J.L.E DREYER "A
HISTORY OF ASTRONOMY FROM THALES TO KEPLER"
THE examination
of the astronomical doctrines of Plato has shown us that
philosophers in the first half of the fourth century before the Christian
era possessed some knowledge of the motions of the planets. No doubt
astronomical instruments, even of the crudest kind, cannot be said to have
existed, except the gnomon for following the course of the sun; but all the same
the complicated movements of the planets through the constellations must have
been traced for many years previously. That the moon, though its motion is not
subject to very conspicuous irregularities, does not pursue the same path among
the stars from month to month and from year to year must also have been
perfectly well known, since Helikon, a disciple of Eudoxus, was able to foretell
the solar eclipse of the 12th May, 361, for which he was rewarded by Dionysius
II of Syracuse with a present of a talent[1].
But the clearest proof of the not inconsiderable amount of knowledge of the
movements of the heavenly bodies, which was available at the time of Plato, is
supplied by the important astronomical system of his younger contemporary,
Eudoxus of Knidus, which is the first attempt to account for the more
conspicuous irregularities of those movements.
Eudoxus was born
at Knidus, in Asia Minor, about the year 408 B.C., and died in his fifty-third
year, about 355[2].At
the age of twenty-three he went to Athens and attended Plato's lectures for some
months, but not content with the knowledge he could attain in Greece, Eudoxus
afterwards proceeded to Egypt, furnished with letters of recommendation from the
Spartan King Agesilaus to Nectanebis, King of Egypt. He stayed at least a year
in Egypt, possibly much
longer (about 378 B.C.), and received instruction from a priest of
Heliopolis. According to
Seneca[3] it
was there that he acquired his knowledge of the planetary motions, but although
this is not unlikely to have been the case, we have no reason to believe that
Eudoxus brought his mathematical theory of these motions home from Egypt, in
which country, as far as we know, geometry had made very little progress[4].
Diogenes of Laerte, who does not say a word about the scientific work of
Eudoxus, does not omit to mention that the Egyptian Apis licked his garment,
after which the priests prophesied that he would be short-lived but very
illustrious. If this prophecy was really uttered it was a true one, as Eudoxus
stands in the foremost rank of Greek mathematicians. Most, if not the whole, of
the fifth book of Euclid is due to him,
as well as the so-called method of exhaustion, by means of which the Greeks were
able to solve many problems of mensuration without infinitesimals. We are told
by Plutarch[5]
that Plato, on being consulted about the celebrated Delian problem of the
duplication of a cube, said that only two men were capable of solving this
problem, Eudoxus and Helikon; and if the story is apocryphal, it shows at any
rate the high renown of Eudoxus as a mathematician. In the history of astronomy
he is also known as the first proposer of a solar cycle of four years, three of
365 and one of 366 days, which was three hundred years later introduced by
Julius Caesar. He was therefore fully capable of grappling successfully with the
intricate problem of planetary motion, which Plato (according to Simplicius) is
said to have suggested to him for solution[6],
and his labours produced a most ingenious cosmical system which represented the
principal phenomena in the heavens as far as they were known in his time.
This system of
concentric spheres, which was accepted and slightly improved by Kalippus, is
known to us through a short notice of it in Aristotle's Metaphysics (A 8), and
through a lengthy account given by Simplicius in his commentary to Aristotle's
book on the Heavens[7].
The systems of Hipparchus and Ptolemy eventually superseded it, and the
beautiful system of Eudoxus was well-nigh forgotten. One historian of astronomy
after another, knowing in reality nothing about it, except that it supposed the
existence of a great number of spheres, contented himself with a few
contemptuous remarks about the absurdity of the whole thing. That the system,
mathematically speaking, was exceedingly elegant does not seem to have been
observed by anybody, until Ideler in two papers in the Transactions of the
Berlin
Academy for 1828 and
1830 drew attention to the theory of Eudoxus and explained its principles. The
honour of having completely mastered the theory and of having investigated how
far it could account for the observed phenomena, belongs, however, altogether to
Schiaparelli, who has shown how very undeserved is the neglect and contempt with
which the system of concentric spheres has been treated so long, and how much we
ought to admire the ingenuity of its author. We shall now give an account of
this system as set forth by Schiaparelli[8].
