To explore how learning really occurred in the pages of journals and letters, let us consider a pair of examples. Both of these involve Gottfried Wilhelm Leibniz (1646–1716).

Leibniz had published the famous first paper on calculus in 1684 in the journal *Acta Eruditorum *(which Leibniz himself had founded); it was this paper that ignited the “priority dispute” (see Note 6) between Leibniz and Newton, since Newton had independently invented a set of methods for differential calculus in the 1660s. Jakob Bernoulli initially had some difficulty in understanding Leibniz’s brief work (see Note 7). Many details were abbreviated and unclear; thus, Jakob wrote to Leibniz asking for clarifications but Leibniz was traveling on business and did not receive the letter in time to help. After years of study, Jakob published papers on Leibniz’s methods in 1691. Jakob Bernoulli learned calculus, not from an intensive university program, but from a brief article which he barely understood at first. He strengthened his understanding by teaching. Jakob taught students as the chair of mathematics at the University of Basel, but he also taught his talented brother Johann.

**Figure 8. **Gottfried Wilhelm Leibniz (1646-1716). Painting by Christoph Bernhard Francke (d. 1729). (Wikimedia Commons, public domain)

The second example involves logarithms. Today, we tell students that the logarithm of a negative argument is “not allowed” or “not a real number,” depending upon the level of the course. Before examining this fascinating episode in history, let us consider how we explain this prohibition to students today. If we informally define a logarithm as the inverse of an exponential function, then we can expect students to see that there is no power to which *e* (or any positive real base) can be raised that results in a negative real number. (We will have already restricted exponential functions to positive real bases.) It is unlikely that students will be first introduced to logarithm functions in a calculus course, so the definition of a logarithm as a definite integral of a function such as \(\frac{1}{x}\) would not be the first definition that students see. However, Leibniz and Johann Bernoulli were not looking at logarithms in quite these ways. To understand what Euler would call their “great controversy” over logarithms of negative numbers, we should consider logarithms as expressions (not functions – this concept was not yet current) which are equal to indefinite integrals (or antiderivatives) of certain algebraic expressions. (The fact that these integrals were not themselves algebraic functions was not clear to either Leibniz or Bernoulli.)

The paper written by Euler which decisively resolved the matter appeared in 1749, thirty-three years after the death of Leibniz. During the late 1600s until his death, Leibniz had certainly been the grand old man of mathematics on the continent, and respect for his achievements remained very high in 1749. He and Johann Bernoulli had corresponded concerning the nature of logarithms with negative real arguments. The field of complex analysis did not yet exist, and both Leibniz and Bernoulli tried to reason from the results of real analysis. Euler wrote:

In the exchange of letters between Messrs. Leibnitz and Jean [Johann] Bernoulli, we find a great controversy over the logarithms of negative and imaginary numbers, a controversy which has been treated by both sides with much force, without however, these two illustrious men having fallen into agreement on this matter, although otherwise we note a very perfect harmony between them on all other points of analysis. [5, p. 1; see Note 8]

Here Euler honored his tutor Johann Bernoulli, not merely by calling him “illustrious,” which must have pleased the vainglorious Bernoulli, but by putting him in a class with the great Leibniz. Euler wrote this paper in September 1747, shortly before Johann died in January 1748.

Today, we would instruct students to consider the domain of \(\ln (x)\); more generally, we would expect them to regard it as a function. Leibniz had not used the language of functions, except intuitively in some places; it would in fact be Euler who would introduce the modern function concept. Indeed, even Euler first regarded functions not in the modern way as relations between sets, but in much the way modern students first regard a function, as “an analytic expression composed in any way whatsoever of the variable quantity and the numbers or constant quantities.” [4, p. 17] Therefore, it is not surprising that Leibniz and Bernoulli would have considered the expression \(\ln (-x), x>0,\) as a combination of symbols rather than in a more rigorous analytical way.

Johann Bernoulli, according to Euler’s summary in the letter [5] cited above, considered logs of negative numbers not only to exist, but to be real numbers. He further concluded that \(\ln (x)=\ln (-x).\) He drew this remarkable, counterintuitive conclusion by applying what we would call the chain rule in taking the derivative of \(\ln (-x).\) (Recall again that, at this time, it was common to speak of taking the derivative of an expression, just as our students do, and not to worry about precisely what that meant.) The derivative of \(\ln (x)\) was known to be \(\frac{1}{x},\) so, applying the chain rule, the derivative of \(\ln (-x)\) would seem to be \({\frac{1}{-x}}\times (-1),\) or, again, \(\frac{1}{x}.\) By the (apparent) equality of their derivatives, Bernoulli concluded that the expressions \(\ln (x)\) and \(\ln (-x)\) were identical.

