Epitome Astronomiae Copernicanae: Liber II, part 3
So the number 360, that is, the division of each sign into 30, also pleased for this reason, that since the solar year has 365 or 366 days, and the lunar year, or twelve natural months, has 354 or 355 days, the arithmetical mean between 354 and 366 is 360, and so also between 355 and 365.
Explain the cause of the division into three hundred and sixty from the nature of the circle. The first figures inscribable in a circle by geometrical determination are the triangle, the quadrangle and the pentagon. The circle is therefore geometrically divisible into three, four and five. Now the difference of the third and fourth, of parts begun from one end, is the twelfth part; by these two figures, then, namely the triangle and the quadrangle, the circle is divisible into 12 parts. Again, the difference of the third and two fifths is the fifteenth part; of the fourth and fifth, the twentieth part. But when the twelfth, fifteenth and twentieth parts are compared among themselves, the difference between each two is the sixtieth part. The three first regular figures, therefore, geometrically demonstrable, teach us to divide the circle into sixty parts.
The sixtieth, geometrically bisected, gives the hundred and twentieth.
Here AB is a fifth, AC a fourth, therefore BC is a twentieth. So since AC is a fourth and AD a third, CD is a twelfth. So since AD is a third and AE two fifths, DE is a fifteenth; and the difference between CD the twelfth and DE the fifteenth, and likewise between this and BC the twentieth, is a sixtieth.
Therefore the division into 60, and further into 120, is natural to the circle, that is, rational and geometrical.
Although it is against the nature of geometrical things that any other part of the circle, besides the whole, the half, the fourth, the eighth, likewise the fifth, the tenth and the other halves of these, should be divided into three (for a nonagon cannot be inscribed in a circle, that is, the third part of the circle cannot be geometrically divided into three, nor the sixth part nor the twelfth, nor the fifteenth, thirtieth, sixtieth or hundred and twentieth), and the continuous rational division of the circle stops at the hundred and twentieth, in which there is a division by five once, by three once, and by two three times; thence, indeed, continued bisection makes parts that are halves of the former without end, as 240, 480, 960; and no rational section goes further into the thirds of these, so that 360 should come from 120: nevertheless the common custom of dividing anything into three, with no regard to the circle but only to length, reigns here also, so that any hundred-and-twentieth part may be understood to be divided into three equal parts, beginning, middle and end, just as if an arc of the circle, or a hundred-and-twentieth part of it, were extended into a perfect straight line. So a hundred and twenty taken three times make also 360 parts.
What name has one of such parts in any circle? The common name through all circles is part; but particularly in the zodiac they are called degrees, because the Sun seems to pass through almost so great an arc within one day; in the equator they are called times, because it is the measure of time; in the horizon they are called by the Arabic word azimuths, in Latin they might be called quarters. But the name of degrees has by use become common through all circles.
In what series of numbers are these parts written in any circle? In the equator and the zodiac the beginning is made from their common vernal section, and those are numbered first which rise first in the temperate zones and the torrid, and in the zodiac those later into which the Sun comes later. In the equator the numbering is continuous from one to 360 through the whole circle, but in the zodiac from one to thirty, which measure makes one twelfth part; thence the numbering returns to the beginning, or unity, until the second twelfth part is finished, and so twelve times. And when the numbering or motion proceeds in this order, it is called the sequence of the signs; the numbering or motion is said to proceed in the contrary way in antecedence.
In those circles which pass through the poles of the world, the beginning from each pole is double, one from one semicircle, the other from the other; and this number is carried up to 90, that is, to the middle circle between the poles; so there are four beginnings in the four quadrants of each. Others make the beginning from the middle circle and end at the poles. Some join or mix both series.
In the parallels, namely in the tropics and polar circles, a division is not thought necessary, because they are of the number of the smaller circles, and the middle one of them, the equator, does the office of all.
