July 7, 1911] 



SCIENCE 



25 



milk for at least three days, and a smaller 

 degree of infection for tea days or even 

 longer. At the same time he shows that 

 blow-flies produce gross infection for six to 

 nine days with non-spore-bearing micro-organ- 

 isms and some degree of infection for three 

 or four weeks. The investigator thinks that 

 it is probable, at any rate in the later stages, 

 that infection is mainly due either to direct 

 infection with the crop contents vomited 

 through the proboscis, or to direct infection 

 by means of the limbs which have been rein- 

 fected with vomited material. 



These experiments were so conducted as to 

 afford no information as to the extent to 

 which house flies bred from larvffi fed on nat- 

 urally infected excreta and similar materials 

 are apt themselves to be infected. 



L. O. Howard 



Lectures on Fundamental Concepts of Alge- 

 hra and Geometry. By J. W. Young. Pre- 

 pared for publication with the cooperation 

 of W. W. Denton, with a note on the 

 growth of algebraic symbolism by W. G. 

 Mitchell. Pp. vii + 247. New York, The 

 Maemillan Company. 1911. 

 While the teacher of secondary mathematics 

 finds a large amount of English literature 

 on the teaching of his subject he looks in 

 vain for much that is well adapted to give 

 him a deep insight into the fundamental 

 theory of the subjects with which he has to 

 deal. The English language contains no 

 encyclopedia on elementary mathematics like 

 Weber and Wellstein's " Enzyklopiidie der 

 Elementarmathematik," or like the new 

 Italian encyclopedia which is being prepared. 

 It has no histories like Cantor's or even like 

 Tropfke's. It has no periodical like L'En- 

 seignement Mathematique, and no large 

 mathematical encyclopedias like the great 

 works which are now being published in Ger- 

 man and in French. 



Although the small size of the book under 

 review precludes any hopes that we, might 

 have here a work to which the teacher of 

 secondary mathematics may turn for an 

 answer to most of his questions, yet he will 



find here an unusually clear exposition of a 

 large number of things relating to the logical 

 foundation of algebra and geometry. The 

 brevity of the exposition will doubtless be 

 welcomed by many who are looking for a first 

 general survey of some basic matters, and it 

 is to be hoped that they may become suffi- 

 ciently interested to pursue the thoughts 

 further, as they are encouraged to do by a 

 fair number of references. 



The book is modern in spirit, and, to a 

 large extent also, in subject matter. Consid- 

 erable attention is given to historical settings 

 but the logical element receives the greatest 

 emphasis. It opens up view points which are 

 of great interest even if they may not always 

 be acceptable to the reader. From the nature 

 of the case many of the questions treated are 

 such as to give rise to difi^erent views, but 

 their fundamental importance justifies in- 

 quiries even if these do not always receive a 

 complete answer. One of the most important 

 lessons for the young mathematician to 

 learn is a keen realization of the narrow 

 limits of the explored parts of mathematics 

 as compared with those regions which invite 

 our inquiry and baflle our efforts. 



The contents of the volume can be readily 

 inferred, in the main, from its title. After 

 a brief consideration of Euclid's elements and 

 non-euclidean geometry, the author considers 

 the logical significance of definitions, axioms 

 and postulates, the consistency, independence 

 and categoricalness of a set of assumptions. 

 This is followed by a consideration of the 

 fundamental notions of class, correspondence 

 and group, and the development of the con- 

 cepts of real and complex numbers. It is 

 pointed out that from the abstract point of 

 view algebra and geometry are identical in 

 the sense that each includes the other, and 

 that this explains the interrelations between 

 these subjects. 



On page 194 the author repeats a histor- 

 ical error which is very wide spread in 

 mathematical literature, as regards the early 

 use of the term function for integral power 

 of a variable. This error seems to have been 

 started by d'Alembert and it has been re- 



