Cavalieri, Buonaventura
,
Geometria indivisibilibvs continvorvm : noua quadam ratione promota
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LIBER I.
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tantum ſui parte, velin vno puncto tantum, prędictas figuras, tan-
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gant in punctis tantum, P, R, & </
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">ab ipſis ducantur parallelę regulis,
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dO, gY, ipſæ, PN, RT, occurrentes incidentibus, LO, uY, in
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punctis, N, T, dico, LN, uY, ſimiliter ad eandem partem ſecari
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in, N, T, ſi.</
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<
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xml:space
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">hoc non ſit, diuidatur, LO, in, M, ſimiliter ad ean-
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dem partem, acdiuiditur, uY, in, T, & </
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<
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">per, M, extendatur, MI,
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parallela, dO, incidentes ambitui figuræ in, I, & </
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<
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xml:space
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">rurſus ſecetur, u
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Y, in, V, ſimiliter ad eandem partem, vt ſecatur, LO, in, N, quia
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ergo, N, eſt intra puncta, M, O, etiam, V, erit inter puncta, T,
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Y; </
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<
s
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xml:space
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">ducatur tandem, VS, parallela, gY, incidens ambitui figurę in,
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S. </
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<
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">Quia igitur, MI, non incidit in punctum contactus rectæ, H o,
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cum figura, erit, MI, maior, NP, eadem ratione oſtendemus, SV,
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fore maiorem ipſa, RT, eſt enim, RT, minima earum, quæ abin-
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cidente, uY, ad ambitum figuræ duci poſſunt æquidiſtanter ipſi, g
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Y. </
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<
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">Cum verò, IM, RT, ſimiliter diuidant, & </
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<
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">ad eandem partem
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ipſas incidentes, LO, uY, erit, IM, ad, RT, vt, LO, ad, uY,
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huius.</
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ideſt vt, PN, ad, SV, ergo, permutando, IM, ad, PN, erit vt,
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RT, ad, SV, eſt autem, IM, maior, PN, ergo etiam, RT, erit
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maior, SV, ſed etiam minor, quod eſt abſurdum, ergo falſum eſt ip-
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ſas, PN, RT, non ſecare ſimiliter ad eandem partem ipſas, LO,
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uY, ſic igitur eaſdem diuidunt, eritque, PN, ad, RT, hoc eſt, H
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L, ad, Xu, vt, LO, ad, uY, idem oſtendemus etiam ſi contactus
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eſſet in parte linearum, Ho, XZ, ſeu in totis eiſdem lineis, vt conſi-
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deranti facilè innoteſcet. </
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<
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">Eadem autem methodo probabimus etiam,
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DL, pu, eſſe vt ipſas, LO, uY, ergo reſiduæ, DH, pX, hoc eſt,
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AC, FK, erunt vt, LO, uY, ideſt vt, E4, ℟ &</
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ſimiliter ſunt diuiſę ab homologis, sl, 7 4, productis, in punctis, B,
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xml:space
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">Defin. 10.
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huius.</
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G, ergo, AB, ad, FG, ideſt, DE, ad, p℟, erit vt, AC, ad, F
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K, ideſt vt, E 4, ad, ℟ &</
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<
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">. Extendantur, NP, TR, quę diuidunt,
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LO, uY, ſimiliter ad eandem partem, ſecentque ipſos, E4, ℟ &</
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<
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in punctis, 2, 3, incidat autem, NQ, in, Q, punctum contactus
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lineæ, Ad, cum figura, oſtendemus, vt factum eſt circa ipſas, NP,
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TR, etiam, T8, incidere in punctum contactus rectę, p g, cum fi-
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gura, quod ſit ipſum, 8, quoniam ergo probatum eſt, DE, ad, p
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℟, eſſe vt, E4, ad, ℟ &</
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<
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℟ &</
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<
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xml:space
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">. Similiter probabimus, 2P, ad, 3R, eſſe vt, E4, ad, ℟ &</
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& </
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<
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xml:space
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<
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xml:space
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viciſſim ſecantur ad eundem angulum ex eadem parte, cum, E4, ℟
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&</
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<
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xml:space
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<
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ſimilium figurarum, PlQs, R487, & </
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<
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DH, do, pX, gZ, quod etiam veriſicaretur de ipſis homologis, ls,
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47, ſi fuiſſent ad oppoſitas tangentes terminatę in punctis, E, 4, </
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