ARISTARCHUS OF SAMOS.
We are told that Aristarchus of Samos was a pupil of Strato of
Lampsacus, a natural philosopher of originality, who succeeded
Theophrastus as head of the Peripatetic school in 288 or 287 B.C., and
held that position for eighteen years. Two other facts enable us to fix
Aristarchus’s date approximately. In 281–280 he made an observation
of the summer solstice; and the book in which he formulated his
heliocentric hypothesis was published before the date of Archimedes’s
_Psammites_ or _Sandreckoner_, a work written before 216 B.C.
Aristarchus therefore probably lived _circa_ 310–230 B.C., that is, he
came about seventy-five years later than Heraclides and was older than
Archimedes by about twenty-five years.
Aristarchus was called “the mathematician,” no doubt in order to
distinguish him from the many other persons of the same name; Vitruvius
includes him among the few great men who possessed an equally profound
knowledge of all branches of science, geometry, astronomy, music, etc.
“Men of this type are rare, men such as were in times past Aristarchus
of Samos, Philolaus and Archytas of Tarentum, Apollonius of Perga,
Eratosthenes of Cyrene, Archimedes and Scopinas of Syracuse, who
left to posterity many mechanical and gnomonic appliances which they
invented and explained on mathematical and natural principles.” That
Aristarchus was a very capable geometer is proved by his extant book,
_On the sizes and distances of the sun and moon_, presently to be
described. In the mechanical line he is credited with the invention of
an improved sun-dial, the so-called _scaphe_, which had not a plane but
a concave hemispherical surface, with a pointer erected vertically in
the middle, throwing shadows and so enabling the direction and height
of the sun to be read off by means of lines marked on the surface of
the hemisphere. He also wrote on vision, light, and colours. His views
on the latter subjects were no doubt largely influenced by the teaching
of Strato. Strato held that colours were emanations from bodies,
material molecules as it were, which imparted to the intervening air
the same colour as that possessed by the body. Aristarchus said that
colours are “shapes or forms stamping the air with impressions like
themselves as it were,” that “colours in darkness have no colouring,”
and that “light is the colour impinging on a substratum”.
THE HELIOCENTRIC HYPOTHESIS.
There is no doubt whatever that Aristarchus put forward the
heliocentric hypothesis. Ancient testimony is unanimous on the point,
and the first witness is Archimedes who was a younger contemporary of
Aristarchus, so that there is no possibility of a mistake. Copernicus
himself admitted that the theory was attributed to Aristarchus, though
this does not seem to be generally known. Copernicus refers in two
passages of his work, _De revolutionibus caelestibus_, to the opinions
of the ancients about the motion of the earth. In the dedicatory letter
to Pope Paul III he mentions that he first learnt from Cicero that one
Nicetas (i.e. Hicetas) had attributed motion to the earth, and that he
afterwards read in Plutarch that certain others held that opinion; he
then quotes the _Placita philosophorum_ according to which “Philolaus
the Pythagorean asserted that the earth moved round the fire in an
oblique circle in the same way as the sun and moon”. In Book I. c. 5 of
his work Copernicus alludes to the views of Heraclides, Ecphantus, and
Hicetas, who made the earth rotate about its own axis, and then goes on
to say that it would not be very surprising if any one should attribute
to the earth another motion besides rotation, namely, revolution in
an orbit in space: “atque etiam (terram) pluribus motibus vagantem et
unam ex astris Philolaus Pythagoricus sensisse fertur, Mathematicus non
vulgaris”. Here, however, there is no question of the earth revolving
round the sun, and there is no mention of Aristarchus. But Copernicus
did mention the theory of Aristarchus in a passage which he afterwards
suppressed: “Credibile est hisce similibusque causis Philolaum
mobilitatem terrae sensisse, quod etiam nonnulli Aristarchum Samium
ferunt in eadem fuisse sententia”.
It is desirable to quote the whole passage of Archimedes in which the
allusion to Aristarchus’s heliocentric hypothesis occurs, in order to
show the whole context.
