William Shanks and His 707 Digits of π

An annotated reading list

Rutherford and Shanks on π

William Rutherford was apparently Shanks's schoolmaster in the 1820s, when Shanks was a boy. In the early 1840s Rutherford published a 208-decimal-place value of π; only 152 places were correct, but that was still a record at the time. A decade later, Rutherford and his former pupil Shanks collaborated on a 441-digit calculation, and Shanks went on alone to 530 decimal places. Just two months later Shanks extended the value to 608 places, but errors marred these figures from the start. After a hiatus of 20 years (and after the death of Rutherford), Shanks went on to 707 decimal places.

Rutherford, William. 1841. Investigation of a new and simple series, by which the ratio of the diameter of a circle to its circumference may easily be computed to any required degree of accuracy. Proceedings of the Royal Society of London 4:302. A brief abstract discussing the merits of various arctan series.

Rutherford, William. 1841. Computation of the ratio of the diameter of a circle to its circumference to 208 places of figures. Philosophical Transactions of the Royal Society of London 131:281–283. Presents the result and includes some discussion of computational procedures.

Rutherford, William. 1853. On the extension of the value of the ratio of the circumference of a circle to its diameter. Proceedings of the Royal Society of London 6:273–275. The paper is signed by Rutherford alone, but it presents joint work with Shanks. They each computed 441 digits of π, using Machin's arctan formula, and cross-checked their results. Shanks went on the 530 digits, all correct apart from minor typographical errors and questionable rounding in the last two digits.

Shanks, William. 1853. Contributions to Mathematics, Comprising Chiefly the Rectification of the Circle to 607 Places of Decimals. London: G. Bell. (Full text available on Google Books.) This small book gives Shanks's most thorough account of his work on π. Most of it is taken up with listing all the terms in the summations for arctan 1/5 and arctan 1/239 in the 530-decimal-place computation of π. Then at the end Shanks extends the two arctangent computations to 609 places, and π to 607 places. However, the individual terms are not listed for these higher-precision values. That's unfortunate, because this is where Shanks began making errors, and they would be much easier to identify if we had all the terms. The book was privately printed; a list of subscribers documents sales of 36 copies.

Shanks, William. 1873. On the extension of the numerical value of π. Proceedings of the Royal Society of London 21:318–319. The extension is to 707 places, but because errors were introduced around decimal place 530, all the later digits are worthless.

Shanks, William. 1873. On certain discrepancies in the published numerical value of π. Proceedings of the Royal Society of London 22:45–46. A note explaining and correcting some isolated errors of transcription. But the major flaw was not discovered until long after Shanks's death.

Other Publications of William Shanks

All of these papers appeared in the Proceedings of the Royal Society of London. If Shanks published elsewhere, I have found no evidence of it.

Shanks, William. 1854. On the extension of the value of the base of Napier’s logarithms; of the Napierian logarithms of 2, 3, 5, and 10; and of the modulus of Briggs’s, or the common system of logarithms; all to 205 places of decimals. Proceedings of the Royal Society of London 6:397–398. Shanks was a pi man, but not just a pi man.

Shanks, William. 1866. On the calculation of the numerical value of Euler’s constant, which Professor Price, of Oxford, calls E. Proceedings of the Royal Society of London 15:429–432. Although Professor Price called it E, everyone now calls it γ. The computation was a troublesome one. The second, third, and fourth "supplementary" papers listed below recount Shanks's less-than-successful struggle to establish the correctness of his figures. His problems were finally explained and resolved by J. W. L. Glaisher in 1870; Shanks's "second paper" in 1871 is a response.

Shanks, William. 1867. Second supplementary paper on the calculation of the numerical value of Euler’s constant. Proceedings of the Royal Society of London 16:154.

Shanks, William. 1868. Third supplementary paper on the calculation of the numerical value of Euler’s constant. Proceedings of the Royal Society of London 16:299–300.

Shanks, William. 1869. Fourth and concluding supplementary paper on the calculation of the numerical value of Euler’s constant. Proceedings of the Royal Society of London 18:49.

Shanks, William. 1871. Second paper on the numerical values of e, log_e(2), log_e(3), log_e(5), and log_e(10); also on the numerical value of M, the modulus of the common system of logarithms, all to 205 decimals. Proceedings of the Royal Society of London 20:27–29. In 1870 J. W. L. Glaisher published a harsh critique of Shanks's computations of e, γ and other constants. This paper and the one that follows constitute Shanks's reply.

Shanks, William. 1871. Second paper on the numerical value of Euler’s constant, and on the summation of the harmonic series employed in obtaining such value. Proceedings of the Royal Society of London 20:29–34.

Shanks, William. 1873. On the number of figures in the period of the reciprocal of every prime number below 20,000. Proceedings of the Royal Society of London 22:200–210. After finishing with π, Shanks embarked on another heavy-duty computational project, compiling a table of the periods of the reciprocals of prime numbers, expressed as decimal fractions. Eventually he got as far as all primes less than 60,000. Shanks's interest in this subject may have grown out of his computations of π and other constants, where the periods of the reciprocals of primes affect how much labor is needed to determine the value of a quatient.

