Alvarus, Thomas
,
Liber de triplici motu
,
1509
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type-free
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capitulum
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<
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Prime partis
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file
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0035
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n
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35
"/>
duplã: manent igitur in eadē ꝓportione / vt ptꝫ ex
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correlario decime ſuppoſitiõis ſecūdi capitꝪ huiꝰ
<
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īueniatur / igitur mediū inter illos duos numeros
<
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et īueniētur due ꝓportiones tnequales in quas di
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uiditur ꝓportio inter illos duos numeros / vt pre-
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oſtenſum eſt. </
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>
<
s
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N13560
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xml:space
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preserve
">Patet igitur vniuerſaliter concluſio
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xml:id
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xml:space
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preserve
">Primuꝫ
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correlari
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um.</
note
>
</
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>
<
s
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N1356A
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xml:space
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preserve
">¶ Ex qua ſequitur primo / quelibet proportio in
<
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infinitas ꝓportiones ſecari valet in numeris ſine
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vnitatis fractione: et capio ly infinitas ſyncathe-
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goreumatice. </
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>
<
s
xml:id
="
N13573
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xml:space
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preserve
">Probatur / qm̄ capta ꝓportione a.
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in numeris manifeſtū eſt / illi numeri ſaltē ꝑ vni-
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/>
tatē diſtabūt / hoc eſt ſaltē maior excedit minorē ꝑ
<
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/>
vnitatē que vnitas eſt pars aliquota minoris: du
<
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/>
pletur igitur vter illoꝝ numeroꝝ: et ſequitur /
<
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/>
adhuc inter illos numeros duplatos manet ꝓpor
<
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/>
tio a. / vt paulo ãte deductū eſt: igitur iam exceſſus
<
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/>
erit in duplo maior: q2 erit pars aliquota eiuſdē
<
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/>
denomīationis numeri in duplo maioris: igitur
<
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/>
iam ibi inter illos duos numeros reperietur vnꝰ
<
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/>
numerus medius vt ſuperiꝰ oſtenſum eſt: et ꝑ cõſe-
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quens due ꝓportiones inequales in quas diuidit̄̄
<
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talis ꝓportio. </
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>
<
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xml:id
="
N1358E
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xml:space
="
preserve
">Iteꝝ duplent̄̄ illi numeri īter quos
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eſt ꝓportio a. et iam inter eos īuenientur tres nu-
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meri intermedii et ſic erūt quatuor ꝓportiões in-
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termedie. </
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>
<
s
xml:id
="
N13597
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xml:space
="
preserve
">Et ſi tertio duplentur illi numeri īueni-
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entur ſeptē numeri intermedii: et ſic erūt .8. ꝓpor-
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tiones: et ſic in infinitū duplando ſemꝑ numeros.
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/>
</
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>
<
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N1359F
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xml:space
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preserve
">Data igit̄̄ quã volueris ꝓportione ipſa vel ſibi e-
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qualis (quod ꝓ eodē reputo) in infinitas ꝓportio-
<
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nes ſecari valet: quod fuit oſtendendū. </
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>
<
s
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="
N135A6
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xml:space
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preserve
">Et ſicut ꝓ-
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batur in numeris: ita et facilius ꝓbabitur in quã-
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titatibus. </
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>
<
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N135AD
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xml:space
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preserve
">Et ſicut ꝓbatur capiēdo primos nume-
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ros excedentes ſe vnitate: ita per locū a maiori ꝓ-
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babitur capiendo numeros excedētes ſe numero:
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vt ſatis conſtat. </
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>
<
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xml:space
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preserve
">Patet igit̄̄ correlariū.
