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(1) When the fence or other boundary is also the boundary of a parish or other civil division which does not follow the centre of the fence, the area is calculated to the parish or other boundary, and not to the centre; (2) when the fences, &c., bounding either side of a railway are included wholly within its area.
In the planimeter (see pages 155, 156) the bar E F can be made either to slide in the box to which the other arm is attached (see fig. 1) or can be permanently fixed to this box. In the latter case it is constructed to record square inches only, and is known as the fixed scale planimeter. In the former case it will record square inches when the mark upon the sliding box is made to come immediately under the mark upon the bar E F, which is situated near the figures 22:174 in fig. 4, and is distinguished as the proportional planimeter. The sliding box is fixed to the sliding arm E F by the clamp screw H, and the two marks above alluded to are made to coincide by the movement of the slow-motion screw F. The horizontal wheel M records the square units in one revolution, the wheel L records square units and tenth parts of a square unit, and the vernier attached outside the wheel L enables hundredths of a square unit to be read. When the needle-point D is placed outside the area to be calculated no account is taken of the figures upon the sliding bar E F, but when the needle-point D is placed inside the area to be calculated, the number engraved upon the bar is to be added to the reading of the instrument after the boundary of the area has been completely traversed by the pointer S, before the first reading of the instrument is deducted.
Goodman's Patent Planimeter is an economical form of instrument, and may be thus explained :- Let it be required to measure the area of the figure, page 158. Choose a point A, as near the centre of the figure as can be judged by eye, and from it draw a line A B to the boundary. Hold the tracing leg of the instrument in the right hand, placing the point at A and the hatchet at X (ie., with the instrument roughly square with A B, see fig. 3), and press the hatchet in order to make a slight dent in the paper at X; then, the finger having been removed
from the hatchet, the tracing point of the instrument is caused to traverse the line A B and the boundary in the direction indicated by the arrows, returning to A via A B, when it will be found that the hatchet has taken up a new position and it must be again lightly pressed (as illustrated in fig. 2) in order to make a fresh dent in the paper at Y (fig. 3). The instrument being held in this position, revolve the paper on which the figure is drawn through about 180° (by eye), using the point of the instrument as a centre (as shown in fig. 4), and taking care that neither the point nor the hatchet shift while the paper is being turned. The line A B will again be roughly at right angles to the axis of the
instrument, but in a reversed position (see dotted figure, fig. 3). Now cause the tracing point to traverse the boundary as before, but in the opposite direction, as indicated by the dotted arrows. The hatchet will take up the new position X, which may or may not coincide with X; then, the mean of X Y and X, Y measured on the scale engraved on the instrument is the area of the figure; this can be readily read off by pricking a central point as shown between X and X, by eye. When it is inconvenient to turn the paper round, the instrument itself may be turned round to form a dent X, on the opposite side of the figure, as shown at fig. 3a. Then by following the boundary in the direction of the arrows Y1 is obtained. The area is the mean of the
lengths X Y and X, Y, measured off on the scale as before, XY + X1 Y1
or the area
When the area is large, the instrument will move through a large angle, and consequently, if approximately square with A B at starting, it will be a long way out at the finish. In such a case all that is necessary is to see that the mean position of the instrument is square with A B.
The following precautions are to be observed:-
smooth writing or drawing papers are suitable; do not work on wood, as the hatchet tends to travel along the grain.
Do not allow the hatchet to go off the edge of the paper or over ridges. Hold the instrument freely so that the motion of the hatchet shall not be interfered with. It is always advisable to use the weight on the hatchet to prevent side slip.
Use the instrument on a flat table, not on a sloping desk. On no account attempt to sharpen the hatchet, either with an oilstone or otherwise.
See that the instrument is held with its legs fairly vertical.
In measuring the area off always work from the zero of the curved scale.
This instrument is an improvement on the Stang planimeter made by Knudsen, of Copenhagen. pamphlet published by him he gives the complete mathe
matical theory of the instrument and arrives at the following result:
C1 + C2 R I = [1 − ( 2 )3] p Where I = the area traced out by the pointer in sq. inches, C1 = the distance between the dents X and Y in inches,
X, and Y p the length of the instrument in inches,
R2= the mean square of the radii of the figure (Le milieu des carrés des distances de T (a) jusqu'à la circonférence)
The making of such a calculation for every area measured is far too tedious an operation. By using Goodman's instrument as described above, first in one direction and then
in the other, the quantity C1+ C is obtained by a direct
C1 + C2
reading. If the instrument were of very great length as compared with the dimensions of the area, the quantity in the brackets would vanish, then putting = c, the area would simply be equal to the product pc, and a scale for measuring the distance in which I square inch was
made equal to inches would at once give the area in square inches. But as the areas generally dealt with are of somewhat large dimensions as compared with the length of the instrument, the quantity in brackets must not be neglected. Assume for a moment that we are measuring the area of a circle with the instrument-its area is proportional to R2, hence as (2p) is a constant for any given instrument, the whole quantity (2) is proportional to the area traced out by the pointer. Thus by making a scale