Hamilton's principle/History/Euler and the Methodus inveniendi - Revision history

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	Euler's ''Methodus inveniendi lineas curvas maximi minimive proprietate		Euler's ''Methodus inveniendi lineas curvas maximi minimive proprietate
	gaudentes'' appeared in 1744, the same year as Maupertuis's memoir. It is the		gaudentes'' appeared in 1744, the same year as Maupertuis's memoir. It is the
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	The priority dispute is at		The priority dispute is at
	[[Hamilton's principle/History/The Maupertuis-Koenig affair]].		[[Hamilton's principle/History/The Maupertuis-Koenig affair]].

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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 ==

[[File:OPL-HP-F-55.png|thumb|right|400px|Three modernised statements attributed to their authors.]]

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 [[Hamilton's principle/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
[[Hamilton's principle/Mathematics/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
[[Hamilton's principle/Mathematics/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 <math>v</math> 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
<math>v</math>, is proportional to the speed.

The Wikisource English translation modernises this. It writes the same statement
with <math>v</math> 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
[[Hamilton's principle/History/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
[[Hamilton's principle/History/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
[[Hamilton's principle/History/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.

<blockquote>
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
[[Hamilton's principle/History/Maupertuis and the 1744 memoir]] is ESTABLISHED;
nothing on this page is.
</blockquote>

== Where to go next ==

The priority dispute is at
[[Hamilton's principle/History/The Maupertuis-Koenig affair]].

← Older revision		Revision as of 04:59, 16 September 2026
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	is not the failure mode that destroyed the companion Maxwell volume. But 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		modernises Euler's notation in a way that matters, described below. The Latin
	original has not been opened. See [[Hamilton's principle/Sources]].		original has not been opened. See [[Hamilton's principle/Sources\|Sources]].

	== The treatise ==		== The treatise ==
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	following decade, replaced it. Euler adopted Lagrange's approach and named it 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		calculus of variations. The equation both are named for is on
	[[Hamilton's principle/Mathematics/The Euler-Lagrange equation]].		[[Hamilton's principle/Mathematics/The Euler-Lagrange equation\|the Euler-Lagrange equation]].

	The action principle is not the point of the treatise. It appears in an appendix.		The action principle is not the point of the treatise. It appears in an appendix.
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	That integral is the momentum accumulated over distance travelled — in modern		That integral is the momentum accumulated over distance travelled — in modern
	terms, the abbreviated action of		terms, the abbreviated action of
	[[Hamilton's principle/Mathematics/Abbreviated action: Maupertuis versus Hamilton]].		[[Hamilton's principle/Mathematics/Abbreviated action: Maupertuis versus Hamilton\|Abbreviated action: Maupertuis versus Hamilton]].

	He notes that the same quantity can be expressed through the kinetic energy		He notes that the same quantity can be expressed through the kinetic energy
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	form as Euler's is the same class of error as putting a mass into Maupertuis's		form as Euler's is the same class of error as putting a mass into Maupertuis's
	1744 definition on		1744 definition on
	[[Hamilton's principle/History/Maupertuis and the 1744 memoir]] — a later		[[Hamilton's principle/History/Maupertuis and the 1744 memoir\|Maupertuis and the 1744 memoir]] — a later
	convention attributed to an earlier author.		convention attributed to an earlier author.

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	Euler nonetheless credited Maupertuis, and later defended his priority publicly.		Euler nonetheless credited Maupertuis, and later defended his priority publicly.
	That is the subject of		That is the subject of
	[[Hamilton's principle/History/The Maupertuis-Koenig affair]].		[[Hamilton's principle/History/The Maupertuis-Koenig affair\|the Maupertuis-Koenig affair]].

	== Why Euler did not claim it ==		== Why Euler did not claim it ==
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	That is correct, and it remains correct: friction is exactly where the variational		That is correct, and it remains correct: friction is exactly where the variational
	formulation in its simple form does not apply, as		formulation in its simple form does not apply, as
	[[Hamilton's principle/History/d'Alembert and virtual work]] observes.		[[Hamilton's principle/History/d'Alembert and virtual work\|d'Alembert and virtual work]] observes.

	Maupertuis's version claims all of nature. Euler's names its own boundary.		Maupertuis's version claims all of nature. Euler's names its own boundary.
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	work. Neither is the Latin original, which has not been opened. No equation is		work. Neither is the Latin original, which has not been opened. No equation is
	attributed to Euler on this page. The Hamilton reference on		attributed to Euler on this page. The Hamilton reference on
	[[Hamilton's principle/History/Maupertuis and the 1744 memoir]] is ESTABLISHED;		[[Hamilton's principle/History/Maupertuis and the 1744 memoir\|Maupertuis and the 1744 memoir]] is ESTABLISHED;
	nothing on this page is.		nothing on this page is.
	</blockquote>		</blockquote>
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	The priority dispute is at		The priority dispute is at
	[[Hamilton's principle/History/The Maupertuis-Koenig affair]].		[[Hamilton's principle/History/The Maupertuis-Koenig affair\|the Maupertuis-Koenig affair]].