Eudoxus made momentous advances in mathematics; he was the world’s greatest mathematician before Archimedes.
Eudoxus founded mathematical astronomy when he created the first mathematical model of the universe, expressing the movement of the heavens in the language of spherical geometry. In doing so he turned physical reality into something more abstract offering a new vantage point from which we could study the universe.
Eudoxus was responsible for the far-reaching theory of proportion expounded in Book V of Euclid’s Elements: Eudoxus gives the first rigorous definition of what today we call real numbers – the set that includes both rational and irrational numbers.
Eudoxus developed the method of exhaustion, giving future mathematicians such as Archimedes a potent tool to calculate areas and volumes. A forerunner of integral calculus, the method of exhaustion allowed Eudoxus to prove the formulas for cone and pyramid volumes. It is the basis of Book XII of Euclid’s Elements.
Elements was a compilation and synthesis of all known mathematics. Eudoxus may have provided more of its contents than any other single person.
Biography – Eudoxus’s Odyssey
Biographical details about Eudoxus are sketchy and must be treated cautiously. What is certain is that he was born about 2,400 years ago and his lifetime coincides approximately with Plato’s and Aristotle’s.
What we know about Eudoxus relies heavily on Lives and Opinions of Eminent Philosophers, written by Diogenes Laërtius six centuries after Eudoxus lived. Laërtius often did not have access to primary sources, and the works he based his story of Eudoxus on have been lost.
A few snippets about Eudoxus are found in the works of other ancient writers, such as Proclus.
Eudoxus was probably born between 410 and 400 BC. His birthplace was Cnidus, a city originally founded by the Spartans. The city’s ruins can still be seen on the Mediterranean coast of south-west Turkey, near the Greek island of Kos.
We do not know what Eudoxus looked like, but we know he lived for 53 years. His father’s name was Aeschines.
- In his early 20s, Eudoxus left Cnidus on a quest for knowledge – his own odyssey.
- Eudoxus learned mathematics in the school of Archytas, one of the greatest mathematicians of his day, in Tarentum, now in southern Italy.
- He learned the art of healing in the school of Philistion of Locri in Sicily.
- At age 23, he visited Athens to attend lectures by Plato. To save money he roomed in the port of Piraeus and made a 14 mile round trip on foot each day of his two month stay.
- Back in Cnidus he raised money from his friends to travel again – ancient crowdfunding! His friends recognized something in the young man that made him worthy of their trust and support.
- In about 381 BC, Eudoxus traveled to the city of Heliopolis in Egypt for 16 months. There he was taught astronomy by Egyptian priests in their observatories. Like them, he shaved off ALL of his hair, including his eyebrows.
- The next stage of his odyssey took Eudoxus to the shores of the sea of Marmaris and the city of Cyzicus, whose ruins lie in modern Turkey. He was now regarded as a wise teacher.
- From Cyzicus he traveled south to the city of Mylasa and joined the court of King Mausolus. The king ruled over the region of Caria, which included Eudoxus’s hometown. King Mausolus became eternally famous when the astounding mausoleum he built at Halicarnassus became one of the seven wonders of the ancient world – we get the word mausoleum from the king’s name. Eudoxus now founded his own school.
- In about 368 BC, aged somewhere in the region of 32-42, Eudoxus took his school to Athens.
- From Athens, Eudoxus’s final move took him home to Cnidus. The people of his hometown quickly voted him into office as one of the city’s governors.
- At some point in his life Eudoxus married, but we do not know his wife’s name. He fathered a son: Aristagoras, and three daughters: Actis, Philtis and Delphis.
- None of Eudoxus’s work survive. We know some of his achievements because of acknowledgements, citations, and references made by the likes of Aristotle, Archimedes, and Hipparchus.
Plato’s Planet Puzzle – Modelling the Heavens
The great philosopher Plato insisted that the planets must move in a uniform, orderly way around Earth. Yet anyone who studied their movements had to admit that they did not seem uniform and orderly. For a start, retrograde motion was perplexing.
Plato challenged the mathematicians and philosophers of Greece to explain the planets’ movements.
Eudoxus responded to Plato by constructing a mathematical model of the solar system. He pictured a central spherical Earth surrounded by a series of 27 nested or concentric rotating spheres.
The outermost sphere carried the fixed stars
Each of the five known planets was carried by four interacting spheres
The sun and moon were each carried by three interacting spheres
Eudoxus gave each rotating sphere its own axis of rotation and these axes pointed in different directions. Different rotation speeds of the spheres could explain the seemingly irregular movements of the planets in the sky.
It was a glorious idea – the first mathematical model of the universe in history. And it succeeded in producing retrograde motion. However, when the numbers it produced were matched against actual planetary movements, there was a mismatch. For all its brilliance, the model did not work. However, the idea inspired generations of Greek philosophers.
