126 Treatment of Geometry as a branch of Analysis. [No 134. 



o 2 

 Commence with the angle 60° or v = ~ when c = 1, and calcu- 

 li 



late successively c [J], c [£], 2 &c. ; and we shall find the series of perime- 

 ters given above approach the limit 1.047197551 1 .. x *•> which is con- 

 sequently the length of the arc of 60° ; call it -. r, then n = 3.14159... 



o 



and the circumference of the circle is 2 7rr, and its area is irr ; prov- 

 ing circles to be as the squares of their radii, (XII. 2 ). 



Now recurring to the general formula for arc and sector ; if 9 = 4, 



the arc becomes 277-?* and the sector v r* ; hence m = ,-, and n = _ ; 



2 4 



consequently / = 7j-r0 and S = ^r 2 Q 



26. In conclusion, by freely applying the principle of limits, the 

 pyramid is treated as the limiting value of a series of inscribed prisms, 



Bh 2 Bh . ,,_ Bh . ... Bh __ Bk io> 



— • «*' -3.(«-l) 2 , -3 («"2) 2 , — . 22 » —.12 



/a 3 n? nr n 6 n 3 



where B is the base of the pyramid, and h its altitude, and n the 

 number of inscribed prisms ; the sum of the series is 



Bh (i» + l)n(2n— 1) B£ 1_ 1 



rc 3 ' 2. 3 ' U ^ ~*~ n)^ n> 



At the limit n is infinite, and the series completes the pyramid. Therefore 

 Pyramid = ^ base X altitude. 

 This involves (XII. 3, 4, 5, 6, 7, 8, 9). The case of similar pyra- 

 mids, (XII. 8,) is done by transformations into similar parallelopipeds. 

 Cylinders are the limits of polygonal prisms inscribed in them ; and 

 cones, those of the inscribed pyramids. Their properties are therefore 

 the same as those of prisms and pyramids, their circular bases permit- 

 ting a definite reference to the homologous lines, the radii. (XII. 10, 11, 

 12, 13, 14, 15.) 



Lastly, the sphere is determined solely by its radius. Hence the 

 volume of the sphere bears a determinate ratio to the cube of its radius. 



27. Modern authors of the highest repute have concurred in deduc- 

 ing the theory of Trigonometry from the definitions of sin and cos, 

 which I have adopted at the commencement of this paper, introducing 

 the functions tan, sec, &c. as convenient abbreviations, but without any 

 reference to their geometric meaning. (See Peacock's Report on Ana- 

 lysis. Brit. Assoc. 1833, page 291.) 



