# .

Griechische Mathematik, Zeitlinie, Ereignisse

**Around 600 BC
**

the Cretan poet Epimenides (Επιμενίδης o Κρης) is attributed to have invented the linguistic paradox with his phrase "Cretans are ever liars" - the Liar's Paradox. 2500 years later, the mathematician Kurt Gödel invents an adaptation of the Liar's Paradox that reveals serious axiomatic problems at the heart of modern mathematics.

**Thales of Miletus** (Θαλής ο Μιλήσιος) He brings Babylonian mathematical knowledge to Greece and uses geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore.

**About 530 BC **

**Pythagoras** (Πυθαγόρας ο Σάμιος) leaved Samos and went to Italy. He discovered the dependence of musical intervals on the arithmetical ratios of the lengths of string at the same tension, 2:1 giving an octave, 3:2 the fifth, and 4:3 the fourth. He is also credited with a general formula for finding two square numbers the sum of which is also a square, namely (if m is any odd number), m2+{1/2(m2-1)}2={1/2(m2+1)}2. "The Pythagoreans and Plato [as well as the Renaissance Neo-Platonists] noted that the conclusions they reached deductively agreed to a remarkable extent with the results of observation and inductive inference. Unable to account otherwise for this agreement, they were led to regard mathematics as the study of ultimate, eternal reality, immanent in nature and the universe, rather than as a branch of logic or a tool of science and technology" (Boyer 1949:1). Consequently, when the Pythagoreans developed the theory of geometric magnitudes, by which they were able to compare two surfaces' ratio, they were led, for lack of a system which could handle irrational numbers, to the 'incommensurability problem': Applying the side of a square to the diagonal, no common rational measure is discoverable.

**About 480 BC
**Parmenides of Elea (Παρμενίδης ο Ελεάτης) founded the Eleatic School where he taught that 'all is one,' not an aggregation of units as Pythagoras had said, and that to arrive at a true statement, logical argument is necessary. Truth "is identical with the thought that recognizes it" (Lloyd 1963:327). Change or movement and non-being, he held, are impossibilities since everything is 'full' and 'nothing' is a contradiction which, as such, cannot exist. "Parmenides is said to have been the first to assert that the Earth is spherical in shape...; there was, however, an alternative tradition stating that it was Pythagoras" (Heath 1913:64).

Corollary to Parmenides' rejection of the existence of 'nothing' is the Greek number system which, like the later Roman system, refused to use the Babylonian positional number system with its marker for 'nothing.' Making no clear distinction between nature and geometry, "mathematics, instead of being a science of possible relations, was to [the Greeks] the study of situations thought to subsist in nature" (Boyer 1949:25). Moreover, "almost everything in [Greek] philosophy became subordinated to the problem of change.... All temporal changes observed by the senses were mere permutations and combinations of 'eternal principles,' [and] the historical sequence of events (which formed part of the 'flux') lost all fundamental significance" (Toulmin and Goodfield 1965:40).

Death of Pythagoras

**480-411 BC **

According to Heath Antiphon (Αντιφών ο Αθηναίος) (a sophist) .. d*eserves an honourable place in the history of geometry as having originated the idea of exhausting an area by means of inscribed regular polygons with an ever increasing number of sides, an idea upon which ... Eudoxus founded his epoch-making method of exhaustion. *

**About 465 BC
**Hippasus ('Ιππασος ο Μεταποντίνος) writes of a "sphere of 12 pentagons", which must refer to a dodecahedron.

**About 450 BC** **
**Greeks begin to use written numerals.

Zeno of Elea (Ζήνων ο Ελεάτης) propounded forty **paradoxes** probably to point out inconsistencies in Pythagorean positions. One of the most famous is this: The fleeing and slower runner can never be overtaken by the faster, pursuer because the faster must first reach the point where the slower is at a that time, but by then the slower will be some distance ahead. Other paradoxes made the same or apposite points, but, in fact, mathematical analysis shows that infinite aggregates and the nature of the continuum are not self-contradictory but only counter to intuition.

Oenopides of Chios (Οινοπίδης ο Χίος) probably created the first three of what became Euclid's 'postulates' or assumptions. What is postulated guarantees the existence of straight lines, circles, and points of intersection. That they needed to be postulated is because they require 'movement,' the possibility of which was challenged by the Eleatics (Szabó 1978:276-279).

**About 440-430 BC
**Hippocrates of Chios squared the lune, a major step toward squaring the circle, probably using the theorem that circles are to one another as the squares of their diameters. He writes the

*Elements*which is the first compilation of the elements of geometry

Hippias of Elis (Ιππίας ο Ηλείος) invents the quadratrix which may have been used by him for trisecting an angle and squaring the circle.

