26 



NA TURE 



[May 1 1, 1905 



great floods of the Seine ; and thirty-one tables aru 

 appended at the end of the volume, g^i'^ing the rain- 

 fall, discharges, and water-levels at different dates in 

 various parts of the Seine basin, and eleven sets of 

 graphic curves indicating the decrease in the dis- 

 charges of the Seine, some of its tributaries, and 

 certain sources, at different periods. Table xxiii., 

 giving the rainfalls of the warm seasons, and the 

 high floods of the following cold seasons, at the Auster- 

 litz Bridge, Paris, and at Mantes, from 1874 to 1900, 

 shows that none of these warm seasons in which the 

 rainfall was below the mean of 14.8S inches, was 

 followed by floods of the Seine rising higher than 

 14.44 feet on the gauge at Paris, and 19-72 feet at 

 Mantes ; and the eight cold seasons in which the 

 Seine reached or exceeded 16.40 feet at Paris, and 

 21.06 feet at Mantes, were all preceded by warm 

 seasons in which the rainfall exceeded the mean. 

 Moreover, with the exception of 1890, when the warm 

 season came between two very dry cold seasons, all 

 the warm seasons having a rainfall above the average 

 have been followed by floods of the Seine, attaining 

 at least 10-17 feet at Paris and 16.40 feet at Mantes; 

 whereas none of the fourteen warm seasons with a 

 rainfall below the average was succeeded by floods 

 in the next cold season, reaching the height attained 

 in eight of the cold seasons preceded by warm seasons 

 in which the rainfall exceeded the average. 



A nKW AMERICAN WORK ON THE 

 CALCULUS. 

 Elements of the Differential and Integral Calculus. 

 By William Anthony Granville, Ph.D., with the 

 editorial cooperation of Percey F. Smith, Ph.D. 

 Pp. xiv + 463. (Boston and London : Ginn and Co.) 

 Price lox. 6d. 



THIS is a book the main object of which seems to 

 be to enable the student to acquire a knowledge 

 of the subject with little or no assistance from a 

 teacher; and, after a very careful study of it, we are 

 enabled to say that the work is admirably constructed 

 for the purpose. There is a complete absence of the 

 stilted formality which is usually supposed to be 

 appropriate to a mathematical treatise. In foot-notes, 

 and sometimes in the text, the student is given scores 

 of useful hints and warnings against errors into which 

 he would probablv fall. Thus the work possesses a 

 verv high value for the student ; and it will be found 

 no less helpful to the teacher, for it contains a very 

 large number of examples in every part of the subject, 

 while it abounds in excellent diagrams. 



The portion on the differential calculus occupies 2S5 

 pages, and terminates with 6 pages containing 

 nothing but figures of all the curves more or less 

 famous which present themselves in the subject, such 

 as the conchoid of Nicomedes, the cycloid, the 

 catenary, the cissoid of Diodes, the probability curve, 

 various spirals, &c. 



The work is very strictly logical in its method — here 

 and there a little too much so, perhaps. 



Thus in p. 97 the proof that the angle between the 

 radius vector and the tangent to a curve has rdBjdr 

 for its tangent is quite unnecessarily accurate, and 

 NO. 1854, VOL. 72] 



has involved an error in work, which, however, is a 

 mere slip. The theorem of mean value is very well 

 explained and used in the deduction of Taylor's 

 theorem for the determination of the remainder, a 

 little geometrical figure assisting the student to under- 

 stand the nature of this remainder. (Correct, how- 

 ever, the errors in sign in the first equation of 

 p. 169.) 



The discussion of the convergency and divergency 

 of series is very good, and a somewhat uninteresting 

 subject is rendered simple and attractive. An in- 

 cautious statement, however, is made with regard to 

 an alternating series, p. 2d.i, according to which if we 

 stop at the nth term of such a series the error made 

 is numerically less than the value of the (n+i)th 

 term. Clearly this is not in general true if the 

 alternating series is one in which the numerical values 

 of the terms increase for a while and then diminish. 

 For example, the series for sin x is an alternating 

 one of this kind. If x = s„ the numerical values do 

 not begin to diminish until after the third term. The 

 property asserted, and the proof in p. 226, must be 

 applied to cases in which we stop after the greatest 

 numerical term has been passed. 



The theory of maxima and minima is well illus- 

 trated by examples taken from various branches of 

 physics. Even at the risk of being a little hyper- 

 critical, we must, however, point out that the time 

 taken by a ball to roll down a plane the base of 

 which is of length a and the inclination of which is 

 <^ is not 2tjajg sin 20, as it is said to be in p. 128. 

 for the simple reason that the acceleration of the 

 centre of the ball (if the ball is solid and homo- 

 geneous) is not g sin <!>, but 5/7 g sin 0. This fact 

 is of importance in dynamics, and the matter should 

 be set right. 



The part of the book dealing with curves is very 

 good, and, in particular, we would commend the 

 systematic manner in which (pp. 267, 268) the student 

 is taught to trace a curve from its equation. 



In the portion dealing with the integral calculus 

 an exhaustive exposition of all the devices used in 

 integrating functions is given. The reduction 

 formulas to be applied to the binomial integral 

 ix^{a+hx^^)Pdx are given in tabular form on p. 345, 



and the student is told very properly that he should 

 not memorise them. Instead of memorising them, 

 he should apply a single simple rule which was given 

 long ago by Hymers in his "Integral Calculus." 

 This rule enables us to obtain, witliout an effort of 

 memory, the exact formula appropriate to the reduc- 

 tion of any given binomial integral. 



Besides areas and volumes (accompanied by excel- 

 lent figures), polar moments of inertia of plane areas 

 are dealt with. The author speaks of these ae 

 moments of inertia about " a point " — an expression 

 which leaves something to be desired, since it is 

 always an axis that is involved. What we always 

 require in this connection in dynamics is the mean 

 square of distance of a body from an axis, and we 

 should look to writers on the calculus to emphasise 

 this notion of a mean square of distance, instead of 

 the "square of the radius of gyration," k-. The 



