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

List of thumbnails

< >
1
1 		2
2
3
3 		4
4
5
5 		6
6
7
7 		8
8
9
9 		10
10
< >
page |< < of 13 > >|
    <archimedes>
      <text>
        <body>
          <chap>
            <p>
              <s id="id.0.4.25.06">
                <pb xlink:href="087/01/008.jpg" n="191"/>
              complicentur: tunc utriusque motus erit impeditus, ergo non aeque velociter movebuntur ut </s>
            </p>
            <p type="margin">
              <s id="id.0.4.25.01.Mg">
                <margin.target id="marg19"/>
              </s>
            </p>
            <p>
              <s id="id.0.4.26.01">Secundo, sint a. et b. duo ignes ascendentes in aere: et congregentur: tunc probatur quod velocius movebuntur quam prius, quia velocius ascendit maior ignis minore: quemadmodum velocius descendit maior lapis minore: sententia est Aristotelis 1. caeli, text. commen. 47 et quarto caeli, tex. commen. </s>
            </p>
            <p>
              <s id="id.0.4.27.01">Tertio, dato a. pedali quadrato: tunc bipedale latum et pedale profundum sed bipedale longum est duplum ad </s>
              <s id="id.0.4.27.02">Et c. bipedale longum, bipedale latum et pedale profundum est duplum ad </s>
              <s id="id.0.4.27.03">Et d. bipedale longum, bipedale latum, et bipedale profundum, est duplum ad </s>
              <s id="id.0.4.27.04">Et tamen congregatis istis duplis, ut in d. tunc ad a. octupla est proportio, et non dupla </s>
              <s id="id.0.4.27.05">Octies enim intrat a. in d. ut </s>
            </p>
            <p>
              <s id="id.0.4.28.01">Ad primum, intelligitur de potentiis non se iuvantibus, neque se impedientibus, qualiter est de complicatione eorum, quae in contraria feruntur: quod si difficilis est imaginatio modi quo conglutinentur: exempli gratia, ignis et terra, imaginentur ambiri vitro: crementum autem motus aut decrementum ratione iuvantis aut impedientis est per </s>
            </p>
            <p>
              <s id="id.0.4.29.01">Ad secundum, caeteris paribus velocius movetur maius elementum minore. sed in proposito non est, scilicet c. quia medium duobus resistens, maiorem facit resistentiam quam uni eorum </s>
            </p>
            <p>
              <s id="id.0.4.30.01">Contra sit aer medium, tunc praecise in duplo plus occurrit de aere aggregato ex duobus, quam uni eorum: ergo quantum resistit aer duobus divisis, tantum resisitit duobus </s>
            </p>
            <p>
              <s id="id.0.4.31.01">Respondeo: Possibile est quod non: ut si primus ignis fuerit pyramidalis: et secundus ignis addatur sub basi prioris ignis: non exeundo a terminis longitudinis prioris </s>
            </p>
            <p>
              <s id="id.0.4.32.01">Ad tertium respondet Calculator in tractatu de motu augumentationis: se iuvant quantitates illae, sed hoc non est verum: non enim apparet via qua una quantitas aliam adiuvet in proportionibus inter eas habendis: sed hoc est, quia non complicantur omnes termini ad quem proportionum: quemadmodum omnes termini a quo computati sunt, quia medii termini computari debent, quibus acceptis habetur proportio 14 pedum ad 7 pedes: quae dupla est sicut prius </s>
              <s id="id.0.4.32.02">Sed limitatio, qua regula indiget, est quam ponit Averrois 7 physic. com. 38 scilicet aequalitas spatii, et temporis et </s>
            </p>
            <p>
              <s id="id.0.4.33.02">
                <arrow.to.target n="marg20"/>
              Quinta </s>
              <s id="id.0.4.33.03">Si duae potentiae inaequaliter movent suas resistentias, tunc congregatae potentiae movebunt resistentias congregatas non ita velociter sicut velocior earum, neque ita tarde sicut tardior earum, quia proportio congregatorum non est ita magna sicut simplicis maioris, neque ita parva sicut simplicis </s>
              <s id="id.0.4.33.04">Istam vult Philosophus, sed implicite septimo physicorum, textu commenti </s>
