Overcoming Greatness. The Decline and Recovery of British Mathematics in the Post-Newton Era.

Robert Bruen Harvard University January 1995

Introduction

"The eighteenth century may be considered a time of consolidation and systematization of the exciting mathematical discoveries of the seventeenth, rather than a period of progress."1

In the late 1600s, Sir Isaac Newton (1642-1727) finalized the work that had been started before him, providing rigor to the mathematics that became known as the calculus. About the same time, Gottfried von Leibniz (1646-1716) did about the same thing. Newton was the Lucasian Professor of Mathematics at Cambridge at the time. Leibniz was mathematician and philosopher of great ability who worked from 1676 to his death as a librarian. Most of his mathematical work was published between 1682 and 1692.2 He claimed to have invented differential and integral calculus in 1674, which he did not publish until 1684. Newton claimed to have invented fluxions(infinitesimal calculus) about 1666. with some manuscripts read by friends in 1669. He finally published his methods in 1693.3

The controversy that ensued has been made famous by the two intellectual giants struggling for supremacy in the arena of first rights. Newton emerged the winner and is now known as the inventor of calculus. During the development of calculus, each created a form of notation to express the differential terms. Newton chose a a form that used dots and Leibniz selected one that used letters, d, for the equivalent term. Over the next century and a half, British mathematicians followed Newton and the mathematicians in the rest of Europe followed the style of Leibniz. The end result was Britain lagging behind mathematical progress. The struggle at Cambridge to break free finally took hold in the nineteenth century, allowing England to once again produce great mathematicians. It was probably the only negative consequence of otherwise overwhelming positive contribution of the great Sir Isaac Newton.

Newton and Leibniz

The controversy came to life when Newton published Opticks (1704), making a reference to someone copying his work, clearly meaning Leibniz, in 1676. The controversy lasted until the year before Newton's death, eight years after Leibniz died, when Newton published his third edition of the Principia, with all references to Leibniz removed.4

For some background, the chronology of events is helpful. In 1674, Leibniz wrote to Henry Oldenburg (1618-1677), secretary of the Royal Society, telling him of his general analytical methods on infinite series. The reply to Leibniz was Newton had already used such methods. Leibniz asked that Newton provide him with a description of his work. In 1676, Newton complied by producing his binomial theorem with examples of it, plus a few related insights. Intrigued by what he saw, Leibniz asked for more. Again, Newton complied, and mentioned that he had used these methods as early as 1665. Moreover, he had given Isaac Barrow additional related methods in 1669. Although he did not describe these methods, he did provide some examples, from which it is inferred that the method was fluxions. At the end of this letter, Newton presented work on tangents, then implied that the questions raised by Leibniz could cause some controversy. Leibniz responded to this letter, in 1677, with a presentation of his own method of handling tangents to curves. He also introduced his own notation of dx, dy . The communication between them stopped and Newton continued on with other work.5 A central point of contention was a visit that Leibniz had with John Collins (1625-1683), who showed Leibniz a copy of Newton's De analysi . This was the work that had been given to Isaac Barrow.6

Later, in 1696, Leibniz solved one of the two challenge problems offered by the John Bernoulli, taking six months to solve it. He then suggested that Newton attempt to solve them. The day after Newton received the problems, he produced solutions to both of them. He had even been able to generalize the second problem.7 Both men were geniuses with different strengths that have helped all who followed.

The evidence appears to support that Leibniz had a copy of one of Newton's manuscripts, had received Newton's letters and made some alterations of his own work (such as dates). These facts place suspicion on Leibniz for an independent invention of calculus. It is also clear that Leibniz presented differential and integral calculus, while Newton presented infinitesimal calculus (fluxions). Leibniz had invented the notation that was adopted by the rest of the world and had a more general form of calculus. The disagreement between them was bitter. The English mathematicians saw Leibniz as a fraud and a thief.8

I do not pretend to know the truth behind the controversy, but what is known is the important consequences that followed. The mathematicians on the continent adopted Leibniz's notation and applied it to the Principia (published 1686). England choose to use the geometrical and fluxional methods of Newton.

