COR. 2. The parameter of the axis is also a third proportional to IE and CK.

The abscisses of any diameter are as the squares of their ordinates.

For, draw the tangent CT, and the externals EI, AT, NO, &c. parallel to the axis, or to the diameter CS.

Then, because the ordinates QE, RA, SN, &c. are parallel to the tangent CT, by the definition of them,

therefore

therefore all the figures IQ, TR, OS, &c. are parallelograms, whose opposite sides are equal,

namely, IE, TA, ON, &c.

are equal to CQ, CR, CS, &c. Therefore, by Prop. IX. CO, CR, CS, &c.

are as CI, CT, CO2, &c.

or as their equals, QE, RA', SN2, &c.

Q. E. D.

COR. Here, as in Prop. II. the difference of the abscisses are as the difference of the squares of their ordinates, or as the rectangles under the sum and difference of the ordinates, the rectangle under the sum and difference of the ordinates being equal to the rectangle under the difference of the abscisses and the parameter of that diameter, or a third proportional to any absciss and its ordinate.

PROPOSITION XI.

If a line be drawn parallel to any tangent, and cut the curve in two points, then if two ordinates be drawn to the intersections, and a third to the point of contact, these three ordinates will be in arithmetical progression, or the sum of the extremes will be equal to double the

Then, by sim. tri. EK:

For, draw EK parallel to the axis, and produce HI to L. HK :: TD or 2AD: CD; : HK KL P the param.

but, by Prop. II. EK
therefore, by equality, 2AD:

::

:

KL :: CD : P.

But, by Prop. I. Cor. 1, 2AD; 2CD

:: CD : P;

therefore, the second terms are equal, KL=2CD, that is, EG+HI2CD.

PROPOSITION XII.

Q. E. D.

Any diameter bisects all its double ordinates, or lines parallel to the tangent at its vertex.

For, to the axis AI draw the ordinates EG, CD, HI, and MN parallel to them, which is equal to CD. Then, by Prop. XI. 2MN, or 2CD=EG+HI, therefore M is the middle of EH.

And for the same reason all its parallels are bisected.

VOL. II.

Ебе

Q. E. D.

SCHOLIUM.

SCHOLIUM. Hence, as the abscisses of any diameter and their ordinates have the same relations as those of the axis, namely, that the ordinates are bisected by the diameter, and their squares proportional to the abscisses; só all the other properties of the axis and its ordinates and abscisses, before demonstrated, will likewise hold good for any diameter and its ordinates and abscisses. And also those of the parameters, understanding the parameter of any diameter as a third proportional to any absciss and its ordinate.

END OF CONIC SECTIONS.

DIALING.

I.

DEFINITIONS,

A DIAL is an instrument, that shews time by means of the rays of light proceeding from some celestial body. When it is constructed to shew time by means of the sun's rays, it is called a sun dial; and when by means of the moon's rays, a moon dial. The former is principally used; and is therefore to be considered as intended by the term dial, when used for a particular instrument without a distinguishing epithet.

2. DIALING, or GNOMONICS, is that branch of Mathe matics, which teaches the construction and use of dials.

3. Dial surface is a surface, on which the hour lines of a dial are drawn. This surface may be plane, and is then denominated a dial plane; or it may be convex, concave, cylindrical, &c. The most common and useful kinds of dials are those, which have the hour lines described on a plane surface.

4. Dial