CONIC SECTIONS. 
Now C V 
C A 2 
CA 2 
sc = 
- P (prob. 8. cor. and def. S.), .'. S V = 
CA 2 —SC 2 SB 2 —SC 2 CB 2 . 
SC " SC 
and by trigonometry, rad. 
BC 2 
MC — SC x coline P S M 
B C “ x cofine P S M 
MC 
SC • sc 
cofine, PSM :: SP, or 
- (propofition 16.), : SN =, 
therefore P L = V N = 
• SC x cofine PSM’ 
SV + SN=£1! + B C 2 x cof. PSM 
s c 
MC X CB 2 
M C — S C x cof. PSM 
; hence, S P : PL 
SCxMC — SCx cof. PSM 
B C 2 _ M C x C B 2 
fquare of the cofine 
CM 2 : CB 2 :: MC 2 — CN 2 
N P 2 = 
C B 2 
C M 2 
X 
MC 2 '—'CN 2 , therefore CP 2 (= CN 2 NP 2 ) = 
r B 2 _:— C B 2 v CNs 
CN ’ + cSP* MC ‘-«i= CB ’- 
CM 2 
C N : 
d- CN 2 ==CB 2 + C M 2 — CB 2 X CM2 
CN 2 
CM + CB X CM — CB X 
CN 2 
C B 2 + i C B x 
C M-CB X 
CM 2 
fquare root, we get CP=CB xCM — CB x 
very nearly, = C B -f CM' 
Q.v ’ 
PC e : CD*, therefore 
Qv< 
CD 
. 85 
X Gy 
Q R P C 2 
— (if Q move to P, in the ultimate Hate of the arc QP) 
CD‘ X 2PC 2 C D' 
JR. 
JB 
D/ 
MC —SCx cof. PSM ' SC X MC —SC X cof. PSM 
:: SC: MC a conftant ratio. 
Cor. Hence, SP : PL :: SA:AV:: SM;MV; 
therefore P L is greater than P S. 
Prop. i8. If the ellipfe ABMD, in fig. 8. be very 
nearly a circle, as P moves from B to M, the in- 
creafe of C P is very nearly in proportion to the 
: of the angle PCM.—'By prop. 
PC 2 PC 
But (Newton’s Prin. 
lib. r. lem. 7. cor.) 
the limit of the ratio 
of Q v : Q P is a ratio 
of equality; therefore *^\ 
2CD 2 QP 2 , . 
~PC“ — QR U tU 
mately; hence (lem. 
2CD* 
p ’ 7,) npc~ =PV 
the chord of curva¬ 
ture palling through the center C. Alfo, if P F L be per;, 
pendicular to PR, or D K, it mud pafs through the cen¬ 
ter of the circle; and, by limilar triangles P C F, P V L, 
P F: rC::Pv(iR§:) ! PL = ±Rih= (he die. 
meter of curvature. Laftly, if PS interfedt the circle in 
a , and La be joined, by limilar triangles PFE, PaL, 
PE (=2 AC by cor. 5. prop. 4.) : PF :: PL^— ^ 
2 C D 
: Pa = 
•) 
AC 
V PF 
= the chord of curvature pafTmg through 
CB 2 -f 
CM : 
very nearly, and by taking the 
C N 2 
CM 2 
- CN 2 
CB x very nearly, 
= CB + CM — CB x col. PCM 2 , radius being uni¬ 
ty ; the quantity therefore CM — C B x cofine P C M% 
by which C P exceeds C B, varies as col. PCM 2 , very 
nearly. 
Prof. 19. If a right line QS P, as in fig. 8, pafs through 
the focus S of an ellipfe, and be terminated by the curve, 
then SP - jTq = s*L‘‘—® raw LF, »QG, TE, P«K, 
perpendicular to xy, and produce (if necelfary) ST to 
meet PK in m ; then (prop. 1.7.) 
SQ:SL::QG: LF, 
SP : ST (SL) :: PK : TE; 
.•. SC.— SQ : SL :: LF — QG (Q») : LF, 
and SP; SP —SL:: PK : PK — TE (Pm) ; 
but SI. : SP :: LF’: PK (prop. 17.) 
.■.by com. S L — SQ ; S P — S L :: Q n : P m 
:: SQ : SP (byAmple 
trigonometry); hence, S L x S I^-S Px S Q —SPx SQ—• 
SLxSQ, therefore SLx SP + sQ = 2S Px SQ, and —- 
O i-j 
SP 4- SQ 
the focus. 
Prop. 21. If a cone ABC, fig. 19, be cut through it& 
twofides by a plane PMNEZ, the 
fedtion will be an ellipfe.—.Bifedt 
PE in K, and let a fedlion LMIZ 
be drawn through K parallel to 
the bafe, and alfo another fedtion 
HNFG parallel to LMIZ, then 
will thefe lections be circles; hence, 
LIG x GF i GN*, and LK x 
K I = KM 1 . Draw PQ, D E, 
parallel to LI, or HF; then by 
limilar triangles, £ 
PK : KL :: PG : GH 
EK: KI :: GE : GF 
.-. PKxEK:KLxKI::PGxGE:GHXGF v 
that is, PK 1 rKM 2 :: PG X GE : GN*, which i,$ 
the property of the ellipfe by prop. 7. Hence, accord¬ 
ing to the definition, the ellipfe is a conic fedtion. 
ON THE HYPERBOLA. 
Definitions.—!. If two right lines, Sa, Hb , fig. zo, 
interfedting each other in P, be made to revolve about 
SP x SQ - SP + Sq‘ 
Prop. 20. To determine the diameter of the circle 
of curvature at any point of an ellipfe, and the chords 
thereof which pafs through the center and focus of the 
ellipfe.—Let PaV L, fig. 18, be the circle of curvature 
at the point P, Qv an ordinate to the diameter PG, and 
QR parallel to PG*; then (prop. 12) PV (QR) X Gw. 
Vol, V. No. 256. 
H and S, fo that HP—TP may be a confbnt quantity, 
their common interfedtion Twill defcribe the hyperbola. 
Z * Ami, 
