base SB and the same height Bc or VC, they have the same area. By symmetry, triangle SBc also has the same area as triangle SAB, therefore the object has swept out equal areas SAB and SBC in equal times. At point C, the object receives another impulse toward S, again deflecting its path during the third interval from d to D. Thus it continues to E and beyond, the triangles SAB, SBc, SBC, SCd, SCD, SDe, SDE all having the same area. Allowing the time intervals to become ever smaller, the path ABCDE approaches indefinitely close to a continuous curve. Note that because this derivation is geometric, and no specific force is applied, it proves a more general law than Kepler's second law of planetary motion. It shows that the Law of Areas applies to any central force, attractive or repulsive, continuous or non-continuous, or zero. Conservation of angular momentum in the Law of Areas The proportionality of angular momentum to the area swept out by a moving object can be understood by realizing that the bases of the triangles, that is, the lines from S to the object, are equivalent to the radius , and that the heights of the triangles are proportional to the perpendicular component of velocity . Hence, if the area swept per unit time is constant, then by the triangular area formula , the product and therefore the product are constant: if and the base length are decreased, and height must increase proportionally. Mass is constant, therefore angular momentum is conserved by this exchange of distance and velocity. In the case of triangle SBC, area is equal to (SB)(VC). Wherever C is eventually located due to the impulse applied at B, the product (SB)(VC), and therefore remain constant. Similarly so for each of the triangles. After Newton Leonhard Euler, Daniel Bernoulli, and Patrick d'Arcy all understood angular momentum in terms of conservation of areal velocity, a result of their analysis of Kepler's second law of planetary motion. It is unlikely that they realized the implications for ordinary rotating matter. In 1736 Euler, like Newton, touched on some of the equations of angular momentum in his Mechanica without further developing them. Bernoulli wrote in a 1744 letter of a "moment of rotational motion", possibly the first conception of angular momentum as we now understand it. In 1799, Pierre-Simon Laplace first realized that a fixed plane