ARISTARCHUS OF SAMOS, THE COPERNICUS OF ANTIQUITY
It appears that Aristarchus was a contemporary of
Archimedes, but the exact dates of his life are not
known. He was actively engaged in making astronomical
observations in Samos somewhat before the
middle of the third century B.C.; in other words, just
at the time when the activities of the Alexandrian
school were at their height. Hipparchus, at a later
day, was enabled to compare his own observations
with those made by Aristarchus, and, as we have just
seen, his work was well known to so distant a contemporary
as Archimedes. Yet the facts of his life are
almost a blank for us, and of his writings only a single
one has been preserved. That one, however, is a most
important and interesting paper on the measurements
of the sun and the moon. Unfortunately, this paper
gives us no direct clew as to the opinions of Aristarchus
concerning the relative positions of the earth and sun.
But the testimony of Archimedes as to this is unequivocal,
and this testimony is supported by other rumors
in themselves less authoritative.
In contemplating this astronomer of Samos, then,
we are in the presence of a man who had solved in its
essentials the problem of the mechanism of the solar
system. It appears from the words of Archimedes that
Aristarchus; had propounded his theory in explicit
writings. Unquestionably, then, he held to it as a positive
doctrine, not as a mere vague guess. We shall
show, in a moment, on what grounds he based his
opinion. Had his teaching found vogue, the story of
science would be very different from what it is. We
should then have no tale to tell of a Copernicus coming
upon the scene fully seventeen hundred years later
with the revolutionary doctrine that our world is not
the centre of the universe. We should not have to
tell of the persecution of a Bruno or of a Galileo for
teaching this doctrine in the seventeenth century of an
era which did not begin till two hundred years after
the death of Aristarchus. But, as we know, the teaching
of the astronomer of Samos did not win its way.
The old conservative geocentric doctrine, seemingly
so much more in accordance with the every-day observations
of mankind, supported by the majority of
astronomers with the Peripatetic philosophers at their
head, held its place. It found fresh supporters presently
among the later Alexandrians, and so fully
eclipsed the heliocentric view that we should scarcely
know that view had even found an advocate were it
not for here and there such a chance record as the
phrases we have just quoted from Archimedes. Yet,
as we now see, the heliocentric doctrine, which we know
to be true, had been thought out and advocated as the
correct theory of celestial mechanics by at least one
worker of the third century B.C. Such an idea, we
may be sure, did not spring into the mind of its
originator except as the culmination of a long series of
observations and inferences. The precise character
of the evolution we perhaps cannot trace, but its
broader outlines are open to our observation, and we
may not leave so important a topic without at least
briefly noting them.
Fully to understand the theory of Aristarchus, we
must go back a century or two and recall that as long
ago as the time of that other great native of Samos,
Pythagoras, the conception had been reached that the
earth is in motion. We saw, in dealing with Pythagoras,
that we could not be sure as to precisely what he
himself taught, but there is no question that the idea
of the world's motion became from an early day a so-called Pythagorean doctrine. While all the other
philosophers, so far as we know, still believed that the
world was flat, the Pythagoreans out in Italy taught
that the world is a sphere and that the apparent motions
of the heavenly bodies are really due to the actual
motion of the earth itself. They did not, however,
vault to the conclusion that this true motion of the
earth takes place in the form of a circuit about the
sun. Instead of that, they conceived the central body
of the universe to be a great fire, invisible from the
earth, because the inhabited side of the terrestrial
ball was turned away from it. The sun, it was held,
is but a great mirror, which reflects the light from the
central fire. Sun and earth alike revolve about this
great fire, each in its own orbit. Between the earth
and the central fire there was, curiously enough, supposed
to be an invisible earthlike body which was given
the name of Anticthon, or counter-earth. This body,
itself revolving about the central fire, was supposed to
shut off the central light now and again from the sun
or from the moon, and thus to account for certain
eclipses for which the shadow of the earth did not
seem responsible. It was, perhaps, largely to account
for such eclipses that the counter-earth was invented.
But it is supposed that there was another reason. The
Pythagoreans held that there is a peculiar sacredness
in the number ten. Just as the Babylonians of
the early day and the Hegelian philosophers of a
more recent epoch saw a sacred connection between
the number seven and the number of planetary
bodies, so the Pythagoreans thought that the universe
must be arranged in accordance with the number
ten. Their count of the heavenly bodies, including
the sphere of the fixed stars, seemed to show
nine, and the counter-earth supplied the missing
body.
