Leibniz: Mathematics

Leibniz had learned simple computation and a little geometry in his elementary studies and in secondary schools, but his interest in mathematics was aroused by the numerous remarks on the importance of the subject that he encountered in his reading of philosophical works. In Leipzig, John Kuhn’s lectures on Euclid left him unsatisfied, whereas he received some stimulation from Erhard Weigel in Jena. During his student years, he had also cursorily read introductory works on Cossist algebra and the Deliciae physicomathematicae of Daniel Schwenter and Philipp Harsdörffer (1636-1653) with their varied and mainly practical content. At this stage Leibniz considered himself acquainted with all the essential areas of mathematics that he needed for his studies in logic, which attracted him much more strongly. The very modest specialized knowledge that he then possessed is reflected in the Dissertatio de arte combinatoria(1666); several additions are presented in the Hypothesis physica nova (1671).

In accord with the encyclopedic approach popular at the time, Leibniz limited himself primarily to methods and results and considered demonstrations nonessential and unimportant. His effort to mechanize computation led him to work on a calculating machine which would perform all four fundamental operations of arithmetic.

Leibniz was occupied with diplomatic tasks in Paris from the spring of 1672, but he continued the studies (begun in 1666) on the arithmetic triangle which had appeared on the title page of Apianus’ Arithmetic (1527) and which was well known in the sixteenth century; Leibniz was still unaware, however, of Pascal’s treatise of 1665. He also studied the array of differences of the number sequences, and discovered both fundamental rules of the calculus of finite differences of sequences with a finite number of members. He revealed this in conversation with Huygens, who challenged his visitor to produce the summation of reciprocal triangular numbers and, therefore, of a sequence with infinitely many members. Leibniz succeeded in this task at the end of 1672 and summed further sequences of reciprocal polygonal numbers and, following the work of Grégoire de St.-Vincent (1647), the geometric sequence, through transition to the difference sequence.

As a member of a delegation from Mainz, Leibniz traveled to London in the spring of 1673 to take part in the unsuccessful peace negotiations between England, France, and the Netherlands. He was received by Oldenburg at the Royal Society, where he demonstrated an unfinished model of his calculating machine. Through Robert Boyle he met John Pell, who was familiar with the entire algebraic literature of the time. Pell discussed with Leibniz his successes in calculus of differences and immediately referred him to several relevant works of which Leibniz was not aware—including Mercator’s Logarithmotechnia (1668), in which the logarithmic series is determined through prior division, and Barrow’s Lectiones opticae (1669) and Lectiones opticae (1670). (Barrow’s works were published in 1672 under the single title Lectiones opticae et geometricae.)

Leibniz became a member of the Royal Society upon application, but he had seriously damaged his scientific reputation through thoughtless pronouncements on the array of differences, and still more through his rash promise of soon producing a working model of the calculating machine. Leibniz could not fully develop the calculator’s principle of design until 1674, by which time he could take advantage of the invention of direct drive, the tachometer, and the stepped drum.

Through Oldenburg, Leibniz received hints, phrased in general terms, of Newton’s and Gregory’s results in infinitesimal mathematics; but he was still a novice and therefore could not comprehend the significance of what had been communicated to him. Huygens referred him to the relevant literature on infinitesimals in mathematics and Leibniz became passionately interested in the subject. Following the lead of Pascal’s Lettres de “A. Delonville” [= Pascal] contenant quelques unes de ses inventions de géométrie (1659), by 1673 he had mastered the characteristic triangle and had found, by means of a transmutation—that is, of the integral transformation, discovered through affine geometry,

for the determination of a segment of a plane curve—a method developed on a purely geometrical basis by means of which he could uniformly derive all the previously stated theorems on quadratures.

Leibniz’ most important new result, which he communicated in 1674 to Huygens, Oldenburg, and his friends in Paris, were the arithmetical quadrature of the circle, including the arc tangent series, which had been achieved in a manner corresponding to Mercator’s series division, and the elementary quadrature of a cycloidal segment (presented in print in 1678 in a form that concealed the method). Referred by Jacques Ozanam to problems of indeterminate analysis that can be solved algebraically, Leibniz also achieved success in this area by simplifying methods, as in the essay on

x + y + z = p², x² + y² + z² = q⁴

(to be solved in natural numbers). Furthermore, a casual note indicates that at this time he was already concerned with dyadic arithmetic.

The announcement of new publications on algebra provoked Leibniz to undertake a thorough review of the pertinent technical literature, in particular the Latin translation of Descartes’s Géométrie (1637), published by Frans van Schooten (1659-1661) with commentaries and further studies written by Descartes’s followers.

His efforts culminated in four results obtained in 1675: a more suitable manner of expressing the indices (ik in lieu of a_ik, for instance); the determination of symmetric functions and especially of sums of powers of the solutions of algebraic equations; the construction of equations of higher degree that can be represented by means of compound radicals; and ingenious attempts to solve higher equations algorithmically by means of radicals, attempts that were not recognized as fruitless because of the computational difficulties involved. On the other hand, Leibniz succeeded in demonstrating the universal validity of Cardano’s formulas for solving cubic equations even when three real solutions are present and in establishing that in this case the imaginary cannot be dispensed with. The generality of these results had been frequently doubted because of the influence of Descartes. Through this work Leibniz concluded that the sum of conjugate complex expressions is always real (cf. the well-known example . The theorem named for de Moivre later proved this conclusion to be correct.

