454 SCIENCE PROGRESS 



and homogeneous continuum and the Archimedean geometry 

 of the straight line. 



C. Caratheodory (Gbtt. Nachr. 191 5, 404) transfers to the 

 concept of length the theories of Lebesgue for the content of 

 point-aggregates and gets a generalised concept of length. 



A. N. Whitehead (Rev. de Metaphys. et de Morale, 191 6, 23, 

 423) develops the relationist theory of space by the help of a 

 fairly frequent use of the symbols of Whitehead and Russell's 

 Principia Mathematica. On p. 427 there is an interesting 

 argument against " transmission of action by the contiguous 

 parts of a continuous medium," which results simply from 

 pointing out that there are no contiguous points in a continuum. 

 But we could in like manner show that " motion from one 

 point on a line to the next one " is impossible ; and yet con- 

 tinuous motion is not impossible in a continuum. However, 

 the real objection to the denial of action at a distance does not 

 depend on the arguments derived from that remark, but lies 

 in the fact that the denial in question is a denial of direct 

 relations between physical objects not occupying the same point 

 — and this implies the negation of the theory of the space- 

 relation. Whitehead gave a short and lucid account of that 

 department of logic with which some of his own studies have 

 been connected in his Presidential Address to Section A of the 

 British Association at Newcastle in 1916. It may be noticed 

 that, in the abridged account of this address given in Nature 

 (191 6, 98, 50), this part is omitted. 



Algebra (including Theoretical Arithmetic). — F. Bernstein 

 and O. Szasz (Math. Ann. 191 5, 76, 295 and 485) give an interest- 

 ing criterion for the irrationality of infinite continued fractions. 



F. Tavani (Giorn. di Mat. 19 14, 52, 204) sets out, in a general 

 theory of series, from the principle of comparing the nth term 

 with the (n + i)th, instead of the usual principle of com- 

 parison. 



Major P. A. MacMahon (Proc. Lond. Math. Soc. 1916, 15, 

 314) gives applications of general theorems in combinatory 

 analysis to (1) the theory of inversions of permutations ; (2) 

 the ascertainment of the numbers of terms in the development 

 of a determinant which has amongst its elements an arbitrary 

 number of zeros. 



Analysis. — Of course there is a wide difference between 

 what we call " limits " and " bounds " (Grenzen and Schranken 





