numbers and counting: Greece The Druids, Celtic spiritual leaders, became quite adept at the mathematical computations required to keep the calendar in order The Celts considered several numbers to be sacred or spiritually significant The number was extremely important Celts saw the world as composed of three components: earth, sea, and sky The human soul also was thought to have three components Celtic deities often appeared as triads, such as the three Irish goddesses of war The number was important as well, because it was three times three and possibly because it was the length of a lunar week The number symbolized the family and, in Ireland, the kingdom The number 27 was the number of warriors in a Celtic king’s royal court and in his war band; it contained the combined spiritual power of and 9, which when multiplied equals 27 The number 33 was also important, representing the total number of gods in the divine court The Celtic peoples used mathematics to create their elaborate knotted decorative patterns, which appeared in Celtic metalwork and stone carving The earliest patterns used parallel lines and carefully drawn circles, but by the first century c.e Celtic patterns were quite complicated and required a solid understanding of geometry Celtic artists designed their patterns on a square grid The number of lines on the grid determined the number of strings or lines in the pattern A two-by-two grid would produce an image from two pieces of string or lines A four-by-four grid required four pieces of string or lines By using a solid geometrical plan, a Celtic artist could create a complex design that was entirely complete within itself, with all lines woven together and no strings left untied Almost all Celtic knot patterns were made on evennumbered grids; it was impossible to tie all ends on patterns made from odd-numbered grids, so they were rarely used Historians believe that knot patterns may have evolved from basket weaving, which also required a practical knowledge of geometry Ancient Celtic and Germanic languages had their own counting words Counting words in the different Celtic languages were similar to one another because of their common origin in Proto-Celtic They also resemble the counting words in other Indo-European languages, including modern languages such as English or French (This is easiest to hear when reading them aloud.) In Proto-Celtic, the numbers from to 10 were as follows: oinos, dwossu, treis, kwetwar, kwenkwe, sweks, sektn, okto, nauin, dekn In Brythonic, spoken in Wales, Brittany, and Cornwall, the numbers were oino, dau, tri, petuar, pempe, hweh, seht, oht, nau, dek In Goidelic, spoken in Ireland, Scotland, and the Isle of Man, these numbers were oino, dassu, triss, keuur, kwessik, swe, sehtn, oht, nowin, dehn And in Gaulish, spoken by the Gauls in France, the numbers ran oino, do, tri, petor, pempe, suekos, sextam, oxtu, nau, decam Numbers in Germanic dialects resembled those in Celtic dialects In Proto-Germanic, through 10 ran ainaz, twai, orijiz, fidwor, fimfi, sehs, sibum, ahto, niwun, tehun In ProtoGerman, the numbers were eins, tswass, drioss, fiossr, fimf, 803 sehs, sibun, ahto, niwun, tsehun In Gothic, spoken by the Visigoths during the early centuries c.e., the sequence was ains, twai, dries, fidwor, fimf, saiehs, sibun, ahtau, niun, taiehun During the classical Greek and Roman periods European peoples began using Greek and Roman number systems Greek numbers were more common in eastern Europe and Roman numbers in the west The most widespread use of numbers was in commerce and in wills and testaments Both Greek and Roman numbers had some deficiencies; neither of them was well suited for computations with very large numbers, and expressing fractions was difficult with both Nevertheless these were the best mathematical systems available at the time and they were certainly an improvement on the existing unwritten European numerical systems GREECE BY J EFFREY S CARNES The Greeks had two number systems in common use: the acrophonic and the alphabetic The acrophonic system used the straight line for the number 1, but the initial letters of the numeral words for higher numbers Thus they had Π (pi for pente, or 5), Δ (delta for deka, or 10), Η (eta for hekaton, or 100), Χ (chi for chilioi, or 1,000); and Μ (mu for myrioi, 10,000) Myrioi was the biggest numerical unit—all higher numbers were expressed as multiples of myrioi Multiples of could be expressed by combining pi with another symbol: thus pi with a delta hanging from it would be 50 and pi with a mu hanging from it, 50,000 Other multiples were expressed by repeating the sign: 42,324, for example, would be ΜΜΜΜΧΧΗΗΗΔΔ|||| This system was used in public inscriptions in Athens until about 100 b.c.e., and until about 200 b.c.e in other cities The alphabetic system is the older of the two, but it was refined and came into widespread use in the fift h century b.c.e It probably originated in Ionia (the area on and around the coast of Asia Minor) and consists simply of the letters of the Ionian alphabet, with the addition of three obsolete letters left over from the Phoenician alphabet: Ϛ (stigma), (koppa), and Ϡ (sampi) This brings the total up to 27, making it usable as a quasi-decimal system: the first nine letters (alpha through theta) represent through 9; the next nine, 10 through 90; and the final nine, 100 through 900 The lack of a zero as placeholder kept it from being a true decimal system, and, in fact, the order of numbers might vary For this reason it was useful to distinguish numbers from letters: in inscriptions this could be done with a space or a raised dot; in print the convention came to be that numbers were written with a following superscript prime mark: thus ψοαʹ, 771 A preceding subscript prime indicated multiplication by 1,000: ψοα, 771,000 There was no zero in common use, though ʹ astronomical treatises sometimes used one consisting of an omicron with a line over it: ō (This, along with the base 60 system for measuring minutes and seconds of angles, was borrowed from the Babylonians.) There existed as well the