In the upper figure it is day, in the lower, night, on the inhabited
side of Earth. The sun is on the equator, as at the time of equinox.]

[Sidenote: Heracleides _c._ 370 B.C.]

But not even here did the Greeks stop. It was taking a less startling
step than they had already taken, to reach the truth that Earth was
merely rotating on her axis once in a day, and so causing the apparent
diurnal revolution of the heavens. This step was taken (it is said)
by a Pythagorean called Hicetas of Syracuse, who is quoted as saying
that the earth, “while it turns and twists itself with the greatest
velocity round its axis, produces all the same phenomena as if the
heavens were moved and the earth were standing still.” We are told also
that “Heracleides of Pontus and Ecphantus the Pythagorean let the earth
move, not progressively, but in a turning manner like a wheel fitted
with an axis, from west to east round its own centre.”

A brilliant guess,[29] which seems fully justified by facts, has
recently explained the personalities of these two mysterious
Pythagoreans of unknown date, Hicetas and Ecphantus, whose names have
been coupled for centuries with that of Heracleides, as teaching the
rotation of Earth on her axis. It seems that they resemble the Shadow
in Hans Andersen’s tale, which became a man and lived apart from the
man to whom it originally owed its existence, for it is now thought
that they were speakers introduced by Heracleides into one of his
dramatic dialogues to discuss astronomy.

[29] See Heath’s _Aristarchus of Samos_, pp. 187-189 and 251, 252.
I very much regret that as Mr. Heath’s book was only published this
year, I have been unable to make use of it while writing of early Greek
astronomy. I can now only advise any readers who may be interested in
my brief sketch of this period to read Mr. Heath’s history, where they
will find the opinions of modern writers summarized and discussed,
and also the full text (in English) of the most ancient and reliable
sources of information. It is a great encouragement to find that my
statements are in agreement with his in nearly all essential points,
but readers will mark the following important differences:—

1. Anaximander’s heavens are said to have been spherical, not
hemispherical, and this seems to be clearly proved by the evidence
quoted from ancient writers.

2. Anaxagoras, not Thales, is said to have been the first to explain
correctly the cause of solar eclipses and of the moon’s phases, viz.
that the moon is an opaque body, shining only by reflected sunlight,
and periodically hiding the sun from us when she passes in front of
it. Mr. Heath regards the authorship of Anaxagoras as conclusively
proved: readers will be able to judge of this from his quotations.
Personally they seem to me to prove no more than that Anaxagoras agreed
with others on this point, and was the first to express it clearly in
writing. It is difficult to see why Mr. Heath denies that Parmenides
held the same views before Anaxagoras: Parmenides’ own words seem to
prove it, and his theory that the moon was composed of air and fire
mingled is rather in favour of it than otherwise. He surely meant that
the moon was not wholly bright, like the sun; yet that she had some
light of her own must have seemed evident from the faint illumination
we see during total lunar eclipses and on the part of her surface not
lighted by the sun. (See Dante’s views, p. 402 of this book.)

The connection between her phases and her distance from the sun in the
sky is so extremely obvious that I can hardly think the Greeks drew no
inference from it until the fifth century B.C., and I cannot see why
we should refuse to credit Thales with the discovery attributed to him
that her light came in some way from the sun. Gruppe acutely observes
that the reason why Thales’ pupil Anaximander did not accept the true
explanation of lunar phases and solar eclipses may have been because
he felt it necessary to have a theory which would apply equally well
to eclipses of the moon; and as he believed in a flat earth he could
not advocate the true explanation here. This was why he invented a new
theory (viz. that both sun and moon were fire shining through holes in
hollow rings, and that the occasional stopping up of these holes caused
both lunar and solar eclipses, and also the lunar phases).

But Parmenides had learned the Pythagorean doctrine of Earth’s
spherical form, hence he was able to accept the older theory that the
moon obtains her light from the sun, and sometimes eclipses the sun by
her opaque spherical body, for he could have added that the moon is
eclipsed in like manner by the opaque spherical body of the earth.

Heracleides, therefore, was the sole author of this remarkable
discovery.

In this way Earth was restored to her central position, but as a
rotating sphere, and the later Pythagoreans apparently tried to
reconcile their new scheme with the old by calling Antichthon the
uninhabited hemisphere of Earth, and placing the central fire within
the earth.

IV. GREEK ASTRONOMY.

SECOND PERIOD. B.C. 400 TO A.D. 150.

    “Chiamavi il cielo, e intorno vi si gira,
    Mostrandovi le sue bellezze eterne.”

1. PLATO.

The Ionian school of philosophy died about the middle of the fifth
century B.C., and the Pythagorean towards the end of the fourth. But
meanwhile a new school of astronomers was growing up. The philosophers
still laid down general principles, founded on abstract reasoning,
which they believed must regulate the nature and movements of the
heavenly bodies, but astronomy began to be regarded as a branch of
mathematics, not of philosophy, and the mathematicians, leaving
problems of ultimate causes to the philosophers, devoted themselves
to observation and calculation. They carefully studied the peculiar
motions of each planet, and their chief aim was to represent these
geometrically by some scheme which should include them all, and make it
possible to predict the places of the heavenly bodies in the sky for
any given date.

One cause of this great progress in methods was no doubt the natural
intellectual growth of the Greek race, as they discovered that their
eager curiosity concerning nature could only be satisfied by patient
investigation. The value of observation was taught, in the latter half
of the fourth century, by the philosophy of Aristotle, and a great
impetus must have been given by the campaigns of Alexander, in which
the Greeks saw distant countries, new climates, strange peoples and
customs.

[Sidenote: Callisthenes _c._ 330 B.C.]

But a potent cause of the advance in astronomy seems to have been the
closer connection between Greek astronomers and those of Egypt and
Babylon. The astronomer Callisthenes went with Alexander to the East,
and received a letter from Aristotle praying him to send to Athens the
Babylonian eclipse records which were centuries old; and Aristotle
mentions, when speaking of the motions of the planets, that the
Babylonians and Egyptians had furnished trustworthy information about
each one of them. Even before this, we find that the Greek descriptive
names of the planets were changed for names of Greek deities which
are believed to correspond with the Babylonian gods and goddesses
who presided over the planets. Thus Plato speaks of “the star sacred
to Hermes” as well as Stilbon the Glitterer, and he is the last to
use commonly the name of Phosphor for the planet which henceforth
was known as Aphrodite among the Greeks, and Venus among the Romans,
corresponding with the Babylonian Ishtar; and so on with the rest.
Instead of vague records of journeys in Egypt or Babylonia, we have a
definite statement that Eudoxus, who was the founder of the new school,
went to Egypt about 378 B.C., with letters from the king of Sparta
to the king of Egypt, and we are told that he studied the planetary
motions under a priest of Heliopolis. It seems highly probable, to say
the least, that Eudoxus was the first Greek to appreciate the value
of those methods of observation and continuous recording of phenomena
which he found among the Egyptians, and to understand the wonderful
regularity which was hidden behind the seeming irregularities of the
“wandering stars.” He was also, apparently, the first Greek to write a
detailed description of the forty-eight ancient constellations.

But if Egypt and Babylonia gave to Greece records of celestial
phenomena, and set the example of accurate and long-continued
observations, Greece made the new knowledge her own, and transformed
it. The legend that Eudoxus applied his mathematical skill to the
ancient monuments of the Egyptians, and showed them how to calculate
the height of the Great Pyramid by measurements of its shadow, is
typical of the history of Greek treatment of Oriental astronomy.
One geometrical theory after another was invented to represent
the planetary motions, was compared with the skies, and rejected
or improved, and meanwhile observation became much more close and
accurate; new instruments were introduced, new methods of calculation
invented, new motions discovered which had to be accounted for;
finally, five hundred years after Eudoxus’ visit to Egypt, the result
of all this labour was summarized in a truly epoch-making work,
which remained the standard treatise on astronomy until the time of
Copernicus.

[Sidenote: Plato _c._ 427-347 B.C.]

Eudoxus was born at Cnidos, in Asia Minor, but at the age of
twenty-three he went to Athens, and studied under Plato. It is said to
have been Plato who inspired the young man with the idea of devoting
his brilliant mathematical powers to solving the problem of celestial
motions, and with this view he went to Egypt. The story is easy
to believe when we recall the many passages in the _Dialogues_ in
which Plato uses the splendid imagery of the skies to illustrate his
philosophic doctrines, dwelling especially on the perfect though little
understood symmetry of the celestial motions, and it will be remembered
that astronomy was one of the subjects to be learned by the rulers of
his ideal state.

It is true that Glaucon is gently but decidedly snubbed by Socrates in
the _Republic_, for suggesting that the study of astronomy is valuable
because of its use in navigation, husbandry, and the arts of war. This
is “vulgar praise,” but has there ever been nobler praise of astronomy
than that which Socrates himself then proceeds to give? Although he
believes that true knowledge, knowledge of realities, is only to be
obtained by the exercise of pure reason without the aid of sense, he
considers that the study of celestial motions is one of the best means
of training the mind to reach those heights, and he does not hesitate
to say that sight was given to us in order that we might look at the
skies. For the embroidery of heaven, says Socrates, though wrought upon
a visible ground, is the fairest and most perfect of visible things;
and it is displayed to our mortal eyes as a pattern of the eternal
realities which are granted to the vision of the soul.

In the _Timaeus_ this idea is elaborately developed, and it undoubtedly
had an effect on Plato’s contemporaries, although his direct influence
on astronomy cannot be compared with that of Aristotle. The _Timaeus_
was widely read also in the Middle Ages, during the long period when
Plato’s other writings were unknown, and it is quoted by Dante.
We are often reminded of him when reading the astronomical and
quasi-astronomical parts.

Timaeus, who is introduced to Socrates by Critias as “the most of an
astronomer among us, and one who has made a special study of the nature
of the Universe,”[30] describes the Creation as he conceives it most
probable that it took place. He assumes a chaos to begin with, where
there is no order, and no matter which can be distinguished by name,
but all is confused and seething with random restless motions.

[30] I follow the translation of Jowett.

Of this, in order to produce something which should express his own
goodness,[31] the Creator formed the four elements,—earth, water,
air, and fire,—and of them he made a world, which became a fair and
intelligent being, animated by a living soul. He made it in the most
perfect form, that of a sphere, polished and smooth on the outside, “as
if from a lathe.” The soul was placed in the centre, and hence diffused
throughout the whole bodily frame. It is the cause of the harmonious
motions of the stars, and of these there are two kinds: the motion
of the Same (the diurnal revolution of the whole heavens) is in the
noblest direction, simple and uniform; the motion of the Diverse is in
the opposite direction and diagonal to the first, and it is divided
into seven parts (the seven orbits of the planets), which bear certain
definite ratios to one another.

[31] Compare _Par._ vii. 64-66.

Timaeus does not name the planets, but in the _Republic_ Socrates
names some, and indicates the rest by their colour or other
characteristic,[32] so we know that the order which he assigns to them,
counting outwards from the central earth, is: Moon, Sun, Mercury,
Venus, Mars, Jupiter, Saturn.

[32] See his famous description of the eight spheres, on each of which
stands a siren, singing, while the whole system turns upon a diamond
spindle, the end of which rests upon the knees of Necessity. This book
was not known in the Middle Ages.

It was in order to make the world like its eternal pattern that the
Creator made a “moving image of eternity,” which we call Time, in the
revolutions of the heavenly bodies; and to make it visible he “lighted
a fire which we now call the sun, in the second of these orbits, that
it might give light to the whole of heaven [note that the stars shine
by reflected sunlight, as well as moon and planets], and that the
animals who were by nature fitted might participate in number: this
was the lesson they were to learn from the revolutions of the Same
and the Like. Thus, then, and by these means, the night and the day
were created, being the period of the one most intelligent revolution.
And the month was created when the moon had completed her orbit and
overtaken the sun; and the year, when the sun had completed his own
orbit. The periods of the other stars [the planets] have not been
understood by men in general, but only by a few, and they have no name
for them, and do not estimate their comparative length by the aid of a
number, and hence they are hardly aware that their wanderings, which
are infinite in number and admirable for their variety, make up time.
And yet there is no difficulty in seeing that the perfect number of
time completes the perfect year when all the eight revolutions, having
their relative degrees of swiftness, are accomplished together, and
again meet at their original point of departure, measured by the circle
of the Same moving equally.”

The heavenly bodies, according to Timaeus, are all divine intelligent
beings. In form they are perfect spheres, like the world of which they
form part, and they are composed of fire. The stars have two motions,
for each rotates on its own axis while it is carried round the centre
on the rotating star sphere.

Earth is also a sphere, immoveable at the centre of the World. Of her
Timaeus says: “The earth, which is our nurse, encircling the pole
which is extended through the universe, he made to be the guardian and
artificer of night and day.”

This passage has given rise to the idea that Plato believed the
apparent diurnal revolution of the heavens to be caused by earth’s
rotation on her axis; but the word here translated “encircling”[33] may
mean—as that does—either motion or situation round about something,
and the whole context ascribes the diurnal movement so clearly and
emphatically to the heavens, that it seems evident Plato could only
have meant that earth was guardian and artificer of day and night by
virtue of her position. The only strong argument in favour of the
other meaning is that Aristotle, when speaking of Earth as supposed
by some to be central in the Universe but moving, quotes Plato and
the _Timaeus_. It might easily happen, however, that Aristotle knew
from other sources, perhaps from conversation with Plato, that at some
time the latter had inclined towards belief in Earth’s motion, and
remembering the ambiguous expression in the _Timaeus_ he quoted it from
memory as a statement of Plato’s belief.