In dividing the horizon also the craftsmen do not follow one method. For since the meridian divides the horizon into two semicircles, the rising and the setting, and the equinoctial into the northern and the southern, some begin from its sections with the meridian, some from its sections with the equator, some from both, and either proceed from 1 to 180 on each side, or from 1 to 90.
Why in the equator and zodiac is the beginning of the numbering made from the vernal section? In a circle by its own nature there is neither beginning nor end. Yet because some beginning must be made, we come, with nature for our guide, to one of the cardinal points, since they are shown by the sections or contacts of the colures, the equinoctial and the tropics, and are placed in an evident and conspicuous place. Now the tropical points are marked more obscurely and lie hidden in some part of the circle in which the change of the Sun’s declination is imperceptible. Of the equinoctial points this one pleased the first inventors of astronomy because it brought back to them in their zone light and heat and opened the beginning of spring, when the face of the Earth is renewed and nature revives. For since all the contrary things happen in the autumnal season, they hold in less esteem that section which the Sun reaches at the time of autumn.
How is one part or degree subdivided? Into 60 parts, which we call scruples or minutes, in Greek lepta; and one minute into 60 seconds, one second into 60 thirds, and so on, as far as there is need of this subtlety.
What is the manner of notating them? To the numbers marking whole parts they commonly add a little circle, to minutes one stroke, to seconds two, and so in order. In this little book the name of the part, degree or time, so of the first minute, second and so on, either whole or abbreviated to its first letter (where the strokes are lacking), is placed indifferently before or after. But if that is not done, the numbers are to be distinguished by their order, and those placed first or alone are to be taken for wholes, those second for first minutes, those third for seconds.
What cause has this sexagesimal division? None natural from motion, nor rational and geometrical, drawn from the nature of the circle, which would of itself concern so small an arc; but only arithmetical. For there is no number within a hundred which has more multiple parts and is therefore more fit to handle, since it has a sixtieth part (1), a thirtieth (2), a twentieth (3), a fifteenth (4), a twelfth (5), a tenth (6), a sixth (10), a fifth (12), a fourth (15), a third (20), a half (30). Add that this division is akin to the former division of the whole circle into 360 parts; for if the circle is divided by the hexagon, which division is the most expeditious, because the side of the hexagon equals the radius, and the same compass which described the circle also divides it: then to one sixth there come ten times six degrees, that is, sixty, whence the sixth part of the circle is also called the sexagene. It is fitting, therefore, that one degree of sixty should also go off into sixty scruples and so on. For if the continuous proportion of the subdivisions is interrupted, as in money, the labour of computing, on account of the necessary resolutions of wholes into parts, or reductions of parts to wholes, is increased immensely.
Of the Division of the Zodiac in particular.
By what names are the twelve equal parts or dodecatemoria of the zodiac distinguished from one another? It is not received by use that we should distinguish them by numbers in common speech, except when we undertake a computation of motions; but we are accustomed to give them names from the constellations which were found in them at the time when the first inventors of astronomy flourished, as was said above. And they are called by the common name signs, no less than the images themselves, outlined among the fixed stars.
Are these signs, then, or configurations of the fixed stars, no longer found today in the dodecatemoria named from them? Today the images outlined among the fixed stars have nearly passed into the following dodecatemoria, yet they have left to their former seats, or dodecatemoria, their names.
But I supposed that the ecliptic, whose parts are the dodecatemoria, does not move among the fixed stars, but remains perpetually fixed to them. This migration of the images out of their dodecatemoria does not come about by a motion of the ecliptic, but the equator, as has been said, migrates from one place of the fixed stars to another, and therefore cuts the ecliptic in different and different places; but the section, as the beginning of the numbering, having been carried back into the antecedents, the dodecatemoria also, as the joints of the numbering, are carried back into the antecedents, both in the ecliptic and among the fixed stars: and so the images seem to be carried into the consequents. The causes of the shifted sections are given in book III, part V, and below from the motions of the secondary movables, chiefly in book VII.
