While the earliest permanent settlements arose in foothills, unified city states first formed along river valleys with better agricultural endowments and attendant animal husbandry.   The importance of astronomy was to fix dates for seasonal planting and harvesting.   And so a better understanding of celestial motions, especially that of the sun, was of considerable practical import.   A secondary benefit was the social cohesion attendant on a common religious view administered by a cadre of priest-leaders who had the leisure time to pursue such interests.


Inevitably, the original sense of the heavens was that view common to all.  Unchanging star patterns appear to revolve in daily circles around a point in the sky near the star Polaris.   This is the celestial north pole and allows one to accurately determine the direction of true north.  Over many days and nights the sun, moon, and five planets visible to the naked eye appear to slowly shift their positions relative to the fixed constellations but all along the same circular path in the sky called the ecliptic.   That is, they all appear to move in about the same plane.


The two inferior planets, Mercury and Venus, never wander far from the sun.  But the three superior planets, Mars, Jupiter, and Saturn, move across the entire ecliptic apparently unconcerned with the sun’s location [1].  In addition, all the planets occasionally appear to stop, to retrace their path along the ecliptic, and then to resume their former motion.   This is called “retrograde” behavior.


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During the day, the sun rises roughly in the east, traces a curved path, and then sets roughly towards the west.  By definition, the sun is highest in the sky at local noon when all shadows lie along a line from north to south.  But four special days each year are notable because the sun does something different to mark the seasons.


The sun is highest in the sky at noon at the summer solstice on June 21 which has the longest day and shortest night.   This is reversed six months later at the winter solstice on December 21 when the noon time sun is at its lowest point resulting in the shortest day and longest night [2].  Note that the path of the sun in the sky is easily measured despite the fearsome glare by simply placing a stick in the ground and periodically recoding the shadow.


The direction of sunrises and sunsets also exhibit a yearly cycle.   In summer the sun rises more and more from the northeast and then moves back to rise from the southeast in winter.  Twice during the year, at the vernal equinox on March 20 and autumnal equinox on September 22, the sun rises everywhere on Earth from exactly due east and sets exactly due west.   When this happens the lengths of those days and nights are equal and the sun is directly overhead at noon as viewed from the Earth’s equator [3].



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Finally, there are rare and frightening heavenly fireworks displays in the form of meteor showers and comets which presage either good or bad luck as imaginations and situations warrant.


What are we to make of all of this?  The obvious explanation is that the stars are, of necessity, attached to a smoothly rotating crystalline sphere else what would keep them from falling out of the sky to Earth or cause them to move unerringly in unison.   A distinction was also made between the unchanging incorruptible heavens and the change and decay found on Earth.  Eventually these observations passed into myth and the oral traditions into written sagas.




Writing was invented in Sumer from about 3500 to 3200 B.C. using cuneiform characters on clay tablets.  The earliest surviving astronomical records are from Mesopotamia and immediately neighboring societies which left extensive records of lunar and solar eclipses and planetary positions against fixed constellations of stars.  One of the earliest surviving records is the “Tablet of Ammisaduqa” (Enuma Anu Enlil) from the mid seventeenth century B.C. listing positions of the planet Venus and perhaps derived from Sumerian observations dating as far back as the third millennium B.C.  Babylonian astronomical diaries notably improved during the reign Nabonassar in the mid seventh century B.C.   


The initial methodology was entirely empirical and relied on ephemeris tables and simple arithmetic notably lacking the theoretical sophistication of later Greek and Christian geometers.  The model of the cosmos was that of a celestial sphere containing both the fixed stars and the moving bodies of the sun, moon, and planets.




The trivial observation that the moon, sun, and planets traverse the same circular path in the sky along the “ecliptic” and the human proclivity for recognizing patterns in random data led to the naming of constellations and astrology.   One of the first divisions of the heavens was the twelfth century B.C. text “Three Stars Each” which divided the heavens into three parts. By the end of the fifth century B.C. Babylonian astrology had improved to invent the twelve signs of the Zodiac which we still know and love today.  The “Magi” of Biblical fame centuries later were Babylonian astrologers.


