VOL. LXVI.] PHILOSOPHICAL TRANSACTIONS. SQ 



1 1 . Lastly, if we take t = -fV, the expression 



I2t — 1 . 11 , 113 41 J 419 4111 o 



gnves a = — -, i) = -^-, c = --, a ^ —— , e = .. .^„ , occ. 



12 + T ° 13' 167 73' 917 11423' 



Here it is evident, that none of these latter cases afford any numbers that are 

 fit for this purpose. And to try any other fractions less than -Jr,- for the value of 

 t, does not seem likely to answer any good purpose, especially as the divisors 

 after 12 become too large to be managed in the easy way of short division in 

 one line. 



By the foregoing means it appears, that we have discovered 5 different forms 

 of the value of a, or ^ of the semicircumference, all of which are very proper 

 for readily computing its length; viz. 3 forms in the first case and its corollary, 

 1 in the 3d case, and 1 in the 4th case. Of these, the first and last are the 

 same as those invented by Euler and Machin respectively, and the other 3 are 

 quite new, as far as I know. 



But another remarkable excellence attending the first 3 of the beforementioned 

 series is, that thev are capable of being changed into others which not only con- 

 verge still faster, but in which the converging quantity shall be W, or some 

 multiple or sub-multiple of it, and so the powers of it raised with the utmost 

 ease. The series, or theorems, here meant are these 3. 



^■^ = 2-x('-^ + I^.-7T' + ^^-) + Jx(i-^4-3-^,-^,^, + &c.) 



^•^= ^-Ii+ .^ - 71^ + ^'^•)- ^ X('-iT9+ 5^ - 7:1^' + ^^-) 



^•- = -3X(»-5^+5T^-7-^ + ^^-)+FX('-n9 + nF-749-' + ^-) 

 Now if each of these be transformed, by means of the differential series in cor. 



3, p. 64, of the late Mr. Thomas Simpson's Mathematical Dissertations, they 



will become of these very commodious forms, viz. 



^- = foX0+3^+3^+fr;+&c.)+,|x(i+5f„+,^,+iL + &c.) 

 ^- ^= 3 X ('+5:^o+3^o+7^+&-)-foX(i+3-;^+i:So+7-^o + &-•) 



3-A = foXO+3-^0+r^ + 74 + &<=-)+foX(' + 34^+5^0 + 7^+^-) 

 Where a, Q, y, &c. denote always the preceding terms in each series. 



Now it is evident that all these latter series' are much easier than the former 

 ones, to which they respectively correspond: for, because of the powers of 10 

 here concerned, we have little more to do than to divide by the series of odd 

 numbers 1, 3, 5, 7, 9, &c. 



Of all these 3 forms, the 2d is the fittest for computing the required propor- 

 tion; because that, of the 2 series' of which it consists, the several terms of the 

 one are found from the like terms of the other, by dividing these latter by 10 and 

 its several successive powers 100, 1000, &c.; that is, the terms of the one consist 

 of the same figures as the terms of the other, only removed a certain number of 



VOL. XIV. N 



