This article is devoted to highlighting Claudius Ptolemy’s enduring contributions to Western civilization. It is hard to overestimate his influence upon the development of both science and religion. His achievements established paradigms that continue to drive the scientific community.

Despite the immensity of his influence, he remains one of the most underappreciated individuals in human culture. Why? He is primarily associated with the geocentric fixed earth system that was replaced by the heliocentric system first developed by Copernicus. Because of his cultural misconceptions, Ptolemy’s greater achievements are tainted and almost always overlooked. Distracted by surface turbulence, writers fail to see the substance of the depths.

Writing in the 2nd century of the Common Era in the midst of Rome’s Golden Age, Ptolemy’s books represented the culmination of nearly 7 centuries of Greek astronomy and mathematics. Employing techniques of spherical trigonometry that he created, Ptolemy developed a mathematical system that could accurately predict planetary position. The implicit beliefs behind the widely used Ptolemaic system inculcated the internal logic of astronomers and scholars – eventually becoming foundational paradigms of Science.

1) There is a natural order that is accessible to human reason. 2) Mathematics is the language of Nature. 3) Elegant simplicity is a feature of the natural order.

Read on for details.

**Section Headings**

- Ptolemy’s 3 bibles
- Astrology drives Astronomy
- Pythagoras (570-500 BCE): Harmony of the Spheres
- Pythagoras: All Phenomena reduced to Mathematics
- The Problem of the Wandering Stars
- Hipparchus (≈ 130 BCE): Father of Systematic Astronomy
- Ptolemy’s Matrix establishes Paradigms of Science

Aesthetics motivated Copernicus to develop his heliocentric conception of our solar system.^{0} Aesthetic considerations continue to drive the scientific community. Elegant simplicity is the standard. Too much complexity is suspicious. Where did this notion arise?

Copernicus firmly believed that the natural order of the firmament is accessible to human reason. This belief is the basis of the scientific endeavor. Where did this notion arise?

Copernicus employed a mathematical system in his attempt to establish the veracity of a solar system. The scientific community continues to rely upon the symbolic language of mathematics to articulate natural law. Where did this notion arise?

We suggest that the answer to all three questions is the same – the geocentric system developed by Ptolemy. Copernicus’ modern sun-centered system eventually replaced Ptolemy’s antiquated earth-centered system of astronomy. If his system is antiquated, how could it exert an enduring effect upon the scientific community? Why was Ptolemy’s contribution so significant? Working nearly 2000 years ago, how was he able to establish foundational beliefs of science?

It is likely that most ancient cultures believed that the heavenly firmament revolved around a fixed earth. After all that is our common perception. The ground beneath us is firm and the Sun, Moon and stars revolve around our seemingly fixed position – rising and setting on the horizon at fairly regular intervals. So why do they call it *Ptolemy’s* geocentric system?

Claudius Ptolemy was a Greek astronomer who lived in Alexandria under Roman rule during the 2nd century of the Common Era. Alexandria with its amazing library was the center of Western scholarship. Due to his writings, Ptolemy was a driving force in the achievements of this culture. In fact, his contributions to astronomy, mathematics, geography, and astrology were so significant that they dominated Western thought for over a millennium. That is a long time. It is difficult to think of many individuals whose influence has been so long-lasting, especially mathematician astronomers. What was it about his contribution that was so unique?

Working during the social stability of Rome’s Golden Age, Ptolemy wrote three books that were to exert an enduring influence upon Roman, Arabic and then Western European civilization: *Geographia, Tetrabiblos*, and* Almagest*. Each of the books was both a consolidation and an extension of what went before. His integration was so thorough and complete that these three works provided the foundation for geography, astronomy and astrology, at least through the European Renaissance^{1}. Due to their enduring persistence and influence for over a thousand years, they could be considered bibles of their respective fields.

