trignometry history

History of Trigonometry Outline

Trigonometry is, of course, a branch of geometry, but it differs from the synthetic geometry of Euclid and the ancient Greeks by being computational in nature. For instance, Proposition I.4 of the Elements is the angle-side-angle congruence theorem which states that a triangle is determined by any two angles and the side between them. That is, if you want to know the remaining angle and the remaining two sides, all you have to do is lay out the given side and the two angles at its ends, extend the other two sides until they meet, and you've got the triangle. No numerical computations involved.

But the trigonometrical version is different. If you have the measurements of the two angles and the length of the side between them, then the problem is to compute the remaining angle (which is easy, just subtract the sum of the two angles from two right angles) and the remaining two sides (which is difficult). The modern solution to the last computation is by means of the law of sines. Details are at Dave's Short Trig Course, Oblique Triangles.

All trigonometrical computations require measurement of angles and computation of some trigonometrical function. The modern trigonometrical functions are sine, cosine, tangent, and their reciprocals, but in ancient Greek trigonometry, the chord, a more intuitive function, was used.

Trigonometry, of course, depends on geometry. The law of cosines, for instance, follows from a proposition of synthetic geometry, namely propositions II.12 and II.13 of the Elements. And so, problems in trigonometry have required new developments in synthetic geometry. An example is Ptolemy's theorem which gives rules for the chords of the sum and difference of angles, which correspond to the sum and difference formulas for sines and cosines.

The prime application of trigonometry in past cultures, not just ancient Greek, is to astronomy. Computation of angles in the celestial sphere requires a different kind of geometry and trigonometry than that in the plane. The geometry of the sphere was called "spherics" and formed one part of the quadrivium of study. Various authors, including Euclid, wrote books on spherics. The current name for the subject is "elliptic geometry." Trigonometry apparently arose to solve problems posed in spherics rather than problems posed in plane geometry. Thus, spherical trigonometry is as old as plane trigonometry.

The Babylonians and angle measurement

The 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.