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Colored diff for /texts/archimedes/xml/carda_propo_015_la_1570.xml between version 1.14 and 1.16

version 1.14, 2004/02/18 15:58:15

version 1.16, 2004/02/23 12:14:23

Line 6515

<s id="id002119">Di <lb/>ces, h&etail;c quid ad proportionem? </s>

<s id="id002120">Imò uelut machina ad <expan abbr="perpendicul&utilde;">perpendiculum</expan> <lb/>librata pauculo illo puluere Pyrio <expan abbr="urb&etilde;">urbem</expan> euertit, ita &longs;cintilla illa infer <lb/>ni ignis &longs;emini magni tyranni indita euertit at que di&longs;&longs;oluit totum re<lb/>gnum &longs;ine machinis, ut dixi, uel exercitibus ullis, & quod maius e&longs;t <lb/>remedio nullo. </s>

<s id="id002121">Sed puerulo indito luxus, ignauiæ, crudelitatis at que<lb/>&longs;tultiti&etail; fontibus, mirabile dictu &longs;anè, & ad proportionem diuino<lb/>rum <expan abbr="in&longs;trumentor&utilde;">in&longs;trumentorum</expan> pertinens. </s>

<s id="id002122">Sed redeamus ad in&longs;titutum: Video <lb/>enim, quid po&longs;sit obijci, &longs;cilicet muros cra&longs;&longs;os, et altiores tueri <expan abbr="urb&etilde;">urbem</expan> <lb/>& ædificia illius po&longs;&longs;e ab&longs;que aggeris erectione, & &longs;i <expan abbr="diruan&ttilde;">diruantur</expan> manere <lb/>etiam nihilominus imo magis, quod e&longs;t terram, u&longs;que <expan abbr="quoniã">quoniam</expan> eadem <lb/>ratione manet, quia concuti non po&longs;sit à machinis: nec ho&longs;tes id cu <lb/>raturos, &longs;perantes hoc <expan abbr="&longs;ol&utilde;">&longs;olum</expan> &longs;ufficere, &qring;d m&oelig;nia &longs;olo <expan abbr="æquen&ttilde;">æquentur</expan>, at que id <lb/><expan abbr="fact&utilde;">factum</expan> e&longs;t Mediolani, & in arce eius, <expan abbr="t&utilde;">tum</expan> Papi&etail; & in Cremonen&longs;i arce. <lb/></s>

<s id="id002122">Sed redeamus ad in&longs;titutum: Video <lb/>enim, quid po&longs;sit obijci, &longs;cilicet muros cra&longs;&longs;os, et altiores tueri <expan abbr="urb&etilde;">urbem</expan> <lb/>& ædificia illius po&longs;&longs;e ab&longs;que aggeris erectione, & &longs;i <expan abbr="diruan&ttilde;">diruantur</expan> manere <lb/>etiam nihilominus imo magis, quod e&longs;t terram, u&longs;que <expan abbr="quoniã">quoniam</expan> eadem <lb/>ratione manet, quia concuti non po&longs;sit à machinis: nec ho&longs;tes id cu<lb/>raturos, &longs;perantes hoc <expan abbr="&longs;ol&utilde;">&longs;olum</expan> &longs;ufficere, quod m&oelig;nia &longs;olo <expan abbr="æquen&ttilde;">æquentur</expan>, at que id <lb/><expan abbr="fact&utilde;">factum</expan> e&longs;t Mediolani, & in arce eius, <expan abbr="t&utilde;">tum</expan> Papi&etail; & in Cremonen&longs;i arce. <lb/></s>

<s id="id002123">Verùm ni fallor, ut paruis arcibus à tanta ui tormentorum nullum <lb/>e&longs;t <expan abbr="præ&longs;idi&utilde;">præ&longs;idium</expan>, aut &longs;alutis &longs;pes, ita neque <expan abbr="cõuenit">conuenit</expan>, ut muris humilibus ag <lb/>geri confidant, nam & pauci homines tanto labori non &longs;ufficerent, <lb/>& agger cum fo&longs;&longs;a effo&longs;&longs;a &longs;cilicet terra defen&longs;ores nimis in <expan abbr="angu&longs;t&utilde;">angu&longs;tum</expan> <lb/>cogeret. </s>

<s id="id002124">At in urbibus contra eueniet: muris enim erectis altius ma <lb/>chinæ lapidum fru&longs;tis hominem <expan abbr="occid&etilde;t">occident</expan>: an percu&longs;&longs;a &longs;uperiore par <lb/>te ob coniunctionem inferior concutitur, & in de <expan abbr="tot&utilde;">totum</expan> &longs;imul cadit, <lb/>ut uidimus Papi&etail;, quo <expan abbr="cad&etilde;te">cadente</expan>, & fo&longs;&longs;a impletur, & <foreign lang="greek">tEIkole/tois</foreign> facilior <lb/>aditus ad &longs;ubruendum reliquas partes <expan abbr="pr&etail;be&ttilde;">pr&etail;betur</expan>: imò percul&longs;i defen

