7 If 1 lemon be worth 3 apples, how many lemons are 6 apples worth? Are 12 apples worth? Are 18 apples worth? Are 24 apples worth? Are 36 apples worth?

8. How many barrels of flour, at 8 dollars a barrel, can you buy for 16 dollars? For 48 dollars? For 96 dollars? For 80 dollars?

3, and 5? Are 4, 2,
Are 10, 8, and 2?
Are 7, 6, 3, and 2?

and 6? Are 8, 3, Are 5, 4, 3, and Are 8,9, and 10?

9. How many are 2, and 2? Are 9, 3, and 4? 2? Are 4, 3, 2, and 1? Are 12, 11, 10, and 9? 10. How many are times 3? 6 times 4? times 8? 9 times 7? 12 times 7? 9 times 5? 7 times 6? 7 times 9? 12 times 11? 8 times 5? 12 times 12?

6

6 times 7? 7

8 times 7?

3 times 7?

11. How many times 2 in 12? 2 in 18? 2 in 24? 3 in 6? 3 in 12? 3 in 36? 4 in 20? 4 in 32? 4 in 48? 5 in 25? 5 in 35? 5 in 60? 6 in 36? 6 in 48? 6 in 72? 7 in 14? 7 in 56? 7 in 84? 22? 11 in 55?

8 in 40? 8 in 96? 9 in 36? 9 in 108?
11 in 132? 12 in 144?

11 in

Note. Younger pupils should be required to review, and dwell on the pre ceding questions for illustration, and the tables, till their solutions be made perfectly familiar.

NUMERATION.

IV. Q. When I say to you, Give me that book, do I mean one book or more than one?

Q. When we speak of a single thing, then, what is it called? A. A unit, or one.

Q. What are one unit and one more, or one and one, called?' Q. What are two units and one more, or two and one, called? Q. What are three units and one more, or three and one, called?

Q. What are four units and one more, or four and one, called r Q. What are five units and one more, or five and one, called? Q. What are six units and one more, or six and one, called? Q. What are seven units and one more, or seven and one, called?

Q. What are eight units and one more, or eight and one, called?

Q. What are nine units and one more, or nine and one, called?

Q. Now, to be obliged always to write these numbers out in words, would be very troublesome; to prevent this, how do we sometimes express the numbers one, two, &c. up to thousands, millions, &c. A. By letters.

Q What does the letter I stand for? A. One.
Q. What does the letter V stand for? A. Five.
What does the letter X stand for? A. Ten.
What does the letter L stand for? A. Fifty.

What does the letter C stand for? A. One hundred. Q. What does the letter D stand for? A. Five hundred. Q. What does the letter M stand for? A. One thousand. Q. You said that V stands for five; suppose you place the letter I before the V, thus, IV, what will both these letters stand for then? A. Only four.

Q. What, then, may be considered as a rule for determining the value of these letters? A. A letter standing for a smaller number, and before a larger, takes out its value from the larger Q. One X stands for ten; what do two XX's stand for? A. Twenty.

Q. What, then, is the value of a letter repeated? 4. It repeats the value as often as it is used.

Q. How many letters do we use for expressing numbers ? A. Seven. Name them. A. I, V, X, L, C, D, M.

Q. What is this method of expressing numbers by letters called? A The Roman method.

Q. Why called Roman ? A. Because the Romans invented and used it.

Repeat the

..IX. Two hundred,

.CC

Eleven,..

...X. Three hundred,
..XI. Four hundred,

..CCC

..CCCC

Twelve,

.XII. F've hundred,.

..D.

XIII. Six hundred,.
.XIV. Seven hundred,

..DC

.DCC.

Fifteen,.

Sixteen,.

Seventeen,

..XV. Eight hundred,
..XVI. Nine hundred,.
.XVII. One thousand,..

.DCCC.

..DCCCC.

Eighteen,.

.XVIII. Fifteen hundred,.

Nineteen,

Twenty,

.XIX. Sixteen hundred,.
.XX. Two thousand,

Eighteen hundred and twenty-eight.................MDCCCXXVIII.

.M.

.MD

.MDC

..MM

TVI. We have a shorter method still, which is in very general use, as will appear by observing what follows:

Q. What are these characters called? A. Figures.

Q. By what other name are they sometimes called? A. The

9 digits.

Q. What is this method of expressing numbers called? A. The Arabic method.

Q. Why so called? A. Because the Arabs are supposed to have invented it.*

Let me see you write down on the slate, in figures, the num bers one, two, three, four, five, six, seven, eight, nine.

Q. To express ten, as we have no one character that will do it, what two characters do we make use of to represent this number? A. The first character, 1, and 0, or cipher; thus, 10 Q. What place does the 0, or cipher, in this case take A. The units' place.

