FOREWORD BY ANDREAS OSIANDER
To the Reader
Concerning the Hypotheses of this Work
There have already been widespread reports about the novel hypotheses of this
work, which declares that the earth moves whereas the sun is at rest in the
center of the universeHence certain scholars, I have no doubt, are deeply
offended and believe that the liberal arts, which were established long ago on a
sound basis, should not be thrown into confusion. But if these men are willing to
examine the matter closely, they will find that the author of this work has done
nothing blameworthy. For it is the duty of an astronomer to compose the history
of the celestial motions through careful and expert study. Then he must conceive
and devise the causes of these motions or hypotheses about them. Since he cannot
in any way attain to the true causes, he will adopt whatever suppositions enable
the motions to be computed correctly from the principles of geometry for the
future as well as for the past. The present author has performed both these
duties excellently. For these hypotheses need not be true nor even probable. On
the contrary, if they provide a calculus consistent with the observations, that
alone is enough. Perhaps there is someone who is so ignorant of geometry and
optics that he regards the epicyclc of Venus as probable, or thinks that it is
the reason why Venus sometimes precedes and sometimes follows the sun by forty
degrees and even more. Is there anyone who is not aware that from this assumption
it necessarily follows that the diameter of the planet at perigee should appear
more than four times, and the body of the planet more than sixteen times, as
great as at apogee? Yet this variation is refuted by the experience of every age.
In this science there are some other no less important absurdities, which need
not be set forth at the moment. For this art, it is quite clear, is completely
and absolutely ignorant of the causes of the apparent nonuniform motions. And if
any causes are devised by the imagination, as indeed very many are, they are not
put forward to convince anyone that are true, but merely to provide a reliable
basis for computation. However, since different hypotheses are sometimes offered
for one and the same motion (for example, eccentricity and an epicycle for the
sun's motion), the astronomer will take as his first choice that hypothesis which
is the easiest to grasp. The philosopher will perhaps rather seek the semblance
of the truth. But neither of them will understand or state anything certain,
unless it has been divinely revealed to him.
Therefore alongside the ancient hypotheses, which are no more probable, let us
permit these new hypotheses also to become known, especially since they are
admirable as well as simple and bring with them a huge treasure of very skillful
observations. So far as hypotheses are concerned, let no one expect anything
certain from astronomy, which cannot furnish it, lest he accept as the truth
ideas conceived for another purpose, and depart from this study a greater fool
than when he entered it. Farewell.
LETTER OF NICHOLAS SCHÖNBERG
Nicholas Schönberg, Cardinal of Capua,
to Nicholas Copernicus, Greetings.
Some years ago word reached me concerning your proficiency, of which everybody
constantly spoke. At that time I began to have a very high regard for you, and
also to congratulate our contemporaries among whom you enjoyed such great
prestige. For I had learned that you had not merely mastered the discoveries of
the ancient astronomers uncommonly well but had also formulated a new cosmology.
In it you maintain that the earth moves; that the sun occupies the lowest, and
thus the central, place in the universe; that the eighth heaven remain
perpetually motionless and fixed; and that, together with the elements included
in its sphere, the moon, situated between the heavens of Mars and Venus, revolves
around the sun in the period of a year. I have also learned that you have written
an exposition of this whole system of astronomy, and have computed the planetary
motions and set them down in tables, to the greatest admiration of all. Therefore
with the utmost earnestness I entreat you, most learned sir, unless I
inconvenience you, to communicate this discovery of yours to scholars, and at the
earliest possible moment to send me your writings on the sphere of the universe
together with the tables and whatever else you have that is relevant to this
subject. Moreover, I have instructed Theodoric of Reden to have everything copied
in your quarters at my expense and dispatched to me. If you gratify my desire in
this matter, you will see that you are dealing with a man who is zealous for your
reputation and eager to do justice to so fine a talent. Farewell.
Rome, 1 November 1536
TO HIS HOLINESS, POPE PAUL III,
NICHOLAS COPERNICUS' PREFACE
TO HIS BOOKS ON THE REVOLUTIONS
I can readily imagine, Holy Father, that as soon as some people hear that in this
volume, which I have written about the revolutions of the spheres of the
universe, I ascribe certain motions to the terrestrial globe, they will shout that I
must be immediately repudiated together with this belief For I am not so enamored
of my own opinions that I disregard what others may think of them. I am aware
that a philosopher's ideas are not subject to the judgement of ordinary persons,
because it is his endeavor to seek the truth in all things, to the extent
permitted to human reason by God. Yet I hold that completely erroneous views
should be shunned. Those who know that the consensus of many centuries has
sanctioned the conception that the earth remains at rest in the middle of the
heaven as its center would, I reflected, regard it as an insane pronouncement if
I made the opposite assertion that the earth moves. Therefore I debated with
myself for a long time whether to publish the volume which I wrote to prove the
earth's motion or rather to follow the example of the Pythagoreans and certain
others, who used to transmit philosophy's secrets only to kinsmen and friends,
not in writing but by word of mouth, as is shown by Lysis' letter to Hipparchus.
And they did so, it seems to me, not, as some suppose, because they were in some
way jealous about their teachings, which would be spread around; on the contrary,
they wanted the very beautiful thoughts attained by great men of deep devotion
not to be ridiculed by those who are reluctant to exert themselves vigorously in
any literary pursuit unless it is lucrative; or if they are stimulated to the
nonacquisitive study of philosophy by the exhortation and example of others, yet
because of their dullness of mind they play the same part among philosophers as
drones among bees. When I weighed these considerations, the scorn which I had
reason to fear on account of the novelty and unconventionality of my opinion
almost induced me to abandon completely the work which I had undertaken.
But while I hesitated for a long time and even resisted, my friends drew me
back. Foremost among them was the cardinal of Capua, Nicholas Schönberg,
renowned in every field of learning. Next to him was a man who loves me dearly,
Tiedemann Giese, bishop of Chelmno, a close student of sacred letters as well
as of all good literature. For he repeatedly encouraged me and, sometimes
adding reproaches, urgently requested me to publish this volume and finally
permit it to appear after being buried among my papers and lying concealed not
merely until the ninth year but by now the fourth period of nine years. The
same conduct was recommended to me by not a few other very eminent scholars.
They exhorted me no longer to refuse, on account of the fear which I felt, to make
my work available for the general use of students of astronomy. Ile crazier my
doctrine of the earth's motion now appeared to most people, the argument ran,
so much the more admiration and thanks would it gain after they saw the
publication of my writings dispel the fog of absurdity by most luminous proofs.
Influenced therefore by these persuasive men and by this hope, in the end I
allowed my friends to bring out an edition of the volume, as they had long
besought me to do.
However, Your Holiness will perhaps not be greatly surprised that I have dared to
publish my studies after devoting so much effort to working them out that I did
not hesitate to put down my thoughts about the earth's motion in written fcrm
too. But you are rather waiting to hear from me how it occurred to me to venture
to conceive any motion of the earth, against the traditional opinion of
astronomers and almost against common sense. I have accordingly no desire to from
Your Holiness that I was impelled to consider a different system of deducing the
motions of the universe's spheres for no other reason than the realization that
astronomers do not agree among themselves in their investigations of this
subject. For, in the first place, they are so uncertain about the motion of the
sun and moon that they cannot establish and observe a constant length even for
the tropical year. Secondly, in determining the motions not only of these bodies
but also of the other five planets, they do not use the same principles,
assumptions, and explanations of the apparent revolutions and motions. For while
some employ only homocentrics, others utilize eccentrics and epicycles, and yet
they do not quite reach their goal. For although those who put their faith in
homocentrics showed that some nonuniform motions could be compounded in this way,
nevertheless by this means they were unable to obtain any incontrovertible result
in absolute agreement with the phenomena. On the other hand, those who devised
the eccentrics seem thereby in large measure to have solved the problem of the
apparent motions with appropriate calculations. But meanwhile they introduced a
good many ideas which apparently contradict the first principles of uniform
motion. Nor could they elicit or deduce from the eccentrics the principal
consideration, that is, the structure of the universe and the true symmetry of
its parts. On the contrary, their experience was just like some one taking from
various places hands, feet, a head, and other pieces, very well depicted, it may
be, but not for the representation of a single person; since these fragments
would not belong to one another at all, a monster rather than a man would be put
together from them. Hence in the process of demonstration or "method", as it is
called, those who employed eccentrics are found either to have omitted something
essential or to have admitted something extraneous and wholly irrelevant. This
would not have happened to them, had they followed sound principles. For if the
hypotheses assumed by them were not false, everything which follows from their
hypotheses would be confirmed beyond any doubt. Even though what I am now saying
may be obscure, it will nevertheless become clearer in the proper place.
For a long time, then, I reflected on this confusion in the astronomical
traditions concerning the derivation of the motions of the universe's spheres. I
began to be annoyed that the movements of the world machine, created for our sake
by the best and most systematic Artisan of all, were not understood with greater
certainty by the philosophers, who otherwise examined so precisely the most
insignificant trifles of this world. For this reason I undertook the task of
rereading the works of all the philosophers which I could obtain to learn whether
anyone had ever proposed other motions of the universe's spheres than those
expounded by the teachers of astronomy in the schools. And in fact first I found
in Cicero that Hicetas supposed the earth to move. Later I also discovered in
Plutarch that certain others were of this opinion. I have decided to set his
words down here, so that they may be available to everybody:
Some think that the earth remains at rest. But Philolaus the Pythagorean believes
that, like the sun and moon, it revolves around the fire in an oblique circle.
Heraclides of Pontus, and Ecphantus the Pythagorean make the earth move, not in a
progressive motion, but like a wheel in a rotation from west to east about its
Therefore, having obtained the opportunity from these sources, I too began to
consider the mobility of the earth. And even though the idea seemed absurd,
nevertheless I knew that others before me had been granted the freedom to imagine any
circles whatever for the purpose of explaining the heavenly phenomena. Hence I
thought that I too would be readily permitted to ascertain whether explanations
sounder than those of my predecessors could be found for the revolution of the
celestial spheres on the assumption of some motion of the earth.
Having thus assumed the motions which I ascribe to the earth later on in the
volume, by long and intense study I finally found that if the motions of the
other planets are correlated with the orbiting of the earth, and are computed for
the revolution of each planet, not only do their phenomena follow therefrom but
also the order and size of all the planets and spheres, and heaven itself is so
linked together that in no portion of it can anything be shifted without
disrupting the remaining parts and the universe as a whole. Accordingly in the
arrangement of the volume too I have adopted the following order. In the first
book I set forth the entire distribution of the spheres together with the motions
which I attribute to the earth, so that this book contains, as it were, the
general structure of the universe. Then in the remaining books I correlate the
motions of the other planets and of all the spheres with the movement of the
earth so that I may thereby determine to what extent the motions and appearances
of the other planets and spheres can be saved if they are correlated with the
earth's motions. I have no doubt that acute and learned astronomers will agree
with me if, as this discipline especially requires, they are willing to examine
and consider, not superficially but thoroughly, what I adduce in this volume in
proof of these matters. However, in order that the educated and uneducated alike
may see that I do not run away from the judgement of anybody at all, I have
preferred dedicating my studies to Your Holiness rather than to anyone else. For
even in this very remote comer of the earth where I live you are considered the
highest authority by virtue of the loftiness of your office and your love for all
literature and astronomy too. Hence by your prestige and judgement you can easily
suppress calumnious attacks although, as the proverb has it, there is no remedy
for a backbite.
Perhaps there will be babblers who claim to be judges of astronomy although
completely ignorant of the subject and, badly distorting some passage of
Scripture to their purpose, will dare to find fault with my undertaking and
censure it. I disregard them even to the extent of despising their criticism as
unfounded. For it is not unknown that Lactantius, otherwise an illustrious writer
but hardly an astronomer, speaks quite childishly about the earth's shape, when
he mocks those who declared that the earth has the form of a globe. Hence
scholars need not be surprised if any such persons will likewise ridicule me.
