Epitome Astronomiae Copernicanae: Liber II, part 2
They are Aries, Taurus, Gemini, Cancer, Leo, Virgo;
And Libra, Scorpio, the Archer, the Goat, the Water-bearer, the Fishes.
Into what parts does the zodiac divide the sphere? Into a northern and a southern hemisphere, as the equator does; of which division more below.
V. VI. The Tropics.
Whence arise the Tropics? The ecliptic is oblique to the equinoctial, and so to the first motion. Imagine, then, a line drawn from the centre of the Earth to the two points of the ecliptic among the fixed stars which are farthest from the equinoctial, one toward the north, the other toward the south, and the Earth meanwhile turned about while this line stays at rest; this line, then, carried round, will cut the surface of the Earth on each side in a circle not greatest but smaller and standing to the side of the equator. The images of these two circles are commonly drawn on the little globe of the material sphere. Next imagine this line, fixed at some point of this terrestrial circle and nevertheless extended as far as the fixed stars, to go round with the Earth, or even to remain at rest with the Earth while the heaven of the fixed stars goes round: this line, then, describes a circle in the highest heaven of the fixed stars, corresponding to the former circle on the Earth, and this in each quarter of the heaven. The images, therefore, of these two circles are those two collateral circles in the material sphere. In figure 23, LS and RQ. Whence is the name of the Tropics? From the Greek word trope, turning: for the Sun, coming to the points of the ecliptic through which these circles pass, namely to the beginnings of Cancer and Capricorn, in some measure turns its course; so that he who before, in going forward, had ranged ever more and more away from the equator to the side, and in our position of the sphere had every day risen higher at midday beyond the tropics, now begins to turn his course again toward the equator and to become lower every day at midday; the contrary happens at the opposite point of the ecliptic: and when this has taken place the seasons also change in Greece, which changes for that reason also are called tropai. And they are called the tropic of Cancer and the tropic of Capricorn, each from its own point of the ecliptic, at which it touches it.
How else are they called? Solstitial, because, before that turning of the Sun’s course, or rather the turning of its declination from the middle line of the equator, is perceived, the Sun for some days is found at the same distance from the equator, and therefore also at the same height at the times of midday, and so, with respect to this unvarying declination and altitude to which he strives every day at midday, he seems in a manner to stand still. Lucan, in book IX of the Pharsalia, calls the tropic of Cancer the circle of the high solstice, and says that it strikes, that is, touches, the middle of the orb of the signs, that is, the ecliptic, drawn through the middle of the breadth of the zodiac. And the place agrees; for the temple of Ammon in Africa, of which Lucan is here speaking, Ptolemy puts at latitude 28 degrees, which Lucan took for 23½. In vain does Sulpitius understand something else, multiplying solstices; he trusts Lucan too much, who, prodigal of words, heaps up here many things which belong partly to the tropic of Cancer itself, partly to the whole torrid zone, most to the middle line of the equator, and some also to the tropic of Capricorn and to the more southern places.
What places of the Earth lie beneath the tropic of Cancer? Its beginning is in the hither part of Africa, beyond Atlas, and it passes through the confines of Libya and through Syene of Ethiopia. Thence, having crossed the Red Sea beyond Mount Sinai and Mecca, the fatherland of Mahomet, it cuts through the middle of Arabia Felix, called Nabataea; and entering the Indian Ocean beyond the Persian Gulf, by the shores of Carmania and the mouths of the river Indus, and having crossed India, it approaches the mouths of the Ganges; and again attacking the mainland, it passes through India beyond the Ganges, and, having greeted the farthest Chinese, crosses the very wide southern ocean, and, now drawing near to America, below California attacks the Mexican kingdom, a little this side of Michoacan and Mexico, and, passing out into the Gulf of Mexico, skirting the northern shores of Cuba, it winds back into the Atlantic Ocean.
