c. 417 – c. 369 BC) Proved that there are exactly five
regularconvexpolyhedra (it is emphasized that it was, in particular, proved that there does not exist any regular convex polyhedra other than these five). This fact led these five solids, now called the
Platonic solids, to play a prominent role in the philosophy of
Plato (and consequently, also influenced later
Western Philosophy) who associated each of the four
classical elements with a regular solid:
earth with the
air with the
water with the
fire with the
tetrahedron (of the fifth Platonic solid, the
dodecahedron, Plato obscurely remarked, "...the god used [it] for arranging the constellations on the whole heaven"). The last book (Book XIII) of the
Euclid's Elements, which is probably derived from the work of Theaetetus, is devoted to constructing the Platonic solids and describing their properties;
Andreas Speiser has advocated the view that the construction of the 5 regular solids is the chief goal of the deductive system canonized in the Elements.AstronomerJohannes Kepler proposed a model of the
Solar System in which the five solids were set inside one another and separated by a series of inscribed and circumscribed spheres.
Aristarchus of Samos (
c. 310 – c. 230 BC) presented the first known
heliocentric model that placed the
Sun at the center of the known universe with the Earth revolving around it. Aristarchus identified the "central fire" with the Sun, and he put the other planets in their correct order of distance around the Sun. In On the Sizes and Distances, he calculates the sizes of the
Moon, as well as their distances from the Earth in terms of Earth's radius. However,
c. 276 – c. 194/195 BC) was the first person to calculate the circumference of the Earth.
c. 135 – c. 51 BC) also measured the diameters and distances of the Sun and the Moon as well as the Earth's diameter; his measurement of the diameter of the Sun was more accurate than Aristarchus', differing from the modern value by about half.
fl. 300 BC) is often referred to as the "founder of
geometry" or the "father of geometry" because of his incredibly influential
treatise called the Elements, which was the first, or at least one of the first, axiomatized deductive systems.
c. 201–215 – c. 285–299 AD) wrote Arithmetica which dealt with solving algebraic equations and also introduced
syncopated algebra, which was a precursor to modern symbolic algebra. Because of this, Diophantus is sometimes known as "the father of
algebra," which is a title he shares with
Muhammad ibn Musa al-Khwarizmi. In contrast to Diophantus, al-Khwarizmi wasn't primarily interested in integers and he gave
an exhaustive and systematic description of solving
quadratic equations and some higher order algebraic equations. However, al-Khwarizmi did not use symbolic or syncopated algebra but rather "
rhetorical algebra" or ancient Greek "geometric algebra" (the ancient Greeks had expressed and solved some particular instances of algebraic equations in terms of geometric properties such as length and area but they did not solve such problems in general; only particular instances). An example of "geometric algebra" is: given a triangle (or rectangle, etc.) with a certain area and also given the length of some of its sides (or some other properties), find the length of the remaining side (and justify/prove the answer with geometry). Solving such a problem is often equivalent to finding the roots of a polynomial.
The conquests of
Alexander the Great around
c. 330 BC led to Greek culture being spread around much of the Mediterranean region, especially in
Alexandria, Egypt. This is why the Hellenistic period of Greek mathematics is typically considered as beginning in the 4th century BC. During the Hellenistic period, many people living in those parts of the
Mediterranean region subject to Greek influence ended up adopting the Greek language and sometimes also Greek culture. Consequently, some of the Greek mathematicians from this period may not have been "ethnically Greek" with respect to the modern
Western notion of
ethnicity, which is much more rigid than most other notions of ethnicity that existed in the Mediterranean region at the time.
Ptolemy, for example, was said to have originated from
Upper Egypt, which is far South of
Alexandria, Egypt. Regardless, their contemporaries considered them Greek.
Straightedge and compass constructions
Creating a regular
hexagon with a straightedge and compass
For the most part,
straightedge and compass constructions dominated ancient Greek mathematics and most
theorems and results were stated and proved in terms of geometry. The straightedge is an idealized
ruler that can draw arbitrarily long lines but (unlike modern rulers) it has no markings on it. A
compass can draw a circle starting from two given points: the center and a point on the circle. Thus a straightedge is used to construct lines while compasses are used to construct circles.
Geometric constructions using lines, such as those formed by the straightedge of a taut rope, and circles, such as those formed by a compass or by a straightedge (such a taut rope) rotated around a point, were also used outside of the Mediterranean region.
Shulba Sutras from the
Vedic period of
Indian mathematics, for instance, contains geometric instructions on how to physically construct a (quality)
fire-altar by using a taut rope as a straightedge. These alters could have various shapes but for theological reasons, they were all required to have the same area. This consequently required a high precision construction along with (written) instructions on how to geometrically construct such alters with the tools that were most widely available throughout the
Indian subcontinent (and elsewhere) at the time. Ancient Greek mathematicians went one step further by
axiomatizing plane geometry in such a way that straightedge and compass constructions became
Euclid's Elements was the culmination of this effort and for over two thousand years, even as late as the 19th century, it remained the "standard text" on mathematics throughout the Mediterranean region (including Europe and the Middle East), and later also in North and South America after
Ancient Greek mathematicians are known to have solved specific instances of
polynomial equations with the use of straightedge and compass constructions, which simultaneously gave a geometric proof of the solution's correctness. Once a construction was completed, the answer could be found by measuring the length of a certain line segment (or possibly some other quantity). A quantity multiplied by itself, such as for example, would often be constructed as a literal square with sides of length which is why the second power "" is referred to as " squared" in ordinary spoken language. Thus problems that would today be considered "algebra problems" were also solved by ancient Greek mathematicians, although not in full generality. A complete guide to systematically solving low-order polynomials equations for an unknown quantity (instead of just specific instances of such problems) would not appear until The Compendious Book on Calculation by Completion and Balancing by
Muhammad ibn Musa al-Khwarizmi, who used Greek geometry to "prove the correctness" of the solutions that were given in the treatise. However, this treatise was entirely rhetorical (meaning that everything, including numbers, was written using words structured in ordinary sentences) and did not have any "algebraic symbols" that are today associated with algebra problems – not even the
syncopated algebra that appeared in Arithmetica.