sents 6 tens, or 6 units each ten times the size or value of a unit of the first place; and the 3 occupying the third place represents 3 hundreds, or 3 units each one hundred times the size or value of a unit of the first place.

ART. 6. The cipher becomes significant when connected with other figures, by filling a place that otherwise would be vacant; as in 10 (ten), where it occupies the vacant place of units; in 120 (one hundred and twenty), where it also occupies the vacant place of units; and in 304 (three hundred and four), where it fills the vacant place of tens.

ART. 7. The simple value of a unit is the value expressed by a figure standing alone; or, in a collection, when standing in the right-hand place. Thus 6 alone, or in 26, expresses a simple value of six single units, or ones.

The local value of a unit is the value expressed by a figure when it is used in combination with another figure or figures, and depends upon the place the figure occupies.

The local values expressed by figures will be made plain by the following table and its explanation.

98

987

9876 98765 987654

9 8 7 6 5 4 3

Ninety-eight.

Nine hundred eighty-seven.

Nine thousand eight hundred seventy-six.
Ninety-eight thousand seven hundred sixty-five.
Nine hundred eighty-seven thousand six hundred
fifty-four.

Nine millions eight hundred seventy-six thousand
five hundred forty-three.

QUESTIONS. Art. 6. When does a cipher become significant?- Art. 7. What is the simple value of a unit? What is the local value of a unit ? What is the design of this table?

It will be noticed in the preceding table, that each figure in the right-hand or units' place expresses the local value of so many units; but when standing in the second place, it expresses the local value of so many tens, each of the value of ten ones; when in the third place, the local value of so many hundreds, each of the value of ten tens; when in the fourth place, the local value of so many thousands, each of the value of ten hundreds; the value expressed by any figure being always made tenfold by each removal of it one place to the left hand.

NUMERATION.

ART. 8. NUMERATION is the art of reading numbers when expressed by figures.

ART. 9. There are two methods of numeration in common use the French and the English.

ART. 10. The French method is that in general use on the continent of Europe and in the United States. It separates figures into groups, called periods, of three places each, and gives a distinct name to each period.

QUESTIONS. Art. 7. What value is expressed by 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 towards the left? Art. 8. What is numeration? What are the two methods of numeration in common use? Where is the French method more generally used? Repeat the French Numeration Table. What are the names of the different periods in the table? What is the value of the numbers represented in the table expressed in words?

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

The names of the periods above Sextillions, in their order, are, Septillions, Octillions, Nonillions, Decillions, Undecillions, Duodecillions, Tredecillions, Quatuordecillions, Quindecillions, Sexdecillious, Septendecillions, Octodecillions, Novemdecillions, Vigintillions, &c.

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 as many units as its name indicates 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 expressed by figures according to the French method.

RULE. Begin at the right hand, and point off the figures into periods of THREE places each.

Then, commencing at the left hand, read the figures of each period, adding the name of each period excepting that of units.

EXERCISES IN FRENCH NUMERATION.

The learner may read orally, or write in words, the numbers represented by the following figures :

QUESTIONS. Art. 10. What are the names of the periods above sextillions? 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 by figures according to the French method, we have the following

RULE. Begin at the left hand, and write in each successive order the figure belonging to it.

If any intervening order would otherwise be vacant, fill the place by a cipher.

EXERCISES IN FRENCH NOTATION AND NUMERATION.

The learner may represent by 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.

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 yon begin to write numbers? Why?

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 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 period designated by a distinct

The value of the figures in the above table, expressed in words according to the English method, is, One hundred thirty-seven thousand eight hundred ninety trillions; seven hundred eleven thousand seven hundred sixteen billions; three hundred seventyone thousand seven hundred twelve millions; four hundred fiftysix thousand seven hundred eleven.

Although there is the same number of figures in the English and in the French table, yet it will be observed that in the French table we have the names of three periods other than in the English. It will also be observel that the variation commences after the ninth place, or the place of hundreds of millions. If, therefore, we would know the value of numbers

QUESTIONS. 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? Give the names of the periods in the English Numeration Table, beginning with the period of units. Repeat the table, giving the names of all the orders or places. What is the value of the num bers in the table expressed in words? How do the figures in the English and French table compare as to numbers? How as to periods? Why is this difference? Has a million the same value reckoned by the French table as when reckoned by the English?