ELLIPSES ARE EPICYCLES

INTRODUCTION

For those interested in the classical debates over planetary motion, it may be surprising to know that ellipses are actually epicycles.  That is to say, every ellipse can be constructed from an epicycle.   But while some epicycles are ellipses, others are not.   The significance of all this is profound and mostly misrepresented in modern popularizations.

Early observers of the night sky noticed that the sun, moon, and five visible planets all move across the constellations of fixed stars in the same direction and along the same circular path called the “ecliptic”.   What mostly distinguishes them are their relative speeds and brightness.  But sometimes they individually stop and move backwards for a short time before resuming their normal travel along this narrow highway in the sky.

Epicycles (literally “on the circle” in Greek) were originally invented in classical Greece c. 200 B.C.  to account for this “retrograde” motion of heavenly bodies.   In their geocentric, or Earth centered, model, two circles rotating at constant speed provides a fairly good approximation.   This was possible only after the stunning intellectual development of axiomatic geometry compiled in Euclid’s Elements about a century earlier c. 300 B.C. which remains a little changed cornerstone of mathematics.

The advantage of epicycles, like sine waves derived from circular harmonic motion in the Fourier Transform, is that circles are mathematically tractable in a simple and straightforward way that no other method or shape can approximate.   But more than that, any orbit regardless of complexity can be modeled to any degree of accuracy by successively adding more and more stages of uniform circular motion.  That the heavens appear to nightly rotate about the earth with the fixed stars attached to a spherical shell was yet another inducement to assume uniform circular motion.

Nevertheless the calculations were complicated requiring extensively tedious computation which alone precluded considering anything other than uniform circular motion.  Indeed, the earliest astronomical texts were notable for their extensive tables of circular cords (sine function in trigonometry) to aid computations.

Modern secular and especially atheist bigotry, against the reasonableness and logic of religious tradition, remains among historical revisionists who attempt to deny these facts, even today in the 21st century.   That ellipses are really epicycles is demonstrated as follows. .   But rather than this being a refutation of uniform circular motion, necessitated by some nebulous and mostly non-existent religious superstition or cultural mythology, it rather is a result of the fact that any repeating path can be mathematically reduced to a series of epicycles.

In the modern heliocentric, or Sun centered, model described by Copernicus and refined by Kepler, they rejected the non-uniform circular motion introduced by Ptolemy which had been the gold standard of predictions unchanged for nearly 1500 years returning to uniform circular motion.    Indeed having the sun,  moon and planets revolve around the sun in ellipses is perhaps the simplest path describable by compound additions of uniform circular motion.

ONE SPECIAL EPICYCLE

Consider the special epicycle described below.  A large circle of radius r1 rotates counterclockwise.   A smaller circle of radius r2 is attached to the larger circle and rotates at the same rate as the larger one but in the opposite, or clockwise, direction.

What is interesting is that any point on the smaller circle will trace out an ellipse.   But there is one point, shown above as a dot, whose ellipse is centered on the x-axis.  Another special point, on the opposite side of the smaller circle, will  trace out the same topological ellipse but rotated by exactly 90°, that is along the y-axis.

EQUATIONS OF MOTION

As previously mentioned, both circles rotate at the same rate making one complete revolution with a period of “T” seconds.   The angle with respect to the x-axis in radians as a function of time “t” is given by

Angle   =  w t  =  (2 π /T)  t

We can then calculate the general (x,y) coordinates of the planet shown above using the following diagram:

x = r1 cos(wt) + r2 cos(wt) = (r1 + r2) cos(wt)

y = r1 sin(wt)  - r2 sin(wt)   = (r1 - r2) sin(wt)

or rearranging

We can square each quantity and add the two equations to get

or if we note that the minor “y” and major “x” axes of the ellipse are

a = (r1 - r2)

b = (r1 + r2)

then finally

which is the well-known equation of an ellipse.

EQUATION OF AN ELLIPSE

The geometric definition of an ellipse is those points which have a constant sum of distances to two foci, which are at coordinates (c, 0) and (-c, 0), as shown below:

From the definition of an ellipse we have for any point (x,y)

We can solve for the constant term by considering the point on the major axis, x, at (b,0) where

And if we consider the point (0, a) on the minor axis, y, we have from two identical right triangles

For the general point (x,y) we have

which we can rearrange and then square both sides as follows

And again rearranging terms, dividing by four, and then squaring we get

We see that the minor “y” axis length of “a” appears on both sides of the above equation, so substituting and normalizing we get

which again from different considerations is the same well known equation of an ellipse.

HISTORICAL COMMENTARY

Starting in ancient Mesopotamia c. 3000 B.C., a cottage industry arose to predict future planetary positions as well as to determine past arrangements at the moment of birth for royal personages.   But these were mostly limited to compiling extensive tables of observation and arithmetically noting repeating patterns.

Epicycles were first used to describe the motions of the moon, sun, and planets by Apollonius of Perga at the end of the 3rd century BC.

Later, epicycle parameters were refined by Hipparchus of Rhodes and then by Ptolemy in his 2nd-century AD astronomical treatise the Almagest.  It is from this work that such orbits are commonly called Ptolemaic though the concepts were actually advanced almost five centuries earlier.  Of note is the fact that the Earth was not at the absolute center of the universe.  Rather the center was “deferred” or displaced by some distance and by a different amount for each heavenly body.

Epicycles approximated the motions of the five planets known at the time within the rough limits of naked eye observation.  They also explained the slight but noticeable change in the apparent distance of the moon as seen from Earth.