[33] ειλλομενη.

There is some evidence that late in life Plato accepted the doctrine
of Philolaus that Earth was not only in motion, but in motion round a
Central Fire. There is a legend that he bought the books of Philolaus
at a great price, and Theophrastus, a disciple of Aristotle, is
reported by Plutarch to have said that “Plato when old assigned to
Earth another place, the central and nobler place being reserved for
something else more worthy of it.” However this may be, he does not
teach either theory in his writings. His views seem to be quite the
same as those of Pythagoreans of the old school, whom he sometimes
quotes.

After describing the creation of the Universe, Timaeus relates that
the Creator deputed the gods whom he had made (including the stars)
to create living beings on the earth, he himself creating directly
only their immortal part, which he made of the same essence as the
World-Soul, but diluted. Then follows the passage which came to the
mind of Dante when he met the first spirits of Paradise in the moon.

    “And when he had framed the Universe, he
    distributed souls in equal numbers to the stars,
    and assigned each soul to a star; and having placed
    them as in a chariot, he showed them the nature of
    the Universe, and the decrees of destiny appointed
    for them, and told them that no one should suffer
    at his hands, and that they must be sown in the
    vessels of the times severally adopted to them....
    He said that he who lived well during his appointed
    time [on earth] would return to the habitation of
    his star, and there have a blessed and suitable
    existence.” If he lived ill, he would be a woman
    at his second birth, if a bad woman, then a beast,
    and as long as he continued to do ill he would “not
    cease from his toils and transformations until he
    followed the original principle of sameness and
    likeness within him.... When he had given all these
    laws to his creatures ... he sowed some of them in
    the earth, and some in the moon, and some in the
    other stars which are the measures of Time.”

The creation of man’s body and all the remainder of the _Timaeus_ does
not concern us here, except that when speaking of the highest use of
man’s faculty of sight, we realize how near Dante and Plato are in
their feeling for the revolving heavens:

“God invented and gave us sight to this end, that we might behold the
courses of intelligence in the heaven, and apply them to the courses
of our own intelligence which are akin to them, the unperturbed to the
perturbed; and that we, learning them and being partakers of the true
computations of nature, might imitate the absolutely unerring courses
of God and regulate our own vagaries.”

2. EUDOXUS.

[Sidenote: Eudoxus 408 B.C. to _c._ 355 B.C.]

With words like these ringing in his ears, Eudoxus went from the
Greek philosopher to the Egyptian priest, and studied “the courses of
intelligence in the heaven.” Legend says that the sacred Egyptian Bull
licked his garment, and the priests no doubt were encouraged by this
omen to divulge their secrets to a person so highly favoured by the
gods. They prophesied that he would have a short but very illustrious
life.

After a year, or perhaps more, spent in Egypt, Eudoxus returned to his
own city, set up an observatory of his own, received pupils, and worked
out an exceedingly ingenious and original planetary scheme. He did not
accept (if he knew of them) the risky theories of the Pythagoreans as
to Earth’s motion, but assumed a central stable earth, round which
circled the stars and the seven planets, according to the teaching of
Plato and the general belief among educated Greeks of his day. But
Egyptian observation and Greek geometry enabled him to describe for the
first time the complicated movements of the planets, and to represent
them by an imaginary mechanism.

This was a series of spheres, or hollow balls, fitting inside one
another, and gradually diminishing in size like the ivory boxes of a
Chinese puzzle, or the coats of an onion. Their size was stupendous,
for the outer one, which contained all the rest, was nothing less
than the sky we see, and was encrusted all over with stars. Of the
inner smaller spheres, one bore, fixed in it like a jewel set in a
ring, the sun; and six others bore, in the same way, the moon and
the planets, one in each. All these hollow spheres were symmetrically
placed so that all centred in a single point, and at this point was a
solid sphere, exceedingly small in comparison, which was the earth. The
star sphere, without moving from its place, rotated round this central
Earth, and this caused the diurnal motion that we see in the stars.
Each planet-bearing sphere rotated also, but the special characteristic
of Eudoxus’ system is that each of these was surrounded by its own
complete set of spheres, bearing no planet, but all attached together,
the poles of one sphere resting on the surface of the next, and moving
with different speeds, in different directions, and with differently
inclined axes: these motions being all communicated to the innermost
sphere on which the planet was fixed, the net result was the movement
of the planet as we see it in the sky. Each planetary set was quite
separate from the rest, and did not interfere with their movements,
although each set was enclosed within the next larger. Since all the
planets have a diurnal motion like the stars, as well as their own
proper motions, each set had to be provided with a sphere which moved
exactly like the great all-enclosing star sphere.

Thus, the sun had one sphere turning like the star sphere, and within
this was a second, on which the sun was fixed, which turned round in
a year, in a west to east direction. The sun, carried along by the
combined motion, travelled through the sky with the daily and yearly
motions, as we see them.

[Illustration: Fig. 19. The spheres of the sun in the system of Eudoxus.

The outer sphere turns on its axis _A A_ in a day and night; the inner
on its axis _a a_ in a year, in the opposite direction.]

The planetary spheres were much more difficult to arrange. Eudoxus used
four spheres for each, and these had in every case to be carefully
adjusted to the very different periods and amplitudes of the planetary
oscillations. It must be confessed that the scheme failed with the
difficult case of Mars, and was not quite satisfactory with Venus, but
it represented remarkably well the movement—so far as then known—of
Sun and Moon, Saturn, Jupiter, and Mercury. It was certainly a feat
for those days, whether we consider it merely as the solution of a
mathematical problem, or as an embodiment of astronomical knowledge.
The periods of the planets as known to Eudoxus, stated in round numbers
only, are given in the following table. They are taken from Simplicius,
who describes the system of Eudoxus, but as in the so-called Papyrus
of Eudoxus the synodical revolution of Mercury is given as 116 days,
the same as the modern value, Eudoxus may have had much more exact
data. It will be seen that his synodic period for Mars is the only one
which is totally wrong, and the large error is difficult to explain.

                                    _Modern     Zodiacal       Modern_
    _Planet.   Synodic Period.       Value.      Period._[34] _Value._

     Mercury    110 days             116 days     1 year        1.0 year
     Venus       19 months           584  ”          ”          1.0  ”
     Mars       8 months, 20 days    780  ”       2 years       1.88 ”
     Jupiter     13 months           399  ”      12   ”        11.86 ”
     Saturn      13 months           378  ”      30   ”        29.46 ”

[34] The period in which a planet is seen to revolve round the zodiac,
and return to the same star, varies greatly, because complicated by
its retrograde movements; but if the average of a sufficient number of
periods be taken, it coincides for Mercury and Venus with the sidereal
year; for Mars, Jupiter, and Saturn, with the period in which each is
actually revolving round the sun (its “sidereal period”).

It is disappointing, after the splendid hypotheses of the Pythagoreans,
to be back again on a central stationary Earth among mechanical
contrivances for moving the heavenly bodies, which remind us of
Anaximander’s series of hemispherical heavens and heavenly wheels, but
at least the earth is spherical, owing to the Pythagoreans, and the sky
extends like a sphere all round, and we shall never have a flat Earth
or a hemispherical sky again among the Greeks. We do not know whether
Eudoxus regarded his spheres as convenient mathematical abstractions
only, or whether he reasoned that the stars were evidently set in an
invisible uniformly rotating sphere, and Plato considered this kind
of movement the most suitable for all heavenly bodies; that therefore
he would try whether a series of similar spheres interacting on one
another would account for the complicated motions of a planet, and
finding that they would, taught that they must truly exist. In any case
the basis of his system was a detailed knowledge of planetary motions
hitherto unapproached by the Greeks, and its chief merit was that it
challenged comparison with the skies.

3. CALIPPUS.

[Sidenote: Calippus _c._ 330 B.C.]

The challenge was soon taken up, for twenty or thirty years later one
of the pupils of Eudoxus, Calippus of Cyzicus, undertook to improve the
system. The defects in the theories of Mars and Venus had evidently
been discovered, for Calippus added another sphere to each of these,
as well as one to Mercury, which would be quite enough to bring the
theories into better agreement with the facts.

With regard to the sun and moon, Calippus had paid special attention to
their movements, for he made an improvement in the old luni-solar cycle
of Meton, to which we shall return later. Eudoxus had ignored a very
important fact discovered by Meton and Euctemon about B.C. 430, viz.:
that the seasons are of unequal length, showing that the sun takes
unequal times to pass over the four arcs of his orbit lying between the
four points of the vernal and autumnal equinoxes and the summer and
winter solstices. “Why,” exclaims a later writer[35] “are there unequal
numbers of days in the four seasons, seeing that the course of the
heavenly bodies must be regular, not being swayed by human passions or
affairs?”

[35] Geminus.

Calippus considered this question very seriously, and made a careful
determination of the length of the four seasons. It seems at first
sight impossible to reconcile their inequality with uniform circular
motion of the sun round the earth, but he found that he could do it by
adding two more spheres to the sun’s set, rotating uniformly but so
arranged that their motion, added to the others, would result in an
actual velocity in the sun itself varying just in the way required by
the facts. The same had to be done for the moon, for the same reason,
so the number of spheres, which Eudoxus had made twenty-seven, was
brought up to thirty-four (including the star sphere). The varying
velocity of the five planets had not yet been perceived.[36]

[36] This varying velocity is due to the fact that all celestial
orbits are not true circles, but ellipses, which was first discovered
by Kepler (1609 A.D.).

4. ARISTOTLE.

[Sidenote: Aristotle B.C. 384-322.]

Calippus went up to Athens about 330 B.C. to lay his scheme before the
great master, Aristotle, and it had his cordial approval. But Aristotle
definitely accepted the spheres as things having a concrete existence,
for he says (_De Cœlo_ II, 12) that we must regard them as heavenly
bodies like the stars and planets, and that they are composed of the
same celestial stuff.

He made one change, when incorporating the system in his scheme of
the Universe. He was not satisfied that each set of spheres should
work quite independently of the rest, and thought that the outermost
sphere of stars ought to communicate its motion to those below
(_i.e._ nearer Earth); and no doubt it did seem rather clumsy to
have a separate sphere in every set rotating in exactly the same
manner as the star sphere. But how could the impulse be communicated
without disturbing the other movements? Aristotle introduced below
each set another set of “unrolling” spheres, as he called them, which
successively neutralized the rotations of all spheres in that set
except the one with diurnal rotation, hence this movement alone was
communicated to the set next below. This seems, however, more clumsy
than the defect it was intended to remedy. Aristotle was perhaps led to
it by his wish to give greatest importance to the star sphere; and if
so, he acted on the principle which he blamed in the Pythagoreans, of
making deductions not from things as they are seen, but as, according
to his own ideas, they ought to be.

For indeed Aristotle, in spite of his own doctrines, and the great
impulse which he gave to truly scientific methods of observation and
experiment, could not rise altogether above the prejudices of his age,
and consequently his Cosmos is a curious mixture of sound reasoning,
based on observation, and of metaphysics, the latter predominating. For
instance, it is only at the end of his second book _On the Heavens_,
after he has “proved,” from purely metaphysical reasons, that Earth
must necessarily be spherical and at the centre of the World, that he
adds in support of his assertions the fact that the curved line of
Earth’s shadow seen on the moon during eclipses is always round, that
stars vary in visibility as we change our horizon, and that astronomers
say that the celestial phenomena occur as they would if Earth were at
the centre of the World.

Nevertheless, Aristotle’s teaching had so overwhelming an influence,
not only throughout this epoch, but in the age of Dante, and the latter
was so greatly influenced by him, both directly and indirectly, that
it is exceedingly interesting to know his ideas about the Cosmos. We
find them in the two books _On the Heavens_, in the _Meteorology_, the
_Metaphysics_, and some other works. A special treatise on Astronomy,
to which he refers[37], is unfortunately not extant.

[37] _De Cœlo_ II. 10.

The form of the Universe, Aristotle says, must be a sphere, because a
sphere is the most perfect of solids and a solid is more perfect than a
surface or a line, because it is in three dimensions, and three means
completion, perfection.[38]

[38] Thus, (he adds) time also is threefold, for we have Beginning,
Middle, and End. Therefore we apply three to Divine things, and also
in common speech we call two “both,” and only say “all” when we reach
three, following Nature’s law. The Pythagoreans say “The all and all
things are bounded by the number three.”—_De Cœlo_ I. 1.

The Universe had no beginning, and will have no end; and this
conclusion, drawn from reasoning, is supported by the belief which all
have who believe in gods, “whether Greeks or not Greeks,” that the
gods, who are immortal, live in the highest heaven, which is therefore
also immortal; and by the fact that no one, throughout the ages, so far
as we know, has ever seen any change in it.

But it is of finite dimensions, for no infinitely great body could
rotate in a finite time; and it is the only universe which exists or
ever can exist: outside is neither space, nor void, nor time. For space
is that which is or may be occupied by matter, and time is the measure
of motion occurring in matter, and no matter exists or can exist there.
Therefore that which exists there is not in space nor is altered by
time, but lives for ever the best and the self-sufficing life (_i.e._
the purely spiritual).

