Jime 6, 1889] 



NATURE 



127 



What boy with this slender knowledge of pure Mathe- 

 matics is likely to have a correct notion of the nature of 

 Acceleration ? And, without knowing the meaning of a 

 sine or a tangent, to what can his knowledge of the 

 Composition and Resolution of Forces and of the method 

 of taking Moments amount ? 



A course of reading in Mechanics and Hydrostatics, 

 specially constructed so as to avoid Trigonometry and 

 the sixth book of Euclid, is a curious object to contemplate. 

 It reminds one of a number of large boulders thrown out 

 into a stream, with painfully and dangerously wide inter- 

 vals between them, their tops barely above the water, to 

 enable a man to get across by a series of courageous 

 jumps ; and, as in the case of such a traveller, not all the 

 caution nor the acrobatic deftness of balancing on the 

 point of one foot which he possesses will prevent him 

 from tumbling off his flimsy and rickety supports, so in 

 the case of the student for whom knowledge has been 

 packed up into a number of scientific boluses, it is im- 

 possible to avoid the acquisition of erroneous notions and 

 fallacious rule-of-thumb principles, which, assimilated 

 thus early, have a strong tendency to remain rooted in 

 the mind. 



In the case of my young friend I found it quite impossi- 

 ble to impart anything that could with propriety be called a 

 knowledge of Mechanics without the aid of Trigonometry. 

 In the Composition and Resolution of Forces a i&w 

 questions made specially to order, which usually depended 

 on the facts that the sine and cosine of 45' are equal, 

 and that the sine of 30' is ^—although, of course, the 

 mention of a sine or cosine was inadmissible — exhausted 

 the field ; and the same restrictions were imposed on the 

 treatment of Moments, so that I was obliged to abandon 

 the task, and to insist on a small knowledge of Trigono- 

 metry and the sixth book. 



In such a cramped and stunted knowledge there is 

 nothing of spontaneity, nothing of power, but much of 

 danger. It is far better to give no encouragement to it, 

 to defer the attempt to study the elements of mathematical 

 physics until the old and well-recognized branches of 

 elementary pure mathematics have been studied with 

 some thoroughness. 



The want of thoroughness which seems to me to be so 

 prominently characteristic of our works on mathematical 

 physics sometimes exhibits itself, quite unconsciously on 

 an authors part, in a ludicrous manner. It may, perhaps, 

 be best described as " Calculus dodging." For some 

 curious reason, which I have never discovered, it has 

 been generally assumed that a student can possess a very 

 extensive knowledge of the results and principles of 

 Dynamics — of the composition and resolution of forces 

 and couples in three dimensions, of the principle of work 

 and energy, of the nature and properties of tubes of force. 

 Potential, &c. — without any knowledge of the Differential 

 or Integral Calculus. This is, surely, a piece of self- 

 deception. The processes of differentiation and of 

 elementary integration are not difficult of acquirement, 

 and it seems to me that they ought to be studied before 

 such an extensive inroid is attempted into Dynamics. 

 But, presuming that such knowledge is not possessed by 

 the reader, we find the author performing such an 



integration, for example, as that of ' between specified 



limits by a process which must strike an intelligent student 

 as at once most ingenious and most unnatural. Special 

 devices exhibited "for this occasion only " confer no inde- 

 pendent power on the student, and, moreover, require 

 him to possess an amount of ability which would be 

 much better and more successfully employed in acquiring 

 a knowledge of the principles of that Calculus which is 

 thus evaded by artifice. 



In all such cases it may be said, I think, that a student 

 who is capable of understanding the notions involved 

 will arrive at the results by other than the special artificial 



methods employed by the author ; while in the case of a 

 student for whom such inethods are necessary, most 

 probably the notions are unsuitable, and the method of 

 proof is apt to puzzle him with what seems to be mathe- 

 matical jugglery. 



If more time were spent in teaching the mathematical 

 principles on which quantitative physics depends, there 

 would be less need for such methods, and in the long run 

 the student of physics would be a gainer. 



But, while advocating a more thorough and leisurely 

 study of the elements of pure mathematics before the 

 study of physics, it seems to me that many of our 

 elementary text-books in mathematical physics make the 

 mistake of occupying the student's attention with 

 questions which, being far removed from physical reality, 

 and being, in fact, merely disguised mathematics, had 

 much better be omitted. We should make the effort to 

 make our works on physics as physical as possible, to use 

 Mathematics for the sake of Physics, and x\o\vice versa; 

 and it is time to recognize that the field of physics is now 

 so extensive as to supply subject-matter for calculation 

 and for illustration of mathematical principles, and to 

 permit us to curtail greatly the space which is now devoted 

 to things comparatively useless. 



Let me take one or two examples. Is anything gained 

 by teaching students the very numerous properties of the 

 curve which a particle would describe if it were projected 

 with any velocity, under the influence of gravity, if the 

 Eaith had no atmosphere ? 



Again, let us turn to any work on Hydrostatics, and we 

 are certain to find a very large number of mathematical 

 trivialities. I say nothing of a number of liquids, whose 

 densities are in some kind of progression, superposed in 

 very fine tubes of peculiar shapes. These are harmless ; 

 but what about that notion oi whole pressure on a curved 

 surface — the sum of all the normal pressures, or, more 

 strictly, the surface-integral of pressure over the curved 

 surface .^ 



Numerous, indeed, are the mathematical problems to 

 which this notion gives rise ; but what about the physical 

 idea involved .'' It is easy to comprehend the trouble which 

 a teacher lays in store for himself if he practises his 

 students in the process of adding together the magnitudes 

 of a number of forces whose lines of action are in all 

 directions in space. And remember that, in most cases, 

 very few students who are taught to calculate the whole 

 pressure of a fluid on a curved surface will subsequently 

 learn that a similar process has a physical meaning only 

 in the case of the normal flux of Newtonian gravitation 

 through a curved surface. Let us see, however, the 

 physical meaning which is actually attributed to the whole 

 pressure of a curved surface by the author of one of our 

 very elementary text-books for students. The following 

 is the literal statement : — " It should be observed that 

 these pressures act in different directions, the pressure at 

 each point being perpendicular to the surface at that 

 point. The whole pressure is the sum of all these 

 pressures, and represents the total strain to which the 

 vessel containing the fluid, or the body immer.-ed, is 

 exposed." Now in this statement, besides the addition of 

 the magnitudes of a number of non-co-planar forces, we 

 have the misuse of the term strain and the completely 

 unintelligible expression " total strain " of the vessel. 

 And, in illustration of this definition, we are given the 

 following example : "Showthat if a sphere or a cube be filled 

 with liquid, the total strain to which it is subjected is three 

 times the weight of the liciuid it contains." You will, I 

 hope, agree with me in holding that such teaching is in 

 the highest degree erroneous and objectionable, and that 

 the efforts of those who have to teach should be directed 

 against it. 



So far as my observation goes, the principle making 

 Physics a mere disguise — and a very unskilful one — for 

 pure mathematics is much too largely adopted by writers 



