Eratosthenes and Aristarchus

We take for granted that science is quantitative.  But the early Greeks were primarily interested in philosophical argument rather than careful measurement.  This began to change in the 3rd century BC when two of the great figures in early science came on the scene.  Eratosthenes would measure the circumference of the Earth, while Aristarchus would be even more ambitious and would attempt to measure the distance to the Sun and the Moon.

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Transcript

Hello and Welcome to the History of Science Podcast.

 Modern science has had a transformative impact on the modern world.  When we think of science, one of the first thing that come to mind are the technological innovations which have improved our lives.  And if we are called to defend science, we often fall back on these technological advancements to argue that science is both practically useful and does indeed actually work.  However, the other consequences of scientific advancement has been more subtle.  Science has not only changed our lives materially, but it has also changed our perceptions of the world we live in, and this is especially true in the case of astronomy.

In the ancient world most people conceived of the universe as having a very limited extent, with the Earth at the centre, and the celestial bodies orbiting relatively close to Earth.  Since then, however, almost every advance in astronomy has seen our conception of the size of the universe expand, and by extension our own place within it shrink. 

We can see this in two ways.  There is a metaphorical shrinking, where we see ourselves as less important and central, because of the enormity of the universe.  But there is also a literal meaning, because every scientific advance in astronomy has also been accompanied by better and more accurate measurements of the distances involved.  

Measurement and quantification have been very important to the development of science; By measuring things, either the very large or very small, we make them tangible to our human senses.  Its no coincidence that many of the earliest units for distance were based on the proportions of the human body.  For example, the hand was a unit literally based on the proportions of an adult male hand.  Similarly, measurements were based on human experience, for example a league was thought to be the distance a person could reasonably walk within an hour.  As astronomy has progressed and our understanding of the world has changed, we’ve created units which are further and further beyond our experience.  For example, the light-year and the Parsec measure distances which are unimaginably large and demonstrate how much our own place in the universe has shrunk compared to the vastness of space.

However, quantification and measurements aren’t necessarily universal desires.  As we discussed in the last episode the very early Greeks weren’t especially interested in measurement, content to make qualitative observations or mathematical models which didn’t necessarily align with the real world.  Nevertheless, an important leap for science in the ancient world was made when the Greeks began to make measurements of the world.  And two of the most important figures in this change were Eratosthenes and Aristarchus.

Eratosthenes, was an Alexandrian Greek, living in the 3rd century BC.  He was the chief librarian of the great library of Alexandria and a man of wide learning, being a mathematician, geographer, historian, poet, as well as an early astronomer.  He was supposedly the second best man at the time in all his intellectual endeavours, which was supposedly an insult, but at least to my ears, sounds incredibly impressive.  

One of Eratosthenes main challenges was that he was keen to measure the circumference of the Earth.  You see the ancient Greeks had known the Earth was a sphere, since at least the 5th century BC, and they had many arguments for this.  One of the simplest was, if you watch a ship sail over the horizon, its sail can be seen first before its hull come into view, implying a curvature to sea level.  The ancient Greeks also put into use their knowledge of astronomy to derive sophisticated arguments for the Earths shape.  For instance, multiple Greek writers point out that the visibility of constellations change depending on the location they are viewed.  Canopus for example, one of the brightest stars in the night sky, was visible from Egypt but not from more northerly Greece, implying at least some curvature from North to South. Aristotle was important for propagating the idea of a spherical Earth throughout the Greek world and had his own arguments for this based on observations of the lunar eclipse.  He noticed that during a lunar eclipse the Earth’s casts a curved shadow upon the moon providing fairly unambiguous evidence of the Earth’s shape.

Despite the overwhelming evidence for the shape of the earth, the task of measuring the size of the Earth seemed hopelessly impractical at this time.  This is even more so when you consider that the desert of the Sahara as well as the Atlantic Ocean, unnavigable to ancient ships, provided hard limits on how far any individual could travel. The ancient Greeks were aware of other places and other cultures, but their primary experience was largely confined to the people around the Mediterranean.  

However, the ancient Greeks were undoubtedly resourceful, and Eratosthenes made his measurement of the Earth using the simplest of scientific devices, the Gnomon.  This was a remarkably versatile tool in ancient astronomy, and as we’ve mentioned in a past episode, was used to accurately calculate the solstice and the equinoxes. The Greeks may have learned of the Gnomon from the Babylonians, who in turn, may have learned about it from the Egyptians.  However, its such a simple device, it could have been invented and re-invented repeatedly across time and space, because a gnomon is a just a vertical stick, set in the Sun.  You use it to measure the length and direction of the shadow cast by the Sun as it passes across the sky. For example, the shortest shadow appears at noon on the Summer solstice, when the Sun is at its highest point in the sky.

Eratosthenes measurement was based on this but also that the fact that the length of the shadow cast also depends on latitude.  Latitude is a measure of how far north you are, which changes the angle at which the suns rays strike the Earth. This means that two measurements made with a Gnomon at the same time at different latitudes will be a different length.

