October 24, 1919] 



SCIENCE 



381 



and will content myself with only a few gen- 

 eral observations. First, then, it would, I 

 think, be conducive to progress if it were more 

 generally recognized that dynamics is a phys- 

 ical subject, and only secondarily a mathe- 

 matical one. Its study should be carried on in 

 the departments of physics, not in those of 

 mathematics or in separate departments of ap- 

 plied mathematics. It is, or ought to be, 

 essentially a subject of the physical lecture- 

 room and the physical laboratory. The stu- 

 dent should be able to handle rotating bodies, 

 to observe and test the laws of precession and 

 nutation, to work himself, in a word, into an 

 instinctive appreciation of at least the simpler 

 results of rotational theory. He should learn 

 to think in vectors, without necessarily refer- 

 ring either to Hamilton or to Grassmann. 

 Some people appear to censure the use of vec- 

 tor ideas without the introduction at the same 

 time of some form of vector notation. I do 

 not feel drawn to any system of vectors in par- 

 ticular — all have their good points, and in 

 some ways for three dimensional work the 

 quaternion analysis is very attractive — but 

 vector ideas are of the very utmost importance. 

 Hence I deprecate the .teaching, however ele- 

 mentary, which as a beginning contents itself 

 with rectilineal motion. The true meaning of 

 rate of change of a directed quantity, even of 

 velocity and acceleration, is missed, and in- 

 stead of having laid a foundation for further 

 progress the teacher, when he desires to go 

 beyond the mere elements, has practically to 

 relay his foundations, has in fact to extract 

 imperfect ideas from his pupils' minds and 

 substitute new ones, with the result that a 

 great deal of avoidable perplexity and vexa- 

 tion is produced. The consideration of the 

 manner of growth of vectors — the resultant 

 vector or it may be component vectors, accord- 

 ing to convenience — is the whole affair. As 

 an illustration of what I mean, take this: A 

 vector quantity has a certain direction, and 

 also a magnitude L. It is turning in a certain 

 plane with angular speed w. This turning 

 causes a rate of production of the vector quan- 

 tity about a line in that plane and perpendic- 

 ular to the former, and towards which the 



former is turning, of amount La. Thus a 

 particle moving in a curve with speed v has 

 momentum mv forwards along the tangent at 

 the position of the particle. The vector is 

 turning towards the principal radius (length 

 B) of curvature at the point at rate v/R. 

 Hence towards the center of curvature mo- 

 mentum is growing up at time rate mv'/R. 



Dealt with in this way, with angular mo- 

 mentum instead of simple momentum, the 

 motions of the principal axes of a rigid body 

 give the equations of Euler instantly and in- 

 tuitively, and all the mind-stupefying notions 

 of centrifugal couples, and the like, are swept 

 away. 



With regard to mathematics, the more the 

 physicist knows the better, and he should con- 

 tinually add to his store by making each phys- 

 ical subject he takes up a starting-point for 

 further acquisition. Some very philistine no- 

 tions as to mathematics prevail, and are very 

 mischievous. For example, I once heard an 

 eminent practical engineer declare that all the 

 calculus an engineering student requires could 

 be learned in an hour or two. This is simply 

 not true, nor is it true, as some exponents of 

 ultrasimplicity seem to suggest, that the pro- 

 fessional mathematical teacher wilfully makes 

 his subject difficult in order to preserve its 

 esoteric character. Like the engineer or physi- 

 cist himself, he is not always so simple as he 

 might be; but the plain truth is that no good 

 progressive mathematical study can be carried 

 out without hard and continued application of 

 the mind of the student to the subject. And 

 why should he depend on the mathematical 

 teacher? Let him be his own teacher ! There 

 are plenty of excellent books. If he has a de- 

 termination to help himself he will, if he 

 makes a practise of reserving diiSculties and 

 returning to them, find them vanish from his 

 path. 



As I have said, I am specially interested in 

 rotational dynamics. In the course of the 

 war I have been appalled by the want of ap- 

 preciation of the principles of this subject, 

 which, in spite of considerable acquaintance 

 with the formal theory, seemed to prevail in 

 some quarters. I don^t refer to mistakes made 



