note on the braikenridge-maclaurin theorem

Note on the Braikenridge-Maclaurin TheoremAuthor(s): Stella MillsSource: Notes and Records of the Royal Society of London, Vol. 38, No. 2 (Mar., 1984), pp.235-240Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/531819 .

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235

NOTE ON THE BRAIKENRIDGE-MACLAURIN THEOREM

By STELLA MILLS

Department of Pure Mathematics, University of Birmingham

N the Mathematical Gazette for I915, there was a paper (I) of Charles Tweedie, which included an account of the Braikenridge-MacLaurin

Theorem: if the sides of a polygon are restricted to pass through fixed points while all the vertices but one lie on fixed straight lines, the free vertex describes a conic section or a straight line. While the account gave a good elucidation of the mathematical problems surrounding the discovery of the theorem, some MS. letters of Colin MacLaurin indicate that Tweedie's historical assumption that priority was the reason for the dispute is not quite correct.

For Tweedie wrote (2):

... the name attached to the theorem is justified by the earlier production of Braikenridge, though MacLaurin's researches seem much more

profound. The dispute furnishes an illustration of the sensitiveness of even great

mathematicians in their claims to the priority of invention.

As Tweedie correctly indicated, in 1733 William Braikenridge's Exercitatio Geometrica (3) was published in London. In the preface Braikenridge wrote (4):

Anno 1727 per tres menses commoratus Londini haec impertivi Celeb Viro D. J. Craig Matheseos Peritissimo. Post paucos dies accidit ut C1. D. Maclaurin Edin. Prof. inviserem qui tunc Londini fuit & ille occasionem nactus retulit se colloquium habuisse cum D. Craig qui Theoremata mea sibi narraverat; dixitique insuper se quaedam invennisse similia & MSS. ostendit quo innuebat inventa sua contineri; sed qua ratione ductus nescio in manus non tradidit nec licuit in illud vel leviter inspicere... [... .in the [following] year, 1727, during a three-months' stay in London, I

imparted these matters to MrJ. Craig, that well-known man very learned in mathematics, [and he urged me to assert my public right to them]. A few days afterwards it chanced that I should pay a visit to Mr Maclaurin, the illustrious Edinburg professor, who [too] was then in London, and he took the opportunity to report that he had had a conversation with Mr Craig, who [then] told him of my theorems; and he stated, moreover, that he himself had discovered certain similar ones, and showed [me] a MS in



236

which he intimated his own findings were contained; but guided by what method I know not: he did not deliver it into my hands, nor allowed [me] even lightly to inspect it.]. In addition, in a later letter (5), undated, but published in the Philosophical

Transactions for March, 1735, Braikenridge wrote:

Nihil enim dubitabam, si qui alii hujus Inventi potirentir, quin, particulari casu edito, occasionem arrepturi essent praesertim lacessiti, generalem Methodum edendi, si talem sibi compertam revera habuissent. [Indeed, I do not doubt that if any others had indeed been master of the discoveries, particularly in the case mentioned, they would have seized the occasion of publishing a general method, especially as challenged, if they had discovered such a method]. One or more of MacLaurin's friends saw or heard of these references to

MacLaurin and told him of what they believed was a slight on his character (6):

I have lately been informed of a piece of injurious treatment I have received in print from one Mr Braikenridge who was formerly a private teacher of mathematics here.

One informant was Robert Simson who wrote (7) on 27July 1736:

... as to the Theorem he brags... I remember yow [sic] told me about it, the time yow mention 1720, and Just now I was looking some Theorems yow wrote down in the first pages of a book of mine that is dated in that year, which yow told me were Lemmas to that proposition concerning the Number of points 2 Curves can intersect one Another in....

This proposition is contained in the first paragraph of MacLaurin's Abstract (8) (published in 1735, but dated 1732) and Simson continued that:

I have . . . asserted Your being undoubtedly the first Inventor, and shall do Yow all justice and what belongs to true friendship in that affair whenever it happens to be mentioned, and have lately in several companies set it in its true light.

