c1 + c2 I = ------------- p [1 - (R / 2p)²] 2 Where I = the area traced out by the pointer in square inches. c1 = the distance between the dents, X and Y, in inches. c2 = the distance between the dents, X1 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: Area of circle, square inches. 10 20 30 40 Error per cent. 0.8 1.6 2.4 3.2 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. The rougher instrument has some advantages, however; there are no delicate moving parts to get out of order, and the cost is but one-fourth.