Q. When a load of wood contains 128 solid feet, what is it culled? A. 1 cord. 3. How many solid feet in a pile of wood 8 feet long, 4 feet wide, and 4 feet high? A. 1281 cord. How many cords of wood in a pile 8 feet long, 4 feet wide, and 8 feet high? A. 256 solid feet = 2 cords. Q. In common language, we say of load of wood brought to market, if it is 8 feet long, 4 feet high, and 4 feet wide, that it is a cord, or it contains 8 feet of wood. But this would make 128 solid feet; what, then, is to be understood by saying of such a load of woood, that it contains 8 feet of wood? or, in common language, "there is 8 feet of it." A. As 16 solid feet, in any form, are of 128 feet, that is, § of a cord, it was found convenient, in reckoning, to call every 16 solid feet 1 cord foot; then, 8 such cord feet will make 123 solid feet, or 1 cord, for 8 times 16 are 128. Q. How, then, would you bring solid feet into cord feet? A. Divide by 16. 4. How many cord feet in a pile of wood 8 feet long, 2 feet high, and I foot wide? How many in a load 8 feet long, 2 feet high, and 2 feet wide? 8 feet long, 4 feet wide, and 2 feet high? 5. If, in purchasing a load of wood, the seller should say that it contains 3 cord feet, how many solid feet must there be in the load? How many solid feet to contain 4 cord feet? 5 cord feet? 6 cord feet? 7 cord feet? 8 cord feet? 9 cord feet? 6. How many cord feet in a pile of wood 8 feet long, 1 foot wide, and 4 feet high? In performing this last example, we multiply 4 feet (the height) by 1 foot (the width), making 4; thon, this 4 by 8 feet (the length), making 32 → 16 (cord feet), 2 cord feet, Ans. But, instead of multiplying the 4 by the 8 feet in length, and dividing by 16, we may simply divide by 2, without multiplying for the divisor, 16, is 2 times as large as the multiplier, 8; consequently, it will produce the same result as before, thus: 4X1=4÷÷2 =2 cord feet, Ans., as before. Q. When, then, a load of wood is 8 feet long, or contains two lengths, each 4 feet (which is the usual length of wood prepared for market,) what easy method is there of finding how many cord feet such a load contains? A. Multiply the height and breadth together, and divide the product by 2. 7. How much wood in a load 8 feet long, 3 feet high, and 2 feet wide? 3 X 2 6÷2 3 cord feet, Ans. 8. How many cord feet in a load of wood 2 feet high, 2 feet wide, and of the usual length? 3 feet high and 2 feet wide ? 3 feet wide and 3 feet high? 4 feet wide and 4 feet high? 4 feet wide and 6 feet high? How many cords in a load 4 feet high, 4 feet wide? high, 9. How wide must a load of wood be, which is 8 feet long and 1 foot high, to make 1 cord foot? How wide to make 2 cord feet? 3 cord feet? 6 cord feet? 10 cord feet? 10. What will a load of wood 8 feet long, 3 feet wide, and 4 feet high, cost, at $1 per foot? The foregoing remarks and illustrations may now be embraced in the following RULES. I. How do you find the contents of any solid or cube? A. Multiply the length, breadth and depth together. II. When the length of wood is 8 feet, how can you find the number of cord feet it contains, without multiplying by 8 and dividing by 16? A. Multiply the breadth and height together, and divide the product by 2; the quotient will be cord feet. III. How do you bring cord feet into cords? A. Divide by 8. Note. If the wood is only 4 feet in length, proceed as last directed; then, as 8 feet in length is 2 times as much wood as only 4 feet in length, hence the result found, as above, will be the answer in cord feet; that is, divide by 2 twice, or once by 4. Exercises for the Slate. 1. How many solid feet in a load of wood 8 feet long, 4 feet wide, and 3 feet high? 4 X 3) = 14 ÷ 27 cord feet, Ans. 2. How many feet in a load of wood 5 ft. 6 in. high, 3 ft. 9 in. wide, and of the usual length ? = (Reduce the inches to the decimal of a foot.) A. 10-325-10% ft. Perform this last example by reducing the inches of a foot to a common fraction. This method, in most cases, will be found preferable: thus, taking the last example : 5 ft. 6 in. 5 ft.