A BITH'HETIC ON THE CANCELLING PLAN: ? EMBRACING THE RULES OF THREE, SINGLE AND DOUBLE, DIRECT AND CIES; INTEREST : AND ALL PROPORTIONAL QUESTIONS IN ONE RULE APPLICABLE TO THE WHOLE. THE PROCESS GREATLY SIMPLIFIED AND ABRIDGED. BY CHARLES G. BURNHAM, A. M. SECOND EDITION. BOSTON. CONCORD. 1841. Entered according to an act of Congress, in the year 18417 BY CHARLES G. BURNHAM, CHARACTERS USED IN THIS WORK. Equality is denoted by two horizontal lines. + Addition; as 4+3=7, which signifies that 4 added to 3 equals 7. X Multiplication; as 4X3=12, which signifies that 4 multi plied by 3 equals 12. Subtraction; as 4—3=1, which signifies that 3 taken from 4 leaves 1. )(, -, 1, 24, Division ; as 2)4(2 and 4:2=2, and 1=2, and 2/4=2; in either case it signifies that 4 divided by 2 equals 2. : : : :, Proportion; as 2:4::6: 12; which is read, 2 is to 4 as 6 is to 12. Vinculum ; as 4+3=7, which is read, the sum of 4 and 3 equals 7, and 4—3=1, is read, the difference of 4 and 3 equals 1. V, Radical Sign; placed before a number, denotes that the square root is to be taken. 42, implies that 4 is to be raised to the second power. 43, 66 3 third power. ASA MCFARLAND, PRINTER. 12-3- 366 33258 INTRODUCTION. l The author of the CANCELLING ARITHMETIC, the second edition of which, enlarged and improved, is now before the public, in his experience of many years, as a teacher of youth of both sexes, has enjoyed ample opportunity of testing the advantages which he claims for his system. The cancelling principle, though recognized in most arithmetical treatises, was by no writer fully illustrated, or to any considerable extent applied, previous to the publication of the first edition of the present work, in 1837. The application of the cancelling principle is not, however, the nyonly peculiar characteristic of this work. In order to lead the pupil readily to perceive, and to follow out, the connexion between the most simple and the most intricate arithmetical operations, great pains have been taken to impress upon the mind fundamental principles. Each of the four fundamental rules is made to illustrate its opposite, by a combination of processes. The rules for Multiplication are presented in a great variety of forms. The rule for Division, after being illustrated in the ordinary mode, is presented as the reverse of Multiplication those factors which were involved in the products, being by, an opposite process, again brought view. As an illustration of the above, we will suppose the pupil is required to divide 16 by 4. He first subtracts the divisor from the dividend, until no remainder is left, thus : 16 By counting the number of subtractions, the scholar per 4 ceives how many times the dividend contains the divisor. If |