RECENT ADVANCES IN SCIENCE 453 



Bopp (Sitzungsber. der K.B. Akad. der Wiss. zu Milnchen, 1914, 

 361) deals with the opinions on it of Lambert's contemporaries. 

 Another landmark in the history of non-Euclidean geometry 

 is formed by the (much earlier) work on parallels of Girolamo 

 Saccheri, and A. Pascal (Giorn. di Mat. 19 14, 52, 229) gives an 

 account of Saccheri's life and work. 



Principles of Mathematics. — Dr. B. A. Bernstein {Bull. 

 Amer. Math. Soc. 1916, 22, 458) simplifies E. V. Huntington's 

 (1904) set of postulates, based on A. N. Whitehead's " formal 

 laws " for Boole's algebra of logic, by reducing the number of 

 postulates from ten to eight and the number of postulated 

 special elements from three to one. 



A new subject whose foundations can be treated by means 

 of a system of axioms is the method of least squares. F. 

 Bernstein and W. S. Baer (Math. Ann. 1915, 76, 284) use, for 

 this purpose, the conception of " weight " introduced by Gauss 

 in his second formulation of the foundations of this method, and 

 make no use of the conceptions of the calculus of probabilities. 



Denes Konig, the late Julius Konig's son, has published 

 (Mathematikai is physikai lapok, Budapest, 19 14, 23, 291) 

 some remarks on his father's last work, Neue Grundlagen der 

 Logik, Arithmetik und Mengenlehre (cf. Science Progress, 

 19 1 6, 11, 92). References to reviews of this work are given in 

 Revue semestrielle (1916, 24 [1], 106). 



K. G. Hagstrom (Arkiv for Math., Stockholm, 10, No. 2) 

 claims to have proved the existence of well-ordered aggregates 

 which are not comparable. This would of course destroy 

 the force of the antinomy of Burali-Forti (1897). 



F. Hartogs (Math. Ann. 191 5, 76, 438) proves, without using 

 Zermelo's principle of selection, that aggregates which are 

 I comparable " (that is, one of the two given aggregates has a 

 part which is equivalent to the other) can be well-ordered. This 

 account (Rev. semest. 191 6, 24 [1}, 32) is not quite clear : it 

 seems possible that a given aggregate may be comparable with 

 a certain second one, but not with a certain third one. Further, 

 it has long been known that Zermelo's axiom, which is neces- 

 sary and sufficient for well-ordering in general, is needed only in 

 that one of the four logically possible cases of the comparison of 

 aggregates which is that of supposed incomparability. 



G. Pucciano (Giorn. di Mat. 1914, 52, 19) continues the work 

 he published in the same journal for 191 3 on the open linear 



