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Colored diff for /texts/archimedes/xml/carda_propo_015_la_1570.xml between version 1.7 and 1.15

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

version 1.15, 2004/02/19 17:23:10

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<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="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="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>

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<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>

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<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>

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<row>

<cell>S<emph type="italics"/>i proportio ex duabus proportionibus in quatuor terminis producatur, ip&longs;a uerò proportio inter duas alias quantitates fuerit con&longs;tituta: con&longs;urgent trecen-ti &longs;exaginta modi productionis proportionis.<emph.end type="italics"/></cell>

<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>

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<cell>I<emph type="italics"/>n modis qui nece&longs;&longs;ariò producuntur ex duabus proportionibus, cum duæ quantitates ex illis quæ modos conficiunt, æquales fuerint: proportio producta ad quatuor quanti-tates omiologas reducetur.<emph.end type="italics"/></cell>

<cell>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 multiplicen-tur atque coniungantur, erit proportio aggregati ad productum ex inferioribus in-uicem proportio, ex primis proportionibus compo&longs;ita.<emph.end type="italics"/></cell>

<cell>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>

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<row>

<cell>S<emph type="italics"/>i duarum proportionum &longs;uperiores numeri alternatim cum inferioribus multiplicen-tur, minusque productum ex maiore detrahatur, erit re&longs;idui ad productum ex in&longs;e-rioribus proportio uelut illa, quæ relinquitur detracta minore proportione ex ma-iore.<emph.end type="italics"/></cell>

<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;ecun-dam quantitatem, erit proportio cuiu&longs;uis quantitatis eiu&longs;dem generis ad &longs;ecundam compo&longs;ita proportio, ex proportionibus eiu&longs;dem quantitatis, a&longs;&longs;umptæ ad utranque partem primæ quantitatis &longs;eor&longs;um.<emph.end type="italics"/></cell>

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

<cell>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>

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<cell>P<emph type="italics"/>roportio confu&longs;a aggregata primæ & tertiæ quatuor quantitatum omiologarum ad aggregatum &longs;ecundæ & quartæ, e&longs;t uelut compo&longs;ita ex ei&longs;dem diui&longs;a per du-plam.<emph.end type="italics"/></cell>

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<cell>S<emph type="italics"/>i fuerint quatuor quantitates proportio confu&longs;a, aggregati primæ & tertiæ, ad aggre-gatum &longs;ecundæ & quartæ, erit ut monadis addito prouentu, qui fit diui&longs;a differentia, differentiarum primæ & &longs;ecundæ, atque quartæ & tertiæ, per aggregatum tertiæ & quartæ ad ip&longs;am monadem.<emph.end type="italics"/></cell>

<cell>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 proportio-nem ad &longs;uam corre&longs;pondentem quàm alia ad aliam, erit proportio confu&longs;a illarum,<emph.end type="italics"/></cell>

<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/>

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<cell>S<emph type="italics"/>i fuerint aliquot quantitates arithmeticæ omiologæ, quarum exce&longs;&longs;us &longs;it æqualis minimè, omnibus autem deficientibus &longs;upplementa ad æqualitatem maximè adiungantur, erunt quadrata omnium quantitatum æqualium, adiecto rur&longs;us quadrato primæ cum eo quod fit ex minima primi ordinis in aggregatum o-mnium quantitatum eiu&longs;dem, tripla aggregato quadratorum omnium quanti tatum primi ordinis pariter acceptis.<emph.end type="italics"/></cell>

<cell>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 quanti tatum primi ordinis pariter acceptis.<emph.end type="italics"/></cell>

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<cell>C<emph type="italics"/>um decu&longs;&longs;atim ducta fuerit prima in quartam, & &longs;ecunda in tertiam, produ-ctumque primæ in quartam, diui&longs;um fuerit per productum &longs;ecundæ in tertiam, erit proportio primæ ad &longs;ecundam, diui&longs;a per proportíonem tertiæ ad quar-tam.<emph.end type="italics"/> E<emph type="italics"/>t &longs;imiliter interpo&longs;ita omiologa.<emph.end type="italics"/></cell>

<cell>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>

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<cell>O<emph type="italics"/>mnis motus uoluntarius æqualis e&longs;t &longs;emper: &longs;impliciter etiam quilibet alius mo-tus.<emph.end type="italics"/></cell>

<cell>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 impel-lunt.<emph.end type="italics"/></cell>

<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>

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<cell>XXXVI.</cell>

<cell>S<emph type="italics"/>i proportio per proportionem minorem æquali ducatur, proportio minor pro-<emph.end type="italics"/></cell>

<cell>S<emph type="italics"/>i proportio per proportionem minorem æquali ducatur, proportio minor pro<emph.end type="italics"/></cell>

<cell/>

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<row>

<cell>XXXVIII.</cell>

<cell>O<emph type="italics"/>mne corpus tantum re&longs;i&longs;tit motui contrario &longs;uo natúrali, quantum mouetur oc-culto motu quie&longs;cendo.<emph.end type="italics"/></cell>

<cell>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>

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

<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>

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<row>

<cell>S<emph type="italics"/>i fuerint duæ quantitates &longs;umaturque toties <expan abbr="aggregat&utilde;">aggregatum</expan> maioris & minoris, quo-ties aggregatum minoris & maioris, erit proportio confu&longs;a maioris aggregati ad minus, minor quam multiplicis maioris ad multiplex minoris.<emph.end type="italics"/></cell>

<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>

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<cell>A<emph type="italics"/>n &longs;it aliqua proportio & qualis inter animam & uitas, & &longs;ua corpora con&longs;ide-rare.<emph.end type="italics"/></cell>

<cell>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, pro-ductum temporis circuituum inuicem, erit æquale producto differentiæ tempo rum circuitus ductæ in tempus coniunctionis primæ.<emph.end type="italics"/></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 tempo rum circuitus ductæ in tempus coniunctionis primæ.<emph.end type="italics"/></cell>

</row>

<row>

<cell>XLVIII.</cell>

<cell>S<emph type="italics"/>i tria mobilia ex eodem puncto di&longs;cedant, fuerintque duorum ac duorum coniun-ctiones in temporibus commen&longs;is, illa tria mobilia denuo coniungentur in tem pore producto ex denominatore diui&longs;ionis temporis maioris per minus in mi-nus aut numeratore in maius.<emph.end type="italics"/></cell>

<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>

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Line 371

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<row>

<cell>LIIII.</cell>

<cell>P<emph type="italics"/>roportio circuli ad &longs;uum diametrum per &longs;imilitudinem e&longs;t quarta pars periphe-riæ.<emph.end type="italics"/> R<emph type="italics"/>ur&longs;usque eiu&longs;dem circuli ad peripheriam diametri quarta pars.<emph.end type="italics"/></cell>

<cell>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 or-dinibus per quantitates & proportiones demon&longs;trare.<emph.end type="italics"/></cell>

<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>

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<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>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/>

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<row>

<cell>LXVII.</cell>

<cell>S<emph type="italics"/>i fuerint aliquot quantitates ab una quantitate aliæque totidem ab eadem analo-gæ, erit proportio tertiæ unius ordinis ad tertiam alterius, ut &longs;ecundæ ad &longs;e-cundum duplicata, & quartæ ad quartam triplicata, quintæ ad quintam quadruplicata, atque &longs;ic de alijs.<emph.end type="italics"/></cell>

<cell>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>

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<cell>S<emph type="italics"/>i fuerint tres quantitates in continua proportione, aliæque totidem in continua proportione poterunt con&longs;tituere tres quantitates in æquali differentia per-uer&longs;im copulatæ.<emph.end type="italics"/></cell>

<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>

<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>

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Line 532

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<cell>LXXXV.</cell>

<cell>P<emph type="italics"/>roportio ponderis unius grauis ad aliud &longs;ub eadem men&longs;ura e&longs;t ueluti eiu&longs;dem ad differentiam ponderis ua&longs;is repleti ex altero graui, & ex ambobus de-tracto priore.<emph.end type="italics"/></cell>

</row>

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<cell>XCIIII.</cell>

<cell>S<emph type="italics"/>i <expan abbr="quãtitas">quantitas</expan> aliqua nota atque proportio erit producta, <expan abbr="quãtitas">quantitas</expan> nota &longs;imiliter.<emph.end type="italics"/> E<emph type="italics"/>t &longs;i duæ proportiones notæ fuerint, erit producta ex his atque diui&longs;a coniunctaque atque detra-cta nota.<emph.end type="italics"/> E<emph type="italics"/>t &longs;i fuerit totius ad partem proportio nota, erit et ad aliam partem nota: & alterius partis ad <expan abbr="alterã">alteram</expan> uno minor.<emph.end type="italics"/> E<emph type="italics"/>t &longs;i fuerit partis ad partem, erit ad totum monade minor atque nota.<emph.end type="italics"/> E<emph type="italics"/>t &longs;i fuerit unius <expan abbr="quãtitatis">quantitatis</expan> ad duas <expan abbr="quãtitates">quantitates</expan> proportio nota, erit & <expan abbr="cõfu&longs;a">confu&longs;a</expan> ex eis nota.<emph.end type="italics"/> E<emph type="italics"/>t &longs;i fuerint trium quantitatum omiologarum, aut quatuor analogarum omnes præter unam cognitæ, erunt & illa alia cognita.<emph.end type="italics"/></cell>

<cell>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>

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Line 638

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<row>

<cell>Q<emph type="italics"/>uales proportiones angulorum doceant laterum proportiones.<emph.end type="italics"/> A<emph type="italics"/>tque uicißim deter-minare.<emph.end type="italics"/></cell>

<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 perpendicu-lum ex diametro exicrint ad circum ferentiam, &longs;ingulæ &longs;upra diametrum erunt ma iores portionibus reliquis diametri &longs;uperioribus, infra autem minores.<emph.end type="italics"/> D<emph type="italics"/>imidium autem portionis &longs;uperioris re&longs;iduum ad centrum maius &longs;agitta habebit.<emph.end type="italics"/> I<emph type="italics"/>n aliqua præterea portionis &longs;uperioris parte, quæ uer&longs;us diametrum tran&longs;uer&longs;um po&longs;ita e&longs;t, maior e&longs;t differentia partis diametri ei <expan abbr="corre&longs;põdentis">corre&longs;pondentis</expan>, <expan abbr="&qtilde;">quae</expan> line æ tran&longs;uer&longs;æ.<emph.end type="italics"/></cell>

<cell>S<emph type="italics"/>i in circulo duæ diametri ad rectum angulum &longs;e &longs;ecauerint: aliæ uerò ad perpendiculum ex diametro exicrint ad circum ferentiam, &longs;ingulæ &longs;upra diametrum erunt ma iores portionibus reliquis diametri &longs;uperioribus, infra autem minores.<emph.end type="italics"/> D<emph type="italics"/>imidium autem portionis &longs;uperioris re&longs;iduum ad centrum maius &longs;agitta habebit.<emph.end type="italics"/> I<emph type="italics"/>n aliqua præterea portionis &longs;uperioris parte, quæ uer&longs;us diametrum tran&longs;uer&longs;um po&longs;ita e&longs;t, maior e&longs;t differentia partis diametri ei <expan abbr="corre&longs;põdentis">corre&longs;pondentis</expan>, <expan abbr="&qtilde;">quae</expan> line æ tran&longs;uer&longs;æ.<emph.end type="italics"/></cell>

</row>

<row>

Line 663

</row>

<row>

<cell>C<emph type="italics"/>ur ex medio tela ualidiorem ictum, & naues in &longs;calmo à remo ac malo recipiant in-de ex puppi explorare.<emph.end type="italics"/></cell>

<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>

Line 683

</row>

<row>

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

<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>

Line 698

</row>

<row>

<cell>CXVIII.</cell>

<cell>Q<emph type="italics"/>uanta proportione decedat ictus in obliquum parietem ab eo qui e&longs;t ad perpendi-culum declarare.<emph.end type="italics"/></cell>

<cell>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>

Line 724

<row>

<cell>CXXIII.</cell>

<cell>C<emph type="italics"/>ognita ratione umbræ ad gnomonem &longs;inum, & arcum altitudinis ab horizon-te, quouis tempore digno&longs;cere.<emph.end type="italics"/></cell>

