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version 1.2, 2003/11/20 17:53:52

version 1.26, 2004/04/06 17:37:36

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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>

<front>

<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 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="id000008">Cum Cæ&longs;. </s>

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

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

<s id="id000025">Si ergo principatum <lb/>cum tot laboribus, curis, periculis, & meritò omnes appetunt: nec <lb/>e&longs;t in eo quicquam præcipuum præter hoc, cui dubium e&longs;t quin <lb/>hoc non &longs;it &longs;ummum huius uitæ hominibus bonum? </s>

<s id="id000026">propter cu<lb/>ius uel dubiam &longs;pem eorum, quæ habent obliti mortales pericli<lb/>tantur. </s>

<s id="id000027">Succedunt inde tot commoda, non &longs;olum utilia, &longs;ed pleraque<pb xlink:href="015/01/007.jpg"/>etiam nece&longs;&longs;aria, quæ nos &longs;apientia docet: huiu&longs;modi ergo omnia <lb/>cùm libris contineantur, meritò optimus qui&longs;que librorum bono<lb/>rum perpetuitati atque in columitati fauere debet. </s>

<s id="id000029">Caligulam exe<lb/>cramur &longs;olum ob id quod Vergilij, & T. </s>

<s id="id000030">Liuij &longs;cripta delere cogi<lb/>tauerit. </s>

<s id="id000031">Quid facturi e&longs;&longs;emus, &longs;i feci&longs;&longs;et quod cogitauerat? </s>

<s id="id000032">E&longs;t in &longs;a<lb/>pientum monumentis bonum &longs;ine malo, mens &longs;ine corporea labe: <lb/>Virtutes ab&longs;que uitijs, gratiæ & iucunditas &longs;ine &longs;orde, & immundi<lb/>tia, uoluptas &longs;ine dolore, conuer&longs;atio ab&longs;que tædio, delitiæ ab&longs;que mi&longs;e<lb/>ria nuda, omnia bona præ&longs;tant, atque laudabilia ab omnibus morta<lb/>litatis exuuijs libera, tantum commodi afferunt libri. </s>

<s 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;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/>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>

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

<table>

<table.target id="table1"/>

<row>

<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>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 trecenti &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 quantitates omiologas reducetur.<emph.end type="italics"/></cell>

</row>

<row>

<cell>S<emph type="italics"/>i duarum proportionum &longs;uperiores numeri alternatim cum inferioribus multiplicentur atque coniungantur, erit proportio aggregati ad productum ex inferioribus inuicem 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 multiplicentur, minusque productum ex maiore detrahatur, erit re&longs;idui ad productum ex in&longs;erioribus proportio uelut illa, quæ relinquitur detracta minore proportione ex maiore.<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;ecundam 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 æqualium <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 duplam.<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 aggregatum &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 proportionem 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 aggregato 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 omnium quantitatum eiu&longs;dem, tripla aggregato quadratorum omnium quantitatum 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, productumque 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 quartam.<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 motus.<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 impellunt.<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;uperficiei, 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> ductas 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 occulto 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 quamcunque 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, quoties 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;iderare.<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, productum temporis circuituum inuicem, erit æquale producto differentiæ temporum 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 coniunctiones in temporibus commen&longs;is, illa tria mobilia denuo coniungentur in tem pore producto ex denominatore diui&longs;ionis temporis maioris per minus in minus 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 peripheriæ.<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 ordinibus 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 tardius 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 mouitur 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 analogæ, erit proportio tertiæ unius ordinis ad tertiam alterius, ut &longs;ecundæ ad &longs;ecundum 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 peruer&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 detracto 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;">aerem</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 manife<expan abbr="&longs;i&utilde;">&longs;tum</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 detracta 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 anguli &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 pretij 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 determinare.<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 perpendiculum ex diametro exierint ad circum ferentiam, &longs;ingulæ &longs;upra diametrum erunt maiores 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 inde 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 inuestigare.<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 declarare.<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 perpendiculum 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 horizonte, 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 æquinoctiali 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 demonstrare.<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 meridiano 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 umbram 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 aggregati ex minore, & adiecta ad ip&longs;am minorem, minor proportio quàm aggregati 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 quædam quantitas, <expan abbr="ead&etilde;que">eadenque</expan> maiori addatur, erit minoris ad re&longs;iduum maior proportio, 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 latere 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 pararallelipedo, 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;uperficiem 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 partium detracta &longs;uperficie unius partis in alteram, e&longs;t æquale aggregato cuborum 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 lineas 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;ecundum <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 duplum 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 additam 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 differentiæ 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 aggregati ex maiore, & dimidio ad ip&longs;am maiorem, uelut ex minore, & aliqua 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 inæqualis 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;umptæ <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 extremo lineæ adiectæ di&longs;tans per lineam mediam.<emph.end type="italics"/> Q<emph type="italics"/>uod &longs;i ab extremo alicuius lineæ æqualis 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"/>uadratorum 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 explicare.<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, & circuli 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 quantitate 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, & concauæ 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 rationem.<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 inuenire.<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;trare.<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 priore illa quantitate, quæ ad duas comparatur, ad eandem priorem quantitatem.<emph.end type="italics"/></cell>

</row>

<row>

<cell>CLXXVIII.</cell>

<cell>P<emph type="italics"/>roportionem mi&longs;tionis metallorum, maximè auri & argenti declarare.<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 partes 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;ecundam & 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 productum 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 proportionum totius ad reliquum re&longs;idui inuenire.<emph.end type="italics"/></cell>

</row>

<row>

<cell>CLXXXIII.</cell>

<cell>S<emph type="italics"/>patium 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, primum 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 oppo&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 quartæ 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 quartum pondus moueatur quibus &longs;ecundum, auxilio primi nece&longs;&longs;e e&longs;t <expan abbr="quart&utilde;">quartum</expan> pondus 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 proportio &longs;it eadem proportioni uirium & duorum ponderum mouentium aggregatum 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 coaptetur, 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, pos&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 moueatur 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 & minore pondere in maius, addiderisque in&longs;uper quantum e&longs;t productum dimidij uirium 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 peripheriam, 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 mutuo 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 coeant 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 ambæ 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 differentia lateris minoris partis à mediæ latere in differentiam maioris partis &longs;uperficiei à media &longs;uperficie bis, & ex differentia amborum laterum inæqualium iunctorum ad ambo latera, æqualia iuncta in minorem partem &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 reflectantur, 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 manife&longs;tum e&longs;t, protractam diametrum angulum &longs;uppo&longs;itum per æqualia diuidere.<emph.end type="italics"/></cell>

</row>

<row>

<cell>S<emph type="italics"/>i duæ lineæ ex duobus punctis peripheriam contingentes, in eandem partem protrahantur, &longs;emper magis di&longs;tabunt inuicem ea ex parte, & nunquam 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 inuenire 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 punctum reflexionis æqualis in medio arcus intercepti inter lineas, quæ à centro 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 lineam, 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 alterutrinque.<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 minore reflexè per magnam partem minoris à maiore perueuire poterunt.<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 tanquam 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 declarare.<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 aliorum ad eam iuxta di&longs;tantiam: ip&longs;ius uerò iuxta rationem pupillæ ad<emph.end type="italics"/> L<emph type="italics"/>unam 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 uitarum 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 differentia 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 quandam habent.<emph.end type="italics"/></cell>

</row>

<row>

<cell>CCXXVIII.</cell>

<cell>P<emph type="italics"/>roportionem &longs;cientiæ futurorum & cæterorum occultorum con&longs;iderare.<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 eos 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;imilflimus 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/>VISQVE 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="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>

<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/>cesse 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="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="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, & diuer&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="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="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="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="id000073">Tertia decima 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="id000076">Quarta decima 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&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="id000081">Quinta decima 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="id000083">Velut in exemplo &longs;uperiore detracta proportione b c ad d ex

<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="id000085">Sexta decima 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">Decima octaua 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">Decima nona 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="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 huc indè: 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="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="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="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="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;ententiæ, 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 inter illas. </s>

<s id="id000116">Sunt enim ambæ detractæ.</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="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="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="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 &etail;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="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="id000140">Nam per tertiam petitionem &longs;i &longs;int duæ quantitates, quæ non ha<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="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="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="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="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="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="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="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="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">Tertia decima 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">Quarta decima 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="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 c ad 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 f ad e, ut c ad 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/>f e&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 f ad 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;ecunda</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, &longs;tatuatur 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="id000201">Si proportio ex duabus proportionibus in quatuor terminis <lb/>producatur, ip&longs;a uerò proportio inter duas alias quantitates fue

<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&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&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>

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<cell>direc.</cell>

<cell>conuer.</cell>

</row>

</table>

<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>

<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>

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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="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>

<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>

<table>

<table.target id="table5"/>

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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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</row>

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<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">& per præcedentem 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 rur&longs;us a ad e ex c <lb/>ad f & b ad d, igitur per quartam eadem produ<lb/>cetur ex c ad d & b ad f. </s>

<s id="id000225">Quare per præceden<lb/>tem c ad f ex a ad e, & d ad b, & ita di&longs;ponemus <lb/>hos modos in tabula. </s>

<s id="id000226">Vides etiam <lb/><arrow.to.target n="table8"/><lb/><figure id="id.015.01.028.4.jpg" xlink:href="015/01/028/4.jpg"/>aliquos modos non produci, ut pri<lb/>mi ad quartum nec ad &longs;extum, & li<lb/>quet, quòd cùm &longs;int quindecim o<lb/>mnes modi qui produci po&longs;&longs;e intelli<lb/>guntur, & nouem tantum producan<lb/>tur &longs;ex e&longs;&longs;e, qui non producuntur, quos <lb/>&longs;eor&longs;um in tabula coniunxi. </s>

<s id="id000227">Et con<lb/>&longs;tat etiam, quod totidem conuer&longs;i &longs;ci<lb/>licet decem octo <expan abbr="produc&utilde;tur">producuntur</expan>, de qui<lb/>bus diximus, ut &longs;int omnes triginta <lb/>&longs;ex, qui con&longs;tat ex duabus propo&longs;i<lb/>tionibus præmi&longs;sis, & hac tertia, <expan abbr="quã">quam</expan> <lb/>adiungemus &longs;cilicet, quòd propor<lb/>tio primi ad tertium producatur ex <lb/>proportionibus <expan abbr="&longs;ec&utilde;di">&longs;ecundi</expan> ad quartum, <lb/>& quinti ad <expan abbr="&longs;ext&utilde;">&longs;extum</expan>. </s>

<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 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="differ&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&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>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>

<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&etail; quantitates ex his, qu&etail; faciunt pro

<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;ciliter 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="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>

<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="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="id000261">Sit a b quantitas diui&longs;a in c, & &longs;i c ut a b ad a c, <lb/>ita b c ad d: eritque iterum permutando a b ad b c, <lb/>ut a c ad d, & &longs;umatur quædam quantitas e eiu&longs;

<pb pagenum="12" 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 tanquam &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="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="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="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="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="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="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 f ex 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;t idem ad eam quantitatem, componunt eandem pro<lb/>portionem.</s>

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

</row>

<row>

</row>

<row>

</row>

<row>

</row>

</table>

<s id="id000300">Propo&longs;itio quarta decima.</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="id000308">Ex quauis ergo illarum data, data erit & reliqua.<lb/><arrow.to.target n="marg46"/></s>

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

<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 f maior 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 coniuncta 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;exta decima.</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 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&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 confu&longs;a 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/>fu&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="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 decima octaua.</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="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="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="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 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="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="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, quæ iunctæ 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 erunt in proportione 11 <lb/>ad 5 experiaris, & inuenies, & demon&longs;tratur ex prioribus.</s>

<s id="id000381"><margin.target id="marg65"/>Q<emph type="italics"/>uæ&longs;tio.<emph.end type="italics"/></s>

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

<s id="id000383">Si fuerint 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&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">aut</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="&etilde;os">eos</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 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, ea ra<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, ante penultima 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/>titatis. </s>

<s id="id000389">primi ordinis, & &longs;upplementis, 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, quæ tanto 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">aut</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">Hæc uolui 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&etail;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&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">aut</expan> &etail;qualis m. </s>

<s id="id000396"><expan abbr="Sequi&ttilde;">Sequitur</expan> ergo quod &longs;umptis duabus quantitatibus <lb/>&longs;ecundi ordinis habentibus <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 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">at uerò 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, pro ut &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;tat igitur, 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 quantitatum, 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 pagenum="20" xlink:href="015/01/039.jpg"/>pli quadratum a, erit æquale producto ex h in omnes quantitates &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 &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 &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;e ip&longs;am h quin<lb/>decies, detractis ergo p <expan abbr="ordin&etilde;">ordinem</expan>, &qring;d fit ex h in a ab utro que aggregato, <lb/>& ex h in b c d e f g h bis <expan abbr="relinque&ttilde;">relinquetur</expan> ex una parte, quae fit ex h in b &longs;emel

<pb pagenum="21" 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 relinquetur, 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;olent re<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 &etail;quales, <expan abbr="cõ&longs;tat">con&longs;tat</expan> per &longs;ecun

<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="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&etail;, <lb/>& &longs;ecundæ. </s>

<s id="id000423">Hæc igitur 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;ima prima.</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="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 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>

<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 tertiam decimam 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/>eandem tertiam decimam huius ergo proportio confu&longs;a a b ad c d <lb/>e&longs;t minor, quàm a ad b.</s>

<s id="id000438">Propo&longs;itio uige&longs;ima tertia.</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;ima quarta.</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&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 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;ima quinta.</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="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="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="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>

<s id="id000473">Secun<lb/>da cum fuerit &longs;ecundum rectam, &longs;ed non ad centrum. </s>

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

<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 proijcit 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>

<s id="id000478">Sed circularis, & uiolentus &longs;oli e&longs;&longs;e non po&longs;&longs;unt: nam uio<lb/>lentus e&longs;t nece&longs;&longs;ariò in corpore graui aut leui: &longs;ed omne corpus gra<lb/>ue aut leue, cùm mouetur, naturaliter mouetur &longs;altem in fine: & per <lb/>totum motum, motu ócculto, qui maximè in hoc libro dignus e&longs;t <lb/>con&longs;ideratione, igitur motus uoluntarius, & uiolentus non po&longs;<lb/>&longs;unt e&longs;&longs;e &longs;imul &longs;oli. </s>

<s id="id000479">Erunt ergo &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="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 trahat, 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="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;ima nona.</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 &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="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&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/>re impetu 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;ima prima.</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/>&longs;a, quæ mouet, &longs;it perpetua, & à principio æterno, quod per dictæ <lb/>æqualiter mouet, igitur motus ille fiet uelocior 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="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="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;t mo<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 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">Et ideo 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="id000527">Omne mobile naturaliter motum, &longs;eu uiolenter uelocius 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>

<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;ima tertia.</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="id000535">Sint duo mobilia a & b magnitudine, & forma omnino paria, <lb/>& &longs;int media c & d, exempli gratia: & pertran&longs;eant æquale &longs;patium <lb/>in utroque in eodem tempore, e dico proportionem ponderis b ad <lb/>pondus a e&longs;&longs;e duplicatam ei quæ e&longs;t raritatis c ad raritatem d. </s>

<s id="id000536">Quia <lb/>enim feruntur æqualiter, nam in æquali tem<lb/><figure id="id.015.01.046.2.jpg" xlink:href="015/01/046/2.jpg"/><lb/>pore, &longs;eu eodem æqualia &longs;patia pertran&longs;e<lb/>unt, erit proportio potentiæ a cum &longs;uo auxi<lb/>lio ad id, quod re&longs;i&longs;tit ex c ut b cum &longs;uo au<lb/>xilio ad id, quod re&longs;i&longs;tit ex d, permutando igi<lb/>tur d ad c, ut b ad a, &longs;ed c ad d proportio rari<lb/>tatis duplicat actionem, tum minus re&longs;i&longs;ten<lb/>do, tum adiuuando motum a, igitur proportio differentiæ motus <lb/>e&longs;t duplicata proportioni raritatis: &longs;ed proportio motus e&longs;t æqua<lb/>lis proportioni ponderis uici&longs;sim per uige&longs;imam &longs;extam &longs;exti Ele<lb/>mentorum b ad a, igitur proportio b ad a ponderis e&longs;t duplicata ei, <lb/>quæ e&longs;t raritatis c ad raritatem d.</s>