While cycles of the moon are obvious to all, the length and timing of the solar year which governs the seasons is not.   This is why all primitive societies base their calendar on the lunar month and then expend no little effort trying to reconcile that to the solar year with its fractional days. The difficulty is that the orbits of the moon, sun, and planets are nearly circular and observations inaccurate.   And small almost undetectable variations from ideal behavior have surprisingly large consequences when trying to create an accurate calendar.   Nevertheless, nothing in life succeeds like persistence.


Extensive tables over many lifetimes and the rise and fall of empires led to the discovery of the 19 year “Metronic” cycle.  This had been well known in Babylonia when Menton of Athens independently published it in 432 B.C.   The discovery was that 19 solar years corresponds almost exactly to 235 lunar months or 6940 days (i.e. synodic months as measured from the Earth against the fixed stars).   This gives the length of a year as 365.25 + 1/76 days whereas the modern value is 365.25 – 1/128 days.  The fractional parts of a day can then be used to reconcile the lunar and solar calendars.


Another discovery sometime before 200-300 B.C. was that lunar and solar eclipses repeated in “saros” cycles lasting 18 years and 11 1/3 days.  They also noted but could not account for the uneven speed of the sun, moon, and planets along the ecliptic. 




The first book of the Bible is Genesis which was in the modern sense constituted the first scientific revolution.  It provided the philosophical bedrock which later enabled the Catholic Church to invent the scientific method.  Indeed almost all fundamental discoveries of cosmological significance were done by Catholics or especially by Catholic Clergy.




While others relied on trigonometric tables, the Greeks were the first to apply purely ratonal analyses and then to invent axiomatic geometry and algebra.  With these, they derived the first models of the cosmos.





where the angle “w t” in radians where 2 π is equivalent to 360° and w = 2 π / T where “T” is the period of one revolution and the “t” is the time. 


Note that the inner or “inferior” planets of Mercury and Venus are never seen to wander far away from the sun.  So the centers of their epicycles needed to be on a straight line between the Earth and Sun.  Also this line, as well as the angle of the outer or “superior” planets of Mars, Jupiter, and Saturn along their epicycles, always have the same angle.


Aristotle (384-322 B.C.) was the star student of Plato (424-348 B.C.) who was a student of Socrates (470-399 B.C.).  Besides being a chronicler of Greek science was an original thinker in his own right.   Being the teacher of Alexander the Great who conquered the known world and spread Greek thinking and culture across the world, his views were more influential than otherwise might have been the case. 


The Greeks knew that the earth is a sphere and had relatively accurate measurements of its size.  They also knew the length of the solar year, and the distances to the moon and sun.  There was no obvious centrifugal force nor any observable stellar parallax.   Nor were there any observable winds from the atmosphere thought to pervade all of space.  A vacuum was thought to be impossible.


Aristarchus of Samos (310-230 B.C.) thought the sun was stationary and the planets revolved around it.   But this was generally rejected on observational grounds. A sun centered system had been proposed earlier and discussed at length by Aristotle.




Hipparchus (120-190 B.C.) noted that their models were much improved if the Earth was moved some distance from the center of the cosmos along a “deferent”.   And the direction and distance of this deferent from the Earth was necessarily different for each planet.  Everyone considered the possibility of the Sun, not the Earth, being stationary at the center of the solar system but rejected it on observational grounds.   His estimate for the length of the solar year was 365.25 – 1/300 days.




A later improvement by Ptolemy (100-170 A.D.) was to vary the speed of the sun or moon or a planet along its orbit by introducing an “equant.”



This proved to be so accurate that it is still the model for all modern planetariums.




Sometime within the decade of 1263 to 1272 A.D., King Alfonso the Tenth (1221-1284 A.D.) who ruled the Spanish Provinces of Castile, Leon, and Galacia, commissioned a set of astronomical tables and algorithms for determining the exact hours for the rising of the fixed starts and wandering planets.   These became known as the “Alphosine” tables.   Considering the complexity of spherical geometry, Kink Alfonso was reputed to have quipped that “had he been present at creation, he would have given some useful and simplifying advice to God.”


