Claudius Ptolemy: The Mathematics of Celestial Motion

Claudius Ptolemy: The Mathematics of Celestial Motion

Although not much is known about the life of Claudius Ptolemy, his legacy is well established. His writings on astronomy and cartography were guided by an advanced understanding of trigonometry, and influenced centuries of math and scientific thought. Ptolemy greatly impacted the work of later Islamic and Byzantine astronomers, and his model of the cosmos was the prevailing one until the 15th- century. 

Although Ptolemy believed that the Earth was the center of the Solar System, he was able to calculate the precise movement of celestial objects with functions. He also provided the most complete and scientifically accurate map of the ancient world that exists. Learn more about this visionary thinker and read on!

Claudius Ptolemy’s Mysteriously Unknown Life 

A portrait of famous mathematician and astronomer Claudius Ptolemy, created by Andre Thevet in the year 1584, via KhanAcademy

Claudius Ptolemy was born in Egypt around 100 AD. He likely spent most of his life living in or around the Egyptian city of Alexandria. Ptolemy is a Greek-Egyptian name, while Claudius is a Roman one. Many historians speculate that he was a Roman citizen, living in Greek-controlled Egypt. Ptolemy was a common name at the time, originating in Greek mythology. He wrote his works in Ancient Greek, while also utilizing Babylonian astronomical data. 

Ptolemy is one of the most famous mathematicians of Antiquity and he had a huge impact on the next several centuries of mathematical and scientific thought. One of his earliest works was a philosophical treatise, his only writing not to deal directly with math. This short essay concerned epistemology, the philosophical study of how humans accumulate knowledge. He outlines his central philosophical belief that people should combine reason and logic with sense perception to arrive at new discoveries. Rather than simply trusting what you see, hear, and feel, Ptolemy believed a rigorous analysis perception combined with analytical reasoning was necessary to arrive at conclusions.

Ptolemy is also known to have believed in the supremacy of mathematics, as the purest form of knowledge. Interestingly, he viewed mathematics as a kind of philosophy, a way to understand and think about the universe.  Unfortunately, not much else is known about Ptolemy’s life. No descriptions of his appearance survive and we don’t know how he died. However, he is thought to have died around 170. 

Claudius Ptolemy’s Mathematical Discoveries and Work 

Ptolemy wrote more than 10 mathematical treatises, two of which were very important to the course of mathematical history. The first is called the Almagest, a critical work of ancient astronomy and one of the few surviving in its entirety. Ancient Babylonian astronomers had discovered how to calculate the path of the planets, Sun, and Moon with simple arithmetic. However, they did not have a model of the Solar System. Greek astronomers had developed a geocentric model of the Solar System, but were unable to calculate the movements of the planets and other celestial bodies. Ptolemy read the work of Hipparchus, a Greek astronomer who attempted to merge the two approaches. 

He took Hipparchus’s work and ran with it, creating a set of geometric models that would predict the movements of the celestial bodies with precision. He drew from Babylonian astronomical data which spanned over 800 years to inform his models. Ptolemy’s calculations were based on an early form of trigonometric functions, called chords. Chords are a line segment joining two points on a curve. To determine the length of chords, Ptolemy visualized them in circles, using the radius of the circle and angle of the chord to determine its length. 

Claudius Ptolemy’s geocentric model of the universe, via

Ptolemy included easy-to-use tables, which aided the reader in calculating the future or past position of different celestial bodies. He also included a list of stars and constellations, called a star catalog. The Almagest was considered to be the foremost astronomical text for centuries, influencing Arabic, African, and European astronomy. The Almagest survives because of its popularity; it was copied into Arabic and Latin several times, with several early copies still in existence. 

The second most important work by Ptolemy is called Geography. It set down Ptolemy’s expansive work on cartography, the study of map drawing. His was the first comprehensive attempt to map the world using scientific methods and mathematics. He incorporated over 8,000 localities in the ancient world, with over 6,300 including coordinates allowing them to be accurately plotted. Ptolemy even set out latitude from the Equator, as it’s measured today. However, he was only able to plot about a quarter of the world, representing the areas known to the Greek and Roman Empires, and their allies. This book represents the most complete snapshot of the world as it existed in Antiquity, thanks to Ptolemy’s rigorous mathematical efforts. It also influenced many later map makers, improving the quality of cartographical projections produced. 

A map of the ancient world from the Geography of Claudius Ptolemy, via the University of Oxford

In addition to astronomy and cartography, Ptolemy wrote about optics and music theory. Although his writing on optics is rudimentary, it explored important topics like refraction from air to water, reflection in various types of mirrors, and a theory of binocular vision. He incorporated mathematical equations and reasoning to back up his claims, laying the groundwork for later discoveries in optics. Ptolemy’s work on music theory was similarly focused on mathematics. He believed in the approach followed by Pythagoras, wherein musical intervals correspond to mathematical ratios. 

Ptolemy also wrote extensively on the subject of astrology, believing that the planets caused various phenomena on Earth, such as heat waves, cold snaps, droughts, and floods. He attempted to apply scientific reasoning to astrology, rejecting earlier astrological assumptions like numerology for being too abstract and fantastical. 

Claudius Ptolemy’s Impact on the Evolution of Mathematics

For over a millennium, the Almagest was the definitive text on astronomy. It spread across Eurasia and Africa, reproduced in several languages. Besides the astronomical implications, the Almagest was also key in spreading trigonometry and geometry. Ptolemy has been cited innumerable times throughout history and his work commented on extensively. Today, there is an asteroid, a Moon crater, and a crater on Mars named after Ptolemy. 

The Brilliant Lives of Famous Mathematicians 

This article is the nineteenth in our series exploring the lives and achievements of famous mathematicians throughout history. (Our last article was about the English mathematician Isaac Newton!) 

Through the lives of these brilliant folks, we hope you’ll find connections, inspiration, and empowerment.

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