Line 6545

<pb pagenum="119" xlink:href="015/01/138.jpg"/>o a, ad o c, igitur clauus e&longs;t longè potentior in comparatione ueli <lb/>d, quam f, ergo uelum d minus agit nauim, quam f. </s>

<s id="id002135">Sed ut extrema <lb/>&longs;e habent, ita medium eorum comparatione, igitur malus b e uali<lb/>dior e&longs;t, multo d a, & infirmior c f. </s>

<s id="id002136">Verùm, ut dixi, ob &longs;itum &longs;impli<lb/>citer ualidius e&longs;t, uelum e quam f, & etiam quia, ut dixi, altior & <lb/>era&longs;sior &longs;olet e&longs;&longs;e, ideo multo ualidior tribus his cau&longs;is, quàm e f: <lb/>adde quartam quòd uelum habet maius, antiquo tempore uoca<lb/>tum acatius. </s>

<s id="id002136">Verùm, ut dixi, ob &longs;itum &longs;impli<lb/>citer ualidius e&longs;t, uelum e quam f, & etiam quia, ut dixi, altior & <lb/>cra&longs;sior &longs;olet e&longs;&longs;e, ideo multo ualidior tribus his cau&longs;is, quàm e f: <lb/>adde quartam quòd uelum habet maius, antiquo tempore uoca<lb/>tum acatius. </s>

<s id="id002137">At ut etiam docui c b non e&longs;t in medio, nec æquidi&longs;tat <lb/>ab a d & c f, &longs;ed in clinatur ad proram ideoque imbecillior: cum ergo <lb/>&longs;it æqualium, & paulo maiorum uirium, quàm c f, & tutior, & me<lb/>lius agatur per <expan abbr="clau&utilde;">clauum</expan> quàm c f, & &longs;it a d nimis iu&longs;to imbecillis, pro<lb/>pterea b e mali, & ueli maximus e&longs;t u&longs;us: adeò mali nomen per an<lb/>tonoma&longs;iam de ip&longs;o &longs;impliciter intelligatur.</s>

Line 6558

<s id="id002140">Sit flabellum a b c appen&longs;um, ut &longs;olet, in a, & moueatur motu </s>

<s id="id002141"><arrow.to.target n="marg413"/><lb/>qua&longs;i circa axem p a q in parte inferiore, & aër comprehen&longs;us &longs;ub <lb/>b h k, & &longs;patium &longs;it 1 m figuræ nauicularis, quæ con&longs;tat e&longs;&longs;e par<lb/>tem cylindri inanis ex formatione ab Euclide &longs;cripta: nam &longs;i pro<lb/>poneretur p a q ad perpendiculum &longs;uper&longs;tans plano, fieret circum<lb/>ducta a b c &longs;uperficie, quæ e&longs;&longs;et lata &longs;uperius, &longs;icut etiam inferius <lb/><arrow.to.target n="marg414"/><lb/>cylindrus: at &longs;uperius a b tenuis e&longs;t, & angu&longs;ta, ergo fiet pars cy<lb/>lindri inanis: quia non circunuoluitur, donecredeat. </s>

<s id="id002142">Ergo per di<lb/>cta &longs;uperius &longs;ectio illius p r q s per axem e&longs;t pars cuiu&longs;dam elly<lb/><arrow.to.target n="marg415"/><lb/>p&longs;is. </s>

<s id="id002143">Et &longs;ectio quæuis planæ &longs;uperficiei æquidi&longs;tans a b cuelut tu, <lb/>item que æquidi&longs;tans axi p a q e&longs;t &longs;uperficies rectangula, quarum <lb/>una e&longs;t &longs;imilis, & æqualis b h k, e&longs;t in una &longs;uperficie cum axe p a q <lb/>alia uerò e&longs;t æquidi&longs;tans eidem axi maior aut minor æquidi&longs;tanti<lb/>um, & ip&longs;a laterum, at que rectangula ac &longs;i cylindrus &longs;tans axi plano <lb/>æquidi&longs;tanti &longs;ecaretur iuxta longitudinem &longs;eu altitudinem &longs;uam: <lb/>& manife&longs;tum e&longs;t, quod i&longs;ta duo plana, & eorum &longs;uperficies &longs;ecant <lb/>&longs;e mutuò ad rectos angulos.</s>

Line 6574

<s id="id002147">Quibus con&longs;titutis, qui &longs;tabunt iuxta l, & m longitudines aëris <lb/>moti, & loci, per quem tran&longs;it flabellum, &longs;entient magnum uentum, <lb/>quoniam cum corpus m x l ab extremis partibus &longs;it elatius a b ex<lb/>tremis, &longs;tantes, & alti tangentur à uento agitato. </s>

<s id="id002148">Si uero &longs;edeant, aer <lb/>primum non attinget illos, ut etiam quia &longs;ur&longs;um pellitur non per<lb/>ueniet ad illos, imò diffugiet, ergo non refrigerabuntur. </s>

