Achillini, Alessandro (Achillinus, Alexander), Alexandri Achillini bononiensis De proportionibus motuum quaestio. , 1545

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              quatuor mediorum ad tria media quam est proportio 4 ad </s>
              <s id="id.0.2.10.05">Secundo miscendo maioritatem cum minoritate. sit a. potentia ut 8 b. potentia ut 2 c. resistentia ut 4, tunc proportio 8 ad 4 est dupla et 2 ad 4 est subdupla et proportio 2 ad medium est quadrupla qualis est proportio 8 ad </s>
              <s id="id.0.2.10.06">Tertio in minoritatibus. sit a. potentia ut 8 b. vero potentia ut 4 c. vero resistentia ut 16 tunc proportio 8 ad 16 est subdupla et proportio 4 ad 16 est subquadrupla proportio vero subduplae ad subquadruplam est dupla proportio, qualis est 8 ad </s>
              <s id="id.0.2.10.07">Quarto miscendo aequalitatem cum minoritate, sit a. potentia ut 8 b. vero potentia ut 4 et c. resistentia ut 8 tunc 8 ad 8 est 1 et 4 ad 8 est subduplum et proportio 1 ad subduplum est dupla, qualis est proportio 8 ad </s>
              <s id="id.0.2.10.08">Quinto miscendo maioritatem cum aequalitate. sit a. potentia ut 8 b. potentia ut 4 c. resistentia sit ut 4 tunc 8 ad 4 est dupla proportio et 4 ad 4 est 1 tunc 2 ad 1 est dupla proportio qualis est 8 ad 4 sic enim universales regulae sunt. non autem particulares et </s>
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              Tertia regula. qualis est proportio denominationum proportionum talis est diversarum resistentiarum cum ad eandem potentiam comparantur. patet in maioritatibus sit a. potentia ut 8 et resistentia b. sicut 4 et resistentia c. sit ut 2 tunc proportio 8 ad 4 est dupla, et propotio 8 ad 2 est quadrupla. tunc quadruplae ad duplam est dupla proportio qualis est inter 4 et </s>
              <s id="id.0.2.11.03">Secundo miscendo aequalitatem cum maioritate, sit a. potentia ut 8 et resistentia b. ut 8 et c. resistentia sit ut 4 tunc proportio 8 ad 8 est 1 et 8 ad 4 est dupla tunc 2 ad 1 est dupla proportio qualis est inter 8 et </s>
              <s id="id.0.2.11.04">Secundo sit a. potentia ut 8 et b. resistentia sit ut 4 c. vero resistentia sit. ut 4 tunc a. ad b. est dupla. et similiter a. ad c. et sicut duplae ad duplam est 1 ita b. ad c. est </s>
              <s id="id.0.2.11.05">Tertio in minoritatibus, sit a. potentia ut 1 et resistentia b. sit ut 8 et c. resistentia sit ut 4 tunc 1 ad 8 est suboctupla et unum ad 4 est subquadrupla. tunc subquadruplae ad octuplam est dupla proportio qualis est resistentiarum b. et </s>
              <s id="id.0.2.11.06">Adverte ad variationem secundae regulae et tertiae quia superior ad inferiorem resistentiam. habet proportionem quaesitam. sed in 3 regula est oppositum scilicet c. ad b. est propositio quaesita, sed accidentalis est differentia, quia aliter poterant signari termini. ut patet. pone enim minores numeros supra, et maiores infra, et erit intentum, et talem indifferentiam terminorum et proportionum mathematica exigit abstractio, et praesertim quia naturali inversioni non </s>
              <s id="id.0.2.11.08">
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              Corollarium 1 octupla est dupla quadruplae quia proportio denominationum earum proportionum est dupla, quia octupla ab 8 denominatur et quadrupla a quatuor. similiter nonupla est tripla triplae. quia denominator nonuplae est triplus ad denominatorem triplae scilicet 9 ad 3 hae regulae modernis fere omnibus quos legerim de proportione proportionum loquentibus contrariae sunt. ideo advertendae, ut Thomae Baduardino et consequenter Suis et Calculatori, Nicolao Orem et </s>
              <s id="id.0.2.11.09">Conformes autem antiquis sunt, ut Aristoteli usque ad Aver. quod si adduxeris mathematicorum compositionem proportionum propter reductionem numeri denominantis in alium denominatorem quam duplicationem appellant sic quod duae quadruplaedecimamsextuplam reddant quia quater quatuor dant 16 et duae triplae dant nonuplam quia ter 3 dant </s>
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              Dicendum hanc reductionem proportionum esse productionem unius proportionis ex altera, non autem
