WITH the elementary knowledge already acquired, you will now be able to enter with pleasure and profit on the various interesting phenomena dependent on the revolution of the earth on its axis and around the sun.
The apparent diurnal revolution of the heavenly bodies, from east to west, is owing to the actual revolution of the earth on its own axis, from west to east. If we conceive of a radius of the earth"s equator extended until it meets the concave sphere of the heavens, then, as the earth revolves, the extremity of this line would trace out a curve on the face of the sky; namely, the celestial equator. In curves parallel to this, called the _circles of diurnal revolution_, the heavenly bodies actually _appear_ to move, every star having its own peculiar circle.
After you have first rendered familiar the real motion of the earth from west to east, you may then, without danger of misapprehension, adopt the common language, that all the heavenly bodies revolve around the earth once a day, from east to west, in circles parallel to the equator and to each other.
I must remind you, that the time occupied by a star, in pa.s.sing from any point in the meridian until it comes round to the same point again, is called a _sidereal day_, and measures the period of the earth"s revolution on its axis. If we watch the returns of the same star from day to day, we shall find the intervals exactly equal to each other; that is, _the sidereal days are all equal_. Whatever star we select for the observation, the same result will be obtained. The stars, therefore, always keep the same relative position, and have a common movement round the earth,--a consequence that naturally flows from the hypothesis that their _apparent_ motion is all produced by a single _real_ motion; namely, that of the earth. The sun, moon, and planets, as well as the fixed stars, revolve in like manner; but their returns to the meridian are not, like those of the fixed stars, at exactly equal intervals.
The _appearances_ of the diurnal motions of the heavenly bodies are different in different parts of the earth,--since every place has its own horizon, and different horizons are variously inclined to each other. Nothing in astronomy is more apt to mislead us, than the obstinate habit of considering the horizon as a fixed and immutable plane, and of referring every thing to it. We should contemplate the earth as a huge globe, occupying a small portion of s.p.a.ce, and encircled on all sides, at an immense distance, by the starry sphere. We should free our minds from their habitual p.r.o.neness to consider one part of s.p.a.ce as naturally _up_ and another _down_, and view ourselves as subject to a force (gravity) which binds us to the earth as truly as though we were fastened to it by some invisible cords or wires, as the needle attaches itself to all sides of a spherical loadstone. We should dwell on this point, until it appears to us as truly up, in the direction B B, C C, D D, when one is at B, C, D, respectively, as in the direction A A, when he is at A, Fig. 14.
Let us now suppose the spectator viewing the diurnal revolutions from several different positions on the earth. On the _equator_, his horizon would pa.s.s through both poles; for the horizon cuts the celestial vault at ninety degrees in every direction from the zenith of the spectator; but the pole is likewise ninety degrees from his zenith, when he stands on the equator; and consequently, the pole must be in the horizon. Here, also, the celestial equator would coincide with the prime vertical, being a great circle pa.s.sing through the east and west points. Since all the diurnal circles are parallel to the equator, consequently, they would all, like the equator be perpendicular to the horizon. Such a view of the heavenly bodies is called a right sphere, which may be thus defined: _a right sphere is one in which all the daily revolutions of the stars are in circles perpendicular to the horizon_.
[Ill.u.s.tration Fig. 14.]
A right sphere is seen only at the equator. Any star situated in the celestial equator would appear to rise directly in the east, at midnight to be in the zenith of the spectator, and to set directly in the west.
In proportion as stars are at a greater distance from the equator towards the pole, they describe smaller and smaller circles, until, near the pole, their motion is hardly perceptible.
If the spectator advances one degree from the equator towards the north pole, his horizon reaches one degree beyond the pole of the earth, and cuts the starry sphere one degree below the pole of the heavens, or below the north star, if that be taken as the place of the pole. As he moves onward towards the pole, his horizon continually reaches further and further beyond it, until, when he comes to the pole of the earth, and under the pole of the heavens, his horizon reaches on all sides to the equator, and coincides with it. Moreover, since all the circles of daily motion are parallel to the equator, they become, to the spectator at the pole, parallel to the horizon. Or, _a parallel sphere is that in which all the circles of daily motion are parallel to the horizon_.