<s id="id002149">Qui uerò <lb/>à lateribus l x m <expan abbr="&longs;tab&utilde;t">&longs;tabunt</expan> hiccinde, uelut in f g, &longs;i &longs;teterint, <expan abbr="nõ">non</expan> refrigeræ <lb/><expan abbr="b&utilde;tur">buntur</expan>, quia <expan abbr="quãdo">quando</expan> flabellum erit in l, uel m aer de&longs;cendet, ergo fugi <lb/>et ab illis, cum autem fuerit in x, erit in loco humiliori, & mouebi

<s id="id002149">Qui uerò <lb/>à lateribus l x m <expan abbr="&longs;tab&utilde;t">&longs;tabunt</expan> hiccinde, uelut in f g, &longs;i &longs;teterint, <expan abbr="nõ">non</expan> refrigera<lb/><expan abbr="b&utilde;tur">buntur</expan>, quia <expan abbr="quãdo">quando</expan> flabellum erit in l, uel m aer de&longs;cendet, ergo fugi <lb/>et ab illis, cum autem fuerit in x, erit in loco humiliori, & mouebi

<pb pagenum="120" xlink:href="015/01/139.jpg"/>tur diuer&longs;a ratione, quippe ab f in h, & non ad latera, ergo ne que <lb/><figure id="id.015.01.139.1.jpg" xlink:href="015/01/139/1.jpg"/><lb/>contactu, neque motu, qui <lb/>fiet per æquidi&longs;tantem f, <lb/>& g non poterunt refrige<lb/>rari. </s>

Line 6664

<s id="id002186">Nam exempli gratia, cum &longs;ol e&longs;t in initio Capricorni, & in C&oelig;li <lb/>medio, minima e&longs;t umbra eius diei, & totius anni. </s>

<s id="id002187">Cum ergo fuerit <lb/>ante meridiem, uel po&longs;t, erit umbra maior ex &longs;uppo&longs;ito &longs;ecudo um<lb/>bra meridiei: at ei æqualis poterit e&longs;&longs;e umbra meridiei alterius diei <lb/>ex primo &longs;uppo&longs;ito, ergo umbræ æquales diuer&longs;orum dierum fi<lb/>unt &longs;ub diuer&longs;o &longs;itu &longs;olis, quo æd circulum meridiei, quod erat de<lb/>mon&longs;trandum.</s>

<s id="id002187">Cum ergo fuerit <lb/>ante meridiem, uel po&longs;t, erit umbra maior ex &longs;uppo&longs;ito &longs;ecudo um<lb/>bra meridiei: at ei æqualis poterit e&longs;&longs;e umbra meridiei alterius diei <lb/>ex primo &longs;uppo&longs;ito, ergo umbræ æquales diuer&longs;orum dierum fi<lb/>unt &longs;ub diuer&longs;o &longs;itu &longs;olis, quo ad circulum meridiei, quod erat de<lb/>mon&longs;trandum.</s>

Line 6858

<s id="id002260">Data regionis altitudine, & loco &longs;olis proportionem gnomo<lb/>nis tam ad umbram rectam, quam uer&longs;am, uel etiam in cylindro de<lb/>terminare.</s>

<s id="id002261">H&etail;c e&longs;t propo&longs;itio illa pulcherrima, quam tot ambagibus tradi<lb/><arrow.to.target n="marg445"/><lb/>dere antiqui cum &longs;uis analematibus, & &longs;cioteris, nec tamen demon <lb/>&longs;trationem, nec rationem exactam in&longs;trumenortum con&longs;tructio<lb/>nem, qua po&longs;&longs;emus per umbras rectas uer&longs;as, & cylindricas &longs;cire ad <lb/>unguem, qualis hora, & minutum, & &longs;ecundum diei e&longs;&longs;et quocun<lb/>que anni tempore. </s>

<s id="id002261">H&etail;c e&longs;t propo&longs;itio illa pulcherrima, quam tot ambagibus tradi<lb/><arrow.to.target n="marg445"/><lb/>dere antiqui cum &longs;uis analematibus, & &longs;cioteris, nec tamen demon <lb/>&longs;trationem, nec rationem exactam in&longs;trumenorum con&longs;tructio<lb/>nem, qua po&longs;&longs;emus per umbras rectas uer&longs;as, & cylindricas &longs;cire ad <lb/>unguem, qualis hora, & minutum, & &longs;ecundum diei e&longs;&longs;et quocun<lb/>que anni tempore. </s>

<s id="id002262">Plerique autem tam laborio&longs;è id conati &longs;unt de<lb/>mon&longs;trare, ut &longs;tudio&longs;os deterruerint ab opere: res autem ip&longs;a facil<lb/>lima e&longs;t. </s>