Euler, however, saw what a modern first-year calculus student would have been taught to recognize: that equality of derivatives does not imply equality of functions. He said, “The argument of M. Bernoulli does not prove what it claims to prove” [5, p. 2] and, a little later in the paper, “M. Bernoulli could have proved by the same reasoning that \(l\,2x=l\,1x\)" *– *in modern notation, \(\ln (2x)=\ln (x)\) *–* since their derivatives are equal as well. It is interesting to note here the difference from modern notation. In the seventeenth and eighteenth centuries, standardization of symbols had not gotten very far, and mathematicians tended to use their own notation or one with which they were familiar from correspondence. Even today, especially in calculus, a wide latitude exists in choice of acceptable notation.

Now, Leibniz had also disagreed with Bernoulli’s argument – but for a different reason. He had, again according to Euler, believed that the differentiation rule which is used for logarithms with positive arguments did not apply to logarithms with negative arguments. Euler would have none of this: “…this response, if it were right, would shake the foundation of all analysis, which consists principally in the generality of the rules…” [5, p. 2; see Note 9]. Sharing this sentiment with students and asking for their response to it in essay form is an excellent and historically rich way of encouraging writing in mathematics.

Bernoulli had put forward a second argument for his claim, one which will please instructors who constantly remind students to look at the graph of a function. For the differential equation \(dy =\frac{dx}{x^n},\) when \(n\) is an odd integer and \(n\ge 3,\) the solution curve has, in Euler’s words, “two equal and similar branches.” In modern terms, the curve is symmetric about the \(y\)*–*axis. Bernoulli apparently felt that this should hold for \(n=1\) as well.

Euler recognized that this simply isn’t the case:

When it is a question, in analysis, of the case of integrability … we rarely find propositions that are general enough, and it is nearly always necessary to except one or several cases which do not apply. [5, p. 4]

Bernoulli had been attempting to draw analytical conclusions by considering the shapes of graphs, which the mathematical world had in large part learned from Leibniz. For example, Leibniz’s differential calculus is based on his original and fearless examination of the geometry of Pascal’s “differential triangle.” (For the story behind this, the reader is encouraged to peruse *Leibniz in Paris 1672-1676: His Growth to Mathematical Maturity,* by Joseph H. Hofmann. [7])

Euler explained further,

One will perhaps object that it is nevertheless the surest means to judge the shape of the curve and the number of its branches by its equation …. To which I respond that this method has a place only when the equation for the curve is algebraic or at least conceived in finite terms, and that a differential equation is never appropriate to this design.

He went on to explain, quite lucidly, that the solutions of differential equations are families of curves.

**Note 6. **The Bernoulli brothers, and Johann in particular, not surprisingly defended Leibniz against Newton in the priority dispute; in fact, they made things far worse than the more diplomatic Leibniz and more retiring Newton would have done. Tent wrote: “Johann saw himself as Horatio, bravely defending Leibnizian calculus from the arrogant, misguided English.” [12, p. 95]

**Note 7.** To see the notation and style used by Leibniz, show your students his early works from the remarkable document, *The Early Mathematical Manuscripts of Leibniz,* translated into English by J.M. Child. [2] A bit of Leibniz's notation is highlighted in A Calculus Collection – Leibniz at the *National Curve Bank* website. Leibniz’s three most important calculus papers are available (in Latin) in Mathematical Treasure: Leibniz’s Papers on Calculus, here in *MAA Convergence*.

**Note 8.** For this and the other quotations by Euler in this section, and for classroom use, explore the letters of Leonhard Euler in the Euler Archive maintained by MAA.

**Note 9. **Indeed, Leibniz had a history of being uncertain about integrals which were equal to logarithms; in July 1676 he wrote “Figures of this kind, in which the ordinates are \(\frac{dy}{y},\) \(\frac{dy}{y^2},\) \(\frac{dy}{y^3},\) are to be sought in the same way as I have obtained those whose ordinates are \(y\,dy, y^2\,dy,\) etc.” [2] Like Bernoulli, Leibniz seems to have felt that there ought to be no significant difference between \(\int\frac{dy}{y}\) and \(\int\frac{dy}{y^n},\) \(n\ge 2.\)