Transcriber’s Notes: Underscores “_” before and after a word or phrase indicate _italics_ in the original text. Equal signs “=” before and after a word or phrase indicate =bold= in the original text. Carat symbol “^” designates a superscript. Small capitals have been converted to SOLID capitals. Illustrations have been moved so they do not break up paragraphs. Old or antiquated spellings have been preserved. Typographical and punctuation errors have been silently corrected. ELEMENTS OF ARITHMETIC. BY AUGUSTUS DE MORGAN, OF TRINITY COLLEGE, CAMBRIDGE; FELLOW OF THE ROYAL ASTRONOMICAL SOCIETY, AND OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY; PROFESSOR OF MATHEMATICS IN UNIVERSITY COLLEGE, LONDON. “Hominis studiosi est intelligere, quas utilitates proprie afferat arithmetica his, qui solidam et perfectam doctrinam in cæteris philosophiæ partibus explicant. Quod enim vulgo dicunt, principium esse dimidium totius, id vel maxime in philosophiæ partibus conspicitur.”--MELANCTHON. “Ce n’est point par la routine qu’on e’instruit, c’est par sa propre réflexion; et il est essentiel de contracter l’habitude de se rendre raison de ce qu’on fait: cette habitude s’acquiert plus facilement qu’on ne pense; et une fois acquise, elle ne se perd plus.”--CONDILLAC. _SEVENTEENTH THOUSAND._ LONDON: WALTON AND MABERLY, UPPER GOWER STREET, AND IVY LANE, PATERNOSTER ROW. M.DCCC.LVIII. LONDON: PRINTED BY J. WERTHEIMER AND CO., CIRCUS-PLACE, FINSBURY-CIRCUS. PREFACE. The preceding editions of this work were published in 1830, 1832, 1835, and 1840. This fifth edition differs from the three preceding, as to the body of the work, in nothing which need prevent the four, or any two of them, from being used together in a class. But it is considerably augmented by the addition of eleven new Appendixes,[1] relating to matters on which it is most desirable that the advanced student should possess information. The first Appendix, on _Computation_, and the sixth, on _Decimal Money_, should be read and practised by every student with as much attention as any part of the work. The mastery of the rules for instantaneous conversion of the usual fractions of a pound sterling into decimal fractions, gives the possessor the greater part of the advantage which he would derive from the introduction of a decimal coinage.