80 
C O N I C 
bccaufe (by the fame lemma) the limit of each to PR is 
a ratio of equality ; alfo (by the above definition,) the 
limit of q R : R Q is a ratio of equality ; hence, when q 
and R move up to P, P v is equal to the ultimate value of 
PQ 2 , ' 
RQ 
Prof. 9. At any point P of a parabola, as in fig. 1. 
the chord P S v of the circle of curvature, palling through 
the focus, is equal to the latus reftum belonging to that 
point.—Draw QR parallel to P S. By prop. 7. 4S Pzr; 
O V 2 ’ O V 2 
-— (as Q R = P X — P V, prop. 4. cor. 3.) ——-; 
PV v > r y + QR 1 
but (Newt. Prin. lib. 1. lent. 7. cor.) the limiting ratio 
of QV to QP is a ratio of equality ; therefore ultimately 
QP 2 
when Q moves up to P, 4SP=^~-—Pt/ by the lemma. 
Cor. Draw »F perpendicular to »P, meeting PF per¬ 
pendicular to PT in F, then P F is the diameter of the 
circle of curvature ; becaufe P v being the chord of that 
circle, and the angle P v F a right angle, it mult be an 
angle in a femicircle. Draw S Y perpendicular to PT ; 
then the triangles SPY, P»F are fimilar, being right 
angled at Y and v, and having the alternate angles Y S P, 
v PF equal; hence, SY:SP::Po (4S P) : PF — 
4SP2 (as SY = ^SPxSA, prop. 8.) 
4 S P2- 
S Y 
V S A 
Prop. 10. Let two parabolas AY, AZ, in fig. 5, 
be defcribed on the fame axis AX, to which NPQ 
A is drawn perpendicular ; then 
if the lines PS, QS, be drawn 
<} \\ to any point S in the axis, the 
area AQS will be to the area 
APS in a given ratio.—For 
(cor. prop. 3.) QN 2 varies 
as A N, and P N 2 varies as 
AN; hence QN 2 varies as 
V 1 'PN 2 , confequently QN va¬ 
ries as PN. Draw qpn indefinitely near to QPN, and 
the areas NQy«, NPp«, will be the increments of the 
areas AQN, APN ; but NQ qn is to NPpw ultimately 
as QN to PN, becaufe their bafes are equal ; hence, the 
areas AQN, APN, beginning together, and always in- 
creafing in the given ratio of QN to PN, the areas them- 
fielves will be in that ratio. But the triangle SQN is to 
the triangle SPN as QN to PN ; hence, AQN—-SQN 
is to A PN —SPN, that is, AQS is to APS, asQNis 
to PN, or in a given ratio. If N be between A and S, 
then AQS — AQN -j-SQN, and APS — APN + SPN, 
and the fame conclufion follows. 
Prop. ii. Let the line QV, as in fig. 6. meet the 
parabola LM in Q, V ; draw the tangent QS, and let 
i» S V be parallel to the axis, 
and from any point T draw 
TPR parallel to SV; then 
SV : TR :: TR : TP.—-For 
(prop. 7. cor. 3.) TP:SV:: 
QT 2 .- QS 2 :: (by Ample trigo¬ 
nometry,) TR 2 : SV 2 , there¬ 
fore TR 2 — TPx SV, and 
S V : TR ;: T R:TP. 
V Cor. 1 .—Becaufe T R : T P 
:: S V : TR :: (by fimple trigo- 
M nometry,) QV : QR, we have, 
by divifion, PR: TP:: RV: RQ; hence, QV, TR, are 
divided at R and P in the fame ratio. Cor. 2. II R bi- 
fed: Q V, then (prop. 7. cor. 1.) PR is an abfcifla to the 
ordinateQ R, and QR is parallel to a tangent PW (def. 6). 
Now as RV — RQ, PR —TP (cor. 1.); hence, TW- 
WQ. Reafoning in the fame manner for a tangent at V 
asat Q, it appears that a tangent at V mull pal's through T. 
Prop. 12. If a cone HXY, as in fig. 7, be cut by a 
plane FZ W parallel to a plane touching the fide XH of 
the cone, the fedion F Z W- will be a parabola.— Let 
SECTIONS- 
A E C V be a fedion of the cone parallel to the bafe,_ 
whofe diameter (the fedion 
being a circle) interfeds the 
axis FR of the figure F Z W 
in B, and whofe circumfe¬ 
rence cuts the curve F Z ter¬ 
minating the fedion F Z W, 
in E, and join B E; then, by 
the property of the circle, 
ABxBCrBE 2 . Draw 
GF parallel to ABC, and 
then AB —GF a conflant 
quantity ; and as HX is that 
fide of the cone to which 
FR is-parallel, the triangles 
H G F, F B C are fimilar, 
G F X E B 
" HG ’ 
therefore 
GF 2 X FB_ 
FIG “ 
GFXBC-ABxBC = BE 2 , or- 
GF 2 
TTg 
X F B — B E 2 ; but G F and H G being conllant quan¬ 
tities, the - abfcifla F B varies as the fquare of the ordi¬ 
nate B E ; therefore the curve is a parabola, as appears 
by cor. prop. 3. Hence, according to the definition, the 
parabola is a conic fedion. 
ON THE ELLIPSE. 
Definitions. —1. If two right lines Sa H b, as in 
fig. 8, revolve a- 
bout two fixed 
points S, H, and 
continually inter- 
fed each other in 
a point P, fo that 
S P + H P may be 
a conllant quan¬ 
tity, the* curve 
defcribed by the 
point P is an el- 
lipfe. 2. The 
points S and FI are 
the foci. 3. The 
right line AM palling through the foci and terminated 
by the curve at A and M, is the axis major. 4. If AM 
be bifeded in C (which is the center,) and BCD be 
drawn perpendicular to the axis major, and terminated 
by the curve, it the axis minor. 5. A right line LST 
palling through the focus S perpendicular to the major 
axis and terminated by the curve, is the latus redum, or 
parameter. 6. If PCG, fig. 9, be a right line palling 
through the 
center C, it 
is a diameter. 
And if DCF 
be drawn pa¬ 
rallel to a tan¬ 
gent R P at P, 
it is a conju¬ 
gate diameter 
to the diame¬ 
ter PCG. 
Alfo, PCG, 
DCF, are conjugate diameters. 7. If QV be drawn 
parallel to a tangent at P, it is an ordinate, and GV, 
PV are the abfciflas. 8. If TV be a tangent at T, 
meeting the axis' MA produced in V, and the line xYy 
be drawn perpendicular fo MV, as in fig. 8, that line is 
the diredlrix. 9. If PG be drawn perpendicular to 
the ellipfe A PM at any point P, and PN perpendicular 
to the axis AM, as in fig. 12, then PG is called the 
normal, and NG the fub-normal. 
Prop. i. The fum of two right lines drawn from the 
foci to any point of the curve, is equal to the major axis. 
For, conceiving the defcribing point P, as in fig. 8, to 
come to A and M, we have, in the former cale, the fum 
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