <expan abbr="quoc&utilde;que">quocunque</expan> numero <expan abbr="circuitu&utilde;">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/></row><pb xlink:href="015/01/011.jpg"/><row><cell/><cell><emph type="italics"/>uitatis centrum.<emph.end type="italics"/></cell><cell>51</cell></row><row><cell>LXI.</cell><cell>P<emph type="italics"/>roportionum ictus ad pondus rei & 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"/>Si fuerint tres quantitates in continua 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="immot&utilde;">immotum</expan> in terra in excipiendo <expan abbr="ict&utilde;">ictum</expan> inuenire.<emph.end type="italics"/></cell><cell>65</cell></row><row><cell>LXXVI.</cell><cell>P<emph type="italics"/>roportionem <expan abbr="duor&utilde;">duorum</expan> mobilium &longs;ibi <expan abbr="inuic&etilde;">inuicem</expan> <expan abbr="concurrenti&utilde;">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;trument&utilde;">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;i&utilde;">&longs;ium</expan> e&longs;t duas naues &longs;ibi <expan abbr="inuic&etilde;">inuicem</expan> occur&longs;antes retrocedere, & <expan abbr="quãt&utilde;">quantum</expan> <expan abbr="retrocedãt">retrocedant</expan> ambæ.<emph.end type="italics"/></cell><cell>86</cell></row><pb xlink:href="015/01/012.jpg"/><row><cell>XCIIII.</cell><cell>S<emph type="italics"/>i <expan abbr="quãtitas">quantitas</expan> aliqua nota atque proportio erit producta, <expan abbr="quãtitas">quantitas</expan> nota &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 totum 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="er&utilde;t">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;picu&utilde;">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="ot&utilde;">otum</expan> inuer&longs;ionis in figuris in <expan abbr="cõparatione">comparatione</expan> ad <expan abbr="mot&utilde;">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 multitudinem &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="ponder&utilde;">ponderum</expan> attractorum per <expan abbr="trochlear&utilde;">trochlearum</expan> <expan abbr="numer&utilde;">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 xlink:href="015/01/013.jpg"/><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 duarum 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="quar&utilde;">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="enomination&utilde;">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="additar&utilde;">additarum</expan> &longs;ingularium<emph.end type="italics"/></cell><cell/></row><pb xlink:href="015/01/014.jpg"/><row><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="aggregat&utilde;">aggregatum</expan> ex una <expan abbr="adiectar&utilde;">adiectarum</expan>, & par te ad <expan abbr="aggregat&utilde;">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="parti&utilde;">partium</expan> <expan abbr="inæquali&utilde;">inæqualium</expan> duplicata, & rur&longs;us ip&longs;um <expan abbr="dimidi&utilde;">dimidium</expan> lineæ a&longs;&longs;um-ptæ <expan abbr="medi&utilde;">medium</expan>, erit proportione inter additas.<emph.end type="italics"/> D<emph type="italics"/><expan abbr="em&utilde;">emum</expan> proportio dimidij <expan abbr="c&utilde;">cum</expan> addita maiore ad <expan abbr="dimidi&utilde;">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="content&utilde;">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 xlink:href="015/01/015.jpg"/><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 f&oelig;mora, & f&oelig;mora 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/></row><pb xlink:href="015/01/016.jpg"/><row><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="quart&utilde;">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 æqualitas, 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;ec&utilde;do">&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, dicetur 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 ferentiam 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 xlink:href="015/01/017.jpg"/><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/></row><pb xlink:href="015/01/018.jpg"/><row><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>

<s id="id000041">FINIS.</s><pb xlink:href="015/01/019.jpg"/></section></front> <body> <chap>

</section>

<s id="id000042">HIERONYMI CAR <lb/>DANI MEDIOLANENSIS, CI<lb/>VI'SQVE BONONIENSIS, MEDICI<lb/>de Proportionibus, &longs;eu Ope<lb/>ris Perfecti <lb/>LIBER QVINTVS.</s>

<s id="id000040">TABVLA PRO<lb/>POSITIONVM DE <lb/>PROPORTIONIBVS.<lb/><arrow.to.target n="table1"/></s>

<table>

<s id="id000043">Prima diffinitio.</s>

<table.target id="table1"/>

<s id="id000044">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>

<row>

<s id="id000045">Secunda diffinitio.</s>

<s id="id000046">Proportiones per &longs;imilitudinem <expan abbr="dic&utilde;tur">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.</s>

<cell>Proportionem <emph type="italics"/>in proportionem duci, e&longs;t &longs;uperiores numeros atque inferiores inuicem ducere.<emph.end type="italics"/></cell>

<s id="id000047">Velut &longs;i a b fingatur monas in comparatione <lb/>ad b c erit rectangulum a c æquale lineæ b c.</s><figure id="id.015.01.020.1.jpg" xlink:href="015/01/020/1.jpg"/>

<cell><emph type="italics"/>pagina<emph.end type="italics"/> 6</cell>

<s id="id000048">Tertia diffinitio.</s>

</row>

<s id="id000049">Proportio æqualis proportioni e&longs;t, cùm eodem modo termini <lb/>&longs;e habent inuicem in utraque</s>

<row>

<s id="id000050">Quarta diffinitio.</s>

<cell>P<emph type="italics"/>roportio extremorum producitur ex intermedijs.<emph.end type="italics"/></cell>

</row>

<row>

<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>

</row>

<row>

<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>

</row>

<row>

<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;ecundum, & quarti ad quintum.<emph.end type="italics"/></cell>

</row>

<row>

<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>

</row>

<row>

<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>

</row>

<row>

<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>

</row>

<row>

<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>

</row>

<row>

<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>

</row>

<row>

<cell>P<emph type="italics"/>roportio aggregati quarumlibet duarum quantitatum ad aggregatum duarum æqua-lium <expan abbr="quantitat&utilde;">quantitatum</expan> e&longs;t, compo&longs;ita ex proportionibus primis, & diui&longs;a per duplam.<emph.end type="italics"/></cell>

</row>

<row>

<cell>P<emph type="italics"/>ropo&longs;itis duabus proportionibus unam alteri iungere ab&longs;que multiplicatione.<emph.end type="italics"/></cell>

</row>

<row>

<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>

</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>

</row>

<row>

<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>

</row>

<row>

<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/>

</row>

<row>

<cell/>

<cell><emph type="italics"/>ut producti ex aggregato primæ & tertiæ, in tertiam ad productum ex aggre gato tertiæ & omiotatæ ad &longs;ecundam in ip&longs;am quartam.<emph.end type="italics"/></cell>

</row>

<row>

<cell>O<emph type="italics"/>mnes duæ proportiones conuer&longs;æ producunt æqualem proportionem.<emph.end type="italics"/></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>

</row>

<row>

<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>

</row>

<row>

<cell>C<emph type="italics"/>um fuerint quatuor quantitates, fueritque <expan abbr="&longs;ec&utilde;da">&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>

</row>

<row>

<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 fuerit 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>

</row>

<row>

<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>

</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>

</row>

<row>

<cell>XXIIII.</cell>

<cell>O<emph type="italics"/>mnis motus circularis uoluntarius e&longs;t.<emph.end type="italics"/></cell>

</row>

<row>

<cell>T<emph type="italics"/>res &longs;unt motus omnino &longs;implices naturalis, uoluntarius, & uiolentus.<emph.end type="italics"/></cell>

</row>

<row>

<cell>M<emph type="italics"/>otus ergo compo&longs;iti quatuor nece&longs;&longs;ariò &longs;unt &longs;pecies.<emph.end type="italics"/></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>

</row>

<row>

<cell>XXVIII.</cell>

<cell>M<emph type="italics"/>otus quilibet uoluntarius aut uiolentus in aliquo medio fit.<emph.end type="italics"/></cell>

</row>

<row>

<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>

</row>

<row>

<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>

</row>

<row>

<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>

</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>

</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>

</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>

</row>

<row>

<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>

</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/>

</row>

<row>

<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 <expan abbr="inuic&etilde;">inuicem</expan> du ctas proportionem minorem unaquaque illarum producere.<emph.end type="italics"/></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>

</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>

</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>

</row>

<row>

<cell>O<emph type="italics"/>mne corpus &longs;phæricum tangens planum in puncto mouetur ad latus per quam-cunque uim, quæ medium diuidere pote&longs;t.<emph.end type="italics"/></cell>

</row>

<row>

<cell>S<emph type="italics"/>i fuerint duæ quantitates &longs;umaturque toties <expan abbr="aggregat&utilde;">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>

</row>

<row>

<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>

</row>

<row>

<cell>XLIII.</cell>

<cell>P<emph type="italics"/>roductionem ad additionem retrahere.<emph.end type="italics"/></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;patium & tempus, nota erit etiam reliquorum nota.<emph.end type="italics"/></cell>

</row>

<row>

<cell>R<emph type="italics"/>ationem &longs;tateræ o&longs;tendere.<emph.end type="italics"/></cell>

</row>

<row>

<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>

</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>

</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>

</row>

<row>

<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="c&utilde;alio">cunalio</expan> mobili in dato puncto <expan abbr="cõueniat">conueniat</expan> &longs;ub <expan abbr="quoc&utilde;que">quocunque</expan> numero <expan abbr="circuitu&utilde;">circuituum</expan> <expan abbr="t&etilde;pus">tempus</expan> quoque <expan abbr="cõiunctionis">coniunctionis</expan>.<emph.end type="italics"/></cell>

</row>

<row>

<cell>O<emph type="italics"/>mnes circuituum portiones in ei&longs;dem temporibus repetuntur.<emph.end type="italics"/></cell>

</row>

<row>

<cell>O<emph type="italics"/>perationes dictas exemplo declarare.<emph.end type="italics"/></cell>

</row>

<row>

<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>

</row>

<row>

<cell>C<emph type="italics"/>irculorum &longs;e in aduer&longs;um mouentium proportionem declarare.<emph.end type="italics"/></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>

</row>

<row>

<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>

</row>

<row>

<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>

</row>

<row>

<cell>M<emph type="italics"/>otus rationem ad pondus inuenire.<emph.end type="italics"/></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>

</row>

<row>

<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>

</row>

<row>

<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/>

</row>

<row>

<cell/>

<cell><emph type="italics"/>uitatis centrum.<emph.end type="italics"/></cell>

</row>

<row>

<cell>P<emph type="italics"/>roportionum ictus ad pondus rei & di&longs;tantiam generaliter con&longs;iderare.<emph.end type="italics"/></cell>

</row>

<row>

<cell>P<emph type="italics"/>roportionem motoris in plano ad motorem, qui eleuat pondus iuxta id quod mouet, inuenire.<emph.end type="italics"/></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>

</row>

<row>

<cell>LXIIII.</cell>

<cell>O<emph type="italics"/>mne mobile quantò latius tanto tardius moustur in plano.<emph.end type="italics"/></cell>

</row>

<row>

<cell>P<emph type="italics"/>roportionem duorum mobilium inter &longs;e cum auxilio medij inuenire.<emph.end type="italics"/></cell>

</row>

<row>

<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>

</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>

</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>

</row>

<row>

<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>

</row>

<row>

<cell>S<emph type="italics"/>i fuerint tres quantitates in continua 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>

</row>

<row>

<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>

</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>

</row>

<row>

<cell>LXXIII.</cell>

<cell>P<emph type="italics"/>roportionem ponderum attractorum penes figuram in plano inuenire.<emph.end type="italics"/></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>

</row>

<row>

<cell>P<emph type="italics"/><expan abbr="roportion&etilde;">roportionem</expan> immoti in aqua, ad <expan abbr="immot&utilde;">immotum</expan> in terra in excipiendo <expan abbr="ict&utilde;">ictum</expan> inuenire.<emph.end type="italics"/></cell>

</row>

<row>

<cell>LXXVI.</cell>

<cell>P<emph type="italics"/>roportionem <expan abbr="duor&utilde;">duorum</expan> mobilium &longs;ibi <expan abbr="inuic&etilde;">inuicem</expan> <expan abbr="concurrenti&utilde;">concurrentium</expan> per <expan abbr="rectã">rectam</expan> inuenire.<emph.end type="italics"/></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>

</row>

<row>

<cell>LXXVIII.</cell>

<cell>P<emph type="italics"/>roportionem nauis ad triremes quotuis concurrentes demon&longs;trare.<emph.end type="italics"/></cell>

</row>

<row>

<cell>LXXIX.</cell>

<cell>P<emph type="italics"/>roportionem medicamentorum purgantium inuicem declarare<emph.end type="italics"/></cell>

</row>

<row>

<cell>P<emph type="italics"/>roportionem motus &longs;ecundum obliquum ad rectum in &longs;pacio declarare.<emph.end type="italics"/></cell>

</row>

<row>

<cell>LXXXI.</cell>

<cell>Q<emph type="italics"/>ualis &longs;it angulus, per quem pote&longs;t moueri nauis ad rectum explorare.<emph.end type="italics"/></cell>

</row>

<row>

<cell>LXXXII.</cell>

<cell>P<emph type="italics"/>roportionem uelorum indagare.<emph.end type="italics"/></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>

</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>

</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>

</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>

</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>

</row>

<row>

<cell>LXXXVIII.</cell>

<cell>I<emph type="italics"/><expan abbr="n&longs;trument&utilde;">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>

</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>

</row>

<row>

<cell>R<emph type="italics"/>ationem impetus uiolenti extra mißi ponderis ad æqualitatem reducere.<emph.end type="italics"/></cell>

</row>

<row>

<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>

</row>

<row>

<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>

</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;i&utilde;">&longs;ium</expan> e&longs;t duas naues &longs;ibi <expan abbr="inuic&etilde;">inuicem</expan> occur&longs;antes retrocedere, & <expan abbr="quãt&utilde;">quantum</expan> <expan abbr="retrocedãt">retrocedant</expan> ambæ.<emph.end type="italics"/></cell>

</row>

<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 totum 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>

</row>

<row>

<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="er&utilde;t">erunt</expan> omnia eius latera cognita.<emph.end type="italics"/></cell>

