The Most Teachable of Mortals

One of Leibniz's first forays into the field of “mathematics” was the creation of a calculating machine in the early 1670's. It was apparently the first machine designed to be capable of multiplication, division, and root extractions, and although it didn’t actually function very well in practice, the design was impressive enough to gain Leibniz election to the Royal Society of London in 1673. He later (in a 1685 manuscript) gave the following account of the moment of inspiration for this invention:

When, several years ago, I saw for the first time an instrument which, when carried, automatically records the number of steps taken by a pedestrian, it occurred to me at once that the entire arithmetic could be subjected to a similar kind of machinery so that not only counting, but also addition and subtraction, multiplication and division could be accomplished by a suitably arranged machine easily, promptly, and with sure results… The calculating box of Pascal was not known to me at that time...  When I noticed, however, the mere name of a calculating machine in the preface of his "posthumous thoughts"...I immediately inquired about it in a letter to a Parisian friend. When I learned from him that such a machine exists I requested the most distinguished Carcavius by letter to give me an explanation of the work which it is capable of performing.

This pattern of discovery seems to have been characteristic of Leibniz. He was a sponge for knowledge and ideas, and he was tireless in ferreting it out of friends, acquaintances, and strangers alike. He tells us that, once he had gotten all the information he could about Pascal's machine, he set himself the task of making an even better machine. However, notice that he begins the discussion by recalling the earlier moment of inspiration when he saw the step-counting machine, and how "it occurred to me at once that the entire arithmetic could be subjected to a similar kind of machinery". In other words, this was the moment when his invention of the calculating machine actually occurred, although he didn't act on the inspiration at the time. It was only later, when he saw the words "calculating machine" in some papers of Pascal, that he was prompted to actually design such a machine - after studying the design and capabilities of Pascal's machine.

There are some interesting parallels between this sequence of events and Leibniz's development of the calculus. In January of 1673, soon after taking up the serious study of mathematics (enlisting Christian Huygens as a tutor!), he visited London and made the acquaintance of Heinrich (Henry) Oldenburg, the secretary of the Royal Society of London. In April of that year Leibniz received from Oldenburg a report drawn up by John Collins on the state of mathematics in England. Among the methodical notes that Leibniz made on this material at the time was the remark

Tangents to figures of all kinds. Development of geometrical figures by the motion of a point in a moving straight line.

This could well be a reference to the work of Isaac Barrow (Newton’s teacher and predecessor as Lucas Ian Professor of Mathematics at Cambridge). Leibniz bought a copy of Barrow’s book during his stay in London that winter, and in that book is to be found (albeit in geometrical garb) most of the basic rules for differentiation and integration, as well as indications of the reciprocal nature of these operations. It is often said that Newton and Leibniz were both greatly indebted to Barrow for the fundamental ideas involved in calculus. Over the next few years Leibniz continued to receive reports on developments in mathematics in England, among which was a list of problems (many involving infinite series) that could be solved by an unspecified method possessed by a secretive man at Cambridge named Isaac Newton. As Leibniz later remembered it, he himself had already had the original inspiration for calculus prior to seeing any of the reports on the work of Barrow, Gregory, and Newton, and there was no actual description of calculus in the 1673 report. Nevertheless, as with the appearance of the "mere name" of a calculating machine in Pascal's papers, it's possible to imagine that the mere mention of a wonderful general method for handling a wide range of mathematical problems of this kind was enough to perk up Leibniz's ears and set him working in earnest on the task of developing his own ideas for the calculus.

