The Greek astronomer Ptolemy’s Almagest was published around the year 145 in Alexandria, Egypt, while the Polish cleric Copernicus’s On the Revolutions (which we will simply call ‘Revolutions’ here) was published in 1543 from Nuremberg, Germany. Ptolemy’s vast synthesis was made possible by two thousand years of systematic observations carried out in Iraq, Syria, and Greece. Copernicus’s monumental analysis became possible thanks to fifteen hundred years of precise observations across India, China, Europe, and the Islamic world. In the five-thousand-year history of astronomy, these two books hold an exceptional place, as their social impact has been the greatest. Perhaps it is because Copernicus’s book was named ‘Revolutions’ that today the word ‘revolution’ has come to mean upheaval—the ‘Revolutions’ of Copernicus gave birth to the greatest scientific and social revolution in history. In this article, we will explore how, from the Almagest to the Revolutions, humans have tried to build a “working model” of the universe, and how the model of the Revolutions ultimately proved itself.

To begin the story of the ‘Almagest’, we have to start with another book: the astrology text ‘Enuma Anu Enlil’ (meaning ‘When Anu and Enlil’) written in Babylon, Iraq, a thousand years before Jesus. This work drew upon nearly two thousand years of the astronomical heritage of Iraq. ‘Enuma’ is a book of omens, meaning portents. Unlike the Greeks, the Babylonians’ focus in astrology was not to discover human fate, but to interpret omens. They identified certain regular events in the sky that could be classified as omens. They believed that if these omens could be understood in advance—that is, predicted—then proper rituals and offerings could avert impending misfortunes.

It was because the authors of ‘Enuma’ closely observed the motions of the Moon, the Sun, and the five visible planets relative to the fixed stars over at least seven centuries that the first cosmological model could be developed—the ‘Almagest’ being its ultimate culmination. They were concerned with reconciling the twelve lunar months (354 days) with the solar year (365 days). Six hundred years before Jesus, the Babylonians discovered the Metonic cycle—realizing that 235 lunar months (29.5 days each, totaling 6,932 days) are nearly equal to 19 solar years (365.2 days each, totaling 6,938 days). And since there are only 228 months in 19 years (which is 7 months fewer than 235), they determined that 7 extra months needed to be added across 19 years, meaning 12 years would have twelve months, and 7 years would have thirteen months.

The practice of creating ephemerides—tables listing the past and future positions of the Moon, Sun, and planets—began in Babylon. For this, they not only measured angular positions but also began calculating velocities. For example, above is a graph showing the change in the Sun’s angular velocity over a year, based on data from a Babylonian tablet dated to 132 BCE. Relative to the background of “fixed” stars, the Sun moves 360 degrees in a year. The graph shows how much distance it covers each month as velocity. We see that in the second month, the Sun’s angular velocity decreases to a minimum (around 28 degrees/month), and then increases to a maximum (30 degrees/month) in the eighth month. For six months, the velocity decreases; for six months, it increases.

They had two methods to explain exactly how the velocity changed. In one method, it was assumed that the Sun’s velocity remained constant for half the year, then shifted once, remaining constant again for the other half. In the other method, it was proposed that the Sun’s velocity steadily increased over half the year and steadily decreased over the other half. Humans first applied calculations of displacement, velocity, and acceleration properly to the Sun. Likewise, the displacement, velocity, and acceleration of the Moon and the five visible planets were also calculated by the authors of ‘Enuma’. However, building a cosmology—a complete model of the universe—based on these calculations happened later, in Greece.

Babylonian astronomers emphasized arithmetic, while the Greeks emphasized geometry. Since the moon, sun, and planets actually follow geometric paths, Greek predictions were easier. Analyzing all the data, the Greeks created a geocentric model of the universe. The first major contribution to this model was made by the Greek philosopher Anaximander from the city of Miletus in Turkey. According to him, the Earth is a cylinder fixed at the center of the universe, with humans living on one flat surface; everything in the sky revolves around it; the sun is the farthest, followed by the moon, then all the stars, and the five planets closest to the Earth.