Shanks, William. 1873. Given the number of figures (not exceeding 100) in the reciprocal of a prime number, to determine the prime itself. Proceedings of the Royal Society of London 22:381–384.

Shanks, William. 1873. On the number of figures in the reciprocal of every prime between 20,000 and 30,000. Proceedings of the Royal Society of London 22:384–388.

Shanks, William. 1875. On the number of figures in the reciprocal of each prime number between 30,000 and 40,000. Proceedings of the Royal Society of London 23:260–261.

Shanks, W. 1876. On the number of figures in the period of each reciprocal of a prime from 53,000 to 60,000. Proceedings of the Royal Society of London 24:392.

Shanks, William. 1876. Remarks chiefly on 487² ≡ 486. Proceedings of the Royal Society of London 25:551–553. The notation used in this paper is oblique and opaque, and I don't understand parts of the argument. But Shanks seems to be pointing out that 487 is a full-repetend prime (i.e., the decimal period of 1/487 is 486) and it has the unusual property that 1/487² also has period 486.

General Works on π

Arndt, Jörg, and Christoph Haenel. 2001. Pi Unleashed. Translated from the German by Catriona and David Lischka. Berlin: Springer. Lots of lore! One of the best popular works.

Bailey, David H., Jonathan M. Borwein, Peter B. Borwein, and Simon Plouffe. 1997. The quest for pi. Mathematical Intelligencer 19(1): 50–57. Preprint at http://www.davidhbailey.com/dhbpapers/pi-quest.pdf. History of π computations, with emphasis on the 20th century.

Bailey, David H., and Jonathan Borwein. 2014. Pi Day is upon us again and we still do not know if pi is normal. American Mathematical Monthly 121(3):191–206. Preprint available at http://www.davidhbailey.com/dhbpapers/pi-monthly.pdf. A normal number has a uniform distribution of digits in any base. Everyone knows that π is normal, but no one has been able to prove it.

Beckmann, Petr. 1971. A History of Pi. New York: St. Martins Press. A well-known and widely available work, but I find it disappointing.

Berggren, Lennart, Jonathan Borwein, and Peter Borwein (editors). 2004. Pi, A Source Book. Third edition. New York: Springer. Huge compendium, reprinting many of the essential documents, including an excerpt from Shanks's 1853 book.

Tweddle, Ian. 1991. John Machin and Robert Simson on inverse-tangent series for pi. Archive for History of Exact Sciences 42(1):1–14. Mostly a polemic arguing for Simson's priority, but along the way there's much useful material on series evaluations.

Wrench, J. W. Jr. 1960. The evolution of extended decimal approximations to π. The Mathematics Teacher 53(8): 644–650. First-rate tutorial on arctan series.

Biographical Material on Shanks and Rutherford

Broadbent, T. A. A. 2008. Shanks, William. Complete Dictionary of Scientific Biography. http://www.encyclopedia.com/doc/1G2-2830903991.html A standard account of Shanks's computation of π, but with too little biographical detail.

Lanagan, Paul. Undated. Maths teacher dedicated to reaching for the pi. The Houghton Star. http://www.houghtonlespring.org.uk/­houghtonstar/­houghtonstar_articles/­houghtonstar_williamshanks_190613.­jpg Shanks's legacy as reported in his home-town newspaper.

O'Connor, J. J., and E. F. Robertson. 2007. William Shanks. In the Mactutor History of Mathematics, University of St. Andrews. http://www-history.mcs.st-andrews.ac.uk/Biographies/Shanks.html The best available biography. O'Connor and Robertson were enterprising enough to explore census records, finding out how many pupils were enrolled in Shanks's boarding school.

Sedgwick, William Fellowes. 1885–1900. Rutherford, William. In Dictionary of National Biography, Vol. 50, Oxford Biography Index Number 101024365. http://www.oxforddnb.com/index/24/101024365/

Commentary on Shanks from his Contemporaries

De Morgan, Augustus. 1866–1873. Table. In The English Cyclopaedia, London: Bradbury, Evans, Section 4, Vol. 7, pp. 976–1016. A long, detailed, fascinating article on all aspects of mathematical tables, with a few paragraphs on Shanks's π calculation. See p. 1006.

De Morgan, Augustus. 1872. A Budget of Paradoxes. London: Longmans, Green and Co. This is De Morgan's catalogue of fallacies and follies. The fact that Shanks is mentioned in this context might lead one to suppose that De Morgan had a low opinion of his work. But in fact De Morgan writes: "These tremendsous stretches of calculation—at least we call them so in our day—are useful in several respects. They prove more than the capacity of this or that computer for labour and accuracy; they show that there is in the community an increase in skill and courage. We say in the community: we fully believe that the unequalled turnip which every now and then appears in the newspapers is a sufficient presumption that the average turnip is growing bigger, and the whole crop heavier. All who know the history of the quadrature are aware that the several increases of numbers of decimals to which π has been carried have been indications of a general increase in the power to calculate, and in courage to face the labour." (The commentary on Shanks appears on pp. 290–293. Much of it is extracted from the Cyclopaedia article on tables.)