What places of the Earth lie beneath the tropic of Capricorn? Here, in the last Ethiopic Ocean, between the island of St. Helena and the Cape of Good Hope, or the southern extremity of Africa, it cuts that tongue of Africa, or the kingdom of Monomotapa, and the southern part of the island of Madagascar, and, traversing the eastern ocean in a very long stretch, again crosses a tongue of the Magellanic coast below the islands of Java, and, after a little of the ocean, another coast of the Magellanic land, called New Guinea; then, entering the southern ocean beyond the islands of Solomon, for a little less than half its length it is immersed in the waves of the ocean, until, issuing into America, it cuts off its last tongue, entering at the side of the province of Chile and passing out into the Atlantic Ocean on this side of the mouths of the river Plate in Brazil; so with a very small part of itself it goes over lands.
VII. VIII. The Polar Circles.
Whence arise the Polar circles? The ecliptic among the fixed stars has its own poles, distant from the poles of the world. Imagine, then, two straight lines drawn from the centre of the Earth to the fixed stars, namely to the poles of the ecliptic, which, remaining at rest while the daily motion of the Earth intervenes, will cut the surface of the Earth, and the sections will be small circles about each pole of the Earth; the images of these are commonly drawn on the middle little globe of the sphere. Let these lines now be fixed at some one point of the terrestrial circle; again, then, the daily motion intervening, that line, carried round together with the rolling of the Earth, will describe beneath the fixed stars circles corresponding to those terrestrial ones. And the images of these are the so-called Polar circles in the material sphere.
In figure 28 they are TO and PY, described about the poles of the equator B and C from the poles of the ecliptic O and P.
What places of the Earth lie beneath the arctic polar circle? It passes through the middle of Iceland, called Thule by the ancients, situated in the northern ocean above Britain. Having traversed that ocean, it enters the farthest part of Norway, thence the innermost recess of the Gulf of Bothnia, into which one sails from the Baltic Sea, and, passing along and traversing Lapland, it will be drawn into the Muscovite gulf; then, touching the farthest Tartary and the shores of Cathay, it crosses the Strait of Anian, contiguous with the Chinese Ocean, and for the rest hides itself in the unknown places of America, from which it emerges again in Davis Strait and crosses the nearest shores of Greenland, returning into itself at Thule.
What places of the Earth fall in the antarctic polar circle? The industry of navigators has not yet reached those places; but all that region of the surface of the Earth is held by the uncertain opinion of a continuous continent, which has begun to be called Magellanic, and which, after our own continent anciently subdivided into three parts, Europe, Africa and Asia, and after America, is the third of those which are separated round about from one another, like islands, by the ocean and its straits.
How is the sphere divided by the tropics and the polar circles, or what do those variously coloured rims on the innermost little globe of the sphere mean? The surface of the Earth is divided by the two tropics and the two polar circles into five parts, of which the three middle ones are circular belts; wherefore all those parts are called the five zones. Now the red colour of the middle zone signifies the torrid zone, and it is that tract of lands which I said above lie beneath the equator and the tropics, with the intervening parts of the Earth. The green colour of the two lateral ones denotes the temperate zones, one of which we Europeans hold, and the chief part of Asia, and no small part of Africa, and finally half of America above New Granada and Mexico; but the colour, blue or white, of the extreme zones, which have rather the appearance of little dishes, signifies the frigid zones and the tracts of lands enclosed by those places which we said lie beneath the polar circles.
What is the cause of the names and colours? The torrid is so called from the immense and intolerable heat which scorches the lands and makes them dry up, so that they take on a brown colour; the frigid from the excess of cold, which kills the lands and makes them pale, or, stripped of trees and plants, whitens them with snow; the temperate from the tempering of heat and cold, pleasant to plants and living creatures, whence all things flourish and are green in their seasons.
Is, then, excess of cold the mark of the frigid zone, and intensity of heat of the torrid? By no means is the matter so simple. For since we are in the temperate zone, yet in summer we are oppressed by heat, in winter by frost, and yet meanwhile we are not carried from one zone into another. And in this same temperate zone there are places, I say, that are coldest throughout the whole year, as New Albion in America; in the frigid zone there are flourishing lands, as Greenland, so named in German from its greenness; and finally in the torrid zone there are most blessed and temperate abodes, as in the westernmost Africa, in Peru and in the islands of the Indian and eastern ocean.