Almost as an illustration of his friendship, Simson gave a synopsis of Braikenridge's paper (9), and although four pages of his letter are missing, Simson actually reproduced Braikenridge's 'Theorema Generale' verbatim from the paper.



237

It is not surprising then that MacLaurin wrote (Io), on 28 July 1736, to Martin Folkes at the Royal Society, asking that his letter (I ), dated I5 November 1732, to John Machin be published, which request Folkes

immediately granted; the letter appearing in the Philosophical Transactions for December I735. This paper contained the Braikenridge-MacLaurin Theorem for a triangle (12), as Tweedie correctly asserted. However, in spite of Simson's suggestion MacLaurin did not dispute priority for in the letter (13) dated 28July 1736, to Folkes, he wrote:

It seems this man published first, an exercitatio geometria where he gives some theorems that I had of old about the description of lines, and have communicated to hundreds of people, and taught here every year since I725.... Mr B_ has since published another piece in the transactions for

January 1735 where he not only publishes theorems I have taught here

every year without naming me (to do which he was welcome) but insults me as if I had them not because I did not publish them quanquam lacessines [although challenged] He does not name me, but I am told (for I have seen neither of his pieces) that it is supposed he means me.

Here then is the reason for MacLaurin's having taken issue with

Braikenridge: not that of priority as such, but the insult that Braikenridge implied that MacLaurin did not know of the theorems, when in fact he had been teaching them since 1725. The 'challenge' was that given in the Preface to Exercitatio Geometrica as given above.

Perhaps one might think that MacLaurin should have waited to see the

printed provocations for himself, but as he wrote (14) to George Scott on 22 November 1736:

All I did in return to MrBraikenridge's double unprovoked attack upon me was to desyre that the paper should be published which I sent in 1732 to the

Society at the desyre of some of the members.

This paper (I5), while not mentioning by name either Braikenridge or the Exercitatio Geometrica, did make remarks such as:

The Author of the Papers given in to the Royal Society, will not refuse that I shewed him the Theorems, I now send you, in 1727.

Indeed, the Editor of the Philosophical Transactions, felt it necessary to subjoin to the list of contents:

N.B. The Papers hinted at by Mr. Mac Laurin p. I43. are printed in a



238

Treatise entituled, Exercitatio Geometrica de Descriptione Curvarum: Authore Gulielmo Braikenridge. Londini, I733. 4to.

and he made a similar remark at the end of MacLaurin's paper. However, before his paper was published, MacLaurin received on 22

November 1736, a 'scurrilous' letter (I6) with a fictitious signature which:

... goes on the supposition that in this paper I am to accuse him of receiv- ing or stealing hints from me.... Could I have shewn better what theorems I had in 1733 than by publishing a paper sent to the Society in 1732.

MacLaurin addressed his reply, now lost, to Braikenridge, who wrote (17) to MacLaurin on I5 January 1737 (New Style). Perhaps it was this letter which

suggested to Tweedie that the controversy was purely over priority of invention for Braikenridge took issue with the words (I8), 'the Author will not refuse that I shewed him the Theorems in I727', which point he would not have raised:

... if you had been so kind as to tell that Mr Craig had told you of my Theorems some days before; of which circumstance to my great surprize you have taken care to be silent....

This letter which continued in a similar style, upset MacLaurin considerably. His draft reply (I9), dated 7 April 1737, indicated that he:

... did indeed take the trouble to write a long answer to your letter, but on reflecting further on your prejudices & your usage of me, I choose not to send it.

MacLaurin relied on his paper (20) for obtaining the justice due to him and

brought the whole controversy to a hasty conclusion (21):

... I intend that this shall be the last letter you shall be troubled with from me....

Braikenridge however, sending his letter, now lost, via Dr Sykes, replied to MacLaurin, requesting him to publish his paper (22) in the 'Republick of Letters' for June 1737. But by 12 August, 1737 (23), Braikenridge did not know if MacLaurin had even received the letter and although he had by now '... quite recovered of that lingring Ilnes [sic] I was seized with in the Spring . . .', it seems that Braikenridge continued with the dispute no longer.

