Aristarchus: The Man Who Introduced Mathematics to Astronomy
Steven R. Edwards
Southern Polytechnic State University
Abstract: We examine the state of astronomical knowledge 300 BC, in particular the contributions of Aristarchus of Samos, who was the first to use astronomical observations and geometry to estimate the relative sizes and distances of the earth, sun, and moon.
After Plato founded his Academy around 365 BC, in addition to his strictly philosophical pursuits, one of the tasks that he proposed to his students was to give an explanation or a mathematical model that would describe the motions of the planets. In this era, Athens was becoming the intellectual center of the Mediterranean, but Greek culture matured in the shadow of the ancient cultures of Babylon and Egypt. Thales of Miletus (621-547 BC), one of the so-called seven wise men of Greece, is credited not only with the first simple theorems of geometry, but also with hypothesizing that earthquakes and other phenomena have natural, i.e. explainable causes, rather than supernatural ones. The shift from putting the gods as the cause of all things to trying to understand the world by cause and effect is one of the reasons that Greek thought is still important to our culture.
Plato’s challenge was answered by Eudoxus of Cnidus (408-355 BC), who devised a system using rotating spheres. Each planet had its motions governed by the motions of four spheres that rotated on various axes, and the combination of the spherical motions gave a model that would produce the movements of the planets, including most importantly retrograde motion. Eudoxus’ model gave a tolerable description of the motions of the three outer planets, but the model did not give an accurate description of the motion of the inner planets. His model also did not give an explanation of the observed variable brightness of Venus and Mars, or of the observed variable size of the moon. Eudoxus’ work does not survive, but his was the world’s first attempt to give a geometrical, rather than supernatural, account of the motion of the heavenly bodies. By contrast, in Egypt at this time, the story was still that the Sun was given birth anew each morning by a goddess, and then the sun died each night. According to the Egyptian priests, the stars and planets were carried across the sky in boats. The Egyptians, on the other hand, had been observing the stars for more than 2500 years, and in fact Plato, Thales, and most of the other luminaries from early Greece visited Egypt to learn from the priests. In modern times, Eudoxus is better known for creating the modern theory of proportion and for the method of exhaustion, which are both in Euclid, than for his astronomical model.
Anaximander of Miletus (610-547 BC) declared that the earth is the center of the universe, and remains at the center because it is equidistant from the rest of the heavenly bodies. Pythagoras of Samos (572-500 BC) is credited with proclaiming the earth to be a sphere, rather than a cylinder floating on water, which was one of the previous theories. Heraclides of Pontus (388-310 BC), a student of Plato, glimpsed the truth. Not only did he propose that the earth rotates daily on its axis, but he also explained the variable brightness of Venus and Mercury by having them rotate about the sun, instead of about the earth. But Heraclides’ theory was dismissed even by Aristotle. Aristotle’s objections included the claims that if the earth were spinning, everything would fly off and the atmosphere would be left behind.
Aristarchus of Samos (310-230 BC) devised the theory of the heliocentric universe. Aristarchus formulated his theory 1800 years before Copernicus. His theory was that the sun and stars are unmoved, that the earth rotates daily about its axis and revolves yearly about the sun. His theory was treated so harshly by his contemporaries that his work on that subject doesn’t survive. We know almost nothing about his life, not even whether he worked in Athens or Alexandria. Aristarchus was a student of Strato of Lampsacus, who was tutor to young Ptolemy II in Alexandria, and who later became head of Aristotle's Lyceum in Athens. It is conjectured that Aristarchus studied with Strato some time after Strato went to Alexandria in 287 BC. Archimedes, who was 25 years younger than Aristarchus, dismissed Aristarchus’ theory in his work “The Sand Reckoner”, in which he calculated how many grains of sand it would take to fill the universe. Aristarchus gave one of the first reasonable calculations of the distances between and sizes of the sun, earth, and moon. He estimated the sun’s volume to be 300 times the earth’s. Aristarchus must have decided that it was more absurd to have an immense sun racing around the earth every day than it was to have us on the surface of a spinning earth.
Aristarchus used four astronomical observations and some geometry and arithmetic to get his estimates. He didn’t have a telescope or a calculator, and given his data, his conclusions are all sound. Let’s look at his methods.
When the moon is half full, there appears to be a line dividing the light half from dark half. Aristarchus realized that when the moon is half full, the line dividing light from dark is a approximately a great circle on the sphere of the moon. Aristarchus hypothesized that when the moon appears to be exactly half full, then the line from our eyes to the moon is in the same plane as the great circle which is the border of light and darkness.
Aristarchus then measured the angle between the lines from the earth to the moon and sun as 87°. The bodies form a right triangle with angles 87° and 3°. In modern terms, to calculate the relative distances of the moon and sun from the earth, he needed to calculate the sine of 3°. Unfortunately, trigonometry had not yet been invented, and there was no such thing as a table of sines, but he was able to use similar triangles to calculate the inequality 1/18 > sin 3° > 1/20. From this, it follows that that the sun is 18 to 20 times as far from the earth as the moon is.
