The instrument we are about to describe is an improvement on the hatchet planimeter and is due to Prof. Goodman, of Leeds. One form of the instrument is intended for the measurement of areas of surfaces, and the other form for the measurement of the mean height of a figure such as an indicator diagram.

London Engineering, to which we are indebted for the cuts and copy, describes the instruments as follows: The method of using the two instruments is practically the same, but for the present we shall confine our remarks to the instrument for measuring areas. In order to familiarize oneself with the peculiar action of the instrument, it will be well to get a large sheet of paper on a drawing board or a large blotting pad, and holding the instrument vertical to the paper, grasp the tracing leg very lightly indeed between the forefinger and thumb of the right hand, with the hatchet toward the left hand, as shown in Fig. 1. Then by moving the tracing point round and round an imaginary figure and allowing the hatchet to go where it pleases, it will be seen that the hatchet moves to and fro along zigzag lines, and travels sideways - the side travel being nearly proportional to the area of the figure described by the tracing point. If the tracing point be too tightly grasped, the hatchet will not move freely, and, will have a side slip. When this occurs the side travel of the hatchet ceases to be proportional to the area traced out.

A loose weight is hung on the hatchet to prevent this side slip, but as soon as a little skill is attained in the use of the instrument, this weight may be dispensed with.

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, X, which may or may not coincide with X; if not, prick a central point between X and X, 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

Where I = the area traced out by the pointer in square inches.

c = the distance between the dents, X and Y, in inches.

c = the distance between the dents, X and Y, in inches.

p = the length of the instrument from center of hatchet to point in inches.

R² = the mean square of the radii of the figure.

The making of such a calculation for every area measured is, of course, quite out of the question. The labor involved would be as great as calculating the area by the ordinate or by Simpson's method; hence it is usual to neglect that part of the formula inclosed within the square brackets, which amounts to assuming the area to be equal to the product of the mean side shift of the hatchet by the length of the instrument; this, however, involves an error too big to be neglected, and, moreover, one that is not a constant fraction of the area measured, thus:

These errors are, however, compensated for in Goodman's improved instrument by making the scale with constantly and regularly increasing divisions. If, however, the area dealt with be not a circle, the error involved in assuming that its R² is equal to the R² of a circle of equal area is so small that it is quite inappreciable on a scale which only reads to one-tenth of a square inch. If the R² for any given area were say 5 per cent. greater than that of the equivalent circle, the error involved would be 0.0016 of the whole quantity when measuring an area of 40 square niches, or 0.064 square inch, a quantity which cannot be measured on the scale. It has been proposed to use a roller and vernier to enable the readings between the dents to be measured with a greater degree of accuracy, but it will be readily seen that the instrument is not reliable to the second place of decimals, hence such refinements are only imaginary. Even with this special scale that we have described above, the inventor does not profess to get as good results as with an Amsler planimeter; he regards his instrument as equivalent to a foot rule in comparison with a micrometric gage as representing Amsler's instrument; but for a great number of purposes the foot rule is sufficiently accurate, and only when great accuracy is required will a micrometer be used, so with the two forms of planimeter.