Although the
various cosmical systems suggested by philosophers from the earliest ages to the
time of Kepler differ greatly from each other both in general principles and in
matters of detail, there is one idea common to them all: that the planets move
in circular orbits. This principle was also accepted by Eudoxus, but he added
another in order to render his system simple and symmetrical. He assumed that
all the spheres which it appeared necessary to introduce were situated one
inside the other and all concentric to the earth, for which reason they long
afterwards became known as the Homocentric spheres. No doubt this added
considerably to the difficulty of accounting for the complicated phenomena, but
the system gained greatly in symmetry and beauty, while it also became
physically far more sensible than any system of excentric circles could possibly
be. Every celestial body was supposed to be situated on the equator of a sphere
which revolves with uniform speed round its two poles. In order to explain the
stations and arcs of retrogression of the planets, as well as their motion in
latitude, Eudoxus assumed that the poles of a planetary sphere are not immovable
but are carried by a larger sphere, concentric with the first one, which rotates
with a different speed round two poles different from those of the first one. As
this was not sufficient to represent the phenomena, Eudoxus placed the poles of
the second sphere on a third, concentric to and larger than the two first ones
and moving round separate poles with a speed peculiar to itself. Those spheres
which did not themselves carry a planet were according to Theophrastus called
avdarpoi, or starless. Eudoxus found that it was possible by a suitable choice
of poles and velocities of rotation to represent the motion of the sun and moon
by assuming three spheres for each of these bodies, but for the more intricate
motions of the five planets four spheres for each became necessary, the moving
spheres of each body being quite independent of those of the others. For the
fixed stars one sphere was of course sufficient to produce the daily revolution
of the heavens. The total number of spheres was therefore twentyseven. It does
not appear that Eudoxus speculated on the cause of all these rotations, nor on
the material, thickness, or mutual distances of the spheres. We only know from a
statement of Archimedes (in his (Ψαμμίτη) that Eudoxus estimated the sun to
be nine times greater than the moon, from which we may conclude that he assumed
the sun to be nine times as far distant as the moon. Whether he merely adopted
the spheres as mathematical means of representing the motions of the planets and
subjecting them to calculation thereby, or whether he really believed in the
physical existence of all these spheres, is uncertain. But as Eudoxus made no
attempt to connect the movements of the various groups of spheres with each
other, it seems probable that he only regarded them as geometrical constructions
suitable for computing the apparent paths of the planets.
Eudoxus explained
his system in a book " On velocities," which is lost,
together with all his other writings. Aristotle, who was only one generation
younger, had his knowledge of the system from Polemarchus, an acquaintance of
its author's. Eudemus described it in detail in his lost history of astronomy,
and from this work the description was transferred to a work on the spheres
written by Sosigenes, a peripatetic philosopher who lived in the second half of
the second century after Christ. This work is also lost, but a long extract from
it is preserved in the commentary of Simplicius, and we are thus in possession
of a detailed account of the system of Eudoxus[9].
While all other
ancient and medieval cosmical systems (apart from those which accept the
rotation of the earth) account for the diurnal motion of sun, moon, and planets
across the sky by assuming that the sphere of the fixed stars during its daily
revolution drags all the other spheres along with it, the system of Eudoxus
provides a separate machinery for each planet for this purpose, thereby adding
in all seven spheres to the number required for other purposes. Thus the motion
of the moon was produced by three spheres; the first and outermost of these
rotated from east to west in twenty-four hours like the fixed stars; the second
turned from west to east round the axis of the zodiac, producing the monthly
motion of the moon round the heavens; the third sphere turned slowly, according
to Simplicius, in the same direction as the first one round an axis inclined to
the axis of the zodiac at an angle equal to the highest latitude reached by the
moon, the latter being placed on what we may call the equator of this third
sphere. The addition of this third sphere was necessary, says Simplicius,
because the moon does not always seem to reach its highest north and south
latitude at the same points of the zodiac, but at points which travel round the
zodiac in a direction opposite to the order of its twelve signs. In other words,
the third sphere was to account for the retrograde motion of the nodes of
the lunar orbit in 18 1/2 years. But it is easy to see (as was pointed out by
Ideler) that Simplicius has made a mistake in his statement, that the
innermost sphere moved very slowly and in the manner described; as the moon
according to that arrangement would only pass once through each node in the
course of 223 lunations, and would be north of the ecliptic for nine years and
then south of it for nine years. Obviously Eudoxus must have taught that the
innermost sphere (carrying the moon) revolved in 27 days[10]
from west to east round an axis inclined at an angle equal to the greatest
latitude of the moon, to the axis of the second sphere, which latter revolved
along the zodiac in 223 lunations in a retrograde direction. In this manner the
phenomena are perfectly accounted for; that is, as far as Eudoxus knew them, for
he evidently did not know anything of the moon's changeable velocity in
longitude, though we shall see that Kalippus about B.C. 325 was aware of this.
But that the motion of the lunar node was known forty or fifty years earlier is
proved by the lunar theory of Eudoxus.
