270

QUICK CONVERGING SERIES TRACT 18 .

that have been given in the introductions to logarithmictables. One of these methods is from the arc of 18°, thetangent of which is V(l — 2VJ-); another is from the arc of22 °a, the tangent of which is and a third is from the

arc of 15°, the tangent of which is 2— V3. All of which areevidently too complex, to afford an easy application to thegeneral series.

In order to a still further improvement of the method bythe above general series, Mr. Machin, by a very singular andexcellent contrivance, has greatly reduced the labour natur-ally attending it. I have given an analysis of his method, ora conjecture concerning the manner in which it is probableMr. Machin discovered it, in my Treatise on Mensuration ;which, I believe, is the only book in which that method hasbeen investigated, as it is repeated in the foregoing Tract.For though the series discovered by that method were pub-lished by Mr. Jones, in his “ Synopsis Palmariorum Ma-theseos,” which was printed in the year 1706, he has giventhem merely by themselves, without the least hint of themanner in which they were obtained- The result shows, thatthe proportion of the diameter to the circumference, is equalto that of 1 to quadruple the sum of the two series,

&c) and

5.239 + 7.239'

9.239'

1

33239 *

The slower of which series converges almost thrice as fast asDr. Halley’s, raised from the tangent of 30°. The latter ofthese two series converges still a great deal quicker; butthen the large prime number 239, by the reciprocals of thepowers of which the series converges, occasions such longand tedious divisions, as to counter-balance its quickness ofconvergency; so that the former series is summed with ra-ther more case than the latter, to the same number of placesof figures. Mr. Jones, in his “ Synopsis,” mentions otherseries besides this, which he had received from Mr. Machinfor the same purpose, and drawn from the same principle-