352 



SCIENCE. 



[N. S. Vol. V. No. 113. 



the method of the latter, and that alone, the 

 source of aesthetic impression. In any case 

 the theory of Groos, which has its roots in the 

 views of Lange and v. Hartmann, is extremely 

 interesting and valuable, especially as con- 

 trasted with the recent psychological theory of 

 Mr. H. E. Marshall. In the present theory, the 

 ' self-exhibition ' of which Mr. Marshall makes 

 so much, enters as the need of impressing others 

 with the play illusion. As to the hedonic ele- 

 ment and its ground, however, the two theories 

 are in sharp contrast, and that of Groos seems 

 to me, on the whole, more adequate. In the 

 wealth of literary reference in his book Mr. 

 Marshall pays singularly little attention to the 

 authors from whom Groos draws, and none to 

 the earlier work of Professor Groos himself, 

 but treats the play theory only in the form of 

 Mr. Spencer's surplus energy construction. As 

 to Groos' theory musical art would present dif- 

 ficulties and so would lower sensuous aesthetic 

 effects generally. 



Genetically art rests upon play, according to 

 Herr Groos, in that the three great motives of 

 art production, ' Self-exhibition ' {Selbstdarstel- 

 lung),' Imitation,' and 'Decoration' (.4 itssc/tTOMcA;- 

 wng), are found in the three great classes of 

 animal plays, respectively, 'Courting,' 'Imita- 

 tion,' and Building Art' {Baukiinste, seen in 

 birds' nest-building, etc.). On the strength of 

 this, Groos finds both sesthetic appreciation and 

 impulse in the animal, and all rests upon the 

 original ' experimenting ' impulse. Of this, 

 however, Professor Groos does not give a satis- 

 factory account. Experimenting is a necessary 

 part of effective learning by ' imitation,' I think, 

 and the use made of it in the selection of move- 

 ments may be its original use. 



On the whole. Professor Groos' book is both a 

 pioneer work and one of great permanent value ; 

 it should be translated into English. It con- 

 tains a good index and a full list of the literary 

 sources. 



J. Maek Baldwin. 



Peinceton. 



A Primer of the History of Mathematics. By W. 

 "W. Rouse Ball. London, The Macmillan 

 Co. 1895. Pp. 148, 16mo. Price, 65 cents. 



A History of Elementary Mathematics, with hints on 



methods of teaching. By Floeian Cajoei. 



New York, The Macmillan Co. 1896. Pp. 



viii-f304, 12mo. Price, $1.50. 



The object of the ' Primer,' as well set forth 

 in its introduction, is "to give a popular account 

 of the history of mathematics, including therein 

 some notice of the lives and surroundings of 

 those to whom its development is mainly due, 

 as well as their discoveries. Such a sketch, 

 written in non-technical language and confined 

 to less than 140 pages, can contain nothing be- 

 yond a bare outline of the subject, and, of 

 course, is not intended for those to whom it is 

 familiar." It consists of the author's larger 

 work* reduced in size by the omission of all 

 detailed and highly technical matter. In a few 

 places the pruning process has been carried too 

 far. For example, on p. 13 we are told that 

 "after the execution of Socrates, in 399 B. C, 

 Plato spent some years in travel * * * " but we 

 are given no clue to the relationship of Socrates 

 to Plato. However, the few instances of this 

 kind which occur do not appreciably detract 

 from the clear, well ordered and interesting 

 style which the ' Primer ' enjoys in common 

 with its source. 



The book aflbrds to students in our high 

 schools and colleges a means of gaining, with a 

 small expenditure of time, a sufficiently com- 

 plete history of the mathematical subjects they 

 are studying, to give them a much greater ap- 

 preciation of and interest for such subjects. 



As its title indicates. Professor Cajori's book 

 does not cover the entire field of mathematics ; 

 he restricts it to arithmetic, algebra, geometry 

 and trigonometry, as presented in undergradu- 

 ate instruction, with a short account of the his- 

 tory of non-Euclidean geometry. The arrange- 

 ment of the material is first under the headings : 

 'Antiquity,' ' Middle Ages,' 'Modern Times;' 

 under each of these are the subdivisions : ' arith- 

 metic,' 'algebra,' 'geometry' and 'trigonom- 

 etry.' For a work of its size it contains a great 

 deal of information, and nearly every statement 

 is supported by a reference either to original 

 sources or to other treatises upon mathematical 

 history. The chapters upon arithmetic are par- 



*A short account of the History of Mathematics. 

 London, the Macmillan Co. 2d edition. 1893. Pp. 

 xxiv-j-520, 16mo. 



