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the indexG. W. Leibniz was one of the most important thinkers of his time.
His contributions to such diverse fields as philosophy, linguistics,
and history are undeniable. And yet although he became acquainted
quite late in his life with the mathematical achieveme nts of his
generation, it will always be his innovations in this field that put
him to the forefront of the enlightened thinkers of his era. These
achievements are especially remarkable considering that Leibniz often
treated the subject as a corollary to his studies in other fields,
notably logic, philosophy, and even law. It was precisely for this
reason that Leibniz had so much success in the field, in that he was
unhampered by much of the dogma that might have hindered its progress.
Leibniz viewed the subject through his own lens, interpreting the
mathematical issues differently from his colleagues. Perhaps it was
the distance from which he viewed the field that allowed Leibniz to be
such an innovator in the rapidly changing subject. He gathered and pr
ocessed as much contem-porary mathematics as possible, reassessed it,
and through his innovative system of notation, repackaged it as a
superior product. It was Leibniz's algebraic symbolism that freed the
subject from much of its rigid verbal structure, allowing it to
develop at an even faster rate. Leibniz's modern mathematical
notation probably represents his greatest single contribution to
mathematics. G.W. Leibniz is generally considered, along with Isaac
Newton, as a cofounder of the differential and integral Calculus. For
this reason Leibniz's achievements are often compared to that of his
subsequent rival from England. It can be said with almost com plete
certainty that he did not have the raw mathematical skill that Newton
possessed. Furthermore, Leibniz, who was a few years younger than
Newton, had very little formal education or experience in the subject.
In fact he had virtually no background in Algebra or Geometry when he
arrived in Paris on a diplomatic post at the relatively developed age
of 24. Leibniz had however shown an aptitude for mathematics prior to
his arrival in France in 1672, namely through his work on
Combinatorics while a univers ity student at Leipzig. Yet virtually
all of his substantial work on issues of analysis occurred during the
four years he spent in Paris. Leibniz's genesis as a mathematician in
such a short time is astounding, even with the backdrop of the second
half of the 17th century, where great strides in mathematics were being made
at a remarkable pace. Leibniz entered this milieu almost completely
ignorant of the recent achievements of his contemporaries. Over the
next few years he would work largely independentl y, developing a
notation that is still in use today. Great strides towards the
development of calculus were made throughout the 17th century.
Mathematicians had developed algebraic methods for finding areas and
volumes of a great variety of geometric figures. This marks one of
the greatest developments in m athematics since the Greeks began using
limits to approximate areas and then find the value of p. It was
Cavalieri (1598-1647) who first introduced the concept of "indivisible
magnitudes" in his Geometry of Indivisibles to study areas under
curves of the form:
At roughly the same time Descartes published his La Giomitrie, in which he showed, somewhat obscurely, how to use Viite's algebra to describe curves and obtain an algebraic analysis of geometric problems. [1, pp. 319-331] The work of these two mathematici ans would have an especially great influence on the development of Leibniz's new calculus. [4, X]
Leibniz meanwhile had scarcely even heard of contemporary mathematicians, including those in England, and despite his great curiosity had only a perfunctory know-ledge of more developed mathematics. In a letter written to his close friend James Bernoulli in 1703, Leibniz wrote "When I arrived in Paris in the year 1672, I was self-taught as regards geometry, and indeed had little knowledge of the subject, for which I had not the patience to read through the long series of proofs. As a youth I consulted th e beginner's Algebra of a certain Lanzius, and afterward that of Clavius; that of Descartes seemed to be more intricate..." [3, p. 12] While his language deliberately expresses his general ignorance towards mathematics, it is undeniable that he had yet to come in contact with recent advances regarding infinite series or inverse-tangent problems. These were of some of the developments that would become the basis of the new calculus.
