William Wallace and the Introduction of Continental Calculus to Britain: a Letter to George Peacock

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HISTORIA MATHEMATICA 14 (1987), 119-132

William Wallace and the Introduction of Continental Calculus to Britain: A Letter to George Peacock

M. PANTEKI

Middlesex Polytechnic, Queensway, EnJield, Middlesex EN3 4SF, United Kingdom

Little is known about William Wallace’ work and even less about his attempts to introduce Continental calculus to Britain in the early 19th century. A letter written by him to George Peacock in 1833 reveals many interesting facts concerning his role in the reform of British mathematics. This paper presents a brief account of his life and work and the full text of the letter, and proposes that Wallace’ contributions deserve to be acknowledged. o 1987 Academic Press, Inc. Wenig ist tiber das Werk von William Wallace bekannt und noch weniger tiber seine Versuche, den kontinentalen Infinitesimalkalktil im frtihen 19. Jahrhundert nach Britannien einzuRihren. Ein Brief, den er 1833 an George Peacock schrieb, deckt viele interessante Tatsachen auf, die seine Rolle bei der Reform der britischen Mathematik betreffen. Dieser Aufsatz gibt eine kurze Darstelhmg seines Lebens und Werks und den vollstandigen Text des Briefes und bitt daftir ein, daO die Beitrage von Wallace verdienen anerkannt zu werden. 0 1987 Academic Press, Inc. On ne sait qu’un peu sur l’oeuvre de William Wallace et encore moins sur ses essais d’introduire le calcul infinitesimal continental dans les Iles britanniques au debut du 1P”’ siecle. Une lettre qu’il avait &rite a George Peacock en 1833 revtle beaucoup de faits interessantes concemant son r81e dans la reforme des mathematiques britanniques. L’article comprend une description courte de sa vie et de son oeuvre et le texte complet de la lettre. It propose que les contributions de Wallace m&tent d’etre appreciees. e 1987Academic Press.Inc. AMS 1980 subject classifications: OlAS5, 2fS03. KEY WORDS: fluxional calculus, differential calculus, British mathematics, mathematical notation, translations, physical sciences.

1. HOW THIS LETTER WAS WRITTEN. QUESTIONS THAT SEEK AN ANSWER William Wallace (1768-1843) was one of the mathematicians who early in the 19th century abandoned the fluxional calculus in an attempt to promote Continen- tal mathematics. AI1 individual attempts of that time to revive British mathematics, together with those of the Analytical Society in 1812 [Enros 19831, constitute precursors of the Cambridge reform that took place in the mid-1810s. In this sense Wallace’ work can be regarded as influential in reforming mathematics at Cam- bridge. Among Wallace’ contemporaries the most well known for his attempts to introduce the differential calculus, and consequently to change Cambridge mathematics, is Robert Woodhouse (1773-1827) [Ball 1889, 113, 117-118; Becher 1980, 9- 119 0315-0860/87 $3.00 Copyright 0 1987 by Academic Press, Inc. AU rights of reproduction in any form reserved. 120 M. PANTEKI HM 14

10; Dubbey 1963, 37-39; Enros 1981, 136; Enros 1983, 261. His first book, Principles of Analytical Calculation, published in 1803, did not have the same success as his Trigonometry published seven years later. Peacock, in his “Report on the Recent Progress of Certain Branches of Analysis” to the British Associa- tion in 1833, stated that Trigonometry was the most revolutionary mathematical work in Britain [Peacock 1833, 2951. Upon reading Peacock’s comments Wallace sent him a letter, bringing to light parts of his own work which he claimed to have been as revolutionary as Woodhouse’ work. His surprise and sorrow at having been unacknowledged by Peacock are obvious throughout the letter. How justifiable was Wallace’ surprise? To what extent could his work be characterized as revolutionary? Should he be regarded as a Cambridge reformer, or should we see him simply as one of those individuals who did their best to introduce French calculus into their country? After giving a brief account of his life and early work, as well as the full text of his letter, I conclude with a commen- tary on the early attempts to revive British mathematics which answers these questions, my aim being to give Wallace the credit he deserves. A list of his main published papers and articles is given at the end of the paper. 2. BIOGRAPHICAL SKETCH OF WALLACE William Wallace was born in September 1768 at Dysart in Fifeshire, Scotland [ 11. His early education was inadequate, for at the age of 11 he was apprenticed to a bookbinder. When he was 16, his family moved to Edinburgh where he continued in the same occupation. During his leisure there he studied mathematics, and before the completion of his apprenticeship he had acquired considerable knowledge of geometry, algebra, trigonometry, and astronomy [2]. An important event was his introduction by a carpenter acquaintance to Dr. John Robison, Professor of Natural Philosophy at the University of Edinburgh. Subsequently, Robison introduced Wallace to John Playfair, Professor of Mathematics at Edinburgh from 1785 to 1803, who became his lifelong friend. Robison and Playfair admired Wallace’ skill in geometry and encouraged him to carry on with his studies. Manual labor which had prevented him from devoting himself to the pursuit of knowledge was gradually replaced by intellectual study, as he was assigned pupils for private teaching. He began to study Latin, and around 1790 he decided to study French also, in order to familiarize himself with Continental mathematics. In 1790 Playfair published a paper “On the Astronomy of the Brahmin” in the Transactions of the Royal Society of Edinburgh, mentioning memoirs by Laplace and Lagrange. The precise influence on Wallace of Play- fair’s awareness of the development of the mathematical sciences in France is not known, however. In 1794, after a period as a bookseller’s assistant and a private teacher, as well as a student at the university, Wallace was appointed assistant mathematical teacher at Perth Academy [3]. In 1803 he exchanged this post for a mathematical mastership at the Royal Military College at Great Marlow in Buckinghamshire (later at Sandhurst), which, with Playfair’s advice and encouragement, he ob- tained as a result of competitive examination. He stayed at the College, with his HM 14 WILLIAM WALLACE: LETTER TO GEORGE PEACOCK 121 wife and children, for 16 years. In 1819 he succeeded Sir John Leslie (1766-1832) as Professor of Mathematics at the University of Edinburgh [Anon. 1872, 4891. There, it may be worthwhile mentioning, he was the tutor of D. F. Gregory (1813- 1844), whose work was influential in promoting the rise of algebra and logic in England in the mid-19th century [4]. Wallace resigned in 1838 because of ill health and died five years later. In a touching letter to Sir John Herschel (1792-1871) in 1840 one can see his desire to be as productive as possible despite his illness. He was the main instrument in securing the erection of two observatories.(one at Sandhurst and the other at Edinburgh) and showed much care for their proper provision of equipment. This was a main subject of discussion with Herschel, whom he often consulted about the quality of the astronomical instruments that it was necessary to order from abroad [5]. Always interested in the practical side of science, Wallace also invented some geodetical instruments.

