Epitome Astronomiae Copernicanae: Liber III, part 1
Book III. On the Doctrine of the First Motion, Called Spherical
Why is the spherical doctrine put before the theoretical? Although the theory of the proper motions of the planets can be delivered and comprehended by itself, from the hypotheses laid down, needing no aid of the daily or first motion, yet the exact explanation of the first motion, which is perfect in all its numbers, is completed only by the knowledge of the secondary motions, above all of the motions of the Sun, and needs many things taken beforehand from the theory: still there are fit reasons why we should begin with the daily motion. For first, the daily motion is better known to sense and more obvious, being swift and daily and answering more in proportion to the needs of our life, and therefore also it became known to the human race earlier; the secondary motions are more hidden, more remote from sense, and on account of their slowness and various intricacies, together with the first, required the long observation of the most learned men, and much more speculation, for the establishing of hypotheses. Rightly, therefore, the delivery of this doctrine follows the order of investigation. Next, the first needs only the instrument of the sphere, which is conformed to the visible heaven and could in a manner be shown even in the heaven itself without an instrument; the causes and reasons of the secondary motions are in no way demonstrated in the visible heaven, but altogether need a plane table, a kind of plane we do not easily imagine in the heaven. But neither is it easy to apply to the heaven itself, as it meets the eyes, what is demonstrated on the table about the secondary motions, unless the first motion has been perceived beforehand, so that the mind can abstract from the secondary.
How many are the parts of the spherical doctrine? Chiefly five. The first and second are more general and contain the preparation for the rest. The first teaches the places of rising and setting and the various positions and delays of the stars above the horizon; where the difference is seen of the position of the right, oblique and parallel sphere in either hemisphere, northern and southern. The second delivers more particularly the risings and settings of any points of the ecliptic whatever, through the seven principal positions of the sphere referred to the ecliptic, in either hemisphere. The third, fourth and fifth are occupied with the explanation of times and what happens to times. For the third treats of the turning year and of days and hours, setting out the various lengths of days and nights through the different progress of the Sun in the ecliptic, and measuring them through different places of the Earth, on which depends the distinction of the Earth into climates. The fourth explains the reasons of the four parts or seasons of the turning year, which are the causes and limits of summer and winter, how great is the variety of the Sun’s altitude and of the length of shadows through the various positions of the sphere, whence comes the distinction of the surface of the Earth into five zones and their diversity in qualities. The fifth contains the other kind of year, called sidereal, and the signs by which the parts both of it and of the turning year are distinguished from one another, that is, the appearances and hidings of the stars through different climates. To these last three parts adheres the geographical distinction of the inhabitants of the Earth into Antoeci, Perioeci and Antipodes, and the method of computing the distances and longitudes of places.
Part One of the Spherical Doctrine. On the Rising and Setting of the Stars.
You said that there are many horizons in the world, distinguished by the inclination of the vertical point to the sides of the world: how, then, do astronomers distinguish the positions of the sphere according to the horizon or vertical point? The horizon is either at right angles to the equator, the path of the daily motion, cutting it at right angles, or oblique to it, cutting it at oblique angles, or does not cut it at all, but coincides with it.
Wherefore the sphere by reason of the first position is called right, by reason of the second oblique, by reason of the third parallel, because the horizon and equator become one of the parallels. It follows, therefore, in the first that the pole of the horizon, that is, the vertical point, falls on the equinoctial, and the poles of the world are cast down equally to the horizon; in the second the vertical point is between the equator circle and one or other of its poles, for of these one is above the horizon, the other below; in the last the poles of the world coincide with the poles of the horizon, so that the world revolves about the zenith.
And the oblique sphere, like the parallel, is twofold, the one northern, in which the northern pole of the world is above the horizon, the other southern, in which the southern pole is seen, the northern being hidden. Of these the one common limit is the right sphere. The right sphere, therefore, is inhabited by all those who are scattered along the length of the terrestrial equator, or in nautical phrase, who dwell on the Line. The northern oblique sphere we Europeans inhabit, and all those who are on this side of the Line, up to that single point of the Earth in which the pole of the Earth is. For an eye placed there alone has the northern parallel sphere. But those who are beyond the Line, which the Portuguese and Belgians frequently cross today in navigating the Ocean, have the southern oblique sphere, and one, the innermost or middle of those places, has the southern parallel sphere.
