Epitome Astronomiae Copernicanae: Liber I, part 2
Part I. Of the figure of the Earth, its magnitude, and the method of measuring it
But how can the order of the heavens be known and its reasons laid open on the Earth? If first the true figure of the Earth, and thereby of the whole world, be investigated.
Is the Earth, then, not flat, nor extended downward by an infinite root, as it commonly seems? Rather it is a body consisting of lands and waters together, cut off from the heaven on every side and turned in upon itself, in short round on every side, in the form of a ball, globe or regular sphere; which can be proved as well from its single parts as from the constitution of the whole itself.
Prove from the parts of the watery surface that they present a convex figure. Concerning the parts of the watery surface, that is, the seas, that they are curved in an arc, we are taught by nautical experience. For when to those voyaging on the deep sea nothing appears but heaven and waves, surrounded by a perfectly plane circle, while the mountains, even the highest, of continents and of islands not very remote are hidden, as if sunk beneath the levelled waves, it happens that as the voyagers advance those mountains seem to emerge from the waves and suddenly to lift themselves to an evident height.
Let D be the mountain, A the first place, its line of sight or horizon BC, a straight line touching the waves at A. This line, continued toward C, passes above D, so that D cannot be seen from A. Let the advance now be made from A to E, so that the line of sight or horizon of the place E is the straight line ED, touching the waters at E, which, continued, falls upon the mountain D, so that whatever of it stands out above FD can all be seen from E.
But if the surface of the water were truly extended in a mere plane, no mountain lifting itself above it would be wholly hidden from a sight placed upon it; those who recede would see the figure of the mountain, with its magnitude, vanish gradually only after an immense interval; and conversely those who approach would by no means see mountains suddenly appear after a short interval, though unseen before; but immense journeys would be needed for this.
As, if the surface of the water were extended without limit along the straight line DF and the mountain D rose above it, since the same line DF would also be the line of sight, always falling upon the mountain D, whether the eye were at E or at F or at any other most remote point of this line.
Then ships or mountains are first seen from the top of the mast, when they cannot yet be seen from the foot of it, even when the waves are roughened by no billows at all, merely by the interposition of their swelling.
Prove it from the whole watery surface. Of the whole, present experience has taught the same, that it is plainly a perfected globe, that is, turned in upon itself on every side. Ferdinand Magellan, setting out from Spain, sailed toward the west, and having found a strait, passed from the Atlantic ocean into the Eastern, and landed at the Moluccas; whence one ship, setting out, with a continuous and direct course (by no means turned back), always toward the west, through the Indian and Atlantic oceans, returned home, as if from the east, to the place whence it had gone out.
It appears, therefore, that the surface of water and of land returns upon itself, drawn round in a circle, and does not adhere, grown to the heaven.
This is so for the water; but do the same things follow for the continents as well? Since the ocean is poured between the lands from east to west and again to the east, and its two vast plains flow together at two straits, one to us in the west, which is called the Strait of Magellan, the other in the east, between New Guinea and the Moluccas, the Earth is therefore divided by the ocean into a northern part, known, and a southern, unknown, which we likewise call Magellanic.
Again the ocean flows between the northern land beyond Tartary, dividing it into two continents, the old and the new, which we call American; thus the ocean flows around three continents of the Earth, as around three vast islands.
Thirdly, although these continents are most ample, yet the ocean insinuates itself into them by parts, so that they are as it were divided into peninsulas, scarcely joined by the narrowest isthmuses. In this way America is, as it were, rent into a southern and a northern part (of which the former is called Peruvian, the latter New Spain); but the continent known of old is divided into Europe, Africa and Asia, by the seas, the Mediterranean, the Ionian, the Euxine and the Red Sea, which are as it were gulfs of one and the same ocean. Therefore nowhere on Earth are gulfs of the one continuous ocean very far away.
These things being so, consider that the seas, where they come nearest together with isthmuses between, by the witness of experience have almost the same height, and so all the shores round about. Then consider that the origins of the rivers which discharge into these seas are for the most part in the highest mountains of the inner continents. For the most part also the sources of several rivers, flowing down into the most diverse regions, are close to one another, with a modest ridge of mountain between. Therefore no part of the Earth is much higher than the highest sources of rivers. If, then, the sources are not much higher than the shores, the summits also of the highest mountains will not much exceed the same height of the shores, and so of the sea itself that surrounds them on every side.