“You are aware [‘you’ being King Gelon] that ‘universe’ is the name
given by most astronomers to the sphere the centre of which is the
centre of the earth, while its radius is equal to the straight line
between the centre of the sun and the centre of the earth. This is the
common account as you have heard from astronomers. But Aristarchus
brought out _a book consisting of certain hypotheses_, wherein it
appears, as a consequence of the assumptions made, that the universe is
many times greater than the ‘universe’ just mentioned. His hypotheses
are that _the fixed stars and the sun remain unmoved, that the earth
revolves about the sun in the circumference of a circle, the sun lying
in the middle of the orbit_, and that the sphere of the fixed stars,
situated about the same centre as the sun, is so great that the circle
in which he supposes the earth to revolve bears such a proportion to
the distance of the fixed stars as the centre of the sphere bears to
its surface.”
The heliocentric hypothesis is here stated in language which leaves
no room for doubt about its meaning. The sun, like the fixed stars,
remains unmoved and forms the centre of a circular orbit in which the
earth moves round it; the sphere of the fixed stars has its centre at
the centre of the sun.
We have further evidence in a passage of Plutarch’s tract, _On the
face in the moon’s orb_: “Only do not, my dear fellow, enter an action
for impiety against me in the style of Cleanthes, who thought it was
the duty of Greeks to indict Aristarchus on the charge of impiety for
putting in motion the Hearth of the Universe, this being the effect of
his attempt to save the phenomena by supposing the heaven to remain at
rest and the earth to revolve in an oblique circle, while it rotates,
at the same time, about its own axis”.
Here we have the additional detail that Aristarchus followed
Heraclides in attributing to the earth the daily rotation about its
axis; Archimedes does not state this in so many words, but it is
clearly involved by his remark that Aristarchus supposed the fixed
stars as well as the sun to remain unmoved in space. A tract “Against
Aristarchus” is mentioned by Diogenes Laertius among Cleanthes’s works;
and it was evidently published during Aristarchus’s lifetime (Cleanthes
died about 232 B.C.).
We learn from another passage of Plutarch that the hypothesis of
Aristarchus was adopted, about a century later, by Seleucus, of
Seleucia on the Tigris, a Chaldæan or Babylonian, who also wrote on
the subject of the tides about 150 B.C. The passage is interesting
because it also alludes to the doubt about Plato’s final views. “Did
Plato put the earth in motion as he did the sun, the moon and the five
planets which he called the ‘instruments of time’ on account of their
turnings, and was it necessary to conceive that the earth ‘which is
globed about the axis stretched from pole to pole through the whole
universe’ was not represented as being (merely) held together and
at rest but as turning and revolving, as Aristarchus and Seleucus
afterwards maintained that it did, the former of whom stated this as
only a hypothesis, the latter as a definite opinion?”
No one after Seleucus is mentioned by name as having accepted the
doctrine of Aristarchus and, if other Greek astronomers refer to it,
they do so only to denounce it. Hipparchus, himself a contemporary of
Seleucus, definitely reverted to the geocentric system, and it was
doubtless his authority which sealed the fate of the heliocentric
hypothesis for so many centuries.
The reasons which weighed with Hipparchus were presumably the facts
that the system in which the earth revolved in a circle of which
the sun was the exact _centre_ failed to “save the phenomena,” and
in particular to account for the variations of distance and the
irregularities of the motions, which became more and more patent as
methods of observation improved; that, on the other hand, the theory
of epicycles did suffice to represent the phenomena with considerable
accuracy; and that the latter theory could be reconciled with the
immobility of the earth.
ON THE APPARENT DIAMETER OF THE SUN.
Archimedes tells us in the same treatise that “Aristarchus discovered
that the sun’s apparent size is about 1/720th part of the zodiac
circle”; that is to say, he observed that the angle subtended at the
earth by the diameter of the sun is about half a degree.
ON THE SIZES AND DISTANCES OF THE SUN AND MOON.
Archimedes also says that, whereas the ratio of the diameter of the sun
to that of the moon had been estimated by Eudoxus at 9 : 1 and by his
own father Phidias at 12 : 1, Aristarchus made the ratio greater than
18 : 1 but less than 20 : 1. Fortunately we possess in Greek the short
treatise in which Aristarchus proved these conclusions; on the other
matter of the apparent diameter of the sun Archimedes’s statement is
our only evidence.
