Euler and the Bernoullis: Learning by Teaching

Paul Bedard (Saint Clair County Community College)


An instructor of mathematics reflects on what we can learn about the teaching and learning of mathematics in classrooms today from Leonhard Euler and three mathematical Bernoullis, Jakob, Johann, and Daniel.


Euler and the Bernoullis: Learning by Teaching - Introduction

Paul Bedard (Saint Clair County Community College)


I soon found an opportunity to be introduced to a famous professor Johann Bernoulli... True, he was very busy and so refused flatly to give me private lessons; but he gave me much more valuable advice to start reading more difficult mathematical books on my own and to study them as diligently as I could; if I came across some obstacle or difficulty, I was given permission to visit him freely every Sunday afternoon and he kindly explained to me everything I could not understand … and this, undoubtedly, is the best method to succeed in mathematical subjects. - Leonard Euler [13, p. 90]

Teaching and learning are part of the same process.  A common refrain among teachers of mathematics is, “I never really understood this material until I taught it.”  Part of the reason for this, of course, is that the repetition and the need to know the material well enough to answer questions solidifies the subject in the mind of the teacher. Another reason is the need to find ways to recast material to make it accessible to learners – to meet them where they are in the process of discovery.  Today, the best way to learn mathematics is to teach it.  And the best way to teach it is to explore with students how the original innovators learned and taught mathematics by investigating its rich history.  This was just as true when modern analysis was getting its start.  In the exciting days of the late seventeenth and early eighteenth centuries, after Gottfried Wilhelm Leibniz had opened the floodgates with a short, difficult article on calculus in his own journal, Acta Eruditorum, two generations of Bernoullis and two generations of Eulers were discovering, and teaching each other, the mathematics that made the modern age possible.  Through formal and informal tutoring, by way of letters and casual discussion, in the absence of input from universities, they encouraged, challenged, instructed, refuted, and inspired each other.

Figure 1. Title page of the 1684 volume of Acta Eruditorum, the journal Leibniz created, published, and made famous with his first work on calculus, which appeared in this volume. (Courtesy of the Lilly Library, Indiana University. You may use this image in your classroom; all other uses require permission from the Lilly Library. See Leibniz’s three most important papers on calculus in their entirety in Mathematical Treasure: Leibniz’s Papers on Calculus here in MAA Convergence.)

Jakob Bernoulli was the tutor of Paul Euler, the father and tutor of Leonhard Euler (see Note 1). Jakob also taught his younger brother Johann Bernoulli. Johann, as noted above, tutored Euler (see Note 2).  Euler in turn befriended and worked with Daniel Bernoulli, the son of Johann. Both Jakob and Johann Bernoulli had learned calculus by reading and corresponding with Leibniz.  In a way, Daniel Bernoulli closed a loop when he abandoned his father’s patriotic dislike of Newton and studied both Newtonian and Leibnizian calculus.

In our classrooms today, do we teach using “the best method to succeed,” according to Euler?  Do we encourage our students to discover obstacles on their own, and bring them to us?  Or do we warn the students of these obstacles beforehand, depriving them of much of the challenge?

Note 1. “I received from my father my first tuition; and because he had been one of the disciples of the world-renowned Jacob Bernoulli, he tried to impart to me the first principles of mathematics, and to this end used Christoph Rudolph’s Coss.” This was the same text Jakob Bernoulli himself had started with! [6, p. 5]

Note 2. In fact, as the opening quotation indicates, Bernoulli refused to make this a formal arrangement because the Eulers could not pay him. Johann’s tutelage consisted of informal discussions and recommendations for reading. See the quote at the beginning of this section. Compare to this passage from John-Jacques Rousseau’s Emile:There is much discussion as to the characteristics of a good tutor. My first requirement, and it implies a good many more, is that he should not take up his task for reward. There are callings so great that they cannot be undertaken for money without showing our unfitness for them; such callings are those of the soldier and the teacher.” [8]

Euler and the Bernoullis: Learning by Teaching - Jakob Bernoulli

Paul Bedard (Saint Clair County Community College)

The Bernoulli family in the eighteenth century produced an astounding number of topflight mathematical minds.  The first of these was Jakob Bernoulli (1654-1705).  Born in Basel in December of 1654, nearly 13 years before his brother Johann, the taste for mathematics was aroused in Jakob from a very young age.

