CONIC SECTIONS. 
70 
PT at P, PV is an ahjcijfa, and Q V an ordinate to the 
diameter PW, Alio, TN is called the fub-tangent. 7. 
Four times the diftance PS, is the lotus re£ium, or para¬ 
meter, to the diameter PW, And the latus reflum to 
the axis is called the/) incipal latus reEfum. 8. If P G be 
perpendicular to the curve, it is called the normal ; and 
N G the J'ub-normal. 
Prop, i . The diftance of the focus from any point of 
the curve, is equal to the fum of the abfeiffa belonging 
to that point, and the diftance of the vertex from the 
focus.—For (definition 1.) SP = PM— (as MPNL is a 
parallelogram by conftrudtion) NL=AN 4- AL = 
AN +AS (def. 1. cor.) 
Prop. 2. The diftance of the vertex from the focus, 
is equal to one-fourth of the principal latus rqffum.—■ 
Draw C S B perpendicular to A Z ; then when P comes 
to B, N coincides with S, and (prop. 1.) SB — SA + 
S A = 2 S A ; therefore B C — (def. 1. cor.) 2 S B — 4S A 
the latus reftum (L) to the axis, or the principal latus 
reftum (def. 7.) 
Prop. 3. The fquare of any ordinate to the axis, is 
equal to the abfeiffa multiplied into the principal latus 
reiStum.—For (prop. 1.) SP=AN + AS — (prop . 2. ) 
AN + )BC; and NS — AN — SA=AN — |BC; 
hence ( Euclid , b. i. p. 47.) N P 2 — S P 2 — S N * 2 — 
AN -f-aj 3 C 2 — AN —^ B C 2 — B C x AN- (prop. 2.) 
4SA X AN — Lx A N.' — Cor. Hence, B C being con- 
ftant, A N varies as N P 2 . 
Prop. 4. If PT bifedt the angle S P M, it will be a 
tangent to the parabola at P.—Join S M, in fig. 2 ; then 
asSP = PM, ZSP«- 
Z. M P n, and n P is com- 
M P n, SrctrrreM, and the 
angles at n are right angles. 
(Euc. b. i. p. 4. d.'io.) Now 
if PT be not a tangent, let 
it cut the curve in fome 
other point p ; join S p, M p, 
and draw mp perpendiciilar 
to ml-. Then as Mn = nS, 
ZMnp— Z§ n fi> and n P i s common to the triangles 
*Mnp, S np, we have Mp — but, by'conftrudhon, 
mp-Sp; therefore M p — mp, or the hypothenufe — the 
perpendicular, which is impoflible ; therefore P T can¬ 
not meet the curve at any other point than P, and confe- 
quently is a tangent to it at that point. 
Cor. 1. A tangent at the vertex A is perpendicular to 
the axis. Cor. 2. AsMP is parallel to TS the 7/MPT — 
/PTS, therefore / PTSr/SPT ; hence, SP-ST. 
Cor. 3. By fig. 1 , as V Q is parallel to P T, the /TPXr 
Z PX V, and /PVXr/MPT^ but /MPT = Z 
TPX; therefore Z PVX-/PXV; hence PX — PV. 
Prop. 5. The fub-tangent is equal to twice the ab- 
feiffa.—-For (cor. 2. prop. 4.) T S — S P — (prop. 1.) A N 
-pAS ; take away AS from both, and T A —AN ; there¬ 
fore T N = 2 A N. 
produced to meet T Z 
in O, then the triangle? 
T PN, O VG are fimilar, 
and NP = GV; therefore 
OG = TN=2AN. 
Prop. 6. The fub-nor- 
mal is equal to half the 
principal latus redlum.— 
For, as in fig.i, the tri¬ 
angles T N P, N G P are 
fimilar, (Euc. b.vi.p.8.) 
and hence, TN : PN :: 
PN 2 
PN: N G —— (prop. 
and 5.) 
TN 
B G x A N 
2 AN : 
principal latus redlum. 
s B C — (prop. 2.) | the 
Cor. As (prop. 4. ccr. 2.) S T = S P, ar.d the angle 
T P G is a right angle, a circle whole center is S, and 
radius ST, will pafs through P and G ; hence, the angle 
TSPr 2/TGP. (Eac.b. iii. p. 20.) 