How do you prove that the sources of rivers are not much higher than the shores of the sea? First, the water-engineers note that a river cannot be navigated without danger whose level falls by one pace in two hundred paces. Now most of the greatest rivers of the whole world are navigable; there is none of them, therefore, that sinks by one mile in two hundred miles of its course. Then consider that the greatest rivers of the whole world, and those that traverse the most land, for the greater part of their course lie stagnant and move on very slowly, and are almost only pushed forward by the oncoming train of waters; so the Ganges, so the Nile through all Egypt, so the Danube through Hungary and almost the whole of Thrace as far as the Euxine. But the height of the shores within which the rivers stagnate is uniform and not sloping. If anywhere they are rolled down in swift fall, it lasts no longer than until they have come out of the narrow places, rough and crowded with rocks on both sides.
Hence it follows that a river is hardly to be found whose sources are raised one German mile above the surface of the sea, even though it traverse a tract of a thousand miles.
It has been demonstrated that the surface of the water is convex, and that the surface of the lands assumes no very different figure from the waters; but how is it proved that this figure is altogether round, since there are many convex figures: the oval, the cylindrical, the toroidal, the swollen, and the like? It is granted indeed of the surface of the Earth that it is not perfectly round, but bulges out here and there into mountains; but of the waters of the ocean, when they are calm, complete roundness is proved, by arguments drawn as well from nature as from sense.
Give the argument from nature. We see that in the body of earth and water there is a corporeal force, of uniting to itself all bodies whatever and attracting them, which force they commonly call gravity. Since, therefore, this whole globe of the Earth is surrounded on every side by waters, a fluid matter not bounding itself at the sides, and since it is not improbable that the Earth also within is everywhere passable by vast channels, so that the Earth is perhaps like a perforated pot, consisting of the continents as of potsherds, filled within with water: surely all the parts of the water round about will be unable to constitute any other figure of the whole than the round, because the uniting force in the waters, hindered neither by itself nor by the earth, produces chiefly one figure, of which kind the round is, having nothing outside itself; whence it comes that no eminence of the waters above roundness, no wave of the ocean, remains long suspended in a heap: it spreads out to establish complete evenness, and the waves yield sideways and give place to the flowing, being driven out by the greater weight of those flowing down.
Could not the same be said also of the nature of the Earth? Indeed the Earth, when it was first created by God, was either itself a fluid and soft mass, and hardened afterward, and then the same things are true of the earth that were said of the waters; or it was created in its primeval solidity and hardness, and even so it is not likely that another figure was given it than it would have taken of itself, had it been soft at the beginning. For although it is hard, it is nevertheless body or matter like water, and, soaked with moisture or melted by fire, can be made fluid in its parts, like water. It is fitting, therefore, that it too received the proper figure of fluid matter, that is, the round.
Why, then, is the Earth not perfectly round like the ocean? The most wise Founder distributed its figure, between the nature of matter and the use of living creatures, for whose sake the Earth was made hard also, in a most considered proportion, so that mountains should be raised somewhat, for pouring forth perpetual waters, and valleys depressed to receive them, yet to the sense, when the whole sphere with its eminences comes into view together, nothing would seem to fall short of perfect roundness.
Give the argument from sense for the perfect roundness of the ocean. When, the sight or the head being brought close to the surface of the waves, some mark is seen at a distance, first as if emerging from the sea, like the flag of another ship or the ball on a tower, wherever it happens over the whole surface of the ocean, it is known at once, as by a universal rule holding in all places of the sea, what is the proportion of the distance of the thing seen to its height.
And conversely, if the sight has this height, it will be able to see a mark floating on the surface of the sea at so great a distance, provided that it also be large enough.
But if both the watcher of our ship and the mark on the opposing ship are raised to the same heights, say each fifteen paces, the thing can be seen from double the interval, namely from 20 Italian or 5 German miles. And if the mountain of some island has risen to a height of the nine-hundredth part of the semidiameter of the Earth, wherever in the ocean it may be, it will begin to be seen at the fortieth German mile from the surface of the sea, at the eightieth from another mountain of the same height.
This equality of the ocean, therefore, detected by sense over the whole circuit of the sea, argues the complete roundness of the ocean not very imperfectly.
