**Edward Waring**'s father, John Waring, was a farmer. Several generations of his family lived at Mytton in Shropshire. John Waring married Elizabeth and their son Edward was educated at Shrewsbury school.

Waring entered Magdalene College, Cambridge on 24 March 1753. He won a Milbridge scholarship and he was admitted as a sizar, meaning that he paid a reduced fee but essentially worked as a servant to make good the fee reduction. He immediately impressed his teachers with his mathematical ability and he graduated B.A. in 1757 as senior wrangler.

On 24 April 1754 Waring was elected a fellow of Magdalene College. Waring's most famous work was *Meditationes Algebraicae* which he worked on during the next few years. He submitted the first chapter of this work to the Royal Society but following this nothing happened for two years. When Waring was nominated for the Lucasian Chair of Mathematics at Cambridge in 1759, the work was distributed as *Miscellanea Analytica* to prove he was qualified for the post despite his youth.

William Powell of St John's College Cambridge had his own ideas about who should fill the Lucasian Chair of Mathematics and attempted to prevent Waring being appointed. He put out a pamphlet entitled *Observations* which criticised Waring and doubted his mathematical abilities. Waring responded to this criticism on 25 January 1760 with the pamphlet *A reply to the observations*. Powell, still following his own agenda, was not going to give up that easily and responded immediately with *Defence of the observations*. John Wilson now wrote *A letter* to support Waring and this was sufficient to see him confirmed as Lucasian professor on 28 January 1760 at the age of 23.

When *Miscellanea Analytica* was published as a complete work in 1762, Waring chose to call it a second edition. Rather strangely, despite it being a second edition, it was given a new title *Meditationes Algebraicae*. We shall comment further below on this important work, covering topics in the theory of equations, number theory and geometry. On 2 June 1763 Waring was elected a Fellow of the Royal Society.

In 1764 Lalande published *Life of Condorcet*. In this work it was claimed that there were no first-class analysts in England. Waring responded quickly to this comment by writing a letter to Nevil Maskelyne, the Astronomer Royal. In the letter Waring pointed out that d'Alembert, Euler and Lagrange had all praised his 1762 work. Waring wrote that in the book he had given:-

However, Waring knew that his work had not been widely read for he added that he:-... somewhare between three and four hundred new propositions of one kind or another, considerably more than have been given by any other English writer.

The reason that so few had read the book was partly because the subject matter was difficult, partly because Waring was a poor communicator, and partly because he did not have a good algebraic notation. It would be reasonable to compare Waring with Ruffini who, about 150 years later, suffered the same fate with his work in algebra for much the same reasons.... never could hear of any reader in England, out of Cambridge, who took pains to read and understand it ...

One would not expect the Lucasian professor of mathematics to take a medical degree but that is exactly what Waring did, graduating with his M.D. in 1767. For a short time he practised medicine in various London hospitals, then Addenbroke hospital in Cambridge and finally at a hospital in St Ives, Huntingtonshire. However, he gave up practising medicine by 1770 [7]:-

One might ask how Waring could practise medicine and hold the Lucasian Chair at the same time. Well, he never lectured as part of his duties. Some claim that it was because his ideas were so profound that they could not be communicated in lectures, but if truth be told it is more likely that the reason was because he was a poor communicator with handwriting which was almost impossible to read. [I will agree that such problems have not stopped others lecturing!]... he was very short-sighted and very shy in manner, so that he quickly abandoned his profession.

In 1776 Waring married Mary Oswell. They lived for a while in Shrewsbury but the town was not to Mary's liking and the couple moved to Waring's estate at Plealey in Pontesbury.

Waring's *Miscellane analytica*... of 1762 formed the basis, as we have noted, of further books. *Proprietates algebraicarum curvarum*, covering geometry, was published in 1772. A further work *Miscellanea Analytica* appeared in 1776 with a new expanded edition in 1785. *Meditationes Algebraicae*, covering the theory of equations and number theory, appeared in 1770 with an expanded version in 1782.

In *Meditationes Algebraicae* Waring proves that all rational symmetric functions of the roots of an equation can be expressed as rational functions of the coefficients. He derived a method for expressing symmetric polynomials and he investigated the cyclotomic equation *x*^{n} - 1 = 0. This work makes Waring one of the earliest contributers to Galois theory. In particular, discussing Problem 22 of Chapter 3, Weeks writes [3]:-

Chapter 4 ofThe 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 Chapter1), a theory whose systematic development was not to appear until the19th century(Lagrange, Gauss, and others)and was ultimately followed by the theory of permutation groups(Galois, Jordan, ...).

*Meditationes Algebraicae*contains results such as:

His result that the product of the degrees of the original equations is the degree of the single reduced equation is known as the Generalised Theorem of Bézout.k equations in k unknowns can be reduced to one equation with one unknown.

The rest of the book deals with number theory, a topic in which Waring made some interesting advances. He stated that any even integer can be written as the sum of two primes and every odd integer is either a prime or the sum of three primes. This result, now known as the Goldbach conjecture, is one of the most famous unsolved problems of mathematics. Although Goldbach proposed his question in a letter to Euler long before Waring published *Meditationes Algebraicae* it is still worth noting that Waring's version was the first to be published.

Waring also stated, without giving a proof, what is now known as 'Waring's theorem':-

In 1909 Hilbert proved that given any integerEvery integer is equal to the sum of not more than9cubes. Also every integer is the sum of not more than19fourth powers, and so on ....

*n*there is an integer

*m*(depending on

*n*) such that every integer is a sum of

*m*

*n*th powers. It is reasonable to assume that Waring had this type of result in mind when he stated 'Waring's theorem'. Hilbert's proof led to major new theorems in number theory.

Waring also wrote on algebraic curves, classifying quartic curves into 12 main divisions with 84551 subdivisions.

Several descriptions of Waring given by authors from his own period are not too flattering. One writes that he was:-

Another says that he is:-... one of the strongest compounds of vanity and modesty which the human character exhibits. The former, however, is his predominant feature.

This last statement may partly explain the strange fact that although Waring was elected a Fellow of the Royal Society in 1763 and had the great distinction of being awarded its Copley Medal in 1784, he resigned from the Society in 1795 claiming poverty. He was awarded other honours, however, such as election to the Royal Society of Göttingen and the Royal Society of Bologna. Perhaps the most accurate assessment of Waring was made by Thomas Thomson:-... one of the greatest analysts that England has produced ...[near the end of his life being]sunk into a deep religious melancholy approaching to insanity.

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.

**Article by:** *J J O'Connor* and *E F Robertson*

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