Since not only astronomers but other writers also are much occupied about the signs, in how many ways are they commonly distinguished? Chiefly in five ways: three indeed on account of the section of the ecliptic with the equator and the colures, where the contiguous ones constitute one class, and two ways where those disjoined in position are brought into one class by the inscription of a figure in a circle.
1. How are those signs distinguished, or the ecliptic itself, by the circle of the equator? Into a semicircle and six northern signs, which decline from the equator and are raised toward the north, as Aries, Taurus, Gemini, Cancer, Leo, Virgo, and into the southern as many, which are depressed from the equator toward the south, as Libra, Scorpio, Sagittarius, Capricorn, Aquarius, Pisces.
Is some ambiguity to be guarded against here in the words northern and southern? Altogether. For the ecliptic also divides the whole sphere into two hemispheres, boreal and austral, by which means the first semicircle of the equator, one tropic and one polar circle and a pole are called boreal, aquilonar, arctic, and the other semicircle of the equator, and so on, austral. Therefore one and the same star, belonging to some sign, is called boreal with respect to the equator, austral with respect to the ecliptic, and conversely, if it is placed between the ecliptic and the equator.
2. Tell the distinction of the signs by the colure of the solstices. Six are called ascending, and their semicircle the ascending, in which the Sun and planets ascend from the south to the north toward the zenith of our zone, as Capricorn, Aquarius, Pisces, Aries, Taurus, Gemini; the other six, or their semicircle, for the contrary reasons are called descending: Cancer, Leo, Virgo, Libra, Scorpio, Sagittarius.
3. How is the ecliptic with the signs distinguished by each colure and the cardinal points? Into four quadrants, corresponding to the four parts of the year, from which they are named.
The vernal are Aries, Taurus, Gemini, in which the Sun ascending from the equator toward the north constitutes spring, and it is the first quadrant. The summer: Cancer, Leo, Virgo, in which the Sun descending from the north toward the equator makes summer; the second quadrant. The autumnal: Libra, Scorpio, Sagittarius, in which the Sun descending from the equator toward the south completes autumn, which is the third quadrant. The winter: Capricorn, Aquarius, Pisces, in which the Sun returning from the south toward the equator measures out winter. This is the fourth quadrant.
4. What is the fourth division and what is its use? This is rather astrological. For three quadrangles are numbered in the zodiac, and in each four signs arranged through the zodiac in tetragonal form, whence the class has its name. It serves nevertheless for comprehending the motions of the Sun and Moon, that we may know that both luminaries, when the Moon is half-full, are in the signs of the same quadrangle.
The first quadrangle is of the cardinal signs, those begun from the cardinal points: Aries, Cancer, Libra, Capricorn. These signs are also called movable by astrologers, because, with the Sun in them, the weather is believed to be generally changeable in the various seasons. The second is of the middle ones between the cardinal and the bicorporeal: Taurus, Leo, Scorpio, Aquarius. The astrologers call them fixed, because, with the Sun in them, the seasons are for the most part thought steadier. The third is of the bicorporeal: Gemini, Virgo, Sagittarius, Pisces. This common name, like each one’s own, they have received from the constellations which were once in these dodecatemoria, which the men of old imagined as having two bodies: Sagittarius composed of half a man and half a horse, the Fishes and the Twins doubled, and the Virgin, in place of a second body, with a sheaf of ears of corn. The astrologers, by comparison with the fixed and the movable, call these common.
5. Tell the fifth distinction. This again is rather astrological, since in that discipline four trigons are numbered, and in each three signs arranged through the zodiac in the form of a triangle, whence the class has its name, triplicity or triangle.
This division is shown, however, by the motions of Saturn and Jupiter, and their meetings every twentieth year, which take place in the signs of one class for nearly two hundred years. For in the year 1603 they met in Sagittarius, in the year 1623 they will meet in Leo, in the year 1643 in Aries, in the year 1663 again in Sagittarius; after 200 years these conjunctions pass into three other signs. So, all four classes being completed, after 600 years a new circle is made.