In the fall of 1675 Leibniz was visited by Tschirnhaus, who, while studying Descartes’s methods (which he greatly overrated), had acquired considerable skill in algebraic computation. His virtuosity aroused admiration in London, yet it did not transcend the formal and led to a mistaken judgment of the new results achieved by Newton and Gregory. Tschirnhaus and Leibniz became friends and together went through the unpublished scientific papers of Descartes, Pascal, and other French mathematicians. The joint studies that emerged from this undertaking dealt with the array of differences and with the “harmonic” sequence …, 1/5, 1/4, 1/3, 1/2, 1/1 and was treated by Leibniz as the counterpart of the arithmetic triangle. They then considered the succession of the prime numbers and presented a beautiful geometric interpretation of the sieve of Eratosthenes-which, however, cannot be recognized from the remark printed in 1678 that the prime numbers greater than three must be chosen from the numbers 6n ± 1.

When Roberval died in 1675 Leibniz hoped to succeed him in the professorship of mathematics established by Pierre de la Ramée at the Collèege de France and also to become a member of the Académie des Sciences. Earlier in 1675 he had demonstrated at the Academy the improved model of his calculating machine and had referred to an unusual kind of chronometer. He was rejected in both cases because his negligence had cost him the favor of his patrons. Nevertheless, his through, critical study of earlier mathematical writings resulted in important advances, especially in the field of infinitesimals. He recognized the transcendence of the circular and logarithmic functions, the basic properties of the logarithmic and other transcendental curves, and the correspondence between the quadrature of the circle and the quadrature of the hyperbola. In addition, he considered questions of probability.

In the late autumn of 1675, seeking a better understanding of Cavalieri’s quadrature methods (1635), Leibniz made his greatest discovery: the symbolic characterization of limiting processes by means of the calculus. To be sure, “not a single previously unsolved problem was solved” by this discovery (Newton’s disparaging judgment in the priority dispute); yet it set out the procedure to be followed in a suggestive, efficient, abstract form and permitted the characterizetion and classification of the applicable computational steps. In connection with the arrangement in undetermined coefficients, Leibniz sought to clarify the conditions under which an algebraic function can be integrated algebraically. In addition he solved important differential equations: for example, the tractrix problem, proposed to him by Perrault, and Debeaune’s problem (1638), which he knew from Descartes’s Lettres (III, 1667); and which required the curve through the origin determined by

He established that not every differential equation can be solved exclusively through the use of quadratures and was immediately cognizant of the far-reaching importance of symbolism and technical terminology.

Leibniz only hinted at his new discovery in vague remarks, as in letters to Oldenburg in which he requested details of the methods employed by Newton and Gregory. He received some results in reply, especially concerning power series expansions-which were obviously distorted through gross errors in copying-but nothing of fundamental significance (Newton’s letters to Leibniz of June and October 1676, with further information on Gregory and Pell supplied by Oldenburg). Leibniz explained the new discovery to him personally, but Tschirnhaus was more precisely informed. He did not listen attentively, was troubled by the unfamiliar terminology and symbols, and thus never achieved a deeper understanding of Leibnizian analysis. Tschirnhaus also had the advantage of knowing the answer, written in great haste, to Newton’s first letter, where Leibniz referred to the solution of Debeaune’s problem (as an example of a differential equation that can be integrated in a closed form) and hinted at the principle of vis viva. Leibniz also included the essential elements of the arithmetical quadrature of the circle; yet it was derived not by means of the general transmutation but, rather, through a more special one of narrower virtue. Tschirnhaus did not know that the preliminary draft of this letter contained an example of the method of series expansion through gradual integration (later named for Cauchy [1844] and Picard [1890] and, in any case, he would not have been able to understand and fully appreciate it. On the other hand, he did see and approve the definitive manuscript (1676) on the arithmetical quadrature of the circle. This work also contains the proof by convergence of an alternating sequence with members decreasing without limit; the rigorous treatment of transmutation and its application to the quadrature of higher parabolas and hyperbolas; the logarithmic series and its counterpart, the arc tangent series, and its numerical representation of (Leibniz series); and the representation of and through omission of members of the series

The planned publication did not occur and subsequently the paper was superseded by Leibniz’ own work as well as by that of others, particularly the two Bernoullis and L’Hospital.

Since there was no possibility of obtaining a sufficiently remunerative post in Paris, Leibniz entered the service of Hannover in the fall of 1676. He traveled first to London, where he sought out Oldenburg and the latter’s mathematical authority, John Collins. He presented them with papers on algebra, which Collins transcribed and which were transmitted to Newton. A longer discussion with Collins was devoted primarily to algebraic questions although dyadics were also touched upon. Leibniz also made excerpts from Newton’s letters and from the manuscript of his De analysi per aequationes numero terminorum infinitas (1669), which had been deposited with the Royal Society, and from the extracts procured by Collins of letters and papers of Gregory, only a small selection of which Leibniz had obtained earlier. He then went to Amsterdam, where he called on Hudde, who informed him of his own mathematical works.