                <expan abbr="componem">compositionem</expan>
              , neque </s>
              <s id="id.0.2.11.12">Hoc sentiebat Campanus, cum super diffinitione 10 quinti geo. Euclidis exposuit duplicata hoc est in semet multiplicata. et diffinitione 11 exponit triplicata hoc est in se postea in productum </s>
              <s id="id.0.2.11.13">Quod autem haec productio non sit compositio, aut maioratio, probo. et suppono quod duplicare est maiorare. quia duplum est species multiplicis, et multiplex est species maioritatis, sed producere proportiones non semper est maius sed aliquando aequale invenire. et aliquando minus invenire. quia reducendo aequalitatem in aequalitatem invenitur aequale. quia unum reductum in unum dat 1 dicere autem quod aequalitas non est proportio, est principia mathematicorum negare scilicet diffinitionem proportionis et </s>
              <s id="id.0.2.11.14">Minoritas autem in se reducta dat minus quam ipsa sit quia medium reductum in medium dat subquadruplum quod est minus quam medium. verum si duplam reduxeris in duplam, habebis praecise duplum ad illam. quia 2 reducta in 2 dant 4 sed maior quam dupla proportio in se reducta dat plusquam duplum illi. quadrupla enim per quadruplam multiplicata dat decimamsextuplam. quae est plusquam duae quadruplae. ut patet ex congregatione denominationum illarum </s>
              <s id="id.0.2.11.15">Secundo suppono principium Arithmaticae et geo. commune, omne totum est maius sua parte, de quantitativis toto et partibus intellige, patet ex 7 elementorum Euclidis. comunis animi conceptio est omnis pars est minor suo </s>
              <s id="id.0.2.11.16">Sed huius oppositum concedunt recentiores mathematici concedentes quod omni aequalitate datur aequalitas dupla tripla et cetera et Campanus concedit. quia unam aequalitatem componunt ex alia et unam concedunt esse aequalem alteri, ergo concedunt totum esse aequali </s>
              <s id="id.0.2.11.17">Item oportet ipsos concedere partem esse maiorem toto, et minoritatem esse partem maioritatis patet producendo nonuplam ex decimaoctupla et medietate quia medium ductum in 18 dat 9 quod et si concedunt minoritatem non esse maioritati </s>
              <s id="id.0.2.11.18">Contra. tunc totius quanti ad certam partem quamvis finitum sit utrunque nulla erit proportio, quod est impossibile. signantur enim minoritates per fractiones notantes partem vel partes unius et aequalitas signatur per unum, et certum est cuiuslibet numeri partem esse unitatem. et non est alia quantitas proportionum ab ea quae ex denominatoribus trahitur, quam magis intima sit </s>
              <s id="id.0.2.11.19">Item oportet ipsos concedere aequalitatem componi ex maioritate et minoritate, quia multiplicatum 4 per unum quartum dat </s>
              <s id="id.0.2.11.20">Si enim 4 inter 1 et 1 interponatur. erit aequalitas composita ex proportione 1 ad 4 quae est subquadrupla et ex proportione 4 ad 1 quae est </s>
              <s id="id.0.2.11.21">Similiter etiam componetur minoritas ex maioritate, et minoritate, patet multiplicando unum quartum per 2 dant unum quartum quod sunt medium patet scribendo inter 1 et 2. 4 pro </s>
              <s id="id.0.2.11.22">Et quemadmodum inter aequalitatis terminos interpositum est medium maius extremis interponi potest medium minus extremis. et sic ex maioritate et minoritate etiam componitur aequalitas. haec autem in numerorum productionibus nullum afferunt inconveniens, dummodo non dicant productionem proportionum esse earundem duplicationum, nisi forte per accidens scilicet in certo dato individuo, neque dicant productionem proportionum esse maiorationem earum. quod si haec media signare voluerint scilicet maiora maiori extremo vel minora minori extremo: tunc mathematicas abstractiones ad certam materiam nimis restringunt scilicet per interfectum extremis intelligunt maius minore et minus maiore. et quod universaliter verum est particulariter </s>
              <s id="id.0.2.11.23">More igitur Arist. mathematicorum excellentissimi, proportiones componant per denominantium numerorum coacervationem si eas vero componere intendunt et </s>