Contrary to existing custom they were written in the vernacular of Castilian Spanish and employed Roman numerals.  Unfortunately the original tables, except for the introduction and a few fragments of data have been lost.   They survive in written manuscripts from Paris which shifted the calculations to the local longitude.  Although widely circulated, they were first printed in Venice only in 1483.   Ironically, the Paris version switched to Latin, which gave it a wider audience among the educated elite of Western Europe, and employed Arabic numerals which greatly simplified the calculations.


They were used in navigation for the voyages of Christopher Columbus and formed the basis for the revolutionary work of Copernicus and the attendant reform of the calendar by the Catholic Church under Pope Gregory.  They were the first such tables produced by Christian Europe and provided the most accurate predictions until superseded by the work of Kepler (1571-1630 A.D.).


They had a very long baseline of observations, the most recent of which was from the Arab astronomer al-Zarqali (1029-1087 A.D. aka Arzachel) but included data from Ptolemy and pre-Christian Greece and even from Mesopotamian records from several millennia B.C.  Corrections were necessary for the shift of the position of celestial north pole around which the starts rotate.  This effect is known as the “precession of the equinoxes” and amounted to about 15 degrees since the time of Ptolemy some 1100 years earlier.


Nevertheless, no one had managed to improve on the simple but extremely efficient model of Ptolemy from the first century A.D.  Consequently these tables relied entirely on his very limited number of epicycles and “equants” without modification.




Nicolaus Copernicus (1473-1543 A.D.) was an ordained minister of the Catholic Church.  He accepted the first of three official “orders” becoming a Deacon which included a vow of chastity but provided a life time sinecure.   The second order would have made him a priest and the third a bishop.  Despite repeated and heartfelt encouragement to progress to higher orders, he was apparently comfortable in his position requiring little work, and never did.    Nevertheless, he was a lifelong devout and believing Catholic in extremely good standing.


His religious duties included conducting a reduced version of the “Liturgy of the Hours” daily in the Cathedral, providing general medical care, and helping to administer Church property.  These left him ample time to minister to the ill and infirm, to advise on monetary reform, to defend fortifications in time of war, to organize rebuilding after the ravages of war, and especially to pursue his studies in astronomy.


Over the lifetime of Copernicus, the Catholic Church was attempting to reform the calendar which was getting out of synchronization with the seasons.  As Copernicus was completing his education in Italy the Western World was using the Julian calendar which had been in general use since its introduction at by Julius Caesar on January 1, 45 B.C.   Caesar assumed the solar year was exactly 365.25 days ignoring the more precise value recorded by Hipparchus of being slightly less by 1/300th of a day.   The modern more precise value is the year less than Caesar’s estimate by 1/128.2th of a day so that the Julian calendar will gain one extra day about every 128 years.  The difference between the exact value and Hipparchus’ estimate amounts to about 6 minutes and 26 seconds per year.


The state of the art at the time of Copernicus was that


1.      No one believed the Earth was flat but rather correctly that it was a sphere with remarkably accurate estimates of its actual size. The shape of the Earth was correctly measured and described by the Greeks before 1000-500 B.C.   A flat earth was the widespread belief in Mesopotamia before 1000 BC and by otherwise literate civilizations such as the Chinese until about the 16th century.


2.      No one took pride in the Earth being at the center of the universe.  The Earth was considered to be corruptible and hell was placed at the lowest levels.  Indeed the further from the center one went, the closer one was to the perfection of heaven.


3.      No one believed or cared if the Earth was at the center of the cosmos because it had to be offset to better explain apparent celestial motions since at least the time of Hipparchus as recorded in his summaries and expanded upon later by Ptolemy.


