number may be multiplied or divided by 4, 9, or any square number, and the result divided or multiplied by 2, 3, or the square root of this number; or, again, the numbers may be both multiplied and divided by any of the same number, and the result divided or multiplied also by the same number, and, in each case, the required mean proportional will be correctly determined. Problem 8. To find the Area of a Board or Plank. First Method.-Set 12 on B to the mean breadth in inches on A, and against the length in feet on в will be found upon A the required area in feet and decimals of a foot. If the plank taper regularly, the mean breadth is half the sum of the extreme breadths; but, if the plank be irregular, several breadths should be measured at equal distances from each other, and their sum divided by their number may be taken as the mean breadth. In the latter case, however, greater accuracy would be obtained by finding separately the areas of portions of the plank, and adding them together for the whole area, or by the following. Second Method. Take the measure in inches of several breadths at equal distances from each other, and add together half the two extreme breadths, and the sum of all the intermediate breadths. Set 12 on B to the sum thus found upon A, and against the distance in feet, at which the breadths have been measured, upon в will be found upon A the required area in feet and decimals of a foot. Example 1.-A board, 15 feet long, being 14 inches broad at one end, and 8 inches broad at the other, required its area. The mean breadth is 11 inches, half the sum of 8 and 14. Set, then, 12 on в against 11 on A, and against 15 on в will be found upon A 1375 or 13 feet, the area required. Example 2.-An irregular board, 18 feet long, being 7 inches broad at one extremity, 11 inches broad at the other, and the intermediate breadths at each 3 feet of the length being 13 inches, 25 inches, 23 inches, 32 inches, and 22 inches, required its area. By the first method, the sum of the seven breadths divided by 7, gives 19 inches for the mean breadth; and, setting 12 on в against 19 on a, against 18 on в will be found upon A 28.5 or 28 feet, the area required. By the second method, half of the two extreme breadths added to the intermediate breadths, gives the sum, 123 inches; and setting 12 on в against 123 upon A, against 3 on B will be found upon A 30%, the area required, a more accurate result than the preceding. Problem 9. To find the solid Content of squared or four-sided Timber, of the same size throughout its entire Length. First Method.-Multiply the breadth by the thickness, and their product again by the length (Problem 1), and the result will be the content required. Second Method. Set the length on c against 12 on D, and against the quarter girt, measured in inches, on D, will be found the approximated content on c in cubic feet: or set the length on c against 10 on D, and against the quarter girt, measured in tenths of feet, on D will be found the approximate content on c. The approximate content thus found is greater than the true content, and the correction to be subtracted to leave the true content is given in the following Table : The fractional portion of the approximate contents in column 3 may be found by dividing the approximate contents by the denomination of the fractions. (Problem 2.) If the excess of the breadth over the thickness be compared with the quarter girt, the correction has to the approximate content the duplicate ratio of half the excess to the quarter girt, as shown in the following Table : 7 The correction may also be found as follows:-Set the length upon c, against 12, upon D, and against half the excess of the breadth over the thickness upon D, will be found upon c, the required correction in cubic feet. As the error of the result obtained with the rule may amount to theth part of the whole, the correction given above may always be neglected, whenever the excess of the breadth over the thickness does not exceed the 4th part of the breadth, or 1 inch for each 12 inches of breadth, and the result may be depended upon to as great an accuracy as can be obtained by the rule. When, however, the excess is more than two inches for each 12 inches of breadth, either the correction should be applied or the first method be used. Example 1.-Required the content of a piece of timber 10 inches broad, 8 inches thick, and 18 feet long. 10 8 80 Since X = set 80 ̊on B, against 144 on A, and against 18 on a, will be found 10 on B, and the content required is 10 cubic feet. Example 2.