We have been particular in describing this process, because so many circular or cylindrical figures come under the mea surer's consideration, whether they be mirrors, arched passages, columns, &c. The contents of a pillar are easily ascer tained, even though its diameter may be perpetually varying ; for if we take the diameter in different parts, and strike a mean between every two adjoined mea surements, and multiply that mean area by the depth,or interval between the two, the solid contents will be found.
The contents of pyramids are measur ed by multiplying the areas of their ba ses by half their lengths : or their lengths by half the areas of their bases. Cones, whose sides are straight, are equal to one the solid contents ofcylinders, equal to them in base and altitude.
Solids, which have a certain degree of regularity, may be easily measured,: thus a cube is computed by multiplying first its width by its length ; then their sum by its height: thus acube, measuring four feet each way, would be 4X4=16x4=-_.
64. This is the meaning of what is called the cube root : see Cram Newark. Parallelopipedons, or solids of a long form, such as squared timbers, are mea sured by the same means: say that a tim ber be seven feet long, and at its ends be six inches by four. The area of either end, which isbere considered as the base, will give 24 squares inches, which multi plied by 84 (the number of inches in 7 feet) will show 2,016 solid inches. Divide by 1728, (the number of solid inches in a solid foot), and the result will be 1 foot 288 solid inches. But we have a short er way, when, as in the above instance, the parts are regular multiples ; for 6 by 4 is the sixth part of a superficial foot; consequently six feet in length of such a beam answers to one foot cube and the remainder will shew the sixth part of a foot cube ; so that we may indicate the %mount, either as above, or by calling it one solid foot and one-sixth. For the mensuration of grow ing timber, various modes have been of fered ; but we know of none more simple than that invented by Captain William son, and exemplified in his "Mathema tics Simplified." His practice has been to fix a short bat ten, at exactly 45 degrees, angular with a staff of about si feet long ; the latter be ing armed with a spike to fix it in the soil, and having a plumb line at one corner. When a sight taken along the batten, (the staff being exactly perpendicular,) points to the highest part of a tree, that is of the main trunk, measure the distance from the place where the staff is fixed to the place where the tree stands : the inter mediate distance, added to the length of the staff, will shew the height to which the timber is marketable. For it is evi
dent, that as an angle of 45 degrees gives equal base and perpendicular, so must the altitude correspondent with the distance between the junction of the batten with the staff to the tree, and a perpendicular from the part cut on the tree, by the line of sight, to the level of that junction, the length of the staff must correspond with the length of stem below that level. We beg leave to refer our readers to the pub lication above quoted for further particu lars on this head, as well as for numerous useful hints in regard to surveying in ge neral. See fig. 9.
After a tree has been felled, its girth is usually taken at each end, and at the mid dle, when there is no particular swell, or that the top extremity does not sud denly decrease. This rule may answer well ; but where the irregularity is great, it is better to take many more girths, and summing up the whole, to divide their amount by the number of girths taken, so as to establish a mean measurement. Divide that mean measurement by 4, to find the side of a square to which the tree will be reduced when prepared for the sawyer. If the whole solid contents are to be estimated, divide by 3, instead of by 4, and taking the third part, thus given, for a diameter, act upon it as alrea dy shewn, to find the side of a square, equal to the circle of which that ascer tamed third part is the diameter.
The greatest portion of mensuration appertains to the contents of solid bodies, or areas, such as hay-stacks, interiors of barns,granaries, &c. ; all of which come under die rule laid down for cubes, &c. When any sides fall in regularly, as in garrets, &c. the inclined part must be treated as a pyramid, or as a quoin, (or wedge), and the whole be summed up together. The contents of casks, tubs, Sic. are treated of under the head of GAUGING, (which see), and that part of our subject which appertains to the ad measurement of lands, as also to the dis tances, heights, &c. of remote objects, accessible or otherwise, will be found un der the head of Stravarraro.