Varia opera mathematica … Accesserunt selectae quaedam eiusdem epistolae, vel ad ipsum à plerisque doctissimis viris Gallicè, Latinè, vel Italicè, de rebus ad mathematicas disciplinas, aut physicam pertinentibus scriptae.

Toulouse: Jean Pech, 1679.

First edition, very rare, with the even rarer portrait, and the Blenhiem Palace copy, of the first publication of the most important works of Pierre de Fermat (1601-65), arguably the greatest French mathematician of the seventeenth century (Descartes and Pascal notwithstanding), “the father of the modern theory of numbers and herald of differential calculus and analytic geometry” (Grolier). “Fermat shares with Descartes the innovation of analytical geometry by applying algebra to geometry. He, independently, represented a curve by an equation defining its characteristic properties. He published little but, in the manner of his times, announced his discoveries in letters to other mathematicians. Among his discoveries was a general method of solving questions of maxima and minima, a method he used in 1629 and one in use today. He contributed basic concepts in the theory of numbers and probability” (Dibner). This volume contains, inter alia, Fermat’s researches in analytic geometry, the methods of maxima and minima, and his techniques of quadrature, together with his correspondence with Pascal, Frénicle, Gassendi, Mersenne, Roberval and other savants. In analytical geometry Fermat, working quite independently, reached much the same results as Descartes, but his presentation was radically different, based as it was on Viète's algebra. In the theory of maxima and minima, on the other hand, Fermat’s and Descartes’ methods were similar, despite a war of words between the two men on the subject in 1638. Fermat’s correspondence with Pascal laid the foundations of modern probability theory, and his letters to Frenicle make him the undisputed father of the modern theory of numbers. Fermat was reluctant to allow any of his work to appear in print, and only two short works were published in his lifetime, most of his work being confined to his correspondence, personal notes, and to marginal jottings in his copy of the 1621 editio princeps, edited by Claude Bachet, of Diophantus’ Arithmetica (the latter famously including his statement of ‘Fermat’s last theorem’). It was only after Fermat’s death that his son, Clément-Samuel (1632-90), was able to make public his father’s important mathematical contributions by editing the present work. No large paper copy has been traced in auction records.

Provenance: The Blenheim Palace copy sold in the Sunderland Library sale 1882 and bought by Quaritch with their label to front paste-down; Royal Society of Edinburgh with their stamp on title and Quaritch’s label. Charles Spencer (1675-1722), third Earl of Sunderland, began collecting in the 1690s and by the time of his death in 1722 had amassed one of the finest private libraries in Europe. Comprising some 20,000 volumes, it was particularly rich in incunables (including numerous works on vellum), Bibles, first editions of the Classics, and fifteenth- and sixteenth-century continental literature. It was located at Sunderland House in Piccadilly, occupying two rooms in the house itself and a further five rooms in a purpose-built library. Following Sunderland's death, the library was inherited by his eldest son Robert, the 4th Earl. After lengthy negotiations he sold the manuscripts to the King of Portugal in 1726. When Robert died in 1729 the Sunderland Library passed to his younger brother Charles Spencer (1706-1758), who succeeded as 5th Earl of Sunderland and inherited the Dukedom of Marlborough on the death of his aunt Henrietta in 1733. Marlborough had the collection moved from London to Blenheim Palace in 1749. The library was dispersed in a series of major sales conducted by Puttick & Simpson between 1881 and 1883. The 13,858 lots brought £56,581, of which around £33,000 was paid by Bernard Quaritch alone.

Fermat was born near Toulouse in the south of France. After spending some years in Bordeaux, in 1631 he became councillor of the High Court of Justice in Toulouse, an office he held until his death. He developed an enduring friendship with Pierre de Carcavi, who became his colleague at the Toulouse High Court in 1632. Carcavi transferred to Paris in 1636, where he formed a close acquaintance with the scientific circle gathered around Mersenne, Etienne Pascal and Roberval. Fermat’s scientific correspondence with the members of that group begins with a letter to Mersenne a few days after Carcavi’s arrival in Paris, continuing until around 1662 when Fermat, perhaps for reasons of ill health, allowed "his Geometry to fall into a deep sleep”. This correspondence provided Fermat with his main outside incentive for pursuing his mathematical work, as he never met other leading mathematicians in person, with the exception of one brief encounter with Mersenne.

