There are three simple ways to find latitude. First: measure the Pole Star’s altitude, φ ≈ hP (error up to about 1°). Second: if a star at upper culmination, south of the zenith, has altitude h and known δ, then φ = 90° – h + δ. Third: from the Sun’s noon altitude h, φ = 90° – h + δ☉, where δ☉ for that date is read from a table. Never measure the Sun’s altitude by looking at the Sun: use a gnomon – a stick standing upright on level ground – with shadow length L and stick height H, tan h = H / L. These methods are old: Ulugh Beg’s zij gives the latitude of Samarkand as about 39°37′, very close to today’s value (≈ 39°40′).
Worked examples
On an equinox day the gnomon’s noon shadow equals its height: tan h = 1, h = 45°. With δ☉ = 0 we get φ = 90° – 45° = 45°.
A star (δ = +20°) culminates south of the zenith at h = 69°. φ = 90° – 69° + 20° = 41°.
Class activity
With an adult: stand a 1 m stick upright on level ground, measure the shadow at noon and find h from tan h = H/L; take δ☉ from a table and estimate your latitude. Looking directly at the Sun is strictly forbidden.
Practice
1
How is latitude found from the Pole Star?
Its altitude above the horizon is measured: φ ≈ hP.
2
A star (δ = +25°) culminates south of the zenith at h = 74°. What is φ?
41
3
Gnomon 1 m, noon shadow 1.5 m. Estimate h (tan h = 0.67 ⇒ h ≈ 34°). At an equinox what is φ?
φ ≈ 90° – 34° = 56°.
4
Why is a shadow used when measuring the Sun’s altitude?
Because looking at the Sun seriously damages the eyes; the shadow is a safe and accurate method.
