Question

How did scientist measure the size of the earth?

Answer 1

From the observable curvature of the Earth's surface and measurement of distance of visible horizon.

Explanation:

When observed from a boat moving away from the Statue of

Liberty on Ellis island, the top of Liberty torch would disappear at

about 22 nautical miles. Maintaining the speed of the boat at v mph,

this distance d nautical miles can be easily approximated, using the

time taken t hours, from d = v t miles.

If the height of the top of the torch, from sea level, is

h = height from the base of the pedestal = the height of the pedestal

from sea level

= 305.5'+ ( my guess ) 25' = 330.5'.

Assuming that the top of the statue is observed, from sea level

near the boat,, the radius r of the earth is given by the (tangent

and secant through the center ) formula for a circle

`PT^2=PA.PB`,

`(r tan (d/r))^2=h(2r+h)`

Using iterative numerical methods with a starter approximation

(say) `r_0=3900` miles, r can be obtained, within the limits of the

sd-precision in h, d and the tangent approximations)

As d is quite small compared to r.,

tan (d/r) is nearly d/r + error `O((d/r)^3)= O(10^(-7))`, for the statue

of Liberty distance d of about 22 miles..

So, the formula for approximation is

(r (d/r))^2=h(2r+h#), nearly.

Explicitly, `r=(d^2-h^2)/(2h)`

For sample data ( having Statue of Liberty in mind)

d=22.27 miles, h = 330.5'= 0.06259 mile.

Radius of the Earth

`r = (22.27^2 - 0.06259^2)/(2 X 0.06259)`

=3962 miles = 6375 km.. .