-Required the content of a piece of timber 15 inches broad, 10 inches thick, and 24 feet long. Set 24 on c, against 12 on D, and against 12.5, or 12, the quarter girt on D, will be found on c, 26.04, the approximate content. The excess of 15 over 10 being d of 15, our table shows the required correction to th of 26:04. Set then 25 on B, against 26-04 on A, and against 1 on B, will be found 104 on A, which subtracted from 26.04 cubic feet leaves 25 cubic feet, the true content. Problem 10. To find the Content of a piece of square timber, which tapers from end to end. Set the length in feet upon c, against 12 upon D, and against half the sum in inches of the quarter girts at the two ends upon D, will be found a content in cubic feet upon c. Again, set one-third of the length in feet upon c, against 12, upon D, and against half the difference, in inches, of the quarter girts at the two ends upon D, will be found a second content in cubic feet upon c. Add together the two contents thus found for the content required. If the breadth exceed the thickness considerably, the same part of the result must be subtracted, as in Problem 9. Example. The quarter girts at the ends of a piece of timber 21 feet long, being 22 inches and 10 inches, respectively, and the breadth not much exceeding the thickness, required the content. Set 21 upon c, against 12 on D, and against 16 upon D, will be found 373 or 37.3 upon c. Again, set 7 upon c, against 12 upon D, and against 6, upon D, will be 1 or 175 upon c. The sum of 37 cubic feet and 1 cubic feet, is then 39 or 391 cubic feet, the whole content required. Problem 11. To find the Content of a round piece of timber of the same size throughout its entire length.-Set the length in feet upon c, against 10.63 (a mark is placed upon the rule at this point, 10-63 being the quarter girt in inches of the circle, whose area is a square foot,) upon D, and against the quarter girt in inches upon D, will be found the content upon c. Example.-Required the content of a round piece of timber 32 feet long, the quarter girt being 11 inches. Set 32 upon c, against 10-63, upon D, and against 11, upon D, will be found upon c, 34-25 or 321, the content required. Problem 12. To find the Content of a round piece of timber, which tapers from end to end. Set the length in feet upon c, against 10-63, upon D, and against half the sum in inches of the quarter girts at the two ends upon D, will be found a content in cubic feet upon c. Again, set one-third of the length in feet upon c, against 10.63 upon D, and against half the difference in inches of the quarter girts at the two ends upon D, will be found a second content in cubic feet upon C. Add together the two contents thus found for the content required. NOTE. In buying rough or unsquared timber, an allowance of about th should be made for the bark. A further allowance should also be made for the loss in against 10-63 squaring down the tree to make useful shaped timber. The whole amount of timber to be taken off to make a square piece from a round piece of timber will be 36 per cent., or more than a third of the whole. The timber so taken off must not, however, be considered completely valueless. If the length upon c be set against 12 upon D, instead of upon 10.63 in the two preceding problems, this will be equivalent to an allowance of about 214 per cent., which may be considered a just allowance. Example 1.-A piece of round tapering timber measures 23 feet in length, the quarter girt at the larger end is 234 inches, and at the smaller end the quarter girt is 15 inches. Required the true content. Set 23 upon c upon D, and against 19, 19-5, or upon D will be found 77.5 upon c. Again, set 7} or 7.66 upon c against 1063 upon D, and against 8 upon D will be found 4·3 Then the sum of 77.5 cubic feet and 4.3 cubic feet is 81.8 cubic feet, the content required. Example 2.-Required the content of a piece of unsquared timber of the same dimensions as in the preceding example, making allowance of 21 per cent. for. loss in squaring down into a useful shape. Set 23 upon c against 12 upon D and against 19, or 195, upon D will be found 60.75 upon c. Again, set 7 or 7.66 upon c against 12 upon D, and against 8 upon D will be found 34 upon c. Then the sum of 60.75 cubic feet and 34 cubic feet is 64.15 cubic feet, the content required. upon c. Problem 13. To find the Content of a Cylindrical Vessel in Gallons.-Set the length of the cylinder in inches upon c against the gauge mark at 1879, marked G, upon D, and against the diameter of the cylinder upon D, will be found the required content in gallons upon c. If the number of inches in the diameter lie beyond c, or if this number be greater than 40, so as not to be contained upon D, theth part, or any part that may be convenient, of the number of inches in the diameter may be taken, and the result thus obtained, multiplied by 100, or the square of the divisor made use of, will give the content required. Example. A circular vat 5 feet in diameter being filled to the depth of four feet, required the quantity of liquor in it. Set 48 upon c against the gauge mark at 18.79 upon D, and against 6, the th part of the diameter in inches, upon D will be found upon c 49; and consequently 4.9 × 100 or 490 gallons, is the quantity of liquor in the vat. |