82 
CONIC SECTIONS. 
Prop. 7. If an ordinate be drawn to the major axis, 
the pr 'u£t of 
the abfciCas is 
to the fquare of 
tire ordinate, as 
the fquare of the 
femi-axis major 
to the fquare of 
the femi-axis mi¬ 
nor.—Draw PN, 
as in fig. 13, 
perpendicular to 
AM, then (def. 
7. and cor. 1. 
prop. 4.) PN is 
an ordinate to 
PS —AC —.CD 
.-.PH — AC q- CD (prop, r.) 
Alfo,. NS-CS-CN 
NH — CH + CN — CS+CN (prob.i, cor. 1.) 
but PH 2 - PN 2 + NH 2 ( F.nc. b. 1. p. 47.) 
PS 2 —PN 2 + N S 2 ; that is, 
j\ C q- C D 2 — P N 2 + C S q- C N 2 
AC —C D 2 —PN 2 q- C S — CN 2 
fub. and AC x C D — C S x CN, 
. • CSxC'N. 
therefore CD — —-: lienee, 
A.C ’ 
CSVCN 
PII-AC + CD-.AC+ - : 
PS — AC — CD — AC- 
C S x C'N _ 
AC — 
(Eve. b. i. 
AC 2 -fCSxCN. 
A C * 
AC 2 —CSxCN 
AC 
But P'S 2 — PN 2 + NS 2 (Eve. b. i. p. 47.) that is,. 
AC 4 — 2AC 2 xCSxCN + CS 2 x CN 2 ' 
AC 2 
— PN 2 q- CS 2 — zCSxCN + CN 2 , 
multiply botli fides of the equation by AC 2 , and we 
get, by tranfpofition, AC 2 X P N 2 — AC 4 — AC 2 x CN 2 
— CS 2 xAC 2 + CS 2 xCN 2 : 
— (as AC —SB) AC 5 
• CN 
AC 2 — C N 2 XAC 2 — CS 2 
X SB 2 — Cb 2 ' (BC 2 ) — 
AC + CN X AC — CN x B C 2 rr ML +CN x AC — CN 
X B C 2 — MN x A N X B C 2 , confequently M N x AN 
; PN 2 :: AC 2 : BC 2 . 
Cor. 1. Draw Pm perpendicular to BCE. Now by the 
propofition, 
AN x NM : NP 2 :: AC 2 : CB 2 
er AC—cPx AC+ n P (AC 2 —n P 2 ) : C ?i 2 :: AC 2 :CB 2 
alt. AC 2 — 72 P 2 : AC 2 :: Cn 2 : CB 2 
div.wP 2 : AC 2 :: C B 2 — C ?; 2 : CB 2 
alt. ?? P 2 : CB q-C b x Cr5 — C u (CB 2 — Cn 2 ) :: AC 2 : CB 2 
inv. E« X Bb : »P 2 :: CB 2 : AC 2 . 
Hence,.there is the fame relation between the abfcifTas 
and ordinates of the minor axis, as between thofe of the 
major axis. Cor. 2. If a circle A R.M K, as in fig. iz, 
be deferibed on the major axis A M of an ellipfe, as a 
diameter, and N P be produced to Q, NP fiiall be in a 
given ratio to N Q. For by the property of the circle 
A N x M N — N Q 2 ; hence, 
N Q 2 : N P 2 AC 2 : BC 2 , 
and NQ: NP:: AC : BC :: AM : BD a given ratio. 
And by cor. 1. it appears, that the fame muff be true, if 
A M be the minor axis cf the ellipfe. Cur. 3. Becaufe 
PN : QN is the- conftant ratio of B D : AM (cor. 2.) 
therefore for the fame reafon as in the parabola, prop. 10. 
the area APN : the area AQN :: P N : Q N B D : 
AM. Hence, the whole area of the ellipfe : the whole 
area of the circle :: B D : A M. Cor. 4. Join Q H, PH, 
then the triangles H N P, H N Q, are to each other as 
N P : N Q, or as B D •- A M ; whence, and cite lull cor. 
we have the area A PH : the area A QH :: B D : A M. 
