-----+------+
| | | | | |
| | | | - | |
| | | | | - | __ |
| ||| | - | | - | || |
| | - | | - | |
| | - | | - | |
| | | | | |
+-------+-------+-------+-------+------+

as in the earlier notation.

Somewhere in the course of these early mathematical activities the
process has changed from the more or less spontaneous operating that led
primitive man to the first enunciation of arithmetical ideas, and has
become a self-conscious striving for the solution of problems. This
change had already taken place before the historical origins of
arithmetic are met. Thus, the treatise of Ahmes (2000 B. C.) contains
the curious problem: 7 persons each have 7 cats; each cat eats 7 mice;
each mouse eats 7 ears of barley; from each ear 7 measures of corn may
grow; how much grain has been saved? Such problems are, however, half
play, as appears in a Leonardo of Pisa version some 3000 years later: 7
old women go to Rome; each woman has 7 mules; each mule, 7 sacks; each
sack contains 7 loaves; with each loaf are 7 knives; each knife is in 7
sheaths. Similarly in Diophantus' epitaph (330 A. D.): "Diophantus
passed 1/6 of his life in childhood, 1/12 in youth, and 1/7 more as a
bachelor; 5 years after his marriage, was born a son who died 4 years
before his father at 1/2 his age." Often among peoples such puzzles were
a favorite social amusement. Thus Braymagupta (628 A. D.) reads, "These
problems are proposed simply for pleasure; the wise man can invent a
thousand others, or he can solve the problems of others by the rules
given here. As the sun eclipses the stars by its brilliancy, so the man
of knowledge will eclipse the fame of others in assemblies of the people
if he proposes algebraic problems, and still more if he solves them"
(Cajori, _Hist. of Math._, p. 92).

The limitation of these early methods is that the notation merely
records and does not aid computation. And this is true even of such a
highly developed system as was in use among the Romans. If the reader is
unconvinced, let him attempt some such problem as the multiplication of
CCCXVI by CCCCLXVIII, expressing it and carrying it through in Roman
numerals, and he will long for the abacus to assist his labors. It was
the positional arithmetic of the Arabians, of which the origins are
o