NA TURE 



361 



THURSDAY, FEBRUARY 16, i! 



KINEMATICS AND DYNAMICS. 

 An Elementary Treatise on Kinematics and Dynamics. 

 By James Gordon MacGregor, M.A., D.Sc, &c , Munro 

 Professor of Physics, Dalhousie College, Halifax, N.S. 

 (London: Macmillan and Co., 1887.) 



THE logical order of arrangement has been carefully 

 attended to in this book : Part I., on " Kinematics," 

 building up a new subject on the foundation of Euclid's 

 axioms in conjunction with the idea of the variables, such 

 as velocity and acceleration, due to the flow of time ; 

 while Part II., on " Dynamics," requires three new axioms 

 — Newton's Laws of Motion — to make a fresh start and 

 connect mechanical effects with their causes. 



But it is doubtful if the strictly logical order is the best 

 order for the student to make his first acquaintance with 

 a new mathematical subject : the ideas must grow in his 

 brain by accretion round simple fundamental problems. 

 A student would master the present treatise more easily by 

 reading Part II. first, and referring back to Part I. as occa- 

 sion required, for the explanation of the details of the 

 mathematical calculations. There is nothing to prevent 

 this order of study here, although the author has, from 

 logical considerations, placed the kinematical part first. 



One defect of the logical system is that it places some 

 of the most difficult parts of the subject in the way of 

 beginners : for instance, the theory of the change of units, 

 a theory of which the importance can only be appreciated 

 by those who have made considerable progress in the 

 subject. 



In Part I., " Kinematics," the treatment is simple and 

 concise, but we should like to see more examples of 

 phenomena on a large scale, such as those of physical 

 astronomy, or even of railway-train problems. 



In questions involving the size of the earth (pp. 74 and 80) 

 it is the circumference and not the diameter which should 

 be given in metres, the circumference being 40,000,000 

 metres, a kilometre being a centesimal minute of latitude. 

 Or, if the size of the earth is given in miles, it is the 

 nautical mile which should be used, the circumference of 

 the earth being 360 X 60 = 21,600 nautical miles, a 

 nautical mile being a sexagesimal minute of latitude. 



The expression " knots an hour " (p. 60) is irritating 

 to a sailor, as emanating from the engine-room ; the 

 proper nautical expression is "knot" simply, a speed of 

 10 knots being 10 nautical miles an hour. 



The formula \v'^ = \v^ + as is to be preferred to that 

 on p. 34, v^ = v^ + 2asj in all cases the factor \ 

 should go with the v"' in the equation of energy, so that 

 the objectionable expression " vis viva " may finally be 

 stamped out from all dynamical treatises. 



In dealing with rotation, in Chapter V., the author 

 would do well to study Maxwell's geometrical representa- 

 tion of the direction by means of the screw, right- 

 handed or left-handed ; and to discard all attempts by 

 comparison with a clock-wise or counter-clock-wise rota- 

 tion, requiring as these do a specification of the aspect of 

 the plane of motion. 



Pure homogeneous strain is analyzed in Chapter VII. 

 as far as is possible by simple geometrical methods ; such 

 a strain may be produced by the superposition of three 

 Vol XXX VI] - Ko. 955. 



linear strains in directions at right angles to one another. 

 In a linear strain the increment of distance of two points 

 in the line of the strain is properly their elongation; 

 while the ratio of the elongation to the original distance 

 is called the extension, not the elongation, as on p. 167. 



In Part II., " Dynamics," we find in Chapter I. the dis- 

 cussion on the units of measurement of weight, mass, 

 and force customary in mathematical treatises, and of the 

 usual unsatisfactory nature. The author, disregarding 

 the vernacular use of the word " weight," defines the 

 weight of a body as the force with which it is attracted by 

 the earth, but is at variance with his own definition in the 

 statement of the majority of the subsequent examples, 

 relapsing into the language of ordinary life. A collection 

 of 500 different ways of spelling the name of the town of 

 Birmingham has been made, and a similar collection 

 could be made from the present treatise of different ways 

 of expressing the simple ideas of the pound weight 

 and the pound force, to use the ordinary language of 

 practical men. The attraction of the earth on a pound is, 

 in the vernacular, " the force of a pound," not the 

 " weight of a pound," the latter implying what the mathe- 

 matician likes to distinguish as the "mass of a pound." 

 Thus a mathematical precisionist, to express the simple 

 idea of a force of 10 pounds, to be consistent should call 

 it " a force equal to the weight of the mass of 10 pound 

 weights,'' the absurdity of which is evident. 



Again, in straining after the equation F = ma, when 

 using the gravitation unit of force, the mathematician in 

 the F.P.S. (foot-pound-second) system of units is obliged 

 to use the variable unit of mass of ^ pounds to measure 

 the invariable quantity, the mass of the body ; while 

 what he calls the weight of the body, and denotes by iv, 

 measuring it in pounds, is, although variable with g, 

 always measured by the same number. 



Next we have the equation w — mg, the source of 

 all the confusion in dynamical teaching, and only to avoid 

 writing the dynamical equation with gravitation units in 

 the form 



This terminology culminates in the solecisms that on 

 p. 477 we must suppose pressure to be measured in 

 poundals on the square foot in hydrostatical problems ; 

 and that if the equation w = mg is supposed to be used 

 with absolute units, that the weight of a body is measured 

 in poundals ; as if a mathematician asked in a shop for 

 " half a poundal of tea, or tobacco." Ordinary people 

 measure weight in pounds, so that if mass is also measured 

 in pounds, then w = m. 



It is time now, as Prof. Minchin has pointed out, that 

 "the astronomical unit of mass," defined in § 315, should 

 disappear, and that in all problems of physical astronomy 

 the gravitation constant k should be retained, while m, 

 the mass, is measured in terms of the ordinary units. 



Although the author does not allow himself the use of 

 the methods and notation of the Calculus, still he has 

 managed to discuss a number of interesting problems 

 in the dynamics of a rigid body, usually proved by the 

 methods of Analytical Mechanics. 



Working under these restrictions, he has given elegant 

 elementary proofs of the chief properties of the common 

 catenary ; but here, again, it is time that the equation 



R 



