Contents

• The 345-Year-Eclipse Great Ratio
• Aristarchus' Great Ratio
• Great Ratios and Àryabhata's Yuga
• Meton's Great Ratio and the Truth

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.

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.

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.

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.

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.

 
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.

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 calendar, realigning it to the solstice. Apparently, Callippus also understood the one day difference between 76 solar orbits and 76 years in four Metonic cycles. Knowledge of the year-orbit difference implies Callippus knew of precession. To keep the solar and Metonic lunar calendars synchronous, one day needs to be deducted every three Callippic cycles.

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:

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).

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.

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.

 
Readings

• Aristarchus of Samos, the Ancient Copernicus, Sir Thomas L. Heath, 1913; reprinted 1981.
• Mesoamerican Archaeoastronomy
• Eclipses, Cosmic Clockwork of the Ancients
• The Dresden Codex Lunar Series and Sidereal Astronomy
• Newark Archaeogeodesy, Assessing Evidence of Geospatial Intelligence in the Americas

Ancient Astronomy, Integers, Great Ratios, and Aristarchus
© 2009.12.21 by James Q. Jacobs
Cite as http://www.jqjacobs.net/astro/aristarchus.html

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