In an effort to produce a model like Eudoxus’s that worked, Aristotle ended up with a model consisting of over 50 spheres. However, the spheres could never account for the known fact that the moon and the planets’ sizes changed as they moved along their orbits. The Greeks realized this meant that sometimes the moon and planets were closer to Earth than at other times and that, without serious modifications, the spheres idea was wrong. Later, Hipparchus abandoned the spheres model in favor of one that explained why the planets’ distances from Earth changed.
Today’s scholars disagree among themselves about the precise details of Eudoxus’s model. We do not know whether Eudoxus believed his spheres really existed or if he proposed them as imaginary aids to calculation.
A century after Eudoxus, Aristarchus hit the nail on the head when he said that all the planets including Earth orbit the sun. Sadly his idea fell on deaf ears. It took close to two millennia before Nicolaus Copernicus resurrected it.
Proportions and Real Numbers – A Giant Leap for Mathematics
Scholars believe that Eudoxus was responsible for the far-reaching rules of proportion contained in Book V of Euclid’s Elements. The rules give us the first relatively rigorous definition of what today we call real numbers – the set that includes both rational and irrational numbers – and a means of dealing with them consistently.
There is a legend that in earlier times Pythagoreans were horrified when they discovered numbers that could not be written as a ratio of whole numbers – in other words irrational numbers, like √2. The legend is echoed in our continuing use of irrational to mean insane or senseless.
Eudoxus’s statement of proportion rescued mathematics from this, perhaps legendary, dead end. Thomas Heath wrote:
Heath“The greatness of the new theory itself needs no further argument when it is remembered that the definition of equal ratios in Euclid Book V, Definition 5 corresponds exactly to the modern theory of irrationals due to Dedekind, and that it is word for word the same as Weierstrass’s definition of equal numbers.”
THOMAS HEATH
A History of Greek Mathematics, Vol. I, 1921
In fact Dedekind, who published his work more than 22 centuries after Eudoxus, said his advances were inspired by Eudoxus’s concepts:
Heath“…this conception from ancient times represents the origin of my theory as well as the theory of Mr. Bertrand and of some other… attempts to introduce irrational numbers into arithmetic.”
RICHARD DEDEKIND
Letter to Rudolf Lipschitz, October 1876
Dedekind pointed out that his theory was more complete than Eudoxus’s because his contained the principle of continuity.
Proto-calculus – The Method of Exhaustion
The Method of Exhaustion was the Ancient Greek forerunner of integral calculus. It involved summing ever smaller magnitudes to obtain results ever closer to the true result.
The Ancient Greek mathematician Antiphon proposed a ‘method of exhaustion’ to find the area of a circle. His method was:
Take a regular polygon, such as a triangle, whose area is easily calculated.
Place it inside a circle, with its vertices touching the circle’s circumference. There will be a considerable difference between the area of the triangle and the circle that encloses it.
Now double the number of sides of the polygon to a hexagon. There is now less of a difference between the hexagon’s area and the circle’s area.
Continue the process for polygons with 12 sides, 24 sides, etc. until the area of the regular polygon of known area is almost identical to the circle’s. You have now exhausted the process and calculated the area of the circle with high accuracy.
Proposition 1 in Euclid’s Elements Book X provides the rigorous mathematical justification of the method of exhaustion. The scholarly consensus is that Proposition 1 is Eudoxus’s work. Proposition 1 begins with the statement below, which is then proved:
The method of exhaustion can be used to find lengths, areas and volumes. It is used in Book XII of Euclid’s Elements to prove, for example, that the area of a circle is proportional to its diameter squared, and also to prove the formulas for volumes of cones and pyramids.
The method of exhaustion is generally completed by a contradiction. The true area is assumed to be greater than area X, and this is proven to be false. The true area is then assumed to be less than area X, and this is also shown to be false. Hence the true area is equal to area X.
The Volumes of Cones and Pyramids
Archimedes tells us that:
the volume of a cone is one-third that of the cylinder with the same base and height
the volume of a pyramid is one-third that of the prism with the same base and height
He says these results were found first by Democritus and the first proofs were published by Eudoxus.
Archimedes made excellent use of the method of exhaustion himself. For example, he used it to compute π to an accuracy of better than one part in 10,000 – his approximate value of 22/7 was used until the digital age.
Observatory
In later life, after returning to Cnidus, Eudoxus established an observatory.
The End
Eudoxus died at age 53, leaving a legacy of major progress in mathematics. He does not seem to have been born wealthy, but as a young man he was regarded so highly that his friends raised money to send him on travels in search of knowledge. In middle-age, he remained popular enough to be elected to serve as a governor when he returned to his hometown.
Aristotle was alive in the same era as Eudoxus, therefore what he said about his character is likely to be true:
aristotle“Eudoxus thought pleasure was the good because he saw all things, both rational and irrational, aiming at it… His arguments were credited more because of the excellence of his character than for their own sake; he was thought to be remarkably self-controlled, and therefore it was thought that he was not saying what he did say as a friend of pleasure, but that the facts really were so.”
ARISTOTLE, TRANSLATED BY W. D. ROSS
Nicomachean Ethics, Book 10
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