I have been intending to answer your letter of the 8th November. So far as it is (if at all) personal to myself, I would remark that the statutory duty of the Sadlerian Professor is that he shall explain and teach the principles of Pure Mathematics and apply himself to the advancement of the Science. As to Partial Differential Equations, they are "high" as being an inverse problem, and perhaps the most difficult inverse problem that has been dealt with. In regard to the limitation of them to the second order, whatever other reasons exist for it, there is also the reason that the theory to this order is as yet so incomplete that there is no inducement to go beyond it; there could hardly be a more valuable step than anything which would give a notion of the form of the general integral of a Partial Differential Equation of the second order. I cannot but differ from you _in toto_ as to the educational value of Analytical Geometry, or I would rather say of Modern Geometry generally. It appears to me that in the Physical Sciences depending on Partial Differential Equations, there is scarcely anything that a student can do for himself:--he finds the integral of the ordinary equation for Sound--if he wishes to go a step further and integrate the non-linear equation (dy/dx)²(d²y/dt²) = a²(d²y/dx²) he is simply unable to do so; and so in other cases there is nothing that he can add to what he finds in his books. Whereas Geometry (of course to an intelligent student) is a real inductive and deductive science of inexhaustible extent, in which he can experiment for himself--the very tracing of a curve from its equation (and still more the consideration of the cases belonging to different values of the parameters) is the construction of a theory to bind together the facts--and the selection of a curve or surface proper for the verification of any general theorem is the selection of an experiment in proof or disproof of a theory. I do not quite understand your reference to Stokes and Adams, as types of the men who alone retain their abstract Analytical Geometry. If a man when he takes his degree drops mathematics, he drops geometry--but if not I think for the above reasons that he is more likely to go on with it than with almost any other subject--and any mathematical journal will shew that a very great amount of attention is in fact given to geometry. And the subject is in a very high degree a progressive one; quite as much as to Physics, one may apply to it the lines, Yet I doubt not thro' the ages one increasing purpose runs, and the thoughts of men are widened with the progress of the suns. I remain, dear Sir, Yours very sincerely, A. CAYLEY. CAMBRIDGE, _6 Dec., 1867_. * * * * * ROYAL OBSERVATORY, GREENWICH, LONDON, S.E. _1867, December 9_.