PROB. XVI. To find the volume of a cylindrical pipe. Multiply the area of the annular surface, formed by the internal and external diameters of the pipe, by the length. Ex. What volume of metal is there in a pipe 12 inches diameter (outside), 1 inch thick, and 12 inches long? 22 × 2 × 785434.5576 sq. inches, area of ring. Volume 34.5576 × 120=4146.912 cubic inches. PROB. XVII. To find the solidity of a prism. Ex. What is the solidity of a triangular prism whose base is an equilateral triangle, each side being 4 feet, and height 10 feet? To find area of base 4+4 +4 = 12, 6-4-2, 6-4 = 2, 6 × 2 × 2 × 2 = 48 cubic feet. sq. 12 = 6, 6-4=2, 2 ft., 48 x 10 = 480 PROB. XVIII. To find the solidity of a cone or pyramid. Multiply the area of the base by one-third of the height. Ex. 1. Find the volume of a cone, the diameter at the base being 10 feet, and height 9 feet. To find area of base. 10 x 10 x 785478·54 sq. feet. Volume 78·54 × 9 ÷ 3 = 2356 2 cubic feet. = Ex. 2. Find the volume of a square pyramid, each side of the base being 4 feet, and height 12 feet. Area of base 4 x 4: = 16 sq. feet. 16 x 12÷ 3 = 64 cubic feet. PROB. XIX. To find the solidity of a sphere. Ex. What is the solid contents of a sphere, the diameter of which is 10 feet? 10 x 10 x 10 x 5236 523.6 cubic feet. CHAPTER V MEASUREMENTS Units of Measurement.—To measure a line, it is first necessary to decide upon some length which is to be considered as a unit, and to which other lengths may be compared. The unit of a yard, a foot, or an inch is in the majority of cases adopted for English measurements, but in nearly all surveying measurements a special unit, known as a link, is taken. It was originally used by a Mr. Gunter, who adopted it to facilitate the working out of areas, and the measure consisting of 100 links is called a chain. A link is 7.92 inches in length, and 100 of these, which go to form a chain, will be 22 yards, or 66 feet in length. The reason for adopting such a unit will be apparent, when it is known that there are 10 sq. chains or 100,000 sq. links in an acre, and that there are 80 chains in a mile (linear), and 640 acres in a square mile. As a matter of fact all chains are "link" chains, but when the term has been applied in this book the Gunter's chain is meant. Methods of Measuring.-There are various methods of measuring lines, viz.-pacing, the measuring wheel, the tape, the chain, the steel band, and rods. Pacing. By walking along the line to be measured, and by counting the number of steps taken, a rough calculation can be made. After a little practice a fairly accurate measurement can be made in this manner, and many persons can measure 100 yards with an error of less than 1 yard. The Measuring Wheel. This instrument consists of a wheel some 2 feet in diameter, which is provided with a handle, by means of which the wheel is pushed along the ground. The number of revolutions which the wheel makes is registered by an index worked by toothed wheels. On even ground it gives fairly good results. The Tape. This is used by many persons for ascertaining lengths, but if great care be not taken, it will sometimes cause considerable errors-as much as 1 foot in 100. When kept dry and constantly checked it is a very simple and convenient method of measuring. By surveyors it is principally used for taking offsets, measuring buildings, or for any short lengths. The Steel Band.- -This is much used in the mines of America, and the accuracy of the measurements effected by it is perhaps greater than by a chain. Its greatest disadvantage is its twisting or kinking propensity, which often causes great annoyance. Thus, Rods.-Rods were formerly much in use, when accurate measurements were required, but of recent years surveyors, for the most part, have considered well-constructed chains satisfactory. The rods were usually constructed of deal, but on occasions requiring great accuracy glass rods were used. in 1874, the base line of the Trigonometrical Survey of the United Kingdom was measured with glass rods. The same line was again measured with a carefully constructed steel chain, and the difference was little more than half an inch in a base line of 27,404 feet. The Chain.-We thus see that a chain of good construction and constantly checked gives an accurate measurement, and, as it is a very convenient method of measuring, is the one adopted for general work by surveyors in this country. The chain is usually 100 links in length, as previously stated, but for surface work a chain consisting of 100 divisions of a foot each is sometimes used. Each division consists of a bar of iron or steel, and the bars are connected together by small coupling links. Thus the length of the bar in a link chain is about 6 inches, and in a foot chain about 10 inches, the remaining length being made up by the coupling links. This allows of the chain being folded up for carrying, as shown in Fig. 64. The reading of both the link and the foot chain is effected in the same manner, and as it is often a source of difficulty to the beginner, an endeavour is made, with the aid of the sketch (Fig. 65), to explain the procedure. At every tenth division on the chain a tag or index of brass about 11 inches in length is suspended, and as the number of tens from the beginning of the chain is known by the shape of the tag, to read off a certain length, all that is required is to find the preceding tag, and knowing the number of tens which it represents, to add on the odd number of feet or links. But from requiring tags to represent certain numbers on the chain arises a difficulty, viz.The same end of the chain should be in front each time, but as it is very convenient to have a chain so that any end may be used as a leader, a compromise is effected, and instead of having nine different kinds of tags to represent the tens, from 10 to 90, there are only five. Thus one tag represents both 10 and 90, another 20 and 80, another 30 and 70, and still another 40 and 60, 50 having a special tag for itself. The index for 10 and 90 has 1 point or finger 80 2 points,, fingers وو دو 20 A 50 is rounded off at the bottom. When measuring, it is very easy to tell whether the index is 10 or 90, 20 or 80, or 30 or 70, by noticing the amount of chain run out; but in case of the index for 40 and 60, unless care is taken a mistake may occur, as there is not as much difference between these lengths as the others. As an example of how the chain is : read if A is the front or leading end of the chain and the measurement is 33, if B be the leading end the same point will be 67. Staffing Out.-It will be apparent that a straight line of any considerable length could not be measured without some marks to keep the direction. In many cases a conspicuous tree, chimney, or steeple serves to keep the direction, but this does not always come in the line of sight, and some other means must be provided. Staffs of from 7 to 10 feet in length, shod with iron, and painted in feet alternately red, white, and black, so as to be seen distinctly, are fixed in the ground at intervals of about 2 chains or more, exactly in the line which it is required to measure. Assume that the distance between A and E (Fig. 66) is required, and that a staff placed at E can just be seen from the point A, but not distinctly enough for the measurers to see quickly, then a few more staffs must be placed in the same line between them. To accomplish this the surveyor stands at the staff at A, and fixes his head so that he can look with an eye on each side of the staff. An assistant then holds a staff at some point between A and E, and is directed by the surveyor to the correct line, as he will be able to see exactly when the staff is in line with A and E; this takes place when the staff E is hidden from sight by the intermediate staff. In the same manner more intermediate staffs may be fixed. Again, assume that a line AB is chosen, and it is required to produce it in the direction of E. The surveyor takes a staff and FIG. 66. goes forward to C, looks back in the direction of A, and places the staff in the position at which B hides A from sight. He |