The use of the double lines, A and B, is for working proportions, and finding the areas of plane figures; and the use of the girt-line D, and the other double line C, for finding the contents of solids.

When 1 at the beginning of any line is accounted 1, then the 1 in the middle will be 10, and the 10 at the end 100; and when 1 at the end is accounted 10, then the 1 in the middle is 100, and the 10 at the end 1000, &c.; and all the smaller divisions are altered in value accordingly.

Upon the girt-line are also marked W G at 17.15, and A G at 18.95, the wine and ale gauge-points, to make the instrument serve the purpose of a Gauging Rule.

On the other part of this face there is commonly either a table of the value of a load, or 50 cubic feet of timber, at all prices, from 6d. to 24d. per foot; or else several plane scales divided into twelfth parts, and marked 1, 2, 1, and , signifying that the inch, inch, &c. are each divided into 12 equal parts.

The edge of the rule is generally divided decimally, or into tenths; namely, each foot into 10 equal parts, and each of those into ten other equal parts. By this scale dimensions may be taken in feet,tenths, and hundredths of a foot, which is a very commodious method of taking dimensions, when the contents are to be cast up decimally.

THE USE OF THE SLIDING RULE.

PROBLEM I.

Multiplication by the sliding rule.

RULE.

Set 1 upon A, to the multiplier upon B; then against the multiplicand on A, will be found the product upon B.

Note. When the third term runs beyond the end of the line, seek it on the other radius or part of the line; and increase the product 10 times, or 100 times, &c. as the case requires.

EXAMPLES.

1. What is the product of 24 multiplied by 12?

As 1 on A: 12 on B :: 24 on A : 288 on B, the Ans. 2. Required the product of 36 multiplied by 18.

Ans. 648.

3. Multiply 12.8 by 6.5.

4. What is the product of 68 multiplied by 35?

PROBLEM II.

Ans. 83.2.

Ans. 2380.

Division by the sliding rule.

RULE.

Set the divisor on A, to 1 upon B; then against the dividend on A, is the quotient on B.

Note. When the dividend runs beyond the end of the line, diminish it by 10 or 100 times, in order to make it fall upon A; and increase the quotient in the same proportion.

EXAMPLES.

1. What is the quotient of 432 divided by 12?

As 12 on A: 1 on B:: 432 on A: 36 on B, the Ans. 2. What is the quotient of 9752 divided by 46?

Ans. 212. 3. If a board be 8 inches broad; what length must be sawn off, to make a foot square? Ans. 18 inches.

Set 1 upon D, to 1 upon C; then against the number upon D, will be found the square upon C.

EXAMPLES.

1. What is the square of 25?

As 1 on D: 1 on C :: 25 on D: 625 on C, the Ans.

2. Required the square of 36.

3. What is the square of 58?

PROBLEM IV.

Ans. 1296.

Ans. 3364.

To extract the square root of any number.

RULE.

Set 1 upon C, to 1 upon D; then against the number on C, is the root on D.

:

Note. In order to value this rightly, you must suppose the 1 on C to be some of these squares, 1, 100, 10000, &c. which is nearest to the given number and then the root corresponding will be the value of the 1 upon D.

EXAMPLES.

1. What is the square root of 144 ?

As 1 on C1 on D:: 144 on C: 12 on D, the Ans. 2. Required the square root of 900.

3. What is the square root of 9586?

PROBLEM V.

Ans. 30. Ans. 97.9.

To find a mean proportional between two numbers.

RULE.

Set one of the numbers on C, to the same on D; then against the other number on C, will be the mean on D.

EXAMPLES.

1. What is the mean proportional between 9 and 16? As 9 on C: 9 on D:: 16 on C: 12 on D, the Ans. 2. Required a mean proportional between 15 and 27. Ans. 20.1. 3. The segments of the hypothenuse of a right-angled triangle made by a perpendicular from the right-angle, are 18 and 32; what is the perpendicular? Ans. 24.

PROBLEM, VI.

To find a fourth proportional to three numbers; or to perform the Rule of Three.

RULE.

Set the first term on A, to the second on B; then against

the third term on A, stands the fourth on B.

Note. The finding of a third proportional is exactly the same; the second number being twice repeated.

Thus, suppose a third proportional was required to 80 and 60.

As 80 on A: 60 on B:: 60 on A: 45 on B, the third proportional sought.

EXAMPLES.

1. What is the fourth proportional to the three numbers, 12, 24, 36 ?

As 12 on A: 24 on B::36 on A: 72 on B, the Ans.

2. If 1 foot of timber cost 3s.; what will 180 feet cost? Ans. 540s.

3. If 50 feet of timber cost 77.; what will 2500 feet cost? Ans. £350.

PROBLEM VII.

To find the areas of plane figures by the sliding rule.

RULES.

1. To find the area of a rectangle or rhomboides. As 1 upon A, is to the perpendicular breadth upon B; so is the length upon A, to the area upon B.

2. To find the area of a triangle.

As 2 upon A, is to the perpendicular upon B; so is the base upon A, to the area upon B.

3. To find the area of a trapezium.

As 2 upon A, is to the sum of the two perpendiculars upon B; so is the diagonal upon A, to the area upon B. 4. To find the area of a regular polygon.

As 2 upon A, is to the perpendicular upon B; so is the sum of the sides upon A, to the area upon B.

5. To find the diameter and circumference of a circle, the one from the other.

As 7 upon A, is to 22 upon B; so is the diameter upon A, to the circumference upon B; and vice versâ.

6. To find the area of a circle.

As 4 upon A, is to the diameter upon B; so is the circumference upon A, to the area upon B.

Or, as 1 upon D, is to .7854 upon C; so is the diameter upon D, to the area upon C.

Note. In order to exemplify the foregoing Rules, the learner may work the questions in the respective Problems of Part II.

PROBLEM VIII.

To find the contents of solids by the sliding rule.

RULES.

1. To find the solidity of a cube.

As 1 upon D, is to the side upon C; so is the side upon

D, to the solidity upon C.

2. To find the solidity of a parallelopipedon.

As 1 upon A, is to the breadth of the base or end upon

B; so is the length of the base upon A, to the area of the base upon B; and, as 1 upon A, is to the length of the solid upon B; so is the area of the base upon A, to the solidity upon B.

3. To find the solidity of a prism, or a cylinder.

Find the area of the base by the last Problem, with which proceed as in the last Rule.

4. To find the solidity of a cone, or a pyramid.

Find the area of the base by the last Problem; then, as 3 upon A, is to the length of the solid upon B; so is the area of the base upon A, to the solidity upon B. Note. Examples for practice may be found in Section I.

SECTION III.

TIMBER MEASURE.

PROBLEM I.

To find the superficial content of a board or plank.

RULE.

Multiply the length by the breadth, and the product will be the superficial content.

Note. If the board taper, add the breadth of the two ends together; and half their sum will be a mean breadth.

By the Sliding Rule.

As 12 upon B, is to the breadth in inches upon A; so is the length in feet upon B, to the content upon A, in feet and fractional parts.

EXAMPLES.

1. If the length of a plank be 12 feet 6 inches, and its breadth 1 foot 3 inches; what is its superficial content? By Cross Multiplication.

By Decimals.