A question of ancient mathematics
After the publication of his article Mathematics in Ancient Egypt, Assem Deif*
received some constructive comments from readers. Here he singles out three, one pertaining to the alignment due north of the Great Pyramid, one to the authenticity of the Greek contribution compared with the ancient Egyptian inheritance, and the third to the squaring of the circle, the most famous unsolved problem of all time
Click to view caption|
Meidum Pyramid in Fayoum; the Red Pyramid of King Senefru and Bent Pyamid in Dahshour
THE EDITOR published in Al-Ahram Weekly on 30 January only parts of my reply to the first question of a reader called Ronald Edge from Chicago asking about how I define directions for determinations of alignment due north of the Great Pyramid, even contrasting it with a famous astronomical site in Paris. But because of precession and the angle of the axis of Earth to the plane of the ecliptic, over time "true north" changes, and is surely now just a few thousand years later, not precisely where it was at the time the Pyramid was built. A reference to the "Spence theory" was also omitted. I here offer the two main theories regarding the alignment techniques of the Great Pyramid, sun shadow and the stellar technique.
The first approach is the simplest; two easy methods are conceivable. One is to use a gnomon (the oldest astronomical device) to record the shortest shadow length at noon when the sun is due south. This will produce a north/ south direction line. In this case the ancients would have intended to orient their pyramids to "Re", the noonday sun, when the sun is at its highest point. Another alternative is to mark the length of the shadow in the morning and draw a circle round the gnomon using the length of the shadow as the radius. In the afternoon, when the shadow reaches the perimeter of the circle, it is again marked. A line through the morning and afternoon marks is an east/west direction line. The sophistication of this method suggests that if the ancients made this alignment they were aligning to the cardinal points and had a good working knowledge of basic geometry. Both methods are a quick and easy way of establishing direction lines on most days of the year. Many scholars support this technique as the most plausible way in which the north pole alignment was calculated; for the ancients had a thorough understanding of the sun -- being their chief god. Egyptian priests would have studied it carefully.
In the stellar alignment technique, however, the idea is to select a star some distance from the north celestial pole and observe the star's rising and setting points in the northeast and northwest. The line bisecting these points is directed due north, and the extension of this line in the opposite direction points due south. The next task would be to determine east and west with the line bisecting the north and south points. We could double-check these findings by observing the rising and setting points of stars we believe to be on the celestial equator. These stars should rise due east and set due west. An alternative technique is to use a plumb bob to measure the transition of a star across the local meridian. If we adjust the relative positions of the observer and plumb line until the star's angle of elevation is greatest, the line connecting the observer's eye and the plumb line points due south, and its extension in the opposite direction points due north. Some scientists went back in history to discover that when the Great Pyramid was built (2570 BC), Sirius (the brightest star in the southern sky) and Dubhe (a prominent star in the northern sky) were simultaneously on the meridian due south and due north. This interesting simultaneous north/south alignment of two prominent stars from the southern and northern skies may also have provided a means of orienting the Pyramids to the north/south direction. Finally, Egyptologist K Spence suggested that the ancients identified true north by using the polar alignment of two northern circumpolar stars, Kochab and Mizar. When a plumb line was set against the vertical alignment of these two stars, it identified the exact point on the horizon which signified true north. The Egyptians could then knock a stake in the ground, in the distance, allowing them to mark out the pyramid's axis with reasonable accuracy.
THE SECOND was from professor R Olley from Reading University in England who described the article as a mean that helped to clear up a certain matter which he found puzzling. If he may explain the mystery: in many mathematics history books, one hears the story of how the Pythagoreans were supposed to have "panicked" as the square root of 2 was irrational. So they fled from number theory to geometry. However, The Mathematics of Plato's Academy: A New Reconstruction by David Fowler put to me the much more satisfactory idea that to the Greeks, number, magnitude and ratio were effectively discrete concepts. And number meant positive integer -- they had no concept of a rational number as we have today, or even of a common fraction. Although this is the idea that I accept, it still left the puzzle of how come, even by the time of Plato, the Greeks had realised that the magnitudes which we would designate as root 2, root 3, root 5, etc, were incommensurable with 1, and had to be expressed by anthyphairesis? The answer presented itself to me a flash, as I saw Gadalla's diagram of "Sacred Square" construction. I would appreciate your opinion on the article "The Pyramids, the Golden Section and 2 PI" by Collyer and Pathan. Are they right in saying that Ahmes refers to the golden section as the "sacred ratio"? Why I worry about their historicity is that they then say that the Greeks called it the "golden ratio". This is not true, since although the Greeks knew it and used it, that name was only invented in the 19th century. But their account of how the ancient Egyptians would have achieved the golden ratio in pyramid construction does sound very plausible.