With regard to
the solar theory, we learn from Aristotle that it also depended on three
spheres, one having the same daily motion as the sphere of the fixed stars, the
second revolving along the zodiac, and the third along a circle inclined to the
zodiac. Simplicius confirms this statement, and adds that the third sphere does
not, as in the case of the moon,' turn in the direction opposite to that of the
second, but in the same direction, that is, in the direction of the zodiacal
signs, and very much more slowly than the second sphere. Simplicius has here
made the same mistake as in describing the lunar theory, as, according to his
description, the sun would for ages have either a north or a south latitude, and
in the course of a year would describe a small circle parallel to the ecliptic
instead of a great circle. Of course the slow motion must belong to the second
sphere and be directed along the zodiac, while the motion of the third sphere
must take place in a year[11]
along the inclined great circle, which the centre of the sun was supposed to
describe. This circle is by the second sphere turned round the axis of the
zodiac, and its nodes on the ecliptic are by Eudoxus supposed to have a very
slow direct motion instead of a retrograde motion as the lunar nodes have. The
annual motion of the sun is supposed to be perfectly uniform, so that Eudoxus
must have rejected the remarkable discovery made by Meton and Euktemon some 60
or 70 years earlier, that the sun does not take the same time to describe the
four quadrants of its orbit between the equinoxes and solstices[12].
It is very
remarkable that although Eudoxus thus ignored the discovery of the variable
orbital velocity of the sun, he admitted as real the altogether imaginary idea
that the sun did not in the course of the year travel along the ecliptic, but
along a circle inclined at a small angle to the latter. According to Simplicius[13], "
Eudoxus and those before him " had been led to assume this by observing that the
sun at the summer- and winter-solstices did not always rise at the same point of
the horizon. Perhaps it did not strike these early observers that neither these
rough determinations of the azimuth of the rising sun nor the observations with
the gnomon were sufficiently accurate; they had without instruments perceived
that neither the moon nor the five planets were confined to move in the ecliptic
(or, as they called it, the circle through the middle of the zodiac), and why
should the sun alone have no motion in latitude, when all the other wandering
stars had a very perceptible one ? This imaginary
deviation of the sun from the ecliptic is frequently alluded to by writers of
antiquity. Thus Hipparchus, who denies its existence, quotes the following
passage from a lost book on the circles and constellations of the sphere, the
Enoptron of Eudoxus : " It seems that the sun also
makes its return (τροπάς, solstices) in different places, but much less
conspicuously[14]."
How great Eudoxus supposed the inclination of the solar orbit to be, or how long
he supposed the period of revolution of the nodes to be, is not known, and he had probably not very precise
notions on the subject. Pliny gives the inclination as 1°, and the point where
the maximum latitude occurs as the 29th degree of Aries[15].
On the other hand, Theon of Smyrna, who goes more into detail on this subject,
states on the authority of Adrastus (who lived about A.D. 100) that the
inclination is 1/2°, and that the sun returns to the same latitude after 3651/8
days, so as to make the shadow of the gnomon have the same length, as he says,
while the sun takes 3651/4 to return to the same equinox or solstice, and 3651/2
days to return to the same distance from us. This shows that the solar nodes
were supposed to have a retrograde motion (not a direct one as assumed by
Eudoxus) and in a period of 3651/4:1/8 = 2922 years[16].
Schiaparelli shows that with an inclination of 1/2° between the axes of the
second and third spheres the solstitial points would oscillate 2° 28'. This of
course influences the length of the tropical year, and it is very possible that
the whole theory of the sun's latitude originally arose from the fact that the
tropical year had been found to be different from the sidereal year, the true
cause of which is the precession of the equinoxes. To whom this theory in the first instance is due is not known.
Notwithstanding the great authority of Hipparchus and Ptolemy the strange
illusion is still upheld by the compiler Martianus Capella in the fifth century[17],
who improves on it by stating that the sun moves in the ecliptic except in
Libra, where it deviates 1/2°! The meaning is probably that the latitude of the
sun was insensible to the instruments of the day except in Libra (and in Aries)
where it reached 1/2°, and consequently the nodes must have been supposed nearly
to coincide with the solstices. It is to be noticed that the precession of the
equinoxes is unknown to all these writers[18].
The solar theory
of Eudoxus was therefore practically a copy of his lunar theory. But the task he
had set himself became vastly more difficult when he took up the theories of the
five other planets, as it now became necessary to account for the stations and
retrograde motions of these bodies. Of the four spheres given to each planet the
first and outermost produced the daily rotation of the planet round the earth in
twenty-four hours; the second produced the motion along the zodiac in a period
which for the three outer planets was respectively equal to their sidereal
period of revolution, while it for Mercury and Venus was equal to a year. From
the fact that the revolution of this second sphere was in all cases assumed to
be uniform, we see that Eudoxus had no knowledge of the orbital changes of
velocity of the planets which depend on the excentricity of each orbit, but that
he believed the points of the zodiac in which a planet was found at successive
oppositions (or conjunctions) to be perfectly equidistant one from the other.
Neither did he assume the orbits to be inclined to the ecliptic, but let the
second sphere of every planet move along this circle, while the latitudes of the
planets were supposed to depend solely on their elongation from the sun and not
on their longitude. To represent this motion in latitude, and at the same time the inequality in longitude
depending on the elongation from the sun, a third and fourth sphere were
introduced for each planet. The third sphere had its poles situated at two
opposite points of the zodiac (on the second sphere), and rotated round them in
a period equal to the synodic period of the planet, or the interval between two
successive oppositions or conjunctions with the sun. These poles were different
for the different planets, but Mercury and Venus had the same poles. The
direction of the rotation of this third sphere is not given by Simplicius except
as being from north to south and from south to north, but it turns out to be
immaterial which of the two possible directions we adopt.