As matter has three dimensions, so motion is of three kinds: viz. (1)
in a straight line down, that is, towards the centre of the World; (2)
in a straight line up, that is, towards the circumference; (3) in a
circle round the centre. Thus simple heavy bodies such as all kinds
of earth, have a simple motion downwards; simple light bodies such as
fire, move upwards; and when they reach their respective goals they
remain where they are, unless disturbed by external force—earthy things
on the earth, fiery vapours in the upper atmosphere. Composite bodies
have composite motions, the motion proper to the predominant substance
predominating. But for the heavenly bodies the only possible motion is
in a circle, where there is neither beginning nor end, no goal and no
limit, hence this motion is eternal.

Thus Aristotle solved for himself the problem of the early
philosophers: how the stars in the sky remain there, for ever circling
round us, and never falling to the ground. There is no need, he says,
to assume an Atlas to support the sky on his shoulders, as in the old
myths, nor a whirlwind such as Empedocles suggested, nor a Soul of the
World, as Plato said; for the heavenly bodies are not heavy things
like Earth to need support, and they are not moved by force, but are
eternally in motion from the nature of their being.

In the same way he disposes of the difficulty of supporting Earth,
having first “proved” that because there is an ever-circling spherical
Heaven, there must also be an ever-resting spherical central Earth;
there must be its opposite, the ever upward-striving Fire; and there
must be the intermediate pairs of opposites, Air and Water. It is
indeed, he says, a strange thing, and one to set any thoughtful man
thinking, that the smallest clod of earth, when thrown up into the air,
immediately falls down, and presumably would never stop falling if the
earth were suddenly removed from beneath it; yet here is Earth herself,
so large and heavy, not falling, but remaining steady in one place. But
the explanations given by philosophers are more difficult than the fact
they seek to explain. Xenophanes of Colophon said that the earth roots
in the infinite, which simply saved him the trouble of considering
further; others that the earth rests on water, which is our oldest
tradition, said to be derived from Thales; but on what then does water
rest and how can water, which is lighter than earth, support it? Do we
not see that even small pieces of earth sink in water, and larger ones
still more quickly? Anaximenes, Anaxagoras, and Democritus said that
Earth rests on air, through her flat shape, as a leaf can float on the
wind, and they added that the air cannot escape because the flat earth
fits close down upon it, like a lid, which is also the reason that
it can support the earth, because it is compressed, and they brought
forward many proofs to show that air, when compressed and still, can
support great weights. Others, like Anaximander, said that Earth rests
because she is in equilibrium, for there is no reason why she should
move in one direction rather than another. But all earthy things (says
Aristotle) do not merely remain at the centre when there, they move
thither whenever displaced. They are not suspended like a hair which
is powerfully but uniformly stretched, and so never breaks, nor like
a man who is equally hungry and thirsty, and has meat and drink at
equal distances from him and therefore starves.[39] No, the truth is
that Earth, and every particle of Earth, tends naturally towards the
centre of the Universe, and rests when at its goal. We must remember
that on every part of the sphere of Earth, heavy bodies fall vertically
to its surface, showing that it is not to the surface in general that
they fall, but exactly to the centre, which is also the centre of the
Universe.

[39] _Par._ iv. 1-3.

Of course it was no solution of the mystery, but only moving it a
step back, to say that the stars circle because it is their nature to
do so, and heavy bodies fall to Earth because that is their nature.
But the interesting point is that the Greeks did reason about these
two motions, and compare them; that they clearly grasped the fact
that “weight” simply means a tendency to move, and that the motion
of falling bodies at Earth’s surface is invariably towards Earth’s
centre, accelerating as it appproaches the surface (_De Cœlo_ I. 8).
It is only the fresh mind of a little child, or of a really intelligent
man, which is forcibly struck by the mystery of everyday sights, such
as stones falling, and stars _not_ falling but eternally moving in the
sky. The force which makes bodies move towards one another we still
call “heaviness,” _i.e._ gravity, and the mystery of its ultimate
nature and mode of action is still unsolved. It is so weak and so often
complicated by other forces that, except in very delicate experiments
or with the mind’s eye, we can only see it in action when bodies fall
to the ground; and thus Nature guarded for centuries the secret that
every tiny particle on the whole earth attracts every other, and also
the earth itself, as surely as the earth attracts them. The further
grand secret concerning this force Eudoxus had unwittingly set out
to discover, with his planetary periods learned from the Egyptians,
and his three motions of the moon. For when this study was far enough
advanced, the necessary data were at hand for Newton, as he pondered
the mystery which had baffled Greece; and he was able, from the
moon’s motions, to verify his guess, that even the heavenly bodies
are in truth always falling, falling, towards one another, exactly as
Aristotle’s “smallest clod of earth” fell to the ground.

From the theory of the three simple motions, it obviously follows that
Earth must be at the centre of the world, that her particles must be
arranged in a spherical form round the central point, also the sphere
must be at rest.

But all are not agreed about this, says Aristotle. All who consider the
Universe finite say Earth is at the centre; but the philosophers in
Italy, the so-called Pythagoreans, on the contrary, say that in the
centre is Fire, and that Earth, which is one of the stars, is in motion
round the centre, and so causes day and night. They also assume a
Counter-Earth, merely from pre-conceived ideas, not from observation of
facts. And some agree about the Central Fire, from pre-conceived ideas,
because they think that the noblest should have the noblest place: fire
is nobler than earth, and boundary nobler than what is bounded, and
circumference and centre are both boundaries; therefore (they say) Fire
and not Earth is at the centre. Moreover, the Pythagoreans say that the
most important part of the Universe is the best guarded, and that the
centre is such a part, and they call it the Watch Tower of Zeus.

To these metaphysical reasons Aristotle replies that the centre is not
a true boundary, it is rather an end than a source, it is the material,
the limited, while it is the circumference which limits, encloses, and
gives the form. Besides, the centre of a thing is not necessarily the
centre of its being; as with animals the centre of their life (meaning
the heart) is not the centre of their body. So the philosophers need
not disturb themselves to put Earth out of her local centre, but they
would be wiser to examine that other centre of the Universe (meaning
the sun), and find out what is its nature and its place, for it also is
a point of origin, and noble.

He continues “Some also assert that though Earth is at the centre,
it is wound and moving round the axis which is extended through the
Universe, as is written in the _Timaeus_.” Plato’s actual words in the
_Timaeus_ will not bear this interpretation, as we have already seen
(p. 85). It is a little surprising that Aristotle does not mention the
names of Ecphantus the Pythagorean and Heracleides of Pontus in this
connection, since the latter was his contemporary, and perhaps the
other also, for they are mentioned together as teaching the doctrine
of Earth’s rotation on her axis.[40] Also Aristotle seems hardly fair
to either this or the Central Fire theory, in that he only answers the
metaphysical reasons of the Pythagoreans, and omits to mention that
either would unify the diurnal celestial motions in a much simpler way
than all his “unrolling” spheres. If he had not especially mentioned
that Earth’s motion was supposed by the Pythagoreans to cause day and
night, we should be inclined to think that he did not understand that
the period was twenty-four hours, and that its effect would be to
produce the apparent diurnal rotation of all the heavens.

[40] See p. 75, note.

The passage, however, has been a cause of endless controversy from
the earliest commentators of Aristotle to the present day, and such a
thorny question would have been avoided altogether in this book were it
not that it is actually quoted by Dante in the _Convivio_.

As to the size of the earth, Aristotle held that it was not a large
sphere, and small when compared with the stars. For, he says, if we
take quite a short journey to north or south, our horizon changes
markedly, so that the stars above us look quite different, and we do
not see the same stars; for some which are well seen in Egypt and near
Cyprus are not visible at all in northern parts, and those which in the
north are always in the sky, set when we go south. And therefore, he
adds, those who say that the regions near the Pillars of Hercules are
connected with India, so that the ocean is one, are not saying anything
altogether incredible; and their proof is that there are elephants
both in the extreme east and the extreme west. He does not mean,
evidently, that there was no sea at all between, but only that one
could quickly travel from one to the other, always going west: there is
no immense stretch of land or sea between west Africa and east India,
nor are they the extremities of a flat disc-like earth. Aristotle
tells us, moreover, that the mathematicians, who have tried to measure
the circumference of the earth, find that it is about 400,000 stadia.
This is the first time we hear of an attempt to measure the earth, but
unfortunately we do not know what stadium was used, nor what was the
method employed.

Aristotle’s Cosmos is arranged as follows:—

Upon the central spherical Earth rests water, and above this is air,
but these intermingle more or less, and are not sharply divided; in
the same way, though fire rises highest of the four elements, there is
not a distinct sphere of fire, but the higher part of the atmosphere
is chiefly composed of it. It is in this upper fiery atmosphere that
shooting stars are produced: hot and dry exhalations rising from Earth
take fire there, but are quickly consumed. Comets have their origin in
the same place, when large masses of vapour rise and are directly below
the sun (had the Greeks noticed that comets’ tails are always streaming
away from the sun?). Aristotle also explains that the Milky Way is
formed from these constantly-rising vapours, but under the influence of
the stars, for it always has the same position amongst them, and that
is where the most numerous and brightest stars congregate.

Thus within and below the fiery atmosphere constant changes are taking
place, and all things are perishable, but as soon as we reach the
lowest of the heavenly spheres, the moon’s, we enter another world.
All is changeless, eternal, divine. Motion is in circles, space is
filled with ether, the heavenly bodies as well as their spheres are of
an ethereal substance.

The Pythagorean idea of music made by the spheres, Aristotle dismisses
as very pretty but unfortunately not true. For if in truth these
immense spheres made a sound as they moved, even if we could not hear
it (as they said) we should feel it, for even earthly thunder bursts
rocks asunder! And there is no reason why they should make any sound,
for nothing moves out of one place: the spheres are simply rotating,
which is the natural movement for a sphere, unless it rolls along,
which they are not doing. If nature had wished the spheres or the stars
and planets to move forward, she would not have treated them worse
than terrestrial animals, in giving them no limbs by which they could
progress! The stars and planets have no motion themselves of any kind,
but are simply carried along by their rotating spheres, as we can
plainly see by the moon turning always the same face towards us: hence
they make no more noise than a ship’s mast set in a ship, or the whole
ship as it glides down a river.

Stress is laid both by Plato and Aristotle on this absence of any
motion of translation in the heavenly bodies and their spheres;
both insist that a movement of rotation, in which the moving body
continually occupies the same place, is the only movement existing in
the heavens. One wonders whether the spheres of Eudoxus suggested or
resulted from this idea.

Aristotle does not enter into detail about the separate planetary
motions, in any extant work, but explains as the general principle that
the outermost, the prime movement of the whole universe, is simple,
and the most rapid, while the inner are complex, slower, and in the
contrary direction; so that the planet nearest to the prime movement
(Saturn) is longest in making his own revolution, because most affected
by it, and the others less so in proportion to their distance. He
refers his readers to the mathematicians, and quotes the Egyptians and
Babylonians as having furnished satisfactory proofs of the relative
positions of the planets, by such observations as occultations of
other planets by the moon, which show that she is below them (_i.e._
nearer to us). The passage is quoted by Dante[41], in which Aristotle
describes how he himself once saw an occultation of Mars. “When the
moon was a half sphere, she passed beneath Mars and he disappeared
under her dark side, but came forth again on her bright illumined side.
And the same kind of thing,” he adds, “is reported to happen with the
other planets also, as those tell us who for a vast number of years
have made observations, viz. the Egyptians and Babylonians.”

[41] _Conv._ II. iii. 59-65.

There is one point which is elaborately discussed in the _De Cœlo_,
which seems very curious to us, but the main point must be noted here,
since it is of some interest to the Dante student. Aristotle tells
us that he considers the sphere of the Universe to have a top and a
bottom, and that the Pole which is not seen by us (the south) is at the
top. One cannot help thinking that Dante had this in mind when he chose
the southern hemisphere for the mount of Purgatory, whither, after all
their mistakes and wrong-doing on this underside of the earth, souls go
to purify themselves on the upper side, under the stars of the southern
pole.

In his book on metaphysics, Aristotle gives a very brief sketch of the
spheres of Eudoxus and his own “unrolling” spheres; and says that all
these planetary movements prove the existence of Essences, eternal
and immoveable themselves, who cause these movements. And it has been
handed down to us in a mythical way, from the most ancient teachers,
that these eternal Essences are gods. Above all these must be a First
Mover, the Primum Movens Immobile, who is one, eternal, and enjoys
for ever the kind of existence which we only experience in our best
moments. Upon this First Mover depend the whole heaven and all nature.

5. ARISTARCHUS.

With a system of revolving spheres accepted by the mathematical
astronomers, and sanctioned by the great philosopher Aristotle, it
may be thought that we are within sight of our goal, the system
of Greek astronomy which was to dominate the scientific world for
many centuries, including the age of Dante. But so exacting had
the careful observers become that the system of Eudoxus must be
completely transformed, by aid of two quite new hypotheses, before
it would satisfy their demands. Also, about half a century after the
consultation of Calippus and Aristotle in Athens, a strange new theory
was propounded, the boldest and strangest of all.