<s id="id000537">SCHOLIVM PRIMVM.</s>

<s id="id000538">Ne tamen &longs;ine exemplo intelligas hanc duplicatam rationem, <lb/>proponatur c raritas quatuor, d unum, a pondus duodecim libra<lb/><figure id="id.015.01.047.1.jpg" xlink:href="015/01/047/1.jpg"/><lb/>rum, tunc c re&longs;i&longs;tit &longs;olum ex quarta parte, & effi<lb/>cit a quadruplo maioris actionis, &longs;cilicet ut qua<lb/>draginta octo, tota igitur proportio, qua mo<lb/>uebitur a in c, erit centum nonaginta duorum, & hoc diuidemus <lb/>per d, quod e&longs;t unum, exibit <expan abbr="põdus">pondus</expan> b centum nonaginta duo. </s>

<s id="id000539">Pro<lb/>portio igitur b ad a e&longs;t &longs;ex de cupla, & hæc e&longs;t duplicata quadruplæ <lb/>raritatis c ad raritatem d.</s>

<s id="id000540">Quòd &longs;i quis neget tantundem augere c actionem a, quanto mi<lb/>nus re&longs;i&longs;tit, &longs;ed aut magis aut minus, & &longs;it proportio b ad a dupli<lb/>cata ip&longs;i f, dico fe&longs;&longs;e proportionem c ad d, nam proportio b ad a <lb/>e&longs;t uelut actionis c ad d per decimam &longs;extam &longs;exti Elementorum, <lb/>ergo ex auxilio c in proportionem a ad c fit proportio b ad a, &longs;ed ex <lb/>fin &longs;e fit proportio b ad a ex diffinitione proportionis duplicatæ. <lb/></s>

<s id="id000541">Sed ex duabus proportionibus a ad c, & actionis ex c ad a produ<lb/>citur proportio b ad a, igitur per <expan abbr="decimam&longs;eptimã">decimam &longs;eptimam</expan> &longs;exti Elemento<lb/>rum proportio c ad d e&longs;t media inter proportiones a ad c, & actio<lb/>nis a in c, quare æqualis f, igitur proportio b ad a duplicata ei, quæ <lb/>e&longs;t c ad d quod erat demon&longs;trandum.</s>

<s id="id000542">SCHOLIVM SECVNDVM.</s>

<s id="id000543">Si autem media fuerint diuer&longs;arum rationum, ut aqua, & aër non <lb/>demon&longs;trat argumentum, quia pondera inter &longs;e non &longs;eruant ratio<lb/>nem. </s>

<s id="id000544">Nam lignum centum librarum ex &longs;alicis arbore, non magis <lb/>de&longs;cendit, quàm lignum libræ unius. </s>

<s id="id000545">Ideò nec in comparatione ad <lb/>medium aëris.</s>

<s id="id000546">Propo&longs;itio trige&longs;ima quarta.</s>

<s id="id000547">Proportio corporis cubi ad &longs;uam &longs;uperficiem quadratam, e&longs;t ue<lb/>lut eiu&longs;dem &longs;uperficiei ad latus, eiu&longs;dem uerò ad monadem.</s>

<s id="id000550">Sit cubus a b c eius quadrata, &longs;uperficies a <lb/><figure id="id.015.01.047.2.jpg" xlink:href="015/01/047/2.jpg"/><lb/>c, latus a b, monas d, dico eas e&longs;&longs;e inuicem <lb/>analogas. </s>

<s id="id000551">Quia enim proportio a b c ad a c <lb/>e&longs;t, ut quoties a&longs;&longs;umitur a c in a b c, & toties <lb/>etiam a&longs;&longs;umitur a b in a c ex diffinitione Eucli </s>

<s id="id000552"><arrow.to.target n="marg90"/><lb/>dis &longs;ecundo Elementorum, &longs;i ergo monas e&longs;t <lb/>in continua proportione, habeo intentum: &longs;i <lb/>non ponatur e media inter a e & d, erit ergo <lb/>per decimam noni Elementorum elatus a c, <lb/>ergo æqualis a b, igitur cum a c, e & d &longs;int analogæ, erunt & a b c, <lb/>a b, & d analogæ, quod fuit demon&longs;trandum.</s>

<s id="id000553"><margin.target id="marg90"/>P<emph type="italics"/>rima ex<emph.end type="italics"/><lb/>C<emph type="italics"/>ampano.<emph.end type="italics"/></s>

<s id="id000554">Propo&longs;itio trige&longs;ima quinta.</s>

<s id="id000555">Vocum magnitudines excre&longs;cunt in acumine non in grauitate, <lb/>finis autem e&longs;t in utroque extremo, propter hoc minima facta uaria<lb/>tione in hypate acutæ uix ferunt.<lb/><arrow.to.target n="marg91"/></s>

<s id="id000557">Quoniam facta uariatione in hypate, quæ e&longs;t <lb/>in Diapa&longs;on, uel bis Díapa&longs;on maiore interual<lb/><figure id="id.015.01.048.1.jpg" xlink:href="015/01/048/1.jpg"/><lb/>lo di&longs;tat, uelut ex a in b in grauiore, maius e&longs;t in<lb/>teruallum ex c in d, igitur maior e&longs;t b d, quàm a c <lb/>ergo &longs;ingulæ uoces inter b & d magis di&longs;tant, <lb/>quàm inter a & c, & quanto magis appropin<lb/>quant ad d, igitur d maius e&longs;t quàm b. </s>

<s id="id000558">Ergo magnitudo e&longs;t ratione <lb/>acuitatis, non grauitatis, cum &longs;uppo&longs;uerimus d e&longs;&longs;e acutiorem b & <lb/>c ip&longs;o a. </s>

<s id="id000559">O&longs;tenditur etiam idem quia uox grauis fit ex priuatione <lb/>motus &longs;icut acuta ex uehementia. </s>

<s id="id000560">Motus autem e&longs;t res, quies, <lb/>priuatio.</s>

<s id="id000561">Secundum &longs;ic: nam remi&longs;sio mota non feriet aurem, ideò &longs;onum <lb/>non pariet ob nimiam tarditatem. </s>

<s id="id000562">At in ueloci&longs;simo motu oportet <lb/>uel fidem uel arteriam contrahi, & non contrahitur ni&longs;i per mu&longs;cu<lb/>los, igitur contentio illa finem habet. </s>

<s id="id000563">Si autem non &longs;it nece&longs;&longs;arium <lb/>habere, uel ualde procul po&longs;sit extendi contentio, ut in machinis <lb/>igneis &longs;trepitus fit maximus, nam motus, ut motus e&longs;t etiam in aëre <lb/>nullum finem per &longs;e habet ni&longs;i ratione in&longs;trumenti, ergo &longs;trepitus <lb/>tantus e&longs;&longs;e pote&longs;t, ut fermè ob&longs;urde&longs;cant, qui audierint, ut ferunt de <lb/>Nili cataractis.</s>

<s id="id000564">Tertium &longs;ic &longs;it a b humi<lb/>lior uox, quæ excre&longs;cat &longs;e<lb/>mitonio minore &longs;olum in <lb/>c, & &longs;it d e dupla ad ab &longs;e<lb/>cundum naturam, ut in uo<lb/>cibus medijs fiet, ut &longs;i e debeat excre&longs;cere &longs;emitonio minore per de<lb/>cimam nonam quinti <expan abbr="Elem&etilde;torum">Elementorum</expan> f e dupla c b, & in acutis ubi ex<lb/>creuerit ad diapa&longs;on quadrupla: pueri autem uox, quæ iam diapa<lb/>&longs;on altior e&longs;t d e, erit bis diapa&longs;on, & ideò quadrupla b c, &longs;ed in acu<lb/>tioribus erit dupla, nullus enim puer e&longs;t adeo fractæ uocis, qui &longs;u<lb/>pra humillimam non a&longs;cendat per diapa&longs;on, igitur interuallum uo<lb/>cum erit octuplum a d, b c, &longs;ed communiter a&longs;cendunt ad bis diapa<lb/>&longs;on, igitur interuallum unius uocis etiam cum &longs;emitonio propor<lb/>tionem habentis e&longs;t æquale fermè toti a b, cum autem in diapa&longs;on <lb/>&longs;int duodecim &longs;emitonia, & duo comata, manife&longs;tum e&longs;t, quod ex<lb/>ten&longs;io illa erit maxima in <expan abbr="cõparatíone">comparatíone</expan> grauioris uo cis a b. </s>

<s id="id000565">Et ideò <lb/>minimum in crementum in humilioribus uocibus, ubi quis coga<pb pagenum="40 [=30]" xlink:href="015/01/049.jpg"/>tur a&longs;cendere, maximum e&longs;&longs;e uidetur, adeò ut ægrè à pluribus fera<lb/>tur, à quibu&longs;dam non omnino feratur.</s>

<s id="id000566">SCHOLIVM.</s>

<s id="id000567">Ob hoc natura fecit, ut non quemadmodum in fidibus uoces ex <lb/>breuitate intenderentur, &longs;ed ex con&longs;trictione ligulæ, ut dicunt, &longs;u<lb/>per a&longs;peram arteriam uox ad diapa&longs;on acueretur addito impetu <lb/>proportione, ut ex con&longs;trictione, & impetu <expan abbr="cõ&longs;urgeret">con&longs;urgeret</expan> dupla pro<lb/>portio. </s>

<s id="id000568">Hoc autem manife&longs;tè experimur in elymis in quibus nullæ <lb/>pror&longs;us facta mutatione in&longs;trumenti con&longs;tantibus digitis omni<lb/>bus præter pollicem &longs;ini&longs;træ uocem exacuimus ad diapa&longs;on, inde <lb/>etiam ad bis diapa&longs;on: &longs;icut declarauimus in commentarijs Epi<lb/>demiorum.</s>

<s id="id000569">Propo&longs;itio trige&longs;ima &longs;exta.</s>

<s id="id000570">Si proportio per proportionem minorem æquali ducatur, pro<lb/>portio minor producetur. </s>

<s id="id000571">Vnde manife&longs;tum e&longs;t duas proportio<lb/>nes minores æqualitate inuicem ductas proportionem minorem <lb/>unaquaque illarum producere.</s>

<s id="id000574">Proportio a b ad c, quali&longs;cunque &longs;it, duca<lb/>tur in proportionem minorem æqualitate <lb/>f ad g, dico quod producta proportio erit <lb/>minor ea, quæ e&longs;t a b ad c fiat d ad a b, ut f <lb/>ad g, et erit per &longs;ecundam huius d ad c pro<lb/>ducta ex proportionibus a b ad c, & f g. </s>

<s id="id000575">Itemque per decimam quar<lb/><arrow.to.target n="marg93"/><lb/>tam quinti <expan abbr="Elementor&utilde;">Elementorum</expan> erit d minor a b, igitur maior a b ad c, quàm <lb/>d ad c. igitur quàm proportio a b ad c in proportionem f ad g. </s>

<s id="id000576">Sit <lb/>autem utraque minor æqualitate ea, quæ a b ad c, & ea quæ f ad g, di<lb/>co productam unaquaque earum e&longs;&longs;e minorem. </s>

<s id="id000577">Quod enim (manen<lb/>tibus his, quæ dicta &longs;unt) minor &longs;it d ad c, quam a b ad c ex prima <lb/>parte o&longs;ten&longs;um e&longs;t. </s>

<s id="id000578">Quòd uerò etiam minor &longs;it d ad c, quàm d ad <lb/>a b, & ex con&longs;equenti quàm f ad g demon&longs;tratur &longs;ic. </s>

<s id="id000579">Quia enim mi<lb/>nor e&longs;t a b ad c, æqualitate erit a b minor c, fiat ergo h æqualis a b, <lb/>erit ergo d ad h, ut d ad a b per &longs;eptimam quinti Elementorum, at d <lb/>ad c minor quàm d ad h per octauam eiu&longs;dem, igitur minor d ad c, <lb/>quàm d ad a b, igitur patet propo&longs;itum.</s>

<s id="id000581">Propo&longs;itio trige&longs;ima &longs;eptima.</s>

<s id="id000582">Si plures homines, quorum nulli per &longs;e nauim mouere po&longs;sint, <lb/>aut pondus ferre &longs;imul iuncti eam moueant, aut pondus ferant, <lb/>erunt illæ proportiones coniunctæ non productæ.<lb/><arrow.to.target n="marg94"/></s>

<s id="id000584">Cùm enim primus non po&longs;sit mouere nec &longs;ecundus, erunt pro<lb/>portiones minores æqualitate, Ideò per &longs;ecundam partem præce<lb/>dentis multo minus mouerent duo, quàm unus. </s>

<s id="id000585">Et &longs;i quatuor mo

<pb pagenum="41 [=31]" xlink:href="015/01/050.jpg"/>uerent unusque per &longs;e mouere non po&longs;&longs;et, adderetur &longs;i proportio <lb/>produceretur, fieret minor, ergo minus mouerent quinque quàm <lb/>quatuor ex ij&longs;dem, quod e&longs;t ab&longs;urdum.</s>

<s id="id000586">Propo&longs;itio trige&longs;ima octaua.</s>

<s id="id000587">Omne corpus tantùm re&longs;i&longs;tit motui contrario &longs;uo naturali quan <lb/>cum mouetur occulto motu quie&longs;cendo.</s>

<s id="id000590">Sit a corpus quie&longs;cens in pauimento b, & mouetur in eo occul</s><s id="id000591"><arrow.to.target n="marg96"/><lb/>to motu uer&longs;us centrum, ut &longs;uprà ui&longs;um e&longs;t, contra<lb/><figure id="id.015.01.050.1.jpg" xlink:href="015/01/050/1.jpg"/><lb/>rius illi &longs;it motus ad c, &longs;i ergo a quie&longs;ceret in c moue<lb/>retur ad b occulto motu certa ui, ergo eadem re&longs;titit, <lb/>ne traheretur ad c. </s>

<s id="id000592">Manife&longs;tum e&longs;t autem, quod hic <lb/><arrow.to.target n="marg97"/><lb/>motus occultus e&longs;t minor manife&longs;to.<lb/><arrow.to.target n="marg98"/></s>

<s id="id000593"><margin.target id="marg96"/>I<emph type="italics"/>n commen.<emph.end type="italics"/><lb/>26. P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/></s>

<s id="id000594"><margin.target id="marg97"/>P<emph type="italics"/>er<emph.end type="italics"/> 30. P<emph type="italics"/>ro <lb/>po&longs;.<emph.end type="italics"/></s>

<s id="id000596">Ex hoc patet cur naues & currus ab initio tardè & difficulter mo<lb/>ueantur, ubi moueri c&oelig;perint motus augetur: quoniam re&longs;i&longs;tunt </s>

<s id="id000597"><arrow.to.target n="marg99"/><lb/>per motum occultum naturalem qui maximus e&longs;t dum quie&longs;cunt, <lb/>ut etiam docebat philo&longs;ophus in mechanicis, nam motus ille natu<lb/>ralis e&longs;t, & ideò contrarius uiolento: Ergo cum iam mouetur uio<lb/>lenter minus, mouetur naturaliter, igitur minus re&longs;i&longs;tit. </s>

<s id="id000598">Declarabi<lb/>tur enim infrà quòd omne quod mouetur duobus motibus tanto <lb/><arrow.to.target n="marg100"/><lb/>minus uno mouetur quanto magis altero.</s>

<s id="id000599"><margin.target id="marg99"/>Q<emph type="italics"/>ue&longs;t.<emph.end type="italics"/> 31.</s>

<s id="id000600"><margin.target id="marg100"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 59.</s>

<s id="id000601">Propo&longs;itio trige&longs;ima nona.</s>

<s id="id000602">Ab æquali aut minore ui, quàm &longs;it <expan abbr="impediment&utilde;">impedimentum</expan>, non fit motus.</s>

<s id="id000603">Sit a quod re&longs;i&longs;tat, ne &longs;ur&longs;um trahatur per decem, dico, quod <expan abbr="nõ">non</expan> <lb/><arrow.to.target n="marg101"/><lb/>&longs;ur&longs;um trahetur neque à decem, neque minore: nam &longs;i impedimen<lb/>tum non e&longs;&longs;et, moueretur infra ut decem, ergo &longs;i traheretur &longs;ur&longs;um <lb/>per decem tantum moueretur &longs;ur&longs;um, <expan abbr="quant&utilde;">quantum</expan> deor&longs;um, ergo quie<lb/>&longs;ceret. </s>

<s id="id000604">Si uerò à minore moueretur à maiore ui deor&longs;um, quam &longs;ur<lb/>&longs;um, ergo deor&longs;um &longs;impliciter non &longs;ur&longs;um.</s>

<s id="id000606">Propo&longs;itio quadrage&longs;ima.</s>

<s id="id000607">Omne corpus &longs;phæricum tangens planum in puncto mouetur <lb/>ad latus per quancunque uim, quæ medium diuidere pote&longs;t.</s>

<s id="id000608">Sit corpus ad unguem &longs;phæricum a tan<lb/><arrow.to.target n="marg102"/><lb/>gens planum b in puncto c (e&longs;t enim hoc <lb/>nece&longs;&longs;arium ex demon&longs;tratis ab Euclide in <lb/>decima &longs;exta Propo&longs;itione tertij Elemento<lb/>rum) dico, quod mouebitur à ui, quæ pote&longs;t <lb/>&longs;cindere aërem. </s>

<s id="id000609">Nam cum non a&longs;cendat, nec <lb/>de&longs;cendat, &longs;ed qua&longs;i in circulo ad centrum <lb/>mundi moueatur, pondus non affert. </s>

<s id="id000610">Neque<lb/> ratione magnitudinis contactus, cum &longs;it in <lb/>puncto &longs;olo, igitur remanet &longs;olum aëris impedimentum.

