# Was English mathematics behind Europe by many years because of Newton's notation?

Below are several quotes suggesting that Newton's notation had the effect of retarding English mathematics by 50 years, 100 years, or even centuries.

Here is my simplistic two-sentence historical account of what happened (based on my reading of the various writers below):

After Newton, English mathematics was in a dark hole years behind the Continent. Then circa 1820, Englishmen adopted Leibnizian notation and by 1830, had mostly caught up and could once again contribute to mathematics.

As a layperson who isn't too familiar with this history, it seems rather incredible that notation alone cost 50+ years of progress.

My questions:

• Was English mathematics ever significantly behind — by say 50 years, 100 years, or even centuries?
• If so, to what extent can this be attributed to Newton's poor notation?

Arturo Magidin on Math.SE (2011):

In fact, Leibniz's notation is so good, so superior to the prime notation and to Newton's notation, that England fell behind all of Europe for centuries in mathematics and science because, due to the fight between Newton's and Leibniz's camp over who had invented Calculus and who stole it from whom (consensus is that they each discovered it independently), England's scientific establishment decided to ignore what was being done in Europe with Leibniz notation and stuck to Newton's... and got stuck in the mud in large part because of it.

Noah Kennedy, The Industrialization of Intelligence: Mind and Machine in the Modern Age (1989):

Disastrously for English mathematics, there was an easy way for an English mathematician to affirm his national allegiance to Newton's claim, a way that seemed innocuous enough but was actually destined virtually to cripple mathematical inquiry in England for more than a century. The two men had quite naturally arrived at two entirely different systems of notation for denoting the central concept of differentiation, and quite naturally the English adopted Newton's and the Germans favoured Leibniz's. The problem for the English was that Leibniz's notation was a far more elegant and evocative expression of the concept and lent itself much more easily to various innovations that rippled through mathematics in the wake of the discovery of the calculus. Primarily because of its utility, Leibniz's notation was in general use throughout Europe and precipitated significant innovation, particularly by the French, while in England, mathematical progress was tortuously slow, in part because of the burden of Newton's notation and in part because English mathematicians had effectively isolated themselves from the common language of continental mathematics.

Morris Kline, Mathematical Thought From Ancient to Modern Times (1972):

England too languished. Brook Taylor, Matthew Stewart (1717–85), and Colin Maclaurin were the only prominent mathematicians. England's poor performance in view of its great activity in the seventeenth century may be surprising, but the explanation is readily found. The English mathematicians had not only isolated themselves personally from the Continentals as a consequence of the controversy between Newton and Leibniz, but also suffered by following the geometrical methods of Newton. The English settled down to study Newton instead of nature. Even in their analytical work they used Newton's notation for fluxions and fluents and refused to read anything written in the notation of Leibniz.

In the first quarter of the nineteenth century the British mathematicians began to take interest in the work on the calculus and its extensions, which had proceeded apace on the Continent. The Analytical Society was formed at Cambridge in 1813 to study this work. George Peacock (1791–1858), John Herschel (1792–1871), Charles Babbage and others undertook to study the principles of "d-ism"—that is, the Leibnizian notation in the calculus, as against those of "dot-age," or the Newtonian notation. Soon the quotient $$dy/dx$$ replaced $$\dot{y}$$, and the Continental texts and papers became accessible to English students. Babbage, Peacock, and Herschel translated a one-volume edition of Lacroix's Traité and published it in 1816. By 1830 the English were able to join in the work of the Continentals. Analysis in England did prove to be largely mathematical physics, though some entirely new directions of work, algebraic invariant theory and symbolic logic, were also initiated in that country.

Jason Bardi, The Calculus Wars: Newton, Leibniz, and the Greatest Mathematical Clash of All Time (2006):

Nor was Newton's notation as useful as the superior notation that Leibniz had invented and the advanced calculus that Johann Bernoulli and the other European mathematicians developed throughout the century. Leibniz had correctly surmised that his symbols would make for the easy development of calculus, and these symbols, which he first penned in his notebooks in Paris in 1675, can still be found to this day in every calculus textbook.

