Alvarus, Thomas, Liber de triplici motu, 1509

Table of Notes

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              <pb chead="Secunde partis" file="0041" n="41"/>
              <p xml:id="N13EE9">
                <s xml:id="N13EEA" xml:space="preserve">Probatur minor / q2 tripla eſt dupla ad ſequial-
                  <lb/>
                terã: et dupla ſexquiquarta eſt minor tripla / ergo
                  <lb/>
                dupla ſexquiquarta eſt minor quã dupla ad ſex-
                  <lb/>
                quialterã. </s>
                <s xml:id="N13EF3" xml:space="preserve">Cõſequentia eſt nota cū minore: et ꝓbat̄̄
                  <lb/>
                maior / qm̄ denominatiõis triple ad denoīationē
                  <lb/>
                ſexquialtere eſt proportio dupla. </s>
                <s xml:id="N13EFA" xml:space="preserve">Triū em̄ ad vnū
                  <lb/>
                cū dimidio eſt proportio dupla: igitur tripla eſt
                  <lb/>
                dupla ad ſexquialterã. </s>
                <s xml:id="N13F01" xml:space="preserve">Patet cõſequētia ex opi-
                  <lb/>
                nione.
                  <note position="left" xlink:href="note-0041-01a" xlink:label="note-0041-01" xml:id="N13F8A" xml:space="preserve">Scḋa cõ-
                    <lb/>
                  firmatio.</note>
                </s>
                <s xml:id="N13F0B" xml:space="preserve">¶ Cõfirmatur ſecūdo / q2 ſi poſitio eſſet vera /
                  <lb/>
                ſequeretur /  aliquid cõtineret alteꝝ pluſ̄ bis: et
                  <lb/>
                tamen eſſet adequate dnplū ad illud quod cõtinet
                  <lb/>
                adequate bis: et aliquid cõtineret alteꝝ minus quã
                  <lb/>
                bis hoc eſt contineret ipſum ſemel et medietatē eiꝰ
                  <lb/>
                p̄ciſe et eſſet duplū ad illud et nõ ſexquialteꝝ. </s>
                <s xml:id="N13F18" xml:space="preserve">Oīa
                  <lb/>
                iſta cõſequentia ſunt cõtra diffinitiões et prīcipia
                  <lb/>
                mathematica / igitur et poſitio. </s>
                <s xml:id="N13F1F" xml:space="preserve">Sūt em̄ cõtra diffi­
                  <lb/>
                nitiones ſexquialtere et duple / vt cõſtat. </s>
                <s xml:id="N13F24" xml:space="preserve">Iã ꝓbatur
                  <lb/>
                ſequela / q2 tripla eſt dupla ad ſexq̇alterã: et tamē
                  <lb/>
                cõtinet bis ſexquialterã: et aliquid vltra puta ſex-
                  <lb/>
                quitertiã: vt ptꝫ in his terminis .12.9.6.4:12. em̄
                  <lb/>
                ad .9. eſt proportio ſexquitertia et .9. ad .6. eſt vna
                  <lb/>
                proportio ſexquialtera et .6. ad 4. vna altera .12.
                  <lb/>
                vero ad .4. eſt tripla ex illis duabus ſexquialteris
                  <lb/>
                et vna ſexquitertia cõpoſita. </s>
                <s xml:id="N13F35" xml:space="preserve">Et ſic ptꝫ ſequela quo
                  <lb/>
                ad primã partē. </s>
                <s xml:id="N13F3A" xml:space="preserve">Secūda pars patet de octupla et
                  <lb/>
                quadrupla: octupla em̄ nõ cõtinet bis quadruplã
                  <lb/>
                et tamen eſt dupla ad illam / vt patet ex poſitione.
