DOCTRINE OF THE SPHERE.
"All are but parts of one stupendous whole, Whose body Nature is, and God the soul."—Pope.
Let us now consider what astronomy is, and into what great divisions it is distributed; and then we will take a cursory view of the doctrine of the sphere. This subject will probably be less interesting to you than many that are to follow; but still, permit me to urge upon you the necessity of studying it with attention, and reflecting upon each definition, until you fully understand it; for, unless you fully and clearly comprehend the circles of the sphere, and the use that is made of them in astronomy, a mist will hang over every subsequent portion of the science. I beg you, therefore, to pause upon every paragraph of this Letter; and if there is any point in the whole which you cannot clearly understand, I would advise you to mark it, and to recur to it repeatedly; and, if you finally cannot obtain a clear idea of it yourself, I would recommend to you to apply for aid to some of your friends, who may be able to assist you.
Astronomy is that science which treats of the heavenly bodies. More particularly, its object is to teach what is known respecting the sun, moon, planets, comets, and fixed stars; and also to explain the methods by which this knowledge is acquired. Astronomy is sometimes divided into descriptive, physical, and practical. Descriptive astronomy respects facts; physical astronomy, causes; practical astronomy, the means of investigating the facts, whether by instruments or by calculation. It is the province of descriptive astronomy to observe, classify, and record, all the phenomena of the heavenly bodies, whether pertaining to those bodies individually, or resulting from their motions and mutual relations. It is the part of physical astronomy to explain the causes of these phenomena, by investigating the general laws on which they depend; especially, by tracing out all the consequences of the law of universal gravitation. Practical astronomy lends its aid to both the other departments.
The definitions of the different lines, points, and circles, which are used in astronomy, and the propositions founded upon them, compose the doctrine of the sphere. Before these definitions are given, I must recall to your recollection a few particulars respecting the method of measuring angles. (See Fig. 1, page 18.)
A line drawn from the centre to the circumference of a circle is called a radius, as C D, C B, or C K.
Any part of the circumference of a circle is called an arc, as A B, or B D.
An angle is measured by an arc included between two radii. Thus, in Fig. 1, the angle contained between the two radii, C A and C B, that is, the angle A C B, is measured by the arc A B. Every circle, it will be recollected, is divided into three hundred and sixty equal parts, called degrees; and any arc, as A B, contains a certain number of degrees, according to its length. Thus, if the arc A B contains forty degrees, then the opposite angle A C B is said to be an angle of forty degrees, and to be measured by A B. But this arc is the same part of the smaller circle that E F is of the greater. The arc A B, therefore, contains the same number of degrees as the arc E F, and either may be taken as the measure of the angle A C B. As the whole circle contains three hundred and sixty degrees, it is evident, that the quarter of a circle, or quadrant, contains ninety degrees, and that the semicircle A B D G contains one hundred and eighty degrees.
Fig. 1.
The complement of an arc, or angle, is what it wants of ninety degrees. Thus, since A D is an arc of ninety degrees, B D is the complement of A B, and A B is the complement of B D. If A B denotes a certain number of degrees of latitude, B D will be the complement of the latitude, or the colatitude, as it is commonly written.
The supplement of an arc, or angle, is what it wants of one hundred and eighty degrees. Thus, B A is the supplement of G D B, and G D B is the supplement of B A. If B A were twenty degrees of longitude, G D B, its supplement, would be one hundred and sixty degrees. An angle is said to be subtended by the side which is opposite to it. Thus, in the triangle A C K, the angle at C is subtended by the side A K, the angle at A by C K, and the angle at K by C A. In like manner, a side is said to be subtended by an angle, as A K by the angle at C.
Let us now proceed with the doctrine of the sphere.
A section of a sphere, by a plane cutting it in any manner, is a circle. Great circles are those which pass through the centre of the sphere, and divide it into two equal hemispheres. Small circles are such as do not pass through the centre, but divide the sphere into two unequal parts. The axis of a circle is a straight line passing through its centre at right angles to its plane. The pole of a great circle is the point on the sphere where its axis cuts through the sphere. Every great circle has two poles, each of which is every where ninety degrees from the great circle. All great circles of the sphere cut each other in two points diametrically opposite, and consequently their points of section are one hundred and eighty degrees apart. A great circle, which passes through the pole of another great circle, cuts the latter at right angles. The great circle which passes through the pole of another great circle, and is at right angles to it, is called a secondary to that circle. The angle made by two great circles on the surface of the sphere is measured by an arc of another great circle, of which the angular point is the pole, being the arc of that great circle intercepted between those two circles.
