Barrow, Isaac - Lectiones opticae & geometricae : in quibus phaenomenon opticorum genuinae rationes investigantur, ac exponuntur: et generalia curvarum linearum symptomata declarantur

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5941 erit. ‖ Sin punctum Y extra datum angulum exiſtat, evidens eſt tan-
tùm uno modo problemati ſatisfactum iri; quódque per alteram in-
erſectionem, & Y, ducta recta ad angulum pertinet dato verticalem.
hæc, inq̀uam, tantillùm attendenti manifeſtè conſtabunt; nihil ut ſit
opus hic plura verba conſumere. verùm ut in horum caſuum primo
conſtet (id quod pro ſequentibus ex uſu erit cognoſcere) quando
dictus circulus _byperbolem_ contingit; ſeu quando tantùm una per
Y recta quantitatis ejuſdem interſeri poſſit, hoc adnectemus _Theo-_
_rema._

IX. Si à puncto quovis Y intra rectum angulum XPF exiſtente
11Fig. 54. demittantur ad ejuſdem anguli latera perpendiculares YB, YD; ac
inter YB, YD proportione mediæ ſint rectæ BN, GD; per puncta
N, Y, G tranſibit recta cunctarum minima, quæ per Y ductæ angu-
lum XPF ſubtendere poſſunt.

Quòd NYG ſit una recta patet, quoniam eſt YB. BN : : GD.
DY (ex conſtructione nimirum) porrò per Y tranſeat alia quæcunque
recta LYM; & NH ad GN, MH ad PF perpendiculares concur-
rant in H. item HA ad NG parallela ducatur; & GS ad PF;
denuóque connectatur GH. Jam patet triangula GDY, YBN,
HMN, HMR ſimilia fore; quódque proptereà eſt MN. MR : :
MN q. MHq : : DGq. YDq. item (ob BN, DG, YD {. ./. .})
eſt BN. YD : : DGq. YDq. hoc eſt YN. YG (vel MN. GS)
: : DGq. YDq. ergò eſt MN. MR : : MN. GS. adeóque MR
= GS. itaque major eſt GS ipsâ MT; abeóque rectæ GH, LM
protractæ concurrent; puta ad Z. ergò LM. GH : : LZ. GZ.
verùm propter angulum LGH recto P majorem, eſt LZ & gt; GZ.
quare LM & gt; GH. aſt ob angulum rectum GNH eſt GH & gt; GN.
quare magìs eſt LM & gt; GN. eodémque modo quævis per Y ducta
major oſtendetur ipsâ GN : Q. E. D.

X. Hinc etiam ſi GN ſit in ratione YB ad YN quarta proportio-
nalis; erit GN minima. nam indè conſequetur fore YB, BN, GD,
YD {. ./. .}. Etenim erit YNq. YBq : : GN. YN. & dividendo
BNq. YBq : : GY. YN : : DY . BN. ac indè YBq x DY =
BNcub; velDY = {BN cub/YBq}. itáque DY eſt quarta proportionalis in ra-
tione YB ad BN.

XI. Subnotari poteſt autem, quòd minimæ GN propiores

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