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In the usual mode of explaining this Notation by the decimal scale, the v is supposed to have sprung from a divided s, and therefore to have been naturally employed in expressing

half as great a number. So I is thought to have come from a divided 1 ; and 15 or p from a divided m or ciɔ. 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 reckon. ing their cycle, (or recurring period, by which they estimate time,) of go years. The same scales were likewise to some extent combined by the ancient Scandinavians.

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 ?

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.

ducks are there in all ? 11 and 3 are how many ?

16. How many are 11 and 5 ? 11 and 6? 11 and 4? 11 and 2 ? 11 and 7 ? 11 and 8? 11 and 10 ? '11 and 9? 11 and 12 ? 11 and 11 ?

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 to? 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 and 0 are 2 | 4and0 are 4 , 6 and 0 are 68 and are 8 2 1 34 1 56 1 ny 1 9 2 2 44 2 66 2 818 2 10 2 3 54 3 ng 6 3 9

3 11 2 4 64 4 86 4

4 12 2 5 74 5 96 5

11

13 2 6 84 6 10 16 6 12 6 14 2 7 914 7 11 16

13

7 15 2 8 10 4 8 126 8 14 8 16 2 9 11 4 9 13 | 6 9 15 9 17

10

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1

3 and 0 are 35 and 0 are 57 and 0 are and 0 are 9 3 1 4 ! 5 1 6 7 1 8 9 1 10 3 2 5 5

2 99 2

11 3 3 65 3 81 3 10 9 3 12 3 4 75 4 97 4 11 9 4 13 3 5 85 5 10 5 12 | 9 5 14 3 6 9 5 6 117 6 13 9 6 15 3 vagy 10 5

127 ng

9 ny 16 3 8

11 5 8 137 8 15 | 9 8 17 3 9 12 | 5 9 147 9 169 9 18

[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 tho regular succession of the numbers.]

14

MENTAL EXERCISES CONTINUED.

are how

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

many - ? 3. James put all his marbles on a bench; 10 have fallen on the floor; 3 are falling ; and I 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 ?

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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 1:1 ? 8 and 11 and 13? n 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

?

18. 10, and 9, and 8, and 7, and 6, and 5, and 4, and 3, and 2, are how many ?

19. 11, and 10, and 9, and 8, and 7, and 6, and 5, and 4, and 3, are how many ?

20. 12, and 11, and 10, and 9, and 8, and 7, and 6, and 5, and 4, and 3, and 2, and 1, are how many ?

21. Add each of these rows of figures, first from the bottom to the top, then from the top to the bottom, then from right to left, and then from left to right.

1 2 3 4 5 6 7 8 9
2 3 4 5 6 7 8 9 10
3 4 6 7 8 9 10 11
4 6 7 8 9 10 11 12

7 8 9 10 11 12 13
6 7 8 9 10 11 12 13 14
7 8 9 10 11 12 13 14 15

8 9 10 11 12 13 14 15 16 (This addition should be made aloud, as should every other calculation in the mental exercises.]

22. How many are 9 and 2? 19 and 2? 29 and 2? 39 and 2 ? 49 and 2 ? 59 and 2 ? 69 and 2 ? 79 and 2 ? 89 and 2 ? 99 and 2 ?

[The teacher should here take 9 and 3, 29 and 3, and so on; after which, he should begin 8 and 2, 18 and 2, &c.; 8 and 3, 18 and 3, &c.; 7 and 2, &c.; 7 and 3, &c., as far as 2 and 2, 12 and 2, 22 and 2, &c.; 1 and 2, &c., up to 91 and 9. This is intended for inexperienced pupils. A few questions are given below, for those in every stage of improvement.]

23. How many are 6 and 6 ? 16 and 6 ? 26 and 6 ? 36 and 6 ? 36 and 7 ? 46 and 7 ? 56 and 8? 76 and 8? 96 and 8? 86 and 9 ? 96 and 9?

24. How many are 25 and 9? 35 and 9 ? 45 and 8? 55 and 8? 65 and 7? 75 and 7 ? 85 and 4? 95 and 4?

25. How many are 3 and 4? 13 and 4 ? 23 and 4 ? 33 and 4? 43 and 5 ? 53 and 5 ? 63 and 6 ? 73 and 7? 83 and 8? 93 and 9? 104 and 10?

26. How many are 4 and 2?. 14 and 2? 24 and 2 ? 34 and 2? 44 and 3 ? 54 and 4? 64 and 5 ? 74 and 6 ? 84 and 7? 94 and 8? 104 and 9 ?

27. How many are 7 and 4? 17 and 4 ? 27 and 5 ? 37 and 5 ? 47 and 6 ? 57 and 6 ? 67 and 7 ? 77 and 8? 87 and 9? 97 and 9?

28. How many are 8 and 2? 18 and 3? 28 and 4 ? 38 and 4 ? 48 and 5 ? 58 and 5 ? 68 and 7? 78 and 7? 88 and 9 ? 98 and 9? 108 and 8? 118 and 8?

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