Epitome Astronomiae Copernicanae: Liber III, part 2
What is to be done with declining planes? Upon these also the altitude of the pole and the angle of the meridians must be sought: but the process for the altitude of the pole above the horizon of the place uses the altitude of the equator, or the distance of the pole from the zenith, for this is the altitude of the pole above the plane of the meridian; in the rest it is plainly the same as before in inclined planes. And this altitude of the pole when found is always less than the distance of it from the zenith which was used.
What is the astronomer to do with the circles of position? The elevation of the pole above each of them must be sought, as above some horizon. Again, that is always less than the elevation of the pole above the horizon of the place.
Whence is this obtained? Either from the inclination of the circle of position to the meridian of the place, as Campanus and Gazulus construct the circles of the houses, and then the process is plainly the same as before, when the altitude of the pole was sought above the circle of an inclined plane in gnomonics; or from the arc of the equator between the meridian and the circle of position, as Regiomontanus constructs the circles of the houses. Then a right-angled triangle is formed of the meridian, the equator and the circle of position, in which the side in the equator is given, the side in the meridian is the altitude of the equator; wherefore the angle opposite to it cannot escape us, which the altitude of the equator measures above the circle of position.
The process is this: let the altitude of the equator of the place be 41° 44′, tangent 89201; let the arc of the equator between the meridian and the circle of position be 30° 0′, sine 50000.
Let it divide: the quotient 89201: 50000 = 178400 is the tangent of the arc 60° 44′ of the altitude of the equator above the circle of position. Therefore 29° 16′ is the altitude of the pole above the same.
By what proof is it established that the altitude of the pole in places on the surface of the Earth is always the same? At Prague 200 years ago the altitude of the pole was observed to be 50° 6′, as it is today.
But it is read that Joh. Maria doubted of this a hundred years ago, from a comparison of the geography of Ptolemy with the modern. The fault is believed to lie in Ptolemy, who did not observe in the places of the west in person, but without doubt copied them from a less accurate geographical table, or investigated them from the length of the summer day, as he had learned it from a rough report.
How do we measure the altitude of a star or its distance from the zenith? With the instrument of the quadrant, whether a separate one, or one that is part of a circle in the astrolabe; one side of which quadrant is to be directed, by the help of a plumb line, to the vertical point, the other to the plane of the horizon, and then the rule is to be directed to the star, with sight for guide and the aid of the sights. As much as it then cuts off of the divided rim, so great is pronounced the altitude of the star, if the quadrant is divided into 90 parts, the numbering proceeding upward from the horizon; but if the order of the numbers proceeds from the top toward the horizon, then the distance of the star from the zenith is cut off.
In figure 31 let AB be directed to the horizon, AF to the zenith, AD the rule to the star: then BC will be reckoned for the altitude of the star, CF for its distance from the zenith.
But how is it known that the star is at this moment in the meridian and highest, since the meridian circle is not open to the eyes in the heaven? By the investigation of the meridian line and the placing of the quadrant upon it or upon a parallel to it; for a star falling in this plane, sight being judge, is in the meridian.
How is the meridian line obtained? This too in various ways, but the mode proper to this place and the most expeditious is this: on a clear night look at the last star in the tail of the Little Bear, for it is near the pole; that quarter, therefore, is the quarter of the north in our hemisphere, and opposite to it is the south, more or less. The southern quarter being known, choose a star which is to the left of the south; toward its rising take the altitude DE by the quadrant TD and fix the rule, but mark the position of the quadrant on that occasion, a line SQ being drawn on the plane of the horizon on which the quadrant stands. Then wait until the star has passed the southern quarter, which will always become higher up to midday, afterwards becomes lower again and at length acquires again that altitude DE which you had noted with the rule fixed. Attend carefully, therefore, when this happens, the plane of the quadrant being always directed to the star, until the star is again seen through the sights of the fixed rule SE, and mark the position in which this happens, another line SO being drawn in the plane of the horizon; then continue both lines until they cut each other in S.
This done, bisect the angle QSO between the two lines, and the bisecting line SP will be the meridian of your place.
The manner by day by means of the Sun is similar, and more popular and easy. On a plane which is parallel to the horizon let a circle ALC be described and at its centre S let a style ST be erected perpendicularly of such a length that its shadow some hours before midday may end at the circumference of the circle CA: when this is exactly done, let that place C of the circle be marked and let the time be awaited after midday, when the shadow of the style again touches the same circle at another place A, which place also let be marked, and the arc CA, intercepted between the two places, being bisected, let a straight line SL be drawn from the centre S through the bisection, which will be the meridian line.
