SS 
fi<v. u.—For (prop. 5.) SW is perpendicular to P Y, 
anR P G is alio perpendicular to PY, it being (def. 9.) 
perpendicular to the curve at P ; therefore the triangles 
H W S, H P G, are fimilar, and HW : H S : :H P : H G— 
H » HP i bu,Cprop.,.)HP = AC + CS X Ctl - 
HW ’ ' vr ‘“ r ' ‘ AC 
alfo (prop. t. cor. 1.) H S =; 2 C S ; and (prop. 5.) HW 
2CS , „ . CS X f N 
CONIC. SECTIONS; 
X KRr=CN, therefore 
triangles CKR,CTr,CK; 
C r , therefore C K x C r 
KR 
CN 
and by fimilar 
cor. 
BC 
AC 
KR::Ct( —— 
C M 
KR ' . 
X 7-,— = C A 
:A2 
prop. 
:C A' 
iAC; therefore HG — —— X A C + 
2 AC 
AC 
^ACH CS^M „ CG=HG_HC 
AC 2 ’ ’ 
„„ „„ CS X AC*+CS“xCN ^. c _CS-xCN 
-HG-CS--^- 
C s 2 v C N 
confequently NG = CN — CG = CN- 
AC 2 —CS 1 
AC 
X CN 
L 
X CNn (as AC = BS prop. 2.)' 
BC< _ v ' BC* 
M> xCN =< as Ac; 
AC 1 
BS-— CS’ 
AC 
'TKc P ro PG’ cor ' 7 -)* 
X CN. 
2 A C 
Cor. By fimilar triangles, as in fig. 14.. GPN, PNT, 
T. 2 A C v P N 2 
-XCN(NG): PN::PN:NTz=- X 
i A C 
A C 2 —CN’ 
L X CN 
= (prop. 7. cor. 8.) - 
Prop. 10. If C H, fig. 14. be a femi-conjugate to C P, 
and HR be an ordinate to the axis A M, then C N 2 =• 
AC 2 —■ C R 2 .—As C H is parallel to PT (def. 6.) the 
triangles C H R, TPN, are fimilar, and T N 2 : N P 2 :: 
C R 2 : RH ! , or (prop. 9. cor. and prop. 7. cor. 8.) 
AC*— CN S -t-yt: - ttyt. L ~ „ L 
C N 2 A jAC 2AC X 
AC 2 —CR 2 , or AC 2 — CN 2 : CN 2 :: CR 2 : AC 2 — 
C R 2 , therefore by comp. AC 2 : CN 2 :: AC 2 ; AC 2 
— CR 2 , confequently CN 2 — AC 2 — CR 2 . 
2 A C 
Cor. Hence, (prop. 7. cor, 8.) —_ xPN’r AC 2 
— CN’^CR 2 ; but (prop.3.) 2 AC : 2 B C :: zBC 
: L, therefore 2 A C : L : : A C 2 : B C ; hence, A C * 
BC 2 
or the fame 
^ : ’ . _ ' * A C N 
X B C, a conftant quantity, and C K X C r — the area 
CKr/P, which is one fourth of a b c d therefore the 
•area abed is conftant. 
Cor. 1. From hence, and cor. 6. prop. 7. the area of the 
ellipfe varies as the area of the circumfcrihing parallelo¬ 
gram abed ; and as the area cPCD of a fourth part of 
the parallelogram is equal to D C x P F (P F being per¬ 
pendicular to DCK), the whole parallelogram varies 
as D C x P F ; therefore tire area of the ellipfe varies as 
D C X P F. Cor. 2. Becaufe the triangles SPY, HPZ a 
PF E, in fig. ii, are fimilar, we have, 
SP.- SY:: PE: PF 
H P : H Z :: P E : P F 
6 P X H P : S Y X H Z :: PE 2 : P t< 2 ; 
but (prop. 6.) SYxHZ = BC 2 , 
P E 2 =r A C 2 ; therefore SP X H P: 
and (cor. 5. prop. 4.) 
BC 2 x AC2 _. DC 3 
PF 2 
1 6{ 
7 PN' = CR ! , and ^ X PN = CR. 
r> C 
A C 
reafon,——- x H R = C N. 
I) C 
Prop. ii. If at the extremities of any two conjugate 
diameters, tangents be drawn to the ellipfe, they will 
forma parallelogram; and all fuch parallelograms will 
fee.equal.—Let PG, DK, fig. 15, be any two conjugate 
by the propofition. 
Now, on the principles of projection, let ARMS', fig. 
be a circle inclined to the ^ 
plane of the paper, and let 
it be projected into the 
curve ABME, by lines 
perpendicular to that 
plane ; and let b be the 
point into which a is pro- A| 
jeCted, and draw a H, b H, 
perpendicular to AM, the 
interfeCtion of the circle 
and the before-mentioned 
plane. Then, by trigo¬ 
nometry, Ha: HL: rad. : 
col. Z. a H b, a conftant ratio ; therefore as, by projection; 
each line is diminifhed in this ratio, the whole area will 
be diminifhed in the fame ratio. And the fame is mani- 
feftly true, whatever be the figure of the plane to be pro¬ 
jected. By the property of the circle, A H X H M — 
H a 2 ; therefore AH X HM varies as Hi 2 , which is the 
property of an ellipfe by prop. 7. hence, AM B E is an 
ellipfe. Now all the Squares deferibed about the fame 
circle are equal ; and if the circle be projected into an 
ellipfe, every fquare will be projected into a circumfcrib- 1 
ing parallelogram ; for the oppofite rides of the fquare 
being parallel and tangents to the circle, they will be 
projected into parallel lilies, which will be tangents to 
the ellipfe ; and thefe lines will be equal ; for as they 
are equal before projection, and equally inclined to the 
plane of projection, they muft be equal in the projection,-. 
from what is proved above. And as every fquare is 
diminished in the conftant ratio of radius : cofine of in¬ 
clination, all the parallelograms muft be equal, as is 
proved in the above prop. 11. 
Prop. 12. If an ordinate be drawn to any diameter, the 
r.eCtangle under the abfeiffas is to the fquare of the.ordi¬ 
nate, as the fquare of the femi-diameter to the fquare of the - 
femi-conjugate diameter.—Draw the ordinate M^figuy, 
diameters; then (prop. 4. cor. 4.) the tangent a P d is 
parallel to the tangent bGc, and the tangent a DC is pa- 
rellel to the tangent dK.c, therefore tided is a parallelo¬ 
gram. Let ACM be the axis major, CB the femi-mi- 
’ ‘ ~ - 
uic iiiajui, uie 1 pirn-mi- 
nor axis ; produce AM to meet the tangent at i* in T, 
and draw C r perpendicular to ad. By prop..10. cor.^-^ 
B C 
parallel to PT or CK, and M'Q } .vn, perpendicular to 
ACE, 
