f)0 
CON 
S A : A V ; and as S A is 
cor. i.) S P is greater than 
.Cor. Hence, SP : PK : 
greater than AV (prop. 8. 
Piv; 
Prop. 20. To determine the diameter of the circle of 
curvature at any point of an hyperbola, and the chords 
thereof which pafs through the center and focus of the 
hyperbola.—Let PaLV, fig. 3 1, be the circle of curva- 
32 E 
c O N 
of the tangent, to determine the conic Te&ion.—The' 
diftance SP, tig. n, and the pofition of the tangent 
being given, the perpendicular S Y will be given ; now 
(cor 1. prop. 6. of the ellipfe and hyperbola, fig. 24,) 
-g“p--—BO; alfo, 2 ACrr;HP±SP, according as 
the curve is an ellipfa or hyperbola ; therefore the latus 
/ 4BC--\ 4SY’ X HP 
\~Tac) = SPxHPibT confequently 
reftum L 1 
2 A< 
L x S ?' 
ture at the point P, Q v an ordinate to the diameter P G, 
and QR parallel to Pt'; then (prop. 13.) P» (QR) X 
O v 2 CD 2 X 
vG : Qn 2 :: PC 2 : CD 2 , therefore —— = —-- 
— (if Q move to P, in the ultimate date of the arc QP) 
C D 2 X 2 P C 2 C D 2 ^ 
- ——• But ( Newton's Prin. lib. 1. 
lem. 7. cor.) the limit of the ratio of Q a : Q P is 
2 C D 2 Q P 1 
a ratio of equality ; therefore -— = ——- ultimate- 
P C Q K 
2 C D 1 
ly; hence (lem. p. 7.) — ^ ^ - — PV the chord of cur¬ 
vature pa fling through the center C. Alfo, if I. P F 
be perpendicular to P R, or C D, it muff pafs through 
the. center of the circle ; hence, by fimilar triangles P V L, 
PCF, PF : PC :: PV : PL = = 
the diameter of curvature. Laftly, if P S (produced if 
neceflary) interfecl the circle in a, and La be joined, by 
Ample trigonometry, P E F, P L a, PE, or A C, (cor. 2. 
W . 4 .,.PF„PL( E =^,P.= 4 ^ = 
the chord of curvature palling tlirough the focus. 
Prop. 21. If a cone RLW, 
fig- 33> be cut through the 
fide R L by a plane A F G, 
which being produced will meet 
the other fide of the cone W R 
produced in any point M, the 
fedtion will be an hyperbola.— 
Let X P Y V be a fedtion of the 
cone parallel to the bafe LW, 
which being a circle, N X x 
NY = N P 2 ; draw R Q paral¬ 
lel to the diameter Y X, and 
the triangles RMQ, NMY, 
£Yy and A Q R, A X N, will be fimi¬ 
lar; hence, 
AQ : QR :: AN : NX 
MQ:QR::NM:NY,_ 
AQ x MQ : QR 2 :: AN X NM : NX X NY, or NP 2 ; 
buf the two firft terms are conftant quantities ; therefore 
A N X N M is to PN ! in a conftant ratio ; confequently 
(cor. 5. prop. 7.) the fedtion is an hyperbola, whole 
major axis is A M. Hence, according to the definition, 
jthe hyperbola is a conic fedtion. 
Prop. 22, Given the diftance of any point of a conic 
fedtion from the focus, the latus redtum, and the pofition 
op__ 
“ 4 Sb 2 — L X SP 
Nowin the ellipfe (cor, 3. prop. 6.) L X SP is lefs 
than 4 S Y 2 ; when therefore Lx SP, in fig. 1 i , is lefs 
than 4 S Y % draw P H on the fame lide of the tangent with 
S P, making an angle HPZ = gSPY, and take H P = 
L x SP a 
~ - g-y_1 x S P ’ anc ^ ^ Be the other focus ; alfo, 
S P 4- PHa: the major axis ; and as H P is known, H Z 
will be given; confequently SYxHZ, orBC % will be 
known, and therefore the ellipfe is determined. 
In the hypeibola (cor. 3. prop. 6.) L x S P is greater 
than 4SY 2 ; when therefore Lx SP is greater than. 
4 S Y 2 fig. 24, draw P H on the other fide of the tangent 
in refpedt- to S P, making an angle KPZ=g SPY, and 
„ , UX3 _ — LxSP 2 Lx SP 2 
kC — 4 S Y 2 —LX SP 1 ° r L X SP — 4 SY 2> 3nd 
II will be the other focus; alfo, H P — S P =: the major 
axis ; and as FI P is known, H Z will be given ; confe¬ 
quently S Y X H Z, or B C 2 , will be known, and there¬ 
fore the hyperbola is determined. 
In the parabola (cor. 3. prop. 8.) L X SP224SY 2 , 
and in this cafe we have the following conftrudtionf 
Make the angle TPM = ^ T P S, as in fig. 1, and take 
PM22PS, draw D M E perpendicular to PM, through 
5 draw Z S L parallel to P M, and bifedt S L in A, and 
A will be the vertex of the parabola, from whence the 
parabola is determined. 
Prop. 23. Let S, in fig. 13, be the focus of a conic 
fedtion, SR, S P, SQ, three lines given in length and 
pofition; to deferibe. a conic fedtion through the three 
given points R, P, Q.—Join PQ, P R, and produce them 
to a and b, fo that Qa:Pa::QS:PS, and P b : R b :: 
P S : R S, and through a b draw xy, perpendicular to 
which draw S V, Q K, PL, R O; and in V S indefinitely 
produced, take V A : A S, and VM : MS, in the ratio 
of Q K : QS, and M A is the major axis of the figure, 
which will be an ellipfe, parabola, or hyperbola, accord¬ 
ing as Q R is greater, equal to, or lefs, than QS; this 
appears by prop. 17. cor. of the ellipfe, def. j. of the 
parabola, and prop. 19. cor. of the hyperbola: and in 
the firft cafe, M lies the fame way from V that A does ; 
in the feeond cafe,. M goes off in infinitum ; and in the 
third cafe, M lies on the other fide of V. For QS : PS 
:: Qa :: Pa : (by Ample trigonometry,) Q‘K : PL; 
hence, PS : PL :: QS : QK 1: S A : AV :: SM:MV. 
Alfo, PS:RS;:Pi:RL::PL:RO; hence, RS : 
RO :: PS : PL:: QS : QK S A: A V :: SM : MV. 
Therefore (by the propofitions quoted above) R, P, Q, 
lie in a conic fedtion., whofe focus is S, and major axis 
AM. By this ^ropofition the orbit of a planet may be 
found, having given three diftances from the fun, and 
the angles between them. 
CO'NICALLY, adv. In form of a cone.—In a water¬ 
ing-pot, ftiaped conically, or like a fugar-loaf, filled with 
water, no liquor falls through the holes at the bottom, 
whilft the gardener keeps his thumb upon the orifice at 
the top. Boyle. 
CO'NICALNESS, f. The ftate or quality of being, 
conical. 
CONICHTHYODON'TES, or PLF.CTRONn\'E,yi one 
of the names by which the foflile teeth of fifties are 
diftinguifhed. 
CO'NICS,/ That part of the higher geometry, or 
geometry 
