Edinburgh: 5 February.
Important autograph letter from the great Scottish mathematician Colin Maclaurin (1698-1746) discussing his most important work, Treatise of Fluxions (1742), “the earliest logical and systematic publication of the Newtonian methods. It stood as a model of rigor until the appearance of Cauchy’s Cours d’analyse in 1821” (DSB). Autograph material by Maclaurin is of great rarity on the market – this is the only example we have traced at auction in the last 40 years.
The letter reads in full, in English translation:
I was honoured to receive your letter dated on the first day of the year, and it has been a long time since I received the copies of the Eloge of Cardinal of Polignac, which you so kindly sent to me.
At that time, I was in the countryside, working on an addendum to my book [i.e. Treatise of Fluxions]. Then I had to take some brief journeys, which prevented me from thanking you as soon as I would have liked. Dr Martin, to whom you addressed one of your pieces, died during the Cartagena expedition.
It seems to me that you have many reasons to be satisfied with the success of the Praise, even if it may be difficult to succeed in that kind of work. As regards my book, I am expecting different feelings about it, especially from foreigners who may not know all the reasons why I had to write about the components of the method in such a detailed manner. I hope that the second volume of the book will better satisfy their expectations, as I discuss the various useful and interesting issues. It would delight me if you found any of it to your taste. I began writing to resolve the objections from subtle metaphysicians, but I wanted a treaty of such a scope that it would also meet the interests of the mathematicians, by addressing previously known issues or by discussing other lesser-known ones. I tried by all means to mention the other authors interested in this subject as is appropriate for persons with the same purpose, even by improving gradually so pleasant and valuable sciences.
I hope you will let me know when your academicians will return from Peru. The members of our society [i.e. the Royal Society] are so very grateful for your honesty and express their high esteem for you. We are attempting to rectify the geography of the north of the island and we are actually going to engrave a map of the northern side of Great Britain with the harbours, which is much more accurate than the one we had, and I will send a copy of it to you by a rider coming to France in few weeks’ time. Do forgive me for the stylistic errors, and if you wish to address some new comments to me, I would be pleased if you would send them.
If you would be so kind, if it is a large parcel, you need only to send it by the coach of Boulogne addressed to Mr Charles Smith, Merchant at Boulogne-Sur-Mer, and with my address on the inside. Or if you have the opportunity to send it to London, you may address the parcel to Mr Mitchell, Under-Secretary of State, Cockpit, Whitehall, London. Mr Mitchell is a close friend of mine and he will take care of it. He is kind and will be happy to help in a situation like this. I take this liberty because the people of the postal service requested 45 Shillings Sterling for carrying the Praises, but after some difficulty I was able to obtain them without charge. I have the honour of being, with the highest esteem and respect,
Your most humble
and very obedient servant
C. Mac Laurin
The letter begins with the usual pleasantries, Maclaurin thanking Mairan for sending him copies of his Eloge de M. Le Cardinal de Polignac, which Mairan had delivered to the Académie the preceding year. Polignac (1661-1742) was one of the most illustrious scholars and courtiers of France in the latter years of Louis XIV, and in the early reign of Louis XV.
Maclaurin turns to a discussion of two-volume Treatise of Fluxions in the third paragraph. “I began writing to resolve the objections from subtle metaphysicians, but I wanted a treatise of such a scope that it would also meet the interests of the mathematicians, by addressing previously known issues or by discussing other lesser-known ones.” One of Newton’s most ardent disciples, Maclaurin was led to write the Treatise as a response to criticisms of Newton’s fluxions made by Bishop Berkeley. “Despite the great progress of analysis during the 18th century, foundational questions remained largely unsolved. … The most influential criticism of the new analysis was put forward by the famous English philosopher George Berkeley [in his pamphlet The Analyst, 1734]. … Berkeley’s criticism was well informed and efficient. Rooted in the tradition of English sensualism, he showed that many definitions in the infinitesimal calculus are paradoxical and cannot be justified by intuition. He explained the success of the new calculus by a repeated neglect of infinitely small quantities leading through a compensation of errors to a correct answer.” (Jahnke: A History of Analysis, 127). “The most authoritative answer to Berkeley came from Maclaurin, one of the most talented of Newton’s disciples. In his Treatise of Fluxions (1742), he tried to show that infinitesimals were used by Newton only to abbreviate proofs; these proofs could be re-expanded, following the style of the De Quadratura (1704), in terms of limits. Furthermore, Maclaurin elaborated the idea that Newtonian proofs in terms of limits were equivalent to the venerated method of exhaustion of the ancient Greek mathematicians. Newton’s ‘method of prime and ultimate ratios’ was just the direct version of the indirect (ad absurdum) method of Archimedes. ” (Grattan-Guinness: Companion Encyclopedia, 314).
The letter seems to indicate that the two volumes of the Treatise were not published at the same time, “I hope that the second volume of the book will better satisfy their expectations, as I discuss the various useful and interesting issues.” Both volumes have title pages dated 1742 but Maclaurin writes as if the second volume is not yet available at the time of writing (February 1743).
Maclaurin then turns to geodetic matters. “I hope you will let me know when your academicians will return from Peru. The members of our society [i.e. the Royal Society] are so very grateful for your honesty and express their high esteem for you.” In the 18th century, a debate raged in the scientific community, especially in the Académie des sciences, as to whether the circumference of the Earth was greater around the equator, as Newton’s theories predicted, or around the poles as held by the Cartesians, who were by far the majority in the Académie. To decide the matter the Académie sent two expeditions, one to Lapland led by Celsius and Maupertuis, the other to what is now Ecuador (then Peru), led by Bouguer and La Condamine. The Lapland expedition reported in 1738 in favour of the Newtonians, a result which did not please the Cartesians of the Académie, which included Mairan himself, something to which Maclaurin perhaps refers, “The members of our society are so very grateful for your honesty and express their high esteem for you.” The Peru expedition left France in 1735 but encountered many difficulties, and Bouguer and La Condamine did not return until 1745; their results confirmed those of the Lapland expedition.
Mairan (1678-1771) was concerned with a wide variety of subjects, including heat, light, sound, motion, the shape of the earth, and the aurora. Although fundamentally a Cartesian, he tried to incorporate some of Newton’s ideas in his work. He became a member of the Academy of Sciences in 1718, serving as secretary from 1741 to 1743, succeeding Fontenelle. He also belonged to the Royal Societies of London, Edinburgh, and Uppsala, the Petersburg Academy, and the Institute of Bologna.
DSB VIII: 609-612 (Maclaurin); DSB IX: 33-34 (Mairan). Norman 1408, Honeyman 2084 (Treatise of Fluxions).
Four pages on a single sheet (375 x 227 mm) folded once vertically (thin paper strip mounted, proably from having been bound), remains of red wax seal on address page, small tears from opening the seal. Edinburgh: 5 February, 1743 (received 16 February).