&c. or by plate I. The former method is preferable, if the pupil be very young, not only for the examples in the first part of this section, but for the first examples in all the sections.

The pupil will probably solve the first examples without any instruction.

Examples in addition, and subtraction may be solved by plate 1. as follows.

How many are 5 and 3?* Select 1 rectangle containing 5 marks, and another containing 3 marks, and ascer tain the number of marks in both.

How many are 8 and 6? Select a rectangle containing 8 marks, and another containing 6 marks, and count them together.

How many are 17 and 5? Keeping 17 in the mind, select a rectangle containing 5 marks, and add them thus: 17 and 1 are 18, and 1 are 19, and I are 20, and 1 are 21, and 1 are 22.

If you take 4 from 9, how many will remain? Select a rectangle containing 9 marks, and take away 4 of them. 18 less 5 are how many? Keeping 18 in mind, select a rectangle containing 5, and take them away I at a time. In this manner all the examples in this section may be solved.

B & C. The articles B and C contain the common addition table as far as the 10 first numbers. In the first the numbers are placed in order, and in the second out of order.

The pupil should study these until he can find the answers readily, and then he should commit the answers to memory.

D. In this article the numbers are larger than in the preceding, and in some instances, three or more numbers are added together. In the abstract examples the numbers from one to ten are to be added to the numbers from ten to twenty.

* Figures are used in the key, because the instructer is supposed to be acquainted with them. They are not used in the first part of the book, because the pupil would not understand them so well as he will the words.

E. This article contains subtraction.

F. This article is intended to make the pupil familiar with adding the nine first numbers to all others. The pupil should study it until he can answer the questions very readily.

G. In this article all the preceding are combined together, and the numbers from 1 to 10 are added to all numbers from 20 to 100; and subtracted in the same manner. 18. 57 and 6 are are 63, and 3 are 66, and 5 are 71, and 2 are 73, less 8 are 65.

H. This article contains practical questions which show the application of all the preceding articles.

6. 37 less 5 are 32, less 8 are 24, less 6 (which he kept himself) are 18; consequently he gave 18 to the third boy.

SECTION II.

Thissection contains multiplication. The pupil will see no difference between this and addition. It is best that be should not at first, though it may be well to explain it to him after a while.

A. This article contains practical questions, which the pupil will readily answer.

1. Three yards will cost 3 times as much as 1 yard.

N. B. Be careful to make the pupil give a similar reason for multiplication, both in this article, and elsewhere. This question is solved on the plate thus; in the second row, count 3 rectangles, and find their sum. 2 and 2 are 4 and two are 6.

11. A man will travel 4 times as far in 4 hours as he will in 1 hour. In the third row count 4 times 3, and ascertain their sum.

15. There are 4 times as many feet in 4 yards as in 1 yard, or 4 times 3 feet.

B. This article contains the common multiplication table, as far as the product of the ten first numbers. The pupil should find the answers once or twice through, until he can find them readily, and then let him commit them to memory.

43. 6 times 3. In the third row count 6 times 3, and then ascertain their sum. 3 and 3 are 6, &c.

59. 7 times 9. In the ninth row count 7 times 9, or 7 rectangles, and ascertain their sum. 9 and 9 are 18, &c.

C. This article is the same as the preceding, except in this the numbers are out of their natural order.

D. In this article multiplication is applied to practical examples. They are of the same kind as those in article A of this section.

12. There are 8 times as many squares in 8 rows, as in 1 row. 8 times 8 are 64.

13. There are 6 times as many farthings in 6 pence, as in 1 penny. 6 times 4 are 24.

17. 12 times 4 are 48.

Note. When a number is taken more than 10 times, as in the above example, after taking it 10 times on the plate, begin at the beginning of the row again, and take enough to make up the number.

23. There are 3 times as many pints in 3 quarts as in 1 quart. 3 times 2 are 6. And in 6 pints there are 6 times 4 gills or 24 gills.