Specifically, Eratosthenes knew that on the summer solstice, the shadow cast at noon in Alexandria will be at its shortest extent, but will still have so me finite and measurable.  However, at the same time, further south in Syene, which is modern day Aswan in South Egypt, the shadow cast by the sun will be close to nothing as the Sun is directly overhead.  This is because of Syene’s latitude, is close to the tropic of cancer.  

Eratosthenes could have left his observations there and taken it simply as another piece of evidence for the sphericity of the Earth.  However, he realised that these observations could instead be used to calculate the circumference of the Earth.  

To understand this it’s helpful to imagine a line stretching directly upwards from the ground, which is called the zenith line.  A Gnomon, properly set up, should point directly upwards towards the zenith.  At most times and in most places the Sun is not directly above us, hence there is some finite angle between the zenith and the suns rays, which causes a shadow.  By measuring the length of the shadow using the Gnomon, the angle between the Zenith and the suns rays can be measured.  However, at noon on the summer solstice at Syene, the sun is directly overhead, implying that the angle between the two is zero.  

Now at this point, you may be asking yourselves so what?  Well Eratosthenes realisation was that on the Summer solstice, the difference in the angle between the zenith and the suns rays, would correspond to the difference in latitude between Alexandria and Syene.  This may be difficult to picture so if you’re interested I have included a picture on the History of Science website.

When Eratosthenes took his measurements at noon on the summer solstice in Alexandria, he calculated that the angle between the suns rays and the zenith was 1/50 of a full circle.  This implied that the distance between Alexandria and Syene was also equivalent to 1/50th of a full circle.  Eratosthenes then had someone measure the distance between Alexandria and Syene, which was measured at 5,000 stades, implying that the circumference of the Earth was 50x5000 or 250,000 stade.  Unfortunately, its difficult to say with any precision quite what this equates to in a modern unit of distance, as their unit, the stade, varied in length between 170 and 210m.  However, taking the range of estimates for the stade, it certainly would be within the right ballpark, corresponding to a measurement 10-30% from the modern accepted value of 40,000km.

This was a remarkable achievement in itself and Eratosthenes methods also formed the basis for the measurement of latitude, a key component of navigation.  From this point measurement began to be seen as a key component of the practice of astronomy.  But while Eratosthenes work was remarkable, he was followed by another early astronomer, who attempted an even more ambitious measurement.  

Aristarchus was a Greek, born around the beginning of the third century BC, on the island of Samos, near Turkey.  As with many figures in this early period of Greek astronomy, very little detail of Aristarchus life is known.  Indeed very little of his work has survived, and only a single original work of his has been reproduced.  However, references to his work of exist in the work of other writers, most notably Archimedes.  We mentioned Aristarchus in the last episode and his most famous contribution to the history of science is that he was the first person to unambiguously suggest the heliocentric model.

As we have discussed, virtually all ancient Greek astronomy followed the general outline set down by Aristotle and Eudoxus.  That is that there are two spheres, the celestial, containing all of the celestial bodies such as the Sun, the Moon and the stars, and the Earthly, containing well…the Earth.  The general picture was then that the heavenly celestial revolved around the earthly, producing day and night, the seasons and so forth.  The so-called geocentric model.  

However, in his book Sand Reckoner, Archimedes, makes reference to a theory of Aristarchus.  To quote:

Aristarchus of Samos brought out a book consisting of some hypotheses,

in which the premises lead to the result that the cosmos is many times

greater than that now so called. His hypotheses are that the fixed stars and

the Sun remain unmoved, that the Earth revolves about the Sun in the

circumference of a circle, the Sun lying in the middle of the orbit, and

that the sphere of the fixed stars, situated about the same center as the

Sun, is so great that the circle in which he supposes the Earth to revolve

bears such a proportion to the distance of the fixed stars as the center of

the sphere bears to its surface.

Here Aristarchus is the first person to seriously consider that the geocentric model might be wrong, and in fact the earth revolves around the sun.  Not only that, but Aristarchus considers the implications of this model, correctly realising that it implied a universe which was many times bigger than the Greeks had imagined.  It’s difficult to appreciate today, but the Greeks imagined that the so-called fixed stars weren’t much further away than the other celestial bodies, such as the Moon and Sun. Aristarchus however, realised that if the Earth was moving then this would imply that the stars had to be much much further away than previously thought.  

To understand this imagine that you are in a field with two trees at the centre.  As you move around the trees your perspective changes, that is they seem to change position with respect to each other, due to a phenomenon known as parallax.  Likewise if the earth is really moving and the stars are relatively close, we would expect parallax to cause the stars position to change relative to one another.  We do not observe this to any great extent on Earth, for example the constellations remain unchanged throughout the year. 

To resolve the apparent inconsistency with his idea, Aristarchus suggested that this implies that the stars are very far away.  Imagine again the two trees, but this time on a distant hilliside. Now as you move around the change in your perspective is very slight, as the further away the objects are, the less parallax you will observe.  The conclusion which follows is that if the Earth is moving, and we don’t observe the parallax of stars, this implies the stars are very far away. 