To render this view of the heavens familiar, I would advise you to follow round in mind a number of separate stars, in their diurnal revolution, one near the horizon, one a few degrees above it, and a third near the zenith. To one who stood upon the north pole, the stars of the northern hemisphere would all be perpetually in view when not obscured by clouds, or lost in the sun"s light, and none of those of the southern hemisphere would ever be seen. The sun would be constantly above the horizon for six months in the year, and the remaining six continually out of sight. That is, at the pole, the days and nights are each six months long. The appearances at the south pole are similar to those at the north.
A perfect parallel sphere can never be seen, except at one of the poles,--a point which has never been actually reached by man; yet the British discovery ships penetrated within a few degrees of the north pole, and of course enjoyed the view of a sphere nearly parallel.
As the circles of daily motion are parallel to the horizon of the pole, and perpendicular to that of the equator, so at all places between the two, the diurnal motions are oblique to the horizon. This aspect of the heavens const.i.tutes an oblique sphere, which is thus defined: _an oblique sphere is that in which the circles of daily motion are oblique to the horizon_.
Suppose, for example, that the spectator is at the lat.i.tude of fifty degrees. His horizon reaches fifty degrees beyond the pole of the earth, and gives the same apparent elevation to the pole of the heavens. It cuts the equator and all the circles of daily motion, at an angle of forty degrees,--being always equal to what the alt.i.tude of the pole lacks of ninety degrees: that is, it is always equal to the co-alt.i.tude of the pole. Thus, let H O, Fig. 15, represent the horizon, E Q the equator, and P P" the axis of the earth. Also, _l l, m m, n n_, parallels of lat.i.tude. Then the horizon of a spectator at Z, in lat.i.tude fifty degrees, reaches to fifty degrees beyond the pole; and the angle E C H, which the equator makes with the horizon, is forty degrees,--the complement of the lat.i.tude. As we advance still further north, the elevation of the diurnal circle above the horizon grows less and less, and consequently, the motions of the heavenly bodies more and more oblique to the horizon, until finally, at the pole, where the lat.i.tude is ninety degrees, the angle of elevation of the equator vanishes, and the horizon and the equator coincide with each other, as before stated.
[Ill.u.s.tration Fig. 15.]
_The circle of perpetual apparition is the boundary of that s.p.a.ce around the elevated pole, where the stars never set._ Its distance from the pole is equal to the lat.i.tude of the place. For, since the alt.i.tude of the pole is equal to the lat.i.tude, a star, whose polar distance is just equal to the lat.i.tude, will, when at its lowest point, only just reach the horizon; and all the stars nearer the pole than this will evidently not descend so far as the horizon. Thus _m m_, Fig. 15, is the circle of perpetual apparition, between which and the north pole, the stars never set, and its distance from the pole, O P, is evidently equal to the elevation of the pole, and of course to the lat.i.tude.
In the opposite hemisphere, a similar part of the sphere adjacent to the depressed pole never rises. Hence, _the circle of perpetual occultation is the boundary of that s.p.a.ce around the depressed pole, within which the stars never rise._
Thus _m" m"_, Fig. 15, is the circle of perpetual occultation, between which and the south pole, the stars never rise.
In an oblique sphere, the horizon cuts the circles of daily motion unequally. Towards the elevated pole, more than half the circle is above the horizon, and a greater and greater portion, as the distance from the equator is increased, until finally, within the circle of perpetual apparition, the whole circle is above the horizon. Just the opposite takes place in the hemisphere next the depressed pole. Accordingly, when the sun is in the equator, as the equator and horizon, like all other great circles of the sphere, bisect each other, the days and nights are equal all over the globe. But when the sun is north of the equator, the days become longer than the nights, but shorter, when the sun is south of the equator. Moreover, the higher the lat.i.tude, the greater is the inequality in the lengths of the days and nights. By examining Fig. 15, you will easily see how each of these cases must hold good.
