[ 38 ] 



In a, the extremity of the diameter, DR is to DO as 2 to i ; 

 as D approaches O, this ratio continually encreafes, and in O this 

 ratio becomes i to nothing. 



Hence if the radius AO be divided into any number of 

 equal parts, and there be conftituted a feries of fradions, whofe 

 numerators are the natural numbers increafing from unity to 

 'that number of parts, and whofe denominators are the continua- 

 tion of that feries ; then the redangles infcribed in the circle 

 will be to the refpedive redlangles in the exterior fpace, ulti- 

 mately as I to the fucceflive terms of this feries beginning with 

 the leaft. Thus, fuppofe the radius divided into 8 equal parts, 

 then the ultimate ratio of the correfponding redtangles from 

 the beginning of the quadrant will be the ratio of i to the 

 terms of the following feries : 



16' 



On fimilar principles we may demonftrate the following theo- 

 rem in the Analyfis Equationum, &c. in a much more fimple and 

 elegant manner than by the method of fluxions. 



Plate III. Let ale be an ellipfe, whofe ^ tranfverfe axis is CL, 



^'°'^" i conjugate AC; let CB=x, BD=y, AC=c, and CL=t. The 



ultimate ratio of DG : GH, is the ratio of DB ; BT, .-. ulti- 

 mately, DG^ -.GW-.-.Y^ :BT\ 



y' = -^xc"-x' from the nature of the figure; 



BT = --x, and KV'^- — 



X X 



7^' 



V DG^ 



