Epitome Astronomiae Copernicanae: Liber III, part 5
Part III. Of the year and its parts, and of the days and their increases and decreases
In how many ways does the astronomer consider times? In two: either by civil custom or by astronomical certainty.
How do you define the civil or political year? It is a space of time, or a fixed number of whole days, at which the motion of the Sun or the Moon or of both stars aims at a distance, as received by the popular institution of each nation.
Which form of the civil year is the most convenient of all for the astronomical discipline? The Julian year of the old Calendar, consisting of 365 days and adding one day in four years, so that after three simple years the fourth has 366 days. For this measure is a mean between the astronomical years; this was the tacit reckoning of times among all nations at least; this has been worn and cultivated among us from the Caesars on by continual observation; and to this form of the year the astronomer must repeatedly return, whatever other he may have taken in hand that is more familiar to the style of his country.
How do astronomers define the year? To them the year is the space of time within which the Sun seems to complete its course in the heavens, which, as will be said in the theoretical part, is brought about by the circuit of the centre of the Earth about the truly immovable Sun.
How many kinds of year are there among astronomers? Two, according to the two termini of the solar revolution: the sidereal and the tropical or turning year.
What are the termini of the sidereal year? The point of the ecliptic on which falls the great circle drawn from the pole of the ecliptic through a certain proposed fixed star, such as Sirius or the horn of Aries, and so on.
State the termini of the turning year. They are the points of the ecliptic at which the equinoctial or one of the colures cuts it, called by one name the cardinal points.
Which kinds of year belong to which parts of this doctrine? Of civil years there is a special discipline, and they cannot be treated universally before the theoretical doctrine of the motion of the Sun, and especially of the Moon, has been explained. Of the sidereal year it will be better to treat in the last part of this third book, although the perfect knowledge of this too must be sought from the theoretical doctrine. There remains, therefore, for this third part the tropical or turning year.
Must we then hold nothing in anticipation, for knowledge of this, concerning the apparent motion of the Sun or the proper motion of the Earth? On the contrary, for the perfect explanation of the turning year not a few things must be sought from the theoretical doctrine.
Why then is it treated of in the spherical doctrine? The turning year has several aspects. For either we divide the course of the Sun, which makes the year, by its natural termini taken from the first motion, and inquire into the various affections of those parts with respect to the unlikeness of days and nights, and so it belongs to the spherical doctrine; or we measure its lengths, different both as a whole and in its several parts, and the causes of this difference must be sought from the theory.
What is the turning year? It is a space of time within which four vicissitudes occur, spring, summer, autumn, winter, the Sun returning from one of the cardinal points to the same.
Whence has the turning year its name? The Greeks called it tropical, from a word which signifies turnings or vicissitudes; the Latin word vertens (turning) corresponds to it word for word. They call it also the seasonal year, because they are accustomed to name these four seasons of the year. It is called also natural, because these vicissitudes touch the whole nature of living things and of what is born of the earth, and even of the elements themselves.
Are not the same four parts, then, in the sidereal year? They are in it too, but by accident. For if sidereal years follow one another in some great number, it will come about at last that its beginning, which was wintry at the outset, will at last fall in summer, and so at times one sidereal year has not four but five seasons, namely one of them double.
By what measure do we measure the length of the year and of its parts? By the days which fall within the termini of one year or part. For this measure is both shorter than the year to be measured, and better known than it, and easy to observe and to count because of the vicissitudes of day and night, and finally sufficiently uniform.
How many days are there in the turning year? Almost as many as we observe in the Gregorian civil year, or that of the new Calendar, namely 365 and a little less than a quarter of a day, less, I say, by about three four-hundredths of one day; for in the theoretical doctrine this small particle will be examined more accurately.
How many senses has the word day among astronomers? Two chiefly. For either it signifies the same as the Greek nychthemeron, namely a space of time within which one day and one night are made, which is usually called the natural day; or day is taken as the opposite of night, or more accurately the time in which the centre of the Sun is above the horizon, and it is called the artificial day.
