Alvarus, Thomas
,
Liber de triplici motu
,
1509
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Secundi tractatus
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per mediam horam, et velocitate vt duo per horam
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<
s
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xml:space
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">Item ſi maior velocitas coextendatur tēpori mīori
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et minor maiori. </
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<
s
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N1E821
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xml:space
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">hoc coutingit tripliciter / quia aut
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ꝓportio tempoꝝ excedit ꝓportionē velocitatū aut
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ꝓportio velocitatū excedit ꝓportionē tempoꝝ aut
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ꝓportiones tempoꝝ et velocitatū ſunt equales. </
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<
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xml:space
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">Exē-
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plum primi / vt ſi aliquod mobile in hora moueatur
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vt duo, et in quarta hore vt quatuor: tunc ꝓportio
<
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tempoꝝ excedit proportionē velocitatū. </
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>
<
s
xml:id
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N1E833
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xml:space
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preserve
">Nam ipſa
<
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tēpoꝝ proportio quadrupla eſt: velocitatū vero du
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pla / vt patet aſpicienti. </
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>
<
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N1E83A
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xml:space
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preserve
">Exemplū ſecundi / vt ſi mo-
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bile moueatur vt vnū per horã, et in media vt .3. / tūc
<
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/>
proportio tempoꝝ eſt dupla, velocitatū o tripla:
<
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exuperat igitur velocitatū proportio tempoꝝ pro-
<
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portionē. </
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>
<
s
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N1E845
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xml:space
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preserve
">Exemplū tertii / vt ſi aliquod mobile mo-
<
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ueatur in hora vt vnū, et aliud in media vt duo: con
<
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ſtat ꝓportionē tempoꝝ ꝓportioni velocitatū equa
<
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ri: vtra em̄ dupla eſt: et velocitatū, et tempoꝝ. </
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>
<
s
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xml:space
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preserve
">Hac
<
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longa diuiſione velocitatū exacta: ipſa velocita
<
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te fruſtrat ī conciſa: opere preciū eſt cuilibet huiꝰ di-
<
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/>
uiſionis fruſto et membro peculiarē propoſitioneꝫ
<
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aſſcriberet. </
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>
<
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xml:space
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">Sit igitur.</
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<
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xml:space
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">Capitalis propoſitio. </
s
>
<
s
xml:id
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N1E860
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xml:space
="
preserve
">Si velocitates
<
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ſint equales equalibus coextenſe temporibus: mo-
<
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/>
bilia in eiſdem mota equalia ſpacia in eiſdē tēpo-
<
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/>
ribus abſoluūt (ceteris aliis deductis) vt puta ra-
<
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/>
refactione condenſatione ſpacii et prepoſtera mo-
<
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tione / vt concluſiones ſexto phiſicoꝝ oſtendunt. </
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>
<
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xml:id
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xml:space
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preserve
">Si
<
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/>
vero velocitates equales per īequalia labantur tē
<
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pora: tunc in ea ꝓportione mobile in maiori tēpo-
<
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re maius ſpaciū pertranſit quam in minori: in qua
<
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/>
ipſū maius tēpus ſe habet ad minus. </
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>
<
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xml:space
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">Prima pars
<
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huiꝰ ꝓpoſitionis patet ex ſe: et ſecunda ꝓbatur: ſup
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poſito / quando aliquid mobile mouetur vnifor-
<
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/>
miter per aliquod tēpus in quacū ꝓportione ſe
<
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/>
habent partes tēporis ad totū: in ea ꝓportione ſe
<
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/>
habent ſpacia pertranſita in illis temporibus ad
<
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/>
ad ſpaciū pertranſitū in toto tēpore: quo ſuppoſi-
<
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/>
to arguitur ſic mobile quod mouetur in maiori tē-
<
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pore et mobile motū in minori tēpore mouētur vni-
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formiter et eque velociter. </