3. WALLACE’ MAIN MATHEMATICAL WORK DURING 1795-1815 Wallace’ main fields of study were geometry and trigonometry. The Mathemati- cal Repository of Thomas Leyboum (1770-1840), to which Wallace contributed from 1795 (first volume of the old series) to 1814 (third volume of the new series), contains many interesting essays, mostly concerning geometrical problems proposed and/or solved by him [6]. This journal and its editor are described in more detail in Sections 4, 5, and 6, below. Wallace submitted his first memoir to the Transactions of the Royal Society of Edinburgh in 1796, on geometrical porisms [ 17981;at the same time, following Dr. Robison’s suggestion, he wrote an article “Porism” [18Olc] for the third edition of the Encyclopaedia Britannica. Wallace’ second paper, which he wrote in 1802 and published three years later, was considered very important. The Edinburgh Review praised it to excess, raising Wallace to the level of Euler and Lagrange [Anon. 1803,510]. It dealt with a formula that expressed the mutual perturbation between planets, including a method for rectifying the ellipse that was different from the methods of Euler and Ivory [Wallace 1805, 2671. Wallace thought that his method was original until he discovered that he had been anticipated in 1794by Legendre, in a memoir “Sur les transcendantes elliptiques.” He seized the opportunity to translate this interesting (and rare) memoir into English. It was published in the Repository in two parts, the first in 1809and the second some time before 1814[7]. What is important about this translation is that, like the original paper, it used differential notation. He next translated a French memoir by Lagrange on spherical triangles [Wallace 1806~1. Wallace, in his letter to Peacock, stressed the importance of these two translations . In 1808 he wrote a paper on the rectification of conic sections, presenting a series for rectifying a circle [Wallace 18121.Once again, it turned out that he had been anticipated, this time by Euler [8]. Most of the rest of his papers dealt with geometrical and trigonometrical problems and their applications (mostly in physics, geodesy, and astronomy). 122 M. PANTEKI HM 14

In addition, Wallace furnished the principal mathematical articles (“Algebra,” “Conic Sections,” “Fluxions,” “Squaring,” etc.) for the Encyclopaedia Britan- nica, fourth edition (published between 1801 and 1810). In particular, “Conic Sections” [1842a] and “Fluxions” [I8551 were revised for the seventh and eighth editions of this encyclopedia [9]. His articles were characterized as of “great and permanent utility” [Napier 1842, xvii, xxiv]. “Conic Sections” was printed sepa- rately as a treatise [Wallace 18371: to his surprise (as he wrote to Herschel in 1840) this book was translated into Russian and was used at the Military School of the Russian Empire. Wallace was also the author of a number of principal article; ii, Brewster’s Edinburgh Encyclopedia from 1808 to 1830. As we see in Section 4, he was proud of having been the first Briton to compose an article on fluxions [1815] which gave a rather complete presentation of the calculus, including not only fluxions and fluents based on limits but also differentials and integrals (but not following Lagrange’s method).

4. WALLACE’ LETTER TO PEACOCK I found Wallace’ letter to Peacock in a copy of the English translation of Lacroix’ Elementary Treatise on the Differential Calculus (1816) from the library of Augustus De Morgan now housed in the University of London Library. On the first page of the letter is the inscription: “from professor Wallace Found among Mr. Galloway’s papers.” The date is missing but seems to be 1833 [IO]. The document appears to be a transcription: an added footnote is signed “T.H.” Months later I discovered by chance another letter in the same handwriting bound in a volume of the Cambridge Mathematical Journal (1839-1841), also from De Morgan’s library. This letter identified T.H. as the astronomer Thomas Hender- son (1798-1844), a close friend of Wallace [ll]. His letter, written in 1844, was addressed to Thomas Galloway (1796-1851), one of Wallace’ beloved students [12]. In his letter Henderson referred to the recent death of D. F. Gregory. All editorial expansions are enclosed in square brackets. The footnotes are my additions, and are incorporated into the system used in this paper. Letter to Peacock, Cambridge. First Copy. A Copy of that sent was preserved. Sir, I have in a cursory way, but with much satisfaction, glanced over your very able Report on certain branches of Analysis which I understand was read at the Meeting of the British Association at Cambridge. I believe it is one object of your Report to trace the progress of improvement in the Mathematical Sciences and as far as possible to ascribe to everyone who has been actively engaged in bringing about the Reformation which has taken place in the Mathematical Science of Britain his first share in the labour which has been accomplished. In the great variety of subjects to which you have adverted it might well happen that the date of some of the steps by which the improvement has been brought about may not be exactly known, and it cannot be supposed that you should have known the efforts that were made by obscure persons living far from Cambridge, the Holy City of Mathematics, more especially if their names were not attached to their labours [131. I trust, therefore, you wih receive with the same good spirit which pervades your Report a few remarks which have occured to me HM 14 WILLIAM WALLACE: LETTER TO GEORGE PEACOCK 123