Pythagoras’ followers were the first to firmly establish that the Earth is a sphere, not a cylinder or any other shape. Aristotle beautifully described this proof through lunar eclipses. During these eclipses, the Earth’s shadow passes over the moon, and the moving edge of the shadow remains round, which is only possible if the Earth is spherical.

By the time of Plato and Aristotle, it was established that there are seven wandering planets around the spherical Earth (moon, sun, Mercury, Venus, Mars, Jupiter, Saturn), and all these are enclosed by a vast sphere in which all the stars are fixed; we can call this the star sphere. The star sphere and all the planets revolve around the Earth once every 24 hours. The star sphere has no other motion; the position of one star relative to another never changes. But the planets revolve at different speeds relative to the star sphere throughout the year. The five planets sometimes stop in their paths, move backward for a few days, and then start moving forward again.

Plato could not accept the motion of the seven planets, especially the strange retrograde motion of the five planets. He tasked his students with proving that the motion of the planets is uniform (constant speed) and circular. The first attempt to prove this was made by Eudoxus from the city of Cnidus in Turkey. His model is shown in the animation above. The planet revolves along the equator of the red sphere, and two rods extending from the poles of this sphere are attached to the outer green sphere. Both spheres rotate, their centers are the same, but their speeds and axes of rotation are different. As a result, the planet moves in a path like the English number ‘eight,’ which sometimes makes it appear to move backward from Earth. This path, resembling a horse’s hoof, was named the hippopede model. For each planet, two more spheres had to be imagined, one for diurnal motion and another for annual motion. Each planet had four spheres, the moon and sun had three each, and the star sphere had one, making a total of 27 spheres. However, Plato’s principle was followed here, explaining everything with uniform circular motion (UCM).

The realist Aristotle could not accept this mathematical model. He first considered the eight spheres shown above, excluding all other spheres. The outermost star sphere (number 8) transmits its 24-hour diurnal motion to all the inner spheres, so no other sphere is needed for diurnal motion. Eight spheres (seven planetary spheres and one star sphere) are sufficient for position and diurnal motion, but additional spheres were needed inside each planetary sphere for annual motion and retrograde motion. Therefore, Aristotle added many more spheres, but unlike Plato and Eudoxus, he was not willing to add anything just for the sake of mathematics; he added what he considered physically real.

For astronomers, this ideal system of philosophers became a burden, whether it was Plato’s geometric or Aristotle’s physical system. Astronomers needed precise predictions the most, which could not be achieved with any specific model at that time. Moreover, explaining everything with circles could not explain why the brightness and size of the planets change. The moon and sun sometimes appear larger, sometimes smaller. The brightness of the five planets sometimes increases, sometimes decreases. The only reason for this could be the change in their distance from the Earth. But if the orbits are circular, the distance can never change.

Due to these problems, Hellenistic Greek astronomers (after Aristotle) began to focus on Babylonian arithmetic calculations rather than geometric systems. Practicality and predictability are paramount in astronomy. Data obtained from observations is more useful than geometry-based models. This is why Aristarchus’ heliocentric model was not popular among the Greeks. Aristarchus believed that the Earth and all other planets revolve around the sun, but this motion does not change the apparent position of the fixed stars because the distance of the stars is much greater compared to the size of the Earth’s orbit. These ideas would be echoed by Copernicus fifteen hundred years later, but at that time, the Greeks needed better observations and predictions, not models. Interestingly, a Babylonian (Seleucus) was the only one who tried to believe and prove Aristarchus’ heliocentric model.

Although Eudoxus’ spheres were not accepted, astronomers started working from Aristotle’s geocentric model and did not dare to abandon Plato’s UCM. About two hundred years before Jesus, Apollonius introduced two new concepts to explain all types of motion of the seven planets using the geocentric model. First, while keeping the orbits of all planets circular, he moved the Earth slightly away from the center of the circle, naming this new eccentric circle the deferent; this explained the changes in the speed and size of the planets. Second, instead of revolving the planets directly on the deferent, he placed them on another circle called the epicycle, and the center of the epicycle was placed on the deferent; the planet revolves on the epicycle, and the center of the epicycle revolves on the deferent. This explained retrograde motion. How this works is shown in the video below.