<s id="id002263">Propo&longs;ita ergo Poli exacta altitudine &longs;olis in Meridie <lb/>declinatione addita uel detracta, habebis re&longs;iduum eius ad qua<lb/>drantem f g, & &longs;imiliter habebis ex declinatione nota loci &longs;olis de<lb/>tracta à quadrante f h & iuxta horam tuam, & minutum multi<lb/><arrow.to.target n="marg446"/><lb/>plicatum per quindecim arcum k e quare angulum f, ex quo arcum <lb/>g h, quare re&longs;iduum h l, igitur punctum umbr&etail; rect&etail;, uel uer&longs;&etail; ip&longs;i<lb/>us gnomonis ad unguem, & ita con&longs;titues horologium exacti&longs;si<lb/>mum &longs;ecundum ea, quæ dixi in Corrolarijs &longs;upradictis, & quia ho<lb/><arrow.to.target n="marg447"/><lb/>rizon a b c d &longs;ecat æquinoctialem in <expan abbr="c&etilde;tro">centro</expan> terræ ducta g h k, erunt <lb/>anguli b h g, & k h l &etail;quales. </s>

<s id="id002264">Igitur po&longs;ito g ortu puncti eclypti<lb/>cæ, erit g b ortus amplitudo nota, & ideò angulus b h g, & k h l

Line 6924

<s id="id002291">Sint duæ line&etail; a b, & c d. </s>

<s id="id002292">& &longs;it c d dupla ad a b, ad datur <expan abbr="cõmunis">communis</expan> <lb/><figure id="id.015.01.145.3.jpg" xlink:href="015/01/145/3.jpg"/><lb/>b e, & uo cetur iuncta c d, d f dico, <lb/>quod proportio e a ad a b, e&longs;t mi<lb/>nor duplicata f c ad c d, adij cia<lb/>tur d f æqualis g f, quia ergo g d <lb/>e&longs;t dupla ad f d, ideo ad e b c d autem e&longs;t du pla ad a b, tota igitur

<s id="id002292">& &longs;it c d dupla ad a b, ad datur <expan abbr="cõmunis">communis</expan> <lb/><figure id="id.015.01.145.3.jpg" xlink:href="015/01/145/3.jpg"/><lb/>b e, & uocetur iuncta c d, d f dico, <lb/>quod proportio e a ad a b, e&longs;t mi<lb/>nor duplicata f c ad c d, adijcia<lb/>tur d f æqualis g f, quia ergo g d <lb/>e&longs;t dupla ad f d, ideo ad e b c d autem e&longs;t dupla ad a b, tota igitur <pb pagenum="127" xlink:href="015/01/146.jpg"/>g c dupla toti e a. </s>

<pb pagenum="127" xlink:href="015/01/146.jpg"/>g c duplatoti e a. </s>

<s id="id002293">quare ut g c ad g d ut e a ad e b <expan abbr="permutãdo">permutando</expan>, & per <lb/>euer&longs;am ut e a ad a b, ita g c ad c d, ut g c ad c d <expan abbr="cõponitur">componitur</expan> ex g e ad <lb/>f e, & f c ad c d, igitur e a ad c b componitur ex ei&longs;dem. </s>

<s id="id002294">Proportio <lb/>autem g c ad f c e&longs;t minor, quam f c ad c d, igitur minor quàm du<lb/>plicata f c ad c d. </s>

<s id="id002295">con&longs;tat uerò ex ei&longs;dem, quod proportio c a ad a b <lb/>maior e&longs;t duplicata g c ad f c.</s>

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Line 7349

<s id="id002405">Pro relata prima ergo ponamus, ut ue<lb/>lim <02> relatam <expan abbr="primã">primam</expan> 25, accipio 5 <02> 25 cuius <02> e&longs;t, ut ui&longs;um e&longs;t, 2 104/441 <lb/>&longs;imiliter <02> cu: 5 fuit 1 69/98 igitur reducam ad unam denominationem, <lb/>& inueniam quatuor numeros in <expan abbr="cõtinua">continua</expan> proportione inter illos, <lb/>& &longs;ecundus po&longs;t minimum ex illis erit <02> relata prima propinqui&longs;<lb/>&longs;ima 25. Quomodo uerò inueniantur facillimè illi termini, do<lb/>cui in &longs;exto libro operis perfecti.</s>

<s id="id002406">Quarta regula e&longs;t utilior, licet minus uideatur nobilis, & e&longs;t &longs;un<lb/>data in hoc, quod &longs;i a b &longs;it maior c & eis ad dantur b e, & d f æqua<lb/>les dico, quod erit minor proportio a c ad c f, quam a b ad c d, & ex <lb/>con&longs;equenti per <expan abbr="uiã">uiam</expan> fracti maior pars unius erit c fip&longs;ius a e, quàm

<s id="id002406">Quarta regula e&longs;t utilior, licet minus uideatur nobilis, & e&longs;t fun<lb/>data in hoc, quod &longs;i a b &longs;it maior c & eis ad dantur b e, & d f æqua<lb/>les dico, quod erit minor proportio a c ad c f, quam a b ad c d, & ex <lb/>con&longs;equenti per <expan abbr="uiã">uiam</expan> fracti maior pars unius erit c f ip&longs;ius a e, quàm <pb pagenum="135" xlink:href="015/01/154.jpg"/>c d ip&longs;ius a f ex Euclide. </s>

