## Ancient Africa: Khemetic Mathematics: Herega Dur DuriiJune 28, 2015

Posted by OromianEconomist in Ancient Egyptian, Ancient Rock paintings in Oromia, Meroetic Oromo.
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EGYPTIAN MATHEMATICS

 Ancient Egyptian hieroglyphic numerals

The early Egyptians settled along the fertile Nile valley as early as about 6000 BC, and they began to record the patterns of lunar phases and the seasons, both for agricultural and religious reasons. The Pharaoh’s surveyors used measurements based on body parts (a palm was the width of the hand, a cubit the measurement from elbow to fingertips) to measure land and buildings very early in Egyptian history, and a decimal numeric system was developed based on our ten fingers. The oldest mathematical text from ancient Egypt discovered so far, though, is the Moscow Papyrus, which dates from the Egyptian Middle Kingdom around 2000 – 1800 BC.

It is thought that the Egyptians introduced the earliest fully-developed base 10 numeration system at least as early as 2700 BC (and probably much early). Written numbers used a stroke for units, a heel-bone symbol for tens, a coil of rope for hundreds and a lotus plant for thousands, as well as other hieroglyphic symbols for higher powers of ten up to a million. However, there was no concept of place value, so larger numbers were rather unwieldy (although a million required just one character, a million minus one required fifty-four characters).

 Ancient Egyptian method of multiplication

The Rhind Papyrus, dating from around 1650 BC, is a kind of instruction manual in arithmetic and geometry, and it gives us explicit demonstrations of how multiplication and division was carried out at that time. It also contains evidence of other mathematical knowledge, including unit fractions, composite and prime numbers, arithmetic, geometric and harmonic means, and how to solve first order linear equations as well as arithmetic and geometric series. The Berlin Papyrus, which dates from around 1300 BC, shows that ancient Egyptians could solve second-order algebraic (quadratic) equations.

Multiplication, for example, was achieved by a process of repeated doubling of the number to be multiplied on one side and of one on the other, essentially a kind of multiplication of binary factors similar to that used by modern computers (see the example at right). These corresponding blocks of counters could then be used as a kind of multiplication reference table: first, the combination of powers of two which add up to the number to be multiplied by was isolated, and then the corresponding blocks of counters on the other side yielded the answer. This effectively made use of the concept of binary numbers, over 3,000 years before Leibniz introduced it into the west, and many more years before the development of the computer was to fully explore its potential.

Practical problems of trade and the market led to the development of a notation for fractions. The papyri which have come down to us demonstrate the use of unit fractions based on the symbol of the Eye of Horus, where each part of the eye represented a different fraction, each half of the previous one (i.e. half, quarter, eighth, sixteenth, thirty-second, sixty-fourth), so that the total was one-sixty-fourth short of a whole, the first known example of a geometric series.

 Ancient Egyptian method of division

Unit fractions could also be used for simple division sums. For example, if they needed to divide 3 loaves among 5 people, they would first divide two of the loaves into thirds and the third loaf into fifths, then they would divide the left over third from the second loaf into five pieces. Thus, each person would receive one-third plus one-fifth plus one-fifteenth (which totals three-fifths, as we would expect).

The Egyptians approximated the area of a circle by using shapes whose area they did know. They observed that the area of a circle of diameter 9 units, for example, was very close to the area of a square with sides of 8 units, so that the area of circles of other diameters could be obtained by multiplying the diameter by 89 and then squaring it. This gives an effective approximation of π accurate to within less than one percent.

The pyramids themselves are another indication of the sophistication of Egyptian mathematics. Setting aside claims that the pyramids are first known structures to observe the golden ratio of 1 : 1.618 (which may have occurred for purely aesthetic, and not mathematical, reasons), there is certainly evidence that they knew the formula for the volume of a pyramid –13 times the height times the length times the width – as well as of a truncated or clipped pyramid. They were also aware, long before Pythagoras, of the rule that a triangle with sides 3, 4 and 5 units yields a perfect right angle, and Egyptian builders used ropes knotted at intervals of 3, 4 and 5 units in order to ensure exact right angles for their stonework (in fact, the 3-4-5 right triangle is often called “Egyptian”).

