Benedetti, Giovanni Battista de
,
Io. Baptistae Benedicti ... Diversarvm specvlationvm mathematicarum, et physicarum liber : quarum seriem sequens pagina indicabit ; [annotated and critiqued by Guidobaldo Del Monte]
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THEOREM. ARIT.
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0119
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tio erit eius differentiæ, quæ eſt inter primam & fecundam ſummam, ad differen-
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tiam quæ eſt inter primas earum partes, quæ illius differentiæ, quæ eſt inter ſecun-
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dam & tertiam ſummam, ad differentiam, quæ eft inter primas illarum partes, ſed
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harum .4. differentiarum, tres nobis cognitæ ſunt, ideft .12. 2. et .9. ergo ex regula de
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tribus ab Eucli. in .20. ſeptì
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mi ſpeculata inueniebatur quarta differentia, quæ eft .1.
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cum dimidio.</
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<
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xml:space
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">A compofitis ſummis idem etiam proueniet, ſed non vt ex proprijs caufis, & per
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ſe, ſedper accidens. </
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<
s
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xml:space
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">Nam quamuis eadem differentia fit inter 71. et .59. quæ in-
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ter .60. et .48. &
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inter .59. et .50. quæ inter .48. et .39. </
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xml:space
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proportio (propriè) ipſius .71. ad .59. quæ ipſius .60. ad .48. nec ea quæ ipſius .59. ad
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50
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50.</
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eft quæ ipſius .48. ad .39: </
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<
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xml:space
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">Vnde non erit eadem proportio ipſius .71. ad .59. quæ
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ipfius .10. ad .8. ne@ea quæ eft ipfius .59. ad .50. quæ ipſius .8. ad .6. cum dimidio. </
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minores illis. </
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<
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xml:space
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">Nam ex æqualibus additamentis diminuuntur proportiones maio-
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ris inęqualitatis.</
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<
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<
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xml:space
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">A fimplicibus igitur ſummis pendet ratio huiuſmodi effectus.</
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<
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<
s
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xml:space
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">Si vero prima pars fecundæ poſitionis effet .4. tunc ſecunda eius pars effet .8. & ter-
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tia .12. quarum ſumma effet .24. (harum fimplicium partium ſeilicet) & minor vera
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(39.) per .15. & differens à ſumma primarum. (60.) per .36. & differentia primarum
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partium effet .6. differentia vero primæpartis ſecundæ poſitionis, a prima parte quę
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fita effet .2. cum dimidio. </
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<
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xml:space
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">Vnde in huiuſmodi exemplo videre eft quare colligan-
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tur errores inuicem, quando alter eorum eccedit, reliquus vero deficit à numero pro
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pofito. </
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<
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xml:space
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">Quod quidem ob aliam caufam non fit, nifi vt cognoſcatur differentia .36.
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differentia ſcilicet ſimplicium ſummarum ipſarum poſitionum.</
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<
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xml:space
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">Secundus autem modus ab antiquis magis exercitatus eſt, quod multiplicabant
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diametraliter errores cum primis partibus, hoc eſt primum errorem cum prima par
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te, hoc eſt cum numero ſecundæ poſitionis, ſecundum vero errorem cum prima
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parte, hoc eſt cum numero primæ poſitionis, differentiam poſteà vel aggregatum
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horum duorum productorum diuidebant per differentiam vel aggregatum dicto-
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rum errorum, proueniens poſteà erat prima pars quæſita numeri propoſiti. </
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de oriebantur tria producta, quorum
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, hoc eſt differentia, ſeu aggregatum il-
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lorum conſtituebatur ex differentia feuaggregato errorum, & ex numero quæ-
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fito.</
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<
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">Vtin præfenti exemplo, primus error eſt .21. qui multiplicatus cum prima par-
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te ſecundæ poſitionis, quæ eſt .8. producit .168.
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verò error eſt .9. qui multi-
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plicatus cum prima parte primę poſitionis producit .90. differentia autem horum
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productorum eſt .78. quæ diuifa per differentiam errorum, quæ eſt 12. dabit .6.
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di
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midio, pro prima parte quæſita dati numeri diuiſibilis, qui erat .50.</
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">Sed, vt ſupra dixi diuiſor non eft per ſe differentia
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errorum, neque etiam differentia per ſe ſummarum compoſitarum, fed bene fim-
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plicium.</
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<
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xml:space
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">Pro cuius rei ſpeculatione, accipiendæ ſunt ſummæ ſimplices, quarum differen-
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tiæ per ſe vtiles ſunt in huiuſmodi operatione; </
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<
s
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xml:space
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">& quia etiam rationes veritatis ex
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iſtis, & non ex illis fluunt; </
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<
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xml:space
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">quamuis tam vnæ, quam aliæ ſint eædem in quantitate,
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ideſt æquales.</
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