Epitome Astronomiae Copernicanae: Liber III, part 8
Part V. Of the appearance and hiding of the stars through the different parts of the year
Of the sidereal year.
Thus far of the turning or tropical year; let us now treat also of the sidereal, of which I ask first whether it is not of the same length as the turning year? It is not of the same length as that, but a little longer: for the stars which we set as the terminus of this year change their places little by little with respect to the ecliptic, and the circle through the poles of the ecliptic and the star cuts the ecliptic always farther from the point of its intersection with the equator.
I thought that with Copernicus the fixed stars truly were what they are called; do these then also move? By no means; but the beginning of the numbering of the parts of the zodiac, or its vernal intersection with the equator, moves backward. The point of the ecliptic, then, on which the perpendicular arc falls from a fixed star is always in fact the same point, but is not always numbered as the same, the beginning of the numbering of the parts of the ecliptic being changed, and always earlier, begun from the wandering intersection of the equator.
Demonstrate from proper causes that the intersections or equinoxes move toward the preceding signs. Indeed incorporeal points are not held to move of themselves, but the Earth, a corporeal thing, with its axis and magnetic fibres, about which as if immovable it revolves by the daily motion, is inclined little by little from one fixed star to another, as was begun to be said in the first book and has been often repeated. This axis, then, so far as within the space of one year it always looks to the same fixed star or point among the fixed stars, makes the fixed stars not appear to move under the ecliptic. But so far as in the succession of centuries it gradually leaves the former fixed star or point and nods toward other neighbouring ones, it makes the imagination as if the pole of the world or sphere passed over to that fixed star. Now it was said in the second book that the colure of the solstices is drawn through the poles of the ecliptic and of the world: the poles of the world, then, going about the poles of the ecliptic with a very slow motion, the colure also follows. But the intersection of the colure of the solstices with the ecliptic is held to be the beginning of Cancer, wherefore the beginning of Cancer seems to move with the inclination of the axis of the Earth under the fixed stars, and so also the beginning of Aries, a quadrant distant, which is the mutual intersection of the equator and the ecliptic, from which intersection the numbering of the parts of the ecliptic begins, as was said in the second book. So it comes about that the number of some point of the ecliptic, in truth at rest, is increased and so that point seems to advance.
Of what use to us in this place is this knowledge? The writings of the ancients on the appearance and hiding of the stars cannot be understood unless we know how great a difference has arisen between us and them from sidereal years as against as many tropical.
How great is the tropical year, how great the sidereal, and the excess of this over that and over the Julian year, and of this over the tropical; and what portion of the circle corresponds to the space of the intervening years? The mean and uniform quantity of the tropical or turning year (for they are not all altogether equal to all, as will be said in the theory of the eighth sphere) has been found to contain 365 days, 5 hours, 49′ 15″ 46‴ by the numbers of Copernicus and the Prutenic Tables, so that it lacks six hours, or a whole day in four years, that is, in the Julian year, 10′ 44″ 14‴ annually. As for the mean quantity of the sidereal year, as far as astronomers have been able by diligence to attain it, they judged that it adds to six hours over 9′ 39″. The difference, therefore, between the tropical and the sidereal is every year 20′ 23″ 14‴, and in 60 years as many hours. So in 70 years and two thirds almost one day intervenes, in 707 years 10 days, in 1413, as much as is from about the age of Ptolemy, 20 days, in 1767, as much as from Hipparchus, 25 days, in which the Sun traverses under the fixed stars as many degrees, less 22 minutes. So the Julian years are longer than the tropical by 13 days, and the sidereal longer than the Julian by 12 days.
How could astronomers observe the length of each year, both tropical and sidereal, since we can see neither the tropical points nor the fixed stars when the Sun is present and applied to them? The length of the tropical year is easily observed in those ways by which we were taught above to observe the meridian altitude of the Sun and the altitude of the pole, and finally the declination of the Sun. For the length of the tropical year is reckoned exact when the Sun again reaches the same greatest declination from the equator of the same kind, or again the mean between the two greatest.
