Page:A History of Mathematics (1893).djvu/91

 disappears from history. It is brought to light again about 100 A.D. by Nicomachus, a Neo-Pythagorean, who inaugurated the final era of Greek mathematics. From now on, arithmetic was a favourite study, while geometry was neglected. Nicomachus wrote a work entitled Introductio Arithmetica, which was very famous in its day. The great number of commentators it has received vouch for its popularity. Boethius translated it into Latin. Lucian could pay no higher compliment to a calculator than this: "You reckon like Nicomachus of Gerasa." The Introductio Arithmetica was the first exhaustive work in which arithmetic was treated quite independently of geometry. Instead of drawing lines, like Euclid, he illustrates things by real numbers. To be sure, in his book the old geometrical nomenclature is retained, but the method is inductive instead of deductive. "Its sole business is classification, and all its classes are derived from, and exhibited by, actual numbers." The work contains few results that are really original. We mention one important proposition which is probably the author's own. He states that cubical numbers are always equal to the sum of successive odd numbers. Thus, $$\scriptstyle{8=2^3=3+5}$$, $$\scriptstyle{27=3^3=7+9+11}$$, $$\scriptstyle{64=4^3=13+15+17+19}$$, and so on. This theorem was used later for finding the sum of the cubical numbers themselves. Theon of Smyrna is the author of a treatise on "the mathematical rules necessary for the study of Plato." The work is ill arranged and of little merit. Of interest is the theorem, that every square number, or that number minus 1, is divisible by 3 or 4 or both. A remarkable discovery is a proposition given by Iamblichus in his treatise on Pythagorean philosophy. It is founded on the observation that the Pythagoreans called 1, 10, 100, 1000, units of the first, second, third, fourth 'course' respectively. The theorem is this: If we add any three consecutive numbers, of which the highest