See more at : –  http://www.storyofmathematics.com/egyptian.html

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## Scientific System, Mathematics and Ancient Kemetic Traditions

https://oromianeconomist.wordpress.com/2013/11/17/kemetic-numerology/

## Oromia: An Awesome Intro about Oromo & the Oromo Gadaa Civilization by Young African-American Scholar at Afric NetworkJune 16, 2015

Posted by OromianEconomist in Africa, Gadaa System, Oromia, Oromo, Oromo Nation.
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## Oromia: The Oromo Heritages: Gadaa, Siiqqee and Irreecha. #Africa.June 13, 2015

Posted by OromianEconomist in Africa, Ancient African Direct Democracy, Ancient Egyptian, Ancient Rock paintings in Oromia, Ateetee, Ateetee (Siiqqee Institution), Boran Oromo, Gadaa System, Irreecha, Irreecha Oromo, Oromia, Oromo, Oromo Nation, Oromo Social System, Oromo Wisdom, Oromo women, Oromummaa, Sirna Gadaa, The Goddess of Fecundity, Waaqeffanna (Oromo ancient African Faith System).
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 The Borana Calendar REINTERPRETED by Laurance R. Doyle Physics and Astronomy Department, University of California, Santa Cruz,at NASA Ames Research Center, Space Sciences Division, M.S. 245-7, Moffett Field, Calif. 94035, U.S. 20 XII 85
 The announcement of a possible first archaeoastronomical site (called Namoratunga II) in sub-Saharan Africa by Lynch and Robbins (1978) and its subsequent reappraisal by Soper (1982) have renewed interest in an East African calendrical system, the Borana calendar, first outlined in detail by Legesse (1973:180-88). I shall here reinterpret the calendar as Legesse describes it in the light of astronomical constraints. The Borana calendar is a lunar-stellar calendrical system, relying on astronomical observations of the moon in conjunction with seven particular stars (or star groups). At no time (except indirectly by way of lunar phase) does it rely upon solar observations. The Borana year is twelve lunar synodic months (each 29.5 days long), 354 days. While it will not correspond to the seasons, this may not be of primary importance for people this close to the equator. There are twenty-seven day names (no weeks), and since each month is either 29 or 30 days long, the first two (or three) day names are used twice in the same month starts on a new day name. The day names are listed in Table 1, the month names in Table 2. The first six months can be identified at the beginning of the month with a particular astronomical observation, whereas the last six months can be so identified only around the middle of the month. The first six months begin with the observation of the new-phase moon in conjunction with six positions in the sky marked by seven particular stars or star groups. Thus the phase of the moon is held constant while its position varies. The last six months are identified by a particular-phase moon seen in conjunction with the first star position. Thus, here, the lunar phase changes and the position is held constant. The seven stars or star groups in order are Triangulum (which I take to mean Beta Trianguli), Pleiades, Aldebarran, Belletrix, central Orion (around the sword), Saiph, and Sirius. They are given in Table 2 next to the months they define. The New Year starts with the observation of the new moon in conjunction with Beta Trianguli. (The term “new moon” here will be taken to be within two days of zero phase, although the Borana allow up to three “leap” days’ leeway, the astronomical observation determining the correct day to start on. This is indicated in the day nomenclature by the assignment of like prefixes to two or three day names before the approximate time an important astronomical observation is to take place.) Since the new moon can be seen only just before sunrise or just after sunset, twilight makes the observation of Beta Trianguli (a third-magnitude star) in conjunction with a new moon impossible with the naked eye. Assuming that such an observation, however, was possible, would the next new moon be in conjunction with the next star group. Pleiades? (Conjunction here is taken to mean “rising with” or “setting with,” having the same right ascension. Legesse says (p. 182), “Let us assume that a new moon was sighted last night and that is appeared side by side with the star Sirius, which the Borana call Basa.”) Since the sidereal period of the moon is 27.3 days long, it will arrive back at the Triangulum position more than two days before completing its synodic month. At the sidereal rate of 13.2° per day, the moon will be within 3° of Pleiades when it rises in the new phase again. However, by the time of the third month it rises, not with Aldebarran, the next star, but a little past Belletrix, the fourth star, which is supposed to start the fourth month. By the fourth month the new moon is rising past Sirius, the sixth start, and the calendar is clearly not working as described. It should be added that the right-ascension positions of the stars in the area from Beta Trianguli to Sirius change with time, at the rate of roughly 15° every thousand years. However, the stars stay in approximately the same configuration, and arguments based on their