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

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[41.] THEOR. IV. PROP. XI.
Page: 43 (23)
[42.] COROLL.
Page: 44 (24)
[43.] MONITVM.
Page: 45 (25)
[44.] LEMMA III. PROP. XII.
Page: 45 (25)
[45.] ALITER idem breuiùs.
Page: 46 (26)
[46.] ITER VM aliter breuiùs, ſed negatiuè.
Page: 47 (27)
[47.] COROLL.
Page: 47 (27)
[48.] THEOR. V. PROP. XIII.
Page: 47 (27)
[49.] COROLL. I.
Page: 49 (29)
[50.] COROLL. II.
Page: 49 (29)
[51.] COROLL. III.
Page: 49 (29)
[52.] THEOR. VI. PROP. XIV.
Page: 49 (29)
[53.] COROLLARIVM.
Page: 50 (30)
[54.] THEOR. VII. PROP. XV.
Page: 51 (31)
[55.] THEOR. VIII. PROP. XVI.
Page: 52 (32)
[56.] THEOR. IX. PROP. XVII.
Page: 55 (35)
[57.] MONITVM.
Page: 55 (35)
[58.] THEOR. X. PROP. XVIII.
Page: 55 (35)
[59.] Definitiones Secundæ. I.
Page: 56 (36)
[60.] II.
Page: 56 (36)
[61.] III.
Page: 57 (37)
[62.] IV.
Page: 57 (37)
[63.] V.
Page: 57 (37)
[64.] VI.
Page: 57 (37)
[65.] VII.
Page: 57 (37)
[66.] VIII.
Page: 58 (38)
[67.] IX.
Page: 58 (38)
[68.] THEOR. XI. PROP. XIX.
Page: 58 (38)
[69.] COROLL. I.
Page: 64 (40)
[70.] COROLL. II.
Page: 64 (40)
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4424 ſtans. Dico ipſam cum ſectione conuenire, eamque omnino ſecare.
Iungatur CS, quæ producta ſectioni occurret, per ſecundam partem 8.
huius, eritque ſectionis diameter: quare per Coroll. 2. præcedentis, ipſa STX
ſectioni occurret, vt in X.
Præterea, cum quæcunque contingenti æquidiſtans GI ſupra RX, inter
aſymptoton, & ſectionem intercepta, maior ſit ipſa RX, ſiue ipſa GZ, pun-
ctum Z cadet extra ſectionem, & ſic de quolibet alio puncto rectæ XTS. E
contra cum quælibet intercepta LY infra RX, parallela ad DB, minor ſit ip-
ſa RX, ſiue LF, punctum F cadet intra ſectionem, idemque de quolibet alio
puncto rectæ XF: vnde recta STX ab ipſo occurſu X cum ſectione, ad partes
verticis tota cadit extra, ad oppoſitas verò partes tota cadit intra ſectionem;
ideoque in vno tantum puncto X Hyperbolen ſecat. Quod erat propoſitum.
19[Figure 19]
COROLL.
HInc eſt, lineam alteri aſymptoton æquidiſtantem per punctum, quod
ſit, vel in ipſa ſectione, vel intra, pariter in vno tantùm puncto ſe-
ctioni occurrere, eamque ſecare.
Nam recta XS ex puncto X, quod cſt in Hyperbola, vel recta FS ex pun-
cto F, quod eſt intra, æquidiſtanter ducta aſymptoto CD, ſi ad partes cen-
tri C producatur, alteri aſymptoto CE omnino occurrit, (quoniam EC, ſe-
cans DC vnam parallelarum ſecat quoque alteram CE) vnde aliqua pars,
ipſius rectæ XS, vel FS cadit in loco ab aſymptotis, & ſectione terminato,
ac ideo ex his, quæ ſuperius oſtendimus, ipſa linea in vno tantùm puncto ſe-
ctioni occurret, ac Hyperbolen ſecabit.
Qua propter, quælibet linea alteri aſymptoton æquidiſtans, dummodo ſit
ducta ex puncto, quod ſit in angulo ab aſymptotis facto, in vno tantùm pun-
cto Hyperbolæ occurrit, atque eam ſecat.

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