circular vault; span 36 feet, height 18 feet, thickness of walls at the spring 6 feet, thickness of crown 4 feet, and length of the vault 100 feet? Ans. { 1884.96 cubic yards solidity of vacuity. 2026 15 cubic yards solidity of materials. To find the surface and solidity of a dome, the height and dimensions of the base being given. RULE. Take twice the area for the surface, and multiply the area of the base by 3rd of the height for the solidity. NOTE. Although these rules are only true when the domes are hemispherical, yet they are sufficiently near the truth for all practical purposes. EXAMPLE. Required the surface and solidity of a hemispherical dome, the diameter of its base being 60 feet. Surface 628.32 square yards. Ans. Solidity 2094 4 cubic yards. NOTE 1. The surface of a saloon is found in the same manner as a vaulted roof. NOTE 2. Rules might have been here given for the measurement of haystacks, coal-heaps, &c; but these may be readily resolved into two or more of those solids, the methods of finding the content of which are given in the Mensuration of Solids, Part IV. Moreover, haystacks are usually sold by weight, and seldom or never by measurement. SPECIFIC GRAVITY. The specific gravity of bodies are their weights when compared with an equal bulk of pure water, which, at the temperature of 40°, weighs 1000 ounces avoirdupois per cubic foot. The following table, therefore, contains the weights of a cubic foot of several bodies in ounces. A TABLE OF THE SPECIFIC GRAVITY OF BOdies. To find the weight of a body from its bulk. RULE.-Multiply the content of the body, in cubic feet, by its tabular specific gravity for its weight in avoirdupois ounces. EXAMPLES. 1. Required the weight of a block of marble, whose length is 63 feet, and breadth and thickness each 12 feet, being the dimensions of one of the stones in the walls of Balbec. 63 x 122 x 2700 16 x 112 x 20 = 683 tons, which is equal to the burden of a second rate East India ship. 2. What is the weight of a block of dry oak, which measures Ans. 4336 lb. nearly. 10 feet long, and 3 feet by 21? To find the magnitude of a body from its weight. RULE. Divide its weight in avoirdupois ounces by its tabular specific gravity for its content in cubic feet. EXAMPLES. . 1. Required the content of an irregular block of freestone, which weighs 1 cwt. 2. How many cubic feet are there in a ton of dry oak? Ans. 381 cubic feet. 3. A cast iron pipe is 6 inches diameter in the bore and 1 inch in thickness; required the weight of a running foot. Ans. 67.45 lbs. LAND SURVEYING. DESCRIPTION OF INSTRUMENTS USED FOR MEASURING AND PLANNING SURVEYS. THE CHAIN. THE chain, usually called Gunter's chain, is almost generally used in the British dominions, for measuring the distances required in a survey. It is 66 feet, or 4 poles, in length, and is divided into 100 links, which are joined by rings. The length of each link, together with half the rings connecting it with the 66 adjoining links, is consequently of a foot, or 100 inches. At every tenth link from each end is attached a piece of brass with notches; that at the tenth link has one notch, that at the 20th two notches, that at the 30th three, that at the 40th four, the middle of the chain, or the 50th link being marked with a large round piece of brass; hence, any distance on the chain may be readily counted. Part of the first link, at each end, is formed into a large ring for the purpose of holding it with the hand. The chain acquires extension by much use, it should therefore, be frequently examined, and adjusted to the proper length by taking out some of the rings between the links: for this purpose, chains having three rings between each link are to be preferred to those having only two. THE OFFSET STAFF. The offset staff is used to measure short distances, called offsets; hence its name. It is usually ten links in length, the links being numbered thereon with the figs. 1, 2, 3, &c. It is usually pointed with iron at one end, for the purpose of fixing it in the ground, as an object for ranging lines, for marking stations, &c. THE CROSS. The cross is an instrument used by surveyors to erect perpendiculars. It is usually a round piece of sycamore, box, or mahogany, about four inches in diameter, with two folding sights at right angles to each other, or more commonly with two fine grooves sawed at right angles to each other, which answer the purpose of sights. It is sometimes fixed on a staff of convenient length for use, pointed with iron at the bottom, that it may be fixed firmly in the ground: but it is found more commodious in practice to have a small pocket cross, which may be readily fitted to the offset-staff, either by an iron spike on the cross being inserted in a hole made in the offset-staff, or the offset-staff being passed through a hole made in the cross, to about the eighth link from the piked end, at which place the staff must be shouldered, that the cross may rest firmly. DIRECTIONS FOR MEASURING LINES ON THE GROUND. Besides the instruments