JM35 · Lesson 35
Application of Trigonometry
Ancient Chinese 圭表 (gnomon): seasons from noon shadow length — using \(\tan\theta\).
Concept & Formula
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圭表 · seasons from a noon shadow
1. Meet the 圭表
How did ancient China track the seasons?
Long before modern clocks and calendars, people used a device called the 圭表 — a gnomon and a measuring scale — to read the time of year from the Sun.
What is a 圭表?
表 = a vertical post of fixed height \(H\) (the gnomon). 圭 = a horizontal scale on the ground. Sunlight hits the tip of the 表 and casts a shadow of length \(s\) onto the 圭.
Must be used at solar noon (正午)
A 圭表 only works at solar noon. At that moment the Sun is due south, so its azimuth (horizontal direction) stays the same every day. Over the year, only the noon altitude \(\theta\) (how high the Sun is) changes. That keeps the shadow on one fixed line along the 圭, so \(s\) can be compared fairly.
Why does shadow length tell the season?
Because \(H\) is fixed and noon azimuth does not change, shadow length \(s\) depends only on altitude \(\theta\): high Sun in summer → short \(s\); low Sun in winter → long \(s\).
\[ \tan\theta = \dfrac{H}{s} \quad\Rightarrow\quad s = \dfrac{H}{\tan\theta} \]
Yes — it’s trigonometry!
Four marks on the 圭 match 冬至, 春分, 秋分, and 夏至. Drag the Sun and watch the single light ray and the shadow move together.
2. Drag the Sun · watch the shadow
Use at solar noon (正午) only — azimuth is fixed, so drag the Sun up/down only. The yellow ray goes through the tip of the 表 (height \(H\)) and hits the 圭 at the tip of shadow \(s\).
\(\tan\theta = \dfrac{H}{s}\)
Drag the Sun until the shadow tip lines up with a season mark.
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Summary
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Quiz
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