When measuring the area of such a surface as that inclosed by the boundary line shown in Fig. 3, a point, A, is chosen somewhat near the center of the figure; the exact position is, however, immaterial. From the point, A, a line, AB, is drawn in any direction to the boundary; the tracing point of the planimeter is now placed at A, with the hatchet at X, Fig. 3, that is, with the instrument roughly square with AB. The hatchet is now lightly pressed in order to mark its position on the paper by making a slight dent, then leaving the hatchet free to move as shown in Fig. 1, the tracing point is caused to traverse the line, AB, and the boundary line in a clockwise direction, as shown by the arrows, returning to A via AB. The hatchet will now be found to have taken up a new position, Y, which must be marked by again pressing the hatchet to make a slight dent in the paper. If the figure under measurement be on a separate sheet of paper, the paper must now be revolved about the point, A, through about 180 deg. (by eye), using the tracing point of the instrument as a center, care being taken that neither the point nor the hatchet be shifted while the paper is being turned. The line, AB, will again be roughly at right angles to the instrument, but in the reverse direction (see dotted lines in Fig. 3). Again cause the tracing point to traverse the line, AB, and the boundary line as before, but this time in a contra-clockwise direction. The hatchet after this backward motion will take up the new position, X1, which may or may not coincide with X; if not, prick a central point between X and X1, as shown, then, of course, the distance of this point from Y is the mean side shift of the hatchet; this distance measured from the zero of the scale on the back of the instrument is the area of the figure in square inches. The scale is read in exactly the same manner as a geometrical scale on a drawing, the whole numbers being read to the right of the zero and the decimals to the left. The instrument does not profess to give results nearer than one-tenth of a square inch. In some cases on large maps, for example, the figure cannot be turned round as indicated above; in that case the instrument itself must be turned round through 180° and two fresh dents, X¹Y¹, obtained; the area of the figure is then the mean of the two readings, XY and X¹Y¹. When the area is large the instrument will move through a large angle, and consequently, if square with AB to start with, it will be considerably out of square at the finish. In such a case it is only necessary to see that the mean position of the instrument is square with AB. By carefully examining the scale it will be observed (see Fig. 1) that the divisions are not equal, but that they gradually increase from zero upward; herein consists the improvement of this instrument over the ordinary hatchet planimeter invented by Knudsen, of Copenhagen, who shows in a pamphlet published by him that