On the surface of
the third sphere the poles of the fourth were fixed, the axis of the latter
having a constant inclination, different for each planet, to the axis of the
third sphere. Round the axis of the fourth sphere the rotation of the latter
took place in the same period, but in a direction opposite to that of the third
sphere. On the equator of the fourth sphere the planet is fixed, and it is thus endowed with four motions, the daily
one, the orbital one along the zodiac, and two others in the synodic period.
 |
What effect will
these two last-mentioned motions have on the apparent position of the planet in
the sky ? In the appended figure a sphere (the third)
rotates round the fixed diameter AB (we may leave the motion of the first, or
daily sphere, altogether out of consideration, and for the present also neglect
that of the second sphere); during this rotation round AB a certain point P, one
of the poles of the fourth sphere, describes the small circle QPR, while this
fourth sphere in the same period, but in the opposite direction, completes a
rotation round P and its other pole P'. The planet is at M in the equator of the
fourth sphere, so that PM = 90°. The problem is now to determine the path
described by M, projected on the plane of the circle AQBR. This is easy enough
with the aid of modern mathematics, but was Eudoxus able to solve it by means of
simple geometrical reasoning? This question has been admirably investigated by
Schiaparelli, who has shown that the solution of the problem was well within the
range of a geometrician of the acknowledged ability of Eudoxus. The result
is that the projected path is symmetrical to the line AB, that it has a double
point in it, and is nothing but the well-known " figure of eight" or lemniscate,
the equation of which is r2 = a2 cos 2θ, or, strictly
speaking, a figure of this kind lying in the surface of the celestial sphere,
for which reason Schiaparelli calls it a spherical lemniscate. The
longitudinal axis of the curve lies along the
zodiac,
and its length is equal to the diameter of the circle described by P, the pole of the sphere
which carries the planet. The
double point is 90o from the two poles of rotation of the
third sphere. The planet describes
the curve by moving in the direction
of the arrow, and passes over the arcs 1-2, 2-3, 3-4, 4-5, etc., in equal
times.
So far we have
only considered the motion of the point M under the influence of the rotations
of the third and fourthsphere. But we must now remember that the axis AB
revolves round the ecliptic in the sidereal period of the planet. During this
motion the longitudinal axis of the lemniscate always coincides with the
ecliptic, along which the curve is carrie with uniform
velocity. We may therefore for the third and fourth sphere substitute the
lemniscate, on which the planet moves in the manner described above. The
combination of this motion with the motion of the curve along the ecliptic gives
the apparent motion of the planet through the constellations. The motion of
the planet on the lemniscate consists in an oscillation forward and backward,
the period being that of the synodical revolution, and during one half of this
period the motion of the planet along the ecliptic becomes accelerated, and
during the second half it becomes retarded, when the two motions are in opposite
directions. Therefore when on an arc of the lemniscate the backward oscillation
is quicker than the simultaneous forward motion of the lemniscate itself, then
the planet will for a time have a retrograde motion, before and after which it
is stationary for a little while, when the two motions just balance each other.
Evidently the greatest acceleration and the greatest retardation occur when the
planet passes through the double point of the lemniscate. The motions must
therefore be so combined that the planet passes through this point with a
forward motion at the time of superior conjunction with the sun, where the
apparent velocity of the planet in longitude is greatest, while it must again be
in the double point, but moving in a retrograde direction, at the time of
opposition or inferior conjunction, when the planet appears to have the most
rapid retrograde motion. This combination of motions will of course be
accompanied by a certain amount of motion in latitude depending on the breadth
of the lemniscate.
This curve was by
the Greeks called the hippopede (ίππου πέδη), because
it was a favourite practice in the riding school to let the horse describe this
figure in cantering; and Simplicius in his account of the planetary theory of
Eudoxus expressly states that a planet describes the curve called by Eudoxus a
hippopede. This word occurs in several places in the commentary to the
first book of Euclid written by
Proklus, in which he describes the plane sections of the solid generated by the
revolution of a circle round a straight line in its plane, assuming that the
line does not cut the circle[19]. A
section by a plane parallel to the line and touching the inner surface of the
" anchor ring " is by Proklus called a hippopede, and
it is therefore proved that Eudoxus and his followers had a clear idea of the
properties of the curve which represents the resultant motion of the third and
fourth sphere. The curve and its application is thus alluded to by Theon of
Smyrna in his account of the astronomical theory of the Platonist Derkyllides:
"He does not believe that the helicoid lines and those similar to the Hippika
can be considered as causing the erratic motions of the planets, for these lines
are produced by chance[20],
but the first cause of the erratic motion and the helix is the motion which
takes place in the oblique circle of the zodiac." After this Theon describes the
helix apparently traced by a planet in the manner of Plato in the Timοeus; but
the opinion rejected by Derkyllides is undoubtedly the motion in the lemniscate
invented by Eudoxus[21].
