Waring, Edward

Edward Waring, (born 1734, Old Heath, near Shrewsbury, Shropshire, England—died August 15, 1798, Pontesbury, Shropshire), English mathematician whose primary research interests were in algebra and number theory.

Waring attended Magdalene College, University of Cambridge, graduating in 1757 as senior wrangler (first place in the annual Mathematical Tripos contest). He was elected a fellow the following year, and Lucasian Professor in 1760. He received an MD from Cambridge (1770) but is believed to have practised medicine only briefly. He was elected a fellow of the Royal Society in 1763, received the Society’s Copley Medal in 1784 but, for reasons that are unclear, took the unusual step of resigning from the Society in 1795.

In 1762 Waring published Miscellanea analytica… (“Miscellany of analysis…”), a notoriously impenetrable work, but the one upon which his fame largely rests. It was enlarged and republished as Meditationes algebraicae (1770, 1782; “Thoughts on Algebra”) and Proprietates algebraicarum Curvarum (1772; “The Properties of Algebraic Curves”). It covers the theory of equations and number theory, as well as what is now known as analytic geometry. Topics discussed include the theory of symmetric functions, included as part of the investigation into the roots of a quartic polynomial and now recognized as a contribution to the prehistory of group theory; imaginary roots; and René Descartes’ rules of signs. Also included is a study of the roots of unity.

Several theorems are stated without proof, including Waring’s problem (or Waring’s theorem), that every positive integer is the sum of not more than nine cubes or the sum of not more than nineteen fourth powers and so on; Wilson’s theorem, if p is a prime number then (p – 1)! + 1 will be divisible by p; and, appearing for the first time in print, the Goldbach conjecture (see Christian Goldbach), that every even number is the sum of two prime numbers.

Waring’s other published works include Meditationes analyticae (1776; 1782 “Thoughts on Analysis”), which contains the ratio test for the convergence of infinite series now attributed to Augustin-Louis Cauchy, On the Principle of Translating Algebraic Quantities into Probable Relations and Annuities (1792), and An Essay on the Principles of Human Knowledge (1794).