present right-ascension relationships will hold over the past several thousand years as well. What happens if we take the term “conjunction,” or “side by side,” as Legesse has it, to mean not “rising with” but “rising single-file,” that is, at the same horizon position (in other words, having the same declination)? Examining the idea that it is not the proximity of the moon to the star that is important but its horizon rising (or setting) position with respect to that star’s horizon rising (or setting) position, we immediately find that the first necessary observation, the new moon rising at the horizon position of Beta Trianguli, is not currently possible. Beta Trianguli rises (at the equator) about 35° north of the east point (0° declination), while the moon (on the northernmost average) rises at 23.5° north of east, never rising farther north than 28.5° from the East Point. The earth’s rotation axis is known to precess over the centuries, and while this does not change the lunar orbital positions significantly, it does change the apparent position of the stars. We can calculate the positions of the seven Borana stars at a time when Beta Trianguli was well within the moon’s declination limits to see if the calendar would have worked then. In 300 BC, Beta Trianguli was rising at a declination of +23° north of east. The right-ascension positions at the time still do not allow a “rising with” interpretation of the calendrical system. We can begin by defining the start of the Borana year as the new moon rising at the rising position of 300 BC Beta Trianguli. (The date of 300 BC was strongly suggested by the preliminary dating of Namoratunga II, but it was chosen because +23°, Beta Trianguli’s declination at the time, is the northern average of the moon’s monthly motion. I will take the moon’s motion, for the example here, from theNautical Almanacs for 1983 and 1984.) The next new moon rises at 14° north of east, which corresponds precisely to the 300 BC horizon rising position of Pleiades, the next Borana star. The next four new moons (starting the next four Borana months) rise at +9 degrees, +1 degree, –11 degrees, and –17 degrees declination. These positions correspond to the 300 BC horizon rising positions of the Borana stars Aldebarran. Belletrix, central Orion—Saiph (taken together), and Sirius, respectively (Table 3). The seventh month should be identifiable 14 or 15 days from its automatic start (about 29 days after the start of the sixth month) by a full moon rising at the Beta Trianguli position, and this is indeed the case. Each subsequent moon rises at this horizon position 27.3 days later (sidereal month) in a phase (synodic month) about two days less waxes (since it is on its way to the full phase again) each time. (Legesse has a waning moon, but this must mean waning with respect to each subsequent monthly observation, not with respect to the Phase State for that month.) On the thirteenth or first month, the moon is seen rising in the new phase again (“new” meaning within a couple of days of zero phase), and another year begins. Tracing the moon’s motion as it arrives at these positions in the sky (which are, however, no longer directly marked by the seven stars), we can derive the calendar (see Table 4). This outline is still general with respect to what is sometimes called the lunar excursion (regression of the line of nodes of the lunar orbit). The three “leap” days the Borana calendar allows for the starting of some of the months just before an important astronomical observation could account for this declination excursion of the moon (± ca. 5° from 23.5° declination on an 18.6-year basis), but this would certainly require confirmation in the field. The Borana calendrical system as described by Legesse is, therefore, a valid timekeeping system, subject to the astronomical constraints outlined here, and the pillars found in northwestern Kenya by Lynch and Robbins and preliminary dates at 300 BC could, as they suggest, represent a site used to derive that calendar. The calendar does not work in right-ascension sense, but it does work if taken as based on declination. It might have been invented around 300 BC, when the declinations of the seven stars corresponded to lunar motion as the calendar indicates, and the star names would therefore apply to the horizon positions as well. Because the horizon rising positions constitute the important observations (over half of which must be made at twilight), some sort of horizon-marking device would seem to be necessary. Since the calendar is still in use, and the horizon-making pillars can no longer be set up by aligning them with the horizon rising positions of these stars, it would seem that the Borana may be using ancient (or replicas of ancient) horizon markers and this possibility should be investigated. I look forward with great interest to a test of these hypotheses.
 Table 1 Borana Day names (Legesse 1973) Bita Kara Gardaduma Bita Lama Sonsa Sorsa Rurruma Algajima Lumasa Arb Gidada Walla Ruda Basa Dura Areri Dura Basa Ballo Areri Ballo Carra Adula Dura Maganatti Jarra Adula Ballo Maganatti Britti Garba Dura Salban Dura Garba Balla Salban Balla Garda Dullacha Salban Dullacha

