GREEK ASTRONOMY.
“The astronomer discovers that geometry, a pure abstraction of the
human mind, is the measure of planetary motion.”
EMERSON.
19. In the earlier period of Greek history one of the chief functions
expected of astronomers was the proper regulation of the calendar.
The Greeks, like earlier nations, began with a calendar based on the
moon. In the time of Hesiod a year consisting of 12 months of 30
days was in common use; at a later date a year made up of 6 =full=
months of 30 days and 6 =empty= months of 29 days was introduced.
To Solon is attributed the merit of having introduced at Athens,
about 594 B.C., the practice of adding to every alternate year a
“full” month. Thus a period of two years would contain 13 months of
30 days and 12 of 29 days, or 738 days in all, distributed among 25
months, giving, for the average length of the year and month, 369
days and about 29-1∕2 days respectively. This arrangement was further
improved by the introduction, probably during the 5th century B.C.,
of the =octaeteris=, or eight-year cycle, in three of the years of
which an additional “full” month was introduced, while the remaining
years consisted as before of 6 “full” and 6 “empty” months. By this
arrangement the average length of the year was reduced to 365-1∕4
days, that of the month remaining nearly unchanged. As, however, the
Greeks laid some stress on beginning the month when the new moon was
first visible, it was necessary to make from time to time arbitrary
alterations in the calendar, and considerable confusion resulted, of
which Aristophanes makes the Moon complain in his play _The Clouds_,
acted in 423 B.C.:
“Yet you will not mark your days
As she bids you, but confuse them, jumbling them all sorts of ways.
And, she says, the Gods in chorus shower reproaches on her head,
When, in bitter disappointment, they go supperless to bed.
Not obtaining festal banquets, duly on the festal day.”
20. A little later, the astronomer _Meton_ (born about 460 B.C.)
made the discovery that the length of 19 years is very nearly equal
to that of 235 lunar months (the difference being in fact less than
a day), and he devised accordingly an arrangement of 12 years of 12
months and 7 of 13 months, 125 of the months in the whole cycle being
“full” and the others “empty.” Nearly a century later _Callippus_
made a slight improvement, by substituting in every fourth period of
19 years a “full” month for one of the “empty” ones. Whether =Meton’s
cycle=, as it is called, was introduced for the civil calendar or not
is uncertain, but if not it was used as a standard by reference to
which the actual calendar was from time to time adjusted. The use of
this cycle seems to have soon spread to other parts of Greece, and it
is the basis of the present ecclesiastical rule for fixing Easter.
The difficulty of ensuring satisfactory correspondence between the
civil calendar and the actual motions of the sun and moon led to the
practice of publishing from time to time tables (παραπήγματα)
not unlike our modern almanacks, giving for a series of years the dates
of the phases of the moon, and the rising and setting of some of the
fixed stars, together with predictions of the weather. Owing to the
same cause the early writers on agriculture (_e.g._ Hesiod) fixed the
dates for agricultural operations, not by the calendar, but by the
times of the rising and setting of constellations, _i.e._ the times
when they first became visible before sunrise or were last visible
immediately after sunset—a practice which was continued long after the
establishment of a fairly satisfactory calendar, and was apparently by
no means extinct in the time of Galen (2nd century A.D.).
21. The Roman calendar was in early times even more confused than the
Greek. There appears to have been at one time a year of either 304 or
354 days; tradition assigned to Numa the introduction of a cycle of
four years, which brought the calendar into fair agreement with the
sun, but made the average length of the month considerably too short.
Instead, however, of introducing further refinements the Romans cut the
knot by entrusting to the ecclesiastical authorities the adjustment of
the calendar from time to time, so as to make it agree with the sun
and moon. According to one account, the first day of each month was
proclaimed by a crier. Owing either to ignorance, or, as was alleged,
to political and commercial favouritism, the priests allowed the
calendar to fall into a state of great confusion, so that, as Voltaire
remarked, “les généraux romains triomphaient toujours, mais ils ne
savaient pas quel jour ils triomphaient.”
A satisfactory reform of the calendar was finally effected by Julius
Caesar during the short period of his supremacy at Rome, under the
advice of an Alexandrine astronomer _Sosigenes_. The error in the
calendar had mounted up to such an extent, that it was found necessary,
in order to correct it, to interpolate three additional months in a
single year (46 B.C.), bringing the total number of days in that year
up to 445. For the future the year was to be independent of the moon;
the ordinary year was to consist of 365 days, an extra day being added
to February every fourth year (our leap-year), so that the average
length of the year would be 365-1∕4 days.
The new system began with the year 45 B.C., and soon spread, under the
name of the =Julian Calendar=, over the civilised world.
22. To avoid returning to the subject, it may be convenient to deal
here with the only later reform of any importance.
The difference between the average length of the year as fixed by
Julius Caesar and the true year is so small as only to amount to about
one day in 128 years. By the latter half of the 16th century the date
of the vernal equinox was therefore about ten days earlier than it was
at the time of the Council of Nice (A.D. 325), at which rules for the
observance of Easter had been fixed. Pope Gregory XIII. introduced
therefore, in 1582, a slight change;, ten days were omitted from that
year, and it was arranged to omit for the future three leap-years in
four centuries (viz. in 1700, 1800, 1900, 2100, etc., the years 1600,
2000, 2400, etc., remaining leap-years). The =Gregorian Calendar=, or
=New Style=, as it was commonly called, was not adopted in England till
1752, when 11 days had to be omitted; and has not yet been adopted in
Russia and Greece, the dates there being now 12 days behind those of
Western Europe.
23. While their oriental predecessors had confined themselves chiefly
to astronomical observations, the earlier Greek philosophers appear to
have made next to no observations of importance, and to have been far
more interested in inquiring into causes of phenomena. _Thales_, the
founder of the Ionian school, was credited by later writers with the
introduction of Egyptian astronomy into Greece, at about the end of the
7th century B.C.; but both Thales and the majority of his immediate
successors appear to have added little or nothing to astronomy,
except some rather vague speculations as to the form of the earth and
its relation to the rest of the world. On the other hand, some real
progress seems to have been made by _Pythagoras_[11] and his followers.
Pythagoras taught that the earth, in common with the heavenly bodies,
is a sphere, and that it rests without requiring support in the middle
of the universe. Whether he had any real evidence in support of these
views is doubtful, but it is at any rate a reasonable conjecture that
he knew the moon to be bright because the sun shines on it, and the
phases to be caused by the greater or less amount of the illuminated
half turned towards us; and the curved form of the boundary between
the bright and dark portions of the moon was correctly interpreted by
him as evidence that the moon was spherical, and not a flat disc, as
it appears at first sight. Analogy would then probably suggest that
the earth also was spherical. However this may be, the belief in the
spherical form of the earth never disappeared from Greek thought, and
was in later times an established part of Greek systems, whence it has
been handed down, almost unchanged, to modern times. This belief is
thus 2,000 years older than the belief in the rotation of the earth
and its revolution round the sun (chapter IV.), doctrines which we
are sometimes inclined to couple with it as the foundations of modern
astronomy.
In Pythagoras occurs also, perhaps for the first time, an idea
which had an extremely important influence on ancient and mediaeval
astronomy. Not only were the stars supposed to be attached to a crystal
sphere, which revolved daily on an axis through the earth, but each
of the seven planets (the sun and moon being included) moved on a
sphere of its own. The distances of these spheres from the earth were
fixed in accordance with certain speculative notions of Pythagoras
as to numbers and music; hence the spheres as they revolved produced
harmonious sounds which specially gifted persons might at times hear:
this is the origin of the idea of the =music of the spheres= which
recurs continually in mediaeval speculation and is found occasionally
in modern literature. At a later stage these spheres of Pythagoras
were developed into a scientific representation of the motions of the
celestial bodies, which remained the basis of astronomy till the time
of Kepler (chapter VII.).
24. The Pythagorean _Philolaus_, who lived about a century later than
his master, introduced for the first time the idea of the motion of
the earth: he appears to have regarded the earth, as well as the sun,
moon, and five planets, as revolving round some central fire, the earth
rotating on its own axis as it revolved, apparently in order to ensure
that the central fire should always remain invisible to the inhabitants
of the known parts of the earth. That the scheme was a purely fanciful
one, and entirely different from the modern doctrine of the motion of
the earth, with which later writers confused it, is sufficiently shewn
by the invention as part of the scheme of a purely imaginary body, the
counter-earth (ἀντιχθών), which brought the number of moving
bodies up to ten, a sacred Pythagorean number. The suggestion of such
an important idea as that of the motion of the earth, an idea so
repugnant to uninstructed common sense, although presented in such a
crude form, without any of the evidence required to win general assent,
was, however, undoubtedly a valuable contribution to astronomical
thought. It is well worth notice that Coppernicus in the great book
which is the foundation of modern astronomy (chapter IV., § 75)
especially quotes Philolaus and other Pythagoreans as authorities for
his doctrine of the motion of the earth.
Three other Pythagoreans, belonging to the end of the 6th century
and to the 5th century B.C., _Hicetas_ of Syracuse, _Heraclitus_,
and _Ecphantus_, are explicitly mentioned by later writers as having
believed in the rotation of the earth.
An obscure passage in one of Plato’s dialogues (the _Timaeus_) has
been interpreted by many ancient and modern commentators as implying a
belief in the rotation of the earth, and Plutarch also tells us, partly
on the authority of Theophrastus, that Plato in old age adopted the
belief that the centre of the universe was not occupied by the earth
but by some better body.[12]
Almost the only scientific Greek astronomer who believed in the motion
of the earth was _Aristarchus_ of Samos, who lived in the first half
of the 3rd century B.C., and is best known by his measurements of the
distances of the sun and moon (§ 32). He held that the sun and fixed
stars were motionless, the sun being in the centre of the sphere on
which the latter lay, and that the earth not only rotated on its axis,
but also described an orbit round the sun. _Seleucus_ of Seleucia, who
belonged to the middle of the 2nd century B.C., also held a similar
opinion. Unfortunately we know nothing of the grounds of this belief in
either case, and their views appear to have found little favour among
their contemporaries or successors.
It may also be mentioned in this connection that Aristotle (§ 27)
clearly realised that the apparent daily motion of the stars could be
explained by a motion either of the stars or of the earth, but that he
rejected the latter explanation.
