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PURCHASED FOR THE
UNIVERSITY OF TORONTO LIBRARY
FROM THE
CANADA COUNCIL SPECIAL GRANT
FOR
HIST 501] 168
ARISTARCHUS OF SAMOS
THE ANCIENT COPERNICUS
A HISTORY OF GREEK ASTRONOMY TO ARISTARCHUS
TOGETHER WITH ARISTARCHUS’S TREATISE
ON THE SIZES AND DISTANCES
OF THE SUN AND MOON
A NEW GREEK TEXT WITH TRANSLATION
AND NOTES
BY
SIR THOMAS HEATH
K.C.B., Sc.D., F.R.S.
SOMETIME FELLOW OF TRINITY COLLEGE, CAMBRIDGE
OXFORD
AT THE CLARENDON PRESS
1913
HENRY FROWDE, M.A.
PUBLISHER TO THE UNIVERSITY OF OXFORD
LONDON, EDINBURGH, NEW YORK, TORONT9
MELBOURNE AND BOMBAY
as en ee
a
« NOV 27 1968
My, 9
<Asiry oF TOROS
᾿ " PREFACE
THIS work owes its inception to a desire expressed to me by
my old schoolfellow Professor H. H. Turner for a translation of
_ Aristarchus’s extant work Ox the sizes and distances of the Sun and
Moon. Incidentally Professor Turner asked whether any light
could be thrown on the grossly excessive estimate of 2° for the
angular diameter of the sun and moon which is one of the funda-
mental assumptions at the beginning of the book. I remembered
_ that Archimedes distinctly says in his Psammites or Sand-reckoner
that Aristarchus was the first to discover that the apparent diameter
of the sun is about 1/720th part of the complete circle described
_ by it im the daily rotation, or, in other words, that the angular
_ diameter is about 4°, which is very near the truth. The difference
_ suggested that the treatise of Aristarchus which we possess was
_ an early work; but it was still necessary to search the history of
Greek astronomy for any estimates by older astronomers that
& be on record, with a view to tracing, if possible, the origin
“Or aie figure of 2°.
Again, our treatise does not contain amy suggestion of any but
the geoeeritric view of the universe, whereas Archimedes tells us
that Aristarchus wrote a book of hypotheses, one of which was
that the sun and the fixed stars remain unmoved and that the
eatth revolves round the sun in the circumference of a circle.
Now Archimedes was a younger contemporary of Aristarchus ;
_ he must have seen the book of hypotheses in question, and we
_ could have no better evidence for attributing to Aristarchus the
_ first enunciation of the Copernican hypothesis. The matter might
_ have rested there but for the fact that in recent years (1898)
Schiapareili, an authority always to be mentioned with profound
respect, has maintained that it was not after all Aristarchus, but
Heraclides of Pontus, who first put forward the heliocentric
ΩΝ PREFACE
hypothesis. Schiaparelli, whose two papers Le sfere omocentriche
di Eudosso, di Callippo e di Aristotele and I precursori di Copernico
nell’ antichita are classics, showed in the latter paper that Heraclides
discovered that the planets Venus and Mercury revolve round the
sun, like satellites, as well as that the earth rotates about its own
axis in about twenty-four hours. In his later paper of 1898 (Origine
del sistema planetario eliocentrico presso ἡ Grect) Schiaparelli went
further and suggested that Heraclides must have arrived at the
same conclusion about the superior planets as about Venus and
Mercury, and would therefore hold that all alike revolved round
the sun, while the sun with the planets moving in their orbits
about it revolved bodily round the earth as centre in a year;
in other words, according to Schiaparelli, Heraclides was probably
the inventor of the system known as that of Tycho Brahe, or was
acquainted with it and adopted it if it was invented by some
contemporary and not by himself. So far it may be admitted
that Schiaparelli has made out a plausible case ; but when, in the
same paper, he goes further and credits Heraclides with having —
originated the Copernican hypothesis also, he takes up much more
doubtful ground. At the same time it was clear that his argu-
ments were entitled to the most careful consideration, and this
again necessitated research in the earlier history of Greek astrorémy”
with the view of tracing every step in the progress towards the
true Copernican theory. The first to substitute another centre for
the earth in the celestial system were the Pythagoreans, who made
the earth, like the sun, moon, and planets, revolve round the central
fire; and, when once my study of the subject had been carried
back so far, it seemed to me that the most fitting introduction to
Aristarchus would be a sketch of the whole history of Greek
astronomy up to his time. As regards the newest claim made by
Schiaparelli on behalf of Heraclides of Pontus, I hope I have
shown that the case is not made out, and that there is still no
reason to doubt the unanimous testimony of antiquity that
Aristarchus was the real originator of the Copernican hypothesis.
In the century following Copernicus no doubt was felt as to
PREFACE } a
_ identifying Aristarchus with the latter hypothesis. Libert Fro-
mond, Professor of Theology at the University of Louvain, who
_ tried to refute it, called his work Avti-Aristarchus (Antwerp,
_ 1631). In 1644 Roberval took up the cudgels for Copernicus in a
_ book the full title of which is Aristarchi Samii de mundi systemate
partibus et motibus eiusdem libellus. Adiectae sunt A. P. de
_ Roberval, Mathem. Scient. in Collegio Regio Franciae Professoris,
_notae in eundem libellum. it does not appear that experts were
_ ever deceived by this title, although Baillet (Fugemens des Savans)
_ complained of such disguises and would have had Roberval call his
work Aristarchus Gallus, ‘the French Aristarchus, after the
“manner of Vieta’s Ajfollonius Gallus and Snellius’s Eratosthenes
_ Batavus. But there was every excuse for Roberval. The times
were dangerous. Only eleven years before seven Cardinals had
forced Galilei to abjure his ‘errors and heresies’; what wonder
_then that Roberval should take the precaution of publishing his
views under another name?
_ Voltaire, as is well known, went sadly wrong over Aristarchus
(Dictionnaire Philosophique, s.v. ‘Systéme’). He said that Ari-
_Starchus ‘is so obscure that Wallis was obliged to annotate him
_ from one end to the other, in the effort to make him intelligible’,
and furer that it was very doubtful whether the book attributed
+o'-\ristarchus was really by him. Voltaire (misled, it is true, by
a wrong reading in a passage of Plutarch, De facie in orbe lunae,
_ ς, 6) goes on to question whether Aristarchus had ever propounded -
_ the heliocentric hypothesis; and it is clear that the treatise which
_ he regarded as suspect was Roberval’s book, and that he confused
_ this with the genuine work edited by Wallis. Nor could he have
_ looked at the latter treatise in any but a very superficial way, or
_ he would have seen that it is not in the least obscure, and that the
_ commentary of Wallis is no more elaborate than would ordinarily
be expected of an editor bringing out for the first time, with the
_aid of MSS. not of the best, a Greek text and translation of a
mathematical treatise in which a number of geometrical propositions
_ are assumed without proof and therefore require some elucidation.
vi PREFACE
There is no doubt whatever of the genuineness of the work.
Pappus makes substantial extracts from the beginning of it and
quotes the main results. Apart from its astronomical content, it is
of the greatest interest for its geometry. Thoroughly classical in ©
form and language, as befits the period between Euclid and
Archimedes, it is the first extant specimen of pure geometry used
with a ¢rigonometrical object, and in this respect is a sort of fore-
runner of Archimedes’ Measurement of a Circle. I need therefore
make no apology for offering to the public a new Greek text with
translation and the necessary notes.
In conclusion I desire to express my best acknowledgements to
the authorities of the Vatican Library for their kindness in allowing
me to have a photograph of the best MS. of Aristarchus which
forms part of the magnificent Codex Vaticanus Graecus 204 of the
tenth century, and to Father Hagen of the Vatican Observatory
for his assistance in the matter. .
1:32, “ἢ;
CONTENTS
PART I
GREEK ASTRONOMY TO ARISTARCHUS OF SAMOS
CHAPTER PAGES
. I. Sources OF THE History . ὃ : : : 1-6
II. Homer anpD HEsiop _.. 3 ; : ἢ Ξ ἡπιτ
SSS et ee τς .χ.22
IV. ANAXIMANDER. ; : ‘ : f ‘ ; 24-39
V. ANAXIMENES . : 4 : : ᾿ Α ‘ 40-45
VI. PyTHAGoRAs . Ξ : ? Ξ - ‘ ; 46-51
_ VII. XENOPHANES . Ξ : - ᾿ - 52-58
_ VIII. Heractitus . : : : ; . : : 59-61
IX. PARMENIDES . . . s ᾿ Ε ; ; 62-77
X. ANAXAGORAS . > : Ξ : : ᾧ . 78-85
ΧΙ. Empepocies . ὃ 5 ‘ ; : ᾿ : 86-93
_ ΧΗ. THe Pyruacoreans : A 2 ς Σ . 964-120
_ XIII. Tue ΑΤΟΜΙΞΊ5, Leucippus ΑΝῸ DEMocrITUS . 121-129
_ XIV. OEnopipes. τὰ i : : ; τὴν + “330-133
XV. Puato . 2 ἥ ν : ἃ : : . 134-189
XVI. THe THErory or ConceNnTRIC SPHERES—EUDOXUS,
| CALLIPPUS, AND ARISTOTLE. ‘ P ? . 190-224
8 XVII. ARISTOTLE (continued) . : y : ‘ . 225-248
XVIII. HeEraciipes or Pontus. . . Ξ ξ . 249-283
XIX. Greek Montus, Years, AND CYCLES . ς . 284-297
PART II
ARISTARCHUS ON THE SIZES AND DISTANCES
OF THE SUN AND MOON
I, ARISTARCHUS OF SaMos. ᾿ : ς ᾿ . 299-316
II. Tue TREATISE ΟΝ SIzEs AND DisTaNceS—HIsTORY
OF THE TEXT AND EDITIONS. : ; . 317-327
III. ConTENT oF THE TREATISE . ‘ . : . 328-336
IV. Later ImprovEMENTs ΟΝ ARISTARCHUS’s CALCULA-
TIONS . ὃ : ἢ : = ; . 337-350
GREEK TEXT, TRANSLATION, AND NoTES . Η . 251-414
. INDEX . : : Ξ d : % ς - : . 415-425
CORRIGENDUM
P. 179, lines 26 and 31. It appears that προχωρήσεις, not προσχωρήσεις, is
the correct reading in 7imaeus 40 C. The meaning of προχωρήσεις is of course
‘forward movements’, but the change to this reading does not make it any
the more necessary to take ἐπανακυκλήσεις in the sense of retrogradations ; on
the contrary, a ‘forward movement’ and a ‘ returning of the circle upon itself’
are quite natural expressions for the different stages of one simple circular
motion. Cf. also Republic 617 B, where ἐπανακυκλούμενον is used of the
‘counter-revolution’ of the planet Mars; what is meant is a simple circular
revolution in a sense contrary to that of the fixed stars, and there is no suggestion
of retrogradations.
PART I
GREEK ASTRONOMY TO ARISTARCHUS OF SAMOS
I
SOURCES OF THE HISTORY
: THE history of Greek astronomy in its beginnings is part of the
history of Greek philosophy, for it was the first philosophers,
Tonian, Eleatic, Pythagorean, who were the first astronomers.
Now only very few of the works of the great original thinkers
of Greece have survived. We possess the whole of Plato and, say,
half of Aristotle, namely, those of his writings which were intended
for the use of his school, but not those which, mainly composed
in the form of dialogues, were in a more popular style. But the
whole of the pre-Socratic philosophy is one single expanse of
ruins ;' so is the Socratic philosophy itself, except for what we
_can learn of it from Plato and Xenophon.
But accounts of the life and doctrine of philosophers begin to
appear quite early in ancient Greek literature (cf. Xenophon, who
was born between 430 and 425 B.C.); and very valuable are the
allusions in Plato and Aristotle to the doctrines of earlier philo-
sophers; those in Plato are not very numerous, but he had the
_ power of entering into the thoughts of other men and, in stating Ὁ
_ the views of early philosophers, he does not, as.a rule, read into
their words meanings which they do not convey. Aristotle, on the
_ other hand, while making historical surveys of the doctrines of his
predecessors a regular preliminary to the statement of his own,
discusses them too much from the point of view of his own system ;
often even misrepresenting them for the purpose of making a contro-
_ versial point or finding support for some particular thesis.
From Aristotle’s time a whole literature on the subject of the
older philosophy sprang up, partly critical, partly historical. This
1 Gomperz, Griechische Denker, i*, Ὁ. 419.
1410 B
2 SOURCES OF THE HISTORY PARTI
again has perished except for a large number of fragments. Most
important for our purpose are the notices in the Doxographi Graeci,
collected and edited by Diels.1_ The main source from which these
retailers of the opinions of philosophers drew, directly or indirectly,
was the great work of Theophrastus, the successor of Aristotle,
entitled Physical Opinions (Φυσικῶν δοξῶν Tm). It would appear
that it was Theophrastus’s plan to trace the progress of physics
from Thales to Plato in separate chapters dealing severally with
the leading topics, First the leading views were set forth on broad
lines, in groups, according to the affinity of the doctrine, after
which the differences between individual philosophers within the
same group were carefully noted. In the First Book, however,
dealing with the Principles, Theophrastus adopted the order of the
various schools, Ionians, Eleatics, Atomists, &c., down to Plato,
although he did not hesitate to connect Diogenes of Apollonia and
Archelaus with the earlier physicists, out of their chronological
order; chronological order was indeed, throughout, less regarded
than the connexion and due arrangement of subjects. This work
of Theophrastus was naturally the chief hunting-ground for those
who collected the ‘ opinions’ of philosophers. There was, however,
another main stream of tradition besides the doxographic; this
was in the different form of biographies of the philosophers. The
first to write a book of ‘successions’ (διαδοχαΐ) of the philosophers
was Sotion (towards the end of the third century B.C.); others
who wrote ‘successions’ were a certain Antisthenes (probably
Antisthenes of Rhodes, second century B.C.), Sosicrates, and
Alexander Polyhistor. These works gave little in the way ot
doxography, but were made readable by the incorporation of
anecdotes and apophthegms, mostly unauthentic. The work
of Sotion and the ‘Lives of Famous Men’ by Satyrus (about
160 B.C.) were epitomized by Heraclides Lembus. Another writer
of biographies was the Peripatetic Hermippus of Smyrna, known as
the Callimachean, who wrote about Pythagoras in at least two Books,
and is quoted by Josephus as a careful student of all history.2_ Our
chief storehouse of biographical details derived from these and all
other available sources is the great compilation which goes by the
1 Doxographi Graeci, ed. Diels, Berlin, G. Reimer, 1879.
* Doxographi Graeci (henceforth generally quoted as D.G.), p. 151.
CE.I SOURCES OF THE HISTORY 3
name of Diogenes Laertius (more properly Laertius Diogenes). It
is a compilation made in the most haphazard way, without the
exercise of any historical sense or critical faculty. But its value
for us is enormous because the compiler had access to the whole
collection of biographies which accumulated from Sotion’s time to
the first third of the third century A.D. (when Diogenes wrote), and
consequently we have in him the whole residuum of this literature
which reached such dimensions in the period.
' ἴῃ order to show at a glance the conclusions of Diels as to the
relation of the various representatives of the doxographic and
biographic traditions to one another and to the original sources
I append a genealogical table’:
: Eusebrus
Gi Cent AND
ua pra TACLO
Bks XIV) ο
Fig. I
__-? Cf. Giinther in Windelband, Gesch, der alten Philosophie (Iwan yon Miiller’s
Handbuch der klassischen Altertumswissenschaft, Band v. 1), 1894, p. 275.
B2
4 SOURCES OF THE HISTORY PARTI
- Only a few remarks need be added. ‘Vetusta Placita’ is the
name given by Diels to a collection which has disappeared, but
may be inferred to have existed. It adhered very closely to
Theophrastus, though it was not quite free from admixture of
other elements. It was probably divided into the following main
sections: I. De principiis; II. De mundo; III. De sublimibus;
IV. De terrestribus; V. De anima; VI. De corpore. The date
is inferred from the facts that the latest philosophers mentioned
in it were Posidonius and Asclepiades, and that Varro used it.
The existence of the collection of Aétius (De placitis, περὶ
ἀρεσκόντων) is attested by Theodoretus (Bishop of Cyrus), who
mentions it as accessible, and who certainly used it, since his
extracts are more complete and trustworthy than those of the
Placita Philosophorum and Stobaeus. The compiler of the Placita
was not Plutarch, but an insignificant writer of about the middle
of the second century A.D., who palmed them off as Plutarch.
Diels prints the Placita in parallel columns with the corresponding
parts of the Aclogae, under the title of Aétz Placita; quotations
from the other writers who give extracts are added in notes
at the foot of the page. So far as Cicero deals with the earliest
Greek philosophy, he must be classed with the doxographers ; both
he and Philodemus (De jietate, περὶ εὐσεβείας, fragments of which
were discovered on a roll at Herculaneum) seem alike to have used a
common source which went back to a Stoic epitome of Theophrastus,
now lost.
The greater part of the fragment of the Pseudo-Plutarchian
στρωματεῖς given by Eusebius in Book I. 8 of the Praeparatio
Evangelica comes from an epitome of Theophrastus, arranged
according to philosophers. The author of the Stromateis, who
probably belonged to the same period as the author of the Placita,
that is, about the middle of the second century A.D., confined
himself mostly to the sections de principio, de mundo, de astris ;
hence some things are here better preserved than elsewhere; cf.
especially the notice about Anaximander.
The most important of the biographical doxographies is that of
Hippolytus in Book I of the Refutation of all Heresies (the sub-
title of the particular Book is φιλοσοφούμενα), probably written
between 223 and 235 A.D. It is derived from two sources. The
᾿
ΟῚ SOURCES OF THE HISTORY 5
_ one was a biographical compendium of the διαδοχή type, shorter
and even more untrustworthy than Diogenes Laertius, but con-
taining excerpts from Aristoxenus, Sotion, Heraclides Lembus,
and Apollodorus. The other was an epitome of Theophrastus.
_ Hippolytus’s plan was to take the philosophers in order and then
_ to pick out from the successive sections of the epitome of Theo-
phrastus the views of each philosopher on each topic, and insert
_them in their order under the particular philosopher. So carefully
was this done that the divisions of the work of Theophrastus can
_ practically be restored.1_ Hippolytus began with the idea of dealing
_with the chief philosophers only, as Thales, Pythagoras, Empedocles,
Heraclitus. For these he had available only the inferior (biographical)
source. The second source, the epitome of Theophrastus, then
came into his hands, and, beginning with Anaximander, he proceeded
to make a most precious collection of opinions.
Another of our authorities is Achilles (not Tatius), who wrote
an Introduction to the Phaenomena of Aratus.* Achilles’ date is
uncertain, but he probably lived not earlier than the end of the
second century A.D., and not much later. The foundation of
Achilles’ commentary was a Stoic compendium of astronomy,
_ probably by Eudorus, which in its turn was extracted from a work
by Diodorus of Alexandria, a pupil of Posidonius. But Achilles
drew from other sources as well, including the Pseudo-Plutarchian
Placita; he did not hesitate to alter his extracts from the latter,
and to mix alien matter with them.
The opinions noted by the Doxographi are largely incorporated
in Diels’ later work Die Fragmente der Vorsokratiker®
For the earlier period from Thales to Empedocles, Tannery gives
a translation of the doxographic data and the fragments in his
work Pour Vhistoive de la science helléne, de Thales ἃ Empédocle,
Paris, 1887 ; taking account as it does of all the material, this work is
᾿ the best and most suggestive of the modern studies of the astronomy
of the period. Equally based on the Dorographi, Max Sartorius’s
dissertation Die Entwicklung der Astronomie bei den Griechen bis
* Diels, Doxographi Graeci, p. 153.
* Excerpts from this are preserved in Cod. Laurentian. xxviii. 44, and are
included in the Uranologium of Petavius, 1630, pp. 121-64, &c.
* Second edition in two vols. (the second in two parts), Berlin, 1906-10.
ό SOURCES OF THE HISTORY
Anaxagoras und Empedokles (Halle, 1883) is a very concise and
useful account. Naturally all or nearly all the material is also to
be found in the monumental work of Zeller and in Professor Burnet’s
Early Greek Philosophy (second edition, 1908); and picturesque,
if sometimes too highly coloured, references to the astronomy of
the ancient philosophers are a feature of vol. i of Gomperz’s
Griechische Denker (third edition, 1911).
Eudemus of Rhodes (about 330 B.C.), a pupil of Aristotle, wrote
a History of Astronomy (as he did a History of Geometry), which
is lost, but was the source of a number of notices in other writers.
In particular, the very valuable account of Eudoxus’s and Callip-
pus’s systems of concentric spheres which Simplicius gives in his
Commentary on Aristotle’s De caelo is taken from Eudemus through
Sosigenes as intermediary. A few notices from Eudemus’s work
are also found in the astronomical portion of Theon of Smyrna’s
Expositio rerum mathematicarum ad legendum Platonem utilium,:
which also draws on two other sources, Dercyllides and Adrastus.
The former was a Platonist with Pythagorean leanings, who wrote
a book on Plato’s philosophy. His date was earlier than the time
of Tiberius, perhaps earlier than Varro’s. Adrastus, a Peripatetic
of about the middle of the second century A.D., wrote historical
and lexicographical essays on Aristotle ; he also wrote a commentary
on the Zzmaeus of Plato, which is quoted by Proclus as well as by
Theon of Smyrna.
1 Edited by E. Hiller (Teubner, 1878).
II
HOMER AND HESIOD
WE take as our starting-point the conceptions of the structure
of the world which are to be found in the earliest literary monuments
of Greece, that is to say, the Homeric poems and the works of
Hesiod. In their fundamental conceptions Homer and Hesiod
_ agree. The earth is a flat circular disc; this is not stated in so
many words, but only on this assumption could Poseidon from
_ the mountains of Solym in Pisidia see Odysseus at Scheria on the
further side of Greece, or Helios at his rising and setting descry
his cattle on the island of Thrinakia. Round this flat disc, on the
horizon, runs the river Oceanus, encircling the earth and flowing
back into itself (ἀψόρροος) ; from this all other waters take their
rise, that is, the waters of Oceanus pass through subterranean
channels and appear as the springs and sources of other rivers.
Over the flat earth is the vault of heaven, like a sort of hemi-
spherical dome exactly covering it ; hence it is that the Aethiopians
_ dwelling in the extreme east and west are burnt black by the sun.
Below the earth is Tartarus, covered by the earth and forming
a sort of vault symmetrical with the heaven; Hades is supposed
to be beneath the surface of the earth, as far from the height of
the heaven above as from the depth of Tartarus below, i.e. pre-
sumably in the hollow of the earth’s disc. The dimensions of the
heaven and earth are only indirectly indicated; Hephaestus cast
down from Olympus falls for a whole day till sundown; on the
other hand, according to Hesiod, an iron anvil would take nine
days to pass from the heaven to the earth, and again nine days
from the earth to Tartarus. The vault of heaven remains for ever
in one position, unmoved ; the sun, moon, and stars move round
under it, rising from Oceanus in the east and plunging into it again
in the west. We are not told what happens to the heavenly bodies
8 HOMER AND HESIOD PARTI
between their setting and rising; they cannot pass round under
the earth because Tartarus is never lit up by the sun; possibly
they are supposed to float round Oceanus, past the north, to the
points where they next rise in the east, but it is only later writers
who represent Helios as sleeping and being carried round on the
water on a golden bed or in a golden bowl.?
Coming now to the indications of actual knowledge of astronomical
facts to be found in the poems, we observe in Hesiod a considerable
advance as compared with Homer. Homer mentions, in addition
to the sun and moon, the Morning Star, the Evening Star, the
Pleiades, the Hyades, Orion, the Great Bear (‘which is also called
by the name of the Wain, and which turns round on the same spot
and watches Orion; it alone is without lot in Oceanus’s bath’ *),
1 Athenaeus, Deipnosoph. xi. 38-9.
2 It seems that some of the seven principal stars of the Great Bear do now
set in the Mediterranean, e.g.,in places further south in latitude than Rhodes
(lat. 36°), y, the hind foot, as well as n, the tip of the tail, and at Alexandria all
the seven stars except a, the head. But this was not so in Homer’s time. In
proof of this, Sir George Greenhill (in a lecture delivered in 1910 to the Hellenic
Travellers’ Club) refers to calculations made by Dr. J. B. Pearson of the effect
of Precession in the interval since 750 B.C., a date taken ‘ without Ὃν pra ;
(Proceedings of the Cambridge Philosophical Soctety, 1877 and 1881), and to the
results obtained in a paper by J. Gallenmiiller, Der Fixsternhimmel jetzt und in
Homers Zeiten mit zwei Sternkarten (Regensburg, 1884/85). Gallenmiiller’s
charts are for the years 900 B.C. and A.D. 1855 respectively, and the chart for
goo B.C. shows that the N.P. Ὁ. of both 8, the fore-foot, and η, the tip of the
tail, was then about 25°. But we also find convincing evidence in the original
writings of the Greek astronomers. Hipparchus (J Avrati et Eudoxi phaeno-
mena commentariorum libri tres, ed. Manitius, 1894, p. 114. 9-10) observes that
Eudoxus [say, in 380 B.C., or 520 years later than the date to which Gallen-
miiller’s chart refers] made the fore-foot (8) about 24°, and the hind-foot (y)
about 25°, distant from the-north pole. This was perhaps not very accurate ;
for Hipparchus says (ibid., p. 30. 2-8), ‘As regards the north pole, Eudoxus is in
error in stating that “there is a certain star which always remains in the same
spot, and this star is the pole of the universe”; for in reality there is no star at
all at the pole, but there is an empty space there, with, however, three stars
near to it [probably a and κ of Draco and β of the Little Bear], and the point at
the pole makes with these three stars a figure which is very nearly square, as
Pytheas of Massalia stated.’ (Pytheas, the great explorer of the northern seas,
was a contemporary of Aristotle, and perhaps some forty years later than
Eudoxus.) But, as Hipparchus himself (writing in this case not later than
134 B.C.) makes the angular radius of the ‘always-visible circle’ 37° at Athens
and 36° at Rhodes (ibid., pp. 112.16 and 114. 24-6), it is evident that in
Eudoxus’s time the whole of the Great Bear remained well above the horizon.
A passage of Proclus (Hyfotyposis, c. 7, δὲ 45-8, p. 234, ed. Manitius) is not
without interest in this connexion. He is trying to controvert the theory of
astronomers that the fixed stars themselves have a movement about the pole
of the ecliptic (as distinct from the pole of the universe) of about 1° in 100 years
CH. II HOMER AND HESIOD 9
Sirius (‘the star which rises in late summer . . . which is called
among men “ Orion’s dog” ; bright it shines forth, yet is a baleful
sign, for it brings to suffering mortals much fiery heat’), the ‘ late-
setting Bodtes’ (the ‘ploughman’ driving the Wain, i.e. Arcturus,
as Hesiod was the first to call it). Since the Great Bear is said
to be the only constellation which never sets, we may perhaps
assume that the stars and constellations above named are all that
_ were definitely recognized at the time, or at least that the Bear
was the only constellation recognized in the northern sky. There
is little more that can be called astronomy in Homer. There are
vague uses of astronomical phenomena for the purpose of fixing
localities or marking times of day or night; as regards the day,
the morning twilight, the rising and setting of the sun, midday,
and the onset of night are distinguished ; the night is divided into
three thirds. Aristotle was inclined to explain Helios’s seven herds
of cattle and sheep respectively containing 50 head in each herd
{i.€. 350 in all of each sort) as a rough representation of the number
of days in a year. Calypso directed Odysseus to sail in such a way
as to keep the Great Bear always on his left. One passage,!
relating to the island called Syrie, ‘which is above Ortygia where
are the turnings (τροπαΐ) of the sun’, is supposed by some to refer
to the solstices, but there is no confirmation of this by any other
‘passage, and it seems safer to take ‘turning’ to mean the turn
which the sun takes at setting, when of course he begins his return
journey (travelling round Oceanus or otherwise) to the place of his
(this is Ptolemy’s estimate). ‘ How is it’, says Proclus, ‘that the Bears, which
have always been visible above the horizon through countless ages, still remain
so, if they move by one degree in 100 years about the pole of the zodiac, which
is different from the world-pole ; for, if they had moved so many degrees as this
would imply, they should now no longer graze (παραξέειν) the horizon but should
partly set’! This passage, written (say) 840 years after Eudoxus’s location of 8 and
y of the Great Bear, shows that the Great Bear was then much nearer to setting
than it was in Eudoxus’s time, and the fact should have made Proclus speak with
greater caution. [The star which Eudoxus took as marking the north pole has
commonly been supposed to be β of the Little Bear; but Manitius (Hipparchi in
Arati et Eudoxi phaen. comment., 1894, p. 306), as the result of studying a
*Precession-globe’ designed by Prof. Haas of Vienna, considers that it was
certainly a different star, namely, ‘Draconis 16,’ which occupies a position
determined as the intersection of (1) a perpendicular from our Polar Star to the
straight line joining κ and of Draco and (2) the line joining y and β of the
Little Bear and produced beyond β.]
1 Odyssey xv. 403-4.
10 HOMER AND HESIOD PARTI
rising, in which case the island would simply be situated on the
western horizon where the sun se¢s.1
Hesiod mentions practically the same stars as Homer, the
Pleiades, the Hyades, Orion, Sirius, and Arcturus. But, as might
be expected, he makes much more use than Homer does of celestial
phenomena for the purpose of determining times and seasons in the
year. Thus, e.g., he marked the time for sowing at the beginning
of winter by the setting of the Pleiades in the early twilight, or
again by the early setting of the Hyades or Orion, which means
the 3rd, 7th, or 15th November in the Julian calendar according to
the particular stars taken ;* the time for harvest he fixed by the
early rising of the Pleiades, which means the Julian 19th of May ;*
threshing-time he marked by the early rising of Orion (Julian gth
of July), vintage-time by the early rising of Arcturus (Julian 18th
of September), and so on. With Hesiod, Spring begins with the
late rising of Arcturus; this would in his time and climate be the
24th February of the Julian calendar, or 57 days after the winter
solstice, which in his time would be the 29th December. He him--
self makes Spring begin 60 days after the winter solstice ; he may
be intentionally stating a round figure, but, if he made an error of
1 Martin has discussed the question at considerable length (‘Comment
Homére s’orientait’ in Mémoires de ? Académie des Inscriptions et Belles-
Lettres, xxix, Pt. 2, 1879, pp. 1-28). He strongly holds that τροπαὶ ἠελίοιο can
only mean the solstice, that by this we must also understand the summer
solstice, and that the expression ὅθι τροπαὶ ἠελίοιο must therefore be in the
direction of the place on the horizon where the sun sets at the summer solstice,
i.e. west-north-west. Martin’s ground is his firm conviction that τροπαὶ nediovo
has mever, in any Greek poet or prose writer, any other than the technical
meaning of ‘ solstice’. This is, however, an assumption not susceptible of proof;
and Martin is not very successful in his search for confirmation of his view.
Identifying Ortygia with Delos, and Syrie with Syra or Syros, he admits that
the southern part of Syra is due west of the southern part of Delos ; only the
northern portion of Syra stretches further north than the northern portion of
Delos; therefore, geographically, either west or west-north-west would describe
the direction of Syra relatively to Ortygia well enough. Of the Greek com-
mentators, Aristarchus of Samothrace and Herodian of Alexandria take rpomai
to mean ‘ setting’ simply; Martin is driven therefore to make the most he can
of Hesychius who (s.v. ’Oprvyin) gives as an explanation τοῦτο δέ ἐστιν ὅπου ai
δύσεις ἄρχονται, ‘This is where the settings commence’, which Martin interprets
as meaning ‘ where the sun sets a¢ the commencement of the Greek year’, which
was about the time of the summer solstice ; but this is a great deal to get out
of ‘commencement of setting’.
2 Ideler, Handbuch der mathematischen und technischen Chronologie, 1825,
i, ΡΡ. 242, 246.
Ibid, p. 242. * Ibid. pp. 246, 247. ©
»
—_ ἈΨΎΥΥ oan ae 7
ΕΝ »
CH. II HOMER AND HESIOD II
three days, it would not be surprising, seeing that in his time there
were no available means for accurately observing the times of the
solstices. His early summer (θέρος), as distinct from late summer
(ὀπώρα), he makes, in like manner, end 50 days after the sum-
mer solstice. Thus he was acquainted with the solstices, but he
says nothing about the equinoxes, and only remarks in one place
that in late summer the days become shorter and the nights longer.
_ From the last part of the Works and Days we see that Hesiod had
an approximate notion of the moon’s period ; he puts it at 30 days,
and divides the month into three periods of ten days each.!
Hesiod was also credited with having written a poem under the
title of ‘Astronomy’. A few fragments of such a poem are pre-
served ;* Athenaeus, however, doubted whether it was Hesiod’s
work, for he quotes ‘the author of the poem “ Astronomy” which
is attributed to Hesiod’ as always speaking of Peleiades. Pliny
observes that ‘Hesiod (for an Astrology is also handed down
under his name) stated that the matutinal setting of the Vergiliae
[Pleiades] took place at the autumnal equinox, whereas Thales
made the time 25 days from the equinox’. The poem was thought
to be Alexandrine, but has recently been shown to be old; perhaps,
if we may judge by the passage of Pliny, it may be anterior to
Thales.
1 Sartorius, op. cit., p. 16; Ideler, i, p. 257.
3 Diels, Vorsokratiker, ii*. 1, 1907, pp. 499, 500.
5. Pliny, WV. H. xviii, c. 25, ὃ 213 ; Diels, loc. cit.
III
THALES
SUCH astronomy as we find in Homer and Hesiod was of the
merely practical kind, which uses the celestial recurrences for the
regulation of daily life; but, as the author of the Epznomis says,
‘the true astronomer will not be the man who cultivates astronomy
in the manner of Hesiod and any other writers of that type, concern-
ing himself only with such things as settings and risings, but the
man who will investigate the seven revolutions included in the eight
revolutions and each describing the same circular orbit [i.e. the
separate motions of the sun, moon, and the five planets combined
with the eighth motion, that of the sphere of the fixed stars, or the
daily rotation], which speculations can never be easily mastered by
the ordinary person but demand extraordinary powers’. The history ἡ
of Greek astronomy in the sense of astronomy proper, the astronomy
which seeks to explain the heavenly phenomena and their causes,
begins with Thales.
Thales of Miletus lived probably from about 624 to 547 B.C.
(though according to Apollodorus he was born in 640/39). Accord-
ing to Herodotus, his ancestry was Phoenician; his mother was
Greek, to judge by her name Cleobuline, while his father’s name,
Examyes, is Carian, so that he was of mixed descent. In 582/1 B.C.
he was declared one of the Seven Wise Men, and indeed his ver-
satility was extraordinary ; statesman, engineer, mathematician and
astronomer, he was an acute business man in addition, if we may
believe the story that, wishing to show that it was easy to get rich,
he took the opportunity of a year in which he foresaw that there
would be a great crop of olives to get control of all the oil-presses
in Miletus and Chios in advance, paying a low rental when there
was no one to bid against him, and then, when the accommodation
was urgently wanted, charging as much as he liked for it, with the
result that he made a large profit For his many-sided culture he
1 Aristotle, Politics i. 11. 9, 1259 a 6-17.
“τ <= ea
eae ae ae ee σον
THALES ; 13
was indebted in great measure to what he learnt on long journeys
which he took, to Egypt in particular ; it was in Egypt that he saw
in operation the elementary methods of solving problems in prac-
tical geometry which inspired him with the idea of making
geometry a deductive science depending on general propositions ;
and he doubtless assimilated much of the astronomical knowledge
which had been accumulated there as the result of observations
recorded through long centuries.
Thales’ claim to a place in the history of scientific astronomy
depends almost entirely on one achievement attributed to him, that
of predicting an eclipse of the sun. There is no trustworthy
evidence of any other discoveries, or even of any observations,
made by him, although one would like to believe the story, quoted
by Plato,! that, when he was star-gazing and fell into a well in con-
sequence, he was rallied ‘by a clever and pretty maid-servant from
Thrace’? for being so ‘eager to know what goes on in the heavens
when he could not see what was in front of him, nay, at his very feet’.
But did Thales predict a solar eclipse? The story is entirely
rejected by Martin.* He points out that, while the references to
the prediction do not exactly agree, it is in fact necessary, if the
oceurrence of a solar eclipse at any specified place on the earth’s
surface is to be predicted with any prospect of success, to know
more of the elements of astronomy than Thales could have known,
and in particular to allow for parallax, which was not done until
much later, and then only approximately, by Hipparchus. Further,
if the prophecy had rested on any scientific basis, it is incredible
that the basis should not have been known and been used by later
Ionian philosophers for making other similar predictions, whereas
we hear of none such in Greece for two hundred years. Indeed,
only one other supposed prediction of the same kind is referred to.
Plutarch* relates that, when Plato was on a visit to Sicily and stay-
ing with Dionysius, Helicon of Cyzicus, a friend of Plato’s, foretold
a solar eclipse (apparently that which took place on 12th May,
1 Theaetetus 174 A;.cf. Hippolytus, Refuz. i. 1. 4 (D. G. p. 555. 9-12).
? There is another version not so attractive, according to which [Diog. Laert.
i. 34], being taken out of the house by an old woman to look at the stars, he fell
into a hole and was reproached by her in similarterms. This version might
suggest that it was the old woman who was the astronomer rather than Thales.
Revue Archéologique, ix, 1864, pp. 181 sq. “ Life of Dion, c. 19, p. 966A.
14 THALES PART I
361 B.C.),1 and, when this took place as predicted, the tyrant was
filled with admiration and made Helicon a present of a talent of
silver. This story is, however, not confirmed by any other evidence,
and the necessary calculations would have been scarcely less im-
possible for Helicon than for Thales. Martin’s view is that both
Thales and Helicon merely explained the cause of solar eclipses
and asserted the necessity of their recurrence within certain limits
of time, and that these explanations were turned by tradition into
predictions. In regard to Thales, Martin relies largely on the word-
ing of a passage in Theon of Smyrna, where he purports to quote
Eudemus; ‘ Eudemus’, he says, ‘ relates in his Astronomies that...
Thales was the first to discover (εὗρε πρῶτος understood) the
eclipse of the sun and the fact that the sun’s period with respect
to the solstices is not always the same’,? and the natural mean-
ing of the first part of the sentence is that Thales discovered
the explanation and the cause of a solar eclipse. It is true that
Diogenes Laertius says that ‘ Thales appears, according to some, to
have been the first to study astronomy and to predict both solar
eclipses and solstices, as Eudemus says in his History of Astronomy ’,®
and Diogenes must be quoting from the same passage as Theon ;
but it is pretty clear, as Martin says, that he copied it inaccurately
and himself inserted the word (προειπεῖν) referring to predictions ;
indeed the word ‘ predict’ does not go well with ‘solstices’, and is
suspect for this reason. Nor does any one credit Thales with having
predicted more than one eclipse. No doubt the original passage
spoke of ‘ eclipses’ and ‘ solstices’ in the plural and used some word
like ‘discover’ (Theon’s word), not the word ‘predict’. And I
think Martin may reasonably argue from the passage of Diogenes
that the words ‘according to some’ are Eudemus’s words, not his
own, and therefore may be held to show that the truth of the
tradition was not beyond doubt.
1 Boll, art. ‘Finsternisse’ in Pauly-Wissowa’s eal-Encyclopidie der
classischen Altertumswissenschaft, vi. 2, 1909, pp. 2356-7; Ginzel, Handbuch
der mathematischen und technischen Chronologie, vol. ii, 1911, p. 527.
3 Theon of Smyrna, ed. Hiller, p. 198. 14-18.
® Diog. L. 1.23 (Vorsokratiker, 15, p. 3. 19-21).
* There is, however, yet another account purporting to be based on Eudemus,
Clement of Alexandria (S¢vomat. i. 65) says : ‘Eudemus observes in his History
of Astronomy that Thales predicted the eclipse of the sun which took place at
the time when the Medes and the Lydians engaged in battle, the king of the
Sa Se
CH, ΠῚ THALES ᾿ 15
Nevertheless, as Tannery observes, Martin’s argument can
hardly satisfy us so far as it relates to Thales. The evidence
that Thales actually predicted a solar eclipse is as conclusive as
ave could expect for an event belonging to such remote times, for
Diogenes Laertius quotes Xenophanes as well as Herodotus as
having admired Thales’ achievement, and Xenophanes was almost
contemporary with Thales. We must therefore accept the fact as
historic, and it remains to inquire in what sense or form, and on
- what ground, he made his prediction. The accounts of it vary.
Herodotus says? that the Lydians and the Medes continued their
war, and ‘when, in the sixth year, they encountered one another, it
fell out that, after they had joined battle, the day suddenly turned
into night. Now that this transformation of day (into night) would
occur was foretold to the Ionians by Thales of Miletus, who fixed
as the limit of time this very year in which the change actually
took place.’* The prediction was therefore at best a rough one,
Medes being Cyaxares, the father of Astyages, and Alyattes, the son of Croesus,
being the king of the Lydians; and the time was about the 5oth Olympiad [58ο-
577].’ The last sentence was evidently taken from Tatian 41 ; but, if the rest of
the passage correctly quotes Eudemus, it would appear that there must have
been two passages in Eudemus dealing with the subject.
1 Tannery, Pour ’histoire de la science helléne, p. 56.
3 Herodotus, i. 74.
3. Other references are as follows: Cicero, De Divinatione i. 49. 112, observes
that Thales was said to have been the first to predict an eclipse of the sun, which
eclipse took place in the reign of Astyages; Pliny, V.H. ii, c. 12, ὃ 53, ‘Among
the Greeks Thales first investigated (the cause of the eclipse) in the fourth year of
the 48th Olympiad [585/4 B.c.], having predicted an eclipse of the sun which
took place in the reign of Alyattes in the year 170 A.U.C.’; Eusebius, Chron.
(Hieron.), under year of Abraham 1433, ‘An eclipse of the sun, the occurrence of
which Thales had predicted: a battle between Alyattes and Astyages’. The
eclipse so foretold is now most generally taken to be that which took place on-
the (Julian) 28th May, 585. A difficulty formerly felt in regard to this date
seems now to have been removed. Herodotus (followed ‘by Clement) says that
the eclipse took place during a battle between Alyattes and Cyaxares. Now,
on the usual assumption, based on Herodotus’s chronological data, that Cyaxares
reigned from about 635 to 595, the eclipse of 585 B.c. must have taken place
during the reign of his son; and perhaps it was the knowledge of this fact which
made Eusebius say that the battle was between Alyattes and Astyages. But it
appears that Herodotus’s reckoning was affected by an error on his part in taking
the fall of the Median kingdom to be coincident with Cyrus’s accession to
the throne of Persia, and that Cyaxares really reigned from 624 to 584, and
Astyages from 584 to 550 B.C. (Ed. Meyer in Pauly-Wissowa’s Real-Encyclo-
padie, ii, 1896, p. 1865, ὅς.) ; hence the eclipse of 585 B.c. would after all come
in Cyaxares’ reign. Oftwo more solar eclipses which took place in the reign of
Cyaxares one is ruled out, that of 597 B.C., because it took place at sunrise, which
would not agree with Herodotus’s story. The other was on 30th September, 610,
and, as regards this, Bailly and Oltmanns showed that it was not total on the
τό THALES PARTI
since it only specified that the eclipse would occur within a certain
year; and the true explanation seems to be that it was a prediction
of the same kind as had long been in vogue with the Chaldaeans.
That they had a system enabling them to foretell pretty accurately
the eclipses of the moon is clear from the fact that some of the
eclipses said by Ptolemy’ to have been observed in Babylon were so
partial that they could hardly have been noticed if the observers had
not been to some extent prepared for them. Three of the eclipses
mentioned took place during eighteen months in the years 721
and 720. It is probable that the Chaldaeans arrived at this method
of approximately predicting the times at which lunar eclipses would
occur by means of the period of 223 lunations, which was doubt-
less discovered as the result of long-continued observations. This
period is mentioned by Ptolemy* as having been discovered by
astronomers ‘still more ancient’ than those whom he calls ‘the
ancients’.. Now, while this method would serve well enough for
lunar eclipses, it would very often fail for solar eclipses, because no
account was taken of parallax. An excellent illustration of the
way in which the system worked is on record; it is taken from
a translation of an Assyrian cuneiform inscription, the relevant words
being the following :
1. To the king my lord, thy servant Abil-istar.
2. May there be peace to the king my lord. May Nebo and
Merodach
3. to the king my lord be favourable. Length of days,
4. health of body and joy of heart may the great gods
presumed field of battle (in Cappadocia), though it would be total in Armenia
(Martin, Revue Archéologiqgue, ix, 1864, pp. 183, 190). Tannery, however
(Pour Phistotre de la science helléne, p. 38), holds that the latter eclipse was that
associated with Thales. The latest authorities (Boll, art. ‘Finsternisse’, in Pauly-
Wissowa’s Real-Encyclopidie, vi. 2, 1909, pp.2353-4, and Ginzel, Spesieller Kanon
der Sonnen- und Mondjinsternisse and Handbuch der mathematischen und tech-
nischen Chronologie, vol. ii, 1911, p. 525) adhere to the date 28th May, 585.
1 Ptolemy, Syntaxis iv, c. 6 sq.
* Ptolemy, Syztaxis iv, c. 2, p. 270, 1 sq., ed. Heiberg.
* Suidas understands the Chaldaean name for this period to have been savos,
but this seems to be a mistake. According to Syncellus (Chronographia, p. 17;
A-B), Berosus expressed his periods in savs, #ers, and sosses, a sar being 3,600
years, while 2267 meant 600 years, and soss 60 years ; but we learn that the same
words were also used to denote the same numbers of days respectively
(Syncellus, p. 32 C). Nor were they used of years and days only; in fact sar,
2167, and 5055 were collective numerals simply, like our words ‘gross’, ‘ score’,
ἄς. (Cantor, Gesch. d. Mathematik, 15, p. 36).
4 See George Smith, Assyrian Discoveries, p. 409.
‘CH. IM THALES 17
5. to the king my lord grant. Concerning the eclipse of the
moon
6. of which the king my lord sent to me; in the cities of
Akkad,
7. Borsippa, and Nipur, observations
8. they made and then in the city of Akkad
9. we saw part. ...
το. The observation was made and the eclipse took place.
17. And when for the eclipse of the sun we made
18. an observation, the observation was made and it did not take
lace.
19. That which I saw with my eyes to the king my lord
20. Isend. This eclipse of the moon
21. which did happen concerns the countries
22. with their god all. Over Syria
23. it closes, the country of Phoenicia,
24. of the Hittites, of the people of Chaldaea,
25. but to the king my lord it sends peace, and according to
26. the observation, not the extending
27. of misfortune to the king my lord
28. may there be.
It would seem, as Tannery says,’ that these clever people knew
how to turn their ignorance to account as well as their knowledge.
For them it was apparently of less consequence that their predic-
tions should come true than that they should not let an eclipse take
place without their having predicted it.*
As it is with Egypt that legend associates Thales, it is natural
to ask whether the Egyptians too were acquainted with the period
of 223 lunations. We have no direct proof; but Diodorus Siculus —
says that the priests of Thebes predicted eclipses quite as well as
the Chaldeans,* and it is quite possible that the former had learnt
from the latter the period and the notions on which the successful
prediction of eclipses depended. It is not, however, essential to
suppose that Thales got the information from the Egyptians; he
_ may have obtained it more directly. Lydia was an outpost of
Ea ea ν Ἔ4 -
1. Tannery, op. cit., p. 57. ob τς :
* Delambre (Hist. de /astronomie ancienne, i, p. 351) quotes a story that in
China, in 2159 B.C., the astronomers Hi and Ho were put to death, according
to law, in consequence of an eclipse of the sun occurring which they had not
3 Cf. Diodorus, i, c. 50; ii, c. 30.
1410 G
18 THALES PARTI _
Assyrio-Babylonian culture ; this is established by (among other
things) the fact of the Assyrian protectorate over the kings Gyges
and Ardys (attested by cuneiform inscriptions); and ‘no doubt the
inquisitive Ionians who visited the gorgeous capital Sardes, situated
in their immediate neighbourhood, there first became acquainted
with the elements of Babylonian science’.? .
If there happened to be a number of possible solar eclipses in the
year which (according ‘to Herodotus) Thales fixed, he was not
taking an undue risk; but it was great luck that it should have
been total.?
Perhaps I have delayed too long over the story of the eclipse ;
but it furnishes a convenient starting-point for a consideration
of the claim of Thales to be credited with the multitude of other
discoveries in astronomy attributed to him by the Doxographi and
others, First, did he know the cause of eclipses? Aétius says
that he thought the sun was made of an earthy substance,® like
the moon, and was the first to declare that the sun is eclipsed
when the moon comes in a direct line below it, the image of the
moon then appearing on the sun’s disc as on a mirror ;* and again ©
he says that Thales, as. well as Anaxagoras, Plato, Aristotle, and
the Stoics, in accord with the mathematicians, held that the moon
is eclipsed by reason of its falling into the shadow made by the
earth when the earth. is between the two heavenly bodies. But, as
regards the eclipse of the moon, Thales could not have given this _
explanation, because he held that theearth floated on the water ; ° from
which it may also be inferred that he, like his successors down. to
Anaxagoras inclusive, thought the earth to be a disc or a short
cylinder. And if he had given the true explanation of the solar
eclipse, it.is impossible that all the succeeding Ionian. philosophers
should have exhausted their imaginations in other fanciful capigee’
tions such as we find recorded.”
We may assume that Thales would regard the sun and the moon
as discs like the earth, or perhaps as hollow bowls which could
1 Gomperz, Griechische Denker, 15, p. 421. ἃ Torey. op. cit., p. 60.
® Aét. li. 20. 9 (D. G. p. 349). * Aét. ii. 24.1 (20. Ὁ. pp. 353, 354).
5 Aét. ii. 29. 6 (D. G. p. 360).
ὁ. Theophrastus.apud Simpl. zz Phys. p. 23. 24 (D.G. p. 475; Vors. i’, p. 9.
22); cf. Aristotle, Metaph. A. 3, 983b 21; De caedo ii. 13, hg 28.
1 Tannery, op. cit., p. 56.
a
- rs
pe ye
CH.III | | THALES 19
turn so as to show a dark side.1 We must reject the statements
of Aétius that he was the first to hold that the moon is lit up by
the sun, and that it seems to suffer its obscurations each month
when it approaches the sun, because the sun illuminates it from
‘one side only.2_ For it was Anaxagoras who first gave the true
Scientific doctrine that the moon is itself opaque but is lit up by
the sun, and that this is the explanation no less of the moon’s
_ phases than of eclipses of. the sun and moon; when we read
in Theon of Smyrna that, according to Eudemus’s History of
Astronomy, these discoveries were due to Anaximenes,* this would
seem to be an error, because the Doxographi say nothing of any
explanations of eclipses by Anaximenes,* while on the other hand
Aétius does attribute to him the view that the moon was made
of fire, just as the sun and stars are made of fire.®
We must reject, so far as Thales is concerned, the traditions that
*Thales, the Stoics, and their schools, made the earth spherical’,’
and that ‘the school of Thales put the earth in the centre’.®
For (1) we have seen that Thales made the earth a circular or
cylindrical disc floating on the water like a log® or a cork; and (2),
so far as we can judge of his conception of the universe, he would
_ appear to have regarded it as a mass of water (that on which the
earth floats) with the heavens superposed in the form of a hemisphere
and also bounded by the primeval water. It follows from this
conception that for Thales the sun, moon, and stars did not, between
their setting and rising again, continue their circular path de/ow the
earth, but (as with Anaximenes later) laterally round the earth.
Tannery *° compares Thales’ view of the world with that found ©
in the ancient Egyptian papyri. In the beginning existed the Vz,
a primordial liquid mass in the limitless depths of which floated
the germs of things. When the sun began to shine, the earth was
flattened out and the waters separated into two masses. The one
gave rise to the rivers and the ocean ; the other, suspended above,
_ formed: the vault of heaven, the waters above, on which the stars
1 Tannery, op. cit., p. 70. * Aét.ii.28.5; 29.6(D. G. p. 358. 19; p. 360. 16).
5 Theon of Smyrna, p. 198. Aes 2
* Tannery, op. cit., pp. 56, I 5 Aét. ii. 25. 2 (D. G. p. 356. 1).
® Aéte ii. 20. 2 (D. G. p. 348. Ὁ; Hippol. Refut. i. 7. 4 (D. G.-p. ey: 3).
7 Aét. iii. το. τ (D. G. p. 376. 22). Aét. iti. 11. 1 (D.G. p. 377. 7).
* Aristotle, De cae/o ii. 13, 294 a 30. 10 Tannery, op. cit., p. 71.
C2
20 THALES . PARTI
and the gods, borne by an eternal current, began to float. The
sun, standing upright in his sacred barque which had endured
millions of years, glides slowly, conducted by an army of secondary
gods, the planets and the fixed stars. The assumption of an
upper and lower ocean is also old-Babylonian (cf. the division in
Gen. i. 7 of the waters which were under the firmament from the
waters which were above the firmament).
In a passage quoted by Theon of Smyrna, Eudemus attributed
to Thales the discovery of ‘the fact that the period of the sun with
respect to the solstices is not always the same’! The expres-
sion is ambiguous, but it must apparently mean the inequality of
the length of the four astronomical seasons, that is, the four parts
of the tropical year? as divided by the solstices and the equinoxes.
Eudemus referred presumably to the two written works by Thales
On the Solstice and On the Equinox,’ which again would seem to be
referred to in a later passage of Diogenes Laertius: ‘Lobon of Argos
says that his written works extend to 200 verses’. Now Hesiod,
in the Works and Days, advises the commencement of certain
operations, such as sowing, reaping, and threshing, when particular
constellations rise or set in the morning, and he uses the solstices
as fixed periods, but does not mention the equinoxes. Tannery ἢ
thinks, therefore, that Thales’ work supplemented Hesiod’s by the
addition of other data and, in particular, fixed the equinoxes in
the same way as Hesiod had fixed the solstices. The inequality
of the intervals between the equinoxes and the solstices in one
year would thus be apparent. This explanation agrees with the
remark of Pliny that Thales fixed the matutinal setting of the
Pleiades on the 25th day from the autumnal equinox. All this
knowledge Thales probably derived from the Egyptians or the
Babylonians. The Babylonians, and doubtless the Egyptians also,
1 Theon of Smyrna, p. 198. 17 (Θαλῆς εὗρε πρῶτος) . . . τὴν κατὰ Tas τροπὰς
αὐτοῦ περίοδον, ὡς οὐκ ἴση ἀεὶ συμβαίνει.
3 The ‘tropical year’ is the time required by the sun to return to the same
position with reference to the equinoctial points, while the ‘sidereal year’ is the
time taken to return to the same position with reference to the fixed stars.
8 Diog. L. i. 23 (Vors. i*, p. 3. 18).
4 Tannery, op. cit., p. 66.
5 Pliny, ΔΝ. H. xviii, c. 25, ὃ 213 (Vors.i?, p.9. 44). This datum points to Egypt
as the source of Thales’ information, for the fact only holds good for Egypt and
not for Greece (Zeller, 15, p. 184; cf. Tannery, op. cit., p. 67).
μ᾿ Ἢ ’
CH. III THALES 21
were certainly capable of determining more or less roughly the
solstices and the equinoxes; and they would doubtless do this
by means of the gzomon, the use of which, with that of the folos,
the Greeks are said to have learnt from the Babylonians.'
Thales equally learnt from the Egyptians his division of the
year into 365 days;* it is possible also that he followed their
arrangement of months of 30 days each, instead of the practice
_ already in his time adopted in Greece of reckoning by lunar months.
The Doxographi associate Thales with Pythagoras and his school
as having divided the whole sphere of the heaven by five circles,
the arctic which is always visible, the summer-tropical, the
equatorial, the winter-tropical, and the antarctic which is always
invisible ; it is added that the so-called zodiac circle passes obliquely
to the three middle circles, touching all three, while the meridian
Ἷ circle, which goes from north to south, is at right angles to all the
five circles.* But, if Thales had any notion of these circles, it
must have been of the vaguest; the antarctic circle in particular
_ presupposes the spherical form for the earth, which was not the
form which Thales gave it. Moreover, the division into zones is
elsewhere specifically attributed to Parmenides and Pythagoras;
and, indeed, Parmenides and Pythagoras were the first to be in
a position to take this step,* as they were the first to hold that
the earth is spherical in shape. Again, Eudemus is quoted® as
distinctly attributing the discovery of the ‘cincture of the zodiac
(circle)’ to Oenopides, who was at least a century later than Thales.
Diogenes Laertius says that, according to some authorities,
Thales was the first to declare the apparent size of the sun (and
the moon) to be 1/720th part of the circle described by it.6 The
version of this story given by Apuleius is worth quoting for a human
touch which it contains:
? Herodotus, ii. 109.
- ® Herodotus (ii. 4) says that the Egyptians were the first of men to discover
the year, and that they divided it into twelve parts, ‘therein adopting a wiser
system (as it seems to me) than the Greeks, who have to put in an intercalary
month every third year, in order to keep the seasons right, whereas the Egyptians
give their twelve months thirty days each and add five every year outside the
4 number (of twelve times 30)’. As regards Thales, cf. Diog. L. i. 27 and 24
(Vors. 15, pp. 3. 27; 4. 9). 5. Aét. ii. 12. 1 (D. G. p. 340. 11 sq.).
* As to Parmenides cf. Aét. iii. 11. 4 (2. G. p. 377. 18-20). Rig τὰ
® Theon of Smyrna, p. 198. 14.
-* Diog. L. i. 24 ( Vorsokratiker, i*, p. 3. 25).
22 THALES “PARTI
‘The same Thales in his declining years devised a marvellous
calculation about the sun, which I have not only learnt but verified
by experiment, showing how often the sun measures by its own
size the circle which it describes. Thales is said to have communi-
cated this discovery soon after it was made to Mandrolytus of
Priene, who was greatly delighted with this new and unexpected
information and asked Thales to say how much by way of fee he
required to be paid to him for so important a piece of knowledge.
“T shall be sufficiently paid”, replied the sage, “1, when you set
to work to tell people what you have learnt from me, you will not
take credit for it yourself but will name me, rather than another,
as the discoverer.” }
Seeing that in Thales’ system the sun and moon did not pass
under the earth and describe a complete circle, he could hardly
have stated the result in the precise form in which Diogenes gives
it. If, however, he stated its equivalent in some other way, it
is again pretty certain that he learnt it from the Egyptians or
Babylonians, Cleomedes,? indeed, says that, by means of a water-
clock, we can compare the water which flows out during the time
that it takes the sun when rising to-get just clear of the horizon
with the amount which flows out in the whole day and night;
in this way we get a ratio of 1 to 750; and he adds that this
method is said to have been first devised by the Egyptians. Again,
it has been suggested® that the Babylonians had already, some
sixteen centuries before Christ, observed that the sun takes 1/30th
of an hour to rise. This would, on the assumption of 24 hours for
a whole day and night, give for the sun’s apparent diameter 1/720th
of its circle, the same excellent approximation as that attributed
to Thales. But there is the difficulty that, when the Babylonians
spoke of 1/goth of an hour in an equinoctial day as being the
‘measure’ (ὅρος) of the sun’s course, they presumably meant 1/30th
of their doudble-hour, of which there are 12 in a day and night,
so that, even if we assume that the measurement of the sun’s
apparent diameter was what they meant by ὅρος, the equivalent
? Apuleius, F/or. 18 (Vors. i*, p. 10. 3-11).
* Cleomedes, De motu circulari corporum caelestium ii. 1, pp. 136. 25-138.
6, ed. Ziegler.
* Hultsch, Poseidonios iiber die Grisse und Entfernung der Sonne, 1897,
pp- 41, 42. Hultsch quotes Achilles, /sagoge in Arati phaen.18(Uranolog. Petavii,
Paris, 1630, p. 137); Brandis, M/iinz-, Mass- und Gewichtswesen in Vorderasien,
p. 17 sq.3 Bilfinger, Die babylonische Doppelstunde, Stuttgart, 1888, p. 21 sq.
The passage of Achilles is quoted 7” extenso by Bilfinger, p. 21.
= Soe ὑμὴν
᾿
—
CH. THALES 23
would be 1°, not 3° as Hultsch supposes.1 However, it is difficult to
believe that Thales could have made the estimate of 1/720th of
the sun’s circle known to the Greeks; if he had, it would be very
strange that it should have been mentioned by no one earlier than
Archimedes, and that Aristarchus should in the first instance have
used the grossly excessive value of 2° which he gives as the angular
diameter of the sun and moon in his treatise On the sizes and
distances of the sun and moon, and should have been left to dis-
cover the value of 4° for himself as Archimedes says he did.?
A few more details of Thales’ astronomy are handed down. He
said of the Hyades that there are two, one north and the other
south. According to Callimachus,* he observed the Little Bear ;
_. ‘he was said to have used as a standard [i.e. for finding the pole]
the small stars of the Wain, that being the method by which
Phoenician navigators steer their course. According to Aratus®
the Greeks sailed by the Great Bear, the Phoenicians by the Little
Bear. Consequently it would seem that Thales advised the Greeks
to follow the Phoenician plan in preference to their own. This use
of the Little Bear was probably noted in the handbook under the
title of Nautical Astronomy attributed by some to Thales, and
by others to Phocus of Samos,*® which was no doubt intended to
improve upon the Astronomy in poetical form attributed to Hesiod,
as in its turn it was followed by the Astrology of Cleostratus.?
1 An estimate amounting to 1° is actually on record in Cleomedes (De motu
circulari, ii. 3, p. 172. 25, Ziegler), who says that ‘ the size of the sun and moon
ike appears to our perception as 12 dactyli’._ Though this way of describing
the angle follows the Babylonian method of expressing angular distances |
between stars in terms of the e// (πῆχυς) consisting of 24 dactyli and equivalent
to 2°, it does not follow that the estimate itself is Babylonian. For the same
system of expressing angles may have been used by Pytheas and was certainly
used by Hipparchus (cf. Strabo, ii. 1. 18, p. 75 Cas., Hipparchiin Arati et Eudoxi
phaenomena
comment. ii. 5. 1, Ὁ. 186. 11, Manit., and Ptolemy, Syzfazis vii. 1,
vol. ii, pp. 4-8, Heib.).
2 Archimedes, ed. Heiberg, vol. ii, p. 248.19; The Works 07 Archimedes, ed.
Heath, p. 223.
3 Schol. Arat. 172, p. 369. 24 (Vors. ii. 1*, p. 652).
* In Diog. L. i. 23 (Vors. i?, p. 3. 14; cf. ii. 2, p. v).
5 Aratus, lines 27, 37-39; cf. Ovid, 77tstia iv. 3. 1-2:
‘ Magna minorque ferae, quarum regis altera Graias,
; Altera Sidonias, utraque sicca, rates’ ;
Theo in Arati phaen. 27. 39: Scholiast on Plat. Rep. 600 A.
δ Diog. L. i, p. 23; Simpl. # Phys.p. 23. 29; Plutarch, Pyth. or. 18, 402 F(Vors.
i?, pp. 3. 125 11. 7, 13). 7 Diels, Vors. ii. 1°, p. 6525 cf. pp. 499, 502.
IV
ANAXIMANDER
ANAXIMANDER of Miletus (born probably in 611/10, died soon
after 547/6 B.C.), son of Praxiades, was a fellow citizen of Thales,
with whom he was doubtless associated as a friend if not as a pupil.
A remarkably original thinker, Anaximander may be regarded as
the father or founder of Greek, and therefore of western, philosophy.
He was the first Greek philosopher, so far as is known, who
ventured to put forward his views in a formal written treatise.
This was a work Adout Nature? though possibly that title was
given to it, not by Anaximander himself, but only by later writers.*
The amount of thought which went to its composition and the
maturity of the views stated in it are indicated by the fact that
it was not till the age of 64 that he gave it. to the world. The
work itself is lost, except for a few lines amounting in no case
to a complete sentence.
Anaximander boldly maintained that the earth is in the centre
of the universe, suspended freely and without support,° whereas
Thales regarded it as resting on the water, and Anaximenes as
supported by the air. It remains in its position, says Anaximander,
because it is at an equal distance from all the rest (of the heavenly
bodies). Aristotle expands the explanation thus:’ ‘for that
which is located in the centre and is similarly situated with
reference to the extremities can no more suitably move up than
1 Themistius, Orationes, 36, p. 317 C (Vors. i*; p. 12. 43).
? Ibid. ; Suidas, 5. Ὁ.
® Zeller, Philosophie der Griechen, ἴδ, p. 197.
* Diog. L. ii. 2 (Vors. i?, p. 12. 7-10).
° Hippol. Refuz. i. 6. 3 (D.G. p. 559. 22; Vors. i*, p. 14. 5).
® Ibid.; cf. Plato’s similar view in Phaedo 108 E-109 A.
7 De caelo ii. 13, 295 Ὁ 10-16. It is true that Eudemus (in Theon of Smyrna,
p- 198. 18) is quoted as saying that Anaximander held that ‘the earth is suspended
freely and moves (κινεῖται) about the centre of the universe’; but there must
clearly be some mistake here ; perhaps κινεῖται should be κεῖται (‘ lies’).
;
,
β
5
3
:
᾿
ἶ
᾿
as
ANAXIMANDER 25
down or laterally, and it is impossible that it should move in
opposite directions (at the same time), so that it must necessarily
remain at rest.’ Aristotle admits that the hypothesis is daring
and brilliant, but argues that it is not true: one of his grounds
is amusing, namely, that on this showing a hungry and thirsty man
with food and wine disposed at equal distances all round him would
have to starve because there would be no reason for him to stretch
his hand in one direction rather than another! (presumably the first
occurrence of the well-known dilemma familiar to the schoolmen
as the ‘ Ass of Buridan’).
According to Anaximander, the earth has the shape of a cylinder,
round, ‘like a stone pillar’;* one of its two plane faces is that on
which we stand, the other is opposite ;* its depth, moreover, is one-
third of its breadth.*
Still more original is Anaximander’s conception of the origin and
substance of the sun, moon, and stars, and of their motion. As
there is considerable difference of opinion upon the details of the
_ system, it will be well, first of all, to quote the original authorities,
beginning with the accounts of the cosmogony.
‘ Anaximander of Miletus, son of Praxiades, who was the successor
and pupil of Thales, said that the first principle (i.e. material cause)
and element of existing things is the Infinite, and he was the first
to introduce this name for the first principle. He maintains that
it is neither water nor any other of the so-called elements, but
another sort of substance, which is infinite, and from which
all the heavens and the worlds in them are produced ; and into
that from which existent things arise they pass away once more, —
“as is ordained ; for they must pay the penalty and make reparation
to one another for the injustice they have committed, according to
the Sequence of time”, as he says in these somewhat poetical
terms.’
1 Aristotle, De cae/o ii. 13, 295 Ὁ 32.
3 Hippol. Refus. i. 6. 3 (D.G. p. 559. 24; Vors. i?, p. 14.6); Aét. iii. το. 2
(D. G. p. 376; Vors. i*, p. 16. 34).
3 Hippol., loc. cit.
* Ps. Plut. Stromat. 2 (D.G. p. 579. 12; Vors. i?, p. 13. 34).
® Simplicius, ix Phys. p. 24. 13 (Vors. 15, p.13.2-9). The passage is from
Theophrastus’s Phys. Ofin., and the words in inverted commas at all events are
_ Anaximander’s own. I follow Burnet (Zarly Greek Philosophy, p. 54) in making
the quotation begin at ‘as is ordained’; Diels includes in it the words just
preceding ‘and into that from which...’
a6 ANAXIMANDER PARTI
‘ Anaximander said that the Infinite contains the whole cause of
the generation and destruction of the All; it is from the Infinite
that the heavens are separated off, and generally all the worlds,
which are infinite in number. He declared that destruction and,
long before that, generation came about for all the worlds, which
arise in endless cycles from infinitely distant ages.’ ὦ
‘He says that this substance [the Infinite] is eternal and ageless,
and embraces all the worlds. And in speaking of time he has in
mind the separate (periods covered by the) three states of coming
into being, existence, and passing away. ἢ
‘Besides this (Infinite) he says there is an eternal motion, in the
course of which the heavens are found to come into being.’ ὃ
‘Anaximander says eternal motion is a principle older than
the moist, and it is by this eternal motion that some things are
generated and others destroyed.’
‘ He says that (the first principle or material cause) is boundless,
in order that the process of coming into being which is set up may
not suffer any check.’ ὅ
‘Anaximander was the first to assume the Infinite as first
principle in order that he may have it available for his new births
without stint.’ ®
‘ Anaximander ... said that the world is perishable.’ ἴ
‘Those who assumed that the worlds are infinite in number, as
did Anaximander, Leucippus, Democritus, and, in later days,
Epicurus, assumed that they also came into being and passed
away, ad infinitum, there being always some worlds coming into
being and others passing away; and they maintained that motion
is eternal; for without motion there is no coming into being or
passing away. ὃ
‘ Anaximander says that that which is capable of begetting the
hot and the cold out of the eternal was separated off during the
coming into being of our world, and from the flame thus produced
a sort of sphere was made which grew round the air about the
earth as the bark round the tree; then this sphere was torn off and
1 Ps. Plut. Stromat.2 (D.G. p. 579; Vors.i*®, p. 13. 29 sq.). This passage
again is from Theophrastus.
2 Hippol. Refut. i. 6.1 (D. G. p. 559; Vors. i*, pp. 13. 44-14. 2).
Z ee 1) 6; ᾿: ive
ermias, /rris. 10 (D. G. p. 653; Vors. i*, p. 14. 21).
5 Aét. i. 3. 3 (D. G. Ὁ. 277 ΑΝ ἐδ; 14. An
δ Simplicius on De caelo, p. 615. 13 (Vors. 13, p. 15. 24). In this passage
Simplicius calls Anaximander a ‘fellow citizen and friend’ of Thales (Θαλοῦ
πολίτης καὶ ἑταῖρος) ; these appear to be the terms used by Theophrastus, to
judge by Cicero’s equivalent ‘ popularis et sodalis’ (Acad. gr. ii. 37. 118).
7 Aét. ii. 4. 6 (D.G. p. 3315 Vors. 13, p. 15. 33).
8 Simplicius, 7 Phys. p, 1121. αὶ (Vors. i*, p. 15. 34-8).
.
CH. IV ANAXIMANDER 27
became enclosed in certain circles or rings, and thus were formed
_ the sun, the moon, and the stars.’!
_ *The stars are produced as a circle of fire, separated off from the
_ fire in the universe and enclosed by air. They have as vents certain
_ pipe-shaped passages at which the stars are seen; it follows that
it is when the vents are stopped up that eclipses take place.’ *
‘ The stars are compressed portions of air, in the shape of wheels,
- filled with fire, and they emit flames at some point from small
_ openings.’ 8
ΠΟ *The moon sometimes appears as waxing, sometimes as waning,
to an extent corresponding to the closing or opening of the
passages.’ *
_ Further particulars are given of the circles of the sun and moon,
including the first speculation about their sizes:
‘The sun is a circle 28 times the size of the earth; it is like
a wheel of a chariot the rim of which is hollow and full of fire,
and lets the fire shine out at a certain point in it through an
_ opening like the tube of a blow-pipe ; such is the sun.’®
_ ‘The stars are borne by the circles and the spheres on which
each (of them) stands.’ ὃ
1 Ps. Plut. Stromat. loc. cit.
" Hippol. ἜΡΟΝ bs 4 (D.G. pp. 559 560; ai i*, p. 14. 8).
Aét. ii. 13. 7 (D.G. p. 342; Vors. i*, Ὁ. 15. 39).
ἢ * Hippol., loc. ἐς : .
᾿ς §& Aé€t. ii. 20. 1 (D. G. p. 348; Vors. i*, p. 16. 8).
ἢ ® Aét. ii. 16. 5 (D.G. p. 345; Vors.i*, p. 15. 43. This sentence presents diffi-
_ culties. It occurs in a collection of passages headed ‘ Concerning the motion of
stars’, and reads thus: ᾿Αναξίμανδρος ὑπὸ τῶν κύκλων καὶ τῶν σφαιρῶν, ἐφ᾽ ὧν
ἕκαστος βέβηκε, φέρεσθαι. If ἕκαστος Means ἕκαστος τῶν ἀστέρων, each of the
_ stars, the expression ἐφ᾽ ὧν ἕκαστος βέβηκε, ‘on which each of them stands’ or
‘is fixed’, is certainly altogether inappropriate to Anaximander’s system; it ~
suggests Anaximenes’ system of stars ‘fixed like nails on a crystal sphere’; I am
therefore somewhat inclined to suspect, with Neuhauser (Anaximander Milesius,
Ρ. 362 note), that the words ἐφ᾽ ὧν ἕκαστος βέβηκε (if not καὶ τῶν σφαιρῶν also)
are wrongly transferred from later theories to that of Anaximander. It occurred
to me whether ἕκαστος could be ἕκαστος τῶν κύκλων, ‘each of the circles’ ; for it
would be possible, I think, to regard the circles as ‘standing’ or ‘ being fixed’
on (imaginary) spheres in order to enable them to revolve about the axis of such
spheres, it being difficult to suppose a wheel to revolve about its centre when it
has no spokes to connect the centre with the circumference.
Diels (‘Ueber Anaximanders Kosmos’ in Archiv fiir Gesch. d. Philosophie, x,
1897, p. 229) suggests that we may infer from the word ‘spheres’ here used that the
_ tings are not separate for each star, but that the fixed stars shine through vents
_ On one ring (which is therefore a sphere); the planets with their different motions
_ would naturally be separate from this. I doubt, however, whether this is
_ correct, since @// the rings are supposed to be like wheels; they are certainly
not spheres. But no doubt the Milky Way may be one ring from which
28 ANAXIMANDER PART I
‘The circle of the sun is 27 times as large (as the earth and that)
of the moon (is 19 times as large as the earth).’ }
‘ The sun is equal to the earth, and the circle from which the sun
gets its vent and by which it is borne round is 27 times the size of
the earth.’ ?
‘The eclipses of the sun occur through the opening by which the
fire finds vent being shut up.’ 8
‘The moon is a circle 19 times as large as the earth; it is
similar to a chariot-wheel the rim of which is hollow and full
of fire, like the circle of the sun, and it is placed obliquely like the
other ; it has one vent like the tube of a blowpipe; the eclipses of
the moon depend on the turnings of the wheel.’ 5
‘The moon is eclipsed when the opening in the rim of the wheel
is stopped up.’®
‘The sun is placed highest of all, after it the moon, and under
them the fixed stars and the planets.’ ®
We are now in a position to make some comments. First, what
is the nature of the eternal motion which is an older principle than
water and by which some things are generated and others destroyed ?
Teichmiiller held it to be circular revolution of the Infinite, which
he supposed to be a sphere, about its axis ;’ Tannery adopted the
same view.® Zeller® rejects this for several reasons. There is no
evidence that Anaximander conceived the spherical envelope of
fire to be separated off by revolution of the Infinite and spread
out over the surface of its mass; the spherical envelope lay, not
round the Infinite, but round the atmosphere of the earth, and it
was only the world, when separated off, which revolved ; it is the
world too, not the Infinite, which stretches at equal distances, and
therefore in the shape of a sphere, round the earth as centre.
Lastly, a spherical Infinite is in itself a gross and glaring contra-
diction, which we could not attribute to Anaximander without
a multitude of stars flame forth at different vents: this may indeed be the idea
from which the whole theory started (Tannery, op. cit., Ρ. 91; Burnet, Zarly
Greek Philosophy, p. 69).
1 Hippol., Refut. i. 6. 5 (D. G. p. 560; Vors. i*, p. 14. 12, and ii. 1°, p. 653).
? Aét. ii. 21.1 (D.G. p. 351; Vors. i®, pe 16. 11).
8 Aét. ii, 24. 2 (D.G. p. 354; Vors. 13, p. 16. 13).
* Aét. ii. 25. 1 (22. α. p. 355; Vors. i*, p. 16. 15).
5 Aét. ii. 29. 1 (D. G. p. 359; Vors. i*, p. 16. 19).
6 Aét. ii. 15.6 (D.G. p. 345; Vors. i*, p. 15. 41).
, Ὁ Teichmiiller, Studien zur Gesch. der Begriffe, Berlin, 1874, pp. 25 564.
® Tannery, op. cit., pp. 88 sqq.
9. Zeller, i°, p. 221.
~~
a le
Se a eee eT eS ae ee ΣΤῊ
CH. IV ANAXIMANDER 29
direct evidence. Tannery! gets over the latter difficulty by the
assumption that the Infinite was not something infinitely extended
in space but qualitatively indeterminate only, and in fact finite in
extension. This is rather an unnatural interpretation, especially
in view of what we are told of the ‘infinite worlds’ which arise
_ from the Infinite substance. The idea here seems to be that the
Infinite is a boundless stock from which the waste of existence is
continually made good? With regard to the ‘infinite worlds’
_ Zeller* held that they were an infinity of successive worlds, not
an unlimited number of worlds existing, or which may exist, at
_ the same time, though of course all are perishable; but in order
to sustain this view Zeller was obliged to reject a good deal of the
evidence. Burnet* has examined the evidence afresh, and adopts
the other view. In particular, he observes that it would be
very unnatural to understand the statement that the Boundless
ἶ ‘encompasses the worlds’ of worlds succeeding one another in
time; for on this view there is at a given time only one world
to ‘encompass’. Again, when Cicero says Anaximander’s opinion
‘was that there were gods who came into being, rising and setting
_ at long intervals; and that these were the ‘innumerable worlds’ ® (cf.
_ Aétius’s statement that,according to Anaximander, the ‘innumerable
_ heavens’ were gods‘), it is more natural to take the long intervals
_ as intervals of space than as intervals of time ;7 and, whether this
is so or not, we are distinctly told in a passage of Stobaeus that
‘of those who declared the worlds to be infinite in number,
Anaximander said that they were at equal distances from one
another’, a passage which certainly comes from Aétius.2 Neu- ~
hauser,? too, maintains that Anaximander asserted the infinity of
worlds in two senses, holding both that there are innumerable
worlds co-existing at one time and separated by equal distances,
and that these worlds are for ever, at certain (long) intervals of
1 Tannery, op. cit., pp. 146, 147.
? Burnet, Zarly Greek Philosophy, p. 55.
5 Zeller, i5, pp. 229-36.
* Burnet, Zarly Greek Philosophy, pp. 62-6.
5 Cicero, De nat. deor. i. το. 25 (Vors. i*, p. 15. 27).
5 Aét. i. 7. 12 (D. G. p. 302; Vors. i*, p. 15. 26).
7 Probably, as Burnet says, Cicero found διαστήμασιν in his Epicurean source.
8. Aét. ii. 1. 8 (D. G. p. 329; Vors. i?, p. 15. 32).
* Neubauser, Anaximander Milesius, pp. 327-35.
30 ANAXIMANDER PART I
time,! passing away into the primordial Infinite, and others con-
tinually succeeding to their places.”
The eternal motion of the Infinite would appear to have been
the ‘separating-out of opposites’,? but in what way this operated
is not clear. The term suggests some process of shaking and
sifting as in a sieve.‘ Neuhduser® holds that it is not spatial
motion at all, but motion in another of the four Aristotelian senses,
namely generation, which takes the form of the ‘separating-out of
opposites’, condensation and rarefaction incidentally playing a part
in the process.
As regards the motion by which the actual condition of the
world was brought about (the earth in the centre in the form of
a flat cylinder, the sun, moon, and stars at different distances from
the earth, and the heavenly bodies revolving about the axis of the
universe), Neuhauser ὃ maintains that it was the motion of a vortex
such as was assumed by Anaxagoras, the earth being formed in
the centre by virtue of the tendency of the heaviest of the things
whirled round in a vortex to collect in the centre. But there is
no evidence of the assumption of a vortex by Anaximander;
Neuhiuser relies on a single passage of Aristotle, which however —
does not justify the inference drawn from it."
1 κατὰ τὴν τοῦ χρόνου τάξιν, Simpl. 7” Phys. p. 24. 20 (Vors. i*, p. 13. 9).
2 Cf. Simpl. 72 Phys. p. 1121. 5 (Vors. i*, p. 15. 34-8, quoted above, p. 26).
® of δὲ ἐκ τοῦ ἑνὸς ἐνούσας τὰς ἐναντιότητας ἐκκρίνεσθαι, ὥσπερ ᾿Αναξίμανδρός φησι,
Aristotle, Phys. i. 4, 187 ἃ 20.
* Burnet, Zarly Greek Philosophy, p. 61.
δ᾽ Neuhauser, Anaximander Milesius, pp. 305-15.
ὁ Neuhduser, Anaximander Milesius, pp. 409-21.
7 The passage is Aristotle, De cae/o ii. 13, 295 ἃ 9sqq. It is there stated that
‘if the earth, as things are, is kept dy force where it is, it must also have come
together (by force) through being carried towards the centre by reason of the
whirling motion; for this is the cause assumed by everybody on the ground of
what happens in fluids and with reference to the air, where the bigger and the
heavier things are always carried towards the middle of the vortex. Hence it is
that all who describe the coming into being of the heaven say that the earth came
together at the centre; but the cause of its remaining fixed is still the subject
of speculation. Some hold...’ Now Neuhauser paraphrases the passage thus:
‘All philosophers who hold that the world was generated or brought into being
maintain that the earth is not only kept 4y force in the middle of the world, but
was, at the beginning, also brought together by force. For all assign as the
efficient cause of the concentration of the earth in the middle of the world a
vortex (δίνη), arguing from what happens in vortices in water or air.” It is clear
that Aristotle says no such thing. He says that the philosophers referred to
assert that the earth comes together at the centre, but not that they hold that it
is kept there 4y force ; indeed he expressly says later (295 b 10-16) that Anaxi-
a ΨΥ
ΨΥ Ἂς
CH.IV ANAXIMANDER 31
We come now to Anaximander’s theory of the sun, moon, and
stars. The idea of the formation of tubes of compressed air within
which the fire of each star is shut up except for the one opening
is not unlike Laplace’s hypothesis with reference to the origin of
Saturn’s rings.’ A question arises as to how, if rings constituting
the stars are nearer than the circles of the sun and moon, they fail
to obstruct the light of the latter. Tannery? suggests that, while
of course the envelopes of air need not be opaque, the rarefied
fluid within the hoops, although called by the name of fire, may
also be transparent, and not be seen as flame except on emerging
at the opening. The idea that the stars are like gas-jets, as it
were, burning at holes in transparent tubes made of compressed
air is a sufficiently original conception.
_ But the question next arises, in what position do the circles,
wheels, or hoops carrying the sun, moon, and stars respectively
revolve about the earth? Zeller and Tannery speak of them as
‘concentric’, their centres being presumably the same as the centre
of the earth ; and there is nothing in the texts to suggest any other
supposition. The hoops carrying the sun and moon ‘lie obliquely’,
this being no doubt an attempt to explain, in addition to the daily
rotation, the annual movement of the sun and the monthly move-
ment of the moon. Tannery raises the question of the heights
(‘hauteurs’) of these particular hoops, by which he seems to mean
their dreadihs as they would be seen (if visible) from the centre.
Thus, if the bore of the sun’s tube were not circular but flattened
(like a hoop), in the surface which it presents towards the earth,
to several times the breadth of the sun’s disc, it might be possible .
to explain the annual motion of the sun by supposing the opening
through which the sun is seen to change its position continually
on the surface of the hoop. But there is nothing in the texts to
support this. Zeller* feels difficulty in accepting the sizes of the
hoops as given, on the supposition that the earth is the centre.
mander regarded the earth as remaining at the centre without any force to keep
it there. Again ‘everybody’ is not ‘all philosophers’, but ‘ people in general’.
Lastly, the tendency of the heavier things in a vortex to collect at the centre
might easily suggest that the earth had come together in the centre because it
was heavy, without its being supposed that a vortex was the only thing that could
Cause it to come together.
* Tannery, op. cit., p. 88. 3. Ibid. p. 92.
* Zeller, i°, pp. 224, 225.
32 ANAXIMANDER PART I
For we are told that the sun’s circle or wheel is 27 or 28 times
the size of the earth, while the sun itself is the same size as the
earth; this would mean that the apparent diameter of the sun’s
disc would be a fraction of the whole circumference of the ring
represented by 1/287, that is, the angular diameter would be about
360°/88, or a little over 4°, which is eight times too large, and
would be too great an exaggeration to pass muster even in those
times. Zeller therefore wonders whether perhaps the sun’s circle
should be 27 times the moon’s circle, which would make it 513
times the size of the earth. But the texts, when combined, are
against this, and further it would make the apparent diameter of
the sun much too small. According to Anaximander, the sun
itself is of the same size as the earth; therefore, assuming d to
be the diameter of the sun’s disc and also the diameter of the earth,
the circumference of the sun’s hoop would be 5137rd, so that the
apparent diameter of the sun would be about 1/1600th part of its
circle, or less than half what it really is. Teichmiiller? and
Neuhduser® try to increase the size of the sun’s hoop 3-1416 times,
apparently by taking the diameter of the hoop to be 28 times
the circumference of the earth, ‘because the measurement clearly
depended on an unrolling’; but this is hardly admissible; the
texts must clearly be comparing like with like. Sartorius* feels
the same difficulty, and has a very interesting hypothesis designed
to include provision for the sun’s motion in the ecliptic as well
as the diurnal rotation. He bases himself on a passage of Aristotle
which, according to a statement of Alexander Aphrodisiensis made
on the authority of Theophrastus, refers to Anaximander’s system.
Aristotle speaks of those who explain the sea by saying that
‘at first all the space about the earth was moist, and then, as it
was dried up by the sun, one portion evaporated and set up winds
and the turnings (τροπαί) of the sun and moon, while the remainder
formed the sea’ ; 5
1 Teichmiiller, Studien zur Geschichte der Begriff, 1874, pp. 16, 17.
? Neuhauser, Anaximander Milesius, p. 371.
5. Sartorius, Die Entwicklung der Astronomte bei den Griechen bis Anaxagoras
und Empedokles, pp. 29, 30.
* Aristotle, Metcorologica ii. 1, 353b 6-9. A note of Alexander (in Meteor.
Ῥ.- 67.3; see D.G. p. 494; Vors. i’, p. 16. 45) explains the passage thus: ‘For, the
space round the earth being moist, part of the moisture is then evaporated by
the sun, and from this arise winds and the turnings of the sun and moon, the
Ε΄
CH. IV ANAXIMANDER 33
and again he says in another place :
‘The same absurdity also confronts those who say that the earth,
_ too, was originally moist, and that, when the portion of the world
_ immediately surrounding the earth was warmed by the sun, air was
produced and the whole heaven was thus increased, and that this is
_ how winds were caused and the turnings of the heaven brought
_ about.’?
It is on these passages that Zeller® grounds his view that the
_ heavens are moved by these winds (πνεύματα) and not by the
eternal rotational movement of the Infinite about its axis assumed
by Teichmiiller and Tannery; accordingly, Zeller cannot admit
that the word τροπαΐ in these passages is used in its technical sense
of ‘solstices’.* Sartorius, however, clearly takes the τροπαί to refer
_ specially to the solstices (so does Neuhauser*), and he shows how
the motions of the sun could be represented by two different but
simultaneous revolutions of the sun’s wheel or hoop. Suppose the
_wheel to move bodily in such a way that (1) its centre describes
a circle in the plane of the equator, the centre of which is the
centre of the earth, while (2) the plane of the wheel is always
at right angles to the plane of the aforesaid circle, and always
_ touches its circumference; lastly, suppose the wheel to turn about
_ meaning being that it is by reason of these vapours and exhalations that the
sun and moon execute their turnings, since they turn in the regions where they
receive abundant supplies of this moisture ; but the part of the moisture which is
left in the hollow places (of the earth) is the sea.’
1 Aristotle, Meteorologica ii. 2, 355 a 21. 3 Zeller, 15, p. 223.
3 Zeller (15, pp. 223, 224) has a note on the meanings of the word τροπή. Even
in Aristotle it does not mean ‘solstice’ exclusively, because he speaks of ‘ rporai
_of the stars’ (De caelo ii. 14, 296 Ὁ 4), “ τροπαί of the sun and moon’ (Meteor. ii.
I, 353 b 8), and ‘rpomai of the heaven’ (according to the natural meaning of ras
τροπὰς αὐτοῦ, 3558 25). It is true that τροπαΐ could be used of the moon in
a sense sufficiently parallel to its use for the solstices, for, as Dreyer says
(Planetary Systems, p. 17, note 1), the inclination of the lunar orbit to that of
the sun is so small (se) that the phenomena of ‘turning-back’ of sun and moon
are very similar. But the use of the word by Aristotle with reference to the
stars and the Aeaven shows that it need not mean anything more than the
‘turnings’ or revolutions of the different heavenly bodies. Zeller’s view is, I
think, strongly supported by a passage in which Anaximenes is made to speak of
Stars ‘executing their turnings’ (τροπὰς ποιεῖσθαι Aét. ii. 23. 1, D. G. p. 352) and
the passage in which Anaximander himself is made to say that the eclipses of
tl “agg rot on ‘the turnings (τροπάς) of its wheel’ (Aét. ii, 25.1, D. G.
355 D 22).
* Neuhdauser, op. cit., p. 403.
1410 D
34 ANAXIMANDER PARTI
its own centre at such speed that the opening representing the sun
completes one revolution about the centre of the wheel in a year,
and suppose the centre of the wheel to describe the circle in the
plane of the equator at uniform speed in one day.
In the figure appended, Z represents the earth, the C’s are posi-
tions of the centre of the sun’s hoop or wheel ;
S, represents the sun’s position at the vernal equinox ;
Se ᾿ “ τῆ οἱ summer solstice ;
Ss δ: Ἄ τ δ autumnal equinox ;
S, " i ;: winter solstice.
Fig. 2.
At the winter solstice the sun is south of the equator, at the
summer solstice north of it, and the diameter of the wheel corresponds
to an angle at E& which is double of the obliquity of the ecliptic,
say 47°. . Now, as the diameter of the sun’s wheel is 28 times the
diameter of the earth, i.e. of the sun itself (which is the same size
as the earth), the angular diameter of the sun at & will be about
47°/28 or 1°41’. This is still far enough from the real approximate
value 3°, but it is much nearer than the 4° obtained from the
hypothesis of a hoop with its centre at the centre of the earth.
CH. IV ANAXIMANDER 35
Let us consider what would be the distance of the sun from the
earth on the assumption that the sun’s diameter (supposed to be
equal to that of the earth) subtends at Z an angle of 13°. If d
be the diameter of the earth, and D the distance of the sun from
the earth, we shall have approximately
360 d/12 = 27D,
or D = 34-4 times the diameter of the earth.
But Sartorius’s hypothesis is nothing more than an ingenious
guess, as the texts give no colour to the idea that Anaximander
Fig. 3.
intended to assign a double motion to the sun, nor is there anything
to suggest that the hoops of the sun and moon moved in any
different way from those of the stars, except that they were both
‘placed obliquely’.
The hypothesis of concentric rings with centres at the centre of
the earth seems therefore to be the simplest.
Neuhiuser,} in his attempted explanation of Anaximander’s theory
_ of the sun’s motion, contrives to give to τροπαὶ ἡλίου the technical
_ meaning of solstices, while keeping the ring concentric with the
earth. The flat cylinder (centre O) is the earth, V.P. and S.P. are
the north and south poles, the equator is the circle about 4A’ as
? Neuhdauser, pp. 405-8 and Fig. 2 at end.
D2
46 ANAXIMANDER PARTI
diameter and perpendicular to the plane of the paper. Neuhduser
then supposes the plane of the sun’s circle or hoop to be differently
inclined to the circle of the equator at different times of the year,
making with it at the summer solstice and at the winter solstice
angles equal to the obliquity of the ecliptic in the manner shown in
the figure, where the circle on 4A’ as diameter in the plane of the
paper is the meridian circle and SS’ is the diameter of the sun’s
ring at the summer solstice, BB’ the diameter of the sun’s ring at
the winter solstice. Between the extreme positions at the solstices
the plane of the sun’s hoop changes its inclination slightly day by
day, its section with the meridian plane moving gradually during
one half of the year from the position S.S’ to the position B&B’, and
during the other half of the year from BB’ back to SS” As it
approaches the summer-solstitial position, it is prevented from
swinging further by the winds, which are caused by exhalations, and
which by their pressure on the sun’s ring force it to swing back again.
The exhalations and winds only arise in the regions where there is
abundant water. Neuhduser supposes that Anaximander had the
Mediterranean and the Black Sea in mind, and that their positions —
sufficiently ‘ correspond’ (?) to the summer-solstitial position SS’ to
enable the winds to act as described. There is no sea in such
a position as would enable winds arising from it to repel the sun’s
ring in the reverse direction from BB’ to SS’; consequently
Neuhdauser has to suppose that the ring has an automatic tendency
to swing towards the position SS’ and that it begins to go back
from BB’, of itself, as soon as the force of the wind which repelled
it from SS’ ceases to operate. There is, however, no evidence in
the texts to confirm in its details this explanation of the working of
Anaximander’s system ; on the contrary, there seems to be positive
evidence against it in the phrase ‘ /yizg obliquely ’, used of the hoops
of the sun and moon, which suggests that the hoops remain at fixed
inclinations to the plane of the equator instead of oscillating, as
Neuhiuser’s theory requires, between two extreme positions rela-
tively to the equator.
In any case Anaximander’s system represented an enormous
advance in comparison with those of the other Ionian philosophers
in that it made the sun, moon, and stars describe circles, passing
right under the earth (which was freely suspended in the middle),
δον ὦ ἃ
ek ίλρων. ςς ο..
—
cH.IV ANAXIMANDER . 37
instead of moving laterally round from the place of setting to the
place of rising again.
We are told by Simplicius that
‘ Anaximander was the first to broach the subject of sizes and
distances ; this we learn from Eudemus, who however refers to the
reans the first statement of the order (of the planets) in
71
space.
This brings us back to the question of the sizes of the hoops of
_ the sun and moon as given by Anaximander. We observe that in
one passage the sun’s circle is said to be 28 times as large as the
earth, while in another the circle ‘from which it gets its vent’ is
27 times as large as the earth. Now, on the hypothesis of
concentric rings, we, being in the centre, of course see the inner
circumference at the place where the sun shines through, the
_ sun’s light falling, like a spoke of the wheel, towards the centre.
The words, then, used in the second passage, referring to the circle
Srom which the sun gets tts vent, suggest that the ‘27 times’ refers
_ to the inner circumference of the wheel, while the ‘28 times’ refers
_ to the outer ;? the breadth therefore of the sun’s wheel measured
_ in the direction from centre to circumference is equal to once the
diameter of the earth. A like consideration suggests that it is the
outer circumference of the moon’s hoop which is 19 times the size
of the earth, and that the zaner circumference is 18 times the size
of the earth ; nothing is said in our texts about the size of the moon
itself. Nor are we told the size of the hoops from which the stars
shine, but, as they are in Anaximander’s view nearer to the earth
1 Simplicius on De caelo, p. 471. 4,ed. Heib. (Vors. i*, p. 15.47). Simplicius
adds: ‘ Now the sizes and distances of the sun and moon as determined up to
now were ascertained (by calculations) starting from (observations of) eclipses,
and the discovery of these things might reasonably be supposed to go back as
far as Anaximander.’ If by ‘these things’ Simplicius means the use of the
phenomena of eclipses for the purpose of calculating the sizes and distances of
the sun and moon, his suggestion is clearly inadmissible. On Anaximander’s
theory eclipses of the sun and moon were caused by the stopping-up of the vents
in their respective wheels through which the fire shone out ; moreover, the moon
was itself bright and was not an opaque body receiving its light from the sun,
notwithstanding the statement of Diogenes Laertius (ii. 1; Vors. i*, pp. 11. 40-
12. 1) to the contrary; it is clear, therefore, that Anaximander’s estimates of
sizes and distances rested on no such basis as the observation of eclipses
afforded to later astronomers.
® Diels, ‘ Uber Anaximanders Kosmos’ in Archiv fiir Gesch. d. Philosophie,
x, 1897, p. 231; cf. Tannery, p. 91.
48 ANAXIMANDER PART I
than the sun and moon are, it is perhaps a fair inference that he
would assume for a third hoop or ring containing stars an inner
circumference representing 9 times the diameter of the earth ; the
three rings would then have inner circumferences of 9, 18, 27,
being multiples of 9 in arithmetical progression, while 9 is the
square of 3; this is appropriate also to the proportion of 1:3
between the depth of the disc representing the earth and the
diameter of one of its faces. These figures suggest that they were
not arrived at by any calculation based on geometrical construc-
tions, but that we have merely an illustration of the ancient cult of
the sacred numbers 3 and ο. 3 is the sacred number in Homer,
g in Theognis, 9 being the second power of 3. The cult of 3 and
its multiples 9 and 27 is found among the Aryans, then among
the Finns and Tartars,and next among the Etruscans (the Semites
connected similar ideas with 6 and 7). Therefore Anaximander’s
figures really say little more than what the Indians tell us, namely
that three Vishnu-steps reach from earth to heaven.
The story that Anaximander was the first to discover the
gnomon*® (or sun-dial with a vertical needle) is incorrect, for
Herodotos says that the Greeks learnt the use of the guomon
and the golos from the Babylonians.* Anaximander may, however,
have been the first to ‘introduce’ * or make known the gnomon in
Greece, and to show on it ‘ the solstices, the times, the seasons, and
the equinox’. He is said to have set it up in Sparta.® He is
also credited with constructing a sphere to represent the heavens,’
as was Thales before him.®
But Anaximander has yet another claim to undying fame. He
was the first who ventured to draw a map of the inhabited earth.
The Egyptians had drawn maps before, but only of particular dis-
tricts ;1° Anaximander boldly planned out the whole world with
‘the circumference of the earth and of the sea’.14 Hecataeus, a
much-travelled man, is said to have corrected Anaximander’s map,
1 Diels, loc. cit., p. 233. 2 Diog. L. ii. 1 (Vors. i*, p. 12. 3).
8. Herodotus, ii. 109. 4 εἰσήγαγε, Suidas (Vors. 15, p. 12. 18).
δ Euseb. Praep. Evang. x. 14. 11 (Vors. i*, p. 12. 24).
δ Diog. L. ii. 1. 7 Ibid. ii. 2. 8. Cic. De rep. i. 14. 22.
* Agathemerus (from Eratosthenes), i. 1 (Vors. i?, p. 12. 36).
19 Gomperz, Griechische Denker, 18, pp. 41, 422.
1 Diog. L. ii. 2 (Vors. 15, Ὁ. 12. 5).
ANAXIMANDER 39
‘so that it became the object of general admiration. According to
another account, Hecataeus left a written description of the world
based on the map. In the preparation of the map Anaximander
_ would of course take account of all the information which reached
his Ionian home as the result of the many journeys by land and
sea undertaken from that starting-point, journeys which extended
to the limits of the then-known world ; the work involved of course
an attempt to estimate the dimensions of the earth. We have,
however, no information as to his results.*
Anaximander’s remarkable theory of evolution does not concern
us here.?
-_10On Anaximander’s map see Berger, Geschichte der wissenschaftlichen
_Erdkunde der Griechen, 2 ed., 1903, pp. 35 544.
__ 53 See Plut. Symp. viii. 8. 4 ( Vors.i*, p. 17.24) ; Aét. v. 19.4 (D. G. p. 430; Vors.
2 B17 18); Ps. Plut. Stromat. 2 (D. G. p. $79) ; Hippol. Refut. i. 6. 6 (D. G.
; 560). According to Anaximander, animals first arose from slime evaporated
the sun; ote τῷ first lived in the sea and had prickly coverings; men
at first resembled fishes.
ν
ANAXIMENES
For Anaximenes of Miletus (whose date Diels fixes at 585/4-
528/4 B.C.) the earth is still flat, like a table,’ but, instead of resting
on nothing, as with Anaximander, it is supported by air, riding
upon it, as it were. Aristotle explains this assumption thus :*
‘Anaximenes, Anaxagoras, and Democritus say that its flatness
is what makes it remain at rest; for it does not cut the air below
it but acts like a lid to it, and this appears to be characteristic of
those bodies which possess breadth. Such bodies are, as we know,
not easily displaced by winds, because of the resistance they offer.
The philosophers in question assert that the earth resists the air
below it, in the same way, by its breadth, and that the air, on the
other hand, not having sufficient space to move from its position,
remains in one mass with that which is below it, just as the water
does in water-clocks.’
The sun, moon, and stars are evolved originally from earth; for
it is from earth that moisture arises ; then, when this is rarefied, fire
is produced, and the stars are composed of fire which has risen
aloft. The sun, moon, and stars are all made of fire, and they ride
on the air because of their breadth. The sun is flat like a leaf ;®
it derives its very adequate heat from its rapid motion.’ The stars,
on the other hand, fail to warm because of their distance.®
The stars are fastened on a crystal sphere, like nails or studs.°
1 Aét. iii. 10. 3 (D.G. p. 377; Vors. 13, p. 20. 26).
® Ps. Plut. Stromat. 3 (1). Ο. p. 580; Vors. i’, p. 18.27); Hippol. Refut. i. 7.
4 (ῦ. G. p. 560; Vors. i*, p. 18. 40); Aét. iii. 15. 8 (D. G. p. 380; Vors. 1,
. 20. 34).
He De caelo ii. 13, 294 Ὁ 13 (Vors. i*, p. 20. 27).
* Ps, Plut. Stromat. 3 (D.G. p. 580; Vors. i*, p. 18.27); Hippol. Refud. i.
7.5 (D.G. p. 561; Vors. i*, p. 18. 42). ν
δ Hippol., loc. cit. (Vors. 15, p. 18. 41).
6 Aét. ii, 22. 1 (D. G. p. 352; Vors. 15, p. 20. 5).
7 Ps. Plut. Stromat. 3 (1). G. p. 580; Vors. i*, p, 18. 28).
8. Hippol., loc. cit. ( Vors. i*, p. 19. 1).
9. Aét. 11, 14.3 (D. G. p. 3445 Vors. 13, p. 19. 38).
ANAXIMENES 41
_ The stars do not move or revolve under the earth as some suppose,
gut round the earth, just as a cap can be turned round the head.
Phe sun i is hidden from sight, not because it is under the earth, but
Ee ecause it is covered by the higher parts of the earth and because
5 distance from us is greater.' With this statement may be com-
d the remark of Aristotle that
δι τ πον of the ancient meteorologists were persuaded that the sun
is ποῖ carried under the earth, but round the earth, and in particular
_our northern portion of it, and that it disappears and produces night
‘because the earth is lofty towards the north.”?
es
*
4
‘Th allusion is also to Anaximenes when we are told that some
(ie. Anaximenes) make the universe revolve like a millstone
(μυλοειδῶς), others (i.e. Anaximander) like a wheel.*
_ Now it is difficult to understand how the stars which, being fixed
ἢ a crystal sphere, move bodily with it round a diameter of the
sphere, and which are seen to describe circles cutting the plane of
e horizon at an angle, can do otherwise than describe the portion
of their paths between their setting and rising again by passing
4 er the earth; and all sorts of attempts have been made to
‘explain the contradiction. Schaubach pojnted out that the circles
‘described by the stars could not all converge and meet, say, on the
‘horizon to the north; for then they could not be parallel.*
Ottingey® supposed that the attachment of the stars to the crystal
sphere only held good while they were above the horizon; then,
when they reached the horizon, they became detached and passed
round in the plane of the horizon till they reached the east again! _
_ Zeller, Martin, and Teichmiiller all have explanations which are
More or less violent attempts to make ‘under’ mean pot exactly
‘under’, but something else. Teichmiiller,* to explain the simile of
the cap, observes that the ancients wore their caps, not as we wear
our hats, but tilted back on the neck. The simile of the cap worn
J Hippol., loc. cit, (Vors. i*, pp. 18. 45-19. 1); cf. Aét. ii, 16. 6 (D.G. p. 346;
Vers i’, Ρ. 19. 39)-
3. Aristotle, Meteorologica ii. 1, 354. 28.
5. Aét. ii. 2. 4 (D. G. p. 329 Ὁ, note; Vors. i*, p. 19. 32).
45 Geschichte der griechischen Astronomie bis auf Eratosthenes,
uoted by Sartorius, op. cit; p, 33.
uller, Studien sur Geschichte der Begriffe, 1874, p. 100.
42 ANAXIMENES Ρ ΤΙ
in this way would no doubt be appropriate if Anaximenes >ied
confined his comparison to some stars only, namely those # ΠῚ
north which are always above the horizon and never set; b» th
does not make this limitation; and this view of the cap doe® ,.ἢ
correspond very well to the revolution ‘like a millstone’. ea
More important is the distinction between the motion of 2
fixed stars, which are fastened like nails on the spheres “ἢ
the motion of the sun and moon. Anaximenes says that ἃ
‘The sun and the moon and the other stars float on the air
account of their breadth.’ 1
This is intelligible as regards the sun, because it is like a leaf; b
as regards ‘the other stars’ it seems clear that floating on the aix
inconsistent with their being fastened to the heavenly sphere ; it,
almost necessary therefore to suppose that ‘the other stars’ q
here, not the fixed stars, but the planets, and that this ‘ floating «
the air’ is a hypothesis to explain the disagreement between t
observed motions of the sun, moon, and planets on the one han
and the simple rotation of the stars in circles on the other. We ai
told in another place that, while Anaximenes said that the stars ar,
fastened like nails on the crystal sphere, ‘some’ say that they arc
‘leaves of fire, like pictures’ ;? it is tempting, therefore, to read}
instead of ἔνιοι in the nominative, the accusative ἐνίους (ἀστέρας),
when the meaning would be ‘ but that some of the stars are leaves
of fire’, &c. The idea that the planets are meant in the above
passage is further supported by another statement that
‘The stars execute their turnings (τὰς τροπὰς ποιεῖσθαι) in conse-
quence of their being driven out of their course by condensed air
which resists their free motion.’ 3:
It seems clear that the ‘turnings’ here referred to are not the
‘solstices’, but simply the turnings of the stars in the sense of their
revolution in their respective orbits, so far as they are not fixed on
the crystal sphere;* that is to say, the statement refers to the
planets only.
1 Hippol. Refut. i. 7. 4
3. Aét. ii. 14. 4 (D.G. p.
»G. p.
3 Aét. ii. 23. 1(D.
* Zeller, 15, p. 250.
(D. 561; Vors. iP. 18. 41).
344 ors. it, p, 19. 38).
seat Vors. i*, p. 20. 5).
Se Ue
cH ANAXIMENES 43
rece’ 7ould seem certain therefore that Anaximenes was the first
one. ‘inguish the planets from the fixed stars in respect of their
οἵ lar movements, which he accounted for in the same way as
_iotions of the sun and moon. This being so, it seems not
wo’ sible that the passages about the sun and the stars not
Ἐπ 6 under, but laterally round, the earth refer exclusively to
ἘΠῚ 4n, moon, and planets;’ the fact of their floating on the air
-* t be supposed to be a reason why they should not ever fall
‘5: w the earth, which itself rests on the air, and in this way the
ee culty with regard to the motion of the fixed stars would
£ ppear.
», snother improvement on the system of Anaximander is the
ΠΣ gation of the stars to a more distant region than that in which
@ sun moves. Anaximander had made the sun’s wheel the most
~~ 0te, the moon’s next to it, and those of the stars nearer still
_ the earth; Anaximenes, however, explains that the stars do not
' e warmth because they are too far off, and with this may be
᾿ς mpared his statement that
_ ‘The rotation which is the furthest away from the earth is (that
_ ) the heaven,’?
‘which view is attributed to him in common with Parmenides.
_ Anaximenes made yet another innovation of some significance.
_ He said that
‘ There are also, in the region occupied by the stars, bodies of an
earthy nature which are carried round along with them,’ *
and that,
‘While the stars are of a fiery nature, they ‘also include (or
_ contain) certain earthy bodies which are carried round along with
them but are not visible.’ 5
Zeller® interprets these passages as ascribing an earthy nucleus
to the stars; and this is not unnaturally suggested by the second
of the two passages. But the first passage suggests another possible
1 This was the suggestion of Heeren (Stobaeus, i, p. 511).
® Aét. ii. 11. 1 (D.G. p. 339; Vors. i*, p. 19. 34).
3 Hippol. Refut. i. 7. 5 (2. σ. p. 561; Vors. i, p. 18. 44).
* Aét. ii. 13. 10 (2. G. p. 342; Vors. i*, p. 19. 36).
5 Zeller, i*, pp. 247, 248.
44 ANAXIMENES PAR'I
interpretation ; bodies of an earthy nature iz the region occufed
by the stars (ἐν τῷ τόπῳ τῶν ἀστέρων) might be separate frm
them and not ‘contained in them’, although carried round wih
them. ‘The stars’ in the two passages no doubt include the sm
and moon; but the sun is flat like a leaf; why then shotd
Anaximenes attach to it an earthy substance as well? The object
of the invisible bodies of an earthy nature carried round along with
‘the stars’ is clearly to explain eclipses and the phases of the moon.
If, then, Anaximenes supposed that one side in both the sun and
the moon was bright and the other dark, his idea would doubtless
be that they might sometimes turn their dark side to us in such
a way as to hide from us more or less the bright side. (This was
the idea of Heraclitus, though with him the heavenly bodies had
not a flat surface but were hollowed out like a basin or bowl.) But
the phenomena of eclipses are more simply accounted for if we
suppose the earthy bodies of Anaximenes to be separate from the
sun and moon, and to get in front of them; we need not therefore
hesitate to attribute to him this fruitful idea which ultimately led
to the true explanation. Anaxagoras said that the moon is eclipsed
because the earth is interposed, but, not being able to account for
all the phenomena in this way, he conceived that eclipses were also
sometimes due to obstruction by bodies ‘below the moon’, which
he describes in almost the same words as Anaximenes, namely as
‘certain bodies (in the region) below the stars which are carried
round with the sun and moon and are invisible to us’. Clearly
therefore Anaxagoras was indebted to Anaximenes for this con-
ception ; and again the réle of the counter-earth in the Pythagorean
system is much the same as that of the ‘earthy bodies’ now in
question.
Tannery ' goes further and maintains that Anaximenes’ hypothesis
was bound to lead to the true explanation of eclipses. ‘ For, if any
one asked himself why these dark bodies were not seen at all, the
question of their being illuminated by the sun would present itself,
and it was easy to recognize that, under the most general conditions,
the phenomena which such a dark body would necessarily present
were really similar to the phases of the moon. From this to the
1 Tannery, Pour l'histoire de la science helldne, pp. 153, 154.
.
. recognition of the fact that the moon itself is opaque there was only
one step more. The réle of the moon in regard to the eclipses
_ of the sun was easy to deduce, while the question of the lighting
_ up of the moon by the sun at night naturally brought into play the
_ shadow of the earth and, through that, led to the discovery of the
cause of eclipses of the moon. The hypothesis then of Anaximenes
_ has a true scientific character, and constitutes for him a title to
fame, the more rare because the conception appears to have been
absolutely original, while his other ideas are not in general of the
same stamp.’ While the successive steps towards the discovery
_ of the truth may no doubt have been taken in the order suggested,
it must, I think, be admitted that, at the point where the question
of the illumination of the opaque bodies by the sun would present
: CH. V ANAXIMENES 45
— μους ὧν ΡῈ
itself (‘se posait’), a very active imagination would be required to
suggest the transition to this question ; and, even after the transition
_ was made, it would be necessary to assume further that the opaque
_ bodies are spherical in form, an assumption nowhere suggested by
_ Anaximenes.
Tannery ' adds that the only feature of Anaximenes’ system that
was destined to an enduring triumph is the conception of the stars
being fixed on a crystal sphere as in a rigid frame. Although
attempts were made later to arrive at a more immaterial and less
gross conception of the substance rigidly connecting the fixed stars,
the character of this connexion was not modified, and the rigidity
of the sphere really remained the fundamental postulate of all
astronomy up to Copernicus. The exceptions to the general
adoption of this view were, curiously enough, the Ionian physicists
of the century immediately following Anaximenes.
It would appear that Anaximenes anticipated the Pythagorean
notion that the world breathes, for he says:
‘Just as our soul, being air, holds us together, so does breath and
air encompass the whole world.’ 3
1 Tannery, op. cit., p. 154.
3 Fragment in Aét. i. 3. 4 (D.G. p. 278; Vors. i*, p. 21. 17).
VI
PYTHAGORAS
PYTHAGORAS, undoubtedly one of the greatest names in the
history of science, was an Ionian, born at Samos about
572 B.C., the son of Mnesarchus. He spent his early manhood
in Samos, removed in about 532 B.C. to Croton, where he founded
his school, and died at Metapontium at a great age (75 years
according to one authority, 80 or more according to others). His
interests were as various as those of Thales, but with the difference
that, whereas Thales’ knowledge was mostly of practical application,
with Pythagoras the subjects of which he treats become sciences
for the first time. Mathematicians know him of course, mostly
or exclusively, as the reputed discoverer of the theorem of Euclid
I. 47; but, while his share in the discovery of this proposition
is much disputed, there is no doubt that he was the first to make
theoretical geometry a subject forming part of a liberal education,
and to investigate its first principles.1 With him, too, began the
Theory of Numbers. A mathematician then of brilliant achieve-
ments, he was also the inventor of the science of acoustics, an
astronomer of great originality, a theologian and moral reformer,
founder of a brotherhood ‘ which admits comparison with the orders
of mediaeval chivalry.’ ?
The epoch-making discovery that musical tones depend on
numerical proportions, the octave representing the proportion of
2:1, the fifth 3:2, and the fourth 4: 3, may with sufficient certainty
be attributed to Pythagoras himself,’ as may the first exposition
of the theory of means, and of proportion in general applied to
commensurable quantities, i.e. quantities the ratio between which
can be expressed as a ratio between whole numbers. The all-
? Proclus, Comm. on Eucl. I, Ὁ. 65. 15-19.
3 Gomperz, Griechische Denker, 15, pp. 80, 81.
8. Burnet, Early Greek Philosophy, p. 118.
PYTHAGORAS 47
pervading character of number being thus shown, what wonder
- that the Pythagoreans came to declare that number is the essence
of all things? The connexion so discovered between number and
music would also lead not unnaturally to the idea of the ‘harmony
of the heavenly bodies’.
Pythagoras left no written exposition of his doctrines, nor did
_ any of his immediate successors in the school; this statement is
_ true even of Hippasus, about whom the different stories arose
_ (1) that he was expelled from the school because he published
_ doctrines of Pythagoras,! (2) that he was drowned at sea for
revealing the construction of the dodecahedron in a sphere and
claiming it as his own,? or (as others have it) for making known
the discovery of the irrational or incommensurable.* Nor is the
absence of any written record of early Pythagorean doctrine to
_ be put down to any pledge of secrecy binding the school; there
_ does not seem to have been any secrecy observed at all unless
perhaps in matters of religion or ritual; the supposed secrecy
_ seems to have been invented to explain the absence of any trace
_ of documents before Philolaus. The fact appears to be merely
that oral communication was the tradition of the school, and the
_ closeness of their association enabled it to be followed without
_ inconvenience, while of course their doctrine would be mainly too
abstruse to be understood by the generality of people outside.
Philolaus was the first Pythagorean to write an exposition of
the Pythagorean system. He was a contemporary of Socrates and
Democritus, probably older than either, and we know that he lived
in Thebes in the last decades of the fifth century.*
It is difficult in these circumstances to disentangle the portions
of the Pythagorean philosophy which may safely be attributed to
the founder of the school. Aristotle evidently felt this difficulty ;
he clearly knew nothing for certain of any ethical or physical
doctrines going back to Pythagoras himself; and, when he speaks
of the Pythagorean system, he always refers it to ‘ the Pythagoreans’,
Ἷ sometimes even to ‘the so-called Pythagoreans’.6 The account
| 2 Clem. Stromat. v. 58 (Vors. i®, p. 30.18); Iamblichus, Vit. Pyth. 246, 247
(Vors. i, p. 30. 10, 14).
* Iamblichus, Vit. Pyth. 88 (Vors. 15, p. 30. 2).
* Ibid. 247 (Vors. i*, p. 30. 17).
* Zeller, i°, pp. 337, 338. 5 Burnet, Early Greek Philosophy, p. 100.
48 PYTHAGORAS PART:
which he gives of the Pythagorean planetary system correspond:
to the system of Philolaus as we know it from the Dorographi.
For Pythagoras’s own system, therefore, that of Philolaus afford:
no guide; we have to seek for traces, in the other writers of the
end of the sixth and the beginning of the fifth centuries, of opinions
borrowed from him or of polemics directed against him. On thess
principles we have seen reason to believe that he was the first tc
maintain that the earth is spherical and, on the basis of thi:
assumption, to distinguish the five zones.
How Pythagoras came to conclude that the earth is spherica
in shape is uncertain. There is at all events no evidence that he
borrowed the theory from any non-Greek source. On the assump.
tion, then, that it was his own discovery, different suggestions ὃ have
been put forward as to the considerations by which Pythagoras
convinced himself of its truth. One suggestion is that he may
have based his opinion upon the correct interpretation of phenomena
and above all, on the round shadow cast by the earth in the eclipses
of the moon. But it is certain that Anaxagoras was the first te
suggest this, the true explanation of eclipses. The second possibility
is that Pythagoras may have extended his assumption of a spherical
sky to the separate luminaries of heaven; the third is that hi:
ground was purely mathematical, or mathematico-aesthetical, and
that he attributed spherical shape to the earth for the simple reason
that ‘the sphere is the most beautiful of solid figures’ I prefe:
the third of these hypotheses, though the second and third have the
point of contact that the beauty of the spherical shape may have
1 Tannery, op. cit., p. 203.
* The question is discussed by Berger (Geschichte der wissenschaftlichen
Erdkunde der Griechen, pp. 171-7) who is inclined to think that, along with the
facts about the planets and their periods discovered, as the result of observations
continued through long ages, by the Egyptians and Babylonians, the doctrine οἱ
a suspended spherical earth also reached the Greeks from Lydia, Egypt, οἱ
Cyprus. Berger admits, however, that Diodorus (ii. 31) denies to the Babylonians
any knowledge of the earth’s sphericity. Martin, it is true, in a paper quoted
by Berger (p. 177, note), assumed that the Egyptians had grasped the idea οἱ
a spherical earth, but, as Gomperz observes (Grtechische Denker, i®, p. 430), this
assumption is inconsistent with the Egyptian representation of the earth's shape
as explained by one of the highest authorities on the subject, Maspero, in his
Hist. ancienne des peuples de l’ Orient classique, Les origines, pp. 16, 17.
8. Gomperz, Griechische Denker, 15, p. 90. :
4 Diog. L. viii. 35 (Vors. 13, p. 280. 1) attributes this statement to the
Pythagoreans,
Ι CH. VI PYTHAGORAS 49
4
ὲ dictated its application doth to the universe and to the earth. But,
bs whatever may have been the ground, the declaration that the earth
‘ is spherical was a great step towards the true, the Copernican
_ view of the universe.!_ It may well be (though we are not told)
that Pythagoras, for the same reason, gave the same spherical
_ shape to the sun and moon and even to the stars, in which case
_ the way lay open for the discovery of the true cause of eclipses and
_ of the phases of the moon.
_ There is no doubt that Pythagoras’s own system was geocentric.
_ The very fact that he is credited with distinguishing the zones is
an indication of this; the theory of the zones is incompatible with
_ the notion of the earth moving in space as it does about the central
_ fire of Philolaus. But we are also directly told that he regarded
_ the universe as living, intelligent, spherical, enclosing the earth
_ in the middle, the earth, too, being spherical in shape.* Further,
_ it seems clear that he held that the universe rotated about an axis
_ passing through the centre of the earth. Thus we are told by
Aristotle that
‘Some (of the Pythagoreans) say that ¢¢me is the motion of the
whole (universe), others that it is the sphere itself’ ; ὃ
and by Aétius that
᾿ς ‘Pythagoras held time to be the sphere of the enveloping
eaven).’ +
᾿ς Alemaeon, a doctor of Croton, although expressly distinguished
_ from the Pythagoreans by Aristotle,° is said to have been a pupil
_ of Pythagoras ;*® even Aristotle says that, in the matter of the
Pythagorean pairs of opposites, Alemaeon, who was a young man
*
_ when Pythagoras was old, expressed views similar to those of the
_ Pythagoreans, ‘ whether he got them from the Pythagoreans or they
| from him’. Hence he was clearly influenced by Pythagorean
i]
δ
* Gomperz, Griechische Denker, i*, p. 90.
® Alexander Polyhistor in Diog. L, viii.
5 Aristotle, Phys. iv. 10, 218 a 33.
* Aét. i. 21. 1 (D.G. p. 318; Vors. i*, p. 277. 19).
® Aristotle, Metaph. A. 5, ee 27-31.
5 Diog. L. viii. 83 (Vors. i*, p. 100. 19); Iamblichus, #4. Pyth. 104.
7 Aristotle, Metaph. i. 5, οἶδα 28
1410 Ἑ
5ο PYTHAGORAS PART I
doctrines. Now the doxographers’ account of his astronomy includes
one important statement, namely that
‘Alcmaeon and the mathematicians hold that the planets have
a motion from west to east, in a direction opposite to that of the
fixed stars.’ ἢ
Incidentally, the assumption of the motion of the fixed stars
suggests the immobility of the earth. But this passage is also the
first we hear of the important distinction between the diurnal
revolution of the fixed stars from east to west and the independent
movement of the planets zz the opposite direction; the Ionians say
nothing of it (though perhaps Anaximenes distinguished the planets
as having a different movement from that of the fixed stars);
Anaxagoras and Democritus did not admit it; the discovery,
therefore, appears to belong to the Pythagorean school and, in view
of its character, it is much more likely to have been made by the
Master himself than by the physician of Croton. For the rest
of Alcmaeon’s astronomy is on a much lower level ; he thought
the sun was flat,? and, like Heraclitus, he explained eclipses and
the phases of the moon as being due to the turning of the moon’s
bowl-shaped envelope.* It is right to add that Burnet® thinks
that the fact of the discovery in question being attributed to ©
Alcmaeon implies that it was zo¢ due to Pythagoras. Presumably
this is inferred from the words of Aristotle distinguishing Alemaeon
from the Pythagoreans; but either inference is possible, and
I prefer Tannery’s. It is difficult to account for Alcmaeon being
credited with the discovery if, as Burnet thinks, it was really Plato’s.
But we have also the evidence of -Theon of Smyrna, who states
categorically that Pythagoras was the first to notice that the
planets move in independent circles :
‘The impression of variation in the movement of the planets
is produced by the fact that they appear to us to be carried through
the signs of the zodiac in certain circles of their own, being fastened
in spheres of their own and moved by their motion, as Pythagoras
1 Aét. ii. 16. 2-3 (D. G. p. 345; Vors. 15, p, 101. 8).
3 Tannery, op. cit., p. 208.
3 Aét. ii. 22. 4 ΤΑ G. p. 352; Vors. i?, p. τοι. 10).
* Aét. ii. 29. 3 (29, G. p. 359; Vors. 15, p. 101. 10-12),
5 Burnet, Zarly Greek Philosophy, p. 123, note.
HI PYTHAGORAS 51
was the first to observe, a certain varied and irregular motion being
thus grafted, as a qualification, upon their simply and uniformly
ordered motion in one and the same sense’ [i.e. that of the daily
_ rotation from east to west].
