called the Axis Major; and if through the center C a line BCO be drawn at right angles to AM, it is called the Axis Minor of the Ellipse.

(34.) From any point P let fall the perpendicular PN upon the axis major AM, and through the focus S draw the straight line LST parallel to it. PN is then called the ordinate to the axis; AN, NM, the Abscissas; and the line LST is called the Latus-rectum, or the Parameter to the Axis.

(35.) Draw any line PCG through the center, and another line DCK parallel to a tangent at P; draw also Qu parallel to DCK. PCG is then called a Diameter, and DCK the Conjugate diameter to PCG; Quis called an Ordinate to the diameter PCG, and Pv, v G, the Abscissas.

VI.

On the Properties of the Ellipse.

PROPERTY I.

(36.) If SB, HB, are drawn from the foci to the extremity of the axis minor, then SB, HB, are each equal to AC. (See Fig. in p. 24.)

Since SC CH, and BC is common to the two rightangled triangles BCS, BCH, SB must be equal to BH; .. SB+BH=2 SB or 2 BH.

Again, by Sect. 2. Art. 2. SP+PH=AM=2AC;

and when P comes to B, SB+BH=2AC; hence 2SB
or 2 BH=2AC; .. SB or BH-AC.

PROPERTY 2.

(37.) MSX SA=BC'.

For BCSB-SC2 (Euc. 47. 1.)

=AC-SC (by Prop'. 1.)

=AC+SCX AC-SC

=CM+SC-AC-SC (for CM=AC)

=MSX SA.

(38.) COR. In the same manner, it might be shewn, that AHX HM-BC.

PROPERTY 3.

(39.) The latus-rectum LST is a third proportional to
the major and minor axis.

For
SL+LH=2AC (by construction),

.. LH=2AC-SL,

and LH2=4ACa—4AC× SL+SL3.

Again,

LH2=SL2+SH (Euc. 47. 1.)
=SL+4SC(for SH=2SC)

=SL +4.SB-BC

=SL2+4.CA – BC2.

Hence

Hence 4AC-4ACX SL+SL=SL'+4AC-4BC'; . 4ACX SL=4BC'.

:

And putting this equa- 2AC 2BC: 2BC: 2SL, tion into a proportion, Jor

we have

or AM: BO :: BO: LT.

PROPERTY 4.

(40.) Produce SP to p; then if YZ bisects the angle HPp, it will be a tangent to the Ellipse in P.

For if YZ does not touch the ellipse, let it cut it in Q; take Pp=PH, and join pH, QS, QH, and Qp. Since

Pp PH, PZ common, and pPZ=HPZ, the side pZ will be equal to ZH; and the 4PZp, PZH, will be equal, and consequently right L'. Again, since pZ =ZH, ZQ common, and 'QZp, QZH right 4, the side Qp is equal to the side QH.

E

Now

Now (by Euc. 20. 1.) SQ+Qp is greater than Sp or SP + Pp or SP+PH; but QH=Qp; therefore SQ+QH is greater than SP+PH; but if Q is a point in the curve, SQ+QH must be equal to SP+PH; Q therefore is not a point in the curve. In the saine manner it might be proved that YZ does not meet the curve in any other point on either side of P, it must therefore be a tangent at P.

SPY=

(41.) From hence it follows, that the LHPZ; for 4 SPY = vertical pPZ; but pPZ= HPZ; .. SPY=HPZ; and this is a distinguishing property of the ellipse; viz. That lines drawn from the foci to any point in the curve make equal angles with the tangent at that point.

(42.) COR. When P comes to A or M, the angle HPp becomes equal to two right angles; at A or M, therefore, the tangent is perpendicular to the axis AM.

PROPERTY 5.

If tangents be drawn at the extremities of any diameter of an Ellipse, they will be parallel to each other.

(43.) Complete the parallelogram SP HG, of which SP, PH are two sides, and join PG; then since the opposite sides of parallelograms are equal to each other, SG+GH is equal to SP+PH, and consequently G is a point in the Ellipse; and since the diagonals of parallelograms bisect

each

each other, SH is bisected in C; therefore C is the center of the Ellipse, and PG a diameter (35).

Now let the tangents ef, gh be drawn at the extremities of the diameter PG; then, by Art. 41. the SPe= HPf; but SPe+HPƒ is the supplement of L SPH;

.. SPesupplement of SPH. For the same reason, the LHGhsupplement of SGH; but the L'SGH, SPH are equal, being opposite of a parallelogram; hence theSPe=LHGh. Again, since SP is parallel to GH, the SPG=LPGH; therefore SPe+SPG=HGh +PGH, or GPe=PGh, and consequently ef is parallel to gh.