The historical digressions
given
below concern the determination of the defining quadratic formula for
the
** Golden Section**:

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
synodic
formula for co-orbital bodies plays a fundamental role:

i.e., where ** T1** and

**HISTORICAL DIGRESSION**
**I**

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

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

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

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?

The modern solution via algebraic
methods leads to a quadratic equation similar to that determined above
for*
k*, but what has this historical aside to do with the synodic
formula
and the quest for the sidereal periods (*T1* and *T3*)
of the planets on either side of Earth (*T2*) ?

Firstly, the period ** T2
= 1** year is not only given, it also provides the frame of
reference
for the sidereal and synodic periods of the planets in question, i.e.,
years.

Secondly, analogous to the procedure carried out for

Thirdly, by setting the product

Lastly (and more to the
point)
this leads in turn to *exactly* the same quadratic equation as
that
obtained above for ** k**, (i.e.,

Remaining with Old Babylonian
methodology to obtain the length and the width of a rectangle with an
area
of 1 and a difference of 1 between the two sides the standard
Babylonian
procedure (firstly in sexagesimal notation) is essentially as follows (the values used here are* not *attested,
as far as I know):

1. Take one half of the difference1, the result is0;30[ Hold the result in your hand ]

2. Take the half-difference and square it, the result is0;15

3. Take the0;15and add it to the area1, the result is1;15

4. Take the square root of1;15, the result is1;7,4,55,20,29,39,6,54

5. Add the half0;30(from step1) to the square root, the result is1;37,4,55,20,29,39,6,54

6. What value when multiplied by1;37,4,55,20,29,39,6,54gives1(the area)?

7.1;37,4,55,20,29,39,6,54multiplied by0;37,4,55,20,29,39,6,54gives1

8.1;37,4,55,20,29,39,6,54is the Length,0;37,4,55,20,29,39,6,54is the Width.

In decimal notation:

1. Take one half of the difference

1, the result is0.5[ carry the result ]

2. Take the half-difference and square it, the square is0.25

3. Take the0.25and add it to the area1, the sum is1.25

4. Take the square root of1.25, which is1.118033989

5. Add the0.5(from step1) to the last square root to obtain1.618033989

6. What value when multiplied by1.618033989results in an area of1?

7.1.618033989multiplied by0.618033989gives1

8.1.618033989is the Length,0.618033989is the Width.

Thus the mean sidereal periods
of the two planets on either side of Earth (with a mean sidereal period
of
**1**) are found to be **0.618033989 **and** 1.618033989 **years
respectively, i.e., the Phi-Series periods for Venus and Mars. The
inclusion
of the period of Earth provides three sequential values and hence the
two
constants of linearity, i.e., Phi and Phi-squared. None of which is to
be
taken as historical fact, though this precise determination remains
feasible.
What it does suggest, though, is that many more possibilities arise
once
this initial triad has been obtained--including the initial step
towards
the One and the Many.

On a technical note, the above
treatment is linguistically imprecise and more verbose than the
instructions
given line-by-line in mathematical texts of the Old Babylonian Era
while
the use of the semi-colon to denote the equivalent of the decimal point
is
a modern addition for clarity. The square root of 5/4 given above as
1;7,4,55,20,29,39,6,54
is accurate to 14 decimal places but nevertheless readily obtainable
from
attested Babylonian procedures of the same era, specifically by the
fourth
iteration of the Babylonian version of "Newton's" method for
approximating
square roots. Rounding at the sixth and fifth places would also provide
useful
if slightly less accurate sexagesimal approximations for Phi of
1;37,4,55,20,29,40 and 1;37,4,55,20,30 respectively. To *One*
sexagesimal place
(1;37) the approximation is 1.61666*.

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

where ** b** is the
difference between the length and the width, and

Which returns us to the present
orientation, bearing with us the method by which two mean proportionals
may
be obtained--values that are not only the theoretical solutions to the
sidereal
periods for the planets on either side of Earth (Venus and Mars), but
also
the sequential values *Phi *^{-1}, Phi ^{0}
and
*Phi *^{1} that belong to both the Phi-Series and
the
Phi-Series exponential planetary framework. As for the latter, although
the
topic belongs to the next section, it can be said that this trio of
numbers
and related periods encapsulate much of what is stated above, for as
discussed
in the later sections, they also appear among the *velocities, *and
other repetitions are evident, especially in the case of Mercury.

