=1746.= In every subject of inquiry there are certain entities, the mutual relations of which, under various conditions, it is desirable to ascertain. A certain combination of these entities are submitted to certain processes or are made the subjects of certain operations. The theory of invariants in its widest scientific meaning determines these combinations, elucidates their properties, and expresses results when possible in terms of them. Many of the general principles of political science and economics can be represented by means of invariantive relations connecting the factors which enter as entities into the special problems. The great principle of chemical science which asserts that when elementary or compound bodies combine with one another the total weight of the materials is unchanged, is another case in point. Again, in physics, a given mass of gas under the operation of varying pressure and temperature has the well-known invariant, pressure multiplied by volume and divided by absolute temperature.... In mathematics the entities under examination may be arithmetical, algebraical, or geometrical; the processes to which they are subjected may be any of those which are met with in mathematical work.... It is the _principle_ which is so valuable. It is the _idea_ of invariance that pervades today all branches of mathematics.--MACMAHON, P. A. _Presidential Address British Association for the Advancement of Science (1901); Nature, Vol. 64, p. 481._ =1747.= [The theory of invariants] has invaded the domain of geometry, and has almost re-created the analytical theory; but it has done more than this for the investigations of Cayley have required a full reconsideration of the very foundations of geometry. It has exercised a profound influence upon the theory of algebraic equations; it has made its way into the theory of differential equations; and the generalisation of its ideas is opening out new regions of most advanced and profound functional analysis. And so far from its course being completed, its questions fully answered, or its interest extinct, there is no reason to suppose that a term can be assigned to its growth and its influence.--FORSYTH, A. R. _Presidential Address British Association for the Advancement of Science (1897); Nature, Vol. 56, p. 378._ =1748.= ... the doctrine of Invariants, a theory filling the heavens like a light-bearing ether, penetrating all the branches of geometry and analysis, revealing everywhere abiding configurations in the midst of change, everywhere disclosing the eternal reign of the law of form.--KEYSER, C. J. _Lectures on Science, Philosophy and Art (New York, 1908), p. 28._