Epitome Astronomiae Copernicanae: Liber I, part 5
Several angles, therefore, having been multiplied into their secants, none is found except 88° 1′, which with its secant 2889400 makes this quotient.
The greatest refraction of the tangent PD being posited as 1°, the inclination in the air ADO of the ray DA, which the rising Sun sends to the Earth, is posited as 88° 1′. This inclination of the ray DA touching the Earth being posited in the air, the height DM of the air is posited, as above, at half a mile, or in more convenient numbers 60 parts of 100000. This height of the air being posited, there is found, according to the doctrine above, for the ray KA, which is inclined upon the Earth at 89° (the sine of the arc 89°, namely 9998477, being divided by 100060, the semidiameter of the air), the inclination upon the air AKO 87° 46′ 40″ whose secant is 2566340.
The product of both, 225267623.
let it be multiplied into the refraction 60.
And let this product be divided by the product 13516057380 of the refraction 60′, the quotient will show 2677 in K. In the same way the refraction of a ray inclined upon the Earth at 70° will be found, by this position of the greatest refraction, to be 1′ 16″.
But Tycho Brahe observed there 21½′, and here, for 1′ 10″, nothing sensible. Again, therefore, the position is false; wherefore the inclinations upon the air are false and the height of the air false. But when we assumed the greatest refraction of the tangent at 30′, the remaining refractions came out less than is just; when here we assumed the greatest at 60′, they came out greater than is just. The truth, therefore, is in the middle and nearer to the second position than to the first. And so at last we learn that, for the refractions which Tycho assigned to the fixed stars, a height of the air is required a little lower than 16 stadia, or half a mile.
Is this lowness of the air confirmed by other proofs also? First, it is natural that most of the mountain ridges which are covered with perpetual snows or ice stand out above the surface of the air for the greater part of the year. For air is an exhaled vapour; in every exhalation there is at least some heat, but where there are snows and ice there is cold, so that the Sun’s rays cannot melt them. But where there is cold, there is no exhalation, or no constant one, nor air. For it is compressed by cold into hoar-frost and snow and falls by its weight, and leaves the ridges of the mountains bare.
Second, on Mount Olympus in Asia, as Aristotle reports, life cannot be continued unless breath is drawn through moist sponges. In America, when the Spaniards were crossing from Nicaragua into Peru, on the highest ridges of the mountains between, a good many suddenly expired together with their very horses. (In the index of Errata we read this: for Nicaragua I think Castilia del Oro or Darien should be read; I have now no author to consult.)
The writer ascribes the cause to a poisonous and penetrating quality of the air, but adds that both the horses and their riders were stiffened with frost, and remained like statues until the return of the others who had escaped. It seems rightly gathered, therefore, that among the causes was a lack of air. So fishes, created to draw in the waters, expire when kept outside the waters.
Third, on the same Olympus Aristotle affirms that there are so few winds that not even letters written in the dust have been disturbed for many years. But where there is no motion of the air, the most mobile of things, it is likely that the air itself is not continued so far.
Fourth, no clouds have been found raised beyond a quarter of a mile, and most much lower, by surveyors on the lowest sea-coasts; wherefore the air also will not be continued much higher. For clouds are exhalations or fresh mists, and therefore are still warm, and are carried farther by the heat than matter that has cooled.
How do we measure the height of a cloud? Not much otherwise than the distances of inaccessible things are usually measured, from two stations, as it were, of known interval; as if two surveyors at one moment, in different places, should take with quadrants the declination of the same cloud from the zenith. But since it rarely happens that two men are so disposed at one time, equipped with instruments and intent on the same cloud, a single surveyor will serve for two, and as it were visit two stations at the same time, by this contrivance. When the Sun shines he will choose a cloud that lies straight toward the Sun or toward the quarter turned away from the Sun, and will note the place on which the shadow of the cloud falls, then with the quadrant he will take the declination from the zenith first of the cloud, then of the Sun. For if another surveyor stood in the place of the shadow, the cloud and the Sun would have for him this same inclination. The rest is despatched by the scale of heights.
In the present figure CE is the perpendicular from the cloud, B the place of the shadow, A the surveyor’s station in the same plane, who must first direct the sights of the quadrant on the line AC and note the arc which the plumb-line cuts off; for this measures the angle ACE; then the sights must be directed on the line AD, so that the Sun shines through the holes; and the arc which the plumb-line cuts off will measure the angle BCE. Third, he must measure the space AB in feet or paces. Then the tangent of the angle ACE is to be taken from the tangent of the angle BCE (if the cloud C is toward the Sun), and the difference dividing the number of paces found, multiplied beforehand into the whole sine, gives the quotient, the altitude CE.