The astrologers gave names to these classes from the four elements. The first triangle is called fiery and has Aries, Leo, Sagittarius. The second earthy, has Taurus, Virgo, Capricorn. The third airy: Gemini, Libra and Aquarius. The fourth watery: Cancer, Scorpio and Pisces.
In how many ways is sign, and being in a sign, used? In three ways. For the sphere of the fixed stars being divided by six circles going through the poles of the ecliptic into 12 parts, one sign is a twelfth part either of the ecliptic, or of the zodiac, or even of the whole sphere of the fixed stars, continued to the poles of the ecliptic on either side; and in the first way the Sun is said to be in a sign, the second way belongs to the planets that wander from the ecliptic and to some fixed stars, the third to the other fixed stars; but all three ways belong to comets as the case may be.
Of the Quarters of the Winds.
How do sailors commonly divide the horizon? In the most ancient times four quadrants of the horizon were recognised, described from the four cardinal points of the world and each marked by the name of a wind. For no more winds are named by Homer than these four: Eurus blowing from the rising, Zephyrus from the setting, Boreas from the north, Notus from the south. But the later Greeks, devoted to seafaring, began to subdivide the horizon more finely, dividing the rising and the setting into the solstitial or summer, the equinoctial and the winter or brumal; after which it followed that the north and the south also were made threefold and so twelve winds came to be, which in the Greek sea, shut in by continents and not so widely open, they named, those between, from the lands lying about, from which they blew. Hence Phoenix, Africus, Lips, Thrascias, Hellespontius, Olympias, Strymonia, Japyx. Vitruvius doubles the number, so that they are 24. But after the whole ocean flowing round the continents began to be navigated, since neither the solstitial risings and settings were equally distant from all the winter ones in all places, nor were names devised by one nation suited to all places, nor easy to remember in so great a number: the Germans introduced a new division of the horizon into 32 winds, by continued doubling of the number of the four cardinal points, and gave them names from their own language, which the other nations, Italians, French, Spaniards and modern Latin writers, mostly imitate by applying the names of the ancients, but by no means with equal success.
Each one of these parts, therefore, occupies equally eleven astronomical parts or degrees and a quarter.
Explain by what names these parts are called. First, by a common word they are commonly called by modern Latin writers rhombi. For the nautical box (called the compass) has a paper card laid upon the magnetic needle, which is always directed toward the north; on this card are painted 32 rays or rhomboid points of different colours, so that when the point of the magnet, with the lily painted over it, is directed toward its natural quarter, every rhombus also indicates its own quarter. All the rhombi are named from the cardinal points of the world, yet differently the cardinal ones themselves and the intermediate.
Which do you call the cardinal points of the world and the cardinal winds? These are shown to us in the northern hemisphere by the motion of the heaven; for we call east, in German Ost, where the Sun rises at the equinoxes, west, where the Sun sets on the same day, north, in German Nord, where the pole of the world is seen, which for another reason also is wont, by a peculiar prerogative, to be called the cardinal point of the world, south, in German Süd, whence the Sun shines at the midday hour. These four points are shown astronomically by the sections of the circles of the meridian and the equinoctial with the horizon; and the north and south, which the meridian marks out, are held to be the chief.
How, then, are the intermediate winds or rhombi named? 1) The four middle ones between the four cardinal winds have names compounded of the names of their lateral cardinal winds; where the Germans put first in the compound the name of the chief cardinal point.
Thus eight winds arise, distinguished by as many names among the Italians.
2) Now between these eight, eight others placed in the middle have names again compounded of the names of the former eight, each from the two lateral ones neighbouring it, the name of the cardinal being put first in the compound, whence it comes about among the Germans that the name of the chief cardinal points is doubled at once from the beginning, while the names of the remaining cardinal points stand at the beginning and the end of the word.
