The Tangent or Velocity Problem

Think of what a tangent means graphically beginning with the tangent to a circle then generalizing to smooth curves.

Leibnitz, used infinitesimals to find tangents using the points (x,y) and (x+dx,y+dy) as two points on a curve an infinitesimal distance apart. An infinitesimal is a positive number that is smaller than all the other positive numbers -- ie an infinitely small, non-zero number. There were historical difficulties with this approach -- folks didn't believe in infinitesimals -- this difficulty was eventually avioded through limits and the work of Cauchy, Bolzano, Weirstrauss & others.

Mathematicians strive for rigorous definitions so that they can evaluate the "truth" of assertions such as Euler's claim that 1 - 1 + 1 - 1 + 1 ... = ½ and so that they know that their methods are reliable.
The problem of finding the instantaneous velocity of a particle is entirely analogous to the tangent problem, as is the problem of finding any instantaneous rate of change. Therefore this problem arises in a myriad of applications
Newton, in his work on Fluxions (what he called the Calculus) was more interested in velocity, accelleration, forces, etc. than in tangents per say.

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