preferable, if the pupil be very young, not only by the examples in the first part of this section, but by the first examples in all the sections.

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

B & C. The articles B and C contain the common addition table as far as the first 10 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.

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 man

ner.

18. 57 and 6 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.

* Figures are used in the key, because the instructor is supposed to be ac quainted 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.

SECTION II.

THIS section contains multiplication. The pupil will see no difference between this and addition. It is best that he 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. 11. A man will travel 4 times as far in 4 hours as he will in 1 hour.

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 first ten numbers. The pupils should find the answers once or twice through, until he can find them readily, and then let him commit them to memory.

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 TOW. 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.

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 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 hours, &c.

SECTION III.

A. THIS section contains division. The pupil will scarcely distinguish it from multiplication. It is not important that he should at first.

The pupil will be able to answer these questions by the multiplication table, if he has committed it to memory thoroughly.

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. If 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

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

names will usually show the pupil how the operation is to be performed. In this section, although the pupil is taught to divide 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.

20. Since 1 is 1 third of 3, 2 must be 2 thirds of 3.

34. Illustrate by grouping the marks or counters by threes.

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 inade perfectly familiar with it. Ask questions like those in the book for large numbers, and also some like the following: What part of 7 is 18? the answer will be 18.

C. The first ten 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 contained in 12, then how many times 3.

D. These examples, which are similar to those in article A of this section, are solved in the same manner.

5. It would take as many hours, as 3 miles are contained in 10 miles. 3 hours and of an hour.

20. They cost as many cents as there are 3 apples in 30 apples; that is, 10 cents.

21. 12 dollars a month: and 12 dollars a month is 3 dollars a week; that is, 18 shillings a week, which is 3 shillings a day.

26. The whole loss was 35 dollars, which was 7 dollars apiece

SECTION IV.

A. THIS article contains multiplication simply. It is repeating a number a certain number of times and a part of another time.

14. 6 times 5 are 30, and g of 5 are 3, which added to 30 make 33.

B. In this article the pupil is taught to change a certain number of twos into threes, threes into fives, &c. This article combines all the preceding operations.

24. 4 cords of wood will cost 28 dollars, and of a cord will cost 2 dollars, which makes 30 dollars. 30 dollars will buy 3 hundred weight of sugar and § of another hundred weight.

29. 7 times 8 are 56, and § of 8 are 5, which added to 56 make 61; 61 are 6 times 9, and of 9.

C. 1. 4 bushels of apples, at 3 shillings a bushel, come to 12 shillings; and 12 shillings are 2 dollars.

2. The two lemons come to 8 cents, and 8 cents will buy 4 apples, at 2 cents a piece.

This is usually called Barter. The general principle is to find what the article will come to, whose price and quantity are given, and then to find how much of the other article that money will buy.

6. If 2 apples cost 4 cents, 1 will cost 2 cents, and 4 will cost 8 cents. Or 4 apples will cost 2 times as much as 2 apples.

22. Find how many times 2 pears are contained in 20 pears, which is 10 times. 10 times 3 cents are 30 cents. Or, first find what 20 pears would come to at 3 cents apiece; and since it is 2 for 3 cents, instead of 1 for 3 cents, the price will be half as much.

23. See how many times you can have 5 cents in 30 cents, and you can buy so many times 3 eggs. 30 is 6 times 5, and 6 times 3 are 18. 18 eggs.

24. 10 dollars a week, and 40 dollars a month.

25. 5 dollars are 30 shillings, which is 10 shillings a day. 6 shillings is equal to one dollar in 14 of the 29 States of the Union.

26. 5 dollars apiece.