Figure 2. Jakob Bernoulli (1654-1705) around 1687 (Wikimedia Commons, public domain), together with the title page of his best-known work, Ars Conjectandi, published in 1713 (courtesy of Butler Rare Book and Manuscript Library, Columbia University Libraries. You may use this image in your classroom; all other uses require permission from Columbia University Libraries. See Jacob Bernoulli's Ars Conjectandi here in MAA Convergence for more information.)

The first text Jakob studied was Die Coss (1525), the first algebra book written in German, by Christoff Rudolph.  Even today, dictionaries define “cossic” as “relating to algebra.”  The term is derived from the Italian cosa or thing, as in ‘unknown quantity.’  Scholars of algebra were called “cossists.”  Jakob attended the University of Basel, but he learned very little mathematics there.  He earned an M.A. at Basel in 1671 at age 16, “having followed a rather classical and humanistic curriculum, strong on languages but basically Aristotelian in its approach to science.” [11, pp. 4-5]  His father, Nicolaus Bernoulli (1623-1708), wanted him to enter the ministry in the Reformed Church, as did Leonhard Euler’s.  Switzerland, the home of John Calvin (1509-1564), was a stronghold of Protestant Christianity, and careers in the ministry were a common goal.  In spite of Nicolaus’s wish, after obtaining degrees in philosophy and theology at Basel, Jakob wanted to travel and to continue to pursue mathematics.  He chose a motto for himself: “Invito patre sidera verso,” or, “Despite my father, I turn to the stars.” [11, p. 7]

Figure 3. View of Basel, Switzerland, and the Rhine River around 1648, copper-plate engraving by Matthäus Merian the Elder (1593-1650) (Wikimedia Commons, public domain) 

To learn more about mathematics than was possible at the University, Jakob traveled to Geneva.  However, before he found a tutor, he became one.  He entered the employ of the Waldkirch family in 1676 as tutor to the young, blind Elizabeth Waldkirch.  His task was to help her learn to read and write – not a common accomplishment for the blind at that time.  He continued in this occupation until 1678.  M.B.W. Tent suggested, in her fictionalized account of the lives of Euler and the Bernoullis, [12] that the elder Waldkirch wanted someone trained in mathematics, since he had learned that the mathematician Girolamo Cardano (1501-1576) had been involved in teaching literacy to the blind.  It is worth noting that it is still true today that many professions seek mathematically trained candidates or use mathematics tests for eligibility, not because the job requires the specific skills involved, but because the assumption is that minds that can grasp mathematics are disciplined and sharp.  This is a fact which the author shares with his students regularly.

What skills as a teacher might Jakob Bernoulli have gained from this experience?  There is a certain poetry in the idea that the man who would bring light to so much that was dark in mathematics began his teaching career by alleviating the disadvantages of physical blindness.  If Tent was correct that he obtained this opportunity due to being a mathematician, then there is a second level of unexpected appropriateness here. 

The early tutoring experiences of Jakob Bernoulli suggested to the author an at-home activity to assign our students.  Rather than merely requiring the students to solve problems, ask them to find volunteers and teach the volunteers how to solve the problems.  Each student will write a brief log entry of how the process goes, what explanations worked or failed, and how her “student” responded.  Even if the person “tutored” in this way is unprepared for this level of mathematics, his response to it may be instructive for, or resonate with, our students.

Euler and the Bernoullis: Learning by Teaching - Johann Bernoulli and Leonhard Euler

Paul Bedard (Saint Clair County Community College)

The first and best tutor the younger Johann Bernoulli (1667-1748) had was likely his brother Jakob.  It is well known of the brothers that they had a stormy relationship filled with bitterness and accusations, but this would come later. Of Johann, William Dunham wrote

Two facts about (Johann) Bernoulli should be noted.  First, he was a proud and arrogant man, as quick to demean the work of others as to praise that of himself.  Second, any such praise was probably deserved. [4, p. xx]

It must have been Jakob who ignited a passion for mathematical learning in his brother (he would certainly later claim credit for many of his brother’s achievements).  As René Descartes wrote, “If one has any sort of skill, I can think of nothing for which a taste, a very passion, cannot be aroused in children,” [3] and Jakob was certainly passionate about mathematics.