Astronomy is written for astronomers. To them my work too will seem, unless I am
mistaken, to make some contribution also to the Church, at the head of which Your
Holiness now stands. For not so long ago under Leo X the Lateran Council
the problem of reforming the ecclesiastical calendar. The issue remained
undecided then only because the lengths of the year and month and the motions of
the sun and moon were regarded as not yet adequately measured. From that time on,
at the suggestion of that most distinguished man, Paul, bishop of Fossombrone,
who was then in charge of this matter, I have directed my attention to a more
precise study of these topics. But what I have accomplished in this regard, I
leave to the judgement of Your Holiness in particular and of all other learned
astronomers. And lest I appear to Your Holiness to promise more about the
usefulness of this volume than I can fulfill, I now turn to the work itself.
REVOLUTIONS Book One
Among the many various literary and artistic pursuits which invigorate men's
minds, the strongest affection and utmost zeal should, I think, promote the
studies concerned with the most beautiful objects, most deserving to be known.
This is the nature of the discipline which deals with the universe's divine
revolutions, the asters' motions, sizes, distances, risings and settings, as well
as the causes of the other phenomena in the sky, and which, in short, explains
its whole appearance. What indeed is more beautiful than heaven, which of course
contains all things of beauty? This is proclaimed by its very names [in Latin],
caelum and mundus, the latter denoting purity and ornament, the former a carving.
On account of heaven's transcendent perfection most philosophers have called it a
visible god. If then the value of the arts is judged by the subject matter which
they treat, that art will be by far the foremost which is labeled astronomy by
some, astrology by others, but by many of the ancients, the consummation of
mathematics. Unquestionably the summit of the liberal arts and most worthy of a
free man, it is supported by almost all the branches of mathematics. Arithmetic,
geometry, optics, surveying, mechanics and whatever others there are all
contribute to it.
Although all the good arts serve to draw man's mind away from vices and lead it
toward better things, this function can be more fully performed by this art,
which also provides extraordinary intellectual pleasure. For when a man is
occupied with things which he sees established in the finest order and directed
by divine mana gement, will not the unremitting contemplation of them and a
certain familiarity with them stimulate him to the best and to admiration for the
Maker of everything, in whom are all happiness and every good? For would not the
godly Psalmist [92:4] in vain declare that he was made glad through the work of
the Lord and rejoiced in the works of His hand s, were we not drawn to the
contemplation of the highest good by this means, as though by a chariot?
The great benefit and adornment which this art confers on the commonwealth (not
to mention the countless advantages to individuals) are most excellently observed
by Plato. In the Laws, Book VII, he thinks that it should be cultivated chiefly
because by dividing time into groups of days as months and years, it would keep
the state alert and attentive to the festivals and sacrifices. Whoever denies its
necessity for the teacher of any branch of higher learning is thinking foolishly,
according to Plato. In his opinion it is highly unlikely that anyone lacking the
requisite knowledge of the sun, moon, and other heavenly bodies can become and be
However, this divine rather than human science, which investigates the loftiest
subjects, is not free from perplexities. The main reason is that its principles
and assumptions, called "hypotheses" by the Greeks, have been a source of
disagreement, as we see, among most of those who undertook to deal with this
subject, and so they did not rely on the same ideas. An additional reason is that
of the planets and the revolution of the stars could not be measured
with numerical precision and completely understood except with the
passage of time and the aid of many earlier observations, through which
this knowledge was transmitted to posterity from hand to hand, so to
say. To be sure, Claudius Ptolemy of Alexandria, who far excels the rest
by his wonderful skill and industry, brought this entire art almost to
perfection with the help of observations extending over a period of more
than four hundred years, so that there no longer seemed to be any gap
which he had not closed. Nevertheless very many things, as we perceive,
do not agree with the conclusions which ought to follow from his system,
and besides certain other motions have been discovered which were not
yet 2 known to him. Hence Plutarch too, in discussing the sun's tropical
year, says that so far the motion of the heavenly bodies has eluded the
skill of the astronomers. For, to use the year itself as an example, it
is well known, I think, how different the opinions concerning it have
always been, so that many have abandoned all hope that an exact
determination of it could be found. The situation is the 1 same with
regard to other heavenly bodies.
Nevertheless, to avoid giving the impression that this difficulty is an
excuse for indolence, by the grace of God, without whom we can
accomplish nothing, I shall attempt a broader inquiry into these
matters. For, the number of aids we have to assist our enterprise grows
with the interval of time extending from 2 the originators of this art
to us. Their discoveries may be compared with what I have newly found. I
acknowledge, moreover, that I shall treat many topics differently from
my predecessors, and yet I shall do so thanks to them, for it was they
who first opened the road to the investigation of these very questions.
THE UNIVERSE IS SPHERICAL Chapter 1
First of all, we must note that the universe is spherical. The reason is
either that, of all forms, the sphere is the most perfect, needing no
joint and being a complete whole, which can be neither increased nor
diminished; or that it is the most capacious of figures, best suited
to enclose and retain all things; or even that all the separate parts of
the universe, I mean the sun, moon, planets and stars, are seen to be of
this shape; or that wholes strive to be circumscribed by this boundary,
as is apparent in drops of water and other fluid bodies when they seek
to be self-contained. Hence no one will question the attribution of this
form to the divine bodies.
THE EARTH TOO IS SPHERICAL Chapter 2
The earth also is spherical, since it presses upon its center from every
direction. Yet it is not immediately recognized as a perfect sphere on
account of the great height of the mountains and depth of the valleys.
They scarcely alter the general sphericity of the earth, however, as is
clear from the following considerations. For a traveler going from any
place toward the north, that pole of the daily rotation gradually climbs
higher, while the opposite pole drops down an equal amount. More stars
in the north are seen not to set, while in the south certain stars are
no longer seen to rise. Thus Italy does not see Canopus, which is
visible in Egypt; and Italy does see the River's last star, which is
unfamiliar to our area in the colder
region. Such stars, conversely, move higher in the heavens for a traveller
heading southward, while those which are high in our sky sink down. Meanwhile,
moreover, the elevations of the poles have the same ratio everywhere to the
portions of the earth that have been traversed. This happens on no other figure
than the sphere. Hence the earth too is evidently enclosed between poles and is
therefore spherical. Furthermore, evening eclipses of the sun and moon are not
seen by easterners, nor morning eclipses by westerners, while those occurring in
between are seen later by easterners but earlier by westerners.
The waters press down into the same figure also, as sailors are aware, since land
which is not seen from a ship is visible from the top of its mast. On the other
hand, if a light is attached to the top of the mast, as the ship draws away from
land, those who remain ashore see the light drop down gradually until it finally
disappears, as though setting. Water, furthermore, being fluid by nature,
manifestly always seeks the same lower levels as earth and pusehs up from the
shore no higher than its rise permits. Hence whatever land emerges out of the
ocean is admittedly that much higher.
HOW EARTH FORMS A SINGLE SPHERE Chapter 3
Pouring forth its seas everywhere, then, the ocean envelops the earth and fills
its deeper chasms. Both tend toward the same center because of their heaviness.
Accordingly there had to be less water than land, to avoid having the water
engulf the entire earth and to have the water recede from some portions of the
land and from the many islands lying here and there, for the preservation of
living creatures For what are the inhabited countries and the mainland itself
but an island larger than, the others?
We should not heed certain peripatetics who declared that the entire body of
water is ten times greater than all the land. For, according to the conjecture
which they accepted, in the transmutation of the elements as one unit of earth
dissolves, it becomes ten units of water. They also assert that the earth bulges
out to some extent as it does because it is not of equal weight everywhere on
account of its cavities, its center of gravity being different from its center of
magnitude. But they err through ignorance of the art of geometry. For they do not
realize that the water cannot be even seven times greater and still leave any
part of the land dry, unless earth as a whole vacated the center of gravity and
yielded that position to water, as if the latter were heavier than itself For,
spheres are to each other as the cubes of their diameters. Therefore, if earth
were the eighth part to seven parts of water, earth's diameter could not be
greater than the distance from [their joint] center to the circumference of the
waters. So far are they from being as much as ten times greater [than the land].
Moreover, there is no difference between the earth's centers of gravity and
magnitude. This can be established by the fact that from the ocean inward the
curvature of the land does not mount steadily in a continuous rise. If it did, it
would keep the sea water out completely and in no way permit the inland seas and
such vast gulfs to intrude. Furthermore, the depth of the abyss would never stop
increasing from the shore of the ocean outward, so that no island or reef or any
form of land would be encountered by sailors on the longer voyages. But
it is well known that almost in the middle of the inhabited
lands barely fifteen furlongs remain between the eastern Mediterranean
and the Red Sea. On the other hand, in his Geography Ptolemy extended
the habitable area halfway around the world. Beyond that meridian, where
he left unknown land, the modem have added Cathay and territory as vast
as sixty degrees of longitude, so that now the earth is inhabited over a
greater stretch of longitude than is left for the ocean. To these
regions, moreover, should be added the islands discovered in our time
under the rulers of Spain and Portugal, and especially America, named
after the ship's captain who found it. On account of its still
undisclosed size it is thought to be a second group of inhabited
countries. There are also many other islands, heretofore unknown. So
little reason have we to marvel at the existence of antipodes or
antichthones. Indeed, geometrical reasoning about the location of
America compels us to believe that it is diametrically opposite the
Ganges district of India.
From all these facts, finally, I think it is clear that land and water
together press upon a single center of gravity; that the earth has no
other center of magnitude; i that, since earth is heavier, its gaps are
filled with water; and that consequently there is little water in
comparison with land, even though more water perhaps appears on the
The earth together with its surrounding waters must in fact have such a
shape as its shadow reveals, for it eclipses the moon with the arc of a
perfect circle. Therefore the earth is not flat, as Empedocles and
Anaximenes thought; nor drum-shaped, as Leucippus; nor bowl-shaped, as
Heraclitus; nor hollow in another way, as Democritus; nor again
cylindrical, as Ansaximander; nor does its lower side extend infinitely
downward, the thickness diminishing toward the bottom, as Xenophanes
taught; but it is perfectly round, as the philosophers hold.
THE MOTION OF THE BEAVENLY BODIES Chapter 4
IS UNIFORM, ETERNAL, AND CIRCULAR OR
COMPOUNDED OF CIRCULAR MOTIONS
I shall now recall to mind that the motion of the heavenly bodies is
circular, since the motion appropriate to a sphere is rotation in a
circle. By this very act the sphere expresses its form as the simplest
body, wherein neither beginning nor end can be found, nor can the one be
distinguished from the other, while the sphere itself traverses the same
points to return upon itself.
In connection with the numerous [celestial] spheres, however, there are
many Motions. The most conspicuous of all is the daily rotation, which
the Greeks call nuchthemeron, that is, the hiterval of a day and a
night. The entire universe, with the exception of the earth, is
conceived as whirling from cast to west in this rotation. It is
recognized as the common measure of all motions, since we even compute
time itself chiefly by the number of days.
Secondly, we see other revolutions as advancing in the opposite
direction, that is, from west to east; I refer to those of the sun,
moon, and five planets. The sun thus regulates the year for us, and the
moon the month, which are also very familiar Periods of time. In like
manner each of the other five planets completes its own orbit.
Yet [these motions] differ in many ways [from the daily rotation or
first motion]. In the first place, they do not swing around the same
poles as the first
motion, but run obliquely through the zodiac. Secondly, these bodies are not seen
moving uniformly in their orbits, since the sun and moon are observed to be
sometimes slow, at other times faster in their course. Moreover, we see the other
five planets also retrograde at times, and stationary at either end [of the
regression]. And whereas the sun always advances along its own direct path, they
wander in various ways, straying sometimes to the south and sometimes to the
north; that is why they are called "planets" [wanderers]. Furthermore, they are
at times nearer to the earth, when they are said to be in perigee; at other times
they are farther away, when they are said to be in apogee.
