Greek mathematics
Greek mathematics refers to mathematics texts and ideas stemming from the Archaic through the Hellenistic and Roman periods, mostly from the 5th century BC to the 6th century AD, around the shores of the Mediterranean.[1][2] 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.[3] The development of mathematics as a theoretical discipline and the use of deductive reasoning in proofs is an important difference between Greek mathematics and those of preceding civilizations.[4][5]
Origins and etymology[edit]
Greek mathēmatikē ("mathematics") derives from the Ancient Greek: μάθημα, romanized: máthēma, Attic Greek: [má.tʰɛː.ma] Koinē 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.[6][7]
The origins of Greek mathematics are not well documented.[8][9] 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.[10][11][8] 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.[12][13]
Achievements[edit]
Greek mathematics constitutes an important period in the history of mathematics: fundamental in respect of geometry and for the idea of formal proof.[44] Greek mathematicians also contributed to number theory, mathematical astronomy, combinatorics, mathematical physics, and, at times, approached ideas close to the integral calculus.[45][46]
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.[47][48][49][50]
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.[51][52][53] Menelaus, a later geometer and astronomer, wrote a standard work on spherical geometry in the style of the Elements, the Spherics, arguably considered the first treatise in non-Euclidean geometry.[54][55]
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),[56] 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).[57] 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).[58]
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.[59][60][61] The methods employed in these works made no explicit use of algebra, nor trigonometry, the latter appearing around the time of Hipparchus.[62][63]
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.[11][64]
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.[65] The two major sources are
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.
Netz (2011) has counted 144 ancient authors in the mathematical or exact sciences, from whom only 29 works are extant in Greek: Aristarchus, Autolycus, Philo of Byzantium, Biton, Apollonius, Archimedes, Euclid, Theodosius, Hypsicles, Athenaeus, Geminus, Hero, Apollodorus, Theon of Smyrna, Cleomedes, Nicomachus, Ptolemy, Gaudentius, Anatolius, Aristides Quintilian, Porphyry, Diophantus, Alypius, Damianus, Pappus, Serenus, Theon of Alexandria, Anthemius, and Eutocius.[66]
The following works are extant only in Arabic translations:[67][68]