Part 9

177. According to [42]Dr. KEILL, and other astronomical writers, it is entirely owing to the Atmosphere that the Heavens appear bright in the day-time. For, without an Atmosphere, only that part of the Heavens would shine in which the Sun was placed: and if an observer could live without Air, and should turn his back towards the Sun, the whole Heavens would appear as dark as in the night, and the Stars would be seen as clear as in the nocturnal sky. In this case, we should have no twilight; but a sudden transition from the brightest sunshine to the blackest darkness immediately after sun-set; and from the blackest darkness to the brightest sun-shine at sun-rising; which would be extremely inconvenient, if not blinding, to all mortals. But, by means of the Atmosphere, we enjoy the Sun’s light, reflected from the aerial particles, before he rises and after he sets. For, when the Earth by its rotation has withdrawn the Sun from our sight, the Atmosphere being still higher than we, has his light imparted to it; which gradually decreases until he has got 18 degrees below the Horizon; and then, all that part of the Atmosphere which is above us is dark. From the length of twilight, the Doctor has calculated the height of the Atmosphere (so far as it is dense enough to reflect any light) to be about 44 miles. But it is seldom dense enough at two miles height to bear up the Clouds.

[Sidenote: It brings the Sun in view before he rises, and keeps him in view after he sets.]

178. The Atmosphere refracts the Sun’s rays so, as to bring him in sight every clear day, before he rises in the Horizon; and to keep him in view for some minutes after he is really set below it. For, at some times of the year, we see the Sun ten minutes longer above the Horizon than he would be if there were no refractions: and about six minutes every day at a mean rate.

[Sidenote: Fig. IX.

PLATE II.]

179. To illustrate this, let _IEK_ be a part of the Earth’s surface, covered with the Atmosphere _HGFC_; and let _HEO_ be the[43] sensible Horizon of an observer at _E_. When the Sun is at _A_, really below the Horizon, a ray of light _AC_ proceeding from him comes straight to _C_, where it falls on the surface of the Atmosphere, and there entering a denser medium, it is turned out of its rectilineal course _ACdG_, and bent down to the observer’s eye at _E_; who then sees the Sun in the direction of the refracted ray _edE_, which lies above the Horizon, and being extended out to the Heavens, shews the Sun at _B_ § 171.

[Sidenote: Fig. IX.]

180. The higher the Sun rises, the less his rays are refracted, because they fall less obliquely on the surface of the Atmosphere § 172. Thus, when the Sun is in the direction of the line _EfL_ continued, he is so nearly perpendicular to the surface of the Earth at _E_, that his rays are but very little bent from a rectilineal course.

[Sidenote: The quantity of refraction.]

181. The Sun is about 32-1/4 min. of a deg. in breadth, when at his mean distance from the Earth; and the horizontal refraction of his rays is 33-3/4 min. which being more than his whole diameter, brings all his Disc in view, when his uppermost edge rises in the Horizon. At ten deg. height the refraction is not quite 5 min. at 20 deg. only 2 min. 26 sec.; at 30 deg. but 1 min. 32 sec.; between which and the Zenith, it is scarce sensible: the quantity throughout, is shewn by the annexed table, calculated by Sir ISAAC NEWTON.

