As one hundred is twice

seventy, LXXX for eighty, and LXXXX for ninety. fifty, it was naturally expressed by L, used twice. These are placed most con. veniently thus, [. This became rounded by writing, to C. Then, two hundred was CC, three hundred CCC, and four hundred CCCC. For the same reason that five and fifty received a particular character, five hundred would probably receive one. Accordingly we find that, as no more figures could be conveniently made with two marks, three were disposed thus, A, for five hundred. This afterwards became D by writing. Six hundred was then written DC, seven hundred DCC, &c. As two V's united signified ten, and two L's, one hundred, it was natural to unite two D's for one thousand. It was done thus CD, or more compactly thus CD: This afterwards became M. These, then, are the charac ters employed by the Romans, for expressing numbers.

Numbers, therefore, came to be expressed, as they are even at the present day, as follows:

It will be observed that four is written IV or IIII. The former method was introduced at a late period, and is not convenient where arithmetical operations, such as Addition and Subtraction, are to be performed with Roman characters. In other cases, it is perhaps to be preferred, as occupying less room. The same may be said of IX, XL, and XC.

We have explained the scale as far as it is in common use at the present day, which is far enough for the ordinary student. It may be well, however, to mention that is sometimes used for D. When this is the case, additional ɔs are sometimes annexed at the right, and each additional o increases the number ten times. Thus 10 is five hundred; 100 is five thousand; 1000 fifty thousand, &c. CIɔ is also put for M, and every additional c at each end, in this case, increases the number, as before, ten times. Thus CIO is one thousand; co ten thousand; ccciɔɔɔ one hundred thousand, &c. A line drawn over a num ber increases its value a thousand times. Thus v is five thousand; x ten thousand; Cone hundred thousand; one million; MM two millions, &c.

Some have supposed that c is taken from the Latin centum, which means one hundred, and M from mille, which means one thousand, but, if this be true, there is no similar mode of accounting for the use of the other letters.

In the usual mode of explaining this Notation by the decimal scale, the v is supposed to have sprung from a divided x, and therefore to have been naturally employed in expressing half as great a number. So L is thought to have come from a divided E; and 10 or D from a divided M or CIO. It is in favor of this explanation, that there exists no single character for twenty-five, which, on the quinary scale, we ought to have expected. On the whole, however, the Notation seems to have been commenced on the quinary, and continued on the deci mal scale, thus forming a mixture of both. In like manner, we find that the Chinese at the present day, employ the decimal and duodecimal scales in reckoning their cycle, (or recurring period, by which they estimate time,) of 60 years. The same scales were likewise to some extent combined by the ancient Scandina vians.

ADDITION.

MENTAL EXERCISES.

§ VII. 1. Two barrels stand in one place, and two in another. How many are there in all; 2 and 2 are how many?

2. A boy is carrying two pails, and three stand on a bench. How many pails in all? 2 and 3 are how many? 3. Two apples are falling from a tree, and four are lying on the ground. How many are there in all? 2 and 4 are how many?

4. Two pens are in an inkstand, and five lie on the table. How many are there in all ? 2 and 5 are how many?

5. John made 3 marks on one leaf of his book, and 6 on another. How many marks did he make? 3 and 6 are how many ?

6. His teacher punished him, for soiling the book, by giving him 4 blows on one hand, and 5 on the other. How many blows did he strike him? 4 and 5 are how many?

7. 7 boys laughed at him on one side of the house, when he was punished, and 2 on the other. How many boys laughed? 7 and 2 are how many?

8. One boy has 6 marbles and another has 7. How many have both? 6 and 7 are how many?

9. A boy bought a story book for 10 cents, and another for 6. How many cents did he give for both? 10 and 6 are how many ?

10. 7 pins were on a pin-cushion, and a girl put on 9 pins more. How many were on it then? 7 and 9 are how many?

11. 8 boys were playing together, and 5 more came to play with them. How many boys were there in all?

8 and 5 are how many?

12. A man gave 9 dollars for a barrel of flour, and 10 dollars for a firkin of butter.

How many dollars did he

give for both? 9 and 10 are how many ?

13. If an orange is worth 7 cents, and a lemon 4 cents, how many cents are both worth? 7 and 4 are how many ? 14. If a pint of filberts is worth 6 cents, and half a pound of figs is worth 9 cents, how many cents are both worth? 6 and 9 are how many ?

15. 11 ducks are on the shore of a pond, and 3 are swimming in it. How many ducks are there in all? 11 and 3 are how many? 16. How many are 11 and 5? 11 and 2? 11 and 7? 11 and 8? 11 and 12? 11 and 11?.

11 and 6? 11 and 4? 11 and 10? 11 and 9?

17. 10 men are going to market, and 5 are going home. How many men are there in all? 10 and 5 are how many?