4.      No one thought there was any religious necessity for circular motion but rather that was the only shape for which they had quantifiable mathematics.  Indeed they added so many epicycle improvements that the overall motion was anything but circular from the very earliest models.  But since the stars all moved in unison and did not fall to earth, it was assumed they were attached to invisible crystalline spheres and there was no little thought given to how the various rotating globes would mesh.


5.      The Bible, in all of its Books, had never taken any position on where the center of everything was.  Rather the Bible from the Book of Genesis claimed nature was orderly reflecting the mind of a single Creator and understandable to the mind of man who was entrusted with its keeping who thus needed to study and quantify it.


There was a miracle described in the Book of Joshua where the Lord lengthened the day to aid the Israelis in a battle.  But whether or not this was allegorical had never been an official teaching of the Catholic Church.  The difference between an Earth centered or Sun centered system meant that the relative motion of one or the other had to appear to stop as they both became stationary with respect to each other.  The Catholic Church has a strict protocol for such official interpretations when it rarely makes them.


The solar year was roughly measurable because one could roughly determine the equinoxes.

Early in his career Copernicus noted the precession of the equinoxes because of earth’s wobble where celestial north rotates in a circle and had advanced some two degrees in two thousand years.

Copernicus taught his ideas to the Catholic Church in Italy around 1500 and in public lectures in Rome.


In 1514, Pope Leo X circulated a letter asking for advice on constructing better models of the heavens and for attendance at a Lateran Council of the Church in Rome to study and defend different concepts.  Eventually Bishop Paul of Fossombrone asked Copernicus to respond and Copernicus forwarded extensive notes on the subject.   Later, Galileo tried to elevate the original contributions of Copernicus by wrongly claiming the Pope had made a direct appeal to Copernicus. [4]


Copernicus’ models of helio-centrism solved several conceptual difficulties.


1.      Because one could now calculate the relative distances of the planets, the change in their apparent brightness was naturally explained and no longer in conflict with the geocentric models.


2.      The observation that the inner planets were never far from the sun and that the outer planets demonstrate retrograde motion was naturally explained rather than being assumed. 


3.      The requirement that the outer planets all rotate in synchronization on their epicycles and which is the same exact angle between the line joining the sun and the epicycle center of the inner planets, and that the period of rotation is exactly one earth year, are naturally explained rather than being an ad hoc assumption.  He noted Ptolemy’s system with the epicycles of the outer planets all moving in lockstep was equivalent to assuming the sun was stationary and the planets all rotated around it.


4.      This model described a more “realistic” system of rotating celestial spheres without the non-physical equant.    There was no easy explanation for the forces involving an “equant.”  No explanations of origins of forces necessary for moving the celestial sphere in non-uniform manner.


The culmination of this effort eventually was the Gregorian calendar after Pope Gregory XIII, who introduced it in October 1582.


The resurrected a much older idea of Aristarchus which was seriously considered by Aristotle and many others but rejected because


1.      There was no evidence for “winds” from the shearing force of the Earth’s rotation.  This was because of an assumption that “air” permeated all of space and calculations indicated the Earth would have to be spinning at thousands of miles an hour and revolving around the sun at many miles per second.


2.      Objects dropped which fell towards the center of the earth did not curve in their downward paths.  There was no sense of centripetal force on a spinning Earth.


3.      There was no observation of stellar parallax which would have been necessary if the stars were roughly the same size and brightness as the sun.


4.      Heliocentric models, including the best of them by Copernicus, were more complicated than that of Ptolemy.  Copernicus initially had 34 circles and his later work, refined with the mathematical books of his young friend, expanded on that in the last two years of his life to 45.  Ptolemy had about 12.  Heliocentric models up to that time were ALL noticeably less accurate than Ptolemy. Indeed, today all modern state of the art planetariums use the Ptolemic system to display planetary motion and not the model proposed by Copernicus because of inaccuracies. 


Copernicus was reluctant to publish for many reasons


1.      Copernicus made perhaps 100 original measurements of which 27 survive in the written record but they were not more accurate than previous tables just a longer baseline.  His measurements were inaccurate.   The deviations from perfectly circular motion are small and require great accuracy in measurements to be able to distinguish between different models.  Tycho actually purchased Copernicius’s instruments and doubted they could measure anything within several degrees.  He recorded his amazement at that fact.