If we now ask how
far this theory could be made to agree with the actually observed motions in the
sky we must first of all remember that we possess no knowledge as to whether
Eudoxus had made observations to ascertain the extent of the retrograde motions,
or whether he was merely aware of the fact that such motions existed, without
having access to any numerical data concerning them. To be able to test the
theory we require to know the sidereal period, the synodic period, and the
distance between the poles of the third and fourth sphere, which Schiaparelli
calls the inclination. The length of this distance adopted by Eudoxus for each
planet is not stated either by Aristotle or Simplicius, and the periods are only
given by the latter in round numbers as follow[22]:
Star
of
|
Synodic
Period
|
Modern
value
|
Zodiacal Period
|
Modern
value
|
Mercury
|
110
days
|
116
days
|
1
year
|
1.0
year
|
Venus
|
19
months
|
584 „
|
1 „
|
1.0 „
|
Mars
|
8
„ 20 days
|
780 „
|
2
years
|
1 .88
years
|
Jupiter
|
13 „
|
399 „
|
12 „
|
11.86 „
|
Saturn
|
13 „
|
378 „
|
30 „
|
29.46
|
With the
exception of Mars these figures show that the revolutions of the planets had
been observed with some care, and Eudoxus may even have been in possession of
somewhat more accurate figures, as the Papyrus of Eudoxus gives the synodic
revolution of Mercury as 116 days, a remarkably accurate value, which he had
most probably obtained during his stay in Egypt[23].
If only we knew the inclination on which the dimensions of the hippopede depend,
we should be able perfectly to reconstruct each planetary theory of Eudoxus. As
the principal object of the system certainly was to account for the retrograde
motions, Schiaparelli has for the three outer planets assumed that the values of
the inclinations were so chosen as to make the retrograde arcs agree with the
observed ones. The retrograde arc of Saturn is about 6°, and with a zodiacal
period of 30 years, a synodic period of 13 months, and an inclination of 6°
between the axes of the third and fourth sphere the length of the hippopede
becomes 12° and half its breadth, i.e. the greatest deviation of the planet from
the ecliptic turns out to be 9', a quantity insensible for the observations
of those days. We have therefore simply a retrograde motion in longitude of
about 6° between two stationary points. Similarly, assuming for Jupiter an
inclination of 13°, the length of the hippopede becomes 26°, and half its
breadth 44', and with periods of respectively 12 years and 13 months this gives
a retrograde arc of about 8°. The greatest distance from the ecliptic during the
motion on this arc, 44', was probably hardly noticeable at that time. For these
two planets Eudoxus had thus found an excellent solution of the problem proposed
by Plato, even supposing that he knew accurately the lengths of the retrograde
arcs.
But
this was not the case with Mars, which indeed is not to
be wondered at, when we remember that even Kepler for a
long time found it hard to make the theory of this planet satisfactory. It is
not easy to see how Eudoxus could put the synodic
period equal to 8 months and 20 days (or 260 days), whereas
it really is 780 days, or exactly three times as long. All
editions of Simplicius give the same figures, and Ideler's suggestion
that we should for 8 months read 25 months seems therefore
unwarranted; besides, it does not help matters in the
least. For with a synodic period of 780 days and putting the inclination equal
to 90° (the highest value reconcilable with
the description of Simplicius), the breadth of the hippopede becomes
60°, so that Mars ought to reach latitudes of 30°. And
even so, the retrograde motion of Mars on the hippopede cannot
in speed come up to the direct motion of the latter along
the zodiac, so that Mars should not become retrograde at all, but
should only move very slowly at opposition. To obtain a retrograde motion the inclination would have to
be greater than 90°; in other words
the third and fourth sphere would have to rotate in the same direction. And even
this violation of the rule would be
of no use, since Mars in that case would reach latitudes greater than 30°, and Eudoxus was
doubtless not willing to accept
this. On the other hand, if we adopt his own value of the synodic period, 260 days, the
motion of Mars on the hippopede
becomes almost three times as great as before, and with an inclination of 34° the retrograde
arc becomes 16° long and the greatest latitude nearly 5°. This is in fair
accordance with the real facts,
but unfortunately this hypothesis gives two retrograde motions outside the
oppositions and four additional
stationary points, which have no real existence. The theory of Eudoxus fails therefore completely
in the case of
Mars.
