DelMonte, Guidubaldo
,
In duos Archimedis aequeponderantium libros Paraphrasis : scholijs illustrata
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nea OX. dicti autem trapezii centrum gauitatis est etiam in li
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nea EF, quare trapezii ABCD centrum grauitatis est punctum
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P. At verò triangulum BCD ad ABD proportionem habet eam,
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OP ad P
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X. cùm ſint puncta OX triangulorum centla graui
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tatis, ac punctum P vtrorum〈que〉 commune centrum.
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Sed vt
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triangulum BDC adtriangulum ABD, ita eſt
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quo〈que〉 baſis
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BC
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ad
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baſim
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AD.
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cùm triangula eandem habeant altitudinem,
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ſiquidem ſunt in ijsdem parallelis AD BC. quare vt BC ad
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AD, ita OP ad PX.
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Sed
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quoniam anguli RPO SPX
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ver
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ticem ſunt ęquales, & angulus PRO ipſi PSX, veluti
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ROP angulo PXS eſt ęqualis, erit triangulum OPR triangu
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lo XPS ſimile; quare
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vt OP ad PX, ſic PR ad PS.
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eſt
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BC ad AD, vt OP ad PX
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; vt igitur BC ad AD, ita RP ad PS.
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& antecedentium dupla, duæ ſcilicet BC ad AD, vt duæ PR
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ad PS. & componendo duæ BC cum AD ad AD; vt
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PR cum PS ad PS. & ad conſe〈que〉ntium dupla, vt ſcilicet
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duæ BC cum AD ad duas AD, ita duæ PR cum PS ad duas
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PS. dictum eſt autem BC ad AD ita eſſe, vt PR ad PS. quare
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conuerrendo AD ad BC erit, vt PS ad PR. &
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dupla. </
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<
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">hoc eſt duæ AD ad BC, vt duæ PS ad PR. Ita〈que〉 in
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eadem ſunt proportione duç BC cum AD ad duas AD, vt
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duę PR
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PS ad duas PS. ſicut verò duę AD ad BC, ita duę
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PS ad PR. antecedentes igitur ad ſuas ſimul conſe〈que〉ntes
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eadem erunt proportione.
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Quare ſicut duæ BC cum AD ad duas
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AD cum BC, ita duæ RP cum PS ad duas P S cum PR,
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verùm duæ quidem RP cum PS eſt vtra〈que〉 ſimul SR RP.
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bis
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enim aſſumitur PR, ſemel verò PS. Cum autem lineæ DH ES
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à lineis diuidantur ęquidiſtantibus ED OT HM, erit DK
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KH, vt ER ad CS; kD verò eſt æqualis KH, erit ER ipſi
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RS ęqualis. </
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<
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">erit igitur ER cum RP,
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hoc est PE
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ipſis SR RP
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ęqualis.
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duæ verò PS cum PR eſt vtra〈que〉 PS SR.
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bis enim aſ
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ſumitur PS, ſemel què PR. & quoniam FS eſt ęqualis ipſi SR.
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quod quidem eodem modo oſtendetur, cùm ſit FS ad SR, vt
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BH ad Hk. </
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">erit FS cum SP,
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hoc est PF
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ipſis PS SR æqualis.
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Quare ita ſehabet PE ad PF, vt duæ BC cum AD ad duas
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AD cum BC. Centrum igitur grauitatis P trapezij ABCD
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in linea eſt EF, quæ
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parallelas AD BC bifariam di </
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