18. 10 and 7 are how many? 10 and 3? 10 and 4? 10 and 2? 10 and 6? 10 and 8? 10 and 9? 10 and 10? 10 and 11? 10 and 12? 10 and 13? 10 and 14?

19. How many are 5 and 2? 5 and 7? 5 and 6 ? 5 and 9? 5 and 12? 5 and 11? 5 and 13? 5 and 14? 5 and 15?

20. How many are 7 and 6? 7 and 7? 7 and 8? 7 and 10? 7 and 9? 7 and 5? 7 and 11? 7 and 12? 7 and 14? 7 and 15?

21. How many are 6 and 6? 6 and 7? 6 and 9? 6 and 10? 6 and 12? 6 and 5? 6 and 8? 6 and 11? 6 and 13? 6 and 15? 6 and 14?

22. How many are 8 and 8? 8 and 7? 8 and 5? 8 and 9? 8 and 6? 8 and 4? 8 and 10? 8 and 11 ? 8 and 13? 8 and 12? 8 and 14? 8 and 15?

23. How many are 9 end 7? 9 and 4? 9 and 3? 9 and 2? 9 and 5? 9 and 6? 9 and 8? 9 and 9? 9 and 10? 9 and 11 9 and 12? 9 and 13? 9 and 15? 9 and 14?

24. How many are 4 and 1? 4and 4? 4 and 3? 4 and 5? 4 and 2? 4 and 7? 4 and 6? 4 and 8? 4 and 10? 4 and 13? 4 and 15? 4 and 17? 4 and 19 ?

25. Recite the

ADDITION TABLE.

~~~~~~~~~~

2

2 2

2

2

2and0 are 2 | 4and0 are 4 6 and 0 are 6 8 and 0 are S

94

10

9 11

444

4 8

6799

10 6

12

2345

14 8

88888

15 8 9

17

3 and0 are 3 5and0 are 57and0 are 7 9and 0 are 9

[The teacher should ask questions upon the table, promiscuously, after it has been recited, in order to render it necessary for the pupil to have each amount ⚫ fixed in his mind, independently of every other. The pupil may recite the table in order, without deriving any benefit from it, merely from his knowledge of the regular succession of the numbers.]

MENTAL EXERCISES CONTINUED.

1. In one ladder there are 7 rounds, in another 8, and in another 6. How many rounds are there in all ? 7 and 8 and 6 are how many?

2. On one end of a table are 5 apples, and on the other 4; and on the floor are 9. How many apples are there

in all? 5 and 4 and 9 are how many?

3. James put all his marbles on a bench; 10 have fallen on the floor; 3 are falling; and 9 remain where he put them. How many marbles has James, in all? 10 and 3 and 9 are how many?

4. One boy has 11 cents, another 2, and another 6. How many have all together? 11 and 2 and 6 are how many?

5. A man bought a barrel of flour for 8 dollars, and a load of wood for 3 dollars, and a plough for 7 dollars. How much did he give for all? 8 and 3 and 7 are how many?

6. A boy gave to one of his companions 9 apples, to another 11, and to another 6. How many did he give away to all? 9 and 11 and 6 are how many?

7. William's father gave him 12 cents, his mother gave him 7, and his uncle gave him 14. How many cents had William? 12 and 7 and 14 are how many?

8. A boy gave 10 cents for a top, and 4 cents for a whistle, and 8 cents for some marbles. How many cents did he spend? 10 and 4 and 8 are how many?

9. 11 and 4 and 3 are how many? 5 and 8 and 2? 7 and 6 and 12? 12 and 9 and 2? 3 and 5 and 7? 6 and 9 and 4? 7 and 8 and 9? 5 and 3 and 10?

10. 8 and 9 and 10 are how many? 10 and 11 and 9? 9 and 12 and 11? 8 and 11 and 13? 7 and 10 and 14? 6 and 8 and 17? 12 and 13 and 14?

11. 5 and 15 and 20 are how many? 20 and 7 and 8? 25 and 5 and 10? 22 and 4 and 11 ? 19 and 20 and 4? 12. 20 and 30 and 40 are how many? 30 and 40 and 10? 20 and 50 and 30? 30 and 60 and 10? 10 and 70 and 20? 40 and 30 and 30?

13. One man owns 3 houses, another 4, another 5, and another 6. How many do they all own? 3 and 4 and 5 and 6 are how many?

14. John has 7 apples, and William 5, and Joseph 9, and James 11. How many have they all? 7 and 5 and 9 and 11 are how many ?

15. Some boys went a fishing. One caught 6 fish, another 8, another 4, another 15, and another 12. How many fish did they all catch? 6 and 8 and 4 and 15 and 12 are how many?

16. Seven men went a hunting. One shot 10 birds, another 13, another 8, another 7, another 5, another 9, and another 11. How many did they all shoot? 10 and 13 and 8 and 7 and 5 and 9 and 11 are how many?

17. 9, and 8, and 7, and 6, and 5, and 4, and 3, and 2, and 1, are how many?