Nova methodus pro maximis et minimis, itemque tangentibus, quae nec fractas nec irrationales quantitates moratur, et singulare pro illis calculi genus.

Leipzig: Christopher Günther for J. Gross & J. F. Gleditsch, 1684.

First edition of Leibniz’s invention of the differential calculus. “Leibniz was an almost universal genius whose place in the history of mathematics depends on his being an independent inventor of the infinitesimal calculus and on his contributions to combinatorial analysis which foreshadowed the development of modern mathematical analysis … The Acta Eruditorum was established in imitation of the French Journal des Sçavans in Berlin in 1682 and Leibniz was a frequent contributor. Another German mathematician (E.W. Tschirnhausen), having published in it his paper on quadratures, based on researches that Leibniz had communicated to him, Leibniz at last decided in 1684 to present to the world the more abstruse parts of his own work on the calculus. His epoch-making papers give rules of calculation without proof for rates of variation of functions and for drawing tangents to curves … The infinitesimal calculus originated in the 17th century with the researches of Kepler, Cavalieri, Torrecelli, Fermat and Barrow, but the two independent inventors of the subject, as we understand it today, were Newton and Leibniz … Although both Newton and Leibniz developed similar ideas, Leibniz devised a superior symbolism and his notation is now an essential feature in all presentation of the subject … With the calculus a new era began in mathematics, and the development of mathematical physics since the 17th century would not have been possible without the aid of this powerful technique” (PMM). Leibniz “applied his new method to the solution of the cubic parabola and the inverse methods of tangents and many problems left unsolved by Descartes” (Dibner). “Although Newton had probably discovered the calculus earlier than Leibniz, Leibniz was the first to publish his method, which employed a notation superior to that used by Newton. The priority dispute between Newton and Leibniz over the calculus is one of the most famous controversies in the history of science; it led to a breach between English and Continental mathematics that was not healed until the nineteenth century” (Norman).

“The invention of the Leibnizian infinitesimal calculus dates from the years between 1672 and 1676, when Gottfried Wilhelm Leibniz (1646–1716) resided in Paris on a diplomatic mission. In February 1667 he received the doctor’s degree by the Faculty of Jurisprudence of the University of Altdorf and from 1668 was in the service of the Court of the chancellor Johann Philipp von Schönborn in Mainz. At that time his mathematical knowledge was very deficient, despite the fact that he had published in 1666 the essay De arte combinatoria. It was Christiaan Huygens (1629–1695), the great Dutch mathematician working at the Paris Academy of Sciences, who introduced him to the higher mathematics. He recognised Leibniz’s versatile genius when conversing with him on the properties of numbers propounded to him to determine the sum of the infinite series of reciprocal triangular numbers. Leibniz found that the terms can be written as differences and hence the sum to be 2, which agreed with Huygens’s finding. This success motivated Leibniz to find the sums of a number of arithmetical series of the same kind, and increased his enthusiasm for mathematics. Under Huygens’s influence he studied Blaise Pascal’s Lettres de A. Dettonville, René Descartes’s Geometria, Grégoire de Saint-Vincent’s Opus geometricum and works by James Gregory, René Sluse, Galileo Galilei and John Wallis.

“In Leibniz’s recollections of the origin of his differential calculus he relates that reflecting on the arithmetical triangle of Pascal he formed his own harmonic triangle in which each number sequence is the sum-series of the series following it and the difference-series of the series that precedes it. These results make him aware that the forming of difference-series and of sum-series are mutually inverse operations. This idea was then transposed into geometry and applied to the study of curves by considering the sequences of ordinates, abscissas, or of other variables, and supposing the differences between the terms of these sequences infinitely small. The sum of the ordinates yields the area of the curve, for which, signifying Bonaventura Cavalieri’s ‘omnes lineae’, he used the sign ‘∫’, the first letter of the word ‘summa’. The difference of two successive ordinates, symbolized by ‘d’, served to find the slope of the tangent. Going back over his creation of the calculus Leibniz wrote to Wallis in 1697: ‘The consideration of differences and sums in number sequences had given me my first insight, when I realized that differences correspond to tangents and sums to quadratures’.

“The Paris mathematical manuscripts of Leibniz … show Leibniz working out these ideas to develop an infinitesimal calculus of differences and sums of ordinates by which tangents and areas could be determined and in which the two operations are mutually inverse. The reading of Blaise Pascal’s Traité des sinus du quart de circle gave birth to the decisive idea of the characteristic triangle, similar to the triangles formed by ordinate, tangent and sub-tangent or ordinate, normal and sub-normal. Its importance and versatility in tangent and quadrature problems is underlined by Leibniz in many occasions, as well as the special transformation of quadrature which he called the transmutation theorem by which he deduced simply many old results in the field of geometrical quadratures. The solution of the ‘inverse-tangent problems’, which Descartes himself said he could not master, provided an ever stronger stimulus to Leibniz to look for a new general method with optimal signs and symbols to make calculations simple and automatic.