</row>

<row>

<cell>C<emph type="italics"/>um in <expan abbr="per&longs;picu&utilde;">per&longs;picuum</expan> den&longs;um radij lumino&longs;i inciderint, quatuor fiunt luminis genera.<emph.end type="italics"/></cell>

</row>

<row>

<cell>XCVII.</cell>

<cell>M<emph type="italics"/><expan abbr="ot&utilde;">otum</expan> inuer&longs;ionis in figuris in <expan abbr="cõparatione">comparatione</expan> ad <expan abbr="mot&utilde;">motum</expan> &longs;phæræ in plano inue&longs;tigare.<emph.end type="italics"/></cell>

</row>

<row>

<cell>XCVIII.</cell>

<cell>P<emph type="italics"/>roportionem ponderum æqualium per differentiam angulorum inuenire.<emph.end type="italics"/></cell>

</row>

<row>

<cell>P<emph type="italics"/>roportionem grauitatum per multitudinem &longs;uppo&longs;itorum orbium o&longs;tendere.<emph.end type="italics"/></cell>

</row>

<row>

<cell>P<emph type="italics"/><expan abbr="roportion&etilde;">roportionem</expan> grauitatis <expan abbr="ponder&utilde;">ponderum</expan> attractorum per <expan abbr="trochlear&utilde;">trochlearum</expan> <expan abbr="numer&utilde;">numerum</expan> inue&longs;tigare.<emph.end type="italics"/></cell>

</row>

<row>

<cell>P<emph type="italics"/>roportionem precij gemmarum ex tribus in eodem genere cognitis inuenire.<emph.end type="italics"/></cell>

</row>

<row>

<cell>P<emph type="italics"/>roportionem motuum inuer&longs;ionis, & attractionis in plano inuenire.<emph.end type="italics"/></cell>

</row>

<row>

<cell>P<emph type="italics"/>roportionem eorundem in accliui demon&longs;trare.<emph.end type="italics"/></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>

</row>

<row>

<cell>P<emph type="italics"/>roportionem ferentium pondus in pertica inuenire.<emph.end type="italics"/></cell>

</row>

<row>

<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>

</row>

<row>

<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>

</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>

</row>

<row>

<cell>R<emph type="italics"/>ationem libræ expendere.<emph.end type="italics"/></cell>

</row>

<row>

<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>

</row>

<row>

<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>

</row>

<row>

<cell>C<emph type="italics"/>ur ex imo leuia longiùs ferantur declarare,<emph.end type="italics"/></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>

</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>

</row>

<row>

<cell>P<emph type="italics"/>roportionem motuum impul&longs;ionis, & attractionis inter &longs;e, ab eadem ui decla-rare.<emph.end type="italics"/></cell>

</row>

<row>

<cell>C<emph type="italics"/>ur machinæ oblongæ igneæ longius emittant &longs;phæram explorare.<emph.end type="italics"/></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>

</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>

</row>

<row>

<cell>Q<emph type="italics"/>uantum ictus machinæ procliuis ad angulum minuatur explorare.<emph.end type="italics"/></cell>

</row>

<row>

<cell>P<emph type="italics"/>roportionem partium nauis ad eundem obliquum uentum explorare.<emph.end type="italics"/></cell>

</row>

<row>

<cell>F<emph type="italics"/>labelli uires atque naturam declarare.<emph.end type="italics"/></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>

</row>

<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>

</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>

</row>

<row>

<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>

</row>

<row>

<cell>CXXVI.</cell>

<cell>C<emph type="italics"/>irculi horarij naturam declarare.<emph.end type="italics"/></cell>

</row>

<row>

<cell>CXXVII.</cell>

<cell>D<emph type="italics"/>ata poli altitudine ortus amplitudinem demonftrare.<emph.end type="italics"/></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>

</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>

</row>

<row>

<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>

</row>

<row>

<cell>CXXXI.</cell>

<cell>S<emph type="italics"/>i lineæ alicui duplum alterius adiungatur, erit proportio duarum ad primam maior quàm dupli cum prima ad primam cum una adiecta.<emph.end type="italics"/></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>

</row>

<row>

<cell>CXXXIII.</cell>

<cell>S<emph type="italics"/>i fuerint duæ quantitates, <expan abbr="quar&utilde;">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>

</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>

</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>

</row>

<row>

<cell>CXXXVI.</cell>

<cell>D<emph type="italics"/>enominationes in infinitum extendere.<emph.end type="italics"/></cell>

</row>

<row>

<cell>CXXXVII.</cell>

<cell>R<emph type="italics"/>ationem numerorum ex progreßione declarare.<emph.end type="italics"/></cell>

</row>

<row>

<cell>CXXXVIII.</cell>

<cell>M<emph type="italics"/>odos u&longs;us horum numerorum declarare.<emph.end type="italics"/></cell>

</row>

<row>

<cell>CXXXIX.</cell>

<cell>R<emph type="italics"/>adices omnes à propo&longs;itis numeris extrahere.<emph.end type="italics"/></cell>

</row>

<row>

<cell>R<emph type="italics"/>adices per numeros fractos determinare.<emph.end type="italics"/></cell>

</row>

<row>

<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>

</row>

<row>

<cell>CXLII.</cell>

<cell>D<emph type="italics"/><expan abbr="enomination&utilde;">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>

</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>

</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>

</row>

<row>

<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>

</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>

</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="additar&utilde;">additarum</expan> &longs;ingularium<emph.end type="italics"/></cell>

<cell/>

</row>

<row>

<cell/>

<cell><emph type="italics"/>& partium &longs;imul iunctarum ad additas &longs;it mutua.<emph.end type="italics"/></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="aggregat&utilde;">aggregatum</expan> ex una <expan abbr="adiectar&utilde;">adiectarum</expan>, & par te ad <expan abbr="aggregat&utilde;">aggregatum</expan> ex alia parte, & adiecta &longs;e habeat, ut &longs;ecunda ad <expan abbr="tertiã">tertiam</expan>.<emph.end type="italics"/></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>

</row>

<row>

<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>

</row>

<row>

<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>

</row>

<row>

<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="parti&utilde;">partium</expan> <expan abbr="inæquali&utilde;">inæqualium</expan> duplicata, & rur&longs;us ip&longs;um <expan abbr="dimidi&utilde;">dimidium</expan> lineæ a&longs;&longs;um-ptæ <expan abbr="medi&utilde;">medium</expan>, erit proportione inter additas.<emph.end type="italics"/> D<emph type="italics"/><expan abbr="em&utilde;">emum</expan> proportio dimidij <expan abbr="c&utilde;">cum</expan> addita maiore ad <expan abbr="dimidi&utilde;">dimidium</expan>, cum addita minore, uelut maioris partis ad <expan abbr="minor&etilde;">minorem</expan>.<emph.end type="italics"/></cell>

</row>

<row>

<cell>CLIII.</cell>

<cell>V<emph type="italics"/>im quamcunque manus multiplicare.<emph.end type="italics"/></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>

</row>

<row>

<cell>Q<emph type="italics"/>uadr atorum numerum proportionem & inuentionem con&longs;iderare.<emph.end type="italics"/></cell>

</row>

<row>

<cell>H<emph type="italics"/>orologiorum tempus multiplicare.<emph.end type="italics"/></cell>

</row>

<row>

<cell>CLVII.</cell>

<cell>H<emph type="italics"/>orologiorum molarium rationem o&longs;tendere.<emph.end type="italics"/></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>

</row>

<row>

<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>

</row>

<row>

<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>

</row>

<row>

<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="content&utilde;">contentum</expan> minorem e&longs;&longs;e.<emph.end type="italics"/></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>

</row>

<row>

<cell>CLXIII.</cell>

<cell>P<emph type="italics"/>roportionem uirium &longs;tellarum per motus &longs;uos indagare.<emph.end type="italics"/></cell>

</row>

<row>

<cell>CLXIIII.</cell>

<cell>S<emph type="italics"/>yderum proportionem in magnitudine o&longs;tendere.<emph.end type="italics"/></cell>

</row>

<row>

<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>

</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>

</row>

<row>

<cell>CLXVII.</cell>

<cell>P<emph type="italics"/>roportionem mu&longs;icam ad &longs;apores & odores coaptare.<emph.end type="italics"/></cell>

</row>

<row>

<cell>CLXVIII.</cell>

<cell>P<emph type="italics"/>icturarum proportiones explicare.<emph.end type="italics"/></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>

</row>

<row>

<cell>C<emph type="italics"/>oniugationes cuiu&longs;uis numeri breuiter inuenire.<emph.end type="italics"/></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>

</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>

</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>

</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>

</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>

</row>

<row>

<cell>CLXXVI.</cell>

<cell>R<emph type="italics"/>ationem centri grauitatis declarare.<emph.end type="italics"/></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>

</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>

</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>

</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>

</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>

</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>

</row>

<row>

<cell>CLXXXIII.</cell>

<cell>S<emph type="italics"/>pacium uitæ naturalis per &longs;pacium uitæ fortuitum declarare.<emph.end type="italics"/></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>

</row>

<row>

<cell>CLXXXV.</cell>

<cell>C<emph type="italics"/>ur homo &longs;edens quanto altius &longs;edet, & quanto magis crura ad f&oelig;mora, & f&oelig;mora 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>

</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/>

</row>

<row>

<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>

</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="quart&utilde;">quartum</expan> pon dus tardius & maiore cum difficultate moueri quàm &longs;ecundum.<emph.end type="italics"/></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 æqualitas, ibi erit maior difficultas.<emph.end type="italics"/></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>

</row>

<row>

<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;ec&utilde;do">&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>

</row>

<row>

<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, dicetur pondus auxiliare æqualis proportionis.<emph.end type="italics"/></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 ferentiam 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>

</row>

<row>

<cell>CXCIII.</cell>

<cell>R<emph type="italics"/>ationem ponderis triplicem explicare.<emph.end type="italics"/></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>

</row>

<row>

<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>

</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>

</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>

</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>

</row>

<row>

<cell>CXCIX.</cell>

<cell>R<emph type="italics"/>ationem remorum nauim impellentium inuenire.<emph.end type="italics"/></cell>

</row>

<row>

<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>

</row>

<row>

<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>

</row>

<row>

<cell>R<emph type="italics"/>ationem &longs;trepitus o&longs;tendere.<emph.end type="italics"/></cell>

</row>

<row>

<cell>CCIII.</cell>

<cell>C<emph type="italics"/>ur &longs;cytalis onera portentur faciliùs, explorare.<emph.end type="italics"/></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>

</row>

<row>

<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>

</row>

<row>

<cell>R<emph type="italics"/>hombi paßiones qua&longs;dam declarare.<emph.end type="italics"/></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>

</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>

</row>

<row>

<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>

</row>

<row>

<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>

</row>

<row>

<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>

</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>

</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>

</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>

</row>

<row>

<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>

</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>

</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>

</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>

</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>

</row>

<row>

<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>

</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>

</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>

</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>

</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/>

</row>

<row>

<cell/>

<cell><emph type="italics"/>uel men&longs;em.<emph.end type="italics"/></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>

</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>

</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>

</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>

</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>

</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>

</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>

</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>

</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>

</row>

</table>

<s id="id000041">FINIS.</s>

</section>

</front>

<body>

<chap>

<s id="id000042">HIERONYMI CAR <lb/>DANI MEDIOLANENSIS, CI<lb/>VI'SQVE BONONIENSIS, MEDICI<lb/>de Proportionibus, &longs;eu Ope<lb/>ris Perfecti <lb/>LIBER QVINTVS.</s>

<s id="id000043">Prima diffinitio.</s>

<s id="id000044">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>

<s id="id000045">Secunda diffinitio.</s>

<s id="id000047">Velut &longs;i a b fingatur monas in comparatione <lb/>ad b c erit rectangulum a c æquale lineæ b c.</s>

<s id="id000048">Tertia diffinitio.</s>

<s id="id000049">Proportio æqualis proportioni e&longs;t, cùm eodem modo termini <lb/>&longs;e habent inuicem in utraque</s>

<s id="id000050">Quarta diffinitio.</s>

<s id="id000051">Proportiones &longs;ecundum genus notæ dicuntur, cùm nouimus, <lb/>quòd &longs;int maiores, aut minores. </s>

<s id="id000052">Nam cùm æquales &longs;unt, &longs;imul ne<lb/>ceffe e&longs;t, ut cogno&longs;camus genus, & &longs;peciem.</s>

<s id="id000052">Nam cùm æquales &longs;unt, &longs;imul ne<lb/>ceffe e&longs;t, ut cogno&longs;camus genus, & &longs;peciem.</s>

<s id="id000053">Quinta diffinitio.</s>

<s id="id000054">Datum po&longs;itione e&longs;t: quod nece&longs;&longs;ariò ex po&longs;itis certam habet <lb/>quantitatem.</s>

<s id="id000055">Sexta diffinitio.</s>

<s id="id000053">Quinta diffinitio.</s>

<s id="id000056">Datum &longs;impliciter dicitur, quod ex propo&longs;itis cogno&longs;ci pote&longs;t, <lb/>quantum &longs;it.</s>

<s id="id000057">Septima diffinitio.</s>

<s id="id000058">Proportiones pote&longs;tate <expan abbr="dicun&ttilde;">dicuntur</expan>, quæ&longs;ub comparatione aliarum <lb/><expan abbr="quantitat&utilde;">quantitatum</expan> nece&longs;&longs;ariam habentium <expan abbr="cõnexionem">connexionem</expan> <expan abbr="&longs;ol&utilde;">&longs;olum</expan> <expan abbr="cogno&longs;cun&ttilde;">cogno&longs;cuntur</expan>.</s>

<s id="id000054">Datum po&longs;itione e&longs;t: quod nece&longs;&longs;ariò ex po&longs;itis certam habet <lb/>quantitatem.</s>

<s id="id000059">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>

<s id="id000060">Octaua diffinitio.</s>

<s id="id000061">Proportio homonyma dicitur duarum quantitatum diuer&longs;i ge</s>