I responded that I don't recall having seen the golden section in the Ahmes papyrus. Still I believe that the Ancient Egyptians knew it. Why?
First, it must have occurred to them to ask the question: what if, in any right-angle triangle ABC at C of dimensions a:b:c, we request that b/a = c/b, then one must reach -- from Pythagoras -- the result that a:b:c = 1:sqrt(Phi):Phi which is easy to prove and by far more plausible as a question than Euclid's assumption that a, b, c lies on one straight line. The latter proportions are indeed those between the side-lengths of Khufu's inner triangle.
Another reason is that the Senefru Pyramid of Meidum (Khufu's father) has as dimensions of its inner triangle 137.5c:175c:222.5c, the latter are again in the same proportions of Khufu's, but in the ratio 1:1.6 in lengths. Thus again Pi = 22/7 and Phi = 89/55. Incidentally, a rule of thumb between both constants is that Pixsqrt (Phi) is almost equal to 4 (this follows from base perimeter of Khufu's divided by the height equals 2Pi). Other historians traced the relation that Pi equals approximately (6/5)xPhi2. If one takes Phi2 = 55/21 (9th/7th terms in the Fibonacci sequence) you end up with Pi = 22/7 exactly.
Another pyramid adhering to the same ratios as Khufu's is the Fifth-Dynasty Nieuserre pyramid at Abu Sir. Egyptologists called the triplet (Meidum, Khufu, Nieuserre) the Pi pyramids. As to the ones of ratios 3:4:5 (half-base: height: slant height) called by Pythagoras himself the "holy triangle", there are at least eight in Egypt. I can send you their names if you wish.
As for the article you sent, I can't see something specific. Besides, Khufu's Pyramid is not composed of four faces, but eight, as you know. and this was discovered in 1940 by a British pilot flying over the Giza Plateau who took a picture of the pyramid showing it to be concave (Petrie knew this, and that it dips of as much as one degree, no definite theory for that). As to the Egyptian figure for Pi being 3.16 which they mentioned, nowhere in the Ahmes papyrus could we find the number 3.16. First, they didn't have decimals; second, Ahmes only reported a remark, i.e. a practical figure of merit to use, that a circle of nine units in diameter is equal in area to a square of side-length eight units.
So to say that Pi and Phi in Egyptian monuments are a matter of coincidence cannot be more distant from the truth. It is acknowledged that Pythagoras is not credited with the theorem, rather with its proof (history tells us that he stayed with the priests in ancient Thebes for 22 years, and announced his theorem only after his departure). However, surfing the Internet reveals at least 17 different ways of proving it. It seems that Pythagoras was the first to have been credited with the proof. The Egyptians being practical people, they could have known the theorem but they didn't need to prove or record it, or perhaps they knew a simple proof which didn't reach us (something like Fermat's last theorem).
So knowing Phi implies knowing Pythagoras, and knowing the latter, the Egyptians must have fumbled with irrationals. History tells us, or so it is said, that the Pythagoreans in 500 BC had proved that sqrt(2) cannot be equal to a/b (a and b integers). But did the Egyptians know irrationals? Gadalla remarked incisions on walls suggesting sqrt(2), then sqrt(3), etc... as you kindly noted. But more important is that they had a unit of length called Remen = 5 palms. It is defined as "half the length of the diagonal of a square having a side- length of one cubit", i.e. a square of side-length seven palms will possess a diagonal of 10 palms in length. This gives a crude value for sqrt(2) as 10/7 with an error of 1 one per cent. In Saqqara's Third Dynasty, a certain area is covered with a grid of squares governed by the two numbers 12 and 17 as side and diagonal. It follows that 17/12 is another approximation with an error of 0.17 per cent. Further, the ratio of the base diagonal of the Great Pyramid relative to the height is 20:9, and since the height measures 280c and the base side-length is 440c, this gives that sqrt(2) = [(20/ 9)x280]/440 = 140/99 being a better approximation with a relative error of 0.005 per cent. Moreover, in the Rhind papyrus, exercise 57 requests the calculation of the Sequed of a pyramid having a square base of side-length 140c and its diagonal is thought to be 198c. This gives 99/70 with the same foregoing accuracy.
