Eratosthenes, Size of the Earth, Radius of the Earth, First Measurement

From the Earth to the Sun in 270 BC

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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.)

Image from: https://thecareyadventures.com/blog/photo-of-the-day-moon-and-the-rockies/

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.

From the Earth to the Moon in 270 BC

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The brain is wider than the sky,

For, put them side by side,

The one the other will include

With ease, and you beside.

Emily Dickinson

How did human beings first work out the distance from the Earth to the Moon?

Aristrarchus of Samos (310 BC – 230 BC) figured out a way to do so in terms of the radius of the Earth in 270 BC. Combined with Eratosthenes’ measurement of the radius of the Earth (c. 240 BC) it enabled people to calculate the actual distance to the Moon. The ancient Greeks used a measurement of distance called stadia (singular: stadium) but we will present the measurements here in terms of kilometres.

Magic with a shadow, not with mirrors

Aristarchus used the fact that the Moon passes through the Earth’s shadow during a total lunar eclipse, which happen once every two to three years on average.

What does a total lunar eclipse look like? Watch this amazing 33 second time lapse video from astrophotographer Bartosz Wojczyński.

https://www.youtube.com/watch?v=LK_44AbfH2Q Note that Mr Wojczyński altered the exposure time of each shot to compensate for the reduced brightness of the Moon as it crossed into the shadow. For reference, the exposure time for the brightly lit Moon was 1/2500 second, and for the dim ‘Blood Moon’ (turned red by sunlight refracted by the Earth’s atmosphere) it was 6 seconds.

The video is sped up so that 1 second of video represents 8 minutes of real time. In the video, the Moon is in shadow for 24 seconds which equates to 8 x 24 = 192 minutes or 3 hours 12 minutes. We will use this later to model Aristarchus’ original calculation.

It’s always Aristarchus before the dawn…

Aristarchus began with the assumption that the Earth of radius r creates a cylinder of shadow that is 2r wide as shown in the diagram below.

The Moon orbits the Earth on a roughly circular path of radius R so it cover a total distance of 2πR. This means that its average speed over its whole journey is 2πR/T where T is the orbital period of the Moon, which is 27.3 days or 27.3 x 24 = 655.2 hours.

The average speed of the Moon as it passes through the Earth’s shadow is 2r / t where t is the time for a lunar eclipse (3 hours 12 minutes, in our example).

The average speed of the speed of the Moon is the same in both instances so we can write:

We can simplify by cancelling out the common factor of two:

Then we can rearrange to make R the subject:

Putting in values for t = 3 hours 12 minutes or 3.2 hours, T = 655.2 hours and Eratosthenes’ value for the radius of the Earth r = 6371 km (which was established a few years later):

So now they do it with mirrors…

Aristarchus’ value is just a shade over 7% too large compared with the modern value of the Earth-Moon distance of 384 400 km, but is impressive for a first approximation carried out in antiquity!

The modern value is measured in part by directing laser beams on to special reflectors left on the Moon’s surface by the Apollo astronauts and also the automated Lunokhod missions. Under ideal conditions, this method can measure the Earth-Moon distance to the nearest millimetre.

Quibbles, Caveats and Apologies

Aristarchus’ estimate was too large in part because of his assumption that Earth’s shadow was a cylinder with a uniform diameter. The Sun is an extended light source so Earth’s shadow forms a cone as shown below.

The value of t is smaller than it would if the shadow was 2r wide, leading to a too-large value of R using Aristarchus’ method.

Also, the plane of the Moon’s orbit is tilted with respect to the plane of the Earth’s orbit. This means that the path of the Moon during an eclipse might not pass through the ‘thickest’ part of the shadow. Aristarchus used the average time t calculated from a number of lunar eclipses.

When timing the lunar eclipse shown in Mr Wojczyński’s excellent video, I started the clock when the leading edge of the Moon entered the shadow, but I confess that I ‘cheated’ a little bit by not stopping the clock when the leading edge of the Moon left the shadow — the error is entirely mine and was deliberate in order to arrive at a reasonable value of R for pedagogic impact.

UPDATE: You could also watch this stunning visualisation of a lunar eclipse from Andrew McCarthy where the shadow of the Earth is tracked rather than the Moon.

This is part 2 of a series exploring how humans ‘measured the size of the sky’.

Part 1: How Eratosthenes measured the size of the Earth

Part 3: How Aristarchus measured the distance from the Earth to the Sun

Measuring the radius of the Earth in 240 BC

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The brain is wider than the sky,
For, put them side by side,
The one the other will include
With ease, and you beside.

