Alvarus, Thomas
,
Liber de triplici motu
,
1509
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Secunde partis
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lia demas etc. ſed modo manet proportio a. ad b.
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et c. ad .d. / ergo ille ſunt equales / quod fuit ꝓbãduꝫ
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</
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<
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xml:space
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">Sꝫ vniuerſaliter probatur / ſi ſicut ſe hꝫ a.b. ad b.
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ita .cd. ad d. / tūc ſic̈ ſe hꝫ a. ad b. ita c. ad .d. </
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<
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xml:space
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">Qḋ ſic
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probatur / qr. ſicut ſe hꝫ a.b. ad b. ita c.d. ad d. / ergo
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ſicut ſe habet a.b. ad c.d. ita b. ad .d. / vt patet ex cõ
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cluſione. </
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<
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xml:space
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">Uolo igit̄̄ / a.b. ꝑdat .b. et c.d. ꝑdat d. ita
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maneãt a. et c. / et tūc arguo ſic / a.b. et c.d. ſe habēt
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in ea proportione in qua ſe habent b. et d. q̄ ſit f.
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gr̄a argumenti: et a.b. terminus maior deperdit
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d. et c.d. terminꝰ minor deperdit d. / ergo inter de-
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perditum a maiori termino et deꝑditū a minori ē
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ꝓportio f. puta īter b. et d. et talis ꝓportio puta f.
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eſt īter a.b. et c.d. / vt ꝓbatū eſt: igit̄̄ facta tali deꝑ-
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ditione vel diminutione inter reſiduū ex a.b. et re-
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ſiduū ex c.d. manet ꝓportio f. / vt ptꝫ ex ſeptīo cor-
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relario quarte cõcluſionis octaui capitis huiꝰ par
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tis: et reſiduū ex a.b. ē a: et reſiduū ex c.d. eſt c. / igit̄̄
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īter a. et c. eſt f. ꝓportio ſicut inter .b. et d. et ꝑ ↄ̨ñs ſi
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cut ſe hꝫ a. ad c. ita b. ad d. puta in f. ꝓportione: et
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ex cõſequēti ſeq̇tur ex cõcluſione / ſicut ſe habet a
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ad b. ita c. ad d. / qḋ fuit probandū. </
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<
s
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xml:space
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">Et eodē mõ pro
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bares ſi a. eēt terminꝰ minor et b. maior. </
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<
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xml:space
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">et ēt c. mi
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nor et d. maior.
<
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xml:space
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">5. correĺ.</
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<
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xml:space
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">¶ Seq̇tur quīto / diſpoſitis ī hac
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medietate q̈tuor terminis: ſicut aggregatū ex q̈r-
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to et tertio ad tertiū ita aggregatum ex ſecūdo et
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prīo ad primū vt diſpoſitis his termīs .8.4.6.3. ſi
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cut ſe hñt 3. et .6. ad .6 ita .4. et .8. ad .8. </
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ſint 4 ṫmini ī hac medietate ↄ̨ſtituti a.b.c.d. / tūc
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ſicut ſe habet d.c. ad c. ita b.a. ad a. </
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<
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q2 bñ ſeq̇tur ſicut ſe habet a. ad b. ita c. ad d. / igitur
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ſicut ſe habet a.b. ad b. ita ſe habet c.d. ad d. / vt ptꝫ
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ex ṗmo correlario huiꝰ concluſionis: et vltra ſicut
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ſe habet a.b. ad b. ita c.d. ad d. / igit̄̄ ſicut ſe hꝫ d. ad
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d.c. ita b. ad b.a. / quod fuit ꝓbandū. </
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<
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quētia ex ꝓbatione tertii correlarii huius ↄ̨cluſio
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nis. </
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<
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xml:space
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">Et ſic patet correlariū.
<
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xml:id
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xml:space
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<
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xml:space
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">¶ Sequitur ſexto / diſ
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poſitis .3. terminis cõtinuo ꝓportionabilibus hac
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medietate: et aliis tribus etiã cõtinuo ꝓportiona-
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bilibꝰ eadē medietate: et eadē ꝓportione qua tres
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priores cõtinuo proportionant̄̄: ſicut ſe habēt ex-
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trema ṗmi ternarii: ita ſe habēt extrema ſecundi.
