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109 



THURSDAY, OCTOBER 9, 1919. 



MATHEMATICAL TEXT-BOOKS. 



(1) Empirical Formulas. By Prof. Theodore R. 

 Running. (Mathematical Monographs, No. 19.) 

 Pp. 144. (New York : John Wiley and Sons, 

 Inc. ; London : Chapman and Hall, Ltd., 1917.) 

 Price 75. net. 



,(2) Differential and Integral Calculus. By Dr. 

 H. B. Phillips. Differential. Pp. v+162. 

 Integral. Pp. v+194. (New York: John 

 Wiley and Sons, Inc. ; London : Chapman and 

 Hall, Ltd., 1916-17.) Price 9^. 6d. net. 



(2) A First Course in the Calculus. Part i. 

 Powers of X. By Dr. W. P. Milne and 

 G. J. B. Wcstcott. (Bell's Mathematical 

 Series.) Pp. xx+196. (London: G. Bell and 

 Sons, Ltd., 1918.) Price 35. 6d. 



(4) Dynamics. Part II. By R. C. Fawdry. 

 (Bell's Mathematical Series.) Pp. viii + 

 179-355 + vii. (London: G. Bell and Sons, 

 Ltd., 1919.) Price 2^. 6d. 



(5) Solid Geometry, including the Mensuration of 

 Surfaces and Solids. By Prof. R. S. Heath. 

 Fourth edition. Pp. iv+123. (London: 

 Rivingtons, 1919.) Price 4s. 



((i) /^NE of the most important activities of 

 ^^ the practical, as well as of the theo- 

 retical, man of science is the discovery of laws. 

 Given a number of observations, the problem is to 

 correlate them in the form of a single analytical 

 expression. The basis of such discovery is the 

 recognition of a curve as being one the equation 

 of which is known. But, strictly speaking, there 

 is only one curve that is really recognisable, and 

 this is the straight line. A piece of a circle can 

 easily be mistaken for a piece of an ellipse, and a 

 parabola for a catenary ; but if a sufficiently long 

 piece of a curve is straight, then the curve can 

 be pronounced to be a straight line. If, then, it 

 is possible to plot the results of observation in 

 such a way that the resulting points lie on a 

 straight line (even if there are some casual, ex- 

 perimental errors and consequent deviations), 

 then we can at once deduce the law. 



This fact underlies the major part of Prof. 

 Running's monograph. The author summarises 

 the most useful types of laws that are reducible to 

 straight-line laws by means of simple transforma- 

 tions. He also gives practical rules for deciding 

 whether such a law is correct for the given data 

 and for the determination of the constants. Nine- 

 teen laws are discussed, and illustrated by means 

 of numerical examples, whilst curves are drawn to 

 show graphically the types of relations given by 

 these laws. 



A twentieth law is the Fourier expansion. 

 Chapters are added on the method of least 

 squares, interpolation, and numerical integra- 

 tion. The result is an eminently useful hand- 

 book for the scientific researcher and the practical 

 engineer, and a highly commendable adjunct to 

 the more theoretical study of mathematics. 

 NO. 2606, VOL. 104] 



The pedagogy is, however, somewhat defective. 

 It is difficult to imagine such a book in the hands 

 of a student. The philosophy of the subject is 

 scarcely entered into at all, and in places, where 

 an attempt at justification is made, the result is 

 not satisfactory. Also, one question remains 

 unanswered : How is one to guess which law to 

 try? Is one to try them all one after the other 

 until the right one is reached? And what if none 

 of those given is correct? Information on this 

 and other points is very desirable, but none is 

 offered. 



One or two definite criticisms must be made. 

 The different schemes in the chapter on Fourier 

 series are not always consistent, and some are 

 incomplete. No explanation is given of the mean- 

 ing of "weights " in the method of least squares. 

 In the chapter on interpolation the difference 

 formula is proved only for integral values of the 

 argument, and then applied to fractional values. 

 In addition, there are a few misprints and some 

 evidences of carelessness. The book thoroughly 

 deserves a second edition, in which, it is hoped, 

 these and other faults will be rectified. 



(2) This is a very good book on the calculus, 

 written in_ the old style with which we have been 

 familiarised by writers like Edwards and William- 

 son. It is very well written and compact in 

 form ; the diagrams are good, and the exercises 

 excellent. Particular attention is paid to ques- 

 tions of a practical nature. The student, who has 

 worked through this book conscientiously will 

 have a good, if dull, appreciation of the subject' 

 and its manipulation. A few of the pages are 

 headed "Unconventional Methods," but the thrill 

 one gets on seeing this only leads to disappoint- 

 ment. There is nothing unconventional in an in- 

 volute or in a parabola rotating about an axis. 



The second part (which is also issued separ- 

 ately) includes the usual chapters on differential 

 equations and the usual box of tricks. 



This book, like so many others, gives the 

 student the impression that there is just one par- 

 ticular integral of a linear differential equation 

 in which the right-hand side is a function of the 

 independent variable. It is more useful to inform 

 the student that there are, of course, an infinite 

 number of particular integrals, but that one of 

 them is obtainable most readily and directly. 



In Ex. 3, p. 10, of the second part it would 

 have been more reasonable to put a negative sign 

 to indicate the retarding effect of friction. 



(3) Dr. Milne and Mr. Westcott have given 

 expression to an important and fundamental prin- 

 ciple in mathematical pedagogy — namely, the 

 secondary nature of the manipulative art, and the 

 first-rate importance of the ideas and methods of 

 mathematics. They have recognised that the 

 main part of the essence of the calculus, and even 

 the most important practical applications of its 

 processes, can be taught and learnt without using 

 anything but the simplest of all functional types 

 --namely, a" and combinations of powers of x. 

 When once the student has learnt to differentiate 

 .x'\ he is ready for much of the mysterious dis- 



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