<s id="id000613">Ex hoc liquet, quod oportet b planum e&longs;&longs;e ex duri&longs;sima mate<lb/>ria, quæ nullo modo cedat, aliter tanget plu&longs;quàm in puncto.</s>

<s id="id000616">Vix fieri pote&longs;t, utin elementaribus &longs;phæra tangat planum in <lb/>puncto. </s>

<s id="id000617">Vel quia planum non erit exactè rectum, uel non durum, <lb/>ut pror&longs;us non cedat, uel non ad æquilibrium po&longs;itum, uel &longs;phæra <lb/>non erit exactè rotunda.</s>

<s id="id000618">Propo&longs;itio quadrage&longs;ima prima.</s>

<s id="id000619">Si fuerint duæ quantitates &longs;umaturque totius aggregatum maio<lb/>ris & minoris, quoties aggregatum minoris, & maioris, erit pro<lb/>portio confu&longs;a maioris aggregati ad minus, minor quàm multipli<lb/>cis maioris ad multiplex minoris.</s>

<s id="id000622">Sint duæ magnitudines a & b, & &longs;it a maior <lb/><figure id="id.015.01.051.1.jpg" xlink:href="015/01/051/1.jpg"/><lb/>b, & &longs;umatur exempli gratia a quater cum b &longs;e<lb/>mel, & b quater cum a &longs;emel, dico, quod propor<lb/>tio (quam confu&longs;am e&longs;&longs;e liquet) aggregati primi ad &longs;ecundum, e&longs;t </s>

<s id="id000623"><arrow.to.target n="marg106"/><lb/>minor quàm quadrupla. </s>

<s id="id000624">Con&longs;tat enim quod proportio quadru<lb/>pli a ad a e&longs;t maior, quam b ad quadruplum b, cum una &longs;it quadru<lb/>pla, alia &longs;ub quadrupla, igitur per uige&longs;imam &longs;ecundam huius ag<lb/>gregati quadrupli a cum b &longs;emel, ad quadruplum b cum a &longs;emel mi<lb/><arrow.to.target n="marg107"/><lb/>nor, quàm quadrupli a ad a, & maior quàm b ad quadruplum b, & <lb/>e&longs;t pro intellectu Archimedis.</s>

<s id="id000626"><margin.target id="marg107"/>I<emph type="italics"/>n<emph.end type="italics"/> 2. <emph type="italics"/>lib.

de<emph.end type="italics"/><lb/>A<emph type="italics"/>tqui pon<lb/>deran.<emph.end type="italics"/><lb/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 10.</s>

<s id="id000627">Propo&longs;itio quadrage&longs;ima &longs;ecunda.</s>

<s id="id000628">Trahentium nauim, ut ferentium pondera proportiones in &longs;e in<lb/>uicem, quomodo ducere oporteat con&longs;iderare.<lb/><arrow.to.target n="marg108"/></s>

<s id="id000630">Hoc quomodo non po&longs;sit fieri &longs;uprà docuimus, nunc etiam ge</s><s id="id000631"><arrow.to.target n="marg109"/><lb/>neraliter dicam, cum con&longs;i&longs;tant hæc in duobus terminis, productio <lb/>uerò præ&longs;upponit quatuor terminos, ut in prima propo&longs;itione, aut <lb/>&longs;altem tres, atque in his medius habet rationem mouentis, & moti, <lb/>ergo cum in huiu&longs;modi <expan abbr="nõ">non</expan> &longs;int quatuor termini, nec tres, è quibus <lb/>unus &longs;it mouens, & motum proportio non poterit produci. </s>

<s id="id000632">Illud <lb/>etiam patet exemplo, nam &longs;i e&longs;&longs;et lapis, aut nauis ob&longs;i&longs;tens ut &longs;ex, & <lb/>e&longs;&longs;ent homines uiribus &longs;inguli, ut quatuor cum dimidio, tres mo<lb/>uerent in proportione dupla &longs;exquiquarta perdicta &longs;uperius eo<lb/>dem loco, at &longs;i proportio duci po&longs;&longs;et aliquorum hominum nume<lb/>rus po&longs;&longs;et mouere in duplicata proportione ad unguem &longs;cilicet <lb/>5 1/16 ut e&longs;&longs;et uix hominum collectorum 30 3/8 at nullus e&longs;t numerus ho<lb/>minum qui collectus faciat hunc numerum, nam &longs;ex homines ex<lb/>plent numerum 27, & &longs;eptem 31 1/2, & ideò non pote&longs;t duci propor<lb/>tio. </s>

<s id="id000633">Et ideò maximus e&longs;t error dicendo decem homines mouent na <lb/>uim proportione tripla, ergo triginta alij additis illis &longs;imiles robo<lb/>re mouebunt à proportione uiginti &longs;eptupla &longs;cilicet ducta nonu<pb pagenum="33" xlink:href="015/01/052.jpg"/>pla in triplam. </s>

<s id="id000634">Sed &longs;umpta proportione alio modo producitur. </s>

<s id="id000635">Ve<lb/>lut &longs;i dicam, homines decem mouent nauim, aut <expan abbr="fer&utilde;t">ferunt</expan> pondus pro<lb/>portione tripla, igitur quadraginta homines idem facient propor<lb/>tione duodecupla &longs;cilicet quadrupla in triplam ducta. </s>

<s id="id000636">Cum ergo <lb/>addo triginta homines, qui mouent in proportione nonupla, non <lb/>oportet ducere nonuplam in triplam, &longs;ed totum numerum accipe<lb/>re, & quam proportionem habet ad partem, tandem habet uis mo<lb/>uens ad uim <expan abbr="mou&etilde;tem">mouentem</expan>. </s>

<s id="id000637">Vnde &longs;i duo moueant in proportione &longs;ex<lb/>quialtera, & &longs;ex in proportione quadrupla cum dimidia, & iungan<lb/>tur, ut fiant octo, non oportebit ducere &longs;exquialteram, in quadru<lb/>plam &longs;exquialteram, &longs;ed cum octo ad duo &longs;it in proportione qua<lb/>drupla, &longs;umemus quadruplam ad &longs;exquialteram, qu&etail; erit &longs;excupla, <lb/>& octo mouebunt, aut pondus gerentin proportione &longs;excupla.</s>

<s id="id000638"><margin.target id="marg109"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 37.</s>

<s id="id000639">Propo&longs;itio quadrage&longs;ima tertia.</s>

<s id="id000640">Productionem ad additionem retrahere.<lb/><arrow.to.target n="marg110"/></s>

<s id="id000642">Sit proportio a ad b dupla pote&longs;tate li<lb/>cet &longs;int quinque homines, & &longs;int quindecim <lb/>homines c, & habebunt ad b &longs;excuplam <lb/>proportionem per præcedentem. </s>

<s id="id000643">Iuncta <lb/>ergo a, & c per octauam huius <expan abbr="moueb&utilde;t">mouebunt</expan> <lb/>b proportione octupla, dico, quod &longs;i du<lb/>xeris <expan abbr="proportion&etilde;">proportionem</expan> c ad a plus uno. </s>

<s id="id000645">qua<lb/>druplam in proportionem a ad b, quæ e&longs;t dupla, proueniet eadem <lb/>octupla. </s>

<s id="id000646">Nam quia in coniunctione &longs;ufficit iungere c cum a, & &longs;u<lb/>mitur &longs;ecundum proportionem a ad b, igitur cum proportio a ad <lb/>b comparata ad proportionem c & a ad b &longs;it, &longs;icut proportio c & a <lb/>ad a, & proportio c & a ad a &longs;it, &longs;icut proportio c ad a, & a ad a, & <lb/>proportio a ad a habet rationem unius, igitur proportio aggregati <lb/>c a ad b e&longs;t producta ex proportione c ad a plus monade in propor<lb/>tionem a ad b, quod erat demon&longs;trandum.</s>

<s id="id000647">Propo&longs;itio quadrage&longs;ima quarta.</s>

<s id="id000648">Si fuerit proportio motoris ad id, quod e&longs;t maximum non mo<lb/>uens & &longs;patium, & tempus, nota erit etiam reliquorum nota.</s>

<s id="id000649">Sæpe contingit, ut quinque homines moueant nauim, & &longs;patium <lb/>ad tempus notum, & etiam cognitum maximum, quod mouere <lb/>non pote&longs;t. </s>

<s id="id000650">Sit ergo a numerus hominum, b na<lb/><figure id="id.015.01.052.2.jpg" xlink:href="015/01/052/2.jpg"/><lb/>uis, c maximum, quod non mouere pote&longs;t, d <lb/>tempus, e &longs;patium, f motor alius &longs;iue numerus <lb/>hominum notus, & g tempus, dico, quod h &longs;patium notum erit, &longs;eu <lb/><expan abbr="not&utilde;">notum</expan> g tempus, & h &longs;patium, dico, quod erit f motor, &longs;eu numerus

<pb pagenum="34" xlink:href="015/01/053.jpg"/>hominum notus. </s>

<s id="id000651">Quoniam ergo notum e&longs;t a & c, quia e&longs;t æquale <lb/>b, igitur proportio a ad b nota e&longs;t: &longs;ed iuxta illam a mouet b in d <lb/>tempore per e &longs;patium, igitur per præcedentem, ut f ad a ita &longs;patij <lb/>ad e in d tempore. </s>

<s id="id000652">Sed per eadem ut temporis d ad &longs;patium illud, <lb/>ita g ad h, ergo cum nota &longs;int d e f g erit etiam h, & ita conuertendo.</s>

<s id="id000653">Propo&longs;itio quadrage&longs;ima quinta.</s>

<s id="id000654">Rationem &longs;tateræ o&longs;tendere.<lb/><arrow.to.target n="marg111"/></s>

<s id="id000656">Archimedes nititur huic fundamento, quod pondera, quæ pro<lb/>portionem mutuam habent, ut di&longs;tantiæ à libella a, quæ &longs;u&longs;pen<lb/>duntur, æqualiter ponderant, &longs;it ergo libella a b, & &longs;u&longs;pen&longs;a in a cen<lb/>trum mundi c, ad quod dirigitur pondus, & liquet, quod ip&longs;um <lb/>non &longs;e inclinabit ex uige&longs;ima tertia propo&longs;itione. </s>

<s id="id000657">Si ergo ponantur <lb/>lo co lineæ b d in e & f, & &longs;it proportio e b <lb/><figure id="id.015.01.053.1.jpg" xlink:href="015/01/053/1.jpg"/><lb/>ad b f, ut g ad h, dico, quòd erit æquili<lb/>brium, per eandem enim h mouebitur in k, <lb/>&longs;cilicet ut perueniat in rectam a d, &longs;i enim <lb/>non e&longs;&longs;et | &longs;u&longs;pen&longs;um h, moueretur in re<lb/>cta e h per eandem, quia ergo retinetur, mo<lb/>uetur per obliquam h k, & &longs;umatur in pro<lb/>pin quum punctum in b e, & n in æquali di<lb/>&longs;tantia in e f, quia ergo e b totum mouetur <lb/>eadem ui in &longs;ingulis partibus, quia a pon<lb/>dere h, & in h mouetur per h k in m per m <lb/>p, ergo qualis e&longs;t proportio magnitudinis h k ad m p, talis e&longs;t uis <lb/>in m p ad uim in h k, & ita in b erit penè infinita: quia quanta ui ex<lb/>tenditur ex h in k tanta puncta b, &longs;e circumuertit ergo propor<lb/>tio hypomochlij ad &longs;patium, uelut roboris ad robur, at eadem n o <lb/>ad h k, e&longs;t enim n o æqualis m p, & n b, & b m æquales, ut uerò g ad <lb/>h, ita e b ad b f: ergo ut e b ad b f, ita uirium n o ad h k, ut igitur g ad <lb/>h, ita uirium m p ad h k: ut etiam g l ad n o, ita uirium f b ad n b. <lb/></s>

<s id="id000658">nam idem pondus &longs;cilicet g mouet totam b f, igitur ut g &longs;e habet </s>

<s id="id000659"><arrow.to.target n="marg112"/><lb/>ad n o, ita h ad m p, &longs;ed m p & n o &longs;unt æquales, ergo tanta e&longs;t uis g <lb/>in f, quanta h in e.<lb/><arrow.to.target n="marg113"/></s>

<s id="id000662">Ex quo patet, quod hypomochlion moueretur infinita ui, &longs;i po&longs;<lb/>&longs;et e&longs;&longs;e punctus: &longs;ed quia in extrema &longs;uperficie cylindri, ideò pote&longs;t <lb/>aliqua ui retineri.</s>

<s id="id000665">Et &longs;i quis po&longs;&longs;et capere ha&longs;tam in extremo puncto, non po&longs;&longs;et <lb/>eam mouere, etiam quod haberet robur infinitum, quia ab æquali <lb/>non fit motus per trige&longs;imam nonam propo&longs;itionem.</s>

<s id="id000668">Et libella nihil retinet ni&longs;i quantum e&longs;t pondus eius quod cu<pb pagenum="35" xlink:href="015/01/054.jpg"/>pit ad centrum peruenire, & pondus ei appen&longs;um non prohi<lb/>bet motum, etiam &longs;i e&longs;&longs;et infinitum, ni&longs;i quatenus non uult recede<lb/>re ex directo centri mundi: & ut grauat hypomochlion faciens im<lb/>pre&longs;sionem.</s>

<s id="id000671">Et &longs;i terra tota e&longs;&longs;et appen&longs;a polo, moueretur magna ui: quoni<lb/>am uis eadem e&longs;t in polo, quæ in circulo toto æquinoctij.</s>

<s id="id000674">Et rota, quanto uelocius mouetur in ambitu, tanto mi<lb/>norem habet uim: &longs;ed propter aërem, qui &longs;ecum circum<lb/><figure id="id.015.01.054.1.jpg" xlink:href="015/01/054/1.jpg"/><lb/>fertur, mouetur magno impetu, & magnas facit læ&longs;iones. <lb/></s>

<s id="id000675">Ideò hoc in cono non accidit.</s>

<s id="id000678">Ex quo patet ratio eleuandi pondera magna per tra<lb/>bem, ut à latere uides.</s>

