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.
In order to ascertain the relative accuracy of various methods of measuring areas, Prof. Goodman has had a large number of irregular areas measured by his first year students within a week or so of their entering the department, before they have attained to any degree of skill in using instruments. The results were as follows. Amsler's planimeter was taken as the standard, the area measured by it being independently checked by an assistant.
| Method. | Measurement of Areas Reduced to 100. |
|---|---|
| Amsler planimeter | 100 |
| Goodman " | 100 + or - 0.6 |
| Simpson's rule. | 100 + or - 1.0 |
| Mean ordinates. | 100 + or - 2.4 |
| Cutting out in cardboard and weighing against piece of known area. | 100 + or - 4.4 |
| Equalizing curved edges by drawing straight lines along boundary and calculating by triangles. | 100 + or - 7.0 |
In the averaging instrument for getting mean heights of figures, the length of the instrument between the hatchet and the pointer is variable. The length is set to the length of the diagram (see Fig. 2); it is then used in precisely the same manner as the planimeter described above. From what we have already said, our diagram in Fig. 5 will be perfectly clear. The mean distance between the dents is in this case the mean height of the diagram, measured on an ordinary scale, or the mean pressure in the case of an indicator diagram measured on a scale to suit the indicator spring.
Knudsen's formula given above applies equally well to this averaging instrument. Neglecting for the moment the quantity in the square brackets, we have I = c p where c = (c1 + c2)/2 but we also have I = h l where h is the mean height and l the length of the figure, therefore h l = c p; but in this instrument we make p = l. Hence h = c, or the mean height of the figure is equal to the mean distance between the dents. The quantity in the brackets is too great to be neglected, however. If we were always dealing with circles, the ratio (R/2p)² would be a constant, and numerically equal to 1/16 or 6.25 per cent. Then all we should have to do would be to use a scale 6.25 per cent. longer than the true scale. But with a long narrow figure such as an indicator diagram, this ratio is much smaller. The measurement of a large number of diagrams gave a mean value of 1/60 for diagrams 4 in. long. It is obvious that, if a diagram be shortened, this ratio will increase, for the value of R does not decrease as rapidly as p, and vice versa; hence this ratio varies approximately inversely as the length of the diagram. Taking the value of 1/60 for the 4 in. diagram, this is equivalent to saying that there is an error of 1 in 60, or 1.67 per cent., in the result, and from the formula it will be seen that the result is too great by this amount; hence, if we make the length, l, between the legs of the instrument 1.67 per cent. of 4 in., or 0.067 in. less than the length from the tracing point to the center of the hatchet, p, we shall compensate for the error on a diagram 4 in. long. But the ratio of this constant quantity 0.067 in. to the length of the diagram also varies inversely as the length in just the same manner as the ratio R/2p, hence this method of correcting the instrument is approximately right for all lengths of diagrams. It must be remembered that if this correction were entirely neglected, it would not exceed two per cent.; hence any inaccuracy in this correction is an exceedingly small quantity, well under 1 per cent.
Whenever errors have been attributed to the instrument, on examination it has always been found that they were due to carelessness in setting the length to the diagram, or to the tracing leg having been grasped so tightly as to cause side slip.
The accuracy of the instrument may be easily demonstrated by drawing a rectangle, say about 4 in. long and 2 in. high, and finding the mean height by the averager, then by doubling the paper over and comparing its height with the mean distance between the dents, it will be found that they agree if the instrument has been carefully used.