Epitome Astronomiae Copernicanae: Liber VII, part 2
What is the proportion of this motion of the pole of the ecliptic to the motion of the pole of the world, or of the precession of the equinoxes? It seems to be to it as 4 to 3 with sufficient precision.
What is the mean obliquity of the ecliptic, what the true, and what the prosthaphaeresis of the obliquity? The mean is the arc of a great circle between the poles, the royal and the mundane, as AC, AE, AF and so on. The true is the arc of the colure of the solstices between the poles of the ecliptic and of the world, as CQ, EO, FP. The prosthaphaeresis is the difference of the two.
What follows from this variation of the obliquity for the precession of the equinoxes? It is rendered unequal, but with a very slow inequality, which is restored together with the anomaly of the obliquity itself. And so the motion of the fixed stars also, though in itself equable from the mean equinox, will by accident be found somewhat unequal from the true equinox.
What is in this hypothesis the prosthaphaeresis of the equinoxes? It is the arc either of the royal road or of the ecliptic, intercepted between two great circles drawn through their poles and their sections with the equinoctial.
What is the true precession? It is the arc of the ecliptic, numbered from the circle of latitude through the first of Aries against the order of the signs to the colure of the equinoxes. Or it can also be numbered in the order of the signs, like the mean precession.
How great is this greatest prosthaphaeresis and how is it to be computed? In the figure let AC be 24° 17′ 40″, AQ 1° 47′ 40″; QCA will then be 4° 22′, and QC, AC continued to the ecliptic will intercept a slightly smaller arc, namely 3° 58′ 45″. So great is the greatest prosthaphaeresis of all, but which in a short number of centuries appears in the smallest part of itself and is not easily perceived.
Is there no greater inequality of the precession of the equinoctial points? From the hypothesis indeed none besides follows, of the points themselves, or the prosthaphaeresis of the sections. Nor can anything certain be drawn from observations, unless something miraculous or wholly irregular and like a physical chance. For from Proclus down to us, through 11 or 12 centuries, the equinoctial points have certainly advanced at an equable pace, to which rule the observations also of Hipparchus and Timocharis from 18 centuries ago conform, if you disregard Ptolemy alone. Wherefore if anything befell this axis of the Earth by which it was enormously removed from its position, it was between Hipparchus and Ptolemy, in a space shorter than 300 years, and it was restored between Ptolemy and Proclus, in an interval again of three centuries. Wherefore it is not unjustly that Ptolemy’s observations of the equinoxes may be doubted. See Tycho Brahe’s Progymnasmata, Volume I, folio 254, and the Commentaries on Mars, chapter LXIX.
Is it possible that Ptolemy erred in the observation of the certain day of the equinoxes, and how? He did not err in the altitude of the pole, which was confirmed by many proofs, nor in the altitude of the Sun, which depends on the altitude of the pole. Perhaps, then, this befell him: when under Augustus the observation of the Egyptian year had been abolished, Ptolemy sought the days of the Egyptian year by the Moon, if it was a question of the Moon, or by the Sun and his calculation left by Hipparchus, if it was a question of an observation of the Sun, and, neglecting then the agreement of the observations of the Moon and trusting too much to the calculation of Hipparchus, he thought that he had only to be concerned about the hour of the entrance into Aries. For Ptolemy could not trust the Roman calendar, which had to be observed in Egypt, because even after Augustus’s correction a day was sometimes, at the pleasure of the pontiffs, taken away in one year and restored in the next. See the example in Dio and the cause: that the Ides might not fall upon market days.
What if he had observed the Sun rising on the days of the equinoxes, choosing a place of the horizon more northerly than was right for the equinoctial rising? Then indeed in all years he would have gathered the vernal equinox later than was right, but the autumnal in turn sooner than was right, the Sun rising from the same place of the horizon. But this was not done. For Ptolemy within the space of one year recorded both equinoxes at the right interval of days. Or if the place of the rising Sun at the vernal equinox had been more northerly than was right, the place of the rising Sun at the autumnal equinox after the right interval would not have been the same, but more southerly.