Ptolemy’s *Geographia* was the bible of geography. It consolidated many different sources from a variety of cultures regarding the world’s geography into one book. Due to its thorough and relatively precise nature, it supplanted preceding works. The book provided the source and authority for geographical conceptions until the Age of Exploration over a millennium later. With the intent of capturing the lucrative spice trade, many Europeans embarked on increasingly long global voyages. From practical necessity, international traders began uncovering and reporting some major discrepancies, especially regarding distance. For instance, Columbus believed the globe to be much smaller due to Ptolemy’s geography. Reality tends to dispel misconceptions.

Ptolemy’s *Tetrabiblos* was and continues to be the bible of astrology. The book consolidated Babylonian, Greek and Egyptian astrology into a cohesive system. Like Copernicus regarding astronomy, Ptolemy had been disturbed by the proliferation of complicated theories of heavenly prognostication. His integration was so thorough that it completely replaced the earlier systems and became the authoritative sourcebook on astrology from then until now. In fact, the *Tetrabiblos* is just as relevant as when it was written – nearly two thousand years ago. Ptolemy compares astronomy’s certainty with astrology’s ambiguity, but states that this difference is appropriate as the heavens are ultimately predictable, while human behavior is inherently ambiguous^{2}.

Ptolemy’s *Almagest* was the bible of astronomy. This is the book that concerns us. He entitled it *Mathematical Treatise*. As an indication of its prestige, it was known as the *Great Treatise* during the Roman and Byzantine era. Arab astronomers called it the *Almagest*. As an indication of the importance of the Muslim contribution, this is the name Western civilization has adopted.

The treatise made a significant contribution to both mathematics and astronomy. Ptolemy’s work laid the foundations for spherical trigonometry. Although improved upon, especially by Arab mathematicians, the mathematics was never disproved. Employing the mathematics of spherical trigonometry, Ptolemy devised a method for correctly predicting all planetary positions, something that had never been possible before. In such a manner, the book conclusively established that the heavenly objects move in regular orbits that are ultimately predictable.

Why the ancient obsession with predicting planetary position? Was it simply a scientific curiosity, even if rudimentary, that drove generations of astronomers to pursue this elusive goal? Or was there a more pragmatic motive?

Indeed there was. From time immemorial, humans have looked to the natural world for omens to guide their behavior. The skies have provided most if not all cultures with communication from the divine world – hence the term divination.

Even Genesis, the first book of the Bible, states that on the 4th day of creation, God provided the heavens as a Sign.

“And God said, Let there be lights in the firmament of the heaven to divide the day from the night; and let them be for *Signs*, and for seasons and for days and years.” Genesis 1:14

People generally believed that accurate knowledge regarding celestial omens, e.g. eclipses, would provide a timing advantage. Indeed much prognostication concerned itself with choosing the perfect moment to begin an endeavor. For instance, rulers consulted diviners as to the proper time to attack and retreat in order to achieve victory and avoid defeat.

The belief gradually arose that aligning personal behavior with celestial behavior produced the best results. The regularity of the starry firmament lent itself to precise predictions. Due to this attribution, it became of utmost importance to know exactly what the celestial objects were doing.

The Greeks followed by the Romans were no different than other ancient cultures. Most in the Greco-Roman world looked to the heavens to provide some type of insight into the future. They had a fairly sophisticated knowledge of the heavens, which was based upon a synthesis of Egyptian, Chaldean, and Babylonian astronomy combined with Greek mathematics and mythology.

Initially, for instance when Genesis was written, people generally believed that the heavenly firmament consisted only of the Stars, the Sun and Moon. Most ancient cultures partitioned the stars into constellations. These constellations revolved around the earth in a circular fashion. Mediterranean cultures called this circle of stars the Zodiac. They further divided the Zodiac into 12 parts based upon the season of the year. They named each of these Signs of the Zodiac by the constellation that rose at sunset during its season. For instance, the constellation of Aires was associated with the first third of spring.

By the Greco-Roman period in history, the ancient astronomers had differentiated the 5 observable planets from the stars. While the seemingly circular progression of the constellations was regular like the Sun and Moon, the planets revolved erratically through the Zodiac, moving gradually from one Sign into another. Due to their seemingly erratic, yet predictable, movement, the planets were ideal omens. Diviners could use their behavior to determine what they considered to be auspicious and inauspicious times.