Gottfried Wilhelm Leibniz was born in 1646 to a semi-noble family in the Protestant city of Leipzig. His father, a professor of Moral Philosophy and a competent scholar, died when the young Leibniz was 6 years old. Leibniz began to teach himself Latin in his father's personal library by reading the inscriptions of illustrated books and comparing them to those in German versions. Meanwhile at school he learned the traditional syllogistic logic of Aristotle and also became proficient in Greek [2, p. 3]. He re he had already envisioned the creation of a new subsection of Aristotelian cate-gories for the ordering of complex expressions and propositions.[12, p. X] He would later state that he was searching for a mathematical means of demonstration, represented by an alphabet of human thought. He believed that through this alphabet letters could be combined into words to express every natural truth. [5, V, p. 3] Leibniz believed there are categories for simple terms, or concepts, which could then be arranged sy stematically to form propositions. Propositions, or complex terms, should then be categorized to form truths, which then could be further arranged to form deductive arguments. Although a similar system was already used in geometric proofs, Leibniz believe d that an improved version could become universally applicable if a greater account were first given to the categories of simple terms. He would later refer to these terms as 'the alphabet of human thought' [12, p. X]. This is in essence Leibniz's vision of a characteristica universalis, an elastic structure in which logic could be conveyed in a more precise manner than through the cumbersome and rigid arrangement governed by words and grammar. Here the complex terms or truths would be represented by a combination of signs corresponding to their various parts. Leibniz believed such a language would be a tool for experiment, with deductive arguments constructed by simply following the language's grammar. Although this 'grammar' would have to be constructed for mathematics, its development would already be greatly facilitated through the language's symbolism itself. This symbolism would be used for calculation without emphasizing the meaning of what was written. The sym bols would represent the processes themselves. Thus, as Leibniz would somewhat naively say elsewhere, the mind 'would be freed from having to think directly of things themselves, and yet everything will turn out correctly.' [12, XVIII]. Although his tech nical knowledge of mathematics was and would for some time remain superficial, Leibniz had already developed an interest towards formal deductive reasoning and systemization.
Leibniz entered the University of Leipzig in 1661 and focused his studies on philosophy. While at Leipzig, Leibniz was influenced greatly by the Mathematician and Philosopher Erhard Weigel, whose work was focused on reconciling Aristotle with contemporar y philosophers through a mathematical method of demonstration patterned on Euclid. Leibniz believed that mathematical demonstration freed philosophy from its verbal limitations and embraced the field, believing its systematic coherence could unify the sci ences. Leibniz would use these theories in virtually every field he studied, including law, where he assigned values of 0, 1, and 1/2 to conditions of laws that were impossible, necessary (absolute), or conditional.
Following the completion of his law degree, Leibniz published his Dissertation De Arte Combinatoria. The work was published in 1666, long before he had any serious formal mathematical training or knowledge of the contemporary literature. [13, p. 423] The Art of Combinatorics was an original work of logic that reconciled his various philosophical interests while showing his penchant for system building. In this work Leibniz finally expressed his vision of 'an alphabet of human thought' where all concepts were just combinations of a relatively small number of simple notions. Leibniz believed that through the combination of these plain concepts one could discover any truth as well as the relationships from which they are created. Leibniz saw his 'art of com binations' as the logic of invention, where the finding of true propositions resulted in concepts that were either 'subjects' or their 'predicates' [12, p. 3].
In a fashion similar to that of medieval Indian mathematicians, Leibniz applies his combinations to examples in law, music, and philosophy. [5, V, p. 8] Mathematically the work was somewhat immature, but its significance lies in its various philosophical applications. As is often the case with such great thinkers, Leibniz's first work unveils his hidden genius. Despite its mathematical primacy, The Dissertation De Arte Combinatoria represents a major first step in Leibniz's mathematical development, long before his arrival in Paris and the subsequent development of his calculus. Mathematical Discoveries
Leibniz arrived in Paris in the spring of 1672 as a diplomatic correspondent to the Count of Mainz. In the fall of that year he met the well-known Dutch scientist Christiaan Huygens, who presided over the Acadimie Royale des Sciences and first introduced him to the works of the great English mathematicians of the time. [5,V, p. 83] It was probably during that meeting that Huygens tested the ambitious German by asking him to find the sum of the infinite series of reciprocal triangular numbers, which he hi mself had already found. [11, p. 29]
He represented his series through a diagram in which each term began at the same point of a line, meaning that instead of placing the value of each successive term next to each other, the diagram portrayed each term as a part of the largest, making the d ifferences of successive terms proportional to the terms of the original series. Starting with the series: AB=1,AC=1/2,AD=1/3,AE=1/5, Leibniz found the sum of their differences BC=1/2,CD=1/6,DE=1/12 and EF=1/20 equaled AB =1. When multiplied by 2, the ser ies becomes Huygen's reciprocal triangle series:1+1/3+1/6+1/10+....=2. This series could be modified and extended indefinitely to find the sums of all reciprocal polygonal numbers. [13, p. 43] Before any formal studies of Mathematics, Leibniz had already demonstrated a certain talent for deduction through the summing of this series.
