OF ELEMENTARY ALGEBRA BY N. F. DUPUIS, M.A., F.R.S.C. PROFESSOR OF PURE MATHEMATICS IN THE UNIVERSITY OF QUEEN'S COLLEGE, KINGSTON, CANADA New York AND LONDON EducT 128.92.34.13 MAYWARD COLLICE LO ANY GIFT OF ME 21 2 COPYRIGHT, 1892, BY MACMILLAN AND CO. TYPOGRAPHY BY J. S. CUSHING & Co., Boston. PRESSWORK BY BERWICK & SMITH, BOSTON. PREFACE. In the following pages I have endeavored to put into form what in my opinion should constitute an Intermediate Algebra, intermediate in the sense that it is not intended for absolute beginners, nor yet for the accomplished algebraist, but as a stepping-stone to assist the student in passing from the former stage to the latter. The work covers pretty well the whole range of elementary algebraic subjects, and in the treatment of these subjects fundamental principles and clear ideas are considered as of more importance than mere mechanical processes. The treatment, especially in the higher parts, is not exhaustive; but it is hoped that the treatment is sufficiently full to enable the reader who has mastered the work as here presented, to take up with profit special treatises upon the various subjects. Much prominence is given to the formal laws of Algebra and to the subject of factoring, and the theory of the solution of the quadratic and other equations is deduced from the principles of factorization. The Sigma notation is introduced early in the course, as being easily understood, and of great value in writ ing and remembering important symmetrical algebraic forms. Synthetic Division is commonly employed, and the principles of its operation are extended to the finding of the highest common factor. Except in the case of surds, no special method is given for finding the square root of an expression which is a complete square, as the operation is only a case of factoring, and a simple case at that. For expressions which are not complete squares, the most rational method is by means of the binomial theorem or of undetermined coefficients, both of which are amply dealt with. Probably the most distinctive feature of the work is the importance attached to the interpretation of algebraic expressions and results. Algebra is an unspoken language written in symbols, of which the manipulation is largely a matter of mechanical method and of the observance of certain rules of operation. The results arrived at have little interest and no special meaning until they are interpreted. This interpretation is either Arithmetical, that is, into ideas involving numbers and the operations performed upon numbers; or Geometrical, that is, into ideas concerning magnitudes and their relations. Both interpretations necessitate observation and the exercise of thought; but the geometrical offers the wider scope for ingenuity, and is the better test of mathematical ability. In several cases, as in that of the quadratic equation, the solution frequently gets its complete explanation only through its geometric interpretation. Hence geometrical problems are freely introduced, and the relations between the symbolism of Algebra and the fundamental ideas of Geometry are discussed at some length. The Graph is freely employed both as a means of illustration and as a medium of independent research; and through these means an effort is made to connect Algebra with Arithmetic upon the one hand, and with Geometry upon the other. The exercises are numerous and varied, and I trust that they will be found to be fairly free from errors. N. F. D. |