It is astonishing that someone in the 3rd century BC not only suggested a heliocentric solar system, but also its implication of a vast universe.  However, his idea was little influence and was largely ignored by later Greek astronomers and it wasn’t until the 16th century that astronomers, who probably knew nothing of Aristarchus’s suggestion, began to rediscover the idea of a heliocentric solar system, and grapple with its implication.  

Indeed, even in Aristarchus’s other work he makes no mention of the heliocentric model.  Instead, his surviving work instead attempts an extraordinary ambitious measurement which was to find the size and distances between the Earth, Moon and Sun.  

He starts by attempting to measure the ratio of the distance between the Sun and the Moon.  To do this he observes the Sun and moon during lunar quadrature, that is when we have a quarter moon.  Aristarchus estimated that this occurs when the angular distance between the Sun and Moon is around 87 degrees. Aristarchus was working before trigonometry had been fully established in the Greek world, but instead uses geometrical constructions to give an upper and lower limit.  He manages to prove that the distance to the Sun is somewhere between 18 and 20 times that of the moon.  This would of course be much easier nowadays and using trigonometry we would find that the ratio is around 19.1, using the estimate of 87 degrees. From Aristarchus’s results its possible to reconstruct his measurements to gain an absolute distance in terms of Earths radius, which would be 20 earth radii for the distance to the moon and some 380 for the distance to the Sun. 

Next up Aristarchus tackles the relative sizes of the Sun and Moon. For this he uses the fact that they each have a roughly similar angular diameter when measured from Earth.  This is obvious when observing a solar eclipse, as the moon almost perfectly covers the area of the sun.  Similar to the last measurement Aristarchus sets up another geometrical construction, similar to when the moon experiences a lunar eclipse.  Again he calculates that the Sun’s diameter is roughly 18-20 times greater than the Moon.  Putting this as an absolute measurement in terms of the Earth’s size, gives an estimate for    the Sun to be 6.67 Earth diameters and the Moon to be 0.351 Earth diameters.  

As you may have noticed by now Aristarchus’s measurements weren’t wholly accurate. Some of his conclusions were in the very general correct, he found that the moon is smaller than Earth, and the sun is considerably larger.  Similarly he found that both the moon and the sun were some sizeable distance away from the earth.  However, the actual values that he arrived at for these sizes and distances, especially of the Sun, are fairly large underestimates of the real values, and there are several reasons for this. 

Firstly the values Aristarchus used in some his calculations were not very accurate. For example, the estimate for the ratio of the distances of the Moon and Sun of around 18-20, is a long way from the true value, where the Sun is actually around 400 times further away than the moon.  However, it is unfair to blame Aristarchus for this. As mentioned Aristarchus estimated that the angle between the moon and Sun at lunar quadrature was around 87 degrees.  In reality it is around 89 degrees and 51 arc minutes, only 9 minutes short of a right angle.  Even at the height of Greek astronomy, such an accurate calculation wouldn’t have been possible with the available equipment.  How for example do you calculate the exact moment that the moon is in quadrature, or decide on where the centres of the moon and sun are with any accuracy?

The second reason is slightly more subtle.  Aristarchus could have improved some of his other estimates fairly easily, for example by taking a better measurement of the angular diameter of the moon, a measurement that can be made fairly easily using just a ruler. The astronomer James Evans has shown that it would in fact be possible gain extremely good estimates for the size and distance to the moon (within 10% of the real values), if a more realistic value of 0.5 degrees were used for the angular diameter of the moon, rather than the 2 degrees used by Aristarchus.  Evans suggests that the reasons for this is Aristarchus didn’t actually take the measurement at all, but instead only estimated it.  This seems likely if we remember that The Greeks in Aristarchus’s time still weren’t that interested in measurement.  In this context Aristarchus was more of a transitional figure, who was interested in showing that measurement was possible, but not wholly invested in actually carrying through with those estimates.  

Despite this failure to fully grapple with measurement as a science, Aristarchus had demonstrated to his successors that Astronomy could be a fully quantitative science.  Greek science after Aristarchus ceased to simply be a matter of philosophical arguments but would start to become dependent on measurement and observation.  This was a great leap in the history of science and Aristarchus’s work would have a lasting influence on the two great Greek scientists who followed, Hipparchus of Nicaea and Claudius Ptolemy.  These two astronomers would improve upon Aristarchus’s methods and more fully accept the need to combine careful measurement into their mathematical models.  

So in the next episode we will move on to the figure I regard as the greatest astronomer of antiquity, Hipparchus of Nicaea.

Description

We take for granted that science is quantitative.  But the early Greeks were primarily interested in philosophical argument rather than careful measurement.  This began to change in the 3rd century BC when two of the great figures in early science came on the scene.  Eratosthenes would measure the circumference of the Earth, while Aristarchus would be even more ambitious and would attempt to measure the distance to the Sun and the Moon.