Important results of calculus before Newton and Leibniz? We have all come to know that calculus was invented by Newton and Leibniz, right? But many calculus results were already proven by the time. I have read that Fermat already found how to  calculate tangent lines, compute maxima and minima, someone else found area under some curves  and yet others found some series directly related to calculus, and found integrals and derivatives of many functions. This may seem reasonable yet, but I cannot understand how then Isaac Barrow proved the fundamental theorem of calculus, without calculus being invented at that time? I guess there is something seriously wrong in my knowledge. Was the work of Newton and Leibniz limited to organizing and formalizing the word done? Can anyone clear my doubts upon this subject?    
 A: Yes, many results were known by that time, but there was no system. Barrow translated geometric -- not algebraic -- problem about areas into tangents, and vice-versa. Altogether before Newton and Leibniz, pre-calculus was very geometrical (and it's worth remembering that anything that we'd recognize as algebra only started with Descartes), and tackled particular problems rather than a general approach. Newton himself was still very grounded in geometry when he invented calculus. However Leibniz was more algebraic and analytical. 
Even in classical times Archimedes had approximated the area of a circle essentially by more and more smaller and smaller pieces. Of all of those before Newton and Leibniz, Fermat arguably has the best claim. He introduced a quantity "e" and then "divided all terms by e" and took out terms in "e" later (this wasn't very rigorous but neither were Newton or Leibniz: the emphasis was on solving problems in this period). Fermat's method for finding a maximum or minimum was essentially saying the derivative was zero, and Lagrange credited Fermat with inventing calculus.
Overall, I don't think you can see a sudden "jump". For example, Descartes was vital in taking algebra forward, Kepler's measurements and laws gave Newton something to apply calculus to in Principia, and so on.