If the speed of the planet is significantly higher than that of the epicycle, then when the planet moves inside the deferent, it will appear to move backward from Earth for some time. Although Apollonius created the model, we do not know if he actually used it to calculate the motion of all the planets, as none of his writings have survived. However, in the next generation, Hipparchus did exactly this. Through Hipparchus, the Greeks’ geometric model began to benefit from Babylonian observations. Hipparchus used the epicycle and deferent to model the moon’s orbit, but he determined the parameters of the size and speed of each circle using Babylonian eclipse records.

The careful integration of Apollonius’ geometry with Babylonian arithmetic by Hipparchus can be understood with an example. From Babylonian records, Hipparchus knew that there are 4,267 synodic months in 126,007 days and 1 hour, during which the moon returns to the same speed 4,573 times and to the same point on the ecliptic 4,612 times. Those who had data for a hundred thousand days must have had a great heritage of sky measurement. However, from Hipparchus to Ptolemy, almost three hundred years passed without any significant work in the Greek world. During that time, much work was done in India. Due to Alexander, a connection had also begun between western India and Greece-Macedonia.

Around 150 AD, Ptolemy published a book called ‘Mathematical Compilation,’ which became known as ‘Almagest’ (The Greatest) due to medieval Muslims. This book synthesized all the work of Apollonius and Hipparchus to create predictive models of the motion of all the planets (including the moon and sun). These models also included some original work by Ptolemy. For example, Ptolemy was the first to add a point called the ‘equant’ (shown in the image above). The equant is exactly as far from the center of the eccentric circle as the Earth is, but in the opposite direction. A planet’s speed in its orbit must be such that its apparent speed relative to the equant is always constant. This constant speed was necessary to preserve Plato’s UCM. The planet will move slowly when far from the equant and quickly when close, so the apparent angular speed will never change. Medieval Islamic scientists were very concerned about this strange point called the equant.

This book also first mentioned the order of distances from the Earth to the moon, sun, planets, and stars. Aristotle and Eudoxus had said five hundred years before Ptolemy that the star sphere is the farthest, and Ptolemy did not change this. He determined the positions of the seven planets based on their periods. The longer it takes for a planet to orbit the Earth, the closer it is to the star sphere. According to this, Saturn comes right after the star sphere (period 21 years), followed by Jupiter (12 years) and Mars (2 years). The moon is placed closest because its period is only 29 days. Three more planets remain: the sun, Mercury, and Venus. Determining their order was difficult because they always stay close to each other and seem to orbit the Earth in about one year. Ptolemy decided to place the sun in the middle, with Mars, Jupiter, and Saturn above it, and Venus, Mercury, and the moon below. There was no logic for why Venus was placed before Mercury.

Another significant contribution of Ptolemy was measuring the size of the universe for the first time. He thought there was no gap between the spheres of one planet and the next. Therefore, by calculating the thickness of each sphere, he found the total distance from the Earth to the star sphere to be 19,865 times the Earth’s radius, or 121 million kilometers. This was, according to him, the size of the cosmos. This number is even less than the distance from the Earth to the sun (150 million km). Nevertheless, Ptolemy’s work must be given importance. He was the first to show how incredibly larger the size of the universe could be compared to everything familiar on Earth.

The Polish priest Copernicus studied in Renaissance Italy and lectured on astronomy in Rome in 1500. He grew up studying ancient Greek and modern (in his time) Islamic knowledge. His main issue with the precise predictive system of the Almagest was that it was like a chimera. A chimera is a mythical creature made by combining parts from different animals. According to Copernicus, Ptolemy’s system was similar, with no unified explanation, as the motion of each planet was explained differently. This resulted in good predictions but made the universe a monstrous chimera.