What arguments have you from sense for the roundness of the Earth and of the whole globe consisting of lands and water? First, if the astronomer posits that the Earth is round, and in his geometrical computations uses not different diameters according to the diversity of mountains and valleys but the same diameter of the Earth everywhere, his operation commonly comes out in such sums as agree with astronomical experience, that is, the celestial phenomena follow. But they would not follow if he did this, while the Earth had diameters different and sensibly differing, that is, if it were not round.
Then, as often as we advance, by land or sea, over equal itinerary spaces straight to the north, so often do we find in astronomy equal increments or decrements of the altitudes of the Sun and stars at noon; as often again as, under one and the same parallel (of which in Book III), we are carried over equal itinerary intervals to the east or west, so often do we count equally more or fewer hours and minutes at the beginnings or ends of lunar eclipses. But unless the Earth were round, that equal proportion would have no place. The Earth, therefore, is round, as well toward north and south as toward east and west.
Finally it is plain to the eye, without long reasoning, that the boundaries of the earth’s shadow on the body of the eclipsed Moon, those toward the north as well as those toward the south, toward the east as well as toward the west, are arcs of a perfect circle. But a body whose shadow, cast by the round Sun, is bounded by a circle must necessarily be circular in that region from which the shadow descends, as optics teaches. Since, therefore, in the course of time, and with many lunar eclipses occurring, all the limits of the body of the Earth cast such shadows, the Earth is therefore round to sense on every side.
What if the Earth had some great eminences, which, though the shadow and astronomy do not easily detect them, the sense itself would easily have detected, if it were possible to survey the whole Earth at one glance? The nature of water denies this, and experience also. For first, if the body of the Earth had notable angles or eminences drawn along the whole length or breadth of the body, the ocean would not be poured around in the form of a circle, but, interrupted, would leave those eminences bare, and so not a continuous ocean would girdle the lands, but a continuous land would girdle the seas; the globe of the Earth could not then have been sailed round from east to west or conversely, which we read has been done in this century more than once.
Next, if one climbs the highest mountains of all the continents, and from them looks round, and even measures with an instrument the whole level of the terrestrial horizon round about: provided such a mountain be the highest of all its neighbours, the observer will find from it some depression of the whole horizon below the level of the water, or the base of the plumb-line, so much the greater in any part of the bounding circle as the other mountains seen there are more remote, but nowhere will he notice a depression greater than two degrees; for so great it is when the prospect over the sea is from a height of half a great mile, which can happen in very few places, and I know not whether anywhere, except perhaps in the region of Chile, the most westerly of all Peru. For mountains of so great a height are for the most part hidden in the middle of the continents, whence there is no prospect to the shores and the ocean.
It is plain, therefore, that the outer circles of the visible earth, lying around the sight everywhere from lofty mountains, appear as nearly as possible plane.
But if the Earth had anywhere notable eminences, and those not of one or two mountains only but of the whole neighbouring continent, it would be necessary that from the high ridge of that eminence the horizon should in some part of itself notably gape open or sink down. Finally, wherever the measurement of the globe of the Earth is made without the help of the heavens, of which presently, provided the distance of the two places required for the work be always known in the same measurement, it always yields very nearly the same semidiameter of the globe of the Earth; which is an argument that no part of the Earth is raised aloft above the rest by any notable quantity.
You always assume this, that the waters are deeper than the shores, yet commonly they seem higher than they. They are not higher, but sensory estimation deceives here, as the opticians teach. But if, standing on the shore, you measure with an instrument the level of the waves, even those farthest of all that the sight reaches, you will by no means find any elevation in them, but always some small declivity, and it will appear that the waves do not hang over the lands but sink below them.
Explain the causes of this error in estimating. The sight looks at the farthest parts of the sea with eyes raised higher than at the near ones; and if those far parts became nearer in these higher visual rays, they would be rendered altogether higher also. But we suppose them to be near, we therefore suppose them also higher than the near parts. And we suppose them near because we view them obliquely, and so we do not rightly comprehend the so long series of intervening matter, from which alone the distance of the far parts ought to be gathered, thinking that in the angles of the higher rays we take in as much of the surface of the water as we take in with equal angles of the lower rays.