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xlink:href
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xml:id
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xml:space
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">Secūduꝫ
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correlar̄.</
note
>
</
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>
<
s
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N135BE
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xml:space
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preserve
">¶ Sequit̄̄
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ſecūdo / capitis tribꝰ terminis cõtinuo ꝓportio-
<
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nabilibus arithmetice: et captis aliis tribus ſic ſe
<
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habentibꝰ / qualis eſt ꝓportio inter duos maio-
<
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res primi ternarii: talis ſit inter duos maiores ſe
<
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/>
cūdi ternarii: et qualis inter duos numeros primi
<
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ternarii: talis etiã ſit inter duos minores ſecundi
<
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/>
ternarii: tūc termini ſecūdi ternarii ſunt ꝓportio-
<
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nabiles arithmetice: ſicut et termini ṗmi ternarii:
<
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vt captis his tribus terminis .4.3.2. qui ſunt pro-
<
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portiõabiles arithmetice: dico / iſti .3. termini .8.
<
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6.4. ſunt etiã arithmetice proportionabiles: qm̄
<
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qualis eſt ꝓportio inter .4. et .3. talis eſt inter .8. et
<
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6. et qualis inter .3. et .2. talis inter .6. et .4. / vt patꝫ
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</
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<
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xml:space
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">Probatur / ſint tres termini a.b.c. ꝓportiõabiles
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arithmetice: et ſint alii trrs d.e.f. et ſit inter d. et e.
<
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talis ꝓportio qualis inter a. et b. et inter e. et f. q̈lis
<
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inter b. et c. </
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>
<
s
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="
N135E5
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xml:space
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">Et tunc dico / d.e.f. ſunt tres termini
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ꝓportionabiles arithmetice: </
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<
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N135EA
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">Ad quod probandū
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volo / exceſſus quo a. excedit b. ſit g. et quo b. exce
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dit c. ſit h. equalis g. / vt oportet: et exceſſus q̊ d. exce
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dit e. ſit i. et quo e. excedit f. ſit k. / et manifeſtū eſt / g.
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eſt tota pars aliquota ipſiꝰ b. vel tote partes q̊ta
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vel quote i. eſt ipſiꝰ e. et eiuſdē denominationis: et
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h. eſt tota pars vel tote partes aliquote et eiuſdeꝫ
<
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denomīationis reſpectu c. ſicut k. reſpectu f. / vt ptꝫ
<
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ex probatione quarte ſuppoſitionis ſecūdi capi-
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tis huiꝰ. </
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>
<
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N135FF
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">Quo ſuppoſito arguit̄̄ ſic / i. quod eſt ex-
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ceſſus inter d. et e. eſt equale ipſi k. / quod eſt exceſſus
<
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inter e. et f. / igit̄̄ illi tres termini d.e.f. ſunt ꝓporti-
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onabiles arithmetice. </
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>
<
s
xml:id
="
N13608
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xml:space
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preserve
">Cõſequentia ptꝫ manifeſte:
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et arguit̄̄ antecedens: q2 ſicut ſe habet b. ad .c. ita e.
<
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ad f. / igit̄̄ ſicut ſe habet b. ad e. ita c. ad f. </
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<
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xml:space
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">Patet cõ-
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ſequentia ex ſecūda cõcluſione tertii capitis huiꝰ:
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et ex ↄ̨ſequenti ſicut ſe habet b. ad e. ita c. ad f. puta
<
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="
Capitulū quartū.
"/>
in l. ꝓportione / igitur g. ſe habet ad i. in l. ꝓporti-
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one et h. ad k. etiã in l. ꝓportione. </
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<
s
xml:id
="
N1361B
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xml:space
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">Patet cõſequen
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tia ex vndecima ſuppoſitione ſecūdi capitis huiꝰ:
<
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ille em̄ ſunt partes aliquote eiuſdē denoīationis
<
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numeroꝝ ſe habentiū in l. ꝓportione: et vltra g. ſe
<
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habet ad i. in l. ꝓportiõe: et h. ad k. etiã in l. pro-
<
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portione: igit̄̄ ſicut ſe habet g. ad h. ita i. ad k. </
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>
<
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="
N13628
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">Ptꝫ
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per locū a. ꝑmutata proportione: ſed g. et h. ſe ha-
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bent in proportione equalitatis: igit̄̄ i. et k. / qḋ fuit
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probandñ. </
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<
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N13631
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xml:space
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">Probatur aliter correlariū tam in nu
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meris quã in quãtitatibus cõtinuis: et retēta eadē
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hypotheſi: manifeſtū eſt / ipſiꝰ a. ad d. et ipſiꝰ b.
