The Development of Trigonometry

Overview

Even in the time of the ancient Babylonians and Egyptians, theorems involving ratios of sides of similar triangles were used extensively for measurement, construction, and for an attempt to understand the movement in the heavens. The Greeks began the systematic study of angles and lengths associated with these angles, again in the service of astronomy. The history of trigonometry is intimately associated with that of astronomy, being its primary mathematical tool. Trigonometry will eventually become its own branch of mathematics, as the study of the modern trigonometric functions.

Background

The people in the ancient civilizations in Egypt and Mesopotamia looked to the heavens. The heavens told the people when to plant and when to harvest. In order to commemorate important events, people needed a yearly calendar. Observing the position of the Sun is necessary for calendar making. In order to tell the time of day, one must look at the lengths of shadows. These shadow lengths are measured using an upright stick in the ground and measuring the length of its shadow. Trigonometry has its origins in the calculation of these measurements. Indeed, the modern degree measurement for arcs and angles has its origins in Babylonian measurement.

The famous Babylonian clay tablet now known as Plimpton 322 dates from approximately 1700 b.c. This tablet is best known for its listing of Pythagorean triples, a listing of sides and the corresponding hypotenuse of right triangles. Less well known is the fact that one of the columns on the tablet contains the square of the ratio of the diagonal to one of the sides, and as one moves down the column this ratio decreases almost uniformly. If one looks at the squares of these ratios as the square of the cosecant of the adjacent angle, this angle increases almost uniformly from around 45° to 58°. Does the Plimpton 322 serve in part as a trigonometric table?

Most of our knowledge of early Egyptian mathematics comes from the Rhind (or Ahmes) Papyrus. The papyrus was bought in 1858 in a Nile resort town by a Scot, Alexander Henry Rhind, hence its name. It is sometimes called the Ahmes papyrus in honor of the scribe who copied it about 1650 b.c. Problems 56 through 60 of this papyrus illustrate the origins of trigonometry in ancient Egypt. These problems concern square pyramids, naturally a subject of great interest to the Egyptians. When constructing a pyramid, the faces must maintain a constant slope. The word seqt or seked appears frequently in these problems, and to the modern reader it means the ratio of the horizontal distance of a slanted line from the vertical to the height, much like run over rise in the modern notion of slope. One can also think of the seqt as the cotangent of the angle between the base of a pyramid to its face. However, the Egyptians did not think of the seqt in this way. To them, it was simply expressed as a length. Trigonometric functions as ratios (as we think of them today) had not appeared at this time.

The use of the geometry of similar right triangles rather than trigonometry is also a feature of the mathematics of early Greece. In fact these ideas may have come directly from the Babylonian civilization. It is said that Thales of Miletus (c. 624-547 b.c.) computed the height of a pyramid by comparing the length of its shadow to that of a vertical rod. Book II of Euclid's Elements (c. 300 b.c.) contains propositions equivalent to the law of cosines, although in geometric language. Archimedes's (c. 287-212 b.c.) theorem on the broken chord can be restated as a formula for the sine of the difference of two angles.

Greek astronomers were also working with these geometric tools. Aristarchus of Samos (310-230 b.c.) observed that when the Moon is half full, the angle between the lines of sight to the Sun and the Moon is almost a right angle, at 87°. Using this observation, Aristarchus estimated that the Sun was between 18 and 20 times as far from Earth as the Moon. While this would be a trigonometry problem for us today (using a 3°-87°-90° right triangle made up of Earth, the Moon, and the Sun), to Aristarchus it was a geometry problem. Using these distance measurements, Aristarchus could also find the ratios of the sizes of the Sun and Moon to Earth. To calculate the sizes of the Sun and the Moon, the size of Earth was needed. This measurement was provided by Eratosthenes of Cyrene (276-195 b.c.), who found it using the relationships between angles and arcs of a circle and angles and chords of a circle.