In the intellectually limited atmosphere of Hannover there was no possibility of serious mathematical discussion. His talks with the Cartesian A. Eckhardt (1678-1679) on Pythagorean triangles with square measures and related questions were unsatisfying. The correspondence with Oldenburg provided an opportunity, in the early summer of 1677, to communicate the determination of tangents according to the method of the differential calculus, but this exchange ended in the autumn with Oldenburg’s death. Huygens was ill and Tschirnhaus was traveling in Italy. Thus, in the midst of his multitudinous duties at court, Leibniz lacked the external stimulus needed to continue his previous studies on a large scale. Instead, he concentrated on symbological investigations, his first detailed draft in dyadics, and studies on pure geometric representation of positional relations without calculation, the counterpart to analytic geometry. Only after the founding of the Acta eruditorum (1682) did Leibniz present his mathematical papers to the public. In 1682 he published “De vera proportione circuli ad quadratum circumscriptum in numeris rationalibus” and “Unicum opticae, catoptricae et dioptricae principium,” concise summaries of the chief results of the arithmetical quadrature of the circle and a hint regarding the derivation of the law of refraction by means of the extreme value method of the differential calculus. These revelations were followed in 1684 by the method of determining algebraic integrals of algebraic functions, a brief presentation of the differential calculus with a hint concerning the solution of Debeaune’s problem by means of the logarithmic curve, and further remarks on the fundamental ideas of the integral calculus.

In 1686 Leibniz published the main concepts of the proof of the transcendental nature of and an example of integration, the first appearance in print of the integral sign (the initial letter of the word summa).

Yet Leibniz did not attract general attention until his public attack on Cartesian dynamics (1686-1688) by reference to the principle of the conservation of vis viva, with the dimensions mv². In the subsequent controversy with the Cartesians, Leibniz put forth for solution a dynamic problem that was also considered by Huygens (1687) and Jakob Bernoulli (1690): Under what conditions does a point moving without friction in a parallel gravitational field descend with uniform velocity? In this connection, Bernoulli raised the problem of the catenary, which was solved almost simultaneously by Leibniz, Huygens, and Johann Bernoulli (published 1691) and which introduced for debate a series of further subjects of increasing difficulty, stemming primarily from applied dynamics. Two are particularly noteworthy: The first was the determination, requested by Leibniz in 1689, of the isochrona paracentrica. Under what conditions does a point moving without fricton under constant gravity revolve with uniform velocity about a fixed point? This problem was solved by Leibniz, Jakob Bernoulli, and Johann Bernoulli (1694). The second was the determination, requested by Johann Bernoulli, of the conditions under which a point moving without friction in a parallel gravitational field descends in the shortest possible time from one given point to another given point below it. This problem, called the brachistochrone, was solved in 1697 by Leibniz, Newton, Jakob Bernoulli, and Johann Bernoulli.

The participants in these investigations revealed only their results, not their derivations. The latter are found, in the case of Leibniz, in the posthumous papers or, in certain instances, in the letters exchanged with Jakob Bernoulli (from 1687), Rudolf Christian von Bodenhausen (from 1690), L’Hospital (from 1692), and Johann Bernoulli (from 1693). These letters, like the papers on pure mathematics that Leibniz published in the scientific journals, were usually hastily written in his few free hours. They were not always well edited and are far from being free of errors. Yet, despite their imperfections, they are extraordinarily rich in ideas. In part the letters were written to communicate original ideas, which were only occasionally pursued later; most of them, however, are drawn from earlier papers or brief notes. The following ideas in the correspondence should be specially mentioned:

(a) The determination of the center of curvature for a point of a curve as the intersection of two adjacent normals (1686-1692). Leibniz erroneously assumed that the circle of curvature has, in general, four neighboring points (instead of three) in common with the curve. It was only after some years that he understood, through the detailed explanation of Johann Bernoulli, the objections made by Jakob Bernoulli (from 1692). He immediately admitted his error publicly and candidly, as was his custom (1695-1696).

(b) The determination of that reflecting curve in the plane by which a given reflecting curve is completed in such a way that the rays of light coming from a given point are rejoined, after reflection in both curves, in another given point (1689).

(c) A detailed presentation of the results of the arithmetical quadrature of the circle and of the hyperbola, combined with communication of the power series for the arc tangent, the cosine, the sine, the natural logarithm, and the exponential function (1691).

(d) The theory of envelopes, illustrated with examples (1692, 1694).

(e) The treatment of differential equations through arrangement in undetermined coefficients (1693).

(f) The determination of the tractrix for a straight path and, following this, a mechanical construction of the integral curves of differential equations (1693), the earliest example of an integraph.

In 1694 Leibniz expressed his intention to present his own contributions and those of other contemporary mathematicians uniformly and comprehensively in a large work to be entitled Scientia infiniti. By 1696 Jakob Bernoulli had provided him with some of his own work and Leibniz had composed headings for earlier notes and selected some essential passages, but he did not get beyond this preliminary, unorganized collection of material. Much of his time was now taken up defending his ideas. Nieuwentijt, for example (1694-1696), questioned the admissibility and use of higher differentials. In his defense, Leibniz emphasized that his method should be considered only an abbreviated and easily grasped guide and that everything could be confirmed by strict deductions in the style of Archimedes (1695). On this occasion he revealed the differentiation of exponential functions such as x^x, which he had long known.