What do the words to rise and to set sound? To rise is to be lifted up from a plain on high, or to rise up, to ascend little by little more and more, to emerge, as mountains seem to sailors on the ocean to emerge from the waves: so that not without reason you may believe that the Greek word oros, which signifies a mountain, is derived from oroein, to rise. The Greek word anatellein, anatole, is akin to the Latin tollere, and sounds properly also of plants, when they raise themselves into the air out of the moist earth. To set and to sink down is to fall forward: it is used of those falling in battle, thence of all that perish. The Greek word dyein is used for a retreat into some chamber, to go under, to enter, to be hidden; because the stars, behind the mountains, withdraw as it were into a bedchamber. The German words Aufgang and Niedergang are manifest, for they signify up and down; untergehen is to be submerged, like ships in the waves.
Do the stars truly become high and low by a daily change, rising and falling alternately? No astronomer will grant this. For even if the daily motion is attributed to the stars while the Earth rests, it will be circular about the centre of the Earth; in a circle, however, no part is lower or higher than another from its centre. All these expressions, therefore, are accommodated to the sense of the eyes, whether they occur in profane writers or in the sacred books, and the very truth of things differs very far from the apparent appearances, expressed in this kind of speech.
What, then, makes the stars seem every day to be lifted up, to emerge from the waves or mountains, as if coming out of a bedchamber, to be born, to rise on high; and in turn to fall, to sink, to go under the Earth, to be hidden behind the mountains, to be plunged in the waves of the ocean? The revolution of the visible horizon about the immovable axis of the Earth, as was said in the first book: for the stars which it uncovers seem to rise, those which it covers to set.
Can this revolution of the Earth represent the daily circular motion of the stars, such as those who observe the stars experience? Altogether. For the apparent circle of a star, as was said in the second book, is described in the sphere which sight imagines, by a straight line drawn from the eye to some star, and fixed immovably to the Earth, and, the star being left, turned round with the body of the Earth itself, or, what is the same, by a parallel to that, drawn from the centre of the Earth. For as on a plane sheet of paper a circle is described with a compass, one foot of which sticks immovable at one point, while the other is carried round on the paper, so here too the place on the Earth, or the eye, turned about the axis of the Earth, represents the joint of the compass, which can be turned by the fingers, the line directed from the eye or centre of the Earth to the pole of the world represents the immovable foot of the compass, the line drawn to the star the foot carried round, and the hollow of the sphere is in the place of the paper, which, since it stands equally round the centre, makes this whole circle, described through the star, be conceived as equally distant everywhere from the Earth.
What are the chief phenomena of this first motion, with respect to the three positions of the sphere? Five: the first is altitude, likewise the ascent and descent of the stars and their culmination or mid-heaven, or in its place a simple and uniform revolution. 2) The quarter toward which the stars seem to move. 3) The distinction of the stars into three classes: those perpetually appearing, those perpetually hidden, and those rising and setting, or grazing the horizon at rising or setting, and how great a breadth of the horizon each star occupies, intercepted between the points of its rising and setting. 4) The overturning of the position of the constellations. 5) The stay of the stars above the horizon and below it.
1. What is, to astronomers, the altitude of a star in this matter? The word altitude is not to be understood popularly of the length of a plumb line let fall from the star to the plane of the Earth, but technically thus, that it is the arc of the vertical circle drawn through the star, intercepted between the star and the rational horizon; the complement of this arc to a quadrant is the distance of the star from the zenith.
Prove the variety about the apparent risings and settings from the revolution of the Earth. Since the Earth turns about an immovable axis in 24 hours, there must be two points in the globe of the Earth, the extremities of the axis, called poles, at which an observer stationed, and turned round with his visible horizon, thinks the heaven and the stars in it, which are immovable, to be seized with a whirl and to go round the outer rim of the visible horizon, and meanwhile through the whole circuit to be always equally distant from the zenith, since the pole of the world is there, so that no star in any quarter becomes higher or lower than itself. On the contrary, all the other visible horizons on the surface of the sphere, and their centres, the eyes of the spectators, are movable about the axis of the Earth in small circles of the Earth, great, or the greatest, according as the places are little or more distant from either pole, or equally distant from both. But the zeniths of the places of the Earth carried round are also carried round beneath the immovable sphere of the fixed stars, and, leaving the stars past which they had gone, come to other stars, approach others, passing near or far. Wherefore, by what was brought from the optics in the first book, the stars themselves will seem now to fall upon the zenith, now to flow away from the zenith and again to approach it; and consequently also to be less or more distant from the rational horizon (which is everywhere a quadrant, and so equally, distant from the zenith), that is, to be reckoned lower or higher. And this happens commonly both to the oblique spheres of either hemisphere and to the right sphere, the middle of all.
