and especially sufficiently varied, to prevent any clear mind from having the least doubt of the exactitude and the generality of this essential property, but which, in a philosophical point of view, do not at all dispense with the research for so important an explanation. The extreme extent of the theorem must make us comprehend both the fundamental difficulties of this research and the high utility for the perfecting of mathematical science which would belong to the general conception of this great truth. This imperfection of theory, however, has not prevented geometers from making the most extensive and the most important use of this property in all parts of concrete mathematics.
It follows from the above general enunciation of the fact, independently of any demonstration, that the property of which we speak must never be applied to magnitudes whose directions are continually varying, without giving rise to a simple opposition of direction; in that case, the sign with which every result of calculation is necessarily affected is not susceptible of any concrete interpretation, and the attempts sometimes made to establish one are erroneous. This circumstance occurs, among other occasions, in the case of a radius vector in geometry, and diverging forces in mechanics.
PRINCIPLE OF HOMOGENEITY.
A second general theorem on the relation of the concrete to the abstract is that which is ordinarily designated under the name of Principle of Homogeneity. It is undoubtedly much less important in its applications than the preceding, but it particularly merits our attention as having, by its nature, a still greater extent, since it is applicable to all phenomena without distinction, and because of the real utility which it often possesses for the verification of their analytical laws. I can, moreover, exhibit a direct and general demonstration of it which seems to me very simple. It is founded on this single observation, which is self-evident, that the exactitude of every relation between any concrete magnitudes whatsoever is independent of the value of the units to which they are referred for the purpose of expressing them in numbers. For example, the relation which exists between the three sides of a right-angled triangle is the same, whether they are measured by yards, or by miles, or by inches.
It follows from this general consideration, that every equation which expresses the analytical law of any phenomenon must possess this property of being in no way altered, when all the quantities which are found in it are made to undergo simultaneously the change corresponding to that which their respective units would experience. Now this change evidently consists in all the quantities of each sort becoming at once m times smaller, if the unit which corresponds to them becomes m times greater, or reciprocally. Thus every equation which represents any concrete relation whatever must possess this characteristic of remaining the same, when we make m times greater all the quantities which it contains, and which express the magnitudes between which the relation exists; excepting always the numbers which designate simply the mutual ratios of these different magnitudes, and which therefore remain invariable during the change of the units. It is this property which constitutes the law of Homogeneity in its most extended signification, that is, of whatever analytical functions the equations may be composed.
But most frequently we consider only the cases in which the functions are such as are called algebraic, and to which the idea of degree is applicable. In this case we can give more precision to the general proposition by determining the analytical character which must be necessarily presented by the equation, in order that this property may be verified. It is easy to see, then, that, by the modification just explained, all the terms of the first degree, whatever may be their form, rational or irrational, entire or fractional, will become m times greater; all those of the second degree, m2 times; those of the third, m3 times, &c.