Why, then, is heat attributed to the torrid zone, cold to the frigid and temperateness to the temperate, by these names and colours? Because, since heat, cold and the fertility or sterility of the lands which accompanies these qualities have various causes, celestial and terrestrial: the celestial are universal and prevail, but the terrestrial are particular and reduced to the narrow limits of certain places and times do not extend widely, but are seen hemmed in by the universal cause. In this innermost little globe of the sphere, therefore, the manner of the celestial cause is depicted for us. Add that in this our Europe, and in Greece, formerly its noblest part, the terrestrial causes were found by the first inventors of these arts to conspire admirably with the celestial. For to those who turn their face to the east, on the right are Syria, Egypt and Africa, fervid regions, and above them Arabia, Ethiopia and Libya, rocky or sandy regions, places which immensely increase the heat of the Sun; on the left are Thrace, Sarmatia and Scythia, from whose marshes and snow-covered mountains the cold winds, blowing out, bind the Ister and the neighbouring rivers with frost in winter, and whence in summer the etesian winds blow, the chief mitigation of the heat.
IX. X. The Colures.
For what reason were two colures constituted in the sphere? The astronomers imagined one circle through the poles of the equator and of the ecliptic, another through the poles of the equator alone and through its sections with the ecliptic, both of the greatest circles, so that by their meeting and mutual section they might figure in the poles of the equator two poles on each side, from which the material sphere, an axis being passed through, might be hung and turned round, and in which the remaining six circles (the horizon and meridian excepted) might be fixed and joined together.
In figure 29, BMCN is the colure of the equinoxes, because M and N are the equinoctial points. And OBLDPC is the colure of the solstices, because L and Q are the solstitial points and O and P the poles of the ecliptic.
How are they called? Colures, from a Greek word meaning the docking of a tail, because their extremities about the lower pole are not seen by us and so are as it were cut off by the horizon. And the former is called the colure of the solstices, because it passes through the solstitial points of the ecliptic, the other the colure of the equinoxes, because it passes through the equinoctial points.
What common name have these sections of the ecliptic? They are called the cardinal points, because on them the hinges of the seasons of the year turn, that is, because, when the Sun is in them, the chief changes of the seasons occur.
Whence are these called equinoctial, those solstitial? Equinoctial they are called because there too the equinoctial cuts the ecliptic, and because the Sun, falling upon them, makes the days equal to the nights. Solstitial and tropical, because at those points the tropics touch the ecliptic and the Sun, having passed them, seems to stand still, about to turn his course.
Of the Distinction of the Circles.
The origin of each having been explained, say now, for the sake of fixing it in the memory, in how many ways these ten circles are distinguished from one another? In six ways: either by reason of the subject, or by reason of the origin, or by reason of the quantity, or by reason of the figure, or by reason of their situation compared with the motion, or by reason of the motion itself.
1. How do they differ among themselves by reason of the subject? Some are proper to certain and separate tracts of the heaven alone, as the zodiac and the two colures; others are proper to certain and separate tracts of the Earth, but are transferred to certain and separate tracts of the heaven, as the equator, the two tropics and the two polar circles; for it must not be thought that there is any fixed star or wandering star, or even the Sun himself, which by its motion crosses that region of the fixed stars to which these circles are transferred; others, finally, belong neither to separate tracts of the heaven nor of the Earth, but agree with all promiscuously by transference, and are proper to the sphere, the instrument of the first motion, as the horizon and the meridian.
Of those which are thus assigned to the heaven, some hold the same seats in it perpetually, as the zodiac, which belongs to the heaven not with respect to the daily motion of the Earth, but on account of the stars alone and their secondary motions; others in the succession of centuries migrate from their seats and pass to neighbouring ones, as those which belong to the heaven with respect to the daily motion of the Earth, such as the colures and all that are transferred to the heaven. So those that inhere in the Earth all inhere in it unchangeably, yet some are in it only on account of its daily motion, as the equinoctial, others on account of the comparison of the daily motion with the celestial circle of the zodiac, as the two tropics and the two polar circles.
