Sacred Number

 

I discovered in 1994 that it is also possible to use N:N+1 triangles, with simple counts of planetary synodic periods, to derive a planet’s sidereal period. A planet’s sidereal period is the time it takes to orbit the Sun whilst its synodic period is the time it takes to repeat its behavior as seen from Earth, the mixture of it’s own orbit superimposed by the orbit of the Earth from which we observe it. The synodic periods seen from Earth can be counted whilst the sidereal, orbital periods were only thought to have become known after the advent of modern astronomical practices and our recent “discovery” of the solar system as being heliocentric, that is Sun centred.

The foregoing work on the Moon and Sun, using N:N+1 triangles, makes it fairly certain that the megalithic astronomers could have deduced the orbital periods of the planets from observation of time through long counts.

The Inner Planets, Mercury and Venus

The synodic and sidereal periods of the inner planets in days are,
                       Mercury    Venus
Sidereal             87.969    224.701
Synodic Period    115.88    583.92

In the case of the inner planets, they are seen to orbit the Sun and from this fact an important fact can be articulated: When observing the synodic frequency (per solar year) of an inner planet, the observation sees one less rotation than that planet’s sidereal frequency because the Earth has completed a full orbit in the same direction. This fact, similar to the difference between the frequencies of lunar orbit and month, makes it possible to apply N;N+1 triangles to deduce the sidereal frequency per year of an inner planet and hence its sidereal period.

Venus is the brightest planet and hence most likely to have been studied in depth. The first step is to note her synodic period as 584 days long and erect this day count as a hypotenuse above the solar year as before, in a right angled triangle. Also as before, these period lengths also represent the relative frequencies of the other body so that the synodic frequency of Venus, per year, is seen as the reciprocal of its period or 1/ 1.599 which equals 0.6255 years.

Adding one to this gives 1.6255 and this base length can then be erected from the opposite end to touch the same third side’s vertical. This is then sidereal frequency and it now stands over a base length of one, representing the frequency of the solar year. The same reciprocation as before means that the sidereal period of Venus is now in proportion to the year, the base beneath it. Indeed, the sidereal period is then 0.615, which is correct, and in days this is 224.69 days – effectively exact.

In our notation this would be,

Venus (sidereal) = 1 / (1 + solar year/ Venus synod) = 0.615 solar years

Where solar year = 365 ¼ days
     and the Venus synod is 584 days
 
The Venus “sailboat” triangle.
It enables the derivation of the Venus sidereal period
from the Venus synodic period.

This indicates how, without modern techniques of notation, algebra and calculation, metrological triangles could discover important information concerning the orbits of the planets. The approach is highly visual and revealing for it shows how Venus hovers about that numerical singularity where the number and its reciprocal have a similar fractional part – the Golden Mean. Venus has a period, relative to the solar year, of the Finonacci approximation to the golden mean of 8/5 or 0.625, where the common unit is 1/5th of a year which is 73 days. As a result, the sidereal period becomes the next higher Fibonacci ratio of 8/13 = 0.615 years and the whole matter is beautifully presented as two triangles representing these two ratios. The dotted line in the first triangle shown above can therefore be seen as length 8, which both ratios have in common, the first as the numerator of 8/5 and the second as the numerator of 8/13 = 0.615.

Using this procedure places the whole explanation before one’s eyes unlike our own mathematics in which a separate explanatory graphic would have to accompany the calculational details. In the Megalithic, the whole of a problem had to remain integral in its solution via geometry.

A different geometry is required to find Mercury’s sidereal period, which we will skip here so as to illustrate how to work with the outer planets

The Outer Planets: Mars, Jupiter and Saturn

Because outer planets orbit further away from the Sun, when the Earth passes closest to them their motion appears retrograde and, because of parallax, the path of the planet then describes a loop in the sky. The time between each loop allows the synodic period of an outer planet to be known, within half a day using a simple day-inch count or a little better, using a long count that covers many synods.

The outer planets again follow a similar rule: When observing the synodic frequency (per solar year) of an outer planet, the observation “sees” one less rotation than that planet’s sidereal frequency because the Earth planet has completed a full orbit in the same direction.

The synodic periods of outer planets are different because they orbit more slowly than the Earth does and this means the Sun has to catch up, after a solar year, with their new position. This catch-up time will divide into the solar year to give a synodic frequency to which needs to be incremented  by one to obtain the true sidereal period for that planet’s orbit.

A triangle is constructed in which the base is the solar year. The planet’s synodic period is then erected above this which, brought down to the base reveals the excess over the year. The base then represents the number of times the excess divides into the solar year, the synodic frequency so to speak. The hypotenuse divided by the excess reveals the true sidereal period in years.

In our notation,

Jupiter’s sidereal period = 1 + solar year / excess

where the solar year is 365 ¼ days

and the excess is 33 ¾ days

This is a really easy task using metrological triangles since the basic N: N+1 triangle already does what is required for an outer planet.