20. The simple value of a figure is the one expressed by it when standing alone or in the units' place of a number.

21. The local value of a figure is that which depends upon the place which the figure occupies in a number; thus, in 37 the figure 3 denotes thirty; and in 904, 9 denotes nine hundred.

22. The naught or cipher (0) has no value, but is used to show that a denomination is wanting; thus, in the number 6074, the hundreds being wanting, the place is occupied by a cipher.

QUESTIONS. Of what does the Arabic notation consist? (19.) What is the simple value of a figure? (20.) What is the local value of a figure? (21.) What is the use of the cipher? (22.) Show on the blackboard what is meant by simple and local values. What is the difference between the Roman and the Arabic notation? (18.) (19.)

LESSON III.

NUMERATION.

23. Numeration is the art of naming in their regular order the places occupied by numbers.

24. Reading numbers is the art of expressing their written value orally.

*From quintillions the table may be extended to sextillions, septillions, octil lions, nonillions, decillions, undecillions, duodecillions, &c.

26. Every vacant place in each period, with the exception of the left hand places of the left hand period, must be filled with a cipher.

27. Every vacant period in each number, with the exception of the left hand period, must be filled with ciphers.

NOTE. (a) In writing numbers periods are usually separated by commas.

Hundreds.
Tens.
Units.

Units.
Tens.

Units.

1st Period.

LESSON IV.

EXERCISES IN NOTATION AND NUMERATION.

Write

Three hundred and eighty-two;

Seven hundred and forty-nine;

Eight hundred and sixty-seven;

One thousand three hundred and ninety-seven;
Fifteen thousand eight hundred and seventy-three;
Eleven thousand and eight.

MODEL OPERATION.

382

749

867

1,397

15,873

11,008

28. RULE FOR NOTATION.-Begin at the left hand, and write the figures belonging to the highest period.

Write the hundreds, tens, and units, of each successive period in their proper order, observing to write a cipher in each vacant place.

NOTE FOR THE TEACHER.-For exercises in notation and numeration the pupil is referred to the First Book of Arithmetical Analysis.

LESSON V.

29. ANALYSIS OF NUMBERS.

MODEL OPERATION,

It is required to analyze the number 3,917,893. (a.) For convenience, every number is divided into periods of three places each, counting from the right.

(b.) The 1st period of every number is called Units;

the 2d, Thousands; the 3d, Millions; the 4th, Billions; the 5th, Trillions; the 6th, Quadrillions; and the 7th, Quintil lions, &c.

(c.) The first place of every period is called units; the second, tens; the third, hundreds.

(d.) The value of the places increases in a tenfold ratio from the right to the left; that is, tens' place has ten times the value of units' place; hundreds' place has ten times the value of tens' place; thousands' place has ten times the value of hundreds' place, &c.

(e.) In the units' period of the given number are written 3 units, 9 tens, 8 hundreds; in the thousands' period are written 7 units, 1 ten, 9 hundreds; in the millions' period are written 3 units.

(f) In the units' period of the given number, 9 tens are equal to 90 units 8 hundreds are equal to 80 tens, or 800 units; in the thousands' period, 7 units of thousands are equal to 70 hundreds, or 700 tens, or 7000 units; 1 ten of thousands is equal to ten units of thousands, or 100 hundreds, or 1000 tens, or 10000 units; 9 hundreds of thousands are equal to 90 tens of thousands, or 900 units of thousands, or 9000 hundreds, or 90,000 tens, &c.

After the model, analyze the following numbers :1. One million three hundred and eighty-five thousand six hundred seventy-four.

2. 19 millions 713 thousand 486.

3. 186 millions 347 thousand 391.

4. 986 trillions 421 million 223 thousand, 86.

5. Five trillions 876 millions 876 thousand.

6. 189 quadrillions 317 thousands 346.
7. 597 trillions 896 millions 348 thousand.
8. 876 millions 384 thousand.

9. 968 millions 117 thousand 18.

10. 678 trillions 347 millions 396 thousand.
11. 876 billions 546 millions 321 thousand.
12. 347 billions 390 millions 416 units.
13. 974 quadrillions 14 trillions 318.
14. 764 millions 39 thousand 340.

15. 97 thousand, four hundred and ninety-five.

QUESTIONS.-What is numeration? (23) What is the reading of numbers? (24) Repeat the numeration table. (25) What is to be done with vacant places? (26) What is to be done with vacant periods? (27) How are periods usually separated? (27 a) What is the rule for notation? (28) Give the analysis of the number 3,917,893. (29) .

SECTION III.
ADDITION.

LESSON I.

30. The Sum of two or more numbers is a number which contains as many units as all the numbers taken together.

'31. Addition is the process of finding the sum of two or more numbers.

32. The SIGN OF ADDITION is a horizontal cross, thus,

(+), and is called plus, which signifies more. When placed between two numbers, it denotes that they are to be added together; as, 4+6 are equal to 10; to be read 4 plus 6 are equal to ten.

33. The SIGN OF EQUALITY is two, short, parallel, horizontal lines; thus, (=), and when placed between two numbers or quantities, it denotes that they are equal; as 8+6=14; to be read, 8 plus 6 equals 14.

34. The DOLLAR SIGN is an S crossed by two perpendicular parallel lines; thus, ($), and when prefixed to num