the property of a right-angled triangle, from which we learn that the angle AMB is always a right angle, in whatever part of the curve M falls (Geom. 128).

that is, the chord AM is a mean proportional between the diameter AB and the segment, or abscissa, AP (Geom. 213).

We might thus find all the other properties of the circle made known in the Elements of Geometry, by setting out always with this supposition, that the ordinates pm are respectively mean proportionals between Ap and p B.

We have counted the abscissas from the point A, the origin of the diameter, and we have had the equation y2 = ax x2. If we would count the abscissas from the centre, or in other words would take CP, Cp, &c., for the abscissas; designating these lines by z, we should have

Putting, therefore, for a this value in the equation

for the equation of the circle, the coordinates being supposed to be perpendicular, and to have their origin at the centre.

In fine, any property, which belongs essentially to every point of the curve, will give, by being translated into algebra, the same equation for the curve, at least so long as we take the same abscissas and the same ordinates; but when we change the origin or direction of the coordinates, or both, we may have a different equation; still it will always be of the same degree. We have just seen the truth of the last part of this proposition in the change of the abscissas, which, instead of y2 = a x-x2, led to

the equation y = a2 — 22; and this, being deduced from the
¦
first, has for its basis the same property. But if we were to set
out with this property, namely, that the several distances MC
are all the same, and each = a; then designating CP by z,
and PM by y, we should have on account of the right-angled
triangle MPC

which gives

y2 + z2 = ÷ a2,

the same equation as that just obtained, although deduced from a different property.

Of the Ellipse.

104. LET us now inquire what would be the curve in which Fig. 59. the sum of the distances MF + Mƒ (fig. 59), of each point from two fixed points F, f, is equal to a given line a.

To find the properties of this curve, which is called an ellipse, an equation is to he sought which shall express the relation, arising from this known property, between the perpendiculars PM, drawn from each point M to a determinate line, as Fƒ, and their distances FP, or AP, from some point F, or A, taken arbitrarily.

For this purpose, I take for the origin of the abscissas the point A, determined by applying from the middle C of Ff the line CA = a; and having made CB = CA, I designate the lines to be used, as follows, namely,

FMz, AF, supposed to be known, = c,

Bf

ƒP PB — Bƒ = AB — AP — Bƒ a — x — c. = This being supposed, the right-angled triangles FPM, ƒ PM,

give

-2

F. = PM + FP, and Mƒ=PM+ƒP,

(*) If the point M had been taken in such a manner, that the perpendicular MP would fall between A and F, then FP would be c-z; but this would produce no change in the final equation, because, in the formation of this equation, we employ only the square of FP, which is always x 2 cx + c2.

C

or

z2 = y2 + x2 and a2 ·2 α z + z2 = y2 + a2.

2 cx + c2,

-

2α x + x2

2ac+2cx + c2. Subtracting the latter of these equations from the former and suppressing a2, which is found in both members of the result, we have

Putting for z this value in the equation

z2 = y2 + x2 - 2 cx + c2,

or, making the denominator to disappear, transposing and reducing,

a2 y2 = (4 a c — 4 c2) a x — (4 a c — 4 c2) x2,

Such is the equation of the curve, in which, any point M being taken, the sum of the distances MF, Mƒ, from two fixed points F, f, is equal to a given line a.

105. This equation would enable us to describe the curve by points, if we were to give successively to x different values, as we have done above, with respect to the circle. But, the mode of proceeding being the same, we shall not repeat the calculation.

106. We can also describe the ellipse by points, in this manner; having made CB (fig. 60) = CA = a, we take Br, equal Fig. 60. to any part of AB less than Af, and from the point f, as a centre, and with a radius equal to Br, we describe arcs above and below AB, which we cut by arcs described from the point F, as a centre, and with the radius Ar; all the points M, M', M", M"", found in this manner, belong to the ellipse.

107. The fundamental property, from which we have derived the equation, furnishes also a very simple method of describing

this curve by a continued motion. Having taken the two points Fig. 60. F,ƒ (fig. 60), át pleasure, and having fixed at these points, by

means of pins, the extremities of a thread of a greater length than the distance Ff, if we stretch this thread by a style M, carrying it round at the same time, the style will trace the curve in question, since the sum of the two distances of the style from the two points F, f, will always be equal to the whole length of the thread.

108. It will be perceived then, that, if the length of the thread is taken equal to AB, the curve will pass through the two points A, B; for, since Cƒ= CF, we have AF Bf, and consequently

and

=

AF+AƒAƒ + Bƒ = a,
BF÷Bƒ = BF + AF = α.

This is made evident also by the equation; for, in order to know where the curve would meet Ff produced, we must make y = 0; now this supposition gives

which can take place in two cases, namely, when x = 0, that is, at the point A, and when x = a, that is, at the point B.

109. It is also evident from the equation, that the curve exFig. 59. tends above, as well as below, the line AB (fig. 59), and that it is the same on each side of the axis. Indeed, the equation gives

from which we learn, that for each value of x, or AP, there are two values of y, or PM perfectly equal; but, having contrary signs, they must be applied in contrary directions.

It is moreover evident, that if from the middle C of AB, we raise the perpendicular DD', the curve will be divided into two parts perfectly equal and similar; this is a consequence of the manner in which it is described; it is also a consequence of the equation. But this will be more easily perceived, after we have further considered this equation.

110. The line AB is called the transverse axis of the ellipse, and the line DD' the conjugate axis. The two points F, ƒ, are called foci; and the points A, B, D, D', the vertices of the axes, and the point C the centre.

111. If we would obtain the value of the ordinate Fm", which passes through the focus, we must suppose AP, or x = AF = ¢. In this case we shall have

This line m"m"" is called the parameter of the ellipse. The parameter, then, is less than the quadruple of the distance c of the

vertex from the focus, since its value

4 (a c- c2)

which is the

α

is obviously less than 4 c.

If we designate this value of the parameter by p, we shall have

tion of the ellipse, therefore, may be changed, by substitution,

into one of a more simple form, namely, y3 =

112. If we would know what is the value have only to suppose, in the equation

· P(α x − x2).

a

of the line CD, we

that AP, or x, is AC, or a; we shall then have

that is,

whence

CD = ac- c2 = c (a — c) = AF × BF,

AF: CD:: CD: BF.

We see, therefore, that CD, or the semiconjugate axis, is a mean proportional between the two distances of the same focus from the vertices A, B.