It appears probable, therefore, that the theory of Pythagoras
himself was that the universe, the earth, and the other heavenly
bodies are spherical in shape, that the earth is at rest in the centre,
at the sphere of the fixed stars has a daily rotation from east to
st about an axis passing through the centre of the earth, and
that the planets have an independent movement of their own in
a sense opposite to that of the daily rotation, i.e. from west to east.
* Theon of Smyrna, p. 150. 12-18.
VII
XENOPHANES
XENOPHANES of Colophon was probably born about 570 an
died after 478 B.c. What we know for certain is that he spoke c
Pythagoras in the past tense,’ that Heraclitus mentions him alon
with Pythagoras,? and that he says of himself that, from the ti
when he was 25 years of age, three-score years and seven hai
‘tossed his care-worn soul up and down the land of Hellas.’
He may have left his home at the time when Ionia became a Persia
province (545 B.C.) and gone with the Phocaeans to Elea,* found
by them in 540/39 B.C., six years after they left Phocaea.6 As he w
writing poetry at 92 and is said to have been over 100 when h
died,* the above dates are consistent with the statement that he w
a contemporary of Hieron, who reigned from 478 to 467 B.C.
According to Theophrastus, he had ‘ heard’ Anaximander. ;
Xenophanes was more a poet and satirist than a natural phil
sopher, but Heraclitus credited him with wide learning,® and h
is said to have opposed certain doctrines of Pythagoras and Thales,!
We are told that he wrote epics as well as elegies and iambic
attacking Homer and Hesiod. In particular, 2,000 verses on th
foundation of Colophon and the settlement at Elea are attribute
to him. He is supposed to have written a philosophical poem ;
Diels refers about sixteen fragments to such a poem, to which th
1 Fr. 7 (Vors. i?, p. 47. 20-23). 2 Heraclitus, Fr. 40 ( Vors. 13, p. 68. το).
3 Fr. 8 (Vors. i’, p. 48. 3-6).
* Gomperz, Griechische Denker, 15, pp. 127, 436.
5 Herodotus, i. 164-7.
® Censorinus, De die natali c. 15. 3, p. 28. 21, ed. Hultsch.
7 Timaeus in Clem. Stromat. i. 14, p.353 (Vors. 15, p. 35. 2).
8. Diog. L. ix. 21 (Vors. i*, p. 34. 35).
9. Heraclitus, loc. cit.: ‘Wide learning does not teach one to have under-
standing ; if it did, it would have taught Hesiod and Pythagoras, and again
Xenophanes and Hecataeus.’ |
10 Diog. L. ix. 18 (Vors. 15, p. 34. 12). 1 Ibid. ix. 20 (Vors. i*, p. 34. 26).
.
=. XENOPHANES 53
name On Nature (Περὶ φύσεως) was given; but such titles are
of later date than Xenophanes, and Burnet? holds that all the
‘fragments might have come into the poems directed against Homer
and Hesiod, the fact that a considerable number of them come
from commentaries on Homer being significant in this connexion.
Xenophanes attacked the popular mythology, proving that God
‘must be one, not many (for God is supreme and there can only
‘be one supreme power),? eternal and not born (for it is as impious
to say that the gods are born as it would be to say that they die;
in either case there would be a time when the gods would not be) ;*
he reprobated the scandalous stories about the gods in Homer and
Hesiod * and ridiculed the anthropomorphic view which gives the
bodies, voices, and dress like ours, observing that the Thracians
made them blue-eyed and red-haired, the Aethiopians snub-nosed
_and black,® while, if oxen or horses or lions had hands and could
ἔχων, they would draw them as oxen, horses, and lions respectively.®
God is the One and the All, the universe ;7 God remains unmoved
in one and the same place ;* God is eternal, one, alike every way,
"finite, spherical and sensitive in all parts,? but does not breathe.?°
‘It is difficult to reconcile the finite and spherical God with
_ Xenophanes’ description of the world, which may be summarized
as follows.
The world was evolved from a mixture of earth and water,"
and the earth will gradually be dissolved again by moisture; this
he infers from the fact that shells are found far inland and on
mountains, and in the quarries of Syracuse there have been found
imprints (fossils) of a fish and of seaweed, and so on, these
imprints showing that everything was covered in mud long ago,
? Burnet, Early Greek Philosophy, p. 128.
* Simpl. iw Phys. p. 22. 31 (Vors. i*, p. 40. 30).
® Aristotle, Rhetoric ii. 23, 1399 Ὁ 6.
* Fr. 11 (Vors. i, p. 48. 13). 5 Fr. 14, 16 (Vors. i?, p. 49. 2, 11).
® Fr. 15 (Vors. 1", Ρ 49. 5).
_," Aristotle, Metaph. A. 5, οΒ6 Ὁ 21 (Vors. i, p. 40. 15); Simpl., loc. cit. (Vors.
ΠΡ. 40. 29); cf. Cicero, De nat, deor.i. ττ. 28 ( Vors. #, p. 41. 44); Acad. pr. ii. 37.
118 (Vors. i*, p. 41. 42).
® Fr. 26 (Vors. i*, p. 50. 22).
" Hippol. Refut. i. 14. 2 (D. G. p. 565; Vors. i*, p. 41. 26).
39 Diog. L. ix. 19 (Vors. i*, p. 34. 18).
4 Fr. 29. 33 (Vors.i*, p. 51. 5, 20).
᾿Ξ I read, with Burnet, after Gomperz φυκῶν (seaweed) instead of φωκῶν.
54 XENOPHANES PARTI
arid that the imprints dried on the mud. All men will disappear
when the earth is absorbed into the sea and becomes mud, after
which the process of coming into being starts again; all the worlds —
(alike) suffer this change.! This is, of course, the theory οὗ
_Anaximander. ᾿
As regards the earth we are told that »
‘This upper side of the earth is seen, at our feet, to touch the air, ©
but the lower side reaches to infinity.’ ?
‘This is why some say that the lower portion of the earth is
infinite, asserting, as Xenophanes of Colophon does, that its roots
extend without limit, in order that they may not have the trouble
of investigating the cause (of its being at rest). Hence Empedocles’
rebuke in the words “if the depths of the earth are without limit
and the vast aether (above it) is so also, as has been said by the
tongues of many and vainly spouted forth from the mouths of men
who have seen little of the whole ”.’*
‘Xenophanes said that on its lower side the earth has roots
extending without limit.’ *
‘The earth is infinite, and is neither surrounded by air nor by the
heaven.’ ὅ
Simplicius ® (on the second of the above passages) observes that,
not having seen Xenophanes’ own verses on the subject, he cannot _
say whether Xenophanes meant that the under side of the earth —
extends without limit, and that this is the reason why it is at rest, or
meant to assert that the space below the earth, and the aether, is
infinite, and consequently the earth, though it is in fact being carried
downwards without limit, appears to be at rest ; for neither Aristotle
nor Empedocles made this clear. Presumably, however, as
Simplicius had not seen Xenophanes’ original poem, he had not
seen Fr. 28, the first of the above passages; for this passage seems
to be decisive ; there is nothing in it to suggest motion downwards,
and, if it meant that there was infinite air below the earth as there
is above, there would be no contrast between the upper and the
under side such as it is the obvious intention of the author to draw.’
1 i ἢ ps . 12 is
: Fes (Vee fie Asha G. p. 566; Vors. i*, p. 41. 33-41).
5 Aristotle, De cae/o ii. 13, 294 ἃ 21-28.
* Aét. iii. 9.45 11. 1,2 (D. G. pp. 376, 377; Vors. i*, p. 43. 33, 35).
5 Hippol. Refut. i. 14. 3 (D. G. p. 565; Vors. i?, p. 41. 29).
® Simplicius on De cae/o, Ρ. 522. 7, ed. Heib. ( Vors. i*, p. 43. 28).
7 As witness the μέν and the δέ and the clear opposition of ‘touching the air’
‘CH.VII XENOPHANES 55
According to Xenophanes the stars, including comets and
meteors, are made of clouds set on fire; they are extinguished
each day and are kindled at night like coals, and these happenings
' constitute their setting and rising respectively.1 The so-called
_Dioscuri are small clouds which emit light in virtue of the motion,
_ whatever it is, that they have.?
_ Similarly the sun is made of clouds set on fire; clouds formed
_ from moist exhalation take fire, and the sun is formed from the
resulting fiery particles collected together.* The moon is likewise
80 formed, the cloud being here described as ‘compressed’
(πεπιλημένον),, following an expression of Anaximander’s for
_ compressed portions of air; the moon’s light is its own.®
| When the sun sets, it is extinguished, and when it next rises, it is
a fresh one; it is likewise extinguished when there is an eclipse.®
_ of the fragment and the passage of Aristotle other than the literal interpretation.
_ The significant words in the passage of Aristotle are ‘saying that it (the earth)
_ is rected ad infinitum (ἐπ᾿ ἄπειρον ἐρριζῶσθαι)᾽. Berger (p.194, note) holds that the
᾿ ession is not used in the literal sense of having roots extending ad infinitum,
that ‘ we use the word ἐρριζῶσθαι only as an expression for a supporting force
_ hot capable of closer definition’; he can only quote in favour of this certain
_ metaphorical uses of ῥίζα ‘ root’ and other words connected with it, ῥιζώματα and
_ pases, which of course do not in the least prove that ἐρριζῶσθαι is used in
a metaphorical sense in our passage; indeed, if it is used in so vague a sense, it
‘is difficult to see how Xenophanes thereby absolved himself from giving a further
explanation of the cause of the earth’s remaining at rest, which, according to
Aristotle, was his object. As regards the fragment from Xenophanes’ own
poem, Berger says that he prefers to regard it as an attempt to give in few
words an idea of the Aorizon which divides earth and heaven into an upper,
visible, half, and an invisible lower half. This again leaves no contrast between
the upper and lower sides of the earth such as the fragment is obviously intended
to draw. On both points Berger’s arguments are of the nature of special
pleading, which can hardly carry conviction.
* A&t. ii. 13. 14, iii. 2. 11 (D. G. pp- 343, 367 ; Vors. i?, pp. 42. 39, 43. 15).
_. * Aét. ii. 18. 1 (2. G. p. 347; Vors.i®, p. 42. 42).
ἥ Coy ii. 20. 3 (D. G. p. 348; Vors. i*, p. 42. 45) ; Hippol. Refut. i. 14. 3 (D.C.
ΟΡ. 505).
* Aét. ii. 25. 4 (D.G. p. 356; Vors. 13, p. 43. 12).
_ + 5 Aét. ii. 28. 1 (2. G. p. 358; Vors. i?, p. 43. 13).
᾿ς * Aét. ii. 24. 4 (2. G. p. 354; Vors. i*, p. 43. 1). The passage, which is under
_ the heading ‘ On eclipse of the sun’, implies that it is an eclipse which comes
about by way of extinguishment (xara σβέσιν), but the next words to the effect
_ that the sun is a new one on rising again suggest that it is ‘setting’ rather
‘than ‘ eclipse’, which should be understood.
56 XENOPHANES PARTI
The phases of the moon are similarly caused by (partial) ex-
tinction.?
According to Xenophanes, the sun is useful with reference to the
coming into being and the ordering of the earth and of living things
in it; the moon is, in this respect, otiose.?
More remarkable are Xenophanes’ theory of a multiplicity of suns
and moons, and his view of the nature of the sun’s motion; and
here it is necessary to quote the actual words of Aétius :
‘Xenophanes says that there are many suns and moons according
to the regions (κλίματα), divisions (ἀποτομαί) and zones of the
earth; and at certain times the disc lights upon some division of
the earth not inhabited by us and so, as it were, stepping on
emptiness, suffers eclipse.
‘The same philosopher maintains that the sun goes forward ad
infinitum, and that it only appears to revolve in a circle owing to it
distance (away from us).’ ὃ
The idea that the sun, on arriving ‘at an uninhabited part of ’
earth, straightway goes out, as it were, is a curious illustratior
the final cause. For the rest, the passage, according to the n _
natural interpretation of it, implies that the sun does not rev
about the earth in a circle, but moves in a straight line ad infini-
that the earth is flat, and that its surface extends without li
On this interpretation we are presumably to suppose that the
of any one day passes out of our sight and is seen successive’
regions further and further distant towards the west until it is f
extinguished, while in the meantime the new sun of the nex ~
follows the first, at an interval of 24 hours, over our part «
earth, and so on, with the result that at any given time the
many suns all travelling in the same straight direction ad injfim
If this is the correct interpretation of Xenophanes’ theory (and
is the way in which it is generally understood), it shows no advan
upon, but a distinct falling off from, the systems of Anaximande.
and Anaximenes. Berger,’ deeming it incredible that Xenophanes
could have put forward views so crude, not to say childish, at
a time when the notion of the sphericity of the earth discovered by
1 Aét. ii, 29. 5 G. p. 360; Vors. i*, p. 43.14).
2 Aét, ii. 30. 8 (D. G. p. 362; Vors. i*, p. 43. 9).
> Aét. ii. 24. 9 (D.G. p. 355; Vors. i*, p. 43. 3-8).
* Tannery, op. cit., p. 133. ° Berger, op. cit., pp. 190 sqq.
.
᾿
Ϊ
Se et aA ae τινα
CH. VII XENOPHANES 57
the earliest Pythagoreans and by Parmenides must already have
spread far and wide, seeks to place a new interpretation upon the
passages in question.
For the Ionians, with their flat earth, there was necessarily one
horizon, so that the solar illumination and the length of the day
were the same for all parts of the inhabited earth. As soon, how-
ever, as the spherical shape of the earth was realized, it would
necessarily appear that there were different horizons according to
the particular spot occupied by an observer on the earth’s surface.
It was then, argues Berger, the different horizons which Xenophanes
_ had in view when he spoke of many suns and moons according to
_ the different regions or climates, divisions and zones of the earth ;
' he realized the difference in the appearances and the effects of the
_ same phenomena at different places on the earth’s surface, and he
Ss “nay have been the first to introduce, in this way, the mode of
' «pression by which we commonly speak of different suns, the
a Spical sun, the Indian sun, the midnight sun, and the like. This
_ ‘ingenious, but surely not reconcilable with other elementary
- hions stated by Xenophanes, such as that there is a new sun
“ty day. Then again, Berger has to explain the sun’s ‘going-
Ward ad infinitum’ as contrasted with circular motion; as, on
"theory, it cannot be motion zz a straight line without limit, he
οἴ it to be the motion in a sfiral which the sun actually exhibits
‘4g to the combination of its two motions, that of the daily
_ ‘fon, and its yearly motion in the ecliptic, which causes a slight
τὰ in its latitude day by day. But in the first place this
ὅπ in a spiral is not motion forward ad infinitum, for the spiral Ὁ
‘ns on itself in a year just as a simple circular motion would in
“hours. Indeed, Berger’s interpretation would make Xeno-
‘ines’ system purely Pythagorean, and advanced at that, for
ve do not hear of the spiral till we find it in Plato.1 And, if
‘Heraclitus’s system also represents (as we shall find it does) a set-
j back in astronomical theory, why should not Xenophanes’ ideas
ie have been equally retrograde?
There remains the story that Xenophanes told of an eclipse of
Mf the sun which lasted a whole month.2_ Could he have intended, by
1 Plato, Timaeus 39 A.
* Aét. il. 24. 4 (D.G. p. 354 ; Vors. i, p. 43. 2-3).
TO eee ee
=
58 XENOPHANES
this statement, to poke fun at Thales?! Berger, full of his theory
that Xenophanes’ ideas were based on the sphericity of the earth,
thinks that he must have inferred that the length of the day would
vary in different latitudes and according to the position of the sun
in the ecliptic, and must have seen that, at the winter solstice for
example, there would be a point on the earth’s surface at which the
longest night would last 24 hours, another point nearer the north
pole where there would be a night lasting a month, and so on, and
finally that at the north pole itself there would be a night six
months long as soon as the sun passes to the south of the equator ;
Xenophanes therefore, according to Berger, must simply have been
alluding to the existence of a place where a night may last a month.
If, as seems certain, Xenophanes’ earth was flat, this explanation
too must fall to the ground.
1 Tannery, op. cit., p. 132.
AS ar er eg ee Heal ge, ---,..ὕ»ὕ. Ξ =:
Vill
HERACLITUS
|
:
|
q
β
᾿
f
IF the astronomy of Xenophanes represents a decided set-back
_ in comparison with the speculations of Anaximander and Anaxi-
_menes, this is still more the case with Heraclitus of Ephesus
(fil. 504/0, and therefore born about 544/0B.C.); he was indeed no
astronomer, and he scarcely needs mention in a history of astronomy
except as an illustration of the vicissitudes, the ups and downs,
through which a science in its beginnings may have to pass. Hera-
clitus’s astronomy, if it can be called such, is of the crudest descrip-
tion. He does not recognize daily rotation; he leaves all the
‘apparent motions of the heavenly bodies to be explained by a
' continued interchange of matter between the earth and the heaven."
His original element, fire, condenses into water, and water into
earth ; this is the downward course. The earth, on the other hand,
may partly melt; this produces water, and water again vaporizes
into air and fire; this is the upward course. There are two kinds
_ of exhalations which arise from the earth and from the sea; the one
kind is bright and pure, the other dark; night and day, the months,
seasons of the year, the years, the rains and the winds, &c., are
_ exhalations. In the heavens are certain basins or bowls (σκάφαι)
_ turned with their concave sides towards us, which collect the bright
alations or-vaporizations, producing flames; these are the
2 The sun and the ὭΡΩΝ are bowl-shaped, like the stars, and
are ee: lit up.2 The flame of the sun is brightest
τ δὰ
60 HERACLITUS PARTI
consequently they give out less light and warmth. The moon,
although nearer the earth, moves in less pure air and is conse-
quently dimmer than the sun; the sun itself moves in pure and
transparent air and is at a moderate distance from us, so that it
warms and illuminates more.’ ‘If there were no sun, it would be
night for anything the other stars could do.* Both the sun and
the moon are eclipsed when the bowls are turned upwards (i.e. so
that the concave side faces upwards and the convex side faces in
our direction); the changes in the form of the moon during the
months are caused by gradual turning of the bowl.’
According to Heraclitus there is a new sun every day,* by which
is apparently meant that, on setting in the west, it is extinguished
or spent,’ and then, on the morrow, it is produced afresh in the
east by exhalation from the sea.°
The question arises, what happens to the bowl or basin supposed
to contain the sun if the sun has to be re-created in this way each
morning? Either a fresh envelope must be produced every day
for the rising of the sun in the east or, if the envelope is supposed
to be the same day after day, it must travel round from the west to
the east, presumably in the encircling water, laterally.’ Diogenes
Laertius (i.e. in this case Theophrastus) complains that Heraclitus
1 Diog. L., loc. cit.; Aét. ii. 28. 6 (D. G. p. 358; Vors. i’, p. 59. 10).
? Plutarch, De fort. 3, p. 98 c ( Vors. 13, p. 76. 8).
5 Diog. ἴω, loc. cit.; Aét. ii. 24. 3 (D. G. p. 354; Vors. i*, p. 59. 5). T
explanation that the hollow side of the basins is turned towards us itself sho
how crude were the ideas of Heraclitus. For it is clear that to account for t
actual variations which we’see in the shape of the moon, it is the ouf¢er side
a hemispherical bowl which should be supposed bright and turned towards
when the moon is full.
* Aristotle, Meteor. ii. 2, 355 a 14.
® Plato, Rep. vi. 498 A.
®° Aristotelian Problems, xxiii. 30, 934b 35. It is true that a certain passage
of Aristotle may be held to imply that Heraclitus did not maintain that the
moon and the stars, as well as the sun, are fed and renewed by exhalations.
Aristotle (AZeZeor. ii. 2, 354 Ὁ 33 5644.) is speaking of those who maintain that
the sun is fed by moisture. He first argues that, although fire may be said to
be nourished by water (the flame arising through continuous alternation between
the moist and the dry), this cannot take place with the sun; ‘and if the sun
were fed in this same way, then it is clear that not only is the sun new every
day, as Heraclitus says, but it is continuously becoming new (every moment) ’
(355 a 11-15). ‘ And,’ he adds (355 a 18-21), ‘it is absurd that these thinkers
should only concern themselves with the sun, and neglect the conservation of
the other stars, seeing that their number and their size is so-great.’
7 Zeller, 1δ, p. 684.
_ . alia « ee a ae eee
2 a Dn rns 2 AG
CH. VIII HERACLITUS 61
gave no information as to the nature of these cups or basins. The
idea, however, of the sun and moon being carried round in these
σκάφαι reminds us forcibly of the Egyptian notion of the sun in
his barque floating over the waters above, accompanied by a host
of secondary gods, the planets and the fixed stars.
Heraclitus held (as Epicurus did long afterwards) that the
diameter of the sun is one foot,? and that its actual size is the
same as its apparent size.* This in itself shows that Heraclitus
was no mathematician; as Aristotle says, ‘it is too childish to
suppose that each of the moving heavenly bodies is small in size
because it appears so to us observing it from where we stand.’ *
He called the arctic circle by the more poetical name of ‘the
Bear’, saying that ‘the Bear represents the limits of morning and
evening’. .. whereas of course it is the arctic circle, not the Bear
itself, which is the confine of setting and rising® (i.e. the stars
_ within the arctic circle never set).
According to Diogenes Laertius, Heraclitus said absolutely
nothing about the nature of the earth;® but we may judge that
_in his conception of the universe he was closer to Thales than to
_ Anaximander; that is, he would regard the universe as a hemi-
' sphere rather than a sphere, and the base of the hemisphere as
"a plane containing the surface of the earth surrounded by the
sea; if he recognized a subterranean region, under the name of
᾿ς “ades, he does not seem to have formed any idea with regard to
3 Ἐν beyond what was contained in the current mythology.’
τ δ ~, When he gave 10,800 solar years as the length of a Great Year,*
᾿ ect no astronomical Great Year, but the period of duration ἢ
᾿ς of the world from its birth to its resolution again into fire and
' vice versa. He arrived at it, apparently, by taking a generation
of 30 years as τ day and multiplying it by 360 as the number of
ἃ yi in a year.”
_ +} See pp. 19, 20 above. 3 Aét. ii. 21. 4 (D. G. p. 351; Vors. 15, p. 62. 7).
:. ; Diog. L. ix. 7 (Vors. 15, p. 55. 12).
__ * Aristotle, 2Ze¢eor. i. 3, 339 534.
5. Strabo, i: 1. 6, p. 3 (Vors. i*, p. 78. 15).
' * Diog. L. ix. 11 (Vors. ®, p. 55. 46). ;
᾿ς ἢ Tannery, op. cit., p. 169.
fe > Aét.’ ii. 32. 3 (D. G. p. 364: Vors. i*, p. 59. 13); Censorinus, De die natalt
᾿ 18. 11 (Vors. i*, p. 59. 16).
5 Tannery, op. cit., p. 168. :
ΙΧ
PARMENIDES
WITH regard to the date of Parmenides there is a conflict of
authority. On the one hand Plato says that Parmenides and Zeno
paid a visit to Athens, Parmenides being then about 65 and Zeno
nearly 40 years of age, and that Socrates, who was then very
young (σφόδρα νέος), conversed with them on this occasion! Now
if we assume that Socrates was about 18 or 20 years of age at
this time, the date of the meeting would be about 451 or 449 B.C.,
and this would give 516 or 514 as the date of Parmenides’ birth. On
the other hand, Diogenes Laertius* says (doubtless on the authority
of Apollodorus) that Parmenides flourished in Ol. 69 (504/0 B.C.),
in which case he myst have been born about 540 B.C. In view of
the number of cases in which, for artistic reasons, Plato indulged in
anachronisms, it is not unnatural to feel doubt as to whether the
meeting of Socrates with Parmenides was a historical fact. Zeller®
firmly maintained that it was a poetic fiction on the part of Plato;
but Burnet, on grounds which seem to be convincing, accepts it
as a fact, exposing at the same time the rough and ready methods
on which Apollodorus proceeded in fixing his dates.‘
1 Plato, Parmenides 127 A-C. 2 Diog. ἵν. ix. 23 ( Vors. i*, p, 106. 10)
ἢ Zeller, ἢ, pp. 555, 556. ,
* Burnet, Early Greek Philosophy, pp. 192,193. The story was early qnestic
Athenaeus (xi. 15,p.505¥; Vors. 12, p. 106, 47) doubted whether the age of Sc
would make it possible for him to have conyersed with Parmenides or at a1
to have held or listened to such a discourse, But Plato refers to the mee
two other places (Zheaet. 183 Ἔ, Sophist 217 Ο), and (as Brandis and I
also pointed out) we should have to assume a deliberate falsification of {
the part of Plato if he had inserted these two allusions solely for the pu
inducing people to believe a fiction contained in another dialogue, ¥
too, independent evidence of the visit of Zeno to Athens. Plutarch
4. 3) says that Pericles ‘heard’ Zeno. The date given by Apollodoru oa
other hand, seems to be based solely on that of the foundation of Elea
adopts that date as the loruit of Xenophanes, so he makes it the δὲ
Parmenides’ birth. In like manner he makes Zeno’s birth contem,
ν
—— Δι με σὠ- νων
4
Γ,
§
PARMENIDES 63
Parmenides is said to have been a disciple of Xenophanes;! he
was also closely connected with the Pythagorean school, being
specially associated with a Pythagorean, Ameinias Diochaites, for
whom he conceived such an affection that he erected a ἡρῷον to
him after his death;? Proclus quotes Nicomachus as authority
for the statement that he actually belonged to the school,* and
Strabo has a notice to the same effect.‘ It is not therefore
unnatural that Parmenides’ philosophical system had points in
common with that of Xenophanes, while his cosmogony was on
Pythagorean lines, with of course some differences. Thus his
Being corresponds to the One of Xenophanes and, like it, is a well-
rounded sphere always at rest; he excluded, however, any idea
of its infinite extension; according to Parmenides it is definitely
limited, rounded off on all sides, extending equally in all directions
_ from the centre. Parmenides differs from Xenophanes in denying
genesis and destruction altogether; these phenomena, he holds,
are only apparent.® Being is identified with Truth; anything else
is Not-Being, the subject of opinion. Physics belongs to the latter
_ deceptive domain.”
_ The main difference between the cosmologies of Parmenides and
the Pythagoreans appears to be this. It seems almost certain that
Pythagoras himself conceived the universe to be a sphere, and
attributed to it daily rotation round an axis® (though this was
denied by Philolaus afterwards); this involved the assumption
ΟΠ that it is itself finite but that something exists round it; the
Pythagoreans, therefore, were bound to hold that, beyond the
finite rotating sphere, there was limitless void or empty space;
h Parmenides’ floruit, thereby making Zeno forty years younger than Par-
ides, whereas Plato makes him about twenty-five years younger. Burnet
gE. Meyer (Gesch. des Alterth. iv. § 509, note) in support of his view.
_ssistotle, Me/aph.a.5,986b 22 ; Simplicius, Jz Phys. p.22. 27(Vors. i2, p. 107.
τ Diog. L. (ix. 21; Vors. 15, p, 105. 26) says that Parmenides ‘heard’
» Ranes but did not follow him.
ig. L. ix. 21 ( Vors. 15, p. 105. 29).
“slus, Jn Parm., i, ad init. ( Vors. 13, p. τοῦ. 30).
bo, vi. I. 1, Ὁ. 252 (Vors. i?, p. 107. 39).
otle, Phys. iii. 6, 207 ἃ 16; Fr. 8, line 42 (Vors. i?, p. 121. 3).
— De caelo iii. τ, 298 Ὁ 14; Aét. 1. 24.1 (D.G. p. 320; Vors. i2,
= τς 50-53 (Vors. i*, p. 121. 11-13),
gry, ΟΡ. cit., p. 123.
64 PARMENIDES PARTI
this agrees with their notion that the universe breathes, a supposition
which Tannery attributes to the Master himself because Xenophanes
is said to have denied it.2 Parmenides, on the other hand, denied
the existence of the infinite void, and was therefore obliged to
make his finite sphere motionless, and to hold that its apparent
rotation is only an illusion.’
As in other respects the cosmology of Parmenides follows so
closely that of the Pythagoreans, it is not surprising that certain
astronomical innovations are alternatively attributed to Parmenides
and to Pythagoras. Parmenides is said to have been the first to
assert that the earth is spherical in shape and lies in the centre ;*
this statement has the great authority of Theophrastus in its favour ;
there was, however, an alternative tradition stating that it was
Pythagoras who first called the heaven κόσμος, and held the earth
to be round (στρογγύλην). As the idea that the earth is spherical
was probably suggested by mathematical considerations, Pythagoras
is the more likely to have conceived it, though Parmenides may
have been the first to state it publicly (the Pythagorean secrecy, |
such as it was, seems to have applied only to their ritual, not to their
mathematics or physics). Parmenides is associated with Democritus
as having argued that the earth remains in the centre because,
being equidistant from all points (on the sphere of the universe),
it is in equilibrium, and there is no more reason why it should
tend to move in one direction than in another. Parmenides
therefore here practically repeats the similar argument used by
Anaximander (see above, p. 24), and we shall find that in other
physical portions of his system he follows Anaximander and other
Ionians pretty closely.
* Aristotle, Phys. iv. 6, 213 Ὁ 24.
2 Tannery, Op. cit., p. 121. Zeller (i5, p. 525), however, does not believe that
the remark μὴ μέντοι ἀναπνεῖν, if Xenophanes really made it, is directed against
the Pythagorean view. He points out, too, that the statement in Diog. L. ix. 19
(Vors. i*, p. 34. 18), so far as these words (‘but that it does not breathe’) are
concerned, may only represent an inference from the fact that Fr. 24 only
mentions seeing, hearing, and thinking. This, however, assumes greater intelli-
gence on the part of Diogenes than we are justified in attributing to him.
3. Tannery, op. Cit., p. 125.
* Diog. L. ix. 21 (Vors. 15, p. 105. 32).
5 Diog. L. viii. 48 (Vors. 2, p- 111. 38).
6 Aét. ili. 15. 7 (D. G. p. 380 ; Vors. 15, p. 111. 40); cf. Aristotle, De caelo, i ii.
13, 295 b 10, and the similar views in Plato, Phaedo 108 E-109 A.
.
ΘΗ. ΙΧ PARMENIDES 65
Secondly, Parmenides is said to have been the first to ‘define
the habitable regions of the earth under the two tropic zones’ : 1
on the other hand we are told that Pythagoras and his school
declared that the sphere of the whole heaven was divided into five
circles which they called ‘zones’.* Hultsch® bids us reject the
attribution to Pythagoras on the ground that these zones would
only be possible on a system in which the axis of the universe
about which it revolves passes through the centre of the earth;
the zones are therefore incompatible with the Pythagorean system,
according to which the earth moves round the central fire.
Hultsch admits, however, that this argument does not hold if the
hypothesis of the central fire was not thought of by any one before
Philolaus; and there is no evidence that it was. As soon as
Pythagoras had satisfied himself that the universe and the earth
were concentric spheres, the centre of both being the centre of the
earth, the definite portion of the heaven marked out by the extreme
deviations of the sun in latitude (north and south) might easily
present itself to him as a zone on the heavenly sphere. The Arctic
Circle, already known in the sense of the circle including within
it the stars which never set, would make another division, while
a corresponding Antarctic Circle would naturally be postulated
by one who had realized the existence of antipodes.* With the
intervening two zones, five divisions of the heaven were ready to
hand. It would next be seen that straight lines drawn from the
centre of the earth to all points on all the dividing circles in the
heaven would cut the surface of the earth in points lying on exactly
corresponding circles, and the zone-theory would thus be transferred
to the earth.2 We are told, however, that Parmenides’ division of
the earth into zones was different from the division which would
be arrived at in this way, in that he made his torrid zone about
1 Aét. iii. 11. 4 (D. G. p. 377).
2 Aét. ii. 12. 1 (D. G. p. 340).
5 Hultsch, art. ‘Astronomie’ in Pauly-Wissowa’s Real-Encyclopddie der
classischen Altertumswissenschaft, ii. 2, 1896, p. 1834.
* Alexander Polyhistor in Diog. L. viii. 1. 26.
5 Aét. iii. 13. 1 (D.G. p. 378), ‘ Pythagoras said that the earth was divided,
correspondingly to the sphere of the universe, into five zones, the arctic, antarctic,
summer and winter zones, and the equatorial zone; the middle of these defines
the middle portion of the earth, and is for this reason called the torrid zone; then
comes the habitable zone which is temperate.’
1410 Ε
66 PARMENIDES PARTI
twice as broad as the zone intercepted between the tropic circles,
so that it spread over each of those circles into the temperate zones.’
This seems to be the first appearance of zones viewed from the
standpoint of physical geography.
Thirdly, Diogenes Laertius says, on the authority of Favorinus,
that Parmenides is thought to have been the first to recognize that
the Evening and the Morning Stars are one and the same, while
others say that it was Pythagoras.2 In this case, although
Parmenides may have learnt the fact from the Pythagoreans, it
is probable that Pythagoras did not know it as the result of
observations of his own, but acquired the information from Egypt
or Chaldaea along with other facts about the planets.®
On the purely physical side Parmenides in the main followed
one or other of the Ionian philosophers. The earth, he said, was
formed from a precipitate of condensed air.4 He agreed with
Heraclitus in regarding the stars as ‘compressed’ fire (literally
close-pressed packs of fire, πιλήματα πυρός).
Parmenides’ theory of ‘wreaths’ (στεφάναι) seems to be directly -
adapted from Anaximander’s theory of hoops or wheels. Anaxi-
mander had distinguished hoops belonging to the sun, the moon,
and the stars respectively, which were probably concentric with
the earth; the hoops were of different sizes, the sun’s being the
largest, the moon’s next, and those of the stars smaller still. These
hoops were rings of compressed air filled with fire which burst out
in flame at outlets, thereby producing what we see as the sun,
moon, and stars. The corresponding views of Parmenides are not
easy to understand ; I will therefore begin by attempting a transla-
tion of the passage of Aétius in which they are set out.®
‘There are certain wreaths twined round, one above the other
[relatively to the earth as common centre]; one sort is made of the
rarefied (element), another of the condensed; and between these
are others consisting of light and darkness in combination. That
1 Posidonius in Strabo, ii. 2. 2, p. 94.
2 Diog. L. ix. 23 (Vors. i*, p. 106. 11).
8. Tannery, op. cit., p. 229.
4 Λέγει δὲ τὴν γῆν πυκνοῦ καταρρυέντος ἀέρος γεγονέναι, Ps. Plut. Stromat. §
(D. G. p. 581.4; Vors. 13, p. 109. 1).
5 Aét. il. 13. 8 (D. G. p. 342; Vors. 13, p. 111. 25). Cf. Anaximander’s
πιλήματα ἀέρος. :
® Aét, ii, 7. 1 (2. G. p. 3353 Vors. 15, p. 111. 5-16).
CH. Ix PARMENIDES 67
which encloses them all is solid like a wall, below which is a wreath
of fire; that which is in the very middle of all the wreaths is solid,
about which (περὶ 6) [under which (ὑφ᾽ 6, Diels)] again is a wreath
of fire. And of the mixed wreaths the midmost is to all of them
the beginning and cause of motion and becoming,‘ and this he calls
the Deity which directs their course and holds sway (κληροῦχον) 5
cooks the keys(xAndodxor, Fiilleborn) |, namely Justice and Necessity.
oreover, the air is thrown off the earth in the form of vapour
owing to the violent pressure of its condensation ; the sun and the
Milky Way are an exspiration 5 of the fire; the moon is a mixture
of both elements, air and fire. And, while the encircling aether
is uppermost of all, below it is ranged that fiery (thing) which we
call heaven, under which again are the regions round the earth.’
But in addition we are told that
‘It is the mixture of the dense and the rarefied which produces
the colour of the Milky Way.’*
‘The sun and the moon were separated off from the Milky Way,
the sun arising from the more rarefied mixture which is hot, and
the moon from the denser which is cold.’®
The fragments of Parmenides do not add much to this. The
relevant lines are as follows:
‘The All is full of light and, at the same time, of invisible
darkness, which balance each other; for neither of them has any
share in the other.’ ®
‘Thou shalt learn the nature of the aether and all the signs in
the aether, the scorching function of the pure clear sun, and whence
they came; thou shalt hear the wandering function and the nature
of the round-eyed moon, and thou shalt learn of the surrounding
heaven, whence it arose, and how Necessity, guiding it, compelled -
it to hold fast the bounds of the stars.’ 7
“(1 will begin by telling) how the earth, the sun and the moon,
the common aether, the milk of the heaven, furthest Olympus, and
the hot force of the stars strove to come to birth.’®
1 I follow the reading adopted by Diels in the Vorsokratiker, ἁπάσαις (ἀρχήν)
τε καὶ Cairiay) κινήσεως καὶ γενέσεως ὑπάρχειν.
2 Burnet (Zarly Greek Philosophy, p. 219) observes that κλῆρος in the Myth
of Er suggests κληροῦχον as the right reading. Fiilleborn suggested κληδοῦχον
_ in view of the use of xAnidas (keys) in Fr. 1. 14.
* The word ἀναπνοή is of course ambiguous ; I follow Diels’ interpretation,
* Ausdiinstung’, ‘evaporation’ or ‘exhalation’. Diels (Parmenides Lehrgedicht,
1897, p. 105) compares ἀναπνοὰς ἴσχον in the Timaecus 85 A.
* Aét. iii. 1. 4 (D. G. p. 365).
5 Aét. ii. 20. 8a (D. G. p. 349; Vors. i*, p. 111. 35).
© Fr. 9 (Vors. i?, p. 122. 11-12).
7 Fr. τὸ von i’, pp. 122. 21-123. 2).
® Fr. 11 (Vors. 15, p. 123. 5-7).
F2
68 PARMENIDES PARTI
Of the wreaths he says that
‘The narrower (wreaths) were filled with unmixed’ fire; those
next in order to them (were filled) with night, and along with them
the share of flame spreads itself. In the middle of these is the
Deity which controls all.’ ?
It is not surprising that there have been a number of interpreta-
tions of these passages taken in combination. To begin with the
outside, there is a doubt as to the relative positions of the ‘ heaven’
and the aether. According to Aétius ‘the encircling aether is
uppermost of all, and below it is ranged that fiery thing which
we call heaven’, whereas the fragments suggest that the ‘common
aether’ is within the ‘encircling heaven’ or ‘furthest Olympus’,
which latter clearly seems to be the solid envelope compared to
a wall. The fragments presumably better represent Parmenides’
own statement, and possibly Aétius’s version (which seems practi-
cally to interchange the ‘ heaven’ and the ‘aether ’) is due to some
confusion.
The next question is, what was the shape of the ‘wreaths’ or
bands?* Zeller, in view of the spherical form of the envelope,
does not see how they can be anything but hollow globes.’ But
surely ‘wreaths’ or ‘garlands’, i.e. bands, would not in that case
be a proper description. Tannery® takes them to be cylindrical
bands fixed one inside the other, comparing with our passage the
description in Plato’s Myth of Er,’ where ‘the distaff of Necessity
by means of which all the revolutions of the universe are kept up’
distinctly suggests that Plato had Parmenides’ system in mind;
Plato there speaks of eight whorls (σφόνδυλοι), one inside the other,
‘like those boxes which fit into one another,’ and of the Zs of
1 Reading dxpyrow, The reading ἀκρίτοιο (literally ‘ confused’ or ‘ undistin-
guishable’, that is to say, dz/u¢ed fire) is impossible, because (1) it does not give
the required sense, and (2) it offends against prosody, since ¢ in ἄκριτος is short
(Diels, Parmenides Lehrgedicht, p. 104).
2 Fr. 12 (Vors. 13, p. 123. 18-20).
® Zeller (i°, p. 573) gives references to the explanations suggested by Brandis,
Karsten, and Krische. More recent views (those of Tannery, Diels, Berger,
and Otto Gilbert) are referred to in the text above. ©
* στεφάνη is sometimes translated as ‘crown’; but this rendering is open to
the objection of suggesting a definite shape. Moreover, it is inapplicable to
a series of wreaths or bands entwined the one within the other.
® Zeller, i°, p. 572. δ. Tannery, op. cit., p. 230.
7 Plato, Republic x. 616 D. Υ
.
CH. Ix PARMENIDES 69
the whorls. In the 7imaeus too there are no spheres, but bands
or strips crossing one another at an angle. We may perhaps take
the bands to be, not cylinders, but zones of a sphere bisected by
a great circle parallel to the bounding circles. Burnet? thinks that
the solid circle which surrounds all the bands cannot be a sphere
either, because in that case ‘like a wall’ would be inappropriate.
I do not, however, see any real difficulty in such a use of ‘like
a wall’, and certainly Parmenides’ All was spherical.*
We now come to the main question of the nature of the bands,
their arrangement relatively to one another, and the meaning to
be attached to them severally. What we learn about them from
Aétius and the fragments taken together amounts to this. First,
the material of which they are composed is of two kinds; one is
alternatively described as the ‘rarefied’ (ἀραιόν), light, flame (φλόξ)
or fire; the other as the ‘condensed’ (πυκνόν), darkness, or night.
The bands are of three kinds, the first composed entirely of the
‘rarefied’ element or fire, the second of the ‘ condensed’ or darkness,
and the third of a mixture of the two. Secondly, as regards their
arrangement, we are told that there is a solid envelope, a spherical
shell, enclosing them all; two bands of unmixed fire are mentioned,
of which one is immediately under the envelope, the other is about
(reading περί with the MSS.) or wnder (reading ὑπό with Diels)
‘that which is in the very midst of all the bands’ and which is
‘solid’; these two bands are also ‘narrower’ (than something),
where ‘ narrower’ means that their radii are smaller, that is to say,
their inner surfaces are nearer (than something) to the centre of.
the earth, which is the common centre of all the bands. The mixed
bands, according to Aétius, are ‘ between’ the bands of fire and the
bands of darkness ; the fragment (12) makes them come next to
both the ‘narrower’ bands, the bands of fire.
There seems to be general agreement that the ‘mixed’ bands
include the sun, the moon, and the planets; it is with regard to
the meaning and position of the bands of fire, and to the place
occupied by the Deity called by the names of Justice and Necessity,
that there has been the greatest difference of opinion. Tannery’s
1 Plato, Zimaeus 36 B. 3 Burnet, Early Greek Philosophy, p. 216.
* *It is complete on every side, like the mass of a well-rounded sphere poised
from the centre in every direction’ (Fr. 8. 42-4; Vors. i*, p. 121. 3-5).
70 PARMENIDES PARTI
view is that the outermost band of fire under the solid envelope (which
envelope may be regarded as one of the bands made of the ‘condensed’
element) is the Milky Way. In that case, however, the fire is not
pure; for ‘it is the mixture of the dense and the rarefied which
produces the colour of the Milky Way’. Tannery would get over
this difficulty by supposing the band to be only fw// of fire, like
the hoops of Anaximander, the almost continuous brightness being
due to exspiration through the covering. But Aétius says that both
the Milky Way and the sun are an exspiration of fire, and the sun
is certainly represented by one of the mixed bands, so that the
Milky Way should also be one of the mixed bands. The band
of fire which (with the reading περί) is about the solid in the very
centre of all the bands (i.e. the earth) Tannery takes to be our
atmosphere. This seems possible, for Parmenides may have re-
garded air it up as being fire. In Diels’ interpretation a similar
view seems to be taken of the outermost band of fire which he
calls ‘aether-fire’; and the assumption that the aether is fire is
perhaps justified by the fact, if true, that Parmenides declared the
heaven to be of fire. The intermediate bands consisting of the
two elements, light and dark, in combination correspond in Tannery’s
view to the orbits of the moon, the sun, and the planets respectively,
which (starting from the earth) come in that order; possibly among
these mixed bands there may be bands entirely dark as well (cf.
Fr. 12).
Diels? takes the bands which consist exclusively of the ‘condensed’
element to be made of earth simply. There are two of these;
one is the solid envelope, the solid firmament, ‘Outer Olympus’ ;
the other is the crust of the earth. Just beneath the solid envelope
comes the outer band of fire, which is the aether-fire. Next within
this come the mixed class of bands which are the bands of stars
containing both elements, earth and fire, not separate from one
another but mixed together. Such dark rings, out of which the
fire flashes out here and there, are the Milky Way, the sun, the
moon, and the planets. After the mixed bands comes the solid
earth-crust, below which again (reading ὑφ᾽ @, which Diels substitutes
1 Aét. ii, 11. 4 (D. G. p . 340; Vors. i*, p. 111. 23).
2 Diels, Vors. ii’, i, p. ‘Gre ; cf. Parmenides οὐ eg εν pp. 104 5644.
CH. Ix PARMENIDES γι
for περὶ 6) comes the inner band of fire, which therefore is inside
the earth and forms a sernel of fire.
It will be seen that the idea of Anaximander that stars are dark
rings with fire shining out at certain points is supposed, both by
Tannery and Diels, to be more or less present in Parmenides’ con-
ception, though Tannery only assumes it as applying to the Milky
Way, which he wrongly identifies with the outer band of undiluted
fire. _ Diels, more correctly, implies that it is the mixed rings made
up of light and darkness in combination which exhibit the pheno-
menon of ‘fire shining out here and there’, these mixed rings
including the Milky Way as well as the sun, moon, and planets.
It is possible that Aétius’s ‘mixed rings’ may be no more than
his interpretation of the line in Fr. 12 which says that after the
‘narrower’ bands ‘filled with unmixed fire’ there come ‘bands
filled with night and wth them (μετά, which Diels translates by
‘between’) is spread (or is set in motion, ferac) a share of fire’.
_ And this line itself may mean either that the bands of night have
a portion of fire mixed in them, or that each of the bands of night
has a stream of fire (its ‘share of fire’) coursing through it. If the
fire were enclosed in the darkness as under the second alternative,
we should have a fairly exact reproduction of Anaximander’s tubes
containing fire; but there is nothing in the fragment to suggest
that fire shines out of vents in the dark covering ; hence the mixture
of light and dark, with light shining out at certain points (without
enclosure in tubes), as assumed by Diels, seems to be the safer
interpretation.
Tannery and Diels differ fundamentally about the inner band |
of fire. According to the former, it is the atmosphere round the
earth, and, if the ‘atmosphere’ be taken to include the empty space
outside the actual atmosphere as far as the nearest of the mixed
bands, this seems quite possible. Diels, however, (reading ὑφ᾽ 4,
‘under which’, instead of περὶ 6,‘ round which’), makes it a kernel
_ of fire zzside the earth and concludes that ‘ Parmenides is for us the -
first who stated the truth not only as regards the form of the earth
but also as regards its constitution, whether he guessed the latter
or inferred it correctly from indications such as volcanoes and hot
springs’. But it seems to me that there are great difficulties in
' Diels, Parmenides Lehrgedicht, pp. 105, 106.
72, PARMENIDES PARTI
the way of Diels’ interpretation. First, it is difficult to regard
a kernel of fire, which ‘would presumably be a solid mass of fire,
spherical in shape, as satisfying the description of a wreath or band.
Secondly, whereas Fr. 12 speaks of the narrower bands as filled
with unmixed fire and then of the mixed bands being ‘next to
these’ (ai δ᾽ ἐπὶ ταῖς νυκτός ...), the mixed bands would, on Diels’
interpretation, be next to only one of the bands of fire (the outer
one) and would not be next to the inner one but would be separated
from it by the earth’s crust. Diels seems to have anticipated this
objection, for he explains that it is doth the unmixed kinds of bands
(i.e. those made of unmixed fire and those of unmixed earth, and
not only the former, the ‘narrower’ bands) on which the mixed
bands follow, in the inward direction starting from the outside
envelope and in the outward direction starting from the centre ;!
but ταῖς would much more naturally mean the narrower bands
only. Thirdly, it seems to me to be difficult to assume that there
is no band intervening between the surface of the earth and the
nearest of the mixed bands; if there were no intervening band,
the nearest mixed band, say that of the moon, would have to be
in contact with the earth, and therefore the moon also, shining out
of it, must practically touch the earth. Therefore there must be
some intervening band. But, if there is an intervening band, it
must be one of three kinds, dense, mixed, or fiery. It cannot
be a dense band, for, if it were, the sun, moon and stars would never
be visible; if it were a mixed band, there would again be some
heavenly body or bodies in the same position of virtual contact
with the earth; therefore the intervening band can only be a band
of fire. I am disposed, therefore, to accept Tannery’s view that the
inner band of fire is our atmosphere with the empty space beyond
it reaching to the mixed bands.
If the above arguments are right, the order would be, starting
from the outside: (1) the solid envelope like a wall; (2) a band
of fire =the aether-fire; (3) mixed bands, in which are included
the Milky Way, the planets, the sun, and the moon; (4) a band
of fire, the inner side of which is our atmosphere, touching the
earth; (5) the earth itself; which is Diels’ solution except as
regards (4).
1 Parmenides Lehrgedicht, p. τοῦ.
ctx PARMENIDES 73
Berger! has an ingenious theory as regards the inner band of
fire round the earth. If I understand him rightly, he argues that
' the bands in the heaven containing the stars were described in one
part of Parmenides’ poem, and the zones of the earth in another,
and that Fr. 12 refers to the zones; that the two descriptions then
got confused in the Dorographi, and that the inner band of fire
is really nothing but the ‘orrid zone, which has no business in the
description at all. Diels has shown that this cannot be correct.?
Gilbert * disagrees with Diels’ view of the inner band of fire as
_ a kernel of fire inside the earth; he himself.thinks that there was
not a band of fire about the earth, but that πυρώδης (with στεφάνη
understood), ‘a band of fire’, is a mistake for mip,‘ fire’, or πυρῶδες
in the neuter, and that the meaning is a fire or a fiery space
_ connected with the earth (περί in that sense being possible) ‘ down-
wards’, which fire or fiery space he says we must suppose to
embrace the under surface of the earth’s sphere.
Lastly, there is a difficulty as to the position occupied by the
‘ goddess who steers all things’, Justice or Necessity. This mytho-
logical personification of Necessity and Justice is, of course, after the
Pythagorean manner,* and reminds us of the similar introduction
_ of Necessity in Plato’s Myth of Er, which has so many other points
of resemblance to Parmenides’ theory. Fragment 12 says that this
_ Deity is ‘in the middle of these’, i.e. presumably ‘these dands’, and
_ Aétius, that is to say Theophrastus, took this to mean in the midst
_ of the ‘ bands filled with night but with a share of fire in them’.
_ Simplicius, on the other hand, takes it to mean ‘in the middle of the
_ whole system (ἐν μέσῳ πάντων)᾽,5 i.e. in the middle of the whole world,
clearly identifying the goddess with the central fire or hearth of the
_Pythagoreans. Diels seems to favour Simplicius’s view, taking the
centre of the universe to be the centre of the earth,® without, how-
* Berger, Geschichte der wissenschaftlichen Erdkunde der Griechen, p. 204 sq.
3 Parmenides Lehrgedicht, p.104. Since the torrid zone, as viewed by Par-
menides, is twice the size of the zone between the tropics, the ‘ narrower’ zones
_must be the temperate zones, which requires the impossible reading ἀκρίτοιο ;
with the true reading dxpyrow, the torrid zone would be ‘broader’, not
_ ‘narrower’, Besides, Aétius’s paraphrase agrees so closely with the fragment,
᾿ς especially in the striking introduction of the Deity, that it cannot be regarded
__as being anything else than Theophrastus’s paraphrase of the verses.
__ ® Gilbert, ‘ Die δαίμων des Parmenides’, in Archiv fiir Gesch. der Philosophie,
᾿ Χχ, 1906, pp. 25-45. * Tannery, loc, cit.
5 Simpl. iz Phys. p. 34. 15 ( Vors. 15, p. 123. 16).
® Diels, Parmenides Lehrgedicht, pp. 107-8.
74 PARMENIDES PART I
ever, attempting to reconcile this with Aétius’s statement that she
is placed in the middle of the mixed bands. It is in any case
difficult to suppose that Parmenides treated his goddess who
‘guides the encircling heaven and compels it to hold fast the
bounds of the stars’ as shut up within a solid spherical earth with
no outlet ; the difficulty is even greater than in the Myth of Er,
where at all events there is ‘a straight light like a pillar which
extends from above through all the heaven and earth’, and which
accordingly passes through the place where Necessity is assumed
to be seated. The statement of Aétius that she is placed in the
middle of the mixed bands suggested to Berger! the possibility
that her place was in the sun, in view of the pre-eminent position
commonly assigned to the sun in the celestial system.? Gilbert
holds that the goddess had her abode in the fiery space under the
earth above mentioned; he quotes from other poets, Hesiod,
Heraclitus, Aeschylus and Sophocles, references to Diké as con-
nected with the gods of the lower world, his object being to show
that, in connecting Justice or Necessity with the earth, night,
and the under-world, Parmenides was only adopting notions
generally current.*. Gilbert (like Diels) is confronted with the
difficulty of Aétius’s location of the goddess ‘in the middle of
the mixed bands,’ and he disposes of this objection by assuming that
the words were interpolated by some one who wished to find her in
the sun.*. This, however, seems too violent.
Both Tannery and Diels specially mention the planets, and Tannery
makes Parmenides arrange the heavenly bodies in the following order,
starting from the earth: moon, sun, planets, fixed stars. There
is, however, nothing in the texts about the bands which distinguishes
the planets from the fixed stars or indicates their relative distances.
1 Berger, op. cit., pp. 204, 205.
2 e.g. Cleanthes (Aét. 11. 4.16) saw in the sun the seat of authority in the
universe (τὸ ἡγεμονικὸν τοῦ κόσμου) : cf. also such passages as Theon of Smyrna,
pp. 138. 16, 140. 7, 187. 16; Plut. De fac. in orbe lunae 30, 945 C; Proclus, zn
Timaeum 258 A, ‘The sun, where the justice ordering the world is placed.’
8 Gilbert, loc. cit., p. 36.
* The text in Diels’ Doxographi (p. 335. 10 sq.) being καὶ τὸ μεσαίτατον
πασῶν περὶ ὃ πάλιν πυρώδης" τῶν δὲ συμμιγῶν τὴν μεσαιτάτην ἁπάσαις τοκέα πάσης
κινήσεως καὶ γενέσεως ὑπάρχειν, ἥντινα καὶ δαίμονα κιτ.ἑ., Gilbert would reject τῶν δὲ
συμμιγῶν τὴν μεσαιτάτην as an interpolation, leaving καὶ τὸ μεσαίτατον πασῶν,
περὶ ὃ πάλιν πυρώδης (ἢ στεφάνη), ἁπάσαις τοκέα πάσης κινήσεως καὶ γενέσεως
ὑπάρχειν κιτ.ἕ,
.
ΠΘΗΟΙΧ PARMENIDES 75
i
| The only passage in the Doxographi throwing light on the matter
_ is a statement that
᾿ς *Parmenides places the Morning Star, which he thinks the same
85 the Evening Star, first in the aether; then, after it, the sun, and
_ under it again the stars in the fiery (thing) which he calls heaven.’ ὦ
Tannery thinks that, if Parmenides distinguished Venus, and
if it was from the first Pythagoreans that he learnt to do so, the
other planets must equally have been known to the Pythagoreans
and therefore to Parmenides. Tannery’s view, however, of
Parmenides’ arrangement of the stars can hardly be reconciled
with the distinct statement of Aétius that, while Venus is outside
the sun, the other stars are below it; this, except as regards Venus,
‘agrees with Anaximander’s order, according to which both the
planets and the other stars are all placed below the sun and moon.
Tannery is therefore obliged to assume that Aétius’s remark is an
‘error based on a too rigorous interpretation of the terms aether
_and heaven ; this, however, seems somewhat arbitrary.
_ It remains to deal with the statement of the Doxographi that
_Parmenides held the moon to be illuminated by the sun:
_ *The moon Parmenides declared to be equal to the sun; for
indeed it is illuminated by it.’ ?
This is the more suspicious because in another place Aétius
attributes the first discovery of this fact to Thales, and adds that
Pythagoras, Parmenides, and Empedocles, as well as Anaxagoras
and Metrodorus, held the same view.* Parmenides was doubtless
credited with the discovery on the ground of two lines from his
poem.* The first is quoted by Plutarch : ὃ
‘For even if a man says that red-hot iron is not fire, or that the
moon is not a sun because, as Parmenides has it, the moon is
“a night-shining foreign light wandering round the earth”,
he does not get rid of the use of iron or of the existence of the
moon.’
? Aét. ii. 15. 7 (2. G. p. 345).
® Aet. ii. 26. 2 (D. G. "ἢ 357; Vors. i, P- 111. 32).
3. Aét. ii. 28. 5 (D. G. p. 358; Vors. i*, p. 111. 33).
4 Fr. 14 and 15 (Vors. 15, p. 124. 6, το).
5. Plutarch, Adv. Colot. 15, p. 1116 A (Vors. i*, p. 124, 4-7).
76 ὶ PARMENIDES PARTI
But, even if the verse is genuine, ‘foreign’ (ἀλλότριον) need not
have meant ‘ borrowed’; the expression ἀλλότριον φῶς is, as Diels
says, a witty adaptation of Homer’s ἀλλότριος φῶς used of persons,
‘a stranger’.2 Tannery thinks that the line is adapted from one
of Empedocles’, and was probably interpolated in Parmenides’
poem by some Neo-Pythagorean who was anxious to refer back
to the Master the discovery which gives Anaxagoras his greatest
title to fame.
Boll,* on the other hand, considers it absolutely certain that
Parmenides knew of the illumination of the moon by the sun.
He admits, however, that we cannot suppose Parmenides to
have discovered the. fact for himself, and that we cannot be
certain whether he got it from Anaximenes or the Pythagoreans.
We have seen (p. 19) good reason for thinking that it was not
Anaximenes who made the discovery; and the only support that
Boll can find for the alternative hypothesis is the statement of
Aétius that Pythagoras considered the moon to be a ‘ mirror-like
_body’ (κατοπτροειδὲς σῶμα). But-this is an uncertain phrase to
build upon, especially when account is taken of the tendency to
attribute to Pythagoras himself the views of later Pythagoreans ;
and indeed the evidence attributing the discovery to Anaxagoras
is so strong that it really excludes all other hypotheses.
The other line speaks of the moon as ‘always fixing its gaze
on the beams of the sun’. This remark is certainly important,
but is far from explaining the cause of the observed fact. But
we have positive evidence against the attribution of the discovery
of the opacity of the moon to Parmenides or even to Pythagoras.
It is part of the connected prose description of Parmenides’
system® that the moon is a mixture of air and fire;’ in other
passages we are told that Parmenides held the moon to be of fire®
? Diels, Vors. 112, τ, p. 675 ; Parmenides Lehrgedicht, p. 110.
3 Homer, ας v. 2143 Od. xviii. 219, &c.
8 Tannery, op. cit., p. 210. The lines are respectively—
Νυκτιφαὲς περὶ γαῖαν ἀλώμενον ἀλλότριον φῶς (Parm.).
Κυκλοτερὲς περὶ γαῖαν ἑλίσσεται ἀλλότριον φῶς (Emped.).
* Boll, art. ‘Finsternisse’ in Pauly-Wissowa’s Real-Encyclopadie der classischen
Altertumswissenschaft, vi. 2, 1909, p. 2342.
5 Aét. ii. 25. 14 (D. G. p. 357).
5 Aét. ii, 7. 1 (D.G. p. 335; Vors. ἴδ, p. 111. 5 sqq.).
7 Ibid. (D. G. p. 335; Vors. i®, p. 111. 13).
8. Aét. ii. 25. 3 (D. G. p. 356; Vors. 13, p. 111. 31).
CHIX PARMENIDES 77
_ and to be an excretion from the denser part of the mixture in the
. Milky Way; which itself (like the sun) is an exspiration of fire.?
- More important still is the evidence of Plato, who speaks of ‘the
fact which Anaxagoras lately asserted, that the moon has its light
from the sun’.* It seems impossible that Plato should have spoken
in such terms if the fact had been stated for the first time by
-Parmenides or the Pythagoreans.
1 Aét. ii. 20.8 a (D. G. p. 349; Vors. i*, p. 111. 35).
5 Aét. ii. 7. 1 (2. σ. p. 335; Vors. #, p. 111. 13).
3 Plato, Cratylus 409 A.
Χ
ANAXAGORAS
ANAXAGORAS was born at Clazomenae in the neighbourhood of
Smyrna about 500B.c. He neglected his possessions, which were
considerable, in order to devote himself to science. Some one once
asked him what was the object of being born, to which he replied,
‘The investigation of sun, moon, and heaven.’? He seems to have
been the first philosopher to take up his abode at Athens, where he
enjoyed the friendship of Pericles, who had probably induced him to
come thither. When Pericles became unpopular shortly before the
outbreak of the Peloponnesian war, he was attacked through his
friends, and Anaxagoras was accused of impiety for holding that
the sun was a red-hot stone and the moon earth. According to
one account he was fined five talents and banished ;* another
account says that he was put in prison and it was intended to put
him to death, but Pericles got him set at liberty ;° there are other
variations of the story. He went and lived at Lampsacus, where he
died at the age of 72.
A great man of science, Anaxagoras enriched astronomy by one
epoch-making discovery. This was nothing less than the discovery
of the fact that the moon does not shine by its own light but
receives its light from the sun. As a result, he was able to give
(though not without an admixture of error) the true explanation of
eclipses. I quote the evidence, which is quite conclusive :
‘, . . the fact which he (Anaxagoras) recently asserted, namely
that the moon has its light from the sun.’ ὃ
‘Now when our comrade, in his discourse, had expounded that
proposition of Anaxagoras, that “the sun places the brightness in
the moon”, he was greatly applauded.’ *
1 Plato, Hippias Major 283 A. 3 Diog. L. ii. 10 (Vors. i*, p. 294. 17).
8 Plato, Apology 26 D. 4 Diog. L. ii. 12 (Vors. i*, p. 294. 32).
δ᾽ Ibid. ii. 13 (Vors. i*, p. 294. 42). 5 Plato, Cratylus, p. 409 A.
7 Plutarch, De facie in orbe lunae 16, p. 929 B (Vors. i*, p. 321. 5-7).
ANAXAGORAS "9
‘The moon has a light which is not its own, but comes from the
sun.’?
‘The moon is eclipsed through the interposition of the earth,
sometimes also of the bodies below the moon’? [i.e. the ‘ bodies
below the stars which are carried round along with the sun and the
moon but are invisible to us’.*]
‘The sun is eclipsed at the new moon through the interposition
of the moon.’* ‘He was the first to set out distinctly the facts
about eclipses and illuminations.’®
‘For Anaxagoras, who was the first to put in writing, most
clearly and most courageously of all men, the explanation of the
moon's illumination and darkness, did not belong to ancient times,
and even his account was not common property but was still a
secret, current only among a few and received by them with caution
or simply on trust. For in those days they refused to tolerate the
physicists and star-gazers as they were called, who presumed to
fritter away the deity into unreasoning causes, blind forces, and
necessary properties. Thus Protagoras was exiled, and Anaxa-
_ goras was imprisoned and with difficulty saved by Pericles.’ ὃ
‘ Anaxagoras, in agreement with the mathematicians, held that
_ the moon’s obscurations month by month were due to its following
_the course of the sun by which it is illuminated, and that the
_ eclipses of the moon were caused by its falling within the shadow
_ of the earth, which then comes between the sun and the moon,
_ while the eclipses of the sun were due to the interposition of the
/ moon,’?
_ ‘Anaxagoras, as Theophrastus says, held that the moon was
_ also sometimes eclipsed by the interposition of the (other) bodies
below the moon.’ ὃ
Here, then, we have the true explanation of lunar and other
eclipses, though with the unnecessary addition that, besides the
earth, there are other dark bodies invisible to us which sometimes
1 Hippolytus, Refuz. i. 8.8 (from Theophrastus: see D.G. p. 562; Vors. i?,
Σ 46).
5 Ibid. i. 8. 9. (D. G. p. 562; Vors. i, p. 301. 47).
5 Tbid. i. 8. 6 (D.G. p. 562; Vors. i*, p. 301. 41).
4 Ibid. i. 8. 9 (D. G. p. 562; Vors. i*, p. 301. 48).
5 Ibid. i. 8. 10 (D. G. p. 562; Vors. 15, p. 302. 3).
§ Plutarch, Vic. 23 (Vors. i?, p. 297. 40-6).
7 A&t. ii. 29. 6 (D. G. p. 360; Vors. 13, p. 308.17). I have in the last phrase
_ translated Diels’ conjecturally emended reading ἥλιον δὲ τῆς σελήνης instead of
᾿ς μᾶλλον δὲ τῆς σελήνης ἀντιφραττομένης (D.G. pp. 53-4). The difficulty, however,
| is that, according to the heading, the passage deals with the eclipses of the
_ moon only.
8 Aét. ii. 29. 7 (2. G. p. 360; Vors. i*, p. 308. 20).
80.” ANAXAGORAS PARTI
obscure the moon and cause eclipses. In this latter hypothesis, as
in much else, Anaxagoras followed Anaximenes.!
Whether Anaxagoras reached the true explanation of the phases
of the moon is much more doubtful. It is true that Parmenides
had observed that the moon has its bright portion always turned in
the direction of the sun; when to this was added Anaxagoras’s
discovery that the moon derived its light from the sun, the explana-
tion of the phases was ready to hand. But it required that the
moon should be spherical in shape; Anaxagoras, however, held
that the earth, and doubtless the other heavenly bodies also, were
1 The same idea is attributed by Aristotle (De caelo ii. 13, 293 Ὁ 21-25) to
certain persons whom he does not name: ‘Some think it is possible that more
bodies of the kind [i.e. such as the Pythagorean counter-earth] may move about
the centre but may be invisible to us owing to the interposition of the earth.
This, they say, is the reason why more eclipses of the moon occur than of the
sun, for each of the bodies in question obscures the moon, and it is not only the
earth which does so.’ An interesting suggestion has been made (by Boll in art.
‘Finsternisse’ in Pauly-Wissowa’s Real-Encyclopadie d. class. Altertumsw, vi. 2,
p- 2351), which furnishes a conceivable explanation of the persistence of the
idea that lunar eclipses are sometimes caused by the interposition of dark bodies
other than the earth. Cleomedes (De motu circulari ii. 6, Ὁ. 218. 8. sqq.)
mentions that there were stories of extraordinary eclipses which ‘the more
ancient of the mathematicians’ had vainly tried to explain; the supposed
‘ paradoxical’ case was that in which, while the sun seems to be still above the
horizon, the ec/ifsed moon rises in the east. The phenomenon appeared to be
inconsistent with the explanation of lunar eclipses by the entrance of the moon
into the earth’s shadow; how could this be if both bodies were above the ©
horizon at the same time? The ‘more ancient’ mathematicians tried to argue
that it was possible that a spectator standing on an eminence of the spherical
earth might see along the generators of a cone, i.e. a little downwards on all
sides, instead of merely in the A/ane of the horizon, and so might see both the
sun and the moon even when the latter was in the earth’s shadow. Cleomedes
denies this and prefers to.regard the whole story of such cases as a fiction
designed merely for the purpose of plaguing astronomers and philosophers ; no
Chaldean, he says, no Egyptian, and no mathematician or philosopher has
recorded such a case. But we do not need the evidence of Pliny (V.H. ii, c. 57,
§ 148) to show that the phenomenon is possible; and Cleomedes himself really
gives the explanation (pp. 222. 28-226. 3), namely, that it is due to atmospheric
refraction. Observing that such cases of atmospheric refraction were especially
noticeable in the neighbourhood of the Black Sea, he goes on to say that it is
possible that the visual rays going out from our eyes are refracted through falling
on wet and damp air, and so reach the sun though it is already below the
horizon ; and he compares the well-known experiment of the ring at the bottom
of a jug, where the ring, just out of sight when the jug is empty, is brought into
view when water is poured in. Unfortunately there is nothing to indicate the
date of the ‘more ancient mathematicians’ who gave the somewhat primitive
explanation which Cleomedes refutes; but was it the observation of the phe-
nomenon, and their inability to explain it otherwise, which made Anaxagoras
and others adhere to the theory that there are other bodies besides the earth
which sometimes, by their interposition, cause lunar eclipses ?
CH. X ANAXAGORAS 81
flat, and accordingly his explanation of the phases could hardly
have been correct.?
Anaxagoras’s cosmology contained other fruitful ideas. Accord-
ing to him the formation of the world began with a vortex set up,
in a portion of the mixed mass in which ‘all things were together’,
by his deus ex machina, Nous.2. This rotatory movement began at
one point and then gradually spread, taking in wider and wider
circles. The first effect was to separate two great masses, one
_ consisting of the rare, hot, light, dry, called the ‘aether’, and the
other of the opposite categories and called ‘air’. The aether or
fire took the outer position, the air the inner.2 The next step is the
successive separation, out of the air, of clouds, water, earth, and
stones. The dense, the moist, the dark and cold, and all the
heaviest things collect in the centre as the result of the circular
motion ; and it is from these elements when consolidated that the
earth is formed.® But, after this, ‘in consequence of the violence of
the whirling motion, the surrounding fiery aether tore stones away
from the earth and kindled them into stars.’ Reading this with
the remark that stones ‘rush outwards more than water’,’ we see
' that Anaxagoras conceived the idea of a centrifugal force as dis-
' tinct from that of concentration brought about by the motion of
_ the vortex, and further that he assumed a series of projections or
*hurlings-off’ of precisely the same kind as the theory of Kant and
_ Laplace assumes for the formation of the solar system.®
Apart from the above remarkable innovations, Anaxagoras did
not make much advance upon the crude Ionian theories; indeed he
showed himself in the main a follower of Anaximenes.
According to Anaxagoras
‘The earth is flat in form and remains suspended because of its
size, because there is no void, and because the air is very strong and
supports the earth which rides upon it.’ ®
‘The sun, the moon, and all the stars are stones on fire, which
are carried round by the revolution of the aether.’ 19
* Tannery, op. cit., p. 278. ? Fragment 13 (Vors. i?, p. 319. 20).
* Fr. 15 (Vors. 15, p. 320. 11). * Fr. 16 (Vors. 15, p. 320. 20).
ἢ Hippol. Refut. i. 8. 2 (from Theophrastus); D.G. p. 562; Vors. ιν", p. 301.
30. ® Aét. ii. 13. 3 (D. G. p. 341 ; Vors. #7, p. 307. 16).
7 Fr. 16 (Vors. 15, p. 320. 22-3).
: * Gomperz, Griechische Denker, i*, p. 176.
_ 2 Hippol. Refut. i. 8. 3 (D.G. p. 562; Vors. i*, p. 301. 31).
_ ” Ibid. i. 8. ὁ (Vors. i?, p. 301. 39).
1410 G
82 ANAXAGORAS PARTI
‘The sun is a red-hot mass or a stone on fire.’ ἢ
‘It is larger’ (or ‘many times larger’*) than the Peloponnese.*®
‘The moon is of earthy nature and has in it plains and ravines.’ *
‘The moon is an incandescent solid, having in it plains, moun-
tains, and ravines.’ ὅ
‘It is an irregular compound because it has an admixture of cold
and of earth. It has a surface in some places lofty, in others low,
in others hollow. And the dark is mixed along with the fiery, the
joint effect being an impression of the shadowy ; hence it is that
the moon is said to shine with a false light.’ ®
Anaxagoras explained the ‘turning’ of the sun at the solstice
thus:
‘The turning is caused by the resistance of the air in the north
which the sun itself compresses and renders strong through its
condensation.’?
‘The turnings both of the sun and of the moon are due to their
being thrust back by the air. The moon’s turnings are frequent
because it cannot get the better of the cold,’ ὃ
Again: :
‘We do not feel the warmth of the stars because they are at
a great distance from the earth; besides which they are not as hot
as the sun because they occupy a colder region. The moon is
below the sun and nearer to us.’ ὃ
‘ The stars were originally carried round (laterally) like a dome,
the pole which is always visible being vertically above the earth,
and it was only afterwards that their course became inclined.’ 19
‘After the world was formed and the animals were produced
from the earth, the world received as it were an automatic tilt
towards its southern part, perhaps by design, in order that some
1 Aét. ii. 20. 6 (D. G. p. 349; Vors. 13, p. 307. 19).
2 Aét. ii. 21. 3 (D.G. p. 3513 Vors. i*, p. 307. 20).
3 Diog. L. ii. 8 (Vors. i*, p. 293. 38).
4 Hippol. Refuz. i. 8. 10 (D.G. p. 562; Vors. i*, p. 302. 4).
5 Aét. ii. 25. 9 (D. G. p. 356; Vors. i, p. 308. 10).
6 Aét. ii, 30. 2 (D. G. p. 361; Vors. 13, p. 308. 12). As Dreyer observes
(Planetary Systems, p. 32, note), the moon has some light of its own which we
see during lunar eclipses ; cf. Olympiodorus on Arist. MZe¢eor. (p. 67. 36, ed.
Stiive ; Zeteor., ed. Ideler, vol. i, p. 200), ‘The moon’s own light is of one kind,
the sun’s of another; for the moon’s own light is like charcoal (av@pax@des), as
we can plainly see during an eclipse.’
7 Aét. ii. 23. 2 (D.G. p. 352; Vors. i*, p. 307. 20).
® Hippol. Refuz. i. 8. 9 (D. G. p. 562; Vors. i*, p. 302. 1).
9 Ibid. i. 8. 7 (D. G. p. 562; Vors. i, p. 301. 42). :
Diog. L. ii. 9 (Vors. 13, p. 294. 3).
CH. xX ANAXAGORAS 83
parts of the world might become uninhabitable and others inhabit-
able, according as they are subject to extreme cold, torrid heat,
or moderate temperature. ἢ
‘ The revolution of the stars takes them round under the earth.’ 2
Gomperz®* finds a difficulty in reconciling the last of these
passages with the other statement that the earth is flat and rests on
air, in which Anaxagoras had followed Anaximenes. Anaximenes
seems to have regarded the basis of air on which the flat earth
rested in the same way as Thales the water on which his earth
floated ; and Anaximenes said that the stars did not pass under the
earth but laterally round it. I do not, however, feel sure that
Anaxagoras could not have supposed the stars to pass in their
revolution through the basis of air under the earth, although no
doubt Thales was almost precluded from supposing them to pass
through his basis of water. If, as Gomperz says, Simplicius * is alone
in attributing to Anaxagoras’s earth the shape of a drum or cylinder,
Aristotle as well as Simplicius seems to imply that at all events
the earth occupied the centre of the universe.®
_ Anaxagoras put forward a remarkable and original hypothesis to
explain the Milky Way. As we have seen, he thought the sun to be
smaller than the earth. Consequently, when the sun in its revolu-
tion passes below the earth, the shadow cast by the earth extends
without limit. The trace of this shadow on the heavens is the
Milky Way. The stars within this shadow are not interfered with
by the light of the sun, and we therefore see them shining; those
stars, on the other hand, which are outside the shadow are over- .
_ powered by the light of the sun, which shines on them even during
the night, so that we cannot see them. Such appears to be the
meaning of the passages in which Anaxagoras’s hypothesis is
explained. According to Aristotle, Anaxagoras and Democritus
both held that
_ ‘The Milky Way is the light of certain stars. For when the sun
is passing below the earth some of the stars are not within its
vision. Such stars then as are embraced in its view are not seen to
? Aét. ii. 8. 1 (D. G. p. 3373 Vors. 15, p. 306. 12).
3 Hippol. Refut. i. 8.8 (D. G. p. 562; Vors. 15, p. 301. 47).
* Gomperz, Griechische Denker, *, pp. 178, 442.
* Simplicius on De cae/o, p. 520. 30.
5 Arist. De caelo ii. 13, 295 a 13.
G2
84 ANAXAGORAS PARTI
give light, for they are overpowered by the rays of the sun ; such of
the stars, however, as are hidden by the earth, so that they are not
seen by the sun, form by their own proper light the Milky Way.’
᾿ς ‘Anaxagoras held that the shadow of the earth falls in this part
of the heaven (the Milky Way) when the sun is below the earth and
does not cast its light about all the stars,’ ?
‘The Milky Way is the reflection (ἀνάκλασις) of the light of the
stars which are not shone upon by the sun,’ ὃ
As Tannery * and Gomperz® point out, this conjecture, however
ingenious, could easily have been disproved by simple observation.
For Anaxagoras might have observed the obvious fact, noted as an
objection by Aristotle,® that the Milky Way always retains the
same position relatively to the fixed stars, whereas the hypothesis
would require the trace of it to change its position along with the
sun; indeed the Milky Way should have coincided with the ecliptic,
whereas it is actually inclined to it. Again, if the theory were true,
an eclipse of the moon would have been bound to occur whenever the
moon passed over the Milky Way, and it would have been easy to
verify that this is not so. As the Milky Way is much longer than
it is broad, it would seem that Anaxagoras thought that the flat
earth was not round but ‘elongated’ (προμήκης), as Democritus
afterwards conceived it to be,’ though Democritus only made its
its length half as much again as its breadth.®
Aristotle ὃ adds an interesting criticism of this theory : ‘ Besides,
if what is proved in the theorems on astronomy is correct, and the
size of the sun is greater than that of the earth, and the distance of
the stars from the earth is many times greater than the distance of
the sun, just as the distance of the sun is many times greater than
the distance of the moon, the cone emanating from the sun and
marking the convergence of the rays would have its vertex not
very far from the earth, and consequently the shadow of the earth,
* Arist. Meteorologica i. 8, 345 a 25-31 (Vors. i’, Pp. 308. 26-31).
2. Aét. iii. 1. 5 (2. G. p . 365; Vors. i*, p. 308. 31
3 Hippol. on i. 8. és (D. G. p. 561; Vors.i’, p. "302, 5); Diog. L. ii. 9 (Vors.
» P. 294. 5
4 Tannery, op. cit., p. 279.
5 Gomperz, Griechische Denker, i*, p. 179.
® Aristotle, Meteorologica i, 8, 345 ἃ 32.
1 Eustathius zm Homer. 11. vii. 446, p. 690 (Vors. ἴδ, p. 367. 42).
8. Agathemerus, i. 2 ( Vors. 13, p. 393. 10).
® Aristotle, Meteorologica i. 8, 345 Ὁ 1-9.
‘
|
.
cH.x ANAXAGORAS 85
which we call night, would not reach the stars at all. In fact the
sun must embrace in his view αὐ the stars and the earth cannot hide
any one of them from him.’
According to Proclus,! who quotes the authority of Eudemus,
Anaxagoras anticipated Plato in holding that in the order of the
revolution of the sun, moon, and planets round the earth the sun
came next to the moon, whereas Ptolemy? says that according
to ‘the more ancient’ astronomers (by which phrase he appears to
mean the Chaldaeans *) the order (starting from the earth) was
Moon, Mercury, Venus, Sun, Mars, Jupiter, Saturn.
It seems clear that Anaxagoras held that there were other worlds
than ours. Aétius,* it is true, includes Anaxagoras among those
who said that there was only one world; but the fragments must
be held to be more authoritative, and one of these leaves no room
for doubt on the subject. According to this fragment
‘Men were formed and the other animals which have life; the
men too have inhabited cities and cultivated fields as with us; they
have also a sun and moon and the rest as with us, and their earth
produces for them many things of various kinds, the best of which
they gather together into their dwellings and live upon.’
Thus much have I said about separating off, to show that it will
not be only with us that things are separated off, but elsewhere
as well.’
Proclus iw Timaeum, p. 258c (on Timaeus 38D); Vors. i*, p. 308. 1-4.
5. Ptolemy, Syntaxis ix. 1, vol. ii, p. 207, ed. Heiberg.
* Tannery, op. cit., p. 261.
* Aét. ii. 1. 2 (D. G. p. 327; Vors. i*, p. 305. 44).
5. Burnet, Early Greek Philosophy, pp. 312, 313.
§ Fr. 4 (Vors. i*, p. 315. 8-16).
XI
EMPEDOCLES
THE facts enabling the date of Empedocles of Agrigentum to be
approximately determined are mainly given by Diogenes Laertius.’
His grandfather, also called Empedocles, won a victory in the
horse-race at Olympia in 496/5 B.C.; and Apollodorus said that
his father was Meton, and that Empedocles himself went to Thurii
shortly after its foundation. Thurii was founded in 445 B.C. and,
when Diogenes Laertius says that Empedocles flourished in Ol. 84
(444/1), it is clear that the visit to Thurii was the basis for this
assumption. According to Aristotle* he died at the age of sixty; —
hence, assuming him to be forty in 444 B.c., we should have 484--
424 B.C. as the date. But there is no reason why he should be
assumed to have been just forty at the date of his visit to Thurii ; and
other facts suggest that the date so arrived at is about ten years too
late. Theophrastus said that Empedocles was born ‘ not long after
Anaxagoras’ ;* according to Alcidamas he and Zeno were pupils
of Parmenides at the same time;* and Satyrus said that Gorgias
was a disciple of Empedocles.2 Now Gorgias was a little older
than Antiphon (of Rhamnus), who was, born in 480 B.c.° It
follows that we must go back az /east to 490 B.C. for the birth of
Empedocles ; most probably he lived from about 494 to 434 B.C."
Empedocles is said to have been the inventor of rhetoric;* as
an active politician of democratic views he seems to have played
1 Diog. L. viii. 51-74 (Vors. ", pp. 149-53).
* In Diog. L. viii. 52 (Vors. i?, p. 150. 15).
* Theophrastus in Simpl. Phys. p. 25. 19 (D. G. p. 477; Vors. i’, p. 154. 33).
* Diog. L. viii. 56 (Vors. i*, p. 150. 41).
5 Diog. L. viii. 58 ( Vors. i2, p- 151. 10).
5 [Plutarch] Vit, X ovat. i. 1. 9, p . 832} (Vors. ii*. 1, p. 546. 25).
7 Cf. Diels’ ‘ Empedokles und egies » 2 (Berl. Sitsungsb, 1884) ; Burnet,
Early Greek rhe GAL . 228-9.
8 Aristotle in Diog. L. vili. 57 (Vors. i?, p. 150. 46).
EMPEDOCLES 87
a prominent part in many a stirring incident; he was a religious
teacher, a physiologist and, according to Galen, the founder of an
Italian school of Medicine, which vied with those of Cos and
Cnidus. That he was no mean poet is sufficiently attested by the
fragments which survive, amounting to 350 (or so) lines or parts
of lines in the case of the poem Oz Nature and over 100 in the
case of the Purifications.
Empedocles followed Anaximenes in holding that the heaven is
a crystal sphere and that the fixed stars are attached to it.2 The
sphere, which is ‘solid and made of air condensed or congealed
by the action of fire, like crystal’? is, however, not quite spherical,
_ the height from the earth to the heaven being less than its distance
from it laterally, and the universe being thus shaped like an egg.*
While the fixed stars are attached to the crystal sphere, the planets
are ἔτεα.
_ The sun’s course is round the extreme circumference of the world
_ (literally ‘is the circuit of the limit of the world’) ;® in this particular
_Empedocles follows Anaximander. The circuit must be just inside
' the circumference because, under the heading ‘ tropics’ or ‘turnings’
of the sun, Aétius says that, according to Empedocles, the sun is
_ prevented from moving always in a straight line by the resistance
_ of the enveloping sphere and by the tropic circles.’
A special feature of Empedocles’ system is his explanation (1) of
day and night, (2) of the nature of the sun.
(1) Within the crystal sphere, and filling it, is a sphere consisting
of two hemispheres, one of which is wholly of fire and therefore
light, while the other is a mixture of air with a little fire, which
mixture is darkness or night. The revolution of these two hemi-
1 Galen, Meth. Med. i. 1 (Vors. i*, p. 154. 19-23).
2 Aét. ii. 13. 11 (D.G. p. 342; Vors. i*, p. 162. 12).
3 Aét. ii. 11. 2 (D. G. p. 339; Vors. i*, p. 161. 40).
* Aét. ii. 31. 4 (D. G. p. 363; Vors. i*, p. 161. 34). The statement as to
_ height and breadth is mathematically inconsistent with the comparison of the
_ figure to an egg, unless we suppose that Empedocles regarded the section of it
_ by the plane containing the surface of the earth as an oval and not a circle,
which does not seem likely. If the said section is a circle, the figure would be
_ what we call an od/ate spheroid (the solid described by the revolution of an
_ ellipse about its nor axis) rather than egg-shaped.
5 Aét. ii. 13. 11 (see above).
* Aét. ii. 1. 4 (D.G. p. 328; Vors. i*, p. 161. 37) τὸν τοῦ ἡλίου περίδρομον εἶναι
Ἐ περιγραφὴν τοῦ πέρατος τοῦ κόσμου.
ΒΕ τ Aét. ii. 23. 3 (2. G. p. 353; Vors. i?, p. 162. 37).
88 EMPEDOCLES PARTI
spheres round the earth produces at each point on its surface the
succession of day and night.1_ The beginning of this motion was
due to the collection of the mass of fire in one of the hemispheres,
the result being that the pressure of the fire upset the equilibrium
of the heaven and caused it to revolve.2 Apparently connected
with this theory of the two hemispheres is Empedocles’ explanation
of the difference between winter and summer. It is winter when
the air (forming one hemisphere) gets the upper hand through
condensation and is forced upwards (into the fiery hemisphere), and
summer when the fire gets the upper hand and is forced downwards
(into the dark hemisphere) ;* that is, in the winter the fire occupies
less than half of the whole sphere of heaven, while in the summer
it occupies more than half. The idea seems to be that the greater
half of the sphere takes longer to revolve about a particular point
on the earth’s surface than the smaller half, and that this explains
why the days are longer in the summer than in the winter. We
are not told what was the axis about which the two hemispheres
were supposed to revolve, but it seems hardly likely that Empedocles
could have assumed a definite axis different from that of the daily
rotation of the heavenly sphere.
According to Empedocles it was the swiftness of the remalition
of the heaven which kept the earth in its place, just as we may
swing a cup with water in it round and round so that in some
positions the top of the cup may actually be turned downwards
without the water escaping.t* The analogy is, of course, not a
good one, because the water in that case is kept in its place by
centrifugal force which throws it, as it were, against the side of the
vessel, whereas the earth is presumably at rest in the centre during
the revolution of the heaven, and is not acted on by such a force.
Empedocles further held that the revolution of the heaven, which
now takes 24 hours to complete, was formerly much _ slower.
At one time a single revolution was only accomplished in a period
equal to ten of our months; later it required a period equal to seven
1 Ps, Plut. Stromat. apud Euseb. Praep. Evang. i. 8. 10 (from Theophrastus) ;
D.G. p. 582; Vors.i*, p. 158. 19-23.
2 Ps, Plut. Stromat., loc. cit. Vors. i”, p. 158. 33-4).
5. Aét. iii. 8. 1 (D. G. P- 375; Vors. i*, p. 163. 16).
* Aristotle, De caelo ii. 13, 295 a 17 (Vors. i’, p. 163. 39).
CH. XI EMPEDOCLES 89
of our months. These views have, however, no astronomical basis ;
they were put forward solely in order to explain the exceptions
to the usual period of gestation afforded by ten-months’ and seven-
months’ children, the period being in each case taken as one day !
Coming now to Empedocles’s conception of the nature of the sun,
we find the following opinions attributed to him:
‘The sun is, in its nature, not fire, but a reflection of fire similar
to that which takes place from (the surface of) water.’?
‘There are two suns; one is the original sun which is the fire
in one hemisphere of the world, filling the whole hemisphere and
always placed directly opposite the reflection of itself; the other
is the apparent sun which is a reflection in the other ‘hemisphere
filled with air and an admixture of fire, and in this reflection what
happens is that the light is bent back from the earth, which is
_ circular, and is concentrated into the crystalline sun where it is
carried round by the motion of the fiery (hemisphere). Or, to
_ state the fact shortly, the sun is a reflection of the fire about the
ΘΑ. ὃ
‘The sun which consists of the reflection is equal in size to
the earth.’ *
__*You laugh at Empedocles for saying that the sun is produced
about the earth by a reflection of the light in the heaven and “ once
more flashes back to Olympus with fearless countenance ”.’®
The second of the above passages is scarcely intelligible at the
point where the reflection is called ‘a reflection iz the other
hemisphere’; it can hardly be in the other hemisphere because
that hemisphere is night. Accordingly Tannery conjectures thatthe |
reading should be ‘a reflection (¢zviszb/e) in the other hemisphere ’.6
The meaning must apparently be that the fire in the fiery hemi-
sphere is reflected from the earth upon the crystal vault, the
reflected rays being concentrated in what we see as the sun. The
equality of the size of the sun and the earth may have been a hasty
inference founded upon the supposition of an analogy with the
recently discovered fact that the moon shines with light borrowed
1 Aét. v. 18. 1 (D. G. p. 427; ΜΝ τ΄, p. 165. 31).
2. Ps. Plut. Stromat., loc. cit. (D. G. p. 582 ; Ba i*, p. 158. 35).
3 Aét. ii. 20. 13 (D. a. Ρ. 350; Vors. i*, p. 162. 18-24).
* Aét. ii. 21. 2 (2. G. p. 351; Vors.i*?, p. 162. 25).
® Plutarch, De Pyth. or. 12, p. emer i*, p. 188. 8-11).
* Tannery, op. cit., p. 323.
90 EMPEDOCLES PART I
from the sun.!_ The theory that the sun which we see is a concen-
tration of rays reflected from the earth upon the crystal sphere
agrees exactly with the statement already quoted that the sun’s
course is confined just within the inner surface of the spherical
envelope. Why it is just confined within the tropical circles and
prevented from deviating further in latitude is not so clear. If, as
Dreyer supposes,” the airy and the fiery hemispheres, which in turn
occupy more than half of the heavenly sphere, ‘thereby make
the sun, the image of the fiery hemisphere, move south or north
according to the seasons’, it would seem necessary to suppose that
the advance of the hemisphere of fire in the summer (and its retreat
in the winter) does not take place uniformly over the whole of its
circular base (which is the division between the two hemispheres),
i.e. in such a way that the base of the new hemisphere is parallel
to the base of the old, but that the advance (or retreat) takes place
obliquely with reference to the circular base, being greatest at a
certain point on the rim of that base and least at the opposite
point, so that the plane base of the new hemisphere is obliquely
inclined to that of the old; in other words, that the avis of the
fiery hemisphere changes its position as the advance (or retreat)
proceeds, and in fact swings gradually (completing an oscillation
in a year) between two extreme positions inclined to the mean
position at an angle equal to the obliquity of the ecliptic. But it is
very unlikely that Empedocles, with his elementary notions hl
astronomy, worked out his theory in this way.
It would appear that- Empedocles’ theory of the sun gave a lead
to the later Pythagoreans, for we shall find Philolaus saying that
‘there are in a manner two suns .. . unless [in Aétius’s words]
1 Cf. Plutarch, De fac. in orbe lunae 16, p. 929 E (Vors. i*, p. 187. 21-6):
‘There remains then the view of Empedocles that the illumination which we
get here from the moon is produced by a sort of reflection of the sun at the
moon [the same word ἀνάκλασιν being used in this case]. Hence we get neither
heat nor brightness from it, whereas we should expect both if there had been
a kindling and mixing of (the) lights, and, just as when sounds are reflected
the echo is less distinct than the original sound, .... ‘‘even so the ray which
struck the moon’s wide orb” passes on tous a reflux which is weak and indistinct,
owing to the loss of power due to the reflection.’ But, if Empedocles spoke in
this way of the moon’s light, he could hardly have conceived the light of the
sun, which is bright and hot, to be a reflection of light in the same sense as the
light of the moon is; the ‘reflection of light’ which constitutes the sun is more
like the effect of a burning-glass than ordinary reflection.
2 Dreyer, Planetary Systems, Ὁ. 25.
CH. ΧῚ EMPEDOCLES gr
_we prefer to say that there are three, the third consisting of the
rays which are reflected again from the mirror or lens [the second
- sun] and spread in our direction’.!
Empedocles does not seem to have mentioned the annual motion
of the sun relatively to the fixed stars, although, as we have seen,
_ he speaks of the tropic circles as limiting its motion (i.e. motion in
latitude).
_ Empedocles, like Anaxagoras, held that the moon shone with
light borrowed from the sun.2_ The moon itself he regarded as
‘a mass of frozen air, like hail, surrounded by the sphere of
fire te or as ‘condensed air, cloudlike, solidified (or congealed)
by fire, so that it is of mixed composition’. This idea may have
put forward to account for the apparent change of shape in
the phases; for we find Plutarch saying that ‘the apparent form
of the moon, when the month is half past, is not spherical, but
lentil-shaped and like a disc, and, in the opinion of Empedocles, its
ctual substance is so too’.®
The stars he thought to be ‘of fire (arising) out of the fiery
(element) which the air contained in itself but squeezed out upwards
the original separation ’.®
_ We are not definitely told whether Empedocles held the earth
_ to be spherical or flat. He might, it is true, have adopted the view
of the Pythagorean school and Parmenides that it was spherical,
but it is more probable that he considered it to be flat. For we
are told that he regarded the moon as ‘like a disc’*; in this he
probably followed Anaxagoras, who undoubtedly thought the earth
flat, and therefore most probably the moon also.
He also shared the view of Anaxagoras that the axis of the
world was originally perpendicular to the surface of the earth, the
north pole being in the zenith, and that it was displaced afterwards.
This view Anaxagoras combined with the hypothesis of a flat
1 Aét. ii. 20, 12 (D.G. p. 349; Vors. i*, p. 237. 39).
_ ? Fr. 43, quoted in note on preceding page; Aét. ii. 28.5 (D.G. p. 358;
WVors. 7, p. 162. 48).
Ε * Plutarch, De fac. in orbe lunae 5, p. 922 C (Vors. 15, p. 162. 43).
4 * Aét. ii. 25. 15 (D. G. p. 357; Vors. i?, p. 162. 41).
_ δ᾽ Plutarch, Quaest. Rom. 101, p. 288 B (Vors. i*, p. 162. 45).
2 5 Aét. ii. 13. 2 (D. G. p. 341; Vors. i, p. 162. το).
_ * Diog. L. viii. 77 (Vors. i, p. 153. 37); Aét. ii. 27. 3 (D.G. p. 358; Vors. 2,
4 162. 44).
92 EMPEDOCLES PART I
earth; indeed, a flat earth is almost necessary if the axis of the
universe was originally perpendicular to its surface. Empedocles,
however, differed from Anaxagoras in his explanation of the cause
of the subsequent displacement ; whereas Anaxagoras could only
account for it tentatively by assuming ‘design’, Empedocles gave
a mechanical explanation : |
‘The air having yielded to the force of the sun, the north pole
became inclined, the northern parts were heightened, and the
southern lowered, and the whole universe was thereby affected.’ ?
‘There are many fires burning beneath the earth’? said
Empedocles. He seems to have inferred this truth from the
existence of hot springs, the water of which he supposed to be
heated, like the water in baths, by running a long course, as it
were in tubes, through fire.®
According to Empedocles the sun is a great collection of fire and
greater than the moon,* and the sun is twice as distant from the
earth as the moon ἰ5.ὅ
He was aware of the true explanation of eclipses of the sun, for
he says that
‘The moon shuts off the beams of the sun as it passes across it,
and darkens so much of the earth as the breadth of the blue-eyed
moon amounts to.’ ®
With this may be compared his description of night as caused
by the shadow of the earth which obstructs the rays of the sun
as the sun passes under the earth.’
Empedocles’ one important scientific achievement, so far as we
know, was his theory that light travels and takes time to pass from
one point to another. The theory is alluded to by Aristotle in
the following passages :
1 Aét. ii. 8. 2 (2. σ. p. 338; Vors. 13, p. 162, 35).
2 Fr. 52 (Vors. i*, p. 189. 14).
3. Seneca, Vaz. Quaest. iii. 24, quoted by Burnet, Early Greek Philosophy, —
ate! eam ΝΣ 6)
log. L. vill. 77 (Vors. i*, p. 153. 36). '
5 Aét. ii. 31.1 (D. G. p. ὩΣ : Tors i’, p. 163. 1-3). I follow the text as”
corrected by Diels after Karsten. The reading of Stobaeus is corrupt. That of
the ἄναξ says that the moon is twice as far from the sun as it is from the
earth.
6 Fr. 42 (Vors. i*, p. 187. 28): cf. Aét. ii, 24. 7 (D. G. p. 354; Vors. i’,
Ρ. 162. 40).
7 Fr. 48 (Vors. 13, p. 188. 31).
[ CH. ΧΙ EMPEDOCLES 93
ὲ
ξ
*Empedocles, for instance, says that the light from the sun
_ reaches the intervening space before it reaches the eye or the earth.
- And this might well seem to be the fact. For, when a thing: is
moved, it is moved from one place to another, and hence a certain
time must elapse during which it is being moved from the one
_ place to the other. But every period is divisible. Therefore there
_ was a time when the ray was not yet seen, but was being trans-
_ mitted through the medium.’
_ *Empedocles represented light as moving in space and arriving
at a given point of time between the earth and that which surrounds
it, without our perceiving its motion.’?
Aristotle of course rejected this theory because he himself held
a different view, namely, that light was not a movement in space
but was a qualitative change of the transparent medium which, he
considered, could be changed all at once and not only (say) half
at a time, just as a mass of water is all simultaneously congealed.*
But he had no better argument to oppose to Empedocles than that
‘though a movement of light might elude our observation within
a short distance, that it should do so all the way from east to west
is too much to assume’.*
* Aristotle, De sensu 6, 446 a 25-b 2.
3 Aristotle, De anima ii. 7. 418 b21.
* Aristotle, De sensu 6, 447 a 1-3.
* Aristotle, De anima ii. 7, 418 b 24.
ΧΙ
THE PYTHAGOREANS
IN a former chapter we tried to differentiate from the astronomical
system of ‘the Pythagoreans’ the views put forward by the Master
himself, and we saw reason for believing that he was the first to give
spherical shape to the earth and the heavenly bodies generally,
and to assign to the planets a revolution of their own in a sense
opposite to that of the daily rotation of the sphere of the fixed
stars about the earth as centre.
But a much more remarkable development was to follow in the
Pythagorean school. This was nothing less than the abandonment
of the geocentric hypothesis, and the reduction of the earth to the
status of a planet like the others. Aétius (probably on the authority
of Theophrastus) attributes the resulting system to Philolaus,
Aristotle to ‘the Pythagoreans’.
Schiaparelli’ sets out the considerations which may have sug-
gested to the Pythagoreans the necessity of setting the earth itself
in motion. If the proper movement of the sun, moon, and planets
along the zodiac had been a rotation about the same axis as that
of the daily rotation of the fixed stars, it would have been easy
to account for the special movements of the former heavenly bodies
by assuming for each of them a daily rotation somewhat slower
than that of the fixed stars; if the movement of each of them
had been thus simple, a moving force at the centre operating
with various degrees of intensity (depending on distance and the
numerical laws of harmony) would have served to explain every-
thing. But, since the daily rotation follows the plane of the
equator, and while special movement of the planets follows the
plane of the ecliptic, it is clear that, with one single moving force
1 Schiaparelli, J precursori di Copernico nell’ antichita (Milano, Hoepli,
1873), Ρ. 4.
THE PYTHAGOREANS 95
situated at the centre, it was not possible to account for both
movements. Hence the necessity of attributing the daily rotation,
which is apparently common to the fixed stars and the planets,
to a motion of the earth itself. But another reason too would
compel the Pythagoreans to avoid attributing to the sun, moon,
and planets the movement compounded of the daily rotation and
the special movement along the zodiac. For such a composite
movement would take place in a direction and with a velocity
continually altering and it would follow that, if at a given instant
the harmonical proportions of the velocities and the distances held
good, these proportions would not hold good for the next instant.
Accordingly it was necessary to assign to each heavenly body one
single simple and uniform movement, and this could not be
realized except by attributing to the earth that one of the compo-
nent movements which observation showed to be common to all
the stars.
Whether the system attributed to Philolaus was really founded
_ on arguments so scientific, combining the data furnished by observa-
_ tions with an antecedent principle based on the nature of things
_ and on a living spirit animating the world, must be left an open
~ question.
_ It is time to attempt a description of the system itself, and
I think that this can best be done in the words of our authorities.
Motion round the central fire.
‘While most philosophers say that the earth lies in the centre...
the philosophers of Italy, the so-called Pythagoreans, assert the
contrary. They say that there is fire in the middle, and the earth,
being one of the stars, is carried round the centre,‘and so produces
night and day. They also assume another earth opposite to ours,
which they call counter-earth,and in this they are not seeking explana-
tions and causes to fit the observed phenomena, but they are rather
trying to force the phenomena into agreement with explanations
and views of their own and so adjust things. Many others might
agree with them that the place in the centre should not be assigned
to the earth, if they looked for the truth not in the observed
facts but in @ priort arguments. For they consider that the
_-worthiest place is appropriate to the worthiest occupant, and fire
is worthier than earth, the limit worthier than the intervening parts,
_while the extremity and the centre are limits; arguing from these
_ considerations they think that it is not the earth which is in the
96 THE PYTHAGOREANS PART I
centre of the (heavenly) sphere but rather the fire. Further, the
Pythagoreans give the additional reason that it is most fitting that
the most important part of the All—and the centre may be so
described—should be safe-guarded ; they accordingly give the name
of “ Zeus’s watch-tower”’ to the fire which occupies this position,
the term “centre” being here used absolutely and the implication
being that the centre of the (thing as a) magnitude is also the centre
of the thing in its nature .... Such are the opinions of certain
philosophers about the position of the earth; and their opinions
about its rest or motion correspond. For they do not all take
the same view; those who say that the earth does not so much
as occupy the centre make it revolve in a circle round that
centre, and not only the earth but the counter-earth also, as we
said before. Some again think that there may be even more
bodies of the kind revolving round the centre; only they are
invisible to us because of the interposition of the earth. This they
give as the reason why there are more eclipses of the moon than of
the sun; for the moon is obscured by each of the other revolving
bodies as well, and not only by the earth. The fact that the earth
is not the centre, but is at a distance represented by the whole
(depth, i.e. radius) of half the sphere (in which it revolves) con-
stitutes, in their opinion, no reason why the phenomena should not
present the same appearance to us if we lived (on an earth) away
from the centre as they would if our earth were at the centre;
seeing that, as it is, we are at a distance (from the centre) repre-
sented by half the earth’s diameter and yet this does not make any
obvious difference.’ ἢ
‘The Pythagoreans, on the other hand, say that the earth is not
at the centre, but that in the centre of the universe is fire, while
round the centre revolves the counter-earth, itself an earth, and
called counter-earth because it is opposite to our earth, and next
to the counter-earth comes our earth, which itself also revolves
round the centre, and next to the earth the moon; this is stated
by Aristotle in his work on the Pythagoreans. The earth then,
being like one of the stars, moves round the centre and, according
to its position with reference to the sun, makes night and day. The
counter-earth, as it moves round the centre and accompanies our
earth, is invisible to us because the body of the earth is continually
interposed in our way... . The more genuine exponents of the
doctrine describe as fire at the centre the creative force which from
the centre imparts life to all the earth and warms afresh the part of
it which has cooled. Hence some call this fire the Tower of Zeus,
as Aristotle states in his Pythagorean Philosophy, others the
1 Aristotle, De caelo ii. 13, 293 a 18-b 30 (partly quoted in Vors. i*, p. 278.
4-20, 38-40).
a
CH. XH THE PYTHAGOREANS 97
Watch-tower of Zeus, as Aristotle calls it here [De caelo ii. 13],
and others again the Throne of Zeus, if we may credit different
authorities. They called the earth a star as being itself too
_ an instrument of time; for it is the cause of days and nights,
_ since it makes day when it is lit up in that part of it which
faces the sun, and it makes night throughout the cone formed by
its shadow.’ !
‘ Philolaus calls the fire in the middle about the centre the Hearth
_ of the universe, the House of Zeus, the Mother of the Gods, the
Altar, Bond and Measure of Nature. And again he assumes another
_ fire in the uppermost place, the fire which encloses (all). Now the
_ middle is naturally first in order, and round it ten divine bodies
move as in a dance, [the heaven] and (after the sphere of the fixed
_stars)* the five planets, after them the sun, under it the moon,
under the moon the earth, and under the earth the counter-earth ;
after all these comes the fire which is placed like a hearth round
the centre. The uppermost part of the (fire) which encloses (all),
in which the elements exist in all their purity, he calls Olympus,
and the parts under the moving Olympus, where are ranged the
five planets with the moon and the sun, he calls the Universe, and
lastly the part below these, the part below the moon and round
_the earth, where are the things which suffer change and becoming,
he calls the Heaven. ?
_ *Philolaus the Pythagorean places the fire in the middle (for this
_ is the Hearth of the All), second to it he puts the counter-earth,
and third the inhabited earth which is placed opposite to, and
_ revolves with, the counter-earth; this is the reason why those who
live in the counter-earth are invisible to those who live in our
earth.’ +
‘The governing principle is placed in the fire at the very centre,
and the Creating God established it there as a sort of keel to the
(sphere) of the All.’®
‘Others maintain that the earth remains at rest. But Philolaus
the Pythagorean held that it revolves round the fire in an oblique
circle in the same way as the sun and moon.’®
* Simplicius on De caelo ii. 13, 293 a 15, pp. 511. 25-34 and 512. 9-17 Heib.
(Vors. i*, p. 278. 20-36).
__# The words are supplied by Diels in view of similar words in a passage of
_ Alexander Aphrodisiensis quoted below (Alex. on Metaph. 985 Ὁ 26, p. 540 Ὁ 4-7
_ Brandis, p. 38. 22-39. 3 Hayduck).
| 5 Aét. ἢ. 7.7 (D. G. p. 336-7; Vors. i*, p. 237. 13 sqq.). This and the next
_ extract probably come from Theophrastus, through Posidonius.
_ * Aét. iii. 11. 3 (D. G. p. 377; Vors. i*, p. 237. 27 sq.).
§ Aét. ii. 4. 15 (D.G. p. 332; Vors. i*, p. 237. 31).
* Aét. iii. 13. 1,2 (2. Ο. p. 378; Vors. i*, p. 237. 46).
ΓῚ
1410 H
98 THE PYTHAGOREANS PARTI
As regards the assumption of tex bodies we have the following
further explanations. In a passage of the Metaphysics Aristotle
is describing how the Pythagoreans find the elements of all existing
things in numbers; he then proceeds thus :
‘They conceived that the whole heaven is harmony and number ;
thus, whatever admitted facts they were in a position to prove in
the domain of numbers and harmonies, they put these together
and adapted them to the properties and parts of the heaven and
its whole arrangement. And if there was anything wanting any-
where, they left no stone unturned to make their whole system
coherent. For example, regarding as they do the number ten as
perfect and as embracing the whole nature of numbers, they say
that the bodies moving in the heaven are also ten in number, and,
as those which we see are only nine, they make the counter-earth
a tenth.’
Alexander adds in his note on this passage :
‘If any of the phenomena of the heaven showed any disagree-
ment with the sequence in numbers, they made the necessary
addition themselves, and tried to fill up any gap, in order to make
their system as a whole agree with the numbers. Thus, considering
the number ten to be a perfect number, and seeing the number
of the moving spheres shown by observation to be nine only, those
of the planets being seven, that of the fixed stars an eighth, and
the earth a ninth (for they considered that the earth too moved
in a circle about the Hearth which remains fixed and, in their view,
is fire), they straightway added to them in their doctrine the
counter-earth as well, which they supposed to move counter to
the earth and so to be invisible to the inhabitants of the earth.’?
Speaking of the sun in an earlier passage, Alexander says:
‘(The sun) they placed seventh in order among the ten bodies
which move about the centre, the Hearth ; for the movement of the
sun comes next after (that of) the sphere of the fixed stars and
the five movements belonging to the planets, while after the sun
the moon comes eighth, and the earth ninth, after which again comes
the counter-earth,’ ὃ
The system may be described briefly thus. The universe is
spherical in shape and finite in size. Outside it is infinite void
1 Aristotle, Metaph. A. 5, 986 a 2-12 (Vors. 15, p. 270. 40-47).
® Alexander on Metaph. 986 a 3 (p. 542 4 35-b 5 Brandis, p. 40, 24-41. I
Hayduck).
* Ibid. 985 Ὁ 26 (p. 540 Ὁ 4-7 Brandis, p. 38. 22-39. 3 Hayduck).
CH. XII THE PYTHAGOREANS 99
which enables the universe to breathe, as it were. At the centre is
the central fire, the Hearth of the universe, called by the various
names, the Tower or Watch-tower of Zeus, the Throne of Zeus,
the House of Zeus, the Mother of the Gods, the Altar, Bond and
Measure of Nature. In this central fire is located the governing
principle, the force which directs the movement and activity of
the universe. The outside boundary of the sphere is an envelope
of fire; this is called Olympus, and in this region the elements
are found in all their purity; below this is the Universe. In the
universe there revolve in circles round the central fire the following
_ bodies. Nearest to the central fire revolves the counter-earth,
which always accompanies the earth, the orbit of the earth coming
next to that of the counter-earth; next to the earth, reckoning
in order from the centre outwards, comes the moon, next to the
moon the sun, next to the sun the five planets, and last of all,
_ outside the orbits of the planets, the sphere of the fixed stars,
_ The counter-earth, which accompanies the earth and revolves
_ina smaller orbit, is not seen by us because the hemisphere of the
earth on which we live is turned away from the counter-earth.
It follows that our hemisphere is always turned away from the
central fire, that is, it faces outwards from the orbit towards
_ Olympus (the analogy of the moon which always turns one side
_ towards us may have suggested this); this involves a rotation of
the earth about its axis completed in the same time as it takes the
earth to complete a revolution about the central fire.
What was the object of introducing the counter-earth which
we never see? Aristotle says in one place that it was to bring
up. the number of the moving bodies to ten, the perfect number
according to the Pythagoreans. But clearly Aristotle knew better ;
indeed he himself indicates the true reason in another passage
where he says that eclipses of the moon were considered to be
due sometimes to the interposition of the earth, sometimes to the
interposition of the counter-earth (to say nothing of other bodies
_ of the same sort assumed by ‘some’ in order to explain why there
_ appear to be more lunar eclipses than solar). The counter-earth,
| 1 Decaelo ii.13,293b21. Cf. Aét.ii.29. 4 (2. σ. Ὁ. 360; Vors.i?, p. 277.46) :
_ *Some of the Pythagoreans, according to the account of Aristotle and the
_ statement of Philippus of Opus, say that the moon is eclipsed through reflection
_ and the interposition sometimes of the earth, sometimes of the counter-earth.’
> Η 2
100 THE PYTHAGOREANS PARTI
therefore, we may take to have been invented for the purpose of
explaining eclipses of the moon, and particularly the frequency
with which they occur.
The earth revolves round the central fire in the same sense as the
sun and moon (that is, from west to east), but its orbit is obliquely
inclined ; that is to say, the earth moves in the plane of the equator,
the sun and the moon in the plane of the zodiac circle. It would
no doubt be in this way that Philolaus would explain the seasons.
Next we are told that the revolution of the earth produces day
and night, which depend on its position relatively to the sun;
it is day in that part which is lit up by the sun and night in the
cone formed by the earth’s shadow. As the same hemisphere is
always turned outwards, it seems to follow from the natural
meaning of these expressions that the earth completes one revolu-
tion round the central fire in a day and a night, or in 24 hours.1
This would, of course, account for the apparent diurnal rotation
of the heavens from east to west; from this point of view it is
equivalent to the rotation of the earth on its own axis in 24 hours
But there is a considerable difficulty here, of which, if we may trust
Aristotle, the Pythagoreans made light. According to him the
Pythagoreans said that whether (1) the earth revolves in a circle
round the centre of the universe or (2) the earth is itself stationary
at that centre could make no difference in the appearance of the
phenomena as observed by us. They argued that, even if we
assume the earth to be at the centre, there is a distance between
the centre and an observer on the earth’s surface equal to the
radius of the earth. On their assumption that the earth revolves
round the centre of the universe, the distance of an observer from
that centre would be greater than the radius of the earth’s orbit;
therefore to assert that the phenomena under the two assumptions
would be exactly the same was to argue in effect that parallax
is as negligible in one case as in the other. This is a somewhat
extreme case of making the phenomena fit a preconceived hypo-
thesis; but we may no doubt infer that the difficulty would lead
the Pythagoreans to maintain that the distance of the earth from
the centre of the universe was very small relatively to the distance
1 Burnet apparently disputes this inference (Early Greek Philosophy, p. 352,
note). We shall return to the point later.
CH. XII THE PYTHAGOREANS 101
of the other heavenly bodies from that centre, and that the radius
_ of the earth’s orbit was not in fact many times greater than the
radius of the earth itself.’
But a still greater difficulty remains. On the assumption that
the earth revolves round the central fire in a day and a night, and
that the sun, the moon, and the five planets complete one revolution
in their own several periods respectively, the observed movements
of these heavenly bodies are accounted for. But, since the apparent
daily rotation of the heavens is due to the revolution of the earth
_ about the central fire in a day and a night, it would follow that the
sphere of the fixed stars does not move at all, and therefore it
could not be said that ‘zen bodies’ (of which that sphere is one)
revolve about the central fire.
Boeckh suggested in his Philo/aus that the motion of the sphere
of the fixed stars could only be the precession of the equinoxes.
This he thought might have been discovered by the Egyptians,”
and Lepsius, later, took the same view, even suggesting that the
_ Egyptians might have communicated the discovery to Eudoxus.®
3
ξ.
Boeckh afterwards, as a result of a study of Egyptian monuments,
_ withdrew his suggestion ;* but, later still, he seems to have taken
3
_ it up again as preferable to the supposition of a very slow movement
serving no purpose and frankly faked. But, so far as we know,
Hipparchus was the first to discover precession. Martin passed
through two stages corresponding to Boeckh’s first and second.
In his commentary on the Zzmaeus of Plato, Martin observed
that precession only ‘required long and steady observations, with-
out any mathematical theory, in order to be recognized’;® but
Martin, too, changed his opinion later and satisfied himself that
precession was not known to any of Hipparchus’s predecessors.®
Schiaparelli thought it probable that Philolaus attributed xo
1 Schiaparelli, 7 frecursori, p. 6.
2 Boeckh, Philolaos des Pythagoreers Lehren, 1819, pp. 118, 119.
5 Lepsius, Chronologie der alten Aegypter, p. 207.
* Boeckh, Manetho und die Hundsternperiode, 1845, p. 54.
᾿ς 5 Martin, Etudes sur le Timée de Platon, ii, p. 98.
5 Martin, ‘La précession des équinoxes a-t-elle été connue des Egyptiens ou
de quelque autre peuple avant Hipparque?’ in vol. viii, pt. 1, of Mémotres de
_ PAcadémie des Inscriptions et Belles-lettres, Savants Etrangers, Paris, 1869 ;
see also Hypothése astronomigque de Philolaus, by the same author, Rome, 1872,
ΠΡ. 14. ἢ
102 THE PYTHAGOREANS PART I
movement to the sphere of the fixed stars,! his ground being the
following. Censorinus attributes to Philolaus the statements that
a ‘Great Year’ consists of 59 years, and that the solar year has
364% days. This gives a Great Year consisting of 21,5052 days,
which period contains very approximately 2 revolutions of Saturn,
_ 5 of Jupiter, 31 of Mars, 59 of the sun, Mercury, and Venus, and
729 of the moon.” If, then, says Schiaparelli, Philolaus had attributed
any movement to the stars, he would probably have included its
period in his Great Year; which apparently he did not. Tannery,
however, has given reason for thinking that the 729 lunations, and
consequently the 3643 days, were not the result of any independent
calculation made by Philolaus, but were an arbitrary variation
from the figures of Oenopides of Chios, of whom we are told by
Censorinus that he made the year to be 36522 days, so that 59
years would give 21,557 days or 730 lunations, not 729. Philolaus
said, as Plato said after him, that the cube of 9 represents the
number of months in a Great Year, and so it does /ess 1; the
arbitrary variation is characteristic of the Pythagorean fanciful
speculations with regard to numbers.®
1 Schiaparelli, 7 precursorz, p. 7.
* Schiaparelli (7 Arecursori, p. 8, note) compares the periods of revolution
based on the figures attributed to Philolaus with the true periods, thus:
Period of revolution.
es 2S i.
Planet. Philolaus. Modern view.
Saturn 1075275 days 10759-22 days
Jupiter 4301-10 ,, 433258 ,,
Mars 693°71 yy 686.98 ,,
Venus
Mercury 364-50 , 365-26,
Sun
Moon 29°50» 29°53»
Schiaparelli admits that the number of days for Mars (693-71) is uncertain,
as it is not clear that Philolaus assumed 31 revolutions of Mars in his Great
Year. But neither does there appear to be any evidence that he definitely fixed
the number of revolutions made by the other planets in the Great Year.
8. Tannery, ‘La grande année d’Aristarque’, in M/émoires de la Société des
sciences physiques et naturelles de Bordeaux, 3° sér. iv, 1888, p. 90.
Tannery holds that Philolaus simply took his Great Year, equal to 59 solar
years, from Oenopides, while Oenopides, arrived at it in a very simple way,
namely, by taking the number of days in the year as 365, and the period of the
moon as 29} days, and observing the natural inference that, in whole numbers,
59 years are equal to 730 lunar months, after which he had only to determine
the number of days in 730 lunar months.
CH. ΧΙ THE PYTHAGOREANS 103
But indeed, as Burnet points out,’ it is incredible that the
Pythagoreans should have put forward the theory that the sphere
of the fixed stars is absolutely stationary. Such a suggestion
would have seemed such a startling paradox that it is inconceivable
that Aristotle should have said nothing about it, especially as he
made the circular motion of the heavens the keystone of his own
system. As it is, he does not attribute to any one the view that
the heavens are stationary; and, in writing of the Pythagorean
system, he makes it perfectly clear that the bodies moving in the
_ heaven are ten in number,” from which it follows that the sphere
οὕ the fixed stars (which is one of the ten) must move. It may
be observed, too, that Alcmaeon, whom Aristotle mentions as
having held views similar to the Pythagoreans, distinctly said that
‘all the divine bodies, the moon, the sun, the stars, and the whole
heaven, move continually ᾿.ὅ
Now, if the Pythagoreans gave a movement of rotation to the
sphere of the fixed stars, there are three possibilities. The first
is that they may have assumed the universe as a whole to share
in the rotation of the sphere of the fixed stars, while the independent
_ revolutions of the earth, sun, moon, and planets were all 272 addition
to their rotation as part of the universe. If this were the assumption,
the rotation of the whole universe might be at any speed whatever
without altering the phenomena as observed by us; the phenomena
would present exactly the same appearance to us as they would
on the assumption that the sphere of the fixed stars is stationary,
and the planets, sun, moon, earth, and counter-earth have only
their own proper revolutions round the central fire; only to ἃ
person situated at the central fire, supposed exempt from the
general movement, would the general movement of the universe
be perceptible. Thus the assumption of such a general movement
would serve no purpose (apart from the objection that it would
leave the speed of the rotation of the whole universe quite
indeterminate); indeed, it would defeat what seems to have been
1 Burnet, Zarly Greek Philosophy, p. 347.
2 Aristotle, Metaph. A. 5,986.4 10 ra Pepopeva κατὰ τὸν οὐρανὸν δέκα μὲν εἶναί φασιν.
Cf. the passages of Alexander, quoted above (p. 98); also Simplicius on De cae/o
293 a 15 (p. 512. 5), ‘ They wished to bring up to ten the number of the bodies
which have a circular motion (κυκλοφορητικῶν).᾽
8. Aristotle, De anima i. 2, 405 a 33. :
104 THE PYTHAGOREANS PARTI
the whole object of Philolaus’s scheme, namely, to separate the
daily rotation from the periodical revolutions of the sun, moon,
earth, and planets, and to account for all the phenomena by simple
motions instead of a combination of two in each case,
The second possibility is only slightly different. The sphere of
the fixed stars might have a movement of rotation and carry with
it all the heavenly bodies except the earth (and of course its
inseparable companion, the counter-earth). The effect would be
that the earth (with the counter-earth) would complete an actual
revolution round the central fire in a period greater or less than
24 hours according to the speed and the direction of the rotation
of the rest of the heavenly bodies ;+ the period would be less than
24 hours if the latter rotation were in the same sense as that of
the earth’s revolution from west to east, and greater if it were in the
opposite sense, from east to west. This alternative is more compli-
cated than the first, and is open to the same or stronger objections.
The third possibility is that the sun, moon, planets, earth, and
counter-earth have their own special movements only, and that
the sphere of the fixed stars moves very slowly, so slowly that its
movement is imperceptible. This is the view of Martin? and of
Apelt,? and it amounts to assuming that Philolaus gave a move-
ment to the sphere of the fixed stars which, though it is not the
precession of the equinoxes, is something very like it. If this is
right, we must suppose that Philolaus gave the sphere of the fixed
stars a merely nominal rotation for the sake of uniformity and
nothing else; and perhaps, as Martin says, to assume an imperceptible
motion would not be a greater difficulty for Philolaus than it was
to postulate an invisible planet or to maintain that the enormous
parallaxes which would be produced by the daily revolution of
the earth about the central fire are negligible.
It is to be feared that a convincing solution of the puzzle will
* Martin (7yfothése astronomique de Philolaiis, Rome, 1872, p. 16) compares
an allusion in Ptolemy's Sy#éaxzs (i.7, p. 24. 11-13 Heib.) to the possibility
of ‘assuming (as an alternative to a scheme in which the fixed stars are station
and the earth rotates on its own axis once in twenty-four hours) that do¢/ the
earth and the sphere of the fixed stars rotate, at different speeds, about one and
the same axis, the axis of the earth.
* Martin, /yfothese astronomigue de Philolaiis, pp. 14-16.
3 Apelt, Untersuchungen iiber die Philosophie und Physik der Alten (Abn.
der Fries’ schen Schule, Heft 1, p. 68),
ν
ha
"
~—_
rs
al
sty.
CH. XII THE PYTHAGOREANS 508
never be found. After all that has been written on the subject,
Gomperz?! still seems to prefer Boeckh’s original suggestion that the
movement attributed by Philolaus to the fixed stars was actually
the precession of the equinoxes, but the new matter contained in
his note on the subject does not help his case. He relies partly
on the ὦ griori arguments originally put forward by Martin; ‘it
is’, he suggests, ‘in itself hardly credible that a deviation in the
position of the luminaries which in the course of a single year
_ amounts to more than 50 seconds of an arc could remain unnoticed
- for long’; he is aware, however, that Martin himself, as the result
of further investigation, could find no confirmation of his earlier
view. He admits, too, that the Babylonians were still unacquainted
_ with precession in the third century B.c.2_ The other main argument
used by Gomperz is that the estimates of the angular velocities
οὗ the planetary movements which go back to Philolaus or other
early Pythagoreans are approximately correct, while only prolonged
observations of the stars could have made them so. But, so far as
Philolaus is concerned, the data are apparently the same as those
_ from which Schiaparelli drew the opposite inference, namely, that
_ Philolaus was not aware of precession and considered the sphere of
_ the fixed stars to be stationary !