-; then, 3 ft. 9 in. =34 ft. = 15 × 11 185 ÷ 2 = 182 = 105, Ans., as before. 165 3. In a block 8 ft. 6 in. in length, 3 ft. 3 in. wide, and 2 ft. 9 in. thick, how many solid feet? A. Decimally 75,96875 feet=75 feet. By common fractions; × 3 × U24317531 feet, Ans., as before. 4 1 4. If a load of wood is 8 feet long and 3 feet wide, how high must it be to make 1 cord? In this example, we know that the height multiplied by the width, and this product divided by 2, must make 8 cord feet, that is, 1 cord or load; hence, 2 = 16÷÷÷ 3 = 5fect, height, Ans. 8 5. If a load of wood is 53 feet high, and 8 feet long, how wide must it be to make 2 cords? 2 cords = 16 cord feet; then, 16 × 2 = 32 ÷ 53 6 feet wide, Ans. 6. If a load of wood is 53 feet high and 8 feet long, how wide must it be to make 3 cords? (9) 4 cords? (12) 8 cords? (24) A. 45 feet. 7. How many solid feet of timber in a stick 8 feet long, 10. inches thick, and 6 inches wide? (3) 10 feet long, 12 inches thick, and 1 ft. 3 in. wide? (123) 20 ft. 6 in. long, 24 inches wide, and 1 ft. 9 in. thick? (713) A. 87 ft. 8. In a pile of wood 10 feet wide, 3 ft. 3 in. high, and 1 mile long, how many cord feet, and how many cords? A. 10725 cord feet = 1340§ cords. 9. How many tons of timber in 2 sticks, each 30 feet long, 20 inches wide, and 12 inches thick? A. 100 feet 502 tons. 10. How many bricks 8 inches long, 4 inches wide, and 21 inches thick, will build a wall in front of a garden, which is to be 240 feet long, 6 feet high, and 1 foot 6 inches wide? A. 51840 bricks. DUODECIMALS. 1 LXXXI. Q. From what is the word duodecimals derived? A. From the Latin word duodecim, signifying twelve. Q. In common decimals, we are accustomed to suppose any whole thing, as à foot, for instance, to be divided into ten equal parts; but how is a foot dividea in duodecimals? and what are the parts called? A. Into twelve equal parts, called inches or primes, and each of these parts into twelve other equal parts, called seconds; also each second into twelve equal parts, called thirds, and each third into twelve equal parts, called fourths, and so on to any extent whatever. A. Fractions of a foot. Q. What, then, are duodecimals? Q. What fraction of a foot is 1 second? A. TE of Th=1}Ţ ft. ΤΣ ΙΣ उठ A. Tz of T'z of 12 of 12 = 20436 A. th= got36 Q. Now, since 12ths multiplied by 12ths make 144ths, and make TZ, 2 1728 12 144 also, 144ths multiplied by 12ths make 1728ths, and make TI, it is plain that we may write the fractions without their denominators, by making some mark to distinguish them. What marks are generally used for this purpose? A. 12ths, inches, or primes, are distinguished by an accent, thus; 8' signifies, 8 inches, or 8 primes; 7" Tx, 144, or 7 seconds; 6" 1728. or 6 thirds, &c. 6 ་ Q. We have seen that 12ths multiplied by 12ths produce 144ths; what, then, is the product of 5' (inches or primnes) multiplied by 7' (inches)? A. 35", that 35 is, 35 seconds, or T44 Q. What is the product of 5" (seconds) multiplied by 7' (Inches)? A. 35", that is, 35 thirds. Q. What is the product of 5'' (seconds) multiplied by 7" (seconds)? A. 35, that is, 35 fourths. Q. How may the value of the product always be determined? A. By placing as many marks or accents at the right of the product as there are marks at the right of both multiplier and multiplicand counted together. Q. What, then, would 7||||| (fifths) multiplied by 8|||||| (sixths) produce? A. 56//////////, that is, 56 elevenths. Q. What would 7" (seconds) multiplied by 5'' (thirds) produce? A. 35, that is, 35 fifths. Q. What would 8" multiplied by 3 produce? A. 24, (fourths.) Q. From the preceding, what appears to be the value of feet multiplied by primes or inches, or what do feet multiplied by primes give? A. Primes. Q. What do primes multiplied by primes give? A. Seconds. Q. What do primes multiplied by seconds give? Q. What do seconds multiplied by seconds give? A. Thirds. A. Fourths. Q. What do seconds multiplied by thirds give ? A. Fifths. Q. What do thirds multiplied by thirds give? A. Sixths. Note. This might be extended in the sume manner to any indefinite length. The following table contains a few of these denominations. Repeat the 127 (seconds) 1" (second.) 