Epitome Astronomiae Copernicanae: Liber III, part 4
For the angle between the ecliptic and the horizon is a right angle at the solstices and most oblique (that is, acute) at the equinoxes, while the angle of the equator and the horizon is always a right angle and greater than that; so a greater arc of the ecliptic is subtended by this angle than an arc of the equator by that.
By what point are the parts of a quadrant that begins at the solstitial point and ends at the equinoctial point, which ascend right, distinguished from those which ascend obliquely? By that point of the ecliptic at which the difference between the arc of the ecliptic and its right ascension is greatest, or which divides the quadrant into two parts, each of them equal to the ascension of the remaining part, so that the arc of the ecliptic together with its ascension makes up the quadrant. This necessarily happens about the middle of the quadrants.
How is this point sought geometrically? Regiomontanus defines it, from the Arab Geber, thus: that the sine of its distance from the pole of the equator is a mean proportional between the sines of the arcs by which the extremities of the quadrant are distant from the same pole.
The beginning of the quadrant is 90° from the pole, and its sine is 100000; the end of the quadrant is 66° 28′ 30″ from it, with sine 91688. Multiplied together, these have the root 95754, whose arc is 73° 14′ 36″; the point sought is therefore so far from the pole, and so is 16° 45′ 24″ from the equator. But the point which is so far from the equator is found, by the doctrine above, to lie 46° 14′ 40″ from the intersection of the ecliptic. So great an arc, then, ascends obliquely from the intersection, and the remaining arc of 43° 45′ 20″ ascends right. And when everything is multiplied by four, the parts that ascend obliquely are 184° 58′ 40″, and those that ascend right 175° 1′ 20″.—But some, accustomed to whole signs, favour the right ascensions, saying that eight signs ascend right and four obliquely, because the right ascensions are found distributed in Taurus, Gemini, Cancer, Leo and Scorpio, Sagittarius, Capricorn, Aquarius; they conceal, however, that the ends of the signs Taurus, Leo, Scorpio and Aquarius ascend obliquely.
There remains in our triangle the angle between the ecliptic and the horizon, the meridian, or the circle of declination, to which the right ascension is subtended. What observations have you on it? This angle, as was already said of the right horizon, is acute, and the smaller the nearer it is to the equinoctial points, yet it never equals the complement of the greatest declination; at the solstitial point itself it is a right angle. Beyond the solstitial point it becomes obtuse with respect to the farther equinox that precedes, and acute as before with respect to the nearer one that follows.
If from the neighbouring solstitial point an arc is counted on the equator equal to the proposed arc of the ecliptic begun from the equinoctial point, and from its end a circle of declination is drawn, its arc between the pole of the sphere and the ecliptic measures the angle which the meridian makes with the point of the ecliptic first proposed.
Thus if the point B is proposed (figure 38), and EQ is taken equal to AB, then, GQ being drawn, the arc GP will be the measure of the angle ABC. Conversely, if the point P is proposed, let EC be taken equal to AP; GC being drawn, the arc GB measures the angle APQ.
How is this angle computed? First, if a table of declinations and a table of right ascensions are at hand, seek among the right ascensions the complement of the arc of the ecliptic begun from the nearer equinox; then opposite it, in the column of declinations, you will find the complement of the angle sought.
Second, if you prefer to use calculation, the procedure will be this.
The arc of the ecliptic from the nearer equinox, 30°; its complement Multiply, cutting off the last five figures.
The complement, 69° 20′ 35″, is the angle sought.
What is the right ascension of a star or of a point in the sphere which lies outside the ecliptic? It is the arc of the equator intercepted between the beginning of Aries and the circle of declination of the star or point, and counted in the order of the signs. But more on this below, in the fifth part of the doctrine of the sphere.
What is akin to the right ascension of a star? The longitude of a place on the Earth.
What is the longitude of a place? It is the arc of the terrestrial equator (or also of the parallel drawn through the place) intercepted between the first terrestrial meridian and the meridian of the place, and counted in the order of the signs. More on this below, in the fifth part.
Of the oblique ascensions of points and arcs of the ecliptic.
How are oblique ascensions investigated? There is need of the declination of the proposed point, its right ascension, and the ascensional difference, which becomes known from the declination. For when this ascensional difference is added to the right ascension or taken from it, the oblique ascension of that point is obtained.
What is the ascensional difference and how is it investigated? When the proposed point is placed on the rising horizon, a right-angled triangle is formed by the horizon, the equator and the circle of declination of the proposed point. In this triangle three things are given: the side on the circle of declination, that is, the declination of the point; the angle of intersection of the equator and the horizon, which is measured by the altitude of the equator; and the angle between the equator and the circle of declination, which is a right angle. The fourth, therefore, namely the side on the equator or the ascensional difference, cannot escape.