**About 425 BC
**Theodorus of Cyrene (Θεόδωρος ο Κυρηναίος) shows that certain square roots are

**irrational**. This had been shown earlier but it is not known by whom.

**About 387 BC
**Plato (Πλάτων ο Αθηναίος) founds his Academy in Athens

**After about 380 BC **

Plato said, in the Timaeus, that "as being is to becoming, so is truth to belief" (Plato 1929:29c). In other words, we can only believe, not know, on the basis of experience. Like, Parmenides, he held being and truth, indeed the world, to be timeless and unchanging, an ideal of which man can only hold the idea. This permitted him a certain amount of flexibility: He was willing to accept objections to his view of the universe, for example, if the new hypothesis would provide a rational explanation or 'save the appearance' presented by the planets. In the Timaeus, he also held that the 'world soul' was constructed according to mathematical principles, and, therefore, these principles are already fixed in the individual. (Forms or ideas that have existence independent of any particular mind came to be called *archetypes*.) He scattered reflections on mathematical issues throughout his dialogues; e.g., in the Meno, he illustrates the difference between a class and its members by reference to the difference between defining 'figure' and enumerating specific figures. References to ratios and proportions are everywhere. The **five regular polygons** he ascribed to the four elements plus the "decoration" of the universe (Plato 1929:55c), probably the animals of the zodiac.

**About 375 BC **

Archytas of Tarentum (Αρχύτας ο Ταραντίνος) develops mechanics. He studies the "classical problem" of doubling the cube and applies mathematical theory to music.

**About 370-360 BC
**

**Eudoxus of Cnidus**(Εύδοξος ο Κνίδιος) invented a model of twenty-seven concentric spheres by which he was able to calculate the sun's annual motions through the zodiac, the moon's motion including its wobble, and the planets' retrograde motion. He used what came much later to be called the 'exhaustion method' for area determination. This method involved inscribing polygons within circles, reducing the difference

*ad absurdum*, and was wholly geometric since there was at that time no knowledge of an arithmetical continuum, at least among the Greeks.

**About 340 BC **

Aristaeus (Αρισταίος ο Κορτωνιάτης) writes Five Books concerning Conic Sections.

**About 335 BC **

Aristotle (Αριστοτέλης ο Σταγειρίτης) settles in Athens, founds Lyceum. He said that universals are abstractions from particulars and that we "have knowledge of a scientific fact when we can prove that it could not be otherwise." But "since observation never shows whether this is the case," he established "reason rather observation at the center of scientific effort" (Park 1990:32). A deductive argument is "a 'demonstration' when the premises from which the reasoning starts are true and primary.... Things are 'true' and 'primary' which are believed on the strength not of anything else but themselves" (Aristotle 1928:100a-100b). Aristotle defined the syllogism as a formal argument in which the conclusion necessarily follows from the premises, and said that the four most common statements of this sort are 'all Subject is Predicate,' 'no S is P,' 'some S is P,' and 'some S is not P.' He also discerned four sorts of 'cause.' The 'formal cause' is the design of a thing. The 'material cause' is that of which it is made. The 'efficient cause' is the maker. And the 'final cause' is the purpose of the thing. Aristotle also insisted on the operational character of mathematics and rejected any metaphysical character of number.

**About 330-310 BC
**Autolycus of Pitane writes

*On the Moving Sphere*which studies the geometry of the sphere. It is written as an astronomy text.

**322 BC
**Death of Aristotle

**About 320 BC **

Eudemus of Rhodes (Εύδημος ο Ρόδιος) writes the *History of Geometry*.

**About 300 BC **

Eukleides, better known as Euclid (Ευκλείδης ο Αλεξανδρεύς), published his *Elements*, a reorganized compilation of geometrical proofs including new proofs and a much earlier essay on the foundations of arithmetic. Elements conclude with the construction of Plato's five regular solids. Euclidean space has no natural edge, and is thus infinite. In his *Optica*, he noted that light travels in straight lines and described the law of reflection.

**About 290-260 BC**

Aristarchus of Samos (Αρίσταρχος ο Σάμιος), in *On the Sizes and Distances of the Sun and Moon*, used trigonometry to estimate the size of the Moon and its distance by the Earth's shadow during a lunar eclipse. Archimedes and others said that he maintained that the Moon revolved around the Earth and the Earth around the Sun which remained stationary like the stars.

**287 BC
**Birth of Archimedes (Αρχιμήδης ο Συρακούσιος)

285 BCPhiletas of Cos - died from considering the Liar Paradox.

**About 280 BC
**The Stoics invent The Crocodile and Baby Paradox.

**280-206 BC
**Chrysippus of Soli, propositional logic, works on logic some considering the Liar Paradox "I'm a liar."

**276 BC
**Birth of Eratosthenes of Cyrene (Ερατοσθένης ο Κυρηναίος)

**About 270 BC
**Death of Euclid

**About 260-250 BC **

Archimedes of Syracuse contributed numerous advances to science including the principle that a body immersed in fluid is buoyed up by a force equal to the weight of the displaced fluid and the **calculation of the value of pi**. "His method was to select definite and limited problems. He then formulated hypotheses which he either regarded, in the Euclidean manner, as self-evident axioms or could verify by simple experiments. The consequences of these he then deduced and experimentally verified" (Crombie 1952:278). Description of the **Loculus** of Archimedes; **Archimedean Polyhedra;** Volume of **Intersection of Two Cylinders; Archimedes' Cattle Problem****.**

**About 230 BC **

**Eratosthenes of Cyrene** develops his **sieve method **for finding all prime numbers.

**About 230 BC
**Nicomedes (Νικομήδης ο Αλεξανδρεύς) writes his treatise

*On conchoid lines*which contain his discovery of the curve known as the "Conchoid of Nicomedes".

