Alvarus, Thomas
,
Liber de triplici motu
,
1509
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Secūde partis
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0030
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30
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a duodenario ſe habeat in proportio dupla a
<
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ſenario: continuo illud quod remanet ex duode-
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nario ſe habet in proportione dupla ad illud qḋ
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remanet a numero ſenario. </
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<
s
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N12C0B
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xml:space
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">Et ſub tenore huiꝰ exē
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pli ego intelligo correlarium </
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<
s
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xml:space
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">Non enī in iſtis exa
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ctꝰ ſenſus dialecticus ex expetendus ſed ipſa ma
<
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thematica ſententia eſt efflagitanda. </
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>
<
s
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xml:space
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rium perinde at primum demonſtrationem con
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cluſionis exquirit. </
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<
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">Applica vt vales.</
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</
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<
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xml:id
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xml:space
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">4. correĺ.</
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<
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<
s
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N12C26
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xml:space
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">¶ Sequitur quarto / quandocun duo numeri ī
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equales continuo creſcunt: et continuo ſe habent
<
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in eadem proportione: oportet / continuo acqui
<
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ſitū maiori numero ſe habeat in eadeꝫ proportio
<
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ne ad acquiſitum minori in qua ſe habent illi nūe
<
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ri creſcentes. </
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>
<
s
xml:id
="
N12C33
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xml:space
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">vt ſi numerus quaternarius et ſena-
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rius continuo creſcant et continuo manent in pro
<
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portione ſexquialtera: oportet / continuo acqui
<
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ſitum ſenario ſe habeat in proportione ſexquial-
<
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tera ad acquiſitum quaternario. </
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>
<
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xml:space
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">Hoc correlariuꝫ
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eadem cum precedentibus demonſtratione oſten-
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ditur. </
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>
<
s
xml:id
="
N12C45
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xml:space
="
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">¶ Sequitur quinto / datis quibuſcun
<
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duobus numeris inequalibus ſe habentibus ī ali
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q̄ ꝓportiõe et ī ea ꝓportiõe ī q̄ mīor excedit̄̄ a maio
<
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re ī eadē cõtinuo tardiꝰ creſcat maiore: cõtinuo ta
<
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les numeri manent in eadem proportione. </
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>
<
s
xml:id
="
N12C50
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xml:space
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">vt da-
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tis: 4. et .6. ſe habentibus in proportione ſexqui-
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altera: ſi quando ſex acquiſiuerint aliquod cremē
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tum. </
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>
<
s
xml:id
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N12C59
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xml:space
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">quatuor acquirant in ſexquialtero minus: ip
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ſi continuo manent in proportione ſexquialtera.
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</
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>
<
s
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="
N12C5F
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xml:space
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">Probatur hoc correlarium / quoniam ſi in eadem
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proportione in qua numerꝰ maior ſe habet ad mi
<
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norem velocius creſcat quaꝫ minor: ſequitur / cõ
<
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tinuo inter acquiſitum minori numero eſt eadem
<
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proportio que eſt inter illos numeros. </
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>
<
s
xml:id
="
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xml:space
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">vt patet ex
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probatioue concluſionis: et per conſequens con-
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tinuo tales numeri manent in eadem proportione
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</
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<
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xml:space
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</
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<
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<
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">Datis tribus nu-
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meris in hac medietate conſtitutis: quod fit ex du
<
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ctu extremi in extremum equale eſt quadrato me-
<
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dii: hoc eſt illi numero qui reſultat ex ductu medii
<
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termiui in ſeipſum. </
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>
<
s
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N12C84
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xml:space
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">vt conſtitutis his tribus termi
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nis .8.4.2. numerus ſexdenarius reſultans ex du-
<
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ctu octonarii in binarium eſt equalis numero qui
<
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fit ex ductu quaternarii in ſeipſū / vt conſtat. </
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>
<
s
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xml:space
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">Pro
<
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batur hec concluſio / ſint tres numeri a.b.c. in hac
<
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medietate conſtituti continuo ꝓportionabiles .g.
<
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ꝓportione: et ſit d. numerus reſultãs ex ductu a. in
<
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b. et e. ſit numerus reſultãs ex ductu b. in idē b. et f.
