NATURE 



265 



THURSDAY, JANUARY 18, 1900. 



THE PRINCIPLES OF MECHANICS. 

 Nnvtoris Lmvs of Motion. By Prof. P. (i. Tait. Pp. 

 viii + 52. (London : A. and C. Black, 1899.) 



HOW is the science of mechanics to be taught to 

 medical students who have to " get up " natural 

 philosophy in three months? If a teacher, confronted 

 with such a problem, took refuge in sheer "cram," his 

 action could cause no astonishment ; an attempt, such 

 as Prof. Tait has made, to provide a solution in which 

 cram has no place commands admiration, even if it 

 cannot be pronounced successful. The author's method 

 is to furnish the student with a set of perfectly recorded 

 lecture notes, and thus leave him free to follow the lec- 

 tures instead of taking notes of them. Perhaps no better 

 method can be devised, if the substance of the lectures 

 is as good as the record in the notes is perfect, and if 

 the student is made to apply the ideas explained in the 

 lectures to simple examples. The latter of these con- 

 ditions is doubtless fulfilled in Prof. Tait's classes ; we 

 are concerned here with what can be made out in regard 

 to the former. .After all the books that have been written 

 on the subject, there was still room for a pointed state- 

 ment of the principles of mechanics, with sufficient 

 detail and sufficient illustration,'but short; and such a 

 summary, if only it were precise and lucid, could not fail 

 to be useful to a class much wider than that immediately 

 in view ; but its value would be diminished in proportion 

 as it was mi-rked by vague statement, inexact definition 

 and loose argument. 



The work before us begins with three pages of intro- 

 ductory remarks, and these are followed by a chapter on 

 kinematics and a chapter on dynamics. In the intro- 

 ductory pages we find a statement of the laws of the con- 

 servation of matter and of energy, and a reference to 

 the laws of the inertia of matter and of the transforma- 

 tion of energy. As a specimen of the dogmatic tone 

 adopted we may quote the following : — 



"The objective realities of the physical world are of 

 two kinds only — matter and energy. Our conviction of 

 their objectivity is based on the experimental fact that 

 we cannot alter the quantity of either.'' 



No exception could be taken to this statement if the 

 evidence for it were going to be adduced ; and indeed 

 the kind of summary that is likely to be most useful is 

 just one that would trace the operation of the laws of 

 conservation and transformation of energy, and of the 

 law of the inertia of matter, in the processes of every- 

 day experience and in easily observed phenomena. In 

 picking out mass and energy as the two fundamental 

 conceptions the author is certainly right, but much 

 depends on the way in which they are discussed. 



The same wisdom in the selection of the topics to be 

 treated is apparent in the chapter on kinematics, and 

 the arrangement also of these topics is excellent ; it is 

 to some of the details that exception must be taken. 

 There is a general discussion of vectors., but the definition 

 which is given of a vector is incomplete, and the neces- 

 sary distmction between a vector associated with a par- 

 NO. 1577, VOL. 61] 



ticular line and one for which all parallel lines are 

 equivalent is not explained. In the definition which is 

 given of the moment of a vector, the fact that the rule of 

 signs is a part of the definition is lost sight of, and the 

 reason why the moment itself should be regarded as a 

 vector is obscure. The proof, on p. 20, that angular 

 velocity is a vector involves a petitio principii. In this, 

 as in the proof of the parallelogram of velocities, what is 

 most required is an explanation of the sense in which a 

 point can be said to have two simultaneous velocities, or 

 a body two simultaneous angular velocities. The defi- 

 nition of velocity is always one of the stumbling-blocks 

 in the way of students ; the author avoids giving a defi- 

 nition ; he says, " Speed need scarcely be defined, as 

 every one knows what it means." It may be that the 

 writers of current text-books know what it means ; they 

 seem quite unable to explain it ; nearly all of them pro- 

 ceed in a vicious circle, saying that the velocity of a 

 point when variable is measured at any instant by the 

 space that would be passed over in a unit of time if the 

 velocity continued the same as it is at the instant— as 

 well might one define the curvature of a curve at a point 

 as the angle that would be contained between the tan- 

 gents at the ends of an arc of unit length if the curvature 

 continued the same all along the arc as it is at the point. 

 If the student is not meek he will ask, " But what is.;^,at 

 the point.'*" It is only because he thinks he "knows 

 what it means " that he does not ask the like question 

 about speed. Prof. Tait gives countenance to the widely- 

 spread vicious definition without reproducing it. Surely 

 he might have spared some space to explain the mathe- 

 matical notion of a limit, and to define velocity as a 

 limit. Why do writers of elementary books treat the 

 student as a baby when any limit is in view, and talk to 

 him, for example, about "the next point" to a point on a 

 curve ? (p. 9). He knows as well as his teacher that there 

 is no next point. The commonly received absurdities 

 about what are really limiting processes secure acqui- 

 escence by frequent repetition, but they foster in the 

 mind of the student a belief in the unreality of the 

 whole business. 



The chapter qn dynamics suffers from defects which 

 are not merely faults of detail, but arise from the position 

 taken up, viz., that Newton's laws of motion still form the 

 simplest foundation of the subject. Some remarks in 

 Prof. Tait's address to Section .A. of the British Associa- 

 tion in 1871, dealing with the use of Euclid's " Elements" 

 by British mathematicians in the teaching of elementary 

 geometry, apply, with at least equal force, to the use of 

 Newton's laws of motion in the teaching of elementary 

 dynamics. He said of the teachers : 



"They seem voluntarily to weight alike themselves and 

 their pupils for the race ; and a cynic might, perhaps 

 without much injustice, say they do so that they may 

 have mere self-imposed and avoidable difficulties to face 

 instead of the new, real, and dreaded ones." 



The defect of the laws of motion as a statement of 

 the principles of dynamics is not that the principles are 

 not implicitly contained in the laws, but that the prin- 

 ciples have to be extracted from the laws, and that the 

 laws themselves are stated in terms of insufficiently de- 

 fined abstractions. How much of the contents of the 



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