Let A be the watcher on the shore, BE the plane surface of the water. AB, AC, AD, AE are the rays to the eye A, and BC, CD, DE the intercepted surfaces, most unequal. The watcher, therefore, supposing CD and DE equal to BC near him, because the angles at the eye are equal, thinks that in the same ray AD he sees F in place of D, in the same ray AE he sees G in place of E, so that BC, CF, FG become equal; by which means the plane surface BCDE seems curved upward and to be BCFG.
You have denied that the eminences of the Earth are notable to sense if they be compared with the whole globe of the Earth; meanwhile you have conceded that most inland places have navels standing out at a height of half a German mile above the surface of the ocean. Is this not a sufficiently notable height? One mile, or half a mile, has to 800 or 900 no sensible proportion. And so great is found the semidiameter of the globe of the Earth.
How can the quantity of this semidiameter be investigated? Since the Earth is round, the plumb-lines hung in various places must incline toward one another, and towers erected to the plumb-line must lean away from one another at their tops, that is, stand farther apart above. Which being so, it is easy, with two mountains, one of which is visible from the other, to examine the plumb-lines against the common line of sight, whence becomes known the angle which the two lines of the two plumb-lines make, continued in imagination as far as the centre of the Earth. Since, then, the distance of those two places is a part of the circuit of the whole globe, as the angle at the centre is a part of four right angles, it is easy from the known length of that part to find the length of the whole circuit in the same measure; or from the known distance of the two places, as from the base of a triangle, and from the known angles, the length of the two legs, or lines meeting from the plumb-lines at the centre of the Earth, which is the quantity of the semidiameter of the Earth, is computed by geometrical precepts.
Teach me the procedure by an example. Let two places B and C be distant 5 common German miles, B a mountain, C a fort. Let the angle at B on the mountain be found 89° 46′, the other at the fort C 89° 55′. The angle BAC will therefore be 0° 19′, since four right angles contain 360°, or 21600 minutes. Now if the part of the Earth’s circuit between the mountain and the fort, which is 19′, is 5 miles of that place, then the 21600′ of the whole circuit will be worth so many miles of this Or multiply the sine of the angle ABC into the known measure of BC itself, and divide by the sine of the angle BAC.
The quotient, 904 with 57313 parts of one divided into 55263, is the length of the line AC, opposite the angle ABC. Therefore the semidiameter of the Earth from the centre to the fort would be 904 such German miles.
Likewise multiply the sine of the angle ACB into the same known measure BC, and divide by the sine of the angle BAC.
The quotient, 904 with 37673 parts of one, is the length of the line AB, opposite the angle BCA, and is the semidiameter of the Earth from its centre to the summit of the mountain. And so that mountain would be raised 360 parts of one higher than the fort, which is a height of 130 geometrical feet in the perpendicular.
Is there another way of measuring the semidiameter of the Earth? None is more readily prepared than this which I have just delivered, since in it nothing is assumed that is not always and everywhere at hand. Yet the method of Clavius comes very near to it, which itself also is twofold; one form of it is as follows.
Let there be on the shore of the sea a promontory of known height above the surface of the sea, on whose summit stands the measurer, directing one side of a quadrant toward the farthest waves, where they seem joined to the heaven, and let him note what angle the plumb-line and the line of sight carried to the farthest waves make. In this way a triangle is formed, with a right angle at the farthest waves, whose perpendicular side is the line of sight, whose base is the line from the centre of the Earth to the farthest waves, and whose hypotenuse is the line from the centre of the Earth to the summit of the promontory and to the sight. Since, therefore, in the table of sines there is set against each angle of the quadrant the proportion of the hypotenuse or secant to the base or radius, and the excess of the former over the latter, to which corresponds the height of the station above the surface of the sea, it is easy, this height being given in German miles, to establish also the number of miles of the radius or semidiameter.
Give an example. Let the height of the promontory CO be one Italian mile, or a quarter of a German, and let G be the extremity of the sea seen from the height C, and let the angle GCO be found 88° 37′. Since, then, CGA is a right angle, inasmuch as CG touches the surface of the globe GO at G, and GA is drawn from the point of contact G to the centre A, GAC will therefore be 1° 23′. But the hypotenuse or secant AC of this angle is 100029. If, then, the excess of CO over the radius OA or GA, which excess is 29, is worth one quarter of a German mile, the whole GA or OA will be worth 862 miles.