In order to fix the position of any place, either on the surface of the earth or in the heavens, both the earth and the heavens are conceived to be divided into separate portions, by circles, which are imagined to cut through them, in various ways. The earth thus intersected is called the terrestrial, and the heavens the celestial, sphere. We must bear in mind, that these circles have no existence in Nature, but are mere landmarks, artificially contrived for convenience of reference. On account of the immense distances of the heavenly bodies, they appear to us, wherever we are placed, to be fixed in the same concave surface, or celestial vault. The great circles of the globe, extended every way to meet the concave sphere of the heavens, become circles of the celestial sphere.
The horizon is the great circle which divides the earth into upper and lower hemispheres, and separates the visible heavens from the invisible. This is the rational horizon. The sensible horizon is a circle touching the earth at the place of the spectator, and is bounded by the line in which the earth and skies seem to meet. The sensible horizon is parallel to the rational, but is distant from it by the semidiameter of the earth, or nearly four thousand miles. Still, so vast is the distance of the starry sphere, that both these planes appear to cut the sphere in the same line; so that we see the same hemisphere of stars that we should see, if the upper half of the earth were removed, and we stood on the rational horizon.
The poles of the horizon are the zenith and nadir. The zenith is the point directly over our heads; and the nadir, that directly under our feet. The plumb-line (such as is formed by suspending a bullet by a string) is in the axis of the horizon, and consequently directed towards its poles. Every place on the surface of the earth has its own horizon; and the traveller has a new horizon at every step, always extending ninety degrees from him, in all directions.
Vertical circles are those which pass through the poles of the horizon, (the zenith and nadir,) perpendicular to it.
The meridian is that vertical circle which passes through the north and south points.
The prime vertical is that vertical circle which passes through the east and west points.
The altitude of a body is its elevation above the horizon, measured on a vertical circle.
The azimuth of a body is its distance, measured on the horizon, from the meridian to a vertical circle passing through that body.
The amplitude of a body is its distance, on the horizon, from the prime vertical to a vertical circle passing through the body.
Azimuth is reckoned ninety degrees from either the north or south point; and amplitude ninety degrees from either the east or west point. Azimuth and amplitude are mutually complements of each other, for one makes up what the other wants of ninety degrees. When a point is on the horizon, it is only necessary to count the number of degrees of the horizon between that point and the meridian, in order to find its azimuth; but if the point is above the horizon, then its azimuth is estimated by passing a vertical circle through it, and reckoning the azimuth from the point where this circle cuts the horizon.
The zenith distance of a body is measured on a vertical circle passing through that body. It is the complement of the altitude.
The axis of the earth is the diameter on which the earth is conceived to turn in its diurnal revolution. The same line, continued until it meets the starry concave, constitutes the axis of the celestial sphere.
The poles of the earth are the extremities of the earth's axis: the poles of the heavens, the extremities of the celestial axis.
The equator is a great circle cutting the axis of the earth at right angles. Hence, the axis of the earth is the axis of the equator, and its poles are the poles of the equator. The intersection of the plane of the equator with the surface of the earth constitutes the terrestrial, and its intersection with the concave sphere of the heavens, the celestial, equator. The latter, by way of distinction, is sometimes denominated the equinoctial.
The secondaries to the equator,—that is, the great circles passing through the poles of the equator,—are called meridians, because that secondary which passes through the zenith of any place is the meridian of that place, and is at right angles both to the equator and the horizon, passing, as it does, through the poles of both. These secondaries are also called hour circles because the arcs of the equator intercepted between them are used as measures of time.
The latitude of a place on the earth is its distance from the equator north or south. The polar distance, or angular distance from the nearest pole, is the complement of the latitude.
The longitude of a place is its distance from some standard meridian, either east or west, measured on the equator. The meridian, usually taken as the standard, is that of the Observatory of Greenwich, in London. If a place is directly on the equator, we have only to inquire, how many degrees of the equator there are between that place and the point where the meridian of Greenwich cuts the equator. If the place is north or south of the equator, then its longitude is the arc of the equator intercepted between the meridian which passes through the place and the meridian of Greenwich.
The ecliptic is a great circle, in which the earth performs its annual revolutions around the sun. It passes through the centre of the earth and the centre of the sun. It is found, by observation, that the earth does not lie with its axis at right angles to the plane of the ecliptic, so as to make the equator coincide with it, but that it is turned about twenty-three and a half degrees out of a perpendicular direction, making an angle with the plane itself of sixty-six and a half degrees. The equator, therefore, must be turned the same distance out of a coincidence with the ecliptic, the two circles making an angle with each other of twenty-three and a half degrees. It is particularly important that we should form correct ideas of the ecliptic, and of its relations to the equator, since to these two circles a great number of astronomical measurements and phenomena are referred.
