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Jīns. $16. Instead of making w = the number of dollars, we might make,
* = the number of dollars;
then x = the number of men resident, &c.
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
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 poighing 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 wo— men. How many were there of each f
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 ió. 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 3
Let a = the number of boys ;
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 a 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. Jins. 5 boys. 2. Some gentlemen made an excursion ; and every one took the same sum of money. Each gentleman had as many seryants 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 f ./lns. 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.
Evtraction 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 roof, 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 must still be found by trial, but rules may be made, which will reduce the number of trials to very few, as has been done above for the second root. * - a In order to find the rules for extracting the third root, it will be necessary to observe how the third power is formed from the first, when the first consists of several figures.
Hence it appears, that the third power of a number consisting of units and tens, contains the third power of the tens, plus three times the second power of the tens multiplied by the units, plus three times the tens multiplied by the second power of the umts, plus the third power of the units. Farther, the third power of 10, which is the smallest number with two places, is 1000, which consists of four places; and the third power of 100, is 1000000, which consists of seven places. Hence the third power of tens will never be less than 1000, nor so much as 1000000. * If therefore, there are tens in the root. their power will not be found below the fourth place; and if the root consists of tens without units, there will be no significant figure below 1000. To trace back again the number 42875, the root of the tens will be found in the 42000, and this must be found by trial.
30 x 30 x 30 = 27000, and 40 × 40 × 40 = 64000.
The largest third power in 42000 is 27000, the root of which is 30. Now I subtract 27000 from 42875, and the remainder is 15875, which contains the product of three times the second power of the tens by the units, plus, &c. If it contained exactly three times the second power of the tens multiplied by the units, the units of the root would be found immediately by