Ostro Sirocco. Ostro Garbino.
Levante Greco. Levante Sirocco.
Ponente Maestro. Ponente Garbino.
In this way sixteen names arise.
3) Between these sixteen, sixteen others interposed have compound German names, each from one of the first eight, to which is joined, with a preposition, the name of the cardinal point toward which it inclines from its leader.
Those who compare the winds of the ancients follow another manner of composition in Greek form, using the word mesos or the preposition hypo, others hyper, but not in an order to be compared with the German nomenclature, nor do they all agree among themselves. The Italians name these 16 from the eight second ones, but call them quarters, because four times eight are 32. For they make from each of these eight names of the second, or compound, winds two names, always putting first the name of the cardinal point toward which each quarter inclines more. So sixteen names arise in naming the quarters, which with the eight compound and the eight simple make 32. The Italian, then, is as follows.
Nord gen Osten or zu Osten. Nord gen Westen.
Quarter of Tramontana Greco.
Quarter of Tramontana Maestro.
Süd gen Osten. Süd gen Westen.
Quarter of Ostro Sirocco. Quarter of Ostro Garbino.
Ost gen Norden. Ost gen Süden.
Quarter of Levante Greco.
Quarter of Levante Sirocco.
West gen Norden. West gen Süden.
Quarter of Ponente Maestro.
Quarter of Ponente Garbino.
Nord-Ost gen Osten. Nord-Ost gen Norden.
Quarter of Greco Tramontana.
Quarter of Garbino Levante.
Quarter of Maestro Ponente.
Quarter of Maestro Tramontana.
Quarter of Sirocco Levante.
Quarter of Levante Ostro.
Quarter of Garbino Ponente.
Quarter of Garbino Ostro.
Whence have the cardinal winds their names? The Homeric cardinals seem to have been named from their qualities; Eurus from moisture that putrefies, or damp putrefaction, Auster from drying, which is auein, and the same Notus from the mists, which are called notides, Boreas from the voracity which its cold brings upon bodies, Zephyrus from heat, which is zein, to boil. Others maintain that Eurus and Zephyrus are named from the quarters, from that line of Homer in the Odyssey, book X:
For we know not where is the gloom, nor where is the dawn.
so that eurus is from eo rhein, to flow from the dawn; zephyros from zophos, which denotes the west.
Do these ten circles of the sphere suffice to explain all the reasons of the first motion? Others also come into consideration, but such as can easily be understood from what has so far been explained.
Review the more important by their kinds. 1) Of the smaller there are several parallels, nay infinite, parallels for the most part to the equator itself, but sometimes also to the ecliptic or the horizon. For through any point of the sphere or any star one parallel to the equator itself is understood to be drawn, on account of the daily motion. 2) Of the greatest are the circles of declination through the poles of the sphere, the circles of latitude through the poles of the zodiac, the circles of position among the astrologers through the sections of the horizon and meridian; finally others drawn through any two points of the sphere whatever.
What do you call longitude and latitude in the sphere? Although the sphere is round, we say nevertheless that its longitude extends along the order of the signs of the zodiac, its latitude toward either pole of the ecliptic, because these are the poles and the circle proper to the heaven; on the Earth longitude is reckoned along the equator, latitude from one pole of the Earth to the other, because again the circle and poles are those proper to the Earth. On both sides longitude is thus double the latitude, since there are 360 degrees there, here 90 degrees toward either side.
What is the kinship of these circles with those commonly expressed in the sphere? Among the parallel circles are numbered the equator, the two tropics and the two polar circles in the sphere; among the circles of declination, the two colures and the meridian, which supplies the place of all those on account of the mobility of the sphere; among the circles of latitude is the colure of the solstices; among the verticals is the meridian, the middle of all; finally among the circles of position the meridian and the horizon.
Which of the many parallels that are not constructed in the sphere are held the more notable? 1) The circles of the natural days, of which in the third book. 2) The arctic and antarctic of each place according to the doctrine of the ancients, which are smaller circles described from either pole through the sections of the horizon and meridian, of which also in the third book.