It was after this initial success that Leibniz would make his original trip to London, where among other things he would unveil a prototype of a calculating machine at the Royal Society. Leibniz was eager to be introduced to the Society, which was headed by fellow German Henry Oldenburg, and to learn of the latest developments in mathematics and science. After showing off an unimpressive prototype of his calculating machine, Leibniz met with John Pell. [13, p. 32] Leibniz claimed to have discovered a gen eral method for finding series through the construction of a series of differences. Pell, who was not impressed by the ambitious young German, stated that the same results had already been given in a work by Gabriel Mouton [V, 1, p. 5]. While Pell's asser tion was correct, Leibniz had come to his results by a different method. Nevertheless to avoid further embarrassment, Leibniz wrote a letter to the Royal Society explaining the method in which he made his discoveries. Despite his obvious ambition, Leibniz exposed his ignorance of the mathematics of the day, leaving open the door to later doubts about the originality of his subsequent work. Before leaving London he promised to complete his calculating machine and achieve significant advances in the develop ment of his mathematics.
Once back in Paris, Leibniz would resume his correspondence with Oldenburg and enter into a dialogue with Pell. Over the next few years Leibniz would work contin-uously, and largely in isolation, in developing his mathematics. He read voraciously, absorb ing as much information as possible, while often giving only secondary thought to the question of proofs. This he would reconcile with his own work, creatively improving his own methods through his correspondence with many of the leading mathematicians of the day. Leibniz perfected his techniques, incorporating many ideas in trying to develop a harmonious whole. His goal was truly to establish a congruent totality in his study of Mathematics. Leibniz, after all, came from the rationalist school, in which all the truths of the physical world are the creation of God, and thus essentially perfect. [6, p. 44]
This represents one of the greatest differences in approach between Newton and Leibniz. Newton, who was probably a far more gifted calculator than Leibniz, was essentially an empirical scientist with little interest in the metaphysics of his German count erpart. Leibniz, on the other hand, was a philosopher, and sought to "reconstruct" the universe through pure reason. [6, p. 45] As a result, Leibniz saw mathematics as a potential link between his interests in other fields. Similarly, he hoped his symboli c language would be applicable to all fields of science. Thus, Leibniz's progress in both symbolism and mathematics were linked to each other, and his wish to "reconstruct" the world into a harmonious whole. Leibniz made significant progress in developing his mathematics during his next three years in Paris. For example, he discovered a way to find the moment of any curve, by building on Pascal's method of finding the moment of a circle's quadrant through the use of a "characteristic triangle". It was also through the influence of Pascal's works that Leibniz discovered his transmutation theorem, for which he gave a detailed proof in 1675 [3, p. 47]. Through this breakthrough, Leibniz claimed he had found a me thod to determine the quadratures of virtually all curves. The method was unusual because instead of using rectangles of infinitesimal width, Leibniz incorporated triangles concurrent at a point. The theorem itself was very similar to the process of integ ration by substitution, in that it created an integral of a rational function, which would then be evaluated by expressing the integrand as a series and integrating each term individually. Around the end of 1673 Leibniz believed he had found the "arithmetical quadrature" of the circle through what is the equivalent of
1 -1/3 +1/5 -1/7 +... = p/4 [7, p. 135]. The above formula, a special case of Gregory's slightly earlier arctangent expansion, was the most elegant series for the quadrature of the circle yet produced. [6, p. 53] Leibniz must have been impressed by the series' esthetic appeal, altho ugh the number of terms needed for any decent degree of accuracy rendered it largely impractical. Nevertheless finding a series equal to the area of a unit circle was a significant achievement for Leibniz, who had only begun his serious studies of mathema tics some one and half years earlier. Up to this point Leibniz had proven to be an able student of mathematics, though most of his discoveries had been found already by others. Even though Paris was the academic center of continental Europe, Leibiniz pursued his research in relative isolation and frequently was unaware of recent progress on the problems of interest to him. At that time mathematicians generally concealed their methods of analysis in their letters, since credit for discovery would usually not be given until a work was published . Publishing, however, was a limited option for mathematicians, and thus the work in their letters was often intentionally ambiguous. Although Leibniz was an eager student of mathematics, he probably could have made a greater effort to follow of the progr ess made by his contemporaries. Having been largely self-taught for most of his life, Leibniz was perhaps unaccustomed to paying close attention to the work of others. Furthermore, Leibniz was not a