<s id="id000055">Sexta diffinitio.</s>

<s id="id000056">Datum &longs;impliciter dicitur, quod ex propo&longs;itis cogno&longs;ci pote&longs;t, <lb/>quantum &longs;it.</s>

<s id="id000057">Septima diffinitio.</s>

<s id="id000060">Octaua diffinitio.</s>

<s id="id000061">Proportio homonyma dicitur duarum quantitatum diuer&longs;i ge</s>

<s id="id000062"><arrow.to.target n="marg1"/><lb/>neris, &longs;ed alterius a b altero dependentium, uelut motus ad tem

<s id="id000063">Dicimus enim motum tardum, uel uelocem in comparatione <lb/>ad tempus.</s>

<s id="id000063">Dicimus enim motum tardum, uel uelocem in comparatione <lb/>ad tempus.</s>

<s id="id000065">Nona diffinitio.</s>

<s id="id000066">Proportionum aliæ dicuntur rhete, aliæ alogæ, rhetæ quæ &longs;unt <lb/>ut numeri ad numerum, alogæ quæ non &longs;unt numeri ad numerum.</s>

<s id="id000067">Decima diffinitio</s>

<s id="id000068">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>

<s id="id000069">Vndecima diffinitio.</s>

<s id="id000065">Nona diffinitio.</s>

<s id="id000070">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>

<s id="id000071">Duodecima diffinitio.</s>

<s id="id000072">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="id.015.01.021.1.jpg" xlink:href="015/01/021/1.jpg"/><lb/>titates a b c dicetur proportio a ad c producta ex pro <lb/>portione a ad b & b ad c, & &longs;imiliter proportio c ad <lb/>a producitur ex proportione b ad a, & c ad b.</s>

<s id="id000066">Proportionum aliæ dicuntur rhete, aliæ alogæ, rhetæ quæ &longs;unt <lb/>ut numeri ad numerum, alogæ quæ non &longs;unt numeri ad numerum.</s>

<s id="id000073">Tertiadecima diffinitio.</s>

<s id="id000074">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>

<s id="id000075">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>

<s id="id000067">Decima diffinitio</s>

<s id="id000076">Quartadecima diffinitio.</s>

<s id="id000077">Additio proportionum intelligitur quotiens duarum quanti<lb/>tatum ad unam tertiam, proportiones per aggregatum ip&longs;arum <lb/>quantitatum ad eandem coniunguntur.</s>

<s id="id000069">Vndecima diffinitio.</s>

<s id="id000070">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>

<s id="id000071">Duodecima diffinitio.</s>

<s id="id000073">Tertiadecima diffinitio.</s>

<s id="id000076">Quartadecima diffinitio.</s>

<s id="id000077">Additio proportionum intelligitur quotiens duarum quanti<lb/>tatum ad unam tertiam, proportiones per aggregatum ip&longs;arum <lb/>quantitatum ad eandem coniunguntur.</s>

<s id="id000078">Velut &longs;i comparentur a b & b c ad d, inde tota <lb/><figure id="id.015.01.021.2.jpg" xlink:href="015/01/021/2.jpg"/><lb/>a c ad d dicemus proportionem, ac ad d 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 id="id000079">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 id="id000079">Hoc & duo &longs;equentes &longs;icut & du&etail; <expan abbr="anteced&etilde;tes">antecedentes</expan> demon<lb/>&longs;trabitur e&longs;&longs;e. </s>

<s id="id000080">nunc &longs;olum quomodo <expan abbr="intelligend&utilde;">intelligendum</expan> &longs;it proponimus.</s>

<s id="id000080">nunc &longs;olum quomodo <expan abbr="intelligend&utilde;">intelligendum</expan> &longs;it proponimus.</s>

<s id="id000081">Quintadecima diffinitio.</s>

<s id="id000082">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>

<s id="id000081">Quintadecima diffinitio.</s>

<s id="id000083">Velut in exemplo &longs;uperiore detracta proportione b c ad d ex

<pb xlink:href="015/01/022.jpg" pagenum="3"/>proportione a c ad d, relinquetur proportio a b ad d. </s>

<pb pagenum="3" xlink:href="015/01/022.jpg"/>proportione a c ad d, relinquetur proportio a b ad d. </s>

<s id="id000084">& probatur <lb/>ex conuer&longs;ione præcedentis.</s>

<s id="id000084">& probatur <lb/>ex conuer&longs;ione præcedentis.</s>

<s id="id000085">Sextadecima diffinitio.</s>

<s id="id000086">Extractio radicum alicuius proportionis fit per extractionem <lb/>radicum quantitatum illius iuxta unam, & eandem rationem.</s>

<s id="id000087">Velut quadratæ, uel cubæ, uel pronicæ, uel uniner&longs;alis, uel alte<lb/>rius modi.</s>

<s id="id000085">Sextadecima diffinitio.</s>

<s id="id000088">Decima&longs;eptima diffinitio.</s>

<s id="id000086">Extractio radicum alicuius proportionis fit per extractionem <lb/>radicum quantitatum illius iuxta unam, & eandem rationem.</s>

<s id="id000087">Velut quadratæ, uel cubæ, uel pronicæ, uel uniner&longs;alis, uel alte<lb/>rius modi.</s>

<s id="id000088">Decima&longs;eptima diffinitio.</s>

<s id="id000089">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 id="id000090">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>

<s id="id000091">Decimaoctaua diffinitio.</s>

<s id="id000092">Confu&longs;a proportio dicitur &longs;implicis, aut compo&longs;itæ quantitatis <lb/>ad compo&longs;itam in comparatione ad proportiones ad partes.</s>

<s id="id000093">Decimanona diffinitio.</s>

<s id="id000091">Decimaoctaua diffinitio.</s>

<s id="id000094">Quantitates qu&ecedil; in continua &longs;unt proportione Analogæ <expan abbr="uocan&ttilde;">uocantur</expan>.</s>

<s id="id000095">Dictum e&longs;t hoc ad fugiendum nomen barbarum, etiam ut bre<lb/>uiter tamen po&longs;&longs;emus &longs;ententiam explicare.</s>

<s id="id000096">Vige&longs;ima diffinitio.</s>

<s id="id000092">Confu&longs;a proportio dicitur &longs;implicis, aut compo&longs;itæ quantitatis <lb/>ad compo&longs;itam in comparatione ad proportiones ad partes.</s>

<s id="id000097">Reflexa proportio dicitur cum trium quantitatum aggregatum <lb/>primæ, & tertiæ &longs;e habet ad &longs;ecundam uelut &longs;ecunda ad tertiam,</s>

<s id="id000098">Vige&longs;ima prima diffinitio.</s>

<s id="id000099">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>

<s id="id000093">Decimanona diffinitio.</s>

<s id="id000094">Quantitates qu&etail; in continua &longs;unt proportione Analogæ <expan abbr="uocan&ttilde;">uocantur</expan>.</s>

<s id="id000095">Dictum e&longs;t hoc ad fugiendum nomen barbarum, etiam ut bre<lb/>uiter tamen po&longs;&longs;emus &longs;ententiam explicare.</s>

<s id="id000096">Vige&longs;ima diffinitio.</s>

<s id="id000097">Reflexa proportio dicitur cum trium quantitatum aggregatum <lb/>primæ, & tertiæ &longs;e habet ad &longs;ecundam uelut &longs;ecunda ad tertiam,</s>

<s id="id000098">Vige&longs;ima prima diffinitio.</s>

<s id="id000100">Velut proportio 6. 4. 3. dupla e&longs;t utrinque, & 6. 3. 2 tripla. </s>

<s id="id000101">& 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 id="id000102">nam in me<lb/>dijs <expan abbr="nõ">non</expan> e&longs;t <expan abbr="uer&utilde;">uerum</expan>, ut in 9. 6. 4 bis diapente, & 16. 12. 9 bis diate&longs;&longs;aron.</s>

<s id="id000102">nam in me<lb/>dijs <expan abbr="nõ">non</expan> e&longs;t <expan abbr="uer&utilde;">uerum</expan>, ut in 9. 6. 4 bis diapente, & 16. 12. 9 bis diate&longs;&longs;aron.</s>

<s id="id000103">Vige&longs;ima &longs;ecunda diffinitio.</s>

<s id="id000104">Quantitates quæ &longs;imilem habent proportionem non continua<lb/>tam, omiologæ appellantur.</s>

<s id="id000105">Vige&longs;ima tertia diffinitio.</s>

<s id="id000103">Vige&longs;ima &longs;ecunda diffinitio.</s>

<s id="id000106">Prima operatione con&longs;i&longs;tere dicuntur proportiones, cùm inter <lb/>primo conflatas quantitates con&longs;titerint.</s>

<s id="id000104">Quantitates quæ &longs;imilem habent proportionem non continua<lb/>tam, omiologæ appellantur.</s>

<s id="id000107">PRIMA Animi communis &longs;ententia.</s>

<s id="id000108">Omnis Proportio e&longs;t, aut æqualitatis, aut maior inæqualis, <lb/>aut minor.</s>

<s id="id000105">Vige&longs;ima tertia diffinitio.</s>

<s id="id000109">Secunda animi communis &longs;ententia.</s>

<s id="id000110">Quilibet numerus tantus dicitur, quanta e&longs;t illius proportio ad <lb/>monadem.</s>

<s id="id000106">Prima operatione con&longs;i&longs;tere dicuntur proportiones, cùm inter <lb/>primo conflatas quantitates con&longs;titerint.</s>

<s id="id000107">PRIMA Animi communis &longs;ententia.</s>

<s id="id000108">Omnis Proportio e&longs;t, aut æqualitatis, aut maior inæqualis, <lb/>aut minor.</s>

<s id="id000109">Secunda animi communis &longs;ententia.</s>

<s id="id000110">Quilibet numerus tantus dicitur, quanta e&longs;t illius proportio ad <lb/>monadem.</s>

<s id="id000111">Dicimus enim quatuor, quod monadem quater contineat. </s>

<s id="id000112">Et <lb/>duo cum dimidio cùm monadem bis & &longs;emis contineat.</s>

<s id="id000112">Et <lb/>duo cum dimidio cùm monadem bis & &longs;emis contineat.</s>

<s id="id000113">Tertia animi communis &longs;ententia.</s>

<s id="id000113">Tertia animi communis &longs;ententia.</s>

<s id="id000114">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 id="id000115">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 id="id000116">Sunt enim ambæ detractæ.</s>

<s id="id000116">Sunt enim ambæ detractæ.</s>

<s id="id000117">Quarta animi communis &longs;ententia.</s>

<s id="id000117">Quarta animi communis &longs;ententia.</s>

<s id="id000118">Inter quantitatem, & defectum minorem quantitate, cuius e&longs;t de <lb/>fectus, e&longs;t proportio, quatenus e&longs;t quantitas. </s>

<s id="id000119">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="id.015.01.023.1.jpg" xlink:href="015/01/023/1.jpg"/><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 id="id000120">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>

<s id="id000121">Quinta animi communis &longs;ententia.</s>

<s id="id000122">Cum proportio producitur ex proportionibus quælibet illa<lb/>rum dicetur producta diui&longs;a per alteram.</s>

<s id="id000123">Sexta animi communis &longs;ententia.</s>

<s id="id000121">Quinta animi communis &longs;ententia.</s>

<s id="id000122">Cum proportio producitur ex proportionibus quælibet illa<lb/>rum dicetur producta diui&longs;a per alteram.</s>

<s id="id000123">Sexta animi communis &longs;ententia.</s>

<s id="id000124">Æqualium quantitatum &longs;eu proportionum ad tertiam compa<lb/>rabilium eadem e&longs;t proportio atque uici&longs;sim. </s>

<s id="id000125">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 id="id000126">Vt &longs;it <lb/>propior animi communi &longs;ententiæ, quàm rei demon&longs;trandæ.</s>

<s id="id000126">Vt &longs;it <lb/>propior animi communi &longs;ententiæ, quàm rei demon&longs;trandæ.</s>

<s id="id000127">Septima animi communis &longs;ententia.</s>

<s id="id000128">Ad quod quantitas proportionem habet infinitam, id in genere <lb/>illius quantitatis non comprehenditur.</s>

<s id="id000129">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>

<s id="id000127">Septima animi communis &longs;ententia.</s>

<s id="id000128">Ad quod quantitas proportionem habet infinitam, id in genere <lb/>illius quantitatis non comprehenditur.</s>

<s id="id000130">PRIMA Petitio.</s>

<s id="id000131">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>

<s id="id000132">Secunda petitio.</s>

<s id="id000133">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>

<s id="id000134">Tertia petitio.</s>

<s id="id000135">Proportionis cuiu&longs;uis nomen à denominatore &longs;uprà &longs;cripto, & <lb/>numeratore infrà &longs;cripto &longs;umitur.</s>

<s id="id000136">Quarta petitio.</s>

<s id="id000130">PRIMA Petitio.</s>

<s id="id000137">Diui&longs;a quauis quantitate per aliam eiu&longs;dem generis, quod exit <lb/>proportio dicitur.</s>

<s id="id000138">Quinta petitio.</s>

<s id="id000139">Qu&ecedil;libet proportio e&longs;t uel inter duas quantitates, uel per unam <lb/>&longs;ignificatur.</s>

<s id="id000131">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 &etail;quale aut <lb/>maius, aut minus tertio, uel quarto eadem ratione & ordine.</s>

<s id="id000140">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>

<s id="id000141">Sexta petitio.</s>

<s id="id000142">Propo&longs;ita proportione quacunque, & monade quantitatem inue <lb/>nire, quæ &longs;e habeat ad monadem in proportione propo&longs;ita.</s>

<s id="id000132">Secunda petitio.</s>

<s id="id000143">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>

<s id="id000144">Septima petitio.</s>

<s id="id000145">Quamlibet quantitatem per aliam eiu&longs;dem generis diuidere <lb/>po&longs;&longs;e.</s>

<s id="id000146">Octaua petitio.</s>

<s id="id000147">Proportionem in proportionem ducere po&longs;&longs;e: quamuis &longs;int in<lb/>ter quantitates diuer&longs;i generis.</s>

