The historical digressions
below concern the determination of the defining quadratic formula for
Golden Section: k2 - k - 1
= 0, thus k = Phi = 1.618033989.
Related material dealing with the determination of the latter in astronomical contexts are given in the following papers:
Part I: Bode's Flaw and the Structure of the Solar System ( http://www.spirasolaris.ca/sbb4a.html )
Part II. The Alternative ( http://www.spirasolaris.ca/sbb4b.html )
Part III. Exponential Order in the Solar System ( http://www.spirasolaris.ca/sbb4c.html )
Part IV: Spira Solaris Archytas-Mirabilis ( http://www.spirasolaris.ca/sbb4d.html ) contains The "Field Problem" referred to below.
In the above the general
formula for co-orbital bodies plays a fundamental role:
i.e., where T1 and T2 are mean sidereal periods of revolution (T2 > T1) and Ts the synodic (difference) cycle, for adjacent planetary periods of Phi 1 (T2 = 1.618033989 years) and reciprocal Phi -1 ( T1 = 0.618033989 years ), TS = Phi 0 = 1 year, thus in this particular astronomical context the relation incorporates three consecutive values of the Phi-Series:
Considering the ubiquitous number Phi = 1.618033989. in both modern and historical contexts it would seem that the relationship between natural growth and the Fibonacci Series is perhaps the best known ( i.e., with Phi the limiting value of the ratios of successive fibonacci numbers 1, 1, 2, 3, 5, 8, 13, etc.).
But Phi can also be defined in simple pythagorean terms relating to three points on a line, namely the division of the said line into two parts such that the ratio of the longer part to the shorter is in the same ratio as the longer part is to the whole. Easy enough to say (?), but how does one determine the precise values and the relationship itself?
It has been suggested in Parts I, II and III that Earth may be occupying an intermediate or synodic position between Venus and Mars (Venus: 0.61521 years; Earth 1 year, and 1.881 years for Mars) whereas the theoretical mean sidereal periods for these three planets form the sequence 0.618033989 years, 1 year, and 1.618033989 years, etc. Bearing in mind that it was synodic relation [1a] that permitted the determination of Phi in the present astronomical context, it is possible to reverse matters and ask what the length of the mean sidereal periods of planets on either side of Earth might be if this was indeed the case. Retaining unity as the standard reference, the sidereal period of Earth is therefore "given" as 1 year, but how does one go about determining the mean sidereal periods of the planets on either side--periods also expressed in years? Here, perhaps surprisingly, it proves useful to travel far back in time--almost four thousand years in fact--to the Old Babylonian Era [ 1900 - 1650 BCE ] to revisit attested Babylonian methodology for solving quadratic equations. Or, perhaps better stated, Babylonian algorithmic procedures applied to solve practical problems of this nature. Such problems in themselves are simple enough, e.g., the area of a rectangular field with sides of unknown length and width is given, along with the difference (i.e., xy = C and x-y = d, where both C and d are given and the dimensions of x and y are to be found). In simple terms the problem is therefore formulated as follows:
A Rectangular Field has an Area of 1
The difference between the Length and the Width is 1.
What are the values for the Length and the Width?
Firstly, the period T2
= 1 year is not only given, it also provides the frame of
for the sidereal and synodic periods of the planets in question, i.e.,
Secondly, analogous to the procedure carried out for k we also know that the synodic period T2 is the product of the planetary periods (T3 x T1) divided by their difference (T3 - T1).
Thirdly, by setting the product T3 x T1 and the difference between T3 and T1 both equal to 1 the problem becomes the Babylonian variant in a simple two-dimensional form.
Lastly (and more to the point) this leads in turn to exactly the same quadratic equation as that obtained above for k, (i.e., k2 - k - 1 = 0).
1. Take one half of the difference 1, the result is 0;30 [ Hold the result in your hand ]
2. Take the half-difference and square it, the result is 0;15
3. Take the 0;15 and add it to the area 1, the result is 1;15
4. Take the square root of 1;15, the result is 1;7,4,55,20,29,39,6,54
5. Add the half 0;30 (from step1) to the square root, the result is 1;37,4,55,20,29,39,6,54
6. What value when multiplied by 1;37,4,55,20,29,39,6,54 gives 1 (the area)?
7. 1;37,4,55,20,29,39,6,54 multiplied by 0;37,4,55,20,29,39,6,54 gives 1
8. 1;37,4,55,20,29,39,6,54 is the Length, 0;37,4,55,20,29,39,6,54 is the Width.
