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" Waring was one of the profoundest mathematicians of the eighteenth century; but the inelegance and obscurity of his writings prevented him from obtaining that reputation to which he was entitled." Thomas Thomson

Edward Waring (1736-1798) was a mathematician considered worthy of the Lucasian Chair, after the somewhat less-than-stellar performances of the Chair holders following Newton. His youth worked against him during the election process, but his brilliance overcame the objection. His major interest was algebra. Waring is noteworthy for contributing problems that were theoretical in nature with no practical application. He expected the problems to be used to help understand mathematics. Unfortunately, his problems were not actually understood for about 150 years, in part because they were difficult and in part because he did not make himself clear. Because he did not communicate effectively, he lost a place in history.

Waring attended Magdalene College, Cambridge, graduating in 1757 as senior wrangler. The following year he was elected a fellow. His M.A. was granted by royal mandate in 1760 in order to qualify him for the Lucasian Chair. He also received his MD degree with a minimal amount of medical work. He was apparently not suited for the demands upon a physician, but did engage in the dissection of a corpse. For brief periods he practiced at two hospitals, Addenbroke in Cambridge and St. Ives in Huntington.¹ Elected to the Royal Society in 1763, he resigned in 1795, one of the few members of the Royal Society to do so, apparently unable to pay his dues. This was all very odd, since he had been awarded the Copley Medal by the Society, its highest honor, in 1784.² One would think that an honored member would not be allowed to resign due to poverty. Membership in the Royal Society of Gottingen and the Royal Society of Bologna were also granted to Waring. He was a commissioner of the Board of Longitude. He mentions in his later papers being a member of the Institute of Bononia, but this is not confirmed elsewhere.

Waring did not lecture while Lucasian professor.³ His ideas were considered to be so profound that they could not be communicated orally, but his poor communication skills were more likely the reason.⁴ Waring acknowledged that he was not aware of anyone outside Cambridge who had read and understood his work.⁵ He served as an examiner for the Smith's Prize, and had a reputation for severity unmatched anywhere in Europe.⁶

Waring wrote in both Latin and English, with little of the Latin translated into English, the exception is his most important work, Meditationes Algebraicae.⁷ This book was originally made available as just its first chapter, submitted to the Royal Society. No action was taken for two years, but when Waring was nominated for the Lucasian Chair, it was distributed as Miscellanea Analytica to prove he was qualified for the post.

William Powell of St. John's College put out a pamphlet⁸ criticizing Waring and his abilities. Waring responded, Powell rejoined, then John Wilson⁹ responded in defense of Waring to put the matter to rest. This series of events helped to ensure Waring's nomination, and to prevent Powell's candidate for Chair from ever being considered again.¹⁰

Miscellanea Analytica was published as a complete work in 1762, with Waring calling it a second edition, but gave it the title Meditationes Algebraicae . The book covered the topics of the theory of equations, number theory and analytic geometry, although the term "analytic geometry" had not been used at the time. In 1782, the third edition was published, but as two separate works, the Meditationes Algebraicae , which was by now about as extensive as the original work, covering the theory of equations and number theory, and the Proprietates algebraicarum curvarum , covering analytic geometry.¹¹

Meditationes Algebraicae has five chapters. In the preface Waring provides a useful history of algebra from its earliest days. Chapter one covers symmetric functions, but at the time he called them simply "... algebraic functions of the roots of a generalized equation." The second chapter deals with "impossible roots," or what we call imaginary roots. The use of i for the the square root of -1 had not yet come about, but he recognized the use of imaginary roots.

