Where to be obtained.-Ordnance maps may be purchased of Edward Stanford, 12, 13, 14, Long Acre, London, W.C. To find the Area, from a plan, of a figure bounded by Irregular Lines. There are various methods : 1. By dividing its boundaries into a number of short lengths, each of which shall be as nearly as possible straight: then treating the whole figure as being bounded by so many straight FIG. 45. lines, dividing it into triangles, and calculating the area of each triangle as described in Chapter 1. 2. By the formation of a polygon whose sides follow the boundaries as nearly as possible, working out the area of the polygon and of the spaces between its sides and the boundaries, and adding or deducting the results as required. 3. By laying down 3.70 4.50 4.35 4.98 5.12 4.60 FIG. 46. equalising' or 'give and take' lines, i.e. straight lines which straighten the boundary by taking in as much as they leave out (see Fig. 45). The area of the polygon will then be equal to the area of the survey. This method is sometimes called casting,' and is frequently adopted. 4. By means of parallel lines, 1 chain apart. (See Fig. 46.) Draw a straight line touching the top of the figure; next draw lines parallel to this line at one chain apart according to the scale of the plan: then draw cross parallels as shown at the bottom of the figure in order to deal with those parts which are less than 1 chain wide. draw give-and-take lines to equalise the boundaries. the lengths between these equalising lines and add them together; and the result is the area in square chains. Now Measure If the drawing of these lines is likely to damage the plan, then a piece of tracing paper can be pinned over the plan and the lines drawn on this. This is an accurate and easy method of ascertaining areas of difficult shape. 5. By Simpson's Rule.-Divide the line AB up into an even number of equal parts, and take offsets A D FIG. 47. from each of these divisional points to the curved boundary, i.e. an odd number of ordinates. (See Fig. 47.) Then Area = (first offset + last offset + 2 x sum of all other odd offsets + 4 x sum of all the even offsets) of the common distance D between the two adjacent offsets. This rule, which is based on the assumption that the curved boundaries of the figure consist of short parabolic arcs, gives accurate results. 6. By the Computing Scale.-This very useful instrument is generally divided so as to measure the areas of figures to one or two scales only, but a Universal Computer has been designed to overcome this difficulty. With the latter only one computing frame is required, and as many scales as are wanted, such as 1, 2, 3, 4, 5, 6, chains to the inch, 6 inches to the mile and 25.344 inches to the mile can be inserted. (See Fig. 48.). To use the instrument a sheet of tracing cloth is required which is ruled with lines at a distance of one B chain apart for scales of 1, 2, 3, 4 chains to the inch. For smaller scales the distance between the lines should be ruled to 4-inch spaces and the computer divided to read to this. This piece of transparent paper (generally made of horn paper, as tracing cloth is not sufficiently firm) is placed over the plan in such a way that the area to be computed lies exactly between the parallel lines (see Fig. 49). Fix the paper with weights or pins so that it does not slip. The area is now divided up so that by adding together the at the area. into strips of 1 chain in width, lengths of each strip we arrive Place the computer upon the horn paper, so that the scale is parallel to the lines on it, and the wire on the slide with its index E F H FIG. 49. at zero is in the position AB, thus equalising the boundary as shown. Now move the index to the position CD. Move the scale bodily, with the reading of the index left upon it, and place it so that the wire is in the position EF sliding it along as before to GH. Proceed thus to the end. If the full length of the scale is used before the area is covered, make a mark on the sheet and note on a piece of paper the area already arrived at. Begin with the index at zero where you left off and proceed as before until the whole area has been covered. Computing sheets can be obtained from the instrument makers for the several scales required, thus saving the trouble of ruling. It is a most useful instrument. By the Planimeter.-This instrument, invented by J. Amsler, is of a delicate nature and does not as a rule find a place in the estate office. If it is of the fixed kind it gives the result in inches, if of the proportional kind the bar must first be set to suit the nature of the computation. The following instructions are given by the makers, and are reproduced here by their kind permission. No. 1 Planimeter (brass) and No. 2 Planimeter (german silver), arranged for measuring areas in square inches (or other unit if specially desired). Needle-point outside the diagram.-Put the instrument on the drawing surface, set the tracing point F at a mark on the outline of the area, press the needle-point E slightly into the paper outside the curve, and read off the roller D and the counting wheel G taking the whole circumference of the recording roller as unit of reading (the roller need not be set to zero). Readings. The drum of the roller is divided into 100 parts. The tenths of a part are read on the vernier. The complete turns of the roller are read on the counting-disc which advances one line at every such turn. The counting-disc performs one revolution at every 10 turns of the roller. FIG. 50 Range: Circle of 18 inches diameter. Each complete reading is a figure of four digits, the units being read on the counting-disc, the tenths and hundredths on a drum, and the thousandths on the vernier. The reading of the roller and disc, shown for example in the adjoining figure, is 5·343. (The zero of the drum and a mark of the disc should simul taneously be opposite the fixed index mark. This is never exactly the case in consequence of the imperfection of the worm wheel-gear, and this should be taken into account in making readings in the same way as with a watch, when the minute-hand points to 12, whilst the hour-hand fails to indicate the exact hour.) Then move the tracing-point F round the area in the direction of the movement of the hands of a watch, till it reaches again the starting point. Now take another reading and subtract the first from the second reading. The difference multiplied by 10 will then be the area of the curve in square inches. Example. To measure the area of a square of 4 inches side. If the reading before starting had been 9.521 then the reading after circumscribing the same square as before would be 1.121. As the travel of the roller is in both cases the same, it is evident that the zero mark of the wheel has passed the fixed index-mark and that, consequently, the second reading has now to be supplied with one more digit on the left before subtracting the first reading. Thus Complete second reading 11.121 9.521 1.600 × 10 = 16 sq. ins. Needle-point inside the diagram.-Circumscribe the diagram with the tracing-point in the direction of the movements of the hands of a watch, watching at the same time the counter in order to see whether the total rotation of the roller is a forward or a back motion. This preliminary rough operation being completed, proceed as before explained, now following the curve carefully with the tracingpoint. If the total rotation of the roller has been a forward motion, subtract the first from the second reading and add the difference to the figure engraved on the top of the small weight used for keeping the needle-point in its place. The sum multiplied by 10 will then be the area of the curve in square inches. Example.-To measure a circle of 18 inches in diameter. (You will see that the total rotation of the roller is forward.) Figure on weight+16-431 (This figure is slightly different for different instruments.) 25.447 × 10 = 254.47 sq. ins. If the total rotation of the roller is a back motion, subtract the second from the first reading and subtract the difference from the figure on the weight. |