“By the time Fermat began corresponding with Mersenne and Roberval in the spring of 1636, he had already composed his ‘Ad locos planos et solidos isagoge’, in which he set forth a system of analytic geometry almost identical with that developed by Descartes in the Géométrie of 1637. Despite their simultaneous appearance (Descartes’s in print, Fermat’s in circulated manuscript), the two systems stemmed from entirely independent research and the question of priority is both complex and unenlightening. Fermat received the first impetus toward his system from an attempt to reconstruct Apollonius’ lost treatise PlaneLoci (loci that are either straight lines or circles). His completed restoration [‘Apollonii Pergaei libri duo de locis planis restituti,’ pp. 12-43], although composed in the traditional style of Greek geometry, nevertheless gives clear evidence that Fermat employed algebraic analysis in seeking demonstrations of the theorems listed by Pappus. This application of algebra, combined with the peculiar nature of a geometrical locus and the slightly different proof procedures required by locus demonstrations, appears to have revealed to Fermat that all of the loci discussed by Apollonius could be expressed in the form of indeterminate algebraic equations in two unknowns, and that the analysis of these equations by means of Viète’s theory of equations led to crucial insights into the nature and construction of the loci. With this inspiration from the Plane Loci, Fermat then found in Apollonius’ Conics that the symptomata, or defining properties, of the conic sections likewise could be expressed as indeterminate equations in two unknowns. Moreover, the standard form in which Apollonius referred the symptomata to the cone on which the conic sections were generated suggested to Fermat a standard geometrical framework in which to establish the correspondence between an equation and a curve. Taking a fixed line as axis and a fixed point on that line as origin, he measured the variable length of the first unknown, A, from the origin along the axis. The corresponding value of the second unknown, E, he constructed as a line length measured from the end point of the first unknown and erected at a fixed angle to the axis. The end points of the various lengths of the second unknown then generated a curve in the AE plane …

“In the years following 1636, Fermat made some effort to pursue the implications of his system. In an appendix to the ‘Isagoge’ (pp. 9-11) he applied the system to the graphic solution of determinate algebraic equations, showing, for example, that any cubic or quartic equation could be solved graphically by means of a parabola and a circle. In his ‘De solutione problematum geometricorum per curvas simplicissimas et unicuique problematum generi proprie convenientes dissertatio tripartita’ (pp. 110-115), he took issue with Descartes’s classification of curves in the Géométrie and undertook to show that any determinate algebraic equation of degree 2n or 2n - 1 could be solved graphically by means of curves determined by indeterminate equations of degree n …

“Although Fermat never found the geometrical framework for a solid analytic geometry, he nonetheless correctly established the algebraic foundation of such a system. In 1650, in his ‘Novus secundarum et ulterioris ordinis radicum in analyticisusus’ (pp. 58-59), he noted that equations in one unknown determine point constructions; equations in two unknowns, locus constructions of plane curves; and equations in three unknowns, locus constructions of surfaces in space. The change in the criterion of the dimension of an equation — from its degree, where the Greeks had placed it, to the number of unknowns in it — was one of the most important conceptual developments of seventeenth-century mathematics” (DSB).

Fermat’s claim to be a ‘herald’ of the differential and integral calculus rests on his works ‘De aequationum localium transmutation, & ennendatione, ad multimodam curvilineorum inter se, vel cum rectilineis comparationem’ (pp. 44-57), ‘Methodus ad disquierendam maximum & minimam’ (pp. 63-73) and ‘De linearum curvarum cum lineis rectis comparatione dissertation geometrics’ (and its Appendix) (pp. 89-109). “When Fermat began, in or about 1629, very little was known about tangents and about maxima and minima. A tangent to a conic was a line which had just one point in common with it. On the other hand, Archimedes had given a brilliant determination of the tangent to a transcendental curve, the spiral, with an ‘apagogic’ proof (i.e. a proof by reductio ad absurdum in the typical Archimedean style). About integration, much was known from Archimedes, who gave ‘apagogic’ proofs for all his theorems, and much work was being done in Italy but did not become known to Fermat until a later date. Every competent mathematician of that time, having studied Archimedes, was expected to be able to construct a rigorous, i.e., Archimedean or ‘apagogic’, proof in each specific case; but this always required the previous knowledge of the result for that case. Thus, what Fermat and others were after was ‘a method’ for obtaining such results; once the result was found, the rest was routine, as they never tired of repeating. 

