EXPLANATION OF ARITHMETICAL SIGNS.

Signs.

Two parallel horizontal lines are the sign of equality. It shows that the number before, is equal to the number after it. Example, 1 dollar 100 cents, is read thus, 1 dollar is equal

to 100 cents.

Two short lines crossing each other at right angles, are the sign of Addition. It shows that numbers with this sign between them, are to be added together. Example, 5+7=12, is read thus, 5 added to 7, or 5 plus 7, is equal to 12.

A short horizontal line is a sign of subtraction. It shows that the number after it, is to be taken from the number before it. Example, 12-7=5, is read thus, 12 less 7, or 12 minus 7, is equal to 5.

X Two short lines crossing each other in the form of an X, are the sign of multiplication. It shows that the number before it, is to be multiplied by the number after it. Example, 6×5=30, is read thus, 6 multiplied by 5, is equal to 30. A short horizontal line between two points, is the sign of division. It shows that the number before it, is to be divided by the number after it Example, 30-6-5, is read thus, 30 divided by 6 is equal to 5.

::: Four points or colons are the sign of proportion; and to show that numbers are proportional, they are written thus, 2:49 16, which are read, 2 is to 4 as 8 is to 16.

This sign signifies the second power or square.

3 This sign signifies the third power or cube.

2 This sign prefixed to any number shows that the square root of the number is required.

3, This sign, prefixed to any number, shows that the cube root of the number is required.

CORRECTIONS AND REMARKS,

Though unwearied pains have been taken to have the typographical execution of this work correct, it is not expected that it is so, in all respects-and the publisher will thank those teachers, or other persons who discover errors, to point them out that they may be corrected in future editions. The following are the only errors we have discovered :

Page 142, Quest. 1.

144, Quest 1.

For the divisor 9, read 3.

Though correctly stated and wrought by one method, is not according to the rule there given.

Page 168. Table of Powers. Square cube of 3, for 728, read 729.

ARITHMETICK.

ARITHMETICK is the science of Numbers ;* and comprises the following principal rules, viz. I. NOTATION or NUMERATION; II. ADDITION; III. SUBTRACTION; IV. MULTIPLICATION; and V. DIVISION. The four last are called simple, when the numbers are all of one denomination; compound, when the numbers are of different denominations.

These five rules are called principal or fundamental, because the whole art of arithmetick is comprehended in their various operations.

I. NUMERATION.

1. NUMERATION teaches us to read or write any sum or number, by means of the following ten characters, called figures :†

Cipher. One.

Two. Three. Four. Five.

2, 3, 4, 5,

6, 7,

8, 9.

0, 1, 2. The value of a figure, when alone, is called its simple value, and is invariable. Figures have also a local value, which varies

* Number is either an unit, or a collection of units. A single thing is an unit, or one. One and one are two. One and two are three. And thus, by constant addition, all numbers are generated.

These figures, which are of Arabic or Indian origin, were introduced into Europe by the Moors, about the year 1150: they were formerly all called ciphers, whence it came to pass that the art of arithmetick was called ciphering.

1. What is Arithmetick?- -2. What are called its fundamental rules?- 3. What does Numeration teach? 4. How many figures are used to represent numbers ? 5. How do you determine their value?

B

according to the place in which they stand. In a combination of figures, reckoning from the right to the left, the figure at the right, or in the first place, represents its simple value; that in the second place, ten times the simple value; that in the third place, ten times the value of that in the second place; and so on, in a tenfold increase.

hous.

[und.

ens.

Units.

EXAMPLE.-Write down the sum 4 444. The first figure at the right, in the place of units, has its simple value, or the same as if standing alone-four. The second, in the place of tens, signifies four tens, or forty. The third figure, in the place of hundreds, signifies four hundred, or ten times its value in the place of tens. The fourth figure is in the place of thousands, bearing ten times its value in the place immediately preceding.

3. A cipher 0, though of no signification itself, when placed on the right hand of figures, in whole numbers, increases their value in the same tenfold proportion. Thus 9, signifies only nine, its simple value. Place a cipher on the right, (90) it becomes ninety; and by placing two ciphers at the right, thus (900) it becomes nine hundred.

Note.-Six places of figures, beginning on the right, are called a period; but they are commonly divided into half periods of three

6. In a combination of figures, how do you ascertain their value ?- -7. What is the nature of the cipher?8. What is a period in Numeration?

figures each. This division enables us to read any number of figures as easily as we can read the first period.

RULE.-Commit the words at the head of the Table, viz. units, tens, hundreds, &c. to memory; then, to the simple value of each figure, join the name of its place, beginning at the left hand and reading towards the right. More particularly-1. Place a dot under the right hand figure of the 2d, 4th, 6th, 8th, &c. half periods, and the figure over such dot will, universally, bave the name of thousands. 2. Place the figures 1, 2, 3, 4, &c. as indices, over the 2d, 3d, 4th, &c. period: These indices will then show the number of times the millions are involved-the figure under 1 bearing the name of millions, that under 2, the name of billions, (or millions of millions); that under 3, trillions, (or millions of millions of millions.)

913,208.000,341. 620,057. 219,356. 819,379. 120,406. 129,763

Thousands

Thousands

Thousands

Note.-Billions is substituted for millions of millions; Trillions, for millions of millions of millions; Quatrillions, for millions of millions of millions of millions. Quintillions, Sextillions, Septillions, Octillions, Nonillions, Decillions, Undecillions, Duodecillions, &c. answer to millions so often involved as their indices respectively denote.

The right hand figure of each half period has the place of units, of that half period; the middle one, that of tens, and the left hand one, that of hundreds.

APPLICATION.-Let the scholar now read, or write down in words at Jength, the following numbers :

Write down, in proper figures, the following numbers :

Fifteen

Two hundred and seventy-nine

9. How are numbers commonly divided?- -10. Of what use is this division ?What is the rule for Numeration?

Three thousand four hundred and three

Thirty-seven thousand five hundred and sixty-}

Four hundred one thousand and twenty-eight Nine millions seventy-two thousand and two hundred

Fifty-five millions three hundred nine thou sand and nine

Eight hundred millions forty-four thousand and fifty-five

Two thousand five hundred and forty-three millions four hundred and thirty-one thousand seven hundred and two

1

NOTATION BY ROMAN LETTERS.

ROMAN NOTATION is the method of representing numbers by Letters; and is now chiefly used to number the chapters of books, &c. Seven letters are used for this purpose, viz. I, V > X, L, C, D, and M.-I, signifies 1; V, 5; X, 10; L, 50; C, 100; D, 500; and M, 1000.

* Sometimes thousands are represented by drawing a line over the top of the numeral letter: thus, Vrepresents five thousand, Lfty thousand, CC two hundred thousand.

12 What is Roman Notation ?. -13. What is its use ?-14. How many letters are used for this purpose ?—15. What number does each represent?