Figure 4.  Painting of Johann Bernoulli (1667–1748) by Johann Rudolf Huber (1668-1748) around 1742 (Wikimedia Commons, public domain)

From his early twenties Johann discussed and worked with his older brother on mathematics; we know this from letters which they exchanged, as discussed in [9] and elsewhere.  Just as his father had wanted Jakob to make his career in the ministry, Nicolaus intended his youngest son for a career in business.  Johann would have none of this.  He intended to follow in his brother’s footsteps.

Figure 5.  Painting of Leonhard Euler (1707–1783) by Jakob Emanuel Handmann (1718-1781) around 1756 (Wikimedia Commons, public domain. See also Leonhard Euler – Image Gallery in the MAA’s Euler Archive.)

Leonhard Euler (1707–1783) was a contemporary and countryman of Jean Jacques Rousseau (1712–1778).  Euler was born in Basel in 1707, five years earlier than Rousseau and 252 kilometers from Rousseau’s birthplace of Geneva; both Basel and Geneva were city-states in the Swiss Confederacy.  This was a time in the history of Europe, the dawning of the Enlightenment, when the world was beginning to be viewed in terms of mathematics (see Note 3).  As René Descartes (1596–1650), the source of the fundamental ideas upon which Euler and the Bernoullis were to expand so significantly, affirmed in his work, Meditations,

I reckoned among the number of the most certain truths those I clearly conceived relating to figures, numbers, and other matters that pertain to arithmetic and geometry, and in general to the pure mathematics. [3]

In his seminal work, Émile, Rousseau said of education,

We are born weak, we need strength; helpless, we need aid; foolish, we need reason. All that we lack at birth, all that we need when we come to man's estate, is the gift of     education. This education comes to us from nature, from men, or from things. The inner growth of our organs and faculties is the education of nature, the use we learn to make of this growth is the education of men, what we gain by our experience of our surroundings is the education of things.  Thus we are each taught by three masters. If their teaching conflicts, the scholar is ill-educated and will never be at peace with himself; if their teaching agrees, he goes straight to his goal, he lives at peace with himself, he is well-educated. [8]

As indicated at the start of this article, the use Euler learned to make of what nature had provided was gained in large part from Johann Bernoulli, the brother of Jakob and father of Daniel.

Note 3. “And with regard to the ideas of corporeal objects, I never discovered in them anything so great or excellent which I myself did not appear capable of originating; for, by considering these ideas closely and scrutinizing them individually, in the same way that I yesterday examined the idea of wax, I find that there is but little in them that is clearly and distinctly perceived. As belonging to the class of things that are clearly apprehended, I recognize the following, viz., magnitude or extension in length, breadth, and depth; figure, which results from the termination of extension; situation, which bodies of diverse figures preserve with reference to each other; and motion or the change of situation; to which may be added substance, duration, and number.” [3]

Euler and the Bernoullis: Learning by Teaching - Daniel Bernoulli and Euler

Paul Bedard (Saint Clair County Community College)

Little of the great learning these formative geniuses of mathematics required was done in a formal university environment.  Instead, they learned from each other, through discussions, letters, reading and criticizing each other’s work, and posing problems for each other to solve.

The author has used this technique to great effect in classes at all levels.  Given a problem and a historical figure associated with it, first, present a brief introduction to the historical figure, be it Bernoulli (any one of them), Newton, Leibniz, Euler, or someone else.  Next, propose whatever version of the problem is appropriate to the level of the students – in the original words if possible.  Students have a right to know that the teaching and learning of mathematics has a history, too.  Seeing the strengths and the weaknesses of older notations, less precise and wordier formulations, and the sort of overgeneralizations to which these early masters of the art were sometimes prone, can bring modern students to a deeper appreciation of the subject as a living, growing, and changing entity.  Finally, challenge the students to address the problem at whatever level is appropriate.  