<s id="id000148">Quod dicitur de multiplicatione intelligendum e&longs;t de alijs ope<lb/>rationibus &longs;uprà enumeratis.</s>

<s id="id000134">Tertia petitio.</s>

<s id="id000149">Nona petitio.</s>

<s id="id000150">Monadem &longs;emper &longs;umere in quo cunque genere po&longs;&longs;e propo&longs;i<lb/>ta proportione.</s>

<s id="id000135">Proportionis cuiu&longs;uis nomen à denominatore &longs;uprà &longs;cripto, & <lb/>numeratore infrà &longs;cripto &longs;umitur.</s>

<s id="id000151">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>

<s id="id000136">Quarta petitio.</s>

<s id="id000152">Decima petitio.</s>

<s id="id000137">Diui&longs;a quauis quantitate per aliam eiu&longs;dem generis, quod exit <lb/>proportio dicitur.</s>

<s id="id000138">Quinta petitio.</s>

<s id="id000139">Qu&etail;libet proportio e&longs;t uel inter duas quantitates, uel per unam <lb/>&longs;ignificatur.</s>

<s id="id000141">Sexta petitio.</s>

<s id="id000142">Propo&longs;ita proportione quacunque, & monade quantitatem inue <lb/>nire, quæ &longs;e habeat ad monadem in proportione propo&longs;ita.</s>

<s id="id000144">Septima petitio.</s>

<s id="id000145">Quamlibet quantitatem per aliam eiu&longs;dem generis diuidere <lb/>po&longs;&longs;e.</s>

<s id="id000146">Octaua petitio.</s>

<s id="id000147">Proportionem in proportionem ducere po&longs;&longs;e: quamuis &longs;int in<lb/>ter quantitates diuer&longs;i generis.</s>

<s id="id000148">Quod dicitur de multiplicatione intelligendum e&longs;t de alijs ope<lb/>rationibus &longs;uprà enumeratis.</s>

<s id="id000149">Nona petitio.</s>

<s id="id000150">Monadem &longs;emper &longs;umere in quo cunque genere po&longs;&longs;e propo&longs;i<lb/>ta proportione.</s>

<s id="id000152">Decima petitio.</s>

<s id="id000153">In quouis genere quantitatum &longs;umere po&longs;&longs;e quantitatem, quæ <lb/><arrow.to.target n="marg2"/><lb/>&longs;e habeat ad monadem in proportione data. </s>

<s id="id000154">Similem huic propo<lb/>nit Euclides in lineis generaliter: nos autem contrà generaliter in <lb/>omnibus quantitatibus, &longs;ed de monade tantum.</s>

<s id="id000154">Similem huic propo<lb/>nit Euclides in lineis generaliter: nos autem contrà generaliter in <lb/>omnibus quantitatibus, &longs;ed de monade tantum.</s>

<s id="id000155"><margin.target id="marg2"/>D<emph type="italics"/>uodecima <lb/>&longs;exti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/>Vndecima petitio.</s>

<s id="id000156">Monadem in quancunque quantitatem ductam æquale ip&longs;i pro<lb/>ducere. </s>

<s id="id000157">Similiter & proportionem æqualem.</s>

<s id="id000157">Similiter & proportionem æqualem.</s>

<s id="id000158">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 id="id000159">Idem dico de diui&longs;ione. </s>

<s id="id000160">Aequalitas etiam ducta, uel diuidens non <lb/><arrow.to.target n="marg3"/><lb/>mutat proportionem: nec quantitatem ip&longs;am, igitur monas æqua<lb/>litatem refert. </s>

<s id="id000161">Quod etiam e&longs;t per&longs;picuum ex &longs;upradictis.</s>

<s id="id000161">Quod etiam e&longs;t per&longs;picuum ex &longs;upradictis.</s>

<s id="id000162"><margin.target id="marg3"/>S<emph type="italics"/>ecunda ani <lb/>mi <expan abbr="cõmunis">communis</expan> <lb/>&longs;ententia.<emph.end type="italics"/></s>

<s id="id000163">Duodecima petitio.</s>

<s id="id000162"><margin.target id="marg3"/>S<emph type="italics"/>ecunda ani <lb/>mi <expan abbr="cõmunis">communis</expan> <lb/>&longs;ententia.<emph.end type="italics"/></s>

<s id="id000163">Duodecima petitio.</s>

<s id="id000164">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 id="id000165">Hæc etiam a&longs;&longs;umitur ab Euclide. </s>

<s id="id000166">Et per <lb/><arrow.to.target n="marg4"/><lb/>hanc intelligimus etiam conuer&longs;am.</s>

<s id="id000166">Et per <lb/><arrow.to.target n="marg4"/><lb/>hanc intelligimus etiam conuer&longs;am.</s>

<s id="id000167"><margin.target id="marg4"/>Q<emph type="italics"/>uinto<emph.end type="italics"/> E<emph type="italics"/>le. <lb/>diff.<emph.end type="italics"/> 6.</s>

<s id="id000168">Tertiadecima petitio.</s>

<s id="id000168">Tertiadecima petitio.</s>

<s id="id000169">Quantitates æquales, atque proportiones in qua&longs;uis quanti<lb/>tates ductæ eandem &longs;eruant rationem. </s>

<s id="id000170">Euclides hanc demon&longs;trat, <lb/>nos autem ad uitandum tædium petimus concedi, &longs;ub qua in<lb/><arrow.to.target n="marg5"/><lb/>cluduntur diui&longs;io etiam additio, detractio, laterum omnium in<lb/>uentio.</s>

<s id="id000171"><margin.target id="marg5"/>Q<emph type="italics"/>uarta quin <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>

<s id="id000172">Quartadecima petitio.</s>

<s id="id000173">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>

<s id="id000171"><margin.target id="marg5"/>Q<emph type="italics"/>uarta quin <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>

<s id="id000174">PROPOSITIO prima.</s>

<s id="id000175">Proportionem in proportionem duci e&longs;t &longs;uperiores nume<lb/>ros atque inferiores inuicem ducere.</s>

<s id="id000172">Quartadecima petitio.</s>

<s id="id000174">PROPOSITIO prima.</s>

<s id="id000175">Proportionem in proportionem duci e&longs;t &longs;uperiores nume<lb/>ros atque inferiores inuicem ducere.</s>

<s id="id000176">Sit proportio lineæ a ad lineam b, ut anguli cad angulum d, &longs;ta<lb/><arrow.to.target n="marg6"/><lb/>tuatur e monas in genere a <lb/><figure id="id.015.01.026.1.jpg" xlink:href="015/01/026/1.jpg"/><lb/>b, & fiat fad e, ut cad d, & du <lb/><arrow.to.target n="marg7"/><lb/>catur a in f & b in e, & pro<lb/>ducantur g & h. </s>

<s id="id000177">Quia ergo <lb/><arrow.to.target n="marg8"/><lb/>fe&longs;t proportio ip&longs;a, erit g ad <lb/><arrow.to.target n="marg9"/><lb/>a ut c ad d, &longs;ed h e&longs;t æqualis <lb/>b, igitur a ad h ut ad b. </s>

<s id="id000178">Du<lb/>cta ergo dicetur proportio a <lb/><arrow.to.target n="marg10"/><lb/>ad b in proportionem c ad d <lb/>ducendo terminos proportionis, &longs;eu quantitatis recta &longs;cilicet &longs;u<lb/>periores cum &longs;uperioribus, & inferiores cum inferioribus. </s>

<s id="id000179">Nam &longs;i <lb/><arrow.to.target n="marg11"/><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"/><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>

<s id="id000184"><margin.target id="marg10"/>P<emph type="italics"/>er<emph.end type="italics"/> 2. A<emph type="italics"/>ni<lb/>mi &longs;entent.<emph.end type="italics"/></s>

<s id="id000187">Propo&longs;itio <expan abbr="&longs;ec&utilde;nda">&longs;ecunnda</expan>.</s>

<s id="id000188">Proportio extremorum producitur ex intermedijs.<lb/><arrow.to.target n="marg13"/></s>

<s id="id000190">Sint a b c quantitates dico proportio<lb/><figure id="id.015.01.026.2.jpg" xlink:href="015/01/026/2.jpg"/><lb/>nem a ad c, produci ex proportione a ad b </s>

<s id="id000184"><margin.target id="marg10"/>P<emph type="italics"/>er<emph.end type="italics"/> 2. A<emph type="italics"/>ni<lb/>mi &longs;entent.<emph.end type="italics"/></s>

<s id="id000187">Propo&longs;itio <expan abbr="&longs;ec&utilde;nda">&longs;ecunnda</expan>.</s>

<s id="id000188">Proportio extremorum producitur ex intermedijs.<lb/><arrow.to.target n="marg13"/></s>

<s id="id000190">Sint a b c quantitates dico proportio<lb/><figure id="id.015.01.026.2.jpg" xlink:href="015/01/026/2.jpg"/><lb/>nem a ad c, produci ex proportione a ad b </s>

<s id="id000191"><arrow.to.target n="marg14"/><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 id="id000192">eritqúe per præce<lb/><arrow.to.target n="marg15"/><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"/><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>

<s id="id000196">Ex hoc &longs;equitur, quòd cùm fuerit quantitas tertia monas ex pro<lb/><arrow.to.target n="marg17"/><lb/>portionibus inuicem ductis producetur prima quantitas.<lb/><arrow.to.target n="marg18"/></s>

<s id="id000199">Ex hoc &longs;equitur, quòd conuer&longs;a proportio producitur ex con<lb/>uer&longs;is proportionibus.</s>

<s id="id000200">Propo&longs;itio tertia.</s>

<s id="id000199">Ex hoc &longs;equitur, quòd conuer&longs;a proportio producitur ex con<lb/>uer&longs;is proportionibus.</s>

<s id="id000200">Propo&longs;itio tertia.</s>

<s id="id000201">Si proportio ex duabus proportionibus in quatuor terminis <lb/>producatur, ip&longs;a uerò proportio inter duas alias quantitates fue

<pb xlink:href="015/01/027.jpg" pagenum="8"/>rit con&longs;tituta: con&longs;urgent trecenti &longs;exaginta modi productionis <lb/>proportionis.</s>

<pb pagenum="8" xlink:href="015/01/027.jpg"/>rit con&longs;tituta: con&longs;urgent trecenti &longs;exaginta modi productionis <lb/>proportionis.</s>

<s id="id000204">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 id="id000204">H&etail;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 id="id000205">Sit ergo proportio a ad b, pro<lb/><arrow.to.target n="table2"/><lb/><figure id="id.015.01.027.1.jpg" xlink:href="015/01/027/1.jpg"/>ducta ex proportione c ad d & e ad f, 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"/><lb/>conuer&longs;æ: erunt ergo triginta. </s>

<s id="id000206">Singulæ autem ha <lb/>rum produci po&longs;&longs;unt duodecim modis: ductis <lb/><figure id="id.015.01.027.2.jpg" xlink:href="015/01/027/2.jpg"/>duodecim in triginta, fiunt trecenti &longs;exaginta mo <lb/>di. </s>

<s id="id000207">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 id="id000207">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&etail; <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 id="id000208">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><table><table.target id="table2"/><row><cell>a</cell><cell>b</cell></row><row><cell>---</cell><cell>---</cell></row><row><cell>c</cell><cell>d</cell></row><row><cell>---</cell><cell>---</cell></row><row><cell>e</cell><cell>f</cell></row><row><cell>---</cell><cell>---</cell></row></table><table><table.target id="table3"/><row><cell>a b</cell><cell>b a</cell></row><row><cell>a c</cell><cell>c a</cell></row><row><cell>a d</cell><cell>d a</cell></row><row><cell>a e</cell><cell>e a</cell></row><row><cell>a f</cell><cell>f a</cell></row><row><cell>b c</cell><cell>c b</cell></row><row><cell>b d</cell><cell>d b</cell></row><row><cell>b e</cell><cell>e b</cell></row><row><cell>b f</cell><cell>f b</cell></row><row><cell>c d</cell><cell>d c</cell></row><row><cell>c e</cell><cell>e c</cell></row><row><cell>c f</cell><cell>f c</cell></row><row><cell>d e</cell><cell>e d</cell></row><row><cell>d f</cell><cell>f d</cell></row><row><cell>e f</cell><cell>f e</cell></row><row><cell>direc.</cell><cell>conuer.</cell></row></table>

<s id="id000208">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>

<s id="id000209">Propo&longs;itio quarta.</s>

<s id="id000210">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>

<table>

<s id="id000211">Sit proportio a b producta ex proportioni<lb/><arrow.to.target n="table4"/><lb/><figure id="id.015.01.027.3.jpg" xlink:href="015/01/027/3.jpg"/>bus c ad d, & e ad f, dico quod etiam erit produ</s><table><table.target id="table4"/><row><cell>a</cell><cell>b</cell><cell/></row><row><cell>c</cell><cell>e</cell><cell>g</cell></row><row><cell>d</cell><cell>f</cell><cell>h</cell></row><row><cell>---</cell><cell>---</cell><cell>---</cell></row><row><cell>c</cell><cell>e</cell><cell>g</cell></row><row><cell>f</cell><cell>d</cell><cell>h</cell></row></table>

<table.target id="table2"/>

<s id="id000212"><arrow.to.target n="marg20"/><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"/><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;it&utilde;">propo&longs;itum</expan>.</s>

<row>

<s id="id000213"><margin.target id="marg20"/>P<emph type="italics"/>er<emph.end type="italics"/> 8. <emph type="italics"/>petit.<emph.end type="italics"/></s>

<s id="id000214"><margin.target id="marg21"/>I<emph type="italics"/>n<emph.end type="italics"/> 13. <emph type="italics"/>petit.<emph.end type="italics"/></s>

<s id="id000215">Propo&longs;itio quinta.</s>

</row>

<s id="id000216">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>

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<table>

<table.target id="table3"/>

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<cell>direc.</cell>

<cell>conuer.</cell>

</row>

</table>

<s id="id000209">Propo&longs;itio quarta.</s>

<table>

<table.target id="table4"/>

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<cell/>

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</table>

<s id="id000212"><arrow.to.target n="marg20"/><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"/><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&etail; ad quartam, quod fuit <expan abbr="propo&longs;it&utilde;">propo&longs;itum</expan>.</s>