Emily Dickinson, ‘The Brain’

Most science teachers find that ‘Space’ is one of the most enduringly fascinating topics for many students: the sense of wonder engendered as our home planet becomes lost in the empty vastness of the Solar System, which then becomes lost in the trackless star-studded immensity of the Milky Way galaxy, is a joy to behold.

But a common question asked by students is: How do we know all this? How do we know the distance to the nearest star to the Sun is 4 light-years? Or how do we know the distance to the Sun? Or the Moon?

I admit, with embarrassment, that I used to answer with a casual and unintentionally-dismissive ‘Oh well, scientists have measured them!’ which (though true) must have sounded more like a confession of faith rather than a sober recounting of empirical fact. Which, to be fair, it probably was; simply because I had not yet made the effort to find out how these measurements were first taken.

The technological resources available to our ancestors would seem primitive and rudimentary to our eyes but, coupled with the deep well of human ingenuity that I like to think is a hallmark of our species, it proved not just ‘world-beating’ but ‘universe-beating’.

I hope you enjoy this whistle stop tour of this little-visited corner of the scientific hinterland, and choose to share some these stories with your students. It is good to know that the brain is indeed ‘wider than the sky’.

I have presented this in a style and format suitable for sharing and discussing with KS3/KS4 students (11-16 year olds).

Mad dogs and Eratosthenes go out in the midday Sun…

To begin at the beginning: the first reliable measurement of the size of the Earth was made in 240 BC and it all began (at least in this re-telling) with the fact that Eratosthenes liked talking to tourists. (‘Err-at-oss-THen-ees’ with the ‘TH’ said as in ‘thermometer’ — never forget that students of all ages often welcome help in learning how to pronounce unfamiliar words)

Alexandria (in present day Egypt) was a thriving city and a tourist magnet. Eratosthenes made a point of speaking to as many visitors as he could. Their stories, taken with a pinch of salt, were an invaluable source of information about the wider world. Eratosthenes was chief librarian of the Library of Alexandria, regarded as one of the Seven Wonders of the World at the time, and considered it his duty to collect, catalogue and classify as much information as he could.

One visitor, present in Alexandria on the longest day of the year (June 21st by our calendar), mentioned something in passing to Eratosthenes that the Librarian found hard to forget: ‘You know,’ said the visitor, ‘at noon on this day, in my home town there are no shadows.’

How could that be? pondered Eratosthenes. There was only one explanation: the Sun was directly overhead at noon on that day in Syene (the tourist’s home town, now known as Aswan).

The same was not true of Alexandria. At noon, there was a small but noticeable shadow. Eratosthenes measured the angle of the shadow at midday on the longest day. It was seven degrees.

No shadows at Syene, but a 7 degree shadow at Alexandria at the exact same time. Again, there was only one explanation: Alexandria was ’tilted’ by 7 degrees with respect to Syene.

Seven degrees of separation

The sphericity of the Earth had been recognised by astronomers from c. 500 BC so this difference was no surprise to Eratosthenes, but what he realised that since he was comparing the length of shadows at two places on the Earth’s surface at the same time then the 7o wasn’t just the angle of the shadow: 7o was the angle subtended at the centre of the Earth by radial lines drawn from both locations.

Eratosthenes paid a person to pace out the distance between Alexandria and Syene. (This was not such an odd request as it sounds to our ears: in the ancient world there were professionals called bematists who were trained to measure distances by counting their steps.)

It took the bematist nearly a month to walk that distance and it turned out to be 5000 stadia or 780 km by our measurements.

Eratosthenes then used a simple ratio method to calculate the circumference of the Earth, C:

Then:

The modern value for the radius of the Earth is 6371 km.

Ifs and buts…

There is still some debate as to the actual length of one Greek stadium but Eratosthenes’ measurement is generally agreed to within 1-2% of the modern value.

Sadly, none of the copies of the book where Eratosthenes explained his method called On the measure of the earth have survived from antiquity so the version presented here is a simplified one outlined by Cleomedes in a later book. For further details, readers are directed to the excellent Wikipedia article on Eratosthenes.

Astronomer Carl Sagan also memorably explained this method in his 1980 TV documentary series Cosmos.

You might want to read…

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

Part 2: How Aristarchus measured the distance between the Earth and the Moon

Part 3: How Aristarchus measured the distance between the Earth and the Sun