PART II.

THEORETICAL ASTRONOMY.

CHAP. I.

GENERAL PHENOMENA OF THE HEAVENS.

Sect. I.—Of the Celestial Sphere.

When a spectator on a clear evening directs his attention to the sky, he perceives a concave hemisphere studded with innumerable brilliant points, all which appear to move in parallel directions, constantly preserving the same distances and relative positions. New groups of stars incessantly follow one another, rising in the east, mounting to a certain height, and then gradually sinking till they disappear in the west. On turning his face towards the north he observes some groups of stars which remain visible during the whole night; wheeling round a certain fixed point, and describing circles of greater or less magnitude, according as they are at a greater or less distance from that point. The same phenomena are observed every successive night; and as the stars present always the same configurations, and succeed each other in the same order, the mind is irresistibly led to the hypothesis of the diurnal revolution of the whole heavens about a fixed axis. This hypothesis implies that the stars move in parallel planes, and that their mutual distances remain invariable. In order, however, to be assured that such is the case, it is necessary to have recourse to more accurate observations, and to refer their successive positions to something that is fixed,—to certain points or planes which do not participate in the general motion.

What is most obviously adapted for this purpose is the Horizon, or the plane which appears extended all round us, and bounded by a circle of which the eye of the spectator occupies the centre. Let the circle SENEW (fig. 1) represent the horizon, C being the centre or place of the observer. Suppose now that a star is observed to rise at a certain point of the horizon A, and, after tracing its circuit in the sky, to set at the point A'; suppose, also, that another star rises at B, and sets at B'; then, if the distances AB, A'B' are measured by means of some angular instrument, it will be found that AB is equal to A'B', and that this is the case with regard to any two stars whatever. Since therefore the arcs AB and A'B' are equal, it follows that the chords AA', BB', joining their extremities, are parallel; whence we infer, that if two stars rise successively at the same point of the horizon A, they will also set at the same point A', a circumstance which is of itself strikingly indicative of the uniformity of the motion of the celestial bodies.

Through C let a perpendicular be drawn to the chord Cardinal AA', meeting the boundary of the horizon in S and N. points. These two points are the South and North points of the horizon; and another straight line drawn through C, parallel to AA', will meet the horizon in E and W, the East and West points. The four together, namely, S, E, N, W, are called the four Cardinal points of the horizon. The horizontal circle is itself denominated the Azimuth Circle, etc. Azimuth distances are measured from the North and South points, so that SA is the azimuth distance of a star's rising at A, and SA' is that of its setting at A'. The complement of the azimuth, or its defect from a right angle, is called the Amplitude, which is consequently measured from the East and West points; thus the arc EA' is the amplitude of a star's rising at A, or its ordinate amplitude, and WA' is that of its setting, or its occasion amplitude.

Conceive a straight line to pass through C, perpendicular to the horizon. This line, which is called the Vertical, will meet the visible hemisphere in a point directly above the observer, which is denominated the Zenith; the Zenith-point diametrically opposite, in which the prolongation of the vertical meets the invisible hemisphere, is the Nadir. The plane determined by the vertical and the straight line NS is called the Meridian, because when the sun reaches it, he is equally distant from the points at which he rises and sets, or, it is mid-day. The great circle formed by the intersection of the meridian with the celestial sphere is the Meridional circle; it divides the sphere into the eastern and western hemispheres. Any circle passing through the Zenith and Nadir is called a Vertical Circle; that which intersects the meridional circle at right angles, or which passes through the points E and W, is called the Prime Vertical.

Let HPZH' (fig. 2) represent a meridional circle passing through P, that point of the sphere which appears to remain immovable; and let Z be the zenith, and HH' the horizon. Join PC, and let it meet the circle in P'. The points P and P' are denominated the Poles of the world, and the straight line PP' is the Axis about which the whole heavens appear to revolve. Let also rr, Hg, Hf, EE', g'H', be projections of the paths of different stars. After having recognised that the celestial bodies move in parallel planes, the next object of the astronomer is to determine whether they revolve with velocities uniform or variable. For this purpose it is necessary to compare, by means of a well-regulated clock, the angular distances between the meridian and the successive positions of a star, with the times elapsed since the star's passage over the meridian. Let \( mn \) (fig. 4) be the parallel described Fig. 4 by a star, and \( s' \) two of its successive positions. The arcs \( ns, ns' \), intercepted between the meridional plane and the planes passing through the axis \( PP' \) and the points \( s, s' \), are measured by the arcs of the equator \( Ca, Cb \). Now, it is the result of constant experience, that the number of degrees in the arc \( Ca \) or \( Cb \) is to the number in the whole circumference as the time employed by the star in describing \( na \) or \( nb \) is to the time in which it completes a revolution. The time occupied by the star in completing its revolution is called a sidereal day; and the uniformity of the diurnal revolution enables us to calculate the angle \( CPs \), which is measured by \( Ca \), without having recourse to any other observation than that of the star's meridional passage. Denoting a sidereal day by \( T \), the interval of time which elapses while the star describes the arc \( ns \) by \( t \), and the angle at the pole by \( P \), we shall have the analogy \( T : t :: 360 : P \); therefore \( P = \frac{370^\circ}{T} \).

A formula which gives the angle at \( P \), or its measure \( Ca \). The angles \( CPs, CPs', \) and \( sPs' \) are called Hour Angles, because the arcs which they intercept on the star's parallel of declination correspond to the hours and fractions of an hour into which the sidereal day is divided. For example, if the hour angle \( sPs' \) contain 15°, the two planes passing through \( s \) and \( s' \) will intercept on \( mn \), the diurnal circle of the star, an arc equal to 15°; consequently there will be 24 such arcs in the whole circumference, each of which will be described in the 24th part of a sidereal day, or in one hour of sidereal time. In consequence of this perfect uniformity of the diurnal motion, the arc which a star describes on its parallel is conveniently measured by the time of its description; and the sidereal day, or the interval between two successive returns of the same star to the plane of the meridian, offers the most perfect unit of time, inasmuch as it is exactly the same to all the inhabitants of the earth, and remains absolutely unalterable in all ages, being one of the very few invariable elements of the system of the world.

In fixing the position of a point on a plane surface, Method of geometeters usually refer it to two straight lines at right angles to each other, the positions of which are supposed to be known. The same practice is imitated by points astronomers, who refer the position of any point on the surface of the sphere to two great circles, which are employed as a system of rectangular co-ordinates, and from which all distances are measured. The equator being a circle of the sphere not subject to any arbitrary condition, but determined by the very nature of the spherical revolution, is obviously well adapted for this purpose. It is therefore universally employed as a term of comparison in assigning the places of the celestial bodies. Suppose an hour circle passing through the star \( s \) to intersect the equator at \( a \); the arc \( sa \), which measures on the sphere the distance of \( s \) from the equator, is called the Declination of the star. All other stars situated on the same parallel \( mo \) have the same declination; and it is invariable for all places of the earth, because it is not affected by the diurnal motion which is performed in the parallel \( mo \). The declination is northern or southern, according as the star is situated in the northern or southern hemisphere. It is expressed in degrees from 0 to 90, proceed- Theoretical Astronomy. Ps is the complement of the declination, because Ps + δ = 90°. In order to distinguish s from all other stars which may be situated on the same parallel, we must also know the point a of the equator to which it corresponds; and for this purpose it is necessary to select a point ρ of the equator for the origin of the co-ordinates. This point ought to be independent of the diurnal motion, otherwise its position could only be determined for a single instant of time; and it must be such that it can be readily found, in whatever part of the earth the observer may be situated. Astronomers have agreed to employ for this purpose the point determined by the intersection of the equator with another very remarkable circle of the sphere, of which we shall speak hereafter, and to count the degrees on the equator, setting out from ρ, from 0 to 360, in the direction opposite to that of the diurnal motion, or from west to east. The arc ρa, which measures the distance between ρ and a, is called the Right Ascension of the star s; it is also the right ascension of any other star situated in the hour circle PsP. It is measured by the interval of time which elapses between the transit of the star s over the meridian, and that of the point ρ, which has been chosen as the point of departure. It may be remarked that the point ρ is also the origin of sidereal time; that is to say, the time is counted 0 hour 0 min. 0 sec. when that point passes the meridian.

From these definitions it is easy to conceive the general method of forming a catalogue of stars. A clock being regulated to sidereal time, and marking 0 hour at the instant of the transit of the point ρ (or any star chosen arbitrarily), the hours, minutes, and seconds, are noted at which the stars successively pass the axis of the transit instrument, together with their respective altitudes at the same instant. These observations give the right ascensions and declinations; for the time is easily converted into degrees, at the rate of 15° an hour, or by the formula given above for determining the hour angle. If the star s pass the meridian one hour after s', the arc ab, which measures the distance between their horary circles, PsP', P'P, is 1 hour or 15°; and the declination of a star s which moves in the parallel Hl (fig. 2) is Hl = Hl - Hl, that is, equal to the difference between its meridional altitude and the complement of the altitude of the pole H.P.

Modern celestial charts are constructed on the principle of assigning to each star the place indicated by the values of these two co-ordinates; and as observation proves that the mutual distances and relative positions of the stars scarcely undergo any sensible variation, a globe or chart once constructed will serve to represent the state of the heavens for at least a long series of ages.

Having now considered the general phenomena of the diurnal revolution, and explained the terms that are technically employed in assigning the positions of the celestial bodies, we may next proceed to inquire how these phenomena are to be accounted for,—whether the stars are really in motion, as they seem to be, or if the apparent motion is only an illusion occasioned by the revolution of our own earth. The perfect uniformity of the motions of the different stars renders it exceedingly improbable that they are disconnected; hence the simplest view of the phenomena is obtained by substituting for each star its projection on the celestial sphere, at an infinite distance, and supposing this sphere, with the projections of all the stars, to revolve in 24 hours from east to west round an immovable axis. But it is easy to see that the phenomena will be equally well explained by supposing that the starry firmament is absolutely at rest, and that the earth revolves in the same time, round the same axis, but in an opposite direction, from west to east. In both cases the stars remain immovable, the phenomena are exactly the same, and, relatively to Spherical Astronomy, in which we only concern ourselves with the apparent motions, it is absolutely indifferent which of the two hypotheses we adopt. They are both only modes of explaining certain appearances; and the one may be employed which renders the explanation most simple and perspicuous. The proofs of the earth's motion will go on accumulating as we proceed. At present we need only remark, that as the organ of sight makes us acquainted with the existence of relative, and not of absolute motion, it is impossible to decide, merely from appearances, whether the motion we perceive is real or otherwise; for whatever motion one body may have in respect of another, it is always possible to explain the phenomena by supposing it to be perfectly at rest, and the other to move in an opposite direction. The conclusions which we draw from the optical effects of motion afford no mathematical certainty with regard to the cause of that motion. The hypothesis of the revolution of the sphere is attended with innumerable and insurmountable difficulties. The distance of the nearest fixed star from the earth is not less than 350,000 times the distance of the sun; a distance which light, prodigious as its velocity is, would not traverse in less than five years. This immense line, therefore, supposing the heavens to revolve round the earth, would form the radius of a circle, the circumference of which, six times larger, would be passed over by the star in the space of 24 hours; its velocity must therefore be $6 \times 5 \times 365 = 10950$ times greater than that of light. This velocity, which is nearly equivalent to 2100 millions of miles in a second of time, is so enormous that it baffles the utmost efforts of the imagination to form any conception of it; and the supposition of its existence will appear still more revolting when we reflect that the distance of the nearest star is probably many thousand times less than that of the Milky Way. If we ascribe the motion to the earth, its velocity, though it may still be supposed great, is moderate in comparison of many well-known phenomena of nature. A point on the equator will describe in 24 hours a circle of about 25,000 miles, or about 17 miles in a minute; a velocity somewhat exceeding that of sound, but 7400 millions of times less rapid than the preceding.

In all that has hitherto been said respecting the apparent motion of the starry sphere, we have only had regard to the innumerable multitude of stars which constantly maintain the same relative situations, and which are in consequence called Fixed Stars. There are, however, several other bodies, some of them remarkable on account of their splendour, which, besides participating in the general motion, have peculiar motions of their own, and are incessantly shifting their positions among the fixed stars. An attentive observation of the state of the sky during a few successive evenings, will suffice to show that there are some stars which have in the mean time changed their places; and, on continuing to observe them, they will be found to separate themselves from particular constellations, and gradually but imperceptibly to approach others, till they at length appear, after unequal intervals of time, in an opposite quarter of the heavens. From this circumstance they were designated by the Greeks Planets, that is, wandering stars, in contradistinction to those which obey only the law of the diurnal motion. Besides the sun and moon, there are five discernible by the naked eye, and which have consequently been known from the remotest ages. These are Mercury, Venus, Mars, Jupiter, and Saturn. Twenty-eight others, including Neptune, Uranus, Theoretical and the small planets, have been discovered by the aid of Astronomy the telescope. Astronomers, with a view to abbreviate their descriptions, have appropriated a certain symbol to each of these planets, as well as to the sun and moon; thus, for the older planets, in the order of distance from the sun, the names and characteristics are as follows:

- The Sun: ☉ - Mercury: ♃ - Venus: ♄ - Mars: ♅ - Jupiter: ♆ - Saturn: ♇ - Uranus: ♈ - Ceres: ♉ - Pallas: ♊ - Juno: ♋

The Earth, which, as we shall afterwards see, is also a planet, and takes its place between Venus and Mars, has for its symbol ⊕, and the moon ♌. Venus, Jupiter, and sometimes Mars, are distinguished by their extraordinary brilliancy. Mercury, on account of its proximity to the sun, is rarely visible to the naked eye. Uranus, discovered by Sir W. Herschel in 1778, can with difficulty, by reason of his great distance, be perceived without a telescope. Ceres, and the other planets discovered since 1801, are extremely small in size, and can only be seen with the aid of the telescope.

Of all the celestial bodies, the most interesting to us are the sun and moon; and their peculiar motions have accordingly, in every age of astronomy, been studied with the greatest attention. The proper motion of the moon is particularly remarkable. In the course of a single night she separates herself very sensibly from the stars in her vicinity, moving over a space nearly equal to her own breadth in an hour, and completing a whole circuit in about 27 days. The sun moves with much less velocity, but his motion is still sufficiently apparent. If we take notice of the stars which immediately follow him when he sinks under the horizon, we shall find that in the course of a few nights they will be no longer visible. Others which, some time previously, did not set till long after him, have taken their places and now accompany the sun. In the morning similar appearances present themselves, but in a contrary order. The stars which appear in the eastern horizon at sun-rise, are, after a few days, considerably elevated above it at the same time. Thus the sun seems to fall behind the stars, by insensible degrees, till at last he appears in the east when they are about to set in the west. To account for these appearances, the ancients supposed the sun's diurnal motion to be really slower than that of the stars; hence they supposed him to be attached to a different sphere. For like reasons they ascribed particular spheres to the moon and each of the planets; and as no trace of these imaginary spheres is perceptible in the heavens, they next supposed them to be crystalline and transparent. The appearances are explained equally well, and with infinitely greater simplicity, by ascribing to the sun a proper motion, in a direction opposite to that of the diurnal rotation of the sphere, in consequence of which he advances to meet the stars, instead of falling behind them.

The greater part of the observations of the early astronomers had for their object the determination of the positions of the stars relatively to the sun at his rising and setting, by which they fixed the seasons, and regulated the operations of agriculture. They distinguished all these phenomena by technical terms, which occur very frequently in the works of the ancient poets, particularly in Hesiod and in Ovid's Fasti. A star which rises at the same time with the sun is effaced by his light, and is said to rise cosmically (ortus cosmicus). Soon after, when the sun by his proper motion has advanced so far to the east that the star can be perceived on the eastern horizon in the morning twilight, it disengages itself from the sun's rays, and is said to rise heliacally (ortus heliacus). At the end of six months, the sun being diametrically opposite to the same star, it sets as the sun rises, and in this case it is said to set cosmically (occultus cosmicus); at nearly the same time it rises when the sun sets, and is said to rise aeronomically (ortus aeronomicus). The sun afterwards begins to approach the star, till he advances so near that it is again about to be effaced by his light; it is now said to set heliacally, or just so long after the sun as to be visible when he has disappeared (occultus heliacus). At the end of a year, the star again rises and sets at the same moment with the sun; it is now said to set aeronomically (occultus aeronomicus). These distinctions, and the ancients had several others of the same kind, which are all defined by Ptolemy, are now scarcely ever mentioned. They have lost the whole of their interest since, in the progress of astronomy, more certain methods have been discovered of determining the commencement of the year and the seasons.

The proper motions of the planets are in general, like Stations that of the sun, in a direction opposite to the diurnal motion, or from west to east, among the stars; but they do not preserve the same character of uniformity; the planet sometimes becomes stationary among the fixed stars, and even advances from east to west in the direction of the diurnal motion. In this case it is said to retrograde. This retrograde motion is, however, not of long continuance. After having been accelerated during a short time, it begins to relax. The planet again becomes stationary, and then resumes its direct motion from west to east. These phenomena were observed in the remotest antiquity, and their explanation formed the principal part of the rational astronomy of the Greeks and the Arabians.

Besides the planets, other bodies occasionally make Comets. Their appearance in the heavens, which, by reason of the extraordinary phenomena they exhibit, have been frequently contemplated with terror and dismay, and regarded by superstitious ignorance as harbingers of calamity, and precursors of the divine vengeance. These bodies shoot down from the remote regions of space, with inconceivable velocity, towards the sun. At their first appearance they are small; their light is feeble and dusky; and they are generally accompanied by a sort of nebulosity or luminous tail, from which they have derived their appellation of Comets (coma, hair). As they approach the sun, their apparent magnitudes and brilliancy greatly increase, and the nebulosity sometimes occupies a large portion of the heavens, presenting a magnificent and astonishing spectacle. Having attained the point of their orbits nearest the sun, they again recede to enormous distances, and vanish by insensible degrees. They differ from the planets not only by the appearances they present, but also by the diversity of their motions; for, instead of being confined to a particular zone, and moving from west to east, they traverse the sky indifferently in all directions. They are visible only in a small part of their orbit, which, being near the sun, is passed over with prodigious rapidity. They seldom continue visible longer than six months. Their number is entirely unknown, but during the last two centuries upwards of 170 have been observed, and their orbits computed.

Thus we recognise three distinct classes of celestial bodies: the planets, the comets, and the fixed stars. It is the business of the astronomer to determine the positions of these bodies in the heavens; to observe their motions, and measure them with precision; to discover the laws by which their courses are regulated; and from these laws to assign the past or future state of the heavens. Theoretical at any given instant of time. The fixed stars form infinitely the most numerous class; but their motions being sensibly uniform, and their mutual distances invariable, the principal object with regard to them is to determine their places and relative positions. The comets in general return only after long periods; and observations are not yet sufficiently numerous to allow any certain deductions to be made with respect to their nature and constitution. The planets present a great variety of curious phenomena; they are always within our view; observations on them may be multiplied indefinitely; and for this reason, as well as on account of their proximity, and their forming a system of which the earth constitutes a part, they present by far the most interesting objects of astronomical study.

Sect. II.—Of the Globular Form of the Earth, Parallax, and Refraction.

Having now considered the general phenomena of the diurnal motion, we proceed next to inquire into the form of the earth, and our situation with respect to the centre round which the celestial sphere appears to revolve. The plane of the horizon seems to be stretched out indefinitely till it actually meets the sky; but this illusion is quickly dissipated by transferring ourselves from one place to another on the surface of the earth, and attending to the phenomena which such a change of place gives rise to. Let an observer, for example, set out from any given point in the northern hemisphere, and proceed directly south. In proportion as he advances, the stars in the southern region of the heavens will be elevated more above the horizon, and describe larger segments of their diurnal circles, while new ones come into view which were invisible at the station he left. On the other hand, the polar star, with those in its vicinity, will be depressed, and some stars which before continued above the horizon during the whole time of their revolution will now rise and set. The planes of the diurnal circles become also more perpendicular to the horizon, so that the aspect of the heavens is entirely changed. If, instead of advancing in the direction of the meridian, the spectator proceeded towards the east or west, he would in this case also remark that his horizon constantly shifted its position. A star will pass his meridian sooner as he advances eastward, or later as he travels westward; in short, by a change of place in any direction whatever, the perpendicular to the horizon, or the plumb-line, will correspond to a different point of the heavens. The plane of the horizon is therefore variable, and its variation by insensible degrees indicates with the greatest evidence the rotundity of the earth. Experience also shows that a spectator sees more of the terrestrial surface in proportion as he is elevated above it; and that on a mountain surrounded by the sea, or standing in the middle of a level plain, the horizon appears equally depressed all round, which is the distinctive feature of a spherical curvature. In fig. 5 let O be the place of a spectator on the summit of a mountain, the straight line HH will represent his horizon, and OH, OH' the directions of visual rays to the remotest visible points on the surface of the earth. The inclinations of these lines to the plane of the horizon, or the angles HOH, HO'H' are called the apparent depression; and these being always observed to be equal to each other, it follows that the curve HAM is a circle. On the Peak of Teneriffe, Humboldt observed the appearance of the sea to be depressed on all sides in an angle of nearly 2°. The sun arose to him 12 minutes sooner than to an inhabitant of the plain; and from the plain, the top of the mountain appeared enlightened 12 minutes before the rising or after the setting of the sun. The same phenomena, though on a smaller scale, are observed with regard to every mountain or elevation on the earth. Another familiar illustration of the globular figure of the earth is derived from the successive and gradual disappearance of a ship which leaves the shore and stands out to sea. The hull disappears before the sails and the rigging, and the top of the mast is the last part that is visible. The ship thus appears gradually to sink under the horizon, exactly in the same manner as she must necessarily do on the supposition that the surface of the sea is spherical. The appearance of the moon at the time she is eclipsed is also demonstrative of the roundness of the earth. When the moon penetrates the shadow of the earth, the line which separates the illuminated from the eclipsed portion of the disk is circular; evidently proving the conical form of the shadow, and consequently the roundness of the body by which the shadow is projected. From all these considerations, it is inferred with the highest certainty that the earth with its waters forms a round mass, isolated in space.

The globular form of the earth is a property common to it with the other bodies which compose the planetary system. The sun and moon are evidently round bodies; and when the planets are examined through a telescope their disks appear sensibly circular; and as they are known to have a motion of rotation, in consequence of which they successively present different points of their surfaces to the earth, the uniform roundness of their disks may be taken as a conclusive proof of their sphericity. We shall by and by have occasion to remark other striking analogies between the planets and the earth.

Since the figure of the earth is spherical, the horizontal plane cannot coincide with any considerable portion and radius of its surface. It may be defined to be the plane which horizontally touches the earth at any given point, and is perpendicular to the vertical line, or the direction of gravity at that point. The Sensible Horizon of any place is the plane which passes through the eye of the spectator, perpendicular to the plumb-line at that place. The Rational Horizon is a plane parallel to this, passing through the centre of the earth. The particular phenomena of different places depend on the position of their horizon with respect to the planes of the apparent diurnal motion of the sun and stars. The rational horizon of a place on the equator passes through the poles, and divides equally the equator and its parallels. Hence the days and nights are always equal in such places, and each of the stars performs one half of its revolution above, and the other below its horizon. The circles of diurnal motion are all perpendicular to the horizon, and therefore the inhabitants are said to be under a Right Sphere. If a spectator could place himself directly under the pole, his horizon would coincide with the equator, and the whole of the northern celestial hemisphere would be within his view, while no part of the southern hemisphere would be visible to him, on account of its being always beneath the horizon. The circles of the diurnal motion being parallel to the equator, and consequently also to the horizon, the fixed stars would never either rise or set. A spectator thus situated is said to be under a Parallel Sphere. In intermediate places the circles of the diurnal motion are oblique to the horizon, one pole being always elevated above it, and the other equally depressed below it. The stars whose distances from the elevated pole are not greater than the arc of its elevation above the horizon never set, while those within the same distance of the depressed pole never rise. These phenomena belong to all places situated between the equator and the poles, and the inhabitants of such places are said to be under an Oblique Sphere. The situation of a place on the surface of the earth is determined by two co-ordinate circles, in the same manner as that of a star on the celestial sphere. Let Z (fig. 2) be a place of which the position is required to be assigned with reference to circles of the terrestrial sphere, analogous to those which we have described as belonging to the sphere of the heavens. Since the axis of the celestial sphere passes through the centre of the earth, we may suppose the poles of the world, the equatorial circle, and the meridians, to be transferred to the earth's surface, so that the same figure may represent either the celestial or terrestrial sphere. The intersection of the plane PZB with the surface of the earth is the terrestrial meridian of the place Z, and the straight lines CZ, CEB, in that plane, intercept an arc of the terrestrial meridian between Z and the equator, containing as many degrees as the arc of the celestial sphere which measures the declination of the zenith of the place Z.

The angle EBCZ which is measured by the meridional arc EBZ is called the Geographical Latitude of Z. But in order to particularize the point at which the equator is intersected by the meridian, it is necessary to assume some point of the equator to which all the other points of that circle may be referred. The meridian passing through the assumed point is called the First Meridian; and the angular distance between the planes of the first and any other meridian measured by the equatorial arc intercepted by these planes on the equator, is called the Geographical Longitude of the place through which the last meridian passes. The longitude is said to be east or west, according as the degrees of the equator are counted from the first meridian towards the east or west. It is usual to reckon the degrees towards the east, all round the globe, or from 0° to 360°. From these definitions it is evident that the geographical latitudes and longitudes are exactly analogous to the declinations and right ascensions on the celestial sphere. The first point of the terrestrial equator cannot be determined by any star or fixed point in the heavens, on account of the diurnal motion; it is therefore necessary to fix its position by means of known places on the earth; and geographers are in the habit of assuming as the first meridian that which passes through the capital city or principal observatory of their country. The choice is of no importance; for in geography, as in astronomy, what is essential to be known is only the difference of longitudes or meridians, in order to reduce the situation of a place, or observations which have been made at it, to any indicated meridian.

The length of the terrestrial radius is a very important element in astronomy, inasmuch as it furnishes the observer with the only scale by which he can estimate the distances of the sun, moon, and planets. On the supposition that the figure of the earth is perfectly spherical, the general principles on which its magnitude may be determined are sufficiently obvious; but the accurate determination of its actual dimensions is attended with great practical difficulties, which can only be overcome by the perfect instruments and refined science of modern times. Eratosthenes seems to have been the first who made use of astronomical methods to determine the circumference of the earth, or the length of the meridian. He remarked that, at Syene, in the Thebais, the sun on the meridian, at the time of the summer solstice, was vertical; and that at Alexandria, at the same time, his zenith distance was $7^\circ 12'$. Now let S (fig. 6) be the sun, vertical to M, and Z the zenith of Alexandria, C being the centre of the earth, and O Alexandria. The angle ZOS is the sun's zenith distance, which was observed to be $7^\circ 12'$. But ZOS = ZCS + OSC; whence, as the angle OSC, which is extremely small on account of the sun's great distance, may be neglected in the calculation, the angle ZCS at the centre is also equal to $7^\circ 12'$, and therefore the whole circumference equals $\frac{360^\circ}{7^\circ 12'} \times M.O.$ To obtain the length of the meridian, it is therefore only necessary to measure the arc M.O. Eratosthenes assumed the distance between Syene and Alexandria to be 5000 stadia; hence the circumference of the earth = $\frac{360^\circ}{7^\circ 12'} \times 5000$ stadia = 250,000 stadia. The uncertainty which exists respecting the length of the Egyptian stade prevents us from deriving any precise information from this rude attempt to estimate the dimensions of the globe.

The method which has just been described takes for granted that the meridian is exactly circular—an assumption which, even supposing the method perfect in all other respects, would lead to erroneous results, especially in an arc of so great a magnitude as seven degrees. In a small arc, of $1^\circ$ for example, the error arising from the non-sphericity of the earth will be insensible; for it is certain, from the phenomena before explained, that the deviation from the spherical figure is not very considerable; and besides, whatever the nature of the meridional curve may be, it will sensibly coincide with its osculating circle, throughout the extent of an arc of $1^\circ$. It has been found by numerous and accurate experiments, that the lengths of arcs of $1^\circ$ on the same meridian are longer in proportion as we advance nearer the pole. Hence, on account of the similarity of the isosceles triangles of which these arcs form the bases, their sides, or the terrestrial radii, must also be longer, and consequently the convexity of the earth is less towards the pole than at the equator. The surface of the earth is extremely irregular, even independently of the inequalities occasioned by mountains and cavities; yet it has been discovered that the meridional curves differ almost insensibly from ellipses; whence it is concluded that the figure of the earth is an ellipsoid of revolution about its shortest axis. In comparing the results of the various measurements which have been made with the formulae belonging to the dimensions of such a body, this conclusion has been fully verified; and the lengths of the arcs, the ellipticity, the distance of the pole from the equator, and, in short, all the elements of the spheroid, have been determined. The results of theory and observation give an ellipticity amounting very nearly to $\frac{1}{305}$; that is to say, the equatorial is to the polar diameter in the ratio of 306 to 305. The following may be regarded as a very near approximation to the dimensions of the earth in English miles:

- Semidiameter of the equator: 3963.7 - Semidiameter of the pole: 3949.8 - Semidiameter at the latitude of $45^\circ$: 3956.2 - Ellipticity: $\frac{1}{305}$ - Length of $1^\circ$ of the meridian: 69.06 - Quarter of the meridian of Paris: 6214.47

The figure of the earth is one of the most difficult and most important questions of astronomy; and as our limits will not permit us to treat it in this place with all the details which its importance renders necessary, we shall reserve, for a separate article, an account of the different geodetical operations which have been undertaken with a view to determine it, together with the development of the mathematical theory of its equilibrium. See Figure of the Earth.

In describing the phenomena of the diurnal revolution, we have supposed the eye of the spectator to be placed that is to say, the parallax of an object is proportional to the sine of its zenith distance. At the zenith, \( Z = 0 \), therefore \( \sin Z = 90^\circ \), and the parallax vanishes; at the horizon, \( Z = 90^\circ \), and \( \sin Z = 1 \); the parallax, therefore, becomes equal to \( \frac{a}{r} \). In this case it is at its maximum, and is denominated the Horizontal Parallax; in all other cases it is called the Parallax of Altitude. Denoting the horizontal parallax by \( P \), we have \( P = \frac{a}{r} \), whence

\[ p = P \sin Z; \]

that is to say, the parallax of altitude is equal to the horizontal parallax, multiplied by the sine of the apparent zenith distance. It is evident that the apparent altitude SOH is always less than the true altitude SBH, by the whole amount of the parallax; the effect of the parallax is therefore to depress the object, or increase its zenith distance; hence, if \( \theta \) be the apparent altitude of the star, and \( \phi \) its true altitude, \( \phi \) will be found from the equation \( \phi = (\theta + P) \sin Z \).

From the above results it is manifest, that if the horizontal parallax can be by any means determined, the parallax at any other altitude will be found at the same time. The determination of the horizontal parallax is, however, attended with considerable difficulty, and various methods have been proposed and practised to ascertain its exact amount, modified by particular circumstances in the cases of the different celestial bodies. The method, however, which serves as the basis of all the others is extremely simple, and exactly analogous to that by which the distance of a remote object is determined on the surface of the earth. Suppose two observers to be stationed at the points O and O' (fig. 6), of which the latitudes are known, and which are both situated on the same meridian, and let them simultaneously observe the zenith distances of the star S; these observations will give the angles ZOS, Z'O'S, whereas their supplements SOC and S'O'C become known at the same time. A third angle of the quadrilateral figure SOCO', namely OCO', is also known, being measured by the meridional arc OMO'; the difference or sum of the latitudes of the two observers, according as they are on the same or opposite sides of the equator. The two sides also, CO and CO', being semidiameters of the earth, are supposed to be known; every part of the quadrilateral figure is therefore determined, and its diagonal CS may be calculated by the rules of plane trigonometry. But when CS is determined, the horizontal parallax is obtained immediately from the formula \( P = \frac{a}{r} \), that is, \( P = \frac{CO}{CS} \). On this principle the horizontal parallax of the moon was determined by Lacaille and Lalande, the former observing at the Cape of Good Hope, and the latter at Berlin; and the parallax of Mars by Lacaille at the Cape, and Wargentin at Stockholm.

Let us suppose two lines to be drawn from the centre of a planet touching the surface of the earth in points diametrically opposite; the inclination of these two straight lines is the double of the horizontal parallax; but the same angle also measures the diameter of the earth as seen from the planet; hence the horizontal parallax of a planet is equal to the apparent semidiameter of the earth at the distance of the planet. From this consideration the true diameter, and consequently the volume, of a planet may be found by measuring its apparent diameter, that is, the optical angle under which its diameter appears when seen from the earth. Let \( d \) be the apparent, and \( D \) the true diameter of a planet; the angle \( \frac{d}{D} \) is comprised between two straight lines drawn from the Theoretical earth, the first of which is directed to the centre of the planet, and the last is a tangent to it; hence \( \frac{1}{2} D = r \tan \frac{1}{2} d \) or \( \sin \frac{1}{2} d \), according as \( r \) is the line which touches the planet or that which is directed to its centre. Now, on account of the smallness of the angle \( \frac{1}{2} d \), the sine and tangent are sensibly equal to the arc; we may therefore suppose in all cases \( D = r d \). Combining this with the equation \( P = \frac{a}{r} \), we deduce \( \frac{D}{a} = \frac{d}{P} \); which gives the ratio between the true diameters of the planet and the earth in terms of the apparent magnitude and horizontal parallax of the planet. It follows also from this last equation that \( \frac{d}{P} \), or the ratio of the apparent diameter and parallax of a planet, is constant; the horizontal parallax may therefore be found at any time whatever by measuring the planet's diameter.

From the equation \( P = \frac{a}{r} \), we have \( r = \frac{a}{P} \); that is, the distance of a planet from the earth is known in terms of its parallax and the earth's semidiameter. The parallaxes, therefore, give the ratio of the distances of all the planets from the earth, and consequently of their distances from one another, and from the sun; hence the radius of the earth furnishes the scale by which the astronomer measures the dimensions of the whole solar system, and the magnitudes or volumes of all the bodies of which it is composed. On this account, the accurate determination of the parallaxes of the celestial bodies is a problem of great importance in practical astronomy.

It is evident from the mere inspection of the figure, that the plane in which the straight lines \( CS \) and \( OS \) are situated is the vertical plane passing through \( S \), consequently the whole effect of the parallax is to diminish the altitude of a planet in its vertical circle. When the observation, therefore, is made in the meridian, the effect of the parallax is to alter the declination, without producing any change whatever in the right ascension of the planet. Out of the meridian it is necessary to have regard to the azimuth or hour angle, as well as to the altitude, in calculating the correction due to the parallax. When we suppose the earth to be spherical, the formulae for the calculation of the parallax are extremely simple, because the radius is constant, and the vertical line, or perpendicular to the surface of the earth, passes through its centre. But in the case of the true or elliptical figure of the earth neither of these circumstances takes place. The radius in this case is variable, and must be determined by a particular process of computation for every point on the meridian; and the vertical line, with reference to which the latitude of the place and the altitudes are determined, does not pass through the centre of the earth, but makes different angles with the axis. These circumstances render the calculation of the parallax a matter of much greater complication and difficulty.

The term parallax in its general signification properly denotes change of place. There are consequently various kinds of parallaxes, such as parallax of right ascension, of declination, longitude, latitude, &c. In what has preceded, we have supposed the earth to be at rest in the centre of the universe, and therefore have had regard solely to the variations produced in the apparent diurnal motions by the eccentric position of the observer on the surface of the earth, and which are comprehended generally under the denomination of the diurnal parallax.

The effects of the diurnal parallax are only sensible with regard to those bodies of which the distance from the earth is not so great as to be incomparable with the earth's semidiameter. That the apparent place of an object must be changed in consequence of a change in the situation of the observer, is a simple geometrical truth which no experiment was required to discover. The first observers must accordingly have anticipated a parallax in all the celestial bodies; and it was doubtless only after considerable experience that they admitted the existence of any exception to the general law. There is, however, another cause of variation in the apparent positions of the celestial bodies also connected with the earth, the existence of which could not be known a priori, but must have been discovered by experience alone. We allude to the refraction of the rays of light in passing through the earth's atmosphere.

The angular distance between two stars is found to undergo very sensible variations at different hours of the day. This phenomenon cannot be explained by any proper motion of the stars, because it evidently depends on their altitude above the horizon; and the differences are found to be the same daily at the same altitude. It is most striking when we compare a star which, without setting, passes the meridian twice a day, once near the zenith, and the second time near the horizon, with another star situated very near the pole, and of which the altitude is consequently nearly invariable. It will be found that at the time of the first transit the distance between the two stars is greater than at the second by nearly half a degree. It is evident, therefore, that the phenomenon consists in diminishing the distance between a star situated in the horizon and the visible pole, that is to say, in elevating the stars, whereas the effect of the parallax is to depress them. The refraction also takes place in an equal degree with regard to the fixed stars, and even the moon and planets, without being in any degree modified by the great differences in the distances of the celestial bodies, contrary in this respect likewise to the parallax, which depends entirely on the distance. The reason of this will be evident from the consideration of the physical cause of the phenomenon.

According to the known principles of optics, a ray of light, in passing obliquely from one transparent medium into another of a different density, does not hold on in its rectilinear course, but is refracted, or bent towards the denser medium. Now, the atmosphere which surrounds the earth may be regarded as composed of an infinity of concentric spherical strata, the densities of which are greater in proportion as they are nearer to the earth's surface. When a ray of light, therefore, proceeding from a star enters the atmosphere, it is inflected towards the earth, or bent so as to form a smaller angle with a perpendicular to the surface of the earth; and this inflection will be increased by every successive stratum of the atmosphere through which the light passes. In fig. 8, let Fig. A, AA', BB', CC', represent the boundaries of the successive strata, which, for the sake of illustration, we here suppose to have a finite thickness. A ray of light proceeding from \( S \) comes in contact with the highest stratum of the atmosphere \( AA' \) at \( a \). The molecular attraction of this atmospherical stratum, acting in the direction of a normal to \( A A' \) at \( a \), causes the luminous ray to deviate from the direction \( S x \), and assume another, \( a y \), in which it would continue to move if the atmosphere were equally dense from \( A A' \) to the earth. But in the course of its progress the ray penetrates another denser stratum at \( b \), and consequently suffers another inflection; so that instead of proceeding in the direction \( a y \), it is bent into a new direction \( b z \), more nearly perpendicular to the concentric strata. A similar effect is produced at \( c \), so that the luminous ray, when it finally reaches the observer at \( O \), has as- Theoretical sumed the direction c O. In its progress from a to O, it has therefore successively moved in the direction of the sides of the polygon a, b, c, O; and to the spectator at O, the star from which it proceeded, instead of appearing in its true place at S, will appear to be at S', or in the last direction of the visual ray. Now, if A A' is the most elevated stratum of the atmosphere into which the ray enters in the direction S a, it is clear that the whole effect is produced by the atmospherical strata situated below A A', and that the length of S a is perfectly indifferent; hence the refraction is entirely independent of the distance of the stars, provided they are beyond the limits of the earth's atmosphere.

The decrease of the density of the atmosphere, from the surface of the earth upwards, follows the law of continuity, or takes place by insensible degrees; so that the luminous ray, in traversing the atmosphere, enters at every instant into a denser medium, and is therefore continually brought nearer and nearer to the vertical direction. Hence the true path of the ray is curvilinear, and concave towards the earth, as represented in fig. 9. This is equivalent to the supposition that the thickness of the different concentric strata of uniform density is infinitely small, and that the light, as it successively penetrates each, deviates from its former path by an infinitely small angle, which may be considered as the differential of the refraction, the total amount of which will therefore be obtained by integration.

The direction of the ray, when it reaches the eye of the observer, is the tangent to the last portion of its curvilinear path; and the apparent zenith distance of the star will be ZOS', while the real zenith distance is ZOS. The difference of these two angles, namely SOS, is what is denominated the Astronomical Refraction. It is evident that the whole path of the ray is confined to the vertical plane, in which the star and the eye of the observer are situated; for the earth and its atmosphere being very nearly spherical, that plane will divide the strata symmetrically; there will therefore be no displacement in a lateral direction, or no refraction out of the vertical plane. When the observed star is due north or south, the vertical plane is the plane of the meridian; hence, in meridional observations, the whole of the refraction, like that of the parallax, takes place in declination, while the right ascension remains unaltered.

It is evident that the amount of the refraction is greater in proportion as the observed star is nearer to the horizon; for in this case the luminous rays strike the tangent planes of the atmospherical strata more obliquely, and have besides to traverse a greater extent of atmosphere before they arrive at the eye of the observer. On determining by experiment the refraction at every altitude from zero to 90°, tables of Refraction may be constructed, which will furnish the means of discovering the law of its diminution; but as such a process would be exceedingly tedious, and likewise subject to lead to erroneous results on account of the inevitable errors of observation, it is found more convenient to assume some hypothesis for a basis of calculation, and to verify the results which it leads to by comparing them with observation. In regard to media which may be said to be permanent, such, for instance, as water and glass, the determination of the refraction is not attended with great difficulty; but the circumstances are greatly altered when we come to make experiments on the atmosphere. In this case the difficulty arises from the incessant changes which the atmosphere is undergoing relatively to its refringent powers; changes which it is impossible for the observer fully to appreciate, inasmuch as he can only determine its physical state within a short distance of the earth, while that of the upper strata remains wholly unknown to him. The refringent power of the atmosphere is affected by its density and temperature. The effects of the humidity are insensible; for the most accurate experiments seem to prove that the watery vapours diminish the density of the air in the same ratio as their refractive power is greater. It is therefore only necessary, even in delicate experiments, to have regard to the state of the barometer and thermometer at the time the observation is made. At a medium density, and at the temperature of melting ice, it was found by Biot and Arago, from a great number of exact experiments, that at any altitude between 10° and the zenith the refraction is very nearly represented by the formula \( r = 60^\circ \tan (Z - 3^\circ 25 \times r) \), in which \( r \) is the refraction corresponding to a given zenith distance \( Z \).

With the exception of the numerical co-efficients, this formula was first given by Bradley; but whether it was deduced from theory by that great astronomer, or was only empirical, is uncertain. Bradley's formula was \( r = 57^\circ \tan (Z - 3^\circ \times r) \). When the direction of the luminous rays makes a smaller angle than 10° with the horizon, it becomes indispensable to take into account, in the calculation of the refraction, the law of the variation of the density of the atmosphere at different altitudes; a law which is subject to incessant variation, from the operation of winds, and other causes which agitate the atmosphere, as well as the decrease of temperature in the superior regions. For this reason all astronomical observations which have not refraction directly for their object, or which are by their nature independent of its influence, are made at an elevation exceeding 10°. For lower altitudes, it is to be feared that no theory will ever be found sufficiently exact to entitle the observations to much confidence.

The existence of the atmospherical refraction was not unknown to the ancient astronomers; but it is only in modern times that the subject has been studied with the requisite of refraction care to admit of its influence being calculated in the reduction of observations. Ptolemy does not allude to the subject in the Almagest, but he has given a sufficiently exact idea of it in his work on Optics. He mentions experiments made to ascertain how far a ray of light is bent from its rectilinear course in passing from air into glass and water; and observes that the astronomical refraction brings a star nearer to the zenith, and that it is feebler in proportion as the star is more elevated. He likewise indicates a method of measuring its effects, although he does not seem to have attempted to practise it. The same notions, but in a less precise form, were reproduced in the Optics of Albazen. Walther began to estimate the effects of refraction near the horizon; and Tycho, a century after, found the means of measuring them with greater accuracy, and was thereby enabled to construct a table. Tycho estimated the horizontal refraction at 34', and supposed it to vanish at the altitude of 45°; which proves that his ideas on the subject were less accurate than those of Ptolemy, who says expressly that it vanished only at the zenith. Dominic Cassini was the first who proposed an hypothesis for calculating the refraction at any altitude; and he computed a table, which was published in 1662. Since that time the subject of refraction has been investigated, theoretically and practically, with the most scrupulous and delicate attention, and tables constructed to exhibit its amount at every altitude, and at the different seasons of the year. (See Mécanique Céleste, tom. iv. p. 231; Mr Ivory's Paper in the Phil. Trans. for 1823, p. 409; De lambre, Astronomie du xviie Siècle, p. 774.)

The refringent power of the atmosphere gives rise to a number of curious phenomena. When the sun appears in Theoretical Astronomy.

The horizon, the rays of light which issue from the extremities of his vertical diameter are refracted unequally on account of the difference of altitude, so that the disk, at other times circular, then assumes an oval appearance; and on measuring the horizontal and vertical diameters by means of the micrometer, the former is sometimes found to exceed the latter by four or five minutes of a degree. This effect is particularly remarkable when the sun is observed at his rising or setting from the top of a mountain, or an elevation near the sea-shore, in which cases the difference of the two diameters sometimes, in particular states of the atmosphere, amounts to upwards of 6', or a fifth of the whole apparent diameter of the sun. In the case of the moon this oval appearance is seldom so perceptible; for unless she happens to be exactly full, or in opposition, at the same time that she is in the horizon—and this can rarely be the case—the defect of the illumination of her disk reduces her figure to those rounder but somewhat irregular shapes which it often exhibits. The horizontal refraction being equal to the apparent diameter of the sun or moon, it happens also that either of these luminaries, and consequently any other of the celestial bodies, may be entirely visible above the plane of the horizon, when in fact it has set below it. A curious instance of this was observed at Paris in 1750. The moon rose eclipsed, while the sun, whose place at the time of a lunar eclipse is diametrically opposite to that of the moon, still appeared above the horizon. Another phenomenon occasioned by the refraction of the atmosphere is the appearance of the moon when she is eclipsed, or immersed in the earth's shadow, and consequently receives no light directly from the sun. On these occasions her disk continues distinctly visible, and is of a dusky red colour, exhibiting the appearance of tarnished copper, or iron nearly red-hot. This phenomenon occasioned great perplexity to the ancient astronomers who were unacquainted with the refractive powers of the atmosphere; and some of them ascribed it to the native light of the moon, in virtue of which she continues visible when hid from the sun's rays. Plutarch ingeniously supposed it to be occasioned by the light of the stars reflected from the dark surface of the moon; but the quantity of this light is greatly too small to produce the effect in question. It is now explained by the scattered beams of light bent into the earth's shadow by refraction in their passage through the atmosphere.

Twilight.

The refraction of the rays of light in traversing the earth's atmosphere is the cause of Twilight, which sensibly lengthens the duration of the day, and prevents a sudden transition from light to darkness on the disappearance of the sun. When the sun is more than 33° below the horizon, the refraction is not powerful enough to bring his rays sufficiently near the earth to reach our eyes; they pass over our heads, and are irregularly reflected by the molecules of the atmosphere. By this means a portion of the celestial vault is enlightened, while the sun is invisible. This illumination of the upper regions is called the twilight. It commences as soon as objects can be distinguished before sun-rise, and terminates when they cease to be visible after the sun has set. The time, however, at which the twilight commences and terminates cannot be assigned with any degree of precision. It is generally supposed to be limited by the depression of the sun 18° below the horizon. Lacaille found the limit in the torrid zone to be between 16° and 17°. According to Lemonnier, it varies in France between 17° and 21°. The duration of the twilight will evidently be longer or shorter according as the inclination of the plane of the sun's orbit to the horizon is more or less oblique.

To assign the time at which it is a maximum or minimum, or of a given length at a given latitude, is a problem of pure geometry, which has been frequently solved since the time of the Bernoullis.

The scintillation or twinkling of the stars is another phenomenon produced by certain modifications of atomization of the spherical refraction. The atmosphere is at all times more stars or less agitated; and on this account the clusters of molecules of which it is composed are constantly undergoing momentary compressions and dilatations, which occasion minute differences of refraction, and consequent changes in the directions of the luminous rays. The minute and rapid variations thus occasioned in the apparent places of the stars produce the twinkling or tremulous appearances which, in certain states of the atmosphere, particularly on the approach of rain after a long drought, are very remarkable. The agitations of the atmosphere are manifest to the eye in the tremulous motion of the shadows cast from high towers, and in looking at objects through the smoke of a chimney, or over beds of hot sand. The scintillation of the planets is much less than that of the stars. This proceeds from the magnitude of their disks, which, although apparently very small, are still far greater than those of the fixed stars, and indeed so considerable, that the accidental variations in the directions of the rays of light are too minute to displace them entirely. The borders of their disks are affected only by a slight undulation; whereas the stars, which are merely brilliant points of insensible magnitude, undergo a total displacement.

The apparent enlargement of the sun and moon in the Horizon-horizon is an optical illusion, connected in some measure with the atmosphere, of which various explanations have been given since the time of Ptolemy. According to the ordinary laws of vision, the celestial bodies, particularly the moon, which is nearest to the earth, ought to appear largest in the meridian, because their distance is then less than when they are near the horizon; and yet daily experience proves that the contrary takes place. To an observer placed at E (fig. 10), the visual angle subtended by Fig. 10, the moon in the horizon at M is somewhat less than that under which she appears in the zenith at O; and this fact, a consequence indeed of her circular motion, is proved by accurate measurements of her diameters in those circumstances by the micrometer. The mean apparent diameter of the moon, at her greatest height, is 31' in round numbers, but in the horizon she seems to the eye two or three times larger. The commonly received explanation of this phenomenon was first given by Descartes, and after him by Dr Wallis, James Gregory, Malebranche, Huygens, and others, and may be stated as follows. The opinion which we form of the magnitude of a distant body does not depend exclusively on the visual angle under which it appears, but also on its distance; and we judge of the distance by a comparison with other bodies. When the moon is near the zenith there is no interposing object with which we can compare her, the matter of the atmosphere being scarcely visible. Deceived by the absence of intermediate objects, we suppose her to be very near. On the other hand, we are used to observe a large extent of land lying between us and objects near the horizon, at the extremity of which the sky begins to appear; we therefore suppose the sky, with all the objects which are visible in it, to be at a great distance. The illusion is also greatly aided by the comparative feebleness of the light of the moon in the horizon, which renders us in a manner sensible of the interposition of the atmosphere. Hence the moon, though seen under nearly the same angle, alternately appears very large and very small. Desaguliers illustrated the doctrine of the horizontal moon on the Theoretical supposition of our imagining the visible heavens to be Astronomy only a small portion of a spherical surface, as mno (fig. 10), in which case the moon, at different altitudes, will appear to be at different distances, and therefore seem to vary in magnitude, as at m, n, o. It is evident, however, that this affords no explanation of the phenomenon; for why does the sky, it may be asked, appear to be a smaller segment than a hemisphere? In the solution of this question the whole difficulty is contained.

It may be remarked that these illusions disappear as soon as the intermediate objects are concealed from view. They may be destroyed by regarding the moon through a tube, which permits her disk alone to be seen, and in this manner she will appear no larger at the horizon than near the zenith. The same effect may be produced by viewing her through a smoked glass, because the dark tint permits only the luminous object to be seen, and conceals all the rest. The only precaution necessary to be observed in making this experiment, is to place the eye in such a position that the surrounding bodies may not be visible.

CHAP. II.

OF THE SUN.

SECT. I.—Of the Apparent Circular Motion of the Sun in the Ecliptic, and Position of the Ecliptic in Space.

The consideration of the diurnal motion common to all the celestial bodies, and the succession of day and night resulting from it, forms the first object of astronomy. The second is to pass from the diurnal motion of the sphere to the proper motion of the sun, and from the vicissitudes of the day to the seasons of the year. The sun, as has already been remarked, is constantly shifting his place among the stars. If we observe the altitude of any star, or group of stars, above the eastern horizon at sun-set, we shall find, on making the same observation a few days afterwards, that its elevation is considerably increased, and that it has approached nearer to the meridian. At the end of three months it will appear at sun-set on the meridian, and from that time continues to advance nearer and nearer to the sun, till it is at last concealed by the splendour of his rays. After remaining for some time invisible, it will again make its appearance in the morning to the westward of the sun, and its distance from him will continue to increase daily, till, at the end of a year, it has made a complete circuit of the sky, and regained the position it occupied at the time of the first observation.

The earliest observers explained this phenomenon by supposing the diurnal motion of the sun to be less rapid than that of the stars, in consequence of which he constantly falls behind them. This supposition would be admissible if the sun remained constantly in the plane of the equator; but as he deviates considerably from that plane, on both sides of it, it is infinitely more simple to ascribe to the sun a motion of his own, independent of the diurnal motion, and performed in a contrary direction, in virtue of which he traces an oblique route among the stars.

The oblique path of the sun may be determined by observing the positions of the stars near which he successively passes in the course of the year. The same object is however accomplished with much greater precision by the modern practice of observing his declinations and right ascensions, that is to say, his distances from the equator, and an arbitrary meridian. In this manner his place may be accurately assigned every day; and, by a repetition of similar observations, a series of points will be obtained on the surface of the celestial sphere, which mark out his annual course. The result of constant experience shows, that the declination reaches its maximum on the south side of the equator about the 22d of December, when it amounts to 23°46′ degrees. From this time it gradually diminishes till about the 21st of March, when the sun reaches the plane of the equator. At this time the days and nights are of equal length all over the earth, and the instant of time at which the sun's centre is in the equatorial plane is called the instant of the equinox. The sun then appears on the opposite side of the equator, and his declination or meridional altitude continues to increase till about the 22d of June, when he becomes stationary, and then again shapes his course towards the equator. His maximum declination on the north side of the equator is exactly equal to that on the south, amounting to 23°46′. The sun now continues to approach the equator till about the 24th of September, when he again reaches that plane, and a second equinox succeeds. Continuing still to move in the same direction, he declines from the equator southward, till he reaches his former limit about the 22d of December, after which he resumes his former course.

The two small circles of the sphere, parallel to the equator, which pass through the two points where the declination is great, are called the Solstices, or the Tropics; that on the northern hemisphere is called the Tropic of Cancer, and the other is called the Tropic of Capricorn. These two parallels, which mark the extreme limits of the sun's declination, are, as has just been stated, equally distant from the equator, with regard to which the variations of declination on either side are perfectly symmetrical and uniform.

If at the time of the vernal equinox we remark the stars which set in the true west while the sun is rising in the east, and which are then separated from him by a semicircumference, it will be found that the difference of their right ascensions, and the right ascensions of those which have precisely a similar situation relatively to the sun at the time of the autumnal equinox, is exactly 180°. It follows, therefore, that the solar orbit intersects the equator in two points diametrically opposite; but we have seen that its northern and southern declinations are equal; hence the orbit projected on the sphere must be a great circle, provided it lies wholly in the same plane. Whether this is the case or not it will be easy to prove by means of a few observations, in the following manner.

Let AQ (fig. 11) be the equator, AE the orbit of the sun, PSM, PTN two circles of declination, drawn through any two points S and T of the orbit. If the sun's path is confined to a plane, then AE must be a great circle, and we shall have the equation $\sin AM = \tan MS$ according to the well-known properties of spherical triangles. Let the cotangent of the unknown but constant angle MAS = n, the declinations MS = D, NT = D', the right ascensions AM = A, AN = A'; then, according to the above formula, we must have $\sin A = n \tan D$, and $\sin A' = n \tan D'$, at whatever points of the orbit S and T may be situated. From these two last equations there result also $\cos A = \sqrt{1 - n^2 \tan^2 D}$, $\cos A' = \sqrt{1 - n^2 \tan^2 D'}$, which, being substituted in the trigonometrical formula $\sin(A' - A) = \sin A' \cos A - \cos A' \sin A$, we shall have $\sin(A' - A) = n \tan D' \sqrt{1 - n^2 \tan^2 D} - n \tan D \sqrt{1 - n^2 \tan^2 D'}$. The observations of the meridional altitudes will give the declinations D and D'; and the difference of right ascensions, A' - A, will be found by comparing the time of the sun's culmination, or transit over Theoretical the meridian, with that of a star. If, therefore, it is found that, by assigning a certain constant value to \( n \), this equation will satisfy all the observations, combined by pairs, of the sun's right ascensions and declinations, it will follow that the plane determined by any two points in the sun's course and in which it intersects the equator, has always the same inclination to the equator; in other words, all the planes so determined are identical.

Now the observations of the sun's right ascensions and meridional altitudes, which have been made daily during so great a number of years, and under so many different meridians, are found to conform entirely with the preceding formulae; they therefore furnish so many proofs that the projection of the sun's orbit is a great circle of the celestial sphere, and that the orbit itself is wholly confined to the same plane.

The great circle which the sun describes in virtue of its proper motion is called the Ecliptic. It has received this name from the circumstance that the moon, during eclipses, is either in the same plane or very near it. These phenomena can in fact only happen when the sun, earth, and moon are nearly in the same straight line, and, consequently, when the moon is in the same plane with the earth and the sun. The angle formed by the planes of the ecliptic and equator, and which is measured by the arc of a circle of declination intercepted between the equator and tropic, is called the Obliquity of the Ecliptic.

The two points in which the equator and ecliptic intersect each other are called the Equinoctial Points; they are also denominated the Nodes of the Equator; and the straight line conceived to join them is the Line of the Equinoxes, or the Line of the Nodes. The node through which the sun passes on coming from the south to the north of the equator is called the Ascending Node, and is usually distinguished by the character \( \odot \); the opposite node is the Descending Node, and is marked by \( \ominus \). A straight line passing through the centre of the earth, perpendicular to the plane of the ecliptic, is called the Axis, and the points in which its prolongation meets the sphere are called the Poles of the Ecliptic; these denominations being analogous to those of the axis and poles of the equator. The two small circles of the sphere which pass through the poles of the ecliptic, and are parallel to the equator, are called the Polar Circles.

The ecliptic has been divided by astronomers, from time immemorial, into twelve equal parts, called Signs, each of which consequently contains 30 degrees. The names and symbols by which they are characterized are as follows:

| North of the Equator | South of the Equator | |----------------------|---------------------| | Aries | Libra | | Taurus | Scorpio | | Gemini | Sagittarius | | Cancer | Capricornus | | Leo | Aquarius | | Virgo | Pisces |

In each of these signs the ancients formed groups of stars, which they denominated asterisms, constellations, animals (Zodiacs), not confined to the ecliptic, but included within an imaginary belt, extending 8° on each side of it, to which they gave the name of Zodiac (Zōōdēs, zōōdēs, circle or zone of the animals). The term sign is now employed only to denote an arc of 30°, and will probably soon be banished entirely from the astronomical tables. It is already confined to the tables of the planets. Thus, to denote that the longitude of a planet is 27° 12', it is usual to write \( 9° 6° 12' \). Formerly it was usual to employ the characteristic symbol, and to write \( \varphi 6° 12' \), meaning that the planet was 12' in the 6th degree of Capricornus, or the tenth sign. This inconvenient practice is now laid aside, and the signs, when they are employed, are simply distinguished by the ordinal numbers.

As the greater part of the celestial phenomena connected with the planetary system take place either in the ecliptic or in planes not greatly inclined to it, it is found to be most convenient to refer the positions of the planets, and frequently those of the stars also, to that plane. The first point of Aries, which is the technical expression for the intersection of the ecliptic and equator, or the place of the sun at the vernal equinox, is assumed as the origin from which the degrees of the equator, as well as of the ecliptic, are counted from west to east, or in the direction of the sun's annual motion. The angular distance of the sun from this point is called his Longitude; and the longitude of a star is the arc intercepted on the ecliptic between the same point and a great circle passing through the star perpendicular to the ecliptic. The arc of this circle intercepted between the star and the ecliptic, or, which is the same thing, the complement of the star's distance from the pole of the ecliptic, is called the Latitude of the star; so that longitude and latitude are with regard to the ecliptic what right ascension and declination are with regard to the equator. Thus, let \( O\gamma Q \) (fig. 12) be the equator, \( L\gamma L \) the ecliptic, \( P \) and \( S \) the north poles of these two circles respectively, and \( S \) the place of a star. Having drawn through the pole of the ecliptic and the star the great circle \( ES M \) perpendicular to the ecliptic in \( M \), then \( \varphi M \) is the longitude of \( S \), counted on the ecliptic from the vernal point \( \varphi \) towards the east; and \( SM \), the distance of the star from the ecliptic, is its latitude; and for this reason all great circles passing through the poles of the ecliptic are called Circles of Latitude. The place of a star is thus determined by its longitude \( \varphi M \), and its latitude \( SM \), as well as by its right ascension \( \varphi R \), and its declination \( SR \); and when the angle \( L\gamma Q \) is known, it is easy to pass from the one system of co-ordinates to the other by means of the formulae of spherical trigonometry. These formulae are of constant use, for it is the declinations and right ascensions only which are directly observed.

The plane of the sun's orbit will be determined completely when its inclination to the equator and the position of the line of the nodes in space have been made known by observation. The declination of the sun twice a year, namely, at the summer and winter solstice, is equal to the obliquity of the ecliptic; whence, if the solstice happened exactly at mid-day, the obliquity would be given directly by an observation of his meridional altitude. This circumstance, however, can happen only for one terrestrial meridian; but as the declination of the sun when he approaches the tropics varies little from one day to another, his greatest observed declination will be a very near approximation to the obliquity, at whatever part of the earth the observation may have been made. It is easy, however, to correct the error which results from the observation being made on a meridian different from the solstitial colure. Taking an example from Woodhouse, let us suppose the sun's declination to be observed on three successive days (the 20th, 21st, and 22nd of June), and found to be on these days respectively,

\( 23° 27' 37'' \), \( 23° 27' 41'' \), \( 23° 27' 20'' \);

then it is obvious, that if the middle observation gave the greatest inclination exactly, the other two would differ from it equally, which they do not. The maximum declination is therefore a quantity somewhat different from \( 23° 27' 41'' \); and it is easy to conclude, from the inspection of the numbers, that it is nearer to \( 23° 27' 41'' \) than to either of the other two. It is obvious, therefore, that that observation must have been made within 12 hours of the Theoretical time when the sun was exactly in the solstitial point. In order to form an exact notion of the amount of error which may possibly arise from this circumstance, let S (fig. 13) be the place of the sun at the time of the observation, and X the true but as yet unknown solstitial point, and S s, X x meridional arcs intersecting the equator in s and x. The arc S X must be less than 30°, for in 12 hours the variation of the sun's longitude does not exceed an arc of that magnitude. Suppose it 30°; then, by Napier's rules,

\[ \sin S s = \sin \varphi \times \sin S \varphi, \]

and

\[ \sin X x = \sin \varphi \times \sin X \varphi; \]

whence, on eliminating \(\sin \varphi\), and observing that \(\sin X \varphi = \sin 90° = 1\), we shall have

\[ \sin X x = \frac{\sin S s}{\sin S \varphi} \cos S X. \]

By taking, according to the observation, \(S s = 23° 27' 41''\), and \(S X = 30°\), we shall find from the logarithmic tables \(X x = 23° 27' 44''\). It will be observed that 30° is the maximum error in longitude; if instead of 30° it had been supposed only 3°, the corresponding error in declination would have amounted only to 0° 0' 05''. In the example chosen, the error of longitude is about 20', whence the error of declination is 1° nearly, and consequently the resulting obliquity differs little from 23° 27' 42'' 5''. This result is, however, to be understood of the apparent obliquity, which is subject to slight variations, depending on the longitudes of the moon's nodes: the mean obliquity, deduced from the comparison of a great number of observations, both of the summer and winter solstice, may be regarded as amounting to 23° 27' 41'' at the commencement of the year 1830. We shall see, when we come to speak of the Nutation, in what the difference between the apparent and mean obliquity consists.

When the mean value of the obliquity of the ecliptic, as determined by the delicate instruments of the present day, is compared with that given by ancient observations, it appears to have undergone a progressive diminution, and is always greater as the observation is more remote. The ancient observers were not, indeed, possessed of the means of determining an element of this sort with great precision; but as all the observations recorded in history agree in making the obliquity greater in former times than it is now, the probability is almost infinite that the angle formed by the planes of the equator and ecliptic has really diminished; for, had the differences of the values assigned to it arisen solely from errors of observation, they would have been in excess and defect indifferently, instead of being, as they are, uniformly in excess. The various observations and traditions by which the progressive diminution of the obliquity is confirmed have been collected by Bailly; in the following table we have inserted those which appear to be the best authenticated, and have added the results of some recent observations, from which can be deduced the present value of the obliquity, and the rate of its diminution.

| Year | Name of Observer | Obliquity | |------|-----------------|-----------| | Before Christ | Eratosthenes, confirmed by Hipparchus and Ptolemy | 23° 51' 15'' | | | The Chinese | 23° 45' 52'' | | After Christ | Arabians at Bagdad | 23° 33' 52'' | | | Albategnius | 23° 35' 40'' | | | Almansor | 23° 33' 30'' | | | The Chinese | 23° 32' 12'' | | | Ulugh Beigh | 23° 31' 58'' | | | Walther | 23° 29' 47'' | | | Tycho | 23° 29' 52'' | | | Riccioli | 23° 30' 20'' | | | Hevelius | 23° 29' 10'' | | | Cassini | 23° 29' 00'' |

Although the comparison of these observations with one another gives very discordant results relatively to the law according to which the obliquity varies, their totality places the fact of its progressive diminution beyond all manner of doubt. Lacaille, who was followed by the greater number of astronomers, estimated the diminution of 44'' in a hundred years; a result to which he was led chiefly by a comparison of his own observations with those of Walther. Lalande, after comparing an immense number of modern observations with those of the 17th, 16th, and 15th centuries, and also with those of the Arabians and Chinese, found the secular diminution to be 50''. Bessel, rejecting the ancient observations as too uncertain, and comparing those only which have been made since the time of Bradley, has fixed the secular diminution at 45'' 7'', which differs inconsiderably from the determination of Lacaille. It cannot yet be determined by observation whether this diminution is uniform, or accelerated, or retarded; but so slow is the rate at which it proceeds, that it may, without any sensible error, be regarded as uniform for many centuries to come.

The gradual diminution of the obliquity of the ecliptic might lead us to suppose that a time will ultimately arrive when that plane will coincide with the equator, and the earth be deprived, in consequence, of the agreeable vicissitude of the seasons. But the theory of universal gravitation, which has revealed the cause of the diminution, has also shown that there are certain limits which the angle of the two planes can never exceed, and between which it must continue for ever to oscillate. Geometers have not yet ventured to assign the precise extent of these limits, but their existence is certain; and the planes of the ecliptic and equator, which have been approaching to each other during the last 2000 years, will, in the course of some thousands of years more, begin to recede.

In what has yet been said respecting the diminution of the obliquity of the ecliptic, no fact has been mentioned from which it can be inferred, whether the phenomenon is... Theoretical occasioned by the displacement of the plane of the ecliptic or that of the equator. This question may also be decided by a comparison of modern with ancient observations; for it is evident, that if the inclination of these two planes becomes less, the stars which are situated between them, particularly those near the solstitial colure, will appear to approach to that plane which changes its position; so that if the ecliptic is displaced, the latitudes of those stars will be diminished, or their declinations if the displacement belongs to the equator. It was first observed by Tycho, and the observation has been confirmed by succeeding astronomers, that the latitudes of the southern stars situated near the solstitial colure, that is, of those stars whose longitudes are nearly 90°, have diminished upwards of 20° since the time of Hipparchus and Ptolemy, while the latitudes of the northern stars have undergone a corresponding augmentation. From this fact it is proved that the diminution of the obliquity is occasioned by the displacement of the ecliptic; and theory has shown that the cause of the displacement is the action of the planets, particularly of Jupiter and Venus, on the earth, by virtue of which the plane of the earth's orbit is drawn nearer to the planes of the orbits of these two planets. This, however, though by far the most considerable, is not the sole cause of the phenomenon; for theory also shows that a slight motion of the plane of the equator is produced by the attraction of the sun and moon, but so very minute that its effects will only become appreciable after a long series of ages.

After determining the inclination of the plane of the ecliptic to that of the equator, the only element requisite to fix its position absolutely in space, is the situation of the straight line formed by its intersection with that plane, that is to say, the line of the nodes. The longitudes of the stars, as has already been mentioned, are counted on the ecliptic from the vernal equinox; and therefore, if the line of the equinoxes, which is the same as the line of the nodes, is invariable, the longitude of any star will always be the same, whatever interval of time may elapse between two observations of that longitude. But on comparing the actual state of the heavens with the observations recorded by ancient astronomers, it is perceived that the longitudes of all the stars are considerably increased; whence we must infer, either that the whole firmament has advanced in the order of the signs, or that the equinoctial points have gone backwards, or retrograded. The latter supposition is infinitely the more probable; for it is inconceivable that the innumerable multitude of stars should have a common motion relatively to points which depend solely on the motion of the earth. The phenomenon is therefore to be explained by attributing to the equinoctial points a retrograde motion from east to west, in consequence of which, the sun, whose motion is direct, arrives at them sooner than if they remained at rest; and therefore the equinoxes, spring, autumn, and the other seasons, happen before the sun has completed an entire circuit. On this account the motion has been denominated the Precession of the Equinoxes. As this motion is extremely slow, its exact amount can be discovered only by a comparison of observations separated from each other by long intervals of time; but the imperfection of instruments prior to the sixteenth century renders the ancient observations of little authority where quantities so minute are concerned, and therefore some discrepancies may be expected in the different determinations of the amount of the precession. The comparison of modern observations with those of Hipparchus gives its annual amount equal to 50.4 seconds, and with those of Ptolemy somewhat greater. The mean result of the observations of Tycho, compared with those of Lacaille, gives 50.1. On comparing modern observations with one another, we find 30°-06'. Delambre, in his solar tables, supposes the annual precession to be equal to 50.1. According to this estimate the equinoctial points go backwards at the rate of one degree in 71.6 years nearly, and therefore will make a complete revolution of the heavens in about 25,868, or nearly 26 thousand years.

The discovery of the precession of the equinoxes is generally attributed to Hipparchus, who, on comparing his own observations with those of Timocharis, more ancient by 160 years, perceived that in this interval the longitudes of the stars had been augmented by about two degrees. It would seem, however, from many proofs collected by Bailly, that this motion, slow as it is, was known to all the ancient nations who cultivated astronomy, long before the time of Hipparchus. It is indeed easy to conceive, from the great attention which they gave to the helical risings of the remarkable stars, that they might observe a gradual change of the seasons at the occurrence of these phenomena, from which they would necessarily be led to conclude a variation of the star's longitude. In consequence of this regression of the equinoctial points, the sun's place among the zodiacal constellations at any given season of the year is now greatly different from what it was in remote ages. Some time prior to Hipparchus, the first points of Aries and Libra corresponded to the vernal and autumnal equinoxes; those of Cancer and Capricorn to the summer and winter solstices: at present these constellations have separated 30 degrees from the same points of the ecliptic. The vernal equinox now happens in the constellation Pisces, the summer solstice in Gemini, the autumnal equinox in Virgo, and the winter solstice in Sagittarius. Astronomers, however, still count the signs from the vernal equinox, which, therefore, always corresponds to the first point of the Sign of Aries. On this account it is necessary to distinguish carefully between the Signs of the Zodiac, which are fixed with regard to the equinoxes, and the Constellations, which are movable with respect to those points.

The diminution of the obliquity of the ecliptic arises Physical from the displacement of the ecliptic itself; the precession cause of the equinoxes is, on the contrary, occasioned by the continual displacement of the plane of the terrestrial equinoxes. This displacement results from the combined action of the sun and moon (for the influence of the planets amounts only to a fraction of a second, and is consequently scarcely sensible,) on the mass of protuberant matter accumulated about the earth's equator, or the matter which forms the excess of the terrestrial spheroid above its inscribed sphere. The attracting force of the sun and moon on this shell of matter may be resolved into two; one parallel to the plane of the equator, the other perpendicular to it. The tendency of this last force is to diminish the angle which the plane of the equator makes with that of the ecliptic; and if the earth had no motion of rotation, it would soon cause the two planes to coincide. In consequence, however, of the rotatory motion of the earth, the inclination of the two planes remains constant; but the effect produced by the action of the force in question is, that the plane of the equator is constantly shifting its place, in such a manner that the line of the equinoxes advances in the direction of the diurnal motion, or contrary to the order of the signs, its pole having a slow angular motion about the pole of the ecliptic, so slow indeed, that it requires nearly 26,000 years to complete its revolution.

If the sun and moon moved in the plane of the equator, there would evidently be no precession; and the effect of their action in producing it varies with their distance from that plane. Twice a year, therefore, the effect of the sun in causing precession is nothing; and twice a year, namely at the solstices, it is a maximum: on no two successive days of the year is it exactly the same, and consequently the regression of the equinoctial points, which results from the sun's action, must be unequal.

On this account the obliquity of the ecliptic is subject to a semi-annual variation; for the sun's force, which tends to produce a change in the obliquity, is variable, while the diurnal motion of the earth, which prevents the change from taking place, is constant. Hence the plane of the equator is subject to an irregular motion, which is technically called the Solar Nutation. The existence of the solar nutation is, however, only a deduction from theory, for its amount is too small to be perceptible to observation; but a similar effect of the moon's action is sufficiently appreciable, and was, in fact, discovered by Dr Bradley before theory had indicated its existence. Its period, however, is different, and depends on the time of the revolution of the moon's nodes, which is performed in 18 years and about 7 months. During this time the intersection of the lunar orbit with the ecliptic has receded through a complete circumference; and the inequality of the moon's action will consequently, in the same time, have passed through all its different degrees.

Bradley observed that the declinations of the stars continued to augment during nine years, that they diminished during the nine years following, and that the greatest change of declination amounted to 15°. He remarked further, that this motion was connected with an irregularity of the precession of the equinoxes, which followed exactly the same period; whence he concluded that the motion of the poles of the equator, occasioned by this vibration of its plane, was not confined to the solstitial colure. A series of observations on stars differently situated proved that all the phenomena could be explained on the hypothesis that the pole of the equator describes in 18 years a small circle of 18° diameter, contrary to the order of the signs; or that the axis of the earth, following the circumference of this circle, describes the surface of a cone, the axis of which forms with its side an angle of 9°. This apparent vibratory motion is significantly denominated the Nutation of the Earth's Axis.

In consequence of the two motions which occasion the precession and the nutation, the true path of the pole of the equator round that of the ecliptic is an epicycloidal curve, which will be understood by referring to fig. 14. Let E be the pole of the ecliptic, round which the pole of the equator P describes, in virtue of the precession, and in a direction contrary to the order of the sines, the circle PQR, of which the radius EP is equal to the obliquity of the ecliptic, or the mean distance of the two poles. While P, the mean place of the pole, moves in the circle PQR, with a velocity equal to the regression of the equinoctial points, or at the rate of 30°·1 a year, the true pole p describes at the same time round P a small circle of 18° diameter in the same direction pqr. The true path of the pole is therefore along the circumference of a circle pqr, the centre of which retrogrades on the circumference of another circle, and consequently moves in an epicycloid abcdefg, the curve which results from the composition of the two motions. Suppose the mean pole at Q, the true pole at a, and qa = 9°. In the course of nine years the mean pole will have retrograded from Q to R, making QR = 9 × 50°·1, while the true pole will have accomplished a semi-revolution in its circle. It will therefore be at e (the being = 9°), and have described the epicycloidal arc abc, the greatest distance of which from the circle PQR is at b, and equal to 9°. At the end of the following nine years the mean pole has retrograded from R to S, or point a of its epicycle; it will therefore be found at e, Se being = 9°. In this interval it has necessarily been within the circle PQR, its greatest distance from which at d is 9°, so that in 18 years it has traced the curve abcde. But ae = QR + aQ + Re = 9 × 50°·1 + 18° = 7°49', and ee = RS - Re - Se = 9 × 50°·1 - 18° = 7°13'. From a to e the motion of the true pole in the epicycle is in the same direction as that of the mean pole; from e to it is in an opposite direction, but as it is always much slower, being only about 3° in a year, while that of the mean motion is 50°·1, the true motion which results from the combination of both will always be in the direction ce, and at d its velocity will be equal to their difference. At b and d the difference between the latitudes of the true and mean pole is a maximum, while the difference of their longitudes is nothing; in other words, the correction of the obliquity is greatest when that of the precession vanishes: at a, c, and e the correction of the obliquity vanishes, and that of the precession is a maximum.

Dr Bradley remarked that the effects of the nutation Pole described by the pole of the equator about its mean place to be an ellipse instead of a circle, the transverse and conjugate axes being 18° and 16° respectively. This is also confirmed by theory, from which Laplace calculated the semi-axes of the ellipse at 9°·63 and 7°·17. The semi-transverse axis of the ellipse described by the pole in virtue of the sun's action alone does not exceed half a second, and is consequently totally imperceptible. The sensible part of the nutation, therefore, follows exactly the period of the revolution of the nodes of the moon. By 603 observations of Polaris, observed at Dorpat between 1822 and 1838, M. Peters has determined the semi-axis major of the ellipse to be 9°·2361, and gives, for his definitive result, 9°·2231.

It is now easy to see the reason of the distinction drawn above between the mean and the true or apparent obliquity. The mean obliquity is represented by the radius EP of the deferent circle PQR, along the circumference of which the centre of the small circle or ellipse is carried, while the true obliquity is that quantity increased or diminished by the nutation. The calculation of the mean obliquity from the true is performed by the aid of the astronomical tables.

The progressive diminution of the mean obliquity and the nutation of the earth's axis are inequalities distinguished from each other, not only by their being derived from different and distinct causes, but still more by the very great difference of time required for their full development. Almost every other element of the planetary system is affected in a similar manner by inequalities of two kinds, which are distinguished by the terms secular and periodic. The secular inequalities proceed with extreme slowness, and continue progressive in the same sense during many centuries; while the periodic are much more rapid in their march, and run through the whole period of their changes in comparatively short intervals of time. The inequalities of the first kind are also periodic; but their periods are vastly longer, and may be reckoned by centuries instead of years; and from this circumstance they derive their name of secular inequalities.

Sect. II.—Of the Orbit of the Sun.

Having now considered the situation of the plane in which the sun is observed to move relatively to the fixed stars, and also the small secular and periodic variations Theoretical to which it is subject, we proceed next to inquire into the nature of his orbit, or the curve which he describes on that plane. The means which we have of determining the solar orbit are the observed variations of the sun's angular velocity, and of his distance from the earth; hence two kinds of observations are necessary,—the first, of his daily meridional altitudes, from which, as the inclination of the orbit is known, his angular velocity is easily found; and the second, of his apparent diameter, the variations of which follow the inverse ratio of his distance.

That the angular velocity of the sun's motion in his orbit is not uniform, is obvious from the fact that he remains 7½ days longer in the northern than in the southern signs, or, which is the same thing, that the interval between the vernal and autumnal equinoxes is 7½ days longer than the interval between the autumnal and vernal. It is proved by numerous observations that the sun moves with the greatest velocity when at a point situated near the winter solstice; while at the opposite point of the orbit, or near the summer solstice, his velocity is the least. At the first point the diurnal motion is 1°·01943, and at the second only 0°·95319. It is constantly varying between these two points; and the variation is observed to be nearly proportional to the sun's angular distance from the point of his orbit where his velocity is a maximum or minimum. The mean velocity is 0°·98632, or nearly 59° 11', which is the rate of the sun's daily motion about the beginning of April and October.

The point of the solar orbit, which is the most remote from the earth, is called the apogee (απογεία, away from the earth); that which is nearest is called the perigee (περιγεία, near the earth). The same points are also called respectively the superior and inferior Apis of the orbit, and the straight line which joins them is called the Line of the Apisides.

Diameter. The exact determination of the sun's diameter is a problem which engaged the earliest astronomers, but of which, before the invention of the telescope and micrometer, it was impossible to obtain a solution sufficiently accurate to make known its variations. Archimedes, by an extremely ingenious though imperfect method, demonstrated that it must be included between the limits 1/10th and 1/12th of a right angle, that is, between 27° and 32° 55' 7". The Egyptians, by observing the time which the sun takes to rise above the horizon, found it to be between a 750th and a 700th part of a circumference, that is, between 28° 48' and 30° 51' 5". Aristarchus of Samos supposed it to be 30°. The precision of modern observations shows that the apparent diameter is greatest about the time of the winter solstice, and least about the summer solstice; but there is some discrepancy among the results of different astronomers with respect to its actual magnitude. According to the tables of Delambre, its greatest value is 32° 25' 6", and its least 31° 31"; the mean apparent diameter, or the diameter at the sun's mean distance, is equal to 32° 2' 9". According to Bessel, the mean diameter is 32° 1' 80", as derived from 1698 transits, and the result from the Greenwich observations is 1°·84 greater than this.

From these remarks it is obvious, that if the orbit of the sun be a circle, the earth is not situated in the centre of that circle, otherwise the distance of the sun from the earth would remain always the same, which is contrary to fact. It is possible, however, that the variation in his angular velocity may not be real, but only apparent. Thus, in fig. 15, let AMPN be the orbit of the sun, C the centre of that orbit, and E the position of the earth at some distance from the centre. It is obvious that P is the sun's perigee, and A his apogee. Now, as the sun's apparent orbit is a circle having the earth in its centre, it is evident that this orbit must be αMPN, and that the angular motion of the sun will be measured upon that circle. Suppose now that the sun in his apogee moves from A to A', it is obvious that his apparent or angular motion will be the segment a a' of the apparent orbit, considerably smaller than AA'; so that at the apogee the angular motion of the sun will be less than his real motion. Again, let the sun in his perigee move from P to P', describing a segment precisely equal to the segment AA'. This segment, as seen from the earth, will be referred to pp', which in that case will be the sun's angular motion, evidently considerably greater than his real motion.

Hence it is obvious, that even on the supposition that the sun moved equably in his orbit, his angular motion as seen from the earth would still vary, that is, would be smallest at the apogee and greatest at the perigee, and that the angular and real motion would only coincide in the points M and N, where the real and apparent orbits intersect each other. From the figure, it is obvious also that the angular velocity would increase gradually from the apogee to the perigee, and diminish gradually from the perigee to the apogee, which likewise corresponds with observation.

