BARTER.

BARTER is the exchanging of one kind of goods for another, in such a way that the value of the goods given away, may be equal to the value of those received.

No general rule can be given for the working of such questions; they must be treated according to the nature of each case. The following will serve as examples:

Example 1.-A and B barter as follows: A has 1385 yards of linen, at 2s. 74d. per yard, for which B gives him £32, 7s. 6d. ready-money, and for the rest printed calicoes at 103d. per yard. How many yards of calico did A receive?

A gives 1385 yards of linen, at 2s. 7 d.
B gives in money,

"

£32

7 6

calicoes at 10 d. a yard, 149 8 1

£181 15 7

181 15 7

This sum of £149, 8s. 14d. divided by 103d., the price per yard, will give 3415 yards, the number required.

Example 2.-A merchant in Hull exported to his correspondent in Oporto 35 pieces of camlet, each 28 yards, at 5s. 21d. per yard, and 80 pieces of serge, each 35 yards, at 3s. 34d. per yard. The correspondent was allowed 24 per cent. commission on the gross amount, and was directed to expend the of the net proceeds on port wine at £48, 16s. 3d. per pipe, and the remainder on raisins at £2, 5s. 74d. per cwt. What quantities of each were imported?

of proceeds = value of the wine, £260 14 817 Remainder = value of the raisins, 434 11 2

Exercises.

1. How much coffee at £7, 9s. 6d. per cwt. should I get in exchange for 897 cwt. 42 lbs. of sugar at 62d. per lb.? Ans. 378 cwt.

2. How much tobacco at £5, 5s. per cwt. must be bartered for 6 cwt. 1 qr. 14 lb. of snuff at 4s. 6d. per lb.? Ans. 30 cwt. 2 qr. 114 lb. 3. I exchanged 172 yards of black cloth at £1, 2s. 8d. per yard, for 688 pair of silk stockings; what was the price of the stockings per pair? Ans. 5s. 8d.

4. A delivers to B 314 yards of cloth at 5s. 6d., and 78 yards of cassimer at 7s. Sd., in barter for wool at 1s. 3d. per lb.; what quantity of wool does A receive? Ans. 615 lb.

Example. Change the numbers (234), (5324),, and (1843)12

to the decimal scale.

3. Change the numbers (31476),, (8196), 2, (2503), to the decimal scale, Ans. (20805)10, (14082) 10, (615)10. 4. How are the numbers (111022)s, (607), represented in the decimal scale? Ans. (3887)10, (391)10

From what has been stated, it is obvious that, in performing Addition or Multiplication in any scale, the number resulting in any column must be divided by the base of the scale, the remainder set down, and the quotient carried to the next column.

N.B.-In Subtraction, when 1 is borrowed, it expresses as many units as there are units in the base in the place immediately to the right; and in Division, a remainder must be multiplied by the base, and the next figure taken in before the next division is performed. By attending to the preceding remarks, no difficulty will be found in performing the following exercises:

Exercises.

107

5. Change (372) 10, (5834)1 (7936)10, to the septenary scale, and find their sum in that scale, Ans. (56142),.

6. Change (39572)10, and (16935)10, to the senary scale, and find their difference in that scale, Ans. (252445)..

7. Change (18394)10, and (375)10, to the nonary scale, and find their product in that scale, Ans. (13872836),.

8. Change (314159)10, and (712)10, to the quinary scale, divide the greater by the less, and find the quotient in that scale, Ans. (32312) 5. 9. Change (76428)10, and (374),, to the duodecimal scale, and find their product in that scale, Ans. (7627060)12.

10. Change (47683)10, and (2130)4, to the senary scale, and divide the former by the latter, Ans. (122525).. 11. Change (54230), and (7213), to the quinary scale, and find their sum and difference,

Ans. Sum, (324112),; difference, (104321),.

INVOLUTION.

INVOLUTION is the term applied to the multiplication of a number one or more times by itself; thus, 2 × 2 = 4. The result is called a POWER of that number.

The first power of a number is the number itself before being multiplied; thus

The second power, termed the SQUARE, is the number multiplied by itself; thus

The third power, termed the CUBE, is the number multiplied by itself, and the product again multiplied by it; thus

2

2×2=4

2 X2 X2=8

And so on, with the higher powers.

