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\begin{document}
\title{Archimedes the astronomer}
\author{Ilan Vardi}
\date{}
\maketitle
Archimedes (287 B.C.--212 B.C.) is widely regarded as the greatest
mathematician of antiquity and one of the greatest mathematicians
of all time. Archimedes lived in Syracuse on the island
of Sicily, and was a protege of its kings Hieron then Gelon.
It is believed that Archimedes' father was an astronomer,
since a passage about the relative sizes of the sun and moon
from The Sand Reckoner reads:
\begin{quotation}
``It is true that, of the earlier astronomers, Eudoxus declared
it to be about nine times as great, and Pheidias, my father
twelve times, while Aristarchus tried to prove that the diameter
of the sun is greater than 18 times but less than 20 times
the diameter of the moon.''
\end{quotation}
\noindent
It is importance to mention that the transmitted Greek text does not
actually read ''my father'' but is instead an ambiguous word in
the transmitted texts. This word was interpreted
as being an erroneous transcription of ``my father'' by F.~Blass
[Astr. Nachr. {\bf 104} (1883), No. 2488, p.~255].
Archimedes was killed by a soldier during the conquest of Syracuse by
the Romans during the Second Punic War between Rome and Carthage.
Episodes in the life of Archimedes have become legendary, the
information coming in large part from Plutarch's account in his
description of the conquest of Syracuse by Rome in his Life of
Marcellus, for a dual language edition, see [Plutarch, {\sl Lives,
Vol.~5,} translated by B.~Perrin, Loeb Classical Library {\bf 87},
Harvard University Press, Cambridge, MA, 1917]. An excellent account
of the life of Archimedes as well as an explanation of his major
results can be found in the work of Dijksterhuis [E.J. Dijksterhuis,
{\sl Archimedes,} Princeton University Press, Princeton 1987]. A good
online account of the life and works of Archimedes is the Archimedes
Home Page by Chris Rorres at {\tt
http://www.mcs.drexel.edu/~crorres/Archimedes/contents.html}.
\medskip The contributions of Archimedes to astronomy are less well
known. Those which have survived are contained in his article The Sand
Reckoner, and will be described below. There was a lost work on
optics, On Catoptrica, some of which is transmitted in a commentary of
Theon of Alexandria on Ptolemy's {\sl Almagest,} see below.
Cicero, who was treasurer of Sicily in 75 B.C. wrote that
spheres built by Archimedes were brought to Rome by Marcellus and that
one of these was a planetarium, a mechanical model showing the motions
of the sun, moon and planets. It is believed that Archimedes wrote a
paper on the construction of his planeteria, On Sphere Making, as is
mentioned, for example, by Pappus of Alexandria. Since lost works of
Archimedes were rediscovered as late as 1900, it is not inconceivable
that these works may eventually be found.
\medskip Most of the works of Archimedes have survived, see
Dijksterhuis' book and Rorres' web site for excellent accounts of the
history of the transmission, and are available in translation. The
foremost English translation is by T.L.~Heath [Archimedes, {\sl The
Works of Archimedes,} edited in modern notation with introductory
chapters by T.L. Heath, Dover, New York, 1953]. Heath decided that,
due to the high level of Archimedes' works, he would present the paper
in modern notation so as to most clearly communicate the ideas of
Archimedes. More faithful literal translations of his works are harder
to find in English, though available in French, e.g., the books of
Mugler [Archim\`ede, {\sl Oeuvres, 4 vol.,} texte \'etabli et traduit
par C.~Mugler, Les Belles Lettres, Paris, 1970--71], which are based
on the edited Greek texts of Heiberg [Archimedes, {\sl Opera Omnia,}
with commentary by Eutocius, edited by I.L.~Heiberg and additional
corrections by E.S.~Stamatis, B.G.~Teubner, Stuttgart, 1972].
\medskip
The surviving astronomical work of Archimedes is contained in
his article The Sand Reckoner, and the rest of this article
will be concerned with this work. Apart from its inherent
contributions, The Sand Reckoner might be the best introduction
to classical science.
\medskip\noindent
\begin{boxedtext}
\noindent
{\bf The Sand Reckoner}
\begin{description}
\item[$\bullet$] is addressed to the King of Syracuse, so may
be the first research--expository paper ever written.
\item[$\bullet$] Its goal of addressing innumeracy is still
relevant to a modern audience.
\item[$\bullet$] The paper contains many details about ancient astronomy
and motivates them by presenting them in the context of solving a
specific problem.
\item[$\bullet$] Contains what might be
the first example of a recorded astronomical experiment.
