Alvarus, Thomas
,
Liber de triplici motu
,
1509
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De motu locali quo ad effectum tempore difformi.
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cium ꝑtranſitum in toto tempore eſt dupluꝫ ſexqui
<
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alterum ad ſpacium pertranſitum in prima parte
<
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proportionali </
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<
s
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xml:space
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">Quod probatur eſto / velocitas ṗ
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me partis ſit vt .4. et pertranſeatur quadrupedale
<
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mediante illa per totam horam exrenſa: et ſic medi
<
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ante illa in prima ꝑte ꝓportionali bipedale / et ar-
<
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/>
guitur ſic / mediãte illa velocitate extenſa ꝑ totã ho
<
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/>
ram mobile ꝑtranſit quadrupedale et mediantibꝰ
<
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/>
exceſſibus quibus velocitates partium proportio-
<
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/>
nalium aliarum a prima excedunt primam pertrã
<
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/>
ſitur vuum: et ſic mediante totali illa velocitate per
<
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/>
tranſeuntur quin pedalia in totali illa hora: et q̇n
<
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/>
tupedalis ad bipedale pertranſitum in prima par
<
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te proportionali hore eſt proportio dupla ſexqui-
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altera. </
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<
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">igitur propoſitum. </
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<
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xml:space
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">Probatur tamen / me
<
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diantibus illis exceſſibus pertranſitur vnum peda
<
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/>
le: quia mediante exceſſu quo velocitas ſecunde par
<
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/>
tis excedit velocitatem prime pertranſeuntur tres
<
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/>
quarte et mediante exceſſu quo tertia excedit primã
<
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/>
pertranſitur ſubquadruplum ſpacium ad tres q̈r-
<
lb
/>
tas / et ſic conſequenter (quia illi exceſſus cõtinuo ſe
<
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/>
habent in proportione ſubdupla / vt facile eſt intue
<
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/>
ri: et continuo coextenduntur tempori in duplo mi
<
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/>
nori) / igitur aggregatum ex omnibus illis ſpaciis
<
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/>
pertranſitis mediantibus illis exceſſibus cõponi-
<
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tur ex infinitis continuo ſe habentibus in propor-
<
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/>
tione ſubquadrupla et ex hoc illud habet ſe ad pri-
<
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/>
mum illoruꝫ ī proportione ſexquitertia / vt patet ex
<
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prima parte capite quinto: et primum illoruꝫ ē tres
<
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quarte pedalis: ergo totum eſt pedale: </
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ſequentia / q2 pedalis ad tres quartas eſt propor-
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tio ſexquitertia. </
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>
<
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xml:space
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">Sed reſtat probare ſpacium per-
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tranſitum ab illo exceſſu quo ſecunda pars propor
<
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tionalis excedit primam eſſe tres quartas quia ve
<
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/>
locitas ṗme partis eſt vt .4. et velocitas ſecunde ꝑ-
<
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/>
tis habet ꝓportionē ſuꝑtripartienteꝫ q̈rtas ad ve-
<
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/>
locitatē prīe / igit̄̄ eſt vt .7. et ſic exceſſus eſt trium gra
<
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/>
duū: ſꝫ mediãte vno gradu in prīa ꝑte ꝓportionali
<
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/>
mobile ꝑtranſibat dimidium pedale vt habetur ex
<
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/>
caſu: igitur mediante vno gradu in ſecunda parte
<
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/>
ꝓportionali que eſt in duplo minor mobile pertrã
<
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/>
ſit vnam quartam et ſunt ibi tres gradus exceſſus:
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igitur mediantibus illis ꝑtranſibit tres quartas /
<
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quod fuit ꝓbandum.