Prop. 7. The lquare of the ordinate to any diameter, 
is equal to its parameter multiplied into the abfeiffa.— 
Through Q draw EQD perpendicular to A Z, meeting 
W P (produced if neceffary) in D, fig. 3. 
Now Q E 2 — O G — G E 2 — 2 i\ iN — G E 2 (prep. 5. ccr. ) 
AE-AN + NG-GE. 
TN 2 — OG 2 — 4AN 2 (prop.5. cor.) 
GV 2 r=NP 2 — 4ASXAN (p rop. 3.) 
EQ 2 —4AS x AE ( pr.3. ) — 4AS x A N 4- N G — GE 
By fimilar triangles T N P) O E Q, 
TN 2 : NP 2 :: OE 2 : EQ 2 , that is, 
4AN 2 : 4ASXAN:: zAN —GE 2 :4 ASxAN+NGTTgE, 
or, 4 AN : 1 :: 2 A N — G E 2 : A N -f-NG — GE; 
Multiply extremes and means, and we get;* 
4AN xNG = GE 2 —'VD 2 . 
By fimilar triangles OGV, VDQ, 
OG 2 : GV 2 :: V D 2 : DQ 2 ; that is, 
4AN 2 :4AS X AN 4ANXNG: D Q 2 — 4ASXNG 4 
hence, QV 2 =VD 2 -fDQ 2 ( Euc. b. i. p. 47.) = 4 A N 
X NG 4- 4 A S XNG-4AN + 4 A S xNG — 4SPX 
PV (prop. 1.) If QV be produced to meet the curve at 
Q', and Q'E' be perpendicular to AEj then O E' — 
O G + G E', and AE'r AN + N G + GE'; and by the 
very fame fteps, we get Q'V 2 — 4SP x PV. 
Cor. 1. Hence, QV — Q'V ; therefore every line Q Q* 
is bifedted by the abfeiffa PV. Cor. 2. When the point 
Pis given, 4SP is conftant; hence, for the fame dia¬ 
meter, the abfeiffa varies as the fquare of tire ordinate. 
Cor. 3. If Ql be drawn parallel to PV, then 4SP x QI 
= PI 2 ; hence, when the point P is given, QI varies 
as PI 2 . 
Prop. 8. If a perpendicular from the focus be let fall 
on a tangent at any point of a parabola, it will be a mean 
proportional between the diftance of that point from the 
focus, and the diftance of the vertex from tire focus.— 
As (cor. 2. prop. 4. fig.i.) SP=ST, the perpendicular 
SY muft bifedt TP, or TY = YP; but (prop. 5.) 
T A —AN ; therefore (Euc. b. vn p. 2.) AY is parallel 
to P N, and confequently perpendicular to the axis. 
Since therefore SY is perpendicular to TY, and YA 
perpendicular to TS, the triangle S Y A is fimilar to 
S Y T (Euc. b. vi. p. 8.) or to S Y P, and S P : S Y :: 
S Y * S A 
Cor. 1. Hence, SP : SA SP 2 : S Y 2 . Cor. 2. As 
S Y 2 
SP=—, and SA is conftant, SP varies as S\ 2 „ 
Cor. 3. As SY 2 —SA x PS, therefore 4SY 2 — 4S Ax 
PS =2 (prop. 2.) BC X PS. 
Definition. —'Let PW, fig. 4. be any curve, Pt)?a> 
circle touching it at P, and , 
T P a tangent at P ; draw R q 4 - 
cutting P W in Q ; then if 
R Q q move up to P, and the 
limiting ratio of R Q : R q be 
a ratio of equality, the circle 
is faid to be a circle of cur¬ 
vature to the curve PW at 
P. 
Lemma. —Draw Pa parallel to R q and join a q, Fy 
then when q and R move up to P,, P v is. equal to the 
r PQ 2 
ultimate value of — 
K. y=l 
The triangles PR?, P 'qv are fimilar, for the-angle 
R q P = the alternate angle q P v, and the angle R P q 
between the chord and tangent — /_ P vq in the alternate 
fegment ; lienee, ?R:?P::?P:Pw = IjlT’ bUt 
fir I. Newton’s Principia, lib. i. lem. 7,. the limit of the 
chord q P to the arc q P is a ratio of equality ; and the 
limit of the arc q P to the. arc P Q is a ratio of equality 
be.c.aufc 
