PREFACE. IN the course of more than fifty years devoted to the instruction of youth, in Languages and Mathematics, the author of this Compendium, has had frequent occasion to remark on the merits, deficiencies, and defects of various books intended for the use of schools. Of systems of Arithmetic, there are many deserving of high commendation. There seems, however, to the author, yet to be wanting, a cheap compendium of short clear rules and easy examples, for the use of junior pupils of both sexes. For this purpose, and to serve as an introduction to large treatises the author has endeavoured to simplify all the rules, to render them concise without obscurity, and intelligible without being too prolix. But, as Iter est breve per exempla, he has adapted to each rule, examples so contrived as not only, not to prevent, as long questions do, the requisite constant exercise of multiplication and division; but to accelerate the progress of the little Tyro in his road to acquire habits of quick and accurate calculation. The work also of many examples is given, in full, where the rules, on the first reading, may not seem easy to be understood. The table of contents shows, that the author differs from others, in respect to the arrangement of some of the rules. For reasons obvious to the tutor, Multiplication comes close after Addition, then Subtraction and Division. Fellowship, Barter, Loss and Gain, and Alligation, follow immediately after the Rule of Three, of which indeed they are little more than exemplifications. By thus classing these five rules, the principle of proportion is kept in view, and the pupil will be led to consider the four last rules as being connected with the Rule of Three, and as only additional exercises on it and the preceding rules. The principle of the Rules of Practice is explained in a copious manner by precepts, and by examples wrought out in full; and is applied to Tare and Tret, Interest, Commission, Insurance, Brokerage, and Exchanges. The latter Rule is treated of very briefly and the examples are few, but the author supposes quite sufficient to exercise a junior pupil. For, although the places, between which and London, exchanges are negotiated, are numerous, and continually increase; the methods of calculation are however always the same, by the Rule of Three or by means of aliquot parts. Connected with the rules of Practice, and grounded upon them, are the short methods of reckoning introduced in the appendix, useful and easy as mental exercises, when the quantity does not exceed 400. Vulgar Fractions, a part of Arithmetic, so abstract, that junior pupils seldom well comprehend their nature, are here taught in a manner different from that of other authors. The 1st. problem shows how a vulgar fraction is produced by the reduction of a given quantity to the fraction of its integer; and, that the pupil may, at once, have a clear idea of the nature of the fractional expression, it is left without reduction to its least terms. The problem for reducing fractions to a common denominator, is here placed last in reduction; and, by its immediate proximity to Addition, will, the author presumes, render the rationale of this part of Arithmetic less difficult to be comprehended. Circulating decimals and contractions of decimal operations are here omitted, to obtain space for more useful matter. For further exercises in decimals, a short table of compound interest is given, and another for the valuation of annuities for ages between 51 and 80. Duodecimals complete the business of the three kinds of fractions usually taught. Involution, with a table of Powers, and Evolution follow in order. The Rules of Trial and Error are briefly introduced, as are Arithmetical and Geometrical Progression. By means of the short rules to the two Problems in the latter, the examples are made much more easy of solution, than by the long rules and directions given by Dilworth, and other authors. The author has purposely avoided to place under any rule, examples that require the management of large numbers, and has not inserted among the miscellaneous examples, any that may not be solved by the rules in this compendium. He has also omitted long operations or such as have reference to statistics or experimental philosophy, with which some books abound. Such exercises may be very proper for studious senior pupils, but as every experienced tutor knows, they are formidable obstacles in the way of the junior Tyro, while endeavouring to become an expert and correct arithmetician. The tutor may perhaps find, that for some pupils there are too few examples to the first rules. As the paper is good, additional figures may be written to supply deficiencies.-As one of the factors, for the multiplications both of whole numbers and decimals, is always divisible by 9, the truth of the products from such factors may easily be known.— The short rule, page 88, for interest at 6 per Cent., is thus discovered. Let P = the Principal, M the Months, R the Ratio, '06. Then PR X the Interest in pounds; in shillings, 20 PR X which reduced, is P M x 1 or PM the Rule. M IQ' In instructing a class, the directions for the examples to the first rules given, viva voce, by a monitor, may be used with much effect. In a short time the whole book may be learned, and its principal parts transcribed, by an industrious pupil of ordinary capacity. BONNYCASTLE, KEITH, or some other scientific work, may then follow, with advantage; and, the author presumes, may be studied with greater success after the use of this easy introduction. But such authors have also a claim to the attention of youth, as they approach to mature years. It is, however, to be regretted, that Arithmetic, replete as it is, with utility and rational entertainment, should be almost quite disregarded after the completion of school instruction: Perhaps one cause of this disregard, is the dislike originally occasioned, by the injudicious, and too early, use of books, containing rules and examples too difficult for the exercise of juvenile capacities. ARITHMETIC. ARITHMETIC teaches how to compute by means of these ten Figures, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. The fundamental rules are NOTATION, NUMERATION, ADDITION, MULTIPLICATION, SUBTRACTION, & DIVISION. Notation teaches how to express Numbers in Figures; Numeration how to read and write them. TABLE. To be read thus 9 86 754 4 THOUSANDS, 365 59 THOUSANDS, 873 869 THOUSANDS, 467 4 MILLIONS, 754 THOUSANDS, 798 23 MILLIONS, 582 THOUSANDS, 462 825 MILLIONS, 687 THOUSANDS, 649 ΝΟΤΑΤΙΟΝ. Observe carefully how the Numbers are divided by Commas in the Table. Express in Figures, Fifty-nine thousand eight hundred and seventy-three. Eight hundred and twenty-five millions six hundred thousand. Four millions and eight. Eight hundred millions and eleven. NUMERATION. Point off the numbers by means of Commas as in the Table. Read and Write the following Numbers: 87680.-67100.-507009.-1687041.-94807.-09467 80000900. B ADDITION OF INTEGERS. Addition teaches how to find the sum or total of several numbers. The Addition Table must be learned by heart. RULE.-Place units under units, tens under tens, hundreds under hundreds, &c., draw a line, and observe the manner of doing the second of the following examples : The sum of the figures in the right hand column is 21, write 1 and carry the 2 for tens to the 7 in the next column, of which the sum will be 31; write 1 and carry 3 for tens to the 4 in the next column, and the sum will be 12; set down 2 and carry 1 to the remaining left hand figures, and their sum will be 16. PROOF.-Divide the example into two parts by means of a line; add the figures below the line together; then add this part and the upper part together, and the sum of the two parts, if each line of the work be right, will be the same as the total. Examples. Add 647, 9678, 49678, 87, and 974 together. What is the sum of 3789, 97167849, 876, and 28 ? Write seven lines of the number 905769 and find their sum. Find the sum of nine lines of the number 678950847. |