2. Set out the variety of the quarters toward which the stars seem to go. Since to all men whatever, whether dwelling in the north or in the south, the right hand is reckoned the same and the left the same, namely that toward which the position of the heart looks, whence the left, as cohering more closely with the source of motion (to which rest belongs) and so as it were reduced to a narrow compass, is less ready for motion, while the right, as more remote from the source of motion, is more expeditious and apt for more offices, since it stretches farther from the heart and reigns in a wider compass: hence arises for the human race the chief and plainly most notable and most popular distinction of the northern and southern hemispheres. For to us in the north, through all the positions of the sphere, and most of all in the parallel sphere, the daily motion of the stars seems to tend from left to right, but to those who are in the southern hemisphere, in the contrary way, from right to left; which to those who pass from us across the Line into the southern ocean produces matter for the greatest wonder, since for them, as our pole is exchanged for the southern pole, so also the right is not exchanged with the left, nor in those who are born in the south does the heart occupy a seat of the body opposite to ours, a very great argument that each race is propagated from one stock. Pliny reports that this was most astonishing to the ambassadors of the king of Taprobane, an island in the south, who came to Rome, namely that their shadows fell toward our heaven (the north), not their own (the south), and that the Sun rose rather on the left (the face being turned to the Sun’s course) and set on the right, rather than the contrary. For to those who are in the south, looking at the tract of the equator and ecliptic which is above the Earth, the signs and parts of the circles seem indeed to rise in the same order among themselves as with us, but with respect to our body to rise on the right and to descend to the left.
Does no greater variety occur here? Nay, this too is notable, that in parallel spheres all the stars seem to pass through the same quarter altogether, because the zenith of the spectator falls on the pole of the revolution; but in oblique spheres the stars which represent circles drawn round between the zenith and the pole, those about the horizon indeed do this very thing, but after they have come into the upper semicircle they seem to tend toward the contrary quarter, because the circle of their apparent motion stands wholly on one side of the zenith, its opposite parts therefore get also opposite motions to sight. Finally in the right sphere all the stars at their first rising rise straight, with a motion inclined toward neither hand, but after they have raised themselves to some altitude, only those which fall on the equator persist in that straightness, tending up to the zenith, which in this position of the sphere falls on the equator, while the rest decline to the sides on which they stand, some to the right, some to the left.
3. What is the variety of the stars rising and not rising through the spheres, and how does it come from the revolution of the Earth about its axis? In parallel spheres nothing sets, but one half of the heavenly host is perpetually seen from the northern pole of the Earth and is hidden from the southern pole, the other half is seen from the southern pole of the Earth and is hidden from the northern; but the stars which are situated on the equator remain perpetually and revolve on the rational horizon, except that refraction raises them somewhat. For the horizon or limiter of sight coincides with the equator, cutting the world into two hemispheres, northern and southern.
In the right sphere all the stars rise and set within the space of one day. For the horizon cuts the sphere, and so all the parallels, through the axis and the poles; which, since it revolves with the globe of the Earth within 24 hours, covers all the parts of the circles within one day with one of its semicircles and uncovers them again with the other: but if ever a star falls on the very pole, it will be seen the whole year and at all hours of the night in the same place of the horizon.
In oblique spheres, since by as much as the zenith declines from the equator, by so much also the horizon sinks below one pole on one side and rises above the other on the other, all the stars, therefore, comprised in the compass of the little circle which the horizon describes about the upper pole by the point of its circumference nearest to it, appear perpetually as in the parallel sphere; all those within the opposite circle which the horizon delineates about the lower pole lie hidden with that pole itself, and the stars through which these circles are drawn touch the horizon once in a day and at once either hide themselves again or withdraw on high. These circles, as was said in the second book, are expressed in some spheres and are called by the name of arctic and antarctic. But the stars between these two circles all rise and set, as in the right sphere, those nearest these circles and the visible pole marking, in rising and setting, points of the horizon very near one another: for hardly have they hidden themselves when they rise again, as if in the same place beneath the pole; those more remote mark more distant places, yet visible from the same quarter, up to those which fall on the equator, for their rising is distant from their setting by an entire semicircle of the horizon. For if the dioptra is directed to the east and fixed in that position, the setting will be seen through the same in the opposite direction, since the eye is the centre of the horizon. The places of rising and setting of these, therefore, cannot be seen at once in one view.
