174 What is the diameter of a circle, whose circumference is 462 feet? Ans. 147 feet. 175. If the distance through the centre of the earth, from side to side, be 7911 miles, how many miles around it?

7911 X 3'14159 = 24853 square miles, nearly, Ans. 176. What is the area or contents of a circle, whose diameter is 7 feet, and its circumference 22 feet?

Note. The area of a circle may be found by multiplying the diameter into the circumference. Ans. 384 square feet. 177. What is the area of a circle, whose circumference is 176 rods? Ans. 2464 rods. 178. If a circle is drawn within a square, containing 1 square rod, what is the area of that circle?

Note. The diameter of the circle being 1 rod, the circumference will be 3'14159. Ans. "7854 of a square rod, nearly. Hence, if we square the diameter of any circle, and multiply the square by 7854, the product will be the area of the circle.

179. What is the area of a circle whose diameter is 10 rods? 102 X 7854 = 78′54. Ans. 78'54 rods. 180. How many square inches of leather will cover a ball 34 inches in diameter ?

Note. The area of a globe or ball is 4 times as much as the area of a circle of the same diameter, and may be found, therefore, by multiplying the whole circumference into the whole diameter. Ans. 38 square inches. 181. What is the number of square miles on the surface of the earth, supposing its diameter 7911 miles?

7911 X 24853 196,612,083, Ans. 182. How many solid inches in a ball 7 inches in diame

ter?

Note. The solid contents of a globe are found by multiplying its area by part of its diameter.

Ans. 179 solid inches. 183. What is the number of cubic miles in the earth, supposing its diameter as above?

Ans. 259,233,031,435 miles. 184. What is the capacity, in cubic inches, of a hollow globe 20 inches in diameter, and how much wine wal it contain, 1 gallon being 231 cubic inches?

Ans. 41888 cubic inches, and 19413+ gallons. 185. There is a round log, all the way of a bigness; the areas of the circular ends of it are each 3 square feet;

how many solid feet does 1 foot in length of this log con2 feet in length?

tain ?
3 feet?
A solid of this form is called a Cylinder.

10 feet?

How do you find the solid content of a cylinder, when the area of one end, and the length are given?

186. What is the solid content of a round stick, 20 feet long and 7 inches through, that is, the ends being 7 inches in diameter ?

Find the area of one end, as before taught, and multiply it by the length. Ans. 5'347 cubic feet. If you multiply square inches by inches in length, what parts of a foot will the product be? if square inches by feet in length, what part?

187. A bushel measure is 18'5 inches in diameter, and 8 inches deep; how many cubic inches does it contain?

Ans. 2150'4+.

It is plain, from the above, that the solid content of all bodies, which are of uniform bigness throughout, whatever may be the form of the ends, is found by multiplying the area of one end into its height or length.

Solids which decrease gradually from the base till they come to a point, are generally called Pyramids. If the base be a square, it is called a square pyramid; if a triangle, a triangular pyramid; if a circle, a circular pyramid, or a cone. The point at the top of a pyramid is called the vertex, and a line, drawn from the vertex perpendicular to the base, is called the perpendicular height of the pyramid.

The solid content of any pyramid may be found by multiplying the area of the base by of the perpendicular height. 188. What is the solid content of a pyramid whose base is 4 feet square, and the perpendicular height 9 feet?

42 X

48.

Ans. 48 feet. 189. There is a cone, whose height is 27 feet, and whose base is 7 feet in diameter; what is its content?

Ans. 3464 feet. 190. There is a cask, whose head diameter is 25 inches, bung diameter 31 inches, and whose length is 36 inches; how many how many wine gallons does it contain ?

beer gallons?

Note. The mean diameter of the cask may be found by adding 2 thirds, or, if the staves be but little curving, 6 tenths, of the difference between the head and bung diame

ters, to the head diameter. The cask will then be reduced to a cylinder.

Now, if the square of the mean diameter be multiplied by 7854, (ex. 177,) the product will be the area of one end, and that, multiplied by the length, in inches, will give the solid content, in cubic inches, (ex. 185,) which, divided by 231, (note to table, wine meas.) will give the content in wine gallons, and, divided by 282, (note to table, beer meas.) will give the content in ale or beer gallons.

In this process we see, that the square of the mean diameter will be multiplied by "7854, and divided, for wine gallons, by 231. Hence we may contract the operation by only multiplying by their quotient (38540034;) that is, by 0034, (or by 34, pointing off 4 figures from the product for decimals.) For the same reason we may, for beer gallons, multiply by (38540028, nearly,) '0028, &c.

