Calculus took two thousand years and many countries before Newton and Leibniz
The word calculus means small pebble in Latin, after the stones once used for counting. The ideas it names were gathered almost as slowly as pebbles: Greek geometers, Chinese and Indian astronomers, Middle Eastern scholars and English logicians each added pieces long before Newton and Leibniz assembled the whole.
Early hints appear in the Egyptian Moscow papyrus of around 1820 BC, which computes volumes, though only for specific numbers and without proof. Greek mathematicians went further. Eudoxus used the method of exhaustion, trapping a shape between ever closer approximations, an idea that foreshadows limits. Archimedes refined it and even found a tangent to a spiral by splitting a point's motion into radial and circular parts. Zeno's paradoxes, meanwhile, made infinitely small quantities look dangerously illogical.
Similar tools surfaced elsewhere. Liu Hui independently invented exhaustion in fourth-century China to find a circle's area, and Zu Chongzhi anticipated what Europe would call Cavalieri's principle. Ibn al-Haytham computed sums of powers to find the volume of a spun parabola. Bhāskara II in twelfth-century India wrote a relation that amounts to cosine being the derivative of sine, and the Kerala school, led by Madhava of Sangamagrama, stated series expansions more than two centuries before Europe. Historian Victor Katz notes they never unified derivative and integral into one problem-solving tool.
In fourteenth-century Europe, the Oxford Calculators and Nicole Oresme studied motion and continuity, and Oresme first proved that the harmonic series grows without limit. The seventeenth century brought a flood: Kepler summed radii to find an ellipse's area and puzzled over wine barrels; Cavalieri treated shapes as stacks of infinitely thin slices; Fermat's adequality found maxima, minima and tangents, and Newton later credited Fermat's way of drawing tangents. Around 1670, Isaac Barrow and James Gregory proved early versions of the fundamental theorem linking areas to antiderivatives.
Some writers have credited Barrow with the invention itself, but the historian Florian Cajori insisted that similar geometric results are not the same thing as the general method. Newton and Leibniz each built that full system independently late in the century, and their priority quarrel ran until Leibniz died in 1716.
Source: History of calculus