By 1675 (or so) he had a method that was, for practical purposes, at least the equal of Newton's fluxions (which Newton, building on the work of Isaac Barrow, had developed in the mid-1660's), although his conceptual justification for it was perhaps less sophisticated than Newton's. In that year, and on into 1676, there was an exchange of letters between Newton and Leibniz (via Oldenburg) in which Newton, although still very reticent about revealing general methods, gave even more explicit hints of calculus, including even an anagram stating the inverse relationship between differentiation and integration. At this point it's clear that Leibniz was already in possession of his own version of calculus. Thus the two men, each having made the same great discovery, neither having published on the subject, were corresponding with each other, and we can imagine that they were each trying to figure out exactly how much the other knew, without revealing too much of what he himself knew. These letters became infamous during the subsequent priority dispute, but it was the Collins report of 1673 (analogous to the mere mention of the words "calculating machine") and Barrow’s book that  represent the most significant influence of British mathematicians on Leibniz's discovery of calculus. The Bernoulli brothers later commented on Leibniz’s indebtedness to Barrow, and Leibniz drafted a reply, which began

Perhaps you will think it small-minded of me that I should be irritated with you, your brother, or any one else, if you should have perceived the opportunities for obligation to Barrow, which it was not necessary for me, his contemporary in these discoveries to have obtained from him.

In this letter Leibniz goes on to give an interesting account of his early mathematical development. He mentions the help he received from Huygens, “who I fully believe saw more in me than there really was”. Huygens gave him a copy of his recently published book on the pendulum, and discussed its contents with him.

This was for me the beginning or occasion of a more careful study of geometry. While we conversed, he perceived that I had not a correct notion of the center of gravity, and so he briefly described it to me; at the same time he added the information that Dettonville [the pseudonym used by Pascal] had worked such things out uncommonly well. Now I, who always had the peculiarity that I was the most teachable of mortals, often cast aside innumerable meditations of mine that were not brought to maturity, when as it were they were swallowed up in the light shed upon them by a few words from some great man, immediately to grasp with avidity the teachings of a mathematician of the highest class; for I quickly saw how great was Huygens. In addition there was the stimulus of shame, in that I appeared to be ignorant with regard to such matters. So I sought a Dettonville from Buotius, a Gregory St. Vincent from the Royal Library, and started to study geometry in earnest.

He might have later regretted characterizing himself as “teachable”, since it seems to support the (unjust) impression that he was strictly a borrower of ideas. Without doubt Leibniz was an enormously creative and brilliant man in his own right, and his ideas ranging from the artificial computer (including the binary number system) to the development of the calculus were immensely important. (It would be difficult to over-state the brilliance of his notation for calculus, and it was surely the good fortune of that subject to have as one of its founders one of the greatest proponents of felicitous notations.) Toward the end of his 1685 computer manuscript, with the characteristically Leibnizian title "Machina arithmetica in qua non additio tantum et subtractio sed et multiplicatio nullo, divisio vero paene nullo animi labore peragantur" he wrote about the value of the computer:

...the astronomers surely will not have to continue to exercise the patience which is required for computation. It is this that deters them from computing or correcting tables, from the construction of Ephemerides, from working on hypotheses, and from discussions of observations with each other. For it is unworthy of excellent men to lose hours like slaves in the labour of calculation which could safely be relegated to anyone else if machines were used.

It's interesting that one of the persistent themes in all of Leibniz's work was the idea of reducing thought processes to automatic and/or mechanical operations that could "safely be relegated to anyone". This was true not only in his development of the computer, but also in his reduction of calculus to a set of straightforward algorithms on a well-defined set of symbols. Likewise his work on the "universal characteristic", which we would now call symbolic logic, was intended to reduce all thought to a set of definite rules, or, as he wrote

...a general method in which all truths of reason would be reduced to a kind of calculation. At the same time, this would be a sort of universal language or script, but infinitely different from all those imagined previously, because its symbols and words would direct the reason, and errors - except those of fact - would be mere mistakes in calculation...

Leibniz was, after all, originally trained as a lawyer, so it may be understandable that he yearned for some automatic way of settling disputes. The youthful optimism of Leibniz in this regard was later satirized by Voltaire in the play Candide ("Come, let us calculate"). On the other hand, is it too farfetched to see in Leibniz's view that we inhabit the "best of all possible worlds" some hint of Maupertius' principle of least action?

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