“The first public presentation of differential calculus appeared in October 1684 in the new journal Acta Eruditorum, established in Leipzig, in only six and an half pages, written in a disorganised manner with numerous typographical errors. In the title, ‘A new method for maxima and minima as well as tangents, which is impeded neither by fractional nor irrational quantities, and a remarkable type of calculus for them’, Leibniz underlined the reasons for which his method differed from—and excelled—those of his predecessors. In his correspondence with his contemporaries and in the later manuscript ‘Historia et origo calculi differentialis’, Leibniz predated the creation of calculus to the Paris period, declaring that other tasks had prevented publication for over nine years following his return to Hannover.

“Leibniz’s friends Otto Mencke and Johann Christoph Pfautz, who had founded the scientific journal Acta Eruditorum in 1682 in Leipzig, encouraged him to write the paper; but it was to be deemed very obscure and difficult to comprehend by his contemporaries. There is actually another more urgent reason which forced the author to write in such a hurried, poorly organised fashion. His friend Ehrenfried Walter von Tschirnhaus (1651–1708), country-fellow and companion of studies in Paris in 1675, was publishing articles on current themes and problems using infinitesimal methods which were very close to those that Leibniz had confided to him during their Parisian stay; Leibniz risked having his own invention stolen from him. The structure of the text, which was much more concise and complex than the primitive Parisian manuscript essays, was complicated by the need to conceal the use of infinitesimals. Leibniz was well aware of the possible objections he would receive from mathematicians linked to classic tradition who would have stated that the infinitely small quantities were not rigorously defined, that there was not yet a theory capable of proving their existence and their operations, and hence they were not quite acceptable in mathematics.

“Leibniz’s paper opened with the introduction of curves referenced to axis x, variables (abscissas and ordinates) and tangents. The context was therefore geometric, as in the Cartesian tradition, with the explicit representation of the abscissa axis only. The concept of function did not yet appear, nor were dependent variables distinguished from independent ones. The characteristics of the introduced objects were specified only in the course of the presentation: the curve was considered as a polygon with an infinity of infinitesimal sides (that is, as an infinitangular polygon), and the tangent to a point of the curve was the extension of an infinitesimal segment of that infinitangular polygon that represented the curve. Differentials were defined immediately after, in an ambiguous way. Differential dx was introduced as a finite quantity: a segment arbitrarily fixed a priori. This definition however would never be used in applications of Leibniz’s method, which was to operate with infinitely small dx in order to be valid. The ordinate differential was introduced apparently with a double definition: ‘dv indicates the segment which is to dx as v is to XB, that is, dv is the difference of the v’.

“In the first part Leibniz establishes the equality of the two ratios (dv : dx = v : XB), the equality deduced by the similitude between the finite triangle formed by the tangent, the ordinate and the subtangent, and the infinitesimal right-angle triangle whose sides are the differentials thereof and is called ‘characteristic triangle’. But the proportion contains a misprint in the expression for the subtangent that would be corrected only in the general index of the first decade of the journal [Acta Eruditorum, 1693], ‘Corrigenda in Schedias- matibus Leibnitianis, quae Actis Eruditorum Lipsiensibus sunt inserta’). The second part (‘dv is the difference of the v’) mentioned the difference between the two ordinates which must lie infinitely close:

dv = v(x + dx) − v(x).

In actual fact, the proportion was needed to determine the tangent line and the definition of dv was consequently the second, as explicitly appeared in three of Leibniz’s Parisian manuscripts. Considering the corresponding sequences of infinitely close abscissas and ordinates, Leibniz called differentials into the game as infinitely small differences of two successive ordinates (dv) and as infinitely small differences of two successive abscissae (dx), and established a comparison with finite quantities reciprocally connected by the curve equation.

“These first concepts were followed, without any proof, by differentiation rules of a constant a, of ax, of y = v, and of sums, differences, products and quotients. For the latter, Leibniz introduced double signs, whereby complicating the interpretation of the operation … Conscious of the criticism that the use of the infinitely small quantities would have had on the contemporaries, Leibniz chose to hide it in his first paper; many years later, replying to the objections of Bernard Nieuwentijt, he showed in a manuscript how to prove the rules of the calculus without infinitesimals, based on a law of continuity. In his ‘Nova methodus’ of October 1684 he would then go onto studying the behaviour of the curve in an interval, specifically increasing or decreasing ordinates, maxima and minima, concavity and convexity referred to the axis, the inflexion point and deducing the properties of differentials …

“After introducing the concept of convexity and concavity referred to the axis and linked to increase and decrease of ordinates and of the prime differentials, Leibniz dealt with the second differentials, simply called ‘differences of differences’ for which constant dx was implicitly presupposed. The inflexion point was thus defined as the point where concavity and convexity were exchanged or as a maximum or minimum of the prime differential. These considerations, burdened by the previous incorrect double implications, would lead him to state as necessary and sufficient conditions which were in fact only necessary. They will be elucidated in l’Hôpital’s textbook of 1696.