while reading the Report. In page 295 you say, the Plane and Spherical Trigonometry of the late Professor Woodhouse appeared in 1810 and more than any other work contributed to revolutionize the Mathematical studies of this country. This is certainly true, but before that period the British Mathematicians had an opportunity of being acquainted with the elegant Memoir of Lagrange which contained A Complete Analysis of Spherical Triangles [14]. This appeared about 1797 in the 6th Cahier of Journal de 1’ Ecole Polytechnique. About 1804 or 1805 while one of the Mathematical Instructors in the R.[oyal] M.[ilitary] C.[ollege] I became acquainted with this memoir and was so delighted with its perusal that I believed I would render an essential service to British Mathematicians by translating it into English. I accord- ingly gave a translation in Leyboum’s Mathematical Repository then coming out in numbers. The Volume which contains it was completed before 1809, but the Number which contained Lagrange’s Memoir was published in 1806 four years before Woodhouse’s Trigonometry came out. I need hardly say that I had no other motive to translate this memoir than an ardent wish to give a beginning to that March of improvement which has been since carried on with such happy effect by the Men of Cambridge although at the time I speak of they were working with the black Triangle [15] and indeed sturdily setting their faces against change. I may mention that the same motive led me to give a translation of another Memoir of Lagrange, viz, An Essay on Numerical Analysis. In the year 1802 while an Assistant Teacher of Mathematics in Perth Academy I communicated to the Royal Society of Edinburgh a Paper on the Rectification of the Ellipse. I had not then seen Legendre’s Memoire [sic] sur les Transcendantes Elliptiques. Afterwards I got a copy of this very interesting work, and to give another impulse to our British Mathematicians I gave an English translation of it in the Repository. This was in the hands of Mathematicians before 1814, and I beg leave to notice that in this Translation I used the foreign Notation. Perhaps this was the first Mathematical Tract in the English language that had been given in the dress of the foreign notation [ 161. But about the year 1807 I had abandoned the English notation, and from that time forward employed the foreign notation in the resolution of problems in the Mathematical Repository. The same thing was done by Mr. Ivory [17]. If you look into the 7th number of the Mathemati- cal Repository (Vol. 2 of that work) you will find problems resolved by Mr. Ivory, viz, Quest. 151, 160, 172 and one by me, viz, 172 in which the foreign notation is employed. These were, I believe, the earliest applications of that notation to our English mathematics, and I can assure you, we employed it in a Revolutionary spirit [ 181. How far our continued efforts had influence in bringing about the happy change at Cambridge which you say took place in 1817 I have no means of knowing, but [I am] certain [that] I lost no opportunity of urging forward the Cambridge men in the good work of Reform; for having engaged to compose a Treatise on Fluxions for Encyclopaedia Britannica [19] I seized what I thought precious opportunity and fearlessly adopted the foreign notation. Thus I was the first to give the British Public a Treatise on the Differential Calculus in the notation of Euler and the Foreign Mathematicians 1201. I humbly trust the good intention manifested by this bold attempt to revolutionize so important a branch of Mathematical Science will atone for the many imperfections of that treatise which was written in England and printed at Edinburgh without my having had an opportunity to correct the Press. I had several years before composed a Treatise for the Encyclopaedia Britannica in the English notation [21] and if there be any merit in having been the first to introduce the subject of the Differential Calculus into our British Encyclopedias as a distinct branch of Mathemat- ics I humbly claim indulgence to these Tracts on that account. I beg to add that the second treatise in the foreign notation appeared in 1815 two years before the Cambridge Mathemati- cians would venture to propose a Senate house Question in the foreign notation. In making this communication it is by no means my wish to reflect on you for overlooking what I have attempted in the way of improvement, you had no knowledge of some of the facts 124 M. PANTEKI HM 14

which I have here communicated. I render now my grateful thanks for the kindness with which my humble efforts have been mentioned by some Cambridge men, particularly Mr Woodhouse and Sir John Herschel, and it gives me much satisfaction to see you on the scene of the labours of the Gregorys, Maclaurin and Stewart. 1 beg to mention that the treatise on Trigonometry appended to the end of Simson’s Euclid was composed by one of his Pupils, the late Dr Trail of Lisburn in Ireland. It is certainly now in harmony with our modern mathematics [22].

