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.