What do you think is the reason of these names, natural and artificial? As a house, a ship, a chest, a table all belong naturally to one kind of thing, in that they are made of wood, but art has made this distinction among woods, so that this should be a ship, that a table; and as all men are of the same species, from which the name of man belongs to them, but mind and custom and the institutions of nations, supervening on nature, distinguish men, different offices and names being distributed among them, so that this one is a king, that one a bishop, another a shepherd, all men of the same nature: so the one and same nychthemeron, arising from nature, is by different horizons, that is, by the different imaginations of different eyes, differently figured into unequal segments of day and night, one way here, another there.
How do astronomers divide the day? They divide it, as is commonly done, into hours, of which the ancients, as is clear from Homer, fixed only four in one natural day, others twelve, and later writers 24, which the astronomers retain, as will be said below.
How do astronomers count hours and days? Either from midnight, from one to 24 of the following midnight, as in the Prutenic Tables; or from noon, counting again 24 up to the following noon, and the day is sometimes called complete, sometimes running, of which the hours expressed are a part.
How many kinds of hours are there? As many as of days: for some are called equinoctial, others temporal, because according to the variation of the times of the whole year they obtain a varying length.
What is the equinoctial hour? It is the twenty-fourth part of the nychthemeron, or of day and night, the natural, taken together.
How is the equinoctial hour divided astronomically? It is divided like any part of a circle, a 360th part, a degree or a time, into sixty minutes, so that 15 minutes make a quarter of an hour; one minute goes into 60 seconds, one second into 60 thirds, and so on continually as far as is needed.
Compare the divisions of the equinoctial hour with the human pulse. In a man who is well, robust and of perfect age, of melancholic complexion or growing old, there is almost one pulse of the artery in each second, with no difference between systole and diastole: so there would be sixty pulses in one minute; but this slowness is rare, and commonly 70 are counted, and in the choleric and in women 80; four in three seconds. In brief, in one hour four thousand more or less.
How is the natural day brought about? By the apparent revolution of the Sun from its rising through noon to its setting, and the lowest point of the heavens, and its return to the rising, which in truth is the revolution of the Earth and, in it, of the horizon, continued in imagination up to the heavens, about the axis of the Earth as if immovable, as was demonstrated in the preceding books.
What are the termini of this revolution? Through the immovable centre of the Sun S, and through the axis PL of the Earth, a plane is conceived in imagination as drawn. Any place, then, on the surface of the Earth, situated in this plane, has the beginning of the natural day, which is either noon or midnight; and this runs through the whole of that place meanwhile, while the place itself, having left the plane PES, from that quarter through the opposite quarter is restored by the rotation of the Earth to the same position and the same plane PLS.
How does this agree with the sphere? What I called the plane PLS is represented in the sphere by the circle of declinations, drawn through the poles of the world and the centre of the Sun, and as if movable with it. But the place of the Earth, truly movable about the axis of the Earth, is represented in the sphere by the immovable meridian. And so the natural day is the space of time within which the centre of the Sun, having departed from a given semicircle of the meridian, is seen to return to the same.
Are all natural days through the whole year equal to one another? First, whole revolutions of the Earth with respect to a plane drawn through the same fixed star, as was said in the first book, are most equal to every subtlety of sense; yet a certain number of several revolutions accumulates from insensible differences something sensible, so that some summer revolutions differ in time from as many winter ones. Second, even if the revolutions of the Earth itself, with respect to a plane drawn through its axis and some fixed star, were perfectly equal, it would not yet follow that natural days are precisely equal to one another to a hair.
