Nicomachus of Gerasa
c. 60-c. 100
Roman Syrian Mathematician and Philosopher
In a play by the satirist Lucian, one character says to another, "You calculate like Nicomachus." The latter, a philosopher of the neo-Pythagorean school, is best remembered for his Arithmetike eisagoge, or Introduction to Arithmetic, a highly influential if rather unusual mathematical text.
Gerasa is now the town of Jerash, Jordan, then part of Roman Syria, and it is likely that Nicomachus was ethnically related to the peoples of this area while being thoroughly Romanized in terms of culture and speech. It can be inferred that he studied in a school that espoused the ideas of Pythagoras (c. 580-c. 500 B.C.), because those ideas pervade his Introduction to Arithmetic.
In the latter text, Nicomachus examined odd, even, prime, composite, and perfect numbers. He also presented an interesting theorem in which he showed that by adding consecutive odd numbers, successively including one additional number, it was possible to produce a series of the sums of all cubed numbers. Thus 13 = 1; 23 = 3 + 5; 33 = 7 + 9 + 11; and so on.
On the other hand, Nicomachus sometimes made false statements that seemed to be true because the information he furnished supported his original assertion. Hence his statement that all perfect numbers—ones in which the divisors add up to the number itself (e.g., 1 + 2 + 3 = 6)—end in 6 or 8 alternately. This is not the case, but it seemed to be so, due to the fact that the only perfect numbers he knew were 6, 28, 496, and 8,128.
The book also shows the strange, highly unscientific, side of Pythagorean mathematics, which for instance assigned personalities to numbers. In discussing an abundant number (one in which the sum of its divisors is greater than the number itself), Nicomachus wrote that these called to mind an animal "with ten mouths, or nine lips, and provided with three lines of teeth; or with a hundred arms, or having too many fingers on one of its hands...." The opposite of an abundant number was a deficient number, whose divisors add up to less than the number, and these resembled a creature "with a single eye...one-armed, or one of his hands has less than five fingers...."
Pappus (fl. c. A.D. 320) and other mathematicians of the late ancient world despised Nicomachus's book, and Boethius (c. 480-524) perhaps revealed himself as a true medieval with his admiration of it. He turned it into a school book, and despite—or perhaps because of—its peculiarities, the Introduction to Arithmetic became the standard arithmetic text of the Middle Ages. Not until the Crusades (1095-1291), when western Europeans increasingly became exposed to Arab versions of more significant ancient works, was it replaced.
In addition to the Introduction to Arithmetic, Nicomachus wrote a book of musical theory, the Manual of Harmonics. Here, too, he applied Pythagorean notions, though in this case he was on much firmer ground with the relation of music to mathematics.
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