5 4 
CONIC SECTION S. 
A .C E, alfo cv parallel to n Q, and Pg perpendicular to 
P I'. Then as Me, vc, Me, are refpe&ively parallel to 
PT, T N, PN, the triangle M cv is fimilar to PNT, it 
is therefore fimilar to the triangles PN«-, and C K R ; 
hence, PN: Ng (~— • ■ , ■, prop. 9.) :: vc:c^Azz 
2 AC 
^ '\'r' ^ 1 alfo r by fimilar triangles CPN, Cvn, 
z A C x N P 1 0 ’ 
C»xNP 
C| 5 
I, x C N x v e 
X 
C N : N P :: C n : nv~- 
-f- c Q ~ c M -f- n v — 
2 A C x N P 
+ 
therefore QM-cM 
C» X NP 
CM 
Alfo, 
CQ~jiQ — Ci 
:ve 
— C n. Now. 
i AC 
X QM — AC 
•—CQ 2 (prop. 7. cor. 8.), in which equation, fubftitute 
for QM and CQ the values found above, and we have, 
2AC /L ! xCN 5 x« 2 C.v’xNP’ L'/»fXC.v) 
X ( + 
L " V 4AC-XNI* r |CN< 1 AC j 
— hQ.-—vc + 2B«X C»— Cn* ; but the lalt term on 
the firft fide, and the third term on the fiecond fide 
deftroy each other, therefore the equation becomes 
L xCN’X lilt ’ 1 :ACxC» ! xNP ! 
— A C 2 —vc - 
L x C N 2 
xPN 2 :z=AC 2 — CN 2 (prop. 7. cor. 8.); 
CN* in the fird 
2 A C X N P 2 
n d 2 A C 
C n 2 . But — j 
in the lad equation therefore, for 
2 A C 
term, put AC 1 -p-XPN', and for NP 1 in the 
fecond term, put - - 
L X A C X »f 2 
and we get 
2AC 
PN 
2ACXPN 2 
(prop. 10. cor.) ; hence, 
= 1, and 
CN 2 
vc 2 Cn 2 
ck 7 + ctT 
CR = x C N - — C n - — vc 2 x C N 2 , therefore CN 2 —C n 2 
: C N 2 :: ve 2 : C R 2 :: (by limple trigonometry,) jM*: 
C K 2 ; alfo, CN ! :C» ! ::CP ! :C» ! , and CN 2 — Cm 2 : 
C N 2 : r C P 2 — C»>:CP’i hence, from this and the 
fird proportion, CP 2 —C»‘ : CP 2 :: v M 2 : C K- ; but 
C P 2 — C v 2 — C P + c v x CP — c ~=G i-x P» ; there¬ 
fore, by alternation, G v X P v : v M 2 :: CP 2 : CK 2 . 
Cor. Produce M v to rn ; then by the fame reafoning 
it appears that G»X P»: C P 2 : C K 2 ; confe- 
quently vM — v m, or any line M v parallel to a tangent 
at P, is bifected by P G. 
Let PVCG fig. 16, be a diameter of the circle, and let 
it be projected into a diameter pvCg of the ellipfe, 
and let V Q, an ordinate of the circle, be projected into 
v q ; then as V Q is parallel to a tangent at P, and paral¬ 
lel lines are projected into parallel lines, qv is parallel 
to a tangent to the ellipfe at p, for a tangent to the circle 
mud evidently be projected into a tangent to the ellipfe; 
hence, qv is an ordinate to the abfeiffas p v, gv , def. 7. 
And if DCK be drawn parallel to qv, it will be conju¬ 
gate diameter to the diameter h g. Now as all the dia¬ 
meters of the circle are bifected l y the center, all the 
diameters of the ellipfe are bifeCted by the center, for 
the different parts of the fame right line in the circle 
afe diminilhed in the fame ratio ; and for the fame rea- 
lon, as P V, V G, are projected into pv, vg , thefe latter 
quantities will be to the former in a given ratio, there¬ 
fore P V X V G varies asp v X vg ; alfo, Q V is to q v 
in a given ratio, therefore Q V- varies as qv 2 . But by 
the property of the circle, PV X V G = Q V 2 ; there¬ 
fore pvXvg varies as qv 2 -, lienee, pvXvg : qv :: 2 pC X op, 
or pC 2 , : CK. 2 , as is proved in the above prop. 12. 