+-------------------------------------------------+ | | | 182. _A_ TABLE _shewing the Refractions | | of the Sun, Moon, and Stars; | | adapted to their apparent Altitudes_. | | | +-------+---------++----+---------++----+---------+ | Appar.| Refrac- ||Ap. | Refrac- ||Ap. | Refrac- | | Alt. | tion. ||Alt.| tion. ||Alt.| tion. | +-------+---------++----+---------++----+---------+ | D. M. | M. S. || D. | M. S. || D. | M. S. | +-------+---------++----+---------++----+---------+ | 0 0 | 33 45 || 21 | 2 18 || 56 | 0 36 | | 0 15 | 30 24 || 22 | 2 11 || 57 | 0 35 | | 0 30 | 27 35 || 23 | 2 5 || 58 | 0 34 | | 0 45 | 25 11 || 24 | 1 59 || 59 | 0 32 | | 1 0 | 23 7 || 25 | 1 54 || 60 | 0 31 | +-------+---------++----+---------++----+---------+ | 1 15 | 21 20 || 26 | 1 49 || 61 | 0 30 | | 1 30 | 19 46 || 27 | 1 44 || 62 | 0 28 | | 1 45 | 18 22 || 28 | 1 40 || 63 | 0 27 | | 2 0 | 17 8 || 29 | 1 36 || 64 | 0 26 | | 2 30 | 15 2 || 30 | 1 32 || 65 | 0 25 | +-------+---------++----+---------++----+---------+ | 3 0 | 13 20 || 31 | 1 28 || 66 | 0 24 | | 3 30 | 11 57 || 32 | 1 25 || 67 | 0 23 | | 4 0 | 10 48 || 33 | 1 22 || 68 | 0 22 | | 4 30 | 9 50 || 34 | 1 19 || 69 | 0 21 | | 5 0 | 9 2 || 35 | 1 16 || 70 | 0 20 | +-------+---------++----+---------++----+---------+ | 5 30 | 8 21 || 36 | 1 13 || 71 | 0 19 | | 6 0 | 7 45 || 37 | 1 11 || 72 | 0 18 | | 6 30 | 7 14 || 38 | 1 8 || 73 | 0 17 | | 7 0 | 6 47 || 39 | 1 6 || 74 | 0 16 | | 7 30 | 6 22 || 40 | 1 4 || 75 | 0 15 | +-------+---------++----+---------++----+---------+ | 8 0 | 6 0 || 41 | 1 2 || 76 | 0 14 | | 8 30 | 5 40 || 42 | 1 0 || 77 | 0 13 | | 9 0 | 5 22 || 43 | 0 58 || 78 | 0 12 | | 9 30 | 5 6 || 44 | 0 56 || 79 | 0 11 | | 10 0 | 4 52 || 45 | 0 54 || 80 | 0 10 | +-------+---------++----+---------++----+---------+ | 11 0 | 4 27 || 46 | 0 52 || 81 | 0 9 | | 12 0 | 4 5 || 47 | 0 50 || 82 | 0 8 | | 13 0 | 3 47 || 48 | 0 48 || 83 | 0 7 | | 14 0 | 3 31 || 49 | 0 47 || 84 | 0 6 | | 15 0 | 3 17 || 50 | 0 45 || 85 | 0 5 | +-------+---------++----+---------++----+---------+ | 16 0 | 3 4 || 51 | 0 44 || 86 | 0 4 | | 17 0 | 2 53 || 52 | 0 42 || 87 | 0 3 | | 18 0 | 2 43 || 53 | 0 40 || 88 | 0 2 | | 19 0 | 2 34 || 54 | 0 39 || 89 | 1 1 | | 20 0 | 2 26 || 55 | 0 38 || 90 | 0 0 | +-------+---------++----+---------++----+---------+

[Sidenote: PLATE II.

The inconstancy of Refractions.

A very remarkable case concerning refraction.]

183. In all observations, to have the true altitude of the Sun, Moon, or Stars, the refraction must be subtracted from the observed altitude. But the quantity of refraction is not always the same at the same altitude; because heat diminishes the air’s refractive power and density, and cold increases both; and therefore no one table can serve precisely for the same place at all seasons, nor even at all times of the same day; much less for different climates: it having been observed that the horizontal refractions are near a third part less at the Equator than at _Paris_, as mentioned by Dr. SMITH in the 370th remark on his Optics, where the following account is given of an extraordinary refraction of the sun-beams by cold. “There is a famous observation of this kind made by some _Hollanders_ that wintered in _Nova Zembla_ in the year 1596, who were surprised to find, that after a continual night of three months, the Sun began to rise seventeen days sooner than according to computation, deduced from the Altitude of the Pole observed to be 76°: which cannot otherwise be accounted for, than by an extraordinary quantity of refraction of the Sun’s rays, passing thro’ the cold dense air in that climate. KEPLER computes that the Sun was almost five degrees below the Horizon when he first appeared; and consequently the refraction of his rays was about nine times greater than it is with us.”

184. The Sun and Moon appear of an oval figure as _FCGD_, just after their rising, and before their setting: the reason is, that the refraction being greater in the Horizon than at any distance above it, the lowermost limb _G_ appears more elevated than the uppermost. But although the refraction shortens the vertical Diameter _FG_, it has no sensible effect on the horizontal Diameter _CD_, which is all equally elevated. When the refraction is so small as to be imperceptible, the Sun and Moon appear perfectly round, as _AEBF_.

[Sidenote: Our imagination cannot judge rightly of the distance of inaccessible objects.]

