# Thomas Bond Sprague

### Quick Info

Born
29 March 1830
London, England
Died
29 November 1920
Edinburgh, Scotland

Summary
Thomas Bond Sprague became the most important actuary of the late 19th Century. He wrote more than 100 papers.

### Biography

Thomas Bond Sprague's father was Thomas Sprague (born about 1801) who was a stationer. His mother was Ellen Sprague (born about 1801). Thomas Bond Sprague had six younger siblings: Mary (born about 1831), William (born about 1833), Robert (born about 1834), John (born about 1835), Frances (born about 1837) and Caroline (born about 1839). Thomas Bond Sprague married Margaret Vaughan Steains, daughter of James Steains, at St Mary, Edge Hill, Lancashire on 13 April 1859. They had six children (Mary, Margaret, James, Charles, William and Alfred)

Thomas Sprague was the dominant actuary of the second half of the 19th century, his supremacy unrivalled.

Dr Sprague was the main person behind the International Actuarial Notation which was adopted unanimously at the 2nd International Congress of Actuaries in 1898 [1]. This established an international actuarial notation.

Dr Sprague was the main person behind a mortality study covering the experience of twenty U. K. life offices. This study resulted in the Institute of Actuaries Life Tables (the so-called Twenty Offices Table) which was published in 1869 [2]. From this study, he produced, in 1879, the first Select Tables of Mortality [3] which were the first two-dimensional mortality tables ever published (the two dimensions being 'insured duration' i.e. the 'select period' and 'age attained'). The 'select period' was five years.

Dr Sprague pioneered the important 1870 Life Insurance Companies Act [4] which was introduced following the notorious insolvencies of both the Albert and the European life assurance companies. The 1870 Act required:-
... an investigation into the financial condition of a life insurance company to be made regularly by an actuary,
required a separate "long-term fund" and required the:-
... preparation of a revenue account and balance sheet every year in prescribed form to be filed with the Board of Trade,
the latter being a public document. Dr Sprague was one of the foremost advocates of the principle of 'Freedom with Publicity' (i.e. documents available to the public) and was opposed to there being any Government regulation prescribing the manner of valuation of policy liabilities. He wrote the major 19th century work on the preparation of life office accounts in conformity with the 1870 Act [5].

Dr Sprague published some eighty papers in the Journal of the Institute of Actuaries and, including articles, Presidential Addresses, translations, comments and reviews for this and other actuarial journals (Transactions of the Faculty of Actuaries, Transactions of the Actuarial Society of Edinburgh), his published contributions to actuarial science must number over one hundred. In addition, he published eight pure mathematical papers: [16] and [17] in the Transactions of the Royal Society of Edinburgh and six papers in the Proceedings of the Edinburgh Mathematical Society: [18],[19], [20], [21], [22] and [23].

A particularly memorable translation, from Danish, is Sprague's translation of Dr T N Thiele's 1871 paper on mortality [6]. Thiele was from 1872 to 1910, among other appointments, the chief actuary of the Danish insurance company Hafnia. Thiele stated, for the first time, a mathematical formula expressing the force of mortality throughout the whole of life. The Thiele 1871 mathematical formula was not bettered until the 1980s [7]. Thiele's law, where $x$ is the age and $\mu_{x}$ is the force of mortality, is:-
$\mu_{x} = Ae^{-Bx} + Ce^{-D(x-E)^{2}} + FG^{x}$
Sprague was elected a Fellow of St John's College following his attainment, in 1853, of the position of Senior Wrangler and First Smith's Prizeman at Cambridge University (first place among those attaining first class honours in mathematics) [8]. The year before Sprague, in 1852, the Senior Wrangler and First Smith's Prizeman had been P G Tait (as well as the Professor of Natural Philosophy at Edinburgh University and the founder of the mathematical theory of knots, Tait was a director of the Scottish Provident Institution, a life insurance company in Edinburgh) and the year after, in 1854, the world-famous physicist, Clerk Maxwell had been second Wrangler (Routh being the Senior), the two Smith's Prizes of 1854 being shared equally between these two eminent men. Dr Sprague qualified as a barrister prior to becoming an actuary and knew both Tait and Maxwell.