The length of the sidereal year is a matter of more delicate observation, and the thing has been attempted by more than one way. The ancient inhabitants of Egypt, enjoying a perpetual clearness of sky, Egypt being flat and having very high pyramids, watched the rising of the greatest of the fixed stars, which they are wont to name the Dog, Sirius, diligently noting on what morning it was first seen before the Sun, and thence counting the days until it again emerged in the morning from the rays of the Sun. Since this observation can take place only about sunrise, the Greeks, lacking so many conveniences, took refuge in the contemplation of lunar eclipses, noting with what fixed stars the Moon appeared when in the shadow of the Earth: for then they knew that the Sun was in the opposite region among the fixed stars. Others first learned to compute the place of the Moon, which is deferred to the theoretical doctrine. Then they observed when the Moon covered some fixed star: for as far as they thought the Moon was from the Sun, by their calculation as index, so far also they set the fixed star to be from the Sun, and if this is done twice with the same fixed star, some just space of years intervening, it discloses the time of the return of the Sun to the same distance from that fixed star.
Hipparchus, not trusting the calculation of the Moon’s motion, began to compare the bare Moon itself with the Sun and with the fixed stars. For by day, the Sun being about to set, he measured by instruments by how great an arc of the ecliptic the Moon was distant from the Sun; an hour after, the Sun being now sunk and the fixed stars uncovered, he inquired by how great an arc of the ecliptic the same Moon was distant from some one of the fixed stars; the distance, there from the Sun, here from the fixed star, being put together, gave the removal of the Sun from the fixed star, and when this is repeated after some centuries, the thing sought is again disclosed. This method of observing, to attain its due subtlety, needs many cautions, and always something small is left uncertain. Therefore Tycho Brahe and others in this age have used Venus in place of the Moon, because it also like the Moon is sometimes seen by day, and yet is not subject to so many intricacies nor so swift, nor of a course so variable by the hour, as the Moon.
What is the beginning of the sidereal year? Nature shows its beginning no more than that of a circle; for any fixed star may be set as the beginning, unless perhaps certain stars at any time are in the cardinal points, which nature commends to us, that we may take from them the beginning of the tropical year. But in this way the fixed stars will be observed as the beginning of the tropical year, not of the sidereal as such. Yet the providence of the supreme Architect has given to our choice certain aids toward choosing one star in preference to another, since by the dissimilarity of the disposition of the fixed stars all parts of the heavens are distinct from all. So in antiquity it came about that the Egyptians began their year from the rising of the Dog, because it fell in summer, the Achaeans from the rising of the Pleiades; but this observation was political.
Into what classes were the stars anciently distributed? First, men, following the outlines of the various disposition of the stars, distributed the whole host of the heavens visible in our temperate zone into 48 images. Then astronomers, not for the sake of place or configuration but for the sake of magnitude, made six classes of those scattered here and there through the whole heavens, of which they called the greatest of the first magnitude, the least of the sixth, those between in the same order.
What are those 48 images? Twelve fall in a manner along the length of the zodiac, which were recorded above in the first book. The rest are placed in the hemispheres distinct from the zodiac, in the northern 21, in the southern 15.
Of these let the well-worn verses be kept in mind:
Toward the parts of the North you will see thrice seven constellations:
the Lesser Bear, the Greater, the Keeper, the Dragon, the Crown and the Kneeler, the Lyre, the Swan, Cepheus and Cassiopeia, Perseus, Andromeda, the Triangle, the Charioteer, the Horse, the Colt, the Dolphin, the Arrow; then the Eagle, the Serpent-bearer, the Serpent.
Then below lies the Sign-bearer, which wheels twice six constellations.
They are the Ram, the Bull, the Twins, the Crab, the Lion, the Virgin, the Balance, the Scorpion, the Archer, the Goat, the Water-jar, the Fishes.
After these, thrice five constellations turn toward the south for you:
the Whale and the River, the Hare and stormy Orion, Sirius and Procyon, the ship Argo and the Water-snake, the Cup, the Raven, the Centaur, the Wolf, the Altar, and the Crown and the Fish.
To these the Portuguese add 12 images about the south pole, which are invisible with us.
Use, if you please, this couplet:
the Phoenix, the Crane, the Indian, the Swordfish, the Peacock, the Goose and the Sea-serpent;
the Sparrow, the Bird of Paradise, the Triangle, the Fly, and the Chameleon.