And the fame muff evidently be true, if H be aflumed' 
anywhere in the axis major. Cor. .5. Conceive a circle 
to be deferibed about C witli a radius — 4/ AC x B C, 
a mean proportional between the femi-axes of the ellipfe, 
and let a be its area ; and let A — the area of the cifcle 
ARMK; then (cor. 4.) 
A : area A B M D of ellipfe :: AC : BC 
<z:A::A-CxBC:AC 2 (Euc. b. xii. p. 2.) 
a : area A B M D of ellipfe :: 1 : 1 ; 
hence, a — the area A B M D of the ellipfe. Cor. 6. As 
(cor. 3.) the whole area (A) of the ellipfe is to the 
whole area (B) of the circle, as the axis minor (a) is to 
the axis major (b)\ we have A =dL 2 Li ; but B varies 
b 
as 1 !/ 2 ; therefore A varies as by. a i that is, the area of 
an ellipfe varies as the product of the axis major and 
minor ; but the area of a parallelogram ciffumfcribing 
the. ellipfe about the axis major and minor, varies in the 
fame ratio hence,, the area of every ellipfe is in a given 
ratio to the area of the parallelogram deferibed about 
the major and minor axes. Cor. 7. If L =: the latus rec¬ 
tum, then (prop. 3.) AM : BD :: BD : L; hence, 
AM : L :: AM 1 ': B D 1 :; A C 1 : B C 1 :: (by this prop.) 
pm. r , T AMxPN 1 
P N , confequently L “ 
M N x A N 
2 A C x PN 
M N x A N 
AN-AC 2 
Cor. 8. Hence, 
: A C 
L 
MN X AN~ 
X PN 1 - MNx 
C N 1 by the demonfiratian of. the propo- 
fition. Cor. 9. If the major axis of an ellipfe be increafed 
without limit, the ellipfe, at all finite difiances, approaches 
to,a parabola as its limit. For if AN remain finite, and 
A M, N M,. be increafed without limit, their difference- 
AN becomes indefinitely fmall in refpeft to the quanti¬ 
ties themfelves; therefore the limit of AM, N M, is a 
„ ,. , T AM X PN 2 
ratio of equality ; but L — ^4 ; therefore the 
limit of. this is L — 
PN ! 
A N 
M N x A N 
or L x A N — P N % which is 
the property of the parabola, by prop. 3. of the para¬ 
bola. 
Prop. 8 . If a circle be deferibed on the major or< 
minor axis of an ellipfe, as in fig. 14, and (when necelfary). 
N P be produ¬ 
ced to meet it 
at Q, the tan¬ 
gents to the 
circle and el¬ 
lipfe at the 
points Q and 
]P will inter- 
feft the axis 
M A produ¬ 
ced, in the 
fame point T. 
For draw QT 
a tangent to 
the circle, at 
Q, and join T P, and if it be not a tangent to the ellipfe, 
let it cut it in fome other point K, and draw the ordi¬ 
nate L K, which (when necelfary) produce to meet the 
circle at I, and its tangent at O, then the triangles 
T N P, TIK, and TNQ, T L O, are fimjlar: lienee, 
TN:TL::NP:LK, 
and TN : TL :: NQ: LO, 
NP : LK :: NQ : LO, 
alt. NP : NQ :: LK : L O, 
but N P : N Q :: L K : L I (cor. 2. prop. 7.) 
therefore LOr LI, which isabfurd; hence, the line 
T P does not cut the ellipfe, and confequently is a 
tangent to it. 
Cor. By Euc. b. vi. p. 8. the triangles C N Q, C Q T, 
are fimilar, therefore CN: CQ (CA) :: CQ (CA) : 
CTj alfo, as C A — C M, CN : CM.:: CM: CT. 
Prop. 9, The fub-normal N G — X C N, as in 
a AC 
fig. 
