Hamilton's principle/History/Least time before least action - Revision history

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	Fermat stated the first variational principle in physics in February 1662, in a		Fermat stated the first variational principle in physics in February 1662, in a
	short paper called ''Synthesis ad refractiones''. It predates the calculus of		short paper called ''Synthesis ad refractiones''. It predates the calculus of
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	editorial summary from secondary sources. See [[Hamilton's principle/Sources]].		editorial summary from secondary sources. See [[Hamilton's principle/Sources]].
	</blockquote>		</blockquote>

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Fermat stated the first variational principle in physics in February 1662, in a
short paper called ''Synthesis ad refractiones''. It predates the calculus of
variations by eighty years and the mechanical action principles by eighty-two.

It is also the origin of a habit this volume spends its length correcting: stating
such a principle with the word ''least''.

== Provenance note ==

[[File:OPL-HP-F-54.png|thumb|right|400px|Four states an electronic source can be in.]]

The Latin text is reachable in the ''Œuvres de Fermat'', transcribed on French
Wikisource under ''Maxima et Minima''.

The transcription '''carries its text''' but is '''visibly OCR-damaged'''. Words
are corrupted, and at least one Greek phrase is reduced to characters that are not
Greek. It is legible and it is not reliable at the level of individual words.

This is a source state the volume's survey did not originally define: not stripped
of content, not a page image, but text-bearing and damaged. It is recorded on
[[Hamilton's principle/Sources]], and it means '''no words are quoted verbatim
from Fermat in this volume'''. His postulate is paraphrased below.

== What he was arguing about ==

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

Descartes had derived the law of refraction, and his derivation required an
assumption: that light moves more easily and quickly through a '''dense''' medium
than through a rare one.

Fermat found that assumption contrary to the nature of light. His paper opens by
saying so — that Descartes needed the claim conceded to him, and that it appears to
run against light's own nature.

So Fermat set out to derive the same law from the '''opposite''' assumption: that
light moves more easily through rare media than dense ones.

And he reports the outcome plainly: proceeding from the contrary axiom, he arrives
at Descartes's own proportion.

That is the paper's point. The same law of refraction follows from either
assumption about the speed, so the law cannot be used to decide between them.

== The postulate ==

Fermat states that his demonstration rests on a single postulate: '''that nature
operates by the easiest and readiest ways'''.

<blockquote>
Paraphrased, not quoted. The Latin is available but the transcription is
OCR-damaged, so this volume does not reproduce his words.
</blockquote>

Read that carefully, because it is not what he is usually credited with.

It does not say '''least time'''. It says easiest and readiest — a claim about the
manner of nature's operation, not a named quantity with a minimum.

'''Least time is a modern gloss on Fermat's postulate, not his statement of it.'''

This is the third instance in this volume of the same error: a later formulation
attributed to an earlier author. The other two are Euler's variable on
[[Hamilton's principle/History/Euler and the Methodus inveniendi]] and the mass in
Maupertuis's 1744 action on
[[Hamilton's principle/History/Maupertuis and the 1744 memoir]].

Three authors, three centuries, one habit. Later readers tidy an earlier statement
into the form they already understand and then cite the earlier author for the
tidy version.

== What the method was ==

Fermat did not have the calculus of variations, and he did not have calculus in
the modern sense.

What he had was his own technique for maxima and minima — the method of
adequality, which he had developed for finding turning points of curves. The
refraction paper is filed in the ''Œuvres'' under that heading, and it is an
application of it.

His treatment is geometrical and, by modern standards, hard going. The
straightforward derivation of the refraction law from least time on
[[Hamilton's principle/Applications/Fermat's principle]] is a modern
reconstruction, not his argument.

== The objection ==

Fermat's principle was controversial immediately, and the objection was not about
mathematics.

It seemed to attribute knowledge and purpose to light. If a ray takes the quickest
route, something must have compared the routes. That looks like intention, and
attributing intention to a beam of light was not acceptable physics.

Descartes's followers pressed exactly this point.

The objection stood for nearly two centuries and it was a reasonable one. It was
answered only when light was understood as a wave: a wavefront spreading from a
point sweeps '''all''' the paths, not one, and the surviving ray is where nearby
paths agree rather than where a choice was made.

'''There is no implication that the ray knew the quickest path.''' The appearance
of purpose is what interference looks like from the outside.

That answer is the same one the quantum account gives for mechanics on
[[Hamilton's principle/Applications/The Feynman path integral]], nearly a century
later again. The two resolutions are the same resolution.

== Fermat was also right about the speed ==

The physical question underneath the dispute was settled experimentally by
Foucault in 1850, who measured light as slower in a dense medium.