<cell>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>

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<row>

Line 734

</row>

<row>

<cell>P<emph type="italics"/>roportionem dimetientis, & peripheriæ cuiuslibet circuli paralleli æquino-ctiali per cognitam partem magni circuli demon&longs;trare.<emph.end type="italics"/></cell>

<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>

Line 754

</row>

<row>

<cell>CXXIX.</cell>

<cell>D<emph type="italics"/>ata altitudine<emph.end type="italics"/> S<emph type="italics"/>olis in quacunque regione, quacunque die di&longs;tantiam<emph.end type="italics"/> S<emph type="italics"/>olis à meri-diano cogno&longs;cere.<emph.end type="italics"/></cell>

<cell>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 um-bram rectam quàm uer&longs;am, uel etiam in cylindro determinare.<emph.end type="italics"/></cell>

<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>

Line 769

</row>

<row>

<cell>CXXXII.</cell>

<cell>S<emph type="italics"/>i ad duas lineas quarum una alteri dupla &longs;it eadem linea addatur, erit aggrega-ti ex minore, & adiecta ad ip&longs;am minorem, minor proportio quàm aggre-gati ex maiore, & adiecta ad ip&longs;am maiorem duplicata.<emph.end type="italics"/></cell>

<cell>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 quæ-dam quantitas, <expan abbr="ead&etilde;que">eadenque</expan> maiori addatur, erit minoris ad re&longs;iduum maior pro-portio, quàm aggregati ad maiorem duplicata.<emph.end type="italics"/> S<emph type="italics"/>i uerò minori addatur, & à maiore detrabatur, erit aggregati ad minorem minor proportio quàm maioris ad re&longs;iduum duplicata.<emph.end type="italics"/></cell>

<cell>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 la-tere quadratæ in re&longs;iduum &longs;uperficiei con&longs;tat, maius e&longs;t quouis corpore ex eadem &longs;uperficies, aliter diui&longs;a con&longs;tituto.<emph.end type="italics"/></cell>

<cell>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 pararalleli-pedo, quod ex diui&longs;ione eiu&longs;dem lineæ creari poßit.<emph.end type="italics"/></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>

Line 824

</row>

<row>

<cell>CXLIII.</cell>

<cell>S<emph type="italics"/>i linea in duas partes diuidatur, corpora quæ fiunt ex una parte in alterius quadratum mutuo æqualia &longs;unt corpori, quod fit ex tota linea in &longs;uperfi-ciem unius partis in alteram.<emph.end type="italics"/></cell>

<cell>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>

Line 839

</row>

<row>

<cell>CXLVI.</cell>

<cell>C<emph type="italics"/>orpus quod fit ex linea diui&longs;a in &longs;uperficiem æqualem quadratis ambarum par tium detracta &longs;uperficie unius partis in alteram, e&longs;t æquale aggregato cubo-rum ambarum partium.<emph.end type="italics"/></cell>

<cell>C<emph type="italics"/>orpus quod fit ex linea diui&longs;a in &longs;uperficiem æqualem quadratis ambarum par tium detracta &longs;uperficie unius partis in alteram, e&longs;t æquale aggregato cuborum ambarum partium.<emph.end type="italics"/></cell>

</row>

<row>

Line 855

</row>

<row>

<cell>CXLVIII.</cell>

<cell>P<emph type="italics"/>ropo&longs;itis tribus lineis primam &longs;ic diuidere, ut adiectis duabus alijs lineis, &longs;ecun-dum <expan abbr="ration&etilde;">rationem</expan> mutuam &longs;ingularum &longs;ingulis, <expan abbr="aggregat&utilde;">aggregatum</expan> ex una <expan abbr="adiectar&utilde;">adiectarum</expan>, & par te ad <expan abbr="aggregat&utilde;">aggregatum</expan> ex alia parte, & adiecta &longs;e habeat, ut &longs;ecunda ad <expan abbr="tertiã">tertiam</expan>.<emph.end type="italics"/></cell>

<cell>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>

Line 865

</row>

<row>

<cell>P<emph type="italics"/>ropo&longs;itis duabus lineis, lineam communem utrique adiungere, ut &longs;it maioris ad ad-ditam proportio, uelut quadratorum minoris, & adiectæ ad duplum unius in alteram.<emph.end type="italics"/></cell>

<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 diffe-rentiæ illarum e&longs;t, uelut totius lineæ ad differentiam.<emph.end type="italics"/></cell>

<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 ag-gregati ex maiore, & dimidio ad ip&longs;am maiorem, uelut ex minore, & ali-qua linea ad ip&longs;am minorem, & rur&longs;us aggregati ex minore, & dimidio ad ip&longs;am minorem, uelut aggregati ex maiore, & alia addita ad ip&longs;am maiorem, erit proportio dimidij ad partem unam inæqualem, uelut alterius partis inæ-qualis ad &longs;uam additam mutuò, & etiam proportio additarum inuicem, uelut proportio <expan abbr="parti&utilde;">partium</expan> <expan abbr="inæquali&utilde;">inæqualium</expan> duplicata, & rur&longs;us ip&longs;um <expan abbr="dimidi&utilde;">dimidium</expan> lineæ a&longs;&longs;um-ptæ <expan abbr="medi&utilde;">medium</expan>, erit proportione inter additas.<emph.end type="italics"/> D<emph type="italics"/><expan abbr="em&utilde;">emum</expan> proportio dimidij <expan abbr="c&utilde;">cum</expan> addita maiore ad <expan abbr="dimidi&utilde;">dimidium</expan>, cum addita minore, uelut maioris partis ad <expan abbr="minor&etilde;">minorem</expan>.<emph.end type="italics"/></cell>

<cell>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>

Line 885

</row>

<row>

<cell>CLIIII.</cell>

<cell>S<emph type="italics"/>i lineæ datæ alia linea adiungatur, ab extremitatibus autem prioris lineæ duæ rectæ in unum punctum concurrant proportionem habentes, quam mediam inter tota m & adiectam, & adiectam erit punctus, concur&longs;us à puncto extre-mo lineæ adiectæ di&longs;tans per lineam mediam.<emph.end type="italics"/> Q<emph type="italics"/>uod &longs;i ab extremo alicuius li-neæ æqua'is mediæ, &longs;eu peripheria circuli, cuius &longs;emidiameter &longs;it media linea duæ lineæ ad prædicta puncta producantur, ip&longs;æ erunt in proportione mediæ ad adiectam.<emph.end type="italics"/></cell>

<cell>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 æqua'is mediæ, &longs;eu peripheria circuli, cuius &longs;emidiameter &longs;it media linea duæ lineæ ad prædicta puncta producantur, ip&longs;æ erunt in proportione mediæ ad adiectam.<emph.end type="italics"/></cell>

</row>

<row>

Line 905

</row>

<row>

<cell>CLVIII.</cell>

<cell>R<emph type="italics"/>ationem indicis mobilis cum rota, qua horarum numerus per ictus indicatur ex-plicare.<emph.end type="italics"/></cell>

<cell>R<emph type="italics"/>ationem indicis mobilis cum rota, qua horarum numerus per ictus indicatur explicare.<emph.end type="italics"/></cell>

</row>

<row>

Line 920

</row>

<row>

<cell>S<emph type="italics"/>i fuerint duo trianguli, quorum ba&longs;es in eadem linea &longs;int con&longs;tituti, & æquales ad unum punctum terminati, & latus unum commune inter reliqua quantita-te medium nece&longs;&longs;e e&longs;t angulum à maioribus lineis <expan abbr="content&utilde;">contentum</expan> minorem e&longs;&longs;e.<emph.end type="italics"/></cell>

<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, & conca-uæ proportiones datæ &longs;int inue&longs;tigare.<emph.end type="italics"/></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>

Line 961

</row>

<row>

<cell>CLXIX.</cell>

<cell>P<emph type="italics"/>roportionem mu&longs;icam in in&longs;trumentis declarare iuxta compo&longs;itionis ra-tionem.<emph.end type="italics"/></cell>

<cell>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>

Line 971

</row>

<row>

<cell>CLXXI.</cell>

<cell>P<emph type="italics"/>ropo&longs;itis duobus quibuslibet numeris, quotuis alios &longs;eu in continuum &longs;eu medios in continua proportione arithmetica, geometrica & mu&longs;ica in-uenire.<emph.end type="italics"/></cell>

</row>

<row>

Line 991

</row>

<row>

<cell>CLXXV.</cell>

<cell>C<emph type="italics"/>au&longs;am uarietatis diametrorum ex &longs;uppo&longs;itis concentricis demon&longs;tra-re.<emph.end type="italics"/></cell>

<cell>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>

Line 1001

</row>

<row>

<cell>CLXXVII.</cell>

<cell>S<emph type="italics"/>i proportio aliqua ex duabus proportionibus eiu&longs;dem quantitatis ad alias duas componatur, erit proportio illarum duarum eadem proportioni producti ex proportione in primam duarum quantitatum, detracta prio-re illa quantitate, quæ ad duas comparatur, ad eandem priorem quanti-tatem.<emph.end type="italics"/></cell>

<cell>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 declara-re.<emph.end type="italics"/></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>

<cell>S<emph type="italics"/>i duobus totis duæ portiones &longs;imiles ab&longs;cindantur ab ei&longs;dem denuò, & ab&longs;cißis portionibus partes eædem auferantur, denuoque ac denuò quoties libuerit à portionibus, & ù re&longs;iduis ip&longs;arum quantitatum partes eædem auferantur, erit re&longs;iduí ad re&longs;iduum, ueluti totius ad totum.<emph.end type="italics"/></cell>

</row>

<row>

<cell>CLXXX.</cell>

<cell>S<emph type="italics"/>i aliqua quantitas in duas partes diuidatur, fueritque alicuius quantitatis ad partes illas compo&longs;ita proportio, non poterit eiu&longs;dem quantitatis ad par-tes alias quantitatis diui&longs;a, aliter proportio eadem componi.<emph.end type="italics"/></cell>

<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;ecun-dam & tertiam, & rur&longs;us quartæ ad quintam & &longs;extam: ita &longs;e habebit proportio &longs;ecundæ ad tertiam, ad proportionem quintæ ad &longs;extam, uelut producti ex proportione in &longs;ecundam detracta prima ad primam ad pro-ductum ex proportione in quintam, detracta quarta ad quartam.<emph.end type="italics"/></cell>

<cell>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 propor-tionum totius ad reliquum re&longs;idui inuenire.<emph.end type="italics"/></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>

Line 1036

</row>

<row>

<cell>CLXXXIIII.</cell>

<cell>Q<emph type="italics"/>uæcunque grauia in uorticibus aquarum, merguntur, in medio uorticis, pri-mum uer&longs;a mergantur.<emph.end type="italics"/></cell>

<cell>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 op-po&longs;ito modo inuicem &longs;e habeant, declarare.<emph.end type="italics"/></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 quar-tæ ad quintam maior quàm &longs;ecundæ ad tertiam.<emph.end type="italics"/> Q<emph type="italics"/>uod &longs;i fuerit maior<emph.end type="italics"/></cell>

<cell>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>

Line 1057

</row>

<row>

<cell>CLXXXVII.</cell>

<cell>S<emph type="italics"/>i ei&longs;dem uiribus & ‘eadem’ proportione cum auxilio ponderis tertij quar-tum pondus moueatur quibus &longs;ecundum, auxilio primi nece&longs;&longs;e e&longs;t <expan abbr="quart&utilde;">quartum</expan> pon dus tardius & maiore cum difficultate moueri quàm &longs;ecundum.<emph.end type="italics"/></cell>

<cell>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> pon dus tardius & maiore cum difficultate moueri quàm &longs;ecundum.<emph.end type="italics"/></cell>

</row>

<row>

<cell>CLXXXVIII.</cell>

<cell>S<emph type="italics"/>i uires aliquæ moueant cum ponderibus aliqua pondera, ut compo&longs;ita pro-portio &longs;it eadem proportioni uirium & duorum ponderum mouentium ag-gregatum æquale duorum ponderum, ubi maior fuerit partium in æqualitas, ibi erit maior difficultas.<emph.end type="italics"/></cell>

<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 coapte-tar, facilius deor&longs;um trahetur quàm quod maius e&longs;t & propius.<emph.end type="italics"/></cell>

<cell>S<emph type="italics"/>i pondus minus ad longitudinem minorem &longs;ub æquali proportione coaptetar, facilius deor&longs;um trahetur quàm quod maius e&longs;t & propius.<emph.end type="italics"/></cell>

