9. Divide 12 into two such parts, that their product shall be equal to eight times their difference. What are the parts?

10. Divide the number 10 into two such parts, that the second power of four times the less part may be 112 more than the second power of twice the greater part?

11. Divide the number 16 into two such parts, that the quotient of the first divided by the second, added to the quotient of the second divided by the first, may be equal to 2.

12. Find two numbers, such that their difference multiplied by their sum, gives 64; and that the square of the less may be 6 more than three times the greater.

13. A man sold a piece of cloth for six times as many cents per yard as there were yards in the piece. If he had sold it for 4 cents more per yard, he would have received for it $2.40. How many yards were there?

14. Says A to B, how many shillings have you? I have so many, says B, that if I multiply the number by itself, and then increase the product by 6 times the number, divide the sum by 4, and add 8 to the quotient, the result will be 36. What is the number?

SECTION XL.

Questions Respecting Squares and Rectangles.

A square whose side is one inch long, is called a square inch. A square whose side is one foot long, is called a square foot, &c.

1. How many square inches are there in a square whose side is 4 inches long?

FIG. 1.

It will be seen, by supposing the side of figure 1. to be 4 inches long, that there are four rows of squares, and 4 square inches in each row. Hence there are 16 square inches in the figure. And, in general, that the second power of the number of units in one side of a square, will give the number of square units in the square.

2. One side of a square is 355 feet; how many square feet does it contain?

3. One side of a square is 400 feet; how many square feet does it contain?

Figure 2. is a rectangle, 6 inches long and 4 inches wide. How many square inches does it contain?

FIG. 2.

We have in figure 2nd 4 rows of squares, and 6 square inches in each row. Hence there are 24 square inches in the rectangle. And, in general, the length of a rectangle multiplied by its breadth, gives the number of square units in its

area.

4. What are the sides of squares of the following areas: 442225? 458329? 488601? 636804?

5. There are two squares, the difference of their sides is 2 inches, and the sum of the square inches in both is 52. What are the lengths of the sides?

6. There are two squares, the difference of whose sides is d yards, and the content a, in square yards. What will represent the sides of each?

7. How do you find the sides of two squares, when you

H

know the difference of the sides, and the sum of the areas of the two squares?

8. The sum of the two sides of two squares is 17, and the sum of their areas is 149. What are the sides of the two squares?

9. The sum of the sides of two squares is s, and the sum of their areas is a. What will represent the sides of the two squares?

10. What is the rule for finding the sides of two squares, when there are given the sum of the sides, and the sum of the areas of the two squares?

11. The difference of the sides of two squares is 5, and the difference of their areas 125. What are the lengths of the sides?

12. In the 11th question, if d were the difference of the sides, and b the difference of the areas; what would represent the sides?

13. What is the rule for finding the sides of two squares, when their difference and the difference of their areas are known?

14. There is a rectangular field, whose length exceeds its breadth by 6 rods, and whose content is just one acre. are the length and breadth of the field?

What

15. If a be the area of a rectangle, and d the difference of the sides, what will represent the lengths of the two sides? and what is the rule for finding the lengths of the sides of a rectangle, when its area and the difference of its sides are given?

16. In a rectangle, the sum of the length and breadth is 80, and its area is 1200. What is its length and breadth?

17. The sum of the length and breadth of a rectangle is «, and its area b. What will represent its sides? And what is the rule for finding the sides?

18. The price of fencing a square piece of land at four dollars per rod, is only 80 dollars less than the price of the content at 160 dollars per acre. What is the side of the square, and what is its content?

19. There is a rectangular field, the length of which exceeds its breadth by 4 rods. The price of the land, at 2 dollars per

square rod, was 312 dollars more than the price of fencing it at 3 dollars per rod. What is the length of the field?

20. There is a looking-glass which has a frame, all parts of which are of equal width. The glass is 18 inches long by 12 wide; and the area of the frame is equal to the area of the glass. What is the width of the frame?

21. There is a rectangular room, the length of which exceeds its breadth by 3 feet. It is 6 feet from the floor to the ceiling. The floor is covered with carpet at one dollar per yard, and the sides with paper at 10 cents per yard. The whole cost of which is $34.40. What are the length and breadth of the room?

SECTION XLI.

Discussion of Affected Equations.

1. If x2+px=q; what will represent the values of x?

Ans. (1) 2 = −1 + (2 + 2) +

x

212 210

2

p2

Here observe, that in both values of x, and q

4

are repre

sented as added together, and the root of the sum taken; and it can be easily seen, that this root will always be greater than

and the root taken, the result will be greater

4'

than

The first value of x is

2

P 2

subtracted from this root;

and as this is subtracting a less from a greater, the remainder will always be plus some number. The second value is minus

Р

2

; and minus this root. The sum is, therefore, minus some number; consequently, if the equation be of this form x2+px =p, one value of x will be positive, and the other negative.

2. If x2-px=q; what will represent the values of x?

Ρ

Ans. (1) x= · (2 + 2) 34.

2

Will one of these values always be plus, and the other minus?

3. If x2-px-q; what will represent the values of x?

Ans. (1) x = 2 — (22 —9)*.

2

It can be easily seen, that there are some cases in which it will be impossible to obtain the value of x. For, we have to

p2

subtract q from and extract the root of the remainder.

If

q should be greater than this remainder would be minus

4

some number; but a minus number has no root. When, there

p2

fore, q is greater than it will be impossible to get the

4

value of x. The value, then, is said to be imaginary. It is evident, that if one value of x is imaginary, the other will be imaginary also; for, this root has to be taken, in order to get either value of x.

Secondly. If the values are not imaginary; that is, if զ is not greater than then both values of x are plus or positive

p2

4

For, this root added to gives one value, and subtracted

p

from gives the other. The first is evidently a positive num

2

唱

the

ber, and as this root, as may be easily seen, is less than

second must, therefore, be positive also.

When the values of x are imaginary, it shows that any question from which the equation could originate, must be ab