y found this
process somewhat wearisome; but this is not an unmixed evil, as the
irksomeness of any manual process has more than once led to the
invention of a valuable machine by the would-be idler. Thus our innate
desire to take things easy is a real incentive to progress. It was
some such desire as this on my part which led me, three years ago, to
inquire whether a practical instrument had not been, or could not be,
constructed to draw sum curves. Such an instrument is an integraph,
and the one I have to describe to you to-night is the outcome of that
inquiry. It is something better than my title, for it is an integraph,
and not an integrator.

[Illustration: A NEW INTEGRATOR]

Before I turn to its claims to be considered new, I must first remind
you of the importance of an instrument of this kind to the
draughtsman. I put aside its purely mechanical applications, where it
has been, or can be, attached to the indicators of steam engines, to
dynamometers, dynamos, and a variety of other instruments where
mechanical integration is of value. These lie entirely outside my
field, and I propose only to refer to a few of the possible services
of the integrator when used by hand, and not attached to a machine.

The simple finding of areas we may omit, as the planimeter will do
that equally well. But of purely graphical processes which the
integraph will undertake for us, I may mention the discovery of
centroids, of moments of inertia (or second moments), of a scale of
logarithms, of the real roots of cubic equations, and of equations of
higher order (with, however, increasing labor). Further, the
calculation of the cost of cutting and embanking for railways by the
method of Bruckner & Culmann, the solution of a very considerable
number of rather complex differential equations, various problems in
the storage of water, and a great variety of statistical questions may
all be completely dealt with, or very much simplified by aid of the
integraph.

In graphical statics proper the integraph draws successively the
curves of shear, bending moment slope, and deflection for simple
beams; it does the like service for continuous beams, after certain
analytical or graphical calculations have first been made; it can
further lighten greatly the graphical work in the treatment of masonry
arches and of metal ribs. In graphical hydrostatics it finds centers
of pressure and gives a complete solution for the shear and bending
moment, cur