FOR THE USE OF TEACHERS. DESIGNED AS A COMPANION AND SUPPLEMENT TO THE AUTHOR'S MANUAL OF ARITHMETIC, FOR THE USE OF PUPILS IN NATIONAL AND OTHER ELEMENTARY SCHOOLS. INCLUDING SOLUTIONS OF THE MORE DIFFICULT EXERCISES IN THE MANUAL. BY THE REV. JOHN HUNTER, VICE-PRINCIPAL OF THE NATIONAL SOCIETY'S TRAINING "That is to muche, and meete for no man,-to bee beleved in all thinges without "Nullâ re juvenum magis vegetatur ingenium, quàm numerorum arte dis- www LONDON: FRANCIS AND JOHN RIVINGTON, ST. PAUL'S CHURCHYARD, AND WATERLOO-PLACE. Sold also at the NATIONAL SOCIETY'S DEPOSITORY, Sanctuary, Westminster. 1847. By the same Author, MANUAL OF ARITHMETIC; for the Use of EXERCISES in the FIRST FOUR RULES of EXERCISES in ENGLISH PARSING; exempli- tons. In Preparation, A TEXT-BOOK of ENGLISH GRAMMAR, for London: Printed by W. CLOWES and SONS, Stamford Street. PREFACE. THERE are two departments of elementary education which eminently conduce to the development of intellect-the sciences of Language and Number; and that they should be so taught as to constitute an intellectual discipline,-that they should not merely fill the memory with dogmatic Rules, but exercise the reason, has for some time been recognised by the advocates and expositors of sound education. In the infancy of any science there is generally a tendency to extravagance and misdirection of Theory, and in its progress there frequently arise a disinclination to Theory, and a disposition to rest satisfied in gathering merely the results of demonstration. Such, at least, has been the case with the subject of the present volume. The mysticism of the Pythagoreans and the abstractions of the Boethian school carried Arithmetic beyond its legitimate province, and refused its application to the ordinary purposes of life; while, on the other hand, it must be admitted that the commercial school of arithmetical writers, which arose in this country in the sixteenth century, too much discouraged attention to Theoretical, and thereby retarded improvement in Practical Arithmetic. During the long prevalence of those unreasoned methods of operation in which a course of arithmetical instruction consisted, there was one circumstance which tended to maintain some recognition of Arithmetic as an intellectual study, the customary practice of constructing and collecting problems which had no immediate reference to buginess, and were designed chiefly to exercise the ingenuity of students. The Recreations of Martin Clare, first published in 1739, in his Youth's Introduction to Trade and Business, and augmented in subsequent editions, formed a rich provision for the compilations of Vyse, Birks, Hay, &c.; and these, with a few additional Recreations by Welsh, Halbert, and others, were appropriated in greater or less quantity, in successive arithmetical productions, to nearly our own day. The uncouth language in which many of the problems were expressed, and especially the rhymes, in which it was often desperately attempted to marshal the terms of a question according to the harmonious numbers of the Muse, were suffered, indeed, to remain too long unimproved, and presented an unintellectual aspect to the eyes of any but those who could "by numbers judge a poet's song." But the elements of scientific amelioration were slowly developed by attempts to improve the solutions of such problems, by employing arithmetically demonstrative methods for many questions which had been committed to the determination of Position or of Algebra. And, eventually, the substratum of improvement has been deprived of superficial deformities, till, like the inscription on the tower of Pharos when the coating of plaster had worn away, Arithmetic has assumed an aspect worthy of science, in the writings of such men as Thomson, De Morgan, Hind, and many others. Perhaps one faulty condition of Arithmetic, as taught at present, is the unnecessary tediousness with which principles are sometimes inculcated, the small degree in which the demonstrative system is extended to the higher applicate forms of the science, and the limits of difficulty to which arithmetical problems are confined. The admirable Treatise by Professor Thomson of Glasgow is free from these objections. His uniform attention to the important object of making the rules of Arithmetic an intellectual discipline, and his admirable collection of Exercises, have left the Author of the present TextBook no other apology for the publication of a new work on Arithmetic, than the warrantableness of an endeavour to carry forward the improvement of the science, by furnishing additional means of intellectual exercise to the more advanced student, and more suitably accommodating the circumstances of the ordinary school-boy. The encouragement which the Pupil's Manual has met with, during the short period of its publication, has greatly exceeded the Author's expectation; but he hopes that that little work will be rendered still more acceptable and useful by the aids furnished in this Companion and Supplement. It will be seen |