**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 West^{1 }gave a concise
introduction
to this complex subject in *Serpent in the Sky*** ^{1}**
as follows:

while more recently further details and insights made available on the Internet, courtesy J.D. Degreef and Guardian's Ancient Egypt Bulletin Board:Measure, volume and the eye

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 thehekat.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.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):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,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 althougha god whose power fluctuates between a seeing / non-seeing or a large-eyed / small-eyed phase.the Eye of Horus may at times have been identified with the SunIMO 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).

As the esteemed history of mathematics journalIn 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?HISTORIA MATHEMATICAreviewed 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: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 2/91a 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 hisAN INTRODUCTION TO THE HISTORY OF MATHEMATICStextbook, 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 inSCIENCE AWAKENING, about 30 years ago.(Milo Rea Gardner, 1995: Breaking The RMP 2/nth Table Code).

What follows next is necessarily hypothetical, but certain implications can hardly be denied, not least of all the likely growth of mathematics and geometry in consort with the rise of agriculture born from the annual inundation of the Nile. As for the eventual awareness of Phi in this context, here again the dimensions and areas of fields might apply, though this need not be the only avenue available. It could, for example, have been perceived more readily and more naturally from nothing more than the observation of nature, as indeed stated by Ovid in a later era:

"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.

Suppose, for example, the ancient Egyptians were indeed aware of the phyllotaxic side of the matter but the knowledge was largely confined to priestly hierachies, etc. If so, the significance of the "Horus-eye" fractions might not be immediately obvious to the uninitiated. However, we do know that the hieroglyphs representing the parts of the Eye of Horus are base-2 fractions that extend out to the sixth place (i.e., 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64) giving a total that is less than unity. In one sense this may be economy of method; but in another it may also point to the

Consider now
the reciprocal of Phi (0.618033989...) to the *sixth*
place *in base-2*:

or better still, to theStage I: 1/2 + 1/16 + 1/32 + 1/64 = 0.609375

Stage II:The latter approximation is still on the low side, but more importantly, expressed1/2 + 1/16 + 1/32 + 1/64 + 1/128 = 0.6171875

Stage II:1/2 +0/4 + 0/8+ 1/16 + 1/32 + 1/64 + 1/128 = 0.6171875

In other words (i.e., in binary
notation:** 0;1,**0,0,**1,1,1,1**) the adjacent Horus-eye
fractions
*1/4* and* 1/8 *are both "missing" in the
result,
while in the "Eye of Horus" the hieroglyphs that correspond to *1/4*
and *1/8* represent *the pupil of the eye and the
eyebrow*
respectively. But clearly, although returning *1/4* and *1/8*
and removing 1/128 would restore the "eye" to the original condition,
this
would hardly be of much value, nor would it require the ministrations
of
Thoth--god of wisdom and learning. Nevertheless, the latter, with the
head
and long curved beak of the sacred Ibis bird--is said to have restored
the
eyes to Horus, who in turn is typified as a falcon--a genus that not
only
possesses a curved beak and talons, but also far smaller eyes and
"eyebrows"
than the original representation. Moreover, as J.D. Degreef noted above:
*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:

(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.Here we receive some guidance, for we can indeed effect a restoration for the "small-eyed" phase by addingThis is a hybrid creation : the eyelids and eyebrow seem human, the tear-shaped part could be from a falcon’s eye..."

(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 thesymbol 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 ++ 1/16 + 1/32 + 1/64 + 1/1280/4 + 0/8+=1/4ro + 1/8ro0.618359375

(or:1/2 + 1/16 + 1/32 + 1/64 + 1/128 +1/1280 + 1/2560)

Neither Stage II nor Stage III
yield the reciprocal of Phi, of course, but while the *addition of
unity
to stage II* produces a fair approximation, *the reciprocal of
Stage
III* (to the seventh decimal place) also produces a comparable
result,
i.e.,

Stage II+1 =Though still not Phi, the above may nevertheless be considered in1.6171875

1/Stage III =1.6171828

=76/471.6170213

=55/341.6176471

as similarly encountered in**
Part IV **(sbb4d.html) noted above.

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.

Copyright ©
1997.
John N. Harris, M.A.(CMNS). Last Updated on March 13, 2007.