Hipparchus and His Contributions to Astronomy and Mathematics

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

By: Sofia Martimianakis

994615387

For: Professor Clarke

SCI199Y1 L0111

Little is known about the life of Claudius Hipparchus (190-120 BCE), but references to his work exist in the texts of Ptolemy, specifically the Almagest, and of Theon of Alexandria (O'Connor, J. J. and Robertson, E. F., 1999). Although he was given much praise during his time, many of Hipparchus’ works are now lost, including those written on optics, arithmetic, geography, astronomy and gravity. It is known that he was born in Nicaea, Bithynia, modern-day Iznik, Turkey. Thus, he is known as Hipparchus of Nicaea or Hipparchus of Bithynia. Much of Hipparchus’ observations were performed in Alexandria and on the island of Rhodes; this has been determined mathematically, since his perception of the positions of celestial bodies suggest what his coordinates were on Earth (O'Connor, J. J. and Robertson, E. F., 1999). Hipparchus’ contributions to mathematics and astronomy provided a basis for future astronomers, such as Ptolemy, and his ideas and methods are common in classical, as well as modern astronomy.

Surveyors, engineers and mathematicians all use trigonometry as a method to solve problems that would otherwise be very complex. It is a common belief that Hipparchus is the father of this branch of mathematics, giving rise to the practical, predictive nature of astronomy as a science (O'Connor, J. J. and Robertson, E. F., 1999). Ptolemy and Theon of Alexandria made references to twelve books written by Hipparchus on chords; the length of a chord with both endpoints on a single circle could be determined by using the chord’s angle and trigonometric calculations. Although these books are now lost, as they were published in 140 BCE, Hipparchus’ function provided grounds for the introduction of other trigonometric functions (O'Connor, J. J. and Robertson, E. F., 1999). Hipparchus’ trigonometric function, the Chord function, is related to the sine function in such a way that half the chord of twice an angle, x, is equal to 3438 times the sine of angle x; or

Crd(2x)/2 = 3438’*sinx.

Furthermore, this value of 3438’ (arc minutes) was determined by dividing the   21 600 arc minutes of a circle by twice the value of pi, yielding a value of approximately 3438’, the radius of a circle whose 360 degrees are each divided into 60 arc minutes (O'Connor, J. J. and Robertson, E. F., 1999).

More importantly, a chord’s length, through a circle, can be determined by multiplying the sine of half that chord’s angle by two; or

Crd(x) = 2sin(x/2).

            It is important to note that within these theorems, Hipparchus introduced another important idea into trigonometry; the model of a circle containing 360 degrees.

Using this data, along with the theorem of Pythagoras (a2 + b2 = c2, where c is the length of the hypotenuse of a right-angled triangle, and a and b are the other sides’ lengths of that right triangle) and several of the half-angle formulae, Hipparchus went on to derive a trigonometric table (O'Connor, J. J. and Robertson, E. F., 1999). The table begins with a chord angle of 7.5°, adding increments of 7.5°, up to 180° (Berggren, J. L., 2005). As the chord angle changes, corresponding values for the length of the chord were given for a circle whose fixed radius is not known for certain. This table was then passed on to Ptolemy, who went on to develop trigonometry further. Remarkably, both astronomers (trigonometry was considered a branch of astronomy at the time) took the value of pi to be 3 and arrived at their accurate conclusions, nonetheless (O'Connor, J. J. and Robertson, E. F., 1996). Mathematics aside, Hipparchus made several other contributions to science, many of which relate to astronomy.

A catalogue of approximately 850 stars, thought to have been compiled in 129 BCE, was made by Hipparchus, denoting the positions of these stars on a unique coordinate system. Hipparchus, having done plenty or work in the field of astrology, used the stars’ positions in accordance with the 12 constellations to indicate their positions on the celestial sphere. He imagined a circle around each of these constellations, which are parallel to the earth’s equator; collectively, these circles occupy all possible right ascension coordinates, since the circles’ perimeters overlap (O'Connor, J. J. and Robertson, E. F., 1999). Although the originality of his catalogue is not known indubitably, Hipparchus makes reference to this book in his only surviving work, Commentary on Aratus and Eudoxus. This catalogue was also used by Ptolemy and other astronomers, emphasizing its significance.

Although the catalogue itself did not survive, a statue known as the Farnese Atlas was recently accredited to Hipparchus. This statue of the Titan Atlas holding a globe over one shoulder has constellations quite accurately plotted over its 65 cm-diameter globe. Calculations done by Dr. B. E. Schaefer, of Louisiana State University, suggest that this statue was sculpted during Hipparchus’ time, 125 BCE, accurate to 55 years (Schaefer, 2005). By using the principle of precession (discussed later), Schaefer calculated that the position of the stars, with respect to the Earth, determine that at the time this statue was sculpted, no other significant astronomer was alive other than Hipparchus. The catalogue itself, which was written in 129 BCE, was written very close to when the sculpture is estimated to have been made.

It is assumed that Hipparchus’ observations were performed at night, when the constellations are most visible. During the night, Hipparchus noted a significant feature of the apparent rotation of the celestial sphere. As the stars moved across the sky, Hipparchus found that this motion could be related to the time of day (O'Connor, J. J. and Robertson, E. F., 1999). For example, the movement of a celestial body through 36° would correspond to a fraction of the celestial sphere’s total rotation, equal to 36°/360° = 0.1 revolutions. By accurately measuring this time interval through the night, Hipparchus determined, mathematically, exactly how long a day lasts (in arbitrary units of time), assuming uniform rotation of the celestial sphere. Referring back to the example of 36°, it would have taken any particular star 2.4 hours to move through this angle of 36°. Thus,

36°/360° = 2.4h/x

Solving for x, Hipparchus would have found that the length of one day is 24 hours. Although these calculations are quite simple, the methods for keeping time during Hipparchus’ era were not nearly as conventional as the accurate stopwatches of today. Sir Isaac Newton, for instance, used the period of his heartbeat as a method of keeping time; Hipparchus would have used a similar technique. Hipparchus found that the position of constellations at night could be used to accurately determine the time, since the Sun would not have been visible.

