Trigonometric Integrals
Historical Perspectives on Trig Intervals
The Babylonians and angle measurements.
Babylonians, sometime before 300 B.C.E. were using degree measurement for angles. The Babylonian numerals were based on the number 60, so it may be conjectured that they took the unit measure to be what we call 60°, then divided that into 60 degrees. Perhaps 60° was taken as the unit because the chord of 60° equals the radius of the circle, see below about chords. Degree measurement was later adopted by Hipparchus. The Babylonians were the first to give coordinates for stars. They used the ecliptic as their base circle in the celestial sphere, that is, the crystal sphere of stars. The sun travels the ecliptic, the planets travel near the ecliptic, the constellations of the zodiac are arranged around the ecliptic, and the north star, Polaris, is 90° from the ecliptic. The celestial sphere rotates around the axis through the north and south poles. The Babylonians measured the longitude in degrees counterclockwise from the vernal point as seen from the north pole, and they measured the latitude in degrees north or south from the ecliptic.
Hipparchus of Nicaea (ca. 180 - ca. 125 B.C.E.)Hipparchus was primarily an astronomer, but the beginnings of trigonometry apparently began with him. Certainly, the Babylonians, Egyptians, and earlier Greeks knew much astronomy before Hipparchus, and they also determined the positions of many stars on the celestial sphere before him, but it is Hipparchus to whom the first table of chords is attributed. It has been hypothesized that Apollonius and even Archimedes constructed tables of chords before him, but there is no reference to any such earlier table. Some of Hipparchus' advances in astronomy include the calculation of the mean lunar month, estimates of the sized and distances of the sun and moon, variants on the epicyclic and eccentric models of planetary motion, a catalog of 850 stars (longitude and latitude relative to the ecliptic), and the discovery of the precession of the equinoxes and measurement of that precession. According to Theon, Hipparchus wrote a 12-book work on chords in a circle, since lost. That would be the first known work of trigonometry. Since the work no longer exists, most everything about it is speculation. But a few things are known from various mentions of it in other sources including another of his own. It included some lengths of chords corresponding to various arcs of circles, perhaps a table of chords. Besides these few scraps of information, others can be inferred from the knowledge that was taken as well-known by his successors.