I grant the right place in the horizon for both equinoctial risings: could not refraction, neglected by Ptolemy, have disturbed something? It would have raised the Sun to the north at both equinoxes, and so the Sun would have been seen swifter at the equator in the vernal season, slower in the autumnal, and the summer would have been too much lengthened. The contrary would have happened, if you suppose the equinoctial settings to have been observed at both.
What will become of the age of Ptolemy, if this astronomy prescribes for it another beginning of the zodiac than he himself followed; shall we still be able to enjoy his other observations? As the entrance of the Sun into the equinoctial signs was in his time noted or even observed in an unusual manner and contrary to the custom of other ages, so also, outside the rule, for the credit of his observations the beginning of the zodiac will have to be put back, against the order of the signs, to his times: which being done, all the rest will be right.
What is the foundation of this confidence? Because the theoretical doctrine and the conjunctions of the stars among themselves could be taught and computed even without any knowledge of the twelfth-parts of the zodiac and their beginning, since that is constituted only on the occasion of the diurnal motion of the Earth. And so Copernicus and some of the ancients teach how to compute the places of the planets, taking the beginning from the first of Aries; then at last they bid us add the precession of the equinoxes, that the place of each star in the zodiac may appear.
What does Copernicus call the simple motion of the Sun, and what the compound motion? That is called simple whose beginning is taken from a fixed point, namely from the first star of Aries, from which the figure the zodiac’s first twelfth-part was once marked out, and the name of which figure that twelfth-part still retains today, after the constellation itself, which had given the name at first, has for the most part emigrated into the neighbouring twelfth-part. The compound motion is that which consists of two parts, 1) of the motion from a fixed point in the order of the signs, 2) of the mean motion of the beginning of the twelfth-part of Aries, or of the vernal section, against the order of the signs, that is, which is numbered from a point not fixed but movable, namely from the equinoctial.
How great is either the mean motion of the fixed stars in the order of the signs, or the mean precession of the equinoxes from the first of Aries against the order of the signs? Hipparchus and Ptolemy thought that it completes from Timocharis on one degree in 100 years and the whole circle in 36000 years. But Tycho Brahe, by comparing his observations with those of Hipparchus and Albategnius, determined that in 70 years a little more than one degree is made. Wherefore one period will have 25410 years, the annual motion of the fixed stars.
By what way was this motion investigated? The simplest manner was this, that, the latitude of a fixed star near the ecliptic being known, they observed its declination (for they supposed the latitude constant, but found the declination varied through the centuries; see Book III, folio 281). From the declination they gathered its place from the intersection. A more laborious way through the Moon and Venus and the Sun, see Book III, folio.
What redounds from the precession of the equinoctial points into the motion of the Sun and the reckonings of the year? 1. Since the Sun advances beneath the fixed stars, while the equinoctial and tropical points go back from the fixed stars, meeting the Sun now approaching: hence it is that the Sun comes sooner to the tropics than to the fixed stars with which the tropical points were at the beginning of the year. And so the tropical year hence becomes shorter than the sidereal, and consequently the stars themselves with their risings and settings desert the former seasons of the year and pass from summer into autumn and so on. On this matter is the last part of Book III.
2. And since the precession of the tropics is unequal, whether little, according to the hypothesis, or much, according to the Ptolemaic observations: hence also the tropical years become somewhat unequal, while the sidereal are equal.