<s id="id000679">Propo&longs;itio quadrage&longs;ima &longs;exta.</s>

<s id="id000680">An &longs;it aliqua proportio, & qualis inter animam, & ui<lb/>tas, & &longs;ua corpora con&longs;iderare.</s>

<s id="id000683">Declarauimus motum c&oelig;li e&longs;&longs;e uoluntarium, ob&longs;equente c&oelig;<lb/>lo per uirtutem in eo infu&longs;am. </s>

<s id="id000684">In animalibus autem, & præcipuè <lb/>in homine notius e&longs;t hoc experientibus nobis in ip&longs;is: &longs;ed motus <lb/>hic, ut dixi &longs;upra, mi&longs;tus e&longs;t, ille uerò c&oelig;le&longs;tis ignotior e&longs;t. </s>

<s id="id000685">Certum </s>

<s id="id000686"><arrow.to.target n="marg120"/><lb/>tamen e&longs;t plenè ob&longs;equi c&oelig;lum uitæ, nec pror&longs;us repugnare. </s>

<s id="id000687">So<lb/>let Ari&longs;toteli imponi, quòd &longs;i adderetur a&longs;trum c&oelig;lo, quòd c&oelig;lum <lb/>aut quie&longs;ceret, aut tardius moueretur: quod e&longs;t, ac &longs;i diceremus, <lb/>quòd homo paruus &longs;i fieret maior, non e&longs;&longs;et adeò agilis, tanquam <lb/>motus ille e&longs;&longs;et ab externa cau&longs;a. </s>

<s id="id000688">Imò perinde e&longs;&longs;et, ac &longs;i quis dice<lb/>ret, quod lapides magni minus uelociter de&longs;cenderent, quam par<lb/>ui. </s>

<s id="id000689">Quin potius ut lapis magnus uelociùs mouetur: quàm par<lb/>uus naturali motu, & tardius præternaturali, ita c&oelig;lum motu uo<lb/>luntario, &longs;i ita dici po&longs;&longs;et æqualius & maiore cum efficacia, quan<lb/>to den&longs;ius. </s>

<s id="id000690">Et ita &longs;i Ari&longs;toteles illud dixi&longs;&longs;et, o&longs;tendi&longs;&longs;et magnam <lb/>imperitiam. </s>

<s id="id000691">Ideò quale iudicium debemus facere de Alexandro, & <lb/><arrow.to.target n="marg121"/><lb/>Aueroe, qui hoc ei tribuunt. </s>

<s id="id000692"><expan abbr="legi&ttilde;">legitur</expan> enim in textu Arabico tale quip<lb/>piam. </s>

<s id="id000693">De Animalibus for&longs;an po&longs;&longs;et hoc dici, <expan abbr="quoniã">quoniam</expan>, ut &longs;uprà dixi<lb/>mus, motus ille mi&longs;tus e&longs;t. </s>

<s id="id000694">Remanet ergo difficultas, <expan abbr="quoniã">quoniam</expan> &longs;i mo<lb/>tus i&longs;te non à proportione fit, quare non e&longs;t infinitus? </s>

<s id="id000695">& dico quae in <lb/>animalibus tres &longs;unt cau&longs;æ, una, quia e&longs;t mi&longs;tus, & habet repugnan<lb/>tiam: &longs;ecunda, quia e&longs;t de loco ad locum, motus autem c&oelig;li e&longs;t in lo<lb/>co: tertia e&longs;t communis etiam c&oelig;lo, et e&longs;t, <expan abbr="quoniã">quoniam</expan> non e&longs;t ratio finis. <lb/></s>

<s id="id000696">Natura enim diuina non appetit mouere <expan abbr="tã">tam</expan> celeriter. </s>

<s id="id000697">Quid e&longs;t ergo <lb/>proportio, <expan abbr="c&utilde;">cum</expan> &longs;it <expan abbr="ultim&utilde;">ultimum</expan> uoluntatis uit&etail;, ut obtemperet primæ cau&longs;æ, <lb/>ideo illud e&longs;t <expan abbr="ultim&utilde;">ultimum</expan>, &qring; mouet. </s>

<s id="id000698">E&longs;t <expan abbr="a&utilde;t">aut</expan> idem uelle, & po&longs;&longs;e. </s>

<s id="id000699">In natura

<pb pagenum="46 [=36]" xlink:href="015/01/055.jpg"/>enim c&oelig;li e&longs;t ille appetitus, cuius principium e&longs;t uita: & eíus uolun<lb/>tatis bonum ip&longs;um. </s>

<s id="id000700">Et ideo hæc proportio <expan abbr="nõ">non</expan> diuiditur. </s>

<s id="id000701">In anima<lb/>libus autem non e&longs;t uis illa ni&longs;i, cum proportione, quia primum in<lb/>&longs;trumentum, quod recipit, & e&longs;t &longs;piritus uim habet determinatam, <lb/>cum &longs;it uirtus in materia: ideo <expan abbr="nõ">non</expan> mouet ni&longs;i cum certa proportio<lb/>ne, uelut lumen in medio in &longs;e non habet proportionem ni&longs;i ad lu<lb/>cem, &longs;ed ut e&longs;t in illo, pote&longs;t e&longs;&longs;e remi&longs;&longs;um, <expan abbr="ob&longs;cur&utilde;">ob&longs;curum</expan> & hebes. </s>

<s id="id000702">Quæ<lb/>ritur ergo quantitas illius? </s>

<s id="id000703">&longs;i dicas, quòd e&longs;t à luce: quæro quanti<lb/>tas lucis, unde &longs;it? </s>

<s id="id000704">for&longs;an dicendum, quòd uelutin motibus, quanto <lb/>den&longs;iora &longs;unt corpora tanto <expan abbr="mouen&ttilde;">mouentur</expan> maiore nixu, & robore. </s>

<s id="id000705">Nam <lb/>calor in materia augetur iuxta illius quantitatem: idem in luce, & <lb/>reliquis. </s>

<s id="id000706">Dico ergo proportionem e&longs;&longs;e infinitam: nam &longs;i corpus e&longs;<lb/>&longs;et infinitum & optimè di&longs;po&longs;itum infinita ui moueretur & agili<lb/>tate, ut enim maius e&longs;t eo maiores uires habet.</s>

<s id="id000707"><margin.target id="marg120"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 27.</s>

<s id="id000708"><margin.target id="marg121"/>T<emph type="italics"/>ex.<emph.end type="italics"/> 71. <lb/>2. <emph type="italics"/>de<emph.end type="italics"/> C<emph type="italics"/>&oelig;lo.<emph.end type="italics"/></s>

<s id="id000709">Propo&longs;itio quadrage&longs;ima &longs;eptima.</s>

<s id="id000710">Si duo mobilia æqualiter in eodem circulo iuxta proprios mo<lb/>tus moueantur, productum temporis circuituum inuicem erit æ<lb/>quale producto differentiæ temporum circuitus ductæ in tempus <lb/>coniunctionis primæ.<lb/><arrow.to.target n="marg122"/></s>

<s id="id000712">Sint duo mobilia a & b in eodem pun<lb/><figure id="id.015.01.055.1.jpg" xlink:href="015/01/055/1.jpg"/><lb/>cto, quæ æqualiter uer&longs;us eandem partem <lb/>moueantur æqualibus in temporibus, inui<lb/>cem tamen in æqualiter, ita quod a in f & b <lb/>in g temporibus ab&longs;oluant circulum, & ho<lb/>rum differentia &longs;it h. </s>

<s id="id000713">Dum itaque a perficit <lb/>circulum b perueniat in c, igitur c d b e&longs;t dif<lb/>ferentia, quæ &longs;uperanda e&longs;t, & proportio <lb/>circuli ad b c ut g ad f, quare reliqui ad reli<lb/>quum, ut re&longs;idui ad re&longs;iduum, &longs;cilicet circu<lb/>li ad c d b, ut g ad h, & b c ad c d b ut f ad h, coniungantur igitur in k <lb/>tempore, eruntque k f g h omiologa, ut productum ex circulo in b c <lb/>diui&longs;o per certam quantitatem & cum circulo & b c & c d b diffe<lb/>rentia, & &longs;it &longs; productum ex f in g, dico quod diui&longs;a &longs; per h exibit k <lb/>tempus coniunctionis primæ, &longs;it itaque d locus coniunctionis, dico <lb/>igitur quod differentia &longs;patij pertran&longs;iti a b, a & a, b in reditu ex con<lb/>iunctione prima ad d e&longs;t unus circulus completus, non enim po&longs;<lb/>&longs;unt e&longs;&longs;e plures, nam &longs;equeretur, quòd a aliquando pertran&longs;i&longs;&longs;et b, <lb/>et &longs;ic non e&longs;&longs;et prima coniunctio, nec pote&longs;t e&longs;&longs;e minus, nam &longs;ic cum <lb/>a & b &longs;int in d ultra perfectas circulationes uterque eorum pertran<lb/>&longs;iuit arcum b c, igitur nullo modo differentia pote&longs;t e&longs;&longs;e minor cir<lb/>culo, neque maior, ut declaratum e&longs;t, igitur e&longs;t unus circulus ad un<pb pagenum="37" xlink:href="015/01/056.jpg"/>guem. </s>

<s id="id000714">Hoch declarato ponatur m spatium compositum ex circulis <lb/>pertran&longs;itis a b a cum &longs;patio b d, etenim &longs;patium, quod pertran&longs;it <lb/>b a coniunctione in a, ad coniunctionem primam in d, & erit ex de<lb/>mon&longs;tratis horum differentia circulus qui uocetur o, & &longs;it p &longs;pa<lb/>tium, quod pertran&longs;it b in tempore eodem, in quo a pertran&longs;it o, & <lb/>&longs;it q differentia o, & p qu&etail; in circulo e&longs;t c d l b, quia igitur in eodem <lb/>tempore a pertran&longs;it m & b, n, erit m ad n, ut a ad b, & eadem ratio<lb/>ne a ad b, ut o ad p, igitur ex undecima quinti Euclidis m ad n, ut o <lb/>ad p, quare cum o &longs;it differentia m & n, & q, differentia o & p erit ex <lb/>decima nona quinti Euclidis, m ad o, ut o ad q, & ita circulus e&longs;t ana<lb/>logus inter &longs;patium pertran&longs;itum à motore uelociori, & inter diffe<lb/>rentiam &longs;patij quæ accidit, dum uelocior motor pertran&longs;it circu<lb/>lum, id e&longs;t quòd circulus a c d e&longs;t analogus inter c d l b, & circulos <lb/>pertran&longs;itos a b a cum portione b d. </s>

<s id="id000715">Reuertor igitur ad propo&longs;i<lb/>tum, cum &longs;it m ad o, ut o ad q, & m ad o, ut n ad p, ex &longs;exta decima <lb/>quinti Euclidis, erit ex undecima eiu&longs;dem n ad p, ut o ad q, quare ex <lb/>&longs;exta decima &longs;exti Elementorum ducto o, id e&longs;t circulo, &longs;eu maiore <lb/>numero in p &longs;patium pertran&longs;itum a b, &longs;eu ducto fin g, & diui&longs;o per <lb/>q differentiam &longs;patiorum, &longs;eu per h exibit n, &longs;eu &longs;patium quod <lb/>pertran&longs;it b ab una coniunctione ad aliam quod erat demon<lb/>&longs;trandum.</s>

<s id="id000718">Ex hoc patet, quod proportio temporis coniunctionis ad tem<lb/>pus tardioris motus circuitionis e&longs;t ueluti temporis circuitus uelo<lb/>cioris motoris ad differentiam temporis motus tardioris, & uelo<lb/>cioris motoris in uno circuitu.</s>

<s id="id000719">Propo&longs;itio quadrage&longs;ima octaua.</s>

<s id="id000720">Si tria mobilia ex eodem puncto di&longs;cedant, fuerintque duorum, ac <lb/>duorum coniunctiones in temporibus commen&longs;is illa tria mobi<lb/>lia denuò coniungentur in tempore producto ex denominatore di <lb/>ui&longs;ionis temporis maioris per minus in minus, aut numeratore <lb/>in maius.</s>

<s id="id000723">Sint tria mobilia a, quod circuat in duobus annis b in quinque, <lb/>c in &longs;eptem. </s>

<s id="id000724">Dico quod primum redibunt in numero producto ex <lb/>&longs;eptem quinque & duobus, qui &longs;unt numeri primi, & erit ille nume<lb/>rus &longs;eptuaginta annorum. </s>

<s id="id000725">Nam in &longs;eptuaginta annis a perficiet tri<lb/>ginta quinque reuolutiones b quatuordecim, c decem: ergo <expan abbr="redib&utilde;t">redibunt</expan> <lb/>per perfectos circuitus ad idem punctum. </s>

<s id="id000726">O&longs;tendo modo quod <lb/>non ante: nam &longs;i &longs;ic: &longs;it, ut in triginta quinque annis igitur b & c per<lb/>ficient perfectos circuitus, ergo <expan abbr="redib&utilde;t">redibunt</expan> ad idem punctum, a autem <lb/>non redibit, quoniam eius circuitus non numerat triginta quinque <lb/>aliter non fui&longs;&longs;et &longs;eptuaginta minimus numeratus ab a b c, cum

<pb pagenum="38" xlink:href="015/01/057.jpg"/>ergo iam &longs;upponatur numerari a b & c non numerabitur a b a, er<lb/>go a non perficiet circuitus, ergo non redibit ad primum <expan abbr="loc&utilde;">locum</expan>, ergo <lb/>non erit iunctus cum b & c. </s>

<s id="id000727">Quod &longs;i dicas a b c coniungi in decem <lb/>&longs;eptem annis numero non numerato ab ali <lb/><figure id="id.015.01.057.1.jpg" xlink:href="015/01/057/1.jpg"/><lb/>quo illorum temporum, auferantur perfe<lb/>ctæ circulationes, & <expan abbr="remaneb&utilde;t">remanebunt</expan> dimidium <lb/>ex a, duæ quintæ ex b, tres &longs;eptimæ ex c, igi<lb/>tur oportebit ut hæ portiones &longs;int æqua<lb/>les, ut po&longs;t perfectas circulationes in idem <lb/>punctum, <expan abbr="cõueniant">conueniant</expan>, ergo 1/2 & 2/5 & 3/7 æqui<lb/>ualebunt, quare proportio 7 ad 3 & 5 ad 2 <lb/>& 2 ad 1, e&longs;t una, quare permutando 3 ad 2 <lb/>ut 7 ad 5, &longs;ed 7 & 5 &longs;unt contra &longs;e primi, ergo in &longs;ua proportione mi <lb/>nimi per dicta in &longs;eptimo Elementorum: ergo tria, & duo non &longs;unt <lb/>in eadem proportione. </s>

<s id="id000728">Rur&longs;us dicantur conuenire in annis qua</s><s id="id000729"><arrow.to.target n="marg125"/><lb/>tuordecim cum dimidio, ergo in uiginti nouem conuenient ite<lb/>rum: ergo per &longs;ecundam partem erit &longs;eptem ad unum, ut duo ad <lb/>unum, igitur permutando unius ad unum, ut &longs;eptem ad duo, &longs;ed <lb/>unum e&longs;t æquale uni, ergo duo erunt æqualia &longs;eptem. </s>

<s id="id000730">Rur&longs;us dica<lb/>mus, quod in tempore annorum <02> quadrata decem &longs;imiliter aufe<lb/>ram integras reuolutiones, quas potero, & erunt <02> 2 1/2 m: 1, & <02> 2/5 & <lb/><02> 10/49 æqualia. </s>

<s id="id000731">Hic uides infinita &longs;equi in conuenientia, quæ longum <lb/>e&longs;&longs;et numerare, nam &longs;eptem e&longs;&longs;et æquale quinque, & proportio reci&longs;i <lb/>ad potentia rethe, ut numeri ad numerum. </s>

<s id="id000732">Igitur non conueniunt <lb/>ante &longs;eptuaginta annos.<lb/><arrow.to.target n="marg126"/></s>