The third chapter shows methods for reducing complex equations to simple forms for solution. In problem XXII of this chapter, Waring again demonstrates the insight that was not understood for over a century. Quoting from Weeks:

"The most significant aspect of Waring's treatment of this example is the symmetric relation between the roots of the quartic equation and its resolvent cubic. This is, in essence, the first result in the theory of symmetric functions (beyond the basic building blocks which appeared in Chapter 1), a theory whose systematic development was not to appear until the 19th century (Lagrange,¹² Gauss¹³, and others) and was ultimately followed by the theory of permutation groups (Galois,¹⁴ Jordan¹⁵ ...)"¹⁶

Chapter four deals with systems of equations having several unknowns. Waring touches on several methods that were not formalized until much later. For example, he concluded that k equations with k unknowns could be reduced to a single equation with just one unknown. He discovered that multiplying the degrees of the original equations gave the degree of the final reduced equation. This is known as the Generalized Theorem of Bezout.¹⁷ The topic of the fifth and last chapter is rational and integral quantities. Among the subtopics is found the theorem that every even number is the sum of two primes and every odd number is either a prime or the sum of three primes.¹⁸ This theorem was independently expressed by Christian Goldbach, a German, and eventually named Goldbach's Theorem. Goldbach had written to Leonard Euler on its development, but it was not published at the time. Waring was the first to publish it.¹⁹

There is a now famous problem, known as Waring's Problem, which stems from a statement in Meditationes Algebraicae that was made without proof, certainly not an act that endears one to mathematicians. The short version is: "... each positive integer is the sum of four squares; nine cubes, 19 fourth powers, and so on ..." ²⁰ The basic conjecture is any given number can be shown to be made up of a group of powers. For example, the number 7 is the sum of 1^2 + 1^2 + 1^2 + 1^2 + 2^2. Multiple uses of a number are permitted and 1 is same no matter what power is used. The implication is that the powers used are limitless, so there is some finite number of 8th power numbers that make other numbers, as well as 10th powers, etc. The interesting part is the top limit and the bottom limit. For example, it is not necessary to use 9 cubes to make up all numbers. This fact was brought to light by G. H. Hardy and E.M. Wright.²¹ They derived the formula (n-2)2^(n-1) + 5, where n represents the power and the result is the maximum needed. It can be seen immediately that if 2is used for n , 4 is the result, if 3 is used, then 9 is the result. The small numbers are easier, but the improvement was in larger numbers. Before Hardy and Wright, the 8th power ceiling was 31,353, but their formula shows the actual number is 425. With improvements in technique, the number for the 8th power is now down to 62. It is still not clear just how low the ceiling will go.²² Although Waring made the statement in 1760, it was not until 1909 that David Hilbert proved that for every integer n there is an integer m such that every integer is a sum of m nth power. ²³

The importance of Waring's formulas and conjectures, beyond the mathematician's view, is that there are applications in computer algorithms involving parallel processing that help speed up the performance of calculations. This naturally translates into efficient use of resources to solve problems that affect our daily lives.²⁴ Waring remained focused on his algebra, producing only one other work outside of his field, the Essay of the Principles of Human Knowledge. ²⁵ It was privately printed and circulated among his friends.

Footnotes

1. Dictionary of National Biography.
2. Dictionary of Scientific Biography.
3. Florian Cajori, A History of Mathematics (New York: Macmillan, 1931), 248.
4. Dictionary of National Biography.
5. Cajori, A History of Mathematics, 248.
6. Dennis Weeks, Meditationes algebraicae, an English translation of the work of Edward Waring (Providence: American Mathematical Society, 1991), 436.
7. Weeks.
8. Powell 1760
9. Wilson 1760
10. Weeks 1991 xii
11. Weeks, xii.
12. Joseph Lagrange (Italy, Germany, France 1736-1813).
13. Carl Friedrich Gauss (Germany 1777-1855).
14. Evariste Galois (France 1811-1832).
15. Camille Jordan (France 1838-1922).
16. Weeks, 182.
17. Etienne Bezout (France 1730-1793).
18. Weeks, 362.
19. Struik, 49.
20. Ian Stewart, "The Waring Experience," Nature 323 (October 1986): 674.
21. Hardy 1960.
22. Stewart, 674.
23. Hilbert 1909 281.
24. Weeks, 34.
25. Waring 1794.