Epitome Astronomiae Copernicanae: Liber III, part 3
Part Two. On the Risings and Settings of the Signs or Points of the Ecliptic.
So far it has been said in general of any points of the sphere whatever. I should like now to be taught in particular what astronomers chiefly inquire about the points and arcs of the ecliptic? For the handling of the following parts of this spherical doctrine astronomers have need above all of the declinations and ascensions, both right and oblique, of the points of the ecliptic, and of the angles which the ecliptic forms at those points with the oblique or right horizon, that is, the meridian.
What is the cause why they consider the ecliptic especially? 1) Because all the planets move perpetually on this side of it and beyond it; 2) in particular the Sun, king of the planets and author of times, is seen to move with his centre perpetually beneath it; 3) finally, because we refer all the fixed stars also to the ecliptic.
The Declinations of the Points of the Ecliptic.
What measures the declinations of the points of the ecliptic? The meridian in the sphere does the office of all the circles of declination, and no less the horizon in the right sphere. The point whose declination is sought being applied to these circles, it is plain to the eye how great an arc lies between the equator and that point of the ecliptic.
Can these declinations of the points of the ecliptic also be investigated by calculation, and from what principles? Since it is presupposed as most true and proved by use itself that the ecliptic is a perfect greatest circle, no less than the equator: its greatest declination from the equator being given, which is the measure of the angle at which the two circles cut each other, the declination of any point of the ecliptic can be investigated by the help of the doctrine of triangles.
What measures the greatest declination of the ecliptic in particular, besides the meridian and the right horizon? The colure of the solstices, since it is drawn through the poles of the ecliptic and of the equator. For as great as is the distance of the poles on either side, so great also is this declination of them.
Tell by what means it is brought about that the ecliptic declines from the equator, and to what end? The genuine and formal cause of this declination is the inclination of that axis, about which as immovable the Earth rotates, upon the plane of the ecliptic, in which the centre of the Earth moves, going round the Sun by the annual motion. For if the axis of the Earth’s daily motion stood erect upon this plane, formed by the Earth’s annual motion, the equator and the ecliptic would coincide. (See the second figure below.) The final cause of this inclination is without doubt the use of men and living creatures scattered over the whole surface of the Earth round about: that the variety of the celestial phenomena might reach all the places of the Earth whatever, which will be explained at greater length below in the fourth part. These things, then, are demonstrated in the Copernican astronomy by most pleasant reasons and most ingeniously, so that in the same Earth are established both the manner of the thing and its end; in the common astronomy we are taught only that it is; why it is and how, is there wrapped in much darkness.
How is the greatest declination of the ecliptic commonly inquired into? Not otherwise than by observing. For 1) astronomers attend on successive days to when the Sun exhibits the greatest altitude of all and again, in the opposite part of the circle and of the year, the least of all, at the time of midday on each side, and they measure each by the instrument of the quadrant. Then they subtract the least from the greatest, and the remainder, cut in two, is taken for the greatest declination.
2) The manner is a little different if the astronomer first observes the altitude of the pole, as we were taught above, by the circumpolar stars, then compares the greatest meridian altitude of the Sun, observed on summer days, with the altitude of the equator; for the difference is again the greatest declination of the ecliptic.
3) Or as above, the altitude of the pole and the meridian line being known, from any azimuth and altitude of the Sun observed together its declination is obtained by calculation, on the day of the solstice indeed the greatest.
How great is this greatest declination of the ecliptic from the equator? Astronomers of all the centuries do not wholly agree in the smallest parts. The most ancient Indians make it precisely 24 degrees, which is the fifteenth part of the whole circle. Eratosthenes, the whole circle being divided into 83 parts, says that eleven of them are equal to twice the greatest declination. So with him the greatest declination is very nearly 23° 51⅓’, as much as Hipparchus and Ptolemy, following Eratosthenes, also retained. Albategnius delivered 23° 35′, Arzachel 23° 34′, Almeon 23° 33′, Prophatius the Jew 23° 32′, Tycho Brahe 23° 31½’, and also by another reckoning 23° 30½’, and Peurbach, Copernicus and the other astronomers of this time in the northern region 23° 28′ or even 23° 27′. So all the disagreement lies within 24 minutes, which are no more than the nine-hundredth part of the circle. We use today, however, in the Tychonic astronomy commonly 23° 31½’.
