28 



SCIENCE 



[N. S. Vol. XLIV. No. 1123 



normal condition on the part of those inter- 

 ested in the history of the human race." We 

 are also told that " with our gradual evolution 

 from the state of barbarism the history of war 

 and bloodshed is being slowly replaced by that 

 of political and intellectual movements." 

 From which we infer that that portion of the 

 preface was composed prior to August, 1914. 



Believing that history courses for secondary- 

 school teachers should be more largely con- 

 cerned with modern developments than is their 

 wont, Professor Miller particularly stresses 

 these developments, though the content of his 

 discourse is in very considerable measure 

 drawn also from ancient and medieval times. 



The 35 pages of the initial chapter are de- 

 voted to sketching the progress made from the 

 beginning of the nineteenth century to the 

 present time in mathematical intelligence, 

 mathematical research, mathematical history 

 and mathematical teaching. In particular the 

 fact is pointed out that the rapid and continu- 

 ously increasing American mathematical activ- 

 ity during the last twoscore years has placed 

 our country among the leading mathematical 

 countries of the world. If we have not yet 

 produced a mathematician of the very first 

 rank, we can at least claim to have produced 

 men of notable ability and productiveness. 



Chapter II. (42 pages) presents a large 

 amount of interesting information respecting 

 types of recent literature, societies, interna- 

 tional congresses, periodicals, works of refer- 

 ence, mathematical tables and collected works. 

 In the 51 pages of the third chapter we have 

 a rather meager discussion of definitions of the 

 term mathematics ; a historical account of the 

 manner in which the science has acquired its 

 grand divisions and subdivisions; a quite too 

 brief but interesting account of the advent, 

 influence and position of a few " dominant 

 concepts " such as irrational quantities, equa- 

 tion solution, function, group, matrix, domain 

 of rationality; some instructive remarks and 

 historical references respecting mathematical 

 terminology and notation; a short section on 

 errors in mathematical literature; a section, 

 entitled living mathematicians, arguing with 

 feeling and good judgment the importance of 



devising suitable means for determining " Who 

 is Who " among mathematicians ; and a final 

 section treating inadequately, hardly more 

 than touching, the now pressing question of 

 mathematics as an educational subject. 



There follow three chapters dealing with 

 " fundamental developments " respectively in 

 arithmetic, in geometry and in algebra. Of 

 these the first (29 pages) opens with Euclid's 

 proof that the number of prime numbers is 

 infinite; explains the Sieve of Eratosthenes; 

 sketches the history and appraises the signif- 

 icance of irrational numbers, giving (doubt- 

 less unintentionally) the impression (p. 133) 

 that these numbers admit of only negative 

 definition; treats briefly the fundamental 

 operations of arithmetic, then of notation sys- 

 tems, and closes with a short and excellent 

 account of the Eermat theorem. The next 

 chapter (23 pages) devotes to " fundamental 

 developments in geometry " three sections, one 

 to the Pythagorean theorem, one to the area 

 and volume of the sphere, and one to the tri- 

 angle. A valuable chapter (22 pages) on 

 algebra is historically rich in its handling of 

 the fundamental theorem of algebra, the no- 

 tion of determinant, numerical equations, do- 

 mains of rationality, the beginnings of invari- 

 ant theory, and the tale of the binomial 

 theorem. 



Chapter VII., somewhat oddly entitled 

 " Twenty-five Prominent Deceased Mathe- 

 maticians," is the largest and most interesting 

 division of Professor Miller's interesting book. 

 It contains a very readable account of the fol- 

 lowing men selected from among the great 

 mathematicians of the world : Euclid, Archi- 

 medes, Apollonius, Diophantus, Vieta, Des- 

 cartes, Eermat, Newton, Leibniz, Euler, La- 

 grange, Gauss, Cauchy, Steiner, Abel, Hamil- 

 ton, Galois, Sylvester, Weierstrass, Cayley, 

 Hermite, Kronecker, Cremona, Lie, Poincare. 



The book closes with a list, accompanied 

 with brief characterizations, of a large num- 

 ber of bibliographies, reference works, and 

 books on the history, and the teaching and 

 philosophy of mathematics. 



Cassius J. Keyser 

 Columbia University 



