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 82 Del. A. A. i. 257.

But the relations of position which astronomy considers, are, for

the most part, angular distances; and these are most simply
expressed by the intercepted portion of a circumference described
about the angular point. The use of the gnomon might lead to the
determination of the angle by the graphical methods of geometry; but
the numerical expression of the circumference required some
progress in trigonometry; for instance, a table of the tangents of
angles.

Instruments were soon invented for measuring angles, by means

of circles, which had a border or limb, divided into equal parts. The
whole circumference was divided into 360 degrees: perhaps
because the circles, first so divided, were those which represented
the sun’s annual path; one such degree would be the sun’s daily
advance, more nearly than any other convenient aliquot part which
could be taken. The position of the sun was determined by means of
the shadow of one part of the instrument upon the other. The most
ancient instrument of this kind appears to be the Hemisphere of
Berosus. A hollow hemisphere was placed with its rim horizontal,
and a style was erected in such a manner that the extremity of the
style was exactly at the centre of the sphere. The shadow of this
extremity, on the concave surface, had the same position with regard
to the lowest point of the sphere which the sun had with regard to the
highest point of the heavens. 163 But this instrument was in fact used
rather for dividing the day into portions of time than for determining
position.

Eratosthenes 83 observed the amount of the obliquity of the sun’s

path to the equator: we are not informed what instruments he used
for this purpose; but he is said to have obtained, from the
munificence of Ptolemy Euergetes, two Armils, or instruments
composed of circles, which were placed in the portico at Alexandria,
and long used for observations. If a circular rim or hoop were placed
so as to coincide with the plane of the equator, the inner concave
edge would be enlightened by the sun’s rays which came under the
front edge, when the sun was south of the equator, and by the rays
which came over the front edge, when the sun was north of the
equator: the moment of the transition would be the time of the
equinox. Such an instrument appears to be referred to by
Hipparchus, as quoted by Ptolemy. 84 “The circle of copper, which
stands at Alexandria in what is called the Square Porch, appears to
mark, as the day of the equinox, that on which the concave surface
begins to be enlightened from the other side.” Such an instrument
was called an equinoctial armil.
83 Delambre, A. A. i. 86.

84 Ptol. Synt. iii. 2.

A solstitial armil is described by Ptolemy, consisting of two circular

rims, one sliding round within the other, and the inner one furnished
with two pegs standing out from its surface at right angles, and
diametrically opposite to each other. These circles being fixed in the
plane of the meridian, and the inner one turned, till, at noon, the
shadow of the peg in front falls upon the peg behind, the position of
the sun at noon would be determined by the degrees on the outer
circle.

In calculation, the degree was conceived to be divided into 60

minutes, the minute into 60 seconds, and so on. But in practice it
was impossible to divide the limb of the instrument into parts so
small. The armils of Alexandria were divided into no parts smaller
than sixths of degrees, or divisions of 10 minutes.

The angles, observed by means of these divisions, were

expressed as a fraction of the circumference. Thus Eratosthenes
stated the interval between the tropics to be 11⁄83 of the
circumference. 85
85 Delambre, A. A. i. 87. It is probable that his observation gave
him 47⅔ degrees. The fraction 47⅔⁄360 = 143⁄1080 = 11∙13⁄1080 =
11⁄ 1 , which is very nearly 11⁄ .
83 ⁄13 83

It was soon remarked that the whole circumference of the circle

164 was not wanted for such observations. Ptolemy 86 says that he
found it more convenient to observe altitudes by means of a square
flat piece of stone or wood, with a quadrant of a circle described on
one of its flat faces, about a centre near one of the angles. A peg
was placed at the centre, and one of the extreme radii of the
quadrant being perpendicular to the horizon, the elevation of the sun
above the horizon was determined by observing the point of the arc
of the quadrant on which the shadow of the peg fell.
86 Synt. i. 1.

As the necessity of accuracy in the observations was more and

more felt, various adjustments of such instruments were practised.
The instruments were placed in the meridian by means of a meridian
line drawn by astronomical methods on the floor on which they
stood. The plane of the instrument was made vertical by means of a
plumb-line: the bounding radius, from which angles were measured,
was also adjusted by the plumb-line. 87
 87 The curvature of the plane of the circle, by warping, was
noticed. Ptol. iii. 2. p. 155, observes that his equatorial circle was
illuminated on the hollow side twice in the same day. (He did not
know that this might arise from refraction.)