“In the search for ‘a method’ for tangents, one could be guided by the case of conics; this led to the purely algebraic problem of determining a parameter so that an algebraic equation acquires a double root. This method, which (in modern terms) belongs to algebraic geometry rather than differential calculus, was the one adopted by Descartes; of course, when he was challenged to find the tangent to the cycloid, he found himself in a quandary and got out of it by being illogical, improvising a beautiful kinematic method and inventing the instantaneous center of rotation. Fermat, led by a surer instinct, developed a method which slowly but surely brought him very close to modern infinitesimal concepts. What he did was to write congruences between functions of x modulo suitable powers of x - x0; for such congruences, he introduces the technical term adaequalitas, adaequare, etc., which he says he has borrowed from Diophantus. As Diophantus V.ll shows, it means an approximate equality, and this is indeed how Fermat explains the word in one of his later writings.

“At first Fermat applies the method only to polynomials, in which case it is of course purely algebraic; later he extends it to increasingly general problems, including the cycloid. A similar development occurs in Fermat's no less outstanding work on integration, where the word ‘adequality’ is again used more and more frequently to denote an increasingly conscious passage to the limit” (Weil, Review, pp. 1145-6).

While several of his contemporaries also made important contributions to the founding of the calculus (Cavalieri, Roberval, Torricelli, etc.), and analytic geometry (Descartes), Fermat is the undisputed founder of modern number theory. His contributions to this field are contained in his correspondence (pp. 121-210), especially in letters to Frénicle and Mersenne. Fermat never gave proofs of any of his number-theoretical results, but he referred to a method of ‘infinite descent’ which was later used by Euler and others.

“Lacking experience and models, Fermat began by studying all kinds of problems with little regard for their possible theoretical value. Thus, in his early career, he paid much attention to questions connected with the function s(n), the sum of the divisors of n other than n; thus the solutions of s(n) = n are the so-called perfect numbers … Although he long retained a fondness for such questions, it is clear that he soon realized their peripheral character. Fairly early, too, he considered problems about the representation of integers by quadratic forms x2 ± Ny2, on which he focussed his attention more and more as time went on. It is idle to doubt that he eventually developed a complete theory for the forms x2 + y2, x2 + 3y2, x2 ± 2y2, with proofs by infinite descent which cannot have been very different from those later found by Euler. The same can plausibly be said of the representation of integers by sums of four squares; on the other hand, no suggestion can be offered at present as to how Fermat could possibly have proved that every integer is a sum of three triangular numbers, and one cannot help thinking that on this point he may have deceived himself. As to ‘Pell's equation’ x2Ny2 = 1 (where N is any non-square positive integer), there is every reason to think that Fermat’s method was in substance much the same as Lord Brouncker's, as described by Wallis in the Commercium epistolicum (1658); one also may well assume that he had found the way to add to this a rigorous proof of existence, since he criticizes Wallis on that point” (Weil, Review, p. 1148).

