Read To Explain the World: The Discovery of Modern Science Online
Authors: Steven Weinberg
To Explain the World: The Discovery of Modern Science (44 page)
The rest is algebra. We can solve the second equation for
d
0
, and find:
Inserting this result in the first equation and multiplying with
D
e
D
s
(
D
e
– 2
D
m
) gives
The terms
d
m
D
s
× (–2
D
m
) and 2
D
m
d
m
D
s
on the right-hand side cancel. The remainder on the right-hand side has a factor
D
e
, which cancels the factor
D
e
on the left-hand side, leaving us with a formula for
D
e
:
If we now use the result of observation 2, that
d
s
/
d
m
=
D
s
/
D
m
, this can be written entirely in terms of diameters:
If we use the prior result
D
s
/
D
m
= 19.1, this gives
De
/
Dm
= 2.85. Aristarchus gave a range from 108/43 = 2.51 and 60/19 = 3.16, which nicely contains the value 2.85. The actual value is 3.67. The reason that this result of Aristarchus was pretty close to the actual value despite his very bad value for
D
s
/
D
m
is that the result is very insensitive to the precise value of
D
s
if
D
s
>>
D
m
. Indeed, if we neglect the term
D
m
in the denominator altogether compared
with
D
s
, then all dependence of
D
s
cancels, and we have simply
D
e
= 3
D
m
, which is also not so far from the truth.
Of much greater historical importance is the fact that if we combine the results
D
s
/
D
m
= 19.1 and
D
e
/
D
m
= 2.85, we find
D
s
/
D
e
= 19.1/2.85 = 6.70. The actual value is
D
s
/
D
e
= 109.1, but the important thing is that the Sun is considerably bigger than the Earth. Aristarchus emphasized the point by comparing the volumes rather than the diameters; if the ratio of diameters is 6.7, then the ratio of volumes is 6.7
3
= 301. It is this comparison that, if we believe Archimedes, led Aristarchus to conclude that the Earth goes around the Sun, not the Sun around the Earth.
The results of Aristarchus described so far yield values for all ratios of diameters of the Sun, Moon, and Earth, and the ratio of the distances to the Sun and Moon. But nothing so far gives us the ratio of any distance to any diameter. This was provided by the fourth observation:
Observation 4
The Moon subtends an angle of 2°.
(See Figure 5d.) Since there are 360° in a full circle, and a circle whose radius is
d
m
has a circumference 2
πd
m
, the diameter of the Moon is
Aristarchus calculated that the value of
D
m
/
d
m
is between 2/45 = 0.044 and 1/30 = 0.033. For unknown reasons Aristarchus in his surviving writings had grossly overestimated the true angular size of the Moon; it actually subtends an angle of 0.519°, giving
D
m
/
d
m
= 0.0090. As we noted in
Chapter 8
, Archimedes in
The Sand Reckoner
gave a value of 0.5° for the angle subtended by the Moon, which is quite close to the true value and would have
given an accurate estimate of the ratio of the diameter and distance of the Moon.
With his results from observations 2 and 3 for the ratio
D
e
/
D
m
of the diameters of the Earth and Moon, and now with his result from observation 4 for the ratio
D
m
/
d
m
of the diameter and distance of the Moon, Aristarchus could find the ratio of the distance of the Moon to the diameter of the Earth. For instance, taking
D
e
/
D
m
= 2.85 and
D
m
/
d
m
= 0.035 would give
(The actual value is about 30.) This could then be combined with the result of observation 1 for the ratio
d
s
/
d
m
= 19.1 of the distances to the Sun and Moon, giving a value of
d
s
/
D
e
= 19.1 × 10.0 = 191 for the ratio of the distance to the Sun and the diameter of the Earth. (The actual value is about 11,600.) Measuring the diameter of the Earth was the next task.