What is akin to the parallels of the sphere? The terrestrial parallels, drawn through the middles and boundaries of the climates, of which also in the third book, in the geographical part.
How are the parallels of the horizon called? The Arabs, devoted to the astrolabe, which is an image of the sphere projected on a plane, call them in their own tongue circles of Almicantarat.
Are some others also akin to the circles of declination? Akin are the circles of the latitudes of places on the Earth, the terrestrial meridians mentioned above.
What name have the verticals among the Arabs? The Arabs call them in their own tongue Azimuths, as the vertex Zenith, and its opposite Nadir. The azimuthal circles, then, pass through the zenith and nadir.
How do astronomers use the azimuth of a star and in what sense? The azimuth of a star is sometimes the quadrant of a greatest circle drawn from the zenith through the star as far as the horizon; so the Arabs. Sometimes it is that by which these verticals are distinguished from one another, namely the angle which that circle makes with the meridian, or the measure of that angle, which is the arc of the horizon intercepted between the vertical and one part or the other of the meridian or even the equator, and they speak of an azimuth from the north toward the rising or setting, from the south toward the rising or setting, an azimuth from the rising or setting toward the north or south, as the case may be and according to the manner of inscribing the number 360 on the horizon.
Which are the more notable of the circles of position? Besides the horizon and meridian, which are expressed in the sphere, there are four others, which are called by astrologers, with the same common name as those, circles of the celestial houses, distinguishing the whole heaven by a certain method into twelve houses, to each of which the astrologers assign its name. Hence the verses:
Life, Wealth, Brothers, Father, Children and Health, Wife, Death, Journey and Kingdom, Good Deeds, Prison.
Are never more circles than ten expressed in the sphere? Nay: 1) in some spheres the arctic and antarctic are expressed according to the opinion of the ancients. But then they are adapted at least to one position of the sphere, not promiscuously to all, like the other circles. 2) In some spheres the meridian has a thrustable roller, which can be applied to the vertical point, and from that roller a quadrant descends to the horizon, which, the roller being unmoved, can be turned round through the whole horizon, representing in any position one vertical or azimuthal. 3) Again there are some spheres which at the two notches of the horizon, by which it takes the meridian, have two poles, from which a semicircle is attached, so that it can be turned about the poles, which semicircle can serve in place of any circle of position. 4) Finally in large spheres or armillary rings there are sometimes added two circles of latitude, passing through the poles of the ecliptic and movable along the ecliptic.
What circles do gnomonists use beyond those which are in the sphere? Gnomonists, besides the planes of the meridian and of all horizons under the meridian of one place, recognise three other kinds of planes, upon which sundials must be described: declining, inclining and declining-inclining, that is, declining and inclining at once. The inclining belong to the same circles which are called circles of position: the declining to the vertical circles, among which gnomonists recognise the primary vertical, which is drawn through the mutual sections of the horizon and equator, having for its poles the sections of the horizon and meridian, so that it is at right angles to the meridian, looking directly south from one quarter, from the opposite directly north. This is that circle in whose plane are drawn the sundials which are called vertical regular; and with respect to this primary vertical the planes of the remaining verticals are said to decline either toward the east or toward the west, except the plane of the meridian itself, which looks directly east and west.
All planes are reckoned declining-inclining which belong to such greatest circles as are drawn neither through the sections of the horizon, nor through those of the primary vertical of any place with the meridian, but which are nevertheless reduced to the inclining planes of an altitude of the pole greater or less than that of the proposed place.
To all kinds of planes their own meridians are assigned, which are of the circles of declination, corresponding to the circles of latitude of places, at right angles to the planes whose meridians they are: where the meridian of the horizontal plane and of the primary vertical plane in any place, and therefore of all the other horizontal and vertical planes under the same meridian, is the meridian of the sphere itself.