into 5 primary divisions of 1 minute each and the left hand division into 12 secondary divisions of 5 seconds each. XI. A sketch has no scale on it. The distance between two points measured on the sketch is 3.6 inches. It is paced on the ground and found to be 1200 paces of 30 inches each. Construct the scale to read yards for the map and determine its R. F. 1000 The unit of measure is yards. 1200 paces of 30 inches each=36,000 inches 1000 yards, represented by 3.6 inches. 1 inch represents as many yards as 1880-277.77 yards, 6 inches would represent 6x277.77 yards=1666.62 yards. Take 1500 yards and substitute in proportion, 277.77 yards: 1500 yards ::1 inch :x=5.40 inches. Draw a line (Fig. 64) 5.40 inches long and divide into 15 primary divisions of 100 yards each, and the left-hand division into 5 secondary divisions of 20 yards each. To determine R. F. 3060-1000. XII. A map is drawn on a scale of R. F. to a supposed length of pace of 32 inches. The length of pace is afterwards found to be only 29 inches. Construct a correct scale for the map to read yards and determine its R. F. In a case of this kind, while the map itself may be correct, the scale of R. F. 5 is not the correct scale for it, since for every pace of only 29 inches an actual representation of 32 inches on the scale of R. F. was plotted, whereas but 29 inches to a scale of some other R. F., as the map stands, should have been plotted. In other words, no distance measured on the map and applied to the scale R. F. 5 will give the true measured distance on the ground between the same points. To determine the true scale it is only necessary to remember that as every pace of 29 inches was represented as 32 inches, the map as constructed will be larger than it would have been had the paces actually been 32 inches long, and consequently the true scale will be larger than R. F., or, as we have already seen, the denominator of its R. F. will be smaller than 500. A larger scale does not mean that more ground has been represented on the same size of paper than could be represented on a smaller scale, but that for the same piece of ground it takes a larger sheet of paper on which to represent it on a large scale than it would with a smaller scale. In the above case, for every pace of 29 inches we have plotted that pace 3 inches, to the scale of R. F., greater than we supposed, consequently the map is that proportion greater than represented by the scale of R. F. do. To determine the correct R. F. for the map, the proportion from the above reasoning is 29:32::50:x=753125, or 32: 29:: 500: x = 453.125. Having found the true R. F. 125, the scale is constructed as in Ex. IV. If the conditions of the problem had been reversed and a 29-inch pace used while the actual length was 32 inches, then the map would have been smaller and the denominator of the true R. F. greater than 500, the proportion being 32: 29:: :x=551.724) or 29: 32:: 500: x=551.724. XIII. A map 8 inches long by 7 inches broad represents 2.2317 acres. Construct its scale to read yards and feet. Determine its R. F. The map being 8 inches by 7 inches, contains 56 square inches. An acre contains 4840 square yards; hence 2.2317 acres will contain 2.2317 times 4840 square yards 10,801.428 square yards. If 56 square inches represents 10,801,428 square yards, 1 square inch will represent of 10,801.428 square yards =192.8826 square yards, 1 linear inch will represent as many linear yards as the square root of 192.882613.89 yards. Now if 1 inch represents 13.89 yards, 6 inches will represent 6x 13.89 yards 83.34 yards; take 80 yards and substitute in proportion, 13.59 yards: 80 yards :: 1 inch x 5.76 inches. Construct scale as shown. To determine R. F. 13.89 yards=13-89x36500.00 1 inch 1 1 XIV. A map on a scale of 3.168 inches to 1 mile is to be copied on a scale of .0079 inches to 1 meter. What ratio exists between the sizes of the maps? Which is the larger? 