Necessity of Circles

 

INTRODUCTION

 

Almost no one on Earth has failed to observe the movement of the heavens.  At first glance the sun, moon, and stars appear to circle the earth in an obvious daily and slightly less obvious yearly cycle.   And the more dedicated observer will note that planetary motion repeats albeit in irregular periods which are yet more complicated and less easily characterized.

 

The earliest records of the heavens come from the kingdom of Sumer beginning with the invention of writing about 3500 B.C.  The motivation for all early civilizations was a dependence on agriculture.   The cultivation of cereal grains, despite the intensive labor of clearing and maintaining fields, permitted permanent settlements and allowed populations to increase many fold.  And so knowing exactly when to plant and harvest crops was crucial for survival.   Knowing the date of the last killing frost at locations removed from the equator is essential even today.

 

And while the monthly cycle of the moon is apparent to all, the solar year which determines the seasons is harder to predict.   The first models employed perfect circles because no other single shape is so efficient a match.  Relatively minor deviations were accommodated by adding yet more perfect circles in epicycles to include as “deferents” and “equants” [1].   But this was not a religious requirement so much as a geometric and algebraic necessity.   Religious apologists may have employed this simple fact as rationalization for various theological propositions, but in truth no other approach had any predictive accuracy.   Claims to the contrary reflect more an inventive bias and blind prejudice against organized religion as they are based entirely on an ignorance of relatively simple mathematics.

 

Today we call such successive approximations of perfect circles the “Fourier Transform” which can be rigorously demonstrated to be able to achieve any desired level of accuracy [2].  Indeed it was only with the advent of desktop calculators in the early 1900s and especially digital computers in the early 1960s when it has been possible to directly integrate the equations of motion.

 

EVOLUTION OF PLANETARY MODELS

Early models were surprisingly sophisticated.  The “Antikythera” mechanism is an ancient Greek analogue computer able to predict solar and lunar eclipses and positions of the sun and moon to amazing accuracy.   It was discovered in a shipwreck off the coast of Greece in 1901.   It has been dated to the first century B.C. [3].

Ptolemy was accurate to within the limits of observational accuracy of his day.  His model from about 100 A.D. remained the simplest and most accurate model for nearly 1500 years.  Unfortunately while mathematically elegant it was physically unrealizable using interlocking crystal spheres to which the stars were supposedly attached to keep them from falling to earth.   In fact this model is so accurate it is still used today in all modern planetariums because it is easy to implement.

 

Unfortunately after nearly a millennia, the earth’s axis had shifted in a precession.   In addition, the calendar of Julius Caesar assumed the length of the year to be 365 and 1/4^th day.   This is actually 11 minutes too long..  So the Catholic Church found itself celebrating the feast of Easter at the wrong time.   Actually each individual year will differ from the last by up to 30 minutes but the long term average is fairly consistent.   Greeks thought the year was 10 minutes too long.  Differences are minute and only add up to any real significance over centuries and millennia.

 

Copernicus postulated a physically simpler heliocentric model in which the earth rotates daily and revolves around the sun.  The kinematics were simpler and physically realizable.  Unfortunately this required the addition of more epicycles than Ptolemy and provided less accurate predictions.  The difficulty was the relative inaccuracy of the recorded lunar and planetary positions.  Copernicus was a mediocre and mostly disinterested experimentalist and recorded less than 100 observations.   Indeed Tyco Brahe purchased his instruments nearly a century after his death and was famously astounded at their inaccuracy.   In any event, some of the most complicated calculations of Copernicus involved trying to curve fit trends in the observational tables that we now know were the result of systematic instrumental bias.

 

 It was only with the phenomenal accuracy of Tycho Brahe that Kepler was able to discover his three laws of planetary motion.  Indeed these tables remain the very best which humanity has ever made with naked eye astronomy.  His measurements were accurate to better than 1/60^th of a degree.   And this model is widely accepted today as the correct description of the solar system.   It is occasionally asserted that some sort of religious bias was responsible for a stubborn insistence on perfect circles which resisted acceptance of the correct shape of ellipses.  But again this demonstrates only an ignorance of mathematics.   In fact ellipses are mathematically identical to the simplest of epicycles.  The insight was to find the correct epicycle rather than trying to remove them entirely.

 

Also no model ever put the Earth at the exact center of the universe but from the earliest times some millennia before Christ shifted centers with joyous abandon in an attempt to accommodate the varying distances and different speeds of everything across the sky.

 

REFERENCES

1.      Ptolemy.

2.      Fourier Transform.

3.      Antikythera Mechanism.