Interestingly enough, F Gnaedinger in 1996 arrived in 1996 at showing that irrational numbers such as sqrt(2), sqrt (3), etc...can be approximated by rational fractions that converge in the limit to these square roots, and this possibly is the method deployed by ancient nations. He considered for the first root the sequence of three banded columns then following a routine similar to the pascal triangle starting by one, one and two then adding up any two consecutive terms of them to obtain the next row and so on, the quotient of any two near by terms is found to converge to the square route of two. The rational 10/7, 17/12, 99/70, 140/99, can be found within these terms. These fractions can be found in Egyptian monuments and the greater number of their occurrence (this needs further investigation) precludes that they are the result of mere coincidence. You must have noticed that he started his table with 1 and 1 the side-lengths of the sacred square.
Please don't take it as if I am trying hard to defend the ancient Egyptian civilisation; for it belongs to the world and not just to their descendants. Only that I claim that maybe it was all in the ancient Alexandria Library and was lost along with all the sciences of antiquity. In the Hellenistic period science was written in Greek; in Europe's renaissance Latin was predominant, in the Middle age Arabic was important; it therefore follows that since everything was in Greek and later written on stronger materials than papyrus, like parchment, then whatever went through Greece was considered to be inherited from the Greeks. Aristotle is quoted as saying that geometry was Egyptian. If this is true, then there is nothing left as Greek. It is also said that Euclid's elements were not all his, so where was it from? Since our written inheritance is so small, the only door open is speculation and the best interpretation. All one can hope for is to put one's hand on facts rather than fiction.
THE THIRD letter was from Wendy Mercer who wrote that:
With all due respect, Squaring the Circle, Trisecting an Angle and Doubling the Cube are attached. Compass and straight-edge only. It's sad that some documents were lost throughout time. My concern is that the ancient Egyptians knew of these problems, likely more than even had the solutions, but as unfolds various new mathematical concepts were introduced which eventually lead up to the impossibility. Did the ancient Egyptians even know about Pi, and if so did they believe Pi was relevant when the problems originated? Pi is important, but it does seem to be the roadblock in the time line. Without Pi squaring the circle seems entirely possible. Sacred geometry! But the latter was suppressed.
In fact I replied that this problem resembles the dream of alchemists trying but failing to transform copper into gold. Comparing such dreams with that of the ancient nations, in trying to square the circle, either by creating a square of perimeter equal to the circumference of a given circle, or having the same area instead, seems to be of a religious aim rather than economic like the previous one. The square represents the physical, whereas the circle represents the spiritual. Squaring the circle is an attempt to unite, i.e. bring together, the physical world with the spiritual one, e.g. the Great Pyramid is a physical place where one can bring one's physical body and simultaneously experience the spiritual.
All sacred geometers have attempted the impossible, however it proved that it cannot be done exactly because we are working with irrational numbers; not only that, but mainly transcendental ones (not algebraic). We can be pretty close, but not exact. My position is that the latter numbers such as Pi or the Euler number e cannot be constructed. True, as you said, one shouldn't worry about Pi if such problem could be solved without it, but constructing the square, it will contain Pi whether we want or not; being implicit in the proof.
The property which Pi possessed has been in the centre of interest from ancient times, and equally with the ancient Egyptians. It was only in 1767 that Lambert was able to prove that Pi was irrational. Then in 1882 Linderman finally proved it was transcendental, and with this, all the dreams that had existed since the antiquity have unfortunately faded away.
The reason why, in my view, geometers have tried endlessly to square the circle, as far as creating, say, a square of perimeter equal to the circumference of a given circle, originates in the Great Pyramid. The perimeter of its base equals the circumference of a circle with a radius equal to the height of the pyramid. But Pi is taken here as the rational value 22/7, or you may equivalently say that the feasibility of equating both is the reason why it was thought that Pi of the ancient Egyptians had this value.
* Assem Deif is a professor of mathematics at Cairo University and MISR University for Science and Technology.