2. Compare them by reason of origin. Some are primary, each arising from its own causes, some secondary, depending on the primary ones joined together with one another.
The primary have either an intrinsic origin of the world, namely motion: as the zodiac, a celestial circle, from the motion of the secondary movables, the equator, a terrestrial circle, from the first motion of the Earth; or they have an extrinsic origin, as the horizon from the sight of man, the inhabitant of the machine of the world. The secondary are those which exist by the mutual comparison of the primary among themselves: on the Earth the two tropics and the two polar circles, by comparison of the zodiac and the equator and the poles, in the heaven the colures, by comparison of the same, and finally in the material sphere the meridian, by comparison of the horizon and the equator and the poles, all of which have been explained so far.
3. Distinguish the circles by reason of quantity. Quantity, as so far, is understood in comparison with the spherical surface, not with the matter either of the world or of the sphere’s construction. Therefore some are greatest, which have the same centre as the sphere, as the zodiac, the equator, the horizon, the meridian and the two colures; others smaller, which have other centres than the centre of the sphere, as the two tropics and the two polar circles.
If six are greatest and therefore equal to one another, why is the meridian in the sphere made larger than the rest, and the horizon alone the largest of all? That comes from the necessity of the matter: for the sphere could not be constructed and turned round by the hands unless the horizon embraced from outside and supported the meridian, and the meridian the sphere; moreover, not the whole surface of these rings, but only the innermost rim of the horizon and meridian, and the outermost of the rest, bears the name of its circle and discharges its office. And although the innermost rim of the outer ones is still broader than the outermost rim of the inner circles, this again is done only to facilitate the motion of the sphere, but by reason it is understood that they ought to be equal, since they were to touch one another; and things that touch one another are geometrically one so far as the contact goes.
What chiefly befalls the circles of the sphere compared with one another in respect of this distinction? Since the planes of the circles of the sphere either meet within or without the sphere and cut one another, or do not meet but go on parallel: in the smaller circles astronomers carefully consider the parallel position, but in the greatest their mutual section. For the greatest cannot be parallel to one another, but either coincide or cut one another; the smaller, though they are not parallel to one another, yet do not always cut one another on the surface of the sphere; nor is the use of those sections great.
What, then, is the use of the sections of the greatest circles with one another? Three greatest circles constitute a spherical triangle, two a spherical angle; in these the doctrine of triangles is occupied, in which all the reasons of the first motion are contained, explained and brought into use.
How many things are considered in each spherical triangle? Six: three sides, namely the arcs of three greatest circles, and three angles.
What measures a spherical angle, that it may be compared with a side? A spherical angle is measured by a circle, likewise a greatest one (or even any smaller), described from the angle as a pole. And a right spherical angle, no less than a rectilinear, is measured by a quadrant of a circle. Therefore if the two legs of the angle are quadrants, the side subtended by the angle is itself the measure of that angle; but if two sides of the triangle are not quadrants of their circles, there will be need for this work of altogether six greatest circles, three which furnish the sides, three others which furnish the measures of the angles.
So, then, the angles contained between circles which pass through the zenith are measured by the horizon, the angles of those which pass through the poles of the world, by the equator or one of the parallels, the angles of those which pass through the poles of the ecliptic, by the ecliptic itself.
Let CAB be a triangle, all three of whose sides are less than a quadrant: if, then, CE and CD are quadrants, ED will be the measure of the angle ACB, because the circle ED is described from the pole C. So if BF and BH are quadrants, the arc FH will be the measure of the angle ABC. So if AI and AG are quadrants, GI will measure the angle CAB.
What is the sum of the doctrine of spherical triangles? If, of the six things of any triangle, three are known, to find out from them any one of the three remaining by the help of the straight lines of the circle. On this doctrine there are special books.
