# William Shanks

### Quick Info

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

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

### Biography

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
$\pi/4 = 4 \tan^{-1}(\large\frac{1}{5}\normalsize ) - \tan^{-1}(\large\frac{1}{239}\normalsize )$
which had been discovered by Machin in 1706 and used by him to correctly calculate to 100 decimal places.

Shanks also calculated $e$ 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 $2^{12m+1}$ for $m = 1, 2, 3, ..., 60$.

In 1944 D F Ferguson calculated π using the formula
$\pi/4 = 3 \tan^{-1}(\large\frac{1}{4}\normalsize ) + \tan^{-1}(\large\frac{1}{20}\normalsize ) + \tan^{-1}(\large\frac{1}{1985}\normalsize )$.
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:
3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706
7982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381
9644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412
7372458700660631558817488152092096282925409171536436789259036001133053054882046652138414695194151160
9433057270365759591953092186117381932611793105118548074462379962749567351885752724891227938183011949
1298336733624406566430860213949463952247371907021798609437027705392171762931767523846748184676694051
3200056812714526356082778577134275778960917363717872146844090122495343014654958537105079227968925892
354201996
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).