<
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ad c. et ipſius c. ad f. eſt eadē ꝓportio: que ſit l. / qm̄
<
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ex hypotheſi ſicut ſe habet a. ad b. ita ſe habet d.
<
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ad e. / ergo per locū a. permutata proportiõe ſicut
<
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ſe habet a. ad d. ita b. ad e. et vltra ſicut ſe habet b
<
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ad c. ita e. ad f. ex hypotheſi: ergo ꝑmutatim: ſicut
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ſe habet b. ad e. ita c. ad f. et a. ad d. eſt etiã ꝓportio
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illa que eſt b. ad c. / igit̄̄ eadē proportio eſt a. ad d. et
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b. ad e. et c. ad f. puta l. </
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>
<
s
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="
N13648
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xml:space
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">Quo ſuppoſito: probatur
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correlariū: q2 i. et k. ſūt equales: igit̄̄ .d.e.f. ſunt ter
<
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mini cõtinuo proportionabiles arithmetice. </
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>
<
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="
N1364F
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xml:space
="
preserve
">Ptꝫ
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cõſequentia ex hypotheſi: iūcta diffinitione ꝓpor
<
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tionalitatis arithmetice. </
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>
<
s
xml:id
="
N13656
"
xml:space
="
preserve
">Probat̄̄ antecedens: q2
<
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ſicut ſe habet g. ad h. ita ſe habet i. ad k. ſed g et h.
<
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/>
ſe habent in proportiõe equalitatis / vt ptꝫ ex hy-
<
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potheſi: igit̄̄ i. et k. ſe habent in proportione equa-
<
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litatis: et ſic ſunt equalia igit̄̄. </
s
>
<
s
xml:id
="
N13661
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xml:space
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preserve
">Probat̄̄ antecedēs /
<
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q2 ſicut ſe habet g. ad i. ita h. ad k. / ergo ꝑmutatim
<
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/>
ſicut ſe habet g. ad h. ita i. ad k. / qḋ fuit probandū.
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</
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>
<
s
xml:id
="
N13669
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xml:space
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preserve
">Probatur antecedens: q2 g. ſe habet ad i. in l. ꝓ-
<
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portione: et h. ſe habet ad k. in eadē l. proportione /
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igitur intentū. </
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>
<
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="
N13670
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xml:space
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">Probat̄̄ maior / q2 g. ſe hꝫ ad i. ſicut
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a. ſe hꝫ ad d. / igitur ſe hꝫ in l. ꝓportione. </
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>
<
s
xml:id
="
N13675
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xml:space
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preserve
">Patꝫ ↄ̨ña
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ex hypotheſi. </
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>
<
s
xml:id
="
N1367A
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xml:space
="
preserve
">Probat̄̄ antecedēs: et volo / a. dimi
<
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nuatur ad equalitatē b. ꝑdendo g. differentiã per
<
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/>
quã excedit ipſum b. ex hypotheſi: et d. diminuatur
<
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ad equalitatē c. ꝑdendo i. differentiã ꝑ quã excedit
<
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e. ex hypotheſi: et manifeſtū eſt / reſidui ex ipſo a. /
<
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qḋ eſt b. ad reſiduū ex ipſo d. / qḋ eſt e. adhuc eſt l. ꝓ
<
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portio: vt ptꝫ ex hypotheſi: g̊ inṫ deꝑditū ab ip̄o a
<
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et deꝑditū ab ip̄o d. eſt etiã l. ꝓportio: et deꝑditū ab
<
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ip̄o a eſt g. et deꝑditū ab ipſo d. eſt i. / g̊ g. ſe hꝫ ad i.