mathematical genius on the order of Newton. He did not h ave the ability to create truly revolutionary work through technical brilliance or powerful calculation. Instead he relied on his ability to simplify complex problems into smaller parts, and to build up a system. He applied his background in logic and phi losophy to creatively reformulate contemporary mathematics into an improved system of notations. Thus Leibniz' main contribution to mathematics would lie in the development of a new notation, which would probably come to be seen as his greatest contribution to mathematics. It was in his private notes from 1675 that Leibniz first introduced the modern symbols for integration and differentiation. [3, p. 74] During the previous two years, he had worked tirelessly on many types of complex problems, including the rectification of curves (arc length) and the problem of inverse tangents (antiderivatives), w hich he had already realized could be reduced to quadratures (area). In a manuscript from 1673 Leibniz writes: "The two questions, the first that of finding the description of the curve from its elements, the second that of finding the figure from the giv en differences, both reduce to the same thing. From this fact it can be taken that almost the whole of the theory of the inverse method of tangents is reducible to quadratures." [3, p. 60] It was during his work on an inverse tangent problem that Leibniz replaced omn, the abbreviation for omnes, with ?, meaning sum. This, he believed, would preserve the "law of homogeneity" and help prevent mistakes. [2, p. 13]Eventually ?y would be replace d by ?y dx. Leibniz recognized that constant factors could be moved before the integral sign, and a sum of integrals equaled the integral of the sum. Subscripts and superscripts were absent because of the general lack of notation for limits, although all integrals at the time were considered definite. Noting that the operation of the summa raised the degree of a function, Leibniz adopted the use of a d in the denominator to represent a lowering of the degree. As a result, ?x=y was also written y=x/d [9, p. 193]. Leibniz later replaced the term x/d with dx, after failing to distinguish between an integrand and base variable in the term ?dx/y, and thus falsely treating it as a logarithm [13, p. 58]. The term dx referred to the infinitely small change or difference in 'x' as one moves along a curve on a C artesian plane. Meanwhile if y was a second variable in a function related to 'x', then dy was any corresponding change in that variable. If dx was constant, then dy defined the slope of the tangent at x. [6, p. 70] Leibniz stated that in a curve defined by some algebraic equation the product of two infinitesimals such as (dx)2 or (dy)2 were small enough to be ignored. From this he demonstrated that the differential of x2 equals 2x dx. Thus the second term of the binomial (x + dx)2 is the "difference" of the first term. Similarly, the second term of any power of the binomial will be the difference of the power. Thus the differential of x3 is 3x2dx [6, p. 71] Leibniz used second derivatives, written ddx, for the determination of "osculating circles", a nd was able to use his calculus with logarithms, a property he claimed made his notation superior to the dots used by Newton [3, p. 42]. He became increasingly aware of the rules and limitations of his new analysis as he developed the notation. In a manuscript from November 1675, for example, Leibniz states a theorem for differentiating a product for any "curves" [9, p. 175]. In 1677 Leibniz again states the rule for the differentiation of products, as well as the rules for the differentiation of Qu otients and Transcendental Functions. [4, XIV] The value of his notation was of supreme importance for Leibniz and for the further development of his analysis. The algebraic symbolism provided him with a sound basis to explore and solve increasingly advanced infinitesimal problems. The new algorithm, as Leibniz himself called it, was a mathematical edifice that provided him a tool vital for accuracy and consistency, while still maintaining a degree of flexibility needed to solve a wide range of equations. [2, p. 13] He also praised the notation's abil ity to free the mind from perpetual reference to diagrams [3, p. 22]. Through his lucid method of presentation Leibniz was able to make rapid progress. He easily found the portion of a circular cone cut off from its base and a plane parallel to the axis o f the cone. This he accomplished after some hours, stating that the solution depended on the quadrature of the circle and that of a hyperbola. He was similarly successful in determining the quadrature of a sector and the tangent of a curve then referred t o as "Bertet's curve". This he did by substituting the circle from which the tangent was generated by an arbitrary curve, and then reducing the quadrature to an integral that could be replaced and solved by an infinite series. [9, p. 197] Although Leibniz was pleased by the ease with which he could work within his system of notatiion, he could not have foreseen how his new symbolism would radically alter mathematics. By eliminating cumbersome rhetoric, mathematicians could now have a commo n apparatus with which they could tackle future problems. Leibniz, for one, could now solve many problems almost effortlessly, and was now well on his way to discovering his calculus. Furthermore, as he developed his algorithm, he created a symbiotic rela