AB, of which A is the vertex; because a straight line touching the parabola in the vertex of the diameter AB, meets, as hath been proved, the diameter drawn through D, in the point where this diameter meets the circumference of the circle, the segment of which described upon the latus rectum of the diameter, passing through A, contains an angle equal to ADH; and since, in the present case, the diameter DE meets the circumference of this segment in H; therefore HA touches the parabola in A: and AB, AH being given in position, and AG given in magnitude, the parabola, according to the 18th proposition, can be described.

The composition of this case is as follows: join AD, and through D draw DE parallel to AB; to DE draw AH, bisecting the angle DAG, and through H to AB draw HG, making the angle AHG equal to ADH or DAB: and let a parabola be described which may have AB for a diameter, and AG for the latus rectum of AB; and which AH may (18. 1.) touch in A: this parabola will pass through D, and DH will touch the circle described about AHG. Draw DK parallel to AH; and since the triangles DAH, AHG are isosceles and equiangular, DH, HA, AG, and consequently KA, KD, AG, are proportionals; the square of DK is, therefore, equal to the rectangle KAG; and DK is parallel to the tangent AH: hence the point D is in the (1. cor. 13. 1.) parabola: and because the angle AHD is equal to AGH in the opposite segment, DH touches (conv. 32. 3. Elem.) the circle in H.

It remains to be inquired whether the parameter AG be greater or less than the parameter of the diameter AB in any other parabola, having AB, for a diameter, and A for the vertex of AB, and which passes through D. Let there be any other parabola admitting of these conditions; and let AE touch it in A and meet the diameter passing through D in the point E, and the circle GHA in L: having joined LG, draw EC parallel to it: draw also DF parallel to EA; DF, therefore, is ordinately applied to the diameter AB: and because the angle ADE is equal to the angle AHG or ALG, that is, to AEC, andthat the angle DEA is equal to AEC, the triangle EDA is equiangular to the triangle AEC: therefore the straight lines DE, EA, AC, that is, AF, FD, AC are proportionals; the square of DF is, therefore, equal to the rectangle FAC: and, for this reason, AC is the parameter of the diameter AB in this parabola. And because DE touches the circlein H, AL is less than AE; and therefore AG is less than AC: therefore AG is the least of all the possible parameters of the diameter AB, in parabolas which have AB for a diameter, and A for the vertex of AB, and which pass through D.

After the same manner it may be shown, that in any parabola whatever, which answers the conditions of the proposition, the latus rectum of the diameter AB is great, er or less, according as the tangent drawn through A, and situated on either side of the tangent AH, is more remote from, or nearer to, the same AH.

To proceed to the composition of what was analysed in the first case: if the proposed parameter be equal to AG, found in the manner above mentioned, the parabola, as hath been shown, may be described, and will be the only one that can fulfil what is required in the problem. If, next, the proposed parameter be less than AG, it is impossible to construct the problem: or if the proposed parameter for example, AC, be greater than AG; upon AC describe the segment of a circle containing an angle equal to ADH, or DAB; and since DH touches the circle AHG, it must cut the segment described upon AC in two points: let E be one of them; and join AE; and let a parabola (18. 1.) be described, having AB for a diameter, and AC for the parameter of AB, and to which the straight line AE may be a tangent; and draw DF parallel to AE; then it may be shown, as above, that DE, EA, AC, that is, that AF, FD, AC, are proportionals and therefore the square of DF is equal to the rectangle FAC, contained by the abscissa FA and the parameter AC; and that, consequently, the parabola passes through the point D. The same thing may be demonstrated with regard to the other parabola, which has for a tangent the straight line joining A, and the other point of intersection e. And as, in the investigation of the problem, it has been proved, that the angles GAH, HAD are equal; the angle AKD is, therefore, equal to ADK; and, of consequence, AK, is equal to AD: but AG is a third proportional to AK, KD, or to AD, DK; that is, the least parameter is a third proportional to the straight line which joins the vertex of the diameter given in position and the given point, and the straight line which is drawn from the same point to the diameter so as to cut off from the diameter a segment equal to the first proportional.

THE FIRST NINE DEFINITIONS IN THE FIRST BOOK OF APPOLLONIUS OF PERGA'S CONIC SECTIONS.

AP. DEF.

1. VIII. If a straight line joining any point and the circumference of a circle not in the same plane with the point, be produced from the point in the opposite direction, and then, while the point remains fixed, be carried round in the direction of that circumference till it return to the place from whence the motion commenced; by the revolution of that straight line, a surface, called the conical surface, and which consists of two surfaces connected together at the fixed point, will be described. The two connected surfaces may each of them be infinitely increased, if the straight line with which they are described be produced both ways to an infinite distance.

2. IX. The fixed point is called the vertex of the coni cal surface.

3. X. The straight line drawn through the point and the centre of the circie, is called the axis.

4. XI. The figure contained by the circle, and the surface which is intercepted between the vertex and the circumference of the circle, is called the cone.

5. XII. The same fixed point, which is the vertex of the surface of the cone, is named the vertex of the cone.

6. XIII. The straight line drawn from the vertex to the centre of the circle, is called the axis of the cone

E

7. XIV. And the circle itself is named the base of the

cone.

8. XV. Cones which have their axis at right angles to the base, are called right-angled cones.

9. XVI. And cones which have not their axis at right angles to the base, are called scalene cones.

PROP. XXI. (PROP. 1. B. 1. APOLL.)

Straight lines drawn from the vertex of the surface of a cone to points in that surface, are in that same surface.

Let there be the surface of a cone: let A (fig. 11.) be its vertex; and having taken any point B in that surface, join AB: the straight line AB is in that same sur

face.

For, if possible, let ACB be a straight line drawn from the vertex A to the point B, and which is not in the surface of the cone; and let DE be the straight line with which the cone is described; and the circle EF the base; and if DE be revolved in the circumference of EF, it will pass through the point B and the vertex A; and thus two straight lines ACB, AGB will have the same extremities: which is absurd. Therefore the straight line drawn from the point A to B, is not without the conical surface; therefore it is in that surface.

COR. A straight line drawn from the vertex of a cone