how many solid feet does 1 foot in length of this log contain? - 2 feet in length? - 3 feet? -10 feet? A solid of this form is called a Cylinder. 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. 168. What is the solid content of a pyramid whose base is 4 feet square, and the perpendicular height 9 feet? Ans. 48 feet 42 X 48. 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, hung diameter 31 inches, and whose length is 36 inches; how many wine gallons does it contain ? how many 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 (1844 = '0034;) 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 (74 = '0028, 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 × 36 in. = 30276. Then, { 30276 × 34 = 1029384. Ans. 10269384 wine gals. 30276 × 28 = 847728. Ans. 84'7728 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 lever as above, what power would it require to raise 1000 pounds? Ans. 1900-250 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 ? 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, = 10, ratio. 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 × 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 × 3 = 377+ cir 3774 cumference, and = 3774, ratio, Ans. 1 200. 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. 1731 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. 6644 inches. FORMS OF NOTES, BONDS, RECEIPTS, AND ORDERS. NOTES. Overdean, Sept. 17, 1802 For value received, I promise to pay to OLIVER BOUNTIFUL, от order, sixty-three dollars fifty-four cents, on demand, with interest after three months. WILLIAM TRUSTY. Attest, TIMOTHY TESTIMONY. 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. For value received, we, jointly and severally, promise to pay to C. D., or order, demand, with interest. Attest, CONSTANCE Adler. dollars cents, on ALDEN FAITHFUL Observations. 1. No note is negotiable unless the words "cr order," otherwise "or bearer," be inserted in it. 2. If the note be written to pay him " or order," (No. I.) then Oliver Bountiful may endorse this note, that is, write his name on the backside, and sell it to A, B, C, or whom he pleases. Then A, who buys the note, calls on William Trusty for payment, and if he neglects, or is unable to pay, A may recover it of the endorser. 3. If a note be written to pay him "or bearer," (No. II.) then any person, who holds the note, may sue and recover the same of Peter Pencil. 4. The rate of interest, established by law, being six per cent. per annum, it becomes unnecessary, in writing notes, to mention the rate of interest; it is sufficient to write them for the payment of such a sum, with interest, for it will be understood legal interest, which is six per cent. 5. All notes are either payable on demand, or at the expiration of a certain term of time agreed upon by the parties, and mentioned in the note, as three months, a year, &c. 6. If a bond or note mention no time of payment, it is always on demand, whether the words "on demand" be expressed or not. 7. All notes, payable at a certain time, are on interest as soon as they become due, though in such notes there be no mention made of interest. This rule is founded on the principle, that every man ought to receive his money when due, and that the nonpayment of it at that time is an injury to him. The law, therefore, to do him justice, allows him interest from the time the money becomes due, as a compensation for the injury. 8. Upon the same principle, a note, payaoit on demand, without any mention made of interest, is on interest after a |