British Progress

"The complete isolation of the English school and its devotion to geometrical methods are the most marked feature in its history during the latter half of the eighteenth century; and the absence of any considerable contribution to the advancement of mathematical sciences was a natural consequence."9

This by necessity an unfortunately short section. In Britain, the followers of Newton comprised a good group of mathematicians, but not great mathematicians, and they supported his fluxional approach. The following men are representative of this group, but it is not an exhaustive list. There were others who had sections of their works devoted to fluxions and others who used Newton's notation in their work. These men were some of the more important writers.

Humphrey Ditton(1675-1715) published a textbook on fluxions in 1706.10 It was of the earliest books treating fluxions. Brook Taylor (1865-1731) was a Cambridge educator. Most of his work was published from 1712 to 1719 and includes book which created the theory of finite differences, as well as the support of fluxions.11 Roger Cotes (1682-1716), the first Plumian Professor of Astronomy and Experimental Philosophy at Cambridge edited the second edition of the Principia in 1713.12 He exchanged numerous letters with Newton and was an ardent supporter. Colin Maclaurin (1698-1746) published his Treatise of Fluxions (1742), the first true systematic description of the method of fluxions. It came into being in response to Berkeley's attacks on the principles of calculus. He was instrumental in convincing British mathematicians to follow Newton.13

European Progress

In this section, some of the most prominent men of mathematics are presented with examples of their achievements. This is not an exhaustive list, but it does represent the best of the eighteenth century European mathematicians. It is very difficult to do justice to all the progress made in this period in so small a space. They are presented to contrast the state of European and British mathematics. There were several reasons for the great progress in Europe, but one that was absent from Britain was the role of patronage. The great courts of Europe thought it to be a matter of status to employ a mathematician.14 It gave the mathematicians an opportunity to do research without distraction. The largest number of top mathematicians come from France, with Switzerland coming next. Some of them, like Euler, were employed by different countries. He worked in both Russia and Germany.

James Bernoulli (1654-1705 Switzerland) and John Bernoulli (1667-1748 Switzerland) were the only other people who really knew calculus besides Newton and Leibniz.15 They supported Leibniz and were instrumental in teaching calculus in Europe. Guillame l'Hospital (1661-1704 France) wrote first treatise explaining principles and methods of calculus published in 1696 as Analyse des infininet petits.16 Jean-le-Rond D'Alembert (1717-1783 France) wrote essays in 1738 and 1740 on calculus good enough to get him elected to the French Academy. He extended Newton's third law of motion to the internal forces of inertia. He also worked on physical astronomy and the great French Encyclopaedia.17 Leonhard Euler (1707-1783 Switzerland) made many contributions, including the symbol e for the base of Napier's logarithms (2.71828...), equations of motion for fluid flow (Euler's equations), and the notation for trigonometric functions. He set the standards for algebra and theory of equations. He wrote the first complete text book on differential calculus in 1775.18 Euler criticized the English notation in his writings, especially when higher powers were involved.19 Joseph Lagrange (1736-1813 Italy, Germany, France) worked in the theory of numbers, analytical geometry, mechanics and astronomy. He created the field of differential equations. He is best remembered for the Lagrangian function, Lagrangian interpolation, Lagrangian method of multipliers and Lagrange's theorem. He authored hundreds of papers.20 Lagrange used Leibniz's notation mainly, but he also used Newton's on occasion. He was comfortable using both.21

Pierre Simon Marquis de Laplace (1749-1827 France)is best known for work in celestial mechanics, but wrote papers in integral calculus, finite differences and differential equations. Laplace's differential equation, the Laplace transformation, Laplace's expansion of a determinant are part of his legacy. In 1816, he was the first to explicitly explain the reason for Newton's theory of vibrational motion producing an incorrect value for the velocity of sound.22 Adrian Legendre (1752-1833 France) One of the greatest mathematicians of all time, he worked in geometry, theory of numbers, calculus, elliptical functions, and prime numbers. He proved \pi to be irrational and is remembered for the Legendre symbol, the Legendre necessary condition, Legendre polynomials, and the Legendre differential equation.23 Jean Fourier (1768-1830 France) was the creator of the Fourier series, the Fourier transform and Fourier Theorem.24 Simeon Poisson (1781-1840 France) has many mathematical ideas named for him including the Poisson distribution, the Poisson differential equation, the Poisson integral, the Poisson process, and the Poisson ratio.25

These men pushed mathematics to new heights in Europe far past the work done in England. The notational difference is not enough to explain the situation. The controversy became a point of national pride until it degenerated into pure stubbornness in England. The men who followed Newton were good mathematicians, but simply tried to defend Newton, rather than progress past him. The men on the continent lined up behind Leibniz, but put their energy into their work to improve what the English had produced. Without the need to retreat into themselves, having a better tool in the notation, plus the patronage system, gave them an edge. This edge would still have been enough to produce the results we have seen, but if Britain had participated in the growth, there is no telling how much more would have been accomplished or how much faster it would have occurred.

