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may be done with advantage, as soon as the learner has advanced as far as the thirteenth section.

In the second part several subjects are introduced, usually found in elementary works on algebra, but which were reckoned too difficult for the learner without the exercises which the previous part of the work furnishes. There are also several subjects not usually found in works on algebra.

The chapters on the lever, the inclined plane, the weights of solids in fluids, and on falling bodies, will not be considered out of place. These subjects are important in themselves, and they also furnish a variety of important practical questions for the exercise of the learner. The subject of maxima and minima belongs, according to custom, to the calculus. It is here treated in a manner purely algebraic; and it is certainly not less interesting, nor more difficult, than many other subjects usually found in works on algebra.

b Algebra is now studied in colleges and academies. It should be studied in common schools. Boys of seven or eight years of

age, after they have understood the first lessons in arithmetic, will learn algebra, when the subject is properly presented to them, with great facility; and in the time usually employed in the study of arithmetic, they will acquire, not only a thorough knowledge of arithmetic, by means of studying algebra, but a thorough knowledge of a science unlimited in its applications to geometry, as well as to most of the physical sciences. Let the intelligent teacher who doubts make the experiment.

The author of this work is satisfied, from long experience in teaching, that the subject of mathematics is not usually presented in text books in the natural order in which it must be learned; so that the teacher is often compelled to adopt a mode of presenting the subject entirely different from that found in the text book, in order to bring the subject within the comprehension of the learner. To obviate this difficulty he has prepared the following work. How far he has succeeded is left for intelligent teachers to decide. Should nothing unforeseen prevent, the algebra will be succeeded by a series of works on the other branches of mathematics.

J. H. HARNEY.

INTRODUCTORY LESSONS

ALGEBRA.

O All the Questions in the Introductory Lessons are

be solved mentally.

SECTION I.

1. Two boys had 12 cents between them one had twice as many as the other. How many had each?

Suppose one had a certain part of 12, then the other had two such parts; and three such parts make 12. Hence one of the parts must be of 12, or 4; two such parts are 8—one boy had 4 cents and the other 8.

Now suppose we take x to represent, or stand for, what one boy had. Then 2 x will stand for what the other had; and 3 x will be equal to 12. If 3 x are equal to 12, 1 x will be equal to į of 12, or 4; and 2 x will be equal to 8.

2. Two boys had 30 marbles between them; and one had twice as many as the other. How many had each?

Let x represent the number one had, then 2 x will represent the number the other had, and 3x will represent the number both had ; then 3 x must be equal to 30. If 3 x are equal to 30, 1 x will be equal to į of 30, or 10, and 2 x equal to 20.

3. Two boys had 15 cents between them ; and one had twice as many as the other. How many cents had each ?

4. Two boys had 27 cents between them; and one had twice as many as the other. How many cents had each?

5. Two numbers make 39; and one is twice as much as the other. What are the numbers ?

6. A man paid a debt of 63 dollars, at two different payments. He paid twice as much at the first as he did at the second. How much did he pay each time?

7. Two numbers added together make 24, and one is three. times as much as the other. What are the numbers ?

8. Two men bought a horse for 36 dollars ; one paid three times as much as the other. What did each pay?

9. A man bought a cow and a sheep. for 16 dollars. He gave three times as much for the cow as he did for the sheep.. What did he give for each?

10. A man bought a saddle and bridle for 45 dollars ; the saddle came to four times as much as the bridle. How much did each cost ?

11. Three boys had 66 cents between them; the second had twice as many as the first, and the third three times as many as the first. How many had each?

12. There are three numbers which together make 54; the second is twice as much as the first, and the third three times, as much as the first. What are the numbers ?

13. There are three numbers which together make 72, the second is twice as much as the first, and the third is as much as both the others. What are the numbers ?

14. A man bought a horse, a cow and a calf, for 64 dollars ; the cow came to three times as much as the calf, and the horse came to as much as the cow and calf both... How much did. each come to ?

15. A man gave to four boys 45 apples; to the second three times as many as the first, to the third as many as to the first and second, and to the fourth as many as to the second and third. How many did he give to each?

16. There are four numbers whose sum is 40, the second is twice, the third three times, and the fourth four times as much as the first. What are the numbers ?

*17. A man gave 38 apples to five persons; to the second twice as many as to the first, to the third as many as to the first and second, to the fourth as many as to the second and third, and to the fifth as many as to the third and fourth. How many did he give to each ?

18. Two men had 42 dollars between them ; one had half as many as the other. How many had each?

19. Two numbers make 48, and one is į of the other What are the numbers?

20. A says to B, I have four times as many dollars as you .

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