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 48:0-4.4=109t a side of sqr- timb. in the stick. 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 560_4.4=1ft. Oʻ874, a sidex1 ft. O'8"=1 ft. 1'4" 5' 4" : mult. by the length 11 ft. 9=13 ft. 1' 1'' 2'' gi" Ans. 4. What solidity of square timber is in a round: stick, that is 31 fte 7 inches long, and girts 50 in. ? 50; 44=114 11'a, side. Ans. 26 ft. 6'5" 7""'. this case, . NOTE.--Having shown by the two first questions that using of the circumference for a side of square timber,. gives too much for the solidity, or more than the stick will measure after it is bewn 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 diamė. ter, 20 in. measured by using of the circumference.. Ans. 34.27 ft. The same stick measured as a cylinder. Ans. 43.5+ ft. Exact solidity of the stick 43.57 ft. Erroneous solidity 34.27 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, ɔr 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, inclue ding 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. 94.2X30+1=706-5X20-144 =98,18 ft. Ans. 2. What is the solidity of a stick of timber, that is 22 ft. long ; and the girt line measuring 31•4 in. ? Ans. 111*1. 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 hewi. 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 cy. pher to the 'girts, or circumferences, and divide each girt by 4 + the quotients will be sides of square timber ; multiply the two sides together ; find the difference between the two sides ; square the difference, and add one third of its square to the product of the two sides, and multiply this sum by the length, the last product is the solidity required. Examples 1. How much hewn timber is in a stick of round timber, that is 24 ft. long, and its circumference at one end is 44 in. and at the other 22 in. ? Cir. 44:0;-4.410 a side of the largest square. square. 8added. 58į sum. 2. How much hewn timber is in a stick that is 21 feet long, and its circumference at one end is 88 in. and at the other 44 in. ? Ans. 34,6. ft. NOTE.-If it is required to ind how much hewn timber is contained in a round stick, allowing the square timber to be all its length of equal bigness, the round stick must be girted only at the smallest end. CASE XX. To find the solidity of a round stick of timber which is of a true fuper from end to end, including the four segments, or slabs. Rule.--Apply the rule for finding the solidity of the frustum of a round cone. (See Frustum of Cones, Case Sixth, Solids.) Eramples 1. What is the solidity of a stick of timber whose largest circumference is 63, and smallest 43 in. ; and is 21 feet long? Cir. 63.00:-3.145 20:07 diam. Cir. 43:00---3•141371 diam. nearly. Cir, and diam. 63x20=1260 =1=315 area, large end. Cir. and diam. 43x137589:1-1147+ area, small ende Areas 315X147=V 46305 = 2154 root add. 677 sum. Sum 677X21=1=4739--144=32411 ft. Ans. 2 Nore.--This method of operation would be too lengthy for common use; and if the following role be adopted the solidity may be found very near the truth (although a little too small) with much more convenience. RULE 2.-Girt the stick near the middle (but rather nearer the but end), the girt is the circumference; annex two cyphers to the circumference; and divide by 3•14 the quotient is the diameter nearly; multiply the circumference and diameter together, and of the product multiplied into the length will give the solidity very near. Examples. 1. What is the solidity of a round stick of timber (including the four segments which is 12 ft. long: and its middle girt, or circumference it 62.8 inches? Cir. 62:80-3•14=20 diam.; and 20 X 62.8=1256*0* 6=314 X12=3768 144=262 ft. Ans. 2. What is the solidity of a round stick of timber that is 10 ft. long; and its mean girt 31-4 in. including the four segments or slabs? Ans. 5,047 ft 3. What is the solidity of a round stick of timber, whose mean girt is 94.2 in.; and 30 ft. long? Ans. 147 4. ft. CASE XXI. To find the solidity of a hewn stick of timber that has all its sider parallel. Rule.-Multiply one side by the other, and the product multiplied by the length, will be the solidity. Examples. 1. What is the solidity of a stick of hewn timber that is 30 ft. 6 in. long ; and one side is 1 ft. 3 in. ; the other side 6 in. ? 30 ft. 6'x1 ft. 3'xo ft. 6'=19 ft. O' 9" or 19 (6) ft. Ans. NOTE.-Stone is measured in the same manner as square or bewn timber. 2. What is the solidity of a stone that is 12 ft. 6 in. long, 4 ft. 10 in. wide, and o ft. 9 in. thick? Ans. 45 ft. 3' 9". Note.--When two sides, of the stick are parallel, multiply the mean width and the other side together, and the product by the length. 3. What is the solidity of a stick of timber that is 12 in. wide at one end, and 8 at the other; and is 8 in. thick, and 3 ft. 4 in. long? Ans. 1 ft. 10° 2" 8'". CASE XXII. To find the solidity of a tapering hewn stick of timber, which has no two sides parallel. RULE.--Apply the rule for finding the solidity of the frustum of a cone or pyramid, (see case sixth;) if the stick is square at each end, measure it as in case 6th, rule 2. |