Category: physics

tan, sin, cos
e.g. tanα = 231 /266

using angle table
534.6m, 401.0m

length of bar = 1.00m, length of shadow = 0.126m, α = 7.2°

Eratosthenes vs Reality
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cos(44 degrees) = 0.7193319

reverse tangent -> arctan(0.932) in degrees -> 42.98
sin(30 degree) = 0.5
tan(79 degree) = 5.14
arcsin(0.62) in degrees = 38.32°
arctan(4/7) in degrees = 29.74°
arctan(1/40) in degrees = 1.42°

α = 7.2°+- 1.4°
αmin = 5.8°, C = 57,400km
αmax = 8.6°, C = 46,250km

Alexandria:α
Syene

Angle α
Distance d
d / c = α / 360°

c = 360°/α * d

Distance d = 5000 stadia

Eratostheness

How long the earth’s circumstance?
Geometry sun’s position

Plato “The earth’s circumference is 400,000 stadia.”
≒ 74,000
1 stadium = 185m

True Circumference is 40,000 Km
Plato’s guess was 85% larger than the true circumference.

Archimedes: circumference of Earth is 300,000 stadia
Archimedes: Look at the geometry of a sphere and find a clever way to calculate the circumference