Greek mathematics refers to
mathematics texts and ideas stemming from the
Archaic through the
Hellenistic and
Roman periods, mostly attested from the late 7th century BC to the 6th century AD, around the shores of the
Mediterranean. Greek mathematicians lived in cities spread over the entire region, from
Anatolia to
Italy and
North Africa, but were united by
Greek culture and the
Greek language.^{
[1]} The development of mathematics as a theoretical discipline and the use of
proofs is an important difference between Greek mathematics and those of preceding civilizations.^{
[2]}^{
[3]}
Origins and etymology
Greek mathēmatikē ("mathematics") derives from the
Ancient Greek: μάθημα,
romanized: máthēma,
Attic Greek: [má.tʰɛː.ma]Koine Greek: [ˈma.θi.ma], from the verb manthanein, "to learn". Strictly speaking, a máthēma could be any branch of learning, or anything learnt; however, since antiquity certain mathēmata (mainly arithmetic, geometry, astronomy, and harmonics) were granted special status.^{
[4]}^{
[5]}
The origins of Greek mathematics are not well documented.^{
[6]}^{
[7]} The earliest advanced civilizations in
Greece and
Europe were the
Minoan and later
Mycenaean civilizations, both of which flourished during the 2nd millennium BC. While these civilizations possessed writing and were capable of advanced engineering, including four-story palaces with drainage and
beehive tombs, they left behind no mathematical documents.
Though no direct evidence is available, it is generally thought that the neighboring
Babylonian and
Egyptian civilizations had an influence on the younger Greek tradition.^{
[8]}^{
[9]}^{
[6]} Unlike the flourishing of
Greek literature in the span of 800 to 600 BC, not much is known about Greek mathematics in this early period—nearly all of the information was passed down through later authors, beginning in the mid-4th century BC.^{
[10]}^{
[11]}
Archaic and Classical periods
Greek mathematics allegedly began with
Thales of Miletus (c. 624–548 BC). Very little is known about his life, although it is generally agreed that he was one of the
Seven Wise Men of Greece. According to
Proclus, he traveled to Babylon from where he learned mathematics and other subjects, coming up with the proof of what is now called
Thales' Theorem.^{
[12]}^{
[13]}
An equally enigmatic figure is
Pythagoras of Samos (c. 580–500 BC), who supposedly visited Egypt and Babylon,^{
[11]}^{
[14]} and ultimately settled in
Croton,
Magna Graecia, where he started a kind of brotherhood.
Pythagoreans supposedly believed that "all is number" and were keen in looking for mathematical relations between numbers and things.^{
[15]} Pythagoras himself was given credit for many later discoveries, including the construction of the
five regular solids. However, Aristotle refused to attribute anything specifically to Pythagoras and only discussed the work of the Pythagoreans as a group.^{
[16]}^{
[17]}
Almost half of the material in
Euclid's Elements is customarily attributed to the Pythagoreans, including the discovery of irrationals, attributed to
Hippasus (c. 530–450 BC) and
Theodorus (fl. 450 BC).^{
[18]} The greatest mathematician associated with the group, however, may have been
Archytas (c. 435-360 BC), who solved the problem of
doubling the cube, identified the
harmonic mean, and possibly contributed to
optics and
mechanics.^{
[18]}^{
[19]} Other mathematicians active in this period, not fully affiliated with any school, include
Hippocrates of Chios (c. 470–410 BC),
Theaetetus (c. 417–369 BC), and
Eudoxus (c. 408–355 BC).
Greek mathematics also drew the attention of philosophers during the
Classical period.
Plato (c. 428–348 BC), the founder of the
Platonic Academy, mentions mathematics in several of his dialogues.^{
[20]} While not considered a mathematician, Plato seems to have been influenced by
Pythagorean ideas about number and believed that the elements of matter could be broken down into geometric solids.^{
[21]} He also believed that geometrical proportions bound the
cosmos together rather than physical or mechanical forces.^{
[22]}Aristotle (c. 384–322 BC), the founder of the
Peripatetic school, often used mathematics to illustrate many of his theories, as when he used geometry in his theory of the rainbow and the theory of proportions in his analysis of motion.^{
[22]} Much of the knowledge about ancient Greek mathematics in this period is thanks to records referenced by Aristotle in his own works.^{
[11]}^{
[23]}
Greek mathematics and astronomy reached its acme during the Hellenistic and early
Roman periods, and much of the work represented by authors such as
Euclid (fl. 300 BC),
Archimedes (c. 287–212 BC),
Apollonius (c. 240–190 BC),
Hipparchus (c. 190–120 BC), and
Ptolemy (c. 100–170 AD) was of a very advanced level and rarely mastered outside a small circle.^{
[27]} There is also evidence of combining mathematical knowledge with technical or practical applications, as found for instance in the work of
Menelaus of Alexandria (c. 70–130 AD), who wrote a work dealing with the geometry of the sphere and its application to astronomical measurements and calculations (Spherica).^{
[28]} Similar examples of
applied mathematics include the construction of analogue computers like the
Antikythera mechanism,^{
[29]}^{
[30]} the accurate measurement of the
circumference of the Earth by
Eratosthenes (276–194 BC), and the mechanical works of
Hero (c. 10–70 AD).^{
[31]}
Several centers of learning appeared during the Hellenistic period, of which the most important one was the
