3. If through the focus of a parabola a semi-ordinate be applied to the axis, and from its extremity a tangent be drawn to meet another semi-ordinate produced; then shall the produced semi-ordinate be equal to the line joining its extremity in the curve and the focus.

4. If a tangent be drawn from any point in the curve, to meet the axis produced, and from the point of contact a perpendicular to the tangent, and a semi-ordinate to the axis, be drawn; then the segment of the axis between the perpendicular and semi-ordinate will be equal to half the parameter of the axis, and the segment between the perpendicular and tangent equal to half the parameter of the diameter which passes through the point of contact.

5. To find the directrix and focus of a parabola given in position.

6. If from the vertex of any diameter, a straight line be drawn to the extremity of a semi-ordinate so as to meet another semi-ordinate, the latter will be a mean proportional between its segments next the diameter and the former.

7. A diameter of a parabola, the tangent at the vertex of that diameter, and a point in the curve, being given, to find the directrix and focus.

8. If from any point in a tangent, a parallel to the diameter passing through the point of contact be drawn to meet an ordinate of the same, the rectangle under the parallel, and its external segment, will be equal to the square of that part of it that is between the curve, and the line joining the vertex of the diameter, and either extremity of its ordinate.

9. The focus and directrix of a parabola being given, to draw a tangent to the curve parallel to a line given in position that is not perpendicular to the directrix.

10. If there be two tangents to a parabola, such, that the straight line joining their points of contact pass through the focus, they will cut each other at right angles, their intersection will be in the directrix, and the line joining it with the focus will be perpendicular to the line joining the points of contact.

11. From a given point, in a given parabola, to draw a tangent without finding the focus.

12. Two parabolas of equal parameters, having their axes in the same line, and in the same direction, but their ver

tices at different points, being produced indefinitely, continually approach, but never meet.

13. In a given parabola to find a diameter that makes a given angle with its ordinates.

14. If from any point in a tangent, a straight line be drawn, to meet the curve and the diameter passing through the point of contact, the square of its segment, between the tangent and the diameter, will be equal to the rectangle under its segments between the tangent and the points in the curve.

15. The focus and directrix of a parabola being given, to draw a tangent to the curve from a given point without it.

16. If from a point in a parabola, a semi-ordinate be applied to a diameter, and from the same point any other straight line be drawn to meet it, the square of this line will be equal to the rectangle under the absciss of that diameter, and the parameter of the diameter to which the straight line, when produced, is an ordinate.

17. If from points in the curve, two tangents be drawn to meet, their squares will be to each other as the parameters of the diameters passing through the points of contact.

18. If two tangents, and the line joining their points of contact, meet the same diameter, the segment of the diameter, intercepted by this line from the vertex, will be a mean proportional between those intercepted by the tangents.

SECOND BOOK.

ELLIPSE.

DEFINITIONS.

1. If two points be given in position, the locus of a point, the sum of whose distances from them is always the same, is a curve called an ellipse.

2. The given points are named the foci, and the middle of the line that joins them, the centre of the ellipse.

3. The distance of the centre from one of the foci is called the eccentricity.

4. A diameter is a straight line drawn through the centre, and terminated on both sides by the curve.

5. The diameter which passes through the foci is named the transverse or major axis, and that which is perpendi cular to it, the conjugate or minor axis.

6. An ordinate to a diameter is a straight line not passing through the centre, but terminated by the curve, and bisected by the diameter.

7. Two diameters are said to be conjugate to one another when each is parallel to the ordinates of the other.

8. The parameter of a diameter is a third proportional to that diameter and its conjugate.

PROPOSITION I.

If from any point in an ellipse, straight lines be drawn to the foci, their sum is equal to the transverse axis.

Let EFGH be an ellipse, of which EF is the transverse, and GH the conjugate axis,

A, B, the foci, and D any point in the curve; then AD + DB-EF.

For EA+EB-FA+FB; E hence AE= BF, and AD + DBEA+ EB (II. Def. 1) =EF.

COR. 1. If two straight

A

H

B

F

lines be drawn to the foci, from a point without the ellipse, their sum is greater than the transverse; but, if from a point within it, less.

