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

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23551

THEOR. XXIII. PROP. XXXX.

Si in Parabola, ex binis ipſius diametris duo æqualia ſegmen-
ta ſint abſciſſa: in Hyperbola verò, Ellipſi, vel circulo duæ ſe-
mi-diametri proportionaliter intra ſectionem ſectę fuerint, & ex
11Schema-
tiſmus 3. terminis æqualium diametrorum in Parabola, vel ex punctis di-
uiſionum, in reliquis ſectionibus, ordinatim applicentur lineæ ad
fuas diametros, & producantur, donec ad vtranque partem ſe-
ctioni occurrant: coni- ſectionum portiones; at in Ellipſi, vel
circulo, minores portiones ijſdem applicatis, tanquam baſibus
inſiſtentes, inter ſe æquales erunt.

ESto A B C Parabole, in prima, ſecunda, & tertia figura, vel Hyper-
bole in quarta, quinta, & ſexta, aut Ellipſis in ſeptima, octaua, &
nona, aut circulus, in reliquis, quarum ſectionum binæ diametri in Pa-
rabola ſint D B, D E, à quibus dempta ſint æqualia ſegmenta B F, E G,
& in reliquis binæ ſemi-diametri D B, D E (quæ primò in Ellipſi, vel
circulo omnino conſtituant angulum B D E) ita intra ſectiones ſectæ ſint
in F, G, vt D B ad B F, ſit vt D E ad E G, & per puncta F, G, in ſin-
gulis figuris ſint ad diametros D B, D E ordinatim ductæ A F C, H G I,
quæ ad vtranque partem ſectioni occurrent in punctis A, C; H, I, & 2219. primi
conic. bifariam in F, G ſecabuntur, cum D B, D G, ſint ipſarum diametri. Di-
co portiones A B C, H E I ſuper ijſdem applicatis, tanquam baſibus inſi-
ſtentes, inter ſe æquales eſſe.

Nam, ductis ex B, E rectis B N, E N ſectionem contingentibus in B,
E; ipſæ occurrent ſimul in N inter diametros D B, D E, & 3358. primi
huius. H I, A C æquidiſtabunt. Iungantur præterea E B, G F.

Iam in Parabolis, cum ſint E G, B F inter ſe æquales, & parallelę, iun-
ctæ quoq; E B, G F inter ſe æquidiſtabunt, & cum ex illarum terminis E,
B, ductæ ſint rectę E N, B N angulum E N B inter eas conſtituentes, atq;
ex reliquis terminis G, F, ſint G I, F A, ipſis E N, B N æqurdiſtantes;
ipſæ G I, F A inter eaſdem E G, B F ſimul conuenient, vt in M, & iuncta
N M ijſdem E G, B F æquidiſtabit, ſiue erit altera Parabolæ diameter. 4438. h. Cum ergo ſit E G parallela ad N M, & E N ad G M, erit E N 5546. pri-
mi conic. G M; eademque ratione B N æqualis F M, quare vt E N ad N B, ita G
M ad M F.

In reliquis verò figuris, cum rectæ D B, D E angulum E D B efficien-
tes, proportionaliter ſectæ, aut productæ ſint in G, F, ſintque ex earum
homologis terminis E, B ductæ E N, B N angulum inter ipſas conſti-
tuentes E N B, & ex reliquis diuiſionum punctis G, F, ſint G I, F A ijſdem
E N, B N parallelę, hæ intra datum angulum E D B ſimul conuenient, vt
in M; & recta iungens puncta D, M, per occurſum M omnino tranſibit,
ſiue erit alia ſectionis diameter. Cumque ob parallelas G M, E N ſit 6638. h. M ad E N, vt M D ad D N, & ob parallelas M F, N B ſit M F ad N B,
77_f_ 47. primi
conic.

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Scan	Original
121	97
122	98
123	99
124	100
125	101
126	102
127	103
128	104
129	105
130	106
131	107
132	108
133	109
134	110
135	111
136	112
137	113
138	114
139	115
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149	125
150	126