The precise genesis and development of this idea
cannot now be followed, but that it was prevalent about
the fifth century B.C. as a Pythagorean doctrine cannot
be questioned. Anaxagoras also is said to have
taken account of the hypothetical counter-earth in his
explanation of eclipses; though, as we have seen, he
probably did not accept that part of the doctrine
which held the earth to be a sphere. The names of
Philolaus and Heraclides have been linked with certain
of these Pythagorean doctrines. Eudoxus, too,
who, like the others, lived in Asia Minor in the fourth
century B.C., was held to have made special studies of
the heavenly spheres and perhaps to have taught that
the earth moves. So, too, Nicetas must be named
among those whom rumor credited with having taught
that the world is in motion. In a word, the evidence,
so far as we can garner it from the remaining fragments,
tends to show that all along, from the time of
the early Pythagoreans, there had been an undercurrent
of opinion in the philosophical world which
questioned the fixity of the earth; and it would seem
that the school of thinkers who tended to accept the
revolutionary view centred in Asia Minor, not far from
the early home of the founder of the Pythagorean doctrines.
It was not strange, then, that the man who
was finally to carry these new opinions to their logical
conclusion should hail from Samos.
But what was the support which observation could
give to this new, strange conception that the heavenly
bodies do not in reality move as they seem to move,
but that their apparent motion is due to the actual
revolution of the earth? It is extremely difficult for
any one nowadays to put himself in a mental position
to answer this question. We are so accustomed to
conceive the solar system as we know it to be, that we
are wont to forget how very different it is from what it
seems. Yet one needs but to glance up at the sky, and
then to glance about one at the solid earth, to grant,
on a moment's reflection, that the geocentric idea is of
all others the most natural; and that to conceive the
sun as the actual Centre of the solar system is an idea
which must look for support to some other evidence
than that which ordinary observation can give. Such
was the view of most of the ancient philosophers, and
such continued to be the opinion of the majority of
mankind long after the time of Copernicus. We must
not forget that even so great an observing astronomer
as Tycho Brahe, so late as the seventeenth century,
declined to accept the heliocentric theory, though admitting
that all the planets except the earth revolve
about the sun. We shall see that before the Alexandrian
school lost its influence a geocentric scheme had
been evolved which fully explained all the apparent
motions of the heavenly bodies. All this, then, makes
us but wonder the more that the genius of an Aristarchus
could give precedence to scientific induction as
against the seemingly clear evidence of the senses.
What, then, was the line of scientific induction that
led Aristarchus to this wonderful goal? Fortunately,
we are able to answer that query, at least in
part. Aristarchus gained his evidence through some
wonderful measurements. First, he measured the
disks of the sun and the moon. This, of course, could
in itself give him no clew to the distance of these bodies,
and therefore no clew as to their relative size; but
in attempting to obtain such a clew he hit upon a
wonderful yet altogether simple experiment. It occurred
to him that when the moon is precisely dichotomized—
that is to say, precisely at the half-the line of vision
from the earth to the moon must be precisely at right
angles with the line of light passing from the sun to
the moon. At this moment, then, the imaginary lines
joining the sun, the moon, and the earth, make a right
angle triangle. But the properties of the right-angle
triangle had long been studied and were well under
stood. One acute angle of such a triangle determines
the figure of the triangle itself. We have already seen
that Thales, the very earliest of the Greek philosophers,
measured the distance of a ship at sea by the application
of this principle. Now Aristarchus sights the
sun in place of Thales' ship, and, sighting the moon at
the same time, measures the angle and establishes the
shape of his right-angle triangle. This does not tell
him the distance of the sun, to be sure, for he does not
know the length of his base-line—that is to say, of the
line between the moon and the earth. But it does
establish the relation of that base-line to the other lines
of the triangle; in other words, it tells him the distance
of the sun in terms of the moon's distance. As Aristarchus
strikes the angle, it shows that the sun is
eighteen times as distant as the moon. Now, by comparing
the apparent size of the sun with the apparent
size of the moon—which, as we have seen, Aristarchus
has already measured—he is able to tell us that, the
sun is "more than 5832 times, and less than 8000''
times larger than the moon; though his measurements,
taken by themselves, give no clew to the
actual bulk of either body. These conclusions, be
it understood, are absolutely valid inferences—nay,
demonstrations—from the measurements involved,
provided only that these measurements have been
correct. Unfortunately, the angle of the triangle
we have just seen measured is exceedingly difficult to
determine with accuracy, while at the same time, as a
moment's reflection will show, it is so large an angle
that a very slight deviation from the truth will greatly
affect the distance at which its line joins the other side
of the triangle. Then again, it is virtually impossible
to tell the precise moment when the moon is at half, as
the line it gives is not so sharp that we can fix it with
absolute accuracy. There is, moreover, another element
of error due to the refraction of light by the
earth's atmosphere. The experiment was probably
made when the sun was near the horizon, at which time,
as we now know, but as Aristarchus probably did not
suspect, the apparent displacement of the sun's position
is considerable; and this displacement, it will be
observed, is in the direction to lessen the angle in
question.
