To find the solidity of the middle frustum of an RULE. To the square of twice a diameter, taken half way between the middle and the end, add the squares of the middle and end diameters; multiply the sum by the length, and again by 1309, the last product is the solidity required. NOTE E F G H in the preceding figure represents the roiddle frustum, of an elliptic spindle. Examples. 1. What is the solidity of the middle frustum E G F H, length 10; diameter at the end 4; diameter C D 6, and diameter taken at O P, equal to 5? Diam. O P 5×2=10×10=100 sq. of twice O P. Diam. C D 6x 6 36 sq. of = C D. Diam. E F or G H 4X 4 16 sq. of E For G H. 152 sum. Sum. 152x10x 1309-198.968 Ans. CASE XVI. Timber measure, the common method. RULE. The common method of measuring round timber, (in order to ascertain the quantity of hewn timber, that is contained in a round stick) has been to girt the stick in the middle with a line, and then to double the line into four equal parts, (and one of these parts is considered equal to one side of the square timber that can be hewn from such a stick); and one of these parts, (or one fourth of the circumference) is multiplied into itself and again by the length of the stick, the last product is called the solidity. If of the girt in inches, is multiplied into itself and again by the length in inches, the last product must be divided by 1728, the quotient is the solidity in feet: if the girt in inches is squared and multiplied into the length in feet, the product must be divided by 144; the quotient will be the solidity in feet, NOTE. This method of measuring round timber is very erToneous; it gives two much for the solidity of square timber in the stick and not enough if the four slabs, or segments are to be included; I shall prove in case 17th. Examples. 1. What solidity of square timber, can be hewn from a round stick that is 21 ft. long, and its circumference, or girt line 48 inches? Girt 4812X12144×21-3024-144-21 ft. Ans. 2. What is the solidity of square timber in a round stick that is 30 ft. long, and its circumference or girt measures 24 inches? 7 ft. Ans. 3. What is the solidity of a stick of timber that girts 56 inches, and its length being 11 ft. 9 inches? Ans. 15 ft. 11' 11", or 15 ft. 4. What is the solidity of square timber in a stick round timber, that girts 50 in. and is 31 ft. 7 in, long Ans. 34 ft. 3' 2" 10" 9""" or 3418 CASE XVII. L868 69T2 A new method of finding the solidity of square timber that is contained in a stick of round timber, or to find how much the stick will measure after it is hewn square. RULE-Girt the stick (after taking off the bark,) and annex a cypher to the girt in inches and divide the whole by 44 the quotient is a side of the greatest square that can be hewn from such a stick; multiply this side into itself, and this product again by the length, and the last product will be the solidity of the stick after it is hewn. Illustration of the preceding rule. Make a circle whose diameter is 20, and of course its circumference will be 62.8†, (which will represent the girt of a stick of timber); annex a cypher to the circumference and divide by 4-4, the quotient is 14 and nearly 3 tenths; now make the largest square in this circle that can be made in it, and you will find that the sides of the square will be 14.3, very nearly. If the same circum. ference were divided by the quotient would be 15.7 for a side of square timber that could be hewn from such a stick. 15·7-14:314 difference. The following figure will further illustrate the preceding rule. Diameter A B or C D 20, and the sides of the square A D, or D B, or B C, or C A 14.3, and are sides of the greatest square that can be made in the circle. Examples. 1. What solidity of square timber can be hewn from a stick that is 20 feet long, and girts 62.8 in.? Girt 62.804414-3 nearly 14-3204-49xby length 20 4089-80 14428-4f solidity, Ans. The same question solved by case sixteenth. Girt 628157X157-246-49Xlength 204929-80÷ 14434-27 solidity, Ans. NOTE.-By comparing the two preceding answers, it is evident that the last method of operation gives too much for the solidity of square timber in such a stick. ft. Solidity found by using the circum. 34-2† 28.41 difference 5.8 2. What is the solidity of square timber in a stick of round timber, that is 21 feet long, and 48. inches in circumference ?. Girt 48-0-44-10.9† a side of sqr- timb. in the stick. 10.9x109x21-2495-01-144 17 32+ solidity, Ans, The same question solved by using of the circum. 12X12×21=3024-144-21 ft. Ans. Girt 48 Difference 3.68 ft. too much. 3. What is the solidity of square timber in a stick of round timber, that is 11 ft. 9' long, and girts 56 inches? Girt 56·0-4-4=1ft. 0' 8't a side×1 ft. 0' 8"-1 ft. 1' 4" 5" 4. mult. by the length 11 ft. 9-13 ft. 1' 1" 2" 8" Ans. 4. What solidity of square timber is in a round stick, that is 31 ft. 7 inches long, and girts 50 in. ? 50-44 11+ 11' a, side. Ans. 26 ft. 6′ 5′′ 7. NOTE.-Having shown by the two first questions in this case, that using of the circumference for a side of square timber,. gives too much for the solidity, or more than the stick will meas~ ure after it is hewn square; It now remains to show that of the circumference multiplied by itself, and then by the length, does not produce the solidity of the stick, if the four segments, or slabs are to be included. Examples.. 1. What is the solidity of a stick of timber whose length is 20 ft. circumference 62.8 in. and diame ter, 20 in. measured by using of the circumference.. Ans. 34.2 ft.. The same stick measured as a cylinder. Ans. 43.5 ft.. Exact solidity of the stick 43.5† ft. diff. 9.3 ft.. NOTE. It is obvious from the preceding examples that the square of the circumference, multiplied by the length, gives too much for the solidity of square timber in the stick, (or more than it would measure if it were hewn square); and not enough if the four slabs, or segments, are to be included: It ought to be a matter of consideration, by those concerned in buying and selling timber, how it should be measured; and if it is agreed by the parties to measure only the square timber, (or what the stick would measure if it were hewn) then apply the rule in case 17th ; but if it is agreed to measure the whole solidity of the stick (including the segments, or four slabs),then apply the rule in the following case. CASE XVIII. To find the solidity of a round stick of timber, including the four slabs, or segments. RULE.-Girt the stick in the middle (after taking off the bark), annex two cyphers to the girt, or circumference, and divide by 3.14 the quotient is the diameter nearly; multiply the girt, or circumference and diameter together, and one fourth part of the product multiplied into the length will be the solidity required. Examples. 1. What is the solidity of a stick of timber, that girts 94.2 in. and length 20 ft.? 94.20 3.14-30 diam. 94.2×30÷4706-5×20÷÷144 98,18 ft. Ans. that 2. What is the solidity of a stick of timber, is 22 ft. long; and the girt line measuring 31.4 in.? Ans. 111. NOTE. In all the preceeding examples in timber measure, the timber has been considered of equal bigness from end to end; it now remains to treat of tapering timber, both round and hewu. CASE XIX. To find the solidity of hewn timber in a round stick, when the stick is tapering from end to end. RULE.-Girt the stick at both ends, annex a cypher to the 'girts, or circumferences, and divide each girt by 44 the quotients will be sides of square |