Hipparchus: The Greatest Ancient Astronomical Observer and the Founder of Trigonometry
Hipparchus, who was known to be the Founder of trigonometry, was born in Nicaea, Bithynia by 190 B.C., but spent much of his life in Rhodes and died there by 120 B.C.
Hipparchus is generally considered to be one of the most influential astronomers of antiquity, yet very little information available about him survives; his only extant work is his commentary on the astronomical poem of Aratus (third century B.C.), the Commentary on the Phainomena of Eudoxus and Aratus.
As a young man in Bithynia, Hipparchus compiled records of local weather patterns throughout the year. Such weather calendars (parapēgmata), which synchronized the onset of winds, rains, and storms with the astronomical seasons and the risings and settings of the constellations, were produced by many Greek astronomers from at least as early as the 4th century B.C.
Most of Hipparchus’s adult life, however, seems to have been spent carrying out a program of astronomical observation and research on the island of Rhodes. Ptolemy cites more than 20 observations made there by Hipparchus on specific dates from 147 to 127, as well as three earlier observations from 162 to 158 that may be attributed to him. These must have been only a tiny fraction of Hipparchus’s recorded observations. In fact, his astronomical writings were numerous enough that he published an annotated list of them.
Hipparchus was recognized as the first mathematician known to have possessed a trigonometric table, which he needed when computing the eccentricity of the orbits of the Moon and Sun. He tabulated values for the chord function, which gives the length of the chord for each angle.
He described the chord table in a work, now lost, called Tōn en kuklōi eutheiōn (Of Lines Inside a Circle) by Theon of Alexandria (4th century) in his commentary on the Almagest I.10; some claim his table may have survived in astronomical treatises in India, for instance the Surya Siddhanta. Trigonometry was a significant innovation, because it allowed Greek astronomers to solve any triangle, and made it possible to make quantitative astronomical models and predictions using their preferred geometric techniques.
Hipparchus was the first to show that the stereographic projection is conformal, and that it transforms circles on the sphere that do not pass through the center of projection to circles on the plane.
Besides geometry, Hipparchus also used arithmetic techniques developed by the Chaldeans. He was one of the first Greek mathematicians to do this, and in this way expanded the techniques available to astronomers and geographers.
There are several indications that Hipparchus knew spherical trigonometry, but the first surviving text of it is that of Menelaus of Alexandria in the 1st century, who on that basis is now commonly credited with its discovery. (Previous to the finding of the proofs of Menelaus a century ago, Ptolemy was credited with the invention of spherical trigonometry.) Ptolemy later used spherical trigonometry to compute things like the rising and setting points of the ecliptic, or to take account of the lunar parallax. Hipparchus may have used a globe for these tasks, reading values off coordinate grids drawn on it, or he may have made approximations from planar geometry, or perhaps used arithmetical approximations developed by the Chaldeans. He might have used spherical trigonometry.
Hipparchus’s most important astronomical work concerned the orbits of the Sun and Moon, a determination of their sizes and distances from the Earth, and the study of eclipses. Like most of his predecessors—Aristarchus of Samos was an exception—Hipparchus assumed a spherical, stationary Earth at the centre of the universe (the geocentric cosmology). From this perspective, the Sun, Moon, Mercury, Venus, Mars, Jupiter, and Saturn (all of the solar system bodies visible to the naked eye), as well as the stars (whose realm was known as the celestial sphere), revolved around the Earth each day.
Every year the Sun traces out a circular path in a west-to-east direction relative to the stars (this is in addition to the apparent daily east-to-west rotation of the celestial sphere around the Earth). Hipparchus had good reasons for believing that the Sun’s path, known as the ecliptic, is a great circle, i.e., that the plane of the ecliptic passes through the Earth’s centre. The two points at which the ecliptic and the equatorial plane intersect, known as the vernal and autumnal equinoxes, and the two points of the ecliptic farthest north and south from the equatorial plane, known as the summer and winter solstices, divide the ecliptic into four equal parts. However, the Sun’s passage through each section of the ecliptic, or season, is not symmetrical. Hipparchus attempted to explain how the Sun could travel with uniform speed along a regular circular path and yet produce seasons of unequal length.