Even the originality of Leibniz’ method was called into doubt. Hence, in 1691, Jakob Bernoulli, who was interested primarily in results and not in general concepts, saw in Leibniz’ differential calculus only a mathematical reproduction of what Barrow had presented in a purely geometrical fashion in his Lectiones geometricae. He also failed to realize the general significance of the symbolism. In England it was observed with growing uneasiness that Leibniz was becoming increasingly the leader of a small but very active group of mathematicians. Moreover, the English deplored the lack of any public indication that Leibniz—as Newton supposed—had taken crucial suggestions from the two great letters of 1676. Representative material from these letters was in Wallis’ Algebra (1685), and it was expanded in the Latin version (Wallis’ Opera, II [1693]). Fatio de Duillier, who shortly before had seen copies of Newton’s letters and other unpublished writings on methods of quadrature (1676), became convinced that in the treatment of questions in the mathematics of infinitesimals Newton had advanced far beyond Leibniz and that the latter was dependent on Newton. Since 1687 Fatio had been working with Huygens, who did not think much of Leibniz’ symbolism, on the treatment of “inverse” tangent problems—differential equations—and in simple cases had achieved a methodical application of integrating factors.

In the preface to Volume I of Wallis’ Opera, which did not appear until 1695, it is stated that the priority for the infinitesimal methods belongs to Newton. Furthermore, the words are so chosen that it could have been—and it fact was—inferred that Leibniz plagiarized Newton. Wallis, in his concern for the proper recognition of Newton’s merits, was unceasing in his efforts to persuade Newton to publish his works on this subject. Beyond this, he obtained copies of several of the letters exchanged between Leibniz and Oldenburg in the years 1673 to 1677 and received permission from Newton and Leibniz to publish the writings in question. He included them in Volume III of his Opera (1699). The collection he assembled was based not on the largely inaccessible originals but, rather, on copies in which crucial passages were abbreviated. As a result, it was possible for the reader to gain the impression that Newton possessed priority in having obtained decisive results in the field of infinitesimals (method of tangents, power series in the handling of quadratures, and inverse tangent problems) and that Leibniz was guilty of plagiarism on the basis of what he had taken from Newton. Fatio pronounced this reproach in the sharpest terms in his Lineae brevissimi descensus investigatio (1699).

Leibniz replied in 1700 with a vigorous defense of his position, in which he stressed that he had obtained only results, not methods, from Newton and that he had already published the fundamental concepts of the differential calculus in 1684, three years before the appearance of the material that Newton referred to in a similar form in his Principia (1687). On this occasion he also described his own procedure with reference to de Moivre’s theorem (1698) on series inversion through the use of undetermined coefficients. Leibniz made his procedure more general and easier to grasp by the introduction of numerical coefficients (in the sense of indices). The attack subsided because Fatio, who was excitable, oversensitive, and given to a coarse manner of expression, turned away from science and became a fervent adherent of an aggressive religious sect. He was eventually pilloried.

Wallis’ insinuations were repeated by G. Cheyne in his Methodus (1703) and temperately yet firmly rebutted in Leibniz’ review of 1703. Cheyne’s discussion of special quadratures was probably what led Newton to publish the Quadratura curvarum (manuscript of 1676) and the Enumeratio linearum tertii ordinis (studies beginning in 1667-1668) as appendices to the Optics (1704). Newton viewed certain passages of Leibniz’ review of the Optics (1705) as abusive attacks and gave additional material to John Keill, who, in a paper published in 1710, publicly accused Leibniz of plagiarism. Leibniz’ protests (1711) led to the establishment of a commission of the Royal Society, which decided against him (1712) on the basis of the letters printed by Wallis and further earlier writings produced by Newton. The commission published the evidence, together with an analysis that had been published in 1711, in a Commercium epistolicum (1713 edition).

The verdict reached by these biased investigators, who heard no testimony from Leibniz and only superficially examined the available data, was accepted without question for some 140 years and was influential into the first half of the twentieth century. In the light of the much more extensive material now available, it is recognized as wrong. It can be understood only in the nationalistic context in which the controversy took place. The continuation of the quarrel was an embarrassment to both parties; and, since it yielded nothing new scientifically, it is unimportant for an understanding of Leibniz’ mathematics. The intended rebuttal did not materialize, and Leibniz’ interesting, but fragmentary, account of how he arrived at his discovery (Historia et origo calculi differentialis [1714]) was not published until the nineteenth century.

The hints concerning mathematical topics in Leibniz’ correspondence are especially fascinating. When writing to those experienced in mathematics, whom he viewed as competitors, he expressed himself very cautiously, yet with such extraordinary cleverness that his words imply far more than is apparent from an examination of the notes and jottings preserved in his papers. For instance, his remarks on the solvability of higher equations are actually an anticipation of Galois’s theory. Frequently, material of general validity is illustrated only by simple examples, as is the case with the schematic solution of systems of linear equations by means of number couples (double indices) in quadratic arrangement, which corresponds to the determinant form (1693).