                  <lb/>
                </s>
                <s xml:id="N13F42" xml:space="preserve">¶ Multa ſimilia poſſunt inferri / que manifeſte ſūt
                  <lb/>
                cõtra dignitates, petitiones, et diffinitiones ma-
                  <lb/>
                thematicas, qui debent ſupponi tan̄ principia
                  <lb/>
                ſcientie mathematice. </s>
                <s xml:id="N13F4B" xml:space="preserve">¶ Sed oīa hec argumenta
                  <lb/>
                facile (quãuis proterue et abſ ratione) reſcindit
                  <lb/>
                baſanus negando illas petitiones et diffinitiões:
                  <lb/>
                eas dūtaxat ad numeros ſiue quantitates conti-
                  <lb/>
                nuas reſtringendo ſiue limitando. </s>
                <s xml:id="N13F56" xml:space="preserve">Sed ꝓfecto et
                  <lb/>
                diminute loquit̄̄ et cõtra rationē: diminute quidē
                  <lb/>
                et inſufficienter, q2 nõ aſſignat diffinitionē ꝓpor-
                  <lb/>
                tions duple, quadruple, aut alterius ſufficienter
                  <lb/>
                que cuilibet cõtento ſub diffinito cõueniat: et cõtra
                  <lb/>
                rationē, qm̄ ſicut ipſe aſtruxit illas diffinitiones
                  <lb/>
                duple, quadruple .etc̈. cõuenire quantitatibꝰ dūta­
                  <lb/>
                xat et nūmeris: pari ꝓteruia quilibet poſſet defen­
                  <lb/>
                ſare at aſſeuerare illas diffinitiones dumtaxat
                  <lb/>
                cõuenire numeris cõpoſitis ex vnitatibus indiui-
                  <lb/>
                ſibilibus puta intelligentiaꝝ aut punctoꝝ: et nul-
                  <lb/>
                lis aliis. </s>
                <s xml:id="N13F6F" xml:space="preserve">Sicut em̄ ipſe negat hanc cõſequentiam
                  <lb/>
                ꝓportio dupla ſexquiquarta cõtinet bis adequa-
                  <lb/>
                te ſexquialterã / ergo eſt dupla ad illã: pari teme-
                  <lb/>
                rario auſu poſſet quilibet hanc cõſequentiã nega­
                  <lb/>
                re bipedale cõtinet bis adequate pedale / ergo eſt
                  <lb/>
                duplū ad pedale: et oī dubio ꝓcul cõtra eū nõ eſſet
                  <lb/>
                diſputandū ſi philoſopho primo phiſicoꝝ credat̄̄
                  <lb/>
                </s>
                <s xml:id="N13F7F" xml:space="preserve">Sed q2 ipſe diceret ſe nõ negare prīcipia mathe-
                  <lb/>
                matica: ſed ea coartare ſiue limitare: qm̄ illa non
                  <lb/>
                ſunt intelligenda in proportionibus.</s>
              </p>
              <note position="left" xml:id="N13F92" xml:space="preserve">3. arguit̄̄</note>
              <p xml:id="N13F96">
                <s xml:id="N13F97" xml:space="preserve">Idco cõtra eū tertio arguo ex prīci-
                  <lb/>
                piis iã limitatis ad ꝓportiones et hoc ſic ꝓportio
                  <lb/>
                ſexdecupla eſt dupla ad q̈druplã: et octupla tripla
                  <lb/>
                ad duplã vt deducã ex mathematicis prīcipiis: et
                  <lb/>
                ſecundū eum proportio ſexdecupla eſt quadrupla
                  <lb/>
                ad quadruplam vt ſuadet proportionū denomi-
                  <lb/>
                natio. </s>
                <s xml:id="N13FA6" xml:space="preserve">Item ſecunduꝫ eum octupla eſt quadrupla
                  <lb/>
                ad duplam / vt denominationes duple et octuple
                  <lb/>
                oſtendunt: igitur ſua poſitio principiis mathema­
                  <lb/>
                ticis ad proportiones limitatis contrariatur / et ꝑ
                  <lb/>
                conſequens falſa. </s>
                <s xml:id="N13FB1" xml:space="preserve">Conſequentia eſt nota cū mino­
                  <lb/>
                re / et maior probatur primo quantum ad priorem
                  <lb/>
                partem / quia capta proportione ſexdecupla inter
                  <lb/>
                16. et .1. ibi reperiūtur .3. termini continuo propor-
                  <cb chead="Capitulum quintū."/>
                  <note position="right" xlink:href="note-0041-02a" xlink:label="note-0041-02" xml:id="N13FF2" xml:space="preserve">Eu. 5. ele.</note>
                tionabiles proportione quadrupla vtpote .16.4:
                  <lb/>
                1. / igitur extremi ad extremū puta .16. ad .1. eſt du-
                  <lb/>
                pla proportio ad proportionē primi ad ſecundū
                  <lb/>
                puta .16. ad .4. / vt patet ex decima diffinitione quī­
                  <lb/>
                ti elementorum euclidis expreſſe: et ex quinta diffi­
                  <lb/>
                nitione ſecundi elementorum iordani.
