616 SCIENCE PROGRESS 



cussion of what we can mean by the question as to whether 

 our space is Euclidean or not, on which Poincare and Russell 

 have come to such different conclusions. Broad's final form 

 of his question is : " Subject to the conditions that space is to 

 be changeless and homogeneous and not to act on matter, and 

 that matter is to move about in space, can we construct a 

 system of physics which assumes Euclidean geometry for space, 

 and enables us to deal consistently and adequately with all 

 the data that scientists agree to be most worthy to be taken 

 into account ? ... Of course the only way to answer such a 

 question as this is actually to try and construct such a system 

 of physics. If you can do it, space is Euclidean ; if not, then 

 space may not be Euclidean. ..." In this connection the 

 reader should refer to an admirable critical notice by Broad 

 {ibid. 555) of A. A. Robb's Theory of Time and Space published 

 at Cambridge in 1914. 



Dr. Robert L. Moore (Bull. Amer. Math. Soc. 191 5, 22, 

 1 1 7) adds a note modifying in certain respects his results about 

 axioms for the linear continuum obtained in the Annals of 

 Mathematics for 191 5 (see Science Progress, 1916, 10, 433). 



Meyer G. Gaba (Trans. Amer. Math. Soc. 1915, 16, 51) 

 gives a set of postulates for general projective geometry in 

 terms of point and transformation. 



The definition of a " plane curve " in terms of the concep- 

 tions of the theory of aggregates is known to be of extremely 

 great importance in the theory of functions of a complex variable . 

 Eric H. Neville (Journ. Indian Math. Soc. 191 5, 7, 175) gives 

 a definition of it as " a multiple perfectly connected plane set 

 which coincides with its own edge," the conceptions used 

 being those familiar to mathematicians from, for example, 

 Dr. W. H. Young and Mrs. Young's Theory of Sets of Points. 

 Neville's definition excludes the space-filling curves of Peano 

 and E. H. Moore. 



Theory of Numbers and Analysis. — G. B. Mathews (Proc. 

 Lond. Math. Soc. 1915, 14, 464) considers the division of the 

 lemniscate into seven equal parts. The same author (ibid. 

 467) gives a direct method in the multiplication theory of the 

 lemniscate functions and other elliptic functions. 



S. Ramanujan (Journ. Indian Math. Soc. 191 5, 7, 173) 

 has an interesting note on finding the sum of the square roots 

 of the first n natural numbers. 