\item[$\bullet$] The first example of psychophysics, the study of
human beings as measuring instruments.
\item[$\bullet$] Faces the problem of naming and manipulating large numbers
without using modern notation.
\end{description}
\end{boxedtext}
\bigskip The paper addresses the problem of innumeracy in
antiquity, in particular, it was not believed that there were numbers
great enough to describe the amount of sand. This belief was so common
that ``sand'' was synonymous with ``uncountable.'' The uncountability
of sand appears 21 times in The Bible, for example, Genesis
32:12: ``And thou saidst, I will surely do thee good, and make thy
seed as the sand of the sea, which cannot be counted for multitude.''
It is a comment on the lack of impact of Archimedes' work that similar
comments appear in the New Testament written centuries later, e.g.,
Hebrews 11:12: ``So many as the stars of the sky in multitude, and as
the sand which is by the seashore innumerable.''
Innumeracy continues to be a problem,
see [J.A. Paulos, {\sl Innumeracy,} Hill \& Wang, New York, 1989].
\medskip In order to rectify this situation, Archimedes sets for
himself the task to name a number larger than the number of sand not
just on a beach, or on all of the surface of the earth, or even the
earth filled with sand, but the idealized question of naming a number
that would be larger than the number of sand that could fill up the
whole universe.
\medskip One reason for this generalization is clear. By taking the
largest amount of sand possible, one can give an upper bound that will
apply to any possible amount of sand, and thus solve the problem
completely. Another possible reason is that Archimedes was very
competitive, he sent his colleague Eratosthenes the intractable
challenge of the Cattle Problem, see [I. Vardi, Archimedes' Cattle
Problem, {\sl Am. Math. Monthly,} American Math. Monthly. {\bf 105}
(1998), 305--319] and in his paper On Spirals, Archimedes says that
he had previously
announced wrong results in order to expose those falsely claiming
that they could independently prove his results.
It is therefore plausible that Archimedes' was desperately
trying to avoid being outnumbered and was
seeking the largest possible amount of sand to estimate.
\medskip In order to solve the problem he has set out for himself,
Archimedes needs to make some physical assumptions, and then apply
mathematical techniques to them. The paper thus has two themes: (a)
physical assumptions based on observational data, (b) mathematical
analysis based on the physical assumptions.
Since Archimedes was a mathematician, and Ancient Greek mathematics
was extremely rigorous (more so than 18th Century mathematics), the
mathematical analysis is very precise, but this is clearly not
possible for the physical part of the paper, a fact which
Archimedes accepts, see below. The physical part of the
paper is written in two different styles. The experiments that
Archimedes is able to perform himself are analyzed with
precision, much more than the other data will allow, while experiments
that he merely reports are overestimated by a factor of 10. This last
strategy is successful in that he actually overestimates the distance
to the sun, even though contemporary estimates of the distance to the
sun were much smaller and estimating this distance is quite difficult.
\medskip\noindent
\begin{boxedtext}
\noindent
{\bf Archimedes' Physical Hypotheses:}
\begin{description}
\item[$\bullet$]
At most 10,000 grains of sand in a spherical poppy seed.
\item[$\bullet$]
At most 40 poppy seed diameters in the diameter of a
finger's breadth.
\item[$\bullet$]
At most 10,000 finger's breadths in a stadium
(ancient measure of distance $\approx$ 200 meters).
\item[$\bullet$]
The earth is a sphere whose perimeter is less than
three million stadia.
\item[$\bullet$]
The distance between the center of the earth and the center of
the sun is less than 10,000 times the radius of the earth.
\item[$\bullet$]
The universe is a sphere with the sun at its center and the
earth rotates around the sun in a circle. The ratio of the
diameter of the universe to the diameter of the earth's orbit
around the sun is less than the ratio of the diameter of the
earth's orbit around the sun to the diameter of the earth.
\end{description}
\end{boxedtext}
\noindent
Given these physical assumptions, it is now simple to compute the
upper bound of $10^{63}$ for the number of grains of sand filling
the universe. However, one must keep in mind that scientific
notation did not exist in Archimedes' time. Indeed, the largest
number expressible in antiquity was a myriad myriads, or
one hundred million, so Archimedes had to invent a system of
expressing large numbers, and also had to discover the law
of exponents $10^a 10^b = 10^{a+b}$. Archimedes not only
succeeded, but managed to name numbers up to $10^{8\cdot 10^{16}}$.
See [I. Vardi, {\sl Archim\`ede face \`a l'innombrable,} Pour La
Science, d\'ecembre 2000, 40--43] for a description and critique.
