Note.-Questions on the foregoing table should be continued till the class becomes familiar with the mode of expressing any number from 1 to 1000. They may be answered orally; but the best way is to let the pupil write the figures denoting the number upon the blackboard, and at the same time pronounce the answer audibly.

OBS. 1. The terms thirteen, fourteen, fifteen, &c., are obviously derived from three and ten, four and ten, five and ten, which by contraction become thirteen, fourteen, fifteen, &c., and are therefore significant of the numbers which they denote. The terms eleven and twelve, are generally regarded as primitive words; at all events, there is no perceptible analogy between them and the numbers which they represent. Had the terms oneteen and twoteen been adopted in their stead, the names would then have been significant of the numbers one and ten, two and ten; and their etymology would have been similar to that of the succeeding terms.

2. The terms twenty, thirty, forty, &c., were formed from two tens, three tens, four tens, which were contracted into twenty, thirty, forty, &c.

3. The terms twenty-one, twenty-two, twenty-three, &c., are compounded of twenty and one, twenty and two, &c. All the other numbers as far as ninety-nine are formed in a similar manner.

4. The terms hundred and thousand are primitive words, and bear no analogy to the numbers which they denote.. The numbers between a hundred and a thousand are expressed by a repetition of the number below a hundred. Thus we say, one hundred and one, one hundred and two, one hundred and three, &c.

ues.

8. It will be perceived from the foregoing table, that the figures standing in different places have different valThus the digits, 1, 2, 3, &c., standing alone or in the right hand place, respectively denote units or ones. But when they stand in the second place, they express tens: thus the 1 in 10, 12, 15, &c., expresses ten, or ten ones; that is, its value is ten times as much as when it stands in the first or right hand place, and it is called a unit of the second order. So the other digits, 2, 3, 4, &c., standing

QUEST. Obs. From what is the term thirteen formed? Fourteen? Sixteen Eighteen? What is said of the terms eleven and twelve? How are the terms twenty, thirty, &c., formed? What is said of the terms hundred, and thousand? How are the numbers between a hundred and a thousand expressed? 8. Does the same figure always express the same value? What does each of the digits, 1, 2, 3, &c., denote, when standing in the right hand place? What does the figure 1 denote when it stands in the second place? What is its value then? What do the other figures denote when standing in the second place?

in the second place, denote two tens, three tens, four tens, &c.

When standing in the third place, they express hundreds: thus the I in 100, 102, 123, &c., denotes a hundred, or ten tens; that is, its value is ten times as much as when it stands in the second place, and it is called a unit of the third order. In like manner, 2, 3, 4, &c., standing in the third place, denote two hundred, three hundred, four hundred, &c.

When a digit occupies the fourth place, it expresses thousands: thus the 1 in 1000, 1845, &c., denotes a thousand, or ten hundreds; that is, its value is ten times as much as when it stands in the third place, and it is called a unit of the fourth order. Thus,

It will be seen that ten units make one ten, ten tens make one hundred, and ten hundreds make one thousand; that is, ten in an inferior order are equal to one in the next superior order. Hence, we may infer universally, that

9. Numbers increase from right to left in a tenfold ratio; that is, each removal of a figure one place towards the left, increases its value ten times.

10. The different values which the same figures have, are called simple and local values.

The simple value of a figure is the value which it expresses when it stands alone, or in the right hand place. The simple value of a figure, therefore, is the number which its name denotes. (Art. 6.)

The local value of a figure is the increased value which

QUEST.-What is a figure called when it occupies the third place? What is its value then? What is it called when in the fourth place? What is its value? What do the other figures denote when standing in the fourth place? How many units are required to make one ten How many tens make a hundred? How many hundreds make a thousand? Generally, how many of an inferior order are required to make one of the next superior order? 9. What is the general law by which numbers increase? What is the effect upon the value of a figure to remove it one place towards the left? 10. What are the different values of the same figure called? What is the simple value of a figure? What the local value? Upon what does the local value of a figure depend? Obs. Why is this system of notation called Arabic? What else is it sometimes called? Why?

it expresses by having other figures placed on its right. Hence, the local value of a figure depends on its locality, or the place which it occupies in relation to other numbers with which it is connected. (Art. 8.)

OBS. 1. This system of notation is called Arabic, because it is supposed to have been invented by the Arabs.

2. It is also called the decimal system, because numbers increase in a tenfold ratio. The term decimal is derived from the Latin word decem, which signifies ten.

11. The art of reading numbers when expressed by figures, is called NUMERATION.

The pupil has already become acquainted with the names of numbers, from one to a thousand. He will now easily learn to read and express the higher numbers in common use, from the following scheme, called the

12. The different orders of numbers are divided into periods of three figures each, beginning at the right hand. The first, which is occupied by units, tens and hundreds,

QUEST.-11. What is numeration? Repeat the Numeration Table, beginning at the right hand. What is the first place on the right called The second place? The third? Fourth? Fifth? Sixth? Seventh? Eighth Ninth? Tenth, &c.? 12. How are the orders of numbers divided? What is the first period called? By what is it occupied? What is the second called? By what occupied? What is the third called? By what occupied? What is the fourth called? By what occupied? What is the fifth called? By what occupied?

is called units' period; the second is occupied by thousands, tens of thousands and hundreds of thousands, and is called thousands' period, &c.

The figures in the table are read thus: Five hundred and sixty-eight quadrillions, three hundred and forty-two trillions, nine hundred and seventy-five billions, eight hundred and ninety-seven millions, six hundred and forty-five thousand, four hundred and thirty-two.

13. To read numbers which are expressed by figures.

Point them off into periods of three figures each; then, beginning at the left hand, read the figures of each period in the same manner as those of the right hand period are read, and at the end of each period pronounce its name.

OBS. 1. The learner must be careful, in pointing of figures, always to begin at the right hand; and in reading them, to begin at the left hand.

2. Since the figures in the first or right hand period always denote units, the name of the period is not pronounced. Hence, in reading figures, when no period is mentioned, it is always understood to be the right hand, or units' period.

EXERCISES IN NUMERATION.

Note.-At first the pupil should be required to apply to each figure the name of the place which it occupies. Thus, beginning at the right hand, he should say, "Units, tens, hundreds," &c., and point at the same time to the figure standing in the place which he mentions. It will be a profitable exercise for young scholars to write the examples upon their slates or paper, then point them off into periods, and read them.

QUEST.-13. How do you read numbers expressed by figures? Obs. Where begin to point them off? Where to read them? Do you pronounce the name of the right hand period? When no period is named, what is understood? 14. In the French method of numeration, how many figures are there in a period? How many in the English method? Which method is preferable?

14. The method of dividing numbers into periods of three figures, is the French Numeration. The English divide numbers into periods of six figures. The French method is the more simple and convenient. It is generally used throughout the continent of Europe, as well as in America, and has been recently adopted by some English authors.

EXERCISES IN NOTATION.

Write the following numbers in figures:

1. Twenty-seven. Ans. 27.
2. Seventy-two. Ans. 72.
3. One hundred and twenty-five.
4. Three hundred and fifty-two.

5. Two hundred and four.

Ans. 204.

6. One thousand and forty-two. Ans. 1042. 7. Thirty thousand nine hundred and seven.

Ans. 30907.

OBS. It will be observed, that in the 5th example no tens are mentioned, in the 6th no hundreds, and that these places in the answers are filled by ciphers. In all cases, when any intervening order is omitted in the given example, the place of that order in the answer must be filled by a cipher. Hence,