The problem before me was, “What is it that makes my engine run so smoothly?” I am not a mathematician, and so could not use his methods. I got along by graphic methods and study of the motion of the piston controlled by the crank. My recollection of the several steps of my progress is quite indistinct. One thing I do remember distinctly, and that is the help that I got from my friend Frederick J. Slade, who was younger than I, but who died several years ago. Mr. Slade was a mathematical genius. The firm of Cooper, Hewitt & Co. were at a later date the pioneer makers in the United States of wrought-iron beams and other structural shapes; and all their designs and computations were the work of Mr. Slade. I had formed his acquaintance in London in ’63. I met him again in Paris in ’67. He was then in France in the employ of Abram S. Hewitt, investigating the Siemens-Martin process of steel manufacture. He took much interest in the engine. One day he brought to me a diagram representing the two now famous triangles, and a demonstration of them which he had made, showing that the ordinates, representing the acceleration or retardation of the piston motion at every point, if erected on the center line of the engine, terminate in a diagonal line, which, with a connecting-rod of infinite length, would cross this center line at its middle point. This exhibited at once the equalizing action of the reciprocating parts in a cut-off engine, absorbing the excessive force of the steam at the commencement and imparting it to the crank at the end of the stroke. I feel myself more indebted to Mr. Slade than to any one else, and would here record the tribute of my grateful acknowledgment. On January 30, 1868, I had the honor of reading a paper on the Allen engine before the Institution of Mechanical Engineers. The discussion of the paper was postponed until the next meeting, April 30, and the paper was ordered meantime to be printed and sent to the members. The result was that on the latter date we had a very interesting discussion. I may mention two things which occurred at the first meeting, but do not appear in the report of the transactions. When the secretary reached the statement that the acceleration of the piston was greatest at the commencement of the stroke, the president of the meeting, Sampson Lloyd, Esq., one of the vice-presidents of the Institution, stopped the reading and said to me, “You do not mean, Mr. Porter, that this is _on_ the commencement of the stroke, but at a point near its commencement.” I was obliged to answer him that I intended to say that precisely on the dead center, at the point where motion in one direction had ceased and that in the opposite direction had not yet commenced, at that precise point the stress on the crank was at its maximum, the crank having brought the reciprocating parts to rest, and then by a continuance of the same effort putting them in motion in the reverse direction. [Illustration: FREDERICK J. SLADE]