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The Interrogations which follow the Rules, are intended as an exercise to direct the attention of the learner in a particular manner to the Rules, and to fix them more permanently in the memory. It will be observed, that the Examples are principally given in Federal money, or dollars and cents, as being more conformable to the currency of our country, and the general mode of keeping accounts throughout the United States.

Observations might be made as to arrangement, &c.; but as those interested in the subject can only judge of the merits of the work by an examination of it, they are respectfully referred to the work itself, and solicited to give it such a perusal as may enable them to decide with impartiality on its claims for admission into schools, in comparison with other works on the same subject

THOS. T. SMILEY. Philadelphia, January, 1828.

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EXPLANATION OF CHARACTERS.
Signs. Significations.

. Equal; as, 20s.=£1.
More; as, 6+2=8..
Less; as, 8—2=6.
Into, with, or multiplied by; as, 6x2=12.

By (i.e. divided by); as, 6-2=3; or, 2)6(3.

: : Proportion; as, 2 : 4 :: 6 : 12.

vor { Square Root; as, 64=8.

& Cube Root; as, 64s4.

Fourth Root; as, V6452, &c.
A Vinculum; denoting the several quantities over

which it is drawn to be considered jointly as a
simple quantity.

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ARITHMETIC.

ARITHMETIC is that part of the Mathematics which teaches the art of computation by numbers.

All operations in Arithmetic are performed by means of the following figures:

Cipher One Two Three Four Five Six Seven Eight Nine
0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

NUMERATION. Numeration teaches the proper disposition of figures to express any proposed number, when that number is too great to be expressed by a single figure.

When a number is expressed by more than one figure, the value of each figure is determined by the situation which it holds in relation to the others, as represented in the following table :

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By the foregoing table, it appears that any figure in the units' place, represents only. its simple value, or so many ones; but, by being placed in the tens' place, represents ten times as much as though it stood in the units' place; by being placed in the hundreds’ place, a hundred times as much as it would if placed in the units' place, and ten limes as much as it would if placed in the tens' place; and so on.

Though it is seldom necessary to make use of more than nine places, as in the table, yet it may be extended to a greater number, by making places for thousands of millions, tens of thousands of millions, hundreds of thousands of millions, &c.

To know the value expressed by any given number of figures.

Rule. 1. Read the figures from right to left: units, tens, hundreds, thousands, &c. as in the Numeration table.

2. To the value of each figure when it stands single, add the name of its place, and read the figures from the left to the right. Example: 321, three hundred and twenty-one.

Questions. What is Arithmetic? By what means are operations in Arithmetic performed? What does Numeration teach?

When numbers are expressed by more than one figure, how is the value of each figure determined?

Recite the Numeration table.

Is it usually necessary to make use of more than nine places to express numbers? When necessary, how is the number of places increased ?

Repeat the Rule to know the value expressed by any number of figures. To write down a proposed number.

Rule. Begin at the right hand, and proceed towards the left, writing units in the units' place, tens in the tens' place, hundreds in the hundreds' place, and so on.

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