We must acknowledge, nevertheless, that their motions are circular or compounded
of several circles, because these nonuniformities recur regularly according to a
constant law. This could not happen unless the motions were circular, since only
the circle can bring back the past. Thus, for example, by a composite motion of
circles the sun restores to us the inequality of days and nights as well as the
is four seasons of the year. Several motions are discerned herein, because a
simple heavenly body cannot be moved by a single sphere nonuniformly. For this
nonuniformity would have to be caused either by an inconstancy, whether imposed
from without or generated from within, in the moving force or by an alteration in
the revolving body. From either alternative, however, the intellect shrinks. It
is improper to conceive any such defect in objects constituted in the best order.
It stands to reason, therefore, that their uniform motions appear nonuniform to
us. The cause may be either that their circles have poles different [from the
earth's] or that the earth is not at the center of the circles on which they
revolve. To us who watch the course of these planets from the earth, it
happens that our eye does not keep the same distance from every part of their
orbits, but on account of their varying distances these bodies seem larger when
nearer than when farther away (as has been proved in optics). Likewise, in equal
arcs of their orbits their motions will appear unequal in equal times on account
of the observer's varying distance. Hence I deem it above all necessary that we
should carefully scrutinize the relation of the earth to the heavens lest, in our
desire to examine the loftiest objects, we remain ignorant of things nearest to
us, and by the same error attribute to the celestial bodies what belongs to the
DOES CIRCULAR MOTION SUIT THE EARTH? Chapter 5
WHAT IS ITS POSITION?
Now that the earth too has been shown to have the form of a sphere, we must in my
opinion see whether also in this case the form entails the motion, and what place
in the universe is occupied by the earth. Without the answers to these questions
it is impossible to find the correct explanation of what is seen in the heavens.
To be sure, there is general agreement among the authorities that the earth is at
rest in the middle of the universe. They hold the contrary view to be
inconceivable or downright silly. Nevertheless, if we examine the matter more
carefully, we shall see that this problem has not yet been solved, and is
therefore by no means to be disregarded.
Every observed change of place is caused by a motion of either the observed
object or the observer or, of course, by an unequal displacement of each. For
when things move with equal speed in the same direction, the motion is
not perceived, as between the observed object and the observer, I mean
It is the earth, however, from which the celestial ballet is beheld in
its repeated performances before our eyes. Therefore, if any motion is
ascribed to the earth, in all things outside it the same motion will
appear, but in the opposite direction, as though they were moving past
it. Such in particular is the daily rotation, since it seems to involve
the entire universe except the earth and what is around it. However, if
you grant that the heavens have no part in this motion but that the
earth rotates from west to east, upon earnest consideration you will
find that this is the actual situation concerning the apparent rising
and setting of the sun, moon, stars and planets. Moreover since the
heavens, which enclose and provide the setting for everything,
constitute the space common to all things, it is not at first blush
clear why motion should not be attributed rather to the enclosed than to
the enclosing, to the thing located in space rather than to the
framework of space. This opinion was indeed maintained by Heraclides and
Ecphantus, the Pythagoreans, and by Eficetas of Syracuse, according to
Cicero. They rotated the earth in the middle of the universe, for they
ascribed the setting of the stars to the earth's interposition, and
their rising to its withdrawal.
If we assume its daily rotation, another and no less important question
follows concerning the earth's position. To be sure, heretofore there
has been virtually unanimous acceptance of the belief that the middle of
the universe is the earth. Anyone who denies that the earth occupies the
middle or center of the universe may nevertheless assert that its
distance (therefrom] is insignificant in comparison with [the distance
of] the sphere of the fixed stars, but perceptible and noteworthy in
relation to the spheres of the sun and the other planets. He may deem
this to be the reason why their motions appear nonuniform, as conforming
to a center other than the center of the earth. Perhaps he can [thereby]
produce a not inept explanation of the apparent nonuniform motion. For
the fact that the same planets are observed nearer to the earth and
farther away necessarily proves that the center of the earth is not the
center of their circles. It is less clear whether the approach and
withdrawal are executed by the earth or the planets.
It Will occasion no surprise if, in addition to the daily rotation, some
other motion is assigned to the earth. That the earth rotates, that it
also travels with several motions, and that it is one of the heavenly
bodies are said to have been the opnions of Philolaus the Pythagorean.
He was no ordinary astronomer, inasmuch as Plato did not delay going to
Italy for the sake of visiting him, as Plato's biographers report.
But many have thought it possible to prove by geometrical reasoning that
the earth is in. the middle of the universe; that being like a point in
relation to the immense heavens, it serves as their center; and that it
is motionless because, when the universe moves, the center remains
unmoved, and the things newest to the center are carried most slowly.
THE IMMENSITY OF THE HEAVENS Chapter 6
COMPARED TO THE SIZE OF THE EARTH
The massive bulk of the earth does indeed shrink to insignificance in comparison
with the size of the heavens. This can be ascertained from the fact that the
boundary circles (for that is the translation of the Greek term horizons) bisect
the entire sphere of the heavens. This could not happen if the earth's size or
distance from the universe's center were noteworthy in comparison with the
heavens. For, a circle that bisects a sphere passes through its center, and is
the greatest circle that can be described on it.
Thus, let circle ABCD be a horizon, and let the earth, from which we do our
observing, be E, the center of the horizon, which separates what is seen from
what is not seen. Now, through a dioptra or horoscopic instrument or water level
placed at E, let the first point of the Crab be sighted rising at point C, and at
that instant the first point of the Goat is perceived to be setting at A. Then A,
E, and C are on a straight line through the dioptra. This line is evidently a
diameter of the ecliptic, since six visible signs form a semicircle, and E, the
[line's] center, is identical with the horizon's center. Again, let the signs
shift their position until the first point of the Goat rises at B. At that time
the Crab win also be observed setting at D. BED will be a straight line and a
diameter of the ecliptic. But, as we have already seen, ABC also is a diameter of
the same circle. Its center, obviously, is the intersection [of the diameters]. A
horizon, then, in this way always bisects the ecliptic, which is a great circle
of the sphere. But on a sphere, if a circle bisects any great circle, the
bisecting circle is itself a great circle. Consequently a horizon is one of the
great circles, and its center is clearly identical with the center of the
Yet a line drawn from the earth's surface [to a point in the firmament] must be
distinct from the line drawn from the earth's center [to the same point].
Nevertheless, because these lines are immense in relation to the earth, they
become like parallel lines [III, 15]. Because their terminus is enormously remote
they appear to be a single line. For in comparison with their length the space
enclosed by them becomes imperceptible, as is demonstrated in optics. This
reasoning certainly makes it quite clear that the heavens are immense by
comparison with the earth and present the aspect of an infinite magnitude, while
on the testimony of the senses the earth is related to the heavens as a point to
a body, and a finite to an infinite magnitude.
But no other conclusion seems to have been established. For it does not follow
that the earth must be at rest in the middle of the universe. Indeed, a rotation
in twenty-four hours of the enormously vast universe should astonish us even more
than a rotation of its least part, which is the earth. For, the argument a that
the center is motionless, and what is nearest the center moves the least, does
not prove that the earth is at rest in the middle of the universe.
To take a similar case, suppose you say that the heavens rotate but the poles are
stationary, and what is closest to the poles-moves the least. The Little Bear,
for example, being very close to the pole, is observed to move much more slowly
than the Eagle or the Little Dog because it describes a smaller circle. Yet all
these constellations belong to a single sphere. A sphere's movement, vanishing at
its axis, does not permit an equal motion of all its parts. Nevertheless these
round in equal times, though not over equal spaces, by the rotation of
the whole sphere. The upshot of the argument, then, is the claim that
the earth as a part of the celestial sphere shares in the same nature
and movement so that, being close to the center, it has a slight motion.
Therefore, being a body and not the center, it too will describe arcs
like those of a celestial circle, though smaller, in the same time. The
falsity of this contention is clearer than daylight. For it would always
have to be noon in one place, and always midnight in another, so that
the daily risings and settings could not take place, since the motion of
the whole and the part would be one and inseparable.
But things separated by the diversity of their situations are subject to
a very different relation: those enclosed in a smaller orbit revolve
faster than those traversing a bigger circle. Thus Saturn, the highest
of the planets, revolves in thirty years; the moon, undoubtedly the
nearest to the earth, completes its course in a month; and to close the
series, it will be thought, the earth rotates in the period of a day and
a night. Accordingly the same question about the daily rotation emerges
again. On the other hand, likewise still undetermined is the earth's
position, which has been made even less certain by what was said above.
For that proof establishes no conclusion other than the heavens'
unlimited size in relation to the earth. Yet how far this immensity
extends is not at all clear. At the opposite extreme are the very tiny
indivisible bodies called "atoms". Being imperceptible, they do not
immediately constitute a visible body when they are taken two or a few
at a time. But they can be multiplied to such an extent that in the end
there are enough of them to combine in a perceptible magnitude. The same
may be said also about the position of the earth. Although it is not in
the center of the universe, nevertheless its distance therefrom is still
insignificant, especially in relation to the sphere of the fixed stars.
WHY THE ANCIENTS THOUGHT THAT Chapter 7
THE EARTH REMAINED AT REST IN
THE MIDDLE OF THE UNIVERSE AS ITS CENTER
Accordingly, the ancient philosophers sought to establish that the earth
remains at rest in the middle of the universe by certain other
arguments. As their main reason, however, they adduce heaviness and
lightness. Earth is in fact the heaviest element, and everything that
has weight is borne toward it in an effort to reach its inmost center.
The earth being spherical, by their own nature heavy objects are carried
to it from all directions at right angles to its surface. Hence, if they
were not checked at its surface, they would collide at its center, since
a straight line perpendicular to a horizontal plane at its point of
tangency with a sphere leads to the [sphere's] center. But things
brought to the middle, it seem to follow, come to rest at the middle.
All the more, then, will the entire earth be at rest in the middle, and
as the recipient of every falling body it will remain motionless thanks
to its weight.
In like manner, the ancient philosophers analyze motion and its nature
in a further attempt to confirm their conclusion. Thus, according to
Aristotle, the motion of a single simple body is simple; of the simple
motions, one is straight and the other is circular; of the straight
motions, one is upward and the other is
downward. Hence every simple motion is either toward the middle, that is,
downward; or away from the middle, that is, upward; or around the middle, that
is, circular. To be carried downward, that is, to seek the middle, is a property
only of earth and water, which are considered heavy; on the other hand, air and
fire, which are endowed with lightness, move upward and away from the middle. To
these four elements it seems reasonable to assign rectilinear motion, but to the
heavenly bodies, circular motion around the middle. This is what Aristotle says
[Heavens, I, 2; II, 14].
Therefore, remarks Ptolemy of Alexandria [Syntaxis, 1, 7], if the earth were to
move, merely in a daily rotation, the opposite of what was said above would have
to occur, since a motion would have to be exceedingly violent and its speed
unsurpassablc to carry the entire circumference of the earth around in
twenty-four hours. But things which undergo an abrupt rotation seem utterly
unsuited to gather [bodies to themselves], and seem more likely, if they have
been produced by combination, to fly apart unless they are held together by some
bond. The earth would long ago have burst asunder, he says, and dropped out of
the skies (a quite preposterous notion); and, what is more, living creatures and
any other loose weights would by no means remain unshaken. Nor would objects
falling in a straight line descend perpendicularly to their appointed place,
which would meantime have been withdrawn by so rapid a movement. Moreover, clouds
and anything else floating in the air would be seen drifting always westward.
