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RECENT ADVANCES IN SCIENCE 



PURE MATHEMATICS. By F. Puryer White, M.A.. St. John's 

 College, Cambridge. 



History and Teaching of Mathematics. — Three articles by Prof. 

 G. A. Miller fall under this head. The first {School and Society, 

 16, 1922, 449-54), under the title " Disagreeing with the Text- 

 book," gives an interesting account of a course which the author 

 gave during the summer session at the University of Illinois — 

 a critical study of Cajori's History of Mathematics. The student 

 was required to find, in a given 25 pages of text, at least one 

 statement with which he did not agree, attention being par- 

 ticularly directed to general statements, such as " Modern higher 

 algebra is especially occupied with the theory of linear trans- 

 formations." In School Science and Mathematics (1921, 715-1 7) 

 Prof. Miller breaks a further lance with Prof. Cajori on the 

 question of the Greek definition of a tangent-line. The third 

 article {Scientific Monthly, 15, 1922, 512-19) is an elementary 

 account of the ideas underlying the Theory of Groups, and its 

 claim to be, in Poincare's words, " entire mathematics, divested 

 of its matter and reduced to a pure form." 



Algebra and Analysis. — If a linear homogeneous transforma- 

 tion s be applied to the variables x,y of a binary form f{x,y), 

 the coefficients a undergo a linear transformation in n + i 

 variables ; these transformations form a group G isomorphic 

 with the linear group of the substitutions s. The characterisa- 

 tion of the group G independent of the transformation of the 

 x,y is given by a theorem due to A. Ostrowski {Math. Ann., 79, 

 1 919* 360-87) — the group G can be defined as the aggregate 

 of all linear transformations of the coefficients which leave the 

 discriminant D of the form f{x,y) unaltered. This theorem 

 cannot, however, be used conversely to characterise the dis- 

 criminant among the invariants of /. We have, in fact, the 

 theorem that every invariant of the form has no further linear 

 transformation beyond those of the group G, except, for n even, 



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