Archimedes
,
Archimedis De iis qvae vehvntvr in aqva libri dvo
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DE IIS QVAE VEH. IN AQVA.
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DE IIS QVAE VEH. IN AQVA.
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ad ſectionem e f g ex parte e linea l m, eidem a c baſi æquidi-
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stans. </
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à ſectione ducuntur: </
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<
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<
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">ſectionis e f c ſit ipſa f o. </
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<
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tur triangula c d b, c f g ſimilia ſunt, erit ut b c ad c f, ita d c
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ad c g; </
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<
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<
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<
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">rurſus quoniam triangula c k b, c l f etiã
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inter ſe ſunt ſimilia, ut b c ad c f, boc eſt ut b d ad f g, ita erit k c
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ad c l; </
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<
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<
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">b K ad f l. </
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<
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">quare K c ad c l, & </
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<
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ad c g: </
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<
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">hoc eſt ut earum duplæ a c ad c e. </
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<
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">ſed ut b d ad f g, ita d c
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ad c g; </
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<
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">hoc ẽ a d ad e g: </
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<
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">& </
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<
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">permutãdo ut b d ad a d, ita f g ad e g.
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</
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<
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">quadratum autem a d æquale eſt rectangulo d b n ex undecima pri
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mi conicorum. </
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<
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">ergo tres lineæ b d, a d, b n inter ſe ſunt proportio
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nales. </
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<
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">eadem quoque ratione cum quadratum e g æquale ſit rectan
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gulo g f o, tres aliæ lineæ f g, e g, f o, deinceps proportionales
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erũt. </
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<
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<
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">ut b d ad, a d, ita f g ad e g. </
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<
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">quare ut a d ad b n, ita e g
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ad f o. </
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<
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">ex æquali igitur, ut d b ad b n, ita g f ad f o: </
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<
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tando ut d b ad g f, ita b n ad f o. </
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<
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ad f l. </
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<
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<
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">permutando, ut b k ad
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bn, ita f l ad f o. </
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<
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">Rurſus quoniá quadratú h K æquale eſt rectan
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gulo k b n: </
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<
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<
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">quadratum m l rectangulo l f o æquale: </
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<
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lineæ b k, k h, b n proportionales: </
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<
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">itémq; </
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<
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">proportionales inter ſe
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f l, l m, f o. </
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<
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">quare ut linea b K ad lineam b n, ita quadratum b K
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ad quadratum h k: </
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">& </
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<
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">ut linea f l ad ipſam f o, ita quadratú f l
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ad quadratum l m. </
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<
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">Itaque quoniam, ut b K ad b n, ita eſt f l ad
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f o; </
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<
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ad l m quadratum. </
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<
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">ergo ut linea b k, ad lineam K h, ita linea f l
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ad ipsã lm: </
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<
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<
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">permutãdo ut b k ad f l, ita k h ad lm. </
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<
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f l erat ut k c ad c l. </
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<
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">ergo k h ad lm, ut K c ad c l. </
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<
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">quare ex eo
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dem lemmate patet lineam h c, & </
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<
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<
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tur K c ad c l: </
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<
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">hoc eſt ut a c ad c e, ita h c ad c m; </
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<
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">hoc eſt ad eam
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ipſius partem, quæ inter c, & </
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<
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">e g c ſectionem interyeitur. </
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<
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demonſtrabimus idem contingere in alijs lineis, quæ à puncto c ad
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a b c ſectionem perducuntur. </
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<
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">At uero b c ad e f eandern propor-
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tionem habere, liquido apparet; </
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<
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">nam b c ad c f, eſt ut d c ad c g;
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</
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<
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">uidelicet ut earum duplæ, a c ad c e.</
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ti.</
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ti.</
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conicorũ</
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xti.</
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