25. _Plato_ (about 428-347 B.C.) devoted no dialogue especially to
astronomy, but made a good many references to the subject in various
places. He condemned any careful study of the actual celestial motions
as degrading rather than elevating, and apparently regarded the subject
as worthy of attention chiefly on account of its connection with
geometry, and because the actual celestial motions suggested ideal
motions of greater beauty and interest. This view of astronomy he
contrasts with the popular conception, according to which the subject
was useful chiefly for giving to the agriculturist, the navigator, and
others a knowledge of times and seasons.[13] At the end of the same
dialogue he gives a short account of the celestial bodies, according
to which the sun, moon, planets, and fixed stars revolve on eight
concentric and closely fitting wheels or circles round an axis passing
through the earth. Beginning with the body nearest to the earth, the
order is Moon, Sun, Mercury, Venus, Mars, Jupiter, Saturn, stars. The
Sun, Mercury, and Venus are said to perform their revolutions in the
same time, while the other planets move more slowly, statements which
shew that Plato was at any rate aware that the motions of Venus and
Mercury are different from those of the other planets. He also states
that the moon shines by reflected light received from the sun.
Plato is said to have suggested to his pupils as a worthy problem
the explanation of the celestial motions by means of a combination
of uniform circular or spherical motions. Anything like an accurate
theory of the celestial motions, agreeing with actual observation, such
as Hipparchus and Ptolemy afterwards constructed with fair success,
would hardly seem to be in accordance with Plato’s ideas of the true
astronomy, but he may well have wished to see established some simple
and harmonious geometrical scheme which would not be altogether at
variance with known facts.
26. Acting to some extent on this idea of Plato’s, _Eudoxus_ of
Cnidus (about 409-356 B.C.) attempted to explain the most obvious
peculiarities of the celestial motions by means of a combination of
uniform circular motions. He may be regarded as representative of the
transition from speculative to scientific Greek astronomy. As in
the schemes of several of his predecessors, the fixed stars lie on
a sphere which revolves daily about an axis through the earth; the
motion of each of the other bodies is produced by a combination of
other spheres, the centre of each sphere lying on the surface of the
preceding one. For the sun and moon three spheres were in each case
necessary: one to produce the daily motion, shared by all the celestial
bodies; one to produce the annual or monthly motion in the opposite
direction along the ecliptic; and a third, with its axis inclined to
the axis of the preceding, to produce the smaller motion to and from
the ecliptic. Eudoxus evidently was well aware that the moon’s path
is not coincident with the ecliptic, and even that its path is not
always the same, but changes continuously, so that the third sphere
was in this case necessary; on the other hand, he could not possibly
have been acquainted with the minute deviations of the sun from the
ecliptic with which modern astronomy deals. Either therefore he used
erroneous observations, or, as is more probable, the sun’s third sphere
was introduced to explain a purely imaginary motion conjectured to
exist by “analogy” with the known motion of the moon. For each of the
five planets four spheres were necessary, the additional one serving
to produce the variations in the speed of the motion and the reversal
of the direction of motion along the ecliptic (chapter I., § 14, and
below, § 51). Thus the celestial motions were to some extent explained
by means of a system of 27 spheres, 1 for the stars, 6 for the sun
and moon, 20 for the planets. There is no clear evidence that Eudoxus
made any serious attempt to arrange either the size or the time of
revolution of the spheres so as to produce any precise agreement with
the observed motions of the celestial bodies, though he knew with
considerable accuracy the time required by each planet to return to
the same position with respect to the sun; in other words, his scheme
represented the celestial motions qualitatively but not quantitatively.
On the other hand, there is no reason to suppose that Eudoxus regarded
his spheres (with the possible exception of the sphere of the fixed
stars) as material; his known devotion to mathematics renders it
probable that in his eyes (as in those of most of the scientific
Greek astronomers who succeeded him) the spheres were mere geometrical
figures, useful as a means of resolving highly complicated motions
into simpler elements. Eudoxus was also the first Greek recorded to
have had an observatory, which was at Cnidus, but we have few details
as to the instruments used or as to the observations made. We owe,
however, to him the first systematic description of the constellations
(see below, § 42), though it was probably based, to a large extent, on
rough observations borrowed from his Greek predecessors or from the
Egyptians. He was also an accomplished mathematician, and skilled in
various other branches of learning.
Shortly afterwards Callippus (§ 20) further developed Eudoxus’s scheme
of revolving spheres by adding, for reasons not known to us, two
spheres each for the sun and moon and one each for Venus, Mercury, and
Mars, thus bringing the total number up to 34.
27. We have a tolerably full account of the astronomical views of
_Aristotle_ (384-322 B.C.), both by means of incidental references, and
by two treatises—the _Meteorologica_ and the _De Coelo_—though another
book of his, dealing specially with the subject, has unfortunately been
lost. He adopted the planetary scheme of Eudoxus and Callippus, but
imagined on “metaphysical grounds” that the spheres would have certain
disturbing effects on one another, and to counteract these found it
necessary to add 22 fresh spheres, making 56 in all. At the same time
he treated the spheres as material bodies, thus converting an ingenious
and beautiful geometrical scheme into a confused mechanism.[14]
Aristotle’s spheres were, however, not adopted by the leading Greek
astronomers who succeeded him, the systems of Hipparchus and Ptolemy
being geometrical schemes based on ideas more like those of Eudoxus.
[Illustration: FIG. 8.—The phases of the moon.]
[Illustration: FIG. 9.—The phases of the moon.]
28. Aristotle, in common with other philosophers of his time, believed
the heavens and the heavenly bodies to be spherical. In the case of the
moon he supports this belief by the argument attributed to Pythagoras
(§ 23), namely that the observed appearances of the moon in its
several phases are those which would be assumed by a spherical body
of which one half only is illuminated by the sun. Thus the visible
portion of the moon is bounded by two planes passing nearly through its
centre, perpendicular respectively to the lines joining the centre of
the moon to those of the sun and earth. In the accompanying diagram,
which represents a section through the centres of the sun (S), earth
(E), and moon (M), A B C D representing on a much enlarged scale a
section of the moon itself, the portion D A B which is turned away from
the sun is dark, while the portion A D C, being turned away from the
observer on the earth, is in any case invisible to him. The part of the
moon which appears bright is therefore that of which B C is a section,
or the portion represented by F B G C in fig. 9 (which represents the
complete moon), which consequently appears to the eye as bounded by
a semicircle F C G, and a portion F B G of an oval curve (actually
an ellipse). The breadth of this bright surface clearly varies with
the relative positions of sun, moon, and earth; so that in the course
of a month, during which the moon assumes successively the positions
relative to sun and earth represented by 1, 2, 3, 4, 5, 6, 7, 8 in
fig. 10, its appearances are those represented by the corresponding
numbers in fig. 11, the moon thus passing through the familiar phases
of crescent, half full, gibbous, full moon, and gibbous, half full,
crescent again.[15]
[Illustration: FIG. 10.—The phases of the moon.]
Aristotle then argues that as one heavenly body is spherical, the
others must be so also, and supports this conclusion by another
argument, equally inconclusive to us, that a spherical form is
appropriate to bodies moving as the heavenly bodies appear to do.
[Illustration: FIG. 11.—The phases of the moon.]
29. His proofs that the earth is spherical are more interesting. After
discussing and rejecting various other suggested forms, he points out
that an eclipse of the moon is caused by the shadow of the earth cast
by the sun, and argues from the circular form of the boundary of the
shadow as seen on the face of the moon during the progress of the
eclipse, or in a partial eclipse, that the earth must be spherical;
for otherwise it would cast a shadow of a different shape. A second
reason for the spherical form of the earth is that when we move
north and south the stars change their positions with respect to the
horizon, while some even disappear and fresh ones take their place.
This shows that the direction of the stars has changed as compared
with the observer’s horizon; hence, the actual direction of the stars
being imperceptibly affected by any motion of the observer on the
earth, the horizons at two places, north and south of one another,
are in different directions, and the earth is therefore curved. For
example, if a star is visible to an observer at A (fig. 12), while
to an observer at B it is at the same time invisible, _i.e._ hidden
by the earth, the surface of the earth at A must be in a different
direction from that at B. Aristotle quotes further, in confirmation of
the roundness of the earth, that travellers from the far East and the
far West (practically India and Morocco) alike reported the presence of
elephants, whence it may be inferred that the two regions in question
are not very far apart. He also makes use of some rather obscure
arguments of an _a priori_ character.
[Illustration: FIG. 12.—The curvature of the earth.]
There can be but little doubt that the readiness with which Aristotle,
as well as other Greeks, admitted the spherical form of the earth and
of the heavenly bodies, was due to the affection which the Greeks
always seem to have had for the circle and sphere as being “perfect,”
_i.e._ perfectly symmetrical figures.
30. Aristotle argues against the possibility of the revolution of the
earth round the sun, on the ground that this motion, if it existed,
ought to produce a corresponding apparent motion of the stars. We
have here the first appearance of one of the most serious of the many
objections ever brought against the belief in the motion of the earth,
an objection really only finally disposed of during the present
century by the discovery that such a motion of the stars can be seen in
a few cases, though owing to the almost inconceivably great distance
of the stars the motion is imperceptible except by extremely refined
methods of observation (cf. chapter XIII., §§ 278, 279). The question
of the distances of the several celestial bodies is also discussed,
and Aristotle arrives at the conclusion that the planets are farther
off than the sun and moon, supporting his view by his observation of
an occultation of Mars by the moon (_i.e._ a passage of the moon in
front of Mars), and by the fact that similar observations had been
made in the case of other planets by Egyptians and Babylonians. It is,
however, difficult to see why he placed the planets beyond the sun,
as he must have known that the intense brilliancy of the sun renders
planets invisible in its neighbourhood, and that no occultations of
planets by the sun could really have been seen even if they had been
reported to have taken place. He quotes also, as an opinion of “the
mathematicians,” that the stars must be at least nine times as far off
as the sun.
There are also in Aristotle’s writings a number of astronomical
speculations, founded on no solid evidence and of little value; thus
among other questions he discusses the nature of comets, of the Milky
Way, and of the stars, why the stars twinkle, and the causes which
produce the various celestial motions.
