Aristarchus of Samos

How to use this course

Aristarchus of Samos is remembered for an idea he probably never fully worked out in writing that survives: that the Earth turns daily on its axis and travels yearly around a motionless Sun. We know of it from a few sentences in Archimedes and from later writers. What does survive in his own words is a short geometric treatise, On the Sizes and Distances of the Sun and Moon, which assumes a stationary Earth and uses the phases of the Moon and the geometry of eclipses to show that the Sun is far larger than the Earth.

The two strands are connected. If the Sun is many times larger than the Earth, it is natural to ask whether the smaller body should circle the larger. Whether Aristarchus reasoned this way is not recorded. This course presents the evidence, separates what each source says from later embellishment, and works through the treatise in detail.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proofs and exercises with folded solutions. Aristarchus worked with ratios of lines and angles, never with trigonometric functions. The course states his results in his form and checks them in ours. The companion course on Claudius Ptolemy shows how his methods were refined by Hipparchus and Ptolemy; the course on Nicolaus Copernicus treats the heliocentric system that was built eighteen centuries later.

Timeline

Only one date in Aristarchus’s life is attested: the solstice of 280 BC. His birth and death years are modern estimates built around it.

Samos and the Lyceum

Aristarchus came from Samos, a prosperous island off the coast of Asia Minor and the birthplace, three centuries earlier, of Pythagoras. His dates are uncertain. The usual estimate, about 310 to about 230 BC, rests on two points: his observation of the summer solstice of 280 BC, which Ptolemy reports, and Archimedes’ discussion of his work in a treatise written before about 216 BC. Archimedes speaks of him as a contemporary or near-contemporary.

Strato

The doxographer Aetius, a compiler of philosophers’ opinions probably of the first or second century AD, calls Aristarchus a pupil of Strato of Lampsacus. Strato had been tutor in Alexandria to the future king Ptolemy II and became head of Aristotle’s school, the Lyceum, in Athens about 287 BC. He was the most empirical of the Peripatetics, known for experiment-based arguments about the void and about falling bodies. Where and when Aristarchus studied with him is not known. Some modern accounts place the instruction in Alexandria, others in Athens.

What is not known

No ancient biography of Aristarchus survives. We do not know his family, where he lived after Samos, whether he held a position in Alexandria, or where he died. Vitruvius, writing in the late first century BC, lists him among the rare men skilled in all the sciences, geometry, astronomy, music and the rest, which tells us of his reputation but nothing of his life. A work on vision, light and colour is also attributed to him by later writers; it is lost.

Astronomy in the age of Euclid

Aristarchus worked in the generation after Euclid, when Alexandria under the first Ptolemies was becoming the centre of Greek mathematics. Several ideas that he used or reacted against were already in circulation.

• Eudoxus of Cnidus (fourth century BC) had described the motions of the Sun, Moon and planets with nested spheres centred on the Earth, and created the theory of proportion that Aristarchus’s treatise depends on (Euclid’s Elements, Book V).
• Philolaus, a Pythagorean of the fifth century BC, had set the Earth in motion around a central fire, not the Sun. The Earth, the Sun, the Moon and the planets all circled this fire.
• Heraclides of Pontus (fourth century BC) had proposed that the Earth rotates daily on its axis. Plutarch says Aristarchus followed him in this.
• Anaxagoras (fifth century BC) had explained that the Moon shines by reflected sunlight and that eclipses are shadows, the two physical facts on which Aristarchus’s treatise rests.
• Euclid’s Phaenomena and Autolycus’s works on the moving sphere gave a geometric treatment of the celestial sphere. Later these were studied together with Aristarchus’s treatise as a preparation for the Almagest.

At Alexandria, Timocharis and Aristyllus measured star positions in the early third century BC; Hipparchus later used their observations to discover precession. Aristarchus belongs to this first generation of systematic observers. Archimedes, younger by a generation, cited him with respect; Eratosthenes, younger still, measured the Earth.

The testimonies

Almost everything known about Aristarchus outside his surviving treatise comes from brief remarks by other authors. They are listed here in chronological order, with what each says.

