# 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?

• You might find the relevant chapters in Morris Kline's Mathematical Thought from Ancient to Modern Times enlightning. here is the google books excerpt. Commented Feb 1, 2014 at 18:22
• @DavidMitra Thanks for the link. :) Commented Feb 2, 2014 at 14:27

• Essentially Newton had a system: given any curve he could get the tangent, and he defined integration to be the inverse of differentiation. It's pretty hard for us to comprehend now (his work in Principia that used his calculus was expressed geometrically). His calculus, published after Principia, talked in terms of "fluxions" and "fluents", essentially concerned with modelling motion. He used a very small quantity $o$ to get a tangent in a similar way to we now use $h$ in saying the tangent to $f$ at $x$ is $(f(x+h)-f(x))/h$ as $h\to0$. Commented Feb 2, 2014 at 14:59