From the Earth to the stars in 1668 AD

It may seem incredible to us, but over a millennium before the invention of the telescope, some of our resourceful ancestors had not only measured the radius of the Earth, but also the distance from the Earth to the Moon, and even the distance from the Earth to the Sun.

However, the ancients still held fast to the idea that the Earth was the centre of the universe. This is often presented unfairly as a failure of intelligence or nerve on their part. However, they believed they had good evidence to support the idea. One of the more convincing arguments advanced in favour of geocentrism was the absence of stellar parallax. Surely, if the Earth described a vast circle around the Sun in one year, then we would observe some shift in the positions of the fixed stars? Just as we would observe the nearer trees shift against the background of the distant hills as we walked through the landscape. The absence of stellar parallax could only be explained by (a) the fixity of the Earth; or (b) the fixed stars were absurdly distant so stellar parallax was too small to be observed.

Robert Hooke’s attempt to measure stellar parallax

In 1674, Robert Hooke (1635-1703) attempted to measure stellar parallax and put an end to one of the more powerful arguments of the anti-Copernicans. He built a zenith telescope (that is to say, a telescope that only points vertically upwards) that extended through the roof of his house in order to observe Gamma Draconis, a star that can be seen directly overhead in London. He recorded a total of four observations and calculated its annual parallax as 30 seconds of arc, that is 30/3600ths of one degree. His contemporaries remained unconvinced, mostly because he had only made four observations.

Modern astronomers have measured the distance of Gamma Draconis to be 153.4 light years, giving it an annual parallax of 0.02 seconds of arc: Robert Hooke’s measurement was 1500 times the actual value, so it would appear that, although his methodology was sound, his instrumentation was not up to the task.

And so it would remain until the 1830s when telescope technology had improved to the point where stellar parallaxes could be accurately measured.

However, Scottish mathematician and astronomer James Gregory (1638-1675) had suggested and acted on an intriguing alternative in 1668.

James Gregory FRS (1638-1675)

Before discussing Gregory’s method, we will look at Christiaan Huygens (1629-1695) later, more famous but — as I shall argue — less accurate method for measuring the distance to Sirius.

Christiaan Huygens’ calculation of the distance to Sirius (1698)

Many astronomers had suggested that we could estimate the distance to the stars if:

  1. we assume that the Sun is a typical star so that other stars have a similar size and luminosity;
  2. we compare the brightness of the Sun to a given star; and
  3. we use the inverse square law to calculate how much further away the star is than the Sun given their relative brightnesses.

The problem was: how can we reliably compare the brightness of the Sun and a given star?

Huygens tried to do this by observing the Sun through a small hole and making the hole smaller until it appeared to match his memory of the brightness of Sirius as observed previously.

Diagram showing Huygens’ method for comparing the luminosity of the Sun and the star Sirius

This is obviously a highly subjective method since Huygens was relying not only on his memory, but also on the memory of an observation made under very different observing conditions. Huygens did take steps to make the observing conditions as similar as possible: for example, he viewed both the Sun and Sirius through a long 4 metre tube to eliminate light from other sources. (Please be aware, that observing the Sun directly, even through a small aperture, is extremely dangerous!)

The difficulty was compounded by the fact that Huygens lacked the technology to make the hole in the brass plate small enough to mimic the appearance of Sirius; he improvised by adding a small microscope(!) lens to scatter the light but this, of course, added another layer of complication. Nonetheless, he was able to estimate the distance to Sirius as being 27664 times the Earth-Sun distance or 0.44 light-years. This is indeed in the right ball park. The modern distance is given as 8.6 light-years which is much further than Huygens’ measurement, partly because Sirius isn’t a star similar to the Sun: Sirius is actually 25.4 times brighter than our local star so is even more distant than Huygens supposed.

James Gregory’s method for calculating the distance to Sirius (1668)

Gregory suggested using a planet as an intermediary in comparing the brightness of the Sun to Sirius.

Diagram showing James Gregory’s method for comparing the luminosity of the Sun and the star Sirius

In 1668, in his ‘Geometriae pars universalis’, James Gregory set out a method for solving the challenging technical problem of actually determining the ratio of the apparent brightness of the Sun as compared with that of a bright star such as Sirius. He proposed using a planet as an intermediary between the Sun and Sirius. We are to observe the planet at a time when its brightness exactly equals that of Sirius, so that the problem then reduces to one of comparing the brightness of the Sun with that of the planet. But the planet’s brightness depends upon the light it receives from the Sun (and therefore upon the brightness of the Sun), and upon quantities such as the size and reflectivity of the planet and distances within the solar system (quantities which we suppose to be accurately known). A simple calculation then yields the required value. Gregory himself obtained [a value of] 83,190 [times the Earth-Sun distance], but he tells us that with more accurate information on the solar system the figure would be greater still

Hoskin (1977)

Gregory’s method is not so subjective as Huygens’ because we would be viewing Jupiter and Sirius under very similar observing conditions and also would not have to rely on our memory of our perception of Sirius’ brightness. His value of 83190 times the Earth-Sun distance equates to 1.32 light-years. However, had he known that Sirius was 25.4 times brighter than the Sun, he would have increased the calculated distance by a factor equal to the square root of 25.4 which would give a value of 6.6 light-years — not bad, considering the modern value is 8.6 light-years!

Also, bear in mind that Gregory had to assume that Jupiter reflected 100% of the sunlight that fell on it since he had no information about Jupiter’s albedo (the proportion of light reflected by Jupiter’s surface) and had only quite sketchy estimates of Jupiter’s diameter. As Gregory correctly surmised, his figure was a lower boundary estimate for the distance of Sirius, which was likely to increase as more information came to light.

Newton and Gregory’s Method

Sir Isaac Newton (1624-1727) possessed a copy of Gregory’s book (Hoskin 1977: 222) and gave a detailed description of the method in The System of The World which, however, was only published in 1728 after Newton’s death.

In consequence, James Gregory’s brilliant proposal of 1668, which so quickly led Newton to a correct understanding of the distances to the nearest stars, was effectively in limbo until the second quarter of the eighteenth century. In its stead, students of astronomy were introduced to the method of Christiaan Huygens, which was based on the same assumptions but used a much inferior technique for comparing the brightness of the Sun and a star.

Hoskin 1977

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’


Hoskin, M. (1977). The English Background to the Cosmology of Wright and Herschel. In: Yourgrau, W., Breck, A.D. (eds) Cosmology, History, and Theology. Springer, Boston, MA.

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]

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


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

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. 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

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:


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