Review the straight lines of each arc that are useful for the solution of spherical triangles. From one end of any arc less than a quadrant two lines are drawn, one to the centre, which is called the radius or whole sine, the other the tangent, toward the direction of the arc; from the other end of the arc likewise two, one a perpendicular upon the radius, or parallel to the tangent, which is called the sine, and divides the radius into the sine of the complement and the arrow, or versed sine; the other inward indeed as far as the centre, outward as far as the tangent, terminating it and in turn terminated by it, which is called the secant; so the more recent writers. Of their predecessors some use Greek names, calling the radius the semidiameter or base, the tangent the cathetus (in Latin perpendicular), the secant the hypotenuse (the subtense, namely, of the right angle); others, looking to use, call the tangents, or rather their numbers, fruitful.
The arc BC; A the centre; BA the radius, the whole sine, semidiameter or base; BD the tangent, cathetus or fruitful; CE the sine; EA the sine of the complement; EB the versed sine or arrow; AD the secant or hypotenuse.
What more is to be observed of the sections of the greatest circles with one another? When one passes through the poles of another, the other in turn also passes through the poles of the first, and consequently, when one passes through the poles of two, it will have its poles in their common sections; and its arc, intercepted between the poles, is equal to the arc between their circles.
So in figure 28 the poles of the equator B and C are in the meridian, and no less the poles of the horizon F and G, that is, the zenith and nadir; therefore also the poles of the meridian M and N (from which it is understood to be as it were described) are in the mutual sections of the equator and the horizon. Wherefore the arc DH of the meridian measures the angle HMD, which the horizon makes with the equator. So LD and BO are equal. The rest is found in the geometers.
4. How are the circles of the sphere distinguished by reason of figure? All the others lack breadth; the zodiac alone has some breadth, in the figure of a spherical rim.
Yet those circular rings all have some thickness, which represents on the outside and the inside the figure of a rim. This again comes of the necessity of the workmanship, and in different materials the thickness is different; but the construction is so adapted that only one side of that thick ring exhibits the circular line to which the name and office of a circle belong.
5. How do the circles differ among themselves by reason of situation with respect to the first motion? The equator, drawn through the middle of the sphere as it moves, is the rule of the motion; the rest are either at right angles to it, as the meridian and the two colures, or oblique, as the zodiac, or parallel to it, as the two tropics and the two polar circles, among which it is itself called the greatest of the parallels; lastly the horizon, according to the different positions of the sphere, that is, according to the different dwellings on the Earth, is now at right angles, now oblique, now coincident with the equator and so among the parallels.
6. Finally, how do the circles differ among themselves by reason of motion? In one way by reason of the material sphere, in another by reason of the world. As to the sphere, the horizon and meridian are immovable, all the rest movable, since sight, the author and origin of these immovable ones, judges itself to be at rest. But as to the world itself the contrary happens, for only the horizon and meridian of the world are moved, while the tracts of the heaven to which the rest belong are at rest. Finally, as to the Earth, the equator, the two tropics and the two polar circles are fixed in it to the meridian and horizon of the place, and so are movable with the whole Earth. These things were explained more clearly above.
What is the cause of this discrepancy between the circles of the Earth? The cause consists in the various and manifold uses of the material sphere. For if its only use were that it, and the other parallels, should show above the zeniths of which places of the Earth those points of the ecliptic pass which fall upon each of the parallels, then in the sphere all the parallels could and ought to be fixed to the meridian and made immovable, as on the Earth they are fixed to it, since these circles belong first to the Earth and are thence transferred to the heaven by imagination. For then the eye would feign to itself another as it were floor, or another vault outside and above the last heaven, under which, as under a resting sphere of the fixed stars, the Sun might run its course; and then upon that outer floor, in which the meridian and horizon of the sphere are imagined to be, they would inscribe for each point of the ecliptic its parallels, corresponding to the parallels of the Earth, likewise as of a resting body; and so by this fiction it would fix them to the meridian. But because then the two points of the ecliptic in which it is cut by the equator would pass in succession over the whole equator, which is immovable, and the equator, on account of other uses, had to take the numbers from 1 to 360, with a beginning not wandering from day to day but certain and constant, namely from the vernal section of the ecliptic: the equator therefore had to be fixed to the ecliptic itself and so to become movable with the ecliptic of the sphere. But one of the parallels being fixed to the movable sphere, there was no reason why all the other parallels should not be fixed to it also and, instead of resting, be made movable, lest their reason should seem different by cause of origin. For to their uses it matters little whether they are movable instead of at rest.
