Ancient Astronomy, Integers, Great Ratios, and Aristarchos (original) (raw)

Ancient Astronomy, Integers, Great Ratios, and Aristarchus

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****** The 345-Year-Eclipse Great Ratio

Eclipse cycles, by definition, express integer intervals of lunar periods. Integers—whole numbers, serve as fractions when any two are used as a ratio. All fractions equate integers. Decimals are fractions, tenths, hundredths, etc. For pi, the ratio 22 : 7 is approximate, 355 : 113 is more accurate. Ratios of large integers express deep fractions accurately. Integer ratios provide evidence of ancient astronomical knowledge.

I noted an Old World 345-year eclipse interval also precisely equals integer numbers of rotations and days. Examination of Aristarchus' 4,876-year "Great Year" revealed a large-integer equation of the three fundamental sidereal motions: solar orbit, lunar orbit, and rotations. This possible import of Aristarchus' interval has gone unnoticed for several millenia, apparently fallen into obscurity along with heliocentrism.

A "great ratio" is herein defined as an accurate integer ratio for three or more astronomical periods. Eclipse intervals inherently express cosmic harmonics of two integer-accurate periods, moons and nodal crossings. The approximate 345-year-eclipse-interval great ratio equating 4,267 synodic periods with 4,573 anomalistic periods was known to the Ancient Greeks and Babylonians. In the Epoch Calcconversion tables, I noticed this interval also accurately equates to more integers; in 126,352 rotations there are 126,007 days, per orbit one less day than rotations, 345 fewer days. This great ratio has five integer periods, six when doubled.

The 345-Year-Eclipse Great Ratio
345 : 4,267 : 4,573 : 4,630.5 : 126,007 : 126,352

Accurate integer equation of lunar synodic with the apogee-perigee period ensures greater interval constancy of eclipse observations. Lunar orbit eccentricity effects speed of orbit during each anomalistic period. Comparing eclipse intervals having whole multiples of the apogee-perigee cycle ameliorates the impact of the inconstancy of orbit speed. Ancient astronomers utilized this knowledge to arrive at more accurate mean period determinations. Hipparchos, who compared eclipses with Babylonian records from 345 years earlier, evidences this astronomical understanding in antiquity.

The well-known Saros Eclipse Cycle expresses a great ratio with anomalistic periods (223 : 239 : 242) and the Triple Saros or Exeligmos with integer days additionally (669 : 717 : 726 : 19,756). Eclipse equation with integer days indicates the interval repeats at near the same time of day. The Saros Great Ratio equates eclipses and lunar orbit eccentricity, thus similar eclipses recur. The Exeligmos Great Ratio additionally roughly equates days, thus the eclispe recurrence is near the same diurnal time. Longer intervals proportionally increase observation accuracy, and more accurately express complex fractions with larger integer ratios. Sufficiently large integer ratios express astronomical values precisely.

Table 1 compares, using 297 B.C.E. ephemeris astronomical values (Universal Time), days per the six integer multiples in the 345-year eclipse interval. One-half a nodal period multiple (4,630.5) is an eclipse integer; ascending and descending nodes cross the ecliptic each period and both can possibly eclipse. Code used as shorthand herein and in the Tables (i.e. d = day) is introduced in the fundamental astronomy section of Eclipses, Cosmic Clockwork of the Ancients.

Two most accurate integer ratios are with earth sidereal rotations. The most accurate integer ratio with moons is rotations. The three most equal periods of this eclipse cycle are 126,352 rotations : 126,007 days : 4,267 moons.

126,352 rotations : 4,267 lunar synodic = 1.0 : 1.000 000 252
126,007 days : 4,267 lunar synodic = 1.0 : 1.000 000 393
126,352 rotations : 126,007 days = 1.0 : 1.000 000 141
4,573 anomalistic : 4,267 lunar synodic = 1.0 : 1.000 000 464

Use of an eclipse interval which equates accurate integer rotations and days implies earth motion and time of day and of rotation may have had a role in the observations—likely augmenting the accuracy capability. The accuracy of integer anomalistic periods makes this interval a favorable observation choice for those who understand lunar elliptical geometry produces variation in orbit speed.

**Aristarchus' Great Ratio

Aristarchus, the earliest known scientific astronomer, is noted as the first Greek to widely teach heliocentrism. Some of his lost work is reported, including a "great year" figure. Heath reports "Aristarchus multiplied ... arrived at 889,020 days containing 2,434 sidereal years, 30,105 lunations, 32,265 anomalistic months, 32,670 draconitic months, and 32,539 sidereal months."