Hipparchus knew of two possible explanations for the Sun’s apparent motion, the eccenter and the epicyclic models. These models, which assumed that the apparent irregular motion was produced by compounding two or more uniform circular motions, were probably familiar to Greek astronomers well before Hipparchus. His contribution was to discover a method of using the observed dates of two equinoxes and a solstice to calculate the size and direction of the displacement of the Sun’s orbit. With Hipparchus’s mathematical model one could calculate not only the Sun’s orbital location on any date, but also its position as seen from the Earth. The history of celestial mechanics until Johannes Kepler (1571–1630) was mostly an elaboration of Hipparchus’s model.
Hipparchus also tried to measure as precisely as possible the length of the tropical year—the period for the Sun to complete one passage through the ecliptic. He made observations of consecutive equinoxes and solstices, but the results were inconclusive: he could not distinguish between possible observational errors and variations in the tropical year. However, by comparing his own observations of solstices with observations made in the 5th and 3rd centuries bc, Hipparchus succeeded in obtaining an estimate of the tropical year that was only 6 minutes too long.
He was then in a position to calculate equinox and solstice dates for any year. Applying this information to recorded observations from about 150 years before his time, Hipparchus made the unexpected discovery that certain stars near the ecliptic had moved about 2° relative to the equinoxes. He contemplated various explanations—for example, that these stars were actually very slowly moving planets—before he settled on the essentially correct theory that all the stars made a gradual eastward revolution relative to the equinoxes. Since Nicolaus Copernicus (1473–1543) established his heliocentric model of the universe, the stars have provided a fixed frame of reference, relative to which the plane of the equator slowly shifts—a phenomenon referred to as the precession of the equinoxes.
Hipparchus also analyzed the more complicated motion of the Moon in order to construct a theory of eclipses. In addition to varying in apparent speed, the Moon diverges north and south of the ecliptic, and the periodicities of these phenomena are different. Hipparchus adopted values for the Moon’s periodicities that were known to contemporary Babylonian astronomers, and he confirmed their accuracy by comparing recorded observations of lunar eclipses separated by intervals of several centuries. It remained, however, for Ptolemy (ad 127–145) to finish fashioning a fully predictive lunar model.
In On Sizes and Distances (now lost), Hipparchus reportedly measured the Moon’s orbit in relation to the size of the Earth. He had two methods of doing this. One method used an observation of a solar eclipse that had been total near the Hellespont (now called the Dardanelles) but only partial at Alexandria. Hipparchus assumed that the difference could be attributed entirely to the Moon’s observable parallax against the stars, which amounts to supposing that the Sun, like the stars, is indefinitely far away. (Parallax is the apparent displacement of an object when viewed from different vantage points). Hipparchus thus calculated that the mean distance of the Moon from the Earth is 77 times the Earth’s radius. In the second method he hypothesized that the distance from the centre of the Earth to the Sun is 490 times the Earth’s radius—perhaps chosen because that is the shortest distance consistent with a parallax that is too small for detection by the unaided eye. Using the visually identical sizes of the solar and lunar discs, and observations of the Earth’s shadow during lunar eclipses, Hipparchus found a relationship between the lunar and solar distances that enabled him to calculate that the Moon’s mean distance from the Earth is approximately 63 times the Earth’s radius. (The true value is about 60 times.)
References:
http://www.hps.cam.ac.uk/starry/hipparchus.html
http://www.gap-system.org/~history/Biographies/Hipparchus.html
http://www.britannica.com/EBchecked/topic/266559/Hipparchus
http://www.astro.cornell.edu/academics/courses/astro201/hipparchus.htm
