Taylor versus Continental mathematicians

Brook Taylor conducted a public controversy with continental mathematicians, particularly with Johann Bernoulli. We reproduce here a few of the exchanges to give a flavour of the rather bitter argument. We also quote from some letters which show even stronger feelings than those expressed in the public dispute. A detailed discussion of Taylor's mathematics and the issues which lay behind the dispute are given in [1].

Brook Taylor published Methodus Incrementorum in 1715. He was a champion of Newton, strongly opposed to the Continental mathematicians who championed Leibniz. It is not surprising, therefore, that his work generated a strong critical attack from Continental mathematicians. Taylor, however, fought back publishing an equally strong response.

First we quote the announcement of the publication of Methodus Incrementorum , written by Taylor himself:-

Methodus Incrementorum Directa, & Inversa.
Author Brook Taylor, L.L.D. and Royal Society Secretary.
In two parts.

In the first part are explained the principles of the new incremental method, and by the means of that the method of fluxions is more fully explained than has yet been done; it being shown how this method is deduced from the former, by taking the first and last ratios of the nascent and evanescent increments. In the second part the usefulness of these two methods is set forth by several examples viz.

1. In the summing of arithmetical series.
2. In finding all the figurate numbers.
3. In the finding of tangents, rays of concavity
4. The quadrature of all sorts of curves.
5. The catenary.
6. The velaria.
6. The fornix.
7. The vibration of a musical string.
8. The centres of oscillation and percussion.
9. The density of the atmosphere.
10. The refraction of light passing through the atmosphere.

Printed for W Innys at the Prince's Arms in St Paul's Church Yard.

Leibniz wrote to Johann Bernoulli about Taylor's book, which he had read. Even before he had seen the book, Johann Bernoulli had heard enough about it to criticise it strongly in his reply written to Leibniz in July 1716:-

I have not yet seen Taylor's book ... You say that while eager to present a sample of his skills he has hardly said anything that was not said before, and that the whole book is written obscurely enough; I myself do not wonder at this, for how can he pass off as his own things which belong to others, unless he affects obscurity on purpose in order to hide the theft ... I can easily believe that Taylor cites no one except Newton: for it is a characteristic of the English that they begrudge everything to other nations and attribute all things to themselves or their nation. Thus Taylor has taken my theory of the centre of oscillation almost complete from the "Acta Eruditorum" of Leipzig, as Mr Hermann writes, and, in order that plagiary should not be excessively obvious, he so wraps up the theory in some cloak of obscurity that what I have most clearly expounded is almost unintelligible.

Leibniz replied:-

I easily judged that the book of Taylor or the English 'Sartorius' would please you very little. It seems to me that such a writer is not at all fit to carry out the office of Secretary of the Royal Society, which requires a man less mathematical perhaps, but more clear and accomplished in literary intercourse.

After he had received a copy of the book and had read through it, Johann Bernoulli wrote again to Leibniz:-

I have finally received Taylor's little book ... what is most new is plagiarism; at the end of the little book he proposes a method for determining the centre of oscillation in compound pendula, which is taken out completely from my new theory inserted in the Acta of 1714.

Still in July 1716, Johann Bernoulli published an anonymous attack on Taylor in an article Epistola pro eminente mathematico Dn Johanne Bernoulli, contra quendam ex Anglia antagonistam scropta. It is reasonably measured compared with what he wrote privately to Leibniz:-

Good God, what does the writer intend by that feigned obscurity of his, in which he cloaks matters extremely clear by their very nature? Doubtless in order to conceal his zeal for stealing: for as far as I understand, as far as I know, I perceive in it through the densest cloud of obscurity nothing except what is ours stolen from us. What he says about isoperimeters is owed to my brother; what he treats concerning the catenaries, sails, cloths filled with water, he has from me.

We will return in a moment to look at further arguments associated with isoperimetric problems but first, maintaining chronological order, we look first at a letter Nicolaus(I) Bernoulli wrote to Leibniz on 24 October 1716 about Taylor's book. As soon as he read it, Nicolaus(I) Bernoulli wrote:-

Taylor's book on the method of increments has arrived here at long last. I have found almost nothing new in it, but most things already known for a long time, however obscurely set forth and excessively abstract so that I believe them to be understood by very few. His third proposition I already solved some time ago in the following way ...

