EXAMPLES.

(29) What is the content of a tree, whose girt is 42 inches,

and length 16 feet

(30) What is the content of a tree, whose compass is 6+

inches, and the length 30 feet

(31) How many loads of timber are there in a hewn tree, whose breadth is 42 inches, depth 30 inches, and length 40 feet?

GAUGING.

PROBLEM XXIX.

To find the area of any triangular tun, back, cooler, or circular and elliptical superficies, in ale gallons, &c.

RULES.

1. Find the area in inches by the different Problems in

Sect. LXXV. and the solidity by Sect. LXXVI.; then,

Ale,

Corn,

and the quotient will be the area in gallons.-Or,

2. If the square of the diameter of any circle;

} for {

ale gallon,
wine

gallon,

corn gallon,

and the quotient will be the area in their respective gallons.

For as, 785398:1:: 282: the square of the diameter of the circle whose area is 282 cubic inches, viz. one ale gallon: and from this proportion arises the preceding divisions:

{ }

268,8

Or, these divisers may be turned into multiplicators, by

dividing unity or 1, or rather by dividing the area in inches,

of that circle whose diameter is 1.

That is,785398 by 282, &c.

The product will be the area in gallons of the same

EXAMPLES.

(32) Suppose the length of a brewer's tun, back, or cooler, be 16 feet 6 inches, and its breadth 7 feet 4 inches, what will be the area in ale or beer gallons, &c. ?

(33) The length of the base of a triangular cooler is 94 inches, and its perpendicular breadth is 58 feet 6 inches. Required its area in ale gallons.

(34) Suppose the longest diameter of a brewer's vessel be 84,5 inches, and the shortest diameter to be 50 inches; what will be the area in ale gallons?

(35) Suppose a tun in the form of the frustum of a pyramid, whose bases are equilateral triangles: let the side of the top be 64 inches, the side of the bottom be 98,6 inches, and its height or depth 36 inches; what is the content of that tun in ale gallons, &c.?

(36) If the diameter of the base of a regular cone be 60,5 inches, and the perpendicular height be 42,8 inches, what will be the content in ale gallons, &c. ?

(37) Suppose the diameter of a frustum of a cone be 84 inches at the top, and the diameter at the bottom be 62 inches, and the height 42 inches: required the content in ale gallons.

The bung diameter EF, head diameter CD, and length of the cask AB (withinside) being given; to find the content of a cylinder nearly equal to it.

1. To twice the area of the circle at the bung, add the area of the circle of the head; multiply the sum by onethird of the length of the cask; the product is the content in cubic inches, which are converted into gallons by dividing by 282 for ale, and 231 for wine gallons.-Or,

2. To the square of the head diameter, add twice that of the bung diameter, and from that sum take of the square of the difference of the said diameters: then multiply the remainder by the length of the cask: then if the product be multiplied

5,00092837-002787

by

2,00113333-003399 x 3, will give the area.

or divided by { 1077,15=359,05

}

882,36-294,12 f

EXAMPLES.

× 3, will give the area.

(38) What is the content of a cask, whose bung diameter, head diameter, and length, are 32, 26, and 40 inches, withinside, respectively?

(39) Suppose the bung diameter of a cask to be 40 inches, head 36, and length 64: required the contents both in ale and wine gallons.

QUESTIONS for EXERCISE in MENSURATION OF SOLIDS.

(1) What is the difference between a solid half foot, and half a foot solid?

(2) What is the proportion, in space, between a room 25 feet 6 inches long, 20 feet two inches broad, 14 feet high, and two others of just half the dimensions?

(3) Another room is 17 feet 7 inches long within, 13 feet 10 inches broad, and 9 feet high; it has a chimney carried up straight in the angle, the plan whereof is just the half of 5 feet, by 4 feet 2 inches. The question is, How many cubic feet of air the same will contain, allowing the content of the fire-place and windows at 4 solid yards?

(4) A ship's hold is 112 feet long, 32 broad, and 51⁄2 deep. How many bales of goods, 3 feet 4 inches long, 2 feet 2 inches broad, and 3 feet deep, may be stowed therein, leaving a gang-way 4+ feet broad?