• Archimedes, The Sand Reckoner (before c. 216 BC). The only report by a contemporary. Aristarchus wrote out certain hypotheses, from which it follows that the universe is many times larger than usually supposed: the fixed stars and the Sun remain unmoved, the Earth revolves about the Sun on the circumference of a circle with the Sun in the middle of the orbit, and the sphere of the fixed stars is so large that the Earth’s circle bears to the distance of the stars the proportion a centre bears to the surface of a sphere. Archimedes also says Aristarchus found the Sun’s apparent diameter to be about 1/720 of the circle of the zodiac, that is \(\tfrac12^\circ\).
• Vitruvius, On Architecture (late first century BC), credits him with two sundials: the hemispherical bowl (scaphe or hemisphaerium) and a flat disc.
• Plutarch, On the Face in the Moon (c. AD 100). A speaker recalls that Cleanthes the Stoic thought the Greeks ought to charge Aristarchus with impiety for setting the hearth of the universe in motion, and explains that Aristarchus supposed the heavens at rest and the Earth revolving in an inclined circle while turning on its axis. Lesson 11 discusses a textual problem in this passage.
• Plutarch, Platonic Questions. Contrasts Aristarchus, who only supposed the Earth’s motion, with Seleucus, who asserted it.
• Aetius (1st–2nd century AD) names Strato as his teacher and reports views on light and colour.
• Censorinus, On the Birthday (AD 238), says Aristarchus reckoned a Great Year of 2484 years.
• Ptolemy, Almagest III (c. AD 150), cites his observation of the summer solstice of 280 BC.

Two points stand out. Archimedes and Plutarch call the heliocentric idea a hypothesis, a supposition from which consequences follow. And the work in which Aristarchus stated it is lost. We cannot tell whether he gave planetary models, or how he answered the objections to a moving Earth.

The heliocentric hypothesis

For the general reader. If the Earth goes around the Sun, the nearer stars should seem to shift back and forth during the year, as a nearby tree shifts against distant hills when you walk past. No such shift was seen. Aristarchus’s answer, as Archimedes reports it, was that the stars are so far away that the Earth’s orbit is a mere point by comparison. The same answer had to be given by Copernicus, and it was confirmed only in 1838.

What the hypothesis explains

Nothing in the sources says what phenomena Aristarchus used his hypothesis to explain. Its consequences are clear, though. The daily rotation of the Earth replaces the daily rotation of the whole sky. The annual revolution replaces the Sun’s yearly motion along the ecliptic. And, if the planets also circle the Sun, the retrograde loops of the planets become an effect of the Earth overtaking them or being overtaken, as Lesson 05 of the Copernicus course explains. Pliny and Seneca later treated retrogradation as merely apparent, which fits this view, but neither attributes the explanation to Aristarchus.

Annual parallax

If observers can detect angles down to \(\delta\), the absence of parallax requires \(d>a/\delta\) roughly. Aristarchus’s hypothesis, that the orbit is to the stars as a centre to a surface, sets \(a/d=0\): no parallax at all. Archimedes objected that a centre has no magnitude and so bears no ratio to anything. He reinterpreted the claim as a proportion: the diameter of the universe is to the diameter of the Earth’s orbit as the diameter of the Earth’s orbit is to the diameter of the Earth. This is equivalent to saying that the stars show the same parallax from the Earth’s orbit as the Sun shows from the Earth’s radius.

Six hypotheses and the lit Moon

For the general reader. Aristarchus’s surviving treatise is written like a book of Euclid: hypotheses first, then numbered propositions with proofs. The hypotheses combine physical facts (the Moon reflects sunlight) with measured values (an angle, a shadow width, an apparent size). From them he derives bounds on the sizes and distances of the Sun and Moon. The first propositions settle a geometric question that might seem pedantic: how much of the Moon does the Sun actually light?

Hypotheses 4, 5 and 6 are measurements. If a sign means \(30^\circ\), Hypothesis 6 gives the Moon an apparent diameter of \(2^\circ\), four times too large. Archimedes says Aristarchus found \(\tfrac12^\circ\) for the Sun, the correct value. Some scholars therefore read the Greek word for “part” as a smaller unit (\(7\tfrac12^\circ\)) that gives \(\tfrac12^\circ\); others think the treatise is an early work and the half degree a later refinement. As Lesson 08 shows, the value matters for the distances in Earth radii but not for the main ratios.

How much of the Moon is lit

Proof. Take the plane through both centres. The lit part of the small sphere is bounded by the points of tangency of the cone that envelops both spheres. Let the spheres have radii \(r<R\) and centres at distance \(D\). The external common tangent makes an angle \(\gamma\) with the line of centres, where \(\sin\gamma=(R-r)/D\). The point of tangency on the small sphere lies at angle \(90^\circ+\gamma\) from the direction to the large sphere, measured at the small sphere’s centre. So the lit cap extends \(\gamma>0\) beyond a hemisphere. Aristarchus argues the same way with Euclid’s theorems on tangents and cones (his Proposition 1 shows unequal spheres are enclosed by a single cone with its vertex beyond the smaller).

