First, I will remark on the study of Partial Differential Equations.

I do not know that one branch of Pure Mathematics can be considered higher than another, except in the utility of the power which it gives. Measured thus, the Partial Differential Equations are very useful and therefore stand very high, as far as the Second Order.

They apply, to that point, in the most important way, to the great problems of nature concerning _time_, and _infinite division of matter_, and _s.p.a.ce_: and are worthy of the most careful study. Beyond that Order they apply to nothing. It was for the purpose of limiting the study to the Second Order, and at the same time working it carefully, philosophically, and practically, up to that point, that I drew up my little work.

On the general question of Mathematical Studies, I will first give my leading ideas on what I may call the moral part. I think that a heavy responsibility rests on the persons who influence most strongly the course of education in the University, to direct that course in the way in which it will be most useful to the students--in the two ways, of disciplining their powers and habits, and of giving them scientific knowledge of the highest and most accurate order (applying to the phenomena of nature) such as will be useful to them through life. I do not think that the mere personal taste of a teacher is sufficient justification for a special course, unless it has been adopted under a consideration of that responsibility. Now I can say for myself that I have, for some years, inspected the examination papers, and have considered the bearing of the course which they imply upon the education of the student, and am firmly convinced that as regards men below the very few first--say below the ten first--there is a prodigious loss of time without any permanent good whatever. For the great majority of men, such subjects as abstract a.n.a.lytical Geometry perish at once. With men like Adams and Stokes they remain, and are advantageous; but probably there is not a single man (beside them) of their respective years who remembers a bit, or who if he remembers them has the leisure and other opportunities of applying them.

I believe on the other hand that a careful selection of physical subjects would enable the University to communicate to its students a vast amount of information; of accurate kind and requiring the most logical treatment; but so bearing upon the natural phenomena which are constantly before us that it would be felt by every student to possess a real value, that (from that circ.u.mstance) it would dwell in his mind, and that it would enable him to correct a great amount of flimsy education in the country, and, so far, to raise the national character.

The consideration of the education of the reasoning habits suggests ideas far from favourable to the existing course. I am old enough to remember the time of mere geometrical processes, and I do not hesitate to say that for the cultivation of accurate mental discipline they were far superior to the operations in vogue at the present day. There is no subject in the world more favourable to logical habit than the Differential Calculus in all its branches _if logically worked in its elements_: and I think that its applications to various physical subjects, compelling from time to time an attention to the elementary grounds of the Calculus, would be far more advantageous to that logical habit than the simple applications to Pure Equations and Pure Algebraical Geometry now occupying so much attention.

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

_Professor Cayley_.

DEAR SIR,

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 a.n.a.lytical 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)(dy/dt) = a(dy/dx) 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 a.n.a.lytical 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_.

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 s.p.a.ce 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 amus.e.m.e.nt. 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 t.i.tle 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 inst.i.tution 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 n.o.ble 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 a.s.sistance. 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_.

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 a.s.sume 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 a.n.a.lytical 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.

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. a.s.suredly 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 send their sons to the University for education. If I had not had your words before me, I should have said that it is impossible to doubt this.

It is much to be desired that Professors and others who exercise no control by force should take every method, not only of promoting science in themselves, but also of placing the promoted science before students: and it is much to be desired that students who have pa.s.sed the compulsory curriculum should be encouraged to proceed into the novelties which will be most agreeable to them. But this is a totally different thing from using the Compulsory Force of Examination to drive students in paths traced only by the taste of the examiner. For them, I conceive the obligation to the nation and the duty to follow the national sense on education (as far as it can be gathered from its best representatives) to be undoubted; and to be, in the intensity of the obligation and duty, most serious.

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

_Professor Cayley_.

1868

"In the South-East Dome, the alteration proposed last year for rendering the building fire-proof had been completely carried out. The middle room, which was to be appropriated to Chronometers, was being fitted up accordingly.--From the Report it appears that 'our subterranean telegraph wires were all broken by one blow, from an accident in the Metropolitan Drainage Works on Groom's Hill, but were speedily repaired.'--In my office as Chairman of successive Commissions on Standards, I had collected a number of Standards, some of great historical value (as Ramsden's and Roy's Standards of Length, Kater's Scale-beam for weighing great weights, and others), &c. These have been transferred to the newly-created Standards Department of the Board of Trade."--In the Report is given a detailed account of the system of preserving and arranging the ma.n.u.scripts and correspondence of the Observatory, which was always regarded by Airy as a matter of the first importance.--From a careful discussion of the results of observation Mr Stone had concluded that the refractions ought to be diminished. 'Relying on this, we have now computed our mean refractions by diminishing those of Bessel's Fundamenta in the proportion of 1 to 0.99797.'--The Magnetometer-Indications for the period 1858-1863 had been reduced and discussed, with remarkable results. It is inferred that magnetic disturbances, both solar and lunar, are produced mediately by the Earth, and that the Earth in periods of several years undergoes changes which fit it and unfit it for exercising a powerful mediate action.--The Earth-current records had been reduced, and the magnetic effect which the currents would produce had been computed. The result was, that the agreement between the magnetic effects so computed and the magnetic disturbances really recorded by the magnetometers was such as to leave no doubt on the general validity of the explanation of the great storm-disturbances of the magnets as consequences of the galvanic currents through the earth.--Referring to the difficulty experienced in making the meteorological observations practically available the Report states thus: 'The want of Meteorology, at the present time, is princ.i.p.ally in suggestive theory.'--In this year Airy communicated to the Royal Astronomical Society a Paper 'On the Preparatory Arrangements for the Observation of the Transits of Venus 1874 and 1882': this subject was now well in hand.--The First Report of the Commissioners (of whom he was Chairman) appointed to enquire into the condition of the Exchequer Standards was printed: this business took up much time.--He was in this year much engaged on the Coinage Commission.

Of private history: There was the usual winter visit to Playford, and a short visit to Cambridge in June.--From about Aug. 1st to Sept. 3rd he was travelling in Switzerland with his youngest son and his two youngest daughters. In the course of this journey they visited Zermatt. There had been much rain, the rivers were greatly flooded, and much mischief was done to the roads. During the journey from Visp to Zermatt, near St Nicholas, in a steep part of the gorge, a large stone rolled from the cliffs and knocked their baggage horse over the lower precipice, a fall of several hundred feet. The packages were all burst, and many things were lost, but a good deal was recovered by men suspended by ropes.

In this year also Airy was busy with the subject of University Examination, which in previous years had occupied so much of his attention, as will be seen from the following letters:

ROYAL OBSERVATORY, GREENWICH, LONDON, S.E.

_1868, March 12_.

MY DEAR MASTER,

I have had the pleasure of corresponding with you on matters of University Examination so frequently that I at once turn to you as the proper person to whom I may address any remarks on that important subject.