In decimal notation:
1. Take one half of the difference 1, the result is 0.5 [ carry the result ]
2. Take the half-difference and square it, the square is 0.25
3. Take the 0.25 and add it to the area 1, the sum is 1.25
4. Take the square root of 1.25, which is 1.118033989
5. Add the 0.5 (from step1) to the last square root to obtain 1.618033989
6. What value when multiplied by 1.618033989 results in an area of 1?
7. 1.618033989 multiplied by 0.618033989 gives 1
8. 1.618033989 is the Length, 0.618033989 is the Width.
As for the Babylonian approach itself, from a modern viewpoint one could suggest that it results from an awareness that for problems of this nature the quadratic formula:
can be split into two parts, i.e., with a = 1, into b/2 to be added to the other part, which (if the 4 inside the square root is brought outside and cancelled by the divisor 2) also includes b/2 already calculated, as expressed in the above procedure. Thus the Babylonian algorithm is essentially the simplified form:
where b is the difference between the length and the width, and c is their product, the area. Which is not to say that this was how it was arrived at by Babylonian mathematicians. But what can be suggested here is that it takes a fair degree of competency whichever way one looks at it, and this proves to be true in other aspects of Babylonian methodology, including the techniques laid out in the Babylonian astronomical cuneiform texts of the much later Seleucid Era [ 310 BCE - 75 CE ], predated in turn by the time of Archytas.
HISTORICAL DIGRESSION II
THE EYE OF HORUS
A further consideration concerning Phi in historical contexts is that binary representations naturally come to mind, especially with respect to ancient Egypt, for Phi is not only implicit in the structure of the pyramids, it is possibly inherent in Egyptian "Horus-Eye" Fractions in addition. In his continuation of Schwaller de Lubicz' symbolist approach to ancient Egypt John Anthony West1 gave a concise introduction to this complex subject in Serpent in the Sky1 as follows:
Measure, volume and the eyewhile more recently further details and insights made available on the Internet, courtesy J.D. Degreef and Guardian's Ancient Egypt Bulletin Board:
Egyptian measures and volumes refer both to man and to the earth, and the symbolic means Egypt chose to express here measures reflects her profound understanding of the relationships between the measures themselves and those human faculties that allow man to measure in the first place. Perhaps the most striking and convincing example of this is the eye, Ouadjit.
The eye gives man access to space, to volume, hence to measure. In Egypt, the symbol of the eye is comprised of those symbols that stand for the various fractions of the hekat. The symbols total up to 63/64... The symbols for the parts derive from the myth in which the eye of Horus is torn to bits by set. Later Thoth miraculously reunites the bits. (John Anthony West, Serpent in the Sky: The High Wisdom of Ancient Egypt, Quest Books, Wheaton,1993:70-71; for the fractional parts of the eye see also: Amun.com: The Eye of Horus)
The Eye of Horus first occurs as an expression in the Pyramid Texts, from the reign of the last 5th dynasty king, Unas, on (we don’t have older offering ritual texts). There it characterizes all kinds of offerings, presented to the deceased or his cult statue. But the theme of the Horus Eye is older. On model offering vases from king Neferirkare’s complex (also 5th dyn.), the Eye of Horus is represented under the form of the Udjat-Eye. This is a hybrid creation : the eyelids and eyebrow seem human, the tear-shaped part could be from a falcon’s eye, and the curl towards the back could derive from a feline eye (not my idea). But on Neferirkare’s vase, there are feather motives showing that the Udjat’s owner there was a bird, very probably the well-known Horus falcon. Now fragments of such incrustations have been found in at least one of the accessory tombs of Djoser’s pyramid (3d dyn.), which would tend to indicate that the theme of the Horus falcon’s Eye already existed at the time, and that it could also be used in the cult of private people. This is very important, for it shows that the dead, not only the kings, became gods of a pre-Osiris type. For when we ask ourselves to which specific Horus the Eye belonged, we must take into account the setting in which the Eye is mentioned in the ritual. And this is filled with allusions to the reconstitution of a dismembered body. In the divine sphere, this could correspond with a corn-god who is threshed (“dismembered”) or to the Moon, whose waning phases could be a nice image of dismemberment –and the waxing to reconstitution !