Harmony and distances.
‘Philolaus holds that all things take place by necessity and by
harmony. ὃ
“ΤῈ is clear too from this that, when it is asserted that the move-
ment of the stars produces harmony, the sounds which they make
being in accord, the statement, although it is a brilliant and remark-
able suggestion on the part of its authors, does not represent the
truth. I refer to the view of those who think it inevitable that,
when bodies of such size move, they must produce a sound;
this, they argue, is observed even of bodies within our experience
which neither possess equal mass nor move with the same speed;
hence, when the sun and moon, and the stars which are so many
_and of such size move with such a velocity, it is impossible that they
* Gomperz, Griechische Denker, i*, p. 93, and note on pp. 430, 431.
3 Gomperz (p. 431) gives this as the opinion of the highest authority on the
_ subject, Pater Kugler, who is to argue the point anew in a forthcoming tract,
_ *Im Bannkreis Babels’. This must be set against the opposite inference drawn
is by Burnet (Zarly Greek Philosophy, p. 25, note) from another work of Kugler’s,
ool apparently confirmed by Hilprecht (Zhe Babylonian Expedition of the
niversity of Pennsylvania, Philadelphia, 1906).
* Diog. L. viii. 84 (Vors. is, Pp. 233. 33).
τού THE PYTHAGOREANS PARTI
should not produce a sound of intolerable loudness, Supposing
then that this is the case, and that the velocities depending on
their distances correspond to the ratios representing chords, they
say that the tones produced by the stars moving in a circle are in
harmony. But, as it must seem absurd that we should not all hear
these tones, they say the reason of this is that the sound is already
going on at the moment we are born, so that it is not distinguishable
by contrast with its opposite, silence ; for the distinction between
vocal sound and silence involves comparison between them; thus a
coppersmith is apparently indifferent to noise through being accus-
tomed to it, and so it must be with men in general.’?
‘For (they said that) the bodies which revolve round the centre
have their distances in proportion, and some revolve more quickly,
others more slowly, the sound which they make during this motion
being deep in the case of the slower and high in the case of the
quicker; these sounds, then, depending on the ratio of their distances,
are such that their combined effect is harmonious. ... They said
that those bodies move most quickly which move at the greatest
distance, that those bodies move most slowly which are at the
least distance, and that the bodies at intermediate distances move
at speeds corresponding to the sizes of-their orbits.’ 3
We have no information as to the actual ratios which the Pytha-
goreans assumed to exist between the respective distances of the
earth, moon, sun, and planets from the centre of the universe.
When Plutarch says that the distances of the ten heavenly bodies
formed, according to Philolaus, a geometrical progression with 3 as
the common ratio,? he can only be referring to some much later
Pythagoreans. For if, on the basis of this progression, the distance
of the counter-earth is represented by 3, that of the earth by
9, and that of the moon by 27, it is obvious that the enormous
parallaxes due to the revolution of the earth round the centre would
1 Arist. De caelo ii. 9, 290 Ὁ 12-29 (Vors. 13, p. 277. 28-42). Yet when
Aristotle is trying to prove his own contention that the stars do not move of
themselves but are carried by spheres which revolve, he does not hesitate to use
the argument that, if the planets moved freely through a mass of air or fire
spread through the universe, ‘as is universally alleged’, they would, in conse-
quence of their size, inevitably produce a sound so overpowering that it would not
only be transmitted to us but would actually shiver things. He maintains, how-
ever, that, if a body is carried by something else which moves continuously and
does not cause actual concussion, it does not produce sound ; hence, in his view,
the fact that we do not hear sounds from the motion of the planets implies that
they have no motion of their own but are carried by something (De cae/o ii. 9,
291 a 16-28).
* Alexander on Mefaph. A. 5, p. 542 ἃ 5-10, 16-18 Brandis, pp. 39. 24-40. I,
40. 7-9 Hayduck. 5. Plutarch, De animae procreatione, Cc. 31, p. 1028 Β,
ν᾿
ἄν ὡΣ δε
= iP? 4 Sees, ae ee
CH. XII THE PYTHAGOREANS 107
be quite inconsistent with ‘saving the phenomena’! Moreover,
the order of the heavenly bodies given in this passage, counter-
earth, earth, moon, Mercury, Venus, Sun, is not the order in
which they were placed by Philolaus (and by Plato later) but the
Chaldaean order, which does not seem to have been adopted by any
Greek before the Stoic Diogenes of Babylon (second century B.C.).
Of the ‘harmony of the spheres’ there are many divergent
accounts,* and it would appear that the places and the number of
the heavenly bodies supposed to take part in it varied at different
periods. Burnet* suggests that we cannot attribute to Pytha-
goras himself more than an identification of his newly-discovered
musical intervals, the fourth, fifth, and octave, with the ¢hree rings
which we find in Anaximander, that of the stars (nearest to the
earth), that of the moon (next) and that of the sun (which is the
furthest from the earth), and that this would be the most natural
beginning for the later doctrine of the ‘harmony of the spheres’.
This is an attractive supposition, but it depends on the assumption
that Pythagoras attributed to the planets and the fixed stars the
same revolution from east to west; whereas he certainly dis-
tinguished the planets from the fixed stars, and he must have
known that their movement was not the same as that of the fixed
stars (this is clear from his identification of the Morning and
Evening Stars), even if he -did not assign to the planets the inde-
pendent movement, in the opposite sense to the daily rotation,
which Alcmaeon is said to have observed. The original form of
the theory of the ‘harmony of the spheres’ no doubt had reference
to the seven planets only (including in that term the sun and moon),
the seven planets being supposed, by reason of their several motions,
to give out notes corresponding to the notes of the Heptachord :
1 Schiaparelli, 7 Zrecursori, pp. 6, 44.
* I must refer for full details to Boeckh, Studien iii, pp. 87 sqq. (Kleine
Schriften, iii, pp. 169 sq.), Carl v. Jan, ῥάζίοί. 1893, pp. 13 sqq., and for a
summary to Zeller, 15, pp. 431-4.
3 Burnet, Early Greek Philosophy, p. 122.
* Cf. Hippol. Refut. i. 2. 2, (D. G. p. 555), ‘Pythagoras maintained that the
universe simgs and is constructed in accordance with a harmony ; and he was
the first to reduce the motion of the seven heavenly bodies to rhythm and song’;
Censorinus, De die matalz 13. 5, ‘ Pythagoras showed that the whole of our
_ world constitutes a harmony. Accordingly, Dorylaus wrote that the world
is an instrument of God ; others added that it is a heptachord, because there
are seven planets which have the most motion.’
108 THE PYTHAGOREANS PART I
it could not have related to the 2272: heavenly bodies of the Pytha-
gorean system, for this would have required ten notes, whereas the
Pythagorean theory of tones only recognized the seven notes of
the Heptachord. This may, as Zeller says,! be the reason why
Philolaus himself, so far as we can judge from the fragments, said
nothing about the harmony of the spheres. Aristotle, however,
clearly implies that in the harmony of the Pythagoreans whom he
knew the sphere of the fixed stars took part ; for he speaks of the
intolerable noise which, on the assumption that the motion of the
heavenly bodies produced sound, would be caused by ‘the stars
which are so many in number and so great.’ Consequently eight
notes are implied: and accordingly we find Plato (in Republic x)
including in his harmony eight notes produced* by the sphere
of the fixed stars and by the seven planets respectively, and
corresponding to the Octachord, the eight-stringed lyre which had
been invented in the meantime. The old theory being that all the
heavenly bodies revolved in the same direction from east to west,
only the planets revolved more slowly, their speeds diminishing in —
the order of their distances from the sphere of the fixed stars,
which rotates once in about 24 hours, it would follow that Saturn,
being the nearest to the said sphere, would be supposed to move the
most quickly; Jupiter, being next, would be the next quickest ;
Mars would come next, and so on; while the moon, being the
innermost, would be the slowest; on this view, therefore, the note
of Saturn would be the highest (νήτη), that of Jupiter next, and so
on, that of the moon being the lowest (ὑπάτη) ; and the speeds
determining this order are absolute speeds in space. Nicomachus,® ἃ
though he mentions that his predecessors assigned notes to the
seven planets in this order, himself took the opposite view,
placing the moon’s tone as the highest and Saturn’s as the lowest
(incidentally he places the sun in the middle of the seven instead
of next to the moon as the older system did). Nicomachus’s order
is explicable if we assume that the independent revolutions of the
planets (in their orbits) was the criterion for the assignment of the
notes; for the moon describes its orbit the quickest (in about
a lunar month), the sun the next quickest (in a year), and so on,
Saturn being the slowest in describing its orbit; these speeds are
1 Zeller, i®. p.432,note2. ἢ Aristotle, De caeloii.9,290b18 es i*, p.277. 33).
* Nicomachus, Harm. 6. 33sq.; cf. Boethius, 7152, AZus, i. 27
— = ee EE ee ee ee
᾿ς CH. XII THE PYTHAGOREANS 109
relative speeds, i.e. relative to the sphere of the fixed stars regarded
as stationary. The adsolute and relative angular speeds of the
planets are of course connected in the following way: for any one
planet its absolute speed is the speed of the sphere of the fixed stars
minus the relative speed of the planet ; hence their order in respect
of absolute speed is the reverse of their order in respect of relative
speed and,so long as only the seven planets (including the sun
and moon) come into the scale of notes, it is possible to assign
notes to them in either order. But this is no longer the case when
the sphere of the fixed stars is brought in as having a note of its
own, making altogether eight notes corresponding to the Octachord.
_ The speed of the fixed stars is of course an absolute speed, and it is
faster than either the absolute or relative speed of any of the
_ planets; it must, therefore, give out the highest note (νήτη). Now,
in assigning the rest of the notes, we cannot take the re/ative speeds
_ of the planets for the purpose of comparison with the absolute
_ speed of the sphere of the fixed stars; we must compare like with
_ like; and indeed, on the hypothesis that the body which moves
_ more swiftly gives out a higher note than the body which moves
more slowly, it is only the absolute speed of the heavenly bodies
im space, and nothing else, which can properly be taken as deter-
Mining the order of their notes. Now Plato says! in the Myth of
Er that eight different notes forming a harmony are given out by
the Sirens seated on the eight whorls of the Spindle, which repre-
sent the sphere of the fixed stars and the seven planets, and that,
while all the seven inner whorls (representing the planets) are carried
round bodily in the revolution of the outermost whorl (representing
the sphere or circle of the fixed stars), each ofthe seven inner
whorls has a slow independent movement of its own in a sense
opposite to that of the movement of the whole, the second whorl
starting from the outside (the first of the seven inner ones) which
represents Saturn having the slowest movement, the third repre-
senting Jupiter the next faster, the fourth representing Mars the
_ next faster, the fifth, sixth, and seventh, which represent Mercury,
_ Venus, and the Sun respectively and which go ‘together’ (i.e. have
_ the same angular speed) the fastest but one, and the eighth repre-
_ senting the moon the fastest of all. Plato, therefore, while speaking
1 Plato, Republic x. 617 A-B.
110 THE PYTHAGOREANS PARTI
of absolute angular speed in the case of the circle of the fixed stars,
refers to the relative speed in the case of the seven planets. To
get the order of his tones therefore we must turn the relative speeds
of the planets into absolute speeds by subtracting them respectively
from the speed of the circle of the fixed stars, and the order of
their respective notes is then as follows:
Circle of fixed stars . . . highest note (νήτη)
Saturn
Jupiter
Mars
Mercury
Venus
Sun
Moon ... . . . lowest note (éaérn).
1 Dr. Adam, in his edition of the Republic (vol. ii, p. 452), supposes that, after
the circle of the fixed stars giving the highest note, the seven planets would
come in the order of their re/ative velocities, thus—
Circle of the fixed stars . . . highest note (νήτη)
6 Moon
EP Sun
” Venus © + = pean
τ: Mercury
sd Mars
ἣν Jupiter
Pe Saturn . . . . lowest note (ὑπάτη)
For the reason given above, I do not think it possible that Plato, who was
a mathematician, would have assigned the notes to the eight circles in this
order, though it is likely enough that, when writing the passage, he had not
in his mind any definite allocation of notes at all. A further difficulty in the
way of Adam’s order is the following. He observes that, if we understand
‘together’ (ἅμα ἀλλήλοις), used of the motion of the sun, Venus, and Mercury,
in a strict sense, there will only be six notes, as the three bodies will have the
same note. He gets over this difficulty quite properly by supposing that Plato
really had in his mind the period taken by the three bodies in describing their
orbits, in other words, their amgu/ar velocity, rather than their linear velocity.
‘In that case the octave will be complete, because, in order to complete their
orbits in the same time, the sun, Venus, and Mercury will have to travel at
different rates of speed.’ True ; but, as the planet with the longer orbit must
have a /inear velocity greater than the planet with the shorter orbit, it follows
that the linear velocity of Venus in the above scheme will be greater than that
of the sun, and the linear velocity of Mercury greater than that of Venus. Thus
the supposed linear velocities, instead of diminishing all the way from the circle
of the fixed stars down to Saturn in the above table, will diminish from the
circle of the fixed stars down to the sun, but will zzcrease after that down to
Mercury, before they diminish again with Mars and the rest; and this upsets
the proper order of the notes altogether. On the other hand, with the arrange-
ment according to absolute speeds, as in the text above, the linear velocities of
Mercury, Venus, and the sun come in the correct diminishing order.
ν
— a ee
a
CH. XII THE PYTHAGOREANS ΤΙΙ
This order agrees with Cicero’s arrangement, in which the highest
circle, that of the fixed stars, has the highest note and the moon
_ the lowest."
_ Although Alexander clearly says that, in the Pythagorean theory
of the harmony of the spheres, the different notes correspond to
the ratios of the distances of the heavenly bodies, we have little
or no authentic information as to how the early Pythagoreans
translated the theory into an actual estimate of the relative
_ distances? It is true that some later writers such as Censorinus
and Pliny give some definite ratios of distances and, as usual, refer
_ them back to Pythagoras himself; but their statements contain
such an admixture of elements foreign to the early Pythagorean
theory that no certain conclusion can be drawn.
Plato implies, in his Myth of Er, that the breadths of the whorls
_ of the spindle represent the distances separating successive planets,
_ but he does not do more than state the order of magnitude in
_ which the successive distances come ; he makes no attempt to give
_ absolute ratios between them.
Tannery* ingeniously conjectures that Eudoxus’s view of the
ratio of the distances of the sun and moon from the earth, which
he put at 9:1, may have been suggested or confirmed by the
theory of the harmony. The original discovery of the octave,
the fourth and the fifth, stated in one of its forms,‘ showed that
they represented ratios of lengths of string assumed to be under
equal tension as follows, namely 1:2, 3:4, and 2:3 respectively.
Bringing these ratios to their least common denominator, we see
that strings at equal tension and of lengths 6, 8, 9, 12 respectively
give the three intervals. The interval between the first and second
strings being a fourth, and that between the first and third a fifth,
the interval between the second and third is a tone, which may
therefore be regarded as represented by the difference between
1 Cicero, Somn. Scip. c. 5.
3 Alexander’s own figures (Alex. on Metaph. 986 a 2, p. 542 a 12-15 Brandis,
Ῥ. 40. 3-6 Hayduck) seem to be illustrations only: ‘The distance of the sun
from the earth being, say [φέρε εἰπεῖν), double the distance of the moon, that of
Aphrodite triple, and that of Hermes quadruple, they considered that there was
some arithmetical ratio in the case of each of the other planets as well.’ The
ratios of I, 2, 3, 4 for the distances of the moon, the sun, Venus, and Mercury
are the same as those indicated by Plato in the 7zmaeus 36D.
3 Tannery, Recherches sur [histoire de l’astronomie ancienne, pp. 293, 328.
* Cf. Theon of Smyrna, pp. 59. 21-60. 6, ed. Hiller; Boethius, /nst. Mus. i. το.
112 THE PYTHAGOREANS PARTI
9 and 8, or 1. Now the Didascalia caelestis of Leptines, known as
Ars Eudoxi, which was written in Egypt between 193 and 165 B.C.
contains a number of things derived from Eudoxus, and the ratio
of the distance of the sun from the earth to the distance of the moon
from the earth is there said to correspond to the relation of the fifth
to the tone.’ If we take the respective notes as represented by the
above numbers, the ratio of the fifth to the tone is 9: (9 --- 8), or 9:1.
It would appear from passages in Theon of Smyrna? and
Achilles,’ doubtless taken in substance from Adrastus or Thrasyllus,
that the harmony was next spoken of in poems by Aratus and
Eratosthenes (third century B.C.); but there is no indication that
they did more than point out the correspondence between -the
planets, in their order from the moon to Saturn or to the sphere
of the fixed stars, and the notes of the heptachord or octachord
from the ὑπάτη, the lowest, to the νήτη, the highest (Etatosthenes
certainly took the octachord for this purpose).*
Achilles tells us that, after Aratus and Eratosthenes, and before
Adrastus and Thrasyllus, Hypsicles the mathematician (the author
of the so-called Book XIV of Euclid) treated of the question of
the harmony of the spheres; and he proceeds to give, as generally
accepted by musicians, a remarkable musical scale in which an
octave is divided into eight intervals and nine notes (including the
two extreme notes of the octave), the nine notes corresponding
to the sphere of the fixed stars, Saturn, Jupiter, Mars, Mercury,
Venus, Sun, Moon, and Earth respectively, in that order, This
scale is the same as that described in verses quoted by Theon
of Smyrna from one Alexander (who was not Alexander of
Aetolia, as Theon wrongly calls him, but Alexander of Ephesus,
a contemporary of Cicero, or possibly, as Chalcidius calls him,
Alexander of Miletus, Alexander Polyhistor). The only difference
The text, indeed, of Leptines has to be filled out in order to get this, and it
is the sizes of the sun and moon, not their distances respectively from the earth,
that are mentioned (though the effect is the same on the assumption that their
apparent angular diameters are equal). The sentence as corrected by Tannery
is ‘Thus the sun is greater than the moon, and the moon greater than ‘he
part of ) the earth (which sees the eclipse); the ratio is that of the fifth to (the
difference between the fifth and) the fourth,’
2 Theon of Smyrna, pp. 105. 13-106. 2; pp. 142. 7 sqq.
8 Petav. Uranolog. p. 136; see Tannery, Recherches sur l’histoire de l’astro-
nomie ancienne, Ὁ. 330.
* Theon of Smyrna, loc. cit.
ΕΟ EE δ δδννυν.. ἰδ ῥενννονμἐ. κ... ὐμμ νμνὰω....-
Tia
CH.XxI THE PYTHAGOREANS 113
_ is that Alexander has the later order for the planets, his order
J
being: sphere of fixed stars, Saturn, Jupiter, Mars, Sun, Venus,
Mercury, Moon, Earth. Tannery infers that this peculiar division
_ of the octave, with the order of the planets as given by Achilles,
is due to Hypsicles.?
Theon of Smyrna criticizes this peculiar scale of nine notes as
described by Alexander. First, he observes that in the last of
the verses Alexander says the heptachord is the image of the
world, whereas he has made an octave, consisting of six tones, out
of wine strings; his notes therefore do not. correspond to the
_ diatonic scale. Again, the lowest note is given to the earth,
_ whereas, being at rest, it gives out no sound. The sun, too, is
given the ‘middle’ note (μέση), whereas the interval from the
lowest (ὑπάτη) to the ‘middle’ is not a fifth but a fourth; and
so on.
_ The scale, however, of nine notes with the sun in the middle,
_ as Alexander has it, is apparently the common foundation of three
| _ scales of eight intervals given by Censorinus,* Pliny, and Martianus
~ Capella’ respectively, who apparently got them from a work of
the encyclopaedic writer Varro (116-27 B.C.). The three scales
_ given by these three authors differ slightly in that Censorinus’s
eight intervals add up to 6 tones (the proper amount), Pliny’s to
7 tones, and Martianus Capella’s to 6% tones; the differences may,
‘Tannery thinks, be due to errors in the MSS. of Varro, whence
the one scale which is the foundation of all three was taken. We
need only set down Censorinus’s version, which is:
From Earth to Moon I tone) ,
Moon to Mercury 3
᾽ν ” 1
». Wenus to Sun ey
» Sun to Mars Tet >}
Mars to Jupiter 3
” ee 1
τ 2
, Jupiter to Saturn ΕΣ -25 tones (a fourth)
» Saturntofixedstars 2 ,
t 6 tones
_ 1 Theon of Smyrna, pp. 140. 5-141. 4. ‘ Tannery, loc. cit.
3 Censorinus, De die matali 13. 3-5. * Pliny, W. H. ii, c. 22, ὃ 84.
® Mart. Capella, De nuptits philologiae et Mercurii, ii. 169-98.
1410 I
114 THE PYTHAGOREANS PARTI
The difference between this and Pliny’s scale is that Pliny takes
the distance from Saturn to the sphere of the fixed stars to be
1% tones instead of half a tone, so that with him the distance
between the sun and the fixed stars is 34 tones, or a fifth instead
of a fourth. Both Censorinus and Pliny make the interval from the
earth to the sun to be a fifth, and from the earth to the moon one tone,
wherein they agree with the view attributed by Tannery to Eudoxus.
Both Pliny and Censorinus add a further detail which apparently
must have come from some source other than the poem of
Alexander; this is that Pythagoras made the actual distance
between the moon and the earth, which he called one tone, to
be 126,000 stades. This would of course enable the other distances
between the heavenly bodies to be calculated on the basis of the
scale; e.g. the distance from the earth to the sun would be 3%
times 126,000 stades, and so on. But this evaluation of the
distance from the earth to the moon, 126,000 stades, is exactly
half of 252,000 stades, which is the estimate of the circumference
of the earth made by Eratosthenes and Hipparchus, This exact
coincidence is enough to make it plain that the 126,000 stades
does not go back to Pythagoras, and can hardly have been
suggested before the second century B.C,
Pliny, however, in a passage immediately preceding that in
which he describes his scale, says that Pythagoras made the
distance from the earth to the moon 126,000 stades, the distance
from the moon to the sun twice that distance, and the distance from
the sun to the sphere of the fixed stars thrice the same distance."
Pliny is here evidently quoting from a quite different authority ;
as he says that Sulpicius Gallus was of the same opinion, he would
appear to be citing some book by Sulpicius Gallus, who may have
got it from some tradition which cannot now be traced.
It is no doubt possible that, if Pythagoras did not estimate the
distance of the moon from the earth in stades, he may have
expressed it in terms of the circumference of the earth. But, seeing
that Anaximander had already estimated the radius of the orbit of
the moon at 18 times the radius of the earth, how could Pythagoras ©
have put the distance of the moon so low as half the circumference
of the earth, or about 3 times the earth’s radius? Tannery
1 Pliny, V.H. ii, c. 21, ὃ 83.
i as
CH, XII THE PYTHAGOREANS 115
conjectures that in the number οὗ stades (126,000) given by Varro
there is a mistake, mz/ia having been written instead of myriads
(pupiddes); in that case the source from which Varro drew might
have given the distance of the moon as Io times the half-
circumference of the earth. Hultsch,) however, thinks it incredible
_ that milia could have been written in error for μυριάδες ; and even
_ if it had been, and the moon’s distance were thus made up to about
_ 30 times the earth’s radius, the absurdity would still be left that
the sun’s distance is only 35 times as great.
It is true, as Martin observes,” that the sounding by the planets
of all the notes of an octave at once would produce no ‘harmony’
in our sense of the word; but the Pythagoreans would not have
‘been deterred by this consideration from putting forward their
fanciful view.* We have, it is true, allusions to other arrangements
of the notes which would make them cover more than an octave,
but these must have been later than Plato’s time. Thus Plutarch
speaks of one view which made the seven planets correspond to
the seven invariable strings of the fifteen-stringed lyre, and of
another which made their distances correspond to the five tetra-
chords of the complete system.* Anatolius® has a peculiar
‘distribution of tones between the heavenly bodies which gives
Ψ
altogether two octaves and a tone. Macrobius® bases his view
on the successive numbers 1, 2, 3, 4, 8, 9, 27 applied to the planets
in the 7imaeus and supposed to represent their relative distances
from the earth; Macrobius makes the first four (from 1 to 4) cover
two octaves, and he seems to make the seven notes cover, in all,
four octaves, a fifth, and one tone.”
The Sun.
‘The Pythagoreans declared the sur to be spherical.’§
*Philolaus the Pythagorean holds that the sun is transparent like
glass, and that it receives the reflection of the fire in the universe
+ Hultsch, Poseidonios iiber die Grisse und Entfernung der Sonne, Berlin,
1897, p. I1, note I.
Martin, Etudes sur le Timée, ii, p. 37. 3 Zeller, 15, p. 432, note.
_ * Plutarch, De animae procr. c. 32, p. 1029 A, B. The five distances are
( Moon to Sun with its concomitants Mercury and Venus, (2) Sun, ἄς. to Mars,
(3) Mars to Jupiter, (4) Jupiter to Saturn, (5) Saturn to sphere of fixed stars.
_ ® Anatolius in Iambl. Theol. Ar. p. 56; cf. Zeller, loc. cit.
_ § Macrobius, Jz Somn. δεῖ. ii, cc. 1, 2.
_ 7 Zeller, ii*, pp. 777 544. ® Aét. ii. 22. 5 (22. G. p. 352).
I2
116 THE PYTHAGOREANS PART I
and transmits to us both light and warmth, so that there are in
some sort two suns, the fiery (substance) in the heaven and the fiery
(emanation) from it which is mirrored, as it were, not to speak of
a third also, namely the beams which are scattered in our direction
from the mirror by way of reflection (or refraction); for we give
this third also the name of sun, which is thus, as it were, an image
of an image.’ !
‘ Philolaus says that the sun receives its fiery and radiant nature
from above, from the aethereal fire, and transmits the beams to us
through certain pores, so that according to him the sun is triple,
one sun being the aethereal fire, the second that which is transmitted
from it to the glassy thing under it which is called sun, and the
third that which is transmitted from the sun in this sense to us.’*
Thus, according to Philolaus, the sun was not a body with light
of its own, but it was of a substance comparable to glass, and it
concentrated rays of fire from elsewhere, and transmitted them to
us. This idea was no doubt suggested in order to give a uniform
nature to all the moving heavenly bodies. But there are difficulties
in the descriptions above given of the sources of the beams οὔ
fire. The natural supposition would be that they would come
from the central fire; in that case the sun would act like a mirror
simply; and the phenomena would be accounted for because the
beams of the fire would always reach the sun except when ob-
structed by the moon, earth, or counter-earth, and, as the earth and
counter-earth move in a different plane from the sun and moon,
eclipses would occur at the proper times. But the first of the above
passages says that the beams come from the fire in the wiverse,
and that one of the suns is the fiery substance in the Aeaven, while
the second passage says that the beams come from adove, from the
fire of the aether. Burnet takes ‘heaven’ in the narrow sense of the
‘ portion of the universe below the moon and round about the earth’
which, according to the Doxographi, was called ‘heaven’,® and he
thinks that ‘the fire in the heaven’ is therefore exclusively the
central fire But this leaves out of account the alternative term
‘the fire in the universe’ and also Achilles’ ‘ fire from adove’; and,
' Aét. ii, 20. 12 (D.G. p. 349, 350; Vors. i*, p. 237. 36).
2 Achilles, Jsugoge in phaenomena (Petav. Uranolog., p. 138; D. G.
ΡΡ. 349, 350). ἃς ahs
3. See above, p. 97 (Aét. ii. 7.7; 22. G. p. 337; Vors. i?, p. 237. 22).
4 Burnet, Zarly Greek Philosophy, p. 348.
ν
eS τὐϑερτάθθνι
Γ
+
CH. XII THE PYTHAGOREANS 117
as the central fire seems in other passages always to be called ‘the
fire in the middle’, Burnet’s interpretation seems scarcely possible.
Boeckh originally took the same view that the beams could only
be those from the central fire, holding to the strict interpretation of
untverse as being below the outer Olympus; but he afterwards
admitted,? with Martin, that the beams might come from the outer
fire, the fire of Olympus, as well. Accordingly the beams coming
from outside would be refracted by the sun, which would act as
a sort of lens.* Tannery* takes a similar view, from which he
_ develops another interesting hypothesis. We are to suppose two
_ cones opposite to one another and each truncated at the sun, where
they meet in a common section; these two cones form a luminous
_ column (that of the Myth of Er) by which a stream of light flows
_ from the fire of Olympus (supposed ἴο θὲ the Milky Way) in the
_ direction of the earth. But there remains a difficulty as regards the
central fire. What is the relation between the central fire and
the fire of the sun, and why does not the central fire always light
_ up the moon sufficiently for us to see it full? The beams of the
central fire must, Tannery conceives, be relatively feeble in com-
_ parison with those from the Milky Way, and though they may
_ suitably light up and warm the side of the counter-earth turned
towards the central fire, they have no appreciable power at the
distance of the moon, still less at the distance of the sun. The
outer cone and the inner cone meeting at the sun are supposed
by Tannery to have a small angular aperture. The base of the
outer cone is therefore presumably a part of the Milky Way;
which part is accordingly the first sun of the texts, and Tannery
suggests that we have in this portion of the Milky Way the ‘enth of
the heavenly bodies which revolve round the central fire, leaving the
sphere of the fixed stars motionless, as the complete system of
Philolaus requires it to be. This suggestion is brilliant but scarcely,
I think, consistent with what we are told of the tenth body; for
? Boeckh, Pihilolaus, pp. 123-30.
3 Boeckh, Das kosmische System des Platon, p. 94:
* Martin, L’hypothese astronomique de Philolaiis, pp. 9, το.
* Tannery, Pour Phistoire de la science helléne, pp. 237, 238.
® Cf. Aét. i. 14. 2 (D. G. p. 312), where it is stated that only the fire in the
very uppermost place is conical. The passage occurs ina section dealing mainly
with the shapes of the e/ements, but it may perhaps have strayed into the wrong
context.
118 THE PYTHAGOREANS PART I
on this assumption it would presumably be, from time to time,
a different portion of the Milky Way varying as the sun revolves.
With Tannery’s idea of the connexion between the sun and the
Milky Way, the following passages should be compared :
‘Of the so-called Pythagoreans some say that this [the Milky
Way] is the path of one of the stars which fell out of their places
in the destruction said to have taken place in Phaethon’s time;
others say that the sun formerly revolved in this circle, and accord-
ingly this region was, so to say, burnt up, or suffered some such
change, through the revolution of the sun.’4
‘Of the Pythagoreans some explain the Milky Way as due to the
burning-up of a star which fell out of its proper place and set on fire
the region through which it circulated during the conflagration
caused by Phaethon; others say that the sun’s course originally lay
along the Milky Way. Some, again, say that it is the mirrored
image of the sun as it reflects its rays at the heaven, the process
being the same as with the rainbow on the clouds.’?
The Moon.
A fanciful view of the moon is quoted by the Doxographi as
held by some of the Pythagoreans, including Philolaus.
‘Some of the Pythagoreans, among whom is Philolaus, say that
the moon has an earthy appearance because, like our earth, it is
inhabited throughout by animals and plants, only larger and more
beautiful (than ours): for the animals on it are fifteen times stronger
than those on the earth ... and the day in the moon is correspond-
ingly longer.’®
No doubt the fact that the animals on the moon are superior to
those on the earth ‘in force (τῇ δυνάμει)᾽ to the extent of fifteen
times is an inference from the fact that the day is fifteen times
longer than ours. Boeckh points out, as regards the day, that the
length of it is clearly meant to be half the time occupied by one
revolution of the moon (in 294 days) round the central fire. Assum-
ing that, as with the earth, the same hemisphere is always turned
outwards (which involves one rotation of the moon round its axis in
1 Aristotle, Aleteorologica i. 8, 345 a 13-18 (Vors. i*, p. 230. 37-41). In the
last words of this passage Diels (loc. cit.) reads φθορᾶς, ‘destruction’ or
‘wasting ’, instead of φορᾶς, ‘ revolution’.
2 Aét. ili, 1. 2 (D. G. p. 364; Vors. 13, p. 278. 42).
8. Aét. ii. 30. 1 (D.G. p. 3613; Vors. δ, p. 237. 42).
CH. XII THE PYTHAGOREANS t19
the same time as it takes the moon to revolve round the central fire),
an inhabitant of that hemisphere would see the sun, that is, it would
be day for him, for roughly half the period of the moon’s revolu-
tion; during the same half of the period an inhabitant of the
hemisphere turned towards the earth would not see the sun, and it
would be night for him; and vice versa. Therefore the ‘day’ for
an inhabitant of the moon, which receives its light from the sun,
would be equal to fifteen of our days and nights added together.
According to the actual wording of the text it should be fifteen
times our day only; this would require that the moon should
revolve on its axis /wice (instead of the once which is automatic,
_ as it were) during a lunation. Martin’ develops this supposition,
_ but it seems clear that the ‘day’ of the inhabitants of the moon
was meant to be equal to fifteen of our days and nights together,
and that the form of the statement in the text is due to
_ inadvertence.
_ According to ‘other Pythagoreans’ what we see on the moon
_ is a reflection of the sea which is beyond the torrid circle or zone in
our earth.”
Eclipses.
We have seen that the counter-earth was probably invented in
order to explain the frequency of eclipses of the moon, and that
there were some who thought there might be more bodies of the
kind which by their interposition caused eclipses of the moon. The
latter bodies would of course, like the counter-earth, be invisible to
the inhabitants of our hemisphere, from which it follows that they
would also, like the counter-earth, revolve along with the earth round —
the central fire and always have the same right ascension with
the earth.
Eclipses of the moon are then caused by the interposition either
of the earth or of the counter-earth (or other similar body) between
the sun and the moon.*
Eclipses of the sun on the other hand are, and can only be, caused
by the moon ‘ getting under the sun’,* i.e. by the interposition of
the moon between the sun and the earth.
Martin, Hyfothése astronomique de Philolaiis, p. 22.
? Aét. ii. 30. 1 (D.G. p. 361 b 10-13).
3. Aét. ii. 29. 4(D. G. p. 360; Vors. i*, p. 277. 46).
* σελήνης αὐτὸν ὑπερχομένης, Aét. ii. 24.6 (D. G. p. 354).
120 THE PYTHAGOREANS
The Phases of the Moon.
In the same passage (under the heading ‘On the Eclipse of the
Moon’) in which Aétius says that ‘some of the Pythagoreans’ give
the explanation of lunar eclipses just referred to, a curious view
is mentioned as having been held by ‘some of the later (Pytha-
goreans)’. The words must apparently (notwithstanding _ their
context) refer to the phases, and not to eclipses, of the moon; the
change is said to come about ‘ by way of spreading of flame, which
is kindled by degrees and in a regular manner until it produces the
perfect full moon, after which again the flame is curtailed by cor-
responding degrees until the conjunction, when it is completely
extinguished’. It would seem that these ‘later’ Pythagoreans had
forgotten the fact that the moon gets its light from the sun, or at
least had no clear understanding of the way in which the variations
in the positions of the sun and moon relatively to the earth produce
the variations in the shape of the portion of the illuminated half
which is visible to us from time to time.
ΧΙ
THE ATOMISTS, LEUCIPPUS AND DEMOCRITUS
LEuciPpus of Elea or Miletus (it is uncertain which’) was
a contemporary of Anaxagoras and Empedocles; and Democritus
_of Abdera was also a contemporary of Anaxagoras, though younger,
for he was, according to his own account,” ‘ young when Anaxagoras
was old’, from which it is inferred that he was born about 460 B.C.
The place of the two Atomist philosophers in the history of astronomy
isnot a large one, for they made scarcely any advance upon their
_ predecessors; most of the views of Democritus are a restatement of
_ those of Anaxagoras, even down to the crudest parts of his doctrine.
_ As Burnet? says, the primitive character of the astronomy taught
by Democritus as compared with that of Plato is the best evidence
of the value of the Pythagorean researches. The weakness of
_ Democritus’s astronomy is the more remarkable because we have
conclusive evidence that he was a really able mathematician.
Archimedes* says that Democritus was the first to state that the
volumes of a cone and a pyramid are one-third of the volumes
of the cylinder and prism respectively which have the same base
and height, though he was not able to prove these facts in the
rigorous manner which alone came up to Archimedes’ standard
of what a scientific proof should be (the discovery of the proofs
of the propositions by the powerful ‘method of exhaustion’ was
3 9am in Phys. p. 28. 4 (from Theophrastus) ; see D. G. p. 483; Vors. 13,
344. 40.
Ps Diog. L. ix. 41 (Vors. 15, p. 387. 12).
3 Burnet, Early Greek Philosophy, p. 345.
* Heiberg, ‘Eine neue Archimedes-Handschrift’ in Hermes, xlii, 1907,
ΡΡ. 245, 246; cf. the translation and commentary by Heiberg and Zeuthen
‘mm Bibliotheca Mathematica, viis, 1906-7, p. 323; The Thirteen Books of
Euclid’s Elements, 1908, vol. iii, pp. 366, 368.
122 THE ATOMISTS PARTI
reserved for Eudoxus). There is evidence, too, that Democritus
investigated (1) the relation in size between two sections of a cone
parallel to the base and very close to each other, and (2) the nature
of the contact of a circle or sphere with a tangent. These facts
taken together suggest that he was on the track of infinitesimals
and of the Integral Calculus.
The Great Diakosmos, attributed by Theophrastus to Leucippus,
is also given in the lists of Democritus’s works;1 indeed no one
later than Theophrastus seems to have been able to distinguish
between the work of Leucippus and Democritus, all the writings
of the school of Abdera being apparently regarded by later authors
as due to Democritus. However, the information which we possess
about the cosmology of the two philosophers goes back to Theo-
phrastus, so that we are not without some guidance as to details
in which they differed. Diogenes Laertius,? in a passage drawn
from an epitome of Theophrastus, attributes the following views
to Leucippus. The worlds, unlimited in number, arise through
‘bodies’, i.e. atoms, falling into the void and meeting one another.
By abscission from the infinite many ‘bodies’ of all sorts of shapes
are borne into a great void, and their coming together sets up
a vortex. By the usual process, in the case of our world, the earth
collects at the centre. The earth is like a tambourine in shape and
rides or floats by virtue of its being whirled round in the centre.
The sun revolves in a circle, as does the moon; the circle of the
sun is the outermost, that of the moon the nearest to the earth, and
the circles of the stars are between. All the stars are set on fire
because of the swiftness of their motion; the sun is also ignited
by the stars; the moon has only a little fire in its composition.
The ‘inclination of the earth’,® ie. the angle between the zenith
1 Vors. i%, p. 357. 21, p. 387. 4; cf. Achilles, Jsagoge i. 13 (Vors. i,
P- 349. 29).
2 Diog. L. ix. 30-33 (Vors. i*, pp. 342. 35 -- 343. 27).
8 The words ‘inclination of the earth’ are missing in the text of Diogenes.
Diels (Vors. i*, p. 343. 22) supplies words thus: {τὴν δὲ AdEwow τοῦ ζῳδιακοῦ
γενέσθαι) τῷ κεκλίσθαι τὴν γῆν πρὸς μεσημβρίαν, ‘{the obliquity of the zodiac
circle is due) to the tilt of the earth towards the south.’ But this can hardly
be right ; the reference must be to the same ‘ inclination of the earth (ἔγκλισις
γῆς), 1.6. the angle between the zenith and the pole or between the earth’s (flat)
surface and the plane of the apparent circular revolution of a star, which is
spoken of in Aét. iii. 12. 1-2 (D. G. p. 377; Vors. 3, pp. 348. 15, 367. 47). The
words which have fallen out may perhaps have been ‘ the obliquity of the circles
ν
ΟΗ ΧΙ LEUCIPPUS AND DEMOCRITUS 123
and the visible (north) pole, or the angle between the (flat) surface
of the earth and the plane of the apparent circular movement of
a star in the daily rotation, is due to the tilt of the earth towards
the south, the explanation of this tilt being on lines which recall
Empedocles rather than Anaxagoras;' the northern parts have
perpetual snow and are cold and frozen.
The sun rarely suffers eclipse, while the moon is continually
darkened, because their circles are unequal.
_ We have here reminiscences of Anaximander in the description
_ of the shape of the earth and partly also in the statement about
_ the relative distances of the sun, moon, and stars from the earth,
while the idea of the earth riding on the air recalls Anaximenes,
with a difference. There are traces of Anaxagoras’s views in the
vortex causing the earth to take the central position, and in the
kindling of the stars due to their rapid motion; but there is the
difference that the atoms take the place of the mixture in which
‘all things are together’, and no force such as Anaxagoras’s Nous
is considered to be required in order to start the motion of the
_ vortex, the atoms being held to have been in motion always.
᾿ς Democritus’s views are much more uniformly those of Anaxagoras.
_ Thus with him the stars are stones,? the sun is a red-hot mass or
ἃ stone on fire;* the sun is of considerable size.* The moon has
_ in it plains, mountains (or, according to one passage, lofty elevations
casting shadows®), and ravines,® or valleys.° Democritus said that
the moon is ‘plumb opposite’ to the sun at the conjunctions, and
of the stars’, or they may have referred to differences of climate in different
parts of the earth. ;
1 Leucippus’s explanation of the tilt (Aét., loc. cit.) is that ‘ the earth turned
sideways towards the southern regions because of the rarefaction (ἀραιότητα) in
those parts, due to the fact that the northern regions became frozen through
excessive cold while the southern parts were set on fire’.
Democritus’s explanation is slightly different: ‘The earth as it grew became
inclined southwards because the southern portion of the enveloping (substance)
is weaker (i.e. presumably weaker in resisting power); for the northern regions
-are intemperate (dxpara), i.e. frigid, the southern temperate (κέκραται) ; hence
it is in the south that the earth sags (βεβάρηται), namely, where fruits and all
wth are in excess.’
2 Aét. ii, 13. 4 e G. p. 341; Vors. 15, Ὁ. 366. 31).
5. Aét. ii. 20. 7 (D. G. p. 349; Vors. i*, p. 366. 35).
* Cicero, De jin. i. 6. 20 (Vors. 13, p. 366. 36).
5 Aét. ii. 30. 3 (D. G. p. 361; Vors. i*, p. 367. 13).
® Aé&t. ii, 25. 9 (D. G. p. 356; Vors. i?, p. 308, 11).
124 THE ATOMISTS PART I
it is evident that he fully accepted the doctrine that the moon
receives its light from the sun.*
As regards the earth, Democritus differed from Anaxagoras in
that, while Anaxagoras said it was flat, Democritus regarded it as
‘disc-like but hollowed out in the middle’? (i.e. depressed in the
middle and raised at the edges); but this latter view was also held
by Archelaus, a disciple of Anaxagoras, and may therefore have
been that of Anaxagoras himself; the proof of the hollowness,
Archelaus thought, was furnished by the fact that the sun does not
rise and set everywhere on the earth’s surface at the same time,
as it would have been bound to do if the surface had been
level.2 How, asks Tannery, did Anaxagoras or Archelaus come
to draw from the observed facts with regard to the rising and setting
of the sun a conclusion the very opposite of the truth ἢ
Again, while Anaxagoras, like Anaximenes, supposed the flat
earth to ride on the air, being supported by it,° Democritus is
associated with Parmenides’ view that the earth remains where
it is because it is in equilibrium and there is no reason why it
should move one way rather than another.®
We are told that the ancients represented the inhabited earth
as circular, and regarded Greece as lying in the middle of it and
Delphi as being in the centre of Greece, but that Democritus was
the first to recognize that the earth is elongated, its length being
1% times its breadth.’ Democritus is also, along with Eudoxus,
credited with having compiled a geographical and nautical survey
of the earth as, after Anaximander, Hecataeus of Miletus and
Damastes of Sigeum had done.®
Democritus agreed with Anaxagoras’s remarkable view of the
Milky Way as consisting of the stars which the sun ‘ does not see’
1 Plutarch, De facte in orbe lunae 16, Ὁ. 929 C (Vors. 15, p. 367. 9-11).
Plutarch is arguing that the moon is made of an opaque substance, like earth.
Were it otherwise, he says, the moon would not be invisible at the conjunctions
when ‘ plumb opposite’ the sun; if, e.g., the moon were made of a transparent
material like glass or crystal, then, at the conjunctions, it should not only be
visible itself, but it should allow the sun’s light to shine through it, whereas it is
in fact invisible at those times and often actually hides the sun from our sight.
2 Aét. iii. 10. 5 (D. G. p. 377; Vors. i®, p. 367. 41).
3 Hippolytus, Refut. i. 9. 4 (D. G. pp. 563-4; Vors. 15, p. 324. 16).
4 Tannery, Pour l’histoire de la science helldne, p. 279.
5 Hippol. Refut. i. 8. 3 (D. G. p. 562. 5-73 Vors. i*, p. 301, 32).
© Aét. iii. 15. 7 (D. G. p. 380; Vors. i*, p. 111. 40).
7 Agathemerus, i. 1. 2 ( Vors. i*, p. 393. 10). 8. Ibid.
8 ( Ρ. 393
CH. XIII LEUCIPPUS AND DEMOCRITUS _ 195
when it is passing under the earth during the night ;! but, at the
same time, he seems to have been the first to appreciate its true
character as a multitude of small stars so close together that the
narrow spaces between them seem even to be covered by the
diffusion of their light in all directions, so that it has the appearance,
almost, of a continuous body of light.”
With Anaxagoras he thought that comets were ‘a conjunction
of planets when they come near and appear to touch one another ’,*
or a ‘ coalescence of two or more stars so that their rays unite ’.*
In his remark, too, about the infinite number of worlds he seems
to have done little more than expand what Anaxagoras had said
about the men in other worlds than ours who have inhabited
cities and cultivated fields, a sun and moon of their own, and
so on.’ It is worth while to quote Democritus’s actual words in
full, in order to see how slight is the foundation for the rhapsodical
estimate which Gomperz gives of his significance as a forerunner
of Copernicus. Hippolytus relates of Democritus that
He said that there are worlds infinite in number and differing
in size. In some there is neither sun nor moon, in others the sun
and moon are greater than with us, in others there are more than
one sun and moon. The distances between the worlds are unequal,
in some directions there are more of them, in some fewer, some are
growing, others are at their prime, and others again declining, in one
direction they are coming into being, in another they are waning.
Their destruction comes about through collision with one another.
Some worlds are destitute of animal and plant life and of all
moisture. . .. A world is at its prime so long as it is no longer
capable of taking in anything from without.’ ®
Let us now hear Gomperz.’? ‘ Democritus’s doctrine was far from
admitting the plausible division of the universe into essentially
different regions. It recognized no contrast between the sublunary
world of change and the changeless steadiness of the divine stars,
important and fatal though that difference became in the Aristotelian
1 Aristotle, Meteorologica i. 8, 345 a 25 (Vors. i*, p. 308. 26).
5 a aa In Somn. Scip. i. 15. 6; Aét. iii. 1.6 (D.G. p. 365; Vors. 13,
Ρ. 367. 21).
5 Aristotle, Meteorologica i. 6, 342 Ὁ 27 (Vors. i*, p. 308. 34).
* Aét. ili. 2. 2 (D. G. p. 366; Vors. i*, p. 308. 37).
5 Anaxagoras, Fr. 4 ( Vors. 15, p. 315. 8-16).
5 Hippolytus, γί. i. 13. 2-4 (D. G. p. 565 ; Vors. i*, p. 360. 10-19).
7 Gomperz, Griechische Denker, 15, pp. 295, 296.
126 THE ATOMISTS PART I
system. At this point Democritus was once more fully in agree-
ment not merely with the opinions of great men like Galilei, who
released modern science from the fetters of Aristotelianism, but even
with the actual results of the investigation of the last three centuries.
It is almost miraculous to observe how the mere dropping of the
scales from his eyes gave Democritus a glimpse of the revelations
which we owe to the telescope and to spectrum analysis. In
listening to Democritus, with his accounts of an infinitely large
number of worlds, different in size, some of them attended by
a quantity of moons [why not suns too, as in the fragment ?], others
without sun or moon, some of them waxing and others waning after
a collision, others again devoid of every trace of fluid, we seem to
hear the voice of a modern astronomer who has seen the moons of
Jupiter, has recognized the lack of moisture in the neighbourhood
of the moon, and has observed the nebulae and obscured stars which
the wonderful instruments that have now been invented have made
visible to his eyes. Yet this consentaneity rested on scarcely any-
thing else than the absence of a powerful prejudice concealing the
real state of things, and ona bold, but not an over-bold, assump-
tion that in the infinitude of time and space the most diverse
possibilities have been realized and fulfilled. So far as the endless
multiformity of the atoms is concerned, that assumption has not
won the favour of modern science, but it has been completely
vindicated in respect to cosmic processes and transformations. It
may legitimately be said that the Democritean theory of the
universe deposed in principle the geocentric point of view. Nor
would it be unfair to suppose that Democritus smoothed the way
for its actual deposition at the hands of Aristarchus of Samos.’.. .
‘Democritus contended that some worlds were without animals
and plants because the requisite fluid was lacking which should
supply them with nourishment. And this dictum of the sage
is especially remarkable inasmuch as it was obviously based on
the assumption of the uniformity of the universe in the substances
composing it and in the laws controlling it, which the sidereal
physics of our own day has proved beyond dispute. He evinced
the same spirit which animated Metrodorus of Chios, himself a
Democritean, in his brilliant parable: “a single ear of corn on
a wide-spreading champaign would not be more wonderful than
ν
CH. XIII LEUCIPPUS AND DEMOCRITUS 127
a single cosmos in the infinitude of space.” The genius of Democritus
did not stop at anticipating modern cosmology.’ . . .
This is a fascinating picture, but surely it is, in any case, much
overdrawn. And, even if it were true, we cannot but ask, why is
Anaxagoras, who, before Democritus, spoke of other worlds than
ours, with their suns and moons, their earths inhabited by men and
animals, where there are cities and cultivated fields, ‘as with us’,
given none of the credit for a theory which ‘ deposed in principle the
_ geocentric hypothesis’? Anaxagoras clearly set no limit to the
_ number of such worlds, and Democritus added little to his statement
_ except the details that at any given time some of the infinite number
_ Of worlds are coming into being, others waxing, others waning, others
being destroyed, and that they represent all possible varieties of
composition (some with suns and moons, some without, &c.), instead
of being more or less on the same plan with ours, as Anaxagoras
perhaps implied. Again, the abandonment of the geocentric hypo-
thesis does not carry us a step towards the Copernican theory
unless some other and truer centre is substituted for the earth.
_ But Democritus's theory of the infinity of worlds does not suggest
_ any such centre, nay, it destroys the possibility of there being such
-acentre at all.
With regard, however, to our sun and moon, Democritus puts
forward a rather remarkable hypothesis connected with the infinite
multiplication of his worlds. With Anaxagoras the stars, and
presumably the sun and moon also, were stones torn from the
earth by the whirling motion of the universe, and afterwards
kindled into fire by the rapidity of that motion. But according to
Democritus the sun and moon, which at the time’of their coming
into being ‘had not yet completely acquired the heat characteristic
of them, still less their great brilliance, but on the contrary were
assimilated to the nature subsisting in the earth’ were then ‘ moving
in independent courses of their own (κατ᾽ ἰδίαν) ; ‘for each of the
‘two bodies, when it first came into being, was still in the nature of
a separate foundation or nucleus for a world, but afterwards, as the
circle about the sun became larger, the fire was caught up in it ’.2
_ 1 Cf. Aristotle’s argument (De cae/o i. 6, 275 Ὁ 13) that the universe cannot
_be infinite because the infinite cannot have a centre.
* Ps. Plut. Stromat. (apud Euseb. Pr. Ev. i. 8. 7); D.G. p. 581; Vors. i?,
ΒΡ. 359. 47.
128 THE ATOMISTS PARTI
The last words appear to relate only to the addition of fire to the
earthy nucleus of the sun, which may be connected with the idea of
Leucippus that ‘the sun was kindled by the stars’: but it seems to
be implied by the whole passage that the sun and the moon, after
beginning to come into being as the nucleus of separate worlds, were
caught up by the masses moving round the earth and then carried
round the earth with them so as to form part of our universe.
As regards the planets, we have seen that Anaxagoras, like
Plato, placed the moon nearest to the earth, the sun further from
it, and the planets further still; Democritus made the order,
reckoning from the earth, to be Moon, Venus, Sun, the other
planets, the fixed stars.1 ‘Even the planets have not all the same
height’ (1.6. are not at the same distance from us).2 Seneca
observes that ‘ Democritus, the cleverest of all the ancients, says
he suspects that there are several stars which have a motion of
their own, but he has neither stated their number nor their names,
the courses of the five planets not having been at that time under-
stood’.2 This seems to imply that Democritus did not even
venture to say how many planets there were; Zeller, however,
holds that he could not but have known of the five planets,
especially as he wrote a book ‘ about the planets’ ;* it may be that
he said in this work that there might perhaps be more planets than
the five generally known, and Seneca, who had this at third hand,
may have misunderstood the observation.®
An interesting remark about Democritus’s views on the motion of
the sun and moon is contained in a passage of Lucretius,® where the
question is raised, why the sun takes a year to describe the full -
circle of the zodiac while the moon completes its course in ἃ month ;
perhaps, says Lucretius, Democritus may be right when he says that
the nearer any body is to the earth, the less swiftly can it be carried —
round by the revolution of the heaven; now the moon is nearer
than the sun, and the sun than the signs of the zodiac; therefore |
the moon seems to get round faster than the sun because, while the
sun, being lower and therefore slower than the signs, is left behind .
d
1 Aét. ii. 15. 3 (D. G. p. 3443 Vors. i*, p. 366. 32).
2 Hippol. Refut. i. 13. 4 (D. G. p. 565; Vors. 13, p. 360. 17)-
3. Seneca, Wat. Quaest. vii. 3. 2 (Vors. 13, p. 367. 29).
* Thrasyllus ap. Diog. L. ix. 46 (Vors. i*, p. 357. 22).
5 Zeller, i°, p. 896 note. ® Lucretius, v. 621 566.
.
ΡΘΗ ΧΠ LEUCIPPUS AND DEMOCRITUS 129
by them, the moon, being still lower and therefore slower still,
_ is still more left behind. Therefore it is the moon which appears
' to come back to every sign more quickly than the sun does, be-
_ cause the signs go more quickly back to her. The view that the
_ bodies which move round at the greatest distance move the most
_ quickly and vice versa is the same as we find attributed by Alexan-
_ der Aphrodisiensis to the Pythagoreans.*
_ Lastly, we are told by Censorinus? that Democritus put the
ot Year at ‘82 years with the same number, 28, of intercalary
τῇ s’, where the ‘ same number ’ is the number of ‘oopacadd
8 years es): which seems probable enough; but, as he says,
_ it is impossible to draw any certain conclusion from the passage.
ie 1 Alexander, Jn metaphysica A. 5, p. 542 a 16-18 Brandis, p. 40. 7-9 Hayduck.
ied Censorinus, De die natali 18. 8 (Vors. i*, p. 390.
, 19).
__ * Tannery in Mém. de la Société des sciences phys. et nat. de Bordeaux, 3° sé.
_ iv, 1888, p. 92.
1410 K
XIV
OENOPIDES
THE date of Oenopides of Chios is fairly determined by the
statement of Proclus that he was a little younger than Anaxagoras.*
He was a geometer of some note; Eudemus credited him with
having been the first to investigate the problem of Eucl. I. 12 (the
drawing of a perpendicular to a given straight line from a given
point outside it), which he ‘thought useful for astronomy’, and
to discover the problem solved in Eucl. I. 23 (the construction on
a given straight line and at a point on it of an angle equal to a
given rectilineal angle). No doubt perpendiculars had previously
been drawn by means of some mechanical device such as a set
square, and Oenopides was the first to give the theoretical con-
struction as we find it in Euclid; and in like manner he probably
discovered, not the problem of Eucl. I. 23 itself, but the particular
solution of it given by Euclid.
In astronomy he is said to have made two discoveries of impor-
tance. The first is that of the obliquity of the ecliptic. It is true
that Aétius says that both Thales and Pythagoras, as well as the
successors of the latter, distinguished the oblique circle of the zodiac —
as touching or meeting three of the ‘five circles which are called
zones’;? Aétius further states that ‘Pythagoras is said to have
been the first to observe the obliquity of the zodiac circle, a fact
which Oenopides put forward as his own discovery’. Now Thales
could not possibly have known anything of the zones, and no doubt
‘Pythagoras and his successors’ may have been substituted for ‘ the
Pythagoreans’ in accordance with the usual tendency to attribute
everything to the Master himself; in like manner the second —
? Proclus, Comm. on Eucl. 7, p. 66. 2 (Vors. i*, p. 229. 36).
5. Aét. ii, 12. 1 (D.G. p. 340).
® Aét. ii. 12, 2 (2. G. p. 340-1; Vors. i*, p. 230. 14).
OENOPIDES 121
__ passage is probably the result of the same jealousy for the reputa-
_. tion of Pythagoras. And for the attribution of this particular
discovery to Oenopides we have the better authority of Eudemus
in a passage taken from Dercyllides by Theon of Smyrna.’
Macrobius observes that Apollo (meaning the sun) is called Loxias,
_as Oenopides says, because he traverses the oblique circle (λοξὸν
κύκλον), moving from west to east.2, The Egyptian priests, we
_ are told, claimed that it was from them that Oenopides learned
_ that the sun moves in an inclined orbit and in a sense opposite
_ to that of the motion of the other stars.* It does not appear that
_ Oenopides made any measurement of the obliquity ; at all events
_ he cannot be credited with the estimate of 24°, which held its own
_ till the time of Eratosthenes (circa 275-194 B.C.).*
1 Theon of Smyrna, p. 198. 14, Hiller ( Vors. 15, p. 230. 11).
3 Macrobius, Sav. i. 17. 31 (Vors. 13, p. 230. 22).
* Diodorus Siculus, i. 98. 2 (Vors. i*, p. 230. 19).
᾿ς * Dercyllides’ quotation from Eudemus (Theon of Smyrna, pp. 198, 199),
_ which states that Oenopides was the first to discover the obliquity of the zodiac
circle, also mentions that it was other astronomers not named in the particular
_ passage who added (among other things) the discovery that the measure of the
obliquity was the angle subtended at the centre of a circle by the side of a
_ regular fifteen-angled figure inscribed in the circle, that is to say, 24°. But this
value was discovered before Euclid’s time, for Proclus, quite credibly, mentions
_ (Comm. on Excl. I, p. 269. 11-21) that the proposition Eucl. ΓΝ. 16, showing how
_ to describe a regular fifteen-angled figure in a circle, was inserted in view of its
use in astronomy. The value was doubtless known to Eudoxus also, if it does
_ not even go back to the Pythagoreans. The angle might no doubt have been
calculated by means of Pytheas’s measurement of the midday height of the sun
_ at Marseilles at the summer solstice. According to Strabo (ii. 5. 8, p. 115, and
ii. 5. 41, p. 134, Cas.), Pytheas found that the ratio of the gnomon to its midday
shadow at the summer solstice at Marseilles was 120: 414 (Ptolemy made it
60:203, or 120: 41%, Synfaztis, ii. 6, p. 110. 5). But we are not told of any
_ value that Pytheas gave for the latitude of Massalia. According to Strabo,
Hipparchus said that the same ratio of the gnomon to the shadow as Pytheas
. found at Massalia held good at Byzantium also, whence, relying on Pytheas’s
accuracy, he inferred that the two places were on the same parallel of latitude.
As, however, Marseilles is 2° further north than Byzantium, it is clear that there
must have been an appreciable error of calculation somewhere. Theon of
Alexandria (On Ptolemy's Syntaxis, p. 60) states that Eratosthenes discovered
the distance between the tropic circles to be 11/83rds of the whole meridian
_circle=47° 42’ 40”, which gives 23° 51’ 20” for the obliquity of the ecliptic. Berger,
however (Die geographischen Fragmente des Eratosthenes, 1880, Ὁ. 131), is
inclined to infer from Ptolemy’s language that it was Ptolemy himself who
invented the ratio 11 : 83, and that Eratosthenes still adhered to the value 24°.
For Ptolemy (Synfaxis i. 12, p. 67. 22 -- 68. 6) says that he himself found the
distance between the tropic circles to lie always between 47° 40’ and 47°45’,
_ *from which we obtain about (σχεδόν) the same ratio as that of Eratosthenes,
which Hipparchus also used. For the distance between the tropics decomes
(or zs found to be, γίνεται) very nearly 11 parts out of 83 contained in the whole
K2
132 OENOPIDES PARTI
The second discovery attributed to Oenopides is that of a Great
Year, the duration of which he put at 59 years! In addition, we
are told by Censorinus that Oenopides made the length of the year
to be 36522 days.2, Tannery ὃ suggests the following as the method
by which he arrived at these figures. Starting first of all with
365 days as the length of a year, and 29% days as the length of the
lunar month, approximate values known before his time, Oenopides
had to find the least integral number of complete years which
would contain an exact number of lunar months; this is clearly
59 years, which contains a number of lunar months represented by
twice 365, or 730. He had then to determine how many days
there were in 730 months. This his knowledge of the calendar
would doubtless enable him to do, and he would appear to have
arrived at 21,557 days as the result,* since this, when divided by 59,
gives 36522 days as the length of the year. Tannery gives good
meridian circle.’ The mean between 47° 40’ and 47° 45’ is of course 47° 42’ 30”,
or only το different from 47° 42’ 40"; but the wording is somewhat curious if
Ptolemy meant to imply that the actual ratio 11:83 represented Eratosthenes’
estimate. For ‘the same ratio’ would them be 11/83 and σχεδόν and ἔγγιστα
would have te mean exactly the same thing. Moreover, in that case, to make
a separate sentence of the comparison with the fraction 11/83 was quite un-
necessary ; all that was necessary was to add to the preceding sentence some
words such as ‘namely 11/83rds of the meridian circle’ in explanation of ‘the
same ratio’, On the other hand, if the intention was to compare the mean
value 47° 42’ 30” with a value 48°, or 2/15ths of a great circle, used by Eratos-
thenes and Hipparchus, there was a sort of excuse for a separate sentence
converting 47° 42’ 30” into a fraction of a great circle as nearly as possible equi-
valent, namely 11/83rds, for the purpose of comparison with 2/15ths, the difference
between the fractions being 1/1245. Hipparchus, in his Commentary onthe
Phaenomena of Aratus and Eudoxus (p. 96. 20-21, Manitius) said that the
summer tropical circle is ‘ very nearly 24° north of the equator’. Another value for
the obliquity of the ecliptic is derivable from an odzter dictum of Pappus (vi. 35,
p. 546. 22-7, ed. Hultsch). Pappus, without any indication of his source, there
says that the value of the ratio which we should call the tangent of the angle is
10/23. We should scarcely have expected a ratio between such small numbers
to give a very accurate value, but 10/23 =.0-4347826, which is the tangent of an
angle of 23° 2955” nearly.
1 Theon of Smyrna, p. 198. 15 (Vors. i*, p. 230. 13): Aelian, V. H. x. 7 (Vors.
i”, p. 230. 27); Aét. ii. 32. 2 (D. G. p. 363; Vors. i*, p. 230. 34).
* Censorinus, De die natali 19. 2.
3 Tannery in A/ém. de la Société des sciences phys. et nat. de Bordeaux,
3° sér. iv. 1888, pp. 90, 91.
4 The true synodic month being 29-53059 days, 730 times this gives, as a
matter of fact, 215574 days nearly.
5 This year of a little less than 365 days 9 hours is slightly more correct than
the average year of the octaéteris of 2923} days, which works out to 365 days
τοῦ hours (Ginzel; Handbuch der mathematischen und technischen Chronologie,
vol. ii, 1911, p. 387).
“᾿ς
CH. XIV OENOPIDES 133
ground for thinking that Oenopides cannot have taken account of
the motion of all the planets as well as of the sun and moon for
the purpose of calculating the Great Year. He would, no doubt,
know the approximate periods of revolution of Saturn, Jupiter,
and Mars, namely 30 years for Saturn, 12 years for Jupiter, and
2 years for Mars, which figures would give roughly, in his great
year of 59 years, 2 revolutions of Saturn, 5 of Jupiter, and 30 or
3t for Mars. Admitting the last number as the more exact, and
_ dividing 21,557 days by these numbers respectively, we obtain
_ periods for the revolution of the several planets which, like the
figures worked out by Schiaparelli for Philolaus, would show errors
not exceeding I per cent. of the true values. But Tannery considers
that this is not the proper way to judge of the error; he would
_ rather judge the degree of inaccuracy by the error in the mean
position of the planet at the end of the period. He finds that,
_ calculated on this basis, the error would not reach as much as
_ 2° in the case of Saturn, and 9° in the case of the sun; but for
_ Mars the error would exceed 107°, which is quite inadmissible.
_ If Oenopides had ventured to indicate the sign of the zodiac in
which each planet would be found at the end of his period, the
_ error in the case of Mars would have been discovered when the
time came.
Aristotle? says that some of the so-called Pythagoreans held
that the sun at one time moved in the Milky Way. This same
view is attributed to Oenopides ; for Achilles says * that ‘ According
to others, among whom is Oenopides of Chios, the sun formerly
moved through this region [the Milky Way], but because of the
Thyestes-feast he was diverted and has (since) revolved in a path
directed the opposite way to the other, that namely which is
defined by the zodiac circle’.
1 Aristotle, Mefeorologica i. 8, 345 a 16 (Vors. i*, p. 230. 39).
2 Achilles, /sagoge ad Arat. 24, p. 55. 18, Maass (Vors. 15, p. 230. 42).
XV
PLATO
IN order to obtain an accurate view of Plato’s astronomical system
as a whole, and to judge of the value of his contributions to the
advance of scientific astronomy, it is necessary, first, to collect and
compare the various passages in his dialogues in which astronomical
facts or theories are stated or indirectly alluded to ; then, secondly,
allowance has to be made for the elements of myth, romance, and
idealism which are, in a greater or less degree depending on the
character of the particular dialogue, invariably found as a setting
and embellishment of actual facts and theories. When these ele-
ments are as far as possible eliminated, we find a tolerably com-
plete and coherent system which, in spite of slight differences of
detail and a certain development and even change of view between
the earlier and the later dialogues, remains essentially the same.
In considering this system we have further to take into account
Plato’s own view of astronomy as a science. This is clearly stated
in Book VII of the Republic, where he is describing the curriculum
which he deems necessary for training the philosophers who are to
rule his State. The studies required are such as will lift up the
soul from Becoming to Being; they should therefore have nothing
to do with the objects of sensation, the changeable, the perishable,
which are the domain of opinion only and not of knowledge. It is
true that sensible objects are useful in so far as they give the
stimulus to the purely intellectual discipline required, in so far, in
fact, as they suffice to show that sensations are untrustworthy or
even self-contradictory. Some objects of perception are adequately
appreciated by the perception; these are non-stimulants; others
arouse the intellect by showing that the mere perception produces
an unsound result. Thus the perception which reports that a thing
is hard frequently reports that it is also soft, and similarly with
PLATO 135
thickness and thinness, greatness and smallness, and the like. In
_ such cases the soul is perplexed and appeals to the intellect for
help; the intellect responds and looks at ‘ great’ and ‘ small’ (e.g.)
as distinct and not confounded; we are thus led to the question
what zs the ‘great’ and what zs the ‘small’. Science then is only
concerned with realities independent of sense-perception ; sensation,
observation, and experiment are entirely excluded from it. At the
beginning of the formulation of the curriculum for philosophers
_ gymnastic and music are first mentioned, only to be rejected at
_ once; gymnastic has to do with the growth and waste of bodies,
that is, with the changeable and perishing; music is only the
_ counterpart, as it were, of gymnastic. Next, all the useful arts are
_ tabooed as degrading. The first subject of the curriculum is then
_ taken,namely the science of Number, in its two branches of ἀριθμη-
_ xh, dealing with the Theory of Numbers, as we say, and of
λογιστική, calculation, with the proviso that it is to be pursued for
_ the sake of knowledge and not for purposes of trade. Next comes
_ geometry, and here Plato, carrying his argument to its logical con-
clusion, points out that the true science of geometry is, in its nature,
directly opposed to the language which, for want of better terms,
geometers are obliged to use; thus they speak of ‘squaring’,
‘applying’ (a rectangle), ‘adding’, &c., as if the object were to do
something, whereas the true purpose of geometry is knowledge.
Geometrical knowledge is knowledge of that which zs, not of that
which becomes something at one moment and then perishes; and,
as such, geometry draws the soul towards truth and creates the
philosophic spirit which helps to raise up what we wrongly keep
down. Astronomy is next mentioned, but Socrates corrects him-
self and gives the third place in ‘the curriculum to stereometry, or
solid geometry as we say, which, adding a third dimension,
naturally follows plane geometry. And fourth in the natural order
is astronomy, since it deals with the ‘motion of body’ (φορὰ
βάθους; literally‘ motion ὁ οὗ depth’ or of the third dimension).
When astronomy was first mentioned, Socrates’ interlocutor
hastened to express approval of its inclusion, because it is proper,
not only for the agriculturist and the sailor, but also for the general,
to have an adequate knowledge of seasons, months, and years ;
whereupon Socrates rallies him upon his obvious anxiety lest the
136 PLATO PART I
philosopher should be thought to be pursuing useless studies.
When the speakers return to astronomy after the digression on
solid geometry, Glaucon tries a different tack: at all events, he says,
astronomy compels the soul to look upward and away from the
things of the earth. But no! he is using the term ‘upward’ in the
sense of towards the material heaven, not, as Socrates had meant
it, towards the realm of ideas or truth; and Socrates at once takes
him up. On the contrary, he says, as it is now taught by those
who would lead us upward to philosophy, it is calculated to turn
the soul’s eye down.
‘You seem with sublime self-confidence to have formed your
own conception of the nature of the learning which deals with the
things above. At that rate, if a person were to throw his head
back and learn something by contemplating a carved ceiling, you
would probably suppose him to be investigating it, not with his
eyes, but with his mind. You may be right, and I may be wrong.
But I, for my part, cannot think any other study to be one that
makes the soul look upwards except that which is concerned with
the real and the invisible, and, if any one attempts to learn anything
that is percezvable, I do not care whether he looks upwards with
mouth gaping or downwards with mouth closed: he will never, as
I hold, learn—because no object of sense admits of knowledge—
and I maintain that in that case his soul is not looking upwards but
downwards, even though the learner float face upwards on land or
in the sea.’ ‘I stand corrected,’ said he; ‘your rebuke was just.
But what is the way, different from the present method, in which
astronomy should be studied for the purposes we have in view δ᾽
‘This’, said I, ‘is what I mean. Yonder broideries in the
heavens, forasmuch as they are broidered on a visible ground, are
properly considered to be more beautiful and perfect than anything
else that is visible; yet they are far inferior to those which are true,
far inferior to the movements wherewith essential speed and essen-
tial slowness, in true number and in all true forms, move in relation
to one another and cause that which is essentially in them to move:
the true objects which are apprehended by reason and intelligence,
not by sight. Or do you think otherwise?’ ‘Not at all,’ said he.
‘ Then’, said I, ‘we should use the broidery in the heaven as illus-
trations to facilitate the study which aims at those higher objects,
just as we might employ, if we fell in with them, diagrams drawn
and elaborated with exceptional skill by Daedalus or any other
artist or draughtsman ; for I take it that any one acquainted with
geometry who saw such diagrams would indeed think them most
CH. XV PLATO 137
beautifully finished but would regard it as ridiculous to study them
seriously in the hope of gathering from them true relations of
equality, doubleness, or any other ratio.’ ‘Yes, of course it would
be ridiculous, he said. ‘Then’, said I,‘ do you not suppose that
one who is a true astronomer will have the same feeling when he
looks at the movements of the stars? That is, will he not regard
the maker of the heavens as having constructed them and all that
is in them with the utmost beauty of which such works admit ; yet,
in the matter of the proportion which the night bears to the day,
both these to the month, the month to the year, and the other stars
to the sun and moon and to one another, will he not, think you,
_ regard as absurd the man who supposes these things, which are
_ corporeal and visible, to be changeless and subject to no aberrations
_ of any kind; and will he not hold it absurd to exhaust every
possible effort to apprehend their true condition?’ ‘ Yes, I for one
certainly think so, now that I hear you state it.’ ‘Hence’, said I,
_ “we shall pursue astronomy, as we do geometry, by means of pro-
_ blems, and we shall dispense with the starry heavens, if we propose
_ to obtain a real knowledge of astronomy, and by that means to
convert the natural intelligence of the soul from a useless to a use-
ful possession.’ ‘The plan which you prescribe is certainly far more
laborious than the present mode of studying astronomy.’ ?
We have here, expressed in his own words, Plato’s point of view,
and it is sufficiently remarkable, not to say startling. We follow
him easily in his account of arithmetic and geometry as abstract
sciences concerned, not with material things, but with mathematical
‘numbers, mathematical points, lines, triangles, squares, &c., as
objects of pure thought. If we use diagrams in geometry, it is only
as illustrations ; the triangle which we draw is an imperfect repre-
sentation of the real triangle of which we ¢/ink. And in the
passage about the inconsistency between theoretic geometry and
the processes of squaring, adding, &c., we seem to hear an echo
of the general objection which Plato is said to have taken to the
mechanical constructions used by Archytas, Eudoxus, and others
for the duplication of the cube, on the ground that ‘the good of
geometry is thereby lost and destroyed, as it is brought back to
things of sense instead of being directed upward and grasping at
eternal and incorporeal images’. But surely, one would say, the
‘case would be different with astronomy, a science dealing with
1 Republic vii. 529 A-530B.
3 Plutarch, Quaest. Conviv, viii. 2.1, p. 718 F (Vors. 15, p.255. 3-5).
--.-
138 PLATO PARTI '
the movements of the heavenly bodies which we see. Not at all,
says Plato with a fine audacity, we do not attain to the real science
of astronomy until we have ‘dispensed with the starry heavens’,
i.e. eliminated the visible appearances altogether. The passage
above translated is admirably elucidated by Dr. Adam in his edition
of the Republic: There is no doubt that Plato distinguishes two
astronomies, the apparent and the real, the apparent being related
to the real in exactly the same way as practical (apparent) geometry
which works with diagrams is related to the real geometry. On the
one side there are the visible broideries or spangles in the visible
heavens, their visible movements and speeds, the orbits which they
are seen to describe, and the number of hours, days, or months
which they take to describe them. But these are only illustrations
(παραδείγματα) of real heavens, real spangles, real or essential speed
or slowness, real or true orbits, and periods which are not days,
months, or years, but absolute numbers. The broideries or span-
gles in both the astronomies are stars, but stars regarded as moving
bodies. Essential speed and essential slowness seem to be, as Adam
says, simply mathematical counterparts of visible stars, because they
are said to be carrie Ah te Eee τοριίουο οἱ real astronomy, and
therefore cannot be the speed and slownéss“of the mathematical
bodies of which the visible stars are illustrations, but must be those
mathematical bodies themselves. The true figures in which they
move are their mathematical orbits,which we might now say are
the perfect ellipses of which the orbits of the visible material planets
are imperfect copies. And lastly, as a visible planet carries with it
all the sensible properties and phenomena which it exhibits, so does
its mathematical counterpart carry with it the mathematical realities
which are in it. In short, Plato conceives the subject-matter of
astronomy to be a mathematical heaven of which the visible heave
is a blurred and imperfect expression in*time ‘and Space ; and the
science is a kind of ideal kinematics, a study in which the visible
movements of the heavenly bodies are only useful as illustrations.
But, we may ask, what form would astronomical investigations
on Plato’s lines have taken in actual operation? Upon this there
is naturally some difference of opinion. One view is that of
1 See, especially, vol. ii, pp. 128-31, notes, and Appendices II and X to Book
VII, pp. 166-8, 186-7.
CH. XV PLATO 139
Bosanquet,' who relies upon the phrase ‘we shall pursue astronomy
as we do geometry by means of problems’, and suggests that the
' discovery of Neptune, picturesquely described by De Morgan as
*Leverrier and Adams calculating an unknown planet into visible
existence by enormous heaps of algebra’,? is the kind of investiga-
tion which ‘seems just to fulfil Plato’s anticipations’. Plato was
a master of method, and it is an attractive hypothesis to picture
him as having at all events foreshadowed the methods of modern
astronomy ; but Adam seems to be clearly right in holding that
the illustration does not fit the language of the passage in the
_ Republic which we are discussing. For Plato says that the person
who thought that the heavenly bodies should always move pre-
cisely in the same way and show no aberrations whatever would
properly be thought ‘absurd’, and that it would be absurd to
_ exhaust oneself in efforts to make out the truth about them ; hence,
on this showing, the visible perturbations of Uranus would scarcely
_have seemed to Plato very extraordinary or worth any very deep
investigation by ‘heaps of algebra’ or otherwise. Besides, the
discovery of Uranus’s perturbations could hardly have been made
without observation, and observation is excluded by the words ‘ we
shall let the heavens alone’. The fact is that, at the time when
our passage was written, Plato’s ‘ problems’ were ὦ griori problems
which, when solved, would explain visible phenomena ; Adams!
began at the other end, with observations of the phenomena, and ©
then, when these were ascertained, sought for their explanation.
It may be that, when Plato is banning sense-perception from the
science of astronomy in this uncompromising manner, he is con-
sciously exaggerating ; it would not be surprising if his enthusiasm
and the strength of his imagination led him to press his point
unduly. In any case, his attitude seems to have changed con-
siderably by the time when he wrote the 7zmaeus and the Laws,
both as regards the use made of sense-perception and the relation.
of astronomy to the visible heaven. In the Republic sense-percep-
tion is only regarded as useful up to the point at which, owing to
its presentations contradicting one another, it stimulates the intellect.
In the 77maeus the senses, e.g. sight, fulfila much more important
1 Bosanquet, Companion to Plato’s Republic, 1895, pp. 292-3.
3 De Morgan, Budget of Paradoxes, p. 53.
arene
a
ὶ
140 PLATO PARTI
réle. ‘Sight, according to my judgement, has been the cause of
the greatest blessing to us, inasmuch as of our present discourse ~
concerning the universe not one word would have been uttered had —
we never seen the stars and the sun and the heavens. But now day
and night, being seen of us, and months and revolutions of years
have made number, and they gave us the notion of time and
the power of searching into the nature of the All; whence we have
derived philosophy, than which no greater good has come nor shall
come hereafter as the gift of the gods to mortal man. This
I declare to be the chiefest blessing due to the eyes.’? In the
Laws Plato makes the Athenian \thenian stranger say that_it is impious
to use the term ‘planets’ oft of the gods s_in heaven as if they and_the
su: on never t kept to one_uniform course, but wandered
hither and thither; the’case is absolutely the reverse of this, ‘ for
each of these bodies follows one and the same path, not many paths —
but one only, which is a circle, although it appears to be borne in
‘many paths.’ Here then we no longer have the view that the
visible heavenly bodies should be neglected as being subject to
perturbations which it would be useless to attempt to fathom,
and that true astronomy is only concerned with the true heavenly
bodies of which they are imperfect copies; but we are told that
the paths of the visible sun, moon, and planets are perfectly uniform, —
the only difficulty being to grasp the fact. Bosanquet observes,
on the passage in the Republic contrasting the visible and the true
heavens, that ‘ Plato’s point is that there are no doubt true laws by
which the periods, orbits, accelerations and retardations of the solids
in motion can be explained, and that it is the function of astronomy
to ascertain them’.? On the later view stated in the Laws this
would be true with ‘the visible rey bodies’ substituted for
‘solids in motion’.
We are told on the authority of Sosigenes,t who had it from
Eudemus, that Plato set it as a problem to all earnest students
to find ‘what are the uniform and ordered movements by the
assumption of which the apparent movements of the planets can
be accounted for’. The same passage says that Eudoxus was the
1 Timaeus 47 A,B. . Laws vii. 822 A.
* Bosanquet, Companion, Ῥ. 291.
* Simplicius on De cae/o ii, 12 (292 Ὁ 10), p. 488. 20-4, Heib.
ν
CH. XV PLATO 141
first to formulate hypotheses with this object ; Heraclides of Pontus
followed with an entirely new hypothesis. Both were pupils of
Plato, and it is a fair inference that the stimulus of the Master’s
teaching was a factor contributing to these great advances, although
it is probable that Eudoxus attacked the problem on his own
| initiative.
_ When we come to extract from the different dialogues the details
of Plato’s astronomical system, we find, as already indicated, that,
_ if allowance is made for the differences in the literary form in which
_ they are presented, and for the greater or less admixture of myth,
“romance, and poetry, the successive presentations of the system
at different periods of Plato’s life merely show different stages of
ment; the system remains throughout fundamentally the
. "same. Some of the passages have nothing mythical about them
at all; e.g. the passage in the Laws, which is intended to combat
- prevailing errors, gives a plain statement of the view which Plato
“thought the most correct. In the passages in which myth has
a greater or less share, that which constitutes the most serious part
is precisely that which relates to astronomy ; and that which proves
that the astronomical part is serious is the fact that, in different
forms, and with more or fewer details in different passages, we have
only one and the same main hypothesis; the variations are on
points which are merely accessory.! Nor was the system revolu-
tionary as compared with previous theories; on the contrary, Plato
evidently selected what appeared to him to be the best of the
astronomical theories current in his time, and only made corrections
which his inexorable logic and his scientific habit of mind could not
but show to be necessary ; and the theory which commended itself
to him the most was that of Pythagoras and the early Pythagoreans-
—the system in which the earth was at rest in the centre of the
universe—as distinct from that of the later Pythagorean school, with
whom the earth became a planet revolving like the others about the
central fire.
Plato's system is set out in its most complete form in the
Timaeus, and on this ground Martin, in his last published memoir
on the subject, began with the exposition in the 7imaeus and then
* Cf. Martin in Mémoires del’ Acad. des Inscriptions et Belles-Lettres, xxx,
1881, pp. 6-13.
142 PLATO - PARTI
added, for the purpose of comparison, the substance of the astrono-
mical passages in the other dialogues. This plan would perhaps —
. enable a certain amount of repetition to be avoided ; but I think
that the development of the system is followed better if the usual —
plan is adopted and the dialogues taken in chronological order.
We begin therefore with the Phaedrus, perhaps the earliest of all
the dialogues. The astronomy in the Phaedrus consists only in the
astronomical setting of the myth about souls soaring in the heaven
_ and then again falling to earth. Soaring in the heaven, they with
difficulty keep up for a time with the chariots of the gods in their
course round the heavens.
‘Zeus, the great captain in heaven, mounted on his winged
chariot, goes first and disposes and oversees all things. Him follows
the army of Gods and Daemons ordered in eleven divisions ; for
Hestia alone abides in the House of God, while, among the other
gods, those who are of the number of the twelve and are appointed
to command lead the divisions to which they were severally
appointed.
Many glorious sights are there of the courses in the heaven
traversed by the race of blessed gods, as each goes about his own
business ; and whosoever wills, and is able, follows, for envy has no
place among the Heavenly Choir...
The chariots of the gods move evenly and, being always obedient
to the hand of the charioteer, travel easily ; the others travel with —
great difficulty ...
The Souls which are called immortal, when they are come to the
summit of the Heaven, go outside and stand on the roof and, as
they stand, they are carried round by its revolution and behold
the things which are outside the Heaven.’!
Here, then, the army of Heaven is divided into twelve divisions.
One is commanded by Zeus, the supreme God, who also commands-
in-chief all the other divisions as well; subject to this, each division
has its own commander. Zeus is here the sphere of the fixed stars,
which revolves daily from east to west and carries round with it
the other divisions except one, Hestia, which abides unmoved in the
middle. Hestia, the Hearth in God’s House, stays at home to
keep house; the other divisions follow the march of Zeus but
perform separate evolutions under the command of their several
leaders. Hestia is here undoubtedly the earth, unmoved in the
1 Plato, Phaedrus 246 E-247 C.
CH. XV PLATO 143
_ centre of the world,’ and is not the central fire of the Pythagoreans.
_ The gods in command of the ten other divisions are, in the first
' place, the seven planets, i.e. the sun and moon and the five planets,
_ and then between them and the earth come the three others which
are the aether, the air, and the moist or water.2 The sun, moon,
_ and planets are all carried round in the general revolution of the
_whole heaven from east to west, but have independent duties and
commands of their own, i.e. separate movements which (as later
_ dialogues will tell us) are movements in the opposite sense, i.e. from
_ west to east.
In the Phaedo Plato puts into the mouth of Socrates his views as
' to the shape of the earth, its position and its equilibrium in the
_ middle of the universe. The first passage on the subject is that in
_ which he complains of the inadequate use by Anaxagoras of his
_ Nous in explaining phenomena.
᾿ς *When once I heard some one reading from a book, as he said, of
_ Anaxagoras, in which the author asserts that it is Mind which dis-
“poses and causes all things, I was pleased with this cause, as it seemed
_to me right in a certain way that Mind should be the cause of all
things, and I thought that, if this is so, and Mind disposes everything,
it must place each thing as is best. ... With these considerations in
view I was glad to think that I had found a guide entirely to my
mind in this matter of the cause of existing things, I mean Anaxa-
goras, and that he would first tell me whether the earth is flat or
round, and, when he had told me this, would add to it an explana-
tion of the cause and the necessity for it, which would be the Better,
that is to say, that it is better that the earth should be as it is; and
further, if he should assert that it is in the centre, that he would
add, as an explanation, that it is better that it should be in the
centre. . . . Similarly I was prepared to be told in like manner,
with regard to the sun, the moon, and the other stars, their relative
speeds, their turnings or changes, and their other conditions, in what
way it is best for each of them to exist, to act, and to be acted upon
so far as they are acted upon. For I should never have supposed
that, when once he had said that these things were ordered by Mind,
he would have assigned to them in addition any cause except the
fact that it is best that they should be as they are. . . . From what
1 Cf. Theon of Smyrna, p. 200. 7; Plutarch, De primo frigido, c. 21, p. 954 F;
Proclus, δὲ Timacum, D. 281 E; ” Chalcidius, Timaeus, c. 122, p. ee and
c. 178, pp. 227-8.
* Chalcidius, loc. cit.; cf. Proclus, Jz remp. vol. ii, p. 130, 6-9, Kroll.
144 PLATO PARTI
a height of hope then was I hurled down when I went on with my
reading and saw a man that made no use of Mind for ordering things,
but assigned as their cause airs, aethers, waters, and any number of
other absurdities.’ [Then follows the sentence stating that it is as if
one were to say that Socrates did everything he did by Mind and
then gave as the cause of his sitting there the fact that his body was
composed of bones and sinews, the former having joints, and the
sinews serving to bend and stretch out the limbs consisting of
the bones with their covering sinews and flesh and skin, and so on.
This inability to distinguish between what is the cause of that —
which is and the indispensable conditions without which the cause
cannot be a cause suggests that most people are fumbling in the
dark.] ‘Thus it is that one makes the earth remain stationary
under the heaven by making it the middle of a vortex, another sets —
the air as a support to the earth, which is like a flat kneading- —
trough.’ ἢ
The last sentence alludes to some of the familiar early views as to
the form of the earth. Only Parmenides and the Pythagoreans
thought it to be spherical, the Ionians and others supposed it to be
flat, though differing as to details ; the theory that it is the motion
of a vortex with the earth in the middle that keeps it stationary is
that of Empedocles, while the idea that it is a disc, or like a flat
kneading-trough, supported by air, is of course that which Aristotle
attributes to Anaximenes, Anaxagoras, and Democritus.”
Plato’s own view is stated later in the dialogue.
‘There are many and wondrous regions in the earth, and it is
neither in its nature nor in its size what it is supposed to be by those ~
whom we commonly hear speak about it; of this I have been con-
vinced, I will not say by whom. ... My persuasion as to the form of
the earth and the regions within it I need not hesitate to tell you...
I am convinced then, said he, that, in the first place, if the earth,
being a sphere, is in the middle of the heaven, it has no need either
of air or of any other such force to keep it from falling, but that the
uniformity of the substance of the heaven in all its parts and
the equilibrium of the earth itself suffice to hold it; for a thing in~
equilibrium in the middle of any uniform substance will not have —
cause to incline more or less in any direction, but will remain as
it is, without such inclination. In the first place I am persuaded
of this.’ ὃ
1 Phaedo 97 B-99 B. 2. Aristotle, De cae/o ii. 13, 294 Ὁ 13.
8 Phaedo 108 C-Icg A.
|
--
ΓΗ. ΧΥ PLATO 145
_ When Socrates says he has been convinced by some one of the
- fact that the earth is different from what it was usually supposed to
be, he is considered by some to be referring to Anaximander who
_ drew the first map of the inhabited earth. But surely Anaxi-
_ mander’s views, no doubt with improvements, would be represented
'in those of his successors, the geographers of the time, whom
‘Socrates considers to be wrong (we are told, for instance, that
Democritus, who, like Anaximander, thought the earth flat, com-
_ piled a geographical and nautical survey of the earth’). ‘Some one :
_may possibly be no one in particular, in accordance with Plato’s
habit of ‘ giving an air of antiquity to his fables by referring them to
_ some supposititious author ’.* On the other hand, the explanation of
the reason why the spherical earth remains in equilibrium in the
“centre of the universe, namely that there is nothing to make it move
“one way rather than another, is sufficiently like Anaximander’s
explanation of the same thing.’
Socrates proceeds :
_ * Moreover, I am convinced that the earth is very great, and that
we who live from the river Phasis as far as the Pillars of Heracles
inhabit a small part of it ; like to ants or frogs round a pool, so we
‘dwell round the sea; while there are many other men dwelling
elsewhere in many regions of the same kind, For everywhere on
the earth’s surface there are many hollows of all kinds both as
regards shapes and sizes, into which water, clouds, and air flow and
are gathered together ; but the earth itself abides pure in the purity
of the heaven, in which are the stars, the heaven which the most
part of those who use to speak of these things call aether, and it is
the sediment of the aether which, in the forms we mentioned, is
| always flowing and being gathered together in the hollow places
) of the earth. We then, dwelling in the hollow parts of it, are not.
| aware of the fact but imagine that we dwell above on its surface;
| this is just as if any one dwelling down at the bottom of the sea
were to imagine that he dwelt on its surface and, beholding the sun
and the other heavenly bodies through the water, were to suppose
the sea to be the heaven, for the reason that, through being sluggish
and weak, he had never yet risen to the top of the sea nor been able,
by putting forth his head and coming up out of the sea into the
place where we live, to see how much purer and more beautiful it is
al
Vas
ant
ἢ
1 Agathemerus, i. 1 (Vors. 15, p. 393. 6, 7).
3 Archer-Hind, Zhe Phaedo of Pilato, p. 161 note.
3 See pp. 24, 25, above : cf. Aristotle, De cae/o ii. 13, 295 Ὁ 11.
1410 L
146 PLATO PARTI |
than his abode, neither had heard this from another who had seen
it. We are in the same case; for, though dwelling in a hollow of —
the earth, we think we dwell upon its surface, and we call the air
heaven as though this were the heaven and through this the stars —
moved, whereas in fact we are through weakness and sluggishness —
unable to pass through and reach the limit of the air; for, if any
one could reach the top of it or could get wings and fly up, then, —
just as fishes here, when they come up out of the sea, espy the
things here, so he, having come up, would likewise descry the things
there, and if his strength could endure the sight would know that
there is the true heaven, the true light, and the true earth. For
here the earth, with its stones and the whole place where we are, ἴδ᾽
corrupted and eaten away, as things in the sea are eaten away by -
the salt, insomuch that there grows in the sea nothing of moment
nor anything perfect, so to speak, but there are hollow rocks, sand,
clay without end, and sloughs of mire wherever there is also earth,
things not worthy at all to be compared to the beautiful objects
within our view ; but the things beyond would appear to surpass
even more the things here.’!.. . :
Then begins the myth of the things which are upon the real
earth and under the heaven.
‘First it is said that, if one saw it from above, the earth is like
unto a ball made with twelve stripes of different colours, each stripe
having its own colour... .’
We need not pursue the picture of the idealized earth with its
varied hues, its precious stones, its race of men excelling us in sight,
hearing, and intelligence in the same proportion as air excels water,
and aether excels air, in purity, and so on. :
Reading the story of the hollows in the earth, we recall the idea
of Archelaus, which he perhaps learnt from Anaxagoras, that the
earth was hollowed out in the middle but higher at the edges. This
shape would correspond to the flat kneading-trough mentioned by
Plato as the form given by some to the earth? Plato, realizing
that certain inhabited regions such as that from the river Phasis
(descending from the Caucasus into the Black Sea) to the Pillars of
Hercules, being partly bounded by mountains, did appear to be
hollows, had to reconcile this fact with his earnest conviction of the
earth’s sphericity. Archelaus regarded the whole earth as one such
1 Phaedo 109 A-110 A. 3. Ibid. 99 8.
CH. XV PLATO 147
hollow; to which Plato replies that the inhabited earth may bea
hollow, but it is not the whole earth. The earth itself is very large
indeed, so that the apparent hollow formed by the portion in which
we live is quite a small portion of the whole. There are any number
of other hollows of all sorts and sizes; these hollows are separated
by the ridges between them, and it is only the tops of these ridges
that are on the real surface of the spherical earth. Consequently
there is nothing in the existence of the hollows that is inconsistent
_ with the earth being spherical; they are mere indentations. The
__ impossibility of our climbing up the sides to the top of the bounding
ridges, or taking wings and flying out of the hollows, and so reach-
_ ing the real surface of the earth and obtaining a view of the real
heavens, is of course poetic fancy and has nothing ,to do with
_ astronomy.
_ The extreme estimate of the size of the earth made by Plato in
_ the Phaedo seems to be peculiar to him. For the sake of contrast,
_ Aristotle’s remarks on the same subject may be referred to! Aris-
_ totle says that observations of the stars show not only that the
| earth is spherical, but that it is ‘not great’. For quite a small
_ change of position from north to south or vice versa involves a
change of the circle of the horizon. Thus some stars are seen in
_ Egypt and Cyprus which are not seen in the northern regions, and
some stars which in the northern regions are always above the
horizon are, in Egypt, seen to rise and set. Such differences for so
small a change in the position of an observer would not be possible
unless the earth’s sphere were of quite moderate size. Aristotle |
adds that the mathematicians of his day who tried to calculate the
circumference of the earth made it approach 400,600 stades. This
estimate had, according to Archimedes,” been reduced in his time
to 300,000 stades, and Eratosthenes made the circumference to be
252,000 stades on the basis of a definite measurement of the arc
separating Syene and Alexandria on the same meridian, compared
with the known distance between those places.
On the negligibility of the height of the highest mountain in
comparison with the diameter of the earth, Theon of Smyrna ὃ has
1 Aristotle, De caelo ii. 14, 297 b 30-298 a 20.
* Archimedes, Sand-reckoner (vol. ii, ed. Heib., p. 246.15; ed. Heath, p. 222).
3 Theon of Smyrna, pp. 124-6, Hiller.
L2
148 PLATO PART I
some remarks based on the estimates of 252,000 stades for the
circumference and of 10 stades (a low estimate, it is true) for
the height of the highest mountain above the general level of the
plains.
Coming now to the Republic, Book X, we get a glimpse of a
more complete system, though again the astronomy is blended with
myth. The story is that of Er, the son of Armenius, who, after
being killed in battle, came to life twelve days afterwards and
recounted what he had seen. He first came with other souls to
a mysterious place where there were two pairs of mouths, one pair
leading up into heaven, the other two down into the earth ; between
them sat judges who directed the righteous to take the road to the
right hand leading up into the heaven and sent those who had
wrought evil down the left-hand road into the earth; at the same
time other souls were returning by the other road out of the earth,
and others again by the other road coming down from the heaven :
the two returning streams met, the former travel-stained after
a thousand years’ journeying under the earth, the latter returning
pure from heaven, and they foregathered in the meadow where
they related their several experiences.
‘Now when seven days had passed since the spirits arrived in the —
meadow, they were compelled to arise on the eighth day and
journey thence; and on the fourth day they arrived at a point from
which they saw extended from above through the whole heaven and
earth a straight light, like a pillar, most like to the rainbow, but
brighter and purer. This light they reached when they had gone
forward a day’s journey; and there, at the middle of the light, they
saw, extended from heaven, the extremities of the chains thereof ;
for this light it is which binds the heaven together, holding together
the whole revolving firmament as the undergirths hold together —
triremes ; and from the extremities they saw extended the Spindle
of Necessity by which all the revolutions are kept up. The shaft
and hook thereof are made of adamant, and the whorl is partly
of adamant and partly of other substances.
Now the whorl is after this fashion. Its shape is like that we use ;
but from what he said we must conceive of it as if we had one great
whorl, hollow and scooped out through and through, into which was
inserted another whorl of the same kind but smaller, nicely fitting
it, like those boxes which fit into one another; and into this again
we must suppose a third whorl fitted, into this a fourth, and after
that four more. For the whorls are altogether eight in number, set
CH. XV PLATO 149
one within another, showing their rims above as circles and forming
about the shaft a continuous surface as of one whorl; while the
shaft is driven right through the middle of the eighth whorl.
The first and outermost whorl has the circle of its rim the
broadest, that of the sixth is second in breadth, that of the fourth
is third, that of the eighth is fourth, that of the seventh is fifth,
_ that of the fifth is sixth, that of the third is seventh, and that of the
_ second is eighth. And the circle of the greatest is of many colours,
_ that of the seventh is brightest, that of the eighth has its colour
_ from the seventh which shines upon it, that of the second and fifth
are like each other and yellower than those aforesaid, the third
_ is the whitest in colour, the fourth is pale red, and the sixth is the
᾿ second in whiteness.
_ The Spindle turns round as a whole with one motion, and within
_ the whole as it revolves the seven inner circles revolve slowly in the
_ Opposite sense to the whole, and of these the eighth goes the most
_ swiftly, second in speed and all together go the seventh and sixth
and fifth, third in the speed of its counter-revolution the fourth
_ appears to move, fourth in speed comes the third, and fifth the
_ second. And the whole Spindle turns in the lap of Necessity.
Upon each of its circles above stands a Siren, carried round with
_ it and uttering one single sound, one single note, and out of all the
notes, eight in number, is formed one harmony.
And again, round about, sit three others at equal distances apart,
-each on a throne, the daughters of Necessity, the Fates, clothed
in white raiment and with garlands on their heads, Lachesis,
Clotho, and Atropos, and they chant to the harmony of the Sirens,
Lachesis the things that have been, Clotho the things that are, and
Atropos the things that shall be.
And Clotho at intervals with her right hand takes hold of the
outer revolving whorl of the Spindle and helps to turn it; Atropos
with her left hand does the same to the inner whorls; Lachesis
with both hands takes hold of the outer and inner alternately
(i.e. of the outer with her right hand and of the inner with
her left).’?
On the precise interpretation of the details of this description
there has been a great deal of discussion and difference of opinion.?
Some of the details are hardly astronomical, and this is not the
place for more than a short statement of the principal points at
issue.
1 Republic x. 616 B-617 Ὁ.
.? Very full information will be found in Adam’s edition of the Republic; see
especially the notes in vol, ii, pp. 441-53, and Appendix VI to Book X,
ῬΡ. 470-9.
150 PLATO PART I
First, what is the form and position of the ‘straight light, like
a pillar’, and at what point is ‘the middle’ of the light where the
souls saw ‘the extremities of the chains’ binding the heavens
together? As early as Proclus’s time one supposition was that
the light was the Milky Way.’ Proclus rejected this view, which
in modern times is represented by Boeckh? and Martin. Boeckh
supposes the souls to be beyond the north pole, outside the circle —
of the Milky Way which, if seen from the outside edgeways, would
look straight ; the middle of the light is for him the north pole,
from which stretch the chains of heaven, ove of which is the —
light. Martin makes the souls see the Milky Way as a straight
column of light from delow ; thence they go quickly up in the day’s
journey to the middle of the light (Martin compares the souls in
Phaedrus 247 B-248B, which get to the outside of the sphere of
the fixed stars); they there see both poles of the sphere, and the ©
curved column is, for them, like a band forming a complete ring —
round the sphere and holding it together; this curved column can |
only be the Milky Way. Martin supports his view by pressing
the comparison of the column to a rainbow, which, he says, must
refer to its form and not to its colours; and for the illusion of
supposing the curved column to be straight he cites the parallel —
of Xenophanes, who thought the stars moved in straight lines
which only appeared to be circles. I agree with Adam’s opinion
that to suppose the column to be curved and only to appear
straight does violence to the language of Plato. Then again, it
would be strange that the souls, one class of which has come back
from a thousand years’ journey in the heaven, and the other from
the same length of journey under the earth, should next be taken
up, all of them, to the top of the heavenly sphere ; there is nothing —
to suggest that, either in the four days elapsing between the time
when they leave the meadow and the time when they first see the
straight column of light, or in the one day following which brings —
them to the middle of the light, they leave the earth at all. The
other alternative is to take the ‘straight light’ to be, in accordance
with the natural meaning of the words, a straight line or straight
* Proclus, Jz remf. vol. ii, p. 194. 19, Kroll. |
2 Boeckh, Kleine Schriften, iii, pp. 266~320. .
8 Martin in M/ém. de l’Acad. des Inscriptions et Belles-Lettres, xxx, 1881,
PP- 94-7.
CH. XV PLATO 151
_ cylindrical column of light passing from pole to pole right through
_ the centre of the universe and of the earth (occupying the centre
of the universe), which column of light symbolizes the axis on
which the sphere of the heaven revolves. Where then is ‘the
middle’ of this column of light which the souls are supposed to
reach one day after they first see the column? Adam thinks
_ it can only be at the centre of the earth, and he seems to base
this view mainly on the fact that, later on, the souls, after passing
' under the throne of Necessity and encamping by the river of
_ Unmindfulness in the plain of Lethe, are said (621 B) to go τ,
_ ‘shooting like stars, to be born again. Here also I cannot but
_ think it strange that all the souls should be brought down to the
_ centre of the earth, seeing that one class of them had just returned
_ from a thousand years’ wandering in the interior of the earth, to
say nothing of the shortness of the time allowed for reaching the
centre of the earth, namely, one day from the time when they first
saw the column of light, while there is nothing in the language
_ describing the five days’ journey to suggest that they did anything
but walk (πορεύεσθαι). Now the place of the judgement-seat which
was between the mouths of the earth and the heaven, and to
which the souls returned after their thousand years in the earth and
heaven respectively, was on the surface of the earth; presumably
therefore the meadow to which they turned aside from that place
was also on the surface of the earth (and not even on the surface
of the ‘True Earth’ of the Phaedo, as Adam supposes); and
Mr. J. A. Stewart' has pointed out that the popular belief as to
the river Lethe made it a river entirely above ground and not one
of the rivers of Tartarus. Hence I am disposed to agree with
Mr. Stewart that the whole journey from the meadow by the
throne of Necessity to the plain of the river Lethe was along the
surface of the earth. Although Adam rightly rejects Boeckh’s
identification of the ‘straight light’ with the Milky Way, he is
induced by the parallel of the ‘undergirths’ (ὑποζώματα) of
triremes to assume, in addition to the straight light forming the
axis of the universe, a circular ring of light passing round it from
pole to pole and joining the straight portion at the poles;? this
1 J. A. Stewart, The Myths of Plato, pp. 154 sqq.
3 Adam, The Republic of Pilato, vol. ii, pp. 445-7, notes.
152 PLATO PARTI
he does because the more proper meaning of ‘undergirths’ appears
to be ropes passed round the vessel outside it and horizontally,
rather than planks passing longitudinally from stem to stern as
Proclus and others supposed.! But there is nothing in the Greek
to suggest the addition of this circle to the straight light; and
the assumption seems, as Mr. Stewart says,? to make too much
of the man-of-war or trireme. Moreover, the ground for assuming
a ring, as well as a straight line, of light vanishes altogether if
the ὑποζώματα are, after all, cables stretched tight, i.e. in straight
lines, inside the ship from stem to stern, as Tannery holds.* It
seems to be enough to regard Plato as saying that the pillar (which
alone is mentioned) holds the universe together in its particular
way as the undergirths do the trireme in their way. I prefer then
to believe that the light is simply a straight column or cylinder
of light, and that the ‘middle of the light’ is the point on the
surface of the earth which is in the centre of the column of light,
i.e. the centre of the circular projection of the cylinder of light on
the earth’s surface. I do not see why the souls, looking from that
point along the cylinder of light in both directions, should not in
this way be supposed to see (illuminated by the column as by a ©
searchlight) the poles of the universe, nor why these should not
be called the extremities of the chains holding the heaven together,
the pillar of light having by a sudden change of imagery become
those chains themselves.
The Spindle of Necessity.
By another sudden change of imagery the chains following the
course of the pillar of light become a spindle which is similarly
extended from the same ‘extremities’ or poles, and the spindle
with its whorls representing the movements of the universe is seen
to turn in the lap of Necessity. The throne of Necessity must on
1 Proclus, 772 remp. vol. ii, p. 200. 25, and scholium, ibid. p. 381. 10.
? Stewart, op. cit., p. 169.
3 Tannery in Revue de Philologie, xix, 1895, p.117: ‘Le Thesaurus constate,
d’ailleurs, que Boeckh a démontré que les ὑποζώματα νεῶν, dont il est assez
souvent fait mention dans les inscriptions, sont des cables, ainsi que du reste
Hesychius [s.v. ζωμεύματα] explique ce mot : σχοινία κατὰ μέσον τὴν ναῦν δεσμευό-
μενα. Ces cables étaient tendus, d’aprés les Origines d’Isidore, entre l’étrave et
’étambot, en tout cas, on ne peut se les figurer tendus autrement que suivant
une ligne droite.’
-_— Te. ΥΥ
CH. XV PLATO Ξ 152
the above view be at the point on the surface of the earth which
is in the middle of the column of light; and on this hypothesis, as
on others, the attempt to translate the details of the poetic imagery
into a self-consistent picture of physical facts is hopeless, for the
simple reason that one thing cannot both be entirely outside
another thing and entirely within it at the same time. Let us
assume with Boeckh that the souls are outside the universe when
they see the apparently straight light; Necessity will then pre-
sumably be outside the universe which in the form of the spindle
and whorls she holds in her lap. It is on this assumption im-
possible to give an intelligible meaning to ‘under the throne of
Necessity’ as an intermediate point on the journey of the souls
from the meadow to the plain of Lethe. The same difficulty
arises if, with Zeller, we suppose Plato to be availing himself of
the external Necessity which, according to Aétius, Pythagoras
regarded as ‘surrounding the world’. Plato’s Necessity is cer-
tainly not outside but in the middle. If, however, Necessity
sits either at the centre of the earth as supposed by Adam, or at
a point on the surface of the earth as supposed by Mr. Stewart,
how can she, being inside the universe, hold the spindle and whorls
forming the universe in her lap? This is no doubt the difficulty
which makes Mr. Stewart infer that Necessity does not hold the
universe itself in her lap, but a model of the universe.”
The whorls.
The real astronomy of the Repudlic is contained in the description
of the whorls and their movements. The first question arising is,
what was the shape of the whorls? They are not spheres because
they have rims (‘lips’, χείλη) one inside the other, which are all
visible and form one continuous flat surface as of one whorl. We
might, on the analogy of Parmenides’ bands, suppose that they
are zones of hollow spheres symmetrical about a great circle, i.e. so
placed that the plane of the great circle is parallel to, and equi-
distant from, the outer circles bounding the zones. Adam supposes
them to be hemispheres, which Plato possibly obtained by cutting
1 Aét. i. 25. 2 (D.G. p. 321).
3 Stewart, op. cit., pp. 152-3, 165.
154 PLATO PARTI
in half the Pythagorean spheres mentioned by Theon of Smyrna.
It is true that there is nothing in the text of Plato requiring them
to be hemispheres, although Proclus regards them as segments of
spheres”; but the supposition that they are hemispheres has the
great advantage that it eliminates all question of the depth of
the whorls measured perpendicularly (downwards, let us say) from
the visible flat surface formed by their rims. Plato says nothing
of the depth of the whorls, but merely gives the rims different
breadths. ‘The moment we suppose the whorls to be zones or rings
we have to consider what depth or thickness (i.e. perpendicular
distance between the two bounding surfaces) must be assigned to
them. The thickness of the rings would presumably be great
enough to hold symmetrically the largest of the heavenly bodies
which the rings carry round with them. Martin* takes the
thickness of the rings to be greater than this; he supposes that
the outer whorl is an equatorial zone of the celestial sphere
included between two equal circular sections ‘which are doubtless
the tropics’. But Martin admits that there is, in the whole passage,
no reference to any obliquity of movements relatively to the
equator, and he can only suppose such obliquity to be Zacitly implied Ὁ
by the thickness of each whorl. I think that this supposition is
unsafe, and that it is better to assume that, at this stage in the
development of his astronomy, or perhaps merely for the purpose
of the imagery of this particular myth, Plato did not recognize
any obliquity, still less any variations of obliquity in the movements
of the planets.6 I prefer therefore to suppose the whorls to be
? Theon of Smyrna, p. 150. 14.
2 Proclus, Ja remp. vol. ii, p. 213. 19-22.
8 The revolving whorls πέριάγουσι τοὺς ἀστέρας (Proclus, Jz vemp. vol. ii,
226, 12).
2 : Martin in Mém. de l’ Académie des Inscriptions et Belles-Lettres, xxx, 1881,
100-1.
᾿ς Yet Berger (Geschichte der wissenschaftlichen Erdkunde der Griechen, 1903,
Ppp. 199-201) still insists on regarding Plato’s ‘ breadths’ as what I have called
depths. According to him a ‘lip’ (χεῖλος) must project (cf. Plato, Critias
115 E); hence he thinks they must project and recede in comparison with one
another. It is difficult, as he sees, to reconcile this with νῶτον συνεχές, ‘a con-
tinuous Zack’ as seen from above, say the pole; he is therefore driven to the
supposition that the words may describe the appearance of the outermost whorl
as seen from a position where it hides all the others, i.e. from a point between
the planes of its bounding circles; but this clearly will not do. The object of
Berger is to make out that Plato wished to distinguish by the ‘ breadths’ of his
rings the inclinations of the movements of the several planets. As I have said
ἘΝ ΡΟ Ψ
_CH. XV PLATO 155
hemispheres, or similar segments of spheres fitting one inside the
other, and having their bases in one plane. The planets, sun, and
moon would perhaps be regarded as fixed in such a position that
their centres would be on the plane surface which is the common
boundary of all the whorls, so that half of each planet would
project above that surface and half of it would be below.
It is not difficult to see what is the astronomical equivalent of
each of the concentric whorls. The outermost (the first) represents
the sphere of the fixed stars; and here we have somewhat the same
difficulty as we saw in the case of Parmenides wreaths or bands.
The fixed stars being spread over the whole sphere, how can that
sphere be represented by a hemisphere, or a segment of a sphere, or
a ring or zone? The answer is presumably that the whorls are
pure mechanism, designed with reference to the necessity of making
the movements of the inner whorls give plane circular orbits to the
seven single heavenly bodies, the sun, the moon, and the five planets.
Mr. Stewart, in accordance with his idea that it is a model which
Necessity holds in her lap, suggests that the model might be an
old-fashioned one with rings instead of spheres, or that, if it were
an up-to-date model, with spheres, it might be one in which only the
half of each sphere was represénted so that the internal ‘works’
might be seen; he compares the passage in the 7zmaeus' where the
speaker says that, without the aid of a model of the heavens, it
would be useless to attempt to describe certain motions.
above, there is nothing in the text to suggest any obliquity in the movements ;
and, if the ‘ breadths’ are defihs, the sizes of the rings as measured by their
inner and outer radii become entirely indeterminate, so that the relative orbital
distances are undistinguished. It 15 quite incredible that Plato should say
nothing about the relative sizes of the orbits while carefully distinguishing their
obliquities relatively to the equator. It is true that Aristotle, M/etaph. A. 8,
1073 Ὁ 17 sq.) and Theon of Smyrna (p. 174. 1-3) admit different obliquities
exhibited by the planetary motions; and Cleomedes (De motu circulari ii. 7,
Ρ. 226. 9-14) gives some estimates of them. These are, however, all obliquities
with reference to the ecliptic, not the equator. Moreover, Cleomedes’ figures
are quite irreconcilable with Plato’s corresponding ‘ breadths’. Cleomedes says
that the obliquity is the greatest in the case of the moon; next comes Venus
which diverges 5° on each side of the zodiac; next Mercury, 4°; next Mars
and Jupiter, 23° each; and last of all Saturn, 1°. Plato places them in
epi order of ‘breadth’ thus: Venus, Mars, Moon, Mercury, Jupiter,
aturn.
1 Timaeus 40D. Cf. Theon of Smyrna, p. 146. 4, where Theon alludes to
the same passage of the 7imaeus, and says that he himself made a model
to represent the system described in the present passage of the Repudlic,
156 PLATO PART I
The second whorl (reckoning from the outside) carries the planet
Saturn, the third Jupiter, the fourth Mars, the fifth Mercury, the
sixth Venus, the seventh the sun, and the eighth the moon. The
earth, as always in Plato, is at rest in the centre of the system.
The outer rim of each whorl clearly represents the path of the
heavenly body which that whorl carries. The breadth of each
whorl, that is, the difference between the radii of its outer and inner
rims respectively (the inner radius of the particular whorl being of
course the outer radius of the next smaller whorl), is the difference
between the distances from the earth of the planet carried by the
particular whorl and of the planet carried by the next smaller
whorl, The rim of the innermost whorl (the eighth) is the orbit of
the moon, the outer rim of the next whorl (the seventh) is the orbit
of the sun, and so on. Proclus! says that there was an earlier
reading of the passage about the breadths of the rims of the
successive whorls which made them dependent on, i.e. presumably
proportional to, the sizes of the successive planets. Professor Cook
Wilson observes that ‘this principle would be a sort of equable
distribution of planetary mass, allowing the greater body more
space. It would come to allowing the same average of linear
dimension of planetary mass to each unit of distance between orbits
throughout the system.’* Adam, however, for reasons which he
gives, decides in favour of our reading of the passage as against the
‘earlier’ reading of Proclus.
As regards the speeds we are told that, while the outermost whorl
(the sphere of the fixed stars) and the whole universe (including
the inner whorls) along with it are carried round in one motion of
rotation in one direction (i.e. from east to west), the seven inner
whorls have slow rotations of their own in addition, the seven
rotations being at different speeds but all in the opposite sense to
the rotation of the whole universe. Hence the quickest rotation is
that of the fixed stars and the whole universe, which takes place
once in about 24 hours; the slower speeds of the rest are speeds
which are not absolute but relative to the sphere of the fixed stars
regarded as stationary, and of these relative speeds the quickest is
that of the moon, the next quickest that of the sun, Venus, and
1 Proclus, /# remp. vol. ii, Ρ, 218,1 5ᾳ. Cf. Theon of Smyrna, p. 143. 14-16,
2 See Adam, Plato’s Repuddic, vol. ii, pp. 475-9.
CH. XV PLATO 157
Mercury, which travel in company with one another, i.e. have the
same angular velocity and take about a year to describe their orbits
respectively ; the next is that of Mars, the next that of Jupiter, and
the last and slowest relative motion is that of Saturn. The speeds
here are all angular speeds because, if the sun, Venus, and Mercury
describe their several orbits in the same time, the sun must have the
least linear velocity of the three, Venus the next greater, and Mercury
the greatest, since the actual length of the orbit of the sun is less
than that of the orbit of Venus, and the length of the orbit of Venus
is again less than that of the orbit of Mercury. To obtain the
absolute angular speeds in the direction of the daily rotation,
i.e. from east to west, we have to deduct from the speed of the
daily rotation the slower relative speeds of the respective planets
in the opposite sense ; the absolute angular speeds are therefore, in
descending order, as follows :
Mercury
Sphere of fixed stars, Saturn, Jupiter, Mars, {venus ᾿ Μοοη.
un
The following table gives the order of orbital distances, or
breadths of rims of whorls, as compared with the order of the
whorls themselves, the order of ve/ative speeds, and the relation of
the colours of the planets respectively :
: Order in
ΟΥ̓ΔῈ tH breadth ofrim Order of partion of
Whort. Planet. as fii Sobiahang. | . —— “ola.
our reading. roclus’s “ὁ speeds.
: reading,
1= Sphere of fixed stars I I _ Spangled.
2= Saturn 8 7 5 Yellower than
sun and moon.
= Jupiter 7 6 4 Whitest.
= Mars 3 5 3 Rather red.
= Mercury 6 8 2 Like Saturn in
colour.
= Venus 2 4 2 Second in white-
ness.
= Sun 5 2 2 Brightest.
= Moon 4 3 I Light borrowed
from sun,
Έ
As, according to either reading, Plato only gives the order of the
_ Successive rims as regards breadth, not the ratios of their breadths,
ts
we cannot gather from this passage what was his view as to the
158 PLATO PART I
ratios of the distances of the respective heavenly bodies from
the earth. Nor can his estimate of the ratios be deduced from the
mere allusion to the harmony produced by the eight notes chanted
by the Sirens perched upon the respective whorls; as to this har-
mony see pp. 105-15 above.
As regards the Sirens, Theon of Smyrna tells us that some sup-
posed them to be the planets themselves ; some, however, regarded
them as representing the several notes which were produced by the
motion of the several stars at their different speeds.’ It is clear
that the latter is the right view ; the Sirens are a poetical expression
of the notes.
It will be noticed that Plato has the correct theory with regard
to the moon’s light being derived from the sun, a fact which, as
before stated, he evidently learned from Anaxagoras.
The Zimaeus is one of the latest of Plato’s dialogues and is the
most important of all for our purpose because in it Plato’s astro-
nomical system is most fully developed and given with the fewest
lacunae. I shall continue to follow the plan of quoting passages in
Plato’s own words and adding the explanations which appear
necessary. First, we are told that the universe is one only, eternal,
alive, perfect in all its parts, and in shape a perfect sphere,? that
being the most perfect of all figures.
‘He (the Creator) assigned it that motion which was proper to its
bodily form, that motion of all the seven which most belongs to
reason and intelligence. Wherefore turning it about uniformly, in
the same place, and in itself, he made it to revolve round and
round; but all the other six motions he took away from it and
stablished it without part in their wanderings.’ ὃ
‘And in the midst of it he put soul and spread it throughout the
whole, and also wrapped the body with the same soul round about
on the outside; and he made it a revolving sphere, a universe one
and alone.’ 5
Here then we have all plurality of worlds denied and the one
universe made to revolve uniformly, carrying with it in its revolution
all that is within it, as in the Republic ; the uniform revolution is of
course the daily rotation. Turning ‘in itself’ means about its own
axis and therefore, so to speak, coincidently with itself, so that one
1 Theon of Smyrna, pp. 146. 8-147. 6. 5 Timaeus 32 C-33 B.
3 Ibid. 34 A. * Jimaeus 348.
OO ee, ee ὦ
“ ε 5 ᾿
CH. XV PLATO 159
position does not overlap another, but in all positions the sphere
occupies exactly the same space and place. The other ‘six motions’
from which it is entirely free are the three pairs of translatory
motions, forward and backward, right and left, up and down.
Next Plato explains how the Creator made the Soul by first
combining in one mixture Same, Other, and Essence, and then
ordering the mixture according to the intervals of a musical scale,
so that its harmony pervaded the whole substance. This 5 :
considered as having taken the form of a bar or band, soul-stri
as it were, he proceeds to divide. : =
‘Next he cleft the structure so formed lengthwise into two halves
and, laying them across one another, middle upon middle in the
shape of the letter X, he bent them in a circle and joined them,
making them meet themselves and each other at a point opposite
to that of their original contact ; and he comprehended them in that
motion which revolves uniformly and in the same place, and one of
the circles he made exterior and one interior. The exterior move-
_ ment he named the movement of the Same, the interior the
movement of the Other. The revolution of the circle of the Same
' he made to follow the side (of a rectangle) towards the right hand,
that of the circle of the Other he made to follow the diagonal and
towards the left hand, and he gave the mastery to the revolution of
the Same and uniform, for he left that single and undivided ; but
the inner circle he cleft, by six divisions, into seven unequal circles
in the proportion severally of the double and triple intervals, each
being three in number; and he appointed that the circles should
move in opposite senses, three at the same speed, and the other
four differing in speed from the three and among themselves, yet
moving in a due ratio.’}
The two circles in two planes forming an angle and bisecting one
another at the extremities of a diameter common to both circles _
are of course the equator and the zodiac or ecliptic. The equator
is the circle of the Same, the ecliptic that of the Other. In the
accompanying figure, AE BF is the circle of the Same (the equator),
CFDE the circle of the Other (the ecliptic), and they intersect at
the ends of their common diameter EF. GH is the axis of the
universe which is at right angles to the plane of the circle AZ AF.
If we draw chords DX, CZ parallel to the diameter 42 common to
the circles AE BF, AGBH, and join CK, DL, we have a rectangle
1 Timaeus 36 B-D.
τόο PLATO PART }
of which KD is a side and CD is a diagonal. As the universe
revolves round GH, each point on the circumference of the circle
AGBH describes a circle parallel to the circle 4.8} i.e. a
G circle about a diameter parallel
to AB or KD; that is, the revo-
lution ‘follows the side’ KD of
GP Pee 27 _~———\p_~—sthe=rectangile. Similarly the
i gered | revolution of the circle of the
᾿ — ὡὦΣΦ ΝΒ Other about an axis perpen-
; dicular to the plane of the circle
ξε΄. ΣΙ -«-Ξ | CFDE ‘follows the diagonal’
CD of the rectangle.
The circle of the Same or the
equator is the outer, and the circle
H of the Other, the ecliptic, is the
Fig. 4. inner. When Plato says that the
Creator ‘comprehended them’ (i.e. both circles) in the motion of
the Same, and then again later that he gave the supremacy to that
circle, he means that the movement of that circle is common to the
whole heaven and carries with it in its motion the smaller circles,
the subdivisions of the circle of the Other, and everything in the
universe ; this he makes still clearer in a later passage where he
speaks of the motion of the planets in the circle of the Other being
‘controlled’ by the motion of the Same, and the motion of the
Same twisting all their circles into spirals.1 The subjection of all
that is in the universe, including all the independent motions of the
planets, to the one general movement of daily rotation is of course
the same as we saw in the Repudlic; but there all the circles were
in one plane, whereas the bodies moving in the opposite sense to
the daily rotation here move in a different plane, that of the ecliptic,
instead of that of the equator.
I have represented the directions of the motions in the two circles
by arrows in the figure. The motion in the circle AZ AF is in the
direction represented by the order of the letters.