</row>

<row>

<cell>S<emph type="italics"/>i fuerit primum graue minus &longs;ecundo, & &longs;ecundum minus tertio, proportio autem primi ad &longs;ecundum multo maior quàm &longs;ecundi ad tertium, po&longs;ibile erit propo&longs;itis uiribus ei&longs;dem addere pondus <expan abbr="&longs;ec&utilde;do">&longs;ecundo</expan>, ut ip&longs;um & tertium mouea-tur faciliùs ab ei&longs;dem uiribus, & primo uel &longs;ecundo quàm antea.<emph.end type="italics"/></cell>

<cell>S<emph type="italics"/>i fuerit primum graue minus &longs;ecundo, & &longs;ecundum minus tertio, proportio autem primi ad &longs;ecundum multo maior quàm &longs;ecundi ad tertium, po&longs;ibile erit propo&longs;itis uiribus ei&longs;dem addere pondus <expan abbr="&longs;ec&utilde;do">&longs;ecundo</expan>, ut ip&longs;um & tertium 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 & mi-nore pondere in maius, addiderisque in&longs;uper quantum e&longs;t productum dimidij ui rium in &longs;e latus aggregati detracto dimidio uirium, dicetur pondus auxiliare æqualis proportionis.<emph.end type="italics"/></cell>

<cell>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 ui rium in &longs;e latus aggregati detracto dimidio uirium, dicetur pondus auxiliare æqualis proportionis.<emph.end type="italics"/></cell>

</row>

<row>

<cell>CXCII.</cell>

<cell>S<emph type="italics"/>i ex medio diametri linea ad perpendiculum erigatur ad circuli peripheri-am, ex eo puncto autem quotlibet lineæ ducantur &longs;eu intus ad circun ferentiam u&longs;que, &longs;eu extra ad diametrum, erit proportio totius lineæ ad totam uelut mu-tuo partis ad partem.<emph.end type="italics"/></cell>

<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>

Line 1127

</row>

<row>

<cell>S<emph type="italics"/>i duæ lineæ non &longs;ecantes circuli peripheriam in unum punctum ex ea coe-ant exterius, nece&longs;&longs;e e&longs;t illas peripheria contenta e&longs;&longs;e maiores.<emph.end type="italics"/></cell>

<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>

Line 1168

<row>

<cell>S<emph type="italics"/>i &longs;uperficies rectangula in duas partes æquales diui&longs;a intelligatur, quæ am-bæ quadratæ &longs;int, itemque in duas inæquales, erit parallelipedum ex latere mediæ partis in totam &longs;uperficiem maius aggregato parallelipedorum ex partibus inæqualibus in latera alterius partis mutuo, in eo, quod fit ex dif ferentia lateris minoris partis à mediæ latere in differentiam maioris par-tis &longs;uperficiei à media &longs;uperficie bis, & ex differentia amborum laterum inæqualium iunctorum ad ambo latera, æqualia iuncta in minorem par-tem &longs;uperficiei.<emph.end type="italics"/></cell>

<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 dif ferentia 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 refle-ctantur, nece&longs;&longs;e e&longs;t angulos cum dimetiente factos æquales e&longs;&longs;e.<emph.end type="italics"/> V<emph type="italics"/>nde ma-nife&longs;tum e&longs;t, protractam diametrum angulum &longs;uppo&longs;itum per æqualia di-uidere.<emph.end type="italics"/></cell>

<cell>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 par-tem protrahantur, &longs;emper magis di&longs;tabunt inuicem ea ex parte, & nun-quam concurrent.<emph.end type="italics"/></cell>

<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 inue-nire lineas, quæ non in alium punctum reflectentur.<emph.end type="italics"/></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>

Line 1193

</row>

<row>

<cell>CCXIIII.</cell>

<cell>S<emph type="italics"/>i extra circulum duo puncta æqualiter à centro di&longs;tantia &longs;ignentur, erit pun-ctum reflexionis æqualis in medio arcus intercepti inter lineas, quæ à cen tro ducuntur ad illa puncta.<emph.end type="italics"/> S<emph type="italics"/>i uerò unum centro proximius fuerit altero, punctum æqualitatis in peripheria tantò longius, uer&longs;us breuiorem line-am, quantò punctum aliud à centro magis di&longs;teterit.<emph.end type="italics"/></cell>

<cell>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 cen tro ducuntur ad illa puncta.<emph.end type="italics"/> S<emph type="italics"/>i uerò unum centro proximius fuerit altero, punctum æqualitatis in peripheria tantò longius, uer&longs;us breuiorem 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 alterutrin-que.<emph.end type="italics"/></cell>

</row>

<row>

<cell>CCXVI.</cell>

<cell>S<emph type="italics"/>i fuerint circuli duo inæquales, & extra utrunqúe punctum ad illud ex mi-nore reflexè per magnam partem minoris à maiore perueuire pote-runt.<emph.end type="italics"/></cell>

<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>

Line 1218

</row>

<row>

<cell>CCXIX.</cell>

<cell>R<emph type="italics"/>ationem eorum quæ apparent circa<emph.end type="italics"/> S<emph type="italics"/>olem &longs;peculo in aqua po&longs;ito decla-rare.<emph.end type="italics"/></cell>

<cell>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>

Line 1228

</row>

<row>

<cell>CCXXI.</cell>

<cell>M<emph type="italics"/>agnitudo<emph.end type="italics"/> L<emph type="italics"/>unæ & cæterorum a&longs;trorum digno&longs;citur ex proportione alio-rum ad eam iuxta di&longs;tantiam: ip&longs;ius uerò iuxta rationem pupillæ ad<emph.end type="italics"/> L<emph type="italics"/>u-nam di&longs;tantiæ ratione.<emph.end type="italics"/></cell>

<cell>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>

Line 1243

</row>

<row>

<cell>CCXXIIII.</cell>

<cell>N<emph type="italics"/>eque temporis totius, ut imaginamur, ip&longs;um e&longs;&longs;e infinitum, neque æui ui-tarum proportio ulla e&longs;t ad tempus, quod pote&longs;tate e&longs;t, utpotè diem<emph.end type="italics"/></cell>

<cell>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>

Line 1259

</row>

<row>

<cell>CCXXVI.</cell>

<cell>P<emph type="italics"/>roportio &longs;ublimis non con&longs;i&longs;tit in magnitudine, &longs;ed ordine, iuxta quem diffe-rentia e&longs;t eius quod e&longs;t ante & po&longs;t.<emph.end type="italics"/></cell>

<cell>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 quand am habent.<emph.end type="italics"/></cell>

<cell>V<emph type="italics"/>itæ iuxta numerum perfectionum in comparatione ad cogitationem no&longs;tram proportionem quand am habent.<emph.end type="italics"/></cell>

</row>

<row>

<cell>CCXXVIII.</cell>

<cell>P<emph type="italics"/>roportionem &longs;cientiæ futurorum & cæterorum occultorum con&longs;idera-re.<emph.end type="italics"/></cell>

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

</row>

<row>

Line 1294

</row>

<row>

<cell>CCXXXIII.</cell>

<cell>Q<emph type="italics"/>uod e&longs;t in mundo incorporeo æternum e&longs;t, beatum, &longs;ecurum, immutabile &longs;ecundum locum, &longs;olum iuxta e&longs;&longs;entiam fit: iuxta quod uelut à leui &longs;u-&longs;urro aquæ & aura æ&longs;tiua demulcetur.<emph.end type="italics"/></cell>

<cell>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>

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

</section>

Line 1308

<chap>

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

<s id="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>

Line 1323

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

Line 1337

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

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

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

Line 1358

<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, & diuen&longs;arum &longs;pecierum, <lb/>ut arcuum per &longs;inus, qua utuntur A&longs;tronomi.</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

Line 1377

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

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

Line 1419

<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="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>

Line 1433

<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>

<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>

Line 1517

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

<s id="id000114">Proportionem defectus, &longs;eu detractæ quantitatis ad defectum <lb/>e&longs;&longs;e po&longs;&longs;e, ut quantitatis ad quantitatem dicuntur communes ani<lb/>mi &longs;entcntiæ, quæ ex intellectu &longs;olo terminorum, quod ueræ &longs;int, <lb/>cogno&longs;cuntur. </s>

<s id="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 interillas. </s>

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

Line 1527

<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>

Line 1584

<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 hæ <lb/>beant unius rationem, nomen &longs;umit proportio à duobus numeris, <lb/>&longs;in autem &longs;it altera monas, erit per &longs;ecundam animi communem &longs;en <lb/>tentiam, proportio numerus ip&longs;e Ideò patet, quod dicitur.</s>

<s id="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>

Line 1628

<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="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>

Line 1648

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

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

Line 1662

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

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

Line 1684

<s id="id000179">Nam &longs;i <lb/><arrow.to.target n="marg11"/><lb/>rur&longs;um con&longs;tituantur fad e ut a ad b cùm f &longs;it proportio, & k ad f ut <lb/><arrow.to.target n="marg12"/><lb/>c ad d, erit k ad e, ut g ad h, k autem fit ex ductu proportionis a ad b, <lb/>quæ e&longs;t fin proportionem c ad d, liquet igitur propo&longs;itum.</s>

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

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

<s id="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, ftatuatur ergo d prima quantitas e &longs;e<lb/>cunda & tertia f quarta. </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>

Line 1754

<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>

Line 1921

<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"/>

Line 2014

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

<s id="id000223">& perpræ cedentem c ad f producitur ex <lb/>a ad b, & d ad e, & per quartam rur&longs;us ex a ad e, <lb/>& d ad b. </s>

<s id="id000224">Et per præcedentem rut&longs;us a ad e ex c <lb/>ad f & b ad d, igitur per quartam eadem produ<lb/>cetur ex c ad d & b ad f. </s>

<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>

Line 2022

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

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

<s id="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="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>

Line 2085

</row>

<row>

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

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

</row>

<row>

<cell/>

Line 2133

</row>

<row>

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

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

</row>

<row>

<cell/>

Line 2213

</row>

<row>

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

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

</row>

<row>

<cell/>

Line 2255

<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"/>

Line 2288

<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;cilicer abiectis amba<lb/>bus æqualibus. </s>

<s id="id000243">Sit gratia exempli prima æqualis quintæ: & quia <lb/>in octauo modo proportio <expan abbr="&longs;ec&utilde;di">&longs;ecundi</expan> ad quartum producitur ex pro<lb/>portione primi ad quintum, & &longs;exti ad tertium, ergo per expo&longs;ita <lb/>proportio &longs;ecundi ad quartum, ut &longs;exti ad tertium, & ita permutan<lb/>do, & conuertendo &longs;ecundi ad &longs;extum, ut quarti ad tertium, & tertij </s>

Line 2305

<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>

Line 2337

<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 cut a b ad a c, <lb/>ita b c ad d: eritque iterum permutando a b ad b c, <lb/>ut a c ad d, & &longs;umatur quædam quantitas e eiu&longs;

<pb pagenum="12" 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>

<pb pagenum="12" xlink:href="015/01/031.jpg"/>dem tamen generis, cum illis dico quòd proportio e ad d e&longs;t com<lb/>po&longs;ita ex proportionibus e ad a c, & e ad b c. </s>

<s id="id000262">Po&longs;ita ergo e tan<08> &longs;u<lb/>periore numero, & a c & c b inferioribus, erit ex octaua propo&longs;itio<lb/>ne huius proportio productorum ex e in a c, & coniunctorum, & <lb/>ex con&longs;equenti per primam &longs;ecundi Elementorum producti ex e in <lb/>a b ad productum ex a c in c b compo&longs;ita ex proportionibus e ad <lb/>a c, & e ad c b: at quod fit ex a c in c b, e&longs;t æquale ei quod fit ex a b in <lb/>d, eo quòd a b, a c, c b & d &longs;unt omiologæ per decimam&longs;extam &longs;exti <lb/><expan abbr="Elem&etilde;torum">Elementorum</expan>: Proportio igitur producti ex e in a b ad productum <lb/>ex d in a b e&longs;t compo&longs;ita ex proportionibus e ad a c, & e ad e b: At <lb/>proportio producti ex e in a b ad productum ex d in a b, e&longs;t uelut e <lb/><arrow.to.target n="marg32"/><lb/>ad d. </s>