“Leibniz then set out the differentiation rules for powers, roots and composite functions. In the latter case, he chose to connect a generic curve to the cycloid because he wanted to demonstrate that his calculus was easily also applied to transcendent curves, possibility that Descartes wanted to exclude from geometry. It was a winning move to attract the attention on one of the most celebrated curves of the time, and his mentor Huygens expressed to him his admiration when in 1690 Leibniz sent him in detail the calculation of the tangent to the cycloid.

“Finally, Leibniz demonstrated how to apply his differential method on four current problems which led him to proudly announce the phrase quoted at the beginning of this paper. The first example, on the determination of a tangent to a curve, was very complex, containing many fractions and radicals. Earlier methods of past and contemporary mathematicians, such as Descartes, P. de Fermat, Jan Hudde and Sluse, would have required very long calculations. The second example was a minimum problem occurring in refraction of light studied by Descartes and by Fermat. Fermat’s method for maxima and minima led to an equation containing four roots, and hence to long and tedious calculations. The third example was a problem that Descartes had put to Fermat, deeming it ‘of insuperable difficulty’ because the equation of the curve whose tangent was to be determined contained four roots. Leibniz complicated the curve whose tangent was sought even more because his equation contained six. He solved a similar problem in a letter sent to Huygens on 8 September 1679. The last argument was the ‘inverse-tangent problem’, which corresponded to the solution of a differential equation, that is, find a curve such that for each point the subtangent is always equal to a given constant. In this case, the problem was put by Florimond de Beaune to Descartes, who did not manage to solve it, while Leibniz reached the goal in only a few steps. By these four examples he demonstrated the power of his differential method …

“From the first, when Leibniz was living in Paris, he had understood that the algorithm that he had invented was not merely important but revolutionary for mathematics as a whole. Although his first paper on differential calculus proved to be unpalatable for most of his readers, he had the good fortune to find champions like the Bernoulli brothers, and a populariser like de l’Hôpital, who helped to promote and advance his methods at the highest level. There was certainly no better publicity for the Leibnizian calculus than the results published in the Acta Eruditorum, and in the Memoirs of the Paris and Berlin Academies. They not only offered a final solution to open problems such as those of the catenary, the brachistochrone, the velary (the curve of the sail when moved by the wind), the paracentric isochrone, the elastica, and various isoperimetrical problems; they also provided tools for dealing with more general tasks, such as the solution of differential equations, the construction of transcendental curves, the integration of rational and irrational expressions, and the rectification of curves. Both the mathematicians and the scholars of applied disciplines such as optics, mechanics, architecture, acoustics, astronomy, hydraulics and medicine, were to find the Leibnizian methods useful, nimble and elegant as an aid in forming and solving their problems” (Roero, pp. 47-55).

These two volumes of Acta Eruditorum contain the following additional papers by Leibniz: ‘De dimensionibus figurarum inveniendis’ (III, pp. 233-6) (Ravier 88); ‘Demonstrationes novae de resistentia solidorum’ (III, pp. 319-25 and Tab. IX) (Ravier 89); ‘Meditatione de cognitione, verite et Ideis’ (III, pp. 537-42) (Ravier 91); ‘Additio ad schedam in Actis proxime antecedentis Maii pa. 233 editam, De dimensionibus curvilineorum’ (III, pp. 585-7) (Ravier 92); ‘Demonstratio geometrica regulae apud staicos receptae de momentis gravium’ (IV, pp. 501-5) (Ravier 93).

Horblit 66a; Norman 1326; PMM 160; Dibner 109; Honeyman 1972; Ravier 88. Roero, ‘Gottfried Wilhelm Leibniz. First three papers on the calculus (1684,1686, 1693).’ Chapter 4 in Grattan-Guinness (ed.), Landmark Writings in Western Mathematics 1640-1940, 2005.

In: Acta Eruditorum, Vol. III (1684), pp. 467-73 and Tab. XII. Bound with Vol. IV (1685). Two vols. bound in one, 4to (209 x 167 mm). Vol. III: pp. [10], 591, [7] with 14 plates; Vol. IV: pp. [6], 595, [16] with 15 plates. Contemporary vellum.

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