 Table 2 Borana Months and Stars/Lunar Phases That Define Them (Legesse 1973) Month Star/Lunar Phase Bittottessa Triangulum Camsa Pleiades Bufa Aldebarran Wacabajjii Belletrix Obora Gudda Central Orion-Saiph Obora Dikka Sirius Birra full moon Cikawa gibbous moon Sadasaa quarter moon Abrasa large crescent Ammaji medium crescent Gurrandala small crescent

 Table 3 Declinations (Degrees) of Borana Stars, 300 BC and Present Star Declination 300 BC Present Beta Trianguli +23 +35 Pleiades +14 +23 Aldebarran +9 +16 Belletrix +1 +6 Central Orion –10 –6 Saiph –13 –10 Sirius –17 –17
 Table 4 Astronomical Borana-Cushitic Calendar (1983-84) Borana-Cushitic Day/Month Gregorian Date Description Bita Kara/ Bittottessa August 7, 1983 New moon rises at Triangulum horizon position Algajima/ Camsa September 6, 1983 New moon rises at Pleiades horizon position Walla/ Bufa October 5, 1983 New moon rises at Aldebarran horizon position Basa Dura/ Wacabajjii November 2, 1983 New moon rises at Belletrix horizon position Maganatti Jarra/ Obora Gudda December 2, 1983 New moon rises at central Orion-Saiph horizon position Salban Dura/ Obora Dikka December 30, 1983 New moon rises at Sirius horizon position Gardaduma/ Birra January 29, 1984 Full moon sets at Triangulum on February 15 Rurruma/Cikawa February 28, 1984 Gibbous moon sets at Triangulum on March 14 Gidada/ Sadasaa March 28, 1984 Quarter moon sets at Triangulum on April 10 Areri Dura/ Abrasa April 26, 1984 Large crescent sets at Triangulum on May 7 Adula Dura/ Ammaji May 25, 1984 Medium crescent sets at Triangulum on June 3 Garba Dura/ Gurrandala June 23, 1984 Small crescent sets at Triangulum on June 30 Bita Kara/ Bittottessa July 28, 1984 “New” moon rises at Triangulum position again, new year starts
 References Cited Legesse, A. 1973. Gada: Three approaches to the study of African Society. New York: Free Press. Lynch, B. M., and L. H. Robbins. 1978. Namoratunga: The first archaeoastronomical evidence in sub-Saharan Africa. Science 200:766-68. Soper, R. 1982. Archaeo-astronomical Cushites: Some comments. Azania 17:145-62

Source:

http://web.archive.org/web/20081029073246/http://www.tusker.com/Archaeo/art.currentanthro.htm

ASTRONOMY IN EAST AFRICA:Borana-Cushitic Calender

ASTRONOMY IN EAST AFRICA
The Borana-Cushitic Calendar and Namoratunga
Laurance Reeve Doyle
Space Sciences Division, N.A.S.A.
Ames Research Center, Moffett Field, California

