numbers and counting: The Americas their sum is a square”), but their solutions lack general application or formal proof Mathematics came to hold tremendous symbolic importance in Greek culture Pythagoras saw in numbers a model for the underlying harmony of the universe, and the mysticism of his worldview influenced many later philosophers, including Plato The famous story that the entrance to Plato’s Academy bore the inscription “Let no one who is ignorant of geometry enter” may be of doubtful authenticity; nevertheless, it reflects well the Platonic attitude toward the centrality of mathematics in the education of the ideal citizen or philosopher ROME BY J USTIN CORFIELD The system of numbers and counting used in Rome and the Roman Empire was developed from Etruscan numerals These numerals, in turn, were adapted from the Greek Attic numerals Although there are only a few archaeological finds that have Etruscan numerals, there are enough to show that the letter I represented 1, an inverted V represented 5, and X represented 10 The first two of these figures can be seen on surviving Etruscan six-sided dice, on which the opposite sides add up to seven, as in the dice of today Several Roman dice also have survived, again with the opposite sides adding up to seven Six small ivory dice were found at Pompeii, all the numbers denoted with small dots; the Romans often played with two dice as opposed to the Greek system, in which three dice were used It has been suggested that the original use of numbers possibly came from notches on a tally stick, with the Romans using a V for Apart from the ordinary people who needed to count out money and commodities as well as tell time and work out the calendar, the Romans also had tax collectors and mathematicians who needed to use larger numbers The Romans used the letters I for 1, V for 5, X for 10, L for 50, C for 100, D for 500, and M for 1,000; they also indicated large numbers by placing a bar over a numeral to indicate a numeral in the thousands Above one million there was no standard format, with five million denoted either by placing a double bar over a letter V or by underlining, as in V There is conflicting evidence regarding the use of some Roman numerals, especially subtractive ones, whereby IV represents one less than (that is, 4) Older Roman material tends to have IIII for 4, making mathematics easier However, gradually the subtractive notion tended to be used more and more, eventually becoming the accepted form despite the fact that IV was also similar to initials representing the god Jupiter (the name in Latin being IVPITER) It was also from Roman numerals that the number 666 became known as the devil’s number In Roman numerals 666 was represented by the letters DCLXVI, using every basic numeral except the letter M Those familiar with this phenomenon must have seen a bad omen when the Great Fire of London broke out in 1666 (MDCLXVI) 805 Interestingly, the Romans did not have a concept of a zero Indeed the first Roman mathematician who is known to have used the concept was Dionysius Exiguus in 525 c.e., though it seems probable that the concept of zero was in use before then Later the letter N for nullus (nothing) was used to denote zero While many Greek mathematicians are household names, there are no details about Roman mathematicians, and it seems likely that the Romans had little interest in pure mathematics Indeed, it was said to have been a Roman soldier who killed the Greek mathematician Archimedes of Syracuse in 212 b.c.e., and most mathematicians in the Roman Empire, such as Diophantus of Alexandria, were actually Greek However, the Romans must certainly have had people who studied applied mathematics to work out architectural plans, load bearings, astronomy, and the supervision of the raising of revenue The poor notation of the Roman numbers did, however, handicap any major advances in mathematics For counting, the Romans used small bronze counters on occasions, these having on them pictures of hands with a particular number of fingers raised Cicero refers to these counters, which effectively were early “counting boards.” Gradually the Romans started using the abacus for more complicated sums, the abacus also often being called a “counting board.” Many mathematical instruments have been found in the excavations at Pompeii Surviving surveying manuals show the combination of arithmetic, geometry, and optics for the Romans’ work Weights and measures followed what now appear to have been arbitrary calculations but must have had some meaning at the time For example, the pes, a Roman foot (in measurement), was just over 11.5 inches, making their pace feet, 10.25 inches With a thousand of these feet making a Roman mile, the mile was 1,618 yards Roman numerals have been used many times since the ancient world; they are regularly found in Christian ecclesiastical documents and in European accounts and are still used in calendars, clocks, and watches THE AMERICAS BY PENELOPE OJEDA DE HUALA Very little information exists regarding the function of numbers and counting in the ancient Americas We know that the inhabitants of the Americas probably used the lunar and solar cycles to track the change of seasons, as these changes were vital to their survival Archaeological remains show evidence of early engineering, which required forms of mathematics However, by the time of European contact, numbering systems existed throughout the Americas The Archaic Period (7000–1800 b.c.e.) in North America saw a near extinction of big-game animals, forcing native groups to rely on local resources to supplement their diets While full-scale agriculture would not be introduced until about 800 c.e., around 3000 b.c.e some forms of early cultivation occurred throughout North America Agricultural