The statement of Plato that the Creator made the circle of the
Same (i.e. the circle of the fixed stars) revolve towards the right
hand and the circle of the Other (comprising the circles of the
1 Timaeus 39 A.
CH. XV PLATO : 161
planets) towards the left hand has given the commentators, from
Proclus downwards, much trouble to explain. It is also in con-
tradiction to the observation in the Laws that motion to the right
is motion towards the eas?,' while the writer of the Epznomis again
represents the independent movement of the sun, moon, and planets
as being to the vigh¢ and not to the left.2 There is of course no
difficulty in the circumstance that Plato has previously said that
the Creator took away from the world-sphere the six motions, up
and down, vight and /eft, forwards and backwards ; for this refers
to movements of translation such as take place zwszde the sphere, not
to the revolution of the sphere itself. The axis of such revolution
being once fixed, the revolution may be in one of two (and only
two) directions;* consequently there is nothing to prevent one of
the two directions being described as 20 the right and the other
as to the left. But why did Plato speak of the revolution from
east to west as being motion to the right? Boeckh has discussed
the question at great length, giving a full account of earlier views
before stating his own.* Martin’s explanation is that Plato is
speaking from the point of view of a spectator looking south, as
he would have to do in northern latitudes in order to see the
apparent revolution of the sun from east to west; that is, the
movement is from /ff to right. Boeckh, however, points out that
the Greeks were accustomed, from the earliest times when diviners
foretold events by watching the flight of birds, to turn their faces
to the north; the east would therefore be on the right hand and
would naturally be regarded as the most auspicious, and therefore
as ‘right’. It is also true that the common view among the Greeks
(we find it later in Aristotle®) would make of the sphere of ἐπε.
universe a sort of world-animal, which would have a right and left
of its own, as it might be a man masked in a sphere put over him;
and no doubt, on such a view, the east would be sure to be
regarded as ‘right’ and the west as ‘left’. Boeckh therefore finds
it difficult to believe that Plato could have represented the east
as /eft. Assuming then that Plato regarded the east as right,
τ Laws vi. 8,760 Ὁ.
: The spheet fan in a ieenaiicas language, only ‘ one ἃ of freedom’
BRN Dac τ λυάψεηο Syeda Pate ge ia μοι
* Aristotle, De cae/o ii. 2, 285 Ὁ 2-3.
1410 M
162 PLATO PARTI
Boeckh thinks Martin’s view untenable, and concludes that the only
possible alternative is to suppose that Plato must have thought,
in the Z7zmaeus, of a movement from the right zo the right again,
i.e. of the whole revolution from east to east instead of the portion
from the east to the west. But the movement, on the assumptions
made, is undoubtedly /eft-wise, and it seems to me that Boeckh’s
explanation is almost as violent as the desperate method of inter-
pretation suggested by Proclus.1. Where Boeckh is in error is,
I think, in supposing that Plato would identify the east in his
world-sphere with the right hand at all; it seems to me that he
could not possibly have done so consistently with the scientific
attitude he adopted in denying the existence of any absolute up
and down, right and left, forward and backward in the spherical
universe. He explains, for example, that ‘up’ and ‘down’ have
only a relative meaning as applied to different parts of the sphere,”
and it is clear that, in the same connexion, he would say the same
of right and left. Now suppose that a particular point on the
equator of the universe is east at a given moment; after about
six hours the same point will be south, after six more wesz, and
so on. The case then is similar to that put by Plato when he says
that a man going round the circumference of a solid body placed.
at the centre of the universe would at some time arrive at the
antipodes of an earlier position and would therefore, on the usual
view of ‘up’ and down, have to call ‘down’ what he had before
described as ‘up’, and vice versa.2 Plato would never, surely,
have made the same mistake in speaking of the universe. On
the contrary, when he spoke of the daily rotation, he properly
ignored all question of a starting-point, whether east or west, right
or left, or of the position of a person setting the sphere in motion,
and confined himself to distinguishing by different names the two
possible directions of motion in order to make it clear that the
circles of the Same and of the Other moved in opposite directions.
The expressions 20 the right and to the left were obviously well
? Proclus (Jz Timaeum 220 £) will have it that ἐπὶ δεξιά does not mean the
same thing as εἰς τὸ δεξιόν, but that, while εἷς τὸ δεξιόν refers to motion im ἃ
straight line, ἐπὶ δεξιά only refers to motion 7” @ circle and means ‘ the place to
which the right moves (anything),’ ἐφ᾽ ἃ τὸ δεξιὸν κινεῖ,
3 Plato, Zimaeus 62 D-63 E.
8 Timaeus 62 E-63 A.
—) μὰ
CH. XV PLATO 163
adapted to express the distinction, and it seems to me that the
reason of Plato’s particular application of them is simply this. He
considered that the circle of the Same must have the superior
motion ; but right is superior to /eft; he therefore described the
revolution of the circle of the Same as being 20 the right, and
the revolution of the circle of the Other as being 20 the /eft, for
this sole reason, without regard to any other considerations, just
as in the Republic he confines himself to saying that Clotho at
intervals, with her right hand, helps to turn the outer whorl of the
spindle, and so on,’ without saying anything about the actual
directions in which the respective whorls revolve. On the other
hand, when he says in the Zaws that revolution from west to east
is to the right and revolution from east to west is to the left, he
is, as Boeckh properly observes, merely using popular language.
The cutting of the circle of the Other into seven concentric
circles (including the original circumference as one of the seven)
produces seven orbits in exactly the same way as the eight whorls
in the Myth of Er give eight orbits, the difference being that the
outermost circle of the Republic, the circle about which the sphere
of the fixed stars moves, is not now in the same plane with the
᾿ other seven, but is the circle of the Same in a different plane.
Plato here says that the seven circles move in opposite directions,
literally ‘in opposite senses to one another’, which, as there are
only two directions, can only mean that a certain number of the
seven revolve in one direction, and the rest in the other ; we shall
return later to this point, which presents great difficulty. The
three which move at the same speed are of course the circles of the
sun, Venus, and Mercury, as in the Republic, the same speed
meaning, as there, not the same linear speed (as they are at
different distances from the earth), but the same angular speed.
The seven circles are said to be ‘in the proportion of the double
and triple intervals, three of each’. The allusion is to the
Pythagorean τετρακτύς represented in the annexed figure, the
numbers on the one side after 1 being successive powers of 2
and those on the other side successive powers of 3. When the con-
centric circles into which the circle of the Other is divided are
said to correspond to these numbers, it is clear that it must be
1 Republic x. 617 C, Ὁ.
M 2
164 PLATO PARTI
the circumferences (or, what is the same thing in other words, the
radii), not the areas, which so correspond ; for, if it were the areas,
the radii would not be commensurable with one another. The
dictum is generally + taken to mean that the radii of the successive
orbits, i.e. the distances between the successive planets and the
earth, are in the ratio of the numbers 1, 2, 3, 4, 8, 9, 27. But
Chalcidius? apparently takes the several numbers to indicate the
successive differences between radii, for he says that, while the
first distance (1) is that between the earth and the moon, the second
(2) is the distance between the moon (not the earth) and the sun;
on this view, the successive radii are 1,1+2 = 3,
1+2+3 =6,&c. Macrobius® says that the Plato-
nists made the distances cumulative by way of
multiplication, the distance of the sun from
ὃ 27 the earth being thus (in terms of the distance of
Fig. 5. the moon from the earth) 1 x 2 or 2, that of Venus
1xX2x3=6, that of Mercury 6x 4= 24, that of
Mars 24x9=216, that of Jupiter 216x8=1,728, and that of
Saturn 1,728x27 = 46,656. (It will be observed that in this
arrangement 9 comes before 8, Macrobius having previously ex-
plained this order by saying that, after 1, we first take the first
even number, 2, then the first odd number, 3, then the second even
number, 4, then the second odd number, 9, then the third even
number, 8, and last of all the third odd number, 27.) But, whatever
the exact meaning, it is obvious that we have here no serious
estimate of the relative distances of the sun, moon, and planets
I
based on empirical data or observations; the statement is a piece
of Plato’s ideal a priorz astronomy, in accordance with his statement
in the Republic, Book VII, that the true astronomer should ‘dis-
pense with the starry heavens’.
Plato goes on to the question of Time and its measurement. As
the ideal after which the world was created is eternal, but no
created thing can be eternal, God devised for the world an image
of abiding eternity ‘moving according to number, even that which
we have named time’.
1 Cf. Zeller, ii‘, p. 779 note.
3 Chalcidius, 7zmaeus, c. 96, p. 167, ed. Wrobel.
8 Macrobius, /” somn, Scip. ii. 3. 14.
CH. XV PLATO 165
‘For, whereas days and nights and months and years were not
before the heaven was created, he then devised their birth along
with the construction of the heaven. Now these are all portions
of time. «1
‘So, then, this was the plan and intent of God for the birth of
time ; the sun, the moon, and the five other stars which are called
planets have been created for defining and preserving the numbers
of time.
‘And when God had made their several bodies, he set them in
the orbits in which the revolution of the Other was moving, in
seven orbits seven stars. The moon he placed in that nearest the
earth, in the second above the earth he placed the sun; next,
the Morning Star and that which is held sacred to Hermes he
placed in those orbits which move in a circle having equal speed
with the sun, but have the contrary tendency to it ; hence it is that
the sun and the star of Hermes and the Morning Star overtake,
and are in like manner overtaken by, one another. And as to the
rest, if we were to set forth the orbits in which he placed them,
and the causes for which he did so, the account, though only by
the way, would lay on us a heavier task than that which is our
chief object in giving it. These things, perhaps, may hereafter,
when we have leisure, find a fitting exposition.’ ?
The crux of this passage is the statement that, while Venus and
Mercury have the same speed as the sun, i.e. have the same angular
speed, describing their orbits in about the same time, ‘they have
the contrary tendency to the sun’; the words are ἐναντίαν δύναμιν,
‘contrary tendency’ or ‘force’. In an earlier passage, as we have
seen, Plato spoke of some of the seven planets moving on the
concentric circles forming part of the circle of the Other as going
‘the opposite way’ (κατὰ τἀναντία) to the others. Now, although
δύναμις need not perhaps here be a ‘principle of movement’ as
Aristotle defines it,* yet if we read the two passages together and
give the most natural sense to the words in both cases, the meaning
certainly seems to be that some of the planets describe their orbits
in the contrary direction to the others, and that those which move,
in the zodiac, the opposite way to the others are Venus and
Mercury ; that is to say, the sun, the moon, Mars, Jupiter, and
Saturn all move in the direction of the motion of the circle of the
Other, i.e. from west to east, while Venus and Mercury move in
the same plane of the zodiac but in the opposite direction, i.e. from
1 Timaeus 37 Ὁ, E. 2 Timaeus 38 C-E.
3 Timaeus 36 Ὁ, p. 159 above. * Aristotle, Meaph. Δ. 12, 1019 a 15.
166 PLATO PARTI
east to west. At the same time we are told that the periods in
which the sun, Venus, and Mercury describe their orbits are the
same. Thus if, say, Venus and the sun are close together at a
particular time, they would according to this theory be nearly
together again at the end of a year; but in the meantime Venus,
moving in a sense contrary to the sun’s motion, i.e. in the direction
of the daily rotation from east to west, would pass through all
possible angles of divergence from the sun and, after gaining a day,
would appear with it again. But, as it is, Venus is never far away
from the sun; and consequently Plato’s statements, thus inter-
preted, are in evident contradiction to the facts, as easily verified
by observation. It is not surprising that commentators have
exhausted their ingenuity to find an interpretation less compromising
to Plato’s reputation as an astronomer. It is true that in the
Republic all the seven planets revolve in one direction ; but Plato
is here referring to a phenomenon which is not mentioned in the
Republic, namely, the fact that Venus and Mercury respectively
on the one hand, and the sun on the other, ‘overtake and are
overtaken by one another’, and the idea of the two planets having
the ‘contrary tendency to the sun’ is clearly put forward for the
precise purpose of explaining this phenomenon. It is accordingly with
reference to the standings-still and the retrogradations of Venus
and Mercury that the commentators try to interpret Plato’s words.
On the first passage (36 D) Proclus gives a number of alternatives,
differing very slightly in substance, some importing the machinery
of epicycles (which, as Proclus says, are foreign to Plato) and others
not, but all designed to make Plato refer to nothing more than the
stationary points and retrogradations; Proclus! on this occasion
rejects them all, observing that the truest explanation is to suppose
that Plato did not mean that there was any opposition of direction
among the seven bodies at all, but only that all the seven, moving
one way, moved in the opposite sense to the general movement of
the daily rotation. This is cutting the knot witha vengeance. On the
second passage (38 D) Proclus has the same kind of discussion,
giving, as an alternative to the importation of epicycles, &c., the
hypothesis that the ‘overtakings’ may be accounted for by the
speeds of the sun, Venus, and Mercury varying relatively to one
1 Proclus, Jz Timaeum 221 D sqq.
es
CH. XV PLATO 167
another at different points of their respective orbits.1 Chalcidius?
has much the same account of the different interpretations, but
fortunately coupled with a precious passage about the view taken
by Heraclides of Pontus of the movements of Venus and Mercury
in relation to the sun: an account which, although it again wrongly
imports epicycles into Heraclides’ theory, as Theon of Smyrna and
others erroneously import them into Plato’s,* enables the true theory
of Heraclides to be disentangled.*
Of modern editors Martin® refuses to accept any of these explana-
tions which give a meaning to the passages other than that which
the words naturally convey, and stoutly maintains that Plato did
actually say that Venus and Mercury describe their orbits the
contrary way to the motion of the sun, and meant what he said,
incomprehensible as this may appear. He quotes in support of his
view the evidence of Cicero in the fragments of his translation of
the Zimaeus. It is true that Cicero fences with the expression ‘ the
contrary tendency’, translating it as ‘vim quandam contrariam’,
where ‘quandam’ has nothing corresponding to it in the Greek, but
merely indicates a certain timidity or hesitation which Cicero felt in
translating δύναμις by vis; Cicero, perhaps, may have had some
idea, such as Proclus had, of a capricious force of some kind causing
the two planets respectively to go faster at one time and slower at
another. But by his translation of the other passage about the
seven smaller circles making up the circle of the Other he shows
that he interpreted Plato as meaning that some of the planets
describing these circles move in the opposite direction to the others:
his words are ‘contrariis inter se cursibus’.
Archer-Hind® maintains that the phrase ‘having the contrary.
tendency to it’ does not mean that Venus and Mercury revolve ‘in
a direction contrary to the sun. He believes that ‘Plato meant the
Sun to share the contrary motion of Venus and Mercury in relation
to the other planets’. ‘It is quite natural,’ he says, ‘seeing that
the sun and the orbits of Venus and Mercury are encircled by the
1 Proclus, 17: Timaeum 259 A-C.
3 Chalcidius, 7imaeus c. 97, pp. 167-8; c. 109, p. 176, Wrobel.
3 Theon of Smyrna, p. 186. 12-24.
* Hultsch, ‘Das astronomische System des Herakleides von Pontos’, in
Jahriuch der classischen Philologie, 1896, pp. 305-16.
5. Martin, Etudes sur le Timée, ii, pp. 66-75.
6 Archer-Hind, 7zmaeus, pp. 124-5 ἢ.
168 PLATO PART I
orbit of the earth, while Plato supposed them all to revolve about
the earth, that he should class them together apart from the four
whose orbits really do encircle that of the earth: his observations
would very readily lead him to attributing to these three a motion
contrary to the rest.’ This seems to be a very large assumption;
and indeed there is no evidence that Plato made any distinction
between the groups of planets which we now call inferior and
superior ; in his system Venus and Mercury were not even inferior
to the sun, but above it. Besides, although Archer-Hind’s view
would satisfy the first passage about some of the seven moving in
the contrary direction to the others, it still does not explain the
second statement that Venus and Mercury have ‘ the contrary ten-
dency to z¢’ (the sun). Accordingly he essays a new explanation.
‘What I believe it’ [the contrary tendency] ‘to be may be under-
stood from the accompanying figure which is copied from part of
a diagram in Arago’s Popular Astronomy.’ It represents the
motion of Venus relatively to the earth during one year as observed
in 1713, and is a sort of epicycloid with a loop. The ‘tendency’,
then, is the ‘tendency on the part of Venus, as seen from the earth,
periodically to retrace her steps’. That is, Archer-Hind’s explana-_
tion is really an explanation of retrogradations by the equivalent of
epicycles, and is therefore no better than the anachronistic explana-
tions by Proclus and others to the same effect.
I do not think that Schiaparellit is any more successful in his
explanations. He suggests that the first passage ‘ seems to allude
to the retrogradations, or perhaps to the opposite positions (with
reference to the sun) in which Mars, "Jupiter, and Saturn on the one
hand, and Mercury and Venus on the other, carry out their stand-
ings-still and their retrogradations’. In the second passage he
translates the words about the ‘contrary tendency’ by ‘receiving
a force contrary to it’? (the sun), and he implies that this force is
really in the sun: ‘it might be interpreted simply as a power, which
the sun seems to have, of making these planets go backward, as if it
attracted them to itself’. This is not less vague than the explana-
tions of Proclus and others; it has the disadvantage also that it is
1 Schiaparelli, 7 Arecursori, p. 16 note.
2 * Ricevendo una forza contraria a lui.’
5 ‘Questo tuttavia si potrebbe interpretare semplicemente di una forza che sem-
bra avere il Sole, di far retrocedere questi pianeti, quasi li attirasse verso di sé.’
CH. XV PLATO : 169
based on a mistranslation of the Greek. The words mean ‘having’
or ‘possessed of (εἰληχότας) the contrary tendency to the sun’, which
clearly shows that the tendency such as it is resides in the planets
themselves.
We pass on to the next passage which is relevant to our subject.
‘But when each of the beings [the planets] which were to join in
creating time had arrived in its proper orbit, and they had been as
animate bodies secured with living bonds and had learnt their
appointed task, then in the motion of the Other, which was oblique
and crossed the motion of the Same and was controlled by it,
one planet described a larger, and another a smaller circle, and
those which described the smaller circle went round it more swiftly
and those which described the larger more slowly ; but because
of the motion of the Same those which went round most swiftly
appeared to be overtaken by those which went round more slowly,
though in reality they overtook them. For the motion of the Same,
which twists all their circles into spirals because they have two sepa-
rate and simultaneous motions in opposite senses, is the swiftest of all,
and displays closest to itself that which departs most slowly from 11.1
The spirals are easily understood by reference to the figure on
p. 160. Suppose a planet to be at a certain moment at the point
fF. It is carried by the motion of the Same about the axis GH,
round the circle FAEB. At the same time it has its own motion
along the circle FDEC. After 24 hours accordingly it is not at the
point F on the latter circle, but at a point some way from F on
the arc FD. Similarly after the next 24 hours, it is at a point
on FD further from 7; and soon. Hence its complete motion is
not in a circle on the sphere about GH as diameter but in a spiral
described on it. After the planet has reached the point on the
zodiac (as D) furthest from the equator it begins to approach the
equator again, then crosses it, and then gets further away from it on
the other side, until it reaches the point on the zodiac furthest from
the equator on that side (as C). Consequently the spiral is included
between the two small circles of the sphere which have KD, CL as
diameters.
The remark about the overtakings of one planet by another is
also easily explained.. Let us consider the matter with reference to
_ two of the seven planets in the wider sense, namely the sun and the
_ moon. Plato says that the moon, which has the smaller orbit,
1 Timaeus 38 E-39 B.
170 PLATO PARTI
moves the faster, that is, the independent movement of the moon
in its orbit is faster than the independent movement of the sun in
its orbit, by which he means that the moon describes its orbit in
the shorter period. Thus the sun describes its orbit in about 365%
days ; the moon returns to the same position relatively to the fixed
stars in 274 days, a sidereal month, and relatively to the sun in 29%
days, a synodic month. Now, if we consider the whole apparent
motion of the sun and moon, i.e. including the daily rotation as
well as the independent motion, the moon appears to go round the
earth more slowly than the sun. For at new moon it sets soon after
the sun. The next day it sets later, the day after later still, and so
on; it appears therefore to be gradually left behind by the sun, or
the sun appears to gain on it daily, that is, the moon ‘appears to be
overtaken’ by the sun. On the other hand, if we consider only the
relative motion of the sun and moon, i.e. if we leave out of account
the daily rotation as common to both, the moon, describing its
orbit more quickly than the sun describes its orbit, gains on. the
sun, that is, ‘in reality it overtakes’ the sun, as Plato says.
‘And that there might be some clear measure of the relative.
slowness and swiftness with which they moved in their eight revo-
lutions, God. kindled a light in the second orbit from the earth,
which we now have named the Sun, in order that it might shine
most brightly through all the heaven, and that living things, so
many as was meet, should possess number, learning it from the
revolution of the Same and uniform. Night then and day have
been created in this manner and for these reasons, making the
period of the one and most intelligent revolution; a month has
passed when the moon, after completing her own orbit, overtakes
the sun, and a year when the sun has completed its own circle.
‘But the courses of the others men have not grasped, save a few
out of many ; and they neither give them names nor investigate the
measurement of them one against another by means of numbers, in
fact they can scarcely be said to know that time is represented by
the wanderings of these, which are incalculable in multitude and
marvellously intricate.
‘None the less, however, can we observe that the perfect number
of time fulfils the perfect year at the moment when the relative
speeds of all the eight revolutions accomplish their course together
and reach their starting-point, being measured by the circle of the
Same and uniformly moving.’?
1 Timaeus 39 Β- Ὁ.
χὰ «dsp ὗν
CH. XV PLATO ; 171
The ‘month’ in the above passage is the σγησας. month, the
period in which the moon returns to the same position relatively
to the sun. ‘The courses of the others’ are the periods of the
planets, which are not called by separate names like ‘ year’ and
‘month’, and which, Plato says, only a very few astronomers had
attempted to measure one against another. The description of the
‘wanderings’ of the planets as ‘incalculable in multitude and mar-
vellously intricate’ is an admission in sharp contrast to the assump-
tion of the spirals regularly described on spheres of which the inde-
pendent orbits are great circles, and still more so to the assertion in
the Laws that it is wrong and even impious to speak of the planets
as ‘wandering’ at all, since ‘each of them traverses the same path,
not many paths, but always one circular path’. For the moment
Plato condescends to use the language of apparent astronomy, the
astronomy of observation; and this may remind us that Plato’s
astronomy, even in its latest form as expounded in the 77mmaeus and
the Laws, is consciously and intentionally ideal, in accordance with
his conception of the true astronomy which ‘ lets the heavens alone ’.
What was the length of Plato’s Great Year? Adam? in his edition
of the Republic, makes it to be 36,000 years, a figure which he bases
on his interpretation of the famous passage in the Republic, Book VIII,
about the Platonic ‘ perfect number’, which is there called the ‘ period
for a divine creature’, just as, in the passage of the 7zmaeus, ‘the
perfect number of time fulfils the perfect year’. The perfect num-
ber of the Republic being, according to both Adam and Hultsch,
the square of 3,600, or 12,960,000, Adam connects the perfect year
with the two periods of the myth in the Po/zticus,? during the first
of which God accompanies and helps to wheel the revolving world,
while during the second he lets it go. Each of these periods
contains ‘many myriads of revolutions’, the word for revolutions
being περιόδων, the same word as is used in the Republic for the
‘period for a divine creature’. Now in the Podliticus περίοδοι,
* periods’ or ‘ revolutions’, refers to the revolutions of the world on
its own axis. Hence Adam infers that the perfect or great year
consists of 12,960,000 daily rotations or 12,960,000 days. Next, he
cites the Laws, in which Plato divides the year into 360 days (which
--1 Laws vii. 22, 821 B—-D, 822 A.
3 Adam’s Republic, vol. ii, pp. 204 sqq. notes, 295-305. * Politicus 2704.
172 PLATO PART I
is, it is true, an ideal division).!. Dividing then 12,960,000 by 360,
we obtain 36,000 years. Adam seeks confirmation of this in the
fact that we find the period of 36,000 years sometimes actually
called the ‘great Platonic year’ in early astronomical treatises.
Thus Sacro-Bosco in his Sphaera says that ‘the ninth circle in a
hundred and a few years, according to Ptolemy, completes one
degree of its own motion and makes a complete revolution in
36,c00 years (which time is commonly called a great year or
Platonic year)’. Since a text-book of Ptolemaic astronomy makes
this statement, Adam infers that Ptolemy or some of his prede-
cessors had understood the Platonic Number, and that we can
perhaps trace the knowledge of the Number as far back as
Hipparchus. For Hipparchus discovered the precession of the
equinoxes and is supposed to have given 36,000 years as the time
in which the equinoctial points make a complete revolution ; and
Adam finds it difficult to believe that Hipparchus was uninfluenced
by Plato’s Number. There is, however, the strongest reason for
doubting this, because Hipparchus’s discovery of precession was
based on something much more scientific than a recollection of the
Platonic Number, namely actual recorded observations. It is true
that Ptolemy estimated the movement of precession at 22° in
265 years, i.e. about 1° in 100 years, or 36” a year,? and it is
commonly supposed that this is precisely Hipparchus’s estimate *.
But it is probable that Hipparchus’s estimate was much more
correct. The evidence of Ptolemy* shows that Hipparchus
found the bright star Spica to be, at the time of his observation
of it, 6° distant from the autumnal equinoctial point, whereas
he deduced from the- observations recorded by Timocharis that
Timocharis had made the distance 8°. Consequently the motion
had amounted to 2° in the period between 283 (or 295) and
129 B.C., a period of 154 (or 166) years; this gives about 46-8”
(or 434”) a year, which is much nearer than 36” to the true
value of 50-3757”. It is true that, in a quotation which Ptolemy
1 Laws vi. 756 B-C, 758 B.
* Ptolemy, Syv¢axts vii. 2, vol.ii, p. 15.9-17 Heib. Yet Ptolemy, in another
place (vii. 3, pp. 28-30), infers from two observations made by Timocharis in
295 and 283 B.C. respectively that the movement amounted to 10’ in about 12
years, which gives 50” a year.
3 See Tannery, Recherches sur l'histoire de l’astronomie ancienne, pp. 265 564.
* Ptolemy, vii. 2, vol. ii, pp. 12, 13, Heib,
CH. XV PLATO ; 173
makes from Hipparchus’s treatise on the Length of the Year,!
1° in 100 years is the rate mentioned; but Tannery points out that
this is not conclusive, because Hipparchus is in the particular
passage only giving a lower limit, for he says ‘az /east one-hun-
dredth of a degree’ and ‘in 300 years the movement would have to
amount to at /east 3°’. It would appear therefore that, if the estimate
of 1°in a hundred years was due to Platonic influence at all, it must
have been Ptolemy who Platonized rather than Hipparchus. And
it seems clear that the Great Year of 36,000 years, if we assume it
to be deducible from the passage of Plato, is certainly not ‘best
explained with reference to precession’ as Burnet supposes.” Indeed
the passage in the 7zmaeus is hardly consistent with this, for the
Great Year is there distinctly said to be the period after which all
the eight revolutions’, i.e. those of the seven ‘planets’ as well as
that of the sphere of the fixed stars, come back to the same relative
positions; and the only revolution of the sphere of the fixed stars
that is mentioned is the daily rotation.
‘The visible form of the deities he made mostly of fire, that
it might be most bright and most fair to behold, and, likening it
to the All, he fashioned it like a sphere and assigned it to the
intelligence of the supreme to follow after it; and he disposed
it round about throughout all the heaven, to be an adornment
of it in very truth, broidered over the whole expanse. And he
bestowed two movements on each, one in the same place and
uniform, as remaining constant to the same thoughts about the
same things, the other a movement forward controlled by the
revolution of the Same and uniform ; but for the other five move-
ments he made it motionless and at rest, in order that each star
might attain the highest order of perfection.
‘From this cause then have been created all the stars that
wander not but remain fixed for ever, living beings, divine, eternal,
and revolving uniformly and in the same place; while those which.
have turnings and wander as aforesaid have come into being on
the principles which we have declared in the foregoing.’ ὃ
The deities are of course the stars, and ‘the intelligence of the
supreme’ which they follow is the revolution of the circle of
the Same which holds the mastery over all. The two movements
common to the fixed stars are (1) rotation about their own axes
_ } Ptolemy, L.c., vol. ii, pp. 15, 16, Heib.
_? Burnet, Early Greek Philosophy, p. 26 note. 8 Timaeus 40 A-B.
174 PLATO PARTI
and (2) their motion as part of the whole heaven in its daily
rotation, the first being a motion in one and the same place, the
other a motion ‘forward’, or of translation, in circles parallel to
the equator, from east to west. The idea that the fixed stars
rotate about their own axes is attributed by Achilles to the
Pythagoreans.! The ‘other five movements’ (in addition to move-
ment forward) are movements backward, right, left, up and down.
Rotation about their own axes is only attributed in express terms
to the fixed stars; but Proclus is doubtless right in holding that
Plato intended to convey that the planets also rotate about their
own axes, the result of which is that, while the fixed stars have
two motions, the planets have three, rotation about their own axes,
revolution. about the axis of the universe due to their sharing in
the motion of the Same, and lastly their independent movements
in their orbits. The ‘turnings’ refer to the fact that, like the sun,
the planets, moving in the circle of the zodiac, go as far from the
equator as the tropic of Cancer and then turn, first approaching
the equator and then passing it, until they reach the tropic of
Capricorn when they again turn back.
‘But the earth our foster-mother, globed round the axis stretched
from pole to pole through the universe, he made to be guardian
and creator of night and day, the first and chiefest of the gods that
have been created within the heaven.
‘But the circlings of these same gods and their comings alongside
one another, and the manner of the returnings of their orbits upon
themselves and their approachings, which of the deities meet one
another in their conjunctions and which are in opposition, in what
order they pass before one another, and at what times they are
hidden from us and again reappearing send, to them who cannot
calculate their movements, terrors, and portents of things to come—
to declare all this without visible imitations of these same move-
ments were labour lost.’ 3
It is mainly upon this passage, combined with a passage of
Aristotle alluding to it, that some writers have based the theory
that Plato asserted the earth’s rotation about its own axis. There
is now, however, no possibility of doubt that this view is wrong,
and that Plato made the earth entirely motionless in the centre
of the universe. This was proved by Boeckh in his elaborate
1 Achilles, /sagoge in Arati Dhaenomena, c. 18 (Uvanologium, p. 138 C).
2 Timaeus 40 B-D.
as
CH. XV PLATO 175
examination of the whole subject? made in reply to a tract by
Gruppe”, and again in a later paper* where Boeckh success-
fully refuted objections taken by Grote to his arguments. The
cause of the whole trouble is the ambiguity in the meaning of
the Greek word which is used of the earth ‘g/obed round the axis’.
It now appears that ἐλλομένην is the correct reading, although there
is MS. authority for εἱλλομένην and εἰλλομένην ; but all three words
seem to be no more than variant forms meaning the same thing
(literally ‘rolled’). _Boeckh indeed seems to have gone too far in
saying that εἱλλομένην can only mean ‘globed round’ in Plato,
because no actual use of εἵλλεσθαι or εἵλεσθαι in the sense of rotation
about an axis or revolution in an orbit round a point can be found
in the Zzmaeus or elsewhere in the dialogues ; for,as Teichmiiller *
points out, εἵλλω is related to ἕλιξ (a spiral) and ἑλίττω (Ionic εἱλίσσω),
to ‘roll’ or ‘ wind’, which latter word is actually used along with the
word στρέφεσθαι (‘to be turned’) in the Theaetetus.® But, while ἐλλο-
μένην does not exclude the idea of motion, it does not necessarily
include it ; ὁ and the real proof that it does not imply rotation here
(but only being ‘rolled round’ in the sense of massed or packed
round) is not the etymological consideration, but the fact that theidea
of the earth rotating at all on its axis is quite inconsistent with the
whole astronomical system described in the 7zmaeus. An essential
feature of that system, emphasized over and over again, is the
motion of the Circle of the Same which carries every other motion
and all else in the universe round with it; this is the daily rotation
which carries round the earth the sphere of the fixed stars, and it is
this rotation of the fixed stars once completed which makes a day
and a night; cf. the passage ‘night and day have been created... -
and these are the revolution of the one and most intelligent circuit ’.”
1 Boeckh, Untersuchungen iiber das kosmische System des Platon, 1852.
3 Gruppe, Die kosmischen Systeme der Griechen, 1851.
3 Boeckh, Kleine Schriften, iii, p. 294 sqq.
* Teichmiiller, Studien zur Geschichte der Begriffe, 1874, pp. 240-2.
5 Theaetetus 194 B.
5 Thus in Sophocles, Antigone 340, ἰλλομένων, used of ploughs, means ‘going to
and fro’; but four instances occurring in Apollonius Rhodius tell in favour of
Boeckh’s interpretation of our passage: i. 129 δεσμοῖς ἰλλόμενον, where (as in
ii. 1249 also) ἰλλόμενος means ‘fast bound’ ; i. 329 ἰλλομένοις ἐπὶ λαίφεσι, ‘ with
sails furled’; ii. 27 ἰλλόμενός περ ὁμίλῳ, ‘hemmed in by a crowd.’ Simplicius
(on De caelo, p. 517, 15) cites Ap. Rh. i. 129 and adds that, even if the word is
spelt εἰλλόμενος, it still means εἰργόμενος (‘ shut in’), as it does once in a play of
Aeschylus (now lost). 1 Timaeus 39 C.
176 PLATO PARTI
If the earth rotated about its axis in either direction, it would not be
the rotation of the sphere of the fixed stars alone which would
make night and day, but the sum or difference of the two rotations
according as the earth rotated in the same or the opposite sense to
the sphere of the fixed stars; but there is not a word anywhere
to suggest any cause but the one rotation of the fixed stars in
24 hours.
This being so, how did Aristotle come to write ‘Some say that,
although the earth lies at the centre, it is yet wound and moves
about the axis stretched through the universe from pole to pole, as
is stated in the Zzmaeus’1? For three MSS. out of Bekker’s five
add the words καὶ κινεῖσθαι, ‘ and is moved’,to ἴλλεσθαι, ‘is wound’,
whereas the actual passage in the Zzmaeus has ἰλλομένην and
nothing more. Alexander? held that Aristotle must have been
right in adding the gloss ‘and moves’ because he could not have
been unaware either of the meaning of ἰλλομένην or of Plato’s
intention. Simplicius* is not so sure, but makes the best excuse he
can. As the word ἰλλομένην might be interpreted by the ordinary
person as implying rotation, Aristotle would be anxious to take
account of the full apparent signification as well as the true one, in
accordance with his habit of minutely criticizing the language of his
predecessors with all its possible implications; he might then be
supposed to say in this passage (which immediately follows his
reference to those who held that the earth is not in the centre but
moves round the central fire): ‘And, if any one were to suppose
that Plato affirmed its rotation in the centre through taking ἰλλο-
μένην (being wound) to mean κινουμένην (being moved), we should
at once have another class of persons coming under the more
general category of those who assert that the earth moves; for the
hypotheses will be that the earth moves in one of two ways, either
round the centre or in the centre; and the person who understands
Plato’s remark in the sense of the latter hypothesis will be proved
to be in error. But Simplicius evidently feels that this is not
a very Satisfactory explanation, for he goes on to suggest the alter-
native that the words καὶ κινεῖσθαι, ‘and moves’, are an interpola-
? Aristotle, De cae/o ii. 13, 293 Ὁ 30.
* Simplicius on De caelo, p. 518. 1-8, 20-21, ed. Heib.
§ Simplicius, loc, cit., pp. 518. 9-519. 8.
—— τὰ ναι
ΡΨ Ρ
CH. XV PLATO ; 177
tion; the passage will then, he says, be easy to understand ; the con-
trast will be a double one, between those who say that the earth (1) is
not in the centre but (2) moves about the centre, and those who say
that (1) it is in the centre and (2) is at rest there. It would not be
unnatural if an unwise annotator had interpolated the words from
the passage at the beginning of the next chapter (14), where the
same remark is made without any mention of the Zzmaeus: ‘for, as
we said before, some make the earth one of the stars, while others
place it in the centre and say that it is wound and moves (ἴλλεσθαι
καὶ κινεῖσθαι, 45 before) about the axis through the centre joining the
poles.’' Archer-Hind 5 is disposed to accept the suggestion that the
words are interpolated from the later into the earlier passage; but the
suggestion only helps if Aristotle is referring in the later passage to
some one other than Plato. Archer-Hind, it is true, thinks that
the added words in the second passage distinguish the theory there
stated from Plato’s; but I think this is not so. The theory
alluded to in both passages is, I think, identically the same, as
indeed we may infer from the words ‘as we said before’. Another
attempted explanation should be mentioned ; it is to the effect that
the words ‘as is stated in the 7imaeus’ in the passage of Aristotle
refer only to the words ‘about the axis stretched through the uni-
verse from pole to pole’ and not to the whole phrase ‘it is yet wound
and moves about the axis, &c.’. This explanation was given, as much
as 600 years ago, by Thomas Aquinas ;* in recent years it has been
independently suggested by Martin* and Zeller,° and Boeckh has
an explanation which comes to the same thing. What seems to
me to be fatal to it is the word ἔλλεσθαι, ‘is wound’, immediately
preceding; this corresponds to Plato’s word ἰλλομένην, and it is ~
impossible, I think, to suppose that ἔλλεσθαι does not, as much as
1 Aristotle, De cae/o ii. 14, 296 ἃ 25. 3 Archer-Hind, Zimaeus, p. 133 note.
5. Dreyer (Planetary Systems, p. 78) was apparently the first to point this out.
The explanation was put forward in Themas Aquinas’s Comment. in libros
Aristotelis de caelo, lib. ii, lect. xxi (in S. Thomae Aguinatis Opera omnia,
ili, p. 205, Rome, 1886): ‘ Quod autem addit, guemadmodum in Timaeo
scripium est, referendum est non ad id quod dictum est, revolvi δέ moveri, sed
_ ad id quod sequitur, guod sit super statutum polum,’
5 ane in Mém. de 1’ Acad. des Inscriptions et Belles-Lettres, xxx, 1881,
ΡΡ- 77; 79.
bi Zeller, ‘Ueber die richtige Auffassung einiger aristotelischen Citate,’ in
Sitzungsber. der k. Preuss. Akad. der Wissenschaften, 1888, p. 1339.
5 Boeckh, Das kosmische System des Platon, pp. 81-3.
1410 N
178 PLATO PARTI
the words about the axis, refer to the Zzmaeus. The only possible
conclusion left is the earlier one of Martin,! in which Teichmiiller?
agrees, namely that Aristotle deliberately misrepresented Plato for
the purpose of scoring a point. There are many other instances in
Aristotle of this ‘ eristic ’ and ‘ sophistical’ criticism, as Teichmiiller
calls it, of Plato’s doctrines.
Other writers seem to have been misled from the first by Aris-
totle’s erroneous description of the theory of the Timaeus. Cicero,
in speaking of the rotation of the earth about its axis, says: ‘And
some think that Plato also affirmed it in the 7zmaeus but in some-
what obscure terms.’* Plutarch* discusses and rejects this inter-
pretation. Proclus is also perfectly clear that Plato made the earth
absolutely at rest: ‘Let’, he says, ‘ Heraclides of Pontus, who was
not a disciple of Plato, hold this opinion and make the earth rotate
round its axis; but Plato made it unmoved.’® Proclus goes on to
support this by a good argument. If, he says, Plato had not denied
motion to the earth, he would not have described his ‘ perfect year’
with reference to eight motions only; he would have had to take
account of the earth’s motion also as a ninth.
The words ‘guardian and creator (φύλακα καὶ δημιουργόν) οἵ
night and day’ have been thought by some to constitute a difficulty
on the assumption that the earth abides absolutely unmoved in the
centre. How, it is asked, can a thing which is purely passive be
said to ‘create’ anything? Martin® furnishes the answer to this.
If the earth were purely passive, it would not be at rest; it would
rotate about its axis once in 24 hours, since it would be carried
round in the daily revolution of the universe. In order to remain
at rest, as Plato requires, it has to exert a force in the opposite
direction equal to that exerted by the daily revolution; it produces
day and night therefore by the energy of its resistance which keeps
it at rest, while it is the ‘guardian’ by virtue of its immobility.
A guardian is, as Boeckh says,’ one who remains on the spot to
watch and ward ; this is the réle of the earth; if it deserted its post,
1 Martin, Etudes sur le Timée, ii, p. 87.
2 Teichmiiller, Studien zur Geschichte der Begriffe, 1874, pp. 238-45.
8 Cicero, Acad. Pr. ii. 39,123. “ Plutarch, Quaest. Plat. viii. 1-3, p. 1006 C-F.
5 Proclus zz Tim. p. 281 E.
6 Martin, Etudes sur le Timée, ii, pp. 88, 90.
7 Boeckh, Das kosmésche System des Platon, p. 69.
ee ae
CH. XV PLATO 179
if it were not there, there would only be light, and not day and
night ; hence it is called the guardian of night and day. Proclus
observes that the earth is of course the ‘creator’ of night because
night is the effect of the earth’s shadow which is cast in the shape
of a cone, and the earth can be said to be the creator of day by
virtue of the day’s connexion with night, although one would say
that the sun rather than the earth is the actual cause of day.!
It is, however, the earth which is the cause of the distinction
between night and day; consequently it may fairly be called the
‘creator’ of both. In the Timaeus Locrus* the earth is called
the ὅρος (boundary, limit, or determining principle) of night and day ;
and Plutarch* aptly compares it to the upright needle of the sundial :
it is its fixedness, he says, which gives the stars a rising and a setting.
Some expressions in the second paragraph of the passage quoted
on p. 174 call for a word or two in explanation. The ‘circlings’,
&c., are of course those of the planets ; the circlings are their revolu-
tions round the earth as common centre, as it were in a round dance
(χορεία), ‘their well-ordered and harmonious revolutions, as Pro-
clus says. The ‘comings alongside one another’ (παραβολαΐί, the
same word as is used in geometry of the ‘application’ of an area to
a straight line) are explained by Proclus as ‘ their comings together
in respect of longitude, while their positions in respect of latitude
or of depth are different, in other words, their rising simultaneously
and their setting simultaneously’® ‘The returnings of their orbits
upon themselves and their approachings’ (ai τῶν κύκλων πρὸς ἑαυ-
τοὺς ἐπανακυκλήσεις Kai προσχωρήσεις) are somewhat differently
interpreted. Proclus understands them as meaning retrogradations
and advance movements respectively: ‘for when they advance they ἢ
are approaching their ἀποκατάστασις (their return to the same place
in the heavens) ; and, when their movement is retrograde, they return
upon themselves.’® Boeckh agrees in taking προσχωρήσεις to mean
their return to the same position in the heavens (ἀποκατάστασις)
but takes ἐπανακυκλήσεις, their return upon one another, to be an
earlier stage of the same motion; they ‘turn upon themselves’ in
? Proclus zz Tim. 282 B,C; cf. Archer-Hind, 7imaeus, p. 134 note.
2 Timacus Locrus 97 Ὁ.
® Plutarch, Quaest. Plat. viii.3, p. 1006 F ; cf. Defac. in orbe lunae, c. 25, p. 938 E.
* Proclus iz Tim. p. 284 B.
5 Ibid., p. 284 6. 5 Ibid.
N2
180 PLATO PARTI
respect of the circular motion tending to bring them round again to
the same point, and the ‘approaching’ is the arrival at the same
point.
Some allusions to the sun, moon, and planets as the ‘ instruments
of time’ (ὄργανα χρόνου) bring us to the end of the astronomy of
the Zimaeus. After a passage about the created gods and other
gods born of them, the Creator makes a second blending of Soul.
‘And when he had compounded the whole, he portioned off souls
equal in number tothe stars and distributed a soul to each star and,
setting them in the stars as in a chariot, he showed them the nature
of the universe and declared to them its fated laws. ... and how
they must be sown into the instruments of time befitting them
severally.’ 3
Archer-Hind explains that the ‘souls’ here distributed among
the stars, one to each, are different from the souls of the stars them-
selves and are rather portions of the whole substance of soul ; this
was so distributed in order that it might learn the laws of the
universe; then finally, he thinks, it was redistributed among
the planets for division into separate souls incorporated in bodies.’
The instruments of time are mentioned again a little later on:
‘And when he had ordained all these things for them. . . God
sowed some in the earth, some in the moon, and some in the other
instruments of time.’ 5
Gruppe seizes upon this passage to argue that the earth is in-
cluded with the moon and the other planets among the ‘ instruments
of time’, and hence that, as a measurer of time, the earth cannot be
at rest but must rotate round its axis. But Boeckh® points out
that even in this passage the earth itself need not be an instrument
of time, for ‘the other instruments of time’ may mean ‘other than
the moon’ just as well as ‘ other than the earth and moon’; and it
is clear from another passage that the earth is ot one of the
‘instruments of time’. For in a sentence already quoted we are
told that ‘ the sun and the moon and five other stars which have the
name of planets have been created for defining and preserving
1 Boeckh, Das kosmische System des Platon, p. 60.
2 Timaeus 41 D, E. 8 Archer-Hind, 7imaeus, p. 141-2 note.
4 Timaeus 42D.
5 Boeckh, Das kosmische System des Platon, pp. 71-3-
——
CH. XV ; PLATO 181
the numbers of time’, i.e. as the instruments of time. It is true
that the remaining ‘instrument’, which measures the day of about
24 hours, is not here mentioned ; but, when it does come to be
mentioned, this instrument is not the earth, but the motion of the
circle of the Same, or the sphere of the fixed stars: ‘Night and day
... are one revolution of the undivided and most intelligent circuit.’ 3
We have next to inquire whether still later dialogues contain or
indicate any modification of the system of the Zzmaeus. We come
then to the Laws.
In Book VII occurs the passage already alluded to above, which
in the first place exposes what appeared to Plato to be errors in the
common notions about the movements of the planets current in his
time, and then states, in a matter of fact way, the view which seems
to him the most correct. After arithmetic and the science of calcu-
lation, and geometry as the science of measurement, with the dis-
tinction between commensurables and incommensurables, astronomy
is introduced as a subject for the instruction of the young, when the
following conversation takes place between the Athenian stranger
and Clinias.
‘ Ath. My good friends, I make bold to say that nowadays we Greeks
all affirm what is false of the great gods, the sun and the moon.
Cl. What is the falsehood you mean ?
Ath. We say that they never continue in the same path, and that
along with them are certain other stars which are in the same case,
and which we therefore call planets.
Cl. By Zeus, you are right, O stranger ; for many times in my
life I too have noticed that the Morning Star and the Evening
Star never follow the same course but wander in every possible way, ©
and of course the sun and the moon behave in the way which is
familiar to everybody.
Ath. These are just the things, Megillus and Clinias, which I say
citizens of a country like ours and the young should learn with regard
to the gods in heaven; they should learn the facts about them all
so far as to avoid blasphemy in this respect, and to honour them
at all times, sacrificing to them and addressing to them pious
prayers.
Cl. You are right, assuming that it is at all possible to learn that
to which you refer; if there is anything in our present views about
1 Timacus 38 C. ? Ibid. 39 B,C.
182 PLATO PARTI
the gods that is not correct, and instruction will correct it, I too
agree that we ought to learn a thing of such magnitude and impor-
tance. Do you then try your best to explain how these things are
as you say, and we will try to follow your instruction.
Ath, Well, it is not easy to grasp what I mean, nor yet is it very
difficult or a very long business. And the proof of this is that,
although it is not a thing I learnt when I was young or have known
a long time, I shall not take long to explain it to you; whereas, if it
had been difficult, I at my age should never have been able to
explain it to you at yours.
Cl. I dare say. But what sort of doctrine is this you speak of,
which you call surprising, and proper to be taught to the young,
but which we do not know? Try to tell us this much about it as
clearly as you can.
Ath, 1 will try. Well, my good friends, this view which is held
about the moon, the sun, and the other stars, to the effect that they
ever wander, is not correct, but the very contrary is the case. For
each of them traverses the same path, not many paths, but always
one, in a circle, whereas it appears to move in many paths, And
again, the swiftest of them is incorrectly thought to be the slowest,
and vice versa. Now, if the truth is one way and we think another
way, it isas if we had the same idea with regard to horses or long-
distance runners at Olympia and were to address the swiftest as the ©
slowest, and the slowest as the swiftest, and to award the praise
accordingly, notwithstanding that we knew that the so-called loser
had really won. I imagine that in that case we should not be.
awarding the praise in the proper way or a way agreeable to the
runners, who are only human. When then we make this very same
mistake with regard to the gods, should we not expect that the
same ridicule and conviction of error would attach to us here and.
in this question as’ we should have suffered on the racecourse?
Cl. Nay, it would be no laughing matter at all.
Ath. No, nor would it be consistent with respect for the gods, if
we repeated a false report against them.’ ὦ
The sentence italicized above is cited by Gruppe as another
argument in favour of his hypothesis that Plato attributed to the
earth rotation about its axis. Plato says that the apparent multi-
plicity of the courses of each planet is an illusion, and that each has
one path only. Now, says Gruppe, this is only true if we reject the
motion of the sphere of the fixed stars as only apparent, and substi-
tute for it the rotation of the earth round its axis; for only then
can it be said, e.g., that the sun and moon have only ove movement
1 Plato, Laws vii. 821 B-822 Ο.
πα να
CH. XV | PLATO - 183
in acirele. If we assume the actual motion of the sun along with
the sphere of the fixed stars, while the earth remains at- rest, the
circle becomes a spiral as described in the 7zmaeus. Schiaparelli,!
influenced also by the passage of Aristotle which he thinks repre-
sents what Aristotle must have known to be the final view of Plato
through hearing the matter discussed in his school, accepts Gruppe’s
conclusion, not apparently having been aware, at the time that he
did so, of Boeckh’s complete refutation of it. Boeckh* answers in
the first place that the unity of the movement of the planets
in single circles is not supposed, here any more than in the 7zmaeus,
to be upset by the fact that the movement of the circle of the Same
turns them into spirals. Thus in the 7zmaeus; in the very next
sentence but one to that about the spirals, Plato speaks of the moon
as describing ‘its own circle’ in a month, and of the sun as describing
‘its own circle’ in a year. Similarly, Dercyllides* says that the
orbits of the planets are primarily simple and uniform circles
round the earth; the turning of these circles into spirals is merely
incidental.
- Gruppe goes so far as to find the heliocentric system in the
passage before us, by means of a forced interpretation of the words
about the planets which are really the quickest being regarded as
the slowest and vice versa. He relies in the first place on two
passages of Plutarch as follows: (1) ‘Theophrastus also adds that
Plato in his old age regretted that he had given the earth the
middle place in the universe, which was not appropriate to it,’ * and
(2) ‘they say that Plato in his old age was moved by these con-
siderations [the Pythagorean theory of the central fire] to regard
the earth as placed elsewhere than in the centre, and the middle -
and chiefest place as belonging to some worthier body ᾿; he then
straightway proceeds to assume the worthier body to be the sun,
and the ambiguity as regards swiftness and slowness to refer to the
stationary points and the retrogradations of the planets. Schia-
parelli,® however, points out, as Boeckh’ had done before him, that
1 Schiaparelli, 7 frecursori, p. 19 54.
3 Boeckh, Das kosmische System des Platon, pp. 52 sqq.
3 Theon of Smyrna, p. 200. 23 sq.
* Plutarch, Quaest. Piat. viii. 1, Ὁ. 1006 C.
5 Plutarch, Vuma, c. 11. Schiaparelli, 7 frecursori, p. 21.
T Boeckh, Das kosmische System des Platon, p. 57.
184 PLATO ' PARTI
this cannot be correct, as it is indicated a little earlier in the same
passage of the Laws. that everybody sees the same phenomena
illustrated in the case of the sun and moon: this clearly implies,
first, that the sun moves and, secondly, that the irregularities
cannot be retrogradations, seeing that they do not exist in the case
of the sun and moon. The fact is that the ambiguity pointed out
in the Laws with regard to the speed of the planets is exactly the
same as that which we have read of in the Zimaeus, and that
the passage in the Laws changes nothing whatever in the system
expounded in the earlier dialogue. The remarks quoted from
Plutarch will be dealt with later.
But we have not even yet finished with the arguments as regards
the supposed rotation of the earth in Plato’s final system. Schia-
parelli? finds another argument in its favour in the Epinomis, a
continuation of the Laws attributed to Philippus of Opus, a disciple
of Plato, who is also said to have revised and published the Laws,
which had been left unfinished. The system described in the
Epinomis is the same as the system of the Zzmaeus. There are
eight revolutions. Two are those of the moon and the sun; ὅ then
come two others, those of Venus and Mercury, of which it is said —
that their periods are about the same as that of the sun,* so that
no one of the three can be said to be slower or faster than the
others ;° after these are mentioned the three revolutions of the other
planets, Mars, Jupiter, and Saturn, which are said to travel in the
same direction as the sun and moon, i.e. from west to east. The
eighth revolution is not that of the earth, so that here, as in the
Timaeus and the Laws, no rotation is attributed to the earth. Of
the eighth revolution we_ read :
‘ And one of the moving bodies, the eighth, is that which it is most
usual to call the universe above [i.e. the sphere of the fixed stars],
which travels in the opposite sense to all the others, while carrying
the others with it, as men with little knowledge of these things would
suppose. But whatever we adequately know we must affirm and we
do affirm.’ ®
Upon this Schiaparelli remarks, adverting to the italicized words :
‘Plato then declares, in the Zpinomis also, that men who under-
1 Laws 821 Ο. 3 Schiaparelli, 7 frecursori, pp. 20-21.
8 Epinomis 986 A,B. * Ibid. 9865, 987B. ὅ Ibid. ο87 Β. 5 Ibid. 987 B.
CH. XV PLATO 185
stand little about astronomy believe in the daily revolution of the
heaven. If he expresses himself according to this system, it is for
the purpose of adapting himself to the intelligence of the ordinary
person. Here we have what Aristotle doubtless had in mind when
he wrote his celebrated remark about the rotation of the earth.’}
But if this is the meaning of the passage, why did the author, after
saying (apparently by way of contrast) ‘but what we adequately
know we must affirm and do affirm’,stop there and say not a single
word of any alternative to the general rotation of the ‘heaven’?
There is still not a word of the earth’s rotation, and indeed it is
excluded by the limitation of the revolutions to eight, as remarked
above. We must therefore, I think, reject Schiaparelli’s interpreta-
tion of the passage and seek another. It occurs to me that the
emphasis is on the word ‘ men’ (ἀνθρώποις without the article), and
that the meaning is ‘so far as mere human beings can judge, who
can have little knowledge of these things ’. The words immediately
_ following are then readily intelligible; they would mean ‘ but if we
are reasonably satisfied of a thing we must have the courage to state
our view ’2
_ One other passage of the Efinomis is quoted by Martin® as
evidence that it only repeats the theory of the 77zsmaeus without
change. All the stars are divine beings with body and soul. A
proof that stars have intelligence is furnished by the fact that ‘they
always do the same things, because they have long been doing
things which had been deliberated upon for a prodigious length of
time, and they do not change their plans up and down, do one
thing at one time and another at another, or wander and change
their orbits’.* Consequently, as the stars include the planets, the -
Epinomis, like the Timaeus,> seems to deny the distinction between
perigee and apogee, all variations of angular speed, stationary posi-
tions and retrogradations, and all movement in celestial latitude.
We have, lastly, to consider the two passages of Plutarch quoted
above (p. 183) to the effect that Plato is said to have repented in his
1 Schiaparelli, 7 frecursori, pp. 20-1.
3 Cf. Laws 716C, to the effect that God is the real measure of all things,
much more so than any man.
3 Martin in Mém. de l’ Acad. des Inscriptions et Belles-Lettres, xxx, 1881, p. 90.
* Epinomis 982 C, Ὁ.
5 Timacus 408; cf. 344, 43 B, ἄς.
186 PLATO PARTI
old age of having put the earth in the centre instead of assigning
the worthier place to a worthier occupant. These passages have
been fully dealt with by Boeckh? and by Martin? after him, and it
is difficult or impossible to dissent from their conclusion, which is
that the tradition is due to a misunderstanding and is unworthy
of credence. To begin with, although the Zaws is later than the
Timaeus and so late that Plato did not finish it, there is in it no sign
of a change of view. Nor is there any sign of such in the Epinomis
written by Plato’s disciple, Philippus of Opus; and it is incredible
that, if the supposed change of view had come out in the last oral
discussions with the Master, Philippus would not have known about
it and mentioned it. Even assuming the tradition to be true, we
can at all events reject without hesitation the inference of Gruppe
that the sun was Plato’s new centre of the universe. If the sun had
been the centre, this would surely have been stated, and we should
not have been put off with the vague phrase ‘some worthier occu-
pant’. As, in the Wma where this expression occurs, Plutarch has
just been speaking of the central fire of the Pythagoreans, the
natural inference is that Plato’s new centre, if he came to assume
one at all, would be either the Pythagorean central fire or some
imaginary centre of the same sort. But from what source did
Theophrastus get the story which he repeats? Obviously from
hearsay, since there is not a particle of written evidence to confirm
it. The true explanation seems to be that some of Plato’s imme-
diate followers in the Academy altered Plato’s system in a
Pythagorean sense, and that the views of these Pythagorizing
Platonists were then put down to Plato himself. In confirmation
of this Boeckh quotes the passage of Aristotle in which, after
speaking of the central fire of the Pythagoreans and the way in
which they invented the counter-earth in order to force the pheno-
mena into agreement with their preconceived theory, he goes on to
indicate that there was in his time a school of philosophers other
than the Pythagoreans who held a similar view: ‘And no doubt
many others too would agree (with the Pythagoreans) that the place
in the centre should not be assigned to the earth, if they looked for
1 Boeckh, Das kasmische System des Platon, pp. 144-50.
2 Martin in “έρι. de l’ Acad. des Inscriptions et Belles-Lettres, xxx, 1881,
pp. 128-32.
Es
CH. XV ’ PLATO 187
the truth, not in the observed facts, but in a priorz arguments. For
they hold that it is appropriate to the worthiest object that it
should be given the worthiest place. Now fire is worthier than earth,
the limit worthier than the things which are between the limits,
while both the extremity and the centre are limits: consequently,
reasoning from these premises, they hold that it is not the earth
which is placed at the centre of the sphere, but rather fire.’?
Simplicius* observes upon this that Archedemus, who was younger
than Aristotle, held this view, but that, as Alexander says, it is
necessary to inquire historically who were the persons earlier than
Aristotle who also held it. As Alexander could not find any such,
he concluded that it was not necessary to suppose that there were any
except the Pythagoreans. But the present indicative ‘ they hold’
makes it clear that Aristotle had certain other persons in mind who,
however, were not philosophers of an earlier time but were contem-
poraries of his own. These may well have been members of, or an
offshoot from, the Academy who expressed the views in question,
not in written works, but in discussion ; and, if this were so, nothing
would be more natural than that a tradition which referred to the
views of these persons should be supposed to represent the views of
Plato in his old age.
Tannery* has a different and very ingenious explanation of
Theophrastus’s dictum about Plato’s supposed change of view.
This explanation is connected with Tannery’s explanation of
another mystery, that of the attribution to one Hicetas of Syracuse
of certain original discoveries in astronomy. Diogenes Laertius+*
says of Philolaus that ‘he was the first to assert that the earth
moves in a circle, though other authorities say that it was Hicetas —
the Syracusan’. Aétius says® that ‘ Thales and those who followed
him said that the earth was one; Hicetas the Pythagorean that
there were two, our earth and the counter-earth’. From these two
passages taken together we should naturally infer that Hicetas was
by some considered to be the real author of the doctrine attributed
1 Aristotle, De cae/o ii. 13, 293 a27-b 1.
? Simplicius on De caelo, p. 513, ed. Heib.
3 amen ‘Pseudonymes antiques’ in Revue des Etudes grecques, X, 1897,
Pp. 127-37.
* Diog. L. viii. 85 (Vors. i’, p. 233. 33).
® Aét. ili. 9. 1. 2 (D. G. p. 376; Vors. i*, p. 265. 25).
188 PLATO PARTI
to Philolaus, in which the earth and counter-earth, along with the
sun, moon, and planets, revolve round the central fire. Cicero,*
however, has a different story: ‘Hicetas of Syracuse, as Theo-
phrastus says, holds that the heaven, the sun, the moon, the stars,
and in fact all things in the sky remain still, and nothing else in
the universe moves except the earth ; but, as the earth turns and
twists about its axis with extreme swiftness, all the same results
follow as if the earth were still and the heaven moved.’ This is
of course not well expressed, because, on the assumption that the
earth rotates about its axis once in every 24 hours, the sun, moon,
and planets would not in fact remain at rest any more than on the
assumption of a stationary earth, for they would still have their
independent movements; but Cicero means no more than that the
rotation of the earth is a complete substitute for the apparent daily
rotation of the heaven as a whole. However, the passage clearly
implies that Hicetas asserted the axial rotation of the earth, and
not its revolution with the counter-earth, &c., round the central
fire. The statements therefore of Cicero on the one side, and of
Diogenes and Aétius on the other, are inconsistent. Tannery
agrees with Martin? that we must accept as the more correct the
version of Diogenes and Aétius identifying Hicetas with the theory
commonly attributed to Philolaus, Now, says Tannery, Aristotle,
when speaking of the doctrine of the central fire as that of ‘the
philosophers of Italy, the so-called Pythagoreans’, clearly shows
that he did not attribute the doctrine to the Pythagoreans in
general or to Philolaus; if he had seen the book of Philolaus of
which our fragments formed part, and if he had referred to that
work in this passage, he would have spoken of Philolaus by name
instead of using the circumlocution; hence Aristotle must have
been quoting from a book by some contemporary purporting to
give an account of Pythagorean doctrines or doctrines claiming
to be such. Tannery supposes therefore that Aristotle was referring
to Hicetas, and that Hicetas was one of the personages in a certain
dialogue, in which Hicetas represented the system known by the
name of Philolaus, while Plato was his interlocutor. This dialogue
* Cicero, Acad. Pr. ii. 39. 123 (Vors. i*, p. 265. 20).
® Martin, Etudes sur le Timée, vol. ii, pp. 101, 125 sq.
CH. XV PLATO 189
would be one of those written by Heraclides of Pontus. One of
Heraclides’ dialogues was ‘On the Pythagoreans’, and an account
of the system of the central fire might easily form part of one of the
others, e.g. that ‘On Nature’ or ‘On the things in heaven’. Now
there was a historical personage of the name of Hicetas of Syracuse
whom Plato might well have known. He was a friend of Dion and
he appears in Plutarch’s lives of Dion and Timoleon as a political
personage of some importance. Faithful to Dion, and for a time to
his family after Dion’s assassination, he threw over that family and
seized the tyranny at Leontini, remaining the principal adversary of
Dionysius the Younger until the arrival of Timoleon, when he was
conquered and killed by the latter. There is nothing to suggest
that he was a physicist or a Pythagorean; but he might quite well
be represented in the dialogue as one who knew by oral tradition
the doctrines of the school, and was therefore a suitable interlocutor
with Plato. Plato’s remarks in the dialogue might no doubt easily
indicate a change from the views which we find in his own dialogues,
and this is a possible explanation of the misconception on the part
of Theophrastus and the Dorographi. Tannery adds that, on his
hypothesis, we can hardly any longer consider the so-called Philolaic
system as anything else but a brilliant phantasy due to that clever
raconteur Heraclides. I do not see the necessity for this, and it is
extremely difficult to believe that Heraclides invented doth the
theory of Philolaus and his own theory of the rotation of the earth
about its axis; I do not see why we should not suppose that the
system known by the name of Philolaus actually belonged to him
or to the Pythagoreans proper, and that Hicetas represented the
Pythagorean view rather than a new discovery of Heraclides.
Tannery's attractive hypothesis is accepted by Otto Voss.’
1 Otto Voss, De Heraclidis Pontici vita et scriptis, Rostock, 1896, p. 64.
XVI
THE THEORY OF CONCENTRIC SPHERES.
EUDOXUS, CALLIPPUS, AND ARISTOTLE
DIOGENES LAERTIUS!? tells us that Eudoxus of Cnidus was
celebrated as geometer, astronomer, physician, and _ legislator.
Philosopher and geographer in addition, he commanded and en-
riched almost the whole field of learning ; no wonder that (though
it was a poor play on his name) he was called ἔνδοξος (‘celebrated ’)
‘ instead of Eudoxus. In geometry he was a pupil of Archytas of
Tarentum, and it is clear that he could have had no better instructor,
for Archytas was a geometer of remarkable ability, as is shown by
his solution of the problem of the two mean proportionals handed —
down by Eutocius.? This solution furnishes striking evidence of
the boldness and breadth of conception which already characterized
Greek geometry, seeing that even in that early time it did not
shrink from the use of complicated curves in space produced by the
intersection of two or more solid figures. Archytas solved the pro-
blem of the two mean proportionals by finding a point in space as
the intersection of three solid figures. The first was an anchor-ring
or tore with centre C, say, inner radius equal to zero, and outward
radius 2a, say ; the second was a right cylinder of radius a so placed
that its surface passes through the centre C of the tore and its axis
is parallel to the axis of the tore or perpendicular to the plane
bisecting the tore in the same way as a split ring is split ; the third
surface was a certain right cone with C as vertex. The intersection
of the first two surfaces gives of course a curve (or curves) of double
curvature in space, and the third surface cuts it in points, one of
which gives Archytas what he seeks. There is, as we shall see,
1 Diog. L. viii. 86-91.
* See Heiberg’s Archimedes, vol. iii, pp. 98-102; or my Afollonius of Perga,
pp. xxii, xxiii.
~EUDOXUS 191
a remarkable similarity between this construction and the way in
which Eudoxus’s ‘ spherical lemniscate’ (Aippopede) is evolved as the
intersection between a sphere, a cylinder touching it internally, and
a certain cone, so that we may well believe that Eudoxus owed
much to Archytas. To Eudoxus himself geometry owes a debt
which is simply incalculable, and it is doubtful, I think, whether, for
originality and power, any of the ancient mathematicians except
Archimedes can be put on the same plane with him. Although no
geometrical work of Eudoxus is preserved, there is, in the first
place, a monument to him aere perennius in Book V of Euclid’s
Elements; it was Eudoxus who invented and elaborated the great
theory of proportion there set out, the essence of which is its
applicability to incommensurable as well as commensurable quan-
tities. The significance of this theory of proportion, discovered
when it was, cannot be over-rated, for it saved geometry from the
impasse into which it had got through the discovery of the irrational
at a time when the only theory of proportion available for geo-
metrical demonstrations was the old Pythagorean numerical theory,
which only applied to commensurable magnitudes. Nor can any
one nowadays even attempt to belittle the conception of equal
ratios embodied in Euclid V, Def. 5, when it is remembered that
Weierstrass’s definition of equal numbers is word for word the same,
_ and Dedekind’s theory of irrational. numbers corresponds exactly
to, nay, is almost coincident with, the same definition. Eudoxus’s
second great discovery was that of the powerful method of °
exhaustion which not only enabled the areas of circles and the
volumes of pyramids, cones, spheres, &c., to be obtained, but is at
the root of all Archimedes’ further developments in the mensuration
of plane and solid figures. It is not then surprising that Eudoxus ~
should have invented a geometrical hypothesis for explaining the
movements of the planets which for ingenuity and elegance yields
to none.
Eudoxus flourished, according to Apollodorus, in Ol. 103 = 368-
365 .8.C., from which we infer that he was born about 408 B.C., and
(since he lived 53 years) died about 355 B.C. In his 23rd year he
went to Athens with the physician Theomedon, and there for two
months he attended lectures on philosophy and oratory, and in
particular the lectures of Plato; so poor was he that he took up his
102 THEORY OF CONCENTRIC SPHERES ΡΑΚΤΙ
abode at the Piraeus and trudged to Athens and back on foot each
day. It would appear that his journey to Italy and Sicily to study
geometry with Archytas and medicine with Philistion must have
been earlier than the first visit to Athens at 23, for from Athens he
returned to Cnidus, after which he went to Egypt with a letter of
introduction to the king Nectanebus, given him by Agesilaus; the
date of this journey was probably 381-380, or a little later, and he
stayed in Egypt sixteen months. After that he went to Cyzicus,
where he collected round him a large school with whom he migrated
to Athens in 468 Β.Ο. or a little later.) There is apparently no
foundation for the story mentioned by Diogenes Laertius that he
took up a hostile attitude to Plato, nor, on the other side, for the
stories that he went with Plato to Egypt and spent thirteen years in
the company of the Egyptian priests, or that he visited Plato when
Plato was with Dionysius, i.e. the younger Dionysius, on his third
visit to Sicily in 361B.C. Returning later to his native place,
Eudoxus ‘was by a popular vote entrusted with legislative office.
When in Egypt Eudoxus assimilated the astronomical knowledge
of the priests of Heliopolis and himself made observations. The
observatory between Heliopolis and Cercesura used by him was
still pointed out in Augustus’s time ;! he also had one built at
Cnidus, from which he observed the star Canopus which was not
then visible in higher latitudes? He wrote two books entitled
respectively the Wirror (ἔνοπτρον) and the Phaenomena: the poem
of Aratus was, so far as verses 19-732 are concerned, drawn from
the Phaenomena of Eudoxus. It is probable that he also wrote
a book on Sphaeric, dealing with the same subjects as Autolycus’s
On the moving sphere and Theodosius’s Sphaerica.
In order to fix approximately the positions of the stars, including
in that term the fixed stars, the planets, the sun, and the moon,
Eudoxus probably used a dioptra of some kind, though doubtless of
more elementary construction than that used later by Hipparchus ;
? Strabo, xvii. 1. 30, pp. 806-7 Cas.
* Strabo, ii. 5.14, p. 119 Cas. Hipparchus (Ja Araté et Eudoxi phaenomena
Commentariorum libri tres, p. 114, 20-28) observes that Eudoxus placed the
star Canopus exactly on the ‘always invisible circle’, but that this is not correct,
since at Rhodes the circumference of this circle is 36° and at Athens 37° from
the South pole, while Canopus is about 384° distant from that pole, so that
Canopus is seen in Greece worth of the said circle. But, at the time when this
was written, Hipparchus had not yet discovered Precession.
CH. XVI EUDOXUS 193
and he is credited with the invention of the arachne (spider’s web),
which, however, is alternatively attributed to Apollonius,’ and which
seems to have been a sun-clock of some kind.”
- But it was on the theoretic even more than the observational side
of astronomy that Eudoxus distinguished himself, and his theory.
of concentric spheres, by the combined movements of which he
explained the motions of the planets (thereby giving his solution
of the problem of accounting for those motions by the simplest of
regular movements), may be said to be the beginning of scientific
astronomy.
Two pupils of Eudoxus achieved fame, one in geometry,
Menaechmus, the reputed discoverer of the conic sections, and the
other in astronomy, Helicon of Cyzicus, who was said to have
successfully predicted a solar eclipse.
The ancient evidence of the details of Eudoxus’s system of con-
centric spheres (which he set out in a book entitled On speeds, Περὶ
ταχῶν, now lost) is contained in two passages. The first is in
Aristotle’s Metaphysics,? where a short notice is given of the num-
bers and relative positions of the spheres postulated by Eudoxus
for the sun, moon, and planets respectively, the additions which
Callippus thought it necessary to make to the numbers of the spheres
assumed by Eudoxus, and lastly the modification of the system
_which Aristotle himself considers necessary ‘if the phenomena are
to be produced by all the spheres acting in combination’. A more
elaborate and detailed account of the system is contained in Sim-
plicius’s commentary on Book II of the De caelo of Aristotle ;*
Simplicius quotes largely from Sosigenes the Peripatetic (second
century A.D., the teacher of Alexander Aphrodisiensis, not the
astronomer who assisted Caesar in his reform of the calendar),
observing that Sosigenes drew from Eudemus, who dealt with the
subject in the second book of his History of Astronomy® Ideler
was the first to appreciate the elegance of the theory and to attempt
1 Vitruvius, De architect. ix. 8 (9). 1.
3 Bilfinger, Die Zeitmesser der antiken Volker, p. 22.
® Aristotle, Metaph. A. 8, 1073 Ὁ 17 - 1074 ἃ 14.
* Simplicii in Aristotelis de caelo commentaria, p. 488. 18-24, PP- 493. 4 — 506.
18, Heiberg; p. 498 a 45-b 3, pp. 498 Ὁ 27 -- 503 a 33, Brandis.
5 Simpl. on De caelo, p. 486, 18-21, Heib. ; p. 498 a 46-8, Brandis.
1410 Oo
194 THEORY OF CONCENTRIC SPHERES ParRTI
to explain its working;! he managed by means of an ordinary
globe to indicate roughly how Eudoxus explained the stationary
points and retrogradations of the planets and their movement in
latitude. E. F. Apelt? too gave a fairly full exposition of the
theory in a paper of 1849. But it was reserved for Schiaparelli to
work out a complete restoration of the theory and to investigate in
detail the extent to which it could account for the phenomena ; this
Schiaparelli did in a paper which has become classical,’ and which
will no doubt be accepted by all future historians (in the absence of
the discovery of fresh original documents) as the authoritative and
final exposition of the system.*
The passages of Aristotle and Simplicius are translated in full
in Appendices I and II to ΡΒΙΒΡΕΓΗΝΙ s paper. The former may
properly be reproduced here.
‘ Eudoxus assumed that the sun and moon are moved by three
spheres in each case; the first of these is that of the fixed stars, the
second moves about the circle which passes through the middle of
the signs of the zodiac, while the third moves about a circle
latitudinally inclined to the zodiac circle; and, of the oblique
circles, that in which the moon moves has a greater latitudinal.
inclination than that in which the sun moves. The planets are
moved by four spheres in each case; the first and second of these
are the same as for the sun and moon (the first being the sphere of
the fixed stars which carries all the spheres with it, and the second,
next in order to it, being the sphere about the circle through
the middle of the signs of the zodiac which is common to all the
planets®); the third is in all cases a sphere with its poles on
the circle through the middle of the signs; the fourth moves about
1 Ideler, ‘ Ueber Eudoxus’ in Adh. der Berliner Akademie, hist.-phil. Classe,
1828, pp. 189-212, and 1830, pp. 49-88.
5 E. Ἐς Apelt, ‘Die Spharentheorie des Eudoxus und Aristoteles’ in the
Abhandlungen der Fries’ schen Schule, Heft ii (Leipzig, 1849).
* Schiaparelli, ‘Le sfere omocentriche di Eudosso, di Callippo e di Aristotele’,
in Pubblicaziont del Καὶ, Osservatorio di Brera in Milano, No. ix, Milano, 1875 ;
German translation by W. Horn, in 4dA. zur Gesch. der Math., τ. Heft, Leipzig,
1877, Pp. 101-98.
4 It is true that Martin (I7ém. de 7 Acad. des Inscr. xxx, 1881) took objection
to Schiaparelli’s interpretation of the theories of the sun and moon, but he was
sufficiently answered by Tannery (‘ Seconde note sur le syst¢me astronomique
d’Eudoxe’ in 77έρι. de la Soc. des sci. phys. et nat. de Bordeaux, 2° série,
v, 1883, pp. 129 sqq., republished in Paul Tannery, Mémoires scientifiques, ed,
Heiberg and Zeuthen, vol. i, 1912, pp. 317-38.
5 ἁπασῶν, with which we must, strictly speaking, understand σφαιρῶν (spheres)
or possibly φορῶν (motions).
CH. XVI EUDOXUS rhe 195
a Circle inclined to the middle circle (the equator) of the third sphere ;
the poles of the third sphere are different for all the planets except
Aphrodite and Hermes, but for these two the poles are the same."
Fuller details are given by Simplicius, but, before we pass to the
details, we may, following Schiaparelli, here as throughout, inter-
pose a few general observations on the essential characteristics of;
the system.” Eudoxus adopted the view which prevailed from the
earliest times to the time of Kepler, that circular motion was suffi
cient to account for the movements of all the heavenly bodies.
With Eudoxus this circular motion took the form of the revolution
of different spheres, each of which moves about a diameter as axis.
All the spheres were concentric, the common centre being the
centre of the earth; hence the name of ‘ homocentric spheres’ used
in later times to describe the system. The spheres were of different
sizes, one inside the other. Each planet was fixed at a point in the
equator of the sphere which carried it, the sphere revolving at
uniform speed about the diameter joining the corresponding poles ;
that is, the planet revolved uniformly in a great circle of the sphere
perpendicular to the axis of rotation. But one such circular motion
was not enough; in order to explain the changes in the speed of
the planets’ motion, their stations and retrogradations, as well as
their deviations in latitude, Eudoxus had to assume a number of
such circular motions working on each planet and producing by
their combination that single apparently irregular motion which
can be deduced from mere observation. He accordingly held that
the poles of the sphere which carries the planet are not fixed,
but themselves move on a greater sphere concentric with. the
carrying sphere and moving about two different poles with a
speed of its own. As even this was not sufficient to explain the
phenomena, Eudoxus placed the poles of the second sphere on
a third, which again was concentric with and larger than the first and
second and moved about separate poles of its own, and with a speed
peculiar to itself. For the planets yet a fourth sphere was required
1 Aristotle, Metaph. A. 8, 1073 Ὁ 17-32.
_? A very useful summary of the results of Schiaparelli’s paper is given in
Dreyer’s History of the Planetary Systems from Thales to Kepler (Camb. Univ.
Press, 1906), pp. 90-103. My account must necessarily take the same line ; and
my apology for inserting it instead of merely referring to Dreyer’s chapter on the
subject must be that a sketch of the history of Greek astronomy such as the
present would be incomplete without it.
O 2
196 THEORY OF CONCENTRIC SPHERES ParRTI
similarly related to the three others; for the sun and moon he
found that, by a suitable choice of the positions of the poles and of
speeds of rotation, he could make three spheres suffice. In the
accounts of Aristotle and Simplicius the spheres are described in
the reverse order, the sphere carrying the planet being the last.
The spheres which move each planet Eudoxus made quite separate
from those which move the others. One sphere sufficed of course
to produce the daily rotation of the heavens. Thus, with three
spheres for the sun, three for the moon, four for each of the planets
and one for the daily rotation, there were 27 spheres in all. It does
not appear that Eudoxus speculated upon the causes of these
rotational motions or the way in which they were transmitted from
one sphere to another; nor did he inquire about the material of
which they were made, their sizes and mutual distances. In the
matter of distances the only indication of his views is contained in
Archimedes’ remark that he supposed the diameter of the sun to be
nine times that of the moon,! from which we may no doubt infer that
he made their distances from the earth to be in the same ratio 9: I.
It would appear that he did not give his spheres any substance or
mechanical connexion; the whole system was.a purely geometrical |
hypothesis, or a set of theoretical constructions calculated to repre-
sent the apparent paths of the planets and enable them to be com-
puted. We pass to the details of the system.
The moon has a motion produced by three spheres ; the first and
outermost moves in the same sense as the fixed stars from east to
west in twenty-four hours; the second moves about an axis per-
pendicular to the plane of the zodiac circle or the ecliptic, and in
a sense opposite to that of the daily rotation, i.e. from west to east ;
the third moves about an axis inclined to the axis of the second, at
an angle equal to the highest latitude attained by the moon, and in
the sense of the daily rotation from east to west ; the moon is fixed
on the equator of this third sphere. Simplicius observes that the
third sphere is necessary because it is found that the moon does not
always reach its highest north and south latitude at the same points
of the zodiac, but at points which travel round the zodiac in the
inverse order of the signs? He says at the same time that
1 Archimedes, ed. Heib., vol. ii, p. 248. 4-8; The Works of Archimedes, p. 223.
* Simplicius on De cae/o, p. 495. 10-13, Heib.
CH. XVI EUDOXUS 197
the motion of the third sphere is slow, the motion of the node being
‘quite small during each month’, while he implies that the monthly
motion round the heavens is produced by the second sphere, the
equator of which is in the plane of the zodiac or ecliptic. The
object of the third sphere was then to account for the retrograde
motion of the nodes in about 184 years. But it is clear (as Ideler saw)
that Simplicius’s statement about the speeds of the third and second
spheres is incorrect. If it had been the third sphere which moved
very slowly, as he says, the moon would only have passed through
each node once in the course of 223 lunations, and would have been
found for nine years north, and then for nine years south, of the
ecliptic. In order that the moon may pass through the nodes as
often as it is observed to do, it is necessary to interchange the
speeds of the second and third spheres as given by Simplicius; that
is, we must assume that the third sphere produces the monthly
revolution of the moon from west to east in 27 days 5h. 5m. 36sec.
(the draconitic or nodal month) round a circle inclined to the
ecliptic at an angle equal to the greatest latitude of the moon,
and then that this oblique circle is carried round by the second
sphere in a retrograde sense along the ecliptic in a period of 223
lunations; lastly, we must assume that both the inner spheres, the
' second and third, are bodily carried round by the first sphere in 24
an
hours in the sense of the daily rotation. There can be no doubt
that this was Eudoxus’s conception of the matter. The mistake
made by Simplicius seems to go back as far as Aristotle himself,
since, in the passage of the J/etaphysics quoted above, Aristotle
clearly implies that the second sphere corresponds to the move-
ment in longitude for all the seven bodies including the sun and —
moon, whereas in fact it only does so in the case of the five planets ;
and no doubt Sosigenes, Simplicius’s authority, accepted the state-
ment of Aristotle, without suspecting that the Master might be an
unsafe guide on such a subject. From the theory of Eudoxus ©
as thus restored we can judge how far by his time the Greeks had |
progressed in the study of the motions of the moon. Observations
had gone far enough to,enable the movement in latitude and the
retrogression of the nodes of the moon’s orbit to be recognized.
Eudokus knew nothing of the variation of the moon’s speed in
longitude, or at least took no account of it, whereas Callippus was
---
198 THEORY OF CONCENTRIC SPHERES Parti
aware of it about 3258B.C., that is, about twenty or thirty years
after Eudoxus’s time.
As regards the sun, we learn from Aristotle that Eudoxus again
assumed three spheres to explain its motion. As in the case of the
moon, the first or outermost sphere revolved like the sphere of the
fixed stars, the second moved about an axis perpendicular tothe plane
of the zodiac, its equator revolving accordingly in the plane of the
zodiac, while the third moved 80 that its equator described a plane
slightly inclined to that of the zodiac, the inclination being less in
the case of the sun than in the case of the moon. Simplicius adds
that the third sphere (which is necessary because the sun does not
at the summer and winter solstices always rise at the same point on
the horizon) moves much more slowly than the second and (unlike
the corresponding sphere in the case of the moon) in the direct
order of the signs.’ Simplicius makes the same mistake as regards
the speeds of the second and third spheres as he made in the case
of the moon. If it were the third sphere which moved very slowly,
the sun would for ages remain in a north or a south latitude and in
the course of a year would describe, not a great circle, but (almost) .
a small circle parallel to the ecliptic. The slow motion must there-
fore belong to the second sphere, the equator of which revolves in
the ecliptic, while the revolution of the third sphere must take place
in about a year (strictly speaking, a little more than a tropic year
in consequence of the supposed slow motion of the second sphere in
the same sense), the plane of its equator being inclined, at the small
angle mentioned, to the plane of the ecliptic. The slightly inclined
great circle of the third sphere which the sun appears to describe
is then carried round bodily in the revolution of the second sphere
about the axis of the ecliptic, the nodes on the ecliptic thus moving
slowly forward, in the direct order of the signs; and lastly both
the second and third spheres are carried round by the revolution of
the first sphere following the daily rotation.
The strange thing in this description of the sun’s motion is the
imaginary idea that its path is not in the ecliptic but in a circle
inclined at a small angle to the latter. Simplicius says that
Eudoxus ‘and those who preceded him’ (τοῖς πρὸ αὐτοῦ) thought
the sun had the three motions described, and that this was inferred
1 Simplicius on De caelo, pp. 493. 15-17, 494. 6-7, 9-11.
CH. XVI EUDOXUS 199
from the fact that the sun, in the summer and winter solstices, does
not always rise at the same point of the horizon.1 We gather from
this that even before Eudoxus’s time astronomers had suspected
a certain deviation in latitude on the part of the sun. Schiaparelli
suggests as an explanation that, the early astronomers having dis-
covered, by comparison with the fixed stars, the deviation of the
moon and the five planets in latitude, it was natural for them to
suppose that the sun also must deviate from the circle of the
ecliptic; indeed it would be difficult for them to believe that
the sun alone was exempt from such deviation. However this may
be, the notion of the nutation of the sun’s path survived for centuries.
Hipparchus? quotes a sentence from the lost Exoptron of Eudoxus
to the effect that ‘it appears that the sun too shows a difference in
the places where it appears at the solstices, though the difference is
much less noticeable and indeed is quite small’; Hipparchus goes
on to deny this on the ground that, if it were so, the prophecies by
astronomers of lunar eclipses, which they made on the assumption
that there was no deviation of the sun from the ecliptic, would
sometimes have proved appreciably wrong, whereas in fact the
eclipses never showed a difference of more than two ‘finger-
breadths’, and only very rarely that, in comparison with the most
accurately calculated predictions. Notwithstanding Hipparchus’s
great authority, the idea persisted, and we find later authors giving
a value to the supposed inclination to the ecliptic. We are not told
what Eudoxus supposed the angle to be, nor what he assumed as
the period of revolution of the nodes. Pliny® gives the inclination
as 1 on each side of the ecliptic; perhaps he misunderstood his
source and took a range of 1° to be an inclination of 1°. For Theon ~
of Smyrna,* on the authority of Adrastus, says that the inclination
is 4 ; Theon also says that the sun returns to the same latitude
after 365% days, whereas it takes 3653 days to return to the same
equinox or solstice and 3653 days to return to the same dis-
tance from us.? This shows that the solar nodes were thought to
have a retrograde motion (not a motion in the order of the signs,
ἢ Simplicius, loc. cit., P- 493. 11-17. ;
: hae wie ἧς ee ey Tae phaenomena, i. 9, pp. 88-92, ed. Manitius.
* Theon of Smyrna, ed. Hiller, pp. 135. 12-14, 194. 4-8
® Ibid., p. 172. 15 -- 173. 16.
200 THEORY OF CONCENTRIC SPHERES ParTI
as assumed by Eudoxus) and a period of 3654+ % or 2922 years.
It is not known who invented this theory in the first instance.
Schiaparelli shows that it was not started for the purpose of
explaining the motion of the equinoctial points, or the precession
of the equinoxes, which was discovered by Hipparchus, but was
unknown to Eudoxus, Pliny, and Theon.
Eudoxus supposed the annual motion of the sun to be perfectly
uniform ; he must therefore have deliberately ignored the discovery,
made by Meton and Euctemon 60 or 70 years before, that the sun
does not take the same time to describe the four quadrants of its
orbit between the equinoctial and solstitial points. LEudoxus, in fact,
seems to have definitely regarded the length of the seasons as being
as nearly as possible equal, since he made three of them ΟἹ days in
length, only giving 92 days to the autumn in order to make up 365
days in the year.t
In the case of each of the planets Eudoxus assumed four spheres.
The first and outermost produced the daily rotation in 24 hours, as
in the case of the fixed stars ; the second produced the motion along
the zodiac ‘in the respective periods in which the planets appear to
describe the zodiac circle’,? which periods, in the case of the superior
planets, are respectively equal to the sidereal periods of revolution,
and in the case of Mercury and Venus (on a geocentric system) one
year. As the revolution of the second sphere was taken to be
uniform, we see that Eudoxus had no idea of the zodiacal anomaly
of the planets, namely that which depends on the eccentricity of
their paths, and which later astronomers sought to account for by
the hypothesis of eccentric circles; for Eudoxus the points on the
ecliptic where successive oppositions or conjunctions took place were
always at the same distances, and the arcs of retrogradation were
constant for each planet and equal at all parts of the ecliptic. Nor
with him were the orbits of the planets inclined at all to the ecliptic;
1 This appears from the papyrus known under the title of Avs Eudoxi,
deciphered by Letronne and published by Brunet de Presle (/Votices et extraits
des manuscrits, xviii. 2, 1865, p. 25 sq.). The papyrus was edited by Blass (Kiel,
1887), and a translation will be found in Tannery’s Recherches sur Phistoire
de lastronomie ancienne, pp. 283-94. Tannery prefers the title restored by
Letronne, Didascalie céleste de Leptine. The document, written in Egypt
between the years 193 and 165 B.C., seems to have been a student’s note-book,
written perhaps during or after a course of lectures.
2 Simplicius, loc. cit., p. 495. 25.
»ὦ ἀἐμππ ee
CH. XVI EUDOXUS : 201
their motion in latitude was believed by Eudoxus to depend exclu-
sively on their elongation from the sun and not on their longitude.
The third sphere had its poles at two opposite points on the zodiac
circle, the poles being carried round in the motion of the second
sphere; the revolution of the third sphere about the poles was
again uniform and took place in a period equal to the synodic
period of the planet or the time which elapsed between two succes-
sive oppositions or conjunctions with the sun. The poles of the third
sphere were different for all the planets, except that they were the
same for Mercury and Venus. The third sphere rotated according
to Simplicius ‘from south to north and from north to south’! (this
followed of course from the position of the poles on the ecliptic) ;
the actual sense of the rotation is not clear from this, but Schia-
parelli’s exposition shows that it is immaterial whether we take the
one or the other. On the surface of the third sphere the poles of
the fourth sphere were fixed, the axis of the latter being inclined to
that of the former at an angle which was constant for each planet
but different for the different planets. And the rotation of the
fourth sphere about its axis took place in the same time as the rota-
tion of the third about its axis but in the opposite sense. On the
equator of the fourth sphere the planet was fixed, the planet having
thus four motions, the daily rotation, the circuit in the zodiac, and
two other rotations taking place in the synodic period.
Simplicius gives the following clear explanation with regard to
the combined effect of the rotations of the third and fourth spheres.
‘The third sphere, which has its poles on the great circle of the
second sphere passing through the middle of the signs of the zodiac,
and which turns from south to north and from north to south, will
carry round with it the fourth sphere which also has the planet
attached to it, and will moreover be the cause of the planet’s move-
ment in latitude. But not the third sphere only; for, so far as it
was on the third sphere (by itself), the planet would actually have
arrived at the poles of the zodiac circle and would have come near
to the poles of the universe ; but, as things are, the fourth sphere,
which turns about the poles of the inclined circle carrying the
planet and rotates in the opposite sense to the third, i.e. from east
to west, but in the same period, will prevent any considerable diver-
_ gence (on the part of the planet) from the zodiac circle, and will
? Simplicius, loc. cit., p. 496. 23.
202 THEORY OF CONCENTRIC SPHERES PArtTi
cause the planet to describe about this same zodiac circle the
curve called by Eudoxus the ippopede, so that the breadth of this
curve will be the (maximum) amount of the apparent deviation of the
planet in latitude, a view for which Eudoxus has been attacked.’ 1
Following up the hint here given, Schiaparelli set himself to
investigate the actual path of a planet subject to the rotations of
the third and fourth spheres only, leaving out of account for the
moment the motions of the first two spheres producing respectively
the daily rotation and the motion along the zodiac. The problem
is, as he says, in its simplest expression, the following. ‘A sphere
rotates uniformly about the fixed diameter AB. P, P’ are two
Fig. 6.
opposite poles on this sphere, and a second sphere concentric with
the first rotates uniformly about P/” in the same time as the former
sphere takes to turn about AB, but in the opposite direction. Misa
point on the second sphere equidistant from the poles P, P’ (in other
words, a point on the equator of the second sphere). Required to find
the path of the point JZ’ This is not difficult nowadays for any
one familiar with spherical trigonometry and analytical geometry ;
but it was necessary for Schiaparelli to show that the solution was
within the powers of Eudoxus. He accordingly develops a solution —
by means of a series of seven propositions or problems involving
only elementary geometrical considerations, which would have
1 Simplicius, loc. cit., pp. 496. 23 -- 497. 5.
{q
“J
CH. XVI EUDOXUS 4% 203
presented no difficulty to a geometer of the calibre of Eudoxus ;
and he finds that, sure enough, the path of 7 in space is a figure
like a lemniscate but described on the surface of a sphere, that is, the
fixed sphere about AZ as diameter. This ‘spherical lemniscate’,
as Schiaparelli calls it, is shown as well as I can show it in the
annexed figure (Fig. 7). Its double point is on the circumference of
the plane section of the sphere which is at right angles to 4B, and
it is symmetrical about that plane as well as about the circumfer-
ence of a circular section which has AZ for diameter and is in what
Schiaparelli calls the ‘fundamental plane’, the plane of the great
circle with diameter AB on which the pole P and the planet // are
Fig. 7.
found at the same moment. The curve is actually the intersection
of the sphere with a certain cylinder touching it internally at the
double point, namely a cylinder with diameter equal to AS,
the sagitta (see Fig. 6) of the diameter of the small circle of
the sphere on which the pole P revolves. But the curve is also
the intersection of ezther the sphere or the cylinder with a certain
cone with vertex QO, axis parallel to the axis of the cylinder
(i.e. touching the circle AOZB at ΟἹ and vertical angle equal to
the ‘inclination’ (the angle AO’P in Fig. 6).} For clearness’ sake
1 Schiaparelli’s geometrical propositions are too long to be quoted here, but
the whole thing can be worked out analytically in a reasonable space. This is
done by Norbert Herz (Geschichte der Bahnbestimmung von Planeten und
204 THEORY OF CONCENTRIC SPHERES Parti
I show in another figure (Fig. 9, p. 206) a right section of the cylinder
by a plane passing through O and perpendicular to AZ in the figure
immediately preceding (Fig. 7).
The arc of the great circle 4OB which bisects the ‘spherical
lemniscate’ laterally is equal in length to the arc QAR of the
great circle dQBR (Figs, 6 and 8) and is of course divided at the
double point O into equal halves of length equal to the are 40.
Kometen, Part I, Leipzig, 1887, pp. 20, 21), and I quote the solution exactly as
he gives it :—
Let AB be the axis of the first sphere, and the circle 4.038 the circle in
which ?, /’, the poles of the second sphere, and 27 the position of the planet,
.are found together at the same moment. Suppose that the motion of the two
spheres is in the direction of the arrows and that, when the circle 4P2 has
moved through an angle 6, PJ/, the circle carrying the planet has also moved
through the same angle, 77 being the position of the planet.
Fig. 8,
Let z be the zuclination AO’P, r the arc of a great circle 4J/, τε (measured
positively downwards) the angle OAM.
Then in the triangle PA we have, since PM = 90°,
cosy = —sinzcos 6,
sinycos(6+) = +coszcos 6,
siny sin (θ- 22) = +sin 6.
Multiplying the second equation by (—sin@) and the third by cos 6, and
adding, we have
sin sin z = sin 8cos θ(1 -- (05 2) = sin?}zsin26.
auleplying the second equation by cos @ and the third by sin 6, and adding,
we have
sin 7.008 # = sin? @+cosi cos? 6
= (cos*}7+sin?} 7) sin? 6+ (cos* 3 7—sin®}2) cos? 4
= cos’}z—sin?$7cos 20
= 1-2sin?4icos* 6,
CH. XVI EUDOXUS 205
The breadth of the ‘lemniscate’, i.e. the Aimear distance between
the two points on either loop of maximum latitude, north and
south, is equal to the diameter of the cylinder, i.e. to the
sagitta AS. The angle at which the curve intersects itself at O is
equal to the inclination (PO’A) of the axes of rotation of the two
spheres. The four points on the curve of greatest latitude, the
double point and the two extreme points at which it intersects
Next, in the triangle 4 OM, if OM = p, and v is the exterior angle at O, we
have, since 40 = 90°,
cosp = sinrcosz,
sinpsiny = 51} 7,31} 24,
sin pcos v = —cos7;
therefore, if £, 7 be the ‘ spherical coordinates’ of J/ with reference to origin O,
we have, in the triangle O17, and by using the results obtained above,
siny = sinvsinp = sin*}/sin2 6,
cot p I—2sin*2Zcos*@
—— = —tanrcosz = — .
cos Vv sinzcos 6
If now we use a system of rectangular coordinates x, y, 2, with origin at Ὁ,
2 being measured along OO’, and x, y being the projections "of the arcs &, on
the plane “4398 at right angles to OO’ (y being positive in the upward direction,
i.e. in the opposite direction to ~, v), we have for the projections ON’, MN,
v', p of ON, MN, v, p respectively
ON =x; MN =-y,
v=,
p = Rsinp,
where 2& is the radius of the sphere.
Consequently x= ρ' ςο5 τ΄ = Rsinpcosv = —Rcos?z,
y =—p'sin’ = —Rsinpsinv = -- ἡ βίῃ 7,31} 26 ;
whence we have
coté =
x= Rsinicos 6,
y = —Rsin*}isin26.
This gives at once the projection of the Aiffofede on the plane AQB as
constructed by Schiaparelli.
So far Norbert Herz. But we can also obtain the remainder of Schiaparelli’s
results, as follows.
We have for z, the third coordinate of 1,
z= R(1—cosp) = R(1—sinr cos 2)
= 2Rsin*}icos?é = Rsin? 47 (1+ cos 26).
Eliminating @ from the equations for y and z, we obtain
(z—Rsin? fz)? +y? = FR sint hi.
Therefore 77 lies on a cylinder which has its axis parallel to 42, touches the
_ Sphere internally at O, and has its radius equal to Rsin?}z, i.e. its diameter
equal to R(1—cos2), which is the sagitta 4S in Fig. 6 and Fig. 8. That is, the
_ Aippopede is the curve of intersection of this cylinder with the sphere.
The sphere being 2°+?+2? = 2z, and the cylinder y*+2* = 2 Rzsin*} ὦ,
the cone is easily found to be
A+ y° +s" = x sec" ὦ
“οὔ THEORY OF CONCENTRIC SPHERES ῬΆΚΤΙ
the ‘fundamental plane’ through AB, divide the curve into eight
arcs which are described by the planet in equal times. Schia-
parelli shows how to construct the projection of the curve upon the
plane through AB perpendicular to the plane which bisects
the curve longitudinally. Describe, he says, a circle with radius
equal to 0.5, the radius of the small circle described by P (Fig. 6).
Then, with the same centre, draw a smaller circle with radius equal
to half the sagitta AS. Divide the lesser circle into any number of
equal parts, say 8, as at the points marked o,1, 2, 3...7 round the
circle, and suppose the same points marked again with the numbers
8,9, 10. ..15 respectively ; divide the greater circle into double the
ο΄
Fig. 9. Fig. Io.
number of equal parts as at the points marked 0, I, 2, 3,4,5...15
(arranging the points so that those marked o are opposite one
another on a common diameter XX, while the numbers go round
in the same sense). Draw YY through the centre perpendicular
to XX, and through the points of division of the outer circle
draw chords parallel to YY, and through the points of division
of the inner circle straight lines parallel to XX. The points of
intersection of the lines give a series of points on the projection
of the ‘spherical lemniscate’. These points are again marked in
the figure by the numbers 0,1, 2....15. The projection of the
position of the planet moves along this curve in the direction indi-
cated by the successive numbers.
CH. XVI EUDOXUS 207
There is no doubt that Schiaparelli has restored, in his ‘ spherical
lemniscate ’, the Aippopede of Eudoxus, the fact being confirmed by
the application of the same term Aippopede (horse-fetter) by Proclus*
to a plane curve of similar shape formed by a plane section of an
anchor-ring or fore touching the tore internally and parallel to its
axis.
So far account has only been taken of the motion due to the com-
bination of the rotations of the third and fourth spheres. But 4, 5,
the poles of the third sphere (Figs. 6-8), are carried round the zodiac
or ecliptic by the motion of the second sphere and in a time equal to
the ‘ zodiacal’ period of the planet. Now the longitudinal axis of
the ‘spherical lemniscate’ (the arc of the great circle bisecting it
longitudinally) always lies on the ecliptic. We may therefore sub-
stitute the ‘lemniscate’ moving bodily round the ecliptic for the
third and fourth spheres, the planet meantime moving round
the ‘lemniscate in the manner described above. The combination
of the two motions (that of the ‘ lemniscate’ and that of the planet
on it) gives the motion of the planet through the constellations.
The motion of the planet round the curve is an oscillatory motion,
now forward in acceleration of the motion round the ecliptic due to
the second sphere, now backward in retardation of the same motion ;
the period of the oscillation is the period of the synodic revolution,
and the acceleration and retardation occupy half the period respec-
tively. When the retardation in the sense of longitude due to the
backward oscillation is greater than the speed of the forward motion
of the ‘ lemniscate’ itself, the planet will for a time have a retrograde
motion, at the beginning and end of which it will appear stationary
for a little while, when the two opposite motions balance each
other. The greatest acceleration of the planet in longitude, and the
greatest retardation (or the quickest rate of retrograde motion),
occur at the times when the planet passes through the double point
ofthe curve. The movements must therefore be so combined that
the planet is at the double point and moving in the forward direction
at the time of superior conjunction with the sun, when the apparent
speed of the planet in longitude is greatest, while it is again at
the double point but moving in the backward direction when it
__ is in opposition or inferior conjunction, at which times the apparent
* Proclus, Comm. on Eucl. I, ed. Friedlein, p. 112. 5.
208 THEORY OF CONCENTRIC SPHERES Parti
retrograde motion of the planet is quickest. This combination of
motions will be accompanied by motion in latitude within limits
defined by the breadth of the lemniscate ; the planet will, during
a synodic revolution, twice reach its greatest north and south lati-
tude respectively and four times cross the ecliptic.
The actual shape of the Azppopede and its dimensions relatively to
the sphere on which it is drawn are fully determined when we know
the inclination of the axis of the fourth sphere to that of the third,
since they depend on this inclination exclusively. In order to test
the working of the theory with regard to the several planets we
need to know three things, (1) the inclination referred to, (2) the
period of the ‘zodiacal’ or sidereal revolution, (3) the synodic
period, in the case of each planet. We are not told what angles of
inclination Eudoxus assumed, but the zodiacal and synodic periods
which he ascribed to the five planets are given in round figures by
Simplicius The following is a comparison of Eudoxus’s figures
with the modern values:
Synodic period Zodiacal period
ὩΣ i ace nic
Eudoxus Modern value _ Eudoxus Modern value
Saturn 13 months 378 days 30 years 29 years 166 days
Jupiter 13 months 399 days. 12 years II years 315 days
Mars 8 months 20 days 780 days 2 years I year 322 days
Mercury 110 days 116 days I year I year
Venus 19 months 584 days I year I year
Except in the case of Mars, these figures are tolerably accurate,
while the papyrus purporting to contain the Avs Eudoxi gives for
the synodic period of Mercury the exact modern figure of 116 days;
it is therefore clear that Eudoxus went on the basis of very careful
observations, whether he obtained the results from Egypt or from
Babylonian sources. As unfortunately the inclinations assumed by
Eudoxus (the third factor required for the reconstruction of the
system) are not recorded, Schiaparelli has to conjecture them for
himself. Assuming that they would be such as to produce ‘lem-
niscates ’ which would give arcs of retrogradation corresponding to
those actually observed, he takes the known retrograde arc of
Saturn (6°) and observes that by the help of the zodiacal period
of 30 years and the synodic period of 13 months, and by assuming
? Simplicius, loc. cit., pp. 495. 26-9, 496. 6-9.
———— ἾΩΝΣ
CH. XVI EUDOXUS : 209
6° as the ‘inclination’, a retrograde arc of about 6° is actually
obtained; the length of the Aippopede (the arc of the great circle of
the sphere bisecting the curve longitudinally) is 12°, and the half of
its breadth about 9’, a maximum deviation from the ecliptic which
would of course be imperceptible to the observers of those days.
In the case of Jupiter, assuming an inclination of 13°, and conse-
quently a Aippopede of 26° in length and twice 44’ in breadth,
with a zodiacal period of 12 years and a synodic period of 13
months, he deduces a retrograde arc of about 8°; and again the
divergence in latitude of 44’ would hardly be noticed. For these
two planets, therefore, Eudoxus’s method gave an excellent solution
of Plato’s problem of finding how the motion of the planets can be
accounted for by a combination of uniform circular motions.
With Mars, however, the system fails. We have no means of
knowing how Eudoxus came to put the synodic period at 8 months
and 20 days, or 260 days, whereas it is really 780 days, or three
times as long. But, whether we take 780 days or 260 days, the
theory does not account for the facts. If the synodic period is 780
days, and we take for the length of the Azppopede the greatest arc
permissible according to Simplicius’s account, namely an arc of
180°, corresponding to an ‘inclination’ of go’, the breadth of the
curve becomes 60°, so that Mars ought to diverge in latitude to
the extent of 30°. Also, even under this extreme hypothesis, the
retrograde motion of Mars on the Aippopede cannot reach a speed
equal to that of the direct motion of the Aippopede itself along the
ecliptic (the zodiacal period being 2 years); consequently Mars
- should not have any retrograde motion at all and should only move
very slowly at opposition. To obtain a retrograde motion at all
we should require an ‘inclination’ greater than go’, and consequently
the third and fourth spheres would rotate in the same instead of the
opposite sense, which is contrary to Simplicius’s statement; and,
even if this were permissible, there is the objection that Mars’s
deviations in latitude would exceed 30°, and Eudoxus would never
have assumed such an amount of deviation. On the other hand, to
assume a synodic period of 260 days would produce a retrograde
motion ; by assuming an inclination of 34° we get 68° as the length
of the Aippopede and a maximum deviation in latitude of 4° 53’,
which is not very far from the true deviation; the retrograde arc
1410 A
210 THEORY OF CONCENTRIC SPHERES parti
becomes about 16°, which is little greater than that disclosed by
observation. This way of producing approximate agreement with
observed facts may perhaps have been what led Eudoxus to assume
a synodic period one third as long as the real period; but unfor-
tunately the hypothesis gives two retrograde motions outside the
oppositions with the sun, and six stationary points, four of which
have no real existence. . .
As regards Mercury and Venus, inasmuch as their mean positions
coincide with the mean position of the sun, Eudoxus must have
assumed that the centre of the Azppopede always coincides with
the sun. This centre being on the ecliptic and at a distance of
go” from each of the poles of rotation of the third sphere, the poles
of the third sphere of Mercury and the poles of the third sphere
of Venus coincide, a fact for which we have the independent
testimony of Aristotle in the passage quoted above. As the mean
position of each of the two planets coincides with that of the sun,
and the greatest elongation of each from the sun is half the length
of the corresponding ippopede, Eudoxus doubtless determined
the ‘inclination’ from the observed elongations. In the case of
Mercury, with a maximum elongation of 23°, the length of the ©
hippopede becomes 46°, and the half of its breadth or the greatest
latitude is 2°14’, nearly as great as the observed deviation. The
retrograde arc for Mercury would be about 6°, which is much
smaller than the true length; but, as this mistake occurs in a part
of the synodic circuit which cannot be observed, the theory cannot
be blamed for this. In the visible portions of the circuit the
longitudes are represented with fair accuracy, though the times
of greatest elongation do not exactly agree with the facts. For
Venus, taking the greatest elongation (and consequently the ‘in-
clination’) at 46°, we have a hippopede 92° in length, and a half-
breadth or maximum latitude of 8°54’, which is roughly in agreement
with the greatest latitude as observed.’ But, since the synodic period
as. given by Eudoxus, 570 days, is more than 14 times the zodiacal
period, Venus, like Mars, can never have a retrograde motion; ©
and this error cannot be avoided whatever value we choose to
substitute for 46° as the inclination. Another serious fault of the
theory is that it requires Venus to take the same time to pass
from the extreme eastern point to the extreme western point
Wee Sag os
CH. XVI EUDOXUS i τι
of the ἀξῤῥοῤεάε as it takes to return from the extreme
western to the extreme eastern point, whereas in fact Venus takes
441 days (out of the synodic period of 584 days) to pass from the
greatest eastern to the greatest western elongation and only 143
days to return from the greatest western to the greatest eastern
elongation. As regards latitude, too, the imperfection of the theory
is more marked in the case of Venus than in that of the other
planets; for the Aippopede intersects the ecliptic four times, once
at each extremity, and twice at the double point ; consequently
the planet ought to cross the ecliptic four times during each synodic
period, which is not the case, as the latitude of Venus is only κα:
twice during each sidereal revolution.
To sum up. For the sun and moon the hypothesis of Eudoxus
sufficed to explain adequately enough the principal phenomena,
except the irregularities due to the eccentricities, which were either
unknown to Eudoxus or neglected by him. For Jupiter and
Saturn, and to some extent for Mercury also, the system was
capable of giving on the whole a satisfactory explanation of their
motion in longitude, their stationary points and their retrograde
motions; for Venus it was unsatisfactory, and it failed altogether
in the case of Mars. The limits of motion in latitude represented
by the various Aippopedes were in tolerable agreement with observed
facts, although the periods of the deviations and their places in
the cycle were quite wrong. But, notwithstanding the imper-
fections of the system of homocentric spheres, we cannot but
recognize in it a speculative achievement which was worthy of the
great reputation of Eudoxus and: all the more deserving of admira-
tion because it was the first attempt at a scientific explanation of
the apparent irregularities of the motions ofthe planets. And,
as Schiaparelli says, if any one, as the result of a superficial study
of the theory, finds it complicated, let him remember that in
none of his hypotheses does Eudoxus make use of more than three
constants or elements, namely the epoch of a superior conjunction,
the period of sidereal revolution (on which the synodic period is
dependent), and the inclination to one another of the axes of the
third and fourth spheres, which inclination determines completely
the dimensions of the £7ppopede ; whereas in our time we require, for
the same purpose, six elements in the case of each planet.
P2
212 THEORY OF CONCENTRIC SPHERES PARTI
Eudoxus died in 355 B.C. at the age of 53. His doctrine of
homocentric spheres was further studied in his school. Menaechmus,
the reputed discoverer of the conic sections, and one of his pupils,
is mentioned as a supporter of the theory.! Polemarchus of
Cyzicus, a friend of Eudoxus, is also mentioned as having studied
the subject, and, in particular, as having been aware of the objection
raised to the system of homocentric spheres on the ground that
the difference in the brightness of the planets, especially Venus and
Mars, and in the apparent size of the moon, at different times,
showed that they could not always be at the same distance from
us; ‘ Polemarchus appears to have been aware of it’ (the variation
in the distances of each planet) ‘ but to have neglected it as not per-
ceptible, because he preferred the assumption that the spheres them-
selves are about the centre of the universe’.? But. it is Callippus
to whom definite improvements in the system are attributed.
Callippus of Cyzicus, the most famous and capable astronomer of
his time, probably lived between 370 and 300 B.C.; he was therefore
perhaps too young to be a pupil of Eudoxus himself; but he studied
with Polemarchus and he followed ὃ Polemarchus to Athens, where
‘he stayed with Aristotle correcting and completing, with Aristotle’s
help, the discoveries of Eudoxus’* This must have been during
the reign of Alexander the Great (336-323 B.C.), at which time
Aristotle was in Athens; it must also have been about the time
when Callippus brought out his improvement of Meton’s luni-solar
cycle, since the beginning of Callippus’s cycle was in 330 B.C.
(28th or 29th June). Aristotle himself gives Callippus the sole
credit for certain improvements on Eudoxus’s system ; immediately
after the passage above quoted from the Metaphysics he says:
‘Callippus agreed with Eudoxus in the position he assigned to
the spheres, that is to say, in their arrangement in respect of
distances, and he also assigned the same number of spheres as
Eudoxus did to Zeus and Kronos respectively, but he thought it
necessary to add two more spheres in each case to the sun and
moon respectively, if one wishes to account for the phenomena,
and one more to each of the other planets.’®
1 Theon of Smyrna, ed. Hiller, pp. 201. 25 — 202. I.
2 Simplicius on De caelo ii. 12, p. 505. 21, Heib.
3. μετ᾽ ἐκεῖνον εἰς ᾿Αθήνας ἐλθών. Schiaparelli translates μετ᾽ ἐκεῖνον as if it were
per’ ἐκείνου, ‘with him’.
4. Simplicius, op. cit., p. 493. 5-8. 5 Aristotle, Metaph. A. 8, 1073 Ὁ 32-8.
¢
§
ἡ
Ε
CH. XVI CALLIPPUS 213
Simplicius says that no book by Callippus on the subject was extant
in his time, nor did Aristotle give any explanation of the reasons why
Callippus added the extra spheres ;
‘but Eudemus shortly stated what were the phenomena in explana-
tion of which Callippus thought it necessary to assume the additional
spheres. According to Eudemus, Callippus asserted that, assuming
the periods between the solstices and equinoxes to differ to the
extent that Euctemon and Meton held that they did, the three
spheres in each case (i.e. for the sun and moon) are not sufficient
to save the phenomena, in view of the irregularity which is observed
in their motions. But the reason why he added the one sphere
which he added in the case of each of the three planets Ares,
Aphrodite, and Hermes was shortly and clearly stated by Eudemus.’!
As regards the planets therefore, although we are informed that
Eudemus gave the reason for the addition of a fifth sphere in each
case, we are not told what the reason was, and we can only resort
to conjecture. Schiaparelli observes that, since Callippus was
content with Eudoxus’s hypothesis about Jupiter and Saturn, we
may conclude that their zodiacal inequality was still unknown to
him, although it can reach the value of 6° in each case, and also
that he regarded their deviations in latitude as non-existent or
negligible. But the glaring deficiencies in the theory of Eudoxus
when applied to Mars would suggest the urgent need for some
improvement which should, in particular, produce the necessary
retrograde motion in this case without the assumption of a synodic
period different from the true one. It is sufficiently probable there-
fore that the fifth sphere was intended for the purpose of satisfying
this latter condition. Schiaparelli observes that, on the assumption
of a synodic period of 780 days, it is possible, by a combination
of three spheres taking the place of Eudoxus’s last two (the third
and fourth), to obtain a retrograde motion agreeing sufficiently with
observed facts, and this can be done in various ways without
involving too considerable deviations in latitude; he gives, as the
simplest arrangement leading to the desired result, the following:
Let 408 (Fig. 11) represent the ecliptic and A, B two opposite
points on it which make the circuit of the zodiac in the zodiacal
period of Mars. Leta sphere (the third of Eudoxus) revolve about
1 Simplicius on De cae/o ii. 12, p. 497. 17-24.
214 THEORY OF CONCENTRIC SPHERES ῬΑΈΤΙ
A, B as poles in the period of the synodic revolution. Take any
point P, on the equator of this sphere as pole of another sphere (the
fourth) rotating about its poles at twice the speed of the third sphere,
in the opposite direction to the latter, and carrying with it P,,
distant from P, by an arc P, P, (which we will call the ‘ inclination’).
About P, as pole, let a fifth sphere rotate at the same speed and
—
Fig. 11.
in the same direction as the third, carrying the planet fixed on
its equator at the point J7. It is easy to see that, if at the
beginning of the motion the three points P,, P,, 27 lie on the
ecliptic in the order AP,P,MB, then at any time afterwards
the angle ¢ at A will be equal to the angle at P, between P,P, and
P,M, while the angle AP,P, at P, will be twice as large. And,
since AP, = P,M = 90’, the planet J will in the synodic period
describe a curve adjacent to the ecliptic and symmetrical about it
which will take a different form according to the value given to the
‘inclination’ P,P,. This curve will for certain values of P,P,
extend considerably in length but little in breadth and, as it has
its centre at O midway between the poles A, BZ, it will, like the
hippopede, produce a direct and retrograde motion alternately, but
CH. XVI ~ CALLIPPUS 215
will have the advantage over the Aippopede that it can give the
planet in the neighbourhood of O a much greater direct and retro-
grade velocity with the same motion in latitude. Hence it is
capable of giving the planet a retrograde motion where the Aippopede
fails to do so. If, for example, P,P, is put equal to 45°, the curve
takes a form like that shown in the figure (in projection). The
greatest deviation in latitude does not exceed 4°11’, the curve has
a length along the ecliptic of 953°, and has two triple points near
the ends at a distance of 45° from the centre O. When the planet
is passing O, its velocity is 1-2929 times the speed of the rotation
of P, about AB. As the period of rotation of P, about AB is
equal to the synodic period, 780 days, the daily motion of P,
is 360°/780 or 0-462, which multiplied by 1-2929 gives 0°-597 as
the daily retrograde motion on the curve at O. And, as O has
a direct motion on the ecliptic of 360° / 686 = 0°-525, the resulting
daily retrograde motion is 0-072, which is a reasonable approxima-
tion to the fact.
Similarly an additional sphere might be made to remove the
imperfection of the theory as applied to Venus. If the ‘inclination’
P,P, is made 45°, the greatest elongation is 473°, which is very near
the truth ; and the different speed of the planet in the four parts of
the synodic revolution is also better accounted for, since, in the
curve above drawn, the passage from one triple point to the other
takes one fourth of the time, the same passage back again another
fourth, while the remaining two fourths are occupied by the very
slow motion round the small loops at the ends. For Mercury the
theory of Eudoxus gave a fairly correct result, and doubtless it
would be possible by means of another sphere to attain a still greater
degree of accuracy.
According to Eudemus, Callippus added two new aliases (making
five) in the case of the sun, in order to account for the unequal
motion in longitude which had been discovered a hundred years
earlier by Meton and Euctemon. Euctemon had made the length
of the seasons (beginning with the vernal equinox) 93, 90, 90, and 92
days respectively, showing errors ranging from 1-23 to 2-01 days;
this was about 430 B.c. Callippus, about 330 B.C., made the cor-
responding lengths 94, 92, 89, 90 days respectively,’ the errors
1 Ars Eudoxi, ὃ 55.
216 THEORY OF CONCENTRIC SPHERES Parti
ranging from oc-08 to 0-44 days only ; this shows the great advance
made in observations of the sun during the century between the two
dates. Now Callippus had only to retain the three spheres assumed
by Eudoxus for the sun and then to add two spheres, (1) a sphere
with its poles on the third sphere of Eudoxus which described the
orbit of the sun at uniform speed in the course of a year, and (2) a
sphere carrying the sun on its equator and having its poles on the
preceding sphere and its axis slightly inclined to the axis of
the same sphere; the second of these spheres would rotate at the
same speed as the first but in the opposite direction. If the inclina-
tion of the axes is equal to the greatest inequality (which was for
Callippus, as it is for us, 2°), the two new spheres give for the sun a
hippopede, the length of which along the ecliptic is 4° and the breadth
nearly 1’ on each side of it; this representation of the motion of
the sun is almost as accurate as that obtained later by means of the
eccentric circle and the epicycle.
Simplicius’s explanation of the reason why Callippus added two
spheres in the case of the moon also is rather confused, because he
tries to deal with the sun and moon in one sentence. But he pre-
sumably meant that the reason in the case of the moon was similar
to the reason in the case of the sun; in other words, Callippus was
aware of the inequality in the motion of the moon in longitude.
This inequality, which often reaches as much as 8°, would neces-
sarily reveal itself as soon as the intervals between a large number
of successive lunar eclipses were noted and compared with the
corresponding longitudes of the moon, which can in this case easily
be deduced from those of the sun. The inclination between the
axes of the two new spheres would in this case have to be taken
equal to the mean inequality of 6°, and a Aippopede of 12° would
mean a maximum deviation from the moon’s path of 9’, so that
the moon’s motion in latitude would not be sensibly affected.
Whether Callippus actually arranged his additional spheres in
the way suggested by Schiaparelli or not, the improvements which
he made were doubtless of the nature indicated above; and his
motive was that of better ‘saving the phenomena ’, his comparison
of the theory of Eudoxus with the results of actual observation
having revealed differences sufficiently pronounced to necessitate a
remodelling of the theory.
i i τ
ee a νν
Ἢ ΨΥ ὙῪ Ὅν
CH. XVI ARISTOTLE 217
We now come to the changes which Aristotle thought it neces-
sary to make in the system of Eudoxus and Callippus. We have
seen that that system was purely geometrical and theoretical ;
there was nothing mechanical about it. Aristotle’s point of view
was entirely different. Aristotle, as we shall see, transformed the
purely abstract and geometrical theory into a mechanical system
of spheres, i.e. spherical shells, in actual contact with one another ;
this made it almost necessary, instead of assuming separate sets of
spheres, one set for each planet, to make all the sets part of one
continuous system of spheres. For this purpose yet other spheres
hhad to be added which Aristotle calls ‘ unrolling ’ or ‘ back-rolling’
(aveXirrovea),' by which is meant ‘reacting’ in the sense of counter-
acting the motion of certain of Eudoxus’s and Callippus’s spheres
which, for the sake of distinction, we may with Schiaparelli call
‘deferent’. Aristotle’s theory and its motive are given quite clearly
in the chapter of the Metaphysics to which reference has already
been made. The words come immediately after the description of
Callippus’s additions to the theory.
‘But it is necessary, if the phenomena are to be produced by all
the spheres acting in combination (συντεθεῖσαι), to assume in the
case of each of the planets other spheres fewer by one [than
the spheres assigned to it by Eudoxus and Callippus] ; these latter
spheres are those which unroll, or react on, the others in such a way
as to replace the first sphere of the next lower planet in the same
position [as if the spheres assigned to the respective planets above
it did not exist], for only in this way is it possible for a combined
system to produce the motion of the planets. Now the deferent
spheres are, first, eight [for Saturn and Jupiter], then twenty-five
more [for the sun, the moon, and the three other planets]; and οὗ.
these only the last set [of five] which carry the planet placed lowest
[the moon] do not require any reacting spheres. Thus the reacting
spheres for the first two bodies will be six, and for the next four will
be sixteen ; and the total number of spheres, including the deferent
spheres and those which react on them, will be fifty-five. If, how-
ever, we choose not to add to the sun and moon the [additional
deferent] spheres we mentioned [i.e. the two added to each by
Callippus], the total number of the spheres will be forty-seven. So
much for the number of the spheres.’ ?
* Theophrastus, we are told (Simplicius, loc. cit., p. 504. 6), called them
>
ἀνταναφέρουσαι.
3 Aristotle, Metaph. A. 8, 1073 b38-1074a15.
218 THEORY OF CONCENTRIC SPHERES ParTI
The way in which the system would work is explained very .
diffusely by Sosigenes in Simplicius;1 Schiaparelli puts the matter
quite clearly and shortly, thus. The different sets of spheres being
merged into one, it is necessary to provide against the motion of
the spheres assigned to a higher planet affecting the motion of the
spheres assigned to a lower planet. For this purpose Aristotle
interpolated between the last (the innermost) sphere of each planet
and the first (or outermost) sphere of the planet next below it
a certain number of spheres called ‘reacting’ spheres. Thus, sup-
pose A, B, C, D to be the four spheres postulated for Saturn, 4
being the outermost and D the innermost on which the planet is
fixed. If inside the sphere D we place a first reacting sphere D’
which turns about the poles of D with equal speed, but in the
opposite sense, to D, the rotations of D and D’ will mutually
cancel each other and any point of D’ will move as though
it was rigidly connected with the sphere C. Again, if we place
inside the sphere D’ a second reagent sphere C’ rotating about
the same poles with C and with equal speed, but in the opposite
sense, the rotations of C and C” cancel each other, and any point |
of C’ will move as if it were rigidly connected with the sphere B.