The equinoctial points, or equinoxes, are the intersections of the ecliptic and equator. The time when the sun crosses the equator, in going northward, is called the vernal, and in returning southward, the autumnal, equinox. The vernal equinox occurs about the twenty-first of March, and the autumnal, about the twenty-second of September.
The solstitial points are the two points of the ecliptic most distant from the equator. The times when the sun comes to them are called solstices. The Summer solstice occurs about the twenty-second of June, and the Winter solstice about the twenty-second of December. The ecliptic is divided into twelve equal parts, of thirty degrees each, called signs, which, beginning at the vernal equinox, succeed each other, in the following order:
| 1. Aries, ♈ | 7. Libra, ♎ |
| 2. Taurus, ♉ | 8. Scorpio, ♏ |
| 3. Gemini, ♊ | 9. Sagittarius, ♐ |
| 4. Cancer, ♋ | 10. Capricornus, ♑ |
| 5. Leo, ♌ | 11. Aquarius, ♒ |
| 6. Virgo, ♍ | 12. Pisces. ♓ |
The mode of reckoning on the ecliptic is by signs, degrees, minutes, and seconds. The sign is denoted either by its name or its number. Thus, one hundred degrees may be expressed either as the tenth degree of Cancer, or as 3s 10°. It will be found an advantage to repeat the signs in their proper order, until they are well fixed in the memory, and to be able to recognise each sign by its appropriate character.
Of the various meridians, two are distinguished by the name of colures. The equinoctial colure is the meridian which passes through the equinoctial points. From this meridian, right ascension and celestial longitude are reckoned, as longitude on the earth is reckoned from the meridian of Greenwich. The solstitial colure is the meridian which passes through the solstitial points.
The position of a celestial body is referred to the equator by its right ascension and declination. Right ascension is the angular distance from the vernal equinox measured on the equator. If a star is situated on the equator, then its right ascension is the number of degrees of the equator between the star and the vernal equinox. But if the star is north or south of the equator, then its right ascension is the number of degrees of the equator, intercepted between the vernal equinox and that secondary to the equator which passes through the star. Declination is the distance of a body from the equator measured on a secondary to the latter. Therefore, right ascension and declination correspond to terrestrial longitude and latitude,—right ascension being reckoned from the equinoctial colure, in the same manner as longitude is reckoned from the meridian of Greenwich. On the other hand, celestial longitude and latitude are referred, not to the equator, but to the ecliptic. Celestial longitude is the distance of a body from the vernal equinox measured on the ecliptic. Celestial latitude is the distance from the ecliptic measured on a secondary to the latter. Or, more briefly, longitude is distance on the ecliptic: latitude, distance from the ecliptic. The north polar distance of a star is the complement of its declination.
Parallels of latitude are small circles parallel to the equator. They constantly diminish in size, as we go from the equator to the pole. The tropics are the parallels of latitude which pass through the solstices. The northern tropic is called the tropic of Cancer; the southern, the tropic of Capricorn. The polar circles are the parallels of latitude that pass through the poles of the ecliptic, at the distance of twenty-three and a half degrees from the poles of the earth.
The elevation of the pole of the heavens above the horizon of any place is always equal to the latitude of the place. Thus, in forty degrees of north latitude we see the north star forty degrees above the northern horizon; whereas, if we should travel southward, its elevation would grow less and less, until we reached the equator, where it would appear in the horizon. Or, if we should travel northwards, the north star would rise continually higher and higher, until, if we could reach the pole of the earth, that star would appear directly over head. The elevation of the equator above the horizon of any place is equal to the complement of the latitude. Thus, at the latitude of forty degrees north, the equator is elevated fifty degrees above the southern horizon.
The earth is divided into five zones. That portion of the earth which lies between the tropics is called the torrid zone; that between the tropics and the polar circles, the temperate zones; and that between the polar circles and the poles, the frigid zones.
The zodiac is the part of the celestial sphere which lies about eight degrees on each side of the ecliptic. This portion of the heavens is thus marked off by itself, because all the planets move within it.
After endeavoring to form, from the definitions, as clear an idea as we can of the various circles of the sphere, we may next resort to an artificial globe, and see how they are severally represented there. I do not advise to begin learning the definitions from the globe; the mind is more improved, and a power of conceiving clearly how things are in Nature is more effectually acquired, by referring every thing, at first, to the grand sphere of Nature itself, and afterwards resorting to artificial representations to aid our conceptions. We can get but a very imperfect idea of a man from a profile cut in paper, unless we know the original. If we are acquainted with the individual, the profile will assist us to recall his appearance more distinctly than we can do without it. In like manner, orreries, globes, and other artificial aids, will be found very useful, in assisting us to form distinct conceptions of the relations existing between the different circles of the sphere, and of the arrangements of the heavenly bodies; but, unless we have already acquired some correct ideas of these things, by contemplating them as they are in Nature, artificial globes, and especially orreries, will be apt to mislead us.