Of the longitude and latitude of the stars.
What do you understand by the distance of two stars? The arc of a great circle of the sphere intercepted between the centres of the two stars is to be understood.
How does the distance of two stars become known? The simplest way is by observation, which is made with an instrument having an arc of a circle and two rules from the centre, furnished with sights. For one rule, which is fixed at the beginning of the arc, is directed to one star, the other, movable, is directed to the other star: the arc intercepted, which is the measure of the angle at the centre, therefore corresponds to the arc of the great circle of the sphere, and expresses it by its engraved numbers of degrees and minutes. See the type of the instrument in the Mechanics of Tycho Brahe, especially of the sextant.
How do astronomers define the longitude of a star? It is the arc of the ecliptic numbered in the order of the signs and intercepted between the equinox and the semicircle of latitude which is drawn through the star: for this arc shows how far the star recedes from the beginning of the ecliptic toward its end according to the longitude of the sphere.
What is the latitude of a star? The arc of the circle of latitude intercepted between the ecliptic and the star; for it shows how far the star has receded from the ecliptic toward the sides of the world by the shortest way.
Compare longitude with right ascension, latitude with declination. Longitude and right ascension begin from the same beginning, but longitude is reckoned in the ecliptic, ascension in the equator; the former is terminated by a circle of latitude, the latter by a circle of declination, of which the former is drawn through the poles of the ecliptic, the latter through the poles of the equator.
How is the longitude and latitude of any fixed star investigated? There are various ways, which cannot all be handed down in an epitome; but the chief are these. For either we use the instrument of the armillary sphere or calculation. If we use the sphere, it is necessary that the place of the Sun in the ecliptic be first known, or in its stead the longitude of one star. Then the sphere must be erected according to the altitude of the pole and have exactly the meridian line of the place, and have two movable circles of latitudes, furnished with movable sights in such a way that the lines of sight directed through the sights are parallel to lines drawn from the centre of the sphere. Look at the zodiacal armillae in Brahe’s Mechanics. All these things being so prepared, one circle of latitudes is applied and fixed upon the known place of the Sun or star in the ecliptic; the sight also is set upon its known latitude and is turned with the sphere together, so that the line of vision falls upon the centre of the Sun or star foreknown: this done, and the material sphere thus corresponding to the heavens, the other circle of latitude, the ecliptic remaining immovable, and its sight in its own circle, is turned until through it the proposed star is seen. Then its longitude is shown in the ecliptic by the circle of latitude, its latitude by the sight in the circle of latitude. But if we lack such a sphere or cannot use it, then with another suitable instrument the distance of the proposed star from the Sun or from another fixed star, whose right ascension and declination are known, is taken, and then the declination of the proposed star itself is to be sought by observation, as we were taught in the first part.
From these data, then, the difference of right ascensions is first to be sought, and by it the right ascension of the star, from which afterward the longitude and latitude are also had.
How is the difference of ascensions of two stars sought from their declinations and distance? A triangle PID is formed between P, the pole of the sphere, and the two stars I and D, of three known sides: one, ID, is the distance, the other two, PI and PD, are the complements of the northern declinations IC and DL; or if the declination is southern, as TS, then the side is composed of the quadrant PT and the declination TS. There is sought in it, therefore, the angle at the pole, IPD or IPS, which is the difference of the right ascensions CL or CT, and which, added to or taken from the ascension previously known, makes the ascension of the proposed star. From the right ascension and the declination the longitude of the proposed star is then sought.
The procedure for inquiring the difference of right ascensions from the declinations and the distance. And first, if both declinations are of the same quarter.
Complement of the lesser declination 60°; lesser declination 30°.
Complement of the greater declination 50°; greater declination 50°.
Sum, more than a quadrant, 110°. Sum, 80°.
Therefore the excess is 20°, sine 98481. (If the sum were less than a quadrant, you would subtract the sine of its complement.) Sum 132683. Half of it, 66342, is the divisor.
Let the distance of the stars be 41°. Subtract the sine 75471 of the complement 49° from the above 98481 (if the distance exceeded a quadrant, you would in this case add the sine of the excess). Remainder 23010.
Therefore the sine 65316 gives the complement of the difference of right ascensions, 40° 47′; the difference of right ascensions is 49° 13′.