Is the natural day, then, not equal to a whole revolution of the Earth? It is a little longer, which is clear from the different termini of the two. For the terminus which determines the whole revolution of the body of the Earth about its axis is a plane drawn invariably through the axis of the Earth, that is, one which, carried round with the axis of the Earth itself by the annual motion (of which in Book VI), does not change the position of its parts but remains parallel to itself, in so far, that is, as the axis itself remains parallel to itself, as you see at PL in all four positions. And a place on the surface of the Earth is then reckoned to have completed a whole revolution when it returns to this plane PSL and to the same part of it. But the terminus which determines the whole natural day, as was said, is the plane PSL drawn indeed through the axis PL of the Earth, but in a variable position; because while it is carried round with the axis of the Earth by the annual motion, one point of it remains fixed to the immovable centre of the Sun S, and so it changes the position of its parts in various ways, so that it neither remains parallel to itself nor always cuts the plane of the ecliptic at the same angle. So let us suppose that, the Earth being placed in Capricorn, whence the Sun appears in Cancer, these two planes coincide; both will then be at right angles to the ecliptic; and as the Earth proceeds toward Aries, the plane PL will first leave S, the centre of the Sun, and remain at right angles to the ecliptic and parallel to itself; but the second plane PLS, clinging to the centre of the Sun, will be separated from the former plane PLS, and its outer parts, stretching beyond the axis of the Earth toward the fixed stars, will run ahead of and flee from the like parts of the former plane, and meanwhile this plane PLS will also be inclined to the plane of the ecliptic, as was said in the second part, until the Earth, placed in Aries, the Sun being seen in Libra, has gone ahead by a whole quadrant and, together with the former plane, has cut the parallels of the Earth into 4 quadrants, the angle SPL being made a right angle. Then, therefore, a place on the surface of the Earth, returning to the former plane PL, is still one quadrant of a whole revolution away from the later plane PLS; and this is how it comes about, in all 4 parts of the year taken together, that the natural days are 365 and a quarter, but the revolutions of the Earth one more, namely 366 and a quarter.
How shall I adapt this to the sphere? As before; the one plane PLS through the Sun and the axis of the Earth is represented in the sphere by a circle of declination drawn through the centre of the Sun and the poles of the sphere; the other PLS is represented by another circle of declination, which passes through a fixed star and the poles of the sphere. That the natural day is therefore more than one revolution of the Earth, and so also of the equator of the sphere, is demonstrated thus through the sphere. For suppose the Sun at the beginning of Cancer and on the meridian; while the sphere and the beginning of Cancer revolve, the Sun has meanwhile departed from the beginning of Cancer to the end of the first degree of Cancer, and so besides the whole revolution there is still need of almost one degree more of revolution, until the Sun returns to the meridian.
What is the name of that small portion which comes in over and above the revolution of the whole equator, and how is it defined? It is called the addition. It is a portion of the equator, namely the right ascension of the Sun’s proper daily motion, or of the arc of the ecliptic which the Sun traverses in one natural day.
If natural days are unequal on account of the unequal additions, what then is the uniform measure that measures them? The revolution of the Earth itself, or of the equator, a mean between the swiftest and the slowest, such as it is today a little after the equinoxes.
What makes the additions unequal? There are two causes. One is to be sought from the theoretical doctrine and the unequal proper motion of the Sun, whence it comes about that the Sun’s daily arcs in the ecliptic are unequal to one another. For today the Sun after the solstice traverses 57′ 5″ in one day, but after midwinter 61′ 21″. And the causes of this inequality are variously given according to the differing opinions of authors, as is taught in its place. A small daily motion has a small right ascension, other things being equal, a great one a great.
The other cause is proper to this place: that although the arcs of the Sun’s daily motion were equal to one another, yet in different places of the ecliptic they have different, and so unequal, right ascensions. Nay, even in the same places of the ecliptic the right ascensions are not perfectly equal in all ages, on account of the variation of the obliquity of the ecliptic, of which something was said above, and more will be said below in Book VII.
How many kinds of additions are there, then, among astronomers? Two: first, the temporal, which at different times are truly now less, now greater; second, the equal, mean between the greater and the less, devised by astronomers for the sake of measuring, which are equal.
How do astronomers establish the mean or equal additions? Because by the accession of the additions over the 365 days and a quarter of the year the supernumerary whole revolution of the equator finally accrues, astronomers divide it by the 365 days and a quarter of the year and call one portion the equal addition; and this is equal to the mean daily motion of the Sun in the ecliptic, namely 59′ 8″.
How many kinds of nychthemera, or natural days, are there? According to the standard of the additions, some are apparent or true, others equal, mean in quantity, and devised by astronomers, which consist, that is, of such an addition.
How great is the mean natural day, how great its hour? The length of the mean natural day has 360° 59′ 8″ of the equator’s times. Its hour is therefore worth almost 15° 2′ 28″ of times.
By how many times do the true natural days differ from one another? The difference between two nearest to one another is unobservable. Even a single one of the shortest compared with a single one of the longest makes no very great difference; but joined together, many in order in one part of the year, and as many joined in the other part of the year, they become shorter or longer by a sufficiently evident difference.
