February 12, 1897.] 



SCIENCE. 



279 



of the author's own investigations on the errors 

 of interpolated values. This chapter will form an 

 excellent supplement to a course on the method 

 of least squares. 



Professor Byerly's chapter on harmonic func- 

 tions is a model of clear and attractive exposi- 

 tion, in a subject by no means easy of approach 

 to the beginner. It is, of course, largely based 

 on the author's more extensive text-book. After 

 showing on three particular examples how the 

 attempt to solve certain physical problems 

 naturally leads to Fourier series, to zonal and 

 cylindrical harmonics, the author discusses each 

 of these three subjects in some detail, illus- 

 trating every method by numerical examples, 

 some of which are worked out even to the 

 arrangement of the logarithmic work. Nothing 

 could be more useful to the student of applied 

 mathematics, while the pure mathematician 

 may regret that the constant occupation with 

 methods of solving certain problems leaves no 

 room for inquiring into the real nature and 

 characteristics of the functions under discussion. 

 But, in a brief chapter, more than is here given 

 could hardly be expected. 



The introduction of numerous applications 

 and exercises, which is a general feature of 

 this volume, is also very prominent in Professor 

 McMahon's chapter on hyperbolic functions. 

 This chapter is perhaps more complete in itself 

 than any other chapter in the book. It gives a 

 very satisfactory exposition of the theory, with 

 graphical representations, seven pages of tables 

 and weli-chosen applied problems. 



Professor Johnson's excellent chapter on dif- 

 ferential equations is naturally one of the longest 

 in the book and also attains a certain degree of 

 completeness. 



On the other hand. Professor Merriman's 

 chapter on ' the solution of equations' appears 

 rather meagre, perhaps, because the author, as 

 one of the editors, felt boUnd to keep strictly 

 within the prescribed limits of space. A some- 

 what remarkable statement about the impos- 

 sibility of the algebraic solution of the general 

 quintic appears at the bottom of p. 22. After 

 referring briefly to the researches of Abel and 

 Galois, the author says: "Although these dis- 

 cussions are complex and not devoid of doubt," 

 (a foot-note gives an inaccurate reference to 



Kronecker and a reference to Cockle), "they 

 have been generally accepted as conclusive. 

 Moreover, the fact that the quintic is still un- 

 solved, in spite of the enormous amouiit of work 

 done upon it during the past two centuries, is 

 strong evidence that the problem is an impos- 

 sible one. ' ' Comment is unnecessary. 



The chapter on determinants contains more 

 than might be expected from its brevity. Pro- 

 fessor Weld's modesty in not referring anywhere 

 to his text-book on the subject is worthy of men- 

 tion. 



The geometrical calculus is represented by 

 two interesting chapters. The elements of 

 Grassmann's methods as applied to plane and 

 solid geometry are set forth at some length by 

 Professor Hj'de, while Professor Macfarlane 

 treats of vector addition and multiplication, 

 with particular reference to their application 

 in mathematical physics. The quaternion 

 proper, although it figures in the title of Chap- 

 ter IX., receives but slight attention. Both 

 chapters are far too brief to show the real 

 power of these methods, which appears espe- 

 cially when geometrical differentiation and in- 

 tegration are introduced. The present writer 

 cannot help regretting that Professor Hyde has 

 not adopted the remarkably elegant and simple 

 treatment of Gi'assmann's fundamental ideas 

 proposed by Peano. From the point of view of 

 pure mathematics Peano's method of laying the 

 foundation for a geometrical calculus can hardly 

 be improved upon. The physicist, however, 

 will probably, for some time to come, prefer to 

 become acquainted with vector analysis in close 

 connection with the development of his physical 

 and mechanical notions, in a manner similar 

 to that pursued by Oliver Heaviside in his 

 'Electro-magnetic Theory,' Vol. I. (1893). For- 

 tunately, Professor Macfarlane's methods and 

 notations do not seem to differ now very much 

 from Mr. Heaviside' s. The peculiarly cum- 

 brous notation for what might be called the po- 

 lar coordinates of a vector is an exception. 



In the last chapter Professor D. E. Smith 

 gives a rapid survey of the historical develop- 

 ment of the various branches of mathematics 

 during the nineteenth century. This rather 

 difficult task seems to be accomplished in a very 

 satisfactory way, the chapter being evidently 



