Rotation now to force convert, And force into rotation; Unchanged the work, we can assert, In spite of transformation. And if two screws no work can claim, Reciprocal will be their name. Five numbers will a screw define, A screwing motion, six; For four will give the axial line, One more the pitch will fix; And hence we always can contrive One screw reciprocal to five. Screws--two, three, four or five, combined (No question here of six), Yield other screws which are confined Within one screw complex. Thus we obtain the clearest notion Of freedom and constraint of motion. In complex III, three several screws At every point you find, Or if you one direction choose, One screw is to your mind; And complexes of order III. Their own reciprocals may be. In IV, wherever you arrive, You find of screws a cone, On every line of complex V. There is precisely one; At each point of this complex rich, A plane of screws have given pitch. But time would fail me to discourse Of Order and Degree; Of Impulse, Energy and Force, And Reciprocity. All these and more, for motions small, Have been discussed by Dr. Ball. --ANONYMOUS. CHAPTER XIX THE CALCULUS AND ALLIED TOPICS =1901.= It may be said that the conceptions of differential quotient and integral, which in their origin certainly go back to Archimedes, were introduced into science by the investigations of Kepler, Descartes, Cavalieri, Fermat and Wallis.... The capital discovery that differentiation and integration are _inverse_ operations belongs to Newton and Leibnitz.--LIE, SOPHUS. _Leipziger Berichte, 47 (1895), Math.-phys. Classe, p. 53._ =1902.= It appears that Fermat, the true inventor of the differential calculus, considered that calculus as derived from the calculus of finite differences by neglecting infinitesimals of higher orders as compared with those of a lower order.... Newton, through his method of fluxions, has since rendered the calculus more analytical, he also simplified and generalized the method by the invention of his binomial theorem. Leibnitz has enriched the differential calculus by a very happy notation. --LAPLACE. _Lés Intégrales Définies, etc.; Oeuvres, t. 12 (Paris, 1898), p. 359._