                  <note position="right" xlink:href="note-0041-03a" xlink:label="note-0041-03" xml:id="N13FF8" xml:space="preserve">Iorda. 2
                    <lb/>
                  ele.</note>
                </s>
                <s xml:id="N13FD1" xml:space="preserve">Secunda
                  <lb/>
                pars maioris probatur / quoniã capta proporti-
                  <lb/>
                one octupla octo ad vnum: ibi reperiuntur quatu­
                  <lb/>
                or termini cõtinuo proportionabiles proportiõe
                  <lb/>
                dupla videlicet .8.4.2.1. / igit̄̄ extremi ad extremuꝫ
                  <lb/>
                puta .8. ad .1. eſt proportio tripla ad proportionē
                  <lb/>
                8. ad .4. que eſt dupla. </s>
                <s xml:id="N13FE0" xml:space="preserve">Patet conſequentia ex ea-
                  <lb/>
                dem decima diffinitione quinti elementoꝝ euclu-
                  <lb/>
                dis: et quinta ſecundi elementoꝝ iordani: </s>
                <s xml:id="N13FE7" xml:space="preserve">Nec ba-
                  <lb/>
                ſanus poſſet hoc argumentū diſſoluere niſi prin-
                  <lb/>
                cipia arithmetica in eum adducta neget.</s>
              </p>
              <p xml:id="N14000">
                <s xml:id="N14001" xml:space="preserve">Quarto  ad opinãtē argr̄ / qm̄ vt ip̄e
                  <lb/>
                ꝓfitet̄̄ in ſui operis ex ordio ſuarū ꝓportionū tra-
                  <lb/>
                ctatus introductorius eſt ad ſuiſethicas calcula-
                  <lb/>
                tiones: ſed ipſe calculator ſuiſeth longe aliter ſen­
                  <lb/>
                tit: et plurimū ab eo diſcrepat in materia de pro-
                  <lb/>
                portione proportionū vt ex quam plurimis locis
                  <lb/>
                eius percipere poſſumus: igitur nec calculatoris
                  <lb/>
                mentem intellexit nec eius tractatus ad eum intel­
                  <lb/>
                ligendum introducit: īmo potius extraducit. </s>
                <s xml:id="N14014" xml:space="preserve">Pro­
                  <lb/>
                bat̄̄ minor.
                  <note position="right" xlink:href="note-0041-04a" xlink:label="note-0041-04" xml:id="N140BF" xml:space="preserve">Cal. ca.