It is noteworthy that in Aristarchus’s extant treatise _On the sizes
and distances of the sun and moon_ there is no hint of the heliocentric
hypothesis, while the apparent diameter of the sun is there assumed to
be, not ½°, but the very inaccurate figure of 2°. Both circumstances
are explained if we assume that the treatise was an early work written
before the hypotheses described by Archimedes were put forward. In the
treatise Aristarchus finds the ratio of the diameter of the sun to the
diameter of the earth to lie between 19 : 3 and 43 : 6; this would
make the volume of the sun about 300 times that of the earth, and it
may be that the great size of the sun in comparison with the earth, as
thus brought out, was one of the considerations which led Aristarchus
to place the sun rather than the earth in the centre of the universe,
since it might even at that day seem absurd to make the body which was
so much larger revolve about the smaller.
There is no reason to doubt that in his heliocentric system Aristarchus
retained the moon as a satellite of the earth revolving round it as
centre; hence even in his system there was one _epicycle_.
The treatise _On sizes and distances_ being the only work of
Aristarchus which has survived, it will be fitting to give here a
description of its contents and special features.
The style of Aristarchus is thoroughly classical as befits an able
geometer intermediate in date between Euclid and Archimedes, and his
demonstrations are worked out with the same rigour as those of his
predecessor and successor. The propositions of Euclid’s _Elements_ are,
of course, taken for granted, but other things are tacitly assumed
which go beyond what we find in Euclid. Thus the transformations of
ratios defined in Euclid, Book V, and denoted by the terms _inversely_,
_alternately_, _componendo_, _convertendo_, etc., are regularly
used in dealing with _unequal_ ratios, whereas in Euclid they are
only used in proportions, i.e. cases of _equality_ of ratios. But
the propositions of Aristarchus are also of particular mathematical
interest because the ratios of the sizes and distances which have
to be calculated are really trigonometrical ratios, sines, cosines,
etc., although at the time of Aristarchus trigonometry had not been
invented, and no reasonably close approximation to the value of π,
the ratio of the circumference of any circle to its diameter, had
been made (it was Archimedes who first obtained the approximation
22/7). Exact calculation of the trigonometrical ratios being therefore
impossible for Aristarchus, he set himself to find upper and lower
limits for them, and he succeeded in locating those which emerge in
his propositions within tolerably narrow limits, though not always
the narrowest within which it would have been possible, even for him,
to confine them. In this species of approximation to trigonometry he
tacitly assumes propositions comparing the ratio between a greater
and a lesser _angle_ in a figure with the ratio between two _straight
lines_, propositions which are formally proved by Ptolemy at the
beginning of his _Syntaxis_. Here again we have proof that textbooks
containing such propositions existed before Aristarchus’s time, and
probably much earlier, although they have not survived.
Aristarchus necessarily begins by laying down, as the basis for his
treatise, certain assumptions. They are six in number, and he refers
to them as _hypotheses_. We cannot do better than quote them in full,
along with the sentences immediately following, in which he states the
main results to be established in the treatise:--
[HYPOTHESES.]
1. _That the moon receives its light from the sun._
2. _That the earth is in the relation of a point and centre to the
sphere in which the moon moves._
3. _That, when the moon appears to us halved, the great circle which
divides the dark and the bright portions of the moon is in the
direction of our eye._
4. _That, when the moon appears to us halved, its distance from the sun
is then less than a quadrant by one-thirtieth of a quadrant._
5. _That the breadth of the (earth’s) shadow is (that) of two moons._
6. _That the moon subtends one-fifteenth part of a sign of the zodiac._
We are now in a position to prove the following propositions:--
1. _The distance of the sun from the earth is greater than eighteen
times, but less than twenty times, the distance of the moon (from the
earth)_; this follows from the hypothesis about the halved moon.
2. _The diameter of the sun has the same ratio (as aforesaid) to the
diameter of the moon._
3. _The diameter of the sun has to the diameter of the earth a ratio
greater than that which 19 has to 3, but less than that which 43 has to
6_; this follows from the ratio thus discovered between the distances,
the hypothesis about the shadow, and the hypothesis that the moon
subtends one-fifteenth part of a sign of the zodiac.
