Hipparchus

(190 BC-120 BC)

Hipparchus was born in Nicaea in Bithynia, but spent much of his life in Rhodes. He is considered to be one of the most influential astronomers of ancient Greece, but very little is known about him. Hispparchus(now lost) made an astronomical calendar, wrote books on optics and arithmetic, a treatise entitled On Objects Carried Down by their Weight, geographical and astrological writings, and a catalogue of his own work. The Almagest written by Ptolemy (second century A.D.) is the source of most of our knowledge about Hipparchus. Ptolemy considered Hipparchus be his most important predecessor. In his own astronomical work, Ptolemy made extensive use of the work by Hipparchus, building on the foundation laid by him. Hipparchus continued to be held in high regard by the various depictions of him on frontispieces of astronomical works published long after his death.

Hipparchus made many important and lasting contributions to astronomy including practical theoretical innovations. He employed geometrical models, including the deferent-epicycle and eccentric previously used by Apollonius (flourished ca. 200 B.C.). Hipparchus is credited with the founding of trigonometry. Hipparchus was very interested in observation; his recorded observations span the years 147 to 127 BC. He used an instrument described by Ptolemy as a dioptra and may have invented the planispheric astrolabe. Hipparchus made extensive observations of star positions, and is credited by some with the production of the first known catalogue of stars. He turned his attention to a wide variety of astronomical questions, including the length of the year, the determination of lunar distance and the computation of lunar and solar eclipses. He developed theories for the Sun and Moon demonstrating that they are represented by uniform circular motions. Hipparchus compiled the planetary observations to which he had access into a more useful arrangement, and demonstrated that the phenomena were not in agreement with the hypotheses of the astronomers of that time. Hipparchus' discussion of the motion of the points of solstice and equinox slowly from east to west against the background of the fixed stars is perhaps his most famous achievement; he has been therefore credited with the discovery of the precession of the equinoxes.

Perhaps most intriguing for historians of astronomy is Hipparchus' use of Babylonian astronomical material, which included methods as well as observations. Many questions remain regarding the relationship between Babylonian and Greek astronomy, but Hipparchus' work provides a clear link. It has been argued that Hipparchus was responsible for the direct transmission of both Babylonian observations and procedures and for the successful binding of Babylonian and Greek astronomy.

Hipparchus compared observations of a solar eclipse in Syene and in Alexandria to Hipparchus also measured the precession of the Earth's rotation axis. Today we know that the precession period is about 26,000 years. While the North Celestial Pole today is near the star Polaris, in 3000 B.C., it was near the star Thuban in the constellation Draco, and in 14,000 A.D. , it will be found near the star Vega in the constellation Lyra.

Hipparchus compared observations of a solar eclipse in Syene and in Alexandria to Hipparchus measured the distance from the Earth to the Moon during a solar eclipse that was a total eclipse at Syene and a partial eclipse at Alexandria. At the same time that an observer at Syene saw the entire Sun blocked by the Moon, one at Alexandria saw 1/5th of the Sun's disk, that is 1/5th of 30 arcminutes of the Sun's disk was visible (The Sun's angular diameter is 30 arcminutes or 1/2 degree). The angular size of the visible Sun seen at Alexandria therefore is 1/10th of a degree (0.1 degree) and this angle, expressed in radians and applying the small angle approximation gives the ratio of the Syene-Alexandria distance to the Earth-Moon distance. Hipparchus compared observations of a solar eclipse in Syene and in Alexandria

Hipparchs’ is also credited for credited for inventing trigonometry usings chords in a circle.

he chord of an angle AOB where O is the center of a circle and A and B are two points on the circle, is just the straight line AB. Chords are related to the modern sine and cosine by the formulas

crd a = d sin (a/2)

sin a = (1/d) crd 2a crd (180° - a) = d cos (a/2)

cos a = (1/d) crd (180° - 2a)

where a is an angle, d the diameter, and crd an abbreviation for chord.

Some properties of chords could not have escaped Hipparchus' notice, especially in a 12-book work on the subject.

For instance, a supplementary-angle formula would state that if AOB and BOC are supplementary angles, then Thales' theorem states that triangle ABC is right, so the Pythagorean theorem says the square on the chord AB plus the square on the chord BC equals the square on the diameter AC. Summarized using a modern algebraic notation

crd2 AOB + crd2 BOC = d2

where d is the diameter of the circle.

Hipparchus probably constructed his table of chords using a half-angle formula and the supplementary angle formula. The half-angle formula in terms of chords is

crd2(t/2) = r(2r - crd (180° - t)

where r is the radius of the circle and t is an angle. Starting with crd 60° = r, Hippocrates could by means of this half-angle formula find the chords of 30°, 15°, and 7 1/2°. He could complete a table of chords in 7 1/2° steps by using crd 90°, the half-angle formula, and the supplementary angle formula.

Sources

http://aleph0.clarku.edu/~djoyce/ma105/trighist.html

http://www.hps.cam.ac.uk/starry/hipparchus.html

http://www.astro.cornell.edu/academics/courses/astro201/hipparchus.htm

http://www.mlahanas.de/Greeks/Hipparchus.htm