But if the variation in the angular motion of the sun were owing alone to the eccentric position of the earth within the solar orbit, it is easy to demonstrate that in that case the diminution of his angular velocity would follow the same ratio as the diminution of his diameter. The fact however is, that the angular velocity diminishes in a ratio twice as great as the diameter of the sun does. The variation of the angular velocity cannot then be owing to the eccentricity alone. Hence it follows that the variation of the motion of the sun is not merely apparent, but real, and that its velocity in its orbit actually diminishes as his distance from the earth increases. Two causes, then, combine to produce the variation in the sun's angular velocity; namely, 1. the increase and diminution of his distance from the earth, and, 2. the real increase and diminution of his velocity in proportion to this variation of distance. These two causes combine in such a manner that the daily angular motion of the sun diminishes as the square of his distance increases, so that the product of the angular velocity multiplied into the square of the distance is a constant quantity.

The observation, that the sun's angular motion in his orbit is inversely proportional to the square of his distance from the earth, is due to Kepler. The discovery was made by a careful comparison of the sun's diurnal motion with his apparent diameter, which is inversely proportional to his distance from the earth. Let ASB (fig. 16) be the sun's orbit, E the earth, and S' the sun. Suppose a line ES joining the centres of the earth and sun to move round along with the sun. It is obvious that when S moves to S', ES, moving along with it, is now in the situation ES', having described the small sector SES'. In the same time that S performs one revolution in its orbit, the radius vector ES will describe the whole area ABS enclosed within the sun's orbit. Let SS' be the sun's angular motion during one day. It is evident that the small sector SES' is proportional to the square of ES, multiplied by SS'; for the radius vector is the sun's distance from the earth, and SS' his angular motion. Hence this sector describes a constant quantity, whatever the angular motion of the areas produced be; and the whole area SEA increases as the number portioned of days which the sun takes in moving from S to A, to the times. Hence results that remarkable law, first pointed out by Kepler, that the areas described by the radius vector are proportional to the times of description. Suppose the sun to describe SS' in one day, and SA in twenty days; then Theoretical the area SES' is to the area SEA as 1 to 20, or the area Astronomy. SEA is 20 times greater than the area SES'.

The knowledge of these facts enables us to draw upon paper, from day to day, lines proportional to the length of the radius vector of the solar orbit, and having the same relative position. If we join the extremity of these lines, by making a curve pass through them, we shall perceive that this curve is not exactly circular. Let E (fig. 17) represent the earth, and E a, E b, E c, E d, E e, &c., the position and length of the radius vector during every day of the year; if we join together the points a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, by drawing the curve e e i m through them, it is obvious that this curve is not a circle, but elongated towards a and i, the points which represent the sun's greatest and least distance from the earth. The resemblance of this curve to the ellipse induced Kepler to compare them together; and he ascertained their identity, and thus proved that the orbit of the sun is an ellipse, having the earth in one of its foci.

Having arrived at the knowledge of the true nature of the sun's orbit, it becomes necessary, in the next place, to determine its position on the plane of the ecliptic, that is, to assign the position of the transverse axis, or line of the apsides, with reference to some other straight line given by position on that plane. The line which it is most convenient to assume for this purpose is the line of the equinoxes; and the position of the apsides may be determined from observations of the time which the sun occupies in performing a semi-revolution, counting from different points of his orbit. Thus, if among the observations of the sun's longitude made daily during the course of a year, we compare, two and two, all those which are diametrically opposite, or which differ by 180°, it will be found that the interval between them will be somewhat less than half a year, if the sun, during that interval, has passed through his perigee, or longer if he has passed through his apogee; and that the time between the two observations will differ less from half the time of a whole revolution, in proportion as the sun's place, at the time of the two observations, was nearer to the apsides. If the difference of time between the two observations is exactly half a year, then the place of the apsides would be obtained at once, because the sun must have been in those points when the observations were made. The probability is, however, infinitely small, that this can ever happen exactly; but as the position of the apsides is known very nearly from the diurnal observations of the variation of the sun's angular velocity, the necessary corrections can easily be supplied by computation.

From the comparison of observations made in different ages, it appears that the position of the apsides is not fixed on the plane of the ecliptic, but that the greater axis of the solar ellipse revolves in the direction of the sun's annual motion. The observations of Hipparchus, compared with those of the present times, show that the apsides have a direct motion at the rate of 65° in a year. According to the observations of Walther, the longitude of the apogee in 1496 was 93° 57' 57"; and according to Lecaille, the same element in 1750 was 98° 37' 28"; whence, dividing the difference by 254, the interval between the two epochs, the annual motion of the line of the apsides amounts to 66°. Delambre found, by the comparison of a great number of modern observations, that the annual motion is 61° 95' a year. The theory of attraction, which, in respect of such slow and minute variations, must be considered as giving more accurate results than can be obtained directly by observation, gives 61° 9' for the yearly progressive motion of the apsides.

In the above determination, the position of the apsidal line referred to the line of the equinoxes, and their motion compared with the sun's tropical revolution. But if we wish to determine their motion with reference to the fixed stars, it is necessary to have regard to the retrograde movement of the equinoctial points; for it is obvious, that if the line of the equinoxes is not fixed, the displacement of the apsides with respect to the stars will be increased or diminished by the whole amount of its motion, according as the two motions are in the same or opposite directions. Now the motion of the line of the apsides is direct, and has just been stated to amount to 61° 9' in a solar year; that of the equinoxes is, on the contrary, retrograde, and amounts to 50° 1' in the same time. Hence the displacement of the apsides, with reference to a star or fixed point in the ecliptic, is 61° 9' - 50° 1' = 11° 8' a year, in the direction of the sun's annual motion. The time, therefore, in which the solar perigee completes a revolution in the heavens, or returns to the same star, is \( \frac{360°}{11° 8'} \) equal nearly to 110,000 years.

Since the greater axis of the solar ellipse has a progressive motion on the plane of the ecliptic, it forms a variable angle with the line of the equinoxes, and at distant epochs will coincide with that line, or be perpendicular to it. The epochs at which these phenomena happen may be easily found by simple proportions, when the longitude of the perigee at any given time, and its annual motion, are known. According to the observations of Lecaille, already quoted, the longitude of the perigee in 1750 was 278° 37' 28"; but when the longitude of the perigee was 270°, the greater axis of the solar ellipse must have been perpendicular to the line of the equinoxes. The difference of these two longitudes is 8° 37' 28", and the number of years requisite to describe that arc, at the rate of 61° 9' annually, is \( \frac{8° 37' 28"}{61° 9'} = 500 \) years nearly; whence the major axis was perpendicular to the line of the equinoxes in the year 1250, when the perigee of the orbit coincided with the winter solstice.

Suppose it were required to assign the epoch at which the major axis coincided with the line of the equinoxes. At the occurrence of this phenomenon the longitude of the perigee was 180°; consequently from that time to 1750 the perigee had advanced 278° 37' 28" - 180° = 98° 37' 28". Now \( \frac{98° 37' 28"}{61° 9'} = 5735 \); therefore 5735 is the number of years intervening between the occurrence of the phenomenon and 1750. Hence about 4000 years before the commencement of our era, the transverse axis of the solar orbit coincided with the line of the equinoxes; and it is a singular coincidence that this epoch is considered by chronologists to be that of the beginning of the world, or, to speak more correctly, of the first traces of the existence of the human race; for numerous physical circumstances attest that the earth itself has existed during an infinitely longer period. The same phenomenon will again occur when the longitude of the perigee shall have reached 360°, or become zero; and by calculating as above, this will be found to take place in the year 6485 of our era. The solar perigee will then coincide with the vernal equinox; in the former case it coincided with that of autumn.

By reason of the progressive motion of the perigee, the length of the seasons is continually varying. Let AP (figs. vary in 18, 19, 20) represent the line of the apsides, S and W the summer and winter solstices, V and O the vernal and autumnal equinoxes. When the greater axis was perpendicular to the line of the equinoxes, as happened in the year 1250 of our era, the perigee P (fig. 18) coincided Fig. 18. Theoretical with W the winter solstice; and on that account the time between the autumnal equinox O, and the winter solstice W, was equal to the time between W and V, or between the winter solstice and the vernal equinox. But in this position the equator, which is here represented by VO, divides the ellipse into two unequal portions, the smaller of which must be described in less time than the greater, because the times of description are proportional to the spaces passed over. The summer was therefore, at that time, longer than the winter, and both were divided into equal parts by the solstices. In any other position the seasons differ in length from each other. In the time of Hipparchus, the longitude of the perigee was less than in 1750 by 32°-292, or was 246°-330. The position of the ellipse with regard to the equinoxes was therefore such as is represented in fig. 19, the angle PEW being 270° - 246°-33 = 23° 40' 12". The interval between V and S was then 94½ days, and that between S and O only 92½ days. The spring was therefore at that time longer than the summer, and the winter longer than the autumn. The position of the ellipse in 1800 is represented in fig. 20. The angle PEW was then 9° 29', and the following were nearly the lengths of the seasons:

| Days | Hours | Min. | |------|-------|-----| | from V to S | 92 | 21 | 45 | | S to O | 93 | 13 | 35 | | O to W | 89 | 16 | 47 | | W to V | 89 | 1 | 42 |

Length of the year, nearly...365 5 49

The spring is therefore at present shorter than the summer, and the autumn longer than the winter.

After describing the position of the transverse axis of the solar ellipse on the plane of the ecliptic, we come next to consider the species of that ellipse, or its eccentricity, on which depend the apparent inequalities of the sun's angular motion. It is evident that this element will be made known if we can by any means determine the ratio of the two segments into which the major axis is divided by the focus. This ratio might seem to be obtainable without any difficulty by comparing the sun's apparent diameters at the perigee and apogee, to which the distances are inversely proportional; but such observations, on account of the irradiation and other difficulties, are liable to considerable uncertainty: it is therefore necessary to have recourse to other methods of estimating the eccentricity, and the variations to which it is subject. These methods are derived from the fundamental law of the elliptic motion, namely, the proportionality of the areas described by the radius vector to the times of description, by which the inequalities of motion and the ellipticity of the orbit are connected with each other; so that when the inequalities are made known by direct observation, the species of the ellipse can be computed from the principles of geometry. It will, however, be necessary to attend more closely than we have yet done to the phenomena resulting from the eccentric position of the earth in the sun's orbit, and to explain some terms technically employed by astronomers to abbreviate and simplify their descriptions.

The sun's motion may be regarded as composed of two distinct parts, namely, a circular uniform motion, which constitutes the principal part of it; and a correction depending on the deviation of the orbit from a circle, which modifies the first, and alternately accelerates and retards the mean angular velocity. To illustrate this, let E (fig. 21) be the earth, PSA the orbit of the sun, AP the line of the apsides, and qφ the intersection of the plane of the orbit with that of the equator. While the true sun moves round his elliptic orbit, describing areas proportional to the times, conceive a fictitious sun s to move round the earth in the circumference of a circle of which the radius is equal to EP the sun's distance at the perigee, and with a uniform motion such, that if the two suns set out at the same instant from P, they may also return to the same point together, after having completed each a revolution. At the perigee the radius vector of the real sun is a minimum; his velocity is consequently a maximum, and he therefore advances before the fictitious sun. But his motion relaxes in proportion as his distance from P becomes greater; and at a certain point S of the orbit it becomes equal to that of the fictitious sun, which has then only reached the point s. After passing the point S, the angular distance SE continues to diminish on account that the motion of the real sun continues to relax, and it vanishes altogether when he reaches his apogee at A, the fictitious sun arriving at a in the same straight line with A and E at the same instant of time. From the apogee to the perigee the phenomena are reversed. The motion of the real sun being then the slowest possible, he at first falls behind the fictitious sun; but his motion continuing to be gradually accelerated, he has again acquired the same velocity when he reaches S, after which his motion is more rapid than that of the fictitious sun, which he at last overtakes at P. If, therefore, we conceive a radius vector to be drawn from the earth to each of the two suns, those lines will form with each other a variable angle, having its maximum value when the velocities are equal, and vanishing at the perigee and apogee. This angle, namely, SEs, is called the Equation of the Centre, or Equation of the Orbit. It is the correction necessary to be made to the longitude of the sun deduced from his equable motion, in order to have his real longitude. Those quantities which form the difference between the true and mean results are, in astronomy, denominated equations. The ancients employed the term Prosthaphaeresis in the same sense.

From the perigee to the apogee, the true longitude of the sun is found by adding the equation of the centre to the mean longitude, and in the other half of the orbit by taking the difference of the same quantities. The equation of the centre is a maximum when the sun is at the points S and S', or when the mean and true motions are equal; and if its value can be obtained from observation at those times, the eccentricity of the orbit may be deduced by means of the geometrical properties of the ellipse. Now the points S and S' may be found very nearly by observing when the diurnal velocity of the sun is equal to the mean motion, or to 0°985647 parts of a degree. By these observations the angle SES, which is the difference of the real longitudes of the sun, or of its distances from qφ, will be given. But sEs' is also given; for it is the angle which would be described by the sun, in virtue of his mean motion, during the interval between the two observations. Hence the difference between sEs' and SES is given, being equal to twice SEs the maximum equation of the centre, in consequence of the symmetrical position of the points S and S', and also of s and s', in respect of the line of the apsides AP. The accuracy of a result obtained in this manner may appear questionable, on account of the impossibility of determining by observation the exact moment at which the sun's true motion in longitude is equal to his mean motion; but as the real motion varies very little during a few days before and after that epoch, the equation of the centre will be scarcely affected by a slight error in the time. Besides, any error in the result is corrected by taking the mean of a great number of similar observations. By a comparison of numerous observations of this kind, Delambre found that the greatest equation of the centre, in the year 1776, amounted to $1^\circ 9234$, or $1^\circ 53' 31''$. In 1801 the same equation was $1^\circ 53' 27''$.

Having obtained the value of the greatest equation of the centre, the eccentricity of the orbit may be computed from the first two terms of the following series,

$$e = \frac{1}{2}E - \frac{11}{2^3}E^3 + \frac{587}{2^6 \cdot 3^5}E^5 - \ldots$$

in which $e$ represents the eccentricity, and $E$ the greatest equation of the centre. By reversing the series the following expression is found for the greatest equation in terms of the eccentricity, viz.

$$E = 2e + \frac{11}{2^3}e^3 + \frac{599}{2^6 \cdot 3^5}e^5 + \ldots$$

For an investigation of the above formulae the reader may consult Biot, *Astronomie Physique*, tom. ii. p. 185.

It has been discovered by observation, that the equation of the centre of the sun's orbit is subject to a nearly uniform secular diminution. This fact is confirmed by theory, from which also the secular diminution has been assigned. It amounts to $8''0047$ in a century. This phenomenon implies a corresponding diminution of the eccentricity of the solar orbit amounting to $000087495$ in the same time, the semiaxis major being unit. This will be conceived more distinctly by converting the above fraction into some familiar expression of linear magnitude. Supposing the mean distance of the sun to be $96,000,000$ miles, the fraction $000087495$ will represent nearly $3700$ miles, so that the annual diminution of the eccentricity is nearly at the rate of $37$ miles—a line which is considerable on the surface of the earth, but which has scarcely an appreciable ratio to the immense distance of the sun. If, however, this diminution, small as it is, were to be continued indefinitely, the eccentricity would ultimately vanish, and the sun's orbit would be changed into a circle; but the theory of universal gravitation proves that, like all the other variations of the elements of the solar system, the variation of the eccentricity is subject to periodic laws. After having continued during a certain time to diminish, the eccentricity will again begin to increase, and will successively pass through all its former values. Thus it will continue to oscillate within certain limits, of which the extent, though not precisely known, cannot be very great; and the solar orbit will eternally preserve its elliptic form, unless the application of some external force shall derange the system of the world, or modify the laws by which it is at present governed.

The true nature of the solar orbit being known, together with its situation on the plane of the ecliptic, and the amount of its eccentricity, as also the time of a revolution, the sun's longitude may be determined at any assigned epoch. This determination is what is usually termed Kepler's Problem, having been first proposed by that great astronomer on the hypothesis of an elliptic orbit. Its solution, which is not susceptible of being exhibited under a finite form, is derived from the principle of the equable description of areas. Let ASP (fig. 22) represent the semi-orbit of the sun, C the centre of the orbit, E the focus occupied by the earth, and let the time and motion be counted from the perigee P. On AP describe a semicircle, on the circumference of which let a point M be supposed to move uniformly, and to describe the semicircle PMA, in the same time in which the sun describes the semi-ellipse PSA. Now, suppose that at the end of a given time $t$, the sun has moved from P to S; then, denoting by $T$ the time of a complete revolution, and putting $\pi$ for the semi-circumference, the place of M, at the end of the same time $t$, will be given by the equation $PM = \frac{T}{\pi}$. Through S let a perpendicular Fig. 22. be drawn to AP, meeting the semicircle in D and AP in N, and join EM, ES, ED, CM, and CD. The different angles of this figure have received certain technical names, which it is necessary to explain. Then S being the true place of sun, the angle PES is called the True Anomaly; PEM is called the Mean Anomaly, and PCD is denominated the Eccentric Anomaly. This last is measured by the arc PMD, which is the perigean distance of a body describing a circle circumscribed about the solar ellipse, and having constantly the same absciss PN as the sun. The word anomaly originally signified inequality; it now simply designates the angular distance of a planet from its perihelion, seen from the sun, or, as in the present case, the angular distance of the sun from his perigee.

According to the law of the equable description of areas, we have the sector PCM to PES as the surface of the semicircle to the surface of the semi-ellipse; that is, as the semi-transverse to the semi-conjugate axis, or as ND to NS, according to a well-known property of the ellipse. Hence $PCM : PES :: PED : PES$, and therefore $PCM = PED = PCD — ECD$. Now, if we express the arc PM by $z$, PD by $x$, the eccentricity EC by $e$, and make the radius equal to unity, we shall have the sector $PCM = \frac{1}{2}z$, the sector $PCD = \frac{1}{2}x$, and the triangle $ECD = \frac{1}{2}e$, $DN = \frac{1}{2}e \sin x$. Substituting, therefore, these expressions in the above equation, it will become, on multiplying both sides of it by $2$,

$$z = x - e \sin x,$$

where $z$ denotes the mean, and $x$ the eccentric, anomaly.

If in this equation $x$ and $e$ were both known, it would be easy to deduce from it the mean anomaly $z$; but the data are $z$ and $e$, and the value of $x$ can only be obtained in a series, the equation being a transcendental one. Its resolution gives the value of the eccentric in terms of the mean anomaly; and if another equation can be found in which the eccentric anomaly is expressed in terms of the true, it will only remain to put these two expressions equal to one another in order to have an equation between the mean and true anomaly.

For this purpose it will be necessary to equate two values of the radius vector ES, expressed in terms of the eccentric and true anomalies respectively. Putting ES = $r$, we shall have, in the first place,

$$r^2 = EN^2 + NS^2;$$

but if the semiaxis minor is represented by $b$, the common equation of the ellipse gives

$$NS^2 = \frac{b^2}{a^2}(a^2 - CN^2) = b^2 - \frac{b^2}{a^2}CN^2;$$

and we have also

$$EN^2 = EC^2 + 2EC \cdot CN + CN^2 = e^2a^2 + 2ea \cdot CN + CN^2;$$

therefore, by addition,

$$NS^2 + EN^2 = b^2 + e^2a^2 + 2ea \cdot CN + \left(1 - \frac{b^2}{a^2}\right)CN^2.$$

Now, $b^2 + e^2a^2 = a^2$, and $1 - \frac{b^2}{a^2} = e^2$; therefore

$$NS^2 + EN^2 = r^2 = a^2 + 2ea \cdot CN + e^2CN^2;$$

consequently

$$r = a + e \cdot CN.$$ To obtain a second value of \( r \), let SF be drawn from S to the other focus of the ellipse, and let SF = \( r' \). The rectangular triangles ESN, FSN give \( SN^2 = r^2 - EN^2 \) and \( SN^2 = r'^2 - (2ea - EN)^2 \); therefore \( r^2 - EN^2 = r'^2 - (2ea - EN)^2 \), whence we derive \( r^2 - r'^2 = 4ea \cdot EN - 4e^2a^2 \). But by the property of the ellipse \( r + r' = 2a \), from which it is easy to deduce \( r^2 - r'^2 = 4a^2 - 4ar = 4ar - 4a^2 \); and hence, by equating these two values of \( r^2 - r'^2 \), \( ea \cdot EN - e^2a^2 = ar - a^2 \). Now, if we denote the angle PES, which is the true anomaly, by \( v \), then \( EN = r \cos v \), and consequently by substitution \( ear \cos v - e^2a^2 = ar - a^2 \), whence

\[ r = \frac{a(1-e^2)}{1+e \cos v} \tag{3} \]

If we now compare the two values of \( r \) given by the equations (2) and (3), we shall find

\[ \cos x = \frac{\cos v + e}{1 + e \cos v}, \]

and therefore, \( 1 - \cos x = (1-e)(1-\cos v) \)

\[ 1 + \cos x = (1+e)(1+\cos v); \]

whence, by dividing,

\[ \frac{1-\cos v}{1+\cos v} = \frac{1+e}{1-e} \cdot \frac{1-\cos x}{1+\cos x}; \]

consequently, by the trigonometrical formula,

\[ \tan \frac{1}{2}v = \left(\frac{1+e}{1-e}\right)^{\frac{1}{2}} \tan \frac{1}{2}x \tag{4} \]

The equations (1) and (2) were first given by Kepler, the last is due to Lacaille. The problem of finding the sun's place after any given time is solved analytically by (1) and (4).

To express \( v \) the true anomaly, in a series involving the powers of \( x \) the mean anomaly, is a problem requiring the aid of the higher analysis, and which it is unnecessary to investigate here. The following are a few of the first terms of the series, in which \( nt \) is substituted for \( x \), \( n \) being the mean motion in the unit of time, and \( t \) the time elapsed since the passage through the perihelion:

\[ v = nt + \left(2e - \frac{1}{2}e^2 + \frac{5}{96}e^3\right) \sin nt \\ + \left(\frac{5}{4}e^2 - \frac{11}{24}e^3 + \frac{17}{192}e^4\right) \sin 2nt \\ + \left(\frac{13}{12}e^3 - \frac{43}{64}e^4\right) \sin 3nt \\ + \text{etc.} \]

The form of the sun's orbit is discovered from the variations of his apparent diameter in passing from the perigee to the apogee; but in order to obtain a knowledge of its real dimensions, or of the mean distance of the sun from the earth, it is necessary to determine his parallax, that is to say, the angle subtended by the earth's semidiameter as seen from the sun. This angle is too small to be measured directly in the manner which was pointed out when treating of parallax; it is most accurately deduced from observations of the time occupied by an inferior planet in passing over the sun's disk, as will be explained afterwards. From such observations it has been found to be less than 9", an angle of which the sine is to radius in the ratio of 1 to 23,000 nearly; whereas the distance of the sun is about 23,000 times the semidiameter of the earth. The following table exhibits more accurately the numerical results of calculation.

| Distance of the Sun from the Earth | |-----------------------------------| | Semidiameters of the Earth | | English Miles | | In Perigee | 23580 | 93280945 | | In Apogee | 24388 | 96478967 | | At Mean Distance | 23984 | 94879956 | | Greatest diameter of his orbit | 47969 | 189759912 |

When the sun's parallax is known, we are enabled not only to estimate his distance, but also to compare his diameter and magnitude with those of the earth. For example, the mean parallax of the sun is 8"56"; his mean apparent semidiameter, seen from the earth, is 961"8; the semidiameters of the sun and earth are therefore to each other in the ratio of 961"8 : 8"56, or of 112 to unity nearly. Supposing the sun and earth to be both spherical bodies, which is a supposition sufficiently accurate for the present purpose, their volumes will be to one another as the cubes of their radii; hence the volume of the sun is to that of the earth as the cube of 112 to unity, or the sun's volume is about 1,405,000 times greater than that of the earth.

Sect. III.—Of the Motion of Translation of the Earth, and the Aberration of Light.

Hitherto we have regarded the sun as being actually in motion round the earth in an elliptic orbit, of which the earth occupies one of the foci. This supposes the motion which we observe to be real. But we have already seen that all the phenomena of the diurnal revolution of the celestial bodies may be equally well explained by supposing these motions to be only optical illusions, of the same nature as those by which a person sailing rapidly along a river, or the sea-shore, is almost irresistibly led to ascribe the motion to the banks on the shore, even when fully aware that the appearance is only occasioned by his own motion in a contrary direction. In the same manner all the phenomena of the annual motion are susceptible of an equally simple explanation, on the hypothesis that the earth revolves in an elliptic orbit, and that the sun is at rest in one of the foci of the ellipse. In fact, if there were no other bodies in the universe than the earth and the sun, it would be matter of absolute indifference which of the two hypotheses we should adopt in order to explain the appearances. Supposing, for example, the earth to revolve in a circular orbit (the eccentricity, on account of its smallness, being here neglected), and the sun to be placed at the centre; a spectator at E (fig. 23) sees the sun S in the heavens at the point A, and corresponding to the star a on the celestial sphere. If the earth now take the place of the sun, and the sun be placed on the curve at A, the spectator will still see the sun correspond to the star a. Let the earth be transferred from E to E'; the spectator will then see the sun in A'. The sun therefore appears to him to have moved from A to A', which is exactly the same appearance as would be presented to a spectator at the centre by a real translation of S from A to A'; so that, in respect of the annual motion, the appearances, when referred to the celestial sphere, are precisely the same on either hypothesis.

If we now consider the phenomena with reference to the earth, we shall find that they may be all equally well explained on the hypothesis of the earth's motion, and the immobility of the sun. The most remarkable phenomenon connected with the annual revolution is the variation of the seasons; and in order to explain their cause, it is only necessary to suppose that the earth, in describing its oblique orbit, always preserves its axis parallel to the same straight line. Let A, B, C, D (fig. 24) repre- Theoretical sent the earth in four different positions of its orbit, as Astronomy being its axis, and n and s its north and south poles respectively. While the earth goes round the sun in its orbit, its axis ns preserves its obliquity, and always continues parallel to its first direction. At A the north pole inclines towards the sun, and brings all the northern places more into the light than at any other season of the year. But when the earth is at C, the opposite point of the orbit, the north pole declines from the sun, and a less portion of the northern hemisphere enjoys the blessings of his light and heat. At B and D the axis is perpendicular to the plane of the orbit, so that the poles are situated in the boundaries of the illuminated hemisphere, and the sun, being directly over the equator, makes the days and nights equal at all places. These phenomena are illustrated in fig. 23, which represents the situation of the north pole with regard to the limits of illumination, in eight different positions of the orbit. In this figure, AE is the terrestrial equator, T the tropic of Cancer, the dotted circle the parallel of London, U the arctic or north polar circle, and P the north pole, where all the meridians or hour-circles meet. The spectator is supposed to be placed at the pole of the ecliptic.

When the earth is at the beginning of Libra, about the 20th of March, the sun, as seen from the earth, appears at the beginning of Aries in the opposite part of the heavens, the north pole is just coming into light, and the sun is vertical to the equator, which, with all its parallels, is divided into two equal parts by the circle which forms the boundary between the dark and illuminated hemispheres, and therefore the days and nights are equal all over the earth. As the earth moves in the ecliptic according to the order of the letters A, B, C, D, &c., the north pole P comes more and more into the light, and the days increase in length at all places north of the equator AE. When the earth comes to the position between B and C, or the beginning of Capricorn, the sun, as seen from the earth, appears at the beginning of Cancer, about the 21st of June; and the north pole of the earth inclines towards the sun, so as to bring into light all the north frigid zone, and more of each of the northern parallels of latitude in proportion as they are farther from the equator. As the earth advances from Capricorn towards Aries, and the sun appears to move from Cancer towards Libra, the north pole recedes from the light, which causes the days to decrease and the nights to increase in length, till the earth comes to the beginning of Aries, and then they are equal as before; the boundary of light and darkness cutting the equator and all its parallels equally. The north pole then goes into the dark, and does not emerge till the earth has completed a semi-revolution of its orbit, or from the 23rd of September till the 20th of March. All these phenomena will be readily understood from the bare inspection of the figure; and it will be perceived that what has been said of the northern hemisphere is equally true of the southern in a contrary sense, that is, at opposite seasons of the year.

The only objection against the annual motion of the earth, which at first sight creates some difficulty, is the enormous distance which, on account of the want of annual parallax, that hypothesis makes it necessary to assign to the fixed stars. Abstracting from the precession, which in the region of the poles amounts only to about 20°, the axis of the earth, in every part of its orbit, appears to be directed towards the same point of the starry sphere. It is certain, however, that the radius of the terrestrial orbit is upwards of 90 millions of miles; and therefore, when the earth is at opposite points of this immense circle, its pole ought to be directed to points in the heavens 180 millions of miles distant from each other. Suppose the theoretical sun to be at rest in S (fig. 26), the centre of the orbit EoB, and the earth to be at the point E. Suppose also A to be the projection of S on the sphere of the fixed stars, Fig. 26, and r the first point of the ecliptic. The longitude of A, as seen from E, is the angle AEq; but when seen from S, the longitude of the same point is ASq. This last is the true longitude of A; AEq is the apparent longitude, and their difference is the angle SqE. Now, in the first half of the ecliptic, the apparent longitude will be less than the true, and in the second half greater; hence there ought to result an apparent annual inequality in the sun's longitude. But as such inequality is shown in very few cases, it becomes necessary either to abandon the hypothesis of the earth's motion, or to suppose the distance of the fixed stars to be so great, that, in comparison of it, the radius ES, which exceeds 90 millions of miles, is altogether insensible; and that the whole orbit of the earth, or the ellipse EoB, is a mere point in comparison of a great circle of the starry sphere.

The most convincing proof of the earth's motion is not to be found in any circumstance of which the senses can take immediate cognizance, but is afforded by the full development of the planetary system, and the mutual connection of all the truths of rational astronomy,—by the clearness, simplicity, and coherence which this hypothesis gives to the most complicated phenomena,—and can therefore only be fully appreciated after an attentive study of the whole series of facts which astronomy makes known. There exists, however, one direct proof of the earth's annual motion, in a phenomenon discovered by the accurate observation and patient sagacity of Bradley, although it is one which, we are almost tempted to think, ought to have been perceived a priori, after Roemer's discovery of the progressive motion of light. It is known by the name of the Aberration of Light, and is manifested in a small difference between the apparent and true places of a star of light occasioned by the motion of light combined with that of the earth in its orbit.

To illustrate this effect, conceive a body to move in the direction EE' (fig. 27), and another to impinge on it in the direction SE'. To find the direction of the resulting motion, take EC and EA proportional to the two velocities respectively, and, having completed the parallelogram EABC, draw the diagonal EB. The combination of the two motions produces an impression on the eye exactly similar to that which would have been produced if the eye had remained at rest in the point E, and the molecule of light had come to it in the direction ES'; the star, therefore, whose real place is at S, will appear to the spectator at E to be situated at S'. The difference between its true and apparent place, that is, the angle SES', is the aberration, the magnitude of which is obtained from the known ratio of EA to EB, or the velocity of light to that of the earth in its orbit. Now, we know from the phenomena of the eclipses of Jupiter's satellites, that a ray of light describes a line equal to the mean radius of the ecliptic in 8 min. 18 sec. or 498 seconds of time. But the arc described by the earth in that time is found from the proportion

\[ \frac{365^{\circ}256}{498^{\circ}} : \frac{360^{\circ}}{x} \]

whence \( x = 20^{\circ}45 \). It is evident, therefore, that when the directions of the two motions are at right angles, the star S always appears in advance of its real place, in a direction parallel to that of the motion of the earth, by a quantity equal to 20°45. This quantity, 20°45, is called the Constant of Aberration; but as it has been obtained on the assumption that the earth moves uniformly in a circular orbit, it is evidently not altogether exact, and recourse must be had to observation to determine its precise amount. Bradley supposed it to be 20°; Dr Brinkley, from the mean of 2653 comparisons of various stars has deduced the value 20°37'. From a series of observations made by M. Struve the Russian astronomer, the constant of aberration has been determined to be 20°445, which is probably an exceedingly close approximation, and may be considered as one of the most certain determinations of astronomy. (Sur le coefficient de l'aberration des étoiles fixes. Petersburg Memoirs, vol. iii.)

It is easy to see that, on the same principles, there ought also to be a diurnal aberration; but the diurnal rotation of the earth being sixty-five times less rapid than the orbital motion, its effect in producing aberration does not amount to more than 0°0208, when reduced to seconds of time. It is, however, taken into account in reducing transit observations, and incorporated with the error of collimation.

The magnitude of the angle of deviation SES depends on the relative directions of the earth and the visual ray, and may have any value from 0 to 20°4. Suppose, for example, we observe a star situated in the plane of the ecliptic. When the earth is at that point of its orbit, between the sun and the star, where the tangent to the orbit is perpendicular to the visual ray (which, on account that the star has no sensible parallax, always maintains a parallel direction), the apparent place of the star will be 20°4 to the eastward of its true place. When the earth is in the opposite point of its orbit the same star will appear to be 20°4 to the westward of its true place; so that it will appear to have an oscillatory motion on the ecliptic, the range of which is 40°8, and the period exactly a year. Half-way between these two points the tangent of the orbit is parallel to the direction of the ray of light, and consequently there is no aberration. When the star is not situated in the ecliptic, it will suffer a displacement in latitude as well as in longitude. To render this more intelligible, let EEE (fig. 28) be the ecliptic, E the earth, and A the true place of a star situated at any altitude above the ecliptic. In the direction EA take Ea to represent the velocity of light, ab that of the earth, and in a parallel direction, that is, parallel to the tangent to the ecliptic at E; the line Eb will now be the apparent visual ray, and the star will seem to be situated at B. Suppose the earth to be placed at different points of its orbit; the lines Ea will be all parallel to each other on account of the infinite distance of the star A; the lines ab will vary little in magnitude, because they are very small in comparison of Ea, but their directions will undergo every possible change, being always parallel to the tangent at E. At the two points of the orbit where the tangent is parallel to EA, the two lines Ea and ab coincide, and consequently there is no aberration. Let us next suppose the star to be situated in the pole of the ecliptic. In this case the visual ray is constantly perpendicular to the direction of the earth's motion, so that the star will always appear at a distance of 20°4 from its true place, or appear to describe a small circle about the pole of the ecliptic. In all other situations, out of the ecliptic, the star's apparent path will be an ellipse, the major axis of which, parallel to the plane of the ecliptic, is always 40°8, while the minor axis varies as the sine of the latitude.

The cause of the aberration being known, we have two methods of measuring its quantity, or the extent of the apparent oscillations, viz., by calculating the angle of deviation SAS (fig. 27), the tangent of which is to radius as AB to AE, that is, as 1 to 10,313; or, by direct observation, when the star is in opposition to the sun. The two methods give sensibly the same result, and the dimensions of the ellipse described by any star not situated in the ecliptic, in consequence of the aberration, are found to be the same when computed from theory or determined by direct observation. The motion of translation of the earth, therefore, receives a mathematical demonstration from this agreement; and the phenomenon of the aberration, otherwise unimportant on account of its minuteness, thus becomes one of the most interesting discoveries ever made in astronomy. The fact of the earth's orbital motion is, however, rendered so probable by other phenomena, that it must have been universally admitted, although the direct proof had never been discovered.

Sect. IV.—Of the Measure of Time. Equation of Time.

The notion of time, or of succession, is generally said to be acquired from the aspect of natural phenomena; but from whatever source it may be derived, it is certain that time itself can only be measured by comparing it with something of which the senses can take cognizance. If we have a series of events, such as the oscillations of a pendulum, or the flux and reflux of the sea, uniformly succeeding each other, then time may be measured by the number of such events that have been observed. In like manner, if a body move uniformly in a certain direction, time may be measured by the spaces over which it successively passes. In strict language, motion and time, being heterogeneous quantities, cannot measure one another; but different times are compared with each other by means of the motions that have taken place in those times respectively; for the motions being supposed uniform, equal spaces are passed over in equal times, or, which is the same thing, the times are directly proportional to the spaces described. If, therefore, we assume for unit the time which is absolved while a certain uniform motion takes place, we may obtain, by a simple proportion, the corresponding time of any other similar motion. In this manner we find the ratio of one portion of time to another, although we can form no idea whatever of its absolute quantity.

In order that motion may be employed as a measure of time, it is indispensably requisite that it be perfectly uniform. The only motions having this property, with which we are acquainted, are those of the rotation of the celestial bodies about their own axes. The motion of the sun and planets in their orbits is irregular from various causes; even the successive returns of the fixed stars to the meridian are rendered unequal on account of the precession and nutation; but the diurnal rotation of the earth appears to be affected by no cause of irregularity whatever. Time, therefore, may be properly measured by this uniform motion; and the method of proceeding is as follows. The culmination of a star, or of a given point of the equator, marks the instant at which the day commences, and at any other instant its horary angle determines the portion of the day which has elapsed at the moment of the observation. In order, therefore, to determine the time of an observation, we must find the horary angle formed by the meridian of the observed star with that which passes through the given point where the motion is supposed to begin; in other words, it is necessary to determine the star's right ascension. This, however, when attempted by direct measurement, is a determination attended with difficulty, and liable to considerable uncertainty. Happily the invention of the pendulum has rendered it unnecessary; and astronomers, instead of deducing the time from the right ascensions, determine, on the contrary, the right ascensions by means of the time indicated by the clock. The clock, of which the motion is supposed to be perfectly regular, is adjusted in such a Theoretical manner that its index describes 24 hours, while a point of the equator describes an arc of 360°, or makes a complete revolution. This is called a sidereal day.

The sidereal hour is divided in the usual manner into minutes, seconds, &c.; whence the corresponding arcs are easily found at the rate of 15° to an hour. The point whose culmination marks the origin of time is arbitrary; but astronomers have agreed to choose for that purpose the equinoctial point of Aries, from which the right ascensions are reckoned, so that the hours of the clock and the degrees of the equator may commence at the same instant. This point, it is true, is not, and cannot be, permanently marked by any star; but the right ascension in time of any star whatever being the hour of its transit over the meridian, the star will be in the plane of the meridian at the instant denoted by its right ascension in time. On this account sidereal time expresses the actual right ascension of the zenith; or, as it is frequently termed, the right ascension of the mid-heaven.

We have already seen, that on account of the precession of the equinoxes, a star employs somewhat more time than the first point of Aries in returning to the meridian. It is therefore not without some violation of language that the interval between two successive transits of a given point of the equator over the meridian is called a sidereal day, which, in its strict acceptation, denotes the time which elapses between two successive transits of the same star; but in this case the difference is so small as to be totally imperceptible. The annual precession in longitude is = 50°1; and that in right ascension is nearly the same, excepting in regard to the circumpolar stars, which, therefore, are not employed in regulating the clocks. This arc of 50°1, converted into time, gives 3½ seconds as the time in which a star will pass the meridian later than the equinoctial point, at the end of the year; and this small quantity being distributed over the whole year, is altogether insensible in short intervals of two or three days. The successive transits of a star, therefore, if we abstract from the nutation and aberration, will mark the sidereal day within the hundredth part of a second of time; and the sidereal year, though not immediately ascertainable by observation, becomes a quantity which may be easily computed. But the regularity of the motion of the stars is deranged by the effects of aberration and nutation; so that in order to measure time with the precision required by modern observers, it is necessary to be acquainted with the minute displacements of the stars. If they seem to return to the meridian after equal portions of absolute time, it is only because our organs are unable to distinguish the hundredths of a second.

The sidereal day is a measure of time which, on account of its uniformity and the facility of observing it, is excellently well adapted for astronomical purposes; but relatively to the ordinary wants of life it is not sufficiently marked,—the culmination of the stars is an event entirely unconnected with civil occupations, and which, for any given star, is even invisible during a great part of the year. The proper motion of the sun causes the sidereal day to commence sometimes by day and sometimes by night, so that great confusion and embarrassment would arise from regulating time and civil affairs by the motions of the stars. On this account the diurnal revolution of the sun has been universally adopted as the measure of time. This is called the civil day, and denotes the interval of time which elapses between two successive transits of the sun over the same hour-circle. Most nations have agreed in reckoning it from the inferior semicircle of the meridian, so that the civil day commences and terminates at midnight; but astronomers, in imitation of Hipparchus and Ptolemy, usually reckon the commencement of the day from the instant of the sun's culmination, that is, from noon; and count through the 24 hours from one noon to the following. Thus 9 o'clock in the morning of February 14th is by astronomers called February the 13th at 21 hours. The day thus determined is called the astronomical or solar day; and being regulated by the true motion of the sun, the time which is measured by it is called true or apparent time.

Astronomical or solar days are not equal. Two causes, Dave vary in particular, conspire to produce their inequality, namely, the unequal velocity of the sun in his orbit, and the obliquity of the ecliptic. The effect of the first cause is sufficiently sensible. At the summer solstice, when the sun's motion is slowest, the astronomical day approaches nearer the sidereal than at the winter solstice, when his motion is most rapid.

To conceive the effect of the second cause, it is necessary to have regard to the motion of the sun in reference to the equator. The sun describes every day a small arc of the ecliptic. Through the extremities of this arc suppose two meridians to pass; the arc of the equator, which they intercept, is the sun's motion for that day referred to the equator, and the time which that arc takes to pass the meridian is equal to the excess of the astronomical day above the sidereal. But it is obvious that at the equinoxes the arc of the equator is smaller than the corresponding arc of the ecliptic, in the proportion of the cosine of the obliquity of the ecliptic to radius: at the solstices, on the contrary, it is greater in the proportion of radius to the cosine of the same obliquity. The astronomical day is diminished in the first case, and lengthened in the second.

To have a mean astronomical day independent of these causes of inequality, astronomers have supposed a second astronomical sun to move uniformly on the ecliptic, and to pass over the extremities of the axis of the sun's orbit at the same instant with the real sun. This removes the inequality arising from the inequality of the sun's motion. To remove the inequality arising from the obliquity of the ecliptic, conceive a third sun to pass through the equinoxes at the same instant with the second sun, and to move along the equator in such a manner that the angular distances of the two suns at the vernal equinox shall be always equal. The interval between two consecutive returns of this third sun to the meridian forms the mean astronomical day. Mean time is measured by the number of the returns of this third sun to the meridian; and true time is measured by the number of returns of the real sun to the meridian. The arc of the equator, intercepted between two meridian circles drawn through the centres of the true sun and the imaginary third sun, when reduced to time, is what is called the Equation of Time. This will be rendered plainer by the following diagram.

Let Zφz (fig. 29) be the earth; ZFrz its axis; Fig. 29. obcd, &c., the equator; ABCDE, &c., the northern half of the ecliptic from φ to ψ, on the side of the globe next the eye; and MNOP, &c., the southern half on the opposite side from ψ to φ. Let the points at A, B, C, D, E, F, &c., mark off equal portions of the ecliptic; gone through in equal times by the real sun, and those at a, b, c, d, e, f, &c., equal portions of the equator described in equal times by the fictitious sun; and let Zφz be the meridian.

As the real sun moves obliquely in the ecliptic, and the fictitious sun directly in the equator, any point between φ and F on the ecliptic must be nearer the meridian Zφz than the corresponding point on the equator from φ to f, that is to say, than the point whose distance from φ is expressed by the same number of degrees; and the more so, as the obliquity is greater; and therefore the true sun comes sooner to the meridian every day whilst he is in the quadrant \( \alpha F \), than the fictitious sun does in the quadrant \( \alpha f \); for which reason the solar noon precedes noon by the clock, until the real sun comes to \( F \), and the fictitious to \( f \); which two points being equidistant from the meridian, both suns will come to it precisely at noon by the clock.

Whilst the real sun describes the second quadrant of the ecliptic FGHIKL from Cancer to \( \alpha \), he comes later to the meridian every day than the fictitious sun moving through the second quadrant of the equator from \( f \) to \( \alpha \); for the points at G, H, I, K, L, being farther from the meridian, their corresponding points at \( g, h, i, k, l \), must come to it later; and as both suns come at the same moment to the point W, they come to the meridian at the moment of noon by the clock.

In departing from Libra through the third quadrant, the real sun going through MNOQP towards \( \gamma \) at R, and the fictitious sun through mnopq towards \( r \), the former comes to the meridian every day sooner than the latter, until the real sun comes to R, and the fictitious to \( r \), and then they come both to the meridian at the same time.

Lastly, as the real sun moves equably through STUVW, from R towards \( \gamma \), and the fictitious sun through stuvw, from \( r \) towards \( \gamma \), the former comes later every day to the meridian than the latter, until they both arrive at the point \( \gamma \), and then they make it noon at the same time with the clock.

Having explained one cause of the difference between true and mean time, we now proceed to explain the other cause of this difference, namely, the inequality of the sun's apparent motion, which is slowest in summer, when the sun is farthest from the earth, and swiftest in winter, when he is nearest to it.

If the sun's motion were equable in the ecliptic, the whole difference between the equal time as shown by the clock, and the unequal time as shown by the sun, would arise from the obliquity of the ecliptic. But the sun's motion sometimes exceeds a degree in 24 hours, though generally it is less; and when his motion is slowest, any particular meridian will revolve sooner to him than when his motion is quickest; for it will overtake him in less time when he advances a less space than when he moves through a larger.

Now, if there were two suns moving in the plane of the ecliptic, so as to go round it in a year, the one describing an equal arc every 24 hours, and the other describing in the same time sometimes a less arc, and at other times a larger, gaining at one time of the year what it lost at the opposite; it is evident that one of these suns would come sooner or later to the meridian than the other, as it happened to be behind or before the other; and when they were in conjunction, they would both come to the meridian at the same moment.

As the real sun moves unequally in the ecliptic, let us suppose a fictitious sun to move equably in a circle coincident with the plane of the ecliptic. Let ABCD (fig. 30) be the ecliptic or orbit in which the real sun moves, and the dotted circle \( \alpha \beta \gamma \delta \) the imaginary orbit of the fictitious sun; each going round in a year according to the order of letters, or from west to east. Let HIKL be the earth turning round its axis in the same direction every 24 hours; and suppose the two suns to start from A and \( a \), at the same moment, A and \( a \) being in the same straight line, and in the plane of the meridian EH. Suppose also that the real sun, when at A, is at his greatest distance from the earth, where his motion is slowest. In the time that the meridian revolves from H to H again, according to the order of the letters HIKL, the real sun has moved from theoretical A to F; and the fictitious sun with a quicker motion from \( a \) to \( f \), through a larger arc; therefore the meridian EH will revolve sooner from H to \( h \) under the real sun at F, than from HE to \( k \) under the fictitious sun at \( f \); and consequently it will then be noon by the sun-dial sooner than by the clock.

As the real sun moves from A towards C, the velocity of his motion increases all the way to C, where it is at its maximum. But notwithstanding this, the fictitious sun gains so much upon the real, soon after his departing from A, that the increasing velocity of the real sun does not bring him up with the equally-moving fictitious sun till the former comes to C and the latter to \( c \), when each has gone half round its respective orbit; and then being in conjunction, the meridian EH, revolving to \( EK \), comes to both suns at the same time, and therefore it is noon by them both at the same moment.

But the increased velocity of the real sun now being at its maximum, carries him before the fictitious one; and therefore the same meridian will come to the fictitious sun sooner than to the real; for whilst the fictitious sun moves from \( c \) to \( g \), the real sun moves through a greater arc from C to G; consequently the point K has its noon by the clock when it comes to \( k \), but not its noon by the sun till it comes to \( l \). And although the velocity of the real sun diminishes all the way from C to A, and the fictitious sun by an equable motion is still coming nearer to the real sun, yet they are not in conjunction till the one comes to A and the other to \( a \), and then it is noon by them both at the same moment.

Mean solar time and sidereal time being both uniform, it is easy to compare the one with the other, and assign the number of degrees, minutes, &c., which the sun and a star will respectively describe in a given portion of sidereal or mean solar time. In a mean solar day the sun's right ascension and mean longitude are increased by \( 59^\circ 8'33'' \); consequently \( 360^\circ 59'8''33'' \) of the equator pass the meridian in 24 mean solar hours. The sidereal time corresponding to this period is 24 hours 3 min. 56'555 sec.; therefore 24 mean solar hours are equal to 24 hours 3 min. 56'555 sec. of sidereal time; and 24 hours of sidereal time are equal to 23 hours 56 min. 4'0907 sec. of mean solar time, or to 24 hours minus 3 min. 55'9093 sec. This difference of 3 min. 55'9093 sec. is called the acceleration of the fixed stars in mean solar time; and the preceding excess of 3 min. 56'555 sec. is the retardation of the sun in sidereal time. Hence the one species of time may be easily converted into the other; and the arc of the equator passed over by the meridian in a given mean time may be calculated. Thus,

\[ \frac{360^\circ 59'8''33''}{24} = 15^\circ 2'27''84708 \]

is the arc described by a star in one hour of mean solar time.

But the process of converting sidereal or mean solar time into true or apparent time, or of computing from the instants of apparent time the corresponding mean solar and sidereal times, is attended with much greater difficulty. The reason is, that the interval between two successive transits of the sun over the meridian, which, in apparent time, measures the day, is a variable quantity; and hence there cannot exist any constant ratio between true and mean time, as there does between mean and sidereal time. The correction or equation by which apparent time is reduced to mean time, is technically called the Equation of Time, and is composed of the aggregate of the several variable terms which denote the inequalities of the sun's motion in longitude. Besides the eccentricity, Theoretical Astronomy.

The obliquity of the ecliptic, and the variations of that obliquity occasioned by the nutation of the earth's axis, is affected also by the small alterations of the sun's right ascension, which result from the effect of the planetary perturbations on the earth; and hence the equation of time cannot be exactly computed without the aid of Physical Astronomy.

It will be evident, from what has preceded, that the equation of time expresses merely the difference between the true and mean right ascensions of the sun, reduced to time. Its different parts may be calculated numerically in the following manner. Let NA (fig. 31) represent the mean motion of the sun during a given interval, then NA is the sun's mean longitude, N being the first point of Aries. Let NA = M, and from this mean longitude subtract the longitude of the apogee, the remainder will be the sun's mean anomaly. From the mean anomaly let the equation of the centre be found and denoted by E. Take AB = E, then NB = M + E is the longitude corrected for the eccentricity.

Let us next suppose BC to be the small quantity by which the sun's longitude is increased in consequence of the perturbations of the planets, and let BC = P; then NC = M + E + P is the true and exact longitude of the sun. Through C let the arc CD be drawn perpendicular to the equator; the point D will be that point of the equator which passes the meridian at the same time with the sun. Let R = NC - ND = the reduction to the ecliptic; we have then the sun's right ascension = M + E + P - R.

Let NF = NA, and F will be the place which the sun would occupy in the equator at the same instant that he occupies the point A in the ecliptic, if he moved uniformly in the former circle; for NF as well as NA will represent the mean diurnal motion of 59° 8' 33" multiplied by the number of days elapsed in the interval between the equinox N and the time of the observation. The mean sun would therefore pass the meridian with the point F, whereas the true sun passes it with the point D; therefore, at the instant of true noon, when the sun C and the corresponding point D are on the meridian, the mean sun is at a distance from D expressed by the arc FD = M + E + P - R - M = E + P - R. Now, the arc of the equator FD measures the horary angle between the mean and the true sun, or the angle at the pole between the meridian and the hour-circle passing through D; it is therefore converted into time by the following proportion:

\[ \frac{360^\circ}{FD} : 24 \text{ mean solar hours} : \text{time from true noon}; \]

consequently the difference between mean and true noon, or the equation of time,

\[ \frac{360^\circ}{FD} = \frac{1}{15}(E + P - R) = dT. \]

But this equation is not yet perfectly accurate: it requires to be corrected for the effects of nutation. Now it is known that the variation of the mean longitude of the sun, arising from the unequal precession of the equinoxes in consequence of the nutation occasioned by the inclination of the lunar orbit, is expressed by the formula

\[ 18^\circ \sin(360^\circ - \text{moon's node}) = 18^\circ \sin N. \]

This variation, reduced to the direction of the equator, will therefore be \(18^\circ \sin N \cos e\) (e being the obliquity of the ecliptic). The difference between the two expressions is

\[ 18^\circ \sin N (1 - \cos e) = 36^\circ \sin^2 \frac{1}{2} \sin N = 1^\circ \cdot 4887 \sin N, \]

which, reduced to time, becomes 0.09925 sec. × sin N. This small correction, amounting to less than a tenth of a second, was long omitted in computing the equation of time. When it is included, that equation becomes

\[ dT = \frac{1}{60}(E + P - R) + 0.09925 \text{ sec.} \times \sin N. \]

The quantities E, P, R, and N, are to be computed separately from the astronomical tables; and it must be observed, that the result will be expressed in mean solar time.

The cosine of the obliquity, that is, \( \cos 23^\circ 28' \), is equal to \( \frac{1}{9173} \) very nearly. Hence, since the equation of time is equal to the sun's true right ascension, diminished by his mean longitude and the effects of nutation in right ascension (\( ND - NA = 18^\circ \sin N \cos e \)), it may, on denoting the true right ascension by A, be expressed as follows:

\[ dT = A - M = 18^\circ \sin N \times \frac{1}{15}. \]

This is the form under which the equation of time was expressed by Dr Maskelyne.

The equation of time is at its maximum about the 3d of November, when it amounts to 16' 16" 7', and is subtractive. At four different times of the year it vanishes, namely, about the 25th of December, the 16th of April, the 16th of June, and the 1st of September. These epochs, however, do not remain constant; for, on account of the change which the line of the apsides is constantly undergoing in reference to the line of the equinoxes, the difference between the true and mean right ascensions of the sun—in other words, the equation of time—varies continually in different years.

Sect. V.—Of the Spots of the Sun, his Rotation, and Constitution.

The sun, the great source of light, heat, and animation, when beheld with the naked eye, appears only as a luminous mass of uniform splendour and brightness; but when examined with the telescope, his surface is frequently observed to be mottled over with a number of dark spots, of irregular and ill-defined forms, and constantly varying in appearance, situation, and magnitude. These spots are occasionally of immense size, so as to be even visible without the aid of the telescope; and their number is frequently so great that they occupy a considerable portion of the sun's surface. Dr Herschel observed one in 1779, the diameter of which exceeded 50,000 miles, more than six times the diameter of the earth; and Scheiner affirms that he has seen no less than 50 on the sun's disk at once. Most of them have a deep black nucleus, surrounded by a fainter shade, or umbra, of which the inner part, nearest to the nucleus, is brighter than the exterior portion. The boundary between the nucleus and umbra is in general tolerably well defined; and beyond the umbra a stripe of light appears more vivid than the rest of the sun.

The discovery of the sun's spots has been attributed to different astronomers. They appear to have been first taken notice of in a work of Fabricius, the friend of Kepler, which was published at Wittenberg in 1611, under the title of Joh. Fabritii Phrygii de Moculis in Sole Observatis, et Apparente carum cum Sole Conversione Narratio. It contains, however, nothing more than a few vague conjectures respecting the spots, the phenomena of which he could not have observed with any degree of accuracy, insomuch as he seems to have been unacquainted with any method of protecting the eye by intercepting a portion of the solar rays; for he recommends to those who should repeat his observations, to admit into the telescope only a small portion of the sun at once, till the eye should by degrees become able to support the full blaze of light. About the same time the discovery was warmly disputed by the illustrious Galileo, and Scheiner, a German Jesuit, professor of mathematics at Ingolstadt. The whole circumstances connected with this dispute are narrated at great length by Galileo in his work entitled Historia e Di- Theoretical mostrazioni intorno alle Macchie Solari, e loro Accidenti.

Astronomy from which it appears certain that he observed the spots so early as April 1611. In a letter published by him in 1612, he remarks that the spots are situated on the surface of the sun, or that at least their distance from it is imperceptible; that the time of their continuance varies from 2 or 3 to 30 or 40 days; that their figures are irregular and variable; that some are seen to separate, and others to unite, even on the middle of the disk; that besides these peculiar motions, they have also a common motion, in virtue of which they traverse the disk in parallel lines. From this general motion he infers that the sun turns on an axis from west to east; and he adds as a curious remark, that the spots are confined within a zone extending only about 28° or 29° degrees to the north and south of the sun's equator. Galileo illustrates all these positions by mathematical reasoning, and by drawings of the spots made on many successive days.

Scheiner's observations were first announced in January 1612, in three letters addressed to his friend Marc Velsler, a magistrate of Augsburg. In the first of these, the date of which is November 1611, he says that he had observed the spots seven months before, but that, having a different object in view, he had given little attention to them. He observed them again in the following October, and at that time imagined the appearance was owing to some imperfection of his telescope; till he was convinced by repeated observations that it was necessary to refer it to the sun. From these remarks it is pretty clear that Scheiner had formed no accurate notions respecting the spots before October 1611, that is, six months after they had been observed by Galileo. Scheiner made the observation of the solar spots his whole occupation during the following eighteen years, in the course of which he discovered the position of the solar equator, and formed a theory much more complete than that of Galileo. The account of his observations was published in 1630, under the title of Rosa Ursina, sive Sol ex admirando Facicularum et Macularum sursum Phænomeno Varius, &c.

The discovery of the solar spots has also been claimed for our countryman Harriot. Amidst these conflicting pretensions it is perhaps impossible to arrive at the truth; but the matter is of little importance; the discovery is one which followed inevitably that of the telescope, and an accidental priority of observation can hardly be considered as establishing any claim to merit.

The solar spots furnish an extensive subject of curious speculation, but in an astronomical point of view they are chiefly interesting on account of their establishing the fact, and affording the means of determining the period of the rotation of the sun. In order to obtain a precise idea of the position of a spot, and the path which it describes, it is necessary to project that path on the plane passing through the centre of the sun, and perpendicular to the visual ray drawn from the earth to the sun's centre.

Suppose the diameter of a circle ASB (fig. 32) to be divided into as many parts of unity as there are seconds in the apparent diameter of the sun. Let CP be taken equal to the number of seconds contained in the difference of the longitudes of the spot and the sun's centre, and the perpendicular PM equal to the number of seconds in the latitude of the spot; then M will represent its position on the surface of the sun. By repeating the same operation a number of days consecutively, a series of points M M' M'', &c., will be obtained in the apparent path of the spot on the sun's disk, or rather in the projection of that path on the plane perpendicular to the visual ray. This projection is in general an oval slightly differing from an ellipse; and it is found that all the spots observed at the same time describe similar and parallel curves. They also return to the same relative positions in the same time, and their period is about 27½ days.

The paths described by the spots undergo very considerable changes, according to the season of the year at which they are observed. About the end of November and beginning of December they appear simply as straight lines Mw, Mw', Mw'' (fig. 33), along which the spots move in the direction Mw, that is, they enter on the eastern and disappear on the western edge of the sun's disk; and the points at which they disappear are more elevated, or nearer the north pole of the ecliptic, than those at which they enter. After a certain time the lines Mw begin to assume a curved appearance, and form ovals, as represented in fig. 34. During the winter and spring Fig. 34, the convexity of the ovals is turned towards the north pole of the ecliptic; but their inclination, or rather the inclination of the straight lines joining their extreme points, to the plane of the ecliptic continues to diminish, and about the beginning of March disappears; so that the points at which they seem to enter and leave the sun's disk are equally elevated, as in fig. 35. From this Fig. 35, time the curvature of the ovals diminishes; they become narrower and narrower till about the end of May or beginning of June, when they again appear under the form of straight lines (fig. 36); but their inclinations to the Fig. 36, ecliptic are now precisely in a contrary direction to what they were six months before. After this they begin again to expand, as in fig. 37, and their convexity is now turned LXXIX, towards the south pole. Their inclinations also vary at Fig. 37, the same time, and about the commencement of September they are seen as represented in fig. 38; the points Fig. 38, at which they enter and disappear being again equally elevated. After this period the ovals begin to contract and become inclined to the ecliptic, and by the beginning of December they have exactly the same direction and inclination as they had the previous year.

These phenomena are renewed every year in the same order, and the same phases are always exhibited at corresponding seasons. Hence it is evident that they depend on a uniform and regular cause, which is common to all of them, since the orbits described by the various spots are exactly parallel, and subject in all respects to the same variations. The simplest method of explaining the phenomena is to suppose with Galileo that the spots are adherent to the surface of the sun, and that the sun uniformly revolves round an axis inclined to the axis of the ecliptic. If the axis of revolution were perpendicular to the plane of the ecliptic, the spots, supposing them to adhere to the sun's surface, would describe circles parallel to that plane, which, seen from the earth, would appear as so many parallel straight lines; but by supposing the axis to have a suitable inclination, all the phenomena become explicable in a very simple manner. While the sun is carried round in his orbit, his axis, constantly preserving its parallelism, will successively assume different positions relatively to the earth; and the planes of the circles described by the spots, which planes are always perpendicular to the axis, will consequently be presented to us under different inclinations; hence the variations of their apparent curvature. In two opposite points of the orbit the visual ray drawn from the earth to the centre of the sun is perpendicular to the axis of rotation. In these two positions the poles of the sun, or the points in which the axis meets the surface, are both visible at the same moment, and the spots appear to move in parallel straight lines. But as the axis retains this perpendicular position only for an instant, and declines from it very sensibly while the spot traverses the sun's disk, the path of Theoretical the spots over the entire disk is neither a straight line nor an ellipse, one of which it would necessarily be if the sun, while revolving about his axis, did not change his place in his orbit. When the axis is not perpendicular to the visual ray, the path of the spots will appear to be a curve of which the concavity is turned towards that pole which is visible from the earth. This inclination of the solar axis to the plane of the ecliptic also explains the reason why the points of the disk at which the spots appear are more elevated during one half of the year, and more depressed during the other half, than those at which they disappear. It will also follow from the same hypothesis, that the curvature of the ovals must be the greatest possible when the straight lines joining their extremities are parallel to the ecliptic; and, on the contrary, least when the same straight lines are most inclined to the ecliptic; all which is exactly conformable to observation.

The various appearances which we have now described may be accurately represented by means of a common celestial or terrestrial globe. Let the wooden horizon of the globe, which is here supposed to represent the sun, be placed horizontally in the same plane with the eye of the spectator, and the pole be inclined about 7° from the zenith. The wooden horizon will now represent the ecliptic; and if the spectator walk round the globe, always keeping his eye in the plane of the wooden horizon, the circles of latitude will appear to him as ellipses of different inclinations and eccentricities: in two opposite points they will appear as straight lines, and, in short, exhibit in their various positions all the phenomena of the oval paths described by the spots of the sun.

The consequences deduced from the hypothesis of the rotatory motion of the sun are so perfectly conformable with observation, as to render the inference inevitable, that the sun revolves from west to east, on an axis inclined about 93° to the plane of the ecliptic. The plane which passes through the centre of the sun, perpendicular to the axis of rotation, is the Equator of the sun; the straight line joining the points in which it intersects the ecliptic is called the Line of the Nodes of the equator. The Nodes themselves are the two opposite points in which this straight line, produced indefinitely, meets the celestial sphere.

In order to determine the situation of the solar axis in space, it is necessary to find its inclination to the ecliptic, and the angle which the line of the nodes makes with any given line on the plane of the ecliptic, for example, with the line of the equinoxes. The requisite data for the solution of this problem are three different positions of the same spot, which must be obtained by observation.

Let S (fig. 39) be the centre of the sun's disk, AB a parallel to the terrestrial equator, and M the place of a spot, the co-ordinates of which, as referred to AB, are SX and MX. In this figure the earth is supposed to be situated in a straight line passing through S perpendicular to the plane of the paper, and it is to be recollected that SX and MX are in fact arcs of the solar globe, though so small when seen from the earth that they may be regarded as straight lines. By comparing the times of the transits of B, the border of the disk, and the spot M, we shall find BX, from which, as SB the semidiameter of the sun is known, we shall have SX the difference of the right ascensions of the spot and sun's centre. The line CY, which is the difference of the declination of the border of the disk and that of the spot, is measured by the micrometer. This will give MX the declination of the spot.

Now, tan. MSX = \frac{MX}{SX}, and SM = \frac{SX}{\cos. MSX}; therefore SM is also a known quantity. Let EE be the ecliptic, e = obliquity of the ecliptic, and O = the longitude of the sun. The angle BSE, which is technically called the Angle of Position, is the complement of the angle made by the ecliptic with the circle of declination passing through the sun, and is therefore given by the formula tan. BSE = tan. e cos. O; hence MSE (=MSX—BSE) is also given. On SE let fall the perpendicular MM, then MM = SM sin. MSE is the geocentric latitude of the spot, and SM = SM cos. MSE is the difference between its geocentric longitude and that of the centre of the sun; the object is now to determine the angles subtended by these lines at the centre of the sun, that is, to convert the geocentric into heliocentric latitudes and longitudes.

Suppose two straight lines to be drawn from the earth, one to the centre of the sun, and the other to the centre of the spot M, and let \( \theta \) be their inclination, which is measured by SM, and is consequently known, being the geocentric distance of the spot from S, the centre of the sun's disk. Let R = distance of the sun's centre from the earth, \( r \) = semidiameter of the sun, \( \phi \) = the angle made at the centre of the sun by the straight lines drawn from it to the earth and the spot M, and let \( \psi \) be the remaining angle of the triangle formed by the lines joining the earth, the centre of the sun, and the spot. We have then \( r : R :: \sin. \theta : \sin. \psi \), whence \( \sin. \psi = \frac{R}{r} \sin. \theta \). But \( R : r \) as radius to the sine of half the sun's true diameter; therefore \( \sin. \psi = \frac{\sin. \theta}{\sin. \frac{1}{2} \text{sun's diameter}} \).

Now \( \phi = 180° - \psi - \theta \), consequently we have the number of degrees, minutes, and seconds in \( \phi \), the heliocentric distance of the spot from the straight line which joins the centres of the earth and sun. Having thus obtained the hypotenuse SM in parts of a great circle of the solar orb, the sides SM and MM will be obtained in similar parts from the common trigonometrical formulae for the resolution of a right-angled spherical triangle. These formulae give

\[ \tan. SM = \tan. SM \cos. MSE, \] \[ \sin. MM = \sin. SM \sin. MSE. \]

It will be remarked that MM is the heliocentric latitude of the spot, SM the difference of its longitude and that of the earth; and as the longitude of the earth is equal to 180° + that of the sun, the heliocentric longitude of the spot will be 180° + O — SM if the spot is behind or to the east of the centre of the sun, and 180° + O + SM if it precedes the centre.

In this manner the heliocentric longitudes and latitudes of the spots are deduced from their observed right ascensions and declinations. The next step is to show how they are employed in determining the position of the solar axis. This problem is in practice somewhat laborious, although the principles on which its solution rests are sufficiently simple. The planes of the circles described by the spots are parallel to the sun's equator; if, therefore, the position of one of them can be found, the position of the equator, and consequently of the axis, will be found at the same time. Now, the position of a plane is determined by three given points through which it is required to pass; consequently, by three observations of the same spot, we shall have three points in its plane, and thence the plane itself. The problem is therefore one of pure geometry, and may be solved in various ways. The results of the most accurate observations make the inclination of the solar equator to the ecliptic amount to 7° 19' 23" or, 7° 1/2 very nearly; and the heliocentric longitude of the ascending node, that is, the point in which the equator of the sun intersects the ecliptic, in passing from Theoretical south to north, 80° 7' 4". The position of the node seems Astronomy to undergo no variation, except such as may be supposed to arise from the precession of the equinoxes.

It has already been mentioned that the mean time in which a solar spot returns to the same position, relatively to the earth, is 27-3 days. This, however, is not the time in which the sun makes a revolution about his axis. In the interval of 27-3 days the sun describes in the ecliptic an arc equal to 26°91 (for 365°25 : 27°3 :: 360° : 26°91), and by virtue of this motion alone he exhibits every day different points of his surface, the whole of which would be successively shown to the earth in the course of a year, independently of the motion of rotation. Hence results an apparent annual rotation round an axis perpendicular to the ecliptic, and we must abstract the effects of this optical illusion in order to arrive at the time of a real rotation. This apparent rotatory motion will be easily understood by referring to fig. 40, in which S is the sun, E the earth, and C the point in which the visual ray ES intersects the surface of the sun. Suppose the sun to have advanced in his orbit from S to S', the visual ray drawn from the sun's centre to the earth will now meet the surface in a different point C', and the angular distance between C and C' will be found by drawing S'E' parallel to SE; for the point c in which SE meets the surface will correspond to C, and the arc Cc, or the angle ESE', will be the apparent rotation, while the sun advances in his orbit from S to S'. Now, the angle ESE' is equal to SES, or the apparent rotatory motion is equal to the angular motion of the sun in his orbit; hence, since the real rotation is in a contrary direction, it is obvious that, if the axis were perpendicular to the plane of the ecliptic, when a spot appears to have made one revolution, it has in reality passed over an arc equal to 360° + 26°91, or 386°91 degrees. By a simple proportion, therefore,

\[ \frac{386°91}{360°} = \frac{27°3}{25°4}; \]

and consequently 25°4 days is the real time of the sun's revolution. This result is, however, not quite accurate, on account of its having been supposed that the axis of rotation is perpendicular to the ecliptic, whereas it differs from a perpendicular to that plane by 7°3; but the correction necessary on account of the difference is so small as to fall far within the limits of the errors to which the observations are liable; it is therefore unnecessary to have regard to it.

Observations of the spots, from which the elements of the sun's rotation are deduced, are attended with a very considerable degree of uncertainty. The semidiameter of the sun, which at the surface of the earth subtends an angle of only 16", or 960°, is equivalent to 90° at the centre of the sun. Hence the error of a single second (and it is impossible to answer for one or two seconds in observations of this sort) corresponds to about 338", or 5° 38", on the surface of the solar globe. An observer may, therefore, notwithstanding the greatest care, be mistaken to the extent of 10" with regard to the length of a heliocentric arc; and when to this we add, that the margins of the spots are ill defined, and even changeable, so that their centres, which it becomes necessary to observe, are not always the same, it will not appear surprising that a considerable discordance exists among the results of different observations. The uncertainty respecting the time of rotation amounts to no less than 10 or 12 hours. From a careful discussion of a numerous set of observations, Delambre found the time of rotation to be only 25°01154 days, instead of 25°4 given above; and he remarks that, in order to render the last-mentioned number admissible, it is necessary to suppose either an error of 2° 22', which is scarcely possible, or else that the spot has a proper mo-

| Node | Inclination | Revolution | Synodic Revol. | |------|-------------|------------|---------------| | 1 | 80°45'7" | 7°19'17" | 25°0°17" | | 2 | 79°21'35" | 7°12'37" | 25°4°17" | | 3 | 80°33'40" | 7°16'33" | Diurnal motion 14°394 |

With regard to observations on the sun's spots, Delambre remarks that he attaches little value to them, first, because it is impossible to make them well; and, secondly, because, even if they were sure, they only lead to results of little importance to astronomy. He discussed a hundred different spots, each observed at least three times by Messier, and deduced thirty different determinations of the elements of rotation. The more he multiplied his calculations, the more certain he became of the impossibility of a good solution; of which, indeed, there is no other chance than in a compensation of errors, little probable on account of their enormous magnitudes. These discrepancies render it probable that the spots, besides partaking of the general motion of the solar globe, have also proper motions either of displacement, or occasioned by a change of form, which may long prevent this part of astronomy from reaching a greater degree of exactness than it has already attained.

From the circumstance that the sun is incontestably endowed with a rotatory motion, Lalande concluded that, according to the rules of probability, he ought also to have a motion of translation in space. This idea was adopted by Herschel, and has since been sufficiently proved. It explains some, though not all, of the proper motions which have been supposed to be observed among the stars. If, indeed, the stars are so many suns, revolving like ours each on its own axis of rotation, it is extremely probable that they have also motions of translation; and thus, there being no fixed points in the heavens, the problem of their proper motions becomes so complicated as to be altogether insoluble.

The only interesting fact which has been deduced from the observations of the solar spots is the rotation of the sun. The curious appearances which they exhibit have, however, attracted great attention, and given rise to numerous theories respecting the constitution of that immense body which governs and vivifies the planetary system. We will now proceed to give a brief account of the appearances, and mention some of the theories that have been founded on them; promising, that after all that has been written on this subject, we are not yet, and probably never will be, in possession of any definite knowledge. The nature of the spots, and the physical constitution of the sun, afford fruitful subjects of harmless conjecture and speculation, but form no part of science.

The phenomena of the solar spots, as delivered by Appear-Scheiner and Hevelius, may be summed up in the following particulars. 1. Every spot which has a nucleus, or the spots comparatively dark part, has also an umbra, or fainter shade surrounding it. 2. The boundary between the nucleus and umbra is always distinct and well defined. 3. The increase of a spot is gradual, the breadth of the nucleus and umbra dilating at the same time. 4. In like manner the decrease of a spot is gradual, the breadth of the nucleus and umbra contracting at the same time. The exterior boundary of the umbra never consists of sharp angles, but is always curvilinear, how irregular soever the outline of the nucleus may be. The nucleus of a spot, whilst on the decrease, often changes its figure by the umbra encroaching irregularly upon it, insomuch that in a small space of time new encroachments are discernible, whereby the boundary between the nucleus and umbra is perpetually varying. It often happens, by these encroachments, that the nucleus of a spot is divided into two or more nuclei. The nuclei of the spots vanish sooner than the umbra. Small umbrae are often seen without nuclei. An umbra of any considerable size is seldom seen without a nucleus in the middle of it. When a spot which consisted of a nucleus and umbra is about to disappear, if it is not succeeded by a facula or spot brighter than the rest of the disk, the place where it was is soon after not distinguishable from the rest.

In the Philosophical Transactions, vol. lixv. Dr Wilson, late professor of astronomy at Glasgow, has given a dissertation on the nature of the solar spots, in which he mentions the following appearances. 1. When the spot is about to disappear on the western edge of the sun's limb, the eastern part of the umbra first contracts, then vanishes, the nucleus and western part of the umbra remaining; then the nucleus gradually contracts and vanishes, while the western part only of the umbra remains. At last this disappears also; and if the spot remains long enough to become again visible, the eastern part of the umbra first becomes visible, then the nucleus; and when the spot approaches the middle of the disk, the nucleus appears environed by the umbra on all sides, as already mentioned. 2. When two spots lie very near to one another, the umbra is deficient on that side which lies next to the other spot; and this will be the case, though a large spot should be contiguous to one much smaller; the umbra of the large spot will be totally wanting on that side next the small one. If there are little spots on each side of the large one, the umbra does not totally vanish, but appears flattened or pressed in towards the nucleus on each side. When the little spots disappear, the umbra of the large one extends itself as usual. This circumstance, he observes, may sometimes prevent the disappearance of the umbra in the manner above mentioned; so that the western umbra may disappear before the nucleus, if a small spot happens to break out on that side.

In the same volume, p. 337, the Rev. Dr Wollaston observes that the appearances mentioned by Dr Wilson are not uniform. He positively affirms that the facula or bright spots on the sun are often converted into dark ones. "I have many times," says he, "observed near the eastern limb a bright facula just come on, which has the next day shown itself as a spot, though I do not recollect to have seen such a facula near the western one after a spot's disappearance. Yet, I believe both these circumstances have been observed by others, and perhaps not only near the limbs. The circumstance of the facula being converted into spots, I think I may be sure of. That there is generally, perhaps always, a mottled appearance over the face of the sun, when carefully attended to, I think I may be as certain. It is most visible towards the limbs, but I have undoubtedly seen it in the centre; yet I do not recollect to have observed this appearance, or indeed any spots, towards the poles. Once I saw, with a twelve-inch reflector, a spot burst to pieces while I was looking at it. I could not expect such an event, and therefore cannot be certain of the exact particulars; but the appearance, as it struck me at the time, was like that of a piece of ice when dashed on a frozen pond, which breaks to pieces and slides in various directions." He also observes, that the nuclei of the spots are not always in the middle of the umbrae, and gives the figure of one seen on the 13th of November 1773, which is a remarkable instance to the contrary.

The faculae, or bright spots, were observed with particular attention by Messier, who frequently saw them enter on the eastern limb of the sun, disappear as they approached the centre of the disk, re-appear on the opposite limb, and continue visible, as they had done at the time of their first appearance, for about three days, till they were carried off the disk by the rotation of the sun. Spots frequently broke out in these faculae, and when this did not happen, they were succeeded by spots which generally became visible on the following day; and from the regularity of this occurrence, he was enabled to predict the appearance of a spot 24 hours before it entered on the sun's disk. He observed, also, that the magnitude of the spots was proportional to the brightness of the antecedent facula. Like the spots, the faculae are generally confined to the equatorial regions of the sun; but they have been occasionally observed by Schroeter on every part of the disk.

To explain these singular appearances, numerous theories, more or less plausible, have been proposed, but all regarding resting on many gratuitous assumptions, and subject to the spots' great difficulties. Scheiner imagined that the spots do not belong to the sun, but supposed them to be inferior planets revolving at no great distance from the central luminary. Galileo, Hevelius, and others, have supposed them to be scoria floating in the inflammable liquid matter of which they imagined the sun to be composed. This opinion, although it accounts for the appearance of the spots in the equatorial regions of the sun, to which such scoria would be carried by the centrifugal force resulting from his rotation, cannot be reconciled with the regularity with which the spots frequently re-appear on the eastern limb of the sun. Dr Wilson, having observed that the spots situated near the edge of the disk are narrow, and without a penumbra on the side next the centre, and that only the central spots are completely surrounded by a penumbra (appearances which would be exactly represented by a conical gulf or cavity presented to us under different aspects by the revolution of the sun), was led to adopt the opinion, that the appearances of the spots are occasioned by real excavations in the solar globe. He supposed the sun to consist of a dark nucleus, covered only to a certain depth by a luminous matter, not fluid, through which openings are occasionally made by volcanic or other energies, permitting the solid nucleus of the sun to be seen; and that the umbra which surrounds the spot is occasioned by a partial admission of the light upon the shelving sides of the precipice opposite to the observer. It is evident that, in proportion as these excavations are seen obliquely, their apparent dimensions will be diminished; one of the edges will disappear as it approaches the sun's limb, or come more into view as it advances towards the middle of the disk; when the spot is about to leave the disk, the bottom of the excavation, or the nucleus seen through it, will first disappear, but a sort of faint or obscure spot will remain visible as long as the visual ray penetrates the cavity. These appearances are all conformable to the laws of perspective; but Dr Wilson, wishing to give a still more palpable demonstration of the accuracy of his theory, fitted up a large globe, into which holes of the proper dimensions were inserted; and this machine being placed at a distance, and made to revolve, was found, when examined through a telescope, to exhibit in the course of its revolution all the phenomena of the solar spots. Dr Wilson's theory was keenly combated by Lalande, who adduced several observations of his own, and some by Cassini, that could not be explained by means of it; and urged with reason, that an hypothesis, founded on the uniformity of appearances which in reality are exceedingly variable, was entitled to little consideration. Lalande himself supposed the spots to be scoriae which have settled or fixed themselves on the summits of the solar mountains; an opinion which he grounded on this circumstance, that some large spots which had disappeared for several years were observed to form themselves again at the identical points at which they had vanished.

The late Sir William Herschel, with a view to ascertain more accurately the nature of the sun, made frequent observations upon it from the year 1779 to the year 1794. He imagined the dark spots on the sun to be mountains, which, considering the great attraction exerted by the sun upon bodies placed at its surface, and the slow revolution it has upon its axis, he thought might be more than 300 miles high, and yet stand very firmly. He says that in August 1792 he examined the sun with several powers from 90 to 500, when it evidently appeared that the dark spots are the opaque ground or body of the sun, and that the luminous part is an atmosphere, through which, when interrupted or broken, we obtain a view of the sun itself. Hence he concluded that the sun has a very extensive atmosphere, consisting of elastic fluids that are more or less lucid and transparent, and of which the lucid ones furnish us with light. This atmosphere, he thought, cannot be less than 1843, nor more than 2765 miles in height; and he supposed that the density of the luminous solar clouds needs not be much more than that of our aurora borealis, in order to produce the effects with which we are acquainted. The sun, then, if this hypothesis be admitted, is similar to the other globes of the solar system with regard to its solidity, its atmosphere, its surface diversified with mountains and valleys, its rotation on its axis, and the fall of heavy bodies on its surface; it therefore appears to be a very eminent, large, and lucid planet, the primary one in our system, disseminating its light and heat to all the bodies with which it is connected.

Herschel supposed that there are two regions or strata of solar clouds: that the inferior stratum is opaque, and probably not unlike our own atmosphere, while the superior is the repository of light, which it darts forth in vast quantities in all directions. The inferior clouds act as a curtain to screen the body of the sun from the intense brilliancy and heat of the superior regions, and, by reflecting back nearly one half of the rays which they receive from the luminous clouds, contribute also greatly to increase the quantity of light which the latter send forth into space, and thereby perform an important function in the economy of the solar system. The luminous clouds prevent us in general from seeing the solid nucleus of the sun; but in order to account for the spots, he supposes an empyrean elastic gas to be constantly forming at the surface, which, carried upwards by reason of its inferior density, forces its way through the planetary or lower clouds, and mixing itself with the gases which have their residence in the superior stratum, causes decompositions of the luminous matter, and gives rise to those appearances which he describes under the name of corrugations. Through the openings made by this accidental removal of the luminous clouds, the solid body of the sun becomes visible, which, not being lucid, gives the appearance of the dark spots or nuclei seen through the telescope. The length of the time during which the spots continue visible renders it evident that the luminous matter of the sun cannot be of a liquid or gaseous nature; for, in either case, the vacancy made by its accidental removal would instantly be filled up, and the uniformity of appearance invariably maintained. Herschel supposed the luminous clouds to be phosphorescent.

It would be a needless waste of time to enter into any discussion of a theory so entirely vague and fanciful, respect- ing and so destitute of all solid foundation. Till we are better acquainted with the nature of light, fire, and heat, and have attained to the knowledge of every possible mode in which these elements can be produced and propagated, all hypotheses respecting the construction of the sun can only be gratuitous and conjectural. Some interesting questions, however, arise out of this subject. Whether the inferior stratum of solar clouds is sufficiently dense, as Herschel imagined, to protect the body of the sun from the scorching effects of the surrounding regions of light and heat, and render it a fit habitation for human beings, is a question of no importance to man, or to any thing pertaining to his planet; but it is interesting for him to know whether the light and heat dispensed by the sun are liable to any variation or secular diminution, either connected with the spots, or resulting from a decrease of the sun's volume.

