Edward Waring
Encyclopedia
Edward Waring was an English
mathematician
who was born in Old Heath
(near Shrewsbury
), Shropshire
, England and died in Pontesbury
, Shropshire, England. He entered Magdalene College, Cambridge
as a sizar
and became Senior wrangler in 1757. He was elected a Fellow of Magdalene and in 1760 Lucasian Professor of Mathematics, holding the chair until his death. He made the assertion known as Waring's Problem
without proof in his writings Meditationes Algebraicae. Waring was elected a Fellow of the Royal Society in 1763 and awarded the Copley Medal
in 1784.
under a Mr Hotchkin and was admitted as a sizar
at Magdalene College, Cambridge
, on 24 March 1753, being also Millington exhibitioner. His extraordinary talent for mathematics was recognized from his early years in Cambridge. In 1757 he graduated BA as senior wrangler and on 24 April 1758 was elected to a fellowship at Magdalene. He belonged to the Hyson Club, whose members included William Paley
.
of mathematics, one of the highest positions in Cambridge. William Samuel Powell, then tutor in St John's College, Cambridge
opposed Waring's election and instead supported the candidacy of William Ludlam. In the polemic with Powell, Waring was backed by John Wilson
. In fact Waring was very young and did not hold the MA, necessary for qualifying for the Lucasian chair, but this was granted him in 1760 by royal mandate. In 1762 he published the full Miscellanea Analytica, mainly devoted to the theory of numbers and algebraic equations. In 1763 he was elected to the Royal Society
. He was awarded its Copley medal
in 1784 but withdrew from the society in 1795, after he had reached sixty, ‘on account of [his] age’. Waring was also a member of the academies of sciences of Göttingen
and Bologna
. In 1767 he took an MD degree, but his activity in medicine was quite limited. He carried out dissections with Richard Watson
, professor of chemistry and later bishop of Llandaff
. From about 1770 he was physician at Addenbrooke's Hospital
at Cambridge, and he also practised at St Ives, Huntingdonshire, where he lived for some years after 1767. His career as a physician was not very successful since he was seriously short-sighted and a very shy man.
, 8 miles out of the town, where Waring owned an estate.
as ‘a work full of excellent researches'. In this work Waring published many theorems concerning the solution of algebraic equations which attracted the attention of continental mathematicians, but his best results are in number theory. Included in this work was the so-called Goldbach conjecture (every even integer is the sum of two primes), and also the following conjecture: every odd integer is a prime or the sum of three primes. Leonard Euler had proved that every positive integer is the sum of not more than four squares; Waring suggested that every positive integer is either a cube or the sum of not more than nine cubes. He also advanced the hypothesis that every positive integer is either a biquadrate or the sum of not more than nineteen biquadrates. These hypotheses form what is known as Waring's problem
. He also published a theorem, due to his friend John Wilson, concerning prime numbers; it was later proved rigorously by Lagrange.
In Proprietates Algebraicarum Curvarum (1772) Waring reissued in a much revised form the first four chapters of the second part of Miscellanea Analytica. He devoted himself to the classification of higher plane curves, improving results obtained by Isaac Newton
, James Stirling
, Leonard Euler, and Gabriel Cramer
. In 1794 he published a few copies of a philosophical work entitled An Essay on the Principles of Human Knowledge, which were circulated among his friends.
Waring's mathematical style is highly analytical. In fact he criticized those British mathematicians who adhered too strictly to geometry. It is indicative that he was one of the subscribers of John Landen
's Residual Analysis (1764), one of the works in which the tradition of the Newtonian fluxional calculus was more severely criticized. In the preface of Meditationes Analyticae Waring showed a good knowledge of continental mathematicians such as Alexis Clairaut, Jean le Rond d'Alembert
, and Euler. He lamented the fact that in Great Britain mathematics was cultivated with less interest than on the continent, and clearly desired to be considered as highly as the great names in continental mathematics—there is no doubt that he was reading their work at a level never reached by any other eighteenth-century British mathematician. Most notably, at the end of chapter three of Meditationes analyticae Waring presents some partial fluxional equations (partial differential equations in Leibnizian terminology); such equations are a mathematical instrument of great importance in the study of continuous bodies which was almost completely neglected in Britain before Waring's researches. One of the most interesting results in Meditationes analyticae is a test for the convergence of series generally attributed to d'Alembert (the ‘ratio test’). The theory of convergence of series (the object of which is to establish when the summation of an infinite number of terms can be said to have a finite ‘sum’) was not much advanced in the eighteenth century.