THE INADEQUACY OF THE PREVIOUS Chapter 8
ARGUMENTS AND A REFUTATION OF TIMM
For these and similar reasons forsooth the ancients insist that the earth remains
at rest in the middle of the universe, and that this is its status beyond any
doubt. Yet if anyone believes that the earth rotates, surely he will hold that
its motion is natural, not violent. But what is in accordance with nature
produces effects contrary to those resulting from violence, since things to which
force or violence is applied must disintegrate and cannot long endure. On the
other hand, that which is brought into existence by nature is well-ordered and
preserved in its best state. Ptolemy has no cause, then, to fear that the earth
and everything earthly will be disrupted by a rotation created through natures
handiwork, which is quite different from what art or human intelligence can
But why does he not feel this apprehension even more for the universe, whose
motion must be the swifter, the bigger the heavens are than the earth? Or have
the heavens become immense because the indescribable violence of their motion
drives them away from the center? Would they also fall apart if they came to a
halt? Were this reasoning sound, surely the size of the heavens would likewise
grow to infinity. For the higher they are driven by the power of their motion,
the faster that motion will be, since the circumference of which it must make the
circuit in the period of twenty-four hours is constantly expanding; and, in turn,
as the velocity of the motion mounts, the vastness of the heavens is enlarged. In
this way the speed will increase the size, and the size the speed, to infinity.
Yet according to the fimiliar ear axiom of physics that the infinite cannot be
traversed 5 or moved in any way, the heavens will therefore necessarily remain
But beyond the heavens there is said to be no body, no space, no void, abso-
lutely nothing, so that there is nowhere the heavens can go. In that
case it is really astonishing if something can be held in check by
nothing. If the heavens are infinite, however, and finite at their inner
concavity only, there will perhaps be more reason to believe that beyond
the heavens there is nothing. For, every single thing, no matter what
size it attains, will be inside them, but the heavens will abide
motionless. For, the chief contention by which it is sought to prove
that the universe is finite is its motion. Let us therefore leave the
question whether the universe is finite or infinite to be discussed by
the natural philosophers.
We regard it as a certainty that the earth, enclosed between poles, is
bounded by a spherical surface. Why then do we still hesitate to grant
it the motion appropriate by nature to its form rather than attribute a
movement to the entire universe, whose limit is unknown and unknowable?
Why should we not admit, with regard to the daily rotation, that the
appearance is in the heavens and the reality in the earth? This
situation closely resembles what Vergil's Aeneas says:
Forth from the harbor we sail, and the land and the cities slip backward
[Aeneid, III, 72].
For when a ship is floating calmly along, the sailors see its motion
mirrored in everything outside, while on the other hand they suppose
that they are stationary, together with everything on board. In the same
way, the motion of the earth can unquestionably produce the impression
that The entire universe is rotating.
Then what about the clouds and the other things that hang in the air in
any manner whatsoever, or the bodies that fall down, and conversely
those that rise aloft? We would only say that not merely the earth and
the watery element joined with it have this motion, but also no small
part of the air and whatever is Raked in the same way to the earth. Ile
reason may be either that the nearby air, mingling with earthy or watery
matter, conforms to the same nature as the earth, or that the air's
motion, acquired from the earth by proximity, shares without resistance
in its unceasing rotation. No less astonishingly, on the other hand, is
the celestial movement declared to be accompanied by the uppermost belt
of air. This is indicated by those bodies that appear suddenly, I mean,
those that the Greeks called "comets" and "bearded stars". Like the
other heavenly bodies, they rise and set. They are thought to be
generated in that region. That part of the air, we can maintain, is
unaffected by the earth's motion on account of its great distance from
the earth. The air closest to the earth will accordingly seem to be
still. And so will the things suspended in it, unless they are tossed to
and fro, as indeed they are, by the wind or some other disturbance. For
what else is the wind in the air but the wave in the sea?
We must in fact avow that the motion of falling and rising bodies in the
framework of the universe is twofold, being in every case a compound of
straight and circular. For, things that sink of their own weight, being
predominantly earthy, undoubtedly retain the same nature as the whole of
which they are parts. Nor is the explanation different in the case of
those things, which, being fiery, are driven forcibly upward. For also
fire here on the earth feeds mainly on earthy matter, and flame is
defined as nothing but blazing smoke. Now it is a property of fire to
expand what it enters. It does this with such great force that it cannot
be prevented in any way by any device from bursting through restraints
and completing its work. But the motion of expansion is directed from
the center to
the circumference. Therefore, if any part of the earth is set afire, it is
carried from the middle upwards. Hence the statement that the motion of a simple
body is simple holds true in particular for circular motion, as long as the
simple body abides in its natural place and with its whole. For when it is in
place, it has none but circular motion, which remains wholly within itself like a
body at rest. Rectilinear motion, however, affects things which leave their
natural place or arc thrust out of it or quit it in any manner whatsoever. Yet
nothing is so incompatible with the orderly arrangement of the universe and the
design of the totality as something out of place. Therefore rectilinear motion
occurs only to things that are not in proper condition and are not in complete
accord with their nature, when they are separated from their whole and forsake
Furthermore, bodies that are carried upward and downward, even when deprived of
circular motion, do not execute a simple, constant, and uniform motion. For they
cannot be governed by their lightness or by the impetus of their weight. Whatever
falls moves slowly at first but increases its speed as it drops. On the other
hand, we see this earthly fire (for we behold no other), after it has been lifted
up high, slacken all at once, thereby revealing the reason to be the violence
applied to the earthy matter. Circular motion, however, always rolls along
uniformly, since it has an unfailing cause. But rectilinear motion has a cause
that quickly stops functioning. For when rectilinear motion brings bodies to
their own place, they cease to be heavy or light, and their motion ends. Hence,
since circular motion belongs to wholes, but parts have rectilinear motion in
addition, we can say that "circular" subsists with "rectilinear" as "being alive"
with "being sick". Surely Aristotle's division of simple motion into three types,
away from the middle, toward the middle, and around the middle, will be construed
merely as a logical exercise. In like manner we distinguish line, point, and
surface, even though one cannot exist without another, and none of them without
As a quality, moreover, immobility is deemed nobler and more divine than change
and instability, which are therefore better suited to the earth than to the
universe. Besides, it would seem quite absurd to attribute motion to the
framework of space or that which encloses the whole of space, and not, more
appropriately, to that which is enclosed and occupies some space, namely, the
earth. Last of all, the planets obviously approach closer to the earth and recede
farther from it. Then the motion of a single body around the middle, which is
thought to be the center of the earth, will be both away from the middle and
also toward it. Motion around the middle, consequently, must be interpreted in a
more general way, the sufficient condition being that each such motion encircle
its own center. You see, then, that all these arguments make it more likely that
the earth moves than that it is at rest. This is especially true of the daily
rotation, as particularly appropriate to the earth. This is enough, in my
opinion, about the first part of the question.
CAN SEVERAL MOTIONS BE ATTRIBUTED Chapter 9
TO THE EARTH? THE CENTER OF THE UNIVERSE
Accordingly, since nothing prevents the earth from moving, I suggest that we
should now consider also whether several motions suit it, so that it can be
regarded as one of the planets. For, it is not the center of all the revolutions.
is indicated by the planets, apparent nonuniform motion and their
varying distances from the earth. These phenomena cannot be explained by
circles concentric with the earth. Therefore, since there are many
centers, it will not be by accident that the further question arises
whether the center of the universe is identical with the center of
terrestrial gravity or with some other point. For my part I believe that
gravity is nothing but a certain natural desire, which the divine
providence of the Creator of all things has implanted in parts, to
gather as a unity and a whole by combining in the form of a globe. This
impulse is present, we may suppose, also in the sun, the moon, and the
other brilliant planets, so that through its operation they remain in
that spherical shape which they display. Nevertheless, they swing round
their circuits in divers ways. If, then, the earth too moves in other
ways, for example, about a center, its additional motions must likewise
be reflected in many bodies outside it. Among these motions we find the
yearly revolution. For if this is transformed from a solar to a
terrestrial movement, with the sun acknowledged to be at rest, the
risings and settings which bring the zodiacal signs and fixed stars into
view morning and evening will appear in the same way. The stations of
the planets, moreover, as well as their retrogradations and [resumptions
of] forward motion will be recognized as being, not movements of the
planets, but a motion of the earth, which the planets borrow for their
own appearances. Lastly, it will be realized that the sun occupies the
middle of the universe. All these facts are disclosed to us by the
principle governing the order in which the planets follow one another,
and by the harmony of the entire universe, if only we look at the
matter, as the saying goes, with both eyes.
THE ORDER OF THE HEAVENLY SPHERES Chapter 10
Of all things visible, the highest is the heaven of the fixed stars.
This, I see, is doubted by nobody. But the ancient philosophers wanted
to arrange the planets in accordance with the duration of the
revolutions. Their principle assumes that of objects moving equally
fast, those farther away seem to travel more slowly, as is proved in
Euclid's Optics. The moon revolves in the shortest period of time
because, in their opinion, it runs on the smallest circle as the nearest
to the earth. The highest planet, on the other hand, is Saturn, which
completes the biggest circuit in the longest time. Below it is Jupiter,
followed by Mars.
With regard to Venus and Mercury, however, differences of opinion are
found. For, these planets do not pass through every elongation from the
sun, as the other planets do. Hence Venus and Mercury are located above
the sun by some authorities, like Plato's Timaeus [38 D], but below the
sun by others, like Ptolemy [Syntaxis, IX, 1] and many of the modems.
Al-Bitruji places Venus above the sun, and Mercury below it.
According to Platos followers, all the planets, being dark bodies
otherwise, shine because they receive sunlight. If they were below the
sun, therefore, they would undergo no great elongation from it, and
hence they would be seen halved or at any rate less than fully round.
For, the light which they receive would be reflected mostly upward, that
is, toward the sun, as we see in the new or dying moon. In addition,
they argue, the sun must sometimes be eclipsed by the interposition of
these planets, and its light cut off in proportion to their size. Since
is never observed, these planets do not pass beneath the sun at all, according to
those who follow Plato.
On the other hand, those who locate Venus and Mercury below the sun base their
reasoning on the wide space which they notice between the sun and the moon. For
the moon's greatest distance from the earth is 64 1/6 earth-radii. This is
contained, according to them, about 18 times in the sun's least distance from the
earth, which is 1160 earth-radii. Therefore between the sun and the moon there
are 1096 earth-radii [~ 1160-641/6]. Consequently,
to avoid having so vast a
space remain empty, they announce that the same numbers almost exactly fill up
the apsidal distances, by which they compute the thickness of those spheres. Thus
the moon's apogee is followed by Mercury's perigee. Mercurys apogee is succeeded
by the perigee of Venus, whose apogee, finally, almost reaches the sun's perigee.
For between the apsides of Mercury they calculate about 177
1/2 earthradii. Then
the remaining space is very nearly filled by Venus' interval of 910 earth-radii.
Therefore they do not admit that these heavenly bodies have any opacity like the
moon's. On the contrary, these shine either with their own light or with the
sunlight absorbed throughout their bodies. Moreover, they do not eclipse the sun,
because it rarely happens that they interfere with our view of the sun, since
they generally deviate in latitude. Besides, they are tiny bodies in comparison
with the sun. Venus, although bigger than Mercury, can occult barely a hundredth
of the sun. So says Al-Battani of Raqqa, who thinks that the sun's diameter is
ten times larger [than Venus'], and therefore so minute a speck is not easily
descried in the most brilliant light. Yet in his Paraphrase of Ptolemy, Ibn Rushd
reports having seen something blackish when he found a conjunction of the sun and
Mercury indicated in the tables. And thus these two planets are judged to be
moving below the sun's sphere.
But this reasoning also is weak and unreliable. This is obvious from the fact
that there are 38 earth-radii to the moon's perigee, according to Ptolemy
[Syntaxis, V, 13], but more than 49 according to a more accurate determination, as will be made
clear below. Yet so great a space contains, as we know, nothing but air and, if
you please, also what is called "the element of fire". Moreover, the diameter of
Venus' epicycle which carries it 45° more or less to either side of the sun, must
be six times longer than the line drawn from the earth's center to Venus'
perigee, as will be demonstrated in the proper place [V, 21]. In this entire
space which would be taken up by that huge epicycle of Venus and which, moreover,
is so much bigger than what would accommodate the earth, air, aether, moon, and
Mercury, what will they say is contained if Venus revolved around a motionless
Ptolemy [Syntaxis, IX, 1] argues also that the sun must move in the middle
between the planets which show every elongation from it and those which do not.
