to the value of rules in an arithmetical treatise. It is not an easy thing for the experienced teacher to express in the most concise and accurate language the method of solving a problem. Much less can such an expression be given by the untrained scholar. Now, as precision in thought is essentially aided by precision in language, it is deemed expedient to furnish the scholar with rules which shall state in the fewest and clearest words the results of previous logical inductions. Moreover, when an intricate reasoning process may have been forgotten and cannot readily be recalled, the brief form of words impressed upon the memory in one's youth may oftentimes enable him in after life to perform an important mathematical operation in which he must otherwise have failed. It will be observed, that, while the author has expressed in rules his modes of operating, he has in every case first given the analysis upon which each rule is based. The author flatters himself that the present edition of the National Arithmetic embraces many improvements on former editions. He has endeavored to present clearer definitions, more rigid analyses, and briefer and more accurate rules. While almost every topic included in earlier editions has been treated in a more elaborate and comprehensive manner, this volume comprises a large amount of new matter, which it is believed will be found useful in business. On comparing this with preceding editions, teachers will find extensive additions and improvements under the heads of Numeration, Addition, and the other fundamental rules, Properties of Numbers, Fractions, Ratio, Percentage, Notes and Banking, Roots, etc. Among the new material will be discovered methods of finding the greatest common divisor and the least common multiple of fractions, of reducing fractions to a common numerator, of contracting the operations in the multiplication and division of decimal fractions, of reducing continued fractions, of averaging accounts, of alligating, of extracting roots to any degree, and of reducing numbers from one system of notation to another. Especial attention is invited to the section on averaging accounts, – a subject rarely taught in schools, though of great importance in the counting-room, to the manner of treating the roots, and to the many new problems which will be found in all parts of the book. In closing these remarks, the author desires to tender his hearty thanks to many teachers who have favored him with valuable sug. gestions; and to acknowledge in an especial manner his indebtedness to Mr. H. B. Maglathlin, who has been constantly associated with him in making this revision, and to whose accurate scholarship and sound judgment the value of the work is largely due. . . . . . Page Page Contractions in Multiplication 56 PROPERTIES OF NUMBERS. 137 UNITED STATES MONEY. 70 Factoring Subtraction of United States Reduction of Common Frac- Multiplication of United States plications 78 Multiplication of Common . . . . . . . . . 177 EQUATION OF PAYMENTS . 320 Notation and Numeration of INSURANCE 198 Fire and Marine Insurance 335 Subtraction of Decimals · 202 CUSTOM-HOUSE BUSINESS . 341 Multiplication of Decimals 203 Transformation of Repetends 222 Arbitration of Exchanges. Decimals 225 Extraction of the Square Root 373 Extraction of the Cube Root 378 232 PROGRESSION, OR SERIES . 395 Arithmetical Progression . 395 237 Compound Proportion 244 PERMUTATIONS AND COM- Discount and Present Worth 286 Multiplication of Duodecimals 421 Stocks 295 . . INTRODUCTION. HISTORY OF ARITHMETIC. It is difficult to determine who was the inventor of Arithmetic, or in what age or among what people it originated. In ordinary history, we find the origin of the science attributed by some to the Greeks, by some to the Chaldeans, by some to the Phænicians, by Josephus to Abraham, and by many to the Egyptians. The opinion, however, which modern investigations have rendered most probable, is, that Arithmetic, properly so called, is of Indian origin, that is, that the science received its first definite form, and became the germ of modern Arithmetic, in the regions of the East. It is evident, from the nature of the case, that some knowledge of numbers and of the art of calculation was necessary to men in the earliest periods of society, since without this they could not have performed the simplest business transactions, even such as are incidental to an almost savage state. The question, therefore, as to the invention of Arithmetic, deserves to be considered only as it respects the origin of the science as we now have it, and which, as all scholars admit, has reached a surprising degree of perfection. In this sense the honor of the invention must be awarded to the Hindoos. The history of the various methods of Notation, or the different means by which numbers have been expressed by signs or characters, is one of much interest to the advanced and curious scholar; but the brevity of this sketch allows us barely to touch upon it here. Among the ancient nations which possessed the art of writing, it was a natural and common device to employ letters to denote what we express by our numeral figures. Accordingly we find, that, with the Hebrews and Greeks, the first letter of their respective alphabets was used for 1, the second for 2, and so on to the number 10,— the latter, however, inserting one new character to denote the number 6, and evidently in order that their notation might coincide with that of the Hebrews, the sixth letter of the Hebrew alphabet having no corresponding one in the Greek. The Romans, as is well known, employed the letters of their alphabet as numerals. Thus I denotes 1; V, 5; X, 10; L, 50; C, 100; D, 500; and M, 1,000. The intermediate numbers were expressed by a repetition of these letters in various combinations ; as, II for 2; VI for 6; XV for 15; IV for 4; IX for 9, &c. They frequently expressed any number of thousands by the letter or letters denoting so many units, with a line drawn above; thus, V, 5,000; VI, 6,000; X, 10,000; L, 50,000 ; T, 100,000; M, 1,000,000. In the classification of numbers, as well as in the manner of expressing them, there has been a great diversity of practice. While we adopt the decimal scale and reckon by tens, the aborigines of Mexico, according to Humboldt, and some of the early nations of Europe, adopted the vicenary, reckoning by twenties; some of the Indian tribes, and several of the African tribes, use the quinary, reckoning by fives; and the Chinese for more than 4,000 years have used the binary, reckoning by twos. The adoption of one or another of these scales has been so general, that they have been regarded as natural, and accounted for by referring them to a common and natural cause. The reason for assuming the binary scale probably lay in the use of the two hands, which were employed as counters in computing; that for employing the quinary, in a similar use of the five fingers on either hand; while the decimal and vicenary scales had respect, the former to the ten fingers on the two hands, and the latter to the ten fingers combined with the ten toes on the naked feet, which were as familiar to the sight of a rude, uncivilized people as their fingers. — It is an interesting circumstance, that in the common name of our numeral figures, digits (digiti) or fingers, we preserve a memento of the reason why ten characters and our present decimal scale of numeration were originally adopted to express all numbers, even of the highest order. It is now almost universally admitted, that our present numeral characters, and the method of estimating their value in a tenfold ratio from right to left, have decided advantages over all other systems, both of notation and numeration, that have ever been adopted. There have been those, as Leibnitz and De Lagni, who have advocated the binary scale; a few, with Claudius Ptolemy, have claimed advantages for the sexagenary scale, or that by sixty; and there are those who think that a duodecimal scale, and the use of twelve numeral figures instead of ten, would afford increased facility for rapid and extensive calculations; but most mathematicians are satisfied with the present number of numerals and the scale of numeration, which have attained an adoption all but universal. It was long supposed, that for our modern Arithmetic the world is indebted to the Arabians. But this, as we have seen, is not the |