Ancient Greek Science: Mathematics, Astronomy and Practical Knowledge

Greek mathematical writers arranged propositions as proofs, and astronomers used geometrical models to describe movement in the sky. Euclid’s Elements assembled earlier results into a sequence of demonstrations. Eratosthenes combined observations at different places to estimate the Earth’s circumference, while Archimedes applied mathematics to balances and floating bodies. This work developed through the Classical and Hellenistic periods in communities around the Mediterranean.

Calculation also served land measurement and commercial transactions. Egyptian and Babylonian specialists had earlier mathematical and astronomical traditions, and Greek scholars worked within networks of exchange. The routes by which individual methods passed between communities are incompletely documented.

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From practical calculation to demonstrative mathematics

Alphabetic Greek writing is attested in the eighth century BCE. Earlier Mycenaean tablets document accounting, while later Greek treatises preserve extended mathematical arguments.

Later writers credited Thales and Pythagoras with foundational mathematical work and reported travel to Egypt or Babylon. Neither left a surviving mathematical book. The theorem bearing Pythagoras’s name concerns a relationship known in earlier mathematical traditions, although the attribution of particular proofs remains uncertain. Pythagorean communities connected arithmetic with geometry and musical intervals. Philolaus and Archytas belong to this longer tradition. Archytas devised a three-dimensional geometrical construction for doubling a cube and investigated mathematical means.

During the fifth and fourth centuries BCE, mathematical investigation became more securely visible in written sources. Hippocrates of Chios wrote an Elements in a tradition later continued by Euclid and investigated geometrical figures related to the problem of squaring the circle. Theodorus of Cyrene was associated with demonstrations concerning the irrationality of square roots, while Theaetetus was later credited with systematic work on incommensurable magnitudes and regular polyhedra. These subjects exposed a difficulty in treating lengths as if they were always expressible as ratios of whole numbers.

Eudoxus of Cnidus developed a theory of proportion capable of handling magnitudes that could not be compared through ordinary whole-number ratios. His work also supported geometrical demonstrations concerning areas and volumes. Archimedes later credited Eudoxus with proving that a cone has one-third the volume of a cylinder with the same base and height.

Geometry, number and proof

Greek arithmetic generally concerned positive whole numbers, while geometry treated lengths and other continuous magnitudes. Euclid’s Elements, compiled at Alexandria, organised earlier results into books covering plane figures, proportions and solids. The Elements also offered procedures that had practical value. Its number-theoretical books included methods for finding the greatest common divisor of two numbers, discussions of divisibility and a proof implying that prime numbers are limitless.

Doubling the cube, trisecting an arbitrary angle and squaring the circle became prominent construction problems. Investigators used curves such as the conchoid or constructions with a marked ruler. Later mathematics established that the general problems cannot be solved using only an unmarked straightedge and compass.

Apollonius of Perga developed the study of conic sections, the curves formed when a plane intersects a cone. His work treated ellipses, parabolas and hyperbolas through geometrical methods rather than modern algebraic equations. The subject was theoretical, but it also provided techniques that could be adapted to astronomy and mechanics. Archimedes extended the study of regular forms to semiregular, or Archimedean, solids.

Observation, calendars and the changing sky

Greek astronomy combined direct observation with geometrical models. Earlier poetic traditions, including the works associated with Homer and Hesiod, named stars and constellations and connected their risings with seasons, navigation and agricultural timing. In the Classical period, astronomers paid closer attention to solstices, equinoxes, the risings and settings of stars, eclipses and the changing positions of planets.

Greek civic calendars were commonly lunisolar, with months adjusted against the solar year. Meton proposed a nineteen-year cycle bringing lunar phases and seasons into approximate alignment. Callippus combined four such cycles and removed a day. Parapegmata recorded astronomical and seasonal observations for reference.

Simple instruments made observation more systematic. A gnomon, a vertical rod whose shadow changes with the Sun’s position, could help identify solstices and equinoxes. Sundials measured daytime hours, while sighting instruments such as the dioptra supported angular observations. Greek astronomers also used or developed devices associated with spherical astronomy, including armillary constructions and instruments for measuring the positions of stars and the Sun.

The concept of the celestial sphere provided a geometrical way to describe the daily movement of the stars. In this model, the observer treats the heavens as a sphere turning around an axis, with the celestial equator and the ecliptic as important circles. The ecliptic is the Sun’s apparent annual path among the stars, inclined to the celestial equator. Autolycus of Pitane’s works on the rotating sphere and on the risings and settings of stars, written around the end of the fourth century BCE, are among the earliest surviving technical treatments of this subject.

