Claudius Ptolemy


Introduction: Greek astronomy

    While most of the mathematics we have studied to this point is mathematics for its own sake, it was seen by most Greeks to be a window into the true nature of things in the real world.  Plato of Athens (427 - 327), one of the great lights of Greek philosophy, developed a very influential theory of forms, which asserts that objects in the real world have an ideal existence accessible to us only through contemplation and thought, and that what we perceive through our senses is just a rough approximation, a "shadow", of their ideal essences.  In this respect, geometrical objects, like "line" and "circle", are ideal, distinguishable from any representation we might given them in our real world drawings of "line" or "circle".  Therefore, the mathematician engaged in geometrical thought would be regarded as accomplishing much the same task as a modern-day physicist in his lab would be regarded today, since the geometer would be seen to be in touch with the world in its ideal form.  Knowledge of mathematical objects therefore informs one about physical objects in their ideal form.  This is the subtext with which Euclid's Elements, for instance, should be read.  Euclid seems to believe that he is laying down in rigorously logical form a physics of the ideal world rather than just a theory of abstract mathematics.
    There were amongst the Greek philosophers those who studied natural philosophy, what we would today call physics, astronomy, biology, chemistry, etc.  Pre-eminent amongst these was Aristotle, whose writings in natural philosophy would be taken as definitive works in the subject for many centuries after his death.
    Many of these thinkers made significant advances in astronomy, as we have seen in the mathematical applications of Eratosthenes and Aristarchus.  From Babylonian astronomers, the Greeks inherited the catalog of constellations, especially the constellations of the zodiac, the twelve regions of the sky through which the sun, moon, and planets travelled in their circuits.  The sun makes a complete circuit of the heavens in one year and the twelve signs of the zodiac correspond roughly to the regions in which the sun lies in successive months.  They also adopted the sexagesimal system of notation for doing astronomy.
    In addition to advances in measurement, other Greek astronomers, convinced that they could understand the underlying forces at work in the cosmos by proposing mathematical models that would describe these motions (instead of attributing them to the action of the gods!), devised explanations for the astronomical phenomena they observed.  Eudoxus, for instance, was as accomplished as an astronomer as he was as a mathematician.  According to Aristotle, he taught (in the fourth century, BC) that the rotating celestial sphere of the fixed stars enclosed a nested sequence of many smaller rotating spheres on which the planetary lights rested; on the innermost sphere rested the moon, then the other planets: the Sun, Mercury, Venus, Mars, Jupiter and Saturn (this is found in book 12 of Artistotle's Metaphysics).  The three outer planets, which are sometimes observed to reverse direction in their motion across the sky, required a series of spheres each, rotating on different axes, to explain this strange behavior.  Significantly, Eudoxus was convinced that the heavenly bodies had to be explained solely by means of spheres and spherical motion, for the sphere was the most perfect three-dimensional shape.
    Apollonius (third century, BC) noticed that this did not explain why the planets wax and wane in brightness, suggesting that their distances from earth vary over time.  He solved this problem by supposing that instead of a homocentric system of spheres which were all centered on the earth, the planets moved along eccentric spheres, in which the centers of the planetary spheres varied to allow them to be sometimes nearer and sometimes farther from earth.
    Hipparchus of Nicaea (180? - 125?) was interested in calculating various important astronomical constants, like the length of the year, which he obtained accurately to within 7 minutes, and the amount of the precession of the equinoxes, that very small amount of drift that the axis of the earth is subject to.  (In Babylonian times, the vernal equinox, or position of the sun at the beginning of spring when day and night are of equal length, was found in the constellation Aries, but by the time of Hipparchus, it had drifted into Pisces; today, it is entering "the age of Aquarius".  This position moves at a rate of 50 seconds of arc--that's one sixtieth of one sixtieth of a degree--each year.  This phenomenon is called the precession of the equinoxes.)  He also published a detailed series of star charts, which required careful calculations, especially the need to measure sides of triangles. To facilitate these calculations, he also drew up a table of chords.  For angles measured at the center of a unit circle, in steps of 71/2°, he calculated geometrically the lengths of the chords that subtend those arcs.

For this reason, Hipparchus is often credited as "the father of trigonometry".  The table of chords allowed astronomers to employ computational methods to make predictions of astronomical events.
    Two centuries later, around the year 147 AD,  Claudius Ptolemy (85? - 165?), an Egyptian-born Greek with Roman citizenship, compiled a systematic summary of mathematical astronomy entitled Syntaxis Mathematica (Mathematical Collection).  In much the same way that Euclid's Elements became the authoritative work in plane and solid geometry and Apollonius' Conics was considered the definitive source for information about the conic sections, this work of Ptolemy's managed to overshadow all preceding works on mathematical astronomy.  His success was an ability to organize a great body of information and present it in a clear and concise manner.  It was so well-respected that it took on the name Megale Syntaxis (Great Collection); in later centuries, when it was translated into Arabic, it became al-Magisti (The Greatest), then in Latin Almagestum, and eventually in English The Almagest.  In the Almagest, Ptolemy works out in detail an extension of the Apollonian system for the planetary motions.  He posits that planets move on eccentric spheres about the earth (to handle their variation in brightness and unevenness in the amount of time they stay in either half of their orbits) and additionally that they travel on smaller cycles centered on these spheres (to explain the retrograde motions).

    The Ptolemaic system of the universe became the dominant cosmological model for centuries thereafter, and was not displaced until the seventeenth century by Kepler and Copernicus.
    While no mention of familiar trigonometric functions like sine or cosine appear in Greek geometry (the modern trigonometric functions of sine, cosine, etc., were first used by Hindu mathematicians as early as the fifth century AD and were transmitted through Arab scholars int he tenth century and later into the West), the ideas are equivalent.  We learn from modern trigonometry that the sine of a central angle in a unit circle is the height of its opposite side.  It follows that the formula

relates the ancient chord value to the modern sine value.

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last modified 10/8/02
Copyright (c) Daniel E. Otero