To interpret these celestial omens, the Greeks ascribed mythical meanings to the planets and their interactions. For instance, Mars the planet symbolized the personality of Mars the war god. In similar fashion, the Greeks assigned a god and personality to the rest of the known planets – Mercury, Venus, Jupiter and Saturn. Modern astronomers employed the same convention to name the outermost planets – Uranus, Neptune, and Pluto. Modern astrologers continue the Greco-Roman tradition of ascribing the personality of its namesake god to the planet. For instance, the planet Venus symbolizes love and beauty just like the god.

As mentioned, the planets move erratically through the Zodiacal Signs. Ancient astrologers began assigning meaning to these planetary movements. It was believed that planetary interactions symbolized or even exerted an effect upon the behavior of kingdoms, kings and individuals. For instance when Mars and Jupiter came together in the sky, astrologers might interpret this as a good time to go to battle.

Eventually, Mediterranean astrologers even assigned meaning to the geometrical relationships (aspects) between planets. For instance, it was considered to be an auspicious time if planets were a third of the way around the Zodiac from each other (trine), and inauspicious if they were only a quarter of the way around (square). This matrix for understanding planetary omens has remained virtually unchanged for millennia, from then until the current day.

Due to the divinatory importance attributed to the relationship between for instance Venus and Jupiter, ancient astrologers required careful observations of the evening sky to make their predictions. Watching the relative movement of the planets was easy if the skies were clear. Anyone can do it, if they want to. However, during a cloudy season or in a generally inclement climate, it was impossible to get a divine message from the sky. A cloudy season could block access to these supposedly divine Signs for months at a time.

Due to the importance attributed to observing the nighttime sky, astronomers began attempting to predict relative planetary position. If effective, the predictive process could be employed to circumvent the effects of bad weather. For instance, if an astrologer could predict when Mars or Venus was going to stop and start, he could more easily suggest times for battle or marriage.

It gradually became evident that mathematics is the most precise symbolic language for making accurate predictions regarding planetary position. For the Greeks, mathematics tended to be geometrical. Generations of astronomers gradually created and then extended a geometric matrix to include most of the planets. Standing on the shoulders of these others, Ptolemy expanded this Math/Astronomy matrix to encompass all the planets.

Ptolemy’s *Almagest* completed a process that had begun many centuries earlier with Pythagoras and his school. Pythagoras had applied mathematics to music. Specifically, he noticed that there were strict and precise mathematical ratios between the lengths of vibrating strings that sound in mutual harmony.

Pythagoras applied this same model to the heavens. By his time, Greek astronomers knew that the firmament was occupied by the Sun, Moon, 5 Planets, and Stars. Making inferences based upon observable phenomena, Pythagoras formulated a theory regarding the firmament.

The Pythagorean model included the following components. 1) Each of the 8 astronomical entities occupies a distinct sphere in which they revolved around a fixed Earth. 2) Each entity is a perfect sphere that moves at a constant velocity. 3) The distances between each of the celestial spheres is based upon a similar harmonic relationship. For this reason, the Pythagorean theory is commonly known as the ‘harmony of the spheres’^{3}.

With this theory, Pythagoras established a conception of the universe that dominated astronomy for over 2000 years. The enduring appeal of the model was at least partially based upon the simplicity of the assumptions, e.g. constant velocity around a fixed circle. The Sun, Moon and Stars provided the basis for the paradigm. The trick was getting the Five Planets to fit the pattern. It took centuries of astronomers culminating in Ptolemy to finally provide what was then conclusive evidence that established the relative truth of the Pythagorean model.

However, mathematics for the Pythagoreans and the Greeks in general was geometrical, not dynamical, in nature. Rather than governed by interacting forces, the heavenly bodies behaved in an idealized geometric fashion. Rather than dynamic relationships between them, the celestial objects were perfect spheres that moved at a constant velocity in perfect circles. This geometric paradigm, first formulated by the Pythagoreans, persisted for over two millennia until Kepler reluctantly overturned the model with his Physics of the Heavens.