Does nothing accrue from the motion of the Sun which helps the inequality of the years? Indeed both the sidereal and the tropical year are varied on account of the progress of the Sun’s apogee in the order of the signs, whence it comes about that now one now another equation of the Sun falls upon the equinoctial and tropical points, and so the sidereal year becomes unequal in one respect, the tropical in two. But this latter inequality obtains only with respect to a certain beginning of the years and is compensated by taking the opposite beginning. For example, from Ptolemy to us the tropical year was somewhat shorter, other things being equal, or also the sidereal, that beginning from the vernal equinox, this from the first of Aries, because the additive prosthaphaeresis at the beginning of Aries has hitherto been increased; but in turn the year beginning from the autumnal equinox, other things being equal, or from the conjunction with the Spike of Virgo, was so much longer, because in that part of the heavens the subtractive prosthaphaeresis was similarly increased: and so the length of the summer has meanwhile grown, at least until about the year 1260. Then the period of this anomaly is very slow, for it runs beyond 25 thousand years, wherefore it is little felt within one thousand years.
How does this inequality of the years differ from that part of the equation of time which is from the same cause, Book III, folio 251, and Book VI, Part I? It differs from that as a year from a day. For there it was shown how great an inequality of 24 hours in length the equation of the Sun would cause; here it is asked how many days accrue from it to some great sum of mean years, or how many others are taken away.
What difference do you make between the precession of the equinoxes and the anticipation of the equinoxes? If we speak accurately, precession is to be understood of motion or quasi-motion, as hitherto, anticipation, in Greek prothesmia, of civil time. And when I say precession, the word equinoxes sounds the certain points of the ecliptic, defined hitherto; but when I name anticipation, the word equinoxes added signifies the Sun’s presence in those sections or equinoctial points, or, more clearly, the day of the civil year on which the Sun makes the days equal to the nights.
The equinoxes anticipate, therefore, in the civil or Julian year, when they desert a certain day of the month and after some centuries are found on the preceding days.
What makes the equinoxes and solstices anticipate? The quantity of the civil year, a little longer than the tropical year: for it comes about hence that our numbering of the days of the year ends after the Sun has already passed beyond the initial point taken on the ecliptic. So, speaking most truly, the time of the equinox does not anticipate, as by some motion, but our numbering comes after.
And what in turn postpones the rising of the stars in the same civil year? The same quantity of the civil year, a little shorter than the sidereal year, for the days of our year end before the Sun has returned to the same fixed stars. It is called hypersthesis. So our civil Julian year is in quantity a mean between the tropical and the sidereal, under which name it was commended above in Book III, folio 246, by the astronomers.
Why, then, do we number what is too much and faulty? Because the units of number ought to be discrete, that is, whole days from the rising of the Sun to the next rising, but the quantity of the year is explained in certain minute parts of hours, which we are not accustomed to number in daily use, being content with whole days. Wherefore we are of necessity compelled to wait so long until the minutes remaining over the tropical year at the end of the civil year (they are 10 minutes 50 seconds 33 thirds of an hour) by accumulation from many years grow into one day.
But even if this were done, yet because this time exceeds the age of man and it is inconvenient to omit a day out of order so rarely, men preferred to keep the civil Julian year unimpaired and to assign in it the equinoxes and solstices anticipating in future times; which is also very convenient for astronomers for the speculation of the equinoxes, to have recorded in the Julian year, as in a little index, the equinoxes of all ages, that it may appear whether they anticipate equally in equal times or otherwise.
How many years does this anticipation require to make up one day? Hipparchus first defined the space as three hundred years, led by the conjecture of his own cycle, or civil lunar year, which period we named above the Calippic, since in it he saw that the lunar motions also anticipate one day in 304 years. He believed therefore that if in 304 civil solar years one day also is taken away, then those solar years are exactly made equal to as many lunar. This opinion of Hipparchus Ptolemy retained too securely, as it seemed to Tycho Brahe, but Ptolemy himself seemed to confirm it by his own observations too; of the credit of which see above. For immediately after Ptolemy it was found that one day is lost much faster. And so if we disregard Ptolemy alone, an equal reckoning will hold good from Hipparchus through Proclus, Albategnius, the Persians, Arabs, Hebrews, Germans, down to our times, which anticipates the equinoxes one day in 134 years, three days in 400; which the civil ordinance of the Gregorian year expresses as nearly as may be, as is in Book III, folio 247.