Prop. 13.—/The fum of the fqtures.of any two femi- 
PS 
and 2 PC’ 
but 2 C D 
P C 2 + 2 C D 2 =;Pri, 4-P b 
•zCS 2 — (prop. 2.) 4SB 
(C H -) — 2 C S (CH) x CN 
pc 2 + ie;s 2 - 
2 C S 2 
cor. 2 .) 
2CS 2 — (prop. 1.) 
+ PH= X 
= PS 2 + PH 
— 2 S P x P H (prop. 11 
: 4SB 2 —2SB 2 
hence, PC 2 -f 
4AC 2 — 2CS 2 — (prop. 2.) 4 SB- — 2CS 
+ 2CB 2 = 2SB 2 + 2CB 2 = 2AC 2 + 2l3C 
CD ! =AC= + BC>. 
Prop. 14.—Tf QV, fig. 9, he an ordinate P G, and a 
tangent Q 1 at tire point Q meet the diameter G P pro¬ 
duced in T ; then CV -. CP :: CP : CT.—Draw Vr per¬ 
pendicular to GP and equal to VQ, then the curve PrG 
pading through all the points r, is an ellipfe ; for GV x 
V P being to QV 2 in a given ratio (prop. 12.), we have 
G V x'V P to V r 2 in a given ratio, therefore (prop. 7. 
and cor. 1.) PrG is an ellipfe, and P G is its major or 
minor axis, the ordinates being perpendicular to it. 
Draw r I a tangent at r ; then (prop. 8. cor.) C V : C P 
:: C P : CT; join T Q, and if it be not a tangent at Q, 
let it cut the ellipfe at fome point in ; draw the ordinates 
m a, o. v, and produce a v to meet T r produced in T. 
Now by fimilar triangles, 
TV : VQ:«8 
TV:Tfl:: V r (VQ) : at. 
Therefore cm — at ; but by con£tru6tion, am — a v ; 
therefore at=.av, which is abfurd ; hence, T Q does 
not cut the curve at Q, and confequently is a tangent to it. 
Prop. 15. — If P F, fig. 15, perpendicular to DC K, 
cut A M in v, then PF X Pc = B C 2 .•—For produce 
C B to meet tlie tangent at P in m, and draw C r perpen¬ 
dicular to the tangent, and P n perpendicular to BC; 
then as C m is parallel to PN, and C x to P v , the tri¬ 
angles m C r, P v N, are fimilar, and C m : C r (P F) :: 
P v : P N (Ci i); therefore PF x P v ~ C my,C n — CB* 
(prop. S. cor.). 
Prop. 16. The diftance SP= MC _ S ”^IoLTS M 
fig. S.—- Draw PN perpendicular to AM. By 
M C 2 : C B 2 :: ANxNM:PN 2 ; but C B 2 
SC 2 , and ANXNM=MC + CN 
as in 
prop. 
= SB 2 .— SC 2 =MC 2 
X MC — cN=MC 2 — CN 2 =MC 2 
M C 2 : M C 2 
■ SNrpSc 2 
S C 2 :: MC 
SN+bb 
; hence, 
PN 2 = 
M C 2 
•SC 2 x MC 2 
MC 2 
■ 15 JN =p 5 C 
MC 4 —MC 2 xSN 2 dr:SNXSCXMC 2 —2MC* XSC 2 SC 2 x su 2 ~,SN x sC 3 -gsc* 
MC 2 r 
add S N 2 to both fides of the equation, and we have 
S P 2 (PN 2 + S N 2 ) — 
MC 4 ±^SNXSCXMC 2 — 2 MC 2 *SC*-}-SC 2 XSK 2 T-,SNXSC 2 -pSC 4 
MC 2 
the fquare root of which is S P = 
MC 2 ±SNxSC —SC 2 
MC 
BC 2 ±SNx SC; 
- (as MC 2 — SC 2 =SB 2 — SC 2 =BC 2 )--? 
but ±SN = SP x cof. P S M ; therefore S P = 
BC + SCxSPx cof. P S M 
MC 
hence, SP = 
B C ’ 
M C — S C x cof. P S M. 
Prox>. 17. If a line be drawn, as in fig. 13, from the 
focus to any point of the curve, and from tiiat point 11 
line be drawn perpendicular to the directrix, thele lines 
are in a conflant ratio.—Draw P L perpendicular to x.y. 
Now 