Figure 6. Daniel Bernoulli (1700–1782), mezzotint engraving by Johann Jakob Haid (1704-1767), after a painting by Johann Rudolf Huber (Wikimedia Commons, public domain)

For mathematicians in the eighteenth century, encouragement, instruction, correction, ‘social networking,’ and career development all happened in the pages of letters. The letters exchanged by the Bernoullis and Euler were sometimes written in the lingua franca of science, Latin, or the true lingua franca, French, and sometimes in German. Such encouragement was vital; other than the Royal Society in England and its pallid copy, founded by Leibniz, the Berlin Society of Sciences, there were no organizations, no Nobel committees, no established network of peers to evaluate and encourage.  A brilliant mathematician like Euler had to take poorly paid positions.  Consider Daniel Bernoulli’s invitation to Euler to join him in Saint Petersburg in 1726:

It is several months since I have written you by order of our President … and invited you in his name to come and take a post in our Academy with a stipend of 200 rubles; I fully understand that this amount is beneath your merit. [1; see Note 4]

This passage of a letter written by Daniel to Euler shows both the efforts Daniel expended to find a post for Euler and the paltry expectations a mathematical genius of that day had for worldly recognition and wealth; perhaps little has changed in this regard since that time.  In the same letter Bernoulli encouraged Euler to toot his own horn:

While waiting [to come to St. Petersburg] don’t forget to send the Academy some of your work, and make them aware by reading it that as well as I have spoken of you, I have not said enough. [1; see Note 5]

Figure 7.  St. Petersburg around 1820 by Angelo Toselli (1762-1839). This is one of ten works making up Toselli’s 1820 “Panorama of St. Petersburg.” (Wikimedia Commons, public domain)

If a particularly enthusiastic student asks a probing question, have her write it up and “mail” it to all the other students, by email or in some other way. This will involve students in the exciting sort of exchange that so inspired Leibniz, Newton, and especially the Bernoullis.

Note 4. Il y a quelque mois que je vous ecrivis par ordre de notre President…et que je vous invitai en son nom de venir prendre la place d’Eleve dans notre Academie avec 200 roubles de pension; je savois fort bien que ce salaire etoit au-dessous de votre merite. [1]

Note 5. En attendant ne manquez pas d’envoyer a l’Academie au plus tot quelque piece de votre facon, et faites lui voir par la que, quelque bien que j’aye dit vous, je n’en ai pas encore assez dit. [1]

Euler and the Bernoullis: Learning by Teaching - Leibniz, the Bernoullis, and Euler

Paul Bedard (Saint Clair County Community College)

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 \(\int\frac{dy}{y^n},\) \(n\ge 2.\)

Euler and the Bernoullis: Learning by Teaching - Teaching the Way Euler and the Bernoullis Learned

Paul Bedard (Saint Clair County Community College)

The foregoing lengthy example of teaching and learning is one the author has used with much success in both precalculus and calculus classes.  First, it is necessary to decide how deeply into the subject to go.  For a precalculus class, obviously the differentiation arguments advanced by Bernoulli would not be appropriate.  But the symmetry arguments are exactly right for such a class (providing that the differential equations can be eliminated or explained in some other manner).  If the historical context, with intriguing asides on personalities and some historical context are first provided, the topic comes to life.  The immediacy and fascination which we wish to inspire in students can be achieved more easily with an honest revelation of the uncertainties, disputes, and corrections of the innovators of the field. Indeed, it seems unlikely that students will ever be inspired to become innovators who think for themselves if mathematics is always presented to them as a brilliant and elegant fait accompli.

Consider how the questions asked by students today recapitulate the history of mathematical discovery far better than the textbooks generally do.  We do a disservice to students when we fail to inform them that the very same questions troubled the great minds who discovered the answers.  For example, when first presented with the concept of negative quantities, students often respond with incredulity or skepticism:  “How can a number be negative?  What can this mean?”  Though Girolamo Cardano (1501–1576) in 1545 laid out the rules for negative numbers, it would be more than a century before coefficients in equations were explicitly permitted to take on negative values, in the work of Jan Hudde (1628–1704) in 1713, in De reduction aequationem. [10]  (Bernoulli met Hudde in 1681; see Note 10.)  Finding ways to answer these questions is just as important to the development of the mathematical intuition of the teacher as it is to the growth of the learner. 