Procedure: let the proposed point be 0° of Gemini or 0° of Aquarius.
Let the altitude of the pole be 48° 30′, the declination 20° 13′.
How does this ascensional difference stand in one and the same position of the sphere, and how is it to be used for the different parts of the ecliptic? First, points of the ecliptic equally distant from the solstitial points have the same ascensional differences, as also the same rising amplitudes; for they have the same declinations, through which the ascensional difference is investigated. Second, when the declination is northern, the triangle falls below the horizon and the ascensional difference is taken away from the right ascension; but if the declination is southern, the triangle is above the horizon and the ascensional difference is added to the right ascension, and so on either side the oblique ascension results.
Here HRI is the horizon, P the pole, PAR the meridian, AEQT the equator, EL a northern part of the ecliptic, EC a southern part of it; the proposed points are C and L, and PQC, PLT are circles of declination; the triangles are LTH and CQH. The ascensional differences HT and HQ are sought from the northern declination TL and the southern declination QC; ET and EQ are the right ascensions, and EH is the oblique ascension common to both sides, which is formed there by taking away TH, here by adding QH.
What general comparison of oblique ascensions for the different points of the ecliptic arises from this? First, two equal arcs of the ecliptic begun from the same equinoctial point, or at least equally distant from it, the one toward the antecedent signs and the other toward the consequent, have equal oblique ascensions. Second, of parts equally distant from a solstitial point, the one forward and the other backward, the oblique ascensions are unequal to one another. Third, hold the same also of equal opposite parts, since this follows from that.
Compare oblique ascensions with right ones in general. Equal portions smaller than a semicircle, begun from the equinoxes, those from the vernal one are quicker in rising in our hemisphere than in the right sphere, and rise more obliquely than in it; those which begin from the autumnal one become slower than in the right sphere, and are therefore said to rise more upright than in the right sphere, though improperly; for seldom in the oblique sphere, and only about the tropics, can the ecliptic rise more upright than in the right. But as for detached arcs, that is, those not begun from the equinoxes, the nearer they are to the vernal equinox, before or after, the more obliquely they ascend, and the nearer to the autumnal, the more upright.
How is oblique descension investigated? Since the ascensional difference subtracted makes the oblique ascension, the same added makes the oblique descension; and conversely, what is added to make the oblique ascension must be subtracted to make the oblique descension.
What analogy arises from this between oblique ascensions and descensions? First, as great as is the ascension of an arc begun from the beginning of Aries, so great is the descension of an equal arc begun from the beginning of Libra, and conversely. The same is true also of detached equal opposite arcs. So no tables of descensions are made; those of ascensions suffice. Second, the same parts that ascend right descend obliquely, and conversely.
How many kinds of positions of the sphere are there, with respect to the equator and ecliptic joined, by which oblique ascensions are varied? There are six positions in each hemisphere, to which the seventh, that of the right sphere, is added. For the zenith of the place falls either under the equator, or between the equator and the tropic, or under the tropic itself, or between the tropic and the polar circle, or under the polar circle itself, or between the polar circle and the pole, or under the pole itself. In the first of these positions the ascensions are only right, and of these we have already treated; in the last there are plainly no ascensions. There remain, therefore, for oblique ascensions five intermediate positions in each hemisphere.
Compare the ascensions and descensions of the two hemispheres with one another. First, as great as is the oblique ascension of any sign, degree or point of the ecliptic at a given elevation of the north pole, so great is the oblique descension of the same sign, degree or point at an equal elevation of the south pole, and as great as the descension is there, so great is the ascension here. Second, whatever is demonstrated of a given sign, degree or point of the ecliptic in the northern hemisphere will be true also of the opposite sign, degree or point in the southern hemisphere, the elevation of either pole being the same on both sides.
How does the ascensional difference stand to the right ascension in the various positions of the sphere? In the right sphere, just as the oblique ascension is nothing, so the ascensional difference of an arc of the ecliptic extended backward or forward from the nearer equinox has no quantity at all; in oblique spheres it acquires some quantity with departure from the equator, and so long as the place is between the equator and the polar circle it is less than the right ascension, under the polar circle it is equal to it, and within the polar circle it is greater than it, by all proportions in succession, the nearer one comes to the pole.