1 mile .0079 in. = The R. F. of the original map is 3.16. in. The R. F. of copied map is 1 meter an examination of the R. F.'s, remembering what was said above, it is seen that the original map is to be copied on a scale four times as large. Comparative or Equivalent Scales are those having the same R. F., but constructed to read different units of measure. Thus, on a foreign map the unit of measure might be meters or anything else, and if the R. F. were given, a scale of yards, feet, and inches, or chains and links, or any other familiar unit of measure could be constructed for the purpose of examining the map, and these scales, though of different units of measure, but of the same R. F., would be comparative scales. Figs. 57, 60, 61, and 63 are comparative scales of R. F. 500, and Figs. 59 and 64 of R. F. To000. Having already constructed a scale of say R. F. 10000, and it is desired to use it when plotting to a scale of R. F. 50, it is only necessary to multiply each measured distance by 2 and take it from the scale and vice versa. Thus any one scale may conveniently be used for another having the same unit of measure, provided the denominator of the R. F. of one is an exact multiple of the denominator of the R. F. of the other. A vernier scale is an auxiliary scale of equal parts constructed on the secondary divisions of another scale of equal parts for reading and setting off fractional parts of the secondary divisions, and is a substitute for the diagonal scale. The principle of its construction is that each division of the vernier scale shall be the amount of the smallest reading desired larger than the divisions to which it is applied. Thus, if it is desired to construct a scale to read .01 of an inch, either a diagonal scale or a vernier scale may be used. The construction of the diagonal scale has been explained. To construct the vernier scale (Fig. 65), first draw Fig. 85. the scale of inches and divide each into tenths, then from the 0 of the scale to the left lay off a distance on the under side equal to 11 of the tenths. Divide this into 10 equal parts and each part will be .01 inch greater than the divisions of the scale, or .11 inches long. Starting from the 0 to the left, the first division will be distant .11 inches, the second .22 inches, etc. To take off from the scale any number of hundredths of an inch, as .07 inch, the distance between the two lines numbered 7 on the scale and vernier scale will be the required amount. If it is desired to take off 24 inches, place the point of the dividers on 4 of the vernier scale and move the other point towards the 0, 2 full divisions on the main scale for the other point of division. If desired to take off 2.85 inches, place one point of dividers on 5 of the vernier scale and move the other point in the direction of 0 over 8 full divisions of scale for the .8, then 2 inches further to the right for the entire distance, 2.85 inches. TABLE OF EQUIVALENTS. ON A SCALE OF 120, 1 SQUARE INCH REPRESENTS 70.859 ACRES. 1 inch represents 21120 inches÷12-1760 feet. 1 inch represents 21120 inches÷36-586.66 yards. 1 inch represents 21120 inches÷39.37-535 897 meters. 1 foot-12 inches÷21120-0.000568 inch. 1 meter-39.37 inches÷21120-0.00186 inch. ON A SCALE OF, 1 SQUARE INCH REPRESENTS 63.72 ACRES. 1 inch represents 20000 inches÷12-1666.66 feet. 1 inch represents 20000 inches÷36-555.55 yards. 1 inch represents 20000 inches÷39.37-507.98 meters. 1 inch represents 20000 inches÷792-25 25 chains. inch represents 20000 inches÷63360-0 316 mile. 1 1 foot-12 inches÷20000 0.0006 inch. ON A SCALE OF т, 1 SQUARE INCH REPRESENTS 17.715 ACRES. 1 inch represents 10560 inches÷12-880 feet. 1 inch represents 10560 inches÷36-293.33 yards. 1 inch represents 10560 inches÷39.37-267.648 meters. 1 inch represents 1056) inches÷792-13.333 chains. 1 inch represents 10560 inches÷63360-0.1667 mile. 1 foot-12 inches÷10560-0.00114 inch. 1 meter 39 37 inches÷10560-0.00372 inch. ON A SCALE OF Todoo, 1 SQUARE INCH REPRESENTS 15.94 ACRES. 1 inch represents 10000 inches÷12-833.33 feet. 1 inch represents 10000 inches÷36-277.77 yards. 1 inch represents 10000 inches÷39 37-254 18 meters. 1 inch represents 10000 inches÷792=12.63 chains. 1 inch represents 10000 inches÷63360—0.158 mile. 1 foot-12 inches÷10000-0.0012 inch. 1 yard=36 inches÷10000-0.0036 inch. 1 meter 39.37 inches+10000-0.003937 inch. 1 chain=792 inches÷10000 0.0792 inch. 1 mile=63360 inches÷10000-6.336 inches. |