Dr Sprague was the only person to have been President of both the Institute of Actuaries (1882-86) in London and the Faculty of Actuaries (1894-96) in Edinburgh. He was also President of the Actuarial Society of Edinburgh on three occasions (1874, 1882 and 1891) and did much to mould the futures of all these bodies.

He was elected a Fellow of the Royal Society of Edinburgh (FRSE) in 1874 [9] and of the Edinburgh Mathematical Society (EMS). The University of Aberdeen made him an LL.D. in 1893.

Dr Sprague was the actuary to the Equity and Law life insurance company from 1861-1873 when he was appointed to be Chief Executive (1873-1900) of the Scottish Equitable Life Assurance Society in Edinburgh (not to be confused with the Equitable Life Assurance Society in London) retiring from business life at age 70. When President of the Institute of Actuaries, from 1882-86, he was resident in Edinburgh but travelled to London regularly to fulfil his role as President of the Institute. He was on the Council of the Institute of Actuaries for thirty-eight years (1863-1900) and editor of the Journal of the Institute of Actuaries for 16 years (1867-83) [10].

Dr Sprague was invited by Clerk Maxwell (the latter being scientific editor, along with T H Huxley, of the 9th Edition of the Encyclopaedia Britannica) to write the article on 'Annuities' [11]. This article was a most thorough analysis of the contributions, including those of de Witt, Halley, and de Moivre, to the advancement of actuarial science.

At the 1874 meeting of the British Association in Belfast (at which occasion Clerk Maxwell wrote his poems entitled 'Notes of the President's Address' and 'Molecular Evolution' [12]), Sprague gave a paper on the solvency of life offices [13] which the 1870 Life Insurance Companies Act (see above) did much to ensure.

Dr Sprague's one peculiarity was his advocacy of spelling reform which caused some difficulties with the Editorial Board of the Journal of the Institute of Actuaries as to whether his special brand of phonetic spelling should be permitted in the papers which he submitted e.g. 'treatis', 'anser', 'articl', 'coud', 'giv', 'conclusiv', 'hav', 'ov' (e.g. the wording in reference [5] is not a mis-spelling). However, such was the eminence of Dr Sprague that the Journal had no alternative but to permit it!

In Maclaurin House (the home of the Faculty of Actuaries, named after Maclaurin, who, in 1744, acted as consultant to the first pension fund in the world established on sound actuarial principles, namely the pension fund for the widows and children of the 'Ministers of the Church of Scotland and the Professors at the Scottish Universities') his bust [14] stands alongside that of William T Thomson (one of the founders of the Institute of Actuaries in 1848 and the Faculty of Actuaries in 1856). A bust [15] of him also stands in the main hall of Staple Inn, the home of the Institute of Actuaries.

A prize for the best performance in actuarial science, finance, insurance, operational research, probability and statistics in Part III of the Cambridge Mathematical Tripos is awarded each year in honour of Dr. Sprague.

His son, Alfred Ernest Sprague, MA, DSc, FFA, FIA was also President of the Faculty of Actuaries from 1919-21.