Those philosophers who adopt the hypothesis of the Mass of production of light and heat by the vibrations of an ethereal fluid, consider the mass of the sun to be invariable. Those, on the contrary, who attribute these effects to an emanation from the sun, think his mass and volume must be diminished by the incessant discharge of torrents of luminous particles from his surface. During the two thousand years which have elapsed since the first astronomical observations, no diminution of the sun's volume has been perceived; but it must be remarked that such an effect may have taken place, though not yet sensible to our instruments. The sun's diameter is nearly 2000'; and at the distance of 95,000,000 miles a second corresponds to 460 miles. Now, supposing the solar diameter to suffer a daily diminution of two feet, which may be considered as enormous, considering the vast magnitude of the sun, and the excessive rarity of light, the diminution would amount to 800 feet in a year, and to 460 miles, or 1°, in 3000 years. Thus, after thirty centuries, the diminution would still be imperceptible, insomuch as our instruments are not sufficiently accurate to enable us to appreciate, in an observation of this sort, so small a variation as one second.

Some astronomers, after Herschel, have imagined that the existence of the solar spots has an influence on the temperature of the seasons. In 1823 the summer was very cold and wet; the thermometer at Paris rose only to 25°-7° of Reaumur, and the sun exhibited no spots; whereas in the summer of 1807 the heat was excessive, and the spots sensibly of vast magnitude. The relation, however, between the temperature and the appearance of the solar spots is not so uniform as to give much weight to this opinion. Warm summers, and winters of excessive rigour, have happened in the presence or absence of the spots. The year 1783 was remarkable for its fertility and the magnitude of the solar spots; a dry fog enveloped the greater part of Europe, and was followed by the earthquake of Calabria. Another opinion entertained by Herschel was, that one hemisphere of the sun emits less light than the other, so that when viewed at a great distance he will resemble some stars of which the brilliancy is subject to periodical variations.

By reason of the globular figure of the sun, his surface towards the border of the disk is seen obliquely, and therefore a much greater portion of it is comprehended.

Theoretical ed under a given visual angle, than when the ray proceeds from his centre. Now, as every point of the sun's surface is supposed to emit an equal quantity of light in all directions, it follows that the light ought to be much more intense near the circumference of the disk, because a greater number of rays will proceed from the larger surface, which forms the oblique base of the luminous cone. Bouguer, deceived by some imperfect experiments, thought the light more intense at the centre of the disk than towards the limb; a circumstance which could only be explained by supposing the light to be diminished by some cause which acts most powerfully with regard to the borders of the disk. Such would be the effect of a dense atmosphere surrounding the sun; for in this case the rays which proceed from the border must traverse a much greater extent of the solar atmosphere, and consequently be absorbed in greater proportion than those which proceed from the central parts and traverse it directly; just as the atmosphere of the earth renders the light of the stars at the horizon much feebler than at the zenith. It has been thought, however, that Bouguer's experiments were inaccurate, and that the light is equally intense at the border and the centre. The existence of a solar atmosphere cannot therefore be demonstrated in this manner; but it is clearly indicated by the faint light which is observable round the sun's limb during a total eclipse.

Another very curious phenomenon connected with the sun, is the faint nebulous aurora which accompanies him, known by the name of the Zodiacaal Light. This phenomenon was first observed by Kepler, who described its appearance with sufficient accuracy, and supposed it to be the atmosphere of the sun. Dominic Cassini, however, to whom its discovery has been generally but erroneously attributed, was the first who observed it attentively, and gave it the name which it now bears. It is visible immediately before sunrise, or after sunset, in the place where the sun is about to appear, or has just quitted, in the horizon. In total eclipses it is seen surrounding the sun's disk, and resembling the beard of a comet. It has a flat lenticular form, and is placed obliquely on the horizon, as represented in fig. 41, the apex extending to a great distance in the heavens. Its direction is always in the plane of the sun's equator, and for this reason it is scarcely visible in our latitudes, excepting at particular seasons, when that plane is nearly perpendicular to the horizon. When its inclination is great, it is either concealed altogether under the horizon, or at least rises so little above it, that its splendour is effaced by the atmosphere of the earth. The most favourable time for observing it is about the beginning of March, or towards the vernal equinox. The line of the equinoxes is then situated in the horizon, and the arc of the ecliptic SS' (fig. 42) is more elevated than the equator SEQ by an angle of 23½ degrees; so that the solar equator, which is slightly inclined to the ecliptic, approaches nearer to the perpendicular to the horizon, and the pyramid of the zodiacal light is consequently directed to a point nearer the zenith, than at any other season of the year. For example, at the summer solstice, ST, a tangent to the ecliptic, is parallel to EQ the tangent to the equator, and the luminous pyramid is in a plane less elevated by 23½ than at the time of the vernal equinox.

Numerous opinions have been entertained respecting the nature and cause of this singular phenomenon. Cassini thought it might be occasioned by the confused light of an innumerable multitude of little planets circulating round the sun, in the same manner as the milky way owes its appearance to the light of agglomerated myriads of stars. Its resemblance to the tails of comets has been noticed by Cassini, Fatio Duillier, and others; and Euler endeavoured to prove that they are both owing to similar causes. Mairan, like Kepler, ascribed it to the atmosphere of the sun; and this hypothesis was generally adopted, till it was shown by Laplace to be untenable for the following reasons. The atmosphere of any planet, endowed with a motion of rotation, cannot extend to an infinite distance: it can only reach to such a height that the centrifugal force is exactly balanced by the force of gravity. Beyond this height the atmosphere would be dissipated by the superior energy of the centrifugal force. Now the height above the sun at which the two forces are equal is that at which a planet, if placed there, would revolve about the sun in the same time in which the sun performs a revolution on his axis. But the orbit of such a planet would be greatly inferior to the orbit of Mercury; for the time in which Mercury makes a revolution in his orbit is eighty-eight days, while the sun revolves about his axis in twenty-five; it is therefore certain that the atmosphere of the sun cannot extend to the orbit of Mercury. Now, the greatest elongation of Mercury does not exceed 28°, and the zodiacal light has been observed to extend to above 100°, reckoning from the sun to the apex of the luminous pyramid. Hence the phenomenon cannot proceed from the sun's atmosphere. Laplace further remarks, that the ratio of the equatorial and polar axes of the solar atmospherical spheroid cannot exceed that of three to two; whence its form would not correspond with the lenticular appearance of the zodiacal light. But to whatever cause this luminous matter is to be attributed, it is certain that it is of extreme rarity, insomuch as it does not intercept the light of the smallest stars which are seen through it without any diminution of splendour.

On the subject of the solar spots and zodiacal light the following works may be consulted:—Galileo, *Istoria e Dimostrazioni intorno alle Macchie Solari*, Rome, 1613; Scheiner, *Rosa Ursina*, Bracciani, 1630; Hevelius, *Selenographia*, Gedani, 1647; Reuschius, *De Maculis et Fasciis Solaribus*, Wittemb., 1661, 4to.; Cassini, *Noue Observ. des Taches du Soleil*; Hooke, *Troctatus de Maculis Solari bus, et Lumine Zodiacali*, in Oper. Posth. Lond. 1705; Weidler, *De Coloribus Mac. Sol.* in his Observ. Meteorol. 1728–9, Wittemb. 1729; Boscovich, *De Mac. Sol. Rom. 1735, 4to.; De Lisle, *Memoires pour servir à l'Histoire de l'Astr. Petersb. 1738; Bernoulli, Lettres Astronomiques, 1771; Wilson, *Phil. Trans*. 1774, vol. xix.; *Ibid*. 1783; Wallaston, *Phil. Trans*. 1774; Lalande, *Phil. Trans*. 1776; *Mém. Acad*. 1776, and *Astronomie*, tom. iii. p. 277; Herschel, *Phil. Trans*. 1795 and 1801; Woodward on the Substance of the Sun, Washington, 1801; Biot, *Traité de l'Astronomie Physique*, tom. ii.; Bohu, *Disput. Astrum de Fascia Zodiacali*; Cassini, *Mém. Acad. Par.*, tom. vii. p. 119, and viii. p. 193; Mairan, *Traité Physique et Historique de l'Aurore Boréale*, 1731; Lalande, *Astronomie*, tom. i. p. 276; Laplace, *Exposition du Système du Monde*; Delambre, *Hist. de l'Astronomie Moderne*, tom. ii. p. 742.

CHAP. III.

OF THE MOON.

Next to the sun, the moon is to mankind the most important and interesting of all the celestial bodies. Her conspicuous appearance in the heavens, the variety of her phases, and the rapidity with which she changes her place among the fixed stars, have rendered her at all times an object of admiration to the vulgar; while her proximity to the earth, her physical effects on the ocean, the intricacy of the theory of her motions, and the vast importance of that theory to navigation and geography, have equally... Theoretical claimed the attention of the observer and mathematician; Astronomy, nor is there any other department of astronomy in which their researches have been crowned with more triumphant success, or been rewarded with more brilliant discoveries.

Sect. I.—Of the Phases, Parallax, and Magnitude of the Moon.

The different appearances or phases of the moon were probably the first celestial phenomena observed with any degree of attention. When the moon, after having been for some days invisible, is again seen on the eastern side of the sun, and at a distance of 20 or 30 degrees from him, she appears as a curved thread of silvery light; and her form is that of a crescent, the horns of which are turned towards the east. The breadth of the crescent increases continually in proportion as she separates herself from the sun, till, having obtained a distance of 90°, she appears under the form of a semicircle. At this point she is said to be in her first quarter. Continuing her motion to the eastward, the line which terminates the eastern side of her disk assumes the curvature of an elliptic arc, and her visible portion continues to increase till she has attained the distance of 180° from the sun, when she appears perfectly round. She is then full, and is said to be in opposition, rising as the sun sets, and consoling us by her pale light for the absence of the great luminary. Having passed this point, she begins to approach the sun; her western side now takes the form of the elliptic arc, and her luminous portion diminishes exactly in the same proportion as it increased through the first half of her orbit. About seven days after the full she again appears as a semicircle, the diameter of which is turned towards the west, and she is now at her third quarter, and at a distance of 90° from the sun. The semicircle after this changes into a crescent, and she continues to approach the sun, till, having advanced to within 20° or 30° from him, she again disappears, being lost in the splendour of his rays. These phases regularly succeed each other, and the time in which they run through all their changes is about 29½ days. When the moon passes the meridian at the same time with the sun, she is said to be in Conjunction. The two points of her orbit in which she is situated when in opposition or conjunction are called the Syzygies; those which are 90° distant from the sun are called the Quadratures; and the intermediate points between the syzygies and quadratures are called the Octants.

A slight attention to the lunar phases during a single revolution will be sufficient to prove that they are occasioned by the reflection of the sun's light from the opaque spherical surface of the moon. This fact, which was recognised in the earliest ages, will be made obvious by the help of a diagram. If the moon is an opaque body we can only see that portion of her enlightened side which is towards the earth. Therefore, when she arrives at that point of her orbit A (fig. 43) where she is in conjunction with the sun S, her dark half is towards the earth, and she disappears, as at a, there being no light on that half to render it visible. When she comes to her first octant, at B, or has gone an eighth part of her orbit from her conjunction, a quarter of her enlightened side is towards the earth, and she appears horned, as at b. When she has gone a quarter of her orbit from her conjunction, to C, she shows us one half of her enlightened side, as at c, and we say she is a quarter old. At D she is in her second octant, and by showing us more of her enlightened side she appears gibbous, as at d. At E her whole enlightened side is towards the earth, and therefore she appears round, as at e, when we say it is full moon. In her third octant, at F, part of her dark side being towards the earth, she again appears gibbous, and is on the decrease, as at f.

At G we see just one half of her enlightened side, and she appears as a semicircle, as at g. At H we only see a quarter of her enlightened side, being in her fourth octant, when she appears horned, as at h. And at A, having completed her course from the sun to the sun again, she disappears, and we say it is new moon. Thus, in going from A to E, the moon seems continually to increase; and in going from E to A, to decrease in the same proportion, exhibiting like phases at equal distances from A and E.

The magnitude of the visible portion of the moon's disk thus depends on the situation of the moon relatively to the sun and the earth, and is easily determined geometrically from her elongation or angular distance from the sun. Let ABCD (fig. 44) be the projection of the lunar Fig. 44. orb on the plane which passes through the centres of the sun, moon, and earth; let S be the place of the sun, E that of the earth, M the centre of the projection, and AB, CD two diameters perpendicular to SM and EM respectively, and let EM meet the circumference in G. It is evident that AGDB represents the hemisphere illuminated by the sun, and CAGD that which is visible from the earth; the whole portion of the visible disk, therefore, is represented by AGD, or by AG and DG. Now, if we conceive the moon's surface to be projected on the plane which passes through her centre perpendicular to the visual ray, the illuminated portion of the disk DG will appear as a semicircle, while the part GA will appear as a semi-ellipse, the minor axis of which is to its major as MF to MC, AF being perpendicular to MC. The eye at E will therefore see the semicircle DMG, together with the semi-ellipse GMA, and the visible part will be to the entire disk as DF : DC, that is, as $1 + \cos \phi : 2$. AME : 2, or, as $1 + \cos \phi : 2$, $\phi$ being equal to EMS the elongation of the earth from the sun as seen from the moon. The geometric elongation of the moon from the sun, or the angle MES = $\phi$, may be substituted instead of $\phi$, when we know the ratio of the distances ES = r and MS = R; for $\sin \phi = \frac{r}{R} \sin \psi$, whence $\cos \phi = \sqrt{1 - \frac{r^2}{R^2}} \sin^2 \psi$, and the above proportion becomes

$$DF : DC :: 1 + \sqrt{1 - \frac{r^2}{R^2}} \sin^2 \psi : 2.$$

When the moon is in opposition the angle $\phi = 0$, whence $1 + \cos \phi = 2$, and the whole disk is visible. At the conjunction $\phi = 180°$, and $1 + \cos \phi = 0$; the moon is consequently invisible.

The triangle EMS likewise furnishes the means of finding the distance of the moon when that of the sun is known; or reciprocally, the distance of the sun from that of the moon; but for this purpose it is necessary to observe accurately the limit of illumination, which is not easily done. Let ADBE (fig. 45) be the illuminated part Fig. 45. of the moon ACBE. The distance between the cusps A and B, measured by the micrometer, gives the major axis of the ellipse ADB, and the semi-minor axis DM is found by measuring in the same manner the distance ED, and subtracting EM, half the semi-minor AB. This gives $\frac{DM}{AM} = \cos \text{EMS}$ (fig. 44), and therefore the angle EMS or M; and from SEM, which is given by observation, we have EMS or S = 180° — E — M; whence M and S are given angles; and from the property of the triangle we have EM = ES $\sin S$ or ES = EM $\sin M$. The calculation would be greatly facilitated if the observation Theoretical were made at the exact instant when the moon is dichotomized, or in her first or third quarter, for the elliptic arc then becomes a straight line and the angle EMS = 90°; so that sin M = 1. In this case therefore we have EM = ES sin S = ES cos E, and ES = \(\frac{EM}{\cos E}\), which gives the ratio of the distances of the sun and moon. In this manner Aristarchus of Samos attempted to compare the distance of the two luminaries; and although his result was extremely erroneous, inasmuch as he found the distance of the sun to be only between eighteen and twenty times greater than that of the moon, whereas it is actually 380 times greater, yet the idea was an ingenious one, and the method perfectly accurate in theory. The error arose from the inaccuracy of the ancient instruments, and the impossibility of observing to within a few minutes the exact time of the dichotomy. Besides, as the angle at the sun is extremely small, a very slight error either with regard to the time or the angle has a great influence on the result.

The moon's absolute distance from the earth is obtained by means of her parallax, which, on account of her proximity, is very considerable, and can therefore be determined by the methods which were explained in chap. I sect. 2. On comparing her parallaxes observed at different times, they are found to differ considerably in value. Thus, according to Lalande, the greatest horizontal parallax at Paris is 61' 29", the smallest 53' 51"; and it assumes every possible value between these two extremes. The horizontal parallax being the angle which the earth's radius subtends at the moon, and consequently equal to the radius of the earth divided by the moon's distance, the perigean and apogean distances corresponding to the extreme parallaxes are respectively 55'916 and 63'842 semidiameters of the earth. The ratio of these two numbers, or of the extreme values of the parallax, is 1:1417, and denotes the greatest variations of the moon's distance from the earth. The ratio of the corresponding quantities in the case of the sun is only 1:034; hence the moon's distance is subject to much greater variations than that of the sun.

These differences in the value of the parallax arise from the variations of the moon's distance from the earth; but it is also observed to differ sensibly at different points of the earth's surface, even at the same instant of time. If the earth were spherical, the horizontal parallax of the moon, supposing her distance to be invariable, would be the same at whatever part of the earth it might be observed. The case is, however, different; for, on account of the spheroidal figure of the earth, the section made by every vertical plane gives a different ellipse, no one of which even passes through the centre of the earth, excepting indeed the cases in which the place of the observer is on the equator or under the pole; for in all other cases the perpendicular to the surface of the earth does not pass through its centre. Hence it is necessary, in speaking of the horizontal parallax, to specify the place of the observation.

Since the parallax of the moon is subject to incessant variation, it is necessary to assume a certain mean value of it, about which the true and apparent values may be conceived to oscillate. This is denominated the constant parallax. It is evident that it cannot be found by taking the arithmetical mean of the extreme values; for by reason of the disturbing force of the sun, the eccentricity of the lunar orbit, and consequently the perigean and apogean distances, are constantly varying; and as the quantity by which one of these distances is increased is not equal to that by which the other is diminished, it follows that the mean of the perigean and apogean parallaxes will not be a constant quantity, but different in successive revolutions of the moon. If we abstract from all the inequalities of the lunar orbit, and suppose the moon to be at her mean distance and mean place, the constant parallax will be the angle under which a given semidiameter of the earth is seen by a spectator at the moon in such circumstances. According to Lalande, the following are the values of the constant parallax:

- Under the equator: 57' 5" - Under the pole: 56' 53'2 - For the latitude of Paris: 56' 58'3 - For the radius of a sphere equal in volume to the earth: 57' 1

The mean equatorial parallax being 57' 5", its double is Magnitude 1' 54" 10", which expresses the angle subtended by the tide of the diameter of the earth at the distance of the moon. The lunar orbit subtended by the moon at the same distance is 31' 26"; whence the diameter of the moon is to that of the earth as 31' 26" is to 1' 54" 10"; that is, as 524 to 1'903, or as 3 to 11 nearly. According to the tables of Burg, the accurate expression of the above ratio is 1: 0·27293; hence the true diameter of the moon is 0·27293 diameters of the terrestrial equator. The surface of the moon is consequently \((0·27293)^2 = 0·0744908 = \frac{1}{13} \text{th}\) of that of the earth, and its volume \((0·27293)^3 = 0·02083308 = \frac{1}{59} \text{th}\), or, in round numbers, \(\frac{1}{59}\)th of the volume of the earth.

**Sect. II.—Of the Elements of the Lunar Orbit.**

1. **Nodes and Inclination of the Lunar Orbit.**—A few simple observations of the right ascensions and declinations of the moon suffice to show that her path is confined nearly to a plane, having only a small inclination to the plane of the ecliptic. If her path were rigorously confined to this plane, a single revolution would be sufficient to determine its inclination; but as numerous disturbing causes exist, which produce inequalities in all the elements of the lunar motion, it is only by taking the mean value of a great number of observations that these elements can be astronomically determined. In fixing the position of the plane of the orbit, the first object is to determine the straight line formed by its intersection with the plane of the ecliptic; that is to say, to determine the line of the Nodes. This would be at once accomplished if it were possible to observe the instant at which the moon's latitude is nothing; for at this instant her centre is in the plane of the ecliptic, and consequently her longitude is the same as the longitude of the node. But as the accuracy of modern practice requires that all important observations be made in the plane of the meridian, and as it could only happen by a very rare coincidence that the latitude deduced from the meridional right ascensions and declinations would be exactly zero, it is necessary to have recourse to a process of calculation to find the precise place of the node.

Let LNL (fig. 46) be a portion of the ecliptic, MNM Fig. 46. of the lunar orbit, N the place of the node, and ML \(= \lambda\), ML' \(= \lambda'\) be two latitudes on opposite sides of the ecliptic. The two right-angled triangles give, according to Napier's Rules,

\[ \tan N = \frac{\tan \lambda}{\sin NL} = \frac{\tan \lambda'}{\sin NL'}; \]

therefore,

\[ \frac{\sin NL}{\sin NL'} = \frac{\tan \lambda}{\tan \lambda'}; \]

whence,

\[ \frac{\sin NL - \sin NL'}{\sin NL + \sin NL'} = \frac{\tan \lambda - \tan \lambda'}{\tan \lambda + \tan \lambda'}; \] Theoretical that is, by the trigonometrical formulae,

\[ \tan \frac{1}{2} (NL - NL') = \frac{\sin (\lambda - \lambda')}{\sin (\lambda + \lambda')} \]

or

\[ \tan \frac{1}{2} (NL - NL') = \tan \frac{1}{2} LL' \cdot \frac{\sin (\lambda - \lambda')}{\sin (\lambda + \lambda')} \]

Having therefore obtained an expression for the difference of \(NL\) and \(NL'\), and knowing also their sum \(LL\), which is the difference of the longitudes of the moon at the time of the two observations, it is easy to deduce \(NL\) or \(NL'\); consequently the longitude of the node is determined.

The Ascending Node of the lunar orbit is that point of the ecliptic through which the moon passes when she rises above the ecliptic towards the north pole; it is distinguished by the character \(Q\). The Descending Node, \(O\), is the opposite point of the ecliptic, through which she passes when she descends below that plane towards the south pole. The nodes of the lunar orbit were anciently called the head and tail of the Dragon.

The position of the nodes is not fixed in the heavens. They move in a retrograde direction, or contrary to the order of the signs; and their motion is so rapid that its effects become very apparent after one or two revolutions. Regulus, a star of the first magnitude, is sometimes eclipsed by the moon; and as the latitude of this star is only about 27° or 28°, it is certain that when that phenomenon occurs the moon is very near her node. Suppose it to be the ascending node; the month following it will be observed that the moon passes to the north of Regulus; and every succeeding month she will pass farther to the north of the same star, till at the end of four or five years, when her meridional altitude will be about 5° greater than that of Regulus. Her latitude on passing Regulus now begins to diminish; and, after a period of about nine years, Regulus is again eclipsed by the moon in her descent to the southern side of the ecliptic. The moon after this passes to the south of the star for the following nine years; and at the end of 18½ years the nodes have returned to their first position, after accomplishing a complete revolution. The same result is found from the observation of eclipses, the magnitude of which depends on the moon's latitude; and therefore, if the nodes remained fixed, the eclipses would always be of the same magnitude in the same quarter of the heavens. This, however, is not the case; and it is only after about 18½ years that they begin to return in the same order.

The mean retrograde motion of the nodes is found, by the comparison of observations made at distant epochs, to amount to 19° 19' 42" 316 in a mean solar year, and the time in which they make a tropical revolution is consequently 6798.179 mean solar days. The longitude of the ascending node on the 1st of January 1801 was 13° 53' 17" 7"; hence its position at any other epoch is easily deduced, since we know the rate of its mean motion.

The inclination of the lunar orbit may be determined either by observing the moon's greatest latitudes, or it may be computed from the formula, \(\tan N = \frac{\tan \lambda}{\sin NL}\), when her latitude and longitude are known. The angle \(N\), or the inclination, is observed to vary between 5° and 5° 17' 35". The mean inclination may therefore be taken at 5° 8' 48".

The retrograde motion of the moon's nodes, as well as all the other inequalities of the lunar motion, is occasioned by the attracting power of the sun, which varies in intensity according to the positions of the earth and moon in their orbits. The principal inequality to which the inclination of the lunar orbit is subject, is proportional to the cosine of double the sun's distance from the moon's ascending node. In fact, the sun and earth being always in the plane of the ecliptic, when the moon is also in that plane the action of the sun will have no tendency to increase or diminish her latitude; it will only affect her radius vector, or distance from the earth. But if the moon is not in the plane of the ecliptic, the sun's attracting force will not only affect her distance from the earth, but also tend to bring her nearer to the ecliptic; and this action will be greater in proportion to her deviation from that plane. When the nodes are in quadratures, and the limits of latitude in the syzygies, the inclination is 5°, as was found by Ptolemy; but when the nodes are in the syzygies, and the limits in quadratures, the inclination is 5° 17' 35". It has therefore a mean value of 5° 8' 48", when the nodes and limits coincide with the octants. Denoting therefore the greatest deviation from the mean values by \(x\), this deviation will pass through all its values between \(+ x\) and \(- x\) while the sun's distance from the ascending node varies from 0° to 90°. It may therefore be represented by a function which varies as the cosine of twice that distance, for the cosine of an angle runs through all its changes from \(+ 1\) to \(- 1\) between 0° and 180°. Hence the equation which expresses the principal inequality of the moon's latitude is

\[ (8° 47") \cos 2 \theta \text{ 's distance from } p \text{'s } Q. \]

The co-efficient 8° 47" is called, in astronomy, the Co-efficient of the Argument, the argument itself being the function by which the inequality is represented.

This periodic inequality of the moon's latitude was discovered by Tycho Brahe, from a comparison of the greatest latitudes at different epochs with the corresponding positions of the moon with reference to her nodes. He observed that these latitudes oscillate about the mean value of 5° 8' 48"; and as the greatest latitudes give immediately the inclination of the moon's orbit to the plane of the ecliptic, it necessarily followed that the inclination is subject to similar variations. The motion of the node of the lunar orbit is also subject to an inequality, depending on the same angle as the preceding, but proportional to its sine. Tycho explained these two inequalities by a very simple hypothesis, similar to that by which he has explained the oscillations of the terrestrial equator, namely, a slight nutation of the earth's axis. In the present case it is only necessary to place the mean pole of the lunar orbit at the distance of 5° 8' 48" from the pole of the ecliptic, and to suppose the true pole to describe a small ellipse about the mean pole in the same time that the sun occupies in making a semi-revolution with regard to the nodes of the moon's orbit, that is, in 173-309 days.

There are various other inequalities which affect the latitude of the moon and the inclination of her orbit, to correct which, a large number of equations are given in the recent and most accurate tables. The greater part of them are, however, only known from theory, and are so small that even their accumulated effects are scarcely perceptible to observation.

The inclination of the lunar orbit to the plane of the terrestrial equator occasions considerable differences in the intervals between the moon's rising or setting on successive harvest days, and gives rise to the phenomenon of the Harvest Moon. As the daily motion of the moon is about 13 degrees from west to east, it follows that if she moved in a plane parallel to the equator, she would rise 50 minutes later every successive evening, because her orbit would then make the same angle with the horizon at all seasons of the year, and the intervals between her consecutive risings would be constant. For the sake of explanation, we may here suppose the moon to move in the plane of Theoretical the ecliptic. Now, the time in which a given arc of the ecliptic rises above the horizon depends on its inclination to the horizon. In our latitudes the inclination of the ecliptic at different points to the horizon varies so much, that at the first point of Aries an arc of 13° becomes visible in the short space of 17 minutes, while at the 23rd of Leo the same arc will only rise above the horizon in one hour and 17 minutes. Hence, when the moon is near the first point of Aries, the difference of the times of her rising on two successive evenings will be only about 17 minutes; and as this happens in the course of every revolution, she will rise for two or three nights every month at nearly the same hour. But the rising of the moon is a phenomenon which attracts no attention, excepting about the time when she is full, that is, when she rises at sunset. In this case she is in opposition to the sun, and consequently, if she is in Aries, the sun must be in Libra, which happens during the autumnal months. At this season of the year, therefore, the moon, when near the full, rises for some evenings at nearly the same hour. This circumstance affords important advantages to the husbandman, on which account the phenomenon attracts particular attention.

It is obvious, that as this phenomenon is occasioned by the oblique position of the lunar orbit with regard to the equator, the effect will be greater than what has just been described if the plane of that orbit makes a greater angle with the equator than the plane of the ecliptic does. But we have seen that the plane of the moon's orbit is inclined to the ecliptic in an angle exceeding 5°; consequently, when her ascending node is in Aries, the angle which her orbit makes with the horizon will be 5° less than that which the ecliptic makes with the horizon; and the difference of time between her risings on two successive evenings will be less than 17 minutes, as would have been the case had her orbit coincided with the ecliptic. On the contrary, when the descending node comes to Aries, the angle which her orbit makes with the horizon will be greater by 5°; and consequently the difference of the times of her successive risings will be greater than if she moved in the plane of the ecliptic. If when the full moon is in Pisces or Aries the ascending node of her orbit is also in one of those signs, the difference of the times of her rising will not exceed one hour and forty minutes during a whole week; but when her nodes are differently situated, the difference in the time of her rising in the same signs may amount to 3½ hours in the space of a week. In the former case the harvest-moons are the most beneficial, in the latter the least beneficial, to the husbandman. All the variations in the intervals between the consecutive risings or settings take place within the period in which the line of the nodes makes a complete revolution.

2. Dimensions, Eccentricity, and Apsidal Lines of the Lunar Orbit; and the Inequalities of the Moon's motion.—Since the moon moves in a plane orbit, the projection of her path on the surface of the celestial sphere will be a great circle. But the variations observable in the magnitude of her apparent diameter, and in the velocity of her motion, prove that if her orbit is a circle, the earth is at least not placed at its centre. The numerous perturbing causes which affect her motion render the exact determination of her orbit a matter of great difficulty; but observation shows that it deviates very little from an ellipse, of which the centre of the earth occupies one of the foci. It has also been found that the spaces passed over by the radius vector of the moon are very nearly proportional to the times of description; which is the distinctive character of the elliptic motion.

In order to obtain a correct idea of the figure and position of the orbit, it is necessary to know its major axis, its eccentricity, and the position of its apsides. The major axis is equal to the sum of the greatest and least distances, and these have already been stated to be respectively 68°842 and 55°916 semidiameters of the earth; whence, supposing the earth's radius to be 4000 miles, the major axis of the lunar orbit will amount in round numbers to 480,000 miles. It may be here remarked that the moon's distance is an element which may be practically determined with great exactness. A variation of 1° in the parallax would occasion an error of only about 67 miles in the determination of the distance; therefore, since the parallax is certainly known to within 4°, the greatest error in the distance deduced from it cannot exceed 280 miles out of about 240,000 miles.

An approximation to the eccentricity of the orbit of the moon is easily obtained from the variations of her apparent diameter. The moon's apparent diameter is observed to vary between 29°30' and 33°30' very nearly. Now, let D denote the mean, D' the apogee, D'' the perigee diameter, and e the eccentricity. We shall then have

\[ D' = \frac{D}{1+e} = 29°30', \] \[ D'' = \frac{D}{1-e} = 33°30', \] \[ D = \frac{D'}{1-e} = 29°30', \] \[ D'' = \frac{D'}{1+e} = 33°30'; \]

whence we deduce

\[ e = \frac{33°30' - 29°30'}{33°30' + 29°30'} = \frac{4}{63} = 0.0635. \]

This eccentricity corresponds to an equation of the centre amounting to 7°16', and is much more considerable than that of the solar orbit, which, as we have before seen, amounts only to 0°0168. The result just given is, however, only to be considered as approximate, and is, in fact, considerably too large. In the best and most recent tables the value assigned to the eccentricity of the lunar orbit is 0°054842, and the corresponding equation of the centre 6°17'12"7. (See Astronomical Tables and Formulae, by Francis Baily, Esq. London, 1827.)

The place of the apsides is likewise found from observations of the moon's apparent diameter, because at these points of the orbit the apparent magnitude has its maximum and minimum values. But as the apparent magnitude varies very slowly at the apsides, it is preferable, as Lalande remarks, to choose the points of mean distance where its variations are most rapid. Having selected two observations which give the same apparent diameter at two opposite points of the orbit, we shall have two points evidently at equal distances from the perigee; the longitude of which will therefore be given at the middle time between the two observations. In comparing the positions of the apsides thus determined at different epochs, it is found that they have a rapid progressive motion, that is, according to the order of the signs. In Mr. Baily's tables the motion of the perigee is stated to be 4069°046278, or 11 revolutions + 109°2'46"6, in 36525 mean solar days. Its mean motion in a mean solar day is consequently 6°41"; and in 365 mean solar days, 40°39'45"36. Hence the time in which it completes a sidereal revolution is 3232°575348 mean solar days. The period of a tropical revolution of the apsides is 3231°4751 mean solar days = 3231 days, 11 hours, 24 minutes, 0.8 seconds, or nearly 9 years. This mean motion of the lunar apsides is, however, subject to periodic inequalities of considerable magnitude, and therefore can only be determined accurately by means of observations separated from Theoretical each other by a long interval of time. In the tables just quoted, the epoch is the commencement of the present century, when the longitude of the perigee was in 266° 10' 7"5. From this and the rate of the mean motion, the mean longitude of the perigee at any other epoch may be easily computed.

If the moon's place in her orbit were not subject to the influence of any disturbing force, her true longitude, or distance from the apsis, would be found at any instant from the theory of the elliptic motion, supposing her mean motion, or the time of a return to the same perigee or apogee, to be accurately known. Thus, supposing E to denote the maximum elliptic equation, which, as we have already stated, amounts to 6° 17' 12"7, we should have the true longitude = mean longitude + E sin A, A being the moon's mean anomaly, and the longitudes counted from the perigee. This first equation or correction of the mean motion results solely from the circumstance that the moon moves in an elliptic orbit, according to the laws of Kepler, in the same manner as the sun and all the planets. Its argument is the mean anomaly A, and its period the anomalistic month. But the moon's longitudes, as determined in this manner, are far from agreeing with observation; and several other corrections must be applied, the accurate determination of which forms the principal object of the lunar theory.

Evolution. After the equation of the centre, the most considerable of the inequalities of the moon's longitude is that to which Boulliau gave the name of the Evolution. It was noticed by Hipparchus; but it was Ptolemy who discovered its law, and gave a construction which represents its general effects with great accuracy. These effects are to diminish the equation of the centre when the line of the apsides lies in syzygy, and to augment it when the same line lies in quadratures. Thus, supposing the apsides to lie in syzygy, and that it is sought to compute the moon's true longitude about seven days after she has left the perigee, by adding the equation of the centre to the mean anomaly, the resulting longitude will be found to be above 80' less than that which is given by observation. But if the line of the apsides lies in quadratures, the place of the moon at about the same distance, that is, 90° from the perigee, computed in the same manner, will be found to be before the observed place by above 80', that is, the computed will be greater than the observed longitude, by more than 80 minutes. After a long series of observations, astronomers have found that the inequality in question is represented with great exactness, by supposing it proportional to the sine of twice the mean angular distance of the moon from the sun, minus the mean anomaly of the moon. In Baily's tables, its maximum value is stated to be 1° 20' 29"9; whence, representing the angular distance of the sun and moon by \( \varphi - \Theta \), and the mean anomaly as before by A, the correction due to the evolution will be

\[ (1° 20' 29"9) \sin [2(\varphi - \Theta) - A]. \]

A third inequality of the lunar motion, called the Variation, was discovered by Tycho Brahe, who found that the moon's place, calculated from her mean motion, the equation of the centre, and the evolution, does not always agree with the true place, and that the variations are greatest in the octants, or when the line of the apsides makes an angle of 45° with that of the syzygies and quadratures. Having observed the moon at different points of her orbit, he found that this correction has no dependence on the position of the apsides, but only on the moon's elongation from the sun. Its maximum value is additive in the octants which come immediately after the syzygies, where the elongation is 45°, or 180° + 45°, and subtractive in the octants which precede the syzygies, where the Theoretical elongation is 360° - 45°, and 180° - 45°. It vanishes altogether in the syzygies and quadratures where the elongation is 0°, 180°, 90°, or 270°; and on this account it was not perceived by the ancient astronomers, who only observed the moon in those positions. It will be readily perceived, from the limits between which it varies, that it is proportional to the sine of twice the angular distance between the sun and moon. Its maximum value, or the coefficient of its argument, is 35° 41' 9"; hence the correction necessary on account of it is represented by the formula

\[ (35° 41' 9") \sin 2(\varphi - \Theta). \]

When this equation is added to the two preceding Annual ones, the differences between the computed and observed equation places of the moon are brought within narrower limits; but they do not yet disappear altogether except in the months of June and December, and about the time of the equinoxes they are found to amount to eleven or twelve minutes. This inequality was observed by Tycho and Kepler, though neither of them determined its magnitude. They supposed it to be an equation of time peculiar to the moon; and under this form Horrox inserted it in his tables, assigning its maximum value at 11° 16", which differs only by four seconds from that which is obtained from the most accurate modern observations. On account that it is regulated by the seasons, and depends not on the lunar orbit, but the anomaly of the sun, Kepler gave it the name of the Annual Equation. By reason of this inequality the motion of the moon is slower than the mean motion during the winter months, when the motion of the sun is most rapid; and, on the contrary, is most rapid in summer, when the sun's motion is slowest. Hence its argument is the same as that of the equation of the centre of the sun, but with a contrary sign. It is therefore proportional to the sine of the sun's mean anomaly; its period is the anomalistic year, and its maximum value is found by observation to be 11° 11"9; hence it is expressed by the formula

\[ (11° 11"9) \sin \Theta's mean anomaly. \]

If we now collect these equations we shall obtain the following corrected expression of the moon's true place:

\[ \varphi's true longitude = \varphi's mean longitude, \]

\[ + (6° 17' 12"7) \sin A, \]

\[ + (1° 20' 29"9) \sin [2(\varphi - \Theta) - A], \]

\[ - (35° 41' 9") \sin 2(\varphi - \Theta), \]

\[ - (11° 11"9) \sin \Theta's mean anomaly. \]

For the physical cause of the three last-mentioned inequalities, viz. the evolution, the variation, and the annual equation, see Physical Astronomy, sect. ii. par. 37, 38, and 39. They are the principal but not the only periodic inequalities which affect the longitude of the moon; for the profound investigations of modern science have detected many others, which, by reason of their small values, and the manner in which they are blended with one another, would have eluded all attempts to discover their period or their law by observation alone. Observation can make known only the joint effect of the independent equations, and thereby indicate their existence. In order to separate them and determine their respective values, it is indispensable to have recourse to the physical theory, from which alone the various circumstances of the lunar motion can be properly determined. Newton was the first who attempted to compute a priori the inequalities of the moon's motion, and thereby led the way in a research which was prosecuted with infinite ingenuity and profound analytical skill by D'Alembert, Clairaut, Euler, and other illustrious mathematicians of the last century, and which was only brought to a successful termination by the labours of Laplace. The formulæ given in the seventh Theoretical book of the Mécanique Céleste, for a great number of new inequalities derived from the theory of attraction, form the basis of the lunar tables of Burg and Burckhardt. It is to these theoretical researches that we are indebted for the precision with which the moon's motion is actually known, and the great advantages which thence result to geography and navigation. The recent tables contain no fewer than twenty-eight equations of longitude, all which may be regarded as corrections of the mean values of the four which we have explained, and which are themselves corrections of the mean motion. It is thus that astronomy approaches nearer and nearer, by every successive step, to the last term of a series which would represent the orbits and motions of the celestial bodies with absolute accuracy.

The lunar inequalities which we have as yet considered are all of a periodic nature, are compensated in the course of a comparatively small number of years, and have passed through many complete revolutions since the commencement of the history of astronomical observations. But there are, as in the case of the sun, others of a different kind, the periods of which are so long that, with reference to the duration of human life, they may be considered as permanently affecting the elements of the lunar orbit. These are the Secular Inequalities, the most remarkable of which is the acceleration of the moon's mean motion.

On comparing the lunar observations made within the last two centuries with one another, there results a mean secular motion greater than that which is given by comparing them with those made by Elbn-Jounis, near Cairo, towards the end of the 10th century, and greater still than that which is given by comparing them with observations of eclipses made at Babylon in the years 719, 720, and 721 before our era, and preserved by Ptolemy in the Almagest. These comparisons, which have been made by different astronomers with the utmost care, prove incontestably the acceleration of the motion of the moon from the Chaldeans to the Arabsians, and from the Arabians to the present times.

The acceleration of the moon's mean motion was first remarked by Dr Halley, and mentioned in his Notes on the observations of Abategnesius, inserted in the Philosophical Transactions for 1693. It was fully confirmed by Dunthorne, who was led by the discussion of a great number of ancient observations of eclipses, to suppose that it proceeded uniformly at the rate of 10" in a hundred years; a supposition which consequently gave him a correction for the mean longitude of the moon proportional to the square of the time, or of 10" multiplied by the square of the number of years elapsed between 1700 and the epoch of the calculation. This was the first attempt to estimate the value of the secular equation, which had hitherto been confounded with the mean secular motion. By a similar discussion of ancient observations, Mayer was likewise led to a secular equation proportional to the square of the time, which, in his first Lunar Tables, published in 1753, he valued at 7" for the first century, counting from the year 1700, but which he advanced to 9" in his last tables, published in 1770. Lalande found it to amount to 9"886, and therefore agreed with Dunthorne in estimating the acceleration at 10" for the first century after 1700. Delambre subsequently undertook to determine accurately the actual motion of the moon in a century, from a comparison of the best modern observations. He found it to amount to 307"53"9" (rejecting the circumferences), while the most ancient observations agree in making it less by 3 or 4 minutes. The equation, which, in consequence of these comparisons, was empirically introduced into the tables, has been confirmed by the theory of gravitation; and the discovery of the physical cause, and of the law of the acceleration, due to Laplace, forms one of the most brilliant triumphs of modern science. In 1786 Laplace demonstrated that the acceleration is one of the effects of the attraction of the sun, and connected with the variations of the eccentricity of the earth's orbit in such a manner that the moon will continue to be accelerated while the eccentricity diminishes, but that it will disappear when the eccentricity has reached its maximum value; and when that element begins to increase, the mean motion of the moon will be retarded.

In order to take into account the effects of the acceleration in the determination of the mean longitude, M. Damoiseau has given the following formula for the secular variation: $10^{-7} \times 232 \times n^2 + 0.019361 \times n$, where $n$ is the number of centuries from 1801; so that if $L$ is the mean longitude of the moon on the 1st of January 1801, and $m$ is the secular motion at that epoch, the mean longitude will be $L + mn + 10^{-7} \times 232 \times n^2 + 0.019361 \times n$ after $n$ centuries. By means of this equation the tables satisfy the most ancient observations, and may be extended to at least a thousand years from the present epoch. It is only necessary to observe, that, in applying the formula, $n$ must be taken negatively, if the epoch is anterior to 1801.

The same cause which gives rise to the acceleration of the mean motion, namely, the diminution of the eccentricity of the earth's orbit, also occasions secular inequalities of the perigee and nodes of the orbit of the moon. These two inequalities are, however, affected with opposite signs to that of the former; that is, while the mean motion of the moon is accelerated, the motion of her perigee and that of her nodes are retarded. They were deduced by Laplace from theory, and the equations by which they are expressed are connected with one another by a very simple proportion. If we take $\Lambda$ to represent the secular acceleration of the mean motion, the secular variation of the perigee found by Laplace is $-4.00052 \Lambda$, and that of the nodes $-0.735452 \Lambda$. From this Laplace concluded that the three motions of the moon, with respect to the sun, to her perigee, and to her nodes, are accelerated, and that their secular equations are in the ratio of the above numbers. By pushing the approximations to a great length, MM. Plana and Carlini, and M. Damoiseau, in Memoirs which obtained the prize of the Academy of Sciences for 1820, have found different numbers; those of Damoiseau are 1, 4702, and 0612. These important results of theory are all confirmed by observation.

The three secular inequalities which have been pointed out will obviously occasion others; for all quantities depending on the mean motion, the motion of the perigee, or of the nodes, must be in some degree modified by them. Thus the mean anomaly, which is the difference of the mean longitude of the moon and the mean longitude of the perigee, is subject to a secular equation equal to the difference of the secular equations affecting the longitudes of the moon and the perigee. The radius vector, the eccentricity and inclination of the orbit, are affected by the secular inequality of the mean motion, which, although too minute to have been hitherto appreciable, will acquire sensible values in the course of ages. The major axis of the ellipse is the only element exempted from inequalities of this sort. It is not probable, however, that the utmost efforts of science will ever make us acquainted with all the irregularities to which the moon's motion is subject. They can only be developed by the complete integration of the differential equations of motion; an integration which is laboriously performed term by term, and which, when attempted to be carried beyond a certain point, transcends Theoretical, the limits of human patience and industry. What is most essential is to select, among the multitude of terms, such as may possibly acquire considerable co-efficients by integration.

The following table (extracted from Bally) exhibits in one view the different elements of the lunar orbit. The epoch is the commencement of the present century.

Mean inclination to the plane of the ecliptic: $5^\circ 8' 47\frac{9}{12}$

Longitude of ascending node: $11^\circ 53' 17\frac{7}{12}$

Motion of node in 365 mean solar days: $19^\circ 19' 42\frac{3}{12}$

Longitude of perigee: $266^\circ 10' 7\frac{3}{12}$

Mean motion of perigee in 365 mean solar days: $40^\circ 39' 45\frac{3}{12}$

Mean longitude: $118^\circ 17' 8\frac{3}{12}$

Greatest equation of the centre: $6^\circ 17' 12\frac{7}{12}$

Eccentricity: $0.0548442$

Mean distance from the earth in diameters of the terrestrial equator: $29^\circ 982175$

Sect. III.—Of the different Species of Lunar Months.

In treating of the sun, we took notice of three different species of revolutions or years, namely, the mean solar year, or the interval of time which the sun employs in performing a complete revolution with regard to the equinoxes; the sidereal year, or the time in which he returns to the same fixed star; and the anomalistic year, or the time in which he returns to the same point of his ellipse. In like manner, if we understand by the term month the time which the moon employs to make an entire revolution relatively to any given point, moveable or fixed, we shall have as many different species of months as there are different motions with which that of the moon can be compared. For example, if we estimate her revolution relatively to the sun, the month will be the time which elapses between two consecutive conjunctions or oppositions. This is called the synodic month, lunar month, or lunation. If we consider her revolution as completed when she has gone through $360^\circ$ of longitude counted from the moveable equinox, we shall have the tropical or periodic month. The interval between two successive conjunctions with the same fixed star is the sidereal month. A revolution with regard to the apsides of her orbit, that is to say, the time in which she returns to her perigee or apogee, gives the anomalistic month; and, finally, the revolution with regard to the nodes is the nodical or draconic month.

Of these different periods the most important to mankind is the synodic month. It is also that which, by reason of the striking manner in which it is marked out by the phases of the moon, would first offer itself to the attention of the observer; and when its period is accurately determined, the other months may be deduced from it without difficulty, when the relative motions of the sun and moon are known with sufficient precision. The eclipses furnish a simple means of determining the synodic revolution with a great degree of accuracy. A few rude observations suffice to show that the period of a lunation is very nearly $29\frac{1}{2}$ days; and with this knowledge we are in a condition to compare two distant eclipses without running any risk of mistaking the number of revolutions that have taken place in the interval. One of the oldest observations recorded by Ptolemy is an eclipse of the moon, which happened 720 years B.C., on the 19th of March, at 6 hours 48 minutes mean time at Paris, according to Lalande. In order to make use of this observation directly for the purpose of determining the synodic revolution, it is necessary to compare it with another of the same kind in which the moon occupied the same point of her ellipse, or the same position relatively to her apsides; for as it is the true place of the moon which is observed, and the mean motion which we are in quest of, the equation of the centre ought to be the same in both observations. Now, a similar observation is furnished by an eclipse which happened in 1717, on the 9th of September, at 6 hours 2 minutes, the moon's anomaly being very nearly the same as in the Babylonian observation. In the interval between the two observations the moon had therefore completed a whole number of revolutions, with regard to her mean as well as her true motion. The interval between the eclipses is 2437 years and 174 days minus 46 minutes, which expressed in days is $\frac{890287 \times 9680555}{30148}$ days. In this interval it is found, from the approximate value of $29\frac{1}{2}$ days, that 30148 synodic revolutions had happened; hence the mean length of the synodic month is $\frac{890287 \times 9680555}{30148} = 29$ days 12 hours 44 min. $28497$ sec.

To deduce the other revolutions, let $N =$ the number of days in a synodic month, $n =$ the number of synodic revolutions in the interval between the two observations, $m =$ the mean motion of the sun, and $T =$ the time of a tropical revolution. We shall then have the proportion $T : N :: 360^\circ : n \times 360^\circ + m$; whence

$$T = \frac{N \times 360^\circ}{n \times 360^\circ + m} = \frac{N}{n + \frac{m}{360^\circ}} = \frac{N}{n} \left(1 - \frac{m}{n \times 360^\circ} + \left(\frac{m}{n \times 360^\circ}\right)^2 + \text{&c.}\right).$$

Let $S =$ the time of a sidereal revolution. The value of $S$ is immediately found from $T$; for let $p =$ the precession of the equinoxes in the time $T$, we shall have $S : T :: 360^\circ : 360^\circ - p$; whence

$$S = \frac{T \times 360^\circ}{360^\circ - p} = \frac{T}{1 - \frac{p}{360^\circ}} = T \left(1 + \frac{p}{360^\circ} + \left(\frac{p}{360^\circ}\right)^2 + \text{&c.}\right).$$

In like manner, if we represent by $u =$ the motion of the apside, and by $v =$ that of the node, in the time $T$, the times of the anomalistic and nodical revolutions will be respectively obtained by the substitution of $u$ and $-v$ (the motion of the node being retrograde) in the last formula in place of $p$. Instead, however, of proceeding in this manner, it is better to assume as the basis of the calculation the mean motion of the moon in longitude, which is accurately known by the observations of 2000 years, and thence to deduce the time of the mean tropical revolution.

The mean motion of the moon in 100 Julian years, or $36525$ days, is found to be $1336$ circumferences $+ 307^\circ 52' 43\frac{3}{5}$; or, by reducing the degrees, &c., to the fraction of a circumference, $1336 \times 35521875$ circumferences. The periodic month is therefore $\frac{35525}{1336 \times 35521875}$ days. In order to reduce the denominator of this fraction to a smaller number of digits, we may divide its terms by 3, multiply them by 32, then divide them by 900, and there will result $432888...$, the denominator of which contains only 7 digits in place of 12. On performing the division, we shall find the periodic month or tropical revolution $= 27^\circ 321582388$ days $= 27$ days 7 hours 43 min. 47183 sec. The mean motion of the moon in 36525 days

Astronomy is \(1336^\circ 307' 52" 43.5'\)

that of the sun \(100^\circ 0 45 45\)

hence the relative motion is \(1236^\circ 307' 6" 58.5'\)

\(= 445267-11625\) degrees. The time of a synodic revolution is therefore \(\frac{360}{445267-11625}\) days; or, multiplying numerator and denominator by 800,

\[ \frac{10519200000}{356213693} \]

whence the synodic month is 29-5305885391 days \(= 29\) days 12 hours 44 min. 2-849778 sec.

In order to obtain the sidereal revolution, we subtract the secular motion of the equinoctial points \(= 5010' = 1^\circ 23' 30"\) from the moon's tropical revolution; the remainder, which gives the sidereal motion of the moon in 36525 days, is \(1336^\circ 306' 29" 13' 5" = 4812664870833\) degrees. Hence the time of a sidereal revolution is \(\frac{360}{4812664870833}\) days.

On multiplying the terms of this fraction by 2400, it is reduced to \(115575600000\), which gives the sidereal month \(= 27\) days 7 hours 43 min. 11-544875 sec.

In like manner, by subtracting the motion of the perigee in 100 years from the secular motion of the moon, we shall find the anomalistic revolution to be 27 days 13 hours 18 min. 34-9488 sec.; and by adding the retrograde secular motion of the node to the secular motion of the moon, the revolution in respect of the nodes is found to be performed in 27 days 5 hours 5 min. 35-60769 sec.

The following table exhibits the different kinds of lunar periods and motions:

| Days, Hr. Min. Sec. | Days | |---------------------|------| | Synodic revolution | 29 12 44 2-84 = 29-5305887 | | Tropical | 27 7 43 4-71 = 27-3215824 | | Sidereal | 27 7 43 11-54 = 27-3216614 | | Anomalistic | 27 13 18 37-40 = 27-5545995 | | Nodal | 27 5 5 35-60 = 27-2122176 | | Tropical revolution of node | 6798 4 17 43-18 = 6798-1789720 | | Sidereal | 6793 6 59 15-34 = 6793-2911498 | | 's mean tropical daily motion | 13° 10' 35" 027 | | 's mean sidereal daily motion | 13° 10' 34" 889 | | 's daily motion in respect of node | 13° 13' 45" 534 |

According to Ptolemy, the synodic month is 29 days 12 hours 44 min. 31 sec., which differs from the above only by half a second. The same great astronomer made the tropical month to consist of 27 days 7 hours 43 min. 73 sec., which exceeds the true time by 2½ seconds; an error into which he was led by assigning too great a value to the mean motion of the sun.

The ancient astronomers paid great attention to these different revolutions, for the purpose of regulating their lunisolar calendar, and of avoiding the calculation of eclipses, which is attended with difficulties that to them must have proved almost insuperable. Their object was therefore to assign composite periods, after the revolution of which the eclipses would again return in the same order. Now, it is easy to see that a period which will bring back eclipses of the same magnitude and duration, on the same day of the year and at the same longitude, must be an exact multiple of the different lunar months. The return of the moon to the same distance from her node will give an eclipse of the same magnitude; if she returns at the same time to the same point of her orbit, the eclipse will also be of the same duration; and if, in addition to these circumstances, she has also returned to the same longitude, the eclipse will take place on the same day of the year. But the numbers in the above table being incommensurable, it is impossible to find any period, however long, that will embrace all these conditions; the ancients therefore formed periods of different lengths, according as they aimed at satisfying the different conditions with a greater or less degree of precision. For an account of some of the most remarkable of the ancient lunisolar periods, see CALENDAR.

On account of the acceleration of the mean motion of the moon, the ratios of the different species of months are constantly undergoing alterations, and therefore the different cycles, supposing them exact at the time of their formation, cannot continue so for an indefinite length of time. This circumstance is, however, little to be regretted; for, in the present state of astronomical science, they are not of any great use, inasmuch as we are in possession of surer methods of predicting eclipses, the calculation of which, from the ephemerides, is now a matter of comparative facility. They are, however, interesting in an historical point of view, and their formation was a principal object of the labours of the early astronomers. (On this subject, see Lalande, Astronomie théorique et pratique, tome ii. p. 185; Delambre, Astronomie théorique et pratique, tome ii. p. 319; Schubert, Traité d'Astronomie théorique, tome ii.; Woodhouse's Astronomy, p. 665.)

SEC. IV.—Of the Rotation and Libration of the Moon.

On account of the proximity of the moon to the earth, the surface of that body is far better known to us than that of any other of the solar system, and in proportion to the increase of optical power which is brought to bear upon it, delineations of the mountains and volcanic craters that occupy the greater portion of it become more minute and accurate. As seen through a good telescope, the dark patches visible with the naked eye assume the appearance of mountainous inequalities, prodigiously increased in number, and with every possible variety of outline. These present no changes of form like those of the sun, but permanently exhibit the same uniform appearances, and retain their relative situations with regard to each other, and also, with some slight variations, to the apparent centre of the moon. The moon, therefore, at all times presents very nearly the same face to the earth. But if this were rigorously the case, it would follow that the moon revolves about an axis, perpendicular to the plane of her orbit, in the same time in which she completes a sidereal revolution about the earth, and that the angular velocities of the two motions are exactly equal. It is, however, proved by observation, that there are some variations in the apparent position of the spots on the lunar disk. Those which are situated very near the border of the disk alternately disappear and become visible, making stated periodical oscillations. But as they suffer no sensible changes in their relative positions, and always re-appear under the same form and magnitude when they return to the same position, it is inferred that they are permanently fixed to the surface of the moon; and their oscillations consequently seem to indicate a sort of vibratory motion of the lunar globe, which is known by the appellation of its libration. This motion has, however, no real existence. The phenomenon is the complicated result of several optical illusions, and does not depend in any degree on the rotation of the moon, which, relatively to us, is perfectly equable; or at least, if it be subject to irregularities, they are too minute to be appreciated.

In order to form a precise idea of the phenomenon of the libration, we must consider that the disk of the moon, seen from the centre of the earth, is terminated by the circumference of a great circle of the moon, the plane of which is perpendicular to a line drawn from the earth's centre to that of the moon. The lunar hemisphere is projected on the plane of this circle turned towards the earth; and if the moon did not revolve round her axis, the projection would incessantly present different appearances to us, inas- Theoretical much as the radius vector drawn from the centre of the earth by which the plane of projection is determined, would intersect the surface of the moon in a different point, at every new position in her orbit. But in consequence of her rotation, the radius vector is always directed to nearly the same point of the lunar surface, and would be always directed exactly to the same point if the angular velocity of rotation corresponded exactly with the angular velocity in the orbit. But the rotation of the moon is sensibly uniform; while the motion of revolution, being affected by the periodic inequalities, is sometimes slower and sometimes more rapid. The apparent rotation occasioned by the revolution of the moon round the earth is consequently in such cases not exactly counterbalanced by the real rotation, which remains constantly the same. Hence the different points of the lunar globe must appear to turn about her centre, sometimes in one direction, and sometimes in the contrary, and the same appearances are produced as would result from a small oscillation of the moon, in the plane of her orbit, about the radius vector drawn from her centre to the earth. The spots near the eastern or western edge of her disk disappear according as her motion in her orbit is more or less rapid than her mean motion. This is called the Libration in Longitude.

Further; the axis of rotation of the moon is not exactly perpendicular to the plane of her orbit. If we suppose the position of this axis fixed, during a revolution of the moon it inclines more or less to the radius vector, so that the angle formed by these two lines is acute during one part of her revolution, and obtuse during another part of it; hence the two poles of rotation and those parts of her surface which are near these poles are alternately visible from the earth. This is the Libration in Latitude.

Besides all this, the observer is not placed at the centre of the earth, but at its surface. It is the radius drawn from his eye to the centre of the moon which determines the middle point of her visible hemisphere. But, in consequence of the lunar parallax, it is obvious that this radius must cut the surface of the moon in points sensibly different according to the height of that luminary above the horizon. An observer at the surface of the earth perceives points on the upper part of the moon's disk, at the time of her rising, which could not be seen from the centre. In proportion as the moon acquires a greater elevation, these points approach the border of the disk, and finally disappear, while new ones become visible on the eastern part of the disk, which increase in number as the moon descends towards the horizon; so that in the course of a day she appears to oscillate about her radius vector in the direction of the earth's rotation. This phenomenon constitutes what is called the Diurnal Libration, and is evidently the effect of the lunar parallax.

The libration in latitude and the diurnal libration were discovered by Galileo soon after the invention of the telescope. It was Hevelius who discovered the libration in longitude, and explained it by the hypothesis of the equable rotatory motion of the moon combined with her unequal velocity in her orbit.

To the inhabitants of the moon, if such there be, the earth will appear as a species of moon, much larger than the moon appears to us, but visible only to that hemisphere which is turned towards the earth. At those places which are situated near the border of her visible disk, the earth will sometimes rise a few degrees above the horizon; and an inhabitant of the moon placed near the middle of the hemisphere presented to the earth will always see the earth near his zenith, making oscillations of only a few degrees in consequence of the libration. But an inhabitant of the other hemisphere will never see the earth at all; so that while one hemisphere of the moon is constantly enlightened, during her long night, by the light reflected from the earth, the other remains in constant darkness. In regard, therefore, to the distribution of light, one of the lunar hemispheres enjoys very great advantages over the other.

The elements of the rotation of the moon, that is to say, the position of her equator, the place of its nodes, and its inclination to the plane of the ecliptic, are found by the same methods which have been explained for determining the corresponding elements relatively to the sun. The geocentric positions of the spots are observed in the same manner; and in converting them into selenocentric latitudes and longitudes, the same formulae may be employed, with a slight modification rendered necessary by the inclination of the lunar orbit to the ecliptic. One circumstance, not less remarkable than the coincidence which obtains between the times of rotation and sidereal revolution, is, that the nodes of the lunar equator coincide with those of the moon's orbit, if not exactly, at least so nearly, that the differences are so small as to fall within the probable errors of observation and calculation. All the observations since the time of Hevelius agree in showing that the longitude of the descending node of the equator is very nearly equal to the mean longitude of the ascending node of the orbit; whence it follows that the nodes of the equator have a retrograde motion equal to that of the nodes of the orbit. With regard to the inclination of the lunar equator to the ecliptic, Mayer states it to be $1^\circ 29'$, and Lalande $1^\circ 43'$. According to the latest computations made from the observations of Bouvard, the mean inclination of the lunar equator to the ecliptic is $1^\circ 28' 42''$. Mr Baily makes it $1^\circ 30' 10''$. Since the descending node of the equator coincides with the ascending node of the orbit, it is evident that its plane must be situated between the planes of the ecliptic and orbit, making an angle of about $1^\circ 30'$ with the first, and of $3^\circ 39'$ with the second.

Suppose three planes to pass through the centre of the moon, one of which represents her equator, the second the mean plane of her orbit, and the third parallel to the ecliptic. It is evident, from what precedes, that these three planes have a common section; that the first falls between the other two, making with them respectively the angles $3^\circ 39'$ and $1^\circ 30'$. In the space of 6793 days, the time of a revolution of the nodes of the lunar orbit, the poles of the first two planes describe about the pole of the ecliptic, in a direction contrary to the order of the signs, two small circles parallel to the ecliptic, and of which the semidiameters are respectively the arcs $1^\circ 30'$ and $5^\circ 9'$. Hence the difference between the longitudes of these two poles is constantly $180^\circ$, and the three poles are situated on the same great circle, that of the ecliptic being between the two others.

These results, which rank among the most curious discoveries of modern astronomy, were first obtained by Dominic Cassini; they were shown by Lagrange to be necessary consequences of the attraction which the earth exercises on the lunar spheroid.

The positions of the spots on the moon's surface are determined by their distance from the lunar equator, and from a conventional meridian, that is, by their selenocentric latitudes and longitudes, after the manner in which the position of places is determined on the surface of the earth. The first meridian is assumed to be that which passes through the pole of the visible hemisphere when the true place of the moon in her orbit is equal to her mean place; hence the first meridian is always very near the middle of the face which the moon turns towards the earth, never deviating from it farther than by a quantity Theoretical equal to the equation of the moon, or her libration in longitude. The rotatory motion being equal to that of revolution, the selenocentric longitude of the first meridian, at any epoch, is found by adding 180° to the mean longitude of the moon; and this gives also the distance of the first meridian from the ascending node of the lunar equator. The position of the equator and first meridian being determined, the co-ordinates of a spot are computed without difficulty; and in this manner catalogues of the spots have been formed, and arranged according to their latitudes and longitudes.

Sect. V.—Of the Nature and Constitution of the Lunar Substance.

It has already been observed, that a slight attention to the different phases of the moon is sufficient to prove that she is an opaque spherical body, shining only by virtue of the light which she receives from the sun. The line bounding the visible part of her surface has exactly the form which would be produced by an illuminated hemisphere brought into different positions with respect to the eye; and the circular contour of the obscured portion of the sun during a solar eclipse could only be caused by the interposition of a spherical body. Besides, the parts of her hemisphere turned towards the earth, which the sun's rays do not reach, are in some circumstances sufficiently discernible, and she has then the same circular appearance which she exhibits when at the full.

But if the light which comes to us from the moon is only that which she receives from the sun and reflects back to the earth, how does it happen, it may be asked, that the portion of her disk not directly exposed to the solar rays is distinctly visible for some days after the new moon? This phenomenon was ascribed by the ancients to the native light of the moon, to which, on account of its pale ashy hue, they gave the name of lumen incinerum. The explanation which is now generally given was first suggested by a celebrated painter, Leonard da Vinci. It consists in supposing that a portion of the light which is reflected from the illuminated hemisphere of the earth to the moon undergoes a second reflection at the lunar surface, and is transmitted back to the earth. The ancients were confirmed in their opinion respecting the native light of the moon, by observing that she is not altogether invisible in her eclipses. Plutarch, indeed, ingeniously ascribes her appearance under these circumstances to the light of the stars reflected from the moon; a cause, however, totally inadequate to produce the effect. This phenomenon is now generally ascribed to the scattered beams of the sun bent into the earth's shadow by the refraction of its atmosphere.

The opinion that the moon's light is chiefly, if not wholly, caused by the reflection of the sun's rays at the lunar surface, has prevailed in all ages; and indeed no other explanation seems to have been thought of till it was suggested by Licetus, professor of philosophy at Bologna, as being more probable that the moon possesses a phosphorescent quality, and that the sun's influence is only wanted to occasion the propulsion of the light which lies absorbed in her substance. This idea has been adopted by Professor Leslie, whose arguments in its support are at least extremely plausible and ingenious. So far as the appearances, and the explanation of the phases, are concerned, it is evidently matter of indifference whether we suppose that the solar rays are reflected from the surface of the moon, or that they exert an action in virtue of which the moon emits rays of her own. In either case it is that part of her surface only which is exposed to the impact of the solar rays that sends forth light to the earth.

The principal argument in favour of the phosphorescent nature of the moon is founded on the quantity of light which proceeds from her surface. It is evident that the moon does not act as a polished speculum, and reflect the whole of the incident rays; for in that case, as is known from the laws of Catoptries, she would merely reflect an image of the sun, equally bright, varying in size according to her different positions relatively to the sun and the earth, and increasing till she arrives at her opposition, when her diameter would appear equal to about the 458th part of its real dimensions. Her phases could never be distinguished, and she would only appear to approach to or recede from the earth, in proportion as her diameter increased or diminished. It follows, therefore, that the moon's surface must be irregular, or what is termed a mat surface, that is, of such a nature, that from every point of it the rays of light are reflected in all directions indifferently. According to the experiments of Bouguer, a white surface of this sort, for instance paper, or plaster of Paris, reflects only about the 150th part of the rays which fall upon it in a perpendicular direction; and the proportion is less as the angle of incidence becomes more oblique. Making allowance for the irregular surface and obscure spots of the moon, Mr Leslie computes that the solar light which she emits to the earth must be attenuated at least 105 million times; but Bouguer's experiments show that the moon's light is between the 250,000th and 300,000th part of the direct light of the sun, or about 350 times greater than the computed amount of reflected light. If every part of the moon's surface reflected the light in the most perfect manner, it may be shown that only the 210,000th part of the rays which she receives from the sun would be thrown off in the direction of the earth; a quantity not much exceeding that which, according to Bouguer's estimate, we actually receive from her. Mr Leslie states that he found the intensity of the moon's light to approach the 150,000th part of the direct light of the sun; a result which, if admitted, must be entirely decisive of the question; for as the utmost possible quantity of reflected light cannot exceed the 210,000th part, it follows that the excess must be owing to the spontaneous light of the moon. Hence this ingenious philosopher concludes that the body of the moon is a phosphorescent substance, like the Bolognian Stone, which possesses the property of shining for some time when carried into a dark room, after having been exposed to the light of the sun. A fact first observed by the celebrated Arago seems to increase the probability of this opinion. All rays reflected from a surface not metallic acquire a peculiar modification, or become polarized; but as the rays of the moon are not so modified, it is inferred that they have not undergone a reflection at her surface.

The secondary light of the moon, of which we have already made mention, affords arguments in favour of her native light precisely similar to the above. If the earth reflected, like a mirror, the whole of the incident rays, the illumination produced by the reflection would amount to about a 16,000th part of that which is caused by the sun; but as the sea reflects only about a 55th part of those rays, and the land a still smaller proportion, we may suppose that the reflected light of the earth does not exceed a millionth part of the direct light of the sun. It is extremely doubtful whether a light so greatly attenuated would suffice to render the moon visible. The lucid bow, or silvery thread of light, which proceeding from the extremities of the lunar crescent, seems to embrace her unenlightened orb, is easily explicable on this hypothesis; Theoretical whereas it can hardly be satisfactorily accounted for by Astronomy, ascribing it to the secondary illumination from the earth.

"I should rather refer it," says Mr Leslie, "to the spontaneous light which the moon may continue to emit for some time after the phosphorescent substance has been excited by the action of the solar beams. The lunar disk is visible although completely covered by the shadow of the earth; nor can this fact be explained by the inflection of the sun's rays in passing through our atmosphere; for why does the rim appear so brilliant? Any such inflection could only produce a diffuse light, obscurely tingling the boundaries of the lunar orb; and, in this case, the earth, presenting its dark side to the moon, would have no power to heighten the effect by reflection. But even when this reflection is greatest about the time of conjunction, its influence seems extremely feeble. The lucid bounding arc is occasioned by the narrow lunula, which, having recently felt the solar impression, still continues to shine; and from its extreme obliquity, glows with concentrated effect." (Inquiry into the Nature and Propagation of Heat.)

Fig. 42. Although these arguments go far to support the ancient opinion of the native light of the moon, they are not entirely conclusive; and indeed cannot be easily reconciled with some of the phenomena. If the moon shines in virtue of her native light, rays will be emitted in all directions from every point of her surface; whence, since a visual angle of a given magnitude includes a much larger portion of a spherical surface near the extremities of its apparent disk than towards the centre, and as the number of rays is proportional to the surface from which they proceed, it follows that the intensity of the moon's light ought to be greater near the border than at the centre of her disk. The reason why this is not the case with regard to the sun is, that a greater proportion of the rays are absorbed in passing through a greater extent of the solar atmosphere; but the moon, having no atmosphere, ought to be sensibly most brilliant near the circumference of her orb. The contrary is, however, the case; her light is greatest at the centre, and less intense towards the circumference, exactly as it ought to be on the supposition of its being occasioned by the reflection of the solar rays. With regard to the ingenious argument of Arago, it cannot be held to be conclusive till we become more certainly acquainted with the nature of the lunar substance. The only property which we can safely ascribe to it as yet is density: whether in its physical properties it resembles the substances with which we are acquainted, is a question hardly within the bounds of legitimate investigation.

The spots of the moon, affording grounds for conjectures relative to her physical constitution and the nature of her surface, have been observed with great interest since the discovery of the telescope; and as they are of some service in the observation of eclipses, astronomers have been at much pains to determine their selenographic positions. On account of their number, it has been found necessary to distinguish them by particular names. Riccioli designated the most conspicuous of them by the names of astronomers, and other eminent men. Hevelius gave them the names belonging to countries, islands, seas, and regions on the earth, without reference to situation or figure. The nomenclature of Riccioli has, however, been deservedly preferred by Schroeter and others who have particularly observed the phenomena of the lunar surface, and is now universally followed. Mayer gave a catalogue of 89 of the most remarkable of the spots, with their selenographic latitudes and longitudes referred to a first meridian, namely, that which passes through the centre of the moon's apparent disk, perpendicular to the theoretical lunar equator, accompanied by an accurate map of her surface. Delineations of the lunar disk have also been given by Hevelius in his Selenographia by Cassini, Russel, Schroeter, Lohrmann, and others. The engraving (fig. 47) which accompanies this article gives a pretty accurate view of the appearance of the moon in her mean libration.

The following table contains the selenographic positions of some of the principal spots. The sign + indicates a northern, and — a southern latitude.

| No. | Riccioli's Names | Long. | Lat. | |-----|-----------------|-------|------| | 1 | Zoroaster | West 72° | +58° | | 2 | Mercurius | 67 | +40 | | 3 | Petavius | 64 | -24 | | 4 | Langrenus | 62 | -8 | | 5 | Endymion | 60 | +53 | | 6 | Cleomedes | 55 | +26 | | 7 | Atlas | 48 | +47 | | 8 | Hercules | 42 | +48 | | 9 | Censorinus | 32 | 0 | | 10 | Fracastorius | 32 | -22 | | 11 | Possidonus | 32 | +31 | | 12 | Theophilus | 27 | -12 | | 13 | Cyrilus | 25 | -13 | | 14 | St Catharina | 24 | -18 | | 15 | Menelaus | 15 | +16 | | 16 | Aristoteles | West 15 | +50 | | 17 | Ptolomaeus | East 2 | -10 | | 18 | Arzachel | 3 | -20 | | 19 | Archimedes | 5 | +28 | | 20 | Tycho | 10 | -43 | | 21 | Plato | 10 | +52 | | 22 | Pitatus | 12 | -29 | | 23 | Eratosthenes | 12 | +14 | | 24 | Clavius | 16 | -60 | | 25 | Copernicus | 19 | +9 | | 26 | Bullialdus | 21 | -21 | | 27 | Biancanus | 25 | -65 | | 28 | Heradilides | 38 | +41 | | 29 | Keplerus | 38 | +7 | | 30 | Gassendus | 39 | -19 | | 31 | Aristarchus | 48 | +24 | | 32 | Hevelius | 67 | -1 | | 33 | Schickardus | 68 | -49 | | 34 | Grimaldus | East 68 | -5 |

That there are prodigious inequalities on the surface of the moon, is proved by looking at her through a telescope at any other time than when she is full; for then there is no regular line bounding the dark and illuminated parts, but the confines of these parts appear as it were toothed and cut with innumerable notches and breaks; and even in the dark part, near the borders of the enlightened surface, there are seen some small spaces enlightened by the sun's beams. Upon the fourth day after new moon there may be perceived some shining points, like rocks or small islands, within the dark body of the moon; but not far from the confines of light and darkness there are observed other little spaces which join to the enlightened surface, but run out into the dark side, which by degrees change their figure, till at last they come wholly within the illuminated face, and have no dark parts round them at all. Afterwards many more shining spaces are observed to arise by degrees, and to appear within the dark side of the moon, which, before they drew near to the illuminated portion of the disk, were invisible, being totally immersed in the shadow. The contrary is observed in the decreasing phases, where the lucid spaces which Theoretical joined the illuminated surface recede gradually from it, and remain for some time visible after they are quite separated from the confines of light and darkness. Now it is impossible that this should be the case, unless these shining points were higher than the rest of the surface, so that the rays of the sun may illuminate their summits before they reach their bases. Portions of considerable extent are also perceived on the lunar surface, which are never brilliant like the other parts, but remain constantly obscure. These are supposed to be deep valleys or cavities: they were formerly supposed to be seas, but, for reasons about to be given, this idea has been abandoned.

These phenomena render it certain that the surface of the moon is covered with mountains of a great height, with rocks or masses of unknown matter, but possessing the property of reflecting the sun's light. The height of the lunar mountains may be determined in the following manner.

Let ABO (fig. 48) be the illuminated hemisphere of the moon, SO the tangential solar ray, and consequently O one of the points of the circle which separates the enlightened from the obscure hemisphere. All the part OD will be in darkness; but if this part contain a mountain aM so elevated that its summit M reaches the solar ray SO, the point M will be enlightened. Now, if the line OM can be determined by observation, it will be easy to deduce the arc Oa, and thence the height of the mountain aM, in terms of the moon's radius. Let E be the place of an observer on the earth; draw the lines EM, EO, EC (C being the centre of the lunar orb), and Om perpendicular to EM. The distance of the moon from the earth being known, we have the distance EO: the angle OEa is measured by the micrometer; therefore Om, which is the projected distance of OM, is a given quantity. Now OM = \(\frac{Om}{\cos(MO)}\); and since OEm is a very small angle, OEm may be considered a right angle, consequently \(OM = MOE - 90^\circ\); therefore \(OM = \frac{Om}{\cos(MOE - 90^\circ)} = \frac{Om}{\sin(MOE)} = \frac{Om}{\sin(EOS)}\), that is, the distance between the summit of the mountain and the illuminated part of the moon's disk is equal to the projected distance measured by the micrometer, divided by the sine of the moon's elongation from the sun. Suppose this distance OM = \(n \times CO\); we shall then have CM = \(CO \sqrt{1 + n^2}\), and the height of the mountain aM = \(CO(\sqrt{1 + n^2} - 1) = \frac{n^2}{2} CO\) (neglecting in the development of the radical the powers of \(n\) which are higher than the square). According to the observations of Hevelius, the greatest value of \(n\) is \(\frac{1}{3}\), which gives the height of the mountain equal to \(\frac{1}{3}\) of the semidiameter of the moon. Schroeter, who estimated the heights of the lunar mountains by measuring the projections of their shadows at the time the sun was near their horizon, makes it in some cases \(\frac{1}{3}\). The highest mountains on the earth do not reach an elevation greater than \(\frac{1}{3}\) of the terrestrial radius; the lunar mountains, therefore, in proportion to the diameters of the earth and moon, are nearly five times higher than those of our globe. Their absolute height is above five English miles. It is easy to see that these determinations are susceptible of very little accuracy.

Not only is the moon's surface rendered irregular by high and precipitous mountains, but numerous cavities appear in every part of her surface, some of which, according to Schroeter, are upwards of four English miles in depth, and forty in circumference at the orifice. An insulated mountain is frequently observed to rise in the centre of these enormous pits or caverns; and they are surrounded by high annular ridges, the masses of which would exactly fill the inclosed cavities. From this circumstance, it is probable that the elevations and hollows which abound on the surface of the moon have been produced by volcanic eruptions. Herschel imagined that he even observed volcanoes in activity. At the time of the new moon he perceived on different parts of her obscure disk three luminous points, resembling pieces of burning charcoal, covered with a thin coat of white ashes, one of which, by a comparison with the third satellite of Jupiter, appeared to be upwards of three miles in diameter. Their brilliancy continued during several days, undergoing variations altogether independent of the increase of the moon's apparent magnitude, and at the end of that time they appeared to become extinct.

The existence of a lunar atmosphere has been a fertile subject of controversy among astronomers. It has been urged that, as the brightness of the moon is sensibly equal at all times when she is not obscured by clouds in the terrestrial atmosphere, she cannot be surrounded by an atmosphere similar to that of our earth, so variable in its density, and so liable to be obscured by clouds and vapours. The assumption of an equable brightness in the appearance of the moon has not, however, been allowed to pass uncontroverted. Hevelius relates that he has several times found in skies perfectly clear, when even stars of the sixth and seventh magnitude were visible, that, at the same altitude of the moon, at the same elongation from the earth, and with the same telescope, the moon and her maculae do not appear equally lucid and clear at all times, but are much brighter and more distinct at some times than at others. Several other observers have noticed that the moon is not always equally conspicuous during total eclipses; a fact which is held to be indicative of accidental variations in the state of the lunar atmosphere.

Another objection against the existence of a lunar atmosphere is derived from the circumstance that, if it existed, its influence would be perceptible in the occultations of the planets, or fixed stars, by the moon. When the moon approaches so near to a star that part of her atmosphere (supposing she has one) is interposed between the star and the eye of the observer, the star, it is contended, ought to suffer a change in its colour in consequence of the absorption of some of its light in traversing the denser parts of the moon's atmosphere. But when we consider that the region of the vapours and clouds in the terrestrial atmosphere does not exceed the height of four miles, or the 880th part of the earth's diameter, and that consequently the obscure part of the lunar atmosphere, supposing it to be similarly constituted, would not subtend an angle of one second (the mean apparent diameter of the moon being 1889 seconds), and that this space is passed over by the moon in less than two seconds of time, it will scarcely be expected that observation will be able to decide whether the supposed obscuration takes place or not. Some observers have, however, remarked instances of stars, when about to be occulted, presenting an evident diminution of light. (See a Paper by Mr Ramage of Aberdeen, in the 2d vol. of the Memoirs of the Astronomical Society.)

The existence of a lunar atmosphere cannot increase or diminish the apparent diameter of the moon, because its effect on the rays of light when they enter it will be exactly counteracted at the time of their emergence. It might seem, therefore, that in order to determine whether the light of a star or planet is inflected when it passes very near the Theoretical border of the lunar disk, it would be sufficient to compare Astronomy, the observed duration of the occultations of the stars, or eclipses of the sun, with the time calculated from the theory of the moon's motion, which she consumes in passing through a space equal to her apparent diameter. But if, as is generally supposed, a certain irradiation is produced around luminous objects, by which their image is dilated, the apparent diameter of the moon will thereby be augmented when she is projected on the dark sky. On the contrary, when the disk of the moon is projected on that of the sun in annular eclipses, the irradiation, by dilating the luminous ring which seems to surround the moon, will cause her apparent diameter to appear smaller than it actually is. On this account it becomes very difficult to ascertain by direct measurement the amount of the inflection of light at the border of the lunar disk, or even to be certain of its existence. An indirect method of arriving at the desired object was, however, imagined by Dionis du Sejour, who regarded the irradiation and inflection as two unknown quantities to be determined simultaneously from the observations of the phases of eclipses, upon which their effects are different.

The observations best adapted for this purpose are those of the magnitude of the luminous crescent, and its successive increase in annular eclipses. Du Sejour calculated with great care the annular eclipse of the sun which took place in the year 1764, and was visible over all Europe; and which, by reason of the extent and variety of its phases, presented a great many points of comparison. He particularly employed the observations made in different places of the instants of the formation and rupture of the ring. He likewise confirmed these observations by measurements made by Mr Short, at London, of the distance between the horns of the crescent at divers instants. It resulted, from the comparison of an immense number of observations, that the measurements could not be reconciled without supposing an irradiation of $3^\circ$ on the semidiameter of the sun, and an inflection of nearly the same amount round the disk of the moon, produced by her atmosphere. According to this result, the horizontal refraction at the surface of the moon amounts to $1^\circ5$. The mean horizontal refraction observed at the surface of the earth is $0^\circ585$, and consequently about 1400 times greater than that at the moon. Hence, supposing the moon's atmosphere of the same nature as that of the earth, its density must be 1400 times less, and consequently rarer than the most perfect vacuum which can be produced by the best pneumatic machines.

