Page:Philosophical Transactions - Volume 037.djvu/163

 an Angle double to that of the Inclination of the two polish'd Surfaces.

Let R F H and R G I represent the Sections of the Plane of the Figure by the polish'd Surfaces of the two Specula B C and D E, erected perpendicularly thereon, meeting in R, which will be the Point where their common Section, perpendicular likewise to the same Plane, passes it, and H R I is the Angle of their Inclination. Let A F be a Ray of Light from any Point of an Object A falling on the Point F of the first Speculum B C, and thence reflected into the Line F G, and at the Point G of the second Speculum D E reflected again into the Line {{nowrap|G K}, produce G F and K G backwards to M and N, the two successive Representations of the Point A; and draw R A, R M, and R N.

Since the Point A is in the Plane of the Scheme, the Point M will be so also by the known Laws of Catoptricks. The Line F M is equal to F A, and the Angle M F A double the Angle H F A or M F H; consequently R M is equal to R A, and the Angle M R A double the Angle H R A or M R H. In the same manner the Point N is also in the Plane of the Scheme, the Line R N equal to R M, and the Angle M R N double the Angle M R I or I R N: Subtract the Angle M R A from the Angle M R N, and the Angle A R N remains equal to double the Difference of the Angles M R I and M R H, or double the Angle H R I, by which the Surface of the Speculum D E is reclin'd from that of B C; and the Lines R A, R M and R N are equal.

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