[ 34 6 ] 
In the general fchemes. Plate xv. and Fig. 5. Plate xiv. 
16. Let a <7, B b y be the longeft and fhorteft axes. 
Draw c? perpendicular to the tangent pt, cutting it in 
Put cp~t ; cq = c; c^rzg. 
Then c ? “ ? ^ 
C/XCE 
cc ty . tc acXcb 
: — X — - — ) — — . 
c c c c 
co. yy 
Hence the parallelogram, under the two axes, is equal to the parallelogram under any 
two diameters. 
17. Draw the tangents an, an y to any vertices a, a , meeting any diameters rp t 
q q, produced in v, u, and v y u , and the tangent pt in n, «. 
AndAV=C c -t^t2 = )2. 
\ CE p \ CD X 
- /AUXCCL , fC A , /CPXCA . /T 
18. Alfo cu =r / — = ) — ► And cv zz ( = ) — . 
\ EQ^ y V CD * 
19. Hence p viz: (cv cacpzz) t x^-^. And />v = t x^tf. 
X X 
Alfo qu zz (cu c«c o =z) c x^df. And yuzrex— ?. 
y y 
20. When a a and b b are the longeft and fhorteft axes; and when y—p r 
Then xx~tt^—yy will become tt=ppt zztl^cc y which call ff. 
And cd — X} wilf become cr~cf = f. 
The points f, f are called the Focii. 
21. Hence AV—af—±t^pf-, Af—a 7 —t+f. 
22« Alfo c F 2 zz cf Z —ff— cc — ztz tf^-pK 
And in the ellipfis, ac 2 -♦ (bc 2 -+-cf z zz ) bf 1 . 
in the hyperbola, cf 2 zz (ac 1 + EC 1 rz ) BA^ 
Hence, a circle deferibed from b, with the diftance AC in the ellipfis, or from c, 
with the diftance ab in the hyperbola, wiU cut the axis a a in the focii F, f. 
23, Draw 