In this manner, the places of the sun and of the moon could be
observed by means of the shadows which they cast. In order to
observe the stars, 88 the observer looked along the face of the circle
of the armil, so as to see its two edges apparently brought together,
and the star apparently touching them. 89
88 Delamb. A. A. i. 185.

89 Ptol. Synt. i. 1. Ὥσπερ κεκολλήμενος ἀμφοτέραις αὐτῶν ταῖς

ἐπιφανείαις ὁ ἀστὴρ ἐν τῷ δι’ αὐτῶν ἐπιπέδῳ διοπτεύηται.

It was afterwards found important to ascertain the position of the

sun with regard to the ecliptic: and, for this purpose, an instrument,
called an astrolabe, was invented, of which we have a description in
Ptolemy. 90 This also consisted of circular rims, movable within one
another, or about poles; and contained circles which were to be
brought into the position of the ecliptic, and of a plane passing
through the sun and the poles of the ecliptic. The position of the
moon with regard to the ecliptic, and its position in longitude with
regard to the sun or a star, were thus determined.
90 Synt. v. 1.

The astrolabe continued long in use, but not so long as the

quadrant described by Ptolemy; this, in a larger form, is the mural
quadrant, which has been used up to the most recent times.

It may be considered surprising, 91 that Hipparchus, after having

165 observed, for some time, right ascensions and declinations,
quitted equatorial armils for the astrolabe, which immediately refers
the stars to the ecliptic. He probably did this because, after the
discovery of precession, he found the latitudes of the stars constant,
and wanted to ascertain their motion in longitude.
91 Del. A. A. 181.

To the above instruments, may be added the dioptra, and the

parallactic instrument of Hipparchus and Ptolemy. In the latter, the
distance of a star from the zenith was observed by looking through
two sights fixed in a rule, this being annexed to another rule, which
was kept in a vertical position by a plumb-line; and the angle
between the two rules was measured.

The following example of an observation, taken from Ptolemy, may

serve to show the form in which the results of the instruments, just
described, were usually stated. 92
92 Del. A. A. ii. 248.

“In the 2d year of Antoninus, the 9th day of Pharmouthi, the sun
being near setting, the last division of Taurus being on the meridian
(that is, 5½ equinoctial hours after noon), the moon was in 3 degrees
of Pisces, by her distance from the sun (which was 92 degrees, 8
minutes); and half an hour after, the sun being set, and the quarter of
Gemini on the meridian, Regulus appeared, by the other circle of the
astrolabe, 57½ degrees more forwards than the moon in longitude.”
From these data the longitude of Regulus is calculated.

From what has been said respecting the observations of the

Alexandrian astronomers, it will have been seen that their
instrumental observations could not be depended on for any close
accuracy. This defect, after the general reception of the Hipparchian
theory, operated very unfavorably on the progress of the science. If
they could have traced the moon’s place distinctly from day to day,
they must soon have discovered all the inequalities which were
known to Tycho Brahe; and if they could have measured her parallax
or her diameter with any considerable accuracy, they must have
obtained a confutation of the epicycloidal form of her orbit. By the
badness of their observations, and the imperfect agreement of these
with calculation, they not only were prevented making such steps,
but were led to receive the theory with a servile assent and an
indistinct apprehension, instead of that rational conviction and
intuitive clearness which would have given a progressive impulse to
their knowledge. 166

Sect. 4.—Period from Hipparchus to Ptolemy.

We have now to speak of the cultivators of astronomy from the

time of Hipparchus to that of Ptolemy, the next great name which
occurs in the history of this science; though even he holds place only
among those who verified, developed, and extended the theory of
Hipparchus. The astronomers who lived in the intermediate time,
indeed, did little, even in this way; though it might have been
supposed that their studies were carried on under considerable
advantages, inasmuch as they all enjoyed the liberal patronage of
the kings of Egypt. 93 The “divine school of Alexandria,” as it is called
by Synesius, in the fourth century, appears to have produced few
persons capable of carrying forwards, or even of verifying, the labors
of its great astronomical teacher. The mathematicians of the school
wrote much, and apparently they observed sometimes; but their
observations are of little value; and their books are expositions of the
theory and its geometrical consequences, without any attempt to
compare it with observation. For instance, it does not appear that
any one verified the remarkable discovery of the precession, till the
time of Ptolemy, 250 years after; nor does the statement of this
motion of the heavens appear in the treatises of the intermediate
writers; nor does Ptolemy quote a single observation of any person
made in this long interval of time; while his references to those of
Hipparchus are perpetual; and to those of Aristyllus and Timocharis,
and of others, as Conon, who preceded Hipparchus, are not
unfrequent.
93 Delamb. A. A. ii. 240.