[Sidenote: Aristarchus _c._ B.C. 281.]

Nearly two thousand years before Galileo was summoned before the
Inquisition, and forced to recant upon his knees his “most damnable
heresy” that the earth goes round the sun, Aristarchus of Samos was
accused of impiety by his countrymen for the same crime. But he met
an even sadder fate than Galileo—neglect. His daring scheme was
almost ignored by his contemporaries, and but for a casual mention by
Archimedes and by Plutarch, we should know nothing about it.[42]

[42] It is also mentioned in a compilation of philosophers’ opinions,
probably made in the fifth century A.D. by Stobæus, who is very likely
quoting Plutarch.

He was, however, very famous as a mathematician, and also as an
observer. Ptolemy quotes his determination of the summer solstice of
the year b.c. 281, and this tells us the date at which he flourished.
He was renowned for a very ingenious method by which he tried to
discover how much further from us the sun is than the moon. When the
moon is half full the angle sun-moon-earth is a right angle, and if
the angle sun-earth-moon be measured, by pointing the astrolabe first
to sun and then to moon, the third angle, at the sun, may be computed,
and then the ratio sun-earth to moon-earth will be known. The method
is perfect theoretically, and if the sun were comparatively near, say
about ten times the moon’s distance, it would be practicable; but the
distance is really so much greater that the angle at the sun almost
vanishes, and a very small error in estimating it causes an error
equal to many millions of miles in the result. It is also impossible
to determine from looking at the moon the exact time when the division
between light and dark is a straight line. Aristarchus made the angle
at Earth 87° instead of 89° 50′, and this gave the sun a distance of
about 19 times as far as the moon, instead of 400 times, which is the
true value.

[Illustration: Fig. 20. Method of Aristarchus for finding the distance
of the sun.]

The foolish fad (as they thought it) about Earth’s motion, held by this
otherwise great man, is described quite clearly by the two writers
above-mentioned. He suggested that the stars might be immoveable, and
Earth be turning on her axis at the same time that she moves in a
circle round the sun. Moreover realizing all that this implies with
regard to the immense distance of the stars, he said that the circle
in which Earth revolves round the sun, compared with the sphere of the
stars, is as the centre of a sphere compared with its circumference.
That is, not only Earth, but Earth’s whole orbit, shrinks to a point
when compared with the infinite distance of the stars.

We have unfortunately absolutely no information as to the way in which
Aristarchus was led to these remarkable truths, and can only make
conjectures from what we know of his times. Evidently the Central
Fire theory was a suggestive step, and so was another theory which
was afloat about this time, and has been called the “Egyptian system”
on the authority of Macrobius in his commentary on Cicero’s _Dream of
Scipio_. No one really knows where it arose, but it is ascribed with
much probability to Heracleides of Pontus. According to this, the
two planets Mercury and Venus circled round the sun, and all three
together circled round Earth, which still remained the centre of the
Universe and of the other celestial motions. It was an idea which might
have occurred to any unprejudiced observer, since the oscillations of
Mercury and Venus from side to side of the sun are more striking than
their movements through the stars. They never go far from him in the
sky, like the other planets, and seem to belong to him.

Further, the clear understanding of the periodic motions of Mars,
Jupiter, and Saturn, and the accurate observation which had been
introduced by Eudoxus, must have revealed the fact that the loops in
the orbits of these planets are connected with the apparent movement of
the sun; and the varying brightness (especially noticeable with Mars)
is inconsistent with the assumption of unvarying distance from Earth.

The great importance of the sun, above all the other planets, had of
course always been recognised, and it is possible that Aristarchus
was struck by an unconscious suggestion in Aristotle’s advice to the
Pythagoreans to examine the nature and place of the sun, “that other
centre of the Universe,” for that was also a point of origin, and
a noble one.[43] His own work, too, would lead him to attribute a
commanding position to the sun, for as it was nineteen times as distant
as the moon, it must also be nineteen times as large (since they appear
equal), so he must have felt certain that it was a very great size,
probably much the largest of all bodies in the Universe.

[43] See page 101.

It was the grandest and truest of all the Greek astronomical theories.
But it was not accepted, it was hardly even discussed, and we can
scarcely be surprised at this. Such an improbable theory needed
many more convincing proofs than Aristarchus could bring forward.
Observation, calculation, comparison of theory with facts, this was
what was needed before safe ground could be won for belief, and in
this Eudoxus was a true pioneer. But freedom of thought and courage in
imagination is needed also in science, and Aristarchus seems to have
been almost the last to possess this.

[Sidenote: Seleucus B.C. 160.]

[Sidenote: Aryabhata born A.D. 467.]

A certain Babylonian, named Seleucus, who lived about B.C. 160, perhaps
on the Tigris, and an Indian astronomer, Aryabhata, of the fifth
century A.D., both taught that Earth turns on her axis, but Aristarchus
stood alone in suggesting that she also has a movement of revolution
round the sun. The possibility of Earth’s motion was alluded to but
seldom by classical and mediæval writers, and then almost always as a
foolish fancy hardly worth discussion;[44] until at last Copernicus,
hardly daring to publish his bold idea in sixteenth century Europe,
for fear of persecution, sought and found among the followers of
Pythagoras, and in Aristarchus of Samos, kindred spirits with his own.

[44] Indian astronomers also refused to accept the doctrine of
Aryabhata. Varâha Mihira (sixth century A.D.) says:—“Others maintain
that the earth revolves and not the sphere: if that were the case,
falcons and other birds could not return from the ether to their nests.”

6. THE SCHOOL OF ALEXANDRIA.

The spheres of Eudoxus would not work. The system was already
overladen, and more variations in velocity were becoming known.
Observation shows, too, that the planets, especially Mars and Venus,
vary greatly in brightness, in regular periods which correspond with
their movements, and this could not be accounted for by any possible
number or arrangement of spheres if all were to remain concentric to
Earth, so that every planet remained always at a uniform distance. Yet
Aristotle had said that Earth must be at the centre of the Universe.
The new philosophers of the Stoic school agreed with him, and among
mathematicians only Aristarchus dared to disagree.

But they were not long at a loss. The homocentric spheres[45] were
thrown aside, and during the third century two new hypotheses were
suggested (beside that of Aristarchus), and although we do not
certainly know by whom, there is little doubt as to the place in which
they originated. By a strange fate, Egypt, the home of astronomy many
centuries before, became the seat of the latest, most brilliant, and
most successful astronomical school of ancient times, and the knowledge
won there in the five centuries between B.C. 300 and A.D. 200 spread,
following in the wake of Alexander’s conquests, over the whole of the
civilized world.

[45] Spheres all centring in one point.

It was a purely Greek school, however. Greece lost her independence
under Alexander, and was finally crushed by Rome in 146 B.C., but never
was Greek learning and culture so much honoured and sought after as
in this age. In Egyptian Alexandria, Greek men of science found a
welcome, and opportunities of research which did not exist in any other
place in the world. In the Museum, founded and liberally endowed by
the royal Ptolemies, was a great library whose custodians were bidden
to obtain every book that had ever been written, and it is said that
when any stranger arrived with a new book it was taken from him and
copied for the Museum, and the copy returned to the owner. Within the
great marble colonnaded building were lecture-halls, and reading rooms,
and laboratories; there were gardens for botanists and zoologists, and
observatories for astronomers. These astronomers were all Greeks, and
though now living in Egypt they do not seem to have learned anything
more from the Egyptians. Perhaps the priests resented the intrusion,
and kept their secrets jealously to themselves; perhaps Eudoxus and
his immediate followers had learned all they had to teach. This seems
the more probable because, although the Greeks of this age did use
Babylonian records of eclipses to form a lunar theory, they complained
of the insufficient accuracy of all available planetary records.
Moreover, they were not hindered from learning astrology: the sacred
books of the Egyptians which taught its principles were translated into
Greek about 300 B.C.; but it was always treated by them as quite a
separate branch of study. Their geometrical methods were entirely their
own. They introduced a system of notation which greatly simplified
calculation[46]; their discovery of the principles of spherical
trigonometry inaugurated a new era in astronomy; and they invented a
new class of astronomical instrument.

[46] Mahaffy, _The Progress of Hellenism in Alexander’s Empire_, p.
119.

We know what these instruments were like, and it is even possible to
give an illustration; for, from descriptions in Ptolemy’s _Almagest_,
we find that the “well-made copper circles,” the gnomons, and the
celestial globes, which were set up in the Square Portico of the
Museum, were of the same pattern as instruments which existed in Pekin,
in the ancient observatory on the ramparts, until they were looted by
the Germans during the late Chinese war. The Alexandrian instruments
were not supported on their stands by beautiful bronze dragons, but on
the other hand the circles were more accurately divided, which after
all was of more importance from the astronomers’ point of view. Fig. 21
gives a general view of the Pekin Observatory, and Fig. 22 one of their
astrolabes dating from the 13th century A.D.

At first glance there seems to be here absolutely nothing like our
modern observatories. Ancient and mediæval astronomers had indeed no
telescopes, being ignorant of the properties of lenses: therefore they
were unable to study the features of any heavenly body except the
moon, and they had no way of finding out anything about their physical
constitution; but they had many ways of measuring their distances
and motions, and even the angular sizes of sun and moon, and their
instruments were the forerunners of our sextants, micrometers, and
transit instruments, our chronometers and sidereal clocks.

[Illustration: PEKIN OBSERVATORY. [_To face p. 114._

_From a photograph taken in 1888, and published in the “Bulletin de la
Société belge d’Astronomie”._]

The gnomon has been already described[47], and it was one of the most
valuable instruments used by the Greeks. The Pekin gnomon at the right
of figure 21 was more than 40 feet in height, and on the top had a
little plate of copper which was pierced by a hole as fine as the eye
of a needle: the observations made with this were much more exact than
observations of the end of the shadow, which must always be vague, and
the Chinese records of the sun’s movements made with this instrument
between 1270 and 1280 A.D. are of great value in modern research.
Ptolemy explains that his gnomon was made accurately vertical by the
use of a plumb-line, and that one way of testing the level of the
surface on which the shadow fell was to flood it with water.

[47] See p. 26.

Clepsydras, or water-clocks, were used by the Greeks, and many kinds
of sundials for telling the time by day. Tables were also made of the
risings of bright stars which served for clocks by night.

The instrument in the middle of the platform is a quadrant, and beyond
this on the left is a large celestial globe, which, however, only
dates from the Jesuit missionaries of the seventeenth century. Ptolemy
says that his globe was made the colour of the night sky; Sirius being
marked in his proper place, all other stars were placed relatively to
him, and in their own colours as nearly as might be; the Galaxy was
drawn, and the figures of the constellations outlined. The globe was
arranged to turn on either the poles of the ecliptic or of the equator;
circles of wood represented the horizon and meridian, and the pole
could be arranged at any altitude according to the latitude of the
place.[48]

[48] _Almagest,_ Bk. viii.

But perhaps the most interesting are the astrolabes, and these owe
their origin to the Greeks. The essential part of any instrument for
determining angular distances is a divided circle and a pointer: the
pointer is directed first to one object then to another, and the angle
between them is then read off on the circle. In the astrolabe, the
pointers themselves were also circles, provided with little perforated
rods for “sights.” (These are not visible on the instrument in Fig.
22, and have probably been broken off.) There were two fixed circles,
set in the plane of the ecliptic and perpendicular to it. Three other
circles could be rotated round the poles of the ecliptic. One of these
was directed (by means of the sights), to some body whose position
was already known, another to the body whose position was to be
ascertained, and the angle between them was read off on the ecliptic
circle; on the third the angular distance north or south of the
ecliptic circle could be read. This last and the ecliptic circle were
both divided into 360 degrees, and as many fractions of a degree as
space and skill would allow.

The equinoctial astrolabe was similar, but the fixed circle was in the
plane of the equator, instead of the ecliptic. One of each of these is
seen in the view of the Pekin Observatory.

But how did the old astronomers know how to find the ecliptic and the
equator in the sky, and set their circles in those planes? This they
did by means of the sun’s motion. The gnomon told them the day of
the equinox (see p. 25), and on that day the sun was in the equator:
therefore, if a circle was set up so that the shadow of the upper part
fell symmetrically upon the lower, with a little line of light each
side, it must be exactly in the plane of the equator. In the Square
Porch such a circle was erected, a large one of copper, and when once
correctly adjusted it was a standard plane, and also showed the date of
the equinoxes, as accurately as the gnomon itself. Since the ecliptic
is the path of the sun as seen in the sky, it is obvious that it could
be determined from a number of different observations of his position
at different times of the year.

[Illustration: A PEKIN ASTROLABE OF THE 13TH CENTURY, A.D. [To face p.
116

_From a photograph taken in 1888, and published in the “Bulletin de la
Société belge d’Astronomie”._]

Finally, accurate solar tables were drawn up, showing the sun’s
position in the sky in degrees for different dates, and then from these
it was possible to find the places of planets and stars. They could
not of course be compared directly, but the position of sun and moon
were compared during the day, when both were in the sky, and then after
dark the planets and stars were compared with the moon, allowing for
her motion among the stars in the meantime. Or secondly, when the moon
was eclipsed, and therefore known to be in the ecliptic and exactly
opposite the sun, the places of stars could be found directly.

