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What number is that, which, being increased by its i, its , and 5 more, will be doubled ? Suppose it is 16

Suppose it is 12
One half is 8

One half is 6
One quarter is 4

One quarter is 3
5

5

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A labourer was hired for 60 days upon this condition, that for every day he wrought, he should receive 75 cents; and for every day he was idle he should forfeit 37} cents; at the expiration of the time he received $18; how many days did he work, and how many was he idle ?

Ans. He wrought 36 days, and was idle 24.

Questions.
What is position ?
What is single position ?
What is the rule for working single position ?
Why does this give the true answer?
What is double position ?
What is the rule for double position ?
On what principle is this rule founded ?

Examples for Practice.

1. A man lent his friend a sum of money unknown, to receive interest for the same at 6 per cent per annam, simple interest, and at the end of 12 years, received for prin. cipal and interest $860; what was the sum lent? Ans.$500.

2. A. B. C. and D. spent $18, at a reckoning, and being a little dipped, they agreed that A. should pay s, B. , C. Į, and D. 1; what did each pay in the above proportion ?

Ans. $6,8571? A's share..

5,1421B's share. 3,4284 C's share.

2,57144 D's share. Proof, $18,000

3. A person, having spent į and of his money, had $180 left; what had he at first?

Ans. $1080. 4. A vessel has 3 cocks, A. B. and C.; A. can empty it in į an hour, B. in $ of an hour, and C. in of an hour; in what time will they all empty it together?

Ans. į of an hour. 5. What number is that, which, being increased by , į, and of itself, the sum shall be 125 ? Ans. 60.

6. Seven eighths of a certain number exceeds four fifths by 6 ; what is that number?

Ans. 80. 7. A gentleman has two horses of considerable value, and a carriage worth $400, now is the first horse be harnessed in it, he and the carriage together will be triple the value of the second ; but if the second be put in, they will be 7 times the value of the first; what is the value of each horse ?

Ans. One, $80, the other $160. 8. Suppose the sea serpent's head is 9 inches long, and his tail is as long as his head, and half the length of his body, and his body is as long as his head and tail; what is the whole length of the monster?

Ans. 6 ft. 9. A farmer having driven his cattle to market receive ed for them all $320, being paid at the rate of $24 per 05, $16 per cow, and 86 per calf; there were as many oxen as cows, and 4 times as many calves as cows ; how many were there of each sort ?

Ans. 5 oxen, 5 cows, and 20 calves. 10. A. and B. laid out equal sums of money in trade; A. gained a sum equal to 4 of his stock, and B. lost $225 ; then A.'s money was double that of B.'s; what did each one lay out?

Ans. 8600

11. A B. and C. built a ship which cost them $5000, of which A. paid a certain sum, B. paid $500 more than A., and C. $500 more than both ; having finished her, they fixed her for sea, with a cargo worth twice the vaive of the ship. The outfits and charges of the voyage, amoonted to f of the ship ; upon the return of which, thei found their clear gain to be sof of the vessel, carge and expenses; what did the ship cost them, severally ; what share had each in her; and what, upon the final adjustment of their accompts, had they severally gained ?

Ans. A. owned of the ship, which cost him $875, and his share of the gain was $1093,75. B. owned which cost him $1375, and his gain was $1718,75. C.

he owned it, which cost $2750, and his gain was $3437,50.

CONVERSATION XI.

INVOLUTION AND EVOLUTION.

Tut. What is involution ?

Pup. I do not know, unless it is multiplying a number by itself.

Tut. You have a correct idea of it. Involution is the multiplying any given number into itself continually, a certain number of times. The numbers which are produced in this way are called powers of the given number. Thus, if 3 is the given number, 3 will be the first power.

3X3 = 9, the second power or square of 3 = 32. 3X3X3 = 27, the third power or cube of 3 = 33. 3X3X3X3 = 81, the 4th power or biquadrate of 3 = 34.

The small figure placed at the right of the given number, shows the order of the power, and is called the index, or the exponent.

To find any power, multiply the given number or first power, continually by itself, till the number of multiplications be 1 less than the index or exponent of the power to be found, and the last product will be the power required.

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What is evolution ?

Pup. Evolution is, I think, the reverse of involution, or it is the finding of the number from which the power was raised.

Tut. Evolution is the extraction of roots. The root of any given number or power, is the number whose continual multiplication into itself produces the power, and is denominated the square, cube, biquadrate, or 2d, 3d, 4th root, &c accordingly as it is, when raised to the seco ond, third, &c. power, equal to that power. Thus 3 is the square root of 9, because 3 x 3 = 9, and 3 is the cube root of 27, because 3 X 3 X 3 = 27, and so on with the other roots.

There is no number, of which we cannot find any power exactly ; but there are many numbers, of which the exact roots can never be obtained. Yet by the aid of deci. mals we can obtain these roots to any necessary degree of exactness. Those roots which cannot be exactly obtained, are called surd or irrational roots, in distinction from those which can be exactly obtained, which are called rational roots.

M

The

square root is denoted by this character , with the index of the root against it, except when the square root is denoted, the character itself denoting the square root. Thus the square root of 24 is expressed 24; the cuhe root of 36 is ✓3 36, and so on.

The second, third, fourth, and fifth powers of the nine digits are expressed in the following

4

8

9 81

Roots, Squares, Cubes, Biquadrates, Sursolids,

TABLE.
I or Ist Powers
1112131 1 S 1

7 1 1 1
| or 2d Powers, 1114 | 9 | 16 | 25 | 36 | 49 | 64 1
or 3d Powers, 11| 8 27 | 64 | 125 | 2:8 | 343 | 512 | 729 |
or 4th Powers, 11 | 16 | 81 | 256 | 625 | 1296 | 2401 | 4096 | 0561 |
| or sth Powers, 11 | 32 | 243 | 1024 3125 1 7776 | 16807132768 | 590491

TO EXTRACT THE SQUARE ROOT.

The extraction of the square root of any number, is the finding another number, which, multiplied into itself, will produce the given number.

Rule.

“ Distinguish the given number into periods of two figures each, by putting a point over the place of units, another over the place of hundreds, and so on, which points show the number of figures the root will consist of.

Find the greatest square number in the first or left hand pe. riod, place the root of it at the right hand of the given number, (after the manner of a quotient in division) for the first figure of the root, and the square number, under the period, and subtract it therefrom, and to the remainder bring down the next period for a dividend.

6 Place the double of the root already found, on the left hand of the dividend for a divisor. Seek how often the divisor is contained in the dividend, (except the right hand figure) and place the answer in the root for the second figure of it, and likewise on the right hand of the divisor ; multiply the divisor with the figure last annexed, by the figure last placed in the root, and subtract the product from the dividend; to the remainder, join the next period for a new dividend.

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