5. REFORMS IN BRITISH MATHEMATICS PRIOR TO THAT AT CAMBRIDGE It is not true that the Cambridge reform of the mid 1810s brought Continental calculus to Britain. A few years earlier the famous Cambridge reform changes had been made at Trinity College, Dublin, mainly by Bartholomew Lloyd (1782-1837), supported by John Brinkley (1763-1835) and others. Lagrange’s and Arbogast’s algebraic approach to calculus had a strong effect on Brinkley’s methodology in his mathematical work, for which Laplace’ Mhnique ce’kste served as another source of inspiration (see [Grattan-Guinness 19871). In Scotland John Toplis (1774-1857), John Playfair (1748-1819), and James Ivory (1765-1842) contributed to the diffusion of Laplace’ work [23]. In particular, Ivory (like Brinkley) not only mastered it but also extended it. Apart from Wallace [24], another Scottish mathematician who should be mentioned is William Spence (1777-1815). De Morgan regarded his essay of 1809 on logarithmic tran- scendents as highly original [De Morgan 1842, 658-659; Enros 1983, 26; Kop- pelman 1971, 1561. Unfortunately it is not within the scope of this paper to exam- ine systematically the fact that during a period of stagnation in the mathematical and physical sciences at Cambridge, Ireland and Scotland kept in contact with their development in Europe. However, elsewhere in England changes were being effected at the two military institutions: the Royal Military Academy at Woolwich and the Royal Military College at Great Marlow (later, at Sandhurst). At the College, besides Wallace, there were Ivory and Leyboum. At the Academy there were Charles Hutton (1737-1823), Olinthus Gregory (1774-1841), and Peter Barlow (1776-1862). Their mathematical contributions, while not of major importance, were of great practical value, especially during the period when mathematical knowledge in England was at an extremely low ebb [25].

6. WALLACE’ MATHEMATICAL WORK; EVALUATION OF HIS ROLE AS A REFORMER Both Wallace and Ivory were interested in modem analysis, which was flourish- ing on the Continent (mainly France) at the end of the 18th century. However, remaining faithful to the traditional Scottish geometrical spirit, they also culti- vated geometry. Ivory’s ability to criticize Laplace’ method on the attraction of spheroids and his application of the differential calculus in papers which concerned the solution of current problems in physical astronomy established his HM 14 WILLIAM WALLACE: LETTER TO GEORGE PEACOCK 125

European reputation by the 1810s. By contrast, Wallace’ career was marked by his devotion to geometry, which was especially pronounced during his professor- ship at Edinburgh; while people at Cambridge were putting aside Greek geometry, Wallace did the very opposite, believing that it was a necessary background for students of Continental analysis [Davie 1961, 1131. From his early work the most important contribution to the development of mathematics was his invention of a series for the rectification of the ellipse [Wallace 18051. Brinkley regarded this memoir as “very ingenious” [Brinkley 1803, 1501, and Herschel mentioned him along with Fagnani, Euler, Landen, Lagrange, Ivory, and Legendre in his article “Mathematics” in the Edinburgh Encyclopedia [Herschel 1830, 3721. Concerning his translation of Legendre’s rare memoir in the Repository, Wal- lace was definitely mistaken in assuming that it had been the first English tract employing differential notation, as Woodhouse’ book in 1803 preceded it. Never- theless, together with Ivory, he was the first to use the differential calculus in journals like the Repository, which was probably read by students more often than Woodhouse’ Analytical Calculation. As for his translation of Lagrange’s memoir on spherical triangles, though devoid of differential calculus, it was still representative of Lagrange’s algebraic methodology, in contrast with the current English treatises on trigonometry, which were purely geometrical in method (see [Woodhouse 1809, Preface]). So, if the Repository was popular, then Wallace contributed significantly to the mathematical science in Britain of the early 19th century, through his signed or unsigned or pseudonymous memoirs and translations. Evidence that students did use the Repository frequently is given by Enros [1979, 24-251. Also of interest is the fact that the Repository was used as a source of geometrical examples by later textbook authors. The role of the Repository is a good topic for research [26]. Finally, though Wallace was not the first to write an article on fluxions for the Encyclopaedia Britannica [1810], he was the first to give a complete presentation of the subject (his article on fluxions for the third edition was only 16 pages long whereas his article for the fourth edition was 81 pages) [27]. Moreover, he was the first to write on this subject for the Edinburgh Encyclopedia [1815] in the notation of Euler and Lagrange, including the doctrine of limits of which the Analytical Society (consisting mainly of Peacock, Babbage, and Herschel) disapproved, The remaining question concerns Peacock’s silence with respect to Wallace in his report of 1833. Was Peacock ignorant of Wallace’ existence as a mathematician, or was he simply indifferent to his work? We have seen that Wallace’ early contributions were not negligible. So if Peacock knew him, why would he omit him from his report? If anonymity were the cause, Peacock could nevertheless have mentioned some of Wallace’ articles or translations [28]. I suspect that Peacock had never heard of Wallace. Herschel, who was in frequent correspondence with both, expressed admiration for Wallace’ article on fluxions of 1815 for “the elegant manner in which the doctrine of limits is laid down” in a letter to Peacock (23 September 1815) [29]. He mildly criticized limits 126 M. PANTEKI HM 14 for their insufficiency to serve as a basis in a less elementary work. Wallace was named as the author of the above-mentioned article by D. M. Peacock (1768-1840) in his pamphlet A Comparative View of the Principles of the Fluxional and Differ- ential Calculus, in which he criticized Wallace’ definition of a fluxion [D. M. Peacock 1819, 25, 431. 1 mention both criticisms to reinforce the view that G. Peacock must have been aware of Wallace and his work. However, Wallace suffered from the silence of mathematicians other than Pea- cock; in his “Dissertation Fourth” for the Encyclopaedia Britannica [ 18421 Leslie mentioned Ivory’s and Playfair’s contributions, omitting those of Wallace. Wal- lace was also overlooked by J. D. Forbes (1809-1868), in his “Dissertation Sixth” [1853]. Forbes disliked not only Wallace but also the entire Scottish approach to mathematics; still, he mentioned Playfair, Ivory, Galloway, and Henderson-all closely related to Wallace [Forbes 1853,803,823,846,863] [30]. In compensation Henry Brougham (1778-1868), in his speech at Edinburgh University in 1860 (see [Davie 1961, 113-117]), acknowledged Ivory and Wallace equally, calling them “distinguished mathematicians who were both deeply imbued with the principles of modem analysis, but diligently cultivating the ancient too.” So Wallace’ surprise is justified to some extent; Peacock, by rejecting the importance of geometry in the teaching of mathematics, disregarded the work of Wallace, who at that time was promoting geometry at Edinburgh University. On the whole Wallace’ work, though not of fundamental importance insofar as its influence on British reforms is concerned, deserves rescue from oblivion. It is hoped that publication of Wallace’ letter to Peacock will further this goal.