What is declination? The word declination takes its origin from the proper apparent motion of the Sun, or from its orbit the ecliptic, which, since it falls on the equator in two places, after those points gradually turns aside and declines from the equator. Declination, therefore, properly means the quantity of the arc of a circle drawn through the poles of the world, by which arc any point of the ecliptic, successively declining from the equator, has receded. Afterwards use prevailed that the distance of any star whatever, even one existing outside the ecliptic, from the equator, on a circle drawn through the poles of the equator, should be called the declination of that star.
Declination, then, in this place is the arc of a circle drawn through the poles of the sphere, intercepted between the equator and a star or any point of the spherical surface whose declination is sought.
How is the declination of any star or point gathered from observations? If a star has come into the meridian from the quarter of the equator, the altitude of the equator and the meridian altitude of the star or point are to be compared with each other. For if the altitude of the star is greater than the altitude of the equator, the declination will be northern, if less, southern. And then the subtraction of the less from the greater gives the quantity of the declination. But if the star falls into the meridian toward the quarter of the north, the altitude of the pole is to be used in place of the altitude of the equator. The difference between this and the greatest or least altitude of the star, taken from a whole quadrant, leaves the declination of the star, northern only in our hemisphere.
How are rising and setting stars distinguished by declination from those that do not rise or do not set? When the declination of a star is greater than the altitude of the equator, the star, if northern, does not set, because, as great as is the altitude of the equator at midday, so great is the depth of the opposite point of the equator in the north beneath the horizon; the star, therefore, being more distant from the equator than the horizon, stands above the horizon when it is lowest. But if a star of so great a southern declination is offered from books, you may know it to be one of those which at the proposed altitude of the pole do not rise or are not seen. Those stars therefore only rise and set whose declination is less than the altitude of the equator.
Can the declination of a star be taken also outside the meridian? If the meridian line is known and established, then from the observed azimuth, the altitude of the pole and that of the star, its declination is computed by the calculation of triangles. For a triangle is constituted, best known in the first motion, whose angles are the pole P, the zenith V, the star S; the angle at the zenith being known from the observation of the azimuth HG, GR, its legs are known. For the one, VP, between the zenith and the pole, is the complement of the altitude of the pole, namely as great as is the altitude of the equator AH, the other, VS, between the zenith and the star, is the complement of the altitude of the star SG, which is called the distance of the star from the zenith. Three things being known, the fourth will be opened, namely the side PS between the pole and the star, from which the declination follows easily. For if this side is found less than a quadrant, its complement to a quadrant, SE, but if greater, its excess over a quadrant, SQ, will be the declination sought, northern in the one case, southern in the other. The rules themselves are to be sought from the geometers: here you have a model of the work.
From the altitude of the equator and the distance of the star from the zenith: the greater 42°, complement 48°; the lesser 30°. The same 30 The sum less than a quadrant, therefore complement 18°, sine 30902 (subtract).
(If the sum had been greater than a quadrant, you would have added the sine of the excess to the remainder 66913) half 33457.
Let the angle at the zenith be 100°.
The versed sine of the angle, 117365; the half above, 33457; (90°; 100000) multiply, the last figures being cast away: the product 39267 is less than the sine of 10°, 17365.
This is the declination of the star, northern, because the quotient is less. If the product had been greater, and the sine subtracted from it, the declination would be southern.
How, the declination of a star being known from the books of some eminent craftsman, and the altitude of the pole, is the meridian line in turn investigated without the tedious waiting of the hours before and after midday? By the observation of the altitude of a star in a certain position of the instrument, and the same triangle. For three sides are given, PV, VS as before, and PS, the northern declination SE being subtracted from the quadrant PE, or the southern declination SQ added to the quadrant PQ. Then the angle SVP, or its measure GR, is sought. Therefore, the position of the instrument being noted, or the point G of the horizon toward which it is directed, it is also disclosed how much the meridian HVR is inclined to its plane, or the angles HCG, GCR in the plane of the horizon.
The first part of the process remains the same as before, the other part is this.
Let the northern declination be 35° 50′, sine; the first sine (you would have added the sine of the southern declination); let the half above divide:
17365 ... is the versed sine of the arc The angle at the zenith is therefore so great, and the exterior angle GYA is 80° 0′.
