Trigonometry
The mathematics that links the angles of a triangle to the lengths of its sides, so that a distance you cannot walk can be computed from an angle you can measure. With a table of these ratios, one measured baseline and two sightings fix a point anywhere.
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✦ 等等，真的吗？
The word "sine" is a thousand-year-old mistranslation. Indian astronomers called the half-chord jya-ardha; Arabic writers transliterated it as jiba, a meaningless string that — written without vowels as "jb" — was later read as jaib, "bosom" or "bay." A 12th-century Latin translator rendered that as sinus, Latin for a fold or bay, and "sine" is that bay, still on every calculator.
它是什么
Trigonometry is the bridge between angle and length. In any right triangle the ratio of two sides is fixed by the angle between them and nothing else, so once those ratios are tabulated you can trade a measurement you *can* make for one you can't. Its ancient form was the table of chords: for a circle of given size, how long is the straight line cutting off an arc of, say, 40°? Hipparchus of Nicaea compiled the first such table around 150 BC; three centuries later Ptolemy, working in Alexandria around AD 150, published a far finer one in Book I of the *Almagest*, tabulating chords at half-degree steps in a circle of diameter 120, in base-60 arithmetic.
它为何重要
It made the unreachable measurable. The height of a mountain, the distance to a ship offshore, the radius of the Earth, the span between planets — none can be paced out, but all reduce to a triangle with one known side and measured angles. Ptolemy's chord table was the computational core of the *Almagest*, the model of the heavens that governed astronomy for fourteen hundred years. Every prediction of a planet's position, every eclipse forecast, ran through those ratios.
它解锁了什么
Trigonometry is the working tool of surveying, navigation, cartography, and astronomy, and the reason a modest baseline plus two angles can triangulate a coastline or a star's parallax. Its tables of sines were exactly the laborious multiplications that later drove Napier to invent logarithms. Recast in algebra and calculus, the sine and cosine become the mathematics of every wave and oscillation — sound, light, tides, alternating current — so that a description invented for triangles now describes the vibration of the physical world.
最简可行版本
A table listing, for each angle, the length of the chord (or half-chord) it subtends in a circle of fixed radius — plus the geometric identities that let you interpolate and combine angles you did not tabulate directly.
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来源
- — Ptolemy, *Almagest*, Book I (c. AD 150)
- — Glen Van Brummelen, *The Mathematics of the Heavens and the Earth: The Early History of Trigonometry* (2009)
- — O. Neugebauer, *A History of Ancient Mathematical Astronomy* (1975)
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