Pythagoras took his chain of reasoning regarding numerical harmonies a step further. He made an inference that continues to dominate scientific thought as an implicit paradigm. After discovering some mathematical ratios that revealed musical harmonies, i.e. octaves, fifths and thirds, the Pythagoreans suggested that music’s logic applied to the heavens – the harmony of the spheres. Not just the heavens, the Pythagoreans proposed that all phenomena could be inevitably reduced to numerical relationships. Just as mathematics was a significant component of musical logic, they asserted that mathematics had an equally significant role in the logic of the heavens and indeed the logic of all phenomena.

When he proposed his theory, Pythagoras had very little evidence backing up his somewhat outrageous theory. However as the centuries passed, Ptolemy illustrated conclusively that mathematics could accurately predict celestial behavior. And then Newton et al successfully applied mathematics to all material behavior.

Many began to believe that the Pythagorean Math Postulate was correct. Although only material behavior had come under the sway of mathematics, many, if not most, mid 20th century scientists came to believe that living behavior could also eventually be reduced to numerical relations, as Pythagoras had predicted.

Their belief was based upon the following syllogism. 1) Material behavior can be reduced to mathematics, as witnessed by their fabulous success in this department. 2) Because their incredibly precise instruments, e.g. electron microscopes, did not reveal anything else, scientists made the reasonable, although improbable, assumption that Life consists solely of Matter. 3) Therefore Life, as a subset of Matter, can also be reduced to mathematical relationships.

However, as the years turned into decades and approached the greater part of a century, the scientific community has yet to apply the mathematics that has been so successful at predicting material behavior to significant swaths of living behavior, especially human. No matter how hard they try, it seems that Life resists a reductionist approach.

Is it possible that the syllogism is based upon a faulty assumption? Could it be that living systems have a non-material component? Might Attention and Choice be a feature of this non-material component? Could Life have features that are innately resistant to a mathematical explanation?^{4}

The jury continues to be out on living behavior. However, scientists have reduced virtually all material phenomena to numerical relationships. It seems safe to say that Pythagoras was certainly correct regarding material behavior.

What evidence led Pythagoras to conceptualize the heavens with his harmony of the spheres? Lacking supporting evidence, why was the Pythagorean model so persistent? Why did the Greeks hold onto this notion for centuries? And how did Ptolemy confirm this conception of the universe?

It is obvious to almost everyone that the heavens are orderly, while life on earth is chaotic. Even to the untrained observer, the undying Sun and Moon follow regular patterns. We depend on these celestial bodies to rise and set at regular times. The seasons follow one upon the other in regular succession. Day, night, day, night follow each other without interruption. Further the Sun and Moon seem to be perfect circles that move at a constant speed throughout the firmament. Even the stars with their constellations are so constant in their progression across the evening sky that ancient sailors depended on their position to navigate the ocean.

Further, our common sense tells us we live on a fixed Earth. For instance, the ground under our feet is stable and dependable. We look up at the stars in the firmament and they revolve around us on a daily level.

It is easy to see how this observational data led Pythagoras to postulate his astronomical theory, i.e. the heavens consist of perfect spheres that revolve in perfect circles at a constant velocity around a fixed earth. In turn, this regularity leads to the notion that the celestial objects are ultimately predictable.

These conclusions were not based in religious beliefs. Rather Pythagoras applied inductive logic to empirical evidence of an observational nature to arrive at his theory regarding planetary motion. Coinciding with common sense, his theory was both compelling and enduring.

The Pythagorean model definitely applied to the Moon, Sun and stars. However at the time, there remained one major unsolved problem regarding the astronomical model – the planets or wandering stars, as they had been called. While the constellations move together in a fixed relationship with each other, the planets wander through the fixed stars of the Zodiac.

It doesn’t take a genius to notice that there are some celestial objects that don’t move at a constant velocity around the earth. Not only do they speed up and slow down, these wandering stars actually move backwards (retrograde) from time to time. They certainly don’t seem to act in a predictable like the stars.