<pb pagenum="135" xlink:href="015/01/154.jpg"/>c d ip&longs;ius a f ex Euclide. </s>

<s id="id002407">Dico ergo quod maior e&longs;t proportio a b <lb/><figure id="id.015.01.154.1.jpg" xlink:href="015/01/154/1.jpg"/><lb/>ad c d, quàm a e ad e f, fiat d g ad quam &longs;it b c ut <lb/><arrow.to.target n="marg467"/><lb/>a b ad c d, eritque a e ad c g ut a b ad c d, minor au<lb/>tem e&longs;t a e ad c f, quam ad c g, igitur minor a e ad <lb/>c f quàm a b ad c d quod fuit propo&longs;itum. </s>

<s id="id002408">Simili <lb/>ter &longs;i fuerint duæ quantitates, a b & c d, quarum a b &longs;it maiore, c d <lb/>autem eadem e minor, dico, quòd dimidium aggregati a b & c d <lb/>maiorem habebit proportionem ad e, quàm c d & minor, nam iun<lb/>cta b f æquali d e ad a b, ita ut f g &longs;it dimidium totius a f, qùia ergo <lb/><figure id="id.015.01.154.2.jpg" xlink:href="015/01/154/2.jpg"/><lb/>f g e&longs;t dimidium f a & fb e&longs;t minor dimidio <lb/><arrow.to.target n="marg468"/><lb/>f a cum &longs;it minor b a, & &longs;imiliter f g e&longs;t mi<lb/>nor a b, quia a b e&longs;t maior dimidio a f, quia <lb/>e&longs;t maior b f, ergo proportio g f ad c e&longs;t ma <lb/>ior quam b f ad e, ita quam c d ad e, & mi<lb/><arrow.to.target n="marg469"/><lb/>nor quàm a b ad e, quod fuit propo&longs;itum. </s>

<s id="id002409">Quo ui&longs;o uolo <02> 1000 <lb/>quadratam, & quòd de quadrata dico, dico etiam de alijs radici<lb/>bus & erit ex &longs;ecunda regula harum 31 39/62 & quadratum erit 1000 <lb/>1521/3844. Iuxta ergo primam partem regulæ 31 38/61 erit minus, & in ueritate <lb/>in eo, quod fit ducendo, ut uides, & hoc e&longs;t pro<lb/><figure id="id.015.01.154.3.jpg" xlink:href="015/01/154/3.jpg"/><lb/>ximum ad 11/160, multiplico igitur duplum 31 39/62, <lb/>quod e&longs;t fermè 63 1/4 in 1/160 fient 63/160 detrahe ex <lb/>1521/3844 hoc modo, diuide 3844 per 160 exit 24 /40 <lb/>diuide 1521 per 24, exit 63 3/8, habes igitur quod <lb/>1521/3844 &longs;unt 63/160, igitur detracto 63/160 ex 63/160 nihil relinquitur, & erit <02> exa<lb/>cta ualde 1000 hoc 31 38/61 cuius quadratum 1000 41/3421 uides breuita <lb/>tem, & propinquitatem in producto differentia e&longs;t 1/100 aut parum <lb/>maius quod ad radicem comparatum cum debeat diuidi per du<lb/>plum eius erit paulo maius 1/6300. Vnde facilior e&longs;t, & breuior hæc <lb/>uia quàm per 00 ad ditus. </s>

<s id="id002410">Rur&longs;us uolo aliquid <expan abbr="adi&mtilde;ere">adimnere</expan> & cum pro <lb/>pinquitate ita facio. </s>

<s id="id002410">Rur&longs;us uolo aliquid <expan abbr="adi&mtilde;ere">adimere</expan> & cum pro<lb/>pinquitate ita facio. </s>

<s id="id002411">Con&longs;idero quòd 31 38/61 e&longs;t maius 1/6300 radice, di<lb/>uido 6300 per 62 exit 103 fermè, neque enim curo in hoc fractiones, <lb/>multiplico ergo 103 in 38/61 & habeo 3914/6283 hic denominator e&longs;t proxi<lb/>mus 6300, aufero ergo 1 ex 3914, habebo ualde proximam <02> 1000, <lb/>31 3913/6283 cuius quadratum e&longs;t 1000 minus 1/1048 hoc ut dixi diui&longs;um <lb/>per duplum <02> quod e&longs;t 63 e&longs;t omnino in&longs;en&longs;ile in radice.</s>

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<s id="id002425">Numeros fractos ad minores in <expan abbr="ead&etilde;">eadem</expan> proportione ualde propinqua</s>

<s id="id002426">Cum plerunque numeri fracti hab cantur per radices, ut aliquan<lb/><arrow.to.target n="marg470"/><lb/>do maiores &longs;int, aut minores eo fit, ut po&longs;sint reduci ad mino<lb/>res numeros, ut melius intelligi po&longs;sint & facilius tractari, &

