Edward Waring was an established 18th-century mathematician and theorist who did groundbreaking work in the areas of imaginary numbers and their roots. He is best known for the Cauchy ratio testand Waring's theorem, which is also known as Waring's Problem. In the former, Waring focused on the convergence and divergence of numerical series. In the theorem named after him, Waring postulated that any positive integer is the sum of not more than nine cubes, and is the sum of not more than 19 fourth powers. The result about cubes was not proved until 1910, and the result about fourth powers was not proved until 1986. Waring also posited that integers are composed of a specific combination of prime numbers. For example, an even integer, he asserted, was the sum of two prime numbers, whereas an odd number was either, in and of itself, a prime number or the sum of three primes. Waring was also known for his work with quartic curves and the approximation of imaginary roots.
Waring was born near Shrewsbury in Shropshire, a borough in western England, to John Waring, a wealthy farmer, and his wife, Elizabeth. Exact details about their oldest son's origins are sketchy: some sources say the mathematician was born in 1734, whereas others say 1736; Plealey has likewise been given as the more exact place of his birth, as has Old Heath.
Waring initially attended school in Shrewsbury before going on to Cambridge University, where he was enrolled as a sizar at Magdalene College in 1753. (A sizar, according to Webster's Dictionary, is a student who receives an academic stipend, often in return for performing work for other students.) In 1757, having established himself and excelled in the field of mathematics, Waring graduated with a bachelor of arts and the title of senior wrangler. Around this time, he was also granted a fellowship, and this enabled him to continue with his studies. He was likewise honored with membership in Cambridge's recently founded Hyson Club.
Fights for Academic Recognition
The death of faculty member John Colson at Cambridge created a job opening for which Waring was nominated. Because of the complaints of some faculty members who faulted him for his age, work, and lack of teaching experience, Waring countered their opposition by showing some of his writings; he was ultimately offered the position, although he did not yet possess his master's degree, a requirement for this position. He eventually received the degree by royal mandate in 1760 and was awarded the title of "Lucasian professor of mathematics," (a position first held by Isaac Newton). Waring held this title for the remainder of his life and it would ultimately enable him to pursue other interests. Additional honors included Waring's induction into the Royal Society of London a few years after his M.A. was conferred upon him, as well as that organization's Copley Medal.
During his tenure at Cambridge, Waring published several nonfiction works and treatises about mathematics, the earliest of which were published in Latin. Portions of his first work, Miscellanea analytica de aequationibus algebraicis, curvarum proprietatibus, fluxionibus et serierum summatione (1759; "Miscellany of Analysis"), were used by Waring when he was attempting to strengthen his position against those critics who deemed him too young to be the Lucasian professor of mathematics and were protesting his nomination. In this work and its 1762 revised version, Miscellanea analytica de aequationibus algebraicis et curvarum proprietatibus, Waring focused in part on his theory of numbers and their composition (as a cube, a fourth power, or some combination of either cubes or fourth powers). Subsequent works published in Latin include the 1770 Meditationes algebraicae("Thoughts on Algebra") and Proprietates algebraicarum curvarum("The Properties of Algebraic Curves"), which was printed in 1772. His later works--notably On the Principle of Translating Algebraic Quantities into Probable Relations and Annuities (1792) and An Essay on the Principles of Human Knowledge (1794)--were first printed in his native language. In addition to his books, Waring wrote numerous other small articles and pieces. Despite the erudition of the ideas presented in his writings, Waring's books were often faulted by critics for their brevity, lack of explanation, and apparent lack, at times, of clarity and organization. For example, Miscellanea analytica was once described as "one of the most abstruse books written in the abstrusest parts of Algebra," a quotation that has been attributed to both Glieg and Charles Hutton, the latter being the English mathematician who had determined the Earth's mean density.
A Second Career and Personal Sacrifice
Perhaps because of the sometimes negative reception of his works upon their publication, it has since been noted that Waring's contributions stem from the originality of his work and research, and not necessarily as a writer or teacher of his ideas. Indeed, during his academic career, Waring spent little time teaching students (reports about his personality indicate that he would not have particularly suited to the job--he has been described as being somewhat arrogant, and it is even rumored that he once claimed that no one in his homeland was competent enough to read, let alone understand, his work) and actually devoted a portion of his life to medicine, eventually earning a medical degree from Cambridge in either the late 1760s or the early 1770s. (Although some sources disagree, it is often accepted that Waring participated in the dissection of corpses and was frequently found at work in the hospitals around London and Cambridge. For a time, it is believed that he actually practiced medicine at various English hospitals before retiring from the profession in the 1770s.)
Whatever the reception of Waring's work in their published format, his ideas about mathematics and science were groundbreaking but frequently misunderstood. Although some of his work still remains unproved and much of it was ignored or disparaged by his contemporaries, his ideas remain highly respected for their innovativeness. Waring died on August 15, 1798, in Pontesbury, Shropshire.
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