<s id="id000733"><margin.target id="marg125"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 23</s>

<s id="id000735">Ex hoc &longs;equitur, quòd nullibi conuenient præterquàm in eo<lb/>dem puncto, &longs;cilicet in quo ab initio coniuncti fuerunt.</s>

<s id="id000739">Sequitur denuo ex propo&longs;itione ip&longs;a repetita, & primo corrola<lb/>rio, quod nullibi alibi conuenient quàm in dato primo puncto, in <lb/>quo coniuncti fuerant ab initio etiam u&longs;que in æternum.</s>

<s id="id000740">Sit rur&longs;us ut a circuat in annis duobus cum dimidio, b in tribus <lb/>cum tertia parte, cin quatuor cum quarta parte ducam per &longs;uos <lb/>denominatores, & erit ut a in quinque annis. </s>

<s id="id000741">b in decem, c in decem<lb/>&longs;eptem circuant, & redeant ad idem punctum, & quia quin que nu<lb/>merat decem, & decem, & decem &longs;eptem &longs;unt numeri inuicem pri<lb/>mi, ducam decem in decem &longs;eptem fiunt centum &longs;eptuaginta. </s>

<s id="id000742">Con<lb/>&longs;tat igitur c quadragíes, b quinquagies &longs;emel, a &longs;exagies octies cir<lb/>cumuerti, & redire ad idem punctum: ergo rur&longs;us coibunt po&longs;t tot <lb/>annos in eo, dico modo, quod non ante: nam &longs;i non &longs;it, ut in trigin<lb/>ta tribus annis. </s>

<s id="id000743">gratia exempli, aufero <expan abbr="decem&longs;ept&etilde;">decem &longs;eptem</expan>, decem, & quin<lb/>que, & relinquentur &longs;ex decim tria & tria, & rur&longs;us ex &longs;ex decim tres <pb pagenum="39" xlink:href="015/01/058.jpg"/>circuitus c, & relinquentur 3 3/4 &longs;equetur igitur, ut &longs;it proportio 17 ad <lb/>13, & 2 1/2 ad 1/2 & 3 1/3 ad 3 eadem, & ita 17/13, 5/2 & 10/9 eadem &longs;i iam &longs;upponi<lb/>mus 17 & 10 e&longs;&longs;e primos inuicem, ut in &longs;ecunda demon&longs;tratione./><lb/></s>

<s id="id000744">Igitur &longs;equuntur eadem corrolaria, quæ dicta &longs;unt.</s>

<s id="id000745">Propo&longs;itio quadrage&longs;ima nona.</s>

<s id="id000746">Propo&longs;ito mobilis in circulo circuitus tempore, dataque ratione <lb/>di&longs;tantiæ ab illo mobilis circuitum inuenire, quod ex eodem pun<lb/>cto di&longs;cedens cum alio mobili in dato puncto conueniat &longs;ub quo<lb/>cunque numero circuituum tempus quoque coniunctionis.</s>

<s id="id000749">Sit in circuli peripheria a <expan abbr="p&utilde;ctus">punctus</expan>, qui cir <lb/>cuat æquali motu (hoc enim &longs;emper intel<lb/>ligitur) in b tempore: & &longs;it datus punctus c <lb/>in quo di&longs;cedens e mobile ex coniunctio<lb/>ne cum a po&longs;t certos circuitus proprios, <lb/>aut etiam. </s>

<s id="id000750">&longs;ine ulla circuitione perfecta de<lb/>beat conuenire. </s>

<s id="id000751">Volo &longs;cire tempus circui<lb/>tionis e: & etiam tempus coniunctionis. <lb/></s>

<s id="id000752">Sit ergo primum ut ab&longs;que circuitione ulla e, a debeat comprehen<lb/>dere e in c po&longs;t numerum circuitionum ip&longs;ius a, qui &longs;it f. </s>

<s id="id000753">nam &longs;i a o c <lb/>currit e in prima circuitione ip&longs;ius e, igitur a mouetur uelocius <lb/>quàm e, cum ergo debeat attingere ip&longs;um e, nece&longs;&longs;e e&longs;t ut a pertran<lb/>&longs;eat prius per punctum ex quo di&longs;ce&longs;sit antequam redeat ad con<lb/>iunctionem e: ergo perficiet &longs;altem unam circuitionem. </s>

<s id="id000754">Ducemus <lb/>ergo f in b, & fiet g tempus circuitus aut circuituum a, & quia &longs;pa<lb/>tium a c datum e&longs;t, &longs;it b temporis circuitus a ad h, uelut circuli to<lb/><arrow.to.target n="marg129"/><lb/>tius ad a c, & iungatur g cum h & fiat k. </s>

<s id="id000755">Fiat quoque, ut monadis <lb/>ad h, ita l ad monadem, & ducatur l in k, & fiat m: dico m e&longs;&longs;e tem<lb/>pus circuitus e. </s>

<s id="id000756">Con&longs;tat enim ex &longs;uppo&longs;ito, quod k e&longs;t tempus to<lb/>tum in quo a peruenit po&longs;t b circuitiones in c, &longs;i ergo e moueretur <lb/>per m tempus totum ex &longs;uppo&longs;ito perficeret circuitum, at quia cir<lb/>cuitus ad a c, ut monadis ad h, igitur etiam ut l ad monadem, ergo <lb/>proportio circuitus ad a c, ut m ad monadem: ergo &longs;i in m tran&longs;it to<lb/>tum circuitum in monade tran&longs;it a c: &longs;ed monas ducta in k facit k, <lb/>igitur e in tempore k perueniet in c, quod erat demon&longs;trandum. <lb/></s>

<s id="id000757">Proponatur modo tempus reuolutionum e ip&longs;um d: eodem mo<lb/><arrow.to.target n="marg130"/><lb/>do agemus ducendo fin b fit g, addatur h & fiat k, diuidatur k per <lb/>aggregatum d & a e, & exeat m, (idem enim e&longs;t diuidere per aggre<lb/>gatum d & h, & multiplicare per l) dico ergo ut in demon&longs;tratione <lb/>priore, quod m e&longs;t tempus circuitus e. </s>

<s id="id000758">Nam cum k &longs;it tempus, in <lb/>quo a po&longs;t circuitus f peruenit ad c, ergo diui&longs;o ip&longs;o toto tempore

<pb pagenum="40" xlink:href="015/01/059.jpg"/>per numerum reuolutionum d, & partem reuolutionis exibit tem<lb/>pus unius reuolutionis.</s>

<s id="id000761">Exemplum primi in re paulò ob&longs;curiore: &longs;it f 4 & b 2 1/2 & a c 4/5, du<lb/>cemus 4 in 2 1/2 fit 10, adde 4/5 6 quod e&longs;t 2 fit 12, diuide per 4/5 &longs;eu mul<lb/>tiplica per 5/4 quod idem e&longs;t, fit 15 circuitus e, in quatuor ergo circui<lb/>tibus, & 4/5 qui &longs;unt duodecim anni perueniet a ad c, & in duodecim <lb/>annis e perueniet ad c, nam 12 &longs;unt 4/5 ip&longs;ius 15. Similiter in &longs;ecundo <lb/>ca&longs;u &longs;it f 4 ut prius b 2 1/3 a c 1/7, ducemus 4 in 2 1/3 fit 9 1/3, addemusque h <lb/>portionem b qualis a c e&longs;t totius circuitus, id e&longs;t 1/7, e&longs;t autem 1/7 2 1/3, 1/3 <lb/>fient 9 1/3, &longs;imiliter ponatur d 5, & quia a c e&longs;t 1/7 erunt 36/7, diuide ergo <lb/>9 2/3 id e&longs;t 29/3 per 36/7 exeunt 203/108 tempus reuolutionis e. </s>

<s id="id000762">Quin que ergo <lb/>reuolutiones e erunt 1015/108 addita &longs;eptima parte, quæ e&longs;t 29/108 fient 2044/108 <lb/>&longs;eu 261/27, & &longs;unt anni 9 18/27 &longs;eu 9 2/3, ergo in tanto tempore a faciet qua<lb/>tuor circuitus, & &longs;eptimam partem, & e quinque circuitus, & &longs;e<lb/>ptimam.<lb/><arrow.to.target n="marg131"/></s>

<s id="id000764">Ex hoc patet, quod non coniungentur in alio loco, neque alio tem<lb/>pore ante prædictum tempus.</s>

<s id="id000765">Propo&longs;itio quinquage&longs;ima.</s>

<s id="id000766">Omnes circuituum portiones in eiu&longs;dem temporibus <expan abbr="repetun&ttilde;">repetuntur</expan>.</s>

<s id="id000767">Sint in circulo a b c d e f g: a & b iuncta, & in primo congre&longs;&longs;u <lb/>iungantur in c, in &longs;ecundo in d, in tertio in e, in quarto in f, in quinto <lb/>in g, in &longs;exto in h, in &longs;eptimo in k, in octauo in l. </s>

<s id="id000768">Et &longs;ic deinceps <expan abbr="c&utilde;quetempora">cuique <lb/>tempora</expan> &longs;int æqualia, erunt & circuitus totidem numero, & exce&longs;<lb/>&longs;us æquales etiam a c, c d, d e, e f, f g, g h, h k, <lb/><figure id="id.015.01.059.1.jpg" xlink:href="015/01/059/1.jpg"/><lb/>k l. </s>

<s id="id000769">Et &longs;i aggregatum a &longs;cilicet circulorum, <lb/>& portionis fuerit commen&longs;um circulo, & <lb/>ita de b erunt omnia <expan abbr="cõmen&longs;a">commen&longs;a</expan> ad circulum, </s>

<s id="id000770"><arrow.to.target n="marg132"/><lb/>& etiam inter &longs;e. </s>

<s id="id000771">Et &longs;i inter &longs;e aggregata, uel <lb/>portiones erunt, & eodem modo reliqua. <lb/></s>

<s id="id000772">Et quoniam circuli circulis commen&longs;i &longs;unt: <lb/>&longs;i portiones erunt inuicem commen&longs;æ <expan abbr="er&utilde;t">erunt</expan>, <lb/>& toti circuitus cum partibus commen&longs;i, & <lb/>&longs;i non commen&longs;i, neque erunt inter &longs;e, neque ad circulum. </s>

<s id="id000773">Et &longs;i totum <lb/>&longs;patium cum circuitibus erit unius generis, erunt duplicata, & tri<lb/>plicata, & quadruplicata eiu&longs;dem generis: quare cum &longs;patia ip&longs;a <lb/>detractis circuitibus uelut rhete habeant naturam reci&longs;i, & &longs;patia <lb/>ip&longs;a tota &longs;int eiu&longs;dem generis, erunt &longs;patia, quæ relinquuntur eiu&longs;<lb/>dem generis. </s>

<s id="id000774">Erunt tamen incommen&longs;a nece&longs;&longs;ariò, &longs;i partes fuerint <lb/>incommen&longs;æ toti. </s>

<s id="id000775">Ponatur a c incommen&longs;a toti circulo dico, quod <lb/>a k <expan abbr="etiã">etiam</expan> e&longs;t incommen&longs;a toti circulo: & <expan abbr="etiã">etiam</expan> a k, & k c. </s>

<s id="id000776">Quia enim a c <lb/>e&longs;t incommen&longs;a circulo, & k a cum toto circulo &longs;emel e&longs;t commen<pb pagenum="41" xlink:href="015/01/060.jpg"/>&longs;a a c, quia multiplex ei. </s>

<s id="id000777">igitur cum circulus, & a k diuidantur in cir<lb/><arrow.to.target n="marg133"/><lb/>culum et a k, & circulus &longs;it incommen&longs;us circulo, cum a k erit aggre<lb/></s><s id="id000778">gatum ex circulo, & a k incommen&longs;um ip&longs;i a k, & a k pariter incom<lb/><arrow.to.target n="marg134"/><lb/>men&longs;a circulo. </s>

<s id="id000779">Rur&longs;us quia a k e&longs;t incommen&longs;a circulo cum a k, & <lb/>circulus cum a k &longs;it multiplex ad a c, erit a k incommen&longs;a a c, quare <lb/><arrow.to.target n="marg135"/><lb/>erit c k incommen&longs;a a k & a c, & circulo ad dita a k. </s>

<s id="id000780">Si ergo a c &longs;it <lb/>commen&longs;a circulo, erunt omnes portiones e genere numeri, & &longs;i <lb/><arrow.to.target n="marg136"/><lb/>potentia rhete erunt omnes, uel potentia rhete, uel circulis detra<lb/>ctis, ut a k & a l reci&longs;a: & a c &longs;it potentia &longs;ecunda rhete, id e&longs;t radix cu<lb/>bica erunt omnes c d, d e, e f, potentia &longs;ecunda rhete, et radices cubi<lb/>cæ numeri, &longs;eu latera corporum rhete, a k uero & a l, & huiu&longs;modi <lb/>in infinitum reci&longs;a potentia rhete.<lb/><arrow.to.target n="marg137"/></s>

<s id="id000781"><margin.target id="marg132"/>P<emph type="italics"/>er<emph.end type="italics"/> C<emph type="italics"/>or<emph.end type="italics"/>^{m}. <lb/><emph type="italics"/>præcedentis.<emph.end type="italics"/></s>

<s id="id000782"><margin.target id="marg133"/>P<emph type="italics"/>er<emph.end type="italics"/> 14. <emph type="italics"/>deci <lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s>

<s id="id000783"><margin.target id="marg134"/>P<emph type="italics"/>er<emph.end type="italics"/> 17. <lb/><emph type="italics"/>eiu&longs;dem.<emph.end type="italics"/></s>

<s id="id000784"><margin.target id="marg135"/>P<emph type="italics"/>er<emph.end type="italics"/> 14. <lb/><emph type="italics"/>rur&longs;us.<emph.end type="italics"/></s>

<s id="id000785"><margin.target id="marg136"/>P<emph type="italics"/>er<emph.end type="italics"/> 17. <lb/><emph type="italics"/>rur&longs;us.<emph.end type="italics"/></s>

<s id="id000787">Ex hoc patet, quod cum circulus po&longs;sit diuidi in infinita gene</s>

<s id="id000788"><arrow.to.target n="marg138"/><lb/>ra quantitatum, quæ non &longs;unt inuicem commen&longs;æ cumque coniun<lb/>ctiones hæ &longs;emper in eodem genere maneant, quod infinita pun<lb/>cta, & infinitis in &longs;peciebus quantitatum remanebunt in quibus a <lb/>& b in perpetuum nunquam conuenient. </s>

<s id="id000789">Velut &longs;i coniunctio pri<lb/>ma fiat in <02> cu. </s>

<s id="id000790">1/2 alicuius circuli, nunquam conuenient, neque in me<lb/>dietate, neque in quarta parte, nec octaua, nec tertia, nec &longs;exta, nec no<lb/>na, nec quinta, nec decima, & &longs;ic de &longs;ingulis in genere commen&longs;a<lb/>rum toti circulo. </s>

<s id="id000791">Neque in <02> quadrata 1/2 uel 1/3 uel 1/5 neque <02> 1/6 uel 1/20, <lb/>neque in <02> 3 m: 1, nec 2 m: <02> 3 nec in <02> <02> 2 aut 3 aut 7 nec in <02> rela<lb/>ta alicuius numeri, nec in 2 m: <02> <02> cub. </s>

<s id="id000793">4, & &longs;ic <lb/>de alijs.</s>

<s id="id000794"><margin.target id="marg138"/>P<emph type="italics"/>er penulti<lb/>mam uige&longs;i<lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s>

<s id="id000795">Propo&longs;itio quinquage&longs;ima prima.</s>

<s id="id000796">Operationes dictas exemplo declarare.<lb/><arrow.to.target n="marg139"/></s>

<s id="id000798">Supponamus in circulo prædicto a c <02> 7 con&longs;tat, quod e&longs;&longs;e non <lb/>pote&longs;t, quia <02> 7 e&longs;t maior monade, ideo toto circulo, quare non po<lb/>terit e&longs;&longs;e pars circuli, &longs;ed referetur ad <expan abbr="quantitat&etilde;">quantitatem</expan> certam, uelut quod <lb/>circulus &longs;it 10. &longs;emper ergo diuidemus <02> 7, &longs;eu eam portionem per <lb/>10 quantitatem circuli & exibit <02> 7/100, & hæc erit portio circuli, & ita <lb/>&longs;i portio &longs;it <02> cub. </s>

