the second? The first in the third? The second in the fifth? &c.

8. Point off and write in words the following figures: 72358, 700, 1245, 604267, 8956238, 284563002, 123456, 7924502, 3824507266.

9. Increase the first two of the above nine numbers tenfold; the second two a hundred fold; the third two a thousand fold; and the remaining three ten thousand fold. In other words, multiply them by 10, by 100, 1000, 10,000.

[The above exercises should be repeated and varied by the teacher till the subject becomes perfectly familiar to the class.] But the right-hand figure does not always represent units. Sometimes it becomes necessary to use or speak of a number less than one. Thus, with respect to the money of the United States, the dollar is considered the unit. But a sum less than a dollar frequently enters into a calculation, cents, for instance, which are hundredths of a dollar; or dimes, which are tenths. These parts of a unit of any kind are called fractions, a word signifying broken into parts. Or, let us suppose an apple to be cut into ten equal parts. One or more of these fractions or tenths of an apple may enter into a calculation. When this is the case, these tenths, as the smallest part of the number, would occupy the rank on the right. But some mark would then be necessary to show which rank was occupied by the units. The character used for this purpose is a reversed comma, called a separatrix,* placed to the right of the rank of units. When a separatrix is used, any number of figures or ciphers can be added to the right of a number without changing the value of the other figures. Thus, if we take the four 3s again, marked with letters as before [see p. 113], and put a separatrix after the 3 marked d, to show that it stands in the place of units, we can add as many figures as we choose on either hand without changing the value of any of the 3s, for this simple reason, that their distance from the rank of units is unchanged by that addition. For example:

* Some writers use a dot, others a comma, for a separatrix. Both are wrong. For, as both these characters are used with figures for other purposes, they thus give rise to much uncertainty and perplexity. For instance, if the dot be used as separatrix, there is no means of ascertain

In the first example, we have the four figures exactly the same as before, the units' rank occupying the right, no separatrix being necessary to designate it. In the second example, we have still the same figures occupying the same stations, with a separatrix at the right of the units. But this is unnecessary. For, as the units occupy the rank on the right, they are sufficiently designated by their local situation. The third example exhibits the same characters as the second, with the addition of three others, 465. Here the separatrix is essential. Without it, the 5 would occupy the place of units, and each of the 3s would be a thousand times greater than in the first and second examples. In the fourth example, three additional figures occur on the left, which of course do not change the value of the others, as has been already sufficiently shown above.

This explanation brings out the fourth principle of decimal arithmetic, as follows:

IV.- When there is a separatrix, the units' place is immediately on its left; when there is none, the right hand figure represents the units.

The fractions, of which two examples are given above, are called Decimal Fractions, or simply Decimals, meaning numbers broken into tenths, or tenths of tenths (hundredths), or tenths of tenths of tenths (thousandths), &c. The value of these fractions depends on the same principle as that of integers or whole numbers; that is, each figure is ten times greater than the same figure on its right, and is only one-tenth of the value of the same figure on its left. The manner of reading them may be learned from the following table:

ing whether 6.5 means six and five tenths, or six times five. If the comma be used, 65,231 may either signify sixty-five thousand two hundred and thirty-one, or sixty-five and two hundred and thirty-one thousandths. By the use of the inverted comma, all uncertainty disappears.

From this table it appears that the same names, with the addition of th, are used for the numbers tenfold, a hundred fold, a thousand fold, &c., less than units, as for those tenfold, a hundred fold, a thousand fold, &c., greater than units. Thus, the figure to the left of units is named tens; that to the right tenths. The second to the left hundreds; the second to the right hundredths, and so on. Observe, however, that the three figures to the right of the units (the fractions) may either be read four tenths, four hundredths, and four thousandths, or four hundred and forty-four thousandths; or four thousand four hundred and forty tens of thousandths; and so on in an infinite variety of expressions. Indeed, this remark may be applied to any number, whether integral or fractional. Take, for instance, the number 538. The usual expression for this is five hundred and thirty-eight. But it might be considered as five hundred and three tens and eight; or fifty-three tens and eight; or five thousand three hundred and eighty tenths; or fifty-three thousand eight hundred hundredths, &c., without end.