Maestlin, following the law of Copernicus’ particular hypotheses concerning the motion of the Sun and the precession of the equinoxes, which are not all received and which belong to the theoretical doctrine, gathers that the difference of the short days of the whole year from the natural or long days is in this our age one hour and one scruple with 2 seconds, and can in other ages run to a third of an hour above the whole hour. This is to be understood thus: that the whole space of the turning year between the two classes of days, of which some are long, fewer in number, and others short, greater in number, is not distributed in proportionable ratio; for one hour and a third is the better part of the fewer in its proportion than the part of the more in its. To Tycho Brahe, who looks to fewer causes, the sum of the difference in this age is one hour and five minutes.
Which are the longer days, which the shorter? And by what argument? Maestlin, from Copernicus, supposing all revolutions of the equator uniform, shows those to be longer which occur while the Sun runs from 11° of Scorpio to 22° of Aquarius, and all the rest of the whole year shorter than the uniform measure of the day. Nor does Brahe vary much from this. For at 8° of Scorpio he takes away at most 24 minutes, at 22° of Aquarius he adds at most eight minutes. The causes belong for the greater part to the theoretical doctrine, where it will appear that, if all the causes are combined, the distribution of the days of the whole year will be far otherwise.
Answer, then, concerning the cause proper to this place, and how much it varies the natural days, and where. The cause proper to this place is the difference of the right ascension from its arc of the ecliptic, as great as it can be at the most, which we indicated above to occur at 16° 4′ 44″ of Taurus and Scorpio and at 13° 45′ 16″ of Leo and Aquarius. In these four places, therefore, the mean and the true natural days are of the same length. So from 17° of Taurus to 14° of Leo longer days arise, because of this cause considered separately, and from 14° of Leo to 17° of Scorpio shorter ones; the gain of the former over the latter is 9° 56′ 20″ of times, or a little less than 40 minutes of one hour. The same reasoning holds in the other semicircle, for again from 17° of Scorpio to 14° of Aquarius the days are longer, and from 14° of Aquarius to 17° of Taurus shorter.
Brahe follows this reasoning in a special way when he gathers the motions of the Moon, just as if the causes proper to the theoretical doctrine were exactly compensated in the motions of the Moon.
What makes the revolutions of the equator unequal to one another? The unequal distance of the Sun from the Earth, by which it comes about that the rotation of the globe of the Earth becomes slower when the Sun is far distant, swifter when it is near. So in summer one revolution lasts a little longer than in winter.
State a general rule, useful also in the theoretical doctrine, for equating time. The time is to be fixed when the apogee of the Sun, of which in Book VI, falls at the beginning of Cancer, so that both inequalities, that of the additions as well as that of the ascensions, and so of whole revolutions, begin from the same beginning. And this time, taken without equation, is to be set down as the root, to which the rest are compared by equation. Then, any apparent time being proposed, the right ascension of the Sun’s place is sought; the mean motion of the Sun from the equinox is also sought; the difference of the two is the equation of time, consisting of the two causes mentioned.
For example, let the apogee of the Sun be at 0° of Cancer at the completion of the year of Christ 1260. And let the time to be equated be the year 1457, 3 September, 11 hours 6 minutes. For this time the place of the Sun is gathered, as we shall learn in Book VI, to be 19° 17′ of Virgo, whose right ascension is 170° 19′. But by mean motion the Sun is distant from the equinox by 171° 27′. Here, then, the difference is 1° 8′ of times, that is, 0 hours 4′ 32″. This much is to be taken away from the apparent time, that it may be known how many times of the equator have elapsed since the year 1260.
Finally, from the annual anomaly of the Sun (of which in Book VI) must be learned the gain or loss of minutes of an hour which whole revolutions undergo; the method belongs to the theoretical doctrine.
Are absolutely all the causes observed by this rule? It is presupposed that the motion of the fixed stars along the longitude of the ecliptic (or, as Copernicus teaches, the precession of the equinoxes) is uniform; if it should have any inequality (of which in Book VII), that, after a due interval of centuries, when this inequality emerges, would have to be applied besides in equating the times of that age. But whatever this inequality may be, it can safely be neglected within these two thousand years, for which written observations exist.
