equal to the portion of the line considered. For the scale assumed it could be Fig. 52, in which the portion of the line 200 150 100 FIGURE 52. 1200 to the right of the 0 one inch long is marked 400, as representing 400 feet; and inch to the left of the 0 marked 100, representing 100 feet, etc., each portion of the line having its origin at the point marked 0 and its other extremity at the number indicated. As the inch is the common unit of measure in the United States by which the eye is accustomed to judge distances on paper, it is usual, when it can be so done, to state the relation as so many inches (on the map) to so many feet, yards, miles, etc. (on the ground). Thus, 3 inches to 1,000 yards, or 5 inches to 3 miles. Frequently, however, maps are drawn on scales, the denominators of the representative fractions of which are multipies of 10, as R. F., R. F. 1, R. F. 5000, in which cases it is not always possible to express it as above. 1 mile 3 ins. 1 mile, 3 ins. 63360 in. The representative fraction for any scale expressed in the manner first indicated, as 3 inches to 1 mile, may be found by first writing it in the form of a fraction with the number of units indicated on the map for the numerator, and the number of units indicated on the ground for the denominator, thus 3 ins, then reducing both terms to the same denomication, because there are 63360 inches in 1 mile, and then reducing the fraction having unity for the numerator, =1120 R. F. 112. So for any other scale. When comparing maps drawn on different scales, the larger the denominator of the R. F. the smaller the scale; that is, any specified dimension on the ground would be represented by a smaller dimension on the map constructed on the smaller scale than it would be on a map constructed on a larger scale. 3 ins. 63360 ins. On a given sheet of paper a greater area of ground can be represented with a small scale than could be represented with a larger scale. If it is desired to construct a new scale a certain number of times larger than some other scale, one has only to divide the denominator of the given R. F. by the multiple of the new scale for the denominator of the new R. F., and conversely if the scale is to be reduced. A graphical or linear scale (Fig 52), in accurate topographic work, should always be constructed on the sheet to contain the map, so that it may change in length with any change in the size of the map due to the effect of changes in temperature and moisture on the paper.* The R. F. should also be given on all maps and plans to facilitate comparison with other maps having different scales and units of measure. When, however, maps are to be reproduced by photography on a larger or smaller scale, the R. F. should be covered so as not to show in the reproduced map, as it would not then be true. The new R. F. could be calculated and put on the reproduced map. The reproduced graphical scale would, however, still be the correct one for the new map. The scale decided upon for any particular work will depend upon the amount of detail to be shown and the nature of the ground. In every case the rule governing is, that the scale should be just large enough to express clearly all the details which it is desired to show. The degree of accuracy to which the work of sketching is carried on in the field depends largely upon the scale selected, since it would be useless to attempt to avoid errors that are *It has been found from experiment that the variation between paper exposed alternately to damp open air and warm dry air of a room, and a boxwood scale is .012, and between paper and ivory, even greater. (Gillespie.) Hence, if a map is constructed on a certain scale with a boxwood or ivory scale and afterwards changes in size due to the weather, then all distances taken from it with the wood or ivory scale would be erroneous, which would not be the case if a graphical scale had been constructed on the paper at the time of making the map, and distances referred to it. too small to appear upon the map, as well as useless to attempt to show details too small to be represented by the scale. If it be assumed that 0.01 of an inch is the smallest division appreciable to the eye, then the length applied to any scale will determine the limit of accuracy of the field plotting or the minimum size of objects that can be represented. Thus, if sketching at the scale of R. F. Toboo, 0.01 inch on paper will represent 200 inches or 16 feet on the ground; hence it will be useless to try to represent to scale objects or independent distances of less dimensions than 16 feet. At R. F. Tobo it will be 8 feet. R. F. 50, 4.16 feet. R. F. 1000 10 inches. R. F.,inch. CONSTRUCTION OF SCALES.-The Unit of Measure of a scale in the linear quantity which the primary divisions of the scale read, as feet, yards, meters, miles, etc. The primary divisions may then be still further subdivided into other units or into fractional parts of the unit of measure. A scale is ordinarily drawn about 6 inches long, but may be either longer or shorter, depending upon the size of the map. For reasons previously stated, one inch is assumed in ali cases as being the numerator of the R. F. for purposes of constructing scales. To construct a scale: First, determine how many of the units of measure are represented by 1 inch. Thus, if the scale is to be R. F. 1 to read yards, then, because 21,120 inches are to be represented by 1 inch, as many yards will be represented by 1 inch as 21,120 inches÷36 inches 586.63. Or, if the scale is to be 6 inches to one mile to read feet, then, because there are 5280 feet in a mile represented by 6 inches, there will be as many feet represented by 1 inch as 5280÷6=880. Second, calculate how many of these units of measure will be represented by about the length in inches that you wish to make the scale; suppose it is desired to make the scale about 6 inches long. In the first case above, as 1 inch represents 586.6 yards, 6 inches will represent 6×586.63 yards=3520 yards; and in the second case, 6 inches will represent 6×880 feet=5280 feet. Third, select the near est exact number of ten, hundred, or thousand units of measure to the number found in about the length you wish the scale, then substitute in the following proportion and solve. This gives the length of the scale (to two places of decimals) in inches. In the first case, select say 3500 yards; then, substituting and solving, one gets 586.63:3500::1:x=5.96 inches. In the second case, selecting 5000 feet, substituting, and solving, one gets 880:5000:: 1:x=5.68 inches. Having found the fourth term of the proportion, or x, lay off this length on a straight line, using a scale of equal parts of inches, and then divide this length of line into the number of equal parts most convenient for use.* EXAMPLES.-I. Construct a scale of 1 inch to 400 feet to read yards. In this the unit of measure is yards. The scale is given in feet. Now 400 feet 133.33 yards, represented by 1 inch. Assuming 6 inches as about the desired length for the scale, it will represent 6 times 133.33 yards= 800.00 yards. 800 being an exact number of hundreds, it is unnecessary to use the proportion, but on a straight line (Fig. 54) lay off AB 6 inches long. From the end A of this line draw another straight line AC a little longer than AB, making any convenient acute angle with AB. On AC lay off 8 equal parts, 1, 2, 3, 4, 5, etc., and join the eighth one C with B. Now from the points on AC draw lines parallel to BC, cutting AB, when AB will be divided into 8 equal parts, each. representing 100 yards and called the primary division. From. *The reason for selecting the nearest exact number of ten, hundred, or thousand units of measure to the number of units of measure found for about the desired length of scale, was because it is easier to divide the length of line obtained by the proportion into a certain number of equal parts than it would have been to attempt to divide the assumed length of the scale into the uneven number of units of meas ure that it would reprsent. |