A Treatise on the Theory of Screws

APPENDIX II. 499 Let the constraints be what they may—let the position B lie anywhere in the close neighbourhood of A—Helix found that he could move the body from A to B by an. extremely simple operation. With the aid of a skilful mechanic he prepared a screw with a suitable pitch, and adjusted this screw in a definite position. The rigid body was then attached by rigid bonds to a nut on this screw, and it was found that the movement of the body from A to B could be effected by simply turning the nut on the screw. A perfectly definite fact about the mobility of the body had thus been ascertained. It was able to twist to and fro on a certain screw. Mr Querulous could not see that there was any simplicity or geometrical clearness in the notion of a screwing movement; in fact he thought it was the reverse of simple. Did not the screwing movement mean a translation parallel to an axis and a rotation around that axis? Was it not better to think of the rotation and the translation sepai’ately than to jumble together two things so totally distinct into a composite notion ? But Querulous was instantly answered by Oue-to-One. ‘ Lamentable, indeed,’ said he, ‘would be a divorce between the rotation and the translation. Together they form the unit of rigid movement. Nature herself has wedded them, and the fruits of their union are both abundant and beautiful.’ The success of Helix encouraged him to proceed with the experiments, and speedily he found a second screw about which the body could also twist. He was about to continue when he was interrupted by Mr Anharmonic, who said, ‘ Tarry a moment, for geometry declares that a body free to twist about two screws is free to twist about a myriad of screws. These form the generators of a graceful ruled surface known as the cylindroid. There may be infinite variety in the conceivable constraints, but there can be no corresponding variety in the character of this surface. Cylindroids differ in size, they have no difference in shape. Let us then make a cylindroid of the right size, and so place it that two of its screws coincide with those you have discovered; then I promise you that the body can be twisted about every screw on the surface. In other words, if a body has two degrees of freedom the cyliudroid is the natural and the perfectly general method for giving an exact specification of its mobility.’ A single step remained to complete the examination of the freedom of the body. Mr Helix continued bis experiments and presently detected a third screw, about which the body can also twist in addition to those on the cylindroicL A flood of geometrical light then burst forth and illuminated the whole theory. It appeared that the body was free to twist about ranks upon ranks of screws all beautifully arranged by their pitches on a system of hyperboloids. After a brief conference with Anharmonic and One-to-One, Helix announced that sufficient experiments of this kind had now been made. By the single screw, the cylindroid, and the family of hyperboloids, every conceivable information about the mobility of the rigid body can be adequately conveyed. Let the body have any constraints, how- soever elaborate, yet the definite geometrical conceptions just stated will be sufficient. With perfect lucidity Mr Helix expounded the matter to the committee. He 32-2