In a letter to Frénicle dated 18 October 1640 (pp. 162-4), Fermat stated his ‘little theorem’, that if a is an integer not divisible by a prime number p, the ap – a is divisible by p. “Fermat had been interested in Euclid's theorem (Elements, Prop. IX, 36) that numbers of the form 2n-1(2" — 1) are perfect … Such prime numbers 2" — 1 Fermat called the radicals of the perfect numbers, and he had sent to Father Marin Mersenne some of his conclusions about these radicals in a letter of June 1640. (If n is not prime, 2" - 1 cannot be prime; if n is prime, 2" – 2 is divisible by n; if n is prime, 2n – 1 is divisible only by prime numbers of the form 2kn + 1). Then, in [the 18 October] letter to Frénicle, Fermat had turned to numbers of the form 2" + 1, writing that he was ‘almost convinced’ that these numbers are prime when n is a power of 2. We now know that, though this is true for n = 2, 4, 8, 16, it stops being true for n = 32, which, as Euler showed (Commentarii Academiae Scientiarum Petropolitanae I, 1732/33, 20 - 48) is divisible by 641 (4294967297 = 641 x 6700417)” (Struik, pp. 27-28). Two proofs of Fermat’s little theorem are known, a group-theoretical one (in modern terms) and one based upon properties of binomial coefficients. “The latter proof occurs in Leibniz’s unpublished manuscripts; Euler discovered it in 1736. He found the other proof somewhat later, and thought it the better one of the two. Looking at Fermat’s formulation of the theorem, it is hardly possible to doubt that this was the proof he had in mind, of course not in group-theoretical language, but in the form in which Euler expressed it (and which became an essential step for the later development of the theory of finite groups)” (Weil, Review, pp. 1147-8).

Italian writers of the fifteenth and sixteenth centuries, notably Pacioli, Tartaglia, and Cardano, had discussed the problem of the division of a stake between two players whose game was interrupted before its close. The problem was proposed to Pascal and Fermat, probably in 1654, by the gambler Chevalier de Méré. The ensuing correspondence between the two men was fundamental to the development of modern concepts of probability. “In regard to what we now know as probability theory, Blaise Pascal’s prime concern was the equitable division of stakes, the “problème des partis”, or, in English idiom, the “Problem of Points” …

“Sometime later in that year (1654) in about early July, he wrote to Fermat almost surely about this problem. That letter has not been found, but the first surviving letter, from Fermat to Pascal, is about a simple version of the problem: if a gambler undertakes to throw a six in eight throws, but stops after the first three throws which have been unsuccessful and does not continue, what proportion of the total stake should he have? Pascal's solution is 125/1296 = (5/6)3(1/6), while Fermat's is 1/6. Fermat's would be correct if a total of 4, rather than 8, throws was originally proposed. With 8 the correct proportion (probability) would be 1−(5/6)5.

“The second surviving letter, the famous one of 29 July, 1654 (pp. 179-183) from Pascal to Fermat, discusses a more sophisticated version of the problem. In the case of two players at each of a number of trials, each has probability 1/2 of winning the trial. It is agreed that the first player with n wins gains the total stake. The game is interrupted when player A needs a trial-wins to gain the stake, and player B needs b. How should the stake be divided? … The solution by Pascal and Fermat (by different methods) even for some particular cases, was a defining epoch in probability theory” (Statisticians of the Centuries, pp. 12-13).

The Varia Opera contains three letters from Pascal to Fermat on the subject of probability (pp. 179-188); four letters from Fermat to Pascal on the same subject were not published until the nineteenth century.

“The question of the publication of Fermat’s work was raised on several occasions during his lifetime … he once expressed a firm resolve of writing up his number-theoretical discoveries in book-form. In 1654 he requested Carcavi’s and Pascal’s cooperation to a more ambitious plan: he wished them to help him prepare the bulk of his mathematical work for publication, while Pascal would have been in sole charge of the number theory, which Fermat despaired of ever writing up in full. This came to naught; but in 1656 Carcavi mentions the matter to Huygens; he does so once more in 1659, in spite of Fermat's rather discouraging and discouraged conclusion to the summary of his arithmetical discoveries (‘mesresveries, as he calls them) in his communication of August 1659. Finally nothing was done before his death.

“Actually, in those days, it was not quite a simple matter for a mathematician to send a work to the press. For the printer to do a tolerable job, he had to be closely supervised by the author, or by someone familiar with the author's style and notation; but that was not all. Only too often, once the book had come out, did it become the butt of acrimonious controversies to which there was no end. Should one wonder, then, if Fermat, whenever the question of publication arose, insisted on anonymity above all? At the same time, it is clear that he always experienced unusual difficulties in writing up his proofs for publication; this awkwardness verged on paralysis when number theory was concerned, since there were no models there, ancient or modern, for him to follow.

