394 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
The protangent to the circle, which is exhibited in this formula, disappears in the 
actual process of integration ; while in the parabola, the protangent which is involved 
in the differential, is evolved by the process of integration. 
As in the parabola, the perpendicular, from the focus on the tangent, bisects the 
angle between the radius vector and the axis of the curve; so in the circle, the 
radius vector OB drawn from the extremity of the diameter, bisects the angle 
between the perpendicular OP and the diameter OA. 
There are some curious analogies between the parabola and the circle, considered 
under this point of view. 
In the parabola, the points T, T^ T^^, which divide the lines 
g (sec6+ tan0), g [sec (0-^-0) + tan (0->-0)], &c. 
into their component parts, are upon tangents to the parabola. The corresponding 
points B, Bp Bp in the circle, are on the circumference of the circle. 
In the parabola the extremities of the lines g (sec0+ tan0) are on a right line AN : 
in the circle, the extremities of the bent lines G (cos^+\/ —1 sin^) are all in the 
point A. 
The locus of the point T, the intersections of the tangents to the parabola with the 
perpendiculars from the focus, is a right line ; or in other words, while one end of a 
protangent rests on the parabola, the other end rests on a right line. So in the circle, 
while one end of the protangent rests on the circle, the other end rests on a cardioide, 
whose diameter is equal to that of the circle, and whose cusp is at O. OPP^A is the 
cardioide. 
The length of the tangent AT to any point T is g tan0. The length of the cardioide 
is 2G sinS. 
It is singular that the imaginary formulae in trigonometry have long been disco- 
vered, while the corresponding real expressions have escaped notice. Indeed, it was 
long ago observed by Lambert, and by other geometers — the remark has been 
repeated in almost every treatise on the subject since — that the ordinates of an equi- 
lateral hyperbola might be expressed by real exponentials, whose exponents are 
sectors of the hyperbola ; but the analogy, being illusory, never led to any useful 
results. And the analogy was illusory from this, that it so happens the length and 
area of a circle are expressed by the same function, while the area of an equilateral 
hyperbola is a function of an arc of a parabola. The true analogue of the circle is 
the parabola. 
LXVI. Let 'a be the conjugate amplitude of a and while u is the conjugate am- 
plitude, as before, of <p and X' 
JCOSW 1C0S<J> ' JCOS;)(^ ' Jcosv 
Then as 
we shall have 
