39 2 Mr. Babbage’s essay towards 
of circles in a semicircle, the following question occurred 
to me. 
If in an hyperbola between its assymptotes a circle be in- 
scribed touching both asymptotes and the curve, and if ano- 
ther circle be inscribed touching the first circle, the curve, and 
one asymptote, and if this be continued as represented in the 
figure, what ratio does the area of the circles bear to that 
of the figure; and conversely, if this ratio is given, what is the 
nature of the curve ? I soon perceived the great difficulty of 
the subject, and that these and other problems similar to the 
latter of them, required the application of methods totally dif- 
ferent from any with which I was then acquainted. 
Hopeless of success, I laid aside the subject until about two 
years after, when the same difficulty recurred under another 
form. I had proposed to myself the following problem : 
What must be the nature of the curve ABC, such that if 
any point C be taken, and the ordinate CD, the normal 
CE, and subnormal DE be drawn, and if the triangle CDE be 
turned into such a position that CD may become the base and 
DE the perpendicular, if DE coincide with some new ordi- 
nate as GB, then the normal CE at the first point shall coin- 
cide with the normal BF at the second? 
