Ancient Astronomy, Integers, Great Ratios, and Aristarchus

Contents


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 Calc conversion 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.

Table 1.  The 345-Year Eclipse Interval, epoch 297 B.C.E.
Period
Code
Value UT1
Multiple
Days UT
Lunar Nodal Period
dn
27.212223
4,630.5
126,006.20
Days
d
1.0
126,007.0
126,007.0
Rotations
dr
0.99726967
126,352.0
126,007.02
Lunar Synodic Period
dm
29.530595
4,267.0
126,007.05
Anomalistic Period
da
27.554583
4,573.0
126,007.11
Lunar Orbit
dl
27.3216678
4,612.0
126,007.53
Years
dy
365.242479
345
126,008.66
Orbits
do
365.256512
345
126,013.50

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.

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' 5,458-moons 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...."



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.

Table 4.  Àryabhata Great Ratio, 500 C.E. UT
1,582,237,500 : 57,753,336 : 4,320,000
Code
Ratio
Accuracy
l : r
57,753,336 : 1,582,237,500
0.0365010537
: 1.0
1.000 000 104
l : r
57,753,339 : 1,582,237,500
0.0365010556
: 1.0
1.000 000 052
l : o
57,753,336 : 4,320,000
1.000 006
l : o
57,753,339 : 4,320,026
1.000 000 008

 
Meton's Great Ratio and the Truth

Another great ratio known to early scientific astronomers was Meton's 19-year eclipse cycle (Table 5). The Metonic great ratio intercalates 254 lunar orbits and the 235 synodic period eclipse interval with only 0.0137 rotations difference (235 m : 254 l = 1 : 1.000 002). Given one less synodic period than lunar orbits per solar orbit, in 19.0 solar orbits there are 254.0062 lunar orbits and 235.0062 moons.

Table 5. Metonic Great Ratio, epoch 2,000 values UT.
255 : 19 : 235 : 254 : 19 = n : y : m : l : o
 
Period, Days
Multiple
Days
Rotations
Lunar Nodal
27.21222
255 n
6939.116
6958.114
Tropical Year
365.24218
19 y
6939.601
6958.601
Lunar Synodic
29.53059
235 m
6939.688
6958.688
Lunar Orbit
27.32166
254 l
6939.702
6958.702
Solar Orbit
365.25635
19 o
6939.871
6958.871

Meton revealed the nineteen-year circuit to the public in Athens, and history remembered the astronomer by naming the Metonic eclipse cycle after him. Archaeoastronomers and science historians consider the Metonic cycle to be noteworthy as a short eclipse cycle with the eclipses on the same day of the calendar year 19 years later. Meton instituted his '19-year-eclipse' calendar in Athens on the summer solstice of 432 B.C.E.

Callippus corrected the Metonic cycle, deducting one day every four Metonic periods, after 76 years comprising 940 months. Apparently, Callippus knew of the one day difference between 76 orbits and 76 years in four Metonic cycles. Knowledge of the year-orbit difference implies Callippus knew of precession.

"the god visits the island every nineteen years, the period in which the
return of the stars to the same place in the heavens is accomplished."
Diodorus Siculus

"the Greeks [who] use the nineteen-year cycle ...
are not cheated of the truth." Diodorus Siculus

76 * 365.25651 days per orbit = 27759.49 days
76 * 365.24248 days per year = 27758.42 days

Callippus began his cycle June 28, 330 B.C., as the beginning
of the lunar cycles nearly coincided with the moment of solstice.

Callipus observed the sequence of the Metonic cycle against sidereal space. Observing the second Metonic eclipse occur, in the context of keeping a lunar and rotations count, one observes the sequence of the events in Table 5 and as follows:

  1. 19 years complete
  2. 235th moon is eclipsed
  3. 254th lunar orbit completes
  4. 19 orbits complete

Given the logic of geometry, the less-nineteen-synodic node cannot precess the nineteenth orbital circumference. The less-nineteen node is the nineteenth solar orbit, when 254.0062 lunar orbits equals (254.0062 less 19.0) lunar synodic periods (254.0062 : 235.0062). Only when 19 solar orbits complete are there 19 more lunar orbits than moons. Eclipse of the 235th lunar synodic occurs just before 254 lunar orbits, revealing that less than 19 orbits equals 235 moons.

The sequence above solves astronomy constants, plus rate of precession is revealed by the orbit-year dichotomy. Eclipses allow timing the relative proportions of lunar synodic and lunar orbits. When 254 orbits complete, there are 235.00046 synodic periods, 18.99954 fewer moons, hence only 18.99954 solar orbits have completed during 254 lunar orbits (254 / 18.99954 = 13.36875 lo). From the tally of rotations in 254 lunar orbits, the value of rotations per orbit also easily computes (6958.70257/18.9995407 = 366.25636) and minus one resolves days per orbit (using TT values).

235 moons = 6,958.688 rotations
254 lunar orbits = 6,958.7016 rotations
235 : 6,958.688 = 235.00046 : 6,958.7016
254.0 - 235.00046 = 18.99954 orbits
6,958.7016 rotations / 18.99954 orbits = 366.25636

For Callipus' Great Ratio interval, 76 orbits does not complete until a full day after 76 years, proportional to precession. Using Meton's calendar and cycles, moons and the orbit cycles move incrementally further from summer solstice with time. Nearly five days after 345 years, 345 orbits complete, and the eclipse of full moon number 4,267 occurs half-a-day before 4,612 lunar orbits complete, an interval of over 6 degrees of lunar orbit and nearly half an earth rotation.

With long observations and eclipse records, ancient astronomers were equipped with the counts needed to resolve the conundrums of cosmology.

"...the fixed stars and the sun remain unmoved ... the earth revolves around the sun in the circumference of a circle."
Aristarchus cited by Copernicus

"Aristarchus ... the earth moves round the sun's circle and is put in shadow according to its inclinations." Aëtius


 

Afterwords

2012.07.25 - Some refinements and corrections to the temporal terms of my astronomy 'constants' formulas have altered the deep decimal expressions on this page. Also, some numbers were converted to Universal Time to better approximate past ephemeris time. Formulas are available in Epoch Calc applet. In particular, even though the epoch was only 1500 years ago, the much larger numbers considered by Indian astronomers were most impacted by the formula refinements.

2012.12.18 - Peter Nockolds corrected a factoral error, 45 Exeligmos Eclipse cycles only approximates 2,434 years rather than 15. Good eye Peter.


 
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Ancient Astronomy, Integers, Great Ratios, and Aristarchus
© 2009.12.21 by James Q. Jacobs
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