This argument carries no conviction because its error is revealed by the fact
that the moon too shows every elongation from the sun.
Now there are those who locate Venus and then Mercury below the sun, or Separate
these planets [from the sun] in some other sequence. What reason will they adduce
to explain why Venus and Mercury do not likewise traverse separate orbits
divergent from the sun, like the other planets, without violating the arrangement
[of the planets] in accordance with their [relative] swiftness and slowness?
Then one of two alternatives will have to be true. Either the earth is
not the center to which the order of the planets and spheres is
referred, or there really is no principle of arrangement nor any
apparent reason why the highest place belongs to Saturn rather than to
Jupiter or any other planet.
In my judgement, therefore, we should not in the least disregard what
was 5 familiar to Martianus Capella, the author of an encyclopedia, and
to certain other Latin writers. For according to them, Venus and Mercury
revolve around the sun as their center. This is the reason, in their
opinion, why these planets diverge no farther from the sun than is
permitted by the curvature of their revolutions. For they do not
encircle the earth, like the other planets, but "have opposite
circles". Then what else do these authors mean but that the center of
their spheres is near the sun? Thus Mercury's sphere will surely be
enclosed within Venus', which by common consent is more than twice as
big, and inside that wide region it will occupy a space adequate for
itself. If anyone seizes this opportunity to link Saturn, Jupiter, and
Mars also to that center, provided he understands their spheres to be
so large that together with Venus and Mercury the earth too is enclosed
inside and encircled, he will not be mistaken, as is shown by the
regular pattern of their motions.
For [these outer planets] are always closest to the earth, as is well
known, about the time of their evening rising, that is, when they are in
opposition to the 2 sun, with the earth between them and the sun. On the
other hand, they are at their farthest from the earth at the time of
their evening setting, when they become invisible in the vicinity of the
sun, namely, when we have the sun between them and the earth. These
facts are enough to show that their center belongs more to the sun, and
is identical with the center around which Venus and Mercury likewise
execute their revolutions.
But since all these planets are related to a single center, the space
between Venus' convex sphere and Mars' concave sphere must be set apart
as also a sphere or spherical shell, both of whose surfaces are
concentric with those spheres. This [intercalated sphere] receives the
earth together with its attendant, 3 the moon, and whatever is contained
within the moon's sphere. Mainly for the reason that in this space we
find quite an appropriate and adequate place for the moon, we can by no
means detach it from the earth, since it is incontrovertibly nearest to
Hence I feel no shame in asserting that this whole region engirdled by
the moon, and the center of the earth, traverse this grand circle amid
the rest of the planets in an annual revolution around the sun. Near the
sun is the center of the universe. Moreover, since the sun remains
stationary, whatever appears as a motion of the sun is really due rather
to the motion of the earth. In comparison with any other spheres of the
planets, the distance from the earth to the sun has a magnitude which is
quite appreciable in proportion to those dimensions. But the size of the
universe is so great that the distance earth-sun is imperceptible in
relation to the sphere of the fixed stars. This should be admitted, I
believe, in preference to perplexing the mind with an almost infinite
multitude of spheres, as must be done by those who kept the earth in the
middle of the universe. On the contrary, we should rather heed the
wisdom of nature. Just as it especially avoids producing anything
superfluous or useless, so it frequently prefers to endow a single thing
with many effects.
All these statements are difficult and almost inconceivable, being of course
opposed to the beliefs of many people. Yet, as we proceed, with God's help I
shall make them clearer than sunlight, at any rate to those who arc not
unacquainted with the science of astronomy. Consequently, with the first
principle remaining intact, for nobody will propound a more suitable principle
than that the size of the spheres is measured by the length of the time, the
order of the spheres is the following, beginning with the highest.
The first and the highest of all is the sphere of the fixed stars, which contains
itself and everything, and is therefore immovable. It is unquestionably the place
of the universe, to which the motion and position of all the other heavenly
bodies are compared. Some people think that it also shifts in some way. A
different explanation of why this appears to be so will be adduced in my
discussion of the earth's motion [I, 11].
[The sphere of the fixed stars] is followed by the first of the planets, Saturn,
which completes its circuit in 30 years. After Saturn, Jupiter accomplishes its
revolution in 12 years. Then Mars revolves in 2 years. The annual revolution
takes the series' fourth place, which contains the earth, as I said [earlier in
I, 10], together with the lunar sphere as an epicycle. In the fifth place Venus
in 9 months. Lastly, the sixth place is held by Mercury, which revolves
in a period of 80 days.
At rest, however, in the middle of everything is the sun. For in this
most beautiful temple, who would place this lamp in another or better
position than that from which it can light up the whole thing at the
same time? For, the sun is not inappropriately called by some people the
lantern of the universe, its mind by others, and its ruler by still
others. [Hermes] the Thrice Greatest labels it a visible god, and
Sophocles' Electra, the all-seeing. Thus indeed, as though seated on a
royal throne, the sun governs the family of planets revolving around it.
Moreover, the earth is not deprived of the moon's attendance. On the
contrary, as Aristotle says in a work on animal , the moon has the
closest kinship with the earth. Meanwhile the earth has intercourse with
the sun, and is impregnated for its yearly parturition.
In this arrangement, therefore, we discover a marvelous symmetry of the
universe, and an established harmonious linkage between the motion of
the spheres and their size, such as can be found in no other way. For
this permits a not inattentive student to perceive why the forward and
backward arcs appear greater in Jupiter than in Saturn and smaller than
in Mars, and on the other hand greater in Venus than in Mercury. This
reversal in direction appears more frequently in Saturn than in Jupiter,
and also more rarely in Mars and Venus than in Mercury. Moreover, when
Saturn, Jupiter, and Mars rise at sunset, they are nearer to the earth
than when they set in the evening or appear at a later hour. But Mars in
particular, when it shines all night, seems to equal Jupiter in size,
being distinguished only by its reddish color. Yet in the other
configurations it is found barely among the stars of the second
magnitude, being recognized by those who track it with assiduous
observations. All these phenomena proceed from the same cause, which is
in the earth's motion.
Yet none of these phenomena appears in the fixed stars. This proves
their immense height, which makes even the sphere of the annual motion, or its
reflection, vanish from before our eyes. For, every visible object has
some measure of distance beyond which it is no longer seen, as is
demonstrated in optics. From Saturn, the highest of the planets, to the
sphere of the fixed stars there is an additional gap of the largest
size. This is shown by the twinkling lights of the stars. By this token
in particular they are distinguished from the planets, for there had to
be a very great difference between what moves and what does not move. So
vast, without any question, is the divine handiwork of the most
PROOF OF THE EARTH'S TRIPLE MOTION Chapter 11
In so many and such important ways, then, do the planets bear witness to
the earth's mobility. I shall now give a summary of this motion, insofar
as the phenomena are explained by it as a principle. As a whole, it must
be admitted to be a threefold motion.
The first motion, named nuchthemeron by the Greeks, as I said [I, 4], is
the rotation which is the characteristic of a day plus a night. This
turns around the earth's aids from west to east, just as the universe is
deemed to be carried in the opposite direction. It describes the
equator, which some people call the "circle of equal
days", in imitation of the designation used by the Greeks, whose term for it is
The second is the yearly motion of the center, which traces the ecliptic around
the sun. Its direction is likewise from west to east, that is, in the order of
the zodiacal signs. It travels between Venus and Mars, as I mentioned [I, 10],
together with its associates. Because of it, the sun seems to move through the
zodiac in a similar motion. Thus, for example, when the earth's center is passing
through the Goat, the sun appears to be traversing the Crab; with the earth in
the Water Bearer, the sun seems to be in the Lion, and so on, as I remarked.
To this circle, which goes through the middle of the signs, and to its plane, the
equator and the earth's axis must be understood to have a variable inclination.
For if they stayed at a constant angle, and were affected exclusively by the
motion of the center, no inequality of days and nights would be observed. On the
contrary, it would always be either the longest or shortest day or the day of
equal daylight and darkness, or summer or winter, or whatever the character of
the season, it would remain identical and unchanged.
The third motion in inclination is consequently required. This also is a yearly
revolution, but it occurs in the reverse order of the signs, that is, in the
direction opposite to that of the motion of the center. These two motions are
opposite in direction and nearly equal in period. The result is that the earth's
axis and equator, the largest of the parallels of latitude on it, face almost the
same portion of the heavens, just as if they remained motionless. Meanwhile the
sun seems to move through the obliquity of the ecliptic with the motion of the
earth's center, as though this were the center of the universe. Only remember
that, in relation to the sphere of the fixed stars, the distance between the sun
and the earth vanishes from our sight forthwith.
Since these are matters which crave to be set before our eyes rather than spoken
of, let us describe a circle ABCD, which the annual revolution of the earth's
center has traced in the plane of the ecliptic. Near its center let the sun be E.
I shall divide this circle into four parts by drawing the diameters AEC and BED.
Let A represent the first point of the Crab, B of the Balance, C of the Goat, and
D of the Ram. Now let us assume that the earth's center is originally at A. About
A I shall draw the terrestrial equator FGHI. This is not in the same plane [as
the ecliptic], except that the diameter GAI is the intersection of the circles, I
mean, of the equator and the ecliptic. Draw also the diameter FAH perpendicular
to GAI, F being the limit of the [equator's] greatest inclination to the south,
and H to the north. Under the conditions thus set forth, the earth's inhabitants
will see the sun near the center E undergo the winter solstice in the Goat. This
occurs because the greatest northward inclination, H, is turned toward the sun.
For, the inclination of the equator to the line AE, through the agency of the
daily rotation, traces the winter solstice parallel to the equator at an interval
subtended by EAH, the angle of the obliquity.
Now let the earth's center start out in the order of the signs, and let F, the
limit of maximum inclination, travel along an equal arc in the reverse order of
the signs, until at B both have traversed a quadrant of their circles. In the
interim the angle EAI always remain equal to AEB, on account of the equality of
their revolutions; and the diameters always stay parallel to each other, FAH to
FBH, and GAI to GBI, and the equator to the equator. In the immensity of the
for the reason already frequently mentioned, the same phenomena appear.
Terefore from B, the first point of the Balance, E will seem to be in
the Ram. The intersection of the circles will coincide with the single
line GBIE, from which [the plane of the axis] win not be permitted by the
daily rotation to deviate. On the contrary, the [axis'] inclination will
lie entirely in the lateral plane. Accordingly the sun will be seen in
the spring equinox. Let the earth's center proceed under the assumed
conditions, and when it has completed a semicircle at C, the sun will
appear to enter the Crab. But F, the southernmost inclination of the
equator, will be turned toward the sun. This will be made to appear in
the north, undergoing the summer solstice as measured by the angle of
the obliquity, ECR Again, when F turns away in the third quadrant of the
circle, the intersection GI will once more fall on the line ED. From
here the sun will be seen in the Balance undergoing the autumn equinox.
Then as H by the same process gradually faces the sun, it will bring
about a repetition of the initial situation, with which I began my
Alternatively, let AEC be in the same way a diameter of the plane under
discussion [the ecliptic] as well as the intersection of that plane with
a circle perpendicular thereto. On AEC, around A and C, that is, in the
Crab and the Goat, draw a circle of the earth in each case through the
poles. Let this [meridian] be DGFI, the earth's axis DF, the north pole
D, the south pole F, and GI the diameter of the equator. Now when F is
turned toward the sun, which is near E, the equator's northward
inclination being measured by the angle IAE, then the axial rotation
will describe, parallel to the equator and to the south of it, at a
distance LI and with diameter KL, the tropic of Capricorn as seen in the
sun. Or, to speak more accurately, the axial rotation, as viewed from
AE, generates a conic surface, having its vertex in the center of the
earth, and its base in a circle parallel to the equator. Also at the
opposite point, C, everything works out in like manner, but is reversed.