“Fermat never bothered to keep copies of his scientific communications to his correspondents. Thus, after his death, it fell to the lot of his son Samuel to bring together, as best he could, the scattered remnants of his father's writings. He began with the Diophantus of 1670, a reprint of Bachet’s Diophantus of 1621 where he inserted the full text of the notes jotted down by Fermat in the margins of his personal copy … This was followed in 1679 by the volume of Varia Opera, the bulk of which consisted of Fermat’s writings on geometry, algebra, differential and integral calculus, together with a number of letters to and from Mersenne, Roberval, Étienne Pascal, Frénicle, Blaise Pascal, Carcavi, Digby, Gassendi. Not a few letters, among them some important ones on ‘numbers’, were not included, obviously because their recipients failed to send them to Samuel; thus they remained unknown until our times” (Weil, pp. 44-46).

“For all its many faults, the Varia remained the only published collection of Fermat’s papers until the nineteenth century. The manuscripts themselves, both autographs and copies, gradually fell into the hands of collectors or, since several were untitled and anonymous, into manuscript collections attributed to other writers. A large number of those in private hands returned briefly to the public domain when, in the Journal des Savants for September 1839, Count Guillaume Libri, the well-known bibliophile and historian of mathematics, announced his purchase in Metz of a collection of manuscripts formerly belonging to Arbogast and containing several hitherto unedited papers of Fermat … In 1848, rumours and some direct evidence led to a warrant for his arrest on charges of having appropriated some 300,000 livres worth of books and manuscripts from French libraries. In the turmoil of the political disorder of the times, Libri managed to escape to Italy with most of his personal library, including the Fermat manuscripts … In 1881, [Charles] Henry received word from Prince Boncompagni that he had acquired two manuscript volumes of Fermat’s works … Boncompagni’s two volumes in fact represented the whole of Libri’s collection of Fermat papers” (Mahoney, pp. 412-3). These two volumes, together with the 1670 Diophantus and the 1679 Varia, formed the basis of the modern edition of Fermat’s works edited by Charles Henry and Paul Tannery (published 1891-1922).

Horblit notes two issues of the Varia Opera, the first with an engraved vignette on the title, the second the with a woodcut vignette; there are also some minor differences in the preliminary leaves (an engraved rather than woodcut headpiece on ê2, and the presence or absence of a headpiece on â2). “In reality there seems to be no priority between the two states and most copies have a mixture of the two” (Norman). For example, Montucla’s copy (Christie’s, Paris, 1999) had the second state of the title but the first state of the preliminary leaves. The frontispiece portrait of Fermat was probably intended only for the very few large-paper copies, the portrait being much larger than the text block of an ordinary paper copy, but it is occasionally found in ordinary paper copies as here (and in Montucla's copy), tipped in and folded. The portrait in this copy seems to have been present since the time of binding. A similar, but not quite identical, portrait was added to large-paper copies of the 1670 edition of Diophantus, edited by Fermat’s son and containing the marginal annotations made by Fermat in his copy of the 1621 edition.

Dibner Heralds of Science 108; En Français dans le Texte 115; Grolier/Horblit 30; Norman 778. Mahoney, The mathematical career of Pierre de Fermat, 1973. Weil, Review of Mahoney (ibid.), Bulletin of the American Mathematical Society 79 (1973), pp. 1138-49. Weil, Number Theory: An Approach Through History from Hammurapi to Legendre, 1984. For a detailed account of the Fermat/Pascal correspondence on probability, see David, Gods, Games & Gambling, Chapter 9 and Todhunter, A History of the Mathematical Theory of Probability, Chapter II.

Folio (335 x 223 mm), pp. [xii], 210, [4], with five engraved plates and engraved frontispiece portrait of Fermat, folded. Light browning on a few gatherings; a large crisp copy in contemporary mottled sheep (with recent and well-done restoration to boards and spine).

Item #5884

Price: $150,000.00