Cambridge

This section presents the story of the overthrow of Newton's notation at Cambridge. The main impetus for the change is found in the Analytical Society (1812-1813), but there were signs early on.

"The efforts of the [analytical] society were supplemented by the rapid publication of good text-books in which analysis was freely used. The employment of analytical methods spread from Cambridge over the rest of Britain, and by 1830 these methods had come into general use."26

John Craig was the first to publish in England using Leibniz's notation, starting in 1685, the year following Leibniz's publication. He wrote several books and a half dozen papers with this notation, then in 1718, he switched to Newton's notation, probably out of loyalty.27 Robert Woodhouse (1773-1827) was both a Lucasian Professor and Plumian Professor. He espoused the continental form, but also criticized its shortcomings. Although he was among the first to publish on this topic, Principles of Analytical Calculations (1803), his only influence appeared to be with several of his students: George Peacock, Charles Babbage and John Herschel.28

George Peacock (1791-1858) eventually became the Lowndean Professor of Geometry, although he left few papers. He was a founding member of the Analytical Society and played a significant role in overturning the use of Newton's notation.29 Another Lucasian Professor and second founding member of the Analytical Society, Charles Babbage (1792-1871) wrote papers comparing the d-ism of Leibniz and the dot-age of Newton. Babbage continued for many years to criticize the state of mathematics and science in Britain as well as Cambridge education. The third founding member of of the Analytical Society, Sir John Herschel (1792-1871) moved from mathematics to follow his father's work in astronomy, limiting his role in the change to his time in the of the Analytical Society.

Another player at Cambridge was Edward Waring (1736-1798), a Lucasian professor of mathematics. He commented in his major work that the notation used by the English was not as good in distinguishing between terms as the Continental notation giving several examples. He used both forms of notation in his work and did not press for one or the other to be adopted.30

The Analytical Society stands out in the history of British mathematics because it was student creation and had a great influence even though it lasted only a short time. It is seen as both the defining moment of the reform and also as just a precursor.31 The Society can be seen as the important event in the history of the reform, but the most important factor is the group of men who made up the Society. They began the reform and continued it for years to come. As with any great reform, the point in time when it occurs is only a point. The real success comes through long, hard efforts. George Peacock and Charles Babbage were the true forces behind the Society and both made important contributions to the reform in later life. In fact, many of the original sixteen members of the Society were outstanding students, with eleven becoming professors at Cambridge. The placement of these students as professors at Cambridge could only have helped as the reforms were undertaken.

The main contribution of the Analytical Society was the translation of the textbook Traite elementaire de calcul differntiel et de calcul integral (1802) by Silvestre Lacroix(1765-1843, France). The Society held numerous meetings to discuss mathematics, especially analytics. The term analytic refers to the formal use of algebra as a basis for work in contrast to synthetic, which implies the geometric basis that characterized early Cambridge mathematics. It should be emphasized that the battle over notation was actually much more. It involved the fundamental approach to all of mathematics.

The actual shift of notational use at Cambridge began in 1817, when George Peacock was appointed moderator for the annual Senate-House mathematics examinations. He favored the differential notation in calculus and wrote his examination questions using it. This caused a major uproar at Cambridge, which included the prediction that he would never again be allowed to moderate an examination. Indeed, in the next year, the mathematical examination returned to its previous use of the fluxional notation. But, in 1819, Peacock was appointed moderator along with Richard Gwatkin (1971-?), who also used the differential notation for his calculus questions. Gwatkin had been one the original members of the Analytical Society. That same year, William Whewell, an earlier critic of Peacock, published the first book on applied mathematics written in English, Elementary Treatise on Mechanics , which used the differential notation exclusively. In 1820, Whewell was appointed moderator along with Henry Wilkinson and this examination fully used differential notation. Peacock was once again appointed moderator in 1821 and the revolution was complete.32

