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

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              natione igitur numerus est. ut in libro de distinctionibus </s>
              <s id="id.0.2.02.26">Ex hoc patet aequivocatio potentiae qua continuum est numerus et qua unitas est numerus. continuum enim est multa postquam facta est continui divisio. sed ante divisionem continuum est unum, possibile tamen dividi, sed unitatis potentia est quia coacervari potest in </s>
              <s id="id.0.2.02.27">Ex his sequitur ubi aliis visum est infinitam esse proportionem inter longitudinem et punctum nullam inter ea esse proportionem iuxta Averroim 6 physicorum commento 29 quia non est proportio inter lineam et punctum. hanc intentionem habet philosophus 7 physicorum textu commento 24 quia aequivoca sunt, non </s>
              <s id="id.0.2.02.28">Corollarium aliquam habet convenientiam unitas cum numero quem principiat, quam non habet punctum cum magnitudine quam principiat. quia unitas est numeri pars. punctum vero magnitudinis non est pars. ut demonstravit Aristoteles 6 physicorum quum igitur unitatem de puncto praedicat Aristoteles, dicens, punctum est unitas positionem habens ex Platone refert, ut metaphorice accipiatur. quomodo autem punctum positionem habeat declaravi in libro distinctionum capite. </s>
              <s id="id.0.2.02.29">Et si numerum unitati comparas totum parti comparas. aut parti aequale, non autem cum lineam puncto comparas. cum etiam finitum infinito comparaveris aut econtra , non habebis proportionem. quia inter ea non est certa habitudo. primo caeli textu com. 52. et. 64. et .8. phy. com. 15. et. 3. phy. textu et com. 43. intellige igitur certam habitudinem in diffinitione proportionis id est determinatam non dico notam. quia licet communicantium quantitatum proportio sit nota ratione numeri eam numerantis et rationalis dicatur. quia minor quantitas pars est aut partes maioris. intellige aliquota vel aliquote, iuxta primam diffinitionem quinti geometriae </s>
              <s id="id.0.2.02.30">Pars autem est quantitas quantitatis minor maioris. quum minor maiorem </s>
              <s id="id.0.2.02.31">Incommensurabilium autem quantitatum proportio quamvis nota esset in potentia non tamen eam ita nobis notam esse oportet quod immediate a numero ipsam nominare possimus, ut proportio diametri ad costam quadrati a numero immediate numerari non potest. quia radix duorum non est numerata sed potentia illius radicis est nota. quia scio quod radix illa in se met reducta dat duo. hoc modo declarat Campanus potentiam lineae 2 geometriae Euclidis propositione 12 quod si Averrois in proemio primi physicorum dicit proportio suae vitae ad tempus sempiternum est sicut puncti ad lineam, vel sicut finiti ad infinitum. et 6 phy. com. 21 minus quam infinitum est finitum necessario, proportione abutitur. quae potius relatio aut comparatio dici debet quam proportio. quia non quantum quanto comparatur. aut finitum infinito, inter quorum nulla proportio proprie </s>
              <s id="id.0.2.02.32">Ex hoc sequitur aliud est esse alicui improportionale, et non incommensurabile. quia diameter vere quadrati proportionatur costae. proportione tamen irrationali. et tamen eidem costae diameter est incommensurabilis ut primo posteriorum exemplificat philosophus. et primo priorum ponit inconveniens quod sequitur ad oppositum scilicet numerum imparem esse aequalem numero pari. et adverte ad deductionem consequentiae quam ponit Campanus super septima propositione decimi elementorum </s>
              <s id="id.0.2.02.33">Sunt autem quantitates commensurabiles quas una conis mensura metitur 10 elementorum Euclidis. neque ex hoc dicendum quod omnis motus omni motui sit commensurabilis. eo quia omnis motus tempore mensuretur quia non omnia tempora sunt commensurabilia immo dantur tempora se excedentia in proportione qua diameter excedit costam </s>
              <s id="id.0.2.02.34">Si enim motus localis penes spatium extrinsece mensuratur. datis spatiis incommensurabilibus etiam motus erunt incommensurabiles, spatium intellige verum. quod enim dicunt moderni primam sphaeram transire spatium imaginarium repugnat Averroi 2 caeli commento 46 quia est magnitudo ipsius caeli. quod si aliquando Averrois infinitum infinito comparat, hoc est improprie secundum quod improprie sunt aequalia. quia negative aequantur. scilicet quod unum non excedit alterum neque exceditur. dixit enim primo caeli commento 33 potentia infiniti maxima est potentiarum omnium </s>
