Page:A History of Mathematics (1893).djvu/92

 is divisible by 3, then add the digits of that sum, then, again, the digits of that sum, and so on, the final sum will be 6. Thus, $$\scriptstyle{61+62+63=186}$$, $$\scriptstyle{1+8+6=15}$$, $$\scriptstyle{1+5=6}$$. This discovery was the more remarkable, because the ordinary Greek numerical symbolism was much less likely to suggest any such property of numbers than our "Arabic" notation would have been.

The works of Nicomachus, Theon of Smyrna, Thymaridas, and others contain at times investigations of subjects which are really algebraic in their nature. Thymaridas in one place uses the Greek word meaning "unknown quantity" in a way which would lead one to believe that algebra was not far distant. Of interest in tracing the invention of algebra are the arithmetical epigrams in the Palatine Anthology, which contain about fifty problems leading to linear equations. Before the introduction of algebra these problems were propounded as puzzles. A riddle attributed to Euclid and contained in the Anthology is to this effect: A mule and a donkey were walking along, laden with corn. The mule says to the donkey, "If you gave me one measure, I should carry twice as much as you. If I gave you one, we should both carry equal burdens. Tell me their burdens, O most learned master of geometry."[6]

It will be allowed, says Gow, that this problem, if authentic, was not beyond Euclid, and the appeal to geometry smacks of antiquity. A far more difficult puzzle was the famous 'cattle-problem,' which Archimedes propounded to the Alexandrian mathematicians. The problem is indeterminate, for from only seven equations, eight unknown quantities in integral numbers are to be found. It may be stated thus: The sun had a herd of bulls and cows, of different colors. (1) Of Bulls, the white (W) were, in number, $$\scriptstyle{(\frac{1}{2}+\frac{1}{3})}$$ of the blue (B) and yellow (Y): the B were $$\scriptstyle{(\frac{1}{4}+\frac{1}{5})}$$ of the Y and piebald (P): the