185. We daily observe, that the objects which appear most distinct are generally those which are nearest to us; and consequently, when we have nothing but our imagination to assist us in estimating of distances, bright objects seem nearer to us than those which are less bright, or than the same objects do when they appear less bright and worse defined, even though their distance in both cases be the same. And as in both cases they are seen under the same angle[44], our imagination naturally suggests an idea of a greater distance between us and those objects which appear fainter and worse defined than those which appear brighter under the same Angles; especially if they be such objects as we were never near to, and of whose real Magnitudes we can be no judges by sight.

[Sidenote: Nor always of those which are accessible.]

186. But, it is not only in judging of the different apparent Magnitudes of the same objects, which are better or worse defined by their being more or less bright, that we may be deceived: for we may make a wrong conclusion even when we view them under equal degrees of brightness, and under equal Angles; although they be objects whose bulks we are generally acquainted with, such as houses or trees: for proof of which, the two following instances may suffice.

[Sidenote: The reason assigned.

PLATE II.]

First, When a house is seen over a very broad river by a person standing on low ground, who sees nothing of the river, nor knows of it beforehand; the breadth of the river being hid from him, because the banks seem contiguous, he loses the idea of a distance equal to that breadth; and the house seems small, because he refers it to a less distance than it really is at. But, if he goes to a place from which the river and interjacent ground can be seen, though no farther from the house, he then perceives the house to be at a greater distance than he imagined; and therefore fancies it to be bigger than he did at first; although in both cases it appears under the same Angle, and consequently makes no bigger picture on the retina of his eye in the latter case than it did in the former. Many have been deceived, by taking a red coat of arms, fixed upon the iron gate in _Clare-Hall_ walks at _Cambridge_, for a brick house at a much greater distance[45].

[Sidenote: Fig. XII.]

Secondly, In foggy weather, at first sight, we generally imagine a small house, which is just at hand, to be a great castle at a distance; because it appears so dull and ill defined when seen through the Mist, that we refer it to a much greater distance than it really is at; and therefore, under the same Angle, we judge it to be much bigger. For, the near object _FE_, seen by the eye _ABD_, appears under the same Angle _GCH_, that the remote object _GHI_ does: and the rays _GFCN_ and _HECM_ crossing one another at _C_ in the pupil of the eye, limit the size of the picture _MN_ on the retina; which is the picture of the object _FE_, and if _FE_ were taken away, would be the picture of the object _GHI_, only worse defined; because _GHI_, being farther off, appears duller and fainter than _FE_ did. But if a Fog, as _KL_, comes between the eye and the object _FE_, it appears dull and ill defined like _GHI_; which causes our imagination to refer _FE_ to the greater distance _CH_, instead of the small distance _CE_ which it really is at. And consequently, as mis-judging the distance does not in the least diminish the Angle under which the object appears, the small hay-rick _FE_ seems to be as big as _GHI_.

[Sidenote: Fig. IX.

Why the Sun and Moon appear biggest in the Horizon.]

187. The Sun and Moon appear bigger in the Horizon than at any considerable height above it. These Luminaries, although at great distances from the Earth, appear floating, as it were, on the surface of our Atmosphere _HGFfeC_, a little way beyond the Clouds; of which, those about _F_, directly over our heads at _E_, are nearer us than those about _H_ or _e_ in the Horizon _HEe_. Therefore, when the Sun or Moon appear in the Horizon at _e_, they are not only seen in a part of the Sky which is really farther from us than if they were at any considerable Altitude, as about _f_; but they are also seen through a greater quantity of Air and Vapours at _e_ than at _f_. Here we have two concurring appearances which deceive our imagination, and cause us to refer the Sun and Moon to a greater distance at their rising or setting about _e_, than when they are considerably high as at _f_: first, their seeming to be on a part of the Atmosphere at _e_, which is really farther than _f_ from a spectator at _E_; and secondly, their being seen through a grosser medium when at _e_ than when at _f_; which, by rendering them dimmer, causes us to imagine them to be at a yet greater distance. And as, in both cases, they are seen[46] much under the same Angle, we naturally judge them to be biggest when they seem farthest from us; like the above-mentioned house § 186, seen from a higher ground, which shewed it to be farther off than it appeared from low ground; or the hay-rick, which appeared at a greater distance by means of an interposing Fog.

[Sidenote: Their Diameters are not less on the Meridian than in the Horizon.]

188. Any one may satisfy himself that the Moon appears under no greater Angle in the Horizon than on the Meridian, by taking a large sheet of paper, and rolling it up in the form of a Tube, of such a width, that observing the Moon through it when she rises, she may, as it were, just fill the Tube; then tie a thread round it to keep it of that size; and when the Moon comes to the Meridian, and appears much less to the eye, look at her again through the same Tube, and she will fill it just as much, if not more, than she did at her rising.

