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?

PROBLEM VI.

Ans. 24.

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 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 one foot of timber cost 3s.; what will 180 feet cost? Ans. 540s.

3. If 50 feet of timber cost 71.; 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 breadths 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?

As 12 on B: 15 on A:: 12.5 on B: 15.6 on A. 2. Required the content of a board whose length is 25 feet 8 inches, and breadth 1 foot 7 inches.

R

Ans. 40 ft. 7 in. 8 pa.

3. What is the content of a board whose length is 18 feet 10 inches, and breadth at the broader end 2 feet 3 inches, and at the narrower 1 foot 7 inches?

Ans. 36 ft. 1 in. 2 pa. 4. If the length of a mahogany plank be 35 feet 9 inches, and its breadth 3 feet 6 inches; what is the value of three such planks at 2s. 9d. per square foot? Ans. £ 51 12s. 31d.

Note. Mahogany is a production of the warmest parts of America. It is also found plentifully in the islands of Cuba, Jamaica, and Hispaniola. This tree grows tall and straight, rising often sixty, and sometimes a hundred feet from the ground to the arms; and is about 4 feet in diameter. The foliage is a beautiful deep green; and the appearance of the whole tree is very elegant. The mahogany brought from Jamaica is most valued, in consequence of its firmness, durability, and beauty of colour.

This wood has been used in medicine with the same effect as Peruvian bark.

PROBLEM II.

To find the solidity of squared or four-sided timber.

RULE.

Multiply the mean breadth by the mean thickness, and this product by the length, and it will give the solidity, according to the customary method.

By the Sliding Rule.

As 1 upon A, is to the mean breadth upon B; so is the mean thickness upon A, to the mean area upon B, or the area of the middle section; and, as 1 upon A, is to the length upon B; so is the mean area upon A, to the solidity upon B.

Note 1. If the tree be equally broad and thick throughout, the breadth and thickness taken in any part, will be the mean breadth and thickness; but if it taper regularly from one end to the other, the breadth and thickness must be taken in the middle. Or, take the dimensions of the two ends, and half their sum respectively will be the mean breadth and thickness.

2. Some measurers multiply the square of one fourth of the circumference, or quarter girt, taken in the midd'e, by the length, for the solidity; but when the ends are not equal squares, this method always gives too much; and the more the breadth and thickness differ, the greater will be the errour.

3. The Rule given in this Problem, is generally used in practice; and when the tree is equally broad and thick throughout, it gives the true content; but in tapering timber, it always gives too little; consequently the method mentioned in the last pote is sometimes nearer to the truth.

When the true content of a tapering tree is wanted, it must be found by the General Rule, for the frustum of a pyramid, given in Problem 6, Section 1. or by the Rule for a prismoid, given in Problem 10.

EXAMPLES.

1. If the length of a piece of timber be 9.8 feet, its breadth 2.6 feet, and its thickness 1.5 foot; what is its solidity?

Feet.

2.6

1.5

130

26

3.90

9.8

3120

3510

38.220 Ans.

By the Sliding Rule.

As 1 on A 2.6 on B: 1.5 on A: 3.9 on B, the

mean area.

As 1 on A 9.8 on B:: 3.9 on A: 38.22 on B, the solidity.

2. The mean breadth of a piece of timber is 2.86 feet, its mean thickness 1.93 foot, and its length 18.64 feet ; what is its solidity? Ans. 102.889072 feet.

3. Each side of the greater end of a piece of squared timber is 28 inches, each side of the less end 14 inches, and its length 18 feet 9 inches; how many solid feet does it contain? Ans. 57.421875 feet.

Note. The true content measured as the frustum of a pyramid, is 59.5486 feet. See Example, Problem VI., Section 1.

4. The length of a piece of timber is 17 feet 3 inches; at the greater end, the breadth is 36 inches, and the thickness 20 inches; and at the less end, the breadth is 18 inches, and the thickness 10 inches: what is the solidity? Ans. 48.515625 feet.

Note. The true content, measured as the frustum of a pyramid, is 50.3125 feet. See Example 3, Problem VI., Section 1.