[ 4'5 ] 



fquare. Make a^ + al> + i>^ = a- — ^ai> + 4l>^. Then will 5^3 = 



3^^ and 5(7=31^. Let 3 = 5 •.•^=3. The produds then with 



their refpedive fadors will he x^ ^ — j{.ox — 1 20 = .v + 3 X .v + 5 



XX — 8 ; and x^ ^ — 49X: + no = x — 3-X x — 5 x x + 8. 



And the fadors of the arithmetical mean x^ — 49X, will be x — 7, 



X, x+ y. 



« 



The fquare of the arithmetical mean, x'^ — 98 x'^ + 2401 x^, 

 differing from the produ£t of the extremes, x^ — 98 a' + 2401 

 x'' — 14400 by the abfolute term 14400. Here then we have 

 two produds of fix dimenfions, which though they have a dif- 

 ference confiderably greater, than the produds already given of 

 lower dimenfions, may yet be of fome ufe in the conftrudion of 

 logarithms. 



Having fhewn different methods of diminifhing the value 

 of the fradion — , in the application of the above general 

 feries to the conftrudion of logarithms, by which means the feries 

 will be made to converge fafter ; I proceed now to an abbrevia-- 

 tion of the feries itfelf, which will ferve to compute the logarithms 

 even of fmall numbers to a much greater degree of accuracy,, 

 with fcarce any increafe of trouble. 



