Proclus was one of the last influential Greek philosophers and mathematicians of ancient times.
He produced an alternative statement of Euclid’s famously problematical parallel postulate: Proclus’s version came to be known as Playfair’s Axiom after it was stated by John Playfair in 1846. When David Hilbert unified two-dimensional and three-dimensional geometry in 1899, he used Playfair’s and hence Proclus’s version of the parallel postulate.
Much of what we know about the history of Ancient Greek geometry before Euclid comes from Proclus’s commentary on Euclid’s Elements.
Proclus was one of the greatest Neoplatonist philosophers and became director of the Academy in Athens founded by Plato 800 years earlier. Proclus supported the idea that everything in the universe has its origins in ‘the One,’ a transcendent god who created the Universe, the Cosmic Soul, and the Divine Mind.
Beginnings
Proclus was born on February 8, 412 into a wealthy, influential family in Constantinople, the Greek-speaking capital of the Eastern Roman Empire. (Today the city is Turkey’s capital, Istanbul).
Proclus’s father, Patricius, was a senior government lawyer. His mother’s name was Marcella.
Education
Proclus began his education in Xanthos, a city in Lycia, now in southern Turkey. He became known as Proclus Lycaeus because of his links to Lycia.
Proclus traveled to Alexandria to complete his formal education. Although the scholars in the Egyptian city were not as brilliant as in the days of Eratosthenes, it still held an almost magnetic attraction for mathematicians, philosophers, and physicians. Proclus studied mathematics, philosophy, and rhetoric.
Law, Mathematics, and Philosophy
Back in his hometown of Constantinople, Proclus began working as a lawyer, but he found it unfulfilling. He decided mathematics and philosophy were his true vocations.
He returned to Alexandria, but soon decided he could do better elsewhere. In 431, age 19, he arrived in the magnificent city of Athens to study mathematics and philosophy at the Academy founded by Plato over 800 years earlier, in 387 BC.
He described his feelings about mathematics:
“She reminds you of the invisible form of the soul; she gives life to her own discoveries; she awakens the mind and purifies the intellect; she brings light to our intrinsic ideas; she abolishes oblivion and ignorance which are ours by birth.”
The Academy
Plato’s Academy was the nerve center of Neoplatonism, a philosophy encompassing a range of concepts with one dominant theme: the idea that everything in the universe has its origins in ‘the One’ – a transcendent god from which the World, the Cosmic Soul, and the Divine Mind have come.
The Academy’s head was Syrianus, a Neoplatonist philosopher whom Proclus revered. When Syrianus died in 437, he was succeeded by Proclus, now age 25.
Proclus became known as Proclus Diadochus, meaning Proclus the Successor.
Lifetimes of Selected Ancient Greek Scientists and Philosophers
The Mathematics and Universe of Proclus
Improving on Euclid
Euclid’s Elements is universally acknowledged as the greatest mathematics book of ancient times. Early in Elements Euclid states five self-evident geometric facts (postulates) – for example, all right angles are equal. From these he developed Euclidean geometry, the self-consistent geometry in two dimensions we still use today.
However, Euclid’s fifth postulate, the parallel postulate, caused mathematicians some discomfort. Euclid considered a straight line crossing two other straight lines. He looked at the situation when the interior angles (shown in the image below) add to less than 180 degrees. In these circumstances, he said that the two straight lines will eventually meet on the side of the two angles that add to less than 180 degrees.
a. When each angle is 90 degrees, the lines are parallel.
b. If one or both of the angles is less than 90 degrees, the lines will meet.
Proclus taught mathematics at the Academy, which led him to write his own commentary on Elements. He had this to say about the parallel postulate:
“[The fifth postulate] ought to be struck out of the Postulates altogether; for it is a theorem involving many difficulties… For the fact that some lines exist which approach indefinitely, but do not cross, although it seems improbable and paradoxical, is nevertheless true and fully ascertained with regard to other species of lines.” [Proclus is referring here to the asymptotes of a curve such as a hyperbola. The curve approaches ever closer to its asymptotes but never touches them.]
The hyperbolas (blue and green) approach but never meet their asymptotes (red). Image courtesy of Drini.
“May not then the same thing be possible in the case of straight lines which happens in the case of the lines referred to? Until the statement in the Postulate is clinched by proof, the facts shown in the case of other lines may direct our imagination the opposite way. And, though the controversial arguments against the meeting of the straight lines should contain much that is surprising, is there not all the more reason why we should expel from our body of doctrine this merely plausible and unreasoned (hypothesis)?”
In 1856, the mathematician John Playfair recast the parallel postulate into a form most mathematicians felt more relaxed about:
“Through a given point only one parallel can be drawn to a given straight line.Alternatively:
Two straight lines which intersect one another cannot both be parallel to one and the same straight line.”
When David Hilbert published Foundations of Geometry in 1899, he unified two-dimensional and three-dimensional geometry. Hilbert used Playfair’s axiom rather than Euclid’s.
Geometry, like arithmetic, requires for its logical development only a small number of simple, fundamental principles. These fundamental principles are called the axioms of geometry.
In fact, Playfair’s axiom was not new. Proclus stated the axiom identically 1,500 years earlier:
“If a straight line intersect one of two parallels, it will intersect the other also.Alternatively:
Straight lines parallel to the same straight line are parallel to one another.”
The Astronomy of Proclus
Proclus taught his students the astronomy of Hipparchus and Ptolemy. Ptolemy’s earth-centered astronomy dominated western and near-eastern astronomy for over a thousand years.
Proclus told his students that Hipparchus’s eccentric, epicycle, and deferent; and Ptolemy’s equant were merely mathematical devices used to calculate the planets’ movements, but they had no satisfactory theory to support them. He was one of many critics of Ptolemy’s work through the ages. Despite the criticism, it took over a thousand years before a clearly superior model to Ptolemy’s emerged in the shape of Johannes Kepler’s laws of planetary motion in the early 1600s.
Ptolemy’s model of planetary motion. The red circle is a planet – Mars, for example. Ptolemy went to rather extraordinary lengths in order to maintain the Platonic-Pythagorean notion that all planetary movements must be based on uniform circular motion.
Neoplatonism
As a follower of Plato, Proclus believed that while mathematics was enormously important, it was subservient to philosophy.
Proclus invented the concept of henads. These were individual ‘ones’ between ‘the One’ and the Divine Mind. Examples of henads might be the Greek gods, such as Athena and Hermes.
The End
Proclus never married and had no children. He was wealthy and his friends regarded him as generous. His lifestyle was similar to that advocated by Pythagoras, whose teachings were respected by the Neoplatonists – hence, Proclus was a vegetarian.
Remarkably, Proclus’s home in Athens was discovered and excavated by archeologists in the 1950s.
The floorplan of Proclus’s home in Athens.
Proclus died age 73 on April 15, 485. He was buried in a tomb near Mount Lycabettus where the body of Syrianus, whom he succeeded as head of the Academy, also rested. On the tomb is engraved:
“I am Proclus, Lycian whom Syrianus brought up to teach his doctrine after him. This tomb reunites both our bodies. May an identical sojourn be reserved to both our souls!”
As a Neoplatonist, Proclus believed the ultimate fate of his soul would be a union with the divine.
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Further Reading
Morris Kline
Mathematical Thought from Ancient to Modern Times
Oxford University Press, New York, 1972
Sir Thomas Heath
The Thirteen Books of Euclid’s Elements
Dover Publications, New York, 1956
Creative Commons
Image of hyperbola asymptotes courtesy of Drini under the Creative Commons Attribution-Share Alike 2.0 Generic license.
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