189. When the full Moon is in _perigeo_, or at her least distance from the Earth, she is seen under a larger Angle, and must therefore appear bigger than when she is Full at other times: and if that part of the Atmosphere where she rises be more replete with vapours than usual, she appears so much the dimmer; and therefore we fancy her to be still the bigger, by referring her to an unusually great distance; knowing that no objects which are very far distant can appear big unless they be really so.

CHAP. IX.

_The Method of finding the Distances of the Sun, Moon, and Planets._

[Sidenote: PLATE IV.]

190. Those who have not learnt how to take the [47]Altitude of any Celestial Phenomenon by a common Quadrant, nor know any thing of Plain Trigonometry, may pass over the first Article of this short Chapter, and take the Astronomer’s word for it, that the distances of the Sun and Planets are as stated in the first Chapter of this Book. But, to every one who knows how to take the Altitude of the Sun, the Moon, or a Star, and can solve a plain right-angled Triangle, the following method of finding the distances of the Sun and Moon will be easily understood.

[Sidenote: Fig I.]

Let _BAG_ be one half of the Earth, _AC_ it’s semi-diameter, _S_ the Sun, _m_ the Moon, and _EKOL_ a quarter of the Circle described by the Moon in revolving from the Meridian to the Meridian again. Let _CRS_ be the rational Horizon of an observer at _A_, extended to the Sun in the Heavens, and _HAO_ his sensible Horizon; extended to the Moon’s Orbit. _ALC_ is the Angle under which the Earth’s semi-diameter _AC_ is seen from the Moon at _L_, which is equal to the Angle _OAL_, because the right lines _AO_ and _CL_ which include both these Angles are parallel. _ASC_ is the Angle under which the Earth’s semi-diameter _AC_ is seen from the Sun at _S_, and is equal to the Angle _OAf_ because the lines _AO_ and _CRS_ are parallel. Now, it is found by observation, that the Angle _OAL_ is much greater than the Angle _OAf_; but _OAL_ is equal to _ALC_, and _OAf_ is equal to _ASC_. Now, as _ASC_ is much less than _ALC_, it proves that the Earth’s semi-diameter _AC_ appears much greater as seen from the Moon at _L_ than from the Sun at _S_: and therefore the Earth is much farther from the Sun than from the Moon[48]. The Quantities of these Angles are determined by observation in the following manner.

[Sidenote: The Moon’s horizontal Parallax, what.

The Moon’s distance determined.]

Let a graduated instrument as _DAE_, (the larger the better) having a moveable Index and Sight-holes, be fixed in such a manner, that it’s plane surface may be parallel to the Plan of the Equator, and it’s edge _AD_ in the Meridian: so that when the Moon is in the Equinoctial, and on the Meridian at _E_, she may be seen through the sight-holes when the edge of the moveable index cuts the beginning of the divisions at o, on the graduated limb _DE_; and when she is so seen, let the _precise_ time be noted. Now, as the Moon revolves about the Earth from the Meridian to the Meridian again in 24 hours 48 minutes, she will go a fourth part round it in a fourth part of that time, _viz._ in 6 hours 12 minutes, as seen from _C_, that is, from the Earth’s center or Pole. But as seen from _A_, the observer’s place on the Earth’s surface, the Moon will seem to have gone a quarter round the Earth when she comes to the sensible Horizon at _O_; for the Index through the sights of which she is then viewed will be at _d_, 90 degrees from _D_, where it was when she was seen at _E_. Now, let the exact moment when the Moon is seen at _O_ (which will be when she is in or near the sensible Horizon) be carefully noted[49], that it may be known in what time she has gone from _E_ to _O_; which time subtracted from 6 hours 12 minutes (the time of her going from _E_ to _L_) leaves the time of her going from _O_ to _L_, and affords an easy method for finding the Angle _OAL_ (called _the Moon’s horizontal Parallax_, which is equal to the Angle _ALC_) by the following Analogy: As the time of the Moon’s describing the arc _EO_ is to 90 degrees, so is 6 hours 12 minutes to the degrees of the Arc _DdE_, which measures the Angle _EAL_; from which subtract 90 degrees, and there remains the Angle _OAL_, equal to the Angle _ALC_, under which the Earth’s Semi-diameter _AC_ is seen from the Moon. Now, since all the Angles of a right-lined Triangle are equal to 180 degrees, or to two right Angles, and the sides of a Triangle are always proportional to the Sines of the opposite Angles, say, by the _Rule of Three_, as the Sine of the Angle _ALC_ at the Moon _L_ is to it’s opposite side _AC_ the Earth’s Semi-diameter, which is known to be 3985 miles, so is Radius, _viz._ the Sine of 90 degrees, or of the right Angle _ACL_ to it’s opposite side _AL_, which is the Moon’s distance at _L_ from the observer’s place at _A_ on the Earth’s surface; or, so is the Sine of the Angle _CAL_ to its opposite side _CL_, which is the Moon’s distance from the Earth’s centre, and comes out at a mean rate to be 240,000 miles. The Angle _CAL_ is equal to what _OAL_ wants of 90 degrees.

