Th e Druids, Celtic spiritual leaders, became quite adept
at the mathematical computations required to keep the cal-
endar in order. Th e Celts considered several numbers to be
sacred or spiritually signifi cant. Th e number 3 was extremely
important. Celts saw the world as composed of three compo-
nents: earth, sea, and sky. Th e human soul also was thought
to have three components. Celtic deities oft en appeared as
triads, such as the three Irish goddesses of war. Th e number
9 was important as well, because it was three times three and
possibly because it was the length of a lunar week. Th e num-
ber 5 symbolized the family and, in Ireland, the kingdom.
Th e 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 3 and 9, which when multiplied equals 27.
Th e number 33 was also important, representing the total
number of gods in the divine court.
Th e Celtic peoples used mathematics to create their elab-
orate knotted decorative patterns, which appeared in Celtic
metalwork and stone carving. Th e earliest patterns used par-
allel lines and carefully drawn circles, but by the fi rst century
c.e. Celtic patterns were quite complicated and required a
solid understanding of geometry. Celtic artists designed their
patterns on a square grid. Th e 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 art-
ist 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 even-
numbered 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 diff erent Celtic lan-
guages were similar to one another because of their com-
mon origin in Proto-Celtic. Th ey also resemble the counting
words in other Indo-European languages, including modern
languages such as English or French. (Th is is easiest to hear
when reading them aloud.) In Proto-Celtic, the numbers
from 1 to 10 were as follows: oinos, dwossu, treis, kwetwar,
kwenkwe, sweks, sektn, okto, nauin, dekn. In Brythonic, spo-
ken 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 num-
bers 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, 1 through 10 ran ainaz, twai,
orijiz, fi dwor, fi mfi , sehs, sibum, ahto, niwun, tehun. In Proto-
German, the numbers were eins, tswass, drioss, fi ossr, fi mf,
sehs, sibun, ahto, niwun, tsehun. In Gothic, spoken by the Vi-
sigoths during the early centuries c.e., the sequence was ains,
twai, dries, fi dwor, fi mf, saiehs, sibun, ahtau, niun, taiehun.
During the classical Greek and Roman periods Euro-
pean peoples began using Greek and Roman number sys-
tems. Greek numbers were more common in eastern Europe
and Roman numbers in the west. Th e most widespread use of
numbers was in commerce and in wills and testaments. Both
Greek and Roman numbers had some defi ciencies; neither of
them was well suited for computations with very large num-
bers, and expressing fractions was diffi cult with both. Nev-
ertheless 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 JEFFREY S. CARNES
Th e Greeks had two number systems in common use: the
acrophonic and the alphabetic. Th e acrophonic system used
the straight line for the number 1, but the initial letters of
the numeral words for higher numbers. Th us they had Π (pi
for pente, or 5), Δ (delt a for deka, or 10), Η (eta for hekaton,
or 100), Χ (ch i 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 5 could be expressed by combining pi with another sym-
bol: thus pi with a delta hanging from it would be 50 and pi
with a mu hanging from it, 50,000. Other multiples were ex-
pressed by repeating the sign: 42,324, for example, would be
ΜΜΜΜΧΧΗΗΗΔΔ||||. Th is system was used in public in-
scriptions in Athens until about 100 b.c.e., and until about
200 b.c.e. in other cities.
Th e alphabetic system is the older of the two, but it was
refi ned and came into widespread use in the fi ft 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 let-
ters left over from the Phoenician alphabet: Ϛ (stigma),
(koppa), and Ϡ (sampi). Th is brings the total up to 27, mak-
ing it usable as a quasi-decimal system: the fi rst nine letters
(alpha through theta) represent 1 through 9; the next nine, 10
through 90; and the fi nal nine, 100 through 900. Th e lack of
a zero as placeholder kept it from being a true decimal sys-
tem, 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:
ʹ
ψοα, 771,000. Th ere was no zero in common use, though
astronomical treatises sometimes used one consisting of an
omicron with a line over it: ō. (Th is, along with the base 60
system for measuring minutes and seconds of angles, was
borrowed from the Babylonians.) Th ere existed as well thenumbers and counting: Greece 803