<s id="id000213"><margin.target id="marg20"/>P<emph type="italics"/>er<emph.end type="italics"/> 8. <emph type="italics"/>petit.<emph.end type="italics"/></s>

<s id="id000214"><margin.target id="marg21"/>I<emph type="italics"/>n<emph.end type="italics"/> 13. <emph type="italics"/>petit.<emph.end type="italics"/></s>

<s id="id000215">Propo&longs;itio quinta.</s>

<s id="id000217">Sit proportio a ad b producta ex proportio<lb/><arrow.to.target n="marg22"/><lb/><arrow.to.target n="table5"/><lb/>nibus c ad d, & e ad f, dico quod proportio c ad <lb/>f producitur ex proportione a ad b & d ad e. </s>

<s id="id000218">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/><figure id="id.015.01.028.2.jpg" xlink:href="015/01/028/2.jpg"/>portionem a ad b, igitur proportio c ad f produ <lb/>citur ex proportionibus a ad b, & e ad f.<lb/><arrow.to.target n="table6"/></s>

<s id="id000220">Propo&longs;itio &longs;exta.</s>

<s id="id000221">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"/></s><table><table.target id="table7"/><row><cell>c</cell><cell>p</cell></row><row><cell>---</cell><cell>---</cell></row><row><cell>a</cell><cell>d</cell></row><row><cell>---</cell><cell>---</cell></row><row><cell>b</cell><cell>e</cell></row><row><cell>---</cell><cell>---</cell></row></table><figure id="id.015.01.028.3.jpg" xlink:href="015/01/028/3.jpg"/>

<table>

<table.target id="table5"/>

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<table>

<table.target id="table6"/>

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<s id="id000220">Propo&longs;itio &longs;exta.</s>

<s id="id000221">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"/></s>

<table>

<table.target id="table7"/>

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</table>

<s id="id000222">Per quartam enim proportio a ad b produ<lb/><arrow.to.target n="marg23"/><lb/>citur bifariam, & ex c ad d, & e ad f, & ex c ad f, & <lb/>e ad d. </s>

<s id="id000223">& 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 id="id000224">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>

Line 276

Line 2021

<s id="id000228">Hoc enim ex præ<lb/>cedentibus non liquet: benè liquet <lb/>permutatis ordinibus, quod &longs;i pro<lb/>portio primi ad tertium producitur,

<pb xlink:href="015/01/029.jpg" pagenum="9 [=10]"/>quod etiam propor<lb/><figure id="id.015.01.029.1.jpg" xlink:href="015/01/029/1.jpg"/><arrow.to.target n="marg24"/><lb/>tio primi ad <expan abbr="quint&utilde;">quintum</expan>. <lb/></s>

<pb pagenum="9 [=10]" xlink:href="015/01/029.jpg"/>quod etiam propor<lb/><figure id="id.015.01.029.1.jpg" xlink:href="015/01/029/1.jpg"/><arrow.to.target n="marg24"/><lb/>tio primi ad <expan abbr="quint&utilde;">quintum</expan>. <lb/></s>

<s id="id000229">Nam tertium, & quin <lb/>tum, item que quartum, <lb/>& &longs;extum non <expan abbr="diffe-r&utilde;t">diffe<lb/>runt</expan> ni&longs;i ordine uolun <lb/>tario. </s>

<s id="id000230">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 id="id000231">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;ec&utilde;dam">&longs;ecundam</expan> propo&longs;itionem. </s>

<s id="id000232">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 id="id000233">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"/></s>

<s id="id000233">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&etail;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"/></s>

<s id="id000235"><margin.target id="marg24"/>Modi qui <expan abbr="nõ">non</expan> <lb/>producuntur <lb/>pri. ad quartu <lb/>pri. ad &longs;extum <lb/>&longs;ec. ad <expan abbr="terti&utilde;">tertium</expan> <lb/>&longs;ec. ad <expan abbr="quint&utilde;">quintum</expan> <lb/>tert. </s>

<s id="id000236">ad quint. <lb/></s>

<s id="id000237">quart. </s>

<s id="id000238">ad &longs;ext.</s><table><table.target id="table8"/><row><cell/><cell>Primi ad &longs;ecundum.</cell></row><row><cell>1</cell><cell>tertij ad <expan abbr="quart&utilde;">quartum</expan>, & quin</cell></row><row><cell/><cell>ti ad &longs;extum.</cell></row><row><cell>2</cell><cell>tertij ad &longs;extum, & quin</cell></row><row><cell/><cell>ti ad quartum.</cell></row><row><cell/><cell>Primi ad tertium.</cell></row><row><cell>3</cell><cell>&longs;ecundi ad quartum, &</cell></row><row><cell/><cell>quinti ad &longs;extum.</cell></row><row><cell>4</cell><cell>&longs;ecundi ad &longs;extum, &</cell></row><row><cell/><cell>quinti ad quartum.</cell></row><row><cell/><cell>Primi ad quintum.</cell></row><row><cell>5</cell><cell>&longs;ecundi ad <expan abbr="&longs;ext&utilde;">&longs;extum</expan>, & ter-</cell></row><row><cell/><cell>tij ad quartum.</cell></row><row><cell>6</cell><cell>&longs;ecundi ad quartum, &</cell></row><row><cell/><cell>tertij ad &longs;extum.</cell></row><row><cell/><cell>Secundi ad quartum.</cell></row><row><cell>7</cell><cell>primi ad tertium, & &longs;ex</cell></row><row><cell/><cell>ti ad quintum.</cell></row><row><cell>8</cell><cell>primi ad quintum, et &longs;ex</cell></row><row><cell/><cell>ti ad tertium.</cell></row><row><cell/><cell>Secundi ad &longs;extum.</cell></row><row><cell>9</cell><cell>primi ad <expan abbr="quint&utilde;">quintum</expan>, & quar</cell></row><row><cell/><cell>ti ad tertium.</cell></row><row><cell>10</cell><cell>primi ad <expan abbr="terti&utilde;">tertium</expan>, & quar-</cell></row><row><cell/><cell>ti ad quintum.</cell></row><row><cell/><cell>Tertij ad quartum.</cell></row><row><cell>11</cell><cell>primi ad &longs;ecundum, &</cell></row><row><cell/><cell>&longs;exti ad quintum.</cell></row><row><cell>12</cell><cell>primi ad quintum, & &longs;ex</cell></row><row><cell/><cell>ti ad &longs;ecundum.</cell></row><row><cell/><cell>Tertij ad &longs;extum.</cell></row><row><cell>13</cell><cell>primi ad &longs;ecundum, &</cell></row><row><cell/><cell>quarti ad quintum.</cell></row><row><cell>14</cell><cell>primi ad quintum, &</cell></row><row><cell/><cell>quarti ad &longs;ecundum.</cell></row><row><cell/><cell>Quarti ad quintum.</cell></row><row><cell>15</cell><cell>&longs;ecundi ad primum, &</cell></row><row><cell/><cell>tertij ad &longs;extum.</cell></row><row><cell>16</cell><cell>&longs;ecundi ad &longs;extum, & ter</cell></row><row><cell/><cell>tij ad primum.</cell></row><row><cell/><cell>Quinti ad &longs;extum.</cell></row><row><cell>17</cell><cell>primi ad &longs;ecundum, &</cell></row><row><cell/><cell>quarti ad tertium.</cell></row><row><cell>18</cell><cell>primi ad <expan abbr="terti&utilde;">tertium</expan>, & quar-</cell></row><row><cell/><cell>ti ad &longs;ecundum.</cell></row></table><table><table.target id="table9"/><row><cell>a</cell><cell>e c</cell><cell>a e</cell><cell>e c</cell></row><row><cell/><cell/><cell>c b</cell><cell>e</cell></row><row><cell/><cell/><cell>f d</cell><cell>c</cell></row><row><cell/><cell/><cell/><cell>f</cell></row></table>

<s id="id000238">ad &longs;ext.</s>

<s id="id000239">Propo&longs;itio &longs;eptima.</s><figure id="id.015.01.029.2.jpg" xlink:href="015/01/029/2.jpg"/>

<s id="id000240">In modis qui nece&longs;&longs;ariò produ<lb/>cuntur ex duabus proportionibus, <lb/>cum du&ecedil; quantitates ex illis, qu&ecedil; mo <lb/><figure id="id.015.01.029.3.jpg" xlink:href="015/01/029/3.jpg"/>dos conficiunt, æquales fuerint: pro<lb/><arrow.to.target n="table10"/><lb/>portio producta ad quatuor quanti<lb/>tates omiologas reducetur.<lb/><arrow.to.target n="marg25"/></s>

<table>

<table.target id="table8"/>

<s id="id000242">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

<row>

<cell/>

<cell>Primi ad &longs;ecundum.</cell>

</row>

<row>

<cell>tertij ad <expan abbr="quart&utilde;">quartum</expan>, & quin</cell>

</row>

<row>

<cell/>

<cell>ti ad &longs;extum.</cell>

</row>

<row>

<cell>tertij ad &longs;extum, & quin</cell>

</row>

<row>

<cell/>

<cell>ti ad quartum.</cell>

</row>

<row>

<cell/>

<cell>Primi ad tertium.</cell>

</row>

<row>

<cell>&longs;ecundi ad quartum, &</cell>

</row>

<row>

<cell/>

<cell>quinti ad &longs;extum.</cell>

</row>

<row>

<cell>&longs;ecundi ad &longs;extum, &</cell>

</row>

<row>

<cell/>

<cell>quinti ad quartum.</cell>

</row>

<row>

<cell/>

<cell>Primi ad quintum.</cell>

</row>

<row>

<cell>&longs;ecundi ad <expan abbr="&longs;ext&utilde;">&longs;extum</expan>, & ter-</cell>

</row>

<row>

<cell/>

<cell>tij ad quartum.</cell>

</row>

<row>

<cell>&longs;ecundi ad quartum, &</cell>

</row>

<row>

<cell/>

<cell>tertij ad &longs;extum.</cell>

</row>

<row>

<cell/>

<cell>Secundi ad quartum.</cell>

</row>

<row>

<cell>primi ad tertium, & &longs;ex</cell>

</row>

<row>

<cell/>

<cell>ti ad quintum.</cell>

</row>

<row>

<cell>primi ad quintum, et &longs;ex</cell>

</row>

<row>

<cell/>

<cell>ti ad tertium.</cell>

</row>

<row>

<cell/>

<cell>Secundi ad &longs;extum.</cell>

</row>

<row>

<cell>primi ad <expan abbr="quint&utilde;">quintum</expan>, & quar</cell>

</row>

<row>

<cell/>

<cell>ti ad tertium.</cell>

</row>

<row>

<cell>primi ad <expan abbr="terti&utilde;">tertium</expan>, & quar-</cell>

</row>

<row>

<cell/>

<cell>ti ad quintum.</cell>

</row>

<row>

<cell/>

<cell>Tertij ad quartum.</cell>

</row>

<row>

<cell>primi ad &longs;ecundum, &</cell>

</row>

<row>

<cell/>

<cell>&longs;exti ad quintum.</cell>

</row>

<row>

<cell>primi ad quintum, & &longs;ex</cell>

</row>

<row>

<cell/>

<cell>ti ad &longs;ecundum.</cell>

</row>

<row>

<cell/>

<cell>Tertij ad &longs;extum.</cell>

</row>

<row>

<cell>primi ad &longs;ecundum, &</cell>

</row>

<row>

<cell/>

<cell>quarti ad quintum.</cell>

</row>

<row>

<cell>primi ad quintum, &</cell>

</row>

<row>

<cell/>

<cell>quarti ad &longs;ecundum.</cell>

</row>

<row>

<cell/>

<cell>Quarti ad quintum.</cell>

</row>

<row>

<cell>&longs;ecundi ad primum, &</cell>

</row>

<row>

<cell/>

<cell>tertij ad &longs;extum.</cell>

</row>

<row>

<cell>&longs;ecundi ad &longs;extum, & ter</cell>

</row>

<row>

<cell/>

<cell>tij ad primum.</cell>

</row>

<row>

<cell/>

<cell>Quinti ad &longs;extum.</cell>

</row>

<row>

<cell>primi ad &longs;ecundum, &</cell>

</row>

<row>

<cell/>

<cell>quarti ad tertium.</cell>

</row>

<row>

<cell>primi ad <expan abbr="terti&utilde;">tertium</expan>, & quar-</cell>

</row>

<row>

<cell/>

<cell>ti ad &longs;ecundum.</cell>

</row>

</table>

<table>

<table.target id="table9"/>

<row>

</row>

<row>

<cell/>

</row>

<row>

<cell/>

</row>

<row>

<cell/>

</row>

</table>

<s id="id000239">Propo&longs;itio &longs;eptima.</s>

<s id="id000240">In modis qui nece&longs;&longs;ariò produ<lb/>cuntur ex duabus proportionibus, <lb/>cum du&etail; quantitates ex illis, qu&etail; mo <lb/><figure id="id.015.01.029.3.jpg" xlink:href="015/01/029/3.jpg"/>dos conficiunt, æquales fuerint: pro<lb/><arrow.to.target n="table10"/><lb/>portio producta ad quatuor quanti<lb/>tates omiologas reducetur.<lb/><arrow.to.target n="marg25"/></s>

<table>

<table.target id="table10"/>

<row>

</row>

<row>

</row>

<row>

</row>

<row>

</row>

<row>

</row>

<row>

</row>

</table>

<pb xlink:href="015/01/030.jpg" 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>

<pb pagenum="11" xlink:href="015/01/030.jpg"/>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 id="id000243">Sit gratia exempli prima æqualis quintæ: & quia <lb/>in octauo modo proportio <expan abbr="&longs;ec&utilde;di">&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>

<s id="id000244"><arrow.to.target n="marg26"/><lb/>ad quartum, ut &longs;exti ad &longs;ecundum.</s>

<s id="id000245"><margin.target id="marg26"/>V<emph type="italics"/>ndecima <lb/>petitione.<emph.end type="italics"/></s>

