Ancient Greek mathematics

Ancient Greek mathematics refers to the history of mathematical ideas and texts in Ancient Greece during classical and late antiquity, mostly from the 5th century BC to the 6th century AD. Greek mathematicians lived in cities spread around the shores of the ancient Mediterranean, from Anatolia to Italy and North Africa, but were united by Greek culture and the Greek language.

One important difference between Greek mathematics and earlier civilizations was the development of mathematics as a theoretical discipline and the use of deductive reasoning in proofs. The earliest complete work on the subject is Euclid's Elements, written during the Hellenistic period. Famous mathematicians like Archimedes and Apollonius also lived during this time.

The works of ancient Greek mathematicians were copied and translated into Arabic and Latin, influencing mathematics in the Islamic world and in Medieval Europe. During the Renaissance, texts by Euclid, Archimedes, and others helped shape the development of early modern mathematics. Some problems from Ancient Greek mathematics were only solved much later, contributing to new areas of math like non-Euclidean geometry.

Etymology

The Greek word mathēmatikē comes from máthēma, meaning "lesson," and ultimately from the verb manthánō, meaning "I learn." In ancient times, certain subjects like arithmetic, geometry, astronomy, and harmonics were considered special branches of learning. These four subjects were later grouped together as the quadrivium. The mathematician Geminus of Rhodes divided these into two parts: one about things we can think about (arithmetic and geometry) and the other about things we can see or measure (astronomy and harmonics). He also added subjects like mechanics, optics, geodesy, and logistics, which are now part of physics or applied mathematics.

Origins

The beginnings of Greek mathematics are still a mystery. Early civilizations in Greece, like the Minoan and Mycenaean cultures, had writing systems, but no mathematical texts from them have been found. Before the Greeks, mathematics in places like Babylon and Egypt mainly helped with measuring land and keeping records. When writing returned to Greece around the 7th century BC, new mathematical ideas began to develop. We know about early Greek math mostly from later writers, like Plato and Aristotle, who talked about mathematicians from the 4th century BC.

Ancient Greeks often said that famous thinkers like Thales of Miletus and Pythagoras of Samos started Greek mathematics, but we don’t have writings from them to prove it. Real progress in writing about math started in the 5th century BC. For example, Hippocrates of Chios wrote the first known book called Elements, and tried to solve tricky problems like squaring the circle. Other mathematicians, such as Philolaus of Croton and Antiphon, also explored geometry and numbers. Later, thinkers like Plato and his friends helped spread mathematical ideas.

Hellenistic and early Roman period

Ancient Greek mathematics reached its highest point during the Hellenistic and early Roman periods. After Alexander the Great's conquests, Greek culture and language spread widely, mixing with other traditions and creating new centers of learning. One of the most important was the Mouseion in Alexandria.

Great mathematicians like Euclid, Archimedes, Apollonius, and Ptolemy made big discoveries. Euclid gathered many math ideas into his book Elements, which became very important for geometry. Archimedes found smart ways to measure shapes and even very big numbers. Trigonometry began to develop, helping astronomers understand the stars and planets better.

Late antiquity

Mathematicians during the later Roman era were mainly known for their writings about earlier works, which helped preserve important ideas from lost texts.

Diophantus, who lived by the 3rd century AD, wrote the Arithmetica, a book with 290 algebra problems. Only part of this book survives today. Pappus of Alexandria wrote the Collection, a survey of earlier math in eight books, some of which still exist. He discussed topics like doubling the cube and squaring the circle.

Many mathematicians from this time wrote commentaries, or explanations, of earlier works. These commentaries often preserved information from books that are now lost.

Reception and legacy

Most of the mathematical writings from Ancient Greece have been lost, but we still know about many Greek mathematicians because their works were mentioned by others. Some of the mathematicians whose works survive include Euclid, Aristarchus of Samos, Archimedes, Apollonius of Perga, Hipparchus, Diophantus, Ptolemy, and Pappus of Alexandria.

Even though many original Greek math books are gone, we have copies from later times. Some of the oldest surviving pieces of Greek math are from the 3rd century BC. These include parts of Euclid’s Elements found on pieces of pottery. Many of the surviving Greek math books we have today were copied much later, during the Byzantine Empire and the Islamic Golden Age. These copies helped keep Greek math ideas alive for future generations.