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This is why Julius Dedekind(1831-1916)took the trouble of formally constructing the real number system based on his idea that is now referred to as Dedekind cuts of the real line. The first mathematician, however, to successfully deal with the dilemma caused by the existence of irrational numbers was Eudoxus of Cnidus(fl 380 BC)whose Theory of Proportions allowed Archimedes to use the so-calledMethod of Exhaustion to rigorously derive results on areas and volumes ofcurved shapes before the advent of calculus some 1,900 years later.