I have spoken about the formal definition of the limit in a previous blog. Today's blog is part of the effort to review the background and details that make up, Euler's Identity, one of the most astounding mathematical formulas of all time:
eiπ + 1 = 0
For those who are not familiar with Euler's Identity, you may wish to start here.
Eudoxus was born around 408 B.C. in Cnidas which is resides in modern Turkey. Not much is known about Eudoxus's early life. We do believe that he studied in Italy with Archytas who was one of the followers of Pythagorus. Archytas was fascinated by the problem of duplicating the cube (the problem of finding a compass/ruler construction for a cube root). It is also probable that Eudoxus studied Pythagorean number theory and music theory.
Later, Eudoxus studied medicine in Sicily under Philiston and later in Athens under Theomedan. While in Athens, he attended lectures by Plato and his followers at the newly created Academy.
Professor Heath writes that Eudoxus at this time was very poor (Heath quoted in MacTutor):
... so poor was he that he took up his abode at the Piraeus and trudged to Athens and back on foot each day.
Later, he went to Egypt to study astronomy with the priests at Heliopolis. He stayed in Egypt for over a year and after this time, he decided to open a school of his own in Cyzikus which is in Asia Minor near the shore the Sea of Marmara. The school became a big success.
There is evidence that after his school became established, Eudoxus went with some of his followers to visit Athens. At this time, there seems to have been a rivalry between Plato and Eudoxus. Eudoxus was critical of Plato's mathematical abilities and Plato seemed jealous of the popularity of Eudoxus's school.
Eudoxus later returned to Cnidas where he came one of the town leaders. He continued to write texts on theology, astronomy, and geography and built an observatory. Hipparchus refers to astronomic data captured by Eudoxus at this observatory that helped explain the rising and setting of constellations.
Eudoxus's contributions to mathematics are legendary. He proposed a theory of proportion to deal with irrational numbers which is found in Euclid's Elements, Book V. All rational numbers can be compared by finding a common unit. The problem is that with irrational numbers there is no such common unit. Eudoxus instead proposed the use of ranges so that a comparison occurs by applying a common multiple and then comparing the resultant value.
In modern terms, for real numbers a,b,c,d: a/b = c/d if for every pair of integers m, n:
(a) if ma is less than nb, then mc is less than nd.
(b) if ma = nb, then mc = nd
(c) if ma is greater than nb, then mc is greater than nd.
This work on proportions was very important in the modern definition of real numbers (see here).
In addition to his theory of proportion, he proposed what is today termed the Method of Exhaustion. This generalized the work done by Antiphon on estimating the area of a circle by using inscribed polygons. Eudoxus's theory is found in Euclid's Elements, Book X, Proposition 1. In Euclid, Eudoxus's Method of Exhaustion is used to show that the area of two circles is in proportion to the square of their diameters.
Eudoxus also offered a solution to the duplication of the cube using curved lines. Eratosthenes who wrote a history of the problem talks about Eudoxus's solution. Unfortunately, the exact details of his solution have been lost. Paul Tannery was able to come up with a proof that is consistent with information known about Eudoxus. The proof uses an algebraic curve known as a kampyle curve. Today, this solution is known as the kampyile of Eudoxus.
In his own lifetime, Eudoxus was most famous for his planetary theory where he was able to emulate the planetary motions using 27 spheres that turn upon each other. These spheres of Eudoxus are mentioned by Aristotle in his famous work Metaphysics.
Although his book Tour of the Earth has been lost, there still exist over 100 quotes in secondary sources. The book surveys all the different peoples known to Eudoxus and reviews their history, their political systems, and their culture. He wrote deeply about Egyptian society and about the Pythagoreans in Italy.