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

Fortyfive 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 2,438orbit period. Regarding the "planets return to the same position in the heavens," both Mercury and Venus are near integer orbits and synodic periods also.
Table
2. The 2,438 Orbit Interval (UT).

Period

Integer

Accuracy

Lunar Orbit

32,593.0

1.000 000 292

Lunar Synodic

30,155.0

1.000 000 315

Mercury Orbit 
10,123.0 
1.000 001 
Venus Orbit 
3,963.0 
1.000 006 
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' 5,458moons eclipse interval with Aristarchus' possible
ratios.
Table
3. Greek Great Ratios. 297 B.C.E., UT

Astronomer

Code

Ratio

Accuracy

Hipparchus

m : n

5,458 : 5,923

1.000 000 041

m : ye

5,458 : 465

1.000 000 521

Aristarchus 2,438 solar

l : o

32,593 : 2,438

1.000 000 292

l : m

32,593 : 30,155

1.000 000
315

"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...."