Whence was the reckoning of the Julian year fetched to Rome? Its quantity of 365 days and a quarter, which in four years completed made one supernumerary day, was observed from the most ancient times by the Greeks, in another form however. Julius Caesar therefore put the Roman toga on it.
What reckoning did Julius Caesar follow in fixing the beginnings of his months? He wished the Kalends of July, the month named by himself, to be that day on which all the Greek nations, from the most ancient rite of the Olympic games, thought the solstice was made, although that had already by the times of Hipparchus anticipated by eight days in the civil calendar of the Greeks, as Hipparchus first taught: asserting that the solstices and equinoxes (understand, of the ancient observation) were made in the eighth parts of the signs. This solstice was called the Chaldaic. So Julius Caesar assigned the true solstices and equinoxes of Hipparchus, which fell at the beginnings of the signs and which he thought still held in his own time, to the 8th before the Kalends of the months.
What, then, is the reason that Pliny elsewhere, and Columella also, in their own times still assign them to the 15th before the Kalends? It is certain that a delusion crept upon them from the confusion of different traditions, of which two likely occasions can be shown. The first is this: when they had drawn from Hipparchus that the solstices, the Chaldaic that is, are made in the eighth parts of the signs, and that therefore the observation of the solstices must be used with correction, they, transferring this doctrine to the Hipparchic solstices, long since corrected by Caesar, that is, assigned to the 8th before the Kalends, as if these still needed correction and had been accommodated to the eighth parts of the signs, thought that these must be carried back to the first or eighth parts of the signs and so that they must ascend from the 8th before the Kalends to the 15th before the Kalends.
The other occasion of error comes from the rude sphere of Eudoxus, in which the colures passed through the middle of the signs of Cancer and Capricorn, Aries and Libra. And that, if you look at the asterisms, was once so, and Eudoxus perhaps had obtained such a sphere, adapted to many centuries back. But grant that the colures passed through the middle not of the asterisms but of the twelfth-parts of the zodiac. It could therefore happen that Columella, reckoning that the Sun on the Kalends is in the colure and that the solstice is made there, but the Chaldaic and obsolete through age, judged that the true solstice of his own time was at the beginning of Cancer, which in the sphere of Eudoxus preceded by 15 degrees; and for that reason he extended the number from the Kalends to the 15th before the Kalends.
What follows from this circuit of the limits of the Earth’s orbit upon the other planets? 1. If the limits of the Earth’s orbit are in a parallel of the royal road, that is, if the poles of the orbit go round in small little circles about the poles of the royal road, then surely also those of all the rest; the poles of the orbits of the primary planets will stand around those royal poles. Wherefore whether they rest beneath the fixed stars, or are moved, provided they go more slowly than the poles of the ecliptic, it is necessary that these now depart from them, now approach them. In the one case, therefore, the greatest inclinations of the limits to the temporary ecliptic will be increased, in the other diminished.
The limit of Mars indeed is about the star of the Heart, the pole therefore of his orbit declines toward ..., about R (figure 98), so that if FP inclines to Qq, PR inclines to ...; but if this pole of the orbit of Mars is moved against the order of the signs, it ascends therefore toward Q, let it ascend more slowly than the pole of the ecliptic descends from O to P, so that GR was of old less than a semicircle, and RO therefore was of old shorter than RP is today. The quantity of the inclination, or of the distance of the poles RP, being varied, the quantity also of the greatest latitude in the northern limit will be varied. And so we also find, if credit is to be given to the Ptolemaic observations. For this greatest latitude about the Heart of ... is today 4° 32′, but Ptolemy gives it as 4° 20′, 12′ less, while he also gives the northern latitude of the Heart of Leo as 10′, 16′ less than it is today. And it may be that the Ptolemaic epicycle stands in the way, so that this latitude could not have become by much yet smaller: for he gives the inclination of Mars as 1° 0′, which today is 1° 50 1/2′, although the change of latitude of the fixed stars cannot effect so much.