              <s id="id.0.2.02.35">Et dixerat primo caeli commen. 48 potentia infiniti est maior potentia finiti duplicibus infinitis id est plusquam bis et plusquam ter et cetera ex hoc intellige Aver. 2 cae. com. 38 cum dixit tempus infinitum dicitur alio modo esse aequale infinito cum infinitum non sit maius </s>
              <s id="id.0.2.02.36">Et primo caeli. com. 38. proportio enim infiniti ad totam magnitudinem quam secat et ad minimam partem eius est eadem </s>
              <s id="id.0.2.02.38">
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              Tripartitur autem proportio per maioritatem, minoritatem, aequalitatem. maioritas est maioris ad minus, minoritas est minoris ad maius. aequalitas est unius aequalis ad aequale sibi proportio. neque sunt haec membra prima genera sub relatione posita, sed sunt species illorum generum. aequiparantia enim sub se continet similitudinem, quae non est aequalitas, et aequalitatem et cetera proportionabiliter de superabundantia, et maioritate, et de suppositione et </s>
              <s id="id.0.2.02.39">Omnis enim minoritas est suppositio, et non econtra. ut </s>
              <s id="id.0.2.02.40">Maioritas et minoritas bipartitur rationali et irrationali </s>
              <s id="id.0.2.02.41">Irrationalis est quantitatum incommensurabilium proportio. incommensurabiles quantitates sunt solum continuae. sed non omnes, sed illae, quarum una est alteri non aequalis. et minor non est pars neque partes scilicet aliquote alterius neque aequalis est ei vel eis totum enim suae parti comparatum eandem habet proportionem quam habet ad aliam quantitatem quae non est pars eius, dummodo illa sit aequalis illi parti </s>
              <s id="id.0.2.02.42">Sed rationalis proportio est quantitatum commensurabilium proportio. et quinque tenet species, tres simplices si multiplicem, super particularem, et superpartientem. et duas compositas scilicet multiplicem superparticularem, et multiplicem superpartientem. ab unitate enim recedendo primus numerus multiplicem causat scilicet duplam. ternarius vero respectu 2 superparticularem facit 5 autem ad 3 superpartientem. et prima multiplicium superparticularium est inter 5 et 2. et prima multiplicium superpartientum est inter 8 et 3. et sic ordo specierum est ex exitu earum priori vel posteriori ab unitate aut numeris comparando simplices species inter se. et compositas inter se. non autem quamlibet compositarum cuilibet simplici, ut multiplicem superparticularem quae primo est inter 5 et 2 quae prior est superpartienti quae primo est inter 5 et </s>
              <s id="id.0.2.02.43">Et minoritas quinque habet species quinque speciebus maioritatum correlativas. ut ex principio 9 arithmeticae colligitur species minoritatum nominant comes addendo nominibus maioritatum. sub. ut subdupla et colligitur ab Euclide 14 elementorum proportione 16 et 17 sed Aristoteles non </s>
              <s id="id.0.2.02.44">Sed quia de rationali ut plurimum loquemur. aequalitatem signamus per 1 minoritatem vero fractione aut fractionibus unius, et sic generaliter a dextro in sinistrum protracta linea supra illam ponitur unitas vel numerus numerans partes, et numerator appellatur. infra vero lineam illam, ponitur numerus denominans partem vel partes. et denominator appellatur. et sic differentia est inter denominatorem proportionis, et est totus numerus ibi existens: et denominatorem fractionis, et est numerus infra lineam </s>
              <s id="id.0.2.02.45">Et tunc quanto numerus fractionis denominator est maior tanto fractio est pars minor. et sic unum quintum est minus quam unum quartum ut ex 2 petitione primi arithmeticae Iordani colligitur. omnis pars minor est quae maiorem habet denominationem. stat tamen propter numerantis disparitatem quod fractio maioris numeri sit maior, ut tria quinta maius sunt, quam medium. et sic patet quam habent proportionem minoritates adinvicem, generaliter tamen.</s>
              <s id="id.0.2.02.46">Maioritatem vero signamus per plus quam </s>
              <s id="id.0.2.02.47">Et hoc dupliciter. aut minus pluries continetur, aut semel tantum si semel tantum est dupliciter. aut super addit maior numerus minori fractionem. aut fractiones ut sesquitertia per unum et unum tertium ut. 3. ad. 3. et superbitertia per unum et duas tertias. ut 5 ad 3 si autem minus pluries continetur, aut pluries praecise, et sic numero signatur maioritas ut dupla per 2 ut 2 ad 1 tripla per 3 ut 3 ad </s>
              <s id="id.0.2.02.48">Si minus continetur non praecise a maiore aut superaddit maius minori partem. aut partes minoris. et sic cum numero signatur fractio aut fractiones ut dupla sesquitertia per 2 per duo tertia. ut septem ad 3 et dupla superbitertia per 2 et duas tertias ut 8 ad </s>