In astronomy, as in other subjects, Aristotle appears to have collected
and systematised the best knowledge of the time; but his original
contributions are not only not comparable with his contributions to
the mental and moral sciences, but are inferior in value to his work
in other natural sciences, _e.g._ Natural History. Unfortunately the
Greek astronomy of his time, still in an undeveloped state, was as it
were crystallised in his writings, and his great authority was invoked,
centuries afterwards, by comparatively unintelligent or ignorant
disciples in support of doctrines which were plausible enough in his
time, but which subsequent research was shewing to be untenable. The
advice which he gives to his readers at the beginning of his exposition
of the planetary motions, to compare his views with those which they
arrived at themselves or met with elsewhere, might with advantage have
been noted and followed by many of the so-called Aristotelians of the
Middle Ages and of the Renaissance.[16]
31. After the time of Aristotle the centre of Greek scientific thought
moved to Alexandria. Founded by Alexander the Great (who was for a
time a pupil of Aristotle) in 332 B.C., Alexandria was the capital
of Egypt during the reigns of the successive Ptolemies. These kings,
especially the second of them, surnamed Philadelphos, were patrons of
learning; they founded the famous Museum, which contained a magnificent
library as well as an observatory, and Alexandria soon became the home
of a distinguished body of mathematicians and astronomers. During the
next five centuries the only astronomers of importance, with the great
exception of Hipparchus (§ 37), were Alexandrines.
[Illustration: FIG. 13.—The method of Aristarchus for comparing the
distances of the sun and moon.]
32. Among the earlier members of the Alexandrine school were
_Aristarchus_ of Samos, _Aristyllus_, and _Timocharis_, three nearly
contemporary astronomers belonging to the first half of the 3rd
century B.C. The views of Aristarchus on the motion of the earth have
already been mentioned (§ 24). A treatise of his _On the Magnitudes
and Distances of the Sun and Moon_ is still extant: he there gives an
extremely ingenious method for ascertaining the comparative distances
of the sun and moon. If, in the figure, E, S, and M denote respectively
the centres of the earth, sun, and moon, the moon evidently appears
to an observer at E half full when the angle E M S is a right angle.
If when this is the case the angular distance between the centres of
the sun and moon, _i.e._ the angle M E S, is measured, two angles
of the triangle M E S are known; its shape is therefore completely
determined, and the ratio of its sides E M, E S can be calculated
without much difficulty. In fact, it being known (by a well-known
result in elementary geometry) that the angles at E and S are together
equal to a right angle, the angle at S is obtained by subtracting
the angle S E M from a right angle. Aristarchus made the angle at S
about 3°, and hence calculated that the distance of the sun was from
18 to 20 times that of the moon, whereas, in fact, the sun is about
400 times as distant as the moon. The enormous error is due to the
difficulty of determining with sufficient accuracy the moment when
the moon is half full: the boundary separating the bright and dark
parts of the moon’s face is in reality (owing to the irregularities on
the surface of the moon) an ill-defined and broken line (cf. fig. 53
and the frontispiece), so that the observation on which Aristarchus
based his work could not have been made with any accuracy even with
our modern instruments, much less with those available in his time.
Aristarchus further estimated the apparent sizes of the sun and moon
to be about equal (as is shewn, for example, at an eclipse of the sun,
when the moon sometimes rather more than hides the surface of the
sun and sometimes does not quite cover it), and inferred correctly
that the real diameters of the sun and moon were in proportion to
their distances. By a method based on eclipse observations which was
afterwards developed by Hipparchus (§ 41), 1∕3 that of the earth, a
result very near to the truth; and the same method supplied data from
which the distance of the moon could at once have been expressed in
terms of the radius of the earth, but his work was spoilt at this point
by a grossly inaccurate estimate of the apparent size of the moon (2°
instead of 1∕2°), and his conclusions seem to contradict one another.
He appears also to have believed the distance of the fixed stars to
be immeasurably great as compared with that of the sun. Both his
speculative opinions and his actual results mark therefore a decided
advance in astronomy.
Timocharis and Aristyllus were the first to ascertain and to record
the positions of the chief stars, by means of numerical measurements
of their distances from fixed positions on the sky; they may thus
be regarded as the authors of the first real star catalogue, earlier
astronomers having only attempted to fix the position of the stars
by more or less vague verbal descriptions. They also made a number
of valuable observations of the planets, the sun, etc., of which
succeeding astronomers, notably Hipparchus and Ptolemy, were able to
make good use.
[Illustration: FIG. 14.—The equator and the ecliptic.]
33. Among the important contributions of the Greeks to astronomy must
be placed the development, chiefly from the mathematical point of
view, of the consequences of the rotation of the celestial sphere and
of some of the simpler motions of the celestial bodies, a development
the individual steps of which it is difficult to trace. We have,
however, a series of minor treatises or textbooks, written for the
most part during the Alexandrine period, dealing with this branch of
the subject (known generally as =Spherics=, or the Doctrine of the
Sphere), of which the _Phenomena_ of the famous geometer _Euclid_
(about 300 B.C.) is a good example. In addition to the points and
circles of the sphere already mentioned (chapter I., §§ 8-11), we now
find explicitly recognised the =horizon=, or the great circle in which
a horizontal plane through the observer meets the celestial sphere, and
its =pole=,[17] the =zenith=,[18] or point on the celestial sphere
vertically above the observer; the =verticals=, or great circles
through the zenith, meeting the horizon at right angles; and the
=declination circles=, which pass through the north and south poles and
cut the equator at right angles. Another important great circle was the
=meridian=, passing through the zenith and the poles. The well-known
Milky Way had been noticed, and was regarded as forming another great
circle. There are also traces of the two chief methods in common use at
the present day of indicating the position of a star on the celestial
sphere, namely, by reference either to the equator or to the ecliptic.
If through a star S we draw on the sphere a portion of a great circle
S N, cutting the ecliptic ♈ N at right angles in N, and another great
circle (a declination circle) cutting the equator at M, and if ♈
be the first point of Aries (§ 13), where the ecliptic crosses the
equator, then the position of the star is completely defined _either_
by the lengths of the arcs ♈ N, N S, which are called the =celestial
longitude= and =latitude= respectively, _or_ by the arcs ♈ M, M S,
called respectively the =right ascension= and =declination=.[19] For
some purposes it is more convenient to find the position of the star
by the first method, _i.e._ by reference to the ecliptic; for other
purposes in the second way, by making use of the equator.
34. One of the applications of Spherics was to the construction of
sun-dials, which were supposed to have been originally introduced
into Greece from Babylon, but which were much improved by the Greeks,
and extensively used both in Greek and in mediaeval times. The proper
graduation of sun-dials placed in various positions, horizontal,
vertical, and oblique, required considerable mathematical skill. Much
attention was also given to the time of the rising and setting of the
various constellations, and to similar questions.
35. The discovery of the spherical form of the earth led to a
scientific treatment of the differences between the seasons in
different parts of the earth, and to a corresponding division of
the earth into zones. We have already seen that the height of the
pole above the horizon varies in different places, and that it was
recognised that, if a traveller were to go far enough north, he would
find the pole to coincide with the zenith, whereas by going south
he would reach a region (not very far beyond the limits of actual
Greek travel) where the pole would be on the horizon and the equator
consequently pass through the zenith; in regions still farther south
the north pole would be permanently invisible, and the south pole would
appear above the horizon.
[Illustration: FIG. 15.—The equator, the horizon, and the meridian.]
Further, if in the figure H E K W represents the horizon, meeting the
equator Q E R W in the east and west points E W, and the meridian H Q Z
P K in the south and north points H and K, Z being the zenith and P the
pole, then it is easily seen that Q Z is equal to P K, the height of
the pole above the horizon. Any celestial body, therefore, the distance
of which from the equator towards the north (declination) is less than
P K, will cross the meridian to the south of the zenith, whereas if
its declination be greater than P K, it will cross to the north of the
zenith. Now the greatest distance of the sun from the equator is equal
to the angle between the ecliptic and the equator, or about 23-1∕2°,
Consequently at places at which the height of the pole is less than
23-1∕2° the sun will, during part of the year, cast shadows at midday
towards the south. This was known actually to be the case not very far
south of Alexandria. It was similarly recognised that on the other side
of the equator there must be a region in which the sun ordinarily cast
shadows towards the south, but occasionally towards the north. These
two regions are the torrid zones of modern geographers.
Again, if the distance of the sun from the equator is 23-1∕2°, its
distance from the pole is 66-1∕2°; therefore in regions so far north
that the height P K of the north pole is more than 66-1∕2°, the sun
passes in summer into the region of the circumpolar stars which never
set (chapter I., § 9), and therefore during a portion of the summer
the sun remains continuously above the horizon. Similarly in the same
regions the sun is in winter so near the south pole that for a time it
remains continuously below the horizon. Regions in which this occurs
(our Arctic regions) were unknown to Greek travellers, but their
existence was clearly indicated by the astronomers.
[Illustration: FIG. 16.—The measurement of the earth.]
36. To _Eratosthenes_ (276 B.C. to 195 or 196 B.C.), another member of
the Alexandrine school, we owe one of the first scientific estimates
of the size of the earth. He found that at the summer solstice the
angular distance of the sun from the zenith at Alexandria was at midday
1∕50th of a complete circumference, or about 7°, whereas at Syene
in Upper Egypt the sun was known to be vertical at the same time.
From this he inferred, assuming Syene to be due south of Alexandria,
that the distance from Syene to Alexandria was also 1∕50th of the
circumference of the earth. Thus if in the figure S denotes the sun, A
and B Alexandria and Syene respectively, C the centre of the earth, and
A Z the direction of the zenith at Alexandria, Eratosthenes estimated
the angle S A Z, which, owing to the great distance of S, is sensibly
equal to the angle S C A, to be 7°, and hence inferred that the arc
A B was to the circumference of the earth in the proportion of 7° to
360° or 1 to 50. The distance between Alexandria and Syene being known
to be 5,000 _stadia_, Eratosthenes thus arrived at 250,000 stadia as
an estimate of the circumference of the earth, a number altered into
252,000 in order to give an exact number of stadia (700) for each
degree on the earth. It is evident that the data employed were rough,
though the principle of the method is perfectly sound; it is, however,
difficult to estimate the correctness of the result on account of
the uncertainty as to the value of the _stadium_ used. If, as seems
probable, it was the common Olympic stadium, the result is about 20 per
cent. too great, but according to another interpretation[20] the result
is less than 1 per cent. in error (cf. chapter X., § 221).