The first assumption is Anaxagoras’s discovery. The second assumption
is no doubt an exaggeration; but it is made in order to avoid having
to allow for the fact that the phenomena as seen by an observer on the
surface of the earth are slightly different from what would be seen
if the observer’s eye were at the centre of the earth. Aristarchus,
that is, takes the earth to be like a point in order to avoid the
complication of parallax.
The meaning of the third hypothesis is that the plane of the great
circle in question passes through the point where the eye of the
observer is situated; that is to say, we see the circle _end on_, as it
were, and it looks like a straight line.
Hypothesis 4. If S be the sun, M the moon and E the earth, the triangle
SME is, at the moment when the moon appears to us halved, right-angled
at M; and the hypothesis states that the angle at E in this triangle is
87°, or, in other words, the angle MSE, that is, the angle subtended
at the sun by the line joining M to E, is 3°. These estimates are
decidedly inaccurate, for the true value of the angle MES is 89° 50′,
and that of the angle MSE is therefore 10′. There is nothing to show
how Aristarchus came to estimate the angle MSE at 3°, and none of his
successors seem to have made any direct estimate of the size of the
angle.
The assumption in Hypothesis 5 was improved upon later. Hipparchus made
the ratio of the diameter of the circle of the earth’s shadow to the
diameter of the moon to be, not 2, but 2½ at the moon’s mean distance
at the conjunctions; Ptolemy made it, at the moon’s greatest distance,
to be inappreciably less than 2⅗.
The sixth hypothesis states that the diameter of the moon subtends at
our eye an angle which is 1/15th of 30°, i.e. 2°, whereas Archimedes,
as we have seen, tells us that Aristarchus found the angle subtended
by the diameter of the sun to be ½° (Archimedes in the same tract
describes a rough instrument by means of which he himself found that
the diameter of the sun subtended an angle less than 1/164th, but
greater than 1/200th of a right angle). Even the Babylonians had, many
centuries before, arrived at 1° as the apparent angular diameter of the
sun. It is not clear why Aristarchus took a value so inaccurate as 2°.
It has been suggested that he merely intended to give a specimen of the
calculations which would have to be made on the basis of more exact
experimental observations, and to show that, for the solution of the
problem, one of the data could be chosen almost arbitrarily, by which
proceeding he secured himself against certain objections which might
have been raised. Perhaps this is too ingenious, and it may be that, in
view of the difficulty of working out the geometry if the two angles in
question are very small, he took 3° and 2° as being the smallest with
which he could conveniently deal. Certain it is that the _method_ of
Aristarchus is perfectly correct and, if he could have substituted the
true values (as we know them to-day) for the inaccurate values which he
assumes, and could have carried far enough his geometrical substitute
for trigonometry, he would have obtained close limits for the true
sizes and distances.
The book contains eighteen propositions. Prop. 1 proves that we can
draw one cylinder to touch two equal spheres, and one cone to touch two
unequal spheres, the planes of the circles of contact being at right
angles to the axis of the cylinder or cone. Next (Prop. 2) it is shown
that, if a lesser sphere be illuminated by a greater, the illuminated
portion of the former will be greater than a hemisphere. Prop. 3
proves that the circle in the moon which divides the dark and the
bright portions (we will in future, for short, call this “the dividing
circle”) is least when the cone which touches the sun and the moon
has its vertex at our eye. In Prop. 4 it is shown that the dividing
circle is not perceptibly different from a great circle in the moon.
If CD is a diameter of the dividing circle, EF the parallel diameter
of the parallel great circle in the moon, O the centre of the moon,
A the observer’s eye, FDG the great circle in the moon the plane of
which passes through A, and G the point where OA meets the latter great
circle, Aristarchus takes an arc of the great circle GH on one side of
G, and another GK on the other side of G, such that GH = GK = ½ (the
arc FD), and proves that the angle subtended at A by the arc HK is less
than 1/44°; consequently, he says, the arc would be imperceptible at
A even in that position, and _a fortiori_ the arc FD (which is nearly
in a straight line with the tangent AD) is quite imperceptible to the
observer at A. Hence (Prop. 5), when the moon appears to us halved, we
can take the plane of the great circle in the moon which is parallel to
the dividing circle as passing through our eye. (It is tacitly assumed
in Props. 3, 4, and throughout, that the diameters of the sun and moon
respectively subtend the same angle at our eye.) The proof of Prop.