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">2. correl.</
note
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<
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">¶ Sequitur ſecundo / parti-
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ta hora ꝑ partes ꝓportionales proportiõe dupla
<
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et in prima illarum mobile aliquod moueatur aliq̈
<
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/>
velocitate: et in ſecunda illarum in ſexquitertio ma
<
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/>
iori: et in tertia in ſexquiſexto maiori ꝙ̄ in prima et
<
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/>
in quarta in ſexquiduodecuplo maiori ꝙ̄ in prima /
<
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et ſic conſequenter aſcendo ꝑ numeros pares conti
<
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/>
nuo ſe habentes in ꝓportione dupla exordiendo a
<
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/>
numero ternario: hoc eſt ꝑ ſpecies ꝓportionis ſuꝑ-
<
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particularis denomīatas a partibus aliquotis de
<
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/>
nominatis ab illis numeris: ſpacium pertranſituꝫ
<
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/>
in totali hora eſt duplum ſuperbipartiens nonas
<
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/>
ad ſpacium pertranſitum in prima parte ꝓportio
<
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nali. </
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<
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">Probatur eſto exempli cauſa / velocitas pri
<
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me partis ꝓportionalis ſit vt .3. et mediante illa mo
<
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/>
bile pertranſeat in prima parte ꝓportionali tripe
<
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/>
dale: et ꝑ conſequens mediante illa extenſa ꝑ totaꝫ
<
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/>
horam ſextipedale: et arguitur ſic mediante illa ve
<
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/>
locitate vt .3: coextenſa toti hore mobile ꝑtranſibit
<
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/>
ſextipedale: et mediantibus excrementis quibus ve
<
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/>
locitates parttium ꝓportionalium aliarum a pri-
<
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/>
ma excedunt primam mobile pertranſit duas ter-
<
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/>
tias pedalis: igitur in totali illa hora pertranſit
<
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/>
ſextipedale cum duabus tertius: ſed ſextipedalis cū
<
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/>
duabus tertiis ad tripedale pertranſituꝫ in prima
<
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/>
parte ꝓportionali eſt proportio dupla ſuperbipar
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De motu locali quo ad effectum tempore difformi.
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tiens nonas: igitur propoſitum. </
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<
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xml:space
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">Sed iam probo /
<
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mediantibus exceſſibus velocitatum quibus alie ꝑ
<
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tes proportionales excedunt velocitatem ṗme mo-
<
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bile pertranſit duas tertias. </
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<
s
xml:id
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xml:space
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">quia velocitas ſecun-
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de partis ꝓportionalis excedit velocitatem prime
<
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ꝑ vnum gradum (eſt enim velocitas prime vt .3. et ſe
<
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/>
cunde ſexquitertia ad illam) et mediante vno gradu
<
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/>
in prima parte ꝓportionali mobile pertranſit vnū
<
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/>
pedale: ergo mediante illo gradu mobile ꝑtranſit
<
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/>
vnum ſemipedale in ſecunda parte proportionali
<
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/>
ſubdupla ad primam: et mediante exceſſu quo tertia
<
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/>
pars excedit primam pertranſit ſubquadruplū ad
<
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/>
illud ſemipedale. </
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<
s
xml:id
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">et mediante exceſſu quo quarta ex
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cedit primam adhuc pertranſit ſubquadruplū ad
<
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precedens / et ſic conſequenter: quia illi exceſſus con-
<
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tinuo ſe habent in ꝓportione ſubpupla / vt patet ex
<
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/>
caſu: et continuo extenduntur in duplo minori par
<
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/>
te: igitur aggregatum ex omnibus illis ſpaciis per
<
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/>
tranſitis mediantibus illis exceſſibus componitur
<
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ex infinitis continuo ſe habentibus in proportione
<
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ſubquadrupla. </
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>
<
s
xml:id
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xml:space
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">igitur ſe habet ad primum illorum
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in ꝓportione ſexquitertia. </
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<
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">Conſequentia ſepiꝰ ar-