What is the cause of this variety? One true celestial cause is in this, that the axis of the Earth is thought to retain its inclination, but the ecliptic to have receded today by a few minutes from those fixed stars by which it was formerly marked. But this cause belongs to book VII. The other cause is terrestrial and peculiar to Germany or the northern region, that the Sun at his lowest meridian altitude, on account of the density of the air, refracts his rays in it and sends them refracted to us, by which means he is reckoned to have another altitude than he truly has. This does not happen so sensibly in the more southern places of the Earth, in which the Sun at midday is not so low.
Teach, then, how to investigate by calculation the declination of any point of the ecliptic. A triangle is to be conceived between the mutual sections of the equator, the ecliptic and a circle of declination drawn through the poles of the equator, in which the angle at the section of the equator and the circle of declination is a right angle; moreover the angle of the section of the equator and ecliptic is known; finally there is given, or taken as known, the side between the nearer section of the equator with the ecliptic and the point whose declination is sought. Three things, therefore, being known in the triangle, the fourth cannot be unknown, namely the declination.
FE is the colure of the solstices, AD the ecliptic, F its pole, AE the equator, G its pole, B the point taken, whose declination BC is sought. DE is the greatest declination, the measure of the angle BAC; ACB is a right angle, AB the side given. The process, if only one declination is sought, is this, the shortest:
I. Let the declination proposed be that of the 17th degree of Taurus, which is distant from the nearer vernal section by 47° 0′.
Side of the ecliptic 47° 0′—AB.
Greatest declination 23° 31′ 30″—BAC or DE.
Its complement, as less than a quadrant, 19° 28′ 30″, sine 33339. Subtract.
Half, 29193, is the sine of the arc 16° 58′ 22″, which is the declination sought (BC).
II. Let the declination proposed be that of the 13th degree of Cancer, to which, since the autumnal section at 0° is nearer, the side of the ecliptic is 77°.
Its excess over the quadrant.
The sine of the arc 22° 53′ 13″, which is the declination sought (BC).
But if many declinations are to be investigated in order, it is better to multiply the sine of the greatest declination by the sines of all the arcs of the ecliptic in order, up to the quadrant. For the quotients, the last 5 figures being cast away, are the sines of the declinations sought.
How, conversely, is the quantity of the arc of the ecliptic to which a declination belongs sought from the declination? The sine of the declination, increased by 5 ciphers, is divided by the sine of the greatest declination; the quotient is the sine of the arc sought.
The sine increased by 5 ciphers.
Sine of the greatest declination.
The quotient is the sine of the arc 46° 14′ 40″.
What is here to be observed, in summary, about the declinations of the different points of the ecliptic? 1) In the quadrants of the ecliptic beginning from the four cardinal points, the points which are equally distant from the equinoctial points, two and two, namely opposite to each other, have equal declinations; so also those 4 among themselves which are equally distant from the solstitial points, with this difference, however, that two points equally distant from the same solstitial point have the denominations of that same solstitial point; those from the same equinox, contrary denominations among themselves.
2) At the thirtieth degree from the equinoxes, before and after, the declination is half the greatest.