<s id="id002426">Cum plerunque numeri fracti habeantur per radices, ut aliquan<lb/><arrow.to.target n="marg470"/><lb/>do maiores &longs;int, aut minores eo fit, ut po&longs;sint reduci ad mino<lb/>res numeros, ut melius intelligi po&longs;sint & facilius tractari, &

<pb pagenum="137" xlink:href="015/01/156.jpg"/>cum hoc &longs;it exactior illa pars exemplum, ergo habeo 2 3829/12566, quem <lb/>uolo certa ratione ad minores diui&longs;iones deducere. </s>

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<s id="id002492">pro<lb/>portio igitur g e ad e f cum &longs;it ut c ad e ex &longs;uppo&longs;ito erit ut ip&longs;i pro<lb/>portioni addamus, & detrahamus ex duplo a b & dimidium re&longs;i<lb/>dui ducamus in &longs;e, & addamus aggregato quadrati a b cum ip&longs;a

<pb pagenum="141" xlink:href="015/01/160.jpg"/>a b, & latus eius detracto dimidio re&longs;idui erit b clinea, quare diui<lb/>&longs;io nota, & e&longs;t ut dicamusu: olo diuidere datam lineam, ut quantita<lb/>tes adiectæ &longs;ub mutua proportione ad unam tertiam cum parti<lb/>bus obtineantinter &longs;e proportionem datam.</s>

<pb pagenum="141" xlink:href="015/01/160.jpg"/>a b, & latus eius detracto dimidio re&longs;idui erit b c linea, quare diui<lb/>&longs;io nota, & e&longs;t ut dicamus : uolo diuidere datam lineam, ut quantita<lb/>tes adiectæ &longs;ub mutua proportione ad unam tertiam cum parti<lb/>bus obtineant inter &longs;e proportionem datam.</s>

<s id="id002493"><margin.target id="marg487"/>P<emph type="italics"/>er<emph.end type="italics"/> 1. <emph type="italics"/>&longs;ecuu <lb/>di<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s>

<s id="id002493"><margin.target id="marg487"/>P<emph type="italics"/>er<emph.end type="italics"/> 1. <emph type="italics"/>&longs;ecun<lb/>di<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s>

<s id="id002494">Propo&longs;itio cente&longs;imaquadrage&longs;imanona.</s>

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<s id="id002525">Propo&longs;itio cente&longs;imaquinquage&longs;ima&longs;ecunda.</s>

<s id="id002526">Si linea in duas partes æquales duas que in æquales diuidatur, fue<lb/>ritque proportio aggregati ex maiore & dimidio ad ip&longs;am maiorem <lb/>uelut ex minore, & aliqua linea ad ip&longs;am minorem, & rur&longs;us aggre<lb/>gati ex minore dimidio ad ip&longs;am minorem, uelut aggregati ex ma<lb/>iore & alia addita ad ip&longs;am maiorem, erit proportio dimidij'ad par <lb/>tem unam inæqualem, uelut alterius partis inæqualis ad &longs;uam ad<lb/>ditam mutuò, & etiam proportio ad ditarum inuicem, uelut pro<lb/>portio partium inæqualium duplicata, & rur&longs;us ip&longs;um dimidium <lb/>lineæ a&longs;&longs;umptæ medium erit proportione inter additas. </s>

<s id="id002526">Si linea in duas partes æquales duas que in æquales diuidatur, fue<lb/>ritque proportio aggregati ex maiore & dimidio ad ip&longs;am maiorem <lb/>uelut ex minore, & aliqua linea ad ip&longs;am minorem, & rur&longs;us aggre<lb/>gati ex minore dimidio ad ip&longs;am minorem, uelut aggregati ex ma<lb/>iore & alia addita ad ip&longs;am maiorem, erit proportio dimidij ad par<lb/>tem unam inæqualem, uelut alterius partis inæqualis ad &longs;uam ad<lb/>ditam mutuò, & etiam proportio additarum inuicem, uelut pro<lb/>portio partium inæqualium duplicata, & rur&longs;us ip&longs;um dimidium <lb/>lineæ a&longs;&longs;umptæ medium erit proportione inter additas. </s>

<s id="id002527">Demum <lb/>proportio dimidij cum ad dita maiore ad dimidium cum addita mi<lb/>nore, uelut maioris partis ad minorem.<lb/><arrow.to.target n="marg496"/></s>

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<pb pagenum="143" xlink:href="015/01/162.jpg"/>c b ad b d, uelut g a ad a d, & hoc e&longs;t primum. </s>