But if one of the declinations is northern, the other southern.
Complement of the lesser declination 48°; lesser declination 42°.
Therefore subtract the sine of the complement, 10°, 17365 (but you would add the sine of the excess as before). Remainder 78761.
Half, 39381, is the divisor.
Let the distance of the stars be 130°, the excess 50°, sine 76604. Let this be subtracted from the above 96126; difference 19522. (If the distance were less than a quadrant, you would here add the sine of its complement, contrary to what was done before.) 19522 to 39381 gives 49572; the quotient is a versed sine, the sine being 50428, complement 30° 17′; here, therefore, the arc 59° 43′ is found according to the analogy of the former case. But because the latitudes here are different, the complement of this arc to a semicircle, 120° 17′, is to be taken for the difference of right ascensions.
Or let the distance be 110°, the excess 20°, sine 34202; let it be subtracted from the above 96126, difference 61924; 61924 to 39381 gives 157243. (Here six figures come out in the division.) The quotient, as a versed sine, shows 124° 55′; its complement to a semicircle in this place, 55° 5′, is the difference of right ascensions.
What is the mediation of the heavens? It is the arc of the ecliptic between the beginning of Aries and the circle of declination drawn through the star.
How is its longitude and latitude now sought from the known right ascension and declination of the star? The angle between the ecliptic and the meridian serves for this. For, the right ascension of the star being given, the point of the ecliptic having the same right ascension is sought, either by calculation or from a table, so that together with the star it comes to the right horizon and to the meridian and so mediates the heavens. The declination of that point of the ecliptic is sought, again either by calculation or from a table, together with the angle which the ecliptic makes with the meridian at that point, which is called the angle of latitude. Then this declination of the point is compared with the declination of the star. For if they were of different names, they are added to each other; if of the same, the less is subtracted from the greater; so the arc of the circle of declination between the star and the ecliptic becomes plain, which may be called the base of the latitude: for it is subtended by the right angle between the ecliptic and the circle of latitude of the star. From this base, then, and the angle of latitude the latitude itself is computed, and the arc between the star and the point of the ecliptic that rises together with it in the right sphere.
The latitude is computed thus: multiply the sine of the base of latitude by the sine of the angle of latitude, five of the last figures being cast off; the sine of the latitude results. If the star and the point of the ecliptic have declinations of the same name and at the same time the declination of the star was greater, the latitude keeps the name of the point of the ecliptic; but if the declinations are contrary, or that of the star was less, the latitude of the star takes the name contrary to that of the point of the ecliptic.
The longitude is computed thus: multiply the tangent of the base of latitude by the sine of the complement of the angle; the tangent of a small prosthaphaeretic arc results. For if the declinations are of the same name and that of the star greater, this arc is added to the mediation of the heavens in the quadrants after the equinoxes, and subtracted in the quadrants after the solstices; otherwise the contrary is done, and so at last the due longitude results.
In this figure S is the star, ON the equator; at N the right ascension of S ends; DO is the ecliptic, at D the mediation of the heavens of S ends. NS is the declination of the star, ND the declination of the co-ascending point D, SD the base of latitude, NDO or SDI the angle of latitude, ESI the circle of latitude, SID a right angle, SI the latitude, ID the prosthaphaeretic arc; at I the longitude of the star S ends. If O were the beginning of Aries and ODIC the order of the signs, OI would be the longitude of the star; examples follow below.
Could not this computation also dispense with the table of the said angle? There is indeed a way very commonly used by Tycho Brahe which does not use this angle, but it is long and more difficult; in it, in place of the triangle DSI the triangle PSE is taken, where P is the pole of the world, E the pole of the ecliptic, EPC the colure of the solstices, PS the complement of the declination SN, PE the distance of the poles, equal to the obliquity of the ecliptic, and the angle EPS known from its measure NOQ, the arc of the equator between the circle of declination of the star and the colure of the solstices: hence ES, the complement of the latitude SI, cannot lie hidden, nor PES, or its measure CI, the removal of the place I of the longitude from the solstitial point C of the ecliptic.
The procedure is this.
From the greatest declination of the ecliptic and the complement of the declination of the star, both being northern.