This very brief description will give some idea of the chief
instruments and methods used, and when we see how very rough and
elementary they were, and remember that the Greeks had to work out
their observations without algebra, or decimal notation, we are amazed
at their results, and their far-reaching ambitions.

[Sidenote: Eratosthenes B.C. 276-194.]

Already in the very early days of the Museum, Eratosthenes, a
celebrated geographer, made a bold attempt to utilize observations
of the sun measuring the size of the earth. It was known that in
Syene (the modern Assuan) on the day of the summer solstice at noon
no shadows were thrown, and the bottoms of wells could be seen:
evidently therefore the sun was in the zenith. Eratosthenes found
that the sun’s distance from the zenith in Alexandria at noon on
the same day was 7° 12′, or one-fiftieth of the circumference of
the heavenly sphere, consequently the two towns must be distant
from one another (assuming them to be nearly in the same meridian)
one fiftieth of the circumference of the earth. The distance from
Alexandria to Meroe was known, and from Meroe to Syene had been paced
by the king’s professional pacers; the whole was 5000 stadia. 50 times
5000 = 250,000. The figure always quoted by the ancients is however
252,000. If the stadium used by Eratosthenes was the measure generally
used for long distances which have been paced, this estimate is equal
to 24,662 miles, only about 200 miles less than the modern value. It
was partly by luck that Eratosthenes got such a good result, for he was
evidently only working with round numbers, and the extra 2000 stadia
seem to have been added in order to make one degree equal to exactly
700 stadia. But in any case it was a highly creditable performance.

[Sidenote: Euclid _c._ B.C. 300.]

[Sidenote: Apollonius _c._ B.C. 270.]

There were celebrated mathematicians and geometers at Alexandria, whose
work was most useful to astronomy, such as Euclid, and Apollonius of
Perge. The latter is specially mentioned by Ptolemy in connection with
the new theory of “moveable eccentrics,” which was invented to account
for the varying brightness of the planets, as well as their peculiar
movements.

Fig. 23 explains this theory. Let P A be a great revolving circle upon
which Mars is fixed. (In the hands of the Alexandrian mathematicians
the spheres almost disappear, and they deal practically only with
circles.) If the earth were at its centre, as Eudoxus demanded,
Mars must always be at the same distance, but if we make the circle
eccentric to Earth, by putting its centre at C while Earth is at E,
then the distance and consequently the brightness will constantly vary,
and Mars will be brightest when at perigee P (point nearest Earth), and
faintest when in apogee A (point furthest from Earth).[49]

[49] Greek _peri_ near, _apo_ away from, _ge_ Earth.

[Illustration: Fig. 23. The Moveable Eccentric.]

But, as the Greeks had discovered, Mars attains his greatest brilliance
at different points of the zodiac, so P must be made moveable, and
it always happens when he is opposite the sun, therefore P must keep
pace with the sun’s apparent motion in the zodiac and P E always point
towards him. This was accomplished by making P C A turn round upon the
fixed point E, so that for instance when the sun had moved through a
quarter of his circle (in three months) P A had moved to P′ A′, and the
whole eccentric had moved into the new position shown in the diagram,
its centre C being now at C′. In other words, the centre of the
eccentric moves round Earth in the same time and in the same direction
as the sun, that is in one year, and “with the signs” (from west to
east).

At the same time, Mars is moving in an opposite direction on the
eccentric, and without entering into all the details of the problem,
we may add that the Greek geometers found that by determining the
proper relative sizes of the large and the small circle they could make
the two motions neutralize one another when the planet reached its
stationary points, and the retrograde motion prevail over the direct
when it retrograded. A similar arrangement was made for Jupiter and
Saturn.

In this very ingenious way the varying brightness as well as the
varying motions of these three planets were accounted for, without
violating the principle of uniform circular motion, and without
removing Earth from the centre of the Universe. She was also still
the centre of planetary motion, in a certain sense; but to place the
true centres of these planets’ spheres _outside Earth and in the
direction of the sun_ was a very suggestive step, and may well have
helped Aristarchus to his bold hypothesis. For he had only to put the
sun, not at some indefinite point along the line E A, but exactly at
the point C, and it became the centre of motion for Mars, Jupiter, and
Saturn, just as in the “Egyptian theory” it was the centre of motion
for Venus and Mercury. In this way he would arrive at the conception
of the sun circling round Earth and carrying all the planets with him,
(a theory which was held by the great astronomer Tycho Brahé in the
sixteenth century A.D.). Then a flash of insight may have revealed to
him the fact that this motion of the sun is apparent only, being but
the reflection of Earth’s own motion; for she is circling round the sun
like all the other planets.

It is, however, only a guess that the Moveable Eccentrics played this
part in the theory of Aristarchus. They did not long hold the field,
because they were not applicable to Venus and Mercury, which are never
seen in opposition to the sun. So they were thrown aside for another
system, the Epicycles, which illustrates much more simply the stations
and retrogressions of the planets, and can be used for them all.

Later on, when more irregularities of motion were discovered, it was
found necessary to combine eccentrics and epicycles, and by means of
this joint system it became possible at last to represent completely,
and as accurately as they could be observed, all the apparent movements
of the heavens. First, however, an immense amount of work had to be
done, and new methods devised, both in observation and mathematics.
The man who contributed most, in both ways, to make it possible, was
Hipparchus.

7. HIPPARCHUS.

[Sidenote: Hipparchus _c._ 140 B.C.]

Of this great man we know scarcely anything but what can be gathered
from the work he did, and this corroborates Ptolemy’s description of
him: “Hipparchus, lover of toil and truth φιλοπονον και φιλαληθεα.”
He lived about B.C. 140, since this is the date of the only book of
his still extant, and his work was not done in Alexandria, though he
may have studied there in his youth, and he used the Museum records.
We count him among the Alexandrians, as he belongs to this era, but
he seems to have been a private astronomer, who set up an observatory
of his own in Rhodes, his native place. Here we seem to see him,
surrounded by his primitive instruments and his papyrus books, patient,
eager, modest, seeking no fame and no reward but the joy of his work.
By day he would keep watch over the sliding shadow of his gnomon,
would write up his observations, make long calculations, and devise
new methods in mathematics, improve and modify his astrolabes and his
clepsydras; at night he would spend long hours with moon, planets, and
stars, making up for the defects and shortcomings of his instruments by
the skill and care with which he applied them to measure positions in
the sky. Nothing but the most loving and conscientious care could have
raised his work to such a pitch of accuracy, and made such rude means
suffice for such splendid achievements.

The book we possess, apparently an early one, is chiefly concerned with
the positions, the risings and the settings, of stars, and at the end
is a list of sixteen which came to the meridian at intervals of an
hour: from this list and the knowledge of spherical trigonometry which
he possessed, it would be possible to calculate the time at night to
within about a minute.

Hipparchus was able to construct a satisfactory theory of the sun
and to some extent of the moon, but he found more irregularities in
the planetary motions than Eudoxus had suspected. The records of his
predecessors were not accurate enough for him to construct a theory
for the planets, and he soon realized that one life-time would not be
long enough to collect all the data necessary, so, as Ptolemy tells us,
“Hipparchus, who loved truth above all things,” quietly set to work to
make as good and as many observations as possible, leaving it to his
successors to complete and explain them.

In the same spirit he undertook the laborious task, of which Pliny
speaks with awe as a presumptuous scheme, even for a god, “_rem etiam
Deo improbam_,” of numbering the stars. Pliny says he was led to do
this by the appearance of a New Star, which blazed out suddenly in
the constellation of Scorpio in b.c. 136, just as Nova Persei did in
Perseus in February 1901. He saw that even in the upper regions of the
eternal heavens, which Aristotle had supposed absolutely changeless,
changes may occur, and in order that even the least of these should
not pass unnoticed, he set to work to note the number, brightness,
and position of all he could see. This great catalogue of 1080 stars,
copied by Ptolemy in his _Almagest_, was the basis for all succeeding
catalogues, from Spain to Turkestan, until quite modern times. In it,
for the first time, the places of the stars were not merely described
according to their position in the constellation figures, but were
noted in degrees on the sphere, as is done to-day.

[Sidenote: Timocharis _c._ B.C. 280.]

One day, when comparing his notes with those of Timocharis, who had
worked at Alexandria about a century and a half earlier, he found that
the brilliant star Spica, the Ear-of-Corn which the Virgin carries
in her hand, had apparently moved nearer to the autumnal equinox by
about 2°. (Two degrees is about four times the angular diameter of the
sun). Of course he or Timocharis might have made a mistake, or Spica
might really be moving among the stars, or she might be carried along
with the rest by a slow movement of the whole star sphere. Apparently
Hipparchus satisfied himself that he could rely upon Timocharis’
observation, and took pains to verify his own; the second hypothesis
could be disproved by the fact that Spica does not change her place
perceptibly among her neighbours; and finally it became clear that her
motion is part of a slow apparent movement of the whole heavens.

[Illustration: Fig. 24. The movement of Spica.]

Here was a discovery of first importance, an unexpected reward of
patient accuracy, of which the white Spica, flashing down from summer
skies, may always remind us. Hipparchus had discovered the grand cycle
which we call the “Precession of the Equinoxes,” and before Spica
returns to the same position in which he saw her then, when she led
him to his great discovery, she will have been watched by generations
of astronomers for another twenty-four thousand years. No notice
of the cycle has been found as yet among the records of any other
nation, although it seems as if the astronomers of Babylon and Egypt,
and other countries where observations had been carried on for many
centuries, must have been aware of it. We can only imagine that at long
intervals of time they found that the stars had somehow changed, and
made corrections accordingly, but without understanding the nature of
the change. What Hipparchus thought about its cause we cannot tell:
probably he left all speculations to future astronomers, and confined
himself to noting the fact.

The displacement of Spica which he observed is shown in the diagram.

Both Timocharis and Hipparchus evidently measured her position
indirectly by comparing it with that of the moon, which was eclipsed at
the time,[50] and therefore known to be in the ecliptic and opposite
the sun. To find the sun’s distance from the equinox was an easy
matter, since his yearly course had long been carefully studied, and
the days on which he passed the equinoxes were regularly observed with
the gnomon. Spica, then, had moved eastward along an arc parallel to
the ecliptic, and since celestial latitude and longitude are referred
to the ecliptic, we may define her apparent movement in astronomical
language by saying that while her latitude had remained constant, her
longitude had increased by about two degrees; and further, as the
celestial equator is oblique to the ecliptic, this implied that her
declination (position north or south of the equator) had also varied.
The diagram shows that she had a less northerly declination than before.

[50] _Syntaxis_, book VII.; Delambre, _Histoire de l’Astronomie
Ancienne_ Vol. II. page 247 (1817 edition).

At this rate, Spica, which was now only 6° from the autumnal equinox,
would reach it in less than five hundred years, and thereafter would
lie east instead of west of it; and that she has in fact done so, may
be seen by consulting a modern star atlas. She is now 22½° of longitude
east of the autumnal equinox, and nearly 11° south of the equator. Her
south declination will continue to increase for about five thousand
years, after which she will come north again.

Ptolemy says that Hipparchus examined other stars, and found that they
also were increasing their longitude at what appeared to be the same
rate as Spica. The yearly amount of the movement, derived from the
Spica observations, is within a few seconds of arc of the true value,
which is 50¼ seconds; but Hipparchus would not fix any value until it
had been tested by further observation, and merely stated that it could
not be _less_ than one degree in a century _i.e._ 36 seconds per annum.

This is a very uncomfortable phenomenon for astronomers, since every
star is for ever changing its measured position on the celestial
sphere. Take three stars, one at the north pole, another on the
equator, and a third in the southern hemisphere. After some years,
the first will no longer be a pole star, the second no longer an
equatorial star, and the third may have so far increased its south
declination that it will be invisible at latitudes in our northern
hemisphere where formerly it used to rise above the horizon. One
compensation for this inconvenience is that if we know what star was
near the pole, or which stars lay along the equator, on any given
occasion, we can calculate the date. Thus it is believed that the Great
Pyramid was built when Alpha Draconis was the Pole Star, that is,
nearly 3000 years B.C.; and by a similar method Mr Maunder determines
the epoch at which the ancient southern constellations were invented,
as we have already seen.

The greatest inconvenience, and also the greatest historical interest,
attaches to stars like Spica which belong to constellations of the
zodiac, for if they are not stationary with regard to the equinoxes and
solstices they are not such simple guides to the length of the solar
year as the ancients supposed them to be. The scheme of the Babylonians
for beginning their month Nisan when the stars of Dilgan rose just
before the sun was an excellent one for a time, but if they had
continued it for many centuries they would have found that their year
was too long, and the months were all falling in the wrong seasons.
This has actually happened with Hindus and Parsis, who now keep their
New Year in the middle of our April, although when their calendar was
fixed, about thirteen hundred years ago, the years began at the spring
equinox. For the sun is like a runner in a circular race-course who
thinks he has completed a lap when he returns opposite a group of
spectators originally standing at the starting-point, but after several
laps he finds that the spectators and the goal no longer coincide;
either they, and also all the others surrounding the course are walking
away from it, or an unseen hand has been moving the flag towards him,
and so shortening the lap.

