III. On certain discrepancies in the published numerical value of π

Abstract

The author’s attention has been recently drawn, by Mr. John Morgan, of Bishopbriggs, Glasgow, to two cases of discrepancy in the author’s published value. of π, and to the misprint of a figure in the value of tan -1 ⅕ (one of the arcs employed in determining π), as given in the author’s paper of April 1873. These two discrepancies in the value of π, which will be described presently, did not appear in the paper presented by the late William Rutherford in 1853, wherein the author’s extension to 530 decimals was correctly given. The source of the two cases of error was easily discovered, on referring to the author’s manuscript of 1852; for it was there found that the 461st and 462nd decimals in the value of tan -1 ⅕ were, in the process of careful revision previous to publication, altered from 88 to 96; and this alteration required the addition of 128 in the corresponding place of the value of π. This addition was rightly made in the value sent to Mr. Rutherford, and given, as above stated, in his paper. It should seem that the author unfortunately did not take a copy (for his own use) of what he forwarded to Mr. Rutherford. At all events, from some strange cause or accident (perhaps from being overworked) the addition of 128 was very soon afterwards made at the 513th, 514th, and 515th, instead of at the 460th, 461st, and 462nd decimal places. Hence arose the double error, which remained, strange to say, undetected for upwards of twenty years! This mistake, however, from its nature, in no wise affects the accuracy of the other figures.