Alvarus, Thomas
,
Liber de triplici motu
,
1509
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Prime partis
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0012
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12
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nitū magnꝰ erit exceſſus quo quantitas maior ex
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cedet minorē. </
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<
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xml:space
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">igitur in infinitū magna erit ꝓpor-
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tio quãtitatis maior ad minorē: et per cõſequens
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illarū infinitarū proportionū in infinitū magna
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erit aliqua: quod fuit probandū. </
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<
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xml:space
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">Et ſic patet con-
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cluſio. </
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xml:space
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">¶ Simile correlariū: correlario ṗme cõclu-
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ſiõis: hic poteris inferre de gñatione huiuſmodi
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proportionū irrationaliū. </
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<
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xml:space
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">¶ Plures adieciſſem
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cõcluſiones et correlaria: niſi obſtaret hanc mate
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riã ex ſecunda parte in vniuerſum dependere. </
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<
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xml:space
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">Nec
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mirari oportet: ſi plurimū in his duobus capitibꝰ
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cõtra morē et ordinē mathematicū: ſequētibꝰ vſus
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fuerim. </
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">Non em̄ potuit hec materia alio mõ īduci</
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<
head
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">Capitulū quintū / in quo agit̄̄ de diuiſione
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corporis in partes proportionales qua pro
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portione rationali quis voluerit.</
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<
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">QUoniam plerū in materia
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triplicis motus occurūt pleri caſus:
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in quibus oportet vti multiplici ſpecie
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diuiſionis corporis in partes ſuas proportiona
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les variis et diuerſis ꝓportionibus rationalibus
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ideo ad vniuerſalē methodū inueniendam ſit.</
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<
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">Prīa ſuppõ. </
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">Nõ oēs ꝑtes alicuiꝰ cor
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poris ī q̈s idē corpꝰ diuidit̄̄ ↄ̨tinuo ſe hñtes ī eadē
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ꝓportiõe: gr̄a exēpli a. ſūt oēs ꝑtes ꝓportionales
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eiuſdē corꝑis eadē ꝓportiõe a. </
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<
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">Probat̄̄ / q2 poſſibi
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le eſt / vna medietas alicuiꝰ corꝑis diuidat̄̄ in oēs
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partes ſuas ꝓportione tripla: et omēs ille partes
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ſunt partes illiꝰ corporis totalis. </
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">in q̈s idē corpꝰ
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diuidit̄̄ hñtes ſe cõtinuo in ꝓportiõe tripla: 2. et tñ
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nõ ſunt oēs partes ꝓportionales illius corporis
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proportione tripla. </
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<
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">Et capio in ſuppoſitiõe ly oēs
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collectiue in primo loco et in ſecundo.</
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">Oēs partes ali
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cuius corporis innuite continue ſe habētes aliq̈
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ꝓportione: puta a. et abſoluentes totū corpꝰ: ſunt
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oēs partes ꝓportionales eiuſdē corporis propor
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tione a. </
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<
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">Et volo dicere / ſi aliquod corpꝰ diuidat̄̄
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in infinitas partes continuo ſe habentes in ꝓpor
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tione a. et abſoluētes totū corpus: ille ſimul ſunt
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oēs partes proportionales proportione a. </
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<
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">Patꝫ
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hec ſuppoſitio: q2 ſic diuidere corpus eſt diuidere
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ipſū in oēs partes ꝓportionales proportione a.
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</
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<
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">Quãdocun ali
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qua cõtinuo ꝓportionãtur aliqua ꝓportione geo
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metrica: qualis eſt ꝓportio inter proportionata:
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talis eſt inter ſuas differētias ſiue exceſſeus: quod
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idem eſt: vt q2 .3. ad .4. ſe habet in ꝓportiõe dupla
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et ſimiliter .4. ad 2. / et cõtinuo proportionant̄̄ eadē
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proportione: ideo differentia ſiue exceſſus inter .8
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et .4. ſe habet ad differãtiã ſiue exceſſum inter .4. et
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2. in proportiõe dupla. </
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">Patet hec ſuppoſitio ex
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quīta proprietate proportionalitatis ſiue medie
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tatis geometrice ex ſecūda parte capitulo ſecūdo</
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">Quarta ſuppoſitio. </
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<
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">Si aliquod cor
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pus diuidatur in infinitas partes: et deperdendo
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primã illarū perdit aliquã ꝓportionē puta a. / hoc
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eſt efficitur in a. ꝓportione minꝰ: et ꝑdendo ſcḋam
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poſt primã iterum efficitur in a. minus: et ꝑdendo
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tertiam poſt ſecūdã iterum efficitur in a. minus. </
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<
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">et
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ſic conſequenter ille partes ſunt oēs partes ꝓpor
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tionales illius corporis ꝓportione a. / ſi vero ꝑden
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do primã illarū non perdit vnam proportionē a. /
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Capitulum quintū.
