ARITHMETICA universalis.

2g c< 5 '+ 2 gcf eft 2gc in gc + cf id eft 2CGF. Ergo i£Gq + Gcq =EC( ?+ 2 cgf. Q. E. D.

Lem. Ii. in conflrudlionis cafu primo ubi circulus pbe tranjitfi er pun£lum d, ejl eg^-gd/7 = 2CGF.

per Lemma primum eft eg^ + gc# = Ecq+ 2CGF, & ablatolinque Gcq y fit EGq = ec^-gc^e 2CGF. Sed Ecq-Gcq idem eftjRiod cd^-gc^, hoc eft idem quod Gi>q. Ergo eg<? = gd<7+2CGF,hibduito utrobique gd<7, fit EGq-Gvq = 2CGF. Q.E.D.

Nem. III. in conflruSlionis cafu fecundo, ubi circulus pbe nontr ^nfit p er pun&um d, ejl eg# + gd<7 = 2Cgf.

Namque in Lemmate primo erat eg#+gc<? - Ecq+ 2cgf. Auferutr inq\i e EC( j, fi et eg^+gc^-ec# = 2cgf. Sed gc = dh S c ec =^ P ' G H : ergo Gcq-Ecq - dh^ - gh q - Gv^, atque adeo EGq+GDq^acGE. Q.E.D.

Lem. IV. Ejl 2CGF in GY - 2CG in AGE.

-namque ob fimilia triangula gef, gya eft gf ad ge ut ag ad^ Y » hoc eft (per Prop. I. lib. VI. Elem.) ut 2 Cgxag ad 2 Cgxgy.

Uc antur extrema 8c media in fe, Lc fiet 2 cg x gy x gf =2Cg x ag x ge. Q. E. D.

tandem ope horum Lemmatum conflrudlio Problematis Jic de -m ° n Jlratur.

. cafu primo eft (per Lem. 2.) eg<7-gd<?= 2Cgf, & duitis orn-atis in gy fit EG(? X GY - GD? x GY = 2CGF X GY (hoc eft per LE. 4.)

2 CG X ^GE. Pro GY fcribe eg+ ey, &: fiet eg cub . + ey x e Gq—GE, ?XEG-gD(7xEY = 2CGAX EG, feu EG CUb. + EY x EG?~ 2GG |x EG

" Gd ?xey = o.

cafu fecundo eft (per Lem. 3.) ef^+gfv - 2CGF, & duitishimbus i n GY Y.Gq x gy + GT>q x gy - 2CGF X gy (hoc eft peretn% 4 -) — 2cg x age. Pro gy fcribe eg+ey, &c fiet eg cub. +ey

+ Eg ? + GD q x EG + GT>q X EY - 2CGA X EG, feu EG Cub. + EY X E Gq"^G\ X EG + GD^X EY - O.

Jam vero erat eg radix aequationis conftruitae diita x ; itemGD ^ V -•) ey = p, icG-n, 8c g a — id eft in cafu primo ubi

filiorum /> 8c r diverfa funt figna: at in cafu fecundo ubi al-tei utrius p vel r mutatur fignum fiet - - + = ga. Scribantur igi—

tur

213

.EqTJATIO-

num Con-structio-nes.