The Power of a number is indicated by writing the number with a small figure above it, called the Index; thus, 22 means the second power of 2-namely, 2 x 2 = 4: 23 means the third power, 2 x 2 x 2 = 8: 21 means the fourth power; and so on.

THE PRODUCT OF TWO POWERS of the same number is equal to that number raised to the power denoted by the sum of their indices.

Take any two powers of the same number, as 5' and 53; then 54 X 53 = 54+3 = 5'. For 5' x 53= 5.5.5.5 X 5.5.5

= 5 X 5 X 5 × 5 × 5 × 5 × 5 = 51.

THE QUOTIENT OF ANY POWER of a number divided by a lower power of the same number, is equal to the number raised to the power denoted by the difference of their indices.

Take any two powers of 5, as 5' and 5, of which 5 is the

57

lower; then =57-4 = 53.

or 57-4.

54

57 54+3 54 X 53

For =
54 54

= 53

ANY POWER OF A COMPOSITE NUMBER is equal to the product of the same powers of its factors.

Take any power of the composite number 35, as 353; then 353 = 73× 53. For 353 35 × 35 X 35 = 7.5 X 7.5

=

X 7.5 = 7 X 7 X 7 × 5 × 5 × 5 = 73 × 53.

ANY POWER OF A POWER of a number is equal to the number raised to that power denoted by the product of the index of the power of the given number, by the index of that power to which the given power is to be raised.

Take any power of any power of the number 5, as (53)2; then (53)2= 53×2 = 56. For (53)2 = 53 × 53 = 5.5.5 × 5.5.5

TO RAISE A GIVEN NUMBER TO ANY POWER.

Find the continued product of the given number, repeated as often as the index points out.

The process may sometimes be shortened by multiplying together powers already found.

A Vulgar Fraction is raised to any power by raising its terms to the given power; thus, the fourth power of 2 is 2 × 2 × 2 × 2

Example 1.-What is the fifth power of 57 ?

Here, by multiplying 57 by 57, we get 3249 for the second power; and 3249, multiplied by 57, gives 185193 for the third power. Now the third power, 185193, multiplied by the second power, 3249, gives 601692057 for the fifth power.

Example 2.-What is the sixth power of?

Here the sixth power of 7 is 117649, and the sixth power of 12 is 2985984; therefore, 27844 is the answer required.

Example 3.-Required the fourth power of 1.035, true to five places of decimals.

9. What is the value of 1·0513 true to 6 places of decimals?

13. If the side of a square table measures 62 inches, how many square inch. are in the surface of the table? Ans. 3844 sq. inches. 14. There is a square court, whose side measures 102 feet; how many square bricks will be required to pave it, supposing the side of a brick to measure 9 inches ? Ans. 18496 bricks.

15. The side of a cubic block of granite measures 6 feet; how many solid feet in the block?-and what is its weight, supposing a cubic foot of granite to weigh 2654 oz.?

Ans. Solidity, 216 feet; weight, 319 cwt. 3 qr. 17 lb. 16. There is a cistern in the form of a cube, and its side measures 3 feet 8 inches; how many imperial gallons will the cistern hold? Ans. 307138637

EVOLUTION, OR THE EXTRACTION OF ROOTS. EVOLUTION is the process of finding or extracting the roots of numbers.

The Root of any number is that number which, on being multiplied one or more times by itself, produces the given number. The Square root is that number which, on being multiplied by itself, produces the given number; thus, 4 is the square root of 16, because 4 multiplied by 4 (4 × 4) produces 16.

The Cube root is that number which, on being multiplied by itself, and the product again multiplied by it, produces the given number; thus, is the cube root of 27, because 3 multiplied by 3, and the product again by 3 (3 × 3 × 3) produces 27.

The sign (termed the radical sign) placed before a number, indicates that the square root of that number is to be extracted; thus 25 5.

The sign placed before a number indicates that the cube root of that number is to be extracted; thus 729 = 9.

The extraction of the square and cube roots serves various useful purposes in connection with the measurement of fields, walls, solid bodies, &c., as will appear from the exercises under the rules for extracting the square and cube roots of numbers.

Note.-The square and cube roots of some numbers can be found exactly-of others, only approximately.