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et ꝑdendo ſecundã poſt primã: vnã alteram, ꝑden-
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do tertiã poſt ſecundã vnã alteram ꝓportionē a. /
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et ſic cõſequenter: tales partes nõ ſunt oēs partes
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ꝓportionales talis corporis ꝓportione a. </
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batur prima pars / q2 ſi nõ: detur oppoſitū: videli
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cet / aliquod corpus diuiditur in aliquas partes
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iufinitas: et ꝑdēdo primã illarum ꝑdit ꝓportionē
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a. etc̈. et tamen nõ ſunt ille oēs partes ꝓportiona-
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les illius corporis ꝓportiõe a. et ſic tale corpus b. /
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et arguitur ſic / b. eſt diuiſum in infinitas partes: et
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ꝑdendo primã illarū in prima parte ꝓportionali
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hore exempli gratia: in fine illius partis eſt in a.
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ꝓportiõe minꝰ: et ꝑdendo ſecundã partē in ſecūda
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parte ꝓportionali tēporis: iterum efficitur in fine
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eiuſdem partis in a. proportione minꝰ quaꝫ erat
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in principio eiuſdē partis: et in tertia parte ꝓpor
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tionali ꝑdēdo terntiã ip̄m efficitur minꝰ / quã erat
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in principio eiuſdē ꝑtis in a. ꝓportione: et ſic con
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ſequēter. </
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<
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">igitur in partibus ꝓportionabilibꝰ illiꝰ
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hore ſunt infinita corpora cõtinuo ſe habentia in
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ꝓportione a. </
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<
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p̄me partis ꝓportionalis: ſe habet in ꝓportione
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a. ad illud quod eſt in prīcipio ſecunde et illud qḋ
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eſt in p̄ncipio ſecunde ſe habet in ꝓportione a. ad
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illud quod eſt in principio tertie: et ſic cõſequēter /
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igitur illa infinta corpora continuo ſe habet in
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ꝓportiõe a. / et ex cõſequēti ſequit̄̄ / exceſſus inter
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illa corpora cõtinuo ſe habēt in ꝓportiõe a. / puta
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exceſſus quo corpus in p̄ncipio ṗme partis ꝓpor
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tionalis excedit corpus in p̄ncipio ſecunde: ſe ha
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bet in ꝓportione a. / ad exceſſum quo corpus in p̄n
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cipio ſecūde excedit corpus in p̄ncipio tertie: et ſic
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cõſequēter. </
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poſitione: et illi exceſſus ſunt ille partes que deper
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dūtur in partibus ꝓportionalibus tēporis: ergo
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ille ꝑtes que deꝑduntur in illis partibus propor-
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tionalibus tēporis ſe habent cõtinuo in ꝓportõe
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a. </
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<
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">Conſequētia patet: et ꝓbatur antecedens: quia
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corpus in principio p̄me partis ꝓportionalis tē-
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poris: exedit corpus in principio ſecunde ꝑ illud
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quod deꝑdit in ip̄a p̄ma parte ꝓportionali tēpo-
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ris: et illud eſt p̄ma illarum partiū in quas diuidi
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tur corpus ex caſu: igitur aſſumptum verum </
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ſic ꝓbabis de quocū alio exceſſu. </
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<
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">et vltra ille par
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tes in quas diuiditur illud corpus b. ſunt infinite
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cõtinuo ſe habentes in ꝓportione a. / et abſoluūt to
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tum corpus: igitur ille ſunt oēs partes ꝓportiona
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les illius corporis ꝓportione a. / quod fuit negatū
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</
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<
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">Quod vero ille partes abſoluant totum corpus
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patet / quia per deperditionem illarū perditur to
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tum corpus ad nõ quantum: cum deperdat infini
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tam latitudinem proportionis: vt conſtat: igitur.
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">Secūda pars patet facile / quia bene ſequitur de-
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perdendo illas partes continuo: tale corpus non
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continuo efficitur minus in proportione a. / ergo
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ſequitur / non ſunt ibi in tali diminutione infini
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ta corpora continuo ſe habentia in proportione
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a. modo ſuperius expoſito: ergo ſequitur / exceſ
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ſus illorum corporum non continuo ſe habent in
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proportione a. </
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<
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poſitione: et illi exceſſus ſunt partes in quas diui
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debatur ipſum corpus b. / igitur ipſe non ſunt par
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tes proportionales corporis b. proportione a. / et
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per conſequens de primo ad vltimum ſequitur il
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la ſecunda pars ſuppoſitionis.</
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