<s id="id000262">Po&longs;ita ergo e tan<08> &longs;u<lb/>periore numero, & a c & c b inferioribus, erit ex octaua propo&longs;itio<lb/>ne huius proportio productorum ex e in a c, & coniunctorum, & <lb/>ex con&longs;equenti per primam &longs;ecundi Elementorum producti ex e in <lb/>a b ad productum ex a c in c b compo&longs;ita ex proportionibus e ad <lb/>a c, & e ad c b: at quod fit ex a c in c b, e&longs;t æquale ei quod fit ex a b in <lb/>d, eo quòd a b, a c, c b & d &longs;unt omiologæ per decimam &longs;extam &longs;exti <lb/><expan abbr="Elem&etilde;torum">Elementorum</expan>: Proportio igitur producti ex e in a b ad productum <lb/>ex d in a b e&longs;t compo&longs;ita ex proportionibus e ad a c, & e ad e b: At <lb/>proportio producti ex e in a b ad productum ex d in a b, e&longs;t uelut e <lb/><arrow.to.target n="marg32"/><lb/>ad d. </s>

<s id="id000263">per &longs;uppo&longs;ita igitur proportio e ad d e&longs;t compo&longs;ita ex propor<lb/>tionibus e ad a c, & e ad b c, quod fuit demon&longs;trandum.</s>

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

Line 2358

<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="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>

Line 2393

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

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

Line 2410

<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="id000287">Quia ergo ex c in b fit f, ex c in d h, erit f ad h, <lb/>ut b ad d, igitur ut e ad c, &longs;ed a ad c, ut g ad h igi<lb/><arrow.to.target n="marg41"/><lb/>tur a e ad c, ut k ad h, &longs;ed k ad h cómponitur ex <lb/>proportionibus a ad c, & b ad d. </s>

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

<s id="id000289">For&longs;an quis dicat hanc eandem e&longs;&longs;e <lb/>octauæ &longs;ed <expan abbr="nõ">non</expan> e&longs;t, in illa enim proportio com<lb/>paratur ad productum, in hac ad unam ex <lb/>quantitatibus.</s>

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

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

Line 2430

<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="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>

Line 2468

<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="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>

Line 2489

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

<s id="id000311">Si fuerint quatuor quantitas-proportio confu&longs;a aggregati pri<lb/>mæ & tertiæ ad aggregatum &longs;ecundæ, & quartæ erit ut monadis <lb/>addito prouentu, qui fit diui&longs;a differentia differentiarum primæ & <lb/>&longs;ecundæ, atque quartæ & tertiæ per aggregatum tertiæ, & quartæ ad <lb/>ip&longs;am monadem.</s>

<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 fmaior d in f g, & <lb/>differentia f g & a h &longs;it a k: dico proportio<lb/>nem a b, & d confu&longs;am ad c & e f, e&longs;&longs;e ut mo <lb/>nadis addito prouentu, uel detracto a k diui&longs;æ per aggregatum c. <lb/>& e f ad ip&longs;am monadem, & manife&longs;tum e&longs;t, quòd pote&longs;t continge<lb/>re pluribus modis: Primus ut a b &longs;it maior c & e f minor d, & tunc <lb/>differentiæ coniungentur, & prouentus, addetur monadi. </s>

<s id="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 coninncta ex monade & proportio<lb/>ne a k ad aggregatum c & e f, quod erat demon&longs;trandum.</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;extadecima.</s>

Line 2524

<s id="id000324">Hæc magis reducit confu&longs;am proportionem ad notitiam, quàm, <lb/>præcedens, quia reducit ad proportionem <expan abbr="productã">productam</expan>, qu&etail; operatio <lb/>e&longs;t &longs;implici&longs;sima, &longs;iue per multiplicationem quantitatum fiat, duæ <lb/>&longs;unt tantum multiplicationes, &longs;iue per eundem terminum &longs;ufficit <lb/>alium addere. </s>

<s id="id000325">Summatur ergo a b, c, d & e, & non &longs;it maior propor<lb/>tio d ad e, quàm a b ad c, & &longs;tatuatur tunc prima a b, &longs;ecunda c, ter<lb/>tia d, quarta e, & po&longs;tquam non e&longs;t minor ratio a b ad c, quàm d ad <lb/>e, &longs;umatur a f ad c, ut d ad e. </s>

<s id="id000326">licet enim hoc facere. </s>

<s id="id000327">Dico quod pro<lb/>portio confufa a b & d ad c & e e&longs;t uelut producti ex aggregato a b <lb/>& d in d ad productum ex aggregato a f & d in e. </s>

<s id="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/>&longs;u&longs;a a b ad c, & d ad e, & e&longs;t proportio a b d ad c & e, producuntur <lb/>ex proportionibus a b d ad a f d, & d ad e. </s>

<s id="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>

Line 2568

<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>

Line 2583

<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>

Line 2597

<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="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>

Line 2615

<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>

Line 2625

<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>

Line 2634

<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>

Line 2643

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

Line 2653

<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="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>

Line 2662

<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>

Line 2674

<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>

Line 2685

<s id="id000380">Si quis dicat in qua proportione &longs;unt infinitæ <lb/>quantitates analogæ cum 12, quæiunctæ efficiunt 10, iunge 10 cum <lb/>12 fit 22, duc 22 in 12 fit 264, diuide 264 per 10, exit 26 2/3, & in ea pro<lb/>portione <expan abbr="er&utilde;t">erunt</expan> illæ quantitates, in qua &longs;unt 26 2/3 ad 12: duc per 5 fiunt <lb/>60, & 132 diuide per 12, exeunt 11 & 5, & ita eruntin proportione 11 <lb/>ad 5 experiaris, & inuenies, & demon&longs;tratur ex prioribus.</s>

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

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

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

Line 2694

<s id="id000383">Si fu erint aliquot quantitates arithmeticæ omiologæ, quarum <lb/>exce&longs;&longs;us &longs;it æqualis minimè, omnibus autem deficientibus &longs;upple<lb/>menta ad &etail;qualitatem maximè adiungantur, erunt quadrata omni<lb/>um quantitatum æqualium adiecto rur&longs;us quadrato primæ cum <lb/>eo quod fit ex minima primi ordinis in <expan abbr="aggregat&utilde;">aggregatum</expan> omnium quan<lb/>titatum eiu&longs;dem tripla aggregato quadra<lb/><figure id="id.015.01.036.1.jpg" xlink:href="015/01/036/1.jpg"/><lb/>torum omnium quantitatum primi ordinis <lb/><arrow.to.target n="marg66"/><lb/>pariter acceptis.</s>

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

<s id="id000386">Arithmetica di&longs;po&longs;it&etail; <lb/>ita ut minima <expan abbr="ear&utilde;">earum</expan> qu&etail; &longs;it h, &longs;it &etail;qualis diffe<lb/>renti&etail; quantitatum <expan abbr="&longs;ecund&utilde;">&longs;ecundum</expan> ordinem di&longs;po <lb/><expan abbr="&longs;itar&utilde;">&longs;itarum</expan>, uelut differentia a & b, & b & c, & c & <lb/>d, et ita de alijs, addantur <expan abbr="a&utilde;t">aunt</expan> <expan abbr="&longs;upplem&etilde;ta">&longs;upplementa</expan> &longs;in <lb/>gulis harum, quæ &longs;int i k l m n o p, ita ut <expan abbr="o&etilde;s">oens</expan> <lb/>fiant &etail;quales <expan abbr="c&utilde;">cum</expan> &longs;uis &longs;upplementis ip&longs;i line&etail; <lb/>à maiori. </s>

<s id="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="o&etilde;s">oes</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

<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>

<pb pagenum="18" xlink:href="015/01/037.jpg"/>tates, & <expan abbr="diuideren&ttilde;">diuiderentur</expan> &longs;ingul&etail; <expan abbr="&longs;ecund&utilde;">&longs;ecundum</expan> numerum <expan abbr="illar&utilde;">illarum</expan>, &longs;i quatuor in <lb/>quatuor partes æquales, &longs;i quinque in quinque, &longs;i decem in decem, eara<lb/>tione ut ultima <expan abbr="diuidere&ttilde;">diuideretur</expan>, ubi e&longs;t finis primæ partis, penultima ubi <lb/>e&longs;t finis &longs;ecundæ partis, antepenultima ubi e&longs;t finis tertiæ, & &longs;ic de <lb/>alijs. </s>

<s id="id000389">primi ordinis, & fupplementis, appellabo quantitates &longs;ecun<lb/>di ordinis: ex quo patet quòd prima <expan abbr="quãtitas">quantitas</expan> erit ex utro que ordine, <lb/>quia non e&longs;t diui&longs;a, reliquæ omnes differunt, quantitates uerò quas <lb/>adiunxi nominabo <expan abbr="&longs;upplem&etilde;ta">&longs;upplementa</expan>, & &longs;unt una minus <expan abbr="quã">quam</expan> quantitates <lb/>ordinum: ut &longs;i <expan abbr="quãtitates">quantitates</expan> ordinum &longs;int octo, erunt &longs;upplementa &longs;e<lb/>ptem, & &longs;i quantitates <expan abbr="ordin&utilde;">ordinum</expan>, e&longs;&longs;ent &longs;eptem e&longs;&longs;ent <expan abbr="&longs;upplem&etilde;ta">&longs;upplementa</expan> &longs;ex, <lb/>quia inter &longs;upplementa <expan abbr="nõ">non</expan> <expan abbr="adnumera&ttilde;">adnumeratur</expan> quantitas indiui&longs;a. </s>

<s id="id000390">Erunt er <lb/>go &longs;upplementa i k l m n o p, quætanto erunt maiora quanto quan <lb/>titates primi ordinis &longs;unt minores, & contrà tanto maiora, quanto <lb/><expan abbr="quãtitates">quantitates</expan> primi ordinis &longs;unt maiores. </s>

<s id="id000391">quantitates <expan abbr="a&utilde;t">aunt</expan> &longs;ecundi ordi <lb/>nis <expan abbr="appellabun&ttilde;">appellabuntur</expan> a, b i, ck, dl, em, fn, go, & hp. </s>

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

<s id="id000393">quæ licet <expan abbr="nõ">non</expan> &longs;it difficilis, e&longs;t <expan abbr="tam&etilde;">tamen</expan> <lb/>confu&longs;a ualde propter multitudinem <expan abbr="quantitat&utilde;">quantitatum</expan> & ordinum. </s>

<s id="id000394">Dico <lb/>ergo &qring;d aggregatum <expan abbr="quadrator&utilde;">quadratorum</expan> quantitatum &longs;ecundi ordinis pri <lb/>mo quadrato bis repetito, &longs;eu uno addito <expan abbr="c&utilde;">cum</expan> eo quod fit ex minima <lb/>in aggregatum quantitatum primi ordinis e&longs;t <expan abbr="tripl&utilde;">triplum</expan> aggregato ex <lb/>quadratis omnibus <expan abbr="quantitat&utilde;">quantitatum</expan> <expan abbr="eiu&longs;d&etilde;">eiu&longs;dem</expan> primi ordinis, & utres exem <lb/>plo facilius innote&longs;cat, &longs;int <expan abbr="quãtitates">quantitates</expan> primi ordinis 8. 7. 6. 5. 4. 3. 2. 1. <lb/>quorum quadrata &longs;int 64. 49. 36. 25. 16. & 9.4 & 1. quæ iuncta <expan abbr="faci&utilde;t">faciunt</expan> <lb/>204, dico quod &longs;i &longs;umamus quadrata omnium <expan abbr="quãtitatum">quantitatum</expan> &longs;ecundi <lb/>ordinis, quæ &longs;unt octies 64, & eis addiderimus unum <expan abbr="quadrat&utilde;">quadratum</expan> ex <lb/>his, ut fiant nouies 64, & erunt 556, &longs;imul iuncta & eis addamus, &qring;d <lb/>fit ex 1 quantitate minima primi ordinis in 36 aggregatum quanti<lb/>tatum omnium primi ordinis, & e&longs;t tale <expan abbr="product&utilde;">productum</expan> 36, ut fiat totum <lb/>612, quod tale 612 e&longs;t triplum 204, aggregati <expan abbr="quadrator&utilde;">quadratorum</expan> primi or<lb/>dinis unius demon&longs;tratio h&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">aunt</expan> &etail;qualis m. </s>