It is clear therefore how the two motions, I mean, the motion of the
the motion in inclination, by their combined effect make the earth's axis remain
in the same direction and in very much the same position, and make all these
phenomena appear as though they were motions of the sun.
I said, however, that the annual revolutions of the center and of inclination are
nearly equal. For if they were exactly equal, the equinoctial and solstitial
points as well as the entire obliquity of the ecliptic would have to show no
shift at all with reference to the sphere of the fixed stars. But since there is
a slight variation, it was discovered only as it grew larger with the passage of
time. From Ptolemy to us the precession of the equinoxes amounts to almost 21°.
For this reason some people believed that the sphere of the fixed stars also
moves, and accordingly they adopted a surmounting ninth sphere. This having
proved inadequate, more recent writers now add on a tenth sphere. Yet they do not
in the least attain their goal, which I hope to reach by the earth's motion. This
I shall use as a principle and hypothesis in the demonstration of the other
[Here Copernicus originally planned to include a little time than two handwritten
pages which he later deleted from his autograph. This deleted material, which was
not printed in the first four editions of the Revolutions (1543, 1566, 1617,
1854), but was incorporated in those published after the recovery of Copernicus'
autograph (1873, 1949, 1972), reads as follows].
The motion of the sun and moon can be demonstrated, I admit, also with an earth
that is stationary. This is, however, lea suitable for the remaining planets .
Philolaus believed in the earth's motion for these and similar reasons. This is
plausible because Aristarchus of Samos too held the same view according to some
people, who were not motivated by the argumentation put forward by Aristotle and
rejected by him [Heavens, II, 13-14). But only a keen mind and persevering study
could understand then subjects. They were therefore unfamiliar to most
philosophers at that time, and Plato does not conceal the fact that there were
then only a few who mastered the theory of the heavenly motions. Even if these
were known to Philolaus or any Pythagorean, they nevertheless were probably not
transmitted to posterity. For it was the Pythagoreans' practice not to commit the
secrets of philosophy to writing nor divulge them to everybody, but to entrust
them only to faithful friends and kinsmen, and pass them on from hand to hand As
evidence of this custom there is extant a letter from Lysis to Hipparchus.
Because of its remarkable opinions and in order to make clear what value was
attached to philosophy among themselves, I have decided to insert it here and to
end this first Book with it. This, then, is a copy of the letter, which I
translate from Greek as follows.
From Lysis to Hipparchus, greetings.
I would never have believed that after Pythagoras' death his followers'
brotherhood would be dissolved. But now that we have unexpectedly been scattered
hither and yon, as if our ship had been wrecked, it is still an act of piety to
recall his godlike teachings and refrain from communicating the treasures of
Philosophy to those who have not even dreamed about the purification of the soul.
For it is indecent to divulge to everybody what we achieved with such great
effort, just as the Eleusinian goddesses' secrets may not be revealed to the
uninitiated. The perpetrators of either of these misdeeds would be condemned as
equally wicked and impious. On the other hand, it is worth considering how much
tune we spent wiping out the stains which clung to our hearts until we became
receptive to his teachings after the course of five years. Dyers, having cleaned
their fabrics, then apply their tincture with a mordant in order to fix the color
indissolubly and prevent it from fading away casily thereafter. That godlike man
prepared the lovers of philosophy in the same way, to avoid being disappointed in
the hope he had conceived for the talents of any one of
them. He did not sell his precepts for a price, and the snares with
which young minds are entangled by many of the sophists were not set out
by him because they are devoid of value. On the contrary, divine and
human doctrines were promulgated by him.
Certain imitators of his teaching, however, perform at great length and
out loud. Their instruction of the young follows a confused and improper
procedure, thereby making their auditors impertinent and brash. For they
mix disorderly and tainted morals with philosophy's lofty precepts. The
result is like pouring pure fresh water into a deep well full of muck,
since the muck is stirred up and the water is wasted. This is what
happens to those who teach and are taught in this manner. For thick,
dark woods obstruct the minds and hearts of those who were not correctly
initiated, and completely damage the gentleness of their spirit and
their reasonableness. These woods are infested with all sorts of vices,
which by flourishing impede thought and prevent it from developing in
As breeders of the interlopers I shall name principally self-indulgence
and greed, both of which are extremely fertile. For, self-indulgence
gives rise to incest, drunkenness, rape, unnatural pleasures, and
certain violent impulses which lead as far as death and destruction. In
fact, passion has inflamed some of these persons to so high a pitch that
they spared neither their mothers nor their daughters. It has even
carried them into conflict with their laws, country, government, and
rulers. It has laid snares such that it brought them bound hand and foot
to the final punishment. Greed, on the other hand, generates mayhem,
murder, temple-robbery, poisoning, and other offspring of that sort. The
lairs in those woods, where these urges lurk, must therefore be
extirpated by fire and sword with all our might. When we have found the
natural reason freed from these lusts, we shall then implant in it a
most excellent and fruitful crop.
You too, Hipparchus, learned these rules with no small zeal. But, my
good man, little did you heed them after you had tasted Sicilian luxury,
for the sake of which you should have abandoned nothing. Many people
even say that you are teaching philosophy publicly. This practice was
forbidden by Pythagoras, who willed his notes to his daughter Damo with
an order not to turn them over to anybody outside the family. Although
she could have sold them for a lot of money, she refused to do so,
considering poverty and her father's commands more precious than gold.
They also say that when Damo, died, she left the same obligation to her
own daughter Bitale. Yet we of the male sex disobey our teacher and
violate our oath. If, then, you mend your ways, I cherish you. But if
you do not, as far as I am concerned, you are dead.
[The foregoing letter, the true nature of which was not suspected by
Copernicus, ended Book I as originally planned. According to that plan,
Book II began immediately after the letter with some introductory
material, which was subsequently deleted. This deleted material, which
was not printed in the first four editions of the Revolutions, but was
included in those published after the recovery of Copernicus' autograph,
reads as follows].
For what I have undertaken to do, those propositions of natural
philosophy which seemed indispensable as principles and hypotheses,
namely, that the universe is spherical, 1 3 and similar to the infinite,
and that the sphere of the fixed stars as the container of everything is
stationary, whereas all the other heavenly bodies have a circular
motion, have been briefly reviewed. I have also assumed that the earth
moves in certain revolutions, on which, as the cornerstone, I strive to
erect the entire science of the stars.
[The rest of the material deleted here in the autograph was printed in
the first four editions of the Revolutions as the following beginning of
The proofs which I shall use in almost the entire work involve straight
lines and arcs in plane and spherical triangles. Although much
information about these topics is already available in Euclid's
Elements, nevertheless that treatise does not contain the answer to what
is the principal question here, how the sides can be obtained from the
angles, and the angles from the sides.
[As the heading of 1, 12, the first edition introduced "The Length of
Straight Lines in a Circle". This caption, for which there is no direct
warrant in the autograph, was repeated in the next three editions of the
HOW MUCH ARE THE ORBITS OF SATURN, Chapter 3
JUPITER, AND MARS INCLINED?
Having explained the theory of the latitudes of the five planets, I must now
turn to the facts and analyze the details. First [I must determine] how much the
individual circles are inclined. We compute these inclinations by means of the
great circle which passes through the poles of the inclined circle at right angles
to the ecliptic. On this great circle the deviations in latitude are determined.
When these arrangements are understood, the road will be open to ascertaining
the latitudes of each planet.
Once more let us begin with the three outer planets. At their farthest southern
limits of latitude, as shown in Ptolemy's Table [Syntaxis, XIII, 5], when they
are in opposition, Saturn deviates 3° 5', Jupiter 2° 7' and Mars 7° 7'. On the
other hand, in the opposite places, that is, when they are in conjunction with
the sun, Saturn deviates 2° 2', Jupiter 1° 5', and Mars only 5', so that it
almost grazes the ecliptic. These values could be inferred from the latitudes
observed by Ptolemy around the time of the planets' disappearances and first
Now that the above assertions have been set forth, let a plane perpendicular to
the ecliptic pass through its center and intersect the ecliptic in AB. But let
its intersection with the eccentric of any of the three outer planets be CD,
passing through the farthest southern and northern limits. Let the ecliptic's
center be E; the diameter of the earth's grand circle, FEG; the southern
latitude, D; and the northern, C. Join CF, CG, DF, and DG.
Now as an example I shall use Mars because it exceeds all the other planets in
latitude. Thus, when it is in opposition at point D, with the earth at G
(corrected from F], angle AFG was known = 7° 7'. But 0 is given as Mars' position
at apogee. From the previously established sizes of the circle,
CE = 1° 22'20",
with FG [a slip for FE] = 1°. In triangle CEF,
the ratio of the sides GE and EF
is given, as well as angle CFE.
Hence we shall also have as given CEF = the
greatest angle of the eccentric's inclination = 5° 11', according to the doctrine
of plane triangles. However, when the earth is in the opposite place, that is, at
G (should have been corrected to F], while the Planet is still at C,
CGF = the angle of the apparent latitude = 4'].
For each planet the ratio of EG, (the radius] of the earth's grand circle, to ED,
[the radius] of the planet's eccentric, has already been shown above for any
given places of earth and planet. But the places of the maximum latitudes are
also given by observation. Therefore BGD, the angle of the greatest southern
latitude, is given as an exterior angle of triangle EGD. In accordance with the
theorems on Plane Triangles, the opposite interior angle GED will also be given
as the angle of the eccentric's maximum southern inclination to the plane of the
ecliptic. By means of the minimum southern latitude we shall likewise demonstrate
the mm inclination, for example, by mean of angle EFD. In triangle EFD, the ratio
of sides EF : ED is given as well as angle EFD. Therefore we shall, have exterior
GED given as the angle of the minimum southern inclination. Accordingly, from the
difference between both inclinations we shall obtain the whole oscillation of the
eccentric in relation to the ecliptic. Furthermore, by means of these angles of
inclination we shall compute the opposite northern latitudes, such as AFC and
EGC. If these agree with the observations, they Will indicate that we have made
However, as an example I shall use Mars, because it exceeds all the other planets
in latitude. Its maximum southern latitude was noted by Ptolemy as about 7° when
Mars was at perigee, and its maximum northern latitude at apogee as 4° 20'
[Syntaxis, XIII, 5]. However, having determined angle BGD = 6° 50',
I found the corresponding angle AFC ~ 4° 30'.
Given EG : ED = 1p : 1p 22' 26" [V, 19],
from these sides and angle BGD we shall obtain angle DEG
of the maximum Southern inclination ~ 1° 51'.
Since EF : CE = 19 : 1° 39' 57" [V, 19] and angle CEF =
DEG = 1° 51', consequently the aforementioned exterior
CFA = 4 1/2° when the planet is in opposition.
Similarly , at the opposite place when it is in conjunction with the sun, suppose
that we assume angle DFE = 5'. From the given sides DE
and EF together with
angle EFD, we shall obtain angle EDF, and exterior angle
DEG of the
inclination ~ 9'. This will furnish us also with angle CGE of the
northern latitude ~ 6'. Hence, if we subtract the minimum inclination
from the maximum,
that is, 1° 51'-9', the remainder ~ 1° 41'. This is the oscillation
of this inclination, and 1/2 [of the oscillation] ~ 50
In like manner the angles of inclination of the other two planets,
Jupiter and Saturn, were determined together with their latitudes. Thus,
Jupiter's maximum inclination = 1° 42'; its minimum inclination = 1°
18'; hence, its entire oscillation comprises not more than 24'. On the
other hand, Saturn's maximum inclnation = 2° 44'; its minimum
inclination = 2° 16'; the intervening oscillation = 28'. Hence, through
the smallest angles of inclination, which occur in the opposite place,
when the planets are in conjunction with the sun, their deviations in
latitude from the ecliptic will emerge as 2° 3' for Saturn and 1° 6'
for Jupiter. These values had to be determined and retained for the
construction of the Tables below [after VI, 8].
