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 Analytical 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 circumstance) 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.

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DEAR SIR,