62 Colorado College Studies. 



The analytical determination of the point G presents no 

 difficulty. If the length of CD be denoted by (/, that of a tan- 

 gent from D to the given circle will be ^/d' — R-, since such a 

 tangent forms a right triangle with CD and the radius to the 

 point of contact. Therefore the distance DG is \/ d' — -B", or 

 CG is d—Vd' — R'. 



We now proceed to the construction of the quadrilateral, and 

 its inscribed circle. 



The construction has already been in substance given. A 

 chord at right angles to AD is drawn through G, and through 

 H, its extremity, a line AH is drawn, and extended to meet the 

 radical axis at Q. The distance QG, laid off on this line from 

 the point Q, fixes /, the point of contact, whence both the centre 

 c and radius ct of the circle become known. Tangents to this 

 circle are drawn from P and from P ' , and these tangents, meet- 

 ing at Pi and P/, compose the quadrilateral. 



Analytically, we apply equation (12) to determine a, which is 

 cC of the figure, having for g the known distance GC, and for jj 

 the radius of the circle at G, which is zero. Having determined 

 a, we find r from equation (11) in which we replace x by its 

 known value, — (/. The formula then is 



7-2 = E' + «•" — 2ad. 



Each process, — the geometrical and the analytical, — may 

 now be applied anew in the case of the octagon. A tangent at 

 right angles to AD is drawn to the circle whose centre is c, and 

 the chord AH' is drawn to the point where this tangent meets 

 the outer circle, etc. For the analytical computation the equa- 

 tions (VI) and (11) are again employed, but now g and p have the 

 values which were previously denoted by a and r, while a and r 

 denote the elements of the new circle inscribed in the octagon. 

 And thus the process may be carried on as many times as 

 desired. 



A practical difficulty occurs in the geometrical construction, 

 on account of the use made of the point in which AH' meets 