It is the flag which must count, in any case, not the spectators, and
with the sun it is the equinox which must count, and not the stars, for
this is the point at which he crosses the equator, making day and night
equal, and from this we count the beginning and ending of our seasons.
So our year is counted from equinox to equinox, and is twenty minutes
shorter than the “sidereal,” or star year, of the ancient Babylonians.
Hipparchus, from observations of equinoxes and solstices, made the year
365 days 5 hours and nearly 55 minutes, which is only 6 minutes longer
than the correct value.

[Illustration: Fig. 25. The Sun and the Equinox.]

After Hipparchus had made his discovery, astronomers agreed upon a
somewhat clumsy and very confusing device, by which the zodiac was
divided into twelve equal “signs” of 30 degrees, which bear the same
names as the zodiacal constellations, but whose beginning is always
reckoned from the vernal equinox. These twelve “signs of the zodiac,”
therefore, do not now agree with the twelve constellations of the
zodiac, and our present “first point of Aries,” which marks the vernal
equinox, is in the _constellation_ of Pisces.

What is the true cause of this strange phenomenon? Are the stars really
all in motion, or is it the equinox which moves?

The successors of Hipparchus, who believed that the stars were fixed
on a sphere, found no great difficulty in conceiving that this sphere
had a very slow easterly motion, round the poles of the ecliptic,
completing a revolution once in 36,000 years (_i.e._ one degree in
century). To us, however, it is impossible to believe that the stars,
which we have found to be at enormous and varied distances, are all
revolving at one rate, parallel to the ecliptic. The ecliptic, to us,
is simply the plane of Earth’s own orbit, and as she moves in it she
has a very slow “wobbling” motion on her axis, as well as the rapid
spinning of the diurnal motion, like the “wobble” of a spinning top.
The top has this motion because gravity is trying to pull it down from
its upright position; the earth because the sun is trying to drag her
slightly protuberant equator into the plane of her orbit.

The resulting motion is not a revolution of the earth, nor an apparent
revolution of the star sphere round Earth: what really happens may be
illustrated with the traditional orange and knitting needle.

Ignoring all motions but the one we are speaking of, let the points of
the knitting needle (Earth’s axis of rotation) trace out small circles
in space, and the equator of the orange will be seen to alter the
direction of its tilt, but without turning round (Fig. 26). Stick a pin
in the equator, and others in north or south latitudes, between equator
and pole; these will always remain facing you, but while the pole makes
its small circle, the equatorial pin will be seen to move up and down,
while the tropical and temperate pins trace out ellipses. These are the
movements which we see reflected in the stars; and if Earth’s diurnal
rotation were suddenly to cease, while her revolution in her orbit
and the movement of “precession” continued, we should see Spica, for
instance, sink slowly lower in the southern sky and after ages rise
again northwards, but there would be very little preceptible movement
east or west.

The movement observed by Hipparchus, then, was not a movement of Spica
and other stars, but a movement of the equinox. For the celestial
equator is simply a reflection of Earth’s equator in the skies, and as
it keeps changing the direction of its tilt in the way described, it
changes the point at which it cuts the ecliptic. This may best be seen
by taking two rings or hoops (two large curtain rings, for instance),
one of which just fits inside the other. Tilt the inner ring, so that
half of it is above and half below the other ring, and they touch
at two points, 1 and 2 (Fig. 27). The outer ring is the Ecliptic,
the inner the Equator, and where they touch each other are the
Equinoxes. Now move the inner ring, not sliding it round, nor making
any difference in the angle between the two, but simply so that they
touch at fresh points, 1′ and 2′. In this way you may make the points
of contact revolve entirely round. This is what the real equinoxes are
doing: while the equator opposite the group of stars in figure 25 rises
and falls, the equinox travels on, and finally returns to the same
place.

[Illustration: Fig. 26. The movement of Earth’s axis, which is the true
cause of Precession.

    _V_ is the vernal equinox, at the intersection
    of the equator and plane of the ecliptic: _APX_
    the earth’s axis, which always preserves the same
    inclination (23½%) to the plane of the ecliptic. As
    _APX_ slowly revolves round _T_ in the direction of
    the arrow, the vernal equinox is gradually shifted
    to _V_′, and so on.

(_From Young’s “Manual of Astronomy,” 1902._)]

The phenomenon is called “precession of the equinoxes,” because they
thus move on to meet the sun in his yearly course.

[Illustration: Fig. 27. Precession of the equinoxes.]

The discovery of precession is what has chiefly made Hipparchus famous,
but the invention of the astrolabe and of spherical trigonometry,
both believed to be due to him, his star catalogue, and his many
observations, more accurate than any made before, were so valuable as
pioneer work that Ptolemy justly called him the Father of Astronomy.
If Hipparchus could visit one of our observatories to-day, and see the
clock-driven equatorials, the transit instruments, the beautifully
divided circles read with microscopes, and the sidereal clocks, one
wonders whether he would be more astonished at the advance on his
astrolabes and clepsydras or at the homage paid to him as one in whose
footsteps all astronomers are proud to tread.

8. PTOLEMY.

    _Claudius Ptolemœus._ I know that I am mortal
    and ephemeral, but when I scan the multitudinous
    circling spirals of the stars, no longer do I touch
    Earth with my feet, but sit with Zeus himself, and
    take my fill of the ambrosial food of the gods.

For more than two centuries after Hipparchus very little original work
was done in astronomy, and no one seems to have had the courage to take
up his unfinished task and study seriously the difficult problem of
planetary motions.

[Sidenote: Posidonius _c._ 135 B.C. to _c._ 50 B.C.]

Posidonius the Stoic, who lived for some years in Rhodes, made a
fresh determination of the earth’s circumference, basing it not on
observations of the sun, like Eratosthenes, but of the star Canopus,
which in his time was just visible at Rhodes while in Alexandria it
rose “a quarter of a sign” (_i.e._ 7½ degrees) above the horizon. By
his method, the earth was a little smaller, (240,000 stadia instead
of 250,000), but it must have been difficult to measure the distance
between Rhodes and Alexandria over the sea, and it is impossible to
say when a star is exactly on the horizon. Posidonius also observed
the tides in the Mediterranean, and showed that “Ocean follows the
movements of the heavens,” and especially of the moon, having daily and
monthly periods.

[Sidenote: Geminus _c._ 70 B.C.]

A little later Geminus wrote an Introduction to Astronomy, which was an
excellent little book as far as it went, but although he was apparently
a native of Rhodes, and speaks of Hipparchus, he seems to know nothing
of his work, for he does not quote his careful determination of the
length of the year, nor his discovery of precession.

[Sidenote: Cleomedes _c._ 20 B.C.]

[Sidenote: Theon of Smyrna _c._ 100 A.D.]

Nor were other writers, such as Cleomedes and Theon of Smyrna, better
informed, and they added nothing new to the advance of astronomical
science.

[Sidenote: Ptolemy _c._ 140 A.D.]

But at last a worthy successor arose at Alexandria, the immortal
Ptolemy, whom Dante met in that Limbo of antique spirits which was
almost Elysium, although on the brink of the Inferno.

We do not know when Ptolemy was born nor when he died, nor where was
his native town: we only know that his first recorded observation was
made in the eleventh year of the Emperor Hadrian, that is A.D. 127, and
his latest in A.D. 150, and that he lived and studied in Alexandria. He
had splendid opportunities for carrying on the work of Hipparchus, for
besides the use of the instruments in the Museum Observatory, he had at
hand all the Museum records, which included the writings of Hipparchus.
Ptolemy was not so painstaking and accurate an observer as Hipparchus,
but he was a very able mathematician to whom it was evidently a joy
to handle figures and work out problems. By examining a number of
observations spread over several centuries, and combining them with his
own, he was able to accomplish the task in which so many others had
failed, and to frame a system which embraced all the celestial motions
then observed. The monumental summary in which he set forth this system
contains a great deal of interesting information about his methods
and instruments, and about the work of Hipparchus, for whom he always
expresses the most generous admiration. The original name of his book
was the “Mathematical System of Astronomy,” but his admirers having
called it the “Great System,” _Megiste Syntaxis_, the Arabs affixed
their article _al_ and gave it the name it preserves to this day, of
_Almagest_. It remained the standard treatise on astronomy until the
_De Revolutionibus Orbium Celestium_ of Copernicus appeared in 1543.

The name of his book indicates the scope of Ptolemy’s work. It was to
represent all the observed motions of the heavenly bodies by means of a
mathematical system, so that they became amenable to calculation; but
to explain the causes of these motions was thought to lie quite outside
an astronomer’s province. It was not for a mere observer and calculator
to determine which motions were real and which apparent, else Ptolemy
must have decided in favour of Earth’s rotation, for he says that
it would be much easier to account for celestial motions on this
assumption. Nor was it his business to investigate the substance of the
stars. Little did he dream that astronomers would one day solve such
problems, and uphold their conclusions in the face of all the world:
for him, as for his contemporaries, the decisions of the philosophers,
and especially of Aristotle, were final, and his task was to describe
what he saw in the light of their teaching.

He brings forward, in the introduction to his book, a few arguments
against the absurd notion, taught by some, that Earth is in motion,
turning on her axis or moving through space; but all that he proves
is the immense difficulty, even to a trained mind, of accepting these
theories, and the great authority of the philosophers who denied
them on abstract principles. Educated Greeks might still discuss the
nature of the heavenly bodies, as in Plutarch’s delightful dialogue
_On the Face of the Moon_, (written about half a century before the
days of Ptolemy), and might ridicule the law of gravity, laughing at
the absurdity of supposing that if the middle of a man’s body were at
the middle of the earth, his feet as well as his head would be “up,”
and that falling weights if they reached this point would stop short,
or oscillate to and fro. Yet even they all agreed that the fixed
stars do most probably move “in a circle of eternal and never-ending
revolution,” and that they are of a pure and eternal substance unlike
Earth; and for the professional astronomers of Alexandria these axioms
were assumed as the basis of all their work. Earth must be immoveable
at the centre of the Universe because the heavy stuff of which she is
made sinks necessarily to the centre and there remains in globular
form without need of support; the heavenly bodies must be in motion,
because, being of ethereal substance, it is their nature to revolve
eternally in circles.

This being granted, we can feel nothing but admiration for the extent
of Ptolemy’s knowledge, the comprehensiveness of his scheme, and the
skill and patience with which he overcame its difficulties.

Earth, according to Ptolemy, is but a point compared with the immense
surrounding sphere in which the stars are set, and this turns always
round us, communicating its motion (he does not inquire how) to sun,
moon and planets, so that day follows night, and the heavenly bodies
daily rise and set. For the slow movement of precession, which also
affects all heavenly bodies, Ptolemy accepted the least value of
Hipparchus, one degree in a century, only testing it in rather a
perfunctory way, which was a great pity, for he might have determined
it much more closely after an interval of some 250 years. But one man
cannot do everything, and he doubtless thought it best to spend more
time on the planets, whose intricacies had baffled Hipparchus and gave
him also a great deal of trouble.

He retained the great spheres which were supposed to carry them round
Earth, inside the star sphere, but the chief feature of his system is
the use of small spheres, which were fixed on the larger, and therefore
called “epicycles,” while the large were known as “deferents” or
carriers. The general principle of epicycles is very simple, as may be
seen by comparing the two diagrams.

Fig. 28 shows the path of Mars as we saw it among the stars of Pisces
in the year 1909. Throughout July the planet was travelling in its
usual direction, “with the signs,” but on August 22nd it came to a
stop, then turned and travelled backwards “against the signs” until
October 26th, when it stopped again, reversed its direction once more,
and during the rest of the year moved rapidly forward.

Fig. 29 shows the principle on which Ptolemy would have explained this
curious track. Each planet was supposed to be fixed on a small circle,
the epicycle, and this was fixed upon a large circle, or deferent, upon
which it travels in the direction shown by the arrow, at a uniform
speed, returning to the same place in the sidereal period of the
planet. Thus Mars, as seen from Earth, which is near C the centre of
the deferent, makes a great circle through all the zodiac in two years,
Jupiter in twelve, and so on. But meanwhile the epicycle is rotating
round its own centre, C1, and when the planet reaches the point
marked S, the two motions neutralize one another, so that it appears
stationary, as Mars did on August 22, 1909. After this, the motion of
the epicycle more than counter-balances the motion of the deferent, and
the planet seems to reverse its direction until it reaches the point on
the epicycle marked S′. After this the two motions are once more in the
same direction, so the planet is seen to move rapidly forward, as Mars
did after October 26.

[Illustration: Fig. 28. The Path of Mars among the Stars, 1909.]

[Illustration: Fig. 29. The Epicycle.

Ptolemy’s method of accounting for movements such as those shown in
Fig. 28.]

[Illustration: Fig. 30. Planet retrograding, and in opposition (_i.e._
in opposition to the sun.)]

In this extremely ingenious way the strange planetary oscillations were
accounted for, without violating the law of uniform circular motion,
and in a more convenient and satisfactory way than by the concentric
spheres of Eudoxus or the moveable eccentrics of Apollonius. Each of
the five planets was provided with an epicycle and a deferent, and
these were made of the proper relative size and given the right speed,
so that the motions should correspond with what we see in the sky.