4 assumes as known the equivalent of the proposition in trigonometry
that, if each of the angles α, β is not greater than a right angle, and
α > β, then
tan α / tan β > α/β > sin α / sin β.
Prop. 6 proves that the moon’s orbit is “lower” (i.e. smaller) than
that of the sun, and that, when the moon appears to us halved, it
is distant less than a quadrant from the sun. Prop. 7 is the main
proposition in the treatise. It proves that, on the assumptions made,
the distance of the sun from the earth is greater than eighteen times,
but less than twenty times, the distance of the moon from the earth.
The proof is simple and elegant and should delight any mathematician;
its two parts depend respectively on the geometrical equivalents of the
two inequalities stated in the formula quoted above, namely,
tan α / tan β > α/β > sin α / sin β,
where α, β are angles not greater than a right angle and α > β.
Aristarchus also, in this proposition, cites 7/5 as an approximation by
defect to the value of √2, an approximation found by the Pythagoreans
and quoted by Plato. The trigonometrical equivalent of the result
obtained in Prop. 7 is
1/18 > sin 3° > 1/20.
Prop. 8 states that, when the sun is totally eclipsed, the sun and moon
are comprehended by one and the same cone which has its vertex at our
eye. Aristarchus supports this by the arguments (1) that, if the sun
overlapped the moon, it would not be totally eclipsed, and (2) that,
if the sun fell short (i.e. was more than covered), it would remain
totally eclipsed for some time, which it does not (this, he says,
is manifest from observation). It is clear from this reasoning that
Aristarchus had not observed the phenomenon of an _annular_ eclipse of
the sun; and it is curious that the first mention of an annular eclipse
seems to be that quoted by Simplicius from Sosigenes (second century,
A.D.), the teacher of Alexander Aphrodisiensis.
It follows (Prop. 9) from Prop. 8 that the diameters of the sun
and moon are in the same ratio as their distances from the earth
respectively, that is to say (Prop. 7) in a ratio greater than 18 : 1
but less than 20 : 1. Hence (Prop. 10) the volume of the sun is more
than 5832 times and less than 8000 times that of the moon.
By the usual geometrical substitute for trigonometry Aristarchus proves
in Prop. 11 that the diameter of the moon has to the distance between
the centre of the moon and our eye a ratio which is less than 2/45ths
but greater than 1/30th. Since the angle subtended by the moon’s
diameter at the observer’s eye is assumed to be 2°, this proposition is
equivalent to the trigonometrical formula
1/45 > sin 1° > 1/60.
Having proved in Prop. 4 that, so far as our perception goes, the
dividing circle in the moon is indistinguishable from a great circle,
Aristarchus goes behind perception and proves in Prop. 12 that the
diameter of the dividing circle is less than the diameter of the moon
but greater than 89/90ths of it. This is again because half the angle
subtended by the moon at the eye is assumed to be 1° or 1/90th of a
right angle. The proposition is equivalent to the trigonometrical
formula
1 > cos 1° > 89/90.
We come now to propositions which depend on Hypothesis 5 that “the
breadth of the earth’s shadow is that of two moons”. Prop. 13 is about
the diameter of the circular section of the cone formed by the earth’s
shadow at the place where the moon passes through it in an eclipse,
and it is worth while to notice the extreme accuracy with which
Aristarchus describes the diameter in question. It is with him “the
straight line subtending the portion intercepted within the earth’s
shadow of the circumference of the circle in which the extremities of
the diameter of the circle dividing the dark and the bright portions in
the moon move.” Aristarchus proves that the length of the straight line
in question has to the diameter of the moon a ratio less than 2 but
greater than 88 : 45, and has to the diameter of the sun a ratio less
than 1 : 9 but greater than 22 : 225. The ratio of the straight line to
the diameter of the moon is, in point of fact, 2 cos² 1° or 2 sin² 89°,
and Aristarchus therefore proves the equivalent of
2 > 2 cos² 1° > ½(89/45)² or 7921/4050.