For AP+PB = AD+ DP+ PB. But DP+PB7 DB; therefore AP+PBAD+ DB, or AP+PB7 EF. And it is similarly shown that AQ + QB <EF. COR. 2.-A point is without, in, or within the curve, according as the sum of the lines drawn to the foci is greater, equal, or less, than the transverse axis. COR. 3.-The distance of either extremity of the conjugate axis, from one of the foci, is equal to half the transverse.

For if GB be joined, the triangles ACG, BCG, are equal (II. Def. 2 and 5, and Pl. Ge. I. 4); therefore AG = GB,

But EF AG + GB=2AG or 2GB. proved that EF = 2AH or 2BH.

It is similarly

COR. 4.-The transverse and conjugate axis are bisected in the centre.

ACCB (II. Def. 2), and EA =BF; therefore ECCF. Also in the triangles ACG, ACH, AG = AH, therefore the angles at G and H are equal, those at C are right angles, and AC is common; hence (Pl. Ge. I. 26) CG = CH. COR. 5.-A perpendicular to the transverse at one of its extremities is a tangent to the ellipse. For if I be a point in that perpendicular, AI BIBF; therefore AI + IB AF + FB or EF. COR. 6.-The square of half the conjugate axis is equal to the rectangle under the segments into which the transverse is divided in one of the foci.

AF,

and

For CG2 AG2 — AC2 — EC2 AC2 EA AF, or EB BF (Pl. Ge. II. 5, Cor.)

COR. 7.-The distance of the foci is a mean proportional between the sum and difference of the transverse and conjugate axis.

For AC2 = AG2 GC2= (AG+GC) (AG — GC) (Pl. Ge. II. 5, Cor), or AG + GC: AC=AC: AG - GC; and the doubles of these terms are also proportional.

PROPOSITION II.

The straight line which bisects the angle adjacent to that which is contained by two straight lines, drawn from any point in the ellipse to the foci, is a tangent to the curve in that point.

Let D be any point in the curve, from which AD, DB, are drawn to the foci, and let the angle BDI adjacent to ADB, be bisected by the line DT; then is DT a tangent to the ellipse in the point D.

For, let any other

E

B

N

point R be assumed in DT, and DI being made equal to DB, let BI, RA, RB, and RI, be drawn. The line DNT, which bisects the vertical angle of the isosceles triangle BDI, also bisects the base BI at right angles. Hence RBRI (Pl. Ge. I. 4), and AR + RB = AR + RI, and therefore greater than AI (Pl. Ge. I. 20) or EF (II. 1). Consequently the point R is without the ellipse (II. 1, Cor. 2). COR. 1.-A perpendicular to the conjugate axis, at one of its extremities, is a tangent to the ellipse.

For this line bisects the angle adjacent to that formed by lines drawn to the foci from this point.

COR. 2. The method of drawing a tangent from a given point in the curve, also of drawing a tangent parallel to a line given in position, is evident.

When the tangent is to be parallel to a given line, draw from the focus B a line BI perpendicular to the line, and make AI EF; join AI, and D will be the point of contact. COR. 3.-There cannot be more than one tangent to the ellipse at the same point.

=

For the sum AD + DB of the lines from A and B, to a point D in RN, which make equal angles with it, is less than the sum of any other two lines drawn from A and B to any other point as R in RN. Now, if another tangent can be drawn through D, AD and DB would not make equal angles with it, but some other two lines AD', D'B, to some point D' in it, would make equal angles with it. But AD' +D'B would be less than AD + DB, and hence the point D' would be within the curve (II. 1, Cor. 2), and the line would not be a tangent.

COR. 4.-Every tangent bisects the angle adjacent to that contained by straight lines drawn to the foci from the point of contact, or, which is the same, these lines make equal angles with the tangent.

COR. 5.-A straight line drawn from the centre to meet a tangent, and parallel to the line joining the point of contact, and one of the foci, is equal to half the transverse axis.

For, since BCCA, and BN = NI, the line that joins CN will be parallel to AD, and equal to the half of AI. COR. 6. A perpendicular to a tangent, from one of the