Ptolemy calls the oscillation a planet’s “anomaly with regard to the
sun,” because (as we have seen when discussing the moveable eccentrics)
it was known to be connected in every case with a planet’s angular
distance from the Sun. On September 24, when Mars was in the middle
of his retrograde arc in Pisces, the sun was exactly opposite, in
the constellation of Virgo. This is found to be always the case, not
only with Mars, but with Saturn and Jupiter too. Whenever one of
these planets has the position O on its epicycle, and therefore is
retrograding, the sun will be found to be exactly opposite in the
sky. Mars comes into this position, and is opposite the sun, once in
780 days; this, therefore, Ptolemy called the period of his epicycle,
while a little less than two years was the period of the deferent. The
two periods of Saturn are 378 days and 29½ years nearly; of Jupiter 399
days and nearly 12 years.

[Illustration: Fig. 31. Venus, Mercury, and the sun.]

Venus and Mercury betray their dependence upon the sun in more striking
fashion, for since these planets simply oscillated from one side to the
other of the sun, their epicycles must be supposed to be keeping pace
with him all the way round the zodiac. Fig. 31 shows their relation
to one another. On September 25, 1911, Mercury was seen from Earth as
a morning star as far west of the sun as it is possible for him to
travel, while Venus, after shining as an evening star all the summer,
had come into line with the sun and become invisible.[51] On December
7 following, the rotation of the epicycles (Ptolemy would say) had
brought both planets to new positions, Mercury now being an evening
star at his “greatest elongation east,” and Venus a morning star. But
the centres of the two epicycles always remain in a line with one
another and the sun, and so their periods on the deferents are the same
as his, viz. one year. The epicyclic periods, or intervals between two
“greatest elongations” west or east, are 116 days for Mercury, 584 days
for Venus.

[51] Venus had passed her “inferior conjunction with the sun” on Sept.
15.

We still explain the complicated course of the planets by resolving
it into two approximately circular motions, but we know now that only
one belongs to the planet itself, the other is Earth’s own motion. The
reason why the sun’s position affects the position of every planet
is simply that the epicycles of Mars, Jupiter, and Saturn, and the
deferents of Venus and Mercury are reflections of Earth’s yearly
journey round the sun.

Ptolemy had by no means finished with the planets when he had provided
each one with an epicycle to represent the “anomaly with regard
to the sun.” Hipparchus had noticed that there was another lesser
irregularity, which seemed to be periodical likewise, although no
former system had taken it into account; and this he called the
“anomaly with regard to the zodiac,” because the speed of each planet
and the amplitude of its loop varies slightly according to the part of
the zodiac it happens to be in. He suggested that this might be dealt
with by combining the two theories of epicycles and eccentrics, and
this suggestion Ptolemy adopted with success. He placed each deferent
with its centre not exactly at the earth, but at a certain small
distance which was different for each planet. (This is not shown on our
small-scale diagrams, therefore Earth appears at the exact centre.)
The true explanation of this irregularity is that each planet’s path
is not strictly circular, but elliptical.

Besides this, Ptolemy had to represent the planet’s movements north and
south (note how this varies in fig. 28.). This was partly managed by
the aid of small wheels, rotating in such a way that they lifted and
lowered the epicycle as required.

Although Ptolemy quotes Babylonian observations of lunar eclipses
dating back as far as the eighth century B.C., the oldest planetary
observations that he uses were made only four hundred years before his
time, and they were probably Greek. Even these were generally very
rough. For instance:—

    In the 496th year of Nabonassar, on the 17th day
    of Choeac, in the morning, Mercury was three
    moon-breadths north of the tail of Capricorn.

    In the same year, Phamenoth the 30th, Mercury was
    three moon-breadths south of the horn of Taurus
    which is also the foot of the Charioteer.

The first year of Nabonassar (a Babylonian epoch) corresponds with
B.C. 747, so the 496th year is B.C. 251. The months used by Ptolemy
are usually Egyptian. The later observations, made with an astrolabe,
were much more precise; Ptolemy quotes one from Theon of Smyrna, which
states that Mercury was 3° 50′ in advance of the Heart of the Lion
(Regulus), and for his own observations he also usually gives the sign,
degree, and minute. Sometimes the planets had been observed so near
stars that their positions could be very accurately determined by the
aid of Hipparchus’ star catalogue. Timocharis, on a certain morning in
the 13th year of Ptolemy Philadelphus (B.C. 273), saw Venus beside the
last star in the wing of the Virgin (Beta Virginis); Ptolemy himself
saw her so close behind a certain star in Aquarius that she seemed
to touch it with her rays; and in the 83rd year after the death of
Alexander, Jupiter had been observed to eclipse the Southern Ass, that
is the southernmost of the pair of stars on either side of the little
cluster in Cancer which the ancients called the Manger.

[Illustration: Fig. 32. The sun’s Deferent. (Its eccentricity is
exaggerated).]

The sun was much easier to manage than the planets. Moving always
in the same direction, he needed no epicycle, and remaining always
in the ecliptic, no wheels. His one irregularity, the varying speed
in different parts of the zodiac, which Meton had discovered and
Calippus confirmed, Hipparchus accounted for by placing him on
eccentric deferent; Ptolemy adopted this without change, and made the
year the same length as Hipparchus had done. As the slowest motion
was observed when the sun was in the sign of Gemini (at the time of
year corresponding with the end of our month of May), the eccentric
was placed as in the diagram; and the sun, supposed to be revolving
uniformly round the centre C, had a slower motion as seen from Earth
when he went north, because he was then more distant.

We know now that it is really Earth which revolves round the sun, but
the elliptical shape of her orbit is not unlike the eccentric deferent
which the Greeks gave to the sun, and they guessed quite rightly from
his motion that he is further from us in one part of the year than
in the other. This is easily proved to-day by the fact that he grows
apparently smaller, as shown by photographs or measurements taken at
different times of year, but the change was not perceptible to the
rougher methods of the ancients.

It was a pity that Ptolemy did not take more pains to verify the work
of Hipparchus on the sun, for the sun’s “apogee,” or point of greatest
distance from Earth, has an extremely slow motion among the stars which
ought to have been quite perceptible in nearly three centuries;[52] but
here again he seems to have thought that where Hipparchus had given so
much attention he might pass on to something else, and so missed an
interesting discovery.

[52] It amounts to 5 degrees of longitude in 300 years.

[Illustration: 1911, July 1. 1912, Jan. 3.

Fig. 33. Apparent Variation in the Size of the Sun. [_To face p 146_

Two half photographs of the Sun, taken at Kodaikanal Observatory. The
smaller was taken two days before apogee, the larger on the day of
perigee.]

Instead, he worked very hard at the moon, and added another periodical
irregularity to those already known. Perhaps his feelings were somewhat
mixed when this happened, his pride and pleasure in his important
discovery being counter-balanced by the consciousness that it would
still further complicate his lunar theory. Our satellite is acted
upon by ourselves as well as by the sun, so that she suffers many
perturbations, for Earth, though so small compared with the sun, is
comparatively near. Ptolemy’s discovery was a difference in her speed
at full and new, as compared with her intermediate phases, and this
periodic difference is called by modern astronomers the “evection.” It
was already known that her nodes, or the points at which she crosses
the ecliptic, are in constant retrogressive motion, just like the
equinoctial points, where the sun crosses the equator; but the moon’s
crossing points, instead of taking thousands of years to circle the
zodiac, run round in about eighteen years. This was discovered early,
because observations were chiefly made during eclipses, and at these
times the moon is always at a node, that is to say, she is crossing the
ecliptic, the sun’s path; otherwise the eclipse could not happen. It
was also known that she has a varying speed in the zodiac, and that her
apogee, where the motion is slowest, instead of being apparently fixed,
like that of the sun, also runs round the zodiac, but with a direct
motion, and in a period of about nine years.

We need not enter into all the details of Ptolemy’s arrangements for
the moon, which are exceedingly complicated, but it is interesting to
note that he does not explain her varying velocity by an eccentric, as
with the sun and the planets. She has an eccentric, but Ptolemy needed
it for representing his own discovery, the evection, so he gave her an
epicycle, using it in quite a different way from the epicycles of the
planets. This epicycle also revolved while moving on the eccentric, but
in the opposite direction, and there was so little difference in speed
between the two motions that it never brought the moon to a stop, nor
reversed her direction, but simply increased and retarded her motion
alternately during her monthly revolution. Thus, when the moon was at
M, in what Ptolemy called the upper apsis (or arc) of her epicycle, or
as we should say in her apogee, the motion on the epicycle was contrary
to her motion on the eccentric, and made it seem slower. When the
epicycle had travelled halfway round the eccentric, it had also made
nearly half a revolution on its own axis: consequently the moon was at
M¹, near the lower apsis, or perigee, and the motion on the eccentric
seemed to be accelerated.

[Illustration: Fig. 34. The moon’s epicycle and deferent.]

The slight difference in speed between the two motions accounted for
the continuous displacement of the apogee in the zodiac, as may be seen
from the diagram. For suppose that when in apogee at M the moon is seen
from Earth among the stars in the middle of Taurus. At the return of
the epicycle to this place next month, she has not yet quite completed
a revolution on her epicycle but is at M², and will not be in apogee
until the centre of her epicycle is advanced 3° further in Taurus.
After some time (five months), apogee does not occur until the moon is
in Gemini, and it will be nine years before it occurs again in Taurus.

These are the leading features of the system by which Ptolemy
represented the motions of the planets, the sun, and the moon. He is
uncomfortably conscious that it may strike us as very complicated, and
in his last book he makes a kind of apology. We must remember, he says,
that we are not dealing with earthly machines which jar and wear, but
with celestial bodies which have no weight, cause no friction, and are
eternally the same. Delambre somewhat cruelly retorts that he need not
have wasted time in writing such rubbish, but have been content to put
his results into tables.

But what true astronomer could be content with tables and nothing
more? He must try to understand their significance. Nearing the end of
his great work, which had cost so much labour, and was so brilliant
a success within its limits, Ptolemy allows us to see in this little
paragraph that he felt he had but touched the hem of Nature’s veil,
and longed in vain to lift it. The circles which he manipulated so
skilfully were only mathematical abstractions to him:[53] what was
the reality behind? What were the stars? whence came their unwearied
strength, their eternal calm? The power of Egypt, of Assyria, and of
Persia had declined, Greece was now laid low, and it was the day of
Rome, but still Venus pursued her ancient path among the little stars
of Aquarius, and when all faded in the solemn dawn, the sun arose in
his ancient majesty. Then Ptolemy looked at his circles and triangles,
and felt how inadequate they were; yet it was the nearest approach he
could make to truth.

[53] This is evident from the way he treats them, picking up an
epicycle or a deferent just as best suits the purpose in hand and
explaining sometimes that either would answer equally well.

They did indeed represent very beautifully the celestial motions, and
also in a general way the variations in brightness or diameter, but
their unreality is betrayed by the fact that the latter was often
grossly exaggerated, as for instance with the moon, whose epicycle had
to be so large, in order to represent her motions, that at perigee
she ought to appear twice as large in diameter as at apogee! Ptolemy
can hardly have failed to notice this, though he does not mention it.
With regard to the planets, he is careful to tell us that he knows no
way of finding their real distances: the ratios of their epicycles
to their deferents he had carefully computed in each case, but to
estimate the diameters in stadia was utterly beyond his powers. He
believed, however, that all were very much nearer than the stars, and
more distant than the moon, and he found universal agreement among
astronomers in placing Saturn, Jupiter, and Mars, beyond the sun, in
order of the periods of their deferents, which were all longer than
a year. Venus and Mercury, however, had periods the same as the sun:
on which side of him, then, must they be placed? Some modern authors,
Ptolemy says, thought they were beyond the sun, but he agrees with
the most ancient, and places them between sun and moon. The general
disposition of the heavenly bodies according to Ptolemy is shown
in Fig. 35, with the periods of epicycles and deferents, and the
directions of the several movements, but epicycles and spaces between
deferents are all made equal. It will be noted that the sun has no
epicycle and that the moon turns on hers in a reverse direction, that
the centres of those of Mercury and Venus are in a line with the sun,
while the lines joining Mars, Jupiter, and Saturn to the centres of
theirs are in each case parallel with the line joining Earth and Sun.
Beyond all the wandering stars is the sphere of the fixed stars, moving
in its vast period of 36,000 years; and the whole system is carried
round Earth in one great revolution of a day and a night.

[Illustration: Fig. 35. The Ptolemaic system.

The arcs of circles are portions of the deferents which carry round the
smaller circles (the epicycles) in the periods named at the side.]

And here we have to note one astonishing fact. Although the planets
were all beyond reach of measurement, it was not so with the moon,
which the Greeks rightly recognized as our nearest neighbour. They had
actually achieved the long-desired feat of measuring her real size and
distance.

The fundamental principle on which they worked is easily explained. If
we look out of a window at a tree in the garden, it appears against
a background of some distant scenery; if we alter our position by
walking to another window, the tree changes its place with regard to
the relatively motionless background, and its apparent change of place
bears a definite proportion to its distance from us. A tree near the
house changes its position greatly, a tree at the far end of the garden
much less. From careful measurements of the angle through which the
tree appears to have moved and the distance we have walked between the
two windows, the distance of the tree from the house may be easily
deduced.