**In the early second century BC
**Diocles (Διοκλής ο Αλεξανδρεύς) in

*On Burning Mirrors*, proved the focal property of a parabola and showed how the Sun's rays can be made to reflect a point by rotating a parabolic mirror (Toomer 1978).

**About 225-210 BC?
**Apollonius of Perga (Απολλώνιος ο Περγαίος)( 262 – 190 BC) writes

*Conics.*He introduced probably first the terms 'parabola' and 'hyperbola,' curves formed when a plane intersects a conic section, and 'ellipse,' a closed curve formed when a plane intersects a cone.

**About 225 BC
**Archimedes treatise

*On Spirals*probably also date of discovery of the Archimedes Screw

**Around 212 BC
**

**Death of Archimedes**

**Around 200 BC
**Death of Eratosthenes

**About 60
**

**Hero of Alexandria**explained that the four elements consist of atoms. He also observed that heated air expanded. In

*Catoptrica*, he demonstrated geometrically that the "path taken by a ray of light reflected from a plane mirror is shorter than any other reflected path that might be drawn between the source and the point of observation" (History of Optics 2001:1). Heron of Alexandria writes

*Metrica*(

*Measurements*). It contains formulas for calculating areas and volumes.

**About 90
**Nicomachus of Gerasa (Νικόμαχος ο Γερασηνός) writes

*Arithmetike eisagoge*(

*Introduction to Arithmetic*) which is the first work to treat arithmetic as a separate topic from geometry.

**About 110 **

Menelaus of Alexandria (Μενέλαος ο Αλεξανδρεύς) writes *Sphaerica* which deals with spherical triangles and their application to astronomy.

**About 250** **
**Diophantus of Alexandria (Διόφαντος ο Αλεξανδρεύς) produces the first book on algebra. He is a pioneer in solving certain indeterminate algebraic equations, i.e., an equation in which the variables can take on integer values and has an infinite but denumerable set of solutions: e.g., x+2y=3.

**About 300**

Pappus of Alexandria (Πάππος ο Αλεξανδρεύς) writes *Synagoge* (*Collections*) which is a guide to Greek geometry. He describes five machines in use: cogwheel, lever, pulley, screw, wedge (c. 285)

**About 325
**Iamblichus: (Ιάμβλιχος ο Χαλκιδηνός)

*On Nicomachus's Introduction to Arithmetic*first mention of Casting Out Nines, first description of the Bloom of Thymarides; first

**Amicable**Numbers.

**About 385**

Aurelius Augustinus, later known as Augustine, a Christian saint, writing in Latin, found the Platonist notion of eternal ideas a certain basis for knowledge which he promulgated in his books *Confessiones* and *Civitas Dei*.

[["The fourth and fifth centuries saw the intellectual triumph of [Roman] Christianity in Europe.... In 389 Christian monks sacked the great Greek library in Alexandria.... Since Greek was the language of a literature whose most famous works expressed a pagan culture [and] by 425 Saint Jerome's [official Latin or] Vulgate Bible was being copied and distributed..., Western scholars no longer needed Hebrew or Greek" (Park 1990:78-79).]]

**About 390**

Theon of Alexandria (Θέων ο Αλεξανδρεύς) produces a version of Euclid's *Elements* (with textual changes and some additions) on which almost all subsequent editions are based.

**About 400**

Hypatia (Υπατία) writes commentaries on Diophantus and Apollonius. She is the first recorded female mathematician and she distinguishes herself with remarkable scholarship. She becomes head of the Neo-Platonist school at Alexandria.

**About 450 or later** **
**Proclus (Πρόκλος ο Λύκιος), the final head of Plato's Academy, said that astronomers "do not arrive at conclusions by starting from hypotheses, as is done in the the other sciences; rather, taking conclusions [the appearance of the heavens] as their point of departure, they strive to construct hypotheses from which effects conformable to the original conclusions follow with necessity" (Proclus, quoted by Duhem 1908:20). The astronomer is only interested in saving the appearance of the phenomena, and whether this conforms to reality is left to the other sciences to decide.

**About 500**

Metrodorus (Μητρόδωρος ο Χίος) assembles the *Greek Anthology* consisting of 46 mathematical problems.

**See also:**

T. L. Heath. Astronomy and Mathematics from the Legacy of Greece

Timeline: Mathematicians from http://www-groups.dcs.st-and.ac.uk/~history/Timelines/index.html

http://www-history.mcs.st-andrews.ac.uk/history/index.html

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