With regard to
Mercury and Venus, we have first to note that the mean place of these planets
always coincides with the sun, so that the centre of the hippopede always lies
in the sun. As this centre is 90° from the poles of rotation of the third
sphere, we see that these poles coincide for the two planets. This deduction
from the theory is confirmed by the remark of Aristotle that "according to
Eudoxus the poles of the third sphere are different for some planets, but
identical for Aphrodite and Hermes," and this supplies a valuable proof of the
correctness of Schiaparelli's deductions. As the greatest elongation of each of
these planets from the sun equals half the length of the hippopede, i.e. the
inclination of the third and fourth spheres, Eudoxus doubtless determined the
inclination by observing the elongations, as he could not make use of the
retrograde motions, which in the case of Venus are hard to see, and in that of
Mercury out of reach. With a hippopede for Mercury 46° in length the half
breadth or greatest latitude becomes 2° 14', nearly as great as that observed.
For Venus we may make the hippopede 92° in length, which gives half its breadth
equal to 8° 54' in good accordance with the observed greatest latitude. But, as
in the case of Mars, Venus can never become retrograde, and no different
assumption as to the value of the inclination can do away with this error of the
theory. And a much worse fault is, that Venus ought to take the same length of
time to pass from the east end of the hippopede to the west end and vice versa,
which is not in accordance with facts, since Venus in reality takes 440 days to
move from the greatest western to the greatest eastern elongation, and only
about 143 days to go from the eastern to the western elongation, a fact which is
very easily ascertained. The theory is equally unsatisfactory as to latitude,
for the hippopede intersects the ecliptic in four points, at the two extremities
and at the double point; consequently Venus ought four times during every
synodic period to pass the ecliptic, which is far from being the
case.
But with all its
imperfections as to detail the system of homocentric spheres proposed by Eudoxus
demands our admiration as the first serious attempt to deal with the
apparently lawless motions of the planets. For Saturn and Jupiter, and
practically also for Mercury, the system accounted well for the motion in
longitude, while it was unsatisfactory in the case of Venus, and broke down
completely only when dealing with the motion of Mars. The limits of motion in
latitude were also well represented by the various hippopedes, though the
periods of the actual deviations from the ecliptic and their places in the
cycles came out quite wrong. But it must be remembered that Eudoxus cannot have
had at his command a sufficient series of observations; he had probably in Egypt
learned the main facts about the stationary points and retrogressions of the
outer planets as well as their periods of revolution, which the Babylonians and
Egyptians doubtless knew well, while it may be doubted whether systematic
observations had for any length of time been carried on in Greece. And if the
old complaint is to be repeated about the system being so terribly complicated,
we may well bear in mind, as Schiaparelli remarks, that Eudoxus in his planetary
theories only made use of three elements, the epoch of an upper conjunction, the
period of sidereal revolution (of which the synodic period is a function), and
the inclination of the axis of the third sphere to that of the fourth. For the
same purpose we nowadays require six elements
!
If, however, the
system was founded on an insufficient basis of observations, it seems that some
of the adherents of Eudoxus must have compared the movements resulting from the
theory with those actually taking place in the sky, since we find Kalippus, of
Kyzikus, a pupil of Eudoxus, engaged in improving his master's system some
thirty years after its first publication. Kalippus is also otherwise favourably
known to us by his improvement of the soli-lunar cycle of Meton, which shows
that he must have possessed a remarkably accurate knowledge of the length of the
moon's period of revolution. Simplicius states that Kalippus, who studied with
Polemarchus, an acquaintance of Eudoxus, went with Polemarchus to
Athens in order to
discuss the inventions of Eudoxus with Aristotle, and by his help to correct and
complete them[24].
This must have happened during the reign of Alexander the Great (336-323), which
time Aristotle spent at
Athens. From the
investigations of Kalippus resulted an important improvement of the system of
Eudoxus which Aristotle and Simplicius describe; and as the former solely
credits Kalippus with it, it does not seem likely that he had any share in it
himself, though he cordially approved of it[25].
Kalippus wrote a book about his planetary theory, but it was already lost before
the time of Simplicius, who could only refer to the history of astronomy by
Eudemus, which contained an account of it.