Hence this concise RULE, for guaging or measuring casks,Multiply the square of the mean diameter by the length; multiply this product by 34 for wine, or by 28 for beer, and, pointing off four decimals, the product will be the content in gallons and decimals of a gallon.

In the above example, the bung diameter, 31 in. - 25 in. the head diameter 6 in. difference, and 3 of 6 = 4 inches; 25 in. + 4 in. 29 in. mean diameter.

Then, 292 = 841, and 841 x 36 in.

Then, {

30276.

30276 × 34 = 1029384. Ans. 102'9384 wine gals. 30276 X 28 = 847728. Ans. 847728 beer gals. 191. How many wine gallons in a cask whose bung diameter is 36 inches, head diameter 27 inches, and length 45 inches? Ans. 166'617.

192. There is a lever 10 feet long, and the fulcrum, or prop, on which it turns, is 2 feet from one end; how many pounds weight at the end, 2 feet from the prop, will be balanced by a power of 42 pounds at the other end, 8 feet from the prop?

Note. In turning around the prop, the end of the lever 8 feet from the prop will evidently pass over a space of 8 inches, while the end 2 feet from the prop passes over a space of 2 inches. Now, it is a fundamental principle in mechanics, that the weight and power will exactly balance each other, when they are inversely as the spaces they pass over. Hence, in this example, 2 pounds, 8 feet from the prop, will balance

Y*

8 pounds 2 feet from the prop; therefore, if we divide the distance of the POWER from the prop by the distance of the WEIGHT from the prop, the quotient will always express the ratio of the WEIGHT to the POWER; 4, that is, the weight will be 4 times as much as the power. 42 × 4 168.

Ans. 168 lbs. 193. Supposing the level as above, what power would it require to raise 1000 pounds? Ans. 1 250 pounds. 194. If the weight to be raised be 5 times as much as the power to be applied, and the distance of the weight from the prop be 4 feet, how far from the prop must the power be applied? Ans. 20 feet. 195. If the greater distance be 40 feet, and the less of a foot, and the power 175 pounds, what is the weight?

Ans. 14000 pounds. 196. Two men carry a kettle, weighing 200 pounds; the kettle is suspended on a pole, the bale being 2 feet 6 inches from the hands of one, and 3 feet 4 inches from the hands of the other; how many pounds does each bear?

Ans. {854 pounds.
S 114 pounds.

197. There is a windlass, the wheel of which is 60 inches in diameter, and the axis, around which the rope coils, is 6 inches in diameter; how many pounds on the axle will be balanced by 240 pounds at the wheel?

Note. The spaces passed over are evidently as the diameters, or the circumferences; therefore, 6210, ratio.

Ans. 2400 pounds. 198. If the diameter of the wheel be 60 inches, what must be the diameter of the axle, that the ratio of the weight to the power may be 10 to 1? Ans. 6 inches. Note. This calculation is on the supposition, that there is no friction, for which it is usual to add to the power which is to work the machine.

199. There is a screw, whose threads are 1 inch asunder, which is turned by a lever 5 feet, 60 inches, long;

what is the ratio of the weight to the power?

Note. The power applied at the end of the lever will describe the circumference of a circle 60 X 2 = 120 inches in diameter, while the weight is raised 1 inch; therefore, the ratio will be found by dividing the circumference of a circle, whose diameter is twice the length of the lever, by the distance

between the threads of the screw. 120 × 343774 cir

377 1

cumference, and =3774, ratio, Ans.

200. There is a screw, whose threads are of an inch asunder; if it be turned by a lever 10 feet long, what weight will be balanced by 120 pounds power? Ans. 30171 pounds.

201. There is a machine, in which the power moves over 10 feet, while the weight is raised 1 inch; what is the power of that machine, that is, what is the ratio of the weight to the power? Ans. 120. 202. A man put 20 apples into a wine gallon measure, which was afterwards filled by pouring in 1 quart of water; required the contents of the apples in cubic inches.

Ans. 173 inches. 203 A rough stone was put into a vessel, whose capacity was 14 wine quarts, which was afterwards filled with 24 quarts of water; what was the cubic content of the stone? Ans. 664 inches.

FORMS OF NOTES, BONDS, RECHIPTS, AND ORDERS.

NOTES.
No. I.

Overdean, Sept. 17, 1802.

For value received, I promise to pay to OLIVER BOUNTIFUL, or order, sixty-three dollars fifty-four cents, on demand, with

interest after three months.

Attest, TIMOTHY TESTIMONY.

WILLIAM TRUSTY:

No. II.

Bilfort, Sept. 17, 1802.

For value received, I promise to pay to O. R., or bearer,

dollars

cents, three months after date.

PETER PENCIL