Another measurement due to Eratosthenes was that of the obliquity of
the ecliptic, which he estimated at 22∕83 of a right angle, or 23° 51′,
the error in which is only about 7′.
37. An immense advance in astronomy was made by _Hipparchus_, whom all
competent critics have agreed to rank far above any other astronomer of
the ancient world, and who must stand side by side with the greatest
astronomers of all time. Unfortunately only one unimportant book of his
has been preserved, and our knowledge of his work is derived almost
entirely from the writings of his great admirer and disciple Ptolemy,
who lived nearly three centuries later (§§ 46 _seqq._). We have also
scarcely any information about his life. He was born either at Nicaea
in Bithynia or in Rhodes, in which island he erected an observatory
and did most of his work. There is no evidence that he belonged to the
Alexandrine school, though he probably visited Alexandria and may have
made some observations there. Ptolemy mentions observations made by
him in 146 B.C., 126 B.C., and at many intermediate dates, as well as
a rather doubtful one of 161 B.C. The period of his greatest activity
must therefore have been about the middle of the 2nd century B.C.
Apart from individual astronomical discoveries, his chief services
to astronomy may be put under four heads. He invented or greatly
developed a special branch of mathematics,[21] which enabled processes
of numerical calculation to be applied to geometrical figures,
whether in a plane or on a sphere. He made an extensive series of
observations, taken with all the accuracy that his instruments would
permit. He systematically and critically made use of old observations
for comparison with later ones so as to discover astronomical
changes too slow to be detected within a single lifetime. Finally,
he systematically employed a particular geometrical scheme (that
of eccentrics, and to a less extent that of epicycles) for the
representation of the motions of the sun and moon.
38. The merit of suggesting that the motions of the heavenly bodies
could be represented more simply by combinations of uniform _circular_
motions than by the revolving _spheres_ of Eudoxus and his school (§
26) is generally attributed to the great Alexandrine mathematician
_Apollonius_ of Perga, who lived in the latter half of the 3rd century
B.C., but there is no clear evidence that he worked out a system in any
detail.
On account of the important part that this idea played in astronomy for
nearly 2,000 years, it may be worth while to examine in some detail
Hipparchus’s theory of the sun, the simplest and most successful
application of the idea.
We have already seen (chapter I., § 10) that, in addition to the
daily motion (from east to west) which it shares with the rest of the
celestial bodies, and of which we need here take no further account,
the sun has also an annual motion on the celestial sphere in the
reverse direction (from west to east) in a path oblique to the equator,
which was early recognised as a great circle, called the ecliptic.
It must be remembered further that the celestial sphere, on which
the sun appears to lie, is a mere geometrical fiction introduced for
convenience; all that direct observation gives is the change in the
sun’s direction, and therefore the sun may consistently be supposed
to move in such a way as to vary its distance from the earth in any
arbitrary manner, provided only that the alterations in the apparent
size of the sun, caused by the variations in its distance, agree with
those observed, or that at any rate the differences are not great
enough to be perceptible. It was, moreover, known (probably long before
the time of Hipparchus) that the sun’s apparent motion in the ecliptic
is not quite uniform, the motion at some times of the year being
slightly more rapid than at others.
Supposing that we had such a complete set of observations of the motion
of the sun, that we knew its position from day to day, how should we
set to work to record and describe its motion? For practical purposes
nothing could be more satisfactory than the method adopted in our
almanacks, of giving from day to day the position of the sun; after
observations extending over a few years it would not be difficult
to verify that the motion of the sun is (after allowing for the
irregularities of our calendar) from year to year the same, and to
predict in this way the place of the sun from day to day in future
years.
But it is clear that such a description would not only be long, but
would be felt as unsatisfactory by any one who approached the question
from the point of view of intellectual curiosity or scientific
interest. Such a person would feel that these detailed facts ought to
be capable of being exhibited as consequences of some simpler general
statement.
A modern astronomer would effect this by expressing the motion of the
sun by means of an algebraical formula, _i.e._ he would represent the
velocity of the sun or its distance from some fixed point in its path
by some symbolic expression representing a quantity undergoing changes
with the time in a certain definite way, and enabling an expert to
compute with ease the required position of the sun at any assigned
instant.[22]
The Greeks, however, had not the requisite algebraical knowledge for
such a method of representation, and Hipparchus, like his predecessors,
made use of a geometrical representation of the required variations in
the sun’s motion in the ecliptic, a method of representation which is
in some respects more intelligible and vivid than the use of algebra,
but which becomes unmanageable in complicated cases. It runs moreover
the risk of being taken for a mechanism. The circle, being the simplest
curve known, would naturally be thought of, and as any motion other
than a uniform motion would itself require a special representation,
the idea of Apollonius, adopted by Hipparchus, was to devise a proper
combination of uniform circular motions.
39. The simplest device that was found to be satisfactory in the case
of the sun was the use of the =eccentric=, _i.e._ a circle the centre
of which (C) does not coincide with the position of the observer on
the earth (E). If in fig. 17 a point, S, describes the eccentric
circle A F G B uniformly, so that it always passes over equal arcs of
the circle in equal times and the angle A C S increases uniformly,
then it is evident that the angle A E S, or the apparent distance of
S from A, does not increase uniformly. When S is near the point A,
which is farthest from the earth and hence called the =apogee=, it
appears on account of its greater distance from the observer to move
more slowly than when near F or G; and it appears to move fastest when
near B, the point nearest to E, hence called the =perigee=. Thus the
motion of S varies in the same sort of way as the motion of the sun as
actually observed. Before, however, the eccentric could be considered
as satisfactory, it was necessary to show that it was possible to
choose the direction of the line B E C A (the =line of apses=) which
determines the positions of the sun when moving fastest and when moving
most slowly, and the magnitude of the ratio of E C to the radius C A of
the circle (the =eccentricity=), so as to make the calculated positions
of the sun in various parts of its path differ from the observed
positions at the corresponding times of year by quantities so small
that they might fairly be attributed to errors of observation.
[Illustration: FIG. 17.—The eccentric.]
This problem was much more difficult than might at first sight appear,
on account of the great difficulty experienced in Greek times and long
afterwards in getting satisfactory observations of the sun. As the
sun and stars are not visible at the same time, it is not possible
to measure directly the distance of the sun from neighbouring stars
and so to fix its place on the celestial sphere. But it is possible,
by measuring the length of the shadow cast by a rod at midday, to
ascertain with fair accuracy the height of the sun above the horizon,
and hence to deduce its distance from the equator, or the declination
(figs. 3, 14). This one quantity does not suffice to fix the sun’s
position, but if also the sun’s right ascension (§ 33), or its distance
east and west from the stars, can be accurately ascertained, its place
on the celestial sphere is completely determined. The methods available
for determining this second quantity were, however, very imperfect. One
method was to note the time between the passage of the sun across some
fixed position in the sky (_e.g._ the meridian), and the passage of a
star across the same place, and thus to ascertain the angular distance
between them (the celestial sphere being known to turn through 15° in
an hour), a method which with modern clocks is extremely accurate,
but with the rough water-clocks or sand-glasses of former times was
very uncertain. In another method the moon was used as a connecting
link between sun and stars, her position relative to the latter being
observed by night, and with respect to the former by day; but owing
to the rapid motion of the moon in the interval between the two
observations, this method also was not susceptible of much accuracy.
[Illustration: FIG. 18.—The position of the sun’s apogee.]
In the case of the particular problem of the determination of the line
of apses, Hipparchus made use of another method, and his skill is shewn
in a striking manner by his recognition that both the eccentricity and
position of the apse line could be determined from a knowledge of the
lengths of two of the seasons of the year, _i.e._ of the intervals into
which the year is divided by the solstices and the equinoxes (§ 11). By
means of his own observations, and of others made by his predecessors,
he ascertained the length of the spring (from the vernal equinox to the
summer solstice) to be 94 days, and that of the summer (summer solstice
to autumnal equinox) to be 92-1∕2 days, the length of the year being
365-1∕4 days. As the sun moves in each season through the same angular
distance, a right angle, and as the spring and summer make together
more than half the year, and the spring is longer than the summer, it
follows that the sun must, on the whole, be moving more slowly during
the spring than in any other season, and that it must therefore pass
through the apogee in the spring. If, therefore, in fig. 18, we draw
two perpendicular lines Q E S, P E R to represent the directions of
the sun at the solstices and equinoxes, P corresponding to the vernal
equinox and R to the autumnal equinox, the apogee must lie at some
point A between P and Q. So much can be seen without any mathematics:
the actual calculation of the position of A and of the eccentricity is
a matter of some complexity. The angle P E A was found to be about 65°,
so that the sun would pass through its apogee about the beginning of
June; and the eccentricity was estimated at 1∕24.
The motion being thus represented geometrically, it became merely a
matter of not very difficult calculation to construct a table from
which the position of the sun for any day in the year could be easily
deduced. This was done by computing the so-called =equation of the
centre=, the angle C S E of fig. 17, which is the excess of the actual
longitude of the sun over the longitude which it would have had if
moving uniformly.
Owing to the imperfection of the observations used (Hipparchus
estimated that the times of the equinoxes and solstices could only be
relied upon to within about half a day), the actual results obtained
were not, according to modern ideas, very accurate, but the theory
represented the sun’s motion with an accuracy about as great as that of
the observations. It is worth noticing that with the same theory, but
with an improved value of the eccentricity, the motion of the sun can
be represented so accurately that the error never exceeds about 1′, a
quantity insensible to the naked eye.
The theory of Hipparchus represents the variations in the distance
of the sun with much less accuracy, and whereas in fact the angular
diameter of the sun varies by about 1∕30th part of itself, or by about
1′ in the course of the year, this variation according to Hipparchus
should be about twice as great. But this error would also have been
quite imperceptible with his instruments.
[Illustration: FIG. 19.—The epicycle and the deferent.]
Hipparchus saw that the motion of the sun could equally well be
represented by the other device suggested by Apollonius, the
=epicycle=. The body the motion of which is to be represented is
supposed to move uniformly round the circumference of one circle,
called the epicycle, the centre of which in turn moves on another
circle called the =deferent=. It is in fact evident that if a circle
equal to the eccentric, but with its centre at E (fig. 19), be taken as
the deferent, and if S′ be taken on this so that E S′ is parallel to C
S, then S′ S is parallel and equal to E C; and that therefore the sun
S, moving uniformly on the eccentric, may equally well be regarded as
lying on a circle of radius S′ S, the centre S′ of which moves on the
deferent. The two constructions lead in fact in this particular problem
to exactly the same result, and Hipparchus chose the eccentric as being
the simpler.