These relationships between angles, arcs, and chords form the basis of observational astronomy. They help give a method for measuring the positions of the stars and planets in the sky, or alternately, on the "celestial sphere," the boundary of the spherical universe, with Earth at its center, which rotates around Earth and where the objects in the sky are located. The Babylonians were responsible for this "ecliptic system" of locating celestial bodies, and the Greek astronomers made use of it as well. In order to work with this system, one needs spherical trigonometry. To understand this, however, it was first necessary to understand plane trigonometry. For example, Menelaus's theorem gives the relationship among arcs of great circles on a sphere and is proven by first showing how the relationship works with segments in the plane.

The subject of trigonometry proper originated with the astronomer Hipparchus of Bithynia (190-120 b.c.) To begin with, Hipparchus introduced the Babylonian degree measurement of angles and arcs of circles to the Greeks. He was also the first to give a tabulation of lengths associated with angles that would allow for the solution of plane triangles. Hipparchus created this table in order to perform his astronomical calculations. Hipparchus's table is unfortunately lost, but from other works that refer to him and his works, we can reconstruct what his table must have looked like.

Hipparchus considered every triangle to be inscribed in a circle with a fixed radius. This meant that that each side of the triangle is a chord of the circle. The only trigonometric function in Hipparchus's trigonometry was the chord function, now abbreviated crd(α), where α is the central angle opposite the chord. In his table, Hipparchus calculated the chord length of every angle from 7.5° to 180° in steps of 7.5°. These chord lengths also vary with the radius of the circle, so a radius had to be fixed. Hipparchus used a fixed radius of R = 57 + 18/60. This radius comes from the calculation 2πR = 360 × 60, which corresponds to the number of minutes in the circumference of a circle. The approximation of 3 + 8/60 + 30/60² = 3.1416... for π is used in the calculation. This means that the chord function is related to the modern sine function by the relation crd(α) = 2R sin(α/2). Note that the chord is a length, not a ratio of lengths.

Hipparchus began his chord table with the chord of a 60° angle. This angle creates an equilateral triangle with a vertex at the center of the circle, so this chord has the same length as the radius of the circle. From there Hipparchus used two results from geometry: crdα²(180 - α) = (2R)² - crdα²(α), which is merely a restatement of the Pythagorean theorem; and crdα²(α/2) = R(2R - crd(180 - α)), which comes from the similarity of a right triangle inscribed in a semicircle with the triangles formed by dropping a perpendicular from a vertex to the hypotenuse.

After Hipparchus, the next major figure in the development of trigonometry was the astronomer Menelaus of Alexandria (c. a.d. 100), who wrote Chords of a Circle, which is lost to us, and Sphaerica, a work on spherical trigonometry. In Sphaerica, Menelaus establishes many propositions for spherical triangles (formed from arcs of great circles on a sphere) that are analogous to propositions for plane triangles. Menelaus's theorem appears in this work, and this theorem is fundamental to the field of spherical trigonometry.

The culmination of Greek trigonometry occurs in a work that is also the culmination of Greek astronomy, the Syntaxis Mathematikos (Mathematical compilation) of Claudius Ptolemaeus, better known at Ptolemy (c. a.d. 100-178). This work was referred to by Islamic scientists many centuries later as al-magisti, "the greatest," and this name translated into Latin became Almagest, the name by which it has been known ever since. Ptolemy realized that the chord in a circle is related to the sides of both spherical and plane triangles, so his work required him to construct a table of chords.

