THE YOUTH'S ASSISTANT IN THEORHETIC AND PRACTICAL ARITHMETIC; DESIGNED FOR THE USE OF SCHOOLS IN THE UNITED STATES. BY ZADOCK THOMPSON, A. M, SIXTH EDITION. Burlington: 1832. The following explanations and table, not being contained in the Written Arithmetic, are inserted here for the convenience of those who have not studied the Mental Arithmetic. =EQUALITY is expressed by two horizontal marks ; thus 100 cts. 1 dollar, signifies that 100 cents are equal to one dollar. +ADDITION is denoted by a cross, formed by one horizontal and one perpendicular line, placed between the number; as 4+5=9, Bignifying that 4 added to 5 equals 9. X MULTIPLICATION is denoted by a cross, formed by two oblique lines placed between the numbers; as 5x3=15, signifying that 5 multiplied by 3, or 3 times 5 are equal to 15. -SUBTRACTION is denoted by one horizontal mark, placed between - the numbers; as 7–4=3, signifying that 4 taken from 7 leave 3. D(, -or DIVISJON is denoted three different ways ; lst. by the re. versed parenthesis ; 2dly. by a horizontal line placed between the numbers with a dot on each side of it; and 3dly. by writing the number to be divided over the other in the form of a fraction ; thus 2)6(3, and 6--2=3 and =3, all signify the same thing, namely that if 6 be divided by 2 the quotient is 3. Entered according to Act of Congress, in the year one thousand aight hundred and twenty 'eight,' by ZADOCK THOMPSON, in the Clerk's Office of the District of Vermont. GEUGE A LA 10.1 JANUAY 26, 1924 (To the fourth edition. ) WHEN the improved edition of this work was published, in 1828, it was intended that the Written Arithmetic, which forms the second and third parts should always be accompanied by the Mental Arith. metic embraced in the first part. Since that time it has, however, been thought best to transpose such tables from the Mental to the Written Arithmetic, as to render the latter complete without the former, in order to lessen the expense of the book to those who do not wish to study mental arithmetic, or who have studied some other treatise ; and, thus prepared, it is now presented to the public. No alteration has been made from the last edition in the arrange. ment of the rules, and the whole of the second part is presented, as before, on the inductive plan of Lacroix. The principles are first developed by the analysis of familiar examples, and the method of applying these principles to the solution of questions is then ex. pressed in general terms, forming a Rule, which is still further il. lustrated by a great variety of practical questions. The analysis is printed in small type, occupies but little space, and may be omittedby those who wish to use rules without understanding them. Addition and Multiplication, both involving the same principles, are presented in connexion, and also Subtraction and Division. A knowledge of decimals being necessary to a good understanding of our Federal currency and this knowledge being easily acquired by such as have learned the notation of whole numbers, decimals and Federal money are introduced immediately after the first section on simple numbers. By acquainting the pupil thus early with dicimals, he will be likely to understand them better and to avail himself of the facilities they afford in the solution of qnestions and the transaction of business. Reduction ascending and descending are arranged in parallel columns and the answers to the questions of one column are found in the corresponding questions of the other. Compound multiplication and division are arranged in the same way, and only one general rule for each is given, which was thought better than to perplex the pupil with a multiplicity of cases. Interest and other calculations by the hundred are all treated decimally, that method being most simple and conformable to the notation of our currency. The nature and principles of proportion are fully developed and the method of applying them to the solution of questions clearly shown. The written arithmetic of fractions being, to young pupils, some. what difficult to be understood, is deferred till they are made fa. mlliar with the most important arithmetical operations performed with whole numbers and decimals. The nature of roots and powers has been more fully explained in the present edition, and several Row diagrams introduced for their elucidation. Throughout the second part, it has been our main object to familiarize the pupil The third part is mostly practical, and composed of such rules CONTENTS. PAGE. SECT. II. SIMPLE NUMBERS. Fractions reduced 25 SECT. VII.-POWERS & ROOTS 106 Multiplication of Decimals 27 Involution 30 Extraction of the Square Root 109 Vulgar Fractions changed to Deci. 31 Extraction of the Cube Root Extraction of Roots in General 116 SECT. IV.-COMPOUND NUM SECT. IX.-MISCELLANEOUS Tables of Money, Weights, &c. 38 Arithmetical Progression 118 Permutation of Quantities 1128 Simple Interest 57 SECT, 1.-EXCHANGE OF OUR- Commission and Insurance 63 SECT. II.-MENSURATION 132 64 Menguration of Superficies 132 Single Rule of Three ** Mechanical Powers SBCT. VII.-FRACTIONS Tablo of Cylindric Measure Table of Square Timber Measure 150 Integers treated as Fractions Log Table-Log Measure Multiplication and Division of Fr. 91 Log Tablo-Board Measure 154 Division by Fractions PRACTICAL FORMS. 242 PART II. SECTION 1. NOTATION AND NUMERATION. 70. An individual thing taken as a standard of compari son, is called unity, a unit, or one. 71. Number is a collection of units, or ones. 72. Numbers are formed in the following manner; one and one more are called two, two and one, three, three and one, four, four and one, five, five and onc, six, six and one, seven, seven and one, eight, eight and one, nine, nine and one, ten; and in this way we might go on to any extent, forming collections of units by the continual addition of one, and giving to each collection a different name. But it is evident, that, if this course were pursued, the names would soon become so numerous that it would be utterly impossible to remember them. Hence has arisen a method of combining a very few names, so as to give an almost infinite variety of distinct expressions. These names, with a few exceptions, are derived from the names of the nine first numbers, and from the names given to the collections of ten, a hundred, and a thousand units. The nine first numbers, whose names are given above, are called units, to distinguish them from the collections of tens, hundreds, &c. The collections of tens are named ten, twenty, thirty, forty, firy, sixty, seventy, eighty, ninety.(6). The intermediate numbers are expressed by joining the names of the units with the names of the tens. To express one ten and four units, we say fourteen, to express two tens and five units, we say twenty-five, and others in like manner. The collections of ten tens, or hundreds, are expressed by placing before them the names of the units; as, one hundred, two hundred, and so on to nine hundred. The intermediate numbers are formed by joining to the hundreds the collections of tens and units. To express two hundred, four tens, and six units, we should |