IN writing numbers we use ten characters, called figures. The first figure, 0, is called a cipher, or nought, and denotes nothing, or the absence of number.

Each of the other figures represents a number of single things, or units.

1. A Unit* is a single thing; as, 1 horse, 1 dog, 1 apple, 1 dozen, 1 fourth.

*NOTE FOR THE TEACHER. - Let this be fully explained, that the pupil may observe the distinction between a hundred as one unit, illustrated by a hundreddollar-note, and as many units, illustrated by one hundred one-dollar-notes.

The teacher should question the class thoroughly on each subject: he should

2. A Number is a unit, or collection of units; as, 2 cows, 1 house, 5 sheep, 1, 3, 4, 9.

3. A Concrete Number is one which indicates some definite kind of thing; as, 1 bird, 7 sheep, 2

ounces.

4. A Denominate Number is one in which the kind of thing named is a unit of weight, measure, or currency; as, 5 pounds, 2 yards, 6 dimes.

5. An Abstract Number is one which does not indicate any definite kind of thing; 1, 5, 7, 6, 9, 14. 6. Arithmetic treats of numbers, and the art of using them.

NOTATION.

7. Notation is a method of expressing numbers by words, letters, or figures. The methods most used are the Roman, which employs letters, and the Arabic, which employs figures.

8. The Roman Notation uses the letters I, V, X, L, C, D, M. The numbers of sections, chapters, and

strive by illustrations and suggestions to develop in the mind of the pupil a clear conception of the idea contained in the definition, proposition, or problem, before requiring the one to be learned, or the other to be solved. The following notes and hints are merely suggestive, and are not intended to be either systematic or exhaustive. No book can take the place of a living teacher; neither can any teacher follow any prescribed method of instruction, for by so doing he sacrifices that individuality which is essential to success.

REVIEW QUESTIONS. *- What is a unit ? (1.) What is a number? (2.) What is a concrete number? (3.) What is a denominate number? (4.) What is au abstract number? (5.) Of what does arithmetic treat? (6.)

to the paragraphs containing the answers.

N.B.-These review questions should by no means be omitted.

other divisions of books, are usually written by this

All other numbers are expressed by the use of these letters repeated or combined; as,—

OBSERVATIONS.

From the above it will be noticed, that

1. The value of a letter is repeated as many times as the letter itself is repeated; as, XXX. thirty, CC. two hundred.

2. If a letter of less value stands before a letter of greater vaiue, the value of the less is subtracted from that of the greater; as, X. ten, IX. nine, L. fifty, XL. forty.

3. If a letter of less value stands after one of greater value, the value of the less is added to that of the greater; as, V. five, VI. six, M. one thousand, MD. fifteen hundred.

4. A dash (—) placed over a letter, or a combination of letters, repeats the value a thousand times; as X. ten, X. ten thousand, XII. twelve, XII. twelve thousand.

NOTE FOR THE TEACHER. -Slate and blackboard drill in writing and reading numbers in Roman notation should be furnished by the teacher.

REVIEW QUESTIONS.-What is the Roman notation? (7.) Why so called? What is its use? How many letters does it use? (8.) Name them. Which Jetter has the greatest value? Which the least? Which letter stands for fifty? Which for ten? Which for one hundred ? Which for five hundred? What effect has the repetition of a letter?

1, 2, 3, 4, 5, 6, 7, 8, and 9 are often called digits.

When we count ten things, as we have no figure to represent the number ten, we unite the ten into a group,✶ and call it ten, or, 1 ten, and represent it by writing the 1 in the second place, thus, 10; the cipher showing, that, while we have 1 ten, we have no additional units.

When we count twelve things, we have one ten and two single things, represented thus; 12.

When we count ten tens, as we have no figure to represent the number, we unite the tens into a group, and call it 1 hundred, and write the 1 in the third place; thus, 100; the ciphers showing that we have no additional tens or units.

10. The Simple Value of a figure is the one expressed by it when standing alone, or in the unit's place.

11. The Local Value of a figure is the one it acquires by standing in a particular place in a number; as, 3 in 30, where it represents three tens, or thirty; 3 in 364, where it represents three hundreds; &c.

12. Numeration is the art of reading numbers expressed by figures.

The different orders of units can be very clearly illustrated by tying ten sticks into a bundle to represent 1 ten, then a package of ten such bundles to represent 1 hundred, &c.

Every place in a number not occupied by digits must be filled with a cipher.

The periods of a number are often separated by

commas.

EXERCISES.

1. Write 4 hundreds and 2 units, commencing at the left hand, and read the number.

Write 3 tens and 7 units in the same manner, and read it.
Write 5 hundreds, 3 tens, and 2 units in the same manner.
Write 4 tens, and 6 units.

Write 5 hundreds and 4 units.
Write 7 hundreds and 2 tens.

2. Write 5 hundreds, 7 tens, and 2 units, and read the number.

Write 3 tens and 1 unit.

Write 5 hundreds and 6 units.

• From quadrillions, the table may be extended to quintillions, sextillions septillions, octillious, nonillions, decillions, undecillions, duodecillions, &c.