This Alexandrian period, so inactive and barren in the history of

science, was prosperous, civilized, and literary; and many of the
works which belong to it are come down to us, though those of
Hipparchus are lost. We have the “Uranologion” of Geminus, 94 a
systematic treatise on Astronomy, expounding correctly the
Hipparchian Theories and their consequences, and containing a
good account of the use of the various Cycles, which ended in the
adoption of the Calippic Period. We have likewise “The Circular
Theory of the Celestial Bodies” of Cleomedes, 95 of which the
principal part is a development of the doctrine of the sphere,
including the consequences of the globular form of the earth. We
have also another work on “Spherics” by Theodosius of Bithynia, 96
which contains some of the most important propositions of the
subject, and has been used as a book of 167 instruction even in
modern times. Another writer on the same subject is Menelaus, who
lived somewhat later, and whose Three Books on Spherics still
remain.
94 b. c. 70.
 95 b. c. 60.

96 b. c. 50.

One of the most important kinds of deduction from a geometrical

theory, such as that of the doctrine of the sphere, or that of
epicycles, is the calculation of its numerical results in particular
cases. With regard to the latter theory, this was done in the
construction of Solar and Lunar Tables, as we have already seen;
and this process required the formation of a Trigonometry, or system
of rules for calculating the relations between the sides and angles of
triangles. Such a science had been formed by Hipparchus, who
appears to be the author of every great step in ancient astronomy. 97
He wrote a work in twelve books, “On the Construction of the Tables
of Chords of Arcs;” such a table being the means by which the
Greeks solved their triangles. The Doctrine of the Sphere required, in
like manner, a Spherical Trigonometry, in order to enable
mathematicians to calculate its results; and this branch of science
also appears to have been formed by Hipparchus, 98 who gives
results that imply the possession of such a method. Hypsicles, who
was a contemporary of Ptolemy, also made some attempts at the
solution of such problems: but it is extraordinary that the writers
whom we have mentioned as coming after Hipparchus, namely,
Theodosius, Cleomedes, and Menelaus, do not even mention the
calculation of triangles, 99 either plain or spherical; though the latter
writer 100 is said to have written on “the Table of Chords,” a work
which is now lost.
97 Delamb. A. A. ii. 37.

98 A. A. i. 117.

99 A. A. i. 249.
 100 A. A. ii. 37.

We shall see, hereafter, how prevalent a disposition in literary

ages is that which induces authors to become commentators. This
tendency showed itself at an early period in the school of Alexandria.
Aratus, 101 who lived 270 b. c. at the court of Antigonus, king of
Macedonia, described the celestial constellations in two poems,
entitled “Phænomena,” and “Prognostics.” These poems were little
more than a versification of the treatise of Eudoxus on the acronycal
and heliacal risings and settings of the stars. The work was the
subject of a comment by Hipparchus, who perhaps found this the
easiest way of giving connection and circulation to his knowledge.
Three Latin translations of this poem gave the Romans the means of
becoming acquainted with it: the first is by Cicero, of which we have
numerous fragments 168 extant; 102 Germanicus Cæsar, one of the
sons-in-law of Augustus, also translated the poem, and this
translation remains almost entire. Finally, we have a complete
translation by Avienus. 103 The “Astronomica” of Manilius, the
“Poeticon Astronomicon” of Hyginus, both belonging to the time of
Augustus, are, like the work of Aratus, poems which combine
mythological ornament with elementary astronomical exposition; but
have no value in the history of science. We may pass nearly the
same judgment upon the explanations and declamations of Cicero,
Seneca, and Pliny, for they do not apprise us of any additions to
astronomical knowledge; and they do not always indicate a very
clear apprehension of the doctrines which the writers adopt.
101 A. A. i. 74.

102 Two copies of this translation, illustrated by drawings of

different ages, one set Roman, and the other Saxon, according to
Mr. Ottley, are described in the Archæologia, vol. xviii.
 103 Montucla, i. 221.