For under the polar circle the ecliptic and the horizon coincide whenever the pole of the ecliptic comes to the zenith. Hence a whole semicircle has either no ascension, since it is all consumed in the subtractive ascensional difference, while the remaining semicircle of the ecliptic adds to the semicircle of the equator rising with it, in the right sphere, the remaining semicircle. And since under the polar circle the angle EQO between the equator and the horizon is equal to the angle between the ecliptic and the equator, QEO, greater toward the right sphere and less toward the parallel, the rising amplitude QO under the polar circle is equal to the proposed arc of the ecliptic EO begun from the neighbouring equinox; outside the polar circle it is less, within it greater. When therefore the circle of declination PTO is drawn to the rising point O of the ecliptic, which separates the right ascension ET from the ascensional difference TQ, these parts so made also follow the proportion of the remaining side of each: ET that of the arc of the ecliptic EO, and TQ that of the rising amplitude QO.
What is the manner of the rising and setting of the signs in these positions of the sphere? From the equator as far as the polar circle all the signs rise, and in the right order; and the horizon cuts the ecliptic within the tropic twice at right angles, namely whenever the pole of the ecliptic comes upon the horizon; under the tropic this happens once; beyond the tropic as far as the polar circle the intersection is more and more oblique. Under the arctic polar circle the ecliptic coincides with the horizon once, and suddenly the whole semicircle, called the ascending, in the middle of which is the vernal equinox, rises at once in a moment; the remaining semicircle, through which the Sun is reckoned to descend, from Cancer through Libra to Capricorn, sets in the same moment; and at the same time, at the northern intersection (R) of the horizon and the meridian, it will begin to rise in the right order, until, the whole equator having revolved, its end, that is, 30° of Sagittarius, comes to the very point of midday (H), as if about to ascend, at the horizon. Thus the ascension of this half of the ecliptic has the whole equator rising with it.
But within the polar circle, since about the lower solstitial point some arc never rises but is always beneath the Earth, and about the upper solstitial point another arc is always above and never sets, an arc always the greater the nearer the pole is to the zenith, until under the pole itself each arc becomes a semicircle; the intermediate arcs rise indeed and set, but the one in which is the vernal equinox (in our hemisphere) in the inverted order, the other, in which is the autumnal, in the right order; and they have the whole equator rising with them, the one that rises in the inverted order, however, having the greater part of it. The rising amplitude itself follows the same variety. For between the equator and the polar circle this rising amplitude widens little by little from the equinoctial rising Q toward the north R and the south H, and the parts ordered from Cancer through Libra to Capricorn have their regions of rising from the northern parts R of the horizon toward the south H, and from Capricorn through Aries to Cancer in retrograde order from the south H toward the north R, at first in modest spaces, until under the polar circle itself and toward the interior this amplitude of risings ranges over the whole semicircle of the horizon, from the very northern point R of the horizon through the rising point Q as far as the very southern point H. But within the polar circle no such retrograde transposition of risings from the south to the north occurs from opposite arcs, but the rising arcs of each semicircle, both that which rises in the right order and that which rises in the inverted, make their earlier risings in the north R, their later ones toward the region of the east Q, and their last at the meridian point H of the horizon.
I speak, however, of the parts of the ecliptic without regard to the motion of the Sun through them. For if the series of risings which the Sun makes is considered, that will be another matter.
Whence is the arc of the ecliptic that is perpetually apparent or hidden made known? From the altitude of the equator, which, being less than the greatest declination of the ecliptic, requires us to seek, by the precepts given, which point of the ecliptic, at what arc from the equinoctial point, has a declination equal to the altitude of the equator. For the complement of that arc is half the arc that does not set, if the declination is northern (with us in the northern hemisphere), or that does not rise, if southern.
Let the altitude of the pole be 80°, and that of the equator therefore 10°. Such a declination is found to belong to 25° 47′ 16″ of Aries. Its complement to the quadrant, 64° 12′ 44″, doubled, makes 128° 25′ 28″. So great an arc does not set.
Compare in particular the whole halves of the ecliptic with their oblique ascensions in these positions of the sphere. Only those halves which lie between the equinoctial points are equal to their ascensions; outside the polar circle, where these halves can ascend, no others, from wherever begun, are so in any oblique position of the sphere.
Compare also the segments of these principal halves with their oblique ascensions. No arc of the ecliptic less than a semicircle, either beyond the tropic or inward to the midpoint between the tropic and the equator, is equal to its oblique ascension. But from the midpoint between the tropic and the equator as far as the equator, a circle VE being drawn bisecting the angle OEQ of the ecliptic and the equator, when the zenith falls in this sector, which happens once at the beginning (namely when the high solstitial point culminates) and twice nearer the equator, then it indeed happens that the arc of the ecliptic EO, begun from the equinox, is equal to its oblique ascension EQ, and so in this case the halves of the principal circles are cut by the horizon into segments, two adjoining ones always equal. And these segments are, at the very midpoint between the tropic and the equator, quadrants, that is, two equal ones of each circle; toward the equator they are more and more unequal.