<s id="id002531">Quia ergo c a e&longs;t æ<lb/>qualis c b, erit c a ad b d, uelut g a ad a d, & iam fuit a d ad c a, ut b d <lb/>ad f b, per conuer&longs;am igitur a d ad b d, ut g a ad a d, & ut b d ad fb, <lb/>interpo&longs;itis ergo a d & d b inter a g & b f cum compo&longs;ita &longs;it pro<lb/>portio a g ad b f ex proportione a g ad a d, & ad d b, & d b <lb/>ad b f, & proportio a d ad d b, &longs;it æqualis proportioni <lb/><figure id="id.015.01.162.1.jpg" xlink:href="015/01/162/1.jpg"/><lb/>a g ad a d, & d b ad b f, igitur proportio a g ad b f. </s>

<s id="id002532">Per de<lb/>mon&longs;trata ab Alchindo e&longs;t duplicata proportioni a d ad <lb/>d b quod e&longs;t &longs;ecundum. </s>

<s id="id002533">Rur&longs;us quia ex primo demon<lb/>&longs;trato, uel eius conuer&longs;o proportio a d ad a c e&longs;t uelut b d <lb/>ad b f, & d b ad a c, ut a d ad a g, proportiones ergo <lb/><figure id="id.015.01.162.2.jpg" xlink:href="015/01/162/2.jpg"/><lb/>a d & d b ad a c componunt proportionem produ<lb/>ducti a d in d b, quod &longs;it h ad quadratum a c quod &longs;it <lb/>k, & &longs;imiliter proportio b d ad b f & a d ad a g com<lb/>ponunt proportionem producti ex b d in a d, quod <lb/>&longs;itl ad productum b f in a g, quod &longs;it m, per demon&longs;trata ab Eucli<lb/>de in &longs;exto Elementorum, igitur proportio h ad k ut l ad m, &longs;ed h & </s>

<s id="id002533">Rur&longs;us quia ex primo demon<lb/>&longs;trato, uel eius conuer&longs;o proportio a d ad a c e&longs;t uelut b d <lb/>ad b f, & d b ad a c, ut a d ad a g, proportiones ergo <lb/><figure id="id.015.01.162.2.jpg" xlink:href="015/01/162/2.jpg"/><lb/>a d & d b ad a c componunt proportionem produ<lb/>cti a d in d b, quod &longs;it h ad quadratum a c quod &longs;it <lb/>k, & &longs;imiliter proportio b d ad b f & a d ad a g com<lb/>ponunt proportionem producti ex b d in a d, quod <lb/>&longs;it l ad productum b f in a g, quod &longs;it m, per demon&longs;trata ab Eucli<lb/>de in &longs;exto Elementorum, igitur proportio h ad k ut l ad m, &longs;ed h & </s>

<s id="id002534"><arrow.to.target n="marg497"/><lb/>l &longs;unt æquales, quia producuntur ex ei&longs;dem, igitur per demon&longs;tra<lb/>ta in quinto Elementorum Euclidis, k e&longs;t æquale m, ergo a c e&longs;t me<lb/>dia pro portione inter b f & g a, quod e&longs;t tertium. </s>

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<pb pagenum="144" xlink:href="015/01/163.jpg"/>una e&longs;t &longs;uperflua, quia ut dixi, una &longs;e quitur ad aliam. </s>

<s id="id002544">Ex hoc pa<lb/>tet cur Dio cles a&longs;&longs;ump&longs;erit lineam unam, quæ e&longs;t a c, quæ &longs;e ha<lb/>bet ad a d, & d b, ut uici&longs;sim a d, & d b ad additas, quod e&longs;t pri<lb/>mum demon&longs;tratum. </s>

<s id="id002545">Sic enim omittit primum quod proponit Ar <lb/>chimedes, & a&longs;&longs;umit quod proximum e&longs;t: & ideò Archimedes non <lb/>pro bat, nec præ&longs;upponit, quod à Diocle probatur, &longs;cilicet datum <lb/>e&longs;&longs;e punctum d in linea a b, &longs;ed &longs;olum in linea g f, ideò cogitur pro<lb/>bare &longs;ecundum quod demon&longs;tratur ab Eutocio, & à nobis demon <lb/>&longs;tratum e&longs;t &longs;uprà. </s>

<s id="id002546">Archimedes <expan abbr="a&utilde;t">aunt</expan> a&longs;&longs;umit <expan abbr="lineã">lineam</expan> extra circulum, <expan abbr="quã">quam</expan> <lb/>uo cat b f, quæ e&longs;t æqualis b c medietati: aliam a&longs;&longs;umit quam uocat <lb/>b h, cuius proportio ad b d e&longs;t &longs;icut quadrati ad a d quadratum a b. <lb/></s>

<s id="id002546">Archimedes <expan abbr="a&utilde;t">aut</expan> a&longs;&longs;umit <expan abbr="lineã">lineam</expan> extra circulum, <expan abbr="quã">quam</expan> <lb/>uocat b f, quæ e&longs;t æqualis b c medietati: aliam a&longs;&longs;umit quam uocat <lb/>b h, cuius proportio ad b d e&longs;t &longs;icut quadrati ad a d quadratum a b. <lb/></s>

<s id="id002547">Con&longs;tat ergo quod proportio g d ad d f e&longs;t data. </s>