Perhaps the most remarkable feature in the two last-named

writers, is the declamatory expression of their admiration for the
discoverers of physical knowledge; and in one of them, Seneca, the
persuasion of a boundless progress in science to which man was
destined. Though this belief was no more than a vague and arbitrary
conjecture, it suggested other conjectures in detail, some of which,
having been verified, have attracted much notice. For instance, in
speaking of comets, 104 Seneca says, “The time will come when
those things which are now hidden shall be brought to light by time
and persevering diligence. Our posterity will wonder that we should
be ignorant of what is so obvious.” “The motions of the planets,” he
adds, “complex and seemingly confused, have been reduced to rule;
and some one will come hereafter, who will reveal to us the paths of
comets.” Such convictions and conjectures are not to be admired for
their wisdom; for Seneca was led rather by enthusiasm, than by any
solid reasons, to entertain this opinion; nor, again, are they to be
considered as merely lucky guesses, implying no merit; they are
remarkable as showing how the persuasion of the universality of law,
and the belief of the probability of its discovery by man, grow up in
men’s minds, when speculative knowledge becomes a prominent
object of attention.
104 Seneca, Qu. N. vii. 25.

An important practical application of astronomical knowledge was

made by Julius Cæsar, in his correction of the calendar, which we
have already noticed; and this was strictly due to the Alexandrian
School: Sosigenes, an astronomer belonging to that school, came
from Egypt to Rome for the purpose. 169
 Sect. 5.—Measures of the Earth.

There were, as we have said, few attempts made, at the period of

which we are speaking, to improve the accuracy of any of the
determinations of the early Alexandrian astronomers. One question
naturally excited much attention at all times, the magnitude of the
earth, its figure being universally acknowledged to be a globe. The
Chaldeans, at an earlier period, had asserted that a man, walking
without stopping, might go round the circuit of the earth in a year; but
this might be a mere fancy, or a mere guess. The attempt of
Eratosthenes to decide this question went upon principles entirely
correct. Syene was situated on the tropic; for there, on the day of the
solstice, at noon, objects cast no shadow; and a well was
enlightened to the bottom by the sun’s rays. At Alexandria, on the
same day, the sun was, at noon, distant from the zenith by a fiftieth
part of the circumference. Those two cities were north and south
from each other: and the distance had been determined, by the royal
overseers of the roads, to be 5000 stadia. This gave a circumference
of 250,000 stadia to the earth, and a radius of about 40,000.
Aristotle 105 says that the mathematicians make the circumference
400,000 stadia. Hipparchus conceived that the measure of
Eratosthenes ought to be increased by about one-tenth. 106
Posidonius, the friend of Cicero, made another attempt of the same
kind. At Rhodes, the star Canopus but just appeared above the
horizon; at Alexandria, the same star rose to an altitude of 1⁄48th of
the circumference; the direct distance on the meridian was 5000
stadia, which gave 240,000 for the whole circuit. We cannot look
upon these measures as very precise; the stadium employed is not
certainly known; and no peculiar care appears to have been
bestowed on the measure of the direct distance.
 105 De Cœlo, ii. ad fin.

106 Plin. ii. (cviii.)

When the Arabians, in the ninth century, came to be the principal

cultivators of astronomy, they repeated this observation in a manner
more suited to its real importance and capacity of exactness. Under
the Caliph Almamon, 107 the vast plain of Singiar, in Mesopotamia,
was the scene of this undertaking. The Arabian astronomers there
divided themselves into two bands, one under the direction of Chalid
ben Abdolmalic, and the other having at its head Alis ben Isa. These
two parties proceeded, the one north, the other south, determining
the distance by the actual application of their measuring-rods to the
ground, 170 till each was found, by astronomical observation, to be a
degree from the place at which they started. It then appeared that
these terrestrial degrees were respectively 56 miles, and 56 miles
and two-thirds, the mile being 4000 cubits. In order to remove all
doubt concerning the scale of this measure, we are informed that the
cubit is that called the black cubit, which consists of 27 inches, each
inch being the thickness of six grains of barley.
107 Montu. 357.

Sect. 6.—Ptolemy’s Discovery of Evection.