1' (inch or prime.) 12' (inches or primes) 1 foot. Q. How may duodecimals be added and subtracted? A. In the same manner as compound numbers; 12 of a less denomination always making 1 of a greater, as in the foregoing table. MULTIPLICATION OF DUODECIMALS. Q. What are duodecimals used for? A. For measuring any thing respecting which length and breadth, also depth, are considered. 1. How many square feet in a board 10 ft. 8 in. long, and 1 ft. 5 in. broad? We have seen how such an example may be performed by common decimals ; we will now perform it by duodecimals. OPERATION. Length, 10 ft. & Breadth, 1 5' 4 5/ 4// 10 8/ Ans., 15 1' 4" then, Xs of a foot, that is, 40" (seconds) = 3′ (inches) and 4" (seconds); we now write down 4 at the right of the inches, reserving the 3' to be carried to the inches. In multiplying 10 feet by or 50' (inches), and the 3' we reserved 4 feet and 5', which we place under feet and inches in their proper places. Then, multiplying 10 ft. 8 hy 1 ft. makes 10 ft 8', which we write under the 4 ft. 5'. We now proceed to add these two products together, which, by carrying 12, after the manner of compound rules, make 15 ft. 1' (inch) 4 (seconds), the Answer. It will be found most convenient in practice to begin by multiplying the multiplicand first by the feet, or highest denomination of the multiplier, then by the inches, c., thus: 4 5/ 4// 15 1/ 1X8 8', and 1 ft. X 10 ft. 10 ft. Then, 5' X 8=40 / 3', (to carry,) and 4", (to write down); 10X550+3' (to carry) 53 4 ft. and 5', which we write down underneath the 10 and 8. Then, the sum of these two products, added together as before, is 15 ft. 14" Ans., the same result as the other. Note. Had we been required to multiply 15 ft. 1/4 by feet and inches again, we should have proceeded in the same manner, carrying (thirds) one place further towards the right, and (fourths) another place still, and so on. From these examples we derive the following RULE. How do you multiply in Duodecimals? A. Begin with the highest denomination of the multiplier and the lowest denomination of the multiplicand, placing the first figure in each product one place further towards the right than the former, recollecting to carry by 12, as in compound rules. More Exercises for the Slate. 2. How many feet in a board 2 ft. G' wide, and 12 ft. 3' long? Ans. 30 ft. 7′ 6′′. 3. In a load of wood 8 ft. 4' long, 2 ft. 6′ high, and 3 ft. 3' wide, how many solid feet? A. 67 ft. 8' 6". Note. Artificers compute their work by different measures. Glazing and mason's flat work are computed by the square foot; painting, paving, plastering, &c. by the square yard; flooring, roofing, tiling, &c. by the square of 100 feet; brick work by the rod of 16 feet, whose square is 2721; the contents of bales, cases, &c. by the ton of 40 cubic feet; and the tonnage of ships by the ton of 95 feet. 4 What will be the expense of plastering the walls of a room 8 ft. 6' high, and each side 16 ft. 3' long, at $,50 per square yard? A. $30,694+. 5. How inany cubic feet in a block 4 ft. 3′ wide, 4 ft. 6' long, and 3 ft. thick? A. 57 ft. 4′ 6′′. 6. How much will a marble slab cost, that is 7 ft. 4' long, and 1 ft. 3' wide, at $1 per foot? foot? A. $9,163. 7. How many square feet in a board 17 ft. 7' long, 1 ft. 5/ #ide? 9. 24 ft. 10′ 11′′. |