\bigskip
The rest of this article will concentrate on how Archimedes
arrived at his physical assumptions regarding the size of the
earth, the distance to the sun, and the size of the universe.
\bigskip
{\bf The size of the earth.}
Archimedes uses known estimates on the perimeter of the earth.
The fact that the earth is round was known is antiquity
will come as a surprise only to those cleaving to current popular
culture. In fact, the earth was known to be round since the time of
Pythagoras (ca. 572 B.C.--500 B.C.). By Archimedes' time, Eratosthenes (ca.
276 B.C.--194 B.C.) had given his celebrated estimate of the earth's
perimeter, coming up with an estimate very close to the correct 40,000
kilometers. Archimedes' upper bound of three million stadia is
therefore consistent with his strategy of giving an estimate at least
10 times larger than the currently accepted figure. This upper
bound is about 600,000 kilometers.
\bigskip
{\bf Distance to the sun.}
Archimedes' estimation of the distance between the earth and
sun is much more interesting, indeed this appears to be one of the
earliest attempts to estimate this distance. The problem of estimating the
distance to the sun is quite difficult and accurate estimates were not
obtained until the 18th century. The method used by Archimedes was to
use contemporary estimates for the size of the moon relative to the
earth (relatively easy) and the size of the sun relative to the moon (very
difficult). Since the sun and moon have the same angular diameter
with respect to a terrestrial observer, as seen during solar eclipses,
it follows that the distances of the earth of the sun and moon are
proportional to their size. The distance to the sun is then computed
once the angular size of the sun, as seen on earth, has been
estimated, a measurement which Archimedes carries out himself.
\medskip Archimedes uses the simplest estimate on the size of the
moon, namely that is smaller than the earth. This is obvious from
observation of lunar eclipses.
The true size of the diameter of the moon is now known to be about
.27 earth diameters.
\medskip Archimedes then uses contemporary estimates on the size of
the sun, these being based on comparing the relative sizes of the sun
and moon. As noted above, Archimedes states that his own father had
given an estimate that the diameter of the sun is twelve times the
diameter of the moon. Archimedes uses the estimate of Aristarchus of
Samos that the sun is between 18 and 20 times the size of the moon.
Since Archimedes only requires a safe upper bound, he overestimates
this to 30 moon diameters. Archimedes' final assumption is that the
sun's diameter is no larger than 30 earth diameters. The actual
diameter of the sun is now known to be approximately 109 earth
diameters.
\bigskip {\bf Archimedes' experiment.} To estimate the distance to
the sun, Archimedes next requires an estimate for
the angular size of the sun. This
is done with extreme care by Archimedes himself, and may be the first
recorded astronomical experiment, see [A.E. Shapiro, Archimedes's
measurement of the sun's apparent diameter, {\sl Journal for the
History of Astronomy,} {\bf 6} (1975), 75--83] for further analysis.
Since Archimedes is a mathematician used to extreme rigor, first admits that
the result will not be exact:
\begin{quotation}
Clearly, the exact evaluation of this angle is not easy since
neither vision, hands, nor the instruments required to measure this
angle are reliable enough to measure it precisely.
\end{quotation}
\noindent
However, Archimedes puts in much effort to make this measurement as
accurate as possible, in fact, much more precision than is called for
by the rest of his estimates, as will be seen below.
Archimedes' basic experiment is the following:
\begin{boxedtext}
\noindent
{\bf Archimedes' measurement of the angular size of the sun:}
\begin{description}
\item[{\bf 1.}]
The measurement is done by observation of the sun at sunrise,
using a long horizontal ruler placed on a vertical stand, and
a cylinder is placed on the ruler.
\item[{\bf 2.}]
The ruler is directed towards the sun, and the eye is placed at
the end of the ruler opposite the rising sun.
\item[{\bf 3.}]
The cylinder is placed so that it blocks the sun from the eye and
moved away from the eye until a small piece of the sun can be seen.
The resulting angle between the sides of the cylinder and the eye,
imagined to be a point at the end of the ruler, is a lower bound
on the angular size of the sun.
\item[{\bf 4.}]
The cylinder placed where it just blocks out the sun will produce
an angle that provides an upper bound on the angular size of the sun.
\end{description}
\end{boxedtext}
\noindent
Archimedes notes that there is an inherent inaccuracy in this experiment,
since the eye does not see from a point at the end of the ruler,
but from a small area, the lower bound of Step~3 is not correct.