If the inclination of the axis of the Earth is the cause of the declination of the ecliptic, and if that inclination of the axis remains constant through the whole year: how does it come about that the declination of all the parts of the ecliptic is not the same? Although the inclination of the axis of the Earth is always the same upon the plane of the ecliptic, and all the positions of this axis are parallel to one another, so that the northern pole of the Earth always inclines toward that part of the sphere of the fixed stars where the beginning of Cancer is reckoned: yet it does not always incline toward the Sun, for the globe of the Earth is carried round with its axis, poles and centre about the Sun, as began to be said in book II and was taken from the theoretical doctrine: by which carrying round it comes about that the northern pole of the Earth, inclining toward Cancer, when the Earth is carried under Capricorn, inclines toward the Sun, since he is seen under Cancer, but the same being transposed under the opposite sign of Cancer, itself inclining toward that sign as before, turns away from the Sun, who then appears to it in Capricorn. It follows, therefore, that under the signs of Aries and Libra the axis of the Earth, inclined indeed toward Cancer as before, but having the Sun at its side, neither inclines toward him nor away from him, but makes a right angle with the line which joins the centres of the Sun and the Earth. By these principles, then, it comes about that the Sun, or the ecliptic beneath which the Sun is perpetually seen, at the beginnings of Aries and Libra is equally distant from either pole of the Earth, that is, falls on the equator, which circle is midway between the poles, in Cancer declines toward the north pole of the Earth, in Capricorn toward the south; and, what follows, that as the Sun is seen to pass from Capricorn to Aries, his southern declination gradually decreases and vanishes, and again from Aries to Cancer the northern declination gradually arises and is completed.
What do astronomers call ascension and descension? The same as the Greeks by compound words synanatole and synkatadysis, as if you said in Latin co-ascension and co-descension. They are arcs of the equator which, with some part of the heaven situated outside the equator, whose certain beginning and certain end are given in the sphere, are seen to ascend above the horizon or to descend beneath it.
Why are arcs of the equator rather than of any other circle observed rising together or setting together? Because of the greatest circles the equator alone is moved uniformly; the equal parts of the others are often moved in unequal times.
Granted this is true of the greatest; but there are also smaller circles parallel to the equator, which are moved no less uniformly than the equator: could we not also measure the times of rising and setting of the others by the uniform motion of these? Not so conveniently, nor always nor everywhere, that is, not in every position of the sphere. For every smaller circle in some position of the sphere stands wholly above the horizon, where its points neither rise nor set. Again, because it is chiefly the times of the arcs of the ecliptic that are to be measured, the other parallels are either not connected with the ecliptic, or not at its cardinal points, except the two tropics, which are connected with it at the cardinal points of the beginnings of Cancer and Capricorn, but not the chief ones. The equinoctial alone rises and sets everywhere in every sphere, wherever anything of the heaven rises and sets, and is connected with the ecliptic at two most opportune beginnings, namely of Aries and Libra, so that it cuts it in the middle.
Prove by an evident argument that equal parts of the ecliptic rise in unequal times. Let whole semicircles be taken, and let a region and time be taken when the day is longer than its night, as in Germany at the time of the solstice the day is of 16 hours, twice as long as the night; and consider that as the Sun rises the part of the ecliptic that is diametrically contrary to him sets, and again, as that part rises, the Sun sets: for the ecliptic and the horizon are greatest circles, cutting each other into equal parts. From that time, therefore, when the Sun rises, until his opposite rises as he sets, a semicircle of the ecliptic has risen successively and 16 hours have meanwhile elapsed. But from the time at which, the Sun setting, the part opposite to him rises, until the Sun rises, the remaining intervening semicircle of the ecliptic again rises successively, and only eight hours elapse meanwhile: one semicircle of the ecliptic, therefore, rises twice as fast as the other.
Why do you say in Germany? Is, then, one and the same arc of the ecliptic, by its single motion, of different speed in different places, and how is this possible? The rising and setting of the points of the heaven, and therefore of the ecliptic itself, depends not on the revolution of the terrestrial sphere alone, but also on the accidents or deceptions of sight and on the imagination of the horizon. One and the same motion, therefore, is not truly made unequal through different places, but the horizons of different places have different termini, distant from one another, of the beginnings and ends of the equinoctial circle, rising and setting together with the same beginnings and ends of arcs of the ecliptic.
But why does not the same happen also to the equinoctial itself through the horizons of different places? Because that motion of the Earth, by which the risings and settings of the stars are represented, is according to the course of the equinoctial, not according to the course of the ecliptic. Hence it comes about that the horizon and the equinoctial cut each other all round at the same point of the horizon, but the ecliptic by its different parts cuts the horizon at different points, and these also differing through the intervals of places or of the positions of the sphere.