Propositions 3 and 4 then show that the circle dividing light from dark on the Moon, as seen from the Earth, differs imperceptibly from a great circle. That justifies Hypothesis 3 and the use of the right angle at the Moon in Lesson 06.

The half moon: the Sun is 18 to 20 times farther

For the general reader. When the Moon is exactly half lit, the Sun, Moon and Earth form a right triangle with the right angle at the Moon. Measure the angle between the Sun and the Moon as seen from the Earth, and the shape of the triangle, and hence the ratio of the distances, is fixed. Aristarchus took the angle to be \(87^\circ\). Without trigonometric tables he had to bound \(1/\cos87^\circ\) by pure geometry, and he invented the inequalities to do so.

The exact ratio

Let \(E\), \(M\), \(S\) be the Earth, Moon and Sun, with \(\angle EMS=90^\circ\) at half moon and \(\angle MES=\theta\). Then

With \(\theta=87^\circ\), this is \(1/\sin3^\circ=19.107\). Aristarchus needed bounds for \(\sin3^\circ\).

Proof. The functions \(\sin x/x\) and \(x/\tan x\) are both strictly decreasing on \((0,\pi/2)\). For the first, the derivative of \(\sin x/x\) is \((x\cos x-\sin x)/x^2<0\) because \(\tan x>x\). For the second, \(x/\tan x=x\cos x/\sin x\) has derivative \((\sin x\cos x-x)/\sin^2x=(\tfrac12\sin2x-x)/\sin^2x<0\). Aristarchus’s proofs are geometric: he compares areas of sectors and triangles in a circle, as Euclid would. Ptolemy used the first inequality four centuries later to find the chord of \(1^\circ\) (see the Ptolemy course, Lesson 04).

Lower bound. By the tangent inequality with \(\alpha=22\tfrac12^\circ\), \(\beta=3^\circ\): \(\tan3^\circ<\tfrac{3}{22.5}\tan22\tfrac12^\circ=\tfrac{2}{15}\tan22\tfrac12^\circ\). Now \(\tan22\tfrac12^\circ=\sqrt2-1=1/(1+\sqrt2)\), and Aristarchus uses \(49>2\cdot25\), that is \(\sqrt2>\tfrac75\), to get \(\tan22\tfrac12^\circ<\tfrac{5}{12}\). Hence \(\tan3^\circ<\tfrac{2}{15}\cdot\tfrac{5}{12}=\tfrac1{18}\), and since \(\sin3^\circ<\tan3^\circ\), \(ES/EM=1/\sin3^\circ>18\).

Upper bound. By the sine inequality with \(\alpha=30^\circ\), \(\beta=3^\circ\): \(\sin30^\circ/\sin3^\circ<10\), so \(\sin3^\circ>\tfrac1{20}\) and \(ES/EM<20\).

Equal discs and the Moon’s diameter

For the general reader. In a total eclipse of the Sun the Moon just covers the Sun’s disc. So the two have almost the same apparent size, and the larger one must be farther away in exactly the proportion of its size. Aristarchus used this to turn the distance ratio of Lesson 06 into a size ratio: the Sun is 18 to 20 times as wide as the Moon.

Proof. Total eclipses show that the Moon and Sun subtend nearly the same angle \(2\theta\) at the eye. Then each radius is \(\sin\theta\) times its distance, and the ratio of radii equals the ratio of distances, bounded by Proposition 7. Volumes of spheres are as the cubes of their diameters (Euclid XII.18).

With an apparent diameter of \(2^\circ\) (Hypothesis 6), the exact ratio is \(2\sin1^\circ=0.0349\). Aristarchus obtains the bounds with the same kind of inequality between ratios of arcs and lines as in Proposition 7.

The eclipse diagram: sizes in Earth units

For the general reader. Ratios between the Sun and Moon do not say how big either is. To connect them to the Earth, Aristarchus used lunar eclipses. The Earth casts a cone of shadow away from the Sun. Where the Moon crosses it, the shadow is about two Moons wide. Because the cone narrows at a rate set by the sizes of the Sun and Earth, this width ties all three bodies together.

