DISTRICT OF MASSACHUSETTS, to wil : District Clerk's Office. BE IT REMEMBEREN, That on the twelfth day of November, A. D. 1823, and in the fortyeighth year of the Independence of the United States of America, LEONARD PEIRCE, of the 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 : Conversations on Arithmetic, with Demonstrations to each Rule, in Easy and Familiar Language; the whole of which, is designed to render the Study of Arithmetic pleasing and instructive to the Pupil. By LEONARD PEIRCE." In conformity to the Act of the 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 time therein mentioned ;” and also to an Act entitled, “ An Act supplementary to an Act, 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 extending the benefits thereof to the Arts of Designing, Engraving and Etching Historical, and other prints. JOHN W. DAVIS, 1 tule ? 11-6-36 33124 PREFACE Some apology will perhaps be required for publishing a work of this kind, when so many arithmetics are already before the public. The only apology which I have to offer, is a desire to render the study of arithmetic more beneficial to scholars. It has become almost a universal practice, particularly in the country, for scholars to pass through their arithmetic, without obtaining any thing more than a practical knowledge of the rules. This is owing, in a great measure, to the multiplicity of studies, which have, of late, been introduced into schools. When our present system of schooling was established, and indeed; till within a few years, the only branches taught, were reading, writing, arithmetic, and En glish grammar. The number of scholars was generally less than at present, and instructers were then enabled to proceed more methodically, and to bestow more attention on each branch, than the present state of schools, in general, will admit of. In most schools at present, the usual studies, in addition to those before studied, are geography, rhetoric, logic, history, and frequently several others, which, with the increase of scholars, compel an instructer to do every thing in a hurry, ard consequently, to da nothing well. These considerations led me to believe, that an arithme. tic on the plan of the following work, would be of public utility. By it, scholars will be able to obtain a knowledge of the first principles of arithmetic, with but very little assistance from an instructer. By this, is not meant, that a thorough knowledge of the theory of arithmetic may be obtained by a mere perusal of the demonstrations, this is not practicable. In order for this, the demonstrations must be attended to closely, and for a considerable time. It should be no discouragement to a scholar, that he can. not at first understand the demonstrations. - If he begins with a determination to understand them, and obtains a thorough knowl. edge of the simple rules, he will have little difficulty in the succeeding rules. In all arithmetics adapted to the use of schools and academies, which I have seen, the demonstrations are given algebraically, or there are no demonstrations at all. In the former case, scholars consider the demonstrations too difficult for them to understand, and in the latter, that nothing more is required of them, than is contained in the book. The authors whom I have consulted, in making this compilation, are, Pike, Webber, Lacroix, and Adams. In many of the rules, particularly Fractions, I have derived more benefit from Lacroix than from the others. Pike and Webber, I have found of much advantage in demonstra ing those rules which are not contained in Lacroix. Adams's demonstration of extracting the Square and Cube Roots, I have found to be superior to any which I have seen in any former author. From all these authors, and from Adams more than either of the others, I have extracted examples when they would an swer my purpose. I have calculated the work entirely in the THE AUTHOR. ON ARITHMETIC. CONVERSATION I. NUMERATION, SIMPLE ADDITION, AND SIMPLE SUBTRACTION. TUTOR AND PUPIL. Tutor. What is Arithmetic ? Pupil. Arithmetic, I think, is about numbers, and shows how to work with them so as to obtain correct an. swers to arithmetical questions. Tut. You are right. Arithmetic is the art or science which teaches how to compute by numbers. Without a knowledge of it, you will not be able to transact business correctly in any calling of life, and you will always be liable to be defrauded by those who are better informed. This ought to excite you to make great exertions to become familiar with every part of Arithmetic, necessary for a man of business. At first it may seem a hard task, but, with a little patience, in learning the fundamental rules, you will find arithmetic to be a very interesting study. Pup. I am always pleased to receive instruction from you, and whatever instruction you may give me, I will not let pass without my utmost endeavours to understand. Tut. Can you tell of how many kinds arithmetic is ? Pup. Arithmetic is of two kinds, theoretical and practical. Tut. Very well. And can you tell how these kinds differ? B Pup. The theory of arithmetic explains the nature and quality of numbers, and shows the reasons of practical operations. The practical part shows the most useful and expeditious method of working by numbers. Tut. What is Numeration ? Pup. Numeration, I think, shows how to read num. bers. Tut. You have a right idea of it, but you ought to have a thorough knowledge of ii before you proceed further. 1. Numeration is that part of arithmetic which teachés how to write any given numbers, by means of certain, characters, and how to read those characters when written. Numbers are expressed by the following characters. 1 2 3 4 5 6 7 8 9 one, two, three, four, five, six, seven, eight, nine. 2. When a number greater than nine is to be expressed, it is done by tens, and these tens are expressed by one of the above characters, having another character placed at its right. If it be required to write twelve, it is done by writing the figure 1, and at its right, write the figure 2, thus, 12. The one at the left hand being removed towards the left, one place, is in the place of tens, and expresses ten units, or ten times as much as 1 expresses when it stands alone ; the 2 at the right expresses only two, it being in the place of units. In this manner twelve is expressed. When it is required to - write eleven, you write two is thus, 11 ; the one at the left expressing ten, and the 1 at the right expressing one. If it be required to write one hundred and eleven, you write three ls, thus 111; the 1 at the right expresses one as in the former case, and likewise the 1 in the middle expresses ten ; but the 1 at the left expresses one hundred, it being removed one place further to the left than the middle figure, which is in the place of tens. The number twenty-two is written 22 ; the 2 at the left expresses twenty, and the 2 at the right expresses two. The expression two hundred and twenty-two is written 222, and all the figures increase in the same manner; every removal to the left increasing the value of the figure len times. |