CONVERGING SERIES EXPRESSING THE RATIO 



BETWEEN THE 



DIAMETER AND THE CIRCUMFERENCE OF A CIRCLE. 



The following method of obtaining converging series expressing the value of 7t, 

 and the series obtained, are thought to be new.* 



Let F x be a function of x, and A 1 , A 2 , A 3 , &c, express the different order of 

 finite differences of F x for the equal intervals of x = o, x = a, x = 2;j, &c. 



Putting n = — , we have the well-known formula, 



to 



„ „ » , n(n — 1) . (n+l)n(n — 1) . 

 (1) F x = Fo + yA. 1 + ±y 2 - A 2 + ^ 1.2.3 A - 



(n+ 1) n(n— l)(n— 2) 

 1.2.3.4 



Let us now suppose that F x is the sine of the arc x, and that Fo = 0. In this 



case A 2 , A \ &c, vanish, and when x is infinitely small F x = x. Hence F s .—= u 



x 



and the preceding equation becomes 



( 2) •-x-£,*t+r&*t-r£&*' ■■■■ 



From the theory of finite differences we have in this case 



(3) A, 2i + 1 = — | — 2 (1 — cos u) 1 * sm w 



If we substitute this in the preceding expression of u, putting 



(4) a = 2 (1 — cos o) 

 we get 



/~\ • /i 1 1.2 o 1 . /£ . O 3 



(D) u=8, " fl ^T3 a + 3X5 a+ oij a ••; 



When the sine of any aliquot part of the circumference of a circle is known, Ave 

 can readily obtain the cosine of that arc, and from (4) the value of a ; and then 

 (5) gives an expression of o, and consequently of the circumference of the circle. 

 Hence, from the computed series of the continued bisections of any aliquot part 

 of the circumference of which the sine is known, an infinite number of such series 

 may be obtained. 



* A number of series of this kind may be found in Davies & Pock's Mathematical Dictionary, 

 under the head of the circle. Also in Chauvenet'a Plane and Spherical Trigonometry. 



(3) 