Fermat's assumption was correct and Descartes's was not — two centuries after
both men were dead, and after Maupertuis had built his own principle on Descartes's
side of the question.

== Why the word had to change ==

Fermat wrote of the easiest way, and later readers wrote ''least''. Both are
wrong, and the correction is the same one.

The path light takes is '''stationary''', not least. A mirror curved sharply
enough gives a real reflected path that takes '''longer''' than nearby paths would,
as worked on
[[Hamilton's principle/Applications/Fermat's principle]]. A maximum is not an edge
case of a minimum.

Fermat's own postulate, being vaguer, is less badly damaged by this than
Maupertuis's. He did not name a quantity and assert its minimality; he described a
manner of operating. The damage is to the version later readers gave him.

== Where to go next ==

The extension of this kind of principle from light to matter is at
[[Hamilton's principle/History/Maupertuis and the 1744 memoir]].

<blockquote>
Provenance: nothing on this page is quoted from Fermat. The account of his paper
is drawn from the OCR-damaged Wikisource transcription of the ''Œuvres''. The
Foucault measurement and the wave-optical resolution of the purpose objection are
editorial summary from secondary sources. See [[Hamilton's principle/Sources]].
</blockquote>

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	This is a source state the volume's survey did not originally define: not stripped		This is a source state the volume's survey did not originally define: not stripped
	of content, not a page image, but text-bearing and damaged. It is recorded on		of content, not a page image, but text-bearing and damaged. It is recorded on
	[[Hamilton's principle/Sources]], and it means '''no words are quoted verbatim		[[Hamilton's principle/Sources\|Sources]], and it means '''no words are quoted verbatim
	from Fermat in this volume'''. His postulate is paraphrased below.		from Fermat in this volume'''. His postulate is paraphrased below.

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	This is the third instance in this volume of the same error: a later formulation		This is the third instance in this volume of the same error: a later formulation
	attributed to an earlier author. The other two are Euler's variable on		attributed to an earlier author. The other two are Euler's variable on
	[[Hamilton's principle/History/Euler and the Methodus inveniendi]] and the mass in		[[Hamilton's principle/History/Euler and the Methodus inveniendi\|Euler and the Methodus inveniendi]] and the mass in
	Maupertuis's 1744 action on		Maupertuis's 1744 action on
	[[Hamilton's principle/History/Maupertuis and the 1744 memoir]].		[[Hamilton's principle/History/Maupertuis and the 1744 memoir\|Maupertuis and the 1744 memoir]].

	Three authors, three centuries, one habit. Later readers tidy an earlier statement		Three authors, three centuries, one habit. Later readers tidy an earlier statement
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	His treatment is geometrical and, by modern standards, hard going. The		His treatment is geometrical and, by modern standards, hard going. The
	straightforward derivation of the refraction law from least time on		straightforward derivation of the refraction law from least time on
	[[Hamilton's principle/Applications/Fermat's principle]] is a modern		[[Hamilton's principle/Applications/Fermat's principle\|Fermat's principle]] is a modern
	reconstruction, not his argument.		reconstruction, not his argument.

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	That answer is the same one the quantum account gives for mechanics on		That answer is the same one the quantum account gives for mechanics on
	[[Hamilton's principle/Applications/The Feynman path integral]], nearly a century		[[Hamilton's principle/Applications/The Feynman path integral\|the Feynman path integral]], nearly a century
	later again. The two resolutions are the same resolution.		later again. The two resolutions are the same resolution.

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	enough gives a real reflected path that takes '''longer''' than nearby paths would,		enough gives a real reflected path that takes '''longer''' than nearby paths would,
	as worked on		as worked on
	[[Hamilton's principle/Applications/Fermat's principle]]. A maximum is not an edge		[[Hamilton's principle/Applications/Fermat's principle\|Fermat's principle]]. A maximum is not an edge
	case of a minimum.		case of a minimum.

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	The extension of this kind of principle from light to matter is at		The extension of this kind of principle from light to matter is at
	[[Hamilton's principle/History/Maupertuis and the 1744 memoir]].		[[Hamilton's principle/History/Maupertuis and the 1744 memoir\|Maupertuis and the 1744 memoir]].

	<blockquote>		<blockquote>
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	is drawn from the OCR-damaged Wikisource transcription of the ''Œuvres''. The		is drawn from the OCR-damaged Wikisource transcription of the ''Œuvres''. The
	Foucault measurement and the wave-optical resolution of the purpose objection are		Foucault measurement and the wave-optical resolution of the purpose objection are
	editorial summary from secondary sources. See [[Hamilton's principle/Sources]].		editorial summary from secondary sources. See [[Hamilton's principle/Sources\|Sources]].
	</blockquote>		</blockquote>