Eudoxus of Cnidus student of Archytas
Life
Eudoxus was born and died in Cnidus (also spelled Knidos), which was a city on the southwest coast of Asia Minor. The years of Eudoxus' birth and death are not fully known but the range may have been c. 408 – c. 355 BC, or c. 390 – c. 337 BC.  
His name Eudoxus means "honored" or "of good repute" (εὔδοξος, from eu "good" and doxa "opinion, belief, fame"). It is analogous to the Latin name Benedictus.
Eudoxus's father, Aeschines of Cnidus, loved to watch stars at night. Eudoxus first traveled to Tarentum to study with Archytas, from whom he learned mathematics. While in Italy, Eudoxus visited Sicily, where he studied medicine with Philiston.
At the age of 23, he traveled with the physician Theomedon—who (according to Diogenes Laërtius) some believed was his lover—to Athens to study with the followers of Socrates. He eventually attended lectures of Plato and other philosophers for several months, but due to a disagreement they had a falling-out. Eudoxus was quite poor and could only afford an apartment at Piraeus. To attend Plato's lectures, he walked the 7 miles (11 km) in each direction each day. Due to his poverty, his friends raised funds sufficient to send him to Heliopolis, Egypt, to pursue his study of astronomy and mathematics. He lived there for 16 months. From Egypt, he then traveled north to Cyzicus, located on the south shore of the Sea of Marmara, the Propontis. He traveled south to the court of Mausolus. During his travels he gathered many students of his own.Around 368 BC, Eudoxus returned to Athens with his students. According to some sources, around 367 he assumed headship (scholarch) of the Academy during Plato's period in Syracuse, and taught Aristotle. He eventually returned to his native Cnidus, where he served in the city assembly. While in Cnidus, he built an observatory and continued writing and lecturing on theology, astronomy, and meteorology. He had one son, Aristagoras, and three daughters, Actis, Philtis, and Delphis.
In mathematical astronomy, his fame is due to the introduction of the concentric spheres, and his early contributions to understanding the movement of the planets.
His work on proportions shows insight into real numbers; it allows rigorous treatment of continuous quantities and not just whole numbers or even rational numbers. When it was revived by Tartaglia and others in the 16th century, it became the basis for quantitative work in science, and inspired the work of Richard Dedekind.Craters on Mars and the Moon are named in his honor. An algebraic curve (the Kampyle of Eudoxus) is also named after him.Mathematics
Eudoxus is considered by some to be the greatest of classical Greek mathematicians, and in all Antiquity second only to Archimedes.  Eudoxus was probably the source for most of book V of Euclid's Elements.  He rigorously developed Antiphon's method of exhaustion,  a precursor to the integral calculus which was also used in a masterly way by Archimedes in the following century.  In applying the method, Eudoxus proved such mathematical statements as: areas of circles are to one another as the squares of their radii, volumes of spheres are to one another as the cubes of their radii, the volume of a pyramid is one-third the volume of a prism with the same base and altitude, and the volume of a cone is one-third that of the corresponding cylinder.Eudoxus introduced the idea of non-quantified mathematical magnitude to describe and work with continuous geometrical entities such as lines, angles, areas and volumes, thereby avoiding the use of irrational numbers. In doing so, he reversed a Pythagorean emphasis on number and arithmetic, focusing instead on geometrical concepts as the basis of rigorous mathematics. Some Pythagoreans, such as Eudoxus's teacher Archytas, had believed that only arithmetic could provide a basis for proofs. Induced by the need to understand and operate with incommensurable quantities, Eudoxus established what may have been the first deductive organization of mathematics on the basis of explicit axioms. The change in focus by Eudoxus stimulated a divide in mathematics which lasted two thousand years. In combination with a Greek intellectual attitude unconcerned with practical problems, there followed a significant retreat from the development of techniques in arithmetic and algebra.The Pythagoreans had discovered that the diagonal of a square does not have a common unit of measurement with the sides of the square; this is the famous discovery that the square root of 2 cannot be expressed as the ratio of two integers. This discovery had heralded the existence of incommensurable quantities beyond the integers and rational fractions, but at the same time it threw into question the idea of measurement and calculations in geometry as a whole. For example, Euclid provides an elaborate proof of the Pythagorean theorem (Elements I.47), by using addition of areas and only much later (Elements VI.31) a simpler proof from similar triangles, which relies on ratios of line segments.
Ancient Greek mathematicians calculated not with quantities and equations as we do today, but instead they used proportionalities to express the relationship between quantities. Thus the ratio of two similar quantities was not just a numerical value, as we think of it today; the ratio of two similar quantities was a primitive relationship between them.
Eudoxus was able to restore confidence in the use of proportionalities by providing an astounding definition for the meaning of the equality between two ratios. This definition of proportion forms the subject of Euclid's Book V.
In Definition 5 of Euclid's Book V we read:
apprentice of disciple of pupil of follower of learner of