Models of the cosmos

By the Classical period, many Greek astronomers and philosophers accepted a spherical Earth, although earlier views had imagined the Earth as flat or cylindrical. The spherical model made it possible to explain differences in the visible sky at different latitudes and to use geometrical relationships in astronomy. It also encouraged attempts to estimate the size of the Earth and the distances of heavenly bodies.

Most Greek astronomical models placed a stationary Earth at the centre. Pythagorean accounts instead included a central fire, and Heraclides was associated with a rotating Earth. In the third century BCE, Aristarchus proposed that Earth orbited the Sun. His heliocentric system remained a minority position.

Eudoxus of Cnidus developed a model of nested, concentric spheres to represent the motions of the Sun, Moon and planets. Each body was carried by several spheres rotating at different speeds and with different axes. The model supplied a geometrical account of apparently irregular planetary paths using combinations of regular circular motions. Its weakness was that concentric spheres kept the heavenly bodies at fixed distances from the Earth, while changes in apparent size and brightness suggested that those distances varied.

Hellenistic astronomers increasingly worked with eccentrics and epicycles. An eccentric circle has a centre displaced from the Earth, while an epicycle is a smaller circle whose centre moves along a larger circle, or deferent. Combining these motions could reproduce changes in speed, retrograde motion and the varying apparent distance of a planet. Apollonius of Perga is associated with the development of this approach, and Hipparchus of Nicaea used geometrical models and observations to describe the motions of the Sun and Moon and to improve eclipse predictions.

Measuring the Earth and the heavens

Aristarchus used the geometry of a right triangle at the Moon’s quarter phase to estimate the relative distances of Sun and Moon. The method depended on measuring the angle accurately, and his resulting ratio placed the Sun much too close by modern measurements.

Eratosthenes of Cyrene, working in Alexandria, estimated the circumference of the Earth by comparing the Sun’s noon altitude at Alexandria with its position at Syene during the summer solstice. He combined the angular difference with an estimated distance between the two places. His surviving result is given as 252,000 stades, although the modern length of the stade is uncertain, making conversion into kilometres problematic.

Hipparchus assembled stellar observations and is credited with a catalogue, known principally through later evidence and surviving fragments rather than a complete original text. Comparing observations from different dates allowed him to identify precession, the changing position of the equinoctial points against the stars.

These problems encouraged the development of trigonometric techniques. Greek astronomers commonly used tables of chords rather than the modern sine function. A chord is the straight line joining two points on a circle, and tables of its length allowed observers to translate measured angles into distances and to determine parameters for astronomical models. Babylonian astronomical practice probably contributed the use of the 360-part division of the circle, though the direction and details of exchange remain difficult to establish.

Practical mathematics and mechanical applications

Theoretical mathematics remained connected to practical work. Land measurement required calculations of lengths and areas, while commerce required procedures for handling quantities and ratios. Extant mathematical problems include computational rules and practical considerations rather than only formal proofs. Geometry also supported astronomical instruments, surveying and the representation of motion.

Archimedes of Syracuse demonstrated how mathematical reasoning could address physical questions. In Measurement of a Circle, he used the method of exhaustion, comparing a curved figure with inscribed and circumscribed polygons to place limits on its area and on the value of pi. In On the Sphere and Cylinder, he studied the areas and volumes of spheres and cylinders. His mechanical investigations used the law of the lever, while The Sand-Reckoner developed a system for expressing extremely large numbers by working with units based on the myriad, or 10,000.

Mechanics was treated as a subject related to geometry. Works associated with Archimedes, Philo of Byzantium and later Hero of Alexandria discussed levers, balances, motion and devices. Water clocks and sundials applied regular movement and the measurement of time. Mechanical models of the heavens also combined calculation with skilled manufacture. The Antikythera mechanism, an assemblage of gears recovered from a shipwreck and generally dated to the Hellenistic or early Roman period, shows that complex astronomical calculations could be embodied in a portable mechanical device.

Observation and treatment of illness are discussed in Greek medicine.

Alexandria and the Hellenistic network

The Ptolemies supported scholars at Alexandria’s Mouseion and library. Greek served as a language for written mathematical work across the Hellenistic kingdoms. Scholars also worked at centres such as Syracuse and Rhodes and communicated through copied texts and letters.

Hellenistic mathematics was produced by a relatively small community of specialists who communicated through letters, teaching and the circulation of copied works. The surviving corpus reflects these networks unevenly. Many treatises are known only through quotations, summaries or later commentaries. Some works were preserved in Byzantine manuscripts, while others survived through translations into Syriac, Arabic and Latin.