To what end tends this admonition? It tends to this, that we may remember that we ought not to imagine, as if the points or poles of the ecliptic, or the Sun falling upon those points, pass in one daily revolution over all those fixed stars which are placed round about in the parallel of each, as it passes over all the places placed in the corresponding parallels of the Earth. For this is false, as will have to be said more often below.
You say that the meridian is immovable with respect to the material sphere: but I see it made thrustable through the notches of the horizon circle. When we make the meridian immovable, we understand it as it is placed in the sphere with respect to the daily motion. For that thrusting and change of position has nothing to do with the first motion.
But to what end is it thrustable? Because the meridian and horizon have sight for their author, and sight, or the men who are observers, are scattered over the whole round surface of the Earth, and therefore have their vertical points scattered over the whole circuit of the heaven, one of which is nearer to the pole of the world, another more remote from it. But it was said that the meridian passes through the poles of the world and the poles of the horizon, or the vertical point and the nadir. That, therefore, for different places the vertical point might approach the pole of the world in different degrees, and so one horizon of the material sphere might serve all places and their natural or visual horizons, the meridian was made removable and thrustable. For by its turning the pole of the sphere is raised above the horizon or depressed.
In this way, however, only that variety of vertical points is served which is toward the poles of the world: but how is the other variety of places, toward the east and west, provided for? With respect to the heaven that variety is nothing, since by the daily motion all points of the circle stretched out to the east and west pass in succession through the zenith of the place: the motion or revolution of the sphere therefore compensates that variety. But with respect to the Earth, the little globe in the middle of the sphere ought not to adhere to an immovable axis wholly immovable, but ought to be made able to be turned (if only the whole image of the terrestrial globe is to be expressed in it), so that it can be twisted and fixed in another position, and so any place of the Earth can be directed upward to the pole of the horizon, or the vertical. For instead of an observer of the stars transferring his body, or his mind and thoughts, from one place of the Earth to another more eastern or western, changing his visual horizon, now in the sphere conversely that place of the Earth, or of the little globe, to which the transition is made, must be applied to the material horizon, which in the sphere is single, that there may be no need of an inclination of the material horizon and an upsetting of the sphere, most inconvenient and absurd, to represent places of the Earth different from and far from the place of the observer.
Of the Division of the Circles.
How do geometers divide a circle? The geometrical division of the circle is first into two semicircles and four quadrants, because each quadrant is the measure of one right angle. When either a semicircle or a quadrant is cut in any way, one part has the name of the arc or angle, the other is called the complement of the arc or angle to the semicircle or quadrant. But the more minute and ordinary subdivisions, by which the quantity of each arc or complement becomes known, the geometers borrow from the astronomers, dividing the circle into 360 equal parts.
What is the cause of this division? The cause is twofold. For first, nature herself and the motions of the Sun and Moon led the way in dividing the zodiac; then reason, contemplating the nature of the circle, supplied what was wanting to the perfection of this division aimed at by the Sun and Moon.
Explain the causes of the division of the zodiac. It was noticed that, while the Sun completes a year, the Moon returns to the Sun twelve times, yet in such a way that after the twelfth return something is still lacking to the Sun for the complete course of the circle. It was noticed secondly that, while the Moon, having departed from the Sun, returns to the Sun, nearly thirty days pass, yet so that a little less than half a day is lacking to this number. If each number were exact, thirty times 12 would then be 360. The zodiac, therefore, was cut precisely into twelve signs, the sign subdivided precisely into thirty degrees, and the ancients had 30 perfect days for a month.
The number twelve also seemed all the more suited to the zodiac because this circle was first divided by the equator into two equal semicircles, then by this and by the colures, one passed through the poles of both, into four equal quadrants. Now by an instinct of nature we are accustomed to divide any quantity into three, into beginning, middle and end, and three times four are twelve. For other conveniences of the number twelve see below, book III, and presently.