There may be one, two, or more tens, just as there are one, two, or more units, or single things; it takes ten cents to make one tencent piece; just so it takes ten single things to make one ten. All figures in the second place express units of the 2d order, that is, units of tens.

One ten and one unit, 11, are called eleven; one ten and two units, 12, twelve, &c. In this way the units of the 1st order are united with the tens, that is, with

Note. Twenty, thirty, &c., are contractions for two tens, three tens, &c.

the units of the 2d order, to form the numbers from 10 to 20, from 20 to 30, to 40, and so on to 99, which is the largest number that can be represented by two figures.

The weeks in a year are 5 tens and 2 units, (5 of the second order and 2 of the first order now described,) and are expressed thus, 52, (fifty-two.) In the same manner express on your slate, or on the blackboard, the two orders united, so as to form all the numbers from 10 to 99.

4. Ten tens are called one hundred, which forms a unit of a still higher, or 3d order, and is expressed by writing two ciphers at the right hand of the unit 1, ... thus,

Note. When there are no units or tens, we write ciphers in their places, which denote the absence of a thing, (2.)

100 one hundred. 200 two hundred. 300 three hundred, &c.

Questions. 3. How is ten represented? What is it considered as forming? Consisting of what? What place does the cipher fill? The one? Where is unit's place, and where ten's place, counting from the right? How much larger is the value of a figure in the place of tens than in the place of units? In which place does it retain its simple value? In ten's place, what is its value called? What is 1 ten and 1 unit called? 1 ten and 2 units? How are the numbers from 10 to 99 expressed? Of what is the number forty made up? Ans. 4 tens and no units. Sixty? What do you unite, to form the number twentythree? thirty-seven? seventy-five? &c. Of what are twenty, thirty, &c., contractions? What is the largest, and what the least, number you can express by one figure? by two figures?

Three hundred sixty-five, the days in a year, are expressed thus, 365; 3 being in the place of hundreds, 6 in the place of tens, and 5 in the place of units.

After the same manner, the pupil may be required to unite the three orders, and express any number from 99 to 999.

5. We have seen that figures have two values, viz., simple and local.

The simple value of a figure is its value when standing alone; thus, the simple value of 7 is seven.

The local value of a figure is its value according to its distance from the place of units; thus, the local value of 7, in the number 75, is 7 tens, or seventy, while its simple value is seven; in the number 756, its local value is seven hundred.

Note. From the fact that 10 is 1 more than 9, it follows, as may be found by trial, that the local value of every figure at the left of units, except 9, exceeds a certain number of nines by the simple value of the figure. Take the number 623; 2 (tens) is 2 more than 2 nines, and 6, (hundreds,) 6 more than a certain number of nines. On this principle is founded a method of proof in the subsequent rules, by casting out the nines.

¶ 6. Ten hundred make one thousand, which is called a unit of the next higher, or 4th order, consisting of thousands, and is expressed by writing three ciphers at the right hand of the unit 1, giving it a new local value; thus, 1000, one thousand.

To thousands succeed tens and hundreds of thousands, forming units of the 5th and 6th orders.

Questions. ¶ 4. What are 10 tens called? What do they form? How many places are required to express hundreds? How much does 1 cipher, placed at the right hand of 1, increase it? 2 ciphers? How do you express two hundred? &c. What are 4 hundreds, 9 tens, and 5 units called? How is one hundred ninety-three expressed? What place does the 3 occupy? the 9? the 1? How do you express the absence of an order? How is the number of days in a year expressed?

5. How many values have figures? What are they? What is the simple value? local value? What is the value of 5 in 59? Is it its simple, or a local value? Is the value of 8, in 874, simple or local? of the 7? of the 4?

¶ 6. How do you express one thousand? seven thousand? A thousand is a unit of what order? How many thousands are 30 hundreds ? What after thousands, and of what order? The 6th order is what? In writing nine hundred and two thousand and nine, where do you place ciphers? Why?

T7. In this table of the six orders now described, you see the unit 1 moving from right to left, and at each removal forming the unit of a higher order. There are other orders yet undescribed, to form which the unit 1 moves onward still towards the left, its value being increased ten times by each removal.

Note 1. The Ordinal numbers, 1st, 2d, 3d, &c., may be called indices of their respective orders.