Most of the appearances of the diurnal revolution can be explained, either on the supposition that the celestial sphere actually turns around the earth once in twenty-four hours, or that this motion of the heavens is merely apparent, arising from the revolution of the earth on its axis, in the opposite direction,--a motion of which we are insensible, as we sometimes lose the consciousness of our own motion in a ship or steam-boat, and observe all external objects to be receding from us, with a common motion. Proofs, entirely conclusive and satisfactory, establish the fact, that it is the earth, and not the celestial sphere, that turns; but these proofs are drawn from various sources, and one is not prepared to appreciate their value, or even to understand some of them, until he has made considerable proficiency in the study of astronomy, and become familiar with a great variety of astronomical phenomena. To such a period we will therefore postpone the discussion of the earth"s rotation on its axis.
While we retain the same place on the earth, the diurnal revolution occasions no change in our horizon, but our horizon goes round, as well as ourselves. Let us first take our station on the equator, at sunrise; our horizon now pa.s.ses through both the poles and through the sun, which we are to conceive of as at a great distance from the earth, and therefore as cut, not by the terrestrial, but by the celestial, horizon.
As the earth turns, the horizon dips more and more below the sun, at the rate of fifteen degrees for every hour; and, as in the case of the polar star, the sun appears to rise at the same rate. In six hours, therefore, it is depressed ninety degrees below the sun, bringing us directly under the sun, which, for our present purpose, we may consider as having all the while maintained the same fixed position in s.p.a.ce. The earth continues to turn, and in six hours more, it completely reverses the position of our horizon, so that the western part of the horizon, which at sunrise was diametrically opposite to the sun, now cuts the sun, and soon afterwards it rises above the level of the sun, and the sun sets.
During the next twelve hours, the sun continues on the invisible side of the sphere, until the horizon returns to the position from which it set out, and a new day begins.
Let us next contemplate the similar phenomena at the _poles_. Here the horizon, coinciding, as it does, with the equator, would cut the sun through its centre and the sun would appear to revolve along the surface of the sea, one half above and the other half below the horizon. This supposes the sun in its annual revolution to be at one of the equinoxes.
When the sun is north of the equator, it revolves continually round in a circle, which, during a single revolution, appears parallel to the equator, and it is constantly day; and when the sun is south of the equator, it is, for the same reason, continual night.
When we have gained a clear idea of the appearances of the diurnal revolutions, as exhibited to a spectator at the equator and at the pole, that is, in a right and in a parallel sphere, there will be little difficulty in imagining how they must be in the intermediate lat.i.tudes, which have an oblique sphere.
The appearances of the sun and stars, presented to the inhabitants of different countries, are such as correspond to the sphere in which they live. Thus, in the fervid climates of India, Africa, and South America, the sun mounts up to the highest regions of the heavens, and descends directly downwards, suddenly plunging beneath the horizon. His rays, darting almost vertically upon the heads of the inhabitants, strike with a force unknown to the people of the colder climates; while in places remote from the equator, as in the north of Europe, the sun, in Summer, rises very far in the north, takes a long circuit towards the south, and sets as far northward in the west as the point where it rose on the other side of the meridian. As we go still further north, to the northern parts of Norway and Sweden, for example, to the confines of the frigid zone, the Summer"s sun just grazes the northern horizon, and at noon appears only twenty-three and one half degrees above the southern.
On the other hand, in mid-winter, in the north of Europe, as at St.
Petersburgh, the day dwindles almost to nothing,--lasting only while the sun describes a very short arc in the extreme south. In some parts of Siberia and Iceland, the only day consists of a little glimmering of the sun on the verge of the southern horizon, at noon.
LETTER IX.
PARALLAX AND REFRACTION.
