H 3M. }8,91. {100925da.}
M.

19,5da. S

answer £.305 0 8,2

2 If 2 persons receive 4,625s. for 1 day's labour, how much should 4 persons have for the work of 10,5 days? answer 4 17s id. 3 If 16s 4d. be the porterage of 5,25 hundred weight for 20 miles; what must be paid for carrying 17,75 hundred weight 7,5 miles? answer il os 8d. 4 How many men should reap 417,6 acres in 12 days, when 5 persons cut down of that quantity in half the time?

answer 20 men.

5 Suppose the interest of 76,941. for 9,5 months to be 15,251. what principal will gain 6. in 12,75 months? answer 221 11s id.4 6 When 12 oxen graze down 16,25 acres, in 20 days; how much, of like pasture, would suffice 24 such cattle for 100 days? answer 162,5 acres. 7. What money, at 3 per cent. per annum, will clear 381 10s. in a year and a quarter?

answer 880 8 A cellar which is 22,5ft. long. 17,3 wide, and 10,25 deep, being dug in 2 days, by six men, working 12,3 hours a day; how many days, of 8,2 hours, should 9 men take to excavate one which measures 45, by 34,6 by 12,3 feet? answer 12 days.

INVOLUTION; OR THE RAISING OF POWERS.

POWER is the product arising from multiplying any

of times; thus,

continually a certain number

2X2= 4 is the second power, or square of 2. 2X2X2 8 the third power, or cube of 2. 2X2X2X2=16 the fourth power of 2, &c.

The number denoting the power is called the index, or the exponent of that power.

If two or more powers are multiplied together, their pro duct is that power whose index is the sum of the exponents of the factors; thus,

2X24 the square of 2; 4X416=4th power of 2; and 16x16-256-8th power of 2, &c. N

TABLE

16384]

65536 262144

78125

390625

390625 1953125

416 64 256 1024 4096 525 125 625 3125 15625 6362161296 7776 46656 279936 1679616 10077696 749 343/240116807 117649 823543 5764801 40353607 8645124096 32768 262144 209715216777216 134217728 981 729 656159049 53144114782969|43046721/387420489

3

EXAMPLES.

What is the fifth power of 7?
7X7×7×7×7=16807-fifth power.
What is the third power or cube of 35?

What is the fourth power of ?
4 What is the fifth power of ,029?
What is the sixth power of 5,03?

,000000020511149.

answer 10190,005304479729. 1785 7 3 0 0 6 23

6 What is the eighth power of 3}?

366816

EVOLUTION: OR THE EXTRACTING
OF ROOTS.

T

HE root of any number, or power, is such a number, as, being multiplied into itself a certain number of times, will produce that power. Thus 2 is the square because 2 X 24; and 4 is the cube root of 64, because 4X4X4=64, and so on.

root of

4,

T

!

THE SQUARE ROOT.
HE square of a number is the product arising from

that number multiplied into itself. Extraction of the square root is the finding of such a number, as, being multiplied by itself, will produce the number proposed.

RULE. 1. Distinguish the given number into periods of two figures each, beginning at the units place, or decimal point, and when the decimal does not consist of an even number of figures, annex a cypher; and equal to the periods of whole numbers and decimals respectively will be the places of each in the root.

2. Dedu&t from the first period the greatest square it contains, setting the root thereof as a quotient figure, and doubling it for a divisor ; and bring down the second period to the remainder, for a dividual.

3. Try how often the said divisor, with the resulting figure of this trial thereto annexed, are contained in the dividual, and set this resulting figure to both the divisor and root ; then multiply and subtract as in division, and bring down the next period.

4. Double the ascertained root for a new divisor, and repeat the process to the end.

PROOF. Square the root, adding in the remainder (if any) which. will equal the number given.

E X A M P L E S. 1 What is the square root of 30138,696025 ?

a

30138,696025(173,605 answer..

1
27) 201

189
343)1238

1029 3466120969

20796 367205)1736025

1736025

Note. When one more than half the figures of the root are found, the reft may be obtained by working as in contracted division of decimals

2 Required the square-root of 14876,2357?

13 If 484 trees be planted in a square orchard, how. many must be in a row each way?

answer 22

Note 1. The fquare of the longeft fide of a right angled triangle is equal to the fum of the fquares of the other two fides; and confequently the difference of the fquare of the longeft, and either of the other, is the fquare of the remaining fide.

2. The fquare root of a vulgar fraction is found by reducing it to its lowest terms, and extracting the root of the numerator, for a numerator, and of the denominator, for a denominator. If it be a furd, reduce it to its equivalent decimal, &c.

3. A mixt number may be reduced to an improper fraction, or a decimal, and the root thereof extracted as before.

14 The wall of a fortress is 17 feet high, which is surrounded by a moat 20 feet in breadth; query the length of a ladder to reach from the outside of the moat to the top of the wall? answer 26,2 feet. 15 A line of 36 yards long will exactly reach from the top of a fort to the opposite bank of a river, known to be 24 yards broad; the height of the wall is required?

answer 26,83+yards." 16 Suppose a ladder 60 feet long be so planted as to reach a window 37 feet from the ground on one side of the street, and without moving it at the foot, will reach a window 23 feet high on the other side; what breadth was the street of?

17 What is the square-root of 2044 18 What is the square-root of

19 What is the square root of

20 What is the

21 What is the

square root of 37

square-root of 17

22. What is the square-root of 76

THE

answer 103,64 feet.

THE CUBE ROOT.

answer

71528

63

4+

8,7649+

HE Cube of a number is the product of that number multiplied into its square.

Extraction of the cube root is the finding of such a num ber, as, being multiplied into its square, will produce the number proposed.

RULE.

First, Distinguish the proposed number into periods of three figures each, beginning at the units place, or decimal

N 2