              <s id="id.0.2.11.24">Omnes igitur proportiones quarum denominationes sunt aequales sunt aequales. et maior est proportio quam maiorem habet denominationem, et minor quam minorem ut 7 geo. Euclidis colligitur in principio, et in principio 2 arithmeticae Iordani.</s>
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              Et qualis est denominationum proportio talis est proportio proportionum, ut sesquialtera est medietas triplae et triplae ad eam est dupla est </s>
              <s id="id.0.2.11.27">Hanc autem proportionum componens naturales experientiae convincunt, et non priorem. patet moveatur a quadrupla proportio lapis super nave mota a proportio quadrupla ad eandem proportionis differentiam, tunc duplum spacium pertransibit lapis ex duarum causarum congregatione ad id quod pertransiret lapis ab una illarum causarum tantum. ex quibus tamen congregatis causis non fit super lapide proportio decimasextupla, unum a quodcunque proportione agant quatuor homines circa quodlibet artificiale, non est inventum alios quam homines 8 facere duplam velocitatem illi quam a quatuor prioribus provenit. stante tamen temporis paritate, vigoris hominum, et iuvamenti unius ad alterum, iuvamentum enim accidentale posset intervenire et </s>
              <s id="id.0.2.11.28">Similiter existente simplici eodem puta gravi in aere aut alio medio cui simplex datum dominetur. patet experientia quod qualis est proportio medii ad medium in spissitudine aut raritate talis est proportio velocitatis ad velocitatem sequentem motores in illis mediis. dato etiam quod Aristoteles illam regulam nunquam dixisset, quam ... dixit et </s>
              <s id="id.0.2.11.29">Confirmatur quia Averrois vult si potentia ut 2 exempli gratia in medio resistente ut 1 transit pedale in hora, quod potentia. ut 3 transit pedale cum dimidio quia qualis est proportio potentiarum talis est proportio spatiorum. ergo potentia ut 6 transibit tres pedes. patet ex eadem regula. quia duplum est 6 ad 3 et duplum est 3 ad unum cum dimidio et sic sextupla est dupla triplae. quod si Alkindus liber de proportionibus diffinitione 3 dixerit proportionem produci aut componi ex proportionibus est denominationem proportionis produci ex denominationibus proportionum altera in alteram ductis, sic quod per produci et componi idem
                <expan abbr="itelligat">intelligat</expan>
              : tunc ab eo accipe verum intelligimus scilicet proportionum productionem. a me autem accipe quod illa non est proportionum vera compositio sed metaphorica est. similiter de aliis mathematicorum auctoritatibus intelligendum est scilicet quod compositio id est productio intelligatur ut particulariter de aliquibus superficiebus tangit proportionem 24 sexti elementorum Euclidis volentis proportionem illarum superficierum esse ex productione proportionum laterum et </s>
              <s id="id.0.2.11.30">Sed quia modernorum aliqui negant proportionem esse inter maioritatem aequalitatem et minoritatem, et aliqui dicunt quod inter illas infinite magna est proportio, adducuntur contra illos mathematicorum </s>
              <s id="id.0.2.11.31">Et primo probo quod maioritas est maior aequalitate utendo 27 propositione 5 geo. </s>
              <s id="id.0.2.11.32.Mg">Quod proportio it finita inter aequalitatem, maioritatem de </s>
              <s id="id.0.2.11.33">Si fuerit quatuor quantitatum maior proportio primae ad duplam quam 3 ad 4 erit permutatim maior proportio primae ad 3 quam 2 ad </s>
              <s id="id.0.2.11.34">Sint quatuor termini 6.3.4.3 tunc maior est proportio primi ad 2 quam 3 ad 4 ergo maior est proportio 1 ad 3 quam 2 ad 4 quod si termini sint 6.3.6.5 sequitur aequalitatem esse maiorem minoritate quod si termini sint 6.3.4.3 sequitur maioritatem esse maiorem </s>
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              Qui libri geometriae Euclidis. </s>
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              </s>
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              Octupla dupla quadruplae </s>
