XXVIII. On the mechanical description of curves

Let A, B, C be three wheels rolling in one another (fig. 1); they may of course be supposed to describe simultaneously the angles mθ, nθ, rθ, when m, n,and r are constant. Let α, β, γ be three nuts situated on A, B, C respectively, at distances a, b, c from their centres. Then if these nuts work in horizontal bars (as exemplified in many sewing-machines), the bars will descend vertically through the spaces a sin mθ, b sin nθ, c sin rθ respectively. We may combine all these vertical motions together; for if vertical rods be attached to the horizontal bars, and a cord fixed at Q, pass over the pulleys a1, A2, a3 b1 , B2, b3, c1, C2, c3, as shown in the figure, the other extremity Q1 will describe the space a sin mθ + b sin nθ + c sin rθ. By this contrivance we are able to combine any number of vertical descents, so that it is readily seen that a sin (mθ + α) + b sin (nθ + β ) + &c. may be described mechanically. A machine on the same principle as this had been previously invented by Mr. Bashforth.