              <s id="id.0.2.02.49">Si autem maius adderet minori aliquid quod non esset pars aut partes intellige aliquote: tunc ibi esset proportio irrationalis ut diameter super costa vere </s>
              <s id="id.0.2.02.50">Et tunc adverte quia Euclides in principio 7 elementorum dixit maioris ad minorem esse proportionem secundum quod maius continet minus et minoris partem vel partes. quod verum est in proportione certa maiori quam dupla. sed in dupla totum continet minus et aequale minori, non autem partem neque partes minoris, nisi in potentia et cetera similiter in certa proportione magisquam dupla, quae tamen rationalis </s>
              <s id="id.0.2.02.51">Proportio igitur duplex. arithmetica et geometrica denominatio sumitur a scientiis proportiones habentibus pro </s>
              <s id="id.0.2.02.52">Arithmetica est proportio excessum ut 4 excedunt 2 per </s>
              <s id="id.0.2.02.53">Geometrica est quantitatum proportio secundum quam una continet, continetur, aut aequat </s>
              <s id="id.0.2.02.54">Non ex his terminorum descriptionibus accipiendum est quod geometra non utatur proportione excessuum. quia hoc falsum est. patet ex 10 elementorum Euclidis propositione </s>
              <s id="id.0.2.02.55">Si fuerint quatuor quantitates differentia primae quarum ad secundam sit sicut tertiae ad quartam erit permutatim differentiae primae ad tertiam sicut tertiae ad quartam. et arithmeticus musicus proportione utitur ut diapason. quae est dupla in sonis et </s>
              <s id="id.0.2.02.56">Proportio etiam duplex </s>
              <s id="id.0.2.02.57">Continua, incontinua, Continua est cum unus terminus habet duorum rationem sine interruptione iuxta diffinitionem quintam quinti elementorum Euclidis. ut 4 ad 2 et 2 ad. </s>
              <s id="id.0.2.02.58">Discontinua est ubi intervenit interruptio ut 16 ad 8 et 2 ad </s>
              <s id="id.0.2.02.59">Proportionalitas autem est proportionum similitudo iuxta diffinitionem quartam quinti elementorum Euclidis. ergo ad minus in quatuor terminis consistit. patet consequentia quia omnis proportio requirit duos terminos, quod si in tribus terminis proportionalitas consistit iuxta diffinitionem nonam quinti elementorum Euclidis. hoc est quia unus terminus duorum terminorum officio </s>
              <s id="id.0.2.02.60">Proportionalitas iuxta Campanum quinto elementorum Euclidis super diffinitione 16 sex habet </s>
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              Proportio [...]</s>
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              Contraria. et est, si una sit proportio primi antecedentis ad primum consequens sicut secundi antecedentis ad secundum consequens, tunc qualis est proportio consequentis primi ad antecedens primum, talis est proportio consequentis secundi ad antecedens secundum. sit primum antecedens 8 sit primum consequens 4 sit secundum antecedens 2 sit secundum consequens 1 tunc si 4 est medietas 8 etiam 1 est medietas duorum. patet ex 12 quinti elementorum </s>
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              Permutata est. si una sit proportio primi antecedentis ad primum consequens. sicut secundi antecedentis ad secundum consequens. tunc qualis est proportio antecedentis primi ad secundum antecedens. talis est proportio consequentis primi ad consequens secundi. patet ex diffinitione 12 quinti elementorum Euclidis. et in 7 eiusdem proportione </s>
              <s id="id.0.2.04.03">Similiter qualis est proportio 2 antecedentis ad 1 antecedens. talis est proportio secundi consequentis ad secundum </s>
              <s id="id.0.2.04.04">Sit primum antecedens 8 primum consequens 4 sit 2 antecedens 2 sit secundum cosequens 1 tunc permutando si 8 est quadruplum ad 2. 4 est quadruplum ad </s>
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              Coniuncta si una sit proportio primi antecedentis ad 1 consequens sicut secundi antecedentis ad 2 consequens, tunc qualis est proportio primi antecedentis et primi consequentis ad 1 consequens, talis est proportio secundi antecedentis et 2 consequentis a 2 consequens. sit 1 antecedens 8 primum consequens 4 sit secundum. antecedens 2 sit 2 consequens 1 tunc si aggregatum ex 8 et 4 est triplum ad 4 etiam 2 et 1 est triplum ad 1 patet ex diffinitione 13 quinti elementorum Euclidis et ex proportione 15 </s>
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              Tertia</s>
