Article Title

Tangent Lines


In [1] Leibniz published the first treatment of the subject of calculus. An English translation can be found in [2]; he says that

to find a tangent is to draw a right line, which joins two points of the curve having an infinitely small difference, or the side of an infinite angled polygon produced, which is equivalent to the curve for us.

Today, according to [3],

a straight line is said to be a tangent line of a curve y = f(x) at a point x = c on the curve if the line passes through the point (c, f (c)) on the curve and has slope f'(c) where f' is the derivative of f.

At first glance, it would appear that this second definition is simply a more precise version of the first; indeed, the cited Wikipedia article states this sentiment explicitly. In this paper we examine cases where Wikipedia's definition is more strict than Leibniz's original one, and present two attempts at formulating a more general, but still precise, definition.

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