At the time when this work was first published, the importance of establishing arithmetic in the young mind upon reason and demonstration, was not admitted by many. The case is now altered: schools exist in which rational arithmetic is taught, and mere rules are made to do no more than their proper duty. There is no necessity to advocate a change which is actually in progress, as the works which are published every day sufficiently shew. And my principal reason for alluding to the subject here, is merely to warn those who want nothing but routine, that this is not the book for their purpose. A. DE MORGAN. _London, May 1, 1846._ [1] Some separate copies of these Appendixes are printed, for those who may desire to add them to the former editions. TABLE OF CONTENTS. BOOK I. SECTION PAGE I. Numeration 1 II. Addition and Subtraction 14 III. Multiplication 24 IV. Division 34 V. Fractions 51 VI. Decimal Fractions 65 VII. Square Root 89 VIII. Proportion 100 IX. Permutations and Combinations 118 BOOK II. I. Weights and Measures, &c. 124 II. Rule of Three 144 III. Interest, &c. 150 APPENDIX. I. On the mode of computing 161 II. On verification by casting out nines and elevens 166 III. On scales of notation 168 IV. On the definition of fractions 171 V. On characteristics 174 VI. On decimal money 176 VII. On the main principle of book-keeping 180 VIII. On the reduction of fractions to others of nearly equal value 190 IX. On some general properties of numbers 193 X. On combinations 201 XI. On Horner’s method of solving equations 210 XII. Rules for the application of arithmetic to geometry 217 ELEMENTS OF ARITHMETIC. BOOK I. PRINCIPLES OF ARITHMETIC. SECTION I. NUMERATION.