Copernicus was inspired by ancient Greek and modern Muslim scholars to destroy this chimera and create a “beautiful” model of the universe. Many ancient Greek philosophers believed that the Earth revolves around the sun. A significant indication of this was that the motion of the five planets inherently included the motion of the sun; the period of each planet could be measured as a multiple of the “year,” which is actually the period of the sun. Copernicus was also aware of Muslim criticisms of the equant. A brief introduction to his heliocentric cosmological model was published in 1539, but the full book, ‘On the Revolutions of the Heavenly Spheres’ (commonly known as ‘Revolutions’), was published in 1543, the year he died.

In the ‘Revolutions’ book, this picture of the universe is presented. From the outside to the inside, respectively, are the star sphere, Saturn, Jupiter (Jove), Mars, Earth (Terra) and Moon (Luna), Venus, Mercury, and at the center, the Sun (Sol). By swapping the positions of the Earth and the sun in the Almagest model, a major problem was solved. Previously, no one understood why, despite the longer periods of Mars, Jupiter, and Saturn, Venus and Mercury had periods similar to the sun’s one year. In the ‘Revolutions’ model, the reason is very clear. The Earth divides the five planets into two groups, three outside and two inside. Since Mercury and Venus are between the Earth and the sun, it appears to us that these two planets also orbit once a year along with the sun. By excluding the Earth’s period, Copernicus was able to measure the actual periods of Venus (7 months) and Mercury (80 days) around the sun. As a result, there was no problem in determining the sequence of the six planets (including Earth) based on their periods. After excluding the Earth’s period from the calculations of all the planets, it was also possible to measure the relative distances of each planet from the sun. However, Copernicus also understood that both the geocentric and heliocentric models could equally explain the universe; their predictive power was the same at that time.

Copernicus’ cosmological vision easily explained the retrograde motion of the planets, as shown in the animation above. Here, the inner blue orbit is Earth’s, and the outer red one is Mars’. When Earth and Mars move in the same direction around the sun, we see Mars moving from west to east in the sky, which is normal prograde motion. However, since Earth’s orbit is inside Mars’, Earth overtakes Mars once a year. Just as when my bus overtakes another bus, it seems to me that the other bus is moving backward against the background of the road, similarly, when we overtake Mars, it appears to move backward from east to west against the background of the stars. This is the reason for retrograde motion, and Copernicus explained the retrograde motion of all planets in this way.

Only the first 5% of ‘Revolutions’ presents this cosmological vision, while the remaining 95% is filled with complex geometric calculations, through which Copernicus tried to prove that in terms of predictive power, his beautiful model was at least equal to, if not better than, the chimera of the Almagest. And this was all Copernicus could say, as he had no definitive proof of the heliocentric model.

We had to wait until the eighteenth century for definitive proof. However, in the meantime, most astronomers began to believe in a heliocentric world without definitive proof because this model was more elegant. The greatest observational astronomer of that time, Tycho Brahe, could not believe it. With royal patronage, Brahe built the largest observatory in history on the island of Ven in Denmark (now Sweden), first Uraniborg in 1576, then Stjerneborg in 1584. Using large sextants and mural quadrants, Brahe measured the positions of over 700 stars with a precision of about 0.5 arcminutes. This rendered the data from the Almagest obsolete for the first time in history.

Every astronomer has a cosmologist within them. Brahe also created a cosmological model with the Earth at the center, the moon and sun revolving around it, and the five planets revolving around the sun. After losing royal favor in Denmark, Tycho moved to Prague and took a job at another royal court. His assistant was Kepler. After Tycho’s death, Kepler used his data to model the entire orbit of Mars. In doing so, he realized that it was impossible to reconcile the calculations if the orbit was circular; the orbit was actually elliptical. This is now known as Kepler’s first law. Battling with the god of war, Mars, Kepler discovered two more laws.

