[ 33 ] 
Let the fine of the angle P C O == C G = r, 
CPrrt.COrrK, (radius being unity) ; draw P O 
perpendicular to CP : then in the right angled plane 
triangle B C O, we have as rad. (i) : CO (») :: 
fine PCO (J) ; BO ( *£) ; and rad. ( i ) : CO («) 
: : cofine PCO (vG — £ x ) : BC (W i — f*) ; but from 
C * 
the nature of the ellipfis we have -j x — x x f 1 
— BCf = jc 1 — k £*-, therefore x = - 
r 
or putting t — c x — f\ 
x=^£-r„ and f z 
r—p ( 
and f — x ~ <p% 
_ **xi — C 
t* c z 
f - Ax £ * 
we have 
PROPOSITION III. 
Fig. 3 . Let BOD be an ellipfis, whofe tranfverfe 
diameter A b makes the angle ACB, with the right 
line BCD, and let T/£G be a tangent to the ellipfis, 
in the point F, making the angle GTC with the 
right line BCD : It is required to find the length of 
the normal Ck, drawn from the center of the el- 
lipfis, to the tangent TG. 
From C, the center of the ellipfis, let CE be 
drawn parallel to the tangent T G, meeting the el- 
lipfis in the point E; and CG perpendicular to the 
line BCD, meeting the tangent in the point G : Put 
the fine of ACB r=r Z, its cofine =_ Z, the fine of 
r 
TGC = V, its cofinc = V (radius being unity) 
A C = t, CO z=s x., and r — k = 0 ; then will 
the fine of O C E (= the fine of 0 C D -|- D C E) 
Vol. LVII. F be 
