THE COMMON SLIDING RULE.

48. The common or carpenter's sliding rule consists of two pieces, a foot long each, connected by a folding joint. It is used for computing the quantity of timber and the work of artificers.

Supposing the rule opened out, one side or face of it is divided into inches and eighths of an inch, with other scales of parts of an inch; and one half of the other side contains several tables of practical use. But the part of it used for performing arithmetical operations is one face of one of the pieces, in the middle of which is a narrow slip of brass, which slides in a groove.

49. On each of the two parts into which the stock is divided by the slider is a scale, and there are also two scales on the slider. The scales on the stock are named A and D, and those on the slider B and C, the scales A and B being contiguous, as are also C and D. The scales A, B, C, are exactly equal, and are just a scale of logarithmic numbers like that in article 177, Part I. The numbers on the scale D are the square roots of those opposite to them on the scale C.

50. Supposing the slider in its place, with 1 on its extremity, coinciding with 1 on the contiguous scales, and let the number d on Ď be opposite to the number c on C, then d2c; and were these numbers on scales of the same standard, then would 2 Ld = Lc; but Ld on the scale D is Lc on the scale C; and hence—

51. The logarithms of the numbers on the scale D are double the logarithms of the same numbers on the scale C.1

In finding any number on any of the scales which is the result of some operation, as of multiplication, division, &c., it is necessary to know previously how many places of figures the number will contain; but this is generally easily known.

52. PROBLEM I.-To find the product of two numbers. 'Set 1 on B to one of the numbers on A, then opposite to the other number on B is the product on A.'

EXAMPLE.-Multiply 24 by 25.

Set 1 on B opposite to 24 on A, then opposite to 25 on B is 600 on A.

For if a, b, and p, are the two numbers and their product,

b p

I a
Lp-La,

Lb LlLp

that is, the extent from 1 to b on the line B is equal to that from a top on the line A; and, therefore, if 1 on B is opposite to a on A, then 6 on B will be opposite to p on A.

53. PROBLEM II.—To divide one number by another. Set 1 on B opposite to the divisor on A, then against the dividend on A is the quotient on B.'

EXAMPLE.-Divide 800 by 32.

Set 1 on B opposite to 32 on A, then opposite to 800 on A is 25 on B, which is the quotient.

Let a, b, and q, be the divisor, dividend, and quotient,

that is, the distance from a to b on A is equal to that from 1 to q on B.

54. PROBLEM III.-To perform proportion.

'Set the first term on B to the second on A, then opposite to the third on B is the fourth term on A."

EXAMPLE. Find a fourth proportional to 20, 28, and 25. Set 20 on B opposite to 28 on A; then opposite to 25 on B is 35 on A, which is the fourth term required.

Let a, b, c, and d, be four terms of a proportion, then a:bc:d, or a:c= =b:d;

hence,

La-LcLb - Ld,

that is, the distance between the numbers a and c on one logarithmic line is equal to the distance between 6 and d on the same or on an equal line, as in article 177, Part I.

55. PROBLEM IV.-To find the square of a number. Set 1 on D to 1 on C, then opposite to the given number on D is its square on C.

EXAMPLE. Find the square of 15.

'Set 1 on D to 1 on C, then opposite to 15 on D is 225 on C.

The reason of the rule is evident from article 9.

The square of a number can also be found by Prob. I.

(52.) For it is just the product of the number by itself. Thus, 15215 × 15, and, by rule in 52, this product is 225.

56. PROBLEM V.-To find the square root of a given number.

'Set 1 on C to 1 on D, then opposite to the given number on C is its square root on D.'

EXAMPLE.-Find the square root of 256.

Set 1 on C to 1 on D, then opposite to 256 on C is 16 on D.

Since the numbers on C are the squares of those opposite to them on D, therefore, according as 1 on D is reckoned 1, 10, 100,... the 1 on C must be reckoned 1, 100, 10,000...

57. PROBLEM VI.-To find a mean proportional between two numbers.

'Set one of the numbers on C to the same on D, then opposite to the other number on C is the mean proportional on D.

EXAMPLE. Find a mean proportional between 9 and 16. Set 16 on C to 16 on D, and opposite to 9 on C will be 12 on D.

58. The mean proportional can also be found by first obtaining the product of the numbers by Prob. I., and then the square root of this product, found by Prob. V., will be the required mean proportional. Thus, if a and b are the given numbers, and a the mean proportional, article 142, Part I.

ab, by

The rule is proved thus:-Since a: x = x: b,

Therefore the distance between the numbers a and b on the line C, will be equal to the distance between a and x on the line D (article 51.)

MEASUREMENT OF TIMBER.

59. The measurement of timber is merely a particular application of the principles of the mensuration of surfaces and solids; but as approximate rules are sometimes adopted on account of their practical utility in measuring timber, it is necessary to treat this subject separately.

60. PROBLEM I.-To find the superficial content of a board or plank.

Multiply the length by the breadth, and the product is the area.

When the board tapers gradually, take half the sum of the two extreme breadths, or the breadth at the middle, for the mean breadth, and multiply it by the length.'

Let

the breadth in inches,

= length in feet,

and s = the superficial content in feet; then s = 11⁄2 bl.

By the sliding rule' Set the breadth in inches on B to 12 on A, and opposite to the length in feet on A will be the content on B in feet and decimal parts of a foot."

EXAMPLE.-How many square feet are contained in the surface of a plank 10 feet 6 inches long and 8 inches broad? a=bl=2× 101 = × 2=7 square feet. Or, set 8 on B to 12 on A, and opposite to 10.5 on A is 7 on B.

The first rule depends on article 250, Part I.

61. The reason of the method by the sliding rule is this:The area or surface abl, b and being the breadth and

b

length in feet. When 6 is given in inches, then a = 12', or bl 12 a, which is convertible into the proportion 12:61: a, and 12, 6 and 7 being given, a is found by Prob. III.

EXERCISES.

1. Find the content of a board 18 inches broad and 16 feet 3 inches long. Ans. 24 feet 54 inches. 2. What is the price of a plank, the length of which is 12 feet 6 inches, and breadth 1 foot 10 inches, at 11d. per square foot?

Ans. 2s. 10d. 3. Find the price of a plank, the length of which is 17 feet, and breadth 1 foot 3 inches, at 21d. a square foot.

Ans. 4s. 5d. 4. What is the superficial content of a board 29 feet long and 22 inches broad? Ans. 531 feet.

5. The length of each of five oaken planks is 17 feet, two of them have the mean breadth of 131 inches, one is 14 inches at the middle, and the remaining two are each 18 inches at the broader end, and 11 inches at the other end; what is their price at 3d. per square foot?

Ans. £1, 5s. 91d.

62. PROBLEM II.-To find the cubic content of squared timber of constant breadth and thickness.

'Find the continued product of the length, breadth, and thickness, and the result is the content.'

Let b, t, l, and v, be the breadth, thickness, length, and volume or solidity, then

v=btl.

By the sliding rule- Find the mean proportional between the breadth and thickness in inches (57), then set the length on C to 12 on D, and opposite to the mean proportional on D is the content on C in feet.'

When the timber is square, the mean proportional is just the side of the square. When the mean proportional is in feet, 1 on D is to be used instead of 12.