This part of the experiment is modified as follows:
\medskip\noindent
\begin{boxedtext}
\noindent
{\bf Archimedes' correction for the size of the eye:}
\begin{description}
\item[{\bf 3a.}] Keeping the cylinder in the same position as in
Step~3 above, one places a small cylinder at the end of the ruler
so that it completely blocks the visual field. The angles formed
by the tangents to the two cylinders will now be a more correct
lower bound to the angular size of the sun.
\item[{\bf 3b.}]
In order to make the lower bound as close as possible, the cylinder
blocking the visual field should be as small as possible in the
following way:
One takes two small cylinders of equal size, one white
and one of a different color, the former being placed at some
distance while the latter as close to the eye as possible. The
smaller these cylinders are relative to the eye, the more the
white cylinder will be seen. The optimal size for the cylinders will
be the smallest size where the white cylinder will be completely
occluded by the cylinder in front of the eye.
\end{description}
\end{boxedtext}
\medskip This is of historical interest as it is the first example of
the science of psychophysics, that is, analyzing the human body as
a measuring instrument, a field subsequently developed by Hermann
Helmoltz (1821--1894).
\medskip
Archimedes experiments yields the following results for the
angular size of the sun:
\begin{description}
\item[{\bf Upper bound:}] 1/164 of a right angle
$\approx .54878$ degrees = $33'$.
\item[{\bf Lower bound}] 1/200 of a right angle
$= .45$ degrees = $27'$.
\end{description}
\noindent
The correct figure is $31'\, 59.3''$, on average, varying
between $31'\, 27.7''$ and $32'\, 31.9''$.
Therefore Archimedes' result is very accurate. Since only the
lower bound is required for his main result, Archimedes adopts this as
his working hypothesis.
\medskip Some criticisms of this experiment come to mind. First, the
near equality of the angular size of the sun and moon would suggest
that the same experiment performed on the rising moon would be
simpler, due to the lesser brilliance of the moon.
Second, Archimedes does not address the question of the moon illusion,
that the sun and moon appear to be larger when on the horizon. Note that
showing that this is in fact an illusion requires an
accurate measurement. Interestingly, Archimedes had written about
the effects of refraction, a possible cause for the moon illusion,
as is indicated in a commentary of Theon of
Alexandria on Ptolemy's {\sl Almagest}, the following being a
translation from Volume~4 of Mugler's Collected Works of Archimedes:
\begin{quotation}
To refute the opinion that celestial bodies appear larger when they
are near the horizon because they are seen from a smaller distance,
Ptolemy proposes here to analyze a phenomenon of this kind and to show
that it does not occur because of the distance between earth and sky
but that due to the very humid emanations that surround the earth,
the visual field encounters a body of air that is denser and
that the rays going to the eye through the air are refracted
and thus make the apparent angle at the eye larger as was shown
by Archimedes in his treatise {\sl On catoptrica,} where he
says that objects submerged in water also seem larger, and the
more so the deeper\ldots
\end{quotation}
\bigskip {\bf Solar parallax.} In order to make an even more accurate
estimate of the distance from the earth to the sun, Archimedes also
takes into account solar parallax, in other words, the fact that his
estimate of the distance to the sun is taken from a measurement on the
surface of the earth, while the actual distance that he is interested
in is from the center of the earth.
The correction for solar parallax can be found by computing the
angular size of the sun as would be seen from the center of the earth.
The object of this computation is to calculate the angular size of the
sun from the center of the earth, given that the diameters of the
sun and earth are known, as well as the angular size of the sun from
the earth's surface at sunrise.
This calculation is quite simple for Archimedes using elementary
geometric arguments, as opposed to trigonometric techniques more usual
for us today. The lower bound for the angular
size of the sun, the estimate required for the final result, obtained
by Archimedes is 1/203 of a right angle,
as opposed to 1/200 of a right angle by his direct observation.
Archimedes' correction for solar parallax is therefore about 1.5\% .
Apparently, this is the first known example of solar parallax being
taken into account. See below for a further comment on this point.
\medskip Archimedes then uses the results of his experiment and the
correction for solar parallax to conclude that the estimate 1/1000 of
360$^\circ$, or $.36$ degrees will be a safe underestimate for the
angular size of the sun. Therefore, a regular polygon of 1,000 sides each of
which has the diameter of the sun will be larger than the disk containing
the earth and sun's orbit around the earth. Given the previous
assumption that the diameter of the sun is no larger than 30 times the
diameter of the earth, this means that the orbit of the sun is
less than 30,000 earth diameters. Using the simple estimate that
$\pi > 3$ (Archimedes in fact proved that rigorously proved that
$3\, \frac{10}{71} < \pi < 3\, \frac{1}{7}$ in his
paper Measurement of the Circle), this leads to the final estimate
that the distance from the center of the
earth to the center of the sun is less than 10,000 times the
radius of the earth.