What do astronomers call the angle at the pole, or between the meridian and the circle of declination of the star? It is called the elongation or distance of the star from the meridian. In the figure it is SPY.
By what means can the quantity of this angle at the pole, that is, its measure in the equator, be investigated? There is need of the knowledge of the altitude of the pole and the declination of the star, to which must be added either the altitude of the star or its azimuth from observation; finally we can do without the declination if we have in its place the altitude and azimuth together: and in the same triangle as before the thing sought is found. But the latter two ways are more laborious and their use is rarer.
The process, then, with a southern declination and the altitude is this. For PS, SV, VP are given, VPS is sought.
greater than a quadrant, therefore the excess 30° 57′ 43″, sine 51449. Add.
(If the sum were less, you would have subtracted the complement.) The aggregate 123348; half 61674 is the divisor.
Let the altitude of the star be 23° 45′, sine 40275. First sine 71899.
The quotient 181882 (6 figures always come out). Let them be added: 112174. Continue with 5 ciphers.
is the versed sine of the arc 144° 58′. The half before, 61674.
Its complement to a semicircle (the division)
35° 2′ is the angle at the pole.
The process with a northern declination and the altitude. From the altitude of the equator and the complement of the declination.
The excess of the greater over the quadrant 15° 7′, sine 26079. Add.
(If the sum had been less, you would have subtracted the sine of the complement.) The aggregate 110527; half 55264.
Let the altitude be 46° 25′, sine 72437. First sine 84448. To be subtracted 12011.
The quotient 21734 (6 figures may come out) is the versed sine of the arc 38° 30′, because 21734 taken from 100000 leaves 78266, the sine of the arc 51° 30′, the complement. This quotient itself, 21734, as the versed sine of the arc 38° 30′, is therefore the angle at the pole sought.
Is the angle at the star also commonly computed, that is, between the vertical and the circle of declination? Altogether; its use occurs in refractions, parallaxes, solar eclipses and elsewhere everywhere. (In the figure VSP.)
Describe its varieties in general. When the star is rising this angle is the least, nor does it ever equal the altitude of the equator, unless the rising star is on the equator. From this the angle grows and becomes a right angle on this side of and beyond the meridian; in a star, therefore, whose declination is greater than the altitude of the pole of the same name, it increases until in the meridian it becomes equal to two right angles. But if the declination is less or even contrary, it diminishes again, until in the meridian it wholly vanishes.
Teach how to compute this angle at the star. We need, in the same primary triangle, the complement of the altitude of the pole, PV, namely the arc of the meridian between the zenith and the pole, the complement or excess of the declination of the star, or the side between the star and the pole, PS, and the complement of the altitude of the star, or the side between the zenith and the star, VS, or in its place the azimuth of the star, HG, GR, or the angle at the zenith SVP, or even the angle at the pole VPS, so that various cases arise.
The process by the three sides. From the complement of the altitude of the star and the distance of the star from the pole.
The sum greater than a quadrant, 148° 45′. Sum 73° 45′; sine 96005.
Its excess, therefore, 58° 45′, sine 85491; add.
(If the sum were less, the complement. The aggregate 181496.
you would have subtracted the sine.) Half 90748.
First sine 69005 (add, if the declination is southern, 147302; you would subtract, if northern).
Let the half above, 90748, divide. The quotient 192072 is the versed sine of the arc 157° 2′. This, because the declination is southern, its complement to a semicircle, 22° 58′, is the angle sought; in the northern case the arc of the quotient itself, as a versed sine, would be the angle sought.
The process by the azimuth in place of the altitude, with the two remaining sides.
Between the zenith and the point of the equinoctial rising or setting (if it were less than a quadrant, you would have added the sine of the complement) the remainder 21043; half 10522.
The distance of the star from the nearer pole 82° 33′, sine 99156. 10522: 99156 = 10616 is the sine of the angle sought, 6° 6′.
Note that in this process it is the same as if you had taken the angle of the azimuth within or without the triangle, whichever is less than a quadrant: 9° 40′, and the distance of the pole from the zenith 38° 48′, the complement.
And, had you dealt with the sines of these complements as above, you see that the same arcs come out. Where, if the sum exceeded a quadrant, you would add the sine of the excess.
What is akin to the declinations of the stars? The latitudes of places on the Earth, of which a little before, because they are defined by subordinate and corresponding circles.
How do astronomers commonly speak of the places of the horizon in which any star seems to rise or set? They use the term ortive amplitude.
