William Shanks

Quick Info

25 January 1812
Corsenside (8 km NE of Bellingham), Northumberland, England
13 June 1882
Houghton-le-Spring, Durham, England

William Shanks is famed for his calculation of π to 707 places in 1873, which unfortunately was only correct for the first 527 places.


William Shanks married Jane Elizabeth Pringle (1815-1904) in London in 1846. In 1847 he moved to Houghton-le-Spring, a small town in the coal-mining area of County Durham. We get more information about him from the census.

In 1851 he was living at Quality Hill, Houghton-le-Spring, with his wife, his widowed mother-in-law Sarah Pringle, and a servant Jane Calbreath. Ten years later, in 1861, he was living at a Private Boarding School in Nesham Place, Houghton-le-Spring, with his wife, mother-in-law, William Routledge (an English teacher), Ann Oliver (cook), Alice Oliver (housemaid) and eleven pupils aged between 7 and 18. In 1871 he was still living at the school with his wife, Jane Monk (cook), Catherine Potts (housemaid) and eleven pupils aged between 9 and 14. In 1881 he was still living at Nesham Place, with his wife and Jane Shaw (domestic servant).

Shanks used his leisure hours working on mathematics, particularly on calculating the decimal expansion of π. He was influenced to undertake this task by William Rutherford from Edinburgh.

In 1853 Shanks published a book entitled Contributions to mathematics, comprising chiefly the rectification of the circle. In the same year William Rutherford gave 440 decimal places in the expansion of π and, later in the same year, Shanks, in a collaboration with Rutherford, gave 530 places. This was a busy year for Shanks, for also in 1853 he gave 607 decimal places in the expansion of π which had been independently checked as correct to the first 500 of those places. At this point Shanks rested in his calculations of the decimal expansion of π, but he continued to write mathematical works. Between 1854 and 1874 Shanks published nine mathematical memoirs in the Proceedings of the Royal Society of London.

Shanks is famed for his calculation of π to 707 places in 1873, which, unfortunately, was only correct for the first 527 places. Despite these errors he did manage to correct some errors in the expansion of to 607 places which he had given twenty years earlier. The method Shanks used in his calculation was based on the formula
π4=4tan1(15)tan1(1239)\large\frac{\pi}{4}\normalsize = 4 \tan^{-1}(\large\frac{1}{5}\normalsize ) - \tan^{-1}(\large\frac{1}{239}\normalsize )
which had been discovered by Machin in 1706 and used by him to correctly calculate to 100 decimal places.

Shanks also calculated ee and Euler's constant γ to a great many decimal places. He published a table of primes up to 60,000, found the natural logarithms of 2, 3, 5 and 10 to 137 places, and the values of 212m+12^{12m+1} for m=1,2,3,...,60m = 1, 2, 3, ..., 60.

In 1944 D F Ferguson calculated π using the formula
π4=3tan1(14)+tan1(120)+tan1(11985)\large\frac{\pi}{4}\normalsize = 3 \tan^{-1}(\large\frac{1}{4}\normalsize ) + \tan^{-1}(\large\frac{1}{20}\normalsize ) + \tan^{-1}(\large\frac{1}{1985}\normalsize ).
He found that his value disagreed with that of Shanks in the 528th place. Ferguson discovered that Shanks had omitted two terms which caused his error. Of course calculating π to 707 places is now a trivial matter using a computer algebra package such as Maple. Shanks spent many long tedious days calculating; he would calculate new digits all morning and then he would spend all afternoon checking his morning's work. Now at a press of a button we get 707 places:
Shanks died in June 1882 and was buried at Houghton Hillside Cemetery on 17 June.

References (show)

  1. T A A Broadbent, Biography in Dictionary of Scientific Biography (New York 1970-1990). See THIS LINK.
  2. A Fletcher, J C P Miller and L Rosenhead, An index of mathematical tables (London, 1962).

Additional Resources (show)

Cross-references (show)

Written by J J O'Connor and E F Robertson
Last Update July 2007