generates an area. If therefore a line OA = QT
is turning about a fixed point O, always keeping parallel to QT, it will
sweep over an area equal to that generated by the more general motion of
QT. Let now (fig. 17) QT be placed on OA, and T be guided round the closed
curve in the sense of the arrow. Q will describe a curve OSB. It may be
made visible by putting a piece of "copying paper" under the hatchet. When
T has returned to A the hatchet has the position BA. A line turning from OA
about O kept parallel to QT will describe the circular sector OAC, which is
equal in magnitude and sense to AOB. This therefore measures the area
generated by the motion of QT. To make this motion cyclical, suppose the
hatchet turned about A till Q comes from B to O. Hereby the sector AOB is
again described, and again in the positive sense, if it is remembered that
it turns about the tracer T fixed at A. The whole area now generated is
therefore twice the area of this sector, or equal to OA. OB, where OB is
measured along the arc. According to the theorem given above, this area
also equals the area of the given curve less the area OSBO. To make this
area disappear, a slight modification of the motion of QT is required. Let
the tracer T be moved, both from the first position OA and the last BA of
the rod, along some straight line AX. Q describes curves OF and BH
respectively. Now begin the motion with T at some point R on AX, and move
it along this line to A, round the curve and back to R. Q will describe the
curve DOSBED, if the motion is again made cyclical by turning QT with T
fixed at A. If R is properly selected, the path of Q will cut itself, and
parts of the area will be positive, parts negative, as marked in the
figure, and may therefore be made to vanish. When this is done the area of
the curve will equal twice the area of the sector RDE. It is therefore
equal to the arc DE multiplied by the length QT; if the latter equals 10
in., then 10 times the number of inches contained in the arc DE gives the
number of square inches contained within the given figure. If the area is
not too large, the arc DE may be replaced by the straight line DE.

To use this simple instrument as a planimeter requires the possibility of
selecting the point R. The geometrical theory here given has so far failed
to give any rule. In fact, every line through any point in the curve
contains such a point. The analytical theory of the inventor, which is very
simil