426 



NA TURE 



{Sept. I, 1887 



with considerable force that a more thorough method of specifying 

 the freedom of the body was in-oitceiTab/e. 



The discovery of the mobility of the body completed the first 

 stage of the labours of the committee, and they were ready 

 to commence the serious dynamical work. Force was now to 

 be used, with the view of experimenting on the behaviour of the 

 body under its influence. lilated by their previous success the 

 committee declared that they would not rest satisfied until they 

 had again obtained the most perfect solution of the most general 

 problem. 



" But what is force? " said one of the committee. " Send for 

 Mr. Cartesian," said the chairman, " we will give him another 

 trial." Mr. Cartesian was accordingly requested to devise an 

 engine of the most ferocious description wherewith to attack the 

 rigid body. He was promptly ready with a scheme, the weapons 

 being drawn from his trusty but old-fashioned armoury. He 

 would erect three rectangular axes, he would administer a 

 tremendous blow parallel to each of these axes, and then he 

 would simultaneously apply to the body a forcible couple around 

 each of them ; this was the utmost he could do, 



" No doubt," said the chairman, "what you propose would 

 be highly effective, but, Mr. Cartesian, do you not think that 

 while you still retained the perfect generality of your attack, you 

 might simplify your specification of it? I confess that these three 

 blows all given at once at right angles to each other, and these 

 three couples which you propose to impart at the same time, 

 rather confuse me. There seems a want of unity somehow. In 

 short, Mr. Cartesian, your scheme does not create a distinct 

 geometrical image in my mind. We gladly acknowledge its 

 suitability for numerical calculation, and we remember its famous 

 achievements, but it is utterly inadequate to the aspirations of 

 this committee. We must look elsewhere." 



Again Mr. Helix stepped forward. He reminded the com- 

 mittee of the labours of Mathematician Poinsot, and then he 

 approached the rigid body. Helix commenced by clearing away 

 Cartesian's arbitrary scaffolding of rectangular axes. He showed 

 how an attack of the most perfect generality could be delivered 

 in a form that admitted of concise and elegant description. " I 

 shall," he said, "administer a blow upon the rigid body from 

 some unexpected direction, and at the same instant I shall apply 

 a vigorous couple in a plane perpendicular to the line of the blow." 



A happy inspiration here seized upon Mr. Anharmonic. He 

 knew, of course, that the efficiency of a couple is measured by 

 its moment — that is, by the product of a force and a linear 

 magnitude. He pro]iosed, therefore, to weld Poinsot's force and 

 couple into the single conception of a ivrench on a screw. The 

 force would be directed along the screw while the moment of the 

 couple would equal the product of the force and the pitch of the 

 screw. "A screw," he said, "is to be regarded merely as a 

 directed straight line with an associated linear magnitude 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 con- 

 sist 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. 

 Every one remembers the two celebrated rules that forces are 

 compounded like rotations and that couples are compounded like 

 translations. These may now be replaced by the single but far 

 more compendious 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 re- 

 marked 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 sairl, " the vigorous language of our friend Mr. One-to-On 

 His faith in geometry is boundless. In fact he is said to beliei 

 that the only real existence in the universe is anharmonic rati' 

 It is also his opinion that if a man travelled sufficiently far alor 

 a straight line in one direction he will ultimately arrive at tl 

 point from which he started. The committee would be glad 1 

 see Mr. Querulous making the trial." 



It was obvious that screws were indispensable alike f 

 the application of the forces and for the observation of the mov 

 ments. Special measuring instruments were devised by whic 

 the positions and pitches of the various screws could be careful 

 a'^certained. All being ready the first experiment was cod 

 menced. 



A screw was chosen quite at random, and a great impulsi\ 

 wrench was administered thereon. In the infinite majority . 

 cases this would start the body into activity, and it would con 

 mence to move in the only manner possible — i.e. it would begi 

 to twist about some screw. It happened, however, that this fir 

 experiment was unsuccessful ; the impulsive wrench failed 1 

 operate, or at all events the body did not stir. " I told you 

 would not do," shouted Querulous, though he instantly subside 

 when One-toOne glanced at him. 



Much may often be learned from an experiment which fail 

 and the chairman sagaciously accounted for the failure, and ; 

 doing so directed the attention of the committee to an importa: 

 branch of the subject. The mishap was due, he thought, ' 

 some reaction of the constraints which had neutralized the effe 

 of the wrench. He believed it would save time in their futu 

 investigations if these reactions could be first studied and the 

 number and position ascertained. 



To this suggestion Mr. Cartesian demurred. He urged th: 

 it would involve an endless task. "Look," he said, "attl 

 complexity of the constraints : how the body rests on these su 

 faces here ; how it is fastened by links to those points then 

 how there are a thousand-and-one ways in which reactions mig' 

 originate." Mr. Commonsense and other members of the cor 

 mittee were not so easily deterred, and they determined to woi 

 out the subject thoroughly. At first they did not see their wj 

 clearly, and much time was spent in misdirected attempts, i 

 length they were rewarded by a curious and unexpected di 

 covery, which suddenly rendered the obscure reactions perfect 

 transparent. 



A trial was being made upon a body which had only 01 

 degree of freedom ; was, in fact, only able to twist about 

 single screw, X. Another .'crew, Y, was speedily found, su( 

 that a wrench thereon failed to disturb the body. It no 

 occurred to the committee to try the effect of interchanging tl 

 relation of these screws. They accordingly arranged that tl 

 body should be left only free to twist about Y, while a wrenc 

 was applied on X. Again the body did not stir. The impoi 

 ance of this fact immediately arrestel the attention of the mo 

 intelligent observers, for it established the following g'ener 

 law : If a wrench on X fails to move a body only free to twi 

 about Y, then a wrench on Y must be unable to move a hoc 

 only free to twist about X. It was determined to speak of tv 

 screws when related in this manner as reciprocal. 



Some members of the committee did not at first realize tl 

 significance of this discovery. Their difficulty arose from tl 

 restricted character of the experiments by which the law of rec 

 procal i-crews had been suggested They said, "You have shov 

 us that this law is observed in the case of a body only free 

 twist about one screw at a time ; but how does this teach an 

 thing of the general case in which the body is free to twist abo 

 whole shoals of screws ? " Mr. Commonsense immediate 

 showed that the discovery could be enunciated in a quite u 

 objectionable form. " The law of reciprocal screws," he sai 

 " does not depend upon the constraints or the limitations of tl 

 freedom. It may be expressed in this way : Two screws c 

 reciprocalivhen a small twist about either can do no worJz again 

 a wrench on the other." 



This important step at once brought into view the who 

 geometry of the reactions. Let us suppose that the freedom 

 the body was such that it could twist about all the screws of 

 system which we shall call U. Let all the possible reactio 

 form wrenches on the screws of another system, V. It th' 

 appeared that every screw upon U is reciprocal to every sere 

 upon V. A body might therefore be free to twist about eve 

 screw of V and still remain in equilibrium, notwithstanding 

 presence of a wrench on every screw of U. A body ffl 



