eighteenth-century British mathematics does not enjoy a good reputation. The eighteenth century, a ‘period of indecision’ as many historians would say, is said to have witnessed ‘the crisis’ or the ‘decline’ of mathematics in the country of Newton, Wallis and Barrow. However, even a glance at the following list of names should be sufficient to refute the prevailing image of eighteenth-century British mathematics. To the imported Abraham de Moivre one can add the native Brook Taylor, James Stirling, Edmond Halley, Roger Cotes, Thomas Bayes, Colin Maclaurin, Thomas Simpson, Matthew Stewart, John Landen and Edward Waring. Through their work they contributed to several branches of mathematics: algebra, pure geometry, physical astronomy, pure and applied calculus and probability.

I devote this work to a theory that all these natural philosophers knew very well: the calculus of fluxions. This was the British equivalent of the more famous continental differential and integral calculus. It is usually agreed that the calculus of fluxions was clumsy in notation and awkward in methodology: the preference given to Newton's dots and to geometrical methods engendered a period which was eventually labelled as the ‘Dot-Age’. Furthermore, the calculus of fluxions is usually indicated as the principal cause of the decadence of British mathematics: the ‘Dot-Age’ was the price paid for a chauvinistic attachment to Newton's theory.

Table of contents of William Emerson's The Doctrine of Fluxions (1743) (2nd edn., 1757)

Preface

The Doctrine of Fluxions

Postulatum

Definitions

Notation

Axioms

Section I The Fundamental Principles and Operations of Fluxions

Prop.I. The fluxion of any fluent or generated quantity is equal to the sum of the fluxions of all the roots or sides, each multiply'd continually by the index of its power, and by the given fluent when divided by the said root or side

Prop.II If two fluents or variable quantities be equal to each other, or in a given ratio; their fluxions will be equal, or in the same given ratio. And if two flowing quantities be equal or in a given ratio, their contemporary fluents will be equal, or in the same given ratio

Prop.III To find the fluxions of quantities Prop. IV–VIII several theorems for finding the fluents of quantities from other given fluents

Prop.IX To transform fluxional quantities into others equal to them

Prop.X To find the fluents of quantities by infinite series

Prop.XI To find fluents by the table

Prop.XII To correct the fluent of a given fluxions

Prop.XIII To investigate a problem by the method of fluxion

Section II The investigation and solution of some of the most general and useful problems in the mathematics

Prob.I To find the maxima and minima of quantities

Prob.II To find the logarithms of numbers

in the two decades which followed the dispute with Berkeley the calculus of fluxions was given shape in a number of ‘treatises’. We have already looked at the best achievement of this generation of textbooks, Maclaurin (1742). In this chapter I have grouped together the other treatises. The first thing which strikes one about them is their quantity rather than their quality. It has been possible to estimate that from 1736 to 1777 about 18000 copies of treatises on fluxions were sold in Great Britain. This situation contrasts sharply with the first three decades of the century, in which the calculus of fluxions was known to very few mathematicians. It was about the middle of the century that the world of ‘philomaths’ and, perhaps, some students in the military academies and universities began to practise with Newton's dots. The form the calculus would take in the second half of the eighteenth century very much depended upon the period covered in this chapter.

Teaching the algorithm of fluxions

The flood of controversial pamphlets on foundations published in the years 1734–6 (see chapter 3) was followed by an intense production of ‘textbooks’ on the calculus of fluxions. They were intended for beginners, as Hayes (1704), Ditton (1706) and Stone (1730) had been: the reader was introduced to the notation, rules and application of the calculus. The only requisite was a knowledge of algebra (but not of series) and geometry (with or without trigonometry).

despite its minor role in the diffusion of Newtonian science, we might expect the calculus of fluxions to have been the main subject of research in early eighteenth-century British mathematics. Scientists of outstanding ability were active in Great Britain. Some of them, such as Cotes, Taylor, Maclaurin and Stirling, were good mathematicians who were able to master every aspect of Newton's work on fluxions. But British mathematicians devoted more attention to other branches of Newton's mathematics: i.e. the geometry of higher order curves and the method of series. The reason why the calculus of fluxions was not considered a fruitful area of research is that it appeared to have been developed by Newton to the highest level of perfection. For instance, in 1721 Colin Maclaurin wrote:

The direct method allowed one to find the fluxion of all the known fluents, whereas the inverse method required term by term integration of power series. One of the problems left open was to speed up the convergence of series; a problem which could be treated by finding appropriate transformations involving finite differences. Newton's ‘Methodus differentialis’ (1711c) appeared more incomplete than ‘De quadratura’ (1704c). Also, Newton's ‘Enumeratio’ (1704b) was more problematic in the classification of cubics.