Perhaps one of Hipparchus’ more complex discoveries would seem more significant. With the help of Babylonian collections of data, Hipparchus was able to estimate, to a respectable degree, the distance between the Earth and its Moon (O'Connor, J. J. and Robertson, E. F., 1999). However, the circumstances required for such an experiment were very specific. Hipparchus needed a solar eclipse; the Sun, Moon and Earth were to be aligned in a plane, with the Moon in the middle, casting a shadow on Earth. The Babylonians determined the period of solar eclipses is 126 007 days, 1 hour. By using only arithmetic, not simple empirical data, Hipparchus found that the period he determined for solar eclipses coincided beautifully with that of the Babylonians (O'Connor, J. J. and Robertson, E. F., 1999).

During that solar eclipse, probably in 129 BCE, Hipparchus stood at Syene, while another observer stood at Alexandria, two cities in close proximity; the distance between these two cities was known (Churchman, S. and Haynes, M., 1999). At the instant that Hipparchus saw a full eclipse (no direct Sunlight was projected onto him), his fellow observer saw a partial eclipse, in which only one-fifth of the Sun was visible. As Hipparchus already knew, the angular size of the Sun is 0.5° (the Sun’s diameter occupies only 0.5° of the 360° of the ecliptic – the circle which the Sun appears to trace out on the celestial sphere). Therefore, the Sun’s visible portion, as observed from Alexandria, had an angular size of 0.2*0.5° = 0.1°. By using trigonometry, Hipparchus derived the ratio of the Syene-Alexandria distance to the Earth-Moon distance. Hipparchus’ estimate of the distance between Earth and Moon is 59 to 67 times the radius of Earth (O'Connor, J. J. and Robertson, E. F., 1999). This is remarkably close to the modern calculation of 60 times the radius of the Earth, even though techniques such as parallax surveying and radar are now used.

Clearly, Hipparchus was very careful to make precise calculations, yielding some very accurate resolutions; his most noteworthy discovery does not fall short. Hipparchus proclaimed that the vernal and autumnal equinoxes (the two instances of the year when the ecliptic intersects the equator; the length of day equals the length of night; the Sun rises precisely in the East and sets precisely in the West), as well as the Earth’s axis, are undergoing a gradual precession (Stern, D. P., 2004). This means that in a slow, periodic manner, the position of the equinoxes is slowly changing, concluding that the Earth’s axis is slowly changing the direction in which it points. Consequently, the North star changes periodically.

The real significance of this discovery is the method Hipparchus used to determine the period of precession, let alone the fact that precession is actually occurring. During a lunar eclipse, when the Sun, Moon and Earth are aligned in a plane, with the Earth in the middle, such that a shadow is cast on the Moon, Hipparchus assumed that this shadow corresponds to the exact position of the Sun on the opposite side of the planetary system (Stern, D. P., 2004). Since the stars are not visible in daylight, the lunar eclipse method was the only suitable way for Hipparchus to determine the position of the Sun with respect to the constellations. As it turns out, the positions of the equinoxes did not match values recorded by the Babylonians 169 years earlier. Hipparchus’ calculated positions of the Sun at the equinoxes were not inaccurate; rather, they seemed to be consistently inconsistent with the Babylonian data, by a factor of approximately 2°. Thus, the annual precession is 2°/169 years ≈ 0.0118°. In arc seconds, this value is 0.0118° * (60’/1°) * (60”/1’) ≈ 42.6”. Hipparchus’ calculated value for Earth’s annual precession is, once again, astonishingly close to the modern value of 50.26” per year.

Having invented the mathematics of astronomy, Hipparchus has the suitable reputation as an innovator in astronomy. Although nearly all of his writings will likely never be recovered, it is clear that Hipparchus’ influence on astronomers after him is central to their findings. Modern methods of calculating unknown values in astronomy are derived from the clever techniques used by Hipparchus over 2000 years ago. His discoveries have not yet failed to surprise us; perhaps, Hipparchus’ lost writings hide discoveries we have yet to make.

References Cited

 

1.         Berggren, J. L. 2005. Trigonometry. MSN Encarta. < http://encarta.msn.com/

encyclopedia_761572350_2/Trigonometry.html>

 

2.         Churchman, S., Haynes, M. 1999. Hipparchus. <http://astrosun2.astro.cornell.edu/

academics/courses/astro201/hipparchus.htm>

 

3.         O’Connor, J. J. and Robertson, E. F. 1996. The Trigonometric Functions. <http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/

Trigonometric_functions.html>

 

4.         O’Connor, J. J. and Robertson, E. F. 1999. Hipparchus of Rhodes. <http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Hipparchus.html>

 

5.         Schaefer, B. E. 2005. Discovery of the Lost Star Catalog of Hipparchus on the Farnese Atlas. Louisiana State University. < http://www.phys.lsu.edu/farnese/>

 

6.         Stern, D. P. 2004. Precession. < http://www-spof.gsfc.nasa.gov/

stargaze/Sprecess.htm>