### References (show)

1. T B Sprague, G King, J Karup, M Am Bégault and M L Marie, The Universal Notation, Transactions of the 2nd International Actuarial Congress (1898), 618.
2. T B Sprague, Select life tables deduced from the Institute of Actuaries' experience (healthy males), with commutation columns and monetary values at 2 1/2, 3, 3 1/2 and 4 per cent (1896). Shelved at: VXL/73 rsl [IF archive] (Lon) [Faculty: Edinburgh University Library, Special Collections Department 072.1942 SPR] [RKN: 4587]
3. D O Forfar, Early Mortality Tables, Encyclopedia of Actuarial Science, (eds. J L Teugels and B Sundt), Vol. 2 (John Wiley, 2004), 597.
4. T B Sprague, On legislation as to life insurance and life insurance companies (c1870). [Faculty: Edinburgh University Library, Special Collections Department Pamphlet file, vol 13] [RKN: 70761]
5. T B Sprague, Treatise on Life Office Accounts: Showing in particular how the annual revenue account and balance sheet ov a company shoud be drawn up, so as to be in strict conformity with the schedules ov the Life Assurance Companies Act 1870; and containing a complete examination ov the accounts deposited with the Board of Trade under that Act : Forming Part II ov 'Life insurance in 1872', (Charles and Edwin Layton, 1874).
6. T N Thiele, On a mathematical formula to express the rate of mortality throughout the whole span of life (translation by T B Sprague in Journal of the Institute of Actuaries 16 (1871), 313-329.
7. D O Forfar, Mortality Laws, Encyclopedia of Actuarial Science (eds. J L Teugels and B Sundt) Vol. 2 (John Wiley, 2004), 1141,
8. University of Cambridge, Historical Register of the University of Cambridge up to 1910.
9. Royal Society of Edinburgh, Archive of Fellows, http://www.royalsoced.org.uk
10. G King, Memoir of Dr Thomas Bond Sprague, Journal of the Institute of Actuaries, 52 (1921), 241-259. Also Obituary Notice, Transactions of the Faculty of Actuaries 9 (1921), 40; 379.
11. T B Sprague, Annuities, article in 9th edition of the Encyclopaedia Britannica (scientific eds, J Clerk Maxwell and T H Huxley) (A and C Black, Edinburgh, 1875).
12. L Campbell and W Garnett, Life of James Clerk Maxwell (1882) page 326. http://www.sonnetusa.com/bio/maxbio.pdf
13. T B Sprague, On the causes of insolvency in life assurance companies and the best means of detecting, exposing and preventing it (1874): [Paper read at the Belfast meeting in 1874 of the British Association]. [Faculty: Edinburgh University Library, Special Collections Department 368.3 SPR Pamphlet file, vol 13 Pamphlet file, vol 44] [RKN: 70199]
14. Bust placed in the Faculty of Actuaries, Transactions of the Faculty of Actuaries 1, 15.
15. Bust placed in the Institute of Actuaries, Journal of the Institute of Actuaries 37, 111.
16. T B Sprague, On the nature of the curves whose intersections give the imaginary roots of an algebraic equation, Transactions of Royal Society of Edinburgh 30 (1882), 467-480.
17. T B Sprague, On a new algebra, by means of which permutations can be transformed in a variety of ways, and their properties investigated, Transactions of Royal Society of Edinburgh 37 (1893), 399-411.
18. T B Sprague, Note on the evaluation of functions of the form 00, Edinburgh Mathematical Society (1884).
19. T B Sprague, On the different possible non-linear arrangements of eight men on a chess board, Edinburgh Mathematical Society (1889).
20. T B Sprague, On the transformation and classification of permutations, Edinburgh Mathematical Society (1890).
21. T B Sprague, On the geometrical interpretation of ii, Edinburgh Mathematical Society (1893).
22. T B Sprague, On the eight Queens' problem, Edinburgh Mathematical Society (1899).
23. T B Sprague, On the singular points of plane curves, Edinburgh Mathematical Society (1902).
24. T B Sprague, Presidential Address to Institute of Actuaries, Journal of the Institute of Actuaries, 24 (1882), 1; 24 (1883), 229; 25 (1884), 65; 25 (1885), 293.
25. T B Sprague, Presidential Address to the Actuarial Society of Edinburgh and to the Faculty of Actuaries, Transactions of the Actuarial Society of Edinburgh, 1, 269-287.
26. Dr. Sprague wrote actuarial papers on mortality, graduation, interpolation, life interests and reversionary interests, combined marriage and mortality tables, vaccination against smallpox, premium rates, distribution of life office surplus, valuation of life office liabilities, life office expenses, legislation of life companies, causes of life office insolvencies, transfer of business from one life office to another, use of probability in actuarial studies. http://ioa.soutron.com/ioa/library/default.htm

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### Honours (show)

Honours awarded to Thomas Bond Sprague

### Cross-references (show)

Written by David O Forfar Heriot-Watt University.
Last Update August 2006