ACKNOWLEDGMENTS For accessto Wallace’ letter I thank the London University Library. I am also grateful to Mr. I. Noble of the Aberdeen University Library and Dr. G. N. Cantor of Leeds University for their information on Brewster’s Edinburgh Encyclopedia; to Mr. Noble for access to Leybourn’s Muthe- matical Repository; and to the librarians of the Royal Society for access to the Herschel manuscripts as well as for their continuous help during this investigation. Mr. N. Guicciardini provided valuable information on the mid-18th-century period. The paper was written with the encouragement of my supervisor, Dr. I. Grattan-Guinness, of Middlesex Polytechnic, who was tireless in correcting my paper. I also thank the two anonymous referees for their helpful comments on a previous draft of this paper.

NOTES 1. Information about Wallace’ life was taken from [Anon. 18451 (an obituary), [Anon. 1872, 1888; Broadbent 1976; Napier 1842; Stronach 18993. 2. In his work as a bookbinder he came across Cunn’s Euclid, Keil’s Astronomy, Emerson’s Fluxions, and other books. For more details see [Anon. 1845, 31-321. 3. This was a higher-class school in the city of Perth built in 1761. It had 80 to 100 students, instructed in various branches of learning; see [Anon. 18191. 4. Gregory was one of Wallace’ favorite pupils [Gregory 1865, xii-xiii; Koppelman 1971, 201. Gregory often referred to his teacher’s work, trying to rescue it from oblivion. About a particular case, HM 14 WILLIAM WALLACE: LETTER TO GEORGE PEACOCK 127

where he defended Wallace’ priority in discovery, see my note 13, below, and also [Gregory 1865,54, 2341. 5. There are seven letters from Wallace to Herschel in the Herschel manuscripts of the Royal Society of London. Their code numbers are H.S. 18:21-27. Two of them (H.S. 18:23 of 16 March 1824; H.S. 18:25 of 1 December 1825) are mainly about the optical instruments that Wallace had undertaken to order from abroad. His last letter (H.S. 18:27 of 21 October 1840) concerns his illness. There are only four letters from Herschel to Wallace (H.S. 20:25, 203, 226; H.S. 22:72). In reading these letters one gains the impression that these two men had a close relationship, though their opportunities to meet were scarce. Two names mentioned frequently in these letters are Babbage and Henderson. 6. The most interesting problems proposed and solved by Wallace in the Repository (with some precaution because of the pseudonyms which almost all authors in this periodical adopted) are found in [Wallace 1795, 15-391; [1801, 80-82, 11l-1191, in which he discovered what was later erroneously called “Simson’s line” (see [Mackay 1871, 83-851); [Wallace 1806, Pt. III, 16-17, 48-52, 169-170; 1809, Pt. I, 65-67, 68-72, 122-124, 156-159, 176-1771. It is important to note the frequent use of fictitious signatures in periodicals of that period. Those usually adopted by Wallace were “Scoticus” and “Edinburgensis.” Ivory’s was “Astronomicus”; see [Wilkinson 1851,55,364,447; 1852,57,483]. Wallace may have used additional pseudonyms; he did not sign his translations, and so even Wilkin- son, who researched this journal in detail, did not identify him. 7. For the particular significance of this memoir see [Wallace 1809, Pt. III, 11. The first part was published in 1809 and the second in the volume of 1814, but as the journal was published by number it is probable that both parts were actually published a bit earlier. See also Wallace’ letter in the next section. 8. This was communicated in the Reposirov by John Dawes [1806. n.s. 1, Pt. Ill] in his answer to problem. 50. 9. “Fluxions” was written in fluxional notation for the fourth (1810), fifth (a reprint of the fourth), and sixth (1823) editions of Encyclopaedia Britannica, with a poor bibliography and no diagrams. For the next two editions the article was revised, with a richer bibliography (containing works by Ivory and Wallace himself) and with diagrams. 10. The handwriting of this inscription is different from that of the letter. The callmark of the letter in the typescript catalog of Additional Autograph Letters is A.L. 483. The date 1833 is given in square brackets. 11. This letter apparently has no callmark. It is properly signed. Wallace, in his letters to Herschel, often referred to Henderson as a very close friend and collaborator. Henderson made a series of very important observations while at the Edinburgh Observatory. A detailed account of his life is given in [Anon. 18461; see also [Clerke 18911. 12. Galloway also taught at the Royal Military College. He contributed many articles on astronomy and similar subjects to encyclopedias. About his life see [Tait 18891. 13. Apart from the frequent omission of his name in the Repository, all of Wallace’ early articles for the two encyclopedias were unsigned. It was only in the Preface of the seventh edition of the Encyclo- paedia Bn’ttanica that his name was included in the list of omissions (1842). Mackay [1871] mentions that the “Wallace Point Theorem” [1806, Pt. I, 22, 1691 was written under the name “Scoticus.” But there was ignorance even of those parts of his work which were properly signed; e.g., his “Property of a Parabola” [1801a] was published as new by Poncelet in Gergonne’s Anna/es de Math& matiques; see [Poncelet 1817, 9; Lubbock 1836, 1001. Comments about priority in discovery of this property are also made by Wilkinson [1851n, 3631, and Wallace was concerned about it; see [Wallace 1837, 1671. This error was pointed out by “G” [Anon. 18371; it is my conjecture that “G” indicates D. F. Gregory. 14. It was published in the Repository in 1806. Comments on this memoir are made in Section 6, below. 128 M. PANTEKI HM 14