In how many ways do we investigate the ascensions of arcs of the ecliptic? In two ways. For either we take those arcs continuous, that is, begun from the common vernal section with the equator, or discrete, that is, not beginning from the vernal section.
Do we not also seek the ascensions of stars or of points of the ecliptic, which are not arcs, but termini of arcs? Whenever it happens that we speak thus, for example the ascension of the 23rd degree of Leo and the like, then there is understood the whole continuous arc of the ecliptic from the beginning of Aries to the named 23rd degree of Leo. It is, therefore, only a compendious way of speaking. Hold the same also when we say the ascension of a star. For it is just as if we took an arc of a great circle, intercepted between the vernal intersection and the star, and sought the ascension of that arc.
How are the ascensions or descensions of arcs of this kind found? The sphere being set as the reason of the place requires, the beginning of the given arc is placed on the rising horizon and the point of the equator through which the horizon then passes is marked with some sign; the same is done with the end of the given arc. The arc of the equator, therefore, intercepted between the two signs made is the ascension of the proposed arc of the ecliptic. If the same is done in the western part of the horizon, the descension of the same given arc will be marked in this way.
In a continuous arc only its end is placed on the horizon. For its beginning and that of the equator coincide at the same point, nor is there need of the indication of the horizon. The number, then, put to the point rising together with the end immediately gives the quantity of the ascension or descension, in parts or times, of which the whole circle has 360.
How many are the kinds of ascensions and descensions? As many as there are kinds of position of the sphere. For as after the right sphere there follow ninety positions of the sphere through as many degrees of elevation of the pole, and infinite intermediate ones could be named, until the last has the pole in the very zenith and the horizon coinciding with the equator in the parallel sphere: so also the ascensions begin from the right, passing through as many oblique, and end in no ascension and descension. For in the parallel sphere the heaven turns in the manner of an upper millstone (or the Earth with the horizon turns, as if someone should turn the lower millstone while the upper stands), for nothing either rises or sets.
In the tables of the first movable of Regiomontanus, Reinhold, Maginus and others there are, after one table of right ascensions, 89 other tables of oblique ascensions for each degree of elevation of the pole. Others also have constructed special tables for intermediate elevations of the pole.
Is there any ambiguity here which can deceive the unwary? Indeed. For in one way the whole circle is said in the right sphere to ascend rightly, in the oblique obliquely. But in another sense the different parts of the whole circle, compared with one another, are said to rise some rightly, some obliquely, in the right sphere as well as in the oblique.
With what regard are arcs said to ascend rightly and obliquely? Arcs of the ecliptic which cut the horizon at more nearly right angles, as being more erect, are said to ascend rightly, those at more oblique angles, as being more inclined, obliquely.
By what argument do we know the greater or less straightness and obliquity of these angles? And which arc ascends rightly, which obliquely? The more oblique the angles, the less ascension belongs to the arcs; therefore the more right, the greater the ascension. In general, then, that arc of the ecliptic is said to rise rightly with which an arc of the equator greater than itself ascends, that one obliquely with which a less.
Of the Right Ascensions of Points and Arcs of the Ecliptic.
Is it that, as to each place or position of the sphere its own table of oblique ascensions is assigned, so right ascensions belong to the right sphere alone, to those namely who dwell under the equator? Nay, right ascensions must be inquired into in every position of the sphere, first because we need them for the investigation of the oblique, which are not known without the right, then for their own sake: because as much as in the single right sphere ascends with any arc of the ecliptic, so much also passes the meridian with it in every sphere. For since the ecliptic passes the meridian unequally also by equal parts of itself, we must needs measure these unequal passages by the uniform rolling of the equator through the celestial meridians of places.