His second observation used the
positions of the sun, moon, and earth at a total solar eclipse. At a total
eclipse, the moon almost exactly obscures the sun. This can happen if the moon
is either relatively small and close to the earth, or large and far away.
In short, by similar triangles, the ratio of distances from earth to sun and earth to moon must equal the ratio of the sun’s diameter to the moon’s diameter. Since the ratio of the distances had been calculated as 19 to 1, this makes the sun’s diameter 19 times the moon’s.
According to modern astronomers, the sun has diameter 1,392,000 km, the moon 3500 km. The distance from the earth to sun is 148,000,000 km, and it is 400,000 km to the moon.
The ratios of distances to diameters are 148,000,000/1,392,000 = 106 and 400,000/3500 = 114. These ratios, 106 and 114, should be equal by Aristarchus’ reasoning. They are obviously not equal, but they are also not hugely different.
The third observation was that of the apparent angular size of the moon. Aristarchus used a measurement of 2 degrees as the angular size of the moon. He then calculated that the distance from us to the moon is between 25 and 33 times the moon’s diameter.
We now have estimates for relative distances between the 3 bodies, and relative sizes for the moon and the sun. The last piece of the puzzle is the size of the earth. No one would get a good and verifiable measurement for the size of the earth until about 100 years after Aristarchus did this work, so Aristarchus settled for a calculation that would give relative values.
Aristarchus didn’t have a method to measure the earth in tangible units such as miles or meters. Instead, he was able to get an estimate for the relative sizes of the sun, moon, and earth. Aristarchus’ fourth observation used the positions of the bodies during a lunar eclipse. He use an estimate based on the length of time the moon is in the shadow of the sun during a lunar eclipse. He estimated that during a lunar eclipse the diameter of the moon is half the width of the shadow of the earth at the moon. This relationship allowed Aristarchus to estimate the size of the earth in relation to the distances to the sun and moon, and he already had a relation between these distances and the relative sizes of the sun and moon.
By Aristarchus’ calculations the sun’s diameter is about 7 times as big as the earth’s. The earth’s diameter is about 3 times as big as the moon’s. Astronomers today estimate that the sun’s diameter is 110 times the earth’s. How did Aristarchus get such a low estimate?
Aristarchus’ measurement of 87° when the moon is half-full is much too small. A better measurement is 89° 50’. If you use Aristarchus’ method with this measurement, then you get that the sun is 344 times as far from the earth as the moon is. Since the distances and diameters are in the same ratio, this also makes the sun much much larger and farther away than Aristarchus’ estimate. The current data support an estimate of 370 times.
In Aristarchus’ time, everyone thought (or knew?) that the earth was the center of the universe, and all other heavenly bodies revolved around the earth. After Aristarchus calculated that the sun’s diameter was seven times as large as the earth’s, he calculated that the volume of the sun would then be more than 300 times the volume of the earth. With the sun so large, he must have wondered how something so large could move with such great speed. In any event, he was the first to come up with the idea of the sun at the center of what we now call the solar system. But his idea was too far out to be believed. No manuscript survives. 1800 years later Copernicus revisited the idea.
Eratosthenes of Cyrene (275-194 BC) was taught in his home town by descendents of colleagues of Socrates, but before long he wound up in Athens with Arcesilaus, current head of Plato’s Academy, and founder of the Skeptics. Before long, Eratosthenes was invited by Ptolemy III to come to Alexandria to tutor his son. When in Alexandria, he heard that on the summer solstice at noon at Aswan in the south of Egypt, the sun would appear directly overhead. He measured the angle of the sun in Alexandria on the summer solstice at noon. He had an estimate for the distance between the two cities, and assuming that Alexandria was directly north of the other spot he got an estimate for the circumference of the earth. Unfortunately, no one is certain how long the units, stadia, were, that he was using, so we’re not sure how good his estimate was. But he was certainly in the right ballpark.
Eratosthenes assumed that the sun was far enough away that at any moment, at two different points on the earth, rays from the sun were close enough to parallel that the difference in direction was negligible. He knew that in Aswan in the south of Egypt the sun was directly overhead at noon on the solstice. So he measured the distance between the city and Alexandria as 4900 stadia, and he measured the inclination of the sun at noon on the solstice in Alexandria as 7°. He reasoned that since the earth is round, 7° on the circumference of the earth is 4900 stadia, giving the circumference of the earth as 252,00 stadia. Unfortunately, no one is sure whether he was using the Attic or the Egyptian stadium. The Attic gives a circumference that is 16% too large, the Egyptian gives 1% too small, when compared to modern measurements.
Apollonius of Perga (262-190 BC) invented epicycles, which were used to model planetary and other celestial motion for the next 1700 years.
Hipparchus of Rhodes (190-120 BC) calculated trigonometric tables, introduced the 360 degree measurement from the Babylonians, made a star catalog, and discovered the precession of the equinoxes.
Ptolemy (85-165 AD) wrote the Almagest, which was the standard reference until Copernicus introduced the heliocentric theory using epicycles in 1543.
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