This erratic behavior is obvious to even the most casual observer of the heavenly firmament. For instance currently (Winter 2019), Venus and Jupiter are going through a planetary dance in the morning sky. As they are brighter than most stars, they catch our attention. Over the weeks, Venus catches and passes Jupiter and then slows down and goes retrograde. Jupiter passes Venus but then slows down to go retrograde. In the meantime, Venus turns around relative to the Zodiac to go direct and will join with Jupiter in the morning sky. All five of the visible planets go through this same backward-forward dance, albeit at different relative speeds with respect to each other. In contrast, the Sun, Moon and Stars never go backwards and instead always move at a constant forward speed through the Zodiac.

For the careful observer, the retrograde motion of the planets appears to occur periodically. The cyclical nature of the planetary motion indicates the possibility of prediction. This became the problem for generations of astronomers – how to predict the back and forth motion of the planets.

With the passage of time, this earlier problem morphed into a more specific form. How to fit the *wandering stars* into the Pythagorean model, i.e. the geometric paradigm? The Sun, Stars and Moon move in perfect circles through the Zodiac at a constant velocity. While the planets move in the same perfect circle, how is it possible to account for their forward and backward motion?

While he provided the ultimate solution to this problem, Ptolemy was part of centuries-long tradition. Ptolemy’s ** Almagest** synthesized and extended Greek astronomical knowledge, especially that of Hipparchus. Most scholars consider Hipparchus to be the father of systematic astronomy. He generated the first star catalog, which included a rating of relative brightness. He also employed geometric considerations to compute the Moon’s size and distance relative to the Earth.

Hipparchus also provided a partial solution to the problem of the planetary retrograde motion. Miraculously, he was able to fit the movements of Mercury and Venus into the Pythagorean model first proposed four centuries prior. He developed an ingenious mathematical system where these planets proceed in perfect circles in a fixed speed about their own orbit – i.e. circles within circles. In such a manner, he was able to make fairly accurate predictions regarding the movement of both Mercury and Venus throughout the Zodiac. Hipparchus’ accomplishment provided crucial support for the assumptions behind the harmony of the spheres.

However the job wasn’t finished. Hipparchus had been unable to account for the retrograde motion of the outer planets – Mars, Jupiter and Saturn. Three centuries later, Ptolemy completed the task by adding yet another internal circle. He developed a mathematical system based in spherical geometry whereby the planets moved at a constant velocity in perfect circles about other perfect circles that revolved around the perfect circles of their orbits around the Earth – circles within circles within circles.

While somewhat complicated, the Ptolemaic system was able to predict planetary position with considerable accuracy. His solution also fit all the planets with their retrograde motion into the Pythagorean’s geometric model. Due to this predictability, it was hard to question the assumption that celestial objects revolve in perfect circles at a constant velocity. Ptolemy’s accomplishment was a definite and major validation of the harmony of the spheres. This Math/Data Matrix regarding the firmament was the primary reason that the Ptolemaic system was so compelling and endured for so long.

Firmly establishing a Math/Astronomy matrix that applied to the entire firmament was probably Ptolemy’s greatest contribution to astronomy in particular and science in general. From this point, Western astronomers knew without a reasonable doubt and were committed to the idea that mathematics could accurately predict celestial behavior. This implicit belief motivated Copernicus, Kepler and onwards to the present.

The Ptolemaic system has a significant, yet implicit, corollary. Nature is confusing to say the least. The ability to predict planetary behavior indicates an incredible possibility: the natural order is accessible to human reason. If humans apply logic to observable phenomena over time, they can actually attain some kind of definitive understanding of this confusion. This belief has been a major driving force for the scientific community.

Ptolemy’s pragmatic matrix was also a major affirmation of the Pythagorean notion that mathematics was at the basis of all phenomena. This astronomical example indicated that mathematics could be effectively employed as a powerful predictive and explanatory tool. Although Pythagoras had suggested this possibility, it was Ptolemy’s *Almagest* that made it a reality^{5}.

