Euler and the Methodus inveniendi
Euler's Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes appeared in 1744, the same year as Maupertuis's memoir. It is the first systematic treatise on the calculus of variations, and its second appendix contains a statement of the action principle that is mathematically far ahead of anything Maupertuis wrote.
Euler did not claim priority for it.
Provenance note
The second appendix, Additamentum II, is reachable in two forms: the Latin original as a page-image scan at the Euler Archive, and an English translation on Wikisource.
The translation was probed for this volume and carries its mathematics — it is not the failure mode that destroyed the companion Maxwell volume. But it modernises Euler's notation in a way that matters, described below. The Latin original has not been opened. See Sources.
The treatise
The Methodus inveniendi is about finding curves that maximise or minimise something. It is the first book-length treatment of that question, and it gives a general method rather than a collection of solved problems.
Euler's method is not the one this volume teaches. He approximates a curve by a polygon with many vertices, requires the quantity to be stationary against moving each vertex, and passes to a limit. It works, and it is laborious.
Lagrange's variational method, developed in correspondence with Euler over the following decade, replaced it. Euler adopted Lagrange's approach and named it the calculus of variations. The equation both are named for is on the Euler-Lagrange equation.
The action principle is not the point of the treatise. It appears in an appendix.
What Euler said
Euler's statement is that among all curves through the same endpoints, the true trajectory of a body is the one making a certain integral least.
That integral is the momentum accumulated over distance travelled — in modern terms, the abbreviated action of Abbreviated action: Maupertuis versus Hamilton.
He notes that the same quantity can be expressed through the kinetic energy instead, and remarks that the principle should therefore appeal both to those who prefer momentum and to those who prefer living force. That even-handedness is worth noticing: the momentum-against-energy dispute was live and bitter, and Euler is pointing out that his principle does not take a side.
A notation trap
Here the volume has to be careful, and the care is the reason this page exists in this form.
Euler's own is not the velocity. It is half the square of the velocity — a quantity a modern reader would call the specific kinetic energy. So the thing he writes under the integral, with a square root over , is proportional to the speed.
The Wikisource English translation modernises this. It writes the same statement with meaning the velocity, so the square root disappears.
The two agree as physics and disagree as attribution. Quoting the modernised form as Euler's is the same class of error as putting a mass into Maupertuis's 1744 definition on Maupertuis and the 1744 memoir — a later convention attributed to an earlier author.
This volume therefore states Euler's integral in his own variable, with his definition of it given alongside, and treats the modern form as a gloss. No equation is attributed to Euler until the Latin original is opened.
Independent, and slightly later
Euler's statement and Maupertuis's are independent. They appeared in the same year, and Euler's came slightly later.
The two are not equivalent in quality. Maupertuis had a quantity and a metaphysical argument. Euler had a quantity, a general method for handling such problems, and the ability to compute with it. He could produce trajectories; Maupertuis could not.
Euler nonetheless credited Maupertuis, and later defended his priority publicly. That is the subject of the Maupertuis-Koenig affair.
Why Euler did not claim it
There is no need to speculate about motives, and this volume does not. Two observations are enough.
Maupertuis was president of the Berlin Academy, where Euler worked. The professional situation is a fact and its influence is not measurable from here.
And Euler's interest was visibly in the method rather than the principle. The Methodus inveniendi is a treatise on how to solve a class of problems. The action principle is one application, in an appendix, and he treats it as such.
The honest limitation
Euler is careful about the reach of his claim in a way Maupertuis is not.
He notes that the principle runs into difficulty for motion in a resisting medium. That is correct, and it remains correct: friction is exactly where the variational formulation in its simple form does not apply, as d'Alembert and virtual work observes.
Maupertuis's version claims all of nature. Euler's names its own boundary.
Least, again
Like Maupertuis, Euler wrote least. Unlike Maupertuis, nothing in his argument depends on it: his method finds curves where the quantity is stationary, and the word minimum is a description of what he expected rather than a load- bearing claim.
The correction, when Hamilton made it ninety years later, cost Euler's mathematics nothing. It cost Maupertuis's metaphysics everything.
That difference is the clearest measure of the two contributions.
Provenance: the characterisation of Euler's statement above is drawn from the Wikisource English translation and from the Euler Archive's own description of the work. Neither is the Latin original, which has not been opened. No equation is attributed to Euler on this page. The Hamilton reference on Maupertuis and the 1744 memoir is ESTABLISHED; nothing on this page is.
Where to go next
The priority dispute is at the Maupertuis-Koenig affair.