(5) I want a rectangular cistern, that, at 16 lb. to the foot square, shall weigh just a fother of lead; it must be

8 feet long, and 44 over; how many hogsheads, wine measure, will this cistern contain, taking it at of an inch from the top?

(6) A log of timber is 18+ feet long, 18 inches broad, and 14 inches thick, die square all through. Now, if 2 solid feet and a half be sawed off the end, how long will the piece then be?

(7) The solid content of a square stone is found to be 1264 feet, its length is 8 feet. What is the area of one end, and what the depth, if the breadth assigned be 38 inches.

(8) The dimensions of the circular Winchester bushel are 181⁄2 inches over, and 8 inches deep. How many quarters of grain will a square bin hold, that measures 7 feet 10 inches long, 3 feet 10 broad, and 4 feet 2 inches deep, within?

(9) Taking the dimensions of the bushel as above, what must the diameter of a circular measure be, which at 12 inches deep will hold 9 bushels of sea coal struck?

(10) A prism of two equal bases, and 6 equal sides, that measures 28 inches across the centre, from corner to corner; the superficial and the solid content is required, taking the length at 134 inches,

(11) I have a rolling stone 44 inches in circumference, and am to cut off three cubic feet from one end. Whereabouts must the section be made?

(12) I would have a syringe, 14 inch in the bore, to hold a pint (wine measure) of any fluid. What must the length of the piston, sufficient to make an injection with it, be?

(13) I would have a cubic bin made capable of receiving just 13 quarters of wheat, Winchester measure; what will be the length of one of its sides?

(14) A Bath stone, 20 inches long, 15 over, and 8 deep, weighs 220 lb. How many cubic feet thereof will freight a ship of 290 tons?

(15) The common way of measuring timber being to girt a round straight tree in the middle, and to take of the girt for the side of a square, equal to the area of the section there; if this be not considered in the price appointed, pray on which side lies the advantage?

(16) The solid content of a globe 20 inches in diameter, a cylinder of the same diameter, 20 inches long, and a cone 20 inches diameter at the base, and 20 inches

high, are severally required; and also what they will cost painting at 8d. per yard.

(17) Our satellite, the moon, is a globe, in diameter 2170 miles. I require how many quarters of wheat she would contain, if hollow, 2150 solid inches being the bushel; and how much yard wide stuff would make her a waistcoat, were she to be clothed.

(18) Suppose the atmosphere, or body of the air and vapours surrounding the globe of the earth and sea, to be 60 miles above the surface, and the earth is 7970 miles in diameter: how many cubic yards of air then hang about and revolve along with this planet?

(19) A square pyramid, whose sides at the base measure 10 inches a-piece, and is 20 feet high by the slope in the middle of each side of the base, is to be sold at 7s. per solid foot; and if the polishing the surface of the sides will be 8d. per foot more; I would know the cost of this stone when finished.

(20) A round mash-vat measures at the top 72 inches over within, at the bottom 54, the perpendicular depth be ing 42 inches; the content in ale gallons is required. (21) The shaft of a round pillar, 16 inches in diameter at the top, is about 8 of the bottom diameters in height, whereof is truly cylindrical, and the other swelling; but we will suppose it tapers straight, and that it is less at the top than at bottom; the price of the stone and workmanship is sought, at 3s. 6d. per cubic foot; and farther, the superficial content, including both ends,

(22) A stick of square timber tapers straight; the side of the greater end is 194 inches, of the less 131⁄2 inches, the length 16 feet 6 inches; the value at 2s. 6d. per foot solid, is demanded,

(23) What quantity of brandy will the distiller's tun contain, that measures 40 inches within at the head, 52 at the bung, and 103 inches long; and how many barrels of London ale would fill it?

(24) Suppose the globe or ball, on the top of St. Paul's Church, to be o feet in diameter; what did the gilding thereof come to at 3td. per inch square?

(25) The famous ton of Heidelberg, that being heretofore annually replenished with Rhenish, had in it some wine that was many ages old before the French de