more wheels arranged symmetrically round a
tracer whose point is guided along the curve; the planes of the wheels all
pass through the tracer, and the wheels can only turn in one direction. The
sum of the readings of all the wheels gives approximately the length of the
curve, the approximation increasing with the number of the wheels used. It
is stated that with three wheels practically useful results can be
obtained, although in this case the error, if the instrument is
consistently handled so as always to produce the greatest inaccuracy, may
be as much as 5%.

[Illustration: FIG. 5.]

Planimeters are instruments for the determination by mechanical means of
the area of any figure. A pointer, generally called the [Sidenote:
Planimeters.] "tracer," is guided round the boundary of the figure, and
then the area is read off on the recording apparatus of the instrument. The
simplest and most useful is Amsler's (fig. 5). It consists of two bars of
metal OQ and QT, [v.04 p.0976] which are hinged together at Q. At O is a
needle-point which is driven into the drawing-board, and at T is the
tracer. As this is guided round the boundary of the figure a wheel W
mounted on QT rolls on the paper, and the turning of this wheel measures,
to some known scale, the area. We shall give the theory of this instrument
fully in an elementary manner by aid of geometry. The theory of other
planimeters can then be easily understood.

[Illustration: FIG. 6.]

Consider the rod QT with the wheel W, without the arm OQ. Let it be placed
with the wheel on the paper, and now moved perpendicular to itself from AC
to BD (fig. 6). The rod sweeps over, or generates, the area of the
rectangle ACDB = lp, where l denotes the length of the rod and p the
distance AB through which it has been moved. This distance, as measured by
the rolling of the wheel, which acts as a curvometer, will be called the
"roll" of the wheel and be denoted by w. In this case p = w, and the area P
is given by P = wl. Let the circumference of the wheel be divided into say
a hundred equal parts u; then w registers the number of u's rolled over,
and w therefore gives the number of areas lu contained in the rectangle. By
suitably selecting the radius of the wheel and the length l, this area lu
may be any convenient unit, say a square inch or square centimetre. By
changing l the unit will be changed.

[Illustration: FIG. 7.]

Again, suppose the rod to turn (fig. 7) about the end Q, th