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 dynamics the applications of the integraph seem still more numerous. It enables us to pass from curves of acceleration to curves of speed, and from curves of speed to curves of position. Applied to the curve of energy of either a particle or the index point of a rigid body, it enables us by the aid of easy auxiliary processes to ascertain speeds and curves of action. In a slightly altered form, that of "inverse summation," we can pass from curves of action to curves of position, and deal with a great range of resisted motions, the analysis of which still puzzles the pure mathematician; the variations of motion in flywheels, connecting rods, and innumerable other parts of mechanism, may all be calculated with much greater ease by the aid of an integraph. Shortly, it is the fundamental instrument of graphic dynamics.
It would be needless to further multiply the instances of its application; the questions we have rather to ask are: Can a practical instrument be made which will serve all these purposes? Has such an instrument been already put upon the market? If I have to answer these questions in the negative, it is rather a doubtful negative, for the instrument I have to show you to-night goes so far, and suggests so many modifications and possibilities, which would take it so much further, that it is very close to bringing the practical solution to the problem.
Let me here lay down the conditions which seem essential to a practical integraph. These are, I think, the following:
1. The price must be such that it is within the reach of the ordinary draughtsman's pocket. The Amsler's planimeter at £2 10s. or £3 may be said to satisfy this first condition. The price for the first complex integraph designed by Coradi was £24 to £30. The modified form in which I show it to-night is estimated to cost retail £14. Till an equally efficient instrument can be produced for £5 I shall not consider the price practical. If the error of its reading be not sensibly greater than that of a planimeter, it is certainly worth double the money.
2. The instrument must not be liable to get out of order by fair handling and a reasonable amount of wear and tear. I cannot speak at present with certainty as to how far our integraph satisfies this condition; it is rather too complex to quite win my confidence in this respect.
3. It must be capable of being used on the ordinary drawing board, and of having a fairly wide range on it, i.e., it must not be limited to working where the primitive is at one part only of the board.
This condition takes out of every day practical drawing use the integraph invented by Professors James and Sir William Thomson, in which the sum curve is drawn on a revolving cylinder. It is essential that the sum curve should be drawn on the board not far from the primitive, and that this sum curve can be summed once or twice again without difficulty. The time involved in drawing the four sum curves, for example, required in passing from the load curve to the deflection curve of a simple beam, if these curves were drawn on different pieces of paper and had to be shifted on and off cylinders, would probably be as long as the ordinary graphical processes. Coradi's integraph works on an ordinary drawing board, but since there are nearly 10 inches between the guide point and tracer, the sum curve is thrown 10 inches behind the primitive in each integration. Thus a double summation requires say 26 inches of board, and it is impossible to integrate thrice without reproducing the primitive. The fact that the primitive and sum curve are not plotted off on the same base is also troublesome for comparison, and involves scaling of a new base for each summation. I have endeavored to obviate this by always drawing the second sum curve on a thin piece of paper pinned to the board, which can then be moved back to the position of the first primitive. But this shifting, of course, involves additional labor, and is also a source of error.
I should like to see the trace and guide chariots on the same line of rails, one below the other, were this possible without producing the bad effect of a skew, pull or push.
4. The practical integraph must not have a greater maximum error than 2 per cent. The mathematical calculations, which are correct to five or six places of decimals, are only a source of danger to the practical calculator of stresses and strains. They tend to disguise the important fact that he cannot possibly know the properties of the material within 2 per cent. error, and therefore there is not only a waste of time, but a false feeling of accuracy engendered by human and mechanical calculation which is over-refined for technical purposes.