IT would be difficult to exaggerate the part which the study of elliptic functions has played in the pure mathematics in the previous century... and this was to be expected; for whether we regard natural science as the application of common sense to the material needs of life, or as the outcome of the need for expansion in the mental world, and whether we consider mathematics as that exact basis without which progress was not permanently possible, or esteem it to be those higher Alps—
Where we can ever climb, and ever
To a finer air—
in either case we must see that a development of integral calculus—a development which was competent to fill so large a part of Legendre's life, which suggested such magnificent algebra as we find in Jacobi's "Fundamenta," which promised, too, in Abel's hands such generalizations as are not even yet brought to perfection, such a theory, surely, was well worthy of persevering pursuit.
And if we attribute the present extent of the theory of curves and of the theory of functions to the day when Riemann stood best man to the ideas of Cauchy and the suggestions of hydrodynamics, we must admit it was because his methods were employed upon the materials left by Abel that such results have come.
The importance of this work lies in its recognition that the theory of elliptic functions arose as a development of integral calculus, and as such may be expected to supply a formulation of the solution of many problems of physics otherwise regarded as unfinished. A. G. Greenhill was well known to be a man who had not allowed his unwearied application to such problems to destroy his sympathy with pure mathematical speculation; on the contrary, he had sought, by every means in his power, to fill the difficult position of apostle to the mathematicians in this respect, by making as many of the results of analysis as are susceptible of application to physics, easily intelligible to students of that subject.