<s id="id000395">Quia ex quarta &longs;ecundi Element. <lb/>Euclidis &longs;ingula quadrata <expan abbr="quantitat&utilde;">quantitatum</expan> <expan abbr="diui&longs;ar&utilde;">diui&longs;arum</expan> &longs;ecundi ordinis con<lb/>&longs;tant ex quatuor partibus quarum du&etail; &longs;unt quadrata partium, reli<lb/>quæ duæ &longs;unt producta ex partibus <expan abbr="inuic&etilde;">inuicem</expan> bis, & quia h fuit æqua<lb/>lis 1, & p &etail;qualis b, quia &longs;upplementa <expan abbr="fuer&utilde;t&etail;qualia">fuerunt &etail;qualia</expan> mutuò quanti<lb/>tatibus, & ita c æqualis o & k æqualis g & d, æqualis n & l, æqualis <lb/>f, e <expan abbr="a&utilde;t">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 hab entibus <expan abbr="&longs;upplem&etilde;ta">&longs;upplementa</expan> mutuò æqualia ip&longs;is quan <lb/>titatibus quod quadrata partium <expan abbr="er&utilde;t">erunt</expan> dupla quadratis primarum <lb/>quantitatum: ueluti capio b i &longs;ecundam & h p ultimam, <expan abbr="quar&utilde;">quarum</expan> qua

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<s id="id000399">atuerò addito quadrato a <lb/>quadratis b c d e f g h, & erunt quadrata omnium quantitatum, & <lb/>quadratis b i, c k, d l, e m, f n, g o, h p, duplo quadrati a &longs;cilicet &longs;emel, <lb/>quia a e&longs;t ex &longs;ecundo ordine quantitatum, & &longs;emel, quia hoc fuit a&longs;<lb/>&longs;umptum in Problemate. </s>

<s id="id000400">Sequitur ut quadrata omnia <expan abbr="quãtitatum">quantitatum</expan> <lb/>&longs;ecundi ordinis, prout &longs;unt diui&longs;a in partes addito quadrato a, &longs;int <lb/>dupla quadratis primarum quantítatum, &longs;imul pariter acceptis. </s>

<s id="id000401">Re <lb/>liquum e&longs;t modo ut o&longs;tendamus dupla <expan abbr="illor&utilde;">illorum</expan> productorum, cum <lb/>eo quod fit ex minima quantitate, &longs;cilicet h in aggregatum ip&longs;arum <lb/>quantitatum primi ordinis e&longs;&longs;e æquale quadratis, <expan abbr="quantitat&utilde;">quantitatum</expan> eiu&longs;<lb/>dem primi ordinis pariter acceptis. </s>

<s id="id000402">Con&longs;tatigitur, quod duplum i<lb/>in b e&longs;t æquale duplo h in ip&longs;um b, quia h & i &longs;unt æquales, & du<lb/>plum k in ip&longs;um c, e&longs;t æquale quadruplo h in idem c, quia k e&longs;t du<lb/>pla h, & &longs;imiliter duplum l in ip&longs;um d e&longs;t æquale &longs;excuplo, h in d, <lb/>quia l e&longs;t tripla h, & ita procedendo erunt illa dupla producta æ<lb/>qualia productis ex h in ip&longs;as quantitates toties &longs;umptis quantus <lb/>e&longs;t numerus, qui prouenit duplicato numero, &longs;ecundum <expan abbr="qu&etilde;">quem</expan> h con <lb/>tinetur in illo &longs;upplemento, exemplum uolo duplum producti lin <lb/>d bis, &longs;cio quòd &longs;upplementum l continet h ter, duplicabo tria & fi<lb/>ent &longs;ex, <expan abbr="igi&ttilde;">igitur</expan> <expan abbr="dupl&utilde;">duplum</expan> lin d æquale e&longs;t &longs;excuplo h in ip&longs;um d. </s>

<s id="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 quantitarum, at quadratum a e&longs;t &etail;quale producto ex h in eam, <lb/>qu&etail; talem habet proportionem ad ip&longs;um a, <expan abbr="qual&etilde;">qualem</expan> habet a ad ip&longs;um <lb/><arrow.to.target n="marg68"/><lb/>h per demon&longs;trata ab Euclide, & pariter de quadrato b, quod e&longs;t &etail;<lb/>quale ei quod fit ex h in eam quæ toties continet b, quotiens b con <lb/>tinet h, & ita quadratum c æquale e&longs;t ei, quod continetur &longs;ub h, & <lb/>habente proportionem ad b eandem, quam b ad h, & &longs;imiliter de <lb/>quadrato c & omnibus reliquis, u&longs;que ad h ip&longs;um. </s>

<s id="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

<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>

<pb pagenum="20" xlink:href="015/01/039.jpg"/>pli quadratum a, erit æquale producto ex h in omnes quatitates &longs;e<lb/>cundas, quia quotus e&longs;t numerus quantitatum, totus e&longs;t numerus <lb/>&longs;ecundum quem a continet h, & &longs;imiliter quotus e&longs;t numerus quan <lb/>títatum incipiendo à b, & quotus e&longs;t numerus quantitatum incipi<lb/>endo à c, toties b uel c <expan abbr="contin&etilde;t">continent</expan> h, & ita de alijs, quadrata ergo om<lb/>nium quantitatum &longs;imul iuncta &longs;unt æqualia productis ex h in &longs;in<lb/>gulas illarum toties &longs;umptis, quoties illæ <expan abbr="cõtinent">continent</expan> h, &longs;eu quotus e&longs;t <lb/>numerus illius quantitatis, incipiendo ab h, & <expan abbr="numerãdo">numerando</expan> uer&longs;us a. <lb/></s>

<s id="id000405">Rur&longs;us dico, quod productum multiplicis cuiuslibet <expan abbr="quãtitatis">quantitatis</expan> in <lb/>minimam, &longs;eu quadratum eiu&longs;dem quantitatis &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>

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<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

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<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="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>

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<s id="id000423">Hæcigitur propo&longs;itio nihil aliud innuit, quàm quod <lb/>in hoc ca&longs;u productio, quæ&longs;olet fieri ex tribus proportionibus fiat <lb/>ex duabus tantum.</s>

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

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

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<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>

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<s id="id000435">Sit proportio a ad b maior quàm c <lb/><arrow.to.target n="marg73"/><lb/>ad d, dico, quod confu&longs;a ex a c ad b d <lb/>e&longs;t maior, quàm c ad d, et minor quàm <lb/>a ad b, ut enim c ad d ita fiat e ad b, erit que per tertiamdecimam ha<lb/><arrow.to.target n="marg74"/><lb/>rum e c ad b d confu&longs;a minor quàm a c ad b d, nam e e&longs;t minor a, <lb/>quia proportionem habent minorem ad b quam a eo quòd e ha<lb/>bet proportionem ad b, quam c ad d, quæ <expan abbr="aut&etilde;">autem</expan> c ad d minor, quám <lb/>a ad b, ut &longs;uppo&longs;itum e&longs;t, igitur e c ad b d minor, quàm a b ad c d, e b <lb/>autem ad c d e&longs;t, ut demon&longs;tratum e&longs;t qualis c ad d, ergo c ad d mi<lb/>nor, quàm confu&longs;a a b ad c d, quod e&longs;t &longs;ecundum per idem proba<lb/>bitur, & primum po&longs;ita f ad d, ut a ad b, eritque a maior c, igitur ma<lb/>ior proportio a f ad b d, quàm a c ad b d, &longs;ed a f ad b d, ut a ad b per <lb/>candem tertiamdecimam huius ergo proportio confu&longs;a a b ad c d <lb/>e&longs;t minor, quàm a ad b.</s>

<s id="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;imatertia.</s>

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<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>

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

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

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<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>

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<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="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>

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<s id="id000487">Sed <lb/>quia fit per mu&longs;culos, qui trahuntur: hic autem motus e&longs;t uiolen<lb/>tus, ideò per con&longs;equentiam fatigat. </s>

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

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

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

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

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

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<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>

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<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>

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

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<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>

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

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<s id="id000518">Cùm enim ex præcedenti augeantur &longs;emper ob medium, & cau<lb/>fa, quæ mouet, &longs;it perpetua, & à principio æterno, quod per dictæ <lb/>æqualiter mouet, igitur motus ille fiet uelo cior in fine quàm in alia <lb/>parte temporis. </s>

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

<s id="id000522">Ex quo patet, quòd motus quadrifariam mi&longs;ti dicuntur, aut &longs;pe<lb/><arrow.to.target n="marg86"/><lb/>cie, ut cùm quis iacit lapidem è turri: uel ex occulto naturali, & uio<lb/>lento manife&longs;to: uelut cùm quis iacit lapidem, & de&longs;cendit po&longs;tmo <lb/><figure id="id.015.01.045.2.jpg" xlink:href="015/01/045/2.jpg"/><lb/>dum ex b in c motu utroque manife&longs;to, &longs;ed ex a <lb/>in b motu uiolento manife&longs;to, & naturali oc<lb/>culto: uel ratione medij, & hoc modo omnis <lb/>motus naturalis etiam non &longs;olum uiolentus e&longs;t <lb/>mi&longs;tus ex proportione uirtutis mouentis, cum motu medij, ad me<lb/>dium ip&longs;um, uel &longs;i uiolentus &longs;it ex proportione uirtutis mouentis,

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

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

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<s id="id000530">A mobile moueatur in b medio rariore, & in c den&longs;io<lb/><arrow.to.target n="marg87"/><lb/>re, igitur b minus re&longs;i&longs;tit, quàm c & magis adiuuat, quia <lb/>uelociùs mouetur: igitur duplici de cau&longs;a a mouebitur <lb/>uelociùs in b quàm in c: & quia per corrolarium trige&longs;i<lb/>mæ, & præcedentis proportio finis (ubi æqualiter moueantur) ad <lb/>&longs;ua principia maior erit in d, quàm in e: ergo per <expan abbr="demõ&longs;trata">demon&longs;trata</expan> à Cam <lb/>pano po&longs;ita d prima, b &longs;ecunda, e tertia, c quarta, maior erit propor<lb/>tio d ad e, quàm b ad c quod fuit propo&longs;itum in naturali.</s>

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

<s id="id000532">Propo&longs;itio trige&longs;ima ertia.</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="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="id000537">SCHOLIVM PRIMVM.</s>

<s id="id000538">Ne tamen &longs;ine exemplo intelligas hanc duplicatam rationem, <lb/>proponatur craritas quatuor, d unum, a pondus duodecim libra<lb/><figure id="id.015.01.047.1.jpg" xlink:href="015/01/047/1.jpg"/><lb/>rum, tunc c 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;exde cupla, & hæc e&longs;t duplicata quadruplæ <lb/>raritatis c ad raritatem d.</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="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>

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<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/>ctiam a&longs;&longs;umitur a b in a c ex diffinitione Eucli </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>

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<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>

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<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="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>

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

<s 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 decimamquar<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="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>

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<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>

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<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="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>

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<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>

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<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>

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

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<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="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>

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<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>

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

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<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 inclin abit ex uige&longs;imatertia propo&longs;itione. </s>

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<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 hypomo chlion moueretur infinita ui, &longs;i po&longs;<lb/>&longs;et e&longs;&longs;e punctus: &longs;ed quia in extrema &longs;uperficie cylindri, ideò pote&longs;t <lb/>aliqua ui retineri.</s>

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<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;imamnonam propo&longs;itionem.</s>

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<s id="id000668">Et libella nihil retinet ni&longs;i quantum e&longs;t pondus eius quod cu

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<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>

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<s id="id000674">Etrota, 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>

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

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<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>

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<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">aunt</expan> idem uelle, & po&longs;&longs;e. </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

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<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>

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<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>

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

<s 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/>decimanona quinti Euclidis, m ad o, ut o ad q, & ita circulus e&longs;t ana<lb/>logus inter &longs;patium pertran&longs;itum à motore uelociori, & inter diffe<lb/>rentiam &longs;patij quæ accidit, dum uelocior motor pertran&longs;it circu<lb/>lum, id e&longs;t quòd circulus a c d e&longs;t analogus inter c d l b, & circulos <lb/>pertran&longs;itos a b a cum portione b d. </s>

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

<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>

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

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

<pb pagenum="38" 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>

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<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>

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<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>

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<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;exdecim tria & tria, & rur&longs;us ex &longs;exdecim 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>