What variety arises through the different spheres in comparing oblique ascensions with right ones? First, between the equator and the polar circle the sum of the right ascensions which two equal opposite arcs have is distributed between their oblique ascensions. Under the polar circle one of the opposites has nothing and the other the whole, that is, the sum of the right ascensions of both its own arc and its opposite, namely twice the right ascension in oblique form. Within the polar circle the arcs rising in the right order have not only twice their right ascension, but add besides the oblique ascension of the opposite arc rising in the inverted order. Outside the polar circle, therefore, the right ascension of one is taken from the sum of the right ascensions of both; under the polar circle nothing is taken away, nothing added; within the polar circle the oblique ascension of the arc rising in the inverted order is added and makes up the oblique ascension of the equal opposite arc.
Second, within the tropics, when the ecliptic passes through the zenith, the right ascension of one part of a quadrant and the oblique ascension of another, put together, themselves also make up a quadrant. Thus if in the nearest figure the zenith were not V but A, and AQ now the ecliptic and VO the equator. For then between the meridian HAR and the horizon HQR there is intercepted a quadrant AQ of the ecliptic, no less than VO of the equator, even though the cardinal points have not fallen on these circles. So the oblique ascension EO of EQ itself, and the right ascension EV of its complement EA, put together, make the quadrant VO.
Whence do we learn the angle at which the ecliptic cuts the horizon? We must conceive the triangle VNM (figure 40) between V the zenith, N the ninetieth degree of the ecliptic from its rising degree, and the point M of the same which is on the midheaven. In this triangle the side in the ecliptic, NM, between the nonagesimal and the meridian, is first investigated. For when the rising point of the ecliptic is given, its oblique ascension is given, and a quadrant counted back from it gives the right ascension of the midheaven. This being given, the point M of the ecliptic that is on the midheaven with it is also given. In the same way a quadrant extended back from the rising point in the ecliptic marks the nonagesimal N of the ecliptic. When therefore the nonagesimal N of the ecliptic and the point M on the midheaven are compared with each other, the intercepted arc NM becomes known.
Second, in this triangle the side in the meridian, VM, between the zenith V and the ecliptic M, is investigated. For when the point M of the ecliptic on the midheaven is given, its declination AM is given, which, taken away from the altitude of the pole VA if northern, added if southern, makes the distance VM of that point of the ecliptic from the zenith; and the angle VNM at the nonagesimal of the ecliptic is a right angle. Three things being given, the fourth can also be had, namely the side in the vertical, or the distance NV of the nonagesimal from the zenith; its complement is the altitude of the nonagesimal above the horizon, the measure of the angle sought.
Form of this procedure.
From this the side VN is computed thus: the altitude of the point on the midheaven is 61° 58′, sine 88267; the complement of the side MN, 83° 36′, has sine 99377. Divide; the quotient 88820 is the sine of 62° 39′. So great is the angle between the horizon and the ecliptic at 7° 5′ of Virgo, and its complement, 27° 21′, is the distance of the nonagesimal from the zenith.
What variety is to be observed in this angle? This angle never becomes greater than the altitude of the summer solstitial point, nor less than the altitude of the winter solstitial point. It has the former magnitude when the autumnal equinoctial point is rising (in our hemisphere), the latter when the vernal point is rising. The nonagesimal is in the eastern quadrant when the northern signs are ascending, in the western when the southern.
Is the angle made by the vertical circle with the ecliptic also usually investigated? Indeed its use is necessary, chiefly in the doctrine of eclipses of the Sun.
How is it investigated? Either from the altitude of the Sun, which always moves in the ecliptic, or from the distance of the point of intersection from the nonagesimal, with the distance of the nonagesimal from the zenith joined on either side. For if the point of intersection is rising, this NV itself measures the angle, and the nearer the intersection is to the nonagesimal, the greater the angle. So, the tangent of NV being extended by five ciphers and divided by the sine NS, the distance of the intersection from the nonagesimal, the tangent of this angle results. Or if the altitude is had, by dividing the sine NV by its sine, the sine of the angle NSV results.
Table of Right Ascensions, Declination and the angle between the Ecliptic and the Meridian.
The point which has a greater declination than this does not reach the horizon, in rising or setting, at the altitude of the pole written below.
Table of Oblique Ascensions, Western Q, and Angles of the rising point.