In several places the metaphysical background is very much in evidence, as in the working out of binary numeration, which Leibniz connected with the creation of the world (indicated by 1) from nothingness (indicated by 0). The same is true of his interpretation of the imaginary number as an intermediate entity between Being and Not-Being (1702). His hope of being able to make a statement about the transcendence of π by employing the dyadic presentation (1701) was fruitless yet interesting, for transcendental numbers can be constructed out of infinite dual fractions possessing regular gaps (an example in the decimal system was given by Goldbach in 1729). The attempts to present (reference in 1682, recognized as false in 1696) and (1696) in closed form were unsuccessful, and the claimed rectification of an arc of the equilateral hyperbola through the quadrature of the hyperbola (1676) was based on an error in computation. Against these failures we may set the importance of the recognition of the correspondence between the multinomial theorem (1676) and the continuous differentiation and integration with fractional index that emerged from this observation. During this period (about 1696) Leibniz also achieved the general representation of partial differentiation; the reduction of the differential equation

a₀₀ + a₁₀x + (a₀₁ + a₁₁x)y’ = 0

by means of the transformation

x = p₁₁u + p₁₂v, y = p₂₁u + p₂₂v;

the solution of

y’ + p(x) y + q(x) = 0

and

(y’)² + p(x)y’ + q(x) = 0

through series with coefficients in number couples; and the reduction of the equation, which had originated with Jakob Bernoulli,

y’ = p(x)y + q(x) yⁿ

to

y’ = p (x)y + Q(x).

Leibniz’ inventive powers and productivity in mathematics did not begin to slow until around 1700. The integration of rational functions (1702-1703) was, to be sure, an important accomplishment; but the subject was not completely explored, since Leibniz supposed (Johann Bernoulli to the contrary) that there existed other imaginary units besides (for example, which could not be represented by ordinary complex numbers. The subsequent study, which was not published at the time, on the integration of special classes of irrational functions also remains of great interest. The discussion with Johann Bernoulli on the determination of arclike algebraic curves in the plane (1704-1706) resulted in both a consideration of relative motions in the plane and an interesting geometric construction of the arclike curve equivalent to a given curve. This construction is related to the optical essay of 1689 and to the theory of envelopes of 1692-1694, but it cannot readily be grasped in terms of a formula (1706). The remarks (1712) on the logarithms of negative numbers and on the representation of Σ(-1)ⁿ by 1/2 can no longer be considered satisfactory. The description of the calculating machine (1710) indicates its importance but does not give the important details; the first machines of practical application were constructed by P. M. Hahn in 1774 on the basis of Leibniz’ ideas.

Leibniz accorded a great importance to mathematics because of its broad interest and numerous applications. The extent of his concern with mathematics is evident from the countless remarks and notes in his posthumous papers, only small portions of which have been accessible in print until now, as well as from the exceedingly challenging and suggestive influential comments expressed brilliantly in the letters and in the works published in his own lifetime.

Leibniz’ power lay primarily in his great ability to distinguish the essential elements in the results of others, which were often rambling and presented in a manner that was difficult to understand. He put them in a new form, and by setting them in a larger context made them into a harmoniously balanced and comprehensive whole. This was possible only because in his reading Leibniz was prepared, despite his impatience, to immerse himself enthusiastically and selflessly in the thought of others. He was concerned with formulating authoritative ideas clearly and connecting them, as he did in so exemplary a fashion in the mathematics of infinitesimals. Interesting details were important—they occur often in his notes—but even more important were inner relationships and their comprehension, as this term is employed in the history of thought. He undertook the work required by such an approach for no other purpose than the exploration of the conditions under which new ideas emerge, stimulate each other, and are joined in a unified thought structure.

Joseph E. Hofmann

BIBLIOGRAPHY

I. Original Works. The following volumes of the Sämtliche Schriften und Briefe, edited by the Deutsche Akademie der Wissenschaften in Berlin, have been published: Series I (Allgemeiner politischer und historischer Briefwechsel), vol. 1: 1668-1676 (1923; Hildesheim, 1970); vol. 2: 1676-1679 (1927; Hildesheim, 1970); vol. 3: 1680-1683 (1938; Hildesheim, 1970); vol. 4: 1684-1687 (1950); vol. 5: 1687-1690 (1954; Hildesheim,1970); vol. 6: 1690-1691 (1957; Hildesheim,1970); vol.7: 1691-1692 (1964); vol. 8: 1692 (1970); Series II (Philosophischer Briefwechsel), vol. 1 : 1663-1685 (1926) ; Series IV (Politischer Schriften), vol. 1: 1667-1676 (1931); vol. 2: 1677-1687 (1963); Series VI (Philosophische Schriften), vol. 1: 1663-1672 (1930); vol. 2: 1663-1672 (1966); vol. 6: Nouveaux essais (1962).