<s id="id000799">16, diuidemus <02> cub. </s>

<s id="id000800">16 per 10 exibit <02> cu 2/125, & <lb/>ita de alijs.</s>

<s id="id000801">Sed cum ex repetitione cre&longs;cat portio illa, donec exuperet mo<lb/>nadem, aut aliquem quemuis numerum detracta monade aut nu<lb/>mero circuituum habebit rationem reci&longs;i. </s>

<s id="id000802">Velut <02> 7/100 quater &longs;um<lb/>pta efficit <02> 112/100. Et hoc e&longs;t potentia rhete, &longs;ed &longs;i quis auferat mona<lb/>dem fiet <02> 112/100 m: 1, & hoc e&longs;t reci&longs;um 1, &longs;cilicet 1 p: <02> v: 23/25 m: <02> 28/25, &longs;ed ta<lb/>men uerè e&longs;t linea media.</s>

<s id="id000803">Quod uerò non contingat coniungi in alio loco, neque tem<lb/>pore &longs;it, ut a b iungantur in c, & &longs;it reuolutio a triplex integra, & b

<pb pagenum="42" xlink:href="015/01/061.jpg"/>&longs;excuplex, & tempus totum decem annorum: ita ut a c &longs;it tertia <lb/>pars circuitus, & a circuitus tres anni, & quia circuitus b &longs;unt &longs;ex <lb/>cum tertia, diuidemus decem per 6 1/3 exit <lb/>1 11/29, dico quod non prius, neque in alio <lb/><figure id="id.015.01.061.1.jpg" xlink:href="015/01/061/1.jpg"/><lb/>puncto. </s>

<s id="id000804">Si enim primùm in eodem pun<lb/>cto, &, gratia exempli, in quatuor annis <lb/>congruit enim, & b dicamus quod per<lb/>egerit duas reuolutiones cum tertia, hoc <lb/>enim e&longs;t nece&longs;&longs;arium, &longs;i debet perueni<lb/>re ad c, & erunt anni tres, & 23/19, non ergo <lb/>anni quatuor. </s>

<s id="id000805">Cum enim tempora di<lb/>uer&longs;a diuiduntur per numeros haben<lb/>tes proportionem erunt, qui prodeunt <lb/><arrow.to.target n="table13"/><lb/>numeri in eadem ratione. </s>

<s id="id000806">Diui&longs;o ergo <lb/>10 per 1 11/19 exit 6 2/3, & diui&longs;o 4 per 1 11/19 exit <lb/>2 8/15, igitur 6 1/3 ad 2 8/15, ut 10 ad 4, igitur 8/25 <lb/>non pote&longs;t e&longs;&longs;e æquale 1/3. Si enim per <lb/>præcedentem repetuntur, ergo non po&longs;<lb/>&longs;unt redire, donec iterum coniungantur in ip&longs;o a. </s>

<s id="id000807">Si enim aliter &longs;it <lb/>ut ex e, igitur e c e&longs;t æqualis a c pars toti, quod contingere non po<lb/>te&longs;t. </s>

<s id="id000808">Sin uerò coniunctio fiat in d, igitur per præcedentem d e e&longs;t <lb/>pars a c &longs;ubmultiplex quomodolibet, quare non fuerunt a&longs;&longs;um<lb/>pti primi numeri. </s>

<s id="id000809">Veluti in exemplo con&longs;tituimus, quod a, & b <lb/>conueniunt in c in decem annis, & a c e&longs;t tertia pars circuitus: er<lb/>go in triginta annis conueniunt in a, & in quadraginta rur&longs;us in c. <lb/>&longs;i ergo quis a&longs;&longs;ump&longs;i&longs;&longs;et quadraginta annos ab initio pro con<lb/>gre&longs;&longs;u, & diui&longs;i&longs;&longs;et per 1 12/19 exiret 25 1/3, & &longs;i per 3 exiret 13 1/3, & mani<lb/>fe&longs;tum e&longs;t, quod uterque numerus pote&longs;t diuidi per eundem nu<lb/>merum, utpote 4 & exit numerus cum eadem parte &longs;cilicet 6 1/3 & <lb/>3 1/3 ergo conuenient ante, non ergo a&longs;&longs;ump&longs;i&longs;ti minimos in ea pro<lb/>portione. </s>

<s id="id000810">Illi autem nequaquam amplius diuidi non po&longs;&longs;unt eo<lb/>dem modo.</s>

<table>

<table.target id="table13"/>

<row>

<cell>Decem</cell>

<cell/>

<cell>Quatuor</cell>

<cell/>

</row>

<row>

</row>

<row>

<cell/>

</row>

</table>

<s id="id000811">Propo&longs;itio quinquage&longs;ima &longs;ecunda.</s>

<s id="id000812">Tria mobilia coniuncta in eodem puncto, quorum duo, & duo <lb/>conueniant in partibus in commen&longs;is inter &longs;e, in perpetuum in nul<lb/>lo unquam puncto conuenient.</s>

<s id="id000815">Sint a b c iuncta, & primo iungantur a & b, iterum in d & b, & <lb/>c in e, & &longs;int a d, a e incommen&longs;æ, dico quòd a b c nunquam con<lb/>uenient in aliquo puncto, &longs;eu primo, &longs;eu alio à primo: &longs;i non con<pb pagenum="43" xlink:href="015/01/062.jpg"/><figure id="id.015.01.062.1.jpg" xlink:href="015/01/062/1.jpg"/><lb/>ueniant in f, erunt ergo in g tempore re<lb/>uolutiones integræ, & portio a f in&longs;uper. <lb/></s>

<s id="id000816">Et quia hæ con&longs;tituuntur per congre&longs;&longs;us <lb/>b cum a, & &longs;unt &longs;patia a d, & b cum c, & <lb/>&longs;unt &longs;patia e f, igitur &longs;patium a f erit ex ge<lb/>nere quantitatis a d, & a e per quinqua<lb/>ge&longs;imam, harum ergo erunt commen&longs;æ: <lb/>quod e&longs;t contra &longs;uppo&longs;itum. </s>

<s id="id000817">Et harum <lb/>propo&longs;itionum principium e&longs;t traditum <lb/>à Campano Nouarien&longs;i Euclidis expo&longs;itore, in quodam libello <lb/>non edito qui diligentia patris mei Facij ad me peruenit.</s>

<s id="id000818">Propo&longs;itio quinquage&longs;ima tertia.</s>

<s id="id000819"><expan abbr="Circulor&utilde;">Circulorum</expan> &longs;e in aduer&longs;um mouentium proportionem declarare.</s>

<s id="id000822">Sit orbis a b cuius cen<lb/><figure id="id.015.01.062.2.jpg" xlink:href="015/01/062/2.jpg"/><lb/>centrum c, manubrium c <lb/>d f e, &longs;eu uero tangat circu<lb/>lum g, &longs;eu more gemmas <lb/>&longs;culpentium aligetur al<lb/>teri orbi funiculo a l b, & <lb/>&longs;it in uertice axis k m or<lb/>biculus &longs;olidus aut &longs;emi<lb/>circulari forma m, dico <lb/>quod proportio motus a <lb/>b ad motum m e&longs;t produ<lb/>cta ex duabus proportio<lb/>nibus c n <expan abbr="&longs;emidimeti&etilde;tis">&longs;emidimetientis</expan>, <lb/>& &longs;emidimetientis m ad k <lb/>o, quare ut rectanguli c n <lb/>in dimidium dimetientis <lb/>m ad quadratum o, ut enim a b ad ol orbem, id e&longs;t <expan abbr="peripheriar&utilde;">peripheriarum</expan> ita <lb/>c n ad o k, quoniam o l mouetur toties in una circuitione a b, quo<lb/>ties <expan abbr="peripheriã">peripheriam</expan> o l <expan abbr="contine&ttilde;">continetur</expan> in peripheria a b, ergo quoties o k con<lb/>tinetur in c n toties in una circuitione a b o l circumuertitur, &longs;ed <lb/>quoties circumuertitur ol, toties etiam m, quia uterque mouetur eo<lb/>dem circuitu k m axis, ergo quoties m circumducitur in circuitu a <lb/>b toties o k continetur in c n, ergo &longs;i fiat comparatio &longs;emidiametri <lb/>m ad c n, erit producta proportio circuitus a b ad circuitum m ex <lb/>proportione c n ad o k, et &longs;emidimetientis m ad <expan abbr="id&etilde;">idem</expan> o k, ergo per 26 <lb/>proportio numeri circuitus unius p <expan abbr="alter&utilde;">alterum</expan> e&longs;t, ut rectanguli &longs;ub c n, <lb/>& &longs;emidimetiente m ad quadratum k o, quod erat <expan abbr="demon&longs;trand&utilde;">demon&longs;trandum</expan>.</s>

<s id="id000823">Manife&longs;tum e&longs;t autem ex ip&longs;a &longs;ola con&longs;titutione, quod &longs;i a b mo</s><s id="id000824"><arrow.to.target n="marg142"/><pb pagenum="44" xlink:href="015/01/063.jpg"/>uetur &longs;ur&longs;um à dextro in &longs;ini&longs;trum in inferiore parte, mouebitur à <lb/>&longs;ini&longs;tro in dextrum, & uterque circulorum g & k in &longs;uperiore parte, <lb/>& in inferiore mouebitur contrario motu, &longs;cilicet in &longs;uperiore à &longs;ini<lb/>&longs;tro in dextrum, & inferiore à dextro in &longs;ini&longs;trum, illi uerò duo or<lb/>bes &longs;imili motu mouebuntur tam in parte &longs;uperiore, quàm inferio<lb/>re, & proportio motuum eorum inter &longs;e erit uelut dimetientium <lb/>eorundem.<lb/><arrow.to.target n="marg143"/></s>

<s id="id000827">Rur&longs;us cum a b circumuertatur cum manubrio c d f e, tanto uelo<lb/>cius circumuertetur, & in ea proportione, qua d f continetur in c n, <lb/>& in eodem tempore, in quo manubrium circumuertitur in eodem <lb/>axis circumuertitur, & orbis, ut dictum e&longs;t, ergo in eodem tempo<lb/>re, in quo axis circumuertitur in eodem orbis: ergo tanto tardius <lb/>uidebitur moueri axis ip&longs;o orbe, quanta e&longs;t proportio minoris in <lb/>æqualitatis ip&longs;ius axis, &longs;eu ambitus, &longs;eu &longs;emidimetientis ad ambi<lb/>tum, &longs;eu &longs;emidimetientem orbis.</s>

<s id="id000828">Propo&longs;itio quinquage&longs;ima quarta.</s>

<s id="id000829">Proportio circuli ad &longs;uum diametrum per <expan abbr="&longs;imilitudin&etilde;">&longs;imilitudinem</expan> e&longs;t quar<lb/>ta pars peripheriæ. </s>

<s id="id000830">Rur&longs;usque eiu&longs;dem circuli ad peripheriam diame<lb/>tri quarta pars.</s>

<s id="id000833">Quoniam enim &longs;uperficies circuli, ut ab <lb/><figure id="id.015.01.063.1.jpg" xlink:href="015/01/063/1.jpg"/><lb/>Archimede demon&longs;tratum e&longs;t, fit ex dimi</s>

<s id="id000834"><arrow.to.target n="marg145"/><lb/>dio diametri in <expan abbr="dimidi&utilde;">dimidium</expan> peripheriæ erit, ut <lb/>eadem fiat ex tota peripheria in <expan abbr="quartã">quartam</expan> par<lb/>tem diametri, & ex tota diametro in quar<lb/>tam <expan abbr="part&etilde;">partem</expan> peripheri&etail;. </s>

<s id="id000835">ergo proportio are&etail; <lb/>circuli ad diametrum per &longs;imilitudinem <lb/><arrow.to.target n="marg146"/><lb/>e&longs;t quarta pars peripheri&etail;, & proportio are&etail; <lb/>ad <expan abbr="peripheriã">peripheriam</expan> e&longs;t quarta pars dimetientis, quod erat probandum.</s>

<s id="id000836"><margin.target id="marg145"/>P<emph type="italics"/>er<emph.end type="italics"/> 16. <emph type="italics"/>&longs;ex <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s>

<s id="id000838">Propo&longs;itio quinquage&longs;ima quinta.</s>

<s id="id000839">Proportionem medicamentorum per ordines &longs;uppo&longs;ita æquali <lb/>proportione in ordinibus per quantitates, & proportiones de<lb/>mon&longs;trare.<lb/><arrow.to.target n="marg147"/></s>

<s id="id000841">Galenus libro quinto de Simplicibus medicamentis, quem &longs;e</s><s id="id000842"><arrow.to.target n="marg148"/><lb/>quuti &longs;unt alij medici, ponit quatuor ordines <expan abbr="medicamentor&utilde;">medicamentorum</expan> iux<lb/>ta qualitates calidi, frigidi, &longs;icci, & humidi, & primus e&longs;t cum <expan abbr="medicament&utilde;">medi<lb/>camentum</expan> non &longs;entitur quale &longs;it licet operetur, uelut cam&etail;melon, ab<lb/>&longs;ynthium, & oriza: &longs;ecundus e&longs;t, cum &longs;entitur, &longs;ed non lædit, ut nux <lb/>myri&longs;tica, &longs;aluia, ozimum: tertius e&longs;t cum &longs;entitur, & lædit, &longs;ed <lb/>non de&longs;truit, neque corrumpit corpus, uelut a&longs;&longs;arum apium &longs;ta<lb/>phi&longs;agria, cappares, myrrha, ruta: quartus e&longs;t, cum de&longs;truit ue<lb/>lut pyretrum, piper, euphorbium cæpe aggre&longs;te, & &longs;inapis, cina

<pb pagenum="45" xlink:href="015/01/064.jpg"/>momum autem, & gingiber numerantur inter medicinas calídas <lb/>tertij gradus, & hoc opus comparatur ad corpus &longs;icut dicit Gale<lb/>nus, & Serapio non ad linguam, ut medici no&longs;tri temporis interpre<lb/>tantur. </s>

<s id="id000843">Ex quo patet, quod aliqua medicina poterit e&longs;&longs;e quarti ordi<lb/>nis, & non lædere linguam in gu&longs;tu, & alia tertij ordinis, quæ non <lb/>&longs;olum lædet linguam, &longs;ed &longs;en&longs;um eius corrumpet, et de&longs;truet, quod <lb/>contingit propter &longs;ub&longs;tantiam tenuem cra&longs;&longs;æ mi&longs;tam cum &longs;iccitate <lb/>pari ip&longs;i calori. </s>

<s id="id000844">Sed non oportet h&etail;c nunc tractar, enon &longs;olum quia <lb/>non &longs;it locus, &longs;ed etiam quòd confu&longs;a &longs;it per &longs;e ip&longs;a materia ab&longs;que <lb/>eo, quod difficultatem difficultati addamus, &longs;olum ergo eas dubita<lb/>tiones adiungemus, quas <expan abbr="uol&etilde;tes">uolentes</expan> declarare propo&longs;itionem præ&longs;en<lb/>tem, neque &longs;uperfugere, neque declinare po&longs;&longs;umus. </s>

<s id="id000845">Nam de &longs;icco, <lb/>& humido, cum &longs;int longè minoris actionis, quàm calidum, & fri<lb/>gidum, & præcipuè humidum, non uideo quomodo po&longs;sit Gale<lb/>nus &longs;tatuere medicinam humidam tertij gradus, nedum quarti, <lb/>cum non po&longs;sit inueniri medicina, quæ de&longs;truat corpus no&longs;trum <lb/>propter humidam qualitatem. </s>

<s id="id000846">Et licet Serapio po&longs;uerit gingiber <lb/><arrow.to.target n="marg149"/><lb/>& enulam & zelim in tertio ordine calidorum & humidorum: & <lb/>inter frigidas, & humidas in tertio portulacam, aizoum, & uirgam <lb/>pa&longs;toris, & fungos. </s>

