THE THEORY OF GRAVITATION

PROPOSITION I. THEOREM I.

That the areas, which bodies, when moving in curves, cut off by radii drawn to a fixed center of force, are in one fixed plane and are proportional to the times.

A

Let the time be divided into equal parts, and in the first period of time let the body driven by one force describe the line AB. In the second period, it would, if nothing hindered it, go on to c, describing the line Bc equal to AB. Then by the radii AS, BS, CS to the center S would be cut off the equal areas ASB, BSc [the bases being equal and the altitude the same]. Now when the body comes to B, a centripetal force [in the direction BS] acts upon it with uniform impulse, and makes it leave the line of direction Bc and pass along the line BC. Let cC be drawn parallel to the direction of the force BS, meeting BC in C. Then at the end of the second (equal) period the body will be found at C, in the same plane with the triangle ASB. Draw SC. Then the triangle SBC, on account of the parallels SB and cC, will be equal to the triangle SBC and therefore to the triangle SAB, etc.-Therefore in equal times equal areas will be described in the same plane.-Let the number of the triangles be increased and their altitude diminished to infinity: their ultimate perimeter will be a curve (Cor. iv. Lem. iii.). And therefore a centripetal force, by which a body is continually drawn from a course tangent to this curve, will act along this radius and whatever areas have been described proportional to the times, will remain proportional to the same times when curvilinear.