“While Western thought has always prided itself on scientific objectivity, it has often been found unprepared for such surprises as an intellectually advanced yet seemingly illiterate society. In the face of apparent primitiveness, the possibility of significant intellectual development may not be fully investigated.
This was certainly the case when, in the early 1970’s, Dr. A. Legesse first found that the Borana people of southern Ethiopia were indeed using a sophisticated calendrical system based on the conjunction of seven stars with certain lunar phases. Previous calendrical investigations into the area up to this time had superficially stated that the Borana “attach magical significance to the stars and constellations,” incorrectly concluding that their calendar was based, as ours is, on solar motion.
What Dr. Legesse found was an amazing cyclical calendar similar to those of the Mayans, Chinese, and Hindu, but unique in that it seemed to ignore the sun completely (except indirectly by way of the phases of the moon). The workings were described to him by the Borana ayyantu (timekeepers) as follows.
There are twelve months to a year, each month being identifiable with a unique (once a year) astronomical observation. The length of each month is either 29 or 30 days – that is, the time it takes the moon to go through all its phases. (This time is actually 291/2 days and is called a synodic month, but the Borana only keep track of whole days). Instead of weeks, there are 27 day names. Since each month is 29 or 30 days long we will run out of day names about two or three days early in the same month. The day names can therefore be recycled and for day 28 we use the first day name again, the second day name for day 29, and start the next month using the third day name. Thus each month will start on a different day name. Whether the particular month is to be 29 or 30 days long would depend on the astronomical observations, which are quite ingeniously defined.
The seven stars (or star groups) used to derive the calendar are, from northernmost to southernmost, 1) Beta Triangulum – a fairly faint navigation star in the constellation of the Triangle, 2) Pleiades – a beautiful, blue star cluster in the constellation of Taurus the Bull, and sometimes referred to as the seven sisters, 3) Aldebarran – a bright, red star that represents the eye of Taurus, 4) Belletrix – a fairly bright star that represents the right shoulder of the constellation Orion the Hunter, 5)Central Orion – the region around Orion’s sword where the Great Orion Nebula may be found, 6) Saiph – the star representing the right knee of Orion, and finally 7) Sirius – the brightest star in the night sky and the head of the constellation Canis Majoris, the Great Dog.
The New Year begins with the most important astronomical observation of the year – a new moon in conjunction with Beta Triangulum. (this day is called Bitotesa, and the next month is called Bitokara). The next month starts when the new moon is found in conjunction with the Pleiades. The third month starts with the new moon being observed in conjunction with the star Aldebarran, the next with Belletrix, then the area in between Central Orion and Saiph, and finally with the star Sirius. So the first six months of the calendar are started by the astronomical observations of the new phase moon found in conjunction with six specific locations in the sky marked by seven stars of star groups.
The method is now switched and the final six months are identified by six different phases of the moon (from full to crescent) being found in conjunction with only one position in the sky – the one marked by Beta Triangulum. Thus the whole Borana year is identified astronomically and when the new phase moon is again finally seen in conjunction with Beta Triangulum the New Year will start again. Since there are 12 such synodic months of 29 ½ days each, the Borana year is only 354 days long.
Now, in the latter part of the 1970’s another interesting development was to take place regarding the astronomy of this region. In 1977 Drs. B.M. Lynch and L.H. Robbins, who were working in the Lake Turkana area of northwestern Kenya, came upon what they believed was the first archaeoastronomical site ever found in sub-Saharan Africa. At Namoratunga, it consisted of 19 stone pillars, apparently man-made, that seemed to align toward the rising positions of the seven Borana calendar stars as they had appeared quite some time ago. (their suggested date from the various archaeological considerations, which still requires corroboration, was about 300 BC). Due to precession (the slow, wobbling of the pointing direction of the rotation axis of the Earth), the stars will seem to move from their positions over the centuries, although the moon’s position would not vary on this time scale. (Such an example is the alignment of certain features of the Egyptian pyramids with the star Thuban in the constellation Draco the Dragon, which was the north polar star about 5000 year ago; today it is Polaris and in several thousand years it will be Vega). If the date that Drs. Lynch and Robbins suggested was correct, the site would then correspond to the time of the extensive kingdom of Cush, referred to as Ethiopia in the Bible but actually centered about present day Sudan. One would then conclude that the Borana calendrical system was old indeed, having been developed by the Cushitic peoples in this area about 1800 years before the development of our present day Western Gregorian calendrical system.
