VOL. LXVI.] PHILOSOPHICAL TKANSACTIONS. 85 



different methods, for determining the length of the circumference to a great 

 number of figures. Among these were, Dr. Halley, Mr. Abraham Sharp, Mr. 

 Machin, and others, of our own country ; and M. De Lagny, M. Euler, &c. 

 abroad. Dr. Halley used the arc of 30°, or Virth of the circumference, the 

 tangent of which being = ^4-, by substituting ^/-^ for t in the above series, 

 and multiplying by 6, the semicircumference is = Ov^i X 



/■i \ I i i- -I — i &c.) ; which series is, to be sure, very 



simple ; but its rate of converging is not very great, on which account a great 

 many terms must be used to compute the circumference to many places of 

 figures. By this very series however, the industrious Mr. Sharp computed the 

 circumference to 72 places of figures; Mr. Machin extended it to 100; and 

 M. De Lagny, still by the same series, continued it to 128 places of figures. 

 But though this series, from the 12th part of the circumference, does not con 

 verge very quickly, it is perhaps the best aliquot part of the circumference 

 which can be used for this purpose ; for when smaller arcs, which are exact ali- 

 quot parts, are used, their tangents, though smaller, are so much more com- 

 plex, as to render them, on the whole, more operose in the application ; this 

 will easily appear, by inspecting some instances, that have been given in the in- 

 troductions ...to logarithmic tables. One of these methods is from the arc of 18°, 

 the tangent of which is \/{l — 2\/^) ; another is from the arc of 22i°, the 

 tano-ent of which is V'2 — 1 ; and a third is from the arc of 15°, the tangent 

 of which is 2 — v^ 3. All of which are evidently too complex to afford an easy 

 application to the general series. 



In order to a still further improvement of the method by the above general 

 series, Mr. Machin, by a very singular and excellent contrivance, has greatly 

 reduced the labour naturally attending it. His method is explained in Mr. 

 Maseres's Appendix to his Dissertation on the Use of the Negative Sign in 

 Algebra ; and I have given an analysis of it, or a conjecture concerning the 

 manner in which it is probable Mr. Machin discovered it, in my Treatise on 

 Mensuration; which, I believe, are the only two books in which that method 

 has been explained, as I never had seen it explained by any, till I met with Mr. 

 Maseres's book abovementioned on the Use of the Negative Sign. For though 

 the series discovered by that method were published by Mr. Jones, in his 

 Synopsis Palmariorum Matheseos, which was printed in the year 1706, he has 

 given them merely by themselves, without the least hint of the manner in which 

 they were obtained. The result shows, that the proportion of the diameter to 

 the circumference, is equal to that of 1 to quadruple the sum of the two series, 



5- X (' - 5^^ + rr^ - 7T» + 9-T« ^^- ^'"^ 4 X ^' - lib^ + ^^ - 



-i— + — — &c 



7.239'' ~ 9.'^39" 



