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version 1.8, 2004/01/29 13:26:37

version 1.9, 2004/01/29 19:37:42

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<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>

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<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>

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<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

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

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

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<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>

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<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>

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<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>

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<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>

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

<s id="id000223">& 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>

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<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>

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

<s id="id000311">Si fuerint quatuor quantitasproportio 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>

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<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>

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<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>

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