As if the cloud had declined 21° 48′, tangent 40000, and the shadow had been distant from the surveyor a thousand paces. Multiply 1000 into the right sine 100000, divide the product 100000000 by 60000, the quotient is 1666 paces. So great would be the height of the cloud.
But if the cloud were in the region of the Sun, then the tangents would have to be added.
Is the height of the air constant? It is not constant, but grows with the heat itself, according to places and times.
Will the quantity of refractions, then, not be constant? Certainly not, but with this difference. In maritime places the refraction is for the most part more constant and almost always uniform, because the surface of the air is higher there; and so the little that is added to this height at different times is the less felt in those places. But on very high mountains the refraction is sometimes almost none, especially in winter, when the air is low and less vaporous; sometimes the refraction is unusual and almost prodigious, which also happens at times on the sea-coasts, but by accident, on account of position; namely when a great force of vapour boils up from the mountains by which the way lies for the rays to that maritime place.
Brahe gives also the solar refractions as several scruples greater than the refractions of the stars, and, what follows, sensible through several degrees higher; by which it seems confirmed, what is also likely on other grounds, that the morning air is augmented and raised by the approach of daylight, and in the night diminished and lowered.
What is twilight? Latin authors use the word for the dusky light, that is, the doubtful, when day dawns. To astronomers it is all that time which intervenes between the first brightness of the air notable to sense and the very rising of the Sun; or conversely what is between the setting of the Sun and the last trace of daylight in the air.
What is the cause of this brightness when the Sun is absent? The causes are many, partly celestial, partly elementary. The first efficient cause, which stirs all the others, is the Sun himself, notwithstanding that he is under the Earth and strikes our eyes with no ray, either direct or refracted. The second, celestial cause is the aethereal breeze, poured next about the Sun to the amplitude of some degrees, shining on account of the neighbourhood of the Sun, now more, now less, which takes up some time in rising, until the Sun himself at length rises.
The third cause is smokes and dry exhalations, excited by the heat of the Sun during the day or by the harmonic radiations of the stars, and carried above the surface of the air by their heat and thinness, which, since they are higher than we, are illuminated sooner by the Sun and by that shining mantle of the Sun than the deep places, and so, raised above our horizon like lamps lit, impart to us the light received from the Sun.
The fourth cause is the air itself, and the manner of the cause is twofold. For the air either grows bright by the direct rays of the Sun, as was just said of the smokes, and sends them back, since it is denser than the aether, as said above; or the air serves the Sun’s rays as a hollow mirror and throws them back. For the rays of the Sun, as soon as they enter the sphere of the air, are refracted at the beginning; then, so refracted, they proceed through the depth of the air, the former strike the Earth, which is from us toward the Sun, the latter pass beyond, tending toward our place, and meet again the hollow surface of the air; which, although it sends outward those twice refracted, yet also throws back a slight part of them inward, since it is smooth and mirror-like. Those, therefore, which it throws back inward, being about to cut off an equal arc of the sphere of the air, strike a second time upon the surface of the air farther on, and so in succession, until they are so weakened by so many attenuations that they can no longer sensibly illuminate the air.
As here FO is refracted at O and becomes OK; meeting the surface at K, it is thinly thrown back to I, and again to H still more thinly, because part of OK itself goes out through K, part of KI goes out through I. But the ray GM, refracted at M, becomes ML and strikes the Earth at L. A place being posited on the surface of the Earth at B, although the Sun has not risen for it, and the Sun’s ray GM by no means reaches B, since it strikes at L, yet its upper ray FO comes through OK, KI, IH to the air at H, illuminating it thinly, which can therefore cause the light of twilight at B, since H is above the horizon of the place B, which the line BI represents.
How did the ancients demonstrate the height of that matter which kindles the twilights? Since they had supposed that this matter is illuminated only by the Sun, and only by direct rays, they therefore, at the moment when first, on the very rising horizon, even the faintest brightness of the air was noticed, investigated from the position of the stars the number of degrees by which the Sun was still sunk below the horizon; they found it to be 16, 18 or 19 degrees; since, then, there lay between just so many degrees up to that place of the Earth where the Sun was rising, grazing the horizon, they concluded that the matter which first gives the sign of twilight hangs over the intermediate place of the Earth. Wherefore the excess of the secant of half the arc over the whole sine, compared with the whole sine, argued the height of that matter above the Earth.
