MATHEMATICIAN

Life
Celebrated Greek mathematician, astronomer, geographer and philosopher, whose genius was apparent from a very young age. He studied first with the famous Pythagorean Archytas of Tarentum; later a wealthy patron, impressed by the ability of this poor student, paid his way to Athens so that he could study at Plato's Academy. Here he was trained in the Platonic method, but continued to work out his own philosophy, which in physics was quite similar to that of Aristippus. He later spent some time in Egypt studying astronomy. The learned men of Egypt so admired his genius that they built him an observatory! When he returned to Greece he founded a school at Cyzicus in the Propontis, which attracted students from all over Greece.


Work
His most important works were:

MATHEMATICS

"Axiom of continuity" (Eudoxus-Archimedes): Discussed by Archimedes and one of the foundations of modern mathematics. The basis of both integral and differential calculus, it was first applied by Eudoxus and later expanded by Archimedes. Centuries later, Newton and Leibnitz based their work on this theorem.

"Method of exhaustion": For the calculation of the volume of the pyramid and cone. Archimedes notes that Eudoxus was the first to prove that the cone and the pyramid are one-third respectively of the cylinder and prism with the same base and height.

"Analysis and synthesis in geometry": Perfected by Eudoxus.

"Delian problem": According to Eutocius, Eudoxus solved it by means of a "curved line". His proof has been lost.

"Incommensurables": Eudoxus developed a general theory of proportion applicable to incommensurable as well as to commensurable magnitudes, as Euclid explains in his Elements (books V and VI). Correlation of straight segments without the use of numbers.

"Theory of irrational numbers": Developed in the 19th century by Richard Wedekind on the basis of Eudoxus' approach.

ASTRONOMY

Eudoxus was the founder of the first known observatory. He described the constellations, and was the first to construct and use a planisphere and to explain the apparent movements of the heavenly bodies by a geometric model: in other words, he introduced mathematics into astronomy. He is therefore considered the founder of mathematical astronomy.

"Theory of concentric spheres": Interpretation of the apparent movement of the planets, using what Eudoxus called an "isopede" (spherical lemniscate), which he devised. This theory became the foundation of the science of astronomy. Eudoxus also wrote a related treatise entitled "On speeds", which studied the movements of the seven celestial bodies: Sun, Moon, Mercury, Venus, Mars, Jupiter, Saturn. His system was universally admired and was accepted by Aristotle, who described it in his "Metaphysics". It was later more fully worked out by his pupil, Callippus.

"On making spheres": Eudoxus constructed a mechanical representation of Autolycus' theory on the movement of the planets.

He was the first to calculate the distance of the Sun and the Moon from the Earth.

"Phenomena and Enoptron": Treatise on astronomy, discussed by Aratus. It describes the position of the constellations in the heavenly sphere, and their risings and settings.

"Mathematical explanation of the observed motions of the stars"

The founder of celestial mechanics, Eudoxus invented a method to calculate the distance of the sun and the moon. Aristarchus of Samos based his work on Eudoxus' method.

He proposed a calendar of the solar year, with 3 three years of 365 days followed by a fourth with 366. This calendar was adopted 300 years later by Julius Caesar (the Julian Calendar).

"Octaeteris": Calendaric treatise, based on an eight-year cycle, on the adaptation of the lunar to the solar year.

GEOGRAPHY - METEOROLOGY

Strabo considered him the fourth great Greek geographer, placing him immediately after Democritus, and noting that Eudoxus was the first to apply mathematical axioms to geography. Eudoxus calculated that the ratio of the length to the width of the world was 2:1.

"Signs and portents - Observations": Observations on the weather and a study of the winds.

"Ges periodos": Treatise containing a wealth of geographical information.

Eudoxus was particularly interested in the climate in various parts of the world, and in the zones of the terrestrial globe with similar astronomical data (appearance of the night sky, length of longest day, etc.).

He constructed a planisphere.

INVENTIONS

The major contribution to the development of mechanics and technology made by this great mathematician and astronomer was the introduction, in about 360 BC, of two particularly important instruments: the astrolabe and the 'polos'. The astrolabe was a genuine Greek invention - not of course in the form in which we know it today - that has been attributed to Eudoxus on the basis of work done by F. Nau in 1899; the first Hellenistic accounts of this device, e.g. in Philoponos of Alexandria, came centuries later (500-550 AD). Vitruvius tells us that Eudoxus used an instrument that he called a 'spider'. Others have argued that the initial form of the instrument, the plane astrolabe, was discovered by Hipparchus in about 150 BC. The 'polos' was a more complicated arrangement of interlinked rings that was used to tell the time. Both were derived from the older gnomon and sundial, which had developed into two different types of instruments: the 'uranosphere', supported by a system of rings, and the plane sundial, which finally supplanted it. The astrolabe began its career as an instrument that could tell the time by night as well as by day; the Arabs developed it into an astronomical instrument in about 810 AD, while in Western Europe it ended up as an instrument for navigation.

The polos was a more complex device. In essence it was a portable time-piece, that followed the path of the shadow of the sun across a circle marked off into segments corresponding to the constellations in the zodiac. Like a modern watch, it indicated both the hour and the month. Indeed, in the measurement of time in particular the ancient engineers produced truly magnificent instruments, that are admired even today.