"We are told by Censorinus that Aristarchus ... gave 2,484 years as the length of the Great Year, or the period after which the sun, the moon, and the five planets return to the same position in the heavens. Tannery shows that 2,484 years is probably a mistake for 2,434 years, and he gives an explanation ... derived from the Chaldaean period of 223 lunations and the multiple of this by 3 ..." —Heath 1913:314

Forty-five Exeligmos Eclipse cycles only approximates 2,434 years or orbits. The number 2,434 has an accurate integral ratio for lunar orbits and rotations (1.0 : 0.0365010). The two visible fundamental motions combine in the ratio 2,434 lunar orbits to 66,683 rotations (1.0 : 1.000 000 086 given 297 B.C.E., UT1). The concept "return to the same position in the heavens" infers sidereal positions return to an original configuration. The number of solar "orbits" meeting this criteria is 2,438.

Following on the above, I considered Aristarchus' Great Year as 2,438 or 4,876 orbits. Aristarchus' interval when reconsidered as representing 2,438 solar "orbits" equates to integer lunar orbits and moons. The nearest integer expression of the three fundamental motions (o : r : l ) is 4,876 solar orbits : 1,785,866 rotations : 65,186 lunar orbits. To distinguish the orbit interval, I term this period a great ratio rather than erroneously label the span as a Great Year.

Aristarchus' "Great Ratio" solar orbits : rotations : lunar orbits 4,876 : 1,785,866 : 65,186

2,438 solar orbits : 32,593.0095 lunar orbits (UT)
32,593.0 : 32,593.0095 = 1.0 : 1.000 000 292

4,876 solar orbits : 64,635.0095 anomalistic periods (UT)
64,635.0 : 64,635.0095 = 1.0 : 1.000 000 159

In Aristarchus' Great Ratio, 2,438 solar orbits is the lowest integer ratio with lunar orbits. Table 2 compares the integer accuracy in the 2,438-orbit period. Regarding the "planets return to the same position in the heavens," both Mercury and Venus are near integer orbits and synodic periods also.

Given integer solar orbits, there is a corresponding integer difference in both the lunar orbit to lunar synodic ratio, in the rotations to days ratio, and in the inner planet sidereal and synodic periods. The accuracy of integer difference of these ratios is a function of integer accuracy of solar orbits. Spatial geometry dictates the x - 1 = y rule for orbital motion. Two independent fundamental motions, lunar orbit and rotations, share their motions with solar orbit. The lunar orbit and rotations ratios both equate to solar orbit with the same integer equation (one less per solar orbit) to produce the number of moons and days (x - solar orbits = y, specifically l - o = m and r - o = d). For the inner planets, their sidereal and synodic difference also equals solar orbits.

x - 1 = y

1 solar orbit = 13.369 lunar orbits
1 solar orbit = 12.369 lunar synodic
1 solar orbit = 366.256 rotations
1 solar orbit = 365.256 days

Table 3 compares the accuracy of Hipparchus' 5,458-moons eclipse interval with Aristarchus' possible ratios.

"Aristarchus has brought out a book consisting of certain hypotheses ... that the fixed stars and the Sun remain unmoved, that the Earth revolves about the Sun on the circumference of a circle, the Sun lying in the middle of the orbit ... the sphere of the fixed stars, situated about the same center as the Sun...."

Great Ratios and Àryabhata's Yuga

As above, again I compared a longer interval for integer ratios (Table 4). Àryabhata wrote 1,582,237,500 rotations of the earth equal 57,753,336 lunar orbits. (57,753,336 : 1,582,237,500 = 0.0365010537 lr). These are larger integers than necessary to express deep decimal numbers precisely. Astronomical "constants" change with time, thus integer ratio accuracy differs with epoch and can be equated to chronology.

Àryabhata's 500 A.D.writing, centuries after Classic Greek astronomy, explicitly presents a lunar orbits to rotations ratio accurate a millenium earlier. This was considered the likely most-accurate ancient astronomical constant in 1997 in the Àryabhatiya of Àryabhata article, albeit the astronomy numbers I used in 1997 require updating with current astronomical constant formulas and with conversion to Universal Time.

"In a yuga the revolutions of the Sun are 4,320,000, of the Moon 57,753,336, of the Earth eastward 1,582,237,500, of Saturn 146,564, of Jupiter 364,224, of Mars 2,296,824 . . . "
(The Àryabhatiya of Àryabhata, An Ancient Indian Work on Mathematics and Astronomy, translated by W. Clark, 1930).

Àryabhata's ratios are not very accurate great ratios (Table 4). Clark (1930) and Kay (1981) present two different lunar numbers. Kay's interval, 57,753,339 lunar orbits, also equates to an integer of 4,320,026 solar orbits and is thus a more precise great ratio. This second ratio, to 4,320,026 solar orbits, lends support to Kay's interval—also the more accurate lunar ratio—as an intended, known ratio. The great ratio 57,753,339 lunar orbits equated accurately to 4,320,026 solar orbits around 400 C.E., just before Àryabhata wrote.