The Parallelogram of Instantaneous Velocities

The kinematical generation of a curve is based on the definition of the motion of a point that describes the curve. During the movement the point at each moment has a directed instantaneous velocity. The seventeenth century geometers realized that the line drawn in the direction of the instantaneous velocity is the tangent to the curve in the position of the point at that instant. Both Evangelista TorricelliTorricelliEvangelista and Gilles Personne de RobervalRobervalGilles Personne de came up with the idea to use the parallelogram of instantaneous velocities to determine tangents. The young Isaac Newton had the same idea. Yet, even in the eighteenth century the parallelogram of instantaneous velocities was still not seen as conceptually unproblematic. A proof given by D’Alembert illustrates this.