The existence of a lunar atmosphere, though of small extent, is indicated with considerable probability by the observations of Schroeter, who perceived that some ranges of lunar mountains, when in the dark hemisphere, are illuminated more feebly in proportion as they recede from the boundary of light and darkness; an effect which would be produced by the partial absorption of the rays of light which pass near the moon's surface. He also remarked the same circumstance with regard to the cusps, and perceived other indications of an atmosphere in some changes of tint, which induced him to think that a twilight might be discernible towards the cusps. With a view to prove the accuracy of this conjecture, he examined the moon with great care under the most favourable circumstances, and at length described a faint glimmering which he took to be crepuscular light, extending into the cusps into the dark body of the moon. Its greatest breadth was $2^\circ$; and it extended $1^\circ20'$ from the cusps, along the circumference of the lunar disk.

From these data he computed that the height of the lunar atmosphere, to the limit where it ceases to inflect the rays of light or diminish the brightness of a star, does not exceed 5742 English feet. A ring of this breadth, at the distance of the moon, will subtend an angle of only $0^\circ94$; hence the almost imperceptible influence of the lunar atmosphere.

Some observations, related by Mr Ramage of Aberdeen, in a memoir already alluded to, of occultations of Jupiter and his satellites, with an excellent reflecting instrument, also tend to confirm the existence of a small lunar atmosphere. On the approach of the satellites no diminution of their light was perceptible. On coming into contact with the moon's limb they did not disappear instantly, like fixed stars, but formed an indentation or notch in the limb, as if imbedded in it, but at the same time separated from it by a fine line of light. The indentation continued visible till about half their diameters were immersed, when it disappeared. (Memoirs of the Astronomical Society of London, vol. ii. p. 87.)

The dark spots on the moon's surface were formerly supposed to be water; but as elevations and cavities are distinctly perceptible in them, that hypothesis is evidently erroneous. Besides, the extreme tenuity of her atmosphere is inconsistent with the existence of water at her surface. It is only by the weight of the terrestrial atmosphere that the liquids at the surface of the earth are prevented from being dissipated in vapours. If the present atmosphere were removed, every liquid would continue to be dissipated in this manner till a new atmosphere was formed, to which each would contribute in proportion to its elastic force; and the evaporation would only cease when the tension of the vapour of each liquid was equal to its elastic force in a vacuum at the same temperature. But if the vapours were removed as they arose, by any absorbing cause, the evaporation would continue till the liquids entirely disappeared. Now we may suppose this to have been the case with respect to the moon, and that at one time she may have had an atmosphere, which the attractive force of the earth, aided by some accidental circumstance, may have swept away and united with our own. (Biot, Astr. Phys., tome ii. p. 413.)

Under these circumstances it is evident, that no animal, similarly constituted to those which inhabit the earth, could respire at the surface of the moon. Everything there appears solid, desolate, and unfit for the production and support of organized substances; and the excessive cold which certainly prevails must be sufficient to destroy every source of animal or vegetable life. May it not then be supposed that the moon is a planet which has not yet reached a state of maturity—a maturity to be prepared by successive volcanic eruptions; or that, having fulfilled its destiny, it is now in a state of decay?

On the subject of this section, see Hevelius, Selenographia; Schroeter, Selenuotopographische Fragmente; Hooke's Micrographia, 1665; Lemonnier, Selenographie; Mayer, Cosmographische Nachrichten, 1748; Boscovich, De Lune Atmosphera, 1753; Dunn, Phil. Trans., 1762; Herschel, Phil. Trans., 1780 and 1787, p. 229; Ferguson's Astronomy, by Brewster; Biot, Astronomie, tome ii. p. 413; Schubert, Traité d'Astronomie Théorique, tome ii. p. 364.

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1 It has always been a favourite opinion with mankind, that the celestial bodies, resembling the earth in some respects, are also, like it, peopled with rational beings. The author of the verses attributed to Orpheus ascribes to the moon many mountains, and cities, and palaces: Sect. VI.—Of Eclipses and Occultations.

In describing these interesting phenomena, we will first consider the eclipses of the moon, which, for any given place on the earth, are much more frequent than those of the sun, and, by reason of certain circumstances about to be explained, can also be computed with much greater facility.

1.—Eclipses of the Moon.

The earth being an opaque, round body, much smaller than the sun, must project behind it in space a conical shadow, limited by straight lines drawn from the extremities of the sun's disk to touch the surface of the earth. When the moon enters this shadow, and a portion of her disk is still enlightened by the sun, the enlightened part will necessarily have the form of a luminous crescent, the concavity of which is turned to the conical shadow of the earth; and this appearance will likewise be exhibited when the moon begins to emerge from the shadow. As she approaches towards the shadow, her light is not suddenly eclipsed, but passes insensibly through all the successive gradations of obscurity, till the darkness attains its greatest intensity. The reason of this will be easily comprehended by considering that when an opaque body is placed between an object and the sun, so as to conceal only a part of his disk, the object is then less enlightened than when none of the solar rays are intercepted; and this in proportion as more or less of the sun is concealed. Between full illumination and total obscurity there are consequently intermediate tints and gradations of light, which are denominated the Penumbra, in contradistinction to the Umbra which covers those places to which the sun's rays are completely intercepted. Let S and E (fig. 49) be the centres of the sun and earth, and ABB'A' any plane whatever passing through the axis SE, in which, let AB, A'B' be tangents to the sun and earth on opposite sides of the axis SE; these tangents will meet the prolongation of the axis in a point C. Let also AB, A'B' touch the sun and earth on opposite sides of SE, and let C' be their intersection. If we now suppose the plane to turn round the axis SC, a conical shadow or umbra, of which the apex is C, will be formed behind BB'; and if the moon is situated within any part of the cone BCB', no light whatever can fall on her. As soon as the moon emerges from this cone, a part at least of the sun's rays will fall upon her, and she will be in the penumbra. If situated, for example, at M, and MN be drawn to touch the earth and meet the solar disk in N, then the part of that disk between A and N will be visible at M. The extent of the penumbra is therefore determined by the angle DBC. When M is situated in the straight line BD, the whole of the sun's disk is visible; when M is in BC the sun entirely vanishes, and his visible portion diminishes from the instant M passes BD till it reaches BC. The intensity of the penumbra, therefore, goes on increasing from the first of these limits to the second, where it is confounded with the total darkness. This explains the progressive obscuration of the moon's disk in her eclipses.

If we now attempt to determine the circumstances under which an eclipse of the moon can take place, it is obviously necessary to inquire, in the first place, into the length of the conical shadow BCB'; for an eclipse can only happen on the supposition of its extending beyond the orbit of the moon. For this purpose let EF (fig. 50) be drawn parallel to AB, and meeting SA in F. We have then sin. ECB = sin. SEF = SF/SE = SA/SE = EB/SE = sin. R — sin. p, R being the apparent semidiameter of the sun, and p his horizontal parallax. By reason of the smallness of these angles the arcs may be substituted for the sines; hence the angle ECB = R — p. But CE = EB/sin. ECB, therefore making EB, the radius of the earth, = a, CE = a/sin.(R — p).

From this expression it is evident that the value of CE depends on the horizontal parallax, and therefore varies with the sun's distance from the earth. On calculating, by means of the values of R, a, and p, given in the preceding sections, the values of CE at the perigee, mean, and apogee distances, the following results will be obtained:

| Lengths of CE | |--------------| | Sun in perigee | 212.896 terrestrial radii. | | at mean distance | 216.531 | | in apogee | 220.238 |

Now the greatest distance of the moon from the earth is less than 64 terrestrial radii (sect. ii.), consequently the shadow of the earth is projected into space between three and four times farther than the distance of the moon. Hence it appears, that if the moon moved in the ecliptic she would traverse the earth's shadow and be eclipsed every revolution. On account of the inclination of her orbit to that plane, the eclipses can only happen when she is in or near her nodes. The greatest distances from the nodes at which they can take place are called the Lunar Ecliptic Limits.

In order to determine these limits, it is necessary first of all to know the apparent diameter of a section of the earth's shadow, at the place where it is traversed by the orbit of the moon. Let lMF (fig. 50) be a part of the moon's orbit, intersecting the lines AC and A'C in m and m'; then m'm' is the geocentric diameter of the shadow, the half of which is measured by the angle m'EC. Now the angle mEC is the difference between the angles EmA and ECA, the first of which, namely, EmA, is the apparent semidiameter of the earth seen from the moon, or, in other words, the moon's horizontal parallax. Let this angle, therefore, be denoted by P. It has already been shown that ECA = R — p; consequently we have mEC = P + p — R in all cases; that is to say, the semidiameter of the shadow is equal to the sum of the horizontal parallaxes of the sun and moon, diminished by the apparent semidiameter of the sun. On calculating the amount of this expression from the values of P and p given above, the following table will be obtained, exhibiting the magnitude of the apparent diameters of the earth's shadow for different distances of the sun and moon.

| Apparent diameters of earth's shadow. | |--------------------------------------| | Sun in perigee | | at mean distance | 1 23 2 31 | | in perigee | 1 30 40 31 | | in apogee | 1 15 56 86 | | Sun at mean distance | | at mean distance | 1 23 34 87 | | in perigee | 1 31 12 87 | | in apogee | 1 16 28 29 | | Sun in apogee | | at mean distance | 1 24 6 30 | | in perigee | 1 31 44 30 |

The greatest apparent diameter of the moon being only 33°31'07", which is about a third part of the diameter of the earth's shadow, it follows that the moon may not only be completely enveloped in the shadow, but since she passes over a space nearly equal to her own breadth in an hour, that she may continue to be totally eclipsed during a space of about two hours.

The above determination refers only to the umbra or cone of total darkness; but the diameter of the penumbra is obtained in a manner exactly similar. In the same Theoretical diagram the semidiameter of the penumbra is measured by the visual angle \( \angle EEC \); but \( \angle EEC = \angle ECA + \angle CEA \), and \( \angle ECA = \angle CEA + \angle CAE \); therefore \( \angle EEC = \angle ECA + \angle CEA + \angle CAE \); or, retaining the same denominations as above, \( \angle EEC = P + p + R \). Hence, the semidiameter of the penumbra is equal to the horizontal parallaxes of the moon and sun, augmented by the apparent semidiameter of the sun.

All the different numerical values of the semidiameter of the penumbra corresponding to particular positions of the sun and moon may be computed exactly in the same manner as in the case of the umbra.

The two expressions for the semidiameters of the umbra and penumbra, viz., \( P + p - R \) and \( P + p + R \), give immediately the distance of the moon's centre from the axis of the cone when her disk comes into contact with the shadow. Representing the apparent semidiameter of the moon by \( r \), her disk will just touch the umbra when the distance of her centre from the axis SE is equal to \( P + p - R + r \); and it will touch the penumbra when the distance is \( P + p + R + r \). But on account of the very feeble obscurity of the penumbra towards its extreme border, it is impossible to observe with any degree of precision the time at which the moon enters it. In computing the ecliptic limits, therefore, it is only necessary to have regard to the umbra.

In the table given above, the extreme values of \( P + p - R \), the apparent semidiameter of the earth's shadow, are \( \frac{1}{2} (15^\circ 13' 24'' 30') = 37^\circ 42' 15'' \), and \( \frac{1}{2} (15^\circ 31' 44'' 30') = 45^\circ 52' 15'' \); and the least and greatest values of \( r \), the apparent semidiameter of the moon, are respectively \( 14^\circ 45' \) and \( 16^\circ 45' \) (sect. ii.); therefore the least distance of the moon's centre from the axis of the shadow at the time of her immersion or emergence is \( 37^\circ 42' 15'' + 14^\circ 45' = 52^\circ 27' 15'' \), and the greatest \( 45^\circ 52' 15'' + 16^\circ 45' = 62^\circ 37' 15'' \). The first is the limit within which an eclipse must necessarily happen; the last that beyond which it cannot happen.

It is now easy to ascertain the limits of the moon's distance from her node, within which the eclipses take place. Let NC (fig. 51) be a portion of the ecliptic, NM part of the moon's orbit, N its node, C the centre of a section of the earth's shadow, M the centre of the moon; the verge of the lunar disk touching, but not penetrating, the shadow at \( a \). It is evident that, if the moon be at a greater distance than CM from NC, there can be no eclipse. The greatest value of CM, as we have just seen, is \( 62^\circ 37' 15'' \), from which the corresponding ecliptic limit NC is easily computed by means of the formula

\[ \sin CM = \sin NC \times \sin CNM. \]

Supposing the angle CNM, that is, the inclination of the moon's orbit to the ecliptic, to be \( 5^\circ \), which is its minimum value, the value of NC is found from the logarithmic tables to be \( 12^\circ 2' \) nearly. Hence an eclipse of the moon can only happen when she is within about \( 12^\circ 2' \) of her node. Under the most favourable circumstances, however, the limits may extend to \( 13^\circ 21' \). A lunar eclipse will certainly take place if the moon's distance from her node, at the time of her mean opposition, is not greater than \( 7^\circ 4' \).

When the moon's disk only comes into contact with the shadow, as in fig. 51, the phenomenon is called an annulus; when the disk only enters into the shadow in part, the eclipse is said to be partial; it is called total if the moon entirely disappears, and central when her centre coincides with the axis of the cone, or if at the time of the eclipse the moon is exactly in her node.

In the preceding determinations the shadow has been supposed to be conical, whereas, on account of the compression of the earth at the poles, it is not exactly a cone, but a conoid on an elliptic base; and it varies at every instant by reason of the earth's rotation. To determine rigorously the figure of the cone at every instant, and the diameter of its section where the moon enters and leaves it, would require calculations of great complication and prolixity. Such precision is, however, unnecessary; and indeed it is impossible to attain to absolute accuracy so long as the figure of the earth is not exactly known. It is usual and natural to employ the largest diameter which the shadow can have at the distance of the moon.

It has also been supposed that the shadow is terminated by tangents to the sun and the earth; that is to say, that all the rays of light which are not obstructed by the globe of the earth pass in straight lines from the sun to the moon. But the earth being surrounded by an atmosphere which near the surface exerts a powerful action on the solar rays, it is to be presumed that those rays which, if unobstructed, would glance by the surface of the earth, are absorbed by the lower strata of the atmosphere. The effect of this will be to enlarge the diameter of the shadow; and the requisite correction may be regarded as an augmentation of the earth's radius, by adding to it a part of the atmosphere; or, which amounts to the same thing, an augmentation of the lunar parallax. Now, it is found by experience that such a correction is necessary; and, according to Mayer, the lunar parallax must be augmented by a 60th part in order to satisfy the observations.

Another effect of the action of the earth's atmosphere on the solar rays is to render the moon dimly visible even when she is totally eclipsed,—a circumstance to which allusion was made in the preceding section. Let the circle \( fghi \) (fig. 52) concentric to the earth include that part of the atmosphere which is sufficiently dense to produce a sensible refraction of the rays of light. All those rays which do not fall within that circle, such as Wf, Vir, proceed in their direct course without suffering any refraction; but those which enter the atmosphere between \( f \) and \( h \), and between \( i \) and \( l \), on opposite sides of the earth, are gradually more bent inward as they go through a greater portion of the atmosphere, until the rays Wk and Vl, touching the earth at \( m \) and \( n \), are bent so much as to meet at \( q \), a little short of the moon; and therefore the dark shadow of the earth is contained in the space \( mopq \), where none of the sun's rays can enter; all the rest, \( R \), being mixed by the scattered rays which are refracted as above, is in some measure enlightened by them; and some of those rays falling on the moon, give her the colour of tarnished copper, or of iron almost red-hot; so that if the earth had no atmosphere, the moon would be as invisible in total eclipses as she is when new. If the moon were so near the earth as to go into its dark shadow, suppose about \( p \) or \( q \), she would be invisible during her stay in it, but visible before and after in the fainter shadow \( RR \).

After having pointed out the general phenomena of the lunar eclipses, and the limits within which they take place, it only remains to show in what manner the time of their commencement, end, and duration, and also their magnitude, may be determined by computation.

Let the line NE (fig. 53) represent the ecliptic, NO the orbit of the moon, C the centre of the terrestrial shadow, and M the centre of the moon at the instant of Fig. 53, the opposition; then CM will be the circle of latitude on which the opposition takes place. The centre of the shadow C being always in opposition with the sun, moves along the ecliptic from west to east, or from \( N \) towards \( E \), with the same velocity as the sun. The moon also, at the same time, moves in her orbit from west to east, or from \( N \) towards \( O \). Now the velocities of these two motions are given by the astronomical tables, and the question is to determine the instant of time at which the circles... Theoretical representing sections of the moon and the earth's shadow meet each other either before or after the opposition.

At the time of an eclipse the apparent distance of the centre of the shadow from the moon is very small, consequently CM, and also the differences of the respective longitudes and latitudes of C and M, may be regarded as straight lines. During the short interval between the commencement and end of an eclipse, the motion of the sun, and consequently that of the centre of the shadow, may likewise be regarded as uniform. By these suppositions, sufficiently accurate for our present purpose, the problem is considerably simplified.

Suppose now that C' and M' are two simultaneous positions of the shadow and moon at any instant before or after the opposition. Let M'P be perpendicular, and MQ parallel, to NE. The velocities of the moon and terrestrial shadow being known from the tables, the lines C'P and QM', which represent the motions of the centre of the moon relatively to that of the shadow in longitude and latitude, are known also; whence C'P and PM' are given, and consequently C'M' the distance of the centres.

Let us assume

\[ \lambda = CM \text{ or } \beta \text{'s latitude when in opposition}, \] \[ s = \beta \text{'s motion in longitude}, \] \[ m = \beta \text{'s horary motion in longitude}, \] \[ n = \beta \text{'s motion in latitude}, \] \[ t = \text{time from M to M'}, \] \[ c = C'M' \text{ the distance of the centres}. \]

Now, since we suppose that CP and QM' are the β's motion in longitude and latitude respectively in the time \( t \), it is evident that \( CP = m \cdot t \) and \( QM' = n \cdot t \). But CC' is the sun's motion, or the motion of the terrestrial shadow in longitude during the same time; therefore \( CC' = st \).

We have then \( CP = m \cdot t - st \), and \( PM' = \lambda + nt \); consequently

\[ c^2 = (m - s)^2 + (\lambda + nt)^2. \]

In this quadratic equation, if \( t \) is regarded as the unknown quantity, the only arbitrary quantity contained in it will be \( c \), the distance of the centres, the others being all determined from the tables. On assigning, therefore, any arbitrary value to \( c \), the resolution of the equation will give the corresponding value of \( t \), and consequently the circumstances or different phases of the eclipse which we may wish to determine.

On arranging the terms of the above equation so as to obtain the resolution relatively to \( t \), we have

\[ [(m - s)^2 + (\lambda + nt)^2]t^2 + 2\lambda \cdot nt = c^2 - \lambda^2, \]

which may be still simplified by introducing an auxiliary angle \( \delta \), such that \( \tan \delta = \frac{n}{m - s} \); for by this substitution there will result

\[ n^2t^2 + 2\lambda \cdot nt \sin^2 \delta \cdot t = (c^2 - \lambda^2) \sin^2 \delta, \]

which gives the two following values of \( t \):

\[ t = \frac{1}{n}(-\lambda \sin^2 \delta \pm \sin \delta \sqrt{c^2 - \lambda^2 \cos^2 \delta}). \]

The first of these denotes the time at which the moon enters, and the second that at which she quits, the umbra or penumbra.

The time at which the different phases of the eclipse happen, are calculated directly from this equation. If, for example, we wish to determine the time at which the moon's disk begins to enter the shadow, we make \( e = P + p - R + r \) (neglecting the small augmentation of the shadow occasioned by the refraction of the atmosphere). In the case of the penumbra we must take \( e = P + p + R + r \); and it is evident that, if in either case \( \lambda \) is of such a magnitude that \( e \) is less than \( \lambda \cos \delta \), the value of \( t \) will be impossible; in other words, no eclipse can take place.

If we suppose \( P + p - R + r = \lambda \cos \delta \), the two values of \( t \) will be equal, and the duration of the phase will only be for an instant, as in the case of the annulus, in which the moon's limb just touches the shadow without entering it.

In general, the portion of the diameter of the eclipsed part is \( P + p - R + r - \lambda \cos \delta \); and consequently the diameter of the part not eclipsed is equal to the diameter of the moon, or \( 2r \), minus this quantity, that is, equal to \( \lambda \cos \delta - P - p + R + r \). When this expression is equal to nothing the eclipse is just a total one; when negative, the upper boundary of the moon's limb will be under the upper boundary of the section of the shadow, and the total eclipse will continue for some time.

The instant at which the middle of the eclipse happens will evidently be that at which the two values of \( t \) are equal, or when the radical disappears, that is, when \( c = \lambda \cos \delta \). In this case \( t = \frac{1}{n} \lambda \sin^2 \delta \), and the instant is called that of the greatest phase. It is usual to express the quantity of the eclipse in digits, or twelfths of the lunar diameter; so that the eclipsed part is represented by \( \frac{12}{2r} (P + p - R + r - \lambda \cos \delta) \). Thus, taking the moon's apparent diameter at \( 33^\circ 18' \), and supposing the eclipsed part to be \( 24' 52'' \), this part expressed in digits will be \( \frac{24' 52''}{33^\circ 18'} \times 12 = 8'96 \) digits.

The obscurity of the penumbra renders observations of the commencement and termination of the lunar eclipses extremely uncertain. To obviate in some degree this inconvenience, care is taken to observe as accurately as possible the instants at which the shadow arrives at or passes different known spots on the moon's disk; so that the same eclipse offers in fact a great number of different observations, the mean of which may be regarded as more certain than any individual one. But after all the precautions that can be taken, the eclipses of the moon are far from affording results equally precise and certain as those of the sun. They were formerly of much greater importance than they are in the present state of astronomy; for the ancients had no other means of determining the geographical longitudes of places on the earth. In fact, as the eclipse is occasioned by the moon's being deprived of her light, the different phases of the eclipse happen at exactly the same physical instant of time to all observers to whom the moon is visible. The difference of the time reckoned by two observers at the instant of the phenomenon will therefore give the difference of the horary angles, or of their meridians; but supposing each to have made a mistake of 4 minutes of time in an opposite sense (and the ancients could scarcely guarantee a greater degree of accuracy), the resulting error in the difference of longitude would amount to \( 2^\circ \). The geographical tables of Ptolemy contain errors of still greater magnitude.

2.—Eclipses of the Sun.

The eclipses of the sun are caused by the interposition of the moon between the sun and the earth, and their general phenomena may be explained in the same manner as those of the moon. When the conical shadow which the moon projects behind her in space reaches the earth, those points of the earth's surface on which it falls are completely deprived of the light of the sun, and involved in total darkness. Those parts of the earth which are covered by the penumbra are only partially deprived of the sun's light, because the moon does not conceal the whole, but only a part of the solar disk. In order to appreciate the different circumstances of a solar eclipse, the procedure to be adopted is in many respects the same as that which has been explained in regard to the eclipses of the moon. The length of the moon's shadow, the first object of inquiry, is found exactly in the same manner as that of the earth; and it is only necessary to substitute in the formula already given the values of the apparent diameter and parallax of the sun which they would have at the surface of the moon. Now, these values are easily found, for the diameter of the sun as seen from the moon is equal to his diameter as seen from the earth, increased in the ratio of the distances of the moon and earth from the sun. In the same manner the parallax of the sun relatively to the moon is equal to his parallax relatively to the earth, augmented in the ratio of the distances, and diminished in the ratio of the diameters, of the moon and earth. Thus, let \( D \) represent the distance of the earth from the sun, \( d \) the moon's distance from the sun, \( m \) the moon's true semidiameter, and \( a \) the semidiameter of the earth; the sun's apparent semidiameter as seen from the moon will be \( R \cdot \frac{D}{d} \) (R being his apparent semidiameter as seen from the earth), and his horizontal parallax will be \( p \cdot \frac{D}{d} \cdot \frac{m}{a} \). The formula then which expresses the length of the terrestrial shadow \( CE \) (fig. 50), namely,

\[ \frac{a}{\sin(R - p)} = \frac{m}{\sin(R - p) \cdot \frac{D}{d}} \]

and expresses the distance between the centre of the moon and the apex of her shadow.

By means of this formula the following results, which refer to the extreme cases in which the length of the shadow is a maximum and minimum comparatively with the moon's distance from the earth, may be computed.

| Sun in apogee, Moon in perigee | Length of shadow | Distance | |-------------------------------|-----------------|----------| | Sun in perigee, Moon in apogee | 59°730 | 55°902 | | | 57°760 | 63°862 |

In the first case the shadow of the moon will reach beyond the centre of the earth; in the second it will not reach even to the surface. It follows, therefore, that even if the orbit of the moon coincided with the ecliptic, she would not produce a total obscurity every time she comes between the sun and the earth. At her greatest distances, where the shadow does not reach the earth, the effect of her interposition would be to conceal only a part of the sun's disk.

By introducing into the other formulae modifications similar to the above, we shall find the apparent diameter of the shadow and the solar ecliptic limits. The apparent semidiameter of the earth's shadow has been shown to be equal to \( P + p - R \); consequently the semidiameter of the lunar shadow at the distance of the earth, as seen by an observer placed on the moon, is equal to the parallax of the earth, plus the parallax of the sun relatively to the moon, minus the apparent semidiameter of the sun seen from the moon. The parallax of the earth means simply the apparent semidiameter of the moon seen from the earth; and if we neglect the parallax of the sun, which cannot influence the result to the extent of half a second, we shall have the following theorem: The semidiameter of the lunar shadow is equal to the excess of the apparent semidiameter of the moon above the apparent semidiameter of the sun. Hence, denoting the moon's apparent semidiameter by \( r \), the semidiameter of the lunar shadow will be expressed by \( r - R \cdot \frac{D}{d} \); or,

taking into account the sun's parallax, \( r + p \cdot \frac{m}{a} \cdot \frac{D}{d} \).

The ratio of the distances \( \frac{D}{d} \) in this and the preceding formula, may be expressed in terms of the parallaxes; for since \( p = \frac{a}{D} \) and \( P = \frac{a}{D - d} \), therefore \( \frac{D}{d} = \frac{P}{P - p} \). Since also \( \frac{m}{a} = \frac{r}{P} \), according to the nature of parallax, we have likewise \( r + p \cdot \frac{m}{a} \cdot \frac{D}{d} = \frac{rP}{P - p} \); therefore, by substituting these values, the expression for the semidiameter of the lunar shadow becomes

\[ (r - R) \cdot \frac{P}{P - p} \]

If to the apparent semidiameter of the shadow at the point where it is touched by the earth, we add the apparent semidiameter of the earth as seen from the moon, that is to say, the moon's horizontal parallax (\( P \)), the distance between the centres of the moon's shadow and of the earth will be

\[ P + (r - R) \cdot \frac{P}{P - p} \]

from which expression the solar ecliptic limits may be readily computed. The result of the computation is, that a solar eclipse may take place if the moon's distance from her node, at the time of her mean conjunction with the sun, does not exceed 19° 44′. If her distance from the node is less than 13° 33′, the sun will certainly be eclipsed in some part of the world.

The solar eclipses present a great variety of appearances, depending on the relative positions of the sun, the moon, and the spectator. If the apparent diameter of the moon happens to surpass that of the sun, the eclipse will be total; but if the moon's diameter be the smaller, the observer will see a luminous ring, formed by that part of the sun's disk which exceeds that of the moon, and the eclipse will in that case be annular. If the centre of the moon is not in the same straight line which joins the observer and the centre of the sun, the eclipse can only be partial, as the moon can only conceal a part of the sun's disk. When the moon merely touches without penetrating the solar disk, the phenomenon is called an annulus; and the eclipse is central if the observer is placed at the centre of the shadow, on the straight line joining the centres of the sun and moon.

When the change happens within 17 degrees of the node, and the moon is at her mean distance from the earth, the point of her shadow just touches the earth, and she eclipses the sun totally to that small spot wherein her shadow falls; but the darkness is not of a moment's continuance.

The moon's apparent diameter, when largest, exceeds the sun's, when least, only two minutes of a degree; so that in the greatest eclipse of the sun that can happen at any time and place, the total darkness continues no longer than whilst the moon passes over two minutes in her orbit, that is, about 3 minutes and 56 seconds of an hour.

The moon's shadow covers only a spot on the earth's surface about 180 English miles broad, when her diameter the moon's appears largest, and the sun's least; and the total darkness shadow can extend no farther than the limits of the dark shadow, penumbra. Yet the partial shadow or penumbra may then cover a circular space 4900 miles in diameter, within all which the sun is more or less eclipsed, as the places are less or more distant from the centre of the penumbra. When the moon changes exactly in the node, the penumbra is circular on the earth at the middle of the general eclipse, because at Theoretical that time it falls perpendicularly on the earth's surface; but at every other moment it falls obliquely, and will therefore be elliptical; and the more so as the time is longer before or after the middle of the general eclipse; and then much greater portions of the earth's surface are involved in the penumbra.

To make several of the above and other phenomena plainer, let S (fig. 52) be the sun, E the earth, M the moon, and AMP the moon's orbit. Draw the straight line Wc from the western side of the sun at W, touching the western side of the moon at c, and the earth at e; draw also the straight line Ve from the eastern side of the sun at V, touching the eastern side of the moon at d, and the earth at e; the dark space ced included between these lines is the moon's shadow, ending in a point at e, on the surface of the earth, because in this figure the moon is supposed to be at her mean distance from the earth. Had the moon been in her perigee, the shadow would have covered a space on the surface of the earth of about 180 miles in diameter, to all places within which space the eclipse would have been total. Had she been in her apogee, the shadow would have terminated in a point above e, and to an observer at e the sun would have been eclipsed annularly. Draw the straight lines WXdh and VXeg, touching the contrary sides of the sun and moon, and ending on the earth at a and b; draw also the straight line SXM from the centre of the sun's disk, through the moon's centre to the earth, and suppose the two former lines WXdh and VXeg to revolve on the line SXM as an axis, and the points a and b will describe the limits of the penumbra TT on the earth's surface, including the large space aboa, within which the sun appears more or less eclipsed, according as the places are more or less distant from the verge of the penumbra ab.

Draw the right line y 12 across the sun's disk, perpendicular to SXM the axis of the penumbra; then divide the line y 12 into twelve equal parts, as in the figure, for the twelve digits or equal parts of the sun's diameter; and at equal distances from the centre of the penumbra at e (on the earth's surface YY) to its edge ab, draw twelve concentric circles.

To an observer on the earth at b, the eastern limb of the moon at d seems to touch the western limb of the sun at W when the moon is at M, and the sun's eclipse begins at b; but at the same moment of absolute time, to an observer at a, the western edge of the moon at c leaves the eastern edge of the sun at V, and the eclipse ends. At the very same instant, to all those who live on the circle next to ab, the moon cuts off or darkens a twelfth part of the sun, and eclipses him one digit; to those who live on the next interior circle, the moon cuts off two twelfths parts of the sun; to those on the following circle, three parts; and so on to the centre at e, where the sun is centrally eclipsed. The different appearances of the eclipse, as seen by spectators in these different situations, with regard to the centre of the shadow, are represented in fig. 54, under which figure there is a scale of hours and minutes, to show at a mean state how long it is from the beginning to the end of a central eclipse of the sun on the parallel of London, and how many digits are eclipsed at any particular time from the beginning at A to the middle at B or the end at C. Thus, in 16 minutes from the beginning, the sun is two digits eclipsed; in an hour and five minutes, eight digits; and in an hour and 37 minutes, 12 digits.

Having determined the diameter of the moon's shadow at the earth, and the limits within which eclipses of the sun can take place, the next object is to determine the time of their commencement and termination. If the position of the observer were on the moon instead of the surface of the earth, our solar eclipses would appear to him as eclipses of the earth, and they would commence at the instant when the earth's disk began to penetrate the lunar shadow. A spectator so situated might therefore compute all the circumstances of a terrestrial eclipse in exactly the same manner in which we compute those of the moon; the same formulæ would suffice, with the slight modifications which have already been made. But relatively to an observer placed on the earth the case is altogether different. To him the eclipse does not begin when the moon's shadow comes into contact with the earth's disk, but when it begins to obscure his station. This, therefore, is one circumstance which renders the computation of solar more complicated and difficult than that of lunar eclipses; for it is necessary not only to determine generally what portion of the terrestrial disk is covered by the shadow, but also its position relatively to the equator and to a given meridian, and likewise the path described by the centre and contour of the umbra and penumbra on the surface of the earth. There is another circumstance which still further augments the difficulty of the computation, namely, the position of the observer, on account of which it is necessary to introduce the particular conditions which depend on the parallax. In the case of a lunar eclipse, it is only necessary that the moon's disk enter the earth's shadow in order that the eclipse may be visible to any part of the terrestrial hemisphere opposite to the moon; but a solar eclipse may happen in some parts of the earth without being visible at others—a circumstance which is occasioned entirely by parallax.

When we abstract from the effects of parallax, or suppose the observer to be placed at the centre of the earth, the problem of determining the different circumstances of a solar eclipse is exactly the same as that relative to an eclipse of the moon. But when the spectator is supposed to be placed at the surface, the latitudes and longitudes corresponding to his situation are different from the geocentric latitudes and longitudes; and in order to adapt the formula from which the time is given to these new circumstances, it is necessary, as a preliminary step, to compute the corrections which must be applied in consequence of the effects of parallax in longitude and latitude. The chief circumstance, therefore, in which the calculation of solar eclipses differs from that of lunar, consists in its being necessary to compute the effect of parallax in the direction of the angular distances which form the data of the problem, in order to apply the requisite correction to the values of those distances furnished by the tables. The development of these computations belongs to Practical Astronomy.

The sun's ecliptic limits exceeding $17^\circ 21'$, while those of the moon are only $11^\circ 26'$, it follows that the eclipses of the sun must be much more frequent than those of the moon. Yet the lunar eclipses being visible to every part of the terrestrial hemisphere opposite to the sun, and those of the sun visible only to the small portion of the hemisphere on which the moon's shadow falls, it happens that for any particular place on the earth the lunar eclipses are much more frequently visible.

In any year the number of eclipses of both luminaries cannot be less than two, nor more than seven: the most usual number is four, and it is very rare to have more than six; for the sun passes through both the nodes but once a year, unless he passes through one of them in the beginning of the year; and if he does, he will pass through the same node again a little before the year is finished; because, as these points move $19\frac{1}{2}$ degrees backwards every year, the sun will come to either of them $173$ days after the other; Theoretical and when either node is within 17 degrees of the sun at the time of new moon, the sun will be eclipsed. At the subsequent opposition, the moon will be eclipsed in the other node, and come round to the next conjunction again ere the former node be 17 degrees past the sun, and will therefore eclipse him a second time. When three eclipses take place about either node, the like number generally happens about the opposite, as the sun comes to it in 173 days afterwards; and six lunations contain but four days more. Thus, there may be two eclipses of the sun and one of the moon about each of her nodes. But when the moon changes in one of the nodes, she cannot be near enough the other node at the next full to be eclipsed; and in six lunar months afterwards she will change nearer the other node; in these cases there can be but two eclipses in a year, and they are both of the sun.

The eclipses of the sun being of great importance for the determination of geographical longitudes, it is of consequence to be in possession of some easy method of assigning the time at which they may be expected to occur, in order to avoid the necessity of long and tedious calculations. This may be done in a very simple manner, by considering that if a time can be assigned after which the sun and moon occupy exactly or nearly the same positions with regard to the nodes of the lunar orbit, their motions after that interval will recommence under the same circumstances, and the eclipses be reproduced in the same order. Now, it has been shown (chap. iii. sect. 2) that the nodes of the lunar orbit retrograde at the rate of 19°3286 in a year, consequently the time in which the sun returns to the moon's node is that which he requires to describe an arc of 360° — 19°3286, or, as is found by a simple proportion, 345619851 days. On comparing this with 295306887 days, the time of a lunation, it will be observed that these numbers are nearly in the ratio of 223 to 19, so that after 223 synodic revolutions the moon has returned 19 times to the same position relatively to the sun. But 223 synodic revolutions are completed in 18 mean solar years and 10 or 11 days, consequently after that interval all the eclipses, whether of the sun or the moon, return again in nearly the same order; which gives a very simple means of predicting them, since only 18 years of observation are required. This period was known to the astronomers of the remotest ages, and is generally supposed to be that which the Chaldeans distinguished by the name of Saros. (See Part I. of this article.)

But the ratio of 223 to 19 is not exact, and it is besides subject to variation from the secular inequalities of the sun and moon, by reason of which the rates of their mean motions are sensibly changed. Discordancies will hence arise; and in the course of time the order of eclipses observed in one of these periods will require correction. But the variations are slow and gradual; the lunisolar periods may, therefore, continue to be employed when approximative results only are required. When rigorous accuracy is wanted, recourse must be had to computation from the astronomical tables.

Occultations of planets and stars by the moon are phenomena of which the calculation depends on exactly the same principles, and is even made by the help of the same formulae, as the eclipses of the sun. Let E (fig. 55) denote the centre of the earth, M that of the moon, and S a star or planet concealed by the moon; the straight line SM will represent the axis of the lunar shadow, that is to say, the portion of space which the rays proceeding from the star cannot reach in consequence of their being intercepted by the moon; and the angle EMO (EO being perpendicular to SM) will be the apparent distance of the centre of the shadow from the centre of the earth.

The expression for this angle in a function of the time is obtained in exactly the same manner as in solar eclipses; and by equating it with the different values of the angle which correspond to the different phases of the occultation, and regarding the time as the unknown quantity, the epochs will be obtained at which the phases take place. There is only one circumstance which renders a slight modification of the formulae necessary. In computing the angle SEM, or the apparent distance of the centres of the star and moon seen from the centre of the earth, that distance may be regarded in solar eclipses as the hypotenuse of a right-angled triangle, the sides of which are respectively the latitude of the moon, and the difference between the longitudes of the moon and sun. But in occultations of the planets or stars, the star may be out of the ecliptic, and consequently its latitude not zero; so that the sides of the right-angled triangle, of which the apparent distance of the centres is the hypotenuse, are the difference of latitude of the star and moon, and the difference of longitude reduced to the moon's place, that is to say, multiplied by the cosine of the moon's latitude. It is evident that any of the planets may suffer an occultation by the moon; but with regard to the fixed stars, it is only those which are situated at a distance from the ecliptic not greater than the moon's extreme latitude, that can ever be hid by the interposition of the lunar disk.

The following is a list of all the solar eclipses that will be visible in this country during the present century. The time of the commencement of the eclipse, and the number of digits eclipsed, are computed for the middle of England. (See Baily's Tables, &c. p. 52.)

| Year | Day and Hour | Digits Eclipsed | |------|--------------|----------------| | 1858 | March 15th 11h. A.M. | 11° 30' | | 1860 | July 18th 2 P.M. | 9 12 | | 1861 | December 31st 2 P.M. | 5 0 | | 1863 | May 17th 6 P.M. | 3 46 | | 1865 | October 19th 4 P.M. | 7 36 | | 1866 | October 8th 5 P.M. | 5 3 | | 1867 | March 6th 8 A.M. | 8 42 | | 1868 | February 23rd 3 P.M. | contact | | 1870 | December 22nd 11 A.M. | 9 36 | | 1873 | May 26th 8 A.M. | 3 43 | | 1874 | October 10th 9 A.M. | 6 18 | | 1875 | September 29th noon | 0 33 | | 1879 | July 19th 7 A.M. | 4 0 | | 1880 | December 30th 2 P.M. | 4 24 | | 1882 | May 17th 6 A.M. | 2 18 | | 1887 | August 19th 3 A.M. | 11 58 | | 1890 | June 17th 8 A.M. | 4 39 | | 1891 | June 6th 5 P.M. | 3 0 | | 1895 | March 26th 9 A.M. | 1 0 | | 1896 | August 9th sunrise | contact | | 1899 | June 8th 5 A.M. | 3 13 | | 1900 | May 28th 3 P.M. | 8 0 | CHAP. IV.

OF THE PLANETS.

Sect. I.—General Phenomena of the Planetary Motions.

Having now described the motions and explained the phenomena of the sun and moon, our attention will be next occupied by the planets, those no less interesting bodies, whose remarkable peculiarities of apparent motion have attracted the curiosity, and formed a principal object of the labours of astronomers in all ages. The sun and moon move among the stars always in the same direction, and with velocities nearly uniform; but the planets, though their apparent motions are most frequently from west to east, sometimes appear to have no proper motions, or to remain stationary among the fixed stars; at other times they appear to move in a contrary direction, or to retrograde; and hence the earth cannot be the centre of the planetary orbits. The determination of that centre, and the order of distance in which the orbits of the different planets are placed around it, is comparatively an easy task since the telescope and micrometer have made us acquainted with the phases and variations of the apparent diameters of the planets; but the ancients, who were guided by the apparent motions alone, found greater difficulty in extricating the elements of their theories from observations, and in framing hypotheses by which the phenomena could be represented with tolerable accuracy. The different hypotheses which have been proposed for this purpose are called, with sufficient impropriety, Systems of the World.

In order to obtain a general notion of the path traced by a planet in the heavens, it is necessary to attend closely to the various phenomena which it exhibits. As an example, we may take Venus, the most brilliant and remarkable of all the planets.

A slight attention to the position of Venus, continued a few days, suffices to show that she changes her place with considerable rapidity among the fixed stars. If we observe her in the evening, we shall soon find that her greatest distance from the sun never exceeds an arc of about 47°; that after attaining this distance she begins again to approach the sun, the time which she continues above the horizon after sunset gradually diminishing, till at last she sets simultaneously with the sun, and is lost in the effulgence of his rays. From the circumstance of her appearing in the evening, and not remaining visible more than about three hours after the sun has descended below the horizon, Venus has obtained the name of Ἐστίας, Hesperus, or the evening star; sometimes also she is called the shepherd's star.

A few days after the evening star has disappeared, a brilliant star is observed in the morning preceding the sun in the east, which was not seen while Venus followed him in the west. At first it rises only a few minutes before the sun, but every succeeding morning somewhat earlier, till its distance from him is between 45 and 47 degrees. It then begins gradually to fall back; its elongation or distance from the sun becomes less and less, till it approaches so near to him as to be again lost in his rays. This has been called Φωσφόρος, Lucifer, or the morning star.

Hesperus and Lucifer were long regarded as different stars. It could not fail, however, to be remarked, that during the time the first continues to shine in the evening, the other is invisible in the morning, and as soon as the bright harbinger of day makes its appearance, the evening star ceases to be visible. It was, besides, observed that the one star disappears within a very short distance from the sun, and that the other, when first seen after its periodical disappearance, is equally near to him on the opposite side. The distance which each recedes from the sun was likewise remarked to be the same; and the times during which they are alternately visible were found to be equal. These phenomena led some bolder genius to affirm that Hesperus and Lucifer are the same star, which is alternately visible in the morning and evening, according as it precedes or falls behind the sun. Obvious as this conclusion is, it was not arrived at till after many ages of reflection and experience.

Since Venus never appears at a greater distance from the sun than about 47°, it is evident that the earth is neither the centre of her orbit, nor included within it; for in either case she would sometimes, like the moon, be seen in opposition to the sun. From the appearances which have as yet been described, it can only be inferred that Venus is a satellite of the sun, and that her orbit is carried along with him in his annual revolution in the ecliptic; but the phenomena which she presents when seen through the telescope afford the means of deducing more definite conclusions.

When Venus, after having been for some time visible in the evening, begins to approach the sun, she appears through the telescope as a fine luminous crescent, the horns of which are turned towards the east, and which becomes narrower as her distance from the sun diminishes. After she has passed the sun, and begins to appear in the morning, the horns of the crescent are turned towards the west, and its breadth gradually enlarges in proportion as the planet recedes from the sun, till she has gained her greatest elongation, when her disk becomes a semicircle. After this she begins to approach the sun with an accelerated motion, and her disk becomes gibbous, the illuminated or visible part being greater than a semicircle; and when she overtakes the sun, her disk has attained the dimensions of a full orb. Having passed the sun, the orb begins again to contract, and passes through the same gradation of changes on the eastern side of the sun, till the planet comes again into conjunction, when it vanishes entirely. All these phases, which nearly resemble those of the moon, are illustrated by fig. 56. When Venus is in the superior conjunction at A, or in the same straight line with the sun, she presents the full orb, because the hemisphere enlightened by the sun is turned directly towards the earth at E. Arrived at B, the illuminated hemisphere is not turned exactly towards the earth, and consequently one side of her orb must appear elliptical, the major axis being to the minor as radius to the cosine of the inclination of the planes of illumination and vision, as was shown in explaining the phases of the moon. (Chap. iii. sect. I.) At C, where the straight lines drawn from the planet to the earth and sun form a right angle, the minor axis of the ellipse vanishes, and we have the half-illuminated orb. At D only a small portion of the enlightened hemisphere is visible from E; and when the planet arrives at F, the inferior conjunction, her dark side is wholly turned towards the earth, and she is invisible.

Mercury exhibits phenomena exactly analogous to those of Venus. Like Venus, he oscillates on opposite sides of the sun; but his oscillations are much quicker, and performed in a much smaller arc. His greatest elongation or distance from the sun does not exceed 28° 20′; so that he never appears above the horizon longer than an hour and 50 minutes after sunset or before sunrise. He emits a very vivid white light; but, by reason of his proximity to the sun, he is seldom visible to the naked eye. His phases resemble those of Venus, and he is frequently seen as a dark spot passing over the sun’s disk. From these phenomena several important conclusions may be drawn. In the first place, Mercury and Venus are opaque bodies, which are only visible in consequence of the sun's rays reflected from their surfaces. In the second place, their orbits are described about the sun, and do not embrace the earth, because both planets pass between the sun and the earth, and their digression from the sun never exceeds a certain limit. For this reason they are called Inferior Planets. In the third place, since the digressions of Venus are much more considerable than those of Mercury, it is obvious that her orbit includes that of Mercury. This last fact is established by other phenomena. Their angular velocities may be compared by means of the times which they respectively employ in returning to their conjunctions; and in this way it is found that the angular velocity of Mercury is nearly three times greater than that of Venus—a circumstance of itself sufficiently indicative of his greater proximity to the sun. But the occultation of one of these planets by the other furnishes a decisive evidence of the disposition of their orbits. On the 17th of May 1787 Mercury was observed to be occulted by Venus near their inferior conjunction; whence it follows that Venus is nearer the earth, and consequently at a greater distance from the sun. This fact was known to the Egyptians, and the name of the Egyptian System given to that theory according to which Mercury and Venus were regarded as satellites of the sun; but as the ancient astronomers were unacquainted with one of its strongest proofs, namely, the transits over the sun, and could only form inferences from the digressions, it was not generally adopted by them.

The revolution of Venus and Mercury about the sun may also be inferred from the variations of their apparent diameters, although, on account of the unequal distances of the sun from the earth, these variations are subject to considerable irregularities. When Venus approaches nearest to the earth, her apparent diameter subtends an angle of about 61°, while it does not amount to 10° when she is at her greatest distance. The apparent diameter of Mercury varies from 12° to 5°, indicating corresponding variations of distance. These variations of his apparent diameter are not sensible to the naked eye, on account of the irradiation which surrounds the disks of the planets, and which renders it impossible to form any correct judgment respecting the magnitudes of small luminous objects seen from so great a distance.

The phenomena of the other planets differ in some respects from those of Mercury and Venus. Instead of remaining constantly within a certain distance from the sun, their angles of elongation assume all possible values, and they are frequently seen in opposition, and consequently more distant from him than the earth is. Their orbits, therefore, embrace the earth; and as they are never observed in their conjunctions to pass, like the inferior planets, over the sun's disk, even when the direction of their motion traverses the sun, it follows that the sun is also included within their orbits. This fact is rendered certain by the appearance of their disks at the time when they are in conjunction. If they were then placed between the sun and the earth, their disks would appear circular, like those of Mercury and Venus; but they uniformly present a full orb at the time of their conjunction, and consequently the same hemisphere is presented to the earth and the sun, at least if, as is certain, they derive their light from him. They must therefore be situated beyond the sun. From the circumstance that their orbits include both the earth and the sun, they are called Superior Planets. In order to determine whether the sun or the earth is the centre of their motions, it will be convenient to have recourse to observations of their apparent diameters, which, if their orbits are circular, and have the earth in their centre, will always be of the same magnitude. But the apparent diameter of Mars gradually increases from his conjunction to his opposition; and therefore, since the distance of the planet is directly proportional to the magnitude of its apparent diameter, Mars is nearer to the earth at his opposition than at his conjunction. The variations of the apparent diameter of this planet are very considerable, the limits being 18°-98 and 3°-6; so that Mars is five times farther from the earth at his greatest distance than at his least. His orbit, therefore, cannot be a circle described about the earth. We might indeed suppose it to be an ellipse, or other elongated curve; but the enormous eccentricity which it would be necessary to assign to it renders this supposition extremely improbable. The analogy of Venus and Mercury will rather lead us to infer that the sun is the centre of his motion; and of this we have a geometrical proof in the circumstance that the difference of his greatest and least distances is equal to the diameter of the earth's orbit. Jupiter, Saturn, and Uranus, present exactly similar phenomena. We therefore conclude in the same manner that they are superior planets, circulating about the sun.

The order of distance in which the superior planets are disposed about the sun and the earth may be inferred either from the rate of their motion when they are in opposition with the sun (it being natural to suppose that their velocities will diminish in proportion to their distances or the magnitude of the orbit which they describe), or it may be determined from the variation of their apparent diameters. Now, the apparent diameters at the conjunctions and oppositions are nearly in the following ratio: those of Mars as 1 to 5, of Jupiter as 10 to 15, of Saturn as 1 to 1:23; hence the diameter of the terrestrial orbit is to the diameters of the orbits of Mars, Jupiter, and Saturn, as the difference of the preceding numbers is to their sum, or as unity to the numbers 1½, 5, and 9½. In this manner we find that Jupiter is at a much greater distance from the earth than Mars, and Saturn than Jupiter.

It has already been remarked that the apparent motions of the planets are not always in the same direction. Through the most considerable part of their orbits they move from west to east, according to the order of the signs; and their motion is most rapid when they are at the greatest distance from the earth. It gradually relaxes until the planet has reached its greatest eastern digression if it is an inferior planet, or its eastern quadrature if a superior one; after which its proper motion is slower than the sun's motion in the ecliptic. In the course of a short time the planet seems stationary among the stars for some days, its right ascension undergoing scarcely any variation. In the course of a few days more, however, it begins again sensibly to change its place, and now moves in an opposite direction, or retrogrades. This retrograde motion continues to be accelerated as the planet approaches its inferior conjunction, or its opposition, at which point it attains its maximum. After this it begins to be retarded; the planet becomes a second time stationary, and then assumes its direct motion, to pass through another series of similar changes. The arc and time of retrogradation are different for each of the planets, being greatest in the case of those which are nearest the earth, and least for those which are at the greatest distance.

These phenomena, which are called the Stations and Retrogradations of the planets, were observed with great attention by the ancients and the astronomers of the middle ages, to whom their explanation gave much embarrassment on account of their being incompatible with... Theoretical Astronomy.

In fact, if the earth is supposed immovable, the path described by the planets is a curve so extremely complicated and irregular, that Aristotle, and even Riccioli, who lived in an age when the celestial motions were much better known, were reduced to the necessity of supposing a genius or angel to reside in each of the planets, directing its motions as the mind of man directs the motions of his body. Suppose the earth to be at rest in E (fig. 57), the orbit of a superior planet will resemble the curve abedf, &c. When the planet arrives at b, before it comes into opposition with the sun at A it becomes stationary. From b to c it retrogrades, and is again stationary at c. Its motion then becomes direct, and its distance from the earth continues to increase while it runs through the arc defg, and till it arrives at h, and is in conjunction with the sun at B. It is then at its greatest distance, and in passing through the arc klm continues to approach the earth till it arrives at m, in opposition with the sun at C. Here it exhibits the same phenomena as at the former opposition, becoming stationary, retrograding, &c.; and at every succeeding opposition describes a sort of node or loop, similar to lmn. Its path is thus made up of an infinity of nodes, and presents a sufficiently striking resemblance to an epicycloid. Fig. 58, which has been copied from Cassini into most of the elementary treatises of astronomy, represents the apparent motions of Saturn, Jupiter, and Mars, in respect of the earth.

Such are the general phenomena presented by the motions of the planets. We must next endeavour to determine the nature of the curve described by each of them, and the law according to which it is described, in order to arrive at the solution of the principal problem of astronomy, viz. to express the position of the heavenly bodies in terms of time reckoned from a given instant.

The first successful attempt to frame a system by means of which the motions of the planets might be numerically calculated, was made by Apollonius of Perga. Apollonius supposed that a planet, instead of describing a circle about the earth, moves in the circumference of a second circle, the centre of which is carried round the circumference of the first. Let us suppose two unequal circles, situated in the same plane, the greater of which ACB (fig. 59), which is called the deferent, carries on its circumference the centre of the smaller PRPS, which is called the epicycle; and let us also suppose that each of these circles turns uniformly about its centre, according to the order of the signs, that is, from west to east. The earth is situated at E the centre of the deferent, and the planet whose motions we consider is placed on the circumference of the epicycle. The phenomena resulting from this disposition will be different according to the ratio of the velocities of the two circles. In the first place, suppose the velocities to be equal, and that at the first instant the planet is placed at P; the extremity of the straight line which joins the centres of the deferent and epicycle: the planet is then at its apogee, and it is evident that its apparent velocity is the greatest possible, being the sum of the velocities in the two circles. It is evident also that the apparent velocity of the planet will diminish as it approaches the lower extremity P' of its epicycle, or its perigee, and that in this point it will vanish altogether, because the perigee is carried backwards in the direction RPS by the motion of the epicycle with a velocity equal to that by which it is carried forward by the motion of the deferent. From this point the apparent motion will receive a gradual augmentation of velocity, till the planet arrives again at its apogee, where it will recommence a course perfectly similar to the preceding. In this case, therefore, the apparent motion of the planet, though alternately accelerated and retarded, will always be direct.

Let us next suppose that the velocity of the epicycle is greater than that of the deferent, which is the hypothesis adopted by Apollonius and Ptolemy. From P to R the motion of the planet is direct. At the point R its path coincides with the tangent RE, and it would be stationary if the epicycle were immovable; but as the epicycle advances according to the order of the signs, the planet will continue to move directly till it arrives at the point π, where its retrograde motion in the epicycle, in the direction πP', is equal to the direct motion in the deferent. At the perigee P' the planet will retrograde, because the directions of the two motions are diametrically opposite, and the retrograde motion in the epicycle is greater than the direct motion in the deferent. At σ' it becomes a second time stationary, after which it resumes its direct motion, precisely according to the actual phenomena.

Lastly, if we suppose the velocity of the deferent to be Ptolemaic greater than that of the epicycle, there will be neither system nor retrogradation, and the planet will always advance in the same direction, but contrary to that of the motion of the deferent. This, in the Copernican system, would be the motion of the moon with reference to a spectator placed on the sun, and regarding the orbit of the earth about the sun as the deferent, and that of the moon about the earth as the epicycle; for then the velocity of the earth about the sun would be about thirty times greater than that of the moon about the earth.

This method of representing the geocentric motions of the planets was adopted and fully developed by Ptolemy, who assigned the ratios of the radii of the epicycles and deferents of each of the planets, and disposed the orbits in the manner most conformable to the apparent motions. It may be remarked that the absolute lengths of the radii are immaterial; it is only their relative lengths to which it is necessary to have regard. As observations were multiplied, and new inequalities in the motions of the planets detected, the system of simple epicycles was in most instances found insufficient to explain the phenomena. Double and triple systems were therefore introduced, in which the first epicycle was regarded as a second deferent carrying its own epicycle, the second epicycle as a third deferent, and so on, till every irregularity of motion was explained. It is easy to conceive all this mechanism, and even to reduce it to general formulae; but although it affords considerable facility for calculation, it is much too complicated to have place in nature. It is now well known that the orbits of the planets are not epicycloids, but ellipses; nevertheless, at a time when the circle and straight line alone were admitted in the solution of geometrical problems, it was extremely natural to inquire whether any construction could be found, without employing other curves, to represent the planetary motions; and the system of epicycles which resulted from this inquiry will remain a perpetual monument of the ingenuity of its authors, Apollonius and Ptolemy.

In arranging the planetary system, Ptolemy placed the earth at the centre of the universe, and nearest to it the moon, whose synodic revolution is the shortest of all, being performed in 29½ days. Next to the moon he placed Mercury, who returns to his conjunctions in 116 days. After Mercury followed Venus, whose periodic time is 584 days. Beyond Venus he placed the sun, then Mars, next Jupiter, and lastly Saturn, beyond which is the sphere of the fixed stars. (See fig. 60.) Plato and some other philosophers had placed the orbit of the sun immediately after that of the moon, and Mercury and Venus beyond the sun, on account that these planets were never seen on the solar disk. Ptolemy, however, remarked that this reason was inconclusive, because the planets might easily be supposed to pass between the sun and the earth, without appearing exactly on the sun's disk, in the same manner as the new moons do not always cause a solar eclipse. In the system of Ptolemy, it is, however, a matter of absolute indifference whether the orbits of these planets are placed above or below that of the sun, insomuch as the phenomena are exactly the same in both dispositions. This fact doubtless furnishes a strong objection to his system, and might have led him, we are apt to suppose, to adopt the system of the ancient Egyptians with regard to Mercury and Venus, and place the sun at the common centre of their orbits. But it must be recollected that Ptolemy had no means of measuring the diameters of the planets, or of forming any accurate notions of their distances; he was unacquainted with the phases of Venus, which demonstrate the revolution of that planet about the sun; and, in short, knew of no phenomenon which could not be reconciled with his system. His object was solely to represent the apparent motions by a geometrical construction, and by such means as geometry at that time could legitimately employ; and this object he fully accomplished. He never regarded his system in any other light than as a mere hypothesis, by means of which the celestial phenomena could be reduced to calculation.

But while thus much must be conceded in favour of the system of Ptolemy, it must be confessed that scarcely anything could well be imagined more complicated and cumbersome; more at variance with the simplicity which pervades the economy of nature, or, in a physical point of view, more absurd. Yet so prone is the human mind to cling to the ideas which have been first presented to it, and with so much difficulty are errors which have once obtained a firm footing eradicated, that, till the beginning of the 16th century, it was implicitly followed by astronomers of all countries. The glory of bursting the fetters of prejudice and authority, and of building the true system of the world on the ruins of a fabric which had been erected with so much address and labour, was reserved for Copernicus. This great man was led, by a profound meditation on the different hypotheses which had been imagined to account for the apparent motions of the heavenly bodies, to adopt the ideas of some of the ancients, and remove the earth from the centre of the world, ascribing to it a double motion of rotation about its own axis and of revolution about the sun. From some scattered hints contained in the writings of the ancient philosophers, Copernicus composed the system which retains his name; a system of the truth of which the complete development of the planetary theory has furnished the most convincing and satisfactory proofs. According to this system the sun is the common centre of the orbits of all the planets, which revolve around him in the following order:—Mercury, Venus, the Earth, Mars, Jupiter, and Saturn. Far beyond the orbit of Saturn he supposed the fixed stars to be placed, which formed the boundaries of the visible creation.

Although the great simplicity and beauty of the Copernican system soon recommended it to the adoption of the most philosophical of the astronomers of that period, yet for a long time it met with considerable opposition. The most eminent of its opponents was the celebrated Tycho Brahe, who could never bring himself to adopt the supposition of the motion of the earth. The principal objections which he urged against it were the immense distance at which it is necessary to suppose the fixed stars to be placed, in order to account for the smallness, or rather the entire absence, of the annual parallax; the improbability that a heavy mass like the earth should have so rapid a motion; and some passages of Scripture which seem to suppose the motion of the sun and the immobility of the earth. Tycho, therefore, proposed another system, in which he endeavoured to retain the most essential advantages of the Copernican theory, and at the same time preserve the earth's stability. In the system of Tycho the earth is supposed to be the centre of the solar and lunar orbits, and the sun to be the centre of all the planetary orbits, the orbits of the two inferior planets being smaller, and those of the superior larger, than the orbit of the sun; a distinction which is necessary for the explanation of the conjunctions and oppositions. This disposition of the orbits will be understood by referring to fig. 61. The system of Tycho is far less philosophical than that of Copernicus, and he has therefore exposed himself to the charge of having made a retrograde movement in science; but it must be confessed that it affords a satisfactory explanation of all the phenomena, and that the objections which can be urged against it are not of an astronomical, but of a physical or mechanical nature. In fact, if the planets are supposed to revolve about the sun, it is absolutely indifferent, so far as regards the phenomena, whether the annual motion is ascribed to the sun or the earth; but it is a physical absurdity to suppose that the sun, with its whole train of attendant planets, revolves about the earth, which, in comparison of them, is a mere atom.

Taking the truth of the Copernican theory for granted, let us consider what effect the motion of the earth has upon the apparent motions of the other planets. Were motion on the earth to stand still in any part of its orbit, the places of conjunction both in the superior and inferior semicircles, as also of the greatest elongation, and consequently the places of direct and retrograde motion, and of the stations of an inferior planet, would always be in the same part of the heavens; whereas, on account of the earth's motion, the places where these appearances happen are continually advancing forward in the ecliptic, according to the order of the signs. In fig. 62 let ABCD be the orbit of the earth: efgh that of Mercury; Θ the sun; GFKI an arc of the ecliptic extended to the fixed stars. When the earth is at A, the sun's geocentric place is at F; and Mercury, in order to be in conjunction, must be in the line AF; that is, in his orbit he must be at f or h. Suppose him to be at f in his inferior semicircle; if the earth stood still at A, his next conjunction would be when he is in his superior semicircle at h; the places of his greatest elongation also would be at e and g, and in the ecliptic at E and G. But supposing the earth to go on in its orbit from A to B; the sun's geocentric place is now at K; and Mercury, in order to be in conjunction, ought to be in the line BK at m. As by the motion of the earth the places of Mercury's conjunctions are thus continually carried round in the ecliptic according to the order of the signs, so the places of his greatest elongations must also be carried forward in the same direction. Thus, when the earth is at A, the places of his greatest elongation from the sun are in the ecliptic E and G; the motion of the earth from A to B advances them forward from G to L and from E to I. But the geocentric motion of Mercury will best be seen in fig. 63. Here we have part of the extended ecliptic marked γ, ξ, π, &c., in the centre LXXXIII. of which S represents the sun, and round him are the orbits of Mercury and the Earth. The orbit of Mercury is divided into 11 equal parts, such as he goes through once in eight days; and the divisions are marked by numeral figures, 1, 2, 3, &c. Part of the orbit of the earth is like- Theoretical wise divided into 22 equal arcs, each arc being as much as the earth goes through in eight days. The points of division are marked with the letters \(a, b, c, d, e, f, \ldots\) and show as many several stations from whence Mercury may be viewed from the earth. Suppose then the planet to be at \(a\) and the earth at \(b\); draw a line from \(a\) to \(b\), and it shows Mercury's geocentric place at \(A\). In eight days he will have advanced to \(2\), and the earth to \(b\); draw a line from \(2\) to \(b\), and it shows his geocentric place at \(B\). In other eight days he will have proceeded to \(3\), and the earth to \(c\); a line drawn from \(3\) to \(c\) will show his geocentric place at \(C\). In this manner, going through the figure, and drawing lines from the earth at \(d, e, f, g, \ldots\) through \(4, 5, 6, 7, \ldots\) we shall find his geocentric places successively at the points \(D, E, F, G, \ldots\) where we may observe, that from \(A\) to \(B\), and from \(B\) to \(C\), the motion is direct; from \(C\) to \(D\), and from \(D\) to \(E\), retrograde. In this figure 22 stations are marked in the earth's orbit from whence the planet may be viewed, corresponding to which there ought to be as many in the orbit of Mercury; but as the periodic time of that planet does not include so many intervals of eight days, his place is marked at the end of every eight days for two of his periodical revolutions; and to denote this, two numerical figures are placed at each division.

The geocentric motion of Venus may be explained in a similar manner; only, as the motion of Venus is much slower than that of Mercury, her conjunctions, oppositions, elongations, and stations, all return much less frequently than those of Mercury.

To explain the stationary appearances of the planets, it must be remembered, that the diameter of the earth's orbit, and even that of Saturn, are but mere points in comparison of the distance of the fixed stars; and therefore any two lines, though absolutely parallel, drawn at the distance of the diameter of Saturn's orbit from each other, would, if continued to the fixed stars, appear to us to terminate in the same point. Let, then, the two circles fig. 64 represent the orbits of Venus and of the Earth; let the lines \(AE, BF, CG, DH, \ldots\) be parallel to \(SP\); we may nevertheless affirm, that if continued to the distance of the fixed stars, they would all terminate in the same point with the line \(SP\). Suppose, then, Venus at \(E\) while the earth is at \(A\), the visual ray by which she is seen is the line \(AE\). Suppose again, that while Venus goes from \(E\) to \(F\), the earth goes from \(A\) to \(B\), the visual ray by which Venus is now seen is \(BF\) parallel to \(AE\); and therefore Venus will be all that time stationary, appearing in that point of the heaven where \(SP\), if extended, would terminate: this station is at her changing from direct to retrograde. Again, suppose, when the earth is at \(C\), Venus is at \(G\), and the visual line \(CG\); if, while the earth goes from \(C\) to \(D\), Venus goes from \(G\) to \(H\), so that she is seen in the line \(DH\) parallel to \(CG\), she will be all that time stationary, appearing in the point of the celestial sphere determined by the prolongation of \(SP\). This station is at her changing from retrograde to direct; and both are in her inferior semicircle.

As the superior planets move in larger orbits than the earth, they can only be in conjunction with the sun when they are on the side opposite to the earth; as, on the other hand, they are in opposition to him when the earth is between the sun and them. They are in quadrature when their geocentric places are 90° distant from that of the sun. In order to understand their apparent motions, we shall suppose them to stand still in some part of their orbit while the earth makes a complete revolution; in which case any superior planet would have the following appearances:

1. While the earth is in its most distant semicircle, the motion of the planet will be direct. 2. While the earth is in its nearest semicircle, the planet will be retrograde. 3. While the earth is near those places of its orbit where a line drawn from the planet would be a tangent, it would appear to be stationary.

Thus, in fig. 65, let \(abcd\) represent the orbit of the plate earth, \(S\) the Sun, \(EFG\) an arc of the orbit of Jupiter, \(ABC\) an arc of the ecliptic projected on the sphere of the fixed stars. Suppose Jupiter to continue at \(F\), while the earth goes round in its orbit according to the order of the letters \(abcd\). While the earth is in the semicircle most distant from Jupiter, going from \(a\) to \(b\) and from \(b\) to \(c\), his motion in the heavens would appear direct, or from \(A\) to \(B\) and from \(B\) to \(C\); but while the earth is in its nearest semicircle \(abc\), the motion of Jupiter would appear retrograde from \(C\) to \(B\) and from \(B\) to \(A\); for \(a, b, c, d\) may be considered as so many different stations from whence an inhabitant of the earth would view Jupiter at different seasons of the year; and a straight line drawn from each of these stations through \(F\), the place of Jupiter, continued to the ecliptic, would show his apparent place there to be successively at \(A, B, C, B, A\). While the earth is near the points of contact \(a\) and \(e\), Jupiter would appear stationary, because the visual ray drawn through both planets does not sensibly differ from the tangent \(Fe\) or \(Fc\). When the earth is at \(b\), a line drawn from \(b\) through \(S\) and \(F\) to the ecliptic shows Jupiter to be in conjunction with the sun at \(B\). When the earth is at \(d\), a line drawn from \(d\) through \(S\), continued to the ecliptic, would terminate in a point opposite to \(B\); which shows Jupiter then to be in opposition to the sun; and thus it appears that his motion is direct when he is in conjunction, but retrograde when he is in opposition with the sun.

The direct motion of a superior planet is more rapid the nearer it is to a conjunction, and slower as it approaches to a quadrature with the sun. Thus, let \(O\) Plate be the sun, the little circle round it the orbit of the plate earth, of which \(abcdefg\) is the most distant semicircle, \(OPQ\) an arc of the orbit of Jupiter, and \(ABCDEFG\) an arc of the ecliptic in the sphere of the fixed stars. If we suppose Jupiter to stand still at \(P\), by the earth's motion from \(a\) to \(g\) he would appear to move directly from \(A\) to \(G\), describing the unequal arcs \(AB, BC, CD, DE, EF, FG\), in equal times. When the earth is at \(d\), Jupiter is in conjunction with the sun at \(D\), and there his direct motion is swiftest. When the earth is in that part of its orbit where a line drawn from Jupiter would touch it, as in the points \(e\) or \(g\), Jupiter is nearly in quadrature with the sun; and the nearer the earth is to any of these points, the slower is the geocentric motion of Jupiter; for the arcs \(CD\) and \(DE\) are greater than \(BC\) or \(EF\), and the arcs \(BC\) and \(EF\) are greater than \(AB\) or \(FG\).

The retrograde motion of a superior planet is more rapid the nearer it is to an opposition, and slower as it approaches to a quadrature with the sun. Thus, let \(O\), fig. 67, be the sun, the little circle round it the orbit of the plate earth, whereof \(ghiklm\) is the nearest semicircle, \(OPQ\) an arc of the orbit of Jupiter, \(NKG\) an arc of the ecliptic. If we suppose Jupiter to stand still at \(P\), by the earth's motion from \(g\) to \(n\) he would appear to move from \(G\) to \(N\), describing the unequal arcs \(GH, HI, IK, KL, LM, MN\), in equal times. When the earth is at \(k\), Jupiter appears at \(K\) in opposition with the sun, and there his retrograde motion is swiftest. When the earth is either at \(g\) or \(n\) (the points of contact of the tangents \(Py\) and \(Pn\)), Jupiter is nearly in quadrature with the sun; and the nearer he is to either of these points the slower is his retrogradation; for the arcs \(IK\) and \(KL\) are greater than \(HI\) or \(LM\), and the arcs \(HI\) and \(LM\) are greater than \(GH\). Since the direct motion is swiftest when the earth is at \(d\) (fig. 66), and continues diminishing till it changes to retrograde, it must be insensible near the time of change; and, in like manner, the retrograde motion being swiftest when the earth is in \(k\) (fig. 67), and diminishing gradually till it changes to direct, must also at the time of that change be insensible; for any motion gradually decreasing till it changes into a contrary one gradually increasing, must at the time of the change be altogether insensible.

The same changes in the apparent motions of this planet will also take place if we suppose him to advance slowly in his orbit; only they will happen successively when the earth is in different parts of its orbit, and consequently at different times of the year. Thus (fig. 65), let us suppose that while the earth goes round its orbit, Jupiter goes from \(F\) to \(G\); the points of the earth's orbit from which Jupiter will now appear to be stationary will be \(x\) and \(y\); and consequently his stations must be at a time of the year different from the former. Moreover, the conjunction of Jupiter with the sun will now be when the earth is at \(f\); and his opposition when it is at \(e\); for which reason these also will happen at times of the year different from those of the preceding opposition and conjunction.

The motion of Saturn is so slow, that it occasions but little alteration either in the times or places of his conjunction or opposition; and the same will take place in a more eminent degree in Uranus; but the motion of Mars is so much swifter even than that of Jupiter, that both the times and places of his conjunctions and oppositions are thereby very much altered.

Fig. 68 exemplifies the geocentric motion of Jupiter in a very intelligible manner. In this figure \(O\) represents the sun; the circle 1, 2, 3, 4, the orbit of the earth, divided into twelve equal arcs for the twelve months of the year; \(PQ\) an arc of the orbit of Jupiter, equal to that which he describes in a year, and divided in like manner into twelve equal parts, each representing the arc he describes in a month. Now suppose the earth to be at 1 when Jupiter is at \(a\), a line drawn through 1 and \(a\) shows Jupiter's place in the celestial ecliptic to be at \(A\). In a month's time the earth will have moved from 1 to 2, Jupiter from \(a\) to \(b\); and a line drawn from 2 to \(b\) will show his geocentric place to be in \(B\). In another month the earth will be in 3, and Jupiter at \(c\); and in like manner his place may be found for the other months at \(D, E, F, \&c.\) It is likewise easy to observe that his geocentric motion is direct in the arcs \(AB, BC, CD, DE\); retrograde in \(EF, FG, GH, HI\); and direct again in \(IK, KL, LM, MN\). The inequality of his geocentric motion is likewise apparent from the figure.

Supposing the orbits of the planets to be circular, the points of station, and the extent and duration of the retrogradations, may be geometrically determined with great facility; but if we attempt to take into consideration all the inequalities of the orbits, the problem becomes one of extreme complication and difficulty. This, however, is the less to be regretted, on account that it is more a problem of curiosity than of any real importance, and the mean values which result from the supposition of circular orbits are more than sufficiently accurate for all the uses which are now made of these phenomena.

Let \(S\), fig. 69, be the sun, \(E\) and \(V\) two planets (which we may suppose to be the Earth and Venus) revolving in their respective orbits in the times \(T\) and \(t\); the problem is, to determine those points at which one of them, as seen from the other, will become stationary, and the extent of the arc through which it will appear to retrograde. Suppose the spectator to be placed at \(E\), and \(V\) to be the point at which Venus is stationary. Also let the semidiameters of their orbits \(SE\) and \(SV\) be \(R\) and \(r\) respectively, the angle of digression \(SEV = \phi\), and the angle \(SVE = \psi\). During a short time, at the station, the lines \(EV\) and \(er\) (\(Ee\) and \(Ve\) being very small arcs) may be considered as parallel; therefore \(S = Sve = Se + Es = \phi + \delta\phi + Es\), whence \(\delta\phi = -Es\). In like manner \(S = Sve = Se - VSe = \psi + \delta\psi - VSe\); hence \(\delta\psi = VSe\). But as the orbits are described uniformly, \(T : t : : VSv : ESv\), and consequently \(T : t : : \delta\phi : \delta\psi\). Now, the triangle \(SEV\) gives the equation \(\sin \psi = \frac{R}{r} \sin \phi\), from which, by differentiating, we get

\[ \delta\psi \cos \psi = \frac{R}{r} \delta\phi \cos \phi, \]

and consequently \(\delta\psi : \delta\phi : : R \cos \phi : r \cos \psi\). Comparing this with the above proportion, we have \(T : t : : R \cos \phi : r \cos \psi\), whence \(r \cos \psi = \frac{T}{t} R \cos \phi\).

Taking the squares, and adding \(r^2 \sin^2 \psi = T^2 \sin^2 \phi\),

\[ r^2 T^2 = R^2 (\ell^2 \cos^2 \phi + T^2 \sin^2 \phi), \]

whence we derive

\[ \sin \phi = \frac{1}{R} \left( \frac{T^2}{T^2 - \ell^2} \right)^{\frac{1}{2}}; \]

or, if we suppose \(\frac{r}{R} = r'\) and \(\frac{t}{T} = \ell'\),

\[ \sin \phi = \frac{(r' + \ell')(r' - \ell')}{(1 + \ell')(1 - \ell')} \frac{1}{2}. \]

From this formula, all the circumstances connected with the stations and retrogradations are easily computed, when the diameter of the orbit and the periodic time of the planet are known. Thus, relatively to Venus, the quantity \(r'\), that is, the ratio of the diameter of her orbit to that of the earth, is, as will be shown in next section, equal to 0.723, and

\[ \frac{224}{365} = 0.6152, \quad \text{whence} \quad \sin \phi = \sqrt{\frac{1}{1.615 \times 385}} = 0.48208; \]

and \(\phi = 28^\circ 49'\). The angle \(\psi\) is obtained from the equation \(\sin \psi = \frac{\sin \phi}{r'}\), which gives \(\psi = 138^\circ 11'\).

The angle \(VSE\) at the sun, or the heliocentric distance of Venus and the Earth, which we shall designate by \(z\), is \(180^\circ - (\phi + \psi) = 13^\circ\). From this angle we likewise obtain the time from the station to the conjunction; for since the mean daily motion of Venus is \(1^\circ 36' 7''8\), and that of the earth \(59'5''3\), the relative daily motion is \(36'59''5\), therefore the time from conjunction will be \(36'59''5\), or about 21 days, and consequently the retrograde motion of Venus will continue about 42 days.

If in the equation \(\sin \phi = \frac{1}{R} \left( \frac{T^2}{T^2 - \ell^2} \right)\) we suppose \(rT = Rt\), the angle \(\phi\) vanishes; consequently there will be no regression, and the station will only be momentary at the inferior conjunction. In this case \(r : R : : t : T\), or the distances of the planets are proportional to their periodic times. If \(Rt\) were greater than \(rT\), \(\phi\) would become imaginary, and the motion of the planet would always be direct. The stations and retrogradations are therefore consequences of that law of the system according to which the distances are to one another in a greater ratio than the periodic times.

The following table, given by Delambre (Astronomie, tome iii. p. 9), exhibits the elements of the stations and retrogradations of the planets, calculated from formulae equivalent to the above. Sect. II.—Of the Orbits of the Planets.

In determining the elements of the orbits of the sun and moon, the labour of the astronomer is facilitated by the circumstance that the earth, at which his observations are made, either in, or may be regarded as, the centre of motion. But in the case of the planets, the sun, and not the earth, is situated in the centre of their orbits, and consequently the elements of those orbits must be determined from the measurement of certain linear and angular distances from the sun; it is therefore necessary, as a preliminary step, to convert the geocentric into heliocentric observations, that is, to deduce the true place of a planet as seen from the sun, from its apparent place as seen from the earth. On this account the determination of the planetary orbits is attended with somewhat greater difficulties than those which present themselves in the cases of the sun and moon.

1. Nodes and Inclinations of the Planetary Orbits.—Conformably with the plan which has been followed in the two preceding chapters, we will first consider those elements which determine the position of the plane of the orbit of a planet; and then those which regard the orbit itself, that is to say, the elements and position of the ellipse which the planet describes. When these are known, and the astronomer is also acquainted with the mean motion, or the time of the periodic revolution of the planet, and the instant at which it occupied any given point of its orbit, he is in a condition to assign the epoch at which it will again occupy the same or any other given point; and this comprehends the complete solution of the problem which he proposes to himself. It is proper, however, to remark here, that although this method of considering the elements separately is extremely convenient, perhaps indispensable, for the purpose of illustration, it is neither that which the practical astronomer follows, nor that by which the various truths which compose the actual system of theoretical astronomy were discovered. No discovery in astronomy has perhaps ever been made by a direct process. The elements first obtained from the observations are imperfect and inexact, and it is only by successive and frequently laborious approximations that they are advanced to accuracy. Hypotheses are first framed to account for or classify the phenomena; the results of these arbitrary suppositions are computed and compared with new observations; and the differences which are found to exist between the computed and observed quantities serve to verify or correct the assumption, or suggest other approximations still nearer the truth. In astronomy, as Woodhouse remarks, scarcely one element is presented simple and unmixed with others. Its value, when first disengaged, must partake of the uncertainty to which the other elements are subject, and can be supposed to be settled to a tolerable degree of exactness only after multiplied observations and many revisions. There are no simple theorems for determining at once the parallax of the sun, or the heliocentric latitude of a planet.

To give a general idea of the method of reducing geocentric to heliocentric longitudes and latitudes, let S (fig. 70) be the sun, E the earth, P the place of a planet in its orbit NP, NA the ecliptic, and N the ascending node, or that point of the ecliptic through which the planet passes when it comes to the north of that plane. Let Sγ, Eγ, be drawn from S and E to the first point of Aries; these lines being parallel, and representing the direction of one of the axes of co-ordinates to which the places of the planet are referred. Let also PL be drawn perpendicular to the ecliptic, and the other lines as in the figure. It is easy to see that as NPC, NLA are regarded as the intersections of the planes of the orbits of the earth and the planet with the celestial sphere, the centre of which is occupied by the sun, they are portions of great circles; and consequently every question relative to the position of the planet may be resolved by the formulas of spherical trigonometry. In fact, PL being a circle of latitude, the arc TNL is the Heliocentric Longitude of the planet, TN the longitude of the ascending node, and the spherical angle PNL the inclination of the orbit. In order that the longitude of the planet in its orbit may not differ greatly from its longitude referred to the ecliptic, the former is not reckoned from the node N, but from another point equally distant from the node with the vernal point γ; so that the longitude of the planet is obtained by adding NP to the longitude of the node. The magnitude of the planet thus reckoned is usually called the longitude in the orbit. The latitude also depends immediately on the distance NP of the planet from the ascending node; hence NP is called the Argument of the Latitude; and the difference between the longitudes in the orbit and ecliptic, that is, NP—NL, is called the Reduction to the Ecliptic. The angle PEL, or the planet's apparent distance from the ecliptic to an observer at E, is the Geocentric Latitude; PSL, its distance from the ecliptic when viewed from the sun, is its Heliocentric Latitude. It is evident from the mere inspection of the diagram, that φEL and φSL are the geocentric and heliocentric longitudes respectively. The angles of the triangle ESL, from which the reductions are computed, being of very frequent use in practical astronomy, have received certain technical denominations with which it is convenient to be acquainted. The angle LES, which is the difference between φEL and φES, the geocentric longitudes of the planet and the sun, is called the angle of Elongation; ESL, the angle at the sun, is called the angle of Commutation; and SLE, which is the difference between the heliocentric and geocentric longitudes, is called the Annual Parallax, or the Prosthapheresis of the Orbit. In the case of an inferior planet, the annual parallax SLE may have any value between zero and 360°; but if the planet be a superior one, the greatest value of the annual parallax has place when the straight line drawn from the planet to the In order to determine the position of the orbit of a planet relatively to the ecliptic, it is necessary to discover in the first place the position of the nodes. The longitude of the node is the heliocentric longitude of the planet at the instant when it is in the ecliptic, and its geocentric and heliocentric latitudes are consequently both equal to zero. If then, at the same instant, the planet should happen to be in opposition with the sun, or (if an inferior one) in its inferior conjunction, its geocentric longitude, which is equal to the longitude of the sun + 180°, would be the heliocentric longitude of the node. But it can happen very rarely that the latitude of a planet is zero at the time of its opposition or conjunction, especially with regard to those of which the periodic times are considerable. Mercury and Venus, indeed, by passing oftener through their nodes within a given time, afford more chances of the occurrence of the phenomenon; and it does occur occasionally under very favourable circumstances when those planets pass over the disk of the sun, for at such times the latitude must be very small, that is, the planet must be very near its node. But this simple method of determining the heliocentric longitude of the node, although it may frequently be had recourse to in the case of Mercury, can seldom be practised with regard to Venus, and is of no use whatever in the cases of the other planets, because, on account of the greater length of their times of revolution, the oppositions and passages through the node take place simultaneously only after very long intervals.

