his book, before he mentions Decimal Fractions. But on the sixtieth page he gives the following question to be answered by the scholar: "A man is indebted to A, $237,62; to B, $350; to C, $86,124; to D, $9,621; and to E, $0,834; what is the amount of his debts?" In this sum, which is seventy pages before Decimal Fractions, and forty pages before Vulgar Fractions, he employs both Vulgar and Decimal Fractions. Nor is this the only question of the kind. Throughout Federal Money, similar ones are proposed, to be answered by the scholar. On one page of Federal Money, the following sum is given:

Directly under the above sum is the following example:

"Mr.

bought of

3 hogsheads new Rum, 118 gal. each, at $0,31 a gal..

2 pipes French Brandy, 126 and 132 gal. $1,12

1 hogshead brown Sugar, 94 cwt... $10,34 cwt..

3 casks of Rice, 269 lb. each,.

5 bags Coffee, 75 lb. each,.

1 chest Hyson Tea, 86 lb...

$0,05 lb.

$0,23
$0,92

66

$706,52"

It is plain, that Decimal Fractions are here employed. Why, then, is not the rule given to regulate their use?

In the present work we have endeavored to be more consistent in our arrangement, than the generality of authors have been. The scholar is here taught to work Decimals, at the same time he ciphers in whole numbers.

HISTORY OF ARITHMETIC.

THE real inventors of Arithmetic are not known. The Egyptians regarded the science of numbers as of divine origin; and, with a superstitious credulity, ascribed human acquaintance with it to the revelation of it by the gods. Arithmetical calculations were early employed in the East, especially in astronomical problems. At the time Alexander entered Babylon, the astronomers of that city claimed that 430000 years had elapsed since the commencement of their astronomical observations.

Ancient writers differ in their accounts of the origin of Arithmetic. While Strabo informs us, that in his time, the Phenicians were considered the inventors of Arithmetic, Josephus maintains that Abraham was the inventor of Arithmetic, and communicated a knowledge of it to the Egyptians. It is certain that the ancient Jews were acquainted with numbers, from the fact, that in Gen. xxiv. 60, the large number of "thousands of millions" is employed. In Leviticus xxvi. 8, and Deut. xxxii. 30, the number "ten thousand" is used. It is affirmed, also, that the Greeks borrowed their alphabet and method of notation from the Hebrews. Thales, a Greek, travelled among the Indians and Egyptians to extend his acquaintance with the sciences, and while in Egypt measured the heights of the pyramids by means of their shadows. This calculation is considered the first in which the principles of arithmetic were applied to geometry.

It has been supposed that the Hebrews, Greeks, and Romans first made use of pebbles, or small stones, in performing their calculations. To this hypothesis, the Greek word psephizein, (Latin, calculare,) to calculate, derived from psēphos, (Latin, calculus,) a small stone, gives the appearance of plausibility. The Chinese and Russians still employ stringed beads in performing arithmetical operations. Pythagoras, a disciple of Thales, gave much time to Arithmetic, and to him the invention of the Multiplication Table is ascribed. The early Greeks were probably acquainted with the fundamental rules of Arithmetic, the method of extracting the Square and Cube Roots, and the theory of Arithmetical and Geometrical Progressions, although their methods of computation were probably tedious and difficult, and unlike those of the present day.

"The Sieve of Eratosthenes deserves to be noticed in a history of Arithmetic, both as an object of curiosity, and as an invention of some importance in the theory of fractions. This Sieve, as it is whimsically called, consists in an easy, though somewhat tedious, method of finding the prime numbers, or those numbers which are divisible only by themselves and unity. For this purpose, Eratosthenes wrote in succession all the odd numbers, (the number 2 being the only even prime number,) as far as he wished his table extended. He then began at unity, and excluded all the multiples of 3; next, those of 5; next, those of 7, &c., cutting them out as he advanced. When he had exhausted the whole of the multiples of the odd numbers by these successive intercisions, the remaining numbers were all prime; and the table itself, on account of the holes which had been made in it, received the name of Sieve, from its resemblance to that utensil." The sexagesimal arithmetic, which is now employed

* Ed. Ency. art. Arithmetic.

among some Eastern nations, was introduced about the second century of the Christian era, and was considered as the invention of Ptolemy. In this system, every unit was divided into sixty parts, and each of these parts was divided into sixty other parts. Sixty was sexagesima prima, and was represented thus, I'; twice 60 was represented thus, II', and thrice 60 thus, III'. This method was pursued till the series closed with 59 times 60; when the second series was commenced with 60 times 60, marked thus, I". A dash at the bottom, thus, I,, or at the top, thus, 'I, denoted.

It remained to a fierce and uncivilized people, in the eighth century, to develop a system of notation, which supplanted the sexagesimal arithmetic, and has been perpetuated to our time, as the most perfect system ever invented. The Arabs expressed numbers by proper signs, and made the value of a figure depend upon its location, but with reference also to its primitive value. To the Arabians we are indebted for our present system of Arithmetic, although it is affirmed that they regarded it to be of Indian origin. The first treatise on decimals was written in 1582; and since that time, decimal fractions have been considered essential to every system of Arithmetic.

PREFACE TO THE SECOND EDITION.

THE utility of this work having become more apparent to the public, and more generally acknowledged; the Publisher having received favorable recommendations of its superiority over old-system arithmetics, has been induced thus early to print a second edition. In this edition, the errors which were found in the first, have been corrected. Some improvements have been made in the first part of the work. Diagrams, illustrating the principles laid down in addition and the rule of parts, have been introduced. These diagrams present the quantities represented by figures, and thus lead the student to fix his mind on the substances treated of, rather than the figures standing for them. Although some alterations have been made in this edition, yet the two can be used in the same class without inconvenience.

It is hoped that the work has now reached such a degree of perfection as will commend itself to teachers, scholars, and the friends of education generally. ZEBULON JONES, Editor.

Peterboro', Sept. 16, 1842.