The questions faced by Leibniz, by Jakob and Johann Bernoulli, and by Leonhard Euler were the same basic questions students face today: 

  • What are these objects we are investigating?  
  • What rules do they follow, and how general are those rules?  
  • What are the underlying principles that unite these investigations?  

They found answers through discussion in journals and letters, and through personal, one-on-one tutelage.  In so doing they established mathematical analysis and set it on a firmer foundation. 

Few students today will be lucky enough to find tutors like Johann Bernoulli, and we would like to boast that university education in mathematics has improved tremendously.  By examining not just the conclusions of the formative minds of mathematics, but how these were reached, might inspire students today not just to read mathematics but also to discuss it, argue about it, and be inspired by it.

Note 10. M.B.W. Tent imagined a discussion between a young Jakob Bernoulli and Hudde, in which Hudde used the tried and true example of credits and debts to show that negative quantities are just as ‘real’ in applications as positive ones.  [12, pp. 61–63]

Euler and the Bernoullis: Learning by Teaching - References and About the Author

Paul Bedard (Saint Clair County Community College)


  1. Bernoulli, Daniel. “Lettre I to Euler, 1726,” Letter number 0090, in The Euler Archive: A Digital Library Dedicated to the Work and Life of Leonhard Euler. Mathematical Association of America (MAA), n.d. Web. Accessed 16 Feb. 2015:  Translated by the author.
  2. Child, J.M.  The Early Mathematical Manuscripts of Leibniz.  Chicago: The Open Court Publishing Company, 1920. Print.  Available from the Hathi Trust Digital Library via permanent url: 
  3. Descartes, Rene. "Meditations on First Philosophy in which are demonstrated the existence of God and the distinction between the human soul and body," in Some Texts from Early Modern Philosophy. 2015. Web. Accessed 16 Feb. 2015:
  4. Dunham, William. Euler: The Master of Us All.  Washington, D.C.: Mathematical Association of America (MAA), 1999. Print.
  5. Euler, Leonhard. “On the Logarithms of Negative and Imaginary Numbers” The Euler Archive, MAA, n.d. Web. Accessed 16 Feb. 2015:  Translation by Todd Doucet (2005) from the French, “Sur les logarithmes des nombres negatifs et imaginaires,” Actor. Acad. Berolinensis tomo V. A. 1749, p. 139. Eneström Number 807
  6. Fellmann, Emil A.  Leonhard Euler. New York: Springer Verlag, 2006.  Print.
  7. Hoffman, Joseph H., Leibniz in Paris 1672-1676: His Growth to Mathematical Maturity. New York: Cambridge University Press, 1974. Print.
  8. Rousseau, Jean-Jacques. Emile. 1762. N.p.: n.p., 2011. Project Gutenberg. Web. Accessed 16 Feb. 2015:
  9. O’Connor, J.J. and E.F. Robertson. “Jacob (Jacques) Bernoulli,” in MacTutor History of Mathematics Archive, 1998:
  10. O’Connor, J.J. and E.F. Robertson. “Johann van Waveren Hudde,” in MacTutor History of Mathematics Archive, 2008:
  11. Sylla, Edith Dudley (translator). The Art of Conjecturing, by Jakob Bernoulli. Johns Hopkins University Press, 2006. Print.
  12. Tent, M.B.W. Leonhard Euler and the Bernoullis: Mathematicians from Basel. Nattick, MA: A.K. Peters, 2009. Print.
  13. Thiele, Rudiger. “The Mathematics and Science Of Leonhard Euler (1707–1783),” in Mathematics and the Historian’s Craft: The Kenneth O. May Lectures (Glen Van Brummelen and Michael Kinyon, eds.). New York: Springer, 2005. Print. 

About the Author

Paul Bedard teaches mathematics at Saint Clair County Community College in Port Huron, Michigan. He received his BS in mathematics from the University of Detroit in 1992 and his MA in Mathematics from Wayne State University in 1995. His article on Leibniz, “Leibniz and the Nature of Innovation,” appeared in the MathAMATYC Educator, volume 5, number 3, May 2014. Fascinated by the many ways that mathematics history and the geniuses who comprised it can be used in the classroom to enrich and add value to any area, Bedard enjoys debating the calculus priority question and, predictably, favors Leibniz.