The shadow cone

Let the Sun, Earth and Moon have radii \(s\), \(t\), \(\ell\), the Sun at distance \(S\) and the Moon at distance \(L\) from the Earth, and let the shadow at the Moon’s distance have radius \(n\ell\) (Hypothesis 5: \(n=2\)). The shadow cone’s edge is a line tangent to the Sun and the Earth. Along it the radius decreases linearly with distance, so

Put \(x=S/L\) and use \(s=x\ell\) (equal apparent sizes). Then \(x\ell-t=x(t-n\ell)\), which gives

With \(n=2\) and \(x=18\), the formula gives \(s/t=\tfrac{19}{3}\), exactly Aristarchus’s lower bound. With \(x=20\) it gives \(7\), inside his upper bound \(\tfrac{43}{6}\approx7.17\). His geometry does not use the simplified tangent line above but works with the actual cone and the chords it cuts on the Moon’s orbit (his Propositions 13 and 14), and he loosens bounds where the exact figure would need more steps. The Earth–Moon ratio follows by combining Proposition 15 with Proposition 9: \(\tfrac{\ell}{t}\) lies between \(\tfrac1{20}\cdot\tfrac{19}{3}=\tfrac{19}{60}\) and \(\tfrac1{18}\cdot\tfrac{43}{6}=\tfrac{43}{108}\).

Distances

The treatise stops at ratios of sizes. But with an apparent radius \(\theta\), the Moon’s distance is \(L=\ell/\sin\theta\), and so

With \(x=19.1\), \(n=2\), \(\theta=1^\circ\), this is about 20 Earth radii for the Moon and 380 for the Sun. With \(\theta=\tfrac14^\circ\) the Moon moves out to about 80. Hipparchus and Ptolemy later reversed the procedure: they measured the Moon’s distance from parallax and used the eclipse relation to find the Sun’s (Ptolemy course, Lesson 07).

Why the numbers were wrong

For the general reader. Aristarchus’s Sun is about twenty times too close. His geometry is flawless. The trouble is that his method asks the eye to judge the exact moment of half moon and to measure an angle that differs from a right angle by a sixth of a degree. This lesson quantifies how sensitive the answer is, and asks whether the 87° was a measurement at all.

Equivalently, the half moon comes slightly before the geometric quarter, by the time the Moon takes to move through \(\varepsilon\) relative to the Sun. The Moon gains on the Sun by about \(12.2^\circ\) per day. With the true \(\varepsilon\approx9'\), the half moon precedes the quarter by about 18 minutes. Aristarchus’s \(3^\circ\) corresponds to about 6 hours. Judging by eye the moment when the terminator is exactly straight is uncertain by hours, because the Moon’s surface is rough and the terminator is never sharp.

Some historians therefore think the 87° was never a careful measurement. Aristarchus may have chosen a round value to illustrate a method, stating only that the angle is less than a right angle by a definite amount, and the treatise reads as an exercise in geometry with given data. Others note that his conclusion, a Sun much larger than the Earth, was robust whatever the exact angle: any \(\varepsilon\) below \(3^\circ\) only makes the Sun larger.

Sundials, solstices and the Great Year

For the general reader. Three further pieces of Aristarchus’s astronomy are known from later writers: a solstice observation, a sundial shaped like a bowl, and a long cycle called the Great Year. Each connects to practical problems of measuring time, and each can be analysed with simple geometry.

The scaphe

The proof is one line: light travels in a straight line through the centre, and a line through the centre of a sphere meets it at antipodal points. The bowl therefore maps the sky onto itself. Lines engraved for the hours and for the solstices turn it into a clock and calendar. Vitruvius credits Aristarchus with this scaphe or hemisphaerium, and also with a flat-disc dial.

The solstice of 280 BC

Ptolemy, in Almagest III, uses a summer solstice observed by Aristarchus in 280 BC as one of the older data for the length of the year. Solstices are hard to time precisely because the Sun’s declination changes very slowly near them.

The Great Year

Censorinus reports that Aristarchus reckoned a Great Year of 2484 years. Its meaning is not explained. Some scholars argue that 2484 is a copying error for 2434, which is 45 times the eclipse cycle called the exeligmos (three saros cycles, 19,756 days).

Seleucus and the road not taken

The heliocentric hypothesis found at least one later supporter. Plutarch, in the Platonic Questions, says that Aristarchus only proposed the Earth’s motion as a hypothesis, while Seleucus of Seleucia, a Babylonian astronomer of the second century BC, asserted it as true. How Seleucus argued is unknown. Later writers say he studied the tides and linked them to the Moon.