From William Emerson's Arithmetic of Infinites London, J. Nourse, 1767.

I The Elements of Trigonometry, by W. Emerson. The second edition, with large additions; together with the tables of sines, logarithms, etc. 8vo. 1764. Price 7s.

II The Elements of Geometry, by W. Emerson 8vo. 1763. Price 5s.

III A Treatise of Arithmetic, by W. Emerson. 8vo. 1763. Price 4s. 6d.

IV A Treatise of Algebra, by W. Emerson. 8vo. 1764. Price 7s.

V Emerson's Navigation. The second edition. 12mo. 1764. Price 4s.

VI A new Method of Increments, by W. Emerson. 8vo. 1763. Price 7s. 6d.

VII A Treatise of Algebra, by T. Simpson. The third edition. 8vo. 1766. Price 6s.

VIII Essays on several curious and useful Subjects in Speculative and Mixed Mathematics, by T. Simpson. 4to. 1740. Price 6s.

IX Mathematical Dissertations on a Variety of Physical and Analytical Subjects, by T. Simpson. 4to. 1743. Price 7s.

X Miscellaneous Tracts on some Curious and very Interesting Subjects, by T. Simpson. 4to. 1757. Price 7s.

XI Trigonometry, Plane and Spherical, with the Construction and Application of Logarithms, by T. Simpson. The second edition. 8vo. 1765. Price 1s. 6d.

XII The Doctrine of Annuities and Reversions, by T. Simpson: 8vo. 1742. Price 3s.

XII Appendix to ditto. 8vo. Price 6d.

XIV The Elements of Geometry, by T. Simpson. The second edition, with large alterations and additions. 8vo. 1760. Price 5s.

XV Select Exercises for Young Proficients in the Mathematicks, by T. Simpson. 8vo. 1752. Price 5s. 6d.

the problem of foundations did not exist in the eighteenth century as we understand it nowadays. Mathematicians were more occupied with defining the ‘principles’ of the calculus. They were concerned with the ontological status of the objects of the calculus and with the correctness of the methods of the calculus according to the standards of Aristotelian logic. In 1734 these issues were raised by Berkeley in The Analyst (1734). His criticisms were particularly devastating in Britain since the terminology of the fluxionists was very loose, being a mixture of Newtonian and Leibnizian ideas. A controversy between Berkeley and some defenders of Newton occupied the years 1734–5. Later the debate divided the fluxionists themselves. These years of debate were of great importance since the British were compelled to reread Newton in order to remove from his calculus the flaws indicated by Berkeley. The great champion of this process of reinterpretation was Maclaurin. After his Treatise (1742) the calculus of fluxions solidified into a theory based on an ‘axiomatization’ of the basic properties of motion and velocity in which geometric limit processes were prominent.

Berkeley's criticisms of the calculus

The most famous aspect of the history of eighteenth-century British calculus is undoubtedly the controversy originated by The Analyst (1734). As is well known, George Berkeley (1685–1753) criticized in this short pamphlet the foundations of the fluxional and the differential calculus.

Cambridge University Library

MS Add 6312: Nicholas Saunderson's lectures on tides, capillary tubes, barometer, vapours, rain, hail, snow, figure of the Earth, thunder and lightening, winds, circulation of sap, rainbow, heat and cold, sound, chronology, astronomy, the horizontal Moon (dated 1737).

MS Add 2977: Nicholas Saunderson's lectures on hydrostatics, sound, optics, mechanics, astronomy, the tides, chronology, the horizontal Moon, heat and cold.

MS Add 589: Nicholas Saunderson's lectures on hydrostatics, sound, optics, mechanics, astronomy, the tides, chronology (dated 1727–9).

MS Add 3444: first part of Nicholas Saunderson's Method of Fluxions (dated 1738).

Gonville and Caius College Library (Cambridge)

MS Add 723/749: John Micklebourgh's mathematical exercise book (1720s).

Watson Library (University College, London)

MSS Graves 3–5: James Bradley's sketch of Oxford lectures

MS Add 243: Nicholas Saunderson's lectures on hydrostatics, the tides, sound, optics, mechanics, astronomy, the rainbow, chronology (dated 1730).

MS Graves 23(2): letters to John Nourse from authors (John Stewart, John Colson, John Robertson, John Landen, William Emerson, Hugh Hamilton, Francis Holliday, Robert Heath, Robert Thorp, Nevil Maskelyne).