Leaving, therefore, out of consideration the circumstance of opposition, let us inquire what other means are in our possession for determining the longitude of the node. It would be by an extraordinary chance that a planet should happen to be observed at the precise instant of its passage through its node; but if observed when very near it, the diurnal variations of latitude afford data for determining that instant by interpolation. Suppose, then, the planet to be in its node at N: the angle φEN is obtained from observation, being the geocentric longitude of the node; and γES, the longitude of the sun, is known from the solar tables. Hence, in the triangle ENS the angle SEN = φEN − γES is known, as is also the side SE, which is the sun's distance from the earth. If, therefore, the angle at the planet, that is ENS, can be found, every part of the triangle will be determined, and we shall then have the heliocentric longitude γSN (equal to 180° − ESN + γES), and also the radius vector SN, or the distance of the planet from the sun. Now, the sides of a triangle being proportional to the sines of their opposite angles,

\[ \frac{SN}{SE} = \frac{\sin SEN}{\sin ENS} = \frac{\sin(\phi EN - \gamma ES)}{\sin(\phi SN - \gamma EN)}; \]

but from this equation nothing can be deduced, inasmuch as it contains two unknown quantities, SN and φSN. Suppose, however, that the planet is observed a second time in its passage through the same node, and that the earth is then in another part of its orbit at e: this second observation will furnish another equation, viz.,

\[ \frac{SN}{Se} = \frac{\sin Sen}{\sin eNS} = \frac{\sin(\phi eN - \gamma eS)}{\sin(\phi eSN - \gamma eEN)}, \]

exactly similar to the former, and in which Se is equal to SE. If, then, we suppose the place of the node has not sensibly changed in the interval between the two observations, this last equation will furnish a second relation between the two unknown quantities SN and φSN, by combining which with the former, the values of both those quantities may be determined. By repeating the same observation, it will be seen whether the place of the node is fixed, or is subject to any other variation than that which arises from the precession of the equinoxes. Its variation, if it does vary, can only be determined by observations made at distant epochs.

When the longitude of the node has been determined in this manner, it will be easy to deduce the inclination of the orbit. For this purpose the planet may be observed at the time when the sun's longitude is equal to that of the node, and the earth is consequently situated in the line of the nodes. By this observation the elongation Plate of the planet, which is then PEN (fig. 71), is given, as LXXXIII, also its geocentric latitude PEL. Now, by reason of the right-angled triangles PEL and PSL, we have PL = EL tan. PEL = SL tan. PSL, consequently tan. PSL = EL tan. PEL; but EL : SL :: sin. LSN : sin. LES, therefore tan. PSL = \[ \frac{\sin LSN}{\sin LES} \] tan. PEL. Now, by Napier's rules for circular parts, the right-angled spherical triangle PNL gives also sin. NL = cot. PNL tan. PL; whence tan. PL = \[ \frac{\sin NL}{\sin PNL} \]; or, since tan. PL = tan. PSL and sin. NL = sin. LSN, therefore tan. PSL = \[ \frac{\sin LSN}{\sin PNL} \]. By equating these two values of tan. PSL, we have

\[ \frac{\sin LSN}{\sin LES} \tan. PEL = \frac{\sin LSN}{\sin LES} \tan. PEL; \]

therefore, ultimately,

\[ \tan. PNL = \frac{\tan. \lambda}{\sin \varphi}; \]

that is, the inclination of the orbit is given in terms of the geocentric latitude and the longitude of the node.

The instant at which the sun is in the node of a planet's orbit cannot be easily seized; but the sun's mean motion being known, if his longitude is observed when it is nearly equal to that of the node, the time at which he passes through it may be determined with all the necessary accuracy by a simple proportion. It requires also to be remarked that the above method of determining the inclination takes it for granted that the position of the node is exactly known; but even should some uncertainty remain regarding this element, the resulting inclination would scarcely be affected in any sensible degree by a slight error, especially if at the time of the observation the planet is not very distant from its quadratures. In fact, if we make PNL = I, PEL = λ, and LES = ϕ, the above equation becomes tan. I = \[ \frac{\tan. \lambda}{\sin \varphi} \], which being differentiated with respect to I and ϕ, gives d tan. I = \[ \frac{d \varphi \cos \varphi \tan \lambda}{\sin^2 \varphi} \].

But d tan. I = \[ \frac{dI}{\cos^2 I} \], and from the equation itself we derive \[ \cos^2 I = \frac{\sin^2 \varphi}{\sin^2 \varphi + \tan^2 \lambda} \]; therefore, by substituting

\[ dI = \frac{d \varphi \cos \varphi \tan \lambda}{\sin^2 \varphi + \tan^2 \lambda}. \]

From this it is evident that the error of inclination resulting from an erroneous position of the node will be smaller in proportion as ϕ is greater, and will disappear altogether when ϕ (that is, LES or the geocentric distance of the planet from its node) is 90°, that is to say, at the quadratures.

The process now explained for determining the position of the nodes gives at the same time the length of Theoretical Astronomy.

The radius vector SN; and, in consequence of certain relations which will be pointed out in the present section, all the other elements of the orbit may be deduced from simple observations of the passages of a planet through its nodes. But when approximate values of these elements are known, the heliocentric longitudes and latitudes may be found from geocentric observations of various kinds. If, therefore, the planet is observed several times before and after its passage through the node, the diurnal variations of longitude and latitude (which we at present suppose to be known very nearly) will give by interpolation the instant at which the latitude was nothing, and the heliocentric longitude, computed for that instant, will give the place of the node. By means of this more accurate determination, the other elements may be corrected; and on repeating the same process with the corrected values, the longitude of the node will be obtained with still greater precision. These indirect methods of correcting the elements by one another were first employed by Kepler, and are of extensive application in every department of astronomy. They lead to more exact values of the elements of the planetary orbits than could be obtained from any direct method whatever.

As the passage of a planet through either of its nodes takes place only once during each periodic revolution, a remote planet, like Uranus, which moves very slowly, can very seldom be observed in those positions. Astronomers have therefore sedulously employed themselves in devising other methods by which all the elements of a planet's orbit may be discovered when a very small portion of it has been made known by observation. But these methods are the results of profound mathematical theory, based on the principle of Universal Gravitation, and could not therefore be properly explained in this place. Their application belongs to Practical Astronomy.

When the longitudes of the nodes of the planetary orbits are deduced by the methods explained above from ancient and modern observations, it is found that the nodes are not altogether fixed, but that, in respect of the fixed stars, they retrograde, or move from east to west; but their motion is so slow, that, although it is in reality, like every motion of the system, alternately accelerated and retarded, it may be regarded as uniform during a very long period of time. The inclinations of the orbits also undergo small variations, scarcely sensible to observation. The retrograde motion of the nodes of the planets, exactly analogous to that of the nodes of the moon, is a necessary consequence of the mutual gravitation of all the planetary bodies. The same is the case with the variations of inclination, of which the period and limits can only be calculated from the same theory to which we owe the knowledge of their existence. It is only indeed by the aid of the theories of Physical Astronomy that the greater part of these small secular variations can be disengaged from the periodic inequalities.

The following table exhibits the inclinations of the orbits of the planets, the positions of their nodes, and the variations of those elements. For the old planets the epoch is the commencement of the present century, that is to say, midnight preceding the 1st of January 1801. For Vesta, Juno, Ceres, and Pallas, the epoch is 1820. The annual increment of the longitude of the node is referred of course to the equinoctial point of Aries: to find its motion in respect of the fixed stars, it is only necessary to subtract the precession of the equinoxes, or $50^\circ 1'$, from the numbers given in the table.

| Names | Inclination | Secular Variation of Inclination | Longitude of Ascending Node | Annual Variation of Longitude of Node | |-----------|-------------|---------------------------------|-----------------------------|--------------------------------------| | Mercury | 7° 0' 9" | + 15° 18' 28" | 45° 57' 30" | + 42° 3' | | Venus | 3 23 28.5 | - 4° 55' 22" | 74° 54' 12" | + 32° 5' | | Mars | 1 51 6.2 | - 0° 15' 23" | 48° 0' 3" | + 26° 8' | | Vesta | 7 8 9.0 | | 103° 13' 18" | + 15° 6' | | Juno | 13 4 9.7 | | 171° 7' 40" | | | Ceres | 10 37 26.2 | | 80° 41' 24" | + 1° 5' | | Pallas | 34 34 55.0 | | 172° 39' 26" | | | Jupiter | 18 15 51.3 | - 22° 60' 57" | 98° 26' 18" | + 34° 3' | | Saturn | 2 29 35.7 | - 15° 51' 31" | 111° 56' 37" | + 30° 7' | | Uranus | 0 46 28.4 | + 3° 13' 31" | 72° 59' 35" | + 14° 2' |

2. Of the Figures of the Orbits.—After the position of the plane of a planet's orbit in space has been determined, it remains to trace its path on that plane, or to determine the figure and elements of the orbit itself. This, it is evident, may be accomplished, if we are in possession of the means of assigning, at any instant, the planet's distance from a fixed point in the plane of its orbit, and likewise the angle formed by the radius vector with a straight line given by position on the same plane. With these data a series of points may be laid down, representing the positions successively occupied by the planet; and the curve formed by joining them together will represent the orbit. Now, when the place of the nodes and the inclination are known, the radius vector and elongation may be computed by the rules of spherical trigonometry from a single observation of the planet's geocentric latitude and longitude at any epoch whatever; but by making the observations when the planet occupies certain situations of its orbit with respect to the earth, the difficulties attending the computation may be in a great measure eluded. If a planet, for example, is observed at the time of its opposition or conjunction, there will be no occasion for a previous deduction of the heliocentric from the geocentric longitude; because, in either of those cases, the radius vector of the planet, and the straight line EP (fig. 72) drawn from it to the observer, are projected on Plate LXXXIV. and consequently the planet, whether seen from the earth or the sun, is referred to the same point of the ecliptic, excepting indeed the inferior planets, which in their inferior conjunctions are referred to points diametrically opposite. Let P, therefore, be the place of a planet in opposition; then, in the spherical triangle PNL, right-angled at L, the side NL, being the longitude of the planet minus that of the node, is given by observation (the longitude of the node being supposed to be previously determined); the inclination PNL is also known; therefore, by Napier's rules,

$$\tan PN = \frac{\tan NL}{\cos PNL};$$

that is, the elongation of the planet is given in terms of its longitude and the inclination of its orbit. The same Theoretical triangle PNL also affords data for determining the radius vector; for tan. PL = tan. PNL sin. NL; but PL is the measure of the angle PSL, and consequently PSL, the heliocentric latitude, also becomes known. Now, in the triangle PES the angle PES is given by observation, therefore EPS (PSL — PES) is also given; but PS : SE :: sin. PES : sin. EPS, whence PS = SE sin. EPS / sin. EPS

or the radius vector is found in terms of the sun's distance from the earth and given angles.

In consequence of the incommensurable relation subsisting between the times of the periodic revolutions of the planets and that of the earth, the successive oppositions and conjunctions never take place at the same points of the orbits; therefore, by making a number of similar observations, as many different angles of elongation and radii vectors will be found, which, unless the orbits are circular, will all have different values. By computing, therefore, the values of these radii vectors in terms of the greater axis of the earth's orbit, and laying them down in their true positions round the sun, we shall obtain an approximation to the curve described by the planet on the plane of its orbit. A few observations of this sort will suffice to show that the orbit is eccentric; and as we have already seen that the solar orbit is an ellipse, analogy will immediately suggest the probability that that of the planet is also an ellipse, having the sun in one of its foci. It is easy to determine whether this be the case or otherwise. Three points given by position on a plane completely determine an ellipse of which a focus is known: having therefore computed an ellipse which satisfies three observations of a planet, its periphery, if the hypothesis of elliptic motion is correct, will comprehend all the places of the same planet computed from any other observations. Now, it is found that this is what takes place with regard to every one of the planets; hence results the first of the three laws discovered by Kepler which form the basis of the whole system of modern astronomy, namely, that the orbits of the planets are ellipses, of which the sun occupies one of the foci.

By a comparison of the sectors formed by two contiguous radii vectors and the arcs included between them, with the time consumed by the planet in describing those arcs, Kepler was led to the discovery of the second great law of the planetary motions, namely, that the areas described by the radius vector of a planet are proportional to the times employed in describing them. The data from which this important conclusion was deduced were not rigorously exact; but the fact itself has been confirmed by an infinity of observations since the time of Kepler; and Newton, by the application of geometry to dynamics, demonstrated that it is necessarily true of all motions regulated by a central force, whatever the nature of that force may be.

Kepler having, by means of the most laborious computations, established the existence of these two laws with regard to each of the planets separately, next undertook to discover whether any analogy could be found regulating their mean distances from the sun. As there were only six planets known in his time, he commenced his investigation by comparing the intervals between their respective orbits with the five regular geometrical solids. Having failed in this speculation, and in various others suggested by a mind equally fertile and persevering, he at length, by a happy inspiration of genius, thought of comparing the mean distances of the planets with their respective periods of revolution, and soon perceived that the numbers which represent the periodic times of the planets, beginning with Mercury and ending with Saturn, increase in a much greater ratio than those representing their mean distances from the sun. Jupiter, for example, is five times more distant from the sun than the earth is; but his period is twelve times greater than that of the earth. Kepler then tried the various powers of the distances and periods. Unfortunately an error of computation for some time concealed the discovery from him; but having resumed the subject, he at length found, on the 15th of May 1618 (and few days, as Mr Playfair has remarked, are more memorable in the annals of science), with all the delight which a great and important discovery gives rise to, that the squares of the numbers which express the times of revolution are to one another as the cubes of Kepler of those which express the mean distances of the planets from the sun.

These three general laws, which are necessary consequences of the law of force varying directly as the masses and inversely as the squares of the distances, greatly facilitate the investigation of the orbit of a newly discovered planet, and may even be regarded as more accurate than the results obtained from any moderate number of observations, however exact. Thus, instead of having recourse to observation for the determination of the radius vector, which cannot be found by direct observation but with considerable difficulty, it is preferable to deduce it by means of the third law from the planet's periodic time, which can always be determined with great precision, and with much greater facility. They also afford a strong analogical proof of the annual motion of the earth. In fact, if we compute from the third law the periodic time of a body placed at the same distance from the sun as the earth is known to be, the result will give exactly the sidereal year. It is therefore certain that the earth obeys the same laws as the planets; and, when clasped among them, the most perfect analogy pervades the whole system.

The periodic time of a planet may be determined by observing the interval which elapses between two consecutive passages through the same node. The retrograde movement of the nodes being, as we have seen, considerable, the planet, in returning to its node, also returns to the same position very nearly with reference to the fixed stars, and therefore has in the interval completed a sidereal revolution. Hence the mean sidereal motion is known, and consequently, from the third law of Kepler, the planet's mean distance from the sun, the double of which is the transverse axis of the orbit. But the observation of the passages through the nodes not only gives the time of a sidereal revolution, but also approximate values of the eccentricity and the position of the line of the apsides; for, unless the orbit is circular, or the line of the apsides coincides with that of the nodes, the time the planet consumes in passing from the ascending to the descending node, or from Ω to Ω', will be different from that in which it passes from Ω to Ω; and, by a comparison of this difference with the theory of elliptic motion, it is easy to deduce the greatest difference between the true and mean anomalies, that is, the greatest equation of the centre, and thence the eccentricity. The method is exactly similar to that which has already been described in chap. ii, sect. 2.

But as these methods lead to results of no great accuracy, and as the passages through the nodes in the case of the distant planets occur only after considerable intervals of time, the astronomer requires some more accurate and expeditious means of determining the elements of an orbit; and three geocentric observations of the planet in any part of its orbit whatever are sufficient to determine these elements, when the nodes and inclination, mean motion and mean distance, are known. Theoretical positions of the sun and planet relatively to the earth, the position of the planet with respect to the sun can be computed. Let then \( v, v', v'' \) represent three heliocentric longitudes of a planet, and \( \pi \) the longitude of the perihelion, all reduced to the plane of the orbit. The true anomalies corresponding to the three observations will consequently be \( v - \pi, v' - \pi, v'' - \pi \). Now, if, as in the series given in chapter ii. section 2, for the true anomaly in terms of the mean, we express the mean motion by \( n \), and the time elapsed since the passage through the perihelion by \( t \), the three corresponding mean anomalies will be respectively \( n t, n t + \epsilon, n t + \epsilon' \); and it will be observed, that although \( t \) is unknown, yet as \( t = t' \) and \( t' = t \) (the intervals between the first and the other two observations) are known, \( t \) and \( t' \) are both given in terms of \( t \).

Omitting the square, and all the higher powers, of the eccentricity \( e \), the series just referred to gives the three following equations,

\[ n t = v - \pi - 2e \sin(v - \pi) \]

\[ n t' = v' - \pi - 2e \sin(v' - \pi) \]

\[ n t'' = v'' - \pi - 2e \sin(v'' - \pi) \]

from which we must deduce the three unknown quantities \( t, e, \sigma \), that is to say, the epoch of the planet's passage through the perihelion, the eccentricity, and the longitude of the perihelion. By subtracting the first of these equations from each of the other two, we obtain

\[ n(t' - t) = v' - v - 2e [\sin(v' - \pi) - \sin(v - \pi)] \]

\[ n(t'' - t) = v'' - v - 2e [\sin(v'' - \pi) - \sin(v - \pi)] \]

and if we assume

\[ n(t' - t) = v' - v = a \]

\[ n(t'' - t) = v'' - v = b \]

(a and \( b \) being thus known quantities), these two equations give us the two following,

\[ a = 2e [\sin(v' - \pi) - \sin(v - \pi)] \]

\[ b = 2e [\sin(v'' - \pi) - \sin(v - \pi)] \]

whence, by division,

\[ \frac{a}{b} = \frac{\sin(v' - \pi)}{\sin(v'' - \pi)} - \frac{\sin(v - \pi)}{\sin(v - \pi)} \]

Now, the numerator of this fraction

\[ = \sin(v - \pi) \left( \frac{\sin(v' - \pi)}{\sin(v - \pi)} - 1 \right) \]

\[ = \sin(v - \pi) \left( \frac{\sin(v' - \pi) \cos(\pi) \sin(\pi)}{\sin(v - \pi) \cos(\pi) \sin(\pi)} - 1 \right) \]

\[ = \sin(v - \pi) \left( \frac{\sin(v' - \pi) \cos(\pi) \sin(\pi)}{\sin(v - \pi) \cos(\pi) \sin(\pi)} - 1 \right) \]

and, similarly, the denominator

\[ = \sin(v - \pi) \left( \frac{\sin(v' - \pi) \cos(\pi) \sin(\pi)}{\sin(v - \pi) \cos(\pi) \sin(\pi)} - 1 \right) \]

therefore

\[ \frac{a}{b} = \frac{\sin(v' - \pi) \cos(\pi) \sin(\pi)}{\sin(v - \pi) \cos(\pi) \sin(\pi)} \]

whence

\[ \tan(\pi) = \frac{a(\sin(v' - \pi) \cos(\pi) \sin(\pi)) - b(\sin(v - \pi) \cos(\pi) \sin(\pi))}{a(\cos(v' - \pi) \cos(\pi) \sin(\pi)) - b(\cos(v - \pi) \cos(\pi) \sin(\pi))} \]

an equation which gives \( \pi \), the longitude of the perihelion reduced to the orbit, the angles \( v, v', v'' \) being given by observation.

Having thus found \( \pi \), and consequently \( v' - \pi, v'' - \pi \), it is easy to determine the eccentricity \( e \). From the equation

\[ a = 2e [\sin(v' - \pi) - \sin(v - \pi)] \]

we obtain

\[ 2e = \frac{a}{\sin(v' - \pi) - \sin(v - \pi)} \]

a formula which may be rendered better adapted for logarithmic calculation by being put under the form

\[ e = \frac{a}{4 \sin(v' - \pi) \cdot \cos(\frac{v' + v}{2} - \pi)} \]

By means of the values of \( e \) and \( v - \pi \), which have now been found, we obtain \( t \), the epoch of the passage of the planet through its perihelion, immediately from the equation

\[ n t = v - \pi - 2e \sin(v - \pi) \]

The only element which now remains undetermined is the radius vector, and this is given by the polar equation of the ellipse, viz.

\[ r = \frac{a(1 - e^2)}{1 + e \cos(v - \pi)} \]

in which \( a \) represents the semi-axis major, or mean distance found by the third law of Kepler.

In this manner approximate values of the different elements are obtained, but no method can be proposed which will give them, without repeated corrections, so accurately as to accord with the precise observations of the present day; and the great object of the practical astronomer is to advance them gradually nearer and nearer to the truth. From the laws of Kepler applied to a few observations, formulae are constructed which represent the mean values of the elliptic elements nearly; every successive observation gives a geocentric latitude and longitude; and by comparing these with the corresponding latitude and longitude computed from the formulae, equations of condition are obtained, by means of which the formulae are corrected and rendered still more exact. New observations give new equations of condition, to be joined with the former; and after a great number of such equations have been obtained, geometry teaches us how to combine them so that each may have its just influence in the determination of the final result. Thus the tables are gradually approximated to perfection, or to such a state that they differ only insensibly from the mean of a great number of observations.

But the great difficulty of obtaining an exact conformity between observation and the results of computation arises from the circumstance that the different elements of the orbits have no fixed values, but are incessantly though slowly varying, in consequence of the mutual disturbances which the planets occasion to the motions of each other. If the planets had no mutual attraction, and obeyed only the central force of the sun, the problem of determining the different circumstances of their motion would be one of easy solution, and each of them would accurately describe an ellipse on the plane of its orbit, according to the laws of Kepler. In consequence, however, of their mutual attraction, every planet is compelled to deviate more or less from its ellipse; so that when the matter is viewed with mathematical precision, the laws of Kepler belong only to an ideal system, and have no actual existence in nature. It is in the determination of the amount of these effects of perturbation, and the secular variations they give rise to, that the most profound and intricate theories of physical astronomy find their application; for it is from theory alone that we can derive any knowledge of the laws and periods of changes which are completed only after many hundreds or thousands of years. The planets whose motions are most disturbed by their mutual attractions are Jupiter and Saturn; and even with respect to them, observation, although it makes known the existence of inequalities, can neither detect their laws nor assign their periods. The only elements which are exempted from secular changes occasioned by the perturbing forces are the greater axes of the orbits, and the mean motions depending on them according to the third law of Kepler. The eccentricities of the orbits vary slowly. With regard to Mercury, Mars, and Jupiter, those elements are at the present time increasing; in the cases of Venus, the Earth, Saturn, and Uranus, they are diminish-

3. Of the real Dimensions of the Planetary Orbits, and the Transits of Venus and Mercury over the Sun’s Disk.—In what has hitherto been said respecting the mean distances of the planets from the sun, those distances have been estimated in parts of the semi-axis major of the earth's orbit; a convenient scale, which enables us to form a very precise idea of the relative dimensions of the several orbits, and likewise serves to express their absolute dimensions, provided we can determine the distance between the sun and the earth in terms of any measure with which we are familiar. It is evident, indeed, in consequence of the relation that subsists between the mean distances and mean motions, that it is only necessary to determine the mean distance of any one of the planets from the sun, in order to determine the mean distances of all the others, and assign the dimensions of the whole solar system. Now there are various ways of determining the sun's distance from the earth in terms of the earth's semidiameter. The distance of a planet from the sun may likewise be obtained if we can find the means of measuring its distance from the earth at any epoch; for the geocentric positions of the sun and the planet being known from the theory of their motions, the radius vector of the orbit, or planet's distance from the sun at that epoch, may be found by a simple trigonometrical computation. To determine the distance of a planet from the earth, it might seem only necessary to determine its horizontal parallax; but in general the parallaxes of the planets are quantities by far too small to be directly observed. That of Mars, however, becomes very appreciable in particular circumstances, that is to say, when Mars is in opposition with the sun, and at the same time near the perihelion of his orbit. Thus, in the year 1751, on the 6th of October, that planet, being near its opposition, was observed at the same instant of time by Laccaille at the Cape of Good Hope, and by Wargentin at Stockholm; and the horizontal parallax deduced from the two observations, in the manner explained in chap. i. sect. 2, was found to amount to $24^\circ.6$. Now the distance of the planet being equal to unity divided by the sine of the horizontal parallax, the distance of Mars from the earth at the time of the observation was consequently

\[ \frac{1}{\sin 24^\circ.6} = \frac{1}{\sin 24^\circ.6} = \frac{1}{0.4146} = 2.4186 \text{ terrestrial radii} \]

But the distance of Mars from the earth at that time, as computed from the theory of his elliptic motion, was $0.435$ parts of the radius of the earth's orbit; consequently the whole length of that radius is

\[ \frac{8381}{0.435} = 19226 \text{ semidiameters of the earth}. \]

Hence the distance of Mars from the sun $= 19226 + 8381 = 27607$ semidiameters of the earth. It will be remarked that these numbers are only approximate, our present object being to explain the method, and not to determine the exact quantities.

But besides the inconvenience attending the determination of a planet's distance by this method, which requires observations to be made simultaneously on opposite sides of the earth, the method is in itself liable to great uncertainty. The error of a result is always in a certain proportion to the error of observation, and in the present case a very large quantity is to be determined from a very small one; hence a very slight error of observation will occasion a very erroneous result in the computation of the mean distance. It would be difficult in a single observation of this nature to answer for an error of $2''$; but here $2''$ is a twelfth part of the whole parallax; consequently an uncertainty amounting to a twelfth part affects the mean distance.

A much more accurate method of determining the sun's distance, and thence the dimensions of the planetary orbits, is afforded, though rarely, by the transits of Venus over the sun's disk. When Venus is at her inferior conjunction, and at the same time very near her node, her body will be projected on the disk of the sun; and through the effect of her proper motion, combined with that of the earth, she will appear as a dark spot passing over the disk, and describing a chord which will be seen under different aspects by spectators placed at different points on the earth, because, by reason of the parallax, they refer the planet to different points on the solar disk. The position of the spectator not only occasions a difference in the apparent path described by the planet, but has also a very sensible influence on the duration of the transit, in consequence of which the parallaxes both of Venus and the sun can be determined with great exactness. In order to illustrate this, let E (fig. 73) represent the earth, V Venus, and S the sun. An observer placed at E, the centre of the earth, would see Venus in the di- Theoretical erection of the visual ray EV: she would consequently appear to him projected on the sun's disk at S, and in her successive positions would appear to describe the line DS. Other observers placed at O' and O'' on the earth's surface would see the planet at V' and V'': to the first she would appear to describe the chord DV', and to the second DV''. This is a necessary result of the difference of the parallaxes of Venus and the sun; and as the chords DV', DV'' differ in length according as they are more or less remote from the centre of the disk, the duration of the transit will be longer or shorter according to the situation of the observer and the geocentric latitude of the planet. If by reason of the relative parallax the time of a transit is longer than the true time in one hemisphere, it will be shorter in the opposite; and hence the difference of the times (which may be observed with great accuracy) at places having very different latitudes may serve to determine the relative parallax, or the difference between the parallax of Venus and that of the sun. But the parallaxes are reciprocally proportional to the distances; and the ratio of the distances being known, therefore the ratio of the parallaxes is also known; and having thus the ratio and the difference of the two parallaxes, it is easy to compute the separate amount of each.

This particular application of the transits of Venus to the determination of the sun's distance was first pointed out by Dr Halley, when he announced the transits of 1761 and 1769. Kepler had before announced the occurrence of a transit, but he regarded it only as a curious, and till that time unobserved, phenomenon.

The transit of Venus which occurred in 1769 was anxiously expected by astronomers, and observed in many different parts of the world. The result of the whole of the observations renders it extremely probable that the parallax of the sun is included within the limits of $8^\circ 5$ and $8^\circ 7$. The mean $8^\circ 6$ has been adopted by Delambre and Lalande. From the following table, computed from the different observations, and published by Delambre in the second volume of his Astronomie, p. 565, an idea may be formed of its probable accuracy. Delambre, indeed, remarks that the sun's parallax is now sufficiently well known for all the practical purposes of astronomy.

| Places of Observation | Deduced Difference of Parallax | Sun's Parallax | |-----------------------|-------------------------------|---------------| | Otaheite, Wardhus | 21°561 | 8°7094 | | Otaheite, Kola | 21°166 | 8°5503 | | Otaheite, Cajaneburg | 20°762 | 8°3865 | | Otaheite, Hudson's Bay| 21°066 | 8°5036 | | Otaheite, Paris and Petersburg | 21°730 | 8°7780 | | California, Wardhus | 21°330 | 8°6160 | | California, Kola | 20°765 | 8°3890 | | California, Cajaneburg| 20°208 | 8°1630 | | California, Hudson's Bay| 20°284 | 8°1521 | | California, Paris and Petersburg | 21°576 | 8°7155 | | Hudson's Bay, Wardhus | 22°592 | 9°1260 | | Hudson's Bay, Kola | 20°941 | 8°4589 | | Hudson's Bay, Cajaneburg| 20°233 | 8°1730 | | Hudson's Bay, Paris and Petersburg | 22°897 | 9°2491 |

Here the mean of the first 5 results is nearly $8°59$ of the next $8°41$ of the next $8°75$ of all $8°57$

Having once obtained the value of the solar parallax, it is easy to deduce the sun's distance, and consequently the dimensions of all the planetary orbits. For this purpose we have

$$\sin 8°6 : 1 :: \text{radius of earth} : \text{sun's distance};$$

that is, on reducing the radius of a circle to seconds,

$$\frac{\text{sun's distance}}{\text{earth's radius}} = \frac{8°6}{\sin 1°} = 23984 \text{ terrestrial radii}. \text{Now, if we assume the semidiameter of the earth to be 4000 miles in round numbers, the distance of the sun, or radius of the earth's orbit, will consequently be } 23984 \times 4000 = 95996000 \text{ miles approximately.}$$

By means of this value the mean distances of the planets from the sun, which in the table given above were expressed in terms of the mean distance of the earth, may be converted into miles. The following are the results in round numbers.

| Planet | Mean Distance from the Sun in Miles | |-----------------|-------------------------------------| | Mercury | 37,000,000 | | Venus | 68,000,000 | | Earth | 95,000,000 | | Mars | 142,000,000 | | Ceres | 262,000,000 | | Jupiter | 485,000,000 | | Saturn | 890,000,000 | | Uranus | 1,800,000,000 |

The transits of Venus being phenomena of great importance, in consequence of their practical application to the problem of the sun's distance, it becomes interesting to determine the periods at which they successively occur. It is evident that, by reason of the inclination of the orbit, they can only take place when Venus is very near one of her nodes. Two conditions must therefore be satisfied: Venus must be within a short distance of her node, and at the same time in her inferior conjunction. Now, the interval between two successive conjunctions, that is, the period of a synodic revolution, is easily deduced from the sidereal revolutions of the planet and the earth. Thus, generally, let A and B be two planets, T and t the times of their sidereal revolutions respectively, and suppose T to be greater than t. In the time T, A describes a complete circumference, therefore $T:t::1:\frac{t}{T}$

= part of a circumference described by A in the time t.

But during the same time t, B describes a whole circumference; therefore $1 - \frac{t}{T}$ is what B gains on A in the time t. But the successive conjunctions will always take place when B has gained a whole circumference; therefore, denoting by S the interval between two successive conjunctions, we have $1 - \frac{t}{T} : 1 : t : S$, whence $S = \frac{Tt}{T-t}$

From this simple formula the synodic revolution of any of the planets is found by substituting for T and t the times of the sidereal revolutions of the earth and planet. Suppose, for example, we wish to find the time of a synodic revolution of Mercury. In this case $T = 365.256$ days, and $t = 87.969$ days; consequently $T-t = 277.287$, and $S = \frac{365.256 \times 87.969}{277.287} = 115.877$ days, which, therefore, is the time of a synodic revolution of Mercury.

In the case of Venus we have $t = 224.700$, whence $T-t = 365.256 - 224.700 = 140.556$, and consequently $S = \frac{365.256 \times 224.700}{140.556} = 583.93$ days; which is the period of her synodic revolution. Let us next attend to the other condition which must be satisfied before a transit can take place, namely, that the planet has returned to its node as well as to its inferior conjunction. If we represent by \( m \) the number of revolutions of the earth in the required period, and by \( n \) the number of synodic revolutions of the planet in the same time, it is evident that we shall have \( mT = \frac{nT}{T-t} \), whence

\[ \frac{m}{n} = \frac{t}{T-t}. \]

In the case of Mercury, therefore, \( \frac{m}{n} = \frac{87969}{277287} \); that is to say, after 87969 years, in the course of which Mercury will have been 277287 times in conjunction, the earth and Mercury will be again in conjunction, occupying the same points of their orbits as at the commencement of the period, supposing the nodes fixed. But periods of such enormous length are of no practical use; it is necessary to find an approximating ratio expressed by smaller numbers. For this purpose it is convenient to have recourse to the method of continued fractions, from which the following series is obtained:

\[ \frac{87969}{277287} = \frac{1}{3 + \frac{1}{6 + \frac{1}{11 + \frac{1}{1998}}}}. \]

Here the first approximating fraction is \( \frac{1}{3} \), which denotes that in one year, during which there will happen three synodic periods, Mercury will not be very far from his conjunction, nor from the same point of his orbit in which he was at the commencement of that time. The next approximation is \( \frac{1}{3 + \frac{1}{6}} = \frac{6}{19} \), showing that after six years, during which there will have been 19 conjunctions, Mercury will be again nearly in conjunction at the same point of his orbit. By continuing the process we obtain the following series of fractions, each approaching nearer to \( \frac{87969}{277287} \), namely,

\[ \frac{1}{3}, \frac{6}{19}, \frac{7}{22}, \frac{13}{41}, \frac{33}{104}, \frac{46}{145}, \text{&c.} \]

of which the numerators express the number of years, and the denominators the corresponding number of synodic revolutions.

Approximative fractions might be found in the same manner to express the ratio of the number of revolutions of Mercury to those of the earth, or of \( \frac{T}{T-t} \); but it is unnecessary to have recourse again to division, insomuch as they are easily obtained from the above. For example, since the fraction \( \frac{1}{3} \) denotes that Mercury has gained three revolutions on the earth in one year, it is evident that he must have completed \( 3 + 1 = 4 \) revolutions. In the same way, in six years Mercury gains 19 on the earth, or completes \( 19 + 6 = 25 \). Hence, in the series required the numerators of the fractions will continue the same as in the series above, while the denominators will be the sums of the terms of the corresponding fractions. The new series will therefore be

\[ \frac{1}{3}, \frac{6}{23}, \frac{7}{29}, \frac{13}{54}, \frac{33}{137}, \frac{46}{191}, \text{&c.} \]

the numerators being the number of years, and the denominators the corresponding periods of Mercury.

Applying to the case of Venus and the earth the formula \( \frac{m}{n} = \frac{t}{T-t} \), we shall have \( t = 2247008 \), \( T - t = 3652563 - 2247008 = 1405555 \); therefore \( \frac{m}{n} = \frac{2247008}{1405555} \).

The series of fractions approximating to this ratio, obtained in the manner indicated above, is

\[ \begin{array}{ccccccc} 1 & 2 & 3 & 8 & 227 & 235 & 243 \\ 1' & 2' & 5' & 142' & 147' & 152' & \text{&c.} \end{array} \]

the numerators of which, as before, express the years, and the denominators the synodic periods. Taking the fourth fraction of the series, it appears that after eight years, in which there are five synodic periods, Venus will again occupy nearly the same position with respect to the earth and the nodes of her orbit. This will take place more nearly after 227 years, and more nearly still after 235; consequently, 235 years after a transit has taken place, the occurrence of another may be expected with great probability. The alteration, however, which takes place in the position of the line of the nodes, which in the preceding computations we have regarded as fixed, renders the numbers a little uncertain.

Since Venus returns to her conjunction at nearly the same point of her orbit after eight years, it may happen, and sometimes indeed actually does happen, that a transit will take place in about eight years after the occurrence of a former one. But in that time the latitude of Venus, in consequence of the inclination of her orbit, undergoes a variation amounting to 20° or 24°; in sixteen years the change of latitude increases to 40° or 48°, which is considerably greater than the diameter of the sun. It cannot happen, therefore, that three transits will take place within sixteen years. From the above series of fractions we might infer that another could not take place before 227 years; but that series was obtained on the supposition that the transits only happen when Venus returns to the same node; and it is evident that they may equally occur when the planet is near the other node, and consequently after an interval of half the length of the former, or 113 years. If, at the occurrence of the first transit, Venus has passed her node, the next will happen eight years sooner; or, if she has not reached the node, eight years later. Hence, after two transits have occurred within eight years, another cannot be expected before 105, 113, or 121 years, that is, 113 = 8 years. But these periods sometimes fail: that of 235 brings about the phenomenon with greater certainty, and 243 (which is the double of 121) is the surest of all. The periods of 235 and 251 are that of 243 diminished or augmented by 8. The whole calculation, therefore, reduces itself to periods of 121 and \( \frac{121}{8} \) years. The last transits took place in 1761 and 1769; the next will not happen till the years 1874 and 1882; and thus the infrequency of these phenomena adds to the interest they derive from their real importance.

Delambre has given a list of all the transits of Venus for a period of 2000 years, from which the following is extracted. (Astronomie, tome ii. p. 473.)

| Year | Month | Mean Time (at Paris) of Conjunction | Node | |------|-------|-----------------------------------|------| | 1631 | Dec. 6 | 17° 28' 40" Ω | | | 1639 | Dec. 4 | 6° 9 40 Ω | | | 1761 | June 5 | 17° 44 34 Ω | | | 1769 | June 3 | 10° 7 54 Ω | | | 1874 | Dec. 8 | 16° 17 44 Ω | | | 1882 | Dec. 6 | 4° 25 44 Ω | | | 2004 | June 7 | 21° 0 44 Ω | | Mercury.

Mercury is a small body, but emits a very bright white light; though, by reason of his always keeping near the sun, he is seldom to be seen; and when he does make his appearance, his return to the sun is so rapid, that he can only be discerned for a short space of time. Delambre was able to observe him only twice with the naked eye.

Mercury is about 3140 English miles in diameter, and his mean distance from the sun about 37 millions of miles. On account of his smallness and brilliancy it is extremely difficult to find any spot on his disk so distinctly marked as to afford the means of determining his rotation. Besides, by reason of his proximity to the sun, an observation of a spot, if made in the evening, can scarcely be well begun before the planet sets; or, if in the morning, before the increasing twilight renders the spot invisible. Hence it is only possible to observe daily a very small arc of a small circle; and if the spot re-appears on the succeeding day, it is doubtful whether the arc which it has passed over exhibits the whole motion, or if one or more circumferences ought to be added. By an attentive observation of the variations of the phases of Mercury, Schroeter has, however, remarked that he revolves about his axis in the space of 24 hours 5 minutes 30 seconds. M. Harding discovered in 1801 an obscure streak on the southern hemisphere of the planet, the observations of which, together with those of a spot discovered by Schroeter, gave the same period of rotation. The results of Schroeter's researches on Mercury may be summed up as follows: 1st, The apparent diameter of Mercury at his mean distance is $6^\circ 02'$; 2d, His form is spherical, exhibiting no sensible compression; 3d, His equator is very considerably inclined to his orbit, and the differences of his days and seasons must consequently be very great; 4th, There are mountains on his surface which cast very long shadows, and of which the height bears a greater proportion to the diameter of the planet than those of the Earth, the Moon, or even of Venus. The height of Chimborazo is $\frac{1}{10}$ of the radius of the earth; one of the mountains in the moon has been estimated at $\frac{1}{10}$ of her radius; the highest in Venus at $\frac{1}{10}$; and one in Mercury at $\frac{1}{10}$. The highest mountains are in the southern hemisphere, which is also the case in respect of the Earth and Venus. There are no observations to prove decisively whether Mercury is surrounded by an atmosphere.

Venus.

Venus, the most beautiful object in the heavens, is about 7700 English miles in diameter, and placed at the distance of 68 millions of miles from the sun. Although the oscillations of this planet are considerably greater than those of Mercury, and she is seldom invisible, yet on account of the uniform brilliancy of her disk, it is extremely difficult to ascertain the period of her rotation. Dominic Cassini, after having long fruitlessly attempted to discover any object on her surface so well defined as to enable him to follow its motions, at length, in 1667, perceived a bright part, distant from the southern horn a little more than a fourth part of the diameter of the disk, and near the eastern edge. By continuing his observations on this spot, Cassini concluded the rotation of Venus to be performed in about 23 hours; but he does not seem to have considered this conclusion as deserving of much confidence. In the year 1726 Bianchini, an Italian astronomer, made a number of similar observations for the same purpose, from which he inferred that the rotation of the planet is performed in 24 days 8 hours. The younger Cassini has shown, however, that the observations of Bianchini, as well as those of his father, could be explained by a rotation of 23 hours and 21 or 22 minutes, whereas the rotation of 24 days 8 hours cannot be reconciled with the appearances observed by the elder Cassini. The determination of Cassini was regarded by astronomers as the more probable of the two, particularly as Bianchini was not able to make his observations in a connected manner, on account that a neighbouring building intercepted his view of the planet, and obliged him to transport his telescope to a different situation. The question of the rotation of Venus was finally settled by Schroeter, who found it to be performed in 23 hours 21 minutes 19 seconds. Each of the three observers found the inclination of the axis of rotation to the axis of the ecliptic to be about 75°. Some doubt, however, still exists with respect to the value of this element.

Schroeter's observations on this planet were principally directed to a mountain situated near the southern horn. The line which joins the extremities of the horns is always a diameter; and the horns of the crescent of a perfect sphere ought to be sharp and pointed. Schroeter remarked that this was not always the case with regard to the horns of Mercury and Venus. The northern horn of the latter always preserved the pointed form, but the southern occasionally appeared rounded or obtuse,—a circumstance which indicated that the shadow of a mountain covered the part Bo (fig. 74), so that the line joining the extremities of the horns appeared to be Ao, and not AB (the diameter of the circular disk); but at d, beyond oB, he remarked a luminous point, which he supposed to be the summit of another mountain, illuminated by the sun after he had ceased to be visible to the rest of that hemisphere. Now, in order that the horn of the crescent may appear obtuse in consequence of the shadow of a mountain falling upon it, and another mountain d present a luminous point, the two mountains must be at the same time both at the edge of the disk and on the line separating the dark from the enlightened part of the planet. But this position cannot be of long continuance; for the rotation will cause d to rise into the enlightened part, or sink into the dark hemisphere, and in either case the mountain will cease to be visible. If, however, the rotation is completed in 23 hours 21 min., the mountain d will appear 39 min. sooner than it did on the previous day; for in the course of a day the boundary of light and darkness will hardly have shifted its position on the surface of the planet through the effects of the orbital motion. Hence it is possible to obtain several consecutive observations, from which an approximate value of the period may be found; and this being once obtained, it may be rendered still more exact by observations separated from each other by a longer interval. Thus Schroeter found that an interval of 20 days 11 hours 15 min. between two apparitions of the mountain being divided by 23 hours 21 min., gave 21-04 revolutions. That intervals of 121 days 14 hours 23 min., 142 days 1 hour 40 min., 155 days 18 hours 11 min., divided each by 23 hours 21 min., gave 125-01, 146-02, 160-09 revolutions respectively. All these comparisons prove that the revolution of 23 hours 21 min. is somewhat too short. They ought to have given 21, 125, 146, and 160 revolutions exactly, supposing the observations to have been perfectly accurate. On dividing the intervals by 21, 125, 146, and 160 respectively, the quotients will be each the time of a revolution very nearly; and by taking a mean among the whole, the most pro- Theoretical table result at least will be obtained. In this manner Schroeter found the period of rotation already stated, namely, 23 hours 21 min. 19 sec.

Since the time of rotation of Mercury and Venus is nearly equal to that of the earth, the compression of these planets at the poles, which results from the centrifugal force, ought also to be nearly in the same proportion. But at the distance of the earth the compression must be imperceptible even in the case of Venus; for supposing it to amount to \( \frac{1}{300} \), the difference between the radius of her poles and that of her equator would only amount to a tenth of a second as seen from the earth.

During the transits of Venus over the sun's disk in 1761 and 1769 a sort of penumbral light was observed round the planet by several astronomers, which was occasioned, without doubt, by the refractive powers of her atmosphere. Wargentin remarked that the limb of Venus which had gone off the sun showed itself with a faint light during almost the whole time of emersion. Bergman, who observed the transit of 1761 at Upsal, says that at the ingress the part which had not come upon the sun was visible, though dark, and surrounded by a crescent of faint light, as in fig. 75; but this appearance was much more remarkable at the egress; for as soon as any part of the planet had disengaged itself from the sun's disk, that part was visible with a like crescent, but brighter (fig. 76). As more of the planet's disk disengaged itself from that of the sun, the part of the crescent farthest from the sun grew fainter, and vanished, until at last only the horns could be seen, as in fig. 77. The total immersion and emersion were not instantaneous; but as two drops of water, when about to separate, form a ligament between them, so there was a dark shade stretched out between Venus and the sun, as in fig. 78; and when this ligament broke, the planet seemed to have got about an eighth part of her diameter from the limb of the sun (fig. 79). The numerous accounts of the two transits which have been published abound with analogous observations, indicating the existence of an atmosphere of considerable height and density. Schroeter calculated that its horizontal refraction must amount to \( 30^\circ 34' \), differing little from that of the terrestrial atmosphere. A twilight which he perceived on the cusps afforded him the data from which he deduced this conclusion.

Cassini and Montaigne imagined that they had observed a satellite accompanying Venus; but this appears to have been an optical illusion arising from the strong light of the planet reflected back from the convex surface of the eye upon the eye-glass of the telescope, and thence reflected a second time back to the eye. This hypothesis at least will explain the appearances which they have described; and although astronomers have sought for this pretended satellite with great care, they have neither observed it on the sun during the transits of Venus in 1761 and 1769, nor in any other part of her orbit.

Mars.

After Venus, Mars is the planet whose orbit is nearest to the earth. His diameter is about one half; and his volume only about one sixth part of that of our globe. He is of a dusky reddish colour, by reason of which he is easily recognised in the heavens. His mean distance from the sun is about 142 millions of miles.

The rotation of Mars was suspected before the year 1643 by Fontana, a Neapolitan astronomer; but it was reserved for Cassini to demonstrate its existence and assign its period. Cassini began to observe the spots on the surface of Mars at Bologna in 1666; and after having continued his observations for a month, he found they returned to the same situation in 24 hours and 40 min. The theoretical planet was observed by some astronomers at Rome with longer telescopes; but they assigned to it a rotation of 13 hours only. This, however, was afterwards shown by Cassini to have arisen from their not distinguishing between the opposite sides of the planet, which, it seems, have spots pretty much alike. He made further observations on the spots of this planet in 1670; which confirmed his former conclusion respecting the time of rotation. The spots were again observed in subsequent oppositions, particularly for several days in 1704 by Maraldi, who took notice that they were not always well defined, and that they not only changed their shape frequently in the interval between two oppositions, but even in the space of a month. Some of them, however, continued of the same form long enough to allow the time of the planet's rotation to be determined. Among these there appeared that year an oblong spot, resembling one of the belts of Jupiter when broken. It did not reach quite round the body of the planet; but had, not far from the middle of it, a small protuberance towards the north, so well defined, that Maraldi was thereby enabled to fix the period of its revolution at 24 hours 39 min., only one minute less than what Cassini had determined it to be.

The near approach of Mars to the earth in 1719 afforded an excellent opportunity of observing him, as he was then within \( 23^\circ \) of his perihelion, and at the same time in opposition to the sun. His apparent magnitude and brightness were thus so much increased, that he was by the vulgar taken for a new star. His appearance at that time, as seen by Maraldi through a telescope of 34 feet long, is represented in fig. 80. There was then a long belt that reached half-way round, to the end of which another shorter belt was joined, forming an obtuse angle with the former, as in fig. 81. This angular point was observed on the 19th and 20th of August, a little to the east of the middle of the disk; and 37 days after, on the 25th and 26th of September, it returned to the same situation. This interval, divided by 36, the number of revolutions contained in it, gives 24 hours 40 minutes for the period of one revolution; a result which was verified by another spot of a triangular shape, one angle whereof was towards the north pole, and the base towards the south, and which on the 5th and 6th of August appeared as in fig. 82. After 72 revolutions it returned to the same situation on the 16th and 17th of October. Some of the belts of this planet are said to be parallel to his equator; but that seen by Maraldi was very much inclined to it.

Besides these dark spots on the surface of Mars, astronomers had noticed that a segment of his globe about spots about the south pole exceeded the rest of his disk so much in the poles brightness, that it appeared to project as if it were the Mars segment of a larger globe. Maraldi informs us that this bright spot had been taken notice of for 60 years, and was more permanent than the other spots on the planet. One part of it is brighter than the rest, and the least bright part is subject to great changes, and has sometimes disappeared.

A similar though less remarkable brightness about the north pole of Mars was also sometimes observed, the existence of which has been confirmed by Sir W. Herschel, who examined the planet with telescopes of much greater power than any former astronomer ever was in possession of. A very full account of Herschel's observations on this planet is given in the 74th volume of the Philosophical Transactions. Some of the remarkable appearances there described are represented in figs. 83-88. The magnifying powers he used were sometimes as high Theoretical as 932; and with this the south polar spot was found to be 41" in diameter. Fig. 96 shows the connection of the other figures marked 89, 90, 91, 92, 93, 94, 95, which complete the whole equatorial succession of spots on the disk of the planet. "The centre of the circle," Herschel observes, "marked 90, is placed on the circumference of the inner circle, by making its distance from the circle marked 92, answer to the interval of time between the two observations, properly calculated and reduced to sidereal measure. The same is done with regard to the circles marked 91, 92, &c.; and it will be found by placing any one of these connected circles in such a manner as to have its contents in a similar situation with the figures in the single representation, which are marked with the same number, that there is a sufficient resemblance between them; though some allowance must be made for the distortions occasioned by this kind of projection."

From these observations Herschel concluded that the diurnal rotation of Mars is accomplished in 24 hours 39 minutes 21½ seconds; that his equator is inclined to his orbit in an angle of 28° 42', and his axis of rotation to the axis of the ecliptic in an angle of 30° 18'. Hence the time of rotation and the seasons of this planet are little different from those of the earth.

The bright appearance so remarkable about the poles of Mars is ascribed by Herschel to the reflection of light from mountains of ice and snow accumulated in those regions. "The analogy between Mars and the earth," says he, "is perhaps by far the greatest in the whole solar system." Their diurnal motion is nearly the same, the obliquity of their respective ecliptics not very different; of all the superior planets, the distance of Mars from the sun is by far the nearest alike to that of the earth; nor will the length of the Martial year appear very different from what we enjoy, when compared to the surprising duration of the years of Jupiter, Saturn, and the Georium Sidus. If we then find that the globe we inhabit has its polar region frozen and covered with mountains of ice and snow, that only partly melt when alternately exposed to the sun, I may well be permitted to surmise, that the same causes may probably have the same effect on the globe of Mars; that the bright polar spots are owing to the vivid reflection of light from frozen regions; and that the reduction of those spots is to be ascribed to their being exposed to the sun."

Since the discovery of the flattened form of the earth, it was to be presumed that the rotation of the other planets would produce a similar effect on their figures, and this supposition has been fully confirmed by observation. The time of the rotation of Mars is nearly equal to that of the earth, but his diameter being only about half that of the earth, the velocity of a point on his equator is consequently only half as great as that of a point on the earth's equator; hence we might expect that the deviation of his figure from a perfect sphere would be much less considerable. The contrary, however, appears to be the case; and his compression seems to be much greater than that of the earth. According to Herschel, the ratio of his equatorial and polar axes is 103 to 98. Schroeter estimates the same ratio to be that of 81 to 80. This remarkable compression at the poles of Mars arises in all probability from considerable variations of density in the different parts of his globe.

It has been commonly related by astronomers, that the atmosphere of this planet is possessed of such strong refractive powers as to render invisible the small fixed stars near which it passes. Dr Smith relates an observation of Cassini, in which a star in the water of Aquarius, at the distance of six minutes from the disk of Mars, became so faint before its occultation, that it could not be seen by the naked eye, nor even with a three feet telescope. This would indicate an atmosphere of a very extraordinary size and density; but the following observations of Herschel seem to show that it is of much smaller dimensions.

"1783, Oct. 26th. There are two small stars preceding Mars, of different sizes; with 460 they appear both dusky red, and are pretty unequal; with 218 they appear considerably unequal. The distance from Mars of the nearest, which is also the largest, with 227, measured 3° 26' 20". Some time after, the same evening, the distance was 3° 8' 55", Mars being retrograde. Both of them were seen very distinctly. They were viewed with a new 20 feet reflector, and appeared very bright. October 27th: the small star is not quite so bright in proportion to the large one as it was last night, being a good deal nearer to Mars, which is now on the side of the small star; but when the planet was drawn aside, or out of view, it appeared as plainly as usual. The distance of the small star was 2° 5' 25". The largest of the two stars," adds he, "on which the above observations were made, cannot exceed the 12th, and the smallest the 13th or 14th magnitude; and I have no reason to suppose that they were any otherwise affected by the approach of Mars, than what the brightness of its superior light may account for. From other phenomena it appears, however, that this planet is not without a considerable atmosphere; for, besides the permanent spots on its surface, I have often noticed occasional changes of partial bright belts, and also once a darkish one in a pretty high latitude; and these alterations we can hardly ascribe to any other cause than the variable disposition of clouds and vapours floating in the atmosphere of the planet."

Ceres, Pallas, Juno, and Vesta.

The commencement of the present century was rendered remarkable in the annals of astronomy by the discovery of four new planets circulating between Mars and Jupiter. Kepler, from some analogy which he found to subsist among the distances of the planets from the sun, had long before suspected the existence of one at this distance; and his conjecture was rendered more probable by the discovery of Uranus, with regard to which the analogy of the other planets is observed. So strongly, indeed, were astronomers impressed with the idea that a planet would be found between Mars and Jupiter, that, in the hope of discovering it, Baron Zach formed an association of 24 observers, who divided the sky into as many zones, and undertook each to explore one carefully. A fortunate accident anticipated a discovery which might have required years of toil. An error in the catalogue of Wollaston, who had laid down a star in a position in which it is not to be found, engaged Piazzi, the superintendent of the observatory at Palermo, to observe for several successive days all the small stars in the neighbourhood of the place indicated. On the first day of the present century, the 1st of January 1801, he observed a small star in Taurus, which, on the day following, appeared to have changed its place. On the 3d he repeated his observation, and was then satisfied that it had a diurnal motion of about 4' in right ascension, and 3½' in declination towards the north pole. He continued to observe it till the 23d, when he communicated his discovery to MM. Bode and Oriani, giving them the positions of the star on the 1st and 23d, and only adding, that between the 11th and 13th its motion had changed from retrograde to direct. Before the communication reached them, however, the planet was lost in the sun's rays; and, owing to its ex- Theoretical extreme smallness, the difficulty of finding it after its emergence was so great, that it was not again seen till the 31st of the following December, when it was detected by Zach. In this was recognised the planet which Kepler had suspected to circulate between Mars and Jupiter. Piazzi, in honour of Sicily, gave it the name of the tutelary goddess of that country, Ceres; and her emblem, the sickle, $\varphi$, has been adopted as its appropriate symbol.

Ceres is of a reddish colour, and appears to be about the size of a star of the eighth magnitude. The eccentricity of her orbit is somewhat greater than that of Mercury; and its inclination to the ecliptic greater than that of any of the old planets. The distance of Ceres from the sun is about 3½ times that of the earth, or nearly 270 millions of miles. Schroeter found her apparent diameter to be 2', corresponding to about 1624 miles; but Herschel reduced this measurement to 0'5, which would indicate a diameter of about 160 miles. The nebulosity which surrounds the planet renders it almost impossible to distinguish the true disk; and hence arises the great discrepancy between the above estimates of its magnitude. From a great number of observations, Schroeter inferred that Ceres has a dense atmosphere, rising to the height of no less than 675 English miles above the planet, and subject to numerous changes. On this account he conceives that there is little chance of discovering the period of its rotation.

The difficulty of finding Ceres induced Dr Olbers of Bremen to examine with particular care the configurations of all the small stars situated near her geocentric path. On the 28th of March 1802 he observed a star of the seventh magnitude, which formed an equilateral triangle with the stars 20 and 191 of Virgo in Bode's catalogue. He was certain that he had never seen a star in that place before, and at first imagined it might be one of those which are subject to periodical changes of brilliancy; but after examining it for two hours, he remarked that its right ascension was diminishing, while its northern declination continued to augment nearly in the same manner as had been the case with Ceres when that planet was first seen by him in almost the same position. On the following day he found its right ascension had diminished 10', while its northern declination had increased 20'. From observations continued during a month, M. Gauss calculated an elliptic orbit, the eccentricity of which amounted to $24764$, much greater than that of any of the other planets. He also found its inclination to be $34° 39'$, exceeding the aggregate inclinations of all the other planetary orbits; and its mean distance $2770552$, almost the same as that of Ceres. On account of these three circumstances, the new planet, otherwise of little importance, became the most singular in the whole system. One planet had been suspected to exist between Mars and Jupiter, and two were now discovered. The great inclination of the last rendered it necessary to enlarge the boundaries of the zodiac; but the extent of the zodiac is entirely arbitrary, and had been limited by the extreme latitudes of Venus. There is no reason, as Delambre remarks, why it may not be extended even to the poles. Dr Olbers gave the new planet the name of Pallas, choosing for its symbol the lance, $\varphi$, the attribute of Minerva.

The most surprising circumstance connected with the discovery of Pallas was the existence of two planets at nearly the same distance from the sun, and apparently having a common node. On account of this singularity Dr Olbers was led to conjecture that Ceres and Pallas are only fragments of a larger planet, which had formerly circulated at the same distance, and been shattered by some internal convulsion. Lagrange made this hypothesis theoretical, the subject of an ingenious memoir, in which he determined the explosive force necessary to detach a fragment from a planet with a velocity that would cause it to describe the orbit of a comet. He found that a fragment detached from the earth in this manner, with a velocity equal to 121 times that of a cannon-ball, would become a direct comet; and if with a velocity equal to 156 times that of a cannon-ball, its motion would be retrograde. For other planets the velocity must be $\frac{121 \text{ or } 156}{\sqrt{\text{mean distance}}}$, and consequently less as the mean distance of the planet from the sun is greater. A smaller velocity would be required to cause the detached fragment to move in an elliptic orbit; and with regard to the four small planets we are now considering, an explosive force less than twenty times that of a cannon-ball would have sufficed to detach them from a primitive planet, and cause them to describe ellipses similar to their actual orbits. This hypothesis served also to explain the great eccentricities and inclinations by which these planets are distinguished from the others belonging to the system; for it is evident that the explosive force must have projected the different fragments in all directions, and with different velocities. It followed also, that other fragments of the original planet might probably exist, revolving in orbits which, however they might differ in respect of inclination and eccentricity, would still intersect each other in the same points, or have common nodes, in which the several fragments would necessarily be found at each revolution. Dr Olbers therefore proposed to examine carefully every month the two opposite parts of the heavens in which the orbits of Ceres and Pallas intersect each other, with a view to the discovery of other planets, which might be sought for in those parts with greater chance of success than in a wider zone embracing the whole limits of their orbits. Subsequent discoveries scarcely support his conjecture, though it has still a great degree of probability.

According to Herschel, the diameter of Pallas is only about 80 English miles, or about one half of that which he assigned to Ceres, while Schroeter estimates it at 2099 miles. Schroeter also found the atmosphere of Pallas to be about two thirds of the height of that of Ceres, or about 450 miles. The light of the planet undergoes considerable variations, the cause of which is uncertain.

While M. Harding, of the observatory of Lilienthal, near Bremen, was engaged in forming complete charts of June, the small telescopic stars near the orbits of Ceres and Pallas, with which these planets were likely to be confounded, he determined, on the 2d of September 1804, the position of a small star, by comparing it with the two stars marked 93 and 98 of Pisces in Bode's catalogue. These two stars are situated very near the equator, and at a small distance from one of the nodes of Ceres and Pallas, and exactly in that sort of defile where, according to Dr Olbers, an observer would be certain of detecting in their passage the other fragments of the original planet of which Ceres and Pallas are parts. On the 4th the star was no longer in the same position, but had moved a little to the south-west. On the 5th and 6th M. Harding, by means of a circular micrometer, determined the rate of its motion to be $12° 42'$ in declination to the south, and $7° 30'$ in right ascension, retrograde, the interval between the observations being 24 hours 14 min. 12 sec. From this it was evident that the body belonged to the planetary system. It had then the appearance of a star between the eighth and ninth magnitudes. It was without any nebulosity, and of a whitish colour. A few days Juno is distinguished from the other planets by the great eccentricity of her orbit, which is so considerable, that she describes that half of it which is bisected by the perihelion in about half the time which she employs to describe the other half. This planet is somewhat smaller than Ceres and Pallas, and, though free from nebulosity, must have, according to Schroeter's observations, an atmosphere of greater density than that of any of the old planets.

The success of M. Harding encouraged Dr Olbers to renew the plan of research which he had pointed out on the discovery of Pallas; and on the 29th of March 1807 he perceived, in the constellation of Virgo, a star of the fifth or sixth magnitude, which he suspected from the first observation to be a new planet. A few subsequent observations rendered this conjecture certain. Dr Olbers left to Gauss the care of giving a name to the new planet, and of determining the elements of its orbit. Gauss named it Vesta, and chose for its symbol ♀, an altar surmounted with a censer holding the sacred fire. Vesta is the smallest of all the celestial bodies known to us. Her volume is only about a fifteen thousandth part of that of the earth, and her surface is about equal to that of the kingdom of Spain. She is distinguished by the vivacity of her light, and the luminous atmosphere with which she is surrounded.

Jupiter.

Jupiter is by far the largest planet in the system. His diameter is about 11 times, and his volume 1281 times, greater than that of the earth. His distance from the sun is 5½ times the radius of the ecliptic, or nearly 125,000 terrestrial semidiameters, and consequently above 490 millions of miles. His apparent diameter, which, at his mean distance, is 36°7', and varies between 45°8' and 50°, would subtend an angle of 3° 17' if seen at the same distance as the sun. From Jupiter the sun will appear under an angle of 6' at most; the sun's disk will appear to be 27 times smaller than when seen from the earth, consequently the light and heat which Jupiter receives from the sun will be only the 27th part of what is received by our globe. His density is -99239, that of the sun being considered as unity, or is about one fourth of the density of the earth; and a body which weighs one pound at the equator of the earth, would weigh 2-444 pounds if removed to the equator of Jupiter.

Jupiter has the same general appearance with Mars, only the belts on his surface are much larger and more permanent. Their usual appearance, as described by Dr Long, is represented fig. 97-100; but they are not to be seen but by an excellent telescope. They are said to have been first discovered by Fontana and two other Italians, but Cassini was the first who gave a good account of them. Their number is very variable, as sometimes only one, but seldom more than three, may be perceived. Messier at one time saw so great a number that the whole disk seemed to be covered by them. They are generally parallel to one another, but not always so; and their breadth is likewise variable, one belt having been observed to grow narrow, while another in its neighbourhood has increased in breadth, as if the one had flowed into the other; and in this case a part of an oblique belt lay between them, as if to form a communication for this purpose. The time of their continuance is very uncertain; sometimes they remain unchanged for three months, at other times new belts have been formed in an hour or two. Theoretical Astronomy.

In some of these belts large black spots have appeared, which moved swiftly over the disk from east to west, and returned in a short time to the same place; whence the rotation of this planet about its axis has been determined. On the 9th of May 1664, Dr Hooke, with a twelve feet telescope, observed a small spot in the broadest of the three obscure belts of Jupiter; and observing it from time to time, found that in two hours it had moved from east to west about half the visible diameter of the planet. In 1665 Cassini observed a spot near the largest belt of Jupiter, which is most frequently seen. It appeared round, and moved with the greatest velocity when in the middle, but appeared narrower, and moved slower, the nearer it was to the circumference; showing that the spot adhered to the body of Jupiter, and was carried round upon it. This principal, or ancient spot as it is called, is the largest and the most permanent of any hitherto known; it appeared and vanished no fewer than eight times between the years 1665 and 1708: from the year last mentioned it was invisible till 1713. The longest time of its continuing to be visible was three years, and the longest time of its disappearing was from 1708 to 1713. It seems to have some connection with the principal southern belt; for the spot has never been seen when that disappeared, though the belt itself has often been visible without the spot. Besides this ancient spot, Cassini, in the year 1699, saw one of less stability, that did not continue of the same shape or dimensions, but broke into several small ones, whereof the revolution was but 9 hours 51 min.; and two other spots that revolved in 9 hours 52½ min. The changes in the appearance of the spots, and the difference in the time of their rotation, make it probable that they do not adhere to Jupiter, but are clouds transported by the winds, with different velocities, in an atmosphere subject to violent agitations.

By means of the spots, which can be easily observed, the rotation of Jupiter has been determined with considerable precision. The time of rotation, according to Cassini, Maraldi, and others, is 9 hours and between 55 and 56 minutes: Schroeter makes it 9 hours 55 min., 33 sec. The inclination of his equator to his orbit is only 3° 5' 30", so that the variations of his seasons must be almost insensible.

The radius of Jupiter being nearly 11 times (10-86) that of our earth, and his rotation being 2½ times more rapid, it follows that the space passed over by a point on his equator is 26 times greater than that passed over by a point of the terrestrial equator in the same time. Hence the centrifugal force is 26 times greater; and if the spheroidal form of the earth is occasioned by the diurnal motion, we may expect to find the same effects on a much larger scale exhibited in the form of Jupiter. And this is in fact observed to be the case; for according to Struve the compression of Jupiter is about ⅓ th of his radius, the diameter of his equator being to that of his poles as 14 to 13 nearly, while that of the earth is only ⅘ th. The equatorial diameter at the mean distance subtends an angle of 38° 32', the polar 35° 33'; and the ellipticity is

\[ \frac{1}{0.0728} = 13.71 \]

The annual parallax of Jupiter is less than 12°, consequently the earth, as seen from Jupiter, will never appear at a greater distance than 11° or 12° from the sun. The digressions of Mars would be 17° 2', those of Venus 8°, and those of Mercury only 4° 16'. An inhabitant of Jupiter must therefore be probably ignorant of the existence of Mercury, which will be almost constantly plunged in Theoretical the sun's rays, and likewise greatly diminished in splendour, on account of his great distance. From this we may infer the possibility of the existence of planets inferior to Mercury, and invisible to us, for similar reasons.

On observing Jupiter through the telescope, he is seen accompanied by four little stars, which oscillate on both sides of him, and follow him in his orbit as the moon follows the earth. On this account they are called satellites or attendants. They were first noticed by Galileo within a year after the discovery of the telescope; and it was soon perceived that they revolve around Jupiter in narrow circles, the planes of which deviate little from that of the equator of the planet. They are distinguished from one another by the denomination of first, second, third, and fourth, according to their relative distances from Jupiter, the first being that which is nearest to him. Their apparent motion is oscillatory, like that of a pendulum, going alternately from their greatest elongation on one side to their greatest elongation on the other, sometimes in a straight line and sometimes in an elliptic curve, according to the different points of view in which we observe them from the earth. They have also their stations and retrogradations, and exhibit in miniature all the phenomena of the planetary system.

Since the satellites revolve in orbits about the huge orb of Jupiter, it is evident that occultations of them must frequently happen, by their going behind their primary, or by coming in between us and it; in the former case when they proceed towards the middle of their upper semicircle, and in the latter when they pass through the same part of their inferior semicircle. Occultations of the former kind happen to the first and second satellites at every revolution; the third very rarely escapes an occultation; but the fourth more frequently, by reason of its greater distance. It is seldom that a satellite can be discovered upon the disk of Jupiter, even by the best telescopes, excepting at its first entrance, when, by reason of its being more directly illuminated by the rays of the sun than the planet itself, it appears like a lucid spot upon it. Sometimes, however, a satellite in passing over the disk appears like a dark spot, and can be easily distinguished. This is supposed to be owing to spots on the body of these secondary planets; and it is remarkable that the same satellite has been known to pass over the disk at one time as a dark spot, and at another appearing so luminous that it could not be distinguished from Jupiter himself, except at its coming on and going off. To account for this diversity of appearance, we must suppose either that the spots are subject to change, or, if they be permanent, like those of our moon, that the different portions of the surfaces of the satellites are not equally luminous, and that at different times they turn different parts of their globes towards us. Possibly both these causes may contribute to produce the phenomena just mentioned. By reason of the spots, also, both the light and apparent magnitude of the satellites are variable; for the fewer spots there are upon that side which is turned towards us, the brighter it will appear; and as the bright parts only can be seen, a satellite must appear larger the more of its bright side it turns towards the earth, and smaller the more it happens to be covered with spots. The fourth satellite, though generally the smallest, sometimes appears larger than any of the rest. The third sometimes seems least, though usually the largest; nay, a satellite may be so covered with spots as to appear less than its shadow passing over the disk of the primary, though we are certain that the shadow must be smaller than the body from which it is projected. To a spectator placed on the surface of Jupiter, each of these satellites would put on the phases of the moon; but as the distance of any of them from Jupiter is but small when compared with the distance of that planet from the sun, the satellites are illuminated by the sun very nearly in the same manner with the primary itself; hence they appear to us always round, having constantly the greater part of their enlightened half turned towards the earth; and, indeed, on account of their small size, their phases can scarcely be discerned even through the best telescopes. Their spots, or rather the observed variations of their brilliancy at different times, have afforded the means of determining the fact and the period of their rotation; and it is a very remarkable circumstance that they all, like the moon, constantly turn the same face towards their primary, or complete a rotation about their respective axes in the same time in which they perform a revolution in their orbits.

When the satellites pass through their inferior semicircles, they may cast a shadow upon their primary, and thus cause an eclipse of the sun; and in some situations this shadow may be observed going before or following the satellite. On the other hand, in passing through their superior semicircles, the satellites may be eclipsed in the same manner as our moon, by passing through the shadow of Jupiter. And this is actually the case with the first, second, and third of these bodies; but the fourth, by reason of the greater magnitude and inclination of its orbit, passes sometimes above or below the shadow, as is the case with the moon. The beginnings and endings of these eclipses are easily seen through the telescope, when Jupiter is at a sufficient distance from the sun. The same satellite disappears at different distances from the planet, according to the relative situations of Jupiter, the sun, and the earth; but always on that side of the disk where the shadow of the planet is known from computation to be. With regard to the first and second satellites, the immersions only are visible while Jupiter is passing from his conjunction to his opposition with the sun, and the emersions while he passes from his opposition to his conjunction. The third and fourth sometimes disappear, and again appear on the same side of the disk; and the time during which the satellite continues invisible is exactly that in which, according to computation, it would pass through the planet's shadow. When Jupiter is near his opposition, the eclipses take place when the satellites are close to his disk; because the eye of the spectator is then nearly in the axis of the dark cone formed by his shadow.

These various phenomena will be better understood by referring to fig. 101, where A, B, C, D represent the earth in different parts of its orbit; J Jupiter in his orbit MN, surrounded by his four satellites, the orbits of which are marked 1, 2, 3, 4. At a the first satellite enters the shadow of the planet; at b it emerges from it, and advances to its greatest eastern elongation at c. It appears to pass over the disk of Jupiter like a dark spot at d, and attains its greatest western elongation at e. Similar phenomena take place with respect to the other satellites. Now, since the shadow of Jupiter is always directed away from the sun, the immersions only will be visible to the earth when the earth passes from the position C to the position A; for the eastern limit of Jupiter conceals the satellite at the time of emersion, as is evident by drawing fg in the direction of the visual ray. For the same reason the emersions only are visible while the earth is passing from A to C, or when Jupiter advances from his opposition to his conjunction. This, however, is only strictly true of the first satellite; for the third and fourth, as we have already remarked, and sometimes even the second, owing to their greater distances from Jupiter, occasionally disappear and re-appear on the same side of the disk.

The disks of the satellites having no sensible magni- Astronomy.

Dimensions of the satellites.

Theoretical tude except in the very best telescopes, their diameters have only recently been determined by direct measurement. Schroeter and Harding attempted to measure them by observing the time which the satellite takes to pass over the disk of Jupiter; but such observations are liable to great uncertainty, by reason of differences in the magnifying power of the telescopes, the sight of the observer, the state of the atmosphere, the distance of the satellites from the primary, their altitude above the horizon, and even on account that, by reason of their rotation, they do not always present to us the same hemisphere. Schroeter estimates their diameters relatively to Jupiter as follows:—That of the first \(= \frac{1}{4}\), of the second \(= \frac{1}{8}\), of the third \(= \frac{1}{16}\), of the fourth \(= \frac{1}{32}\). The following are the results of a series of micrometrical measurements made by Professor Struve at the Dorpat observatory, with the great refractor of Fraunhofer:

- Diameter of the first \(= 1°015\) - Diameter of the second \(= 0°911\) - Diameter of the third \(= 1°488\) - Diameter of the fourth \(= 1°273\)

These dimensions are adapted to the mean distance of Jupiter, namely 520279. Compared with the earth, the diameters of the satellites are approximately as follows:—

I. \(= \frac{1}{3}\), II. \(= \frac{1}{4}\), III. \(= \frac{1}{5}\), IV. \(= \frac{1}{6}\).

As seen from Jupiter, the apparent magnitude of the first will be nearly equal to that of our moon seen from the earth; the second and third somewhat greater than half; and the fourth nearly equal to a quarter of that of the moon. These four moons must present to the inhabitants of Jupiter a spectacle of endless variety, on account of the rapid rotation of the planet, the short period of their revolutions, and their eclipses, which happen almost daily.

Saturn.

Saturn, the remotest of the planets known to the ancient astronomers, circulates round the sun at a distance equal to about 94 times the semidiameter of the terrestrial orbit, or nearly 900 millions of miles. His apparent diameter at his mean distance is only about 10°2, yet his true diameter is nearly 10 times, and his volume about 995 times, that of our globe. The area of the sun's disk, as seen from Saturn, is only \( \frac{1}{36} \) of its apparent magnitude as seen from the earth; consequently the light and heat which any point on his surface receives from the great luminary is 80 times less than that which we enjoy. His density, compared with that of the sun considered as unity, is supposed to be \( \frac{55}{3} \), or about \( \frac{1}{4} \) of the density of the earth; and a body which weighs one pound at the equator of the earth, would weigh about 1°01 pound if transferred to the equator of Saturn.

This planet, in consequence of a luminous double ring with which he is surrounded, presents one of the most curious phenomena in the heavens. This singular appendage was first noticed by Galileo, to whom the planet presented a triple appearance, the large orb being situated between two small bodies or ansae. Sometimes the ansae are so enlarged as to present the appearance of a continuous ring; at other times they entirely disappear, and Saturn appears round like the rest of the planets. After a certain time they again become visible, and gradually increase in magnitude; and they evidently do not adhere to the surface of the planet, inasmuch as a vacant space between them is distinctly perceived even in ordinary telescopes.

These curious appearances were shown by Huygens to be occasioned by an opaque, thin, circular ring, surrounding the equator of Saturn, and at a considerable distance from the planet. Saturn moving in the plane of his orbit carries the ring along with him, which, presenting itself to the earth under different inclinations, occasions all the phenomena which have been described. The ring being only luminous in consequence of its reflecting the solar light, it is evident that it can be visible only when the sun and the earth are both on the same side of it; if they are on opposite sides it will be invisible. It will likewise be invisible in two other cases, namely, 1st, when its plane produced passes through the centre of the earth, for then none of the light reflected from it can reach us; and, 2d, when its plane passes through the sun, because its edge is then only enlightened; and being very thin, the whole quantity of reflected light will scarcely be sufficient to render it visible. It is, however, evident that in these two cases the effect will be modified in some degree by the power of the telescope. In ordinary telescopes the ring disappears sometimes before its plane comes into either of the situations mentioned; but Herschel never lost sight of it, either when its plane passed through the earth or the sun. In the last case the edge of the ring appeared as a luminous line on the round disk of the planet, measuring scarcely a second in breadth; but at the distance of Saturn a second corresponds to 4000 miles, which is equal to the semidiameter of the terrestrial globe. The reason of the ring's disappearance will be easily understood by referring to fig. 102, where the circle \(a b c d\) represents the orbit of the earth, \(A B C D\) that of Saturn \(9 \frac{1}{2}\) times more distant from the sun. When Saturn is at \(A\), the earth and sun are both in the plane of the ring; its edge is consequently turned towards us, and it will be invisible unless telescopes of very high power are used. As Saturn advances from \(A\) to \(B\) the ring gradually opens, and it attains its greatest breadth at \(C\), where its face is turned more directly towards us, or a straight line perpendicular to its plane makes a more acute angle with the visual ray than in any other situation. As the planet advances towards \(D\), the plane of the ring becomes more oblique to the visual ray; the breadth of the ring consequently contracts, and it again disappears at \(E\). From \(E\) to \(F\), \(G\), and \(A\), the same phenomena will be repeated, only in this case it is the southern side of the ring which is visible to the earth, whereas, while Saturn was in the other half of its orbit, it was the northern side.

The successive disappearances of the ring form a period of about 15 years, with some variations arising from the different positions of the earth in its orbit. Before its disappearance in 1833, its south side was presented to us; its northern surface became visible in 1838; it again disappeared in 1848, and it will show its southern side in 1855.

Sir W. Herschel's observations have added greatly to our knowledge of Saturn's ring. According to him, the ring is separated into two annular portions by a dark belt or zone, which he has constantly found on the north side. As this dark belt is subject to no change whatever, it is probably owing to some permanent construction of the surface of the ring; and it is evidently contained between two concentric circles, for all the phenomena correspond with the projection of a circular zone. The matter of the ring Herschel thinks no less solid than that of Saturn, and it is observed to cast a strong shadow upon the planet. The light of the ring is also generally brighter than that of the planet; for it appears sufficiently luminous when the telescope affords scarcely light enough for Saturn. It is remarkable that the outer ring is much less brilliant than the inner. Herschel concludes that the edge of the ring is not flat, but spherical or spheroidal. The dimensions of the ring, or of the two rings with the space between them, he gives as follows:— Inner diameter of smaller ring ........................................... 146345 Outside diameter of ditto .................................................. 184393 Inner diameter of larger ring .............................................. 190248 Outside diameter of ditto .................................................. 204883 Breadth of the inner ring .................................................... 20000 Breadth of the outer ring .................................................... 7200 Breadth of the vacant space, or dark zone ......................... 2839

The following measures were taken by Professor Struve, at Dorpat in Russia, in 1828, with a repeating wire-micrometer attached to the large refracting telescope of Fraunhofer, belonging to the observatory at that place, and may be regarded as decidedly the most accurate of any that we possess. (See Memoirs of the Astronomical Society, vol. iii. p. 301.)

Outer diameter of the outer ring ........................................ 40°095 Inner diameter of the outer ring .......................................... 35°289 Outer diameter of the inner ring ......................................... 34°475 Inner diameter of the inner ring ......................................... 26°608 Breadth of the outer ring .................................................... 2°403 Breadth of the division between the rings ......................... 0°408 Breadth of the inner ring .................................................... 3°903 Distance of the ring from the ball ....................................... 4°839 Equatorial diameter of Saturn ............................................. 17°991

These dimensions are adapted to the mean distance of Saturn, 9-53877. According to the same excellent astronomer, the inclination of the plane of the ring to the ecliptic is 28° 5′9″.

In observing the ring with very powerful telescopes, some astronomers have remarked, not one only, but several dark concentric lines on its surface, which divide it into as many distinct circumferences. In common telescopes these are not perceptible; for the irradiation, by enlarging the space occupied by each ring, causes the intervals between them to disappear, and the whole seems blended together in one belt of uniform appearance. (See fig. 103.) Struve, however, noticed no trace of the division of the ring into many parts.

By means of some spots observed on the surface of the ring, Herschel found that it revolves in its own plane in 10 hours 32 minutes 15′4 seconds; and Laplace arrived at the same result from theory. It is particularly worthy of remark, that this is the period in which a satellite, having for its orbit the mean circumference of the ring, would complete its revolution according to the third law of Kepler. This circumstance furnishes a physical explanation of the reason why the ring is able to maintain itself about the planet without touching it; or at least brings the fact within the general law by which the planets are sustained in their orbits. The centrifugal force resulting from its rotation, and the attraction of the planet, suffice to maintain its equilibrium.

From observations of some obscure belts, and a very conspicuous spot on the surface of Saturn, Herschel concluded that his rotation is performed in 10 hours 16 minutes, on an axis perpendicular to the belts and to the plane of the ring; so that the planes of the planet's equator and ring coincide. According to the same astronomer, the ratio of the equatorial and polar diameters of Saturn is 2281 to 2061, or nearly 11 to 10. He also believed that the globe of Saturn appeared to be flattened at the equator as well as at the poles. The compression he thought to extend to a great distance over the surface of the planet, and the greatest diameter to be that of the parallel of 43° of latitude, where, consequently, the curvature of the meridians is also the greatest. The disk of Saturn, therefore, resembled a square of which the four corners have been rounded off. According to the more recent measures of Bessel and Main, this idea is proved to be erroneous, and the shape of the planet is that of an exact spheroid of revolution. (See Theoretical Astronomy Supplement to Part II.)

Saturn is attended by eight satellites, but so small that they can only be seen by the help of powerful telescopes. Huygens first discovered one of these satellites in 1655, the sixth in the order of distance, and the largest. Four others were discovered about twenty years afterwards by Dominic Cassini; and Sir W. Herschel, in 1789, discovered two new satellites, at a time when the ring was visible only in a telescope of forty feet. The orbits of these are interior to those of the five satellites formerly discovered, but exterior to the ring, though so near to it that it is only when the ring disappears that they can be seen. Lastly, an eighth satellite was discovered almost simultaneously by two observers in 1848. All the satellites appear to revolve in the plane of the ring, with the exception of the two nearest.

