# Calculus without functions (or, how did Newton differentiate?)

I was recently reading about how functions did not really exist at the time of Newton and Leibniz; They thought in terms of geometry. That makes me curious.

I can understand that derivation would be slopes, and integration would be areas/volumes. Their biggest contribution may be the fundamental theorem of calculus. However, to find an antiderivative, you need to know the derivative of that function.

That makes me wonder. What is the derivative of a line that has no function? How do you find the derivative of a polynomial if it doesn't have the function $\sum_{i=0}^{m} a_n x^n$?

I know mathematics of the time would be very difficult for individuals taught by modern books. However, is there some simple example that can give me some idea of how they worked?

Actually, is there some book that explains it relatively well without me having to read Descartes and Euclid?

• I'm not sure that this question should be moved, but just in case you were unaware, there is a History of Science and Mathematics SE. – user137731 Mar 26 '15 at 0:26
• In the circle case, just geometry tells you that the tangent is perpendicular to the radius. In general I don't know the answer to your question, though. – Ian Mar 26 '15 at 0:27
• Ian - Yes, bad example from me. I'll edit it to something that makes more sense. BW - Sorry, did not know that. I'll remember that for next time (or this, if it gets moved). – Avatrin Mar 26 '15 at 0:29

They didn't have the general concept of a function in our modern sense, but they certainly did have particular functions such as $\sin$ and $\cos$, as well as algebraic expressions and equations. Newton did a lot of work with what we nowadays would call Taylor series.
Looking into Newton's The Method of Fluxions and Infinite Series, on p. 21 I see equations in $x$ and $y$, so at least implicit functions $F(x,y) = 0$ are used.