By referring, in this place, to the last-mentioned measure of the

earth, we include the labors of the Arabian as well as the
Alexandrian astronomers, in the period of mere detail, which forms
the sequel to the great astronomical revolution of the Hipparchian
epoch. And this period of verification is rightly extended to those later
times; not merely because astronomers were then still employed in
determining the magnitude of the earth, and the amount of other
elements of the theory,—for these are some of their employments to
the present day,—but because no great intervening discovery marks
a new epoch, and begins a new period;—because no great
revolution in the theory added to the objects of investigation, or
presented them in a new point of view. This being the case, it will be
more instructive for our purpose to consider the general character
and broad intellectual features of this period, than to offer a useless
catalogue of obscure and worthless writers, and of opinions either
borrowed or unsound. But before we do this, there is one writer
whom we cannot leave undistinguished in the crowd; since his name
is more celebrated even than that of Hipparchus; his works contain
ninety-nine hundredths of what we know of the Greek astronomy;
and though he was not the author of a new theory, he made some
very remarkable steps in the verification, correction, and extension of
the theory which he received. I speak of Ptolemy, whose work, “The
Mathematical Construction” (of the heavens), contains a complete
exposition of the state of astronomy in his time, the reigns of Adrian
and Antonine. This book is familiarly known to us by a term which
contains the record of our having received our first knowledge of it
from the Arabic writers. The “Megiste Syntaxis,” or Great
Construction, gave rise, among them, to the title Al Magisti, or
Almagest, by which the work is commonly described. As a
mathematical exposition of the Theory of Epicycles and Eccentrics,
of the observations and calculations which were employed in 171
order to apply this theory to the sun, moon, and planets, and of the
other calculations which are requisite, in order to deduce the
consequences of this theory, the work is a splendid and lasting
monument of diligence, skill, and judgment. Indeed, all the other
astronomical works of the ancients hardly add any thing whatever to
the information we obtain from the Almagest; and the knowledge
which the student possesses of the ancient astronomy must depend
mainly upon his acquaintance with Ptolemy. Among other merits,
Ptolemy has that of giving us a very copious account of the manner
in which Hipparchus established the main points of his theories; an
account the more agreeable, in consequence of the admiration and
enthusiasm with which this author everywhere speaks of the great
master of the astronomical school.

In our present survey of the writings of Ptolemy, we are concerned

less with his exposition of what had been done before him, than with
his own original labors. In most of the branches of the subject, he
gave additional exactness to what Hipparchus had done; but our
main business, at present, is with those parts of the Almagest which
contain new steps in the application of the Hipparchian hypothesis.
There are two such cases, both very remarkable,—that of the
moon’s Evection, and that of the Planetary Motions.

The law of the moon’s anomaly, that is, of the leading and obvious
inequality of her motion, could be represented, as we have seen,
either by an eccentric or an epicycle; and the amount of this
inequality had been collected by observations of eclipses. But
though the hypothesis of an epicycle, for instance, would bring the
moon to her proper place, so far as eclipses could show it, that is, at
new and full moon, this hypothesis did not rightly represent her
motions at other points of her course. This appeared, when Ptolemy
set about measuring her distances from the sun at different times.
“These,” he 108 says, “sometimes agreed, and sometimes disagreed.”
But by further attention to the facts, a rule was detected in these
differences. “As my knowledge became more complete and more
connected, so as to show the order of this new inequality, I perceived
that this difference was small, or nothing, at new and full moon; and
that at both the dichotomies (when the moon is half illuminated) it
was small, or nothing, if the moon was at the apogee or perigee of
the epicycle, and was greatest when she was in the middle of the
interval, and therefore when the first 172 inequality was greatest
also.” He then adds some further remarks on the circumstances
according to which the moon’s place, as affected by this new
inequality, is before or behind the place, as given by the epicyclical
hypothesis.
108 Synth. v. 2.

