10. Divide the product of 32, 18, 72, 81, 7, 56, 63, by the product of 24, 36, 162, 63, 2, 32. Quo. 147. 11. Divide the product of 96, 54, 108, 72, 56, 18, and 21, by the product of 27, 81, 324, 28, 72, 3. Quo. 1991. 12. Divide the product of 16, 27, 42, 44, 55, and 66, by the product of 48, 88, 22, 33, and 49: Quo. 194. 13. Divide 6, 8, 10, 5, 20, 25, 36, 48, and 60, by the product of 9, 12, 15, 28, 36, 42, and 9. Quo. 2013. 14. If 12 horses eat 15 bushels of oats, how many would 36 horses eat in the same time? In solving this sum, if I divide 15 bushels by 12, it will evidently give me what one horse will eat, viz. 14 bushels; and if I multiply this by 36, I shall obtain the quantity that 36 horses will eat. This sum may then be easily solved by the principle here introduced. The statement would be thus: 15.36 12' and 15x3=45 bushels. But we will not anticipate: enough has already been done to show that the principle is a practical one. We will leave its application, for future consideration. QUESTIONS.-What is the object of the rule of canceling? What problems may be solved on this principle? What is the first of the facts on which this principle is founded? Illustrate. What is the second? Illustrate. What is another advantage? What is the third fact? What is the fourth? Illustrate. What is the first step of the rule? What is the second step? The third? The fourth? The fifth? The sixth? The seventh, and last? What is note first? What note second? When the product of the remaining numbers above the line is less than the product of those below, in what form will the answer be? What does one number standing directly over another, with a line between them, always indicate? COMPOUND NUMBERS. We have thus far been operating with numbers, of the same denomination, and increasing in the constant ratio of 10. There is, however, another class of numbers composed of several denominations, increasing in no uniform ratio, and requiring to be separately denoted or expressed. These are called Compound Numbers. Under this head are included all those denominations employed to express measures of any defi nite kind; such as length, breadth, solidity, weight, time, money, capacity, &c. The following are the tables of these denominations. They require to be made very familiar, as they show how many units of each lower denomination, are equal to a unit of the next higher denomination. I. ENGLISH MONEY. The denominations of English Money are pounds, shillings, pence, and farthings. TABLE. 4 farthings (marked qr.) make 1 penny, marked d. By this weight, the precious metals, such as gold and silver, also jewels and precious stones, are weighed. The following are the denominations : TABLE. 24 grains (gr.) make 1 pennyweight, marked pwt. By this weight, all coarse materials, such as hay and grain, and also the baser metals, such as copper, are weighed. 1 ton-T. 1: 16= 256. gr. 1 28 448 7168. 1 hun. wght.-cwt. 1= 4= 112— 1792— 28672. cwt. T. 1-20-80-2240-35840-573440 IV. APOTHECARIES' WEIGHT. This weight is used by apothecaries and physicians in mix ing and preparing medicines. This is the measure used for measuring all kinds of cloth. This measure is used for measuring all liquors, with the exception of beer and ale. The gallon contains 231 cubic inches. VII. ALE OR BEER MEASURE. Besides ale and beer, milk is measured by this measure. The gallon contains 282 cubic inches. This measure is used for measuring all kinds of grain, fruit, salt, coal, &c. The following denominations and numbers are used for measuring distance. 40 rods 1 furlong-fur. 8 furlongs 1 mile-m. L. m. 18-320-1760 5280 63360— 190080. 3 miles 1 league-L. 1-3-24-960-5280-87120-1045440-3136320. 60 geographic, or 691 statute miles, make 1 degree on the earth's surface. 360 degrees make the earth's circumference. X. LAND OR SQUARE MEASURE. TABLE. Sq. ft. 144 square inches (Sq. in.) make 1 square foot-Sq. ft. 1 = Sq. in. 144. 2724= 39204. rd. 1 30+ square yards 1 square rod-Sq. rd. 1304 R. 40 square rods-1 square rood-R. 1= 40—1210 =10890 1568160. A. 4 square roods-1 Acre-A. 1-4-160-4840 640 square acres-1 square mile. Land is usually measured by Gunter's chain, which is 4 rods or 66 feet in length. The whole chain is divided into 100 equal parts, called links. The link is therefore part of the rod, and is 79 inches in length. 80 chains, or 320 rods, make 1 mile in length. 1 square chain makes 16 square rods, and 10 square chains make 1 acre. XI. SOLID MEASURE. This measure is employed in measuring substances which have three dimensions; viz. length, breadth, and thickness. Timber, stone, &c., are among these substances. 40 feet of round or 50 feet of hewn timber make 1 ton-T. 128 solid feet make 1 cord-C. A pile of wood 8 feet long, 4 feet wide, and 4 feet high, contains just one cord, since 8X4X4 128. |