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</
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<
s
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">vt cõſtitutis .4. et .1.21.6.3. ſicut ſe habēt .4. ad .1. ita
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21. ad .3. </
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">Sint ſex termini a.b.c.d.e.f. et continuo
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ꝓportionētur tres primi termini ꝓportione g. / et
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eadē ꝓportione cõtinuo ꝓportionent̄̄ alii tres pu
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ta d.e.f. / et ſit ꝓportio cõpoſita adequate ex dupli
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ci g.h. / tūc dico / eadē eſt ꝓportio a. ad c. q̄ eſt d. ad
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f. </
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">q2 ꝓportio a. ad .c. eſt .h. et ea-
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dē eſt d. ad .f. / igit̄̄ eadē eſt proportio a. ad c. q̄ eſt d.
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ad f. / qḋ fuit ꝓbãdū. </
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batur maior: quia proportio a. ad c. cõponitur ex
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duplici g. proportione adeq̈te puta ex proportio-
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ne que eſt a. ad b. q̄ eſt g. et b. ad .c. q̄ etiã eſt g. / igitur
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illa proportio a. ad c. eſt h. </
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<
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proportio h. vt ponit̄̄ cõponitur ex duplici g. ade-
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quate. </
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<
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">Et iſto mõ probabis minorē: qm̄ proportio
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d. ad f. componitur ex duplici g. puta ex proportio
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ne g. q̄ eſt d. ad e. et ex proportione g. que eſt e. ad f.
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adequate. </
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<
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xml:space
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<
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xml:space
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">Et pari demon-
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ſtratione oſtendes: conſtitutis tribus quaterna
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riis continuo proportionabilibus eadem propor
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tione: et quin quinariis: et in quo volueris nūe-
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ro: in quacun proportione ſe habent extrema
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vniꝰ in eadē ſe habent extrema cuiuſuis alterius.</
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medieta
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tis geo-
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metrice.</
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<
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medietate geometrica terminis conſtitutis conti
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nuo proportionabilibus:: qualis eſt illoruꝫ termi
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norū cõtinuo ꝓportio: talis eſt inter eorū differen
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Capitulum tertiū.
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tias ſiue exceſſns. </
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<
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">vt conſtitutis his terminis .16.8
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4.2.1. qualis eſt ꝓportio .6. ad .8. talis eſt exceſſus
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quo .16. excedunt .8. ad exceſſum quo .8. excedūt .4.
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et exceſſus quo .4. excedunt .2. ad exceſſum quo duo
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excedunt vnum / vt patet. </
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ſus proportio dupla quēadmodū īter terīos </
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<
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bat̄̄ / ſint .3. ṫmini cõtinuo ꝓportionabiles .f. ꝓpor-
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tione puta .ab.cd.e. et exceſſus quo primus excedit
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ſecunduꝫ ſit a: et exceſſus quo ſecundus excedit ter
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tium ſit c. / tūc dico / ſicut f. ꝓportio eſt inter illos
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terminos: vcꝫ īter primum et ſecundum et inter ſe-
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cundum et tertium. </
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<
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et c. exceſſus ita a. ad c. eſt proportio f. </
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<
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oſtendit̄̄ / q2 b. ad d. eſt ꝓportio f. et a. ad .c. eſt eadeꝫ
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proportio / igitur a. ad c. eſt f. proportio / quod fuit
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ꝓbandū. </
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<
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">Probatur maior / quia b. eſt equale c.d.
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q2 a.b: excedebat preciſe per a. ipſum .cd. et ſic re-
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moto exceſſu .b. manebit equale c.d. et d. eſt equale
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e. eadem rõne: et inter .cd. et e. eſt f. proportio / vt po
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nitur: ergo inter b: et d. eſt eadem f. proportio </
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tet conſequentia / q2 oīm equaliū eſt eadem ꝓpor-
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tio </
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<
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a. habeat ꝓportionē f. qui ſit g. / et arguo ſic / ſicut ſe
<
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habet b. ad d. ita ſe habet a. ad g. puta in f. propor
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tione: ergo ſicut ſe habet b. ad d: puta in f. ꝓportio
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ne ita ſe habet a.b. ad g.d. puta in f. proportione.