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              Disiuncta est si una fuerit proportio primi antecedentis et primi consequentis ad 1 consequens sicut secundi antecedentis et secundi consequentis ad 2 consequens. tunc si qualis est proportio primi antecedentis ad 1 consequens, talis est proportio 2 antecedentis ad 2 consequens sit primum antecedens 8 sit 1 consequens 4 sit 2 antecedens 2 sit 2 consequens 1 tunc si 8 et. 4. est triplum ad 4 et 2 et 1 triplum ad 1 tunc talis est proportio 8 ad 4 sicut 2 ad 1 patet ex diffinitione 14 quinti et ex propositione 15 </s>
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              Eversa est si una fuerit proportio 1 antecedentis et 1 consequentis ad 1 consequens sicut secundi antecedentis et secundi consequentis ad 2 consequens. tunc qualis est proportio primi antecedentis et primi consequentis ad 1 antecedens, talis est proportio secundi antecedentis et secundi consequentis ad 2 antecedens. sit primum antecedens 8 sit 1 consequens 4 sit 2 antecedens 2 sit 2 consequens </s>
              <s id="id.0.2.07.03">Tunc si aggregati ex 8 et 4 ad 8 est sesquialtera etiam aggregati ex 2 et 1 est sesquialtera ad 2 patet consequentia ex 15 quinti elementorum Euclidis. et propositione 15 septimi </s>
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              <s id="id.0.2.08.02">Aeque proportionalitas est datis duobus ordinibus quantitatum qualis fuerit proportio 1 ad 2. 1 ordinis. talis sit proportio 1 ad 2. 2 ordinis similiter qualis est proportio 2 ad 3. 1 ordinis. talis est 2 ad 3. 2 ordinis. tunc qualis est proportio 1 ad 3. 1 ordinis. talis est proportio 1 ad 3. 2 ordinis. sit 1 ordo 8. 4. 2 sit 2 ordo 12. 6. 3. tunc si est similitudo proportionum 1 et 2. 1 ordinis. sicut est 1 et 2 secundi ordinis. similiter sit similitudo 2 et 3. 1 ordinis sicut est 1 et 3. 2 ordinis. similiter sit similitudo proportionum primi et 3 primi ordinis sicut est primi et 3. 2 ordinis. tunc inter illa est aequa proportionalitas patet 7 elementorum Euclidis propositione 15 et diffinitione 16. </s>
              <s id="id.0.2.08.03">Et si moderni alia addant aut aliis nominibus utantur, non mihi est </s>
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              His praemissis Aristotelicum et Averroisticum fundamentum in comparandis proportionibus tribus regulis </s>
              <s id="id.0.2.09.03">Prima est qualis est proportio denominationum a quibus proportio nomen accipit, talis est proportio proportionum probatur quia nullam quantitatem sibi intimiorem habet proportio quam illam a qua proportio nomen accipit quia quantitas fundamenti non est ei ita intima, quoniam ad varios terminos comparari potest quantitas fundamenti ad quos in alia et alia proportione, se habet. </s>
              <s id="id.0.2.09.04">Neque quantitas termini est illi ita intima. quia varia sunt quae illi termino comparari possunt, quantitas autem denominationis proportionis utrunque comprehendit. fundamentum scilicet et </s>
              <s id="id.0.2.09.05">Irrationabilium autem proportionum denominationem notam non habentium non erit nota proportio ut ex Campano super decimasexta diffinitione quinti geometriae Euclidis colligitur, nisi eam in potentia notam dixeris. ut proportionis diametri vere quadrati ad costam eius denominator est radix 2 ut praedictum </s>
              <s id="id.0.2.09.06">Scias quod quamvis libros Euclidis geometriae appellem, non intelligo omnes esse geometriae. quia septimus, octavus et nonus arithmetice sunt, sed a maiori parte denominatio accipitur, primi enim sex libri et ultimi geometriae </s>
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              Regulae [...] proportio [...] </s>
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              Secunda regula, qualis est proportio denominationum proportionum, talis est proportio diversarum potentiarum cum ad eandem resistentiam comparantur, patet primo in </s>
              <s id="id.0.2.10.03">Sint a. et b. duae potentiae a. ut 8 b. ut 6 et comparentur ad resistentias ut 1 tunc proportio 8 ad 1 est octupla 6 ad 1 est sextupla. octuplae ad sextuplam est proportio sesquitertia quae est proportio 8 ad 6 et resolvendo ad primos numeros est proportio 4 ad </s>
              <s id="id.0.2.10.04">Secundo. sit a. potentia ut 8 b. potentia ut 6 c. resistentia sit ut 4 tunc proportio 8 ad 4 est dupla. et 6 ad 4 est sesquialtera. tunc dupli ad sesquialteram proportio est sesquialtera qualis est proportio 8 ad 6 patet reducendo illos numeros ad eandem fractionem puta in medium: erit enim proportio </s>
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