The principles, definitions, rules, and applications, so far as developed in this work, coincide with the other books of the same series. Many of the Contractions, and special applications of the rules, particularly those that are at all difficult, have been omitted, and also the treatment of Denominate Fractions, and Decimals, all of which are fully and practically treated in the Progressive Practical, and the Higher Arithmetic. A few easy and practical applications of Cancellation, Analysis, Percentage, and Simple Interest have been given, and a very large number of easy examples. IMPROVED EDITION. The plates of this book having become much worn, and its circulation requiring the printing of large editions, the book has been newly electrotyped, in the latest and best style of typography. The text of the previous editions has been preserved, unchanged, page for page, and line for line, as nearly as possible, except to expunge from the tables, and some other portions of the book, such matter as has become entirely obsolete and useless. No changes have been made, that will at all interfere in its use, with any of the previous editions. April, 1877. RUDIMENTS OF ARITHMETIC. DEFINITIONS. 1. Quantity is any thing that can be increased, dimin ished, or measured; as distance, space, weight, motion, time. 2. A Unit is one, a single thing, or a definite quantity. 3. A Number is a unit, or a collection of units. 4. An Abstract Number is a number used without reference to any particular thing or quantity; as 3, 24, 756. 5. A Concrete Number is a number used with reference to some particular thing or quantity; as 21 hours, 4 cents, 230 miles. 6. A Simple Number is either an abstract number, or a concrete number of but one denomination; as 48, 52 pounds, 36 days. 7. A Compound Number is a concrete number expressed in two or more denominations; as 4 bushels 3 pecks, 8 rods 4 yards 2 feet 3 inches. 8. An Integral Number, or Integer, is a number which expresses whole things; as 5, 12 dollars, 17 men. 9. A Fractional Number, or Fraction, is a number which expresses equal parts of a whole thing or quantity; as, of a pound, of a bushel. 10. Like Numbers have the same kind of unit, or express the same kind of quantity. Thus, 74 and 16 are like numbers; so are 74 pounds and 16 pounds; also, 4 weeks 3 days, and 16 minutes 20 seconds, both being used to express units of time. 11. Unlike Numbers have different kinds of units, or are used to express different kinds of quantity. Thus, 36 miles, and 15 days; 5 hours 36 minutes, and 7 bushels 3 pecks. 12. Arithmetic is the Science of numbers, and the Art of computation. 13. The Five Fundamental Operations of Arith metic are, Notation and Numeration, Addition, Subtrac tion, Multiplication, and Division. NOTATION AND NUMERATION, 14. Notation is a method of writing or expressing numbers by characters; and, 15. Numeration is a method of reading numbers expressed by characters. 16. Two systems of Notation are in general use the Roman and Arabic. THE ROMAN NOTATION 17. Employs seven capital letters to express numbers, thus : Letters. Values. I V X L C D M 18. The Roman notation is founded upon the following principles: 1st. Repeating a letter repeats its value. Thus, Il represents two, XX twenty, CCC three hundred. 2d. If a letter of any value be placed after one of greater value, its value is to be added to that of the greater. Thus, XI represents eleven, LX sixty, DC six hundred. 3d. If a letter of any value be placed before one of greater |