Ptolemy's table was more complete than that of Hipparchus. Ptolemy's table calculated chords of arc from 0.5° to 180° in steps of 0.5°. Ptolemy used the value of 60 for his fixed radius. Ptolemy started his table by calculating the chords using regular polygons inscribed in a circle of radius 60. The side of a regular triangle gives crd(120°), that of a regular square gives crd(90°), a regular pentagon, crd(72°), and a regular decagon, crd(36°). Ptolemy then developed formulas for crd(180 - α ), crd(α/2), and crd(α±β) (the last using Ptolemy's theorem) and used these formulas to complete his table. This table allowed Ptolemy to solve plane triangles, much as trigonometric tables allow us to solve plane triangles now. He could also calculate the values necessary in the eccentric model of the heavens, where the Sun revolves around Earth not in a circle, but around a circle with center away from Earth. With this calculation, Ptolemy could predict the position of the Sun and other celestial objects in the sky at various times. Although Ptolemy's trigonometry would be modified by mathematicians in the East, his concepts of astronomy held for another 1,400 years, until the time of Copernicus (1473-1543).

During Ptolemy's time, northern India was conquered and ruled by the Kushan Empire which established trade routes with Rome. Historians believe that Greek astronomy was transmitted to India over these routes. The Gupta rulers of India (fourth to the sixth century a.d.) also had regular communication with the inheritors of Greco-Roman culture. The earliest known Hindu work with trigonometry is the Paitāmmaha Siddhānta, written early in the fifth century. This work contains a table of half-chords, or in Sanskrit jyā-ardha. Interestingly, this word eventually would become our modern term sine. The word jyā-ardha was shortened to jya or jiva. When this was translated later in Arabic by Islamic scholars, they translated it as jiba, and wrote it jb, since vowels were not written. This was later interpreted at jaib, meaning "bosom" or "fold," and was translated into Latin as sinus. This table of half-chords seems to be derived from the table of chords of Hipparchus, since it uses the same value of the radius as he did.

The first work to mention the sine function (although still as a length in a circle of fixed radius, not as a ratio of lengths) was the Aryabhatiya of the Hindu astronomer Aryabhata I, written around 510. His sine table was actually a table of values of Rsin(α). Aryabhata was aided in the completion of his table by developing the equivalent of the modern identity sin[(n + 1)α] - sin(nα) sin(nα) sin[(n - 1)α]-1/225 sin(nα). Hindu tables are constructed using such recursion relations for sine. Aryabhata also introduced the cosine and versine functions, again as lengths and not as ratios; using the modern Rcos(α) for their cosine function. The versine or "versed sine" function is defined as vers(α) = R(1 - cos(α)). Once again, the emphasis in this work was using trigonometry in service of astronomy and the making of calendars.

These Greek and Hindu works were translated into Arabic in the eighth century and afterward. Islamic astronomers furthered the study of trigonometry for astronomical use as well as for religious use, since Muslims must know the direction of Mecca from any place they find themselves. These works in turn made their way back into the West, and trigonometry developed into the branch of mathematics that is studied today.

Impact

With the development of trigonometry, astronomy was transformed from an observational qualitative science to a predictive quantitative science. This transformation promoted the idea that a mathematical description of natural phenomena is possible. For the first time, observational data could be converted into mathematical models. Ptolemy's Almagest was a milestone event in this regard, serving for applied mathematics the role Euclid's Elements played for theoretical mathematics. Mathematics had been applied to solve problems before, but trigonometry was developed to do applied mathematics, and has continued to do so in more and more sophisticated ways.

Although the trend towards applications coincided with the decline of Greek mathematics, these applications attracted Hindu and Islamic scholars, mainly because they were relevant to religion and calendar making. Mathematics in these two cultures was on the rise, and thanks to them much of the heritage of the classical Greeks was returned to the West.

As trigonometry and astronomy developed as a predictive science in both East and West, it was important that calculating tables became more and more accurate, and that calculations became less cumbersome. As a direct result, later centuries witnessed advances in Hindu-Arabic numeration, decimal fractions, and the invention of logarithms, all designed to ease the computational burden. The French astronomer and mathematician Pierre-Simon de Laplace (1749-1827) said that the invention of logarithms, "by shortening the labors, doubled the life of the astronomer."

GARY S. STOUDT