6 9 o SCIENCE PROGRESS 



meaning. In mathematics, Euclid is the classical example of the textbook which 

 is both science and art, and the proof of this is that it remains our textbook to the 

 present day in spite of frequent attacks which have failed to dislodge it from its 

 position. 



Prof. Dickson's little book is an excellent one in the class to which it belongs. 

 It is well but not too well condensed ; it covers the field usually covered by these 

 books ; the author's meaning is seldom in doubt ; important propositions are not 

 often relegated to examples, and so on. In spite of the Preface, however, from 

 which we gather that the author has designed his book not only for specialists, it 

 will be more useful to them than to those who occasionally dip into the theory of 

 equations. But this is not the fault of the author so much as that of academical 

 custom in the writing of these mathematical textbooks. Really, such books on 

 the " Theory of Equations " are quite wrongly named. They should be called 

 books " On the Theory of Rational Integral Equations " only, and it would 

 generally be right to add the words l< Of Low Degree." They seldom make any 

 reference whatever to transcendental equations, or even to rational integral ones 

 of high degree but of few terms, or to other equations which can easily be solved 

 without being put into the rational integral form. Yet such equations are of 

 frequent occurrence in, let us say, chemistry, physics, or statistical work. The 

 labourer in these fields, who may not be himself a very expert mathematician, 

 when he is investigating some problem is often confronted and sometimes defeated 

 by such equations ; yet when he " looks up " a textbook on the subject he can 

 get no help whatever, and, instead of finding assistance, is supplied only with 

 purely academical though beautiful theorems, such as those on the square root 

 of minus one. In fact, these academical works would seem to be designed 

 principally for examination purposes, and contain scarcely any business-like 

 dealing with the whole theme. There is no wide survey of algebraic equations 

 in general ; little reference to the literature, and no list of previous textbooks ; 

 and, while problems which have been solved are triumphantly given, those which 

 continue to foil the human intellect are too frequently left quite unmentioned — 

 much to the discomfiture of the non-expert reader. At the same time, these 

 books really contain much matter which properly belongs to other branches of 

 mathematics, or, indeed, which are of very little practical importance. After all, 

 our main concern with an equation is to solve it, and as quickly as possible ; but 

 our textbooks too often fail to help us in this respect, and give us endless invariants 

 and complex numbers in place of the bread which we seek. 



Is it not time that some change be made ? For example, Prof. Dickson's book 

 commences very wisely with a chapter on the graph of an equation. After that, 

 the humble amateur will think that he should have proceeded at once to the 

 analysis and solution of cubic and quartic equations, and then to that of numerical 

 equations of any degree, with at least some survey of the classes of equations 

 mentioned above. Instead of this, Prof. Dickson's second chapter plunges at 

 once into complex numbers, and his fifth chapter rushes at once upon the 

 theorems of Gauss and Cauchy on the existence of a root, this being called the 

 fundamental theorem of algebra — a doubtful proposition, since much work was 

 done on algebra by Descartes, Newton, and a few other inferior individuals before 

 this theorem was invented. The seventh chapter deals with symmetric functions, 

 and the solution of numerical equations is not considered until the ninth and tenth 

 chapters are reached. Surely it would be better to remove the academical parts 

 of the subject to the end of the work, or even to remove them entirely into general 

 algebra or double algebra, and to replace them by a sufficient study of equations 