Bodleian Library (Oxford)

MS Rigaud 3–4: Nicholas Saunderson's lectures.

MS Radcliffe Trust: Thomas Hornsby Oxford lectures.

Royal Society Library (London)

MS LBC.16,412,426,456: Robert Simson's letters.

British Library (London)

MS Add Eg.834: Nicholas Saunderson's lectures.

MS Royal 487.b. 17: first part of Nicholas Saunderson's Method of Fluxions.

the military schools played an interesting role in the reform of the British calculus. At Woolwich we find a group of engineers including Hutton, Barlow, Gregory and Bonnycastle. They had at their disposal the dockyard, the arsenal and nautical instruments, and so were in a position to investigate the strength of materials, magnetism and ballistics. Their knowledge of similar work in France was extensive, while their own contributions to engineering were known on the continent. They did not excel as mathematicians, and, with the exception of Bonnycastle, they did not break with the fluxional tradition. As teachers they could not introduce any sophisticated innovations into the curriculum for the ‘raw and inexperienced’ cadets. However, with their textbooks and essays they greatly contributed to improving the knowledge of continental science in Britain. At Sandhurst more basic research on integration was carried out by the two Scots, Ivory and Wallace, assisted by Leybourn and several other colleagues. They contributed to a Mathematical Repository, where differential notation, partial derivatives and difference equations appeared as early as 1806. Their work marked a significant step towards the continental calculus. Certainly it was not possible for them to introduce their students to more than algebra and trigonometry. The masters at the military schools also participated actively in the astonishing number of scientific journals and encyclopaedias published during the first decades of the nineteenth century. Their contribution to the development of British science has been unjustly neglected.

has my research been successful in refuting the accepted views on the crisis of the Newtonian calculus? None of these views corresponds to the image we obtain from a close scrutiny of the fluxional texts. Nevertheless, the label ‘dotage’ still attaches to the treatises on fluxions we have encountered. A crisis did occur, but it set in later than is usually thought.

During the first four decades of the eighteenth century the calculus of fluxions was developed by men such as Cotes, Taylor, Stirling and Maclaurin. With their work they made original and important contributions to the fields of integration, series and applied mathematics. The British began losing ground in the 1740s. In the mid-eighteenth century only Simpson and Landen achieved new results. Very soon many British mathematicians realized the gap which separated them from the continentals and several attempts were made to change this situation of isolation. We have seen how slow and complex was this process of reform, which began with Simpson, Landen and Waring, and was continued especially by Playfair, Ivory, Wallace, Woodhouse, Brinkley and the Analytical Society's fellows.

The era of the Newtonian calculus cannot be simply described as a period of decline. It was a period of the history of British mathematics which began with successes, suffered a period of crisis, and ended with serious attempts to reform.

the period which followed the death of Newton did not witness only the disputes about foundations: with Maclaurin's Treatise of Fluxions (1742) and Thomas Simpson's Mathematical Dissertations (1743) and Miscellaneous Tracts (1757) we find the best examples of successful applications of the calculus of fluxions. As was typical in the eighteenth century, Maclaurin's and Simpson's mathematical works were motivated to a great extent by mechanics and physical astronomy. The Principia constituted a field of research rich in problems which demanded improvements in mechanical concepts as well as in mathematical techniques.

Basically, all the natural philosophers who decided to accept the idea of universal gravitation had to cope with three great problems: the mathematical treatment of fluids, the determination of the shape of the Earth (and the motion of the tides), and the determination of the Moon's orbit and the perturbations of the planetary motions. What lay behind these problems was the necessity of extending mechanics from the study of point masses to continuous bodies. Both the study of the Earth, conceived as a rotating fluid, and that of the Moon were connected with the development of the mechanics of rigid and fluid bodies. In order to develop this programme in mechanics and astronomy, eighteenth-century mathematicians were compelled to reshape the calculus. New concepts, such as the concept of a function of many variables, and new theories, such as the theory of partial differential equations and the calculus of variations, were created in this context.

when at the beginning of the eighteenth century the Newton–Leibniz controversy exploded, the great majority of British scientists declared their loyalty to Isaac Newton. In their opinion the calculus of fluxions had been invented long before the differential calculus of Leibniz and had been stolen by the German philosopher. It was also generally maintained that Newton's calculus was more firmly grounded than Leibniz's: not only was it thought that Newton had been the first to invent the calculus, but also that he had laid its scientific foundation. It was not clear, however, which was the genuine form of the calculus of fluxions since it was difficult to derive a coherent idea of it from Newton's work.

