Or 
On 
in Mathematical Reasoning. 3 
whence 
Oy + v 
Nn = 2; anaes 7 ae 
under which circumstance the value of sin 6, becomes really in- 
determinate, not depending even on the vajue of 7 the radius of 
the circle. 
This indeterminate case suggests the following porism: 
Any three points in a straight line being given, another point 
may be found about which as a centre if a circle with any radius 
be drawn, and if through the three given points, three chords be 
drawn in any direction, but parallel to each other; then the sum 
of the squares of two of them shall be always equal to the square 
of the third. 
It is to be observed, that the origin of the lines denoted by 
v, %, v2, may be changed by removing it to the distance a, then 
the latter of the two conditions which rendered the problem in- 
definite becomes 
2(v, + a)? = (v + a) + (v + a)’, 
whence 
ed 20, > or — 8 j 
20+ 2%, — 4%, 
Before I conclude my observations on this subject, which 
may perhaps be considered as a digression from that which the 
title prefixed to this Essay would seem to imply, I shall offer 
one more illustration of the division of a problem into the several 
stages which I have pointed out. 
This examination of all the circumstances attending the 
equation containing the solution, is stilk more necessary when 
that equation is a differential one: if it be only capable of in- 
tegration by means of transcendents or by approximating series, 
it sometimes happens that some relation amongst the data may 
be assumed, by which in the one case the transcendents shall 
Vol. If. Part IL. Zz 