              <s id="id.0.4.33.05">Exemplum: Movent 2. 1 et 3. 1 proportio congregatorum est 5 ad 2 quae non est ita magna sicut tripla, neque ita parva sicut </s>
            </p>
            <p type="margin">
              <s id="id.0.4.33.01.Mg">
                <margin.target id="marg20"/>
              </s>
            </p>
            <p>
              <s id="id.0.4.34.01">Contra: Sit a. potentiae, ut 6 ad descendum in aere: et resistentiae, ut 2 sit b. potentiae, ut 6 et resistentiae, ut 3 sit medium aer resistentiae, ut 1 tunc a. movet a proportione dupla: et b. a proportione sesquialtera : et tamen a. et b. congregata movebunt ita velociter sicut velocius eorum scilicet a. quia a proportione dupla: quia potentia a. et b. congregatorum est, ut 12 resistentia a. est, ut 2 resistentia b. est, ut 3 medii vero resistentia est, ut 1 et sic 6 resistunt: et 12 movent: ergo a dupla proportione est motus, sicut erat prius: quod si conceditur </s>
            </p>
            <p>
              <s id="id.0.4.35.01">Contra: sequitur quod tantae velociter movet agens, cui parum resistitur, sicut agens, cui multum resistitur: patet de a. quod per se movetur a. dupla, et coniunctum similiter: et ipsi a. coniunctior resistit b. quia est b. tardius mobile quam a. ergo b. resistit ei: patet consequentia, quia suppono quod a. sit superpositum ipsi </s>
            </p>
            <p>
              <s id="id.0.4.36.01">Respondeo: Non est omnis resistentia congregata, quia medium duobus resistens resistit, ut 2 separatum: sed illis coniunctis ipsum non resistit, nisi ut </s>
            </p>
            <p>
              <s id="id.0.4.37.02">
                <arrow.to.target n="marg21"/>
              Sexta </s>
              <s id="id.0.4.37.03">Non si aliqua potentia movet aliquod mobile: medietas potentiae movebit illud in duplo tardius, quia potest esse quod proportio medietatis motoris ad mobile sit aequalitas vel </s>
              <s id="id.0.4.37.04">Hanc voluit Aristoteles 7. physicor. tex. com. </s>
              <s id="id.0.4.37.05">Esto enim quod 100 moveant navem per 50 leucas in die, non oportet 50 eam movere per 25 leucas: immo stat quod neque per unam, puta cum resistentia navis erit 50 vel </s>
              <s id="id.0.4.37.06">Sed quia illa potentia posita est motor in potentia, quia non dividitur virtus divisione motoris: ideo ponatur quod resistentia sit, ut 4 et potentia sit ut 6 et </s>
              <s id="id.0.4.37.07">Si tamen fuerit maioritas medietatis motoris supra mobile: tunc potest medietas motoris movere mobile in tempore aequali per medietatem illius spatii: ut dictum est, ut 10 ad 4habent duplam sesquialteram 5 vero ad 4 habent sesquiquartam: patet autem quod duaesesquiquartae, congregatae dant duplam </s>
            </p>
            <p type="margin">
              <s id="id.0.4.37.01.Mg">
                <margin.target id="marg21"/>
              </s>
            </p>
            <p>
              <s id="id.0.4.38.02">
                <arrow.to.target n="marg22"/>
              Septima </s>
              <s id="id.0.4.38.03">Non si aliqua potentia movet mobile, illa movet mobile in duplo plus resistens, quia potest esse quod data potentia habeat ad resistentiam duplam priori resistentiae aequalitatem: ut si a dupla proportione esset motus, aut minoritatem, ut si a proportione minori quam dupla esset </s>
              <s id="id.0.4.38.04">Hanc voluit Philosophus 7 physicor. tex. com. 37 quod si maioritatem habeat data potentia supra id mobile in duplo maius, ipsa movet illud in duplo </s>
              <s id="id.0.4.38.05">Unde quancunque maioritatem habeat medietas potentiae super aliquid, supra id habebit potentia duas </s>
              <s id="id.0.4.38.06">Et quancunque proportionem habuerit potentia supra aliquid, habebit medietas potentiae medietatem eius supra </s>
              <s id="id.0.4.38.07">Similiter quancunque proportionem habeat potentia super aliquid: habet medietatem eius supra resistentiam in duplo maiorem: unde quatuor ad quatuor est aequalitas: et 4 ad octo est </s>
