The Project Gutenberg EBook of A Scrap-Book of Elementary Mathematics, by William F. White This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org Title: A Scrap-Book of Elementary Mathematics Notes, Recreations, Essays Author: William F. White Release Date: August 30, 2012 [EBook #40624] Language: English Character set encoding: ISO-8859-1 *** START OF THIS PROJECT GUTENBERG EBOOK A SCRAP-BOOK *** Produced by Andrew D. Hwang, Joshua Hutchinson, and the Online Distributed Proofreading Team at http://www.pgdp.net (This file was produced from images from the Cornell University Library: Historical Mathematics Monographs collection.) Transcriber’s Note Minor typographical corrections, presentational changes, and no- tational modernizations have been made without comment. In- ternal references to page numbers may be off by one. All changes are detailed in the LATEX source file, which may be downloaded from www.gutenberg.org/ebooks/40624. This PDF file is optimized for screen viewing, but may easily be recompiled for printing. Please consult the preamble of the LATEX source file for instructions. NUMERALS OR COUNTERS? From the Margarita Philosophica. (See page 48.) A Scrap-Book of Elementary Mathematics Notes, Recreations, Essays By William F. White, Ph.D. State Normal School, New Paltz, New York Chicago The Open Court Publishing Company London Agents Kegan Paul, Trench, Trübner & Co., Ltd. 1908 Copyright by The Open Court Publishing Co. 1908. CONTENTS. PAGE Preface. viii The two systems of numeration of large numbers. .1 Billion. .1 Repeating products. .2 Multiplication at sight: a new trick with an old principle. .5 A repeating table. .7 A few numerical curiosities. .9 A curious property of 37 and 41..............................9 Numbers differing from their logarithms only in the position of the decimal point. .9 Fourier’s method of division. 13 Nine. ........................................................... 15 Familiar tricks based on literal arithmetic. 17 General test of divisibility. 19 Test of divisibility by 7....................................... 20 Test of divisibility by 7, 11, and 13........................... 20 Miscellaneous notes on number. 22 The theory of numbers. 22 Fermat’s last theorem. 22 Wilson’s theorem. 23 Formulas for prime numbers. 23 A Chinese criterion for prime numbers. 23 Are there more than one set of prime factors of a number? . 24 Asymptotic laws. 24 Growth of the concept of number. 24 Some results of permutation problems. 24 Tables. ....................................................... 26 Some long numbers. 27 How may a particular number arise? . 28 Numbers arising from measurement. 29 Decimals as indexes of degree of accuracy of measure. 30 iv CONTENTS. v PAGE Some applications. 30 Compound interest. 32 If the Indians hadn’t spent the $24........................... 32 Decimal separatrixes. 34 Present trends in arithmetic. 36 Multiplication and division of decimals. 42 Arithmetic in the Renaissance. 48 Napier’s rods and other mechanical aids to calculation. 51 Axioms in elementary algebra. 55 Do the axioms apply to equations? . 58 Checking the solution of an equation. 63 Algebraic fallacies. 65 Two highest common factors. 72 Positive and negative numbers. 73 Visual representation of complex numbers. 75 Illustrations of the law of signs in algebraic multiplication. 79 A geometric illustration. 79 A more general form of the law of signs. 81 Multiplication as a proportion. 82 Gradual generalization of multiplication. 82 Exponents. 83 An exponential equation. 84 Two negative conclusions reached in the nineteenth century. 85 The three parallel postulates illustrated. 87 In trigonometry. 88 Parallels meet at infinity. 89 The three postulates again. 89 Geometric puzzles. 91 Another puzzle. 95 Paradromic rings. 99 Division of plane into regular polygons. 100 A homemade leveling device. 102 A SCRAP-BOOK OF ELEMENTARY MATHEMATICS. vi PAGE “Rope stretchers.” . 103 The three famous problems of antiquity. 104 The circle-squarer’s paradox. 107 The instruments that are postulated. 110 The triangle and its circles. 113 Collinearity of centers. 113 The nine-point circle. 113 Linkages and straight-line motion. 116 The four-colors theorem. 120 Parallelogram of forces. 122 A question of fourth dimension by analogy. 123 Symmetry illustrated by paper folding. 124 Apparatus to illustrate line values of trigonometric functions. 125 “Sine.” . 127 Growth of the philosophy of the calculus. 128 Some illustrations of limits. 130 Law of commutation. 132 Equations of U.S. standards of length and mass. 133 The mathematical treatment of statistics. 134 Mathematical symbols. 138 Beginnings of mathematics on the Nile. 140 A few surprising facts in the history of mathematics. 141 Quotations on mathematics. 142 Autographs of mathematicians. 143 Bridges and isles, figure tracing, unicursal signatures, labyrinths. 145 Unicursal signatures. 150 Labyrinths. 151 The number of the beast. 154 Magic squares. 156 Domino magic squares. 160 Magic hexagons. 160 The square of Gotham. ..