15. The following is from Peacock’s report [1833, 295, footnote]: “The author of this report well recollects a treatise of this kind which was extensively used when he was a student at the University, in which the proposition for expressing the sine of an angle in terms of the sides of the triangle, was familiarly denominated the black triangle, in consequence of the use of thick and dark lines to distinguish the primitive triangle amidst the confused mass of other lines in which it was enveloped, for the purpose of obtaining the required result by means of an incongruous combination of geometry and algebra.” 16. Woodhouse [1803] had used this notation, as had Ivory [18W] in his paper on ellipsoids. 17. Ivory was one of the main contributors to this journal; see [Wilkinson 1852 57, 4831. Wilkinson gives a detailed account of the contents of the Repository and also mentions articles and problems which were published in The Gentleman’s Mathematical Companion, to which Wallace contributed from time to time. 18. It is striking that even in the 1814 volume of the Repository other mathematicians continued to use fluxional notation. Problems solved using differential notation in the New Series, Volume 2, are on pages 151, 160, 172, 188. Solutions given by Ivory and Wallace are on pages 65-68,99, 118-124, 156- 159. Strangely enough, of the two solutions given to Pappus’ problem (p. 151), one used fluxional notation (Wallace, pp. 65-67) and the other made no use whatsoever of fluxional calculus (Ivory. pp. 67-68). but they were followed by a scholium (written by either Wallace or Ivory) on max- ima and minima which, for the first time in the Repository, made use of differential calculus [ 1809. Pt. I, 68-721. 19. It is here that T.H. added a footnote pointing out that Wallace meant Brewster’s Edinburgh Encyclopedia. This encyclopedia was issued as a series of parts between 1808 and 1830. See also [Walsh 1968, 4-121. 20. He meant the article in the above-mentioned encyclopedia. See also my comments in Section 6, below. 21. Here he meant his [1810b]. 22. The 1830 edition of this book was revised extensively, but it was still less up to date than Woodhouse’ treatise [1809]. 23. See the Preface to Toplis’ translation of Laplace’ Analytical Mechanics [ 1814, iii-vii]; and also [Enros 1981, 137, 1471. On Playfair see [Anon. 1824; Enros 1981, 136-137, 140, 1471. About Ivory see [Anon. 18421 (an obituary), [Dubbey 1963, 39; Enros 1983, 26; Koppelman 1971, 1561. 24. Wallace is mentioned by Enros [1979, 222; 1983, 261. Dubbey [1963, 19781 ignores him. Kop- pelman mentions only his work on functional equations [1971, 2081 and Grattan-Guinness [1985, 90, 1031 mentions him briefly as a principal author in the Transactions ofthe Royal Society ofEdinburgh. 25. Unfortunately there is no useful information at present on these institutions. N. Guicciardini is currently studying them as the concluding section of his doctorate on the history of the fluxional calculus (his supervisor is I. Grattan-Guinness). 26. [Salmon 1852,221; Walton 18511. Apart from [Wilkinson 1851,1852], information on the Reposi- tory can be found in [Anderson 1910,466] and [Archibald 1929, 390-3911 (see also p. 392 for information regarding The Gentleman’s Mathematical Companion). Both these articles contain valuable information on the obscure British journals and carefully distinguish them. 27. By contrast, an article on differential calculus in the supplement was only five pages long and was representative of Lagrange’s method of expansion in series [Bromhead 18241. 28. One can see, for example, that Peacock overpraised Ivory’s article “Equations” [1824] on three occasions [Peacock 1833, 229, 302-303, 3171. 29. The code number of this letter is H.S. 20-25 R.S. 30. Even more striking is that Forbes said much about Playfair’s contributions, calling him of “excellent mathematical capacity” [Forbes 1853, 603, footnote]. HM 14 WILLIAM WALLACE: LETTER TO GEORGE PEACOCK 129

