From the Earth to the Sun in 270 BC

The sun is only 32 miles across and not more than 3000 miles from the Earth. It stands to reason it must be so. God made the sun to light the earth, and therefore must have placed it close to the task it was designed to do. What would you think of a man who built a house in Zion, Illinois and put the lamp to light it in Kenosha, Wisconsin?

Wilbur Glen Voliva c.1915 (quoted in Morgan and Langford 1982: 106)

Contrary to the above quote from noted ‘Flat Earther’ Wilbur Glen Voliva (1870-1942), we have very good reason to believe that the Sun is very far away from both the Earth and the Moon.

The argument was first put forward by Aristarchus (310 – 230 BC) and it relies on shadows and geometry.

The daylight moon

A surprisingly large proportion of people are unaware that the crescent or gibbous Moon is often visible in the daylight sky. (‘Gibbous’ = less than full, more than half.)

It’s actually only a completely Full Moon that is visible only at night since, almost by definition, it will rise at sunset and set at sunrise. (Which I find strange, because a common symbol for ‘night’ or ‘sleep’ is a stylised cresent Moon — but I digress…)

It’s only a phase . . .

A daylight Moon can provide a memorable demonstration of why the Moon has phases. Just stand in a patch of sunlight and hold up a ball when the Moon is in the sky…

The ping pong ball has the same ‘phase’ as the Moon. The Moon is also a ball partially lit by the Sun but much further away… [Image from https://skyandtelescope.org/astronomy-resources/what-are-the-phases-of-the-moon/]

By the light of the silvery (half) Moon…

Aristarchus realised that when the Moon was half-lit by the Sun as viewed from the Earth (the phases known as ‘First quarter’ and ‘Last quarter’) then a line drawn from the centre of the Earth to the Moon would be at 90 degrees to a line connecting the Moon to the Sun as shown below.

To an observer on Earth, the angular distance θ measured between the Moon and the Sun would be small if the Sun was close to the Earth; conversely, the angle θ would be large if the Sun was far away from the Earth.

Aristarchus realised that if he measured the angle θ between the Moon when it was half lit (i.e. during First Quarter or Last Quarter) and the Sun, then he would be able to find the ratio between the Earth-Moon distance and the Earth-Sun distance. Since he had previously worked out a method to measure the Earth-Moon distance, this meant that he could calculate the distance from the Earth to the Sun.

Modern measurements of the angle θ produce a mean value of 89 degrees and 51.2 minutes of arc (it does vary as the Moon has an elliptical rather than a circular orbit).

Using some trigonometry we calculate that the Earth-Sun distance (ES) is 400 times the Earth-Moon distance.

Quibbles and Caveats

Aristarchus measured an angle of 87 degrees for θ which meant that he calculated that the Sun was only 20 times further away from the Earth than the Moon. Also, trigonometrical techniques were not available to him which meant he had to use a geometrical method to calculate the Earth-Sun distance. However, Aristarchus achievement is still worth celebrating!


This is part 3 of a series exploring how humans ‘measured the size of the sky’.

Part 1: How Eratosthenes measured the size of the Earth

Part 2: How Aristarchus measured the distance from the Earth to the Moon

REFERENCE

Langford D & Morgan C.. (1982), Facts and Fallacies: A book of definitive mistakes and misguided predictions. Corgi Books.

6 thoughts on “From the Earth to the Sun in 270 BC

  1. @IanYorston June 20, 2021 / 1:51 pm

    Typo in:

    “ Aristarchus measured an angle of 87 degrees for θ which meant that he calculated that the Sun was only 20 times further away from the Earth than the Sun.”

    • e=mc2andallthat July 22, 2021 / 9:42 pm

      D’oh! *Now* I see it — and it only took me 4 weeks(!)

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