26 



SCIENCE 



[ST. 8. Vol. XLIV. No. 1123 



son's book gives the beginner an admirable 

 introduction. It is, I say, for beginners, for 

 it presupposes only a fair knowledge of analyt- 

 ical geometry and the differential calculus. 

 The book is much larger than it appears 

 to be, being very compactly written, the 

 author having the art of getting a maximum of 

 results with a minimum of talk. Yet the 

 exposition is remarkably clear, uniting the two 

 stylistic virtues of precision and conciseness. 

 The work is composed of three parts. The 

 symbolic notation is reserved for part III. 

 Geometric interpretation is emphasized. In 

 part I. linear transformation is alternately in- 

 terpreted non-pro jectively and protectively; 

 that is, on the one hand as working a change 

 of reference configuration, and on the other 

 as merely effecting a lawful transfer of atten- 

 tion from old loci (or envelopes) to new ones 

 referred to the old configuration. Part II., 

 which is mainly concerned with the properties 

 of binary forms, deals with such matters as 

 homogeneity, weight, transformation products, 

 annihilators, linear independence, Hermite's 

 reciprocity law, etc. The canonical form of 

 the quartic is found and the equation is solved. 

 Part III., which occupies 38 of the book's 

 100 pages, is devoted to a presentation and use 

 of the symbolic method of Aronhold and 

 Clebsch. In passing the author notes that this 

 method is equivalent to the previously invented 

 but relatively cumbrous hyperdeterminant 

 method of Cayley. This part and indeed the 

 book may be said to culminate in Hilbert's 

 theorem regarding the expressibility of the 

 forms of a system in terms of a finite number 

 of them and the use of the theorem in proving 

 the finiteness of a fundamental system of co- 

 variants of a set of binary forms. 



It seems unfortunate that Professor Dick- 

 son did not deem it wise or find it practicable 

 to set forth the- matter of this volume in its 

 natural relation to the theory of groups. Per- 

 haps some one will some time write for begin- 

 ners a book on transformations, groups and in- 

 variants with applications. 



Professor Glenn's treatise is somewhat 

 more extensive than Professor Dickson's. It, 

 too, is introductory, beginning with a variety of 



simple considerations. Both the symbolic and 

 the non-symbolic methods are explained and 

 employed. Geometric interpretations are given 

 and some connection with the group concept is 

 made. The book comprises the following nine 

 chapters: the principles of invariant theory 

 (32 pages) ; properties of invariants (7 pages) ; 

 the processes of invariant theory (40 pages), 

 dealing with operators, the Aronhold sym- 

 bolism, reducibility, concomitants in terms of 

 roots, and geometric interpretations ; reduction 

 (46 pages), concerned with Gordan's series, 

 the quartic, transvectant systems, syzygies, 

 Hilbert's theorem, Jordan's lemma, and 

 grade; Gordan's theorem (16 pages), giving 

 proof of the theorem and illustration by the 

 cubic and quartic; fundamental systems (16 

 pages) ; combinants and rational curves (13 

 pages) ; seminvariants and modular invariants 

 (32 pages) ; and invariants of ternary forms 

 (25 pages). There is added an appendix of 

 ten pages devoted to exercises. 



With access to the foregoing books, to 

 Salmon's classic book, and to such recent Brit- 

 ish works as that by Grace and Young and 

 Elliott's "Algebra of Quantics," the English- 

 speaking student can not complain of having 

 to resort to other languages for a knowledge 

 of this classic branch of modern algebra. 



Many readers, including mathematicians and 

 philosophers, will be grateful to Mr. Jourdain 

 for his excellent translation of Cantor's 

 famous memoirs of 1895 and 1897. These were 

 published in the Mathematische Annalen 

 under the title, " Beitrage zur Begrtindung der 

 transfiniten Mengenlehre." The translator's 

 rendering of the title is justified by the con- 

 tent of the memoirs. This content is not 

 likely to be fully intelligible to any but such 

 as have mastered Cantor's earlier works begin- 

 ning in 1870. The value of the volume is much 

 increased by the translator's Introduction of 

 82 pages sketching the development of func- 

 tion theory in the course of the last century 

 and by the notes he has appended dealing with 

 the growth of the theory of transfinite numbers 

 since 1897. 



Dr. Leib's collection of problems presents 

 a good list under each important theme dealt 



