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TO THE

SECOND (STEREOTYPE) EDITION.

The first edition of this work having been favorably received by the public, the author is now induced carefully to revise it, and make a few additions. It is believed, that, in the present edition, all the answers to the questions will be found correct.

Great pains have been taken to make the rules and demonstrations intelligible.

In revising his work, the Author has availed himself of the aid and suggestions of many practical teachers ; among whom he would particularly acknowledge his obligations to two distinguished teachers in Newburyport, David P. Page, Esq., of the English High School, and Mr. Joseph Williams, of the Grammar School.

BENJAMIN GREENLEAF.

BRADFORD SEMINARY,

July 1st, 1843.

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$ Contraction, for U. S., United States' currency, and

is prefixed to dollars and cents. Sign of equality ; as 12 inches = 1 foot, signifies,

that 12 inches are equal to one foot. + Sign of addition ; as 8+6=14, signifies, that 8 add

ed to 6 is equal to 14. Sign of subtraction ; 8-6=2, that is, 8 less 6 is

equal to 2. X Sign of multiplication ; as 7X6= 42, that is, 7 multi

plied by 6 is equal to 42. • Sign of division ; as 42-6=7, that is, 42 divided by

6 is equal to 7. za Numbers placed in this manner imply, that the upper

line is to be divided by the lower line. ::: : Signs of proportion ; thus, 2:4:: 6 : 12, that is,

2 has the same ratio to 4, that 6 has to 12 ; and

such numbers are called proportionals.
15 -5+3=13. Numbers placed in this manner show,

that 5 is to be taken from 15, and 3 added to the
remainder. The line at the top is called a vincu-
lum, and connects all the numbers, over which it

is drawn.
9 Implies, that 9 is to be raised to the second power ;

that is, multiplied by itself. 8 Implies, that 8 is to be multiplied into its square,

2

3

ARITHMETIC.

Section 1.

ARITHMETIC is the art of computing by numbers. Its five principal rules are Numeration, Addition, Subtraction, Multiplication, and Division.

NUMERATION.

Numeration teaches to express the value of numbers either by words or characters.

The numbers in Arithmetic are expressed by the following ten characters, or Arabic numeral figures, which the Moors introduced into Europe about nine hundred years ago ; viz. 1 one, 2 two, 3 three, 4 four, 5 five, 6 six, 7 seven, 8 eight, 9 nine, 0 cipher, or nothing.

The first nine are called significant figures, as distinguished from the cipher, which is, of itself, insignificant.

Besides this value of those figures, they have also another, which depends on the place in which they stand, when connected together; as in the following table.

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Here any figure in the first place, reckoning from right to left, denotes only its simple value ; but that in the second place, denotes ten times its simple value ; and that in the third place a hundred times its simple value ; and so on ; the value of any successive place being always ten times its former value.

Thus in the number 1831, the 4 in the first place denotes only four units, or simply 4; 3 in the second place signifies three tens, or thirty; 8 in the third place signifies eighty tens or eight hundred ; and the 1, in the fourth place, one thousand ; so that the whole number is read thus, - one thousand eight hundred and thirty-four.

As to the cipher, 0, though it signify nothing of itself, yet, being joined to the right hand of other figures, it increases their value in a tenfold proportion ; thus 5 signifies only five, but 50 denotes 5 tens or fifty ; 500 is five hundred ; and so on.

Note. – The idea of number is the latest and most difficult to form. Before the mind can arrive at such an abstract conception, it must be familiar with that process of classification, by which we successively remount from individuals to species, from species to genera, from genera to orders. The savage is lost in his attempts at enumeration, and significantly expresses his inability to proceed, by holding up his expanded fingers, or pointing to the hair of his head. See Lacroix.

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