Lastly, if inside ( a third reagent sphere B’ is introduced which
rotates about the same poles with B and at the same speed but in
the opposite sense, the rotations of B and J’ will cancel each other
and any point of B’ will move as if it were rigidly connected with
the sphere A. But, as A is the outermost sphere for Saturn, A is the
motion of the sphere of the fixed stars; hence B’ will move in
the same way as the sphere of the fixed stars; and consequently
Jupiter's spheres can move inside B’ as if the spheres of Saturn did
not exist and as if B’ itself were the sphere of the fixed stars.
Hence it is clear that, if ~ is the number of the deferent spheres
of a planet, the addition of ~—1 reacting spheres inside them
neutralizes the operation of ~—1 of the original 7 spheres and pre-
vents the inner set of spheres from being disturbed by the outer
set. The innermost of the »—1 reacting spheres moves, as above
shown, in the same way as the sphere of the fixed stars. But the
first sphere of the next nearer planet (as of all the planets) is also a
sphere with the same motion as that of the sphere of the fixed
? Simplicius on De cae/o ii. 12, pp. 498. 1 - 503. 9.
CH. XVI ARISTOTLE Ξ 219
stars, and consequently we have two spheres, one just inside the
other, with one and the same motion, that is, doing the work of one
sphere only. Aristotle could therefore have dispensed with the
second of these, namely the first of the spheres belonging to
the inner planet, without detriment to the working of his system ;
and, as the number of ‘ planets’ inside the outermost, Saturn, is six,
he could have saved six spheres out of his total number.
Aristotle omits, as unnecessary, any reacting spheres for the
last and innermost planet, the moon. Yet, as Martin points out,'
Aristotle should have realized that, strictly speaking, the account
which he gives in the Mefcorologica of shooting stars, comets, and
the Milky Way necessitates the introduction of four reacting
spheres below the moon. For, according to Aristotle, these
phenomena are the effects of exhalations rising to the top of the
sublunary sphere and there coming into contact with another
warm and dry substance which, being the last layer of the sublunary
sphere and in contact with the revolution of the outer heavenly
sphere, is carried round with it ; the rising exhalations are kindled
by meeting and being caught in the other substance and are carried
round with it. Hence there must be a sphere below the moon
which has the same revolution as that of the sphere of the fixed
stars, in order that comets, &c., may be produced and move as they
are said todo. The four inner spheres producing the moon’s own
motion should therefore be neutralized as usual by the same
number of reacting spheres.
As it is, however, the hypotheses of Callippus, with the additions
of spheres actually made by Aristotle, work out thus:
Deferent spheres Reacting spheres
For Saturn 4 aa
» Jupiter 4 3
» Mars 5 4
» Mercury 5 4
3, Venus 5 4
» Sun 5 4
» Moon 5 ο
Total 33 + 22 = 55
-In saying that, if Callippus’s additional spheres for the sun and
moon are left out, the total number of spheres becomes 47, it would
1 Mémoires de PAcad. des Inscr, et Belles-Letires, xxx. 1881, pp. 263-4.
220 THEORY OF CONCENTRIC SPHERES PARTI
seem that Aristotle made an arithmetical slip;1 for the omission
would reduce the number 55 by 6 (4 deferent and 2 reacting), not
by 8, and would leave 49, not 47. The remark would also seem
to show that Aristotle did not feel quite certain that the two
additional spheres assumed by Callippus for the sun’ and moon
respectively were really necessary. We may compare the passage
in the De caelo where he definitely regards the sun and moon as
having fewer motions than some of the planets; in that passage
he endeavours to explain two ‘difficulties’ (ἀπορίαι), one of which
is stated as follows:
‘ What can be the reason why the principle that the bodies which
are at a greater distance from the first motion [the daily rotation
of the sphere of the fixed stars] are moved by more movements
does not apply throughout, but it is the middle bodies which have
most movements? For it would appear reasonable that, as the
first body [the sphere of the fixed stars] has one motion only,
the nearest body to it should be moved by the next fewest move-
ments, say two, the next to that by three, or in accordance with |
some other similar arrangement. But in practice the opposite is
what happens; for the sun and moon are moved by fewer move-
ments than some of the planets, and yet the latter are further from
the centre and nearer the first body [the sphere of the fixed stars]
than the sun and moon are. ‘In the case of some planets this is
even observable by the eye; for, at a time when the moon was
halved, we have seen the star of Ares go behind it and become
hidden by the dark portion of the moon and then come out at the
bright side of it. And the Egyptians and Babylonians of old
whose observations go back a great many years, and from whom
we have a number of accepted facts relating to each of the stars,
tell us of similar occultations of the other stars.’ 2
1 There are other explanations, but they are all somewhat forced, and involve
greater difficulties than they remove (see Simplicius on De caelo, pp. 503. 10 --
504. 3, and Martin, loc. cit., pp. 265-6). A further reduction of the number 49
to 47 which might have been, but obviously was not, made by Aristotle, is
indicated by Martin (loc. cit., p. 268) and by Dreyer (Planetary Systems,
p- 114 note). Aristotle might have abolished the ‘¢hzrd’ of the sun’s spheres
(as well as the fourth and fifth); this would have been a real improvement,
since the ‘third’ sphere was meant to explain a movement which did not exist,
namely, the supposed movement of the sun in latitude; the number of the
spheres would thus have been reduced by two (one deferent and one reacting).
But Aristotle had not the knowledge necessary to enable him to suggest this
improvement.
2 De caelo ii. 12, 291 Ὁ 29-292a9.
CH. XVI ARISTOTLE 221
Aristotle’s explanation is teleological, based on comparison with
things which have life and are capable of action. We may perhaps
say that that thing is in the best state which possesses the good
without having to act at all, while those come nearest the best
state which have to perform the fewest acts.‘ Now the earth is
in the most happy state, being altogether without motion. The
bodies nearest to it have few movements; they do not attain the
ideal, but come as near as they can to ‘the most divine principle’.
The ‘first heaven’ [the sphere of the fixed stars] attains it at once
by means of one movement only ; the bodies between the first and
the last [the last being the sun and the moon] attain it but only by
means of a greater number of movements.”
The theory of concentric spheres was pursued for some time after
Aristotle. Schiaparelli conjectures that even Archimedes still held
to it. Autolycus, the author of the treatises On the moving sphere
and Ox risings and settings, who lived till the end of the fourth
or the beginning of the third century B.C., is said to have been
the first to try, presumably by some modification of the theory,
to meet the difficulties which had been seen from the first and
were doubtless pointed out with greater insistence as time went on.
What was ultimately fatal to it was of course the impossibility
of reconciling the assumption of the invariability of the distance
of each planet with the observed differences in the brightness,
especially of Mars and Venus, at different times, and the apparent
difference in the relative sizes of the sun and moon. The quotation
by Simplicius from Sosigenes on this subject is worth giving in full.*
‘Nevertheless the theories of Eudoxus and his followers fail to
save the phenomena, and not only those which were first noticed
at a later date, but even those which were before known and
actually accepted by the authors themselves. What need is there
for me to mention the generality of these, some of which, after
Eudoxus had failed to account for them, Callippus tried to save,—
if indeed we can regard him as so far successful? I confine my-
self to one fact which is actually evident to the eye; this fact
no one before Autolycus of Pitane even tried to explain by means
of hypotheses (διὰ τῶν ὑποθέσεων), and not even Autolycus was
able to do so, as clearly appears from his controversy with
* De caelo ii. 12, 292 a 22-4. ? Ibid. 292 b 10-25.
3 Simplicius on De caedo, pp. 504. 17 -- 505. 11, 505. 19-506. 3.
222 THEORY OF CONCENTRIC SPHERES ParTI
Aristotherus!. I refer to the fact that the planets appear at
times to be near to us and at times to have receded. This is
indeed obvious to our eyes in the case of some of them; for the
star called after Aphrodite and also the star of Ares seem, in
the middle of their retrogradations, to be many times as large, so
much so that the star of Aphrodite actually makes bodies cast
shadows on moonless nights. The moon also, even in the perception
of our eye, is clearly not always at the same distance from us, because
it does not always seem to be the same size under the same
conditions as to medium. The same fact is moreover confirmed
if we observe the moon by means of an instrument; for it is at
one time a disc of eleven fingerbreadths, and again at another
time a disc of twelve fingerbreadths, which when placed at the
same distance from the observer hides the moon (exactly) so that
his eye does not see it. In addition to this, there is evidence for
the truth of what I have stated in the observed facts with regard
to total eclipses of the sun; for when the centre of the sun, the
centre of the moon, and our eye happen to be in a straight line,
what is seen is not always alike; but at one time the cone which
comprehends the moon and has its vertex at our eye comprehends
the sun itself at the same time, and the sun even remains invisible
to us for a certain time, while again at another time this is so far —
from being the case that a rim of a certain breadth on the outside
edge is left visible all round it at the middle of the duration of the
eclipse. Hence we must conclude that the apparent difference in
the sizes of the two bodies observed under the same atmospheric
conditions is due to the inequality of their distances (at different
times). ... But indeed this inequality in the distances of each star
at different times cannot even be said to have been unknown to
the authors of the concentric theory themselves. For Polemarchus
of Cyzicus appears to be aware of it, but to minimize it as being
imperceptible, because he preferred the theory which placed the
spheres themselves about the very centre in the universe. Aristotle
too, shows that he is conscious of it when, in the Physical Problems,
he discusses objections to the hypotheses of astronomers arising
from the fact that even the sizes of the planets do not appear to be
the same always. In this respect Aristotle was not altogether
satisfied with the revolving spheres, although the supposition that,
being concentric with the universe, they move about its centre
attracted him. Again, it is clear from what he says in Book A
of the Metaphysics that he thought that the facts about the move-
ments of the planets had not been sufficiently explained by the
1 Apparently a contemporary of Autolycus and, like him, a mathematician.
The famous poet Aratus appears to have been a pupil of Aristotherus (Buhle’s
Aratus, Leipzig, 1793, vol. i, p. 4).
CK. XVI EARLY CRITICISMS 223
astronomers who came before him or were contemporary with him.
At all events we find him using language of this sort: “(on the ques-
tion how many in number these movements of the planets are), we
must for the present content ourselves with repeating what some
of the mathematicians say, in order that we may form a notion and
our mind may have a certain definite number to apprehend ; but
for the rest we must investigate some matters for ourselves and
learn others from other investigators, and, if those who study these
questions reach conclusions different from the views now put forward,
we must, while respecting both, give our adherence to those which
are the more correct ”’.?
Schiaparelli observes that we must not be misled by these
attempts to father on Aristotle doubts as to the truth of the theory
of homocentric spheres; the object is to make an excuse for the
line taken by the later Peripatetics in getting away from the
revolving spheres of Aristotle and going over to the theory of
eccentric circles and epicycles.
The allusion by Sosigenes to annular eclipses of the sun is
particularly interesting, as it shows that he had much more correct
notions on this subject than most astronomers up to Tycho Brahe.
Even at the beginning of the seventeenth century, says Schiaparelli,
some persons doubted the possibility of a total eclipse. Proclus
points out that the views of Sosigenes are inconsistent with the
opinion of Ptolemy that the apparent diameter of the sun is always
the same, while that of the moon varies and is only at its apogee
the same as that of the sun. ‘If the latter contention is true,’
says Proclus,? ‘then that is not true which Sosigenes said in his
work On the revolving (or reacting) spheres, namely, that in eclipses
at perigee the sun is seen to be not wholly obscured, but to overlap
with the edges of its circumference the circle of the moon, and to
give light without hindrance. For if we accept this statement,
then either the sun will show variation in its apparent diameter,
or the moon will not, at its apogee, have its apparent diameter,
as ascertained by observation, the same as that of the sun.’
Cleomedes, too, alludes to the views of some of the more ancient
astronomers who held that in total eclipses of the sun a bright rim
1 Aristotle, Mefaph. A. 8, 1073 Ὁ 10-17.
* Proclus, Hyfotyposis astronomicarum positionum, c. 4, δὲ 98, 99, p-130,
16-26, ed. Manitius.
224. THEORY OF CONCENTRIC SPHERES
of the sun was visible all round (Cleomedes’ words would imply
that they asserted this to be true for αὐ total eclipses, which is
presumably a misapprehension), but adds that the statement has
not been verified by observation.’ Schiaparelli infers that Sosigenes
was aware of the variations of the apparent diameter of the sun,
as well as of the moon, and thinks that his object in alluding
to annular eclipses in the above passage quoted from Simplicius,
where the subject is again that of revolving spheres, was to use
as an argument against that theory the fact that the distance
of the sun from us is variable.
1 Cleomedes, De motu circulari ii. 4, p. 190, 19-26, Ziegler.
XVII
τὴς ARISTOTLE (continued)
IT was convenient to give Aristotle’s modified system of concen-
tric spheres in close connexion with the systems of Eudoxus and
Callippus, and to reserve the rest of his astronomy for separate
treatment. While his modification of the beautiful theory of
Eudoxus and Callippus was far from being an improvement,
Aristotle rendered real services to astronomy in other respects.
Those services consisted largely of thoughtful criticisms, generally
destructive, of opinions held by earlier astronomers, but Aristotle
also made positive contributions to the science which are of sufficient
value to make it impossible to omit him from a history of Greek
astronomy.
We have seen that he modified the purely geometrical hypotheses
of Eudoxus and Callippus in a mechanical sense. A purely
geometrical theory did not satisfy him; he must needs seek to
assign causes for the motions of the several concentric spheres. We
may therefore conveniently begin this chapter with an account of
his views on Motion. Motion, according to Aristotle, is, like Form!
and Matter,” eternal and indestructible, without beginning or end.*-
Motion presupposes a primum movens which is itself unmoved ;* for
that which is moved, being itself subject to change, cannot impart
an unbroken and uniform movement ;° the primum movens, then,
must be one,® unchangeable, absolutely necessary ;7 there is nothing
merely potential about it, no unrealized possibility ;§ it must there-
fore be incorporeal,° indivisible,!° and unconditioned by space,! as
1 Metaph. Z. 8, 1033b 16, Z. 9, 1034 Ὁ 7, A. 3, 1069 Ὁ 35, ἄς.
2 Phys. i. 9, 192 a 22-32. 8 Metaph. A. 6, τογι Ὁ 7.
* Ibid. 1071 b 4. 5 Phys. viii. 6, 259b 22; c. 10, 2678 24.
δ Metaph. Δ. 8, 1073a 25, 1074 a 36, τε. 7 Ibid. A. 7, 1072 b 7-11.
® Ibid. A. 6, 1071 b 12. 9 Tbid. A. 6, 1071 Ὁ 20.
10 Ibid. A. 9, 1075 a 7.
™ De caelo i. 9,279 a 18 sq.; Phys. viii. 10, 267 b 18.
1410 Q
226 ARISTOTLE PARTI
well as motionless and passionless ;’ it is absolute Reality and pure
Energy,” that is,God.* In another aspect the primum movens is the
Final Cause, pure Being, absolute Form, the object of thought and
desire ;* God is Thought, self-sufficient,® contemplating unceasingly
nothing but itself,’ the absolute activity of Thought, constituting
absolute reality and vitality and the source of all life.* The primum
movens causes all the movements in the universe, not by any activity
of its own °—for that would be a movement and, as immaterial, it
can have no share in movement—but by reason of the fact that all
things strive after it and try to realize, so far as possible, its Form ;19
it operates like a beloved object, and that which is moved by it
communicates its motion to the rest."
Motion takes place only by means of continuous contact between
the motive principle and the thing moved. Aristotle insists upon
this even in a case where the contact might seem to be only
momentary, e.g. where a thing is zirown. The motion in that case
seems to continue after contact with the thrower has ceased, but
Aristotle will not admit this; he assumes that the thrower moves
not only the thing thrown but also the medium through which the —
thing is thrown, and makes the medium able to act as moved and
movent at the same time (i.e. to communicate the movement); and
further that the medium can continue to be movent even after it
has ceased to be moved.!2 God then, as the first cause of motion,
must be in contact with the world,” though Aristotle endeavours to
exclude contiguity in space from the idea of ‘contact’, which he
often uses in the sense of immediate connexion, as of thought with
its object.4 The primunt movens operates on the universe from the
circumference, because the quickest motion is that of the (outermost
limit of the) universe, and things move the quickest which are
nearest to that which moves them. Hence in a sense it could be
1 De anima iii. 2,426a10. * Metaph. A.7,1072a25. * De caelo, loc. cit.
* Metaph. A. 7, 1072426; De anima iii. 10, 4338 18.
5 Eth. N. x. 8,1178b21; Metaph. A. 9, 1074b 25.
® De caelo ii, 12, 29265; Politics, H. 1, 1323 Ὁ 23.
7 Metaph. A. 9, 1075a 10. 8 Metaph. A. 7, 1072 Ὁ 28.
® De caelo ii. 12, 292a 22; Eth. N.x. 8, 1178 Ὁ 20.
0 Metaph. A. 7, 1072 a 26. 1 Ibid. 1072 Ὁ 3.
12. Phys. viil. το, 266 Ὁ 27 - 267 ἃ 18. 3 De gen. et corr.i. 6, 323 a 31.
4 Metaph. 8. 10, 1051b24; A. 7, 1072 Ὁ 21.
15 Phys. vili. 10, 267 Ὁ 7-9.
CH. XVII ARISTOTLE 227
said that God is to Aristotle ‘the extremity of the heaven’;! but
Aristotle is careful to deny that there can be any body or space or
void outside the universe ; what is outside is not in space at all; the
‘end of the whole heaven’ is life (αἰών), immortal and divine’.?
Motion in space is of three kinds, motion in a circle, motion in
a straight line, and motion compounded of the two (‘mixed’).*
Which of these can be endless and continuous? The ‘mixed’
would only be so if both the two components could; but move-
ment in a straight line cannot have this character, since every finite
rectilinear movement has terminal points at which it must turn
back,* and an infinite rectilinear movement is impossible, both in
itself,> and because the universe is finite; hence circular motion is
the only motion which can be without beginning or end.* Simple
bodies have simple motions ; thus the four elements tend to move
in straight lines; earth tends downwards, fire upwards ; between
the two are water, the relatively heavy, and air, the relatively light.
Thus the order, beginning from the centre, in the sublunary sphere
is earth, water, air, fire.” Now, says Aristotle, simple circular
motion is more perfect than motion in a straight line. As, then,
there are four elements to which rectilinear motion is natural and
circular motion not natural, so there must be another element,
different from the four, to which circular motion is natural.* This
element is superior to the others in proportion to the greater perfec-
tion of circular motion and to its greater distance from us ;° circular
motion admits no such contraries as ‘up’ and ‘down’; the superior
element therefore can neither be heavy nor light;!° the same absence
of contrariety suggests that it is without beginning or end, im--
perishable, incapable of increase or change (because all becoming
involves opposites and opposite motions).11_ This superior element
which fills the uppermost space is called ‘aether’,!” the ‘first ele-
1 Sextus Emp. Adv. Math. x. 33; Hypotyp. iii. 218.
2 De caelo i. 9, 279a 16-28. 3 Phys. viii. 8, 261 Ὁ 29.
* Phys. viii. 8, 261 b 31-4. ® Ibid. iii. 5, 206a7; c. 6, 206a 16.
® Ibid. viii. 8, 261 a 27-263 a 3, 2644 7 sqq.; c. 9, 265a 13 sq.
7 Aristotle is careful, however, to explain that the division between air and fire
is not a strict one, as between two definite layers ; there is some intermixture
(cf. Meteor. i. 3, 341a 1-9). Further, the ‘fire’ is what from force of habit we
call fire; it is not really fire, for fire is an excess of heat, a sort of ebullition
(ibid. 340 b 22, 23). ® De caelo i. 2, 268 Ὁ 26-269 Ὁ 17. 9 Ibid.
15. De caelo i. 3, 269 Ὁ 18-33. Ἡ bid. 270a 12-35.
#2 Ibid. 270 Ὁ 1-24.
Q2
228 ARISTOTLE PARTI
ment ’,! or ‘a body other and more divine than the four so-called
elements’ ;? its changelessness is confirmed by long tradition, which
contains no record of any alteration in the outer heaven itself or in
any of its proper parts.2 Of this element are formed the stars,*
which are spherical,® eternal,® intelligent, divine.’ It occupies the
whole region from the outside limit of the universe down to the
orbit of the moon, though it is not everywhere of uniform purity,
showing the greatest difference where it touches the sublunary
sphere. Below the moon is the terrestrial region, the home of the
four elements, which is subject to continual change through
the strife of those elements and their incessant mutual transfor-
mations.°
There is, Aristotle maintains, only one universe or heaven, and
that universe is complete, containing within it all the matter there
is. For, he argues, all the simple bodies move to their proper
places, earth to the centre, aether to the outermost region of the
universe, and the other elements to the intervening spaces. There
can be no simple body outside the universe, for that body has its
own natural place inside, and, if it were kept outside by force, the
place occupied by it would be the zatural place for some other
body; which is impossible, since a// the simple bodies have their
proper places inside. The same argument holds for mixed bodies ;
for, where mixed bodies are, there also are the simple bodies of
which they are composed. Nor can there be any space or void out-
side the universe, for space or void is only that in which a body is
or can be.!° Another argument is that the primum movens is single
and complete in itself; hence the world, which derives its eternal
motion from the primum movens, must be so too." If it be sug-
gested that there may be many particular worlds as manifestations
of one concept ‘world’, Aristotle replies that this cannot be; for
the heaven is perceptible to our senses; hence it and other heavens
1 De caelo iii. 1, 298b6; Meteor. i. τ, 338b 21, ἅς.
® De gen. an. ii. 3, 736 Ὁ 29-31. 8 De caelo i. 3, 270 Ὁ 11-16,
* De gen. an. ii, 3, 737 41. 5 De caelo ii. 8, 290 ἃ 7- 11.
® Metaph. A. 8, 1073 a 34.
7 Ibid. 1074a 38-b3; δᾶ. XN. vi. 1143 b 1. 8 Meteor. i. 3, 340 Ὁ 6-10.
® Meteor. ii. 3, 357 Ὁ 30.
10 De caelo i. 9, 278 Ὁ 8 -- 279 a 14.
1 Metaph. A. 8, 1074 a 36-8.
a ". ᾿νΤω τ ὐἱ.-
CH. XVII ARISTOTLE 229
(if any) must contain mazter ; but our heaven contains all the matter
there is, and therefore there cannot be any other.?
Next, the universe is finzte. In the Physics Aristotle argues that
an infinite body is inconceivable, thus. An infinite body must
either be simple or composite. If composite, it is composed of
elements ; these are limited in number; hence an infinite body
could only be made up of them if one or more were infinite in
magnitude ; but this is impossible, because there would then be no
room for the rest. Neither can it be simple; for no perceptible
simple body exists except the elements, and it has been shown that
none of them can be infinite. In the De caelo he approaches the
subject from the point of view of motion. A body which has
a circular motion, as the universe has, must be finite. For, if it is
infinite, the straight line from the centre to a point on its circum-
ference must be infinite; now if, as being infinite, this distance can
never be traversed, it cannot revolve in a circle, whereas we see
that in fact the universe does so revolve.* Further, in an infinite
body there can be no centre; hence the universe which rotates about
-its centre cannot be infinite.*
Aristotle’s arguments for the spherical shape of the universe are
of the usual kind. As the circle, enclosed by one line, is the first of
plane figures, so the sphere, bounded by one surface, is the first
of solid figures; hence the spherical shape is appropriate to the
‘first body’, the subject of the ‘outermost revolution’. Next, as
there is no space or void outside the universe, it must, as it revolves,
continually occupy the same space ; therefore it must be a sphere;
for, if it had any other form, this condition would not be satisfied.®
[Aristotle is not strictly correct here, since any solid of revolution
revolving round its axis always occupies the same space, but it is
true that onlya sphere can remain in exactly the same position when
revolving about any diameter whatever.] Further, we may infer the
spherical form of the universe from the bodies in the centre. We
have first the earth, then the water round the earth, air round the
water, fire round the air, and similarly the bodies above the fire;
Pes tg ged δα ρρδνες 278 a 28. is ne ie
LYS. Il. 5, 204 Ὁ 3-3 δ caelo 1. 5, 271 Ὁ 28 -- 272 8 7.
* De εαείο i. 2, 275 b 12-15. 5 Ibid. ii. 4, 286 b 10 -- 287 ἃ 5.
5. Ibid. ii. 4, 287 a 11-22.
230 ARISTOTLE PARTI
now the surface of the water is spherical ; hence the surfaces of the
layers following it, and finally the outermost surface, correspond."
The fabric of the heavens is made up of spherical shells, as it
were, one packed inside the other so closely that there is no void
or empty space between them ;” this applies not only to the astral
spheres,’ but right down to the earth in the middle ;* it is necessary
so far as the moving spheres are concerned because there must
always be contact between the moving and the moved.°®
We have above described the working of Aristotle’s mechanical
system of concentric spheres carrying the fixed stars and producing
the motions of the planets respectively, and it only remains to add
a word with reference to the motive power acting on the spheres
other than that of the ‘outermost revolution’, The outermost
sphere, that of the fixed stars, is directly moved by the one single
and eternal primum movens, Divine Thought or Spirit. Only one
kind of motion is produced when one movens acts on one object ;°
how then do we get so many different movements in the spheres
other than the outermost? Aristotle asks himself this question:
Must we suppose that there is only one unmoved movens of the
kind, or several, and, if several, how many are there? He is obliged
to reply that, as eternal motion must be due to an eternal movens,
and one such motion to one such movens, while we see that, in addi-
tion to the simple revolution of the whole universe caused by the
unmoved primum movens, there exist other eternal movements,
those of the planets, we must assume that each of the latter move-
ments is due to a substance or essence unmoved in itself and
eternal, without extension in space.’ The number of them must be
that of the separate spheres causing the motion of the separate
planets. The number of these spheres he had, as we have seen
(Ρ. 217), fixed provisionally, while recognizing that the progress of
astronomy might make it necessary to alter the figures.? Of the
several spheres which act on any one planet, the first or outermost
alone is moved by its own motion exclusively ; each of the inner
spheres, besides having its own independent movement, is also
1 De caelo ii. 4, 287 a 30-b 4.
2 Cf. Phys. vii. 2, 243 ἃ 5. 8 De caelo i. 9, 278 Ὁ 16-18.
* De caelo ii. 4, 287 a 5-11. 5 Phys. vii. 1, 242 Ὁ 24-6, vii. 2, 243 a 3-5.
6 Phys. viii. 6, 259 a 18. 7 Metaph. A. 8, 1073 a 14-b το.
8 Metaph. A. 8. 1074 13-16. ® Ibid. 1073 Ὁ 10-17.
.
CH. XVII ARISTOTLE | 231
carried round in the motion of the next sphere enveloping it, so that
all the inner spheres, while themselves movent, are also moved by
the eternal unmoved movent.!
In a chapter of the De caelo* Aristotle discusses the question
which is the vighz side of the heaven and which the ft. The dis-
quisition is not important, but it is not unamusing.* He begins with
a reference to the view of the Pythagoreans that there is a right
and a left in the universe, and proceeds to investigate whether the
particular distinction which they draw is correct or not, ‘ assuming
that it is necessary to apply such principles as “right ” and “ left”’
to the body of the universe’.* There being three pairs of such
opposites, up and down (or upper and lower), right and left, before
and behind (or forward and backward), he begins with the distinc-
tions (1) that ‘up’ is the principle of length, ‘right’ of breadth, and
‘before’ of depth, and (2) that ‘up’ is the source of motion (ὅθεν ἡ
κίνησις), ‘right’ the place from which it starts (ἀφ᾽ οὗ), and ‘ fo the
front’ (εἰς τὸ πρόσθεν) is the place to which it is directed (ἐφ᾽ 8).
Now the fact that the shape of the universe is spherical, alike in all
its parts, and continually in motion, is no obstacle to calling one
part of it ‘right’ and the other ‘left’. What we have to do is to
think of something which has a right and left of its own (say a man)
and then place a sphere round [ὃ Now, says Aristotle, I call the
diameter through the two poles the /ength of the universe (because
only the poles remain fixed), so that I must call one of the poles
the upper, and the other the /ower. He then proceeds to show
that the proper relativities can only be preserved by calling the
south (the invisible) pole the upper and the xorth (the visible)
pole the /ower, from which it follows that we live in the ower and
left hemisphere, and the inhabitants of the regions towards the south
pole live in the upper and right hemisphere; and this is precisely
the opposite of what the Pythagoreans hold, namely that we live
in the upper and right hemisphere, and the antipodes represent
the Jower and /eft. The argument amounts to this. ‘Right’ is the
place from which motion in space starts; and the motion of
1 Phys. viii. 6, 259 Ὁ 29-31. 2 De caelo ii. 2, 284 Ὁ 6 -- 286 42.
3 This matter also is fully discussed by Boeckh, Das kosmische System des
Platon, pp. 112-19.
* De caelo ii. 2, 284 Ὁ 9-10. 5 Ibid. 285 b 1-3.
232 : ARISTOTLE PART I
the heaven starts from the side where the stars rise, i.e. the east ;
therefore the east is ‘ right’ and the west is ‘left’. If now (1) you
suppose yourself to be lying along the world’s axis with your head
towards the zorth pole, your feet towards the south pole, and your
right hand towards the east, then clearly the apparent motion of the
stars from east to west is over your Jack from your right side towards
your left ; this motion, Aristotle maintains, cannot be called motion
‘to the right’, and therefore our hypothesis does not fit the assump-
tion from which we start, namely that the daily rotation ‘ begins
from the right and is carried round towards the right (ἐπὶ τὰ degra)’.
We must therefore alter the hypothesis and suppose (2) that you
are lying with your head towards the south pole and your feet
towards the zorth pole. If then your right hand is to the east, the
daily motion begins at your right hand and proceeds over the front
of your body from your right hand to your left. We should nowa-
days regard this as giving precisely the wrong result, since motion
round us zz front from right to left can hardly be described as ἐπὶ
τὰ δεξιά, ‘to the right’; so that hypothesis (1) would, to us, seem
preferable to hypothesis (2). But Aristotle’s point of view is fairly
clear. We are to suppose a man (say) standing upright and giving
a horizontal turn with his right hand to a circle about a vertical
diameter coincident with the longitudinal axis of his body. Aris-
totle regards him as turning the circle towards the right when he
brings his right hand towards the front of his body, although we
should regard it as more natural to apply ‘ towards the right’ to a
movement of his right hand szz// more to the right, i.e. round by
the right to the back. The (to us) unnatural use of the terms
by Aristotle is attested by Simplicius who says that motion ἐπὲ
δεξιά is in any case towards the front (πάντως εἰς τὸ ἔμπροσθέν
éo7t),? and it is doubtless due to what Aristotle would regard as the
necessity of making frout (in the dichotomy front and back, or
before and behind) correspond to rvigh¢ (in the dichotomy right and
left), just as wp (in the dichotomy up and down) must also corre-
spond to right; this is indeed clear from his own statement quoted
above that, as ‘the right’ is the place from which motion starts, so
‘to the front’ is the place towards which it is directed.
We come next to Aristotle’s view as to the shape of the heavenly
1 De caelo ii. 2, 285 Ὁ 20, 2 Simplicius on De caelo, p. 392, 1, Heib.
CH. XVII ARISTOTLE 233
bodies and the arguments by which he satisfied himself that they
do not move of themselves but are carried by material spheres.
He held that the stars are spherical in form. One argument in
support of this contention is curious. Nature, he says, does nothing
without a purpose; Nature therefore gave the stars the shape
most unfavourable for any movement on their own part; she denied
to them all organs of locomotion, nay, made them as different as
possible from the things which possess such organs. With this end
in view, Nature properly made the stars spherical ; for, while the
spherical shape is the best adapted for motion in the same place
(rotation), it is the most useless for progressive motion.’ This is
in curious contrast to the view of Plato who, with more reason,
regarded the cube as being the shape least adapted for motion
(ἀκινητότατον).3 The second argument is from analogy. Since
the moon is shown by the phases to be spherical, while we see
similar curvature in the lines separating the bright part of the sun
from the dark in non-total solar eclipses, we may conclude from this
that the sun and, by analogy, the stars also are spherical in form.*
With regard to the spheres carrying the stars round with them,
we note first that the ‘heaven’, in the sense of the ‘outermost
heaven’ or ‘the outermost revolution of the All’ (ἡ ἐσχάτη περι-
φορὰ τοῦ παντός), which is the sphere of the fixed stars, was with
Aristotle a material thing, a ‘ physical body’ (σῶμα φυσικόν)."
Now, says Aristotle,> seeing that both the stars and the whole
heaven appear to change their positions, there are various a priori
possibilities to be considered ; (1) both the stars and the heaven
may be at rest, (2) both the stars and the heaven may be in motion,
or (3) the stars may move and the heaven be at rest, or vice versa.
Hypothesis (1) is at once ruled out because, under it, the
observed phenomena could not take place consistently with
the earth being at rest also; and Aristotle assumes that the earth
is at rest (τὴν δὲ γῆν ὑποκείσθω ἠρεμεῖν). Coming to hypothesis (2),
we have to remember that the effect of a uniform rotation of the
‘heaven about an axis passing through the poles is to make par-
ticular points on this spherical shell describe parallel circles about
1 De caelo ii. 8, 290a 31- 5 ; c. 11, 291 b11-17.
2 Plato, Timaeus 55 Ὁ, E. 8 De caelo ii. 11, 291 Ὁ 17-23.
* De caelo i. 9, 278 Ὁ 11-14. ® Ibid. ii. 8, 289 b1 sqq.
234 ARISTOTLE PARTI
the axis ; suppose then that the heaven rotates in this way, and that
the stars also move. Now, says Aristotle, the stars and the circles
cannot move independently ; if they did, it is inconceivable that the
speeds of the stars would always be exactly the same as the speeds
of the circles ; for, while the speeds of the circles must necessarily
be in proportion to their sizes, i.e. to their radii, it is not reasonable
to suppose that the stars, if they moved freely, would revolve at
speeds proportional to the radii of the circles; yet they would have
to do so if the stars and the circles are always to return to the same
places at the same times, as they appear to do. Nor can we, as
in hypothesis (3), suppose the stars to move and the heaven to be
at rest ; for, if the heaven were at rest, the stars would have to move
of themselves at speeds proportional to the radii of the circles they
describe, which has already been stated to be an unreasonable sup-
position. Consequently only one possibility remains, namely that
the circles alone move, and the stars are fixed on them and carried
round with them ;} that is, they are fixed on, and carried round with,
the sphere of which the circles are parallel sections.
Again, says Aristotle, there are other considerations which sug-
gest the same conclusion. If the stars have a motion of their own,
they can, being spherical in shape, have only one of two movements,
namely either (1) whirling (δίνησις) or (2) rolling (κύλισι5). Now
(1), if the stars merely whirled or rotated, they would always
remain in the same place, and would not move from one position
to another, as everybody admits that they do. Besides, if one
heavenly body rotated, it would be reasonable to suppose that they
all would. But, in fact, the only body which seems to rotate is the
sun and that only at the times of its rising and setting ; this, how-
ever, is only an optical illusion due to the distance, ‘for our sight,
when at long range, wavers’ (literally ‘turns’ or ‘ spins’, ἑλίσσεται).
This, Aristotle incidentally observes, may perhaps be the reason
why the fixed stars, which are so distant, twinkle, while the planets,
being nearer, do not. It is thus the tremor or wavering of our
sight which makes the heavenly bodies seem to rotate.” In thus
asserting that the stars do not rotate, Aristotle is of course opposed
to Plato, who held that they do.°
* De caelo ii. 8, 289b 32. 2. Ibid. ii. 8, 290 ἃ 9-23. ὃ Plato, Zimaeus 40 A.
CH. XVII ARISTOTLE 235
Again (2), if the stars rolled (along, like a wheel), they would
necessarily turn round ; but that they do not turn round in this
way is proved by the case of the moon, which always shows us one
side, the so-called face.'
It is for a particular reason that I have reproduced so fully
Aristotle’s remarks about rotation and rolling as conceivable move-
ments for stars as spherical bodies. It has been commonly re-
marked that Aristotle draws a curious inference from the fact that
the moon has one side always turned to us, namely that the moon
does not rotate about its own axis, whereas the inference should be
the very opposite.? But this is, I think, a somewhat misleading
statement of the case and less than just to Aristotle. What he says
is that the moon does not turn round in the sense of rolling along ;
and this is clear enough because, if it rolled along a certain path, it
would roll once round while describing a length equal to 3-1416
times its diameter, but it manifestly does not do this. But Aris-
totle does not say that the moon does not rotate; he does not, it is
true, say that it does rotate either, but his hypothesis that it is
fixed in a sphere concentric with the earth has the effect of keeping
one side of the moon always turned towards us, and therefore Ζ7:ε2-
dentally giving it a rotation in the proper period, namely that of its
revolution round the earth. I cannot but think that the fact of the
moon always showing us one side was one of the considerations, if
not the main consideration, which suggested to Aristotle that the stars
were really fixed in material spheres concentric with the earth.
We pass to matters which are astronomically more important.
And first as to the spherical shape of the earth. Aristotle begins.
by answering an objection raised by the partisans of a flat earth,
namely that the line in which the horizon appears to cut the sun as
it is rising or setting is straight and not curved.* His answer is
confused ; he says that the objectors do not take account of the
distance of the sun from the earth and of the size of its circum-
ference,* the fact being that you can have an apparently straight line
1 De caelo ii. 8, 2908 26.
2 Cf. Martin, ‘Hypothéses astronomiques grecques’ in Mémoires de P Acad.
des Inscriptions et Belles-Lettres, xxx. 1881, p. 287; Dreyer, Planetary Systems,
p. 111, note. 3 De caelo ii. 13, 294a1.
* τῆς περιφερείας seems Clearly to be the circumference of the sun (not that of
the horizon which cuts the solar disc).
286 ARISTOTLE PARTI
as a section when you see it from afar in a circle which on account
of its distance appears small. He should no doubt have said, first,
that the sun, as we see it, looks like a flat disc of small size on
account of its distance, and then that the section of an object
apparently so small by the horizon is indistinguishable from a
section by a plane through our eye, so that the section of the disc
appears to be a straight line. He has, however, some positive
proofs based on observation. (1) In partial eclipses of the moon
the line separating the bright from the dark portion is always
convex (circular)—unlike the line of demarcation in the phases of
the moon, which may be straight or curved in either direction—
this proves that the earth, to the interposition of which lunar
eclipses are due, must be spherical! He should. no doubt have
said that a sphere is the ovly figure which can cast a shadow such
that a right section of it is always a circle; but his explanation
shows that he had sufficiently grasped this truth, (2) Certain stars
seen above the horizon in Egypt and in Cyprus are not visible
further north, and, on the other hand, certain stars set there which |
in more northern latitudes remain always above the horizon. As
there is so perceptible a change of horizon between places so near
to each other, it follows not only that the earth is spherical, but
also that it is not a very large sphere. He adds that this makes it
not improbable that people are right when they say that the region
about the Pillars of Heracles is joined on to India, one sea connect-
ing them. It is here, too, that he quotes the result arrived at
by mathematicians of his time, that the circumference of the earth
is 400,000 stades.? He is clear that the earth is much smaller
than some of the stars.2 On the other hand, the moon is smaller
than the earth.* Naturally, Aristotle has a prior reasons for the
sphericity of the earth. Thus, using once more his theory of heavy
bodies tending to the centre, he assumes that, whether the heavy
particles forming the earth are supposed to come together from all
directions alike and collect in the centre or not, they will arrange
themselves uniformly all round, i.e. in the shape of a sphere, since, if
there is any greater mass at one part than at another, the greater
1 De caelo ii. 14, 297 b 23-30. 2 Ibid. 297 Ὁ 30 -- 298 a 20,
3 Ibid. 298a19; Meteor. i. 3, 339b7-9.
* Aétius, ii. 26. 3 (D. G. p. 357 Ὁ 11).
χέων δια ον κα ον, δε e
ΜΝ Ψ ΨὉ
CH, XVII } ARISTOTLE 237
mass will push the smaller until the even collection of matter all
round the centre produces equilibrium.’
Aristotle's attempted proof that the earth is in the centre of the
universe is of course a fetitio principiz. He begins by attempting
to refute the Pythagorean theory that the earth, like the planets and
the sun and moon, moves round the central fire. The Pythagoreans,
he says, conceived the central fire to be the abode of sovereignty in
the universe, the Watchtower of Zeus, while others might say that
the centre, being the worthiest place, is appropriate for the worthiest
occupant, and that fire is worthier than earth. To this he replies
that the centre of a thing is not so worthy as the extremity, for it
is the extremity which limits or defines a thing, while the centre is
that which is limited and defined, and is more like a termination
than a beginning or principle.2_ When Aristotle comes to state his
own view, he rightly says that heavy bodies, e.g. parts of the earth
itself, tend towards the centre of the earth; for bodies which fall
towards the earth from different places do not fall in parallel lines
but ‘at equal angles’, i.e. at right angles, to the (spherical) surface
of the earth, and this proves that they fall in the direction of its
centre. Similarly, if a weight is thrown upwards, however great the
force exerted, it falls back again towards the centre of the earth.* But,
he asks, do bodies tend towards the centre because it is the centre
of the universe or because it is the centre of the earth, ‘ since both
have the same centre’?* He replies that they must tend towards
the centre of the universe because, in the reverse case of the light
elements, e.g. fire, it is the extremities of the space which envelops
the centre (i.e. the extremities of the universe) to which they.
naturally tend.® Even the show of argument in the last sentence
does not prevent the whole of the reasoning from being a fetitio
principit. For it is exclusively based on the original assumption
that, of the four elements, earth and, next to that, water tend to
move in a straight line ‘downwards’, i.e, on Aristotle’s view,
towards the centre of the universe,® the effect of which is that not
1 De caelo ii. 14, 297 ἃ 8-b 18.
® Ibid. ii. 13, 293 a 17- 15. 5 Ibid. ii. 14, 296 Ὁ 18-25.
* Ibid. 296 b 9-12. 5 Ibid. 296 Ὁ 12-15.
δ Phys. iv. 4,212a26; c.8,214b14; and especially De cae/o iv. 1, 308 a 15--
31. In the last-cited passage Aristotle, without mentioning Plato by name,
attacks Plato’s doctrine that, in a perfect sphere such as the universe is, you
238 ARISTOTLE PART I
only do the particles of the earth tend to the centre of the universe,
but @ fortiori the earth itself, which must therefore occupy the
centre of the universe.!
Another argument is that, according to the astronomical views of
the mathematicians, the phenomena which are observed as the
heavenly bodies change their positions relatively to one another are
just what they should be on the assumption that the earth is in the
centre.2 The answer to this is, as Martin says, ‘How do you
know? And how can you use the argument when you have quoted,
without stating any objection to it, the argument of the Pytha-
goreans that their theory of the motion of the earth need cause no
sensible difference of parallax in comparison with the theory that
the earth is at the centre?’
The earth being in the centre of the universe, what keeps it
there? Dealing with this question, Aristotle again begins by a
consideration of the views of earlier philosophers. He rejects
Thales’ view that the earth floats on water as contrary to experi-
ence, since earth is heavier than water, and we see water resting or
riding on earth, but not the reverse. He rejects, too, the view of
Anaximenes, Anaxagoras, and Democritus that it rides on the air
because it is flat and, acting like a lid to the air below it, is sup-
ported by it. Aristotle points out first that, if it should turn out
that the earth is round and not flat, it cannot be the flatness which
is the reason of the air supporting it; according to the argument it
cannot properly describe one part rather than another as ‘up’ or ‘ down’; on the
contrary, says Aristotle, I call the centre, where heavy bodies collect, ‘down’,
and the extremities of the sphere, whither light bodies tend to rise, ‘up’. But,
as usual, there is less difference between the two views than Aristotle would
have us believe. Plato (7imaeus 62 Ὁ) said, it is true, that, as all points of the
circumference are equidistant from the centre, it is incorrect to apply the terms
‘up’ and ‘down’ to different specific portions of that circumference, or to
call any portion of the sphere ‘up’ or ‘down’ relatively to the centre, which
is neither ‘up’ nor ‘down’, but simply the centre. But he goes on to say
(63 B-E) that you can use the terms in a purely ve/ative sense; any two
localities may be ‘up’ and ‘down’ relatively to one another, and Plato proposes
a criterion. Ifa body tends to move to a certain place by virtue of seeking for
its like, this tendency is what constitutes its eaviness, and the place to which
it tends is ‘down’; and the opposite terms have the opposite meanings. The
only difference made by Aristotle is in definitely allocating the centre of the
universe as the place of the heaviest element, earth, and arranging the other
elements in order of lightness in spherical layers round it, so that on his system
the centre of the universe becomes ‘down’, and amy direction outwards along
a radius is ‘up’.
1 De caela ii. 14, 296 Ὁ 6-9. 2 Ibid. 297 a 2-6.
.
CH. XVII ARISTOTLE 239
must rather be its size than its flatness, and, if it were large enough,
it might even be a sphere. With this he passes on Nor does
Empedocles’ theory meet with more favour; if the earth is kept
in its place in the same way as water in a cup which is whirled
round, this means that the earth is kept in its place by force, and to
this view Aristotle opposes his own theory that the earth must
have some natural tendency, and a proper place, of its own. Even
assuming that it came together by the whirling of a vortex, why do
all heavy bodies now tend towards it? The whirling is at all events
too far away from us to cause this. And why does fire move
upwards? This cannot be through the whirling either ; and, if fire
naturally tends to move to a certain region, surely the earth should
too. But, indeed, the heavy and the light were prior to the whirling,
and what determines their place is, not whirling, but the difference
between ‘up’ and ‘down’. Finally he deals with the view of
Anaximander (followed by Plato) that the earth is in equilibrium
through being equidistant from all points of the circumference, and
therefore having no reason to move in one direction rather than
another. Incidentally comparing the arguments (1) that, if you
pull a hair with force and tension exactly equal throughout, it will
not break, and (2) that a man would have to starve if he had
victuals and drink equally disposed all round him, Aristotle again
complains that the theory does not take account of the natural
tendency of one thing to move to the centre, and of another to
move to the circumference. It happens incidentally to be true that
a body must remain at the centre if it is not more proper for it to
move this way or that, but whether this is so or not depends on the -
body ; it is not the equidistance from the extremities which keeps
it there, for the argument would require that, if fire were placed in
the centre, it would remain there, whereas in fact it would not,
since its tendency to fly upwards would carry it uniformly in all
directions towards the extremities of the universe; hence it is not
the equidistance, but the natural tendency of the body, which
determines the place where it will rest.*
In setting himself to prove that the earth has no motion what-
ever, Aristotle distinguishes clearly between the two views (1) of
1 De caelo ii. 13, 294 a 28-b 30, ? Ibid. 295 a 16-b 9.
5 Ibid. 295 Ὁ lo-— 296 ἃ. 21.
240 ARISTOTLE PARTI
those who give it a motion of translation or ‘make it one of the
stars’, and (2) of those who regard it as packed aud moving about an
axis through itscentre.1. Though he arbitrarily adds the words ‘ and
moves’ (kai κινεῖσθαι) to the phraseology of the 7zmacus, thereby
making it appear that Plato attributed to the earth a rotation about
its axis, which, as we have seen, he could not have done, the second
of the two views was actually held by Heraclides Ponticus, who
was Aristotle’s contemporary. It seems likely, as Dreyer suggests,”
that, in speaking of a motion of the earth ‘at the centre itself’,®
Aristotle is not thinking of a rotation of the earth zz twenty-four
hours, i.e. a rotation replacing the apparent revolution of the fixed
stars, as Heraclides assumed that it did; for he does not mention
the latter feature or give any arguments against it; on the con-
trary, he only deals with the general notion of a rotation of the
earth, and moreover mixes up his arguments against this with his
arguments against a translation of the earth in space. He uses
against both hypotheses his fixed principle that parts of the earth,
and therefore the earth itself, move naturally towards the centre.* —
Whether, he says, the earth moves away from the centre or a7 the
centre, such movement could only be given to it by force; it could
not be a natural movement on the part of the earth because, if
it were, the same movement would also be natural to all its parts,
whereas we see them all tend to move in straight lines towards the
centre; the assumed movement, therefore, being due to force and
against nature, could not be everlasting, as the structure of the
universe requires.®
The second argument, too, though directed against both hypo-
theses, really only fits the first, that of motion in an orbit.
‘ Further, all things which move in a circle, except the first (outer-
most) sphere, appear to be left behind and to have more than one
movement; hence the earth, too, whether it moves about the
centre or in its position at the centre, must have two movements.
Now, if this occurred, it would follow that the fixed stars would
exhibit passings and turnings (παρόδους καὶ τροπάς). This, how-
De caelo ii. 13, 293 a 20-3, Ὁ 18-20; c. 14, 296 ἃ 25-7.
* Dreyer, Planetary Systems, pp. 116, 117.
8 De caelo ii, 14, 296 ἃ 29, Ὁ 2. 4 Ibid. 296 b 6-8,
5 Ibid. 296 a 27-34.
re ov
CH. XVII ARISTOTLE 241
ever, does not appear to be the case, but the same stars always rise
and set at the same places on the earth.’?
The bodies which appear to be ‘left behind and to have more
movements than one’ are of course the planets. The argument
that, if the earth has one movement, it must have two, is based upon
nothing more than analogy with the planets. Aristotle clearly
inferred as a corollary that, if the earth has two motions, one must
be oblique to the other, for it would be obliquity to the equator in
at least one of the motions which would produce what he regards
as the necessary consequence of his assumption, namely that the
fixed stars would not always rise and set at the same places. As
already stated, Aristotle can hardly have had clearly in his mind
the possibility of one single rotation about the axis iz twenty-four
hours replacing exactly the apparent daily rotation ; for he would
have seen that this would satisfy his necessary condition that the
fixed stars shall always rise and set at the same places, and there-
fore that he would have to get some further support from elsewhere
to his assumption that the earth must have ¢wo motions. Still
less could he have dreamt of the possibility of Aristarchus’s later
hypothesis that the earth has an annual revolution as well as a
daily rotation about its axis, which hypothesis satisfies, as a matter
of fact, both the condition as to two motions and the condition as
regards the fixed stars.
The Meteorologica deals with the sublunary portion of the
heavenly sphere, the home of the four elements and their combina-
tions. Only a small portion of the work can be said to be astrono- _
mical, but some details bearing on our subject may be given. We
have seen the four elements distinguished according to their relative
heaviness or lightness, and the places which are proper to them
respectively; in the MWeteorologica they are further distinguished
according to the tangible qualities which are called their causes
(αἴτια). These tangible qualities are the two pairs of opposites,
hot—cold, and dry—moist ; and when we take the four combinations
of these in pairs which are possible we get the four elements ; hot
and dry = fire, hot and moist = air (air being a sort of vapour),
cold and moist = water, cold and dry =earth.? Of the four qualities
1 De caelo ii. 14, 296 a 34-b 6. 2 De gen. et corr. ii. 3, 3308 30-b7.
1410 R
242 ARISTOTLE ’ PARTI
two, hot and cold, are regarded as active, and the other two, dry
and moist, as passive.’ Since each element thus contains an active
as well as a passive quality, it follows that all act upon and are acted
upon by one another, and that they mingle and are transformed into
one another.? Every composite body contains all of them ;* they
are never, in our experience, found in perfect purity. Elemental fire
is warm and dry evaporation,’ not flame; elemental fire is a sort of
‘inflammable material ’ which ‘ can often be kindled by even a little
motion, like smoke’;*® but flame, or fire in the sense of flame, is
‘an excess of heat or a sort of ebullition’,’ or an ebullition of dry
wind 8 or of dry heat®; again, flame is said to be a fleeting, non-
continuous product of the transformation of moist and dry in close
contact.!° The reason for this distinction between ‘fire’ and flame
is obvious, as Zeller says; for Aristotle could not have made the
outer portion of the terrestrial sphere, contiguous to the aether, to
consist of actual burning flame. According to Aristotle, the stars
are not made of fire (still less all the spaces between them); in
themselves they are not even hot; their light and heat come from |
friction with the air through which they move [notwithstanding
that they are in the aethereal sphere]; the air in fact becomes fire
through their impact on it ; the stratum of air which lies nearest to
them underneath the aethereal sphere is thus warmed. Especially
is this the case with the sun; the sun is able to produce heat in
the place where we live because it is not so far off as the fixed
stars and it moves swiftly (the stars, though they move swiftly,
are far off, and the moon, though near to us, moves slowly); further,
the motion often causes the fire surrounding the atmosphere to
scatter and rush downwards."
Such phenomena as shooting stars (didrrovres or διαθέοντες
ἀστέρες) and meteors (of the two kinds called δαλοί and αἶγες) are
next dealt with. These are due to two kinds of exhalation, one
more vaporous (rising from the water on and in the earth), the other
1 Meteor. iv. 1, 378 Ὁ 10-13, 21-5.
® De gen. et corr. ii. 2, 329 Ὁ 22 sq. 5 Ibid. c. 8, 334 Ὁ 31 sq.
* Ibid. c. 3, 330 b 21; Meteor. ii. 4, 359 b 32, &c.
° Meteor. i. 3, 340 Ὁ 29; c. 4, 341 Ὁ 14. 6 Ibid. 341 Ὁ 19-21.
7 Ibid. c. 3, 340 b 23. 8 Ibid. c. 4, 341 Ὁ 21-2.
® De gen. et corr. ii. 3, 330 Ὁ 29. 10 Meteor. ii. 2, 355 ἃ 9.
De caélo ii. 7, 289 a 13-35; Meteor. i. 3, 340a1, 341 a 12-36,
CH. XVII ARISTOTLE 243
dry and smoke-like (rising from the earth); these go upwards, the
latter uppermost, the former below it, until, caught in the rotation
at the circumference of the sublunary sphere, they take fire. The
particular varieties of appearance which they present depend on
the shape of the rising exhalations and the inclination at which
they rise. Sometimes, however, they are the result, not of motion
kindling them, but of heat being squeezed out of air which comes
together and is condensed through cold; in this case their motion
is like a throw (ῥῖψις) rather than a burning, being comparable to
kernels or pips of fruits pressed between our fingers and so made to
fly to a distance; this is what happens when the star falls down-
wards, since but for such compelling force that which is hot would
naturally always fly upwards. All these phenomena belong to the
sublunary sphere. The aurora is regarded as due to the same
cause combined with reflection lighting up the air.? |
Aristotle has two long chapters on comets.* He begins, as
_ usual, by reviewing the opinions of earlier philosophers and so
clearing the ground. Anaxagoras and Democritus had explained
comets as a ‘ conjunction of the planets when, by reason of coming
near, they seem to touch one another’. Some of ‘the so-called
Pythagoreans’ thought that they were one planet, which we
only see at long intervals because it does not rise far above the
horizon, the case being similar to that of Mercury, which, since it
only rises a little above the horizon, makes many appearances
which are invisible to us and is actually seen at long intervals only.
Hippocrates of Chios and his pupil Aeschylus gave a similar
explanation but added a theory about the tail. The tail, they said, -
does not come from the comet itself, but the comet, as it wanders
through space, sometimes takes on a tail ‘ through our sight being
reflected, at the sun, from the moisture attracted by the comet’.
Explanations by Hippocrates and Aeschylus follow, of the reasons
(1) of the long intervals between the appearances of a comet: the
reason in this case being that it is only left behind by the sun very
slowly indeed, so that for a long time it remains so close to the sun
as not to be visible ; (2) of the impossibility of a tail appearing
when the comet is between the tropical circles or still further
1 Meteor. i. 4, 341 Ὁ -- 342 a. 53 Ibid. c. 5, 3428 34-b 24.
§ Ibid. cc. 6, 7, 342 b 25 — 345 a Io.
R2
244 ARISTOTLE PARTI
south: in the former position the comet does not attract the mois-
ture to itself because the region is burnt up by the motion of the
sun in it, and, when it is still further south, although there is plenty
of moisture for the comet to attract, a question of angles (only a
small part of the comet's circle being above the horizon) precludes
the sight being reflected at the sun in this case, whether the sun be
near its southern limit or at the summer solstice ; (3) of the comet’s
taking a tail when in a northerly position: the reason here being
that a large portion of the comet’s circle is above the horizon, and
so the reflection of the sight is physically possible. Aristotle
states objections, some of which apply to all, and others to some
only, of the above views. Thus (1) the comet is not a planet,
because all the planets are in the zodiac circle, while comets are
often outside it ; (2) there have often been more than one comet at
one time; (3) if the tail is due to ‘ reflection’, and a comet has not
a tail in all positions, it ought sometimes to appear without one;
but the five planets are all that we ever see, and they are often all
of them visible above the horizon ; and, whether they are all visible,
or some only are visible (the others being too near the sun), comets
are often seen in addition. (4) It is not true that comets are only
seen in the region towards the north and when the sun is near the
summer solstice; for the great comet which appeared at the time
of the earthquake and tidal wave in Achaea [ 373/2 B.C.] appeared
in the region where the sun sets at the equinox, and many comets
have been seen in the south. Again (5) in the archonship of Eucles,
the son of Molon, at Athens [427/6 B.c.], a comet appeared in the
north in the month of Gamelion, when the sun was at the winter
solstice, although, according to the theory, reflection of the sight
would then be impossible. Aristotle proceeds:
‘It is common ground with the thinkers just criticized and the
supporters of the theory of coalescence that some of the fixed
stars, too, take a tail; on this we must accept the authority of the
Egyptians (for they, too, assert it), and moreover we have ourselves
seen it. For one of the stars in the haunch of the Dog got a tail,
though only a faint one; that is to say, when one looked intently
at it, its light was faint, but when one glanced easily at it, it
appeared brighter.
‘Moreover, all the comets seen in our time disappeared, without
setting, in the expanse above the horizon, fading from sight by
.
CH. XVII ARISTOTLE 245
slow degrees and in such a way that no astral substance, either
one star or more, remained. For instance, the great comet before
mentioned appeared in the winter of the archonship of Astaeus
[373/2 B.C. en in clear and frosty weather, from the beginning of the
evening ; the first day it was not seen because it had set before the
sun, but on the following day it was visible, being the least distance
behind the sun that allowed of its being seen at all, and setting
directly ; the light of this comet stretched over a third part of the
heaven with a great /eap as it were (οἷον ἅλμα), so that people
called it a street. And it went back as far as the belt of Orion and
there dispersed.
* Nevertheless Democritus forone stoutly defended his own theory,
asserting that stars had actually been seen to remain on the disso-
lution of comets. But in that case it should not have sometimes
happened and sometimes failed to happen; it should have hap-
pened always. The Egyptians, too, say that conjunctions take
place of planets with one another and of planets with the fixed stars ;
we have, however, ourselves seen the star of Zeus twice meet one of
the stars in the Twins and hide it, without any comet resulting.’
Aristotle adds that this explanation of comets is untenable on
general grounds, since, although stars may seem large or small,
they appear to be indivisible in themselves. Now, if they were really
indivisible, they would not produce anything bigger by coming in
contact with one another; therefore similarly, if they only seem
indivisible, they cannot seem by meeting to produce anything bigger.
Aristotle’s own theory of comets explains them as due, much
like meteors, to exhalations rising from below and catching fire
when they meet that other hot and dry substance (also here called
exhalation) which, being the first (i.e. outermost) portion of the
sublunary sphere and in direct contact with the revolution of
the upper (aethereal) part of the heavenly sphere, is carried round
with that revolution and even takes with it part of the contiguous
air. The necessary conditions for the formation of a comet, as
distinct from a shooting star or meteor, are that the fiery principle
which the motion of the upper heaven sets up in the exhalation
must neither be so very strong as to produce swift and extensive
combustion, nor yet so weak as to be speedily extinguished, but of
moderate strength and moderate extent, and the exhalation itself
must be ‘ well-tempered ’ (εὔκρατος); according to the shape of the
kindled exhalation it is a comet proper or the ‘bearded’ variety
246. ARISTOTLE PARTI
(πωγωνίας). But two kinds of comets are distinguished. One is —
produced when the origin of the exhalation is in the sublunary
sphere ; this is the independent comet (καθ᾽ ἑαυτὸν κομήτης). The
other is produced when it is one of the stars, a planet or a fixed
star, which causes the exhalation, in which case the star becomes a
comet and is followed round in its course by the exhalation, just as |
haloes are seen to follow the sun and the moon. Comets are thus
bodies of vapour in a state of slow combustion, moving either freely
or in the wake of a star. Aristotle maintains that his view that
comets are formed by fire produced from exhalations in the manner
described is confirmed by the fact that in general they are a sign of
winds and droughts. When they are dense and there are more
of them, the years in which they appear are noticeably dry and
windy ; when they are fewer and fainter, these characteristics are
less pronounced, though there is generally some excess of wind
either in respect of duration or of strength. He adds the following
remarks on particular cases:
‘On the occasion when the (meteoric) stone fell from the air at
Aegospotami, it was caught up by a wind and was hurled down in ~
the course of a day;! and at that time too a comet appeared from
the beginning of the evening. Again, at the time of the great comet
[373/2 B.C., see pp. 244, 245 above] the winter was dry and arctic,
and the tidal wave was caused by the clashing of contrary winds ;
for in the bay the north wind prevailed, while outside it a strong
south wind blew. Further, during the archonship of Nicomachus .
at Athens [341/0 B.C.] a comet was seen for a few days in the
neighbourhood of the equinoctial circle; it was at the time of this
comet, which did not rise with the beginning of the evening, that
the great gale at Corinth occurred.’
1 This appears to be the earliest mention of the meteoric stone of Aegospotami
by any writer whose works have survived. The date of the occurrence was
apparently in the archonship of Theagenides [468/7 B.c.]. The story that
Anaxagoras prophesied that this stone would fall from the sun (Diog. L. ii. 10)
was probably invented by way of a picturesque inference from his well-known
theory that the fiery aether whirling round the earth snatched stones from the
earth and, carrying them round with it, kindled them into stars (Aét. ii. 13. 3;
D.G.p.341; Vorsokratiker, i*, p. 307.16), and that one of the bodies fixed in the
heaven might break away and fall (Diog. L. ii. 12; Plutarch, Lysander 12;
Vorsokratiker, 15, pp. 294. 29, 296. 34). Diogenes of Apollonia, too, a contempo-
rary of Anaxagoras, said that along with the visible stars there are also stones
carried round, which are invisible, and are accordingly unnamed; ‘and these
often fall upon the earth and are extinguished like the stone star which made
a fiery fall at Aegospotami’ (Aét. ii. 13. 9; D. G. p. 3423. Vorsokratiker,
15, p. 330. 5-8).
Se
so ΎΝΝΝ a ϑϑων ἐν.
CH, XVII ARISTOTLE 247
It has been pointed out that Aristotle’s account of comets held
its ground among the most distinguished astronomers till the time
of Newton.!
Passing to the subject of the Milky Way,” Aristotle again begins
with criticisms of earlier views. The first opinion mentioned is
that of the Pythagoreans, some of whom said that it was the path
of one of the stars which were cast out of their places in the
destruction said to have occurred in Phaethon’s time; while others
said that it was the path formerly described by the sun, so that this
region was, so to speak, set on fire by the sun’s motion. But,
Aristotle replies, if this were so, the zodiac circle should be burnt
up too, nay more so, since it is the path not only of the sun but of
the planets also. But we see the whole of the zodiac circle at one
time or another, half of it being seen in a night; and there is no
sign of such a condition except at points where it touches the
Milky Way. The remarkable hypothesis of Anaxagoras and
Democritus is next controverted ; we have already (pp. 83-5) quoted
Aristotle’s criticisms. Next,a third view is mentioned according to
which the Milky Way is ‘a reflection of our sight at the sun’, just
as comets had been declared to be. Aristotle refutes this rather
elaborately. (1) If, he says, the eye, the mirror (the sun) and the
thing seen (the Milky Way) were all at rest, one and the same part
1 Ideler, Aristotelis Meteorologica, vol. i, p. 396. Yet Seneca (Nat. Quaest.
vii) had much sounder views on comets. He would not admit that they could
be due to such fleeting causes as exhalations and rapid motions, as of whirlwinds,
igniting them ; if this were their cause, how could they be visible for six months
at a time (vii. 10.1)? They are not the effects of sudden combustion at all, but
eternal products of nature (22. 1). Nor are they confined to the sublunary.
sphere, for we see them in the upper heaven among the stars (8. 4). If it
is said that they cannot be ‘wandering stars’ because they do not move in
the zodiac circle, the answer is that there is no reason why, in a universe
so marvellously constructed, there should not be orbits in other regions than
the zodiac which stars or comets may follow (24. 2-3). It is true, he says, that,
owing to the infrequency of the appearances of comets, their orbits have not
as yet been determined, nay, it has not been possible even to decide whether
they keep up a definite succession and duly appear on appointed days. In
order to settle these questions, we require a continuous record of the appearances
of comets from ancient times onwards (3.1). When generation after generation
of observers have accumulated such records, there will come a time when the
mystery will be cleared up; men will some day be found to show ‘in what
regions comets run their courses, why each of them roams so far away from the
others, how large they are and what their nature ; let us, for our part, be content
with what we have already discovered, and let our posterity in their turn contribute
to the sum of truth (25. 7).’ ® Meteor. i. 8, 345 a 11 - 346 Ὁ 10,
248 ARISTOTLE
of the reflection would belong to one and the same point of the
mirror ; but if the mirror and the thing seen move at invariable
distances from our eye (which is at rest), but at different speeds and
distances relatively to one another, it is impossible that the same
part of the reflection should always be at the same point of the
mirror. Now the latter of the two hypotheses is that which corre-
sponds to the facts,’because the stars in the Milky Way and the sun
respectively move at invariable distances from us, but at different
distances and speeds in relation to one another ; for the Dolphin
rises sometimes at midnight, and sometimes at sunrise, but the
parts of the Milky Way remain the same in either case ; this could
not be so if the Milky Way were a reflection instead of a condition
of the actual localities over which it extends. (2) Besides, how can
the visual rays be reflected at the sun during the night? Aristotle’s
own explanation puts the Milky Way on the same footing as the
second kind of comets, those in which the separation of the vapour
which takes fire on coming into contact with the outer revolution is —
caused by one of the stars; the difference is that what in the case
of the comet happens with one star takes place in the case of the
Milky Way throughout a whole circle of the heaven and the outer
revolution. The zodiac circle, owing to the motion in it of the sun
and planets, prevents the formation of the exhalations in that neigh-
bourhood ; hence most comets are seen outside the tropic circles.
The sun and moon do not become comets because they separate
out the exhalation too quickly to allow it to accumulate to the
necessary extent. The Milky Way, on the other hand, represents
the greatest extent of the-operation of the process of exhalation ;
it forms a great circle and is so placed as to extend far beyond the
tropic circles, The space which it occupies is filled with very great
and very bright stars, as well as with those which are called ‘ scat-
tered’ (σποράδων); this is the reason why the collected exhalations
here form a concretion so continuous and so permanent. The
cause is indeed indicated by the fact that the brightness is greater
in that half of the circle where it is double, for it is there that the
stars are more numerous and closer together than elsewhere.
oe |
XVIII
HERACLIDES OF PONTUS
THE Pythagorean hypothesis of the revolution of the earth with
the counter-earth, and of the sun, moon, and planets, about the
central fire disappeared with the last representatives of the Pytha-
gorean school soon after the time of Plato. The counter-earth was
the first part of the system to be abandoned; and it is suggested
that this abandonment was due to the extension of the geographical
horizon. Discoveries were made both to the east and to the west.
Hanno, the Carthaginian, had made his great voyage of discovery
beyond the Pillars of Hercules, and on the other (the eastern) side
India became part of the known world. It would naturally be
expected that, if journeys were made far enough to the east and
west, points would be reached from which it should be possible
to see the counter-earth, but, as neither the counter-earth nor the
central fire proved in fact to be visible, this portion of the Pytha-
gorean system had to be sacrificed.!
We hear of a Pythagorean system in which the central fire was
not outside the earth but in the centre of the earth itself. Simplicius,
in a note upon the passage of Aristotle describing the system of ‘the
Italian philosophers called Pythagoreans’ in which the earth revolves |
about the central fire and so ‘makes day and night’, while it has
the counter-earth opposite to it, adds that this is the theory of the
Pythagoreans as Aristotle understood it, but that those who repre-
sented the more genuine Pythagorean doctrine ‘describe as fire at
the centre the creative force which from the centre gives life to all
the earth and warms afresh that part of it which has cooled down.
... Lhey called the earth a star, as being itself too an instrument of
time. For the earth is the cause of days and of nights, since it makes
day when it is lit up in that part which faces the sun, and it makes
1 Gomperz, Griechische Denker, i*, pp. 97, 98 ; Schiaparelli, J precursori di
Ci ico nell’ antichita, pp. 22, 25.
Simplicius on De cae/lo, p. 512. 9-20, Heib.
250 HERACLIDES OF PONTUS | PARTI
night throughout the cone formed by its shadow. And the name
of counter-earth was given by the Pythagoreans to the moon, just
as they also called it “earth in the aether” (αἰθερίαν γῆν), both
because it intercepts the sun’s light, which is characteristic of the
earth, and because it marks a delimitation of the heavenly regions,
as the earth limits the portion below the moon.’
It is no doubt attractive to suppose, as Boeckh! does, that we
have here a later modification of the system of Philolaus. But,
as Martin? points out and Boeckh ὃ admits, the earth in the system
described by Simplicius is not in motion but at rest. For Simplicius,
so far from implying that the earth rotates, thinks it necessary to
explain how the Pythagoreans to whom he refers could, notwith-
standing the earth’s immobility, call it a ‘star’ and count it, exactly
as Plato does, among the ‘instruments of time’. The fact is that
the system, except for the detail of the term ‘counter-earth ’ being
applied to the moon, agrees with the Platonic system as described
in the Z7zmaeus, and, as we have seen, there is nothing to suggest
that Plato was acquainted with the Philolaic system at all; he was
rather basing himself upon the views of Pythagoras and the first
Pythagoreans.
A scholiast, writing on the same passage of Aristotle and
describing the views of the Pythagoreans in almost the same
words as those used by Simplicius, does, however, attribute motion
to the earth. They put, he says, the fire at the centre of the
earth. ‘They said that the earth was a star as being itself too an
“instrument”. The counter-earth for them meant the moon... .
And this star [i.e. evidently the earth] dy its motion (φερόμενον)
makes night and the day, because the cone formed by its shadow
is night, while day is the illuminated part of it which is in the
sun.’* The attribution of motion to the earth may be due to
a misapprehension by the scholiast, just as Boeckh himself had
at first assumed the earth’s rotation to be indicated in the passage
of Simplicius,
However this may be, if the system of Philolaus be taken, and
the central fire be transferred to the centre of the earth (the
1 Boeckh, Das khosmische System des Platon, p. 96. ,
2 Martin, Etudes sur le Timée, ii, p. 104. 83 Boeckh, loc, cit.
* Scholia in Aristotelem (Brandis), pp. 504 Ὁ 42 - 505 ἃ 5.
CH. XVIII HERACLIDES OF PONTUS © 251
counter-earth being also eliminated), and. if the movements of the
earth, sun, moon, and planets round the centre be retained without
any modification save that which is mathematically involved by the
transfer of the central fire to the centre of the earth, the daily revo-
lution of the earth about the central fire is necessarily transformed
into a rotation of the earth about its own axis in about 24 hours.
All authorities agree that Heraclides of Pontus affirmed the daily
rotation of the earth about its own axis; but the Doxographi
associate with this discovery another name, that of ‘ Ecphantus
the Pythagorean’. Thus we are told of Ecphantus that he asserted
‘that the earth, being in the centre of the universe, moves about
its own centre in an eastward direction’.t Again, ‘ Heraclides of
Pontus and Ecphantus the Pythagorean make the earth move, not
in the sense of translation, but by way of turning as on an axle,
like a wheel, from west to east, about its own centre.’ Who then
is this Ecphantus, described in another place in Aétius as Ecphantus
the Syracusan, one of the Pythagoreans? His personality is even
more of a mystery than that of Hicetas. The Doxographi, however,
tell us of other doctrines of his; Hippolytus* devotes a short
paragraph to him, between paragraphs about Xenophanes and
Hippon, which shows that Theophrastus must have spoken of him
at length. Some of his views were quite original, particularly on
the subject of atoms. Holding that the universe was made up of
indivisible bodies separated by void, he was the first to declare
that the monads of Pythagoras were corporeal; he attributed to
the atoms, besides size and shape, a motive force (δύναμις) ; the
atoms were moved, not by their weight or by percussion, but by -
a divine force which he called mind and soul. The universe was
a type of this, and accordingly the divine motive force created it
spherical. Now it is remarkable that Ecphantus’s views all agree
with Heraclides’ so far as we know them; Heraclides has the same
divine force moving the universe, which he also calls mind and soul ;
he has the same theory of atoms, which he calls masses* (ὄγκοι).
And the two hold the same view about the rotation of the earth.
1 Hippolytus, Refut. i. 15 (D. G. p. 566; Vors. i*, p. 265. 35).
2 Aét. iii. 13. 3 (D. G. p. 378; Vors. i*, p. 266. 5). 8 Hippolytus, loc. cit.
+ * Galen, Histor. phil. 18 (D.G. p. 610. 22); Dionysius episcop. ap. Euseb.,
P.E. xiv. 23. See Otto Voss, De Heraclidis vita et scriptis, p.64; Tannery,
Revue des Etudes grecques, x, 1897, pp. 134-6
252 HERACLIDES OF PONTUS PART I
Zeller observes, in addition, that the remark about the universe
being made spherical reminds us of Plato.’ Just as in the case of
Hicetas, the natural conclusion is that the views attributed by the
Doxographi to Ecphantus were expressed in a dialogue of Heraclides
and put into the mouth of Ecphantus represented as a Pythagorean.
Theophrastus may then have said something of this sort: ‘ Hera-
clides of Pontus has developed the following theories, attributing
them to a certain Ecphantus’; and this would account for the
Doxographi citing the doctrines sometimes by the name of Heraclides,
sometimes by the name of Ecphantus.?
Heraclides, son of Euthyphron, was born at Heraclea in Pontus,
probably not many years later than 388 B.c. He is said to have
been wealthy and of ancient family. He went to Athens not later
than 364, and there met Speusippus, who introduced him into the
school of Plato. Proclus, it is true, denied that he was a pupil
of Plato,? but this was because Proclus resented his contradiction
of the Platonic view of the absolute immobility of the earth in the
centre of the universe. Diogenes Laertius,* Simplicius,® Strabo,® .
and Cicero” leave us in no doubt on the subject; and we may
also infer his relation to Plato from words of his own quoted
elsewhere by Proclus, according to which he was sent by Plato
on an expedition to Colophon to collect the poems of Antimachus.
Suidas® says that, during a journey of Plato to Sicily, Heraclides
was left in charge of the school. After the death of Plato in 347,
Speusippus was at the head of the school for eight years, and on
his death in 338 B.C. Xenocrates was elected his successor,
Heraclides and Menedemus, who were also candidates, being beaten
by a few votes.!° Heraclides then returned to his native town,
where he seems to have lived till 315 or 310 B.c. While at
Athens he is said to have attended the lectures of Aristotle also ;"
but Diogenes’ statement that he also ‘heard the Pythagoreans’
1 Zeller, ἰδ, pp. 494, 495. 3 Tannery, loc. cit., p. 136.
8 Proclus, iz 7272. 281 E. 4 Diog. L. iii. 46, v. 86.
δ Simpl. im Ar. Phys. iii. 4 (p. 202 Ὁ 36), p. 453. 29, Diels.
® Strabo, xii. 3. 1, p. 541.
7 Cic. De nat. deor. i. 13. 34; De legg. iii. 6. 14; Tusc. Disp. v. 3.8; De
Divin. i. 23. 46. 8 Proclus, 272 Tim. 28 C.
® Suidas, s.v. Ἡρακλείδης. Zeller and Wilamowitz adduce confirmatory#
evidence. Voss alone disputes the statements; for references see Voss, pp. 11-12.
0 Ind, Acad. Hercul. vi (Voss, Ρ. 7). 1! Sotion in Diog. L. v. 86.
—
SO ὅσων τ νμννν.
CH. XVIII HERACLIDES OF PONTUS 253
is no doubt incorrect ; for by that time the Pythagoreans had left
Greece altogether. The words were probably interpolated in the
passage of Diogenes by some one who inferred first-hand ac-
quaintance with Pythagorean doctrines on the part of Heraclides
from the fact, among others, that he wrote a book ‘concerning
the Pythagoreans’.
Diogenes Laertius tells us that Heraclides wrote works of the
highest class both in matter and style. The remark is followed
by a catalogue covering a very wide range of subjects, ethical,
grammatical, musical and poetical, rhetorical, historical, with a note
that there were geometrical and dialectical treatises as well. His
dialogues are classified as (1) those which were by way of comedy,
e.g. those on Pleasure and on Prudence, (2) those which were
tragic, such as those on Things in Hades and on Piety, and (3) in-
termediate in character, familiar dialogues between philosophers,
soldiers, and statesmen. They were very varied and very persuasive
in style, adorned with myth and full of imagination, so original as to
make Timaeus describe their author as παραδοξολόγος throughout,
while Epicurus and the Epicureans, who attacked his physical
theories, spoke of him as ‘cramming his books with puerile stories’.
There seems to have been more action in his dialogues than in
Plato’s;* his prologues generally had nothing to do with what
followed ;* there were usually a number of characters, and he
was fond of introducing as interlocutors personages of ancient
times. He was much read and imitated at Rome, e.g. by Varro
and Cicero; Cicero, for example, modelled upon Heraclides his
dialogue De republica. Two of his dialogues at least, those ‘On -
Nature’ and ‘ On the Heavens’, may have dealt with astronomy.
He naturally had enemies, who not only impugned his doctrines
but took objection to his personality. We are told that he was
effeminate in dress and over-corpulent, so that he was called,
not Ponticus, but Pompicus (IIopmixés); his gait was slow and
stately.®
Several of the fragments of Heraclides recall passages in Plato.
Thus Heraclides represents souls as coming down, for incarnation,
ὲ 1 Voss, pp. 12-13. 3 Ibid., pp. 26, 27.
3 Proclus, iz Plat. Parmenidem, Book i, ad jin.
* Voss, p. 22. 5 Diog. L. v. 86.
- 2,54. HERACLIDES OF PONTUS » PARTI
from regions in the heaven, which he places in or about the Milky
Way 1 (cf. the Phaedrus myth). The universe is a god ; so are the
planets, the earth, and the heaven.?. Other views of his about the
universe and what it contains may also be referred to before we
pass to the great discoveries in astronomy on which his fame rests.
The universe is infinite ;* each star is also a universe or world, sus-
pended in the infinite aether and comprising an earth, an atmosphere
and an aether.t* The moon is earth surrounded with mist.2 Comets
are clouds high in air lit up by the fire on high; he accounts
similarly for meteors and the like; their different forms follow that
of the cloud.®
We now pass to the first of Heraclides’ great discoveries, that
of the daily rotation of the earth about its axis. Besides the
passages above quoted, in which ‘ Ecphantus’ is also credited with
the discovery, we have the following clear evidence on the
subject :
‘He (Aristotle) thought it right to take account of the hypothesis
that doth (i.e. the stars and the heaven as a whole) are at rest— Ὁ
- although it would appear impossible to account for their apparent
change of position on the assumption that both are at rest—because
there have been some, like Heraclides of Pontus and Aristarchus,
who supposed that the phenomena can be saved if the heaven
and the stars are at rest while the earth moves about the poles of
the equinoctial circle from the west (to the east), completing one
revolution each day, approximately ; the ‘approximately’ is added
because of the daily motion of the sun to the extent of one degree,
For of course, if the earth did not move at all, as he will later
show to be the case, although he here assumes that it does for the
sake of argument, it would be impossible for the phenomena to be
saved on the supposition that the heaven and the stars are at
rest,’7
‘But Heraclides of Pontus supposed that the earth is in the
centre and rotates (lit. ‘moves in a circle’) while the heaven is
at rest, and thought by this supposition to save the phenomena.’ ὃ
‘Heraclides of Pontus supposed that the earth moves about the ©
? Iamblichus in Stobaeus, F/or., p. 378, ed. Wachsmuth.
2. Cicero, De nat. dor. i. 13. 34 (D. G. p. 541. 3-13).
8 Aét. ii. 1. 5 (D. G. p. 328 Ὁ 4). * Aét. ii. 13. 15 (D. G. p. 343).
5 Aét. ii. 25. 13 (D. G. p. 356). ® Aét. ili. 2. 5 (D. G. pp. 366, 367).
7 Simplicius on De cae/o ii. 7 (289 Ὁ 1), pp. 444. 31 -- 445. 5, Heib.
§ Ibid. (on c. 13, 293 Ὁ 30), p. 519. 9-11, Heib.
CH. XVIII HERACLIDES OF PONTUS 255
centre, while the heaven is at rest, and thought in this way to
save the phenomena.’
‘This would equally have happened [i.e. the stars would have
seemed to be at different distances at different times instead of,
as now, appearing to be always at the same distance, whether at
rising or at setting or between these times, and the moon would
not, when eclipsed, always have been diametrically opposite the sun,
but would sometimes have been separated from it by an arc less
than a semicircle] if the earth had a motion of translation; but
if the earth rotated about its centre while the heavenly bodies were
at rest, as Heraclides of Pontus supposed, then (1), on the hypo-
thesis of rotation towards the west, the stars would have been seen
to rise from that side, while (2) on the hypothesis of rotation towards
the east, (a) if it so rotated about the poles of the equinoctial circle
(the equator), the sun and the other planets would not have risen
at different points of the horizon [!], and, (4) if it so rotated about
the poles of the zodiac circle, the fixed stars would not always have
risen at the same points, as in fact they do; so that, whether
it rotated about the poles of the equinoctial circle or about the
poles of the zodiac, how could the translation of the planets in
the direct order of the signs have been saved on the assumption of
the immobility of the heavens?’ ?
-* How can we, when we are told that the earth is wound round,
reasonably make it turn round as well and give this as Plato’s
view? Let Heraclides of Pontus, who was not a disciple of Plato,
hold this opinion and move the earth round and round (κύκλῳ) ;
but Plato made it unmoved.’ ὃ
The second great advance towards the Copernican system made
by Heraclides was his discovery of the fact that Venus and Mercury
revolve round the sun as centre. Some of the passages alluding to
Heraclides’ recognition of this fact import the later doctrine of ©
epicycles; but it is not difficult to eliminate this anachronism and
to arrive at Heraclides’ true theory. In some of the references
the name of Heraclides is not mentioned. Vitruvius* describes the
hypothesis thus :
‘The stars of Mercury and Venus make their retrograde motions
and retardations about the rays of the sun, forming by their courses
a wreath or crown about the sun itself_as centre. It is also owing
to this circling that they linger at their stationary points in the
spaces occupied by the signs.’
' Schol. in Arist. (Brandis), p. 505 b 46-7.
? Simpl. on De cae/o ii. 14 (297 a 2), pp. 541. 27 — 542. 2, Heib.
3. Proclus, ἐς Tim. 281 E. * Vitruvius, De architectura ix. τ (4). 6.
256 HERACLIDES ΟΕ PONTUS PART I
Next Martianus Capella}, who drew from Varro’s work on astro-
nomy, mentions the same hypothesis, but again without the name
of its discoverer.
‘For, although Venus and Mercury are seen to rise and set daily,
their orbits do not encircle the earth at all, but circle round the sun
in a freer motion. In fact, they make the sun the centre of their
circles, so that they are sometimes carried above it, at other times
below it and nearer to the earth, and Venus diverges from the sun
by the breadth of one sign and a half [45°]. But, when they are
above the sun, Mercury is the nearer to the earth, and when they
are below the sun, Venus is the nearer, as it circles in a greater
and wider-spread orbit .....
‘The circles of Mercury and Venus I have above described as
epicycles. That is, they do not include the round earth within
their own orbit, but revolve laterally to it in a certain way.’
‘Cicero says that the courses of Venus and Mercury ‘follow the
sun as companions’,? but has nothing about their revolving round
the sun.
It is in Chalcidius*® that we find the name of Heraclides con- .
nected with the revolution of the planets Mercury and Venus round
the sun as centre; but, like Adrastus in Theon of Smyrna, he
erroneously imputes to Heraclides, as to Plato in the Z77maeus, the
machinery of epicycles. His words are:
‘Lastly Heraclides Ponticus, when describing the circle of Lucifer
as well as that of the sun, and giving the two circles one centre and
one middle, showed how Lucifer is sometimes above, sometimes
below the sun. For he says that the position of the sun, the moon,
Lucifer, and all the planets, wherever they are, is defined by one
line passing from the centre of the earth to that of the particular
heavenly body. ‘There will then be one straight line drawn from
the centre of the earth showing the position of the sun, and there
will equally be two other straight lines to the right and left of it
respectively, and distant 50° from it, and 100° degrees from each
other, the line nearest to the east showing the position of Lucifer
or the Morning Star when it is furthest from the sun and near the
eastern regions, a position in virtue of which it then receives the
name of the Evening Star, because it appears in the east at evening
after the setting of the sun.’. . . . (And so on.)
1 Martianus Capella, De nuptiis Philologiae et Mercurit, viii. 880, 882.
3 Cicero, Somn. Scip. c. 4. 2.
5 Chalcidius, Zimaeus, c. 110, pp. 176-7, Wrobel.
CH. XVIII HERACLIDES OF PONTUS 257
Chalcidius only mentions Venus in this passage, but he has just
previously indicated a similar relation between Mercury and the
sun. Reading this passage and the explanation, illustrated by
a figure, which follows, together with supplementary particulars
given in a passage of Macrobius presently to be mentioned, we can
easily realize Chalcidius’s conception. According to this we are
to suppose a point which revolves uniformly about the earth from
west to east ina year. This point is the centre of three concentric
circles (epicycles) on which move respectively the sun (on the
innermost), Mercury (on the middle circle), and Venus (on the
outermost) ; the sun takes, of course, a year to describe its epi-
cycle That the epicycle for the sun is wrongly imported into
Heraclides’ true system is confirmed by the next chapter of Chal-
cidius, with its illustrative figure, where he imports epicycles into
Plato's system also. According to him, Plato used, not one principal
circle with three epicycles having as their common centre a point
describing that principal circle, but three principal circles, each with
one epicycle.; two circles, namely a principal circle and an epicycle,
being used for each of the three bodies, the sun, Mercury, and Venus.
But we know that in Plato’s system the sun, Mercury, and Venus
described three simple circles of which the earth is the centre.
Hence the epicycles must be rejected altogether so far as Plato’s
System is concerned. Similarly, we must eliminate the sun’s epi-
cycle from the account of Heraclides’ system, and we must suppose
that he regarded Mercury and Venus as simply revolving in con-
centric circles about the sun.
The same contrast as is drawn by Chalcidius between Heraclides’
system and Plato’s system, as he represents them respectively, is
drawn by Adrastus* between two possible theories, the authors of
1 Chalcidius indicates (cc. 81, 109, and 112) that the sun’s motion on its epi-
cycle (which is from east to west) is in the contrary sense to the motion (from
west to east) of Mercury and Venus on their epicycles respectively (cf. Adrastus
in Theon of Smyrna, p. 175, 13-15, who says that the motion of the sun and moon
on their epicycles is in the sense of the daily rotation from east to west, while the
motion of the five planets on their epicycles is in the opposite sense). The
commentators did not fail to see in this fact a possible explanation of Plato’s
remark that Mercury and Venus have ‘the contrary tendency to the sun’ (Chal-
Cidius, c. 109, p. 176); and the explanation would be quite satisfactory zf Plato
could be supposed to have been acquainted with the theory of epicycles (cf.
pp- 165-9 above).
5 Adrastus in Theon of Smyrna, pp. 186. 17 - 187. 13.
1410 5
258 HERACLIDES OF PONTUS PART I
which he does not specify. The first possibility corresponds to
Chalcidius’s version of Plato’s system ; only Hipparchus’s epicycles
are, in agreement with Eudoxus’s theory of spheres, represented
by ‘solid’ spheres as distinct from ‘hollow’. We are to conceive,
in the plane of the ecliptic, three concentric circles with the earth as
common centre ; on each circle there moves, in one and the same
direction, the centre of an immaterial sphere at such speed that the
centre of the earth and these three centres are always in a straight
line. As the plane of the ecliptic cuts the three immaterial spheres,
this determines three circles which, with Hipparchus, we distin-
guish from the principal circles as epicycles. The sun moves on
the epicycle of the circle nearest the earth, Mercury on that of the
next, Venus on that of the outer circle. This is, therefore, precisely
the Platonic system as conceived by Chalcidius. The second possi-
bility, says Adrastus, is that the three principal circles may coalesce
into one. Thus the three epicycles are reduced to sections of three
concentric spheres, and the whole system of these spheres revolves
about the earth, their common centre describing a circle about the
earth. Here we have Heraclides’ system as described by Chalcidius ;
but Adrastus’s version is better, in that, evidently relying on an older
source, he hints that what moves on the main circle is not an
immaterial point but the ‘true solid sphere of the sun’; that is to
say, it is only Mercury and Venus which move on epicyecles, i.e. in
circles about the sun as centre."
Martin? exposed the error of those who inferred from the passage
of Macrobius already alluded to that the Egyptians were acquainted
with the fact thus stated by Heraclides. Macrobius observes that
Cicero, in placing the sun fourth in the order of the planets reckon-
ing from the earth, i.e. after the moon, Venus, and Mercury, followed —
the order adopted by the Chaldaeans and Archimedes, while
‘Plato followed the Egyptians, the parents of all branches of
philosophy, who, while placing the sun between the moon and
Mercury, yet have detected and enunciated the reason why the sun
is believed by some to be above Mercury and above Venus; for
eis are those who hold this view far from the apparent
truth. °...
1 Hultsch, ‘Das astronomische System des Herakleides von Pontos’ in Jahré,
fiir class. Philologie, 1896, pp. 305-16.
3 Martin, Etudes sur le Timée, ii, Pp. 130-3. Cf. Boeckh, Das kosmische
System des Platon, pp. 142, 143. 8. Macrobius, /m# somn, Scip. i. 19. 2.
CH. XVIII HERACLIDES OF PONTUS 259
Then, after explaining that Saturn is as far from Jupiter as is
indicated by the difference between their periods, 30 years and
12 years respectively, and again, that Jupiter’s distance from Mars
corresponds to the difference between their periods of 12 and 2 years
respectively, he observes that Venus is so much below Mars as
corresponds to the shorter period of Venus, one year, while Mercury
is so near to Venus, and the sun to Mercury, that they all describe
their orbits in one year, more or less, so that, as Cicero says, Venus
and Mercury are companions of the sun. There was, therefore, no
dispute about the order of the superior planets, Saturn, Jupiter, and
Mars, nor about the relative position of the moon as the lowest
of all;
‘But the proximity of the three others which are the nearest to
one another, namely Venus, Mercury, and the sun, has caused
uncertainty as regards their order, though only in the minds of
others, not of the Egyptians ; for the true relation did not escape the
penetration of the Egyptians, and it is as follows. The circle on
which the sun moves |‘ circulus, per quem sol discurrit’= the sun’s
epicycle| is lower than, and encircled by, the circle of Mercury;
above the circle of Mercury, and including it, is the circle of Venus;
hence it is that, when the two planets are describing the upper
portions of their circles, they are regarded as placed above the sun,
but when they are traversing the lower portions of their circles, the
sun is considered to be superior to them.’!. . .
Macrobius’s main object may have been to put the Egyptians on
a level with the Chaldaeans, the oldest cultured Asiatics.? But,
though the Chaldaeans arranged the planets in an order different
from that adopted by Plato, the idea of Mercury and Venus revolv- .
ing round the sun was certainly not Chaldaean but Greek, and
originated with Heraclides. If Macrobius really intended to attri-
bute Heraclides’ discovery to the Egyptians, it must be because
the theory had perpetuated itself as a tradition of the Alexandrine
astronomers anterior to our era.* And if the Egyptians had
really regarded Mercury and Venus as being in the relation of
satellites to the sun, it is not easy to understand why they placed
Mercury and Venus above the sun, since they might equally well
have placed them below it.
Hultsch explains the evolution of the Heraclides-epicyclic system
1 Macrobius, /# somn. Scip. i. 19, 5-6. * Hultsch, loc, cit.
3 Tannery, Recherches sur P histoire de Pastronomie ancienne, pp. 260, 261.
52
260 HERACLIDES OF PONTUS PARTI
in the following way. The axial rotation of the earth was rejected
by Hipparchus. Hence the occasion, for some one living after
Hipparchus’s time, of modifying Heraclides’ system and grafting
on to it the theory of epicycles. Or perhaps the post-Hipparchian
inventor of the Heraclides-epicyclic blend wished to oppose to some
enthusiastic champion of Hipparchus the authority of Heraclides,
but could not get rid of epicycles.
The next question which arises is this. Having made Mercury
and Venus revolve round the sun as satellites, did Heraclides
proceed to draw the same inference with regard to the other, the
superior, planets? When it was once laid down that all the five
planets alike revolved round the sun, and this hypothesis was com-
bined with that of the revolution of the sun round the earth as
centre, the result was the system of Tycho Brahe, with the improve-
ment, already made by Heraclides, of the substitution of the daily
rotation of the earth for the daily revolution of the whole system
round the earth supposed at rest. Schiaparelli, who added to his
first tract, J precursori di Copernico nell’ antichita, a further ex- —
tremely elaborate study! dealing at length with the above question
among others, came to the conclusion that it was probably Hera-
clides himself who took the further step of regarding all the five
planets alike as revolving round the sun, but that, if it was not
Heraclides, it was at all events some contemporary of his who did
so. This conclusion represents a certain change of view on the
part of Schiaparelli after the date of 7 precursori, where he says, ©
‘it appears that Heraclides Ponticus, as the evidence cited indicates,
limited to Venus and Mercury the revolution round the sun, and it
seems that he retained the earth as the centre of the movements
of the superior planets’.? Schiaparelli’s later view is based upon
presumption rather than upon direct evidence, which indeed does
not exist. His argument is a tour de force, but, although opinions
will differ, I for my part think that he trusts too much to the testi-
mony of late writers as to the supposed very early discovery of the
machinery of eccentrics and epicycles, and his case does not seem
to me to be made out.
1 Schiaparelli, Origine del sistema planetario eliocentrico presso ἡ Grect, 1898
(in Memorte del R. Istituto Lombardo di scienze e lettere, vol. xviii, pp. 61 sqq.).
2. Schiaparelli, 7 precursori, pp. 27, 28.
ESE, νοδι
CH. XVIII HERACLIDES OF PONTUS 261
Schiaparellis arguments are, however, well worthy of considera-
tion, and I will represent them as completely and fairly as I can.
Having hit upon the hypothesis of the revolution of Mercury and
Venus round the sun, and not the earth, as centre, Heraclides had
found a possible explanation of the varying degrees of brightness
shown by the two inferior planets and of the narrow limits of their
deviation from the sun; he would also easily see that the hypo-
thesis gave a solution of the difficulty of the stationary points and
the retrogradations in the case of these planets. Eudoxus had
tried to solve the latter difficulty by ingenious and elegant combina-
tions of concentric spheres; but he only succeeded with Jupiter and
Saturn. Callippus went further on the same lines and succeeded
to a certain extent with Mars; probably, too, he came nearer to
accounting for the movements of Mercury and Venus. The most
formidable objection to the explanation of the planetary move-
ments by means of concentric spheres was the fact that, on this
hypothesis, the distance of each planet from the earth, and conse-
quently its brightness, should be absolutely invariable, whereas
mere ocular observation sufficed to prove that this is not so. This
difficulty was, as we have seen, very early realized; Polemarchus,
a friend of Eudoxus himself, was aware of it, but tried to make out
that the inequality of the distance was negligible and of no account
in comparison with the advantage of having all the spheres about
one and the same centre ; Aristotle, too, in his Physical Problems
(now lost) discussed the same difficulty. The first who tried to get
over the difficulty was Autolycus of Pitane, the author of the tract
On the moving sphere, but even he was not successful.2 Now Hera-
clides, departing altogether from the system of spheres, to which
the Aristotelian school doggedly adhered, and adopting a system of
circles more akin to Pythagorean ideas, had suggested a sufficient
explanation with regard to Venus and Mercury; and, as Mars was
seen, equally with Venus, to vary in apparent size and brightness,
it was natural for the same school of thought to try to find an
explanation of the similar phenomena with regard to Mars on ¢heir
lines as opposed to those which found favour with Aristotle.
Now, with regard to Mars, it would be seen that the times of its
1 Sosigenes in Simplicius on De caelo (293 a 4), p. 505. 21-7, Heib.
3 Ibid., p. 504. 22-5, Heib.
262 HERACLIDES ΟΕ PONTUS PART I
greatest brightness corresponded with the times when it was in
opposition and not in conjunction ; that is to say, it is brightest
when it occupies a position in the zodiac opposite to the sun; it
must therefore be nearest the earth at that time, and consequently
the centre of its orbit cannot be the centre of the earth, but must
be on the straight line joining the earth to the sun. The analogy
of Venus and Mercury might then suggest that perhaps Mars, too,
might revolve round the sun, I do not attach much importance in
this connexion to a passage from Theon of Smyrna quoted by
Schiaparelli. Theon, in the passage contrasting two hypotheses
(the supposed Platonic and supposed Heraclidean) with regard to
the movements of Venus and Mercury, adds:
‘And one might suspect that this [the Heraclidean view] repre-
sents the truer view of their relative position and order, the effect
of it being to make this region the abode of the animating principle
in the universe, regarded as a living thing, the sun being as it were
the heart of the All in virtue of its great heat and in consequence
of its motion, its size, and its connexion with the bodies about it. |
For in animate beings the centre of the thing, that is, of the animal
as animal, is different from the centre of it regarded as a magnitude ;
thus with ourselves as men and living beings one centre is the
region about the heart, the centre of the vital principle .. . the other
is that of the body as a magnitude. .. . Similarly, if we may extend
to the greatest, noblest and divine the analogy of the small, insig-
nificant and mortal, the centre of the universe as a magnitude is
the region about the earth which is cold and destitute of motion ;
while in the universe as universe and living thing the region about the |
sun is the centre of its animating principle, the sun being as it were
the heart of the All, which is also, as we are told, the starting-point
whence the soul proceeds to permeate the whole body spread over
it from the extremities inwards.’ 1
The argument of Theon seems rather to be offered as a plausible
defence of the new theory of Venus and Mercury as satellites of
the sun, after the event as it were, than as an ὦ friori ground for
putting forward that hypothesis or for extending it to Mars and
the other superior planets.
When the possibility of Mars revolving round the sun came to
1 Theon of Smyrna, pp. 187. 13 - 188..7. Cf. Plutarch, De fac. in orbe lunae,
c. 15, p. 928 B,C; Macrobius, 27: somn. Scip. i. 20. 1-8.
CH. XVIII HERACLIDES OF PONTUS 263
be considered, it would be at once obvious that the precise hypo-
thesis adopted for Mercury and Venus would not apply, because
the circles described by those planets about the sun are relatively
small circles and are entirely on one side of the earth, whereas the
circle described by Mars comprehends the earth which is inside it.
The next possibility that would present itself would be that the
planet might move uniformly round an eccentric circle of some
kind, a circle passing round the earth but with some other point
not the earth as centre. Suppose £ is the earth, fixed at the centre
of the universe, QR an eccentric circle with centre O. Draw the
diameter QR through £, Ὁ. Then Q represents the perigee of
ὯΝ
7
Fig. 12.
a planet moving on the eccentric circle. In opposition, therefore, .
Mars will be at Q, and the sun will be opposite to it, i.e. at some
point on ZR. If now the oppositions always occurred in the same
place in the zodiac, i.e. in the same direction ZQ, this hypothesis
would explain the differences of brightness. But the oppositions
do not always take place in the same direction; they may take
place at any part of the zodiac. Consequently, the direction of
opposition is not constant, as EQ, but the diameter RQ must
move round the centre Z& in such a way that the perigeal point Ὁ
is always opposite to the sun. Therefore Q, the point of opposition,
revolves round E in the space of a year along the ecliptic in the
direct order of the signs. Hence O, the centre of the eccentric, also
264 HERACLIDES OF PONTUS PARTI
revolves round £ ina year in such a way that it is always in the
direction of the sun. We suppose, therefore, that the whole
eccentric circle moves bodily round £ as centre, as if it were
a material disc attached to £& as a sort of hinge. If now we
suppose Mars to move uniformly round the circumference of the
eccentric in the zzverse order of the signs, completing the circuit
from perigee to perigee, or from apogee to apogee, in a time equal to
the period of its synodic revolution, the opposition will occur at the
right places and the brightness will then be greatest. Further
(and this is the most important point) if the distance ZO (the
‘eccentricity ᾽) is chosen in the proper ratio to the radius OR, the
irregular movements of the planet, its stationary positions, and its
retrogradations will be explained also (this would. be clear to any
one who was enough of a geometer, though the corresponding facts
are easier to see when the hypothesis is that of epicycles). By
means of observations it would be possible to deduce the ratio
of the radius to the ‘eccentricity’, but not their absolute magni-
tudes. But the centre O is always in the direction of the sun; |
it only remained to fix its distance (ZO). The natural thing in
the case of Mars would be to make the material sun the centre, just
as had been done with the epicycles of Venus and Mercury. The
use of ideal points as centres for epicycles and eccentrics was no
doubt first thought of, at a later stage, by some of the great
mathematicians such as Apollonius.
The next link in Schiaparelli’s chain of argument is the fact
that the same movement as is represented by movable eccentrics
of the sort just described can equally well be represented by means
of epicycles, a fact which is proved by Theon of Smyrna and
others. Let us then see how the motion of Mars, as above repre-
sented by means of a movable eccentric, can be represented by
means of an epicycle. Let Figure 13 (A) represent a movable
eccentric, & being the earth, S the centre of the eccentric which
moves round the circle SS’ in the direction shown by the arrow,
in such a way that ZS is always in the direction of the sun and
moves in the direct order of the signs. Let CC’ be the eccentric
with centre S. Produce ZS to meet the eccentric in C, which will ς΄
then be the position of the apogee of the eccentric. Let the planet —
be then at the point D describing the circle CC’ in the inverse
.
ἂ :
CH. XVIII HERACLIDES OF PONTUS 265
order of the signs. The angle CSD or the arc CD reckoned from
the apogee in the zvverse order of the signs will be the argument
of the anomaly, or shortly the axomaly; the planet will be seen
from the earth in the direction ZED.
Now [ Fig. 13 (B)], on the hypothesis of the epicycle, let @ be the
centre of the earth. About Θ as centre describe the circle 22’ equal
to the eccentric circle of the other figure, and draw the radius O&
parallel (and equal) to SD in the other figure. Take 3 as the centre
of the epicycle, and about it describe the circle 44’ equal to the
circle SS’ in the other figure. If we produce OF to K,X will
be at the moment the apogee of the epicycle. Make the angle
(A) : (Β)
Fig. 13. ;
KZA equal to the anomaly (i.e. the angle CSD in the other figure)
but reckoned in the opposite sense (i.e. in the direct order of the
signs). Suppose then that the planet is at 4 and seen from the -
earth in the direction ΘΖ.
In the triangles ESD, ΘΣΖ, DS is equal and parallel to 30,
and the angles 2.58, OX4 are equal; therefore ES, ΔΣ are
parallel. But 5.5, 4 are also equal; therefore the two sides
ES, SD are equal to the two sides 42, YO respectively. And
the included angles are equal ; therefore the triangles ESD, 430
are equal in all respects. And, since the two sides ES, 45 are
equal and parallel, and the sides SD, @ are also equal and
parallel, it follows that the third sides ED, ΘΖ will be equal and
parallel, i.e. the planet will be seen in the same direction and at
the same distance under either hypothesis.
“66 HERACLIDES OF PONTUS PARTI
The conditions necessary in order that this may be true at any
instant are two: (1) the radii SD, OX of the eccentric and the
deferent circle respectively must always remain parallel; (2) the
anomaly CSD in the eccentric must be equal to the anomaly K 4
in the epicycle, while the anomaly must in the first case be reckoned
in the zzverse order, and in the second case in the direct order of the
signs. It is evident also that the proof still holds if, instead of
making the radii of the two circles in each hypothesis equal, we
suppose them proportional only and change the dimensions of
either figure as we please.
It is clear why the Greek mathematicians preferred the epicycle
hypothesis to the eccentric. It was because the former was applic-
able to all cases ; it served for the inferior as well as the superior-
planets, whereas the eccentric hypothesis, as then conceived, would
not serve for the inferior planets ; moreover, the epicycle hypothesis
enabled the phenomena of the stationary points and retrograda-
tions to be seen almost by simple inspection, whereas on the
eccentric hypothesis a certain amount of geometrical proof would —
be necessary to enable the effect in this respect to be understood.
But it will be observed that in the above figures the motion of S
round the circle S’S may be the motion of the material sun in its
orbit but, when this is so, the point 3 which is the centre of the
epicycle in the other case is not a material but an zdeal point.
Hence, before geometers had fully developed the theory of revo-
lution about ideal points, the eccentric hypothesis was the only
practicable way of representing the movements of the superior
planets, Mars, Jupiter, and Saturn. :
Now we infer from a passage of Ptolemy’ that, while Apollonius
understood the theory of epicycles in all its generality, he only
knew of the particular class of eccentrics in which the movable
centre of the eccentric moves at an angular speed equal to that
of the swz describing its orbit about the earth. The description
by Apollonius of the two hypotheses is in these words:
(1) The epicycle hypothesis: ‘Here the epicycle’s advance in
longitude is in the direct order of the signs round the circle con-
centric with the zodiac, while the star moves on the epicycle about
its centre at a speed equal to that of the anomaly and in the direct
* Ptolemy, Synzazis xii. 1 (vol. ii, pp. 450. 10-17, 451. 6-14, Heib.).
CH. XVIII HERACLIDES OF PONTUS 267
order of the signs in that part of the circumference of the epicycle
which is furthest from the earth.’
(2) The eccentric hypothesis : ‘ This is only applicable to the three
planets which can be at any angular distance whatever from the sun,
and here the centre of the eccentric circle moves about the centre
of the zodiac in the direct order of the signs and at a speed equal
to that of the sun, while the star moves on the eccentric about its
centre in the inverse order of the signs and at a speed equal to
that of the anomaly.’
What makes Apollonius say that the eccentric hypothesis is not
applicable to the inferior planets is the fact that, in order to make it
apply to them, we should have to suppose the circle described by the
centre of the eccentric to be greater than the eccentric circle itself.
The object of the passage of Ptolemy is to explain the stationary
points and retrogradations on either hypothesis, and he reproduces
in his own form two propositions which, he says, had been proved
“by other mathematicians as well as by Apollonius of Perga with
reference to one of the anomalies, the anomaly in relation to the
sun.’ It is from the passage in question that it has commonly been
inferred that Apollonius of Perga was the inventor of epicycles.
I agree, however, with Schiaparelli that, if we read the passage
carefully, we shall find that it does not imply this. It is at least
as easy to infer from the language of Apollonius that, in the case of
the epicycle-hypothesis at all events, he was only stating formally
what was already familiar to those conversant with the subject.
Now the eccentric hypothesis, which is, in the proposition with
regard to it proved by Apollonius, limited to the particular case -
of the three superior planets, was evidently generalized at or
before the time of Hipparchus. This is clear from passages of
Ptolemy and Theon of Smyrna quoted by Schiaparelli. (1) Ptolemy
says that Hipparchus was the first to point out that it is necessary
to explain how there are two kinds of anomaly in the case of each
of the planets, the solar (ἡ παρὰ τὸν ἥλιον ἀνωμαλία) and the
zodiacal, or how the retrogradations of each planet are unequal
and of such and such lengths, whereas all other mathematicians
had based their geometrical proofs on the assumption that the
. anomaly and the retrogradation were one and the same respec-
tively. Hipparchus added that these phenomena were not accounted
268 HERACLIDES OF PONTUS PARTI
for either by eccentric circles, or by circles concentric with the
zodiac carrying epicycles, or even by a combination of both
hypotheses, (2) ‘Hipparchus says it is worthy of investigation
by mathematicians why on two hypotheses so different from one
another, that of eccentric circles and that of concentric circles with
epicycles, the same results appear to follow.’? A further allusion
to the same remark of Hipparchus shows that the identity of the
results following from the two hypotheses was shown with regard
to the “δι, which is the case for which Adrastus proved it.*
Again, Theon of Smyrna says that ‘ Hipparchus prefers the hypo-
thesis of the epicycle which he claims as his own, asserting that
it is more natural that all the heavenly bodies should be properly
balanced, and connected together in the same way, about the
centre of the universe; and yet, because he was not sufficiently
equipped with physical knowledge, even he did not know for
certain which is the natural and therefore true movement of the
planets and which the incidental and apparent ; but he, too, supposes
that the epicycle of each planet moves on the encentric circle and
the planet on the epicycle’® (3) In a famous passage where
Simplicius reproduces a quotation by Alexander from Geminus or
Posidonius (if Geminus was actually copying Posidonius) we read,
‘Why do the sun, moon, and planets appear to move irregularly?
Because, whether we suppose that their circles are eccentric or that
they move on epicycles, their apparent irregularity will be saved ;
and it will be necessary to go further and consider in how many ways
these same phenomena are capable of being explained, in order that
our theory of the planets may agree with that explanation of the
causes which proves admissible.’ ®
The theory of eccentrics had therefore been generalized by
Hipparchus’s time, but with Apollonius was still limited to the case
of the three superior planets. This indicates clearly enough that
it was invented for the specific purpose of explaining the movements
of Mars, Jupiter, and Saturn about the sun, and for that purpose
alone. Who then took this step in the formulation of a system
1 Ptolemy, Syutaxts ix. 2 (vol. ii, pp. 210. 19-211. 4, Heib.).
3 Theon of Smyrna, p. 166. 6-10. 8 Ibid., p. 185. 13-19.
* Ibid., pp. 166. 14-172. 14. 5 Ibid., p. 188. 15-24.
ὁ Simplicius zm Phys., p. 292. 15-20, ed. Diels.
CH. XVII HERACLIDES OF PONTUS : 269
which is the same as that of Tycho Brahe? Tannery! thinks it was
Apollonius, and in that case Apollonius, coming after Aristarchus
of Samos, would be exactly the Tycho Brahe of the Copernicus of
antiquity.
Schiaparelli, however, as I have said above, will have it that
it was not Apollonius, but Heraelides or some contemporary of his,
who took the final step towards the Tychonic system. In order
to prove this it is necessary to show that epicycles and movable
eccentric circles were both in use by Heraclides’ time, and Schia-
parelli tries to establish this by quotations from Geminus, Proclus,
Theon of Smyrna, Chalcidius, and Simplicius ; but it is here that
he seems to me to fail. The passages cited are as follows.
(1) Geminus: ‘It is a fundamental assumption in all astronomy
that the sun, the moon, and the five planets move in circular
orbits at uniform speed in a sense contrary to that of the
universe. For the Pythagoreans, who were the first to apply
themselves to investigations of this kind, assumed the movements
of the sun, the moon, and the five planets to be circular and
uniform. They would not admit, with reference to things divine
and eternal, any disorder such as would make them move at one
time more swiftly, at one time more slowly, and at another time
stand still, as the five planets do at their so-called stationary points.
For such irregularity of motion would not even be expected of
a decent and orderly man in his journeys. With men, of course,
the necessities of life are often causes of slowness and swiftness ; but
with the imperishable stars it is not possible to adduce any cause
of swiftness or slowness. Accordingly, they proposed the problem,
how the phenomena could be accounted for by means of circular
and uniform movements.’? Geminus goes on, it is true, to explain
why the sun, although moving at uniform speed, describes equal
arcs in unequal times, and explains the fact by assuming the sun
to move uniformly in an eccentric circle, i.e. a circle of which the
earth is an internal point but not the centre. But there is nothing
to suggest that this was the Pythagorean answer to the problem,
Geminus says ‘ We shall give the explanation as regards the other
* Tannery, Recherches sur Phistoire de [astronomie ancienne, c. 14, pp. 245,
253-9.
* Geminus, /sagoge, c. 1. 19-21, p. 10, 2-20, ed. Manitius.
270 HERACLIDES OF PONTUS PARTI
stars in another place ; but ze τοῦ show at once with regard to the
sun how....
(2) Theon of Smyrna says, quoting Adrastus:' ‘The apparent
intricacy of the motion of the planets is due to the fact that they
seem to us to be carried through the signs of the zodiac in circles
of their own, being fixed in spheres of their own and moved along
the circles, as Pythagoras was the first to observe, a certain intricate
and irregular movement being thus incidentally grafted on to their
simple and uniform motion, which remains the same.’
(3) Chalcidius says:* ‘Yet all the planets seem to us to move
unequally and some even to show disordered movements. What
then shall we give as the explanation of this erroneous supposition ?
That mentioned above, which was also known to Pythagoras,
namely that, while they are fixed in their own spheres and so
carried round, they appear, owing to our feebleness of vision, to
describe the circle of the zodiac.’ .
Schiaparelli adds: ‘We cannot attribute any historical value to
this notice unless we admit that by “ Pythagoras” are to be under-
stood those same Pythagoreans of whom Geminus speaks. And
it would follow that those Pythagoreans had explained the irregu-
larity of the planetary movements by means of the combination
of two circular movements, one with the earth as centre, the other
having its centre outside the earth (eccentric or epicycle).’ But
there is nothing whatever in these passages to suggest eccentrics
or epicycles. Theon follows up his remark by referring to the
combination of movements as explained by Plato in the 7) imacus,
i.e. the supposition that, while the sun, moon, and planets have
an independent circular movement of their own in the zodiac about
the earth as centre, they also share in the movement of the fixed
stars (the daily rotation about the axis of the universe) The
passage of Chalcidius seems to mean the same thing. Martin
interprets the passages of Geminus and Chalcidius as saying that
Pythagoras denied the irregularity of the movement of the stars
called planets, considering it an optical illusion.* Zeller observes
that the passage of Theon indicates that the early Pythagoreans
1 Theon of Smyrna, p. 150. 12-18.
? Chalcidius, 7imaeus, c. 77, 78, pp» 145, 146, ed. Wrobel.
3 Martin, Etudes sur le Timée, ii, p. 120. .
CH. XVIII HERACLIDES OF PONTUS 271
developed the doctrine of Anaximander into a theory of spheres
carrying round stars which are made fast to them, and that this
is confirmed by the occurrence of the same conception in Parmenides
and Plato. Whether all the stars are carried by spheres of their own,
i.e. hollow spheres, or only the fixed stars are carried by one sphere,
while the planets, as with Plato, are fixed on hoop-like circles, is not
clear. But Zeller rejects altogether the view that the Pythagoreans
assumed eccentrics and epicycles as not only unsupported by trust-
worthy evidence but as inconsistent with the whole development of
the old astronomy.!
But we have not done with the evidence cited by Schiaparelli.
(4) Proclus says?: ‘The hypotheses of eccentrics and epicycles com-
mended themselves also, so history tells us, to the famous Pythagoreans
as being more simple than all others—for it is necessary in dealing
with this question, and Pythagoras himself encouraged his disciples,
to try to solve the problem by means of the fewest and most simple
hypotheses possible.’ This passage, as Schiaparelli says, attributes
the first idea of movable eccentrics as well as of epicycles to the Pytha-
goreans. But it has tobe considered alongwith a passageof Simplicius
which Schiaparelli regards as the most important notice of all ;
(5) Simplicius says, after speaking of the system of concentric
spheres: ‘Later astronomers then, rejecting the hypothesis of
revolving spheres, mainly because they do not suffice to explain
the variations of distance and the irregularity of the movements,
dispensed with concentric spheres and assumed eccentrics and
epicycles instead—if indeed the hypothesis of eccentric circles
was not invented by the Pythagoreans, as some tell us, in- |
cluding Nicomachus and Iamblichus who followed him.’* This
passage, it is true, may indicate that it was only eccentric circles,
and not epicycles also, which the Pythagoreans discovered ; but
Schiaparelli regards it as conclusive with reference to movable
eccentrics. Unfortunately, he has not allowed for the fact that
it was the habit of the neo-Pythagoreans to attribute, so far as
possible, every discovery to the Pythagoreans, and even to Pytha-
goras himself. The evidence of Nicomachus would therefore
1 Zeller, 15, p. 415 2.
3 Proclus, Hypotyposis astronomicarum positionum, c. 1, ὃ 34, p. 18, ed.
Manitius. * Simplicius on De caelo, p. 507. 9-14, Heib.
272 HERACLIDES OF PONTUS PARTI
be worthless even if it could not easily be accounted for; but, as
Hultsch says,! the statement is easily explained as a reminiscence
of the Pythagorean central fire, for of course in that system each
planet moved in a circle about the central fire as centre and, as the
earth also moved round the same central fire, the orbit of the planet
would be eccentric relatively to the earth. _The passage of Proclus
may be based on the authority of Nicomachus ; or it may be a case
of a wrong inference, thus: the Pythagoreans sought the simplest
hypothesis because they held that that would be the best; the
simplest is that of eccentrics and epicycles; therefore the Pytha-
goreans would naturally think of that hypothesis.
But, even on the assumption that ‘the Pythagoreans’ are to be
credited with the invention of eccentrics and epicycles, the difficul-
ties are great, as Schiaparelli himself saw.?, Who are the particular
Pythagoreans who made the discovery? The problem which,
according to Geminus, the Pythagoreans propounded of finding
‘how the phenomena could be accounted for by means of circular
and uniform motions’ is almost identical with that which Sosigenes,
on the authority of Eudemus, says that Plato set, ‘What are the
uniform and ordered movements by the assumption of which the
facts about the movements of the planets can be accounted for?’.
If now the Pythagoreans had, by Plato’s time, discovered the solu-
tion by means of movable eccentrics and epicycles, Plato could not
have been unaware of the fact, and he would not then have set the
problem again in almost the same terms; Plato, however, makes
no mention whatever of epicycles or eccentrics. Hence the Pytha-
goreans in question could not have been the early Pythagoreans or
any Pythagoreans up to the time of Philolaus (who was about half
a century earlier than Plato); they must therefore be sought among
the contemporaries of Plato or in the years immediately after his
death ; indeed, if the hypothesis had been put forward in his life-
time, we should have expected to find some allusion to it in his
writings. We are, therefore, brought down to the period of Philip of
Macedon and Alexander the Great. But it was in these reigns
that the Pythagorean schools gradually died out, leaving the
1 Hultsch, art. ‘Astronomie’ in Pauly-Wissowa’s Real-Encyclopiidie, ὃ 14.
? Schiaparelli, Origine del sistema planetario eliocentrico presso t Greci,
pp. 81-2.
CH. XVIII HERACLIDES OF PONTUS 273
name to certain fraternities whose objects were rather ascetic
and religious than philosophical ;' according to Diodorus the last
Pythagorean philosophers lived about 366 B.c.* Schiaparelli is
therefore obliged to assume that, ‘if the schools ceased, their
doctrines were not entirely lost,’ and his whole case for crediting
Heraclides or one of his contemporaries with the complete anticipa-
tion of the system of Tycho Brahe really rests on this assumption
combined with the statement of Diogenes Laertius that Heraclides
‘also heard the Pythagoreans’.* It is true that Schiaparelli has one
other argument, which however seems to be an argument of despair.
It is based on the passage, already quoted above (pp. 186-7), in which
Aristotle, after speaking of the central fire of the ‘so-called Pytha-
goreans’, says :* ‘And no doubt many others, too, would agree (with
the Pythagoreans) that the place in the centre should not be assigned to
the earth, if they looked for the truth, not in the observed facts, but
in ὦ priori arguments. For they hold that it is appropriate to the
worthiest object that it should be given the worthiest place. Now
fire is worthier than earth . . “ Schiaparelli adds, ‘On this passage
Boeckh rightly observes that the reference is not to the past, but to
opinions held in the time of Aristotle. The Pythagorean doctrines
had ceased to be the object of teaching in special schools, but they
survived in the opinions of many and in part found favour even in
the Academy. From these reflections we draw the conclusion that
the first idea of epicycles and of eccentrics was conceived towards
the time of Philip or of Alexander, not among the pure Academics,
nor in the Lyceum, but among those more independent thinkers
who, like Heraclides, without forming a separate school, had
remained faithful, at least so far as regards natural philosophy, to
Pythagorean ideas, and for that reason could still with some truth
be called Pythagoreans, especially by writers of a much later date.’
That is to say, Nicomachus must, when claiming the discovery of
eccentrics for the Pythagoreans, have been referring to certain
persons whom Aristotle expressly distinguishes from that school,
his ground for claiming those persons as Pythagoreans being that
they were imbued with Pythagorean doctrines. It seems to me
1 Zeller, i*, pp. 338-42, iii. 25, pp. 79 544.
2 Diodorus, xv. 76; Zeller, i*, p. 339, note 2. 3 Diog. L. v. 86.
* Aristotle, De cae/o ii. 13, 293 a 27-32.
1410 T
474 HERACLIDES OF PONTUS PART I
that, by this desperate suggestion, Schiaparelli practically gives
‘away his case so far as it is based on Nicomachus. But, even if
we assume Nicomachus to have been referring to the independent
persons who, according to Aristotle, agreed in the theory of a
central fire, this does not help Schiaparelli’s argument, because in
Aristotle’s account of those persons’ views there is no hint what-
ever of eccentrics or epicycles.
It is no doubt possible that Heraclides or one of his contem-
poraries may, in the manner suggested, have arrived at the Tychonic
system; but I think that Schiaparelli has failed to establish
this, and the probabilities seem to me to be decidedly against it.
I judge mainly by the passage of Ptolemy (xii. 1) about the two
propositions proved by Apollonius and other geometers. Apol-
lonius was born, probably, 125 years later than Heraclides. Now
Heraclides certainly originated a particular hypothesis of epicycles,
namely epicycles described by Venus and Mercury about the
material sun as centre. By Apollonius’s time the hypothesis of
epicycles had become quite general, and such a generalization
might easily come about in a period of a century and more. But
the hypothesis of eccentrics had, by Apollonius’s time, advanced only
a very short way indeed towards a corresponding generality. Started
to explain the movements of the superior planets, the hypothesis
originally made the material sun the centre of the eccentric circle,
and by Apollonius’s time it had been only so far generalized as to
allow the sun to be anywhere on the line joining the centre of the
earth to the moving centre of the eccentric circle. This represents
very little progress for a hundred years; and the fact suggests that
nothing like a hundred years had passed since the first formulation
of the hypothesis in its most simple form, corresponding to the first
form of the epicycle hypothesis. In other words, the Tychonic
system was most probably completed by some one intermediate
between Heraclides and Apollonius and nearer to Apollonius than
Heraclides, if it was not actually reserved for Apollonius himself.
And that there is a fair probability in favour of attributing the step
to Apollonius himself seems to me to follow from two considera-
tions. It is a priori less likely that the ‘great geometer’ should
merely have proved two geometrical propositions to show the effect
of two hypotheses formulated by some of his predecessors, than that
CH. XVIII HERACLIDES OF PONTUS 275
he should have attached the propositions to hypotheses, or to a
comparison of hypotheses, which he was himself the first to develop ;
and the fact that he takes the trouble to mention that the eccentric
hypothesis only applies to the case of the three superior planets is
more intelligible on the assumption that the hypothesis was at the
time a new one, than it would beif the hypothesis had been familiar
to mathematicians for some time.
