TITLE: What did Newton and Leibniz actually discover?
QUESTION [26 upvotes]: Most popular sources credit Newton and Leibniz with the creation and the discovery of calculus. However there are many things that are normally regarded as a part of calculus (such as the notion of a limit with its $\epsilon$-$\delta$ definition) that seem to have been developed only much later (in this case in the late $18$th and early $19$th century).
Hence the question - what is it that Newton and Leibniz discovered?

REPLY [2 votes]: I'm not a historian nor have studied these things deeply. So, please make corrections in commentaries/edits when necessary.
From what I've read, it was more or less like this:
Calculus was discovered/created independently by Newton and Leibniz.

Leibniz started with sequences of numbers, and after discovering some results about them he sought a tutor in mathematics. He found Huygens and told him about his discoveries and asked to be tutored. Huygens tested him with a problem of finding the sum of a given sequence. Leibniz easily solved the problem proposed. Huygens accepted to tutor him and taught him analytic geometry and presented texts of Pascal and Fermat to him. Then Leibniz discovered/created calculus.
Newton was tutored by Barrow, who was a very competent classical geometer (i.e. in the style of the Ancient Greeks). Barrow was very interested in chronology and because of that he studied kinematics, optics and mathematics. Barrow discovered the fundamental theorem of calculus, but he did not know much of algebra and analytic geometry. Newton, being very comfortable with algebra and analytic geometry, after having learned the "geometric calculus" of Barrow soon turned it into a machine of solving problems. In this sense, Newton discovered/created calculus. 

Note: It's very interesting that Newton developed many "versions" of calculus: one time he even sought to divorce it from algebra and create a "geometric calculus" more powerful than that of Barrow, by using things like similar triangles and "ultimate ratios", but he also had one more "algebraic" with relied more on polynomials and series, and he experimented many ways of presenting his calculus from "the start".
Much before: Eudoxus created the method of exhaustion which can be used to prove some things that today are proved via the theory of limits. Archimedes discovered infinitesimals and used them to obtain many results in geometry, but his published proofs did not mention infinitesimals, and his letter to a friend, in which he explained infinitesimals, was lost until recently and so infinitesimals had to be rediscovered in Europe (much later!) and were then used to calculate areas and volumes. Descartes and Fermat developed analytic geometry which was maybe crucial for Newton and Leibniz to discover the calculus. Limit theory was developed much later. More recently, infinitesimals were finally accepted after a long history of rejection.