ART. 4. The Arabic or Indian notation employs ten distinct characters or figures, sometimes called digits, viz.:

1, 2, 3, 4, 5, 6, 7, 8, 9, 0. one, two, three, four, five, six, seven, eight, nine, cipher.

The first nine are called significant figures, because each one has a value of itself when standing alone. The cipher is also sometimes called naught or zero; and, when standing alone, it has no value and signifies nothing.

NUMERATION.

ART. 5. NUMERATION is the art of reading numbers, or naming the value of figures in the order of their places.

ART. 6. The Arabic figures have two values, a simple and a local, and, from their convenience, are now universally used in arithmetical calculations.

ART. 7. The simple value of a figure is the value it has when standing alone, thus, 6; or when standing in the right-hand place of whole numbers, thus, 26. In either case the 6 denotes six units or ones.

ART. 8. The local value of a figure is the value it has when it is removed from the right-hand place toward the left, and depends on the place the figure occupies.

For example, 6 standing at the left hand of 5, thus, 65, expresses ten times the value it does when standing alone, or in the right-hand place, and denotes six tens or sixty; the five at the right hand of it denotes five units, and the two figures together express sixty-five. When placed at the left of two figures, thus, 678, it expresses one hundred times its simple value, or ten times its value when standing in the second or tens' place; its value being always increased tenfold, when it is removed one place to the left. Therefore, while the 8 denotes eight units, and the 7, seven tens, the 6 denotes six hundreds, and the whole together, 678, six hundred and seventy-eight.

QUESTIONS.-Art. 4. How many characters are employed in the Arabic or Indian notation? What are the first nine called? Why? What is the tenth called? What does it represent or signify when standing alone? - Art. 5. What is numeration?-Art. 6. What two values have the Arabic figures? — Art. 7. What is the simple value of a figure ? - Art. 8. What is the local value? Why is this value called its local value? What effect has the removal of a figure one place to the left upon its value? Two places? &c.

ART. 9. The cipher becomes significant when connected with other figures; as in 10 (ten), where it gives a tenfold value to the 1; and 120 (one hundred and twenty), where it gives a tenfold value to the 12; and 304 (three hundred and four), where it has the same influence on the 3, causing it to represent three hundreds instead of three tens.

The local value of figures will be made plain by the following table and its explanation.

9 8 7 6 5 Ninety-eight thousand seven hundred sixty-five.

9 8 7 6 5 4{fifty-four. S Nine hundred eighty-seven thousand six hundred

9 8 7 6 5 4 3 { five hundred forty-three. Nine millions eight hundred seventy-six thousand

It will be noticed in the above table, that each figure in the right-hand or units' place expresses only its simple value, or so many units; but, when standing in the second place, it denotes so many tens, or ten times its simple value; and when in the third place, so many hundreds, or one hundred times its simple value; when in the fourth place, so many thousands, or a thousand times its simple value, and so on; the value of any figure being always increased tenfold by each removal of it one place to the left hand.

QUESTIONS. Art. 9. When does the cipher become significant? What is its effect, when placed at the right hand of a figure? What is the design of this table? What value has a figure standing in the right-hand or units' place? What, in the second place? What, in the third? How do figures increase from the right toward the left?

ART. 10. There are two methods of numeration in common use; the French and the English. The former is more generally used on the continent of Europe and in the United States. In the French method, a new name is given to every third figure above millions, and in the English, to every sixth figure above millions.

127, 894, 237, 867, 123, 678,

478, 63 8.

The value of the numbers in this table, expressed in words, is, One hundred twenty-seven sextillions, eight hundred ninetyfour quintillions, two hundred thirty-seven quadrillions, eight hundred sixty-seven trillions, one hundred twenty-three billions, six hundred seventy-eight millions, four hundred seventy-eight thousand, six hundred thirty-eight.

The preceding table may be extended to any number of figures by supplying the names of the periods above sextillions, in their order; viz. Septillions, Octillions, Nonillions, Decillions, Undecillions, Duodecillions, Tredecillions, Quatuordecillions, Quindecillions, Sexdecillions, Septendecillions, Octodecillions, Novemdecillions, Vigintillions, &c.

QUESTIONS. -Art. 10. What are the two methods of numeration in common use? Where is the French method more generally used? How does the French method differ from the English? Repeat the French Numeration Table, giving the names of all the places or orders, beginning at the right. What are the names of the different periods in the table? What is the value of the numbers in the table expressed in words? Repeat the names of the periods above sextillions.

ART. 11. The successive places occupied by figures are often called orders. Hence, a figure in the right-hand or units' place is called a figure of the first order, or of the order of units; a figure in the second place is a figure of the second order, or of the order of tens; in the third place, of the order of hundreds, and so on; thus, in the number 1847, the 7 is of the order of units, 4 of the order of tens, 8 of the order of hundreds, and 1 of the order of thousands, each figure expressing so many units of that order to which it belongs; so that we read the whole number one thousand eight hundred and forty

seven.

ART. 12. From the preceding table and explanation, we deduce the following rule for numerating and reading numbers according to the French method.

RULE. Begin at the right hand, and divide the number into periods of THREE figures each, remembering the name of each period. Then, commencing at the left hand, read the figures of each period in the same manner as the period of units, giving the name of each period excepting the last.

EXERCISES IN FRENCH NUMERATION.

The learner may read orally, or write in words, the following

QUESTIONS. Art. 11. What are the successive places of the figures in the table called? Of what order is the first or right-hand figure? The second? The third? &c. Art. 12. What is the rule for numerating and reading numbers according to the French method?

ART. 13. To write numbers according to the French method, we have the following

RULE. - Begin at the left, and write the figure of the highest order to be written, and place in each successive order the figures belonging to it, observing to fill the place by a cipher, when no number is mentioned to be written.

EXERCISES IN FRENCH NOTATION AND NUMERATION.

The learner may write in figures, and read, the following numbers:

1. Forty-seven.

2. Three hundred fifty-nine.

3. Six thousand five hundred seventy-five.
4. Nine hundred and eight.

5. Nineteen thousand.

6. Fifteen hundred and four

7. Twenty-seven millions five hundred.

8. Ninety-nine thousand ninety-nine.

9. Forty-two millions two thousand and five.

10. Four hundred eight thousand ninety-six.

11. Five thousand four hundred and two.

12. Nine hundred seven millions eight hundred five thousand and seventy-four.

13. Three hundred forty-seven thousand nine hundred and fifteen.

14. Eighty-nine thousand forty-seven.

15. Fifty-one thousand eighty-one.

16. Seven thousand three hundred ninety-five.

17. Fifty-seven billions fifty-nine millions ninety-nine thousand and forty-seven.

ART. 14. The following table exhibits the English method of numeration, in which it will be observed that the figures are separated by commas into divisions or periods of six figures each. The first or right-hand period is regarded as units and thousands of units; the second as millions and thousands of millions; and so on, six places being assigned to each division designated by a distinct name.

QUESTIONS.-Art. 13. What is the rule for writing numbers according to the French method? At which hand do you begin to numerate figures? Where do you begin to read them? At which hand do you begin to write numbers? Why?- Art. 14. How many figures in each period in the English method of numeration? What orders are found in the English method

that are not in the French?