<
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numerus reſultans ex ductu a. in c. / tunc dico / e. et
<
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f. ſunt equales. </
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>
<
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">Qḋ ſic ꝓbatur: qm̄ d. ad e. eſt ꝓpor
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tio g. et d. ad f. eſt eadē ꝓportio g. / ergo e. et f. ſunt
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equalia / quod fuit ꝓbandū. </
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<
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maior oſtenditur. </
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<
s
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xml:space
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">quia ſicut ſe hꝫ d. ad a. ita ſe ha-
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bet e. ad .b. q2 toties adeq̈te a. cõtinet̄̄ in d. quoties
<
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eſt vnitas in b. et toties continetur b. in e. quoties
<
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eſt vnitas in b. cum d. fiat ex ductu a in b. et e. ex du
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ctu b. in b. / igit̄̄ ſicut ſe habet d. ad a. ita e. ad b. </
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">Con
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ſequētia claret ex tertia ſuppoſitione huius capi-
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tis: et ex conſequēti ſicut ſe hꝫ d. ad a. ita e. ad b: er
<
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go ſicut ſe habet d. ad e. ita ſe habet a. ad b. ſed a.
<
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ad b. eſt g. proportio / ergo .d. ad e. eſt g. ꝓportio / qḋ
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fuit ꝓbandū. </
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<
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<
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">Iã probat̄̄ mi-
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nor. </
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<
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">q2 d. in g. ꝓportione pluries cõtinet a. quaꝫ f.
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contineat idē a. adeq̈te: ergo d. ſe habet ad f. in g.
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ꝓportione </
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ctione p̄allegata. </
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<
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">Probatur antexedens / q2 d. toti
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es continet a. quoties eſt vnitas in b. cuꝫ a. in b. du
<
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catur et inde reſultat d. et f. toties continet a quo
<
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="
Capitulum ſecundum
"/>
ties eſt vnitas in c. eadē rõue: ſꝫ in g. ꝓportiõe plu
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ries continetnr vnitas in b. quã in c. cū b. et c. ſe ha
<
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beant in g. ꝓportione: ergo in g. ꝓportione pluri
<
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es cõtinetur a. in d. quã in f. / qḋ fuerat oñdendū. </
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<
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">Et
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ſic patet cõclnſio q̄ ꝓfecto pulchra eſt et induſtria q̄
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ſit huius medietatis. </
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<
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<
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">1. correĺ.</
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<
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">¶ Ex hac
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concluſione ſeq̇tur ṗmo / ī hac medietate id quod
<
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fit ex ductu vnius extremi ad triū termīorū alterū
<
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extremū eſt numerꝰ quadratꝰ: </
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<
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N12CFC
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">Probatur / q2 talis
<
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numerus eſt equalis quadrato medii termini / g̊ eſt
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numerus quadratus </
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<
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">Cõſequētia patet de ſe et añ
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cedens ex concluſione.
<
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="
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xlink:href
="
note-0030-02a
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xlink:label
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note-0030-02
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xml:id
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xml:space
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">2. correĺ.</
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>
</
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<
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xml:id
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xml:space
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">¶ Sequitur ſecundo / ſi cõ
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ſtitutis duobus numeris ſe habentibus in aliqua
<
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ꝓportione maioris ineqnalitatis rationali. </
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<
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xml:id
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rus q̇ fit ex ductu vnius extremi in alterū non eſt q̈
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dratꝰ: inter tales termīos nõ eſt medium ꝓportio
<
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nabile ꝓportione rationali: ita ṗmi ad illḋ me-
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diū ſit eadē ꝓportio rationalis que eſt illius me-
<
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dii ad tertiuꝫ. </
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>
<
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N12D21
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xml:space
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">Probatur hoc correlarium / qnia ſi
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inter tales numeros reperiatur mediū ꝓportiõa-
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bile ꝓportione rationali: puta aliquis numerus
<
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medio loco proportionabilis: iam ſequitur / ibi
<
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dē reperiuntur tres numeri cõtinuo ꝓportionabi
<
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les hac medietate: et ꝑ cõſequēs numerꝰ q̇ fit ex du
<
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ctu extremi in extremum eſt equalis quadrato me
<
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dii / vt patet ex coucluſione: igitur talis numerus ē
<
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quadratus / vt patet ex primo correlario / quod eſt
<
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oppoſitū añcedetis correlari ꝓbãdi. </
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<
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">īfert igir̄ cor
<
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relarii oppoſitū conſequentis oppoſitū añceden-
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tis / et ꝑ conſequēs correlarium verum.