Until publication of the Sämtliche Schriften und Briefe is completed, it is necessary to use earlier editions and recent partial editions. The most important of these editions are the following (the larger editions are cited first): L. Dutens, ed., G. W. Leibnitii opera omnia, 6 vols. (Geneva, 1768); J. E. Erdmann, ed., G. W. Leibniz. Opera philosophica quae extant latina gallica germanica omnia (Berlin, 1840; Aalen, 1959); G. H. Pertz, ed., G. W. Leibniz. Gesammelte Werke, Part I: Geschichte, 4 vols. (Hannover, 1843-1847; Hildesheim, 1966); A. Foucher de Careil, ed., Leibniz. Oeuvres, 7 vols. (Paris, 1859-1857; Hildesheim, 1969); C. I. Gerhardt, ed., G. W. Leibniz. Mathematische Schriften, 7 vols. (Berlin-Halle, 1849-1863; Hildesheim, 1962). An index to the edition has been compiled by J. E. Hofmann (Hildesheim, 1971); C. I. Gerhardt, ed., Die philosophischen Schriften Von G. W. Leibniz, 7 vols. (Berlin, 1875-1890; Hildesheim, 1960-1961); G. E. Guhrauer, ed., G. W. Leibniz. Deutsche Schriften, 2 vols (Berlin, 1838-1840; Hilesheim, 1966); J. G. Eckhart, ed., G. W. Leibniz. Collectanea etymological (Hannover, 1717; Hiledesheim, 1970); A. Foucher de Careil, ed., Lettres et opuscules inédits de Leibniz (paris, 1854; Hildesheim, 1971); A. Foucher de Careil, ed., Nouvelles Lettreset opuscules inédits de Leibniz (Paris, 1857; Hildesheim, 1971); C. Hass, ed. and tr., Theologisches system (Tübingen, 1860; Hildensheim, 1966); C. L. Grotefend, ed., Briefwechsel zwischen Leibniz Arnaud and dem Landgrafen Ernst von Hessen-Rheinfels (Hannover, 1846); C. I. Gerhard, ed., Briefwechsel zwischen Leibniz and Christian Wolf (Halle, 1860; Hildesheim, 1963); E. Bodemann, ed., Die Leibniz-Handschriften der Königlichen öffentlichen Bibliothek zu Hannover (Honnover, 1889; Hildesheim’ 1966); and Der Briefwechsel des G. W. Leibniz in der Königlichen öffentlichen Bibliothek zu Hannover (Hannover, 1895; Hildesheim, 1966); C. I. gerharhardt, ed., Der Brief-wechsel; von G. W. Leibniz mit Mathematikern (Berlin, 1899; Hildeshemis, 1962); L. Couturat, ed., Opusculas et fragments inédits de Leibniz (Paris, 1903; Hildesheim, physikalischen, mechanischen and technischen Inhales(Leibzig, 1906); H. Lestienne, ed., G. W. Leibniz. Discours de métaphysique (Paris, 1907, 2nd ed., 1929; paris, 1952); I. Jagodinskij, ed., Leibnitiana elementa philosophize areanae de summa serum (Kazan, 1913); P. Schrecker, ed., G. W. Leibniz. Lettres et fragments inédits sur les problèmes philosophiques, théologiques de la réconciliation des doctrines protestantes (1669-1704) (Paris, 1934); G. Grua, ed., G. W. Leibniz. Texts inédits, 2 vols. (Paris, 1948); W. Von Engelhardt, ed. and tr., G. W. Leibniz. Protoraea, in Leibniz. Werke (W. E. Peuckert, ed.), vol. I (Stuttgart, 1949); A. Robinet, ed., G. W. Leibniz. Prinicipes de la nature et de la grÂave fondeées en raison. Principes de la philosophize ou monadolorie. Publicés intégralement d’après les manuscripts de Hanovre, Vienne et Paris et présentés d’après des letters inédits (Paris, 1954); and Correspondences Leibniz-Clarke. Présentés d’aprè les manuscripts originates originals des bibliothèques de Hanovre et de Londres (paris, 1957); G. le Roy, ed., Leibniz. Discours de méthphysiques et correspondance avec Arnauld. (Paris, 1957) ; O. Saame, ed. and tr., G. W. Leibniz. Confessio philosophi. Ein Dialog (Frankfurt, 1967); J. Brunschwig, ed., Essais de Théodicée sur la bonté de Dieu, la libertyé de l’homme et l’origine du mal (paris, 1969).

There are translations of single works, especially into English and German. English translations (including selections): Philosophical Works, G. M. Duncan, ed. and tr. (New Haven, 1890); The Monadology and Other Philosophical Writings, R. Latta, ed. and tr. (Oxford, 1898); New Essays Concerning Human Understanding, A. G. Langley, ed. and tr. (Chicago, 1916, 1949); Discourse on Metaphysics, P. G. Lucas and L. Grint, ed. and tr. (Mancheater, 1953; 2nd ed. 1961); Philosophical writings, M. Morris, ed, and tr. (London, 1934, 1968); Theodicy, E. M. Huggard and A. Farrer, ed. and tr. (London, 1951): Selection, P. p. Wiener, ed. (New York, 1951; 2nd ed. 1971); Philosophical Papers and Letters, L. E. Loemker, ed. and tr., 2 vols. (Chicago, 1956; 2nd ed. Dordrecht, 1969); The Leibniz-Clarke Correspondence,, H. G. Alexander. ed. (Manchester, 1956); Monadology and Other philosophical Essays, P. and A. Schrecker, ed. and tr. (Indianapolis, 1965); Logical Papers, G. H. R. Parkinson, ed. and tr. (oxford, 1966); The Leibrniz-Arnauld Correspondence, H. T. Manson, ed. Tr. (Manchester-New York, 1967); General Investigations Concerning the Analysis of Concepts and Truths, W. H. O’Brain, ed. and tr. (Athens, Ga., (1968).

