522 TRANSACTIONS OF SECTION A. 



training which should be imparted by a study of geometry is vitiated. If this 

 objection really implies a demand for a purely deductive treatment of the subject, 

 T think some of those who raise it hardly realise all that would be involved in 

 the complete satisfaction of their requirement. I have already remarked that 

 Euclid's treatment of the subject is not rigorous as regards logic. Owing to the 

 recent exploration of the foundations of geometry we possess at the present time 

 tolerably satisfactory methods of purely deductive treatment of the subject; in 

 regard to mechanics, notwithstanding the valuable work of Mach, Herz, and 

 others, this is not yet the case. But, in the schemes of purely deductive geometry, 

 the systems of axioms and postulates are far from being of a very simple 

 character; their real nature, and the necessity for many of them, can only be 

 appreciated at a much later stage in mathematical education than the one of 

 which I am speaking. A purely logical treatment is the highest stage in the 

 training of the mathematician, and is wholly unsuitable — and, indeed, quite im- 

 possible — in those stages beyond which the great majority of students never pass. 

 It can then, in the case of all students, except a few advanced ones in the univer- 

 sities, only be a question of degree how far the purely logical factor in the 

 proofs of propositions shall be modified by the introduction of elements derived 

 from observation or spatial intuition. If the freedom of teaching which I have 

 advocated be allowed, it will be open to those teachers who find it advisable in 

 the interests of their students to emphasise the logical side of their teaching to do 

 so ; and it is certainly of value in all cases to draw the attention of students to 

 those points in a proof where the intuitional element enters. I draw, then, the 

 conclusion that a mixed treatment of geometry, as of mechanics, must prevail in 

 the future, as it has done in the past, but that the proportion of the observational 

 or intuitional factor to the logical one must vary in accordance with the needs 

 and intellectual attainments of the students, and that a large measure of freedom 

 of judgment in this regard should be left to the teacher. 



The great and increasing importance of a knowledge of the differential and 

 integral calculus for students of engineering and other branches of physical science 

 has led to the publication during the last few years of a considerable number of 

 text-books on this subject intended for the use of such students. Some of these 

 text-books are excellent, and their authors, by a skilful insistence on the principles 

 of the subject, have done their utmost to guard against the very real dangers 

 which attend attempts to adapt such a subject to the practical needs of engineers 

 and others. It is quite true that a great mass of detail which has gradually come 

 to form part — often much too large a part — of the material of the student of 

 Mathematics, may with great advantage be ignored by those whose main study is 

 to be engineering science or physics. Yet it cannot be too strongly insisted on that 

 a firm grasp of the principles, as distinct from the mere processes of calculation, is 

 essential if Mathematics is to be a tool really useful to the engineer and the 

 physicist. There is a danger, which experience has shown to be only too real, 

 that such students may learn to regard Mathematics as consisting merely of 

 formulae and of rules which provide the means of performing the numerical com- 

 putations necessary for solving certain categories of problems which occur in the 

 practical sciences. Apart from the deplorable effect, on the educational side, of 

 degrading Mathematics to this level, the practical effect of reducing it to a number 

 of rule-of-thumb processes can only be to make those who learn it in so unintelli- 

 gent a manner incapable of applying mathematical methods to any practical 

 problem in which the data differ even slightly from those in the model problems 

 which they have studied. Only a firm grasp of the principles will give the neces- 

 sary freedom in handling the methods of Mathematics required for the various 

 practical problems in the solution of which they are essential. 



The following Papers wore then read : — ■ 



1. Positive Fays. By Professor Sir J. J. Thomson, F.F.S. 



The investigation of the positive rays which are produced when the electric 

 discharge passes through a tube filled with gas at a low pressure is much 

 facilitated by using very large tubes. With large tubes, in which there is room 



