234 PHILOSOPHICAL TRANSACTIONS. [aNNO 1668. 



Therefore, in the first series, half the first term is greater than the sum of 

 the two next ; and half the sum of the second and third greater than the sum 

 of the four next ; and half the sum of those four greater than the sum of the 

 i;iext eight, &c. in infinitum. For -^ dD = br •\- hn; but hn z^fG^ therefore 

 ^dD>br -{-/Gf &c. And in the second series, half the second term is less 

 than the sum of the two next, and half this sum less than the sum of the four 

 next, &c. in infinitum. 



Then the first series are the even terms, viz. the 2d, 4th, 6th, 8th, lOth, 

 &c. and the second, the odd terms, viz. the 1st, 3d, 5th, 7th, gth, &c. of 

 the following series, viz. ^, ji^, ■^^, ji-., ^., _I^, &c. m nifini- 

 turn = 1 . Whereof a being put for the number of terms taken at pleasure, 

 — - is the last, ~-^ is the sum of all those terms from the beginning, and 



the sum of the rest to the end. 



c + I 



Also that 4- of the first term in the third series, is less than the sum of the 

 two next ; and a quarter of this sum, less than the sum of the four next ; and 

 one fourth of this last sum, less than the next eight, I thus demonstrate. 



Let a = the 3d or last number of any term of the first column, viz. of divisors. 



1 1 i(3Q? — 48a' + 56 a — 24 _ . 



«xa— 1 X« — 2 a' — 3 a* +2 a l6" a^ — 96 «' 4- 033 a-*— 288 a^ + 184 a* — 48 a 



1 1 



2ax2o— lx2a — 2 80^ — 12 a* + 4a ( l6a' — 48a* + 5(ja— 24 



\ 64a^'— 38^ 



,1 I f ~64a^'— 384a5 + 880a4— 96'0a^ + 496a— C)6" 



2fl— 2x2a— 3x2a— 4 8a3— 36a* + 52a— 24 



64 a^^— 384a^ + 928 a^ — 1152 a^ + 72,6a ^— 192 a ^ . p* 



6"4 a^ — 384 aS + 880 a* — 960a' + 496 a* — 96" a ^ "T A — D. 



And 48 a'* — 192, a^ + 240 a^ — 96 a = excess of the numerator above the 

 denominator. 



But the affirmatives >► the Negatives "J 

 That is, 48 a* + 240 a^ >► I92 a^ -\- gOal.r. 

 Because a" + 5 a^ > 4a^+ laC ^^^' 

 Or a^-l- 5a > 4 a^ -j- 2 J 

 Therefore B >- ^ A. 



respondence on mathematical subjects with Dr. Walhs, by whom this correspondence was published 

 in his Commercium Epistolicum, l658. Lord Brounker was one of the personages who signed the 

 remarkable declaration concerning King Charles II, in l66'0. After tlie restoration, he was ap- 

 pointed Chancellor and Keeper of the Great Seal to the Queen, also a Commissioner of the Navy, 

 and Master of St. Katharine's Hospital, near the Tower of London. He was one of those learned 

 men who laid the foundation of the Royal Society, of which he was, by charter in 1662, appointed 

 the first President : which office he held, witli great honour to himself and benefit to the Society, 

 till the anniversary election, Nov. 30, 1677, when he was succeeded in the presidency by Sir 

 Joseph Williamson. He died April 5, l684. — Lord Brounker made several communications to the 