Tycho Brahe notes something similar also of the Moon (Progymnasmata, Volume I, folio 27 of the Lunar Insertions), but from which you cannot gather a conjecture of a changed greatest latitude of the Moon: for it was not fitting that the Moon, a secondary and earthly planet, should look constantly to anywhere else than to the temporary orbit of the Earth, in whatever situation under the fixed stars it might be at any time; rather, hence the quantity of the obliquity of the ecliptic, handed down by Eratosthenes and Ptolemy, ought to be called in doubt, if it were permitted to argue from the very difficult observations of the Moon’s latitudes against the far easier observations of the obliquity of the ecliptic.
2. Nay more, hence it will follow that the limits and nodes of Saturn, Jupiter, Mars and Venus seem to go backward, even if they stuck immovable beneath the fixed stars. For let the poles of the ecliptic be turned between Q and P about the line AC, extended into Libra. Today the limit of Mars is in ..., of Venus in ...; wherefore the poles will be from P toward ... and .... Let them be in R; PR will therefore be the circle of latitude through the limits of Mars or Venus. P therefore descending from O against the order of the signs, whether the poles R rest, or themselves also go toward the line AL against the order of the signs, or even are carried in the order of the signs toward the line AD, provided this is slower than QP against the order of the signs: in all three cases the parts of the line PR which are beyond R will ascend toward AL against the order of the signs. So when the limits of Saturn and Jupiter are in Libra, the poles of the orbits will be from the pole of the temporary ecliptic toward Aries in a line parallel to CA or nearly, as about B, and again for the same causes, P going from O, they will seem to go backward. The limit of Mercury is in ..., the pole therefore is from P toward Q, namely about S (put S in the line AE between A and E), for he has a very great inclination. In this planet, therefore, the contrary will happen, that, the pole of the ecliptic descending from O toward P, the pole of Mercury S, whether it rests or goes more slowly against the order of the signs, seems to advance in the order of the signs, the line PS being inclined toward PC. All which things, since they agree with the observations, set forth in Book VI, Parts II and III, especially about Mercury, it wants little but that, having cast off timidity, I should cry out in victory, using the places mentioned, before the numbers and measures are known.
3. Whether the poles of the planets rest beneath the fixed stars, or themselves also move, it follows from the motion of the pole of the ecliptic OP that the motions of the planetary poles, and therefore of the limits and nodes, beneath the fixed stars seem unequal. For unless the pole of some planet coincides with the very royal pole A, the centre, namely, of the little orbit OP, or has a motion most equable with the pole P about A in the same direction, its distance from the pole of the ecliptic P will always be changed; whatever arcs, therefore, this one or that makes at any time, these will make at the pole of the ecliptic P a greater angle from near, a smaller from far, wherefore the legs of the angle, produced, will also intercept unequal arcs at the limits, wherefore unequal arcs will also correspond at the nodes, always a quadrant distant.
What is to be gathered hence? Since the quantity of the little circle OP about the royal pole A is not from observation, the magnitude of the motion OP unknown, the quarter of the motion not most certain, whether against the order of the signs at OP or in the order of the signs at B, wherefore twenty centuries, or less, from which the memory of cultivated astronomy lasts, by no means suffice for founding a universal astronomy: but a temporary one at least (through the temporary ecliptic, namely) is practised by men in any age. The truest inclinations, therefore, of the planets to the royal road, and the causes and quantities and quarters of the motions, of the limits and of the nodes, these, I say, and the rest of the kind, lie hidden in the pandects of the following age, not to be learned before God, the arbiter of the ages, shall have unsealed this book to mortals, himself immortal, to whom be praise, honour and glory for ever and ever. Amen.