On the Sizes and Distances (Aristarchus)

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.

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:

Symbol	Meaning
s	Radius of the Sun
S	Distance to the Sun
l	Radius of the Moon
L	Distance to the Moon
t	Radius of the Earth
D	Distance to the vertex of Earth's shadow cone
n	d/l, a directly observable quantity during a lunar eclipse
$\phi$	Directly observed.
x	S/L, which is derived from $\phi$

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 Moon. 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.

From the diagram and trigonometry, we can calculate that

${\frac {S}{L}}={\frac {1}{\cos \varphi }}=\sec \varphi .$

The diagram is greatly exaggerated, because in reality, $S=390L$, and $\phi$ is extremely close to a right angle (only 10' shy). Aristarchus observed an angle as a thirtieth of a quadrant, or in modern terms, three degrees. When Aristarchus wrote trigonometric functions had not yet been invented, but using geometrical analysis in the style of Euclid, he determined that

$18<{\frac {S}{L}}<20.$

Lunar eclipse

Aristarchus then used another construction based on a lunar eclipse:

By similarity of triangles, ${\frac {D}{S}}={\frac {t}{s-t}}\$ and $\ {\frac {d}{t}}={\frac {D-L}{D}}.$

Since the apparent sizes of the Sun and Moon are the same, it follows that ${\frac {L}{S}}={\frac {\ell }{s}}$. Now

${\frac {D}{L}}={\frac {t}{t-d}},{\frac {D}{S}}={\frac {t}{s-t}}\Rightarrow {\frac {L}{S}}={\frac {t-d}{s-t}}\Rightarrow {\frac {\ell }{s}}={\frac {t-d}{s-t}}\Rightarrow 1-{\frac {t}{s}}={\frac {t}{\ell }}-{\frac {d}{\ell }}\Rightarrow {\frac {t}{\ell }}+{\frac {t}{s}}=1+n.$

We can rewrite several variables in terms of x:

$\ell ={\frac {s}{x}}$, and $s=\ell x.$

Combining this with the previous equation gives:

${\frac {tx}{s}}+{\frac {t}{s}}=n+1\Rightarrow {\frac {t}{s}}={\frac {1+n}{1+x}}\Rightarrow {\frac {s}{t}}={\frac {1+x}{1+n}}$

${\frac {t}{\ell }}+{\frac {t}{\ell x}}=n+1\Rightarrow {\frac {t}{\ell }}={\frac {1+n}{1+x}}x$

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:

${\frac {L}{t}}=\left({\frac {\ell }{t}}\right)\left({\frac {180}{\pi \theta }}\right)$

${\frac {S}{t}}=\left({\frac {s}{t}}\right)\left({\frac {180}{\pi \theta }}\right)$

It is unlikely that Aristarchus used these exact formulae, since he would have lacked a precise value for $\pi$. However a simple approximation $\pi =3$ 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
$\theta$	n/a	1	0.259
$s/t$	$(1+x)/(1+n)$	6.67	109
$t/\ell$	$x(1+n)/(1+x)$	2.85	3.67
$L/t$	$(\ell /t)(180/\pi$$\theta )$	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 $\theta$. The poor value for $\theta$ 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 $\theta =0.25$, and a corresponding distance to the moon of 80 earth radii, a much better estimate.

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.

Works cited

• Heath, T. L.. Aristarchus of Samos. Oxford, 1913. This was later reprinted, see (ISBN 0486438864).
• van Helden, A. Measuring the Universe: Cosmic Dimensions from Aristarchus to Halley. Chicago: Univ. of Chicago Pr., 1985. ISBN 0-226-84882-5.

External links

• The Moon's Distance (1)   How Aristarchus estimated the distance to the Moon, using a lunar eclipse.

• The Moon's Distance (2)   How Hipparchus estimated the distance to the Moon, using a different method and a solar eclipse (similar to that of August 1999).

• How Aristarchus estimated the Sun's distance   The reasoning behind his heliocentric theory.