The second law states that a planet sweeps out equal areas ($A$) in equal times ($t$) in its elliptical orbit. When the planet is farther away, the area is narrower; when it is closer, the area is wider, but the value of the area does not change if it takes the same time to traverse the arc of the area. In the animation above, the purple area is always equal. In the language of calculus:

$$ \frac{dA}{dt} = \frac{J}{2m}$$

where $J$ is the angular momentum of the planet, and $m$ is its mass. This means the right side is always constant. The third law states that the square of a planet’s period ($T$) is proportional to the cube of its average distance ($r$) from the center of its orbit, which Newton used in his theory of gravitation as follows:

$$ T^2 \propto r^3 \Rightarrow \left(\frac{r}{v}\right)^2 \propto r^3 \Rightarrow \frac{v^2}{r} \propto \frac{1}{r^2} \Rightarrow a_c \propto \frac{1}{r^2} $$

where $v$ is the velocity of the planet and $a_c$ is its centripetal acceleration. When a stone tied to one end of a rope is swung around the head for a long time and then released, the stone flies away like a slingshot. This centripetal acceleration was first mathematically understood by Huygens of the Netherlands. This centripetal acceleration can thus be called Huygens’ pull. Multiplying the last equation by the mass ($m$) of the planet and using Newton’s second law of motion, the gravitational force ($F_c$) appears on the left side:

$$ F_c = m a_c = GMm \frac{1}{r^2} $$

where $G$ is Newton’s gravitational constant, and $M$ is approximately the mass of the sun. However, the intuition of Newton was greatly influenced by the work of several others, especially Digges, Descartes, and Galileo.

Although Copernicus was revolutionary, he could not abandon two things of Aristotle: he could not eliminate the fundamental difference between the Earth and the sky, and he could not think of the geometry of the universe as anything other than circular. These two ideas were dispelled by Descartes’ ‘Principles of Philosophy,’ published exactly one hundred years after ‘Revolutions.’ However, long before that, in 1576, Thomas Digges of England did a revolutionary work. Copernicus, following ancient tradition, thought of all the stars on the surface of the star sphere, and his universe was bounded by a circle. Digges liberated the stars, thinking that after Saturn’s orbit, there were no more circular orbits, but countless stars like the sun were scattered across infinite space. Digges’ model of the universe is shown above.

In the work of transforming astronomy from mathematics to physics through cosmology, Galileo’s name must be mentioned. Following the Dutch, Galileo made a telescope and first pointed it at the sky in 1610, seeing many unseen stars, mountains and valleys on the moon, the phases of Venus, and four satellites of Jupiter. By observing the phases of Venus, he understood that Venus indeed revolves around the sun, and Jupiter’s satellites (observations from January 1610 are shown above) made him realize that it was possible for something other than the Earth to be orbited in the universe. Although not definitive proof, these two pieces of information made Galileo the greatest propagandist for Copernicus, for which he was sentenced to lifelong house arrest.

By the time Newton published Principia in 1687, there was little doubt among scientists and philosophers about the heliocentric model of ‘Revolutions,’ although it was not yet proven beyond all reasonable doubt. However, it is ironic that when Einstein published his modified gravity through the theory of general relativity, we somewhat returned to Ptolemy and Copernicus, to the Almagest and Revolutions. Because Einstein’s relativity ultimately showed that there is no absolute motion; all motion ultimately depends on the coordinate system. To calculate the motion of the solar system, we can take any point as a reference, whether it is the Earth, the sun, or the center of mass of the entire system.

To understand the first observational proof of the ‘Revolutions’ model, we need to go back to the eighteenth and nineteenth centuries from Einstein in the twentieth century. In the eighteenth century, the direct proof of Earth’s motion around the sun came from measuring the aberration of light. The significant proof of the nineteenth century came from stellar parallax. The amount of aberration is about 20 arcseconds, while parallax is only 1 arcsecond. It took almost a hundred years for telescope precision to improve from 20 to 1 arcsecond, which is why the discovery of aberration (1720s) and parallax (1830s) are about a hundred years apart.