Uranus.

Uranus is the remotest planet belonging to the system, and is scarcely visible excepting through the telescope. His distance from the sun is nineteen times the mean radius of the orbit of the earth, or about 1800 millions of miles; and his sidereal revolution is performed in rather more than 84 years. His diameter is about 35,112 English miles, or nearly six and a half times that of the earth, and, seen from the earth, subtends an angle of only 4″, even at the time of his opposition. The apparent diameter of the sun, seen from this planet, is 1° 40″; consequently the surface of the sun will there appear 400 times less than it does to us, and the light and heat which is received will be less in the same proportion. Analogy leads us to infer that Uranus is opaque and revolves on his axis, but of this there is no direct proof. Laplace has concluded from theory, that the time of his diurnal rotation cannot be much less than that of Jupiter and Saturn, and that the inclination of his equator to the ecliptic is very inconsiderable. His density is supposed to exceed somewhat that of the earth. Schroeter thinks that certain variations in the appearances of his disk indicate that great changes are going on in his atmosphere.

This planet was discovered by Sir W. Herschel at Bath on the 13th of March 1781. His attention was attracted to it by the largeness of its disk in the telescope, which exceeded that of stars of the first magnitude, while to the naked eye it was scarcely, if at all, visible. In the course of a few days its proper motion became sensible; consequently it could not be a fixed star. Herschel at first took it for a comet, but it was soon perceived that it described a path which, instead of resembling the eccentric orbits of the comets, was almost circular, like that of the planets. It was then recognized to be one of the principal planets of the solar system; and the observations of the last fifty years have not only confirmed this fact, but afforded data from which the elements of its orbit have been determined with great precision. Herschel gave it the name of the Georgium Sidus, in honour of his royal patron George III. Foreigners for some time generally called it the Herschel, after its discoverer; but the mythological name of Uranus, suggested by the late Professor Bode of Berlin, is now generally adopted. In 1787 Herschel discovered that it was attended by two satellites; he subsequently discovered four others, but they have been never reobserved, and the exact number of the satellites of Uranus is still uncertain. Two additional satellites interior to all the others have been recently discovered. (See Supplement to Part II.)

Before concluding this section, it will be proper to take notice of the following curious relation of the numbers Theoretical which express, approximately at least, the distances of the planets from the sun. It was first pointed out by Bode, and, though purely empirical, and not even very accurate, served to confirm the German astronomers in their anticipations of the discovery of a new planet between the orbits of Mars and Jupiter.

Let the number 10 be assumed to represent the semidiameter of the earth's orbit; then the semidiameters of the orbits of the other planets may be expressed in round numbers as follows:

- Mercury...4 - Venus....7 = 4 + 3^2 - Earth....10 = 4 + 3^2 - Mars....16 = 4 + 3^2 - Ceres....28 = 4 + 3^2 - Jupiter....52 = 4 + 3^2 - Saturn....100 = 4 + 3^2 - Uranus....196 = 4 + 3^2

It will be remarked that every succeeding term of this series of numbers, after the second, is the double of the preceding, minus 4; the general term being $4 + 3^{n-2}$, commencing with Venus, and $n$ indicating the rank of the planet.

A view of the proportional magnitudes of the orbits and disks of the planets, as also of the comparative magnitudes of the sun seen from each planet, is given in fig. 104.

On the subjects contained in this section the following works may be consulted: Galileo, *Nuntius Sidereus*; Simon Marius, *Mundus Jovialis anno 1609 detectus*, &c., 1614; Cassini, *Martis circa proprium axem revolvi* Observationalis Bononienses, Bonon, 1665; Idem, *Disseptatio Apologetica de Maculis Jovis et Martis*, Bonon, 1667; Idem, Nouvelles Découvertes dans le Globe de Jupiter, Paris, 1690; Bianchini, *Hesperi et Phosphori Nova Phaenomena*, 1728; Cassini, *Elementa Astronomiae*, 1740; Schroeter, *Aphroditeographische Fragmente*, Helmstadt, 1796; Idem, *Lithographische Beobachtungen der neu entdeckten planeten*, Göttingen, 1805; Gauss, *Theoria Motus Corporum Celestium*, Hamburgi, 1801; Idem, *Journal of Gotha*, 1811; Laplace, *Mécanique Céleste*, tome iv. p. 135; Idem, *Système du Monde*; Delambre, *Astronomie*, tome ii. chap. xxvii.; Schubert, *Traité d'Astronomie Théorique*, tome ii. Petersburg, 1822; and the numerous papers of Sir W. Herschel in the Philosophical Transactions.

**Sect. IV.—Of the Orbits of the Satellites.**

In order to establish a theory of the motions of the satellites, the first thing necessary is to ascertain the directions in which they move with reference to the primary planet. Now, it is observed that their motion is sometimes towards the east, and at other times towards the west; but that the satellites are never occulted except in passing from the west to the east of the planet. When an occultation occurs the satellite is always moving eastward; on the other hand, the satellite is always moving westward when it appears on the planet's disk. From this it results that the true motions of the satellites around the planets are from west to east, according to the order of the signs, or in the same directions as the motions of the planets about the sun. This fact, which holds true of the moon, and the satellites of Jupiter and Saturn, is one of the most remarkable in the planetary system. With regard to the satellites of Uranus, their motions are performed in orbits almost perpendicular to the ecliptic; they cannot therefore with propriety be said to be either direct or retrograde.

The eclipses of the satellites of Jupiter present an easy method of determining their mean motions and periodic times; and, by reason of the small inclinations of the orbits, these are of very frequent occurrence; for the first three satellites traverse the shadow of Jupiter in every revolution, and the fourth only passes by it sometimes, in consequence of the greater inclination of its orbit. If the instants can be observed at which a satellite enters and emerges from the shadow, the middle point of time between these two instants will be that of the heliocentric conjunction of the satellite with its primary. The interval between two central eclipses gives the synodic period of the satellite; whence, since the motion of the primary is known, the sidereal period of the satellite, and its mean angular motion with regard to the straight line joining the centres of the sun and the planet, are easily deduced. Instead of two successive eclipses, it is preferable to compare two that are separated from each other by a long interval of time; the interval divided by the number of sidereal revolutions will give the mean time, unaffected by any periodic inequalities which may exist in consequence of the mutual action of the satellites on one another. In order to render the result as accurate as possible, those eclipses are chosen which take place when the planet is nearly in opposition.

The distances of the satellites from their primary are ascertained by measurement with the micrometer, at the time of the greatest elongations. On comparing the distances with the times of revolution, the beautiful law of Kepler is found to prevail; and, as in the system of the planets, so in the various systems of the satellites, the squares of the periodic times vary as the cubes of the mean distances from the central body. The distances of the satellites of Jupiter and Saturn, compared with the diameters of their respective primaries, are represented in fig. 105.

The inclinations of the orbits of the satellites of Jupiter, and the positions of their nodes, together with the other elements of their elliptic motion, are determined by means of their eclipses. The plane of the orbit of the first satellite nearly coincides with the plane of the equator of Jupiter, the inclination of which to the plane of his orbit is $3^\circ 5' 30''$. The inclination of the orbit of the second to the plane of the planet's equator is $27^\circ 49' 2''$. Its nodes have a retrograde motion on that plane, and go through an entire circuit in the space of 30 years. The orbit of the third is inclined to the equator of Jupiter in an angle of $12^\circ 20''$; and the line of the nodes retrogrades through a whole circumference in 142 tropical years. Hence the inclinations of the orbits of these two satellites to the orbit of Jupiter are variable; that of the second varying between $3^\circ 19' 24''$ and $2^\circ 51' 35''$, and that of the third between $3^\circ 17' 50''$ and $2^\circ 58' 10''$. The inclination of the orbit of the fourth satellite to the orbit of Jupiter is also variable. Its nodes have a retrograde motion; and complete a revolution in 531 years. Since the middle of the last century the inclination of this satellite has been observed to increase, and the motion of its nodes to diminish. (Laplace, *Mécanique Céleste*, tome iv. livre viii.)

The orbits of the four satellites are doubtless elliptical, but those of the first and second are so small that it has been found impossible to determine their eccentricity. The eccentricity of the third is perceptible, that of the fourth much more so. According to Laplace, the greatest equation of the centre of the third, at its maximum in 1682, amounted to $13' 16''$, and at its minimum in 1777, only to $5' 7''$. The eccentricity of the orbit of the fourth is still greater, and also subject to considerable variations. The line of the apsides has a direct motion, amounting to $42' 55''$ annually. The great in- Theoretical fluence of the compressed figure of Jupiter on these elements gives each of the orbits an eccentricity peculiar to it; but each is also affected by the eccentricities of the others. The mutual perturbations of the satellites greatly affect their motions, and render their analytical theory exceedingly complicated and difficult.

On comparing the mean longitudes with the mean motions of the first three satellites of Jupiter, Laplace discovered the two following relations, which, by reason of their remarkable simplicity, may perhaps be regarded as among the most curious discoveries ever made in astronomy. Denoting by \( m_1, m_2, m_3 \), the mean motions of the three satellites respectively, and their mean longitudes by \( l_1, l_2, l_3 \), these laws are expressed by the formulae,

\[ m_1 + 2m_2 - 3m_3 = 0 \\ l_1 + 2l_2 - 3l_3 = 180^\circ; \]

that is to say, the mean sidereal motion of the first, added to twice that of the third, is equal to three times that of the second; and the mean longitude of the first satellite, plus twice that of the third, minus three times that of the second, is always equal to a semicircumference. The first of these relations is true of the synodical as well as of the sidereal revolutions; and it follows from the second, that the first three satellites can never be eclipsed at the same time, because in that case their longitudes would be equal, or \( l_1 + 2l_2 - 3l_3 = 0 \). These results of theory agree so nearly with observation, that we are tempted to regard them as rigorously exact, and to ascribe the slight differences that may be perceptible to the unavoidable errors of observation, and to the periodic inequalities in the motions of the satellites, by reason of which their true motions are alternately greater and smaller than their mean motions. We must therefore infer that these relations depend on a physical cause, by which they will be preserved for ever, or at least during a long series of ages, notwithstanding the small oscillations to which, from various sources of perturbation, the mean longitudes of the satellites are subject.

From observations of the eclipses of Jupiter's satellites, Roemer was led to the very important discovery of the successive propagation of light. The times at which these eclipses happen are found to differ from the times computed from the sidereal revolutions of the satellites, being sometimes earlier and sometimes later, according to the position of Jupiter relatively to the sun and the earth. When Jupiter is in opposition with the sun, and his distance from the earth consequently less than his distance from the sun by the whole radius of the earth's orbit, the satellites are eclipsed sooner than they ought to be according to computation. On the contrary, when Jupiter is in conjunction, and his distance from the earth greater than his distance from the sun by the same quantity, the eclipses happen later. These differences, which are exactly the same for all the satellites, cannot be ascribed either to the eccentricity of the orbit of Jupiter, or to inequalities in their motion; for the oppositions and conjunctions of the planet correspond successively to all the different points of his orbit, and the eclipses also happen when the satellite is at different points of its own orbit. The simplest and most natural way of explaining the phenomenon is to suppose that the light reflected from the satellites is not transmitted to the earth instantaneously, but occupies a sensible portion of time in traversing the diameter of the terrestrial orbit. When Jupiter is near his conjunction, the eclipses are observed to happen about 16 minutes 26 seconds later than when he is near his opposition; the difference between his distances from the earth in these two positions is equal to the diameter of the earth's orbit, supposing the orbits to be circular: it follows, therefore, that light employs 16 minutes 26 seconds in traversing the terrestrial orbit, and consequently the half of that time, or 8 minutes 13 seconds, in coming from the sun to the earth. The exact agreement of this hypothesis with observation renders its truth unquestionable. The fact of the successive transmission of light led to another discovery of the utmost importance in astronomy, namely, the aberration.

The eclipses of Jupiter's satellites are useful in determining the longitude of places on the earth, and on this account the theory of the motions of these bodies has been cultivated with the most laborious care. The epochs at which the eclipses take place are calculated in advance, and inserted in the Ephemerides. On comparing these epochs, computed for a given meridian, with the immediate results of observation made in another place at a given hour, the difference of time is obtained, whence the difference of longitude is immediately deduced. The method is the same as for the eclipses of the moon. Unfortunately, by reason of the magnifying power required to render the satellites visible, it cannot be employed at sea, the instability of the vessel rendering the telescopes unserviceable. The tables of Delambre, which were computed from the theory of Laplace, and the comparison of an immense number of observations, give the places of the satellites with all the precision which it is perhaps possible to obtain.

The satellites of Saturn have not the practical utility of those of Jupiter, because, by reason of their great distance, their eclipses are invisible; and indeed some of them cannot be perceived at all, excepting through telescopes of extraordinary power. Their periods, mean motions, and some of the other elements of their orbits, are determined by the micrometrical measurement of their greatest digressions from their primary. At the time of their greatest digressions they are always situated in the same straight line with the greater axis of the ring, and their distances from Saturn are then equal to the semi-transverse axes of their apparent orbits. In their conjunctions the minor axes of their apparent ellipses seem only half as great as their transverse axes, whence it is inferred that the sine of their inclination is one half, and consequently that the inclination itself is about 30°, which is nearly the inclination of the ring. Hence the satellites seem to move in the plane of the ring. The only one which deviates considerably from that plane is the seventh. From certain observations made by Bernard in 1787, Launde makes the inclination of this satellite 22° 42' to the orbit of Saturn, or 24° 45' to the ecliptic. There exists, however, considerable uncertainty with regard to the inclination of the orbits of the satellites, as well as with regard to that of the ring.

According to the theory of Laplace, the spheroidal figure of Saturn, of which the compression is very considerable, must maintain the ring, and the orbits of the inferior satellites, in the plane of his equator. All that is certain is, that the inclination of the orbits of the first five satellites to the equator of the planet is insensible at the distance of the earth; consequently all the satellites, excepting the seventh, and probably also the sixth, will appear, as well as the ring, to move in the plane of the equator. The orbit of the seventh satellite preserves the same mean inclination to the plane of the equator of the planet; and the line of its nodes has a retrograde motion nearly uniform.

The orbit of the sixth satellite is elliptical. For the meridian of Paris its longitude in 1800 was 67° 25' 47"; that of its inferior apside 203° 35' 7". Its mean motion in 36525 days is 2290 revolutions + 202° 12'; in one day Theoretical 22° 34' 37"186. The eccentricity is 0.04887; and its greatest equation 5° 36' 8". By particular observations on this satellite, Bessel found its inclination to the ecliptic to be 24° 30', or 25° 53'; differing very sensibly from that of the ring or the equator of the planet. It is, however, not improbable that these two planes may have some inclination to each other. (Delambre, Astronomie Théorique et Pratique, tome iii. p. 510.)

The satellites of Uranus can be perceived with still greater difficulty than those of Saturn. The orbits of these satellites are almost perpendicular to the ecliptic. The elements of the second and fourth have been determined by actual measurement; the periods of the two others have been theoretically deduced from the third law of Kepler. The inclination of the fourth is 89° 30', or 90° 30', and the ascending node 171° or 249°, according as it is conceived to be direct or retrograde.

The following table exhibits the mean distances and sidereal revolutions of the satellites of Jupiter, Saturn, and Uranus (all, except the 2d and 4th, very doubtful).

| MEAN DISTANCES. | SIDEREAL REVOLUTIONS. | |-----------------|-----------------------| | (The radius of the planet being = 1.) | According to Laplace. | According to Delambre. | | Jupiter. | Days. | d. h. m. s. | | 1st satellite | 5-61296 | 17691378 | 1 18 28 35 94537 | | 2d | 9-24568 | 355511810 | 3 13 17 53 73010 | | 3d | 14-75240 | 71545528 | 7 3 59 35 82511 | | 4th | 25-94686 | 166887697 | 16 18 5 7 02098 | | Saturn. | Days. | d. h. m. s. | | 1st satellite | 3-080 | 094271 | 0 22 37 32 9 | | 2d | 3-952 | 137024 | 1 8 53 8 9 | | 3d | 4-893 | 188780 | 1 21 18 26 2 | | 4th | 6-268 | 273948 | 2 17 44 51 2 | | 5th | 8-754 | 451749 | 4 12 25 11 1 | | 6th | 20-295 | 1594530 | 15 22 41 13 1 | | 7th | 59-154 | 7932960 | 79 7 53 42 8 | | Uranus. | Days. | d. h. m. s. | | 1st satellite | 13-120 | 589928 | 5 21 25 0 | | 2d | 17-022 | 87068 | 8 17 11 9 | | 3d | 19-845 | 109611 | 10 23 4 | | 4th | 22-752 | 134559 | 11 11 5 15 | | 5th | 45-507 | 380750 | 38 1 49 | | 6th | 91-008 | 1076944 | 107 16 40 |

CHAP. V.

OF COMETS.

The comets form a class of bodies belonging to the solar system, distinguished from the planets by their physical appearances and the great eccentricities of their orbits. By reason of the smallness of their diameters, and a nebulosity which renders them ill adapted to reflect the rays of light, the greater part of them are only visible in the telescope, and continue to be so only during a short period of time; for as they advance to and recede from the sun almost in straight lines, and with prodigious velocities, they are soon carried far beyond the limits of vision. They have received the name of comets (coma, hair) from the bearded appearance which they frequently exhibit.

Sect. I.—Of the Orbits of Comets.

The comets are not more remarkably contrasted with the planets in the singularity of their physical appearances, than in the directions of the paths which they follow in space. While the orbits of all the planets are confined within a narrow zone, or to planes not greatly inclined to the ecliptic, those of the comets are inclined in all possible angles, and some of them are even observed to be perpendicular to the ecliptic. Nor is the contrast less striking with regard to the figures of the orbits, which, instead of being nearly circular, like those of the planets, have the appearance of being almost rectilinear. Kepler was of opinion that the cometary orbits are straight lines; Cassini supposed them to be very eccentric circles; and Tycho was for some time of the same opinion, but he afterwards found that the hypothesis of the eccentric circle would not satisfy the observations of the comet of 1577. Hevelius seems to have been the first who discovered, by means of a geometrical construction, that the orbits might be represented by parabolas; and Dörfler first calculated their elements on this hypothesis. After it was known, however, that certain comets return to the sun in the same orbits, it became necessary to adopt an opinion, already probable from analogy, that, in conformity with the laws of Kepler, the cometary orbits are ellipses having the sun in one of the foci. This hypothesis is now universally admitted; but as the ellipses are in general extremely elongated, and the comets are only visible while they describe a small portion of their orbits on either side of their perihelion, their paths during the time of their appearance differ very little from parabolas; whence it is usual, on account of the facility of computation, to assume that they really move in parabolic curves. Newton employed the hypothesis of an elliptic motion to compute the orbit of the famous comet of 1680. Since that time the orbits of more than a hundred and fifty different comets have been computed on the elliptic hypothesis, and their elements determined so as to satisfy all the observations. It is possible that the orbits of some comets may be in reality parabolic; but in this case the comet, after having passed its perihelion, would recede to an infinite distance from the sun, and never again visit our system. Burckhardt imagined that the observations of the comet of 1771 were best represented by supposing the orbit to be an hyperbola. In fact, it is demonstrated in the Principia, and every treatise on Physical Astronomy, that the species of curve which one body describes about another, in virtue of an attractive force varying inversely as the square of the distance, depends only on its velocity of projection. The curve must necessarily be a conic section; but it may be an ellipse, a parabola, or an hyperbola, according as the primitive impulsive force falls within or exceeds certain assignable limits.

If the comets moved in parabolas or hyperbolae, and had consequently only a temporary connection with the solar system, the determination of their orbits would be a matter of mere curiosity, and of no consequence whatever to astronomy. But it is only by the accurate determination of the elements of their orbits that it can be discovered whether those bodies ever revisit the system; for the appearances which they exhibit, depending on the situation of the earth in its orbit with relation to them at the time they are visible, are far too variable and uncertain to afford any sure means of recognising them. The comet of 1811 was scarcely visible in the months of April and May; it was subsequently lost in the sun's rays, and, having passed its perihelion, reappeared in August with a splendour and magnificence that rendered it an object of admiration. It is therefore only by observing that a comet follows the same orbit in its successive returns to the sun that we can be assured of its identity. But even the determination of the orbit is not always sufficient to lead to the detection of a comet in a Theoretical subsequent revolution; for if, in the course of the previous one, it came within the sphere of attraction of Jupiter, or any of the larger planets, the elements of its orbit may have been greatly or entirely changed. The orbit of the comet of 1770 was calculated by Lexell, and subsequently by Burckhardt, and both these astronomers found that the observations could only be represented by an ellipse in which the time of revolution was five years and a half; yet the comet has never been seen since, or at least seen moving in the same orbit. Hence it is concluded with certainty that the attraction of Jupiter, near which planet it approached, was so great as to compel it to move in a totally different ellipse. Other causes also conspire to render the chances of the discovery of periodic comets extremely small. In the first place, it is only within a very small portion of their orbits that they are visible; and this, on account of their proximity to the sun, is passed over with inconceivable rapidity. But in proportion as they recede from their perihelia the solar action diminishes; and, towards the aphelion of its orbit, a comet may be almost motionless, and for this reason not return for thousands of years. In the second place, it may happen, that during the greater part of the time the comet continues in the visible portion of its orbit, it may rise above the horizon only during the day, in which case it will be invisible, and may consequently pass through our system without being observed. The comet of 1818 was present in all its splendour long before it became visible, but in full day. Seneca relates a very curious instance of one having been seen during a total eclipse of the sun, in the year 60 before our era. A third cause of uncertainty consists in the difficulty of observing their true places with sufficient precision to enable the elements of their orbits to be exactly determined. The small comets are only nebulous points, which can be distinguished with difficulty; the larger ones are surrounded with a variable, ill-defined, and indistinct nebulosity. The comet of 1729 continued visible during six months; its orbit was computed by three different astronomers, whose results were far from coinciding. The same uncertainty exists with regard to the orbits of several other comets. When, therefore, all these circumstances are taken into consideration, it will not appear surprising that, although the elements of above 150 comets have been computed, there are only three which are certainly known to have been observed in their successive revolutions.

The most remarkable periodic comet with which we are acquainted was made known to astronomers by Dr Halley. That active and indefatigable genius, having perceived that in 1682 the elements of its orbit were nearly the same as those of two comets which had respectively appeared in 1531 and 1607, concluded that the three orbits belonged to the same identical comet, of which the periodic time was about 76 years. After a vague estimate of the perturbations it must sustain from the attraction of the planets, Dr Halley predicted its return for 1757,—a bold prediction at that time, but justified by the event, for the comet again made its appearance as was expected, though it did not pass through its perihelion till the month of March 1759, the attraction of Jupiter and Saturn having caused, as was computed by Clairaut previously to its return, a retardation of 618 days. This comet had been observed in 1066; and the accounts which have been preserved represent it as having then appeared to be four times the size of Venus, and to have shone with a light equal to a fourth of that of the moon. History is silent respecting it from that time till the year 1456, when it passed very near to the earth: its tail then extended over 60° of the heavens, and had the form of a sabre. An object so striking and so terrific could not fail, in a superstition age, to excite universal dismay, and be regarded as portentous of the greatest calamities to the human race, if not of the destruction of the world. Accordingly Pope Calixtus ordered public prayers to be said over Christendom, in which he exorcised the comet, and the Turks, who had at that time made themselves masters of Constantinople, and overthrown the eastern empire. Dr Halley's comet returned to its perihelion in 1835; and the splendour of its appearance rendered it once more an object of universal interest; it was well observed in almost every observatory. The following table of its elements in 1835 is given by Poisson (Théorie Analytique du Système du Monde, tome ii. p. 147):

| Instant of the passage through the perihelion | October 1835 | |-----------------------------------------------|-------------| | Semiaxis major | 3142 | | Ratio of the eccentricity to the semiaxis major | 0.967453 | | Place of the perihelion on the orbit | 304° 34' 19"| | Longitude of the ascending node | 55° 6' 59" | | Inclination | 17° 46' 50" |

Two other famous comets whose periodic returns have been verified by observation have received the names of Encke and Biela, the astronomers who first computed their orbits, or recognised them as having been observed in their previous revolutions. The first returns to its perihelion in 1208 days, and the second in 2440 days. Encke's comet, although its identity was not discovered till 1818, has been frequently observed, as in 1789, 1795, 1801, and 1805, and on these occasions it exhibited very different appearances, having been seen with and without a nucleus, with and without a tail,—circumstances which account for its having so long escaped being recognised as a regular attendant on the sun. In its returns to its perihelion in 1808, 1812, and 1815, it escaped detection; but it reappeared in 1818, and it was from the observations of this year that Encke computed the elliptic elements of its orbit. On its next return, in 1822, it was invisible in Europe; but it was observed at Paramatta, in New South Wales, during the whole month of June, and the time of its perihelion passage was found to differ only by about three hours from that previously computed by Encke. On its returns in 1825 and 1828 its observed and computed places agreed equally well. The following, according to Poisson, are its elements for 1829–30, computed from the observations at Paramatta:

| Passage through the perihelion, 1829, January 10°4573 | |------------------------------------------------------| | Mean diurnal motion | 1069° 5570 | | Eccentricity | 0.8446852 | | Place of perihelion | 157° 18' 35"| | Longitude of ascending node | 334° 24' 15"| | Inclination | 13° 22' 34"|

Encke's comet presents in some respects a considerable analogy with the planet Ceres, the inclination and greater axis of its orbit being the same, while its sidereal revolution is only 46 days shorter than that of Vesta. The orbit is, however, greatly more elongated, for its perihelion falls within the orbit of Mercury, and its aphelion is situated between Jupiter and the new planets. The perturbations it sustains are chiefly occasioned by the attraction of Jupiter, that of the earth and Venus being extremely small, while the action of Mercury is insensible.

A third comet of short period with which our knowledge of the solar system has been enriched receives its name from Biela, by whom it was first perceived in Bohemia, on the 27th of February 1826. The parabolic elements computed from the first observations presented a striking resemblance with those of two comets which had been observed in 1772. Theoretical and 1806, which induced MM. Clausen and Gambart, the astronomers at Marseilles, and the second at Altona, to compute the elements of the three comets on the hypothesis of elliptic orbits; and, after some attempts, each found an ellipse which represented all the observations so accurately as to leave no doubt of the identity of the comet. Its period is six years and about nine months, and it returned to its perihelion in November 1832, about the same time with Encke's. The following table of its elements was computed from the observations of 1826, and the theory of the perturbations (Pontécoulant, tome ii. p. 168):

| Passage through the perihelion 1832, November | 2744808 | | Eccentricity | 0.7517481 | | Place of the perihelion | 109° 56' 45" | | Longitude of ascending node | 248° 12' 24" | | Inclination | 13° 13' 13" | | Semiaxis major | 3.363683 |

Besides the periodical comets which have been mentioned, there are two others of long period, of which the orbits are supposed to be known, though their returns to their perihelia have not yet been verified. The first is that which appeared in 1680, and of which Newton computed the period to be 575 years. It may therefore be identical with those which are recorded in history to have appeared in 1106, 531, 34 B.C., and 619 B.C. The second is that which appeared in 1556, and is supposed to have made a former visit in 1264. This comet is expected to appear again in 1858.

The following table, taken from one given by Delambre in the third volume of his Astronomie, shows the comparatively small distances within which the greater part of the comets hitherto observed approach to the sun, and the apparently fortuitous inclinations of the planes of their orbits to that of the ecliptic. Supposing the sun's distance from the earth to be unity, then, of 120 comets, there are

- 5 whose perihelion distance is less than...0·1 - 3 between...0·1 and 0·2 - 6 between...0·2 and 0·3 - 11 between...0·3 and 0·4 - 10 between...0·4 and 0·5 - 22 between...0·5 and 0·6 - 12 between...0·6 and 0·7 - 11 between...0·7 and 0·8 - 8 between...0·8 and 0·9 - 9 between...0·9 and 1 - 21 between...1 and 2 - 1 equal to 2·293 - 1...4·069.

Of the same number there are

- 4 whose inclination is between...1° and 5° - 3 between...5° and 10° - 4 between...10° and 15° - 3 between...15° and 20° - 2 between...20° and 25° - 2 between...25° and 30° - 7 between...30° and 35° - 2 between...35° and 40° - 4 between...40° and 45° - 1 between...45° and 50° - 3 between...50° and 55° - 4 between...55° and 60° - 7 between...60° and 65° - 3 between...65° and 70° - 8 between...70° and 75° - 3 between...75° and 80° - 4 between...80° and 85° - 2 between...85° and 90°.

The motions of the above 61 are direct; those of the remaining 59 are retrograde, and their orbits are distributed over the whole quadrant in the same random manner. This circumstance sufficiently indicates that the mechanical causes, whatever they were, which gave the same direction to the two motions of translation and rotation of all the planets and satellites, exercised no influence on the comets. Hence many astronomers have entertained the idea that these bodies have only a casual or transient connection with the planetary system.

Sect. II.—Of the Appearances and Physical Constitution of Comets.

Of all the celestial phenomena, those of the comets are the most striking, and the most calculated to impress the ignorant with the idea of supernatural agency. Appearing suddenly in the heavens, and under aspects the most uncommon and terrific, they have been almost universally regarded as visible demonstrations of the wrath, and harbingers of the vengeance, of offended deities. These superstitious terrors, arising from that vain propensity of the mind of man to regard the universe as created for himself alone, have only been dissipated by the progress of sound philosophy, and a more extended acquaintance with the riches of nature, and the endless variety of her productions.

The appearances exhibited by the comets are exceedingly diversified, and sometimes extremely remarkable. That which appeared in the year 134 B.C., at the birth of Mithridates, is said to have had a disk equal in magnitude to that of the sun. Ten years before this, one was seen, which, according to Justin, occupied a fourth part of the sky, that is, extended over 45°, and surpassed the sun in splendour. Another, equally remarkable, appeared in the year 117 of our era; and in 479 there was one of which the disk, according to Ferret, was of such magnitude that it might have occasioned the extraordinary eclipse of the sun which took place about that time. In 400 one was observed, which is said, on the authority of Gaimas, to have resembled a sword, and to have extended from the zenith to the horizon. That of 531 was of greater magnitude still, and its appearance more terrific. Those which appeared in 1066 and 1505 exhibited disks larger than that of the moon. It is, however, highly probable that all these accounts have been greatly exaggerated, through the ignorance and credulity of the historians by whom they are related; for, since comets have been observed by astronomers, no instances have been occurred in which their magnitudes and appearances have been so extraordinary. The most remarkable among those of which we possess accurate accounts appeared in the years 1456, 1618, 1680, 1744, 1759, 1769, 1807, and 1811.

Fig. 106 is a representation of the celebrated comet of Plate 1680, taken from Lemonnier's Histoire Céleste. It exhibits the nucleus or disk with its surrounding atmosphere. Above is a sort of ring, wider at the summit, and narrower towards the sides. A coma or beard succeeds the ring; and lastly, an immense train of luminous matter, somewhat less vivid than the nucleus. This luminous train, or tail as it is called, is by far the most singular and striking feature presented by the comets. That of the comet of 1744 was one of the most remarkable. It was divided into six branches, all diverging, but curved in the same direction; and between the branches the stars were visible. It is represented in fig. 107. The tail of the comet of 1811 was composed of two diverging parts inclined to each other in an angle which varied from 90 to 15 or 20 degrees. These branches were curved in opposite directions, and descended from the nucleus like a veil: between the The nucleus, which is the densest and most luminous part, may be said to form the true body of the comet. It is, however, so far from having the dense and solid appearance of the planets, that some astronomers have imagined it to be diaphanous, and even supposed that they have observed stars through it. But supposing such an observation certain, it may be accounted for with much greater probability by the effects of refraction; and it is besides extremely difficult to distinguish the nucleus from the surrounding nebulosity. If the nucleus were an opaque globular body, it would exhibit phases like Venus or Mars, according to its different positions with relation to the sun and the earth; and such were observed, or at least were supposed to be observed, in the case of the comet of 1682, by Hevelius, Picard, and Lhütre. But the nebulosity renders the phases exceedingly obscure and indistinct, and prevents the true body of the comet from being seen; in the same way as a globe of roughened glass prevents us from distinguishing the form of the flame of an inclosed lamp. The real nucleus has probably never been observed by any astronomer; and, from the appearances, we are led to infer that a comet, at least near its perihelion, is only an agglomerated mass of vapours. As it recedes from the perihelion, the vapours may be condensed by cold into a solid substance. This hypothesis is also favoured by the extreme smallness of the density of the comets, which is known certainly from the circumstance that they produce no appreciable effect on the motions of the planets. The comet of 1770 traversed the system of Jupiter's satellites without causing any sensible perturbation of those small bodies. This comet also passed very near the earth; and Laplace calculated, that if its mass had been equal in density to that of the earth, the effect of its attraction would have increased the length of the sidereal year by two hours and twenty-eight minutes. But since its influence was altogether insensible, it is certain that its mass was not equal to the five thousandth part of that of the earth, and probably much inferior even to this quantity.

If the nuclei of comets are solid, the matter of which they are composed must be extremely fixed in order to enable them to resist the intense heat they necessarily experience in their approaches to the sun. According to the computation of Newton, the great comet of 1680, at its perihelion, was only distant from the sun by the 163rd part of the semidiameter of the earth's orbit, where it would be exposed to a heat above 2000 times greater than that of red-hot iron,—a temperature of which we can form no conception, and which would instantly dissipate any substance with which we are acquainted.

In order to explain the singular phenomena of the train of light which frequently attends the comets, the following theory was proposed by Newton. The comets move in very eccentric orbits, and consequently, towards their perihelia, approach very near to the sun. The excessive degree of heat they sustain near this point of their orbits must convert into vapour every substance capable of vaporization; and hence the prodigious extent of their atmosphere in comparison of the smallness of their nuclei. When this atmosphere has acquired all the volatility of which it is susceptible, the impulsion its vapours receive from the solar rays, however feeble that force may be conceived to be, is sufficient to put them in motion, and drive them off in a direction opposite to the sun. Thus it is remarked, that the tail becomes most conspicuous after the comet has passed the perihelion, and that its direction, as was first observed by Appian, is the straight line joining the centres of the sun and comet. The slight curvature which is generally observed may be accounted for by combining the motion given to the vapours by the impulsion of the sun's rays with the motion of the comet in its orbit; for the detached vapours are driven by the impact of the luminous particles beyond the sphere of the comet's attraction, and consequently cease to follow the direction of the nucleus. Hence the curvature of the tail must be greatest towards its extremity; and this is observed to be actually the case. It may be remarked, however, that although this hypothesis serves to explain the phenomena when all the branches of the tail are bent in the same direction, it is inapplicable when the directions of the curvature are opposite, as was the case with regard to the comet of 1811. In fact, no ultimate reasons for several of the phenomena exhibited by comets have yet been given which can be considered as entirely satisfactory.

The discovery of the periodic returns of certain comets necessarily put an end to the apprehensions and terrors which their unusual appearances were well calculated to excite, and proved them to be permanent bodies belonging to the same system, and acted on by the same laws, as the planets. But this very discovery gave rise to apprehensions of another kind, more natural, though, when closely examined, hardly more reasonable. Since the comets are so numerous, and their orbits traverse the planetary system in all directions, and come within the orbit of the earth, is there not a probability that some of them may come into contact with our globe, and destroy it by the direct collision; or at least approach so near as to produce the most disastrous effects by their attraction? Halley found that the comet of 1680 had approached its perihelion about the time of the universal deluge, and thought it probable that that great catastrophe might have been immediately occasioned by the earth's being enveloped in the aqueous vapours of its tail,—an idea which was afterwards more fully developed by Whiston. Lalande and Maupertuis have minutely detailed the terrible effects which might be produced by the shock of a comet, or even by its approach to our earth. The vapours brought by the tail would mingle with the atmosphere, and render it less respirable. The attraction of the nucleus would destroy the equilibrium of the ocean, and cause extraordinary inundations, which might sweep off the greater part of the human species. The direct shock might change the position of the earth's axis, or even cause the earth to leave its present orbit. It might then become a satellite of the comet, and be carried away with it to the extreme limits of the sun's attractive influence; or, as the mass of the comet would probably be inferior to that of the earth, the earth would carry the comet along with it in its orbit, and thus acquire a second moon; and it has even been surmised that the moon we actually enjoy may owe its origin to an accidental occurrence of this kind. But all these reveries have disappeared before the calculus, by which it is demonstrated that the orbit of the moon can never at any time have been greatly different from what it now is. The collision of a comet with the earth is not an impossible event; though so infinitely little probable, that it can never excite any just cause of alarm. The conjunctions of the planets anciently caused terrors still more unreasonable; and the eclipses, which now scarcely attract the notice of the vulgar, long rivalled the comets in the terrors which they occasioned to the inhabitants of the earth.

Various opinions have been entertained respecting the nature, and formation, and uses of the comets. Newton supposed that, as some of them pass so near to the sun as Theoretical to be involved in all probability within his atmosphere, the resistance they must consequently experience will cause them to approach nearer and nearer to the great luminary at every successive revolution, till at last they are precipitated into his substance; hence their use in the system may possibly be to repair the losses which the sun sustains from the constant emission of light. Whatever destinies they may be appointed to fulfil, the recent discovery of several comets of short periods must be regarded as of great importance to astronomy, inasmuch as the frequency of their appearance will enable observers to take notice of any great changes with which either their masses or orbits may be affected.

With regard to the actual number of comets belonging to the solar system, there are no data from which we can form any probable conjecture. Those only (with two or three exceptions) which come within the orbit of the earth are visible to us, and nearly 200 have been observed since the discovery of the telescope. If, then, we suppose them to be equally distributed throughout the whole system, it would follow that the number of comets coming within the sphere of Uranus, the radius of which is twenty times that of the earth's orbit, amounts to 1,200,000 (the cube of 20 multiplied into 150), assuming the average period of their revolutions to be that in which 150 have come within the sphere of the earth. Such computations, however, are scarcely deserving of notice.

On the subject of comets, the reader may consult Aristotle, Meteorol. lib. i. cap. vi.; Seneca, Quest. Natural. vii.; Hevelii Cometographia; Newton, De Mundo Syste- mate, and Princip. lib. iii. prop. 42; Halley, Synopsis Astronomiae Cometicae, and Phil. Trans. tom. xxiv.; Euler, Mem. Acad. Berlin, 1756; Dionis du Sejour, Essai sur les Comètes, Paris, 1775; Lexell, Phil. Trans., 1779; Clairaut, Mem. Acad. Paris, 1760; Lambert, Lettres Cosmologiques, and Mem. Acad. Berlin, 1771; Bode, ibid., 1786, 1787; Sir H. Englefield on the Orbits of Comets, 4to, London; Pingré, Cométographie, 2 vols. 4to, Paris, 1784; Laplace, Mécanique Céleste, tome iv. p. 193, and Système du Monde, 4to, p. 127; Delambre, Astronomie Théorique et Pratique, tome iii. chap. xxxiii.; also the Connaissance des Temps, Memoirs of the Astronomical Society, and the various scientific Journals. (See also Supplement.)

CHAP. VI.

OF THE FIXED STARS.

After having treated of the different classes of bodies which compose the solar system, it only remains for us to inquire what observation has been able to discover respecting that innumerable host which "studs the galaxy,"—that multitude of brilliant points which, on account of their always sensibly retaining the same relative positions, have received the name of Fixed Stars.

Sect. I.—Of the Arrangement of the Fixed Stars.

The great multitude of stars visible even to the naked eye renders it impossible to distinguish each by a particular name: astronomers have accordingly, for the sake of reference, formed them into groups, to which they give the name of Constellations or Asterisms. To the different constellations the early astronomers gave the names of men, and animals, and other familiar objects, from some fancied resemblances or analogies, which, for the most part, are not easily traced. These denominations, consecrated by ancient usage, are preserved in modern catalogues; and the practice of delineating the object itself on celestial globes and charts has been only recently, if it has yet altogether, been abandoned. The ancients likewise distinguished some of the brightest stars in the different constellations by particular names; but when it was wished to include others less conspicuous, it became necessary to have recourse to a different mode of proceeding. According to the usual method, first introduced by Bayer in his Uranometria, each of the stars in every constellation is marked by a letter of the Greek alphabet, commencing with the most brilliant, which is designated by α, the next most conspicuous is called β, the third γ, and so on. When the Greek letters are exhausted, recourse is had to the Roman or Italic; but even with the help of these the nomenclature cannot be extended far, and the simplest and most comprehensive method is undoubtedly to employ the ordinal numbers to particularize the stars belonging to each constellation. It requires to be remarked, that the order of the letters indicates only the relative brilliancy of the stars in the same constellation, without any reference to those in other parts of the heavens. Thus α Aquarii is a star of the same order of brightness as γ Virginis. The stars of the first order of brightness are likewise denominated stars of the first magnitude; those of a degree inferior in brightness are said to be of the second magnitude; and so on with the third, fourth, &c. Below the sixth the same denominations are continued; but the stars of the seventh and inferior magnitudes are no longer visible to the naked eye, and are therefore called telescopic stars. It is obvious that, in conferring these denominations, it has been assumed that the brilliancy of the stars is proportional to their magnitudes,—an hypothesis at least extremely doubtful. The terms are however only used for the sake of distinction, and no exact ideas can be attached to the numbers. Observers are even at variance on the subject, certain stars being regarded by some of them as being of the first, which are considered by others as being only of the second magnitude.

A few stars have preserved the names conferred on them by the Greek or Arabian astronomers. Some of those names, belonging to stars of the first magnitude, are the following:—Sirius, in the right shoulder of Orion; Rigil, in his left foot; Aldebaran, or the eye of the Bull; Capella; Lyra; Arcturus; Antares; Spica Virginis; Regulus, or the heart of the Lion; Canopus; Pomahadu; Acharnar, &c.

The number of constellations given by Ptolemy is 48. They do not comprehend all the stars in his catalogue, and those not included in the figures are called by him unformed stars (ἀγαπητοὶ, informes), and given at the end of that constellation to which they are nearest. The following table includes Ptolemy's constellations, and those which have been added by the moderns.

PTOLEMY'S CONSTELLATIONS.

North of the Zodiac.

1. Ursa Minor, The Little Bear. 2. Ursa Major, The Great Bear. 3. Draco, The Dragon. 4. Cepheus, Cepheus. 5. Bootes, Arctophila. 6. Corona Borealis, The Northern Crown. 7. Hercules, Engonasin, Hercules kneeling. 8. Lyra, The Harp. 9. Cygnus, Gallina, The Swan. 10. Cassiopeia, The Lady in her Chair. 11. Perseus, Perseus. 12. Auriga, The Waggoner. Theoretical 13. Serpentarius, Ophiuchus, Serpentarius. Astronomy 14. Serpens, 15. Sagitta, 16. Aquila, Vultur, et Antinous, 17. Delphinus, 18. Equinus, Equi Sectio, 19. Pegasus, Equus, 20. Andromeda, 21. Triangulum,

In the Zodiac.

22. Aries, 23. Taurus, 24. Gemini, 25. Cancer, 26. Leo, 27. Virgo, 28. Libra, Chelos, 29. Scorpio, 30. Sagittarius, 31. Capricornus, 32. Aquarius, 33. Pisces,

Southern Constellations.

34. Cetus, 35. Orion, 36. Eridanus, Fluvius, 37. Lepus, 38. Canis Major, 39. Canis Minor, 40. Argo Navis, 41. Hydra, 42. Crater, 43. Corvus, 44. Centaurus, 45. Lupus, 46. Ara, 47. Corona Australis, 48. Piscis Australis,

The constellations added by Hevelius are the following:

1. Antinous, 2. Mons Menelai, 3. Asterion et Chara, 4. Camelopardalus, 5. Cerberus, 6. Coma Berenices, 7. Lacerta, 8. Lynx, 9. Scutum Sobieski, 10. Sextans, 11. Triangulum, 12. Leo Minor,

The constellations added by Halley in the southern hemisphere are,— 1. Columba Noachi, 2. Robur Carolinum, 3. Grus, 4. Phoenix, 5. Pavo, 6. Apus, Avis Indica, 7. Apis, Musca, 8. Chameleon.

One of the most important objects of practical astronomy is the formation of catalogues of the fixed stars, in which their positions are determined for a given epoch; theoretical for it is only by means of registered observations that the Astronomy state of the heavens can be compared at different times, and any changes which take place be detected. The apparent place of a star is easily determined by observation; but in order to render such observations available for the purposes of comparison, the mean place of the observed star must be computed and reduced to a given epoch; and this reduction, which involves a knowledge of the precession, nutation, aberration, and in general of all the motions which affect the star's apparent place, is only accomplished by a laborious process of calculation.

The principal catalogues of the stars which we possess are the following:—

Ptolemy's catalogue, which contains 1022 stars. The positions are referred to the ecliptic, and the longitudes are for the year 137 of our era. It is supposed that the greater part of the observations on which it is founded were made and computed by Hipparchus 267 years before, and that Ptolemy merely reduced them to his epoch by adding to each of the longitudes 2° 40', which, according to him, was the amount of the precession of the equinoxes in that interval. This catalogue forms part of the Almagest.

The catalogue of Ulugh Beigh, containing 1017 stars.

Tycho's catalogue, which contains only 777 stars, in 45 constellations.

Riccioli's catalogue, which contains 1468 stars. Part of it, however, was merely copied from more ancient catalogues.

Bayer's catalogue, containing 1762 stars, in 72 constellations. It was published in his Uranometria in 1603. The third edition of this work appeared at Ulm in 1661.

The catalogue of Hevelius, which contains 1888 stars, of which it gives the latitudes, longitudes, right ascensions, and declinations, for the year 1661. Published in his Prodromus Astronomicus in 1690.

Flamsteed's catalogue, containing 2884 stars. Published in the Historia Coelestis Britannicae in 1725. A less perfect edition was given by Halley in 1712.

Catalogues of Lacaille.—The first of these, published in his Astronomiae Fundamenta, contains 397 stars; the second, which is given in his Caelum Australe Stelliferum, contains 1942 of the stars in the southern hemisphere; and the third, which was reduced from his observations by the celebrated Bailly, contains the places of 515 zodiacal stars.

Mayer's catalogue, containing 998 zodiacal stars. It appeared in his Opera Inedita, Göttingen, 1775, and was reprinted in the Connaissance des Temps for 1778.

Bradley's catalogue, containing 587 stars. This was published in the first volume of his observations, edited by Hornsby, in 1798. The positions of 389 stars, calculated from Bradley's observations, had been given by Mason in the Nautical Almanack in 1773. Bradley's observations extended to 3000 stars, but the greater part of them remained useless to astronomy till they were reduced and made the subject of discussion by Bessel in his Fundamenta Astronomiae, Regiomonti, 1818.

Maskelyne's catalogue of 36 stars.

Cagnoli's catalogue, containing 501 stars. Published in the Memoirs of the Italian Society.

Bode's catalogue, which contains 17,240 stars, reduced from the observations of various astronomers.

Piazzi's catalogue, which contains 6748 stars, reduced to the year 1800. In 1814 Piazzi published a new catalogue, comprising 7646 stars.

Zach's catalogue, inserted in his Tabulae speciales Aberrationis et Nutationis, Gothae, 1806. Catalogue of the Astronomical Society of London, containing 2881 stars, published in the second volume of their Memoirs. Almost all the stars comprised in this catalogue are to be found in the catalogues of Bradley or Piazzi, from which they have been reduced to the year 1830.

In addition to the above, the records of great masses of observations may be found in the Philosophical Transactions, the Connaissance des Temps, and the various astronomical and scientific Journals. Lalande has registered in the Memoirs of the Academy of Paris, and his Histoire Céleste, the positions of no fewer than 50,000; and Bessel, of the Königsberg Observatory, who continues to explore the heavens with unabated zeal, has already examined an equal number. Astronomers are now aware of the importance of extending their researches to the most minute sidereal objects. (See Supplement.)

Sect. II.—Of the Parallax, Distance, Magnitude, and Number of the Fixed Stars.

The fixed stars being the points of departure from which all the celestial motions are estimated, one of the first objects in astronomy is to determine the amount and law of all the minute variations of position, real or apparent, to which they are subject. One of the most obvious consequences of the hypothesis of the annual motion of the earth is the existence of an annual parallax of the stars; but on account of the enormous distances of these bodies, this effect of the earth's motion is so small that it cannot be easily measured; and there are even now very few cases in which, with the utmost refinements of methods and instruments, a measurable parallax has been detected. The longest line which nature has furnished us with the means of actually measuring, is the circumference of our own globe. From this geometry teaches us how to find its diameter; and the diameter we employ as a scale with which to compare the distances of the sun and moon, and the other bodies of the solar system. But experience shows us that this scale, large as it is in our conceptions, is only an insensible point in comparison of the distances of the fixed stars. Astronomy has furnished us with another base, about 24,000 times longer than the former, or above 190 millions of miles. This is the diameter of the earth's orbit, which is most conveniently used for expressing the distances of the planets and comets from the sun. Yet even this line is in general insensible when compared with the distances of the stars; for, on observing the same star from its two extremities, at the end of six months, no variation whatever is perceptible in the star's position, after the proper corrections have been made for the small effects produced by different and known causes. The limits of the errors of modern observations cannot well be supposed to exceed $1''$. It follows, therefore, that, seen from the distance of the fixed stars, the diameter of the ecliptic, which exceeds 190 millions of miles, subtends an angle of less than $1''$. Had the annual parallax exceeded this small quantity, it could scarcely have escaped the multiplied efforts that have been made to detect it, not only by Bradley, whose observations, undertaken for the express purpose of determining the parallax of the stars, conducted him to the grand discoveries of the aberration and nutation, but also by other observers furnished with the more delicate instruments of the present day; and particularly the observations made with the splendid instruments of the Royal Observatory of Greenwich. A long controversy was carried on, extending from the year 1810 to 1824, between Dr Brinkley and Mr Pond, on the subject of the annual parallax of some of the brightest stars, Brinkley asserting that he had found large parallaxes, and Pond denying the existence of any measurable parallaxes. Brinkley's conclusions are now generally acknowledged to have been erroneous. Astronomy Brinkley did not observe any parallax in the case of the stars of the constellation $\beta$ Cygni, some of which, having a very sensible proper motion, may, with great probability, be considered as being at a less distance from the earth; but in Wega (a Lyre) he found a parallax of $1''\cdot13$, and one of $1''\cdot42$ in the star Altair, in Aquila. Bradley supposed the parallax of Sirius to amount to $1''$. These facts are, however, disproved by other eminent observers. In a series of 14 stars Struve found the parallax to be negative; that is, the small change of position which the observations seemed to indicate was in a direction contrary to what it would have been if it had arisen from the annual motion of the earth. Mr Pond, the Astronomer Royal, thought the probable value of the parallax could not exceed $0''\cdot018$, a quantity so extremely minute as to be altogether lost in the uncertainties of instrumental errors, and the errors of refraction, which are at least 20 times greater.

Let us concede for a moment the disputed parallax of $1''$, and inquire what must be the corresponding distance of the star. The semidiameter of the terrestrial orbit being taken at 95 millions of miles, the distance of a star whose parallax is $1''$ will be expressed by $\frac{1}{\sin 1''} \times 95000000 = 206264 \times 95000000 = 19,595,080,000,000$, or about 20 trillions of English miles. To assist the imagination in forming some idea of this almost inconceivable distance, we may calculate that a ray of light which darts from the sun to the earth in the space of 8 min. 7 sec., would require $206264 \times 8$ min. 7 sec., or 3 years and 216 days, to reach us from the star. A spider's thread before the eye of a spectator placed at the same distance would suffice to conceal the orbit of the earth; and the breadth of a hair would blot out the whole planetary system. But a star having a parallax of $1''$ is at a moderate distance in comparison of innumerable others, in which no parallactic motion whatever can be distinguished. Supposing the distance of one of these to be only a thousand times greater, a ray of light darted from it would travel between 3000 and 4000 years before it reached the earth; and if the star were annihilated by any sudden convulsion, it would appear to shine in its proper place during that immense period, after it had been extinguished from the face of the heavens. Pursuing speculations of this kind, we may conceive, with Huygens, that it is not impossible that there may exist stars placed at such enormous distances, that their light has not yet reached the earth since their creation.

When viewed with the naked eye, the magnitudes of the Magellanic stars appear to be very different,—a circumstance which may be attributed either to a real diversity of brightness, or, which is more probable, to the great differences in their distances. The sensible magnitudes which they exhibit when viewed in this manner are owing only to the numerous reflections of the rays of light from the aerial particles surrounding the eye; as is proved by looking at them through a long tube, which prevents any rays from reaching the eye excepting those which come directly from the star. In the telescope their dimensions are entirely inappreciable: the greater the power with which they are viewed, the smaller are their apparent diameters, because they are then more completely divested of the effects of irradiation. A star having a diameter of $1''$, and an annual parallax of $1''$, would be more than a million of times larger than the sun. Nevertheless, Sir W. Herschel assures us, that, by means of the great powers which his telescopes carried, he had seen the disks of some stars perfectly round, and had even succeeded in measuring Theoretical their apparent diameters. He found the diameter of Wega to be \( \frac{1}{4} \), that of Alderbaran \( \frac{1}{5} \), and that of Capella \( \frac{2}{5} \). Supposing the measurement accurate, and the annual parallax of this last not to exceed \( \frac{1}{7} \), its volume would be equal to 20 million times that of the sun.

With regard to the number of the stars, it is altogether impossible to form any satisfactory conjecture. Of those which are visible to the naked eye, the number does not, probably, at any time, exceed 1000, although, from the effect of their twinkling, and the confused manner in which they are seen, one is apt to suppose them to be much more numerous; but in the telescope they are prodigiously multiplied. Within the limits of a space extending \( 15^\circ \) by \( 2^\circ \), Herschel counted no fewer than 50,000. In the single constellation of the Pleiades, instead of 6 or 7, which can be distinguished by the unassisted vision, Hooke, with a telescope of 12 feet, counted 78; and in telescopes of greater power the number appears to be vastly larger. Although, by reason of their very unequal distribution, no accurate estimate can be formed of the number contained in the whole sphere by the examination of any small portion of it, yet there is some reason to conclude that there cannot be less than 75 millions of stars altogether visible in a good telescope. Baron Zach estimates that there may be at least a thousand millions of stars in the entire heavens, without reckoning (what may probably exist) opaque bodies which cannot be perceived, and stars whose light has been extinguished. It is, however, evident that all estimates of this sort have no other limit than such as is imposed by the imagination. If an observer could be transported to the remotest star visible in his telescope, he would probably see extending before him in the same direction, a firmament equally rich and splendid as that which he beholds from our own insignificant planet.

**Sect. III.—Of the Proper Motions of the Stars.**

In Chap. II, four different causes of apparent motion with regard to the stars have been explained; namely, the precession of the equinoxes, the nutation of the earth's axis, the secular diminution of the obliquity of the ecliptic, and the aberration of light; all which are occasioned by the various motions proper to the earth, and for that reason are called apparent motions. But if, after due allowance has been made for the effects produced by them, it should happen that the observed place of any star, or number of stars, does not correspond with former observations, a new and peculiar motion would be indicated, which must be explained by a new hypothesis. Now there are obviously two ways of accounting for such a phenomenon, namely, a parallactic motion and a proper motion, from either of which the observed variation of position may result. It has already been shown that the annual parallax, or that apparent alteration in the places of the fixed stars which might be expected from the motion of the earth in its orbit, is insensible; but we may suppose, and indeed observation has even rendered it probable, that the sun, accompanied by his whole train of planets and comets, is in motion in space; and if this is the case, the stars must appear to change their relative situations as soon as the sun has described a space bearing a sensible ratio to their distance. The resulting motion would thus partake of the nature of parallax, and is hence termed parallactic. The other method of accounting for any observed alteration in the mutual position of the stars, is to ascribe to them a proper motion, instead of supposing them to be absolutely fixed. On this supposition the motion is real; yet it may happen that the observed changes of position result from the combined effects of a real and parallactic motion. The effects of these motions are in their nature sufficiently distinct, although, on account of the extremely minute quantities to be determined, and the uncertainty that still exists as to the exact amount of some of the apparent motions with which they are blended, it may for centuries to come be found impossible to separate them. If all the stars forming a group, or situated in the same quarter of the heavens, appear to recede from or approach to each other, their motions may with reason be ascribed to the translation of the solar system in space, and consequently be parallactic; but if, on the contrary, some appear stationary, while others appear to move in different directions, the phenomena will indicate a real change in the positions of the stars.

From the analogy of our own system we are naturally induced to extend the principle of gravity to the sphere of the stars, and to suppose, as a necessary consequence, that none of the celestial bodies are in a state of absolute repose. Their mutual attraction must communicate to them a motion which would end in uniting them all in the same mass unless it were counteracted by a centrifugal force; hence the stars are supposed to move about distant centres in orbits analogous to those of the planets. But if the stars are not absolutely at rest in space, their motions must be extremely slow, inasmuch as the actual state of the heavens corresponds entirely with the descriptions that have been given of it by Hipparchus and Ptolemy. Ptolemy has transmitted to us a great number of observations on the relative situations of the fixed stars, made by Hipparchus, whose method was to observe those which are situated in an arc of the same circle, or which can be intersected by the same straight line; and he assures us that he himself, after an interval of 260 years, could perceive no alteration. In order to furnish posterity with the means of pursuing similar inquiries, Ptolemy added the positions of many other stars determined by his own observations; and we owe to his labours the certain knowledge that the relative positions of the stars, notwithstanding the numerous displacements of the ecliptic and equator, are at present nearly the same as they were 2000 years ago. The ancient observations were, however, of too rude a nature to admit of any satisfactory conclusion being deduced from them respecting the minute quantities in question. The accurate instruments of the present day have enabled observers to remark some changes of position too decided to admit of doubt; hence astronomers have inferred a proper motion in several stars, as well as a translation of the sun and the planets in space. From the comparison of a great number of observations, Sir W. Herschel was of opinion that many of the proper motions might be explained by supposing the solar system to have a motion directed towards the star \( \lambda \) in the constellation Hercules. Bessel, having subjected to a rigorous comparison a much greater number of the proper motions indicated by comparing the catalogues of Bradley, Mayer, Piazzi, &c., arrived at the conclusion that many points might be assigned in the sphere, some of them even diametrically opposite to each other, situated in the direction of those motions; but that, in whatever direction the sun is supposed to move, so many proper motions will remain unaccounted for, that there is no reason for preferring one point to another. It has since been demonstrated, however, by astronomers, that Bessel's conclusion was erroneous, and that Herschel's idea was correct. The conspicuous proper motions of the stars which have hitherto been remarked are subject to an assignable law, and are not directed to arbitrary points in space; and it is only after eliminating the effects of the solar motion, that the real proper motions can be inferred. It is, how- Theoretical ever, on the whole, probable that they are due, in part at least, to a real displacement of the stars, and not to a general translation of the solar system.

As early as the beginning of the present century, astronomers had detected proper motions in a great number of stars, the rates of which they inserted in their catalogues along with those of the precession; but great discrepancies existed in their conclusions. Baron Zach, comparing Maskelyne's observations of the right ascensions of the Greenwich stars, reduced to 1802, with those of Bradley, reduced to 1760, found results which differ from those of Maskelyne himself, not only in amount, but in several instances even in the direction of the supposed proper motions. For example, the proper motions (in right ascension) of γ Pegasi, α Ceti, Rigel, Sirius, Spica, γ and β Aquilae, α Cygni, η Aquarii, and a Pegasi, are all positive according to Zach, while Maskelyne considered them as being all negative. Such was the uncertainty respecting the proper motions even of the Greenwich stars, which, by reason of the frequent observations they had undergone, and the rigorous scrutinies to which they had been subjected, were probably those whose places had been the most accurately determined of any in the whole heavens. But the motions are so slow that they must remain, for a considerable number of years at least, blended with the errors of observation. The double star 61 Cygni is indeed demonstrated to have an annual proper motion of +5°06' in right ascension, and 3°34' in declination; but in general the rates are confined within much narrower limits, and appear to amount only to a fraction of a second. The number of stars in which this proper motion has been supposed to be observed is, as we have already noticed, very considerable. M. Bessel, by a comparison of 2959 stars out of Bradley's catalogue, with the same stars in the catalogue of Piazzi, found that 425 of them had an annual proper motion, amounting to more than 0°2', in the arc of a great circle. The following table, published by Mr Bailly in the second volume of the Memoirs of the Astronomical Society, contains a list of all those stars observed by Bradley and Mayer, whose annual proper motions, according to Piazzi, as given in his catalogue, amount to 0°5 either in right ascension or declination. The positive sign, in the column of declinations, denotes a motion towards the north, the negative a motion towards the south. The numbers prefixed are those of Flamsteed, unless when inclosed within a parenthesis, in which case they are those of Piazzi.

| Star | Proper motion in | |------|-----------------| | | N. | D. | | 11 β Cassiopeiae | +0°82 | | 24 Ditto | +1°78 | -0°72 | | 37 μ Andromedae | +1°20 | | 1 Polaris | +1°47 | | 37 δ Cassiopeiae | +0°64 | | 107 Pisces | -0°57 | | 52 τ Ceti | -1°86 | +0°84 | | 13 δ Persei | +0°67 | | 12 Eridani | +0°64 | +0°82 | | 23 Ditto | -0°60 | | 27 μ Ditto | -0°59 | | 40 d Ditto | -2°21 | -3°60 | | 1 Orionis | +0°54 | | 104 μ Tauri | +0°69 | | 15 δ Leporis | -0°62 | | 9 α Can. Maj. | -0°51 | -1°14 | | 10 α Can. Min. | -0°71 | -0°98 | | 78 β Geminorum | -0°72 | | 15 ξ Cancri | -0°60 | | 9 ι Ursae Maj. | -1°05 | | 81 π Cancri | -0°55 | | 25 δ Ursae Maj. | -1°80 | -0°60 | | 29 v Ditto | -0°60 | | 7 α Crateris | -0°59 | | 63 ζ Leonis | -0°53 | | 33 ε Ursae Maj. | -0°52 | -0°64 | | 94 ζ Leonis | -0°53 | | 5 β Virginis | +0°76 | | 16 c Ditto | -0°55 | | 5 x Draconis | -0°50 | | 3 Canum Venaticorum | -1°02 | | 29 γ Virginis | -0°72 | | 43 δ Ditto | -0°65 | | 43 Com. Ber. | -1°19 | +0°94 | | 61 Virginis | -1°30 | -1°08 | | 70 Ditto | -0°53 | | 85 η Ursae Maj. | -0°50 |

The proper motions of the stars are not susceptible of direct observation or measurement: they are only indicated by the minute differences which remain after applying to the observed position of any given star the corrections due to all the apparent motions with which we are acquainted. Thus, let M and M' be the mean places of any given star... Theoretical at two different epochs, the interval between which is \( t \) years; \( P \) the annual precession of the equinoxes, and \( x \) the annual motion of the star in right ascension. It is obvious that \( x \) will be given by the formula

\[ x = \frac{M - M'}{\epsilon} - P; \]

so that the value of \( x \), or the amount of the proper motion, is dependent not only on the accuracy of the observations, and on the computations from which \( M \) and \( M' \), the mean places of the star, have been deduced, but also on the quantity \( P \), which is calculated by a method of approximation not entirely exact. For all these reasons it is easy to see that the determination of the proper motions must be attended with considerable uncertainty, and that hundreds, probably thousands, of years will be required to develop their rates and directions, and assign the distant centres round which they are performed. (See Supp.)

**Sect. IV.—Of Variable and Double Stars, Nebulae, and the Milky Way.**

The periodical variations of brilliancy to which some of the fixed stars are subject, may be reckoned among the most remarkable of the phenomena exhibited by those bodies. Several stars, formerly distinguished by their splendour, have entirely disappeared; others are now conspicuous which do not seem to have been visible to the ancient observers; and there are some which alternately appear and disappear, or of which the light at least undergoes great periodic variations. Some seem to become gradually more obscure, as \( \delta \) in the Great Bear; others, like \( \beta \) in the Whale, to be increasing in brilliancy. Some stars have all at once blazed forth with great splendour, and, after a gradual diminution of their light, have again become extinct. The most remarkable instance of this sort is that of the star which appeared in 1572, in the time of Tycho. It suddenly shone forth in the constellation Cassiopeia with a splendour exceeding that of stars of the first magnitude, even of Jupiter and Venus at their least distances from the earth, and could be seen with the naked eye on the meridian in full day. Its brilliancy gradually diminished from the time of its first appearance, and at the end of sixteen months it entirely disappeared, and has never been seen since. During the whole time of its apparition its place in the heavens remained unaltered, and it had no annual parallax; its distance was consequently of the same order as that of the fixed stars. Its colour, however, underwent considerable variations. Tycho describes it as having been at first of a bright white; afterwards of a reddish yellow, like Mars or Aldebaran; and, lastly, of a leaden white, like Saturn. Another instance of the same kind was observed in 1604, when a star of the first magnitude suddenly appeared in the right foot of Ophiuchus; it presented phenomena analogous to the former, and disappeared in like manner after some months. Kepler wrote a book on this singular apparition. These instances sufficiently prove that the stars are subject to great physical revolutions.

A great number of stars have been observed whose light seems to undergo a regular periodic increase and diminution, and these are properly called variable stars. One in the Whale has a period of 334 days, and is remarkable for the magnitude of its variations. From being a star of the second magnitude, it becomes so dim as to be seen with difficulty through powerful telescopes. Some are remarkable for the shortness of the period of their variation. Algol has a period of between two and three days, \( \delta \) Cephei one of \( 5\frac{1}{2} \) days, \( \beta \) Lyrae one of \( 6\frac{1}{2} \) days, \( \mu \) Antliae one of 7 days. The regular succession of these variations precludes the supposition of their being occasioned by a real or permanent destruction of the stars; neither can they be supposed to arise from a change of distance; for as the stars invariably retain their apparent places, it would be necessary to suppose that they approach to and recede from the earth in straight lines,—an hypothesis which is at least extremely improbable. The most probable supposition is, that the stars revolve, like the sun and planets, about an axis; and that the surfaces of the variable stars are unequally covered with dark spots, or unequally fitted to emit light; whence their dark sides will be turned towards us after certain intervals by the effect of rotation. In this way Newton accounted for the phenomenon. Other astronomers have devised different explanations. Maupertuis supposed that the figure of the stars is not globular, but flat, and that the variations of brilliancy depend on the angle which their flat sides make with the visual ray,—an angle which will be constantly varying if the stars are endowed with a rotatory motion. Others, again, have imagined that the partial obscuration of the stars may be occasioned by their being eclipsed by opaque bodies or planets revolving round them.

On examining the stars with telescopes of considerable power, many of them are found to be composed of two or more stars placed contiguous to each other, or of which the distance subtends a very minute angle. This appearance is probably in many cases owing solely to the optical effect of their position relative to the spectator; for it is evident that two stars will appear contiguous if they are placed nearly in the same line of vision, although their real distance may be immeasurably great. There are, however, many instances in which the angle of position of the two stars varies in such a manner as to indicate a motion of revolution about a common centre; and in this case the two stars form a binary system, performing to each other the office of sun and planet, and connected together by gravity or some equivalent principle. The recent observations of Herschel, Dawes, South, and Struve, have placed this fact beyond doubt. Motions have been detected so rapid as to become measurable within very short periods of time; and at certain epochs the satellite or feebler star has been observed to disappear, either on passing behind or before its primary, or by approaching so near to it that its light has been absorbed by that of the other. The most remarkable instance of a regular revolution of this sort is that of the double star \( \xi \) Ursae Majoris, in which the angular velocity is \( 6^\circ 4' \) annually, so that the two stars complete a revolution about one another in the space of 60 years; and above three fourths of a circuit have been already described since its discovery in 1781. The double star \( p \) Ophiuchi presents a similar phenomenon, and the satellite has a motion in its orbit still more rapid. \( \alpha \) Castoris, \( \gamma \) Virginis, \( \zeta \) Cancri, \( \xi \) Bootis, \( \delta \) Serpentis, and that remarkable double star \( 61 \) Cygni, together with several others, exhibit similar variations in their respective angles of position.

The examination of double stars was first undertaken by Sir W. Herschel, with a view to the question of parallax; for it is evident, and indeed had been remarked by Galileo, that the apparent distance of two stars which are very near each other will vary with the position of the earth in its orbit, unless they are both so remote that, in comparison of their distance from the earth, the diameter of the terrestrial orbit is insensible. His attention was, however, soon arrested by the new and unexpected phenomena which these bodies presented. Sir W. Herschel observed in all 2400 of them. Messrs South and Herschel have given a catalogue of 380 in the Transactions of the Theoretical Royal Society for 1824, and South added 458 to the list Astronomy, in the volume for 1826. Mr Herschel has published three series of observations in the Memoirs of the Astronomical Society, containing altogether 1000 double stars; and the catalogue of M. Struve of Dorpat contains 3063 of the most remarkable. The object of these catalogues is not merely to fix the place of the star within such limits as will enable it to be easily discovered at any future time, but also to record a description of the appearance, position, and mutual distances, of the individual stars composing the system, in order that subsequent observers may have the means of detecting their connected motions, or any changes with which they may be affected. M. Struve has also taken notice of 52 triple stars, among which No. 11 of the Unicorn, ξ of Cancer, and ξ of the Balance, appear to be ternary systems in motion. Quadruple and quintuple stars have likewise been observed, which also appear to revolve about a common centre of gravity. Every region of the heavens furnishes examples of these curious phenomena, especially those which abound in stars; though M. Struve remarks that some parts of the milky way contain very few, while others present them in great abundance.

Some of the double stars present curious instances of contrasted colours, and generally assume the complementary tints,—a circumstance which, Mr Herschel thinks, may be owing in some degree, at least in cases where red and green, yellow and blue stars are combined in a double star, to the influence of optical deception. "When I first observed," says that ingenious philosopher, "the double star No. 881, R. 19 hours 8 minutes 56 seconds, P.D. 95° 45' 33", I remarked it as a case of contrasted colour, the large star being ruddy, and the small one blue. But on closer attention I perceived the small star itself to be double; yet each of the two very minute stars of which it consists appeared equally blue while the eye continued under the influence of the large one; but when this was withdrawn from the field, they appeared of no particular colour, but just like other small stars in the neighbourhood. It may be remarked further, too, that yellow stars are, generally speaking, accompanied by blue small ones, blue being the complementary tint of yellow; but when the large star has an excess of red rays, the blue verges to green, as it ought on the hypothesis of contrast. A most remarkable instance is that of No. 895 (Struve's catalogue), R. 6 hours 12 minutes, P.D. 84° 12', in which the large star is of a full ruby red, and the smaller one of a fine green, but which colour it also loses when the large star is concealed behind the cross of the wires." (Memoirs of the Astronomical Society, vol. iii. p. 186.)

The Nebulae, so called from their dim cloudy appearance, form another class of objects which furnish matter for curious speculation and conjecture respecting the formation and structure of the sidereal heavens. When examined with a telescope of moderate powers, the greater part of the nebulae are distinctly perceived to be composed of clusters of little stars, imperceptible to the naked eye, because, on account of their apparent proximity, the rays of light proceeding from each are blended together, through the effects of irradiation, in such a manner as to produce only a confused luminous appearance. In others, however, no individual stars can be perceived, even through the best telescopes; and the nebula exhibits only the appearance of a self-luminous or phosphorescent patch of matter in a highly dilated or gaseous state; though it is possible that even in this case the appearance may be owing to a congeries of stars so minute, or so distant, as not to afford singly sufficient light to make an impression on the eye. In some instances the nebula presents the appearance of a faint luminous atmosphere, of a circular form, and of large extent, surrounding a star of considerable brilliancy. One of the most remarkable nebulae is that which is situated in the sword-handle of Orion. It was discovered by Huygens in 1656, and described and figured by him in his Systema Saturnium. Since that time it has been examined and described by various observers, particularly Fouchy, Mairan, Le Gentil, and Messier, who have given engravings of it; and if any trust can be placed in their descriptions of so indistinct and difficult an object, it must have undergone great changes in its form and physical appearances. Unfortunately, however, no satisfactory inference can be drawn from the comparison of the different descriptions; for it is found that the same nebula, viewed on the same night with different telescopes, presents appearances so different as to be scarcely recognizable as the same object. The effects of atmospherical variations also cause great differences in its appearance, even when it is viewed through the same telescope at different times; so that it is scarcely possible that any two observers will be found to agree in their delineations of its outline. Sir J. Herschel, in the second volume of the Memoirs of the Astronomical Society, has given a detailed description of this nebula as it appeared in his twenty feet reflector in 1824, together with a drawing which, on account of the superiority of his telescope, is probably a much more correct representation of the object than any which previously existed. Fig. 108 is copied from that drawing. Of that portion of the nebula which he calls the Huygencian region Mr Herschel gives the following account: "I know not how to describe it better than by comparing it to a curdling liquid, or a surface strewed over with flocks of wool, or to the breaking up of a mackerel sky when the clouds of which it consists begin to assume a curious appearance. It is not very unlike the mottling of the sun's disk, only (if I may so express myself) the grain is much coarser, and the intervals darker; and the flocculi, instead of being generally round, are drawn out into little wisps. They present, however, no appearance of being composed of small stars, and their aspect is altogether different from resolvable nebulae. In the latter we fancy that we see stars, or that, could we strain our sight a little more, we should see them; but the former suggests no idea of stars, but rather of something quite distinct from them."

Another very remarkable nebula is that in the girdle of Andromeda, which, on account of its being visible to the naked eye, has been known since the earliest ages of astronomy. It was re-discovered in 1612 by Simon Marius, who describes it as having the appearance of a candle seen through horn, that is, a diluted light, increasing in density towards a centre. Le Gentil mentions that its figure had appeared to him for many years round, but that in 1757 it had become oval. He also remarks that its light was perfectly uniform in all parts,—a fact which is quite at variance with its present appearance, and which, if true, argues that the nebulous matter is in a rapid state of condensation. "At present," says Sir J. Herschel, in the volume above referred to, "it has not, indeed, a star or any well-defined disk in its centre; but the brightness, which increases by regular gradations from the circumference, suddenly acquires a great accession, so as to offer the appearance of a nipple, as it were, in the middle, of very small diameter (10" or 12"), but totally devoid of any distinct outline, so that it is impossible to say precisely where the nucleus ends and the nebula begins. Its nebulosity is of the most perfect milky, absolutely irresolvable kind, without the slightest tendency to that separation..." Sir W. Herschel, who devoted himself to the examination of every uncommon appearance in the sidereal heavens, has given catalogues of 2000 nebulae and clusters of stars discovered by him, and has shown that the nebulous matter is distributed through the immensity of space in quantities inconceivably great, and in separate parcels of all shapes and sizes, and of all degrees of brightness between a mere milky appearance and the condensed light of a fixed star. Finding that the gradations between the two extremes were tolerably regular, he thought it probable that the nebulae form the materials out of which nature elaborates suns and systems, and conceived that, in virtue of a central gravitation, each parcel of nebulous matter becomes more and more condensed, and assumes a rounder form; that from the eccentricity of its shape, and the effects of the mutual gravitation of its particles, it acquires gradually a rotatory motion; that the condensation goes on increasing till the mass acquires consistency and solidity, and all the other characters of a comet or planet; that by a still further process of condensation the body becomes a real star, self-shining; and that thus the waste of the celestial bodies, by the perpetual diffusion of their light, is continually compensated and restored by new formations of such bodies, to replenish ever the universe with planets and stars. (See the Philosophical Transactions for 1811.) These hypotheses or conjectures give no doubt a mechanical reason for the formation of stars, but the answer to them is exceedingly obvious. Has any instance yet been observed of a nebula being succeeded by a star, or cluster of stars, or even of becoming so much more condensed as absolutely to change its form? Till a change of this sort has been observed, the inferences are drawn from analogies too slender to entitle them to be regarded as anything more than mere fancies and speculations. Even if every link in the chain were perfect, and the gradation distinctly traced from the most diffuse nebula to the most compact star, the facts would still be insufficient to warrant the conclusion that the celestial matter had actually undergone a transition from the nebulous to the stellar state, or that any star or nebula in the heavens ever existed in a state different from its present. Though there is little reason to hope that we shall ever obtain a full knowledge of the mysterious processes which the great Architect of the universe has employed in the formation of the celestial bodies, yet a long series of observations with such instruments as are now constructed may lead to the detection of changes sufficiently indicative of the nature of the forces by which they are produced. These cosmological speculations, however, it may be well to remark, are not of the slightest value to astronomy.

That great luminous tract which encompasses the sky Milky-like a girdle, and is called the Galaxy or Milky-Way, is supposed by Sir W. Herschel to be a nebula of which the sun forms one of the component stars; and hence it appears immensely greater than other nebulae only in consequence of our situation with respect to it, and of its greater proximity to our system. On examining any part of it with a good telescope, we perceive a prodigious multitude of small stars, whose blended light occasions the whitish appearance which forms so remarkable an object in the heavens. Yet notwithstanding the apparent contiguity of the stars which crowd the galaxy, it is certain that their mutual distances cannot be less than a hundred thousand times the radius of the terrestrial orbit. From this we may attempt to form some notion of the inconceivable distances of the other nebulae, some of which, probably not inferior in magnitude to the milky-way, appear only as small luminous patches in the telescope.

See Tycho de Nova Stella anni 1572; Hevelius, Historiola mirae Stella in Collo Ceti, anno 1660; Mauupertuis, Figure des Astres, Éuvres, tome i. Lyon, 1756; Michel, Phil. Trans. 1784; Pigott, Phil. Trans. 1785, 1797; Lommier, Mem. Acad. Par. 1789; Lalande, Astronomie (885); Herschel, Phil. Trans. passim. (r. o.)

Supplement to Part II.

Since the last revision of the article Astronomy in this work, the discoveries in many of the branches of the science have been so various and of so much importance, that it would have been useless to attempt their incorporation in the body of the work without a total alteration of its construction. It has been, therefore, judged more convenient, as well as more useful, to add in a supplement such discoveries and improvements as have had an undoubted influence on the science in its present condition, and which hold out hopes of still greater extension for the future.

We shall preserve the same order as that used in the preceding part devoted to Theoretical Astronomy, and describe as briefly as possible the chief additions which have been made to the respective departments of the Planetary, Cometary, and sidereal branches of the subject.

CHAP. I.

OF THE SUN.

Sec. I.—Of the Motion of the whole Solar System in Space.

It has been previously mentioned (vol. iii. p. 807) that the idea of a proper motion of the sun had been embraced by Sir W. Herschel, but had been rejected as improbable by Bessel and other astronomers. The fact, however, of the existence of this motion has been now rendered incontestible by the independent labours of several astronomers, and we propose to give briefly the grounds of their several determinations.

First, with regard to theory, let us imagine that the sun has a motion (carrying, of course, the earth with it) which, for the small interval of time included by our modern observations which are of sufficient accuracy to decide the point, may be considered as uniform and rectilinear, towards a point of the heavens denoted by right ascension A and declination D. It is evident, in the first place, that the general effect of such a motion would be to produce an apparent displacement of all the stars in the heavens of such a nature as to increase the apparent angular distances of all stars towards which the motion is described, and to diminish the apparent distances of all those in the opposite quarter of the heavens. With regard to the right ascensions of all stars, then, the effect would be to produce an apparent proper motion in that element, such that the right ascensions to the left of the point towards which the sun is moving would be increased, and those to the right would be diminished; while those lying in the meridian of the solar motion would remain unaltered. Similarly with regard to the stars lying in the latter plane, the north polar distance of all to the north of the point of solar motion would be diminished, while all to the south of it would be increased.

To determine accurately the amount of apparent motion of any star in right ascension and declination, let \( \alpha \) and \( \delta \) be the right ascension and declination of any star at a linear distance \( r \) from the sun or earth. Then, if the line joining the earth or sun, and the star, make with the line of direction of the supposed motion, an angle \( \theta \); and if, in the unit of time, the linear solar motion be \( a \), the angle of apparent displacement of the star in this plane will be \( \frac{a}{r} \sin \theta \). If, now, we imagine the Theoretical spherical triangle formed by the great circles joining the north pole of the heavens, the apex of solar motion, and the star, its three sides will be $90^\circ - D$, $90^\circ - J$, and $s$; and the angle opposite to $s$ will be $\alpha - A$. From this it may be readily shown that the apparent motion in R. A. is $+ \frac{a}{r} \cos D \sin (\alpha - A)$ (expressed in time), and the motion in declination (northwards) is $+ \frac{a}{r} \sin D \cos s - \cos D \sin J \cos (\alpha - A)$. These, then, are the formulae by which the proper motions of the stars, when deduced from comparison of observations made at distant epochs, are to be tested; and, assuming that the solar motion does really exist, and that, on the whole, the proper motions of the stars are to be attributed to it, the most probable values of $A$ and $D$ are to be deduced; and it is evident that, since there is antecedent probability of the reality of the motions of many of the stars, the most probable values can only be determined by the use of a tolerably large catalogue in all parts of the heavens, so that their real proper motions having no determinate direction, the effects will be eliminated, or nearly so, in the final result. Another difficulty arises from the circumstance that the formulae involve the distances of the stars which, except in a very few exceptional cases, are totally unknown. This can only be obviated either by assuming, with Argelander, that, on the average, stars having the greatest proper motions are nearest to the sun; or, by introducing, with Otto Struve, some empirical law of distance derived from the apparent magnitudes. Both hypotheses are precarious; but it is likely that, with either, the average result for the position of the apex of solar motion, deduced from a large catalogue of proper motions, will be tolerably correct; and it is also not improbable that in large numbers of stars the distances will so far agree with the law of apparent magnitudes, as to give a result for the actual amount of solar motion (based on the few cases of distances actually known) not very far distant from the truth.

