As P recedes from A, it is evident that the angles P D L, P L
D, will increase, until D L assumes the position D T tangent to the
given circle, when they will become right angles. P will therefore be
infinitely remote, and the point I having then reached t, where D T
touches the smaller circle, C t S will be an asymptote to the curve.
This shows that the measurements from the convex arc, for the
construction of A P, are made only from the portion F T of the given
circumference.
In the diagram the point Q is so chosen that D L produced passes
through E, so that Q J, the tangent at Q, is parallel to P I. It will
thus be seen that the measurements from the concave arc, for the
construction of B Q, are confined to the portion G T of the given
circumference. As D L E rises, the points P and Q recede from A and B,
the points L and E approach each other, finally coinciding at T; at
this instant I and J fall together at t, so that S S is the common
asymptote to A P and B Q.
In Fig. 2 the given point D lies within the circumference of the given
circle. Bisect D F at A, and D G at B; about D describe an arc with
any radius D P greater than D A, and about O another, with radius O P
= O F--D P, these arcs intersect in P, and producing O P to cut the
circumference in L, we have P D = P L. Similarly E D = E H, U D = U W,
etc. And since P D + P O = L P + P O, D E + E O = H E + E O, and so
on, the curve is obviously the ellipse of which the foci are D and O,
and the major axis is A B = F O, the radius of the given circle.
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