Autobiography of Sir George Biddell Airy, George Biddell Airy [dark books to read TXT] 📗

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) squared(d squaredy/dt squared) = a squared(d squaredy/dx squared) 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_.

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ROYAL OBSERVATORY, GREENWICH,
LONDON, S.E.
_1867, December 9_.

MY DEAR SIR,

I have received with much pleasure your letter of December 6. In this University discussion, I have acted only in public, and have not made private communication to any person whatever till required to do so by private letter addressed to me. Your few words in Queens' Hall seemed to expect a little reply.

Now as to the Modern Geometry. With your praises of this science--as to the room for extension in induction and deduction, &c.; and with your facts--as to the amount of space which it occupies in Mathematical Journals; I entirely agree. And if men, after leaving Cambridge, were designed to shut themselves up in a cavern, they could have nothing better for their subjective amusement. They might have other things as good; enormous complication and probably beautiful investigation might be found in varying the game of billiards with novel islands on a newly shaped billiard table. But the persons who devote themselves to these subjects do thereby separate themselves from the world. They make no step towards natural science or utilitarian science, the two subjects which the world specially desires. The world could go on as well without these separatists.

Now if these persons lived only for themselves, no other person would have any title to question or remark on their devotion to this barren subject. But a Cambridge Examiner is not in that position. The University is a national body, for education of young men: and the power of a Cambridge Examiner is omnipotent in directing the education of the young men; and his responsibility to the cause of education is very distinct and very strong. And the question for him to consider is--in the sense in which mathematical education is desired by the best authorities in the nation, is the course taken by this national institution satisfactory to the nation?

I express my belief that it is _not_ satisfactory. I believe that many of the best men of the nation consider that a great deal of time is lost on subjects which they esteem as puerile, and that much of that time might be employed on noble and useful science.

You may remember that the Commissions which have visited Cambridge originated in a Memorial addressed to the Government by men of respected scientific character: Sabine was one, and I may take him as the representative. He is a man of extensive knowledge of the application of mathematics as it has been employed for many years in the science of the world; but he has no profundity of science. He, as I believe, desired to find persons who could enter accurately into mathematical science, and naturally looked to the Great Mathematical University; but he must have been much disappointed. So much time is swallowed up by the forced study of the Pure Mathematics that it is not easy to find anybody who can really enter on these subjects in which men of science want assistance. And so Sabine thought that the Government ought to interfere, probably without any clear idea of what they could do.

I am, my dear Sir,
Yours very truly,
G.B. AIRY.

_Professor Cayley_.

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DEAR SIR,

I have to thank you for your last letter. I do not think everything should be subordinated to the educational element: my idea of a University is that of a place for the cultivation of all science. Therefore among other sciences Pure Mathematics; including whatever is interesting as part of this science. I am bound therefore to admit that your proposed extension of the problem of billiards, _if it_ were found susceptible of interesting mathematical developments, would be a fit subject of study. But in this case I do not think the problem could fairly be objected to as puerile--a more legitimate objection would I conceive be its extreme speciality. But this is not an objection that can be brought against Modern Geometry as a whole: in regard to any particular parts of it which may appear open to such an objection, the question is whether they are or are not, for their own sakes, or their bearing upon other parts of the science to which they belong, worthy of being entered upon and pursued.

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_.

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_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.

You state your idea that the educational element ought not to be the predominating element in the University. "I do not think that every thing should be subordinated to the educational element." I cannot conceal my surprise at this sentiment. Assuredly the founders of the Colleges intended them for education (so far as they apply to persons in statu pupillari), the statutes of the University and the Colleges are framed for education, and fathers