We have lastly to deal with a still greater claim put forward by
Schiaparelli on behalf of Heraclides ; this is nothing less than the
claim that it was Heraclides, and not Aristarchus of Samos, who
first stated as a possibility the Copernican hypothesis. Schia-
parelli’s argument rests entirely on one passage, a sentence forming
part of a quotation from Geminus which Simplicius copied from
Alexander and embodied in his commentary on the Physics of
Aristotle ;1 and, inasmuch as this passage, as it stands in the MSS.,
is not only unconfirmed by any other passage in Greek writers, but
is in direct conflict with other passages found in Simplicius himself,
it calls for the very closest examination. As the context is itself
important, I shali give a translation of the whole quotation from
Geminus according to the text of Diels; I shall then discuss the
text and the interpretation of the particular sentence relied upon
by Schiaparelli. The passage then of Simplicius is as follows:
‘ Alexander carefully quotes a certain explanation by Geminus
taken from his summary of the Meteorologica of Posidonius.
Geminus’s comment, which is inspired by the views of Aristotle,
is as follows:
*“ Tt is the business of physical inquiry to consider the substance
of the heaven and the stars, their force and quality, their coming into
being and their destruction, nay, it is in a position even to prove
the facts about their size, shape, and arrangement ; astronomy, on
the other hand, does not attempt to speak of anything of this kind,
but proves the arrangement of the heavenly bodies by considera-
tions based on the view that the heaven is a real κόσμος, and
further, it tells us of the shapes and sizes and distances of the earth,
sun,and moon, and of eclipses and conjunctions of the stars, as well
as of the quality and extent of their movements. Accordingly, as
it is connected with the investigation of quantity, size, and quality of
form or shape, it naturally stood in need, in this way, of arithmetic
1 Simplicius, 7x Phys. (ii. 2, 193 Ὁ 23), pp. 291. 21 -- 292. 31, ed. Diels (1882).
T2
276 HERACLIDES OF PONTUS PART I
and geometry. The things, then, of which alone astronomy claims
to give an account it is able to establish by means of arithmetic and
geometry. Now in many cases the astronomer and the physicist
will propose to prove the same point, e.g., that the sun is of great
size or that the earth is spherical, but they will not proceed by the
same road. The physicist will prove each fact by considerations of
essence or substance, of force, of its being better that things should
be as they are, or of coming into being and change; the astronomer
will prove them by the properties of figures or magnitudes, or by
the amount of movement and the time that is appropriate to it.
Again, the physicist will in many cases reach the cause by looking
to creative force; but the astronomer, when he proves facts from
external conditions, is not qualified to judge of the cause, as when, for
instance, he declares the earth or the stars to be spherical; some-
‘times he does not even desire to ascertain the cause, as when he
discourses about an eclipse; at other times he invents by way
of hypothesis, and states certain expedients by the assumption of
which the phenomena will be saved. For example, why do the
sun, the moon, and the planets appear to move irregularly? We
may answer that, if we assume that their orbits are eccentric circles
or that the stars describe an epicycle, their apparent irregularity
will be saved ; and it will be necessary to go further and examine
in how many different ways it is possible for these phenomena to be
brought about, so that we may bring our theory concerning the
planets into agreement with that explanation of the causes which
follows an admissible method. Hence we actually find a certain
person, Heraclides of Pontus, coming forward and saying that, even
on the assumption that the earth moves in a certain way, while the
sun 15 in a certain way at rest, the apparent irregularity with reference
to the sun can be saved. For itis no part of the business of an astro-
nomer to know what is by nature suited to a position of rest, and
what sort of bodies are apt to move, but he introduces hypotheses
under which some bodies remain fixed, while others move, and then
considers to which hypotheses the phenomena actually observed
in the heaven will correspond.. But he must go to the physicist for
his first principles, namely that the movements of the stars are simple,
uniform and ordered, and by means of these principles he will then
prove that the rhythmic motion of all alike is in circles, some being
turned in parallel circles, others in oblique circles.” Such is the
account given by Geminus, or Posidonius in Geminus, of the dis-
tinction between physics and astronomy, wherein the commentator
is inspired by the views of Aristotle.’
The important sentence for our purpose is that which I have
italicized, and the above translation of it is a literal rendering of the
.
CH. XVIII HERACLIDES OF PONTUS 277
reading of the MSS. and of Diels (διὸ καὶ παρελθών τίς φησιν
Ἡρακλείδης ὁ Ποντικός, ὅτι καὶ κινουμένης πως τῆς γῆς K.T. é.).
The reading and possible emendations of it will have to be discussed,
but it will be convenient first of all to dispose of a question arising on
the interpretation of the context. What is meant by ‘the apparent
irregularity with reference to the sun ( περὶ τὸν ἥλιον φαινομένη
ἀνωμαλία) δ Can this be so interpreted as to make it possible to
take the motion of the earth to be rotation about its axis and not a
motion of translation at all? Boeckh?! took the πως (‘in a certain
way’) used of the sun’s remaining at rest (as it is also used of the
earth’s motion) to signify that the sun is not guite at rest; and he
thought that Heraclides meant that the sun and the heaven were
only at rest so far as the general daily rotation was concerned, while
the earth rotated on its own axis from west to east in 24 hours, but
that the sun still performed its yearly revolution in the zodiac
circle. This, however, does not account for the ‘apparent zrregu-
larity or anomaly with reference to the sun’, which expression could
not possibly be applied to the daily rotation.
Martin? and Bergk® took the irregularity to be the irregularity
of the sun’s own motion in the ecliptic, by virtue of which the sun
seems to go quicker at one time than at another, and the four
seasons differ in length. But if, as Bergk apparently supposed, the
two hypotheses which are contrasted are (1) the sun moving irregu-
larly as it does and the earth completely devoid of any motion of
translation, (2) the sun completely at rest and the earth with an
irregular motion of translation, it is, as Schiaparelli says, impossible
to get any plausible sense out of the passage. For the problem of .
explaining the irregularity of the sun’s motion presents precisely the
same difficulties on the one hypothesis as it does on the other;
the substitution of one hypothesis for the other does not advance
the question in any way, and it explains nothing. Martin saw this,
and tried another explanation based on the use of the word πως,
‘in a certain way’. The mean speed of the sun, says Martin, is one
thing, its anomaly is another; the former is accounted for by the
? Boeckh, Das kosmische System des Platon, pp. 135-40.
3 Martin in Mémoires de [ Académie des Inscriptions et Beiles-Lettres, xxx.
Pack pp. 26 564.
Fiinf Abhandlungen zur Gesch. der griechischen Philosophie und
Astronomie, Leipzig, 1883, p. 151.
278 HERACLIDES OF PONTUS PARTI
annual revolution; the latter had to be otherwise accounted for, and
one way of accounting for it was that of Callippus, who gave the
sun two spheres more than Eudoxus assigned to it. Take now
from the sun the small movement (of irregularity) only, thus leaving
it at rest only ‘in a certain sense’, and give the earth a small
annual movement sufficient to explain the apparent anomaly of the
sun. A mere rotation of the earth on its axis would not suffice;
the movement must be one of translation in the circumference of a
circle, the result of which would be that, for the inhabitants of the
earth, the solar anomaly would be the effect of a parallax, not
daily, but annual, and dependent on the radius of the circle
described by the earthinayear. That is, the earth must be supposed
to accomplish, on the circumference of a small orbit round the centre
of the universe, an annual revolution at a uniform speed from east
to west, while the sun accomplishes from west to east its annual
revolution about the same centre in a great orbit enveloping that of
the earth. Schiaparelli shows the impossibility of this explanation.
If we take from the sun only the small zvregularity of its movement .
and leave it its mean movement in an enormous circle round the
earth, how could any one properly describe this as making the sun
‘ stationary in a certain sense’, when at the same time the earth,
which is made to describe a small orbit, is said to ‘ move in a certain
sense’? Moreover, it is inadmissible to suppose that, in Heraclides’
time, any one could have assumed that the place in the centre of
the universe was occupied by xothing, and that both the sun and
the earth revolved about an ideal point ; the conception of revolu-
tion about an immaterial point appeared later, in the generalized
theory of epicycles and eccentrics, and we find no mention of it
before Apollonius.
But indeed there is nothing to suggest that Heraclides was aware
of the small irregularities of the sun’s motion, and it is. therefore
necessary to find another meaning for the expression ‘the apparent
irregularity with reference to the sun’ (ἡ περὶ τὸν ἥλιον φαινομένη
ἀνωμαλία). I agree with Schiaparelli’s view that it must be the
same thing as Hipparchus and Ptolemy in the Syxtaxis commonly
describe as ‘the irregularity re/atively to the sun’ (ἡ πρὸς τὸν ἥλιον
ἀνωμαλία or ἡ παρὰ τὸν ἥλιον ἀνωμαλία), that is to say, that great
inequality in the apparent movements of the planets, which alone
CH. XVIII HERACLIDES OF PONTUS 279
was known in the time of Heraclides and which manifests itself
principally in the stationary points and retrogradations. It is true
that in the particular sentence the planets are not mentioned, but
they are mentioned in the sentence of Geminus which immediately
precedes it, and, if our sentence is a quotation of words used by
Heraclides, they no doubt followed upon a similar reference to the
planets by Heraclides.
If then the text as above translated is right, there is no escape from
the conclusion that Heraclides actually put forward the Copernican
hypothesis as a possible means of ‘saving the phenomena’. But
it is precisely the text of the sentence referring to Heraclides that
gives rise to the greatest difficulties. The reading of the MSS.
followed in the translation above is διὸ καὶ παρελθών τίς φησιν
ἩΗρακλείδης ὁ Ποντικός, ὅτι καὶ κινουμένης πως τῆς γῆς K.T-é. Diels!
is satisfied with this reading, which, he thinks, renders unnecessary
the many scruples felt by scholars, and the emendation proposed
by Bergk.2 Gomperz,* on the other hand, says that after the most
careful consideration he finds himself compelled to dissent from
Diels’ view of the passage. Schiaparelli observes that it is really
impossible to suppose that a historian of sciences such as Geminus
could have used the word τις, and said ‘a certain Heraclides of
Pontus’,in speaking of a philosopher who was celebrated through-
out antiquity and whom Cicero, a contemporary of Geminus, read
and spoke of with great respect. This consideration may have
been one of those which induced the editor of the Aldine edition
to insert the word ἔλεγεν before ὅτι, a reading which involves the
punctuation of the passage thus: διὸ καὶ παρελθών τις, φησὶν “Hpa-.
κλείδης ὁ Ποντικός, ἔλεγεν ὅτι. I think there is no doubt that
Boeckh is right in his interpretation of παρελθών as ‘having come
forward’, which he supports by quoting a number of passages con-
taining the same use of the word. According, therefore, to the
reading of the Aldine edition we have a quotation from one of
Heraclides’ dialogues introduced by the parenthetical words in
oratio recta,‘says Heraclides of Pontus,’ and the translation will
1 Diels, ‘ Uber das physikalische System des Straton’ in Berliner Sitzungs-
Berichte, 1893, p. 18, note I.
3 Bergk, op. cit., p. 150.
3 Gomperz, Griechische Denker, ἰδ, p. 432.
280 HERACLIDES OF PONTUS PARTI
be, ‘ This explains too why “ some one came forward”, as Heraclides
of Pontus says, “and said that....”’ Bergk objects that, while
παρελθών, ‘coming forward, is used of one who comes forward in
a public assembly, it is not, so far as he can find, used of the
interlocutors in a dialogue.’ This is, however, not conclusive, as
such expressions marking the interposition of a new speaker may
have been common in Heraclides’ dialogues ; indeed we gather that
there was a great deal of action in them.? A more substantial
objection is the form of the quotation, the plunging direct, after
the words διὸ καί, ‘For which reason also,’ into the actual words
of Heraclides ‘some one came forward and said’. This is, it must
be admitted, extremely abrupt and awkward; if Geminus had
been quoting in this way, it would have been more natural to put
the sentence in a different form, such as‘ This is the reason too
why, in a dialogue of Heraclides of Pontus, some one came forward
and said that...’. .
Bergk’s own suggestion for emendation is to omit the 71s, alter
παρελθών into προελθών, and write the sentence thus, διὸ καὶ προελ-.
θών φησιν: Ἡρακλείδης ὁ Ποντικὸς ἔλεγεν. ‘For this reason too,
he goes on to say “Heraclides of Pontus said that...”’ The
words διὸ καὶ προελθών φησιν would thus be the words, not of
Geminus, from whom the whole passage is quoted, but of
Alexander, who is quoting ; these words would therefore come
in between one textual citation from Geminus and another, I think
this reading has nothing to commend it; the omission of τις is an
objection to it, and the net result is a perfectly unnecessary
interposition by Alexander, which moreover spoils the sense;
‘this is the reason, too, why Geminus goes on to say that Heraclides
declared ...’ is not so good as ‘this is the reason, too, why some
one, according to Heraclides of Pontus, said .. .’
I omit a number of suggestions for replacing παρελθών by some
other word or words; they have no authority and, so long as
Heraclides of Pontus remains in the sentence either as having
himself held the view in question, or as having attributed it to some
one else unnamed, they do not really affect the issue.
I now come to Tannery’s view of the passage, which is not only
1 Bergk, op. cit., p. 149. 2 Otto Voss, Heraclides, p. 27.
»
mens
—— να τ΄
CH, XVIII HERACLIDES OF PONTUS 281
that suggested by the ordinary principles of textual criticism, but
furnishes a solution of the puzzle so simple and natural that it
should, as it seems to me, carry conviction to the mind of any
unbiased person. As Tannery says, from the*moment when it is
realized that the insertion of the word ἔλεγεν does not, after all is
said, suffice to remove all difficulties, we are thrown back upon
the text of the MSS. as established by Diels, διὸ καὶ παρελθών τίς
φησιν Ἡρακλείδης ὁ Ποντικός, ὅτι καὶ κινουμένης πως τῆς γῆς K.T-é.,
and we have to consider in what way an error could have crept
into the text. Now it ‘leaps to the eyes’ that, if the original text
said simply διὸ καὶ παρελθών τίς φησιν ὅτι Kai κινουμένης πως τῆς
γῆς κοιτ.ιἕ, it was the easiest thing in the world for a glossarist to
insert in the margin, in explanation of τις, the name of Heraclides
of Pontus, which would then naturally find its way into the text.
If the name is left out, everything is in perfect order. The passage
in its context is then as follows: ‘Why do the sun, moon, and
planets appear to move irregularly? We may answer that, if we
assume that their orbits are eccentric circles or that the stars
describe an epicycle, their apparent irregularity will be saved; and
it will be necessary to go further and examine in how many different
ways it is possible for these phenomena to be brought about,
so that we may bring our theory concerning the planets into
agreement with that explanation of the causes which follows an
admissible method. This is why one astronomer has actually
suggested that, by assuming the earth to move in a certain way
and the sun to be in a certain way at rest...’ Nothing clearer
or more correct could possibly be desired ; the different hypotheses —
would then all alike be stated in general terms without the names
of their authors, whereas nothing could be more awkward than
that Geminus, after speaking of the hypotheses of eccentrics and
epicycles in this way, should change the form of statement and
bring in quite abruptly an historical fact about one particular person
by name, or a textual citation from a work of his. Even Gomperz?
admits that it is possible that Ἡρακλείδης 6 Ποντικός may have
been inserted ‘ by a (well-informed) reader’ ; but the ‘ well-informed’
1 Tannery, ‘Sur Héraclide du Pont,’ in Revue des Etudes grecques, xii, 1899,
_ pp. 30: ἼΙ.
3 Gomperz, loc. cit.
282 HERACLIDES OF PONTUS PART I
is a pure assumption on his part, due, I think, merely to bias, for
how can we possibly pronounce the reader to have been ‘ well-
informed’ when there is absolutely no other evidence telling in his
favour ? :-
If then, as seems to me inevitable, the words Ἡρακλείδης ὁ
ITovrikés are rejected as an interpolation, the view of Schiaparelli
based upon the passage must be given up, and there remains no
ground for disputing the accuracy of the other definite statement
by Aétius to the effect that ‘ Heraclides of Pontus and Ecphantus
the Pythagorean make the earth move, yet zo¢ in the sense of
translation but with a movement of rotation’,! confirmed as it is
by the sharp distinction drawn in one place by Simplicius between
those who supposed the earth to have a motion of translation and
Heraclides who supposed it to rotate about its axis.?
If it is asked whom Geminus had in mind when using the
expression τίς φησιν, we can have no hesitation in answering that
it. was Aristarchus of Samos, for it is to him that all ancient
authorities agree in attributing the suggestion of the heliocentric
system. :
It is not possible to say at what time the interpolation of the
name of Heraclides into the passage took place. There is nothing
to show, for instance, whether it was made in the archetype of the
MSS. of Simplicius or in the sources from which he drew. As he
did not quote Geminus directly, but copied the quotation of
Alexander, it is a question whether the gloss is earlier or later than
Alexander, or even due to Alexander himself. I agree with Tannery
that an annotator of the second or third century of our era, at
a period when Heraclides was sufficiently well known through the
Doxographi as having attributed a movement to the earth, might
very well, on reading the first words, κινουμένης πὼς τῆς γῆς (‘if
the earth moves in a certain way’), have immediately thought of
Heraclides rather than Aristarchus, and have written the name
of the former in the margin without looking forward to see what
were the words immediately following these. ‘In any case,’ Tannery
concludes, ‘ the attribution to Heraclides Ponticus of the heliocentric
1 Aét. iii, 13. 3 (D. G. p. 378; Vors. i?, p. 266. 5-7).
2 Simplicius on De caelo ii, 14 (297 a2), Ρ. 541. 27-9, Heib. See above,
Ρ. 255-
CH. XVIII HERACLIDES OF PONTUS 283
system does not in any way rest on the authority of Posidonius
or of Geminus ; it is the act of an anonymous annotator of uncertain
date, and probably the result of a simple inadvertence only too
easy to commit ; it must therefore be considered as null and void.’
It may be added that it is hardly a griorz surprising that such
extensive claims on behalf of Heraclides should prove, on examina-
tion, to be in part unsustainable. It was much to discover, as he
did, the rotation of the earth about its axis and the fact that Venus
and Mercury revolve round the sun like satellites; it would seem
@ priorz almost incredible that the complete Tychonic system should
have been evolved in Heraclides’ lifetime and ‘perhaps’ by
Heraclides himself, and that he should a/so have suggested the
Copernican hypothesis.
ΧΙΧ
GREEK MONTHS, YEARS, AND CYCLES?
ALTHOUGH there is controversy as to whether in the earliest
times (e.g. with Homer and Hesiod) the day was supposed to
begin with the morning or evening, it may be taken as established
that in historic times the day, for the purpose of the calendar, began
with the evening, both at Athens and in Greece generally. As
regards Athens the fact is stated by Gellius on the authority of
Varro, who, in describing the usage of different nations in this
respect, said that the Athenians reckoned as one day the whole
period from one sunset to the next sunset ;? the testimony of Pliny ὅ
and Censorinus* is to the same effect. The practice of regarding
the day as beginning with the evening is natural with a system of
reckoning time by the moon’s appearances; for a day would
naturally be supposed to begin with the time at which the light of
the new moon first became visible, i.e. at evening.
There is no doubt that, from the earliest times, the Greek month
(μήν) was lunar, that is, a month based on the moon’s apparent
motion. But from the first there began to be felt, among the
Greeks as among most civilized peoples, a desire to bring the
reckoning of time by-the moon into correspondence with the
seasons of the year, for the sake of regulating the times of sacri-
fices to the gods which had to be offered at certain periods in the
year; hence there was from the beginning a motive for striving
after the settlement of a luni-solar year. The luni-solar year thus
had a religious origin. This is attested by Geminus, who says:°
‘The ancients had before them the problem of reckoning the
months by the moon, but the years by the sun. For the legal and
1 For the contents of this chapter I am almost entirely indebted to the
exhaustive work of F. K. Ginzel, Handbuch der mathematischen und technischen
Chronologie, vol. ii of which appeared in the nick of time (1911).
2 Gellius, oct. Az. iii. 2. 2. 8. Pliny, WV. H. ii. c. 77, ὃ 188.
* Censorinus, De die natali, c. 23. 3.
5 Geminus, /sagoge, c. 8, 6-9, p. 102. 8-26, Manitius.
GREEK MONTHS, YEARS, AND CYCLES 285
oracular prescription that sacrifices should be offered after the
manner of their forefathers was interpreted by all Greeks as mean-
ing that they should keep the years in agreement with the sun and
the days and months with the moon. Now reckoning the years
according to the sun means performing the same sacrifices to the
gods at the same seasons in the year, that is to say, performing
the spring sacrifice always in the spring, the summer sacrifice
in the summer, and similarly offering the same sacrifices from year
to year at the other definite periods of the year when they fell due.
For they apprehended that this was welcome and pleasing to the
gods. The object in view, then, could not be secured in any other
way than by contriving that the solstices and the equinoxes should
occur in the same months from year to year. Reckoning the days
according to the moon means contriving that the names of the
days of the month shall follow the phases of the moon,’
At first the month would be simply regarded as lasting from
the first appearance of the thin crescent at any new moon till the
next similar first appearance. From this would gradually be
evolved a notion of the length of a moon-year. A rough moon-year
would be 12 moon-months averaging 294 days; but it was necessary
that a month should contain an exact number of days, and it was
therefore natural to take the months as having alternately 29 and 30
days. These ‘hollow’ and ‘full’ months are commonly supposed
to have been introduced at Athens by Solon (who was archon in
594/3-B.C.), since he is said to have ‘taught the Athenians to
, reckon days by the moon’. But it can hardly be doubted that
‘full’ and ‘hollow’ months were in use before Solon’s time;
Ginzel therefore thinks that Solon’s reform was something different.
We shall revert to this point later.
At the same time, alongside the ‘full’ and ‘ hollow’ months of the
calendar, popular parlance invented a month of 30 days, as being
convenient to reckon with. Hippocrates makes 280 days = 9
months τὸ days;? Aristotle speaks of 72 days as 1/5th of a year ;*
the riddle of Cleobulus implies 12 months of 30 days each;* the
original division of the Athenian people into 4 φυλαΐ, 12 φρατρίαι,
and 360 γένη is explained by Philochorus as corresponding to the
seasons, months, and days of the year.° In the Courts a month
1 Diog. L. i. 59. ? Hippocrates, De carnibus, p. 254.
3. Aristotle, Hzst. an. vi. 20, 574 ἃ 26. * Diog. L. i. 91.
0g: 9
. Suidas, 5.0. γεννῆται.
286 GREEK MONTHS, YEARS, AND CYCLES parti
was reckoned at 30 days, and wages were reckoned on this basis,
e.g. daily pay of 2 drachmae makes for 13 months 780 drachmae
(2x 30x 13).! From such indications as these it has been inferred
that the Greeks had at one time years of 360 days and 390 days
respectively. Indeed, Geminus says that ‘the ancients made the
months 30 days each, and added .the intercalary months in alter-
nate years (παρ᾽ ἐνιαυτόν)᾽.2 Censorinus has a similar remark ; when,
he says, the ancient city-states in Greece noticed that, while the sun
in its annual course is describing its circle, the new moon sometimes
rises thirteen times, and that this often happens in alternate years,
they inferred that 125 months corresponded to the natural year, and
they therefore fixed their civil years in such a way that they made
years of 12 months and years of 13 months alternate, calling each
of such years ‘annus vertens’ and both years together a great year.®
Again, Herodotus* represents Solon as saying that the 70 years of
a man’s life mean 25,200 days, without reckoning intercalary
months, but, if alternate years are lengthened by a month, there
are 35 of these extra months in 70 years, making 1,050 days more
and increasing the total number of days to 26,250. But under this
system the two-years period (called nevertheless, according to
Censorinus, ¢rieteris because the intercalation took place ‘every
third year’) would be more than 7 days too long in comparison
with the sun, and in 20 years the calendar would be about 25 months
wrong in relation to the seasons. This divergence is so glaring that
Ginzel concludes that the system cannot have existed in practice.
He suggests, in explanation of Geminus’s remark, that Geminus is
not to be taken literally, but is in this case merely using popular
language (cf. his remark that go days = 3 months®); he regards
Censorinus’s story as suspicious because in the following sentence
Censorinus says that the next change was to a pentaéteris of four
years each, which involves the supposition that the Greeks of, say,
the eighth or ninth century B.c., had already anticipated the Julian
system; moreover, Geminus says nothing of a four-years period at
all (whether called stetraéteris or pentaéteris) but passes directly to
the octaéteris which, according to him, was the first period that the
ancients constructed.
1 Corp. Inscr. Att. ii. 2, no. 834 c, 1.60 (p. 532). * Geminus, Jsagoge, c. ὃ. 26.
5 Censorinus, De die natali, c. 18. 2. * Herodotus, i. 32.
5 Geminus, /sagoge, c. 8. 30, p. 112. 7, 10.
CH.xIx GREEK MONTHS, YEARS, AND ΟΥ̓ΓΙΕῈΒ 287
On the alternation of ‘full’ and ‘hollow’ months an apparently
interpolated passage in Geminus says :?
‘The moon-year has 354 days. Consequently they took the
lunar month to be 293 days and the double month to be 59 days.
Hence it is that they have hollow and full months alternately,
namely because the two-months period according to the moon is
59 days. Therefore there are in the year six full and six hollow
months. This then is the reason why they make the months full
and hollow alternately.’
The octaéteris.
Geminus’s account of the eight-years cycle follows directly on
what he says of the supposed ancient system of alternating years
of 12 and 13 months of 30 days each.
‘Observation having speedily proved this procedure to be incon-
sistent with the true facts, inasmuch as the days and the months
did not agree with the moon nor the years keep pace with the sun,
they sought for a period which should, as regards the years, agree
with the sun, and, as regards the months and the days, with the
moon, and should contain a whole number of months, a whole
number of days, and a whole number of years. The first period
they constructed was the period of the octaéteris (or eight years)
which contains 99 months, of which three are intercalary, 2922 days,
and 8 years. And they constructed it in this way. Since the year
according to the sun has 365% days, and the year according to the
moon 354 days, they took the excess by which the year according
to the sun exceeds the year according to the moon. This is 114
days. If then we reckon the months in the year according to the
moon, we shall fall behind by 114 days in comparison with the solar
year. They inquired therefore how many times this number of
days must be multiplied in order to complete a whole number
of days and a whole number of months. Now the number [114]
multiplied by 8 makes 90 days, that is, three months. Since thén
we fall behind by 113 days in the year in comparison with the sun,
it is manifest that in 8 years we shall fall behind by 90 days, that
is, by 3 months, in comparison with the sun. Accordingly, in each
period of 8 years, three intercalary (ἐμβόλιμοι) months are reckoned,
in order that the deficiency which arises in each year in comparison
with the sun may be made good, and so, when 8 years have passed
from the beginning of the period, the festivals are again brought
into accord with the seasons in the year. When this system is
followed, the sacrifices will always be offered to the gods at the
_ same seasons of the year. 4
1 Geminus, Jsagoge, c. 8. 34-5, pp. 112. 28 -- 114. 7.
“88 GREEK MONTHS, YEARS, AND CYCLES parti
‘ They now disposed the intercalary months in such a way as to
spread them as nearly as possible evenly. For we must not wait
until the divergence from the observed phenomena amounts to
a whole month, nor yet must we get a whole month ahead of the ©
sun’s course. Accordingly they decided to introduce the inter-
calary months in the third, fifth, and eighth years, so that two of
the said months were in years following two ordinary years, and
only one followed after an interval of one year! But it is a matter
of indifference if, while preserving the same disposition of (i.e. inter-
vals between) the intercalary months, you put them in other years.’*
Here then we have an account which purports to show how the
octaéteris was first arrived at, the supposition being that it was
based on a solar year of 365% days, Ginzel, however, thinks it
impossible that this can have been the real method, because the
evaluation of the solar year at 365% days could hardly have been
known to the Greeks of, say, the 9th and 8th centuries B.C. ; this,
he thinks, is proved by the erroneous estimates of the length of the
solar year which continued to be put forward much later.
Ginzel considers that the octaéteris was first evolved as the result
of observation of the moon’s motion, which was of course easier to
approximate to within a reasonable time. The alternation of 6 full
with 6 hollow months gives a moon-year of 354 days; but the true
moon-year exceeds this by 0-36707 day, and hence, after about 23
moon-years, a day would have to be added in order to keep the
months in harmony with the phases; that is to say, at such inter-
vals, there would have to be a year of 355 days. Now this rate
of intercalation corresponds nearly to the addition of 3 days in
a period of 8 moon-years, i.e. to a cycle of 8 moon-years in which
5 have 354, and 3 have 355 days, each. (And, as a matter of fact,
the same proportion of 5 : 3 serves very roughly to bring the moon-
year into agreement with the solar year, for we have only to reckon,
in a cycle of 8 solar years, 5 moon-years of 354 days and 3 of
384 days.)* Ginzel cites evidence showing that particular years
actually had 355 days and 384 days, e.g. Ol. 88, 3 = 355 days,
Ol. 88, 4 = 354 days, Ol. 89, 1 = 384 days, and Ol. 89, 2 = 355
1 8¢ ἣν αἰτίαν τοὺς ἐμβολίμους μῆνας ἔταξαν ἄγεσθαι ἐν τῷ τρίτῳ ἔτει καὶ πέμπτῳ
καὶ ὀγδόῳ, δύο μὲν μῆνας μεταξὺ δύο ἐτῶν πιπτόντων, ἕνα δὲ μεταξὺ ἐνιαυτοῦ ἑνὸς
ἀγομένου. }
2 Geminus, /sagoge 8. 26-33, pp. 110. 14 -- 112. 27. 8 Ginzel, ii. 330-1.
ee
CH. XIX THE OCTAETERIS ὁ 289
days. The method by which the octaéterts was evolved is, he thinks,
something of this sort. Having from observation of the moon con-
structed an 8-years period containing 5 moon-years of 354 days and
3 intercalated years of 355 days each, making a total of 2,835 days,
the Greeks, by further continual observation directed to fixing the
duration of the phases exactly, would at last come to notice that,
after 8 returns of the sun to the same azimuth-point on the horizon,
the phases fell nearly on the same days once more, and also that the
sun returned to the same azimuth-point for the eighth time after about
99 lunar months. Now, if the ancients had divided the 2,835 days of
_ 8 moon-years by οὔ, they would have found the average lunar month
_ to contain 2937 days ; and again, if they had multiplied this by 99,
_ they would have obtained 2,92335 or nearly 2,9235 days. But the
<
first inventors of the octaéteris certainly did not make the 8 solar years
contain 2,9235 days; this, we are told, was a later improvement on
the 2,922 days which, according to Geminus, the first octaéteris con-
tained. No doubt the first discoverers of it would notice that 99
times 293 days is 2,9204 days, that is to say, approximately 8 years
of 365 days (= 2,920 days). This may have been what led them
to construct a luni-solar octaéteris. But why did they give it 2,922
days? Ginzel suggests that, as the octaéteris was thus shown to be
very useful for the purpose of bringing into harmony the motions
of the sun and moon, the Greeks would be encouraged to try to
obtain a more accurate estimate of the average length of the lunar
month. If then, for example, they had assumed 293§ days as the
average length, they would have found, at the end of an octaéteris,
that they were only wrong by 0-3 of a day relatively to the moon,
but were nearly two days ahead in relation to.the sun.* This
might perhaps lead them to conjecture that the solar year was
a little longer than 365 days; and they may have hit upon 3653
days by a sort of guess. This would give 293$ days as the length
of the lunar month. Ginzel thinks that the gradual process by
which the Greeks arrived at the 2,922 days may have lasted from
the 9th or 8th century into the 7th.* This, he suggests, may explain
1 Ginzel, ii. 341-3.
2 2046 x 99 = 2923-8 days (against 2923-528, the correct figure) ; 8 solar years
have 8 x 365-2422 = 2921-938 days.
* Ginzel, ii. 376, 377.
1410 U
290 GREEK MONTHS, YEARS, AND CYCLES Parti
the fact that we find mentions or indications of eight-years periods
going back as far as the mythical age. Thus Cadmus passed an
‘eternal’ (ἀΐδιος) year (i.e. says Ginzel, an 8-year year) in servitude
for having slain the dragon of Ares ; similarly Apollo served 8 years
with Admetus after he killed the dragon Python. The Daphne-
phoria were celebrated every 8 years; in the procession connected
with the celebration an olive staff was carried with a sphere above
(the sun), a smaller one below (the moon), and still smaller spheres
representing other stars, while 365 purple bands or ribbons were
also attached, representing the days of the solar year. The Pythian
games were also, at the beginning, eight-yearly. Kingships were
offices held for eight years (thus Minos spoke with Zeus, the great
God, ‘nine-yearly’).1 According to Plutarch the heaven was
observed at Sparta by the Ephors on a clear night once in eight
years.” These cases, however, though showing that 8-years periods —
were recognized and used in various connexions, scarcely suffice, I
think, to prove the existence in such very early times of an accu-
rately measured period of 2,922 days, Ginzel, in arguing for so
early a discovery of the octaéteris of 2,922 days, departs consider-
ably from the views of earlier authorities on chronology. Boeckh
thought that the octaéferis was introduced by Solon, and that the —
first such period actually began with the beginning of the year at
the first new moon after the summer solstice in Ol. 46, 3, i.e. 7th
July, 594 B.c.° As regards the period before Solon, Boeckh went,
it is true, so far as to suggest that, as early as 642 B.C., there may
have been a rough octaéteris in vogue which was not actually fixed
or exactly observed ; this, however, was only a conjecture. Ideler+
argued that the octaéteris could not be as old as Solon’s time
(594/3 B.C.) or even as old as Ol. 59 (544-540), because so accurate
a conception is in too strong a contrast to what we know of the
state of astronomical knowledge in Greece at that time. As regards
Solon’s reforms, we are told ὅ that he prescribed that the day in the
1 Odyssey xix. 178, 179. * Plutarch, Agis, c. 11.
8 The practice of beginning the year in the summer (with the month
Hecatombaion) is proved by Boeckh to have existed during the whole of the
fifth century. It was probably much older in Attica; the transition (if the Attic
year previously began in the winter) may have taken effect in the time of Solon.
* Ideler, Historische Untersuchungen uber die astronomischen Beobachtungen
der Alten, p. 191.
® Plutarch, So/on, c. 25.
‘CH. XIX THE OCTAETERIS 291
course of which the actual conjunction at the new moon took place
should be called ἕνη καὶ νέα, the ‘last and new’ or ‘old-new’,
and that he called the following day νουμηνία (new moon), which
therefore was the first day belonging wholly to the new month.
Diogenes Laertius says that Solon taught the Athenians ‘to
reckon the days according to the moon’ ;! and Theodorus Gaza,
a late writer, it is true, says that Solon ‘ordered everything in
connexion with the year generally better’.2 Boeckh, as already
stated, thought that Solon’s reform consisted in the introduc-
tion of the octazteris. Ginzel, however, holding as he does that
the octaéteris of 2,922 days was discovered much earlier, considers
that Solon’s reform had to do with the improvement on this
figure by which 99 lunations were found to amount to 2,923%
days, a discovery which led to the formulation of the 16-years and
160-years periods presently to be mentioned ; this may be inferred,
according to Ginzel, from the fact that the accounts show Solon’s
object to have been the bringing of the calendar specially into
accordance with the moon. But it is difficult to accept Ginzel’s
view of the nature of Solon’s reform in the face of another statement
as to the authors of the octaéteris. Cemsorinus says:
‘This octaéteris is commonly attributed to Eudoxus, but others
say that Cleostratus of Tenedos first framed it, and that it was
modified afterwards by others who put forward their octaéterides
with variations in the intercalations of the months, as did Harpalus,
Nauteles, Menestratus, and others also, among whom is Dositheus,
who is most generally identified with the octaéteris of Eudoxus.’*
Now we know nothing of the date of Cleostratus, except that
he came after Anaximander; for Pliny says that Anaximander is
credited by tradition with having discovered the obliquity of the
zodiac in Ol. 58 (548-544 B.C.), after which (ἐσίγα) Cleostratus
distinguished the signs in it.* Thus Cleostratus may have lived
soon after 544B.C. Ginzel seems to admit that Cleostratus was
the actual founder (‘eigentliche Begriinder’) of the octaéteris.5 Of
Harpalus, who was later than Cleostratus but before Meton
(432 B.C.), we only know that he formed a period which brought
the moon into agreement with the sun after the latter had revolved
1 Diog. L. i. §9. ;
3 Theodorus Gaza, c. 8 and 15. * Censorinus, De die natali, 18. 5.
* Pliny, NV. HZ. ii. c. 8, ὃ 31. 5 Ginzel, ii, p. 385.
U2
202 GREEK MONTHS, YEARS, AND CYCLES ΡΑΚΤῚ
‘through nine winters’,) which statement must, as Ideler says, be
due to a misapprehension of the meaning of the words ‘ nono quoque
anno’. According to Censorinus, Harpalus made the solar year
consist of 365 days and 13 equinoctial hours.?, Eudoxus’s variation
will be mentioned later.
The 16-years and 160-years cycles.
After describing the octaéteris of 2,922 days, Geminus proceeds
thus:
‘If now it had only been necessary for us to keep in agreement
with the solar years, it would have sufficed to use the aforesaid
period in order to be in agreement with the phenomena. But as we
must not only reckon the years according to the sun, but also the
days and months according to the moon, they considered how this
also could be achieved. Thus the lunar month, accurately measured,
having 294 35 days, while the octaéteris contains, with the inter-
calary months, 99 months in all, they multiplied the 294 34 days of
the month by the 99 months; the result is 2,9234 days. Therefore
in eight solar years there should be reckoned 2,929} days according
to the moon. But the solar year has 3654 days, and eight solar
years contain 2922 days, this being the number of days obtained by
multiplying by 8 the number of days in the year. Inasmuch then
as we found the number of days according to the moon which are
contained in the 8 years to be 2,923%, we shall, in each octaéteris,
fall behind by 14 days in comparison with the moon. Therefore in
16 years we shall be behind by 3 days in comparison with the
moon. It follows that in each period of 16 years three days have
to be added, having regard to the moon’s motion, in order that we
may reckon the years according to the sun, and the months
and days according to the moon. But, when this correction is
made, another error supervenes. For the three days according to
the moon which are added in the 16 years give, in ten periods
of 16 years, an excess (with reference to the sun) of 30 days, that is
to say, a month. Consequently, at intervals of 160 years, one of
the intercalary months is omitted from (one of) the octaéterides ;
that is, instead of the three (intercalary) months which fall to be
reckoned in the eight years, only two are actually introduced.
Hence, when the month is thus eliminated, we start again in agree-
ment with the moon as regards the months and days, and with the
sun as regards the years.’ *
1 Festus Avienus, Prognost. 41, quoted by Ideler, op. cit., p. 191.
? Censorinus, De die natali, 19. 2.
5 Geminus, /sagoge, 8. 36-41, pp. 114. 8-116, 15.
Sa er ee τατον"
CH. XIX METON’S CYCLE 293
This passage explains itself; it is only necessary to add that
there is no proof that the 16-years period was actually used. The
160-years period was, however, presupposed in Eudoxus’s octaéteris,
the first of which, according to Boeckh, may have begun in 381 or
373 B.C. (Ol. 99, 4 or Ol. 101, 4) on 22/23 July, the ‘first day of
Leo’, i.e. the day on which the sun entered the sign of Leo; the
effect was that, after 20 octaéterides and the dropping of 30 days,
the beginning of the solar year was again on ‘the first of Leo’.
In Eudoxus’s system, then, the luni-solar reckoning was independent
of the solstices.1 According to the Eudoxus-Papyrus (Ars Eudoxi)
the intercalary months came in the 3rd, 6th, and 8th years of
Eudoxus's octaéteris.
Meton’s cycle.
Curiously enough, Meton is not mentioned by Geminus as the
author of the I9-years cycle; his connexion with it is, however,
clearly established by other evidence. Diodorus has the following
remark with regard to the year of the archonship of Apseudes
(Ol. 86, 4 = 433/2 B.C.).
‘In Athens Meton, the son of Pausanias, and famous in astro-
nomy, put forward the so-called 19-years period (ἐννεακαιδεκα-
ernpida); he started (ἀρχὴν ποιησάμενος) from the 13th of the
Athenian month Skirophorion.’?
Aelian says that Meton discovered the Great Year, and ‘ reckoned
it at τὸ years’,® and also that ‘the astronomer Meton erected
pillars and noted on them the solstices’. Censorinus, too, says that
Meton constructed a Great Year of 19 years, which was accordingly
called exneadecaeteris* Euctemon, whom Geminus does mention, :
assisted Meton in the matter of this cycle.
Geminus’s account of the cycle shall be quoted in full:
‘Accordingly, as the octaéteris was found to be in all respects
incorrect, the astronomers Euctemon, Philippus, and Callippus [the
phrase is of περὶ Εὐκτήμονα xré., as usual] constructed another
period, that of 19 years. For they found by observation that in 19
years there were contained 6940 days and 235 months, including
the intercalary months, of which, in the 19 years, there are 7.
[According to this reckoning the year comes to have 36 5τ5 days.]
1 Boeckh, Ueber die vierjahrigen Sonnenkreise der Alten, 1863, pp. 159-56.
2 Diodorus Siculus, xii. 36. 3. Aelian, V. H. x. 7.
* Censorinus, De die natalz, 18. 8.
204 GREEK MONTHS, YEARS, AND CYCLES parti
And of the 235 months they made rio hollow and 125 full, so that
hollow and full months did not always follow one another alter-
nately, but sometimes there would be two full months in succession.
For the natural course of the phenomena in regard to the moon
admits of this, whereas there was no such thing in the octaéteris.
And they included 110 hollow months in the 235 months for the
following reason. As there are 235 months in the 19 years, they
began by assuming each of the months to have 30 days; this gives
7,050 days: Thus, when all the months are taken at 30 days, the
7,050 days are in excess of the 6,940 days; the difference is (110
days), and accordingly they make 110 months hollow in order to
complete, in the 235 months, the 6,940 days of the 19-years period.
But, in order that the days to be eliminated might be distributed as
evenly as possible, they divided the 6,940 days by 110; this gives
63 days.! It is necessary therefore to eliminate the [one] day after
intervals of 63 days in this cycle. Thus it is not always the 30th
day of the month which is eliminated, but it is the day falling after
each interval of 63 days which is called ἐξαιρέσιμος. ἢ
The figure of 3655; days = 365 days 6" 18™ 56-98, and is still 30
minutes 11 seconds too long in comparison with the mean tropic —
year; but the mean lunar month of Meton is 29 days 12° 45™ 573%,
which differs from the true mean lunar month by not quite 1 minute
54 seconds. When Diodorus says that, in putting forward his 19-
years cycle, Meton started from the 13th of Skirophorion (which
was the 13th of the last month of Apseudes’ year=27th June, 432),
he does not mean that the first year of the period began on that
date; this would have been contrary to the established practice.
The beginning of the first year (the 1st Hekatombaion of that year)
would be the day of the first visibility of the new moon next after
the summer solstice, i.e. in this case 16th July, 432. The 13th
Skirophorion was the day of the solstice, and we have several
allusions to Meton’s observation of this;* presumably, therefore,
1 What should really have been done is to divide 7,050 by 110; this would
give 64 as quotient, and the result would be that every 64th day would have to
be eliminated, i.e. the day following successive intervals of 63 days. This fact
would easily cause 63 to be substituted for the quotient, and this would lead to.
6,940 being taken as the number to be divided by 110.
3. Geminus, J/sagoge, c. 8. 50-6, pp. 120. 4-122. 7.
5. Philochorus (Schol. ad Aristoph. Aves 997) says that, under Apseudes,
Meton of Leuconoé erected a e/iotropion near the wall of the Pnyx, and it was
doubtless there that he observed the solstice. Ptolemy says of this observation
that it was on the 21st of the Egyptian month Phamenoth in Apseudes’
year (Syntaxis, iii. 2, vol. i, p. 205, Heib.), “This is confirmed by the discovery
1 tite μδ.
CH. XIX CALLIPPUS'S CYCLE __ 295
Diodorus meant, not that the first year of Meton’s cycle began on
that day, but that it was on that day that Meton began his para-
pegma (or calendar).1 Ginzel? gives full details of the many divergent
views as to the date from which Meton’s cycle was actually intro-
duced at Athens. Boeckh put it in ΟἹ. 112, 3 = 330/29 B.C., Unger
between Ol. 109, 3 (342/1 B.C.) and Ol. 111, 1 (336/5 B.C.). Schmidt
holds that Meton’s cycle was introduced in 342 B.C., but in a
modified form. The 235 months of the 19-years cycle contained,
according to the true mean motion of the moon, 235 x 29-53059 days,
or 6,939 days and about 163 hours. Consequently after 4 cycles
there was an excess of four times the difference between 6,940 days
and 6,939 days 16% hours, or an excess of 1 day 6 hours; after 10
cycles an excess of 3 days 3 hours, and so on. The Athenians,
therefore, according to Schmidt, struck out one day in the 4th, 7th,
10th, 13th, 16th, 2oth, and 23rd cycles, making these cycles 6,939
days each. But, as Ginzel points out, the confusions in the calendar
which occurred subsequently tell against the supposition that such
a principle as that assumed by Schmidt was steadily followed in
Athens from 342 B.C.
Callippus’s cycle of 76 years.
Geminus follows up his explanation with regard to the Metonic
cycle thus:
‘In this cycle [the Metonic] the months appear to be correctly
taken, and the intercalary months to be distributed so as to secure
agreement with the phenomena; but the length of the year as
taken is not in agreement with the phenomena. For the length of
the year is admitted, on the basis of observations extending over ©
many years, to contain 365% days, whereas the year which is
obtained from the 19-year period has 365°; days, which number of
of a fragment of a Jarafegma at Miletus which alludes to the same observation
of the summer solstice on 13th Skirophorion or 2Ist Phamenoth, and adds that
in the year of ...«vxros the solstice fell on 14th Skirophorion or the Egyptian
11th Payni. Diels showed from another fragment that the archon must have
been Polykleitos (110/109 B.c.), so that the second observation of the solstice
mentioned in the fragment must have been on 27th June, 109, i.e. in the last
(19th) year of the 17th Metonic cycle (Ginzel, ii, pp. 423, 424).
1 The παράπηγμα was a posted record (παραπήγνυμι), a sort of almanac giving,
for a series of years, the movements of the sun, the dates of the phases of the
moon, the risings and settings of certain stars, besides ἐπισημασίαι or weather
indications.
2 Ginzel, op. cit., ii. 418, 430, 431, 442 sqq.
296 GREEK MONTHS, YEARS, AND CYCLES Parti
days exceeds 365% by gth of a day. On this ground Callippus and
the astronomers of his school corrected this excess of a (fraction of
a) day and constructed the 76-years period (ἑκκαιεβδομηκονταετη-
ρίδα) out of four periods of 19 years, which contain in all 940
months, including 28 intercalary, and 27,759 days. They adopted
the same arrangement of the intercalary months. And this period
appears to agree the best of all with the observed phenomena.’
With Meton’s year of 3653 4, days (6,940 divided by 19), four
periods of 19 years amount of course to 27,760 days, and the effect
of Callippus’s change was to reduce this number of days by one.
27,759 days divided by 940 gives, for the mean lunar month, 29
days 125 44™ 252°, only 22 seconds in excess of the true mean
length. —
Callippus was probably born about 370 B.C.; he came to Athens
about 334 B.C.; the first year of the first of his cycles of 76 years
was Ol. 112, 3 = 330/29 B.C., and probably began on the 29th or
28th of June. His cycles never apparently came into practical use,
but they were employed by individual astronomers or chronologists
for fixing dates; Ptolemy, for example, gives various dates both »
according to Egyptian reckoning and in terms of Callippic cycles.*
Hipparchus’s cycle.
It is only necessary to add that yet another improvement was
made by Hipparchus about 125 B.c. Ptolemy says of him:
‘ Again, in his work on intercalary months and days, after pre-
mising that the length of the year is, according to Meton and
Euctemon, 365% τῆς days, and according to Callippus 3653 days
only, he continues in these words: “We find that the number of
whole months contained in the 19 years is the same as they make
it, but that the year in actual fact contains less by 335th of a day
than the odd 4 of a day which they give it, so that in 300 years
there is a deficiency, in comparison with Meton’s figure, of 5 days,
and in comparison with Callippus’s figure, of one day.” Then,
summing up his own views in the course of the enumeration of his
own works, he says: “I have also discussed the length of the year
in one book, in which I prove that the solar year—that is, the
length of time in which the sun passes from a solstice to the same
1 Geminus, /sagoge, 8. 57-60, p. 122. 8-23.
2 Ptolemy, Syv/azxzs, iti. 1, vol. i, p. 196.6; iv. 11, vol. i, pp. 344. 14, 345.
12, 346.14; v. 3, vol.i, p. 363. 16; vii. 3, vol. ii, pp. 25. 16, 28 12, 29. 13,
32. 5.
CH, XIX HIPPARCHUS’S CYCLE 297
solstice again, or from an equinox to the same equinox—contains
365 days and i, less very nearly 35th of a day and night, and not
the exact 4 which the mathematicians suppose it to have in addition
to the said whole number of days.”’?
‘Censorinus gives Hipparchus’s period as 304 years, in which there
are 112 intercalary months.* Presumably, therefore, Hipparchus
took four times Callippus’s cycle (76 x 4= 304) and gave the period
111,035 days instead of 111,036 (-- 27,759 Χ 4). This gives, as the
length of the year, 365 days 5° 55™ 15°88, while 3654-335 days =
365 days 5" 55™ 128, the excess over the true mean tropic year
being about 64 minutes. The number of months in the 304 years
is 304 X 12+28 x 4=3,760, whence the mean lunar month is, accord-
ing to Hipparchus, 29 days 12" 44™ 23°, which is very nearly correct,
being less than a second out in comparison with the present accepted
figure of 29-53059 days!
1 Ptolemy, Syztaxis, iii. 3, vol. i, pp. 207. 7 -- 208. 2.
2 Censorinus, De die natali, 18. 9.
PART II
ARISTARCHUS OF SAMOS
ON THE SIZES AND DISTANCES OF THE SUN
AND MOON
I
ARISTARCHUS OF SAMOS
WE are told that Aristarchus of Samos was a pupil of Strato
of Lampsacus,' a natural philosopher of originality,2, who suc-
ceeded Theophrastus as head of the Peripatetic school in 288 or
287 B.C. and held that position for eighteen years. Two other
facts enable us to fix Aristarchus’s date approximately. In
281/280 B.C. he made an observation of the summer solstice ;*
and the book in which he formulated his heliocentric hypothesis
was published before the date of Archimedes’ Psammites or Sand-
veckoner, a work written before 216 B.c. Aristarchus therefore
probably lived czrca 310-230 B.C., that is, he came about 75 years
later than Heraclides and was older than Archimedes by about
25 years. ;
Aristarchus was called ‘the mathematician’, doubtless in order
to distinguish him from the many other persons of the same
name; he is included by Vitruvius among the few great men who
possessed an equally profound knowledge of all branches of science,
geometry, astronomy, music, &c. ‘Men of this type are rare,
men such as were, in times past, Aristarchus of Samos, Philolaus
and Archytas of Tarentum, Apollonius of Perga, Eratosthenes
of Cyrene, Archimedes and Scopinas of Syracuse, who left to
? Aétius, i. 15. 5 (2. G. p. 313).
3 Galen, Histor. Philos. 3 (D.G. p. 601. 1).
® Ptolemy, Syzfazxis, iii, 2 (i, pp. 203, 206, ed. Heib.).
300 ARISTARCHUS OF SAMOS PART IT
posterity many mechanical and gnomonic appliances which they
invented and explained on mathematical (lit. numerical) and natural
principles.’! That Aristarchus was a very capable geometer is
proved by his extant work Ox the sizes and distances of the ~
sun and moon, translated in this volume: in the mechanical line —
he is credited with the discovery of an. improved sun-dial, the ©
so-called σκάφη, which had, not a plane, but a concave hemi-
spherical surface, with a pointer erected vertically in the middle
throwing shadows and so enabling the direction and the height
of the sun to be read off by means of lines marked on the surface
of the hemisphere. He also wrote on vision, light, and colours.’
His views on the latter subjects were no doubt largely influenced
by his master Strato; thus Strato held that colours were emanations
from bodies, material molecules, as it were, which imparted to ~
the intervening air the same colour as that possessed by the body,* —
while Aristarchus said that colours are ‘ shapes or forms stamping
the air with impressions like themselves as it were’,® that ‘colours —
in darkness have no colouring’,® and that light is ‘ the colour
impinging on a substratum’.’ It does not appear that Strato ~
can be credited with any share in his astronomical discoveries:
of Strato we are only told (1) that, like Metrodorus before him,
he held that the stars received their light from the sun (Metrodorus —
alleged this of ‘all the fixed stars’, and it is not stated that Strato —
made any limitation);* (2) that he held a comet to be ‘the light —
of a star enclosed in a thick cloud, just as happens with λαμπτῆρες
(torches) ’;® (3) that, like Parmenides and Heraclitus, he considered —
the heaven to be of fire;!° (4) that he regarded time as ‘quantity —
in (i.e. expressed by) things in motion and at rest’; (5) that
he said the divisions of the universe were without limit;’ and —
(6) that he maintained that there was no void outside the universe, —
though there might be within it.
1 Vitruvius, De architectura, i. τ. τό, 3 Ibid. ix. 8 (9). I.
3. Aét. i, 15. 5 (2. G. p. 313), iv. 13. 8 (D. G. pp. 404 and 853).
_* Aét. iv. 13. 7 (D. G. p. 403).
δ Ibid. iv. ἦς (D. G. pp. 404 and 853).
“δ Ibid. i. 15. 9 (D. G. p. 314). Τ Ibid. i. 15. 5 (2. G. p. 313).
8 Ibid. ii. 17. 1-2 (D. δ, p- 346). 9. Ibid. iii. 2. 4 (D. G. p. 366).
30 Ibid. ii. 11. 4 (D. G. p. 340). 1. hid. i. 22. 4 (D. G. p. 318).
1 Epiphanius, Adv. haeres. iii. 33 (D. G. p. 592).
8 Aét, i. 18, 4 (2. G. p. 316).
cH.I ARISTARCHUS OF SAMOS δ 301
The Heliocentric Hypothesis.
There is not the slightest doubt that Aristarchus was the first to put
_ forward the heliocentric hypothesis. Ancient testimony is unanimous
_ on the point, and the first witness is Archimedes, who was a younger
contemporary of Aristarchus, so that there was no possibility of
a mistake. Copernicus himself admitted that the theory was
_ attributed to Aristarchus, though this does not seem to be generally
known. Thus Schiaparelli quotes two passages from Copernicus’s
work in which he refers to the opinions of the ancients about the
- motion besides rotation, namely revolution in an orbit i
motion of the earth. One is in the dedicatory letter to Pope
Paul III, where Copernicus mentions that he first found out from
Cicero that one Nicetas (1.6. Hicetas) had attributed motion to the
earth, and that he afterwards read in Plutarch that certain others
held that opinion; he then quotes the /%acifa, according to
which ‘ Philolaus the Pythagorean asserted that the earth moved
round the fire in an oblique circle, in the same way as the sun
and moon’! The other passage is in Book I, c. 5, where, after
an allusion to the views of Heraclides, Ecphantus, Nicetas
(Hicetas), who made the earth rotate about its own
centre of the universe, he goes on to say that it woul
‘atque etiam (terram) pluribus motibus vagantem et unam ex astris
Philolaus Pythagoricus sensisse fertur, Mathematicus non vulgaris.’
_ Here, however, there is no question of the earth revolving round the
sun,and there is no mention of Aristarchus. But it is a curious fact
_ that Copernicus did mention the theory of Aristarchus in a passage
_ which he afterwards suppressed: ‘Credibile est hisce similibusque
causis Philolaum mobilitatem terrae sensisse, quod etiam nonnulli
Aristarchum Samium ferunt in eadem fuisse sententia.’?
I will now quote the whole passage of Archimedes in which
the allusion to Aristarchus’s heliocentric hypothesis occurs, in order
to show the whole context.®
1 Ps. Plutarch, De plac. phil.= Aé&t. iii. 13. 2 (D. G. p. 378).
2 De Revolutionibus Caelestibus, ed. Thorun., 1873, p. 34 note, quoted in
Gomperz, Griechische Denker, i*, p. 432.
3 Archimedes, ed. Heiberg, vol. ii, p. 244 (Avenarius 1. 4-7); The Works of
Archimedes, ed. Heath, pp. 221, 222.
302 ARISTARCHUS OF SAMOS PART II
‘You are aware [‘you’ being King Gelon] that “universe” is
the name given by most astronomers to the sphere, the centre
of which is the centre of the earth, while its radius is equal to
the straight line between the centre of the sun and the centre of
the earth. This is the common account (τὰ γραφόμενα), as you
have heard from astronomers. But Aristarchus brought out a do0k
consisting of certain hypotheses, wherein it appears, as a con-
sequence of the assumptions made, that the universe is many
times greater than the ‘ universe” just mentioned. His hypotheses
are that the fixed stars and the sun remain unmoved, that the
earth revolves about the sun tn the circumference of a circle, the
sun lying tn the middle of the orbit, and that the sphere of the fixed
stars, situated about the same centre as the sun, is so great that
the circle in which he supposes the earth to revolve bears such
a proportion to the distance of the fixed stars as the centre of
the sphere bears to its surface. Now it is easy to see that this is
impossible ; for, since the centre of the sphere has no magnitude,
We cannot conceive it to bear any ratio whatever to the surface
of the sphere. We must, however, take Aristarchus to mean this:
since we conceive the earth to be, as it were, the centre of the
universe, the ratio which the earth bears to what we describe as
the “universe” is equal to the ratio which the sphere containing
the circle in which he supposes the earth to revolve bears to the
sphere of the fixed stars. For he adapts the proofs of the pheno-
mena to a hypothesis of this kind, and in particular he appears
to suppose the size of the sphere in which he makes the earth
7}
move to be equal to what we call the “ universe”.
We shall come back to the latter part of this passage ; at present
we are concerned only with the italicized words. The heliocentric
hypothesis is stated in language which leaves no room for dispute
as to its meaning. The sun, like the fixed stars, remains unmoved
and forms the centre of a circular orbit in which the earth revolves
round 1:1 the sphere of the fixed stars has its centre at the
1 There is only one slight awkwardness in the language. The sentence is
ὑποτίθεται yap τὰ μὲν ἀπλανέα τῶν ἄστρων καὶ τὸν ἅλιον μένειν ἀκίνητον, τὰν δὲ γᾶν
περιφέρεσθαι περὶ τὸν ἅλιον κατὰ κύκλου περιφέρειαν, ὅς ἐστιν ἐν μέσῳ τῷ δρόμῳ
κείμενος, and it would be natural to suppose that the relative ὅς would refer to
the masculine substantive nearest to it, 1.6. κύκλου, ‘circle,’ rather than τὸν ἅλιον,
‘the sun’; but ‘ which is situated in the middle of the (earth’s) course’ cannot
possibly refer to the circle, i.e. to the course itself, and must refer to the sun.
The awkwardness suggested to Bergk (Fiinf Abhandlungen, 1883, p. 162) that
Archimedes wrote és ἐστιν ἐν μέσῳ τῷ οὐρανῷ, ‘which is in the middle of the
heaven’ This would enable ὅς to refer to the ‘ circle’, but there seems to be no
sufficient ground for changing the reading δρόμῳ.
CH.I ARISTARCHUS OF SAMOS 303
centre of the sun. But a question arises as to the form in which
Aristarchus’s hypotheses were given out. The expression used
by Archimedes is ὑποθεσίων τινῶν ἐξέδωκεν γραφάς, ‘put out
ypagai of certain hypotheses.’ I take it in the sense of bringing
out a tract or tracts consisting of or stating certain hypotheses ;
for one of the meanings of the word γραφή is a ‘writing’ or
a written ‘description’. Heiberg takes γραφαί in this sense, but
regards ὑποθεσίων as the title of the book (‘ libros quosdam edidit,
qui hypotheses inscribuntur’?). Hultsch,? however, takes γραφαί
in its other possible sense of ‘drawings’ or figures constructed
to represent the hypotheses; and Schiaparelli* suggests that the
word γραφή here used seems not only to signify a verbal description
but to include also the idea of explanatory drawings. I agree
that it is probable enough that Aristarchus’s tract or tracts included
geometrical figures illustrating the hypotheses, but I still think
that the word γραφαί here does not itself mean ‘figures’ but
means written statements of certain hypotheses. This seems to
me clear from the words immediately following γραφάς, namely
ἐν ais ἐκ τῶν ὑποκειμένων συμβαίνει x.7.é., ‘7m which it results
from the assumptions made that the universe is many times greater
than our “universe” above mentioned’; ‘in which’ can only refer
to γραφάς or ὑποθεσίων, and it cannot refer to ὑποθεσίων because
what /o/ows from the assumptions made cannot be zz those
assumptions which are nothing but the hypotheses themselves;
therefore ‘in which’ refers to γραφάς, but a result following from
assumptions does not follow zz figures illustrating those assump-
tions but in the course of a description of them or an argument
about them. The words ‘in which it results . . .’ also show clearly
enough that the tract or tracts did not merely state the hypothesis
but also included some kind of geometrical proof. I need only
1 Archimedes, ed. Heiberg, vol. ii, p. 245.
3 Hultsch, art. ‘Aristarchus’ in Pauly-Wissowa’s Real-Encyclopéddie, ii, p. 875.
* Schiaparelli, Origine del sistema planetario eliocentrico presso 2 Grect, p. 95.
* Bergk (Fiinf Abhandlungen, p. 160) thinks that ‘ γραφάς cannot be taken as
synonymous with γράμματα : this would be a somewhat otiose circumlocution :
but it means the “ outline” {Umriss), like xaraypagy. Archimedes chooses this
expression because Aristarchus had rather indicated his hypotheses than worked
them out and established them.’ I do not think this inference necessary ;
is may be quite colourless without being otiose, a sufficient reason for its
insertion being the fact that some word other than ὑποθεσίων is necessary as an
804 ARISTARCHUS OF SAMOS PART II
add that there are other cases of the use of γραφή in the sense
of ‘writing’; cf. an expression in Eutocius, ‘I have come across
writings (γραφαῖς) of many famous men which give this problem’
[that of the two mean proportionals].!
Our next evidence is a passage of Plutarch :
‘Only do not, my good fellow, enter an action against me for
impiety in the style of Cleanthes, who thought it was the duty
of Greeks to indict Aristarchus of Samos on the charge of impiety
for putting in motion the Hearth of the Universe, this being the
effect of his attempt to save the phenomena by supposing the
heaven to remain at rest and the earth to revolve in an oblique
circle, while it rotates, at the same time, about its own axis.’ ‘
Here we have the additional detail that Aristarchus followed
Heraclides in attributing to the earth the daily rotation about its
axis; Archimedes does not state this in so many words, but it
is clearly involved by his remark that Aristarchus supposed that
the fixed stars as well as the sun remain unmoved in space. When
Plutarch makes Cleanthes say that Aristarchus ought to be indicted
for the impiety of ‘ putting the Hearth of the Universe in motion’,
he is probably quoting the exact words used by Cleanthes, who »
doubtless had in mind the passage in Plato’s Phaedrus where
‘Hestia abides alone in the House of the Gods’. A similar ex-
pression is quoted by Theon of Smyrna from Dercyllides, who
‘says that we must suppose the earth, the Hearth of the House
of the Gods according to Plato, to remain fixed, and the planets
with the whole embracing heaven to move, and rejects with
abhorrence the view of those who have brought to rest the things
which move and set in motion the things which by their nature and
position are unmoved, such a supposition being contrary to the
hypotheses of mathematics’;* the allusion here is equally to
Aristarchus, though his name is not mentioned. A tract ‘ Against
Aristarchus’ is mentioned by Diogenes Laertius among Cleanthes’
works; and it was evidently published during Aristarchus’s lifetime
(Cleanthes died about 232 B.C.).
antecedent to the relative sentence ‘7m which it follows from the assumptions
made, ἄς.
1 Archimedes, ed. Heiberg, vol. iii, p. 66. 9.
3 Plutarch, De facie in orbe lunae, c. 6, pp. 922 F — 923A.
8 Theon of Smyrna, p. 200, 7-12, ed. Hiller.
“-
CH. I ARISTARCHUS OF SAMOS 305
Other passages bearing on our present subject are the fol-
lowing.
‘ Aristarchus sets the sun among the fixed stars and holds that
the earth moves round the sun’s circle (i.e. the ecliptic) and is put in
shadow according to its (i.e. the earth’s) inclinations.’*
One of the two versions of this passage has ‘¢he dzsc is put
in shadow’, and it would appear, as Schiaparelli says, ‘that the
words “the disc” were interpolated by some person who thought
that the passage was an explanation of solar eclipses.’ It is indeed
placed under the heading ‘Concerning the eclipse of the sun’ ;
but this is evidently wrong, for we clearly have here in the
concisest form an explanation of the phenomena of the seasons
according to the system of Copernicus.”
‘Yet those who did away with the motion of the universe and
were of opinion that it is the earth which moves, as Aristarchus
the mathematician held, are not on that account debarred from
having a conception of time.”*
‘Did Plato put the earth in motion, as he did the sun, the moon,
and the five planets, which he called the instruments of time on
account of their turnings, and was it necessary to conceive that the
earth “ which is globed about the axis stretched from pole to pole
through the whole universe” was not represented as being held
together and at rest, but as turning and revolving (στρεφομένην
καὶ ἀνειλουμένην), as Aristarchus and Seleucus afterwards main-
tained that it did, the former stating this as only a hypothesis
(ὑποτιθέμενος μόνον), the latter as a definite opinion (καὶ ἀπο-
φαινόμενος) ὃ“
‘Seleucus the mathematician, who had written in opposition to
the views of Crates, and who, himself too affirmed the earth’s motion,
says that the revolution (περιστροφή) of the moon resists the rota-
tion {and the motion]° of the earth, and, the air between the two
bodies being diverted and falling upon the ng ocean, the sea
is correspondingly agitated into waves.’®
When Plutarch refers to Aristarchus as only putting forward the
double motion of the earth as a yfotheszs, he must presumably
1 Aét. ii. 24. 8 (D. G. p. 355. 1-5).
3 Schiaparelli, 7 frecursori, Ὁ. 50.
5 Sextus Empiricus, Adv. Mathematicos, x. 174, p. 512. 19, Bekker.
* Plutarch, Plat. guaest, viii. 1, 1006 C.
5 The Ps. Plutarch version has the words καὶ τῇ κινήσει; ‘and the motior,’
after αὐτῆς τῇ δίνῃ φησί; Stobaeus omits them, and has τῷ dive instead of τῇ δίνῃ.
5. Aét. ili. 17. 9 (D. G. p. 383).
1410 x
306 ARISTARCHUS OF SAMOS PART IT
be basing himself on nothing more than the word hyfotheses used
by Archimedes, and his remark does not therefore exclude the
possibility of Aristarchus having supported his hypothesis by some
kind of argument ; nor can we infer from Plutarch that Seleucus
went much further towards proving it. Plutarch says that
Seleucus declared the hypothesis to be true (ἀποφαινόμενος), but
it is not clear how he could have attempted to prove it. Schiapa-
relli suggests that Aristarchus’s attitude may perhaps be explained
on the basis of the difference between the rdles of the astronomer
and the Zhyszczs¢ as distinguished by Geminus in the passage quoted
above (pp. 275-6). Aristarchus, as the astronomer and mathe-
matician, would only be concerned to put forward geometrical
hypotheses capable of accounting for the phenomena; he may
have left it to the physicists to say ‘which bodies ought from
their nature to be at rest and which to move’. But this is only
a conjecture.
Seleucus, of Seleucia on the Tigris, is described by Strabo' as
a Chaldaean or Babylonian; he lived about a century after Aris-
tarchus and may have written about 150 B.C. The last of the
above quotations is Aétius’s summary of his explanation of.
the tides, a subject to which Seleucus had evidently given much
attention ;* in particular, he controverted the views held on this
subject by Crates of Mallos, the ‘grammarian’, who wrote on
geography and other things, as well as on Homer. The other
explanations of the tides summarized by Aétius include those of
Aristotle and Heraclides, who sought the explanation in the sun,
holding that the sun sets up winds, and that these winds, when
they blow, cause the high tide and, when they cease, the low tide ;
Dicaearchus who put the tides down to the direct action of the
sun according to its position; Pytheas and Posidonius who con-
nected them with the moon, the former directly, the latter through
the setting up of winds; Plato who posited a certain general oscil-
lation of the waters, which pass through a hole in the earth;*
Timaeus who gave as the reason the unequal flow of rivers from
the Celtic mountains into the Atlantic; then, immediately before
Seleucus, are mentioned Crates ‘the grammarian’ and Apollo-
2. Strabo, xvi. 1. 6, 1.1.9. ων tiie we 2 Cf, Strabo, iii. 5. Ὁ.
. Phaedo R
CH.I ARISTARCHUS OF SAMOS © 307
dorus of Corcyra, the account of whose views is vague enough,
the former attributing the tides to ‘the counter-movement (ἀντι-
σπασμός) of the sea,”! and the latter to ‘the refluxes from the Ocean ’°.
When Aétius adds, in introducing Seleucus’s views, that ‘he too
made the earth move’, we should expect that he had just before
mentioned some one else who had done the same. But Crates
adhered to the old view and did not make the earth move ;? nor is
there anything to suggest that Apollodorus attributed motion to
the earth. Consequently Bergk supposes that, just before the
‘notice of Seleucus’s explanation of the tides with reference to
the earth’s motion, there must have been a notice of a different
explanation of them by a person who also attributed motion to the
earth, and that, as we know of no other person by name who
adopted Aristarchus’s views, except Seleucus, the notice which has
dropped out must have given a different explanation of the tides
by Aristarchus himself? But, as the motion of the earth referred
to in Seleucus’s explanation may be rotation only (δίνη or δῖνος), it
seems possible that Heraclides (who made the earth rotate) is the
other person referred to. in the collection of notices as having
‘made the earth move’, although he is mentioned some way back,
‘To judge by Seleucus’s explanation of the tides, he would seem to
have supposed that the atmosphere about the earth extended as
far as the moon and rotated with the earth in 24 hours, and that
the resistance of the moon acted upon the rotating atmosphere
either by virtue of the relative slowness of the moon’s revolution
about the earth or of its motion perpendicular to the equator ;*
Strabo tells us that Seleucus had discovered periodical inequalities
in the flux and reflux of the Red Sea which he connected with the
position of the moon in the zodiac.
No one after Seleucus is mentioned by name as having accepted
the doctrine of Aristarchus, and if other Greek astronomers refer to
it, they do so only to denounce it, as witness Dercyllides.6 The
rotation of the earth is, howeyer, mentioned as a possibility by
Seneca.
? Some details of Crates’ views are also given in Strabo, i. 1. 8.
* Bergk (Fiinf Abhandlungen, p. 166) quotes from Strabo, i. 2. 24, the words
τὴν πάροδον τοῦ ἡλίου. 3. Bergk, op. cit., p. 167.
* Schiaparelli, 7 Zrecursori, p. 36. ® Strabo, iii. 5. 9.
® Theon of Smyrna, p. 200. 7-12: see above (p. 304).
xX 2
308 ARISTARCHUS OF SAMOS PART II
‘It will be proper to discuss this, in order that we may know
whether the universe revolves and the earth stands still, or the
universe stands still and the earth rotates. For there have been
those who asserted that it is we whom the order of nature causes to
move without our being aware of it, and that risings and settings do
not occur by virtue of the motion of the heaven, but that we ourselves
rise and set. The subject is worthy of consideration, in order that
we may know in what conditions we live, whether the abode allotted
to us is the most slowly or the most quickly moving, whether God
moves everything around us, or ourselves instead.’ ἢ
Hipparchus, himself a contemporary of Seleucus, reverted to the
geocentric system, and it was doubtless his great authority which
sealed the fate of the heliocentric hypothesis for so many centuries.
The reasons which weighed with Hipparchus were presumably (in
addition to the general prejudice in favour of maintaining the earth
in the centre of the universe) the facts that the system in which the
earth revolved in a circle of which the sun was the exact centre
failed to ‘save the phenomena’, and in particular to account for the
variations of distance and the irregularities of the motions, which
became more and more patent as methods of observation improved;
that, on the other hand, the theory of epicycles did suffice to repre- Ὁ
sent the phenomena with considerable accuracy ; and that the latter
theory could be reconciled with the immobility of the earth.
We revert now to the latter part of the passage quoted above
from Archimedes, in which he comments upon the assumption of
Aristarchus that the sphere of the fixed stars is so great that the
ratio in which the earth’s orbit stands to the said sphere is such
a ratio as that which the centre of the sphere bears to its surface.
If this is taken in a strictly mathematical sense, it means of course
that the sphere of the fixed stars is infinite in size, a supposition
which would not suit Archimedes’ purpose, because he is under-
taking to prove that he can evolve a system for expressing large
numbers which will enable him to state easily in plain words the
number of grains of sand which the whole universe could contain ;
hence, while he wishes to base his estimate of the maximum size of
the universe upon some authoritative statement which will be
generally accepted, and takes the statement of Aristarchus as suit-,
1 Seneca, Wat. Quaest. vii. 2. 3.
ἘΠ δὰ ee
CH. 1. ARISTARCHUS OF SAMOS 309
able for his purpose, he is obliged to interpret it in an arbitrary
way which he can only justify by somewhat sophistically pressing
the mathematical point that Aristarchus could not have meant to
assert that the sphere of the fixed stars is actually zz/mzte in size
and therefore could not have wished his statement to be taken
quite literally ; consequently he suggests that a reasonable inter-
pretation would be to take it as meaning that
(diameter of earth) : (diameter of ‘ universe’) =
(diam. of earth’s orbit): (diam. of sphere of fixed stars),
instead of
0: (surface of sphere of fixed stars) =
(diam. of earth’s orbit): (diam. of sphere of fixed stars),
and he explains that the ‘ universe’ as commonly conceived by the
astronomers of his time (he refers no doubt to the adherents of
the system of concentric spheres) is a sphere with the earth as
centre and radius equal to the distance of the sun from the earth,
and that Aristarchus seems to regard the sphere containing (as a
great circle) the orbit in which the earth revolves about the sun as
-equal to the ‘ universe ’ as commonly conceived, so that the second
and third terms of the first of the above proportions are equal.
While it is clear that Archimedes’ interpretation is not justified,
it may be admitted that Aristarchus did not mean his statement to
be taken as a mathematical fact. He clearly meant to assert no
more than that the sphere of the fixed stars is zzcomparably greater
than that containing the earth’s orbit as a great circle; and he was
shrewd enough to see that this is necessary in order to reconcile
the apparent immobility of the fixed stars with the motion of
the earth. The actual expression used is similar to what was
evidently a common form of words among astronomers to ex-
press the negligibility of the size of the earth in comparison with
larger spheres. Thus, in his own tract Ox the sizes and distances
of the sun and moon, Aristarchus lays down as one of his assump-
tions that ‘the earth is in the relation (λόγον ἔχειν) of a point and
centre to the sphere in which the moon moves’. In like manner
Euclid proves, in the first theorem of his Phaenomena, that ‘ the
earth is in the middle of the universe (κόσμος) and holds the
310 ARISTARCHUS OF SAMOS PART II
position (τάξιν) of centre relatively to the universe’. Similarly
Geminus! describes the earth as ‘in the relation of a centre to the
sphere of the fixed stars’; Ptolemy? says that the earth is not
sensibly different from a point in relation to the radius of the
sphere of the fixed stars; according to Cleomedes*® the earth is
‘in the relation of a centre’ to the sphere in which the sun moves,
and a fortiort to the sphere of the fixed stars, but of to the
sphere in which the moon moves.
In Aristarchus’s extant treatise Ox the sizes and distances of the
sun and moon there is no hint of the heliocentric hypothesis, but
the sun and moon are supposed to move in circles round the earth
as centre. From this we must infer either (1) that the work in
question was earlier than the date at which he put forward the
hypotheses described by Archimedes, or (2) that, as in the tract the
distances of the sun from the earth and of the moon from the earth
are alone in question, and therefore it was for the immediate pur-
pose immaterial which hypothesis was taken, Aristarchus thought
it better to proceed on the geocentric hypothesis which was familiar
to everybody. Schiaparelli+ suggests that one of the reasons which
led Aristarchus to place the sun in the centre of the universe was ~
probably the consideration of the sun’s great size in comparison
with the earth. Now in the treatise referred to Aristarchus finds
the ratio of the diameter of the sun to the diameter of the earth to
lie between 19:3 and 43:6; this makes the volume of the sun
something like 300 times the volume of the earth, and, although
the principles of dynamics were then unknown, it might even in
that day seem absurd to make the body which was so much larger
revolve round the smaller.
There is no reason to doubt that, in his heliocentric system,
Aristarchus retained the moon as a satellite of the earth revolving
round it as a centre; thus even in his system there was one epi-
cycle, that described by the moon about the earth as centre.
1 Geminus, Jsagoge, 18. 16, p. 186. 16, ed. Manitius.
? Ptolemy, Syntaxts, i. 6, p. 20. 5; Heib.
5. Cleomedes, De motu circulari, i. 11. * Schiaparelli, 7 Jrecursor?, p. 33.
CH.I ARISTARCHUS OF SAMOS 211
The apparent diameter of the sun.
Another passage of the Saud-reckoner of Archimedes states that
* Aristarchus discovered that the sun’s apparent size is about one
720th part of the zodiac circle.’?
This, again, is a valuable contribution to our knowledge of
Aristarehus, for in the treatise Oz the sizes and distances of the
sun and moon he makes the apparent diameter not ~3>th of the
zodiac circie, or $°, but one-fifteenth part of a sign, that is to say 2°,
which is a gross over-estimate. The nearest estimate to this which
we find recorded appears to be that mentioned by Macrobius,? who
describes an experiment made with a hemispherical dial by marking
the points on which the shadow of the upright needle fell at the
moments respectively when the first ray of the sun as it began to
rise fell on the instrument and when the sun just cleared the horizon
respectively. The result showed that the interval of time was 3th
of an hour, which gave as the apparent diameter of the sun 5},th
of 360° or 13. Macrobius would apparently have us believe that
this very inaccurate estimate was due to the Egyptians. We have,
however, seen reason to believe that Macrobius probably attributed
to the ‘Egyptians’ the doctrines of certain Alexandrian astro-
-nomers,’ and in the present case it would seem that we have to do
with an observation very unskilfully made by some even less com-
petent person.t The Babylonians had, however, many centuries
before arrived at a much closer approximation; they made the
time which the sun takes to rise - ἢ of an hour, and, even if
the hour is the double hour (one-twelfth of a day and night), this
gives 1° as the apparent diameter of the sun. How, then, did
Aristarchus in his extant work come to take 2° as the value?
Tannery has an interesting suggestion, which is however perhaps
too ingenious.® ‘If Aristarchus chose for the. apparent diameter
of the sun a value which he knew to be false, it is clear that his
treatise was mainly intended to give a specimen of calculations
1 Archimedes, ed. Heiberg, ii, p. 248. 19; Zhe Works of Archimedes, ed.
Heath, p. 223.
2 Macrobius, 77: somn. Scip. i. 20. 26-30. 3. See p. 259 above.
* Hultsch, Poseidonios tiber die Grosse und Entfernung der Sonne, p. 43.
5 Tannery, ‘Aristarque de Samos’ in M¢m. de la Soc. des sciences phys. et
nat. de Bordeaux, 2° sér. v. 1883, p. 241; Mémoires scientifiques, ed, Heiberg
and Zeuthen, i, pp. 375-6.
312 ARISTARCHUS OF SAMOS PART II
which require to be made on the basis of more exact experimental
observations, and to show at the same time that, for the solution
of the problem, one of the data could be chosen almost arbitrarily.
He secured himself in this way against certain objections which
might have been raised. According to the testimony of Macrobius,
it seems that in fact the Egyptians had, by observations completely
erroneous, fixed the apparent diameter of the sun at 5},th of the
circumference, i.e. 12. Aristarchus seems to have deliberately
chosen to assign it a still higher value; but it is beyond question
that he was perfectly aware of the consequences of his hypothesis.’
Manitius? suggests that the ‘one-fifteenth part (πεντεκαιδέκατον
μέρος)᾽ of a sign of the zodiac in Aristarchus’s treatise should be
altered into ‘ one-jiftieth part’ (πεντηκοστὸν μέρος), which would
give the quite acceptable value of οὗ 36’. But the propositions in
the treatise in which the hypothesis is actually used seem to make
it clear that ‘ one-fifteenth’ is what Aristarchus really wrote. Unless
therefore we accept Tannery’s suggestion, we seem to be thrown
back once more on the supposition that the treatise was an early
work written before Aristarchus had made the more accurate
observation recorded by Archimedes. From the statement of |
Archimedes that Aristarchus dzscovered (εὑρηκότος) the value of
zioth, I think we may infer with safety that Aristarchus was at
least the first Greek who had given it, and we have therefore an
additional reason for questioning the tradition which credits Thales
with the discovery. How Aristarchus obtained his result we are
not told, but, seeing that he is credited with the invention of an
improved sun-dial (σκάφη), it is possible that it was by means of
this instrument that he made his observations. Archimedes himself
seems to have been the first to think of the better method of using
an instrument for measuring angles ; by the use of a rough instru-
ment of this kind he made the apparent angular diameter of the sun
lie between the limits of τέ χίῃ and 535th of a right angle. Hippar-
chus used for the same purpose a more elaborate instrument, his
dioptra, the construction of which is indicated by Ptolemy,? and
is more fully described by Pappus in his commentary on Book V of
1 Proclus, Hyfotyposis, ed. Manitius, note on p. 292.
* Ptolemy, Sywtuxis, v. 14, p. 417. 2-3 and 20-23, ed. Heib.
oe
~~ —
en δου. Be
ΘΗ ARISTARCHUS OF SAMOS 313
Ptolemy, quoted by Theon of Alexandria; Proclus describes it
somewhat differently.2 Though we gather that Hipparchus made
many observations of the apparent diameters of the sun and
moon,’ only one actual result is handed down; he found that the
diameter of the moon was contained about 650 times in the circle
described by it. This would no doubt be the mean of the different
observations of the moon at its varying distances; it is of course
equivalent to nearly οὐ 33° 14”. Ptolemy complains that the
requisite accuracy could not be secured by the dioptra; he there-
fore checked the observations as regards the moon by means of
‘certain lunar eclipses’, and found Hipparchus’s values appreciably
too high. Ptolemy ὅ himself made the apparent diameter of the
moon to be (a) at the time when it is furthest from the earth
οὗ 31’ 20”, and (δ) at its least distance οὗ 35’ 20”. The mean of
these figures being οὗ 33’ 20”, and the true values corresponding
to Ptolemy’s figures being 29’ 26” and 32’ 51”, it follows that
Hipparchus’s mean value is actually nearer the true mean value than
Ptolemy’s.® Aristarchus, as we shall see, took the apparent dia-
meters of the sun and moon to be thesame. Sosigenes (2ndc. A.D.)
showed that they are not always equal by adverting to the pheno-
menon of annular eclipses of the sun,’ and doubtless Hipparchus
had observed the differences; Ptolemy found that the apparent
diameter of the sun was approximately constant, whenever observed,
its value being the same as that of the moon when at its greatest
distance, not (‘as supposed by earlier astronomers’) when at its
mean distance. Another estimate of the apparent diameter of the
sun, namely ;2,th of the complete circle described by the sun, or
29’, is given by Cleomedes as having been obtained by means of
a water-clock ; he adds that the Egyptians are said to have been the
first to discover this method.* Yet another valuation appears in
? Theon, in Piolem. magn. construct. p. 262.
? Proclus, Hypotyposis, ed. Manitius, pp. 126. 13-128. 13.
3 Ptolemy, Syufazxis, loc. cit.
* Ptolemy, Synfazis, iv. 9, p. 327. 1-3, Heib.
® Ptolemy, v. 14, p. 421. 3-5; Pappus, ed. Hultsch, vi, p. 556. 17-19.
5 On the whole of this subject see Hultsch, ‘Winkelmessungen durch die
Hipparchische Dioptra’ in Abhandlungen zur Gesch. d. Math. ix (Cantor-
Festschrift), 1899, pp. 193-209.
7 Simplicius on De caelo, p. 505. 7-9, Heib.
® Ptolemy, Syntaxis, v. 14, p. 417. 3-11, Heib.
* Cleomedes, De motu circular, ii. 1, pp. 136-8, ed. Ziegler.
814 ARISTARCHUS OF SAMOS PART II
Martianus Capella;! the diameter of the moon is there estimated as
sooth of its orbit or 36’. This estimate was probably quoted from
Varro, and belongs to a period anterior to Hipparchus.?
The Year and the Great Year of Aristarchus.
We are told by Censorinus that Aristarchus added τ -τὰ of a
day to Callippus’s figure of 3654 days for the solar year,* and that
he gave 2,484 years as the length of the Great Year, or the period
after which the sun, the moon, and the five planets return to the
same position in the heavens. Tannery® shows that 2,484 years
is probably a mistake for 2,434 years, and he gives an explanation,
which seems convincing, of the way in which Aristarchus arrived
at his figures. They were doubtless derived from the Chaldaean
period of 223 lunations and the multiple of this by 3, which was
called ἐξελιγμός, a period defined by Geminus as the shortest time
containing a whole number of days, a whole number of months
(synodic), and a whole number of anomalistic months. The Greeks
were by Aristarchus’s time fully acquainted with these periods,
which were doubtless known through the Chaldaean Berosus,
who flourished about 280 B.C., in the time of Alexander the Great, .
and founded an astronomical school on the island of Cos opposite
Miletus. Ptolemy,’ too, says of the first of the two periods (which
he attributes to ‘ the ancients’, not the Chaldaeans specifically) that
it was estimated at 6,5854 days, containing 223 lunations, 239
‘restorations of anomaly’ (i.e. anomalistic months), 242 ‘restora-
tions of latitude’ (i.e. draconitic months, the draconitic month—a
term not used by Ptolemy—meaning the period after which the
moon returns to the same position with respect to the nodes), and
241 sidereal revolutions A/zs 102° which the sun describes in the time
in addition to 18 sidereal revolutions. The eve/igmus then, which
was three times this period, consisted of 19,756 days, containing
669 lunations, 717 anomalistic months, 726 draconitic months, and
1 Martianus Capella, De nuptits philologiae et Mercurit, viii. 860.
2 Tannery, Recherches sur [histoire de l’astronomie ancienne, Ὁ. 334.
8 Censorinus, De die natali, c. 19. 2. 4 Ibid., c. 18. 11.
5 Tannery, ‘La Grande Année d’Aristarque de Samos’ in M/ém. de la Soc.
des sciences phys. et naturelles de Bordeaux, 3° série, iv. 1888, pp. 79-96.
9. Geminus, /sagoge, c. 18, pp. 200 sqq., ed. Manitius.
? Ptolemy, Syvtaxis, iv. 2, pp. 269. 21-270, 18, Heib.
νυν να ΣΎ ΑΝ Ν
CH.I ARISTARCHUS OF SAMOS 315
723 sidereal revolutions A/zs 32° described by the sun in the period
over and above 54 sidereal revolutions.
It follows that the number of days in the sidereal year is
19756 _ 19756 _ 45-19756 _ 889020
54 τσ 54 τ ΖΞ 2434 2434
Now 4888 = 1623 -- 4. Thus, in replacing the complementary
ases bY τεῖςΞ Aristarchus followed the fashion of only admitting
fractions with unity as numerator, and thereby only neglected
the insignificant fraction agg3-gse5 OT τοσύτες᾿
It is clear that Aristarchus multiplied by 45 so as to avoid all
fractions, and so arrived at 889,020 days containing 2,434 sidereal
years, 30,105 lunations, 32,265 anomalistic months, 32,670 draconitic
months, and 32,539 sidereal months.
Tannery gives good reason for thinking that this evaluation of
the solar year at 3653 - ὃ days was really a sort of argument in
a circle and was therefore worthless. The Chaldaean period was
obtained from the observation of eclipses ; those which were similar
were classified, and it was recognized that they returned at the end
of a period estimated at 6,5854 days. Ifthe theory of the sun had
been sufficiently established, or if the difference of longitude between
the positions of two similar eclipses had been observed and allow-
ance made for the solar anomaly, it would have been possible to
evaluate with precision the number of degrees traversed during
the period by the sun over and above the whole number of its
revolutions. But this precision was beyond the powers of the
Chaldaeans, and the estimate of the excess of 102° was probably
obtained by means of the simple difference between 65854 days and
18 years of 365% days or 6,5744 days. This difference is 103 days, -
and, if this is turned into degrees by multiplying by 360/3653, we
have about 103 εἶτ΄ ; the complementary fraction 3, would then be
neglected as unimportant. Thus Aristarchus’s estimate of 3653 τοῖς
days was valueless, as the Chaldaean period itself depended on
a solar year of 365% days.
The question remaining is whether Aristarchus’s Great Year was
intended to be the period which brings all the five planets as well
as the sun and moon back again to the same places, as appears to
be implied by Censorinus, who mentions different estimates of the
= 305% + πεξε
316 ARISTARCHUS OF SAMOS
Great Year (including Aristarchus’s) just after an explanation that |
‘there is also a year which Aristotle calls the greatest rather than
the great year, which is completed by the sun, the moon, and the five
planets when they return together to the same sign in which they
were once before simultaneously found’. As Tannery observes, if
Aristarchus’s Great Year corresponded to an effective determination
of the periods of the revolutions of the planets, it would have
a particular interest because Aristarchus would have, in accordance
with his system, to treat the revolution of Mercury and Venus as
heliocentric, whereas in the earlier estimates of Great Years, e.g.
that of Oenopides, the revolution of these planets was geocentric
and of the same mean duration as that of the sun, so that they could
be left out of account. But, just as we were obliged to conclude
that Oenopides could not have maintained that his Great Year of
59 years contained a whole number of the periods of revolution of
the several planets, so it seems clear that Aristarchus could hardly
have maintained that his Great Year exactly covered an integral
number of the periods of revolution of the five planets. For suppose
that his Great Year of 889,020 days is divided by the respective
periods of their sidereal revolutions, and that we take the nearest Ὁ
whole numbers to the quotients—say 10,106 for Mercury, 3,950 for
Venus, 1,294 for Mars, 206 for Jupiter, 83 for Saturn—the errors
as regards the positions at the end of the period would amount,
according to Tannery’s calculation, to 133° for Saturn, 7o° for
Jupiter, 25° for Mars, 171° for Venus, and 11° for Mercury. This
being so, it is difficult to believe that the period of Aristarchus is
anything more than a luni-solar cycle.!
? Tannery, loc. cit., pp. 93-4.
a
II
ARISTARCHUS ON THE SIZES AND DISTANCES OF
THE SUN AND MOON
HISTORY OF THE TEXT AND EDITIONS.
AT the beginning of Book VI of his Syxagoge, Pappus refers to
want of judgement (as to what to include and what to omit) on the
part of ‘many of those who teach the Treasury of Astronomy (τὸν
ἀστρονομούμενον τόπον). The marginal note of the contents of the
Book, written in the third hand in the oldest MS., says that it
contains solutions of difficulties ἐν τῷ μικρῷ ἀστρονομουμένῳ, which
words, with τόπῳ understood, indicate that the collection of trea-
tises referred to by Pappus was known as the ‘ Little Astronomy ’,
as we might say. The collection formed a sort of preliminary
course, introductory to what would presumably be regarded as the
* Great Astronomy ’, the Syz¢axzs of Ptolemy. From Pappus’s own
references in the course of Book VI we may infer that the Little
Astronomy certainly included the following books :
Autolycus, On the moving sphere (περὶ κινουμένης chaipas).
Euclid, Optics,
a Phaenomena.
Theodosius, SAhaerica,
“ Ou days and nights.
Aristarchus, Oz the sizes and distances of the sun and moon.
No doubt Autolycus’s other treatise, Ou rzsings and settings,
Theodosius’s Ox habitations, and Hypsicles’ ἀναφορικός (De ascen-
stontbus) were also included ; they duly appear in MSS. containing
the whole collection. All these treatises are extant in Greek as
well as in Arabic. Not so another important work, the Sphaerica
1 Heiberg, Literargeschichtliche Studien tiber Euklid, 1882, p. 152.
98:18 TREATISE ON SIZES AND DISTANCES parti
of Menelaus, which has only survived in the Arabic and in transla-
tions therefrom, but seems to have belonged to the collection, since
Pappus gives four propositions found in Menelaus;! this treatise
was important for the study of the Syzfaxzis, as is proved by the
fact that Ptolemy takes for granted certain propositions of
Menelaus.? :
As regards some of these treatises it is certain that they were by
no means the first or the only works dealing with the same subjects.
Thus Euclid’s Phaenomena is closely akin to Autolycus’s Ox the
moving sphere, and both assume as well known a number of
propositions which are found in Theodosius’s Sphaerzca, a work
much later in date.* It is certain therefore that before the date of
Autolycus (latter half of fourth century B.C.) there was in existence
a body of sphaeric geometry; and indeed it would appear to have
contained fully half of the propositions subsequently incorporated
in Theodosius’s Sphaerica. This early sphaeric may have origin-
ated with Eudoxus and his school or may have been older still.
Its object was purely astronomical ; it did not deal with the geometry
of the sphere as such, still less did it contain anything of the nature
of spherical trigonometry (this deficiency was afterwards made good
by Menelaus’s SAhaerzca) ; it was designed expressly for such pur-
poses as fixing the sequence of the times of rising and setting of
different heavenly bodies, comparing the durations of the risings
and settings of particular constellations, comparing the apparent
speeds of the motion of the heavenly bodies at different points
in their daily revolution, and so on. Perhaps it may best be
1 A. A. Bjérnbo, Studien tiber Menelaos’ Spharik. Beitrige zur Geschichte
der Spharik und Trigonometrie der Griechen (in Abhandlungen zur Geschichte
der mathematischen Wissenschaften, Heft xiv, 1902), pp. 4, 51, 55-
2 Bjérnbo, op. cit., p. 51.
$ On the question of Theodosius’s date we know little except that he was
before Menelaus’s time. Menelaus made observations in the first year of
Trajan’s reign (A.D. 98); and Theodosius, probably of Bithynia, lived before our
era. Vitruvius (first century B.C.) mentions (ix. 8) a Theodosius who invented
a sun-dial for all climates, and he may have been contemporary with Hipparchus
or a little earlier (Tannery, Recherches sur l’histoire de l’astronomie ancienne,
ῬΡ. 36, 37; Bjérnbo, op. cit., pp. 64, 65).
* The sort of thing may be illustrated by the following enunciations of
propositions :
Autolycus, On the moving sphere, 9. ‘If in a sphere a great circle oblique to
the axis defines the visible and the invisible (halves) of the sphere [the great
circle is of course the horizon], then of those points which rise at the same time
ee ΆΒΨΗ
CH. II HISTORY OF TEXT AND EDITIONS 319
described as the theoretical equivalent of a material sphere or
combination of spheres (such as are said to have been constructed
by many astronomers from Anaximander onwards) which should
exactly simulate the motions of the heavenly bodies and yisualize
the order, &c., of the phenomena as they occur. The special
necessity for theoretical works of this kind was of course due to the
obliquity, with reference to the circle of the equator, of (1) the
horizon at any point of the earth’s surface, and (2) the plane of
the ecliptic in which the independent motions of the sun, moon,
and planets were supposed to take place.
We may assume that this mathematical side of astronomy began
to be studied very early. We know that Oenopides studied certain
geometrical propositions with a view to their application to astro-
nomy; and, whether he brought his knowledge of the zodiac and its
twelve signs from Egypt or not, he was apparently the first to state
the theory of the oblique movement of the sun. The application of
mathematics to astronomy may therefore have begun with Oeno-
pides; but it had evidently made progress by the time of Archytas,
Eudoxus’s teacher, for Archytas expresses himself, at the beginning
of a work On Mathematics, thus:
‘The mathematicians seem to me to have arrived at correct con-
clusions, and it is not therefore surprising that they have a true
conception of the nature of each individual thing; for, having
reached such correct conclusions as regards the nature of the whole
universe, they were bound to see in its true light the nature of
those which are nearer the visible pole set later, and of those which set at the
same time those which are nearer to the visible pole rise earlier.’
Euclid, Phaenomena, 8. ‘The signs of the zodiac rise and set in unequal -
segments of the horizon, those on the equator in the greatest, those next to them
in the next smaller, those on the tropic circles in the smallest, and those equi-
distant from the equator in equal segments.’
Theodosius, Sphaerica, iii. 6. ‘If the pole of the parallel circles be on the cir-
cumference of a great circle and this great circle be cut at right angles by two
great circles, one of which is one of the parallel circles [i. e. the equator], while the
other is oblique to the parallel circles [say the ecliptic]; if then from the oblique
circle equal arcs be cut off adjacent to one another and on the same side of the
greatest of the parallel circles [the equator]; and if through the points so deter-
mined and the pole great circles be drawn; the arcs which they will intercept
between them on the greatest of the parallel circles will be unequal, and the
intercept which is nearer to the original great circle will always be greater than
that which is more remote from it.’
+ Tannery, op. cit., p. 33.
39. TREATISE ON SIZES AND DISTANCES parti
particular things as well. Thus they have handed down to us clear
knowledge about the speed of the stars, their risings and settings,
and about geometry, arithmetic, and sAhaerzc, and last, not least,
about music; for all these branches of knowledge seem to be
sisters.’ ?
We must suppose, then, that Theodosius’s compilation (long-
winded and dull as it is) simply superseded the earlier text-books
on Sphaeric, which accordingly fell into disuse and so were lost,
just as the same fate befell the works of the great Hipparchus as the
result of their being superseded by Ptolemy’s Syz¢axzs.
Why Euclid’s Ofzics was included in the Little Astronomy is
not clear. It was a sort of elementary theory of perspective and
may have been intended to fore-arm students against the propoun-
ders of paradoxes such as that of the Epicureans, who alleged that
the heavenly bodies must δέ of the size which they affear ; it would
also serve to justify the assumption of circular movement on the
part of the stars about the earth as centre.®
It was a fortunate circumstance that Aristarchus’s treatise found
a place in the collection; for presumably we owe it to this fact that
the work has survived, while so many more have perished.
Whether Aristarchus had any predecessors in the mathematical
calculation of relative sizes and distances cannot be stated for
certain. We hear, indeed, of a book by Philippus of Opus (the
editor of the Zaws of Plato and the author of the EAznomzs)
entitled Ox the size of the sun, the moon, and the earth, which is
mentioned by Suidas directly after another work, Oz the eclipse of
the moon, attributed to the same author; but we know nothing of
the contents of these treatises.
Like the other books included in the Little Astronomy, our
treatise passed to Arabia and took its place among the Arabian
‘middle’ or ‘intermediate books’, as they were called. It was
translated into Arabic by Qusta Ὁ. Liga al-Ba‘labakki (died about
912), who was also the translator of the three works of Theodosius,
1 In connexion with the remark that the mathematicians had investigated the
speed of the stars, it is perhaps worth while to recall that Eudoxus’s great theory
of concentric spheres was set out in a book Ox Speeds, περὶ ταχῶν (Simplicius on
De caelo, p. 494. 12, Heib.).
® Porphyry, /z Plolem. Harm., p. 236; Nicomachus, /utrod. Arithm. i. 3. 4;
pp. 6.17 --7. 2; Vorsokratiker, 15, p. 258. 4-12.
8 Tannery, op. cit., p. 36.
CH. II HISTORY OF TEXT AND EDITIONS 321
Autolycus’s On risings and settings, and Hypsicles’ Avagopixés."
A recension of it, as of all the books contained in the Little
Astronomy, including the Spiaerzca of Menelaus, which had been
translated by Ishaq Ὁ. Hunain, was made by Nesiraddin at-Tasi,’
famous as the editor of Euclid and for an attempt to prove Euclid’s
Parallel-Postulate. There are MSS. of this collection, including
of course Aristarchus, in the India Office (743, 744) and in the
Bodleian Library (Nicoll and Pusey, i. 875, i. 895, and ii. 279).
The first published edition of Aristarchus’s treatise was a Latin
translation by George Valla, included in a volume which appeared
first in 1488 (‘per Anton. de Strata’) and again in 1498 (‘per
Simonen Papiensem dictum Bevilaquam’).®
It next appeared in a Latin translation by that admirable and
indefatigable translator Commandinus, under the title:
Aristarchi de magnitudinibus et distantits solis et lunae liber
cum Pappi Alexandrini explicationibus guibusdam a Federico
Commandino Urbinate 272 datinum conversus et commentarits
wdlustratus, Pisauri, 1572.
Commandinus complains of the state of the text, which made the
task of translation difficult, but he does not mention Valla’s earlier
translation and was presumably not acquainted with it.
The honour of bringing out the edztio princeps of the Greek
text belongs to John Wallis. The title-page is as follows:
Ὁ Suter, Die Mathematiker und Astronomen der Araber und ihre Werke
(Abh. zur Gesch. a. math. Wissenschaften, x. Heft, 1900), p. 41.
3. Suter, p. 152.
5. Fabricius, Bibliotheca Graeca, iv. 19, Harles.
1410 V
ΑΡΙΣΤΑΡΧΟΥ ΣΑΜΙΟΥ͂
Πριεὶ μεγεθῶν καὶ Sogn HAY καὶ Dealwns,
BiB A TO
ΠΑΠΠΟΥ AAEZANAPEOQS
Τῇ ὁ Σιωαγωγῖς BIBAIOY B
Απύσπασμωα.
ARISTARCHI SAMII
De Magnitudinibus & Diftantiis Solis ὅς Lunz,
ob oR:
Nunc primum Grace editus cum Federici Com-
mandini verfione Latina, notifq; ilius 8 Editors.
PAPPI ALEX ANDRINI
SECUND1 LIBR1
MATHEMATICH COLLECTIONIS,
Fragmentum,
Hactenus Defideratum.
E Codie MS. edidit, Latinum fecit,
Notifgue illuftravit
JOHANNES WALLIS, §.'T. Ὁ. Geometriz
Profeffor Savilianus ; ὅς Regalis Societatis
Londini , Sodalis.
OXONTLEA,
E THEATRO SHELDONIANQO,
1688.
ee νιν: .. .
ee ee τὰ
CH. II HISTORY OF TEXT AND EDITIONS 323
-The book was reprinted in the collected edition of Johannzs
Wallis Opera Mathematica, 1693-1699, vol. iii, pp. 565-94.
Wallis states in his Preface that he used for the preparation of his
text (1) a Greek MS. (which he calls B) belonging to Edward
Bernard, Savilian Professor of Astronomy, who had copied it from
the Savile MS., and (2) the Savile MS. itself (S). The Savile MS.
was copied by Sir Henry Savile himself from another (presumed
by Wallis to have been one of the Vatican MSS.), and had (as
appeared from notes in the margin) been collated with a second
MS. vaguely described as Codex Vetus. Wallis preferred Com-
mandinus’s translation to Valla’s, and retained the former version
intact because it agreed so closely with the Greek MSS. of
Savile and Bernard that it seemed to have a common source with
them; Wallis also incorporated Commandinus’s notes along with
his own.
Wallis adds that there are two Selden MSS. in the Bodleian
Library containing Aristarchus’s treatise in Arabic, and that Bernard
had noted in the margin of his MS. (B) anything in the Arabic
version which seemed of moment, as well as some things from
Valla’s translation.
In 1810 there appeared the edition by the Comte de Fortia
d’Urban,
Histoire d@Aristarqgue de Samos, sutvte de la traduction de son
ouvrage sur les distances du Soleil et de la Lune, de l'histoire
de ceux gut ont porté le nom d’Aristarque avant A ristarque
de Samos, et le commencement de celle des Philosophes gut
ont paru avant ce méme Aristargue. Par M. de F* * * *,
Paris, 1810.
There follows, as a separate title-page for the work of Aristarchus,
Ἀριστάρχου περὶ μεγεθῶν καὶ ἀποστημάτων ἡλίου καὶ σελήνης,
followed by the Latin equivalent. Pages 2-87 contain the Greek
text along with Commandinus’s Latin translation (altered in places).
On p. 88 is a note referring to the MSS. used by the editor in pre-
paring the Greek text of the treatise and the scholia. The scholia
in Greek and Latin occupy pages 89-199, and are followed by the
critical notes, which extend from p. 201 to p. 248. Particulars of
the MSS. used will be found in a later paragraph.
Y2
324 TREATISE ON SIZES AND DISTANCES ῬΑΒΤῚΣ
This Greek text of Fortia d’Urban was issued prematurely and
without any diagrams ; an explanation on the subject is contained
in the editor's preface to his French translation published thirteen
years later,
Traité ad’ Aristarque de Samos sur les grandeurs et les distances
du Soleil et de la Lune, tradutt en francats pour la premiére
Sots, Bar M. le Comte de Fortia d’' Urban. Paris, 1823.
The Preface to this translation, with the omission of an explana-
tion of the lettering in the figures (which is double, to correspond
to the Greek text and the Latin and French translations), runs as
follows : .
‘Le texte de l’ouvrage d’Aristarque de Samos, que j'avais revu
sur huit manuscrits de la bibliothéque du Roi, et que j’avais fait
imprimer en France ou il n’avait point encore été publie, avec des
scholies absolument inédits, ayant été mis en vente sans mon
autorisation, a paru d'une maniére presque ridicule. On y trouve
citées, a toutes les pages, des planches que j’avais fait graver, mais
que des circonstances facheuses ont fait disparaitre pendant mon
séjour en Italie. Je vais tacher ἀν suppléer par la publication de
cette traduction qui sera accompagnée de nouvelles planches ou
j'ai fait graver les lettres grecques pour ceux qui voudront joindre
cette traduction δὰ texte .. . Je donnerai d’abord l’ouvrage d’Aris-
tarque de Samos, tel qu'il nous est parvenu; je traduirai ensuite les
scholies, suivant ainsi l’ordre observé pour l'impression du texte
grec. J’avertis que les démonstrations d’Aristarque s’appuient sur
la Géométrie d’Euclides, qu’il suppose connue de ses lecteurs.
Paris, 2 avril 1823.’
The French translation is a meritorious and useful book.
There is yet another Greek text, besides those of Wallis and
Fortia d’'Urban, namely i
Δριστάρχου Σαμίου βιβλίον περὶ μεγεθῶν καὶ ἀποστημάτων ἡλίου
καὶ σελήνης, mit kritischen Berichtigungen von ΚΕ. Nizze.
Stralsund, 1856.
This text is, however, untrustworthy, not having been prepared
with sufficient care. It was based on the texts of Wallis and Fortia
d’Urban without, apparently, any recourse to MSS.
A German translation also exists,
Artstarchus tiber die Groéssen und Entfernungen der Sonne
und des Mondes, tibersetzt und erliéutert von A. Nokk.
Freiburg i. B., 1854.
er
ΠΤ Ὺ
cH. HISTORY OF TEXT AND EDITIONS 325
We come now to the MS. authority for our Greek texts. It
would appear! that our treatise is included in five MSS. in the
Vatican, namely Vat. Gr. 204 (1oth cent.), 191 and 203 (13th cent.),
192 (14th cent.), and 202 (14th-15th cent.), and in eight at Paris,
namely Paris. Gr. 2342 (14th cent.), 2363 (15th cent.), 2364, 2366
(16th cent.), 2386 (16th cent.), 2472 (14th cent.), 2488 (16th cent.),
and Suppl. Gr. 12 (16th cent.). There are others at Venice, Mar-
cian. 301 and 304 (15th cent.); at Milan, Ambros. A rot sup. (14th
cent.); at Vienna, Vindobon. Suppl. Gr. 9 (17th cent.); and so on.
The oldest of all these MSS. and by far the best is the beautiful
Vaticanus Graecus 204, of the roth century; indeed it seems to be
the ultimate source of all the others, and so much superior that the
others can practically be left out of account. This great MS. is
described by Menge.? Its contents are: fol. 1-36* Theodosius,
Sphaerica, i, ii, iti; 377-42%, Autolycus, Ox the moving sphere;
42°-58", Prolegomena to Eucld’s Optics (τὰ πρὸ τῶν Εὐκλείδου
ὀπτικῶν) ; 5 58'-76", Euclid’s Phaenomena ; 76°-82*, Theodosius, Ox
habitations ; 83-95", Theodosius, On nights and days; 957-108",
Theodosius, Oz days and nights, ii; 108*-117*, Aristarchus, Ox the
stzes and distances of the sun and moon; 118'-132", Autolycus, Ox
risings and settings, i, ii; 132°-134", Hypsicles, Avagopixés ; 1357-
143’, Euclid’s Cafoptrica ; 1447, figures to the Cafopirica; 144” blank;
1457-172", Eutocius, Commentary on Books I-III of Apollontus’s
Conics; 172*-194*, Euclid’s Data; 195"-197", Marinus, Commen-
tary on Euclia’s Data; 198'-205", Scholia to Euclid’s Elements.
The MS. is of parchment, incomplete at the end, and the 206
leaves are preceded by three more, the first of which is empty, the
second has a πίναξ, and the third, a sheet of paper fastened in later,
contains a Latin index. The first two leaves, containing the begin-.
ning of Theodosius’s Sphaerica, are written byalater hand who
1 | have collected these particulars, except as regards three of the MSS. used
by Fortia d’Urban, from the introductions to Heiberg’s editions of Euclid and
Apollonius in Greek, the same scholar’s Literargeschichtliche Studien iiber
Exuklid, 1882, and Om Scholierne til Euklids Elementer, 1888, and from the
introductions to one or two other Greek mathematical texts.
3 Addendum to a review of Hultsch’s Autolycus, Neue Jahrbiicher fiir
Philologie, 1886, pp. 183, 184.
3 Fol. 42’-58" contain Theon’s recension of Euclid’s Optics, with a preface
which was apparently written by some pupil of Theon’s. It is to this preface
that the title refers.
326 TREATISE ON SIZES AND DISTANCES parti
has cleverly imitated the handwriting of the rest of the MS., which
is by one hand. The figures, drawn in red, are clear and adequate.!
Many things in the text are struck out, erased, and changed by
different hands. The MS. is rich in old and new scholia. It has
on it the stamp of the Bibliotheque Nationale, having been, like the
famous Peyrard MS. of Euclid (Vat. Gr. 190), among the MSS.
which were taken to Paris in 1808 and restored to the Vatican after
the Congress of Vienna. ε
In settling a text to translate from, I have mainly relied on a
photograph of Vat. Gr. 204 together with Wallis’s text, though
I have had Nizze’s text by me and have also consulted Fortia
d’Urban’s edition of 1810. The occasional references to the
Paris MSS. in my critical notes are taken from Fortia d’Urban.?
It is not clear from which of the Vatican MSS. Savile copied his
own (Wallis’s S); it cannot, however, have been Vat. Gr. 204, because
(a) nearly all the words and sentences which Wallis supplied, on the
basis of Commandinus's translation, in order to fill up gaps in his
two MSS., are actually found (either exactly or with no more
variation than would naturally be expected between a re-translation
into Greek and the original Greek text) in 204, and (4) a scholium |
from S added by Wallis at the end of Prop. 7 does not appear in
204. Fortia d’Urban suggests, as a possibility, that the MS. of
which Wallis had a copy was Paris. 2366, but it seems clear that it
cannot have been any of the Paris MSS., and therefore was pre-
sumably (as Wallis thought) one of those in the Vatican.* There
1 The words used by Menge are ‘klar und genau’, but I think the figures can
hardly be called ‘ accurate’ or ‘exact’.
2 In Fortia d’Urban’s critical notes there are several references to the
reading of a MS. which he quotes as 2483. But Paris. Gr. 2483 is not included
in his list of the MSS. of Aristarchus used by him; and it appears to contain,
not Aristarchus, but Nicomachus’s /utroductio arithmetica with scholia (Omont,
Inventaire sommaire des manuscrits grecs de la Bibliotheque Nationale, ii).
It would seem, from internal evidence, that the references should be to Paris.
Gr. 2472, not 2483, in these cases.
3. Fortia d’Urban observes that Paris. 2366 alone omits a sentence in Prop. 1
(πολλῷ ἄρα ἡ BY τῆς BA ἐλάσσων ἐστὶν ἢ pe’ μέρος) which Wallis likewise omits,
whereas Paris. 2342, 2364, 2488 and Commandinus all have it; hence he thinks
that Wallis’s MS. may have been a copy of Paris. 2366. But, on the other hand,
a sentence in Prop. 13 which is absent from Wallis’s text (καὶ ἡ ΒΝ ἐφάπτεται...
λαμπρόν) is, according to Fortia d’Urban, found in all the Paris MSS. except
2342; presumably therefore Paris. 2366 has it. These two cases create a strong
presumption that Wallis’s MS. was not a copy of any of the Paris MSS.
CH. II HISTORY OF TEXT AND EDITIONS 327
is apparently no clue to the identity of the ‘Codex Vetus’ with
which S was collated.
We are better informed as to the MSS. used by Fortia d’Urban.
He tells us, in the note on p. 88 of the edition of 1810, that they
were Codd. Paris. 2342, 2363, 2364, 2366, 2386, 2472, and 2488,
and one Vatican MS. The particular Vatican MS. had, he observes,
just been brought to Paris; it was therefore presumably Vat. Gr.
204. He does not, however, seem to have collated the latter MS.
with sufficient care; for example, he says that some words! in the
‘setting-out οὗ Prop. 3 and a whole sentence? occurring later in
the proposition are wanting in the MS., though, as a matter of fact,
they are there in full; when, therefore, on the occasion of the first
of these supposed omissions, he says that the Vatican MS. does not
seem to him in any way superior to ‘our own’, he is apparently
allowing his patriotism to get the better of his judgement. For the
scholia he says that he relied mostly upon Paris. 2342 and 2488 ;
but the scholia in Vat. Gr. 204 seem to correspond exactly. He
does not seem to have found in any of his eight MSS. the particular
scholium to Prop. 7 taken by Wallis from S; for, while he gives it
in his French translation, he says it comes, through Wallis, from S.
1 σελήνης δὲ κέντρον, ὅταν 6 περιλαμβάνων κῶνος
3 καὶ διελόντι, ὡς ἡ BI πρὸς τὴν IA, οὕτως ἡ BA πρὸς τὴν AO.
III
CONTENT OF THE TREATISE
THE style of Aristarchus is thoroughly classical, as befits an able
geometer intermediate in date between Euclid and Archimedes,
and his demonstrations are worked out with the same rigour as
those of his predecessor and successor. The propositions of
Euclid’s Evements are, of course, taken for granted, but other things
are tacitly assumed which go beyond what we find in Euclid.
Thus the transformations of ratios defined in Eucl, V and indicated
by the terms zzversely, alternately, componendo, convertendo, &c.,
are regularly and naturally used in dealing with wmeguad ratios,
whereas in Euclid they are only used in proportions, i.e. cases of
equality of ratios. But the propositions of Aristarchus are also of
particular mathematical interest because the ratios of the sizes and
distances which have to be calculated are really “rigonometrical
ratios, sines, cosines, &c., although at the time of Aristarchus trigono-
metry had not been invented, while no reasonably close approxima-
tion to the value of 7, the ratio of the circumference of a circle to its
diameter, had been made-(it was Archimedes who first obtained the
value 22/7). Exact calculation of the trigonometrical ratios being
therefore impossible for Aristarchus, he set himself to find upper
and lower limits for them, and he succeeded in locating those which
emerge in his propositions within tolerably narrow limits, though
not always the narrowest within which it would have been possible,
even for him, to confine them.’ In this species of approximation to
trigonometry he tacitly assumes propositions comparing the ratio
between a greater and a less amg/e in a figure with the ratio
between two straight lines in the figure, propositions which are
CONTENT OF THE TREATISE 329
formally proved by Ptolemy at the beginning of his Synfazxzs.
Here, again, we have a proof that text-books containing such
propositions existed before Aristarchus’s time, and probably much
earlier, although they have not survived.
The formal assumptions of Artstarchus and
their effect.
One of the assumptions or hypotheses at the beginning of the
treatise, the grossly excessive estimate of 2° for the apparent
angular diameter of the moon, has already been discussed (pp. 311,
312 above). We proceed to Hypotheses 4 and 5, giving values
for a certain ratio and a certain other angle respectively.
In Hypothesis 5, Aristarchus takes the diameter of the earth's
shadow (at the place where the moon passes through it at the time
of an eclipse) to be twice that of the moon. The figure 2 for this
ratio was presumably based on the observed length of the longest
eclipses on record.!_ Hipparchus, as we learn from Ptolemy,’ made
the ratio 24 for the time when the moon is at its mean distance in
the conjunctions; Ptolemy chose the time when the moon is at its
greatest distance, and made the ratio insensibly less than 23 (a
little too large).$
Tannery * shows in an interesting way the connexion between
(1) the estimate (Hypothesis 4) that the angular distance between
the sun and moon viewed from the earth at the time when the
moon appears halved is 87°, the complement of 3°, (2) the estimate
(Hypothesis 5) of 2 for the ratio of the diameter of the earth’s
shadow to that of the moon, and (3) the ratio (greater than 18 to 1
and less than 20 to 1) of the diameter of the sun to the diameter of
the moon as obtained in Props. 7 and 9 of our treatise.
The diagram overleaf (Fig. 14) will serve to indicate very roughly
the relative positions of the sun, the earth, and the moon at the
moment during a lunar eclipse when the moon is in the middle of
the earth’s shadow.
? Tannery, Recherches sur histoire de l’astronomie ancienne, p. 225.
53 Ptolemy, Syntaxis, iv. 9, p. 327. 3-4, Heib.
3. Ibid., v. 14, p. 421. 12-13.
* Tannery in Mémoires de la Société des sciences physiques et naturelles de
Bordeaux, 2° série, v, 1883, pp. 241-3 ; Mémoires scientifiques, ed. Heiberg and
Zeuthen, i, 1912, pp. 376-9.
330 TREATISE ON SIZES AND DISTANCES ΡΑΒΤῚΙ
SJ
Sun Earth eee See
Fig. 14.
Let .S be the radius of the sun’s orbit,
‘B : : ς moon’s orbit,
s the radius of the sun,
Z ᾿ ; : moon,
ee : earth,
. D the distance from the centre of the earth to the vertex of
the cone of the earth’s shadow,
and. d the radius of the earth’s shadow at the distance of the moon.
Then we have, approximately, by similar triangles,
a a 3 a D-L,
ag a ον a 2
ς ΤΣ ad :
whence, if we suppose that $= 5: and put z= 7» we easily
; Beare,
derive ga ERE ΣΝ : ae ate 6)
and Pe a an Ξ ;
yaaa BAY ΤΩ 7 Ὡς BM (2)
nt cd Σ
Now, since eclipses of the sun occur through the interposition of
the moon, S > Z,so that 5.» 2 The ancients knew, too, that the
sun is larger than the earth, so thats >z. It follows from (1) that
Ζ ,
7} 5, 80 that the moon is smaller than the earth.
Now suppose ὃ to be the angle subtended at the centre of the sun
by the distance between the moon and the earth at the time when
the moon appears halved, i.e. when the earth, sun, and moon form
CH. ΠῚ CONTENT OF THE TREATISE 331
a right-angled triangle with its right angle at the centre of the
moon.
δου ἃ I
eee aL ane
We have then from (1), substituting s/ for ὦ,
ic ΟΣ S 2X+1
-- —— I, OF -= ,
mies WSs i
and, substituting 4 for s, we have
Fig. 15.
: ee ‘ :
Now if x (=<) is taken at 19, Aristarchus’s mean value, and
Z
u = 2, these formulae give
7 = 19, = τ᾿ = 6-6, : = 2-85, é= sin = c= >. ie ai
Tannery’s object is to prove that the method of our treatise was
not invented by Aristarchus but by Eudoxus. We know in the
first place, from Aristotle, that by the middle of the fourth century
mathematical speculations on the sizes and distances of the sun and
moon had already begun. Aristotle’ says:
‘ Besides, if the facts as shown in the theorems of astronomy are ©
correct, and the size of the sun is greater than that of the earth,
while the distance of the stars from the earth is many times greater
than the distance of the sun, just as the distance of the sun from the
earth is many times greater than that of the moon, the cone marking
the convergence of the sun’s rays (after passing the earth) will have
its vertex not far from the earth, and the earth’s shadow, which we
call night, will therefore not reach the stars, but all the stars will
necessarily be in the view of the sun, and none of them will be
blocked out by the earth.’
1 Arist. Mefeorologica, i. 8, 345 Ὁ 1-9.
332 TREATISE ON SIZES AND DISTANCES parti
Now Eudoxus was the first person to develop scientifically the
hypothesis that the sun and moon remain at a constant distance
from the earth respectively, and this is the hypothesis of Aristar-
chus. Further, we are told by Archimedes that Eudoxus had
estimated the ratio of the sun’s diameter to that of the moon at
9:1, Phidias, Archimedes’ father, at 12:1, and Aristarchus at a
figure between 18:1 and 20:1. Accordingly, on the assumption
that Eudoxus and Phidias took # = 2 in the above formulae, as
Aristarchus did, we can make out the following table :
5 Ss t ὃ
a Ζ 7 (calculated value)
Eudoxus 9 4:8 2:7 6° 22’ 46”
Phidias 12 4:2 2:76923 4° 46’ 49”
Aristarchus | 19 6-6 2:85 ΠΝ Ue τὰ
(mean)
Hence, says Tannery, while Aristarchus took 3° as the value of
6, Eudoxus probably took 6° or } of a sign of the zodiac, and
Phidias 5° or 4 ofasign. ‘Icannot believe that these values were
deduced from direct observations of the angular distance. The
necessary instruments were in all probability not in existence in the
fourth century. But Eudoxus could, on the day of the dichotomy,
mark the positions of the sun and the moon in the zodiac, and try
to observe at what hour the dichotomy took place. The evaluations
involve an error of about twelve hours for Eudoxus, ten for Phidias,
and six for Aristarchus. It seems that all of them sought upper
limits for 6. It will be noticed that the value of ὃ especially affects
the values of the ratios s//, s/f; the ratio 7//on the contrary depends
mostly on the value of z.’1 Seeing, however, that the only figures
in the above tables which are actually attested are the three in the ©
first column, the 3° of Aristarchus, and the results obtained by
Aristarchus on the basis of his assumptions, it seems a highly
speculative hypothesis to suppose that Eudoxus started with 6°,
and Phidias with 5°, as Aristarchus did with 3°, and then deduced
the ratio of the diameter of the sun to that of the moon by precisely
Aristarchus’s method.
* Tannery, Mémoires de la Société des sciences phys. et nat. de Bordeaux,
2° série, v, 1883, pp. 243-4; Mémoires scientifiques, ed. Heiberg and Zeuthen,
1, Ρ. 379.
