descriptions of them, and the rules for calculating their volumes.

1. A spherical segment is any part of a sphere cut off by a plane.

To find the volume of a spherical segment, add together three times the square of the radius of its base and the square of its height; multiply this sum by the height, and the product by 5236. This may be more conveniently

2. The zone of a sphere is a part intercepted between two parallel planes.

If these planes be equally distant from the centre, it is called the middle zone of the sphere.

To find the volume of a spherical zone, add together the square of the radius of each end, and one-third of the square of their distance (i.e., of the height of the zone); multiply the sum by the height of the zone, and this product by 1.5708.

This rule may be expressed as a formula, thus-

1. A ball is 3 ft. 6 in. in diameter, and I cut from it a segment 15 in. high; find its volume.

2. A spherical segment is 24.8 in. high, and its diameter 5 ft. 1.2 in.; find its volume.

3. The height of a zone is 1 ft. 3 in., and the diameters of its ends 2 ft. 4 in. and 1 ft. 8 in.; find its volume.

4. The diameters of the ends of a zone are 9 ft. 3 in. and 6 ft. 9 in., and its height is 5 ft. 6 in.; find its volume.

5. From a sphere whose radius is 6 ft., I cut off a segment 4 ft. high; find its volume.

6. A sphere whose radius is 4 ft. 6 in., is divided into three parts of equal height by two parallel planes; find the volume of each part.

LESSON XXIII.-ARTIFICERS' WORK.

In nearly all trades the common standards of measurement are the foot and yard, and calculations may readily be made by practically applying the rules laid down in the preceding lessons of this book. In some trades, however, workmen are accustomed to use terms, and to calculate by rules which we have not as yet explained; and, accordingly, this lesson will be devoted to the explanation of a few such simple terms and rules.

I. In calculations of area for flooring, roofing, plastering, and tiling, the standard measure commonly taken is called a square," "and contains 100 square ft., i.e., it is the same size as a square, measuring 10 ft. each way.

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In many instances it would be inconvenient or dangerous to get upon the top of a house, in order to measure the roof; and, hence, when the pitch of the roof is known, the calculation is generally made by measuring only the base of the house. Builders adopt various pitches for roofs, either according to the fancy of the designer, or with regard to the material of which the roof is to be made; but there are three common pitches which have received special names, and are more commonly adopted than any others:

1. The common pitch, in which the length of the rafters on each slope of the roof is three-fourths of the breadth of the building, so that the area of the whole roof, including both slopes, is equal to half as much again as the area of the base.

2. The Gothic pitch, in which the length of the rafters is equal to the breadth of the building; in this case the area of the whole roof is equal to twice the area of the base.

3. The pediment pitch, which is very much flatter, the perpendicular height of the roof being only of the breadth of the building, so that the length of the rafters is nearly of the breadth of the building, and the area of the whole roof is then of the area of the base.

II. By professional engineers the solidity of brickwork is generally calculated by the cubic yard, but workmen com

monly calculate by what is called a standard rod, which consists of a piece of brickwork, a brick-and-a-half thick, and of the area of one square rod, i.e., 301 square yds., or 2724 square ft. This is by no means a convenient mode of calculating work, and in practice 272 square ft., is commonly taken instead of 2724, which, however, does not simplify it very much.

For convenience, all building bricks are made of one uniform size, viz., 81⁄2 in. in length, 4 in. broad, and 21⁄2 in. thick. But when built in a wall the bricks do not lie close to each other, a layer of mortar encasing them all round; and this layer is supposed to be in. thick, so that in measuring brickwork the dimensions usually taken for each brick are 9 in. long, 4 in. broad, and 3 in. thick. If 44 bricks be placed flat upon the ground and side by side, with

in. of mortar between them they will reach exactly one rod (5 ft.); if the wall be then built 66 bricks high, the height also will be exactly one rod; and in the whole square rod 2904 bricks will have been used; but the wall being then only one brick thick, instead of a brick-and-a-half, we must allow for 1452 more bricks, making altogether a total of 4356. Builders, in calculating, usually make reasonable allowance for waste, and reckon that it will take 4500 bricks for every standard rod.

III. Pieces of timber are usually measured by the rules laid down in previous lessons for calculating the dimensions of regular bodies, and for practical purposes the results obtained are tolerably accurate, even when the timber is not perfectly regular in its shape.

The volume of squared (or four-sided) timber is generally found by multiplying the length by the mean breadth and

the mean thickness, these last being found either by measuring the actual breadth and thickness in the middle, or else by taking several measurements at various equi-distant points in the length, and dividing the sum of them by the number of measurements.

The volume of round (or unsquared) timber is generally calculated by the following rule: multiply the square of the mean quarter-girt by the length, and the product will be the volume of the timber. This rule is, however, by no means accurate, and gives a result which is only about three-fourths of the true result, as you will see at once if you consider (p. 30) that the area of the circle representing the middle, or mean, section of the timber is considerably more than the square of a quarter of its circumference. But as this rule is used for calculating the volume of rough pieces of timber, from which a great deal is necessarily lost by cutting and squaring, it gives a result which is generally accepted as fair and reasonable.

IV. The contents of casks are generally measured by a process called gauging.

Calculations of this kind are most frequently made by excisemen, for the purpose of ascertaining the amount of duty payable on spirituous liquors kept for sale. Excisemen make their calculations expeditiously by the aid of gauging instruments, which, however, it is not necessary to explain or describe now.

Casks vary in shape, and hence no rules can be laid down

down on p. 53.

with regard to estimating their volumes, which are rigidly accurate. But for practical purposes a cask may be supposed to consist of two similar and equal frustrums of a cone, placed together base to base, as in the figure, and the volume may then be calculated by the rule laid

The method commonly adopted is, however, much simpler than this, and serves tolerably well for all shapes of casks; to apply it three internal measurements must be thrown

in inches, viz., the length of the cask, the diameter at one end, called the head-diameter (d), and the diameter at the middle, called the bung-diameter (D). The rule can be stated in words, thus:—

Add together 39 times the square of the bung-diameter, 25 times the square of the head-diameter, and 26 times the product of these diameters; multiply this sum by the length of the cask, and the product by 000031473, and the result will represent very nearly the volume of the cask in gallons.

Perhaps you will find it less difficult to remember this rule in the form of a formula, thus:

Number of gallons a cask will hold

=

1 (39R2+25r2+26Rr) × 000031473.

If a cask be only partly filled, that portion filled by the liquid is called the wet ullage; the portion left unfilled is called the dry ullage; and the dimensions of these are expressed respectively in wet and dry inches.

EXERCISES (S).

1. A wooden partition is 45 ft. 5 in. long, and 8 ft. 2 in. high; find its cost at £13, 10s. per square.

2. The length of a house is 80 ft., and its breadth 35 ft.; find the cost of roofing it, common pitch, at 7s. 6d. per square.

3. A piece of timber is 24 ft. long, and its mean girt is 3 ft. 4 in.; find its solidity.

4. The trunk of a fir tree when felled is 40 ft. long, and the circumference of its ends is 7 ft. and 4 ft.; what would be the common estimate of its value at 1s. per cubic foot.

5. The length of a cask is 30.5 in., and the bung and head diameters 26.5 in. and 23 in. respectively; how many gallons will the cask hold?