(If the sum were greater than a quadrant, you would add the sine of the excess to the remainder, 62503.) Half, 31252.
Now let the right ascension of the star be 275° 52′; therefore the angle between the circle of declination and the colure of the solstices is 5° 52′ (less than a quadrant).
Its versed sine is 524; multiply by the half above, 31252, five of the last figures being cast off; the product, less, is 164; let it be subtracted from the first sine, 88287.
This, then, is the latitude of the star, northern because the product was less; if it had been greater, the latitude would have been southern; this being found, its longitude also, numbered from the same semicircle of the colure, that is, from the beginning of Capricorn (whose right ascension is 270°), becomes plain thus.
Let it be multiplied by the sine, 10221, of the angle at the pole, 5° 52′.
Five last figures being cast off, there results 16931, the sine of an arc of 9° 45′; this is the longitude of the star from 0° of Capricorn.
This part can be abbreviated thus.
Half, 8003. (If the sum exceeded a quadrant, you would add the sine of the excess.)
This half divided by 47268, the sine of the complement of the latitude, gives also 16931, the sine of the angle of longitude, as before.
Since, however, slips are easy in this procedure of Tycho’s, it is advisable to fit to each case its own proper figure. For if the declination is northern, the northern poles are to be used, if southern, the southern: on either side the star either precedes the neighbouring solstice or follows it, and on either side either the beginning of Cancer is taken or that of Capricorn.
If the latitudes of two stars and their distance were known, how is the difference of longitudes thence made known? By exactly the same procedure by which the difference of right ascensions was found above, only that in place of the word ascension you understand longitude, for declination, latitude, for the pole of the world, the pole of the ecliptic, and so on.
Example. Complements of the latitudes 89° 0′ and 83° 29′. Sum 172° 29′, greater than a quadrant, therefore the excess is 82° 29′.
The lesser latitude itself is 1° 0′, the complement of the greater 83° 29′. Sum 84° 29′ (let the latitudes be of different quarters). Add the sine of 84° 29′ (99537) to the sine of 82° 29′ (99141). Half of the aggregate 198678 is 99339. (If the sum were less, you would subtract the sine of the complement.)
Let now the distance be 33° 15′, less than a quadrant, as almost always. Add the sine 83629 of the complement 56° 45′ to 99537 because of the different latitudes; otherwise you would subtract. (If the distance exceeded a quadrant, you would subtract the sine of the excess for different latitudes, and add it for latitudes of the same name.)
183166 to 99339 gives 184385. The quotient is the versed sine of an arc of 147° 33′; therefore its complement to a semicircle, 32° 27′, is the difference of longitudes sought, because the latitudes are different. (Had the latitudes not been different, not the complement but the arc itself of this versed sine would have given the thing sought.)
By the same procedure the other part of the preceding operation is also dispatched, when from the given latitude and declination of a star with its right ascension its longitude is sought. For the pole of the world of the same name as the latitude is taken for the other star, whose longitude from the pole is at the beginning of Cancer or Capricorn; for its latitude there is the complement of the greatest declination, for its distance from the star the complement of the declination of the star of the same name, or, if of different quarters, an arc composed of a quadrant and the declination.
The example above (page 278). From the greatest declination of the ecliptic and the complement of the northern latitude.
Complement 38° 16′; subtract the sine 61932.
Of the remainder, 37734, the half is 18867.
This is therefore the interior angle sought of the triangle at the pole of the ecliptic, showing that the star is distant backward from the beginning of Cancer by so great an arc of the ecliptic, that is, in 9° 47′ of Capricorn (nearly as before).
How, conversely, are the right ascension and declination of a star sought from the longitude and latitude? By the same method by which we previously did the contrary, and that likewise twofold. First, by the table of the angle, only the names of longitude and latitude being changed into ascension and declination, of the ecliptic into the equator, and the angle between the ecliptic and the circle of declination turned into the angle between the equator and the circle of latitude. Examples with the stars of the Dog follow below.
Second, but lest confusion arise from this diverse use of the same table, it is better to use the other way, because it is general and uses the same figure. Nor is more needed than a single example.
From the greatest declination and the complement of the latitude.
The sum being less than a quadrant, subtract the sine 13096 of the complement 7° 31′.
