Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/410

318 which they hould converge; CDE the curve line which by its revolution round the axis AB decribes the uperficies ought; 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 coalece with the point E; and the ultimate ratio of the line DF by which AD is increaed, to the line DG by which DB is diminihed, will be the ame as that of the ine of incidence to the ine 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 mut pas, 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 decribed two circles cutting each other in D; that point D will touch the curve ought CDE, and by touching it any where at pleaure, will determine that curve. Q. E. I.

By cauing the point A or B to go off ometimes in infinitum, and ometimes to move towards other parts of the point C, will be obtained all thoe figures which Carteuis has exhibited in his Optics and Geometry relating to refractions. The invention of which Carteius having thought fit to conceal, is here laid open in this propoition.

If a body lighting on any uperficies CD (Pl. 25. Fig. 9.) in the direction of a right line AD, drawn according to any law, hould 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