<s id="id002548">Et &longs;imiliter f g ad <lb/>g d, & e&longs;t 1 præ proportione data. </s>

<s id="id002549">Vnde notandum quod datum <lb/>dicitur, &longs;impliciter cognitum alio modo, dicitur datum po&longs;itione, <lb/>quod e&longs;t certum & tale, uelut &longs;i quis dicat, diuide 10 in duos nume<lb/>ros quadratos: hoc non e&longs;t datum, pote&longs;t enim diuidi pluribus mo <lb/>dis. </s>

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<s id="id002570">H&etail;c propo&longs;itio e&longs;t admirabilis: & etiam de&longs;crip&longs;i, ut multa &longs;ecre<lb/>ta Dialecticæ potius <expan abbr="aperiren&ttilde;">aperirentur</expan> quam quod huic propo&longs;ito <expan abbr="mult&utilde;">multum</expan> <lb/>congrueret. </s>

<s id="id002571">Ideò potius &longs;cholij cau&longs;a po&longs;ita e&longs;t quam ip&longs;ius tracta<lb/>tionis: ut <expan abbr="mod&utilde;">modum</expan> demon&longs;trandi magis quam id, &qring;d <expan abbr="demon&longs;tra&ttilde;">demon&longs;tratur</expan>, re<lb/>&longs;picere oporteat. </s>

<s id="id002572"><expan abbr="Con&longs;titua&ttilde;">Con&longs;tituatur</expan> ergo (per uiam problematis) linea a b <lb/>& proportio c ad d, & fiat d e ad c, ut c ad d, & a b ad e ut b f ad d, & <lb/>ut g ad c, eritque g media inter a f & f b, quod licet &longs;olum &longs;upponatur <lb/>ab Appollonio, <expan abbr="tam&etilde;">tamen</expan> facilè demon&longs;tratur & à Commandino adie<lb/>cta e&longs;t <expan abbr="demõ">demom</expan> &longs;tratio. </s>

<s id="id002573">Concurrant ergo ex a & b du&etail; line&etail; in aliquod </s>

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<s id="id002583">Cum ergo h&etail;c demon&longs;tratio &longs;it ex &longs;en&longs;u in uno puncto h, ideò ad <lb/>quælibet puncta traduci pote&longs;t, quæ potero imaginari, & ita pri<lb/>ma uo cabitur &longs;en&longs;us, <expan abbr="&longs;ec&utilde;da">&longs;ecunda</expan> imaginandi: Et <expan abbr="quoniã">quoniam</expan> in demon&longs;tran<lb/>do non a&longs;&longs;umimus aliquid, quod &longs;it proprium alicui puncto, ni&longs;i <lb/>proportionem h a ad h b &longs;imilem e&longs;&longs;e c ad d, ideo hoc pertinet ad <lb/>intellectum, & e&longs;t tertium. </s>

<s id="id002584">Etidem dico &longs;i k e&longs;&longs;et ultra h quod po<lb/>te&longs;t contingere. </s>

<s id="id002585">modò k a ad k b &longs;it ut c ad d & k f &longs;it &etail;qualis g idem <lb/>&longs;equetur, & comprehenditur &longs;ub tertio & pertinet ad intellectum, <lb/>& quoniam demon&longs;tratur quod punctum k ubicun que &longs;umatur, e&longs;t <lb/>in &etail;quali <expan abbr="di&longs;tãtia">di&longs;tantia</expan> à puncto f&longs;cilicet per g lineam, erit &longs;emper in peri<lb/>pheria circuli, & hoc pote&longs;t e&longs;&longs;e in infinitis locis &longs;impliciter & extra <lb/>infinitum nihil e&longs;t, igitur &longs;ub hoc continetur conuer&longs;um &longs;cilicet, <lb/>quod a quolibet puncto circuli ductis lineis ad a & b ip&longs;&etail; erunt in <lb/>proportione c ad d. </s>

<s id="id002586">Et ita ab&longs;que principijs Geometricis concluditur <lb/>propo&longs;itio Geometrica & hoc e&longs;t <foreign lang="greek">w_erila/mpousin</foreign> & fermè &longs;ummum in<lb/>tellectus humani. </s>

<s id="id002586">Et ita ab&longs;que principijs Geometricis concluditur <lb/>propo&longs;itio Geometrica & hoc e&longs;t <foreign lang="greek">perila/mpousin</foreign> & fermè &longs;ummum in<lb/>tellectus humani. </s>

<s id="id002587">Et pote&longs;t demon&longs;trari Geometricè duobus uer<lb/>bis. </s>

<s id="id002589"><expan abbr="f&longs;upponi&ttilde;">f&longs;upponitur</expan> æqualis g eo quòd h e&longs;t in peripheria circu<lb/>li erit media inter a f & f b, quare cum angulus f &longs;it communis, erit <lb/>proportio a h ad h b, laterum re&longs;picientium angulum f in utroque </s>

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