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SCIENCE 



[N. S. Vol. XXVII. No. 703 



mastered in a three-hour course of one year. 

 In view of the high excellence of the book one 

 hesitates to note so minor an infelicity as the 

 recurring phrase, tangent in a point; or to 

 query why the notion of limit is not defined 

 instead of being presupposed; or to question 

 the scientific or the didactic value of the cau- 

 tionary note (p. 5), for while it is tnie that 

 two coincident points of a curve do not de- 

 termine a secant, is it not also true that, if P 

 be a point of a curve admitting a tangent T 

 at P and if P' be a second point of the curve, 

 the secant PP', if P' move along the curve 

 into coincidence with P, at the same time 

 rotates about P into the definite position of 

 coincidence with T? The tangent T is not, 

 indeed, then determined by the mere coinci- 

 dence of P and P', but by that coincidence 

 regarded as having resulted from P' moving 

 along the curve. 



Professors Townsend and Goodenough's 

 " Course " is a notable contribution to the 

 text-book literature of the calculus. It is too 

 large by a hundred pages to admit of the 

 satisfactory presentation of the whole of it 

 in the time usually allotted to the subject 

 even in the best schools. The thickness of the 

 volume is partly due, however, to the presence 

 of a chapter dealing with ordinary differential 

 equations, an excellent table of integrals, a 

 table of answers, and a good index that ren- 

 ders the book a convenient work of reference. 

 The method of limits is employed exclusively. 

 The notion of integration is introduced at an 

 early stage, and topics are in general arranged 

 in the order of increasing difficulty, such 

 topics as infinite series, expansion of func- 

 tions, singularities of plane curves, envelopes 

 and the like being reserved for treatment when 

 the reader shall have had time to confirm his 

 grasp of fundamentals. It is especially note- 

 worthy that the book is a joint product of a 

 professional mathematician, who is chiefly re- 

 sponsible for the theory, and a professional 

 teacher of mechanical engineering, who is 

 largely responsible for the practical aspects of 

 the work. Indeed, the applications of the 

 subject are about equally distributed between 

 geometry and mechanics, a fact that should be 

 of interest to the student of engineering. 



though the book is by no means written for 

 him alone. 



Professors Woods and Bailey's book is the 

 first volume of a work in course of preparation 

 which is designed to present together — that is, 

 in a single course — so much of algebra, an- 

 alytical geometry, calculus and differential 

 equations as is usually required of engineering 

 students in the first two years of their pro- 

 fessional study. The attempt represents a 

 wholesome reaction against the long-prevailing 

 practise of presenting these subjects in as 

 many separate courses, and of thus incident- 

 ally giving the student the impression that the 

 several doctrines are essentially insulated and 

 independent, instead of being, as in fact they 

 are, but different parts of one complicate in- 

 strument or different organs of a single body 

 of doctrine. The experiment sufficiently com- 

 mends itself a priori to deserve a fair trial, 

 though this will not be easy in view jf the 

 readjustment of programs and schedides neces- 

 sarily involved. The danger of the reaction 

 lies, of course, in the opposite extreme, name- 

 ly, of so presenting a group of interpenetrating 

 disciplines that they shall produce the effect 

 of a mere melange. 



Professor Schultze's " Graphic Algebra " is 

 an excellent introduction to the plotting of 

 equations and therewith to the graphical rep- 

 resentation of functional dependence in gen- 

 eral. The method is illustrated in connection 

 with equations of the first four degrees in two 

 variables. An appendix extends the method 

 to other than equational relationships, and 

 furnishes for practise some tables of data 

 drawn from a considerable variety of fields. 

 The book is timely and should be interesting 

 to many, for this is indeed the age of coordi- 

 nates and graphical depiction, the method long 

 familiar in analytical geometry having proved 

 its availability in almost every field of study, 

 including even the critical study of biblical 

 literature. 



Professor Johnson's " Integral Calculus " 

 treats more fully than his earlier one on the 

 same subject of reduction formula and of 

 multiple integrals. It contains, besides, new 

 chapters dealing with mean values, probability, 

 definite integrals (including the Eulerian), 