In 1982, a number of significant questions arose concerning the site, the calendar, and archaeoastronomy of East Africa in general. The pillars were remeasured by an anthropologist in Kenya (Mr. Robert Soper) and found to be magnetic in nature. The original measurements had to be modified but, again, alignments with the seven Borana stars were found. However, this brought up the question of whether pillar alignments are significant at all, since the Borana ayyantu certainly can recognize the phases of the moon and when it is in conjunction with the appropriate seven stars. It was time to approach the question astronomically, and ask the moon and the stars how the calendar worked.
First, we could take the New Year’s observations, a new moon in conjunction with the faint star Beta Triangulum. What is meant by the term “conjunction” which is astronomically defined as the closest approach between two celestial objects? A new moon means that the moon is very close to the sun, being at best only a very small crescent, and therefore can only be seen just before sunrise or just after sunset. Interestingly enough, it turns out that during this twilight time the sky is too bright to be able to see the star Beta Triangulum so that seeing the new moon next to Beta Triangulum, the most important observation of the Borana calendar, was impossible!
In addition, assuming that the new moon and Beta Triangulum could be somehow seen rising together, the next month’s new moon rises significantly behind Pleiades, the newt conjunction star group. The third new moon rises with Belletrix, having skipped the third star, Aldebarran, completely. This is certainly not how the Borana described their calendar. If we were to continue to try to work the calendar in this way, by the start of the sixth month the new moon would be rising almost four hours after Sirius.
How could the calendar work then? Suppose (as we did), that one takes the term “conjunction” to mean “rising at the same horizon position” instead of “rising horizontally next to at the same time.” Thus one could mark the horizon rising position of Beta Triangulum, with pillars for instance, and once a year a new moon will rise at that position on the horizon. Let us suppose that this astronomical event marks the start of the New Year. We must add that we are taking the horizon rising position of these seven stars as they were in or around 300 BC, since present day Beta Triangulum has precessed too far to the north over the centuries and the moon will never rise there. However, the position of 300 BC Beta Triangulum, as well as the other Borana stars, was quite within the realm of the moon’s orbit.
Now where will the next new moon rise? It turns out to rise at precisely the rising position of Pleiades! The next new moon, marking the start of the third month, rises at the Aldebarran horizon position, the next at Belletrix, the next in between Central Orion and Saiph, and finally the sixth new moon rises at the horizon position that Sirius rose at during the night. During the next six months one can tell what month it is only in the middle of the month, since one has to wait to see what phase the moon is in when it appears at the Beta Triangulum horizon position. During the seventh month, as described, a full moon will be observed at the Beta Triangulum position. The next month a gibbous waxing moon, then a quarter moon, and successively smaller crescents will be seen there until, at the time when the 13th or first month should start the new year again (exactly 354 days later), a new moon is again seen rising at the Beta Triangulum position on the horizon.
It is interesting that one can draw some significant anthropological results from the astronomical derivation of this calendrical system. It would appear that the calendar would have had to have been invented (to use the stars correctly) sometime within a few hundred years of 300 BC, a time when the Cushitic peoples were dominant in this part of the world. Hence we would call it the Borana-Cushitic calendar. In addition, although the seven Borana-Cushitic stars no longer rise in the correct horizon positions to be correctly marked by pillars for observing the monthly rising position of the new moon, the present day Borana people nevertheless use this system of timekeeping. The implication is that the Borana require ancient horizon markers in their present derivation of the calendar.
Concerning the site at Namoratunga, and considering that the use of pillars is apparently necessary to the derivation of the calendar, such horizon markers as are found there may, indeed, have been an ancient observatory. Petroglyphs on the pillars at Namoratunga may also hold the possibility of being ancient and, if Cushitic, may represent the alignment stars or moon. Cushitic script has never been deciphered and any hints as to the meaning of tits symbols could be significant clues with very exciting prospects indeed!

Thus, archaeoastronomy in East Africa is still quite new and many discoveries await. From coming to understand, even in a small way, the calendrical reckoning and observational abilities of the ancient and modern astronomer-timekeepers of this region, Western thought should certainly not again underestimate the ingenuity and intellect present there. As for this Western thinker, this study continues to be a welcome lesson in perspective and humility, taught to him by his astronomical colleagues of long ago.”

This is a summary of a talk delivered at Caltech for Ned Munger’s African Studies class.

http://www.africaspeaks.com/reasoning/index.php?topic=2194.0;wap2