Biancani, Giuseppe
,
Aristotelis loca mathematica
,
1615
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121
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probare P, eſſe polum prædicti ambitus, ſic. </
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<
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id
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s.002106
">Primò enim ſciendum in præ
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miſſa conſtructione eſſe, vt F, ad G K, & B, ad K P, ita D, ad P M. nam ſi non
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ſit eadem ratio D, ad P M, cum alijs prædictis, erit eadem ratio eiuſdem D,
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ad aliam maiorem, vel minorem ipſa P M. ſit ad minorem P R. nihil enim
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refert ſiue dixeris habere eandem rationem ad minorem, ſiue ad maiorem,
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ergo permutando erunt G K, K P, P R, proportionales cum F, B, D. ſed li
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neæ F, B, D, erant proportionales
<
expan
abbr
="
componẽdo
">componendo</
expan
>
hoc modo, vt F B, ad D, ita
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D, ad B: quare ſimiliter erunt vt G P, ad P R, ita P R, ad P K. per 18. 5. ſi igi
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tur à punctis G, & K, figuræ nu. </
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>
<
s
id
="
s.002107
">164.
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expan
abbr
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iungãtur
">iungantur</
expan
>
lineæ ad R, quæ ſint G R, K R,
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erit vt G R, ad K R, ita G P, ad P R. quia orta
<
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abbr
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sũt
">sunt</
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duo
<
expan
abbr
="
triãgula
">triangula</
expan
>
G P R, K P R,
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quæ habent eundem angulum ad P. & latera proportionalia circa dictum
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angulum. </
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<
s
id
="
s.002108
">eſt etiam vt G P, ad P R, in maiori triangulo, ita P R, ad K P, in
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minori, ex conſtructione, quare per 6. 6. erunt illa duo triangula æquian
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gula; ergò per 4. 6. erunt latera circum æquales angulos proportionalia;
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quare erit vt G P, ad P R. ita G R, ad R K: erat autem vt K M, ad G M, ita
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B, ad D. & ita etiam G P, ad P R; ergò per 11. 5. vt K M, ad M G. ita K R,
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ad R G, intra eandem circunferentiam, & in eodem plano: quod eſſe im
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poſſibile ſupra oſtendimus, hoc autem impoſſibile, ſequitur ſi neges eſſe vt
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F, ad G K; & B, ad K P, ita D, ad P M.</
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167</
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<
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">Ibidem
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(Quoniăm igitur quæ D,
<
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abbr
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neq;
">neque</
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>
ad minorem ea, quæ P M,
<
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abbr
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neq;
">neque</
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>
ad maiorem
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(ſimiliter enim demonſtrabimus) palam eſt, quod ad ipſam
<
expan
abbr
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vtiq;
">vtique</
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>
erit, in qua P M,
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quare erit, quod quæ M P, ad P K, quæ P G, ad M P. </
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<
s
id
="
s.002112
">Si igitur eo in quo P, polo
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vtens, diſtantia autem ea, in qua P M, circulus deſcribatur, omnes angulos attin
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get, quos reflexæ faciunt, quæ à K, G. ſi autem non, ſimiliter oſtendentur eandem
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babere rationem, quæ alibi, quam in ſemicirculo conſtituuntur; quod quidem erat
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impoſſibile)
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type
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italics
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quoniam igitur, inquit, linea D,
<
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abbr
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neq;
">neque</
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ad minorem,
<
expan
abbr
="
neq;
">neque</
expan
>
ad ma
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iorem quam P M, habet eam rationem, quæ eſt ipſius F, ad G K, aut ipſius
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B, ad K P. ſimiliter enim demonſtratur abſurdum ſequi. </
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<
s
id
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s.002113
">palàm eſt, quoniam
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erit D, ad P M, vt prædictæ ad prædictas: quare componendo, & permu
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tando, erunt tandem vt G P, ad P M, ita P M, ad P K, & ita G M, ad M K,
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aſſumpſimus enim in conſtructione eſſe G M, ad M K, ita F B, ad D, & D, ad
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B. quare cum ſit vt G M, ad M K, ita F B, ad D. & G P, ad P M. & P M, ad
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K P; erunt per 11. 5. vt G M, ad M K. ita G P, ad P M. & P M, ad P K. ſi quis
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igitur vtens puncto P, tanquam polo, & interuallo P M, circulum deſcribat,
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omnes angulos reflexionis attinget, quos faciunt lineæ productæ à K, & re
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flexæ ab M, ad G. harum enim infinitam multitudinem debemus imaginari
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à K, ad infinita puncta M, produci in ambitu illo conſtituta,
<
expan
abbr
="
reſlectiq́
">reflectique</
expan
>
; ad G.
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ſi enim non attingat omnes illos angulos, ſequitur, vt ſupra, in eodem ſemi
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circulo
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abbr
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cõſtitui
">conſtitui</
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>
poſſe duas alias rectas proportionales prioribus G M, M K,
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quod eſt impoſſibile. </
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<
s
id
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s.002114
">Porrò ſub angulo G M K, linearum G M, M K, Iris
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apparet: quare apparebit etiam ſub alijs omnibus, quæ à punctis G K, duci
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poſſunt ad extremum lineæ P M, quia erunt in eadem ratione cum illis; cum
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non deſinant in eundem
<
expan
abbr
="
ſemicirculũ
">ſemicirculum</
expan
>
, ſed in ambitum Iridis M N, in quo M,
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punctum imaginamur circumduci. </
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<
s
id
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s.002115
">Ex quibus pater P, eſſe polum Iridis, ex
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quo per puncta M, vbi ſit reflexio, deſcribitur arcus attingens omnes Iridis
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reflexiones.</
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