              <s id="id.0.4.38.08">Similiter 6 ad 8 est tres quartae: et 6 ad 16 est tres </s>
              <s id="id.0.4.38.09">Naturalis tamen Philosophus in comparandis motibus proportiones illas considerat, ad quas motus consequatur: cuiusmodi non sunt aequalitas, neque minoritas: maioritatem enim motus sequitur, seu dominium: quod si aliquod est dominium, ad quod motus non sequatur, hoc est per accidens, puta quia tantam pravitatem non tolerat natura: sit enim unus gradus motus qui sit minimus potens per se seorsum existere, et sequatur duplam proportionem: tunc a nulla superparticulari proportione motus fieri posset: non quin esset ibi maioritas, sed quia tanta parvitas seorsum existere non posset, similiter a nulla proportione superpartiente: patet consequentia, quia omnis superparticularis, et omnis superpartiens est minor </s>
            </p>
            <p type="margin">
              <s id="id.0.4.38.01.Mg">
                <margin.target id="marg22"/>
              </s>
            </p>
            <p>
              <s id="id.0.4.39.02">
                <arrow.to.target n="marg23"/>
              Octava </s>
              <s id="id.0.4.39.03">Uniformiter crescente potentia movente, stante resistentia, difformiter crescit motus, quia difformiter crescit dominium agentis supra resistentiam: intellige crescere quando crescit in virtute secundum quam movet.</s>
              <s id="id.0.4.39.04">Similiter decrescente uniformiter movente, difformiter decrescit motus, usque ad primum instans aequalitatis virtutis cum resistentia: quod est primum non esse ipsius motus, quemadmodum ad imaginationem loquendo, aut usque ad instans, in quo esset motus sub minimo gradu eius, secundum naturam loquendo: exemplum sit virtus, ut 3 quae uniformiter crescat in virtute, exempli gratia in 5 horis ad 8 tunc in prima hora aliquantus motus acquiritur, quantus acquiritur in quatuor horis post: ergo difformiter crescit motus: patet antecedens ex regulis de dupla </s>
              <s id="id.0.4.39.05">Potentia igitur intendens motum ex cremento eius, tardius et tardius </s>
              <s id="id.0.4.39.06">Et potentia remittens ex eius remissione velocius et velocius remittit: quia aequalis excessus in minori maiorem facit proportionem quam in maiori: addendo, si additur, et minuendo si minuitur: supponitur caeterorum paritas ut in applicatione agentis ad passum: in adventu impedimenti, aut recessu et cetera 18 et 19 Calculatoris de motu locali: ergo uniformiter crescens tardius proportionaliter crescit in secunda parte temporis quam in prima illi aequali: ergo motu sequente proportionem tardius et tardius continue crescit motus et cetera econtra autem est de </s>
            </p>
            <p type="margin">
              <s id="id.0.4.39.01.Mg">
                <margin.target id="marg23"/>
              </s>
            </p>
            <p>
              <s id="id.0.4.40.02">
                <arrow.to.target n="marg24"/>
              Nona </s>
              <s id="id.0.4.40.03">Ubi duae potentiae inaequales cum resistentia aequali moveant: eis aeque velociter crescentibus: non aeque velociter crescit motus, sed velocius crescit cum potentia minor: et tardius cum potentia maiori, quia maiori et minori aequaliter crescentibus: minus in ea proportione, quae minus est, velocius proportionabiliter crescit: quia in ea proportione qua minus est, minus distat a suo proportionali, ut duplo, triplo, et cetera quam maius, ut 2 ad 4 distant per 2 et 4 ab 8 distant per 4 sexta Calculatoris de motu </s>
            </p>
            <p type="margin">
              <s id="id.0.4.40.01.Mg">
                <margin.target id="marg24"/>
              </s>
            </p>
            <p>
              <s id="id.0.4.41.02">
                <arrow.to.target n="marg25"/>
              Decima </s>