<
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ſicut a. ad d. puta in l. ꝓportione. </
s
>
<
s
xml:id
="
N1368F
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xml:space
="
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">Ptꝫ tamen ↄ̨ña
<
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ex primo correlario quinte cõcluſionis ſecūdi ca-
<
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pitis huiꝰ partis. </
s
>
<
s
xml:id
="
N13696
"
xml:space
="
preserve
">Et ſic ptꝫ maior. </
s
>
<
s
xml:id
="
N13699
"
xml:space
="
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">Iam ꝓbo mi-
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norē / q2 h. ſe hꝫ ad k. ſicut b. ſi ſe hꝫ ad e. / igr̄ ꝓpoſitū
<
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/>
</
s
>
<
s
xml:id
="
N1369F
"
xml:space
="
preserve
">Probat̄̄ ãtecedēs: et volo / b. diminuat̄̄ ad equa-
<
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litatē c: perdendo h. differentiã: et e. diminuat̄̄ ad
<
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/>
equalitatē f. perdendo k. differentiã: et manifeſtuꝫ
<
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/>
eſt / reſidui ex ipſo b. / qḋ eſt c. ad reſiduū ex ipſo e.
<
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/>
qḋ eſt f. eſt adhuc l. ꝓportio: vt patet ex hypotheſi:
<
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/>
igitur inter h. deperditū a b. termino maiori, et
<
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k. deꝑditū ab c. ṫmīo minori eſt ēt l ꝓportio: vt ſu-
<
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pra argutū eſt / igr̄ h. ſe hꝫ ad k. ſicut b. ad e. puta in
<
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l. ꝓportione: qḋ fuit probandū. </
s
>
<
s
xml:id
="
N136B2
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xml:space
="
preserve
">Et ſic ptꝫ correla-
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riū.
<
note
position
="
right
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xlink:href
="
note-0035-03a
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xlink:label
="
note-0035-03
"
xml:id
="
N13713
"
xml:space
="
preserve
">Calcu. de
<
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īduc. gra
<
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dus ſūmi</
note
>
</
s
>
<
s
xml:id
="
N136BC
"
xml:space
="
preserve
">Et hec ē ſuppoſitio quã calculator ponit ī ca-
<
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/>
pitulo de inductione gradus ſummi circa princi-
<
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piū ſub iſta forma. </
s
>
<
s
xml:id
="
N136C3
"
xml:space
="
preserve
">Si ſint tria cõtinuo ꝓportio-
<
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nabilia ꝓportione arithmetica: et ſint alia tria cõ
<
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ſimiliter ꝓportionabilia proportiõe geometrica
<
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ſicut prima tria: illa etiã ſunt cãtinuo ꝓportiõabi
<
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/>
lia proportiõe arithmetica.
<
note
position
="
right
"
xlink:href
="
note-0035-04a
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xlink:label
="
note-0035-04
"
xml:id
="
N1371D
"
xml:space
="
preserve
">Tertium
<
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correlar̄.</
note
>
</
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>
<
s
xml:id
="
N136D3
"
xml:space
="
preserve
">¶ Sequit̄̄ ex hoc ter-
<
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tio / ſi ſint tres termini arithmetice proportiõa-
<
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/>
biles: et quilibet illoꝝ dupletur, aut tripletur, aut
<
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/>
ſexquialteretur .etc̈. ſemꝑ ꝓportio extremi ad ex-
<
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/>
tremū manet equalis: et cõtinuo manebūt illi tres
<
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/>
termini arithmetice ꝓportiõabiles: et in ea ꝓpor-
<
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/>
tiõe in qua termini augmētant̄̄ exceſſus augmētat̄̄
<
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/>
</
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>
<
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xml:id
="
N136E3
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xml:space
="
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"/>
</
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