<s id="id000847">Primum non au&longs;us e&longs;t ponere medicinas ullas <lb/>calidas, aut frigidas in quarto ordine, qu&etail; &longs;int humidæ. </s>

<s id="id000848">&longs;ecundum, <lb/>quando dicit medicinas calídas, aut frigidas, atque humídas in ter<lb/>tio ordine, intelligit &longs;olum de qualitate actiua &longs;cilicet caliditate, uel <lb/>frigiditate, & non de humida qualitate, quod o&longs;tendit de gingibe<lb/>re, & enula, dicens, quod &longs;unt calidæ in tertio ordine, & humidæ <lb/>humido crudo, non au&longs;us addere ordinem, quia non uídit ratio<lb/>nem, qua po&longs;&longs;ent dici humidæ in tertio. </s>

<s id="id000849">Et clarius in capite de zei<lb/>len, quem &longs;tatuerat inter medicinas calidas, & humidas in tertio, di<lb/>cit quod e&longs;t calida in tertio, & humida in primo, ergo non intelligit <lb/>per medicinas calidas & humidas in tertio ordine, quod &longs;int humi<lb/>dæ in tertio ordine. </s>

<s id="id000850">Clarius etiam de frigidis & humidis, nam por<lb/>tula cam dicit e&longs;&longs;e frigidam in tertio, humidam in &longs;ecundo, & quod <lb/>maius, e&longs;t cum collo ca&longs;&longs;et aizoum inter medicinas frigidas, & hu<lb/>midas in tertio ordine, dicit, quod e&longs;t frigidum in tertio ordine, ad<lb/>ijcit, quod e&longs;t &longs;iccum parum, & de uirga pa&longs;toris nihil dicit de hu<lb/>mido, &longs;ed dicit, quod a&longs;tringit, ex quo concludo, quod &longs;ecun<lb/>dum mentem Serapionis nulla e&longs;t medicina humidior portulaca, <lb/>etiam uidetur innuere de fungis, &longs;atis e&longs;t quod non excedunt &longs;ecun<lb/>dum ordinem in humido neque calida neque frigida, &longs;ed frigida &longs;unt <lb/>humidiora, ut fungi, & portulaca, quia frigiditas in generatione <lb/>humidum magis admittit, quàm caliditas, & calida magis hu<pb pagenum="46" xlink:href="015/01/065.jpg"/>mectant, quia magis penetrat uis medicamenti, & hæc regula de <lb/>humido, & &longs;icco e&longs;t generalis apud Serapionem, quod non intelli<lb/>gitur ordo in pa&longs;siuis, ni&longs;i &longs;pecialiter exprimatur, nam de &longs;iccitate <lb/>non nego, quin inueniantur medicinæ &longs;iccæ in tertio, & for&longs;an in <lb/>quarto ordine, &longs;ed de hac Galeni o&longs;citantia, quæ in illo peculiaris <lb/>e&longs;t dum uult &longs;equi &longs;uas methodos &longs;ine alio di&longs;crimine, medicis con<lb/>&longs;iderandum relinquo.</s>

<s id="id000854">Secunda difficultas e&longs;t maior, & magis pertinet ad nos, & e&longs;t, <lb/>quòd non declarauit an i&longs;ti ordines inter &longs;e <expan abbr="aliquã">aliquam</expan> proportionem <lb/>&longs;eruarent, an omnino nullam, &longs;i enim nulla proportio &longs;eruatur, fieri <lb/>nullo modo pote&longs;t, ut per cognitionem temperaturæ &longs;implicium <lb/>medicamentorum cogno&longs;camus temperaturam compo&longs;itorum ex <lb/>illis ratione ulla, &longs;ed oportebit &longs;olum experiri. </s>

<s id="id000855">Sed &longs;i ordines &longs;er<lb/>uant proportionem, adhuc relinquitur dubium, an illa proportio <lb/>&longs;it Arithmetica, uel Geometrica, uel Mu&longs;ica, & nihil mirum e&longs;&longs;et, <lb/>quod e&longs;&longs;et Mu&longs;ica, ut aliâs docuimus, ubi tractauimus de differen<lb/>tia inter &longs;en&longs;um auditus, et ui&longs;us. </s>

<s id="id000856">Sed quia de hac nullus medicus ui <lb/>detur intellexi&longs;&longs;e, omittam hanc tractationem. </s>

<s id="id000857">Et quanquàm Gale<lb/>nus po&longs;sit uideri non exi&longs;tima&longs;&longs;e, quòd hi ordines non &longs;eruent <lb/>proportionem ullam, quia non au&longs;us e&longs;t tractare de temperamen<lb/>to medicamentorum compo&longs;itorum per rationem temperamen<lb/>ti &longs;implicium, nihilominus &longs;uppo&longs;ito quod ita e&longs;&longs;et, quod &longs;eruetur <lb/>altera proportionum, uolo o&longs;tendere rationem componendi in <lb/>utraque proportione & Arithmetica, & Geometrica. </s>

<s id="id000858">Ex quo &longs;e<lb/>quitur, quod Aueroes quàm o&longs;citanter tractauerit in quinto &longs;uo<lb/>rum collectaneorum de hoc, & non di&longs;tinguit, neque docet pri<lb/>mum an &longs;it aliqua proportio, deinde &longs;i qua &longs;it, cuius generis &longs;it, & <lb/>cum in re tam clara pugnet pror&longs;us, ut c&oelig;cus ictus maximos eden<lb/>do, &longs;ed in ca&longs;&longs;um plero&longs;que, quàm malè agant qui ei in arduis tan<lb/>tum tribuunt fidei, & authoritatis, &longs;ed hæc e&longs;t infelicitas no&longs;tra, & <lb/>ira Deorum. </s>

<s id="id000859">Suppo&longs;ito ergo quod primò ordines di&longs;tinguantur <lb/>per proportionem arithmeticam, &longs;it &longs;uperficies a b pro quantitate, <lb/><figure id="id.015.01.065.1.jpg" xlink:href="015/01/065/1.jpg"/><lb/>& a &longs;it calida in primo gradu, & b in ter<lb/>tio, erit ergo perinde ac &longs;i duo corpora <lb/>e&longs;&longs;ent unum altitudinis unius cum ba&longs;i <lb/>quadrilatera rectangula a, aliud altitu<lb/>dinis trium, ba&longs;i autem quadrilatera &longs;u<lb/>perficie rectangula b, hoc igitur erit to<lb/>tum mi&longs;tum, & quia quantitas medicamenti non mutatur quæ e&longs;t <lb/>a, b, ergo talia corpora æquantur uni corpori, cuius ba&longs;is e&longs;t a b, <lb/>cum ergo talia corpora producantur ex a in unum, & b in tria, ergo

<pb pagenum="47" xlink:href="015/01/066.jpg"/>diui&longs;o aggregato per a b prodibit altitudo, &longs;eu ordo qualitatis to<lb/>tius medicamenti, iuxta quod con&longs;tituitur regula prima libri artis <lb/>medendi paruæ huiu&longs;modi, & reliquæ, traduxi autem illas ad hunc <lb/>locum, “quia pendent ex demon&longs;tratione hac: “duc numerum ordi<lb/>nis &longs;ingulorum medicamentorum in numerum quantitatis, &longs;imilia <lb/>iunge, di&longs;similia detrahe, quod fit, diuide per aggregatum, quanti<lb/>tatum, exibit numerus ordinis compo&longs;iti. </s>

<s id="id000860">Sic mi&longs;cendo calidum in <lb/>&longs;ecundo ordine cum duplo pondere temperati conflabit calidum <lb/>in be&longs;&longs;e. </s>

<s id="id000861">Secunda &longs;i ex pluribus diuer&longs;arum, qualitatum, & ordi<lb/>num temperatum efficere uelis, duc quæ &longs;unt eiu&longs;dem qualitatis in <lb/>&longs;uas quantitates, & iunge, quod fit, diuide per numerum ordinis <lb/>medicamenti contrarij, exibit quantitas illius, &longs;ub qua &longs;i iungatur, <lb/>fiet medicamentum temperatum. </s>

<s id="id000862">Tertia cum nolueris ex tempera<lb/>to, & alio cuiu&longs;cunque ordinis medicamen conficere ordinis re<lb/>mi&longs;sionis, detrahe numerum ordinis eius, quod conficere uis ex nu<lb/>mero ordinis eius, quod habes, & cum re&longs;iduo diuide numerum <lb/>medicaminis, quod conficere uis, quod exit e&longs;t numerus quantita<lb/>tis medicamenti non temperati in comparatione ad temperatum.” <lb/>Ex his potes propo&longs;itis quibu&longs;cunque medicamentis conficere <lb/>antidotum &longs;ub quo cunque ordine remi&longs;siore potenti&longs;simo ex il<lb/>lis. </s>

<s id="id000863">Quarta in compo&longs;itione, quæ non fermente&longs;cit calida, calidis <lb/>iuncta &longs;emper opus augent, ut mel cum pipere. </s>

<s id="id000864">Quæ autem &longs;ub mi<lb/>nore quantitate exhibentur non &longs;ub remi&longs;siore ordine agant, &longs;ed <lb/>uel facilius impediuntur, uel minorem corporis partem, uel leuius <lb/>immutant.</s>

<s id="id000865">Quod &longs;i &longs;tatuamus proportionem e&longs;&longs;e Geometricam, modus <lb/>erit idem in omnibus, & quo ad numerum etiam in primo, & &longs;ecun<lb/>do ordine, quia in proportione dupla Geometrica &longs;ecundus ordo <lb/>tantundem di&longs;tat à primo, quantum primus ab æqualitate, quia <lb/>unum & duo &longs;eruant proportionem, & æqualem di&longs;tantiam, &longs;ed in <lb/>cæteris ordinibus non ita erit, quia qui e&longs;&longs;et trium in Arithmetica, <lb/>&longs;cilicet totius ordo e&longs;t, quatuor in Geometrica, & quartus ordo, <lb/>qui e&longs;&longs;et quatuor in Arithmetica, e&longs;&longs;et octo in Geometrica, ideo <lb/><figure id="id.015.01.066.1.jpg" xlink:href="015/01/066/1.jpg"/><lb/>&longs;cribemus ordines hoc modo, & operabimur cum <lb/>numeris loco ordinum, exemplum ergo primum <lb/>&longs;it medicina calida in tertio ordine quatuor uncia<lb/>rum, & medicina frigida in <expan abbr="&longs;ec&utilde;do">&longs;ecundo</expan> ordine duarum <lb/>unciarum, duco quatuor in tria, &longs;i proportio &longs;it Arithmetica, fit <lb/>duodecim, duco duo in duo fit quatuor, detraho quatuor in duo<lb/>decim, quia omnis medicina tantum retondit de contrario, &longs;eu mi<lb/>nuit relinquuntur octo &longs;cilicet caliditatis, diuido per &longs;ex ag<pb pagenum="48" xlink:href="015/01/067.jpg"/>gregatum unciarum exit unum, & tertia, ergo erit calida in princi<lb/>pio &longs;ecundi ordinis. </s>

<s id="id000866">Secundum exemplum &longs;int eædem medicinæ, <lb/>& &longs;it proportio Geometrica, ducemus ergo quatuor in quatuor, & <lb/>fiunt &longs;exdecim, & duo in duo fiunt quatuor, detrahe quatuor ex &longs;ex<lb/>decim, & remanent duodecim, diuide per &longs;ex, ut prius, exeunt duo, <lb/>ergo erit calida in fine &longs;ecundi gradus uides ergo di&longs;crimen. </s>

<s id="id000867">rur&longs;us <lb/>&longs;int ambæ medicinæ calidæ, & ducemus, ut prius in tertio exem<lb/>plo, ubi proportio &longs;it Arithmetica iungendo duodecim cum qua<lb/>tuor, & fient &longs;exdecim, diuide per &longs;ex, exeunt duo, & duæ tertiæ, er<lb/>go erit calida in medio tertij gradus, rur&longs;us in quarto exemplo iun<lb/>gemus &longs;edecim cum quatuor, & fient uiginti, diuide per &longs;ex exi<lb/>bunt tria & tertia, & ita erit in medio tertij gradus, ut prius, &longs;ed &longs;i <lb/>ille quatuor unciæ e&longs;&longs;ent calidæ in quarto gradu, & illæ duæ unciæ <lb/>in &longs;ecundo gradu, ut prius ducendo quatuor in quatuor fiunt &longs;ex<lb/>decim, & duo in duo fiunt quatuor, iunge, & fient uiginti, diuide <lb/>per &longs;ex exeunt tria cum tertia, ergo erit calida in principio quarti <lb/>gradus &longs;ecundum proportionem Arithmeticam, &longs;ed &longs;ecundum <lb/>Geometricam duc quatuor in octo, fiunt triginta duo, adde qua<lb/>tuor ut prius, &longs;cilicet productum duorum in duo fiunt triginta &longs;ex, <lb/>diuide per &longs;ex, exeunt &longs;ex, & quia &longs;ex ad quatuor maiorem habent <lb/>proportionem, quàm octo ad &longs;ex ideo hæc medicina erit calida ul<lb/>tra medium quarti gradus, iam ergo uides rationem, & differen<lb/>tiam horum.</s>

<s id="id000868">Quod &longs;i quis dicat, an debeat attendi Geometrica proportio in <lb/>medicamentis, an Arithmetica, re&longs;pondeo, quòd ueri&longs;imilius e&longs;t de <lb/>Arithmetica, quia illa proportio etiam quod &longs;it minor quatuor ad <lb/>trium, quàm trium ad duo, & multò minor quàm duo ad unum ni<lb/>hilominus longè plus operatur, quia tertius ordo iam incipit e&longs;&longs;e <lb/>præter naturam, & uidemus, quod læ&longs;io facta in uulnerato, etiam <lb/>quòd &longs;it quadruplo minor, plus nocet longè, quàm in &longs;ano qua<lb/>druplo maior: quia termini præter naturam &longs;unt ualdè angu&longs;ti in <lb/>comparatione ad latitudinem naturalem, &longs;icut etiam uidemus in<lb/>tendendis chordis &longs;corpionum, quod ultima pars e&longs;t breuis & ta<lb/>men homini tantam difficultatem adijcit. </s>

<s id="id000869">Notandum e&longs;t etiam, <lb/>quòd ob hoc diui&longs;erunt ordines in tres partes, uelut gingiber e&longs;t <lb/>calidum in fine tertij ordinis, origanum in medio, cinamomum in <lb/>principio, & ita euphorbium e&longs;t calidum in principio quarti gra<lb/>dus, &longs;ed in fine principij piper, in principio principij aqua &longs;epara<lb/>tionis in medio quarti ordinis, &longs;ed oleum chalcanthi factum ea ar<lb/>te, ut exurat paleas, &longs;icut ignis e&longs;t calidum in fine quarti ordinis, & <lb/>ita &longs;ufficiet diuidere propter eandem cau&longs;am primum, & &longs;ecun

<pb pagenum="49" xlink:href="015/01/068.jpg"/>dum ordinem in duas tantum partes non ratione latitudinis, quæ <lb/>e&longs;t æqualis, uel etiam for&longs;an maior, &longs;ed ratione uarietatis operatio<lb/>nis quæ minus &longs;entitur, & maximè in primo ordine.</s>

<s id="id000870">Propo&longs;itio quinquage&longs;ima&longs;exta.</s>

<s id="id000871">Proportio cuiu&longs;uis binomij ad &longs;uum reci&longs;um, uel ei commen<lb/>&longs;um e&longs;t duplicata ei, quæ ad numeri latus.<lb/><arrow.to.target n="marg150"/></s>

<s id="id000873">Cum enim proportionis medium &longs;it latus numeri eo quod ex bi<lb/>nomio in reci&longs;um &longs;uum fit numerus ex his, quæ demon&longs;trata &longs;unt <lb/>generaliter in tertio Arithmeticæ de omnibus binomijs cum &longs;uis </s>