Excepting incidental remarks of no immediate consequence, there are, in



239

the extant correspondence of MacLaurin, no more references of importance to this unhappy episode in MacLaurin's life. There is no evidence thatJohn Craig showed MacLaurin anything, as Braikenridge alleged, or that Braikenridge had the theorem prior to 1733. Indeed, contemporary mathematicians had little

opinion of Braikenridge's mathematical prowess (24) and Tweedie rightly concluded that the works of MacLaurin are much more profound.

Indeed, it has been shown above that, while Braikenridge was concerned with priority of discovery, MacLaurin only took issue with the implied comment by Braikenridge that, after teaching the theorem for eight years, MacLaurin did not know of it. Thus MacLaurin was not concerned with

priority as such; he was only trying to obtain justice after a false accusation of

ignorance.

ACKNOWLEDGMENT

The author wishes to record that her work concerning Colin MacLaurin has been greatly facilitated by that of the late John Eaton, M.A. She is

particularly grateful to Dr D. T. Whiteside for the translation of the first Latin

quotation and to the referee for helpful comments. She is also grateful to the libraries of The Royal Society, the Universities of Aberdeen and Glasgow and the British Museum for kindly allowing reproduction of MSS.

NOTES

(I) Tweedie, C., 'A Study of the Life and Writings of Colin MacLaurin', The Mathematical Gazette, 8 (1915), I32-IS I. This paper was supplemented with notes in The Mathematical Gazette, 9 (1917), 303-306, and IO (1918), 209.

(2) In (I). (3) Braikenridge, W., Exercitatio Geometrica de Descriptione Curvarum, (London, 1733). (4) From MS. 34. 2, Library of University of Aberdeen, now published in S. Mills,

ed., The Collected Letters of Colin MacLaurin, (Nantwich, I982). This is the only form of the preface that MacLaurin saw, and is an accurate reproduction of the published version except for 'qui me hortatus est ut publici juris facerem' inserted after 'J. Craig Matheseos peritissimo'. ['[Craig] urged me to assert my public right to them'.].

(5) Braikenridge, W., 'A general Method of describing Curves, by the Intersection of Right-Lines; moving about Points in a given Plane. In a Letter to Dr. Hoadly, by the Rev. Mr. Braikenridge', Phil. Trans., 39 (I735-36), 25-36.

(6) Correspondence of Colin MacLaurin to Martin Folkes, No. 3 (28 July 1736), Library of the Royal Society, London. (Fo.4.3).



240

(7) MS. Gen. 1378/4, Library of University of Glasgow. This MS. was discovered in a private attic in I977.

(8) MacLaurin, C., 'A Letter from Mr. Colin Mac Laurin, Math. Prof. Edinburg. F.R.S. to Mr. John Machin, Ast. Prof. Gresh. & Secr. R.S. concerning the Description of Curve Lines', Phil. Trans. 39 (1735-36), I43-65.

(9) (5). (IO) (6). (II) i.e. (8). (12) See Proposition II of the Paper dated 27 November 1722 contained in (8). (I3) i.e. (6). (I4) MS. 206.I9, Library of University of Aberdeen.

(IS) i.e. (8). (I6) This letter, now lost, is mentioned in (14) and also MS. 206.20, Library of

University of Aberdeen. (17) (4). (I8) In (8). ( 9) MS. 206. 16, Library of University of Aberdeen.

(20) i.e. (8). (21) From (I9). (22) i.e. (8). (23) See Sloane MS. 4301, f. 257 in the British Museum. This MS. is a letter from

Braikenridge to Thomas Birch, later a Secretary of the Royal Society. (24) For example, Robert Simson expressed this view in MS. 206.9, and MS. 206.10,

Library of University of Aberdeen, and MS. Gen. 1378/4, Library of University of Glasgow. In this last-named letter, Simson mentioned that George Campbell also could demonstrate the Braikenridge-MacLaurin Theorem.



note on the braikenridge-maclaurin theorem

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