"Go, wondrous creature! mount where science guides, Go measure earth, weigh air, and state the tides; Instruct the planets in what orbs to run, Correct old Time, and regulate the sun."--_Pope._
I THINK you must have felt some astonishment, that astronomers are able to calculate the exact distances and magnitudes of the sun, moon, and planets. We should, at the first thought, imagine that such knowledge as this must be beyond the reach of the human faculties, and we might be inclined to suspect that astronomers practise some deception in this matter, for the purpose of exciting the admiration of the unlearned. I will therefore, in the present Letter, endeavor to give you some clear and correct views respecting the manner in which astronomers acquire this knowledge.
In our childhood, we all probably adopt the notion that the sky is a real dome of definite surface, in which the heavenly bodies are fixed.
When any objects are beyond a certain distance from the eye, we lose all power of distinguishing, by our sight alone, between different distances, and cannot tell whether a given object is one million or a thousand millions of miles off. Although the bodies seen in the sky are in fact at distances extremely various,--some, as the clouds, only a few miles off; others, as the moon, but a few thousand miles; and others, as the fixed stars, innumerable millions of miles from us,--yet, as our eye cannot distinguish these different distances, we acquire the habit of referring all objects beyond a moderate height to one and the same surface, namely, an imaginary spherical surface, denominated the celestial vault. Thus, the various objects represented in the diagram on next page, though differing very much in shape and diameter, would all be _projected_ upon the sky alike, and compose a part, indeed, of the imaginary vault itself. The place which each object occupies is determined by lines drawn from the eye of the spectator through the extremities of the body, to meet the imaginary concave sphere. Thus, to a spectator at O, Fig 16, the several lines A B, C D, and E F, would all be projected into arches on the face of the sky, and be seen as parts of the sky itself, as represented by the lines A" B", C" D", and E" F". And were a body actually to move in the several directions indicated by these lines, they would appear to the spectator to describe portions of the celestial vault. Thus, even when moving through the crooked line, from _a_ to _b_, a body would appear to be moving along the face of the sky, and of course in a regular curve line, from _c_ to _d_.
[Ill.u.s.tration Fig. 16.]
But, although all objects, beyond a certain moderate height, are projected on the imaginary surface of the sky, yet different spectators will project the same object on _different parts_ of the sky. Thus, a spectator at A, Fig. 17, would see a body, C, at M, while a spectator at B would see the same body at N. This change of place in a body, as seen from different points, is called parallax, which is thus defined: _parallax is the apparent change of place which bodies undergo by being viewed from different points_. [Ill.u.s.tration Fig. 17.]
The arc M N is called the _parallactic arc_, and the angle A C B, the _parallactic angle_.
It is plain, from the figure, that near objects are much more affected by parallax than distant ones. Thus, the body C, Fig. 17, makes a much greater parallax than the more distant body D,--the former being measured by the arc M N, and the latter by the arc O P. We may easily imagine bodies to be so distant, that they would appear projected at very nearly the same point of the heavens, when viewed from places very remote from each other. Indeed, the fixed stars, as we shall see more fully hereafter, are so distant, that spectators, a hundred millions of miles apart, see each star in one and the same place in the heavens.
It is by means of parallax, that astronomers find the distances and magnitudes of the heavenly bodies. In order fully to understand this subject, one requires to know something of trigonometry, which science enables us to find certain unknown parts of a triangle from certain other parts which are known. Although you may not be acquainted with the principles of trigonometry, yet you will readily understand, from your knowledge of arithmetic, that from certain things given in a problem others may be found. Every triangle has of course three sides and three angles; and, if we know two of the angles and one of the sides, we can find all the other parts, namely, the remaining angle and the two unknown sides. Thus, in the triangle A B C, Fig. 18, if we know the length of the side A B, and how many degrees each of the angles A B C and B C A contains, we can find the length of the side B C, or of the side A C, and the remaining angle at A. Now, let us apply these principles to the measurements of some of the heavenly bodies.
[Ill.u.s.tration Fig. 18.]
[Ill.u.s.tration Fig. 19.]