The first astronomer who successfully attempted a solution of the problem was Argelander, in a paper published in 1837 in the Mémoires présentés par divers Savans of the Imperial Academy of St Petersburg. The proper motions employed are those derived from his own catalogue of 560 stars observed at Åbo, by comparison with Bradley's observed places, as given in Bessel's Fundamenta. The only peculiarity which we have space to notice in Argelander's treatment of the problem is, that, assuming stars with the greatest proper motions to be nearest to us, he divides them into three classes determined by the greatness of the proper motions, and deduces a separate result from each class. The final result which he deduces is, when reduced to 1800,

$$A = 259^\circ 51' 8''; \quad D = + 33^\circ 29' 1''.$$

In No. 398 of the Astronomische Nachrichten, Argelander gives a result obtained by Lundahl, based on a comparison of 147 stars of Pond's catalogue of 1112 stars with the Fundamenta, each of the stars having an annual proper motion not less than $0^\circ 09'$. The result arrived at is for 1792-5,

$$A = 252^\circ 24' 4''; \quad D = + 14^\circ 26' 1''.$$

The most important paper is that of M. Otto Struve, printed in the fifth volume of the Petersburg Transactions for 1842. The title of the paper is Bestimmung der Constante der Precession mit Berücksichtigung der eigenen Bewegung der Sonnensysteme, and for it the author received the gold medal of the Royal Astronomical Society in 1850. The investigation is grounded on the proper motions of about 400 stars (chiefly double) observed at Dorpat; for determining the weights of his equations, he assumes that the distances of the stars are inversely as the apparent magnitudes; and for his final result he obtains, as the most probable values of $A$ and $D$ for 1790,

$$A = 261^\circ 21' 8''; \quad D = + 37^\circ 36' 0''.$$

And, by combining his observations with those of Argelander and Lundahl with proper weights he obtains, finally, for 1792-5,

$$A = 259^\circ 9' 4''; \quad D = + 34^\circ 36' 5''.$$

The most recent investigation of the direction of the solar motion is by Mr Galloway, printed in the Philosophical Transactions for 1847, Part I., and for which the royal medal was awarded to him. His object was to show whether the general drift of the southern stars indicated a motion of the sun towards the same point of the heavens as that indicated by the northern stars. For this purpose he employed the catalogue of 606 stars observed at St Helena by Mr Johnson, and the catalogue of 172 stars observed by Mr Henderson at the Cape of Good Hope; Mr Galloway's method of investigation is precisely the same as that of Argelander, excepting that he does not attempt to assign weights to his equations by any hypothesis whatever with regard to the distances of the fixed stars, considering that no greater probable accuracy could be obtained in his inquiry by the adoption either of Argelander's or of Otto Struve's assumption of the criterion of distance. The result which he obtains for the values of $A$ and $D$ is, for the epoch 1790,

$$A = 260^\circ 0' 6''; \quad D = + 34^\circ 23' 4''.$$

The essential identity of this position of the apex of solar motion, derived from the southern stars, with that deduced from three separate sets of northern stars, confirms the fact of the existence of this motion, and shows that we have arrived at a very close approximation to its direction. The investigation of Otto Struve has also shown that its amount ($0^\circ 3$ in a year) is too large to be neglected in any future cosmical speculations. The subject is therefore one of very considerable importance, and must form the basis of any future inquiries respecting the connexion of our sun and planets with that sidereal system of which they form a part.

Section II.—Of the time of rotation of the Sun, and the physical peculiarities of his surface.

In the preceding part of this article, the method has been explained by which astronomers have determined, with various degrees of accuracy, the time of the sun's rotation, and the position of his equator with regard to the ecliptic, by means of observations of the spots which are frequently observed on his disk. The chief observations and investigations having this object in view, were made by Scheiner, Cassini, and La Lande, at epochs included between the years 1626 and 1776. Recently several attempts have been made to obtain results of greater accuracy, by M. Langlier, Dr Petersen, and Dr Bohm of Vienna. The dissertation of the last-named astronomer has been recently printed in vol. iii. of the Denkschriften der Mathematisch-Naturwissenschaftlichen Classe of the Academy of Sciences of Vienna; and this paper is well worthy of the attention of the student, from the elaborate care which has been bestowed upon the mathematical treatment of the subject, as well as for the excellence of the observations carried on for some years on which the results depend.

The results deduced by Dr Bohm are, for the epoch 1834-6:

- Longitude of ascending node of sun's equator, $= 76^\circ 46' 9''$. - Inclination of sun's equator to ecliptic, $= 6^\circ 56' 6''$. - Time of sun's rotation, $= 25$ days 12 hours 30 minutes.

The times of rotation, according to M. Langlier and Dr Petersen, are respectively,

$$25^\circ 8' 10'' \text{ and } 25^\circ 4' 30''.$$

With regard to the physical peculiarities of the surface of the sun, some important additions have been recently made by Mr Dawes, by means of an eye-piece devised by himself, which enables the observer to examine minutely, and with comfort, any separate portion of the disk. Mr Dawes' apparatus is very simple, and consists of a metallic perforated slide (the perforations being of a different size for different purposes) which crosses the eye-tube at right angles, in place of the fixed bar; that is, in the plane of the focus of the object-glass. By this means the field of view can be rendered as small as is desired for examination of a minute portion of the solar disk, the only light and heat reaching the eye being that transmitted from that portion. This eye-piece has since been adopted by other astronomers, and Mr Dawes has been rewarded almost immediately after its appearance by the discovery of two facts of considerable importance in the theory of the solar spots.

The first discovery relates to the existence of a stratum of comparatively faint luminosity, which he denominates a cloudy stratum, giving the impression of considerable depth below the second luminous stratum, which forms the shadow or pen- Theoretical memoirs usually seen round the nucleus of a spot. This stratum appears to be not self-luminous, but of such a nature as to absorb a vast quantity of light, and to reflect very little. The faint illumination is rarely uniform, presenting rather a mottled or cloudy surface, and occasionally some very small patches are very decidedly more luminous than the rest. In all spots of considerable size, a black opening is perceptible in the closely stratum, which is proved to be, if luminous at all, less so than our own atmosphere when illuminated by the direct rays of the sun.

The second curious fact discovered by Mr Dawes is that of the rotation of the spots. This phenomenon was seen remarkably in the case of a spot sketched on January 17 and January 23 of the year 1852; by which it became evident that in this interval the spot had rotated through 160°. If we accept the usual hypothesis, that the spots are vast disturbances in the atmosphere of the sun, revealing to us for a time part of his surface, what an idea does this present to us of the magnitude of the operations of nature in this stupendous globe! A mass of aeriform or gaseous fluid, whose diameter is frequently more than 100,000 miles, is agitated by a rotating storm similar probably to those which sometimes devastate the surface of our own planet. In the above instance, this mass must have taken about twelve days to complete a revolution; and therefore, though the angular velocity is moderate, yet the outer circle would be moving with the enormous velocity of about 600 miles per hour, a velocity in fact ten times greater than that of the fiercest hurricanes which have at times laid waste portions of the surface of the earth.

With regard to the faculae, Mr Dawes has had opportunities of proving the correctness of Sir J. Herschel's supposition, that they are in reality great waves or undulations projecting beyond the general surface of the solar disk. He says, "The faculae are best seen near the east and west edges of the sun's disk, where they give the impression of narrow ridges, whose sides are thus presented to view. They are rarely seen as actual projections from the limb. On one occasion, however, I had an opportunity of observing a satisfactory confirmation of the idea that they are ridges or heappings up of the luminous matter. A large bright streak or filament was observed to run, as usual, nearly parallel to the sun's edge for some distance, and very near it; and then to turn rather abruptly towards the edge and pass over it. The limb was at this time very well defined; and, when it was most sharp and steady, the bright streak was seen to project slightly beyond the smooth outline of the limb, in the manner of a mountain ridge nearly parallel to the sun's equator."

Considerable information respecting the solar disk has been obtained by the organized observations of the total solar eclipses which occurred on July 8, 1842, and July 28, 1851. The former of these phenomena was admirably observed by Mr Baily at Pavia, and by Mr Airy at Turin, and detailed accounts of their observations and impressions are given in vol. xv. of the Memoirs of the Royal Astronomical Society, accompanied by drawings of the different phases of the eclipse. The attention of astronomers having been, by these and other accounts, directed to the extreme importance of observing with every possible accuracy the total eclipse of 1851, an organization was formed under the auspices of the British Association, and Suggestions were published previously, giving minute directions as to the observation of every possible phenomenon which might occur. The observations were remarkably successful, and the accounts of the British observers alone form the first part of vol. xxi. of the Memoirs of the Royal Astronomical Society.

An abstract of the results of the observations with regard to the points of greatest interest, such as Baily's beads, usually seen on the breaking up of the last narrow annulus of light; the corona or faint ring of light which becomes visible when the moon is quite hidden; and the red or rose-coloured prominences which are then seen outside, and in contact with the black lunar disk, will be found in the Annual Report of the Council of the Royal Astronomical Society, for February 1852.

In looking for proximate causes of these phenomena, it would naturally occur to any inquirer to seek for their origin, either in the spots on the surface of the solar disk, or in the faculae or wave-like ridges of light that are visible near the borders, and to this probable identification the attention of astronomers was directed previously to the eclipse of 1851. In particular, M. Schweizer, director of the Observatory of Moscow, took the precaution of having drawings carefully made of all the faculae which were visible for several days previous and subsequent to the eclipse; and he has come to some remarkable conclusions, which render it very probable that the red prominences are identical with them, and not with the spots.

Thus, though we cannot say that very much has been done in establishing a satisfactory theory respecting the constitution of the various layers forming the solar atmosphere, yet every possible advantage has been taken of all the circumstances by which, during the last few years, additional knowledge could be gained. Several circumstances which were matters of speculation have become now well-ascertained matters of fact, and materials are laid up which may at some time not far distant tend to the formation of a satisfactory theory based on observation.

It has been tolerably well confirmed by recent observations, and especially by photographic representations of the sun, that the light at the centre is considerably more intense than that near the borders of the disk; and Professor Secchi has recently been engaged in a series of experiments, which seem to prove very distinctly that the same is true with regard to heat.

Secchi's experiments were made with a thermo-electric pile belonging to a Melloni's apparatus, attached to the telescope of the equatorial of the Collegio Romano at Rome, and some of his conclusions are very satisfactory. Amongst these, the most important conclusion is, that the heat of the solar image is at the centre almost twice as great as at the borders, whether estimated in the direction of right ascension or of polar distance. A second conclusion satisfactorily arrived at is, that the point of greatest heat was not exactly at the centre of the disk, but about 3° above it in declination; and on constructing a curve of intensity, and considering the position of the sun's equator at the different times at which the experiments were made, the maximum line of heat appeared to be coincident with the equator. Professor Secchi also thinks it not improbable that the two solar hemispheres possess different temperatures, as seems to be the case with the earth; and if so, he considers that his researches, if continued, will throw some light on the climatology of our planet; since the heat of the sun would be different, accordingly as one or the other of its poles is turned towards the earth. He also had occasion to remark, that the influence of solar spots upon the temperature was very striking; so that sometimes a spot which did not occupy more than one-hundredth part of the aperture of the pile, caused the temperature to fall 3°, or about one-fiftieth of the whole intensity.

CHAP. II.

OF THE MOON.

Section I.—Correction of the Elements of the Lunar Orbit.

The greatest work of the present century, tending to bring to perfection the theory of the motions of the moon, is the reduction, under the direction of Mr Airy, of the ancient lunar observations made at Greenwich. These observations extend from the time of Bradley to that of Pond, and embrace a period of eighty years, during which the nodes or the orbit in which the cycle of many of the perturbations is completed have made more than four revolutions. The number of observations made in this interval, and thoroughly discussed, amount to about 9000.

It has been fortunate also to the science of astronomy, that the same great astronomer who had courage to undertake the vast amount of labour connected with the reductions, has found leisure to deduce from the results the corrections of the elements of the orbit. The paper containing the investigation connected with the corrections of the elements, is printed in vol. xvii. of the Memoirs of the Royal Astronomical Society, from which we extract the following curious and important results.

Correction of the Epoch of Mean Longitude.

The correction of this element proved to be of very great service to the lunar theory, by eliciting from Professor Hansen the discovery of a new inequality, depending upon the motion of Venus.

By dividing the observations into groups of about five years, Theoretical it was found that the corrections to Damoiseau's epoch of longitude were neither constant, implying that the mean motion was correct, nor uniformly increasing or diminishing, but that a periodic inequality was very clearly exhibited, showing the result of some unknown cause of disturbance. On the results being shown in MS. to M. Hansen, he very quickly traced the origin of the perturbation, and proved it to be due to the unequal action of Venus at different distances from the earth and moon. On the application of these inequalities for the respective epochs, the periodical inequality disappeared, and the successive corrections to the epochs of longitude for the different epochs showed an evident error of assumed mean motion, the secular correction to which was found to be \(+39^\circ\).

Another important result deduced by Mr Airy, was the correction of Bureckhardt's value of the moon's parallax, which he found ought to be increased by \(\frac{1}{10}\)th part.

Subsequently, however, Mr Adams (the English discoverer of the planet Neptune) has entered into a minute examination of Bureckhardt's tables, especially with regard to the parallax; and has found errors of such a magnitude as to enforce the necessity of applying the requisite corrections to all the observed places of the moon in which this tabular parallax has been applied, including all for which the data of the Nautical Almanac have been employed in the reductions. Mr Adams's paper on the lunar parallax is added to the volume of the Nautical Almanac for 1856, and the requisite tables are given for deducing the corrections to be applied to the observed places of the moon which have been reduced with the erroneous parallax.

In Mr Airy's corrections of the elements of the lunar orbit, there is an inequality in the latitude of the form \(+2^\circ-17\times\cos.\) moon's true longitude, which had not been detected by theory, but which clearly resulted from the observations. Professor Hansen, soon after the publication of Mr Airy's results, showed this also to be a legitimate deduction from theory, his calculated coefficient of the inequality being \(+1^\circ-38\). The cause of the inequality he found to be due essentially to the circumstance, indicated by Laplace, of the invariability of the mean inclination of the orbit of the moon to the plane of the earth's orbit, as affected by the action of the planets. The development of the expression for the latitude, according to this law of the inclination, with a slight modification, was found to contain a term of the form required. The same train of investigation has given a new term in the expression for the moon's longitude \(=0^\circ-50\times\cos.\) long., of node, while that deduced from observation and hitherto unexplained \(=0^\circ-97\times\cos.\) long. of node.

Again, in a subsequent communication to Mr Airy, M. Hansen announced that he had verified by theory another small term discovered by observation. According to his statement, he found the correction to Damoiseau's secular motion of the node to be \(+1^\circ-643\), while Mr Airy's investigations had given \(+1^\circ-721\).

Section II.—Of Selenography, or Observations and Delineations of the Surface and Physical Peculiarities of the Moon.

The surface of the moon has received that attention which might be looked for from the vast increase of optical power which has been recently brought into action, and from the zeal and talent which have been displayed in every department of physical research.

The most important delineation of the whole surface of the moon, as derived from observation, is that contained in the admirably executed map, accompanying the elaborate work by Beer and Mädler, entitled Der Mond. This map is the result of some years' careful study and micrometrical measurement of the surface of the moon, and every part discovered by the telescope is laid down with all available precision.

A raised model of the whole surface, deserving especial notice, was executed by Madame Witte, a Hanoverian lady, and exhibited in England in 1845. It is composed of a mixture of mastic and wax, forming a globe of about 12 inches in diameter, on which the positions and general outlines of the craters and other remarkable features were, in the first instance, laid down from their latitudes and longitudes, as given by Beer and Mädler, and the modelling was afterwards performed by the aid of a magnifying-glass from the actual appearance of the objects as presented in the telescope. Another original model, Astronomy: executed by the same lady, is laid up in the Royal Museum of Berlin.

Of English observers of the lunar surface, the most successful has been Mr James Nasmyth, a gentleman well known to the general, as well as the scientific public by his mechanical inventions. In the year 1844 he exhibited, and subsequently presented to the Royal Astronomical Society, a model and drawing of a portion of the surface, occupying a space of about 190 by 160 miles, the result of four years' careful and continuous labour. The observations were made with two reflecting telescopes, one of 53 inches aperture and 9 feet focal length, and the other of 12 inches aperture, and 13 feet focal length. Mr Nasmyth has continued up to the present time to direct his attention to the lunar surface, and in the memorable year 1851 he exhibited, at the Crystal Palace in Hyde Park, some excellent models and charts of various portions of it. He has recently announced that he is still engaged in sketching and modelling the surface, employing for the purpose an excellent reflecting telescope of 20 inches aperture, with a power of 410.

For Mr Nasmyth's ingenious speculations on the formation of the moon's craters, and on the apparent identity of the volcanic action with that existing on the surface of our own planet, the reader may consult the Memoirs and Notices of the Royal Astronomical Society.

Professor Secchi, to whom we are indebted for some valuable contributions to our knowledge of the heating properties of the solar surface, and who, in conjunction with Professor Piazzi, a distinguished Italian geologist, devoted considerable attention to the study of the surface of the moon, and furnished some remarkable speculations respecting the successive periods of volcanic action. We have not space for any detailed account of these speculations, for which the reader is referred to vol. xiii. of the Notices of the Royal Astronomical Society.

CHAP. III.

OF THE PLANETS.

Mercury.

The theory of the motion of Mercury has recently been most thoroughly investigated, and new tables have been constructed by M. Leverrier.

The investigation of the corrected orbit, which is contained in the additions to the Connaissance des Temps for 1848, is one of the most profound and laborious, and at the same time original, contributions to planetary astronomy, which have appeared during the last few years, and will well repay the perusal, both of the student and the accomplished astronomer.

We have space only for a brief explanation of the processes employed, and for a recapitulation of the principal results.

His object was to obtain new elliptic elements of the orbit from the Paris Observations made between 1801 and 1828, and between 1836 and 1842, and also totally to revise the whole theory of the perturbations of the planets, using the best modern elements, and taking in every term in the developments of the disturbing function, which could possibly exercise the smallest influence on the place of the planet.

Using Lindemann's elements as the basis of his investigations, he first proceeds to prepare the expressions for the equation of the centre and the radius-vector, to the ninth power of the eccentricity. The next and most laborious part of the work is the original redetermination of the perturbations of the elements of the orbit. This process is described most elaborately, and the action of each of the planets is given. Assuming the provisional values for the masses of the disturbing planets, he gives the numerical values of the secular variations of the eccentricity, perihelion, inclination, and node of the orbit, and points out a serious error committed by Laplace in the determination of the secular variation of the equation of the centre. Lastly, after having in a most elaborate way calculated the periodical variations of the elements of the orbit, and of the heliocentric co-ordinates produced by each planet separately, he sums up the provisional expressions of the elliptic elements, and the perturbations, with every detail.

The next part of the work is the reduction of the observa- Theoretical tions, and the comparison of the observed and tabular places.

Astronomy: For the reduction of the observations of R.A., his fundamental Supplement stars are taken from the Tabulae Regiomontanae; the right ascensions of the sun are taken from the Compendium des Temps, but are corrected by the Paris Observations; the radii-vectors of the Earth's orbit are taken from Bessel's tables; and Bessel's obliquity of the ecliptic is adopted in the reduction to geocentric longitude and latitude. In the reduction of the observations of declination, the planet is compared with stars lying nearly in the same parallel, and observed by the same observer—the zenith point apparently not having been determined independently. The mean parallax of the sun is assumed to be $8^{\circ}6$.

The observed right ascensions and declinations are then reduced to geocentric longitudes and latitudes; and the tabular heliocentric places are reduced by appropriate formulae to geocentric longitudes and latitudes; and these, when corrected for aberration, are immediately comparable with the observed geocentric longitudes and latitudes, and the errors of the tabular places are thus found. These errors are then converted into errors of heliocentric longitudes and latitudes by the use of formulae similar to those given in the Introductions to the Greenwich Observations, and enable the error of radius-vector. But the errors of longitudes can be immediately expressed in terms of the errors of the speed, mean motion, eccentricity, and perihelion, by differentiation of the expression for the true longitude, in terms of the mean anomaly and eccentricity; and the errors of latitude can be easily expressed in terms of the errors of inclination and node of the orbit; hence the two sets of equations are formed for the determination of the six elements of the orbit.

The next part of M. Leverrier's process is the collection and discussion of all the observations which had been made of the transits of the planet across the sun's disk, since those alone are capable of giving the exact position of the planet for more remote epochs.

Finally, he solves the equations resulting from the Paris Observations by the method of least squares, and obtains the errors of the elements.

One of the most important results is the correction which he finds it necessary to apply to the annual mean motion, namely, $+0^{\circ}424$. "The correction," he observes, "is large; but it is quite impossible without it to represent the ancient and modern observations completely."

Mr. Leverrier, for the purpose of testing the accuracy of his elements, applies them to the recomputation of the transits across the disk of the sun from the year 1631 to 1832; and the tabulated errors arising from the use of these elements, and those of Lindeman, being placed in adjacent columns, leave no doubt of the superior accuracy of the former.

He finally constructs two distinct sets of tables of the motion of the planets; one according to the old form, and the other according to a form totally original, and of which the chief peculiarity consists in taking the time for the sole argument throughout. The latter he considers to possess very great advantages, and enters into a minute critical investigation of the defects of all planetary tables constructed in the ordinary way.

Venus.

The planet Venus being so close a neighbour to the Earth and her satellite the Moon, it might be expected that she would produce serious disturbances on the orbits of these bodies. It has been seen that this is the case with regard to the Moon; and long before the discovery of this inequality, a very important inequality, of very long period, in the orbits of the Earth and Venus, produced by their mutual action, was detected by Mr. Airy.

This investigation (one of the most laborious and remarkable in physical astronomy since the time of Laplace) is contained in Part I. of the volume of the Philosophical Transactions for 1832; and we will give a few of the leading features of the investigation, without any explanation of the abstruser parts of the discussion.

In Mr. Airy's paper in the Philosophical Transactions for 1832, "On the Correction of the Elements of Delambre's Solar Tables," he had stated that the comparison of the corrections late observations, with the corrections given by the observations of the last century, appeared to indicate the existence of some inequality not included in the arguments of those tables; and as soon as he had convinced himself of the necessity of seeking for some inequality of long period, he commenced an examination of the mean motions of the planets, with the view of finding one whose ratio to the mean motion of the earth could be expressed very nearly by a proportion whose terms were small.

Venus is precisely in this predicament with regard to the Earth, since eight times the mean motion of Venus is so nearly equal to thirteen times the mean motion of the Earth, that the difference does not exceed $\frac{1}{4}$th part of the Earth's mean motion. This would imply an inequality whose period is about 240 years—a longer period with regard to the periodic time of the planets disturbed, than any for which a term of perturbation had been calculated.

The coefficient of the term of the expansion of the disturbing function in which this inequality exists, is of the fifth order with regard to the eccentricities and inclinations, and therefore would be exceedingly small; but on the other hand it would be multiplied in integration by a quantity amounting to $3 \times 13 \times (240)^2$ or about 2,200,000, and on this circumstance depended the chance of its becoming sensible.

The orbit of Venus has also received considerable attention during the present century, and the whole series of the Greenwich observations from 1750 to 1840 have been made available, and been employed to obtain the corrections of Lindeman's elements. These investigations, conducted successively by Mr Main, Mr Glaisher, and Mr H. Breen of the Greenwich Observatory, will be found in various volumes of the Memoirs of the Royal Astronomical Society, to which the reader is referred, in connexion with the Introductions to the Greenwich Observations, for explanation of the peculiarities of the methods employed.

Mars.

The orbit of Mars has received at the hands of Mr H. Breen a revision similar to that bestowed on the orbit of Venus, by a comparison of the places computed from Lindeman's tables, with the Greenwich observed places from 1750 to 1830. The investigation is precisely similar to that for the orbit of Venus, and will be found in vol. xix. of the Memoirs of the Royal Astronomical Society.

Next to the transits of Venus across the sun's disk (which afford absolutely the best means of determining the solar parallax), the observations of Mars at his opposition are the most useful for determining this element. The observatory at the Cape of Good Hope is admirably situated relatively to Greenwich for such determinations, and, ever since the time of La Caille, comparisons of observations between the two observatories have been of great service to astronomy. Notwithstanding the apparently definitive value of the sun's parallax obtained from the transits of Venus in 1761 and 1769, astronomers have been so impressed with the importance of establishing the correctness of this element by all other available means, that, at every opposition of Mars, the places of stars suitable for comparison with the planet are published in the Nautical Almanac, and observed at Greenwich and the Cape, and at other places. At the opposition of 1851, a good series of comparative observations of the planet was made at the two observatories, and, as soon as the transit circle now in preparation is established at the Cape, we may hope for some comparative results which will be of still greater service for the determination of the parallax.

The shape of the planet has been well measured, and it appears certain that it has a measurable ellipticity, though there is still some doubt as to its amount. A good set of measures made at Greenwich with a double-image micrometer in 1845, gives $\pm 0.4$ for the ellipticity, and another set made in 1852, gives $\pm 0.6$. M. Arago contends for a much larger ellipticity, namely $\pm 0.7$, as the result of the Paris measures, and some observers have not been able, after careful measures, to detect any certain difference between the polar and equatorial diameters.

The physical peculiarities of the planet have been well studied. On this head the reader may consult various papers in the The discovery of so many of these small planets has made it necessary to seek for some more convenient method of computing their perturbations than that of mechanical quadratures formerly in use. The difficulty of the calculations consists in the largeness both of the eccentricities and inclinations of the orbits, since it is by powers of these quantities or functions of them that the effects of perturbation are expressed in the ordinary methods.

In a paper published by Professor Hansen in 1843, in the Transactions of the Royal Academy at Berlin, and since translated into the French language by Mauvais, and published in the Connaissance des Temps for 1847, a method is given by which the variations of these elements can be expressed in general terms by means of series converging with a sufficient degree of rapidity.

Professor Encke has for several years devoted considerable attention to the same subject, and has recently devised a method of remarkable simplicity for effecting the summations of the perturbations. The pamphlet containing the exposition and application of the method has been translated from the German by Mr Airy, with additional notes and illustrations, and is appended to the Nautical Almanac for 1856.

A remarkable paper by Mr G. P. Bond, on the same subject, is printed in the fourth volume of the Memoirs of the American Academy, entitled "On some applications of the method of mechanical quadratures."

Jupiter.

On account of the vast size of the planet Jupiter, the determination of his mass is a matter of very great importance, and on account of the brightness of his satellites, this determination is easily effected by means of their observed elongations.

Till the year 1832, however, no observations of the satellites were extant which were sufficiently accurate for the purpose. At this time, Professor Airy made an excellent series of observations of the extreme elongations of the fourth satellite, and obtains for the value of the mass \( \frac{1}{10} \). In the years 1833 and 1834 he repeated the observations and found for the mass the values \( \frac{1}{10} \) and \( \frac{1}{10} \). He finally caused a similar set of observations to be made with the Sluckburgh equatorial at Greenwich, in 1836, and found for the value \( \frac{1}{10} \). The weights which he assigns to these results are 2, 2, 3, and 8, and his definitive value is \( \frac{1}{10} \). The separate investigations will be found in vols. vi., viii., ix., and x. of the Memoirs of the Royal Astronomical Society.

About the time of the commencement of Mr Airy's observations, Bessel had undertaken the same problem on a more extended plan, including not only the determination of the mass of Jupiter, but a complete revision of the elements of the orbits of the satellites, by means of observations of their elongations. For this purpose he commenced with the Königsberg heliometer a very elaborate series of measures of distance and angle of position of all the satellites relatively to their primary, which was continued from 1832 to 1835, and then again taken up and completed in 1839. His resulting values of the mass are as follow:

From the observations of the 1st satellite, \( \frac{1}{10} \) \( \frac{1}{10} \) \( \frac{1}{10} \) \( \frac{1}{10} \) \( \frac{1}{10} \) \( \frac{1}{10} \) \( \frac{1}{10} \)

and from these values he deduces for the definitive value of the mass of the Jovial system \( \frac{1}{10} \).

Bessel's dissertation on the mass of Jupiter is contained in the second volume of the Astronomische Untersuchungen, a work containing several of the most profound researches of this eminent astronomer, published in 1842. The dissertation is accompanied by new tables of the satellites.

An investigation of the value of the mass of Jupiter was also made by Professor Santini at Padua in 1835, by observations of distance of the fourth satellite from the centre of the primary, made with a double-image micrometer furnished by Amici. The definitive result for the mass is \( \frac{1}{10} \), and this dissertation also is accompanied by tables of the motion of the fourth satellite. (Memorie della Societa Italiana delle Scienze, vol. xxl.)

With regard to the form of the planet Jupiter, the received value of the ellipticity for many years was the one given by Struve at Dorpat—viz., \( \frac{1}{10} \). (See Memoirs of the Royal Astronomical Society, vol. iii.) This value has, however, been found to be sensibly erroneous, by means of measures taken in successive years at the Greenwich Observatory, with a double-image micrometer applied to the Sheepshanks equatorial. The value of the ellipticity, deduced from nearly fifty sets of measures averaging ten measures in each set, is \( \frac{1}{10} \).

We may finally mention a valuable paper by Mr Woolhouse, on the satellites of Jupiter, which will be found in the appendix to the Nautical Almanac for 1835. It contains new tables for the calculation of the occultations of the satel- The additions made to our knowledge of the theory and physical constitution of this important member of the solar system have been very great. In treating of them, we will take in order the orbit, the uranography of the body, the form of the body, the rings, and finally the satellites.

The tables representing the motion of Saturn, which have been used to the present time, are those of Bouvard, and in the construction of them a curious error has been detected by Mr Adams, in his course of investigations concerning the orbit.

Mr Adams detected a periodical error of latitude of considerable amount, which, he felt convinced, could not arise from any imperfection in the theory of the perturbations; and on examining the law of the error, of which the period appeared to be nearly twice that of Saturn in his orbit, he traced it to Table xlii. in Bouvard's Tables. The formula given in the introduction is correct, but the computation of the whole table is totally erroneous.

Mr Adams gives, in the 14th number of vol. vii. of the Notices of the Royal Astronomical Society, a correct table of the values of the inequality in question, to which we would direct the attention of the reader.

In considering the various phenomena which must be constantly occurring before the eyes of a spectator situated on the surface of Saturn, attention is naturally directed to the effects which would be produced by the projection of the rings on the firmament of the spectator, and to the consequences resulting from its interposition for a long period between him and the sun. It has been generally considered, without entering into calculation, that, to the inhabitants below a certain latitude of the planet, the rings, stretching in a circular shape across the heavens, would present a magnificent spectacle for a certain number of years, or as long as the sun should continue on the same side of the ring with the spectator; but that an equally long eclipse, with total absence of light, would be suffered by the inhabitants after the passage of the sun to the other side of the ring.

Dr Lardner has recently shown, in an elaborate paper printed in vol. xxii. of the Memoirs of the Royal Astronomical Society, that this view of the subject is to a great extent erroneous; and that even Mädler, who, in his Popular Astronomie, has entered more minutely into the consideration of the uranography of Saturn, has made some grave errors in treating the subject. For the details of the investigation by Dr Lardner, we must refer to his paper.

The form of the planet Saturn has been the subject of considerable discussion at various times, but its strictly elliptical figure has been recently established beyond controversy, and the amount of the ellipticity has been very accurately determined.

At the time of the disappearance of the ring in the years 1832 and 1833, Bessel took the opportunity of measuring the planet with the heliometer, with the express view of deciding upon the correctness of Sir William Herschel's opinion, "that at middle latitudes the figure of Saturn deviated considerably from an ellipse," and also of determining accurately the amount of the ellipticity. He measured, therefore, in the directions of the polar and equatorial diameters, and of diameters lying nearly in latitude 45°, and found that the shape was strictly elliptical, and that the ellipticity, deduced from all the measures taken when the ring was either very small or totally invisible, was \( \frac{1}{4} \).

Recently, at the last disappearance of the ring in 1848, a series of measures was made with a similar object at Greenwich, by the Rev. R. Main, which give results almost identical with those of Bessel. The shape is proved, by a careful discussion of all the measures, to be strictly elliptical, and the deduced ellipticity is \( \frac{1}{4} \). Mr Main's paper is printed in vol. xix. of the Memoirs of the Royal Astronomical Society.

With regard to the present physical condition of the surface of Saturn as well as of Jupiter, Mr Nasmyth has thrown out some ingenious conjectures based on the knowledge which the rapid advance of geological science has gained for us with regard to the former conditions of the surface of our own planet. Such inquiries are fair subjects of speculation in the present state of the physical sciences, and Mr Nasmyth's paper in No. 2 of vol. xiii. of the Notices of the Royal Astronomical Society is well worth perusal.

Amongst the most remarkable discoveries of recent times with regard to the rings of Saturn, is that of the inner dusky or semi-transparent ring, sufficiently obvious to any observer capable of using well a moderately good telescope, but which, previously to the year 1830, was scarcely suggested by any astronomer. It appears, however, by a paper by Dr Galile of Berlin, published in the Nachrichten, No. 756, and discussed with his usual judgment by Mr Dawes in the Monthly Notices of the Royal Astronomical Society, vol. xi. p. 184, that this interesting phenomenon was seen by him with the large Berlin refractor in the year 1838, though the observations were not published at the time in the Nachrichten, and the attention of the scientific world was not generally drawn to it. The account of Galile's observations, accompanied by drawings exhibiting the trace of the dusky ring as it crosses the body of the planet, was given by Encke in the Transactions of the Berlin Academy for 1839.

This was the only indication of the existence of the inner ring received by the scientific world from that time till the year 1850, when its existence was recognized almost simultaneously by two observers, namely, by Professor Bond of the Cambridge Observatory, U.S., and by the Rev. W. R. Dawes at Wateringbury, near Maidstone. Since that time there has been no difficulty in seeing this curious appendage to Saturn, though it still requires a practised eye and a good telescope.

At the time of the discovery of the dusky ring, Mr Dawes also satisfactorily established the fact of the outer division of the exterior ring near its outer extremity; and subsequently he observed a series of discontinuous gradations of colour or intensity of brightness in a portion of the inner bright ring. He observes, that "the exterior portion of the inner bright ring to about one-fourth of its whole breadth was very bright, but that interior to this, the shading-off did not appear, as under ordinary circumstances, to become deeper towards the inner edge without any distinct or sudden gradations of shade; on the contrary, it was clearly seen to be arranged in a series of narrow concentric bands, each of which was darker than the next exterior one. Four such were distinctly made out; they looked like steps leading down to the black chasm between the ring and the ball. The impression I received was, that they were separate rings, but too close together for the divisions to be seen in black lines." This curious phenomenon was confirmed afterwards by Professor Bond.

Many valuable observations of Saturn and his rings have been made recently by Mr Lassell, who, in the autumn of 1852, established himself with his 20-foot reflecting telescope at Malta, for the purpose of observing in a purer atmosphere. Amongst his discoveries may be mentioned the fact of the semi-transparency of the dusky ring, which curious fact had been previously recognized by Captain Jacobs at the Madras Observatory.

Mr De la Rue has also devoted considerable attention to the system of Saturn, and has recently published an admirably executed drawing, which embodies the results of his observations.

We would, before concluding our remarks on the rings of Saturn, draw attention to a remarkable paper by M. Otto Struve, printed in vol. v. of the Petersburg Memoirs. M. Struve shows with tolerable certainty that the inner or dusky ring is not a modern appendage to the planet, as might almost be suggested by the fact of its remaining so long undiscovered; but that at the beginning of the eighteenth century, the dark line thrown by it across the planet was not confounded with the shadow of the bright rings, but was known by the name of the equatorial belt.

But one of the most curious results of M. Struve's researches is, that by a comparison of the micrometrical measures of Huyghens, Cassini, Bradley, Herschel, W. Struve, Encke, Galile, and himself, he finds that the inner edge of the interior bright ring is gradually approaching the body of the planet, while at the same time the total breadth of the two bright rings is constantly increasing.

An eighth satellite of Saturn was discovered in 1848, independently, and almost simultaneously by two observers, Professor Bond and Mr Lassell. The names of the seven satel- For many years the orbit of Uranus has been the occasion of great embarrassment to astronomers, from the impossibility of adequately reconciling the ancient and modern observations by any one set of men, and from the rapid increase of the tables from year to year.

Amongst the astronomers who entered seriously upon the task of tracing the course of the irregularities in the motions of the planet we can only mention those who arrived at a successful solution of the problem; namely, M. Leverrier and Mr Adams, each of whom has attained to a world-wide celebrity by the profound and masterly analysis which led to the discovery of the disturbing planet Neptune.

In France the attention of M. Arago had been particularly directed to the subject; and about the year 1845 he earnestly represented to M. Leverrier the importance of the question, and induced him to lay aside the cometary researches on which he was engaged, and to enter upon the painful and laborious calculations which the problem demanded.

The first part of M. Leverrier's work consists of a complete revision of the theory of the perturbations of Uranus; and his first memoir, which was presented to the Institute in November 1845, is devoted to the exact redetermination of the perturbations produced by Jupiter and Saturn; and this profound investigation is conducted with the same masterly skill which distinguished his researches on the orbit of Mercury.

The second memoir was read before the academy on June 1, 1846, and has for its chief object the comparison of the observed place of Uranus with the tabular places computed by Bouvard's tables, corrected according to the previous investigations for the new inequalities produced by Jupiter and Saturn, and for such other errors as had been discovered in the progress of the work. He arrives at the important conclusion, that though large errors existed in the tables, and through this cause alone they could not correctly represent the observations, yet that the difference existing between theory and observation could not be adequately accounted for by such imperfections, and that the cause of the "discrepancies" must be sought for elsewhere.

It is but just to Mr Adams to state, that he also, before entering upon the task of accounting for the discrepancies observed, by the hypothesis of an exterior disturbing planet, recomputed all the principal inequalities produced by the action of Jupiter and Saturn, and satisfied himself that errors in the tables would not account for the failure in the agreement between theory and observation.

With regard to the satellites of Uranus some considerable discoveries have been made, though modern research has failed to identify the greater number of those seen by Sir W. Herschel. However, two additional satellites certainly (and probably three) were found in 1847 by the exertions of Mr Lassell and M. Otto Struve, each of which is nearer to the planet than any of those discovered by Sir W. Herschel. In the Notice of the Royal Astronomical Society for January 1848, will be found the account given by each of these astronomers of the discovery of at least one satellite interior to the second of Sir W. Herschel. Mr Lassell observed the same object for several nights, and on one occasion he observed an additional one, the positions of the object observed being always on the north side of the planet. M. Otto Struve, on the contrary, observed an object always on the south side only of the same object. In the Notice for March 1848, will be found a paper by Mr Dawes, in which he discusses very thoroughly the whole of the observations made by both astronomers, and arrives at the conclusion that the objects observed by them are not identical, and consequently that two satellites must have been discovered.

Mr Lassell has since that time been able to follow up the observations of both these bodies, and by a comparison of the observations made by him in 1847, 1851, and 1852, he finds theoretical for the period of the inner satellite which he calls Ariel, Astronomy: 2:520378 days, and for the period of the exterior one (Um: Supplement brief) he deduces 4:144537 days.

By a rigorous scrutiny of all the observations made by the two Herschels, Dr Lammont, and Mr Lassell, of the two bright satellites of Sir W. Herschel, Mr Adams finds for their periods 84° 16' 56" = 24° 28', and 13° 11' 6" = 55° 21'.

Neptune.

In whatever light we regard the discovery of the planet Neptune, it must certainly rank amongst the most brilliant of the discoveries of the present century;—the highest achievement in an age abounding with every resource of science and intellectual cultivation.

It has been stated that Leverrier undertook the task of the revision of the theory of Uranus at the instance of M. Arago, that his first memoir on the subject was read before the French Academy in November 1845, and that his second memoir was read in June 1846. These memoirs were printed in the Comptes Rendus, vols. xxi. and xxii. In the second memoir Leverrier demonstrates that an account of the motions of Uranus can only be given by the introduction of the disturbing force of a new exterior planet, and he fixes its position for 1847, January 1, at 32° heliocentric longitude.

The third memoir, entitled "Sur la planète qui produit les anomalies observées dans le mouvement d'Uranus—Determination de sa masse, de son orbite, et de sa position actuelle," was read on August 31, 1846. In it the position of the disturbing planet is fixed more exactly at 326° 39' heliocentric longitude for 1847, January 1.

The three memoirs mentioned above were read before the discovery of the planet;—the fourth memoir, containing the remaining part of M. Leverrier's investigations, was read after its discovery, on October 5, 1846, and both this and the third are printed in vol. xxiii. of the Comptes Rendus. The fourth memoir is entitled "Sur la planète qui produit les anomalies observées dans le mouvement d'Uranus; cinquième et dernière partie, relative à la détermination de sa position au plan de l'orbite."

Finally, the whole investigation was printed as one continuous treatise in the Additions to the Connaissance des Temps for 1849.

The discovery of the disturbing planet followed almost immediately after the publication of the third memoir of Leverrier. In acknowledging the receipt of a paper from his friend Dr Galle of Berlin, he took the opportunity of requesting him to search for the planet with the large refracting telescope of the Berlin observatory, at the position which he indicated to him. This letter reached Berlin on September 23, and on the same evening Galle had an opportunity of complying with the request contained in it. He observed all the stars in the neighborhood of the place indicated, and compared their places with those given in Bremiker's Berlin Star-Map (Hora xxii.). This map, it is necessary to say, had not yet reached England, and on this circumstance probably depended the priority of discovery at Berlin. He very quickly found a star about the eighth magnitude, nearly in the place pointed out, which did not exist in the map. Little doubt was entertained at the time that this was the planet, and the observations of the next two days confirmed the discovery.

It will now be necessary to explain in few words the attempts made in England during the same period of time to secure the discovery of the planet. Mr Adams had, ever since the year 1841, determined on attempting the solution of the problem relating to the unknown disturbances of Uranus, and in 1843 he began his investigations. In September 1845, he communicated to Professor Challis the values of the elements of the orbit of the supposed disturbing planet, and, in the following month, he communicated to the astronomers-royal the same results slightly corrected. Mr Airy entered into correspondence with him, and requested to be informed whether the assumed perturbations would explain the errors of radius-vector as well as of longitude. From some unexplained cause, no answer was received to this letter, and no steps were taken in England to secure by observation the discovery of the planet till the summer of 1846, after the publication of Leverrier's The resources of the Cambridge Observatory were now brought into use for the discovery of the planet, and a systematic search was begun by Professor Challis with the great Northumberland telescope. With regard to the success of the system devised for conducting the observations, it is sufficient to say that the sweeps of the portion of the heavens in which it was supposed the planet would be found, were begun on July 29, and the planet was actually observed on August 4, but without recognition. After the discovery of the planet at Berlin, it was found that the planet had also been observed on August 12. If, therefore, Bremicker's map had been in the hands of Professor Challis, or if he had found leisure for the mapping of his observations from night to night, Neptune would have been infallibly detected within a very few days of the commencement of the search, and the whole glory of the discovery would have belonged to the English geometer.

The subsequent history of the planet may be summed up in a few words. An ancient observation in Lalande's catalogue was discovered almost simultaneously by Mr Walker, of the Washington Observatory; and by Dr Petersen at Altona; two observations, the one in October 1845, and another in September 1846, were also detected by Mr Haid in Lamont's Zones; and these (especially the former) contributed greatly to the construction of an accurate orbit. An excellent orbit was computed by the joint labours of the American astronomers Peirce and Walker, which, up to the present time, continues to represent perfectly the motions of the planet.

Our sketch of the history of Neptune is necessarily so imperfect, through want of space for details, that we feel it necessary to add a list of the principal works to be consulted by those who are desirous of obtaining correct notions on all points relating to this brilliant discovery—1. Airy's Account of some circumstances historically connected with the Discovery of the Planet exterior to Uranus.—Notices of the Royal Astronomical Society, vol. vii., p. 121; 2. Gould's Report to the Smithsonian Institution on the History of the Discovery of Neptune; 3. Lindemann's Beitrag zur Geschichte der Neptun-Entdeckung, in the Ergänzung-Heft zu den Astr. Nachr., published in 1849; 4. Sir J. Herschel's account in his Outlines of Astronomy.

Almost immediately after the discovery of Neptune, it was found to be attended by one satellite. This discovery was made by Mr Lassell in October 1846, and the orbit of the satellite is found to be inclined to the ellipse at an angle of about 35°. By observations of this satellite at Cambridge in America, and at Pulkowa, two separate values of the mass, namely, \( \frac{1}{7} \) and \( \frac{1}{11} \), have been deduced; but the object is so difficult to observe that a considerable time must elapse before any very accurate determination will be arrived at.

CHAP. IV.

OF COMETS.

On the chief Discoveries recently made in Cometary Astronomy.

The discoveries in Cometary Astronomy have kept pace with those in the other departments of the science.

With regard to theory, it has become of great importance, on account of the great number of comets which are discovered, to ease as much as possible the practical difficulties of the methods of computing the orbits, and many excellent papers will be found in the Astronomische Nachrichten, and in the Memoirs of the Royal Astronomical Society.

In particular, we would refer to a paper by Mr Airy, in vol. xi. of the Memoirs, the object of which is to render Laplace's method of computing the orbit of a comet easier of application, by enabling the computer to deduce the orbit immediately from the observed right ascensions and declinations, without the conversion of these quantities into longitudes and latitudes. This method evidently requires that the co-ordinates of the sun referred to the equator should also be computed; and these co-ordinates have therefore been given for some years past in the Nautical Almanac.

For an excellent method of calculating an ephemeris of a theoretical comet from its elements, we would also refer to a paper by Mr Woolhouse, in the appendix to the Nautical Almanac for 1835.

We will now proceed to give brief notices of some of the most remarkable comets which have appeared during the present century. For full information, and for complete catalogues of comets, the following works may be referred to—1. Jahns's Verzeichnisse aller bis zum Jahre 1847 berechneten Cometenbahnen, Leipzig, 1847; 2. Hind's work, entitled The Comets, London, 1832; 3. Cooper's Cometic Orbits, Dublin, 1862; 4. Encke's valuable edition of Olbers's Abhandlung, Berlin, 1847.

Encke's Comet, or Comet of Pons.—This comet was discovered in October 1805, almost simultaneously by Pons and other astronomers, being the third observed appearance. At this apparition it was sufficiently bright to be seen with the naked eye. Its next apparition was in 1818, when it was again discovered by Pons on November 26. By calculations according to Gauss's methods, based upon the great number of observations made at this apparition, Encke proved the orbit to be an ellipse, with a period of about 23 years, and established its identity with the comets which appeared in 1786, 1795, and 1805. To do this it was necessary to calculate the whole of the perturbations which it had experienced from the large planets, from the time of its apparition backwards to the dates of the other apparitions. This enormous labour Encke satisfactorily accomplished in a very short space of time; and the comet has ever since been called most commonly after his name. Since that time it has been observed at most of its apparitions, of which one of the most favourable for observation was in 1838. It was in the discussion of the elements of the orbit from the observations made at this apparition, compared with the previous ones, that Encke was induced to enter seriously upon the celebrated problem of the retardation probably experienced by the comet by a highly attenuated resisting medium. The value of the constant of resistance was afterwards definitely determined by him, and the disturbance produced by this singular cause has been introduced into every subsequent discussion of the orbit.

In the year 1805 appeared the comet now known as Biela's Comet, also of short period. It was discovered by Pons on November 10. It was afterwards observed at the apparitions of 1826, 1832, and 1846, and at the last appearance presented the remarkable phenomenon of a double comet, both components being together in the field of the same telescope. Its period is rather less than seven years.

In 1807 appeared a great comet, surpassing in splendour any which had appeared since 1769. One of the most remarkable phenomena attending it was a double tail.

In 1811 appeared a very splendid comet, whose appearance is still remembered by many persons yet living. Its orbit was so situated that for many months it was circumpolar, and this added to the lustre of its appearance. In October the tail extended over an arc of 25° in length.

In 1812 appeared a fine comet, at one time visible to the naked eye. It was discovered by Pons on July 20. Encke calculated elements, giving a period of about 71 years.

In 1819 a great comet suddenly made its appearance at the beginning of July. The tail was about 7° in length, and at one time it must have transited the sun's disk.

In 1826 appeared a comet worthy of notice, from the fact of its exhibiting two tails, of which one was turned towards the sun, and the other in the opposite direction.

In 1825 appeared a great comet, which was discovered independently by Pons and Biela in July, and continued visible till the same season of the following year. The tail, which was double, extended over an arc of 15°.

In 1830 appeared two comets, visible to the naked eye, the first of which was discovered by Gambart in April. Each was attended by a tail, and the nucleus of each was bright.

In 1835 appeared, according to prediction, the famous comet of Halley. It was first detected at Rome by Dumouchal, on the 5th of August 1835, and was observed till May 1836; and excellent series of observations of it were made at the greater number of the observatories both of the northern and southern hemisphere. For particulars of its appearance, the reader Theoretical may consult Sir J. Herschel's Results of Observations made Astronomy at the Cape of Good Hope; the Memoirs of the Royal Astronomical Society; and Struve's Beobachtungen des Halleischen Cometen, where admirable drawings will be found of its appearance at various phases. To these may be added Denys's drawings in the Astronomische Nachrichten, No. 362.

Wolfe's elliptic orbit corresponds to a period of about 70 years.

In 1843 appeared the most remarkable comet of the present century, familiarly known as the great comet of 1843. This comet came suddenly upon astronomers in all its brightness, the immense tail being first seen in England in the month of March, like a long band of light cirrus cloud, not very high above the south-west horizon soon after sunset.

For accounts of its physical appearance, we are indebted chiefly to accounts given by officers of ships on their passage to England from the southern seas, and a great number of such accounts will be found in vols. v. and vi. of the Notices of the Royal Astronomical Society. As seen by northern observers, the spectacle presented by it must have been very grand. The tail extended over an arc varying from 30° to 45°, and its breadth was so great that it produced the effect of two distinct streams of light united at the head of the comet, but separated throughout nearly the whole length by a dark interval.

But the most remarkable observations of this comet were made in America at the time of its nearest approach to the sun's disk, in broad daylight, by Mr J. G. Clarke, at Portland, in the State of Maine. He measured the distance of the nucleus from the sun (about 3½) in strong sunshine, and states that it and the tail were as well defined as the moon in a clear day, and resembled a perfectly pure white cloud.

In 1843 was also discovered a comet of short period, by M. Paye, which has been the subject of some remarkable papers by Leverrier, in vols. xxx., xxxi., and xxvi. of the Comptes Rendus. The period of this comet is about 7½ years, and it was observed on its return in 1859.

In 1844 another periodic comet was discovered by De Vico at Rome. The period is about 1903 days, but it was not observed at its return in 1850, owing to its unfavourable position.

In 1845 a bright comet was discovered by Colla at Parma, on June 2. It was at one period visible to the naked eye, and exhibited a tail.

In 1846, on February 26, an interesting periodic comet was discovered by Brooman. The period is about 53 years. It was not seen at its return in 1851, being very near the sun at the time of its perihelion passage.

In 1847, a remarkable comet was discovered by Mr Hind on February 6. Before its perihelion passage, it became bright enough to be seen in the morning twilight, and it was observed close to the sun on March 30, at noonday.

In 1850, on May 1, a comet was discovered by Petersen, which became bright enough to be visible to the naked eye, with a tail several degrees in length.

In 1851, Brooman discovered a bright comet on October 22, which exhibited a double tail.

In the present year 1853, a comet was discovered by Klinkerfus at Göttingen, on June 10, which in August became very bright, and for several evenings was a splendid object in the west, seen after the setting of the sun. It exhibited a tail perfectly straight, of several degrees in length.

CHAP. V.

OF THE FIXED STARS.

Sect. I.—Star Catalogues.

Within the last few years very many valuable Star Catalogues have been added to the list, through the combined labours of public and private observers. We can give only the names of a few of the most important, with such notices of their construction as seem necessary.

1. The Catalogues of Ptolemy, Ulugh Begh, Tycho Brahe, Halley, and Flamsteed, have been admirably edited by Mr Baily, and form vol. xiii. of the Memoirs of the Royal Astronomical Society.

2. Flamsteed's Catalogue in the Historia Cælestis has been also edited by Mr Baily, and was published in 1835, together with his life of Flamsteed.

3. A Catalogue of the Stars in Lalande's Histoire Céleste Theoretical (about 40,000 in number), edited and compiled by Mr Baily Astronomy; at the expense of the British Association, and printed at the expense of the Government in 1847.

4. The Catalogue of Stars of the British Association, containing 3877 stars, edited by Mr Baily. 1845.

5. Lalande's Catalogue of 9766 Stars, reduced at the expense of the British Association, under the superintendence of Professor Henderson, and printed at the expense of the Government, under the direction of Mr Baily, with Preface by Sir J. Herschel. 1847.

6. Sir Thomas Brisbane's Catalogue of 7335 Stars, chiefly in the Southern Hemisphere, observed at Parramatta. 1835.

7. Grooteveld's Catalogue of Circumpolar Stars, edited by Mr All. 1838.

8. Johnson's Catalogue of 606 Principal Stars in the Southern Hemisphere. 1835.

9. Argelander's Catalogue of 500 Stars exhibiting Proper Motions observed at Abo. 1835.

10. The Greenwich Twelve-Year Catalogue of 2156 Stars. 1849.

11. Taylor's Madras Catalogue of 11,015 Stars. 1844.

12. Weise's Catalogue of those Stars in Bessel's Zones, situated in a Zone extending 15° North and South of the Equator. 1846.

13. Olbrin's Catalogue of the Stars in Argelander's Zones, situated between 65° and 80° North Declination. 2 vols. 1851 and 1852. (This valuable Catalogue forms vols. i. and ii. of the Third Series of the Annalen des k. k. Sternwarte in Wien.)

14. Rümker's Catalogue of 12,000 Stars. 1843-1851.

15. Cooper's two Catalogues of 30,667 Stars near the Elliptic, observed at Markree. 1851 and 1853.

Several other Catalogues will be found in the Memoirs of the Royal Astronomical Society, but we have not space for separate accounts of them.

The following Catalogues of Double Stars must be mentioned:

1. Struve's Great Catalogue of Double and Multiple Stars, observed at Dorpat. 1837.

2. Struve's Catalogue of 514 Double and Multiple Stars observed at Pulkowa. 1843.

3. Herschel and South's Catalogue of 360 Double and Triple Stars. (Phil. Trans. for 1825.)

4. Dawes's Catalogue of 121 Double Stars, observed at Ormskirk in the years 1831-1833. (Memoirs of the Royal Astronomical Society, vol. viii.)

5. Smyth's Cycle of Celestial Objects. 2 vols. 1844.

6. Struve's Catalogue of the Mean Places of the Stars observed at Dorpat from 1822 to 1843. 1852.

Sect. II.—Of the Apparent Magnitudes, Number, and Distribution of the Stars.

In any scheme for the re-arrangement of the stars in new constellations or asterisms, according to the views of Sir John Herschel (Memoirs of the Royal Astronomical Society, vol. xii.), or for thoroughly correcting the old arrangement of the constellations, a knowledge of their relative brightness or apparent magnitudes is absolutely necessary; yet this department of the science remains, even at present, in a very unsatisfactory state; and the estimated magnitudes set down in our best catalogues are always vague and frequently inaccurate.

Notwithstanding this, however, several successful attempts have been made by eminent astronomers to put this branch of science on a better footing; and, under this head, its progress is mainly due to Argelander, Herschel, Johnson, and Dawes.

Argelander's remarkable work, the Uranometria Nova, published in 1843, gives for the northern heavens the most accurate scale of magnitudes for all stars visible to the naked eye, though the estimations do not profess greater accuracy than the descending scale of thirds of magnitudes employed by him will furnish.

Sir J. Herschel, while at the Cape, made the estimation of the magnitudes of the stars on a better system than any formerly employed—one of his most important subjects; and by naked-eye observations alone has assigned magnitudes to nearly 500 stars on a systematic plan, and with a degree of demon- Theoretical stralble accuracy far exceeding anything which had been previously attempted. For a detail of his methods we must refer Supplement to his Observations at the Cape of Good Hope.

Mr Dawes has also been successful in devising a method for observing the magnitudes of telescope stars, which is very easily applicable, and requires only ordinary perseverance and laborious observing, to establish, on a perfectly satisfactory basis, a scale of magnitudes for all telescope stars in the heavens. (See Monthly Notices of the Royal Astronomical Society, vol. xii.)

Mr Johnson has also, in the course of his re-observation of the stars of Groombridge's Catalogue, paid great attention to the correct estimation of their magnitudes, and has lately devised a method, by the use of the heliometer, for comparing directly the magnitudes of such stars as lie within its range, so as to be brought into the same field. The results of his researches are published in a pamphlet appended to his last published volume of observations.

On the subject of the number and distribution of the stars, a most valuable work was published by the elder Struve in the year 1847, entitled Études d'Astronomie Stellaire : sur la Voie Lactée et sur le distance des Étoiles Fixes; which gives a most lucid account of his own elaborate researches on this subject distributed throughout his other works, and especially of his preface to Weisse's Catalogue. Of this treatise an excellent abstract was given by Mr Airy, and is included in the Report of the Council of the Royal Astronomical Society of February 11, 1848, printed in the Monthly Notices. To this extract we must refer the reader for further information.

The almost-demonstration of the means of finding the distances of some of the fixed stars, has been completely solved during the present half-century, and in no branch of astronomy has the perfection, both of the theoretical and practical methods adopted by modern astronomers, been more severely tested, or more triumphantly successful.

The star 61 Cygni, a binary system of two stars of very nearly equal magnitudes, connected by gravity, and having a large proper motion, is that of which the parallax was first detected and measured by Bessel, by means of comparative measures of distance made with the heliometer, extending from the year 1837 to 1840, of which the whole series will be found in vols. xvi. and xvii. of the Astr. Nachr. The definitive value of the parallax arrived at is 0°3483, which corresponds to a distance of about 600,000 radii of the earth's orbit.

The bright southern star α Centauri, has also had its parallax determined with great care by several series of meridian observations of zenith distance made by Professor Henderson and Mr Maclear. The resulting parallax amounts to very nearly 1°.

The parallax of α Lyrae was determined by micrometrical measurements, made by M. Struve, of its distance from a neighbouring faint star. The series of observations extended from November 1835 to August 1838, and the resulting value of the parallax is 0°2619.

But the most remarkable paper on annual parallax is that by M. Peters of the Pulkowa Observatory. He gives in this paper the results of the meridian observations made with Eratosthenes' circle, to determine the parallaxes of the stars Polaris, Capella, Urse Majoris, Groombridge 1830 (having large proper motion), Arcturus, Lyrae, Cygni, and 61 Cygni. The results he arrives at are as follow:

| Star | Parallax | With prob. error | |---------------|----------|-----------------| | Polaris | +0°067 | 0°012 | | Capella | +0°046 | 0°200 | | Urse Majoris | +0°133 | 0°106 | | Groombridge 1830 | +0°226 | 0°141 | | Arcturus | +0°127 | 0°073 | | Lyrae | +0°103 | 0°053 | | Cygni | -0°082 | 0°043 | | 61 Cygni | +0°349 | 0°080 |

We have endeavoured in the preceding pages to give a few of the most prominent of the researches and discoveries which have enriched astronomical science in the present century; but the very small space which could be allotted for this extension renders it necessary the accounts very imperfect, and forbids any details. It is hoped, however, that the references given will enable the reader to obtain for himself the full details of investigations of which we have only indicated the outlines, and to study fully the methods by which the results given in the preceding pages have been arrived at.

(P.M.—N.)

PART III.

PHYSICAL ASTRONOMY.

The whole science of astronomy may be reduced to two general problems. The first is to express the position of all the heavenly bodies in terms of the time reckoned from a given instant, either in the past or the future duration of the world. The same may be otherwise stated by saying, that the thing required is, to express the position of any one of the heavenly bodies in a function of the time, the time being considered as the only variable quantity, though combined with other known quantities, which enter into the function as the co-efficients of the different terms. This is the most general view of that which is usually called descriptive, or sometimes geometrical astronomy. The solution of this problem enables us to determine for any time the places of the heavenly bodies, relatively to one another, and relatively to any point on the earth's surface. It contains under it an endless variety of subordinate problems, embracing a long series of successive generalizations, from the first observations to the determination of the orbits of the heavenly bodies, and the final reduction of all that concerns their motions into the form of astronomical tables. The second problem is, to compare the laws of motion in the heavens, as discovered from the preceding investigations, with the laws of motion as already known on the surface of the earth, in order to find out whether or not they are the same; and, if not, in what their difference consists. The solution of this problem constitutes what is called Physical Astronomy; it is the same with inquiring into the causes of the celestial motions; for by causes we mean the general facts concerning the motion of bodies which are observed to take place on the surface of the earth.

Though the first of these two problems goes necessarily before the second, for the solution of which it affords the data, yet, after this solution is obtained, it affords great assistance to many of the researches involved in the first, and exemplifies, in a most remarkable manner, the use of theory in the investigation of facts, and the re-action, as it were, of the second problem on the first.

Taking for granted the solution of the first problem, as given under the other Parts of this article, we are now to

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1This portion of the present treatise is reprinted from the article on Physical Astronomy contributed by the late Professor Playfair to the Supplement to the fourth, fifth, and sixth editions of this work. The chief alteration made in this, the eighth edition, is the substitution of the differential for the exploded fluxional notation. consider the second, and to explain the manner in which it has been resolved by Newton and the philosophers who have come after him.

The history of the first of these two problems is long and interesting, beginning from the remotest period to which the records or the traditions of mankind have ventured to ascend, and coming down to the present time; and, in the ages to come, it is never likely to know any limit but the movable instant which separates the past from the future,—as long, at least, as science and civilization are inhabitants of the earth.

The history of the second comes within small compass; because, between the first rude effort and the last refined investigation there is hardly any intermediate step but one.

The concentric orbs of the ancient philosophers were an attempt at an explanation of the physical causes of the celestial motions, or at an assimilation of those motions to such as we are accustomed to see on the surface of the earth. The great phenomenon to be explained was the diurnal motion of the heavens, by which so many bodies, very distant from one another, all describe circles round the earth, keeping time so precisely with one another, that the revolutions, whether great or small, are accomplished in the same interval. This could not be, unless a connection subsisted between those bodies; and the most simple idea of that connection was, that the bodies were fixed in the surface of a sphere which revolved on an axis, and carried them along with it.

If the whole of astronomy had been confined to the single fact of the diurnal revolution of the fixed stars, the hypothesis just mentioned would have been quite satisfactory. But as some of the heavenly bodies, such as the sun and planets, did not revolve precisely in the same time with the rest, it was necessary to assign to them particular spheres of their own. Those spheres, therefore, must be transparent: light must find an easy passage through them, and hence they must be crystalline. By degrees, as more accurate knowledge was obtained of the motion of the planets, it was found necessary so to increase the number of the spheres, that the complication of the structure was burdensome to the imagination; the hypothesis did not answer the very first object of a theory, that of connecting the facts together; and it was so unlike any process of nature with which we are acquainted, that it was highly improbable. The hypothesis of the homocentric orbs therefore fell into discredit, and, after the discovery of the earth's motion, was entirely abandoned.

The next attempt to explain the whole system of the celestial motions was that of Descartes, by means of vortices of subtle matter, and the pressure which, by the centrifugal force of those vortices, was produced on the grosser bodies of the stars. But as a taste for accurate knowledge increased, and as men reflected more on the true objects of philosophic theory, the system of vortices appeared more and more defective, and at length ceased to have any followers.

Newton succeeded, who, rejecting all the cumbersome machinery, both solid and fluid, of his predecessors, adopted a plan far more philosophical in the design, and far more difficult in the execution, than any thing yet known in the physical or mathematical sciences. Assuming as true the three general facts concerning the planetary system known by the name of the laws of Kepler, he proceeded to inquire by what sort of action on one another the planets could be made to describe orbits having the properties indicated by these three general facts. The general facts to which we now refer, are,

1. That every planet moves so, that the line drawn from it to the sun describes about the sun areas proportional to the times.

II. That the planets describe ellipses, each of which has one of its foci in the same point, viz., the centre of the sun.

III. That the squares of the times of the revolutions of the planets are as the cubes of their mean distances from the sun.

Sect. I.—Of the Forces which retain the Planets in their Orbits.

1. If a body gravitating to a fixed centre have a motion communicated to it in a direction not passing through that centre, it will move in a curve, and the straight line drawn from the body to the centre will describe areas proportional to the times.

Let S (fig. 109) be the centre to which the body A gravitates, at the same time that a motion is communicated LXXXIX. to it in the direction AB. And first, let the gravitating or centripetal force be supposed to act, not continually, but at intervals, producing instantaneously, at the beginning of each interval, the same velocity that it would have produced by acting continually during the whole of that time: let AC be the space which the body would describe by the action of this force alone; also let AB be the space which it would describe in the same time by the projectile force acting on it alone. It will therefore describe the line AD, the diagonal of the parallelogram contained by AB and AC, and at the end of the first interval will be in D. If, then, no new impulse of gravity were to act on it, it would in the second interval of time go on in the direction AD, and describe DF equal to AD. But if, at the beginning of the second interval, an impulse of the centripetal force be instantaneously impressed, sufficient to carry the body in that time from D to E in the line DS, it will describe the line DG, the diagonal of the parallelogram contained by DE and DE. The same is true of the third interval, in which the body will go from G to L, and of every subsequent interval. Join SB, SE, SK, &c. The areas of the triangles ABS, ADS are equal, the triangles being on the same base AS, and between the same parallels AS and BD. For the same reason, the triangles DGS, DFS are equal, and DFS is equal to ADS, because they have equal bases and the same altitude. For the same reason, the triangle SGL is equal to SDG, or to ADS; and the same is true of all the other triangles that are described in the equal intervals of time by the line drawn from the body to the centre S. This holds, however short the intervals may be, and however great their number; and therefore it is true when the intervals are indefinitely small, and their number infinitely great, that is, when the action of the centripetal force is continued.

But when the intervals of time become indefinitely small, the rectilineal figure ADGL passes into a curve. For when these intervals diminish, the lines AB, DF, &c. the lengths of the parallelograms, diminish in the same proportion; but the lines AC, DE, &c. the breadths, diminish in a greater proportion, viz. in that of the squares of these intervals. Hence, the angles which AD, DG, GL, the diagonals, make with the sides AB, DF, GK, continually diminish; and therefore the angles ADG, DGL, or the angle which each diagonal makes with that which is contiguous, increases without limit, so that, as the diagonals diminish in length, the angles they make with one another become greater than any finite rectilineal angles, and therefore the figure becomes a curve line.

That the lines AC, &c. or the supposed effect of the centrifugal force, diminish as the squares of the times, is evident from the laws of the descent of heavy bodies, as explained under the head of Dynamics.

2. Hence, conversely, if a body move in a curve, so that the line drawn from it to a fixed point describe areas proportional to the times, the body gravitates to that point, or tends continually to descend to it. For, since it does not move in a straight line, it must be continually acted on by a deflecting force; and the direction of the deflecting force must always pass through the same point, otherwise the areas described about that point would not be proportional to the time.

3. Corollary. The velocities of a body in different points of the curve which it describes about a centre of force, are inversely as the perpendiculars drawn from the centre to the tangents of the curve at these points. Let $ACA'$, fig. 110, be the curve which a body describes about the centre $S$. Let $Aa$ and $A'a'$ be two arches of the curve, described in the same indefinitely small portion of time. Join $Sa$, $Sa'$, then the areas of the triangles $ASa$, $ASa'$ are equal by this proposition. At $A$ and $A'$, draw the tangents $AB'$, $A'B$, and from $S$ let fall on them the perpendiculars $SB$ and $SB'$. Because the areas of the triangles $ASa$, $ASa'$, are equal, $Aa \times SB = A'a' \times SB'$, or $Aa : A'a' :: SB : SB'$; but $Aa$ is to $A'a'$ as the velocity of the body describing the curve at $A$ to its velocity at $A'$, therefore these velocities are inversely as the perpendiculars $SB$, $SB'$.

The straight line $AB$ (fig. 109), according to which the projectile motion was impressed on the body, is a tangent to the curve at the point $A$.

4. On comparing the first and second of these propositions with the first of Kepler's laws, as just enumerated, it is evident that the primary planets all gravitate to the sun, and that the secondary planets gravitate every one to its primary. The next thing, therefore, is to discover the law observed by this force, or the function of the distance to which it is proportional; and also, whether, in that function, other variable quantities are not involved beside the distance. The general fact that the orbits, or curves described by the planets round the sun, are ellipses, may assist in this investigation, and in expressing the velocity of a planet, in terms of the radius vector, or its distance from the sun.

5. Let $ADBE$ (fig. 111) be the orbit of a planet, $S$ the focus in which the sun is placed, $AB$ the axis major, and $DE$ the axis minor, $C$ the centre, and $F$ the superior focus. Let the planet be anywhere at $P$; draw a tangent to the orbit in $P$, on which from the foci let fall the perpendiculars $SG$, $FH$. Draw also $DK$ touching the orbit in $D$, and let $SK$ be perpendicular to it. Let the velocity of the planet, when at the mean distance, or at $D$, be $c$, and when at $P = e$. Join $SP$, $FP$. Then, by the corollary to the last proposition, the velocity at $D$ is to the velocity at $P$ as $SG$ to $SK$, that is, $c : v : : SG : DC$, or $v = c \cdot \frac{DC}{SG}$.

But because the triangles $SGP$, $FHP$, are equiangular, having right angles at $G$ and $H$, and also the angles $SPG$, $FPH$ equal, from the nature of the ellipse, $SP : PF :: SG : FH$, and therefore also $SP : PF : SG^2 : SG \times FH$. But $SG \times FH = CD^2$, therefore $SP : PF :: SG^2 : CD^2$, and $\frac{CD^2}{SG^2} = \frac{PF}{SP}$. Now $e = c \cdot \frac{CD}{SG}$, or $v^2 = c^2 \cdot \frac{CD^2}{SG^2}$, and therefore $v^2 = c^2 \cdot \frac{PF}{SP}$.

Hence, as the distance of a planet from the sun, at any point in its orbit to its distance from the superior focus, so the square of its velocity at its mean distance from the sun to the square of its velocity at the point just mentioned.

6. If $SL$ be taken in the greater axis equal to $SP$, and $FN = PF$, so that $SN = \text{the transverse axis } AB$, $v^2 = c^2 \cdot \frac{NL}{LS} = c^2 \cdot \frac{SN - SP}{SP}$. Then as $SN$ is a given line, $v$ is expressed in terms where $SP$, the distance from the sun, is the only variable quantity.

If, from the velocity of the revolving body thus expressed in terms of the distance, a transition can be made to that of a body descending in a straight line, the law of the centripetal force will be easily investigated. This will be facilitated by the following proposition:

An equal approach to the centre of force produces an equal increase of the square of the velocity, whether the body revolve in a curve about the centre, or descend to it in a straight line. In like manner, equal recesses from the centre of force produce equal diminutions of the square of the velocities, in whatever lines the bodies move.

Let $ABC$ (fig. 112) be a curve which a body describes about a centre, $S$, to which it gravitates, while another body descends in a straight line $AS$, to that centre. Let $BC$ be any arch of the curve $ABC$, and let $BD$, $CH$, be arches of circles described from the centre $S$, intersecting the line $AS$ in $D$ and $H$; the square of the velocity of the body, which describes the arch $BC$, will be as much increased as the square of the velocity of that which falls through $DH$.

From the centre $S$ describe the arch $bd$, indefinitely near to $BD$, and draw $Ef$ perpendicular to the arch $bd$. Also let the centripetal force at $B$ or $D$ be called $G$. Now, the part of this force which is in the direction $bd$, and which is employed in accelerating the body moving in that line, is $G \times \frac{Bf}{BE}$; and the increment of the space being $Bb$, therefore $2G \times \frac{Bf \times Bb}{BE}$ is the momentary increment of the square of the velocity of the body at $B$. But $Bf \times Bb = BE^2$, because $BE$ is a right-angled triangle, and $Ef$ the perpendicular on the hypotenuse. Therefore $2G \times \frac{Bf \times Bb}{BE} = 2G \times \frac{BE^2}{BE} = 2G \times BE = 2G \times Dd$. But $2G \times Dd$ is the momentary increment of the square of the velocity of the body at $D$, or the increment of that square while the body falls from $D$ to $d$. These momentary increments therefore are equal; and as the same may be shown for the next and every subsequent instant, the whole increase of the square of the velocities of the bodies in moving over $BC$ and $DH$ are equal.

If the bodies moved in the opposite directions, the one from $C$ to $B$, and the other from $H$ to $D$, it would be proved, in the same manner, that the squares of their velocities would be equally diminished.

7. Hence it is evident, that, if the velocities of the revolving and of the falling body are equal in any one instance when they are equally distant from the centre, their velocities will always be equal when they are equally distant from that point; for equal quantities receiving equal increments continue equal.

8. Suppose now that a planet revolves in the elliptical orbit $APB$ (fig. 111), it will have at $A$, the higher apsis, a velocity $= c \times \sqrt{\frac{AF}{AS}}$, or (if $AN$ in the axis produced be taken equal to $AE$) $= c \times \sqrt{\frac{AN}{AS}}$. Let a body at $A$ begin to descend towards $S$ with the same velocity; then if $SL = SP$, the velocity of the planet at $P$ will be the same with that of the falling body at $L$. But the velocity of the planet at $P$ is $c \times \sqrt{\frac{PF}{PS}} = c \times \sqrt{\frac{NL}{SL}}$, therefore, a body descending from $A$, and falling directly to the sun under the action of the same centripetal force which urges the planet, would at any point $L$ in its fall have its velocity $= c \times \sqrt{\frac{LN}{LS}}$. Hence, at the point $N$... its velocity would be equal to 0, or the body must begin to fall from N, in order that its velocity may be everywhere equal to that which the planet has in its orbit, when at the same distance from the sun.

The law, therefore, according to which the planets gravitate is such, that any body under the influence of the same force, and falling direct to the sun, will have its velocity at any point equal to a certain velocity, multiplied into the square root of the distance it has fallen through, divided by the square root of the distance between it and the sun's centre.

This is a fact with respect to the law of gravity in the solar system, of which, though there be no direct example, yet it is no less certain than the ellipticity of the planetary orbits, of which it is a necessary consequence.

9. From the law thus found to regulate the velocity of bodies falling in straight lines to the sun, the law of the force by which that velocity is produced may be derived by help of reasoning which is quite elementary.

Let C (fig. 113) be the centre to which the falling body gravitates, A the point from which it begins to fall, and let its velocity at any point B, be to its velocity in the point G, which bisects AC, as \( \frac{AB}{BC} = 1 \); it is required to find the law of the force with which the body gravitates to C.

Let DEF be a curve, such, that if AD be an ordinate or a perpendicular to AC, meeting the curve in D, and BE any other ordinate, AD is to BE as the force at A to the force at B, then will twice the area ABED be equal to the square of the velocity which the body has acquired in B. If, therefore, the velocity at B be \( v \), that at the middle point G being \( c \), \( v = c \sqrt{\frac{AB}{BC}} \), by hypothesis, and therefore \( 2ABED = c^2 \cdot \frac{AB}{BC} \); and since \( AB = AC - BC \), \( 2ABED = c^2 \cdot \frac{AC - BC}{BC} = c^2 \left( \frac{AC}{BC} - 1 \right) \). For the same reason \( 2BbE = c^2 \left( \frac{AC}{BC} - 1 \right) \), and therefore the difference of these areas, or \( 2BbE \), that is, \( 2EB \times Bb = c^2 \left( \frac{AC}{BC} - \frac{AC}{BC} \right) = c^2 \cdot \frac{AC \cdot Bb}{BC^2} \). Therefore, dividing by Bb, \( 2EB = c^2 \cdot \frac{AC}{BC} \); or \( EB = c^2 \cdot \frac{AG}{BC} \); now \( c^2 \) and AG are given, therefore EB is inversely as BC, that is, the centripetal force at B is inversely as the square of BC, the distance from the centre of force. In the planetary system, therefore, the force with which any planet gravitates to the sun varies in the inverse ratio of the square of the distance of the planet from the sun's centre.

10. The line CG is the same with the mean distance of the planet, in an orbit of which AC is the length of the major axis, and if the gravitation at that distance \( F \), and the mean distance itself \( a \), \( F = c^2 \times \frac{a}{c^2} = \frac{c^2}{a} \), or \( aF = c^2 \).

Let it next be required, the elliptic orbit of a planet being given, to find the time in which the planet will revolve round the sun.

If \( a \) be the mean distance, or the semimajor axis, \( b \) the semiminor, then \( \pi ab = \text{the area of the orbit.} \) But as \( c \) is the velocity at the mean distance, or the elliptic arch which the planet moves over in a second when it is at D, the vertex of the minor axis, therefore \( \frac{1}{2} bc \) is the area described in that second by the radius vector; and since this area is the same for every second of the planet's revolution, therefore the area of the orbit divided by \( \frac{1}{2} bc \) will give the number of seconds in which the revolution is completed, which is therefore \( \frac{\pi ab}{\frac{1}{2} bc} = \frac{2\pi a}{c} \), or since \( c^2 = aF \), the time of a revolution \( = \frac{2\pi a}{\sqrt{aF}} = 2\pi \sqrt{\frac{a}{F}} \).

11. Hence it is easy to compare the times of the revolutions of any two planets of which the mean distances are known. Let \( t \) and \( t' \) be the times of revolution for two different planets, of which the mean distances are \( a \) and \( a' \), and the gravitation at those distances \( F \) and \( F' \), and, by what has just been shown, \( t : t' :: \frac{a}{F} : \frac{a'}{F'} \) or \( t^2 : t'^2 :: \frac{a^2}{F^2} : \frac{a'^2}{F'^2} \). But \( F : F' :: a^2 : a'^2 \), by what is already shown (Art. 9), therefore \( t^2 : t'^2 :: \frac{a^2}{a'^2} : \frac{a^2}{a'^2} \), or \( t^2 : t'^2 :: a^2 : a'^2 \), that is, the squares of the times of revolution of any two planets are as the cubes of their mean distances from the sun. Thus the third law of Kepler is explained by the conclusions deduced from the other two.

12. The share which this third law has in establishing the principle of universal gravitation does not seem to have been always clearly apprehended. From the elliptical orbit of a planet, it is fairly inferred that, over all the circumference of that orbit, gravitation is inversely as the square of the distance from the centre of the sun. That force is shown to be \( \frac{c^2}{x^2} \) (\( x \) being the distance from the centre of force), and the same is true of every individual planet; but whether \( c^2 \) was a constant quantity, or one which retained the same value through the whole planetary system, could not be known without comparing the periods of different planets with their distances from the sun. It was indeed highly probable that \( c^2 \) was a given quantity, or the same for every part of our system; but it could not be considered as a thing demonstrated till the evidence of the third law was introduced.

13. These laws hold of the secondary planets relatively to their primary, just as with the primary planets relatively to the sun. Each system of secondary planets, however, has a different numerator to the fraction which expresses gravity; that is, the quantity \( c^2 \) is the same for all the satellites of Jupiter, but it is a different quantity from that which belongs to the satellites of Saturn, and different from that which belongs to the primary planets. The quantity \( c^2 \) seems therefore to depend on the central body of each system of planets, and the precise nature of this connection requires to be farther examined into.

14. Let the centripetal force tending to a given centre \( S \) be inversely as the squares of the distances, and let the intensity of that force at any given distance from the centre be also given; then, if a body be projected from a given point, with a given velocity, and in a given direction, it is required to determine the conic section which it will describe.

Let the semimajor axis, or the mean distance to be found, \( = a \), the semiminor \( = b \), the velocity at the distance \( = c \), and at the given distance \( d \) let the centripetal force \( = f \); and first let the direction of the initial motion be at right angles to the radius vector, so that the point of projection is either the higher or the lower apsis. Let the velocity of the projection \( = v \), and the radius vector at the point of projection \( = r \).

Because the areas described in equal times are equal, Physical Astronomy.

be \( rv \); and if \( F \) denote the centripetal force at the distance \( a \), \( c^2 = aF \), and \( F = \frac{d^2f}{a^2} \). But \( F = \frac{d^2f}{a^2} \), therefore

\[ \frac{c^2}{a} = \frac{d^2f}{a^2}, \quad \text{and} \quad c = d\sqrt{\frac{f}{a}}. \]

Hence, by substituting for \( c \),

\[ bd\sqrt{\frac{f}{a}} = re, \quad \text{and} \quad b^2d^2f = ar^2v^2. \]

But \( b^2 = AS \times SB = r(2a - r) \), wherefore \( r(2a - r) \cdot d^2f = ar^2v^2 \), and \( a = \frac{r^2d^2f}{r^2v^2} \).

Thus \( a \), the semitransverse axis, and therefore the transverse axis itself, is found; and thence with the focus \( S \) and the apsis \( A \), the conic section may be described.

15. The conic section will be a circle, when \( a = r \), that is, when \( 2d^2f - v^2r = d^2f \), or when \( d^2f = v^2r \), and \( v^2 = \frac{d^2f}{r} \).

16. But if \( 2d^2f > v^2r \), the denominator vanishes, and \( a \) becomes infinite, so that the trajectory is a parabola, of which the focus is \( S \), the vertex \( A \), and the parameter \( 4r \). The square of the velocity which determines the trajectory to be a parabola is, therefore, double of the square of the velocity which determines it to be a circle.

17. When \( 2d^2f < v^2r \), the value of \( a \) is affirmative and the conic section is an ellipse, and this ellipse has its higher apsis at \( A \), if \( v^2 < \frac{d^2f}{r} \); but when \( v^2 > \frac{d^2f}{r} \), and less \( 2d^2f \), \( A \) is the lower apsis.

18. When \( v^2 \) goes beyond this last limit, or when \( v^2 > \frac{2d^2f}{r} \), the value of \( a \) is negative, and the trajectory becomes an hyperbola.

