Scales of Construction.

—Scales of construction are intended to afford means of measuring more minute quantities than scales of distances. Of the former there are two kinds, known respectively as the Diagonal and the Vernier scale. The diagonal is the more frequently employed. Its construction involves no peculiar difficulty, as it consists simply of an ordinary scale of distances, with the addition of a number of parallel lines crossed by other parallel lines drawn diagonally from the smaller points of division. An example will best show the construction and mode of using this scale. Suppose it to be required to construct a scale of 10 miles to the inch, showing furlongs diagonally; the scale to measure 50 miles. Here 1 : 10 :: x : 50, whence x = 5 inches. Divide this length of 5 inches into five equal parts, and the first part into tenths to show miles, in the manner already described for scales of distances. Then, since it is required to show furlongs or eighths of a mile, eight equidistant parallel lines must be drawn above the scale, at a convenient interval apart, as shown in [Fig. 80]. Produce the primary points of division to meet the top parallel; and from the last secondary point of division draw a line to the point in which the extreme primary division meets the top parallel. Draw from the other points of division, lines parallel to this one, and the scale will be complete. It will be seen that the inclined lines are the diagonals of the rectangular figures formed by the top and bottom parallels and vertical lines drawn from the smaller points of division.

Fig. 80.

[Larger illustration] (31 kB).

To use this scale, suppose a length of 24 miles 5 furlongs is required. Place one leg of the dividers upon the point in which the fourth diagonal intersects the fifth parallel, and extend the other to the point in which the primary division marked 20 intersects the same parallel. In like manner, if the distance required be 33 miles 3 furlongs, it must be taken from the intersection of the third diagonal with the third parallel, to the intersection of the primary division marked 30 with the same parallel.

It is obvious that if a scale of feet showing inches diagonally be required, twelve equidistant parallel lines must be drawn instead of eight as in the foregoing example where furlongs are required. The diagonal scale possesses the important advantages of accuracy and distinctness of division which render it very suitable as a scale of construction. Another practical advantage is that it is less rapidly defaced by use than the other kinds, in consequence of the measurements being taken on so many different lines.

The construction of the vernier scale is similar to that of the graduated arcs of surveying and astronomical instruments. The principle of the vernier is as follows. If a line containing n units of measurement be divided into n equal parts, each part will, of course, represent one unit; and if a line containing n + 1 of these units be also divided into n parts, each part will be equal to n + 1n units; and the difference between one division of the latter and one of the former will be x + 1n - 1 = 1⁄n of the original unit. Similarly, the difference between two divisions of the one and two of the other will be 2⁄n of a unit, between three of the one and three of the other, 3⁄n, and so on. Hence, to obtain a length of x⁄n of a unit, we have only to make a division on one scale coincide with one on the other scale; the space between the two corresponding xth divisions from this on both scales will be the required length of 2⁄n of a unit. The same reasoning will evidently hold good if a length equal to n - 1 be taken.

Fig. 81.