How is the right ascension of any arc of the ecliptic sought by geometrical calculation? We use the same triangle ABC (figure 38) as before, when the declination was sought, because the circle of declinations BC also represents the right horizon, since each is drawn through the poles of the sphere. In this triangle, then, three things are given: the angle C between AC, the equator, and BC, the circle of declinations, a right angle; the angle BAC between the equator and the ecliptic; and AB, the arc of the ecliptic proposed, the side namely opposite the right angle, which begins at the vernal section and ends at a point expressed by the numbers of degrees and the names of the signs. Sometimes over and above a fourth also is given, namely BC, the declination of that point of the ecliptic: AC cannot then escape us, the arc of the equator terminated between the neighbouring equinoctial point A and the circle of declination BC, which arc AC is the right ascension of the arc of the ecliptic AB.
The process by the angle of the section, or the greatest declination, is this:
Greatest declination—DE—23° 31′ 30″.
Its complement GD 66° 28′ 30″, sine 91688. Let there be put the 13th degree of Aquarius, whence to the neighbouring section, 0° of Aries, there are 47°, tangent 107237.
Let them be multiplied, the last 5 figures being cast away; there comes out 98323, the tangent of the arc 44° 31′ 6″.
So as in this instance 47° taken from the end of the ecliptic, or 30° of Pisces, leave 13° of Aquarius, so also 44° 31′ 6″, taken from the end of the equator, or times 360, leave the ascension.
How, conversely, is the arc of the ecliptic that ascends together with it investigated, a given arc of the equator being given, in the right sphere, or mid-heavening in every non-parallel sphere? By the contrary process, which stands thus:
Let the arc AC of the equator be 44° 31′ 6″. Tangent 983320.
Continue with 5 ciphers. The sine GD, the complement of the greatest declination, 91683 (divide).
The quotient 107236 is the tangent of AB, 47°, the arc of the ecliptic ascending together.
The more notable and easily remembered rules of right ascensions.
1. Compare right ascensions with descensions. The right ascension is equal to the descension of the same point: because each semicircle of the horizon cuts the equator at a right angle, and the same arc of the ecliptic remains and the same angle between the equator and the ecliptic: three things, therefore, remaining in the triangle, the other three must also remain, among which is the arc of the equator, which there is the right ascension, here the right descension.
2. Compare the ascensions of opposite equal parts and semicircles. The ascensions of these also are equal to one another, however begun, because the angle of the vernal section is equal to the angle of the autumnal section, the rest as before. Either, then, they begin from the equinoctial points, and the proposition is plain of itself, or not from the equinoctial points, then what remains to the neighbouring equinoxes on either side is the same on either side: equals, therefore, being taken from equals, equals are left; or if they are semicircles, these are cut through the equinoctial points, and again the ascensions of the single equal parts on either side are equal, wherefore also of those joined, that is, of the whole semicircles.
3. Compare whole quadrants of the ecliptic with their ascensions. With a whole quadrant of the ecliptic from a cardinal point there ascends a quadrant of the equator.
For the horizon FE, drawn through the solstitial points D, E, passes also through F, the pole of the ecliptic: it cuts it, therefore, at a right angle no less than the equator. Since, then, ADE and AED are equal, AD and AE will also be equal.
If a quadrant of the ecliptic does not begin from a cardinal point, it is not equal to its ascension, but either greater or less.
The horizon GC, not drawn through the solstitial point D, wanders also from the pole F of the ecliptic AD, and cuts it, therefore, obliquely at B, the equator at C rightly, and the meridian does the same. The parts of the equator between the horizon and the meridian are, therefore, quadrants; but the parts of the ecliptic, whose pole is intercepted at the same time, are less than a quadrant, the rest greater.
4. Compare the ascensions of different smaller parts with each other. Parts of quadrants are not equal to their ascensions, and those which begin from the equinoctial points ascend obliquely, having ascensions less than themselves, but those which begin from the solstitial points ascend rightly and have ascensions greater than themselves. When of two arcs of the ecliptic one, begun from an equinoctial point, is equal to the ascension of the other terminated at a solstitial point, or conversely: the difference between the arcs and their ascensions is likewise equal. Discrete parts, the nearer they are to the equinoctial points, ascend the more obliquely, the nearer to the solstitial, the more rightly.