He then observes (without explanation) that 7921/4050 > 88/45 (an
approximation easily obtained by developing 7921/4050 as a continued
fraction (= 1 + (1 1 1)/(1 + 21 + 2))); his result is therefore
equivalent to
1 > cos² 1° > 44/45.
The next propositions are the equivalents of more complicated
trigonometrical formulæ. Prop. 14 is an auxiliary proposition to Prop.
15. The diameter of the shadow dealt with in Prop. 13 divides into two
parts the straight line joining the centre of the earth to the centre
of the moon, and Prop. 14 shows that the whole length of this line is
more than 675 times the part of it terminating in the centre of the
moon. With the aid of Props. 7, 13, and 14 Aristarchus is now able,
in Prop. 15, to prove another of his main results, namely, that the
diameter of the sun has to the diameter of the earth a ratio greater
than 19 : 3 but less than 43 : 6. In the second half of the proof he
has to handle quite large numbers. If A be the centre of the sun, B the
centre of the earth, and M the vertex of the cone formed by the earth’s
shadow, he proves that MA : AB is greater than (10125 × 7087) : (9146
× 6750) or 71755875 : 61735500, and then adds, without any word of
explanation, that the latter ratio is greater than 43 : 37. Here again
it is difficult not to see in 43 : 37 the continued fraction 1 +
11/(6+6); and although we cannot suppose that the Greeks could actually
develop 71755875/61735500 or 21261/18292 as a continued fraction (in
_form_), “we have here an important proof of the employment by the
ancients of a method of calculation, the theory of which unquestionably
belongs to the moderns, but the first applications of which are too
simple not to have originated in very remote times” (Paul Tannery).
The remaining propositions contain no more than arithmetical inferences
from the foregoing. Prop. 16 is to the effect that the volume of the
sun has to the volume of the earth a ratio greater than 6859 : 27 but
less than 79507 : 216 (the numbers are the cubes of those in Prop.
15); Prop. 17 proves that the diameter of the earth is to that of the
moon in a ratio greater than 108 : 43 but less than 60 : 19 (ratios
compounded of those in Props. 9 and 15), and Prop. 18 proves that the
volume of the earth is to that of the moon in a ratio greater than
1259712 : 79507 but less than 216000 : 6859.
ARISTARCHUS ON THE YEAR AND “GREAT YEAR”.
Aristarchus is said to have increased by 1/1623rd of a day Callippus’s
figure of 365¼ days as the length of the solar year, and to have given
2484 years as the length of the Great Year or the period after which
the sun, the moon and the five planets return to the same position
in the heavens. Tannery has shown reason for thinking that 2484 is a
wrong reading for 2434 years, and he gives an explanation which seems
convincing of the way in which Aristarchus arrived at 2434 years as
the length of the Great Year. The Chaldæan period of 223 lunations was
well known in Greece. Its length was calculated to be 6585⅓ days, and
in this period the sun was estimated to describe 10⅔° of its circle in
addition to 18 sidereal revolutions. The Greeks used the period called
by them _exeligmus_ which was three times the period of 223 lunations
and contained a whole number of days, namely, 19756, during which the
sun described 32° in addition to 54 sidereal revolutions. It followed
that the number of days in the sidereal year was--
19756/(54 + 32/360) = 19756/(54 + 4/45) = (45 × 19756)/2434
= 889020/2434 = 365¼ + 3/4868.
Now 4868/3 = 1623 - ⅓, and Aristarchus seems to have merely replaced
3/4868 by the close approximation 1/1623. The calculation was, however,
of no value because the estimate of 10⅔° over 18 sidereal revolutions
seems to have been an approximation based merely on the difference
between 6585⅓ days and 18 years of 365¼ days, i.e. 6574½ days; thus
the 10⅔° itself probably depended on a solar year of 365¼ days, and
Aristarchus’s evaluation of it as 365¼ 1/1623 was really a sort of
circular argument like the similar calculation of the length of the
year made by Œnopides of Chios.
LATER IMPROVEMENTS ON ARISTARCHUS’S FIGURES.