<pb pagenum="39" xlink:href="015/01/058.jpg"/>cir cuitus c, & relinquentur 3 3/4 &longs;equetur igitur, ut &longs;it proportio 17 ad <lb/>13, & 2 1/2 ad 1/2 & 3 1/3 ad 3 eadem, & ita 17/13, 5/2 & 10/9 eadem &longs;i iam &longs;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>

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

<s 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>

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<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">cunque<lb/>tempora</expan> &longs;int æqualia, erunt & circuitus totidem numero, & exce&longs;<lb/>&longs;us æquales etiam a c, c d, d e, e f, f g, g h, h k, <lb/><figure id="id.015.01.059.1.jpg" xlink:href="015/01/059/1.jpg"/><lb/>k 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>

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<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="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>

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<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>

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<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>

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<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="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;imaprima.</s>

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<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>

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<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 fex <lb/>cum tertia, diuidemus decem per 6 1/3 exit <lb/>1 11/29, dico quod non prius, neque in alio <lb/><figure id="id.015.01.061.1.jpg" xlink:href="015/01/061/1.jpg"/><lb/>puncto. </s>

<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, doneciterum coniung antur 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="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>

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<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 inconimen&longs;æ, dico quòd a b c nunquam con<lb/>uenient in aliquo puncto, &longs;eu primo, &longs;eu alio à prim o: &longs;i non con

<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>

<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>

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<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 product a proportio circuitus a b ad circuitum m ex <lb/>proportione c n ad o k, et &longs;emidimetientis m ad <expan abbr="id&etilde;">idem</expan> o k, ergo per 26 <lb/>proportio numeri circuitus unius p <expan abbr="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>

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<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/>corundem.<lb/><arrow.to.target n="marg143"/></s>

<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>

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<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>

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<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>

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<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="medi-cament&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

<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 ordl <lb/>nis, & non lædere linguam in gu&longs;tu, & alia tertij ordinis, quæ non <lb/>&longs;olum lædet linguam, &longs;ed &longs;en&longs;um eius corrumpet, et de&longs;truet, quod <lb/>contingit propter &longs;ub&longs;tantiam tenuem cra&longs;&longs;æ mi&longs;tam cum &longs;iccitate <lb/>pari ip&longs;i calori. </s>

<s 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 con&longs;u&longs;a &longs;it per &longs;eip&longs;a materia ab&longs;que <lb/>eo, quod difficultatem difficultati addamus, &longs;olum ergo eas dubita<lb/>tiones adiungemus, quas <expan abbr="uol&etilde;tes">uolentes</expan> declarare propo&longs;itionem præ&longs;en <lb/>tem, neque &longs;uperfugere, neque declinare po&longs;&longs;umus. </s>

<s 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>

<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;i derandum relinquo.</s>

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<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/>locuin, “quia pendent ex demon&longs;tratione hac: “duc numerum ordi<lb/>nis &longs;ingulorum medicamentorum in numerum quantitatis, &longs;imilia <lb/>iunge, di&longs;similia detrahe, quod fit, diuide per aggregatum, quanti<lb/>tatum, exibit numerus ordinis compo&longs;iti. </s>

<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 or dinis <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>

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<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 relin quuntur octo &longs;cilicet caliditatis, diuido per &longs;ex ag

<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>

<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;ecund i 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>

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<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 crunt 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="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>

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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>

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<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>

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<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, quarndoquidem fortuiti mo<lb/>tus, qui &longs;unt multo tardiores non latentnos. </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, ita que<lb/>cum hic multo tardior etiam in &longs;ublimi, dum e&longs;t, nos latere non <lb/>queat, multo minus po&longs;&longs;et naturalis latere, &longs;i adeò uelox & in ea<lb/>dem parte <expan abbr="a&etilde;ris">aerris</expan> e&longs;&longs;et at que continuus. </s>

<s 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, Capado cia, Sy <lb/>ria, Babylonia, Hyrcanomarí, Bactrianis, Sacís, Scythis, ac Seris, to<lb/>to præterea Oceano orientali tam ua&longs;to, & Gallica noua, terra que flo <lb/>rida non &longs;olum res e&longs;t admirabilis', & incredibilis, &longs;ed etiam aliena <lb/>à &longs;en&longs;u, & ab his, quæ eueniunt. </s>

<s 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="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>

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<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>

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<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="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="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 eretar dabit motum ad d, tanto motus a c erit tardí<lb/>or ab&longs;olutè motu ad 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/>rum ob cau&longs;am dictam. </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>

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<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>

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<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;cu forma, &longs;i ita excutiatur, ut pars grauior <expan abbr="nõ">non</expan> &longs;it, infrà opor<lb/>tet, ut circumuoluatur.</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;imaprima.</s>

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

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<s id="id000944"><margin.target id="marg168"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 60.</s>

<s id="id000946">Primùm &longs;i quod incidit, molle fuerit, non uulneratur caput, uel <lb/>pars &longs;ubiecta, quia re&longs;ilit in corpus molle: nec à molli, quia retundi<lb/>tur, pote&longs;t uulnerari: ergo nullo modo. </s>

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<s id="id000950">Secundum in omni colli&longs;ione &longs;eu duri, &longs;eu mollis, &longs;ed magis du<lb/>ri, dilabuntur partes aëris ad latera, ideo quod partes mediæ pre<lb/>muntur. </s>

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<s id="id000954">Tertium in motu uelo ci fit maior ictus & læ&longs;io, & maiora omnia <lb/>quam proproportione motus: quoniam ob uelo <expan abbr="citat&etilde;">citatem</expan> minus diffu <lb/>git aëris. </s>

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<s id="id000958">Quartum res latæ, duræ concutiunt, & non uulnerant ni&longs;i &longs;int <lb/>cum magno impetu, aut ualde graues: acutæ autem uulnerant, &longs;ed <lb/>non concutiunt, ni&longs;i parti acutæ lata &longs;uccedat.</s>

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<s id="id000960"><arrow.to.target n="marg173"/><lb/>tur, mollia autem à tenuibus, quia diuiduntur: nam mollitie excipi<lb/>unt aërem, & ita à latis non adeò patiuntur, & etiam, quoniam nec <lb/>franguntur, nec &longs;ponte &longs;cinduntur.</s>

<s id="id000962">Sextum, etiam in duris penetrat aliquid aëris, aliter tota frange<lb/><arrow.to.target n="marg174"/><lb/>rentur. </s>

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<s id="id000965">Hoc etiam uidetur &longs;en&longs;i&longs;&longs;e Philo <lb/>&longs;ophus, qui uult, quòd res franguntur ob poros.</s>

<s id="id000967">Propo&longs;itio &longs;exage&longs;ima&longs;ecunda.</s>

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<s id="id000969">Con&longs;titutum e&longs;t inuenire proportionem uirium, quæ eleuant <lb/><arrow.to.target n="marg175"/><lb/>pondus ad uires, quæ ip&longs;um in plano leui trahere po&longs;<lb/><figure id="id.015.01.072.1.jpg" xlink:href="015/01/072/1.jpg"/><lb/>&longs;unt. </s>

<s id="id000970">Vires enim, quæ eleuant pondus a &longs;unt eædem <lb/>puta b, quæ uero trahunt c, &longs;ed hæ po&longs;&longs;unt uariari, nam <lb/>quanto uinculum altius, aut decliuis locus magis, aut <lb/>a&longs;pera &longs;uperficies &longs;eu ponderis &longs;eu plani, tanto difficilius trahitur, <lb/>& maiores expo&longs;cit uires: hoc enim experimento deprehenditur. <lb/></s>

<s id="id000971">Duæ uerò po&longs;tremæ cau&longs;æ etiam per &longs;e per&longs;picuæ &longs;unt, nec demon <lb/>&longs;tratione indigent: ni&longs;i quod &longs;i planum &longs;it duri&longs;simum, ac leui&longs;si<lb/>mum, quod e&longs;t a&longs;perum facilius trahitur, quia minore &longs;ui parte pla<lb/>num tangit. </s>

<s id="id000972">Nos præterea &longs;upponimus planum æquale undique <lb/>leue durum, & corpus undique &longs;ibi &longs;imile, id e&longs;t cubi formam refe<lb/>rens, & uinculum in imo: Demon&longs;trare igitur expedit primum, <lb/>quòd in hoc ca&longs;u b e&longs;t duplum ad c. </s>

<s id="id000973">Quia enim cum a eleuatur b ui <lb/>res &longs;uperant motum ob&longs;curum &longs;eu occultum, &longs;eu pondus a, & &longs;i <lb/>permitteretur &longs;ine eo, quod &longs;u&longs;tineret, de&longs;cenderet iuxta pondus <lb/>&longs;uum, quod &longs;it d: nititur ergo per pondus d, at quia trahendo duci<lb/>tur circa medium, nam plana &longs;uperficies parum differt à rotunda <lb/>terræ ob terræ magnitudinem, media erit repugnantia: in eo enim <lb/>quod mouetur, grauitatem habet d in eo, quod <expan abbr="nõ">non</expan> remouetur nul<lb/>lam habet grauitatem, mediam ergo retinet grauitatem, quare ut b <lb/>ad d, ita c ad dimidium, grauitatis a, at b e&longs;t primum, quod pote&longs;t <lb/>mouere d, igitur c e&longs;t primum, quod pote&longs;t mouere dimidium a, ut <lb/>ergo dimidium a ad d, ita c ad b, e&longs;t igitur c dimidium b.</s>

<s id="id000975">Propo&longs;itio &longs;exage&longs;imatertia.</s>

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<s id="id000978">Sit graue a b c alligatum funibus in d ef, dico, <lb/><figure id="id.015.01.073.1.jpg" xlink:href="015/01/073/1.jpg"/><lb/>quòd facilius trahetur per fe quàm c b & e b, quàm <lb/>d a, quia &longs;i debet trahi ex a uel b, aut cadet, aut uis ex <lb/>a & b communicabitur c, igitur erit minor quàm in <lb/>c, & hoc naturaliter. </s>

<s id="id000979">Mathematica autem ratione quoniam ex a tra<lb/>hetur c, qua&longs;i per lineam d c: at attractio recta e&longs;t ualidior obliqua<lb/>igitur attractio c per d e&longs;t debilior, quàm per f. </s>

<s id="id000979">Mathematica autem ratione quoniam ex a tra<lb/>hetur c, qua&longs;i per lineam d c: at attractio recta e&longs;t ualidior obliqua <lb/>igitur attractio c per d e&longs;t debilior, quàm per f. </s>

<s id="id000980">Rur&longs;us &longs;i e trahitur <lb/>per d cùm a peruenerit in d, erit perinde ac, &longs;i attractum e&longs;&longs;et per li<lb/>neam c d, &longs;ed linea c d mouet duobus motibus, uno ad &longs;uperiora, al </s>

<s id="id000981"><arrow.to.target n="marg177"/><lb/>tero ad latus, ergo lentius ad f per d c quàm f c, quod erat demon<lb/>&longs;trandum.</s>

<s id="id000982"><margin.target id="marg177"/>P<emph type="italics"/>er<emph.end type="italics"/> 59. <emph type="italics"/>buius.<emph.end type="italics"/></s>

<s id="id000983">Propo&longs;itio &longs;exage&longs;imaquarta.</s>

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<s id="id000984">Omne mobile quanto latius tanto tardius mouetur in plano.<lb/><arrow.to.target n="marg178"/></s>

<s id="id000986">Demon&longs;tratum e&longs;t &longs;uperius quòd &longs;i mobile &longs;it &longs;ph&etail;ricum, & tan </s>

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<s id="id000994">Proportionem duorum mobilium inter &longs;e cum auxilio medij <lb/>inuenire.<lb/><arrow.to.target n="marg181"/></s>

<s id="id000996">Graue de&longs;cendit naturaliter quatuor cau&longs;is: prima e&longs;t ponderis <lb/>magnitudo, unde quod grauius e&longs;t celerius de&longs;cendit. </s>

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<s id="id000999"><arrow.to.target n="marg182"/><lb/>materia con&longs;tat, &longs;emper de&longs;cendit parte acutiore &longs;uprapo&longs;ita, ne aër <lb/>cogatur celerius ferri: & quanto diutius de&longs;cendit, tanto magis in<lb/>tenditur motus, at que augetur, ut &longs;uprà de claratum e&longs;t. </s>

