I had been under the impression before reading Suzuki’s book that curved-space geometry was a modern invention. However, Suzuki informs us that Menelaos of Alexandria was working with triangles in curved space as early as the first century AD– curved-space triangles being important in astronomical studies.
Speaking of curves, another important tool for early astronomers was the determination of chord lengths (lines passing from one point of the circle to another without going through dead center). A table of chord-lengths was compiled by Ptolemy. He associated each chord length with the central angle defined by the line segment’s endpoints (a picture would be worth a thousand words; perhaps you can google for one). Ptolemy divided the circle up into half-degree central angles, so that he had 720 entries. Suzuki informs us that this table of chord-lengths is “the precursor to the modern table of sines,” and that thus “the fundamental principles of constructing a table of trigonometric values were established by Ptolemy.”
According to Suzuki, the Sine of Arc AB is the length of the line drawn from A to D, with D being on the radius line that is perpindicular to line AD and whose endpoint is Point B. The center of the circle is Point C. The angle formed by ADC is a right angle. This means the whole triangle ADC is a right triangle, with AC not only being a radius of the circle, but also being the hypoteneuse of our triangle. From the perspective of the central angle ACD, the side AD is the “opposite” angle. The ratio of this side to the hypoteneuse (radius of the circle) is the equivalent of the modern day “sine”… in other words, the sine of an angle is equal to the “opposite” side divided by the “hypoteneuse.”
It wasn’t until Rheticus in the 1500s that trig tables began to be based on the sides of triangles rather than on the chords of a circle. Rheticus, says Suzuki, “was the first author to speak of the sine of an angle in our modern sense.” And it was the Arabs who really invented Trigonometry as we know it today, says Suzuki, developing “the five remaining trigonometric quantities” after the Sine.
The origin for the term “Sine” is sorta interesting. According to Suzuki, the word is a mistranslation of the Arabic term for the chord of an arc, “jiva” or “jiba.” But the translator mistook the term for “jaib,” the Arabic word for “bay,” which he then translated into the Latin word “sinus,” which also means “bay.”
Suzuki points out that until the 1600s, “the Sine (and other trigonometic functions) were generally considered to be the length of a line segment.” Today, he says, the Sine is thought of, not as a length, but as a ratio (that opposite-over-hypoteneuse thing I mentioned earlier).
In recent times, the physics of the vibrating string has come to play a prominent role in physics, and for this reason, the trig functions associated