2.      Ptolemy was the most accurate model and no improvements since improved on it.


3.      Copernicus added many epicycles but didn’t get better accuracy.  He stared with 34 cycles in an initial papter.  But in the last two years of his life, was exposed to more mathematics and greatly improved his accuracy by increasing the circles and epicycles to 45.  


The contributions of Copernicus were


1.      His real contribution was that he reanalyzed Ptolemy and set forth the mathematical tools necessary to refine the estimates of eccentrics and the size of deferents.  Ptolemy had optimized the geocentric.   Copernicus optimized the heliocentric.   Copernicus used the long baseline since Caesar and Ptolemy to describe methods to estimate the length of the year and the precession of the equinox.   He described for future mathematicians the mathematics of the Greeks lost because of Islamic invasions.  The net result was that Copernicus could explain the uneven speed of planets and the sun and moon without equants although less accurately.   But Copernicus considered more realistic as it eliminated the “equant” which was thought to be a mathematical but physically inexplicable artifice.

2.      Copernicus described how to calculate the precession of the axis of the earth.

3.      Copernicus derived some new formulae for spherical trigonometry.

4.      Copernicus laid out all the mathematics necessary for improving the calendar.

5.      Copernicus didn’t necessarily want the Sun to be at the center but needed to do that to remove the equant which he thought to be a mathematical contrivance and not physically realizable.  He introduced more epicycles than Ptolemy had (that is 34 in total), did not make the calculations any easier, and got slightly less accuracy than Ptolemy.  He relied on uniform circular motion and DOUBLE epicycles.   Many more epicycles than Ptolemy but amount the same amount of work to calculate. 

6.      Relied on tabulated observations by others.  Rather Copernicus studied the Algamist by Ptolemy and attempted to simplify and improve the calculations.

7.      Copernicus rejected the non-uniform motion of Ptolemy which was wrong and a step backward.





He circulated a summary of his ideas in a pamphlet “Commentarious” to many other astronomers and Catholic Church officials who clamored for a more complete and extensive publication in their attempts to correct the calendar.  Later in life he was reluctant to publish, despite wholesale support by fellow astronomers and the Catholic Church, because he thought it was not yet finished and inaccurate.


He received official permission to publish in the form of an “Imprimatur” from the Catholic Church and dedicated it to the Pope. “On the Revolution of Heavenly Spheres” in (1543).

The local Catholic Bishop Thiedemann Giese who was also a professor of mathematics and Copernicus' pupil, Joacim Rheticus both strongly encouraged him to publish his ideas.


His problem was a lack of accurate observations.  Copernicus was strongly supported only by the Catholic Church. The lifelong concern of Copernicus was not religious controversy but rather scientific accuracy.  The printing of Copernicus’s work was large by contemporary standards as nearly 500 copies of the original survive today.  It was widely circulated among astronomers across all of Europe.


A Lutheran asked to help with preparing Copernicus’ work, added an unauthorized preface which claimed this was a mathematical and not a real model.  The local Catholic Bishop so violently disagreed with this that he formally asked the civil authorities of Nuremberg to require the publisher to retract and republish without the preface.  In addition, Copernicus’s younger friend wrote strong letters of disagreement and crossed out the preface of every copy he could get in red ink.




His ideas were strongly condemned by the burgeoning Protestant movement especially Martin Luther (1483-1546 A.D.), Phillip Melancthon (1497-1560 A.D.), and John Calvin (1509-1564 A.D.). 

"There is talk of a new astrologer who wants to prove that the earth moves and goes around instead of the sky, the sun, the moon… The fool wants to turn the whole art of astronomy upside-down."

Elsewhere Luther refers to Copernicus as "a fool who went against Holy Writ" which in particular is Joshua 10:10-15 in which the sun seemed to stand still and aid the battle in favor of the Israelites.

John Calvin also condemned Copernicus.