40. The motion of the moon being much more complicated than that of
the sun has always presented difficulties to astronomers,[23] and
Hipparchus required for it a more elaborate construction. Some further
description of the moon’s motion is, however, necessary before
discussing his theory.
We have already spoken (chapter I., § 16) of the lunar month as the
period during which the moon returns to the same position with respect
to the sun; more precisely this period (about 29-1∕2 days) is spoken
of as a =lunation= or =synodic month=: as, however, the sun moves
eastward on the celestial sphere like the moon but more slowly, the
moon returns to the same position with respect to the _stars_ in a
somewhat shorter time; this period (about 27 days 8 hours) is known as
the =sidereal month=. Again, the moon’s path on the celestial sphere is
slightly inclined to the ecliptic, and may be regarded approximately
as a great circle cutting the ecliptic in two =nodes=, at an angle
which Hipparchus was probably the first to fix definitely at about 5°.
Moreover, the moon’s path is always changing in such a way that, the
inclination to the ecliptic remaining nearly constant (but cf. chapter
V., § 111), the nodes move slowly backwards (from east to west) along
the ecliptic, performing a complete revolution in about 19 years.
It is therefore convenient to give a special name, the =draconitic
month=,[24] to the period (about 27 days 5 hours) during which the moon
returns to the same position with respect to the nodes.
Again, the motion of the moon, like that of the sun, is not uniform,
the variations being greater than in the case of the sun. Hipparchus
appears to have been the first to discover that the part of the moon’s
path in which the motion is most rapid is not always in the same
position on the celestial sphere, but moves continuously; or, in other
words, that the line of apses (§ 39) of the moon’s path moves. The
motion is an advance, and a complete circuit is described in about nine
years. Hence arises a fourth kind of month, the =anomalistic month=,
which is the period in which the moon returns to apogee or perigee.
To Hipparchus is due the credit of fixing with greater exactitude than
before the lengths of each of these months. In order to determine them
with accuracy he recognised the importance of comparing observations of
the moon taken at as great a distance of time as possible, and saw that
the most satisfactory results could be obtained by using Chaldaean and
other eclipse observations, which, as eclipses only take place near the
moon’s nodes, were simultaneous records of the position of the moon,
the nodes, and the sun.
To represent this complicated set of motions, Hipparchus used, as
in the case of the sun, an eccentric, the centre of which described
a circle round the earth in about nine years (corresponding to the
motion of the apses), the plane of the eccentric being inclined to the
ecliptic at an angle of 5°, and sliding back, so as to represent the
motion of the nodes already described.
The result cannot, however, have been as satisfactory as in the case of
the sun. The variation in the rate at which the moon moves is not only
greater than in the case of the sun, but follows a less simple law,
and cannot be adequately represented by means of a single eccentric;
so that though Hipparchus’ work would have represented the motion of
the moon in certain parts of her orbit with fair accuracy, there must
necessarily have been elsewhere discrepancies between the calculated
and observed places. There is some indication that Hipparchus was aware
of these, but was not able to reconstruct his theory so as to account
for them.
41. In the case of the planets Hipparchus found so small a supply of
satisfactory observations by his predecessors, that he made no attempt
to construct a system of epicycles or eccentrics to represent their
motion, but collected fresh observations for the use of his successors.
He also made use of these observations to determine with more accuracy
than before the average times of revolution of the several planets.
[Illustration: FIG. 20.—The eclipse method of connecting the distances
of the sun and moon.]
He also made a satisfactory estimate of the size and distance of the
moon, by an eclipse method, the leading idea of which was due to
Aristarchus (§ 32); by observing the angular diameter of the earth’s
shadow (Q R) at the distance of the moon at the time of an eclipse, and
comparing it with the known angular diameters of the sun and moon, he
obtained, by a simple calculation,[25] a relation between the
distances of the sun and moon, which gives either when the other is
known. Hipparchus knew that the sun was very much more distant than
the moon, and appears to have tried more than one distance, that of
Aristarchus among them, and the result obtained in each case shewed
that the distance of the moon was nearly 59 times the radius of the
earth. Combining the estimates of Hipparchus and Aristarchus, we
find the distance of the sun to be about 1,200 times the radius of
the earth—a number which remained substantially unchanged for many
centuries (chapter VIII., § 161).
42. The appearance in 134 B.C. of a new star in the Scorpion is said
to have suggested to Hipparchus the construction of a new catalogue of
the stars. He included 1,080 stars, and not only gave the (celestial)
latitude and longitude of each star, but divided them according to
their brightness into six magnitudes. The constellations to which he
refers are nearly identical with those of Eudoxus (§ 26), and the
list has undergone few alterations up to the present day, except for
the addition of a number of southern constellations, invisible in the
civilised countries of the ancient world. Hipparchus recorded also a
number of cases in which three or more stars appeared to be in line
with one another, or, more exactly, lay on the same great circle, his
object being to enable subsequent observers to detect more easily
possible changes in the positions of the stars. The catalogue remained,
with slight alterations, the standard one for nearly sixteen centuries
(cf. chapter III., § 63).
The construction of this catalogue led to a notable discovery, the best
known probably of all those which Hipparchus made. In comparing his
observations of certain stars with those of Timocharis and Aristyllus
(§ 33), made about a century and a half earlier, Hipparchus found that
their distances from the equinoctial points had changed. Thus, in the
case of the bright star Spica, the distance from the equinoctial points
(measured eastwards) had increased by about 2° in 150 years, or at
the rate of 48″ per annum. Further inquiry showed that, though the
roughness of the observations produced considerable variations in the
case of different stars, there was evidence of a general increase in
the longitude of the stars (measured from west to east), unaccompanied
by any change of latitude, the amount of the change being estimated
by Hipparchus as at least 36″ annually, and possibly more. The
agreement between the motions of different stars was enough to justify
him in concluding that the change could be accounted for, not as a
motion of individual stars, but rather as a change in the position
of the equinoctial points, from which longitudes were measured. Now
these points are the intersection of the equator and the ecliptic:
consequently one or another of these two circles must have changed.
But the fact that the latitudes of the stars had undergone no change
shewed that the ecliptic must have retained its position and that the
change had been caused by a motion of the equator. Again, Hipparchus
measured the obliquity of the ecliptic as several of his predecessors
had done, and the results indicated no appreciable change. Hipparchus
accordingly inferred that the equator was, as it were, slowly sliding
backwards (_i.e._ from east to west), keeping a constant inclination to
the ecliptic.
[Illustration: FIG. 21.—The increase of the longitude of a star.]
The argument may be made clearer by figures. In fig. 21 let ♈ M
denote the ecliptic, ♈ N the equator, S a star as seen by Timocharis,
S M a great circle drawn perpendicular to the ecliptic. Then S M is
the latitude, ♈ M the longitude. Let S′ denote the star as seen by
Hipparchus; then he found, that S′ M was equal to the former S M, but
that ♈ M′ was greater than the former ♈ M, or that M′ was slightly to
the east of M. This change M M′ being nearly the same for all stars,
it was simpler to attribute it to an equal motion in the opposite
direction of the point ♈, say from ♈ to ♈′ (fig. 22), _i.e._ by a
motion of the equator from ♈ N to ♈′ N′, its inclination N′ ♈′ M
remaining equal to its former amount N ♈ M. The general effect of this
change is shewn in a different way in fig. 23, where ♈ ♈′ ♎ ♎′ being
the ecliptic, A B C D represents the equator as it appeared in the
time of Timocharis, A′ B′ C′ D′ (printed in red) the same in the time
of Hipparchus, ♈, ♎ being the earlier positions of the two equinoctial
points, and ♈′, ♎′ the later positions.
[Illustration: FIG. 22.—The movement of the equator.]
[Illustration: FIG. 33.—The precession of the equinoxes.]
The annual motion ♈ ♈′ was, as has been stated, estimated by Hipparchus
as being at least 36″ (equivalent to one degree in a century), and
probably more. Its true value is considerably more, namely about 50″.
[Illustration: FIG. 24.—The precession of the equinoxes.]
An important consequence of the motion of the equator thus discovered
is that the sun in its annual journey round the ecliptic, after
starting from the equinoctial point, returns to the new position of the
equinoctial point a little before returning to its original position
with respect to the stars, and the successive equinoxes occur slightly
earlier than they otherwise would. From this fact is derived the name
=precession of the equinoxes=, or more shortly, =precession=, which is
applied to the motion that we have been considering. Hence it becomes
necessary to recognise, as Hipparchus did, two different kinds of year,
the =tropical year= or period required by the sun to return to the same
position with respect to the equinoctial points, and the =sidereal
year= or period of return to the same position with respect to the
stars. If ♈ ♈′ denote the motion of the equinoctial point during a
tropical year, then the sun after starting from the equinoctial point
at ♈ arrives—at the end of a tropical year—at the new equinoctial point
at ♈′; but the sidereal year is only complete when the sun has further
described the arc ♈′ ♈ and returned to its original starting-point ♈.
Hence, taking the modern estimate 50″ of the arc ♈ ♈′, the sun, in
the sidereal year, describes an arc of 360°, in the tropical year an
arc less by 50″, or 359° 59′ 10″; the lengths of the two years are
therefore in this proportion, and the amount by which the sidereal year
exceeds the tropical year bears to either the same ratio as 50″ to
360° (or 1,296,000″), and is therefore (365-1∕4 × 50)∕1296000 days of
about 20 minutes.
Another way of expressing the amount of the precession is to say
that the equinoctial point will describe the complete circuit of the
ecliptic and return to the same position after about 26,000 years.
The length of each kind of year was also fixed by Hipparchus with
considerable accuracy. That of the tropical year was obtained by
comparing the times of solstices and equinoxes observed by earlier
astronomers with those observed by himself. He found, for example, by
comparison of the date of the summer solstice of 280 B.C., observed
by Aristarchus of Samos, with that of the year 135 B.C., that the
current estimate of 365-1∕4 days for the length of the year had to
be diminished by 1∕300th of a day or about five minutes, an estimate
confirmed roughly by other cases. It is interesting to note as an
illustration of his scientific method that he discusses with some
care the possible error of the observations, and concludes that the
time of a solstice may be erroneous to the extent of about 3∕4 day,
while that of an equinox may be expected to be within 1∕4 day of
the truth. In the illustration given, this would indicate a possible
error of 1-1∕2 days in a period of 145 years, or about 15 minutes in
a year. Actually his estimate of the length of the year is about six
minutes too great, and the error is thus much less than that which he
indicated as possible. In the course of this work he considered also
the possibility of a change in the length of the year, and arrived at
the conclusion that, although his observations were not precise enough
to show definitely the invariability of the year, there was no evidence
to suppose that it had changed.
