The Age of Application
(–300 to +400)
In about –300, Alexander the Great, the pupil of Aristotle, had conquered Greece, the Near East, and Egypt. He founded Alexandria at the junction of Asia, Africa, and Europe. In this cosmopolitan city, native Egyptians mingled on the streets with Ethiopians, Greeks, Persians, Jews, Arabs, Syrians, and Romans. Trading took place via ship to points all over the world known to them. The city became full of fancy buildings, baths, parks, and theaters.
Most important to the development of mathematics were the Museum (university) and library at Alexandria. The library grew to hold over 600,000 volumes from all over the world, including the works of the Classical Greeks. Mathematicians, poets, philosophers, astronomers, and other scholars from many countries flocked to this cultural center. The complex was like a modern university. The emphasis on learning represented by this institution is one reason historians often call the first part of this period the Hellenistic, or Greek-like, age.
The new city was less socially segregated than Athens had been. Through their interactions with tradespeople and navigators and builders and generals, mathematicians found applications of Classical Greek mathematics. The culture benefited from such devices as improved water clocks and fantastic machines (including automobiles) run by steam. The mathematician Eratosthenes, who was a director of the library, developed a calendar that was very similar to the one we use today. It consisted of 365 days per year except for a leap year every four years.
One of the greatest scholars in Alexandria was the native Egyptian Ptolemy. In about the year +150, Ptolemy wrote a great work on astronomy, called the Almagest, which built on data collected by the improved astronomical instruments of the period. A word about the Ptolemaic theory is in order.
To understand Ptolemy's scheme for the universe, imagine that planets did not follow circular paths around the earth. Rather, imagine that they followed circular paths around invisible points that were following circular paths around the earth. Then sometimes they would appear to be going forward, and sometimes in reverse. This would explain the retrograde motion of the wandering planets, observed since ancient times as in the top picture above. Observational data of the time were accurate enough, however, that Ptolemy knew that this explanation of the planets motion wouldnt quite work. Instead, he needed planets moving around invisible points that were moving around invisible points that were moving around the earth. It was quite complicated, and involved many different circles to describe the motion of the five planets. The scheme did such a good job of predicting planetary orbits, though, that it lasted for 1500 years.
Brief mention should also be made of the earlier theory by the Alexandrian Aristarchus (about –250) that the planets moved in circles around the sun. Ptolemy did not take this nonsense seriously. Even the ancient Mesopotamians realized that if this were so then the stars would not be fixed.
Besides Eratosthenes, major mathematicians of the Hellenistic period include Apollonius, who studied the conics extensively; and Hipparchus, best known for his development of trigonometry. The most influential mathematicians of the period were Euclid and Archimedes. We consider the end of the age to be with the death in 415 of Hypatia, the first known woman mathematician.
Some problems of the period show how these mathematicians combined the theory of their Classical Greek ancestors with an interest in calculations and more immediately practical mathematics:
How can we prove formulas for areas and volumes of curved figures?
How can we find and prove formulas for centers of mass of different solids?
How can we represent varying relationships among numbers?