(You would add the sine of the excess.) Half, 34196.
Let the star be distant backward from the beginning of Capricorn by 108° 48′, that is, 90° 0′, sine 100000, and the remainder, sine 32227. (This, then, is the angle at the pole of the ecliptic.) Its versed sine 132227.
Therefore multiply the versed sine of the angle into the half above, five last figures being cast off; there results 45216; the first sine is 81488; the difference, 36272, is the sine of an arc of 21° 16′; this is the declination of the star.
For the angle at the pole, or the difference of right ascensions.
The angle of difference of longitude 108° 48′, or its complement to a semicircle 71° 12′. Complement of the latitude 58° 57′. Sum 130° 9′, greater than a quadrant. Excess 40° 9′, sine 64479.
Add the sine 97723 of the complement 77° 45′ of the difference 12° 15′. Half of the sum 162202 is 81101 (a sum less than a quadrant would require the complement to be subtracted). Let the sine 93190 of the complement of the declination 68° 44′ divide it. The quotient 87028 is the sine of an arc of a little over 60°. This, then, is the difference of the right ascensions of the star and 0° of Capricorn, to be extended backward, because the longitude also was taken from 0° of Capricorn extended backward, so that the right ascension of the star is 209°.
To what end does it conduce to search out the places of the fixed stars according to longitude? There are three chief utilities of this. The first is in the part of the spherical doctrine handed down thus far, that by observation of the fixed stars we may know what hour of the night it is, and consequently describe the face of the whole heavens at any moment, by erecting a figure or scheme of the heavens. For this cause the places of the fixed stars are inserted in the net of the astrolabe. The second is in the theoretical doctrine, that by comparing the planets with the fixed stars we may know their places at any moment, the places of the fixed stars through which the planets pass being first known. For unless the planets had first been observed, in what places they stood at certain times, the reasons of their motion could not be brought into the open. The third and chief use of this place is this: because the risings and settings of the stars mark certain seasons of the year, and of these signs the ancient husbandmen, and from their traditions the writers on rural affairs, and also physicians, poets and historians, have made use, and they cannot be understood without this part of the doctrine.
Would it not suffice to refer the stars to the equator? By no means. For first, the equator is not in origin a celestial but only a terrestrial circle, as was said above in many places. Second, the distance of the fixed stars from the equator, or their declinations, change very sensibly in the succession of centuries; the latitude, or distance from the ecliptic, remains as nearly as possible the same in all ages. Third, the apparent progress of the fixed stars along the length of the equator is found unequal in different centuries; along the ecliptic the progress observes a far more perfect uniformity. Fourth, which is the chief point in this place: if I have the right ascension and declination of a star, I can indeed compute its rising above any horizon, and its setting below it, but not yet its appearance and hiding under the rays of the Sun, which moves under the ecliptic, unless I refer the Sun also to the equator, where every position of the sphere has something peculiar. But if the places of the fixed stars are described by the ecliptic, the norm and rule of all the varieties through the various positions of the sphere is established, passing through the middle of them. Fifth, the places also of the other planets are perpetually ordered this side of and beyond the ecliptic and depart with the arcs of the ecliptic itself from the equator. It is expedient, therefore, to bring all the stars to one ecliptic, as to a common order.
Of the points of the ecliptic that rise together.
How is it known with what degree of the ecliptic any star rises and sets? There must be foreknown the altitude of the pole, the declination and the right ascension of the star, and by these, from the second part, the ascensional difference, which, added to or taken from the right ascension, according as the declination is southern or northern, makes the oblique ascension of the star. The oblique ascension of the star being given, the point of the ecliptic rising with this oblique ascension, and so also with the proposed star, is easily given either from tables or from the doctrine of the second part; or, if it is a descension, the point setting with it.
Of horizontal variations, refraction and parallax.
What happens to stars rising or setting? Refraction of the rays, which at the very rising or setting raises the stars (by estimation of sight) in the vertical circle by 30 minutes, which amount quickly decreases, so that at an altitude of 20 degrees it wholly vanishes.