The principle of
the homocentric spheres, as we shall see in the next chapter, fitted in well
with the cosmological ideas of Aristotle, and had therefore to be preserved, so
that Kalippus was obliged to add more spheres to the system if he wished to
improve it. He considered the theories of Jupiter and Saturn to be sufficiently
correct and left them untouched, which shows that he had not perceived the
elliptic inequality in the motion of either planet, though it can reach the
value of five or six degrees. But the very great deficiencies in the theory of
Mars he tried to correct by introducing a fifth sphere for this planet in order
to produce a retrograde motion without making a grave error in the synodic
period. This is only a supposition, as we are not positively told why Kalippus
added a sphere each to the theories of Mars, Venus, and Mercury[26],
but Schiaparelli has shown how the additional sphere can produce retrogression
without unduly adding to the motion in latitude. Let AOB represent the ecliptic,
A and B being opposite points in it which make the circuit of the zodiac in the
sidereal period of Mars. Let a sphere (the third of Eudoxus) rotate round these
points in the synodic period of the planet, and let any point P1 in
the equator of this sphere be the pole of a fourth sphere which rotates twice as
fast as the third in the opposite direction carrying the point P2
with it, which is the pole of a fifth sphere rotating in the same direction and
period as the third and carrying the planet at M on its equator. It is easy to
see that if at the beginning of motion the points P1, P2,
and M were situated in the ecliptic in the order A P2
P1MB, then at any time the angles will be as marked in the figure,
and as AP1 = MP2 = 90°, the planet M will in the synodic
period describe a figure symmetrical to the
ecliptic which alters its
form with the adopted length of the arc P1P2, and, like
the hippopede, may produce retrograde motion. And it has this advantage over the
hippopede, that it can give the planet in the neighbourhood of Ο a much greater
direct and retrograde velocity with the same motion in latitude. It can
therefore make the planet retrograde even in the cases where the hippopede of
Eudoxus failed to do so. Thus, if P1 P2
is put equal to 45°, the curve assumes a figure like that shown;
the greatest digression in latitude is 4° 11', the length of the curve along the
ecliptic is 95° 20', and it has two triple points near the ends, 45° from the
centre. When the planet passes 0, its velocity is 1-293 times the velocity of
P1 round the axis AB, and as the period of the latter rotation is 780
days, the daily motion of P1 is 360°/780 = 0°.462, which number
multiplied by 1.293 gives 0°.597 as the daily velocity of the retrograde motion
on the curve at 0. But as Ο has a direct motion on the ecliptic of 360°/686 =
0°.525, the resulting daily retrograde motion of the planet in the heavens is
0o.072, which is a sufficient approximation to the real motion of
Mars at opposition. It must however be remembered that we have no way of knowing
what value Kalippus assumed for the distance P1P2 ; but
that the introduction of another sphere could really make the theory
satisfactory has been proved by Schiaparelli's investigation.
In the same way
an additional sphere removed the errors in the theory of Venus. If
P1P2 is = 45°, the greatest elongation becomes 47°40',
very nearly the true value; and the different velocity of the planet in the four
parts of the synodic revolution is also accounted for; as in the curve depicted
above the passage from one triple point to the other takes one fourth of the
period, the same passage back again another fourth, while the very slow motion
through the small loops at the end of the curve occupies the remaining time. In
the case of Mercury the theory of Eudoxus was already fairly correct, and no
doubt the extra sphere made it better still.
In the solar
theory Kalippus introduced two new sphere in order to
account for the unequal motion of the sun in longitude which had been discovered
about a hundred years previously by Meton and Euktemon through the unequal
lengths of the four seasons. The so-called papyrus of Eudoxus, to which we have
already referred, gives us the values adopted by Kalippus for the lengths of the
seasons (taken from the Parapegma, or meteorological calendar of Geminus), and
though only given in whole numbers of days (95, 92, 89, 90, beginning with the
vernal equinox), the values are in every case less than a day in error, while
the corresponding values determined by Euktemon about B.C. 430 are from 11/4 to
2 days wrong[27].
The observations of the sun had therefore made good progress in
Greece during the
century ending about B.C. 330. By adding two more spheres to the three spheres
of Eudoxus, Kalippus had only to follow the same principle on which Eudoxus had
represented the synodic inequalities of the planets, and a hippopede 4° in
length and 2' in breadth gives in fact the necessary maximum inequality of 2° in
a perfectly satisfactory manner. Similarly the number of lunar spheres was
increased by two, and though Simplicius is not very explicit, we can hardly
doubt that he means us to understand the cause to be similar to that which he
has just stated in the case of the sun. In other words, Kalippus must have been
aware of the elliptic inequality of the moon. Indeed he can hardly have failed
to notice it, even if he merely confined his attention to lunar eclipses without
watching the motion of the moon at other times, since the intervals between
various eclipses compared with the corresponding longitudes (deduced from those
of the sun) at once show how far the moon's motion in longitude is from being
uniform. A hippopede 12° in length would only be twice 9' in breadth, and would
therefore not sensibly affect the latitude, while it would produce the mean
inequality of 6°. The improved theory was therefore quite as good as any other,
as long as the evection had not been discovered.
Such then was the
modified theory of homocentric spheres, as developed by Kalippus. Scientific
astronomy may really be said to date from Eudoxus and Kalippus, as we here for
the first time meet that mutual influence of theory and observation on each
other which characterizes the development of astronomy from century to century.
Eudoxus is the first to go beyond mere philosophical reasoning about the
construction of the universe; he is the first to attempt systematically to
account for the planetary motions. When he has done this the next question is
how far this theory satisfies the observed phenomena, and Kalippus at once
supplies the observational facts required to test the theory and modifies the
latter until the theoretical and observed motions agree within the limit of
accuracy attainable at the time. Philosophical speculation unsupported by
steadily pursued observations is from henceforth abandoned; the science of
astronomy has started on its career.