In this sense, the high esteem in which Newton was held in Britain was not always a good thing, because, many of the mathematicians and scientists living there in the eighteenth century were behind the iron curtain of Newton's fame and glory. Ironically, as much as Leibniz's reputation suffered in Great Britain, the whole country may have suffered a self-inflicted wound by so underappreciating him. After the calculus wars, British mathematicians were prevented from learning calculus using Leibniz's notations, which were largely in use elsewhere, and they were not finally accepted in that country until the early nineteenth century.

Christopher D. Green, "Charles Babbage, the Analytical Engine, and the Possibility of a 19th-Century Cognitive Science" (2001):

The upholding of the Newtonian notation was a matter of some national pride for the British, for the nasty priority dispute between Newton's and Leibniz's partisans over the discovery of calculus still echoed in the very traditional halls of Cambridge. Unfortunately for the British, Newton's notation was difficult to manipulate algebraically, and they were now some 50 years behind the mathematical developments of their Continental colleagues, who had of course used Leibniz's notation from the first. Babbage, Herschel, and Peacock aimed to put a stop to what they called the "dot-age" of Cambridge (a satirical reference to the dots used to indicate derivatives in the Newtonian notation) and replace it with the "pure d-ism" of Leibniz (who used the letter "d" to indicate the same).

Gerald L. Alexanderson, "About the Cover - Voltaire, du Châtelet, and Newton" (2014):

Newton’s clumsy notation may have hampered progress in England.

• In Morris Kline's quote above (the only Quote for a "professional" math historian) we can find several related "causes" only one of which is the notation issue. For sure, the success of Leibniz's notation (with his quasi-algebraic flavor) was due some "continental" mathematicians : Bernoulli's, Euler, Lagrange. To assertion that their ability was due only to the capability of managing the symbolsim is - IMO - untenable. Sep 17, 2018 at 10:42
• For a good study, see Niccolò Guicciardini, The Development of Newtonian Calculus in Britain : 1700-1800 (2003). See Conclusion : "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." 1/2 Sep 17, 2018 at 11:50
• "The era of the Newtonian calculus cannot be simply described as a period of decline. It was a period of the history of British math which began with successes, suffered a period of crisis, and ended with serious attempts to reform. In what exactly consisted the crisis? At the beginning of the century British math was in close contact with the rest of Europe: but by the middle of the century it was almost completely separated from the continent. The works of continental mathematicians were not understood in Britain, while the works of the British aroused little interest on the continent." Sep 17, 2018 at 11:54
• Actually, when you look at modern mathematics textbooks they often warn students not to take dx/dy literally... Sep 17, 2018 at 14:32
• This reads like an answer rather than a question. I would also caution against taking rhetorical phrasing (like "fell behind all of Europe for centuries") at face value, or turning a single issue into an explanatory magic bullet. Newton's kinematic approach to calculus was considered in 18th century to be conceptually superior to infinitesimals even on the continent, for example. Sep 17, 2018 at 22:17

Several factors come together to suggest that the idea that "English mathematics [was] ever significantly behind -- by say 50 years, 100 years, or even centuries" (i.e. in the post-Newtonian 18th or early 19th centuries) is at best a sweeping over-generalization, although something very like it has clearly become a received view.

Two recent valuable studies in particular shed some light on the question: Judith V Grabiner ('...The Continental Influence of Maclaurin's Treatise...' American Mathematical Monthly, 104 (1997), 393-410), and by Niccolo Guicciardini ('...Newton's Mathematical Legacy...', in 'Early Science and Medicine' 9 (2004), 218-256).

They show for example:

(1) that Colin Maclaurin's mathematical work (in fluxions) up to the 1740s was well-received by continental mathematicians (Grabiner, 1997); Maclaurin was also awarded two prizes by the Academie royale des sciences in Paris.

{Edit begins:} In particular, Maclaurin's work gained a special contemporary continental appreciation: he was credited for his important contribution, in his work of 1742, of placing the infinitesimal calculus on a rigorous mathematical foundation, which the methods of Leibnitz in themselves did not provide. Thus he definitely and satisfactorily answered the attacks on the foundations of the calculus that had arisen repeatedly during the 18th century (see also Did Michel Rolle say that the calculus is "a collection of ingenious fallacies"?).