In Fig. 19, let A represent the earth, C H the horizon, and H Z a quadrant of a great circle of the heavens, extending from the horizon to the zenith; and let E, F, G, O, be successive positions of the moon, at different elevations, from the horizon to the meridian. Now, a spectator on the surface of the earth, at A, would refer the moon, when at E, to _h_, on the face of the sky, whereas, if seen from the centre of the earth, it would appear at H. So, when the moon was at F, a spectator at A would see it at _p_, while, if seen from the centre, it would have appeared at P. The parallactic arcs, H _h_, P _p_, R _r_, grow continually smaller and smaller, as a body is situated higher above the horizon; and when the body is in the zenith, then the parallax vanishes altogether, for at O the moon would be seen at Z, whether viewed from A or C.
Since, then, a heavenly body is liable to be referred to different points on the celestial vault, when seen from different parts of the earth, and thus some confusion be occasioned in the determination of points on the celestial sphere, astronomers have agreed to consider the true place of a celestial object to be that where it would appear, if seen from the centre of the earth; and the doctrine of parallax teaches how to reduce observations made at any place on the surface of the earth, to such as they would be, if made from the centre.
When the moon, or any heavenly body, is seen in the horizon, as at E, the change of place is called the horizontal parallax. Thus, the angle A E C, measures the horizontal parallax of the moon. Were a spectator to view the earth from the centre of the moon, he would see the semidiameter of the earth under this same angle; hence, _the horizontal parallax of any body is the angle subtended by the semidiameter of the earth, as seen from the body_. Please to remember this fact.
It is evident from the figure, that the effect of parallax upon the place of a celestial body is to _depress_ it. Thus, in consequence of parallax, E is depressed by the arc H _h_; F, by the arc P _p_; G, by the arc R _r_; while O sustains no change. Hence, in all calculations respecting the alt.i.tude of the sun, moon, or planets, the amount of parallax is to be added: the stars, as we shall see hereafter, have no sensible parallax.
It is now very easy to see how, when the parallax of a body is known, we may find its distance from the centre of the earth. Thus, in the triangle A C E, Fig. 19, the side A C is known, being the semidiameter of the earth; the angle C A E, being a right angle, is also known; and the parallactic angle, A E C, is found from observation; and it is a well-known principle of trigonometry, that when we have any two angles of a triangle, we may find the remaining angle by subtracting the sum of these two from one hundred and eighty degrees. Consequently, in the triangle A E C, we know all the angles and one side, namely, the side A C; hence, we have the means of finding the side C E, which is the distance from the centre of the earth to the centre of the moon.
[Ill.u.s.tration Fig. 20.]
When the distance of a heavenly body is known, and we can measure, with instruments, its angular breadth, we can easily determine its _magnitude_. Thus, if we have the distance of the moon, E S, Fig. 20, and half the breadth of its disk S C, (which is measured by the angle S E C,) we can find the length of the line, S C, in miles. Twice this line is the diameter of the body; and when we know the diameter of a sphere, we can, by well-known rules, find the contents of the surface, and its solidity.
You will perhaps be curious to know, _how the moon"s horizontal parallax is found_; for it must have been previously ascertained, before we could apply this method to finding the distance of the moon from the earth.
Suppose that two astronomers take their stations on the same meridian, but one south of the equator, as at the Cape of Good Hope, and another north of the equator, as at Berlin, in Prussia, which two places lie nearly on the same meridian. The observers would severally refer the moon to different points on the face of the sky,--the southern observer carrying it further north, and the northern observer further south, than its true place, as seen from the centre of the earth. This will be plain from the diagram, Fig. 21. If A and B represent the positions of the spectators, M the moon, and C D an arc of the sky, then it is evident, that C D would be the parallactic arc.
[Ill.u.s.tration Fig. 21.]
These observations furnish materials for calculating, by the aid of trigonometry, the moon"s horizontal parallax, and we have before seen how, when we know the parallax of a heavenly body, we can find both its distance from the earth and its magnitude.