Take it to the limit one more time – a history of calculus V

We have been looking at the historical development of the method of exhaustion in Ancient Greece and China. Parallels are often drawn, not incorrectly, between the method of exhaustion and integral calculus. However, there are important differences between the two mathematical methods. One very significant difference is in the area of mathematics utilised. The method of exhaustion was carried out entirely geometrically, whereas calculus is arithmetical and algebraic. The Greeks couldn’t really cope with irrational, or as they put it incommensurable, numbers and turned instead to magnitudes. Calculus uses the base ten place-value number system with the full spectrum of real numbers–rational, irrational, algebraic, transcendental–and at the highest level complex numbers. Neither algebra nor the decimal place-value number system existed when mathematicians such as Eudoxus and Archimedes were plying their trade. Taking algebra in its most basic form as the solution of equations, Euclid’s…