Aristarchus's 3rd century BC calculations on the relative sizes of the Earth, Sun and Moon, from a 10th century CE Greek copy

On the Sizes and Distances [of the Sun and Moon] is the only extant work written by Aristarchus of Samos, an ancient Greek astronomer who lived circa 310 BC - 230 BC. In this work, he calculates the sizes of the Sun and Moon, as well as their distances from the Earth in Earth radii. Image File history File links Aristarchus_working. ... Image File history File links Aristarchus_working. ... Statue of Aristarchus at Aristoteles University in Thessaloniki, Greece Aristarchus (310 BC - c. ... Statue of Aristarchus at Aristoteles University in Thessaloniki, Greece Aristarchus (310 BC - c. ... Centuries: 5th century BC - 4th century BC - 3rd century BC Decades: 360s BC 350s BC 340s BC 330s BC 320s BC 310s BC 300s BC 290s BC 280s BC 270s BC 260s BC Years: 315 BC 314 BC 313 BC 312 BC 311 BC _ 310 BC _ 309 BC... Centuries: 4th century BC - 3rd century BC - 2nd century BC Decades: 280s BC 270s BC 260s BC 250s BC 240s BC - 230s BC - 220s BC 210s BC 200s BC 190s BC 180s BC Years: 235 BC 234 BC 233 BC 232 BC 231 BC - 230 BC - 229 BC 228 BC... The Sun (Latin: Sol) is the star at the center of the Solar System. ... Apparent magnitude: up to -12. ... Adjectives: Terrestrial, Terran, Telluric, Tellurian, Earthly Atmosphere Surface pressure: 101. ...

Symbols

His method relied on several observations:

• The apparent size of the Sun and the Moon in the sky (this is easy to measure).
• The size of the Earth's shadow in relation to the moon during a lunar eclipse (this is harder to measure, but can be done with a little effort)
• The angle between the Sun and Moon when the Moon is exactly half lit (this is very hard to measure precisely enough as it is very close to 90 degrees, and is the main reason why the method is not all that accurate)

This construction uses the following variables: Time lapse movie of the 3 March 2007 lunar eclipse A lunar eclipse occurs whenever the Moon passes through some portion of the Earths shadow. ...

The Sun (Latin: Sol) is the star at the center of the Solar System. ... The Sun (Latin: Sol) is the star at the center of the Solar System. ... Apparent magnitude: up to -12. ... Apparent magnitude: up to -12. ... Adjectives: Terrestrial, Terran, Telluric, Tellurian, Earthly Atmosphere Surface pressure: 101. ... Time lapse movie of the 3 March 2007 lunar eclipse A lunar eclipse occurs whenever the Moon passes through some portion of the Earths shadow. ...

Half-lit Moon

Aristarchus began with the premise that, when the moon was exactly half-lit, it forms a right triangle with the Sun and Earth. By observing one of the other angles in this right triangle, Aristarchus could deduce the ratio of the distances to the Sun and Moon using trigonometry. For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... Wikibooks has a book on the topic of Trigonometry Trigonometry (from Greek trigōnon triangle + metron measure[1]) is a branch of mathematics that deals with triangles, particularly triangles in a plane where one angle of the triangle is 90 degrees (right angled triangles). ...

Image File history File links AristarchusTriangleConstruction. ...

From the diagram and trigonometry, we can calculate that

The diagram is greatly exaggerated, because in reality, S = 390L, and φ is extremely close to a right angle (only 10' shy). Aristarchus determined φ to be a thirtieth of a quadrant (in modern terms, three degrees) less than a right angle: in current terminology, 87˚. Trigonometric functions had not yet been invented, but using geometrical analysis in the style of Euclid, Aristarchus determined that Euclid (Greek: ), also known as Euclid of Alexandria, was a Hellenistic mathematician who flourished in Alexandria, Egypt, almost certainly during the reign of Ptolemy I (323–283 BC). ...

In other words: that the distance to the Sun was somewhere between 18 and 20 times greater than the distance to the Moon. This value (or values approximately close to it) was accepted by astronomers for the next two thousand years, until the invention of the telescope permitted a more precise estimate of solar parallax. Motion Parallax (Greek: παραλλαγή (parallagé) = alteration) is the change of angular position of two stationary points relative to each other as seen by an observer, due to the motion of an observer. ...

He also reasoned that as angular size of the Sun and the Moon were the same, but the distance to the Sun was between 18 and 20 times further than the Moon, the Sun must therefore be 18-20 times larger. Angular size is a measurement of how large or small something is using rotational measurement (degrees of arc, arc_minutes, and arc-seconds). ...

Lunar eclipse

Aristarchus then used another construction based on a lunar eclipse:

Image File history File links AristarchusConstruction. ...

By similarity of the triangles, and

Since the apparent sizes of the Sun and Moon are the same, it follows that . Now

We can rewrite several variables in terms of x:

, and

Combining this with the previous equation gives:

These give the radii of the sun and moon entirely in terms of observable quantities. Along with a value for the apparent size of the sun and moon (in degrees), these formulae give the distances to the sun and moon in terrestrial units:

It is unlikely that Aristarchus used these exact formulae, since he would have lacked a precise value for π. However a simple approximation π = 3 will incur in a relative error smaller than 5%, well bellow experimental errors in measurements at the time.

These formulae are likely a good approximation to those of Aristarchus.

Results

His values, then, are computed as:

Quantity	Formula	Value	Actual value
x	n/a	~19	390
n	n/a	2	2.587
θ	n/a	1	0.259
s / t	(1 + x) / (1 + n)	6.67	109
	x(1 + n) / (1 + x)	2.85	3.67
L / t	θ)	20	60.32
S / t	(L / t)(S / L)	380	23,500

The error in this calculation comes primarily from the poor values for x and θ. The poor value for θ is especially surprising, since Archimedes writes that Aristarchus was the first to determine that the sun and moon had an apparent diameter of half a degree. This would give a value of θ = 0.25, and a corresponding distance to the moon of 80 earth radii, a much better estimate. Archimedes (Greek: c. ...

A similar procedure was later used by Hipparchus, who estimated the mean distance to the moon as 67 earth radii, and Ptolemy, who took 59 earth radii for this value. On Sizes and Distances [of the Sun and Moon] (Peri megethoon kai apostèmátoon) is a text by the ancient Greek astronomer Hipparchus. ... Hipparchus. ... A medieval artists rendition of Claudius Ptolemaeus Claudius Ptolemaeus (Greek: ; ca. ...

Works cited

• Heath, T. L.. Aristarchus of Samos. Oxford, 1913. This was later reprinted, see (ISBN 0-486-43886-4).
• van Helden, A. Measuring the Universe: Cosmic Dimensions from Aristarchus to Halley. Chicago: Univ. of Chicago Pr., 1985. ISBN 0-226-84882-5.