into the house, and got twelve more, and gave each of the three boys two apiece. How many had he then left for himself?

11. A boy had seventeen nuts; another gave him three; another seven; another five; and another gave him enough to make his number forty. How many did this last boy give him?

12. Six men bought a horse for seventy dollars. The first gave twenty-three dollars; the second, fifteen; the third, twelve; the fourth, nine; the fifth, seven. How much did the sixth give? and how much did the first give more than he?

13. A man bought a horse for eighty dollars, and paid fifteen dollars for keeping him. He let the horse enough to receive twenty dollars, and then sold him for eighty-three dollars. Did he gain or lose by the bargain, and how much?

14. A man bought a horse for a hundred and twenty dollars; a wagon for fifty dollars; a harness for the same, for twenty-five dollars. He afterwards sold the whole for two hundred dollars. Did he lose or gain, and how much?

SECTION XIV. - Increase and Decrease by Equal Numbers; or, Multiplication and Division.

[IN the four lessons that follow, show the aggregate numbers on the Frame, and let the pupils divide them by the eye alone. But, where this does not suffice, the teacher, or one of the class, may occasionally separate them by the fingers. The smaller numbers can be readily separated by the eye, and this should be the chief resort, especially in reviewing.]

1. How many twos in four? How many are twice two, then? How many twos in six? Three twos, then? Two threes? How many twos in eight? Four twos, then? Two fours? How many twos in ten? Five twos, then? Two fives? How many twos in twelve? Six twos, then? Two sixes? How many twos in fourteen? Seven twos, then? Two sevens? How many twos in sixteen? Eight twos, then? Two eights? How many twos in eighteen? Nine twos, then? Two nines? How many twos in twenty? Ten twos, then? Two tens? How many twos in twenty-two? Eleven twos, then? Two elevens? How many twos in twenty-four? Twelve twos, then? Two twelves?

2. How many threes in six? Two threes, then? Three

Threes in

Threes in

Threes in Threes in

twos? How many threes in nine? Three threes, then? How many threes in twelve? Four threes, then? Three fours? Threes in fifteen? Five threes, then? Three fives? Threes in eighteen? Six threes, then? Three sixes? twenty-one? Seven threes, then? twenty-four? Eight threes, then? twenty-seven? Nine threes, then? thirty? Ten threes, then? Three three? Eleven threes, then? Three elevens? Threes in thirty-six? Twelve threes, then? Three twelves?

Three sevens?
Three eights?
Three nines?

tens? Threes in thirty

3. How many fours in eight? Two fours, then? Four twos? Fours in twelve? Three fours, then? Four threes? Fours in sixteen? Four fours, then? Fours in twenty? Five fours, then? Four fives? Fours in twenty-four? Six fours, then? Four sixes? Fours in twenty-eight? Seven fours, then? Four sevens? Fours in thirty-two? Eight fours, then? Four eights? Fours in thirty-six? Nine fours, then? Four nines? Fours in forty? Ten fours, then? Four tens? Fours in forty-four? Eleven fours, then? Four elevens? Fours in forty-eight? Twelve fours, then? Four twelves?

4. How many fives in ten? Two fives, then? Five twos? Fives in fifteen? Three fives, then? Five threes? Fives in twenty? Four fives, then? Five fours? Fives in twentyfive? Five fives, then? Fives in thirty? Six fives, then? Five sixes? Fives in thirty-five? Seven fives, then? Five sevens? Fives in forty? Eight fives, then? Five eights? Fives in forty-five? Nine fives, then? Five nines? Fives in fifty? Ten fives, then? Five tens? Fives in fifty-five? Eleven fives, then? Five elevens? Fives in sixty? Twelve fives, then? Five twelves?

5. Two sixes are how many? Three sixes; another six? Four sixes; another six? Five sixes; another six? Six sixes; another six? Seven sixes; another six? Eight sixes; another six? Nine sixes; another six? Ten sixes; another six? Eleven sixes; another six? Twelve sixes; another six? 6. Two sevens, how many? Three sevens; another seven? Four sevens; another seven? Five sevens; another seven? Six sevens; another seven? Eight sevens; another seven? Nine sevens; another seven? Ten sevens; another seven ? Eleven sevens; another seven? Twelve sevens; another seven ? 7. Two eights, how many? Three eights; another eight? Four eights; another eight? Five eights; another eight?

Six eights; another eight?
Eight eights; another eight?
Ten eights; another eight?
Twelve eights; another eight?

Seven eights; another eight?
Nine eights; another eight?
Eleven eights; another eight?

8. Two nines, how many? Three nines; another nine? Four nines; another nine? Five nines; another nine? Six nines; another nine? Seven nines; another nine? Eight nines; another nine? Nine nines; another nine? Ten nines; another nine? Eleven nines; another nine? Twelve nines; another nine? 9. Two tens?

