Eudoxus of Cnidus


Mathematician, Astronomer, Philosopher, Inventor, Meteorologist, Geographer, Physician, Rhetorician, Lawmaker (408 BC – 355 BC)

The greatest mathematician of antiquity second to Archimedes and Apollonius of Perga. He was the one who “rescued” the ancient Greek mathematics from the dead-end they had reached after the discovery of asymmetry during the 5th century BC. As an eminent astronomer he created the geocentric model and devised an astronomical instrument to interpret the planets’ movements. He is acknowledged worldwide as one of the greatest mathematicians who ever lived.

Eudoxus studied mathematics in Archytas’ school in Sicily and at the age of 23 he enrolled in Plato’s Academy where he studied medicine. He travelled to Egypt where he was taught astronomy by the priests of Heliopolis and later founded the Eudoxian Observatory. He returned to Greece where he founded the School of Cyzicus and a school in Athens. On Plato’s request he became a professor of sciences in Plato’s Academy and taught for most of his life, until he moved back to Cnidus and founded an observatory.

Eudoxus’ work on mathematics is titanic. He is credited by Archimedes as the founder of the Archimedean Property or Eudoxian Theorem on the Axioms of Continuity, which today dominate in modern higher mathematics. The Eudoxus Axiom forms the basis of calculus, which Eudoxus first discovered and applied and later Archimedes expanded. These axioms were later developed by Newton and Leibnitz. The discovery of the Method of Exhaustion belongs to Eudoxus. With it he was able to calculate areas and volumes of geometrical shapes which were unknown at the time. He was the first to prove that the volume of the pyramid is equal to 1/3 the volume of the prism that has the same height and base as the pyramid. In addition, he proved that the volume of the cone is 1/3 the volume of the cylinder with the same height and base.

Eudoxus perfected the analysis and synthesis in geometry, expanded the theory of proportionality and overcame the asymmetric problem that had been troubling ancient Greek mathematics, formulated the bisection principle, solved the Delian problem of the doubling of the cube using curved lines, even though unfortunately the solution was lost, developed the theory of irrational magnitudes and made important discoveries on the Theorem of the Golden Ratio. He also proved that the areas of circles are proportional to the squares of their diameter.

As a pioneer in astronomy, Eudoxus is considered the founder of mathematical astronomy. He was the first to introduce the concept of celestial bodies and explain their movement using a geometric model based on mathematic principles. With his observations of the star sky, he described the constellations, their position, the dates when the stars are visible, and the weather associated with each of their phases. He understood the need of mathematical analysis on interpreting astronomical findings. Moreover, he researched the sizes of the Sun, the Moon and the Earth, introduced the Οκταετηρίς (Octaeteris) and eight-year cycle calendar with 365 days of the year which 300 later was introduced by Julius Cesar.

His Theory of the Homocentric Spheres interpreted the movement of the planets using hippopedes, complex geometric curves, an invention of Eudoxus that formed the basis of mathematical astronomy. Using this theory he created the Homocentric Sphere System, a planetary system consisting of a number of rotating spheres depicting the 7 planets Mercury, Venus, Earth, Mars, Zeus, Saturn, the Moon and the Sun. His geocentric planetary system achieved considerable acclaim, was accepted by Aristotle and by the scientific community was later perfected by his student Callipus and survived until the Renaissance. He wrote a treaty named On Velocities, which concerned the planetary movements. In his treaty Sphaeropoia Eudoxus first calculated the distances of the Sun and the Moon from Earth using his own genious methods. Eudoxus is therefore the founder of theoretical astronomy and celestial mechanics.

The introduction of mathematical axioms in geography is attributed to Eudoxus. He wrote studies on the weather, the climate and the wind. Numerous treaties concerning the climate in various places of the Earth and the zones of the globe sharing common astronomical data.

As an inventor, Eudoxus invented the planetary simulator, an instrument demonstrating the ostensible motion of the planets. It consisted of two concentric rings that when rotated simultaneously at opposite directions caused the planets to move in an octant motion. The system rested on a third ring that produced the complex orbit of the planet. His second invention, the “Spider” was a map of the celestial sphere. According to others, it was a sundial or some kind of astronomical instrument. The astrolabe was an instrument originally designed by Eudoxus and was later perfected by Hipparchus. In turn, the dioptres, instruments used for astronomical observations were perfected by Heron of Alexandria. Finally, the Pole was an instrument much more complex than Hipparchus’ astrolabe. It was a portable sundial which contained markings corresponding to the zodiac cycle. The movement of the sun caused its shadow to fall on one of the corresponding markings indicating the time, the day and the month of the year.

During his own lifetime, Eudoxus’ fame had reached its apogee. He was widely known as “Eudoxus the Endoxus” (Eudoxus the Glorious). Eratosthenes called him θεοειδῆ (God-like) and Strabo mentions him as the fourth greatest geographer. Sadly, absolutely none of Eudoxus’ works survived. Today a crater in the moon’s surface bears his name.


  1. “Eudoxos o Knidios”. Helios New Encyclopaedic Dictionary. Passas, I. Athens: 1946. Print.
  2. Georgakopoulos, Konstantinos. Ancient Greek Scientists. Athens: Georgiades, 1995. Print.
  3. Kotsanas, Kostas. Ancient Greek Technology: The Inventions of the Ancient Greeks.  Pyrgos: Kostas Kotsanas, 2013. Print.
  4. Koutoulas, Diamantis. The Ancient Greek Religion and the Mathematics. Thessaloniki: Psaras, 2001. Print.
  5. The Editors of Encyclopaedia Britannica. ”Eudoxus of Cnidus” Encyclopaedia Britannica. Web. 23 August. 2016.
Eudoxus of Cnidus

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