ARITHMETIC, IN WHICH THE PRINCIPLES OF OPERATING BY NUMBERS ARE ANALYTICALLY EXPLAINED, AND SYNTHETICALLY APPLIED ; THUS COMBINING THE ADVANTAGES TO BE DERIVED BOTH FROM THE INDUCTIVE AND SYNTHETIC THE WHOLE MADE FAMILIAR BY A GREAT VARIETY OF USEFUL AND INTER- LIFE. DESIGNED FOR THE USE OF SCHOOLS AND ACADEMIES IN THE UNITED STATES. BY DANIEL ADAMS, M. D. AUTHOR OF THE SCHOLAR'S ARITHMETIC, SCHOOL GEOGRAPHY, KEENE, N. H. PUBLISHED BY J. & J. W PRENTISS. 1840. Be it remembered, That on the eighteenth day of September, A. D. 1827, in the fifty second year of the Independence of the United States of America, Daniel Adams of said district, has deposited in this office the title of a book, the right whereof he claims as author, in the words following, to wit "Arithmetic, in which the Principles of operating by Numbers are analytically explained, and synthetically applied: thus combining the Advantages to be derived both from the inductive and synthetic Mode of instructing: the whole made familiar by a great Variety of useful and interesting Examples, calculated at once to engage the Pupil in the Study, and to give him a full Knowledge of Figures in their Application to all the practical Purposes of Life. Designed for the Use of Schools and Academies in the United States. By Daniel Adams. M. D. Author of the Scholar's Arithmetic, School Geography, &c." In conformity to the act of Congress of the United States, entitled, "An Act for the encouragement of learning, by securing the copies of maps, charts, and books, to the authors and proprietors of such copies during the times therein mentioned ;" and also to an act, entitled "An Act supplementary to an act for the encouragement of learning, by securing the copies and maps, charts, and Books, to the authors and proprietors of such copies during the times therein mentioned; and extending the benefits thereof to the arts of designing, engraving and etching historical and other prints." CHARLES W. CUTTER, Clerk of the District of New-Hampshire. A true copy. Attest, C. W. CUTTER, Clerk. 36926 PREFAC THERE are two methods of teaching, the synthetic and the analytic. n the synthetic method, the pupil is first presented with a general view of the science he is studying, and afterwards with the particulars of which it consists. The analytic method reverses this order: the pupil is first presented with the particulars, from which he is led, by certain natural and easy gradations, to those views which are more general and comprehensive. The Scholar's Arithmetic, published in 1801, is synthetic. If that is a fault of the work, it is a fault of the times in which it appeared. The analytic or inductive method of teaching, as now applied to ele. mentary instruction, is among the improvements of later years. Its introduction is ascribed to PESTALOZZI, a distinguished teacher in Switzerland. It has been applied to arithmetic, with great ingenuity, by Mr. COLBURN, in our own country. The analytic is unquestionably the best method of acquiring knowledge; the synthetic is the best method of recapitulating, or reviewing it. In a treatise designed for school education, both methods are usedful. Such is the plan of the present undertaking, which the author, occupied as he is with other objects and pursuits, would willingly have forborne, but that, the demand for the Scholar's Arithmetic still continuing, an obligation, incurred by long-continued and extended patronage, did not allow him to decline the labour of a revisal, which should adapt it to the present more enlightened views of teaching this science in our schools. In doing this, however, it has been necessary to make it a new work. In the execution of this design, an analysis of each rule is first given, containing a familiar explanation of its various principles; after which follows a synthesis of these principles, with questions in form of a supplement. Nothing is taught dogmatically; no technical term is used till it has first been defined, nor any principle inculcated without a previous developement of its truth; and the pupil is made to understand the reason of each process as he proceeds. The examples under each rule are mostly of a practical nature, beginning with those that are very easy, and gradually advancing to those more difficult, till one is introduced containing larger numbers, and which is not easily solved in the mind'; then, in a plain, familiar manner, the pupil is shown how the solution may be facilitated by figures. In this way he is made to see at once their use and their ap plication. At the close of the fundamental rules, it has been thought advisable to collect into one clear view the distinguishing properties of those rules, and to give a number of examples involving one or more of them These exercises will prepare the pupil more readily to understand the application of these to the succeeding rules; and, besides, will serve to interest him in the science, since he will find himself able, by the application of a very few principles, to solve many curious questions. The arrangement of the subjects is that, which to the author has appeared most natural, and may be seen by the Index. Fractions have received all that consideration which their importance demands. The principles of a rule called Practice are exhibited, but its detail of cases is omitted, as unnecessary since the adoption and general use of federal money. The Rule of Three, or Proportion, is retained, and the solution of questions involving the principles of proportion, by analysis, is distinctly shown. The articles Alligation, Arithmetical and Geometrical Progression, Annuities and Permutation, were prepared by Mr. IRA YOUNG, a member of Dartmouth College, from whose knowledge of the subject, and experience in teaching, I have derived important aid in other parts of the work. The numerical paragraphs are chiefly for the purpose of reference these references the pupil should not be allowed to neglect. His attention also ought to be particularly directed, by his instructer, to the illustration of each particular principle, from which general rules are deduced for this purpose, recitations by classes ought to be institute in every school where arithmetic is taught. The supplements to the rules, and the geometrical demonstrations of the extraction of the square and cube roots, are the only traits of the old work preserved in the new. Mont Vernon, (N. H.) Sept. 29, 1827. DANIEL ADAMS. - Fractions arise from Division, Miscellaneous Questions, involving the Principles of the preceding Rules, Pago 12 19 26 42 52 COMMON, OF VULGAR. Their Notation, Proper, Improper, &c. a Mixed Number to an Improper Fraction, To reduce a Fraction to its lowest Terms, Greatest common Divisor, how found, To divide a Fraction by a Whole Number; two ways, General Rule for the Multiplication of Fractions, FRACTIONS. 101 102 To change an Improper Fraction to a Whole or Mixed Number, 103 104 105 To reduce Shillings, &c., to the Decimal of a Pound, by Inspection, 146 the three first Decimals of a Pound to Shillings, &c., by Inspection, 157 General Rule for the Division of Fractions, Common Denominator, how found, Rule for the Addition and Subtraction of Fractions, |