How do we learn from the heavens themselves what hour of the day it is, in the astronomical reckoning? First, knowledge of the altitude of the pole is needed. Second, then by day we use the Sun alone, by night besides some fixed star whose declination and right ascension are known. The right ascension of the Sun is easily obtained by the doctrine of the second part, from the known place of it in the ecliptic; but the fifth part will teach how to find the right ascension of a star. Third, if these are at hand, the altitude of the Sun or star is taken at the proposed moment. Fourth, then, according to the doctrine of the first part, the elongation of the Sun or star from the meridian circle is sought from the declination and altitude. By night, through the elongation of the star from the meridian, the elongation of the Sun itself from the same is also sought, the elongation of the star being taken from the difference of the right ascensions if the Sun and the star are on opposite sides of the meridian, but added if on the same: so the distance of the Sun from the meridian also becomes plain. Fifth, this elongation of the Sun, since it is an arc of the equator intercepted between the circle of declination of the Sun or star and the meridian, is resolved into hours, 15° 2′ 30″ of times being taken for one, if the right ascension is had of that place which the Sun has held or will hold at noon itself. But if you have used the place of the Sun for seeking the hour itself, foreknown in a rough way, then 15° of times exactly make one hour. Sixth, by day, therefore, these hours, the Sun still rising, are taken from 12, that it may be known how many hours have elapsed since midnight; but with the Sun already falling there is no need of subtraction, for the hours which result are counted from noon in the astronomical manner.
How conversely is the right ascension of the midheaven, the oblique ascension of the horoscope, the point of the ecliptic on the midheaven and the rising point, and finally the oblique ascensions of the houses of heaven and their beginnings in the ecliptic, sought from a given hour? Before all things there is need of knowledge of the true place of the Sun in the ecliptic for the year, day, hour and minute of the proposed apparent time. The right ascension of that place is sought from what has been set forth, and to it, for each hour counted from noon, 15° of times are added, for 4 minutes one degree of time, and so on. Thus the right ascension of the midheaven is established.
For the oblique ascensions there is further need of knowledge of the altitude of the pole above the horizon of each place and of the remaining 4 circles of position, which are distant from one another by thirty degrees of the equator (according to Regiomontanus) or of the vertical (according to others), the beginning being taken from the meridian; which method was handed down in the first part. Then, therefore, to the right ascension of the midheaven, or tenth house, 30, 60, 90, 120, 150 degrees of the equator’s times being added, the oblique ascensions of the houses XI, XII, I, that is, of the horoscope, II and III, are established. The points of the ecliptic rising with these oblique ascensions, each at its own altitude of the pole, are found by the doctrine of the second part. The opposite points of the ecliptic hold the beginnings of the opposite houses IV, V, VI, VII, VIII, IX. Thus the whole face of the heavens, or the celestial figure, is erected and the planets are inserted in it in their places.
If the hour is the 24th part of the natural day, and that is worth 360° 59′ 8″ of the equator’s times, then the hour seems also to be worth more than 15° of times. Indeed that addition of 59′ 8″, by which any day exceeds the whole equator, must also be divided among the 24 hours, if they are counted from the invariable point of the equator which was on the meridian with the Sun. But when the true place of the Sun and its right ascension is computed not simply to noon but exactly to the hour itself, then by that very fact there has already accrued to the midheaven as much as was due to as many hours out of the addition; it is sufficient, therefore, then to reckon 15° of times for one hour.
Of artificial days and nights.
By what properties are the different parts of the turning year distinguished from one another? By two chiefly: the length and shortness of the artificial days and nights, and the vicissitude of heat and cold.
What is properly an artificial day or night among astronomers? The artificial day is the space of time in which the centre of the Sun can be seen with free and unrefracted rays above the rational horizon; the night, as long as it is below, although the twilight at the ends of the night is great and almost equal to the daylight.
Are, then, the artificial day and night parts of one natural day? Speaking accurately, one artificial day, in which the Sun indeed rises and sets, is divided in its very middle between two natural days, of which one began at the preceding noon and the other ends at the following noon. But the artificial night is part of only one of these two natural days, namely the preceding part of the preceding day, and the following part of the following. And then, namely when the Sun can set and rise, the artificial day is certainly less than the natural day and has the character of a part, no less than its companion, the artificial night. But that artificial day in which the Sun cannot go below the horizon is composed of some whole natural days. And that artificial night in which the Sun, by its daily revolution, cannot rise above the horizon is similarly composed of some whole natural days.