Such is the announcement of the celebrated discovery of the

moon’s second inequality, afterwards called (by Bullialdus) the
Evection. Ptolemy soon proceeded to represent this inequality by a
combination of circular motions, uniting, for this purpose, the
hypothesis of an epicycle, already employed to explain the first
inequality, with the hypothesis of an eccentric, in the circumference
of which the centre of the epicycle was supposed to move. The
mode of combining these was somewhat complex; more complex we
may, perhaps, say, than was absolutely requisite; 109 the apogee of
the eccentric moved backwards, or contrary to the order of the signs,
and the centre of the epicycle moved forwards nearly twice as fast
upon the circumference of the eccentric, so as to reach a place
nearly, but not exactly, the same, as if it had moved in a concentric
instead of an eccentric path. Thus the centre of the epicycle went
twice round the eccentric in the course of one month: and in this
manner it satisfied the condition that it should vanish at new and full
moon, and be greatest when the moon was in the quarters of her
monthly course. 110
109 If Ptolemy had used the hypothesis of an eccentric instead of
an epicycle for the first inequality of the moon, an epicycle would
 have represented the second inequality more simply than his
method did.

110 I will insert here the explanation which my German translator,

the late distinguished astronomer Littrow, has given of this point.
The Rule of this Inequality, the Evection, may be most simply
expressed thus. If a denote the excess of the Moon’s Longitude
over the Sun’s, and b the Anomaly of the Moon reckoned from her
Perigee, the Evection is equal to 1°. 3.sin (2a − b). At New and
Full Moon, a is 0 or 180°, and thus the Evection is − 1°.3.sin b. At
both quarters, or dichotomies, a is 90° or 270°, and consequently
the Evection is + 1°.3.sin b. The Moon’s Elliptical Equation of the
centre is at all points of her orbit equal to 6°.3.sin b. The Greek
Astronomers before Ptolemy observed the moon only at the time
of eclipses; and hence they necessarily found for the sum of these
two greatest inequalities of the moon’s motion the quantity
6°.3.sin b − 1°.3.sin b, or 5°.sin b: and as they took this for the
moon’s equation of the centre, which depends upon the
eccentricity of the moon’s orbit, we obtain from this too small
equation of the centre, an eccentricity also smaller than the truth.
Ptolemy, who first observed the moon in her quarters, found for
the sum of those Inequalities at those points the quantity
6°.3.sin b + 1°.3.sin b, or 7°.6.sin b; and thus made the
eccentricity of the moon as much too great at the quarters as the
observers of eclipses had made it too small. He hence concluded
that the eccentricity of the Moon’s orbit is variable, which is not
the case.

The discovery of the Evection, and the reduction of it to the 173

epicyclical theory, was, for several reasons, an important step in
astronomy; some of these reasons may be stated.

1. It obviously suggested, or confirmed, the suspicion that the

motions of the heavenly bodies might be subject to many
inequalities:—that when one set of anomalies had been discovered
and reduced to rule, another set might come into view;—that the
discovery of a rule was a step to the discovery of deviations from the
rule, which would require to be expressed in other rules;—that in the
application of theory to observation, we find, not only the stated
phenomena, for which the theory does account, but also residual
phenomena, which remain unaccounted for, and stand out beyond
the calculation;—that thus nature is not simple and regular, by
conforming to the simplicity and regularity of our hypotheses, but
leads us forwards to apparent complexity, and to an accumulation of
rules and relations. A fact like the Evection, explained by an
Hypothesis like Ptolemy’s, tended altogether to discourage any
disposition to guess at the laws of nature from mere ideal views, or
from a few phenomena.

2. The discovery of Evection had an importance which did not

come into view till long afterwards, in being the first of a numerous
series of inequalities of the moon, which results from the Disturbing
Force of the sun. These inequalities were successfully discovered;
and led finally to the establishment of the law of universal gravitation.
The moon’s first inequality arises from a different cause;—from the
same cause as the inequality of the sun’s motion;—from the motion
in an ellipse, so far as the central attraction is undisturbed by any
other. This first inequality is called the Elliptic Inequality, or, more
usually, the Equation of the Centre. 111 All the planets have such
inequalities, but the Evection is peculiar to the moon. The discovery
of other inequalities of the moon’s motion, the Variation and Annual
Equation, made an immediate sequel in the order of the subject to
174 the discoveries of Ptolemy, although separated by a long interval
of time; for these discoveries were only made by Tycho Brahe in the
sixteenth century. The imperfection of astronomical instruments was
the great cause of this long delay.
 111 The Equation of the Centre is the difference between the
place of the Planet in its elliptical orbit, and that place which a
Planet would have, which revolved uniformly round the Sun as a
centre in a circular orbit in the same time. An imaginary Planet
moving in the manner last described, is called the mean Planet,
while the actual Planet which moves in the ellipse is called the
true Planet. The Longitude of the mean Planet at a given time is
easily found, because its motion is uniform. By adding to it the
Equation of the Centre, we find the Longitude of the true Planet,
and thus, its place in its orbit.—Littrow’s Note.
I may add that the word Equation, used in such cases, denotes
in general a quantity which must be added to or subtracted from a
mean quantity, to make it equal to the true quantity; or rather, a
quantity which must be added to or subtracted from a variably
increasing quantity to make it increase equably.

