The reasoning faculties of youth should be brought into exercise on every subject susceptible of argument; and as the powers of the mind expand, this habit of reasoning will be applied to more abstruse disquisitions with great facility. The foregoing objects have been attempted in the following treatise on the science of numbers. Every term belonging to the science, which has been brought into use, has been clearly defined. As far as the subject would admit, but one new principle has been introduced at the same time; and each principle has been theoretically and practically illustrated in a manner capable of being comprehended by most scholars at the age of nine or ten years. The ten sections which comprise the body of the work, embrace all which is generally included in a more numerous class of divisions. The advantage of this arrangement is, that the learner will not be perplexed with a multiplicity of divisions founded upon the same general principle. When a student arrives at a new subject, he expects to find some new principles, but if mone exist there different from what is contained in preceding divisions, he is required, not only to waste his time in a fruitless search, but will generally pass, unnoticed, principles with which he has been previously acquainted. One design in writing this work, has been to furnish a system of Arithmetick suited to the wants of those who wish to pursue the higher branches of Mathematicks. TO INSTRUCTERS. No exertions of him who gives written instructions, can supersede the necessity of great pains in the superintending instructer. The peculiar expression of the countenance, the power of emphasis, and the use of numerous familiar examples to illustrate the same idea, are some of the advantages which belong to the acting instructer. When very young scholars commence the study of Arithmetick, they should for a considerable time be confined to short and easy questions, proposed by the instructer, or taken from books written for that purpose. Afterwards, they may be allowed to proceed with a book as their general guide. Scholars should not be allowed to leave one subject until they have acquired a thorough knowledge of the principles contained in that subject, otherwise their knowledge of what follows must be superficial. When a scholar is unable to operate a question, the solution should not be immediately given by the instructer, but he should question the student concerning the principles upon which the operation is founded. This, in most cases, will give the learner light enough upon the subject to enable him to solve the question himself. When this method fails, the instructer should point out. some collateral circumstances connected with the operation. Scholars should frequently be examined in what they have passed over, and more attention should be paid to a knowledge of principles, than to any mechanical method of solving questions ; and numerous practical questio" should be given by the instructer. The principal part of the examination should be confined to the illustrations. The given rules may be of use in commencing a new division or subject, but the student should not be required to place his dependence upon them. Let him understand the principles, which must be derived from the illustrations, and he will be able to solve questions with greater accuracy and expedition, and will always have more confidence in the correctness of the results, CONTENTS. PAGE. Explanation of arithmetical characters, - - 13 Combination of figures, - - - - 15 Numeration, or reading figures, - - - 17 Addition of simple numbers, . - - 23 Multiplication of simple numbers, - - 32 Subtraction of simple numbers, - - - 49 Division of simple numbers, - - - 57 Compound numbers, - . . . - 73 Addition of compound numbers, - - - 75 • Multiplication of compound numbers, . - 87 SECTION W. Subtraction of compound numbers, - - 100 Division of compound numbers, - - ... 109 Reduction of compound numbers, - - 117 Promiscuous questions in per cent., . - - 225 Questions to be solved by the principles al- ready illustrated, . - - , 232 Extracting the square root, . e - . . .247 Progression, . - - - - - . 266 Arithmetical progression, . - - . . 267 Arithmetical proportion, - - - . 274 Geometrical progression, - - - . 276 Geometrical proportion, - - - e 279 |