Jean-Étienne Montucla's Histoire des Mathématiques, 2nd ed. vol.3, was mostly completed by the end of Montucla's life and published 1802 shortly after his death by Jérôme de Lalande. It contains appreciations of the defence of the calculus by a number of British mathematicians, but gives this special acknowledgement to Maclaurin (in passages from pp.116 and 118, here in my translation):--

"Nobody is unaware these days that the infinitesimal calculus is in its fundamentals absolutely the same as what Newton has called the calculus of fluxions. Now, this latter has nothing that is not in conformity with the most rigorous principles of Geometry, as has been shown at full length. Accordingly, both the one and the other must enjoy the same degree of certainty."

[...] "It was seemingly in answer to the attacks of Berkeley that Mr Maclaurin undertook his 'Treatise of Fluxions' that appeared in 1742. There the method of Newton is fully demonstrated without any assumption of infinitesimals or anything else capable of lending itself to controversy ... Mr Maclaurin's demonstrations are of prodigious length ... he could have limited himself to some examples ... . [But] however that may be, one can say that if any doubts could remain about the solidity of Newton's method, they are entirely dissipated by this work of Maclaurin ... ."

Also shown by the studies cited above is that {edit ends:}

(2) Maclaurin's was not an isolated example. Guicciardini (2004) discusses the work and influence of a number of 18th-century mathematicians writing in English, including Brook Taylor, James Stirling, Abraham De Moivre, Thomas Simpson, William Emerson, and others, as well as Maclaurin himself.

Another work by Guicciardini offers strong arguments that the differing notational preferences between 18th-century Newtonians and Leibnizians were not as big of an issue as has often since been made out: they were interconvertible and actually interconverted: ('Reading the Principia', Cambridge, 1999; e.g. points referred to in chapter 9, p.250 et seq.). Guicciardini recommended that "it is more fruitful, and more adherent to historical evidence, to focus on the amount of shared knowledge between the two [Newtonian and Leibnizian] schools."

Guicciardini (2004, at 220) also notes that the received view "can easily be traced back to the irreverent writings of reformers such as John Playfair, John Toplis, and Robert Woodhouse, but even more so to the fellows of the Cambridge Analytical Society who, at the beginning of the nineteenth century, tried to introduce the algebraic methods of Joseph Louis Lagrange and F. A. Arbogast into Great Britain. Like all reformers, they offered a pessimistic view of the past. Since then, this received view of eighteenth-century Newtonian mathematics has prevailed in the histories of mathematics."

On the other hand, it may be suspected that there is often 'no smoke without fire', and the 'received view' of English mathematics may have some source in institutional rivalries, in which 18th-century British mathematicians lost out to representatives of competing branches of science. After Newton's death in 1727, as Guicciardini (2004 at 250) noticed, there was a contest in the Royal Society between the 'philomaths', who sympathised with Newton's view of the primacy of mathematics, and those who saw themselves as 'naturalists'. The election of Hans Sloane in 1727 as President to succeed Newton, "marked a defeat of the philomaths". The primacy of the 'naturalists' and relative discouragement of mathematicians in the Royal society seems to have continued for a long time. Thus, Marie Boas Hall's 'All Scientists Now' (Cambridge, 1984, especially at chapter 1, 'The eighteenth-century legacy') recorded that under the 42-year presidency of botanist Joseph Banks, there was only one attempt to challenge Banks' dominance, and that came in the 1780s from two men of mathematical sympathy, Charles Hutton and Samuel Horsley. Hutton had been foreign secretary of the Society but was dismissed by Banks for reasons not altogether clear. Horsley "tried to make it a revolt of the mathematical scientists against ... the biological scientists".

What may perhaps be truer than the 'received view' is that mathematics in Britain had a period of some relative institutional discouragement after the death of Newton: and this may have had a repressive effect on the numbers of practicing mathematicians and the opportunities open to them, even while those who were active were in touch and interactive with mathematical trends and mathematicians outside their own country, and not 'behind' by whatever time-units have been suggested.