Stars situated beyond the equator, yet less than the zenith is distant from it, have the places of their rising and setting both in the southern quarter, notwithstanding that they themselves are seen highest in the northern quarter. For their circles are wholly beyond the equator, and therefore also their sections are beyond the sections of the equator with the horizon. Hence the farther the stars are from the equator, the nearer in the southern quarter the points of rising and setting come together, and at last you will see some stars which, hardly have they emerged, hide themselves again, as if in the same place of the horizon toward the quarter of the midday Sun.
Do all these things appear every night together, each in its own star? By no means; rather the observation of these things requires time and diligence and a stay in one place. For seldom can one and the same star within the space of one night be seen both rising and setting, because in most stars the light of day hides one or other of these, either the rising or the setting, and therefore another day must be waited for, when it falls also in the night.
How, then, can these phenomena be investigated otherwise than by observing, so that afterwards, being investigated, each may be compared with the heaven itself at its own times? We have need of an inquiry into the altitude of the pole in any place of the Earth, the altitude of the equator, the altitude of each meridian star and its declination from the equator.
What is the meridian altitude of the pole, the equator or a star? It is the arc of the meridian circle intercepted between the neighbouring part of the horizon and the pole, or the equator, or the star.
Whence is this altitude of a star called meridian? Not always from the southern quarter, but from the meridian circle, to whose northern part also certain stars are seen to come, and indeed twice, some of those which do not set, whence of some there is a double meridian altitude, one greatest, the other least.
Explain this variety of meridian altitudes more particularly. At an altitude of the pole of 45°, which quantity is half a quadrant, stars grazing the horizon pass also through the zenith; all those nearer the pole than these, therefore, come twice into the meridian from the quarter of the visible pole; none of those more remote from the pole seems to come twice to the meridian, for the horizon hides the lower approach; nor does any of these more remote from the visible pole come to the meridian toward the quarter of the visible pole.
At a smaller altitude of the pole, those which are between the vertical stars and the grazing ones come to the meridian from the quarter of the pole once, those nearer the pole twice from the quarter of the pole.
At a greater altitude of the pole, whatever stars are nearer the pole than the horizon-grazing ones seem to come twice; those which are nearer the pole than the verticals, twice from the same quarter, but those which are between the verticals and the horizon-grazing ones, once from the quarter of the pole, once from the contrary quarter of the meridian.
How is it known how great this arc of the altitude of the pole is in any place? In various ways, but the mode proper to this place is this: when the night is longer than the day, so that within one night more than half a revolution of the Earth can be made, then we choose a star near the pole of the world, which does not set, and which at the beginning and end of the night is observed in the meridian, once when it appears highest, again when it appears lowest. Now the parallel of the vertical point, by the definition of parallels, has the same poles as the sphere, and therefore the meridian circle, passing through the pole, cuts this parallel in two opposite points, one of which (the point of the zenith falling on it) is nearest the star, the other farthest from it. When, therefore, that semicircle of the meridian which passes through the zenith touches the star, the star appears highest, when the contrary semicircle, then the star appears lowest. And then the arithmetical mean between the two meridian altitudes of the star is the altitude of the pole.
Let V be the zenith, P the pole, SI the little circle of the motion of the apparent star, for instance the polar star (for the figures express the motion of the stars, not the motion of the vertical point, whose motion would be VH); let the greatest altitude of the star S, the arc SR, be 51° 3′, the least IR 45° 29′. Take IR from SR, there remains SI 5° 34′, half of which is PI 2° 47′. Add PI to IR, there results PR, the altitude of the pole 48° 16′. The same would happen if S were an immovable star and V a movable zenith, for first, if the semicircle PYL, having the zenith V, passes through the star S, the star will be distant from the zenith by YS, then let the zenith be in H, and its semicircle PHL and the opposite PVL pass through the star, therefore the distance of the star from the zenith will be HS. VS or its equal HI being taken from HS, SI remains, as before.
How is the altitude of the equator obtained? The altitude of the pole and the altitude of the equator together make a whole quadrant. Wherefore, the altitude of the pole being taken from a quadrant, the altitude of the equator is left.