-, or of both of the above. There are later myths related with the bringing back of the god’s solar eye, but I don’t feel the link with dismemberment would be as natural in that case. Also, the deceased dwells in a nightly country, which is more the Moon’s than the Sun’s realm. The link of the Udjat with corn is shown by the fact that during the New Kingdom the subdivisions of the bushel are indicated by pieces of the Eye. That this is a much more ancient theme is shown by the occurrence of the tear part of the Udjat on a box from the cache of Hetepheres, Kheops’ mother (4th dyn.). It is there associated with the symbol of Horus of Letopolis, a god whose power fluctuates between a seeing / non-seeing or a large-eyed / small-eyed phase. This is a god of the Haroeris class, i.e. a homologue of Osiris. The Letopolitan’s symbol also occurs on Neferirkare’s vase. So although the Eye of Horus may at times have been identified with the Sun IMO it was at first mainly if not exclusively lunar. In Egypt everything is assimilated with anything else anyhow. (J.D. Degreef, January 16, 2001, Guardian's Ancient Egypt Bulletin Board; italics supplied).In searching for mathematical details to examine this matter further, however, one comes to realizes that--for whatever reason--there is a strange dearth of information concerning both ancient Egyptian astronomy and Egyptian mathematics. Although largely and perhaps unjustly down-graded in the past, the asssociated complexities are nevertheless clearly apparent in an upward revision of the latter provided by Milo Rea Gardner in 1995 (emphases supplied):
As the esteemed history of mathematics journal HISTORIA MATHEMATICA reviewed in its Feb. 1995 issue, HM 22, Sylvia Couchoud's 208 page amateur paper was worthy of a closer look by professional mathematical historians. Maurice Caveing's review of Sylvia Couchoud's work is totally in French, I may have been able to read about 75% of Maurice's comments, but even on that level I found a review of 2,000 BC Egyptian fractions has been grossly under valued by Egyptologists and math historians. You may be able to appreciate the following facts:In the present context it is the earliest material associated with the Horus-Eye fractions followed by the references to the Golden Proportion that are of primary interest; but why was this particular set of base-2 fractions chosen in the first place, why was so much importance attached to the "Eye of Horus" historically, and what can be made of the legend that the "eye" is first torn to bits and then restored by Thoth?
1. Prior to 2,000 BC Egyptian fractions followed a binary structure, with the notation being called Horus-Eye, as noted by 1 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + ... + 1/2n + ...
2. Babylonian base 60 followed a very similar structure such that zero was not required to be used. Only the fractions needed were listed. No zero place holders were required, as our base 10 decimal system required.
3. By 2,000 BC Babylonian algebra has been reported by the majority of mathematical historians, such as Boyer in his popular text, that this rhetorical algebra is equivalent to our modern algebra I.
4. Yet even the algebraic geometry listed in the Moscow Papyrus, 2,000 BC, as noted by Couchoud, continues to under valued, as connected to an unworthy form of Egyptian hieratic arithmetic.
5. By 1850 BC, as recorded in 1650 BC by Ahmes in the Rhind Mathematical Papyrus a hieratic form of fractions alters greatly from the earlier Horus -Eye hieroglyphic fractions.
6. History of Science authors like Neugebauer, Gillings and Knorr have cited a consistent composite number pattern, as I prefer to write as: 2/pq = (1/q+1/pq)2/(p+1) where p and q are prime with p >q.
7. Neugebauer notes the general algorithmic aspect of the composite form, as does Gillings for the multiple of 3 case, and as does Knorr.
8. Disagreement between scholars is noted on two levels for the exceptions 2/35, 2/91 and 2/95.
... Concerning 2/35 and 2/91 a well known pattern does emerge, refuting the conclusions set down by scholars, a form that is clearly an inverted Greek Golden Proportion, the product of the arithmetic mean and the harmonic mean. Note that the arithmetic mean A = (p + q)/2 and the Harmonic mean H = 2pq/(p + q) can be seen as 2/AH = 2/pq = (1/p + 1/q)2/(p + q). Fill in the values for p = 5, q = 7 for 2/35 and p = 7 and q = 13 for 2/91 and see what I mean. As a 500 AD to 800 AD Akhmim Papyrus points out, the Egyptian inverted Golden Proportion seems to be improved upon, as Howard Eves noted in his AN INTRODUCTION TO THE HISTORY OF MATHEMATICS textbook, by: z/pq = 1/pr + 1/qr where r = (p + q)/2.
2.Second is the exception 2/95, which is really: 2/19 stated as a prime unit fractions time 1/5. Here the prime unit fraction algorithm is revealed by: 2/p = 1/a + (2a - p)/ap where a is a highly divisible number, about 2/3rds the value of p....
It should be noted at the prime number pattern point that two aspects are significant. First, and most importantly, all prime numbers in the RMP 2/nth table follow this rule (RMP = Rhind Mathematical Papyrus. Since there are no exceptions could it be that the famous Sieve of Eratosthenes was anticipated by over 1,500 years? Second, the essentials of the aliquot part algorithm, divisors of the first partition, was noted by B.L. van der Waerden in SCIENCE AWAKENING, about 30 years ago.(Milo Rea Gardner, 1995: Breaking The RMP 2/nth Table Code).