The length of the tropical year being thus evaluated at 365 days 5
hours 55 minutes, and the difference between the two kinds of year
being given by the observations of precession, the sidereal year was
ascertained to exceed 365-1∕4 days by about 10 minutes, a result
agreeing almost exactly with modern estimates. That the addition of two
erroneous quantities, the length of the tropical year and the amount
of the precession, gave such an accurate result was not, as at first
sight appears, a mere accident. The chief source of error in each
case being the erroneous times of the several equinoxes and solstices
employed, the errors in them would tend to produce errors of opposite
kinds in the tropical year and in precession, so that they would
in part compensate one another. This estimate of the length of the
sidereal year was probably also to some extent verified by Hipparchus
by comparing eclipse observations made at different epochs.
43. The great improvements which Hipparchus effected in the theories of
the sun and moon naturally enabled him to deal more successfully than
any of his predecessors with a problem which in all ages has been of
the greatest interest, the prediction of eclipses of the sun and moon.
That eclipses of the moon were caused by the passage of the moon
through the shadow of the earth thrown by the sun, or, in other words,
by the interposition of the earth between the sun and moon, and
eclipses of the sun by the passage of the moon between the sun and the
observer, was perfectly well known to Greek astronomers in the time of
Aristotle (§ 29), and probably much earlier (chapter I., § 17), though
the knowledge was probably confined to comparatively few people and
superstitious terrors were long associated with eclipses.
The chief difficulty in dealing with eclipses depends on the fact that
the moon’s path does not coincide with the ecliptic. If the moon’s path
on the celestial sphere were identical with the ecliptic, then, once
every month, at new moon, the moon (M) would pass exactly between the
earth and the sun, and the latter would be eclipsed, and once every
month also, at full moon, the moon (M′) would be in the opposite
direction to the sun as seen from the earth, and would consequently be
obscured by the shadow of the earth.
[Illustration: FIG. 25.—The earth’s shadow.]
[Illustration: FIG. 26.—The ecliptic and the moon’s path.]
As, however, the moon’s path is inclined to the ecliptic (§ 40), the
latitudes of the sun and moon may differ by as much as 5°, either when
they are in =conjunction=, _i.e._ when they have the same longitudes,
or when they are in =opposition=, _i.e._ when their longitudes differ
by 180°, and they will then in either case be too far apart for an
eclipse to occur. Whether then at any full or new moon an eclipse will
occur or not, will depend primarily on the latitude of the moon at the
time, and hence upon her position with respect to the nodes of her
orbit (§ 40). If conjunction takes place when the sun and moon happen
to be near one of the nodes (N), as at S M in fig. 26, the sun and moon
will be so close together that an eclipse will occur; but if it occurs
at a considerable distance from a node, as at S′ M′, their centres are
so far apart that no eclipse takes place.
Now the apparent diameter of either sun or moon is, as we have seen (§
32), about 1∕2°; consequently when their discs just touch, as in fig.
27, the distance between their centres is also about 1∕2°. If then
at conjunction the distance between their centres is less than this
amount, an eclipse of the sun will take place; if not, there will be
no eclipse. It is an easy calculation to determine (in fig. 26) the
length of the side N S or N M of the triangle N M S, when S M has this
value, and hence to determine the greatest distance from the node at
which conjunction can take place if an eclipse is to occur. An eclipse
of the moon can be treated in the same way, except that we there have
to deal with the moon and the shadow of the earth at the distance of
the moon. The apparent size of the shadow is, however, considerably
greater than the apparent size of the moon, and an eclipse of the moon
takes place if the distance between the centre of the moon and the
centre of the shadow is less than about 1°. As before, it is easy to
compute the distance of the moon or of the centre of the shadow from
the node when opposition occurs, if an eclipse just takes place. As,
however, the apparent sizes of both sun and moon, and consequently also
that of the earth’s shadow, vary according to the distances of the sun
and moon, a variation of which Hipparchus had no accurate knowledge,
the calculation becomes really a good deal more complicated than at
first sight appears, and was only dealt with imperfectly by him.
[Illustration: FIG. 27.—The sun and moon.]
[Illustration: FIG. 28.—Partial eclipse of the moon.]
[Illustration: FIG. 29.—Total eclipse of the moon.]
Eclipses of the moon are divided into =partial= or =total=, the former
occurring when the moon and the earth’s shadow only overlap partially
(as in fig. 28), the latter when the moon’s disc is completely
immersed in the shadow (fig. 29). In the same way an eclipse of the sun
may be partial or total; but as the sun’s disc may be at times slightly
larger than that of the moon, it sometimes happens also that the whole
disc of the sun is hidden by the moon, except a narrow ring round the
edge (as in fig. 30): such an eclipse is called =annular=. As the
earth’s shadow at the distance of the moon is always larger than the
moon’s disc, annular eclipses of the moon cannot occur.
[Illustration: FIG. 30.—Annular eclipse of the sun.]
Thus eclipses take place if, and only if, the distance of the moon from
a node at the time of conjunction or opposition lies within certain
limits approximately known; and the problem of predicting eclipses
could be roughly solved by such knowledge of the motion of the moon
and of the nodes as Hipparchus possessed. Moreover, the length of
the synodic and draconitic months (§ 40) being once ascertained, it
became merely a matter of arithmetic to compute one or more periods
after which eclipses would recur nearly in the same manner. For if any
period of time contains an exact number of each kind of month, and if
at any time an eclipse occurs, then after the lapse of the period,
conjunction (or opposition) again takes place, and the moon is at the
same distance as before from the node and the eclipse recurs very much
as before. The saros, for example (chapter I., § 17), contained very
nearly 223 synodic or 242 draconitic months, differing from either by
less than an hour. Hipparchus saw that this period was not completely
reliable as a means of predicting eclipses, and showed how to allow for
the irregularities in the moon’s and sun’s motion (§§ 39, 40) which
were ignored by it, but was unable to deal fully with the difficulties
arising from the variations in the apparent diameters of the sun or
moon.
An important complication, however, arises in the case of eclipses
of the sun, which had been noticed by earlier writers, but which
Hipparchus was the first to deal with. Since an eclipse of the moon is
an actual darkening of the moon, it is visible to anybody, wherever
situated, who can see the moon at all; for example, to possible
inhabitants of other planets, just as we on the earth can see precisely
similar eclipses of Jupiter’s moons. An eclipse of the sun is, however,
merely the screening off of the sun’s light from a particular observer,
and the sun may therefore be eclipsed to one observer while to another
elsewhere it is visible as usual. Hence in computing an eclipse of the
sun it is necessary to take into account the position of the observer
on the earth. The simplest way of doing this is to make allowance for
the difference of direction of the moon as seen by an observer at the
place in question, and by an observer in some standard position on the
earth, preferably an ideal observer at the centre of the earth. If, in
fig. 31, M denote the moon, C the centre of the earth, A a point on the
earth between C and M (at which therefore the moon is overhead), and B
any other point on the earth, then observers at C (or A) and B see the
moon in slightly different directions, C M, B M, the difference between
which is an angle known as the =parallax=, which is equal to the angle
B M C and depends on the distance of the moon, the size of the earth,
and the position of the observer at B. In the case of the sun, owing to
its great distance, even as estimated by the Greeks, the parallax was
in all cases too small to be taken into account, but in the case of the
moon the parallax might be as much as 1° and could not be neglected.
[Illustration: FIG. 31.—Parallax.]
If then the path of the moon, as seen from the centre of the earth,
were known, then the path of the moon as seen from any particular
station on the earth could be deduced by allowing for parallax, and the
conditions of an eclipse of the sun visible there could be computed
accordingly.
From the time of Hipparchus onwards lunar eclipses could easily be
predicted to within an hour or two by any ordinary astronomer; solar
eclipses probably with less accuracy; and in both cases the prediction
of the extent of the eclipse, _i.e._ of what portion of the sun or moon
would be obscured, probably left very much to be desired.
44. The great services rendered to astronomy by Hipparchus can hardly
be better expressed than in the words of the great French historian of
astronomy, Delambre, who is in general no lenient critic of the work of
his predecessors:—
“When we consider all that Hipparchus invented or perfected, and
reflect upon the number of his works and the mass of calculations
which they imply, we must regard him as one of the most astonishing
men of antiquity, and as the greatest of all in the sciences which are
not purely speculative, and which require a combination of geometrical
knowledge with a knowledge of phenomena, to be observed only by
diligent attention and refined instruments.”[26]
45. For nearly three centuries after the death of Hipparchus, the
history of astronomy is almost a blank. Several textbooks written
during this period are extant, shewing the gradual popularisation of
his great discoveries. Among the few things of interest in these books
may be noticed a statement that the stars are not necessarily on the
surface of a sphere, but may be at different distances from us, which,
however, there are no means of estimating; a conjecture that the sun
and stars are so far off that the earth would be a mere point seen
from the sun and invisible from the stars; and a re-statement of an
old opinion traditionally attributed to the Egyptians (whether of the
Alexandrine period or earlier is uncertain), that Venus and Mercury
revolve round the sun. It seems also that in this period some attempts
were made to explain the planetary motions by means of epicycles, but
whether these attempts marked any advance on what had been done by
Apollonius and Hipparchus is uncertain.
It is interesting also to find in _Pliny_ (A.D. 23-79) the well-known
modern argument for the spherical form of the earth, that when a
ship sails away the masts, etc., remain visible after the hull has
disappeared from view.
A new measurement of the circumference of the earth by _Posidonius_
(born about the end of Hipparchus’s life) may also be noticed; he
adopted a method similar to that of Eratosthenes (§ 36), and arrived at
two different results. The later estimate, to which he seems to have
attached most weight, was 180,000 stadia, a result which was about as
much below the truth as that of Eratosthenes was above it.