<s id="id001000">Quarta cau&longs;a <lb/>e&longs;t, quod non impediatur ab aëre tran&longs;uerfim moto, et à latere: ideo <lb/>leuia mobilia & magna non &longs;olum lentius de&longs;cendunt, quoniam <lb/><arrow.to.target n="marg183"/><lb/>paruam uim habeant, & magnam repugnantiam, &longs;ed quia tran&longs;uer <lb/><arrow.to.target n="marg184"/><lb/>&longs;im impul&longs;a minus mouentur motu recto, ut &longs;upra ui&longs;um e&longs;t. </s>

<s id="id001000">Quarta cau&longs;a <lb/>e&longs;t, quod non impediatur ab aëre tran&longs;uer&longs;im moto, et à latere: ideo <lb/>leuia mobilia & magna non &longs;olum lentius de&longs;cendunt, quoniam <lb/><arrow.to.target n="marg183"/><lb/>paruam uim habeant, & magnam repugnantiam, &longs;ed quia tran&longs;uer<lb/><arrow.to.target n="marg184"/><lb/>&longs;im impul&longs;a minus mouentur motu recto, ut &longs;upra ui&longs;um e&longs;t. </s>

<s id="id001001">Por

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<s id="id001004"><margin.target id="marg184"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 62.</s>

<s id="id001005"><margin.target id="marg185"/>P<emph type="italics"/>er<emph.end type="italics"/> 42. <emph type="italics"/>harum.<emph.end type="italics"/></s>

<s id="id001006"><margin.target id="marg186"/>I<emph type="italics"/>n<emph.end type="italics"/> 61. <emph type="italics"/>harum.<emph.end type="italics"/></s>

<s id="id001007">Propo&longs;itio &longs;exage&longs;ima&longs;exta.</s>

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<s id="id001008">Proportionem laterum eptagoni, & &longs;ubten&longs;arum con&longs;iderare, <lb/>& quæ à reflexa proportione pendent.<lb/><arrow.to.target n="marg187"/></s>

<s id="id001010">Sit eptagonus a b d f g e c, & &longs;ubten&longs;æ b <lb/><figure id="id.015.01.074.1.jpg" xlink:href="015/01/074/1.jpg"/><lb/>c, & f e duobus lateribus, tribus autem d c <lb/>d e, & erunt (quia intelligitur eptagono æ<lb/>quilatero, & æquiangulo) b c & e finuicem <lb/>æquales: & item d c, & d e æquales: & &longs;i du<lb/>cerentur b e & c f inuicem æquales: & ad a c <lb/>& d g: quare cum angulus cb d con&longs;i&longs;tatin </s>

<s id="id001011"><arrow.to.target n="marg188"/><lb/>arcu c e g f d, & angulus b d c in arcu b a c, <lb/>& angulus b c d in arcu b d; & &longs;it arcus c e g <lb/>f d duplus arcus b a c, quia c e g f d &longs;ubtendit quatuor latera epta<lb/>goni, & arcus b a c duo, & ita arcus etiam b a c duplus arcui b d <lb/>erit angulus d b e duplus angulo c d b, & angulus c d b duplus an<lb/><arrow.to.target n="marg189"/><lb/>gulo b c d, quare per demon&longs;trata à nobis proportio laterum b d, <lb/>b c, c d, e&longs;t reflexa, igitur proportio d b & b c, ad d c, ut d e ad b c, & <lb/><arrow.to.target n="marg190"/><lb/>rur&longs;us proportio b d & d e ad b e, ut b e ad b d. </s>

<s id="id001012">Quare &longs;uppo&longs;ita <lb/>d b 1, b c 1 po&longs;itione, erit d c latus 1 quad. </s>

<s id="id001013">p: 1 po&longs;itione. </s>

<s id="id001014">Proportio <lb/><arrow.to.target n="marg191"/><lb/>uerò, ut dictum e&longs;t b d & d c ad b c, id e&longs;t p: <02> 1 quad. </s>

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<s id="id001030">æqualem 1 3/4 pos p: 7/8.</s>

<s id="id001031"><margin.target id="marg188"/>P<emph type="italics"/>er<emph.end type="italics"/> 28. & 29. <emph type="italics"/>tertij<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>

<s id="id001033">&longs;exti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>

<s id="id001033">&longs;exti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>

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<s id="id001062">Igitur a c e&longs;t <02> 4 1/4 m: 1/2, & ita <lb/>de alijs.</s>

<s id="id001063"><margin.target id="marg192"/>P<emph type="italics"/>er<emph.end type="italics"/> 42. <emph type="italics"/>pri mi<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s>

<s id="id001064"><margin.target id="marg193"/>P<emph type="italics"/>er<emph.end type="italics"/> 32. <emph type="italics"/>pri mi<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s>

<s id="id001065"><margin.target id="marg194"/>P<emph type="italics"/>er &longs;extam eiu&longs;dem.<emph.end type="italics"/></s>

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<pb pagenum="57" xlink:href="015/01/076.jpg"/>ab eadem analo gæ, erit proportio tertiæ unius ordinis ad tertiam <lb/>alterius, ut &longs;ecundæ ad &longs;ecundam duplicata, & quartæ ad quartam <lb/>triplicata, quintæ ad quintam quadruplicata, at que &longs;ic de alijs.<lb/><arrow.to.target n="marg195"/></s>

<s id="id001069">Sint quantitates b c d e f, ab a in continua proportio<lb/><figure id="id.015.01.076.1.jpg" xlink:href="015/01/076/1.jpg"/><arrow.to.target n="table14"/><lb/>ne, & aliæ totidem g h k l m, dico quod proportio h c e&longs;t <lb/>duplicata ei, quæ e&longs;t g ad b, & k ad d triplicata, & l ad e <lb/>quadruplicata, & &longs;ic deinceps, &longs;umatur enim unum, & ab </s>

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</row>

</table>

<s id="id001070"><arrow.to.target n="marg196"/><lb/>co o p q r s in proportione b ad a, & tuxyz in propor<lb/>tione g ad a, erit igitur p quadratum o, & u quadratum t, <lb/>& q cubus o, & x cubus t, & ita de alijs: ergo proportio <lb/><arrow.to.target n="marg197"/><lb/>n ad p duplicata ei, quæ t ad o, & x ad q triplicata ei, quæt <lb/>ad o, & pote&longs;t etiam demon&longs;trari generaliter ultra qua<lb/><arrow.to.target n="marg198"/><lb/>dratum, & cubum: nam &longs;i ducatur t in o, fiat que <foreign lang="greek">a</foreign> erit, pro<lb/>portio enim ad <foreign lang="greek">a</foreign> eadem quæ t ad o, & proportio a ad p, <lb/>ut t ad o, igitur per diffinitionem proportionis duplicatæ <lb/><arrow.to.target n="marg199"/><lb/>po&longs;itam in quinto libro ab Euclide u ad p duplicata ei, <lb/>quæ t ad o, & &longs;imiliter ex t in p fit <foreign lang="greek">b</foreign> ex o in u, <foreign lang="greek">g</foreign> eruntque<lb/><arrow.to.target n="marg200"/><lb/>q <foreign lang="greek">b g</foreign> x in continua proportione per eandem. </s>

<s id="id001070"><arrow.to.target n="marg196"/><lb/>eo o p q r s in proportione b ad a, & t u x y z in propor<lb/>tione g ad a, erit igitur p quadratum o, & u quadratum t, <lb/>& q cubus o, & x cubus t, & ita de alijs: ergo proportio <lb/><arrow.to.target n="marg197"/><lb/>n ad p duplicata ei, quæ t ad o, & x ad q triplicata ei, quæ t <lb/>ad o, & pote&longs;t etiam demon&longs;trari generaliter ultra qua<lb/><arrow.to.target n="marg198"/><lb/>dratum, & cubum: nam &longs;i ducatur t in o, fiat que <foreign lang="greek">a</foreign> erit, pro<lb/>portio enim ad <foreign lang="greek">a</foreign> eadem quæ t ad o, & proportio a ad p, <lb/>ut t ad o, igitur per diffinitionem proportionis duplicatæ <lb/><arrow.to.target n="marg199"/><lb/>po&longs;itam in quinto libro ab Euclide u ad p duplicata ei, <lb/>quæ t ad o, & &longs;imiliter ex t in p fit <foreign lang="greek">b</foreign> ex o in u, <foreign lang="greek">g</foreign> eruntque<lb/><arrow.to.target n="marg200"/><lb/>q <foreign lang="greek">b g</foreign> x in continua proportione per eandem. </s>

<s id="id001071">Quia ergo propor<lb/>tio q ad <foreign lang="greek">b</foreign> e&longs;t ut o ad t, patet, quod x ad q e&longs;t triplicata ei, quæ e&longs;t t ad <lb/>o, & ita de reliquis, cum ergo proportio p ad o &longs;it, ut e ad b, & o ad <lb/><arrow.to.target n="marg201"/><lb/>n, ut b ad a, & n ad t, ut a ad g, & t ad u, ut g ad h, &longs;equitur ut &longs;it t ad a, <lb/>ut g ad b, & u ad p, ut h ad c, igitur cum &longs;it ut u ad p duplicata ei, qu&etail; <lb/>e&longs;t t ad o erit h ad e, duplicata ei quæ e&longs;t g ad b, & ita de reliquis, & <lb/>no&ngrave; refert, &longs;eu dicas u ad p duplicatam ei, quæ e&longs;t t ad o, &longs;eu dicas p <lb/><arrow.to.target n="marg202"/><lb/>ad u duplicatam ei, quæ e&longs;t o ad t. </s>

<s id="id001072">Aliter & euidentius in duabus <lb/>&longs;oleo demon&longs;trare: cum enim &longs;it e & h duplicata ei quæ e&longs;t b & g <lb/>ad a, ut &longs;upra, & quadrati b ad quadratum a, & quadrati g ad qua<lb/><arrow.to.target n="marg203"/><lb/>dratum a duplicata his quæ b & g ad a erunt b & g quadratorum <lb/>ad quadratum a, uelut c & h ad a. </s>

<s id="id001073">Et conuertendo qua<lb/><arrow.to.target n="table15"/><lb/>drati a ad quadratum g, ut a ad h, con&longs;tituantur ergo <lb/><figure id="id.015.01.076.2.jpg" xlink:href="015/01/076/2.jpg"/>hic & erit quadrati b ad <expan abbr="quadrat&utilde;">quadratum</expan> g, ita c ad h: &longs;ed qua<lb/>drati b ad quadratum g, ut b ad g proportio duplicata <lb/>igitur e ad h, ut b ad g duplicata.</s>

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<s id="id001076"><margin.target id="marg198"/>P<emph type="italics"/>er<emph.end type="italics"/> 23. <emph type="italics"/>&longs;ex ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/> & 33. <emph type="italics"/>undeci-mi.<emph.end type="italics"/></s>

<s id="id001077"><margin.target id="marg199"/>P<emph type="italics"/>er<emph.end type="italics"/> 17. <emph type="italics"/>&longs;e-ptimi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>

<s id="id001077"><margin.target id="marg199"/>P<emph type="italics"/>er<emph.end type="italics"/> 17. <emph type="italics"/>&longs;eptimi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>

<s id="id001079"><margin.target id="marg201"/>P<emph type="italics"/>er<emph.end type="italics"/> 24. <emph type="italics"/>quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>

<s id="id001081">quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>

<s id="id001081">quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>

<s id="id001082"><margin.target id="marg203"/>P<emph type="italics"/>er<emph.end type="italics"/> 20. <emph type="italics"/>&longs;ex ti<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s>

<table>

<table.target id="table15"/>

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<s id="id001094">Quilibet &longs;ector &longs;phæræ æqualis e&longs;t cono, cuius ba&longs;is e&longs;t circu<lb/>lus æqualis &longs;uperficiei eiu&longs;dem portionis, altitudo uerò &longs;phæræ &longs;e<lb/>midiameter. </s>

<s id="id001095">Proportio &longs;phæræ ad &longs;ectorem datum, e&longs;t duplica<lb/>ta ei, qu&etail; e&longs;t dimetientis ad lineam, quæ à uertice portionis ad lim<lb/>bum. </s>

<s id="id001096">Cum enim &longs;phæra &longs;it æqualis cono, cuius ba&longs;is e&longs;t maior cir<lb/>culus, altitudo uerò dupla dimetienti per tertiam harum, quæ hic <lb/><arrow.to.target n="marg209"/><lb/>proponuntur: erit &longs;phæra æqualis cono ba&longs;im habenti circulum, <lb/>cuius &longs;emidiameter &longs;it æqualis diametro &longs;phæræ, altitudo uerò &longs;e<lb/>midiameter &longs;phæræ. </s>