tionship that was greatly appreciated by a rationalist philosopher like himself. It seems that there existed a symbiotic relationship between the rate in which he developed his new analysis and the symbolism itself. Newton, on the other hand, remained largely unimpressed by Leibniz's notation. He would even refer to Leibniz's calculus as "more obscure" because of its "less apt notation" [6, p. 72] (in 1712). Although he no doubt exaggerated its deficiencies because of the ongoing priority debate, Newton was in general less receptive to any discussion on notation, which he viewed as trivial. The development of a system of algorithms and such had no significance on his mathematics. Newton, after all, was an empiricis t far more interested in rigorous calculation then the development of an abstract symbolic logic. Indeed because his aptitude for pure mathematics was probably far greater than any of his rivals, he may have found it simply vexing to concentrate on matte rs of notation. His method of analysis was a means to an end in solving a new set of complex problems, and not a part of a greater scientific order as envisioned by Leibniz. This is perhaps one reason that Newton did not publish his work until much later: his focus was on mathematics, and his progress in the field was so rapid, that he perhaps was not afforded the opportunity to consider the utility of a formal calculus, and its many potential applications. Leibniz, on the other hand, would become more aware of the requirements of this analysis, and its notation. He believed that the formalism of the algorithm, which was far more conceptual than any previous notation, should take some of burden of thought off the user [9, p. 299]. Leibniz first published his work in his own journal, the Acta Eroditorum, in 1684, nine years after his discovery. Later, of course, an argument developed between Newton and Leibniz, regarding charges of plagiarism. This debate w ould escalate into the 18th century, influenced as much by nationalistic ambitions, other completely unrelated scientific issues, and bruised egos, as by any underlying issue of plagiarism. Conclusion
After leaving Paris in 1676, Leibniz took up a variety of diplomatic posts and pursued his many intellectural passions in a variety of locations. Mathematics would never again occupy as central a role in his studies as it had during those fruitful years in Paris. Leibniz never treated mathematics as a subject in isolation. Instead he viewed it as a means toward the development of his symbolic language, leading to a better understanding of the universe as a whole. He considered his calculus as a tool for discovery, an apparatus that should be simplified and standardized to be used for as many applications as possible. Newton, on the other hand, had envisioned his calculus to be used mostly for problems of velocity and acceleration in physics. Whereas Newton may simply have been a genius, Leibniz ( to use a clichi ( was a universal genius. His interests were great and varied, and he could not have spent as much time with mathematics as his rivals in England. Nevertheless he continued to keep u p his correspondence with many of Europe's great thinkers. Many of these of course were mathematicians who helped promote his calculus on the continent. Among these were the Bernoulli brothers, who would greatly advance Leibniz's analysis, and succeed in describing the motion of a vibrating string mathematically. Unfortunately the priority debate would increasingly consume Leibniz before his death in 1716. In any case Leibniz has today been largely exonerated from the allegations of plagiarism, with an im plicit cognizance of the individuality of his methods and the superiority of his notation.
While Leibniz lacked the mathematical aptitude of many of his contemporaries, he was able to contribute greatly to the field through his skills in synthesizing information and building up systems. He would continuously improve his own techniques; callin g on his knowledge of other disciplines to best feed his needs in mathematics. In the end Leibniz created simpler and more general methods, which could be harnessed with the help of his algebraic symbolism to greatly elaborate on the new analysis. He beli eved that his improvements would not only benefit techniques in computation, but lead to deeper conceptual penetration. [9, p. 298] His notation was far more versatile than that of Newton, and could be continuously enhanced to deal with increasingly compl ex forms of analysis. While initially used for the solving of inverse tangent problems, the new method quickly developed into a useful tool for all types of higher analysis. This was in essence his contribution to Calculus. As stated by Gerhardt in the pr eface of Der Briefwechsel von Gottfried Wilhelm Leibniz mit Mathematikern (Leibniz' correspondence with mathematicians): Leibniz wies mit Recht darauf hin, dass es sich nicht um das Prinzip der hvheren Analysis handle, das seit Archimedes bekannt war; es kam vielmehr darauf an, f|r dieses Prinzip eine Rechnung aufzustellen. [4, XIX] (Leibniz pointed out, correctly, that the question is not the principle of higher analysis, known since Archimedes, but rather the creation of a Calculus for this principle.) Leibniz had in essence accomplished his life long goal of developing an analytical language, an ingenious edifice unifying disciplines as diverse as calculus' own applications.
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