THE ELEMENTS OF ARITHMETIC. BY AUGUSTUS DE MORGAN, OF TRINITY COLLEGE, CAMBRIDGE; ONE OF THE VICE-PRESIDENTS OF THE FOURTH EDITION. "Ce n'est point par la routine qu'on s'instruit, c'est par sa propre réflexion; LONDON: PRINTED FOR TAYLOR AND WALTON, BOOKSELLERS AND PUBLISHERS TO UNIVERSITY COLLEGE, M.DCCC.XL. PREFACE ΤΟ THE SECOND EDITION. I HAVE added several new chapters in this Edition, particularly on the Square Root, Proportion, and Permutations and Combinations, with many occasional articles, principally intended to give such ideas on the subject of Algebra as a young arithmetical student may be able, with a little assistance, to comprehend. I have also added six or seven Examples to each Rule, accompanied by the answers. These would be enough for any single pupil, but may not be considered sufficient for a school. To obviate this objection, I proceed to collect some expeditious modes of forming questions, of which the answers shall be readily known. I am aware that the publication of these methods, in a preface equally open to the master and the learner, is something like calling the enemy to council; nevertheless, as the following abbreviations all contain some mathematical principle, and as some facility of computation will be necessary even to make use of them, the master may depend upon it that a pupil who discovers and applies the way to make the answer to any one rule, is fit to pass on to the next. Addition.-Let a series of numbers be taken, each of which is the complement to 10" of the preceding. Strike out one or more, and arrange the rest miscellaneously. It will be evident how to ascertain whether these have been added up correctly. Many arrangements may be made for recollecting which numbers were struck out: for example, their complements may be made to begin with a given figure, and to be the only ones which begin with that figure. Another method, preferable perhaps to the former, is the following: Let any series of numbers be taken, such as a, b, c, &c. each of which exceeds the following; let the master form a-b, b-c, c-d, &c. and give the results to the pupil to add together, annexing to them the last number which he used. The answer will be a, which number cannot possibly be recovered by the pupil from the data, except by the very operation which he is required to perform. The continued subtractions may be done by one pupil, and the addition made by another; and thus the process may afford examples in the first two rules. Subtraction. In addition to the method just explained, the following may be used: Instead of giving one number to be subtracted from another once only, let it be required to subtract the first time after time from the second, until it can no longer be subtracted, as in the examples of article 46. This being, in point of fact, a question of division, may be proved by casting out the nines, and this after any number of steps, using the number of subtractions performed as the quotient. Or questions might be formed thus: Subtract 1259 from 12590, until this can no longer be done; or, multiplication by 25, 5, or 9, being very expeditiously done, the minuend might be 25 x, 5 x, or 9 x, and the subtrahend x. Multiplication and Division.- For these a table of squares and cubes is amply sufficient. The most useful of the kind is "Barlow's Tables," which gives at one glance the square and cube, square and cube roots, factors, and reciprocal of any number under 10,000. From such a table, many thousands of examples in multiplication and division may be derived immediately, with the answers, and many hundreds of thousands more may be obtained from the formula, where a and b are both even, or both odd. Greatest Common Measure, and Least Common Multiple. In "Barlow's Tables" is given a list of all prime numbers under 100,000. The multiplication of any two of these by the same number, will give a question in the first rule, with its answer. For the second rule, take any low prime numbers a, b, c, d, &c. and multiply one or more of them by each of the low prime numbers e, f, g, &c. Then will a b c d, &c. × efg, &c. be the least common multiple of the products above mentioned. All that has been said on the first four rules, applies equally to Common and Decimal Fractions. In the former case, with a table of squares, the formula |