TEANSACTIONS OF SECTION A. 573 



called the pitch. The screw has for us a dual aspect of much significance. No 

 small movement of the body is conceivable which does not consist of a twist 

 about a screw. No set of forces could be applied to the body which were not 

 equivalent to a wrench upon a screw. Everyone remembers the two celebrated 

 rules that forces are compoimded like rotations and that couples are compounded 

 like translations. These may now be replaced by the single but far more com- 

 pendious rule which asserts that wrenches and twists are to be compounded by 

 identical laws. Would you unite geometry with generality in your dynamics ? 

 It is by screws, and screws only, that you are enabled to do so,' 



These ideas were rather too abstract for Cartesian, who remarked that, as 

 D'Alembert's principle provided for everything in dynamics, screws could not be 

 needed. Mr. Querulous sought to confirm him by saying that he did not see how 

 screws helped the study either of Foucault's Pendulum or of the Precession of the 

 Equinoxes. 



Such absurd observations kindled the intellectual wrath of One-to-One, who 

 rose and said, ' In the development of the natural philosopher two epochs may be 

 noted. At the first he becomes aware that problems exist. At the second he 

 discovers their solution. Querulous has not yet reached the first epoch ; he cannot 

 even conceive those problems which the " Theory of Screws " proposes to 

 solve. I may, however, inform him that the " Theory of Screws " is not a general 

 dynamical calculus. It is the discussion of a particular class of dynamical 

 problems which do not admit of any other enunciation except that which the 

 theory itself provides. Let us hope that ere our labours have ended Mr. 

 Querulous may obtain some glimmering of the subject.' The chairman happily 

 assuaged matters. ' We must pardon,' he said, ' the vigorous language of our friend 

 Mr. One-to-One. His faith in geometry is boundless — in fact he is said to believe 

 that the only real existence iu the universe is anharmonic ratio. It is even his 

 opinion that if a man travelled sufficiently far along a straight line in one direction 

 he would ultimately arrive at the point from which he started. 



It was thus obvious that screws were indispensable alike for the application 

 of the forces and for the observation of the movements. Special measuring instru- 

 ments were devised by which the positions and pitches of the various screws 

 could be carefully ascertained. All being ready the first experiment was 

 commenced. 



A screw was chosen quite at random, and a great impulsive wrench was ad- 

 ministered thereon. In the infinite majority of cases this would start the body 

 into activity, and it would commence to move in the only manner possible — i.e., it 

 would begin to twist about some screw. It happened, however, that this first 

 experiment was unsuccessful ; the impulsive wi-ench failed to operate, or at all 

 events the body did not stir. ' I told you it would not do,' shouted Querulous, 

 though he instantly subsided when One-to-One glanced at him. 



Much may often be learned from an experiment which fails, and the chairman 

 sagaciously accounted for the failure, and in doing so directed the attention of the 

 committee to an important branch of the subject. The mishap was due, he 

 thought, to some reaction of the constraints which had neutralised the effect oi 

 the wrench. He believed it would save time in their future investigations if these 

 reactions could be first studied and their number and position ascertained. 



To this suggestion Mr. Cartesian demurred. He urged that it would involve 

 an endless task. ' Look,' he said, ' at the complexity of the constraints : how the 

 body rests on these surfaces here ; how it is fastened by links to those pomts there ; 

 how there are a thousand-and-one ways in which reactions might originate.' Mr. 

 Commonsense and other members of the committee were not so easily deterred, 

 and they determined to work out the subject thoroughly. At first they did not see 

 their way clearly, and much time was spent in misdirected attempts. At length 

 they were rewarded by a curious and unexpected discovery, which suddenly 

 rendered the obscure reactions perfectly transparent. 



A trial was being made upon a body which had only one degree of freedom ; 

 was, in fact, only able to twist about a single screw, X. Another screw, Y, was 

 speedily found, such that a wrench thereon failed to disturb the body. It now 