              <s id="id.0.4.41.03">Ubi resistentia minor quam potentia crescat ad aequalitatem potentiae uniformiter: tunc difformiter remittitur motus scilicet tardius et tardius sit potentia, ut 4 resistentia vero ut unum: cum crescit resistentia ad duo, iam perditur medietas motus: cum vero crescit ad tria, minoratur motus per quantum sesquitertia est minor quam dupla: et sic minus decrescit motus quam prius vigesima Calculatoris de motu locali: et similiter vigesimaprima, de resistentia decrescente quod velocius et velocius crescit motus potentiae non </s>
            </p>
            <p type="margin">
              <s id="id.0.4.41.01.Mg">
                <margin.target id="marg25"/>
              </s>
            </p>
            <p>
              <s id="id.0.4.42.02">
                <arrow.to.target n="marg26"/>
              Undecima </s>
              <s id="id.0.4.42.03">Motis inaequalibus uniformiter et aequaliter crescentibus aut decrescentibus respectu potentiarum aequalium stantium, difformiter intenditur, aut remittitur motus: et inaequaliter, quia cum minore resistentia plus acquiritur, vel perditur, quam cum maiore: quia aequale maiorem facit proportionem cum minori quam cum maiori, ut dictum est iam, et dicendum esset ad verificandum sequentem partem conclusionis, et plus in prima parte temporis, quam in secunda, cum crescit </s>
              <s id="id.0.4.42.04">Plus vero in secunda parte temporis, cum decrescit resistentia, ex nona </s>
              <s id="id.0.4.42.05">Et si dicas, quomodo possibile est quod regulas Calculatoris admittas, cummodum eius proportionandi non </s>
            </p>
            <p type="margin">
              <s id="id.0.4.42.01.Mg">
                <margin.target id="marg26"/>
              </s>
            </p>
            <p>
              <s id="id.0.4.43.01">Respondeo, aliquae regulae illius sunt sibi et mihi communes, licet sit aliqua differentia in aliis proportionibus a dupla, quia apud utrunque uniformiter crescens potentia tardius et tardius intendit, velocius et velocius decrescens </s>
              <s id="id.0.4.43.02">Sed differentia est, quia maior est proportio tarditatis vel velocitatis apud unum quam alterum, quia si 3 ad 1 sit proportio ad hoc ut dupletur proportio, apud me in duplo citius duplabitur quam apud ipsum, quia apud me satis est ut crescat ad 6 et apud ipsum oportet ut crescat ad 9 si autem sit quadrupla, ut 4 ad 1 in triplo citius duplabitur apud me, quia satis est ut crescat ad 8 ubi apud ipsum oportet crescere ad 16 et cetera proportionabiliter de decrementis et </s>
            </p>
            <p>
              <s id="id.0.4.44.02">
                <arrow.to.target n="marg27"/>
              Duodecima </s>
              <s id="id.0.4.44.03">Datis duabus potentiis aequalibus, sint ut quatuor: moventibus resistentias aequales sint, ut unum remittentibus motus suos a. per remissionem potentiae ad unum: b. vero per intensionem resistentiae ad quatuor, tunc difformiter perditur motus, et non aeque velociter, sed bene aeque cito: quia a. velocius et velocius remittit: b. vero tardius et tardius: et utrunque exempli gratia in hora perdit motum: in quo tempore fit aequatio potentiae activae cum resistentia, et ad non quantum minoratur motus, quia ad non quantum minoratur maioritas respectu huius servando mathematicam imaginationem, sed naturalem veritatem imitando: minoratu: motus ad minimum gradum ad quem gradum citius attingit quod velocius remittit: non tamen in infinitum minoratur, neque ad non quantum minoratur proportio: quia a principio aliquanta fuit proportio: ut quadrupla: et in fine, quando nulla est maioritas respectu huius, quia tanta est potentia activa, quanta resistentia, aliquanta est proportio, quia </s>
              <s id="id.0.4.44.04">Aliquas istarum regularum dixit Aver. 1. caeli, com. 64 et physi. com. 71 esse per se notas, quemadmodum est illa quam ponit Aver. 8 physic. com. 78 scilicet omnis proportio composita ex duabus proportionibus finitis est finita </s>
            </p>
            <p type="margin">
              <s id="id.0.4.44.01.Mg">
                <margin.target id="marg27"/>