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>
<
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xml:space
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">ergo in eq̈libꝰ tēporibus
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equalia ſpacia ꝑtranſeunt / vt patet ex priori parte:
<
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ergo quantū ſpaciū mobile motū in minori tēpore
<
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ꝑtranſit in totali ſuo tēpore: tantū adequate per-
<
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trãſit mobile motū in maiori tēpore in tēpore ſibi
<
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equali: ergo qualis eſt ꝓportio illius tēporis ma-
<
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/>
ioris ad tēpus minus talis eſt ꝓportio ſpacii ꝑtrã
<
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/>
ſiti in tēpore maiori ad ſpaciū ꝑtranſitū in tēpore
<
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minori / quod fuit ꝓbandū: et cõſequentia patet ex
<
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/>
ſuppoſito hoc adiecto / qualis eſt ꝓportio totius
<
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/>
tēporis ad illam ſuã partem equalē tēpori minori
<
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/>
talis eſt proportio ipſius maioris temporis ad il-
<
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lud minus tempus / vt patet de ſe.</
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<
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<
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">Quãdo inequales
<
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velocitates equalibus tēporibus coextendūtur: tūc
<
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mobile quod maiore velocitate mouetur in ea pro
<
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portione maius ſpaciū ꝑtranſit ꝙ̄ alterum / mobile
<
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in qua ſe habet velocitas maior ad minorē. </
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batur hec ꝓpoſitio (̄uis facilis ſit) / quia ſi mobile
<
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motū velocitate maiori in tēpore a. moueretur ade
<
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/>
quate equali velocitate ſicut mouetur aliud mobile
<
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/>
motū velocitate minori in eodē a. tempore / tūc illa
<
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duo mobilia equalia ſpacia ꝑtranſirent in a. tēpo
<
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re / vt ptꝫ ex priori parte p̄cedentis ꝓpoſitionis: ſed
<
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modo illud mobile mouetur in aliqua proportione
<
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puta in f. velocius quã tunc: ergo in f. proportione
<
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maius ſpaciū pertranſit quã tunc: et per conſequēs
<
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maius ſpaciū ꝑtranſit in eodem tēpori in f. ꝓpor-
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Capitulū tertiū.
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tione quã alterū mobile motum in eodem tempore
<
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/>
velocitate in f. proportione minori.</
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>
</
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<
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">Tertia ꝓpoſitio. </
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<
s
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xml:space
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">Si inequales velo-
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citates in equalibus tēporibus coextendunt̄̄, et ma
<
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ior velocitas maiori tempori coextendatur: et mīor
<
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minori: tunc mobile quod mouetur in maiori tem-
<
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pore maius ſpaciū ꝑtranſit in ꝓportione cõpoſita
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temporis maioris ad tempus minus: et velocitatis
<
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maioris ad velocitatē minorē. </
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<
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">Exemplū / vt ſi mobi
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le a. moueatur per horã vt quatuor, et b. per mediã
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horã, vt .2. / tunc dico / a ꝑtranſit maius ſpacium
<
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quã b. in ꝓportione cõpoſita ex ꝓportione hore ad
<
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mediã horam: et velocitatis vt .4. ad velocitatem vt
<
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duo. / et cū vtra illaꝝ ꝓportionū ſit dupla: conſe-
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quens eſt / cõpoſita ex eis ſit quadrupla / vt patet
<
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ex ſecunda parte: et per conſequens in quadruplo
<
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maius ſpaciū ꝑtranſit a. in hora quam b. in media
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hora. </
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>
<
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xml:space
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">Probatur hec concluſio / quia ſi a. et b. moue-
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rentur equaliter in illis duobus temporibus ine-
<
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/>
qualibus: tunc a. ꝑtranſit maius ſpaciū quam b. in
<
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/>
illa ꝓportione in qua ſe habent tēpora / vt patet ex
<
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ſecunda parte prime ꝓpoſitionis: et modo a. in ali-
<
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qua proportione que ſit f. maiori velocitate mouet̄̄
<
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quã tunc: ergo in f. proportione maius ſpaciū per-
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tranſit quã tunc. </
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>
<
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">Patet conſequētia / quia quanto
<
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in eodem tempore velocitas eſt maior: tanto in eo-
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dem tempore per eandeꝫ maius ſpaciū ꝑtranſitur.
