8. Two men, A and B, lay out some money on speculation. A disposes of his bargain for £11, and gains as much per cent. as B lays out; B's gain is £36, and it appears that A gains four times as much per cent. as B. Required the capital of

each.

9. There is a rectangular field containing 360 square rods, and whose length is to its breadth as 8 to 5. Required the length and breadth.

10. There are two square fields, the larger of which contains 13941 square rods more than the smaller, and the proportion of their sides is as 15 to 8. Required the sides.

11. There is a rectangular room, the sum of whose length and breadth is to their difference as 8 to 1; if the room were a square whose side is equal to the length, it would contain 128 square feet more than it would, if it were only equal to the breadth. Required the length and breadth of the room.

12. There is a rectangular field, whose length is to its breadth in the proportion of 6 to 5. A part of this, equal to of the whole, being planted, there remain for ploughing 625 square yards. What are the dimensions of the field?

13. A charitable person distributed a certain sum amongst some poor men and women, the number of whom were in the proportion of 4 to 5. Each man received one third as many shillings as there were persons relieved; and each woman received twice as many shillings as there were women more than men. The men received all together 18s. more than the women. How many were there of each?

14. A man purchased a field whose length was to the breadth as 8 to 5. The number of dollars paid per acre was equal to the number of rods in the length of the field; and the number of dollars given for the whole, was equal to 13 times the number of rods round the field. Required the length and breadth of the field.

15. There is a stack of hay whose length is to its breadth as 5 to 4, and whose height is to its breadth as 7 to 8. It is worth as many cents per cubic foot as it is feet in breadth; and the whole is worth, at that rate, 224 times as many cents as there are square feet on the bottom. Required the dimensions of the stack.

16. There is a field containing 108 square rods, and the sum of the length and breadth is equal to twice the difference. Required the length and breadth.

17. There are two numbers whose product is 144, and the quotient of the greater by the less is 16. What are the numbers?

XXXI. Questions producing Pure Equations of the Third
Degree.

1. A number of boys set out to rob an orchard, each carrying as many bags as there were boys in all, and each bag capable of containing 8 times as many apples as there were boys. They filled their bags, and found the whole number of apples was 1000. How many boys were there?

In this equation, the unknown quantity is raised to the third power; and on this account is called an equation of the third degree.

In order to find the value of x in this equation, it is necessary to find what number multiplied twice by itself will make 125. By a few trials we find that 5 is the number; for

therefore

5 X 5 X 5 = 125
x = 5.

Ans. 5 boys.

2. Some gentlemen made an excursion; and every one took the same sum of money. Each gentleman had as many servants attending him as there were gentlemen; and the number of dollars which each had, was double the number of all

the servants; and the whole sum of money taken out was $1458. How many gentlemen were there?

Ans. 9 gentlemen.

3. A poulterer bought a certain number of fowls. The first year each fowl had a number of chickens equal to the original number of fowls. He then sold the old ones. The next year

each of the young ones had a number of chickens equal to once and one half the number which he first bought. The whole number of chickens the second year was 768. What was the number of fowls purchased at first?

It appears that in equations of the third degree, as in those of the second degree, the power of the unknown quantity must first be separated from the known quantities, and made to stand alone in one member of the equation, by the same rules as the unknown quantity itself is separated in simple equations. When this is done, the first power or the root must be found, and the work is finished.

Extraction of the Third Root.

The third power of a quantity is easily found by multiplication, but to return from the power to the root, is not so easy. It must be done by trial, in a manner analogous to that employed for the root of the second power.

We shall hereafter have occasion to speak of the root of the fourth power, of the fifth power, &c. In order to distinguish them the more readily, we shall call the root of the second power, the second root of the quantity; that of the third power, the third root, that of the fourth power, the fourth root, &c. To preserve the analogy, we shall sometimes call the root of the first power, the first root.

N. B. The first power, and the first root, are the same thing, and the same as the quantity itself.

It always has been, and is still the practice of mathematicians, to call the second root the square root, and the third root the cube root, and sometimes, though not so universally, the fourth root the bi-quadrate root. But as these terms are unappropriate, they will not be used in this treatise.

When the root consists of but one figure, it must be found by trial. When the root consists of more than one place, it

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