Shape and space
c. 150 BC
The first trigonometry table
Hipparchus of Nicaea
How do you turn an angle you can see into a distance you can’t reach?
To predict where the Sun and Moon would be, Hipparchus needed to turn angles into lengths. He built a table of chords: the straight line across a circle for each angle. It is the ancestor of the sine button on your calculator.
- Angle θ
- 0.00 rad (0.00 turn)
- e^{iθ}
- 1.000 + 0.000 i
The point is e^{iθ}. Its height is sin θ and its sideways position is cos θ, so spinning in a circle makes two waves. Add growth or decay and the circle becomes a spiral, the waves grow or fade: e^{(σ + iω)t}. That one idea describes guitar strings, circuits and black holes.
How they figured it out
Hipparchus wanted to predict eclipses, and that meant converting the angles he measured in the sky into lengths and distances. So he built a table: for each angle, the length of the straight chord it cuts across a circle. It was the first trigonometric table we know of.
Comparing his star positions with ones recorded about 150 years earlier, he noticed that every star had shifted by roughly two degrees in the same direction. The stars hadn’t moved: the Earth’s axis slowly wobbles, like a spinning top, once every 26,000 years. Almost none of his writing survives; we know him mostly through Ptolemy, who used his work.
The equation
The straight line across a circle depends only on the angle it spans, so you can list it once and look it up forever.
See it in real life
Measure a tree without climbing it
Stand 20 m from a tree and measure the angle up to its top: say 35°. Its height is m. The measuring app on your phone does the same calculation every time it measures a room.
Where engineers use it
Signals
Every alternating current, radio wave and sound is described with sine and cosine.
Forces on structures
Resolving a force into components uses and in every statics problem.
What it led to
- Circles on circles c. AD 150
Built on
Ptolemy’s model of the heavens ran on Hipparchus’s chord tables.
- Infinite sums in Kerala c. 1400
Built on
Madhava wrote the sine, the function astronomers had tabulated since Hipparchus, as an endless sum.