Three tens? Four tens? Five tens? Six
Eight tens? Nine tens? Eleven tens?

tens? Seven tens? Twelve tens?

10. Two elevens? elevens? Six elevens? elevens ? Ten elevens?

Three elevens? Four elevens? Five Seven elevens? Eight elevens? Nine Eleven elevens? Twelve elevens ? 11. Two twelves? Three twelves; another twelve? Four twelves; another twelve? Five twelves; another twelve? Six twelves; another twelve? Seven twelves; another twelve? Eight twelves; another twelve? Nine twelves; another twelve? Ten twelves; another twelve? Eleven twelves; another twelve? Twelve twelves; another twelve?

[This section will require more frequent repetition than the others. Omit, in reviewing, the words " another six," "another seven," &c.]

SECTION XV.- Explanatory.

NUMBERS are not always expressed in words. What are called figures, are frequently used for that purpose. These figures are only nine in number, as may be seen below. They should be well studied, so as to be readily known, wherever they may appear.

1, stands for one.

6, stands for six.

5, 66

"five.

These are all the figures that stand for numbers. But how, then, do we manage, when we wish to use a number larger than nine? The same figures are used, only they are put in a dif

ferent place. Every figure becomes ten-fold greater by being removed one place to the left. Thus, the figure 1 stands for one, when alone, or at the right hand of other figures; for ten, when placed the second from the right; and for a hundred, or ten times ten, when it stands the third from the right. Thus, the three figures below,

111

stand for one hundred and eleven (or one-teen): the first figure on the left standing for one hundred, the second for one ten, the third for a single one. It is the same with all the other figures. Thus,

444

stands for four hundred and forty- (or four tens) four; and 666

stands for six hundred and sixty-six.

These places for the figures are called ranks, or orders, and are reckoned from the right. Every figure placed in the first order, stands for as many ones, or units, as it represents; when placed in the second order, for as many tens, or teen; and for as many hundreds when it stands in the third order. A figure placed in the next order to the left (the fourth order) would stand for so many thousands, each of which is equal to ten hundred. Thus, in the following number,

4536

the 4 stands for so many thousands, the 5 for hundreds, the 3 for ty, or tens, the 6 for units, or ones. The whole number should be read thus: four thousand, five hundred, and thirtysix.

This is very much like the arrangement of the Frame. [Exemplify on the frame.] A single bead on any of the upper ten wires stands for one. Each row of beads stands for ten, any one of which is called teen, if units be added to it. Each bead on the lower row stands for 100, and the whole row, of course, for ten hundred, which is a thousand. Each of these numbers increases tenfold, just as the figures do from the place in which they stand.

[Let the following figures now be written vertically on the slate or blackboard, and named repeatedly by the class till they are familiar. 7 9 6 3 8 5 1 4 2.]

But it is frequently necessary to write a number in which one or more of the orders is wanting: for example, two thousand and fifty-four. Here we must have four places, or orders,

to represent thousand, and yet we have only three figures, viz., 2 for two thousand, 5 for fifty, and 4 for four. In all such cases, we use this character, 0, which is called cipher or nothing, because it stands for nothing. Our number, two thousand and fifty-four, becomes 2054. There are no hundreds, you perceive, and the O fills that place. Had it not been put there, the 2 would have stood in the third order, and thus represented 2 hundred instead of two thousand. The cipher, accordingly, is sometimes called figure of place, because it is only used to show the place of the other figures.

Take notice, however, that a cipher is useless unless it occupies the place of units, or stands between a significant figure and the place of units. Thus, if we wish to write three hundred and seventy-four, the cipher is not wanted, although there are only three figures, because each figure can stand in its proper order, 374, without any cipher. But a cipher must be used in expressing two hundred and five, since we have only two figures, while the hundred is in the third order. Accordingly the number is written 205. For a similar reason, the number three thousand and forty-five must be written with a cipher, 3045.

Write the following numbers in figures on the slate or blackboard, and then read them over without the book:

1. Four hundred and thirty-five.

2. Two thousand, six hundred, and four.

3. Three thousand, and forty-two.

4. Six thousand, three hundred, and seventy-six.

5. Four thousand, four hundred, and forty-four.

6. Two hundred and three.

7. One thousand and twelve.

Sometimes one thousand is considered as ten hundred, as in the following:

8. Fifteen hundred and sixty.

9. Eighteen hundred and two.

[Specimen of questions to the class on the above numbers, when they have changed them from words to figures on the blackboard or slate.]

For No. 1. What does the 4 stand for? [Point to the figures as they are spoken.] Why hundreds? The 5? Why units? The 3? Why ty, or tens?

No. 2. What is the value of the 2? The 6? The 4?