              </s>
            </p>
            <p>
              <s id="id.0.4.45.02">
                <arrow.to.target n="marg28"/>
              Secunda </s>
              <s id="id.0.4.45.03">Si aliqua est proportio inter potentias moventes, non dico motivas: non enim a motivo habetur motus, si non movet, talis est proportio inter velocitates provenientes ab eis super aequali resistentia: stat enim maiorem potentiarum certam resistentiam movere posse, quam minor potentia movere non potest, et stat super diversis resistentiis diversas potentias aequalem habere proportionem, et inaequalem etiam: super aliis resistentiis </s>
              <s id="id.0.4.45.04">Hanc voluit Philosophus 1 caeli, tex. com. 51. 52. 53 oportet secundum excellentias moveri, super resistentias moveri secundum excellentias potentiarum inter se supra </s>
              <s id="id.0.4.45.05">Ubi Averrois proportio gravis ad grave est sicut proportio tarditatis ad velocitatem: tarditas enim a potentia minore fit, velocitas vero a </s>
              <s id="id.0.4.45.06">Et Aver. 2 caeli, com. </s>
              <s id="id.0.4.45.07">Et quia causa in terminatione proportionum quae sunt inter potentias motorum et rerum motarum ab eis, est diversitas formarum: contingit ut haec finitas super proportionis sit communis formis, quae sunt in materia, et formis quae non sunt in </s>
              <s id="id.0.4.45.08">Item 4 physic. com. 71 et </s>
              <s id="id.0.4.45.09">Omnis diversitas motuum in velocitate et tarditate est secundum proportionem, quae est inter duas </s>
              <s id="id.0.4.45.10">Volunt tamen Aristo. et Averrois salvari caeterorum paritatem, ut in figura et magnitudine: unde 2 caeli, tex. com. </s>
              <s id="id.0.4.45.11">Quanto magis aer fuerit maior, tanto citius movebitur ad </s>
              <s id="id.0.4.45.12">Et dixit Aristoteles 4 caeli, tex. com. 9 quanto plus est, tanto est </s>
              <s id="id.0.4.45.13">Item in 4 caeli, tex. com. 26 et </s>
              <s id="id.0.4.45.14">Et 4 caeli, tex. com. 42.</s>
              <s id="id.0.4.45.15">Figura non est causa motus, sed bene est causa velocioris et tardioris, quia facit occurere maiori vel minori parti medii resistentis, vel passi: et sic agenti plus vel minus resistitur ratione </s>
              <s id="id.0.4.45.16">Idem Aver. ibi com. </s>
              <s id="id.0.4.45.17">Ad idem Philosophus 8 physi. text. com. 80 potentia in duplo maior, idem mobile movebit in medietate </s>
              <s id="id.0.4.45.18">Et dixit ibi Aver. maior motor idem movet mobile in minori tempore, et temporis ad tempus est, sicut proportio motoris ad </s>
              <s id="id.0.4.45.19">Ad idem est Arist. 4 caeli, textu ultimo: dixit enim ibi </s>
              <s id="id.0.4.45.20">Cum imaginati fuerimus duo corpora gravia dividentia idem corpus, tunc proportio velocitatis divisionis ab altero eorum ad velocitatem divisionis a secundo est sicut proportio gravitatis ad gravitatem: patet autem quod gravitates sunt motrices: et sic proportio gravitatum est proportio moventium et </s>
              <s id="id.0.4.45.21">Contra moveant 4. 2 tunc possunt 2 moveri in duplo tardius quam 4 moveant 2 ut 4 physicor., tex. com. 96 et 6 physi., tex. commen. 15 aut igitur a duobus, et tunc aequale movebit aequale: quod est impossibile, aut a minori quam 2 et tunc maius movebitur [= movebitur] a minori, quod est impossibile: aut a maiori quam 2 movebuntur, et in duplo tardius quam a. et tunc dupla est proportio velocitatum per casum et non potentiarum, ut sequitur conclusio ergo falsa.</s>
            </p>
            <p type="margin">
              <s id="id.0.4.45.01.Mg">
                <margin.target id="marg28"/>
              Qualis est proportio potentiae inter se sap i ilo, talis est proportio celo </s>
            </p>
            <p>
              <s id="id.0.4.46.01">Secundo possunt 4 in duplo tardius moveri ab aliquo quam moveantur </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
Impressum