Why did two mathematicians both claim to have invented calculus?

Calculus is the branch of mathematics built to study continuous change, split into two linked halves: differential calculus, which handles instantaneous rates of change and the slopes of curves, and integral calculus, which handles accumulated quantities and the areas under curves. The two halves are tied together by the fundamental theorem of calculus, and together they've earned calculus the nickname 'the basic instrument of physical science.' In the late 1600s, two men independently worked out the whole system: the Englishman Isaac Newton and the German Gottfried Wilhelm Leibniz. Neither copied the other, but the overlap sparked a bitter priority dispute that lasted until Leibniz died in 1716.

Newton actually got there first. He formulated his calculus between 1664 and 1666, later calling those years the prime of his life for invention, and he first wrote it down during a plague-induced isolation. As with many of his works, though, he delayed publication, and his key treatise on the method wasn't published until 1736, seventy years after he'd worked it out. Leibniz, working entirely independently, developed his own version later and published first, which meant that to the outside world, he looked like the inventor.

That gap between who discovered it and who published it first lit the fuse. At first, students and supporters of the two men traded accusations on their behalf, but after 1711 Newton and Leibniz themselves jumped into the fight directly, each accusing the other of plagiarism. Today both men are credited with developing calculus independently of each other.

The idea of calculus, as it happens, is far older than either Newton or Leibniz. Ancient Greek mathematicians Eudoxus and Archimedes had already invented methods, centuries before Christ, that foreshadowed the modern idea of a limit and resembled the techniques of integral calculus. Those ancient ideas lay mostly dormant until the 17th century, when the Italian Cavalieri formalized a version of them as the 'method of indivisibles,' a method Newton would later fold into his own, far more general framework. Even the very word 'calculus' has a quiet, ancient origin of its own: it's Latin for 'pebble,' a nod to the Romans' habit of counting with small stones on their abacuses.