Στ τ δ,
δε λόδν, ....:...
CH. ΠῚ CONTENT OF THE TREATISE 333
Trigonometrical equivalents.
Besides the formal Assumptions laid down at the beginning of
the treatise, there lie at the root of Aristarchus’s reasoning certain
propositions assumed without proof, presumably because they were
generally known to mathematicians of the day. The most general
of these propositions are the equivalent of the statements that—
If « is what we call the circular measure of an angle, and a is
less than ἔπ, then
(1) The ratio sina/x decreases as « increases from Ὁ to ἔπ,
but (2) the ratio tan οὐχ zzcreases as a increases from 0 to ἔπ.
Tannery! took pains to set out the trigonometrical equivalents
of the particular results obtained by Aristarchus in the several
propositions.
If we bear in mind that
. 7 T
Sin- =tan-=I,
2 +
sin? = 4,
ae I
8 W241’
and if we substitute for ν΄ 2 the approximate value 3 which is
assumed by Aristarchus, we can deduce the following inequalities :
(1) If #2 > (en's,
2m
m
- τ 722-τ
or (2) cos ——= sin (3. —)> :
2m 2 2m 772
* π 2
(3) If me > 2,sin "_ ««δἡ-“.- «-,
2772 2m mt
If # > 3,sin 7 >
(4) 3» a ee
τ
(5) Mine age χὴν; LW es:
2772 2m 55:
Ὁ Tannery, Mémoires de la Soc. des sciences phys. ct nat. de Bor deaux, 2° série,
v, 1883, pp. 244 sq.; Mémoires scientifiques, i, pp. 380 sqq.
334 TREATISE ON SIZES AND DISTANCES partu
. . . Tv .
The narrowest limits for sin τ obtained by means of these
inequalities are
(6) oS Gin ».5. :
3m 202° 2m
whereas, if Aristarchus had known the approximate value 2? for 7,
he could have obtained the closer upper limit
Now, for example, in Prop. 7, Aristarchus has to find limits for
sin 3°, that is to say sin zi thus # = 30, and the formula (6)
above gives his result
I ne gical
13 > sin3z > art
In Prop. 4 Aristarchus proves the negligibility of the maximum
angle (ε) subtended at the centre of the earth by a certain arc (a) on
the surface of the moon subtended at the centre of the moon by an
angle equal to half the apparent angular diameter of the moon.
From the figure of the proposition it is easy to see that, taking the
radius of the moon to be unity,
. . fo
sin & sin -
tan
. α
Ι - 81 αἱ COS 2
For, if 17 be the foot of the perpendicular from H on 4B,
he
ise HM HM cs BH sin >
a a ee ἀν Be
BD sin > sin & sin ®
2 2
a 3 α
4" -»» cos = I—sin ἃ cos >
ΟΣ ΜΕΤ. 4
This would give, for α = 1°, ε = 0° 1’ 3”.
CH. ΠῚ CONTENT OF THE TREATISE 335
What Aristarchus in fact does is to prove that
ὡ» ἡ,
ΡΠ, π΄ BG ἘΠῚ ce fa.
fe oe es. ΤΑ isin x
Now, if « = 2/2 (m> 4), formula (5) above would give
ee 5. and if z= 90 se ees
ΐ- 3m@—5 ᾿ "© — 4770
but Aristarchus is content with the equivalent of using formula (3)
which gives
exo 18:
Se ta Le Ὁ Ὁ γε οὗ» 22”
᾿. Ὡ μδ ἃ 3960" ς
In Prop. 11 Aristarchus uses the equivalent of formulae (3) and
(4), proving that
— >sin 1°> >
45 60
Prop. 12 is the equivalent of using formula (2) to prove that
o. 89
Le COS) > -ξος
go
From formula (2) we deduce
cos? "_ >
2m m* m
and, for #z= 90, this gives the equivalent of the first part of
Prop. 13, namely Ss bok tl
45
In Prop. 14 Aristarchus determines a lower limit for the ratio
L/c, where Z is the radius of the moon’s orbit and ς the distance of
the centre of the moon from the centre of the circle of the shadow
at the middle of an eclipse. The arithmetical value of the limit
depends of course on the particular assumptions which he makes
as to the angles subtended at the centre of the earth by the
diameter of the moon and by the diameter of the circle of the
shadow. If these angles be 2a, 2y respectively, we see from
the figure of Prop. 14 that
BR= BMcosxa=Lcos?*a, BS= Lcosacosy, RC=Lsin? a.
336 TREATISE ON SIZES AND DISTANCES part Il
Therefore SR: RC = Lcos« (cosa—cos y) : LZ sin? a,
and CR: CS = Lsin? «: L (sin? «+ cos? «—cos οἱ cos y)
= Lsin*a:Z (1—cosacos y).
Now BC:CR=(BC: CM) x (CM: CR)
= (1:sina) x (1:sin a)
er} Sint oO,
Therefore, ex aegualz,
BC: CS=L:L(1—cosa cos γ),
or L:c=1:(1—cosacos y)
= 1: (sin? «+cos? ~—cos acos y)
> 1: {sin?a+cos?a (I—cos y)}.
If y = 2.4, as assumed by Aristarchus, this becomes
L:e>(1:sin? a). {1:(1+2 008. a)}.
The corresponding inequality obtained by Aristarchus, who
assumes that a = 1°, is
LSB HAS ATP SS)
> 675: 1.
The generalized trigonometrical equivalent of Prop. 15 is more
complicated and need not be given here. Tannery has an inter-
esting remark, which was however anticipated by Fortia d’Urban,'
upon one of the arithmetical results obtained by Aristarchus in that
proposition. If y be the ratio of the sun’s radius to the earth’s
radius, his result is
a 5 75755875
y= ~ 61735500"
He replaces this value by a merely remarking that ‘ 71755875
has to 61735500 a ratio greater than that which 43 has to 37’. It
is difficult, says ee not to see in $3 the expression 1 +41
which suggests that $3 was obtained by -devetaates Ὁ} 755815 or
24384 as a continued fraction: ‘We have here an important proof
of pre employment by the ancients of a method of calculation, the —
theory of which unquestionably belongs to the moderns, but the
first applications of which are too simple not to have originated in
very remote times.’
Fortia d’Urban, Traité d’ Aristarque de Samos, 1823, p. 86, note.
IV
LATER IMPROVEMENTS ON ARISTARCHUS’S
CALCULATIONS
WHILE it would not be consistent with the plan of this work to
carry the history of Greek astronomy beyond Aristarchus, it will be
proper, I think, to conclude this introduction with a few particulars
of the improvements which later Greek astronomers made upon
Aristarchus’s estimates of sizes and distances.
We have already spoken of Aristarchus’s assumption of 87° as the
angle subtended at the centre of the earth by the line joining
the centres of the sun and moon at the time when the moon
appears halved. The true value of this angle is 89 50’, so
that Aristarchus’s estimate was decidedly inaccurate; no direct
estimate of the angle seems to have been made by his successors.
Aristarchus himself, as we have seen, afterwards corrected ἰοὸ 2“ the -
estimate of 2° for the apparent angular diameter of the sun and
moon alike. His assumption of 2 as the ratio of the diameter of
the circle of the earth’s shadow to the diameter of the moon was
‘improved upon ἣν Hipparchus and Ptolemy. Hipparchus made
it 24 at the moon’s mean distance at the conjunctions; Ptolemy
made it atthe moon's greatest distance ‘ inappreciably less than 23’.*
Coming now to estimates of absolute and relative sizes and
distances, we find some data in Archimedes ;* according to him
Eudoxus had declared the diameter of the sun to be nine times the
diameter of the moon, and Phidias (Archimedes’ father) twelve times; _
most astronomers, he says, agreed that the earth is greater than the
moon, and ‘ some have tried to prove that the circumference of the
earth is about 300,000 stades and not greater’. It seems probable
that it was Dicaearchus who (about 300 B.C.) arrived at this value,*
* Ptolemy, Synfazxis, iv. 9, vol. i, p. 327. 3-4, Heib.
2 Ibid. v. 14, vol. i, p. 421. 12-14, Heib.
3. Archimedes, Sand-reckoner (Archimedis opera, ed. Heib., vol. ii, p. 246 sqq ):
The Works of Archimedes, pp. 222, 223.
* Berger, Geschichte der wissenschaftlichen Erdkunde der Griechen,
PP. 37° sqq-
1410 vA
338 LATER IMPROVEMENTS ON PART II
and that it was obtained by taking 24° (1/15th of the whole meri-
dian circle) as the difference of latitude between Syene and Lysi-
machia (on the same meridian), and 20,000 stades as the actual
distance between the two places.! Archimedes’ own estimates are
scarcely estimates at all; they are intentionally exaggerated, as, his
object being to measure the number of grains of sand that would
fill the universe, he adopts what he considers maximum values in
order to be on the safe side. Thus he says that, whereas Aristar-
chus tried to prove that the ratio of the diameter of the sun to that
of the moon is between 18:1 and 20:1, he himself will take the
ratio to be 30:1 and not greater, in order that his thesis may be
proved ‘beyond all cavil’; in the case of the earth he actually
multiplies the estimate of the perimeter by 10, making it 3,000,000
instead of 300,000 stades.
Before passing on to later writers, it will be convenient to re-
capitulate Aristarchus’s figures; and for brevity I shall use the
letters by which Tannery denotes the various distances and radii,
namely .S for the distance of the centre of the sun, Z for the
distance of the centre of the moon, from the centre of the earth,
and s,/,¢ for the radii of the sun, moon, and earth respectively.
Aristarchus’s figures then are as follows:
L/2zl > 224 but < 30 (Prop. 11).
S/E > 18 but < 20 (Prop. 7).
25/2t or s/t > δὲ but < γᾷ (Prop. 15).
2d/at or Ut > ἐδ but < 74% (Prop. 17).
We may with Hultsch,? for convenience of comparison with other
calculations, take figures approximating to the mean between the
higher and lower limits; and it will be convenient to express
the various diameters and distances in terms of the diameter of the
earth. We may say then, roughly, that
ee eh: Te ee Α
2b/2t. = τσ = ἔξ ;.
25/2t = 63;
L/2d = 264;
S/L = 19:
1 Cf. Cleomedes, De motu circulari, i. 8, p. 78, Ziegler.
2 Hultsch, Poseidonios «ber die Grosse und Entfernung der Sonne,
1897, Pp» 5.
CH. IV ARISTARCHUS’S CALCULATIONS 339
whence
Lat = 395.2 = of, say οἱ;
S/2t = 182 .19 = 17933, say 180.
We are not told what size Aristarchus attributed to the earth, but
presumably, like Archimedes, he would have accepted Dicaearchus’s
estimate of 300,000 stades for its circumference.
Eratosthenes (born about eleven years after Archimedes, say
276 B.C.) is famous for a measurement of the earth based on
scientific principles. He found that at noon at the summer solstice
the sun threw no shadow at Syene, while at the same hour at
Alexandria (which he took to be on the same meridian) it made
the gnomon in the scaphe cast a shadow showing an angle equal
to one-fiftieth of the whole meridian circle; assuming, further, that
the sun’s rays at Syene and Alexandria are parallel in direction,
and that the known distance from Syene to Alexandria is 5,000
stades (doubtless taken as a round figure), Eratosthenes arrived by
an easy geometrical proof at 50 times 5,000 or 250,000 stades as
the circumference of the earth. This is the figure given by
Cleomedes ;' but Strabo quite definitely says that, according to
Eratosthenes, the circumference is 252,000 stades,* and this is the
figure which is most generally quoted in antiquity. The reason
for the discrepancy has been the subject of a good deal of discus-
sion ;* it is difficult, in view of Cleomedes’ circumstantial account,
to suppose that 252,000 was the original figure arrived at by
Eratosthenes ; it seems more likely that Eratosthenes himself cor-
rected 250,000 to 252,000 for some reason, perhaps in order to get
a figure divisible by 60 and, incidentally, a round number (700) of
stades for one degree. There is some question as to the length of
the particular stade used by Eratosthenes, but, if Pliny is right in
saying that Eratosthenes made 40 stades equal-to the Egyptian
σχοῖνος," then, taking the σχοῖνος at 12,000 Royal cubits of 0-525
metres,° we get 300 such cubits, or 157-5 metres, as the length ot
the stade, which is thus equal to 516-73 feet. The circumference
of the earth, being 252,000 times this length, works out to about
1 Cleomedes, De motu circulari, i. 10, especially p. 100. 15-23, ed. Ziegler.
2 Strabo, ii. 5. 7, p. 113 Cas. 3 Berger, op. cit., pp. 410, 411.
* Pliny, WV. H. xii. c. 13, ὃ 53.
5 Hultsch, Griechische u. rimische Metrologie (Berlin, 1882), p. 364. -Cf.
Tannery, Recherches sur l'histoire de l’astronomie ancienne, pp. 109, 110.
Z2
340 LATER IMPROVEMENTS ON PART II
24,662 miles, and the diameter of the earth on this basis is about
7,850 miles, only 50 miles shorter than the true polar diameter,
a surprisingly close approximation, however much it owes to happy
accidents in the calculation."
We have no trustworthy information as to evaluations by Erato-
sthenes of other dimensions and distances. The Doxographz, it is
true, say that Eratosthenes made JZ, the distance of the moon from
the earth, to be 78 myriads of stades, and \S, the distance of the
sun, to be 80,400 myriads of stades? (the versions of Stobaeus and
Joannes Lydus admit of 408 myriads of stades as an alternative
interpretation, but this figure obviously cannot be right). Tannery*
considers that none of these figures can be correct. He suggests
that Z was put by Eratosthenes at 278 myriads of stades, not 78 ;
but I am not clear that 78 is wrong. We have seen that, if we
take mean figures, Aristarchus made the distance of the moon from
the earth to be about οἱ times the earth’s diameter. Now 252,000/r,
approximately 252,000/33, is about 80,180, or roughly 8 myriads
of stades; 93 times this is 76 myriads, and Eratosthenes’ supposed
figure of 780,000 is sufficiently close to this. According to
Tannery’s conjecture of 2,780,000 stades, the ratio Z/2¢ would be
nearly 34-7, which is greater than the values given to it by Hippar-
chus, Posidonius, and Ptolemy, and also greater than the true value.
With regard to Eratosthenes’ estimate of S, Tannery points to
Macrobius’s statement that Eratosthenes said that ‘the measure
(mensura) of the earth multiplied 27 times will make the measure
of the sun’.t The question here arises whether it is the solid
contents of the two bodies or their diameters which are compared.
Tannery takes the latter to be the meaning. If this is right, and
if Eratosthenes took the value of #° for the apparent angular
diameter of the sun discovered by Aristarchus, the circumference
2mS of the sun’s orbit would be equal to 27. 2Ζ. 720, which, if we
put 34 for π, would give
S' = 6185 2 = 24,800 myriads of stades, nearly.
1 Cf. Dreyer, Planetary Systems, p. 175.
2 Aét. ii. 31. 3 (D. G. 362-3).
3 Tannery, ‘Aristarque de Samos’ in Wém. de la Soc. des sci. phys. et nat. de
Bordeaux, 2° sér., v, 1883, pp. 254,255 3 Wémoires scientifiques, ed. Heiberg and
Zeuthen, i, pp. 391-2.
* Macrobius, /# somn. Scip. i. 20. 9.
CH. IV ARISTARCHUS’S CALCULATIONS 341
But Hultsch! shows reason for believing that ‘mensura’ in the
statement of Macrobius means solid content. One ground is the
further statement of Macrobius that Posidonius'’s estimate of the size
ofthe sun in terms of the earth was ‘many many times’ greater than
that of Eratosthenes (‘ multo multoque saepius ’, sc. ‘ multiplicata ’).
But we shall find that Posidonius’s figures lead to only about 393 as
the ratio of the diameter of the sun to that of the earth, which is
not ‘many many times’ greater than 27. It seems therefore
necessary to conclude, if Macrobius is to be trusted, that according
to Eratosthenes s/f was equal to 3, not 27. This would divide
the value of S by 9, and S/2¢ would be equal to 343% instead of
30923.
We are much better informed on the subject of Hipparchus’s
estimates of sizes and distances, thanks to the investigations of
Hultsch,2 who found in the commentaries of Pappus and Theon
on chapter 11 of Book V of Ptolemy’s Syz¢axzs particulars as to
which Ptolemy himself leaves us entirely in the dark. Ptolemy
States that there are certain observed facts with regard to the
sun and moon which make it possible, when the distance of one
of them from the centre of the earth is known, to calculate the
distance of the other, and that Hipparchus first found the dis-
tance of the sun on certain assumptions as to the solar parallax,
and then deduced the distance of the moon. According to the
value assumed for the solar parallax (and Ptolemy says that there
was doubt as to whether it was the smallest appreciable amount
or actually negligible), Hipparchus deduced, of course, different
figures for the distance of the moon.* Ptolemy does not state these
figures, but Pappus supplies the deficiency. Pappus begins by
saying that Hipparchus’s calculation, depending mainly on the sun,
was ‘not exact’. Next, he observes that, if the apparent diameter
of the sun is taken to be very nearly the same as that of the moon
at its greatest distance at the conjunctions, and if we are given the
relative sizes of the sun and moon and the distance of one of them,
the distance of the other is also given; then, after paraphrasing
1 Hultsch, Poseidonios tiber die Grosse und Entfernung der Sonne, pp. 5, 6.
3 Hultsch, ‘ Hipparchos iiber die Grésse und Entfernung der Sonne’ (Berichte
der philologisch-historischen Classe der Kgl. Sachs. Geselischaft der Wissen-
schaften zu Leipzig, 7. Juli 1900).
5 Ptolemy, Synfaxis, v. 11, vol. i, p. 402, Heib.
242 LATER IMPROVEMENTS ON PART II
Ptolemy’s remarks above quoted, he proceeds.as follows: ‘ In his first
book about sizes and distances Hipparchus starts from this observa-
tion: there was an eclipse of the sun which was exactly total in
the region about the Hellespont, no portion of the sun being seen,
whereas at Alexandria in Egypt about. four-fifths only of its
diameter was obscured.' From the facts thus observed he proves
in his first book that, if the radius of the earth be the unit, the least
distance of the moon contains 71, and the greatest 83 of these units ;
the mean therefore contains 77. After proving these propositions,
he says at the end of the first book: “ In this treatise I have carried
the argument to this point. Do not, however, suppose that the
theory of the distance of the moon has ever yet been worked out
accurately in every respect; for even in this question there is an
investigation remaining to be carried out, in the course of which the
distance of the moon will be proved to be less than the figure just
calculated,” so that he himself admits that he is not quite in a
position to state the truth about the parallaxes. Then, again, he
himself, in the second book about sizes and distances, proves from
many considerations that, if we take the radius of the earth as the
unit, the least distance of the moon contains 62 of these units, and its:
mean distance 674, while the distance of the sun contains 2,490. It
is clear from the former figures that the greatest distance of the
moon contains 722 of these units.’ The figure of 2,490 for the
distance of the sun has to be obtained by a correction of the Greek
text. The later MSS. have ς or go, but one MS. has υς or 490.
The 2,490 is credibly restored by Hultsch on the following grounds..
Adrastus* and Chalcidius* tell us that. Hipparchus made the sun
nearly 1880 times the size of the earth,* and the earth about 27 times
the size of the moon. The size means the solid content, and, the
cube root of 1880 being approximately 12}, we have approximately
f:d:8 = 1i1shs12h
Roe be the
1 This same eclipse is also mentioned by Cleomedes, De motu circulari, ii. 3,
pp. 172. 22 and 178. 14, ed. Ziegler.
2 Theon of Smyrna, p. 197. 8-12, ed. Hiller.
8. Chalcidius, 7zmaeus, c. ΟἹ, p. 161.
‘ A less trustworthy authority, Cleomedes (De motu circulari, ii. τ, p. 152. 5-7),
mentions a tradition that Hipparchus made the sun 1050 times as large as the
earth.
CH. IV ARISTARCHUS’S CALCULATIONS 343
Now the mean distance of the moon is, according to Hipparchus,
674 times the earth’s radius; assuming then that the apparent
angular diameter of the sun and moon as seen from the earth is
about the same, we find that
S = 673 2. 37 = 24915 7.
That is to say, S= 2490 4, nearly. It is clear, therefore, that 8
has fallen out of the text before vG, and the true number arrived at
by Hipparchus was 2490.
Thus Hipparchus made the distance of the moon equal, at the
mean, to 332 times the dzamefer of the earth, and the distance of the
sun equal to 1,245 times the diameter of the earth. As we said above,
Ptolemy does not mention these figures of Hipparchus, much less
does he make any use of them. Yet they are remarkable, because
not only are they far nearer the truth than Aristarchus's estimates,
but the figure of 1,245 for the distance of the sun is much better
than that of Ptolemy himself, namely 605 times the earth’s
diameter, or less than half the figure obtained by Hipparchus.
Yet Hipparchus’s estimate remained unknown, and Ptolemy’s held
the field for many centuries; even Copernicus only made the
distance of the sun to be equal to 750 times the earth’s diameter,
and it was not till 1671—3 that a substantial improvement was made,
observations of Mars carried out in those years by Richer enabling
Cassini to conclude that the sun’s parallax was about 9”-5, corre-
sponding to a distance of the sun from the earth of 87,000,000 miles."
Hultsch shows that the particular solar eclipse referred to by
Hipparchus was probably that of 20th November in the year
129 B.C.,? and he concludes that the following year (128 B.C.) was
the date of Hipparchus’s treatise in two books ‘On the sizes and
distances of the sun and moon’.
Hipparchus, in his Geography, definitely accepted the estimate
of 252,000 stades obtained by Eratosthenes for the circumference
of the earth ;* if there is any foundation for the statement of Pliny *
that he added a little less than 26,000 stades to this estimate, making
nearly 278,000, the explanation must apparently be that he stated
1 Berry, A Short History of Astronomy, 1898, pp. 205-7.
2 Cf. Boll, Art. ‘ Finsternisse’ in Pauly-Wissowa’s Real-Encyclopidie, vi. 2,
1909, p. 2358. ; e
8 Strabo, ii. 5. 34, ps 132 Cas. * Pliny, WV. H. ii. c. 108, § 247.
344 LATER IMPROVEMENTS ON PART II
this number as a maximum, allowing for possible errors resulting
from Eratosthenes’ method;! but Berger considers that Pliny’s
statement is based on a misapprehension.?
Posidonius of Rhodes (135-51 B.C.) cannot be reckoned among
astronomers in the strict sense of the term, but he dealt with astro-
nomical questions in his work on Meteorology, and he wrote a
separate tract on the size of the sun.* It was presumably in the
latter work that he put forward a bold hypothesis as to the distance
of the sun, which has the distinction of coming far nearer to the
truth than the estimates of Hipparchus and all other ancient writers
had done.t Cleomedes tells us that Posidonius supposed the circle
in which the sun apparently moves round the earth to be 10,000
times the size of a circular section of the earth through its centre.
With this hypothesis he combined (says Cleomedes) the assumption
which he took from Eratosthenes that at Syene (which is under
the summer tropic) and throughout a circle round it with a diameter
of 300 stades the upright gnomon throws no shadow (at noon).
It follows from this that the diameter of the sun occupies a portion
of the sun’s circle 3,000,000 stades in length; in other words, the
diameter of the sun is 3,000,000 stades.° If we only knew the
Jraction of the circumference of the sun’s circle occupied by the sun
itself, we could calculate the circumference of the earth, and the
absolute distance of the centre of the sun from the centre of the
earth ; but Cleomedes gives us no information on this, and we have
to go elsewhere for what we want—in this case to Pliny. Now Pliny
says that according to Posidonius there is round the earth a height
of not less than 40 stades, which is the region of winds and clouds,
and beyond which there is pure air; the distance from the belt of
clouds, &c., to the moon is 2,000,000 stades, and the further distance
from the moon to the sun is 500,000,000 stades.* This would give
£—t = 2,000,040 stades,
S'—Z = 502,000,040 stades.
1 Tannery, Recherches sur l’hist. de Pastronomie ancienne, p. 116,
? Berger, Gesch. der wissenschaftlichen Erdkunde der Griechen, pp. 413-14.
5 Cleomedes, De motu circulart, i. 11, p. 118. 4-6.
* On the whole of this subject, see Hultsch, ‘ Poseidonios iiber die Grésse
und Entfernung der Sonne’ (46h. der Kgl. Gesellschaft der Wissenschaften su
Gottingen, Phil.-hist. Klasse, Neue Folge, Bd. I, Nr. 5), 1897.
5 Cleomedes, ii. I, p. 144. 22-146. 16; ibid. i. 10, pp. 96. 28-98, 5.
® Pliny, ii, c. 23, ὃ 85.
CH. IV ARISTARCHUS’S CALCULATIONS 345
Dividing the latter figure by 10,000 we obtain, approximately, for
the radius of the earth
Ζ = 50,200 stades.
Hultsch gives reason for thinking that the 500,000,000 stades
should be the distance from the centre of the earth to the centre of
the sun, not the distance from the moon to the sun; the 40 stades
representing the depth of the region of clouds, &c., is clearly
negligible ; and, as Posidonius dealt in round figures, we may infer
that his estimate of the earth’s diameter would be 100,000 stades.
If now we use the Archimedean approximation of 3} for z, the
circumference of the earth would on this basis be 314,285 stades;
but we may, with some probability, suppose that Posidonius would
take the round figure of 300,000 stades corresponding to 7 = 3,
an approximation used by Cleomedes in another place.
On the other hand, Cleomedes gives 240,000 stades as Posidonius's
estimate of the earth’s circumference based on the following assump-
tions, (1) that the star Canopus, invisible in Greece, was just seen to
graze the horizon at Rhodes as it rose and set again immediately,
whereas its meridian altitude at Alexandria was ‘a fourth part of a
sign, that is, one forty-eighth part of the zodiac circle’, (2) that the
distance between the two places was considered to be 5,000 stades.?
The circumference of the earth was thus made out to be 48 times
5,000 Or 240,000 stades. But the estimate of the difference of lati-
tude at 1/48th of a great circle, or 73°, was very far from correct
(the true difference of latitude is 53° only); indeed the effects of
refraction at the horizon would inevitably vitiate the result of such
an attempt at measurement of the angle in question as Posidonius
was in a position to make. Moreover, the estimate of 5,000 stades
for the distance was also incorrect; it was the maximum estimate —
put upon it by mariners, while some put it at. 4,000 only, and
Eratosthenes, by observations of the shadows cast by gnomons,
found it to be 3,750 stades only.* The existence of the latter
estimate of the distance between Rhodes and Alexandria seems to
account for Strabo’s statement that Posidonius favoured ‘ the latest
of the measurements which gave the smallest dimensions to the
1 Cleomedes, De motu circulari, i. 8, Ὁ. 78. 22-3.
2 Ibid. i. 10, pp. 93. 26 -- 94. 22.
3. Strabo, ii. 5.24, pp.125-6Cas.; Berger, Gesch. der wissenschaftlichen Erdkunde
der Griechen, p. 415.
346 LATER IMPROVEMENTS ΟΝ. PART II
earth’, namely about 180,000 stades ;1 for 180,000 is 48 times 3,750,
just as 240,000 is 48 times 5,000. Now Eratosthenes must presum-
ably have arrived at his distance of 3,750 stades by means of
a calculation based on his own estimate of the total circumference
of the earth (250,000 or 252,000) and the observed angle represent-
ing the difference of the inclination of the shadows thrown by the
gnomon at the two places respectively.2,_ We are not told what
the angle was, but it can be inferred that it was 52° or 5,°;', because
250,000 (252,000) : 3,750 = 360°: 53° (52% ).
It is nothing short of extraordinary that Posidonius should have
used the 3,750 stades without a thought of its origin and then
calculated the circumference of the earth by combining the 3,750
with an estimate of the corresponding angle which is so grossly
erroneous (74°). It may seem not less extraordinary that Ptolemy
(following Marinus of Tyre) should have accepted without any
argument or question Posidonius’s figure of 180,000 stades. But
the explanation doubtless is that Ptolemy’s stades were Royal
stades of 210 metres (nearly 3th of a Roman mile) instead ot
Eratosthenes’ stades of 1574 metres ; for Ptolemy in his Geography
says that the length of a degree is 500 stades,? whereas Eratosthenes
made a degree contain about 700 stades. Thus, as Ptolemy’s
stades were to Eratosthenes’ as 4 to 3, Ptolemy’s estimate of the
circumference of the earth would, in stades of Eratosthenes, be
240,000, the same as the estimate attributed by Cleomedes to
Posidonius.
As we have seen, Pliny’s account of Posidonius’s estimates of the
distances of the sun and moon leads to about 300,000 stades, and
not 240,000, as the circumference of the earth. What is the
explanation of the discrepancy? Hultsch takes the 300,000 stades
and the assumption that the sun’s circle is 10,000 times as large as
the circumference of the earth to be part of a calculation of the
minimum distance of the sun, on the ground that Cleomedes goes
on to say that ‘it is indeed plausible that the sun’s circle is wot ess
than 10,000 times the circumference of the earth, seeing that the
earth is to it in the relation of a point; but it may also be greater
1 Strabo, ii. 2. 2, p. 95 Cas. 2 Berger, op. cit., pp. 579, 580.
5. Ptolemy, Geography, vii. 5. 12.
CH. IV ARISTARCHUS’S CALCULATIONS 347
still without our knowing it’ But it is somewhat awkward to
suppose with Hultsch that Posidonius is arguing, ‘I take the earth
to be of the size attributed to it by Dicaearchus, namely 300,000
stades in circumference, although this figure exceeds the truth ;
_ but I am satisfied that, even if I take the circumference to be μῶν
300,000 stades, I shall ye¢ arrive at an estimate of the sun’s distance
which is less than the true distance.’ The italics are mine, and
represent the part of Hultsch’s argument which seems to me
_ doubtful. The use of an exaggerated estimate of the earth's
circumference with a view to a #zuitmum estimate of the sun’s
distance is so strange that I prefer to suppose that, in the develop-
ment of the hypothesis about the sun’s distance, Posidonius simply
_ used Dicaearchus’s figure for the earth’s circumference without any
| arritre-pensée at all.
In considering the origin of the bold hypothesis of Posidonius
with regard to the sun’s distance, it is necessary to refer to the
hypotheses of Archimedes with regard to the size of the universe, on
which in his Sand-reckoner he bases his argument that it is possible
to formulate a system for expressing numbers as large as we please,
say a number such as the number of the grains of sand which would
be required to fill an empty space as large as our ‘universe’. For
the purpose which he has in view, Archimedes has of course to
take what he considers to be outside or maximum measurements.
Thus, whereas his predecessors had tried to prove the perimeter
of the earth to be 300,000 stades, he will allow it to be as much
as ten times that ‘and not greater’, viz. 3,000,000 stades. Next,
whereas Aristarchus had made the sun between 18 and 20 times as
large as the moon, he will take it to be 30 times, but not greater, so
that (the earth being greater than the moon) the sun will be less
than 30 times the size of the earth. Archimedes proceeds to con-
sider the size of the so-called ‘ universe’ and of the sun. He has
explained that the ‘ universe’ as commonly understood by astrono-
mers is the sphere which has for its centre the centre of the earth
and for its radius the distance between the centre of the earth and
the centre of the sun, but that the sphere of the fixed stars is much
_ greater than this so-called ‘universe’. Considering now the sun
1 Cleomedes, ii. 1, p. 146. 12-16. The text has μείζονα αὐτὸν ὄντα ἢ πάλιν
_ μείονα, ‘it may be greater, or again it may be less’; Hultsch rejects ἢ πάλιν
᾿ μείονα as a gloss inconsistent with the trend of Cleomedes’ argument.
448 LATER IMPROVEMENTS ΟΝ PART II |
in relation to its orbit, a great circle of the so-called ‘ universe’,
Archimedes found by a rough experiment (in confirmation of
Aristarchus’s discovery that the apparent angular diameter of the
sun is ξσί of four right angles) that the angle subtended by
the sun’s diameter is between ;4,th and 335th part of a right angle,
or between gigth and 53,th part of four right angles. By means
of this result he proves that the diameter of the sun is greater than
the side of a chiliagon (or a regular polygon with 1,000 sides)
inscribed in its orbit. The proof of this is very interesting because
we there see Archimedes abandoning the traditional view that
the earth is a point in relation to the sphere in which the sun
‘moves! (Aristarchus assumed it to be so in relation even to the
mtoon’s sphere), and recognizing parallax in the case of the sun,
apparently for the first time ; for, from the fact that the apparent
diameter of the sun, as seen at its rising by an observer on the
surface of the earth, subtends an angle less than εἰς and greater
than εἰσίῃ of four right angles, he proves geometrically that the
arc of the sun’s orbit subtended by a chord equal to the diameter of
the sun subtends at the centre of the earth an angle greater than
gigth and @ fortior? greater than z 55th of four right angles.
Now, says Archimedes, since
(perimeter of chiliagon inscribed in sun’s orbit)
<1,000 (diam, of sun)
< 30,000 (diam. of earth),
while the perimeter of any regular polygon of more than six sides
is greater than 3 times the diameter of the circle in which it
is described, it follows that '
(diameter of sun’s orbit) < 10,000 (diam. of earth).
Posidonius assumed, not that the diameter of the sun’s orbit was
Zess than 10,000 times the diameter of the earth, but that it was
equal to (or not less than) 10,000 times the earth’s diameter. But
the origin of his ratio of 10,000 : 1 is sufficiently clear; he took it
from Archimedes. Similarly, the combination of the estimate of
300,000 stades for the circumference of the earth with Erato-
sthenes’ assumption that the shadowless circle at Syene was 300
* Cf. Cleomedes, De motu circulari, i. 11, pp. 108-12, ed. Ziegler.
CH.IV ARISTARCHUS'’S CALCULATIONS 349
stades in diameter suggests that Posidonius likewise adopted from
Archimedes the ;,4,th part of four right angles as the apparent
_ angular diameter of the sun, being satisfied to take Archimedes’
minimum estimate as the actual figure.
It remains to express Posidonius’s estimates of the sun’s and
_ moon's sizes and distances in terms of the earth’s diameter. On
_ the basis of his estimate of 240,000 stades for the circumference ot
_ the earth, the earth’s diameter, which we will call D, is 240,000/r
_ Stades, or about 76,400 stades.
Distance of sun = 500,000,000 D/76,400 = about 6,545 D.
Diameterofsun = 3,000,000 D/76,400 = 393 D.
Distance of moon = 2,000,000 D/76,400 = 263 D.
Diameter of moon = ;2, (diameter of sun) = 0-157 D, nearly.
As Ptolemy gives none of the estimates which Pappus’s com-
_ mentary on the Syzfaxzs quotes from Hipparchus's treatise on the
_ sizes and distances of the sun and moon, it was not unnatural to
suppose, as Wolf did,’ that the elaborate calculations in Ptolemy
_ (v. 13-16) were referable to Hipparchus. This cannot be so as
regards the results, as Hultsch has shown by means of Pappus’s
commentary, though doubtless Ptolemy may have been at least
partially indebted to Hipparchus for the methods which he fol-
lowed. The following are Ptolemy’s results:
The mean distance of the moon = _ 59 times the earth's radius.”
= is ‘s ν sun = 1,210 Ἔ = Ἢ
The diameter of the earth = 32 times the diameter of the moon.’
» » » sun=18 ,, » τον, oom ἐς
It follows that :
the diam. of the sun = about 53 times the diam. of the earth.
I will conclude with Hultsch’s final comparative table* of sizes
1 Wolf, Geschichte der Astronomie, pp. 174 564.
5 Ptolemy, Synfazis, v.15, p. 425. 17-20, Heib.
5. Ibid., v. 16, p. 426. 12-15, Heib.
* Hultsch, Hipparchos iiber die. Grosse und Entfernung der Sonne, Pp- 199.
350
COMPARISON OF CALCULATIONS
and distances in terms of the earth’s mean diameter (=1,716
geographical miles) :
Mean dis- Mean dis-
tance of | Diameter | tance of | Diameter
moon from | of moon | sun from of sun
earth earth τ
According to Aristarchus οἱ os = 0°36 180 62
” ” Hipparchus 333 B= 033 1245 128
» » Posidonius 26% = | fe = 0157/6545 39%
ewer Ptolemy 293 ἡ = 0°29 605 53
In reality 3 30:2 Ο᾽27 11726 108-9
ARISTARCHUS OF SAMOS
ON THE SIZES AND DISTANCES OF
THE SUN AND MOON
TEXT, TRANSLATION, AND NOTES
ΑΡΙΣΤΑΡΧΟΥ ΠΕΡῚ ΜΕΓΈΘΩΝ ΚΑΙ
AIIOZTHMATON HAIOY ΚΑΙ ΣΕΛΗΝΗΣ
ΑἸΠΟΘΕΣΕΙΣ)
α΄. Τὴν σελήνην παρὰ τοῦ ἡλίου τὸ φῶς λαμβάνειν.
5. β΄. Τὴν γῆν σημείου τε καὶ κέντρου λόγον ἔχειν πρὸς τὴν
τῆς σελήνης σφαῖραν.
γ΄. Ὅταν ἡ σελήνη διχότομος ἡμῖν φαίνηται, νεύειν εἰς
τὴν ἡμετέραν ὄψιν τὸν διορίζοντα τό τε σκιερὸν καὶ τὸ
λαμπρὸν τῆς σελήνης μέγιστον κύκλον.
10 ὃ΄. Ὅταν ἡ σελήνη διχότομος ἡμῖν φαίνηται, τότε αὐτὴν
ἀπέχειν τοῦ ἡλίου ἔλασσον τεταρτημορίου τῷ τοῦ τεταρ-
τημορίου τριακοστῷ. : |
ε΄. Τὸ τῆς σκιᾶς πλάτος σεληνῶν εἶναι δύο.
΄ ‘ U4 ς 4 ς x va ,
ς. Τὴν σελήνην ὑποτείνειν ὑπὸ πεντεκαιδέκατον μέρος
15 ζῳδίου.
᾿Επιλογίζεται οὖν τὸ τοῦ ἡλίου ἀπόστημα ἀπὸ τῆς γῆς τοῦ τῆς
σελήνης ἀποστήματος μεῖζον μὲν ἢ ὀκτωκαιδεκαπλάσιον, ἔλασσον
δὲ ἢ εἰκοσαπλάσιον, διὰ τῆς περὶ τὴν διχοτομίαν ὑποθέσεως" τὸν
[W = Wallis. F = Fortia d’Urban. Vat. = Cod. Vaticanus Graecus 204.]
I. APISTAPXOY] APISTAPXOY SAMIOY W 3. (YIIOCESEIS) addidi
(cf. ὑποθέσεως 1.18 infra ; ὑποτίθεται Pappus) : OESEIS W 4. τὸ] om. Pappus
8.re]om. Pappus 12. τριακοστῷ] τριακοστημορίῳ Pappus 16. οὖν] δὴ Pappus
16,17. τὸ τοῦ ἡλίου... ἀποστήματος] τὸ τοῦ ἡλίου ἀπόστημα τοῦ τῆς σελήνης ἀποστή-
ματος πρὸς τὴν γῆν Pappus 18. εἰκοσαπλάσιον»] εἰκοσιπλάσιον W διὰ τῆς
: ἐποδέννωαϊ τοῦτο δὲ διὰ τῆς περὶ τὴν διχότομον ὑποθέσεως post 1. 1, p. 354
σελήνης διάμετρον posuit Pappus
ARISTARCHUS ON THE SIZES AND DISTANCES
OF THE SUN AND MOON
[HYPOTHESES ]
1. That the moon receives tts light from the sun.
2. That the earth ts tn the relation of a point and centre to the
Sphere in which the moon moves.
3. That, when the moon appears to us halved, the great circle
| which divides the dark and the bright portions of the moon ἐς
tn the direction of our eye.*
4. That, when the moon appears tous halved, its distance from
the sun ts then less than a quadrant by one-thirtieth of a
guadrant®
5. That the breadth of the (earth's) shadow ts (that) of two
moons.
6. That the moon subtends one fifteenth part of a sign of the
_ zodtac.*
Ave
Ca PS ἀκ Ὺ
poms
We are now ina position to prove the following propositions :—
1. The distance of the sun from the earth ts greater than
eighteen times, but less than twenty times, the distance of the
moon ( from the earth); this follows from the hypothesis about
the halved moon.
1 Literally ‘the sphere of the moon’.
* Literally ‘verges towards our eye’, the word νεύειν meaning to ‘verge’ or
‘incline’. What is meant is that the plane of the great circle in question passes
through the observer's eye or, in other words, that his eye and the great circle
are in one plane (cf. Aristarchus’s own explanation in the enunciation of Prop. 5).
* T.e. is less than go® by 1/30th of 90° or 3°, and is therefore equal to 87°.
* T.e.1/15th of 30°, or 2°. Archimedes in his Sand-reckoner (Archimedes, ed.
Heiberg, ii, p. 248, 19) says that Aristarchus ‘discovered that the sun appeared
to be about 1/720th part of the circle of the zodiac’; that is, Aristarchus dis-
covered (evidently at a date later than that of our treatise) the much more
correct value of 4° for the angular diameter of the sun or moon (for he maintained
that both had the same angular diameter: cf. Prop. 8). Archimedes himself
in the same place describes a rough method of observation by which he inferred
that the diameter of the sun was less than 1/164th part, and greater than
1/2ooth part, ofa right angle. Cf. pp. 311-12 ante.
1410 Aa
354 ON THE SIZES AND DISTANCES
αὐτὸν δὲ λόγον ἔχειν τὴν τοῦ ἡλίου διάμετρον πρὸς τὴν τῆς σελήνης
διάμετρον: τὴν δὲ τοῦ ἡλίου διάμετρον πρὸς τὴν τῆς γῆς διάμετρον
μείζονα μὲν λόγον ἔχειν ἢ ὃν τὰ LO πρὸς γ, ἐλάσσονα δὲ ἢ ὃν py
πρὸς ς, διὰ τοῦ εὑρεθέντος περὶ τὰ ἀποστήματα λόγου, τῆς (τε
5 περὶ τὴν σκιὰν ὑποθέσεως, καὶ τοῦ τὴν σελήνην ὑπὸ πεντεκαιδέκατον
μέρος ζῳδίου ὑποτείνειν.
’,
α.
c
Δύο σφαίρας ἴσας μὲν ὁ αὐτὸς κύλινδρος περιλαμβάνει,
ἀνίσους δὲ ὁ αὐτὸς κῶνος τὴν κορυφὴν ἔχων πρὸς τῇ
τοἐλάσσονι σφαίρᾳ καὶ ἡ διὰ τῶν κέντρων αὐτῶν ἀγομένη
ἐὐθεῖα ὀρθή ἐστιν πρὸς ἑκάτερον τῶν κύκλων, Kab’ ὧν
ἐφάπτεται ἡ τοῦ κυλίνδρου ἢ ἡ τοῦ κώνου ἐπιφάνεια τῶν
σφαιρῶν.
Ἔστωσαν ἴσαι σφαῖραι, ὧν κέντρα ἔστω τὰ A, Β σημεῖα, καὶ
15 ἐπιζευχθεῖσα ἡ AB ἐκβεβλήσθω, καὶ ἐκβεβλήσθω διὰ τοῦ AB
ἐπίπεδον." ποιήσει δὴ τομὰς ἐν ταῖς σφαίραις μεγίστους κύκλους.
Cy : Fe
ae D G Bis A
εἴ Ηθ
Fig. 16.
ποιείτω οὖν τοὺς ΓΔΕ, ΖΗΘ κύκλους, καὶ ἤχθωσαν ἀπὸ τῶν A, B
τῇ AB πρὸς ὀρθὰς αἱ ΓΑΕ, ZBO, καὶ ἐπεζεύχθω ἡ ΓΖ. καὶ ἐπεὶ
I. ἔχειν τὴν] ἔ ἔχει καὶ ἡ Pappus διάμετρον] διάμετρος Pappus 3: μείζονα
μὲν λόγον ἔχειν] ἐν μείζονι λόγῳ Pappus a] om. Pappus ΣΝ δὲ] ἐν
ἐλάσσονι δὲ λόγῳ Pappus py] τὰ μγ Pappus 4. τῆς (re)] 3 addidi: καὶ τῆς
Pappus 6. ὑποτείνειν ante ὑπὸ posuit Pappus 16. δὴ] δὲ W
OF THE SUN AND MOON 355
2. The diameter of the sun has the same ratio (as aforesaid )
to the diameter of the moons
3. The diameter of the sun has to the diameter of the earth
a ratio greater than that which 19 has to 3, but less than that
which 43 has to 6; this follows from the ratio thus discovered
between the distances, the hypothesis about the shadow, and the
hypothesis that the moon subtends one fifteenth part of a sign of
the zodiac.
PROPOSITION 1.
Two egual spheres are comprehended by one and the same
cylinder, and two unequal spheres by one and the same cone which
has tts vertex in the direction of the lesser sphere, and the
straight line drawn through the centres of the spheres ts at right
angles to each of the circles in which the surface of the cylinder,
or of the cone, touches the spheres.
Let there be equal spheres, and let the points 4, B be their
centres.
Let 4B be joined and produced ;
let a plane be carried through 4 &; this plane will cut the spheres
in great circles.?
Let the great circles be CDE, FGH.
Let CAE, FBH be drawn from 4, 8 at right angles to 42;
and let CF be joined.
1 Pappus gives this second result immediately after the first result, i.e. before
the parenthesis ‘this follows from the hypothesis .. .’._ This arrangement seems
at first sight more appropriate, and Nizze alters his text accordingly. But
I think it better to follow the above order which is that of the MSS. and Wallis.
One consideration which weighs with me is that the second result does not
follow from the hypothesis of the halved moon alone; it depends on another ~
assumption also, namely, that the sun and the moon have the same apparent
angular diameter (see Prop. 8). ,
Literally ‘it will make, as sections in the spheres, great circles’, and
then, in the next sentence, ‘let it then make the circles CDE, FGH.’ In
translating these characteristic phrases, which occur very frequently, I wish
I could have reproduced the Greek exactly, keeping the word ‘sections’, but it
becomes impossible to do so when the phrase is extended so as to distinguish
several sections made by one plane, e.g. one section in one sphere, one section
in another sphere, and one section in a cone: Thus ‘let it make, as sections, in
_ the spheres, the circles CDE, FGH, and, in the cone, the triangle CEK’
(Prop. 2) would be intolerable, with or without the multitude of commas,
whereas clearness and conciseness is easily secured by saying ‘let it cut the
spheres in the circles CDE, FGH and the cone in the triangle CEX’.
Aaz
356 ON THE SIZES AND DISTANCES
ai TA, ZB ἴσαι τε καὶ παράλληλοί εἰσιν, καὶ ai ΓΖ, AB ἄρα ἴσαι
τε καὶ παράλληλοί εἰσιν. παραλληλόγραμμον ἄρα ἐστὶν τὸ ΓΖΑΒ,
καὶ αἱ πρὸς τοῖς Γ,, Ζ γωνίαι ὀρθαὶ ἔσονται: ὥστε ἡ ΓΖ τῶν ΓΔΕ,
ΖΗΘ κύκλων ἐφάπτεται. ἐὰν δὴ μενούσης τῆς AB τὸ AZ παραλ-
5 ληλόγραμμον καὶ τὰ KT'A, HZA ἡμικύκλια περιενεχθέντα εἰς τὸ
αὐτὸ πάλιν ἀποκατασταθῇ ὅθεν ἤρξατο φέρεσθαι, τὰ μὲν Κὶ ΓΖ,
ΗΖ. ἡμικύκλια ἐνεχθήσεται κατὰ τῶν σφαιρῶν, τὸ δὲ AZ παραλ-
ληλόγραμμον γεννήσει κύλινδρον, οὗ βάσεις ἔσονται οἱ περὶ δια-
μέτρους τὰς TE, ΖΘ κύκλοι, ὀρθοὶ ὄντες πρὸς τὴν AB, διὰ τὸ ἐν
τοπάσῃ μετακινήσει διαμένειν tas TE, ΘΖ ὀρθὰς τῇ AB. καὶ
φανερὸν ὅτι ἡ ἐπιφάνεια αὐτοῦ ἐφάπτεται τῶν σφαιρῶν, ἐπειδὴ ἡ ΓΖ
κατὰ πᾶσαν μετακίνησιν ἐφάπτεται τῶν ΚΙΖ, HZA ἡμικυκλίων,
Ἔστωσαν δὴ αἱ σφαῖραι πάλιν, ὧν κέντρα ἔστω τὰ A, Β, ἄνισοι,
καὶ μείζων ἧς κέντρον τὸ 4" λέγω ὅτι τὰς σφαΐρας ὁ αὐτὸς κῶνος
15 περιλαμβάνει τὴν κορυφὴν ἔχων πρὸς τῇ ἐλάσσονι σφαίρᾳ.
Ἐπεζεύχθω ἡ AB, καὶ ἐκβεβλήσθω διὰ τῆς AB ἐπίπεδον"
ποιήσει δὴ τομὰς ἐν ταῖς σφαίραις κύκλους. ποιείτω τοὺς TALE,
ΖΗΘ' μείζων ἄρα ὁ TAE κύκλος τοῦ ΗΖΘ κύκλου" ὥστε καὶ ἡ
ἐκ τοῦ κέντρου τοῦ ΓΔΕ κύκλου μέΐζων ἐστὶ τῆς ἐκ τοῦ κέντρου
λοτοῦ ΖΗΘ κύκλου. δυνατὸν δή ἐστι λαβεῖν τι σημεῖον, ὡς τὸ Καὶ, iv
ἦ, ὡς ἡ ἐκ τοῦ κέντρου τοῦ ΓΔῈ κύκλου πρὸς τὴν ἐκ τοῦ κέντρου
CY
4
ο αἰ IL δ᾽. Ὁ M_\N
ἕξ α λ ὃ 7 B |e /v «
Ὥ
Ee
Fig. 17.
τοῦ ΖΗΘ κύκλου, οὕτως ἡ AK πρὸς τὴν KB. ἔστω οὖν εἰλημμένον
τὸ K σημεῖον, καὶ ἤχθω ἡ ΚΖ ἐφαπτομένη τοῦ ΖΗΘ κύκλου, καὶ
ἐπεζεύχθω ἡ ΖΒ, καὶ διὰ τοῦ A τῇ ΒΖ παράλληλος ἤχθω ἡ AT,
6. ἀποκατασταθῃ] ἀποκαταστῇ W το. ΘΖ] ZOW τι. ἐφάπτεται] ἐφάπτηται W
13. B ad init. Vat. εἰ codd. Paris. δὴ] δὲ W 17. τομὰς] corr. W: τομὴν
Vat. 18. κύκλος] om. W ΗΖΘ] ZHO W 20, τὸ K] τὸ KE ναι.
OF THE SUN AND MOON 357
Then, since CA, FB are equal and parallel, therefore CF, 4.8
are also equal and parallel.
Therefore CFA B is a parallelogram,
and the angles at C, F will be right;
so that CF touches the circles CDE, FGH.
If now, “4.8 remaining fixed, the parallelogram AF and the
semicircles KCD, GFL be carried round and again restored to the
position from which they started, the semicircles KCD, GFL will
move in coincidence with the spheres’; and the parallelogram 4 F
will generate a cylinder, the bases of which will be the circles about
CE, FH as diameters and at right angles to 4 B, because, through-
out the whole motion, CE, HF remain at right angles to 4 2.
And it is manifest that the surface of the cylinder touches the
spheres,
since CF, throughout the whole motion, touches the semicircles
KCD, GFL.
Again, let the spheres be unequal, and let 4, 2 be their centres ;
let that sphere be greater, the centre of which is 4.
I say that the spheres are comprehended by one and the same
cone which has its vertex in the direction of the lesser sphere.
Let AB be joined, and let a plane be carried through 42;
this plane will cut the spheres in circles.
Let the circles be CDE, FGH;
therefore the circle CDE is greater than the circle GH; so that
the radius of the circle CDZ is also greater than the radius of the
circle FGH.
Now it is possible to take a point, as X (on 47 produced), such
that, as the radius of the circle CDZ is to the radius of the circle -
FGH, so is AK to KB.
Let the point X be so taken, and let KF be drawn touching the
circle FGH;
let FB be joined, and through 4 let 4 C be drawn parallel to BF;
1 The force of xara here is very difficult to render. The Greek phrase
ἐνεχθήσεται κατὰ τῶν σφαιρῶν means ‘ will be carried, or move, 7# the spheres’,
that is, the circumferences of the semicircles will pass neither over nor under the
surfaces of the spheres, but in coincidence with them throughout, in other words,
atl will by their revolution describe (as we say) the actual surfaces of the
spheres.
ἊΝ
2
x
᾿ ᾿
Ww FN
358 ON THE SIZES AND DISTANCES
καὶ ἐπεζεύχθω ἡ ΓΖ. καὶ ἐπεί ἐστιν, ὡς ἡ AK πρὸς τὴν KB, ἡ
AA πρὸς τὴν BN, ἴση δὲ ἡ μὲν AA τῇ AT, ἡ δὲ ΒΝ τῇ ΒΖ,
ἔστιν ἄρα, as ἡ AK πρὸς τὴν ΚΒ, ἡ AT πρὸς τὴν ΒΖ. καὶ ἔστιν
παράλληλος ἡ AI τῇ BZ: εὐθεῖα ἄρα ἐστὶν ἡ TZK. καὶ ἔστιν
5 ὀρθὴ ἡ ὑπὸ τῶν ΚΖΒ' ὀρθὴ ἄρα καὶ ἡ ὑπὸ τῶν KTA. ἐφάπτεται
ἄρα ἡ KI τοῦ TAE κύκλου. ἤχθωσαν δὴ αἱ TA, ΖΜ ἐπὶ τὴν
AB κάθετοι. ἐὰν δὴ μενούσης τῆς ΚΗ τά τε ἘΓΖ, HZN
ἡμικύκλια καὶ τὰ KIA, ΚΖΜ τρίγωνα περιενεχθέντα εἰς τὸ αὐτὸ
πάλιν ἀποκατασταθῇ ὅθεν ἤρξατο φέρεσθαι, τὰ μὲν BTA, ΗΖΝ
το ἡμικύκλια ἐνεχθήσεται κατὰ τῶν σφαιρῶν, τὸ δὲ KT'A τρίγωνον καὶ
τὸ ΚΖΜ γεννήσει κώνους, ὧν βάσεις εἰσὶν οἱ περὶ διαμέτρους τὰς
TE, ΖΘ κύκλοι, ὀρθοὶ ὄντες πρὸς τὸν KA ἄξονα: κέντρα δὲ αὐτῶν
τὰ A, Μ' καὶ ὁ κῶνος τῶν σφαιρῶν ἐφάψεται κατὰ τὴν ἐπιφάνειαν,
ἐπειδὴ καὶ ἡ KZI ἐφάπτεται τῶν ἘΓΖ, HZN ἡμικυκλίων κατὰ
15 πᾶσαν μετακίνησιν.
β΄.
Ἐὰν σφαῖρα ὑπὸ μείζονος ἑαυτῆς σφαίρας φωτίζηται,
μεῖζον ἡμισφαιρίου φωτισθήσεται.
Σφαῖρα γάρ, ἧς κέντρον τὸ Β, ὑπὸ μείζονος ἑαυτῆς σφαίρας
ao φωτιζέσθω, ἧς κέντρον τὸ A> λέγω ὅτι τὸ φωτιζόμενον μέρος τῆς
σφαίρας, ἧς κέντρον τὸ Β, μεῖζόν ἐστιν ἡμισφαιρίου.
Y
εἰ
Α 5. 15 Β
δὺ Ἢ β
Fig. 18.
᾿Επεὶ γὰρ δύο ἀνίσους edutbes ὁ αὐτὸς κῶνος περιλαμβάνει τὴν
κορυφὴν ἔχων πρὸς τῇ ἐλάσσονι σφαίρᾳ, ἔστω ὁ περιλαμβάνων τὰς
σφαίρας κῶνος, καὶ ἐκβεβλήσθω διὰ τοῦ ἄξονος ἐπίπεδον" ποιήσει
15 δὴ τομὰς ἐν μὲν ταῖς σφαίραις κύκλους, ἐν δὲ τῷ κώνῳ τρίγωνον.
8. ΚΓΔ4] KTAVat. ο. ἀποκατασταθῇ] ἀποκαταστῇ 14. BIA) ΖΓΔ Vat.
16. β Γ Vat. 17. φωτίζηται] φωτίζεται 22. κῶνος] κόνος Vat.
OF THE SUN AND MOON 359
let CF be joined.
Then since, as 4X is to KB, so is AD to BN,
while 4 D is equal to AC, and BN to BF,
therefore, as 4K is to KB, so is AC to BF.
And AC is parallel to BF;
therefore CF is a straight line.
Now the angle XF 2 is right ;
therefore the angle XC is also right:
therefore XC touches the circle CDE.
Let CZ, FM be drawn perpendicular to 4B.
If now, KO remaining fixed, the semicircles OCD, GFN and the
triangles KCL, KM be carried round and again restored to the
position from which they started, the semicircles OCD, GFN will
move in coincidence with the spheres; and the triangles KCZ and
KFM will generate cones, the bases of which are the circles about
CE, FH as diameters and at right angles to the axis KZ, the
centres of the circles being Z, /.
And the cone will touch the spheres along their surface, since
KFC also touches the semicircles OCD, GFN throughout the
whole motion.
PROPOSITION 2.
Tf a sphere be tlluminated by a sphere greater than itself,
the illuminated portion of the former sphere will be greater than
a hemisphere.
For let a sphere the centre of which is B be illuminated by -
a sphere greater than itself the centre of which is 4.
I say that the illuminated portion of the sphere the centre ot
which is B is greater than a hemisphere.
For, since two unequal spheres are comprehended by one and
the same cone which has its vertex in the direction of the lesser
sphere, [Prop. 1]
let the cone comprehending the spheres be (drawn), and let a plane
be carried through the axis ;
this plane will cut the spheres in circles and the cone in a triangle.
360 ON THE SIZES AND DISTANCES
ποιείτω οὖν ἐν μὲν ταῖς σφαίραις κύκλους τοὺς TAE, ΖΗΘ, ἐν δὲ
τῷ κώνῳ τρίγωνον τὸ TEK. φανερὸν δὴ ὅτι τὸ κατὰ τὴν ZHO
περιφέρειαν τμῆμα τῆς σφαίρας, οὗ βάσις ἐστὶν ὁ περὶ διάμετρον
τὴν ZO κύκλος, φωτιζόμενον μέρος ἐστὶν ὑπὸ τοῦ τμήματος τοῦ
5 kata τὴν ΓΔΕ περιφέρειαν, οὗ βάσις ἐστὶν 6 περὶ διάμετρον τὴν
TE κύκλος, ὀρθὸς ὧν πρὸς τὴν AB εὐθεῖαν: καὶ γὰρ ἡ ΖΗΘ
περιφέρεια φωτίζεται ὑπὸ τῆς TAE περιφερείας: ἔσχαται γὰρ
ἀκτῖνές εἰσιν αἱ ΓΖ, ΕΘ' καὶ ἔστιν ἐν τῷ ΖΗΘ τμήματι τὸ
κέντρον τῆς σφαίρας τὸ Β' ὥστε τὸ φπτιζόμεην μέρος τῆς σφαίρας
10 μεῖζόν ἐστιν ἡμισφαιρίου.
,
γ΄.
Ἔν τῇ σελήνῃ ἐλάχιστος κύκλος διορίζει τό τε σκιερὸν
καὶ τὸ λαμπρόν, ὅταν ὁ περιλαμβάνων κῶνος τόν τε ἥλιον
καὶ τὴν σελήνην τὴν κορυφὴν ἔχῃ πρὸς τῇ ἡμετέρᾳ ὄψ ει.
13 Ἔστω γὰρ ἡ μὲν ἡμετέρα ὄψις πρὸς τῷ A, ἡλίου δὲ κέντρον τὸ
Β, σελήνης δὲ κέντρον, ὅταν μὲν ὁ περιλαμβάνων κῶνος τόν τε
ἥλιον καὶ τὴν σελήνην τὴν κορυφὴν ἔχῃ πρὸς τῇ ἡμετέρᾳ ὄψει, τὸ T,
ὅταν δὲ μή, τὸ A> φανερὸν δὴ ὅτι τὰ A, I, Β ἐπ᾽ εὐθείας ἐστίν.
ἐκβεβλήσθω διὰ τῆς AB καὶ τοῦ A σημείου ἐπίπεδον" ποιήσει δὴ
2ο τομάς, ἐν μὲν ταῖς σφαίραις κύκλους, ἐν δὲ τοῖς κώνοις εὐθείας.
ποιείτω δὲ καὶ ἐν τῇ σφαίρᾳ, καθ᾽ ἧς φέρεται τὸ κέντρον τῆς σελήνης,
κύκλον τὸν I'd: τὸ Α ἄρα κέντρον ἐστὶν αὐτοῦ" τοῦτο γὰρ ὑπόκειται"
ἐν δὲ τῷ ἡλίῳ τὸν EZP κύκλον, ἐν δὲ τῇ σελήνῃ, ὅταν μὲν ὁ
περιλαμβάνων κῶνος τόν τε ἥλιον καὶ τὴν σελήνην τὴν κορυφὴν ἔχῃ
25 πρὸς τῇ ἡμετέρᾳ ὄψει, κύκλον τὸν KOA, ὅταν δὲ μή, τὸν MNE,
ἐν δὲ τοῖς κώνοις εὐθείας τὰς ΕΑ, AH, ΠΟ, ΟΡ, ἄξονας δὲ τοὺς
AB, ΒΟ. καὶ ἐπεί ἐστιν, ὡς ἡ ἐκ τοῦ κέντρου τοῦ EZH κύκλου
πρὸς τὴν ἐκ τοῦ κέντρου τοῦ OKA, οὕτως ἡ ἐκ τοῦ κέντρου τοῦ EZH
κύκλου πρὸς τὴν ἐκ τοῦ κέντρου τοῦ MNE- ἀλλ᾽ ὡς ἡ ἐκ τοῦ
4. τὴν ZO] ZOW 11. γΊ A Vat. 15. ἡλίου δὲ] ἡλίου W
16. μὲν] om. W 421. δὲ] δὴ W 25. KOA] OKA W 26. τοὺς] om. W
27. κύκλου] om. W
a ee ee |!
OF THE SUN AND MOON 361
Let it cut the spheres in the circles CDE, FGH, and the cone in
the triangle CE-K.
It is then manifest that the segment of the sphere towards the
circumference /GH, the base of which is the circle about 7H as
diameter, is the portion illuminated by the segment towards the
circumference CDE, the base of which is the circle about CZ as
diameter and at right angles to the straight line 4B;
for the circumference /GH is illuminated by the circumference
CDE, since CF, EH are the extreme rays.!
And the centre B of the sphere is within the segment />GZ;;
so that the illuminated portion of the sphere is greater than a
hemisphere.
PROPOSITION 3.
The circle in the moon which divides the dark and the bright
Portions ts least when the cone comprehending both the sun and
the moon has tts vertex at our eye.
For let our eye be at 4, and let 3 be the centre of the sun ;
let C be the centre of the moon when the cone comprehending both
_ the sun and the moon has its vertex at our eye, and, when this is
_ not the case, let D be the centre.
It is then manifest that 4, C, B are in a straight line.
Let a plane be carried through 4 # and the point D; this plane
_ will cut the spheres in circles and the cones in straight lines.
Let the plane also cut the sphere on which the centre of the
moon moves in the circle CD;
therefore 4 is the centre of this circle, for this is our hypothesis
[Hypothesis 2].
Let the plane cut the sun in the circle ER, and the moon, when
the cone comprehending both the sun and the moon has its vertex _
at our eye, in the circle KAZ and, when this is not the case, in th
circle UNO ;
and let it cut the cones in the straight lines EA, 4G, OP, PR, the
axes being 4B, BP.
Then since, as the radius of the circle EFG is to the radius of
the circle HXZ, so is the radius of the circle EFG to the radius of
the circle “NO,
Ὁ In Wallis’s figure the letters F, H are interchanged. With his lettering, the
extreme rays should be CH, EF. I have given F, # the positions necessary to
_ suit the text, and my figure agrees with that of Vat.
Βα"
462 ON THE SIZES AND DISTANCES
κέντρου τοῦ EZH κύκλου πρὸς τὴν ἐκ τοῦ κέντρου τοῦ OAK κύκλου,
οὕτως ἡ ΒΑ πρὸς τὴν AI ὡς δὲ ἡ ἐκ τοῦ κέντρου τοῦ EZH κύκλου
πρὸς τὴν ἐκ τοῦ κέντρου τοῦ ΜΝΈ, κύκλου, οὕτως ἐστὶν ἡ ΒΟ πρὸς
τὴν OA: καὶ ὡς ἄρα ἡ ΒΑ πρὸς τὴν AT, οὕτως ἡ ΒΟ πρὸς τὴν
504. καὶ διελόντι, ὡς ἡ ΒΓ πρὸς τὴν TA, οὕτως ἡ BA πρὸς τὴν
Fig. 19.
AO, καὶ ἐναλλάξ, ὡς ἡ BI πρὸς τὴν BA, οὕτως ἡ ΓΑ πρὸς τὴν AO.
καὶ ἔστιν ἐλάσσων ἡ BI' τῆς BA: κέντρον γάρ ἐστι τὸ A τοῦ TA
κύκλου: ἐλάσσων ἄρα καὶ ἡ ΑΓ τῆς 40. καὶ ἔστιν ἴσος 6 OKA
κύκλος τῷ MNE κύκλῳ: ἐλάσσων ἄρα ἐστὶν καὶ ἡ OA τῆς ΜΈΪ,
το διὰ τὸ λῆμμα] ὥστε καὶ ὁ περὶ διάμετρον τὴν OA κύκλος
γραφόμενος, ὀρθὸς ὧν πρὸς τὴν AB, ἐλάσσων ἐστὶν τοῦ περὶ διά-
μετρον τὴν ME κύκλου γραφομένου, ὀρθοῦ πρὸς τὴν ΒΟ. ἀλλ᾽ ὁ
μὲν περὶ διάμετρον τὴν OA κύκλος γραφόμενος, ὀρθὸς ὧν πρὸς τὴν
ΑΒ, ὁ διορίζων ἐστὶν ἐν τῇ σελήνῃ τό τε σκιερὸν καὶ τὸ λαμπρόν,
15 ὅταν ὁ περιλαμβάνων κῶνος τόν τε ἥλιον καὶ τὴν σελήνην τὴν
1. τοῦ ΕΖΗ) ΕΖΗ W τοῦ OAK] OKA Ὺ 5. διελόντι] διαιρεθέντι
W, qui lacunam post καί ope versionis Commandini expleverat
OF THE SUN AND MOON 363
while, as the radius of the circle HFG is to the radius of the circle
HLK, so is BA to AC,
and, as the radius of the circle HFG is to the radius of the circle
MNO, so is BP to PD,
therefore, as BA is to 4C,so is BP to PD,
and, separando, as BC is to CA, so is BD to DP;
therefore also, alternately, as BC is to BD, so is CA to DP.
And BC is less than BD, for A is the centre of the circle CD;
therefore 4 C is also less than DP.
And the circle XZ is equal to the circle MNO;
therefore HZ is also less than 170 [by the Lemma?].
Accordingly the circle drawn about #Z as diameter and at right
angles to 4B is also less than the circle drawn about W/O as
diameter and at right angles to BP.
But the circle drawn about HZ as diameter and at right angles
to 4B is the circle which divides the dark and the bright portions
in the moon when the cone comprehending both the sun and the
moon has its vertex at our eye;
1 The promised Lemma (the equivalent of which is stated, rather than proved,
in Euclid’s Optics, 24) does not appear. Some of the MSS. have a scholium
containing a rather clumsy proof. A shorter proof is that of Nizze. We can
use one circle instead of two equal circles; and we have to prove that, if 4, P
are points on the radius produced, P being further from the centre (C) than 4
Fig. 20.
is, and if AH, AZ be the pair of tangents from 4, and PM, PO the pair of
tangents from P, then ZO>AL.
By Eucl. vi. 8 and 17, CM@*?=CT.CP, and CH*=CS.CA; therefore
_CT.CP=CS.CA, or CA: CP=CT:CS. But CA< CP; therefore C7 < CS,
so that the chord AZ is less than the chord WO.
ΕΟ γα.
364 ON THE SIZES AND DISTANCES
κορυφὴν ἔχῃ πρὸς τῇ ἡμετέρᾳ ὄψει: ὁ δὲ περὶ διάμετρον τὴν MA
κύκλος, ὀρθὸς ὧν πρὸς τὴν ΒΟ, ὁ διορίζων ἐστὶν ἐν τῇ σελήνῃ τό τε
σκιερὸν καὶ τὸ λαμπρόν, ὅταν ὁ περιλαμβάνων κῶνος τόν τε ἥλιον
καὶ τὴν σελήνην μὴ ἔχῃ τὴν κορυφὴν πρὸς τῇ ἡμετέρᾳ ὄψει: ὥστε
5 ἐλάσσων κύκλος διορίζει ἐν τῇ σελήνῃ τό τε σκιερὸν καὶ τὸ λαμπρόν,
ὅταν ὁ περιλαμβάνων κῶνος τόν τε ἥλιον καὶ τὴν σελήνην τὴν
κορυφὴν ἔχῃ πρὸς τῇ ἡμετέρᾳ ὄψει.
δ:
‘O διορίζων κύκλος ἐν τῇ σελήνῃ τό TE σκιερὸν καὶ τὸ
ιολαμπρὸν ἀδιάφορός ἐστι τῷ ἐν τῇ σελήνῃ μεγίστῳ κύκλῳ
πρὸς αἴσθησιν.
"Ἔστω γὰρ ἡ μὲν ἡμετέρα ὄψις πρὸς τῷ A, σελήνης δὲ κέντρον τὸ
B, καὶ ἐπεζεύχθω ἡ AB, καὶ ἐκβεβλήσθω διὰ τῆς AB ἐπίπεδον"
ποιήσει δὴ τομὴν ἐν τῇ σφαίρᾳ μέγιστον κύκλον. ποιείτω τὸν
τ ΕΓΔΖ, ἐν δὲ τῷ κώνῳ εὐθείας τὰς AT, AA, 4Τ' ὁ ἄρα περὶ
Fe
D
Fig. 21.
διάμετρον τὴν ΓΖ, πρὸς ὀρθὰς ὧν τῇ AB, ὁ διορίζων ἐστὶν ἐν τῇ
σελήνῃ τό τε σκιερὸν καὶ τὸ λαμπρόν. λέγω δὴ ὅτι ἀδιάφορός ἐστι
τῷ μεγίστῳ πρὸς τὴν αἴσθησιν."
Ἤχθω γὰρ διὰ τοῦ Β τῇ TA παράλληλος ἡ ΕΖ, καὶ κείσθω
,οτῆς AZ ἡμίσεια ἑκατέρα τῶν HK, HO, καὶ ἐπεζξύχθωσαν αἱ KB,
ΒΘ, ΚΑ, ΑΘ, BA. καὶ ἐπεὶ ὑπόκειται ἡ σελήνη ὑπὸ ιε΄ μέρος
1. τὴν] τὸν Vat. 2. τὴν] τὸν Vat. 43, 4. τόν τε ἥλιον καὶ τὴν σελήνην] om. W
8. δΊ Ε Vat. 12. τῷ] τὸ W
OF THE SUN AND MOON 365
and the circle about 4/O as diameter and at right angles to BP is
the circle which divides the dark and the bright portions in the
moon when the cone comprehending both the sun and the moon
has not its vertex at our eye.
Accordingly the circle which divides the dark and the bright
portions in the moon is less when the cone comprehending both the
sun and the moon has its vertex at our eye.
PROPOSITION 4.
The circle which divides the dark and the bright portions in
the moon 5 not percepithly different from a great circle in the
For let our eye be at 4, and let B be the centre of the moon.
Let AB be joined, and let a plane be carried through 42;
this plane will cut the sphere in a great circle.
Let it cut the sphere in the circle CDF and the cone in the
straight lines 4C, 4D, DC.
Then the circle about CD as diameter and at right angles to 4 B
is the circle which divides the dark and the bright portions in the
moon.
I say that it is not perceptibly different from a great circle.
For let EF be drawn through BZ parallel to CD;
let GX, GH both be made (equal to) half of DF;
and let KB, BH, KA, AH, BD be joined.
Then since, by hypothesis, the moon subtends a fifteenth part of
a sign of the zodiac,
466 ON THE SIZES AND DISTANCES
ἑῳδίου ὑποτείνουσα, ἡ dpa ὑπὸ TAA γωνία βέβηκεν ἐπὶ ιε΄ μέρος
(odiov. τὸ δὲ ιε΄ τοῦ ἑῳδίου τοῦ τῶν ἑῳδίων ὅλου κύκλου ἐστὶν pm’,
ὥστε ἡ ὑπὸ τῶν TAA γωνία βέβηκεν ἐπὶ pr’ ὅλου τοῦ κύκλου"
τεσσάρων ἄρα ὀρθῶν ἐστιν ἡ (ὑπὸ TAA ρπ΄. διὰ δὴ τοῦτο ἡ ὑπὸ
8 ΓΑΔ γωνία ἐστὶν με΄ ὀρθῆς" καὶ ἔστιν αὐτῆς ἡμίσεια ἡ ὑπὸ ΒΑΔ
γωνία: ἡ ἄρα ὑπὸ τῶν BAA ἡμισείας ὀρθῆς ἐστι (με) μέρος, καὶ
ἐπεὶ ὀρθή ἐστιν ἡ ὑπὸ τῶν 448, ἡ ἄρα ὑπὸ τῶν BAA γωνία πρὸς
ἥμισυ ὀρθῆς μείζονα λόγον ἔχει ἤπερ ἡ BA πρὸς τὴν AA, ὥστε ἡ
BA τῆς 4A ἐλάσσων ἐστὶν ἢ με΄ μέρος, ὥστε καὶ ἡ BH τῆς BA
το πολλῷ ἐλάσσων ἐστὶν ἢ με΄ μέρος. διελόντι ἡ ΒΗ͂ τῆς HA
ἐλάσσων ἐστὶν ἣ μδ΄ μέρος, ὥστε καὶ ἡ BO τῆς AO πολλῷ
6. ἡμισείας corr. 6 μιᾶς, ut videtur, Vat. et Paris. 2342: μιᾶς F Paris. 2366,
2472 (?), 2488 {με') om. Vat. et alii codd. μέρος] με Paris. 2342 erasis
litteris pos 10. δεελόντι] καὶ διαιρεθέντι W, qui lacunam post 10 ἢ ope
versionis Commandini expleverat
1 This is a particular case of the more general proposition (similarly —
assumed by Archimedes in his Sand-reckoner) which amounts to the statement
that, if each of the angles Οἱ, 8 is not greater than a right angle, and & >, then
ἴλη ἃ Οἱ
tan β »Β ᾿
The proposition is easily proved geometrically (cf. Commandinus on the
passage of the Sand-reckoner).
Let BC, BA make with ACD the angles “,8 respectively, and let BD be
perpendicular to AD.
At ER ἢ
Fig. 22.
Now ἰδ αἱ τε BD/CD, tanB= BD/AD.
We have therefore to prove that
AD: CD>4:8.
OF THE SUN AND MOON 367
therefore the angle C4 D stands on a fifteenth part of a sign.
But a fifteenth part of a sign is 1/180th of the whole circle of the
zodiac,
so that the angle CAD stands on 1/180th of the whole circle;
therefore the angle C4 D is 1/180th of four right angles.
It follows that the angle C4 D is 1/45th of a right angle.
And the angle 24 D is half of the angle C4 D;
therefore the angle BA D is 1/45th part of half a right angle.
Now, since the angle 4 DB is right,
the angle BAD has to half a right angle a ratio greater than that
which BD has to DA.
Accordingly BD is less than 1/45th part of DA.
_ Therefore BG is much less* than 1/45th part of BA, and,
separando, BG is less than 1/44th part of GA.
Accordingly BZ is also much less than 1/44th part of 4 Z.
Cut off AF equal to CD, and draw FE at right angles to 4D and equal to
BD. Join 45.
Then LEAF=ZBCD=4.
Let EF meet AB in G.
Since AE > AG >AF, the circle with 4 as centre and AG as radius will
cut AZ in H and AF produced in Κ΄.
Now LEAG: £GAF = (sector HAG) : (sector GAK)
< AEAG: AGAF
< EG: GF.
Componendo, LEAF :LGAF< EF: GF.
But EF:GF=BD:GF=AD:AF=AD:CD.
Therefore a:B<AD: CD,
or AD:CD>a: 8.
In the particular application above made by Αὐπίαξονυς &=3R, so that
CD = BD.
In this case therefore AD: DB >4R:2ZBAD,
or BD: DA<LZBAD:}R,
that is to say, LBAD:4R>BD: DA.
2 “Much less’, πολλῷ ἐλάσσων = ‘less by much’. πολλῷ μείζων and πολλῷ
ἐλάσσων are the traditional expressions used by Euclid and Greek geometers in
general for ‘a fortiori greater’ and ‘a fortiori less’. In Euclid the expressions
have generally been translated ‘ much more then is. . . greater, or less, than’.
But there is no double comparative in the Greek. The idea is that, if a is, let
us say, a /:¢¢/e greater than 4, and if c is greater than a, then ς must be much
greater than ὁ.
ΞΟ νυ
468 ON THE SIZES AND DISTANCES
ἐλάσσων ἐστὶν ἢ μδ΄ μέρος. καὶ ἔχει ἡ BO πρὸς τὴν OA μείζονα
λόγον ἤπερ ἡ ὑπὸ τῶν BAO πρὸς τὴν ὑπὸ τῶν 4ΒΘ' ἡ ἄρα ὑπὸ
τῶν ΒΑΘ τῆς ὑπὸ τῶν ABO ἐλάσσων ἐστὶν ἢ μδ΄ μέρος. καὶ ἔστιν
τῆς μὲν ὑπὸ τῶν BAO διπλῆ. ἡ ὑπὸ τῶν KAO, τῆς δὲ ὑπὸ τῶν
5 ABO διπλῆ ἡ ὑπὸ τῶν ΚΒΘ' ἐλάσσων ἄρα ἐστὶν καὶ ἡ ὑπὸ τῶν
ΚΑΘ τῆς ὑπὸ τῶν KBO 4 τεσσαρακοστοτέταρτον μέρος. ἀλλὰ ἡ
᾿
ὑπὸ τῶν ΚΒΘ ἴση ἐστὶν τῇ ὑπὸ τῶν ABZ, τουτέστιν τῇ ὑπὸ τῶν
TAB, τουτέστιν τῇ ὑπὸ τῶν BAA: ἡ ἄρα ὑπὸ τῶν KAO τῆς ὑπὸ
τῶν BAA ἐλάσσων ἐστὶν ἢ μδ΄ μέρος. ἡ δὲ ὑπὸ τῶν BAA (ἡμισείας),
10 ὀρθῆς ἐστιν (με) μέρος, ὥστε ἡ ὑπὸ τῶν KAO ὀρθῆς ἐστιν ἐλάσσων
5,6. ἐλάσσων... : ἢ] (ὥστε ἡ KAO γωνία τῆς KBO γωνίας ἐλάσσων ἐστὶν }) W
9. (ἡμισείας, 10. (ue’), supplevit W Io. (τουτέστι τῆς ὀρθῆς ς' μέρος) post μέρος
addidit W
1 This is immediately deducible from a proposition given by Ptolemy
(Syntaxis, 1. Lo, pp. 43-4, ed. Heiberg).
If two unequal chords are drawn in a circle, the greater has to the lesser
a ratio less than the circumference (standing) on the greater chord has to the
circumference (standing) on the lesser.
That is, if CB, BA be unequal chords in a circle, aad CB > BA, then
(chord CB) : (chord BA) < (arc CB) : (arc BA).
Ptolemy’s proof is as follows.
Bisect the angle 4 BC by the straight line BD, meeting the circle again at D.
Join AEC, AD, CD
D
Fig. 23.
Then, since the angle ABC is bisected by BD,
CD=AD. [Eucl., iii. 26, 29.)
And CE>E£EA. [Eucl., vi. 3.]
Draw DF perpendicular to 4.50.
OF THE SUN AND MOON 369
And BF has to HA a ratio greater than that which the angle
BAZ has to the angle 4 2H
Therefore the angle B47 is less than 1/44th part of the angle
ABH.
And the angle X-4 7 is double of the angle B4Z,
while the angle XZ is double of the angle 4 BH;
therefore the angle X47 is also less than 1/44th part of the angle
KBH.
But the angle KPH is equal to the angle DZF, that is, to the
angle CDB, that is, to the angle BAD.
Therefore the angle X_4 Z is less than 1/44th part of the angle
BAD.
But the angle BAD is 1/45th part of half a right angle.
Accordingly the angle X-4 is less than 1/3960th ofa right angle.”
Now, since DA > DE > DF, the circle described with D as centre and DE
as radius will cut AD between 4 and D, and will cut DF produced beyond F
Let the circle be drawn.
Since the triangle AZD is greater than the sector DEG, and the triangle
_DEF is less than the sector DEH,
4 DEF: 4 DEA < (sector DEH) : (sector DEG).
_ Therefore FE:EA<ZFDE:ZEDA. [{Eucl.,vi.tand 33.]
Componendo, FA:EA< LFDA:LEDA.
Doubling the antecedents, we have
CA: AE<LZCDA:LADE,
and, separando, CE: EA <ZCDE:LEDA.
But CE: EA = CB: BA,
and LCDE: L£EDA = (arc CB) : (arc BA).
Therefore CB: BA < (arc CB) : (arc BA).
[The proposition is easily seen to be equivalent to the statement that, if «
is an angle not greater than a right angle, and 8 another angle less than αὶ, then
sin &X | a ]
sin § < β᾽
Now, since CDE =ZCAB and 4ADE=ZACB, in the same segments,
we have
CB: BA < ZLCAB:ZACB,
ΟΥ̓́, inversely, AB:BC>LZACB:LBAG,
which is the property assumed by Aristarchus.
hod d= sits:
1410 Bb
470 ON THE. SIZES AND DISTANCES
K
ἢ yA. τὸ δὲ ὑπὸ τηλικαύτης γωνίας ὁρώμενον μέγεθος ἀνεπαί-
σθητόν ἐστιν τῇ ἡμετέρᾳ ὄψει: καὶ ἔστιν ἴση ἡ ΚΘ περιφέρεια τῇ
AZ περιφερείᾳ: ἔτι ἄρα μᾶλλον ἡ AZ περιφέρεια ἀνεπαίσθητός
ἐστι τῇ ἡμετέρᾳ ὄψει. ἐὰν γὰρ ἐπιζευχθῇ ἡ AZ, ἡ ὑπὸ τῶν Ζ44
5 γωνία ἐλάσσων ἐστὶ τῆς ὑπὸ τῶν ΚΑΘ. τὸ A ἄρα τῷ Ζ τὸ αὐτὸ
δόξει εἶναι. διὰ τὰ αὐτὰ δὴ καὶ τὸ Τ' τῷ Ε δόξει τὸ αὐτὸ εἶναι:
ὥστε καὶ ἡ TA τῇ EZ ἀνεπαίσθητός ἐστιν. καὶ ὁ διορίζων ἄρα ἐν
τῇ σελήνῃ τό τε σκιερὸν καὶ τὸ λαμπρὸν ἀνεπαίσθητός ἐστι τῷ
μεγίστῳ.
΄,
1ο ε.
Ὅταν ἡ σελήνη διχότομος ἡμῖν φαίνηται, τότε 6
μέγιστος κύκλος ὁ παρὰ τὸν διορίζοντα ἐν τῇ σελήνῃ τό
τε σκιερὸν καὶ τὸ λαμπρὸν νεύει εἰς τὴν ἡμετέραν ὄψιν,
τουτέστιν, ὁ παρὰ τὸν διορίζοντα μέγιστος κύκλος καὶ ἡ
15 ἡμετέρα ὄψις ἐν ἑνί εἰσιν ἐπιπέδῳ.
᾿Επεὶ γὰρ διχοτόμου οὔσης τῆς σελήνης φαίνεται ὁ διορίζων τό τε
λαμπρὸν καὶ τὸ σκιερὸν τῆς σελήνης κύκλος νεύων εἰς τὴν ἡμετέραν
ὄψιν, καὶ αὐτῷ ἀδιάφορος ὁ παρὰ τὸν διορίζοντα μέγιστος κύκλος,
ὅταν ἄρα ἡ σελήνη διχότομος ἡμῖν φαίνηται, τότε ὁ μέγιστος κύκλος
20 ὁ παρὰ τὸν διορίζοντα νεύει εἰς τὴν ἡμετέραν ὄψιν.
’
Se
Ἡ σελήνη κατώτερον φέρεται τοῦ ἡλίου, καὶ διχότομος
οὖσα ἔλασσον τεταρτημορίου ἀπέχει ἀπὸ τοῦ ἡλίου.
Ἔστω γὰρ ἡ ἡμετέρα ὄψις πρὸς τῷ 4, ἡλίου δὲ κέντρον τὸ Β, καὶ
as ἐπιζευχθεῖσα ἡ AB ἐκβεβλήσθω, καὶ ἐκβεβλήσθω διὰ τῆς AB καὶ
τοῦ κέντρου τῆς σελήνης διχοτόμου οὔσης ἐπίπεδον: ποιήσει δὴ
τομὴν ἐν τῇ σφαίρᾳ, καθ᾽ ἧς φέρεται τὸ κέντρον τοῦ ἡλίου, κύκλον
1. γ Ae] W’p'd’ Vat.: yy Ab’ péposW 7. ἀνεπαίσθητός] sic Vat. 7,8. καὶ
ὁ διορίζων dpa ἐν... ἀνεπαίσθητός ἐστι] (ὁ dpa διορίζων κύκλος ἐν .. . ἀδιάφορός
ἐστι πρὸς αἴσθησιν) supplevit W, qui lacunam in suo codice animadverterat
10. εἼ ς Vat. 13. λαμπρὸν] λαμπρὸν αὐτοῦ W: λαμπρὸν αὐτῆς Nizze
18. ἀδιάφορος] ἀδιάφορός ἐστιν W 19. φαίνηται] WF: φανῆται Vat. ;
21... om. Vat. 22. φέρεται] WF Paris. 2364, 2472 (?): φαίνεται Vat. (in
ras. sed ν quasi in p mutato) Paris. 2366. 24. τῷ] ro W
OF THE SUN AND MOON 371
But a magnitude seen under such an angle is imperceptible to
our eye.
And the circumference XH is equal to the circumference DF;
therefore still more is the circumference DF imperceptible to
our eye;
for, if 4 F be joined, the angle #4 D is less than the angle K4 H.
Therefore D will seem to be the same with F.
For the same reason, C will also seem to be the same with £.
Accordingly CD is not perceptibly different? from EF.
Therefore the circle which divides the dark and the bright por-
tions in the moon is not perceptibly different from a great circle.
PROPOSITION 5.
When the moon appears to us halved, the great circle parallel
to the circle which divides the dark and the bright portions in
the moon ts then in the direction of our eye; that is to say, the
great circle parallel to the dividing circle and our eye are in one
plane.
For since, when the moon is halved, the circle which divides the
bright and the dark portions of the moon is in the direction of our eye
[Hypothesis 3], while the great circle parallel to the dividing circle
is indistinguishable from it,
therefore, when the moon appears to us halved, the great circle
parallel to the dividing circle is then in the direction of our eye.
PROPOSITION 6.
The moon moves (in an orbit) lower than (that of) the sun, and,
when wt ts halved, ts distant less than a quadrant from the sun.
For let our eye be at 4, and let B be the centre of the sun; let
AB be joined and produced, and let a plane be carried through
A B and the centre of the moon when halved;
this plane will cut in a great circle the sphere on which the centre
of the sun moves.
1 Pappus (pp. 560-8, ed Hultsch) gives an elaborate proof of this proposition
depending on two lemmas ; the proof, however, in the text as we have it, contains
a serious flaw (p. 568. 2-3). But the truth of the assumption in Aristarchus’s
particular case is so obvious as scarcely to require proof.
3 ἀνεπαίσθητος is strangely used with dat. as if equivalent to ἀνεπαισθήτως
_ διάφορος or ἀδιάφορος πρὸς αἴσθησιν, ‘imperceptibly different from’.
Bb2
472 ON THE SIZES AND DISTANCES
μέγιστον. ποιείτω οὖν τὸν ΓΒΔ κύκλον, καὶ ἀπὸ τοῦ A τῇ AB
πρὸς ὀρθὰς ἤχθω ἡ TAA: τεταρτημορίου ἄρα ἐστὶν ἡ BA περι-
φέρεια. λέγω ὅτι ἡ σελήνη κατώτερον φέρεται τοῦ ἡλίου, καὶ
διχότομος οὖσα ἔλασσον τεταρτημορίου ἀπέχει ἀπὸ τοῦ ἡλίου, τουτ-
5 éorw, ὅτι τὸ κέντρον ἐστὶν αὐτῆς μεταξὺ τῶν BA, AJA εὐθειῶν
καὶ τῆς AEB περιφερείας. |
Εἰ yap μή, ἔστω τὸ κέντρον αὐτῆς τὸ Ζ μεταξὺ τῶν 44, AA
εὐθειῶν, καὶ ἐπεζεύχθω ἡ ΒΖ. ἡ ΒΖ ἄρα ἄξων ἐστὶν τοῦ περι-
Fig. 24.
λαμβάνοντος κώνου τόν τε ἥλιον καὶ τὴν σελήνην, καὶ γίνεται ἡ
το ΒΖ ὀρθὴ πρὸς τὸν διορίζοντα ἐν τῇ σελήνῃ τό τε σκιερὸν καὶ τὸ
λαμπρὸν μέγιστον κύκλον. ἔστω οὖν ὁ μέγιστος κύκλος ἐν τῇ
σελήνῃ ὁ παρὰ τὸν διορίζοντα τό τε σκιερὸν καὶ τὸ λαμπρὸν ὁ ΗΘΚ.
καὶ ἐπεὶ διχοτόμου οὔσης τῆς σελήνης ὁ μέγιστος κύκλος ὁ παρὰ
τὸν διορίζοντα ἐν τῇ σελήνῃ τό τε σκιερὸν καὶ τὸ λαμπρὸν καὶ ἡ
15 ἡμετέρα ὄψις ἐν ἑνί εἰσιν ἐπιπέδῳ, ἐπεζεύχθω ἡ 4Ζ' ἡ AZ ἄρα ἐν
I. τὸν] τὸ Vat. 4. ἔλασσον] ἔλαττον Vat.
OF THE SUN AND MOON 373
Let it cut it in the circle CBD; and from A let CAD be drawn
at right angles to 4B.
Then the circumference BD is that of a quadrant.
I say that the moon moyes (in an orbit) lower than (that of) the
sun, and, when halved, is distant less than a quadrant from the sun ;
that is to say, its eentre is between the straight lines 24, 4D and
the circumference DEB.
For, if not, let its centre / be between the straight lines D4,
AL, and let BF be joined ;
then BF is the axis of the cone which comprehends both the sun
and the moon,
_ and BF is at right angles to the great circle’ which divides the
_ dark and the bright portions in the moon.
Let the great circle in the moon parallel to the circle which
_ divides the dark and the bright portions be GHX;* then since,
when the moon is halved, the great circle parallel to the circle which
divides the dark and the bright portions in the moon and our eye
are in one plane [Prop. 5], let 4 be joined.
Therefore AF is in the plane of the circle KGH.
1 It is of course not actually a great circle, but a circle parallel to a great
circle, which is however so close to it as to be indistinguishable from a great
circle so far as our vision of it is concerned [Prop. 4]. The expression is there-
fore excusable, as in Hypothesis 3; there is no need to omit μέγιστον from the
text as Nizze does.
2 T have drawn the circle GH< and the other circles representing the sections
of the moon as they are drawn in Wallis’s figures ; but I think the circles in the
moon defining the dark and bright portions and, by hypothesis, in the same
plane with our eye would be better represented by the dotted circles which
I have added to the figure.
374 ON THE SIZES AND DISTANCES
τῷ τοῦ ΚΗΘ κύκλου ἐστὶν ἐπιπέδῳ. καὶ ἔστιν ἡ BZ τῷ KOH
κύκλῳ πρὸς ὀρθάς, ὥστε καὶ τῇ 4Ζ: ὀρθὴ ἄρα ἐστὶν ἡ ὑπὸ ΒΖΑ
γωνία. ἀλλὰ καὶ ἀμβλεῖα ἡ ὑπὸ τῶν ΒΑΖ' ὅπερ ἀδύνατον. οὐκ
ἄρα τὸ Ζ σημεῖον ἐν τῷ ὑπὸ τὴν AAA γωνίαν τόπῳ ἐστίν.
5. Λέγω ὅτι οὐδὲ ἐπὶ τῆς AA. εἰ γὰρ δυνατόν, ἔστω τὸ Μ, καὶ
πάλιν ἐπεζεύχθω ἡ ΒΜ, καὶ ἔστω μέγιστος κύκλος ὁ παρὰ τὸν
διορίζοντα, οὗ κέντρον τὸ Μ. κατὰ τὰ αὐτὰ δὴ δειχθήσεται ἡ ὑπὸ
BMA γωνία ὀρθὴ πρὸς τὸν μέγιστον κύκλον: ἀλλὰ καὶ ἡ ὑπὸ τῶν
BAM: ὅπερ ἀδύνατον. οὐκ ἄρα ἐπὶ τῆς AA τὸ κέντρον ἐστὶ τῆς
το σελήνης διχοτόμου οὔσης" μεταξὺ ἄρα τῶν AB, AA ἐστίν.
Λέγω δὴ ὅτι καὶ ἐντὸς τῆς BA περιφερείας. εἰ γὰρ δυνατόν,
ἔστω ἐκτὸς κατὰ τὸ Ν, καὶ τὰ αὐτὰ κατεσκευάσθω. δειχθήσεται δὴ
ἡ ὑπὸ τῶν BNA γωνία ὀρθή: μείζων ἄρα ἐστὶν ἡ BA τῆς AN.
ἴση δὲ ἡ BA τῇ AE: μείζων ἄρα ἐστὶν καὶ ἡ AE τῆς AN: ὅπερ
15 ἀδύνατον. οὐκ ἄρα τὸ κέντρον τῆς σελήνης διχοτόμου οὔσης ἐκτὸς
ἔσται τῆς BEA περιφερείας. ὁμοίως δειχθήσεται ὅτι οὐδὲ ἐπ᾽
αὐτῆς τῆς BEA περιφερείας" ἐντὸς dpa. ἡ ἄρα σελήνη κατώτερον
φέρεται τοῦ ἡλίου, καὶ διχότομος οὖσα ἔλασσον τεταρτημορίου
ἀπέχει ἀπὸ τοῦ ἡλίου.
1. ΚΘΗ] KHO W 2. ὑπὸ] ὑπὸ {τῶν W 4. τὴν JAA γωνίαν
inusitato sane dicendi more: τὴν ζὑπὸ τῶν) 4444 γωνίαν W, sed dubito an ipse
Aristarchus ὑπὸ τὴν ὑπὸ τῶν scripserit 11. Β4] ΒΕΔ W 12. κατε-
σκευάσθω] κατασκευάσθω Vat. 13. γωνία] om. W 14. καὶ] om. W
17. ἐντὸς] W F Paris. 2364, 2472 (3) : ἐκτὸς Vat. in ras., Paris. 2363
1 The phrase in the Greek text, κατὰ τὰ αὐτὰ δὴ δειχθήσεται ἡ ὑπὸ BMA γωνία
ὀρθὴ πρὸς τὸν μέγιστον κύκλον, is strange. Literally this would appear to mean
‘In the same way it can be proved that the angle BMA is at right angles to
the great circle’, but this is intolerable. If we took ‘the angle 217A’ to be
the A/ane of the angle, the expression would be possible, but it would not give
the meaning which is required, namely that the angle 247A is a right angle
because 577 is at right angles to the plane of the circle and therefore to any
straight line in the plane of the circle, such as A, passing through 47, The
OF THE SUN AND MOON 375
And BF is at right angles to the circle XHG, and therefore to
AF;; therefore the angle BFA is right.
But the angle 4 is also obtuse: which is impossible.
Therefore the point 2515 not in the space bounded by the angle
DAL.
I say that neither is it on 4D.
For, if possible, let it be 17; and again let B// be joined, and let
the great circle parallel to the dividing circle be taken, its centre
being J.
Then, in the same way as before, it can be proved that the angle
BMA [made with the great circle]? is right.
But the angle 5.4 Π is so also: which is impossible.
Therefore the centre of the moon, when halved, is not on 4D.
Therefore it is between 42 and 4D.
Again, I say that it is also within the circumference BD.
For, if possible, let it be outside, at V;
and let the same construction be made.
It can then be proved that the angle B14 is right; therefore BA
is greater than 4 JV.
But BA is equal to 44;
therefore 4Z is also greater than 4 iV: which is impossible.
Therefore the centre of the moon, when halved, will not be out-
side the circumference BED.
Similarly it can be proved that neither will it be on the circum-
ference BED itselt.
Therefore it will be within.
Therefore, &c.
words πρὸς τὸν μέγιστον κύκλον are in fact not wanted, and, if they are
retained, cannot be taken with ὀρθή in the sense of ‘at right angles to the great
circle’; they can only be taken closely with γωνία and as meaning ‘ towards the
great circle’, or ‘made with the great circle’. But, as the words do not occur in
the corresponding passage about the angle BNA _ lower down, I think they
should be struck out, as an interpolation by some one who thought the inference
wanted some further explanation but failed to supply it intelligibly.
476 ON THE SIZES AND DISTANCES
ζ.
Τὸ ἀπόστημα ὃ ἀπέχει ὁ ἥλιος ἀπὸ τῆς γῆς τοῦ ἀπο-
στήματος οὗ ἀπέχει ἡ σελήνη ἀπὸ τῆς γῆς μεῖζον μέν
ἐστιν ἢ ὀκτωκαιδεκαπλάσιον, ἔλασσον δὲ ἣ εἰκοσαπλάσιον.
5 Ἔστω γὰρ ἡλίου μὲν κέντρον τὸ A, γῆς δὲ τὸ B, καὶ ἐπιζευχθεῖσα
ἡ AB ἐκβεβλήσθω, σελήνης δὲ κέντρον διχοτόμου οὔσης τὸ T, καὶ
ἐκβεβλήσθω διὰ τῆς AB καὶ τοῦ I’ ἐπίπεδον, καὶ ποιείτω τομὴν ἐν
τῇ σφαίρᾳ, καθ᾽ ἧς φέρεται τὸ κέντρον τοῦ ἡλίου, μέγιστον κύκλον
τὸν 44Ε, καὶ ἐπεζεύχθωσαν ai AT, ΓΒ, καὶ ἐκβεβλήσθω ἡ ΒΤ'
10 ἐπὶ τὸ 4. ἔσται δή, διὰ τὸ τὸ Γ' σημεῖον κέντρον εἶναι τῆς σελήνης
διχοτόμου οὔσης, ὀρθὴ ἡ ὑπὸ τῶν ΑΓΒ. ἤχθω δὴ ἀπὸ τοῦ Β τῇ
BA πρὸς ὀρθὰς ἡ ΒΕ. ἔσται δὴ ἡ EA περιφέρεια τῆς EAA
περιφερείας X+ ὑπόκειται γάρ, ὅταν ἡ σελήνη διχότομος ἡμῖν pat-
νηται, ἀπέχειν ἀπὸ τοῦ ἡλίου ἔλασσον τεταρτημορίου τῷ τοῦ
15 Τεταρτημορίου λ΄: ὥστε καὶ ἡ ὑπὸ τῶν EBT γωνία ὀρθῆς ἐστι X.
συμπεπληρώσθω δὴ τὸ AE παραλληλόγραμμον, καὶ ἐπεζεύχθω ἡ
ΒΖ. ἔσται δὴ ἡ ὑπὸ τῶν ΖΒΕ γωνία ἡμίσεια ὀρθῆς. τετμήσθω
ἡ ὑπὸ τῶν ZBE γωνία δίχα τῇ BH εὐθείᾳ: ἡ ἄρα ὑπὸ τῶν HBE
γωνία τέταρτον μέρος ἐστὶν ὀρθῆς. ἀλλὰ καὶ ἡ ὑπὸ τῶν ABE
20 γωνία λ΄ ἐστι μέρος ὀρθῆς: λόγος ἄρα τῆς ὑπὸ τῶν HBE γωνίας
πρὸς τὴν ὑπὸ τῶν ABE γωνίαν (ἐστὶν) ὃν (Exe) τὰ te πρὸς τὰ δύο"
οἵων γάρ ἐστιν ὀρθὴ γωνία £, τοιούτων ἐστὶν ἡ μὲν ὑπὸ τῶν HBE te,
ἡ δὲ ὑπὸ τῶν ABE δύο. καὶ ἐπεὶ ἡ ἨῈ πρὸς τὴν EO μείζονα
λόγον ἔχει ἤπερ ἡ ὑπὸ τῶν HBE γωνία πρὸς τὴν ὑπὸ τῶν ABE
3. οὗ] ὃ W F Nizze, sed nihil mutandum 4. εἰκοσαπλάσιον) εἰκοσιπλάσιον
WwW 6. τὸ ΓῚ (ἔστων τὸ I’ Nizze 9. Br] ΓΒᾺΝ 12. BE] add.
καὶ ἐκβεβλήσθω ἡ BI" ἐπὶ τὸ A Vat. Paris. 2364, 2366, 2472 (ἢ) 13. λΊ τρια-
κοστόν W 14. τῷ] om. W 15. λΊ τριακοστῷ W τῶν] τὴν Vat.
λΊ τριακοστόν W 18. ὑπὸ τῶν (ad init.)] ὑπὸ W 20. λΊ τριακοστόν W
21. γωνίαν] γωνίαν (ἐστὶν) Nizze ἔχει] om. Vat. 23. ἡ δὲ ὑπὸ τῶν] ἡ δὲ W
OF THE SUN AND MOON 377
PROPOSITION 7.
The distance of the sun from the earth ts greater than eighteen
times, but less than twenty times, the distance of the moon from
the earth.
For let 4 be the centre of the sun, B that of the earth.
Let AB be joined and produced.
Let C be the centre of the moon when haived ;
let a plane be carried through 4 BZ and C, and let the section made
by it in the sphere on which the centre of the sun moves be the
great circle 4 DE.
Let AC, CB be joined, and let BC be produced to D.
Then, because the point C is the centre of the moon when
halved, the angle 4 CB will be right.
Let BE be drawn from # at right angles to BA ;
then the circumference ED will be one-thirtieth of the circumterence
EDA;
for, by hypothesis, when the moon appears to us halved, its dis-
tance from the sun is less than a quadrant by one-thirtieth of a
quadrant [Hypothesis 4].
Thus the angle EBC is also one-thirtieth of a right angle.
Let the parallelogram 4 Z be completed, and let BF be joined.
Then the angle FBZ will be half a right angle.
Let the angle BE be bisected by the straight line BG;
therefore the angle GZ is one fourth part of a right angle.
But the angle 29 is also one thirtieth part of a right angle;
therefore the ratio of the angle GE to the angle DBZ is that
which 15 has to 2:
for, if a right angle be regarded as divided into 60 equal parts, the
angle GE contains 15 of such parts, and the angle DBE
contains 2.
Now, since GE has to EH a ratio greater than that which the
angle GBE has to the angle DBE}
1 The proposition assumed is again the equivalent of the fact that =a » ξ >
where each of the angles &, 8 is not greater than a right angle and a>. (Cf.
note on pp. 366-7, above.) Let the angles a, 8 be the angles GRE, HBE
respectively in the subjoined figure (Fig. 26). Let GE be perpendicular to BE
378 ON THE SIZES AND DISTANCES
γωνίαν, ἡ ἄρα HE πρὸς τὴν EO μείζονα λόγον ἔχει ἤπερ τὰ τε
πρὸς τὰ β. καὶ ἐπεὶ ἴση ἐστὶν ἡ ΒΕ τῇ ΕΖ, καὶ ἔστιν ὀρθὴ ἡ
πρὸς τῷ E, τὸ ἄρα ἀπὸ τῆς ΖΒ τοῦ ἀπὸ BE διπλάσιόν ἐστιν"
ὡς δὲ τὸ ἀπὸ ΖΒ πρὸς τὸ ἀπὸ BE, οὕτως ἐστὶ τὸ ἀπὸ ZH πρὸς τὸ
5 ἀπὸ HE: τὸ ἄρα ἀπὸ ΖΗ τοῦ ἀπὸ HE διπλάσιόν ἐστι. τὰ δὲ pO
τῶν κε ἐλάσσονά ἐστιν ἢ διπλάσια, ὥστε τὸ ἀπὸ ΖΗ πρὸς τὸ ἀπὸ
Α | F
" ζ
με
K fe 5."
Fig. 25.
HE μείζονα λόγον ἔχει ὴ (ὃν τὰν μθ. πρὸς Ke καὶ ἡ ZH
ἄρα πρὸς τὴν HE μείζονα λόγον ἔχει ἢ (ὃν) τὰ ᾧ πρὸς τὰ ε' καὶ
συνθέντι ἡ ΖΕ ἄρα πρὸς τὴν EH μείζονα λόγον ἔχει ἢ ὃν τὰ ιβ
το πρὸς τὰ ε, τουτέστιν, ἢ ὃν (τὰ) ἃς πρὸς τὰ te. ἐδείχθη δὲ καὶ ἡ
2. β] δύο W 3. ΒΕ] τῆς ΒΕΝ 7. (ὃν ra)] om. Vat. ke] τὰ
κε W 8. {év)] om. Vat. 9. ἔχει] ἔχουσα W 1ο, {τὰν}
om. Vat. δὲ] δὴ W
OF THE SUN AND MOON 379
therefore GZ has to £/a ratio greater than that which 15 has to 2.
Next, since BE is equal to ZF, and the angle at Z is right,
therefore the square on 7 is double of the square on 52.
But, as the square on /P is to the square on BZ, so is the
square on /G to the square on GZ;
therefore the square on FG is double of the square on GE.
Now 49 is less than double? of 25,
so that the square on /G has to the square on GZ a ratio greater
than that which 49 has to 25;
therefore /G also has to GZ a ratio greater than that which 7
has to 5. ἢ
Therefore, componendo, FE has to EG a ratio greater than that
which 12 has to 5, that is, than that which 36 has to 15.
and let it meet BHin AH. Leta circle be described with 2 as centre and BY
as radius, meeting BG in P and BE produced in Q.
6
=z
EQ
Fig. 26.
Then 4 GBH: 4 HBE > (sector PBH) : (sector HBQ);
therefore GH:HE>LGBH:LHBE,
and, componendo, GE: HE>ZGBE:2ZHBE.
Ὁ Aristarchus here uses the well-known Pythagorean approximation to “2,
namely {, one of the first of the successive approximations obtained by the
development of the system of ‘ side-’ and ‘diagonal-’ numbers (as to which
see Theon of Smyrna, pp. 43, 44, ed. Hiller, and Proclus, Comm. in Platonis rem-
publicam, ed. Kroll, vol. ii, pp. 24, 25, 27-9, 393-400). The approximation
$ is alluded to by Plato in the Republic, 546c. Plato there speaks of the
_ diagonal of the square, the side of which contains 5 units, and contrasts the
‘irrational diameter of 5’ (ἄρρητος διάμετρος τῆς πεμπάδος), which is of course
4/ (50), with the ‘rational diameter’ (ῥητὴ διάμετρος), which is the square root of
50 less a single unit, i.e. the square root of 49.
280 ΟΝ THE SIZES AND DISTANCES
HE πρὸς τὴν EO μείζονα λόγον ἔχουσα ἢ ὃν τὰ ve πρὸς τὰ δύο"
δι’ ἴσου ἄρα ἡ ΖΕ πρὸς τὴν ΕΘ μείζονα λόγον ἔχει ἢ ὃν τὰ AS
πρὸς τὰ δύο, τουτέστιν, ἢ ὃν τὰ in πρὸς a ἡ ἄρα ZE τῆς EO
μείζων ἐστὶν ἢ ιη. ἡ δὲ ZE ἴση ἐστὶν τῇ BE: καὶ ἡ ΒΕ ἄρα τῆς
5 EO μείζων ἐστὶν ἢ in: πολλῷ ἄρα ἡ ΒΘ τῆς OE μείζων ἐστὶν ἢ
in. ἀλλ᾽ ὡς ἡ ΒΘ πρὸς τὴν ΘΕ, οὕτως ἐστὶν ἡ AB πρὸς τὴν BI,
διὰ τὴν ὁμοιότητα τῶν τριγώνων: καὶ ἡ. AB ἄρα τῆς ΒΓ μείζων
ἐστὶν ἢ in. καὶ ἔστιν ἡ μὲν AB τὸ ἀπόστημα ὃ ἀπέχει ὁ ἥλιος ἀπὸ
τῆς γῆς, ἡ δὲ ΓΒ τὸ ἀπόστημα ὃ ἀπέχει ἡ σελήνη ἀπὸ τῆς ys τὸ
10 ἄρα ἀπόστημα ὃ ἀπέχει ὁ ἥλιος ἀπὸ τῆς γῆς τοῦ ἀποστήματος, οὗ
ἀπέχει ἡ σελήνη ἀπὸ τῆς γῆς, μεῖζόν ἐστιν ἢ ιη.
Λέγω δὴ ὅτι καὶ ἔλασσον ἢ kK. ἤχθω γὰρ διὰ τοῦ A τῇ EB
παράλληλος ἡ AK, καὶ περὶ τὸ AKB τρίγωνον κύκλος γεγράφθω ὁ
4ΚΒ.' ἔσται δὴ αὐτοῦ διάμετρος ἡ AB, διὰ τὸ ὀρθὴν εἶναι τὴν πρὸς
τι τῷ Καὶ γωνίαν. καὶ ἐνηρμόσθω ἡ BA ἑξαγώνου. καὶ ἐπεὶ ἡ ὑπὸ
τῶν ABE γωνία λ΄ ἐστιν ὀρθῆς, καὶ ἡ ὑπὸ τῶν BAK ἄρα X ἐστιν
ὀρθῆς: ἡ ἄρα BK περιφέρεια ξ΄ ἐστιν τοῦ ὅλου κύκλου. ἔστιν δὲ
καὶ ἡ BA ἕκτον μέρος τοῦ ὅλου κύκλου: ἡ ἄρα BA περιφέρεια τῆς
ΒΚ περιφερείας ι ἐστίν. καὶ ἔχει ἡ ΒΑ περιφέρεια πρὸς τὴν ΒΚ
20 περιφέρειαν μείζονα λόγον ἤπερ ἡ BA εὐθεῖα πρὸς τὴν BK εὐθεῖαν"
ἡ ἄρα BA εὐθεῖα τῆς BK εὐθείας ἐλάσσων ἐστὶν ἢ κι. καὶ ἔστιν
αὐτῆς διπλῆ ἡ BA: ἡ dpa BA τῆς ΒΚ ἐλάσσων ἐστὶν i κε ὡς δὲ
ἡ BA πρὸς τὴν ΒΚ, ἡ AB πρὸς (τὴν) BI, ὥστε καὶ ἡ AB τῆς
BI’ ἐλάσσων ἐστὶν ἢ Kx. καὶ ἔστιν ἡ μὲν AB τὸ ἀπόστημα ὃ
25 ἀπέχει ὁ ἥλιος ἀπὸ τῆς γῆς, ἡ δὲ ΒΓ τὸ ἀπόστημα ὃ ἀπέχει
ἡ σελήνη ἀπὸ τῆς γῆς: τὸ ἄρα ἀπόστημα ὃ ἀπέχει ὁ ἥλιος ἀπὸ τῆς
γῆς τοῦ ἀποστήματος, οὗ ἀπέχει ἡ σελήνη ἀπὸ τῆς γῆς, ἔλασσόν
ἐστιν ἢ κι ἐδείχθη δὲ καὶ μεῖζον ἢ tn.
1. HE] FHW 4, 5, 6. 8. wn] ὀκτωκαιδεκαπλασίων W 8. τὸ] om. W
10. οὗ] ὃ WF Nizze, sed cf. 1. 3, p. 376 11. μεῖζον] μείζων W tn] ὀκτω-
καιδεκαπλάσιον W 12. x] εἰκοσιπλάσιον W τοῦ] {τὸ W 13. ar
ABK W 15. τῷ] ὸ ΝΝ ΚΓ Vat. 16. λ' (bis)] τριακοστόν W 17. &
ἑξακοστόν W 19, 21. ¢] δεκαπλασίων W 22, 24. x] εἰκοσιπλασίων W
22. was] ὥστε W 23. ἡ AB (prius)| οὕτως ἡ ABW <riv)] om. Vat. Paris.
2366, leg. Paris. 2364, 2488 27. οὗ] WF Nizze, sed cf. 1. 3, p. 376
28. x] εἰκοσιπλάσιον W μεῖζον] μείζων W Vat. in] ὀκτωκαιδεκαπλάσιον W
OF THE SUN AND MOON 381
But it was also proved that GZ has to Ha ratio greater than
that which 15 has to 2;
therefore, ex aegualt, FE has to EH a ratio greater than that
which 36 has to 2, that is, than that which 18 has to 1;
therefore FZ is greater than 18 times EH.
And ΖῈ is equal to BZ;
therefore BZ is also greater than 18 times EH;
therefore BH is much greater than 18 times HE.
But, as BH is to HE, so is AB to BC, because of the similarity
of the triangles ;
therefore 4B is also greater than 18 times BC.
And AB is the distance of the sun from the earth, while CP is
the distance of the moon from the earth; therefore the distance of
_ the sun from the earth is greater than 18 times the distance of the
moon from the earth.
Again, I say that it is also less than 20 times that distance.
For let DX be drawn through D parallel to 2, and about the
triangle DXB let the circle DXB be described; then DZ will be
its diameter, because the angle at X is right.
Let BZ, the side of a hexagon, be fitted into the circle.
Then, since the angle DBE is 1/3oth of a right angle, the angle
_ BDK is also 1/3;0th of a right angle;
therefore the circumference BX is 1/60th of the whole circle.
But BZ is also one sixth part of the whole circle.
Therefore the circumference 2.2 15 ten times the circumference BK.
And the circumference BZ has to the circumference BX a ratio
ter than that which the straight line BZ has to the straight
ine BK ;}
therefore the straight line BZ is less than ten times the straight
line BX.
And BD is double of BZ;
therefore 2D is less than 20 times BX.
But, as BD is to BK, so is AB to BC;
therefore 4 BZ is also less than 20 times BC.
And AB is the distance of the sun from the earth,
while AC is the distance of the moon from the earth ;
therefore the distance of the sun from the earth is less than 20 times
the distance of the moon from the earth.
_ And it was before proved that it is greater than 18 times that
distance.
2 By the proposition proved in Ptolemy’s Synfaxis, i, 10, pp. 43-4, ed.
Heiberg. See, for his proof, the note on pp. 368-9, above.
482 ON THE SIZES AND DISTANCES
΄
η΄.
Ὅταν ὁ ἥλιος ἐκλείπῃ ὅλος, τότε ὁ αὐτὸς κῶνος περι-
λαμβάνει τόν τε ἥλιον καὶ τὴν σελήνην, τὴν κορυφὴν
ἔχων πρὸς τῇ ἡμετέρᾳ ὄψει.
5 Ἐπεὶ γάρ, ἐὰν ἐκλείπῃ ὁ ἥλιος, 8: ἐπιπρόσθεσιν τῆς σελήνης
ἐκλείπει, ἐμπίπτοι ἂν ὁ ἥλιος εἰς τὸν κῶνον τὸν περιλαμβάνοντα τὴν
σελήνην τὴν κορυφὴν ἔχοντα πρὸς τῇ ἡμετέρᾳ ὄψει. ἐμπίπτων δὲ
ἤτοι ἐναρμόσει εἰς αὐτόν, ἢ ὑπεραίροι, ἢ ἐλλείποι: εἰ μὲν οὖν
ὑπεραίροι, οὐκ (ἂν) ἐκλείποι ὅλος, ἀλλὰ παραλλάττοι αὐτοῦ τὸ
I
ο
ὑπεραῖρον. εἰ δὲ ἐλλείποι, διαμένοι ἂν ἐκλελοιπὼς ἐν ὅσῳ διεξ-
ἔρχεται τὸ ἐλλεῖπον. ὅλος δὲ ἐκλείπει καὶ οὐ διαμένει ἐκλελοιπώς"
τοῦτο γὰρ ἐκ τῆς τηρήσεως φανερόν. ὥστε οὔτ᾽ ἂν ὑπεραίροι, οὔτε
ἐλλείποι. ἐναρμόσει ἄρα εἰς τὸν κῶνον, καὶ περιληφθήσεται ὑπὸ
τοῦ κώνου τοῦ περιλαμβάνοντος τὴν σελήνην τὴν κορυφὴν ἔχοντος
15 πρὸς τῇ ἡμετέρᾳ ὄψει.
θ΄
Ἡ τοῦ ἡλίου διάμετρος τῆς διαμέτρου τῆς σελήνης
μείζων μέν ἐστιν ἣ in, ἐλάσσων δὲ ἢ κ.
Ἔστω γὰρ ἡ μὲν ἡμετέρα ὄψις πρὸς τῷ Α, ἡλίου δὲ κέντρον τὸ
20 Β, σελήνης δὲ κέντρον τὸ Τ', ὅταν ὁ περιλαμβάνων κῶνος τόν τε
ἥλιον καὶ τὴν σελήνην τὴν Κορυφὴν ἔχῃ πρὸς τῇ ἡμετέρᾳ ὄψει,
τουτέστιν, ὅταν τὰ A, Τ', Β σημεῖα ἐπ᾽ εὐθείας 7, καὶ ἐκβεβλήσθω
διὰ τῆς ΑΓΒ ἐπίπεδον: ποιήσει δὴ τομάς, ἐν μὲν ταῖς σφαίραις
I. 7] om. Vat. 7. πρὸς τῇ ἡμετέρᾳ ὄψει] πρὸς τὴν ἡμετέραν ὄψιν Vat.
8, ἐναρμόσει] ἐναρμώσει Vat. ἐλλείποι] ἐλλείπει Paris. 2364: ἐκλείποι F Paris.
2488 9. οὐκ Cav) ἐκλείποι] συνεκλείποι Vat. p.m. : οὐκ ἐκλείποι Vat. corr.
(οὐκ supra lineam scripto) F Paris. 2488 : οὐσυνεκλείποι Paris. 2366: οὐκ ἐκλείπει
Paris. 2342, 2364: οὐκ ἐλλείπει W 10. ἐλλείποι] ἐλλείπει W Paris. 2364:
ἐκλείποι F Paris. 2342, 2488 13. ἐλλείποι] ἐκλείποι F Paris. 2488: ἐλλείπει
Paris. 2364
16. 6] H Vat. 18. in] ὀκτωκαιδεκαπλασίων W k] εἰκοσιπλασίων W
21. ἔχῃ] ἔχει Vat.
OF THE SUN AND MOON 383
PROPOSITION 8.
When the sun ts totally eclipsed, the sun and the moon are then
comprehended by one and the same cone which has tts vertex at
our eye.
For since, if the sun is eclipsed, it is eclipsed because the moon
is in front of it,
the sun must fall into the cone comprehending the moon and having
its vertex at our eye.
And, if it falls into it, either it will exactly fit into it, or it must
overlap it or fall short of it.
If now it should overlap it, the sun would not be totally eclipsed,
but the portion which overlaps would be unobstructed.
If, however, it should fall short, the sun would remain eclipsed
for the time which it takes to pass through the portion by which
it falls short.
But it is in fact totally eclipsed and does not remain eclipsed: for
this is manifest from observation.”
Hence it can neither overlap nor fall short;
therefore it will exactly fit into the cone and will be comprehended
by the cone comprehending the moon and having its vertex at
our eye.
PROPOSITION 9.
The diameter of the sun ts greater than 18 times, but less than
20 t2mes, the diameter of the moon.
For let our eye be at 4, let B be the centre of the sun, and C the
centre of the moon when the cone comprehending both the sun
and the moon has its vertex at our eye, that is, when the points
A, C, # are ina straight line.
Let a plane be carried through 4C2;
this plane will cut the spheres in great circles and the cone in
straight lines.
* Gr. παραλλάττοι. As in Euclid, i. 7 and iii. 24, παραλλάττειν means to “ fall
beside ’ or ‘ awry’, to ‘ miss’, to ‘ pass by without touching’.
? It is evident from this that Aristarchus had not observed the phenomenon
of an axnular eclipse of the sun. The first mention of annular eclipses on
record appears to be that quoted by Simplicius (on De caelo, ii. 12, p. 505, 7-9,
Heiberg) from Sosigenes, the teacher of Alexander Aphrodisiensis (end of
second century A.D.).
384 ON THE SIZES AND DISTANCES
μεγίστους κύκλους, ἐν δὲ τῷ κώνῳ εὐθείας. ποιείτω οὖν ἐν μὲν ταῖς
σφαίραις μεγίστους κύκλους τοὺς ΖΗ͂, ΚΑΘ, ἐν δὲ τῷ κώνῳ
εὐθείας tas AZO, AHK, καὶ ἐπεζεύχθωσαν αἱ TH, ΒΚ. ἔσται
A
\ τ.
ζ
᾿ ‘ nie aes
| aise! Sie iB
7
G K
K
Fig. 27.
δή, ὡς ἡ BA πρὸς τὴν AT, ἡ BK πρὸς TH. ἡ δὲ BA τῆς AT
5 ἐδείχθη μείζων μὲν ἢ in, ἐλάσσων δὲ ἢ κα καὶ ἡ ΒΚ ἄρα τῆς TH
μείζων μέν ἐστιν ἢ tn, ἐλάσσων δὲ ἣ κ. ᾿
lA
lt.
Ὁ ἥλιος πρὸς τὴν σελήνην μείζονα μὲν λόγον ἔχει ἢ
ὃν τὰ εωλβ πρὸς a, ἐλάσσονα δὲ ἢ ὃν τὰ ἡ πρὸς a
το Ἑστω ἡ μὲν τοῦ ἡλίου διάμετρος ἡ 4, ἡ δὲ τῆς σελήνης ἡ B.