Although it took a millennium and a half to take it another step, Ptolemy’s validation of the Pythagorean Math Postulate laid another foundational belief of modern science. In addition to celestial behavior, the scientific community eventually came to believe that mathematics could also be employed to accurately predict the behavior of all phenomena, both material and living. Stated another way, mathematics is the language of the natural order. This implicit belief has driven legions of scientists for centuries and continues to drive them.

There is yet another feature of the Ptolemaic system that the scientific community has adopted: Nature has an elegant simplicity that is revealed by mathematics. An assumption behind the Pythagorean model that Ptolemy proved was that celestial bodies move in perfect circles at a constant velocity. Although astronomers have disproved these particular notions, scientists have retained and established over and over again that natural laws have an economy and simplicity about them. Too much complexity is suspicious. This aesthetic feature of the scientific endeavor was a driving force behind many discoveries, for instance Copernicus’ heliocentric system and Mendeleyev’s periodic table of elements. It was Ptolemy that first validated the scientific belief that elegant simplicity is a feature of the laws of nature.

Summarizing, Ptolemy’s pragmatic validation of the Pythagorean model suggested at least three implicit beliefs that have driven and continue to drive scientists everywhere. 1) The natural order is accessible to human reason. 2) Mathematics is the symbolic language of this order^{6}. 3) Elegant simplicity is a feature of Nature.

We can derive a few lessons from the Ptolemy example. First, even if a mathematical system can make accurate predictions, this does not imply that the assumptions are necessarily true. Despite the Ptolemaic system’s ability to predict planetary position, Venus does not move in perfect circles at a constant velocity. Data’s increasing precision has regularly forced the scientific community to revise assumptions. For instance due to better data, scientists believe that light consists of quantized entities, i.e. photons, rather than being a wave.

Second, even if the assumptions are wrong, we can still learn some valuable lessons from the process. Even though Ptolemy’s geometric paradigm was flawed, his process conclusively revealed the predictive power of mathematics. False assumptions don’t imply that everything associated with thought chain is false. Despite their misguided motivations, our enemies might have something to teach us. Or, don’t throw out the baby with the bathwater.

In the next chapter, we will examine how the implicit logic behind the Ptolemaic system could have been a major contributing factor in the development of Christian monotheism and predestination.

^{0} Lehman, *Aesthetics drives Copernican Discoveries*

^{1} As an indication of his diverse brilliance, Ptolemy also made significant contributions to the field of optics. In fact during certain historical periods, his books were thought to be written by different men with the same name.

^{2} The author suggests that this is due to Life’s reflexivity. He develops this notion fully in his *Theory of Attention*.

^{3} Ironically there is a kind of harmony between the planetary distances from the sun. Although not quite as Pythagorus conceptualized it, the radiuses of the planetary orbits from Mercury to Pluto have an exponential relationship that is step-like from 1 to 9. While a mathematical, rather than musical, harmony, I think Pythagoras would have been pleased by this vindication of his seemingly antiquated theory.

^{4} I have developed a mathematically based *Theory of Attention* that applies exclusively to living behavior. However, the mathematics is of a different kind than material mathematics. Rather than set-based like material mathematics, living mathematics is process-based and reflexive. Determined by context, the whole cannot be broken (reduced) into individual parts. This inherent resistance to reductionism indicates that Life’s non-material component has features that are non-mathematical. If so, then Pythagoras was wrong regarding living behavior.

^{5} Similarly although Copernicus suggested a heliocentric system, it was Kepler that turned the tide.

^{6}Some believe like Pythagoras that all phenomena can be ultimately characterized by mathematics. While mathematics does an almost perfect job of characterizing material behavior, an abundance of living phenomena resists a mathematical explanation. We suggest that the reason for the inadequacy of mathematics to accurately characterize living behavior concerns our capacity for Attention. Living systems focus their Attention upon images, in the general sense. For instance, single celled amoebae focus awareness upon their image of food. While the rhythms of Attention are revealed by mathematics, our image-making capacity is reliant upon innate primordial feelings or derivatives of such. These images resist a mathematical explanation. (For more see author’s *Theory of Attention*.)