RECENT ADVANCES IN SCIENCE 277 



In all Wiener's papers, the symbolism of Whitehead and Russell 

 is used. In his second paper, Wiener shows that we can regard 

 the series of the instants of time as a construction from the 

 non-serial relation of complete temporal succession between 

 events in time. In a note to this paper, and in the third paper, 

 he is occupied with methods for constructing series from non- 

 serial relations quite in general. The method last given bears 

 much the same relation to the various series of senstion-in- 

 tensities that the method of the paper just referred to bears to 

 the series of instants that constitutes one sort of extension, 

 time. 



Another short paper by Wiener in these Proceedings for 

 191 5 (18, 56) contains a solution of the question to find the 

 shortest line dividing an area in a given ratio. The proof is 

 based entirely on first principles, and does not use any higher 

 branch of technical mathematics. Another work on the 

 principles of geometry is Dr. M. J. M. Hill's study of the fifth 

 Book of Euclid's Elements from the point of view of its relation 

 to the principle known as " the axiom of Archimedes," which 

 is, however, assumed in the fourth definition of Euclid's fifth 

 book {Trans. Camb. Phil. Soc. 191 5, 22, 87). The results of 

 Hill's previous (1897, l 9° 2 ) investigations on the subject of 

 Euclid's fifth book are given in his Theory of Proportion, which 

 will be reviewed in the January number of this Quarterly. We 

 may also notice that the well-known logical method of treating 

 geometry on the basis of axioms about indefinables such as 

 " point " and " is collinear with " is exemplified in Prof. H. P. 

 Manning's Geometry of Four Dimensions (New York : The Mac- 

 millan Co. ; London : Macmillan & Co., Ltd., 1914, 8s. 6d. net). 



Very closely connected with the principles of mathematics 

 is the enormously important theory of aggregates, and a careful 

 translation, with a long historical introduction and notes by the 

 present writer, of Georg Cantor's famous memoirs of 1 895 and 

 1897 on transfinite numbers has just been published (London : 

 Open Court Co., 191 5, 3s. 6d. net). 



Arithmetic and Theory of Numbers. — The invention of loga- 

 rithms simplified multiplication and division by reducing 

 them to addition and subtraction respectively ; but involu- 

 tion and evolution were only replaced by the still possibly 

 laborious multiplication and division. E. Chappell, in a paper 

 read to the Royal Society in March of this year, describes a 