________________
1 Boeokh,
Ueber die vierjahrigen Sonnenkreise der Alten, besonders der
Eudoxischen.
Berlin, 1863,
p. 153. For an account of the life of Eudoxus see
ibid. p. 140, and about the geographical researches attributed to him see
Berger's Erdkunde d. Gr. n. pp. 68-74.
2 Strabo (p. 119)
mentions the observatory of Eudoxus (at Knidus) as not having been much higher
than the houses, but still he was able to see the star
Canopus from it.
4
Cantor,
Gesch. der Math. chap.
2. Whatever the Egyptians may have known of
geometry, there is no doubt that the Greeks had long before the time
of
Eudoxus outstripped them completely.
5 De genio
Socratis, cap.
viii.
6 Simpl. De
Cmlo, p. 488 (Heib.).
8 Schiaparelli: "Le sfere omocentriche di Endosso, di Callippo e di Aris-totele,"
Pubblicazioni del R. Osservatorio di Brera in Milano, No. ix. Milano, 1875. German translation in Abhandlungen zur Geschichte der Mathematik,
Erstes Heft.
Leipzig,
1877.
Schiaparelli does not mention a paper by E. P. Apelt:
" Die Spharentheorie des Eudoxus und Aristoteles," in
the Abhandlungen der
Gries'schen Schule, Heft n.
(Leipzig, 1849),
which gives a fairly full exposition of
the theory. Later than Schiaparelli's paper appeared one by Th. H. Martin in
the Mem. de l' Acad. des Inscr. t. xxx. 1881. In 'this objections
are raised to
Sehiaparelli's interpretation of the theories of the sun and moon, but they have
been sufficiently refuted by Tannery in the Mem. de
la Soc. des sc. phys. et nat. tie Bordeaux,
2e
Serie, t. v. 1883, pp. 129-147.
9 Simplicius also
quotes in the course of his account Alexander of Aphrodisias and
Porphyrius, the Neo-Platonic philosopher (p. 503 Heib.).
10
More
accurately in 27d 5h 5m 36", the draconitic or
nodical month
11
Strictly speaking in a period slightly longer than a tropical year, owing to
the
supposed slow, direct motion of the second sphere.
12 This
agrees with the statement in the so-called Papyrus of Eudoxus, that this
astronomer gave the length of the autumn as 92 days, and that of each of
the
three other seasons as 91 days. This papyrus was written about the year
190 b.c, and seems to have been a student's
note-book, perhaps hastily written during
or after a series of lectures. See Boeckh, Ueber die vierjdhrigen
Sonnen-kreise.
der Alten, p. 196
and foil. It was published by F. Blass (Eudoxi Art astronomica,
Kiel,
1887, 25 pp. 4°), and translated by Tannery, Recherchet tur L' Attr. ancienne, pp. 283-294.
14 That
is, the maximum latitude is much less than that of the moon.
Hipparchus
adds, that observations with the gnomon show no
latitude, and lunar
eclipses calculated without assuming any solar latitude agree with observations
within at most two digits. Hipparchi ad Arati et Eudoxi Phenomena, lib.
I.; ed.
Manitius, pp. 88-92.
15
Hist. Nat.
ii.
16
(67).
He has doubtless misunderstood his source andtaken a
range of 1° to mean an inclination of 1°.
16 Astronomia,
ed. Th. H.
Martin, pp. 91, 108, 175 (cap. xii.),
263 (cap.xxvii.),
314 (cap.
xxxviii.).
17 De
nitptiis Phiilologie et Mercurii, lib. vm. 867, on the authority of a book by Tereutius
Varro.
18 Schiaparelli
(I.e. p. 17) shows that Theon's theory cannot have been designed to explain the
motion of the equinoxes discovered by Hipparchus. He also gives a langthy
refutation of the assertion of Lepsius, that the third
solar
sphere of Eudoxus proves that Eudoxus knew precession and had received
his
knowledge of it from the Egyptians (1. e. pp. 20-23). This had, however,
already
been refuted by Martin, "Menoire sar cette question: La
precession dee
Equinoxes
a-t-elle1 conuue avant Hipparche" {Hem. par divers savans,
(t. vii.
1869, pp. 303-522).
19 Cantor, Gesch. dcr
Math. 1. pp. 229-30 (2nd cd.).
20 Does this allude to the loops described by the
planets about the time of opposition,
and not to the machinery supposed to produce them?
21 Theon,
ed. Martin, p. 328,
23
This
papyrus gives the zodiacal periods of Mars and Saturn as two years and
thirty years, in perfect accordance with Simplicius (Blass, p. 16; Tannery,p.
287).
25 Metaph. xi. 8, p. 1073
b.
26 Simplicius merely says that Eudemus has clearly
and shortly stated the reasons for
this addition (De Coelo, ed. Heiberg, p. 497, 1. 22).
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