In the fifth figure the scale of 5 chains to 1 inch is subdivided diagonally to only every quarter chain, or 25 links. The distance between the upper pair of arrow-heads on it is 121 chains, or 12.25; between the next pair of arrow-heads, it is 6.50; and between the lower pair, 14.75. A diagonal scale for dividing an inch, or a half inch, into 100 equal parts, is found on the "Plain scale" in every case of instru ments. (56) Vernier Scale. This is an ingenious substitute for the diagonal scale. The one given in the following figure divides an inch into 100 equal parts, and if each inch be supposed to represent a chain, it gives single links. Make a scale of an inch divided into tenths, as in the upper scale of the above figure. Take in the dividers eleven of these divisions, and set off this distance from the 0 of the scale to the left of it. Divide the distance thus set off into 10 equal parts. Each of them will be one tenth of eleven tenths of one inch; i. e. eleven hundredths, or a tenth and a hundredth, and the first division on the short, or vernier scale, will overlap, or be longer than the first division on the long scale, by just one hundredth of an inch; the second division will overlap two hundredths, and so on. The principle will be more fully developed in treating of "Verniers," Part IV, Chapter II. Now suppose we wish to take off from this scale 275 hundredths of an inch. To get the last figure, we must take five divisions on the lower scale, which will be 55 hundredths, for the reason just given. 220 will remain which are to be taken from the upper scale, and the entire number will be obtained at once by extending the dividers between the arrow-heads in the figure from 220 on the upper scale (measuring along its lower side) to 55 on the lower scale, 254 would extend from 210 on the upper scale to 44 on the lower. 318 would extend from 230 on the upper scale to 88 on the lower. Always begin then with subtracting 11 times the last figure from the given number; find the remainders on the upper scale, and the number subtracted on the lower scale. (51) A plat is sometimes made by a rcminally reduced scale in the following manner. Suppose that the scale of the plat is to be ten chains to one inch, and that a diagonal scale of inches, divided into tenths and hundredths, is the only one at hand. By dividing all the distances by ten, this scale can then be used without any further reduction. But if the content is measured from the plat to the same scale, in the manner explained in the next chapter, the result must be multiplied by 10 times 10. This is called by old Surveyors "Raising the scale," or "Restoring true measure." 66 Fig. 24. (52) Sectoral Scales. The Sector, (called by the French Compass of Proportion"), is an instrument sometimes convenient for obtaining a scale of equal parts. It is in two portions, turning on a hinge, like a carpenter's pocket rule. It contains a great number of scales, but the one intended for this use is lettered at its ends L in English instruments, and consists of two lines running from the centre to the ends of the scale, and each divided into ten equal parts, each of which is again subdivided into 10, so that each leg of the scale contains 100 equal parts. To illustrate its use, suppose that a scale of 7 chains to 1 inch is required. Take 1 inch in the dividers, and open the sector till this distance will just reach from the 7 on one leg to the 7 on the other. The sector is then "set" for this ONE INCH 10 L to find the scale on which a map is drawn. Let area of ABC represent the true area the plot. Let ADE represent Also, for the same reason, ADE represent the calculated area by the assumed scale. being triangles with a 4 ABE: A ABC :: AE: AC '.' A ADE: A ABC :: A D x AE AB x AC. By similar triangles, AD: ABAEAC we have also AD: ABAD: AB, we have multiplying these together A D': AB' :: AD AE: AB AC * A ADE: 4 ABC :: AD: AB", or, as the calculated area : the assumed scale: The the true area :: the square of the correct scale. com. square of (54) Scale omitted. It may be required to find the unknown scale to which a given map has been drawn, its superficial content being known. Assume any convenient scale, measure the lines of the map by it, and find the content by the methods to be given in the next chapter, proceeding as if the assumed scale were the true one. Then make this proportion, founded on the geometrical principle that the areas of similar figures are as the squares of their corresponding sides: As the content found Is to the given content Se is the square of the assumed scale To the square of the true scale. its use, suppose that a scale of 7 chains to 1 inch is required. Take 1 inch in the dividers, and open the sector till this distance will just reach from the 7 on one leg to the 7 on the other. The sector is then "set" for this ONE INCH scale, and the angle of its opening must not be again charged. Now let a distance of 580 links be required. Open the dividers till they reach from 58 to 58 on the two legs, as in the dotted line. in the figure, and it is the required distance. Again, suppose that a scale of 23 chains to one inch is desired. Open the sector so that 1 inch shall extend from 25 to 25. Any other scale may be obtained in the same manner. Conversely, the length of any known line to any desired scale can thus be readily determined. (53) Whatever scale may be adopted for platting the survey, it should be drawn on the map, both for convenience of reference, and in order that the contraction and expansion, caused by changes in the quantity of moisture in the atmosphere, may affect the scale and the map alike. When the drawing paper has been wet and glued to a board, and cut off when the map is completed, its contractions have been found by many observations to average from one-fourth to one-half per cent. on a scale of 3 chains to an inch, (1:2376), which would therefore require an allowance of from one-half perch to one perch per acre. A scale made as directed in Art. (49), if used to make a plat on unstretched paper, and then kept with the plat, will answer nearly the same purpose. Such a scale may be attached to a map, by slipping it through two or three cuts in the lower part of the sheet, and will be a very convenient substitute for a pair of dividers in measuring any distance upon it. (54) Scale omitted. It may be required to find the unknown scale to which a given map has been drawn, its superficial content being known. Assume any convenient scale, measure the lines of the map by it, and find the content by the methods to be given in the next chapter, proceeding as if the assumed scale were the true one. Then make this proportion, founded on the geometrical principle that the areas of similar figures are as the squares of their corresponding sides: As the content found Is to the given content So is the square of the assumed scale To the square of the true scale. |