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              Productio proportionis unius ex altera non est
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              neque </s>
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              Maior est proportio quae maiorem h3 </s>
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              <s id="id.0.2.12.01">Secundo. utendo propositione 29 quinti geo. Euclidis si fuerint quatuor quantitates quarum 1 et 2 ad 2 fit maior proportio quam tertiae et quartae ad 4 erit quoque disiunctim proportio primae ad 2 maior quam tertiae ad </s>
              <s id="id.0.2.12.02">Probatur. maioritatem esse maiorem </s>
              <s id="id.0.2.12.03">Sint quatuor termini 6.2.3.3 tunc primi et 2 ad 2 est maior proportio quam tertii et 4 ad 4 ergo proportio primi ad 2 est maior proportione 3 ad 4 quod si termini sint 6.2.2.3 sequitur maioritatem esse maiorem </s>
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              <s id="id.0.2.13.01">Tertio. utendo propositione 31 quinti geo. </s>
              <s id="id.0.2.13.02">Si fuerit tres quantitates in uno ordine, itemque tres in alio, fueritque primae priorum ad 2 maior proportio quam primae posteriorum ad </s>
              <s id="id.0.2.13.03">Itemque secundae priorum ad 3 maior quam secundae posteriorum ad 3 erit quoque primae priorum ad 3 maior proportio quam primae posteriorum ad </s>
              <s id="id.0.2.13.04">Sit primus ordo 9.3.6 secundus ordo 7.3.7 tunc maior est proportio primi primi ordinis ad 2.1 ordinis quam 1.2 ordinis ad 2 secundi. et maior est proportio 2.1 ordinis ad 3 eiusdem quam 2.2 ordinis ad 3 eiusdem, ergo maior est proportio primi ad 3 in 1 ordine quam primi ad 3 in 2 ordine ergo maioritas est maior aequalitate. quod si primus ordo sit </s>
              <s id="id.0.2.13.05">Et secundus sit 6.3.7 sequitur maioritatem esse maiorem </s>
            </p>
            <p>
              <s id="id.0.2.14.01">Quarto. utendo propositione 32 quinti geometriae Euclidis si fuerint tres quantitates in uno ordine, itemque tres in alio, fueritque proportio secundae priorum ad 3 maior quam primae posteriorum ad 2 itemque primae priorum ad 2 maior quam secundae posteriorum ad 3 erit maior proportio primae priorum ad 3 quam primae posteriorum ad 3 probatur. maioritatem esse maiorem </s>
              <s id="id.0.2.14.02">Sit primus ordo 9.3.6 sit 2 ordo 3.7.3 quod si ordines sint 9.3.6 et 3.7.4 sequitur maioritatem esse maiorem </s>
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            <p>
              <s id="id.0.2.15.01">Quinti. utendo propositione 27 secundi arithmeticae </s>
              <s id="id.0.2.15.02">Si proportio primi ad 2 est maior quam 3 ad 4 qui ex 1 in 4 producitur est maior producto ex 2 in 3 quod si productus numerus ex 1 in 4 maior fuerit numero producto ex 2 in 3 etiam proportio primi ad 2 maior erit quam proportio 3 ad 4 tunc si sint termini 6.2.2.2 sequitur maioritatem esse maiorem aequalitate. quod si sint termini 6.2.1.2 sequitur maioritatem esse maiorem </s>
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            <p>
              <s id="id.0.2.16.01">Sexto. utendo 19 propositione secundi arithmeticae </s>
              <s id="id.0.2.16.02">Si primus terminus fuerit maior 3 compositusque ex primo et 2 aequalis composito ex 3 et 4 maior erit proportio 1 ad 2 quam 3 ad </s>
              <s id="id.0.2.16.03">Si sint termini 6.2.4.4 sequitur quod maioritas est maior aequalitate. quod si termini sint 6.2.3.5 sequitur quod maioritas est maior </s>
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            <p>
              <s id="id.0.2.17.01">Septimo. utendo diffinitione 8 quinti geo. </s>
              <s id="id.0.2.17.02">Cum fuerint primae et tertiae aeque </s>
              <s id="id.0.2.17.03">Item 2 et quartae aeque multiplices addetque multiplex primae super multiplicem </s>
              <s id="id.0.2.17.04">Non addet autem multiplex 3 super multiplicem 4 dicetur prima maioris proportionis ad 2 quam 3 ad </s>
