Ex. The diameter of the sphere being 12, and the height of segment 6, then

6 x 12 x 3.1416226.1952 = surface of spheric segment.

6. For the content of a sphere and spheric segment.

Ex. If the diameter of a sphere be 2 inches, then 28 X 0.5236 = 4.1888 = the content.

(radius of segment's base × 3 + height of segment3) × height x 5236 = content of segment.

Ex. If the height of a spheric segment be 2, and radius of base 6, then

(63 × 3 + 2o) × 2 × ·5236 = 117-2864 = content 7. For the solidity of a spheroid.

Revolving axis 2× fixed axis × ·5236 = content. NOTE. If the spheroid revolve round the greater axis, it is said to be oblate; if round the lesser, oblong.

Ex. The two axes of a spheroid are 24 and 18; therefore,

24 X 18 X 5236 = 5428.56 = content if it be oblate. 18 2x 24 x ⚫5236 =4071.5 = content if it be oblong.

8. For the solidity of a parabolic conoid.

Area of base × half the height =content.

Ex. The height being 18, and the diameter of base 24, then the area of the base therefore is 452.39; hence 452.39 × 9 = 4071.51 the content.

9. For the frustum of a cone or pyramid. (perim. of one end + perim. of the other end) × slant height

= surface.

2

Ex. In the frustum of a triangular pyramid the perimeter of one end is 25, that of the other 36, and the slant height is 10; therefore,

(2536) × 10
2

305 the surface.

✔(area of one end + ar. of other) + area of one end+ar. of other

× height =content.

3

Ex.-A log of wood is 20 feet long; its ends are squares,

of which the sides are respectively 12 and 16 inches; there

fore,

✔(12+16o)+122+162

3

-× 240 = 33600 inches = content

TIMBER MEASURE.

EXAMPLES of timber measuring have already been given in the department allotted to arithmetic, but it is necessary to be here somewhat more particular. The surface of a plank is found:

1st. By multiplying the length by the breadth. When the board tapers gradually, add the breadth at both ends together, and take the half of this sum for the mean breadth.

2d. By the sliding rule.-Set the length in inches on B to 12 on A, and against the length in feet on B will be the area in square feet and decimals on A.

Ex. A board is 12 feet 6 inches long and 1 foot 3 inches broad; hence,

1st. For the content of squared timber, length mean breadth x mean thickness

= content.

2d. By the sliding rule.-Find the mean proportional between the breadth and thickness, then set the length on C to 12 on D, and against the mean proportional on D the solid content on C. If the mean proportional be in feet, reduce to inches.

Ex.-A log is 24 feet long, the mean depth and breadth being each 13 inches.

For round timber.-1st. Take one-fourth of the mean girth and square it, this multiplied by the length will give the content.

2d. By the sliding rule.—Set the length in feet on C to 12 on D, then against the quarter girth in inches on D, will be the content on C.

This gives no allowance for bark, but there is usually a deduction made of about an inch to the foot of quarter girth. The rule given above gives the customary, but not the truc ontent; the following gives the true content.

One-fifth of the girth squared and multiplied by twice the length = content.

Ex.-The mean girth of a tree being 5 feet 8 inches, and its length 18 feet, the two rules will apply as below:5) 58 (1:1: 7

4) 5:8 (15

1: 5

201

18

36 1 6

1

: 1 : 7

Trees very seldom have an equal girth throughout, one end being generally much smaller than the other: the girth taken above is the mean girth; that is to say, the girths of both ends added together, and their sum halved for the mean girth. It is to be observed, however, that, if the difference of the girths is great, it will be best to find the content of the tree as if it were a conic frustum.-The method of using the sliding rule in the measurement of tim ber has been given before.

ARTIFICERS' WORK.