It is also plain from the last table that, by changing the place of the separatrix, the value of every figure is changed; being increased tenfold, a hundred fold, &c., by removing it one, two, &c., places to the right; and decreased tenfold, a hundred fold, &c., by removing it one, two, &c., places to the left. And this will evidently be the case whatever may be the figures employed. For instance, in the number 42'56, which reads forty-two and fifty-six hundredths, if the separatrix be removed one place to the right, we shall have 425'6, which reads four hundred and twenty-five and six tenths. If it be now removed two places to the left, we shall have 4.256, which reads four and two hundred and fifty-six thousandths. And lastly, by removing the separatrix altogether, the number becomes 4256, four thousand two hundred and fifty-six. Thus, it appears that the principles regulating the notation and numeration of decimal fractions are precisely the same as in whole numbers, as exemplified in first and second principles, which see (pp. 114 and 115).

Observe, however, that should there be so many decimal places as to require division into periods for the sake of easy reading, the period adjoining the units should only consist of two figures, the rank of units of course being wanting in fractions, as 436,427 38,945. Observe, also, in forming any num

ber, whole or fractional, into periods, always to begin with units, whether proceeding to the right or to the left, or both ways. Division of fractions into periods, however, will rarely be necessary.

Exercises for the Black-board or Slate.

1. Point off and write in words the following numbers: 54326 48; 8043 805; 2769 0072; 5214'3724.

2. Point off and write in words the same figures, increasing the first tenfold, the second a hundred fold, the third and fourth a thousand fold, by changing the place of the separatrix ; in other words, multiply them by 10, 100 and 1000.

3. Use the figures in the first exercise once more, decreasing them tenfold, &c.; that is, dividing them by 10, 100, and 1000, by changing the place of the separatrix.

4. Increase '092 tenfold; that is, multiply it by 10, and then decrease it 100 fold; that is, divide it by 100.

5. Mention separately the effect that would be produced on each of the following numbers by a removal of the separatrix : 25'07; 38 206; '525; 92'3.

6. Mention separately the effect that would be produced on each of the following numbers by placing a separatrix after the first figure on the left: 2346; 18; 398; 27945.

7. Mention what numbers are superfluous in the four numbers that follow, and why: 600; 006; ‘006; 600.

8. Name the value of the 4 and of the 6 in the following number: 4060. If the 4 were removed, would the value of the 6 be changed? Why? If the 6 were removed, would the value of the 4 be changed? Why? If the cipher between the two significant figures were removed, what change, if any, would it effect upon the 4? Why? Upon the 6? Why? If the cipher occupying the place of units were removed, what effect would be produced upon the number? Why?

9. Fifty-two millions six thousand and twenty. How many figures are necessary to represent this number? [The number is not to be written in figures?] How many of them are significant figures?

10. Express in words 637 in four different ways. Ans. Six hundred and thirty-seven; six hundred thirty and seven; sixty-three tens and seven; six thousand three hundred and seventy tenths.

11. Express in words the following numbers, each in six different ways: 5326; 9478; 124,679; 38,472; 5'4; 7360; 9024; 573 7.

The fifth principle of decimal arithmetic will now be understood without further explanation, namely,

V. The cipher is superfluous, except where it intervenes between a significant figure and the place of units.

Exercises, continued.

12. Increase 245 tenfold, or, which is the same thing, multiply it by 10.

13. Decrease 2540 tenfold, or, which is the same thing, divide it by 10.

14. Multiply 3'532 by 100.

15. Divide 453'2 by 100.

16. Multiply 1768 by 10, and then divide it by 100, removing superfluous figures, if any.

17. Multiply 17'68 by 1000.

18. Divide 17680 by 100, removing superfluities, if any. 19. Write down in figures, in separate lines, the following numbers, first determining in your own mind how many figures are necessary for each number, and how many of them are significant :

Thirty-six.

Two hundred and seven.

Two hundred and seventy.

Two hundred and seven tenths.

Two hundred and seven hundredths.

Twenty-five thousand nine hundred and twenty-six. Twenty-five thousand and nine hundred and twenty-six. thousandths.

Three thousand and four and eighteen hundredths.

Four million six thousand and thirty-seven.

Three hundredths.

- Fifty-nine thousandths.

Nine millions and nine thousandths.

Thirty-four billions thirty-seven thousand and fifty.

Eleven thousand eleven hundred and eleven (a puzzle.)

Four millions four hundred thousand and forty and four hundredths.