What is the ortive amplitude? It is the arc of the horizon intercepted between the equator and the point of the star’s rising. Although the name seems first to have come from whole constellations, and it was asked how wide a space of the horizon any constellation, with all the stars assigned to it, occupies in rising.
How is this ortive amplitude known, or the arc of the horizon by which the points of the star’s rising and setting are distant? In the sphere it is plain to the eye, more or less, how great an arc of the horizon is intercepted between the equator and the points of the star’s rising or setting, the pole of the sphere being erected to the right altitude and the star brought round to the horizon.
But if one wishes to examine this accurately by calculation, since the sphere cannot be so fine: it can be done in the same triangle as hitherto, but by an easier method. For, PV being given, the arc between the pole and the zenith, which is called the altitude of the equator, the arc PS between the pole and a star of the same hemisphere, which is the complement of the declination, and finally the arc VS between the zenith and the star, which is always a quadrant, since the star is placed in the horizon, there is sought the angle SVP or SVA at the zenith, which measures the arc of the horizon between the star and the nearer semicircle of the meridian, by this process.
Declination of the star 40°, sine 64279. Altitude of the equator 42°, sine 66913.
The division (5 ciphers being added) gives the sine 96064 of the arc 73° 52′, which is the ortive amplitude, whose complement 16° 8′ is the arc between the star and the meridian, and its double 32° 16′ is the distance of the rising and setting in the northern part of the horizon.
If the declination is southern, what comes out also takes its name from the southern quarter; the rest is the same on either side.
Another triangle can be formed for this, below the Earth, when the star is northern, or above the Earth, when southern, between the circle of declination SQ, the horizon ST and the equator QT, with a right angle, and the process remains the same. For the same quantities are given: one side SQ, the declination, the angle Q a right angle, and the angle STQ between the arc of the horizon, or ortive amplitude, sought, ST, and the equator, whose measure is the altitude of the equator HA.
4. How does it come about that the position of some constellations is seen overturned by the daily motion, of others not at all? This happens to the stars on account of the turning of the face of the spectator, standing erect, toward contrary quarters, into which the stars pass, or toward a quarter always the same.
How manifold is the overturning of position? Twofold, according to the twofold difference of the stars in the oblique sphere; one complete, the other half; as when those which seemed to stand upright afterwards appear laid down. For of those constellations which do not set at an elevation of the pole greater than 45°, which are outside the compass of the parallel circle VQ (figure 34), which the vertical point V describes in the first motion, these do not seem to be overturned. For the parts or members of the figures are always ordered in the same way toward the pole of the world, the zenith, therefore, constituted at whatever point of its parallel, always inclines from the constellation toward the quarter in which P the pole is, whether the constellation appears above the pole or below it. The observer standing, therefore, beholds the figures as if he had his zenith beneath the pole itself in the parallel sphere; he sees them placed in the same way, as well high in the southern quarter as low in the northern quarter. Here, therefore, the turning of the face following the star prevents the position from being overturned. But those which are within the compass of the parallel VQ through the zenith, so that they come twice to the meridian in the northern quadrant of the meridian, once when they are highest at S, once when lowest at I, because then in both places the face of the spectator is turned toward the same quarter of the pole, their position must necessarily be completely overturned; for the parts of others farthest from the pole are the uppermost, those of the low ones, on the contrary, the parts nearest the pole.
The contrary happens in the stars which rise and set. For the turning of the face, following the constellation toward contrary quarters, represents their position as overturned. For of the rising ones the preceding parts are the upper, of the setting ones the following parts. In the right sphere, therefore, a complete overturning happens in this way also, in oblique spheres a half one; for those which rise upright set laid down, prone or supine, and that variously according to the greater or less obliquity of the sphere and the position of the constellations on the surface of the sphere of the fixed stars.
5. How is the stay of any star or point above the horizon known? By the help of the parallel drawn through the star or point: for the horizon cuts it, and so the part lying hidden beneath the horizon is the proof of the absence of the star below the horizon, or, if we are speaking of the Sun, of night, and is called the nocturnal arc; the part standing out is the proof of the stay above the horizon, or, when we speak of the Sun, of day, and is called the diurnal arc, which one may even measure with a compass and compare with the whole parallel.
This would follow if the star or Sun described such a parallel by its motion; but you said above that the star or Sun does not come to other points of that circle, but is as it were fixed to it at a single point. This is no hindrance, for, as you were told in the second book, another immovable parallel is imagined, standing above this movable parallel in the same continued plane, which parallel the star describes as it were on some hollow floor which covers the fixed stars. Such an immovable one, therefore, this movable parallel here represents.