Andreas Oisander (1498-1552) was one of the first and most ardent supporters of Martin Luther.  Against the express wishes of Copernicus, a Bishop of the Catholic Church, and Copernicus’ young admirer and assistant Georg Joachim Rheticus, he inserted an unauthorized preface stating the Copernicus’ helio-centric model was a mathematical artifice not reflecting physical reality.




Bruno (1548-1600 A.D.) was a self serving demagogue who claimed to be able to work magic spells and eschewed the scientific method.  His famous quote was that nature was inexplicable and that geometry and mathematics cannot be used to explain the movements of the heavens.   He claimed Jesus Christ was a magician, and not divine, who had discovered magic chants and potions and used them to fake his miracles which anyone with enough learning and intelligence could duplicate.


He claimed visions of other worlds filled with people like us beamed to him from the Egyptian god “Thoth” and spread throughout an infinite cosmos.  The difficulty with this was not cosmological but rather theological because, as he claimed in voluminous publications, this meant Christ’s sacrifice for humanity here on Earth had no meaning whatever.  For his whacky views he was defrocked from the Catholic Church, later excommunicated and charged with heresy by the Lutherans, and finally excommunicated and charged with heresy by the Calvinists forcing him to hastily flee each time.  He was finally burned at the stake in Rome for spreading social unrest and by refusing to recant during the Protestant Revolution despite nearly seven years of warnings.




Johannes Kepler (1571-1630 A.D.) noted that about 5% of the First Book of Copernicus was devoted to the heliocentric hypothesis and 95% to a critique and revision of Ptolemy’s calculations.  His quote on Copernicus is that “Further, while he strives to outdo Ptolemy in the uniformity of motion, he is in turn outdone by him by the perfection of the planetary path.”



and that Copernicus would “better have emulated nature than Ptolemy.” 


Kepler was able, by adding yet more epicycles, to both the geocentric and heliocentric models to predict the position of Mars to within about 8 minutes of arc.   These were the most accurate yet developed and easily within the margins of all prior measurements.   But Tycho Brahe had made the most accurate naked eye measurements ever done by humanity even to the present day.  And Tycho’s measurements were accurate to one arc minute or about 1/60th of a degree.  It was only with this precision, and on the most extreme movements of all the planets, namely Mars, that Kepler went on to postulate two epicycles for each planet, i.e. effectively an ellipse, with the Sun off center from the cosmos.





Galileo Galilei (1564-1642) claimed to be able to prove the heliocentric system but then could not.   He falsely claimed to have invented the telescope and tried to sell the idea as his own.   He was also a correspondent of Kepler and incorporated much at that thinking into his writings.


He started to teach a revision of sacred scripture.   He was not so much punished for his science, imperfect as it was, as for usurping the right to reinterpret scripture.

After he ridiculed the Pope in print and not in Latin but in the vernacular of the time, he was put under house arrest and his writings and even reference material were put on a restricted list requiring special approval to read.  This did not, however, stop Catholics from pursuing science which continued unabated sponsored by the Catholic Church which invented it.


Galileo was among the first to use the telescope to study the heavens.  He noted that Venus went through phases of partial illumination exactly like the moon.  Because Venus was sometimes fully illuminated, meant that it had to be on the other side of the sun as seen from the Earth.  This meant that and the standard geocentric models were wrong.




Sir Isaac Newton (1642-1726 A.D.), almost single handedly, invented modern physics.




Albert Einstein (1879-1955 A.D.) revolutionized the theory of gravity.




Edwin Hubble (1899-1953 A.D.) discovered the universe is expanding.  He removed the solar system from the center of the cosmos.














3.      Technically speaking, this is where the plane of the ecliptic intersects the plane of the Earth’s equator the tilt between the two varying from 22.1 to 24.5 degrees with a current value of 23.4 degrees. 


4.      “Galileo’s Miss-Statements about Copernicus’, Rosen, Edward, Isis 49, no. 3, (Sept. 1958), pages 319 ff.


APPENDIX: Epicycles and Fourier Analyses