Cavalieri, Buonaventura
,
Geometria indivisibilibvs continvorvm : noua quadam ratione promota
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LIBER I.
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AP, KX, quidem ipſis, VG, Λ Υ, &</
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<
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xml:space
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& </
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<
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xml:space
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">Y, perpendiculares, iunganturque, PE, XT, PF, XZ, &</
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<
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xml:space
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E, ZT. </
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<
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xml:space
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">Quoniam ergo, APG, eſt angulus rectus, erit quadra-
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Elem.</
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tum, AG, æquale quadratis, GP, PA, quadratum verò, PA,
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æquatur duobus quadratis, PE, EA, propter angulum rectum, A
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xml:space
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">Defin. 3.
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Vndec.
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Elem.</
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EP, ergo quadratum, AG, hoc eſt duo quadrata, GE, EA, ęqua-
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buntur tribus quadratis, GP, PE, EA, & </
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<
s
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">ablato communi qua-
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drato, EA, quadratum, GE, æquabitur quadratis, GP, PE, er-
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go, EP, erit perpendicularis ipſi, PV, cui etiam eſt perpendicula-
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<
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">48. Primi
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ris, AP, ergo, APE, erit inclinatio planorum, AV, VH. </
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dem modo oſtendemus, KXT, eſſe inclinationem planorum, Κ Λ,
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xml:space
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">Defin. 6.
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Vndec.
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Λ &</
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<
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">angu os, EFG, TZY, eſſe rectos. </
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<
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">Quoniam verò angu-
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lus, AGV, æquatur ipſi, Κ Υ Λ, (ſunt. </
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<
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">figuræ, AV, Κ Λ, ſimi-
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les ex hypoteſi) etiam, AGP, æquabitur, KYX, &</
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<
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Y, recti ſunt, ergo triangula, APG, KXY, ſimil a erunt. </
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<
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">Eodem
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modo probabimus etiam triangula, AGF, KYZ, eſſe ſimilia, er-
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go, PG, ad, GA, erit vt, XY, ad, YK, &</
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<
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xml:space
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">, GA, ad, GF, vt, Y
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K, ad, YZ, ergo ex æqual@, PG, ad, GF, erit vt, XY, ad, YZ,
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& </
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<
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">ſunt latera proportionalia circa æquales augulos, PGF, XYZ,
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(ſunt.</
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<
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<
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">æquales ijs, qui ſunt ad verticem, nempè, HGV, & </
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<
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">Υ Λ,
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qui adęquantur, cum ſint ſimilium figurarum, HGV, & </
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<
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">Υ Λ,) er-
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Elem.</
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go triangula, PGF, XYZ, erunt ſimilia, & </
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<
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vt &</
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<
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">, GFP, YZX, inter ſe æquales, ergo ipſi, FPE, ZXT; </
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FE, ZXT, inter ſe quoque erunt æquales, cum ſint reſiduirectc-
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rum, GPE, GFE, YXT, YZT; </
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<
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pariter ſimilia erunt. </
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<
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">Erit ergo, AP, ad, PG, vt, KX, ad, XY;
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</
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Elem.</
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PG, ad, PF, vt, XY, ad, XZ; </
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<
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">&</
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<
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">, PF, ad, PE, vt, XZ, ad, X
<
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T, ergo ex ęquali, AP, ad, PE, erit vt, KX, ad, XT, & </
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<
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guli, AEP, KTX, rect@, ergo triangula, APF, KXT, ſitnilia
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erunt, & </
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">angu@i, APE, KXT, ęqual@s, qu@@unt inclinationes pla-
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norum, AV, Κ Λ, ad plana, VH, Λ &</
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oſſendendum erat.</
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<
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ſimiles quaſcumque rectihneas ſiguras, AV, Κ Λ, interſe, nec-
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S@xti El.</
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non, HV, & </
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munibus, GV, Υ Λ, ſint autem homologæ inter ſe, AG, KY; </
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G, & </
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<
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">ipſæ figuræ æquè ad eandem partem inuicem inclinatæ.
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<
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<
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pr@portionalia, quod etiam de angulis, DVN, Q Λ ℟, pariter ve-
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rum eſſe oſtendemus.</
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