transcendental functions only when expressed in a series of powers. Irrational quantities would be represented approximately.
To express the complicated relations among the various parts of the machine, Babbage invented what he called a "mechanical notation" explained in a paper published in the Philosophical Transactions for 1826, entitled "On a method of expressing by signs the action of machinery." It consists of three divisions; first, Notation for the parts; second, Representation of trains; third, Representation of cycles. He denoted pieces and points of the frame by upright letters, the former capitals and the latter small letters; movable pieces and their points by slant letters, capitals and small letters respectively. On account of the great number of movable pieces he employed indices, placing them to the left above the letters. The train is designed to show how motion is transmitted from the prime motor to the final driven piece. The several pieces are marked on a diagram by trial so that each pair of driver (point) and driven (piece) may be connected by arrows; after a number of trials the pieces are so placed as to make the connecting arrows the shortest. In a cycle he aimed at representing the time during which each piece moved and the time of action of each of its working points. The period of the machine is represented by a vertical line divided into proper subdivisions on the nature of the machine; to each piece and to each working point is allotted a parallel line, and those portions of the period are marked off during which there is no movement of the piece or the point, thus giving a synoptic view of the motion of the machine. To make drawings, perfect the notations, and test mechanical contrivances, he turned his coach house into a forge and foundry, transformed his stables into a workshop, and expended a large sum in employing skilled workmen.
In 1840 he received a letter from M. Plana, nephew of Lagrange, urging him to come to a meeting of Italian philosophers which was to be held in Turin. Babbage went, furnished with models, drawings, and notations of his Analytical Engine, and explained them to the Italian mathematicians, among whom was M. Menabréa. Subsequently Menabréa wrote an