REFERENCES Part 1. Selected List of Wallace’ Published Works This list includes all of Wallace’ contributions to major societies and the principal encyclopedia articles which relate to the calculus. The dates of reading are added to the extent that details are known. He also contributed to another periodical besides the Repository, The Gentleman’s M&rem&- ical Companion. This journal is today as fugitive as the Repository. 1.1. Transactions of the Royal Society of Edinburgh 1798. Some geometrical porisms, with examples of their application to the solution of problems. 4, 107-134 [Read 7 March 17961. 1805. A new method of expressing the coefficients of the development of the algebraic formula ((I? + h? - 2ab cos 4)“, by means of the perimeters of two ellipses, when n denotes the half of any odd number; together with an appendix, containing the investigation of the formula for the rectification of any arch of an ellipse. 5, 253-293 [Read 11 January 18021. 1812. New series for the quadrature of the conic sections, and the computation of logarithms. 6, 269- 344 [Read 27 June 18081. 1826a. Investigation of formulae, for finding the logarithms of trigonometric quantities from one another. 10, 148-167 [Read 3 November 18231. 1826b. A proposed improvement in the solution of a case in a plane trigonometry. 10, 168-170 [Read 23 November 18231. 1836a. Account of the invention of the pantograph, and a description of the Eidograph, a copying instrument invented by William Wallace. 13, 418-439 [Read 13 January 18311. 1840a. Investigation of analogous properties of coordinates of elliptic and hyperbolic sectors. 14,431- 447 [Read 15 April 18391. 1840b. Solution of a functional equation, with its application to the parallelogram of forces, and to curves of equilibration. 14,625-676 [Read 2 December 18391. I .2. Transactions of the Philosophical Society of Cambridge 1838. Geometrical theorems, and formulae. particularly applicable to some geodetical problems. 6, 107-140 [Read 30 November 18351. I .3. Memoirs of the Royal Astronomical Society 1836b. The elementary solutions of Kepler’s problem by the angular calculus. 9, 185-192 [Read 18351. I .4. Cambridge Mathematical Journal 1839. On a property of the triangle. 1, 21-22. 1841. Demonstrations of expressions for the area of a triangle. 2, 35-37.

1S. Leybourn’s Mathematical Repository Entries with an asterisk are unsigned and those with a dagger are under a pseudonym; J trust that my identifications of the latter are correct. 1795*. Eight propositions of Stewart’s general theorems. 1, 15-38. 1801a. On the description of parabolic trajectories. 2, 80-82. 1801b. Mathematical lucubrations. 2, 1 I I-I 19. 1806at. A geometrical porism, with two examples of its application to the solution of problems. New Series 1, Pt. II, 16-17. 1806bt. Problems relating to the twilight of shortest durations. New Series 1, Pt. II, 48-52. I30 M. PANTEKI HM 14

1806c.* “Solutions of some problems relative to spherical triangles; together with a complete analysis of these triangles” by Lagrange. New Series 1, Pt. III. l-23. 1806d.* “ An essay on numerical analysis on the transformation of fractions” by Lagrange. New Series 1, Pt. III, 24-40. 1809.* “A memoir on elliptic transcendentals” by Legendre (first part). New Series 2, Pt. 111, l-34. l814.* ‘&A memoir on elliptic transcendentals” by Legendre (second part). New Series 3, Pt. III, l-45. I .6. Brewster’s Edinburgh Encyclopediu 1815. Fluxions. 9, 382-467. I. 7. Encyclopaedia Britannica 1801~. Porism. 3rd ed. 15, 394-400. 18lOa. Conic sections. 4th ed. 6, 519-548. 18lOb. Fluxions. 4th ed. 8, 697-778. 1810~. Squaring. 4th ed. 19, 614-622. 18lOd. Trigonometry. 4th ed. 28, 477-488. 1823. Fluxions. 6th ed. 8, 697-778. 1842a. Conic sections. 7th ed. 7, 218-262. 1842b. Fluxions. 7th ed. 9, 631-708. 1842~. Trigonometry. 7th ed. 21, 370-385. 1855. Fluxions. 8th ed. 9, 670-677. 1.8. Textbooks 1837. A geometrical treatise on the conic sections. Edinburgh: Adam & Charles Black. 1839. Geometrical theorems and analytical formulae. with their application to the solution of certain geodetical problems with an appendix conruining a description of trio copying instruments. Edinburgh. Part 2. Other Items Anderson, P. J. 1910. Mathematical periodicals: T. Leybourn’s Mathematical Repository.Notes Queries (11) 2, 466-467. Anonymous. 1803. A new method by W. Wallace. Edinburgh Review 1, 506-510. - 1819. Perth. Rees Cyc/opedia 26 [no pagination]. - 1837. On the property of the parabola demonstrated by Mr. Lubbock. Philosophical Magazine 10, No. 3, 32-35. - 1839. Essay review of “Geometrical theorems and analytical formulae” by W. Wallace. Philo- sophical Magazine 14, No. 3.467-468. - 1842. Obituary of J. Ivory. Proceedings of the Royal Society of London 4, 406-412. - 1845. Obituary of W. Wallace. Monthly Notes of the Royal Astronomical Society 6, 31-41. - 1846. Obituary of Th. Henderson. Memoirs of the Royal Astronomical Society 15, 368-395. - 1872. W. Wallace. Biographical Dictionary of Eminent Scotsmen 3, 489-490. - 1888. W. Wallace, Encyclopaedia Britannicu, 9th ed. 24, 327-328. Archibald, R. C. 1929. Notes on some English mathematical serials. Mathematical Gazette 14, 379-400. Ball, W. W. R. 1889. A history of the study of mathematics at Cambridge. Cambridge: Cambridge Univ. Press. HM 14 WILLIAM WALLACE: LETTER TO GEORGE PEACOCK 131