<s id="id000874"><arrow.to.target n="marg151"/><lb/>reci&longs;is, uel in quadratis lateribus erit <02> numeri media proportione <lb/>inter binomium, & &longs;uum reci&longs;um, igitur cum proportio producto<lb/>rum ex binomio in commen&longs;a reci&longs;o &longs;it, ut commen&longs;orum ad reci<lb/><arrow.to.target n="marg152"/><lb/>&longs;a erunt omnia producta ex binomio in commen&longs;a reci&longs;o &longs;uo <02> nu<lb/><arrow.to.target n="marg153"/><lb/>meri, igitur proportio binomij ad reci&longs;um &longs;uum, & omnia com<lb/>men&longs;a illi, e&longs;t duplicata ei quæ ad <02> numeri.<lb/><arrow.to.target n="marg154"/></s>

<s id="id000875"><margin.target id="marg151"/>P<emph type="italics"/>er<emph.end type="italics"/> 6. P<emph type="italics"/>ro<lb/>po&longs;. </s>

<s id="id000876">lib.

de<emph.end type="italics"/><lb/>A<emph type="italics"/>liza.<emph.end type="italics"/></s>

<s id="id000877"><margin.target id="marg152"/>P<emph type="italics"/>er<emph.end type="italics"/> 17. <emph type="italics"/>&longs;ex <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s>

<s id="id000878"><margin.target id="marg153"/>P<emph type="italics"/>er<emph.end type="italics"/> 17. <lb/><emph type="italics"/>&longs;eptimi <lb/>eiu&longs;dem.<emph.end type="italics"/></s>

<s id="id000879"><margin.target id="marg154"/>P<emph type="italics"/>er<emph.end type="italics"/> 6. <emph type="italics"/>deci<lb/>mi<emph.end type="italics"/> E<emph type="italics"/>lement:<emph.end type="italics"/></s>

<s id="id000880">Propo&longs;itio quinquage&longs;ima &longs;eptima.</s>

<s id="id000881">Motus rationem ad pondus inuenire.</s>

<s id="id000884">O&longs;ten&longs;um e&longs;t antea, quod motus naturalis uelocior fit in fine, ac <lb/>magis augetur ob aëris motum, ubi uerò hæret e&longs;t ac &longs;i quie&longs;cat. <lb/></s>

<s id="id000885">Eadem autem e&longs;t ratio in motis uiolenter, & naturaliter dum &etail;qua<lb/>li impetu feruntur. </s>

<s id="id000886">Sed &longs;ubitò po&longs;t etiam, quod motus æqualiter <lb/>augerentur minus tamen cre&longs;cit proportio uiolenti &longs;cilicet ob im<lb/><figure id="id.015.01.068.1.jpg" xlink:href="015/01/068/1.jpg"/><lb/>pedimentum naturale. </s>

<s id="id000887">Sed &longs;i uis mouens fuerit <lb/>adeò ualida ut proportio incrementi ex aëre &longs;it <lb/>maior, quàm impedimentum, & in crementum al<lb/>terius mobilis naturaliter moti, motus ille uelo<lb/>cior fiet naturali, ut in &longs;phæris ferreis ex machina <lb/>igne excu&longs;sis, quod ergo attinet ad præ&longs;entem <lb/>motum ratio e&longs;t eadem. </s>

<s id="id000888">Quicunque ergo motus <lb/>minoris grauis cogit de&longs;cendere lancem ex ad<lb/>uer&longs;o proportionem habet eandem ad &longs;uum mo <lb/>bile quam habet graue æquiponderans. </s>

<s id="id000889">Sit ergo <lb/>ut a ex b, c, d, e, eleuet eodem ordine pondera e, f, <lb/>g, h, erit ergo ponderum h, g, f, e, ad &longs;e inuicem, & ad a qualis mo<lb/>tuum ob di&longs;tantiam intentorum. </s>

<s id="id000890">Experimentum ergo docet, quòd <lb/>dimidium ponderis æquilibrium facit ex palmo minoris dimidio <lb/>motum manife&longs;tum, & ex palmo quarta pars ponderis, ergo &longs;e ha<lb/>bent prope portionem.</s>

<s id="id000891">Propo&longs;itio quinquage&longs;ima octaua.</s>

<s id="id000892">Qu&etail; ex alto de&longs;cendunt cur non eandem pro di&longs;tantia motus ra<lb/>tionem in libero aëre &longs;eruent con&longs;iderare.</s>

<s id="id000893">Aër in &longs;ublimiore eius regione &longs;emper naturali motu fertur ex <lb/>Oriente in Occidentem, &longs;ed & infra uerum minus manife&longs;tè. </s>

<s id="id000894">At ca<lb/>&longs;u plerun que contingit, ut moueatur longè uehementius, &longs;eu ad ean<lb/>dem partem, &longs;eu aliam. </s>

<s id="id000895">Qui uerò naturalis e&longs;t, debilis <lb/><figure id="id.015.01.069.1.jpg" xlink:href="015/01/069/1.jpg"/><lb/>e&longs;t, quoniam in tenui ualde &longs;ub&longs;tantia e&longs;t: nec <expan abbr="cõtinuus">continuus</expan> <lb/>&longs;ed in&longs;tar motus aquæ maris fluit ac refluit: aliter ne<lb/>ce&longs;&longs;e e&longs;&longs;et, ut &longs;ingulis horis per mille milliaria procede<lb/>ret, ut &longs;ic ne que latere po&longs;&longs;et, quandoquidem fortuiti mo<lb/>tus, qui &longs;unt multo tardiores non latent nos. </s>

<s id="id000896">Nam tardiores illos <lb/>e&longs;&longs;e <expan abbr="cõ&longs;tat">con&longs;tat</expan>, cum in hora &longs;int pul&longs;us arteriarum, quatuor millia <expan abbr="ictu&utilde;">ictuum</expan> <lb/>in homine prope temperamentum: &longs;i igitur motus naturalis aëris <lb/>e&longs;&longs;et continuus, in hora aër procederet ob ambitum terræ millies <lb/>mille pa&longs;&longs;us, <expan abbr="igi&ttilde;">igitur</expan> in ictu pul&longs;us &longs;uperaret pa&longs;&longs;us 250. At experimur <lb/>nullum uentum aut procellam &longs;uperare quinquaginta pa&longs;&longs;us, cum <lb/>etiam continuus e&longs;&longs;e nunquam &longs;oleat, imò ne po&longs;sit quidem, itaque <lb/>cum hic multo tardior etiam in &longs;ublimi, dum e&longs;t, nos latere non <lb/>queat, multo minus po&longs;&longs;et naturalis latere, &longs;i adeò uelox & in ea<lb/>dem parte <expan abbr="a&etilde;ris">aeris</expan> e&longs;&longs;et at que continuus. </s>

<s id="id000897">Præterea tantus impetus nun<lb/>quam à minore motu, aut cau&longs;a &longs;uperaretur, adeò ut &longs;emper flatum <lb/>aëris orientalem &longs;entiremus. </s>

<s id="id000898">Quotidie etiam aduenire ad nos aë<lb/>rem ex Illyrico, Macedonia, My&longs;ia, Ponto, Bythínia, Capadocia, Sy <lb/>ria, Babylonia, Hyrcanomarí, Bactrianis, Sacís, Scythis, ac Seris, to<lb/>to præterea Oceano orientali tam ua&longs;to, & Gallica noua, terraque flo<lb/>rida non &longs;olum res e&longs;t admirabilis', & incredibilis, &longs;ed etiam aliena <lb/>à &longs;en&longs;u, & ab his, quæ eueniunt. </s>

<s id="id000899">A' &longs;en&longs;u quidem, quoniam nebul&etail;, <lb/>quæ in aëre mouentur, primùm non in eandem partem &longs;emper mo<lb/>uentur: nun quam autem adeò celeriter: at &longs;i aër &longs;ic circumuoluere<lb/>tur, mouerentur & illa, qu&etail; in eo continentur, quotidieque aërem ex<lb/>periremur & nubilo&longs;um, & madidum propter mare. </s>

<s id="id000900">Nechis, quæ <lb/>eueniunt hoc &longs;atis re&longs;pondet, nec nobis id contingeret, ut &longs;i pe&longs;ti <lb/>aliqua in regione no&longs;tra directa &longs;æuiret, ut aër &longs;ingulis diebus la<lb/>be ea infectus ad nos deferretur. </s>

<s id="id000901">Moueri uerò aërem &longs;emper mani<lb/>fe&longs;ti&longs;simum e&longs;t tum experimento, tum ratione: ratione &longs;iquidem, <lb/>quod aqua & c&oelig;lum naturaliter perpetuò mouentur, quare etiam <lb/>aër. </s>

<s id="id000902">Experimento, quòd ubi hiant o&longs;tia, & ianuæ, ibi perpetuus &longs;en<lb/>titur flatus. </s>

<s id="id000903">Ergo &longs;i a pondus de&longs;cendat in c, ex alto fertur rectà, &longs;ed <lb/>&longs;i ex &longs;ublimi transferetur in b, & indirecta, & ad latus, unde ex <lb/>hoc &longs;equitur.</s>

<s id="id000904">Propo&longs;itio quinquage&longs;ima nona.</s>

<s id="id000907">Omne mobile motum duobus motibus non ad idem tendenti<lb/>bus, utro que &longs;eor&longs;um tardius mouetur &longs;imili motu.</s>

<s id="id000908">Sit a mobile, quod moueatur per a b c impul&longs;u uenti aut uiolen</s>

<s id="id000909"><arrow.to.target n="marg157"/><lb/><figure id="id.015.01.070.1.jpg" xlink:href="015/01/070/1.jpg"/><lb/>to cum naturali coniuncto: & &longs;it terminus naturalis e, <lb/><arrow.to.target n="marg158"/><lb/>& uiolenti d: uter que in directo c, dico, quod tardius per<lb/>ueniet ad c quam d, uel e. </s>

<s id="id000910">De e manife&longs;tum e&longs;t, quoniam <lb/>motus aëris, qui intendit motum a, diuíditur in partem, <lb/>quæ iuuat motum ad d, & partem, quæ mouetur ad e, <lb/>igitur fit minor adiectio. </s>

<s id="id000911">Et etiam quia a c e&longs;t longior <lb/>a e ex diffinitione rectæ: quare tardius perueniet ad c quàm ad e du<lb/>plici ratione. </s>

<s id="id000912">Dico etiam, quod tardius ad c quàm d. </s>

<s id="id000913">Quia enim <lb/>uis, quæ fert ad d repugnat ei, quæ fert ad e, & uis, quæ fert ad e, re<lb/>pugnat ei quæ fert ad d, igitur tardius perueniet ad c, quàm d. </s>

<s id="id000914">Nec <lb/>potes dicere, quòd uis, quæ fert ad c adiuuet ad motum è regione <lb/>d, nam cum unus motus non po&longs;sit perfici &longs;ine altero, igitur quan<lb/>tum motus ad e retardabit motum ad d, tanto motus a c erit tardí<lb/>or ab&longs;olutè motu ad d. </s>

<s id="id000915">Verum etiam e&longs;t, quod c e breuior erit a d, <lb/>quia motus ad e &longs;emper contrahit motum ad d naturalis uiolen<lb/>tum ob cau&longs;am dictam. </s>

<s id="id000916">Vtrùm uerò motus ad c ab&longs;olutè &longs;it tardi<lb/>or, quàm ad d, non &longs;uppo&longs;ito, quod c e &longs;it æqualis a d, &longs;ed minor, <lb/>nunc non e&longs;t locus determinandi.</s>

<s id="id000918"><margin.target id="marg158"/>P<emph type="italics"/>er<emph.end type="italics"/> 20. <emph type="italics"/>buius.<emph.end type="italics"/></s>

<s id="id000919">Ex hoc patet, quod motus æquidi&longs;tantis mobilis, finis e&longs;t mini<lb/><arrow.to.target n="marg159"/><lb/>mus omnium: quoniam mobile qua&longs;i quie&longs;cit in illo. </s>

<s id="id000920">Velut &longs;i a mo<lb/>ueatur ad b, inde deflectat ad c minimus motus erit in b, ubi incipit <lb/>naturalis: nam cum incipiat, erit debili&longs;simus, quia non <lb/><figure id="id.015.01.070.2.jpg" xlink:href="015/01/070/2.jpg"/><lb/>e&longs;t motus actu: uiolentus autem æqualis e&longs;t naturali, <lb/>dum minimus e&longs;t: ergo cum ex di&longs;tantia medij palmi <lb/>duplicetur, naturalis erit motus in b minimus, ni&longs;i b c <lb/><arrow.to.target n="marg160"/><lb/>e&longs;&longs;et minor dimidio palmi. </s>

<s id="id000921">Et etiam quòd e&longs;&longs;et minor, quia ut di<lb/>ctum e&longs;t, uter que &longs;imul iunctus e&longs;t æqualis uni eorum non impedito <lb/>uel minor.</s>

<s id="id000923"><margin.target id="marg160"/>P<emph type="italics"/>er<emph.end type="italics"/> 57. <emph type="italics"/>buius.<emph.end type="italics"/></s>

<s id="id000924">Propo&longs;itio &longs;exage&longs;ima.</s>

<s id="id000925">Omne mobile motu naturali de&longs;cendens parte, de&longs;cendit gra<lb/>uiore &longs;ecundum grauitatis centrum.</s>

<s id="id000926">Sit a mobile, grauitatis centrum b, cuius pars ei pro<lb/><arrow.to.target n="marg161"/><lb/><figure id="id.015.01.070.3.jpg" xlink:href="015/01/070/3.jpg"/><lb/>ximior &longs;it c a, dico quod de&longs;cendat motu naturali c a, <lb/>parte tangendo terram, quia enim totum a non pote&longs;t <lb/>de&longs;cendere ad centrum de&longs;cendit b, quia eadem e&longs;t na<lb/>tura partis, & totius: totius autem terræ natura e&longs;t ut <lb/>centrum, totius &longs;it centrum grauitatis, quare b breuiore uia fertur <lb/><arrow.to.target n="marg162"/><lb/>ad centrum, ergo per c d proximiorem partem ip&longs;i b. </s>

<s id="id000927">Sed pars pro<lb/>ximior nece&longs;&longs;ariò e&longs;t grauior, quia centrum e&longs;t in medio grauita<pb pagenum="52" xlink:href="015/01/071.jpg"/>tis, ergo omne mobile de&longs;cendit motu naturali per &longs;ui grauio<lb/>rem partem.<lb/><arrow.to.target n="marg163"/></s>

<s id="id000929"><margin.target id="marg162"/>P<emph type="italics"/>er<emph.end type="italics"/> 23. <emph type="italics"/>buius.<emph.end type="italics"/></s>

<s id="id000931">Ex hoc &longs;equitur, quòd graue habens partes inæquales, &longs;eu &longs;ub<lb/>&longs;tantia, &longs;eu forma, &longs;i ita excutiatur, ut pars grauior <expan abbr="nõ">non</expan> &longs;it, infrà opor<lb/>tet, ut circumuoluatur.</s>

<s id="id000932">Propo&longs;itio &longs;exage&longs;ima prima.</s>

<s id="id000933">Proportionem ictus ad pondus rei, & di&longs;tantiam generaliter <lb/>con&longs;iderare.</s>

<s id="id000936">Dictum e&longs;t &longs;uperius de proportione de&longs;cen&longs;us ad grauitatem: </s>

<s id="id000937"><arrow.to.target n="marg165"/><lb/>& quòd &longs;i graue de&longs;cendat ex alto impeditur à motu aëris: & quòd <lb/><arrow.to.target n="marg166"/><lb/>res, quæ mouetur duobus motibus non ad idem tendentibus tar<lb/><arrow.to.target n="marg167"/><lb/>dius mouetur, quam motus &longs;it unu&longs;qui&longs;que. </s>

<s id="id000938">Demùm quòd graue <lb/><arrow.to.target n="marg168"/><lb/>de&longs;cendens circumuoluitur, &longs;i pars grauior non &longs;it, deor&longs;um: & an<lb/>tea