In point of fact, Aristarchus estimated the angle at
eighty-seven degrees. Had his instrument been more
precise, and had he been able to take account of all the
elements of error, he would have found it eighty-seven
degrees and fifty-two minutes. The difference of
measurement seems slight; but it sufficed to make the
computations differ absurdly from the truth. The
sun is really not merely eighteen times but more than
two hundred times the distance of the moon, as Wendelein
discovered on repeating the experiment of Aristarchus
about two thousand years later. Yet this discrepancy
does not in the least take away from the
validity of the method which Aristarchus employed.
Moreover, his conclusion, stated in general terms,
was perfectly correct: the sun is many times more
distant than the moon and vastly larger than that
body. Granted, then, that the moon is, as Aristarchus
correctly believed, considerably less in size than the
earth, the sun must be enormously larger than the
earth; and this is the vital inference which, more than
any other, must have seemed to Aristarchus to confirm
the suspicion that the sun and not the earth is the
centre of the planetary system. It seemed to him
inherently improbable that an enormously large body
like the sun should revolve about a small one such as
the earth. And again, it seemed inconceivable that a
body so distant as the sun should whirl through space
so rapidly as to make the circuit of its orbit in twenty-four hours. But, on the other hand, that a small body
like the earth should revolve about the gigantic sun
seemed inherently probable. This proposition granted,
the rotation of the earth on its axis follows as a necessary
consequence in explanation of the seeming motion
of the stars. Here, then, was the heliocentric
doctrine reduced to a virtual demonstration by Aristarchus
of Samos, somewhere about the middle of the
third century B.C.
It must be understood that in following out the,
steps of reasoning by which we suppose Aristarchus
to have reached so remarkable a conclusion, we have
to some extent guessed at the processes of thought-development; for no line of explication written by the
astronomer himself on this particular point has come
down to us. There does exist, however, as we have
already stated, a very remarkable treatise by Aristarchus
on the
Size and Distance of the Sun and the
Moon, which so clearly suggests the methods of
reasoning of the great astronomer, and so explicitly
cites the results of his measurements, that we cannot
well pass it by without quoting from it at some length.
It is certainly one of the most remarkable scientific
documents of antiquity. As already noted, the
heliocentric doctrine is not expressly stated here.
It seems to be tacitly implied throughout, but it is
not a necessary consequence of any of the propositions
expressly stated. These propositions have to do with
certain observations and measurements and what
Aristarchus believes to be inevitable deductions from
them, and he perhaps did not wish to have these
deductions challenged through associating them with
a theory which his contemporaries did not accept.
In a word, the paper of Aristarchus is a rigidly scientific
document unvitiated by association with any
theorizings that are not directly germane to its central
theme. The treatise opens with certain hypotheses
as follows:
- "First. The moon receives its light from the sun.
- "Second. The earth may be considered as a point
and as the centre of the orbit of the moon.
- "Third. When the moon appears to us dichotomized
it offers to our view a great circle [or actual meridian]
of its circumference which divides the illuminated part
from the dark part.
- "Fourth. When the moon appears dichotomized its
distance from the sun is less than a quarter of the
circumference [of its orbit] by a thirtieth part of that
quarter.''
That is to say, in modern terminology, the moon
at this time lacks three degrees (one thirtieth of ninety
degrees) of being at right angles with the line of the
sun as viewed from the earth; or, stated otherwise,
the angular distance of the moon from the sun as
viewed from the earth is at this time eighty-seven
degrees—this being, as we have already observed, the
fundamental measurement upon which so much depends.
We may fairly suppose that some previous
paper of Aristarchus's has detailed the measurement
which here is taken for granted, yet which of course
could depend solely on observation.
"Fifth. The diameter of the shadow [cast by the
earth at the point where the moon's orbit cuts that
shadow when the moon is eclipsed] is double the
diameter of the moon.''