<s id="id000246">Propo&longs;itio octaua.</s>

<s id="id000244"><arrow.to.target n="marg26"/><lb/>ad quartum, ut &longs;exti ad &longs;ecundum.</s>

<s id="id000247">Si duarum <expan abbr="proportion&utilde;">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><figure id="id.015.01.030.1.jpg" xlink:href="015/01/030/1.jpg"/>

<s id="id000245"><margin.target id="marg26"/>V<emph type="italics"/>ndecima <lb/>petitione.<emph.end type="italics"/></s>

<s id="id000246">Propo&longs;itio octaua.</s>

<s id="id000248">Sit proportio una a ad b, alia c ad d, ducatur b in <lb/><arrow.to.target n="marg27"/><lb/>c, fiatque e & a in d, & fiat f, iunganturque e & f & fiat h, <lb/>& ducatur b in d et fiat g: dico <expan abbr="proportion&etilde;">proportionem</expan> h g com<lb/>po&longs;itam e&longs;&longs;e ex proportione a ad b, & c ad d. </s>

<s id="id000249">Quia <lb/><arrow.to.target n="marg28"/><lb/>enim ex b in c fit e, & ex b in d fit g, erit proportio e <lb/>ad g, ut c ad d, & &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 id="id000250">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"/><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>

<s id="id000252"><margin.target id="marg28"/>E<emph type="italics"/>x<emph.end type="italics"/> 13 <emph type="italics"/>peti<lb/>tione.<emph.end type="italics"/></s>

<s id="id000253"><margin.target id="marg29"/>P<emph type="italics"/>er<emph.end type="italics"/> 14 <emph type="italics"/>diffi <lb/>nitionem.<emph.end type="italics"/></s>

<s id="id000254">Propo&longs;itio nona.</s>

<s id="id000255">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>

<s id="id000256">Hæc eodem modo probatur, ut præcedens, ni&longs;i quod h fit de<lb/><arrow.to.target n="marg30"/><lb/>tracto è minore: gratia exempli ex f, & ita ex diffinitione patet pro<lb/>po&longs;itum.</s>

<s id="id000252"><margin.target id="marg28"/>E<emph type="italics"/>x<emph.end type="italics"/> 13 <emph type="italics"/>peti<lb/>tione.<emph.end type="italics"/></s>

<s id="id000258">Propo&longs;itio decima.</s>

<s id="id000259">Si fuerit alicuius quantitatis ad unam partem proportio uelut al <lb/>terius partis ad <expan abbr="&longs;ec&utilde;dam">&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"/></s>

<s id="id000253"><margin.target id="marg29"/>P<emph type="italics"/>er<emph.end type="italics"/> 14 <emph type="italics"/>diffi <lb/>nitionem.<emph.end type="italics"/></s>

<s id="id000254">Propo&longs;itio nona.</s>

<s id="id000258">Propo&longs;itio decima.</s>

<s id="id000261">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 xlink:href="015/01/031.jpg" 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>

<pb pagenum="12" xlink:href="015/01/031.jpg"/>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 id="id000262">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"/><lb/>ad d. </s>

<s id="id000263">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>

<s id="id000263">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>

<s id="id000265">Propo&longs;itio undecima.</s>

<s id="id000266">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"/></s>

<s id="id000265">Propo&longs;itio undecima.</s>

<s id="id000268">Sit proportio a ad c, & b ad d, & &longs;int c & d <lb/><figure id="id.015.01.031.1.jpg" xlink:href="015/01/031/1.jpg"/><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 id="id000270"><arrow.to.target n="marg34"/><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"/><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"/><lb/>&longs;tatuatur ergo e æqualis c d media inter a b & c. </s>

<s id="id000271">Et erit per &longs;ecun<lb/>dam propo&longs;itionem proportio aggregati a b ad c producta ex <lb/><arrow.to.target n="marg37"/><lb/>proportione aggregati a b ad c, & e ad c, igitur proportio a b ad e <lb/>erit proportio a b ad c, diui&longs;a per proportionem e ad c, &longs;ed e ad c e&longs;t <lb/><arrow.to.target n="marg38"/><lb/>dupla: igitur proportio a b ad c d e&longs;t proportio a b ad c diui&longs;a per <lb/>duplam.</s>

<s id="id000272"><margin.target id="marg34"/>E<emph type="italics"/>x &longs;exta<emph.end type="italics"/> A<emph type="italics"/>nim. <lb/>com. </s>

<s id="id000273">&longs;ententia.<emph.end type="italics"/></s>

<s id="id000273">&longs;ententia.<emph.end type="italics"/></s>

<s id="id000274"><margin.target id="marg35"/>D<emph type="italics"/>ecimaquarta<emph.end type="italics"/></s>

<s id="id000274"><margin.target id="marg35"/>D<emph type="italics"/>ecimaquarta<emph.end type="italics"/></s>

<s id="id000277"><margin.target id="marg38"/>P<emph type="italics"/>er quintam<emph.end type="italics"/><lb/>A<emph type="italics"/>nim. </s>

<s id="id000279">&longs;en <lb/>tentiam.<emph.end type="italics"/></s>

<s id="id000279">&longs;en <lb/>tentiam.<emph.end type="italics"/></s>

<s id="id000280">Propo&longs;itio duodecima.</s>

<s id="id000281">Propo&longs;itis duabus proportionibus unam alteri iungere ab&longs;que <lb/>multiplicatione.<lb/><arrow.to.target n="marg39"/></s>

<s id="id000280">Propo&longs;itio duodecima.</s>

<s id="id000283">Sint propo&longs;itæ proportiones a ad c & <lb/><figure id="id.015.01.031.2.jpg" xlink:href="015/01/031/2.jpg"/><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>

<s id="id000284"><arrow.to.target n="marg40"/><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>

<s id="id000281">Propo&longs;itis duabus proportionibus unam alteri iungere ab&longs;que <lb/>multiplicatione.<lb/><arrow.to.target n="marg39"/></s>

<s id="id000285"><margin.target id="marg40"/>E<emph type="italics"/>x generali <lb/>com.<emph.end type="italics"/> A<emph type="italics"/>nim. &longs;en <lb/>tentia.<emph.end type="italics"/></s>

<s id="id000286">Aliter ex b in c fiat fex a in d, g ex c in d h coniunctum ex f g, k.</s>

<s id="id000286">Aliter ex b in c fiat fex a in d, g ex c in d h coniunctum ex f g, k.</s>

<s id="id000287">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"/><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 id="id000288">Ex octaua ha <lb/>rum igitur proportio a c ad c compo&longs;ita e&longs;t ex <lb/>ei&longs;dem. </s>

<s id="id000289">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>

<s id="id000291">Ex hoc &longs;equitur quòd: Quælibet duæ quantitates quarum ag<lb/><arrow.to.target n="marg42"/><lb/>gregatum e&longs;tidem ad eam quantitatem, componunt eandem pro<lb/>portionem.</s>

<s id="id000293">Propo&longs;itio tertiadecima.</s>

<s id="id000294">Proportio confu&longs;a aggregati primæ & tertiæ quatuor quantita<lb/>tum omiologarum ad <expan abbr="aggregat&utilde;">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"/></s>

<s id="id000293">Propo&longs;itio tertiadecima.</s>

<s id="id000296">Sint a ad b, ut c ad d, dico, quòd erit confu&longs;a <lb/><figure id="id.015.01.032.2.jpg" xlink:href="015/01/032/2.jpg"/><arrow.to.target n="table11"/><lb/>proportio a c aggregati ad <expan abbr="aggregat&utilde;">aggregatum</expan> b d, com <lb/>po&longs;itæ ex his proportionibus diui&longs;æ per du<lb/>plam æqualis. </s>

<s id="id000297">Erit enim aggregati ex a c ad aggregatum ex b d, ue<lb/>lut a ad b per 18 quinti Elementorum. </s>

<s id="id000298">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="product&utilde;">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 id="id000299">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><table><table.target id="table11"/><row><cell>a</cell><cell>c</cell></row><row><cell>-----</cell><cell>-----</cell></row><row><cell>b</cell><cell>d</cell></row><row><cell>---</cell><cell>---</cell></row></table>

<s id="id000299">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>

<s id="id000300">Propo&longs;itio quartadecima.</s>

<s id="id000301">Proportiones confu&longs;æ, & coniunctæ in tribus quantitatibus in<lb/>uicem commutantur.</s><figure id="id.015.01.032.3.jpg" xlink:href="015/01/032/3.jpg"/>

<table>

<s id="id000302">Sint tres quantitates, dico, quod proportio c </s>

<table.target id="table11"/>

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<row>

</row>

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</row>

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<s id="id000300">Propo&longs;itio quartadecima.</s>

<s id="id000301">Proportiones confu&longs;æ, & coniunctæ in tribus quantitatibus in<lb/>uicem commutantur.</s>

<s id="id000302">Sint tres quantitates, dico, quod proportio c </s>

<s id="id000303"><arrow.to.target n="marg44"/><lb/>ad a b confu&longs;a e&longs;t, conuer&longs;a coniunctæ a & b ad <lb/><arrow.to.target n="marg45"/><lb/>c. </s>

<s id="id000304">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 id="id000305">Igitur pro<lb/>portio confu&longs;a in tribus quantitatibus e&longs;t contraria coniunctæ in <lb/>ei&longs;dem.</s>

<s id="id000305">Igitur pro<lb/>portio confu&longs;a in tribus quantitatibus e&longs;t contraria coniunctæ in <lb/>ei&longs;dem.</s>

<s id="id000308">Ex quauis ergo illarum data, data erit & reliqua.<lb/><arrow.to.target n="marg46"/></s>

<s id="id000310">Propo&longs;itio quintadecima.</s>

<s id="id000311">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><figure id="id.015.01.033.1.jpg" xlink:href="015/01/033/1.jpg"/>

<s id="id000308">Ex quauis ergo illarum data, data erit & reliqua.<lb/><arrow.to.target n="marg46"/></s>

<s id="id000310">Propo&longs;itio quintadecima.</s>

<s id="id000312">Sint quatuor quantitates a b, c, d, e f, & <lb/><arrow.to.target n="marg47"/><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/>& 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 id="id000313">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 id="id000314">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 id="id000315">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 id="id000316">uno ergo demon&longs;trato <lb/>ut pote primo intelligentur reliqui. </s>

<s id="id000317">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 id="id000318">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>

<s id="id000320">Ex hoc patet quod proportionum confu&longs;io <lb/><arrow.to.target n="marg48"/><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>

<s id="id000322">Propo&longs;itio &longs;extadecima.</s>

<s id="id000322">Propo&longs;itio &longs;extadecima.</s>

<s id="id000323">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="id.015.01.033.3.jpg" xlink:href="015/01/033/3.jpg"/><lb/>erit proportio confu&longs;a illarum, ut pro<lb/>ducti ex aggregato primæ & tertiæ in

<pb xlink:href="015/01/034.jpg" pagenum="15"/>tertiam, ad productum ex aggregato tertiæ & omiotatæ ad &longs;ecun<lb/>dam in ip&longs;am quartam.</s>

<pb pagenum="15" xlink:href="015/01/034.jpg"/>tertiam, ad productum ex aggregato tertiæ & omiotatæ ad &longs;ecun<lb/>dam in ip&longs;am quartam.</s>

<s id="id000324">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 id="id000324">Hæc magis reducit confu&longs;am proportionem ad notitiam, quàm, <lb/>præcedens, quia reducit ad proportionem <expan abbr="productã">productam</expan>, qu&etail; 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 id="id000325">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 id="id000326">licet enim hoc facere. </s>

<s id="id000327">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 id="id000328">Statuatur aggre<lb/><arrow.to.target n="marg49"/><lb/>gatum a b & d linea a d prima quantitas, & aggregatum a f & d, <lb/><figure id="id.015.01.034.1.jpg" xlink:href="015/01/034/1.jpg"/><lb/>a d &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"/><lb/>ducta ex proportionibus a b d ad <lb/>a f d, & d ad c. </s>

<s id="id000329">Sed proportio a f d <lb/>ad aggregatum c e, e&longs;t uelut d ad <lb/>e. </s>

<s id="id000330">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 id="id000331">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>

<s id="id000331">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>

<s id="id000334">Propo&longs;itio decima&longs;eptima.</s>

<s id="id000335">Omnes du&ecedil; proportiones conuer&longs;æ producunt æqualem pro<lb/>portionem.<lb/><arrow.to.target n="table12"/></s><table><table.target id="table12"/><row><cell>a</cell></row><row><cell>-----</cell></row><row><cell>b</cell></row><row><cell>---</cell></row><row><cell>c</cell></row><row><cell>----</cell></row></table>

<s id="id000334">Propo&longs;itio decima&longs;eptima.</s>

<s id="id000335">Omnes du&etail; proportiones conuer&longs;æ producunt æqualem pro<lb/>portionem.<lb/><arrow.to.target n="table12"/></s>

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<s id="id000336">Sint duæ proportiones a ad b & b ad a conuer&longs;a, <lb/><figure id="id.015.01.034.2.jpg" xlink:href="015/01/034/2.jpg"/><arrow.to.target n="marg51"/><lb/>dico, quòd producunt proportionem æqualem. </s>

<s id="id000337">fiat <lb/>enim b ad c, ut b ad a, erit igitur a æqualis c & b c con <lb/><arrow.to.target n="marg52"/><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>

<s id="id000339"><margin.target id="marg52"/>P<emph type="italics"/>er<emph.end type="italics"/> 6. A<emph type="italics"/>ni<lb/>mi <expan abbr="commun&etilde;">communem</expan> <lb/>&longs;ententiam.<emph.end type="italics"/></s>

<s id="id000340">Propo&longs;itio decimaoctaua.</s>

<s id="id000340">Propo&longs;itio decimaoctaua.</s>

<s id="id000341">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.

