The mathematical rule may be described in words as follows: Required the
area between a curve, the axis of x and two ordinates; it is necessary
to draw a new curve, such that its steepness, as measured by the tangent
of the inclination, may be proportional to the ordinate of the given
curve for the same value of x, then the _ascent_ made by the new curve
in passing from one ordinate to the other is a measure of the area
required.

The figure shows a plan and side elevation of a model of the instrument,
made merely to test the idea, and the arrangement of the details is not
altogether convenient. The frame-work is a kind of T square, carrying a
fixed center, B, which moves along the axis of x of the given curve, a
rod passing always through B carries a pointer, A, which is constrained
to move in the vertical line, ee, of the T square, A then may be made
to follow any given curve. The distance of B from the edge, ee, is
constant; call it K, therefore, the inclination of the rod, AB, is such
that its tangent is equal to the ordinate of the given curve divided
by K; that is, the tangent of the inclination is proportional to the
ordinate; therefore, as the instrument is moved over the paper, AB has
always the inclination of the desired curve.

The part of the instrument that draws the curve is a three-wheeled cart
of lead, whose front wheel, F, is mounted, not as a caster, but like the
steering wheel of a bicycle. When such a cart is moved, the front wheel,
F, can only move in the direction of its own plane, whatever be the
position of the cart; if, therefore, the cart is so moved that F is in
the line, ee, and at the same time has its plane parallel to the rod,
AB, then F must necessarily describe the required curve, and if it is
made to pass over a sheet of black tracing paper, the required curve
will be _drawn_. The upper end of the T square is raised above the