MADE EASY, BY A COMPLETE DEVELOPMENT OF ITS PRINCIPLES, AND SIMPLIFICATION OF ITS RULES, ILLUSTRATED BY NUMEROUS PRACTICAL EXAMPLES, ADAPTED TO SCHOOLS AND PRIVATE TUITION, AND CALCULATED TO EXPEDITE THE PROGRESS OF THE TO WHICH ARE ADDED, NUMEROUS SHORT METHODS OF ARRIVING AT RESULTS, NOT TO BE FOUND IN ANY OF THE ELEMENTARY THE WHOLE FOUNDED ON A SYSTEM OF EQUATIONS BOD BY A. TURNBULL. LONDON: PUBLISHED BY SIMPKIN AND MARSHALL, AND SOLD BY ALL BOOKSELLERS. 1836. ENTERED AT STATIONERS' HALL. NEWCASTLE: PRINTED AT THE COURANT OFFICE, PILGRIM STREE? BY J. BLACKWELL AND CO. PREFACE. THE number of Elementary Works on Arithmetic being so great, it will be expected that some reason should be given for obtruding another on the public. The Author has no desire to detract from the merits of other publications, but it is his wish to supply in this treatise what he deems a great omission in almost every work hitherto published on this important branch of education. These works are not sufficiently explanatory. The principles of the science are not clearly unfolded to the understanding of the pupil, and he is taught to work arithmetical questions mechanically rather than rationally. He is taught the how, but not the why of the science. The great Pascal invented a machine for the working of all questions in arithmetic. The greater portion of students are only made calculating machines, who can solve long and difficult questions in figures, multiplying and dividing according to rule, without really understanding on what principles they have been proceeding in the solution of the question proposed. They attain correctness in the answer by a strict observance of certain given rules, but the principle of the rule, the reason why the question is worked in such a manner, they do not understand, because it is not explained. Hence the difficulties of arithmetic are multiplied by the unnecessary multiplication of rules, through which the teacher appeals rather to the memory, which he overburthens, than to the understanding of the student, which he ought to enlighten with principles. This is the error which the Author wishes to avoid. He is desirous of rendering Arithmetic easy, by the development of its principles; and, by addressing the understanding of the student, to render the task that has hitherto been imposed upon his memory less. From Addition to the most involved and difficult processes in Arithmetic-from the simplest rule to the most complicated-the clue of principle will be given to guide him through the most direct passages, as well as through the most intricate labyrinths of the science. Every thing will be explained, and the student will not be suffered to proceed a step without understanding the reason of every operation he performs. Nor is the above-mentioned defect the only one to be met with in School Arithmetics; there is also, generally, a defect in arrangement. Fractions and Progression, for instance, are usually deferred until after the student has passed through Single and Double Proportion, Practice, Interest, Fellowship, Exchange, &c., though all these rules are but so many applications of the principles of Fractions; and as Fractions must unavoidably occur frequently in the the working of these rules, boys have very often to add, subtract, multiply, and divide fractions, without having any knowledge of the principles of this department of Arithmetic. Progression, too, is the very basis of Proportion, and therefore ought to precede it by all means. The plan adopted in the present work is intended to lead the student on, by easy gradations, from that which is simple and easy to that which is more complicated and difficult. The Author has endeavoured to anticipate such obstacles as may occur; and to show the way to surmount them, he has developed the principles on which every rule is founded, explained the method of applying them, and endeavoured to make the student perceive that the course he is pursuing must necessarily lead to the right result. The different subjects comprised in the work are treated of in the following order :-Notation, Addition, Subtraction, Multiplication, and Division of WHOLE NUMBERS-Notation, Reduction, Addition, Subtrac tion, Multiplication, and Division of COMPOUND NUMBERS- -Notation, Reduction, Addition, Subtraction, Multiplication, and Division of FRACTIONS, Vulgar and Decimal- ALGEBRA ARITHMETICAL AND GEOMETRICAL PROGRession-PropORTION, as founded on Geometrical Progression - PRACTICE-Involution and Evolution of Numbers. In Notation it is explained how figures change their value by a change in their position, and thus the reason for carrying and borrowing in Addition and Subtraction is made easily perceptible to the student, when he enters on these rules. These rules are treated of conjointly, and made subservient in proving each other; and here the reason for carrying is explained: Multiplication and Division are also made to prove each other, and are shown to be repeated Additions and Subtractions. The principle of carrying, and the relatative value of figures, is further explained. Measure is explained, and the principles of the rules for finding the greatest common measure, and the least common mul. tiple, are made plain. The relationship of multiple and sub-multiple is fully considered. The Notation of simple and compound numbers is compared. The principle of carrying in compound numbers is explained. The origin and nature of Fractions are considered, and their connexion with Proportion and Algebraic Equations is shown. The powers and roots of numbers are treated of scientifically, and illustrated by Arithmetical and Geometrical Progression. Proportion is also treated of with relation to Geometrical Progression. Under the head of General Proportion are included, the Rule of Three, Double Rule of Three, The Chain Rule, Interest, Exchange, Barter, Loss and Gain, Fellowship, Discount, and some others; also, the method is shown of working the same questions by Equations, and the two modes are compared. In the course of the work are exhibited many concise modes of calculation, adapted to matters of business, and tending to facilitate mercantile calculations. The method of calculating Interest, Commission, Discount &c. is |