3. The Epicyclical Hypothesis was found capable of

accommodating itself to such new discoveries. These new
inequalities could be represented by new combinations of eccentrics
and epicycles: all the real and imaginary discoveries by astronomers,
up to Copernicus, were actually embodied in these hypotheses;
Copernicus, as we have said, did not reject such hypotheses; the
lunar inequalities which Tycho detected might have been similarly
exhibited; and even Newton 112 represents the motion of the moon’s
apogee by means of an epicycle. As a mode of expressing the law of
the irregularity, and of calculating its results in particular cases, the
epicyclical theory was capable of continuing to render great service
to astronomy, however extensive the progress of the science might
be. It was, in fact, as we have already said, the modern process of
representing the motion by means of a series of circular functions.
112 Principia, lib. iii. prop. xxxv.
 4. But though the doctrine of eccentrics and epicycles was thus
admissible as an Hypothesis, and convenient as a means of
expressing the laws of the heavenly motions, the successive
occasions on which it was called into use, gave no countenance to it
as a Theory; that is, as a true view of the nature of these motions,
and their causes. By the steps of the progress of this Hypothesis, it
became more and more complex, instead of becoming more simple,
which, as we shall see, was the course of the true Theory. The
notions concerning the position and connection of the heavenly
bodies, which were suggested by one set of phenomena, were not
confirmed by the indications of another set of phenomena; for
instance, those relations of the epicycles which were adopted to
account for the Motions of the heavenly bodies, were not found to fall
in with the consequences of their apparent Diameters and
Parallaxes. In reality, as we have said, if the relative distances of the
sun and moon at different times could have been accurately
determined, the Theory of Epicycles must have been forthwith
overturned. The insecurity of such measurements alone maintained
the theory to later times. 113
113 The alteration of the apparent diameter of the moon is so
great that it cannot escape us, even with very moderate
instruments. This apparent diameter contains, when the moon is
nearest the earth, 2010 seconds; when she is furthest off 1762
seconds; that is, 248 seconds, or 4 minutes 8 seconds, less than
in the former case. [The two quantities are in the proportion of 8 to
7, nearly.]—Littrow’s Note. 175

Sect. 7.—Conclusion of the History of Greek Astronomy.

I might now proceed to give an account of Ptolemy’s other great

step, the determination of the Planetary Orbits; but as this, though in
itself very curious, would not illustrate any point beyond those
already noticed, I shall refer to it very briefly. The planets all move in
ellipses about the sun, as the moon moves about the earth; and as
the sun apparently moves about the earth. They will therefore each
have an Elliptic Inequality or Equation of the centre, for the same
reason that the sun and moon have such inequalities. And this
inequality may be represented, in the cases of the planets, just as in
the other two, by means of an eccentric; the epicycle, it will be
recollected, had already been used in order to represent the more
obvious changes of the planetary motions. To determine the amount
of the Eccentricities and the places of the Apogees of the planetary
orbits, was the task which Ptolemy undertook; Hipparchus, as we
have seen, having been destitute of the observations which such a
process required. The determination of the Eccentricities in these
cases involved some peculiarities which might not at first sight occur
to the reader. The elliptical motion of the planets takes place about
the sun; but Ptolemy considered their movements as altogether
independent of the sun, and referred them to the earth alone; and
thus the apparent eccentricities which he had to account for, were
the compound result of the Eccentricity of the earth’s orbit, and of the
proper eccentricity of the orbit of the Planet. He explained this result
by the received mechanism of an eccentric Deferent, carrying an
Epicycle; but the motion in the Deferent is uniform, not about the
centre of the circle, but about another point, the Equant. Without
going further into detail, it may be sufficient to state that, by a
combination of Eccentrics and Epicycles, he did account for the
leading features of these motions; and by using his own
observations, compared with more ancient ones (for instance, those
of Timocharis for Venus), he was able to determine the Dimensions
and Positions of the orbits. 114