The Work of Leonhard Euler and the Bernoulli Family Ensured the Success of Newtonian Mechanics

In my previous post Pierre Louis Maupertuis and the Principle of Least Action, I wrote about the letter that Leonhard Euler (1707-1783) penned in defence of Pierre Louis Maupertuis (1698-1759), against Johann Samuel König (1712-1757), accusing the latter of having forged a letter written in 1707 by Gottfried Wilhelm Leibniz (1646-1716) to Jacob Hermann (1678-1733). Although that specific episode does not present Euler in a positive light, it should be understood that for much of the 18th century, he was Europe’s leading mathematician.

I concluded in that post that I would write about the assessment by Adolf Kesner (1862-1930) in his Das Prinzip der Kleinsten Wirkung von Leibniz bis zur Gegenwart¹ [The Principle of Least Action from Leibniz to the Present] that the Principle of Least Action is directly derivable from the teleology of Leibniz. However, as I started to prepare that post, I realized that I could not write it until I had at least given a (very) cursory overview of 18th century physics and mathematics as a whole.

Cover page of Truesdell’s collection of essays

A good introduction to this difficulty can be found in an essay written by Clifford Ambrose Truesdell III (1919-2000), “A Program toward Rediscovering the Rational Mechanics of the Age of Reason”², which covers the century leading from the publication in 1687 by Isaac Newton (1643-1727) of his Philosophiæ Naturalis Principia Mathematica, or Principia for short, to the publication in 1788 by Joseph-Louis Lagrange (1736-1813) of his Méchanique analitique [Analytical Mechanics].

An example of the problem studying this century can be seen in the following paragraph, in which Truesdell is focusing on the study of the three-body problem by Newton in the second half of Book I of the Principia. Truesdell writes that it took more than half a century to go from Newton’s work until “Newton’s equations” were first published by Euler:

But surely the masterpiece of Book I is NEWTON’S treatment of the problem of three bodies. Here the modern scientist who does not know NEWTON’S work at first hand must be initiated into the facts. While he may regard NEWTON’S laws as equivalent to the differential equations called “NEWTON’S equations” in modern textbooks, there is no evidence that NEWTON himself thought of or ever used his principles in any general mathematical form. Earlier in Book I, the problem of two bodies is skilfully reduced to an equivalent problem of one body attracted to a fixed center. Problems of this kind NEWTON could indeed formulate by means of differential equations, expressed in his usual style in terms of components tangent and normal to the path, and then solve. But the three-body problem cannot be so reduced. For it, not only does NEWTON give no solution or approximate solution in the modern sense, but also he shows no sign of any attempt even to set up equations of motion. That he obtained some correct inequalities and groped his way to the major approximate results under this handicap is one more tribute to his peerless grasp of the physical essence of mechanics and to the might of his brain. It does not show, however, that his formulation of the general laws of mechanics was adequate. History proves the contrary. It was not so much additional mathematics that was needed to get further, for NEWTON was a master at approximate solutions, quadratures, and series expansions in definitely set mathematical problems. Rather, the fact that fifty years passed before any improvement over his results on the three-body problem was made shows that he himself had gotten the most out of the subject that his own methods and concepts could produce; to wrest so much from so primitive a formulation of mechanics required the genius of a NEWTON; none of his disciples, who might reasonably have been expected to build on his foundation, could raise the structure an inch higher. Before the next real advance, a half century of abstraction, precision, and generalization of Newtonian concepts was necessary. The first to go substantially beyond NEWTON in the three-body problem was the man who found out how to set up mechanical problems once and for all as definite mathematical problems, and this man was EULER. The year in which the “Newtonian equations’ for celestial mechanics were first published is not 1687 but 1749, as we shall see. [p.90, my emphasis]

Leonhard Euler

So where can we find out about the work of Euler? Well, it is a daunting task: he is by far the most prolific mathematician in history. In 2023, the last of 81 volumes of the Leonhardi Euleri Opera omnia³ [Collected Works of Leonhard Euler] was published; the project had been initiated in 1908! Six of these volumes were edited by Truesdell. And there is in fact further correspondence that will only be published online. A huge proportion of the mathematical notation—as well as mathematical techniques—that we use today was introduced by none other than Euler. There is also a massive biography of Euler by Ronald S. Calinger.⁴

So what exactly did Euler publish around 1749? Precisely those equations that all of us learnt in high school to be Newton’s equation, F = M a, i.e., force = mass times acceleration! Here is Truesdell:

EULER himself was not yet so far along in 1747. He had reached clarity on discrete systems, but for solid or fluid media he still failed of a general and unifying method. After experience using a similar approach to the vibrating string and to hydraulics, in 1750 he finally saw that the principle of linear momentum applied to mechanical systems of all kinds, whether discrete or continuous. His paper called “Discovery of a new principle of mechanics”, published in 1752, presents the equations

