RECENT ADVANCES IN SCIENCE 523 



to the theory of prime numbers of Stifel, Fermat, Lambert, 

 Legendre, Gauss, Dirichlet, and Tschebychef. 



Logic, Principles, and Theory of Aggregates. — J. S. Taylor 

 (9-10) gives a complete existential theory of Bernstein's set 

 of four postulates for Boolean algebras. D. Hilbert (25) makes 

 what seems to be an exaggerated claim that " everything 

 which can be an object of scientific thought falls, as soon as 

 it is ripe for the formation of a theory, to the axiomatic method." 

 L. E. J. Brouwer (48) gives a foundation of the theory of 

 aggregates which is independent of what he thinks is the 

 (mathematically) unpermissible logical theorem of the excluded 

 middle : apparently this point of view is due to the fact that 

 it appears to some that this theorem leads to contradictions in 

 mathematics. B. Russell (Monist, 191 8, 28, 49SS 2 7) gives 

 two lectures of his on the " philosophy of logical atomism," 

 to which mathematical considerations have led him, and deals 

 with facts and propositions, particulars, predicates, and 

 relations. It seems to be unfortunate that Russell persists in 

 regarding a proposition as a " symbol," although he admits 

 that propositions are either true or false. Miss D. M. Wrinch 

 (ibid. 620-3) gives an account of recent work in mathematical 

 logic. Philip E. B. Jourdain (Science Progress, 191 8, 13, 

 299-304, cf. 178) shows clearly that his method does not 

 depend on arbitrary selections. 



On the theory of relativity, A. Einstein (n-12) gives some 

 cosmological considerations, and C. Cailler (6y) applies quater- 

 nions to Lorentz's transformation. 



H. M. Westergaard (Monist, 191 8, 28, 613-20) gives a very 

 useful comparison of work on the conception of probability and 

 adheres to the " statistical method " due to Montessus. A. 

 Berger (62) shows that, for an infinity of cases of observation, 

 it follows from the existence of a most probable value that 

 this value is the arithmetical mean. R. Schumann (56-7) 

 has a long paper on the determination of a straight line by 

 the method of least squares. G. Polya (58) writes on geomet- 

 rical probabilities. S. D. Wicksell (66-7) writes on the genetic 

 theory of frequency. 



N. Lusin and W. Sierpinski (33) write on a property of the 

 continuum, and also (32) on a decomposition of an interval into 

 a non-enumerable infinity of non-measurable aggregates. 

 H. Rademacher (60) gives a theory of measurable correspon- 