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guta eſt. </
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<
s
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xml:space
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">et cum primum illorum ſit ſemipedale: con
<
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ſequens eſt vt aggregatum ex omnibus illis ſit due
<
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/>
tertie (ſiquidem duarum tertiarum ad ſemipedale
<
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/>
ſit ſexquitertia proportio) </
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>
<
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xml:space
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">Et ſic patet probandum
<
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et totum correlarium. </
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>
<
s
xml:id
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xml:space
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">¶ Innumera talia correlaria
<
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poſſunt inferri diuidendo horam aliis ſpeciebus ꝓ
<
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poportionis: et faciendo exceſſus quibus alie par-
<
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tes excedunt primam in certa ꝓportiõe continue ſe
<
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/>
habere: vt ſi hora diuidatur per partes proportio
<
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/>
nales ꝓportione tripla: et in prima illaruꝫ aliquod
<
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/>
mobile moueatur aliquanta velocitate et in ſecun-
<
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/>
da in duplo ſexquialtero maiori: et in tertia in du-
<
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/>
plo ſexquiſexto: et in quarta in duplo ſexquidecimo
<
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/>
octauo maiori ꝙ̄ in prima: et in quinta in duplo ſex
<
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/>
quiquīquageſimo quarto maiori ꝙ̄ in prima: et ſic
<
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/>
conſequenter procedendo ex parte ꝓportionis mul
<
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/>
tiplicis ſuperparticularis per numeros ſe haben-
<
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/>
tes continuo in ꝓportione ſubtripla </
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>
<
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">Ibi enim ex-
<
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ceſſus ſe habent in proportione ſubtripla </
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>
<
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">Itē ſi ho
<
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ra partiatur per partes ꝓportionales ꝓportione
<
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/>
ſuperbipartiente tertias et a. mobile in prima mo-
<
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/>
ueatur aliquanta velocitate et in ſecunda in triplo
<
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/>
ſexquiquinto velocius: et in tertia in triplo ſexqui-
<
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/>
decimo velocius ꝙ̄ in prima: et in quarta in triplo
<
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/>
ſexqui viceſimo velocius ꝙ̄ in prima: et in quinto in
<
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/>
triplo ſexquiquadrigeſimo progrediendo per ſpe
<
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/>
cies denomīatas a numeris imparibus ſiue ab vni
<
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/>
tate et partibus aliquotis denominatis ab illis nu
<
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/>
meris continuo ſe habentibus in ꝓportione dupla
<
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exordiendo a quinario. </
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>
<
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xml:space
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">Et ſic conſequenter poteris
<
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infinita ſimilia ponere</
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</
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<
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">Undecima concluſio </
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>
<
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">Diuiſa hora per
<
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/>
partes proportionales quacun libuerit propor-
<
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/>
tione et in prima mobile moueatur aliquanta velo
<
lb
/>
citate et in ſecunda in ſexquialtero maiori: et in ter-
<
lb
/>
tia in ſexquitertia maiori ꝙ̄ in ſecunda: et in quarta
<
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/>
in ſexquiquarta maiori ꝙ̄ in tertia et in quinta ī ſex
<
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quiquinto maiori ꝙ̄ in quarta / et ſic conſequenter.
<
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/>
</
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>
<
s
xml:id
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xml:space
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preserve
">et ſi nõ valeat regula vniuerſalis ſignari ad reperi
<
lb
/>
endum ſpacium pertranſitum in totali hora: nichi
<
lb
/>
lominus tamen qualibet ſpecie diuiſionis hore ſi-
<
lb
/>
gnata poteſt certitudinaliter inueſtigari ſpaciuꝫ ꝑ
<
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/>
tranſitum in tota hora: et ꝓportio eius ad ſpacium
<
lb
/>
pertranſitum in prima parte ꝓportionali. </
s
>
<
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">Proba
<
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tur hec concluſio / et primo probatur ſecunda ꝑs eiꝰ.
<
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/>
</
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<
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xml:space
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">quia ſit hora fuerit diuiſa per partes proportiona </
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