What then especially comes into consideration concerning these days and nights? The very great inequality of days and nights, through the different parts of the year as well as the positions of the sphere.
By what measure do we measure this inequality? We measure it by the circles of the natural days and their arcs; and we express the length of each by the number of equinoctial or mean hours, or also of natural days.
What circles of the natural days do you mean, and how many? The parallels of the equator, 180; of which the extremes are the two tropics, and each of the others is drawn through the ends of two degrees of the ecliptic, equally distant from the tropical points.
How are these circles constituted, and on what occasion? Nearly in the same way as above in the first part, and above in the second book, the circle of a star; except that here it is made by two motions of the Earth compounded with each other, one of rotation, the other of circumlation (in which, as was said in the second part, the axis of rotation remains carried over parallel to itself in every position), so that one circle is joined to another.
For imagine, first, that the axis and centre of the Earth remain in their place and are connected with the centre of the Sun by a straight line, which will be drawn across the surface of the Earth; the body of the Earth, rotating in this disposition, will be cut on its surface all round by this line in a perfect circle, so that the section returns to where it began. Admit, second, this consideration, that the axis of the Earth is meanwhile shifted slightly, and so is now inclined a little otherwise to the Sun than at the beginning of one rotation; by which means, as we learned in the second part, the Sun is brought a little nearer to one of the poles. Therefore at the end of the revolution undertaken the line connecting the centres of the Sun and Earth will cut the globe of the Earth nearer the pole, and so will stray from the beginning of the section and give a beginning to a new circle, joining one circle to another, as happens in the cylinders on which threads are wound. Such coils, then, the series of places over whose zeniths the Sun seems to pass in the courses of the day and the year makes on the surface of the Earth, leaving these places and coming upon those. For example, the Molucca islands are arranged in this series from north to south between the great islands Gilolo and Celebes: first Ternate, second Tidore, third Machian, fourth Bachian. Now at noon on 22 September let the Sun be vertical over the island of Ternate; thence departing toward Celebes it passes over Borneo, Sumatra, the Maldives in the Indian Ocean, the Ethiopia of Africa, the Atlantic Ocean, in America Guiana and Manoa on Lake Parime, and the Southern Ocean, until, 24 hours being completed, on 23 September it returns from the east; no longer over Ternate but over Tidore, and on the 24th passing lower over Bachian and making beginnings with new circles.
And on these circles on the Earth the circles of the natural days are imagined to stand perpendicularly in the sphere, as taught in the second book; not that in the heavens also the Sun, leaving the ecliptic, wanders into other regions of the fixed stars along the course of the circle of the natural day, but because, if in some higher hollow vault at rest anyone could imitate and express with white lead the tracks of the Sun and of its ecliptic point, such a circle would be expressed by this marking.
But the circles of the natural days in the heavens do not correspond, as you have described them, to those circles on the Earth, either in figure or in number. For they are perfect circles and not joined to one another, not exceeding the length of a perfect circle by the quantity of the addition which comes to the whole rotation of the Earth that the day may be complete; and they are 180 in number, while the pairs of days of the year are 182 or 183. It is familiar to geometers to adapt things which are irregular to regular ones coming nearest to the proposed figure, for the sake of art and measuring; for no art can be practised without fixed rules. So here also, although the natural day runs on meanwhile, while there is not only a whole revolution of the Earth but also a small particle of the next revolution coming on, yet astronomers compare the whole time of the day to a perfect circle returning into itself, which is drawn through the end of the Sun’s degree nearest to the Sun’s place through the whole day, as if nothing were added to that circle, or as if the Sun, by reason of its proper motion, rested at some one beginning of a degree of the ecliptic through the whole day, and afterwards suddenly and in a moment made a leap to the beginning of the following degree.
But is not the certainty of the comparison and of astronomical reckonings disturbed in this way? Whatever error is made by this fiction is wholly insensible, and in one day; wherefore the ancients judged it not to be minded, because here each single day is considered separately, and artificial days are not accumulated in some number here as the natural ones were before.