The Roman notation is now but little used, except in number. ing sections, chapters, and other divisions of books.

EXERCISES IN ROMAN NOTATION.

Write the following numbers in letters :—

1. Ninety-six.

2. Eighty-seven.

3. One hundred and ten.

4. One hundred and sixty-nine.

5. Two hundred and seventy-five.

6. Five hundred and forty-two.

7. One thousand three hundred and nineteen.

8. One thousand eight hundred and fifty-eight.

Ans. XCVI.

4. The Arabic Notation, or that made known through the Arabs, employs in expressing numbers ten characters or figures, 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 digits, from digitus, the Latin signifying a finger, because of the use formerly made of the fingers in reckoning. The cipher is called naught, or zero, from its expressing the absence of a number, or nothing, when standing alone.

5. The particular position a figure occupies with regard to other figures is called its PLACE; as in 32 (thirty-two), counting from the right, the 2 occupies the first place, and the 3 the second place.

The digits have been denominated significant figures, because each of itself always represents so many units, or ones, as its name indicates. But the size or value of the units represented by a figure differs according to the place occupied by it.

Thus, in 366 (three hundred and sixty-six), each of the figures, without regard to its place, represents units, or ones; but the 6 occupying the first place represents 6 single units; the 6

3. What use is now made of Roman notation? —4. How many characters are employed in the Arabic notation? What are the first nine called, and why? The cipher? What does it represent when standing alone? 5. What is meant by the place of a figure? What have the digits been denominated? Why? How does the size or value of units represented by figures differ?

occupying the second place represents 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.

6. The cipher, when connected with other figures, occupies a place that otherwise would be vacant; as in 10 (ten), where it occupies the vacant place of units; and in 304 (three hundred and four), where it occupies the vacant place of tens.

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.

The figures in this table are read thus : —

Nine.
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.

6. What does a cipher occupy when written in connection with other figures? 7. What is the simple value of a unit? The local value of a unit? The design of the table?

In the table, any figure in the right-hand or units' place expresses the local value of so many units; but the same in the second place expresses the local value of so many tens, each of the value of ten ones; in the third place, the local value of so many hundreds, each of the value of ten tens; in the fourth place, the local value of so many thousands, each of the value of ten hundreds; and, in general,

The value expressed by any figure is always made tenfold by each removal of it one place to the left hand.

NUMERATION.

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

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

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.

7. What value is expressed by a figure standing in the right-hand or units' place? In the second place? In the third? How do figures increase from the right towards the left? 8. What is numeration? -9. What are the two methods of numeration in common use?-10. Where is the French method more generally used? Repeat the French Numeration Table. Name the different periods in the table.

The value of the number represented in the table is, One hundred twenty-seven sextillions, eight hundred ninety-four quinillions, 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 periods above Sextillions, in their order, are, Septillions, Octillions, Nonillions, Decillions, Undecillions, Duodecillions, Tredecillions, Quatuordecillions, Quindecillions, Sexdecillions, Septendecillions, Octodecillions, Novemdecillions, Vigintillions, &c.

11. The successive places occupied by figures are often called Orders. 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 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, so that we read the whole, one thousand eight hundred and forty

seven.

12. To numerate and read figures according to the French method.

RULE.

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

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

EXERCISES IN FRENCH NUMERATION.

Read or write in words the numbers represented by the follow

10. What is the value of the number represented in the table expressed in words? What are the names of the periods above sextillions? -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. - 12. The rule for numerating and reading numbers according to the French method?

13. To write numbers by figures according to the French method.

RULE. – Begin at the left, and write in each successive order the fig ure belonging to it.

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

EXERCISES IN FRENCH NOTATION AND NUMERATION. 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.

13. The rule for writing numbers according to the French method? At which hand do you begin to numerate? Where do you begin to read? At which hand do you begin to write numbers? Why?