German translations (including selections) are Hand schriften zur Grundlegung der Philosphie,, E. Cassirer, ed., A. Buchenau, tr., 2 vols. (Hamburg, 1904; 3rd ed. 1966); Neue Abhandlungen über den menshclichen Verstand, E. Classier, ed. and tr. (3rd ed., 1915; Hamburg, 1971); the same work, edited and translated by H. H. Holz and W. von Engelhardt, 2 vols. (Frankfurt, 1961); Die Theodizee, A. Buchenau, ed. and tr. (Hamburg, 2nd ed., 1968); schöphferishe Vernunft, W. von Engelhardt, ed. and tr. (Marbung,1952); Metaphysische Abhandlung, H. Herring ed. and tr. (Hamburn,1958); Fragmente zur Logik, F. Schmidt, ed. and tr. (Berlin, 1960) ; Vernunftprinzipien der Natur and der Gnade. Monadologie, H. Herring, ed., A. Buchenau, tr. (Hamburg, 1960); Kleine Schriften zur Metaphysik, H. H. Holz, ed., 2 vols. (Frankfurt-Vienna, 1966-1967).

Bibliographical material on the works of Leibniz and the secondary literature can be found in E. Ravier, Bibliographic des oeuvres de Leibniz (paris, 1937; Hildesheim, 1966), additional material in P. Schrecker, “une bibilographie de Leibniz,” in Revue philosophique de la France et de l’Éranger, 126 (1938), 324-346: Albert Rivaud, Catalogue critique des manuscrits de Leibniz Fasc. II (Mars 1672-November 1676) (Polities, 1941-1924: rept. New york-Hildesheim, 1972); K. Müller, Leibniz-Bibolographie. Verzeichnis der Literature über Leibniz (Frankfurt, 1967). Bibloigraphical supplements appear regularly in Studia Lelbnitiana. Vierteljahresschrift für Philographie and Geschichte der Wissenschaften, K. Müller and W. Totok, eds., 1 (1969); G.Uyermö;hlen, “LeibnizBibliographic 1967-1968; 1 (1969), 293-320 G. Utermöhlen, and A. Schmitz, “Leibniz-Bibliographie. Neue Title 1968-1970,” 2 (1970), 302-320; A. Koch-Klose and A. Hölzen, “Leibniz-Bibliographie. Naue Titel 1969-1971,” 3(1971), 309-320.

II. Secondary Literature. The literature on Leibniz’ philosophy and science is so extensive that a complete presentation cannot be given. In the following selection, more recent works have been preferred, especially those which may provide additional views on the philosophy and science of Leibniz.

