cause o a and OA are sections of parallel planes, by the plane V O A, oa is parallel to OA (IV. 12.), and consequently (II. 30. Cor. 2.), o a is to OA as Voto V O. And, in the same manner, it may be shown that o b is to OB as Vo to VO. Therefore (II. 12.) o a is to OA as ob to OB; and, because OA is equal to O B, oa is equal to o b (II. 18. Cor.). In like manner, if c be any other point in the circumference abc, and if oc be joined, it may be shown that o c is equal to o a or o b. Therefore, every point in the circumference abc is at the same distance from the point o; that is (I. def. 24.), abc is a circle of which o is the

centre.

Therefore, &c.

PROP. 12.

Every conic section QPR is the perspective projection of a circular section qpr, upon the plane of the conic section, by straight lines drawn from the vertex V; and the vertical plane of such perspective projection is the vertical plane of the conic section.

For, every straight line which is drawn from V through a point of the circumference pqr to meet the plane of the conic section, meets that plane in some point of the conic section; and there is no point of Q PR which is not in a straight line with V and some point of qpr; therefore (def. 1.), QPR is the perspective pro

R

P

jection of qpr by straight lines drawn from V. And, because the vertical plane of the conic section QPR is parallel to the plane QPR (def. 9.), that vertical plane is also the vertical plane of projection (def. 1.).

Therefore, &c.

Cor. 1. In like manner, also, every circular section qpr may be considered as the perspective projection of the conic section QPR by straight lines drawn from the vertex V.

Cor. 2. The projection of every point in the conic section may be found in the circular section, whether it be an ellipse, or a parabola, or an hyperbola (1.): for the plane which passes through V pa

rallel to the circular section, falls entirely without the cone, so that no point of the conic section is found in that plane.

Cor. 3. And so, the projection of every point in the circular section may be found in the conic section; except, in the case of the parabola, the projection of the point in which the vertical plane touches the circular section, and except, in the case of the hyperbola, the projections of the two points in which the vertical plane cuts the circular section.

PROP. 13.

Every conic section is symmetrically divided by a straight line, which is the common section of the cutting plane, and a plane which passes through the axis of the cone perpendicular to the cutting plane.

Let V be the vertex, and VO the axis of the cone, and let PQR be the conic section; from V draw VU, perpendicular to the plane PQR, and let the a plane UVO, which passes through VO, and (IV. 18.) is perpendicular to the plane PQR, cut the lat

R

F

ter plane in the straight line AM: the conic section PQR shall be divided symmetrically by the straight line A M.

Through the point A let there be drawn a plane perpendicular to the axis V O, and let it cut the cone in the circular section pqr, having the centre O (11.), and the plane QPR in the straight line AF (IV. 2.); through V draw VD parallel to AF (I. 48.). Take any point P in the conic section, join V P, and let the plane DVP cut the planes of the conic section and circle in the straight lines PQ, and p q respectively (IV. 2.); also, let these straight lines cut A M, AO in the points M, m respectively. Then, because VD is parallel to AF, it is parallel (IV. 10. Cor. 1.) to PQ and to p q, which are the common sections of planes passing through AF with the plane DVP which passes through DV: therefore, also, PQ and pq are parallel to one another (IV. 6.). Now, because the plane pqr of the circle is perpendicular to the axis V O, it is perpendicular to the plane UV O, which passes through VO (IV. 18.); and the plane P Q R is

by the construction perpendicular to the same plane UVO: therefore AF, the common section of the planes pqr, PQR, is perpendicular to the same plane (IV. 18. Cor. 2.), and, consequently (IV. def. 1.), perpendicular to each of the straight lines AM, AO which meet it in that plane. Therefore, because PQ is parallel to AF, PQ is perpendicular to AM (I. 14.); and, for the like reason, p q is perpendicular to A O. And, because the chord p q of the circle pqr is perpendicular to the radius A O, it is bisected by AO in the point m (III. 3.); but P Q is parallel to Pq; therefore (II. 30.) PQ is bisected by AM in M. And in the same manner it may be shown, that all other straight lines which are drawn in the conic section parallel to A F, or (which is the same thing) perpendicular to AM, are bisected by A M. Therefore, the straight line A M divides the conic section P Q R symmetrically.

Therefore, &c.

Cor. 1. The straight line AF, to which the bisected chords are parallel, touches both the circular section and the conic section at the point A.

