_The Linear Constructions of Equations; Universal Arithmetic (London, 1769), Vol. 2, p. 468._ =1893.= The pursuit of mathematics unfolds its formative power completely only with the transition from the elementary subjects to analytical geometry. Unquestionably the simplest geometry and algebra already accustom the mind to sharp quantitative thinking, as also to assume as true only axioms and what has been proven. But the representation of functions by curves or surfaces reveals a new world of concepts and teaches the use of one of the most fruitful methods, which the human mind ever employed to increase its own effectiveness. What the discovery of this method by Vieta and Descartes brought to humanity, that it brings today to every one who is in any measure endowed for such things: a life-epoch-making beam of light [Lichtblick]. This method has its roots in the farthest depths of human cognition and so has an entirely different significance, than the most ingenious artifice which serves a special purpose.--BOIS-REYMOND, EMIL DU. _Reden, Bd. 1 (Leipzig, 1885), p. 287._ =1894.= _Song of the Screw._ A moving form or rigid mass, Under whate’er conditions Along successive screws must pass Between each two positions. It turns around and slides along-- This is the burden of my song. The pitch of screw, if multiplied By angle of rotation, Will give the distance it must glide In motion of translation. Infinite pitch means pure translation, And zero pitch means pure rotation. Two motions on two given screws, With amplitudes at pleasure, Into a third screw-motion fuse, Whose amplitude we measure By parallelogram construction (A very obvious deduction). Its axis cuts the nodal line Which to both screws is normal, And generates a form divine, Whose name, in language formal, Is “surface-ruled of third degree.” Cylindroid is the name for me. Rotation round a given line Is like a force along, If to say couple you decline, You’re clearly in the wrong;-- ’Tis obvious, upon reflection, A line is not a mere direction. So couples with translations too In all respects agree; And thus there centres in the screw A wondrous harmony Of Kinematics and of Statics,-- The sweetest thing in mathematics. The forces on one given screw, With motion on a second, In general some work will do, Whose magnitude is reckoned By angle, force, and what we call The coefficient virtual.