<s id="id000343">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 id="id000344">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, ut</s>

<s id="id000344">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, ut</s>

<s id="id000345"><arrow.to.target n="marg54"/><lb/>d ad e, &longs;ed e gnomo cum quadrato d efficit qua<lb/><figure id="id.015.01.035.1.jpg" xlink:href="015/01/035/1.jpg"/><lb/>dratum e, igitur ut c quadrati ad d & eiuncta, ita <lb/>d ad e. </s>

<s id="id000346">Rur&longs;us, quia b quadrati ad c quadratum, <lb/><arrow.to.target n="marg55"/><lb/>ut c ad d erit gnomonis b ad quadratum c, ut <lb/>gnomonis c ad quadratum d, & ita d ad e, igitur <lb/><arrow.to.target n="marg56"/><lb/>gnomonum b c cum quadrato d ad aggrega<lb/>tum c d e quadratorum, ut d ad e, &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 id="id000347">Et ita <lb/>repetendo de quotuis quantitatibus in infi <lb/>nitum u&longs;que. </s>

<s id="id000348">Hæc proponitur ab Archimede in libro de quadrato <lb/>æquali parabolæ, & minus generaliter & pluribus demon&longs;tratur. <lb/></s>

<s id="id000349">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"/></s>

<s id="id000349">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"/></s>

<s id="id000350"><margin.target id="marg54"/>13. P<emph type="italics"/>ropo&longs;. <lb/>quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>

<s id="id000352"><margin.target id="marg56"/>P<emph type="italics"/>er<emph.end type="italics"/> 12. <emph type="italics"/>quin <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s>

<s id="id000354">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>

<s id="id000357">Hæc enim e&longs;t euidens, quia conuenit ei demon&longs;tratio propo&longs;ita. <lb/><figure id="id.015.01.035.2.jpg" xlink:href="015/01/035/2.jpg"/><lb/>exemplo autem in numeris à latere <lb/>po&longs;ito uides declarationem. </s>

<s id="id000358">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>

<s id="id000358">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>

<s id="id000361">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>

<s id="id000364">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>

<s id="id000367">Velut collectæ in &longs;e&longs;quialtera duplæ in &longs;exquitertia triplæ in <lb/>&longs;exqui&longs;eptima &longs;eptuplæ. </s>

<s id="id000368">Vt capio 512 448 392 343, & ita deinceps <lb/>u&longs;que in infinitum aggregatum omnium earum erit 3584. Septu

<pb xlink:href="015/01/036.jpg" pagenum="17"/>plum 512, & aggregatum 18. 12. 8. 5 2/3, & ita deinceps in &longs;exquialtera <lb/>erit 54 duplum 27 primæ in eo ordine.</s>

<pb pagenum="17" xlink:href="015/01/036.jpg"/>plum 512, & aggregatum 18. 12. 8. 5 2/3, & ita deinceps in &longs;exquialtera <lb/>erit 54 duplum 27 primæ in eo ordine.</s>

<s id="id000369">SCHOLIVM.</s>

<s id="id000370">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>

<s id="id000371"><arrow.to.target n="marg62"/><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"/></s>

<s id="id000369">SCHOLIVM.</s>

<s id="id000374">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>

<s id="id000377">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 id="id000378">Et facile po</s>

<s id="id000378">Et facile po</s>

<s id="id000379"><arrow.to.target n="marg65"/><lb/>re&longs;t demon&longs;trari. </s>

<s id="id000380">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="er&utilde;t">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>

<s id="id000381"><margin.target id="marg65"/>Q<emph type="italics"/>uæftio.<emph.end type="italics"/></s>

<s id="id000382">Propo&longs;itio decimanona.</s>

<s id="id000383">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="aggregat&utilde;">aggregatum</expan> omnium quan<lb/>titatum eiu&longs;dem tripla aggregato quadra<lb/><figure id="id.015.01.036.1.jpg" xlink:href="015/01/036/1.jpg"/><lb/>torum omnium quantitatum primi ordinis <lb/><arrow.to.target n="marg66"/><lb/>pariter acceptis.</s>

<s id="id000381"><margin.target id="marg65"/>Q<emph type="italics"/>uæftio.<emph.end type="italics"/></s>

<s id="id000382">Propo&longs;itio decimanona.</s>

<s id="id000383">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 &etail;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="aggregat&utilde;">aggregatum</expan> omnium quan<lb/>titatum eiu&longs;dem tripla aggregato quadra<lb/><figure id="id.015.01.036.1.jpg" xlink:href="015/01/036/1.jpg"/><lb/>torum omnium quantitatum primi ordinis <lb/><arrow.to.target n="marg66"/><lb/>pariter acceptis.</s>

<s id="id000385">Sint aliquot quantitates a b c d e f g h in <lb/>continua proportione. </s>

<s id="id000386">Arithmetica di&longs;po&longs;it&ecedil; <lb/>ita ut minima <expan abbr="ear&utilde;">earum</expan> qu&ecedil; &longs;it h, &longs;it &ecedil;qualis diffe<lb/>renti&ecedil; quantitatum <expan abbr="&longs;ecund&utilde;">&longs;ecundum</expan> ordinem di&longs;po <lb/><expan abbr="&longs;itar&utilde;">&longs;itarum</expan>, uelut differentia a & b, & b & c, & c & <lb/>d, et ita de alijs, addantur <expan abbr="a&utilde;t">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="c&utilde;">cum</expan> &longs;uis &longs;upplementis ip&longs;i line&ecedil; <lb/>à maiori. </s>

<s id="id000386">Arithmetica di&longs;po&longs;it&etail; <lb/>ita ut minima <expan abbr="ear&utilde;">earum</expan> qu&etail; &longs;it h, &longs;it &etail;qualis diffe<lb/>renti&etail; quantitatum <expan abbr="&longs;ecund&utilde;">&longs;ecundum</expan> ordinem di&longs;po <lb/><expan abbr="&longs;itar&utilde;">&longs;itarum</expan>, uelut differentia a & b, & b & c, & c & <lb/>d, et ita de alijs, addantur <expan abbr="a&utilde;t">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 &etail;quales <expan abbr="c&utilde;">cum</expan> &longs;uis &longs;upplementis ip&longs;i line&etail; <lb/>à maiori. </s>

<s id="id000387">E&longs;tque <expan abbr="id&etilde;">idem</expan> ac &longs;i e&longs;&longs;ent aliquot quanti

<pb xlink:href="015/01/037.jpg" pagenum="18"/>tates, & <expan abbr="diuideren&ttilde;">diuiderentur</expan> &longs;ingul&ecedil; <expan abbr="&longs;ecund&utilde;">&longs;ecundum</expan> numerum <expan abbr="illar&utilde;">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>

<pb pagenum="18" xlink:href="015/01/037.jpg"/>tates, & <expan abbr="diuideren&ttilde;">diuiderentur</expan> &longs;ingul&etail; <expan abbr="&longs;ecund&utilde;">&longs;ecundum</expan> numerum <expan abbr="illar&utilde;">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 id="id000388">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 id="id000389">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="ordin&utilde;">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 id="id000390">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 id="id000391">quantitates <expan abbr="a&utilde;t">aunt</expan> &longs;ecundi ordi <lb/>nis <expan abbr="appellabun&ttilde;">appellabuntur</expan> a, b i, ck, dl, em, fn, go, & hp. </s>

<s id="id000392">Hæcuolui pluribus <lb/>agere, ut dilucidior e&longs;&longs;et propo&longs;itio. </s>

<s id="id000393">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="quantitat&utilde;">quantitatum</expan> & ordinum. </s>

<s id="id000394">Dico <lb/>ergo &qring;d aggregatum <expan abbr="quadrator&utilde;">quadratorum</expan> quantitatum &longs;ecundi ordinis pri <lb/>mo quadrato bis repetito, &longs;eu uno addito <expan abbr="c&utilde;">cum</expan> eo quod fit ex minima <lb/>in aggregatum quantitatum primi ordinis e&longs;t <expan abbr="tripl&utilde;">triplum</expan> aggregato ex <lb/>quadratis omnibus <expan abbr="quantitat&utilde;">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="faci&utilde;t">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="quadrat&utilde;">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="product&utilde;">productum</expan> 36, ut fiat totum <lb/>612, quod tale 612 e&longs;t triplum 204, aggregati <expan abbr="quadrator&utilde;">quadratorum</expan> primi or<lb/>dinis unius demon&longs;tratio h&ecedil;c e&longs;t. </s>

<s id="id000395">Quia ex quarta &longs;ecundi Element. <lb/>Euclidis &longs;ingula quadrata <expan abbr="quantitat&utilde;">quantitatum</expan> <expan abbr="diui&longs;ar&utilde;">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="fuer&utilde;t&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="a&utilde;t">aunt</expan> &ecedil;qualis m. </s>

<s id="id000395">Quia ex quarta &longs;ecundi Element. <lb/>Euclidis &longs;ingula quadrata <expan abbr="quantitat&utilde;">quantitatum</expan> <expan abbr="diui&longs;ar&utilde;">diui&longs;arum</expan> &longs;ecundi ordinis con <lb/>&longs;tant ex quatuor partibus quarum du&etail; &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 &etail;qualis b, quia &longs;upplementa <expan abbr="fuer&utilde;t&etail;qualia">fuerunt&etail;qualia</expan> mutuò quanti <lb/>tatibus, & ita c æqualis o & k æqualis g & d, æqualis n & l, æqualis <lb/>f, e <expan abbr="a&utilde;t">aunt</expan> &etail;qualis m. </s>

<s id="id000396"><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="er&utilde;t">erunt</expan> dupla quadratis primarum <lb/>quantitatum: ueluti capio b i &longs;ecundam & h p ultimam, <expan abbr="quar&utilde;">quarum</expan> qua

<pb xlink:href="015/01/038.jpg" pagenum="19"/>drata partium &longs;unt quadrata b & i, & h & p, &longs;ed b e&longs;t æqualis p, & h <lb/>æqualis i. </s>

<pb pagenum="19" xlink:href="015/01/038.jpg"/>drata partium &longs;unt quadrata b & i, & h & p, &longs;ed b e&longs;t æqualis p, & h <lb/>æqualis i. </s>

<s id="id000397">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"/><lb/>&longs;unt æquales. </s>

<s id="id000398">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 id="id000399">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 quadrati 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 id="id000400">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 id="id000401">Re <lb/>liquum e&longs;t modo ut o&longs;tendamus dupla <expan abbr="illor&utilde;">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="quantitat&utilde;">quantitatum</expan> eiu&longs;<lb/>dem primi ordinis pariter acceptis. </s>

<s id="id000402">Con&longs;tatigitur, quod duplum i<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="dupl&utilde;">duplum</expan> lin d æquale e&longs;t &longs;excuplo h in ip&longs;um d. </s>

<s id="id000403">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="product&utilde;">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="aggregat&utilde;">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"/><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 id="id000403">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="product&utilde;">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="aggregat&utilde;">aggregatum</expan> ip&longs;a<lb/>rum quantitarum, at quadratum a e&longs;t &etail;quale producto ex h in eam, <lb/>qu&etail; 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"/><lb/>h per demon&longs;trata ab Euclide, & pariter de quadrato b, quod e&longs;t &etail;<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 id="id000404">Gratia ergo exem

<pb xlink:href="015/01/039.jpg" 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>

<pb pagenum="20" xlink:href="015/01/039.jpg"/>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 id="id000405">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="reliquar&utilde;">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="c&utilde;">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 id="id000405">Rur&longs;us dico, quod productum multiplicis cuiuslibet <expan abbr="quãtitatis">quantitatis</expan> in <lb/>minimam, &longs;eu quadratum eiu&longs;dem quantitatis &etail;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 &etail;quales ip&longs;i a, & duplo <expan abbr="reliquar&utilde;">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="c&utilde;">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 &etail;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&etail; 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 id="id000406">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 id="id000407">hoc enim e&longs;t, quod nuper demon&longs;traui<lb/>mus. </s>

<s id="id000408">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="c&utilde;">cum</expan> producto h in <expan abbr="aggregat&utilde;">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

<s id="id000408">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="c&utilde;">cum</expan> producto h in <expan abbr="aggregat&utilde;">aggregatum</expan> a b c d e f g h <lb/>erat &etail;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 xlink:href="015/01/040.jpg" 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>

<pb pagenum="21" xlink:href="015/01/040.jpg"/>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 id="id000409">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 id="id000410">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 id="id000411">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>

<s id="id000414">Propo&longs;itio uige&longs;ima.</s>

<s id="id000415">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"/></s>

<s id="id000417">Cùm enim quantitates hæ non fuerint &ecedil;quales, <expan abbr="cõ&longs;tat">con&longs;tat</expan> per &longs;ecun

<s id="id000414">Propo&longs;itio uige&longs;ima.</s>

<s id="id000417">Cùm enim quantitates hæ non fuerint &etail;quales, <expan abbr="cõ&longs;tat">con&longs;tat</expan> per &longs;ecun

<pb xlink:href="015/01/041.jpg" 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>

<pb pagenum="22" xlink:href="015/01/041.jpg"/>dam harum, quod proportio primæ ad <expan abbr="quartã">quartam</expan> producitur ex pro<lb/>portione primæ ad &longs;ecundam, &longs;ecund&etail; ad tertiam, & terti&etail; 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&etail; ad &longs;e<lb/>cundam, & tertiæ ad quartam producunt proportionem producti <lb/>primæ in &longs;ecundam ad productum tertiæ in quartam. </s>

<s id="id000418">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 id="id000419">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="id.015.01.041.1.jpg" xlink:href="015/01/041/1.jpg"/><lb/>tio a ad d producta ex proportioni<lb/>bus a ad b, b ad c, & c ad d, producan<lb/>tur igitur ex proportionibus a ad b, c <lb/>ad d. </s>

<s id="id000420">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>

<s id="id000420">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>

<s id="id000421"><arrow.to.target n="marg70"/><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 id="id000422">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 id="id000423">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>

<s id="id000423">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>

<s id="id000425">Propo&longs;itio uige&longs;imaprima.</s>

<s id="id000425">Propo&longs;itio uige&longs;imaprima.</s>

<s id="id000426">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 id="id000427">Et &longs;imiliter interpo&longs;ita <lb/>omiologa.<lb/><arrow.to.target n="marg71"/></s>

