XLII. TIMBER MEASURE.

369. If a piece of timber be in the shape of any of the bodies considered in the Fourth Section, the volume can be determined by the appropriate Rule there given. If no exact Rule is applicable we may in some cases use with advantage the method of equidistant sections which is given in Art. 307. In two cases which often occur in practice rules are adopted, which although not exact, are recommended by their simplicity: these rules we shall now give.

370. To find the volume of squared or four-sided timber.

RULE. Multiply the mean breadth by the mean thickness, and the product by the length; and take the result for the volume.

In order to obtain the mean breadth, the actual breadth should be measured at various equidistant points, and the sum of the results divided by the number of the measurements and in the same way the mean thickness should be obtained.

:

371. Examples.

(1) The length of a piece of timber is 24 feet, the mean breadth 1 foot 9 inches, and the mean thickness 1 foot 6 inches.

Thus we obtain 63 cubic feet for the volume.

(2) The length of a piece of timber is 16 feet, the thickness at one end is 1 foot, and at the other end 1 foot 8 inches: the breadth is 2 feet.

Here we take for the mean thickness in feet half the sum of 1 and 13, that is 13.

372. If the piece of timber tapers regularly from one end to the other it is usual to take for the mean breadth the breadth at the middle, or, which is the same thing, half the sum of the breadths at the ends; and similarly the mean thickness is estimated. But in this case the piece of timber is really a prismoid, and so the volume might be determined exactly by the Rule of Art. 283. The approximate Rule has the advantage of being simpler than the exact Rule.

If, as in Example (2) of Art. 371, it tapers regularly as to its thickness, and is constant as to its breadth, the Rule gives the exact result. The piece of timber is in this case a prism, the ends of the prism being trapezoids, and the height of the prism being the breadth of the piece of timber. Of course a similar remark applies to the case in which the thickness is constant and the breadth tapers regularly.

373. To find the volume of round or unsquared timber.

RULE. Multiply the square of the mean quarter girt by the length, and take the product for the volume.

374. Examples.

(1) The length of a piece of unsquared timber is 32 feet, and the girt is 6 feet.

3 2

3

The quarter girt is feet; the square of is

2

x 32 72. Thus we obtain 72 cubic feet for the volume.

(2) The length of a piece of unsquared timber is 24 feet; the girt at one end is 5 feet, and at the other end 6 feet.

Thus we obtain 453 cubic feet for the volume.

375. If the piece of timber be exactly a cylinder in shape, we can determine its volume exactly by the Rule of Art. 246. We shall find that in the case of a right circular cylinder the Rule of Art. 373 gives a result which is rather more than three-fourths of the true result; perhaps the Rule was constructed with the design of making some allowance for the loss of timber which occurs when the piece is reduced by squaring in the ordinary way; see the last Exercise of Art. 313.

If the piece of timber be not a circular cylinder the result given by the Rule will generally approach nearer to the true result.

376. Dr Hutton proposed to use the following Rule instead of the Rule of Art. 373: Multiply the square of one-fifth of the mean girt by twice the length. Dr Hutton's Rule makes the volume times as great as the ordinary

32

25

Rule, and gives a result which is very nearly exact when the piece of timber is exactly a circular cylinder.

377. If the piece of unsquared timber tapers regularly from one end to the other it is usual to take as the mean girt the girt at the middle, or, which is the same thing, half the sum of the girts at the ends. If the ends are exact circles and the piece of timber tapers regularly, it is really a frustum of a cone, and so the volume might be determined exactly by the Rule of Art. 268. The approximate Rule has the advantage of being simpler than the exact Rule.

EXAMPLES. XLII.

Find the number of cubic feet in pieces of timber of the following dimensions:

1. Length 22 feet; breadth at one end 2 feet 9 inches, at the other 2 feet 3 inches; thickness at one end 1 foot 10 inches, at the other 1 foot 6 inches.

2. Length 27 feet; mean breadth 3 feet, mean thickness 1 feet.

3. Length 32 feet; mean breadth 23 feet, mean thickness 1 feet.

4. Length 56 feet; mean girt 5 feet.

5. Length 32 feet; girt at one end 25 inches, at the other 35 inches.

6. Length 24 feet; mean girt 40 inches.

7. A piece of timber is 36 feet long and tapers regularly; its breadth and thickness at one end are 30 inches and 20 inches respectively, and at the other end 24 inches and 18 inches respectively: find the number of cubic feet in the piece by the rule of Art. 370.

8. Find the number of cubic feet in the piece of timber of the preceding Example by the exact rule of Art. 283.

9. A piece of timber is 40 feet long and tapers regularly; one end is a circle 7 feet in circumference, and the other end is a circle 4 feet in circumference: find the number of cubic feet in the piece by the rule of Art. 373.

10. Find the number of cubic feet in the piece of timber of the preceding Example by the rule of Art. 376.

11. Find the number of cubic feet in the piece of timber of Example 9 by the exact rule of Art. 268.

12. If the piece of timber of Example 9 is squared, the ends being made squares as large as possible: find the number of cubic feet in the piece reduced. Examples 51 and 52 of Chapter XVI.]

[See

XLIII. GAUGING.

378. By gauging is meant estimating the volumes of casks, that is, the volumes of liquids which the casks would hold.

Casks differ in shape, and various rules have been given for estimating their volumes according to the shape which they take exactly or approximately. For example, suppose a cask to be formed of two equal frustums of a cone united at their bases; we can determine the volume exactly by Art. 268: and if a cask be very nearly of this shape, we may estimate the volume approximately by proceeding as if the cask were exactly of this shape.

379. But it is found that a Rule may be given which will serve tolerably well for all the shapes of casks which occur in practice. To apply this Rule we must know three internal measurements of the cask, namely the length, the diameter at one end, which is called the head diameter, and the diameter at the middle, which is called the bung diameter.

380. The dimensions of a cask will always be taken to be expressed in inches.

381. To find the volume of a cask.

RULE. Add into one sum 39 times the square of the bung diameter, 25 times the square of the head diameter, and 26 times the product of those diameters; and multiply the sum by the length of the cask: multiply the product by 000031473, and the result may be taken for the volume of the cask in gallons.