Ptolemy also found it necessary to remove the earlier assumption of uniform circular motion by inventing an “equant” which was a displaced center from the center of motion.  This gave much better agreement with observation but was very difficult to calculate.   The modern thinking is that uniform circular motion (of two epicycles forming an ellipse) forms the true trajectories of planetary motion.  Atheists revising history and modern mathematics typically get this exactly backwards.

In 1542, Copernicus, who had taken minor Holy Orders in the Catholic Church, to include a vow of chastity observed more in the breach, noted that his calculations to determine the date of Easter were simpler (because all motion was uniform) if one assumed a sun centered rather than an earth centered arrangement.  Copernicus was reluctant to publish because his improved circular orbits, with a significant increase in the number of epicycles, were not more accurate than the simpler geocentric system of Ptolemy.   But considering its lifelong sponsorship of his work, the insistence of his superiors in the Catholic clergy on publication, especially as the Pope had asked for his assistance to correct the calendar and had praised his initial heliocentric models, finally overcame his reluctance.  [1]

This revived a well known earlier speculation, put forward for similar reasons by the early Greek astronomer Aristarchus of Samos (c. 310 – c. 230 BC) who was also said to have postulated a sun-centered solar system.

Galileo was an early admirer of the Copernican system as was the reigning Pope Urban VIII who was a mathematician and had earlier defended Galileo’s ideas and his right to publish.  Unfortunately Galileo came to believe his fame as an early, even revolutionary, astronomer gave him license to ridicule the current Pope in print for his own aggrandizement and to boast that his speculations had solid observational and theoretical foundation.  But while annoying, this alone was not considered sufficient to forbid his speculations.  In particular, the heliocentric theory of Copernicus had received the formal approval of the Catholic Church in the form on an “Imprimatur” some seventy three years earlier.

Unfortunately, Galileo had no such observations or calculations and his challenge to Church authority was punished only when he began to demand that the Catholic Church make a formal interpretation of sacred scripture supporting his theories.   In particular, he noted that the Catholic Church had never taken a formal stand on the miracle of Joshua where the sun supposedly stood still in the sky.    He then demanded that the Pope declare that verse an allegory supporting a moral truth rather than a literal fact the result of which would be to endorse his ideas.

The difficulty was that despite careful observation and scientific experiment, neither the required stellar parallax nor any centrifugal force appeared to exist.   The suggestion that perhaps the stars were too far away, even though correct, seemed too much an ad hoc excuse.  Nor was there any known framework to calculate the miniscule kinetic forces operating at the surface of a spinning earth.   In fact, Galileo’s fervent belief that the rotation of the earth was proved by the tides, is now known to be absolutely incorrect.  In the time of severe attacks on long standing Christian theology by mere men such as the Protestants Martin Luther and John Calvin and others, who were literally re-writing their versions of the Bible, Galileo and his source material were put under restrictions.

Interestingly, assorted musings and speculations aside, the first solid evidence for a sun centered solar system was not experimental but theoretical.  In 1609 and 1619 by analyzing the meticulous observations of his mentor, Tycho Brahe, Kepler finally made the definitive discoveries that the orbits of planets are best described as ellipses, i.e. special cases of epicycles.  It was however necessary to put the sun “off-center” at one of the foci of the ellipse in a similar manner in which the Earth was off-center in all previous models.

This conceptual breakthrough allowed extremely accurate predictions of planetary motion even when compared with the best measurements made with modern telescopes and lead to its adoption by all Western scholars to include the Catholic Church, which as previously noted, invented and has always sponsored modern science.   Protestants, however, took hundreds of years afterward to even accept the resulting improvements of the calendar by Pope Gregory.

Despite widespread acceptance, experimental evidence for the stars being at great distance and the forces due a spinning earth being miniscule were a long time in coming.  It finally arrived with a report by Giuseppe Calandrelli of Italy who observed stellar parallax in the star α-Lyrae and in 1838 by Friedrich Bessel of Germany with the first quantitative measurements on the star 61-Cygni.  The first physical demonstration of the kinetic forces for a spinning earth was the Foucault pendulum introduced in Paris in 1851.   Unfortunately these validations of earlier insightful speculation only came centuries after Aristarchus of Samos, Copernicus, Galileo, and others had long passed.

It is thus an entirely false criticism of early efforts to decipher planetary motion that there was some inherent religious bias that stymied intellectual inquiry.   Rather the Catholic Church went to great lengths to sponsor astronomical studies and in the process invented the scientific method.   At the time of Kepler, the best models of the day contained tens of epicycles tweaked to fit ever more accurate observations.  Kepler’s insight, which had escaped both Copernicus and Galileo whose models were more complicated and less accurate than Ptolemy more than 1400 earlier, was that only two epicycles, but the right two, were sufficient.

Despite widespread modern acceptance, a stationary sun and rotating earth apparently remains a hard pill to swallow because it is so contrary to common sense appearance.

REFERENCES

1.      Repcheck, Jack, “Copernicus’ Secret”, Simon and Schuster, (2007), pp xiv, 156.

Despite what many simply assume to have been true, a detailed examination of Copernicus’s letters and the history of Christian Churches who held a passionate regard for the advancement of human knowledge, the Catholic Church rather than objecting, actually encouraged Copernicus to publish.

“Copernicus … made no move to finish it or submit it to a publisher, despite strenuous urgings from friends and colleagues in high places.  He was not afraid of being declared a heretic, so many assume; rather, he was worried that parts of the theory were simply wrong, of if not wrong, incomplete.”

2.      xxx