> *Much* more serious problem for agriculturists, I think, is when
> to put that seed in the ground early enough to give you food
> before you run out of last year's store and end up eating that
> seed, and late enough so the seed won't freeze. But calendar
> math seems to be already around rudimentarily before writing. I
> think it's astronomical calendar calculation that must have
> given early farmers the greatest need for the most demanding
> math.
Yes, I overlooked the need of agriculturists for some kind of
calendar (at least in the temperate zones, where there are
winters). Thanks for reminding me.
At URL http://www.math.buffalo.edu/mad/AMU/amu_chma_19.html
I found an interesting example of an isolated agricultural
society and its math and astronomy:
Jose/ Barrios Garci/a defended successfully (Apt cum Laude) his
doctoral thesis, entitled "Number systems and calendars of the
berber populations of Grand Canary and Tenerife in the 14th-15th
centuries" at the Faculty of History (Department of
Anthropology) of the University of La Laguna (Tenerife), on June
5, 1997.
the thesis abstract:
Grand Canary and Tenerife were inhabited in 14-15th centuries by
Berber populations, called Canarians in Grand Canary and Guanches
in Tenerife, coming from the nearby continent presumably on
different occasions between the first millennium BC and at least
the first millennium AD. These populations remained relatively
isolated until the European rediscovery of the Islands
in late 13th century. At this time the population of each Island
were of about 40-60,000 inhabitants, sustaining a notably
developed agricultural (barley, wheat) and stock raising (goats,
sheep, pigs) economy. After two centuries of struggles the
Islands were finally incorporated to the Spanish crown in late
15th century.
The above mentioned economical characteristics necessarily
involved a certain number of arithmetical and calendrical
activities, and ethnographic written sources available from c.
1300 AD on certainly certified it. With this in mind, in 1990 I
began a research on the mathematical and astronomical
practices of these people, who have finally crystallised in the
doctoral dissertation I have just presented at the University of
La Laguna (Tenerife).
[2 paragraphs deleted]
My analysis of the notices preserved about the number systems
lead in both Islands to a pure 10-based system, deeply related
with both proto-Berber and ancient Egyptian numeral systems
(without discarding a possible concurrent use of a 12-based
system related with calendrical counts), as well as to
the existence of systematic census of the inhabitants of each
Island, due to the economical constraints imposed by isolation.
With respect to the calendars, drawing evidence from ethnographic
written sources I establish for both Islands the existence of
systematic records of lunar, solar and sidereal counts.
The research for Grand Canary is complemented with an
archaeo-astronomical study of the mountain of Cuatro Puertas,
usually considered of a very great religious importance. From the
collected evidence I infer that at its top there is a summer
solstice marker which works by mean of the shadow a certain
rock casts at sunrise upon a great sign carefully carved on the
opposite wall.
On the other hand, collected archaeological, ethnohistorical and
linguistic evidences led me to propose that the Canarians
systematically recorded numerical, astronomical and calendrical
data by mean of geometrical figures (squares, triangles, circles,
etc.) painted in white, red and black on wood planks
and on the walls of certain caves. Evidence from the decoration
of the Painted Cave of Galdar (the main preserved painted cave of
the Island), led me to propose they use a chessboard of 3
(vertical) x 4 (horizontal) squares, named acano, to represent 12
moons. On this base, I proceed to study the acano as a lunar
calendar, showing how the vertical numeration of its squares
force the solstitial, equinoctial and eclipse moons to move
across the board with very simple and stable patterns. These
patterns provide a safe and clear mnemonic guide for performing
on the acano an easy arithmetical calculus of seasonal and
eclipse moons over extended periods of time, just using
the difference in days of the lunar year with either the solar
year or the eclipse year to perform an elementary saw function on
the squares.
The proposed calculus establish the octaeteris and the 135-moon
eclipse cycle as basic periods of the acano. It is well known
that the Canarians observed the summer solstice and had important
festivals on the crescent moon that followed, so to complete the
evidence I present two notices from ancient written sources
supporting that they measured one and half eclipse year as 520
days. The proposed calculus on the acano would reveal an
unsuspected high level of Canarian mathematical astronomy and
pose the question of the origin of this set of techniques.
In contrast with some (weak) notices supporting the existence of
a Sirius calendar in Grand Canary, my main Thesis with respect to
the Guanche calendar is the fundamental role played in it by the
phases of the star Canopus. From my analysis its helical rise
about middle August fixed the first moon of the Guanche lunar
calendar, while its helical set on late April and its
acronical rise on late January fixed the two other well
documented feasts of the Island. The Guanche cult to this star
was later transferred to what have been by far the main Catholic
cult of the Island after the conquest: the Virgin of Candelaria.
Additional evidence drawn from continental Berbers supports the
antiquity and widespread of a Canopus cosmological system in
Northwest Africa.
With respect to Guanche record keeping, written sources point out
the use of tally woods and, very specially, small clay beads
joined with a string to form a sort of necklaces, very usually
found in funerary Guanche caves. Nevertheless, the absence of
well preserved examples found in a reliable archaeological
context completely impede, for the moment, to check these
written notices.
end of abstract
#129 Barreto, Manuel Cabrera: Die Zahlwoerter der Altkanarier [The
number words of the ancient Canarians], in: Almogaren, Hallein,
1971, vol. II, 151-167
"The author examines critically the information available
regarding the numerals in the language of the natives of the
Canary Islands. Their basis of counting is the decimal system,
which is clearly proven by all recent critical and historical
studies. Ancient Canarian and Berber numerals are closely akin as
regards language and counting, which shows the North African
origin of the ancient Canary islanders also in this
domain. Apparently Semitic traits can be explained by the
presence of Negro and Berber slaves in the Canary Islands, as
stated by dr. Bosch Millares, an assumption which is better
established than that of a linguistic hybridization of the
Canarian natives" (p.167).