l; 



SCIENCE, 



[N". S. Vol. XXIV. No. 601. 



dent patriot, this independent and indom- 

 itable worker, this genuine democrat— 

 Pasteur: "The true democracy is that 

 v/hich permits each individual to put forth 

 his maximum of effort." 



Charles W. Eliot. 



SCIENTIFIC BOOKS. 

 A College Algehra. By Henry Burchabd 



Fine. Ginn & Company. 1905. 



The present day is remarkable for its pro- 

 duction of large numbers of mathematical 

 text-books. In most cases the aim of the 

 writers of these books seems to be to convince 

 the student that the subject treated is devoid 

 of any element of interest, that it possesses 

 no logical sequence, and that memory of a 

 large assortment of unconnected facts is the 

 only requisite for a sound mathematical train- 

 ing. One meets with proofs of theorems 

 divided into first, second, etc., steps — an obvi- 

 ous attempt to burden the memory at the 

 expense of the reasoning faculty, and stress 

 is laid on the fact that all problems are ' easy,' 

 in fact on examination they appear scarcely 

 worth the name of problems. There is not 

 the slightest doubt that these harmful books 

 are one of the causes of the decrease in mathe- 

 matical students at our colleges and univer- 

 sities. The books are, unfortunately, given 

 a trial somewhere, no matter how bad they 

 may be, and one can conceive of no surer way 

 of destroying the interest of the young stu- 

 dent in the subject. For those who are 

 merely general students they are equally de- 

 fective. In the forefront of an author's mind 

 should be a desire to develop the reasoning 

 faculties. Let us have easy exercises by all 

 means, but let us also have exercises which 

 will make students think for themselves. Let 

 us develop our subject along the easiest se- 

 quence, but let us develop it logically. 



Professor Fine's ' College Algebra ' is in 

 refreshing contrast to such books as I have 

 mentioned. He aims at giving an exposition 

 at once logical and easy to understand. The 

 result is a book that must make the subject 

 interesting to the ordinary college student. 

 The work is divided into two parts. The first 



consists of 78 pages devoted to the ideas at 

 the base of the notion of number, a develop- 

 ment of those ideas which are associated with 

 the names of Cantor, Dedekind and others. 

 This difficult subject has been handled by the 

 author with conspicuous clearness, and every 

 student of it should make himself familiar 

 with these first 78 pages. It is questionable, 

 however, whether, even with Professor Fine's 

 exposition, it is possible to make this subject 

 really understood by a studesnt who is just 

 beginning his college algebra course, and pos- 

 sibly the author in later editions may decide 

 to present this section as a separate book, 

 under a separate title. 



The second part, some 500 pages, is con- 

 cerned with algebra proper. It is ' meant to 

 contain everything relating to algebra that a 

 student is likely to need during his school and 

 college course.' Even this wide ideal is given 

 a wide interpretation, and the last chapter, 

 Properties of Continuous Functions, is a fit- 

 ting introduction to the calculus. The chap- 

 ters on the solution of equations are of special 

 interest. The author makes much use of 

 graphs, the only way to make clear to the 

 student what is implied by the solution of a 

 set of equations. It would have been of ad- 

 vantage to give a brief account of the gen- 

 eralization of the use of graphs to the case of 

 three variables, and thus to prepare the mind 

 for the idea of a space of more than three 

 dimensions. Particularly noteworthy in con- 

 nection with graphs is the discussion of in- 

 equalities. The idea of a graph as dividing 

 the plane into two regions, in one of which 

 f(x, 2/) > in the other < 0, should certainly 

 be emphasized in ordinary algebra, before the 

 introduction of analytic geometry, as alge- 

 braic questions, otherwise unintelligible to the 

 learner, become almost intuitive. Observe, 

 for instance, the illuminating example on 

 page 341. 



The general theory of the solution of equa- 

 tions is developed in very effective form; in 

 particular the treatment of symmetric equa- 

 tions. The important idea is the taking of 

 the various simple symmetric functions as 

 new auxiliary variables and, after salving for 

 these, finding the solutions of a set such as. 



