642 PHILOSOPHICAL TRANSACTIONS. [anNO 1742-3. 



tact contained by the curve and circle of curvature, in the same manner as the 

 curvature itself is measured by the angle of contact contained by the curve and 

 tangent. The ground of this remark will better appear from an example: ac- 

 cording to Sir Isaac Newton's explication, the variation of curvature is uniform 

 in the logarithmic spiral, the fluxion of the ray of curvature in this figure being 

 always in the same ratio to the fluxion of the curve; and yet while the spiral is 

 produced, though its curvature decreases, it never vanishes; which must appear 

 strange to such as do not attend to the import of his definition. — It is easy, 

 however, to derive one of these measures of this variation from the other, and 

 because Sir Isaac Newton's is, generally speaking, assigned by more simple ex- 

 pressions, the author has the rather conformed to it in this treatise, but thought 

 it necessary to give the caution we have mentioned. 



The greatest part of this chapter is employed in treating of useful problems, 

 that have a dependence on the curvature of lines. First, the properties of the 

 cycloid are briefly demonstrated, with the application of this doctrine to the 

 motion of pendulums, by showing that when the motion of the generating 

 circle along the base is uniform, and therefore may measure the time, the mo- 

 tion of the point that describes the cycloid, is such as would be acquired by a 

 heavy body descending along the cycloidal arc, the axis of the figure being sup- 

 posed perpendicular to the horizon. In the next place, the caustics, by re- 

 flexion and refraction, are determined. If perpendiculars be always drawn from 

 the radiating point to the tangents of the curve, and a new curve be supposed 

 to be the locus of the intersections of the perpendiculars and tangents, then 

 the line, by the evolution of which that new curve can be described, is similar 

 and similarly situated to the caustic by reflexion. The doctrine of centripetal 

 forces is treated at length from art. 4l6 to 493. 



First, a body is supposed to descend freely by its gravity in a vertical line, 

 and because the gravity is the power which accelerates the motion of the body, 

 it must be measured by the fluxion of its velocity, or the second fluxion of the 

 space described by it. When the vertical line is supposed to move parallel to 

 itself with an uniform motion, the body will descend in it in the same manner 

 as before; and the gravity will be still measured by the second fluxion of the 

 descent, or the second fluxion of the ordinate of the curve that is traced in this 

 case by the body on an immoveable plane, and therefore is as the square of the 

 velocity, which is measured by the fluxion of the curve, directly, and the chord 

 of the circle of curvature that is in the direction of the gravity inversely, by a 

 proposition mentioned above. When the gravity acts uniformly, and in parallel 

 lines, the projectile, in describing any arc, falls below the tangent drawn at the 

 beginning of the arc, as much as if it had fallen perpendicularly in the ver- 



