976 



SCIENCE 



[N. S. Vol. XXIX. No. 755 



modern critical movements in logic and math- 

 ematics. 



The remaining 2Y3 pages of the first volume 

 are devoted to books I. and II. of the " Ele- 

 ments." The entire discourse is conducted in 

 the most admirable manner. First are given 

 in clear, bold, beautiful type the definitions, 

 axioms and postulates of book I. Each of 

 these is then taken up in order, restated in 

 Greek followed by the English equivalent, and 

 made the subject of an elaborate and lucid 

 historical and critical commentary. The ex- 

 tent of Dr. Heath's commentaries may be in- 

 ferred from the fact that no less than eighty- 

 five large pages are occupied by discussion of 

 the thirty-three definitions, axioms and postu- 

 lates of the first book, the immense wealth of 

 the material presented being gathered from all 

 the principal sources from pre-Euclidean times 

 to the present. Then follows the statements 

 and demonstrations of the propositions of 

 book I. with critical discussion of each. The 

 remaining twelve books are treated in similar 

 fashion. The third volume closes with the so- 

 called "Book XIV." by Hypsicles, a note on 

 the so-called " Book XV.," a carefully prepared 

 list of addenda et corrigenda, an index of 

 Greek words and forms, and an excellent gen- 

 eral index to the entire work. 



Dr. Heath has been animated by no narrow 

 or partisan spirit. Like every other well-in- 

 formed student of mathematics, he is well 

 aware of the fact that the time is long since 

 gone when the reading of Euclid was indis- 

 pensable to one who would become a learned 

 or a productive geometrician. And in so far 

 as this new edition may be regarded as a plea 

 for the " Elements," it is an enlightened 

 plea, one entirely worthy of its great theme, 

 the noblest plea in English speech for the 

 Alexandrine classic. 



The content of Professor Murray's book has 

 been taken chiefly from his " Infinitesimal 

 Calculus," though some matter relating to 

 indeterminate forms, solid geometry and ap- 

 plications to motion has been added, and the 

 treatment of several topics has been revised. 

 Among the noteworthy features of the work 

 may be mentioned that it contains more than 

 can be properly read in the time usually given 



in the best schools to a first course in the cal- 

 culus ; that in view of this fact certain articles 

 have been indicated as constituting a course 

 suitable for " students having a minimum of 

 time " ; that the book begins with a discussion 

 of certain problems designed to foreshadow 

 the nature of the calculus and a presentation 

 of such algebraic notions as are of most fre- 

 quent use in the calculus; that the subject is 

 presented with modem regard — rather too 

 much than too little, if it errs in either re- 

 spect — for precision and rigor; that, though 

 the subject is divided into differential calculus 

 and integral calculus, the notion of anti-dif- 

 ferentiation is presented in connection with 

 that of the direct process ; that, with a view to 

 facility of applications, the view of integra- 

 tion as summation is put in advance of the 

 other view; that numerous examples including 

 simple applications to geometry, physics and 

 mechanics, with illustrative solutions, are in- 

 serted in connection with the development of 

 cardinal ideas and processes, a table of answers 

 being given at the end of the volume; that 

 unusual care has been given to accentuation of 

 important matters ; that especial attention has 

 been accorded to the concepts of speed, 

 velocity and acceleration; that, for alternative 

 or fuller treatment of numerous ideas, the 

 reader is referred to a good deal of the better 

 and more accessible literature; and that there 

 is given a brief introduction to differential 

 equations and a table of integrals. The work 

 is adequate to the needs of the college student, 

 to the technological student and to the rarer 

 spirit preparing for higher flights. 



Professor Johnson's book is an abridgment 

 of his " Differential Calculus " and is based 

 on the method of rates. It is distinguished 

 by the absence of the histological methods of 

 the rigorists. Things are presented pretty 

 much at their face values, being shown graph- 

 ically rather than laboriously proved by help 

 of the refined logical machinery brought in 

 from across the sea. The English is excellent, 

 everything moves along with the calmness, 

 dignity and facility of an elder day, and the 

 reader, while acquiring much useful knowl- 

 edge, will acquire also a degree of confidence 

 that in these critical times is apt to be rare 