What does this refraction of altitude effect in oblique ascension and declination, in longitude and latitude? It diminishes ascension and longitude at rising, and declination and latitude on the meridian; on the contrary it increases descension and longitude at setting, and northern declination and latitude, and by diminishing distances, in so far as they are at right angles to the horizon, if we use the place of one fixed star as certain and unrefracted, it brings the said errors into a calculation drawn up from the distance.
How do we learn by computation how much any refraction of altitude effects in length and breadth, so that the place of a star spoiled by refraction may be corrected? The general method is by the principal oblique-angled triangle of the first motion, VPS, in whose side VS, an arc of the vertical, let a higher point R be chosen, for the place of the refracted star S, and an arc being drawn from the pole P through R and continued, let a perpendicular SI fall on it. In the triangle PVR, therefore, the sides are known. For the quantity RS of the refraction being given, VR, the distance of the star from the zenith, or its complement, the altitude RO, to which that quantity belongs, is also given: from the altitude affected by refraction (or refracted, as Tycho says) the calculation described in the first part yields PR, the refracted declination, and the angle VRP at the refracted place R of the star. A perpendicular being let fall from S, the unrefracted place, upon PR, the circle of the refracted declination continued, two right-angled triangles are made, RIS and PIS, in the lower of which, RIS, the angle at R is given with the base RS, which is the quantity of the refraction, wherefore the sides also can be computed: one, RI, in the circle of declination, which is the refraction of declination, the other, IS, in the perpendicular let fall, the index of the refraction of right ascension, which is so much the more nearly equal to that which it indicates the less the declination of the star is. But if the declination is great, we come to the other right-angled triangle PIS, whose sides are known, PI the arc of the circle of declination increased by the refraction RI found, and IS this perpendicular; by these the angle P at the pole is sought, which is the corrected refraction of right ascension.
The shortest procedure. Multiply the sine of the arc RS (or rather the arc itself, because it differs in nothing sensible from the straight line) by the sine of the angle IRS or VRP, and five last figures being cast off there results the sine of the arc IS (or the arc IS itself). Then divide the secant of the arc RS, continued with 5 ciphers, by the secant of the arc IS; there results the secant of the arc RI, which with PR makes PI. Divide, therefore, the tangent of IS (or even the arc IS itself), continued with 5 ciphers, by the sine of the arc PI; there results the refraction of right ascension.
So from the corrected declination and right ascension both the longitude and the latitude result, free from refractions.
But what if not the right ascension and declination of the star are given, but longitude and latitude: must we always use these roundabout ways, and have recourse to ascension and declination? If the longitude and latitude of the star are given from the place of the Sun and the hour, the nonagesimal of the ecliptic and its altitude must be sought, so that P is the pole of the ecliptic, PR and PS are circles of latitudes, and RS will be the refraction of latitude, IPS the refraction of longitude, and the procedure will remain the same in the rest as before.
What is akin to refractions? The parallaxes of certain stars, of which in the sixth book. For the parallax of any altitude being given is resolved into length and breadth by exactly the same method, but with contrary effect; for where refraction diminishes, parallax, if there is any, increases, and conversely: therefore before all things subtraction of the less from the greater is made, and with the remainder, which keeps the nature of the greater, one must proceed as hitherto with either element of it. Understand this of the stars which have parallax.
Is there any gain here if the stars are near the equator or the ecliptic? If parallax is proposed alone, there is some saving, in that there is no need of the angle at the star; but since parallax chiefly concerns the Moon, which also is always near the ecliptic, especially in eclipses of the Sun, when parallax is most needed, this saving is deferred to the motions of the Moon.
What, then, are the chief differences of refractions and parallaxes? The first difference, in effect, has already been stated: parallax depresses, refraction raises. Second, all stars suffer refraction of altitude equally, whichever of them has come, on the same day, or on different days of the same quality, at the same place of the horizon, or of the same quality to the same altitude, because the cause of refractions is with us, in the air; sensible, perpetual diurnal parallax is made only by the Moon and some of the comets; Mercury, Venus, the Sun and Mars are indeed suspected of parallax, but the possession is disputed, the sense nil. Of the superior planets there is not even a suspicion of any sensible parallax, much less of the fixed stars. Third, refractions soon cease to be sensible at an altitude of 20° or 30°, as is to be seen in the first book; portions of parallaxes remain up to the zenith itself, proportional to the sines of the distance from the zenith.