As one example of the climb out the hole, the Lucasian Chair of Mathematics shows the difficulty Cambridge experienced. Woodhouse became the Lucasian Professor in 1820, but moved on to the Plumian Chair after only two years. He was followed by Thomas Turton who spent his four years writing religious tracts. Next came George Airy, an outstanding astronomer, but he only spent four years in the Chair. Charles Babbage assumed the Professorship in 1828. Because he was instrumental in forcing the shift in notational methodology one would have expected him to be a wonderful leader in mathematics. The truth is that he did not even reside at Cambridge, but rather spent his time in London working on his computing engines. These engines eventually became the computer industry today and the basis for our technological revolution, certainly a great achievement. Sadly, it did not help British mathematics very much. Joshua King held the Chair from 1839 to 1849. He was last of the Lucasian professors that represented the old guard. He published nothing while in the Chair, instead spent his time on administrative tasks. It was not until Sir George Stokes assumed the Chair in 1849 that the Lucasian Chair and British mathematics returned to heights of Barrow and Newton.33

This Chair is representative of the struggle of Britain to regain its place in the world of mathematics. From the point of the change in the Tripos examinations in 1820 to Stokes taking the Chair, was about thirty years, but the Chair was more like a musical chair with seven professors in the Chair including Woodhouse and Stokes. In contrast the eighteenth century, while not a great time of mathematical progress, still had only four professors. The twentieth century has also only had four professors, each of whom has been outstanding. The instability of the Chair in the early part of the nineteenth century reflected the turmoil of the fundamental change that effecting the mathematics of Britain. The return of stability and excellence with Stokes, also reflected the state of British mathematics as a whole.

Conclusion

What had started with Woodhouse in 1803 was more or less finished by 1830. England had returned to the fold.34 There were three phases of this story after the Newton-Leibniz controversy began. The first was the retrenchment of British mathematicians in support of Newton and the forward movement of the rest of Europe. The second was the reform movement started by Woodhouse and finished by Peacock. The last was the gradual improvement of British mathematics. It was challenging time for Britain, but one that was met with success. Newton's negative affect took almost a century and half to undo to return to his level, or as close as one can come to such a genius, but it was returned.

Footnotes

  1. J. M. Dubbey, Development of Modern Mathematics (London: Butterworths 1970), 66.
  2. W. W. Rouse Ball A Short Account of the History of Mathematics (New York: Dover, 1960), 353.
  3. Ball, 351.
  4. Eli Maor. e The story of a number (Princeton: Princeton University 1994), 91.
  5. Ball, 326.
  6. Maor, 84.
  7. Ball, 350.
  8. Ball, 360.
  9. Ball, 438.
  10. Florian Cajori, A History of the Conceptions of Limits and Fluxions in Great Britain from Newton to Woodhouse (Chicago: Open Court, 1919), 43.
  11. Ball, 381. Ball, 382.
  12. Ball, 386.
  13. Dubbey, Development of Modern Mathematics, 68.
  14. Ball, 366.
  15. Ball, 370.
  16. Ball, 374.
  17. Ball, 393.
  18. Florian Cajori, History of Mathematical Notations (Chicago: Open Court, 1929), 213.
  19. Ball, 401.
  20. Cajori, History of Mathematical Notations, 214.
  21. Ball, 412.
  22. Robert James Mathematics Dictionary (New York: Van Nostrand Reinhold 1992).
  23. Ball, 432.
  24. Ball, 433.
  25. Ball, 442.
  26. Cajori, of Limits, 37.
  27. Cajori, History of Mathematical Notations, 211.
  28. Ball, 441.
  29. Cajori, History of Mathematical Notations, 211.
  30. Philip Enros, "The Analytical Society (1812-1813): The Precursor of the Renewal of Cambridge Mathematics," Historica Mathematica 10 (1983): 24-47.
  31. J. M. Dubbey, The Mathematical Work of Charles Babbage (Cambridge: Cambridge University Press, 1978), 41.
  32. Robert Bruen. The Lucasian Legacy: The Lucasian Professorship of Mathematics at Cambridge University 1663-1993 Ph.D. diss. Boston College, 1995.
  33. Morris Kline, Mathematical Thought from Ancient to Modern Times (Oxford: Oxford University Press, 1972), 622.

Bibliography

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