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.
[Illustration: GOODMAN'S HATCHET PLANIMETER.]
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
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.
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.
Measurement of Areas
Method. 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.
In many quarters we know that there is a great deal of prejudice
against instruments of this kind. We are quite sure, however, that if
only draughtsmen and others would spend half an hour in trying them
over, they would save themselves many hours of tedious labor in
calculating areas by methods which are seldom as accurate as the
results obtained by a planimeter in the same number of minutes.
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