July 7, 1916] 



SCIENCE 



25 



by the Standard Chemical Company for the 

 past four and a half years, and on the basis of 

 figures published by Dr. Charles L. Parsons 

 in the May number of the Journal of Indus- 

 trial and Engineering Chemistry, it is not evi- 

 dent that the method is satisfactorily efficient, 

 when applied to the treatment of low-grade 

 carnotite ore. 



Charles H. Viol 

 Pittsburgh, Pa., 

 June 3, 1916 



SCIENTIFIC BOOKS 



RECENT BOOKS IN MATHEMATICS 



Algebraic Invariants. By Leonard Eugene 

 Dickson, Professor of Mathematics, Univer- 

 sity of Chicago. New York, John Wiley and 

 Sons, 1914. Pp. 100. $1.25. 

 A Treatise on the Theory of Invariants. By 

 Oliver E. Glenn, Ph.D., Professor of Mathe- 

 matics in the University of Pennsylvania. 

 Boston, Ginn and Company, 1915. Pp. 245. 

 Contributions to the Founding of the Theory 

 of Transfinite Numbers. By Georg Cantor. 

 Translated and Provided with an Introduc- 

 tion and Notes by Philip E. B. Jourdain. 

 Chicago and London, The Open Court Pub- 

 lishing Company, 1915. Pp. 211. $1.25. 

 Problems in the Calculus. With Formulas and 

 Suggestions. By David D. Leib, Ph.D., In- 

 structor in Mathematics in the Sheffield 

 Scientific School of Yale University. Bos- 

 ton and New York, Ginn and Company, 

 1915. Pp. 224. 

 Diophantine Analysis. By Robert D. Car- 

 michael, Assistant Professor of Mathe- 

 matics in the University of Illinois. New 

 York, John Wiley and Sons, 1915. Pp. 118. 

 Historical Introduction to Mathematical Liter- 

 ature. By G. A. Miller, Professor of 

 Mathematics in the University of Illinois. 

 New York, The Macmillan Company, 1916. 

 Pp. 295. 



An invariant is any thing — a property or a 

 relation or an expression or a configuration — 

 that remains unaltered when other things con- 

 nected with it suffer change. In this very 

 comprehensive but essential meaning of the 

 term, the notion is probably as ancient as the 



human intellect. Certainly in historic time 

 the appeal of the idea has been universal. It 

 has been said that science may be defined as 

 the quest of invariance. Doubtless that quest 

 is an essential mark of science but it is not 

 peculiar to science. For the problem of invari- 

 ance, the problem of finding permanence in the 

 midst of change, arises out of the flux of 

 things to confront man in all departments of 

 life. And so it is that the search for what 

 abides is not confined to science but is and 

 always has been the chief enterprise of philos- 

 ophy and theology and art and jurisprudence. 

 It is, however, in mathematics that the notion 

 of invariance has come to the clearest recogni- 

 tion of its character and significance. In this 

 respect the notion in question has had a his- 

 tory like that of all other great ideas that 

 have slowly and at length become available for 

 the processes of logic. 



The oldest and now most elaborate portion 

 of the mathematical doctrine of invariance is 

 about as old as American independence. 

 Though now an imposing theory, its begin- 

 ning was like a mustard seed. It began, not 

 in ratiocination, but in an observation — mathe- 

 matics indeed depends even more upon obser- 

 vation than upon formal reasoning. It began 

 in what was in itself a very small observation, 

 an observation (1773) by Lagrange that the 

 discriminant of the quadratic form ax 2 -f- 

 Sbxy -\- cy 2 remains unaltered on replacing x 

 by x -\- Xy. The next important step was 

 taken by Gauss in 180i and the next by Boole 

 in 1841. Incited by Boole's beautiful results, 

 the English mathematicians, Cayley and 

 Sylvester, entered the field, the former pro- 

 ducing in rapid succession his great memoirs 

 on Quantics and the latter his brilliant inves- 

 tigations in what he conceived more poetically 

 as the Theory of Forms. The interest so 

 aroused quickly passed to the continent en- 

 gaging the great abilities of such mathe- 

 maticians as Aronhold, Hermite, Clebsch, 

 Gordan and others. The result is the colossal 

 doctrine variously styled the algebra of quan- 

 tics, the theory of algebraic invariants and 

 covariants, and the theory of forms. 



It is to this doctrine that Professor Dick- 