28. In 3 gallons there are 12 quarts, and in 12 quarts there are 24 pints.

31. In 2 gallons are 8 quarts, in 8 quarts 16 pints; in 16 pints 64 gills. 16 times 4 are 64.

35. In one gallon are 32 gills; and 32 times 2 cents are 64 cents. Or, 1 pint will cost 8 cents, and there are 8 pints in a gallon. 8 times 8 are 64.

38. They will be 2 miles apart in 1 hour, 4 miles in 2 bours, &c.

SECTION III.

A. This section contains division. scarcely distinguish it from multiplication. portant that he should at first.

The pupil will
It is not im-

Though the pupil will be able to answer these questions by the multiplication table, if he has committed it to memory thoroughly; yet it will be better to use the plate for some time.

16. As many times as 3 dollars are contained in 15 dollars, so many yards of cloth may be bought for 15 dollars. On plate I, in the third row, count fifteen and see how many times 3 it makes. It is performed very nearly like multiplication.

B. In this article, the pupil obtains the first ideas of fractions, and learns the most important of the terms which are applied to fractions.* The pupil has already been accustomed to look upon a collection of units, as forming a number, or as being itself a part of another number. He knows, therefore, that one is a part of every number, and that every number is a part of every number larger than itself. As every number may have a variety of parts, it is necessary to give names to the different parts in order to distinguish them from each other. The parts receive their names, according to the number of parts which any number is divided into. If the number is divided into two equal parts, the parts are called halves, if it is divided into three equal parts, they are called thirds, if into four parts, fourths, &c.; and having divided a number into parts, we can take as many of the parts as we choose. a number be divided into five equal parts, and three of the parts be taken, the fraction is called three fifths of the number. The name shows at once into how many parts the number is to be divided, and how many parts are taken. The examples in this book are so arranged that the names will usually show the pupil how the operation is to be formed. In this section, although the pupil is taught to diper

If

* As soon as the terms applied to fractions are fully comprehended, the operations on them are as simple as those on whole numbers.

vide numbers into various parts, he is not taught to notice any fractions, except those where the numbers are divided into their simple units, which is the most simple kind.

It will be best to use beans, pebbles, &c. first; and then plate I.

4. Show the pupil one of the rectangles in the second row, and explain to him that one is 1 is 1 balf of 2.

7. In the second row count 3 units; it will take all the marks in the first, and 1 in the second rectangle. Consequently it is 1 time 2, and 1 half of another 2.

15. In the second tow count 9. It will take all the marks in the four first rectangles, and 1 in the fifth. Therefore 9 is 4 times 2 and 1 half of another 2.

18. Show the pupil a rectangle in the third row, and ask him the question, and explain to him that 1 is 1 third of 3. 20. Since 1 is 1 third of 3, 2 must be 2 thirds of 3.

34. In the third row count 11. It will take 3 rectangle and 2 marks in the fourth. Therefore 11 is 3 times 3, and 2 thirds of another 3.

Proceed in the same manner with the other divisions.

This being one of the most useful combinations, and one but very little understood by most people, especially when applied to large numbers, the pupil must be made perfec ly familiar with it. Ask questions like those in the book for large numbers, and also some like the following: Whit part of 7 is 18? the answer will be 18-7ths.

C. The ten first figures are here explained. They are used as an abridged method of writing numbers, and not with any reference to their use in calculating.

This article is only a continuation of the last. All the numbers from 1 to 100 are introduced into the two articles, and are divided by all the numbers from 1 to 10; except that some of the largest are not divided by some of the smallest.

2. The pupil answers first, how many times 2 is contain. ed in 12, then how many times 3?

45. 63 are how many times 5? In the fifth row count 63. It will take 12 rectangles and 3 marks in the 13th. It will be necessary to count once across the plate and begu again, and take 2 rectangles and a part of the third. 69 is 12 times 5 and 3 fifths of another 5.