A RITHMETIC, DESIGNED FOR THE USE OF HIGH SCHOOLS, ACADEMIES, AND COLLEGES; IN WHICH SOME ENTIRELY NEW PRINCIPLES ARE DEVELOPED, AND MANY BEFORE APPEARED IN ANY ARITHMETIC. By GEO. R. PERKINS, A. M., PRINCIPAL, AND PROFESSOR OF MATHEMATICS, UTICA ACADEMY. PUBLISHED BY BENNETT, BACKUS, & HAWLEY, UTICA ; GOULD, NEWMAN, & SAXTON, & Co., ROCHESTER. By George R. PERKINS, in the Clerk's Office of the Northern District of New York, in the year 1840. BENNETT, BACKUB, & HAWLEY, Franklin Square, Utica, N. Y. . PRE FACE. ARITHMETIC is a subject of so much importance, being that branch of mathematics upon which all the others are based, that any attempts to elucidate its rules of operation must be considered as worthy of commen. dation, even should those endeavors fail of their object. I am well aware that our schools are already flooded with books on this subject, but among the multiplicity of works which have appeared within a few years, there seems not to have been any material change ; they all wear nearly the same aspect. Whilst all other school books have been rapidly improving, our Arithmetic has remained nearly sta- tionary. This work is not designed to teach the fundamental, or ground rules, of the science, but is intended for such pupils as have already pursued some simple elementary Arithmetic as far as the Rule of Three. I know of no elementary work better adapted for this purpose, than “DAVIES, MENTAL AND PRACTICAL ARITHMETIC.” I have endeavored to simplify many of those rules, which hitherto have been considered as the most difficult. Under Chapter I., will be found many important properties of num. bers, demonstrated by the aid of prime numbers ; under this Chapter will also be found an exposure of the erroneous rule, given in nearly all our Arithmetics, for finding the least common multiple of several numbers. Under Chapter III., will be found some entirely new things in reference to that class of repetends, which I denominate Perfect Repetends. The rule for extracting the cube root, as well as the general rule for roots of all powers, as given under Chapter XI., has been drawn from MR. HOLDRED's method of solving Algebraic Equations ; which was first published in 1820. I am also indebted, for the arrangement of the numerical work, to the “ Root Extractor;" a work in pamphlet form, by TIMOTHY Clowes, LL. D., published in 1831. These rules have been universally admired by all who have used them. Under Chapter XV., I have given Analytical Solutions to that class of questions, which, by most authors, are solved by Position; which rule, in my opinion, should never be used when a direct solution can be ob. tained. This method of solving questions is preferable to every other, since it appeals to the reasoning powers of the mind, and is not, like many arithmetical rules,a mere mechanical method of operation. Many of these questions have been made expressly for this work, others have been copied from standard works on Algebra. GEO. R. PERKINS. Utica, April, 1841. . CONTENTS. PAGE. Examples illustrating definitions and symbols, . Multiplication of compound expressions, .. Interesting properties of numbers, Table of Prime numbers, . . . . . First method of finding the greatest common measure, .. First method of finding the least common multiple, To find all the divisors of any number given, The number of divisors given, without exhibiting then, . Vulgar fractions defined, . . . To reduce simple fractions to their lowest terms, To reduce improper fractions to mixed numbers, To reduce mixed numbers to improper fractions, To reduce compound fractions to simple ones, .. To reduce fractions to a common denominator, To reduce fractions to the least common denominator, Addition of fractions, · · · · Multiplication of fractions, . . . . . |