That isoperimetric problems now became a Taylor versus the Bernoullis argument is rather ironical since Johann Bernoulli and Jacob Bernoulli had themselves been involved in a bitter dispute over this matter. Jacob Bernoulli had published a correct solution in 1701 but Johann Bernoulli's solution, obtained at the same time, was not satisfactory. When Johann Bernoulli published a correct version in 1718 he began by writing:-

... I don't think that anyone would accuse me of doing something already done if in a matter as difficult as this, I demonstrate a short, clear, and easy method ... without engaging in the lengthy calculations of my brother or in the obscurity of that of Mr Taylor.

Johann Bernoulli then launched into a bitter attack on Taylor:-

Mr Taylor - a man of acuteness, and a very skilful geometer, who has successfully penetrated even our most profound discoveries, as it would appear from his book on "Methodus Incrementorum" - well aware of the tedious length of my brother's analysis, and wishing to render it shorter and a little more clear, has himself spread such obscurity over this matter (as well as other in which he has wished to be brief) that he seems to take pleasure in it, and I doubt there be anyone, no matter how penetrating he might be, that would understand all of it, even did he already know that matter in another way. To say nothing here of the short and condensed calculation that this author employs according to his custom, if one examines it closely, one finds it still rather long and confused.

In 1718 Taylor replied, although as Johann Bernoulli had done, it was published anonymously. In the article, which appeared in Bibliothèque Angloise ou Histoire Littéraire de la Grande Bretagne, he wrote:-

... it is not reasonable to expect of an author that he mention expressly all of the luminaries from whom he could have borrowed, and those who have written before him, principally when their works are so well known. Mr Bernoulli in particular hasn't the least cause to complain about the author in this regard, since he has sufficiently demonstrated the inclination he has had to render him justice, by the manner in which he mentions him in an extract he has given of his book in the "Philosophical Transactions".

Taylor, of course, is correct in saying that although he didn't credit the Bernoullis in the Methodus Incrementorum, he did credit them for their work on the isoperimeter problem on page 345 of his article An account of a book entitled Methodus Incrementorum which appeared in the October 1715 part of the Philosophical Transactions of the Royal Society.

Taylor's 1718 reply continues, using the standard ploy that attack is the best form of defence:-

The author has confined himself solely to his subject; he has avoided express mention of what has been done by others because that would necessarily have forced him to note several mistakes and imperfections found in their solutions: for that reason he has not judged it apropos to speak of the solut)rowed from these solutions the analysis he uses to resolve these problems, these solutions having the defect of being restricted to particular cases, although the problems are proposed in general terms.

The author thought that his silence concerning the Memoires de l'Academie Royale des Sciences
would be regarded as a favour done to the author of this solution, because, if he had mentioned it, he would not have been able to dispense with censuring three or four very considerable mistakes which one encounters there.

Taylor did not end the matter there. In the following year he published Apologia D Brook Taylor in the May part of the Philosophical Transactions of the Royal Society. In it he continues with his attack on Johann Bernoulli:-

In an Epistola pro eminente mathematico Dn Johanne Bernoulli Acta Eruditorum of Leipzig 1716, among others, I am accused of plagiarism, as if I arrogated to myself the inventions of Mr Johann Bernoulli and others. Let them produce their examples, and then they shall have an answer. I have indeed treated of many things in common with others; but I have by no means used other men's inventions as my own. I have everywhere used my own analysis, except in the problem of the isoperimeters; so that I have no ways defrauded others. ... What problems I have treated of in common with Bernoulli, are on the funicularia, on the centre of oscillation, and on isoperimeters. In the two former of these I have used my own analysis entirely. In the isoperimeters I used that of the author Jacob Bernoulli, a man very deserving in mathematics, to whom I now pay his due honour.

Let us declare the result of the bitter argument an honourable draw!


1. L Feigenbaum, Brook Taylor and the method of increments, Arch. Hist. Exact Sci. 34 (1-2) (1985), 1-140.

JOC/EFR March 2006

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