I trust you will be able to obtain the use of a globe,[1] to aid you in the study of the foregoing definitions, or doctrine of the sphere; but if not, I would recommend the following easy device. To represent the earth, select a large apple, (a melon, when in season, will be found still better.) The eye and the stem of the apple will indicate the position of the two poles of the earth. Applying the thumb and finger of the left hand to the poles, and holding the apple so that the poles may be in a north and south line, turn this globe from west to east, and its motion will correspond to the diurnal movement of the earth. Pass a wire or a knitting needle through the poles, and it will represent the axis of the sphere. A circle cut around the apple, half way between the poles, will be the equator; and several other circles cut between the equator and the poles, parallel to the equator, will represent parallels of latitude; of which, two, drawn twenty-three and a half degrees from the equator, will be the tropics, and two others, at the same distance from the poles, will be the polar circles. A great circle cut through the poles, in a north and south direction, will form the meridian, and several other great circles drawn through the poles, and of course perpendicularly to the equator, will be secondaries to the equator, constituting meridians, or hour circles. A great circle cut through the centre of the earth, from one tropic to the other, would represent the plane of the ecliptic; and consequently a line cut round the apple where such a section meets the surface, will be the terrestrial ecliptic. The points where this circle meets the tropics indicate the position of the solstices; and its intersection with the equator, that of the equinoctial points.
The horizon is best represented by a circular piece of pasteboard, cut so as to fit closely to the apple, being movable upon it. When this horizon is passed through the poles, it becomes the horizon of the equator; when it is so placed as to coincide with the earth's equator, it becomes the horizon of the poles; and in every other situation it represents the horizon of a place on the globe ninety degrees every way from it. Suppose we are in latitude forty degrees; then let us place our movable paper parallel to our own horizon, and elevate the pole forty degrees above it, as near as we can judge by the eye. If we cut a circle around the apple, passing through its highest part, and through the east and west points, it will represent the prime vertical.
Simple as the foregoing device is, if you will take the trouble to construct one for yourself, it will lead you to more correct views of the doctrine of the sphere, than you would be apt to obtain from the most expensive artificial globes, although there are many other useful purposes which such globes serve, for which the apple would be inadequate. When you have thus made a sphere for yourself, or, with an artificial globe before you, if you have access to one, proceed to point out on it the various arcs of azimuth and altitude, right ascension and declination, terrestrial and celestial latitude and longitude,—these last being referred to the equator on the earth, and to the ecliptic in the heavens.
When the circles of the sphere are well learned, we may advantageously employ projections of them in various illustrations. By the projection of the sphere is meant a representation of all its parts on a plane. The plane itself is called the plane of projection. Let us take any circular ring, as a wire bent into a circle, and hold it in different positions before the eye. If we hold it parallel to the face, with the whole breadth opposite to the eye, we see it as an entire circle. If we turn it a little sideways, it appears oval, or as an ellipse; and, as we continue to turn it more and more round, the ellipse grows narrower and narrower, until, when the edge is presented to the eye, we see nothing but a line. Now imagine the ring to be near a perpendicular wall, and the eye to be removed at such a distance from it, as not to distinguish any interval between the ring and the wall; then the several figures under which the ring is seen will appear to be inscribed on the wall, and we shall see the ring as a circle, when perpendicular to a straight line joining the centre of the ring and the eye, or as an ellipse, when oblique to this line, or as a straight line, when its edge is towards us.
Fig. 2.
It is in this manner that the circles of the sphere are projected, as represented in the following diagram, Fig. 2. Here, various circles are represented as projected on the meridian, which is supposed to be situated directly before the eye, at some distance from it. The horizon H O, being perpendicular to the meridian, is seen edgewise, and consequently is projected into a straight line. The same is the case with the prime vertical Z N, with the equator E Q, and the several small circles parallel to the equator, which represent the two tropics and the two polar circles. In fact, all circles whatsoever, which are perpendicular to the plane of projection, will be represented by straight lines. But every circle which is perpendicular to the horizon, except the prime vertical, being seen obliquely, as Z M N, will be projected into an ellipse, one half only of which is seen,—the other half being on the other side of the plane of projection. In the same manner, P R P, an hour circle, is represented by an ellipse on the plane of projection.