ARTIFICERS Compute the contents of their works by several different measures; as, glazing and masonry by the foot; painting, plastering, paving, &c., by the yard of 9 square feet; flooring, partitioning, roofing, tiling, &c., by the square of 100 square feet; and brickwork, either by a yard of 9 square feet, or by the perch, or square rod o pole, containing 2724 square feet, or 30 square yards, being the square of the rod or pole of 16 feet of 5 yards long. As this number 2724 is troublesome to divide by, the is often omitted in practice, and the content in feet divided only by the 272. But when the exact divisor

272 is to be used, it will be easier to multiply the feet by 4, and then divide successively by 9, 11, and 11. Also to divide square yards by 304, first multiply them by 4, and then divide twice by 11.

BRICKLAYERS' WORK.-Brickwork is estimated at the rate of a brick and a half thick. So that, if a wall be more or less than this standard thickness, it must be reduced to it, as follows:-Multiply the superficial content of the wall by the number of half bricks in the thickness, and divide the product by 3. The dimensions of a building are usually taken by measuring half round on the outside, and half round on the inside; the sum of these two gives the compass of the wall,—to be multiplied by the height, for the content of the materials. Chimneys are by some measured as if they were solid, deducting only the vacuity from the hearth to the mantel, on account of the trouble of them. And by others they are girt or measured round for their breadth, and the height of the story is their height, taking the depth of the jambs for their thickness. And in this case, no deduction is made for the vacuity from the floor to the mantel-tree, because of the gathering of the breast and wings, to make room for the hearth in the next story. To measure the chimney shafts, which appear above the building, gird them about with a line for the breadth, to multiply by their height. And account their thickness half a brick more than it really is, in consideration of the plastering and scaffolding. All windows, doors, &c., are to be deducted out of the contents of the walls in which they are placed. But this deduction is made only with regard to materials; for the whole measure is taken for workmanship, and that all outside measure too, namely, measuring quite round the outside of the building, being in consideration of the trouble of the returns or angles. There are also some other allowances, such as double measure for feathered gable ends, &c.

Ex. The end wall of a house is 28 feet 10 inches long, and 55 feet 8 inches high, to the eaves; 20 feet high is 2 bricks thick, other 20 feet high is 2 bricks thick, and the remaining 15 feet 8 inches is 1 brick thick; above which is a triangular gable, 1 brick thick, and which rises 42 courses of bricks, of which every 4 courses make a foot. What is the whole content in standard measure? Ans. 253 626 yards.

MASONS' WORK.-To masonry belong all sorts of stonework; and the measure made use of is a foot, either super ficial or solid. Walls, columns, blocks of stone or marble, &c., are measured by the cubic foot; and pavements, slabs, chimney-pieces, &c., by the superficial or square foot. Cubic or solid measure is used for the materials, and square measure for the workmanship. In the solid measure, the true length, breadth and thickness, are taken, and multiplied continually together. In the superficial, there must be taken the length and breadth of every part of the projection, which is seen without the general upright face of the building.

Ex. In a chimney-piece, suppose the

Length of the mantel and slab, each 4 feet 6 inches;
Breadth of both together,

Length of each jamb,•
Breadth of both together,

Required the superficial content.

3

2

•4

4

CARPENTERS' AND JOINERS' WORK.-To this branch belongs all the wood-work of a house, such as flooring, partitioning, roofing, &c. Large and plain articles are usually measured by the square foot or yard, &c., but enriched mouldings, and some other articles, are often estimated by running or lineal measures, and some things are rated by the piece.

In measuring of joists, it is to be observed, that only one of their dimensions is the same with that of the floor; for the other exceeds the length of the room by the thickness of the wall and of the same, because each end is let into the wall about of its thickness.

No deductions are made for hearths, on account of the additional trouble and waste of materials.

Partitions are measured from wall to wall for one dimension, and from floor to floor, as far as they extend, for the other. No deduction is made for door-ways, on account of the trouble of framing them.

In measuring of joiners' work, the string is made to ply close to every part of the work over which it passes.

The measure for centering for cellars is found by making a string pass over the surface of the arch for the breadth, and taking the length of the cellar for the length; but in groin centering, it is usual to allow double measure, on account of their extraordinary trouble.