hours in a day, how many rods can 6 men build in 16 days, working 12 hours in a day? Operation. 3 912 4 12 4 64, Answer. Operation. 44 124 203 The mode of stating questions, here adopted, is of great advantage when there is a reduction to be made of the terms of the Proportion. Example. If one nail cost 3 farthings, how many pounds will 40 yards cost? It will be seen that the yards are here reduced to nails, and multiplied by 3 farthings, the cost of 1 nail. We then reduce the farthings to pounds, by dividing by 4, 12 and 20. By writing down the factors of dividend and divisor before commencing the operation, opposing factors are at once seen, and can be excluded from the operation. But in the ordinary mode the factors of the dividend would be lost in the process of reduction, and we should be obliged to divide by the same factors, to obtain the answer. The process of cancelling is thus explained: 4, on the right of the line, cancels 4 on the left, and 3×4, on the right, cancels 12 on the left. Then 20 in 40, twice; or, rejecting the factor, 10, we say, 2 in 4, twice. (For Illustration of similar questions, see page 149.) 2 £. Ans. The cancelling mode of stating questions, as applied in Proportion, is useful to the scholar, in enabling him to know where to commence the statement of a question. He here learns that the difficulty lies in the demand of the question, and that the terms of the condition are the facts in the case by which the difficulty presented in the demand is to be removed, and the question answered. This mode of teaching arithmetic is found to prepare the scholar better than any other for the study of Algebra, and the higher branches of Mathematics. Having a thorough knowledge of the cancelling system the scholar is prepared to pursue the study of Algebra with comparatively little assistance. It is in Algebra that a knowledge of the use of factors is indispensable. Suppose the scholar is required to divide ab+ad by a. He writes the question as he has been accustomed, and excludes from the dividend a factor equal to the divisor. Thus: a ab+ad b+d, Answer. Again: Suppose the student be required to compound the ratios of the following couplets : He is already familiar with analogous processes, which may be shown by writing numerals instead of letters: Thus We have now given a brief view of the Cancelling System, and the mode of treating it, which the writer has for several years practised. It is confidently believed that this mode of presenting the subject of arithmetic is better calculated to induce a fondness for the study that it unfolds more of the science, and brings out principles more clearly, than any other system now before the public. With these views the author submits this second edition of his work to the candid perusal of all who are interested in the progress of knowledge. CHARLES G. BURNHAM. Pembroke, N. H., November 27, 1841. The author of this work acknowledges himself indebted to several teachers who have used the first edition, for many profitable suggestions, especially to Mr. WILLIAM A. BURNHAM, teacher of Mathematics in the Burr Seminary, Vermont. Multiplication and Division by Ratio, or the Relation of Numbers, To reduce a Whole Number to an To reduce Improper Fractions to Mixed Numbers, and the reverse, To reduce a Fraction to its lowest General Rule for the Multiplication 50 Multiplication of Decimals, 54 Subtraction of Federal Money, 57 Multiplication of Federal Money, Reduction Descending and Ascend- 22, for Examples read Exercises. 48, twelfth line, for divide read separate. 49, tenth line from the bottom, for divided read separated. 72, thirteenth line above "RULE," after the word multiplicand, in- ARITHMETIC. ARITHMETIC is the science of numbers. It explains their properties, and teaches how to apply them to practical purposes. The principal, or fundamental rules, are, Notation, Numeration, Addition, Subtraction, Multiplication, and Division. These are called fundamental rules, because all questions in Arithmetic are solved by one or more of them. 7 Notation is the expressing of any number or quantity by figures; thus, 1 one; 2 two; 3 three; 4 four; 5 five; 6 six; seven; 8 eight; 9 nine; 0 cipher. The nine first figures are sometimes called digits, from the Latin word digitus, which means a finger. In the early stages of society people counted by their fingers; they were also formerly all called ciphers-hence the art of Arithmetic was called ciphering. There are two methods of Notation-the Arabic, as above, and the Roman, which is expressed by the following seven letters of the Alphabet. I, V, X, L, C, D, M. 1 2 3 4 5 6 7 S 9 10 20 30 40 I, II, III, IV, V, VI, VII, VIII, IX, X, XX, XXX, XL. 50 60 70 When a letter of less, is placed before one of a greater value, it diminishes the value of the greater, by the value of itself-thus, X signifies ten, but IX is only nine. When a letter of less, is placed after one of greater value, it increases the value of the greater by the value of itself. This method is seldom used except in numbering chapters, sections, &c. QUESTIONS. 1. What is Arithmetic? 2. What are the principal, or fundamental rules? 3. Why so called? 4. What is Notation? 5. What are the nine first figures sometimes called? 6. What were they all formerly called? 7. How many methods of Notation, and what are they? 8. How many are the Arabic characters, or figures? 9. By what is the Roman method expressed? 10. How is a letter affected when one of less value is placed before it? 11. How when one of less value is placed after it? 12. For what is the Roman method of notation principally used? B |