But admitting (as I do not) that Pure Mathematics are only to be studied with a view to Natural and Physical Science, the question still arises how are they best to be studied in that view. I assume and admit that as to a large part of Modern Geometry and of the Theory of Numbers, there is no present probability that these will find any physical applications. But among the remaining parts of Pure Mathematics we have the theory of Elliptic Functions and of the Jacobian and Abelian Functions, and the theory of Differential Equations, including of course Partial Differential Equations. Now taking for instance the problem of three bodies--unless this is to be gone on with by the mere improvement in detail of the present approximate methods--it is at least conceivable that the future treatment of it will be in the direction of the problem of two fixed centres, by means of elliptic functions, &c.; and that the discovery will be made not by searching for it directly with the mathematical resources now at our command, but by "prospecting" for it in the field of these functions. Even improvements in the existing methods are more likely to arise from a study of differential equations in general than from a special one of the equations of the particular problem: the materials for such improvements which exist in the writings of Hamilton, Jacobi, Bertrand, and Bour, have certainly so arisen. And the like remarks would apply to the physical problems which depend on Partial Differential Equations. I think that the course of mathematical study at the University is likely to be a better one if regulated with a view to the cultivation of Science, as if for its own sake, rather than directly upon considerations of what is educationally best (I mean that the best educational course will be so obtained), and that we have thus a justification for a thorough study of Pure Mathematics. In my own limited experience of examinations, the fault which I find with the men is a want of analytical power, and that whatever else may have been in defect Pure Mathematics has certainly not been in excess. I remain, dear Sir, Yours sincerely, A. CAYLEY. CAMBRIDGE, _10th Dec., 1867_. * * * * * _1867, December 17_. MY DEAR SIR, Since receiving your letter of 9th I positively have not had time to express the single remark which I proposed to make on it.