The impiety story

Plutarch’s dialogue On the Face in the Moon contains the one hint of hostility. In the text printed since the seventeenth century, a speaker says that Cleanthes, the head of the Stoic school, thought the Greeks ought to indict Aristarchus for impiety for moving the hearth of the universe. In the manuscripts, the grammatical cases are reversed, so that Aristarchus is the one who says Cleanthes should be indicted. The scholar Gilles Ménage changed the reading in the seventeenth century, soon after the trials of Galileo and Bruno. Lucio Russo has argued that the image of a persecuted Aristarchus rests on that emendation. No source records any charge or trial.

Why the hypothesis did not prevail

Hipparchus, the greatest astronomer of the next century, built geocentric models. Ptolemy rejected the Earth’s motion on physical grounds in Almagest I. Several reasons are given by historians:

• No parallax. The stars showed no annual shift, which required an enormous and empty universe (Lesson 04).
• Physics. Aristotle’s physics gave the Earth a natural place at the centre and could not explain why bodies on a moving Earth were not left behind.
• No predictive gain. A Sun-centred system with circular orbits does not predict positions better than a geocentric one with epicycles; Lesson 05 of the Copernicus course shows they are equivalent for directions.

The “ancient Copernicus”

On the Sizes and Distances survived because it became a school text. In late antiquity it was read with Euclid’s Phaenomena and other short works as preparation for the Almagest, and Pappus of Alexandria discussed it in the fourth century. Nasir al-Din al-Tusi made an Arabic recension in the thirteenth century. Giorgio Valla published a Latin translation in 1488, Federico Commandino a Latin translation with commentary in 1572, and John Wallis the first printed Greek text in 1688.

Copernicus

In the autograph manuscript of De revolutionibus, kept in the Jagiellonian Library in Kraków, Copernicus mentioned Philolaus’s belief in the mobility of the Earth and named Aristarchus in the same passage. The passage was struck out and does not appear in the printed book of 1543. Owen Gingerich judged the deletion editorially sensible and found no evidence that Copernicus knew the Sand Reckoner, which was printed only in 1544. Copernicus could have learned of Aristarchus’s view from Plutarch or from compilations of ancient opinions. Gingerich concluded that he reached heliocentrism independently. The Copernicus course, Lesson 03, sets out the postulates of the Commentariolus, the closest modern parallel to the hypotheses Archimedes reports.

Legacy

• Method: the first known attempt to measure the solar system with geometry, using the Moon’s phases and eclipses. Hipparchus and Ptolemy refined the eclipse method, and it remained the standard route to the Sun’s distance until the seventeenth century. Values for the Sun’s distance of the order of a thousand Earth radii, a solar parallax of a few minutes of arc, persisted until the seventeenth century.
• Mathematics: the inequalities between ratios of arcs, sines and tangents, used again by Ptolemy for his chord table.
• Cosmology: the first known Sun-centred system, with the right answer to the parallax objection.
• Names: Thomas Heath’s 1913 edition called him the ancient Copernicus. A bright lunar crater, the asteroid 3999 Aristarchus and the Aristarchos telescope in Greece bear his name.

Sources and further reading

Biography and testimonies

1. J. J. O’Connor and E. F. Robertson, Aristarchus of Samos, MacTutor History of Mathematics.
2. Aristarchus of Samos, Wikipedia (with extensive references).
3. M. E. Mickelson, Aristarchus of Samos, Biographical Encyclopedia of Astronomers, Springer, 2007.
4. The Sand Reckoner, Wikipedia; T. L. Heath, The Works of Archimedes, Cambridge, 1897.
5. Great Year, Wikipedia (Censorinus’s report).
6. L. Russo, The Forgotten Revolution, Springer, 2004.

The treatise and its mathematics

1. T. L. Heath, Aristarchus of Samos, the Ancient Copernicus, Oxford, 1913; Dover reprint, 1981.
2. On the Sizes and Distances (Aristarchus), Wikipedia.
3. Aristarchus, On the Sizes and Distances, Greek text with English translation, R. Cottrell.
4. J. L. Berggren and N. Sidoli, Aristarchus’s On the Sizes and Distances of the Sun and the Moon: Greek and Arabic texts, Archive for History of Exact Sciences 61 (2007).
5. A. Van Helden, Measuring the Universe, University of Chicago Press, 1985.
6. J. Evans, The History and Practice of Ancient Astronomy, Oxford University Press, 1998.
7. O. Gingerich, Did Copernicus owe a debt to Aristarchus?, Journal for the History of Astronomy 16 (1985).