Let C be our place, CD the horizon, and in it, not above it, D the whitening air, when the Sun is 16, 18 or 19° below the horizon. Then CB is 16, 18 or 19°, and B is the place for which at that time the Sun is rising, grazing with the ray BD and illuminating by this ray the matter at D. Since, therefore, both lines, CD and BD, touch the horizon, E and A being joined, the arcs CE, EB will be equal and each 8, 9 or 9½ degrees. Therefore the secant AD is 100983 or 101247 or 101391. Of such parts, therefore, as AE is 100000, the height DE of the matter is 983 or 1247 or 1391, that is, 8½ miles, or 10, or 11½.
But this demonstration omits all causes other than the Sun, such as refraction, reflection and the bright matter about the Sun, which being posited, the air becomes much lower; wherefore this demonstration does not infer the necessity of so great a height of the air.
By what argument do you prove that bright matter about the Sun concurs in forming the light of twilight? Because for the most part, especially at the colder dawns, the aurora shines forth in a manifest circular figure (as the painters paint it), rising into brightness high in the middle, curved on both sides toward the horizon, the proportion of the altitude to the part of the horizon intercepted not being great.
Prove that this arched appearance cannot be represented by illuminated air. The illumination of the air made by the Sun ends in a great circle whose diameter has 1740 miles, the height of the air being posited at 10 miles. When, therefore, the horizon begins to touch this circle, that will be the beginning of twilight, and the contact of this circle will be 9° from us, raised above the surface of the Earth by 1247 parts, of which the semidiameter is 100000, because the secant of 9° is 101247. From that moment let a very few minutes of time pass, within which the Sun becomes one degree higher; the circle of illumination of the air will then be 8° from us, whose secant is 100983, and so 264 parts of the height of the illuminated air will stand above the horizon. But a segment of the circle whose semidiameter is 101247, occupying 264 parts in its height, occupies in the length of the section 14416, which is 55 times the height. Behold the dawn within the first 6 minutes from its beginning 55 times broader than high, if the air alone (even 10 miles high) were the cause of twilights, and these two lines, the altitude and the breadth of the segment, from the interval of 14054 parts, which is the tangent of 8°, would appear under angles such that the breadth would be 26 times and more spread out on the horizon than the altitude above the horizon. How much more absurd things follow if we take the just height of the air, namely half a mile? The illumination of the air alone cannot, therefore, save for us this phenomenon of the arched dawn.
Perhaps, then, this height of the rising dawn can be demonstrated from the reflection alone of the solar rays in the hollow of the air? By no means; for if only by repeated reflections rays came to the eye from the Sun not yet risen, the appearance represented would be of a bright straight line directed toward the Sun, no broader than the Sun himself appears, as happens on the surface of a rippled river opposite the Sun.
We use reflections, therefore, not for conveying the Sun’s rays to the eye, but for lighting the whole body of the air nearer to us, so that, brightening on, it shines of itself with a second light.
But since the matter is certain of the bright substance about the Sun, what need is there to bring the air into the consideration of twilights? There are many offices of the air here, which the celestial cause cannot perform of itself.
First, if a secondary light came from one quarter only, the east, so that the air did not shine on every side in all directions, it would never grow light in chambers before sunrise, and the western side of a tablet turned toward the east would be in mere darkness. Second, that bright substance about the Sun sometimes fails, twilights never. Third, that substance occupies the region near the Sun, while the light of twilight a little before sunrise, especially in summer, is seen diffused through the whole sky; the whole air, therefore, is illuminated, even if not yet touched on every side by the Sun’s rays. Fourth, there is seen in the region opposite the Sun a manifest, almost semicircular distinction between the upper part of the air, illuminated, and the lower, either not yet illuminated in the morning or in the evening already deserted by the Sun’s rays, which Virgil expressed with emphasis in three words: night rushes from the ocean. But this appearance the bright substance about the Sun cannot express. Fifth, twilights vary in length, even when the celestial cause does not vary. For in some lands, as Josephus Acosta testifies of Chile, twilights are very short, so that within a quarter of an hour from deep night the clearest day comes, bringing forth the Sun himself. Beyond the more direct ascensions, therefore, of which in Book III, the air also must be admitted among the causes, which in those places is purer, more transparent and less capable of illumination than with us.
Sixth, the different parts of the year also bear witness to this. For on winter days twilights are short, because by lack of heat the air is low; on summer days, on the contrary, they are long and last through the night, because by excess of heat the air is higher. For the bright substance about the Sun, although variable, does not follow the circuit of the elementary heat and cold, being itself celestial; nor are the astronomical causes alone, to be explained in Book III, sufficient to produce this variety.
