TKANSACTIONS OF SECTION A. 571 



it rotate around a; nor //, nor z ; but I could push it an inch parallel to .!', provided 

 that at the same time 1 pushed it a foot parallel to y and a yard backwards parallel 

 to s, and that it was also turned a degree around .r, half a degree the other way 

 around i/, and twenty-three miuutes and nineteen seconds around =.' 



' Is that all ? ' asks the chairman. ' Oh, no,' replied Mr. Cartesian, ' there are 

 other proportions in which the ingredients may he combined so as to produce 

 a possible movement,' and he was proceeding to state them when Mr. Common- 

 sense interposed. ' Stop ! stop ! ' said he, ' I can make nothing of all these figures. 

 This jargon about .r, i/, and s may suffice for your calculations, but it fails to 

 convey to my mind anj- clear or concise notion of the movements which the body 

 is free to make.' 



Many of the committee sympathised with this view of Commonsense, and they 

 came to the conclusion tbat there was nothing to be extracted from poor old 

 Cartesian and his axes. They felt that there must be some better method, and 

 their hopes of discovering it were raised when they saw Mr. Helix volunteer his 

 services and advance to the rigid body. Helix brought with him no cumbrous 

 rectangidar axes, but commenced to try the mobility of the body in the simplest 

 manner. He found it lying at rest in a position we may call A. Perceiving that 

 it was in some ways mobile, he gave it a slight displacement to a neighbouring 

 position B. Contrast the procedure of Cartesian with the procedure of Helix. 

 Cartesian tried to force the body to move along certain routes which he had 

 arbitrarily chosen, but which the body had not chosen ; in fact the body would not 

 take any one of his routes separately, though it, would take all of them together in 

 a most embarrassing manner. But Helix had no preconceived scheme as to the 

 nature of the movements to be expected. He simply found the body in a certain 

 position A, and then he coaxed the body to move, not in this particular way or 

 in that particular way, but any way the body liked to any new position B. 



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 has thus been ascertained. It is able to twist to and fro ou 

 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.' 



