Maupertuis On The Quantity Of Action (1744)

  Here's a rough translation of a 1744 passage by Maupertuis containing his statement of the quantity of action.

 "It was by this principle that Fermat solved the problem, by this principle so plausible, that light which, in its propagation & in its reflection always goes in the shortest possible time, still follows that same law in its refraction; & he did not hesitate to consider that light does not move with greater ease and more swiftly in rarer mediums than in those where, for the same space, there is found a greater quantity of matter: in fact, couldn't one assume at the onset that light traverses with greater ease & more swiftly in crystal and water than in air and the void?
  Also many of the most famous mathematicians are known to embrace the feelings of Fermat; Leibnitz is the one who has made the most use of it, and by his name and by a more elegant analysis which he has given of this problem: he was so charmed by the metaphysical principle, & here to find its final causes to which he was considerably attached, that he regarded, as an unmistakable fact, that light moves faster in air than in water or glass.
  It is however the opposite. Descartes had advanced the first, that the light moves most rapidly in the densest mediums, and although the explanation of the refraction which he had deduced from it was insufficient, his fault did not come from the supposition that he had made. All systems that give some plausible explanation for the phenomena of refection, assume the paradox, or confirm it.
  If one now supposes, that light moves most rapidly in the densest mediums, the whole edifice which Fermat & Leibnitz had built, is destroyed: the light, when it crosses different mediums, neither by the shortest way, or by the quickest way; the ray which passes from the air into the water making the greater part of its path in the air, arrives later than if it did not make the slightest difference. We can see in the Memoir that Mr. de Mairan gave on Reflection and Refraction, the history of the dispute between Fermat & Descartes, & the difficulty & impotence to which we have so far been able to grant the law of refection with metaphysical principle.
  While meditating deeply on this matter, I thought that the light, when it passes from one medium to another, already abandoning the shortest path, which is that of the straight line, could well also not follow that at the most rapid time: indeed, what preference should there be here of time over space? the light being unable to go at once by the shortest way, and by that of the quickest time, why should it go by one of these paths rather than by the other? so it does not follow either of the two, it takes a path that has a more real assertion: the path it takes is that by which the amount of action is a minimum.
  Now I have to explain what I mean by the quantity of action. When a body is carried from one point to another, it requires a certain action, this action depends on the speed of the body and the space it travels, but it is neither the speed nor the space taken separately. The quantity of action is all the greater as the speed of the body is greater, and the path which it traverses is longer, it is proportional to the sum of the spaces multiplied each by the speed with which the body travels. It is this, that quantity of ation which is here the true expenditure of Nature, and which it spares as much as possible in the movement of light.
  Let two different media, separated by a common surface be represented by the line CD, such that the speed of light in the medium which is above, set = V, & the speed in the medium which is below, set = W. Let a ray of light AR, which from a given point A must reach the given point B.

  To find the point R where it must break, I search for the point where the ray breaks, the amount of action is the least, & I have V.AR + W.RB which must be a minimum, or V.√(AC²+CR²) + W.√(BD²+CD²-2CDxCR+CR²) = min. So AC, BD & CD being constant, I have V.CR.dCR/√(AC²+CR²)-W.(CD-CR).dCR/√(BD²+DR²)=0, or V.CR/AR=W.DR/BR. CR/AR:DR/BR::W:V, that is, the sine of incidence to the sine of refraction is in inverse ratio to the speed that light has in each medium.
  All the phenomena of refraction now agree with the great principle, that Nature in the production of its effects always acts in the simplest ways. From this principle follows that when the light passes from one medium to another, the sine of its angle of refraction is to the sine of its angle of incidence in inverse ratio to the velocities of light in each medium."