Becher, H. 1980. William Wheweh and Cambridge mathematics. Historical Studies in the Physid Sciences 11, l-48. Brinkley, J. 1803. A theorem for finding the surface of an oblique cylinder. Trcmsoctions ofthe Royul Irish Academy 9, 145-158. Broadbent, T. A. 1976. W. Wallace. Dictionary of ScientiJc Biogruphy 14, 140-141. Bromhead, E. F. 1824. Differential calculus. Encyclopuediu Brirunnicu Supplementa 3, 568-572. Clerke, A. M. 1891. Th. Henderson. Dictionury of Nutionul Biogruphy 9,404-406. Davie, G. E. 1961. The democratic intellect. Edinburgh: Edinburgh Univ. Press. De Morgan, A. 1842. The differentiul und inregrul calculus. London: Baldwin & Cradock. Dubbey, J. 1963. The introduction of the differential notation to Great Britain. Annals of Science 19, 37-48. - 1978. The mathematical work of Churles Bubbuge. Cambridge: Cambridge Univ. Press. Enros, P. 1979. The Analytical Society: Mathematics at Cambridge University in the early nineteenth century. Ph.D. thesis, University of Toronto. - 1981. Cambridge University and the adoption of analytics in early 19th century England. In Social history of mathematics, H. Mehrtens, H. Bos, & 1. Schneider, Eds., pp. 135-147. Boston: Birkhauser. - 1983. The Analytical Society (1812-1813): Precursor of the renewal of Cambridge mathematics. Historia Mathemutica 10, 24-47. Forbes, J. D. 1853. Dissertation sixth. Encyclopaediu Britunnicu, 8th ed. 1, 795-996. Grattan-Guinness, I. 1985. Mathematics and mathematical physics at Cambridge, 1815-40: A survey of the achievements and of the French influences. In Wranglers und Physicists, P. M. Harman, Ed., pp. 84-l 11. Manchester: Manchester Univ. Press. - 1987. Mathematical research and instruction in Ireland, 1782-1840. In Trudition und reform: Science and engineering in Ireland 1800-1930, N. MacMillan & J. Nudds, Eds. (to appear). Gregory, D. F. 1865. The mathemutira/ writings, W. Walton, Ed. Cambridge: Deighton, Bell. Herschel, J. 1830. Mathematics. Edinburgh Encyclopedia 13, 359-383. Ivory, J. 1798. A new series for the rectification of the ellipsis; together with some observations on the evolution of the formula (u? + b’ - 2ub cos 4P. Trunsuctions of the Royal Society of Edinburgh 4, 177-190. - 1809. On the attractions of homogeneous ellipsoids. Transactions of the Royal Society of London 99, 345-372. - 1824. Equations. Encyclopaediu Britannica Supplementa 4, 669-708. Jeffrey, F. 1824. J. Playfair. Encyclopaediu Britannica Supplementu 6, 198-204. Koppelman, H. 1971. The calculus of operations and the rise of abstract algebra. Archiw for History of Exact Sciences 8, 155-242. Lacroix, S. F. 1816. An elementary treatise on the differentiul und integrul culculus, translated by G. Peacock and J. F. W. Herschel. Cambridge: Deighton. Lagrange, J. L. 1798. Essai d’analyse numerique sur la transformation des fractions. In Oeuures de Lagrange, Vol. 7, pp. 291-313. Paris: Gauthier-Villars. - 1799. Solutions de quelques problemes relatifs aux triangles spheriques. In Oeuvres de La- grange, Vol. 7, pp. 331-359. Paris: Gauthier-Villars. Laplace, P. S., Marquis de 1814. Analytical mechanics, Books I, 2, translated by J. Toplis. Notting- ham: Bamett. Legendre, A. M. 1794. Memoire sur les transcendantes elliptiques. Memoires de I’AcadPmie Royule des Sciences, I-102. Leslie, J. 1842. Dissertation fourth. Encyclopaedia Britannica. 7th ed. 1, 575-677. 132 M. PANTEKI HM 14

Lubbock, J. W. 1836. On a property of the parabola. Philosophical Magazine 9, No. 3, 100-104. Mackay, J. S. 1871. The Wallace line and the Wallace point. Proceedings ofthe Edinburgh Mathemati- cal Society 9, 83-91. - 1905. Bibliography of the envelope of the Wallace line. Proceedings of/he Edinburgh Murhe- muticul Society 23, 80-85. Napier, M. 1842. “Preface” of Ent~yclopuediu Britunnica, 7th ed. I. Peacock, D. M. 1819. A compurutive view of the principles of rhejuxional und differentiul cu/cu/us. Cambridge: Deighton. Peacock, G. 1833. Report on certain branches of analysis. Report of the British Associution for the Advuncement of Science. 185-352. Playfair, J. 1790. On the astronomy of the Brahmin. Trun.suction.s of the Royul Society of Edinburgh 2, 135-192. Poncelet, J. V. 1817. Gtometrie des courbes: Theorbmes nouveaux sur les lignes du second ordre. Annu/e.s de MuthPmutiques Purrs et AppliyuPs 8, I-13. Salmon, G. 1852 A rreutise on the higher plune curues. Dublin: Hodges & Smith. Simson. R. 1830. The elements of plane and spherical trigonometry. In Euclid, 23rd ed. London. Stronach, G. 1899. W. Wallace. Dictionary of Nurionul Biogruphy 20, 572-573. Tait, J. 1889. Th. Galloway. Dicfionury of National Biography 7, 827-828. Walton, W. 185 I. Problems in plune wordinute geometry. Cambridge: Deighton. Wilkinson, T. W. 1851. 1852. On Leybourn’s Muthemuticul Repository. Mechanics Muguzine 55 (1851), 264-266, 306-310, 363-365, 445-448; 56 (1852), 134-135, 145-147, 445-447; 57 (1852), 7-9, 64-66, 245-247, 291-294. Walsh, S. P. 1968. Anglo-Americun genc~rul encyc/opediu.s. New York/London: Bowker. Woodhouse, R. 1803. Principles of unulyticul cu/cu/ution. Cambridge: Cambridge Univ. Press. - 1809. A rreurise on plane und sphericul trigonometry. Cambridge: Black, Pary & Kingsbury.