which they ſhould converge; CDE the curve line which by its revolution round the axis AB deſcribes the ſuperficies ſought; D, E, any two points of that curve; and EF, EG perpendiculars let fall on the paths of the bodies AD, DB. Let the point D approach to and coaleſce with the point E; and the ultimate ratio of the line DF by which AD is increaſed, to the line DG by which DB is diminiſhed, will be the ſame as that of the ſine of incidence to the ſine of emergence. Therefore the ratio of the increment of the line AD to the decrement of the line D8 is given; and therefore if in the axis AB there be taken any where the point C through which the curve CDE muſt paſs, and CM the increment of AC be taken in that given ratio to CN the decrement of BC, and from the centres A, B, with the intervals AM, BM there be deſcribed two circles cutting each other in D; that point D will touch the curve ſought CDE, and by touching it any where at pleaſure, will determine that curve. Q. E. I.
Cor. 1. By cauſing the point A or B to go off ſometimes in infinitum, and ſometimes to move towards other parts of the point C, will be obtained all thoſe figures which Carteſuis has exhibited in his Optics and Geometry relating to refractions. The invention of which Carteſius having thought fit to conceal, is here laid open in this propoſition.
Cor. 2. If a body lighting on any ſuperficies CD (Pl. 25. Fig. 9.) in the direction of a right line AD, drawn according to any law, ſhould emerge in the direction of another right line DK; and from the point C there be drawn curve lines CP, CQ always perpendicular to AD, DK; the increments of the lines PD, QD, and therefore
