88 PHILOSOPHICAL TKANSACTIONS. [aNNO 1776. 



, ' ' i ■■ ■ ' ' t* 3t ^ 11 



2. If / be taken = 4-, then the expression -r '- gives a =: -, Z> = -. Here 



the value of a = -i- gives the same expression for the value of a as the first in 

 the foregoing case, and the value o( b = ^ gives the value of a the very same as 

 in the corollary to the case above. 



„ n^ 1 . ^1 . 4t — 1 . 3.7 5 , 79 



3. Taking t = x, the expression ^-^-^ givesa = ^, b z= -, c - -, d = - —. 



Here it is evident that the value oi' c = ^ is the fittest number afforded by this 

 case; and hence it appears, that the arc of 45° is equal to the sum of the arc of 

 which the tangent is -g^-^-, and the triple of the arc of which the tangent is -i. 

 Or that 



3 ill 5 i^ S* d*" 



^ = i ^ ^^ ~ SAG '^ sTG'' ~ TIP + ^^-^ "^ 99 ^ ^^ ~ JW "^ 639'~7l9''^ 

 &c.) Which is the best theorem that we have yet found, because the number 



99 resolves into the two easy factors 9 and 11. 



4. Let now l be taken = -L; then the expression ^i ^ t . 



^~ ^ will give a = -, b = ^, c = ■^^, d = — -r--. Where it' is' evident that 

 the last number, or the value of d, is the fittest of those produced in this case; 

 and from which it appears, that the arc of 45" is equal to the difference between 

 the arc of which the tangent is ^J-^-, and quadruple the arc of which the tangent 



isx. Or that ^^ | . y ;^^^__; ^ ^^ ^ _ _ ^^.^ 



Which is the very theorem that was invented by Mr. Machin, as we have be- 

 fore mentioned. 



5. Take now t =■ i-; then the expression 



^' will give a = J, /. = e, . = |L, rf = ^', . = - ^,. Of which nam- 

 hers, it is evident, that none are fit for our purpose. 



6. Again, take t = i-, and the expression 



'^ will give a = 2, « = L;, . = il, , = H, , = .;|, , = _ .||. Neither 



are any of these numbers fit for our purpose. 



7. In like manner take r= i, so shall — ' •'■ 

 8T - 1 . _7 ,_-l7 _ 297 ,_'0"97 _ 7S^J_ /•_ '-^-O-^^ 

 -g-p— give a —-,J — ^^,c— ^.j,^, a — ^^.^g, e — ^^^^^,J 388079' 



8. And if I be taken = J,-, the expression 



qt - 1 .,, . 4 , 31 115 , 799 246'7 o 



^—-^y,A\giwea=-^,b=-,c = ^.,d=^^ye=j^^, he. 



9. Also, if we take t = ■^, the expression 



lOT-l .„ . 9 , 79 ,_ 671 ._ 5441 •H049 n 



10. Further, if we take t = -'-> the expression _ , ' 



llT-l . :■- , 49 234 , 2159 9475 „ - I) X " — > 



—— - gives a = /; = ^. c' =? jyr, d = j^, e _ ^^, &c. 