Waring's work was known both in Britain and on the continent, but it is difficult to evaluate his impact on the development of mathematics. His work on algebraic equations contained in Miscellanea Analytica was translated into Italian by Vincenzo Riccati
in 1770. Waring's style is not systematic and his exposition is often obscure. It seems that he never lectured and did not habitually correspond with other mathematicians. After Jérôme Lalande
in 1796 observed, in Notice sur la vie de Condorcet, that in 1764 there was not a single first-rate analyst in England, Waring's reply, published after his death as ‘Original letter of Dr Waring’ in the Monthly Magazine, stated that he had given ‘somewhere between three and four hundred new propositions of one kind or another’.
, on 15 August 1798. He was buried in the churchyard at Fitz, Shropshire
.
England
England is a country that is part of the United Kingdom. It shares land borders with Scotland to the north and Wales to the west; the Irish Sea is to the north west, the Celtic Sea to the south west, with the North Sea to the east and the English Channel to the south separating it from continental...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
who was born in Old Heath
Shrewsbury
Shrewsbury is the county town of Shropshire, in the West Midlands region of England. Lying on the River Severn, it is a civil parish home to some 70,000 inhabitants, and is the primary settlement and headquarters of Shropshire Council...
(near Shrewsbury
Shrewsbury
Shrewsbury is the county town of Shropshire, in the West Midlands region of England. Lying on the River Severn, it is a civil parish home to some 70,000 inhabitants, and is the primary settlement and headquarters of Shropshire Council...
), Shropshire
Shropshire
Shropshire is a county in the West Midlands region of England. For Eurostat purposes, the county is a NUTS 3 region and is one of four counties or unitary districts that comprise the "Shropshire and Staffordshire" NUTS 2 region. It borders Wales to the west...
, England and died in Pontesbury
Pontesbury
Pontesbury is a large village and civil parish in Shropshire and is approximately eight miles southwest of the county town of Shrewsbury. The village of Minsterley is just over a mile further southwest. The A488 road runs through the village, on its way from Shrewsbury to Bishop's Castle...
, Shropshire, England. He entered Magdalene College, Cambridge
Magdalene College, Cambridge
Magdalene College is a constituent college of the University of Cambridge, England.The college was founded in 1428 as a Benedictine hostel, in time coming to be known as Buckingham College, before being refounded in 1542 as the College of St Mary Magdalene...
as a sizar
Sizar
At Trinity College, Dublin and the University of Cambridge, a sizar is a student who receives some form of assistance such as meals, lower fees or lodging during his or her period of study, in some cases in return for doing a defined job....
and became Senior wrangler in 1757. He was elected a Fellow of Magdalene and in 1760 Lucasian Professor of Mathematics, holding the chair until his death. He made the assertion known as Waring's Problem
Waring's problem
In number theory, Waring's problem, proposed in 1770 by Edward Waring, asks whether for every natural number k there exists an associated positive integer s such that every natural number is the sum of at most s kth powers of natural numbers...
without proof in his writings Meditationes Algebraicae. Waring was elected a Fellow of the Royal Society in 1763 and awarded the Copley Medal
Copley Medal
The Copley Medal is an award given by the Royal Society of London for "outstanding achievements in research in any branch of science, and alternates between the physical sciences and the biological sciences"...
in 1784.
Early years
Waring was the eldest son of John and Elizabeth Waring, a prosperous farming couple. He received his early education in Shrewsbury SchoolShrewsbury School
Shrewsbury School is a co-educational independent school for pupils aged 13 to 18, founded by Royal Charter in 1552. The present campus to which the school moved in 1882 is located on the banks of the River Severn in Shrewsbury, Shropshire, England...
under a Mr Hotchkin and was admitted as a sizar
Sizar
At Trinity College, Dublin and the University of Cambridge, a sizar is a student who receives some form of assistance such as meals, lower fees or lodging during his or her period of study, in some cases in return for doing a defined job....
at Magdalene College, Cambridge
Magdalene College, Cambridge
Magdalene College is a constituent college of the University of Cambridge, England.The college was founded in 1428 as a Benedictine hostel, in time coming to be known as Buckingham College, before being refounded in 1542 as the College of St Mary Magdalene...
, on 24 March 1753, being also Millington exhibitioner. His extraordinary talent for mathematics was recognized from his early years in Cambridge. In 1757 he graduated BA as senior wrangler and on 24 April 1758 was elected to a fellowship at Magdalene. He belonged to the Hyson Club, whose members included William Paley
William Paley
William Paley was a British Christian apologist, philosopher, and utilitarian. He is best known for his exposition of the teleological argument for the existence of God in his work Natural Theology, which made use of the watchmaker analogy .-Life:Paley was Born in Peterborough, England, and was...
.
