Apkil 6, 1900.] 



SCIENCE. 



531 



larly in algebra, we are laying more and 

 more stress upon a distinct statement of the 

 number world (the domain of rationality) 

 in which we are operating. Such specifica- 

 tions add a clearness and rigor to our work 

 which would otherwise scarcely be possible. 



This refinement which the mathematical 

 thought of to-day is so actively cultivating 

 is not restricted to the finite region. Math- 

 ematical infinity is receiving its share of at- 

 tention. It is well known that it is some- 

 times desirable to regard the infinite region 

 as a single point. This is, for instance, the 

 case in the transformation known as inver- 

 sion. Again, in ordinarj' projective geom- 

 etry it is generally convenient to regard the 

 infinite region as of one lower dimension 

 than the finite, so that the infinite region 

 of a plane is merely a line and the infinite 

 region of space is a plane. The student of 

 diiferential calculus is, moreover, familiar 

 with the infinite variable and the many 

 simplifications which its uses make possible. 



The most fruitful investigations along 

 this line are those on multiplicities (Meng- 

 enlehre, ensembles). Any total of definite 

 and clearly defined elements is said to form 

 a multiplicity. If two multiplicities are 

 simply isomorphic, i. e., if there is a 1,1 

 correspondence between the elements of the 

 multiplicities, they are said to be equiva- 

 lent, or to have the same power. For ex- 

 ample, it is easy to prove that all the posi- 

 tive rational numbers are equivalent to the 

 natural numbers. To do this we may asso- 

 ciate all the rational numbers p/q for which 

 the sum p -\- q^ n any positive integer. 

 We thus have the n — 1 numbers. 



n — 1 n— 2 n — 3 ... 2 1 



1 ' ~^~ ' ~3"' 



' )i — 2 ' » — 1 ■ 



We may let 1 correspond to 1 ; the num- 

 bers for which n = 3 correspond to 2 and 3 ; 

 the numbers for which n = 4 correspond to 

 4, 5 and 6, etc. We thus obtain the fol- 

 lowing 1, 1 correspondence between all the 



rational numbers and the positive integers ; 



1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ■• 



It may be observed that we do not need 

 to reduce the rational fractions to their 

 lowest terms to effect this correspondence. 

 This method of proof is due to Cantor, who 

 has also proved that all algebraic numbers 

 are equivalent to the natural numbers.* 

 How important and far-reaching the inves- 

 tigations along this line are may be inferred 

 from the fact that Jordan has employed 

 them to serve as a foundation of the ele- 

 mentary part of the second edition of his 

 magistral ' Cours d'analyse.' 



A large number of mathematical prob- 

 lems may be reduced to equations involving 

 a single unknown. The solution of such 

 equations has occupied a prominent place 

 in the mathematical literature for centuries. 

 The problem is so difficult that it has been 

 attacked from a number of different points 

 and by means of a large variety of instru- 

 ments. The instrument which has proved 

 to be the most powerful and far-reaching is 

 substitution groups. By means of it Abel 

 succeeded in 1826 to prove that an equa- 

 tion whose degree exceeds four cannot gen- 

 erally be solved by the successive extrac- 

 tion of roots t and Gralois a few years later 

 sketched a far-reaching theory of equations 

 which rests upon the theory of these groups. 



In recent years it has been recognized 

 (especially tlirough the labors of Sophus 

 Lie) that the theory of groups has very ex- 

 tensive and fundamental application in a 

 large number of the other domains of 

 mathematics. About a year ago the great 

 French mathematician H. Poincare showed 

 in an article, published in the Chicago 

 Monist X how this concept may be employed 

 in laying the foundations of elementary 



* Cantor, CreUe, vol. 77, 258; cf. vol. 84, p. 250. 

 t Crelle, vol. 1. 

 t October, 1898. 