H. Aarsleff, “Leibniz on Locke and Language,” in American Philosophical Quarterly, 1 (1964), 165-188; E. J. Aiton, “The Harmonic Vortex of Leibniz,” in The Vortex Theory of Planetary Motions (London–New York, 1972); and “Leibniz on Motion in a Resisting Medium,” in Archive for History of Exact Sciences, 9 (1972), 257-274; W. H. Barber, Leibniz in France, From Arnauld to Voltaire (Oxford, 1955); Y. Belaval, Leibniz critique de Descartes (Paris, 1960); and Leibniz, Initiation à sa philosophie (Paris, 1962; 3rd ed. 1969); A. Boehm, Le “vinculum substantiale” chez Leibniz. Ses origines historiques (Paris, 1938; 2nd ed. 1962); F. Brunner, Études sur la signification historique de la philosophie de Leibniz (Paris, 1950); G. Buchdahl, Metaphysics and the Philosophy of Science. The Classical Origins-Descartes to Kant (Oxford, 1969); P. Burgelin, Commentaire du Discours de métaphysique de Leibniz (Paris, 1959); H. W. Carr, Leibniz (London, 1929, 1960); E. Cassirer, Leibniz’ System in seinem wissenschaftlichen Grundlagen (Marburg, 1902; Darmstadt, 1962); P. Costabel, Leibniz et la dynamique. Les textes de 1692 (Paris, 1960); L. Couturat, La logique de Leibniz (Paris, 1901; Hildesheim, 1961); L. Davillé, Leibniz historien. Essai sur l’activité et les méthodes historiques de Leibniz (Paris, 1909); K. Dürr, Neue Beleuchtung einer Theorie von Leibniz. Grundzüge des Logikkalküls (Darmstadt, 1930); K. Dürr, “Die mathematische Logik von Leibniz,” in Studia philosophica, 7 (1947), 87-102; K. Fischer, G. W. Leibniz. Leben, Werke und Lehre (Heidelberg, 5th ed. 1920, W. Kabitz, ed.); J. O. Fleckenstein, G. W. Leibniz. Barock und Universalismus (Munich, 1958); G. Friedmann, Leibniz et Spinoza (Paris, 2nd ed. 1946, 3rd ed. 1963); M. Gueroult, Dynamique et métaphysique leibniziennes suivi d’une note sur le principe de la moindre action chez Maupertuis (Paris, 1934,1967); G. E. Guhrauer, G. W. Freiherr von Leibniz. Eine Biographie, 2 vols. (Wroclaw, 1846; Hildesheim, 1966); G. Grua, Jurisprudence universelle et théodicée selon Leibniz (Paris, 1953); and La justice humaine selon Leibniz (Paris, 1956); H. Heimsoeth, Die Methode der Erkenntnis bei Descartes und Leibniz, 2 vols. (Giessen, 1912—1914); A. Heinekamp, Das Problem des Guten bei Leibniz(Bonn, 1969; K. Hildebrandt, Leibniz und das Reich der Gnade (The Hague, 1953); H. H. Golz, Leibniz (Stuttgart, 1958); K. Huber, Leibniz (Munich, 1951); J. Jalabert, La théorie leibnizienne de la substance (Paris, 1947); J. Jalabert, Le Dieu de Leibniz (Paris, 1960); J. Guitton, Pascal et Leibniz (Paris, 1951); M. Jammer, Concepts of Force. A Study in the Foundations of Dynamics (Cambridge, Mass., 1957); W. Janke, Leibiz. Die Emendation der Metaphysik (Frankfurt, 1963); H. W. B. Joseph, Lectures on the Philosophy of Leibniz, J. L. Austin, ed. (Oxford , 1949); W. Kabitz, Die Philosophie des jungen Leibniz. Untersuchungen zur Entwicklungsgeschichte seines Systems (Heidelberg, 1909); F. Kaulbach, Die Metaphysik des Raumes bei Leibniz und Kant (Cologne, 1960); R. Kauppi, über die leibnizsche Logik. Mit besonderer Berücksichtigung des Problems der Intension und der Extension (Acta Philosophica Fennica XII; Helsinki,1960); W. Kneale and M. Kneale, The Development of Logic (Oxford, 1962); L. Krüger, Rationalismus und Entwurf einer universalen Logik bei Leibniz (Frankfurt, 1969); D. Mahnke, “Leibnizens Synthese von Universalmathematik und Individualmetaphysik,” in Jahrbuch für Philosophie und Phänomenologische Forschung, 7 (1925), 305-612 (repr. Stuttgart, 1964); G. Martin, Leibniz. Logik und Metaphysik (Cologne, 1960; 2nd ed., Berlin, 1967; English tr. by P. G. Lucas and K. J. Northcott from the 1st ed.: Leibniz. Logic and Metaphysics (Manchester-New York, 1964); J. T. Mertz, Leibniz (Edinburgh—London, 1884; repr. New York, 1948); R. W. Meyer, Leibniz and the 17th Century Revolution (Glasgow, 1956); J. Mittelstrass, Neuzeit und Aufklärung. Studien zur Entstehung der neuzeitlichen Wissenschaft und Philosophie (Berlin, 1970); J. Moreau, L’niverse leibnizien (Paris, 1959); E. Naert, Leibniz K. Müller and G. Krönert, Leben und Werk von G. W. Leibniz. Eine Chronik (Frankfurt, 1969); E. Naert, Leibniz et la querelle du pur amour (Paris 1959); E. Naert, Mémoire et conscience de soi selon Leibniz (Paris, 1961); G. H. R. Parkinson, Logic and Reality in Leibniz’ Metaphysics (Oxford, 1965); G. H. R. Parkinson, Leibniz on Human Freedom (Studia Leibnitiana Sonderheft 2; Wiesbaden, 1970); C. A. van Peursen, Leibniz (Baarn, 1966), tr. into English by H. Hoskins (London, 1969); H. Poser, Zur Theorie der Modalbegriffe bei G. W. Leibniz (Studia Leibnitiana Suppl. VI; Wiesbaden, 1969); N. Rescher, “Leibniz’ Interpretation of His Logical Calculi,” in Journal of Symbolic Logic, 19 (1954), 1-13; N. Rescher, The Philosophy of Leibniz (Englewood Cliffs, N.J., 1967); W. Risse, Die Logik der Neuzeit II: 1640—1780 (Stuttgart, 1970); A. Robinet, Malebranche et Leibniz. Relations personelles (Paris,1955); A. Robinet, Leibniz et la racine de l’existence (Paris, 1962); B. Russell, A Critical Exposition of the Philosophy of Leibniz (Cambridge, 1900; 2nd ed., London, 1937); H. Schiedermair, Das Phänomen der Macht und die Idee des Rechts bei G. W. Leibniz (Studia Leibnitiana Suppl. VII; Wiesbaden, 1970); H. Scholz, “Leibniz” (1942), reprinted in H. Scholz, Mathesis universalis, H. Hermes, F. Kambartel and J. Ritter, eds. (Basel, 1961); L. Stein, Leibniz und Spinoza. Ein Beitrag zur Entwicklungsgeschichte der leibnizischen Philosophie (Berlin, 1890); G. Stieler, Leibniz und Malebranche und das Theodizee problem (Darmstadt, 1930); W. Totok and C. Haase, eds., Leibniz. Sein Leben, sein Wirken, seine Welt (Hannover, 1966); A. T. Tymieniecka, Leibniz’ Cosmological Synthesis (Assen, 1964); P. Wiedeburg, Der junge Leibniz, das Reich und Europa, pt. I, 2 vols. (Wiesbaden, 1962).

E. Hochstetter, ed., Leibniz zu seinem 300 Geburtstag 1646-1946, 8 pts. (Berlin, 1946-1952); G. Schischkoff, ed., Beiträge zur Leibniz-Forschung (Reutlingen, 1947); E. Hochstetter and G. Schischkoff, eds., Zum Gedenken an den 250 Todestag von G. W. Leibniz (Zeitschrift für Philosophische Forschung, 20 . nos. 3-4 (Meisenheim, 1966), 377-658; and Zum Gedenken an den 250 Todestag von G. W. Leibniz (Philosophia naturalis, 10 , no.2 (1968), 134-293); Leibniz (1646-1716). Aspects de l–homme et de l’oeuvre (Journées Leibniz, organis. au Centre Int. de Synthèse, 28-30 mai 1966) (Paris,1968); Studia Leibnitiana Supplementa, vols. I-V (Akten des Int. Leibniz-Kongresses Hannover, 14—19 November 1966) (Wiesbaden, 1968-1970).