Note 2. Various Readings. In the number 546873, the left hand figure 5 expresses 5 units of the 6th order, or it may be rendered in the next lower order with the 4, and together they may be read 54 units of the 5th order, (ten thousands,) and connecting with the 6, they may be read, 546 units of the 4th order, or 546000. Hence, units o any higher order may be rendered in units of any lower order.

To hundreds of thousands succeed units, tens, and hundreds of millions.

TS. To millions succeed billions, trillions, quadrillions, quintillions, sextillions, septillions, octillions, nonillions, decillions, undecillions, duodecillions, tredecillions, &c., to each of which, as to units, to thousands, and to millions, are assigned three places, viz., units, tens, hundreds, as in the following examples:

Questions.-17. How is the unit 1 of the 1st order made a unit of the 2d order? of the 3d order, &c., to the 6th order? What may the ordinal numbers, 1st, 2d, 3d, &c., be called? 7 units of the 6th order are how many units of the 4th order? The teacher will multiply such questions. What is the least, and what the largest, number which can be expressed by 2 places? 3 places? &c. What after hundreds of thousands? Of what order will millions be? tens of millions? hundreds of millions?

¶ 8. What after millions? How many places are allotted to billions? to trillions? &c. Give the names of the orders after trillions. In reading large numbers, what is frequently done? Why? The 1st period at the right is the period of what? the 2d? the 3d? the 4th?

&c.

2

3, 0 8 2,7 1 5, 2 0 3, 1 7 4, 5 9 2, 8 3 7, 4 6 3, 5 1 2

To facilitate the reading of large numbers, we may point them off into periods of three figures each, as in the 2d example. The names and the order of the periods being known, this division enables us to read numbers consisting of many figures as easily as we can read those of only three figures. Thus, in looking at the above examples, we find the first period at the left hand to contain one figure only, viz., 3. By looking under it, we see that it stands in the 9th period from units, which is the period of septillions; therefore we read it 3 septillions, and so on, 82 sextillions, 715 quintillions, 203 quadrillions, 174 trillions, 592 billions, 837 millions, 463 thousands, 512.

T9. From the foregoing we deduce the following principles :

Numbers increase from right to left, and decrease from left to right, in a ten-fold proportion; and it is

A FUNDAMENTAL LAW OF THE ARABIC NOTATION; that,

Questions. T 9. How do numbers increase? how decrease, and in what proportion? To what is 1 ten equal? 1 hundred? 1 thousand? &c. To what are 10 units equal? 10 hundreds? &c. What is a fundamental law of the Arabic notation? What is notation? numeration? How do you write numbers? read numbers? If you were to write a number containing units, tens, hundreds, and millions, but no thousands, how would you express it?

I. Removing any figure one place towards the left, increases its value ten times, and

II. Removing any figure one place towards the right, decreases its value ten times.

The expressing of numbers as now shown is called Notation. The reading of any number set down in figures is called Numeration.

To write numbers. — Begin at the left hand, and write in their respective places the units of each order mentioned in the number. If any of the intermediate orders o units be omitted in the number mentioned, supply their respective places with ciphers.

To read numbers.-Point them off into periods of three figures each, beginning at the right hand; then, beginning at the left hand, read each period separately.

Let the pupil write down and read the following numbers:

Two million, eighty thousand, seven hundred and five.
One hundred million, one hundred thousand and one.
Fifty-two. million, sixty thousand, seven hundred and three.
One hundred thirty-two billion, twenty-seven million.
Five trillion, sixty billion, twenty-seven million.
Seven hundred trillion, eighty-six billion, and nine.
Twenty-six thousand, five hundred and fifty men.
Two million, four hundred thousand dollars.
Ninety-four billion, eighty thousand minutes.
Sixty trillion, nine hundred thousand miles.

Eighty-four quintillion, seven quadrillion, one hundred million grains of sand.

T10. Numbers are employed to express quantity.

Quantity is anything which can be measured. Thus, Time is quantity, as we can measure a portion of it by days, hours, &c. Distance is quantity, as it can be measured by miles, rods, &c.

By the aid of numbers quantities may either be added together, or one quantity may be taken from another.

Arithmetic is the art of making calculations upon quantities by means of numbers.

Questions. T 10. Numbers are employed to express what? What is quantity? By what is a quantity of grain measured? a quantity of cloth? What is arithmetic? What is an abstract number? a denominate number? What is the unit of a number? What is the unit value of 8 bushels? of 16 yards? of 20 pounds of sugar? of 3 quarts of milk? of 9 dozen of buttons? of 18 tons of hay? of 16 hogs. heads of molasses?