ἡ A ἄρα πρὸς τὴν Β μείζονα λόγον ἔχει ἢ ὃν τὰ tn πρὸς α,
ἐλάσσονα δὲ ἢ ὃν τὰ κ πρὸς a. καὶ ἐπειδὴ ὁ ἀπὸ τῆς A κύβος πρὸς
τὸν ἀπὸ τῆς Β κύβον γ λόγον ἔχει ἤπερ ἡ A πρὸς τὴν Β, ἔχει δὲ
καὶ ἡ περὶ διάμετρον τὴν A σφαῖρα πρὸς τὴν περὶ διάμετρον τὴν Β
15 σφαῖραν y λόγον ἤπερ ἡ A πρὸς τὴν B, ἔστιν ἄρα, ὡς ἡ περὶ
διάμετρον τὴν 4 σφαῖρα πρὸς τὴν περὶ διάμετρον τὴν Β σφαῖραν,
οὕτως ὁ ἀπὸ τῆς A κύβος πρὸς τὸν ἀπὸ τῆς Β κύβον. ὁ δὲ ἀπὸ τῆς
A κύβος πρὸς τὸν ἀπὸ τῆς Β κύβον μείζονα λόγον ἔχει ἣ ὃν τὰ
τωλβ πρὸς a, ἐλάσσονα δὲ ἢ ὃν τὰ ἡ πρὸς a, ἐπειδὴ ἡ A πρὸς τὴν
20 B μείζονα λόγον ἔχει ἢ ὃν τὰ in πρὸς a, ἐλάσσονα δὲ ἣ ὃν τὰ κ
πρὸς ἕν: ὥστε ὁ ἥλιος πρὸς τὴν σελήνην μείζονα λόγον ἔχει ἤπερ τὰ
,“ωλβ πρὸς a, ἐλάσσονα δὲ ἢ ὃν τὰ ἡ πρὸς a.
1. εὐθείας] W Paris. 2364: εὐθεῖαν F Vat. Paris. 2366, 2488 μὲν] om. W
2. τῷ κώνῳ] τοῖς κώνοις Vat. Paris. 2366, 2488 4. ἡ BK] (οὕτως) ἡ ΒΚ Nizze
ΓΗῚ τὴν ΓΗ ἊΝ 5. Hon] me Vat. 6. ἢ wn] & Vat.
7. «| © Vat. 13, 15. y] sic Vat. pro τριπλασίονα : τριπλασίονα W
14. πρὸς τὴν] πρὸς τὴν B Vat. 21. ἕν] aW |
OF THE SUN AND MOON 385
Let it cut the spheres in the great circles F/G, KZLH, and the
cone in the straight lines 4-H, AGK,
and let CG, BX be joined.
Then, as BA is to AC, so will BX be to CG.
But it was proved that 24 is greater than 18 times, but less
than 20 times, 4 C. [Prop. 7]
Therefore BX is also greater than 18 times, but less than 20
times, CG.
PROPOSITION το.
The sun has to the moon a ratio greater than that which 5832
has to τ, but less than that which 8000 has fo τ.
Let 4 be the diameter of the sun, 2 that of the moon.
A
ed
Fig. 28.
Then 4 has to #a ratio greater than that which 18 has to 1,
but less than that which 20 has to 1.
Now, since the cube on 4 has tothe cube on # the ratio triplicate
of that which 4 has to 2,
while the sphere about 4 as diameter also has to the sphere about
B as diameter the ratio triplicate of that which 4 has to B,
therefore, as the sphere about 4 as diameter is to the sphere about ~
B as diameter, so is the cube on 4 to the cube on BZ.
But the cube on 4 has to the cube on Z a ratio greater than
that which 5832 has to 1, but less than that which 8000 has to 1,
since 4 has to # a ratio greater than that which 18 has to 1, but
less than that which 20 has to 1.
Accordingly the sun has to the moon a ratio greater than that
which 5832 has to 1, but less than that which 8000 has to 1.
1410 Cec
486 ON THE SIZES AND DISTANCES
,
ια΄.
Ἡ τῆς σελήνης διάμετρος τοῦ ἀποστήματος, οὗ ἀπέχει
τὸ κέντρον τῆς σελήνης ἀπὸ τῆς ἡμετέρας ὄψεως, ἐλάσσων
μέν ἐστιν ἣ δύο pe, μείζων δὲ ἣ Χ΄.
5 Ἔστω γὰρ ἡ μὲν ἡμετέρα ὄψις πρὸς τῷ A, σελήνης δὲ κέντρον
τὸ Β, ὅταν ὁ περιλαμβάνων κῶνος τόν τε ἥλιον καὶ τὴν σελήνην τὴν
κορυφὴν ἔχῃ πρὸς τῇ ἡμετέρᾳ ὄψει. λέγω ὅτι γίγνεται τὰ διὰ τῆς
προτάσεως.
᾿Ἐπεξεύχθω γὰρ ἡ 48, καὶ ἐκβεβλήσθω τὸ διὰ τῆς AB ἐπίπεδον"
το ποιήσει δὴ τομὴν ἐν μὲν τῇ σφαίρᾳ κύκλον, ἐν δὲ τῷ κώνῳ εὐθείας.
ποιείτω οὖν ἐν μὲν τῇ σφαίρᾳ κύκλον τὸν ΓΕΖ, ἐν δὲ τῷ κώνῳ
εὐθείας τὰς AA, AT, καὶ ἐπεζεύχθω ἡ ΒΓ, καὶ διήχθω ἐπὶ τὸ E,
᾿φανερὸν δὴ ἐκ τοῦ προδεδειγμένου ὅτι ἡ ὑπὸ τῶν ΒΑΓ γωνία,
ἡμισείας ὀρθῆς ἐστι με΄- καὶ κατὰ τὰ αὐτὰ ἡ ΒΓ τῆς ΓΑ ἐλάσσων
τῷ ἐστὶν ἢ pe. πολλῷ ἄρα ἡ ΒΓ τῆς ΒΑ ἐλάσσων ἐστὶν ἢ pe
μέρος. καὶ ἔστι τῆς ΒΓ διπλῆ ἡ ΓῈ: ἡ ΓΕ ἄρα τῆς AB ἐλάσσων
ἐστὶν ἢ δύο μέ. καὶ ἔστιν ἡ μὲν TE ἡ τῆς σελήνης διάμετρος, ἡ
δὲ BA τὸ ἀπόστημα ὃ ἀπέχει τὸ κέντρον τῆς σελήνης ἀπὸ τῆς
ἡμετέρας ὄψεως: ἡ ἄρα διάμετρος τῆς σελήνης τοῦ ἀποστήματος, οὗ
20 ἀπέχει τὸ κέντρον τῆς σελήνης ἀπὸ τῆς ἡμετέρας ὄψεως, ἐλάσσων
ἐστὶν ἡ δύο pe. a
Λέγω δὴ ὅτι καὶ μείζων ἐστὶν ἡ TE τῆς BA ἣ λ' μέρος.
ἐπεζεύχθω γὰρ ἡ AE καὶ ἡ AD, καὶ κέντρῳ μὲν τῷ A, διαστήματι
δὲ τῷ AI’, κύκλος γεγράφθω 6 TAZ, καὶ ἐνηρμόσθω εἰς τὸν TAZ
a3 κύκλον τῇ ΑΓ ἴση ἡ AZ. καὶ ἐπεὶ ὀρθὴ ἡ ὑπὸ τῶν EAT ὀρθῇ τῇ
I. ια I Vat. 2. οὗ] ὃ WF Nizze, sed nihil mutandum 4. με τεσσαρα-
κοστόπεμπτα W X'] τριακοστόν W 6. περιλαμβάνων] παραλαμβάνων W
ἡ. ἔχῃ] ἔχει Vat. γίγνεται] γίνεται W διὰ] om. W 13. ὑπὸ]
ἀπὸ W 14, 15. με τεσσαρακοστόπεμπτον W 14. ΓΑ] BAW 15-16.
πολλῷ dpa... pe’ μέρος] om. W Paris. 2366 17. με] τεσσαρακοστόπεμπτα W
19. ob] ὃ WF Nizze, sed cf. 1. 2 supra 21. με] τεσσαρακοστόπεμπτα W
OF THE SUN AND MOON 387
PROPOSITION It.
The diameter of the moon ts less than 2/45ths, but greater than
1/30th, of the distance of the centre of the moon from our eye.
For let our eye be at 4, and let @ be the centre of the moon
when the cone comprehending both the sun and the moon has its
vertex at our eye.
_ I say that the above proposition is true.
Let 4B be joined, and let the plane through 4B be drawn ;
this plane will cut the sphere [i.e. the moon] in a circle and
the cone in straight lines.
Let it cut the sphere in the circle CED and the cone in the
straight lines 4D, AC; ;
let BC be joined and produced to £.
Then it is manifest from what has before been proved [Prop. 4]
that the angle BAC is 1/45th part of half a right angle;
and, in the same way as
before, BC is less than
1/45th part of C4 ;
therefore BC is much less
than 1/45th part of BA.
And CE is double of
7a Of
therefore CZ is less than
2/45ths of 4B.
Now CZ is the diameter
of the moon,
; H ο
while BA is the distance
of the centre of the moon
from our eye. rE
Therefore the diameter Fig. 29.
of the moon is less than
2/45ths of the distance of the centre of the moon from our eye.
I say next that CZ is also greater than 1/30th part of BA.
For let DE and DC be joined, and with centre 4 and distance
A C let the circle CDF be described ; let DF equal to 4 C be fitted
into the circle CDF.
mr,
εὐ
488 ON THE SIZES AND DISTANCES
ὑπὸ τῶν BIA ἐστὶν ἴση, ἀλλὰ καὶ ἡ ὑπὸ τῶν BAI τῇ ὑπὸ τῶν
ΘΙΓΒ ἐστὶν ἴση, λοιπὴ ἄρα ἡ ὑπὸ τῶν AET λοιπῇ τῇ ὑπὸ τῶν
OBI ἐστὶν ἴση: ἰσογώνιον ἄρα ἐστὶν τὸ AE τρίγωνον τῷ ABI
τριγώνῳ. ἔστιν ἄρα, ὡς ἡ ΒΑ πρὸς AT, οὕτως ἡ ΕΓ πρὸς ΓΖ'
5 καὶ ἐναλλάξ, ὡς ἡ AB πρὸς TE, οὕτως ἡ AI πρὸς ΓΖ, τουτέστιν,
ἡ 4Z πρὸς TA. ἀλλ’ ἐπεὶ πάλιν ἡ ὑπὸ τῶν AAT γωνία με΄ μέρος
ἐστὶν ὀρθῆς, ἡ ΓΖ ἄρα περιφέρεια pr’ μέρος ἐστὶ τοῦ κύκλου" ἡ δὲ
AZ περιφέρεια ἕκτον μέρος ἐστὶν τοῦ ὅλου κύκλου: ὥστε ἡ ΓΖ
περιφέρεια τῆς 4Ζ περιφερείας λ' μέρος ἐστίν. καὶ ἔχει ἡ ΓΖ
10 περιφέρεια, ἐλάσσων οὖσα τῆς AZ περιφερείας, πρὸς αὐτὴν τὴν AZ
περιφέρειαν ἐλάσσονα λόγον ἤπερ ἡ T'A εὐθεῖα πρὸς τὴν ZA
εὐθεῖαν: ἡ ἄρα TA εὐθεῖα τῆς AZ μείζων ἐστὶν ἢ λ΄. ἴση δὲ ἡ
ZA τῇ AT: ἡ ἄρα AT τῆς ΓΑ μείζων ἐστὶν ἣ XN, ὥστε καὶ ἡ TE
τῆς ΒΑ μείζων ἐστὶν ἣ Χ. ἐδείχθη δὲ καὶ ἐλάσσων οὖσα ἣ δύο pe’.
15 51.
ἭἫ διάμετρος τοῦ διορίζοντος ἐν τῇ σελήνῃ τό τε
σκιερὸν καὶ τὸ λαμπρὸν "τῆς διαμέτρου τῆς σελήνης
ἐλάσσων μέν ἐστι, μείζονα δὲ λόγον ἔχει πρὸς αὐτὴν 7
ὃν τὰ πθ πρὸς ς.
20 Ἔστω γὰρ ἡ μὲν ἡμετέρα ὄψις πρὸς τῷ A, σελήνης δὲ κέντρον
τὸ Β, ὅταν ὁ περιλαμβάνων κῶνος τόν τε ἥλιον καὶ τὴν σελήνην τὴν
κορυφὴν ἔχῃ πρὸς τῇ ἡμετέρᾳ ὄψει, καὶ ἐπεζεύχθω ἡ 48, καὶ
ἐκβεβλήσθω διὰ τῆς AB ἐπίπεδον: ποιήσει δὴ τομὰς ἐν μὲν τῇ
σφαίρᾳ κύκλον, ἐν δὲ τῷ κώνῳ εὐθείας. ποιείτω (ἐν μὲν τῇ σφαίρᾳ
as κύκλον τὸν AET, ἐν δὲ τῷ κώνῳ εὐθείας) τὰς 44, AT, TA.
2. ΘΓΒῚ ETAW 6. με] τεσσαρακοστόπεμπτον W 9. λΊ τριακοστὸν W
10. περιφέρεια] add. πρὸς τὴν AZ περιφέρειαν Vat. II. λόγον] λόγον ἔχει Vat.
12, 14. A’] τριακοστόν W 12-13. ἴση δὲ ἡ... ἢ Δ om. W 14. με τεσ-
σαρακοστόπεμπτα W we
15. «8’] IA Vat. 21. περιλαμβάνων] παραλαμβάνων W 24-25. (ἐν μὲν...
εὐθείας) supplevit W : om. codd.
OF THE SUN AND MOON 389
Then, since the right angle EDC is equal to the right angle BCA,
while the angle BA C is also equal to the angle HCB,
therefore the remaining angle DEC is equal to the remaining
angle HBC.
Therefore the triangle CDZ is equiangular with the triangle 4 BC.
Therefore, as BA is to AC, 5015 EC to CD;
and, alternately, as 42 is to CE, so is AC to CD, that is, DF
to CD.
But again, since the angle DAC is 1/45th part of a right angle,
the circumference CD is 1/18oth part of the circle;
and the circumference DF is one sixth part of the whole circle ;
thus the circumference CD is 1/30th part of the circumference DF.
And the circumference CD, being less than the circumference
DF, has to the circumference DF itself a ratio less than that which
the straight line CD has to the straight line FD."
Therefore the straight line CD is greater than 1/30th of DF.
But FD is equal to 4C;
therefore DC is greater than 1/30th of C4,
so that CZ is also greater than 1/30th of BA [see above].
But it was before proved to be also less than 2/45ths.
PROPOSITION 12.
The diameter of the circle which divides the dark and the bright
portions in the moon ts less than the diameter of the moon, but
has to tt a ratio greater than that which 89 has to go.
For let our eye be at 4, and let B be the centre of the moon
when the cone comprehending both the sun and the moon has its
vertex at our eye; .
let AB be joined, and let a plane be carried through 42; EN
plane will cut the sphere [i.e. the moon] in a circle and the cone in
straight lines.
Let it cut the sphere in the circle DEC and the cone in the
straight lines 4D, AC, CD.
1 For, by the proposition proved by Ptolemy (see note on Prop. 4, above),
FD: DC < (arc FD) : (arc DC),
and, by inversion, (arc CD): (arc DF) < CD: DF.
890 ON THE SIZES AND DISTANCES
ἡ ΓΖ dpa διάμετρός ἐστι τοῦ κύκλου τοῦ διορίζοντος ἐν τῇ σελήνῃ
τὸ σκιερὸν καὶ τὸ λαμπρόν. λέγω δὴ ὅτι ἡ ΓΖ τῆς διαμέτρου τῆς
σελήνης ἐλάσσων μέν ἐστι, μείζονα δὲ λόγον ἔχει {πρὸς αὐτὴν) ἢ ὃν
τὰ πθ πρὸς ς.
5 “Ore μὲν οὖν ἡ ΓΖ ἐλάσσων ἐστὶ τῆς διαμέτρου τῆς σελήνης,
φανερόν. λέγω δὴ ὅτι καὶ μείζονα λόγον ἔχει (πρὸς αὐτὴν) ἢ ὃν
τὰ πθ πρὸς ς.
Ἤχθω γὰρ διὰ τοῦ Β τῇ TA παράλληλος ἡ ZH, καὶ ἐπε-
ζεύχθω ἡ BI. ἔσται δὴ πάλιν κατὰ τὰ αὐτὰ ἡ ὑπὸ τῶν AAT
το γωνία ὀρθῆς pe’ μέρος, ἡ δὲ ὑπὸ τῶν BAT ὀρθῆς ς΄ μέρος. καὶ
a
5»»
Fig. 30.
ἔστιν ἡ ὑπὸ τῶν BAT γωνία ion τῇ ὑπὸ τῶν TBZ: καὶ ἡ ὑπὸ τῶν
ΓΒΖ ἄρα γωνία ὀρθῆς ἐστιν ς΄, τουτέστιν, τῆς ὑπὸ τῶν ZBE
γωνίας ζ΄, ὥστε καὶ ἡ ΓΖ περιφέρεια τῆς ZTE περιφερείας ἐστὶν
ς΄. ἡ ΤῈ ἄρα περιφέρεια πρὸς τὴν ΕΓΖ περιφέρειαν λόγον ἔχει
15 ὃν τὰ πθ πρὸς ς. καὶ ἔστι τῆς TE β ἡ AET, τῆς δὲ ΕΓΖ β ἡ
HEZ- ἡ ἄρα SET περιφέρεια πρὸς τὴν ΗΕΖ περιφέρειαν λόγον
ἔχει ὃν τὰ πθ πρὸς ς. καὶ ἔχει ἡ ΔΓ εὐθεῖα πρὸς (τὴν) HZ
εὐθεῖαν μείζονα λόγον ἤπερ ἡ JET περιφέρεια πρὸς τὴν HEZ
περιφέρειαν: καὶ ἡ AT ἄρα εὐθεῖα πρὸς τὴν HZ εὐθεῖαν μείζονα
20 λόγον ἔχει ἢ ὃν τὰ πθ πρὸς ς.
2. δὴ] δὲ W 3, 6. (πρὸς αὐτὴν)] addidi 10. pe’] τεσσαρακοστό-
meumtov W τῶν] τὸν Vat. 10, 12, 14. (7 ἐννενηκοστόν W 13. γωνίας] γωνίας,
γωνίας W 15, 8 bis] διπλῆ W 17. καὶ ἔχει] ἔχει post λόγον (1. 18) posuit W
OF THE SUN AND MOON 39%
Therefore CD is a diameter of the circle which divides the dark
and the bright portions in the moon.
I say that CD is less than the diameter of the moon, but has to it
a ratio greater than that which 89 has to go.
Now, that CD is less than the diameter of the moon is manifest.
I say, then, that it also has to it a ratio greater than that which
80 has to go.
For let FG be drawn through # parallel to CD,
and let BC be joined.
Then again, in the same way as before, the angle DAC will be
1/45th part of a right angle,
and the angle 2A C will be 1/goth part of a right angle ;
_ but the angle BAC is equal to the angle CBF;
_ therefore the angle CBF is also 1/goth of a right angle,
that is, 1/goth of the angle 55 8;
so that the circumference CF is also 1/goth of the circumference
FCE.
Therefore the circumference CZ has to the circumference ECF
the ratio which 89 has to go.
Now DEC is double of CZ, and GEF double of ECF;
therefore the circumference DEC has to the circumference GE F
the ratio which 89 has to go.
And the straight line DC has to the straight line GF a ratio
greater than that which the circumference DEC has to the
circumference GEF\
Therefore also the straight line DC has to the straight line GF
a ratio greater than that which 89 has to go. |
* By the proposition quoted from Ptolemy, i. 10, pp. 43-4, ed. Heiberg. 5
note on Props. 4 and 11, above. δἰ celts τὴ
392 ON THE SIZES AND DISTANCES
ιγ΄.
‘H ὑποτείνουσα εὐθεῖα ὑπὸ τὴν ἀπολαμβανομένην ἐν
τῷ σκιάσματι τῆς γῆς περιφέρειαν τοῦ κύκλου, καθ’ οὗ
φέρεται τὰ ἄκρα τῆς διαμέτρου τοῦ διορίζοντος ἐν τῇ
σελήνῃ τό τε σκιερὸν καὶ τὸ λαμπρόν, τῆς μὲν διαμέτρου
τῆς σελήνης ἐλάσσων ἐστὶν ἢ διπλῆ, μείζονα δὲ λόγον
ἔχει πρὸς αὐτὴν ἢ ὃν τὰ πὴ πρὸς με, τῆς δὲ τοῦ ἡλίου
διαμέτρου ἐλάσσων μέν ἐστιν ἢ ἔνατον μέρος, μείζονα
δὲ λόγον ἔχει πρὸς αὐτὴν ἣ dv KB πρὸς σκε, πρὸς δὲ τὴν
τοἀπὸ τοῦ κέντρου τοῦ ἡλίου ἠγμένην πρὸς ὀρθὰς τῷ ἄξονι,
συμβάλλουσαν δὲ ταῖς τοῦ κώνου πλευραῖς, μείζονα λόγον
ἔχει ἣ ὃν τὰ YOO πρὸς κε.
Ἔστω γὰρ ἡλίου μὲν κέντρον πρὸς τῷ Α, γῆς δὲ κέντρον τὸ Β,
σελήνης δὲ τὸ I, τελείας οὔσης τῆς ἐκλείψεως καὶ πρώτως ὅλης
15 ἐμπεπτωκυίας εἰς τὸ τῆς γῆς σκίασμα, καὶ ἐκβεβλήσθω διὰ τῶν A,
B, TI ἐπίπεδον: ποιήσει δὴ τομὰς ἐν μὲν ταῖς σφαίραις κύκλους, ἐν
δὲ τῷ κώνῳ εὐθείας τῷ περιλαμβάνοντι τόν τε ἥλιον καὶ τὴν γῆν.
ποιείτω ἐν μὲν ταῖς σφαίραις μεγίστους κύκλους τοὺς ΔΕΖ, ΗΘΚ,
ΛΜΝ, ἐν δὲ τῷ σκιάσματι τῆς γῆς κύκλον, καθ᾽ οὗ φέρεται τὰ ἄκρα
20 τῆς διαμέτρου τοῦ διορίζοντος ἐν τῇ σελήνῃ τό τε σκιερὸν καὶ τὸ
λαμπρόν, τὸν BAN, ἐν δὲ τῷ κώνῳ εὐθείας τὰς ΔΗΞ, ΖΚΝ'
ἄξων δὲ ἔστω 6 ABA. φανερὸν δὴ ὅτι 6 ABA ἄξων ἐφάπτεται
τοῦ AMN κύκλου, διὰ τὸ τὸ σκίασμα τῆς γῆς σεληνῶν εἶναι δύο, καὶ
δίχα διαιρεῖσθαι τὴν NAB περιφέρειαν ὑπὸ τοῦ ABA ἄξονος, καὶ
25 ἔτι τὴν σελήνην πρώτως ἐμπεπτωκέναι εἰς τὸ τῆς γῆς σκίασμα.
ἐπεζεύχθωσαν δὴ αἱ ἘΝ, NA, BN, AB. ἡ ΑΝ dpa ἐστὶν ἡ
διάμετρος τοῦ διορίζοντος ἐν τῇ σελήνῃ τό τε σκιερὸν καὶ τὸ
λαμπρόν, καὶ ἡ ΒΝ ἐφάπτεται τοῦ ANOM κύκλου, διὰ τὸ εἶναι τὸ
Β πρὸς τῇ ἡμετέρᾳ ὄψει, καὶ τὴν AN διάμετρον τοῦ διορίζοντος ἐν
3. τῇ σελήνῃ τό τε σκιερὸν καὶ τὸ λαμπρόν. καὶ ἐπεὶ αἱ BA, AN
I. ty] IB Vat. 42. τὴν] om. 7. pe] τὰμε W 8. ἔνατον] ἔννατον W
14. ἐκλείψεως] ἐκλίψεως Vat. 16. δὴ] δὲ 18. rovs]om.W 109. κύκλον]
κύκλων Vat. 22. ὁ 27] ἡ ἐφάπτεται] ἐφάπτηται W a πρώτως] πρώτως
(ὅλην) Nizze 28-30. καὶ ἡ ΒΝ ἐφάπτεται... λαμπρόν] om. soli, ut videtur, codd.
Savilianus et Paris. 2342 28. ἐφάπτεται εὐθεῖα ἐφάπτεται W ΑΝΟΜΊῈ
Paris. 2364, 2488: ANON Vat.: AMN W 28-9. εἶναι τὸ B] τὸ Β σημεῖον
εἶναι W, qui lacunam post 1. 28 λαμπρόν ope versionis Commandini expleverat
29. διάμετρον] τὴν διάμετρον εἶναι W
OF THE SUN AND ΜΟΟΝ 393
PROPOSITION 13.
The straight line subtending the portion tntercepted within the
earth's shadow of the circumference of the circle tn which
the extremities of the diameter of the circle dividing the dark
and the bright portions tn the moon move ts less than double
of the diameter of the moon, but has to tt a ratio greater than
that which 88 has to 45; and it ts less than 1/oth part of the
diameter of the sun, but has to tt a ratio greater than that which
22 has to 225. But it has to the straight line drawn from the
centre of the sun at right angles to the axis and meeting the sides
of the cone a ratio greater than that which 979 has to 10125.
For let the centre of the sun be at 4, let B be the centre of the
earth, and C the centre of the moon when the eclipse first becomes
total through the moon having fallen wholly within the earth’s
shadow.
Let a plane be carried through 4, 2, C;
this plane will cut the spheres in circles and the cone comprehending
both the sun and the earth in straight lines.
Let it cut the spheres in the great circles DEF, GHK, LMN,
the earth’s shadow in the circle OZ NV in which the extremities of
the diameter of the circle dividing the dark and the bright portions
in the moon move, and the cone in the straight lines DGO, ΕΑΝ.
Let A BL be the axis.
Then it is manifest that the axis 4 BZ touches the circle LIN,
because the shadow of the earth is of two moon-breadths, [Hyp.5] .
that the circumference VVZO is bisected by the axis 4 PL,
and further that the moon has for the first time fallen within the
earth’s shadow.
Let ON, NL, BN, LO be joined.
Therefore ZN is the diameter of the circle dividing the dark
and the bright portions in the moon.
And ΔΜ touches the circle ΣΙ ΔΩ͂,
because the point Z is at our eye, and ZW is the diameter of the
circle dividing the dark and the bright portions in the moon.
394 ON THE SIZES AND DISTANCES
ἴσαι εἰσίν, Simdacioves dpa εἰσὶ τῆς AN, ὥστε ἡ ἘΝ τῆς AN
ἐλάσσων ἐστὶν ἢ δι-
πλῆ. ἐπεζεύχθωσαν
δὴ αἱ AI, ΓΝ, καὶ
5 διήχθω ἡ AT ἐπὶ τὸ
Ο' πολλῷ ἄρα ἡ ἘΝ
τῆς AO ἐλάσσων ἐσ-
τὶν ἡ β. καὶ ἐπεὶ ἡ
TA κάθετός ἐστιν ἐπὶ
10 τὴν BA, παράλληλος
ἄρα ἐστὶν τῇ BN: ἴση
ἄρα ἐστὶν ἡ ὑπὸ τῶν
AEN τῇ ὑπὸ τῶν
TAN γωνίᾳ. καὶ ἔσ-
15 τιν ἴση μὲν ἡ NA τῇ
ΛΈ, ἡ δὲ ΛΓ τῇ TN:
ὅμοιον ἄρα ἐστὶν τὸ
ENA τρίγωνον τῷ
ANT τριγώνῳ" ἔστιν
20 ἄρα,ὡς ἡ ἘΝ πρὸς τὴν
NA, οὕτωςἡ ΝΑπρὸς
τὴν AT. ἀλλ᾽ ἡ ΝΑ
πρὸς τὴν ΔΓ μείζονα
λόγον ἔχει ἢ ὃν τὰ πθ
a5 πρὸς τὰ με, τουτέστι,
τὸ ἀπὸ NA πρὸς τὸ
ἀπὸ AI μείζονα λό-
γον ἔχει ἤπερ τὰ
ζϑκα πρὸς τὰ Bre
30 καὶ τὸ ἀπὸ ἘΝ ἄρα
πρὸς τὸ ἀπὸ NA μεί-
(ova λόγον ἔχει ἤπερ
τὰ ( δκαπρὸς τὰ Bre,
καὶὴἡ ἘΝ πρὸςτὴνΛ 0
8. β] διπλῆ W
Fig. 31.
30. EN] ris BN W 31. NA] τῆς NA W
OF THE SUN AND MOON 395
Now, since OZ, ZN are equal, their sum is double of ZN, so
that ON is less than double of ZV.
Let ZC, CN be joined ; and let ZC be carried through to P.
Therefore ON is much less than double of LP.
And, since CZ is perpendicular to BZ,
therefore it is parallel to OW.
Therefore the angle LON is equal to the angle CLN.
And WZ is equal to ZO, and LC to CN;
therefore the triangle 0.172 is similar to the triangle ZNC;
therefore, as OW is to NZ, so is WZ to LC.
But VZ has to ZC a ratio greater than that which 89 has to 45 ;'
that is, the square on ΛΖ has to the square on ZC a ratio greater
_ than that which 7921 has to 2025.
Therefore the square on OV also has to the square on WZ
a ratio greater than that which 7921 has to 2025,
and (therefore) ON has to ZP a ratio greater than that which 7921
has to 4050.”
1 For WZ : LP > 89: 90, by the preceding proposition.
3 We have ON: NL=NL:LC;
therefore ON : LC = (sq. on ON): (sq. on WZ)
> 7921 : 2025,
whence ΟΝ: LP >7921 : 4050.
406 ON THE SIZES AND DISTANCES
μείζονα λόγον ἔχει ἤπερ TA ¢%Ka πρὸς ὃν. ἔχει δὲ καὶ τὰ (Axa
πρὸς ὃν μείζονα λόγον ἤπερ τὰ πη πρὸς pe ἡ NE ἄρα πρὸς AO
μείζονα λόγον ἔχει ἣ ὃν τὰ πη πρὸς τὰ με. ἡ ἄρα ὑποτείνουσα ὑπὸ
τὴν ἀπολαμβανομένην ἐν τῷ σκιάσματι τῆς γῆς περιφέρειαν τοῦ
5 κύκλου, καθ᾽ οὗ φέρεται τὰ ἄκρα τῆς διαμέτρου τοῦ διορίζοντος ἐν TH
σελήνῃ τό τε σκιερὸν καὶ τὸ λαμπρόν, τῆς διαμέτρου τῆς σελήνης
ἐλάσσων μέν ἐστιν ἢ β, μείζονα δὲ λόγον ἔχει (πρὸς αὐτὴν) ἢ ὃν τὰ
πὴ πρὸς με.
Τῶν αὐτῶν ὑποκειμένων, ἤχθω ἀπὸ τοῦ A τῇ AB πρὸς ὀρθὰς
10% IIAP: λέγω ὅτι ἡ ἘΝ τῆς διαμέτρου τοῦ ἡλίου ἐλάσσων μέν
ἐστιν ἢ 6 μέρος, μείζονα δὲ λόγον ἔχει πρὸς αὐτὴν 4 ὃν τὰ KB πρὸς
τὰ σκε, πρὸς δὲ τὴν ΠΡ μείζονα λόγον ἔχει ἢ ὃν τὰ AO πρὸς
Μ ρκε. ἐπεὶ γὰρ ἐδείχθη ἡ ἘΝ τῆς διαμέτρου τῆς σελήνης ἐλάσσων
οὖσα ἢ B, ἡ δὲ διάμετρος τῆς σελήνης τῆς διαμέτρου τοῦ ἡλίου
15 ἐλάσσων ἐστὶν ἢ ιη΄ μέρος, ἡ ἄρα ἘΝ τῆς διαμέτρου τοῦ ἡλίου
σι
ἐλάσσων ἐστὶν ἢ θ΄ μέρος. πάλιν ἐπεὶ ἡ ἘΝ πρὸς τὴν διάμετρον
τῆς σελήνης μείζονα λόγον ἔχει ἢ ὃν τὰ πὴ πρὸς τὰ με, ἡ δὲ
διάμετρος τῆς σελήνης πρὸς τὴν τοῦ ἡλίου διάμετρον μείζονα λόγον
ἔχει ἢ ὃν τὰ με πρὸς δ᾽ ἐπεὶ γὰρ ἡ τῆς σελήνης διάμετρος πρὸς
20 τὴν τοῦ ἡλίου μείζονα λόγον ἔχει ἣ ὃν α πρὸς κ, καὶ πάντα
τεσσαρακοντάκις καὶ πεντάκις" ἕξει ἄρα ἡ ἘΝ πρὸς τὴν διάμετρον
τοῦ ἡλίου μείζονα λόγον ἢ ὃν τὰ πη πρὸς τὰ ὃ, τουτέστιν, ἢ ὃν
τὰ κβ πρὸς τὰ σκε. ἤχθωσαν δὴ ἀπὸ τοῦ Β τοῦ 4Ε κύκλου
ἐφαπτόμεναι αἱ BY, BOT, καὶ ἐπεζεύχθω ἡ TH καὶ ἡ TA.
as ἔσται δή, ὡς ἡ διάμετρος τοῦ διορίζοντος ἐν τῇ σελήνῃ τό τε σκιερὸν
καὶ τὸ λαμπρὸν πρὸς τὴν διάμετρον τῆς σελήνης, οὕτως TH πρὸς
τὴν διάμετρον τοῦ ἡλίου, διὰ τὸ τὸν αὐτὸν κῶνον περιλαμβάνειν τόν
2. ἡ NE ἄρα] ἄρα ἡ ΝΒ Vat. 3. τὰ με] με W ὑποτείνουσα]
ἀποτείνουσα Vat. 7. B| διπλῇ W (πρὸς adriy)| addidi 11, 16, 4]
ἔννατον W 12. oxe] κε Vat. ἔχει] ἔχει πρὸς αὐτὴν W 14. β]
διπλασίων W 15. ἢ en] ey! Vat. 17. τὰ pe] pe W 20. a] τὸ α W
21. τεσσαρακοντάκις καὶ πεντάκις] τεσσαρακοντακαιπεντάκις ἊΝ 22. τὰ A]
AW 23. ΔΕ) 4EZW
OF THE SUN AND MOON 397
_ But 7921 also has to 4050 a ratio greater than that which 88 has
to 45;
therefore WO has to ZP a ratio greater than that which 88 has
to 45.
Therefore the straight line which subtends the portion inter-
cepted within the earth’s shadow of the circumference of the circle
in which the extremities of the diameter of the circle dividing the
dark and the bright portions in the moon move is less than double
of the diameter of the moon, but has to it a ratio greater than that
which 88 has to 45.
The same suppositions being made, let 04 be drawn ftom 4
at right angles to 4B.
I say that OW is less than 1/oth part of the diameter of the sun,
but has to it a ratio greater than that which 22 has to 225, and has
to OR a ratio greater than that which 979 has to 10125.
For, since it was proved that OJ is less than double of the
diameter of the moon,
while the diameter of the moon is less than 1/18th part of the diameter
of the sun, [Prop. 9]
_ therefore OW is less than 1/oth part of the diameter of the sun.
_ Again, since OW has to the diameter of the moon a ratio greater
than that which 88 has to 45,
while the diameter of the moon has to the diameter of the sun
a ratio greater than that which 45 has to goo:
for, since the diameter of the moon has to the diameter of the sun
a ratio greater than that which 1 has to 20, we have only to
multiply throughout by 45:
therefore (ex aegualz) ON has to the diameter of the sun a ratio
greater than that which 88 has to goo, that is, than that which 22.
has to 225.
Now let BUS, BVT be drawn from B touching the circle DE:
and let UV, UA be joined.
Then, as the diameter of the circle dividing the dark and the bright
portions in the moon is to the diameter of the moon, so is VV
to the diameter of the sun, because the sun and the moon are
* If we develop {5s 2S 4 continued fraction, we easily obtain the approximation
τὴξ His Σ, which is in fact 82. See the similar case in Prop. τς, p. a
the observation thereon, p. 336 aa fit P- 15, Ρ. 407,
398 ON THE SIZES AND DISTANCES
τε ἥλιον καὶ τὴν σελήνην τὴν κορυφὴν ἔχοντα πρὸς TH ἡμετέρᾳ ὄψει.
ἡ δὲ διάμετρος τοῦ διορίζοντος ἐν τῇ σελήνῃ τό τε σκιερὸν καὶ τὸ
λαμπρὸν πρὸς τὴν διάμετρον τῆς σελήνης μείζονα λόγον ἔχει ἣ ὃν τὰ
πθ πρὸς τὰ ς᾽ καὶ ἡ TD ἄρα πρὸς τὴν τοῦ ἡλίου διάμετρον μείζονα
5 λόγον ἔχει ἢ ὃν τὰ πθ πρὸς G καὶ XT ἄρα πρὸς τὴν TA μείζονα
λόγον ἔχει ἢ ὃν τὰ πθ πρὸς ς. ὡς δὲ ἡ XT πρὸς τὴν TA, οὕτως ἡ
TA πρὸς τὴν AX, διὰ τὸ παραλλήλους εἶναι τὰς FA, ΥΧ' καὶ ἡ
TA ἄρα πρὸς τὴν AX μείζονα λόγον ἔχει ἢ ὃν τὰ πθ πρὸς τὰ ς΄
πολλῷ ἄρα ἡ TA πρὸς τὴν AP μείζονα λόγον ἔχει ἢ ὃν τὰ πθ πρὸς —
ιοτὰς. καὶ τὰ β' ἡ ἄρα διάμετρος τοῦ ἡλίου πρὸς τὴν ΠΡ μείζονα
λόγον ἔχει ἢ ὃν τὰ πθ πρὸς τὰ ς. ἐδείχθη δὲ καὶ ἡ ἘΝ πρὸς τὴν
διάμετρον τοῦ ἡλίου μείζονα λόγον ἔχουσα ἢ ὃν τὰ KB πρὸς τὰ σκε.
δ ἴσου πολλῷ ἄρα ἡ ἘΝ πρὸς τὴν ΠΡ μείζονα λόγον ἔχει ἢ (ὃν)
ς la a ~ ‘ Ν ἈΝ ? “- .
-6 συνηγμένος ἔκ τε τῶν KB Kal πθ πρὸς τὸν ἐκ τῶν ς Kal σκέ,
β ἢ
15 τουτέστιν, τὰ a Avy πρὸς τὰ Mov καὶ τὰ ἡμίση, τουτέστιν, τὰ
α
‘00 πρὸς τὰ Moke.
ιδ΄.
Ἡ ἀπὸ τοῦ κέντρου τῆς γῆς ἐπὶ τὸ κέντρον τῆς σελήνης
ἐπιζευγνυμένη εὐθεῖα πρὸς τὴν εὐθεῖαν, ἣν ἀπολαμβάνει
2. ἀπὸ τοῦ ἄξονος πρὸς τῷ κέντρῳ τῆς σελήνης ἡ ὑπὸ τὴν
ἐν τῷ σκιάσματι τῆς γῆς ὑποτείνουσα εὐθεῖα, μείζονα
λόγον ἔχει ἢ ὃν τὰ χοε πρὸς α.
Ἔστω τὸ αὐτὸ σχῆμα τῷ πρότερον, καὶ ἡ σελήνη οὕτως ἔστω
ὥστε τὸ κέντρον αὐτῆς εἶναι ἐπὶ τοῦ ἄξονος τοῦ κώνου τοῦ περι.
4. τ c] ¢ W 10, B] διπλάσια W 12. ra oxe] oxe W
B a
15. Mov] M8.cv W 16. τὰ Mpxe] Ma. ρκε W 17. ιδ΄] IF Vat.
OF THE SUN AND MOON 399
comprehended by one and the same cone haying its vertex at
our eye.
But the diameter of the circle dividing the dark and the bright
portions in the moon has to the diameter of the moon a ratio
greater than that which 89 has to 90; [Prop. 12]
therefore 7/V also has to the diameter of the sun a ratio greater
than that which 89 has to go.
Therefore WU also has to A a ratio greater than that which
89 has to go.
But, as WU is to VA,so is UA to AS, because S4, UW are
parallel ;
therefore (7A also has to 4S a ratio greater than that which 89
has to 90;
therefore (7A has to 4R a ratio much greater than that which
89 has to go.
The same is true of the doubles ;
therefore the diameter of the sun has to QF a ratio greater than
that which 89 has to go.
But it was proved [above] that OW has to the diameter of the
sun a ratio greater than that which 22 has to 225 ;
therefore, ex aeguali, ON has to QR a ratio much greater than
that which the product of 22 and 8g has to the product of go and
225, that is, 1958 to 20250, or, if the halves be taken, 979 to
10125.
PROPOSITION 14.
The straight line joined from the centre of the earth to the
centre of the moon has to the straight line cut off from the axis
towards the centre of the moon by the straight line subtending
the (circumference) within the earth's shadow a ratio greater
than that which 675 has fo τ.
For let the same figure be drawn as before; Ὁ
and let the moon be so placed that its centre is on the axis of the
cone comprehending both the sun and the earth ;
* The proof, which is given by Commandinus, is obvious. The fact cannot
be seen from our figure, which, owing to exigency of space, could not be drawn
so as to make the angles LBN, UBV equal,
400 ON THE SIZES AND DISTANCES
λαμβάνοντος τόν τε ἥλιον Kai τὴν γῆν, καὶ ἔστω τὸ Γ', μέγιστος δὲ
τῶν ἐν τῇ σφαίρᾳ κύκλος ὁ ΠΟΜ ἐν τῷ αὐτῷ ἐπιπέδῳ ὧν αὐτοῖς,
καὶ ἐπεζεύχθω ἡ ΜΟ' ἡ ΜΟ ἄρα διάμετρός ἐστι τοῦ διορίζοντος ἐν
τῇ σελήνῃ τό τε σκιερὸν καὶ τὸ λαμπρόν. ἐπεζεύχθωσαν δὴ αἱ
5 MB, BO, ΛΞ, ἘΒ, MI: ἐφάπτονται ἄρα τοῦ MOII κύκλου αἱ
ΜΒ, ΒΟ, διὰ τὸ τὴν ΟΜ διάμετρον εἶναι τοῦ διορίζοντος ἐν τῇ
σελήνῃ τὸ σκιερὸν καὶ τὸ λαμπρόν. καὶ ἐπεὶ ἴση ἐστὶν ἡ ἘΔ τῇ
MO: ἑκατέρα γὰρ αὐτῶν διάμετρός ἐστι τοῦ διορίζοντος ἐν τῇ σελήνῃ
τό τε σκιερὸν καὶ τὸ λαμπρόν: ἴση ἄρα καὶ ἡ BMA περιφέρεια τῇ
10 MAO περιφερείᾳ, καὶ ἡ ἘΜ ἄρα ἴση ἐστὶν τῇ AO. ἀλλ ἡ ΛΟ
τῇ ΔΜ ἴση ἐστίν: καὶ ἡ ἘΜ ἄρα ἴση ἐστὶν τῇ AM. ἔστι δὲ καὶ
ἡ ἘΒ ἴση τῇ ΒΑ, διὰ τὸ τὸ Β σημεῖον κέντρον εἶναι τῆς γῆς, καὶ
2. ἐν τῇ σφαίρᾳ] ἐν τῇ τῆς σελήνης σφαίρᾳ Nizze, suadente F, qui lectionem
cod. Parisiensis 2488 σελήνῃ ante σφαίρᾳ in σελήνης correxit ; mallem ἐν τῇ oa '
pro ἐν τῇ σφαίρᾳ, sed cf. 1.14, p. 364; 1. 10, ee 1, 24, p. 388 4. δὴ] δὲ
7. τὸ σκιερὸν] τό τε oKtepov ΝΥ ἐστὶν] om.
β
: joined.
itl a ΨΥ ee ee
OF THE SUN AND MOON 401
Jet its centre be C, and let the great circle OPM in the sphere
[i.e. the moon] be in the same plane with the rest of the
figure."
Let MP be joined ;
therefore 1/P is a diameter of
the circle which divides the dark
and the bright portions in the
moon.
Let UB, BP, LO, OB, MC be
Therefore 1/2, BP touch the
circle PQ,
because PJ is a diameter of the
circle which divides the dark and
the bright portions in the moon.
And, since OZ is equal to ZP—
for each of them is a diameter of
the circle which divides the dark
and the bright portions in the
moon—
therefore the circumference OML
is equal to the circumference
MLP;
therefore ΟἿ is also equal to ZP.
But ZP is equal to Z1/;
therefore ΟΠ is also equal to 2.17.
And OB is equal to BZ,
because the point 2 is the centre of the earth, and the earth has
Fig. 33.
1 Literally ‘in the same plane with ¢em’ (αὐτοῖς), which no doubt means the
axis and the sections of the sun and moon made by the plane originally assumed,
which also contains the circle in which the diameter of the ‘ dividing circle ’ in
the moon moves while the moon is passing through the earth’s shadow.
1410 pd
402 ON THE SIZES AND DISTANCES
(τὴν γῆν) σημείου καὶ κέντρου λόγον ἔχειν πρὸς τὴν τῆς σελήνης
σφαῖραν, καὶ τὸν MOII κύκλον ἐν τῷ αὐτῷ ἐπιπέδῳ εἶναι: ἡ ἄρα
BM καθετός ἐστιν ἐπὶ τὴν BA. ἔστιν δὲ καὶ ἡ ΓΜ κάθετος ἐπὶ
τὴν ΒΜ' παράλληλος ἄρα ἐστὶν ἡ ΓΜ τῇ BA. ἔστι δὲ καὶ ἡ
5 ΣΈ τῇ MP παράλληλος: ὅμοιον ἄρα ἐστὶ τὸ ABS τρίγωνον τῷ
MPT τριγώνῳ: ἔστιν ἄρα, ὡς ἡ ΣΈ, πρὸς τὴν MP, οὕτως ἡ YA πρὸς
τὴν PT. ἀλλ᾽ ἡ ΣΈ τῆς MP ἐστὶν ἐλάσσων ἣ β, ἐπεὶ καὶ ἡ ἘΝ
τῆς MO ἐλάσσων ἐστὶν ἣ β' καὶ ἡ Σ Δ ἄρα τῆς ΤΡ ἐλάσσων
ἐστὶν ἢ β' ὥστε ἡ ΣΡ τῆς PI πολλῷ ἐλάσσων ἐστὶν 7 B. ἡ ΣΤ
το ἄρα τῆς ΓΡ ἐλάσσων ἐστὶν ἣ τριπλασίων: ἡ ΤΡ ἄρα πρὸς τὴν ΤΣ
μείζονα λόγον ἔχει ἢ ὃν a πρὸς γ. καὶ ἐπεί ἐστιν, ὡς ἡ ΒΤ' πρὸς
ΓΜ, οὕτως ἡ ΓΜ πρὸς τὴν IP, ἡ δὲ BI πρὸς τὴν ΓΜ μείζονα
λόγον ἔχει ἢ ὃν με πρὸς a, καὶ ἡ ΓΜ ἄρα πρὸς ΤΡ μείζονα λόγον
ἔχει ἢ ὃν με πρὸς α. ἔχει δὲ καὶ ἡ ΤΡ πρὸς τὴν ΓΣ μείζονα
15 λόγον ἢ ὃν α πρὸς γ' St ἴσου ἄρα ἡ ΓΜ πρὸς τὴν ΤΊΣ μείζονα
λόγον ἔχει ἣ ὃν με πρὸς γ, τουτέστιν, (ἢ) ὃν ce πρὸς a. ἐδείχθη δὲ
καὶ ἡ ΒΓ πρὸς τὴν ΓΜ μείῤονα λόγον ἔχουσα ἢ ὃν με πρὸς. ar
de ἴσου ἄρα ἡ BI’ πρὸς τὴν TH μείζονα λόγον ἔχει ἢ ὃν τὰ χοε
πρὸς α.
20 ιε΄.
‘H τοῦ ἡλίου διάμετρος πρὸς τὴν τῆς γῆς διάμετρον
μείζονα λόγον ἔχει ἣ ὃν τὰ ιθ πρὸς γ, ἐλάσσονα δὲ ἣ ὃν
τὰ μγ πρὸς τὰ ς.
Ἔστω γὰρ ἡλίου μὲν κέντρον τὸ 4, γῆς δὲ κέντρον τὸ Β, σελήνης
a, δὲ κέντρον τὸ I’, τελείας οὔσης τῆς ἐκλείψεως, τουτέστιν, ἵνα τὰ
A, Β, Τ' én’ εὐθείας ἢ, καὶ ἐκβεβλήσθω διὰ τοῦ ἄξονος ἐπίπεδον,
1. (τὴν γῆν)}] haec verba prorsus necessaria solus habet Paris. 2364 :
5. ANS] ASH W γ,8,9. β] διπλασίων 12. τὴν ΓΡῚ TP W 13. a]
μίαν Vat. Paris. 2366, 2488: ἕν Paris. 2342, 2364 15. 1M] ΜΓ W τὴν]
om. W 16. δὲ] δὴ W
20. ve] IA Vat. 23. Tas] ¢ W
OF THE SUN AND MOON 403
the relation of a point and centre to the sphere in which the moon
moves [Hyp. 2], while the circle PQ is in the same plane ;
therefore BV is perpendicular to OL.
But CJ is also perpendicular to BY;
therefore ΟἿ is parallel to OL.
And SO is also parallel to WR;
therefore the triangle ZOS is similar to the triangle RC.
Therefore, as SO is to WR, so is SZ to RC.
But SO is less than double of YR,
since ON is also less than double of 272; [Prop. 13]
therefore SZ is also less than double of CR,
so that SR is much less than double of RC.
Therefore SC is less than triple of CR;
therefore CR has to CS a ratio greater than that which 1 has to 3.
And since, as BC is to CY, so is CY to CR,
while BC has to CW a ratio greater than that which 45 has to 1,
[see Prop. 11]
therefore ΟΜ also has to CR a ratio greater than that which
45 hastor. |
But CR also has to CS a ratio greater than that which 1
has to 3;
therefore, ex aeguali, CM has to CS a ratio greater than that
which 45 has to 3, that is, than that which 15 has to 1.
And it was proved that BC has to ΟἿ a ratio greater than that
which 45 has to 1;
therefore, ex aegualz, BC has to CS a ratio greater than that which
675 has to 1.
PROPOSITION 15.
The diameter of the sun has.to the diameter of the earth
a ratio greater than that which 19 has to 3, but less than that
which 43 has to 6.
For let 4 be the centre of the sun, B the centre of the earth,
C the centre of the moon when the eclipse is total, so as to secure
that 4, B, C may be in a straight line.
Dd2
404 ON THE SIZES AND DISTANCES
"καὶ ποιείτω τομὰς ἐν μὲν TO ἡλίῳ τὸν ΔΕΖ κύκλον, ἐν δὲ TH γῇ τὸν
HOK, ἐν δὲ τῷ σκιάσματι τὴν NE περιφέρειαν, ἐν δὲ τῷ κώνῳ
εὐθείας τὰς AM, ΖΜ, καὶ ἐπεζεύχθω ἡ ΝΈ, καὶ ἀπὸ τοῦ A τῇ AM
πρὸς ὀρθὰς ἤχθω OAII. καὶ
5 ἐπεὶ ἡ NE τῆς διαμέτρου τοῦ Me
ἡλίου ἐλάσσων ἐστὶν ἢ θ΄ μέρος,
ἡ ΟΓ ἄρα πρὸς τὴν NE πολλῷ
μείζονα λόγον ἔχει 7) ὃν τὰ θ
πρὸς α' καὶ ἡ ΑΜ ἄρα πρὸς
10 THY MP μείζονα λόγον ἔχει ἢ
ὃν τὰ θ πρὸς α. καὶ ἀνα-
στρέψαντι ἡ ΜΑ πρὸς AP
ἐλάσσονα λόγον ἔχει ἢ ὃν τὰ
θ πρὸς η. πάλιν ἐπεὶ ἡ AB
ι5 τῆς ΒΓ μείζων ἐστὶν ἢ ιη,
πολλῷ ἄρα τῆς ΒΡ μείζων
ἐστὶν ἢ in? ἡ AB ἄρα πρὸς τὴν
ΒΡ μείζονα λόγον ἔχει ἢ ὃν
τὰ in πρὸς a. ἀνάπαλιν ἄρα
2. ἡ ΒΡ πρὸς τὴν ΒΑ ἐλάσσονα
λόγον ἔχει ἢ ὃν α πρὸς ιη.
καὶ συνθέντι ἡ ῬΑ ἄρα πρὸς
τὴν AB ἐλάσσονα λόγον ἔχει
39
ἡ ὃν τὰ ιθ πρὸς τὰ Ln. ἐδείχθη
25. δὲ καὶ ἡ ΜΑ πρὸς τὴν ΑΡ
ἐλάσσονα λόγον ἔχουσα ἢ ὃν τὰ
θ πρὸς τὰ n° ἕξει ἄρα St ἴσου
ἡ MA πρὸς τὴν AB ἐλάσσονα
λόγον. ἢ ὃν τὰ ροα πρὸς ρμδ,
Fig. 34.
8ο καὶ ὃν τὰ ιθ πρὸς Ist τὰ
γὰρ μέρη τοῖς ὡσαύτως πολλαπλασίοις τὸν αὐτὸν ἔχει λόγον"
6. 8] ἔννατον W μέρος] ἡ ἄρα διάμετρος τοῦ ἡλίου μείζονα λόγον ἔχει πρὸς τὴν
NZ ἢ ὃν τὰ θ πρὸς ar καὶ add. W 9. πρὸς a] ἀλλ᾽ ὡς ἡ OT πρὸς τὴν NE, τουτέστιν,
ὡς ἡ AO πρὸς τὴν PN, οὕτως ἡ AM πρὸς τὴν PM, 8v ὁμοιότητα τριγώνων add, W
καὶ] om. W 10. MP] PMW_ μείζονα] πολλῷ μείζονα W 14. ἢ] ran W
15,17. 1η] ὀκτωκαιδεκαπλασίων W 431, ὡσαύτως] ὡσαυτῶι Vat.
OF THE SUN AND MOON 405
Let a plane be carried through the axis,
and let it cut the sun in the circle DEF, the earth in GHK, the
shadow in the circumference VO, and the cone in the straight
lines DM, FM.
Let WO be joined, and from 4 let PAQ be drawn at right
_ angles to 4M.
Then, since WO is less than 1/9th part of the diameter of the
sun, [Prop. 13]
therefore PQ has to VO a ratio much greater than that which
9. has to 1.
Therefore AMZ also has to MR a ratio greater than that
which g has to 1;
and, convertendo, MA has to AR a ratio less than that which
9 has to 8.
_ Again, since 4B is greater than 18 times BC, [Prop. 7]
therefore it is much greater than 18 times BR ;
therefore 428 has to BR a ratio greater than that which 18
has to 1;
therefore, inversely, BR has to BA a ratio less than that which
1 has to 18;
therefore, componendo, RA has to AB a ratio less than that
which το has to 18.
But it was proved that 1/4 also has to 4 R a ratio less than
that which 9 has to 8;
therefore, ex aeguali, MA will have to 47 a ratio less than that
which 171 has to 144, and therefore less than that which 19 has
to 16: for parts have the same ratio as the same multiples ot
them :
406 ON THE SIZES AND DISTANCES
ἀναστρέψαντι dpa ἡ AM πρὸς BM μείζονα λόγον ἔχει ἣ ὃν τὰ ιθ
πρὸς τὰ y. ὡς δὲ ἡ ΑΜ πρὸς MB, οὕτως ἡ διάμετρος τοῦ ΔΕΖ
κύκλου πρὸς τὴν διάμετρον τοῦ HOK κύκλου: ἡ ἄρα τοῦ ἡλίου
διάμετρος πρὸς τὴν τῆς γῆς διάμετρον μείζονα λόγον ἔχει ἣ ὃν τὰ ιθ
5 πρὸς γ.
Λέγω δὴ ὅτι ἐλάσσονα λόγον ἔχει (πρὸς αὐτὴν) ἢ ὃν τὰ μγ πρὸς
ς. ἐπεὶ γὰρ ἡ ΒΓ πρὸς τὴν ΓΡ μείζονα λόγον ἔχει ἢ ὃν τὰ χοε πρὸς
a, ἀναστρέψαντι ἄρα ἡ ΓΒ πρὸς τὴν ΒΡ ἐλάσσονα λόγον ἔχει ἣ ὃν
τὰ χοε πρὸς τὰ χοῦ. ἔχει δὲ καὶ ἡ AB πρὸς τὴν BI ἐλάσσονα
10 λόγον ἢ ὃν τὰ κ πρὸς a ἕξει ἄρα δι’ ἴσον ἡ AB πρὸς τὴν ΒΡ
ἐλάσσονα λόγον ἢ ὃν τὰ M yd πρὸς τὰ χοδ, τουτέστιν, ἣ ὃν τὰ
ay πρὸς τὰ τλῷ ἀνάπαλιν ἄρα καὶ συνθέντι ἡ PA πρὸς τὴν AB
μείζονα λόγον ἔχει ἢ ὃν τὰ (mg πρὸς ayy. καὶ ἐπεὶ ἡ Ν ἘΞ πρὸς
τὴν ΟΠ μείζονα λόγον ἔχει ἢ (ὃν τὰ) 00 πρὸς Mpxe, ἀνάπαλιν
τι ἄρα ἡ ΟΠ] πρὸς NE ἐλάσσονα λόγον ἔχει ἢ (ὃν Tx) Mpxe πρὸς
Hob: ὡς δὲ ἡ ΟΠ] πρὸς NZ, οὕτως ἡ ΑΜ πρὸς MP: καὶ ἡ AM
ἄρα πρὸς ΜΡ ἐλάσσονα λόγον ἔχει ἢ (ὃν 7a) Mpxe πρὸς ὃοθ'
ἀναστρέψαντι ἡ ΜΑ ἄρα πρὸς τὴν ΑΡ μείζονα λόγον ἔχει ἣ ὃν τὰ
Moke πρὸς τὰ Opus. ἔχει δὲ καὶ ἡ ῬΑ πρὸς AB μείζονα λόγον ἢ
20 ὃν τὰ mg πρὸς τὰ σψν' Ov ἴσου ἄρα ἕξει ἡ MA πρὸς τὴν AB
μείζονα λόγον ἢ ὃν ὁ περιεχόμενος ἀριθμὸς ὑπὸ τῶν Moke καὶ τῶν
(rg πρὸς τὸν περιεχόμενον ἀριθμὸν ὑπό τε τῶν Opus καὶ τῶν
‘ $poe “Ξρογ ie Sp0e ἡ
sw, τουτέστιν, ὁ M ewoe πρὸς Mcp. ἔχει δὲ καὶ 6 M woe
/Spoy
πρὸς M ep μείζονα λόγον ἣ ὃν τὰ μγ πρὸς λῷ Kal ἡ ΜΑ dpa πρὸς
as τὴν AB μείζονα λόγον ἔχει ἢ dv py πρὸς λῴ: ἀναστρέψαντι ἄρα ἡ
1. AM] ABW 6. δὴ] δὲ (πρὸς αὐτὴν) addidi δ8Β, ἄρα] ἕξει ἄρα W,
qui lacunam post |. 7 χοε expleverat ἔχει} om. W 11. My] Ma.yob W
12. TAC] τὰν Vat. 13. SW] ra ov W 14,15, 17, 19, 21. Mpxe] Μαιρκε W
16. 06] ra Ad W 17. 06] ra Aod W 21-2. τῶν <r] τὸν ζπῷ Vat.
Spe ᾿ς Spoy : Ξ
23, 24. M,ewoe] M {poe καὶ woe W, bis Med] MSpoy καὶ <p W, bis (haud recte)
OF THE SUN AND MOON 407
therefore, convertendo, AM has to BM a ratio greater than that
which το has to 3.
But, as 4.17 is to MB, so is the diameter of the circle DEF to
the diameter of the circle GHXK;
therefore the diameter of the sun has to the diameter of the earth
a ratio greater than that which το has to 3.
Again, I say that it has to it a ratio less than that which 43
has to 6.
For, since BC has to CR a ratio greater than that which 675
has to 1, [Prop. 14]
therefore, convertendo, CB has to BR a ratio less than that which
675 has to 674.
But ABP also has to BC a ratio less than that which 20
has to 1; [ Prop. 7]
therefore, ex aeguali, AB will have to BR a ratio less than that
which 13500 has to 674, that is, than that which 6750 has to 337;
therefore, inversely and componendo, RA has to AB a ratio
greater than that which 7087 has to 6750.
Now, since VO has to PQ a ratio greater than that which 979
has to 10125, [ Prop. 13]
‘therefore, inversely, PQ has to VO a ratio less than that which
10125 has to 979.
And, as PQ is to VO, so is AM to MR;
therefore 4M also has to WR a ratio less than that which 1o125
has to 979;
therefore, convertendo, MA has to AR a ratio greater than that
which 10125 has to 9146.
But RA also has to 4,18 a ratio greater than that which 7087
has to 6750;
therefore, ex aegualz, MA will have to AB a ratio greater than
that which the number representing the product of 10125 and 7087
has to the number representing the product of 9146 and 6750,
that is, 71755875 to 61735500.
But 71755875 has to 61735500 a ratio greater than that which
43 has to 37;*
therefore 174 also has to 4 fa ratio greater than that which 43
has to 37;
? As to this approximation see p. 336 ad jin.
5
408 ON THE SIZES AND DISTANCES
AM πρὸς τὴν MB ἐλάσσονα λόγον ἔχει ἣ ὃν TA py πρὸς G. ὡς δὲ
ἡ ΑΜ πρὸς τὴν ΒΜ, οὕτως ἐστὶν ἡ διάμετρος τοῦ ἡλίου πρὸς τὴν
διάμετρον τῆς γῆς᾽ ἡ ἄρα διάμετρος τοῦ ἡλίου πρὸς τὴν διάμετρον
τῆς γῆς ἐλάσσονα λόγον ἔχει ἢ ὃν py πρὸς ς. ἐδείχθη δὲ καὶ
μείζονα λόγον ἔχουσα ἢ ὃν τὰ ιθ πρὸς τὰ γ.
: ἐφ΄.
Ὁ ἥλιος πρὸς τὴν γῆν μείζονα λόγον ἔχει ἢ ὃν σωνθ
πρὸς Kg, ἐλάσσονα δὲ ἣ ὃν Mog¢ πρὸς σις.
Ἔστω γὰρ ἡλέου μὲν διάμετρος ἡ A, γῆς δὲ ἡ Β. ἀποδείκνυται
10 δὲ ὅτε ἐστίν, ὡς ἡ τοῦ ἡλίου σφαῖρα πρὸς τὴν τῆς γῆς σφαῖραν,
15
20
A ε > Ν ~ 4 ~ ς "2 4 Ν Ν > ~
οὕτως ὁ ἀπὸ τῆς διαμέτρου τοῦ ἡλίου κύβος πρὸς τὸν ἀπὸ τῆς
διαμέτρου τῆς γῆς κύβον, ὥσπερ καὶ ἐπὶ τῆς σελήνης, ὥστε ἐπεί
> « «ε 2 oe ~ 4 x XX + eA ~ ,᾿ cv
ἐστιν, ὡς ὁ ἀπὸ τῆς A κύβος πρὸς τὸν ἀπὸ τῆς Β κύβον, οὕτως
ὁ ἥλιος πρὸς τὴν γῆν, ὁ δὲ ἀπὸ τῆς A κύβος πρὸς τὸν ἀπὸ τῆς
Β (κύβον) μείζονα λόγον ἔχει 7) ὃν τὰ σωνθ πρὸς Kg, ἐλάσσονα δὲ
¢ “
ἢ ὃν M Oh¢ πρὸς σις" καὶ yap ἡ A πρὸς τὴν Β μείζονα λόγον ἔχει
ἢ ὃν ιθ πρὸς γ, ἐλάσσονα δὲ ἢ ὃν py πρὸς g ὥστε ὁ ἥλιος πρὸς
τὴν γῆν μείζονα λόγον ἔχει ἢ ὃν σωνθ πρὸς Kg, ἐλάσσονα δὲ ἢ ὃν
ζ
Ν
M 6¢6¢ πρὸς σις.
ἐξ.
Ἢ διάμετρος τῆς γῆς πρὸς τὴν διάμετρον τῆς σελήνης
ἐν μείζονι μὲν λόγῳ ἐστὶν ἣ ὃν (ἔχε py πρὸς py, ἐν
ἐλάσσονι δὲ ἢ ὃν ἃ πρὸς 18,
1. τὴν MB] MB W 2. 7 AM] pax
6. ις΄] IE Vat. 8. κῷ τὰ κα W MOC] μυριάδες ζ καὶ OE W σις] ις Vat.
εἰ codd. Paris., excepto 2364 9. γῆς δὲ] γῆς W 11. διαμέτρου] διαμέτρου
tis Ν 14. τὸν] τὴν Vat. 15. (κύβον)] om. Vat. εἰ codd. Paris. 16. M,6p{]
Μῷ θφ Ws ots] «5 Vat. 17. ὥστε] apodosis hic desideratur ; exspectaveris
διὰ ταῦτα δὴ ὁ ἥλιος vel ὁ ἥλιος ἄρα 19. MOp¢] μυριάδες ζᾧ καὶ θφζ W
20, (1 19 Vat. 22. py] τὰ pn W
"ΠΥ ὙΡῚ we ee Po. ey Ψονν aA
OF THE SUN AND MOON 409
therefore, convertendo, AM has to MB a ratio less than that
which 43 has to 6.
But, as 4M is to BM, so is the ΤΌ ΞΩΡΕΑ of the sun to the
diameter of the earth;
therefore the diameter of the sun has to the diameter of the earth
a ratio less than that which 43 has to 6.
And it was before proved that it has to it a ratio greater than
that which 19 has to 3.
PROPOSITION 16.
The sun has to the earth avratio greater than that which 6859
has fo 27, but less than that which 79507 has to 216.
For let 4 be the diameter of the sun, # that of the earth.
Now it is proved that, as the sphere of the sun is to the sphere
of the earth, so is the cube on the diameter of the sun to the cube
A
B
Fig. 35.
on the diameter of the earth, just as in the case of the moon [cf.
Prop. ro].
Thus, since, as the cube on 4 is to the cube on Z, so is the sun
to the earth,
while the cube on 4 has to the cube on Δ᾽ a ratio greater than that
which 6859 has to 27, but less than that which 79507 has to 216:
for 4 has to 2 a ratio greater than that which 19 has to 3, but less
than that which 43 has to 6: τς [Prop. 15]
it follows that the sun has to the earth a ratio greater than that
which 6859 has to 27, but less than that which 79507 has to 216.
PROPOSITION 17.
The diameter of the earth ts to the diameter of the moon
in a ratio greater than that which 108 has to 43, but less than
that which 60 has to το.
410 ON THE SIZES AND DISTANCES
Ἔστω yap ἡλίου μὲν διάμετρος ἡ A, σελήνης δὲ ἡ B, γῆς de
ἡ T. καὶ ἐπεὶ ἡ A πρὸς τὴν Γ' ἐλάσσονα λόγον ἔχει ἢ ὃν τὰ μγ
πρὸς ς, ἀνάπαλιν ἄρα ἡ Τ' πρὸς τὴν A μείζονα λόγον ἔχει ἢ ὃν σ'
πρὸς py. ἔχει δὲ καὶ ἡ A πρὸς τὴν Β μείζονα. λόγον ἢ ὃν τὰ
51n πρὸς a: Ov ἴσου ἄρα ἡ I’ πρὸς τὴν Β μείζονα λόγον ἔχει ἢ ὃν
Α ΐ
α
Β
Got
4
Fig. 36.
τὰ pn πρὸς τὰ py. πάλιν ἐπεὶ ἡ A πρὸς τὴν Τ' μείζονα λόγον ἔχει
ἢ ὃν τὰ ιθ πρὸς τὰ γ, ἀνάπαλιν ἄρα ἡ I’ πρὸς τὴν A ἐλάσσονα
λόγον ἔχει ἢ ὃν τὰ γ πρὸς τὰ ιθ. ἔχει δὲ ἡ A πρὸς τὴν Β ἐλάσσονα
λόγον ἢ ὃν τὰ κ πρὸς a: δι’ ἴσου ἄρα ἡ Τ' πρὸς τὴν Β ἐλάσσονα λόγον
το ἔχει ἢ ὃν ἕ πρὸς ιθ.
ιη΄.
‘H γῆ πρὸς τὴν σελήνην ἐν μείζονι μὲν λόγῳ ἐστὶν
“ ὃν (éxet) Μθψιβ πρὸς M696, ἐν ἐλάσσονι δὲ ἢ ὃν
Μ Ss πρὸς ove.
15 “στῶ γὰρ γῆς μὲν διάμετρος Ἧ A, σελήνης δὲ ἡ Β' ἡ 4 ἄρα
ig τὴν B μείζονα λόγον ἔχει ἢ ὃν τὰ py πρὸς τὰ py, ἐλάσσονα
δὲ ἢ ὃν τὰ E πρὸς ιθ' καὶ ὁ ἀπὸ τῆς A ἄρα κύβος ΤῊΝ τὸν ἀπὸ τῆς
Β κύβον μείζονα λόγον ἔχει ἢ ὃν ΜΌθψιβ πρὸς M θφς, ἐλάσσονα
δὲ ἢ ὃν Μ ς πρὸς σωνθ. ὡς δὲ ὁ ἀπὸ τῆς A κύβος πρὸς τὸν ἀπὸ
20 79S Β κύβον, οὕτως ἐστὶν ἡ γῆ πρὸς τὴν σελήνην" ἡ γῆ ἄρα πρὸς
κε ¢
τὴν σελήνην μείζονα μὲν λόγον ἔχει ἢ ὃν M byi8 πρὸς M O¢¢,
κα
ἐλάσσονα δὲ ἢ ὃν M ς πρὸς “ωνθ.
I σελήνης δὲ ἡ Β] σελήνης ἡ BW 5. ἄρα] γὰρ Vat. 8. 10] 6 Vat.
ἡ] καὶ ἡ W
11. ιη} oF Vat. 13. (ἔχει) MOv8] oa μυριάδες ρκε καὶ θψιβ W
13, 18, 21. ἡ φῇ Mé. bot W 14, 19, 22. Ms] Mxa.> W 16. ra py]
py W 18, 21. ΜΑΨΙΒῚ Μρκε. θψιβ W 21. πρὸς] πρὸς μὲν W
OF THE SUN AND ΜΟΟΝ 4τπτ
For let 4 be the diameter of the sun, B that of the moon, C that
of the earth.
Then, since 4 has to ( ἃ ratio less than that which 43 has to 6,
[Prop. 15]
therefore, inversely, C has to 4 a ratio greater than that which
6 has to 43.
But 4 also has to Z a ratio greater than that which 18 has to 1 ;
[Prop. 9]
_ therefore, ex aeguali, C has to B a ratio greater than that which
ο΄ 108 has to 43.
Again, since 4 has to C a ratio greater than that which 19
has to 3, [Prop. 15]
therefore, inversely, C has to 4 a ratio less than that which
3 has to το.
But 4 also has to # a ratio less than that which 20 has to 1;
[Prop. 9]
therefore, ex aegualz, C has to Ba ratio less than that which 60
has to 19.
PROPOSITION 18.
The earth ts to the moon tn a ratio greater than that which
1259712 has to 79507, but less than that which 216000 has 20
68509.
For let 4 be the diameter of the earth, 2 that of the moon;
therefore 4 has to # a ratio greater than that which 108 has to
43, but less than that which 60 has to 19. [Prop. 17]
Therefore also the cube on 4 has to the cube on BZ a ratio
A
B
Fig. 37.
greater than that which 1259712 has to 79507, but less than
that which 216000 has to 6859.
But, as the cube on 4 is to the cube on B, so is the earth to
the moon;
therefore the earth has to the moon a ratio greater than that which
1259712 has to 79507, but less than that which 216000 has to 6859.
412 ON THE SIZES AND DISTANCES
COMMENTS OF PAPPUS.!
‘In his book on sizes and distances Aristarchus lays down these
six hypotheses :
1. That the moon receives light from the sun.
2. That the earth is in the relation of a point and centre to
the sphere in which the moon moves.?
3. That, when the moon appears to us halved, the great
circle which divides the dark and the bright portions of the moon
is in the direction of our eye.
4. That, when the moon appears to us halved, its distance
from the sun is then less than a quadrant by one-thirtieth of a
quadrant.’
5. That the breadth of the (earth’s) shadow is (that) of two
moons.
_ 6, That the moon subtends one fifteenth part of a sign of the
zodiac.
Now the first, third, and fourth of these hypotheses practically
agree with the assumptions of Hipparchus and Ptolemy. For the
moon is illuminated by the sun at all times except during an
eclipse, when it becomes devoid of light through passing into the
shadow which results from the interception of the sun’s light by
the earth, and which is conical in form; next the (circle) dividing
the milk-white portion which owes its colour to the sun shining
upon it and the portion which has the ashen colour natural to the
moon itself is indistinguishable from a great circle (in the moon)
when its positions in relation to the sun cause it to appear halved,
at which times (a distance of) very nearly a quadrant on the circle
of the zodiac is observed (to separate them) ; and the said dividing
circle is in the direction of our eye, for this plane of the circle
if produced will in fact pass through our eye in whatever position
the moon is when for the first or second time it appears halved.
1 Pappus, vi, pp. 554. 6-560. το (Hultsch).
? Literally, ‘the sphere of the moon.’
5. Hultsch brackets, as an obvious interpolation, words added here in the
Greek text ‘instead of (saying) that its distance is 87°: for this is less than
a quadrant, or 90°, by 3°, which is 1/30th of 90°’,
OF THE SUN AND MOON 413
But, as regards the remaining hypotheses, the aforesaid mathe-
maticians have taken a different view. For according to them the
earth has the relation of a point and centre, not to the sphere in
which the moon moves, but to the sphere of the fixed stars, the
breadth of the (earth’s) shadow is not (that) of two moons, nor does
the moon’s diameter subtend! one fifteenth part of a sign of the
zodiac, that is, 2°. According to Hipparchus, on the one hand, the
circle described by the moon is measured 650 times by the diameter
of the moon, while the (earth’s) shadow is measured by it 24 times at
its mean distance in the conjunctions; in Ptolemy’s view, on the
other hand, the moon’s diameter subtends, when the moon is at its
greatest distance, a circumference of οὗ 31’ 20”, and when at its
least distance, of οὗ 35 20”, while the diameter of the circular
section of the shadow is, when the moon is at its greatest distance,
οὗ 40° 40”, and when the moon is at its least distance, οὗ 46’.
Hence it is that the authors named have come to different
conclusions as regards the ratios both of the distances and of the
sizes of the sun and moon.
Now Aristarchus, after stating the aforesaid hypotheses, proceeds
in a passage which I will quote word for word.?
* We are now in a position to prove that the distance of the sun
from the earth is greater than 18 times, but less than 20 times, the
distance of the moon, and the diameter of the sun also has the same
ratio to the diameter of the moon: this follows from the hypothesis
about the halved (moon). Again, we can prove that the diameter
of the sun is to the diameter of the earth in a greater ratio than
that which 19 has to 3, but in a less ratio than that which 43 has
to 6: this follows from the ratio thus discovered as regards the
distances, from the hypothesis about the shadow and from the
hypothesis that the n-oon subtends one fifteenth part of a sign
of the zodiac.” .
He says “ We are in a position to prove that the distances”, &c.,
1 Hultsch brackets, as an interpolation, some clumsy words in the Greek text,
the object of which is to qualify ‘diameter’ and make it mean the diameter of
the moon when it is ‘at the same mean distance’; there are no such words in
Aristarchus.
? GK. λέγων κατὰ λέξιν οὕτως. Although Pappus professes to quote Aristarchus
word for word, he shows some slight variations from the text of Aristarchus
as we have it; where the changes are for the worse, however, they may be due
to copyists rather than to Pappus himself.
------------- -. -.
414 COMMENTS OF PAPPUS
implying that he will prove the properties after giving such pre-
liminary lemmas as are of use for the proofs of them. As the renee
of the whole investigation he concludes that
(1) the sun has to the earth a greater ratio than that which
6859 has to 27, but a less ratio than that which 79507 has to 216;
(2) the diameter of the earth is to the diameter of the moon in
a greater ratio than that which 108 has to 43, but in a less (ratio)
than that which 60 has to 19; and 3
(3) the earth is to the moon in a greater ratio than that which
1259712 has to 79507, but in a less (ratio) than that which
216000 has to 6859.
But Ptolemy proved in the fifth book of his Syntaxis? that, if the
radius of the earth is taken as the unit, the greatest distance of
the moon at the conjunctions is 64%3 of such units, the greatest
distance of the sun 1210, the radius of the moon 3% 38. the radius
of the sun 522. Consequently, if the diameter of the moon is
‘taken as the unit, the earth’s diameter is 32 of such units, and the
sun’s diameter 184. That is to say, the diameter of the earth is
32 times the diameter of the moon, while the diameter of the sun
is 184 times the diameter of the moon and 5% times the diameter of
the earth.
From these figures the ratios between the solid contents are
manifest, since the cube on 1 is 1 unit, the cube on 32 is very
nearly 391 of the same units, and the cube on 184 very nearly
66442, whence we infer that, if the solid magnitude of the
moon is taken as a unit, that of the earth contains 39% and that
of the sun 66444 of such units; therefore the solid magnitude of
the sun is very nearly 170 times greater than that of the earth.’
1 Ptolemy, Sya¢axis, v, 14-16, vol. i, pp. 416-427, Heib.
INDEX
Achilles (not Tatius) 5, 112, 116, 174.
Adam, Dr. 110 #., 138-9, 149-52, 153,
156, 171-2.
Adrastus (in Theon of Smyrna) 6, 112,
199, 256, 257-8, 268, 270, 342.
Aeschylus, pupil of "Hippocrates of
Chios, 243-4
‘Aétius, περὶ ἀρεσκόντων (De placitis), 4
and passim.
‘Agesilaus 192.
Alcmaeon of Croton 49-50, 103, 107.
Alexander Aphrodisiensis 98, 111, 129,
176, 187, 275, 282.
Alexander of Ephesus (or Miletus)
112-14.
Alexander Polyhistor 2, 65 7:.
Anatolius 115.
Anaxagoras 18, 30, 40, 44, 48, 50, 78-85,
143: cosmogony through motion of
vortex and ‘hurlings-off’ (centri-
fugal force) 81-2 : other worlds than
ours 85 : stars are stones on fire 81-2,
kindled by ry ae motion 81: stars
first revolved horizontally, then axis
tilted 82-3, 91, 92: stars more dis-
tant than sun and moon 82: remark-
able theory of Milky Way 83-5:
order of planets 85, 128: sun a red-
hot mass or stone on fire 82, larger
than the Peloponnese #é7d.: earth
(and probably other bodies) flat 81,
QI-2, 144, 146: earth rests on air 83,
144, 238 : discovered that moon is lit
up by sun 19, 76, 77, 78-9, 91, Sek
on substance and light of moon 82:
other dark bodies besides earth and
moon cause eclipses 79-80: on
comets 125, 243: story of meteoric
stone 246.
Anaximander 4, 24-39, 55, 114: work
About Nature 24 : cosmogony 25-7,
28-30: innumerable worlds 29-30:
sun moon and stars hoops or wheels
with vents 27, 28, 31: positionof hoops
31, 33-6: sizes of hoops of sun and
1410
moon, and of sun and moon them-
selves (first speculation on sizes and
distances) 27, 28, 32, 37, 38, 114:
stars nearer than sun and moon 28:
earth a short cylinder 25, and sus-
pended freely 24, 25, 64, 145: drew
first map of inhabited earth 38, 39,
124, 145 : said to have introduced
gnomon 38 and constructed sphere
38, 319: evolutionary theory 39 z.
Anaximenes 19, 27 #., 33 %., 40-5:
on nature of heavenly bodies 40:
stars fixed like nails on crystal
sphere 40, 45, except (presumably)
planets 42, 43, 50: stars more dis-
tant than sun 43: stars revolve
laterally round, not under, earth 41-
3, 83: earth flat and rides on air 40,
83, 144, 238: earthy bodies moving
about among stars 43, 44, significant
with reference to eclipses 44-5 (cf.
Anaxagoras) : universe breathes 45.
*Avadopixos (De ascensionibus) of
Hypsicles 317, 321, 325.
Antipodes 65.
Antisthenes, author of διαδοχαί, 2.
Apelt, E. F. 104, 194.
Apollodorus of Athens, grammarian, 5.
Apollodorus of Corcyra 307.
Apollonius of Perga 193, 299: had
theory of epicycles in all its generality
266-7, 274, but thought eccentrics
only applicable to superior planets
266-8, 274: possibly originated
hypothesis of Tycho Brahe 269, 274.
Aquinas, Thomas, 177.
Arabic versions of Aristarchus 320-1:
MSS. of, 321, 323.
Aratus 23, 112, 222 721.
Archedemus 187.
Archelaus 124, 146.
Archer-Hind 145 #., 167-8, 177, 180.
Archimedes 23, 221, 299, 366 #.: on
Democritus’s discovery of volume of
pyramid and cone 121: on Aristar-
416
chus’s heliocentric hypothesis 302-4,
on other hypotheses of Aristarchus
302, 308-9: on apparent diameter
of sun (a) as discovered by Aristar-
chus 311, 353 #., (2) as measured by
himself 312, 348: on estimates of
relative size of sun and moon 332,
337 : on estimates of earth’s circum-
ference 147, 337-8, 347: allowed for
parallax in case of sun 348.
INDEX
40, 238, Xenophanes 54, Empedocles
88, 239: on Heraclitus’s sun 60 %. :
on Anaxagoras’s and Democritus’s
theory of Milky Way 83-5, 247:
on Empedocles’ theory that light
travels 93: on Pythagorean system
of ten ‘bodies’ revolving round
central fire 95-6, 98, 99, 100, 103,
237 (‘outermost revolution ’ a ‘physi-
cal body’ 233): reference to others
who displaced earth from centre
186-7, 273 : on supposed rotation
of earth in 7imaeus 174, 176-8, 240:
on Eudoxus’s system of concentric
Archytas, as geometer, 190, 191, 192,
299: on mathematics 319-20.
ἀριθμητική 135.
Aristarchus of Samos 282 : ‘ the mathe-
matician,’ pupil of Strato 299: on
vision, light and colours 300: cos- |
mology zd¢d.: inventor of a sun-dial |
(σκάφη) 312: originator of heliocen-
tric hypothesis 301-6, attacked for
it by Cleanthes 304 : earth negligible
in size compared with universe 302,
308-9, and (in extant treatise) even |
with orbit of moon 309, 353, 412:
moon a satellite of earth 310: found
angular diameter of sun to be }° but
in treatise assumes value 2°, 23, 311--
12, 353, 412: ‘Great Year’ of 2434
years and solar year of 3654 +4!
days 314-16, explanation of these
figures zéd.: treatise on sizes and
distances 317, Arabic translations
320-1, editions 321-4, MSS. 325-7,
style 328, trigonometrical bearing of
geometry in, 328, contains no trace
of heliocentric hypothesis 310,
assumptions in, as to apparent dia-
meter of sun 353, 412, as to equality
of apparent diameters of sun and
moon 383, 412, as to diameter of
earth’s shadow (= 2 diameters of
moon) 329 sq., 337, 353, 412, as to
angle subtended at centre of sun by
line joining centres of earth and
moon at half-moon 329, 337, 353,
412: trigonometrical equivalent of
propositions 333-6: summary of
main results 338, 350, 413-14: text
spheres 193-5, 196, 197, 198, and
Callippus’s improvements 212: A.’s
own modification of system of con-
centric spheres in mechanical sense
by means of reacting spheres (ἀνελίτ-
Tovoat) 217-21, 225: supposed doubts
as to concentric system 222, 223,
261: A.’s own views, on motion and
primum movens 225-7, kinds of
motion appropriate to elements 227,
aether 227-8: universe one 228-9,
finite 229, spherical 229-30: eternal
movements other than that of whole
universe 230-1: stars spherical 233:
stars do not move of themselves but
are fixed on spheres which move
106 #., 233-4: on question whether
moving heavenly bodies produce
sound 105-6, 108: heavenly bodies
do not rotate or vol/ 234-5 (A. does
not deny z#cidenta/ rotation of moon
235): on planets’ obliquities 155 7. :
earth a sphere, reasons for this view
235-7: attempt to prove earth at
rest in centre 237-41: on size of
earth 147, 236: on ‘right’ and ‘left’
in universe 161, 231-2: on ‘up’ and
‘down’ 237-8 21.,) 239: on four ele-
ments, their ‘causes’ and inter-
changes 241-2: on shooting stars
and meteors 219, 242-3: on comets
219, 243-7: on Milky Way 219,
247-8: on the tides 306.
Aristoxenus 5.
Ars Eudoxi, or Didascalia caelestis of
Leptines 112, 200 #., 208, 293.
Assyrian predictions of eclipses, 16, 17.
Astronomy, ‘Treasury’ of (ἀστρονο-
povpevos τόπος), or ‘Little Astro-
nomy’ (μικρὸς ἀστρονομούμενος) 317--
20: Arabic translations of, 320-1.
Athenaeus II.
and translation of treatise 351-411.
Aristarchus of Samothrace Io 2.
Aristotherus 222.
Aristotle 1, 9, 12 ., 18, 30, 32, 41,
47, 48, 49, 50, 61, 83: criticisms
of views of predecessors regarding
the earth, Thales 18 ., 19 2., 238,
Anaximander 24-5, 239, Anaxi-
menes, Anaxagoras and Democritus
INDEX
Autolycus of Pitane 221, 261: works
On the moving sphere and On risings
and settings 192, 317-18, 321, 325.
Babylonians, used and polos
38 : unacquainted with sphericity of
earth 48 #.: unacquainted with pre-
cession in third cent. B.C., 105 : esti-
mate of apparent diameter of sun
22, 311: reference in Aristotle to,
220: see also Chaldaeans.
Bear, Great, non-setting in Homer's
time, but not so now in Mediterra-
nean 8-9 #.: Eudoxus and Proclus
on position of, zézd.
Bear, Little, observed by Thales 23:
Phoenician navigators set their
course by, 23.
Berger 39 7., 481., 55 71.) 56-8, 154-5 71.,
337 71.,) 344, 345 #.
Bergk 277, 279-80, 302 71.) 303-4 7, 307.
Bernard, Edward, 323.
Berosus 16 #., 314.
Berry, A. 343 7.
Bilfinger 22 7., 193 722.
a egg by Satyrus, Heraclides
Lembus, Hermippus, Laertius Dio-
genes 2-3.
Bjornbo, A. A. 318 2.
Boeckh i101, 105, 117, 118, 150-1, 1527.,
153, 161-2, 163, 175, 177, 178, 179,
180, 183, 186, 231 22.) 250, 258 #.,
277, 290, 291, 293, 295.
Boll, F. 14 22.) 16 2., 76, 80 721.) 343 71.
Bosanquet 139-40.
Brandis 22 2.
Burnet (Zarly Greek Philosophy) 6,
25 71... 29, 30 71.) 46 71., 47 71.) 50, 53,
62, 67 #., 69 #., 85 2., 92 7., 100 7.,
103, 105 #., 107, 116, 117, 121, 173.
Callippus 129, 193, 197, 200, 293: im-
provements on Eudoxus’s theory of
concentric spheres 212-16, 217, 219,
220, 221, 225, 261, 278: on length
of seasons 215-16: cycle of 76 years
(ἑκκαιεβδομηκονταετηρίς) 295-6.
Canopus, the star, 192 #., 345.
Cantor, M. 16 5.
. Cassini 343.
inus 107 #., 113, 114, 129, 132,
284, 286, 291, 292, 293, 314, 315.
Chalcidius 164, 167, 270, 342: on
Heraclides’ theory of Venus and
Mercury 256-8.
Chaldaeans: predictions of eclipses
1410
417
by means of periods of 223 luna-
tions 16 (cf. 314, 315): eclipses
observed in Babylon in 721-0 B.c.,
16: Chaldaean order of planets
258, 259.
Cicero 4, 15 #., 29, 111, 167, 178, 188,
252, 253, 256, 258, 259.
Cleanthes, the Stoic, 74 #.: attacked
Aristarchus for his heliocentric
hypothesis 304. :
Cleomedes 22, 23, 80 22.) 155 22.) 223-4,
310, 313, 339) 342 %-, 344-7.
Cleostratus, Astrology 23: connected
with octaéteris 291.
Comets, views on: Anaxagoras and
Democritus 125, 243, 245, certain
Pythagoreans 243-4, Hippocrates of
Chios and Aeschylus 243-4, Aris-
totle 243-7, Heraclides 254, Seneca
247 71. : particular comets mentioned
by Aristotle 244-6.
Commandinus, translation of Aristar-
chus 321, 323.
Continued fractions, application of,
probably as early as Aristarchus
336.
Cook Wilson 156.
Counter-earth of Pythagorean system
96-8: invented to explain eclipses
99-100, 119, abandoned early 249:
identified with moon 250.
Copernicus : allusions of, to Philolaus,
Heraclides, Ecphantus, and Hicetas
301: was aware of Aristarchus’s
hypothesis, zézd.: on distance of sun
343-
Crates of Mallos 305, 306-7.
Cycles (of years) : reputed ¢rieferis and
tetraéteris or pentaéterzs (q.v.) 286:
octaéteris 287-92: 16-years’ (éxxat-
Sexaernpis) and 160-years’ periods
292-3: Meton’s cycle of 19 years (ἐν-
veaxatdexaetnpis) 293-5: Callippus’s
cycle (76 years) 295-6: Hipparchus’s
cycle (304 years) 296-7.
Damastes, of Sigeum, 124.
Day, in Greece, was from one sunset
to next 284.
Delambre 17 #.
Democritus 26, 40, 50, 64: discovered
volumes of pyramid and cone, and
was on track of infinitesimals 121-2:
in astron. generally followed Anaxa-
goras 123: sun and stars red-hot
stones, moon has light from sun
Ee
418 INDEX
123-4: ΟἹ comets 125, 243,245: on
Milky Way 83, 124-5: infinite num-
ber of worlds 125 : oursun and moon
originally nuclei for other worlds
127-8: on planets and their order
128: on relative speeds of sun, moon,
and planets 128-9: earth a disc
hollowed out in middle 124, and
elongated (προμήκης) 84, 124: earth
in equilibrium without support (cf.
Anaximander and Parmenides) 64,
124, or rides on air 144,238: ‘ Great
Year’ 129: geographical and nauti-
cal survey of inhabited earth 124,
145: Gomperz’s rhapsodical estimate
of Democritus’s astronomy 125-7.
De Morgan 139.
Dercyllides (in Theon of Smyrna) 6,
131, 183, 304, 307.
διαδοχαί, ‘ Successions,’ authors of, 2.
Diakosmos, Great, of Leucippus 122.
Dicaearchus 306, 337, 339, 347-
Diels: Doxographi Graect 2-5, and
‘Fragmente der Vorsokratiker 5,
passim: 27 t, 37-8, 52, 67, 70-4,
76, 79 M5 92 μ΄, 97 ὦ), 122 %., 277,
279, 281, 295 2.
Diodorus, of Alexandria, Astronomy 5.
Diodorus Siculus 17, 131, 293, 294,
295.
Diogenes of Apollonia 2, 246 .
Diogenes of Babylon 107.
Diogenes Laertius 2-3, and quoted
passim.
Dioscuri 55.
Dositheus 291.
Doxographi Graeci, Diels’ account of,
2-5: genealogical table 3.
Dreyer, J. L. Ἐς 33 #, 82 2. 90,
177 My 195 21., 220 72.) 235 71.) 240,
340 2.
Earth: sphericity of, first maintained
by Pythagoras and Parmenides 21,
48, 49, 64, unknown to Babylonians
and Egyptians 48 2.: rotation of,
affirmed by Heraclides 251, 254-5
and Aristarchus 304 (discovery ques-
tionably attributed to Hicetas 187-9,
and to Ecphantus 251-2, 282),admit-
ted as possibility by Seneca 307-8:
size of, earliest estimates 147, 236,
Eratosthenes’ measurement, 114,
147, 339-40, Posidonius’s measure-
ments 345-7 : size relatively to sphere
of universe, &c. 308-10: as ‘ instru-
ment of time’ 250.
Earth’s shadow, diameter of, at dis-
tance of moon, as estimated by
Aristarchus, Hipparchus, and Pto-
lemy 329, 337, 353s 412, 413.
Eccentrics, movable, 263 sq.: equiva-
lent to epicycles 264-6: with Apollo-
nius application limited to superior
planets 266-7, 274: theory gene-
ralized by Hipparchus’s time 267--
8: not invented by Pythagoreans
270-4.
Ecphantus, credited with theory of
earth’s rotation 251, 282: probably
a personage in one of Heraclides’
dialogues 251-2.
Eclipses : supposed prediction of solar
eclipse by Thales 13-18, by Helicon
13-14, 193: date of Thales’ eclipse
15-16: period of recurrence (223
iamatianey discovered by Chaldaeans
16, and known to Thales zd7d., ar
haps to Egyptians 17: cuneiform
inscription on observations of pre-
dicted eclipses 16-17 : causes known
to Pythagoreans 119: ‘ paradoxical
case,’ and true explanation given by
Cleomedes 80 #.: Sosigenes on ᾿
annular eclipses of sun 222-4, 313,
383 #.
Ecliptic (or zodiac circle): discovery
of obliquity attributed to Oenopides
21, 130-1, 319, possibly learnt from
Egyptians 131, 319: estimate of
obliquity 24°, discovered before
Euclid’s time 131 #., and possibly
still used by Eratosthenes and Hip-
archus zzd.: later estimate, half of
τ of 360°, attributed to Eratos-
thenes but probably Ptolemy’s zd¢d.:
another value incidentally given by
Pappus 132 77.
Egyptians: predicted eclipses 17:
possibly discovered obliquity of
ecliptic 131, 319: unaware of spheri-
city of earth 48 ~.: on order of
planets 258: Heraclides’ theory
regarding Venus and Mercury
wrongly attributed to, 259: had year
of 365 days and month of 30 days
21: drew maps 38.
ἑκκαιδεκαετηρίς (16 years’ period) 292-3.
ἑκκαιεβδομηκονταετηρίς (76 years’ cycle
of Callippus) 295-6.
Ell {πῆχυς), Babylonian: use of, as
το τῷῳ Po
INDEX 419
. astronomical measure of angles
(= 2°) 23».
Empedocles : date, ἅς. 86 : poems 87 :
stars fixed on crystal sphere, which |
explanation of |.
is egg-shaped 87:
night and day by dark and light
hemispheres revolving 87-8: earth
kept in place by swift revolution of |
heaven 88, 144, 239: sun explained
as reflection of fire in universe 80- |
90: on tropic circles limiting motions
of sun 87, 91: substance of moon
and stars 91: moon lit up by sun
zbid.: earth flat (probably) 91: axis
of earth originally perpendicular to |
surface 9I-2, subsequently became
tilted 92: fires in centre of earth 92:
sun most distant of heavenly bodies
87, and twice as distant as moon 92:
gave true explanation of eclipses 92 :
greatest scientific achievement was
theory that light travels 92-3.
ἐννεακαιδεκαετηρίς (19 years’ period,
Meton’s cycle) 293-5.
Epicurus, on size of sun 61, 253: on
Heraclides 253.
Epicycles : equivalence of, to movable
eccentrics 264-6 : theory understood
in all its generality (including epi-
cycles with ideal points as centres)
by Apollonius 266-7: not invented
by Pythagoreans 270-4: wrongly
imported into explanations of Plato’s
and Heraclides’ systems 166-7,
256-8, 260.
Efpinomis, astronomy of, 184-5.
Eratosthenes 112, 299, 345, 348: on
measure of obliquity of ecliptic 131-
2 21. : measurement of circumference
of earth 114, 147, 339-40: on dis-
tances of sun and moon from earth
340-1.
Euclid: works included in ‘ Little
Astronomy’, Phaenomena 309-10,
317-19, Optics 317, 320.
Euctemon : on length of seasons 200,
' 213, 215: associated with Meton’s
cycle 293, 296.
Eudemus, History of Astronomy, 6,
14, 19, 20, 21, 24 71.. 37, 131, 140,
213, 272.
Eudorus 5.
Eudoxus 131 #., 141, 200, 278, 331: par-
ticulars of life 190-2: pupil of Archy-
tas 190, 192: discoverer of general
theory of proportion expounded in
Eucl. V, and of method of exhaus-
tion 191: wrote Mirror (ἔνοπτρον)
192, 199, Phaenomena 192, On
Speeds 193, 320 #., and probably a
work on Sfhaeric 192, 318 : invented
arachne 193: varied, if he did not
invent,ocfaéteris (8-years’ cycle) 291,
293: views on position of north pole
8-9 #., on ratio of sizes of sun and
moon (9:1) III, 332, 337, possible
explanation of latter ratio 111-12 :
theory of concentric spheres, de-
scribed by Aristotle and Simplicius
193-4,196-202, Schiaparelli’srestora-
tion of system 202-11: hippopede
described by planets 202, proof of
shape 204-5 #.: supposed sun to
deviate in latitude 198-9: ignored
differences in length of seasons 200:
estimates of zodiacal and synodic
periods of planets 208: Ars Eudoxt
112, 200 #., 208, 293: perhaps
originated method of Aristarchus’s
treatise on sizes and distances 331-2.
Eusebius 4.
Exeligmus 314-15.
Fortia d’Urban, Comte de: Greek text
and Latin translation of Aristarchus’s
treatise 323-4, 326: French transla-
tion of same 324: MSS. used by,
326-7 : 336.
Fiilleborn 67.
Galilei 126.
Gallenmiiller 8 5.
Geminus 268, 269, 270, 272, 284, 286,
287-8, 289, 292, 293, 294, 295, 296,
310, 314-15 : famous passage quoted
from, by Simplicius through Alex-
ander 275-6.
Gilbert, Otto 73, 74.
Ginzel, F. K. 16 2., 132 7., 284 sq.
Gnomon, introduced into Greece from
Babylon 21, 38.
Gomperz, Griechische Denker, 1 n., 6,
18 2., 38 2., 46 2., 48 2., 49 71.) 52 2,
53 #., 81 #., 83, 84, 105, 125-7, 249 71.»
279, 281, 301 2.
‘Great Year,’ of Heraclitus 61, Philo-
laus 102, Democritus 129, Oeno-
pides 102, 132-3, Plato 171-3, Aris-
tarchus 314-16: see also Cycles.
Greenhill, Sir Ὁ. 8-9 2.
Gruppe 175, 180, 182, 183, 186.
Giinther, Siegmund 3 2.
Ee2
420 INDEX
Hanno 249.
Harmony of spheres, in Pythagorean
system 98, 105-15, not mentioned by
Philolaus 108: possibly Pythagoras
distinguished only three notes 107,
but first form of harmony was of seven
notes (heptachord) 107 : complete Py-
thagorean systemimplied eight notes
(cf. Plato) 108-11: how heavenly
bodies corresponded to eight notes
(Plato, Nicomachus, Cicero, Boe-
thius) zézd.: other lengths of scale,
(a) octave of nine notes (Hypsicles)
112-13, (4) scales of eight intervals
adding up to 6, 63 or 7 tones (Cen-
sorinus, Pliny, Martianus Capella)
113-14,(c) otherdistributions, making
up more than one octave (Plutarch,
Anatolius, Macrobius) 115 : difficulty
in deducing definite ratios of dis-
tances 111, but Pythagoras said to
have made one tone (= 126,000
stades) separate moon and earth
“114-15.
Harpalus 291--2.
Hearth of universe 97-9, 142, 304.
Hecataeus 38, 39, 124.
Heeren 43 7.
Heiberg, J. L. 121 ., 303, 317 γεν
325 71.
Heraclides Lembus, epitome of bio-
graphies, 2, 5.
Heraclides of Pontus 141, 167: pupil
of Plato 252: particulars of life 252—
3: characteristics of dialogues 253:
Hicetas and Ecphantus probably
personages in dialogues 188-9,
251-2: views of H. on universe
251, 252,254: universe infinite 254:
each star a universe zdzd, : on moon,
comets, and meteors zézd.: on the
tides 306: affirmed rotation of earth
about its axis 251, 254-5, 282: dis-
covered that Mercury and Venus
revolve round sun 255-60: may pos-
sibly have extended this theory to
other planets and so invented system
of Tycho Brahe 260-75, but evidence
does not confirm this 269-75: con-_
tention of Schiaparelli that H. was
first enunciator of heliocentric hypo-
thesis 275-9, but words in passage
of Geminus relied on are clearly
interpolated 280-2, so that Schia-
parelli’s argument fails 283.
Heraclitus 50,66: on Xenophanes and
Pythagoras 52: crude astronomy
59-61: sun, moon, and stars are
bowls collecting bright exhalations
59-60: eclipses due to turning of
bowls upwards 44, new sun
every day 60: sun a foot in diameter,
really as well as apparently 61:
‘Great Year’ 61.
Hermippus the ‘ Callimachean ’, writer
of biographies, 2.
Herodian of Alexandria 10 71.
Herodotus 15, 21 2., 38, 286.
Herz, Norbert, 203-5 22.
Hesiod: astronomy in, 7-11, 12: con-
stellations mentioned by, 10: times
and seasons determined by risings
and settings of stars 10-1ἃ : men-
tions solstices but not equinoxes 11,
20: supposed author of poem
‘Astronomy’ II, 23.
Hicetas, of Syracuse, credited with
discovery of rotation of earth about
axis 187-8, or alternatively with
Pythagorean system attributed to
Philolaus zézd.: probably appeared
as one of interlocutors in a dialogue
of Heraclides 188-9.
Hilprecht 105 7.
Hipparchus : expressed anglesinterms -
of e// 23 2. : observation of gnomon’s
shadow at Byzantium at summer
solstice 131 #.: on position of north
pole 8 #.: on latitude of star Cano-
pus 192 #.: discovery of precession
101, 172-3, 200, and estimate of its
rate 172-3: denied that sun’s orbit
was inclined to ecliptic 199: distin-
guished two anomalies, solar and
zodiacal 267-8: on solar parallax
13, 341 : upheld geocentric view
308: on epicycles and eccentrics
267-8: on apparent diameter
of moon 313, 413: on diameter of
earth’s shadow 329, 337, 413: esti-
mates of distances of sunand moon,
and of sizes relatively to earth 342-3,
350: view as to circumference of
earth 114, 343-4, and measure of
obliquity of ecliptic 132 ”.: 258,
278, 344, 412.
Hippasus, the Pythagorean, 47.
Hippocrates of Chios, on comets
243-4.
Hippopede of Eudoxus 202 sq.
Homer: astronomy in, 7~9: stars and
constellations mentioned by, 8-9;
μα i se
<_< en ee
INDEX
Great Bear non-setting in his time
8-9 and note: τροπαὶ ἤελίοιο 9-10,
10 #.: divisions of day and night 9.
Hultsch 22 #., 23, 65, 115, 167 #., 171,
258 ., 259, 272, 303, 311 #., 313 71.)
338, 339 %-, 341, 342, 343, 344 1.»
345; 346, 347; 349-50, 412 #., 413 #.
Hypsicles : * 317, 321, 325:
wrote on harmony of spheres 112-13.
Iamblichus 271.
Ideler 10 #., 11 #., 193-4, 197, 247 71.»
290.
ἰλλομένην (in Timacus 40 B) 175-7.
Ishag b. Hunain 321.
Josephus 2.
Ludoxi 112, 200, 208, 293.
Leucippus 26, 121-3, 128: cosmogony
122: earth like tambourine and kept
in position by virtue of whirling
motion 122 : on sun, moon, and stars,
their nature and motion z4id.: moon
nearest earth, stars next, and sun
furthest away zdid.: explanation of
tilt of earth's axis 122-3.
‘Little Astronomy,’ The: a collection
of astronomical treatises 317-20.
λοξὸς κύκλος, the zodiac circle or eclip-
tic 131.
Lucretius 128-9.
Macrobius 115, 131, 164, 257, 258-9,
311, 340-1.
Manitius 9 #., 312.
Marinus of Tyre 346.
Martianus Capella 113, 256, 314.
Martin, T. H. 10 #., 13-14, 16 #., 41,
48 #., ΤΟΙ, 104, 105, 115, 117, 119,
141, 150, 154, 161-2, 167, 177, 178,
185, 1 188, 194 #., 219, 220 7.,
235 #., 238, 250, 258, 270, 277.
Maspero 48 2.
Menaechmus, pupil of Eudoxus, 193,
212.
Menedemus 252.
Menelaus, SpAaerica 318, 321.
Menestratus 291.
Menge, H. 325, 326 .
421
Meteoric stone, supposed prediction of
fall of, by Anaxagoras 246.
Meton : on length of seasons 200, 213,
215: Great Year of 19 years (ἐννεα-
καιδεκαετηρίς) or Meton’s cycle 293-5,
296: observation of summer solstice
in 432 B.C., 294.
Metrodorus of Chios, on infinity of
worlds 126-7.
Milky Way: views on, of Parmenides
67, 7°-2, 77; Anaxagoras 83-5 and
Democritus 83-5, 124-5, Pytha-
goreans 118, 133, Oenopides 133:
another view (‘reflection’) 247-8 :
Aristotle on, zééd.: by some con-
nected with Plato’s ‘straight light’
150-1.
Month: Greek month lunar 284:
‘hollow’ and ‘full’ months 287,
probably in use before Solon 285:
popular month of 30 days 285-6:
Solon’s reform 285, 291: intercala-
tions of menths 286, 287, 288, 293,
296. ,
Moon: substance and light of, 82,124. :
gets its light from the sun (disco-
very due to Anaxagoras) 19, 76, 77,
78-9, 91, 158 : phases of, 60, 80,120:
apparent diameter of, 313, 413 (see
also under Sun): estimates of size
II 1-12, 332, 337, 338, 340-1, 342, 349,
350: estimates of distance 28, 37-8,
114-15, 335-6, 338-9, 340, 342, 343,
344, 349, 350.
Nasiraddin at-Tisi, editor of Aristar-
chus 321.
Nauteles 291.
Nectanebus 192.
Ner, Chaldaean collective numeral
(= 600) 16 #. ;
Neuhauser 27 71.) 29, 30, 32, 33, 35-6.
Nicomachus 108-9, 271-4.
Nizze, editor of Greek text of Aristar-
chus 324.
Nokk 324.
North Pole, position of: views of
Eudoxus, and of Hipparchus after
Pytheas 8 2.
Octaéteris, or 8-years’ cycle of 2,922
days, and how evolved 287-92 :
improvement to 2,923} days 291,
292: attributed to Cleostratus and
Eudoxus 291 : varied as regards in-
tercalations of months by others ἐξα,
422
Oenopides of Chios: applied mathe-
matics to astronomy 130, 319: cre-
dited with two of Euclid’s proposi-
tions (I. 12 and 23) 130: reputed
discoverer of obliquity of ecliptic
21, 130-1, 319: ‘Great Year’ of 59
years 102, 132-3: on Milky Way
133.
ὅρος of sun’s course 22.
Ottinger 41.
Pappus 132 ., 317, 341, 371 #.: com-
ments on Aristarchus’s treatise 412--
14.
Parallax, solar: allowed for by Archi-
medes 348: Hipparchus and Pto-
lemy on, 341.
Parapegma 295.
Parmenides 21, 62-77: date, &c. 62-3:
cosmology of Parm. and Pythagoras .
compared 63-6: held universe to |
be spherical, finite, and motionless
63-4: earth spherical 64, and in
equilibrium without support 64: on
zones 65-6 : recognized Morning and
Evening Stars to be one 66, 75: stars
compressed fire, earth a precipitate
of condensed air 66: theory of
wreaths and interpretations 66-74:
position of goddess Justice or Neces-
sity 73-4: parallel of Myth of Er
zbid.: doubt as to view of planets
74-5, and as to whether Parm. held
moon to be lit up by sun 75-7: on
Milky Way 67, 70-2, 77.
Pearson, Dr. J. Β. 8 21.
Pentaéteris =four (not five) years’
period 286.
Petavius, Uranologium 5 n.
Phidias, father of Archimedes, 332,
337:
Philippus of Opus 99 2., 184, 186, 293 :
astronomical works attributed to,
320.
Philistion 192.
Philochorus 285, 294 71.
Philodemus, De pietate 4.
Philolaus 299: first to write exposition
of Pythagorean doctrines 47, 48:
credited with ‘Pythagorean’ astro-
nomical system (see Pythagoreans)
48, 94, 97, 187-9: on order of planets
107: on sun as a sort of crystal
lens concentrating and transmitting
rays of light 115-16, ‘two suns if
not three’ 90-91, 116: as to original
INDEX
source of beams concentrated in sun
116-17.
Phocus of Samos 23.
Placita philosophorum, not by Plu-
tarch 4.
Planets: independent motion first
asserted by Alcmaeon (probably
after Pythagoras) 49, 50: Platonic
order of, 85, 108-10, 258: later order
(Chaldaean) with sun in middle 107,
adopted by Diogenes of Babylon
107, Nicomachus 108, Alexander of
Ephesus 112-13 : obliquities of or-
bits 155 21. : distances from earth 106,
113-15, 164: relative and absolute
speeds 108-10, 156-7: notes assigned
to, 110-11, 112-15.
Plato 1, 13, 18, 233, 237-8 #., 370 #.:
view of astronomy as a sort of ideal
kinematics 134-40, actual appear-
ances being imperfect illustrations
like diagrams in geometry 136-8:
astronomy as ‘motion of body’
follows stereometry in curriculum
135: Plato’s objection to mechani-
cal constructions in geometry 137 ;
question how Plato’s real astronomy
would actualy work 138-40: view
slightly modified in 7zmaeus and
Laws 140: mixture of myth with
astronomy 134,141: problem ve mo-
tion of planets 140-1, 209,272: astro-
nomy of Phaedrus, heavenly host in
12 divisions performing separate
evolutions under command-in-chief
-of Zeus (sphere of fixed stars) 142-3,
Hestia abiding at home (= Earth)
142, 304: in Phaedo, complaint of
Anaxagoras 143-4, earth in equili-
brium without support 24 2., 144, of
extreme size (contrast Aristotle) 145,
147, a sphere, with hollows, in one of
which we live 145-7: astronomy of
Republic, Myth of Er, 68, 73-4, 109,
111, 117, 148-58, the ‘straight light’
150-2, the spindle 152-3, the whorls
68, 153-8, speeds of sun, moon and
planets, absolute and relative 108-
10, 156-7, and their order in celes-
tial harmony 108-10: Plato’s astro-
nomy in final form in Zimmaeus 158-
181, equator and ecliptic 159-60,
motion along equator to right 160-3,
seven circles of sun, moon and
planets in ecliptic 163-9, which are
twisted into spirals 169, harmonic
ΘΝ ΝΣ ΡΠ
᾿.
>
INDEX
intervals 163-4, Venus and Mercury
have ‘ contrary tendency’ to sun 165-
9, 257 #., ‘overtakings’ of planets
169-70, and other apparent irregu-
larities 171, 179-80 (cf. 182-3), stars
rotate about their axes 174, but earth
in centre does of rotate 174-9, 180—
5, 240, 305, earth guardian and
creator of night and day 178-9,
Time 164-5, 170, 180,‘ instruments of
time’ 180-1, Great Year 171-3: astro-
nomy in Zaws 181-4, comparison
with Efinomis 184-5: Plato sup-
posed in old age to have deposed
earth from centre 183-9 : did not use
epicycles 257: on use of models of
heavens 155: on the tides 306.
Pliny 11, 20, 80 7., 113, 114, 199, 200,
284, 339, 343, 344- ᾿
Plutarch 13,90 7., 91, 106, 115, 124 22.)
178, 179, 183, 185, 290, 304, 305-6.
Polemarchus of Cyzicus 212, 222, 261.
Polos, introduced from Babylon, 21,
3 .
Posidonius 66 #,, 275: on size of sun
341, 344: on distance of sun 344-5,
346-9: different estimates of earth’s
circumference attributed to, 345-7:
adopted certain figures from Archi-
medes 348-9: summary of estimates
of sizes and distances 349, 350: on
the tides 306.
Precession : not motion of ‘tenth body’
in Pythagorean system IoI, 104-5:
unknown to Egyptians 101, and to
Babylonians in third century B.C,
105 : discovered by Hipparchus ΙΟῚ,
172-3, 200: estimates of rate of the
motion by Hipparchus and Ptolemy
172-3: effect on position of Great
Bear now as compared with Homer’s
time 8 2.
Proclus 8-9 #., 130, 131 #., 150, 152,
154, 156, 161, 162, 166, 168, 174, 178,
179, 207, 223, 252, 271, 379 #.
Pseudo-Plutarch: Placita philosopho-
7UM 4, στρωματεῖς 1011.
Ptolemy 16, 266, 267, 278, 294 ., 296,
310, 312, 368-9 ., 381 2., 389 #.,
. 301 #., 412: estimate of obliquity
of ecliptic 131-2 #.: wrong esti-
mate of rate of precession 172-3:
on apparent diameters of sun and
moon 223, 313, 413: on diameter of
earth’s shadow 329, 337: on dis-
tance of sun 343: on circumference
423
of earth 346: estimates of sizes and
distances of sun and moon summa-
rized 349, 350,414: on solar parallax
431: on exeligmus 314-15.
Pythagoras 21, 46-51: first to make
geometry a science, originated Theory
of Numbers and theory of propor-
tion, and discovered dependence of
musical intervais on numbers 46,
47: left no written works 47: sup-
posed secrecy of doctrines 47, 64:
probably thought universe a sphere,
rotating about an axis 63, outside
spherical universe limitless void
(universe dreathes) 63-4: first to
maintain sphericity of earth and to
distinguish zones 21, 48-9, 64-5:
system certainly geocentric 49: pro-
bably first to assert independent
motion of planets from west to east
50-51: recognized identity of Morn-
ing and Evening Stars 66, 107:
moon a ‘ mirror-like body’ 76: sup-
posed estimate of distance of moon
114-15: credited with discovery of
obliquity of ecliptic 130.
Pythagoreans 94-120: abandoned
geocentric hypothesis 94: motion of
earth (with counter-earth) as well as
sun, moon, and planets about central
fire 95-100: names given to central
fire 96-7: position of counter-earth
relatively to earth 96-7, 99 : counter-
earth invented to explain eclipses of
moon 99-100: ten‘ bodies’ altogether
moving in heaven 98-9: has tenth
body (sphere of fixed stars) a slow
imperceptible movement or zo move-
ment? (views of Boeckh, Martin,
Apelt, Schiaparelli) 101-5 : harmony
of spheres 105-15: earth revolves
from west to east, with same hemi-
sphere always turned outwards, in
24 hours, making night and day, loo:
neglect of consequent parallax, zé7d. :
on eclipses 119: on phases of moon
120: animals in moon fifteen times
stronger than ours 118: on Milky
Way 118,133: supposed Pythagorean
system with central fire in centre of
earth 249-50: movable eccentrics
and epicycles not invented by Py-
thagoreans 270-4: Pythagorean ap-
proximation to 4/2 (1), 379 x.
Pytheas : on position of north pole 8 z.:
observed ratio of gnomon to midday
424
shadow at summer solstice at Mar-
seilles 131 ~.: on the tides 306.
Qusta b. Liga al-Ba‘labakki, Arabian
translator, 320.
Refutation of all heresies, by Hippo-
lytus, 4-5.
Richer 343.
Sacro-Bosco 172.
Sar (Gk. σάρος or σαρός), Chaldaean
collective numeral (3,600), not period
of 223 lunations 16 22.
Sartorius, M. 5-6, 11 #., 32, 33-5, 41 #.
Satyrus, author of Zzves, 2.
Savile, MS. of Aristarchus, 323, 326-7.
Schaubach 41.
Schiaparelli 94, 101-2, 133, 168-9,
183, 184, 185, 194, 195, 200-11,
213-16, 217, 221, 223, 224, 249 #.,
260-2, 264, 267, 269-75, 278-9, 282,
301, 303, 305, 307, 310.
Schmidt, Ad., 295.
Selden (Arabic) MS. of Aristarchus
323.
Seleucus : supporter of Aristarchus’s
heliocentric system 305-6, 307: on
the tides 305, 306, 307.
Seneca: on Empedocles’ assumption
of fires inside the earth 92%.; on
Democritus and planets 128: on
comets 247 #.: on rotation of earth
as possibility 307-8.
Simplicius 6, 83, 187, 220 #., 252, 271,
383 ~.: on Anaximander 25, 26, 37:
on Xenophanes’ earth ‘ rooted to in-
finity’ 54: on goddess Necessity or
Justice in Parmenides 73 : on Pytha-
gorean system 96-7: on a ‘more
genuine’ Pythagorean system 249-
50: on ἰλλομένην applied to earth in
Timaeus 175 2., 176-7: on Eudoxus’s
system of concentric spheres 193,
196-8, 201-2, 209, 221-2, and Cal-
lippus’s improvements 213, 216,
218, 221: on Eudoxus’s estimates
of synodic and zodiacal periods of
planets 208 : famous passage quoted
from Geminus purporting to attri-
bute heliocentric hypothesis to
Heraclides 275-6: on Heraclides
254, 255, 282: on Aristarchus 254.
Sizes and distances: first speculations
27, 28, 32, 37-8, 114-15, 331: later
estimates, by Eudoxus 111-12, 332,
INDEX
337, Platonists 164, Phidias 332, 337,
Aristarchus 332, 338, Eratosthenes
339-41, Hipparchus 341-4, Posido-
nius 344-9, Ptolemy 343, 346, 349,
414: summary of estimates 350.
σκάφη, a sundial invented by Aristar-
chus, 312.
Smith, G. 16 #.
Solon 285, 286, 291.
Sosicrates, διαδοχαί, 2.
Sosigenes 6, 140, 221, 272: account
of Eudoxus’s system of concentric
spheres (see Simplicius) : on annular
eclipses of sun 222-4, 313, 383 2.
Soss, Chaldaean collective number
(= 60), 16 #.
Sotion, διαδοχαί, 2, 5.
Speusippus 252.
Stewart, J. A. 151-3, 155.
Stobaeus, clogae, 4.
Strabo 131 7., 252, 339, 343 %., 345 71.
Strato of Lampsacus 299, 300.
Sulpicius Gallus 114.
Sun: apparent angular diameter 21,
Egyptian and Babylonian measure-
ments of, 22, 311: Aristarchus’svalues
23, 311-12, measurements by Archi-
medes 312, 348, Ptolemy and Hip- .
parchus 313, other estimates 313-4:
supposed deviation in latitude from
ecliptic (Eudoxus, Pliny, Adrastus)
198-200, denied by Hipparchus 199:
estimates of size 111-12, 332, 337,
338, 342, 344, 349, 350, 414: esti-
mates of distance 27, 28, 32, 37-8,
338-9, 340-1, 342, 343, 344-5, 349,
350, 414. unit
Syrie and Ortygia in Homer 9, Io.
Tannery, P. 5, 15, 16 #., 17, 18 #., 19,
20, 28, 29, 31, 33, 37 71.) 44, 45, 50,
56 2. 58 71.) 59 21.) ΟΣ #., 63 2, 64,
66 My 68-72, 74; 75; 76, 81 2... 84,
85 2., 89, 102, I11, 112, 113, 114,
117-18,124, 129, 132, 133, 152, 172-3,
187-9, 194 #., 200 #., 251 722.) 252 #.,
259%., 269, 280-2, 311-12, 314-16,
318 ., 319 22.) 320 %., 329-32, 333;
336, 338, 340, 344 #.
Teichmiiller 28, 32, 33, 41, 175, 178.
Tetraéteris, four-years’ period (reputed)
286.
Thales: date, &c. 12-13: story of oil-
presses 12, and of fall into well 13:
prediction of solar eclipse 13-18,
date of eclipse 15-16: could not
INDEX
have known cause of eclipses 18:
earth (and probably sun and moon)
a disc 18-19: earth floats on water 18,
19: view of universe compared with
Egyptian and Babylonian 19-20:
wrote on solstices and equinoxes 20:
recognized inequality of astronomical
seasons 14, 20: discoveries wrongly
attributed to, 21-3, 130: year of
365 days 21: observed Little Bear
23: supposed author of Nautical
Theodosius of Tripolis: Sphaerica
192, 317-18, 319 #., 320: On Days
and Nights and On Habitations 317.
Theon of Alexandria 341.
Theon of Smyrna 6, 113, 147-8, 154,
155 #., 158, 200, 262, 379 n. (see also
=e Adrastus, Dercyilides, Eude-
mus).
hrastus, Physical Opinions
eaten, δοξῶν ιη), apr source of
phy 2, 4,5, plan of, 2: quoted
from, 25, 26, 32, 60, 64, 73, 79, 94,
97 #., 122, 183, 186, 187, 188, 189,
"217 ., 251-2.
Thrasyllus 112, 128 x.
ἜΜΕΝ explanations of, by Plato,
Timaeus, Aristotle, Dicaearchus,
Pytheas, Heraclides, Posidonius 306,
Crates and Apollodorus 306-7,
Seleucus 305, 307.
Timaeus, on the tides, 306.
Timaeus Locrus 179.
Timocharis 172.
Trieteris (reallytwo-years’ period) 286.
τροπή: use in Homer 9-10: not always
used in technical sense of solstice
33 #.: meaning in Anaximander 33 :
τὰς τροπὰς ποιεῖσθαι in Anaximenes
33 #., 42.
Tycho Brahe 223, 260, 269.
Unger 295.
ὑποζώματα of triremes (Republic 616 C)
151-2.
Valla, G.: first edition (Latin transla-
tion) of Aristarchus 321, 323.
425
Varro 4, 113, 253, 284.
Vaticanus Graecus 204, best MS. of
Aristarchus: description 325-6.
Vetusta Placita, assumed compilation
adhering closely to Theophrastus :
date and divisions of, 4.
Vitruvius 299, 318 #.: on Heraclides’
theory of Venus and Mercury re-
volving round sun 255.
Voss, Otto, 189, 251 7., 252 #., 253 71.»
280 γι.
Wallis, John, editio princefs of Greek
text of Aristarchus 321-3: MSS.
used by, 323, 326-7.
Wilamowitz-MGllendorff, U. von, 252 2.
Wolf 349.
Xenocrates 252.
Xenophanes 15, 52-8: poet and satirist
52-3: attacked popular mythology
53: evolution of world, evidence of
fossils, 53-4: earth flat, with roots
extending ad infinitum, 54-5 : nature
of stars (clouds set on fire) 55: new
sun every day 55-6: multiplicity of
suns and moons 56: sun really
moves in straight line, which only
seems to be a circle 56-7, 150: on
eclipse ‘ lasting a month’ 57-8.
Year, views of length of: 365 days
(Egyptians and Thales) 21: 3652%
days (Oenopides) 102, 132: 365y5
days (Meton and Euctemon) 295
365 days 13 hours (Harpalus) 292:
365 days (Callippus) 296: 365}
days (Aristarchus) 314-15:
Mi i soo OF 365 {ym days (Hippar-
chus) 296-7.
Zeller 6, 28, 29, 31, 32, 33, 41, 43, 60 #.,
62, 64 #., 68, 108, 115 #., 128, 153,
177, 242, 252, 270, 271, 273 ne
Zodiac: see Ecliptic.
Zones: distinction of, alternatively
attributed to Pythagoras and Par-
menides 21, 65-6.
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