Q. What place does the figure 1 take? A. A new place. Q. What is this new place called? A. The tens' place.

Q. Write down in figures, on the slate, the number ten; now take away the 1, and what will be left? A. Nothing but 0, or cipher.

Q. What is the value of this 0, or cipher, thus standing alone? A. No value.

Q. Now place the 0 at the right of the figure 1, and what will t become? A. Ten, (10.)

Q. How many times is the figure 1 increased by the 0, or cipher? A. Ten times.

Q. What effect, then, has a cipher in all cases when placed at the right of figures? A. It increases the value ten times. Q. In what proportion is this increase said to be? A. Tenfold proportion.

Q. How was it obtained from the Arabs? A. The Moors communicated to the Spaniards, and John of Basingstoke, Archdeacon of Leicester, introduced it into England; hence its introduction into our own country.

Q. About what time was it introduced into England? A. About the middle of the eleventh century.

Q How extensively is it now used? A. All over the civilized orld

As you have probably learned by this time how to write dow ten in figures, by the help of a cipher, and learned also the value of this cipher, we will now proceed to higher numbers: and to begin: let me see you write down in figures, on the slate, the following numbers, viz.

One ten and one unit, or eleven,..
One ten and two units, or twelve,
One ten and three units, or thirteen,
One ten and four units, or fourteen,
One ten and five units, or fifteen,
One ten and six units, or sixteen,
One ten and seven units, or seventeen,
One ten and cight units, or eighteen,
One ten and nine units, or nineteen,

.11

.12

.13

.14

.15

.16

.17

..18

..19

..70

..80

..90

..100

Six tens,

Seven tens,.

Eight tens,.

Nine tens,

........or seventy,

.......or eighty,..

............or ninety,

Ten tens, ... ... ... ... ... ... ... ... ... ... ... ... ... or one hundred,.

Q. Here we see the value of the cipher again; for, by placing a cipher at the right of ten, it becomes one hundred, (100,) tha is, ten tens should we place another cipher still at the right of the 100, (thus, 1000,) what would it become? A. One thousand, (1000).

Q. From what you have now seen of the value of figures, what may 2 and 5 be made to stand for? A. 25 or 52.

Q. What is this different value called, which arises from the figures being placed or located differently? A. Their local

value.

Q. What would be the value of the five written alone? A. Simply 5.

Q. What is the value, then, of a figure standing alone! A The simple value.

Q. How many values do figures appear to have? A. Two. Q. What are they? 4. Simple and local

Q. Now, as it takes 10 units to make one ten, or one in the next left hand place, and 10 tens to make 100, how do figures appear to increase by being removed one place farther to the left?. In a tenfold proportion from right to left.

You must have acquired, by this time, some considerable knowledge of figures: let me examine you a little; and, in the first place, let me see you write down on the slate the figure 8

Q. What do you call it? A. 8 units.
Write at the left of the 8 the figure 2, (thus.)

Q. What do you call them both, and how are they read 4. 8 units and 2 tens read twenty-eight.

Write at the left of the 28 the figure 8, (thus, 828.)

Q. What do you call the three figures now, and how are they read? A. 8 units, 2 tens, and 8 hundreds, read eight hundred and twenty-eight.

Write at the left of 828 the figure 1, (thus, 1828.)

Q. What do you call the 4 figures now, and how read?

A. 8 units, 2 tens, 8 hundreds, and 1 thousands, read one thousand eight hundred and twenty-eight.

Q. We have now been combining, or placing figures together till we have obtained the number 1828, representing the number of years it is since Christ appeared on earth, to the present time. We might continue to put figures together in this way that would express higher numbers still, up to billions, &c. That you may be able to form some idea of the power of figures, let me tell you that there is not a billion of seconds in thirty thousand years; notwithstanding there are 60 seconds in every minute, 60 minutes in every hour, 24 hours in every day, and in a solar year, 365 days, 5 hours, 48 minutes, and about 48 seconds. Should we continue to go on as we began, in combining more figures still, it would be very inconvenient; to avoid this we have a rule by which we can read almost any number of figures, ever so large. What is this rule called. A. Numeration.

Q. What is the reading, or expressing a number by figures as now shown, called? A. Notation or Numeration.

RULE.

I. From the above illustrations, how does it appear that you must begin to numerate? A. Begin at the right hand.

II. At which hand would you begin to read? A. The left. III. What is the first figure at the right hand, or first place, called? . Units.

What is the second figure, or second place, called? A. Tens What is the third place called? A. Hundreds.

What is the fourth place called? A. Thousands.

IV. In reading, what value do you give those figures which were called units in numerating? A. Units.

V. What value do you give tens? A. Tзns.

VI. What value do you give hundreds, thousands, & A. Hundreds, thousands, &c.

1. Repeat the Numeration Table