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Euclid's Elements is by far the most famous mathematical work of classical
antiquity, and also has the distinction of being the world's oldest continuously used
mathematical textbook. Little is known about Euclid, beyond the fact that he lived in
Alexandria around 300 BCE. The main subjects of the work are geometry, proportion,
and number theory. Euclid’s Elements (comprised of 13 books) form one of the most
beautiful and influential works of science in the history of humankind. Its beauty lies in
its logical development of geometry and other branches of mathematics. It has
influenced all branches of science, but none so much as mathematics and the exact
sciences. The Elements have been studied 24 centuries in many languages starting, of
course, in the original Greek, then in Arabic, Latin, and many modern languages.
Trigonometry is a branch of geometry, but it differs from the 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 trigonometric 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 cosines. All trigonometric computations require
measurement of angles and computation of some trigonometric function. The modern
trigonometric 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
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, sometime before 300 B.C., 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. 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 (about 180 B.C. – 125 B.C.) is usually credited as being the
father of trigonometry. 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
a 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 knowledge that was taken as well-known by his successors.
The name sine came to us from the Latin sinus, a term related to a curve, fold, or
hollow. It is often interpreted as the fold of a garment, which was used as we would use
a pocket today. The use in mathematics probably comes about through the incorrect
translation of a Sanskrit word. Mathematician Aryabhata used jya-ardha (Sanskrit for
“chord half”), shortened later into jya, for what we now call “sine.” Arab
mathematicians in the eighth century transliterated the Sanskrit word jya into the
proximate sound of jiba and then later changed it into jaib (with the same consonants as
jiba), which is a good Arabic word, meaning a bay or a cove, and it was this word that
was later translated by Gherardo of Cremona (circa 1150) into its equivalent Latin word
for a bay or a cove, in other words, sinus, from which the modern term “sine” is derived.
When Leonardo de Fibonacci used the term in his writing, it became permanent.