Cor. 2. The

straight line AM, which divides a parabola symmetrically, is parallel to the slant side V L, in K which the vertical plane of the parabola touches the surface of the cone.

For if the vertical plane cuts the plane

L

cipal vertices of the ellipse or hyperbola, and the straight line A A', which is intercepted between them, is called the principal diameter or transverse axis of the ellipse or hyperbola.

In the parabola, the axis cuts the curve in one point, A, only; and this point is called the principal vertex of the parabola.

15. The centre of an ellipse or hyperbola, is the middle point C of the transverse axis A A'.

A parabola has no centre.

See the figures of defs. 10, 11, 12.

PROP. 14.

The hyperbola has two asymptotes, which pass through the centre, and make equal angles with the axis upon opposite sides of it.

Let V be the vertex of the cone, V O its axis, VA and V A' the slant sides in

F

E

of the circle pqr in the straight line KL, KL touches the circle, and is therefore (III. 2.) perpendicular to the radius OL; and A F was shewn in the proposition to be perpendicular to AO; also, KL is parallel to A F, because they are sections of parallel planes by the plane of the circle (IV. 12.); there fore (I. 14.) OL and OA are in the same straight line, and V L is in the plane VA O, that is, in the same plane with AM. And, because VL and AM are sections of parallel planes by the same plane, they are parallel to one another.

Def. 13. The straight line A M, which divides a conic section symmetrically, is called the axis of the conic section.

14. In the ellipse and hyperbola, the axis A M cuts the curve in two points, A, A'; these points are called the prin

M

which the conical surface is cut by a plane perpendicular to the plane of the hyperbola, A and A' the principal vertices of the hyperbola, A ef the circular section passing through the point A, O its centre, A O a' its diameter, and E AF the common section of the planes of the circle and hyperbola which, as has been already seen (13. Cor. 1.), touches both the circle and the hyperbola in the point A; also, let the vertical plane of the hyperbola cut the circular section A ef in the chord ef, and at the points e, f, let the straight lines c E, F be

drawn touching the circle; let c E and C F meet one another in c and the straight line EF in the points E and F respectively, and let the chord ef cut the diameter A O a' in L. Then, because ef and E F are sections of parallel planes by the plane of the circle Aef, ef is parallel to E F (IV. 12.); but E F is perpendicular to the diameter AO a', because it touches the circle in A (III. 2.); therefore e ƒ is perpendicular to the same diameter (I.14.), and consequently (III.3.) is bisected by it in the point L; also, because ce, cf, are tangents drawn from c to the circle A ef, they are equal to one another (III. 2. Cor. 3.), and the triangle cef is an isosceles triangle: therefore the straight line A O a', which bisects eƒ at right angles, passes through the point c (I. 6. Cor. 3.).* Join c V, and produce it to cut A A' in C (I. 14. Cor. 3.), and from C, through the points E and F, draw the straight lines CE, CF: the point C shall be the centre of the hyperbola, and the straight lines CE, CF produced, shall be asymptotes, making equal angles with the axis A A' upon either side of it.

е

Join Oe and V L. Then, because ce touches the circle in e, the triangle O ec is right-angled at e (III. 2.); and it has been shown that e L is perpendicular to the opposite side Oc; therefore, O e or O a' is a mean proportional between O L and O c (II. 34. Cor.). But OA is equal to Oa'. Therefore, the straight line Ac is harmonically divided in the points L and a' (II. 46.), and, consequently (II. def. 20. page 68.), the four straight lines VA, VL, V a', and V c, are harmonicals. But, because VL and A A' are sections of parallel planes by the plane V A A', A A' is parallel to VL (IV. 12.). Therefore (II. 49. Cor. 1.), A A' is bisected by Vc produced, and the point C is the centre of the hyperbola (def. 15.).

Again, because the hyperbolic arcs AP, A' P', are the projections of the circular arcs Apf, a'p'f, by straight lines drawn from V (12.), and that the point f of the arcs Apf, a'p'f, lies in the vertical plane Vef; the projection of the tangent cf, that is, the straight line CF produced, is an asymptote to the hyperbolic arcs A P, A' P' (6.). And, for the like reason, CE produced is an asymptote to the hyperbolic arcs A Q, A'Q'.