Newton's first published work on fluxions, ‘De quadratura’ (1704c), appeared in 1704 as an appendix to Opticks (1704a). This was followed in 1711 by the ‘De analysi’ (1711b), a tract written in c. 1668/9. And then, of course, there were the Principia (1687) with their huge apparatus of mathematical techniques. It was not until 1736 that there appeared in English, with the title The Method of Fluxions and Infinite Series, the translation of a long and comprehensive treatise which Newton had composed in 1671. All the Newtonians agreed that the calculus of fluxions was contained in these works. However, in the Principia there was no proper algorithm of fluxions, even though series and some algebraical manipulations with ‘moments’ occurred, especially in the second book.

the next three chapters are devoted to the attempts at reforming the calculus which were made in the late eighteenth century and early nineteenth century. The first location to which our attention is drawn is Scotland, where it was more likely that there were contacts with the continent. A geometric tradition was alive in Glasgow and Edinburgh, but this did not hinder several Scottish mathematicians from appreciating the importance of the ‘analysis’ elaborated by the continentals. John Playfair played a significant role in promoting the study of analytical mathematics. In his lectures on physical astronomy in Edinburgh he certainly introduced a taste of the new techniques of Laplace. Perhaps more influential were his reviews and historical accounts concerning the development of mathematics in which he was extremely critical of the fluxional tradition. His call for a change was taken up by two young Scots, Wallace and Ivory. Their researches on elliptic integrals, motivated by Landen (1775), led them to appreciate Legendre's work. I conclude the chapter with a few notes on a quite obscure outsider, William Spence, who adopted in the early nineteenth century the differential notation and, more importantly, some algebraical methods of Arbogast.

The universities and the Royal Society of Edinburgh

Mathematics was taught in all four of the Scottish universities. There were Chairs of Mathematics at Glasgow, Aberdeen, St Andrews and Edinburgh.

at the beginning of the eighteenth century, Newtonian Natural Philosophy began to spread all over Europe. In Great Britain an astonishing amount of interest was shown in the new science. A great number of lectures were given both in and outside London. The role of the fluxional calculus in this process of reinterpretation and popularization was negligible. Quite understandably, a technical subject did not attract the audiences who gathered to attend the lectures on Natural Philosophy.

However, a group of ‘philomaths’ was already active in the seventeenth century. They were most likely interested in Newton's mathematical theory; and, when properly instructed, they were probably able to develop the calculus of fluxions. But in the later years of the seventeenth century the Newtonian calculus could only have been known to Newton's correspondents, such as Collins, Oldenburg and Wallis. Manuscript copies of parts of Newton's work on fluxions were in circulation. The few who were allowed to glance at them, such as Craige, Fatio and David Gregory, were able to grasp just a fraction of Newton's achievements.

Needless to say, the first persons to try and systematize this information in the form of a treatise had to use as their sources the Acta Eruditorum and L'Hospital's Analyse (1696). Particularly remarkable were the attempts of Cheyne (1703) and Hayes (1704).

In later years the calculus enjoyed a time of great popularity thanks to the priority controversy with Leibniz.

in this final chapter we will consider the important contributions to the reform of the calculus which took place in Cambridge and Dublin. There is a link between the two universities, since John Brinkley, one of the most influential Dublin reformers, was educated at Cambridge and brought the heritage of Maskelyne and Waring to Ireland. Furthermore, both the Dublin and the Cambridge reformers were deeply concerned with the teaching of mathematics. Their attempt to reform the teaching of mathematics (and the calculus in particular) was much bolder than that of Playfair in Edinburgh, while – as we have seen – in the military schools such a project could not be implemented. The Dublin group insisted more on the teaching of applied mathematics (mechanics, physical astronomy, optics, etc.), whereas the Cambridge group was definitely purist–algebraist. A distinction must also be made at the level of research: in Ireland the stimulus came from Laplace, while in Cambridge the reformers were followers of Lagrange. Scholars of William Rowan Hamilton's optics, quaternions and mechanics, as well as scholars of the algebras of Peacock, Boole and De Morgan, will find this distinction quite significant.

Fluxions in Cambridge

During the late eighteenth century, Cambridge did not appear a promising centre of mathematical reform. Notwithstanding the fact that mathematics had become the most important subject in the education and in the ranking of students, their curriculum did not include any of the advances made after the 1720s.