Alvarus, Thomas
,
Liber de triplici motu
,
1509
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Secundi tractatus
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velocitates ſe habeant in ꝓportione ſexquiquarta
<
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omnium vero parium in ꝓportione ſexquiquinta:
<
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excedat velocitas ṗme partis paris velocitatem
<
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ṗme partis imparis in proportiõe ſexquiſexta: tūc
<
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inueſtiga ꝓportionem totius ſpacii ad ſpaciuꝫ per
<
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tranſitū in prima innitendo p̄cedentibus. </
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<
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xml:space
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">Itē parti
<
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ta hora in partes ꝓportionales ꝓportiõe quadru
<
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pla: mobili in omni īpari ſequente mouēte in ſex
<
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/>
quiſexto velocius ꝙ̄ in immediate ꝓcedente impari
<
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/>
et in omni pari ſequente in ſexquiſeptimo velocius
<
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quã in pari immediate precedente: ſuperet veloci
<
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/>
tas prime partis paris velocitatem prime imparis
<
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/>
in ꝓportione ſexquioctaua: tunc cõmenſura totale
<
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ſpacium ſpacio prime partis ꝓportionalis precedē
<
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tibus ſuffultus </
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<
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xml:space
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">Et ſic aſcendendo per ſpecies ꝓpor-
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tionis multiplicis in diuidenda hora velocitatibꝰ
<
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/>
ſe habentibus continuo in diuerſis ꝓportionibus
<
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/>
ſuperparticularibus infinitam multitudinem ſe ſe
<
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ↄ̨ſequētiū cõcluſionum inferre valebis. </
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<
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xml:space
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">Deinde diui
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ſa hora aliqua multipli ſimplici vel compoſita ve-
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locitatibus partiuꝫ imparium cõtinuo ſe habētibꝰ
<
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/>
in ꝓportione aliqua ſuprapartiente: et partium pa
<
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/>
riū immediatarum velocitatibus continuo ſe habē
<
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/>
tibus in aliqua alia ꝓportione ſuprapartiente: ex-
<
lb
/>
cedente velocitate prime partis paris velocita-
<
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/>
tem prime partis imparis in aliqua alia propor-
<
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tione ſuperpartiente infinita correlaria inferre po
<
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teris. </
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<
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portionales ꝓportione multipici: quarūcun dua
<
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rum ꝑtium ꝑ .4. partes ꝓportionales diſtantiū ve-
<
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/>
locitatibus ſe habentibus in aliqua ꝓportione ſu
<
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/>
perparticulari vel ſuperpartiente ita vt ṗme diſtã
<
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/>
tes ꝑ .4. partes ꝓportionales vt puta prima et ſex
<
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/>
ta ſe habeant in velocitate in ꝓportione ſexquial-
<
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tera: et ſeptime velocitas ad velocitatem ſecunde in
<
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ꝓportione ſexquitertia: et octaue velocitas ad velo
<
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/>
citatem tertie in ꝓportione ſexquiq̈rta: et none ve-
<
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/>
locitas ad velocitatem quarte in ꝓportione ſexqui
<
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/>
quinta: et decime velocitas ad velocitatem quinte ī
<
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/>
ꝓportione ſexquiſexta: et vndecime velocitas ad ve
<
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locitatem ſexte in ꝓportione ſexquialtera: et ſic ite-
<
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/>
rum aſcendendo vſ ad proportionem ſexquiſextã
<
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et deinde redeundo vſ ad ꝓportionem ſexquial-
<
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teram / et ſic conſequenter: ita omnes diſtantes ꝑ
<
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4. incipiendo a ṗma ſe habeant in ꝓportione ſexq̇
<
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altera in velocitate: et incipiendo a ſecunda in ſexq̇
<
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tertia: et a tertia in ſexquiquarta: et a quarto in ſex
<
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quiquinta: et a quinta in ſexquiſexta: et non plus.</
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<
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">Ita poteris facere de partibus inter quas cõtinuo
<
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mediant octo ꝑtes aſcendendo a prima vſ ad de-
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cimã: et ſic in infinitum poteris variare caſus reten
<
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ta ſemper aliqua vniformiter ꝓportionum </
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<
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xml:space
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">Et ſi-
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cut inferuntur multa correlaria quando velocitas
<
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maior coextenditur ꝑtibꝰ minoribus. </
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<
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xml:space
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">ita plura alia
<
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poſſunt inferri quando continuo velocitas maior
<
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coextenditur partibus minoribus que omnia ex ṗ-
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oribus facile inducuntur. </
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<
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xml:space
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">Et quia nimium in iſtis
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immorari vltra modum eis inherere, eſt a melio
<
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ribus ſublimioribuſ ꝓſtergari: </
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<
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xml:space
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">Ideo calculator
<
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his dedaleis laberinthulis implicitꝰ: verbiſ mul
<
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tiplicibus multiformibuſ ꝓportionibus implica
<
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tus: inflate bucce garritum contineat.</
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">Decima concluſio </
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<
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">Diuiſa hora ꝑ par
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tes ꝓportionales ꝓportione dupla et a. mobile in
<
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prima ꝑte ꝓportionali moueatur aliquantula ve-
<
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locitate: et in ſecunda in ſexquialtero maiori veloci