already described, ten arrows must be provided, about 12 inches long, pointed at the end, so as to be readily pressed into the ground, and turned at the other end, so as to form a ring to serve for a handle. In using the chain, marks are to be set up at the extremities of the line to be measured, as well as its intermediate points, if its extremities cannot be seen from one another, on account of hills, woods, hedges, or other obstructions. Two persons are then required by the surveyor to perform the measurement. The chain leader starts with the ten arrows in his left hand, and one end of the chain in his right; while the follower remains at the starting point, who, looking at the staff or staves, that mark the line to be measured, directs the leader to extend the chain in the direction of the staff or staves. The leader then puts down one of his arrows, and proceeds a second chain's length in the same direction, while the follower comes up to the arrow first put down. A second arrow being now put down by the leader, the first is taken up by the follower; and the same operation is repeated till the leader has expended all his arrows. Ten chains, or 1000 links, having now been measured and noted in the field book, the follower returns the ten arrows to the leader, and the same operation is repeated as often as necessary. When the leader arrives at the end of the line, the number of arrows in the follower's hand shows the number of chains measured since the last exchange of arrows noted in the field book, and the number of links extending from the last arrow to the mark or staff at the extremity of the line, being also added, gives the entire measurement of the line. Thus, if the arrows have been exchanged seven times, and if the follower have six arrows, and from the arrow last put down to the end of the line be 83 links, the whole measurement will be 7683 links, or 76 chains 83 links, which is usually written thus-76.83 chains, the two last figures being decimals of a chain. 7000 600 83 7683 In using the chain, care must be taken to stretch it always with the same tension, as it will extend by much use, and will therefore require to be examined occasionally, and shortened, if necessary. But a good chain may be used several days on tolerably smooth ground, without any material extension. The surveyor must mark, or caused to be marked, every station on the line, while it is being measured, with a staff or cross on the ground, entering its distance in the field book. When a survey is made for a finished plan, all remarkable objects should be noted down; as buildings, roads, rivers, ponds, footpaths, gates, &c. The boundary of the estate measured ought to be carefully observed. If the ditch be outside the boundary fence, it usually belongs to the estate, and vice versd; although this is not uniformly the case; therefore, inquiry ought to be made with respect to the real boundary. In some places five links from the hedge-post or roots of the quickwood are allowed for the breadth of the ditch, but this breadth varies to as far as even ten links, especially in swampy countries. All ditches and fences must be measured with the fields to which they belong, when the full quantity on the plan is required: but when the growing crops only are to be measured, only so much as is occupied by the crops. INSTRUMENTS FOR LAYING DOWN OR PLOTTING SURVEYS. THE COMMON DRAWING COMPASSES. This instrument consists of two legs moveable about a joint, so that the points at the extremities of the legs may be set at any required distance from one another: it is used to transfer and measure distances, and to describe arcs and circles. NOTE. There are several other kinds of compasses, used for planning, as those with moveable points, for the introduction of black lead pencils, or ink points, beam compasses for taking large distances, proportional compasses, &c., the uses of which are easily learned. (See Heather's Treatise on Mathematical Instruments.) PLOTTING SCALES. Plotting scales, also called feather-edged scales, are straight rulers, usually about 12 inches long. Each ruler has scales of equal parts, decimally divided, placed on its edges, which are made sloping, so that the extremities of the strokes marking the divisions lie close to the paper. The primary divisions represent chains, and the subdivisions ten links each, the intermediate links being determined by the eye. Plotting scales may be procured in sets, each with a different number of chains to the inch. They are usually made of ivory or box, and each provided with a small scale called an offset scale for laying down the offsets. In using these scales, the first division or zero, on the plotting scale, is placed coincident with the beginning of the line to be plotted, and so as just to touch that line with the featheredge: the end of the offset scale is then placed in contact with the edge of the plotting scale; and thus the offsets may be expeditiously pricked off: for which purpose an instrument called |