              <s id="id.0.2.17.05">Sint termini 8.4.4.4 tunc duplum ad 1 scilicet 16 est maius duplo ad 3 scilicet quam 8 et triplum ad 2 non est maius quam triplum ad 4 scilicet 12 quo maius est duplum ad 1 scilicet 16 ergo proportio primi ad 2 quae est maioritas est maior proportione 3 ad 4 quam est </s>
              <s id="id.0.2.17.06">Item sumatur duplum ad 1 et ad 3 et quadruplum ad 2 et ad 4 tunc maior est proportio multiplicis primi ad multiplex 2 quam multiplicis 3 ad 4 ergo maior est proportio 1 ad 2 quam 3. ad </s>
              <s id="id.0.2.17.07">Item probatur maioritatem esse maiorem </s>
              <s id="id.0.2.17.08">Sint termini 8.4.3.4 tunc duplum ad 1 est 16. 6 est duplum 3 et 8 est duplum 2 et 4 tunc aeque multiplex 1 est maius quam sit aeque multiplex 2 scilicet 16 est maius quam 8 et multiplex 3 scilicet 6 non est maius multiplici ad 4 ergo maior est proportio 1 ad 2 quam 3 ad 4.</s>
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            <p>
              <s id="id.0.2.18.01">Octavo. respondeo propositione 33 quinti geo. Euclidis. probatur. quod maioritas est maior </s>
              <s id="id.0.2.18.02">Sint quatuor termini 8.8.1.2 et similiter 1 duo </s>
              <s id="id.0.2.18.03">Alii duo sint abscissa ab eis 3 a 1 et 4 a 2 tunc proportio totorum est maior proportione abscissorum quia aequalitas est minor maioritate. ergo residui ad residuum est maior proportio quam totius ad totum. sed totorum fuit aequalitas. residuorum vero maioritas. ergo maioritas est maior aequalitate. quod si dicitur quod petitur minoritatem esse minorem aequalitate, dicendum quod non aliter signari potest minor proportio aequalitate, quod si tota sint 4. 8 abscissa vero 1.5 tunc maior est proportio totorum quam abscissorum, quia plus est esse medium quam quintum. ergo proportio residuorum est maior proportione totorum, sed proportio residuorum est aequalitas scilicet 3 ad 3 ergo aequalitas est maior minoritate quae fuit inter 4 et </s>
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            <p>
              <s id="id.0.2.19.01">Nono. utendo propositione 26 quinti geometriae Euclidis probatur aequalitatem esse maiorem </s>
              <s id="id.0.2.19.02">Sint termini 2.1.1.1 tunc maior est proportio primi ad secundum quam tertii ad quartum, ergo proportio secundi ad primum est minor quam quarti ad tertium. </s>
              <s id="id.0.2.19.03">Sed proportio secundi ad primum est minoritas et proportio quarti ad tertium est aequalitas, ergo equalitas est maior </s>
              <s id="id.0.2.19.04">Si dicitur quod petitur principium tunc probatur antecedens. quia ad eandem quantitatem scilicet 1 comparantur duae quantitates scilicet 2 et 1 ergo cum 2 sint maius quam </s>
              <s id="id.0.2.19.05">Maiorem habent 2
                <expan abbr="proprtionem">proportionem</expan>
              ad 1 quam habeat 1 ad 1 et patet probatio ex 8 propositione quinti geo. Euclidis si autem sint termini 1.1.1.3 probatur maioritatem esse maiorem aequalitate. quia proportio primi ad secundum est maior quam tertii ad quartum. ergo proportio secundi ad primum est minor quam quarti ad tertium. sed proportio secundi ad primum est aequalitas, et proportio quarti ad tertium est maioritas. ergo aequalitas est minor </s>
              <s id="id.0.2.19.06">Similiter si propositione 12 quinti elementorum Euclidis utamur scilicet si fuerit proportio primi ad secundum sicut tertii ad quartum. tertii vero ad quartum. maior quam quinti ad 6 erit proportio primi ad 2 maior quam 5 ad </s>
              <s id="id.0.2.19.07">Si termini sint 1.1.1.1.1.2 tunc petitur aequalitatem esse maiorem </s>
              <s id="id.0.2.19.08">Si autem sint termini 2.1.2.1.1.1 tunc petitur maioritatem esse maiorem </s>
              <s id="id.0.2.19.09">Contra hanc determinationem concedentem maioritatem esse maiorem aequalitate. et aequalitate esse maiorem minoritate arguitur, quia sequitur propositionem 34 quinti geo. Euclidis esse </s>
              <s id="id.0.2.19.10">Si quotlibet quantitates ad totidem alias comparentur fueritque cuiuslibet praecedentis ad </s>
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