It may interest the reader to know how far Aristarchus’s estimates of
sizes and distances were improved upon by later Greek astronomers. We
are not informed how large he conceived the earth to be; but Archimedes
tells us that “some have tried to prove that the circumference of
the earth is about 300,000 stades and not greater,” and it may be
presumed that Aristarchus would, like Archimedes, be content with this
estimate. It is probable that it was Dicaearchus who (about 300 B.C.)
arrived at this value, and that it was obtained by taking 24° (1/15th
of the whole meridian circle) as the difference of latitude between
Syene and Lysimachia (on the same meridian) and 20,000 stades as the
actual distance between the two places. Eratosthenes, born a few years
after Archimedes, say 284 B.C., is famous for a better measurement of
the earth which was based on scientific principles. He found that at
noon at the summer solstice the sun threw no shadow at Syene, whereas
at the same hour at Alexandria (which he took to be on the same
meridian) a vertical stick cast a shadow corresponding to 1/50th of the
meridian circle. Assuming then that the sun’s rays at the two places
are parallel in direction, and knowing the distance between them to
be 5000 stades, he had only to take 50 times 5000 stades to get the
circumference of the earth. He seems, for some reason, to have altered
250,000 into 252,000 stades, and this, according to Pliny’s account
of the kind of stade used, works out to about 24,662 miles, giving
for the diameter of the earth a length of 7850 miles, a surprisingly
close approximation, however much it owes to happy accidents in the
calculation.
Eratosthenes’s estimates of the sizes and distances of the sun and moon
cannot be restored with certainty in view of the defective state of
the texts of our authorities. We are better informed of Hipparchus’s
results. In the first book of a treatise on sizes and distances
Hipparchus based himself on an observation of an eclipse of the sun,
probably that of 20th November in the year 129 B.C., which was exactly
total in the region about the Hellespont, whereas at Alexandria about
⅘ths only of the diameter was obscured. From these facts Hipparchus
deduced that, if the radius of the earth be the unit, the least
distance of the moon contains 71, and the greatest 83 of these units,
the mean thus containing 77. But he reverted to the question in the
second book and proved “from many considerations” that the mean
distance of the moon is 67⅓ times the radius of the earth, and also
that the distance of the sun is 2490 times the radius of the earth.
Hipparchus also made the size (meaning thereby the solid content) of
the sun to be 1880 times that of the earth, and the size of the earth
to be 27 times that of the moon. The cube root of 1880 being about 12⅓,
the diameters of the sun, earth and moon would be in the ratio of the
numbers 12⅓, 1, ⅓. Hipparchus seems to have accepted Eratosthenes’s
estimate of 252,000 stades for the circumference of the earth.
It is curious that Posidonius (135–51 B.C.), who was much less of an
astronomer, made a much better guess at the distance of the sun from
the earth. He made it 500,000,000 stades. As he also estimated the
circumference of the earth at 240,000 stades, we may take the diameter
of the earth to be, according to Posidonius, about 76,400 stades;
consequently, if _D_ be that diameter, Posidonius made the distance of
the sun to be equal to 6545_D_ as compared with Hipparchus’s 1245_D_.
Ptolemy does not mention Hipparchus’s figures. His own estimate of
the sun’s distance was 605_D_, so that Hipparchus was far nearer the
truth. But Hipparchus’s estimate remained unknown and Ptolemy’s held
the field for many centuries; even Copernicus only made the distance
of the sun 750 times the earth’s diameter, and it was not till 1671–3
that a substantial improvement was made; observations of Mars carried
out in those years by Richer enabled Cassini to conclude that the sun’s
parallax was about 9·5″ corresponding to a distance between the sun
and the earth of 87,000,000 miles.
Ptolemy made the distance of the moon from the earth to be 29½ times
the earth’s diameter, and the diameter of the earth to be 3⅖ times that
of the moon. He estimated the diameter of the sun at 18⅘ times that of
the moon and therefore about 5½ times that of the earth, a figure again
much inferior to that given by Hipparchus.
BIBLIOGRAPHY.
On the astronomy of the early Greek philosophers much information is
given by Aristotle (especially in the _De Caelo_); for Aristotle was
fortunately in the habit of stating the views of earlier thinkers as
a preliminary to enunciating his own. Apart from what we learn from
Aristotle, we are mainly dependent on the fragmentary accounts of
the opinions of philosophers which were collected in the _Doxographi
Graeci_ of Hermann Diels (Berlin, 1879), to which must be added, for
the period before Socrates, _Die Fragmente der Vorsokratiker_ by the
same editor (2nd edition, with index, 1906–10, 3rd edition, 1912).