"The three-fold number is present in all things whatsoever; nor did we ourselves discover this number, but rather nature teaches it to us"This observation leads naturally enough to the phyllotaxic side of the matter (for details on this topic see part D above and the related links); in other words, the acute observation of nature might reasonably lead to an early awareness of the various ratios known to exist in plants, etc.--ratios that often embrace the first dozen or so Fibonacci numbers, i.e., 8:13, 13;21, 34:55, 55:89, etc. Thus there is more than one way to become aware of the "three-fold number." Where such an awareness might lead is another matter, but we can at least explore the possibility in the earlier Egyptian context with respect to the Horus-Eye Fractions.
Consider now the reciprocal of Phi (0.618033989...) to the sixth place in base-2:
Stage I: 1/2 + 1/16 + 1/32 + 1/64 = 0.609375or better still, to the seventh place, which is:
Stage II: 1/2 + 1/16 + 1/32 + 1/64 + 1/128 = 0.6171875The latter approximation is still on the low side, but more importantly, expressed fully in base-2 the complete sequence is in fact:
Stage II: 1/2 + 0/4 + 0/8 + 1/16 + 1/32 + 1/64 + 1/128 = 0.6171875
(1) " On model offering vases from king Neferirkare’s complex (also 5th dyn.), the Eye of Horus is represented under the form of the Udjat-Eye.This is a hybrid creation : the eyelids and eyebrow seem human, the tear-shaped part could be from a falcon’s eye..."Here we receive some guidance, for we can indeed effect a restoration for the "small-eyed" phase by adding smaller fractions of "1/4" and "1/8" from the mathematical relationship 5ro = 1/64 hekat (Gillings, 1972:210),2 i.e., by adding 1/4ro + 1/8ro = 1/1280 + 1/2560 to Stage II to obtain the following result:
(2) " The link of the Udjat with corn is shown by the fact that during the New Kingdom the subdivisions of the bushel are indicated by pieces of the Eye. That this is a much more ancient theme is shown by the occurrence of the tear part of the Udjat on a box from the cache of Hetepheres, Kheops’ mother (4th dyn.). It is there associated with the symbol of Horus of Letopolis, a god whose power fluctuates between a seeing / non-seeing or a large-eyed / small-eyed phase." (italics supplied).
Stage III: 1/2 + 0/4 + 0/8 + 1/16 + 1/32 + 1/64 + 1/128 +1/4ro + 1/8ro = 0.618359375
(or: 1/2 + 1/16 + 1/32 + 1/64 + 1/128 +1/1280 + 1/2560)
Stage II+1 = 1.6171875Though still not Phi, the above may nevertheless be considered in phyllotaxic terms, specifically approximations that lie between phyllotaxic ratios of 76/47 (Lucas) and 55/34 (Fibonacci), i.e.,
1/Stage III = 1.6171828
76/47 = 1.6170213
55/34 = 1.6176471
The inclusion of Lucas and Fibonacci ratios here can hardly be considered that surprising given that the former is readily apparent via the "rectangles and areas" Phi-series route, and the latter from the observation of nature, etc. Moreover, for the Fibonacci and Lucas series there are also ways and means to simplify matters, i.e., approaches that deal with integers rather than fractions, as alluded to by Plutarch with respect to the "First of Unities" described in an obscure work entitled Timeus the Locrian, etc. (for details, plus pythagorean and related aspects see Section IVc: The Fourth Planet and the Fifth Element).
With respect to ancient Egypt further complexities no doubt arise from the inundative and regenerative elements of the Nile on one hand and the associated pantheon of Egyptian gods and goddesses on the other. Then again other difficulties arise from a largely modern dichotomy that relegates Egyptian science and methodology while elevating the achievements of the later Greek commentators and philosophers. Yet historically this dichotomy may well be incorrect given the acknowledged Egyptian influence on Pythagoras, the apparent Egyptian background to the Timeus of Plato, and the continuity provided by the likes of Proclus and later commentators. The critical question here is not rejection of the premise itself, but to what degree the extant Pythagorean material and other historical sources can be applied to regain a better understanding of ancient Egyptian wisdom.
In closing, it seems entirely possible a clearer understanding can be achieved, though not without the reinstatement of Mother Earth in her rightful position. To which may also be added the re-apportionment of the numbers and the genders of the four primary elements along with the recognition of the seven visible planets and the nine luminaries.
John N. Harris, M.A.(CMNS). Last Updated on March 13, 2007.
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