Sept. I, 1887] 



NATURE 



425 



were of a simple character, and the possible movements of the 

 body were obvious. But the constraints in the present case were 

 of puzzling complexity. There were cords and links, moving axes, 

 surfaces with which the body lay in contact, and many other 

 j^eouietrical constraints. Experience of ordinary problems in 

 mechanics would be of little avail. In fact, the chairman truly 

 appreciated the situation when he said that the constraints were 

 of a perfectly general type. 



In the dismay with which this announcement was received 

 Mr. Commonsense advanced to the body and tried whether 

 it could move at all. Yes, it was obvious that in some ways the 

 body could be moved. Then said Commonsense, " Ought we 

 not first to study carefully the nature of the freedom which the 

 body possesses? Ought we not to make an inventory of 

 every distinct movement of which the body is capable? Until 

 this has lieen obtained I do not see how we can make any 

 progress in the dynamical part of our business. " 



Mr. Querulous ridiculed this proposal. " How could you," he 

 said, "make any geometrical theory of the mobility of a body 

 without knowing all about the constraints? And yet you are 

 attempting to do so with perfectly general constrainis of which 

 you know nothing. It must all be waste of time, for though 

 I have read many books on mechanics, I never saw anything 

 like it." 



Here the gentle voice of Mr. Anharmonic was heard. " Let 

 us try, let us simply experiment on the mobility of the body, and 

 let us faithfully record what we find." In justification of this 

 advice Mr. Anharmonic made a remark which was new to most 

 members of the committee ; he asserted that though the con- 

 straints may be of endless variety and complexity there can 

 be on'y a very limited za'iety in the types of possible mobility. 



It was therefore resolved to make a series of experiments with 

 the simple object of seeing how the body could be moved. Mr. 

 Cartesian, having a repuiation for such work, was requested to 

 undertake the inquiry and to report to the committee. Cartesian 

 commenced operations in accordance with the well-known 

 traditions of his craft. He erected a cumbrous apparatus which 

 he called his three rectangular axes. He then attempted to push 

 the body parallel to one of these axes, but it would not stir. He 

 tp>d to move the body parallel to each of the other axes, 

 but was again unsuccessful. He then attached the body 

 to one of the axes and tried to effect a rotation around that 

 axis. Again he faihd, for the constraints were of too elaborate 

 a type to accommodate themselves to Mr. Cartesian's crude 

 notions. 



We shall subsequently find that the movements of the body 

 are necessarily of an exquisitely simple type, yet such was the 

 clumsiness and the artificial character of Mr. Cartesian's 

 machinery that he failed to perceive the simplicity. To him it 

 appeared that the body could only move in a highly complex 

 manner ; he saw that it could accept a composite movement con- 

 sisting of rotations about two or three of his axes and simultaneous 

 trjnstaiions also parallel to two or three axes. Cartesian was a 

 very skilful calculator, and by a series of experiments even with 

 his unsympathetic apparatus he obtained some knowledge of the 

 subject, sufficient for purposes in which a vivid comprehension of 

 the whole was not required. The inadequacy of Cartesian's 

 geometry was painfully evident when he reported to the com- 

 mittee on the mobility of the rigid body. "I find," he said, 

 *' that the body can neither move parallel to x, nor to y, nor 

 to z ; neither can I make it rotate around x, nor y, nor z ; but 

 I could push it an inch parallel to x, provided that at the same 

 time I pushed it a foot parallel to y and a yard backwards 

 parallel to z, and that it was also turned a degree around x, half 

 a degree the other way around y, and twenty-three minutes and 

 nineteen seconds around z." 



"Is that all?" asks the chairman. "Oh no," replied Mr. 

 Cartesian, "there are other proportions in which the ingredients 

 may be combined so as to produce a possible movement," and 

 he was proceeding to state them when Mr. Commonsense 

 interpo-ed. "Stop! stop!" said he, "I can make nothing 

 of all these figures. This jargon about x, y, and z may suffice 

 for your calculations, but it fails to convey to my mind any clear 

 or concise notion of the movements which the body is free 

 to make." 



Many of the committee sympathized with this view of Common- 

 sense, and they came to the conclusion that 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 dis- 

 covering it were raised when they saw Mr. Helix volunteer his 



services and advance to the rigid body. Helix brought with him • 

 no cumbrous rectangular axes, but commenced to try the mobility 

 of the body in the simplest manner. He found it lyin;^ 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 neighV)ouring 

 position, B. Contrast the procedure of Cartesian with the pro- 

 cedure 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 the 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 M 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 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 s-imple. Did not 

 the screwing movement mean a translntion parallel to an axis 

 afid 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 happy 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 perfect 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 al.-^o ■ 

 twist in addition to those on the cylindroid. A flood of 

 geometrical light then burst forth and illuminated the whole 

 theory. It appeared th.at the body was free to twist about ranks 

 upon ranks of screws all beautifully arranged by their pitches on 1 

 a system of hyperboloids. After a brief conference with Anhar- 

 monic and One-to-One, Helix announced that sufficient experi- 

 ments 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, . 

 however elaborate, yet the definite geometiical conceptions just 

 stated will be sufficient. 



With perfect lucidity Mr. Helix expounded the matter to the 

 committee. He exhibited to them an elegant fabric of screws, 

 each with its appropriate pitch, and then he summarized his • 

 labours by saying, " About every one of these screws you can 

 displace the body by twisting, and what is of no less importance • 

 it will not admit of any movement which is not such a twist." 

 The committee expressed their satisfaction with this information. 

 It was both clear and complete. Indeed, the chairman remarked 1 