GENERAL EXPLANATION OF ANY OTHER Chapter 4
LATITUDES OF THESE THREE PLANETS
From what has been expounded above, the particular latitudes of these
three planets will likewise be clear in general. As before, conceive the
intersection AB of the plane perpendicular to the ecliptic and passing
through the limits of their farthest deviations, with the northern limit
at A. Also let straight line CD be the intersection of the planet's
orbit [with the ecliptic], and let CD intersect AB in point D.
With D as
cemer, describe EF as the earth's grand circle. From E, where the earth
is allgued with the planet in opposition, take any known arc EF. From F
and from Q, the place of the planet, drop CA and FG perpendicular to AB.
Join FA and FC.
In this situation we first seek the size of ADC, the angle of the
eccentric's inclination. It has been shown [VI, 3] to be at its maximum
when the earth is in point E. Its entire oscillation, moreover, as is
required by the oscillation's nature, was revealed to be commensurate
with the earth's revolution on circle EF, as determined by diameter BE.
Therefore, because arc EF is given, ratio ED : EG will be given, and
this is the ratio of the entire oscillation to that which was just
detached from angle ADC Hence in the present situation angle ADC is
Consequently, in triangle ADC, the angles being given, all its sides are
given. But ratio CD : ED is given by the foregoing. Also given,
therefore, is [the ratio of CD] to DG, the remainder [when EG is
subtracted from ED].
Consequently the ratios of both CD and AD to GD are
known. Accordingly, AG, the remainder
[when GD is subtracted from AD],
is also given. From this information FG is likewise given, since it is
half of the chord subtending twice EF. Therefore, in right triangle AGF,
two sides [AG and FG] being given,
hypotenuse AF is given, and so is
ratio AF: AG. Thus, finally, in right triangle ACF,
two sides [AF and
AC] being given, angle AFC will be given, and this is the angle of the
apparent latitude, which was sought.
Again I shall exemplify this analysis with Mars. Let its maximum limit
of southern latitude, which occurs near its lower apse, be in the
vicinity of A. How
ever, let the place of the planet be C where ADC, the angle of the
inclination, was shown [VI, 3] to be at its maximum, namely, 1° 50',
when the earth was at point E. Now let us put the earth at point F, and
the motion in parallax, along arc EF = 450. Therefore, straight line FG
is given = 7071p whereof ED = 10,000
p, and GE, the remainder (when GD =
FG = 7071p is subtracted] from the radius
[= ED = 10,000p] = 2929p. But
half of ADC, the angle of the oscillation, has been shown = 0° 50 1/2'
[VI, 3]. In this situation its ratio of increase and decrease = DE : GE
= 50° 1/2' : 15'.
When we subtract this latter quantity from 1° 50',
the remainder = 1° 35' = ADC, the angle of the inclination in the
present situation. Therefore, the angles and sides of triangle ADC will
be given. CD was shown above to be = 9040p
whereof ED = 6580p [V, 19].
Hence, in those same units FG = 4653p;
AD =9036p; AEG, the remainder
[when GD = FG = 4653P is subtracted from
AD 9036p] = 4383p,
and AC = 249 1/2p. Therefore, in right triangle
AFG, perpendicular AG = 4383p, and
base FG = 4653p; hence, hypotenuse AF
= 6392p. Thus, finally, triangle
ACF has CAF as a right angle, together with given sides
AC and AF [= 2491/2p,
6392p]. Hence, angle AFC is given = 2° 15'
= the apparent latitude when the earth is situated at F.
We shall pursue the analysis in the same way for the other two planets,
Saturn and Jupiter.
THE LATITUDES OF VENUS-AND MERCURY Chapter 5
Venus and Mercury remain. Their deviations in latitude, as I said [VI,
1], win be demonstrated jointly by three interrelated latitudinal
excursions. In order to be able to separate these from one another, I
shall begin with the one called the "declination", since it is simpler
to treat. It is the only one which sometimes happens to be separated
from the others. This [separation occurs] near the middle longitudes and
near the nodes when, as reckoned by the corrected motions in longitude,
the earth is located a quadrant's distance from the planet's apogee and
perigee. When the earth is near the planet, [the ancients] found 6° 22'
of southern or northern latitude in Venus, and 4° 5' in Mercury; but
with the earth at its greatest distance [from the planet], 1° 2'
in Venus, and 1° 45'in
Mercury [Ptolemy, Syntaxis, XIII, 5]. Under these circumstances the
planets' angles of inclination are made known through the established
tables of corrections [after VI, 8]. Therein, when Venus is at its
greatest distance from the earth with its latitude = 1° 2, and at its
least distance [from the earth with its latitude =] 6° 22', an arc of
approximately 2 1/2° of orbital [inclination]
fits both cases. When
Mercury is most remote (from the earth], its latitude = 1° 45', and
when it is closest [to the earth, its latitude =] 4° 5' require an arc
of 6 1/4° [as the inclination] of its orbit. Hence, the orbits' angles
of inclination = 2° 30' for Venus, but for Mercury 6 1/4°, with 360° = 4
right angles. Under these circumstances each of their particular
latitudes in declination can be explained, as I shall presently
demonstrate, and first for Venus.
Let the ecliptic be the plane of reference. Let a plane perpendicular to
it and passing through its center intersect it in ABC. Let (the
ecliptic's] intersection with Venus, orbital plane be DBE. Let the
earth's center be A; the center of the a planet's orbit,
B; and the angle
of the orbit's inclination to the ecliptic, ABE. With B as center,
describe orbit DFEG. Draw diameter FBG perpendicular to
diameter DE. Let the orbit's plane be conceived to be so related to the
assumed perpendicular plane that lines drawn therein perpendicular to DE
are parallel to one another and to the plane of the ecliptic, in which
FBG is the only [such perpendicular].
From the given straight lines AB and BC, together with ABE,
the given angle of inclination, it is proposed to find how much the planet
deviates in latitude. Thus, for example, let the planet be at a distance
of 45° away from E, the point nearest to the earth. Following Ptolemy
[Syntaxis, XIII, 4], I have chosen this point in order that it may be
clear whether the inclination of the orbit produces any variation in the
longitude of Venus or Mercury. For, such variations would have to be
seen at their maximum about halfway between the cardinal points D, F, E,
and G. The principal reason therefor is that when the planet is located
at these four cardinal points, it experiences the same longitudes as it
would have without any declination, as is self-evident.
Therefore, let us take arc EH = 45° as was said.
Drop HK perpendicular to BE.
Draw KL and HM perpendicular to the ecliptic as the plane of
reference. Join HB, LM, AM, and AH.
We shall have LKHM as a
parallelogram with 4 right angles, since HK is parallel to the plane of
the ecliptic [KL and HM having been drawn perpendicular to the
ecliptic]. The side [LM of the parallelogram] is enclosed by LAM, the
angle of the longitudinal prosthaphaeresis. But angle HAM embraces the
deviation in latitude, since HM also falls perpendicularly on the same
plane of the ecliptic. Angle HBE is given = 45'. Therefore, HK = half
the chord subtending twice HE = 7071p
whereof EB = 10,000p.
Similarly, in triangle BKL, angle KBL is given = 2° 1/2'
[VI, 5, above],
ELK is a right angle, and hypotenuse BK= 7071p whereof
BE = 10,000p. In
the same units, the remaining sides KL = 308p and
BL = 7064p.
But, as was shown above [V, 21], AB : BE
~10,000p : 7193p.
In the same units,
therefore, the remaining sides HK = 5086p; HM = KL = 221P; and BL =
5081p. Hence LA, the remainder (when BL = 5081p is subtracted from AB =
10,000p] = 4919p. Now once more, in triangle ALM, sides AL and LM = HK
are given [= 4919p, 5086p], and ALM is a right angle. Hence we shall
have hypotenuse AM = 7075p, and angle MAL = 45° 57' = Venus'
prosthaphaeresis or great parallax, as computed.
Similarly, in triangle [MAH], side AM is given = 7075p,
and side MH = KL [= 221p].
Hence, angle MAH is obtained = 1° 47' = the latitudinal
declination. But if it is not boring to consider what variation in
longitude is produced by this declination of Venus, let us take triangle
ALH, understanding LH to be a diagonal of parallelogram LKHM
= 5091p where of AL = 4919p.
ALH is a right angle. From this information
hypotenuse AH is obtained = 7079p.
Hence, the ratio of the sides being
given, angle HAL = 45° 59'. But MAL was shown
= 45° 57'. Therefore, the excess is only 2'. Q. E. D.
Again, in like manner I shall demonstrate the latitudes of declination
in Mercury by a construction similar to the foregoing. Therein assume
arc EH = 45° so that each of the straight lines HK and KB
is taken, as
before, = 7071p whereof hypotenuse HB = 10,000p.
In this situation, as
can be inferred from the differences in longitude as shown above [V, 27],
radius BH = 3953p and AB = 9964
In such units, BK and KH will both be
= 2795p. ABE, the angle of inclination, was shown [VI, 5, above] = 6°
15', with 360° = 4 right angles. Hence, in right
triangle BKL the angles are given. Accordingly, in the same units base
KL = 304p,
and the perpendicular BL = 2778p.
Therefore, AL, the remainder
(when BL = 2778p is subtracted
from AB = 9964p) = 7186p.
But LM = HK =
2795p. Hence, in triangle ALM,
L is a right angle, and two sides, AL and
LM, are given [= 7186p,
2795p]. Consequently, we shall have hypotenuse
AM = 7710p, and angle LAM = 21° 16'
= the computed prosthaphaeresis.
Similarly, in triangle AMH, two sides are given: AM [= 7710p], and MH =
KL [= 304p], forming right angle M.
Hence, angle MAH is obtained = 2° 16' = the latitude we were seeking. It may be asked how much [of the
latitude] is owing to the true and apparent prosthaphaeresis. Take LH,
the diagonal of the parallelogram. From the sides we obtain it = 2811p,
and AL = 7186p. These show angle LAH = 21° 23' = the apparent
prosthaphaeresis. This exceeds the previous calculation [of angle LAM =
21° 16'] by about 7'. Q. E. D.
VENUS' AND MERCURYS SECOND Chapter 6
LATITUDINAL DIGRESSION, DEPENDING
ON TBE INCLINATION OF TBEIR
ORBITS AT APOGEE AND PERIGEE
The foregoing remarks concerned that latitudinal digression of these
planets which occurs near the middle longitudes of their orbits. These
latitudes, as I said [VI, 1], are called the "declinations". Now I must
discuss the latitudes which happen near the perigees and apogees. With
these latitudes is mingled the deviation or third [latitudinal]
digression. Such a deviation does not occur in the three outer planets,
but [in Venus and Mercury] it can more easily be distinguished and
separated out in thought, as follows.
Ptolemy observed [Syntaxis, XIII, 4] that these (perigeal and apogeal]
latitudes appeared at their maximum when the planets were on the
straight lines drawn from the center of the earth tangent to their
orbits. This happens, as I said [V, 21, 27], when the planets are at
their greatest distances from the sun in the morning and evening.
Ptolemy also found [Syntaxis, XIII, 3] that Venus' northem latitudes
were 1/3° greater than the southern,
but Mercury's southern latitudes
were about 11/2° greater than the northern.
However, out of a desire to
take into account the difficulty and labor of the computations, he
as a sort of average quantity for the varying values of
the latitude, mainly because he believed that no perceptible error would
thereby arise, as I too shall soon show [VI, 7]. These degrees are
subtended by the latitudes on the circle around the earth and at right
angles to the ecliptic, the circle on which the latitudes are measured.
If we now take 21/2°
as the equal digression to either side of the
ecliptic and for the time being exclude the deviation, our
demonstrations will be simpler and easier until we have ascertained the
latitudes of the obliquations.