The tree in the garden is the moon, the distant landscape is the
star-spangled sky. The space between the two windows must be increased
to thousands of miles, but the astronomer need not walk it. If he makes
his first observation when the moon has just risen, carefully measuring
her position among the stars, the revolving Earth will carry him to
a new position in a few hours, when with the moon high in the sky he
can once more compare her position with the same stars, and from the
change he then finds he can deduce her distance. The Greeks thought
it was the sky, and not the earth which moved, but this makes no
difference, as the question is one of relative motion only.

The problem, however, when applied to the moon, is a complicated one,
implying not only skill in trigonometry and the possession of suitable
instruments for measuring the necessary angles, but also accurate
knowledge of the size of the earth and the motions of the moon, since
her progress eastward among the stars during the interval between the
two observations must be allowed for. Hipparchus and the astronomers
of Alexandria were the first to qualify themselves for attacking this
difficult problem, and the proof of their success is that Ptolemy’s
value for the distance of the moon is very near the truth as obtained
by modern methods. The same method is now used, only it is found
better to do what was not possible for the Greeks, namely to compare
observations made at places several thousands of miles distant, for
instance Greenwich and the Cape, instead of allowing the same place to
be moved by the earth’s rotation.

When the distance of the moon had been thus discovered, it was a very
simple matter to find her real size from her apparent size.

The distances of all other heavenly bodies are too great to be
determined by naked eye methods, for the displacement (technically
called “parallax”) is so small that it is quite invisible without a
telescope. Hipparchus had tried repeatedly to measure the distance of
the sun, but recognized that neither the method of Aristarchus nor any
other was really conclusive. It was, however, the best attempt that had
been made, and Ptolemy assumed that he knew the distance to be about
twenty times that of the moon, so he gave the sizes and distances of
these two bodies as follows, taking Earth’s semi-diameter as unit:—

              According to Ptolemy.            Modern Values.
          _Semi-diameter.   Distance.   Semi-diameter.   Distance._
    Moon   ⁵/₁₇ (=0·290)      59               0·273       60·3
    Sun     5½               1210              109·4      23,439

In Books VII and VIII of the _Almagest_ Ptolemy describes the
48 ancient constellations and the course of the Milky Way among
them. The position of each star is noted as it appears in its
constellation-figure, but the celestial latitude and longitude is also
given, and this great catalogue is evidently taken from Hipparchus.

        PTOLEMY’S FORTY-EIGHT CONSTELLATIONS.[54]

    [54] Taken from the _Almagest_ catalogue, as given in Delambre’s
    History.

        Ursa Minor                Pegasus
        Ursa Major                Andromeda
        Draco                     Triangulum
        Cepheus                   Cetus
        Auriga                    Orion
        Corona Borealis           The River (Eridanus)
        The Kneeler (Hercules)    Lepus
        Lyra                      The Dog (Canis Major)
        The Bird (Cygnus)         Canis Minor (Procyon)
        Cassiopeia                Argo
        Perseus                   Hydra
        Böotes                    Crater
        Ophiuchus                 Corvus
        Serpens                   Centaur
        Sagitta                   The Wild Beast (Lupus)
        Aquila                    Ara
        Delphinus                 Corona Australis
        Equuleus                  Piscis Australis,
        Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpio,
        Sagittarius, Capricornus, Aquarius, Pisces.

Ptolemy sometimes remarks on the colour of the brighter stars, and
always mentions the brightness, or “magnitude” as we call it now, for
the classification of Ptolemy (or Hipparchus?) was found convenient
and accurate enough to be retained by modern astronomers, and the same
system is now continued with the faint telescopic stars.

The stars ranked as “first-magnitude,” or brightest of all, are fifteen
in number, and as these are evidently the “quindici stelle”[55] alluded
to by Dante in _Par._ xiii. 4, it will be interesting to give a list of
them here.

[55] Fifteen stars.

                   PTOLEMY’S FIRST-MAGNITUDE STARS.

      _Name and Description            Modern Name and Meaning._
        _in the Catalogue._
     1. Arctouros, fire-coloured       Arcturus (the Bear-Watcher).
     2. The brightest star in the      Vega, in Lyra. (Falling Eagle,
          Lyre                           Arabic).
     3. Aichs (the Goat)               Capella (Little Goat, Latin).
     4. The brightest star of the      Aldebaran. (The follower,
          Hyades, fire-coloured          because following the
                                         Pleiades, Arabic).
     5. Basiliskos (the royal), on     Regulus (Latin equivalent).
          the heart (of Leo).
     6. The tip of the tail (of Leo )  Denebola (Lion’s Tail, Arabic).
     7. Stachys (Ear-of-Corn)          Spica (Latin equivalent).
     8. The last of the Water and      Fomalhaut. (Mouth of the
          Mouth of the Southern          Fish, Arabic).
          Fish.
     9. The fire-coloured bright       Betelgueux. (Shoulder of
          star on the shoulder           Giant, Arabic).
          of Orion.
    10. On the left foot of Orion,     Rigel. (Foot of the Giant,
          common to the Water.           Arabic).
    11. The last of the River          Achernar (Arabic equivalent).
    12. The very brilliant star,       Sirius (Greek _Seirios_, or
          fire-coloured, at the          _Sothis_, from the Egyptian
          mouth of the Dog,              _Sept_),in Canis Major.[56]
          called the Dog.
    13. Prokuon (the preceding         Procyon (in Canis Minor), which
          Dog), on the thigh of          rises before Canis Major.
          Prokuon
    14. Canopos, on the rudder         Canopus (an Egyptian god, and
          (of Argo)                      a town on the Nile delta).
    15. The tip of the right forefoot  Alpha Centauri.
          (of the Centaur).

    Altair and Antares were counted as second magnitude,
    though we now class them among the first.

[56] Lockyer, _Dawn of Astronomy_, p. 196.

It will be seen from the above that only two of these fifteen
brightest stars of Ptolemy’s still bear their Greek names, Arcturus,
and Procyon; but most of the other modern names are direct translations,
either into Latin or Arabic, of Ptolemy’s description or name for the
star, while Vega and Aldebaran have preserved their original Arabic
names (much corrupted), and Canopus and Sirius are derived from the
Egyptian. Regulus-Basiliskos apparently comes from Babylon, for the
name of this star on tablets of the second century B.C. was Shar-ru,
which means “royal.”

Ptolemy’s “Last of the River” was taken by his Arab and other
commentators to be the first-magnitude star which consequently they
named Achernar, but this was too far south to be visible at Alexandria,
though possibly he heard of it or saw it himself at Syene, where it
rose over the horizon in his times. Delambre thinks it is the same as
Fomalhaut, which already belongs to two other constellations in the
star catalogue; Brown suggests it was Theta Eridani, which may have
been brighter sixteen centuries ago. Perhaps it was the star we now
call Alpha in the Phoenix, for this lies between the last bend of the
River and the Water of Aquarius, and its magnitude is between first and
second. The stars of the Southern Cross are included by Ptolemy in the
stars about the hind feet of the Centaur, though it is difficult to
identify each one with certainty: the positions are not very accurately
given.

Ptolemy wrote also on Astrology, or as it was then called Judicial
Astronomy, and his work on the subject, which is in four books,
received the name of the _Tetrabiblios_. It seems that astrology was
not very much believed in by the Greeks, for he protests that the
influences of the stars are real, and can be known and predicted, but
allows that this part of astronomy is much more difficult than the
mathematical part, with which he had dealt in the _Syntax_, and that
it has not yet been perfected, and therefore it is sometimes slandered
as untrue. Astrologers sometimes make mistakes, like doctors, yet both
astrology and medicine are useful arts. He mentions especially the
Egyptians as practising it.

Only second in importance to the _Almagest_, and even better known in
old days, was Ptolemy’s work on Geography. In this he shows how the
size of the earth and the latitudes and longitudes of places on Earth,
can be discovered by observations of the heavens. For Earth’s diameter
he adopts the value found by Poseidonius. His terrestrial globe had a
moveable half-circle attached to the poles, which was divided into 90°
from the equator in both directions, so that by placing this against
any spot on the globe, its latitude north or south of the equator
could be immediately read off. Only half the equator was marked on
this globe, and divided into 180°, for the known part of the earth
all lay within these limits. The parallels of latitude marked were
not the same as we use, though the most northerly fell very close to
the Arctic Circle: it showed the latitude of the Island of Thule, far
away “towards the Bears,” as Ptolemy expresses it, meaning that it lay
under the constellations near the North Pole. The most southerly was
not far below the equator, and others were marked between these which
divided the known Earth into “climates,” according to the height of the
Pole and the length of the longest day in each. Among the many towns
which figured on this globe we may mention Alexandria, Rome, Athens,
Jerusalem, Florence, Cadiz, Paris, Strasburg, London, Bath. Meridians
were marked at every 5 degrees. From this globe Ptolemy shows how a
plane map may be constructed. He took Rhodes as the central meridian,
because it had a nearly central position among the “climates,” and was
the place in which Hipparchus had observed.

Ptolemy also wrote books on Optics, on the theory and construction of
Dials, and on Music!

V. RETROSPECT.

The _Almagest_ is the last word of Greek astronomy. We have seen how
from the dawn of Greek history, as far as we can trace it back in
literature, the Greeks were familiar with the skies, and how the most
ancient of their philosophers had tried, chiefly by abstract reasoning,
to discover the cause of the circling motions—never changing, never
ceasing—of sun and stars. When the seemingly unruly movements of the
planets were known, they still felt sure that there must be some
underlying principle which would bring all into harmony. Inspired by
Plato, and helped by the Egyptians, Eudoxus took the first step towards
finding this principle by careful study of the planetary motions,
and at last, after many generations of observers and mathematicians,
Ptolemy was able to describe these motions as accurately as was
possible with the methods available.

At first the heavenly bodies had been thought of as gods, then as
worlds like our own, then as spheres of ethereal fire. They were
swept round by a mighty wind, they ran on wheels, they floated in the
ether, they were set in crystal spheres; the controlling force was
the principle of number or harmony, an all-pervading World-Soul, a
host of immaterial intellectual Beings subject to one eternal First
Mover. It was also suggested that the greater part of the motions were
apparent only, that Earth was really in motion, spinning on her axis,
revolving round a Central Fire, or revolving round the Sun. But these
ideas had not enough evidence to support them when suggested. Had a
great imaginative thinker, a Pythagoras or an Aristarchus, arisen
after Ptolemy, he could have shown in detail how by assuming these
two motions the phenomena could be more simply accounted for, and he
could have made out a very good case for their probability.[57] But the
time was past then for such bold originality, and the explanation of
Aristotle was universally adopted. He, as we saw, placed the abode of
the gods, the rulers of the universe, beyond the outermost sphere, and
found the principle for which all were seeking, which should be the key
to all celestial motions, in the law of circular motion.

[57] The changing latitudes of the planets, for instance, which gave
Ptolemy much trouble, are much more easily explained when it is granted
that they partly depend upon Earth’s motion in an orbit whose plane is
slightly inclined to the planes of their orbits.

The Greeks had at first thought that the earth was a disc, under a
tent-like sky; then it was a cylinder, with the sky in tiers above;
then a huge hemisphere filling half the universe. But they discovered
that it was a sphere, surrounded on every side by the heavens; and they
found its true size, and that the portion of it which was known to
them was less than a quarter of its whole extent. They explained the
marvel of Earth’s remaining unsupported in space by the known facts of
gravity, arguing that the falling of every particle of earth towards
Earth’s centre proved that it was also the centre of the Universe, and
that every heavy thing tended thither by a law of nature.

They also knew the size and the distance of the moon; they realized
that the planets were all immeasurably remote, and the stars vastly
more distant still. They believed that there were great intervals
between the planets, corresponding to the differences in their periods
of revolution, but the stars were always thought of as set in a great
sphere, and therefore all at the same distance from Earth at its
centre. The relative positions of the stars on this great sphere,
and times of rising and setting of many of them had been known in
a rough way for many ages, through the familiar appearance of the
constellations, named we know not by whom; but Hipparchus measured
their positions in degrees, and was able to foretell where any star
would be in the sky for any place or time. He also discovered the
Precession of the Equinoxes, and this was thought to prove a slow
rotation of the star sphere.

Astrology was learned by the Greeks in Egypt and the East, but was
never practised with the enthusiasm shown by their teachers.

As regards the division of time, the Greeks adopted from the
Babylonians the twenty-four hour day which we still use, and their
months began with the young moon. Their year therefore had to contain
a whole number of months, and it had sometimes twelve, and sometimes
thirteen, as with the ancient Babylonians, but the system by which
this was arranged was quite different, as they did not depend upon
observations of the stars, but counted the number of days between
equinox and equinox by means of gnomons. Great difficulties were
found in trying to reconcile the lunar and solar periods. The ancient
12-month year, used in the time of Hesiod, was found to be much too
short, and alternate years of 12 and 13 months made the period too
long; then an eight-year cycle was invented, but this had to be
constantly corrected, which led to great confusion. As we have seen how
carefully Meton and Euctemon had determined the length of the tropical
year, even before the days of Eudoxus, surprise may be felt that a
calendar year was not fixed to correspond accurately with the movements
of the sun, ignoring the irreconcileable movements of the moon, but it
is difficult for us to realize in these days how wrong and strange it
seemed if a new moon occurred in the middle or at the end of a month,