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is easy to work with. In Athens in the fifth century B.C., for instance, at the height of Greek civilization, a person who wanted to write down a number used a kind of alphabetic code.9 The first nine of the twenty-four letters in the Greek alphabet stood for the numbers we call 1 through 9. The next nine letters stood for the numbers we call 10, 20, 30, and so on. And the last six letters plus three additional symbols stood for the first nine hundreds (100, 200, and so on, to 900). If you think you have trouble with arithmetic now, imagine trying to subtract from ! To make matters worse, the order in which the ones, tens, and hundreds were written didn’t really matter: sometimes the hundreds were written first, sometimes last, and sometimes all order was ignored. Finally, the Greeks had no zero.

The concept of zero came to Greece when Alexander invaded the Babylonian Empire in 331 B.C. Even then, although the Alexandrians began to use the zero to denote the absence of a number, it wasn’t employed as a number in its own right. In modern mathematics the number 0 has two key properties: in addition it is the number that, when added to any other number, leaves the other number unchanged, and in multiplication it is the number that, when multiplied by any other number, is itself unchanged. This concept wasn’t introduced until the ninth century, by the Indian mathematician Mahāvīra.

Even after the development of a usable number system it would be many more centuries before people came to recognize addition, subtraction, multiplication, and division as the fundamental arithmetic operations—and slowly realized that convenient symbols would make their manipulation far easier. And so it wasn’t until the sixteenth century that the Western world was truly poised to develop a theory of probability. Still, despite the handicap of an awkward system of calculation, it was the civilization that conquered the Greeks—the Romans—who made the first progress in understanding randomness.

THE ROMANS generally scorned mathematics, at least the mathematics of the Greeks. In the words of the Roman statesman Cicero, who lived from 106 to 43 B.C., “The Greeks held the geometer in the highest honor; accordingly, nothing made more brilliant progress among them than mathematics. But we have established as the limit of this art its usefulness in measuring and counting.”10 Indeed, whereas one might imagine a Greek textbook focused on the proof of congruences among abstract triangles, a typical Roman text focused on such issues as how to determine the width of a river when the enemy is occupying the other bank.11 With such mathematical priorities, it is not surprising that while the Greeks produced mathematical luminaries like Archimedes, Diophantus, Euclid, Eudoxus, Pythagoras, and Thales; the Romans did not produce even one mathematician.12 In Roman culture it was comfort and war, not truth and beauty, that occupied center stage. And yet precisely because they focused on the practical, the Romans saw value in understanding probability. So while finding little value in abstract geometry, Cicero wrote that “probability is the very guide of life.”13

Cicero was perhaps the greatest ancient champion of probability. He employed it to argue against the common interpretation of gambling success as due to divine intervention, writing that the “man who plays often will at some time or other make a Venus cast: now and then indeed he will make it twice and even thrice in succession. Are we going to be so feeble-minded then as to affirm that such a thing happened by the personal intervention of Venus rather than by pure luck?”14 Cicero believed that an event could be anticipated and predicted even though its occurrence would be a result of blind chance. He even used a statistical argument to ridicule the belief in astrology. Annoyed that although outlawed in Rome, astrology was nevertheless alive and well, Cicero noted that at Cannae in 216 B.C., Hannibal, leading about 50,000 Carthaginian and allied troops, crushed the much larger Roman army, slaughtering