Aberration of light occurs due to the change in Earth’s velocity around the sun. If we assume that light from a distant star is falling vertically on Earth, as shown in the image above, we will see this light bending differently in different seasons. The light ray bends in the direction of Earth’s velocity. In September, Earth is moving to the right, so the light bends to the right; in December, it does not bend left or right as Earth is moving inward on this page; in March, the light bends to the left. Due to this bending, the position of all stars changes periodically. A star’s position increases from its average value in June, reaches its farthest point in September, then decreases back to its original position in December, and then increases again to its farthest point in the opposite direction in March. This simple harmonic motion (SHM) can be modeled with a sine curve. The simple equation to measure its amplitude is:

$$ \alpha = \theta-\phi = \frac{v}{c} $$

where $\alpha$ is the aberration, $\theta$ is the star’s declination in the rest frame, $\phi$ is the star’s declination in the moving frame, $v$ is Earth’s velocity, and $c$ is the speed of light.

One of the greatest astronomers of the eighteenth century, England’s James Bradley, published this change in the position of stars due to Earth’s velocity in 1727. The observations of the aberration of Gamma Draconis (blue curve) and 35 Camelopardalis (red curve) are shown above. It is seen that the aberration of Gamma Draconis is at its maximum of 20 arcseconds in March and September. After Bradley’s discovery, astronomers began to calculate the proper motion of stars by subtracting the effect of aberration caused by Earth’s orbit from the star’s position. This greatly increased the precision of observations.

Interestingly, Bradley actually started his observations to measure the parallax of stars. The ancient Greeks knew that if Earth revolved around the sun, an observer on Earth would see the position of stars in the sky change as shown in the image above. This effect is called parallax. It can also be modeled with a sine curve. If a star is directly overhead, its position will change in a circular path due to parallax, and the closer a star is to the horizon, the more elliptical its parallax path will be. Parallax is directly related to distance. The greater the distance of a star, the smaller its parallax.

From geometric calculations, Bradley knew that due to parallax, Gamma Draconis should be at its southernmost point of its annual path in December, and its position should not change much within a month. But when measuring, he saw that the star continued to move south after December and reached its southernmost point in March, three months late. The reason came to him while boating on the Thames River. He saw that the direction of the boat’s weather vane depended not only on the direction of the wind but also on the direction of the boat’s velocity. Replacing the wind with light and the boat with Earth, it is understood that the direction of a star’s light depends not only on the direction of the star but also on the direction of Earth’s velocity. However, for this, the speed of light must be finite, which Ole Rømer had proven in the 1670s. Therefore, Bradley had no doubt that the position of Gamma Draconis was changing due to aberration, not parallax.

He also understood why the maximum of aberration would be three months before or after the maximum of parallax. Parallax depends on Earth’s position, which is the end point of the radius of Earth’s orbit, meaning it works along the radius. On the other hand, aberration depends on Earth’s velocity, which is tangent to the orbit. Since there is a 90-degree angle between the radius and the tangent, there will be a 90-degree phase difference between them. If a circle’s 360-degree rotation takes twelve months, the 90-degree difference will relate to three months.

Since no parallax was detected in any of Bradley’s observations, the parallax must be less than 1 arcsecond. And indeed it is. As shown in the adjacent image, the parallactic angle

$$ \tan p \approx p = \frac{a}{r} $$

where $a$ is the distance from Earth to the sun, i.e., 1 astronomical unit, and $r$ is the distance to a star at point $S$. Substituting the distance of the nearest star, Proxima Centauri, which is 4.2 light-years, the parallax is only 0.76 arcseconds. The parallax of all other stars will be even less. Understanding this, astronomers did not attempt to measure parallax for a long time after Bradley. Almost a hundred years later, in 1835, Friedrich Struve measured the parallax of Vega to be 1/8 arcsecond, but when further observations two years later found the same parallax value to be 1/4 arcsecond (twice as much as before), many were unwilling to believe Struve’s discovery.

In Germany, Friedrich Bessel began trying to measure the parallax of the star 61 Cygni in 1834. Encouraged by Struve’s claim, Bessel began more intensive observations and announced the parallax of 61 Cygni to be 1/3 arcsecond in 1838. This was the first effective method for humans to measure the distance to stars.