<s id="id001097">At per &longs;extam harum &longs;ector &longs;phæræ e&longs;t æqua<lb/>lis cono habenti altitudinem &longs;cmidiametrum &longs;phær&etail;, ba&longs;im autem <lb/><arrow.to.target n="marg210"/><lb/>ip&longs;am portionis &longs;uperficiem: igitur proportio &longs;phæræ ad &longs;ecto<lb/>rem, uelut circuli cuius diameter e&longs;t dupla dimetienti &longs;phæræ ad <lb/>círculum æqualem &longs;uperficiei portionis: at &longs;uperficies portionis <lb/>per quintam harum e&longs;t æqualis circulo, cuius &longs;emidiameter e&longs;t li<lb/>nea à uertice portionis ad limbum eiu&longs;dem: ergo proportio &longs;phæ<lb/>ræ ad &longs;uum &longs;ectorem e&longs;t uelut circuli, cuius dimetiens e&longs;t duplus di <lb/>metienti &longs;phæræ, aut &longs;emidimetiens e&longs;t æqualis dimetienti &longs;phæræ <lb/>ad circulum, cuius &longs;emidimetiens e&longs;t linea à uertice portionis ad <lb/>limbum. </s>

<s id="id001097">At per &longs;extam harum &longs;ector &longs;phæræ e&longs;t æqua<lb/>lis cono habenti altitudinem &longs;emidiametrum &longs;phær&etail;, ba&longs;im autem <lb/><arrow.to.target n="marg210"/><lb/>ip&longs;am portionis &longs;uperficiem: igitur proportio &longs;phæræ ad &longs;ecto<lb/>rem, uelut circuli cuius diameter e&longs;t dupla dimetienti &longs;phæræ ad <lb/>círculum æqualem &longs;uperficiei portionis: at &longs;uperficies portionis <lb/>per quintam harum e&longs;t æqualis circulo, cuius &longs;emidiameter e&longs;t li<lb/>nea à uertice portionis ad limbum eiu&longs;dem: ergo proportio &longs;phæ<lb/>ræ ad &longs;uum &longs;ectorem e&longs;t uelut circuli, cuius dimetiens e&longs;t duplus di <lb/>metienti &longs;phæræ, aut &longs;emidimetiens e&longs;t æqualis dimetienti &longs;phæræ <lb/>ad circulum, cuius &longs;emidimetiens e&longs;t linea à uertice portionis ad <lb/>limbum. </s>

<s id="id001098">Sed proportio talium circulorum e&longs;t duplicata propor<lb/><arrow.to.target n="marg211"/><lb/>tioni &longs;emidimetientium, igitur proportio &longs;phæræ ad &longs;uum &longs;ecto<lb/>rem e&longs;t ueluti dimetientis &longs;phæræ ad lineam, quæ á uertice portio<lb/><arrow.to.target n="marg212"/><lb/>nis ad limbum duplicata. </s>

<s id="id001099">Cuicunque portioni &longs;phæræ conus ille <lb/>habetur æqualis, qui ba&longs;im hab eat eandem cum portione, altitudi<lb/>nem uerò lineam rectam, quæ ad altitudinem portionis eandem <lb/>habeat proportionem, quam &longs;emidiametros &longs;phæræ unà cum alti<lb/>tudine reliquæ portionis habet ad eandem reliquæ portionis alti<lb/><arrow.to.target n="marg213"/><lb/>tudinem. </s>

<s id="id001100">Earum &longs;phæræ portionum, quæ æqualibus &longs;uperfi<lb/><arrow.to.target n="marg214"/><lb/>ciebus continentur medietas &longs;phæræ maxima exi&longs;tit. </s>

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<s id="id001109">Ex hoc rur&longs;us &longs;equitur quod ellip&longs;is ad ellip&longs;im, ut re<lb/><arrow.to.target n="marg221"/><lb/>ctanguli ex diametris unius ad rectangulum ex diametris alterius.</s>

<s id="id001110"><margin.target id="marg209"/>P<emph type="italics"/>er<emph.end type="italics"/> 14. & 15. <emph type="italics"/>duodeci mi<emph.end type="italics"/> E<emph type="italics"/>le.<emph.end type="italics"/> E<emph type="italics"/>ucl.<emph.end type="italics"/></s>

<s id="id001111"><margin.target id="marg210"/>P<emph type="italics"/>er<emph.end type="italics"/> 11. <emph type="italics"/>duo decimi<emph.end type="italics"/> E<emph type="italics"/>le.<emph.end type="italics"/></s>

<s id="id001111"><margin.target id="marg210"/>P<emph type="italics"/>er<emph.end type="italics"/> 11. <emph type="italics"/>duodecimi<emph.end type="italics"/> E<emph type="italics"/>le.<emph.end type="italics"/></s>

<s id="id001112"><margin.target id="marg211"/>P<emph type="italics"/>er<emph.end type="italics"/> 2. <emph type="italics"/>duode cimi<emph.end type="italics"/>, & 20. <emph type="italics"/>&longs;exti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>

<s id="id001112"><margin.target id="marg211"/>P<emph type="italics"/>er<emph.end type="italics"/> 2. <emph type="italics"/>duodecimi<emph.end type="italics"/>, & 20. <emph type="italics"/>&longs;exti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>

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<s id="id001116"><margin.target id="marg215"/>P<emph type="italics"/>er<emph.end type="italics"/> 22. <emph type="italics"/>quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>

<s id="id001117"><margin.target id="marg216"/>P<emph type="italics"/>er<emph.end type="italics"/> 20. <emph type="italics"/>&longs;ex ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>

<s id="id001118"><margin.target id="marg217"/>P<emph type="italics"/>er<emph.end type="italics"/> 11. <emph type="italics"/>quinti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>

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<s id="id001163">Si in parabole contingente ad diametrum ducta ex alio puncto <lb/>ei æquidi&longs;tans ducatur ex ip&longs;a &longs;ectione, ubi iterum &longs;ecat &longs;ectionem/><lb/>intercepta per æqualia diuidetur linea à puncto contingentis dia</s>

<s id="id001163">Si in parabole contingente ad diametrum ducta ex alio puncto <lb/>ei æquidi&longs;tans ducatur ex ip&longs;a &longs;ectione, ubi iterum &longs;ecat &longs;ectionem <lb/>intercepta per æqualia diuidetur linea à puncto contingentis dia</s>

<s id="id001164"><arrow.to.target n="marg238"/><lb/>metro æquidi&longs;tanti ducta. </s>

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<pb pagenum="62" xlink:href="015/01/081.jpg"/><arrow.to.target n="marg248"/><lb/>miles, &longs;ed unam per&longs;æpe in utri&longs; que e&longs;&longs;e uult. </s>

<s id="id001179">Sed & hoc Archime<lb/>des dicere uidetur: lineæ ductæ à uertice coni&longs;caleni ad perpendi<lb/>culum &longs;uper ba&longs;es &longs;ingulas omnium triangulorum per axem/> coni <lb/>tran&longs;euntium in peripheriam unius circuli cadunt.</s>

<s id="id001179">Sed & hoc Archime<lb/>des dicere uidetur: lineæ ductæ à uertice coni&longs;caleni ad perpendi<lb/>culum &longs;uper ba&longs;es &longs;ingulas omnium triangulorum per axem coni <lb/>tran&longs;euntium in peripheriam unius circuli cadunt.</s>

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<s id="id001192">Si fuerint tres quantitates in continua proportione, aliæque toti<lb/>dem in continua proportione, poterunt con&longs;tituere tres quantita<lb/>tes in æquali differentia peruer&longs;im copulatæ.<lb/><arrow.to.target n="marg249"/></s>

<s id="id001194">Velut &longs;int a b c primi ordi<lb/><figure id="id.015.01.081.1.jpg" xlink:href="015/01/081/1.jpg"/><lb/>nis, & d ef &longs;ecundi, & &longs;it 28, </s>

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<s id="id001203">Propo&longs;itio &longs;eptuage&longs;imaprima.</s>

<s id="id001204">Proportionem leuitatis ponderis per uirgam torcularem attra<lb/>cti ad rectam &longs;u&longs;penfionem inuenire.</s>

<s id="id001204">Proportionem leuitatis ponderis per uirgam torcularem attra<lb/>cti ad rectam &longs;u&longs;pen&longs;ionem inuenire.</s>

<s id="id001205">Sit torcularis uirga, cuius &longs;piræ a b per circui<lb/><arrow.to.target n="marg252"/><lb/>tum &longs;int centuplæ ad altitudinem a b, & axis d c <lb/><arrow.to.target n="marg253"/><lb/>&longs;emidiametro b c centupla, & quoniam per &longs;upe<lb/>rius a&longs;&longs;umpta, qualis e&longs;t proportio &longs;patij ad &longs;pa<lb/>tium, talis leuitatis ad <expan abbr="leuitat&etilde;">leuitatem</expan>, <expan abbr="igi&ttilde;">igitur</expan> e pondus a&longs;cen <lb/>dens per a b leuius quam per b <expan abbr="crectã">crectam</expan> centuplo, et <lb/>&longs;imiliter cum circuitus b c, & d c &longs;int in eodem tem <lb/>pore, & circuitus d c, &longs;it centuplus ad &longs;piralem b c <lb/>per demon&longs;trata ab Euclide, ergo e erit centuplo <lb/>leuius circum ductum per d quàm b, &longs;ed per b circumductum cen<lb/>tuplo leuius e&longs;t, quàm per rectam, igitur e ponderat folum particu<lb/>lam ex decem millibus recti ponderis.</s>

<s id="id001205">Sit torcularis uirga, cuius &longs;piræ a b per circui<lb/><arrow.to.target n="marg252"/><lb/>tum &longs;int centuplæ ad altitudinem a b, & axis d c <lb/><arrow.to.target n="marg253"/><lb/>&longs;emidiametro b c centupla, & quoniam per &longs;upe<lb/>rius a&longs;&longs;umpta, qualis e&longs;t proportio &longs;patij ad &longs;pa<lb/>tium, talis leuitatis ad <expan abbr="leuitat&etilde;">leuitatem</expan>, <expan abbr="igi&ttilde;">igitur</expan> e pondus a&longs;cen<lb/>dens per a b leuius quam per b <expan abbr="crectã">c rectam</expan> centuplo, et <lb/>&longs;imiliter cum circuitus b c, & d c &longs;int in eodem tem<lb/>pore, & circuitus d c, &longs;it centuplus ad &longs;piralem b c <lb/>per demon&longs;trata ab Euclide, ergo e erit centuplo <lb/>leuius circum ductum per d quàm b, &longs;ed per b circumductum cen<lb/>tuplo leuius e&longs;t, quàm per rectam, igitur e ponderat &longs;olum particu<lb/>lam ex decem millibus recti ponderis.</s>

<s id="id001207"><margin.target id="marg253"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 45.</s>

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<s id="id001213"><margin.target id="marg255"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 40. 7</s>

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<s id="id001215">Proportionem ponderum attractorum penes figuram in pla<lb/>no inuenire.<lb/><arrow.to.target n="marg256"/></s>

<s id="id001217">Sint duo pondera æqualia in plano a & b, & &longs;it <lb/><figure id="id.015.01.083.1.jpg" xlink:href="015/01/083/1.jpg"/><lb/>a &longs;uperficies qua planum tangit dupla b &longs;uperfi<lb/>ciei, qua planum tangit: dico quod &longs;i trahantur ab <lb/>imo, quod erunt æqualia: &longs;u&longs;pendantur, & erunt <lb/>æqualia ex &longs;uppo&longs;ito, &longs;ed a quie&longs;cens in plano e&longs;t <lb/>dimidium a &longs;u&longs;pen&longs;i, & b quie&longs;cens in plano e&longs;t di <lb/>midium b &longs;u&longs;pen&longs;i ex demon&longs;tratis &longs;uperius, igi<lb/>tur per communem animi &longs;ententiam a & b in pla<lb/>no &longs;unt æqualia.</s>

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<s id="id001220">Ex hoc manife&longs;tum e&longs;t, quod proportio uirium trahentium pon<lb/>dera in plano eadem e&longs;t, quæ ip&longs;orum ponderum dum &longs;u&longs;pendun<lb/>tur. </s>