<s id="id000427">Et &longs;imiliter interpo&longs;ita <lb/>omiologa.<lb/><arrow.to.target n="marg71"/></s>

<s id="id000429">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"/><lb/>ergo per <expan abbr="&longs;ec&utilde;dam">&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 id="id000430">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 xlink:href="015/01/042.jpg" pagenum="23"/>c, ergo k producetur ex c d in b, ergo ex c d in a fit h, ex c d in b fit k. <lb/></s>

<pb pagenum="23" xlink:href="015/01/042.jpg"/>c, ergo k producetur ex c d in b, ergo ex c d in a fit h, ex c d in b fit k. <lb/></s>

<s id="id000431">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>

<s id="id000433">Propo&longs;itio uige&longs;ima&longs;ecunda.</s>

<s id="id000434">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><figure id="id.015.01.042.1.jpg" xlink:href="015/01/042/1.jpg"/>

<s id="id000435">Sit proportio a ad b maior quàm c <lb/><arrow.to.target n="marg73"/><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"/><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>

<s id="id000433">Propo&longs;itio uige&longs;ima&longs;ecunda.</s>

<s id="id000438">Propo&longs;itio uige&longs;imatertia.</s>

<s id="id000439">Omnis motus naturalis ad locum &longs;uum e&longs;t: ideo per rectam li<lb/>neam fit.<lb/><arrow.to.target n="marg75"/></s>

<s id="id000438">Propo&longs;itio uige&longs;imatertia.</s>

<s id="id000439">Omnis motus naturalis ad locum &longs;uum e&longs;t: ideo per rectam li<lb/>neam fit.<lb/><arrow.to.target n="marg75"/></s>

<s id="id000441">Motus naturalis e&longs;t, ut con&longs;eruetur corpus, & conueniat locus <lb/>corpori, igitur fit ad &longs;uum locum. </s>

<s id="id000442">Locus autem dicitur in compara<lb/>tione ad uniuer&longs;um. </s>

<s id="id000443">ideo omnis motus naturalis e&longs;t à centro mun<lb/>di &longs;ur&longs;um, uel ad centrum deor&longs;um. </s>

<s id="id000444">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>

<s id="id000445"><arrow.to.target n="marg76"/><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>

<s id="id000446"><margin.target id="marg76"/>D<emph type="italics"/>i&longs;t. tertia <lb/>primi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>

<s id="id000447">Propo&longs;itio uige&longs;imaquarta.</s>

<s id="id000447">Propo&longs;itio uige&longs;imaquarta.</s>

<s id="id000448">Omnis motus circularis uoluntarius e&longs;t.</s>

<s id="id000449">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,

<s id="id000448">Omnis motus circularis uoluntarius e&longs;t.</s>

<s id="id000449">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&etail;cedentem nece&longs;&longs;e <lb/>e&longs;t, ut recta feratur ad c, uel recedat, igitur motus a e&longs;t uoluntarius,

<pb xlink:href="015/01/043.jpg" pagenum="24"/><figure id="id.015.01.043.1.jpg" xlink:href="015/01/043/1.jpg"/><lb/>non naturalis. </s>

<pb pagenum="24" xlink:href="015/01/043.jpg"/><figure id="id.015.01.043.1.jpg" xlink:href="015/01/043/1.jpg"/><lb/>non naturalis. </s>

<s id="id000450">nam &longs;i uiolentus e&longs;&longs;et, non <lb/>e&longs;&longs;et perpetuus. </s>

<s id="id000451">Omnia ergo a&longs;tra feruntur <lb/>circa centrum mundi. </s>

<s id="id000452">Sit modo rota e f g, di<lb/>co e non moueri motu circulari nam linea <lb/>e c <expan abbr="lõgior">longior</expan> e&longs;t g c, ergo recta mouetur ad cen<lb/>trum non circa centrum. </s>

<s id="id000453">Indicio etiam id <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 id="id000454">nihil etiam hoc modo &longs;ponte mouetur. </s>

<s id="id000455">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>

<s id="id000456">Propo&longs;itio uige&longs;imaquinta.</s>

<s id="id000457">Tres &longs;unt motus omnino &longs;implices naturalis, uoluntarius & <lb/>uiolentus.<lb/><arrow.to.target n="marg77"/></s>

<s id="id000456">Propo&longs;itio uige&longs;imaquinta.</s>

<s id="id000457">Tres &longs;unt motus omnino &longs;implices naturalis, uoluntarius & <lb/>uiolentus.<lb/><arrow.to.target n="marg77"/></s>

<s id="id000459">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 id="id000460">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 id="id000461">Hæc autem diui&longs;io e&longs;t &longs;olum propria <lb/>non prima. </s>

<s id="id000462">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 id="id000463">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 id="id000464">Sunt enim quatuor genera mo</s>

<s id="id000464">Sunt enim quatuor genera mo</s>

<s id="id000465"><arrow.to.target n="marg78"/><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 id="id000466"><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>

<s id="id000466"><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>

<s id="id000467"><margin.target id="marg78"/>7. P<emph type="italics"/>hy&longs;. <lb/>cap.<emph.end type="italics"/> 2.</s>

<s id="id000468">Propo&longs;itio uige&longs;ima.</s>

<s id="id000468">Propo&longs;itio uige&longs;ima.</s>

<s id="id000469">Motus ergo compo&longs;iti quatuor nece&longs;&longs;ariò &longs;unt &longs;pecies.</s>

<s id="id000469">Motus ergo compo&longs;iti quatuor nece&longs;&longs;ariò &longs;unt &longs;pecies.</s>

<s id="id000470">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 id="id000471">Di&longs;quiramus ergo an &longs;int natura<lb/>liter tot &longs;pecies, for&longs;an enim repugnabit aliquis alicui. </s>

<s id="id000472">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>

Line 575

Line 2884

<s id="id000474">Tertia cum <lb/>fuerit in recta ad centrum, &longs;ed contrario modo, uelut terræ &longs;ur&longs;um.

<pb xlink:href="015/01/044.jpg" pagenum="25"/>Quarta cùm in recta ad centrum, &longs;ecundum naturam, &longs;ed <expan abbr="nõ">non</expan> à prin <lb/>cipio naturali. </s>

<pb pagenum="25" xlink:href="015/01/044.jpg"/>Quarta cùm in recta ad centrum, &longs;ecundum naturam, &longs;ed <expan abbr="nõ">non</expan> à prin <lb/>cipio naturali. </s>

<s id="id000475">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 id="id000476">Hic igi<lb/>tur motus e&longs;t compo&longs;itus ex naturali, & uiolento. </s>

<s id="id000477">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>

Line 583

Line 2892

<s id="id000479">Eruntergo &longs;ecundum naturam tantùm tres &longs;pe<lb/>cies. </s>

<s id="id000480">Velut cùm quis &longs;candit, aut&longs;alit: E&longs;t enim motus naturalis &longs;al<lb/>tem in fine, & uoluntarius, & uiolentus. </s>

<s id="id000481">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 id="id000482">Motum autem occultum uocamus grauita<lb/>tem aut leuitatem.</s>

<s id="id000482">Motum autem occultum uocamus grauita<lb/>tem aut leuitatem.</s>

<s id="id000483">Propo&longs;itio uige&longs;ima&longs;eptima.</s>

<s id="id000484">Motus uoluntarius e&longs;t in loco: naturalis ad locum: uiolentus <lb/>exloco.</s>

<s id="id000483">Propo&longs;itio uige&longs;ima&longs;eptima.</s>

<s id="id000484">Motus uoluntarius e&longs;t in loco: naturalis ad locum: uiolentus <lb/>exloco.</s>

<s id="id000485">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/>c&oelig;lo, corpora autem animalium in eodem loco feruntur: quia in <lb/>eodem orbe nata redire ad proprium locum. </s>

<s id="id000486">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 id="id000487">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 id="id000488">Qui uerò naturalis, e&longs;t ut re<lb/>deat corpus ad &longs;uum locum, igitur naturalis e&longs;t ad locum. </s>

<s id="id000489">Sed <lb/>uiolenti finis e&longs;t, ut protrudatur ex loco in quo e&longs;t, non habens cer<lb/>tum finem. </s>

<s id="id000490">licet enim qui trahit, ad &longs;uum locum trabat, non tamen <lb/>ad locum mobilis.</s>

<s id="id000490">licet enim qui trahit, ad &longs;uum locum trabat, non tamen <lb/>ad locum mobilis.</s>

<s id="id000491">Propo&longs;itio uige&longs;imaoctaua.</s>

<s id="id000492">Motus quilibet naturalis aut uiolentus in aliquo medio fit.<lb/><arrow.to.target n="marg79"/></s>

<s id="id000491">Propo&longs;itio uige&longs;imaoctaua.</s>

<s id="id000494">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>

<s id="id000495">Propo&longs;itio uige&longs;imanona.</s>

<s id="id000496">Omnis motus uoluntarius æqualis e&longs;t &longs;emper: &longs;impliciter etiam <lb/>quilibet alius motus.</s>

<s id="id000492">Motus quilibet naturalis aut uiolentus in aliquo medio fit.<lb/><arrow.to.target n="marg79"/></s>

<s id="id000495">Propo&longs;itio uige&longs;imanona.</s>

<s id="id000496">Omnis motus uoluntarius æqualis e&longs;t &longs;emper: &longs;impliciter etiam <lb/>quilibet alius motus.</s>

<s id="id000499">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 id="id000500">neque enim e&longs;t, ut dixi, per medium corpus. <lb/></s>

<s id="id000501">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>

<s id="id000501">Naturalis quoque, & uiolentus cum ratione proportionis mouentis <lb/>&longs;upra mobile per&longs;e non uarientur, & ab &etail;quali proportione &etail;qua<lb/>lis uelo citas proueniat, igitur natura tales motus &longs;unt &etail;quales, nam <lb/>in utroque mouens, mouet &longs;ecundum ultimam &longs;uam uim.</s>

<s id="id000502">Propo&longs;itio trige&longs;ima.</s>

<s id="id000503">In omni corpore mobili in medio, partes medij re&longs;i&longs;tunt obuiæ, <lb/>aliæ impellunt.</s>

<s id="id000502">Propo&longs;itio trige&longs;ima.</s>

<s id="id000503">In omni corpore mobili in medio, partes medij re&longs;i&longs;tunt obuiæ, <lb/>aliæ impellunt.</s>

<s id="id000506">Sit mobile a cui partes &longs;ubiaceant directæ b, & &longs;it graue. </s>

<s id="id000507">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="id.015.01.045.1.jpg" xlink:href="015/01/045/1.jpg"/><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>

<s id="id000507">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&etail;rentes &longs;ecum, atque ita e c d f <lb/><figure id="id.015.01.045.1.jpg" xlink:href="015/01/045/1.jpg"/><lb/>adiuuant ad 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>

<s id="id000510">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>

<s id="id000513">Et ideò etiam bellicæ machinæ cuiu&longs;cunque generis certam exi<lb/>gunt di&longs;tantiam, ut uiolentius feriant.</s>

<s id="id000514">Propo&longs;itio trige&longs;imaprima.</s>

<s id="id000515">Omnis motus naturalis in æquali medio ualidior e&longs;t in fine, <lb/>quàm in principio: uiolentus contrà.</s>

<s id="id000510">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>

<s id="id000513">Et ideò etiam bellicæ machinæ cuiu&longs;cunque generis certam exi<lb/>gunt di&longs;tantiam, ut uiolentius feriant.</s>

<s id="id000514">Propo&longs;itio trige&longs;imaprima.</s>

<s id="id000515">Omnis motus naturalis in æquali medio ualidior e&longs;t in fine, <lb/>quàm in principio: uiolentus contrà.</s>

<s id="id000518">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 id="id000519">In uiolento autem, cùm perueniat ad finem de&longs;init </s>

<s id="id000519">In uiolento autem, cùm perueniat ad finem de&longs;init </s>

<s id="id000520"><arrow.to.target n="marg85"/><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>

<s id="id000521"><margin.target id="marg85"/> 29. P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/></s>

<s id="id000521"><margin.target id="marg85"/> 29. P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/></s>

<s id="id000522">Ex quo patet, quòd motus quadrifariam mi&longs;ti dicuntur, aut &longs;pe<lb/><arrow.to.target n="marg86"/><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="id.015.01.045.2.jpg" xlink:href="015/01/045/2.jpg"/><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 xlink:href="015/01/046.jpg" pagenum="27"/>& medij ad mobile, ac medium, quod re&longs;i&longs;tit. </s>

<pb pagenum="27" xlink:href="015/01/046.jpg"/>& medij ad mobile, ac medium, quod re&longs;i&longs;tit. </s>

<s id="id000523">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 id="id000524">Etideo de<lb/>mon&longs;trationes illæ Ari&longs;totelis quoad u&longs;um nihil iuuant nos.</s>

<s id="id000524">Etideo de<lb/>mon&longs;trationes illæ Ari&longs;totelis quoad u&longs;um nihil iuuant nos.</s>

<s id="id000526">Propo&longs;itio trige&longs;ima&longs;ecunda.</s>

<s id="id000526">Propo&longs;itio trige&longs;ima&longs;ecunda.</s>

<s id="id000527">Omne mobile naturaliter motum, &longs;eu uiolenter uelo cius moue<lb/>tur in medio rariore, quàm den&longs;iore. </s>

<s id="id000528">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 id="id000529">In uiolento autem celeriùs perueniet ad finem mo<lb/>tus in corpore den&longs;iore.</s><figure id="id.015.01.046.1.jpg" xlink:href="015/01/046/1.jpg"/>

<s id="id000529">In uiolento autem celeriùs perueniet ad finem mo<lb/>tus in corpore den&longs;iore.</s>

<s id="id000530">A mobile moueatur in b medio rariore, & in c den&longs;io<lb/><arrow.to.target n="marg87"/><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>

<s id="id000532">Propo&longs;itio trige&longs;imatertia.</s>

<s id="id000533">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"/></s>

<s id="id000532">Propo&longs;itio trige&longs;imatertia.</s>