In the language of the mathematician, the integrator gives only that miserly result, a definite integral, but the integraph yields an indefinite integral, a picture of the result at all times or all points—­a much greater boon in most mechanical and physical investigations.  Members of our Society as students of University College have probably become acquainted with a process termed “drawing the sum curve from the primitive curve.”  Many have probably 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, curves in ships, besides curves for their stability.  In graphical