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 separately than to jumble together two things so 

 totally distinct into a composite notion? 



But Querulous was instantly answered by One-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 cylindroid 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 his experiments and presently detected a third screw, about 

 which the body can also twist in addition to those on the cylindroid. 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 



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