Astronomy and trigonometry
Calculation
In an inspired use of their geometry, the Greeks did what no earlier people seems to have done: they geometrized the heavens by supposing that the Sun, Moon, and planets move around a stationary Earth on a rotating circle or set of circles, and they calculated the speed of rotation of these supposititious circles from observed motions. Thus they assigned to the Sun a circle eccentric to the Earth to account for the unequal lengths of the seasons.
Ptolemy (flourished 127â145 ce in Alexandria, Egypt) worked out complete sets of circles for all the planets. In order to account for phenomena arising from the Earthâs motion around the Sun, the Ptolemaic system included a secondary circle known as an epicycle, whose centre moved along the path of the primary orbital circle, known as the deferent. Ptolemyâs Great Compilation, or Almagest after its Arabic translation, was to astronomy what Euclidâs Elements was to geometry. Contrary to the Elements, however, the Almagest deploys geometry for the purpose of calculation. Among the items Ptolemy calculated was a table of chords, which correspond to the trigonometric sine function later introduced by Indian and Islamic mathematicians. The table of chords assisted the calculation of distances from angular measurements as a modern astronomer might do with the law of sines.
Epistemology
The application of geometry to astronomy reframed the perennial Greek pursuit of the nature of truth. If a mathematical description fit the facts, as did Ptolemyâs explanation of the unequal lengths of the seasons by the eccentricity of the Sunâs orbit, should the description be taken as true of nature? The answer, with increasing emphasis, was âno.â Astronomers remarked that the eccentric orbit representing the Sunâs annual motion could be replaced by a pair of circles, a deferent centred on the Earth and an epicycle the centre of which moved along the circumference of the deferent. That gave two observationally equivalent solar theories based on two quite different mechanisms. Geometry was too prolific of alternatives to disclose the true principles of nature. The Greeks, who had raised a sublime science from a pile of practical recipes, discovered that in reversing the process, in reapplying their mathematics to the world, they had no securer claims to truth than the Egyptian rope pullers.
Ancient geometry: cosmological and metaphysical
Pythagorean numbers and Platonic solids
The Pythagoreans used geometrical figures to illustrate their slogan that all is numberâthus their âtriangular numbersâ (n(nâ1)/2), âsquare numbersâ (n2), and âaltar numbersâ (n3), some of which are shown in the . This principle found a sophisticated application in Platoâs creation story, the Timaeus, which presents the smallest particles, or âelements,â of matter as regular geometrical figures. Since the ancients recognized four or five elements at most, Plato sought a small set of uniquely defined geometrical objects to serve as elementary constituents. He found them in the only three-dimensional structures whose faces are equal regular polygons that meet one another at equal solid angles: the tetrahedron, or pyramid (with 4 triangular faces); the cube (with 6 square faces); the octahedron (with 8 equilateral triangular faces); the dodecahedron (with 12 pentagonal faces); and the icosahedron (with 20 equilateral triangular faces).
The cosmology of the Timaeus had a consequence of the first importance for the development of mathematical astronomy. It guided Johannes Kepler (1571â1630) to his discovery of the laws of planetary motion. Kepler deployed the five regular Platonic solids not as indicators of the nature and number of the elements but as a model of the structure of the heavens. In 1596 he published Prodromus Dissertationum Mathematicarum Continens Mysterium Cosmographicum (âCosmographic Mysteryâ), in which each of the known six planets revolved around the Sun on spheres separated by the five Platonic solids. Although Tycho Brahe (1546â1601), the worldâs greatest observational astronomer before the invention of the telescope, rejected the Copernican model of the solar system, he invited Kepler to assist him at his new observatory outside of Prague. In trying to resolve discrepancies between his original theory and Braheâs observations, Kepler made the capital discovery that the planets move in ellipses around the Sun as a focus.
Measuring the Earth and heavens
Geometry offered Greek cosmologists not only a way to speculate about the structure of the universe but also the means to measure it. South of Alexandria and roughly on the same meridian of longitude is the village of Syene (modern AswÄn), where the Sun stands directly overhead at noon on a midsummer day. At the same moment at Alexandria, the Sunâs rays make an angle α with the tip of a vertical rod, as shown in the . Since the Sunâs rays fall almost parallel on the Earth, the angle subtended by the arc l (representing the distance between Alexandria and Syene) at the centre of the Earth also equals α; thus the ratio of the Earthâs circumference, C, to the distance, l, must equal the ratio of 360° to the angle αâin symbols, C:l = 360°:α. Eratosthenes made the measurements, obtaining a value of about 5,000 stadia for l, which gave a value for the Earthâs circumference of about 250,000 stadia. Because the accepted length of the Greek stadium varied locally, we cannot accurately determine Eratosthenesâ margin of error. However, if we credit the ancient historian Plutarchâs guess at Eratosthenesâ unit of length, we obtain a value for the Earthâs circumference of about 46,250 kmâremarkably close to the modern value (about 15 percent too large), considering the difficulty in accurately measuring l and α. (See Sidebar: Measuring the Earth, Classical and Arabic.)
Aristarchus of Samos (c. 310â230 bce) has garnered the credit for extending the grip of number as far as the Sun. Using the Moon as a ruler and noting that the apparent sizes of the Sun and the Moon are about equal, he calculated values for his treatise âOn the Sizes and Distances of the Sun and Moon.â The great difficulty of making the observations resulted in an underestimation of the solar distance about 20-foldâhe obtained a solar distance, Ï, roughly 1,200 times the Earthâs radius, r. Possibly Aristarchusâ inquiry into the relative sizes of the Sun, Moon, and Earth led him to propound the first heliocentric (âSun-centredâ) model of the universe.
Aristarchusâ value for the solar distance was confirmed by an astonishing coincidence. Ptolemy equated the maximum distance of the Moon in its eccentric orbit with the closest approach of Mercury riding on its epicycle; the farthest distance of Mercury with the closest of Venus; and the farthest of Venus with the closest of the Sun. Thus he could compute the solar distance in terms of the lunar distance and thence the terrestrial radius. His answer agreed with that of Aristarchus. The Ptolemaic conception of the order and machinery of the planets, the most powerful application of Greek geometry to the physical world, thus corroborated the result of direct measurement and established the dimensions of the cosmos for over a thousand years. As the ancient philosophers said, there is no truth in astronomy.