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Let x = the number in front;
then x + 5 = the number in depth;
** + 5x = the whole number of men.
Again x + 845 = the number in front after the movement;
And 5 x + 4225 = the whole number.

za + 5 x = 5x + 4225
22 = 4225

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The number of men = 5x + 4225 = 4550.

3. A piece of land containing 160 square rods, is called an acre of land. If it were square, what would be the length of one of its sides? Let x = one side.

2 = 160

x = 12649 + Ans. The side is 12.649 + rods. It cannot be found exactly, because 160 is not an exact 2d

power. This is exact within less than oto of a rod. It might be carried to a greater degree of exactness if necessary.

4. What is the side of a square field, containing 17 acres ?

5. There is a field 144 rods long and 81 rods wide; what would be the side of a square field, whose content is the same?

6. A man wishes to make a cistern that shall contain 100 gallons, or 23100 cubic inches, the bottom of which shall be square, and the height 3 feet. What must be the length of one side of the bottom ? 4. A certain sum of money was divided every

week

among the resident members of a corporation. It happened one week that the number resident was the root of the number of dollars to be divided. Two men however coming into residence the week after, diminished the dividend of each of the former individuals 1 dollars. What was the sum to be divided ?

Let x = the number of dollars to be divided ;

then x

1

= the number of men resident, and also the sum each received.

The root of x is properly expressed by the fractional index 1. For it has been observed, that when the same letter is found in two quantities which are to be multiplied together, the multiplication is performed, as respects that letter, by adding the exponents. Thus a X a = a +1 = a; 2 x 20 = 22 +: = x, &c. Applying the same rule; ", if

represents a root or first power, the second power or x2 x , mit

3+ =X or x.

The second power of a letter is formed from the first by multiplying its exponent by 2, because that is the same as adding the exponent to itself. Thus a xa = a += a' X: =a'. This furnishes us with a simple rule to find the root of a literal quantity; which is, to divide its exponent by 2. Thus the root of a' is a = a'; the root of a* = ai

== a*; the root of a® is až = a?, &c. By the same rule, the root of a' is at; the root of « is ał ; the root of a is aš, the root of a is až, &c. In the above example x = the number of dollars to be divided; 1

= the number of men resident;

and

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+2 = the number of men the succeeding week;

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= the number of dollars each received the latter week;

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Hence by the conditions

4

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Instead of making x = the number of dollars, we might
make,

= the number of dollars ;
then x = the number of men resident, &c.
Then we have

x2
3 x + 2

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X

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12.

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 3 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.

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

The men received all together 18s. more than the wo-
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 päid 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.

men. men.

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

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?

Let x = the number of boys;
then x x x = x = the number of bags;
and 8x x x = 8.30 = the number of apples.
By the conditions

8 202 =1000

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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

5 X 5 X 5 = 125 therefore

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

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