Explain also the other form of the Clavian measurement. This form has no need of climbing a mountain, but requires in its place the knowledge both of the height of the mountain CO and of the distance of the ship G from the mountain C. For the square of this GC, divided by OC, gives the whole diameter of the watery globe increased by OC.
As if GC be 21 miles and CO a quarter of one. The square of 21 is 441, which divided by CO, a quarter of one mile, makes the quotient 1764; therefore the whole diameter of the watery globe would be 1763 miles and three quarters.
But by what method are astronomers wont to measure the globe of the Earth? Astronomers employ the heavens, that is, the altitude of the pole in two places under the same meridian, and by it they first measure the circuit of the Earth, then from the circuit they draw out also its diameter. But the knowledge of the spherical doctrine is first required, which will be delivered only in what follows.
Yet show the matter by an example. At Prague the altitude of the pole is 50° 6′. At Linz, under the same meridian, 48° 16′.
Now from Linz to Prague there are commonly reckoned 26 miles. If, then, 1° 50′ is worth 26 miles, the whole circuit, of 360 degrees, will be worth 5105 miles. But the circuit is to the diameter as 22 to 7. If, then, the circuit 22 is worth 5105 miles, the diameter 7 will be worth 1615 miles and the semidiameter 807 miles.
How great, then, is the semidiameter of the Earth reckoned today? Commonly today 15 middling German miles are counted to each degree, so that the whole circumference comes to 5400, the semidiameter nearly 860.
We reckon, moreover, for one German mile, 4 Italian miles, or 4 thousand geometrical paces, each of which has 5 feet, the foot being 4 palms. In the Italian mile there are 8 stadia, in the German 32, each of 125 paces. Thus, according to the moderns, one degree occupies 480 stadia and the whole circumference 172800 stadia.
What did the ancients hand down on this? The ancients, in the beginnings of nascent astronomy, were less accurate. For Eratosthenes, who lived before Christ, gives 250000 stadia.
His reasoning is this. At Syene the Sun at the summer solstice, at the hour of noon, lights the bottoms of the wells; it becomes, therefore, precisely vertical there. But at Alexandria it then stands off from the zenith by one fiftieth part of the circle, that is, 7° 12′. And the itinerary interval between the two places is reckoned at 5000 stadia, which taken fifty times makes 250000.
Posidonius, about the time of Christ, took off a twenty-fifth part, so that there are 240000 stadia. His reasoning is nearly like the former. The star Canopus, in his time, at the island of Rhodes just grazed the horizon and did not rise higher. At Alexandria it rose to a fourth part of one sign, that is, 7° 30′, which is the forty-eighth part of the whole circumference. But the itinerary interval, or space of sea, between the two places is reckoned at 5000 stadia, which taken 48 times makes 240000.
Ptolemy, who lived after Christ, comes nearer to us; for he assigns to one degree 500 stadia, which taken 360 times make 180000.
On the authority of Alfraganus, in the time of Almamon the Arabs, having compared opinions, determined that to one degree of a great circle on the Earth belong 1360000 palms, of which 6 make a cubit. Of these, 4 are reckoned by us for a foot, 20 for a geometrical pace; so there will come 68 thousand paces to each degree, that is, 17 German miles, 544 stadia, so that the whole circuit is 195840 stadia.
Albategnius, or Mahomet of Aracta, estimates a half degree at the day’s journey of an unencumbered man, or 42 Arabic miles of his age.
Why is the measurement of the Earth inserted here, a geographical matter, when we are engaged in astronomy? Although it belongs to geography to measure the circuit of the lands, the distances of places, the convex area of the surface of the Earth, and the very bulk of the whole globe, astronomy nevertheless cannot do without this knowledge.
For first, according to the number of degrees of longitude and latitude of the Earth, the celestial phenomena vary in the different places of the Earth. But from the itinerary distance the number of degrees of longitude and latitude can be gathered, if we have known in that measure the whole circumference of the greatest terrestrial circle.
Second, since the Earth is our dwelling, we use the semidiameter of the Earth as a measuring-rod for measuring the heavenly bodies and their distance from the Earth. It is fitting, therefore, that our measure be known to us, that is, weighed against the size of the stature, the fathom, the cubit, the foot, the span, the palm, the thumb’s breadth and the finger’s breadth in our own little bodies.
Third, the very method of demonstration demanded that this manner of measuring be inserted in this very place, because by it the perfect roundness of the Earth was examined and proved.