Career
At the end of 1759 Waring published the first chapter of Miscellanea Analytica. On 28 January the next year he was appointed Lucasian professorLucasian Professor
The incumbent of the Lucasian Chair of Mathematics, usually called the Lucasian Professor, is the holder of a mathematics professorship at the University of Cambridge, England. It is widely regarded as one of the world's most prestigious academic posts...
of mathematics, one of the highest positions in Cambridge. William Samuel Powell, then tutor in St John's College, Cambridge
St John's College, Cambridge
St John's College is a constituent college of the University of Cambridge. The college's alumni include nine Nobel Prize winners, six Prime Ministers, three archbishops, at least two princes, and three Saints....
opposed Waring's election and instead supported the candidacy of William Ludlam. In the polemic with Powell, Waring was backed by John Wilson
John Wilson (mathematician)
John Wilson was an English mathematician. The theorem, Wilson's Theorem, named after him for its discovery from Ibn al-Haytham, not its proof....
. In fact Waring was very young and did not hold the MA, necessary for qualifying for the Lucasian chair, but this was granted him in 1760 by royal mandate. In 1762 he published the full Miscellanea Analytica, mainly devoted to the theory of numbers and algebraic equations. In 1763 he was elected to the Royal Society
Royal Society
The Royal Society of London for Improving Natural Knowledge, known simply as the Royal Society, is a learned society for science, and is possibly the oldest such society in existence. Founded in November 1660, it was granted a Royal Charter by King Charles II as the "Royal Society of London"...
. He was awarded its Copley medal
Copley Medal
The Copley Medal is an award given by the Royal Society of London for "outstanding achievements in research in any branch of science, and alternates between the physical sciences and the biological sciences"...
in 1784 but withdrew from the society in 1795, after he had reached sixty, ‘on account of [his] age’. Waring was also a member of the academies of sciences of Göttingen
Göttingen
Göttingen is a university town in Lower Saxony, Germany. It is the capital of the district of Göttingen. The Leine river runs through the town. In 2006 the population was 129,686.-General information:...
and Bologna
Bologna
Bologna is the capital city of Emilia-Romagna, in the Po Valley of Northern Italy. The city lies between the Po River and the Apennine Mountains, more specifically, between the Reno River and the Savena River. Bologna is a lively and cosmopolitan Italian college city, with spectacular history,...
. In 1767 he took an MD degree, but his activity in medicine was quite limited. He carried out dissections with Richard Watson
Richard Watson (bishop)
Rt Rev Richard Watson was an Anglican clergyman and academic, who served as the Bishop of Llandaff from 1782 to 1816. He wrote some notable political pamphlets....
, professor of chemistry and later bishop of Llandaff
Llandaff
Llandaff is a district in the north of Cardiff, capital of Wales, having been incorporated into the city in 1922. It is the seat of the Church in Wales Bishop of Llandaff, whose diocese covers the most populous area of South Wales. Much of the district is covered by parkland known as Llandaff...
. From about 1770 he was physician at Addenbrooke's Hospital
Addenbrooke's Hospital
Addenbrooke's Hospital is an internationally renowned teaching hospital in Cambridge, England, with strong links to the University of Cambridge. It was founded in 1766 on Trumpington Street with £4,500 from the will of Dr John Addenbrooke, a fellow of St Catharine's College...
at Cambridge, and he also practised at St Ives, Huntingdonshire, where he lived for some years after 1767. His career as a physician was not very successful since he was seriously short-sighted and a very shy man.
Personal life
Waring had a younger brother, Humphrey, who obtained a fellowship at Magdalene in 1775. In 1776 Waring married Mary Oswell, sister of a draper in Shrewsbury; they moved to Shrewsbury and then retired to PlealeyPlealey
Plealey is a small village in Shropshire, England. It is located between Pontesford and Longden.The village lies in the parish of Pontesbury and has a population of 122 .There is a post box and a Methodist Chapel...
, 8 miles out of the town, where Waring owned an estate.
Work
Waring wrote a number of papers in the Philosophical Transactions of the Royal Society, dealing with the resolution of algebraic equations, number theory, series, approximation of roots, interpolation, the geometry of conic sections, and dynamics. The Meditationes Algebraicae (1770), where many of the results published in Miscellanea Analytica were reworked and expanded, was described by Joseph Louis LagrangeJoseph Louis Lagrange
Joseph-Louis Lagrange , born Giuseppe Lodovico Lagrangia, was a mathematician and astronomer, who was born in Turin, Piedmont, lived part of his life in Prussia and part in France, making significant contributions to all fields of analysis, to number theory, and to classical and celestial mechanics...
as ‘a work full of excellent researches'. In this work Waring published many theorems concerning the solution of algebraic equations which attracted the attention of continental mathematicians, but his best results are in number theory. Included in this work was the so-called Goldbach conjecture (every even integer is the sum of two primes), and also the following conjecture: every odd integer is a prime or the sum of three primes. Leonard Euler had proved that every positive integer is the sum of not more than four squares; Waring suggested that every positive integer is either a cube or the sum of not more than nine cubes. He also advanced the hypothesis that every positive integer is either a biquadrate or the sum of not more than nineteen biquadrates. These hypotheses form what is known as Waring's problem
Waring's problem
In number theory, Waring's problem, proposed in 1770 by Edward Waring, asks whether for every natural number k there exists an associated positive integer s such that every natural number is the sum of at most s kth powers of natural numbers...
