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

Table of contents

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[81.] PROBL. VIII. PROP. XXIII.
Page: 71 (47)
[82.] PROBL. IX. PROP. XXIV.
Page: 73 (49)
[83.] PROBL. X. PROP. XXV.
Page: 74 (50)
[84.] PROBL. XI. PROP. XXVI.
Page: 75 (51)
[85.] SCHOLIVM I.
Page: 76 (52)
[86.] SCHOLIVM II.
Page: 76 (52)
[87.] PROBL. XII. PROP. XXVII.
Page: 77 (53)
[88.] PROBL. XIII. PROP. XXVIII.
Page: 78 (54)
[89.] PROBL. XIV. PROP. XXIX.
Page: 80 (56)
[90.] PROBL. XV. PROP. XXX.
Page: 81 (57)
[91.] PROBL. XVI. PROP. XXXI.
Page: 83 (59)
[92.] THEOR. XIII. PROP. XXXII.
Page: 84 (60)
[93.] THEOR. IV. PROP. XXXIII.
Page: 85 (61)
[94.] MONITVM.
Page: 85 (61)
[95.] THEOR. XV. PROP. XXXIV.
Page: 87 (63)
[96.] THEOR. XVI. PROP. XXXV.
Page: 88 (64)
[97.] THEOR. XVII. PROP. XXXVI.
Page: 88 (64)
[98.] COROLL.
Page: 89 (65)
[99.] THEOR. XIII. PROP. XXXVII.
Page: 90 (66)
[100.] THEOR. XIX. PROP. XXXVIII.
Page: 91 (67)
[101.] LEMMA IV. PROP. XXXIX.
Page: 92 (68)
[102.] THEOR. XX. PROP. XXXX.
Page: 92 (68)
[103.] COROLL.
Page: 93 (69)
[104.] THEOR. XXI. PROP. XXXXI.
Page: 94 (70)
[105.] COROLL.
Page: 95 (71)
[106.] THEOR. XXII. PROP. XXXXII.
Page: 95 (71)
[107.] ALITER.
Page: 97 (73)
[108.] COROLL. I.
Page: 98 (74)
[109.] COROLL. II.
Page: 98 (74)
[110.] LEMMA V. PROP. XXXXIII.
Page: 99 (75)
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THEOR. XIII. PROP. XXXVII.
Hyperbolæ concentricæ per eundem verticem ſimul adſcriptæ,
quarum recta latera ſint inæqualia, ſunt inter ſe nunquam coeuntes,
& ſemper magis recedentes, & in infinitum productæ, ad interual-
lum perueniunt maius quolibet dato interuallo, & aſymptotos in-
ſcriptæ ſecat Hyperbolen circumſcriptam.
SInt duę Hyperbolę ABC, DBE per
60[Figure 60] eundem verticem B ſimul adſcri-
pte, quarum idem centrum ſit F, idem-
que tranſuerſum BFG, ſed tamen Hy-
perbolæ ABC rectum latus ſit BH, ma-
ius recto BI Hyperbolæ DBE. Dico
primùm eas ſimul eſſe non coeuntes.
Cum enim Hyperbolæ DBE, ABC
ſint per verticem ſimul adſcriptæ cum
eodem tranſuerſo BG, ipſa DBE, cuius
rectum minus eſt, inſcripta erit 112. Co-
roll. 19. h. perbolæ ABC, cuius rectum maius eſt,
hoc eſt, ſi iſtæ ſimul in infinitum produ-
cantur, erunt ſimul non coeuntes.
Iam dico, has etiam eſſe ſemper in-
ter ſe recedentes. Ductis enim, & pro-
tractis regulis; GH, GI, & applicatis
duabus vbicunque rectis ADL, MON; quæ regulas ſecent in Q, S, T, V,
cum ſit vt quadratum MN ad quadratũ NO, ita recta VN ad NT, vel 226. Co-
roll. 19. h. SL ad SQ, vel quadratum AL ad LD, erit etiam recta MN ad NO, vt AL ad
LD, & per conuerſionem rationis, & permutando MN ad AL, vt MO ad
AD, ſed eſt MN maior AL, quare, & MO erit maior AD; ſimiliter 3332. h. ſtrabitur quamlibet aliam interceptam applicatę portionem inter Hyperbo-
las infra MO, maiorem eſſe ipſa MO, & hoc ſemper, quare huiuſmodi Hy-
perbolæ ſunt ſemper inter ſe recedentes. Quod ſecundò, & c.
Ampliùs dico, has ſectiones in infinitum productas, aliquando perueni-
re, ad interuallum maius quolibet dato interuallo X. Hoc autem, eadem
penitùs arte, ac in 33. huius fieri poſſe demonſtrabitur. Quod tertiò, & c.
Tandem ſit FP aſymptotos inſcriptæ DBC, & FR aſymptotos circumſcri-
pte, & contingens HB producatur, vtranque aſymptoton ſecans in P, R: erit
ergo quadratum BP ęquale quartę parti figurę GBI, & quadratũ BR 448. huius. parti figuræ GBH, ſed rectangulum GBI maius eſt rectangulo GBH, cum ſit
BI minor BH, ergo BP minor eſt BR; hoc eſt FP aſymptoton inſcriptæ cadit
infra FR aſymptoton circumſcriptæ diuidens angulũ ab ipſius aſymptotis fa-
ctum, ex quo ipſa FP producta ſecabit Hyperbolen circumſcriptam ABC. 55ibidem. Quod erat vltimò, & c.

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