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The first descriptions of function concepts were linked to mechanical or geometrical ideas. For example, a logarithmic function was thought of as hyperbolic area; an elliptic function was the arc of a conic section; in calculus, integrals were distances, areas, arcs, and volumes among other things. "Newton chooses time as a universal argument and interprets variables as continuously flowing quantities possessing some velocity of change."

Leibniz developed his function ideas from the geometry of curves. The word "function" first appeared in Leibniz's manuscripts of 1673. Leibniz introduced the words "constant" and "variable," "coordinates" and "parameter" in terms of an arbitrary constant segment or quantity. He classified functions and curves into two categories: (1) Algebraic: those represented by an equation of a certain order, and (2) Transcendental: those represented by an equation of an indefinite or infinite order. He thought of functions as any parts of straight lines or points of curves. This still did not correspond to a broader, more analytical definition. Johann Bernoulli and Euler would be the ones to do that.