left of the first, and this first figure still expresses only one; its value is still unaffected by the approach of the other figure. But let him place this second figure at the right of the first, and then the value of the first is changed; standing before another figure, it is advanced, what may be called, one step, by which advance its value is increased tenfold; so that, instead of one, it now stands for ten. Let the scholar advance this figure another step, by placing a third figure after it, (let it be the 3,) and then the 1 stands for one hundred; being ten times as much as it represented before the addition of the last figure. Another figure joined to the line, in the same manner, increasing all the figures which stand before it tenfold, makes the first figure stand for one thousand. Such is the effect of a change in the station of a figure. This process may be performed on a slate by placing the numbers as stated aboveerasing one, and writing another, either at the left or right hand, instead of making use of cards.

Here are four figures placed in a line thus:

1 2 3 4

The first stands for one thousand, the second for two hundred, and the third for thirty, to which amounts they have been raised, from their simple value of one, two, and three, merely by the circumstance of there being other figures placed after them; but the fourth figure, having none after it, and standing in the first and lowest station, represents merely its simple number, four; the whole line representing one thousand, two hundred, and thirty-four. Such is the manner in which large sums are represented by a few simple figures.

Thus we see, that it is the rank, or station, in which the figure is placed, that determines its value; and the next thing the scholar has to do, is, to fix in his mind the exact value, and also the proper name, of each of these ranks or stations; or, as they are usually called, of these PLACES of figures, as a clear

understanding of them will greatly assist him in all which is to follow.

It has been already stated that there are four figures, namely, 1, 2, 3, 4, severally occupying the first, second, third, and fourth stations, or places, as I shall henceforth call them. The first place at the right hand, (for it is there we begin to count,) is called the place of UNITS, that is of ones; the second place is called the place of TENS; the third place is called the place of HUNDREDS; and the fourth place is called the place of THOUSANDS; representing, as every figure written in this place does, just so many thousands as it would represent units, if written in the first place. Thus we have ascertained the first four places of figures, and have, thereby, learned to write UNITS, TENS, HUNDREDS, and THOUSands.

In the same manner we proceed to write larger numbers, every place of figures being ten times the value of that next before it. The fourth being the place of THOUSANDS, the fifth is the place of TENS of THOUSANDS, the sixth is the place of HUNDREDS of THOUSANDS, the seventh is the place of MILLIONS, the eighth is the place of TENS of MILLIONS, and the ninth is the place of HUNDREDS of MILLIONS, and

so on.

I shall take notice of one other particular before I conclude this branch of the subject; and that is, the use of the cipher, or naught.

The use of this figure, for figure it is, although it does not represent any value, is to fill up the place, or places that would oftentimes, but for some such thing, be left vacant by the advancement of any other figure, or line of figures. Thus, for instance, we have occasion to write down the number ten. This number is represented by the figure 1 advanced

one step. This advance, however, can not be made otherwise than by placing some other figure after it. If we write after this 1, a figure which represents a number, we not merely advance the one into ten, but add unto it the number, or value of the additional figure. Suppose this figure to be 2, and that we write them thus, 12; we shall then have the numbers ten and two, or twelve; and not ten, the number which we wish to express. But, if we add the cipher to the 1, writing them thus, 10, then we shall have the desired number. Twenty is thus expressed, 2 and a cipher, 20; thirty, 3 and a cipher, 30: forty, thus, 40; fifty, thus, 50; sixty, thus, 60; and so on. By adding another cipher these numbers are raised to hundreds; as,

100, one hundred.

200, two hundred.

300, three hundred.

400, four hundred, and so on.

By adding a third cipher these numbers are raised to thousands; as,

1000, one thousand.

2000, two thousand.

3000, three thousand, and so on.

Q. What is the first place of figures on the

right hand called?

A. It is the place of UNITS.

Q. What is the second place called?

A. The place of TENS.

Q. What is the third place called?

A. The place of HUNDREDS.
Q. What is the fourth place?

A. The place of THOUSANDS.

Q. What is the fifth?

A. The place of TENS of THOUSANDS.

Q. What is the sixth?

A. The place of HUNDREDS of THOUSANDS.

Q. What is the seventh?

A. The place of MILLIONS.

Q. What is the eighth?

A. The place of TENS of MILLIONS.
Q. What is the ninth?

A. The place of HUNDREDS of MILLIONS.
Q. What is the tenth?

A. The place of THOUSANDS of MILLIONS.

Note.-To TEACHERS. To be satisfied that the learner thoroughly understands the NoTATION of figures, which he will find of the utmost value in his future study, let him write down, in figures, the following sums.

EXERCISES.

Three hundred and eighty-six.

Nine thousand, three hundred and seventy-three. Forty-three thousand, four hundred and eighteen. One hundred and thirty-four thousand, six hundred and thirty.

Three hundred and sixty-eight thousand, five hundred and four.

One million, three hundred and eighty thousand, four hundred and twenty-six.

Seven millions, three hundred and forty-seven thousand, two hundred and twenty-two.

Nineteen millions, one hundred and seventy-four thousar 1, eight hundred and seven.

Forty-six millions, five hundred and thirty thousand, six hundred and fourteen.

One hundred and eighty-four millions, nine hundred and fifty-three thousand, six hundred and two. Four hundred thirteen millions, eight hundred and sixty thousand, two hundred and seventy-five.

Two hundred and six millions, nine hundred and thirty-five thousand, four hundred and eighty-one. Seven hundred and forty-two millions, ninetythree thousand, eight hundred and sixty-five.

One thousand two hundred and fifty-seven millions, eight hundred and forty-three thousand, three hundred and seventy-nine.

Six hundred and fifty millions, four thousand five hundred and fifty.

Six thousand, five hundred and thirteen millions, two hundred and eighty thousand, three hundred and fourteen.

Q. What is NUMERATION?

A. Numeration is the reciting or reading, in words, any number or series of figures; and a proper attention to what has been said in treating of NOTATION, will render this next step, and, indeed, every future step of the study of Arithmetick, comparatively easy.

EXPLANATIONS.

To proceed to an example, let the scholar numerate the following:

3457

The scholar will remember, that the value of every →re depends almost entirely on the place in which