F_x = M a_x, F_y = M a_y, F_z = M a_z,

where the mass M may be either finite or infinitesimal, as the axioms which “include all the laws of mechanics.” Later he called them “the first principles of mechanics”. These are the famous “Newtonian equations”, here proposed for the first time as general, explicit equations for mechanical problems of all kinds. The discovery of this principle seems so easy, from the Newtonian ideas, that it has never been attributed to anyone but NEWTON; such is the universal ignorance of the true history of mechanics. It is an incontestable fact that more than sixty years of research using more complicated methods even for rather simple problems took place before this “new principle” was seen. [pp.116-117]

So what about Book II of the Principia? Truesdell writes that this is Newton’s most exploratory work, and often wrong:

While NEWTON’S Book I is the achievement of KEPLER’S program, NEWTON had a program of his own. Bodies on the earth, as any Aristotelian physicist was ready to say, did not in actual experiments obey GALILEO’S orders, and the ideal medium or vacuum in which those proclamations are valid exactly does not, as the Aristotelian was all the readier to say, exist anywhere on this old, practical, physical earth. Seeing clearly that to justify neglect of friction one must first estimate its effects precisely, NEWTON set himself the problem of determining mathematically the nature of motion in resisting media. This, with some excursions, is the subject of Book II.

To make any solid study of fluid resistance, NEWTON had first to learn the laws of fluid motion. Here he had no foregoing pathfinder; here his program of mathematical deduction broke down. Book II is almost entirely original, and much of it is false. New hypotheses start up at every block; concealed assumptions are employed freely, and the stated assumptions sometimes are not used at all. Little from Book II has found its way into either texts or histories; what has, is often misrepresented. To create theory for the spring of air, for the flow of fluid from an orifice, for the progress of waves in water, for the oscillation of water in a tube, for the resistance suffered by bodies in rare or dense fluids, for the propagation of sound in air, for the internal friction of fluids, NEWTON created new concepts. That few of his concepts have been retained and few of his solutions are right is less remarkable, considering the entire freshness of the subjects, than that he should have been able, in each case, to grope or beat his way to some definite answer. The brilliantly ingenious but, in the end, largely unsatisfactory Book II laid out the areas and defined the problems for many of the mechanical researches of the next century. [pp.90-91, my emphasis]

So who took up the gauntlet of developing the ideas in Book II of the Principia?

Jacob, Johann and Daniel Bernoulli

Truesdell continues looking at Book II in another essay, “Reactions of Late Baroque Mechanics to Success, Conjecture, Error, and Failure in Newton's Principia”⁵. Therein, he focuses on the fact that all of the loose ends left by Newton were picked up by the Basel school of mathematicians, whose most prominent members were Euler and three members of the Bernoulli family, Jacob I (1654-1705), Johann I (1667-1748), and Daniel (1700-1782). Truesdell writes:

It was these two flimsy and unmathematical props, the “cataract” and the “solid particles of the air”, no better than the vortices of the Cartesians, that drew the strongest criticism upon the Principia. However successful was Book I, Book II was a failure as an essay toward a unified, mathematical mechanics.

With its bewildering alternation of mathematical proof, brilliant hypothesis, pure guessing, bluff, and plain error, Book II has long been praised, in whole or in part, and praised justly, as affording the greatest signs of NEWTON’S genius. To the geometers of the day, it offered an immediate challenge: to correct the errors, to replace the guesswork by clear hypotheses, to embed the hypotheses at their just stations in a rational mechanics, to brush away the bluff by mathematical proof, to create new concepts so as to succeed where NEWTON had failed. It is not an exaggeration to say that rational mechanics, and hence mathematical physics as a whole and the general picture of nature accepted today, grew from this challenge as it was accepted by the Basel school of mathematicians: the three great BERNOULLIS and EULER, on the basis of whose work LAGRANGE, FOURIER, POISSON, NAVIER, CAUCHY, GREEN, STOKES, KELVIN, HELMHOLTZ, KIRCHHOFF, MAXWELL, and GIBBS constructed what is now called classical physics. Between 1700 and 1750 the Basel school met occasional competition from TAYLOR, MACLAURIN, CLAIRAUT, and D’ALEMBERT, though of these only the last did work of any breadth. [p.149, my emphasis]

So to properly address the development of 18th–century physics means to also understand the development of the mathematics supporting said physics. So, at some point, for this blog, I will have to focus on some of the developments of 18th-century mathematics.

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1. Adolf Kneser. Das Prinzip der Kleinsten Wirkung von Leibniz bis zur Gegenwart. Springer Fachmedien Wiesbaden, 1928.

2. C. Truesdell. A Program toward Rediscovering the Rational Mechanics of the Age of Reason. In Essays in the History of Mechanics. Springer, 1968, pp.84-137.

3. Opera Omnia Leonhard Euler. https://en.wikipedia.org/wiki/Opera_Omnia_Leonhard_Euler

4. Ronald S. Calinger. Euler: Mathematical Genius in the Enlightenment. Princeton University Press, 2016.

5. C. Truesdell. Reactions of Late Baroque Mechanics to Success, Conjecture, Error, and Failure in Newton's Principia. In Essays in the History of Mechanics. pp.138-183.