But this fiction fits only the motion of the heaven; you, however, wish the Earth to move. It has often been answered that the reason is plainly the same. For such as is the immovable parallel here imagined on some floor above the fixed stars, such also answers on the Earth a parallel, subordinate to the celestial parallel, as appears from the genesis of the parallels. Imagine, then, that it comes about that by the rolling of the Earth the star passes on the very surface of the Earth through the places set in that parallel, but the spectator is not on the surface of the globe but within, at the centre, and has a horizon parallel to that which touches the surface at the place of the spectator: plainly the same appearance will arise as before, when we suppose that the star goes round in the lofty heaven and the spectator stands on the surface of the globe.
I should like to measure this upper arc of the star or point not mechanically with a compass but by accurate calculation. This is done again in that first triangle of the first motion, which is between the pole, the zenith and the star. For there must be given the side PV (figure 37) between the upper pole and the zenith, the side PS between the pole and a star of the same hemisphere, which is the complement of the declination, and finally the side VS between the zenith and the star, which is always a quadrant, since in this process the star must always be placed in the rising or setting. From these three things known beforehand the angle VPS at the pole is sought, measuring the semidiurnal arc SM of the star. Since, then, the declination is less than the altitude of the equator (otherwise the star would not rise and set, as was said before), the process is this:
The northern declination of the star 40°, tangent 83910; let it be increased by the ciphers of the radius.
The altitude of the equator 42°, tangent 90040 (let it divide).
There comes out the sine 93192 of the arc 68° 44′; add a quadrant, 90, it makes 158° 44′, the angle at the pole and so the semidiurnal arc of the star. Therefore the double, 317° 28′, is the upper arc of the star, or in the Sun the diurnal.
But if the star is placed on the equator itself, its parallel is one of the greatest, wherefore it will be cut by the horizon into two semicircles, and the day, or presence of the star, is equal to its night, or absence.
If the declination of the star is southern, the opposite triangle beneath the Earth, LSN, is to be taken, whose angles are L the lower pole, N the nadir, S the star; and the same process remaining, the lower arc of the star comes out, or in the Sun the nocturnal, which being subtracted from the whole circle leaves the diurnal arc.
Again here another triangle, SQT, can be formed, described a little before, between the declination SQ, the ortive amplitude ST and the equator QT, which in a northern star is beneath the Earth, in a southern above. For there comes out QT, an arc of the equator, which below will be called the ascensional difference, measuring the excess of the semidiurnal arc of the parallel over a quadrant, for the process is the same.
Review all the varieties of these stays through all three positions of the sphere. In the right sphere, or among those who dwell under the equator, the upper arcs of all the stars are equal to the lower, so that every star stays 12 hours above the horizon, 12 below. In oblique spheres, in which the altitude of its own pole is the same for each, as stars not rising to one do not set to the other, and conversely, so also stars grazing the horizon for one as if about to set, since they do not set, graze the horizon for the other likewise as if about to rise, since they do not rise; but of the rising stars, those of greater declination, northern on one side, southern on the other, have longer days, shorter nights; those of contrary declination the contrary, up to those which fall on the equator, which alone make the days equal to the nights, thence the greater declination of the contrary quarter they have, the longer is their night, the shorter their day. Finally, those which have equal declinations of contrary quarters have in the same place of the Earth the reckonings and measures of days and nights exchanged, so that one day is equal to the other’s night. Again, the same star at the same altitude of the contrary poles, as long as it stays here above the horizon, so long below there, and conversely.
In parallel spheres nothing rises, nothing sets, wherefore half the stars have in one a continuous day, in the other a continuous night; the other half the contrary.
You said that the rays of the stars are refracted about the horizon: does not this, then, disturb the doctrine so far delivered? Since the stars seem by refraction to be raised higher than is just in the vertical circle, both at rising and at setting, therefore their declinations in the right sphere are not changed in any way that can be felt, in oblique spheres they are changed sensibly; and of those which are of the same name as either hemisphere the declinations are represented greater than is just, of the others less than is just, whence it follows that the diurnal arcs of the former are greater than is just, of the latter less, which difference is greatest in the right sphere, none in the parallel; nay, the ortive amplitude is everywhere altered, not at all in the right and parallel sphere, but most about an altitude of the pole of 45°.