Then we must first show that this latitude's digression reaches its
near the eccentric's point of tangency, where the longitudinal
prosthaphaereses are also at their peak. Let the planes of the ecliptic
and the eccentric, whether Venus' or Mercury's, intersect [in a line]
through the [planet's] apogee and perigee. On the intersection take A as
the place of the earth, and B as the center of the
eccentric circle CDEFG, which is inclined to the ecliptic. Hence, [in
the eccentric] any straight lines drawn perpendicular to CG form angles
equal to the inclination
[of the eccentric to the ecliptic]. Draw AR tangent to the eccentric,
and AFD as any secant. From points D, E, and F,
furthermore, drop DH, EK, and FL Perpendicular to
CQ; and also DM, EN, and FO perpendicular to
the horizontal plane of the ecliptic. Join MH, NK, and OL,
as well as AN and AOM. For, AOM is a straight line,
since three of its points are in two planes, namely,
the plane of the ecliptic, and the plane ADM perpendicular to the plane
of the ecliptic. For the assumed inclination, then, angles HAM and KAN
enclose the longitudinal prosthaphaereses of these planets, whereas
their digressions in latitude are embraced by angles DAM and EAN.
I say, first, that the greatest of all the latitudinal angles is EAN,
which is formed at the point of tangency, where the longitudinal
prosthaphaeresis also is nearly at its maximum. For, angle EAK is the
greatest of all [the longitudinal angles]. Therefore RE: EA > HD : DA
and LF : FA. But EK : EN = RD : DM = LF : FO, since the angles subtended
(by the second members of these ratios) are equal, as I said. Moreover,
M, N, and O are right angles. Consequently,
NE : EA > AW : DA and OF :
FA. Once more, DMA, ENA, and FOA are light angles.
Therefore, angle EAN is greater than DAM and all the [other]
angles which are formed in tins way.
Of the difference in longitudinal prosthaphaeresis caused by this
obliquation, consequently, clearly the maximum is also that which occurs
at the greatest elongation near point B. For on account of the
equality of the angles subtended [in the similar triangles], HD : HM =
KE : KY = LF : LO. The same ratio holds good for their differences [HD -
HM, KE - KN, LF - LO]. Consequently, the difference EK-KN has a greater
ratio to EA than the remaining differences have to sides like AD.
Hence it is also clear that the ratio of the greatest longitudinal 25
Prosthaphaeresis to the maximum latitudinal digression will be the same
as the ratio of the longitudinal prosthaphaereses of segments of the
eccentric to the latitudinal digressions. For, the ratio of KE to EN is
equal to the ratio of all the sides like LF and RD
to the sides like FO and DM. Q. E. D.
THE SIZE OF THE OBLIQUATION ANGLES Chapter 7
OF BOTH PLANETS, VENUS AND MERCURY
Having made the foregoing preliminary remarks, let us see how great an
angle is contained in the inclination of the planes of both these planets. Let
us recall what was said above [VI, 5], that each of the planets, when
[midway] between its greatest and least distances [from the sun],
becomes farther north or south
at the most by 5° in opposite directions depending on its position in
its orbit. For, at the eccentric's apogee and perigee Venus' digression
makes a deviation imperceptibly greater or smaller than 5° from which
Mercury departs by more or less.
As before, let ABC be the intersection of the ecliptic and the
eccentric. Around B as center, describe the planet's orbit inclined
to the plane of the ecliptic in the Manner explained [Previously]. From
the center of the earth draw straight line AD tangent to the [planets]
orbit at point D. From D drop perpendiculars, DF on
CBE, and DG on the horizontal plane of the ecliptic.
Join BD, FG, and AG. Also assume that
in the caw of both planets angle DAG, comprising half of the
aforementioned ditference in latitude, = 21/2°,
with 4 right angles = 360°.
Let it be proposed to find, for both planets, the size of the angle of
inclination of the planes, that is, angle DFG.
In the case of the planet Venus, in units whereof the orbit's radius =
7193p, the planet's greatest distance [from the earth],
which occurs at the apogee, has been shown = 10,208p,
and its least distance, at perigee, = 9792p
[V, 21-22 : 10,000±208].
The mean between these values = 10,000p,
which I have adopted for the purposes of this demonstration.
Ptolemy wanted to take laboriousness into account and, as far as
possible, seek out short cuts [Syntaxis, XIII, 3, end]. For where the
extreme values did not produce a manifest difference, it was better to
accept the mean value.
Accordingly, AB : BD = 10,000p :
7193p, and ADB is a right angle. Then
we shall have side AD = 6947p in length. Similarly, BA : AD = BD : DF,
and we shall have DF = 4997p in length. Again, angle DAG is assumed = 2
1/2° and AGD is a right angle. In triangle [ADG, then], the angles being
given, side DG = 303p whereof AD = 6947p. Thus also (in triangle DFG]
with two sides, DF and DG, being given [= 4997,303], and DGF a right
angle, DFG, the angle of inclination or obliquation, = 3° 29'. The excess
of angle DAF over FAG comprises the difference in longitudinal parallax.
Then the difference must be derived from the known sizes [of those
It has already been shown that in units whereof DG = 303p,
hypotenuse AD = 6947p,
and DF = 4997p,
and also that (AD)2- (DG)2= (AG)2,
and (FD)2-(DG)2 = (GF)2.
Then as a length AG is given = 6940p,
and FG = 4988p. In
units whereof AG = 10,000p,
FG = 7187p, and angle FAG = 45° 57'.
In whereof AD = 10,000p, DF = 7193p,
and angle DAF~ 46°. In the greatest
obliquation, therefore, the parallactic prosthaphaeresis is diminished
by about 3' [= 46° - 45° 57']. At the middle apse, however, clearly the
angle of the inclination between the circles was 2 1/2°. Here, however,
it has increased [to 3° 29'] by nearly a whole degree, which was added
by that first libratory motion which I mentioned.
For Mercury the demonstration proceeds in the same way. In units
whereof the orbit's radius 3573p,
the orbit's greatest distance from the
earth = 10,948p; its least distance
9052p; and between these values the
mean = 10,000p [V, 27].
AB : BD = 10,000p : 3573p.
Then (in triangle ABD] we shall have the third side
AD = 9340p.
AB : AD = BD : DF.
Therefore DF = 3337p in length.
DAG = the angle of the latitude, is
assumed = 2 1/2°. Hence DG = 407p
whereof DF = 3337p. Thus in triangle
DFG, with the ratio of these two sides being given, and with G a right
angle, we shall have angle DFG ~ 7°. This is the angle at which
Mercury's orbit is inclined or oblique to the plane of the ecliptic.
Near the middle longitudes at a quadrant's [distance from apogee and
perigee], however, the angle
of inclination was shown = 6° 15' [VI, 5]. Therefore, 45' [=7'- 6° 15']
have now been added by the motion of the first libration.
Similarly, for the purpose of ascertaining the angles of
prosthaphaeresis and their difference, it may be noticed that straight
line DG has be= shown = 407p
whereof AD = 9340p and DF = 3337p.
(AD)2-(DG)2 = (AG)2,
and (DF)2 - (DG)2 = (FG)2.
Then we shall have as a length AG = 9331p,
and FG = 3314p.
From this information is obtained GAF = the angle of the
prosthaphaeresis = 20° 48', whereas DAF = 20° 56',
than which GAF, which
depends on the obliquation, is about 8' smaller.
It still remains for us to see whether these angles of obliquation and
the latitudes connected with the orbit's maximum and minimum distance
[from the earth] are found to conform with those obtained by
observation. For this purpose in the same diagram again assume, in the
first place, for the greatest distance of Venus' orbit [from the earth]
that AB : BD = 10,208p : 7193p.
Since ADB is a right angle, as a length
AD = 7238p in the same units. AB : AD = BD : DF.
Then in those units DF
= 5102p in length. But DFG = the angle of the obliquity,
was found = 3° 29' [earlier in VI, 7]. The remaining side DG = 309p whereof AD = 7238p.
Then, in units whereof AD = 10,000p,
DG = 427p. Hence, angle DAG is
inferred = 2° 27' at the [planet's] greatest distance from the earth.
However, in units whereof BD = the orbit's radius = 7193p,
AB = 9792p [=
10,000-208] at the [planet's] least [distance from the earth]. AD,
perpendicular to BD, = 6644p. AB : AD = BD : DF.
Similarly, as a length DF is given = 4883p in those units.
But angle DFG has been put = 3° 29'.
Therefore, DG is given = 297p whereof AD = 6644p. Consequently in
triangle [ADG], the sides being given, angle DAG is given= 2° 34'.
However, neither 3' nor 4' [2° 30' = 3'+2° 27' = 2° 34'-4'] are large
enough to be registered instrumentally with the aid of astrolabes.
Hence, what was regarded as the maximum latitudinal digression in the
planet Venus stands up well.
In like manner assume that the greatest distance of Mercury's orbit
[from the earth is to the radius of Mercury's orbit], that is, AB : BD =
101948p :3573p [V, 27]. Thus,
by demonstrations like the foregoing, we
obtain AD = 9452p,
and DF = 3085p. But here again we have DFG,
the angle of the inclination [between Mercury's orbit and the plane of the
ecliptic] known = 7° and for that reason straight line DG = 376p whereof
DF = 3085p or DA = 9452p.
Hence in right triangle DAG, whose sides are
given, we shall have angle DAG ~ 2° 17' = the greatest digression in
At the [orbits] least distance [from the earth], however, AB : BD is put
= 9052p :3573p.
Hence, in those units AD = 8317p,
and DF = 3283p. However,
on account of the same inclination [ = 7°] DF : DG
is put = 3283p : 400p
whereof AD = 8317p. Hence, angle DAG = 2°45'.
The latitudinal digression associated with the mean value [of the
distance of Mercury's orbit from the earth) is here too assumed = 2
1/2°. From this quantity the latitudinal digression at apogee, where it
reaches its minimum, differs by 13'
[=2° 30'- 2° 17']. At perigee, however, where the latitudinal digression
attains its maximum,
it differs (from the mean value] by 15' [= 2°45'-2° 30']. Instead of
these [apogeal and perigeal differences], in computations based on the
mean value, above it and below it I shall use 1/4°, which does not
differ perceptibly from the observations.
As a result of the foregoing demonstrations, and also because the
greatest longitudinal prosthaphaereses have the same ratio to the
greatest latitudinal digression as the partial prosthaphaereses in the
remaining portions of the orbit have to the several latitudinal
digressions, we shall obtain all the latitudinal quantities occurring on
account of the inclination of the orbits of Venus and Mercury. But only
the latitudes midway between apogee and perigee, as I said [VI, 5], are
available. It has been shown that of these latitudes the maximum 21/2°
[VI, 6], while Venus' greatest prosthaphaeresis = 46°, and Mercury's ~ 22°
[VI, 5 : 45° 57', 21° 16']. And now in the tables of their nonuniform
[after V, 33] we have the prosthaphacreses alongside the individual
portions of the orbits. To the extent that each of the prosthaphacreses
is smaller than the maximum, I shall take the corresponding part of
those 2 1/2° for each planet. I shall record that part numerically in
the Table which is to be set out below [after VI,
8]. In this way we shall have in detail every individual latitude of
obliquation which occurs when the earth is at the higher and lower
apsides of these planets. In like manner I have recorded the latitudes
of their declinations [when the earth is] at a quadrant's distance
(midway between the planets' apogee and perigee], and Me planets are] at
their middle longitudes. What occurs between these four
critical points [higher, lower, and both middle apsides] can be derived
by the subtlety of the mathematical art from the proposed system of
circles, not without labor, however. Yet Ptolemy was everywhere as
compact as possible. He recognized [Syntaxis, XIII, 4, end] that by
themselves both of these kinds of latitude [declination, obliquation] as
a whole and in all their parts increased and decreased pro
portionally like the moon's latitude. He therefore multiplied each of
their parts by twelve, since their maximum latitude = 5° =
1/12 x 60°. He
made these [products] into proportional minutes, which he thought should
be used not only in these two planets but also in the three outer
planets, as will be explained below [VI, 9].