Geometrical Optics as a part of the mathematical curriculum in schools, universities, and technical colleges is now a dead subject, yet the generalised study of ray-tracing in optics started by Sir William Rowan Hamilton in 1824 is well worth attention on all sides. He writes: “Those who have meditated on the beauty and utility in theoretical mechanics of the general method of Lagrange, who have felt the power and dignity of that central dynamical theorem which he deduced in the Mécanique Analytique must feel that mathematical optics can only then attain a coordinate rank with mathematical mechanics when it shall possess an appropriate method and become the unfolding of a central idea.” With this end in view, Sir Wm. Hamilton submits his three papers on the Theory of Systems of Rays to the Royal Irish Academy, which, as Lord Rayleigh said, bear the mark of genius throughout. The enormous task that Sir Wm. Hamilton set himself will be evident when we discuss the wanderings of rays through optical systems in an elementary way to-night leading up to Hamilton’s general method of the Characteristic Function and its development.