<
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tate: et in tertia in ſexquiquarto maiori velocitate
<
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̄ in prima: et in quinta in ſexquiſexdecimo maiori
<
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quã in prima / et ſic conſequenter aſcendendo ꝑ ſpe-
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cies ꝓportionis ſuperparticularis denominatas
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a numeris pariter paribus </
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<
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xml:space
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">(Meliꝰ tñ diceret̄̄ deſcē
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dēdo: q2 ꝓportiões ſuꝑparticulares ſūt mīores quã
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to a maiori numero denominantur hoc eſt a parte
<
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aliquota denominata a maiori numero) ſpacium ꝑ
<
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trauſitum in totali hora ſe habet ad ſpacium per-
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tranſitum in prima ꝑte ꝓportionali in ꝓportione
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dupla ſexquitertia. </
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>
<
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">Probatur et ſit gratia exempli
<
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velocitas ṗme ꝑtis ꝓportionalis vt duo, ꝑtrãſeat
<
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a. mobile mediante illa velocitate in prima ꝑte ꝓ
<
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portionali bipedale: et arguitur ſic / illa velocitas vt
<
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duo coextenditur toti hore, quia in qualibet parte
<
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/>
ꝓportionali hore velocitas eſt maior quam vt duo
<
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vt habetur ex caſu et tota hora eſt dupla ad primaꝫ
<
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partem ꝓportionalem eius in qua mobile pertran
<
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/>
ſit bipcdale mediante velocitate vt duo: igitur me-
<
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diante illa velocitate coextenſa toti hore pertran-
<
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ſit quadrupedale: et mediantibus exceſſibus parti-
<
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um ꝓportionalium ſupra illam velocitatem vt duo
<
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pertranſit duas tertias pedalis que faciūt vnã ter
<
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tiam duorum pedalium: igitur totuꝫ ſpacium ſe ha
<
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/>
bebit ad ſpacium pertranſitum in prima parte ꝓ-
<
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/>
portionali in proportione dupla ſexquitertia cuiuſ
<
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/>
modi eſt ꝓportio ipſoꝝ quatuor cum duabus ter-
<
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tiis vnius ad duo </
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>
<
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">Probo tamen / mediantibꝰ il-
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lis exceſſibus ꝑtranſeat duas tertias pedalis: quo
<
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/>
niam cum velocitas ſecunde ꝑtis ꝓportionalis ſit
<
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ſexquialtera ad velocitatem prime que eſt vt duo ſe
<
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quitur / exceſſus velocitatis ſecunde ad velocitatē
<
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prime eſt vnus gradus et quia tertia excedit primã
<
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/>
in ꝓporeione ſexquiquarta / ſequitur / exceſſus eius
<
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eſt medietas vnius gradus quoniam duorum cū di
<
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midio ad duo eſt proportio ſexquiquarta, et veloci
<
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tas quarte partis ſe habet ad velocitatem prime ī
<
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ꝓportione ſexquioctaua: igitur exceſſus eius ē vna
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quarta: igitur in illo caſu exceſſus ſecunde ad exceſ
<
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ſum tertie eſt ꝓportio dupla et exceſſus tertie ad ex-
<
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ceſſum quarte dupla ſimiliter: et ſic conſequenter re
<
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peries illos exceſſus ſe habere in ꝓportione ſubdu
<
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pla et ſubdupla. et coextenduntur partibus cõtinuo
<
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ſe habentibus in ꝓportione ſubdupla et ſubdupla /
<
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igitur continuo illa ſpacia mediantibus illis velo
<
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/>
citatibus ꝑtranſita ſe habet in ꝓportione ſubqua-
<
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drupla / et ꝑ conſequens aggregatum ex omnibꝰ eis
<
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ſe habebit ad primum illorum in ꝓportione ſexqui
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tertia et ṗmum illorum eſt vnum ſemipedale: ergo
<
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totum erit vnum ſemipedale cum vna ſexta peda-
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lis: et ꝑ conſequens due tertie vnius pedalis / qḋ fuit
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ꝓbandum. </
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<
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">Sed iam ꝓbo / p̄mum illorum ſit vnum
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ſemipedale quoniam primum illorum ꝑtranſit̄̄ me
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diante exceſſu ſecunde ꝑtis ꝓportionalis ſupra pri
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mam qui exceſſus eſt vnus gradus mediante quo ī
<
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prima parte ꝓportionali pertranſitur vnum peda
<
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le: igitur mediante illo in ſecunda parte ꝓportiõa-
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li ſubdupla ad illam pertranſitur vnum ſemipeda
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le / quod fuit ꝓbandum. </
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>
<
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">Patet conſequentia ex ſecū
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da ꝑte prime ꝓpoſitionis ſecundi notabilis.</
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<
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">¶ Ex hac concluſione ſequitur primo / ſi fuerit ho
<
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ra diuiſa ꝓportione dupla: et in prima illarum par
<
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tium moueatur aliquod mobile aliquanta velocita
<
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/>
te, et in ſecunda in ſupertripartiente quartas maio
<
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ri velocitate, et in tertia in ſupertripartiente octa-
<
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uas maiori velocitate ꝙ̄ in prima: et in quarta in ſu
<
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ꝑtripartiente ſexdecimas maiori ꝙ̄ in prima et in
<
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quinta in ſuꝑtripartiente triceſimas ſecundas ma
<
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/>
iori velocitate ꝙ̄ in prima / et ſic conſequenter ꝓcedē
<
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do per ſpecies ꝓportionis ſupertripartientis de-
<
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nominatas a numeris pariter paribus ſiue a par-
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tibus aliquotis denominatis ab illis numeris: ſpa </
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