Glaisher, J. W. L. 1870. On the calculation of Euler’s constant. Proceedings of the Royal Society of London 19:514–524. J. W. L. Glaisher was the enfant terrible of British mathematics in the later 19th century. At the time he wrote this polite but merciless critique of Shanks's work, he was 21 or 22 and still an undergraduate at Cambridge. He demolishes all of Shanks's claims about the accuracy of his calculation of Euler's γ constant.

Erwin Engert's Commentary on Shanks and π

The last time I mentioned William Shanks, I heard from Erwin Engert, who has taken a do-it-yourself approach to understanding Shanks's work; he has performed hand calculations of π to 20 and 40 decimal places. He also has a careful analysis of Shanks's errors. Engert's papers are on his website at http://www.engert.us/erwin/Miscellaneous.html

Engert, Erwin. Undated manuscript. William Shanks 707 digits.

Engert, Erwin. Undated manuscript. Hand calculation of pi to 20 digits.

Engert, Erwin. Undated manuscript. How to calculate pi to 40 digits.

Other Computations of π

Selected works up to 1950, when electronic computers utterly changed the game.

Ballantine, J. P. 1939. Questions, discussions, and notes: The best (?) formula for computing pi to a thousand places. American Mathematical Monthly 46(8):499–501. A follow-up to the 1938 paper by D. H. Lehmer listed below. When Ballantine speaks of a formula suitable "for machine computation," he's talking about mechanical desk calculators.

Camp, C. C. 1926. Questions and discussions: A new calculation of π. American Mathematical Monthly 33(9):472–473. Camp uses a desk calculator to get 56 digits from an arctan formula, showing the individual terms of the series and discussing how best to compute them.

Ferguson, D. F. 1946. Value of π. Nature 157:342. A very brief note reporting a discrepancy between Shank's value of π and the new computation Ferguson was then completing.

Ferguson, D. F. 1946. Evaluation of π. Are Shanks' figures correct? The Mathematical Gazette 30(289):89–90. A fuller account of Ferguson's confrontation with Shanks. Working with a desk calculator, it took him a year to get up to 530 decimal places. "Up to this point, whenever I had disagreed with Shanks' figures (and this has occurred from time to time, owing to copying errors, etc.), I had never had any real difficulty in finding where I had gone wrong. But at this point I not only found my figures differing completely from those of Shanks, but all my efforts to find my mistake failed." He spent another four months checking his work by means of a different series summation before venturing the opinion that Shanks might have erred.

Ferguson, D. F., and John W. Wrench, Jr. 1948. A new approximation to pi (conclusion). Mathematical Tables and Other Aids to Computation 3(21):18–19. Follow-up and corrections to the paper by Smith, Wrench, and Ferguson of 1947; see below.

Lehmer, D. H. 1938. On arccotangent relations for π. American Mathematical Monthly 45(10):657–664. Defines a figure of merit for arctangent formulas for high-precision computations of π.

Reitwiesner, George W. 1950. An ENIAC Determination of π and e to more than 2000 decimal places. Mathematical Tables and Other Aids to Computation 4:11–15. Welcome to the machine age: 2,035 digits over the Labor Day weekend in 1949.

Shelburne, Brian J. 2012. The ENIAC’s 1949 determination of π. IEEE Annals of the History of Computing 34(3):44–54. A retrospective with much interesting analysis of what numerical computing was like at the dawn of the electronic age.

Smith, L. B., J. W. Wrench, and D. F. Ferguson. 1947. A new approximation to pi. Mathematical Tables and Other Aids to Computation 2(18):245–248. This paper is the culmination of the pre-electronic-computer era in pi-making. The authors report the results of two independent calculations of π to more than 800 digits. But there is an ironic echo of the Shanks affair: An error corrupted the digits beyond the 722nd decimal place. (The error is corrected in the paper by Ferguson and Wrench listed above.)

Cyclic Numbers

Cyclic numbers and the primes whose reciprocals generate them enter this story through their apparent involvement in one of Shanks's mistakes. But it should be noted that Shanks himself was interested in the topic, as evidenced by his tables of the periods of the reciprocals of primes up to 60,000.

Gardner, Martin. 1970. Mathematical games: Cyclic numbers and their properties. Scientific American 222(3):121–124. The focus is on magic tricks and other recreational aspects, but Gardner also explains what it's all about. Shanks is mentioned.

Glaisher, J. W. L. 1878. On circulating decimals with special reference to Henry Goodwyn’s “Table of circles” and “Tabular series of decimal quotients.” Proceedings of the Cambridge Philosophical Society 3:185–206.

Murty, M. Ram. 1988. Artin’s conjecture for primitive roots. Mathematical Intelligencer 10(4):59–67. For the most part, we can study Shanks's work while safely splashing in the shallowest tidepools of mathematics. But here we go right off the deep end.

Guttman, Solomon. 1934. On cyclic numbers. American Mathematical Monthly 44:159–166.

Rao, K. Subba. 1955. A note on the recurring period of the reciprocal of an odd number. American Mathematical Monthly 62:484–487.