46. The last great name in Greek astronomy is that of Claudius
Ptolemaeus, commonly known as _Ptolemy_, of whose life nothing is known
except that he lived in Alexandria about the middle of the 2nd century
A.D. His reputation rests chiefly on his great astronomical treatise,
known as the _Almagest_,[27] which is the source from which by far
the greater part of our knowledge of Greek astronomy is derived, and
which may be fairly regarded as the astronomical Bible of the Middle
Ages. Several other minor astronomical and astrological treatises are
attributed to him, some of which are probably not genuine, and he was
also the author of an important work on geography, and possibly of a
treatise on _Optics_, which is, however, not certainly authentic and
maybe of Arabian origin. The _Optics_ discusses, among other topics,
the =refraction= or bending of light, by the atmosphere on the earth:
it is pointed out that the light of a star or other heavenly body S,
on entering our atmosphere (at A) and on penetrating to the lower
and denser portions of it, must be gradually bent or =refracted=,
the result being that the star appears to the observer at B nearer
to the zenith Z than it actually is, _i.e._ the light appears to
come from S′ instead of from S; it is shewn further that this effect
must be greater for bodies near the horizon than for those near the
zenith, the light from the former travelling through a greater extent
of atmosphere; and these results are shewn to account for certain
observed deviations in the daily paths of the stars, by which they
appear unduly raised up when near the horizon. Refraction also explains
the well-known flattened appearance of the sun or moon when rising or
setting, the lower edge being raised by refraction more than the upper,
so that a contraction of the vertical diameter results, the horizontal
contraction being much less.[28]
[Illustration: FIG. 32.—Refraction by the atmosphere.]
47. The _Almagest_ is avowedly based largely on the work of earlier
astronomers, and in particular on that of Hipparchus, for whom Ptolemy
continually expresses the greatest admiration and respect. Many of its
contents have therefore already been dealt with by anticipation, and
need not be discussed again in detail. The book plays, however, such
an important part in astronomical history, that it may be worth while
to give a short outline of its contents, in addition to dealing more
fully with the parts in which Ptolemy made important advances.
The _Almagest_ consists altogether of 13 books. The first two deal with
the simpler observed facts, such as the daily motion of the celestial
sphere, and the general motions of the sun, moon, and planets, and
also with a number of topics connected with the celestial sphere and
its motion, such as the length of the day and the times of rising and
setting of the stars in different zones of the earth; there are also
given the solutions of some important mathematical problems,[29] and
a mathematical table[30] of considerable accuracy and extent. But the
most interesting parts of these introductory books deal with what may
be called the postulates of Ptolemy’s astronomy (Book I., chap. ii.).
The first of these is that the earth is spherical; Ptolemy discusses
and rejects various alternative views, and gives several of the usual
positive arguments for a spherical form, omitting, however, one of the
strongest, the eclipse argument found in Aristotle (§ 29), possibly as
being too recondite and difficult, and adding the argument based on the
increase in the area of the earth visible when the observer ascends to
a height. In his geography he accepts the estimate given by Posidonius
that the circumference of the earth is 180,000 stadia. The other
postulates which he enunciates and for which he argues are, that the
heavens are spherical and revolve like a sphere; that the earth is in
the centre of the heavens, and is merely a point in comparison with the
distance of the fixed stars, and that it has no motion. The position
of these postulates in the treatise and Ptolemy’s general method of
procedure suggest that he was treating them, not so much as important
results to be established by the best possible evidence, but rather as
assumptions, more probable than any others with which the author was
acquainted, on which to base mathematical calculations which should
explain observed phenomena.[31] His attitude is thus essentially
different from that either of the early Greeks, such as Pythagoras, or
of the controversialists of the 16th and early 17th centuries, such
as Galilei (chapter VI.), for whom the truth or falsity of postulates
analogous to those of Ptolemy was of the very essence of astronomy and
was among the final objects of inquiry. The arguments which Ptolemy
produces in support of his postulates, arguments which were probably
the commonplaces of the astronomical writing of his time, appear to us,
except in the case of the shape of the earth, loose and of no great
value. The other postulates were, in fact, scarcely, capable of either
proof or disproof with the evidence which Ptolemy had at command. His
argument in favour of the immobility of the earth is interesting, as
it shews his clear perception that the more obvious appearances can be
explained equally well by a motion of the stars or by a motion of the
earth; he concludes, however, that it is easier to attribute motion to
bodies like the stars which seem to be of the nature of fire than to
the solid earth, and points out also the difficulty of conceiving the
earth to have a rapid motion of which we are entirely unconscious. He
does not, however, discuss seriously the possibility that the earth or
even Venus and Mercury may revolve round the sun.
The third book of the _Almagest_ deals with the length of the year
and theory of the sun, but adds nothing of importance to the work of
Hipparchus.
48. The fourth book of the _Almagest_, which treats of the length of
the month and of the theory of the moon, contains one of Ptolemy’s most
important discoveries. We have seen that, apart from the motion of
the moon’s orbit as a whole, and the revolution of the line of apses,
the chief irregularity or inequality was the so-called equation of
the centre (§§ 39, 40), represented fairly accurately by means of an
eccentric, and depending only on the position of the moon with respect
to its apogee. Ptolemy, however, discovered, what Hipparchus only
suspected, that there was a further inequality in the moon’s motion—to
which the name =evection= was afterwards given—and that this depended
partly on its position with respect to the sun. Ptolemy compared the
observed positions of the moon with those calculated by Hipparchus
in various positions relative to the sun and apogee, and found that,
although there was a satisfactory agreement at new and full moon, there
was a considerable error when the moon was half-full, provided it was
also not very near perigee or apogee. Hipparchus based his theory of
the moon chiefly on observations of eclipses, _i.e._ on observations
taken necessarily at full or new moon (§ 43), and Ptolemy’s discovery
is due to the fact that he checked Hipparchus’s theory by observations
taken at other times. To represent this new inequality, it was found
necessary to use an epicycle and a deferent, the latter being itself a
moving eccentric circle, the centre of which revolved round the earth.
To account, to some extent, for certain remaining discrepancies between
theory and observation, which occurred neither at new and full moon,
nor at the =quadratures= (half-moon), Ptolemy introduced further a
certain small to-and-fro oscillation of the epicycle, an oscillation to
which he gave the name of =prosneusis=.[32] Ptolemy thus succeeded in
fitting his theory on to his observations so well that the error seldom
exceeded 10′, a small quantity in the astronomy of the time, and on the
basis of this construction he calculated tables from which the position
of the moon at any required time could be easily deduced.
One of the inherent weaknesses of the system of epicycles occurred
in this theory in an aggravated form. It has already been noticed in
connection with the theory of the sun (§ 39), that the eccentric or
epicycle produced an erroneous variation in the distance of the sun,
which was, however, imperceptible in Greek times. Ptolemy’s system,
however, represented the moon as being sometimes nearly twice as far
off as at others, and consequently the apparent diameter ought at some
times to have been not much more than half as great as at others—a
conclusion obviously inconsistent with observation. It seems probable
that Ptolemy noticed this difficulty, but was unable to deal with it;
it is at any rate a significant fact that when he is dealing with
eclipses, for which the apparent diameters of the sun and moon are of
importance, he entirely rejects the estimates that might have been
obtained from his lunar theory and appeals to direct observation (cf.
also § 51, note).
49. The fifth book of the _Almagest_ contains an account of the
construction and use of Ptolemy’s chief astronomical instrument, a
combination of graduated circles known as the =astrolabe=.[33]
Then follows a detailed discussion of the moon’s parallax (§ 43), and
of the distances of the sun and moon. Ptolemy obtains the distance of
the moon by a parallax method which is substantially identical with
that still in use. If we know the direction of the line C M (fig. 33)
joining the centres of the earth and moon, or the direction of the moon
as seen by an observer at A; and also the direction of the line B M,
that is the direction of the moon as seen by an observer at B, then the
angles of the triangle C B M are known, and the ratio of the sides C B,
C M is known. Ptolemy obtained the two directions required by means
of observations of the moon, and hence found that C M was 59 times C
B, or that the distance of the moon was equal to 59 times the radius
of the earth. He then uses Hipparchus’s eclipse method to deduce the
distance of the sun from that of the moon thus ascertained, and finds
the distance of the sun to be 1,210 times the radius of the earth. This
number, which is substantially the same as that obtained by Hipparchus
(§ 41), is, however, only about 1∕20 of the true number, as indicated
by modern work (chapter XIII., § 284).
[Illustration: FIG. 33.—Parallax.]
The sixth book is devoted to eclipses, and contains no substantial
additions to the work of Hipparchus.
50. The seventh and eighth books contain a catalogue of stars, and
a discussion of precession (§ 42). The catalogue, which contains
1,028 stars (three of which are duplicates), appears to be nearly
identical with that of Hipparchus, It contains none of the stars
which were visible to Ptolemy at Alexandria, but not to Hipparchus at
Rhodes. Moreover, Ptolemy professes to deduce from a comparison of
his observations with those of Hipparchus and others the (erroneous)
value 36″ for the precession, which Hipparchus had given as the least
possible value, and which Ptolemy regards as his final estimate. But
an examination of the positions assigned to the stars in Ptolemy’s
catalogue agrees better with their actual positions in the time of
Hipparchus, _corrected for precession at the supposed rate of 36″
annually_, than with their actual positions in Ptolemy’s time. It is
therefore probable that the catalogue as a whole does not represent
genuine observations made by Ptolemy, but is substantially the
catalogue of Hipparchus corrected for precession and only occasionally
modified by new observations by Ptolemy or others.
