REVIEWS 685 



the invariants of certain corresponding linear groups or the groups themselves 

 (p. 174). The theory has since that time been greatly developed in scattered 

 memoirs. Thus, the present exceedingly able exposition will be of great value. 

 The eight chapters deal with elementary properties of linear groups, groups of 

 operators, and substitution groups, the linear groups in two variables, advanced 

 theory of linear groups, the linear groups in three variables, the theory of group 

 characteristics, the linear groups in four variables, and the history and applications 

 of linear groups. There is an appendix and indexes. 



Philip E. B. Jourdain. 



Lectures on Ten British Mathematicians of the Nineteenth Century. By 

 Alexander Macfarlane, Late President of the International Association 

 for Promoting the Study of Quaternions. (Pp. 148.) (New York : John 

 Wiley & Sons ; London : Chapman & Hall, Ltd., 1916. Price 5s. 6d. net.) 



This volume is No. 17 of the " Mathematical Monographs," edited by Mansfield 

 Merriman and Robert S. Woodward. During the years 1901-4 the late 

 Dr. Macfarlane delivered at Lehigh University some lectures on twenty-five 

 British mathematicians of the nineteenth century, and in this book ten lectures on 

 ten pure mathematicians are given in essentially the same fcrm as delivered. 

 " In a future volume it is hoped to issue lectures on ten mathematicians whose 

 main work was in physics and astronomy " (p. 3). A short biography and portrait 

 of Macfarlane (1851-1913) are given (p. 4), and the volume is adorned by a page 

 of portraits of the ten mathematicians mentioned in the excellent and fairly 

 popular lectures. The mathematicians are George Peacock (1791-1858), Augustus 

 De Morgan (1806-1871), Sir William Rowan Hamilton (1805 -1865), George Boole 

 (181 5-1864), Arthur Cayley (1821-1895), William Kingdon Clifford (1845-1879), 

 Henry John Stephen Smith (1826-1883), James Joseph Sylvester (1814-1897), 

 Thomas Penyngton Kirkman (1806-1895), and Isaac Todhunter (1820-1884). 



The work of many of the British mathematicians of the early half of the 

 nineteenth century forms in a sense an almost complete whole : the work of 

 Peacock, Gregory, Hamilton, Boole, De Morgan, and a few others, on the 

 foundations of algebra, symbolic logic, and generalisations of algebra form a 

 closely connected chapter in science in which foreign influences are not very 

 perceptible. It was lucky that Macfarlane, whose interest largely lay in the 

 directions just mentioned, should have given these excellent lectures. It may 

 be noticed that the works of Lagrange and Laplace were of direct influence 

 in the beginning of the work of Hamilton, Boole, and Cayley (p. 65). The 

 foundations of algebra and the calculus of logic are treated, and some critical 

 remarks are made, on pp. 14-18, 24-30, 41-42, 54-63; and these passages will be 

 found exceedingly useful contributions to the history of mathematics. It is perhaps 

 rather disappointing that neither the work of Jevons nor Hankel is mentioned, 

 while surely the names of Gauss and Weierstrass might have been mentioned on 

 p. 29. It is a mistake to attribute to Kant the idea of Hamilton of algebra as the 

 science of pure time (pp. 41, 85). Surely some mention of Woodhouse should 

 have been made on p. 10 (cf. p. 136). The remark about Newton on p. 9 seems to 

 be an error, and De Morgan's attitude can hardly be described as " morbid " 

 (p. 24). The remark about Boole and Cambridge on pp. 51-2 is not quite 

 accurate, and Hamilton's biographer is wrongly stated to be Charles Graves on 

 pp. 34 and 54 : it was Robert Perceval Graves. There is a misprint for 

 " rigorousness " on p. 95. The extract from Todhunter on p. 140 is perfectly 

 delicious. Philip E. B. Jourdain. 