In the figure PQ is the altitude of the pole, AH the altitude of the equator: since, therefore, HQ, the line of the horizon, passes through the centre of the sphere, HVQ will be a semicircle, but PA is a quadrant, because the equator is midway between the poles: the quadrant PA, therefore, being taken from QH, the semicircle, the remaining arcs PQ and AH will also make a quadrant.
How do geographers commonly call the altitude of the pole? To geographers it is the latitude of the place: for by as great an arc as the pole is raised above the horizon, by so great an arc the place is distant from the equator.
What is the latitude of a place? It is the arc of the terrestrial meridian intercepted between the place and the terrestrial equator.
Prove that the arc of the altitude of the pole and the latitude of the place are proportional. From the equator to its pole is a quadrant, from the zenith to the horizon is likewise a quadrant in the meridian circle: these arcs, therefore, are equal. Take away the common arc from the zenith to the pole, therefore the remainders are equal, there from the equator to the zenith, here from the pole to the horizon. Now the arcs of the celestial and terrestrial meridians, cut off by two straight lines from the centre, the plumb line and the axis, are proportional.
In the former figure PA and VQ are quadrants and equal; take away the common VP: AV and PQ will be equal, and so also on the Earth NL and OD, but NL is the latitude of the place and PQ the altitude of the pole.
What do gnomonists call the declination of a plane? The arc of the horizon circle intercepted between the primary vertical and the circle of the declining plane.
What is to them the inclination of an inclined plane? It is the arc of the primary vertical intercepted between the meridian and the circle of the inclined plane.
What is the inclination of a declining-inclining plane? It is the arc of the vertical at right angles to the circle of the declining-inclining plane, intercepted between it and the zenith.
What is incumbent on gnomonists about the declining-inclining plane? That it be referred first of all to the inclined, the altitude of the pole being sought, under which any declining-inclining plane can be computed as a simply inclined one, and the angle of inclination above that altitude of the pole.
How can this be done? A right-angled triangle is formed between the horizon, the meridian and the circle of the declining-inclining plane; in it there is given the side in the horizon, which is the complement of the declination of the plane, there is given also the angle between the circle of the declining-inclining plane and the horizon, which is the complement of the inclination. The angle between the horizon and the meridian is a right angle; thence is sought the side in the meridian, which is between the horizon and that point where the circle of the declining-inclining plane cuts the meridian, and the angle of that section: this side, compared with the altitude of the pole of your place, discovers the altitude of the pole sought.
The process for the side is this: let the complement of the inclination be 60°, tangent 173205; let the complement of the declination of the declining-inclining plane be 70°, sine 93969; multiply, the last 5 figures being cut off: 5196, 1558.
The tangent of the side in the meridian, or of the arc 58° 26′, is 162759.
Let the altitude of the pole be 48° 16′. Take it away, because it is less, there remain 10° 10′. This is the altitude of the pole under which this declining-inclining plane is among the simply inclined.
For the angle the process is this: let the complement of the declination of the declining-inclining plane be 70°.
The sine of the side found in the meridian is 85203.
The quotient (322463) is the tangent of the arc 72° 46′ of the inclination to the meridian of the altitude of the pole found.
What is incumbent on the astronomer about the circle of the inclined plane? The elevation of the pole above it, as above some horizon, is to be sought, which is always less than the altitude of the pole above the horizon of the place; likewise also the arc of the same circle intercepted between the meridians, one of the plane itself, the other of that elevation of the pole under which this plane is reckoned among the simply inclined. For this arc the gnomonists call the angle of the meridian lines.
By what means are these investigated? A right-angled triangle is formed between the altitude of the pole, under which the plane of the circle is held for inclined, or the meridian of that place, then between the meridian of the inclined plane itself and the circle of the inclined plane, in which there is given the angle, or its measure the inclination, also the side in the meridian of that place, that is, the said altitude. Wherefore the side cannot lie hid, neither the side in the proper meridian, that is, the altitude of the pole sought, nor the other side in the meridian of the place.
The process is this for the former side. The altitude of the pole under which the plane is held for Let them be multiplied, the last 5 figures being cast away.
Inclination 72° 44′, sine.
The arc of the altitude of the pole above the inclined plane 9° 42′, sine 16856.
The process for the angle between the meridian lines is this: add 5 ciphers. The altitude of the pole, under which the plane is held for inclined, 10° 10′, secant 101595.
The altitude of the pole above the plane 9° 42′, secant 101451. Let it divide 100; the quotient (100142) is the secant of the angle between the meridians, 3° 3′.