Huygens, Christiaan - Christiani Hugenii opera varia; Bd. 2: Opera geometrica. Opera astronomica. Varia de optica

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114393ILLUST. QUORUND. PROB. CONSTRUCT.

Probl. III.

Datis duabus rectis duas medias propor-
tionales invenire.

VEterum Geometrarum ad hoc Problema conſtructiones
complures retulit Eutocius ad lib. 2. Archimedis de
Sphæra & Cylindro, at non omnes inventione diverſas, uti
recte quoque ipſe animadvertit. Heronis enim inventionem
ſecuti videntur Apollonius & Philo Byzantius: quanquam
Heronem Apollonio ætate poſteriorem nonnulli exiſtiment.
Dioclis modum Pappus & Sporus. Nicomedea autem con-
ſtructio præ cæteris ſubtilis ibidem extat, quam Fr. Viëta
paulò aliter concinnatam ſuo Geometriæ ſupplemento inſe-
ruit. R. Carteſii egregia eſt & nova per paraboles & cir-
cumferentiæ interſectionem, cujus demonſtratio legitur in
libris Harmonicôn M. Merſenni. Noſtræ autem ſequen-
tes.

Sit datarum linearum major A C, quæ bifariam ſecetur
11TAB. XLI.
Fig. 4. in E. Minor autem ſit A B, quæ ſic conſtituatur ut trian-
gulus E A B habeat crura æqualia A E, E B. Et perficia-
tur parallelogrammum C A B D. Et producantur A C,
A B. Porro applicetur regula ad punctum D, & moveatur
quousque poſitionem habeat G F, abſcindens nimirum E F
æqualem rectæ E G; (Hoc autem vel ſæpius tentando aſſe-
quemur, vel deſcriptâ hyperbole, uti poſtea oſtendetur)
Dico inter A C, A B medias duas inventas eſſe B G,
C F.

Sit enim E K ipſi A B ad angulos rectos. Quia igitur
B E æqualis E A, dividetur A B in K bifariam: adjecta au-
tem eſt linea B G. Ergo rectangulum A G B cum quadrato
ex K B, æquabitur quadrato K G. Et addito utrimque qua-
drato K E, erit rectangulum A G B unà cum quadratis
B K, K E, hoc eſt unà cum quadrato B E, æquale qua-
drato E G. Similiter quia A C bifariam dividitur in E, &

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Scan	Original
61	349
62	350
63
64
65
66
67
68
69
70	358
71	359
72	360
73	361
74	362
75	363
76	364
77	365
78	366
79
80
81
82	367
83	368
84	369
85	370
86	371
87	372
88	373
89	374
90	375