Mouseion in
Alexandria,
Egypt, which attracted scholars from across the Hellenistic world (mostly Greek, but also
Egyptian,
Jewish,
Persian, among others).^{
[32]}^{
[33]} Although few in number, Hellenistic mathematicians actively communicated with each other; publication consisted of passing and copying someone's work among colleagues.^{
[34]}
Later mathematicians in the Roman era include
Diophantus (c. 214–298 AD), who wrote on
polygonal numbers and a work in pre-modern algebra (Arithmetica),^{
[35]}^{
[36]}Pappus of Alexandria (c. 290–350 AD), who compiled many important results in the Collection,^{
[37]}Theon of Alexandria (c. 335–405 AD) and his daughter
Hypatia (c. 370–415 AD), who edited Ptolemy's Almagest and other works,^{
[38]}^{
[39]} and
Eutocius of Ascalon (
c. 480–540 AD), who wrote commentaries on treatises by Archimedes and Apollonius.^{
[40]} Although none of these mathematicians, save perhaps Diophantus, had notable original works, they are distinguished for their commentaries and expositions. These commentaries have preserved valuable extracts from works which have perished, or historical allusions which, in the absence of original documents, are precious because of their rarity.^{
[41]}^{
[42]}
Most of the mathematical texts written in Greek survived through the copying of manuscripts over the centuries, though some fragments dating from antiquity have been found above all in
Egypt, and to a much lesser extent in
Greece,
Asia Minor,
Mesopotamia, and
Sicily.^{
[27]}
Eudoxus of Cnidus developed a theory of proportion that bears resemblance to the modern theory of
real numbers using the
Dedekind cut, developed by
Richard Dedekind, who acknowledged Eudoxus as inspiration.^{
[46]}^{
[47]}^{
[48]}^{
[49]}
Euclid, who presumably wrote on optics, astronomy, and harmonics, collected many previous mathematical results and theorems in the Elements, a canon of geometry and elementary number theory for many centuries.^{
[50]}^{
[51]}^{
[52]}
Archimedes made use of a technique dependent on a form of
proof by contradiction to reach answers to problems with an arbitrary degree of accuracy, while specifying the limits within which the answers lay. Known as the
method of exhaustion, Archimedes employed it in several of his works, including an approximation to
π (Measurement of the Circle),^{
[53]} and a proof that the area enclosed by a
parabola and a straight line is 4/3 times the area of a
triangle with equal base and height (Quadrature of the Parabola).^{
[54]} Archimedes also showed that the number of grains of sand filling the universe was not uncountable, devising his own counting scheme based on the
myriad, which denoted 10,000 (The Sand-Reckoner).^{
[55]}
The most characteristic product of Greek mathematics may be the theory of
conic sections, which was largely developed in the
Hellenistic period, starting with the work of
Menaechmus and perfected primarily under Apollonius in his work Conics.^{
[56]}^{
[57]}^{
[58]} The methods employed in these works made no explicit use of
algebra, nor
trigonometry, the latter appearing around the time of
Hipparchus.^{
[59]}^{
[60]}
Ancient Greek mathematics was not limited to theoretical works but was also used in other activities, such as business transactions and in land mensuration, as evidenced by extant texts where
computational procedures and practical considerations took more of a central role.^{
[9]}^{
[61]}
Transmission and the manuscript tradition
Although the earliest
Greek language texts on mathematics that have been found were written after the Hellenistic period, many of these are considered to be copies of works written during and before the Hellenistic period.^{
[62]} The two major sources are
Byzantine codices, written some 500 to 1500 years after their originals, and
Nevertheless, despite the lack of original manuscripts, the dates of Greek mathematics are more certain than the dates of surviving Babylonian or Egyptian sources because a large number of overlapping chronologies exist. Even so, many dates are uncertain; but the doubt is a matter of decades rather than centuries.
^Knorr, W. (1981). On the early history of axiomatics: The interaction of mathematics and philosophy in Greek Antiquity. D. Reidel Publishing Co. pp. 145–186. Theory Change, Ancient Axiomatics, and Galileo's Methodology, Vol. 1
^Kahn, C. H. (1991). Some remarks on the origins of Greek science and philosophy. Science and Philosophy in Classical Greece: Garland Publishing Inc. pp. 1–10.
^
^{a}^{b}Jones, A. (1994).
"Greek mathematics to AD 300". Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences: Volume One. pp. 46–57. Retrieved 2021-05-26.
^"Hellenistic Mathematics". The Story of Mathematics - A History of Mathematical Thought from Ancient Times to the Modern Day. Retrieved 2023-01-07.
^J J O'Connor and E F Robertson (October 1999).
"How do we know about Greek mathematics?". The MacTutor History of Mathematics archive. University of St. Andrews. Archived from
the original on 30 January 2000. Retrieved 18 April 2011.
^Netz, Reviel (27 September 2011). "The Bibliosphere of Ancient Science (Outside of Alexandria)". NTM Zeitschrift für Geschichte der Wissenschaften, Technik und Medizin (in German). 19 (3): 239–269.
doi:
10.1007/s00048-011-0057-2.
ISSN1420-9144.
PMID21946891.
^Toomer, G. J. (January 1984). "Lost greek mathematical works in arabic translation". The Mathematical Intelligencer. 6 (2): 32–38.
doi:
10.1007/BF03024153.