Lastly, because CE and Ve are sections of parallel planes by the plane C Ve, CE is parallel to V e (IV, 12.), and, for

the like reason, CF is parallel to Vƒ; and it has been already shown that A A or CA is parallel to V L; therefore the angles ECA and FCA are equal to the angles e V L and ƒV L respectively (IV. 15.); but, because the slant sides Ve, Vf, are equal to one another (IV. 8.), and that ef is bisected in L, the triangles Ve L, VfL, have the three sides of the one equal to the three sides of the other, each to each, and, consequently, the angle eV L equal to the angle V L (I. 7.); therefore, also, the angle E CA is equal to the angle FCA, that is, the angles which CE and CF make with the axis A A' upon either side of it, are equal to one another.

Therefore, &c.

Cor. 1. The asymptotes of the hyperbola are parallel respectively to the slant sides in which the vertical plane of the hyperbola cuts the surface of the cone.

Cor. 2. And as the existence of asymptotes to the hyperbola has been inferred from Proposition 6, so from Cor. 1., of the same proposition, it follows that the infinite arcs of the parabola do not admit of asymptotes; because the tangent of the circle at the point a' (see the figure of 13. Cor. 2.) lies in the vertical plane of the parabola.

PROP. 15,

A conic section cannot be cut by a straight line in more than two points; and, if a straight line touches a conic section, it shall meet it in one point only, viz. the point of contact.

For, the circular section being considered (12. Cor. 1.) as the projection of the conic section, the projection of any point, P, of the conic section, may be found in the circular section (12. Cor. 2.); and the projection of a straight line which cuts any curve in a point that may be projected, is a straight line which cuts the projection of the curve in the projection of that point (6. Cor. 2.); therefore, if it were possible that a straight line could cut a conic section in more points than two, the straight line which is its projection (2.) would cut the circular section in more points than two, which is impossible (III. 1.).

And, in the same manner, we may reason with regard to the tangent. If it were possible that a straight line which touches a conic section should meet it in any other point besides the point of contact; then, because the projections of the point of contact and any such

other point may, both of them, be found in the circular section (12. Cor. 2.), and because the projection of a straight line which touches a curve in a point that may be projected is a straight line which touches the projection of that curve (6.), the projection of such tangent would be a straight line touching the circular section, and meeting it also in another point besides the point of contact, which is impossible (III. 2.). Therefore, &c.

PROP. 16.

A straight line which is parallel to the axis of a parabola cannot cut the parabola in more than one point; and a straight line which is parallel to either of the asymptotes of an hyperbola cannot cut the hyperbola in more than one point; but, with these exceptions, if a straight line cuts a conic section in any point, it may be produced to cut it in a second point.

Let PAQ be a parabola, A M its axis, and PN any straight line which is parallel to the axis A M, and cuts the parabola in P: PN shall not cut the parabola in any other point, to what. ever distance it may be produced.

Let V be the vertex of the cone, V L the slant side in which the vertical

P

plane of the parabola touches the surface of the cone, and Ap Lq the circular section passing through A. Then, because V L (13. Cor. 2.) and PN are each of them parallel to AM, VL is pa rallel to PN (IV. 6.). And, because VL is parallel to PN, the projection pn of the straight line PN on the plane Ap Lq will, if produced, pass through the point L (2. Cor. 4), and the projection of every point in PN produced will, it is evident, be found between p and L, that is, within the surface of the cone. Therefore, every point of PN produced falls within the surface of the cone, and consequently, since the curve of the parabola is always in the surface of the cone, PN cannot be produced to meet the parabola in a second point.

But, if PN' be not parallel to the axis of the parabola, then neither is it parallel to V L, and consequently the projection pn' does not pass through the

point L. But, because PN' cuts the parabola in P, pn' likewise cuts the circle in p(6.Cor. 2.). Therefore, pn', if produced, will cut the circle in a second point q different from the point L. And, because the projection of every point, except L, may be found in the parabola (12. Cor. 3.), Q the projection of q may be found, and consequently PN' produced will meet the parabola in the point Q.

And the same reasoning, in every respect, applies to the case of the hyperbola. For, if a straight line PN be parallel to one of the asymptotes CE, it will also be parallel to the slant side Ve, to which (14. Cor. 1.) the asymptote is parallel, and, therefore, its projection pn passing through the point e, every point of PN produced will fall

N

within the surface of the cone. But if a straight line PN' be not parallel to either of the asymptotes, its projection pn' will not pass through either of the points e, f, but, because P N' cuts the hyperbola, pn' cuts the circle in p, and will cut it if produced in some other point q, the projection Q of which may be found in the hyperbola, and, conse. quently, P N produced will meet the hyperbola in a second point Q.