. He also published a theorem, due to his friend John Wilson, concerning prime numbers; it was later proved rigorously by Lagrange.
In Proprietates Algebraicarum Curvarum (1772) Waring reissued in a much revised form the first four chapters of the second part of Miscellanea Analytica. He devoted himself to the classification of higher plane curves, improving results obtained by Isaac Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
, James Stirling
James Stirling (mathematician)
James Stirling was a Scottish mathematician. The Stirling numbers and Stirling's approximation are named after him.-Biography:...
, Leonard Euler, and Gabriel Cramer
Gabriel Cramer
Gabriel Cramer was a Swiss mathematician, born in Geneva. He showed promise in mathematics from an early age. At 18 he received his doctorate and at 20 he was co-chair of mathematics.In 1728 he proposed a solution to the St...
. In 1794 he published a few copies of a philosophical work entitled An Essay on the Principles of Human Knowledge, which were circulated among his friends.
Waring's mathematical style is highly analytical. In fact he criticized those British mathematicians who adhered too strictly to geometry. It is indicative that he was one of the subscribers of John Landen
John Landen
John Landen was an English mathematician,He was born at Peakirk near Peterborough in Northamptonshire, and died at Milton in the same county...
's Residual Analysis (1764), one of the works in which the tradition of the Newtonian fluxional calculus was more severely criticized. In the preface of Meditationes Analyticae Waring showed a good knowledge of continental mathematicians such as Alexis Clairaut, Jean le Rond d'Alembert
Jean le Rond d'Alembert
Jean-Baptiste le Rond d'Alembert was a French mathematician, mechanician, physicist, philosopher, and music theorist. He was also co-editor with Denis Diderot of the Encyclopédie...
, and Euler. He lamented the fact that in Great Britain mathematics was cultivated with less interest than on the continent, and clearly desired to be considered as highly as the great names in continental mathematics—there is no doubt that he was reading their work at a level never reached by any other eighteenth-century British mathematician. Most notably, at the end of chapter three of Meditationes analyticae Waring presents some partial fluxional equations (partial differential equations in Leibnizian terminology); such equations are a mathematical instrument of great importance in the study of continuous bodies which was almost completely neglected in Britain before Waring's researches. One of the most interesting results in Meditationes analyticae is a test for the convergence of series generally attributed to d'Alembert (the ‘ratio test’). The theory of convergence of series (the object of which is to establish when the summation of an infinite number of terms can be said to have a finite ‘sum’) was not much advanced in the eighteenth century.
Waring's work was known both in Britain and on the continent, but it is difficult to evaluate his impact on the development of mathematics. His work on algebraic equations contained in Miscellanea Analytica was translated into Italian by Vincenzo Riccati
Vincenzo Riccati
Vincenzo Riccati was an Italian mathematician and physicist. He was the brother of Giordano Riccati, and the second son of Jacopo Riccati....
in 1770. Waring's style is not systematic and his exposition is often obscure. It seems that he never lectured and did not habitually correspond with other mathematicians. After Jérôme Lalande
Jérôme Lalande
Joseph Jérôme Lefrançois de Lalande was a French astronomer and writer.-Biography:Lalande was born at Bourg-en-Bresse...
in 1796 observed, in Notice sur la vie de Condorcet, that in 1764 there was not a single first-rate analyst in England, Waring's reply, published after his death as ‘Original letter of Dr Waring’ in the Monthly Magazine, stated that he had given ‘somewhere between three and four hundred new propositions of one kind or another’.
Death
During his last years he sank into a deep religious melancholy, and a violent cold caused his death, in PlealeyPlealey
Plealey is a small village in Shropshire, England. It is located between Pontesford and Longden.The village lies in the parish of Pontesbury and has a population of 122 .There is a post box and a Methodist Chapel...
, on 15 August 1798. He was buried in the churchyard at Fitz, Shropshire
Fitz, Shropshire
Fitz is a small village in Shropshire, England.The village is situated close to the River Severn, downstream from Montford Bridge and upstream of Shelton, near Shrewsbury.It lies in the parish of Pimhill....
.