This distance is now known to be approximately $23,455$ times the
radius of the earth. Note that in this calculation,
Archimedes does not use his overestimate for the
size of the earth. Plugging in this estimate gives an upper bound on the
distance from the earth to the sun of 1,000,000,000 kilometers, that is,
$10^9$ kilometers.
The actual distance is approximately 150,000,000 kilometers.
\bigskip {\bf The universe of Archimedes.} In his initial definition
of the universe, Archimedes takes this to be the sphere with center at
the earth, and radius the distance from the center of the earth to the
center of the sun. This appears to be inconsistent with the standard
astronomy of Archimedes' time, where the universe is taken to be the
sphere of fixed stars with the earth as center, and the moon, sun, and
planets turning around smaller spheres contained in the sphere of
fixed stars. However, Archimedes makes no further mention of this.
\medskip {\bf The heliocentric theory.} Perhaps the most surprising
assumption is the adoption of the heliocentric theory of Aristarchus.
Indeed, the original work of Aristarchus on his heliocentric theory
has been lost and the Sand Reckoner is only one of a few ancient
sources citing his work. A comprehensive account of Aristarchus' work
can be found in [T.L. Heath, {\sl Aristarchus of Samos, the Ancient
Copernicus,} Dover, New York 1981]. This work is also one of the
best references for Ancient Greek astronomy.
One can wonder why Archimedes chose this model of the universe, since
there was no evidence for this at the time. The explanation
may lie in the belief that Archimedes was very keen on not being
outnumbered in his attempt to estimate the number of grains of sand, so
not only chose the largest venue to be entirely filled with sand, the
universe, but the largest model of the universe that existed at the
time, the heliocentric model.
Indeed, the heliocentric model requires a larger universe so that the
motion of the earth around the sun will not cause stellar
parallax, undetectable at the time (in fact, not detected until the 19th
Century). In order to eliminate any possibility of stellar parallax,
Aristarchus assumed that the distance to the stars was such that
the sun was a point with respect to the sphere of fixed stars,
so essentially infinitely large.
Archimedes strongly criticizes this assumption, on logical grounds that
the ratio of a surface to a point makes no sense. Moreover,
it is clear that Archimedes requires the universe to be finite,
since an infinite {\sl would\/} contain an infinite amount
of sand. Archimedes therefore gives an explicit estimate for the size of
Aristarchus' universe:
\begin{quotation}
``We must take Aristarchus to mean this: since we conceive
the earth to be, as it were, the centre of the universe,
the ratio which the earth bears to what we describe as the
`universe' is the same as the ratio which the sphere containing
the circle in which he supposes the earth to revolve bears
to the sphere of fixed stars.''
\end{quotation}
In other words, the ratio of the diameter of the universe (sphere
of fixed stars) to the diameter of the orbit of the earth around
the sun equals the ratio of the diameter of the orbit of the earth
around the sun to the diameter of the earth.
Symbolically, this can be written as
$$
\frac{r_u}{r_{es}} = \frac{r_{es}}{r_e}\,,
$$
where $r_u$ is the radius of the universe, i.e., the distance to
the stars, $r_{es}$ is the distance from the earth to the sun, and
$r_e$ is the radius of the earth.
A simple explanation for
Archimedes' amendment to Aristarchus' theory is that it simply states
\begin{center}
Stellar Parallax = Solar Parallax.
\end{center}
\medskip\noindent
Indeed, since Archimedes' initial assumption is that the Sun moves around
the earth in a perfect circle around the centre of the earth,
it is reasonable to infer that one should not expect to see a variation
in the sun's position depending on where the sun is viewed from
different places on the earth (solar parallax). Note that Archimedes'
previous computation correcting for solar parallax makes it clear that
he was well aware of this phenomenon.
It then follows that if the ratio of the radius of the universe to the
ratio of the earth's orbit around the sun is the same as the ratio of
the earth's orbit around the sun to the radius of the earth, then
no noticeable solar parallax will imply no noticeable stellar parallax.
Since Archimedes' measurements gave an estimate of 5,000 earth
diameters for the distance from the earth to the sun, the ratio
$\frac{r_{es}}{r_e}$ is 10,000. Given the previous estimate of
$10^9$ kilometers for $r_{es}$, the distance from the earth to the
sun, the upper bound for $r_u$, the radius of Archimedes'
universe, is therefore $10^{13}$ kilometers. This is slightly more
than one light year, which is about $9.5 \times 10^{12}$ kilometers.
\end{document}