[Sidenote: The Sun’s distance cannot be yet so exactly determined as the Moon’s;

How near the truth it may soon be determined.]

191. The Sun’s distance from the Earth is found the same way, but with much greater difficulty; because his horizontal Parallax, or the Angle _OAS_ equal to the Angle _ASC_, is so small as, to be hardly perceptible, being only 10 seconds of a minute, or the 360th part of a degree. But the Moon’s horizontal Parallax, or Angle _OAL_ equal to the Angle _ALC_, is very discernible; being 57ʹ 49ʺ, or 3469ʺ at it’s mean state; which is more than 340 times as great as the Sun’s: and therefore, the distances of the heavenly bodies being inversely as the Tangents of their horizontal Parallaxes, the Sun’s distance from the Earth is at least 340 times as great as the Moon’s; and is rather understated at 81 millions of miles, when the Moon’s distance is certainly known to be 240 thousand. But because, according to some Astronomers, the Sun’s horizontal Parallax is 11 seconds, and according to others only 10, the former Parallax making the Sun’s distance to be about 75,000,000 of miles, and the latter 82,000,000; we may take it for granted, that the Sun’s distance is not less than as deduced from the former, nor more than as shewn by the latter: and every one who is accustomed to make such observations, knows how hard it is, if not impossible, to avoid an error of a second; especially on account of the inconstancy of horizontal Refractions. And here, the error of one second, in so small an Angle, will make an error of 7 millions of miles in so great a distance as that of the Sun’s; and much more in the distances of the superiour Planets. But Dr. HALLEY has shewn us how the Sun’s distance from the Earth, and consequently the distances of all the Planets from the Sun, may be known to within a 500th part of the whole, by a Transit of Venus over the Sun’s Disc, which will happen on the 6th of _June_, in the year 1761; till which time we must content ourselves with allowing the Sun’s distance to be about 81 millions of miles, as commonly stated by Astronomers.

[Sidenote: The Sun proved to be much bigger than the Moon.]

192. The Sun and Moon appear much about the same bulk: And every one who understands Geometry knows how their true bulks may be deduced from the apparent, when their real distances are known. Spheres are to one another as the Cubes of their Diameters; whence, if the Sun be 81 millions of miles from the Earth, to appear as big as the Moon, whose distance does not exceed 240 thousand miles, he must, in solid bulk, be 42 millions 875 thousand times as big as the Moon.

193. The horizontal Parallaxes are best observed at the Equator; 1. Because the heat is so nearly equal every day, that the Refractions are almost constantly the same. 2. Because the parallactic Angle is greater there as at _A_ (the distance from thence to the Earth’s Axis being greater,) than upon any parallel of Latitude, as _a_ or _b_.

[Sidenote: The relative distances of the Planets from the Sun are known to great precision, though their real distances are not well known.]

194. The Earth’s distance from the Sun being determined, the distances of all the other Planets from him are easily found by the following analogy, their periods round him being ascertained by observation. As the square of the Earth’s period round the Sun is to the cube of it’s distance from him, so is the square of the period of any other Planet to the cube of it’s distance, in such parts or measures as the Earth’s distance was taken; see § 111. This proportion gives us the relative mean distances of the Planets from the Sun to the greatest degree of exactness; and they are as follows, having been deduced from their periodical times, according to the law just mentioned, which was discovered by KEPLER and demonstrated by Sir ISAAC NEWTON.

_Periodical Revolution to the same fixed Star in days and decimal parts of a day._

Of Mercury Venus The Earth Mars Jupiter Saturn

87.9692 224.6176 365.2564 686.9785 4332.514 10759.275

_Relative mean distances from the Sun._

38710 72333 100000 152369 520096 954006

_From these numbers we deduce, that if the Sun’s horizontal Parallax be 10ʺ, the real mean distances of the Planets from the Sun in English miles are_