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</
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<
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xml:space
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">Patet hec conſequentia ex ſecundo correlario q̈r
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te concluſionis: et .ab. etiam ad .cd. eſt proportio
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f. / vt ponitur igitur g.d. et c.d. ſunt equalia. </
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<
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conſequentia / quia idem tertium eandē ꝓportiõeꝫ
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hꝫ ad vtrum illoꝝ: et vltra .gd. et c.d. ſūt eq̈lia: g̊
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eodē cõi dēpto puta d. r̄ſidua manebūt eq̈lia / ſꝫ re
<
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ſidua ſunt g. et c. / g̊ g. et c. ſunt eq̈lia et a. ad g. eſt f.
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proportio / vt poſitū eſt / ergo a. ad c. eſt f. proportio /
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quod fuit ꝓbandum </
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<
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">Patet hec conſequētia / quia
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eiuſdē tertii ad vtrū duorū equaliū eſt eadem ꝓ
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portio. </
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<
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<
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">Qm̄ eo modo quo pro
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batū eſt in illis tribus terminis probabitur quot
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cnn diſpoſitis cõtinuo proportionabilibus hac
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medietate. </
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<
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xml:space
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">Et hec ſit quinta proprietas medieta-
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tis geometrice.
<
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xml:id
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xml:space
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">1. correĺ.</
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>
</
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<
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xml:space
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">¶ Ex hac concluſione ſequitur pri-
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mo / ſi duo numeri inequales continuo diminuã
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tur continuo in eadem ꝓportione manentes: con-
<
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tinno deperditū maiori numero ſe habet in eadeꝫ
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ꝓportione ad deperditū minori numero in qua cõ
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tinuo ſe habent illi numeri qui diminuuntur. </
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<
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">vt ſi
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numerus octonarius et quaternarius continuo di
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minuantur continuo manētes in ꝓportiõe dnpla:
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continuo deperditum ab octonario ſe habebit in
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ꝓportione dupla ad deperditum a quaternario.
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<
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">Hoc correlarium facile ex demonſtratione conclu
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ſionis probatur.
<
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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:space
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">¶ Sequit̄̄ ſecundo / ſi nõ conti-
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nuo deperditum maiori numero ſe habeat ad de
<
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perditum a minori numero in eadem proportiõe:
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in qua continuo ſe habent illi numeri q̇ diminuun
<
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tur: illi duo numeri inequales q̇ cõtinuo diminuū
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tur non ſe habent in eadem ꝓportione etc. </
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<
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hoc correlarium ex priori / qm̄ p̄cedens correlariū
<
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eſt vna conditionalis a: igitur ex oppoſito ↄ̨ñtis
<
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eius ſequit̄̄ oppoſitum añcedentis: et ꝑ conſequēs
<
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conditionalis in qua arguitur ex oppoſito conſe
<
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quentis illius ad oppoſitum añtis eſt vera: et ta-
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lis eſt correlarium / igitur correlarium verum.</
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xml:space
="
preserve
">3. correĺ.</
note
>
<
p
xml:id
="
N12BEF
">
<
s
xml:id
="
N12BF0
"
xml:space
="
preserve
">¶ Sequitur tertio / ſi continuo deperdita a duo
<
lb
/>
bus numeris inequalibus manent in eadem pro-
<
lb
/>
portione in qua ſe habent illi numeri in principio
<
lb
/>
deperditionis: numeri remanētes cõtinuo manēt
<
lb
/>
in eadem ꝓportione. </
s
>
<
s
xml:id
="
N12BFB
"
xml:space
="
preserve
">vt ſi numerus duodenarius
<
lb
/>
et ſenarius diminuantur: et continuo deperdituꝫ </
s
>
</
p
>
</
div
>
</
div
>
</
div
>
</
text
>
</
echo
>