Viviani, Vincenzo - De maximis et minimis, geometrica divinatio : in qvintvm Conicorvm Apollonii Pergaei

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176152 atque ex N agatur N P O ipſi G L parallela, ex O verò recta O Q R paral-
lela ad L N, & R S A ad N O, atque A D C ipſi O R, & hoc fiat quoties li-
buerit: patet, ſi per puncta I, P, S, interſectionum ipſarum diagona-
lium cum diametro, agantur applicatæ T V, X Y, Z K, & c. & circa diametri
ſegmenta F M, M Q, Q D, & c. & per extrema prædictarum applicatarum
deſcribantur Ellipſes T F V M, X M Y Q, Z Q K D, has omnes eſſe Para-
bolæ A B C inſcriptas, & ſimiles inter ſe, ac ſe mutuò ſucceſſiuè 11@. h. gentes. Iam dico eaſdem Ellipſes, primæ B F ſimiles eſſe, atq; inter ſe eam
rationem habere, ac numeri quadrati diſparium numerorum ab vnitate: ni-
mirum eſſe in progreſſione numerorum 1. 9. 25. 49. & c.

Quoniam igitur eſt M B ad B I, vt B I 221. Co-
roll. 13. h.

141[Figure 141] B F, erit diuidédo M I ad I B, vt I F ad F B,
ſed eſt I F æqualis F B, ex conſtructione,
quare M I ipſi I B æqualis erit, ac ideo in
Ellipſi T F V M, erit quadratum T I ad re-
ctangulum M I F, hoc eſt rectum eius 3321. pri-
mi Conic. ad tranſuerſum, vt idem quadratum T I, vel
rectangulum ſub I B, & recto B E, ad 44Coroll.
1. huius. gulum ſub eadem I B, & ſub I F, hoc eſt vt
linea B E ad I F, (cum ſit I B communis re-
ctangulorum altitudo) vel ad ei æqualem B
F, nempe vt rectum ad tranſuerſum Ellipſis
BF: quapropter Ellipſis B F ipſi T F V M eri
ſimilis, ſed vnaqueque aliarum inſcriptarum
Ellipſium circ? diametri ſegmenta M Q, Q
D, & c. eidem T F V M eſt ſimilis, vt ſupra
monuimus, quare omnes huiuſm odi inſcri
ptæ Ellipſes erunt ſimiles inter ſe. Et cum ſit
M I ęqualis I B, & I B dupla F B, erit to-
ta M B quadrupla B F. Si ergo B F conci-
piatur vt vnum, erit B I vt 2, & M B vt 4,
atque M I vt 2, & M F vt 3.

Cumq; in triangulis P M N, I F G ſint anguli ad M, F inter ſe æquales,
ob æquidiſtantes applicatas M N, F G; atque anguli ad P, I item ęquales,
ob parallelas diagonales N P, G I, erunt reliqui ad N, G pariter æquales,
ſiue ipſa triangula inter ſe ſimilia, vnde latus N M ad M P erit, vt latus G F
ad F I, & permutando N M ad G F, vt M P ad F I, vel quadratum N M ad
G F, ſiue recta M B ad B F; hoc eſt 4. ad 1, vt quadratum M P ad 5520. pri-
mi Conic. dratum F I; vnde quadratum M P quadruplum erit quadrati F I, ſiue linea
M P dupla F I, ſiue dupla ad B F, ſed B F ponitur vt vnum, ergo M P erit
2; eſtque B M 4, ergo B P erit 6, eſtque B M ad B P, vt eſt B P ad 661. Co-
roll. 13. h. Q, quare B Q erit vt 9, ſed B M eſt vt 4, ergo M Q erit 5. Præterea, ea-
dem ratione, ac ſupra, oſtendetur triangulum R Q S ſimile triangulo G F I,
& quadratum R Q ad G F eſſe vt quadratum Q S ad F I, ſed eſt 7720. pri-
mi Conic. tum R Q nonuplum quadrati G F, cum ſit recta Q B nonupla B F, vt mo-
dò oſtendimus, ergo, & quadratum Q S erit nonuplum quadrati F I, ſiue
quadrati B F, hoc eſt linea Q S tripla B F, quare tota B S erit vt 12; eſtq;

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