by David J. Thomas

© Oxford University Press 2004 All rights reserved

**Raphson, Joseph** *(fl.* 1689-1712), mathematician and writer, was possibly born in 1648, though this date, found in some standard biographies, must be regarded as speculative. No details of his early life have been found.

Since the middle of the sixteenth century, mathematicians had been able to solve analytically simple polynomial equations of a single variable up to the fourth order, usually known simply as quartic equations. More complicated equations were generally intractable. Mathematicians tried to resolve these using numerical methods, the standard but slow one being that of François Viète. In 1689 Raphson replaced the divisor in Viète's method by what is now called the first derivative or slope of the function in the equation to be solved, though he did so without reference to the calculus of either Newton or Leibniz. His formula was novel in that it could be used repeatedly, making the method iterative. He showed that on each step the number of correct decimal places of the solution doubles, which means that convergence to the solution is rapid. He published his work in 1690 in the form of a book, *Analysis aequationum universalis.*

On 27 November 1689 Edmond Halley proposed Raphson for fellowship of the Royal Society, and he was elected just three days later. On 4 December he was admitted and signed the charter book and a bond guaranteeing payment of dues, on which he described himself as being 'of London, Gent.'; this might indicate that he was a landowner, but no corroborating records have been found. These are the only known signatures, and like many other fellows he seems not to have taken the second very seriously--surviving records show payments only in 1692 and 1693. By 1708 he had been excused such, possibly because he was unable to attend the society's meetings.

In the summer of 1691 Halley and Raphson met Newton with the intention of publishing Newton's work on the quadrature (integration) of curves, though ultimately Newton published it himself. The following year a royal mandate issued on 30 March instructed the University of Cambridge to confer on Raphson the degree of MA, and he was admitted to Jesus College as a fellow-commoner on 31 May. It was customary to donate silver to the college in these circumstances, and his unnecessarily extravagant gift was a monteith weighing no less than 46 oz 10 dwt. He gave his address at this time as Middlesex.

Raphson was well educated, but no details of his schooling have been traced. He wrote up his own work consistently in Latin, even at a time when the use of the vernacular was becoming popular. His second, more learned, book was *De spatio reali,* published in 1697 as an annex to the second edition of the *Analysis.* An English translation of *Mathesis enucleata* by Johan Christoph Sturm appeared in 1700 under the abbreviated Latin form 'J.R. A.M. & R.S.S.', which is presumed to mean that Raphson was the translator. His translation of Ozanam's *Dictionnaire mathématique* appeared under his full surname in 1702, and in November he even took the trouble of advertising in the press that the publishers had added material that he wished to disclaim, and had misspelt his name.

The full extent of Raphson's scholarship became apparent with the publication in 1710 of the heavy-going *Demonstratio de Deo,* which shows that his knowledge even extended to the Kabbalah, which together with his name suggests that he was probably of Jewish extraction and of an Irish immigrant family.

Roger Cotes reported in a letter dated 15 February 1711 that he met Raphson in the summer of 1709 and was surprised by his lack of interest in some of Newton's papers relevant to the writing of Raphson's *Historia fluxionum.* When this was published posthumously in 1715 it turned out to be a biased history of Newton's development of the differential calculus. No direct record of Raphson's death is known; conflicting circumstantial arguments place it in either 1715 or, more probably, late in 1712, the last year for which he was listed as belonging to the Royal Society.

Raphson's works continued to be published and reprinted after his death, and indeed his translation of Newton's *Arithmetica universalis* appeared for the first time as late as 1720. His method was formally related to Newton's fluxional calculus in 1740 by Simpson, and survives to this day in a form due to Lagrange (1797) but often attributed to Newton:

DAVID J. THOMAS

**Sources **

D. J. Thomas and J. M. Smith, 'Joseph Raphson', *Notes and Records of the Royal Society,* 44 (1990), 151-67

journal book (copy), 1689-1715, RS

B. P. Copenhaver, 'Jewish theologies of space in the scientific revolution: Henry More, Joseph Raphson, Isaac Newton and their predecessors', *Annals of Science,* 37 (1980), 489-548

E. G. R. Taylor, *The mathematical practitioners of Tudor and Stuart England* (1954), 418

*The correspondence of Isaac Newton,* ed. H. W. Turnbull and others, 5-7 (1975-7)

M. Hunter, *The Royal Society and its fellows, 1660-1700: the morphology of an early scientific institution* (1982)

*The correspondence of Isaac Newton,* ed. H. W. Turnbull and others, 2-3 (1960-61)

R. S. Westfall, *Never at rest: a biography of Isaac Newton* (1980)

D. Alexander, 'Newton's method: or is it?', *Focus,* 16/5 (1996), 32-3

T. J. Ypma, 'Historical development of the Newton-Raphson method', *SIAM Review,* 37 (1995), 531-51

N. Kollerstrom, 'Thomas Simpson and "Newton's method of approximation": an enduring myth', *BJHS,* 25 (1992), 347-54

F. Cajori, 'Historical note on the Newton-Raphson method of approximation', *American Mathematical Monthly,* 18 (1911), 29-32

CUL, department of manuscripts and university archives

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