392 



SCIENCE 



[N. S. Vol. XXIX. No. 740 



honor have not appeared to make rigid quota- 

 tion essential. Theo. B. Comstock 

 Los Angeles, Cal., 

 January 15, 1909. 



Veber das Wesen der MathematiJc. Eede 

 gehalten am 11 Marz, 1908, in der offent- 

 lichen Sitzung der k. Bayerischen Akademie 

 der Wissensehaften. Von Dr. A. Voss, 

 Professor der Mathematik in Miinchen. 

 Pp. 98. Leipzig tind Berlin, B. G. Teubner. 

 1908. 



The numerous and valuable earlier publica- 

 tions of the author of the present address 

 inspire confidence in his ability to treat such 

 a general subject in a scholarly and helpful 

 manner. The reader will find that this confi- 

 dence has not been misplaced, for the address 

 is not only replete with important suggestions 

 in regard to fundamental questions in mathe- 

 matics, but it also emphasizes those elements 

 which point towards rapid progress in the 

 near future and thus awaken a healthy op- 

 timism. It seems especially suited to widen 

 the outlook and to arouse energizing enthu- 

 siasm on the part of the young mathematician 

 who may fail to appreciate the dignity and the 

 beauty of abstract thought. 



The author begins his address by the state- 

 ment that we are living in the epoch of nat- 

 ural sciences and technology, and he quotes 

 approvingly the remarks of Galileo : 



True philosophy explains nature, but no one 

 can understand her except those who have learnt 

 the language and the symbols by means of which 

 she speaks. This language is mathematics and 

 the symbols are mathematical figures. 



The bearing of mathematics just mentioned 

 tends to explain why this subject is constantly 

 taking deeper root in the educational systems 

 of the world, notwithstanding the fact that 

 it is " the most unpopular of all the sciences ; 

 it is a part of the essence of a true science to 

 be unpopular." 



The brief introductory remarks are followed 

 by a rapid sketch of some fundamental facts 

 in the history of mathematics. Beginning 

 with the Egyptian work, written by Ahmes 

 nearly four thousand years ago, which claims 

 to give " Directions to obtain a knowledge of 



all dark things, all secrets contained in the 

 things," our author considers the historical 

 development of a number of fundamental 

 mathematical concepts and symbols. He gen- 

 erally follows the " Prince of mathematical 

 historians," Moritz Cantor. In one instance, 

 however, he adopts a view which is not in 

 accord with the most recent work of Cantor,- 

 viz., as regards the question of the origin of 

 zero and the positional arithmetic. Ten years 

 ago it was generally believed that these dis- 

 coveries were due to the Hindus, while the 

 most recent work of Cantor makes a Baby- 

 lonian origin appear much more plausible. 



As may be inferred from the heading of the 

 address, emphasis is placed upon those mathe- 

 matical concepts which border on the domain 

 of philosophy. Among the questions which 

 receive considerable attention are the follow- 

 ing: definitions of mathematics, relations be- 

 tween mathematic and logic, the development 

 of the concept of number, higher complex 

 number systems and different points of view 

 as regards ordinaTy complex numbers, differ- 

 ent theories in regard to ordinary fractions 

 and irrational numbers, continuity and limit, 

 importance of the concept of function, and 

 suggestions as to changes in the subject-matter 

 to be used for instruction in secondary schools. 

 The address is written in a popular style and 

 should interest the man of general culture 

 as well as the professional mathematician. 

 G. A. Miller 



Univeesity of Illinois 



SCIENTIFIC JOURNALS AND ARTICLES 

 The Journal of Experimental Zoology, Yol. 

 VI., No. 1 (January, 1909), contains the fol- 

 lowing papers: "A Study of Growth in the 

 Salamander, Diemictylus viridescens," by Ada 

 Springer. " Studies on Chromosomes — IV., 

 The Accessory Chromosome in Syi'omastes 

 and Pyrrochoris, with a Comparative Review 

 of the Types of Sexual Differences of the 

 Chromosomes," by Edmund B. Wilson. This 

 paper is devoted to a reexamination of two 

 forms heretofore studied by Gross. It shows 

 that sex-production in these forms agrees in 

 principle with that seen in other insects. In 

 Pyrrochoris the spermatogonial number is 23 