<
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position
="
right
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xlink:href
="
note-0030-03a
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xlink:label
="
note-0030-03
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xml:id
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xml:space
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">3. correĺ.</
note
>
</
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<
s
xml:id
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N12D42
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xml:space
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">¶ Sequitur
<
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tertio / ſi medium proportionabile īter duos nu
<
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meros ſe habētes in proportione maioris inequa
<
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litatis nõ ſit latus numeri contenti ſub extremis:
<
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tunc numerus qui fit ex ductu vnius extremi in al-
<
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terū nõ eſt quadratus. </
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<
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xml:id
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xml:space
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">Probatur / ſint a.c. duo nu
<
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meri ſe habentes in proportione maioris inequa
<
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litatis a. maior c. minor: et numerus qui fit ex du
<
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ctu a. in c. ſit d. et e. ſit medium ꝓpõrtionale inter a
<
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et c. / tunc dico / ſi e. non ſit latus ipſius d: d. nõ eſt
<
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numerus quadratus. </
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<
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xml:space
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">Quod ſic oñditur: q2 ſi d: ſit
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numerus quadratus ſequitur / eius latus eſt e. /
<
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igitur ex oppoſito ſequitur oppoſitum: et per con
<
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ſequens correlarium verum. </
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<
s
xml:id
="
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xml:space
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">Probatur antece-
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dens / quia ſi d. eſt numerus quadratus cum nõ ſit
<
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quadratus a. nec quadratus ipſius c. / vt conſtat:
<
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qm̄ quando duo numeri inequales in ſeipſos du
<
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cuntur quod inde ſit neutrius illoꝝ eſt quadratuꝫ:
<
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ſed eſt alicuius numeri minoris maiore illorum et
<
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maioris minore: ſit igitur talis numerus b. cuius
<
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d. eſt quadratum / et ſequitur / a. ad b. eſt aliqua ꝓ
<
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portio: conſtituo igitur tres terminos continuo ꝓ
<
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portionabiles illa proportione a. ad b. que ſint a.
<
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b.h. / et ſequitur ex cõcluſione / numerus qui fit ex
<
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ductu a. in h. eſt equalis ipſi d. et per te numerꝰ qui
<
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fit ex ductu a. in c. eſt equalis ipſi d. </
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<
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="
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xml:space
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">Imo eſt ipſum
<
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d. / igitur h. et c: ſunt numeri equales. </
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<
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xml:id
="
N12D85
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xml:space
="
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">Patet hec cõ
<
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ſequentia / q2 ex ductu vuius tertii in vtrū illorū
<
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reſultat idem numerus. </
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>
<
s
xml:id
="
N12D8C
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xml:space
="
preserve
">et ſic tot vnitates continet
<
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c. ſicut h. / et per conſequens ſunt equales. </
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>
<
s
xml:id
="
N12D91
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xml:space
="
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">ſed inter
<
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a. et h. eſt mediū ꝓportionale quod eſt latus qua-
<
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drati quod fit ex ductu a. in h. quod latus eſt b. / igi
<
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tur inter a. et c. eſt mediū ꝓportionale quod eſt la-
<
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tus quadrati quod fit ex ductu a. in h. / et per conſe
<
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quens medium e. inter a. et c. eſt latus numeri d. q̇
<
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fit ex ductu a. in c. / quod fuit probandum. </
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>
<
s
xml:id
="
N12DA0
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xml:space
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">Et ſic pa
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tet correlarium.
<
note
position
="
right
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xlink:href
="
note-0030-04a
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xlink:label
="
note-0030-04
"
xml:id
="
N12E45
"
xml:space
="
preserve
">4. correĺ.</
note
>
</
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>
<
s
xml:id
="
N12DAA
"
xml:space
="
preserve
">¶ Sequitur quarto / conſtitutis
<
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duobus terminis ſe habentibus in aliqua ꝓpor-
<
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tione maioris inequalitatis rationali ſi numerus
<
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qui fit ex ductu vnius extremi in alterum ſit qua- </
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