Seventh, even in the same place, at the same times of the year, twilights vary with the changes of the air. For in winter time after frost, when at the stimulus of aspects a sudden loosening of the cold comes on, day dawns earlier. For the vapour boiling out of the earth, which loosens the cold, which incrusts walls with hoar-frost and moistens them, which begets winds and rains, is also carried to a greater height by the force of heat drawn in under the earth, so that it is both earlier and more strongly illuminated by the Sun and illuminates in turn. But the celestial cause, or the bright matter about the Sun, is not so easily and quickly changed as this effect would demand, nor does it feel that stimulus of aspects, because an aspect is formed only on the Earth, not at the same time also at the Sun.
But is not the reflection of the rays in the hollow surface of the air to be admitted? Indeed, of the seven offices of the air now explained, the second is so arranged that without this reflection it cannot be fulfilled. For the air, since it is certainly so low, the bright substance about the Sun failing, could not be illuminated at so great a depth of the Sun as 18° unless the light of the Sun were conveyed to us as if through many hands, following one another in order, that is, through many points to be reflected.
Use, however, if you please, in place of mirror-like reflections, the re-vibration, or illumination of the more remote particles of the air by those nearer the Sun; as when the Sun illuminates a cloud, the cloud a plastered wall in the street, the wall the inner floors of a house, the floors the face of one writing, the face the paper, and so on. Hold only this from optics, that however the vibration of borrowed light may differ from reflection, it is certainly derived from it.
What is to be thought of the shooting stars, and those seen to fall to the Earth, mention of which was made at the beginning? Shooting stars are nothing other than a flame feeding on dry matter, sent forth from the earth and stretched out in length, as certain mists or little clouds are wont to be; which matter the surrounding cold of the upper region sets on fire. Falling stars are viscous matter inflamed. Some of them are consumed in falling, some truly fall to the Earth, drawn by their weight. Nor is it unlikely that some are conglobated from turbid matter mingled in the aethereal breeze itself and pass in a rectilinear course from the region of the aether through the air, like tiny comets, the cause of the motion of both being hidden.
But since all these things are momentary and do not delay the revolution of the first motion, which is the swiftest of all, they belong to physicists and to meteorology, and are to be removed from astronomy and the spherical doctrine, in which nothing momentary, but only what rises and sets with the revolution of the day, is thought worthy of consideration.
By what reason does it happen that at times the fixed stars all appear so large, and sparkle so with twinklings? Not even this befalls the stars on account of any astronomical motion, but only by reason of the air. For when rains come on, there first arises in the bowels of the Earth a vapour far denser than common air, which touches and as it were floods the eyes: the usual difference, therefore, between the transparent coats of the eye and the transparent medium of the air no longer holds; and the proportion of the media through which the rays pass being changed, those refractions by which vision is made are changed; these refractions being changed, the cones of the visible points, which we are wont to call pencils, become blunter where they touch and paint the retina of the eye, and so not single points in the eye from single points of visible things, but little surfaces in the eye are tinged from several neighbouring points of the visible thing, the painting of the bright ones prevailing, that of the darker being obliterated. Thus with the broader part of the sensory instrument the stars are represented not only larger than they are, but their real twinklings also move the sight more strongly; all which things in the optics and in the true and genuine mode of vision rest on most clear and pleasant demonstrations.
But why are the Sun, Moon and constellations seen so large at the horizon especially, and daily; whereas before, refractions produced the contrary, things appearing smaller than themselves at rising or setting? Very different in optics are the measurement of visual angles by instruments and the estimation. The Sun, Moon and constellations, if we measure their apparent magnitudes, that is, their visual angles, by instruments, are by no means found greater at the horizon than in the middle of the heavens, but rather, along the erect diameters, somewhat smaller, as was rightly shown hitherto. But they are estimated greater along all diameters when they rise and set. For estimation gathers the magnitude of a visible thing from its distance, and the distance from comparison with other bodies. Now when the constellations are in the middle of the heavens, the body of the air is neither deep in respect of the visible breadth of the lands, nor, if it were deep, can it be seen; common sense therefore reasons by error, from the absence of intervening bodies, that the stars, when they are in the middle of the heavens, are nearer to us than when they rise and set; for then they are judged more remote, because valleys and mountains lie open to the eyes, stretched out between us and the rising stars through a very long tract. Another error therefore follows, that the Sun, which remains of the same magnitude, is judged to culminate quite small, but to rise huge, like a giant. For of things seen under the same angle of vision, those are greater which are more remote, as we are taught in optics.