2.  If P moves along a vertical line, P’ does not change, but Q R turns round it, remaining parallel to O B.

[Illustration:  Fig. 1, 2, 3]

Without taking the trouble, as I ought to have done, to inquire what previous investigations had achieved in this matter, I thought, three years ago, I could get an apparatus to save me the trouble of drawing sum curves, made somewhat after the following fashion.

P (Fig. 2) is the guide or point to be taken round the primitive.  It is attached to a block, D, which works along the bar, B C, which in its turn moves on the four wheels, e e f f, upon the frame R S U T fixed upon the drawing board.  O A is fixed perpendicular to R U, and is such that O may be fixed at various points to determine the polar distance.  O B D is a light bar passing freely through B and forming one side of a parallel ruler of two or more points, g g, h h, i i.  Along i i is a slot and in this works a loaded block containing a wheel P’, whose plane is always parallel to i i.  This block also passes through a slot in D E, an arm at right angles to B C. A little consideration will show that P’, if worked at all, would trace out the sum curve of P.

It was only when I showed the rough idea of this to Professor Kennedy, with the view of ascertaining what would be the amount of back-lash and friction, that I learned that Mr. Boys had already invented a very similar integrator.  In his model the double parallel ruler is replaced by two endless strings and pulleys, and the bar, B C, by a T square.

Although this integrator was afterward made in a less crude form, I do not think it has ever been a practical instrument for the draughtsman.  Shortly afterward I came across a work by Abdank-Abakanowicz, entitled “Les Integraphes,” being a study of a “new kind of mechanical integrator.”

The new kind of integrator was really only an independent version of Boys’ instrument, but in many respects a great improvement.  The real merit will ultimately belong to the scientific instrument maker who constructs an instrument reasonably cheap and capable of efficient practical service.  Abdank-Abakanowicz’s integrator however certainly went further in the practical direction than any previously constructed.  The drawing board machines, it is true, of rather a complex nature, were actually exhibited to the Paris Academy, but no more have been made.  The instrument before me was made by Coradi, of Zurich, on conditions laid down by me, namely, that the cost should not exceed L14, and that polar distances should range between one and ten half-inches.  The first machine made by Coradi on these lines was, by a misunderstanding, sold in Germany, but the one I exhibit is the first, I believe, that has reached England, and to this extent I may, perhaps, be permitted to call it new.  I look upon it rather as a suggestion upon which a still more practical instrument can be made in this country than as a perfect model.  I believe there would be a wide sale for such an instrument were it once generally known to exist, and, what is more to work efficiently.  It remains for me to point out in what the Abdank-Abakanowicz, or, rather, Coradi, integraph differs from Boys’ instrument.