Hamilton's principle/History/Least time before least action
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
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
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.
Paraphrased, not quoted. The Latin is available but the transcription is OCR-damaged, so this volume does not reproduce his words.
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.
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.