A Treatise on the Theory of Screws

500 THE THEORY OF SCREWS. exhibited to them an elegant fabric of screws, each with its appropriate pitch, and then he summarised 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 with considerable force that a more thorough method of specify- ing the freedom of the body was inconceivable. 1 he 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. Elated by their pi'evious 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 1 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. Ihere 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 i'emember 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 committee of the labours of Mathematician Poinsot, and then he approached the rigid body. Helix com- menced by clearing away Cartesian’s arbitrary scaffolding of rectangular axes. He showed how an attack of the most pei'fect generality could be delivered in a form that admitted of concise and elegant description. ‘I shall,’ he said, ‘admin- ister 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 proposed, therefore, to weld Poinsot’s force and couple into the single conception of a wrench 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