                    <lb/>
                  de aug.</note>
                </s>
                <s xml:id="N1401E" xml:space="preserve">Tū primo / quoniã calculator in quīta
                  <lb/>
                concluſione prime opinionis de augmentatione
                  <lb/>
                dicit /  ſi aliquid augeatur in duplo velocius al-
                  <lb/>
                tero: et illud acquirat vnam proportionē f. in ali-
                  <lb/>
                quo tēpore neceſſe eſt in eodeꝫ tempore illud quod
                  <lb/>
                in duplo velocius augetur proportionem compo­
                  <lb/>
                ſitam ex duplici f. acquirere: cum in caſu calcula-
                  <lb/>
                toris ibidem illud quod in duplo velocius auge-
                  <lb/>
                tur continuo in duplo velocius augetur: ſed illa
                  <lb/>
                conſequentia nichil penitus valeret ſi baſani po-
                  <lb/>
                ſitio eſſet vera. </s>
                <s xml:id="N14035" xml:space="preserve">qm̄ quando a. acquireret propor-
                  <lb/>
                tionem quadruplam et b. in eodem tempore in du­
                  <lb/>
                plo velocius augeretur adequate non eſſet neceſſe
                  <lb/>
                 b. in eodem tempore acquireret proportionem
                  <lb/>
                compoſitam ex duabus quadruplis: īmo neceſſe
                  <lb/>
                eſſet /  non acquireret tantum: ſed acquireret cõ-
                  <lb/>
                poſitã ex quadrupla et dupla que eſt octupla que
                  <lb/>
                ſecundū baſanū eſt dupla ad quadruplam.
                  <note position="right" xlink:href="note-0041-05a" xlink:label="note-0041-05" xml:id="N140C7" xml:space="preserve">Cal. de
                    <lb/>
                  diffi. ac.</note>
                </s>
                <s xml:id="N1404B" xml:space="preserve">Tum
                  <lb/>
                ſecundo / quia idem calculator in capitulo de diffi­
                  <lb/>
                cultate actionis in primo argumento quo impu-
                  <lb/>
                gnat tertiam poſitiouem aſſumit potentiam mo-
                  <lb/>
                uentem a proportione ſexquialtera in aliquo me-
                  <lb/>
                dio: et dicit /  ſi illa potētia augeatur ad ſexquial­
                  <lb/>
                terum preciſe ſtante reſiſtentia medii  ipſa po-
                  <lb/>
                tentia mouebitur in duplo velocius adequate: ex
                  <lb/>
                quo immediate ſequitur /  proportio potentie ad
                  <lb/>
                reſiſtentiã fuit effecta in duplo maior. </s>
                <s xml:id="N14060" xml:space="preserve">Patet con­
                  <lb/>
                ſequentia / quoniã ſecundū eum velocitas motuum
                  <lb/>
                ꝓportionū ꝓportionē inſequit̄̄ / vt ptꝫ ex principio
                  <lb/>
                capituli de motu locali: ſed cū potētia illa, habēs
                  <lb/>
                ꝓportionē ſexquialterã ad ſuã reſiſtētiã acquirit
                  <lb/>
                ſupra ſe proportionem ſexquialteram tota pro-
                  <lb/>
                portio componitur adequate ex duabus ſexquial­
                  <lb/>
                teris et efficitur dupla ſexquiquarta qualis eſt .9
                  <lb/>
                ad .4. / igitur dupla ſexquiquarta ſecundum calcu­
                  <lb/>
                latorem eſt dupla ad ſexquialteram: et ſecundum
                  <lb/>
                baſanum tripla eſt dupla ad ſexquialteram: igi-
                  <lb/>
                tur ſua poſitio, ſuuſ ſuarum proportionuꝫ tra-
                  <lb/>
                ctatus non ad intelligendam calculatoris ſenten­
                  <lb/>
                tiam introducit ſed ei aduerſatur.
                  <note position="right" xlink:href="note-0041-06a" xlink:label="note-0041-06" xml:id="N140CF" xml:space="preserve">Calcu. de
                    <lb/>
                  me. nõ re­
                    <lb/>
                  ſiſ. capite
                    <lb/>
                  ſecūdo.</note>
                </s>
                <s xml:id="N14082" xml:space="preserve">Tum tertio / q2
                  <lb/>
                idem calculator in vltimo capitulo de medio non
                  <lb/>
                reſiſtente concluſione octaua dicit expreſſe in pro­
                  <lb/>
                batione illius concluſionis /  ſexdecupla eſt du-
                  <lb/>
                pla ad quadruplã: et ſi ſic non eſſet. </s>
                <s xml:id="N1408D" xml:space="preserve">concluſio eſſet </s>
              </p>
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