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</
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>
<
s
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xml:space
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">Ergo proportio ſpacii ꝑtranſiti a mobili quod ve-
<
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lociꝰ mouetur ad ſpaciū ꝑtranſitū a mobili / quod
<
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tardius mouetur componitur adequate ex propor
<
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tione tēporū: et ex proportione velocitatū que eſt f. /
<
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quod fuit probandū. </
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>
<
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xml:space
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">Patet / quia inter terminos il
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/>
lius proportionis reperiūtur iſti termini puta ſpa
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/>
ciū ꝑtranſitum ab illa velocitate maiori in maiori
<
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/>
tempore et ſpaciū pertranſitū in eodem maiori tē-
<
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pore a velocitate equali velocitate minoris tempo
<
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ris: et ſpacium ꝑtranſitum a velocitate minoris tē-
<
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poris in minori tempore: ſed primi termini ad ſe-
<
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/>
cundū eſt proportio f. que eſt proportio velocitatū
<
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/>
et ſecundi ad tertiū eſt proportio tempoꝝ: et totalis
<
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/>
illa ꝓportio q̄ cõponit̄̄ ex illis duabꝰ eſt proportio
<
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ſpacii ad ſpaciū: g̊ ꝓportio ſpacii ꝑtrãſiti a mobi
<
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li velociori ad ſpaciū ꝑtrãſitū a mobili tardiori cõ-
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ponitur ex ꝓportiꝰe velocitatis ad velocitatē: et tꝑs
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ad tēpus / quod fuit probandū: et ſic ptꝫ propoſitio
<
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<
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xml:space
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<
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xml:space
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">¶ Ex hac propoſitione ſequitur primo / ſi a. mo-
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ueatur per vnã horã velocitate vt .6. et b. ꝑ mediam
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horã velocitate vt .4. / ſpaciū ꝑtranſitū ab a. erit
<
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triplū ad ſpaciū ꝑtranſitū a.b. </
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>
<
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xml:id
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xml:space
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">Ptꝫ / qm̄ ex propor
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tione tēporis ad tēpus, et velocitatis ad velocitatē
<
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quaꝝ prima eſt dupla: et ſecūda ſexq̇altera cõponit̄̄
<
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tripla proportio / vt ptꝫ in his terminis .6. ad .4. et
<
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/>
4. ad .2. et in illa proportione a. mouet̄̄ velociꝰ b. / vt
<
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ptꝫ ex precedenti propoſitione: igitur propoſitum.</
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<
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<
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xml:id
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xml:space
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">Sequit̄̄ ſcḋo / ſi a. mobile moueat̄̄ ꝑ
<
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horã velocitate vt .6. et b. ꝑ duas tertias hore velo-
<
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/>
citate vt .4. / in minori proportione maiꝰ ſpacium
<
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ꝑtrãſit a. ꝙ̄ b. ꝙ̄ in priori caſu. </
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>
<
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xml:id
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xml:space
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preserve
">Ptꝫ / q2 tūc ſpaciū ꝑ-
<
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trãſitū ab a. erit duplū ſexquiq̈rtū ad ſpaciū ꝑtrã-
<
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ſitū a.b. et in priori caſu erat triplū: g̊ in minori pro
<
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/>
portione maiꝰ ſpaciū ꝑtraſit a. quã b. in iſto caſu ̄
<
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/>
in priori. </
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>
<
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N1E978
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xml:space
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">Ptꝫ ↄ̨ña / q2 tripla eſt maior ꝙ̄ dupla ſex-
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quiquarta proportio. </
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>
<
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xml:space
="
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">Probo tamen maiorē / quia
<
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proportio tēporis ad tēpus eſt ſexq̇altera: et ſimili
<
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/>
ter velocitatis ad velocitatē: ergo ſpaciū ꝑtrãſitū
<
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/>
ab a. eſt maiꝰ ſpacio ꝑtrãſito a b. in proportione cõ
<
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/>
poſita ex duabꝰ ſexq̇alteris, q̄ eſt dupla ſexquiq̈rta /
<
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/>
vt patet in his terminis: 9.6.4. auxiliantibus his </
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