The doxographic data and the fragments for the period from Thales to
Empedocles were translated and explained by Paul Tannery in _Pour
l’histoire de la Science Hellène_ (Paris, 1887). A History of Astronomy
was written by Eudemus of Rhodes, a pupil of Aristotle; this is lost,
but a fair number of fragments are preserved by later writers. Of the
theory of concentric spheres we have a short account in Aristotle’s
_Metaphysics_, but a more elaborate and detailed description is
contained in Simplicius’s commentary on the _De Caelo_; Simplicius
quotes largely from Sosigenes the Peripatetic (second century A.D.) who
drew upon Eudemus.
There are a number of valuable histories of Greek astronomy. In German
we have Schaubach, _Geschichte der griechischen Astronomie bis auf
Eratosthenes_, 1809; R. Wolf, _Geschichte der Astronomie_, 1877;
and two admirable epitomes (1) by Siegmund Günther in Windelband’s
_Geschichte der alten Philosophie_ (Iwan von Müller’s _Handbuch der
klassischen Altertumswissenschaft_, Vol. V, Pt. 1), 2nd edition, 1894,
and (2) by Friedrich Hultsch in Pauly-Wissowa’s _Real-Encyclopädie der
classischen Altertumswissenschaft_ (Art. Astronomie in Vol. II, 2,
1896).
In French, besides the great work of Delambre, _Histoire de
l’astronomie ancienne_, 1817, we have the valuable studies of Paul
Tannery in _Recherches sur l’histoire de l’astronomie ancienne_, 1893,
and Pierre Duhem, _Le Système du Monde_, Vol. I, 1913.
In English, reference may be made to Sir G. Cornewall Lewis, _An
Historical Survey of the Astronomy of the Ancients_, 1863; J. L. E.
Dreyer, _History of the Planetary Systems from Thales to Kepler_,
Cambridge, 1906; and the historical portion of Sir Thomas Heath’s
_Aristarchus of Samos, the ancient Copernicus_, Oxford, 1913.
Aristarchus’s treatise _On the Sizes and Distances of the Sun and
Moon_ first appeared in a Latin translation by George Valla in 1488
and 1498, and next in a Latin translation by Commandinus (1572). The
_editio princeps_ of the Greek text was brought out by John Wallis,
Oxford, 1688, and was reprinted in _Johannis Wallis Opera Mathematica_,
1693–1699, Vol. III, in both cases along with Commandinus’s
translation. In 1810 there appeared an edition by the Comte de Fortia
d’Urban, _Histoire d’Aristarque de Samos_ ... including the Greek text
and Commandinus’s translation but without figures; a French translation
by Fortia d’Urban followed in 1823. The treatise was translated into
German by A. Nokk in 1854. Finally, Sir Thomas Heath’s work above cited
contains a new Greek text with English translation and notes.
CHRONOLOGY.
(_Approximate where precise dates are not known._)
B.C.
624–547 Thales.
610–546 Anaximander.
585–526 Anaximenes.
572–497 Pythagoras.
born 516 or 514 Parmenides.
(possibly 540).
500–428 Anaxagoras.
494–434 Empedocles.
5th century { Œnopides of Chios.
{ Philolaus.
427–347 Plato.
408–355 Eudoxus.
388–315 Heraclides of Pontus.
384–322 Aristotle.
370–300 Callippus.
310–230 Aristarchus of Samos.
287–212 Archimedes.
284–203 Eratosthenes.
265–190 Apollonius of Perga.
3rd century Aratus.
fl. 150 Hipparchus.
135–51 Posidonius.
A.D.
50–125 Plutarch.
100–178 Ptolemy.
ABERDEEN: THE UNIVERSITY PRESS
Transcriber’s Notes
Punctuation and spelling were made consistent when a predominant
preference was found in this book; otherwise they were not changed.
Simple typographical errors were corrected. One unbalanced parenthesis
was remedied.
Italics are indicated by _underscores_.
End of Project Gutenberg's The Copernicus of Antiquity, by Thomas Little Heath