.wo.H a ' OME UNIVERSITY LIBRARY MODERN KNOWLEDGE F AN INTRODUCTION TO MATHEMATICS BY A. N. WHITEHEAD, Se.D., F.R.S. LONDON WILLIAMS & NORGATE HKNRY HOLT & NEW YORK Co., CANADA: WM. BRIGGS, TORONTO INDIA : R. & T. WASHBOURNE, LTD. HOME UNIVERSITY LIBRARY OF MODERN KNOWLEDGE Editor*t HERBERT FISHER, M.A.. F.B.A. PROF. GILBERT MURRAY, D.LlTT., LL.D., F.B.A. PROF. J. ARTHUR THOMSON, M.A. PROF.WILLIAM T. BKEWSTER, M.A. COLUMBIA.UNIVERSITY, U.S.A.) iifrA/uilfJ I IRV^T AN INTRODUCTION TO MATHEMATICS BY A. N. WHITEHEAD, SC.D- F.R.S., AUTHOR OF "UNIVERSAL ALGEBRA," JOINT AUTHOR OF "I'RINCIPIA MATHBHATICA" NEW AND REVISED EDITION PRINTED BT HAKELJy, WATSON AND VINET, LONDON AND AYLBSBUBT. CONTENTS OH&P. PASH ..... I THE ABSTRACT NATURE OF MATHE- MATICS 7 H VARIABLES . . , . . 15 m METHODS OF AP.PLIC.ATIO.N .. 25 . . IV DYNAMICS 42 V THE SYMBOLISM OF MATHEMATICS 58 . VI GENERALIZATIONS OF NUMB.ER.. .. 71 VH IMAGINARY NUMBERS 87 VIH IMAGINARY NUMBERS (CONTINUED) . 101 IX COORDINATE GEOMETRY . . .112 X CONIC SECTIONS .128 . . . XI FUNCTIONS . . . . . 145 XII PERIODICITY IN NATURE . . .164 v CONTENTS vi CHAP. PAGE TRIGONO.METR,Y ...,.173 . . . XIV SERIES 194 XV THE DIFFERENTIAL CALCULUS 217 . . XVI GEOMETRY 236 . . . . XVH QUANTITY... .. .. .. .. 245 NOTES 250 BIBLIOGRAPHY .251 . . INDEX 253 AN INTEODUCTION TO MATHEMATICS CHAPTER I THE ABSTRACT NATURE OF MATHEMATICS THE study of mathematics is apt to com' mence in disappointment. The important applications of the science, the theoretical interest of its ideas, and the logical rigour of its methods, all generate the expectation of a speedy introduction to processes of interest. We are told that by its aid the stars are weighed and the billions of molecules in a drop of water are counted. Yet, like the ghost of Hamlet's father, this great science eludes the efforts of our mental weapons " to grasp it 'Tis here, 'tis there, 'tis gone " and what we do see does not suggest the same excuse for illusiveness as sufficed for the ghost, that it is too noble for our gross methods. '* A show of violence," if ever excusable, may surely be " offered " to the trivial results which occupy the 7 INTRODUCTION TO MATHEMATICS 8 pages of some elementary mathematical treatises. The reason for this failure of the science to live up to its reputation is that its funda- mental ideas are not explained to the student disentangled from the technical procedure which has been invented to facilitate their exact presentation in particular instances. Accordingly, the unfortunate learner finds himself struggling to acquire a knowledge of a mass of details which are not illuminated byanygeneral conception. Withoutadoubt, technical facility is a first requisite for valu- able mental activity : we shall fail to appre- ciate the rhythm of Milton, or the passion of Shelley, so long as we find it necessary to spell the words and are not quite certain of the forms of the individual letters. In this sense there is no royal road to learning. But it is equally an error to confine attention to technical processes, excluding consideration of general ideas. Here lies the road to pedantry. The object of the following Chapters is not to teach mathematics, but to enable students from the very beginning of their course to know what the science is about, and why it is necessarily the foundation of exact thought as applied to natural phenomena. All allu- sion in what follows to detailed deductions in any part of the science will be inserted