51. The last five books deal with the theory of the planets, the
most important of Ptolemy’s original contributions to astronomy. The
problem of giving a satisfactory explanation of the motions of the
planets was, on account of their far greater irregularity, a much more
difficult one than the corresponding problem for the sun or moon. The
motions of the latter are so nearly uniform that their irregularities
may usually be regarded as of the nature of small corrections, and
for many purposes may be ignored. The planets, however, as we have
seen (chapter I., § 14), do not even always move from west to east,
but stop at intervals, move in the reverse direction for a time, stop
again, and then move again in the original direction. It was probably
recognised in early times, at latest by Eudoxus (§ 26), that in the
case of three of the planets, Mars, Jupiter, and Saturn, these motions
could be represented roughly by supposing each planet to oscillate to
and fro on each side of a fictitious planet, moving uniformly round the
celestial sphere in or near the ecliptic, and that Venus and Mercury
could similarly be regarded as oscillating to and fro on each side of
the sun. These rough motions could easily be interpreted by means of
revolving spheres or of epicycles, as was done by Eudoxus and probably
again with more precision by Apollonius. In the case of Jupiter,
for example, we may regard the planet as moving on an epicycle, the
centre of which, _j_, describes uniformly a deferent, the centre of
which is the earth. The planet will then as seen from the earth appear
alternately to the east (as at J_{1}) and to the west (as at J_{2}) of
the fictitious planet _j_; and the extent of the oscillation on each
side, and the interval between successive appearances in the extreme
positions (J_{1}, J_{2}) on either side, can be made right by choosing
appropriately the size and rapidity of motion of the epicycle. It is
moreover evident that with this arrangement the apparent motion of
Jupiter will vary considerably, as the two motions—that on the epicycle
and that of the centre of the epicycle on the deferent—are sometimes
in the same direction, so as to increase one another’s effect, and at
other times in opposite directions. Thus, when Jupiter is most distant
from the earth, that is at J_{3}, the motion is most rapid, at J_{1}
and J_{2} the motion as seen from the earth is nearly the same as that
of _j_; while at J_{4} the two motions are in opposite directions,
and the size and motion of the epicycle having been chosen in the way
indicated above, it is found in fact that the motion of the planet in
the epicycle is the greater of the two motions, and that therefore the
planet when in this position appears to be moving from east to west
(from left to right in the figure), as is actually the case. As then
at J_{1} and J_{2} the planet appears to be moving from west to east,
and at J_{4} in the opposite direction, and sudden changes of motion
do not occur in astronomy, there must be a position between J_{1} and
J_{4}, and another between J_{4} and J_{2}, at which the planet is
just reversing its direction of motion, and therefore appears for the
instant at rest. We thus arrive at an explanation of the stationary
points (chapter I., § 14). An exactly similar scheme explains roughly
the motion of Mercury and Venus, except that the centre of the epicycle
must always be in the direction of the sun.
[Illustration: FIG. 34.—Jupiter’s epicycle and deferent.]
Hipparchus, as we have seen (§ 41), found the current representations
of the planetary motions inaccurate, and collected a number of fresh
observations. These, with fresh observations of his own, Ptolemy now
employed in order to construct an improved planetary system.
As in the case of the moon, he used as deferent an eccentric circle
(centre C), but instead of making the centre _j_ of the epicycle move
uniformly in the deferent, he introduced a new point called an =equant=
(E′), situated at the same distance from the centre of the deferent as
the earth but on the opposite side, and regulated the motion of _j_ by
the condition that the apparent motion _as seen from the equant_ should
be uniform; in other words, the angle A E′ _j_ was made to increase
uniformly. In the case of Mercury (the motions of which have been found
troublesome by astronomers of all periods), the relation of the equant
to the centre of the epicycle was different, and the latter was made to
move in a small circle. The deviations of the planets from the ecliptic
(chapter I., §§ 13, 14) were accounted for by tilting up the planes of
the several deferents and epicycles so that they were inclined to the
ecliptic at various small angles.
[Illustration: FIG. 35.—The equant.]
By means of a system of this kind, worked out with great care, and
evidently at the cost of enormous labour, Ptolemy was able to represent
with very fair exactitude the motions of the planets, as given by the
observations in his possession.
It has been pointed out by modern critics, as well as by some mediaeval
writers, that the use of the equant (which played also a small part in
Ptolemy’s lunar theory) was a violation of the principle of employing
only uniform circular motions, on which the systems of Hipparchus and
Ptolemy were supposed to be based, and that Ptolemy himself appeared
unconscious of his inconsistency. It may, however, fairly be doubted
whether Hipparchus or Ptolemy ever had an abstract belief in the
exclusive virtue of such motions, except as a convenient and easily
intelligible way of representing certain more complicated motions,
and it is difficult to conceive that Hipparchus would have scrupled
any more than his great follower, in using an equant to represent
an irregular motion, if he had found that the motion was thereby
represented with accuracy. The criticism appears to me in fact to be
an anachronism. The earlier Greeks, whose astronomy was speculative
rather than scientific, and again many astronomers of the Middle Ages,
felt that it was on _a priori_ grounds necessary to represent the
“perfection” of the heavenly motions by the most “perfect” or regular
of geometrical schemes; so that it is highly probable that Pythagoras
or Plato, or even Aristotle, would have objected, and certain that the
astronomers of the 14th and 15th centuries ought to have objected (as
some of them actually did), to this innovation of Ptolemy’s. But there
seems no good reason for attributing this _a priori_ attitude to the
later scientific Greek astronomers (cf. also §§ 38, 47).[34]
It will be noticed that nothing has been said as to the actual
distances of the planets, and in fact the apparent motions are
unaffected by any alteration in the scale on which deferent
and epicycle are constructed, provided that both are altered
proportionally. Ptolemy expressly states that he had no means of
estimating numerically the distances of the planets, or even of knowing
the order of the distance of the several planets. He followed tradition
in accepting conjecturally rapidity of motion as a test of nearness,
and placed Mars, Jupiter, Saturn (which perform the circuit of the
celestial sphere in about 2, 12, and 29 years respectively) beyond the
sun in that order. As Venus and Mercury accompany the sun, and may
therefore be regarded as on the average performing their revolutions in
a year, the test to some extent failed in their case, but Ptolemy again
accepted the opinion of the “ancient mathematicians” (_i.e._ probably
the Chaldaeans) that Mercury and Venus lie between the sun and moon,
Mercury being the nearer to us. (Cf. chapter I., § 15.)
52. There has been much difference of opinion among astronomers as to
the merits of Ptolemy. Throughout the Middle Ages his authority was
regarded as almost final on astronomical matters, except where it was
outweighed by the even greater authority assigned to Aristotle. Modern
criticism has made clear, a fact which indeed he never conceals, that
his work is to a large extent based on that of Hipparchus; and that
his observations, if not actually fictitious, were at any rate in most
cases poor. On the other hand his work shews clearly that he was an
accomplished and original mathematician.[35] The most important of his
positive contributions to astronomy were the discovery of evection
and his planetary theory, but we ought probably to rank above these,
important as they are, the services which he rendered by preserving
and developing the great ideas of Hipparchus—ideas which the other
astronomers of the time were probably incapable of appreciating, and
which might easily have been lost to us if they had not been embodied
in the _Almagest_.
53. The history of Greek astronomy practically ceases with Ptolemy.
The practice of observation died out so completely that only eight
observations are known to have been made during the eight and a half
centuries which separate him from Albategnius (chapter III., § 59). The
only Greek writers after Ptolemy’s time are compilers and commentators,
such as _Theon_ (_fl._ A.D. 365), to none of whom original ideas of
any importance can be attributed. The murder of his daughter Hypatia
(A.D. 415), herself also a writer on astronomy, marks an epoch in the
decay of the Alexandrine school; and the end came in A.D. 640, when
Alexandria was captured by the Arabs.[36]
54. It remains to attempt to estimate briefly the value of the
contributions to astronomy made by the Greeks and of their method of
investigation. It is obviously unreasonable to expect to find a brief
formula which will characterise the scientific attitude of a series
of astronomers whose lives extend over a period of eight centuries;
and it is futile to explain the inferiority of Greek astronomy to our
own on some such ground as that they had not discovered the method of
induction, that they were not careful enough to obtain facts, or even
that their ideas were not clear. In habits of thought and scientific
aims the contrast between Pythagoras and Hipparchus is probably greater
than that between Hipparchus on the one hand and Coppernicus or even
Newton on the other, while it is not unfair to say that the fanciful
ideas which pervade the work of even so great a discoverer as Kepler
(chapter VII., §§ 144, 151) place his scientific method in some
respects behind that of his great Greek predecessor.
The Greeks inherited from their predecessors a number of observations,
many of them executed with considerable accuracy, which were nearly
sufficient for the requirements of practical life, but in the matter of
astronomical theory and speculation, in which their best thinkers were
very much more interested than in the detailed facts, they received
virtually a blank sheet on which they had to write (at first with
indifferent success) their speculative ideas. A considerable interval
of time was obviously necessary to bridge over the gulf separating such
data as the eclipse observations of the Chaldaeans from such ideas as
the harmonical spheres of Pythagoras; and the necessary theoretical
structure could not be erected without the use of mathematical methods
which had gradually to be invented. That the Greeks, particularly
in early times, paid little attention to making observations, is
true enough, but it may fairly be doubted whether the collection of
fresh material for observations would really have carried astronomy
much beyond the point reached by the Chaldaean observers. When once
speculative ideas, made definite by the aid of geometry, had been
sufficiently developed to be capable of comparison with observation,
rapid progress was made. The Greek astronomers of the scientific
period, such as Aristarchus, Eratosthenes, and above all Hipparchus,
appear moreover to have followed in their researches the method
which has always been fruitful in physical science—namely, to frame
provisional hypotheses, to deduce their mathematical consequences, and
to compare these with the results of observation. There are few better
illustrations of genuine scientific caution than the way in which
Hipparchus, having tested the planetary theories handed down to him
and having discovered their insufficiency, deliberately abstained from
building up a new theory on data which he knew to be insufficient, and
patiently collected fresh material, never to be used by himself, that
some future astronomer might thereby be able to arrive at an improved
theory.
Of positive additions to our astronomical knowledge made by the Greeks
the most striking in some ways is the discovery of the approximately
spherical form of the earth, a result which later work has only
slightly modified. But their explanation of the chief motions of
the solar system and their resolution of them into a comparatively
small number of simpler motions was, in reality, a far more important
contribution, though the Greek epicyclic scheme has been so remodelled,
that at first sight it is difficult to recognise the relation between
it and our modern views. The subsequent history will, however, show
how completely each stage in the progress of astronomical science has
depended on those that preceded.
When we study the great conflict in the time of Coppernicus between the
ancient and modern ideas, our sympathies naturally go out towards those
who supported the latter, which are now known to be more accurate,
and we are apt to forget that those who then spoke in the name of
the ancient astronomy and quoted Ptolemy were indeed believers in
the doctrines which they had derived from the Greeks, but that their
methods of thought, their frequent refusal to face facts, and their
appeals to authority, were all entirely foreign to the spirit of the
great men whose disciples they believed themselves to be.