Here again a knowledge of previously established
measurements is taken for granted; but, indeed, this is
the case throughout the treatise.
"Sixth. The arc subtended in the sky by the moon
is a fifteenth part of a sign'' of the zodiac; that is
to say, since there are twenty-four, signs in the zodiac,
one-fifteenth of one twenty-fourth, or in modern terminology,
one degree of arc. This is Aristarchus's measurement
of the moon to which we have already referred
when speaking of the measurements of Archimedes.
"If we admit these six hypotheses,'' Aristarchus
continues, "it follows that the sun is more than
eighteen times more distant from the earth than is the
moon, and that it is less than twenty times more
distant, and that the diameter of the sun bears a
corresponding relation to the diameter of the moon;
which is proved by the position of the moon when
dichotomized. But the ratio of the diameter of the
sun to that of the earth is greater than nineteen to
three and less than forty-three to six. This is demonstrated
by the relation of the distances, by the position
[of the moon] in relation to the earth's shadow, and by
the fact that the arc subtended by the moon is a
fifteenth part of a sign.''
Aristarchus follows with nineteen propositions intended
to elucidate his hypotheses and to demonstrate
his various contentions. These show a singularly
clear grasp of geometrical problems and an altogether
correct conception of the general relations as to size
and position of the earth, the moon, and the sun.
His reasoning has to do largely with the shadow cast
by the earth and by the moon, and it presupposes a
considerable knowledge of the phenomena of eclipses.
His first proposition is that "two equal spheres may
always be circumscribed in a cylinder; two unequal
spheres in a cone of which the apex is found on the
side of the smaller sphere; and a straight line joining
the centres of these spheres is perpendicular to each
of the two circles made by the contact of the surface
of the cylinder or of the cone with the spheres.''
It will be observed that Aristarchus has in mind here
the moon, the earth, and the sun as spheres to be
circumscribed within a cone, which cone is made
tangible and measurable by the shadows cast by
the non-luminous bodies; since, continuing, he clearly
states in proposition nine, that "when the sun is
totally eclipsed, an observer on the earth's surface is
at an apex of a cone comprising the moon and the
sun.'' Various propositions deal with other relations
of the shadows which need not detain us since they
are not fundamentally important, and we may pass
to the final conclusions of Aristarchus, as reached in
his propositions ten to nineteen.
Now, since (proposition ten) "the diameter of the
sun is more than eighteen times and less than twenty
times greater than that of the moon,'' it follows
(proposition eleven) "that the bulk of the sun is to
that of the moon in ratio, greater than 5832 to 1, and
less than 8000 to 1.''
"Proposition sixteen. The diameter of the sun is
to the diameter of the earth in greater proportion than
nineteen to three, and less than forty-three to six.
"Proposition seventeen. The bulk of the sun is to
that of the earth in greater proportion than 6859 to
27, and less than 79,507 to 216.
"Proposition eighteen. The diameter of the earth
is to the diameter of the moon in greater proportion
than 108 to 43 and less than 60 to 19.
"Proposition nineteen. The bulk of the earth is to
that of the moon in greater proportion than 1,259,712
to 79,507 and less than 20,000 to 6859.''
Such then are the more important conclusions of
this very remarkable paper—a paper which seems to
have interest to the successors of Aristarchus generation
after generation, since this alone of all the writings
of the great astronomer has been preserved.
How widely the exact results of the measurements of
Aristarchus, differ from the truth, we have pointed
out as we progressed. But let it be repeated that this
detracts little from the credit of the astronomer who
had such clear and correct conceptions of the relations
of the heavenly bodies and who invented such correct
methods of measurement. Let it be particularly
observed, however, that all the conclusions of Aristarchus
are stated in relative terms. He nowhere
attempts to estimate the precise size of the earth, of
the moon, or of the sun, or the actual distance of one
of these bodies from another. The obvious reason
for this is that no data were at hand from which to
make such precise measurements. Had Aristarchus
known the size of any one of the bodies in question, he
might readily, of course, have determined the size of
the others by the mere application of his relative
scale; but he had no means of determining the size
of the earth, and to this extent his system of measurements
remained imperfect. Where Aristarchus halted,
however, another worker of the same period took the
task in hand and by an altogether wonderful measurement
determined the size of the earth, and thus brought
the scientific theories of cosmology to their climax.
This worthy supplementor of the work of Aristarchus
was Eratosthenes of Alexandria.