twice in the length of a pole, or 163 feet. What then is the diameter ?

(19) I would turf a round plot, measuring 130 feet about, and would know the charges, at 4d. per yard square. (20) I want the length of a line, by which my gardener may strike a round aurangerie, that shall contain just half an acre of land.

(21) Agreed for an oaken curb to a round well at 8d. per square foot; it is exactly 42 inches in diameter, within the brick work, and the breadth of the curb is to be 144 inches. What will it come to?

(22) It is observed, that the extreme end of the minute-hand of a public dial moves just 5 inches in the space of 34 minutes. The question is, what is the length of that index?

(23) A. B. and C. join for a grinding stone, 36 inches in diameter, value 20s. towards which A. paid 7s. B. 8s. and C. 5s. the waste hole through which the spindle passed was 5 inches square. To what diameter ought the stone to be worn, when B. and C, begin severally to work with it?

(24) I demand what difference there is in the area of the section of a round tree, 20 inches over, and its inscribed and circumscribed squares.

(25) Having paved a semicircular alcove with black and white marble, at 2s, 4d. per foot, the mason's bill was just 107. What then was the arch in front?

(26) What proportion is there between the arpent of France, which contains 100 square poles of 18 feet each, and the English acre, containing 160 square poles of 161 feet each, considering that the length of the French foot is to the English as 10 to 15?

(27) In turning a one-horse chaise within a ring of a certain diameter, it was observed that the outer wheel made two turns, while the inner made but one. The wheels were equally high; and supposing them fixed at the statutable distance, or 5 feet asunder on the axle-tree, pray what was the circumference of the track described by the outer wheel?

(28) Required the area of a sector (supposing one of the di visions of a wilderness), which, being struck from a centre with a line 30 yards long, makes the sweep or circular part 63 feet?

(29) If the chord or line drawn through the two ends of the above curve be 15 inches shorter than the arch-line, I demand the segment.

(30) Suppose the ellipse in Grosvenor-square measure 840 links the longest way, and 612 across, within the rails, and if the curb stones be 14 inches thick, what ground do they inclose, and what do they stand upon? Note. The dimensions of all similar figures are in proportion to their areas, as the squares of their respective sides, et contra.

着

(31) If a round pillar, 7 inches over, have 4 feet of stone in it, of what diameter is the column, of equal length, that measures ten times as much?

(32) A pipe of 6 inches bore will be 3 hours in running off a certain quantity of water. In what time will 4 pipes, each 3 inches bore, be in discharging double the quantity?

(33) Suppose a yard of rope, 9 inches round, weigh 22 lb. what will a fathom of that weigh which measures a

foot about?

(34) If 20 feet of iron railing shall weigh half a ton when the bars are an inch and quarter square, what will 50 feet

of ditto come to, at 31d. per lb. the bars being but of an inch square?

(35) A sack that holds three bushels of corn is 224 inches broad when empty; what would the sack contain,

that, being of the same length, had twice its circumference, or twice its breadth?

(36) My plumber has set me up a cistern; and his shop-book being burnt, he has no means of bringing in the charge; and I do not choose to take it down to have it weighed; but by measure he finds it contains 64 square feet, and that it was of an inch precisely in thickness. Lead was then, wrought at 217. per fother. Let the accomptant, from these items, make out the poor man's bill, considering further, that 4 oz. is the weight of a cubic inch of lead.

LXXVI. MENSURATION of SOLIDS.

PROBLEM XIII.

To find the solidity of a cube, prism, or right cylinder.

Multiply the area of the base into the height or altitude, and the product will be the solidity.

EXAMPLES.

(1) What is the solid content of a cube, whose side is 24feet?

(2) How many ale gallons of water will a cistern hold, whose length, breadth, and depth, are 4 feet 9 inches, 3 feet 6 inches, and 2 feet 10 inches?

(3) What is the content of a cylinder, whose diameter is 41 feet, and 8 feet high?

PROBLEM XIV.

To find the convex surface of a prism, or a right cylinder.

A GENERAL RULE.

Find the area of each side and end separately, then add those areas together, and their sum will be the whole surface of any prism or body whatever.

A PARTICULAR RULE.

Multiply the circumference of the base by the altitude of one cylinder, and the product will give the convex surface.

EXAMPLE.

(4) What is the convex surface of a right cylinder, whose circumference is 104 feet, and height 7 feet?

PROBLEM XV.

To find the solidity of a pyramid, or right cone.

Fig. 15.

Fig. 16.

B

RULE.

Multiply the area of the base by a third part of the valtitude, and the product will be the content required.

EXAMPLES.

(5) Required the solidity of a square pyramid, each side of whose base is 12 feet, and the slant height 25 feet, (6) What is the solid content of a triangular pyramid, whose height is 30 feet, and each side of its base 51⁄2?

(7) What is the solidity of a cone, whose base is 31 feet diameter, and altitude 6 feet?

PROBLEM XVI.

To find the convex surface of a pyramid or cone (as Fig. 15 and 16).

RULE.

Multiply the perimeter or circumference of the base by the slant height or length of the side (AC) and half the product will be the area.

EXAMPLES.

(8) What is the surface of a triangular pyramid, including the base, the slant height being 20 feet, and each side of the base 3?

(9) What is the convex surface of a right cone, whose base is 45 feet in circumference, and slant side is 20 feet in length?

PROBLEM XVII.

To find the solidity of a frustum of a pyramid or cone.

Fig. 17.

Fig. 18.

C

Add into one sum the areas of the two ends, and the mean proportional between them; multiply the sum by the