19. Next, let the body be projected from \( B \) (fig.114) with the velocity \( v \), in the direction \( BD \), oblique to \( BS \). Find the distance from which a body must fall to acquire at \( B \) the velocity \( v \), and let \( OB \), taken in \( SB \) produced, be equal to this distance; then is \( SO \) equal to the major axis. Let \( BE \) be drawn, making with \( BD \) the same angle that \( SB \) makes with \( BG \), and let \( BE = BO \), then is \( E \) the higher focus. Produce \( SE \) to \( N \), so that \( SN = SO \), and bisect \( EN \) in \( A \), then is \( A \) the higher apsis; and if \( SP \) be made equal to \( EA \), \( P \) is the lower apsis, and \( AP \) the major axis; and therefore the foci and the major axis being given, the elliptic orbit may be described.

20. From what has been shown at Art. 9, it is evident that the primary planets gravitate to the sun with forces that are inversely as the squares of the distances, and that the secondary gravitate toward the primary, according to the same law. This inference, however, does not apply exactly to the moon, which, being a single satellite, does not by comparison with any other afford a proof that, in bodies revolving round the earth, the squares of the periodic times are as the cubes of the mean distances. The centripetal force at the moon, however, from our knowledge of her periodic time, may be compared with the force of gravity at the earth's surface, and will determine whether that force decreases as we recede from the earth in the inverse ratio of the squares of the distances.

Let \( a \) be the distance of the moon from the centre of the earth, \( r \) the radius of the earth, \( g \) the velocity acquired by a heavy body at the earth's surface by falling during one second; let \( t \) be the period of the moon's revolution in seconds, and \( e \) the velocity of her motion.

Then, by Art. 14, \( ae^2 = r^2g \), and therefore \( e = r\sqrt{\frac{g}{a}} \).

Now, the circumference of the circle described by the moon is \( 2\pi a \), and this, divided by \( e \), gives the periodic time of the moon in seconds, or \( \frac{2\pi a}{e} \times \sqrt{\frac{a}{g}} = t \), so that

\[ t = \frac{4\pi^2a^3}{r^2g}, \quad \text{and} \quad a^3 = \frac{r^2gt^2}{4\pi^2}. \]

Hence \( \frac{a^3}{r^3} = \frac{gt^2}{4\pi^2r} \), and \( \frac{a}{r} = \left( \frac{gt^2}{4\pi^2r} \right)^{\frac{1}{3}} \).

Hence, as \( g \), \( r \), and \( t \) are known, we may find \( \frac{a}{r} \) or the ratio of the moon's distance to the radius of the earth, which, if it come out the same that it is known to be from observations of the moon's parallax, will prove that the force which retains the moon in her orbit is the same that causes bodies to fall at the surface of the earth, but diminished in the same ratio that the square of the moon's distance is greater than the square of the radius of the earth.

Now \( g = 32-166 \) feet, \( r = 3481279-4 \) fathoms or 20887676-4 feet, and \( t = 2360591-5 \) seconds. Hence \( \frac{a}{r} = 60-218 \).

Now the mean equatorial parallax of the moon is found by observation \( = 57'0''9 \), from which the mean distance, in semidiameters of the equator, is found \( = 59-964 \).

But it is in mean semidiameters of the earth that the moon's distance is given in the former computation; therefore, to reduce the last measure to the same scale, it must be increased by a 600th part, as the mean radius of the globe is about that much less than the radius of the equator; the distance 59-964 then becomes 60-063, which agrees with the former number to the small fraction .003 of the earth's radius.

Thus, from the theory of gravity, combined with the time of the moon's sidereal revolution, her distance from the earth is found to within a very small fraction of the whole.

21. It is therefore a general proposition, derived from the most rigorous induction, that the primary planets gravitate to the sun, and the secondary planets to the primary, with forces which are inversely as the squares of the distances.

But since, in all communication of motion, the reaction is equal to the action, when a planet gravitates to the sun, analogy forces us to conclude that the sun gravitates to the planet, in such a manner, that if the momentary approach of the planet to the sun, and of the sun to the planet, were respectively multiplied by the quantity of matter in those bodies, the products, or the quantities of motion, would be equal. Such a mutual tendency, therefore, of the great bodies of our system to the sun, and of the sun to them, doubtless takes place; but whether this be in consequence of an attractive force residing in their centres, as the magnetic force does in certain parts of the lodestone, or if it arise from the mutual attraction of all the particles of the one for all the particles of the other, does not appear from the phenomena hitherto examined.

We may, however, observe that the bodies between which this attraction in the inverse ratio of the squares of the distances takes place, are all of a round form, and are either accurately spherical, or nearly approaching to that shape. It will therefore be of use for resolving this question, to inquire whether, if the particles of matter did attract one another with forces inversely as the squares of their distances, the spherical bodies, compounded of such particles, would attract one another according to the same law. If this is found to be the case, it will be reasonable to conclude that the gravitation of large bodies to one another arises from the mutual attraction of their particles to one another.

22. In order to determine the relation between the attraction of a sphere and that of the particles of which it consists, we may consider the sphere as made up of plates Physical or imaginary infinitely great in number, and infinitely small in thickness. The attraction of each of these is to be estimated, and from thence the attraction of the whole may be computed. Let AFBG (fig. 115), therefore, be a circular plate, of which the centre is C; CE a straight line passing through C, and perpendicular to the plane AFBG; E any particle in that line attracted by each particle of the circular plate, as D, with a force inversely as the square of DE, the distance between the particles; it is required to find the whole force with which E is attracted in the direction EC.

If DC be drawn, the force with which D attracts E in the direction ED is inversely as \(DE^2\) or as \(\frac{1}{DE^2}\) and that same force, reduced to the direction EC, is as \(\frac{1}{DE^2} \times \frac{EC}{DE} = \frac{EC}{DE^2}\). From the centre C, with the radius DC, let a circle DKH be described, and indefinitely near it the circle dkh; then, since every particle in the ring of matter contained between these circles has its attraction proportional to \(\frac{EC}{ED^2}\), the attraction of the whole ring will be as \(\frac{EC}{ED^2}\) multiplied into the number of particles, or into the solidity of the ring. But if EC = a, ED = x, and AC = r, CD² = \(x^2 - r^2\), and the surface of the ring = \(2\pi x dx\). If then the thickness of the plate AFBG = m, the solidity of the ring = \(2\pi mx dx\), and its attraction in the direction EC is \(\frac{2\pi mx dx \times a}{x^3} = \frac{2\pi mx dx}{x^2}\), the integral of which taken so as to vanish when DC = 0, or when \(x = a\), is \(2\pi m - \frac{2\pi ma}{x} = 2\pi m \left(1 - \frac{a}{x}\right)\) = the attraction of the circle DKH. Therefore, when \(x = AE\), the whole attraction of the plate, or the whole force which it exerts on the particle E, is \(2\pi m \left(1 - \frac{EC}{EA}\right)\).

23. Next, let ABD (fig. 116) be a circle of which the centre is C, and E a particle of matter anywhere in the diameter AB produced. Draw ED to any point D in the circumference; draw also DC, and let DF be at right angles to AB. Then, when the whole figure revolves about EB, the semicircle ADB will generate a sphere, and DF a circle perpendicular to the plane ABD, and having its centre in F. If all the particles of the sphere attract the particle E with forces inversely as the squares of their distances from it, then, by the last proposition, the attraction of the circular plate, of which the centre is F, will be \(2\pi m \left(1 - \frac{EF}{ED}\right)\).

Let CE = a, AC = r, ED = x, EF = y, and the attraction above will be \(2\pi m \left(1 - \frac{y}{x}\right)\); and if \(x\) and \(y\) be variable, the quantity \(m\) in this formula, or the thickness of the circular plate will be \(dy\), and therefore the attraction of the plate \(= 2\pi dy \left(1 - \frac{y}{x}\right)\). In order to integrate this quantity, \(y\) must be expressed in terms of \(x\), or \(x\) in terms of \(y\).

Now, because AE = \(a - r\), and AF = \(y - a + r\), FB = \(2r - y + a - r = a + r - y\), and AF × FB = \((r - a - y)(r + a - y) = r^2 - a^2 + 2ay - y^2 = DI^2 = x^2 - y^2\). Hence \(r^2 - a^2 + 2ay = x^2\), or \(y = \frac{x^2 - r^2 + a^2}{2a}\), and therefore

\[dy = \frac{x dx}{a}.\]

By substituting these values of \(y\) and \(dy\) in the expression for the attraction of the circular plate, that attraction \(= \frac{2\pi dx}{a} \left(1 - \frac{a^2 - r^2 + x^2}{2ax}\right)\)

\[= \frac{2\pi dx}{a} \left(\frac{a^2 - r^2 + x^2}{2ax}\right).\]

But the attraction of this circular plate may be considered as the differential of the attraction of the spherical segment, generated by the revolution of the arch AD, and therefore the integral of the above differential quantity will give the attraction of that segment. Now, this integral

\[= \frac{\pi}{a} \left(ax^2 - a^2x + r^2x - \frac{1}{3} x^3\right) + C.\]

Here C must be so determined that the integral may be equal to 0, when the arch AD = 0, or when \(x = y = a - r\). Therefore C

\[= \frac{\pi}{a} \left(\frac{a^3 - a^2 + \frac{3}{2} r^2}{a^2}\right);\] and the attraction

\[= \frac{\pi}{a} \left(ax^2 - a^2x + r^2x - \frac{1}{3} x^3 + \frac{1}{3} a^3 - ax^2 + \frac{3}{2} r^2\right).\]

This is the attraction of the spherical segment generated by the arch AD, and will become equal to that of the whole sphere when AD = the semicircle ADB, or when \(x = a + r\). This substitution being made, and the terms reduced, the attraction is found \(\frac{4\pi m^2}{3a^2}\). But \(\frac{4\pi m^2}{3a^2}\) is the solid content of the sphere; therefore the attraction of the sphere, on any particle E, is as the quantity of matter in the sphere, divided by the square of the distance of its centre from E. Hence also the sphere attracts any particle without it, as if all its matter were united in its centre. The sphere, it is also obvious, would attract another sphere just in the ratio of its quantity of matter, divided by the distance of the centres of the spheres.

24. Thus, supposing that the particles of matter attract one another with forces which are inversely as the squares of the distances, it is certain that the spherical bodies composed of these particles would do so likewise, or would attract one another with forces directly as their quantities of matter, and inversely as the squares of the distances of their centres. Since, therefore, it has been found that round or spherical bodies, such as the sun and the planets, do attract other bodies with forces that are inversely as the squares of the distances, it is reasonable to suppose that these bodies are composed of particles gravitating towards one another, or attracting one another with forces inversely as the squares of the distances. Gravitation, therefore, is not to be considered as a force residing in the centres of the planets, but as a force belonging to all the particles of matter, and as universally diffused throughout the universe.

And as it has been shown that between spherical bodies constituted of such particles, the force of attraction is as the quantity of matter in the attracting body, divided by the square of the distance between its centre and that of the attracted body; if \(m\) be the mass or quantity of matter in the former body, and \(x\) the distance of the centres, \(\frac{m}{x^2}\) is the value of \(f\), the accelerating force with which it attracts the other body.

25. Hence the masses of any two planets which have bodies revolving round them may be compared with one another. Let \(a\) and \(a'\) be the mean distances at which satellites revolve about any two planets, \(m\) and \(m'\) the quantities of matter in those planets, \(t\) and \(t'\) their periods. of revolution; it has been shown that \( t = \frac{2\pi}{d} = \frac{2\pi}{m^2} \),

and consequently \( t : t' = \frac{a^3}{m^3} : \frac{a'^3}{m'^3} \), and \( m : m' = \frac{a^3}{c^3} : \frac{a'^3}{c'^3} \).

The masses, therefore, of any two planets are as the cubes of the mean distances at which their satellites recede, divided by the squares of the periodic times of those satellites.

26. In this way the masses of the four planets which have satellites may be compared with one another, and with the mass of the sun.

When this calculation is undertaken with the requisite data, it is found that, making

Mass of the Sun ........................................... 1 that of the Earth ........................................... \( \frac{1}{354936} \) of Jupiter .................................................. \( \frac{1}{10705} \) of Saturn .................................................. \( \frac{1}{3512} \) of Uranus .................................................. \( \frac{1}{17918} \)

Or if we make the mass of the Earth 1, that of the Sun \( = 329630 \), of Jupiter \( = 3304 \), of Saturn \( = 101-06 \), and of Uranus \( = 20-3 \). From this also may be derived the densities of the sun and of the four planets just mentioned. Seen from a distance equal to the mean radius of the earth's orbit, the diameter of the sun subtends an angle of \( 1923^\circ \), that of the earth would subtend \( 177^\circ 4 \), of Jupiter \( 186^\circ 8 \), of Saturn \( 177^\circ 7 \), and of Uranus \( 74^\circ \). The real diameters, therefore, are in the proportion of these numbers, and the bulk in the proportion of their cubes. By dividing the quantities of matter by the bulks, we have the densities; and if that of the earth be \( 4713 \), which is its mean density, that of water being \( = 1 \), then

Density of the Sun ........................................... \( = 1775 \) of the Earth .................................................. \( = 4713 \) of Jupiter .................................................. \( = 11678 \) of Saturn .................................................. \( = 04055 \) of Uranus .................................................. \( = 10348 \)

The mean density of the earth, in respect of water, is here taken from the experiments made at Schehallien. (Phil. Trans. 1811, p. 376.)

27. It has been already observed, that because action is always accompanied by an equal re-action, when the sun attracts a planet, the planet also attracts the sun, and that the velocities impressed on the bodies by their mutual attraction are in the inverse ratio of their masses.

In consequence of this mutual action the sun and the planet must both move, and must describe orbits about their common centre of gravity, the only point which the mutual action of those bodies has no tendency to put in motion.

In the solar system, therefore, the centre of gravity of the whole is the focus about which all the orbits are described. Thus, if C be that centre (fig. 117), S the sun, and P a planet; while P describes the elliptic arch PP' about C, S describes the arch SS' similar to PP', and having to it the ratio that SC has to CP, or the ratio which the mass of the planet has to the mass of the sun.

The true orbits, therefore, are all described about the same immovable point; but the orbit of any of the planets may be referred to the sun as a centre, by supposing a body placed in that centre equal to the sum of the masses of the sun and of the planet. This is true, because the bodies appear to approach one another, or to recede from one another, with a force that is equal to the sum of the forces with which they tend towards their centre of gravity. Thus, if S denote the mass of the sun, and E that of the earth, the distances from the centre being CP and CS, the orbit which each of the two bodies will appear to describe round the other, is that which would be described about an immovable centre C, with a centripetal force

\[ S + E = \frac{S}{SP^2}. \]

Thus we have arrived at the knowledge of the principle of universal gravitation, a power which pervades all nature, extending to an unlimited distance, and determining the condition of every body in the universe at any instant, from its state in the former instant, and from the relations in which it stands to all other bodies. Whether this force can be explained upon any principle more general than itself, is yet undecided, though, from the bad success which has hitherto attended all attempts towards that object, it seems probable that such explanation is not within the reach of the human understanding. Thus much, however, we know with certainty, that the law of gravity, as just announced, may be considered as a very accurate expression of all the phenomena of the planetary motions.

Sect. II.—Of the Forces which disturb the Elliptic Motion of the Planets.

1. Of the forces by which the Sun disturbs the motion of the Moon round the Earth.

28. The motion of the moon in an elliptic orbit round the earth is disturbed by the action of the sun: the gravity of the moon to the earth is increased at the quadratures, and diminished at the syzygies; and the areas described by the radius vector, except near these four points, are never exactly proportional to the times.

Let ADBC (fig. 118) be the orbit, nearly circular, in which the moon M revolves, in the direction CADB, round the earth E. Let S be the sun, and let SE, the radius of the earth's orbit, be taken to represent the force with which the earth gravitates to the sun.

Then \( \frac{1}{SE^2} : \frac{1}{SM^2} : : SE : SM = \text{the force by which the sun draws the moon in the direction MS}. \)

Take MG \( = \frac{SE^2}{SM^2} \) and let the parallelogram KF be described, having MG for its diagonal, and having its sides parallel to EM and ES. The force MG may be resolved into the two, MF and MK, of which MF, directed towards E, the centre of the earth, increases the gravity of the moon to the earth, and does not hinder the areas described by the radius vector from being proportional to the times.

The other force MK draws the moon in the direction of the line joining the centres of the sun and earth. It is, however, only the excess of this force above the force represented by SE, or that which draws the earth to the sun, which disturbs the relative position of the moon and earth. This is evident, for if KM were just equal to ES, no disturbance of the moon relatively to the sun could arise from it. If, then, ES be taken from MK, the difference HK is the whole force in the direction parallel to SE, by which the sun disturbs the relative position of the moon and earth. Now, if in MK, MN be taken equal to HK, and if NO be drawn perpendicular to the radius vector EM produced, the force MN may be resolved into two, MO and ON, the first lessening the gravity of the moon to the earth; and the second, being parallel to the tangent of the moon's orbit in M, accelerates the moon's motion from C to A, retards it from A to D, and so alternately in the other two quadrants.

Thus the whole solar force directed to the centre of the earth is composed of the two parts MF and MO, which are sometimes opposed to one another, but which never affect the uniform description of the areas about E. Near the quadratures the force MO vanishes, and the force MF, which increases the gravity of the moon to the earth, coincides with CE or DE. As the moon approaches the conjunction at A, the force MO prevails over MF, and lessens the gravity of the moon to the sun. In the opposite point of the orbit, when the moon is in opposition at B, the force with which the sun draws the moon is less than that with which the sun draws the earth, so that the effect of the solar force is to separate the moon and earth, or to increase their distance; that is, it is the same as if, conceiving the earth not to be acted on, the sun's force drew the moon in the direction from E to B. This force is negative, therefore, in respect of the force at A, and the effect in both cases is to draw the moon from the sun, in a direction perpendicular to the line of the quadratures.

29. The analytical values of these forces must be found if a more exact estimate is to be made of their effects. Let SE, considered as constant, = a; r, the radius vector of the moon's orbit, = r; the angle CEM = ϕ; the mass of the sun = m. The force SE, then, which retains the earth in its orbit, is \( \frac{m}{a^3} \), and the sun's force in the direction SM, if ML be drawn perpendicular to ES, is

\[ \frac{m}{SM^2} = \frac{m}{SL^2 + LM^2 - (a - r \sin \phi)^2 + r^2 \cos^2 \phi} = \frac{m}{a^2 - 2ar \sin \phi + r^2}. \]

The part of this force, which is in the direction ES or MK, is therefore \( \frac{ma}{(a^2 - 2ar \sin \phi + r^2)^2} \). By raising the denominator to the power \( \frac{1}{2} \), rejecting the terms which involve the higher powers of r, and multiplying me by those that are left, the force MK comes out \( \frac{ma}{a^2} \left( 1 + \frac{3r}{a} \sin \phi \right) \) nearly. Taking away from this ES or MH = \( \frac{m}{a^2} \), there remains the force MN = \( \frac{m}{a^2} \times 3r \sin \phi \).

Hence the force MO = \( \frac{m}{a^2} \cdot 3r \sin^2 \phi \); and the force NO at right angles to the radius vector = \( \frac{m}{a^2} \cdot 3r \sin \phi \times \cos \phi = \frac{m}{a^2} \cdot \frac{3r}{2} \sin 2\phi \); also the force MF = \( \frac{mr}{a^2} \), rejecting such terms as involve the square and higher powers of r. Therefore MF—MO, or the whole solar force increasing or diminishing at any point, the moon's tendency to the earth is \( \frac{mr}{a^2} \left( 1 - 3 \sin^2 \phi \right) \).

30. At the quadratures where ϕ vanishes, this force is \( \frac{mr}{a^2} \) and is affirmative, increasing the moon's gravity to the earth. At a certain point, between the quadratures and the syzygies, when \( 3 \sin^2 \phi = 1 \), or \( \sin \phi = \frac{1}{\sqrt{3}} \), that is, when \( \phi = 35^\circ 15' 5'' \), the same force becomes equal to 0, and at this point in each quadrant the moon's gravity to the earth is neither increased nor diminished. From these points to the conjunction and opposition, as \( \sin \phi \) increases, the quantity \( 1 - 3 \sin^2 \phi \) is negative, and the moon's gravity to the earth suffers a diminution. At the opposition and conjunction \( \sin \phi = 1 \), and therefore the disturbing force is \( \frac{2mr}{a^2} \), and by this quantity the moon's gravitation is diminished.

The mean quantity of the force which is thus continually directed to or from the centre of the earth may also be easily computed. Since for any point in the moon's orbit, where the radius vector makes the angle ϕ with the line of the quadratures, this force \( \frac{mr}{a^2} \left( 1 - 3 \sin^2 \phi \right) \); multiplying by \( d\phi \), we have \( \frac{mr}{a^2} \left( d\phi - 3d\phi \sin^2 \phi \right) \), the integral of which \( \frac{mr}{a^2} \left( -\frac{1}{2} \phi + \frac{3}{8} \sin \phi \cos \phi \right) \), and this, when ϕ is an entire circumference or four right angles, is \( \frac{mr}{a^2} \times \pi \). This is the sum of the forces for an entire revolution, and when divided by \( 2\pi \), gives the mean force \( \frac{mr}{2a^2} \), which being negative, shows that the solar force, on the whole, diminishes the gravitation of the moon to the earth.

Thus it appears, that at the quadratures the gravity of the moon to the earth is increased by a quantity equal to the mass of the sun, multiplied into the radius of the moon's orbit, and divided by the cube of the sun's distance from the earth; at the syzygies it is diminished by twice this quantity; and the effect on the whole is a diminution by one half of the same quantity.

If \( \frac{mr}{a^2} \) be reduced to its numerical value, supposing the moon's gravitation to the earth to be 1, it is found \( \frac{1}{174} \) nearly. Hence the mean disturbing force of the sun is nearly \( \frac{1}{348} \) of the moon's gravity to the earth.

31. From the disturbing force of the sun arise two kinds of inequalities which affect the lunar motions; the one kind affects the form and position of the orbit of that planet, the other immediately affects the motion of the planet in the orbit. When any of these inequalities is expressed numerically, the measure of it so obtained is, in the language of astronomy, called an Equation.

32. The line in which the plane of the moon's orbit cuts the ecliptic is called the line of the nodes; and this line is subject to change its position continually, in such a manner as to go back annually \( 19^\circ 19' 42'' \). The way in which this effect is produced may be thus conceived. That part of the solar force which is parallel to the line joining the centres of the sun and earth, is not in the plane of the moon's orbit, except when the sun itself is in that plane, or when the line of the nodes, being produced, passes through the sun. In all other cases it is oblique to the plane of the orbit, and may be resolved into two forces, one of which is at right angles to that plane, and is directed towards the ecliptic. This force of course draws the moon continually towards the ecliptic, or produces a continual deflection of the moon from the plane of her own orbit towards that of the earth. Hence the moon meets the plane of the ecliptic sooner than it would have done if that force had not acted. At every half-revolution, therefore, the point in which the earth meets the ecliptic advances in a direction contrary to that of the moon's motion, or contrary to the order of the signs. This retrograde motion is such that, in its mean quantity, it amounts to \( 19^\circ 19' 42'' \) in a year. The manner of de- producing it from the theory of gravity is explained by Newton, Princip. lib. iii. prop. 31. This motion is subject to many inequalities, depending on the changes in the quantity and direction of the solar force.

If the earth and the sun were at rest, the effect of the deflecting force just described would be to produce a retrograde motion of the line of the nodes till that line was brought to pass through the sun, and of consequence the plane of the moon's orbit to do the same, after which they would both remain in their position, there being no longer any force tending to produce a change in either. The motion of the earth carries the lines of the nodes out of this position, and produces, by that means, its continual retrogradation.

33. The same force produces a small variation in the inclination of the moon's orbit, giving it an alternate increase and decrease within very narrow limits.

34. The line of the moon's apsides, that is, the longer axis of her orbit, has also a slow angular motion round the centre of the earth, which is progressive, or in the same direction with the motions of the moon itself. To conceive the cause of this phenomenon, we may begin with supposing the moon at the lower apsis, or perigee; and it is plain, if that planet were urged by no other force than its gravitation to the earth, that after the radius vector had moved over $180^\circ$, the moon would be at the higher apsis, where its motion would be at right angles to the said radius. But as the mean disturbing force in the direction of the radius vector tends, on the whole, to diminish the gravitation of the moon to the earth, the portion of her path, described in any instant, will be less bent or deflected from the tangent, than if this disturbing force did not exist. The actual path of the moon, therefore, will be less incurvated than the elliptic orbit that would be described under the influence of gravity alone, and will not be brought to intersect the radius vector at right angles, till this last have moved over an arch of more than $180^\circ$.

Hence the solar force, by lessening the moon's gravity to the earth, produces a progressive motion in the apsides of the lunar orbit. If the disturbing force had increased the moon's gravity to the earth, the motion of the apsides would have been in antecedentia.

The precise quantity of the motion of the apsides is not however easily determined. Newton left this part of the lunar theory almost untouched; and the only investigation he has entered into having any reference to it, assigned a measure only the half of that which is known from observation to belong to it. Several years afterwards, when Clairaut attempted a more accurate investigation of the lunar inequalities than was to be obtained by the method which Newton had followed, he at first encountered the same difficulty, and found that his calculus gave the motion of the apogee only half of the real quantity. He began, therefore, to suspect that gravity does not follow so simple a law as the inverse of the squares of the distances, but one which is more complex, and such as cannot be expressed but by a formula of two terms. The second of these terms he supposed to be inversely as the fourth power of the distance, and proceeded to inquire what must be the co-efficient of that term, in order to make this new supposition represent the true motion of the apsides. In order to this, he found it necessary to carry his approximation farther than he had yet done, and to include terms which he had before neglected. When these terms were included, he found that the co-efficient he was seeking for came out equal to $0$; the necessary inference from which was, that there was no such term; and that the Newtonian law of gravity, when the approximation was carried far enough, was quite sufficient to explain the motion of the apsides. This doubt concerning the law of gravity terminated, therefore, in the confirmation of it.

35. When the doubts excited by Clairaut's first attempt were made known, and before his final solution of the difficulty was fully understood, there were several mathematicians who, still following the method of Newton, endeavoured to deduce the true motion of the moon's apsides from the theory of gravity. Among those who were most successful in this attempt were Dom. Walmsley, and afterwards Dr Matthew Stewart, professor of mathematics in the university of Edinburgh. In his Mathematical and Physical Tracts he has demonstrated this remarkable theorem:

Let $r$ be the radius of the moon's orbit, supposing it to be a circle, and the moon to be acted on only by $F$, her gravity to the earth. If the mean disturbing force by which the sun diminishes the moon's gravity be $f$, then will the greatest distance to which the moon will recede from the earth be $r \times \frac{F}{F-5f}$; and the cube of this distance will be to the cube of $r$, in the duplicate ratio of the angle described by the moon from one apsis to the same apsis again, to four right angles.

Hence the angle described by the radius vector from one apsis to the same apsis, is $360^\circ \times \left(\frac{F}{F-5f}\right)^{\frac{3}{2}}$.

This proposition, which is demonstrated by Dr Stewart in the fourth of his Tracts, in a manner somewhat prolix, on account of his rigorous adherence to the methods of the ancient geometry, but in a way perfectly clear and elementary, is employed by him to deduce the mean disturbing force from the motion of the apsides as ascertained by observation. But when the mean disturbing force is known from other phenomena, the same proposition may be employed to deduce the motion of the apsides from that force. Accordingly, if the disturbing force be taken $\frac{1}{357}$, the motion of the apsides will come out $= 3^\circ 1' 20''$ for a sidereal revolution of the moon, very near the quantity actually observed.

36. Having determined the sun's mean disturbing force from the motion of the apsides, Dr Stewart proceeded to determine from the former of these the sun's distance from the earth. The result of a very nice investigation gave the sun's parallax $6^\circ 9'$, a quantity that is no doubt too small, and makes of course the sun's distance too great. It is indeed but an inconsiderable part of the sun's disturbing force into which the parallax enters as an element, and therefore any deduction founded on it must be liable to this inaccuracy, that a small error in the data will produce a great one in the result.

37. After the inequalities which are conceived as belonging to the moon's orbit, come those which directly affect the place of the moon in that orbit. The most considerable of these, after what is called the equation of the centre, arising from the elliptic figure of the lunar orbit, and independent of all disturbance, is the equation or inequality called the excentric, which was discovered by the Greek astronomers. This depends on the position of the transverse axis of the moon's orbit in respect of the line of the syzygies. When that axis is in the line just mentioned, because the quantity by which the solar force diminishes the gravitation of the moon in the syzygies is, ceteris paribus, proportional to her distance from the earth, it is greatest when the moon is in the apogee, and least when in the perigee. In this situation of the orbit, therefore, the greatest diminution is made from the quantity of the moon's gravitation which is already the least, and the least from that which is already the greatest; the gravitation at the perigee, and therefore the difference, is augmented, and the orbit appears to have its eccentricity increased. When the line of the apsides is in the quadratures, the contrary happens; the gravitation at the apogee is most augmented, and at the perigee least; the difference is therefore diminished, and the eccentricity of the lunar orbit seems also to be diminished. This is conformable to observation; and when the eversion is accurately deduced from the theory of gravitation, it appears

\[ = (1° 20' 29" \cdot 9) \sin \left( 2 (\epsilon - \Theta) - a \right) \]

where \( \epsilon \) is the mean longitude of the moon, \( \Theta \) that of the sun, and \( a \) the mean anomaly of the moon counted from the perigee.

38. The moon's variation is an inequality which was discovered by Tycho, and found to depend on the angular distance of that planet from the sun. It is derived from that part of the sun's disturbing force which is at right angles to the radius vector, and which accelerates the motion of the moon from the quadratures to the syzygies, and retards it from the syzygies to the quadratures. The effect of this force is found, from the theory of gravity, to be represented by three terms, which, if \( \Delta \) be the angular distance of the moon from the sun, are,

\[ + (35' 41" \cdot 9) \sin 2 \Delta \] \[ + (0' 2" ) \sin 3 \Delta \] \[ + (0' 14" ) \sin 4 \Delta \]

39. The lunar inequality, called the annual equation, arises from the variation of the sun's disturbing force according to the place which the earth occupies in its orbit. It is shown above that the sun's disturbing force is, ceteris paribus, inversely as the cube of its distance; so that when the earth is in its perihelion this force is the greatest, and at the aphelion the least, its effect varying at the same rate with the equation of the sun's centre, or having everywhere the same ratio to that equation. Hence this equation is nearly \((11' 12") \times \sin\) sun's mean anomaly, with a contrary sign to the equation of the sun's centre.

40. These inequalities are all phenomena which were observed before the explanation of them was known. To them may be added a fourth inequality, known by the name of the moon's acceleration. It appeared to astronomers as a continual increase in the velocity of the moon, or in the rate of her mean motion, amounting to about \(10''\) in a century, and its effect, like that of all other constant accelerations, accumulating as the squares of the times. It did not seem to be periodical, like the other lunar inequalities, but to be a constant increase of the velocity, and a corresponding diminution of the periodical time of the moon, which must in the end change entirely the relation of that body to the earth.

It is but within these few years that Laplace discovered it to be a periodic inequality, though requiring, in order to accomplish the series of its changes, a length of time which science has not yet ventured to calculate. For many centuries to come it may be expressed by this formula, taking \(n\) to denote the number of centuries reckoned from the year 1700, viz.

\[ 10^6 \times 7232 \times n^2 + 0^6 \times 019361 \times n^3. \]

The first term includes all that was known from observation previously to the discovery of Laplace. This, however, must be considered not as the true form of the equation, which must include the sines or cosines of certain angles, but merely a provisional formula, to serve till the true one can be rigorously assigned.

This inequality has in its cause a great affinity to the annual equation.

Whatever changes the form of the earth's orbit, has an effect on the disturbing force of that body on the moon, which is in the inverse ratio of the cube of the distance between the sun and earth. But it is found that though the mean distance remains invariable, the eccentricity of the earth's orbit changes, on account of the action of the other planets, and in fact has been diminishing, from a more remote antiquity than that to which the history of astronomy extends. From this cause Laplace has deduced the supposed acceleration of the moon's mean motion.

41. All these inequalities have been pointed out by observations, and have been explained in the most satisfactory manner by the principle of universal gravitation. But when all these were reduced into equations and arranged in tables, yet the places of the moon calculated from them were never quite exact; and there seemed a cause of error or a mass of small inequalities unknown in their magnitude and form, to which this inaccuracy was to be ascribed, and which operated, as it may be said, like a mist which concealed the true place of the moon from the calculator, and prevented his results from agreeing completely with those of the observer. The most likely way to discover these inequalities, if they arose from gravity, was to push the approximation to the moon's place still farther, and to try if the terms hitherto neglected in the approximation would not, when taken into account, afford a complete analysis of the circle of confusion which might be said to surround the moon on all occasions.

The problem on which mathematicians now entered, and which Clairaut, already mentioned, Euler, and D'Alembert, all three resolved nearly about the same time, has been called the Problem of Three Bodies. The thing proposed is, three bodies which attract one another with forces directly as their quantities of matter, and inversely as the squares of their distances, being given, and any motions whatever being impressed on them, to find the orbits which they will describe round their common centre of gravity. It is, however, only in certain cases that this general problem admits of solution, and one of these is, when one of the bodies is at a vast distance from the other two. This is exactly the case with the moon and earth in respect of the sun, the orbit of the earth being nearly the same as if there only existed the sun and earth, and the orbit of the moon relative to the earth being nearly the same if there were only the moon and earth. This solution of the problem, however, in this direct way, leads to far more exact conclusions than can be obtained from the more simple but more indirect method which Newton followed. The general view which leads to the most exact estimate of the merit of the two solutions is, that the motions of the moon, when analytically and fully expressed, necessarily form a number of different series, each of which converges with more or less rapidity. The prosecution of the direct method allows the terms of these series to be computed to an indefinite extent, or till the quantities omitted are too small to affect observation. The method of Newton can go no farther than to compute the first, or at most a few of the leading terms of each of the series. Its accuracy is therefore limited; that of the other knows no limits. Though this be a true estimate of the value of the methods, yet that of the original inventor possesses infinite merit, as having first led the way to this arduous investigation, and as still serving to carry the imagination better along with it than the other, and to keep the mechanical principles more directly in view.

The complete solution of the problem of the three bodies has accordingly discovered a great number of new equations, each individually small, which would sometimes nearly destroy one another, and, at other times, having many of them the same sign, would accumulate to a considerable amount. This was the triumph of the theory, and the strongest evidence of its truth. The effect of these irregularities varied so much, and depended on so many elements, that it may be doubted whether the most accurate and most constant observation would ever have enabled astronomers to discover their precise quantities, and to separate them from one another.

The tables of the moon, in the state to which they are now brought, contain twenty-eight equations for the longitude, twelve for the latitude, and thirteen for the horizontal parallax of the moon. Of the first of these, twenty-three have been deduced from theory alone; of the second, nine; and of the third, eleven. This applies to the tables of Burg; those since published by Burckhardt contain more equations, and are still more accurate.

2. Of the Disturbance in the Motion of the Primary Planets, produced by their action on one another.

42. It is evidently necessary, in this inquiry, to know the quantities of matter in the different planets, or, which comes to the same, the intensity of the attraction of each at a given distance from its centre. With respect to those planets which have satellites, the Earth, Jupiter, Saturn, and Uranus, their masses or quantities of matter have been already determined. The masses of Venus and Mars have been estimated by Laplace, from the effects which they appear to produce on the earth's motion. The mass of Mercury has been estimated on the supposition that the densities of that planet and of the earth are inversely as their mean distances from the sun. This law holds with respect to the Earth, Jupiter, and Saturn, and analogy renders it probable that the same law includes the other planets. Thus, the mass of the Sun being 1, that of Mercury is $\frac{1}{2025810}$ of Venus $\frac{1}{405871}$, and of Mars $\frac{1}{2546320}$; the masses of the other planets being as already stated.

43. The effects of the action of the planets on one another is more difficult to be investigated than the effects of the sun's action on the moon, because the disturbing forces are not only more numerous, but because the distance of the disturbing from the disturbed body is not so great that the quantities divided by higher powers of that distance can be so safely rejected. The general principle, however, according to which the solar action on the moon was resolved into forces either in the direction of the radius vector or at right angles to it, is applicable to both questions.

Thus, supposing P and P' (fig. 119) to be two planets revolving in orbits, nearly circular, about the sun at S; in order to find how the motion of P' is affected by the action of P, let PP', PS, and PS be drawn, and let the line A denote the force with which P attracts a particle of matter at the distance PS, then the force with which it attracts a particle at the distance PP', will be $A \times \frac{PS^2}{PP'^2}$. Let

$$PR = A \times \frac{PS^2}{PP'^2};$$

and if PR be resolved into two forces, PM and PN, the one in the direction of the radius vector PS, and the other parallel to PS, take NO = A; then the remaining forces OP' and PM are those which disturb the motion of P', as was proved in the case of the moon. The former of these, OP', may be resolved into OQ and PQ, of which PQ diminishes the gravity of the planet to the sun, and OQ accelerates its motion in a direction perpendicular to the radius vector. Therefore, as the force PM always increases the planet's gravity to the sun, PM—PQ is the whole force increasing or diminishing the gravity of P' to S; and the force directly employed in increasing or diminishing the angular motion of P about S, is OQ or PT.

The analytical values of these quantities may be found, as in the theory of the moon, though not with equal simplicity, because SP cannot always be supposed great in respect of SP'.

44. In consequence of these actions, the orbit of every planet may be considered as an ellipse, which is undergoing slowly certain changes in its form, magnitude, and position, or in what are called its elements. By the elements of the orbit of any heavenly body, are meant the quantities that determine the position and magnitude of that orbit, viz. the position of the line of the nodes, the inclination of the plane of the orbit to the plane of the ecliptic, the position of the line of the apsides, the eccentricity, and the mean distance. These are all quantities independent of one another, and from them may be deduced all other circumstances with respect to the elliptic orbit. Of these five elements, which would be invariable if the planet only gravitated to the sun, all except the mean distance are subject to slow but perpetual changes.

45. The line of the nodes, in every one of the planets, has a retrograde angular motion, which goes on continually, and of which the amount, when calculated as due to each planet, agrees very well with observation. The plane of the orbit also varies its inclination, by certain small periodical changes, which alternately increase and diminish it, as in the case of the moon. The line of the apsides, from the same cause as in the planet just mentioned, has a continued motion forward, or according to the order of the signs. Thus, in Mercury the node goes back about 7°82 annually. The aphelion goes forward about 3°84 in a year, and the inclination of the orbit in the course of a century increases about 18°18, which, in the course of succeeding ages, will be compensated by an equal diminution, so as to preserve it always nearly of the same quantity. In the same planet the equation of the centre, which depends on the eccentricity, increases about 1°6 in a century, indicating a small increase of eccentricity. These variations in the orbit of Mercury arise from the action of Venus, the Earth, Mars, Jupiter, and Saturn; the effects of the first of these planets, on account of its vicinity, being by much the most considerable. The mean distance, however, of Mercury from the Sun, does not, any more than that of the other planets, undergo any change whatever.

46. Similar conclusions apply also to the orbit of Venus. The orbit of the earth also is subject to similar changes, the line of the apsides moving forward annually at the rate of 11°8, in respect to the fixed stars. The earth's eccentricity is also diminishing, and the secular variation of the greatest equation of the centre is — 17°66.

The motion of the earth is subject to another inequality on account of the action of the moon; for, to speak strictly, it is not the centre of the earth, but the centre of gravity of the moon and earth, which describes equal areas in equal times about the centre of the sun. It is evident that, on this account, the earth will be sometimes advanced before, and sometimes will fall behind, the point which describes the circumference of the ellipse, in conformity with the general law of the planetary motions. From the same cause also, as the moon does not move in the plane of the ecliptic, the earth will be forced out of that plane, in order to preserve a position diametrically opposite to the moon. These irregularities, however, are inconsiderable. By observers on the earth's surface, they are transferred to the sun, but in an opposite direction. The sun, therefore, has a motion in longitude, by which he alternately advances before the point which describes the elliptic orbit in the heavens, and falls behind it; and also a motion in latitude, by which he alternately ascends above and descends below the plane of the ecliptic. As the mass of the moon, however, is not more than $\frac{1}{79}$ part of that of the earth, the distance of the centre of gravity of the moon and earth from the centre of the latter must be less than a semidiameter, and therefore the inequality thus produced in the sun's longitude must be less than his horizontal parallax. The alteration in latitude can hardly amount to a second. This inequality in the sun's motion is called the menstrual parallax, and was first mentioned by Smeaton, Phil. Trans. 1768.

47. In the orbit of Mars the node moves backward $23^\circ 3$ annually, and the line of the apsides moves forward $15^\circ 8$, both in respect of the fixed stars. The eccentricity of the orbit is increasing, and the secular variation of the greatest equation of the centre is $+37''$.

In the case of this planet, however, the elliptic orbit is not only changed by these quantities, but the place of the planet in that orbit is sensibly affected by the action of Venus, Jupiter, and the Earth. The effect of the action of Venus is expressed by this formula, $5^\circ 7 \sin (\text{long.} \varphi - 3 \text{long.} \delta)$; of the earth, $7^\circ 2 \sin (\text{long.} \varphi - \text{long.} \delta)$. Several inequalities are produced in Jupiter.

48. The inequalities of the small planets Vesta, Juno, Ceres, and Pallas, have also been computed. The disturbances which they must suffer from Venus, Mars, and Jupiter are considerable, and, on account of their vicinity, though their masses are small, they may somewhat disturb the motions of one another. Their action on the other bodies in the system is probably insensible.

As Pallas and Ceres have nearly the same periodic time, they must preserve nearly the same distance and the same aspect with regard to one another. This offers a new case in the computation of disturbing forces, and may produce equations of longer periods than are yet known in our system.

The motion of the apsides and the change of eccentricity in the orbits of Jupiter and Saturn are chiefly produced by their action on one another, but a part also depends on the action of the other planets. The node of Jupiter moves backward annually $15^\circ 8$, and his aphelion forward $6^\circ 96$. The secular change in the inclination of the orbit is $22^\circ 6$, and in the first and last of these inequalities the action of Venus has the principal share. The equation of the centre increases $56^\circ 25$ in a century, of which nearly the whole arises from the action of Saturn. In Saturn again the node goes back at the rate of $19^\circ 4$ annually, and the aphelion forward at the rate of $19^\circ 4$; the secular change of the inclination is $-15^\circ 5$, and the secular diminution of the equation of the centre $2^\circ 1$.

There is, besides these variations in the orbits, an inequality in the motion of each of these planets, which for a long time was difficult to explain, and was ultimately fully accounted for, according to the theory of gravity, by the profound investigations of Laplace. These inequalities are both of a long period, viz. $91876$ years, which is the time that they take to run through all their changes. If $n$ express a number of years reckoned from the beginning of 1750, $S$ the mean longitude of Saturn, and $I$ that of Jupiter, reckoned from the same time, then the equation which must be applied to the mean longitude of Jupiter, or the amount of this inequality, is

$$+ (20^\circ 49'' - n \times 0^\circ 042733) \times \sin ((5S - 2I + 5^\circ 34' 8'' - n \times 58^\circ 88));$$

and that which must be applied to $S$ is

$$-(48^\circ 44'' - n \times 0^\circ 1) \times \sin ((5S - 2I + 5^\circ 34' 8'' - n \times 58^\circ 88)).$$

These two equations are to one another nearly in the ratio of 3 to 7. The reason of the long period above mentioned is, that the argument $5S - 2I - n \times 58^\circ 88$ requires all that time to increase from $0$ to $360^\circ$.

Uranus, on account of his great distance, suffers hardly any disturbance in his motion, but from Saturn and Jupiter. The node moves backward at the rate of $36^\circ$ annually, and the aphelion forward at that of $2^\circ 55$. The eccentricity is diminishing, and the secular variation of the greatest equation of the centre is $11^\circ 03$.

There is also an inequality in the longitude of this planet, depending on the action of Saturn. If $S$ be the longitude of this last planet, $U$ the longitude of Uranus, and $A$ the longitude of the aphelion of Saturn, the inequality in question amounts to $2^\circ 30'' \times \sin ((S - 2U + A))$.

49. Of all these inequalities, and of many other smaller ones which theory has discovered, it must be observed that they are periodical, each returning after a certain time to run through the same series of changes which it had formerly exhibited.

Another remark is, that one element in every orbit, viz. the mean distance, is exempted from all excepting periodical changes; and since on the mean distance depends the time of revolution, that time remains also unchanged. From the invariability of the mean distance, and the periodical revolution of all the inequalities, it follows that the actual condition of the planetary system can never deviate far from the mean, about which we may, therefore, conceive it to be continually making small oscillations, which in the course of ages compensate one another, and therefore produce nothing like disorder or permanent change. It is in this manner that the stability of the planetary system is provided for by the wisdom of its Author.

50. Comets, in describing their elliptic orbits round the sun, have been found to be disturbed by the action of the larger planets, Jupiter and Saturn; but the great eccentricity of their orbits makes it impossible, in the present state of mathematical science, to assign the quantity of that disturbance for an indefinite number of revolutions, though it may be done for a limited portion of time, by considering the orbit as an ellipsis, the elements of which are continually changing. This is the method of Lagrange, and is followed in the Mécanique Céleste, Part ii. chap. ix. Dr Halley, when he predicted the return of the comet of 1682, took into consideration the action of Jupiter, and concluded that it would increase the periodic time of the comet a little more than a year; he therefore fixed the time of the re-appearance to the end of the year 1758, or the beginning of 1759. He professed, however, to have made this calculation hastily, or, as he expresses it, levi calamo. (Synopsis of the Astronomy of Comets.)

The effects both of Jupiter and Saturn on the return of the same comet were afterwards calculated more accurately by Clairaut, who found that it would be retarded 511 days by the action of the former planet, and 100 by the action of the latter; in consequence of which, the return of the comet to its perihelion would be on the 15th of April 1759. He admitted at the same time that he might be out a month in his calculation. The comet actually reached its perihelion on the 13th of March, just 33 days earlier than was predicted; affording, in this way, a very striking verification of the theory of gravity, and the calculation of disturbing forces. The same comet appeared again, according to prediction, in 1835.

In some instances, the effect which the planets produce on the motion of comets are far more considerable than in this example. A comet which was observed in 1770 had a motion which could not be reconciled to a parabolic orbit, but which could be represented by an elliptic orbit of no great eccentricity, in which it revolved in the space of five years and eight months. This comet, however, which had never been seen in any former revolution, has never been seen in any subsequent one. On tracing the path of this comet, Mr Burckhardt found that between the year 1767 and 1770 it had come very near to Jupiter, and had done so again in 1779. He therefore conjectured, that the action of Jupiter may have so altered the original orbit as to render the comet for a time visible from the earth; and that the same cause may have so changed it, after one revolution, as to restore the comet to the same region in which it had formerly moved. This is the greatest instance of disturbance which has yet been discovered among the bodies of our system, and furnishes a very happy, as well as an unexpected, confirmation of the theory of gravity.

Though the comets are so much disturbed by the action of the planets, yet it does not appear that their reaction produces any sensible effect. The comet of 1770 came so near to the earth as to have its periodic time increased by 2-246 days, according to Laplace's computation; and if it had been equal in mass to the earth it would have augmented the length of the year by not less than two hours and forty-eight minutes. It is certain that no such augmentation took place, and therefore that the disturbing force by which the comet diminished the gravity of the earth is insensible, and the mass of the comet, therefore, less than \( \frac{1}{3} \)th of the mass of the earth. The same comet also passed through the system of the satellites of Jupiter without causing any derangement of their motions. Hence it is reasonable to conclude, that no material or even sensible alteration has ever been produced in our system by the action of a comet.

3. Of the disturbances which the satellites of Jupiter suffer from their action on one another.

51. The same resolution of the forces by which one satellite acts upon another, into two, one directed to the centre of the primary, and the other at right angles to it, serves to explain the irregularities which had been observed in their motions, and to reduce under known laws several other inequalities, of which the existence only is indicated by observation.

An instance of this we have in the very remarkable relation which takes place between the mean motions of the first three satellites; the mean motion of the first satellite, together with twice the mean motion of the third, being equal to three times the mean motion of the second. Laplace has shown that, if the primitive mean motions of these satellites were nearly in this proportion, the mutual action of these bodies on one another must in time have brought about an accurate conformity to it.

The first moves nearly in the plane of Jupiter's equator, and the orbit has no eccentricity, except what is communicated from the third and fourth, the irregularities of one of these small bodies producing similar irregularities in those that are contiguous to it. The first satellite has, beside, an inequality, chiefly produced by the action of the second, and circumscribed by a period of 437-659 days.

52. The plane of the orbit of the second satellite is inclined to a determinate fixed plane at an angle of 27° 13', and on which its nodes have a retrograde motion, so that they complete a revolution in 299-914 years. The motion of the nodes of this satellite is one of the principal data used for determining the masses of the satellites themselves, which are so necessary to be known for computing their disturbances. This satellite has no eccentricity but that which it derives from the action of the third and fourth.

The plane of the orbit of the third satellite is inclined to a determinate fixed plane at an angle of 12° 20', and its nodes make a tropical revolution backwards in 141-739 years. The equator of Jupiter is inclined to the plane of his orbit at an angle of 3° 5' 30'. The fixed planes on which the planes of the orbits move are determined by theory, and could not have been discovered by observation alone.

The orbit of the third satellite is eccentric, but appears to have two distinct equations of the centre; one which really arises from its own eccentricity, and another which theory shows to be an excentration from the equation of the centre of the fourth satellite. The first equation is referable to an apsis which has an annual motion of 2° 36' 39" forward in respect of the fixed stars; the second equation is referable to the apsides of the fourth satellite.

These two equations may be considered as forming one equation of the centre, referable to an apsis that has an irregular motion. The two equations coincided in 1682, and the sum of their maxima was 13° 16". In 1777 the equations were opposed, and their difference was 5° 6".

The last two inequalities were perceived by Mr Wargentin, by observation alone; but their exact amount, and the law which they observe in their changes, he could not discover. The plane of the orbit of the fourth satellite is inclined to a determinate fixed plane at an angle of 14° 58'; and its nodes complete a sidereal revolution backward in 531 years. The fixed plane on which the orbit moves is inclined at an angle of 24° 33' to the equator of Jupiter; the orbit is sensibly elliptical, and its greater axis has an annual motion of 42° 58' 7". The motion of this axis is one of the principal data from which the quantities of matter of the different satellites have been determined.

If the mass of Jupiter be supposed unity, the mass of the 1st satellite = -0000173281 of the 2d = -0000232355 of the 3d = -0000884972 of the 4th = -0000426591

If the mass of the earth be supposed unity, that of the third satellite will be found = -027837; and as the mass of the moon is \( \frac{1}{79-89} = -012517 \), the quantity of matter in the third satellite is about twice as great as that in the moon. The fourth satellite is therefore nearly equal to the moon, the second about one half, and the first somewhat more than one third.

53. The general result of this investigation concerning the inequalities in the motion of the planets, both primary and secondary, is, that in every one of these orbits two things remain secure against all disturbance, the mean distance and the mean motion, or, which is the same, the transverse axis of the orbit, and the time of the planet's revolution. Another result is, that all the inequalities in the planetary motions are periodical, and observe such laws that each of them, after a certain time, runs through the same series of changes. This last conclusion follows from the fact, that every inequality is expressed by terms of the form \( A \sin nt \) or \( A \cos nt \), where \( A \) is a constant co-efficient, and \( n \) a certain multiplier of \( t \) the time, so that \( nt \) is an arch of a circle, which increases proportionally to the time. Now, in this expression, though \( nt \) is capable of indefinite increase, yet, since \( nt \) never can exceed the radius, or 1, the maximum of the inequality is \( A \). Accordingly, the value of the term \( A \sin nt \) first increases from 0 to \( A \), and then decreases from \( A \) to 0; after which it becomes negative, extends to — \( A \), and passes from thence to 0 again. If, when the inequality was affirmative, it was an addition to the mean motion, when negative it will become a diminution of it; and the sum of all these increments and decrements, after \( nt \) has passed over an entire circumference, or 360°, is equal to 0; so that at the end of that period the planet is in the same position as if it had moved on regularly all the while at the rate of the mean motion. As this happens to every one of the inequalities, the deviation of the system from its mean state can never go beyond certain limits, each inequality in a certain course of time destroying its own effect.

It would be far otherwise if into the value of any inequalities a term entered of the form \( A \times nt \), \( A \tan nt \), The inequalities so expressed would continually increase with the time, so as to go beyond any assignable limit, and of consequence to destroy entirely the order of any system to which they belonged.

Lagrange and Laplace, who discovered and demonstrated that no such terms as these last can enter into the expression of the disturbances which the planets produce by their action on one another, made known one of the most important truths in physical science. They proved that the planetary system is stable, and that it does not involve any principle of destruction in itself; but is calculated to endure for ever, or till the action of an external power shall put a period to its existence. After the knowledge of the principle of gravitation, this may be fairly considered as the greatest discovery to which men have been led by the study of the heavens.

The accurate compensation, just remarked, depends on three conditions, belonging to the primitive or original constitution of our system, but not necessarily determined, as far as we know, by any physical principle. The first of these conditions is, that the eccentricities of the orbits are all inconsiderable, or contained within very narrow limits, not exceeding in any instance one tenth or one eighth part of the mean distance. The second condition is, that the planets all move in the same direction, or from west to east. This is true both of the primary and secondary planets, with the exception only of the satellites of Uranus, which may be accounted retrograde; but their planes being nearly at right angles to the orbit of their primary, the direction of their motion, whether retrograde or otherwise, can have little effect. Lastly, the planes of the orbits of the planets are not much inclined to one another. This is true of all the larger planets, though it does not hold of some of the new and smaller ones; of which, however, the action on the whole system must be altogether insensible.

Unless these three conditions were united in the constitution of the solar system, terms of the kind just mentioned, admitting of indefinite increase, might enter into the expression of the inequalities, which would indicate a gradual and unlimited departure from the original order and constitution of the universe. Now, the three conditions just enumerated do not necessarily arise out of the nature of motion, or of gravitation, or from the action of any physical cause with which we are acquainted. Neither can they be considered as arising from chance; for the probability is almost infinity to one, that, without a cause particularly directed to that object, such a conformity could not have arisen in the motions of so large a number of bodies, scattered over the whole extent of the solar system. The only explanation, therefore, which remains is, that all this is the work of intelligence and design, directing the original constitution of our system, and impressing such motions on the parts as are calculated to give stability to the whole.

For some further discussions connected with Physical Astronomy, see the articles Comets; Earth, Figure of; Precession of the Equinoxes; and Tides.

PART IV.

PRACTICAL ASTRONOMY.

INTRODUCTION.

Practical Astronomy may be considered as comprehending the observations which must originally have been made to determine the facts which have now been embodied in a system, as well as those which are constantly being made for its further extension and improvement; the observations which are required to make the science useful in the affairs of life; and the rules and calculations which must be applied to the observations, to obtain from them the required results.

Of the first class of observations mentioned above, the most important are those which relate to general surveys of the heavens, and the extension of cosmical research amongst those bodies that lie beyond the solar system. The sweeps of the stars, and the observations of the nebulae begun by Sir W. Herschel in the last century, have been carried forward with additional vigour in this. Star maps and star catalogues for almost every portion of the heavens have been accumulating during the present century, till the places and magnitudes of very nearly one hundred thousand stars have been accurately laid down; speculation has closely followed practice; and not only has our own position with respect to the star cluster of which our solar system forms a part, been pretty accurately defined, but clearer views have been obtained of the probable distances and even of the constitution of some others of those systems which, though seen by us under the aspect of a feeble nebulous light, scarcely larger than the apparent disk of a planet of our system, yet probably occupy individually a space equivalent to that of our galaxy, and fill the firmaments of other worlds like our own, with their light and their glory.

Even with regard to observations connected with the solar system merely, observation and theory combined have achieved conquests, since the last edition of this work was published, not inferior in interest or importance. Since the beginning of the present century, the telescope, added by the improved methods of modern research, has discovered twenty-two of those small fragmentary bodies moving in planetary orbits between Mars and Jupiter; and theory alone, without the aid of the telescope, using only the laws of mechanical action and the results of previous observation, has extended the bounds of the known solar system, by predicting the place of a new body, and calling in the aid of the observer only to verify the fact, and to assist in the inquiry. The assumed well-marked distinction between the planetary and cometary orbits has been disproved by the discovery of several comets moving in ellipses with a periodical term shorter than that of Jupiter; curious physical facts have been elicited by the motions of some of them, while several new members of this class of bodies are yearly added to the list, and their orbits eagerly computed by new aspirants for astronomical fame.

Yet while the science has been thus followed up in its more speculative departments with all the success that the united zeal, talent, and energy of a highly intellectual age would give us reason to expect, and with an accuracy of detail in the observations which brings into play all the resources of pure practical astronomy, it has been cultivated with equal ability in those departments to which the term practical more peculiarly belongs, and has lent its aid to the most recent and distant branch of navigation in every which will soon render the art of guiding a ship through the trackless ocean as perfect both in theory and practice as anything depending upon human faculties can possibly be. The problem of determining the position of a ship at sea depends upon three things mainly, namely, 1st, an accurate knowledge of the lunar orbit, and the consequent motions of the moon; 2dly, upon the indisputable accuracy of the places of certain fundamental stars and bright planets with which her place is compared; and, 3dly, upon the accuracy of performance of chronometers and nautical observing instruments. Now, in all these departments, the progress towards perfection during the last twenty years has been immense. The lunar orbit has been nearly cleared of all its theoretical irregularities, or those depending Astronomy.

Practical upon unknown causes of perturbation, and, very recently, important errors in the tables ordinarily in use have been detected, and corrected; ephemerides of the motions of the moon are computed with all attainable accuracy both in England and America; and finally, by the extra-meridional observations begun at Greenwich in 1847, and continued without intermission to the present time, the errors due to want of observations in certain parts of the orbit (those near conjunction namely) will be altogether got rid of. Again, the Greenwich Catalogues alone furnish the places of large numbers of stars with an accuracy indispensable for lunar purposes only in the case of a few, and every fundamental result of observations connected with the theory of the moon is supplied from the Greenwich Observatory with an accuracy and steadiness even now unrivalled. Finally, the construction of chronometers and of nautical instruments has been brought to almost as much perfection as the case will admit of. Our government has encouraged the artists to place in competition every year a certain number of chronometers; and the ingenuity of this whole class of persons, brought to bear upon this branch of art, has been the means of eliciting some important improvements in the methods of compensating for temperature and in general construction, at the same time that their rivalry has been the means of reducing the price of the article, and bringing it more within the reach of the ordinary merchant captains of vessels. Various improvements have been at the same time made in surveying instruments and hydrographical methods. The sextant has been simplified and improved; charts have been made in almost every shore in our dominions; the tides have been studied with great attention; the union of high analytical investigations, and the practical results of observations, which have put astronomy on so sound a basis; the laws of the currents of various seas have recently been made the subject of special investigation on a scale equal to its importance; and, in fact, there is not a single branch of nautical astronomy which has not been benefited in an equal degree with the more theoretical or elevated branches of the science.

The science of electricity, in its application to the electric telegraph, has been already of considerable service to astronomy, by affording the opportunity of making simultaneous signals at any two observatories situated on a line of railway, however distant they may be from each other. By this means the differences of longitude of three principal British observatories—namely, Greenwich, Cambridge, and Edinburgh—have already been obtained with an accuracy unattainable by any other method, and the difference of longitudes of Greenwich and Oxford will shortly be determined. The galvanic connexion between Greenwich and Paris is also complete, by means of the submarine communication between Dover and Calais; and experiments will shortly be tried for the determination of the difference of longitudes of these two observatories; and, after this has been accomplished, there will be no difficulty whatever in determining the longitudes of all the continental observatories situated near the grand lines of railroad or telegraph wires. Hopes are beginning to be entertained also of carrying a wire across the Atlantic, and thus bringing the American observatories into connexion with Europe; and the project is not in reality very chimerical.

We need not dwell specifically upon the means by which these great results have been attained. Our two supplements historical and theoretical will have given ample information concerning every available source of improvement in the methods of making observations, and of the chief improvements which have been devised by the eminent men conducting the establishments which have been described. There is scarcely a discovery of importance, or a new method deserving of explanation, in any one of the branches of pure astronomy, concerning which the reader will not find information, and guidance for future research; many of the books which the accomplished astronomer, as well as the student, will require for his pursuit of the science are indicated, and some account of the contents of some of them, where it is necessary to elucidate the subject, has been given.

Such information, indeed, as relates exclusively to the allied sciences of surveying and navigation, must of course be sought for under their respective titles, since the space allotted us would not allow of the extension which such subjects would have required; but, in everything that relates to the observations made in fixed observatories, and to the results deduced from them as applied to the use of these sciences, it is hoped that some instruction has been given.

There is one subject which we hope to have entered into with more minuteness, but which we are prevented from doing through want of space, namely the method of reducing as well as making the observations necessary for the determination of the position of a body in the heavens; and we were the more reluctant to relinquish our design because it is a subject on which most elementary books are either deficient or silent. It has been stated again and again, that the two elements which determine the position in question, are the right ascension and declination of the body, and the instruments which ordinarily determine these elements are the transit instruments (assisted by an astronomical clock) and the mural circle, if only two instruments be used, or the transit circle, if only one instrument be employed. Under this head it has been deemed sufficient to retain the accounts of the transit instrument constructed by Troughton for Sir James South, which is still mounted and in excellent condition; and of Troughton's mural circle, erected at the Royal Observatory of Greenwich, in 1812, which was used with excellent effect till the end of the year 1850, when it was replaced by the great transit circle now in use. Each of these instruments is probably one of the best existing specimens of its class, and neither has been excelled, in perfection of workmanship, even to the present time. In addition to the instruments which determine the place of a body by observations made on the meridian, a third instrument, namely the equatorial, is necessary for various purposes, but especially for comparing planets and comets, of which a sufficient number of observations cannot be obtained on the meridian, with neighbouring stars; of this class of instruments we have retained the accounts of the celebrated Dorpat telescope already mentioned in the account of that observatory, and of the three-foot equatorial in the possession of Sir James South, which was used with such effect by him and Sir John Herschel in the measurement of double stars.

With regard to the tables printed in this department, we have retained those relating to the calculation of eclipses, as interesting to the student, though it must be borne in mind that much better elements exist at present for such calculations than those which were used in these tables.

The catalogue of stars has been replaced by a more modern catalogue accurately compiled, by the use of the best elements, from the Greenwich Twelve-Year Catalogue of 21,56 stars. This list includes all the stars down to the fifth magnitude inclusive, which are found in that catalogue, and the places are reduced to the epoch 1850, by the use of the geometrical precessions given in the British Association Catalogue, and by the proper motions given in Mr Main's paper on Proper Motions, (Memoirs of the Royal Astronomical Society, vol. xix.) Where no proper motion is given by Mr Main,—that is for stars not observed by Bradley or not given in the Fundamenta,—the annual variation is taken altogether from the British Association Catalogue, except in a few cases specified in the notes.

Finally, we have retained, as useful to the student, the small table of refractions given in the former edition, not as insisting upon its accuracy (since the whole subject has been since revised by Bessel), but as enabling the student to see by compendious tables and examples how this element of correction is taken account of in the reduction of observations.

A few problems given in the former edition have been retained, as being good exercises for the students, and amongst the most interesting and important that belong to the science of plane astronomy, though it is evident that in the systematic study of the science they should be read in their proper order in works treating of the details of the science.

CHAP. I.

PROBLEMS IN PRACTICAL ASTRONOMY.

Problem I.—Given the right ascension and declination of a star or planet, together with the obliquity of the ecliptic, to find the star's longitude and latitude.

Let P (fig. 120, Plate XC.) be the pole of the equator O Q, E the pole of the ecliptic I L, and S the place of the star. Let PR, the circle of latitude passing through Astronomy

S, meet the equator in R; and EM, the circle of declination, meet the ecliptic in M; also let \( \alpha E, \alpha P \) be arcs of great circles passing through the equinoctial point \( \alpha \) and the poles of the ecliptic and equator respectively. Let us now make

\( L = \alpha M \) the longitude of S, \( l = SM \) the latitude, \( \alpha R = \alpha R \) the right ascension, \( D = SR \) the declination, \( \omega = LQ \) the obliquity of the ecliptic.

Since \( \alpha M = \alpha L - ML = 90^\circ - ML \), and ML is the measure of the angle SEP, we have \( \alpha M = L = 90^\circ - SEP \), and, consequently, \( \sin L = \cos SEP \). Again, since \( \alpha R = \alpha Q - RQ = 90^\circ - RQ \), and RQ is the measure of the angle RPQ, we have \( \alpha R = AR = 90^\circ - RPQ \); whence \( \sin AR = \cos RPQ = \cos EPS \). In like manner we have \( l = SM = 90^\circ - ES \), whence \( \sin l = \cos ES \); and also \( D = SR = 90^\circ - PS \), whence \( \sin D = \cos PS \). Thus the problem depends on the solution of the oblique angled spherical triangle PES.

From the known properties of spherical triangles, we have

\[ \cot PS \sin EP = \cot PES \sin EPS + \cos EP \cos EPS; \]

therefore, by substituting the values just found, and observing that \( EP = \omega \), we have

\[ \tan D \sin \omega = \tan L \cos AR - \cos \omega \sin AR; \]

whence

\[ \tan L = \frac{\tan D \sin \omega + \sin AR \cos \omega}{\cos AR}. \]

Assume an angle \( \phi \) such that \( \tan \phi = \sin AR \cot D \), then

\[ \sin D \tan \phi = \sin AR \cos D, \]

or \( \tan D = \frac{\sin AR}{\tan \phi} = \frac{\sin AR \cos \phi}{\sin \phi} \);

whence, by substituting,

\[ \tan L = \tan AR \left( \frac{\sin \omega}{\tan \phi + \cos \omega} \right), \]

and by reducing,

\[ \tan L = \frac{\sin (\omega + \phi)}{\sin \phi} \tan AR...........(1) \]

a formula from which the longitude \( L \) is easily computed by means of the logarithmic tables.

To find the latitude \( l \), we have

\[ \cos ES = \cos EP \cos PS + \sin EP \sin PS \cos EPS, \]

which gives

\[ \sin l = \cos \omega \sin D - \sin \omega \cos D \sin AR. \]

Assume as before, \( \tan \phi = \cot D \sin AR \), then, by substituting,

\[ \sin l = \sin D (\cos \omega - \sin \omega \tan \phi), \]

that is,

\[ \sin l = \sin D \left( \frac{\cos \omega \cos \phi - \sin \omega \sin \phi}{\cos \phi} \right), \]

whence

\[ \sin l = \frac{\sin D \cos (\omega + \phi)}{\cos \phi}...........(2). \]

Problem II.—Given the longitude and latitude of a star or planet, together with the obliquity of the ecliptic, to determine its right ascension and declination.

The spherical triangle EPS gives the relation

\[ \cot ES \sin EP = \cot EPS \sin PES + \cos EP \cos EPS, \]

that is,

\[ \tan l \sin \omega = \tan AR \cos L + \cos \omega \sin L, \]

whence

\[ \tan AR = \frac{\cos \omega \sin L - \tan l \sin \omega}{\cos L}. \]

Assume \( \tan \psi = \sin L \cot l \); then

\[ \tan l = \frac{\sin L}{\sin L \cos \psi}, \]

and by substituting this value of \( \tan l \) in the above equation, it becomes

\[ \tan AR = \tan L \left( \cos \omega - \sin \omega \frac{\cos \psi}{\sin \psi} \right), \]

that is,

\[ \tan AR = \tan L \left( \cos \omega \sin \psi - \sin \omega \cos \psi \right) \frac{\sin \psi}{\sin \psi}, \]

whence

\[ \tan AR = \frac{\tan L \sin (\psi - \omega)}{\sin \psi}...........(1). \]

To find the declination, we have

\[ \cos PS = \cos EP \cos ES + \sin EP \sin ES \cos PES; \]

that is,

\[ \sin D = \cos \omega \sin l + \sin \omega \cos l \sin L, \]

\[ = (\cos \omega + \sin \omega \cot l \sin L) \sin l, \]

and by substituting \( \tan \psi \) for \( \cot l \),

\[ \sin D = (\cos \omega + \sin \omega \tan \psi) \sin l, \]

\[ = \frac{(\cos \omega \cos \psi + \sin \omega \sin \psi) \sin l}{\cos \psi}, \]

therefore

\[ \sin D = \frac{\cos (\psi - \omega)}{\cos \psi} \sin l...........(2). \]

Corollary.—In the case of the sun, the latitude \( l \) becomes zero, and the formulae are considerably simplified. Thus, let \( T \) be the sun's place in the ecliptic; the relations subsisting between the longitude, right ascension, declination, and obliquity, are given by means of the right-angled spherical triangle \( \alpha TR \). The following are the formulae:

\[ \begin{align*} \sin L &= \frac{\sin D}{\sin \omega}, \\ \tan L &= \frac{\tan AR}{\cos \omega}, \\ \sin AR &= \cot \omega \tan D, \\ \tan AR &= \cos \omega \tan L, \\ \sin D &= \sin \omega \sin L, \\ \tan D &= \tan \omega \sin AR. \end{align*} \]

Problem III.—Given the latitude of the observatory, the polar distance of a star, and its hour-angle at the pole, to find its zenith distance, azimuth, and angle of variation.

Let \( P \) (fig. 121) be the pole of the equator, \( A \) the zenith of the place, and \( V \) the position of the star: we have then given the two sides, \( AP \) and \( PV \), of an oblique angled spherical triangle, together with the included angle \( APV \), to determine the third side and the remaining angles.

Make \( \lambda = AP \) the complement of the latitude,

\( \Delta = PV \) the polar distance of the star, which is here supposed to be north,

\( P = APV \) the hour-angle at the pole,

\( Z = AV \) the zenith distance,

\( A = VAP \) the azimuth,

\( V = AVP \) the angle of variation.

I. To find \( Z \) we have

\[ \cos Z = \cos \lambda \cos \Delta + \sin \lambda \sin \Delta \cos P. \]

Assume \( \tan \psi = \tan \Delta \cos P \); then by substituting,

\[ \cos Z = \cos \lambda (\cos \lambda + \sin \lambda \tan \phi); \]

whence, by reducing, 2. To find \( \Lambda \), we have from spherical trigonometry the formula

\[ \cot \Lambda = \frac{\cot \Delta \sin \lambda}{\sin P} - \cot P \cos \lambda. \]

Assume \( \psi = \cot \Delta \cos \phi \) whence \( \cot \Delta = \cot \psi \cos P \),

and, by substituting,

\[ \cot \Lambda = \cot P (\cot \psi \sin \lambda - \cot \psi \cos \lambda), \]

whence

\[ \cot \Lambda = \cot P \sin \lambda \frac{\lambda - \psi}{\sin \psi}. \]

3. To find \( V \) we have

\[ \cot V = \cot \lambda \sin \Delta - \cos \Delta \cot P. \]

Assume \( \chi = \cot \lambda \) or \( \cot \lambda = \cot \chi \cos P \), then,

by substituting and reducing as above, we have

\[ \cot V = \cot P \sin (\Delta - \chi). \]

**Problem IV.**—Given the declination and zenith distance of a star, and the latitude of the observatory, to determine the hour-angle.

This is the case in the solution of spherical triangles, in which the three sides are given to find one of the angles. The formula, therefore, from which \( P \) is computed (see Spherical Trigonometry) is

\[ \tan^2 \frac{1}{2} P = \frac{\sin \frac{1}{2}(Z + \lambda - \Delta)}{\sin \frac{1}{2}(Z + \lambda + \Delta)} \cdot \frac{\sin \frac{1}{2}(Z - \lambda + \Delta)}{\sin \frac{1}{2}(Z - \lambda - \Delta)}. \]

**Problem V.**—Given the latitude of a place, and the sun's declination, to find, 1st, the time of sunrise; 2d, the sun's amplitude at rising; 3d, the time when the sun is due east; 4th, the sun's altitude when in that position.

In fig. 122 let \( AZP \) be the meridian, in which \( Z \) is the zenith, and \( P \) the pole; let \( ZO \) be the prime vertical, and \( AOB \) the horizon, \( A \) being the south, \( O \) the east, and \( B \) the north points; let \( SS' \) be the part of the parallel described by the sun between his rising and passing the prime vertical, and \( PS \) \( PS' \) hour circles.

Let \( L \) = the latitude of the place \( = PB \), \( D \) = the sun's declination \( = 90^\circ - PS \), \( r \) = the time from midnight to sunrise \( = \text{hour angle SPB} \), \( x \) = the sun's amplitude at rising \( = OS \), \( y \) = the time from midnight to the sun's coming on the prime vertical \( = \text{hour-angle SPB} \), \( z \) = the sun's altitude when in the prime vertical \( = OS' \).

In the spherical triangle \( PBS \), right angled at \( B \),

\[ \cos \text{SPB} = \tan \text{PB} \cdot \cos \text{BS} = \cos \text{PS} \cdot \cos \text{PB}, \]

and in the spherical triangle \( PZS' \), right angled at \( Z \),

\[ \cos \text{ZPS} = \tan \text{PZ} \cdot \cos \text{ZS'} = \cos \text{PS} \cdot \cos \text{PZ}. \]

Now \( PB = L, PZ = 90^\circ - L, PS = PS' = 90^\circ - D, BS = 90^\circ - x, ZS' = 90^\circ - z \); hence we have these four formulæ:

\[ \begin{align*} \cos e &= \tan L, \tan D, \\ \sin x &= \cos L, \\ \cos y &= \cot L, \tan D, \\ \sin z &= \sin L. \end{align*} \]

The hour-angles \( v \) and \( y \) must be converted into time by allowing \( 15^\circ \) to one hour.

**Problem VI.**—To find how much the rising of the sun or a star is advanced by refraction.

Let \( ZPP' \) be the meridian (fig. 123), \( Z \) the zenith, \( P \) the pole, and \( OB \) the horizon. Let \( SS' \) the parallel described by the sun, meet the horizon in \( S \). Were it not for the effect of refraction, the sun or star would appear to rise at \( S \), and at rising the hour-angle from midnight would be \( SPB \); but it is elevated by refraction (which takes place in a vertical circle), and appears to rise at \( D \), while in fact it is below the horizon at \( S' \) somewhere in the vertical circle \( ZD \) produced downward, and the time of rising is accelerated by the small angle \( SPS' \).

Let \( L \) denote the latitude of the place \( = 90^\circ - PZ \); \( D \) the declination of the star \( = 90^\circ - PS \); \( H \) the hour-angle \( BPS \), reckoned from midnight; \( r \) the arc \( DS \) of the vertical \( ZS \), the effect of refraction; and \( x \) the angle \( SPS \), the acceleration of time of rising.

In the spherical triangle \( ZPS' \),

\[ \cos ZS' = \cos ZP \cos PS' + \sin ZP \sin PS' \cos ZPS', \]

Now, \( \cos ZS' = \cos (90^\circ + r) = -\sin r \),

and \( \cos ZPS' = -\cos BPS' = -\cos (H - x) \);

therefore

\[ -\sin r = \sin L \sin D - \cos L \cos D \cos (H - x). \]

In like manner, in the triangle \( ZPS \), in which \( \cos ZS = \cos 90^\circ = 0 \),

\[ 0 = \sin L \sin D - \cos L \cos D \cos H; \]

hence, by subtraction,

\[ \sin r = \cos L \cos D [\cos (H - x) - \cos H], \]

and \( \cos (H - x) - \cos H = \frac{\sin r}{\cos L \cos D} \);

but \( \cos (H - x) - \cos H = 2 \sin \frac{1}{2} x \sin \left( H - \frac{x}{2} \right) \)

(Algebra, § 240, D);

therefore, \( 2 \sin \frac{1}{2} x \sin \left( H - \frac{x}{2} \right) = \frac{\sin r}{\cos L \cos D} \).

Now, \( x \) and \( r \) being small angles, we may consider

\[ 2 \sin \frac{1}{2} x = x, \quad \sin r = r, \quad \sin \left( H - \frac{x}{2} \right) = \sin H. \]

We have then \( x = \frac{\sin H}{\cos L \cos D} \),

and here \( H \) is determined by the formula

\[ \cos H = \tan L \tan D. \]

The value of \( x \) just found is only a near approximation; let it be denoted by \( x' \), and let \( H' = H + x' \), and we shall have more nearly

\[ x = \frac{r}{\sin H' \cos L \cos D}. \]

Since \( \cos H = \tan L \tan D = \frac{\sin L \sin D}{\cos L \cos D} \),

therefore, \( \sin^2 H = \frac{\cos^2 L \cos^2 D - \sin^2 L \sin^2 D}{\cos^2 L \cos^2 D} \).

Again, \( \cos^2 L \cos^2 D - \sin^2 L \sin^2 D \)

\( = (\cos L \cos D - \sin L \sin D)(\cos L \cos D + \sin L \sin D) \)

\( = \cos (L + D) \cos (L - D) \). Therefore, \( r^2 \mathbf{H} = \frac{\cos(L + D) \cos(L - D)}{\cos^2 L \cos^2 D} \), and \( x = \frac{\sqrt{[\cos(L + D) \cos(L - D)]}}{r} \).

The value of \( r \) is variable; but in general it is about \( 33 \) minutes of a degree, or \( 132 \) seconds of time. Therefore the acceleration by refraction is nearly, in time,

\[ \frac{132}{\sqrt{[\cos(L + D) \cos(L - D)]}} \]

**Problem VII.**—To find the length of the twilight.

Let ZPB (fig. 123) be the meridian, Z the zenith, P the pole, and SDB the horizon. Let SS' be the arc of the parallel described by the sun between the beginning of the twilight, when he is at S', and its end at sunrise, when he reaches the horizon at S.

The twilight begins when DS', the sun's depression below the horizon, is about \( 18^\circ \).

Let \( a \) denote the arc, DS the depression, L the latitude of the place \( = 90^\circ - ZP \), D the sun's declination \( = 90^\circ - PS \), H the hour angle SPB (from midnight), or the angle SPZ:

Then, \( \cos ZZS' = \cos ZP \cos PS' + \sin ZP \sin PS' \cos ZPS' \);

that is, because \( \cos ZZS' = -\sin a \), and \( \cos ZPS' = -\cos SPB, \)

\[-\sin a = -\sin L \sin D = \cos L \cos D \cos(H - x); \]

hence we obtain

\[ \cos(H - x) = \tan L \tan D + \frac{\sin a}{\cos L \cos D}. \]

Now, by Problem V, \( \cos H = \tan L \tan D \). \[........(1)\]

Let \( \phi \) be such an angle, that

\[ \cos \phi = \frac{\sin a}{\cos L \cos D}; \]

then \( \cos(H - x) = \cos H + \cos \frac{\phi}{2}(H - \phi) \). \[........(2)\]

Now H is determined by formula (1); therefore x, the angle described about the pole while the twilight lasts, is determined by formula (2).

**Problem VIII.**—Given the right ascensions and declinations of two celestial bodies, to compute their angular distance.

Let S and M (fig. 124) be the two stars (for example the sun and moon), and let Z be the zenith, and P the pole of the equator. In the triangle SPM, the side SM represents the angular distance of the two stars; PS and PM are respectively their polar distances, or the complements of their declinations, and therefore given; and the angle SPM is also given, being the difference of ZPS and ZPM, the right ascensions of the given stars. Let PS = A, PM = \( \Delta \), the angle SPM = \( \beta \), and SM, the distance sought, = d. We have then

\[ \cos d = \frac{\cos A \cos(\Delta - \phi)}{\cos \phi} \]

Assume \( \tan \phi = \tan A \cos \beta \); then, by substituting and reducing, as in Problem II., we obtain

\[ \cos d = \frac{\cos A \cos(\Delta - \phi)}{\cos \phi}. \]

**Problem IX.**—To determine the latitude of a place.

1. Of the various methods which are employed for determining the latitude, that which depends on the observation of the double transits, or upper and lower culmination of a circumpolar star, is perhaps the best; being inde-

pendent of the star's declination, and of the effects of practical aberration and nutation. The accuracy of the result depends, indeed, on the allowance made for refraction; but unless the observed star at the lower culmination passes within \( 15^\circ \) or \( 20^\circ \) of the horizon, the errors of the tables will be very inconsiderable.

Let Z be the observed zenith distance, and R the refraction of the star at its lower culmination; \( Z' \) and \( R' \) the same quantities at its upper culmination, and \( \lambda \) the correct zenith distance of the pole, or the co-latitude; then

\[ \lambda = \frac{1}{4}(Z + Z') + \frac{1}{4}(R + R'). \]

The quantities R and R' must be taken from the tables, regard being had to the state of the barometer and thermometer.

2. Another method of determining the latitude, which, by reason of the facility of observation and computation, is extremely commodious, and therefore much employed, especially by voyagers, depends on observations of the meridional zenith distances of the sun or a star. Let P (fig. 125) be the pole, Z the zenith, C the intersection of the meridian and equator, and S the sun or star; then, l being the required latitude.

\[ l = \frac{1}{4}(ZC - ZS) \pm SC = Z \pm D, \]

according as the star is situated S or SS, that is, above or below the equator.

If the star is to the north of the zenith, and above the pole, as at a, then

\[ l = ZC = Ca - Za = D - Z, \]

and if below the pole, at \( a' \), then Ca' = 180° - D, and

\[ l = ZC = Ca' - Za' = 180° - (Z + D). \]

**Problem X.**—To determine the difference of longitude between two points on the earth's surface.

The different methods which have been proposed for the solution of this problem, one of the most difficult in practical astronomy, are the following: 1st, The eclipses of Jupiter's satellites; 2d, the eclipses of the moon; 3d, the eclipses of the sun; 4th, the occultations of fixed stars by the moon; and, 5th, the comparison of the moon's transits over the meridian with those of certain fixed stars selected for the purpose. Of these five methods, the two first give results affected by many causes of uncertainty; and the third can seldom be practised, because solar eclipses occur very rarely for any given point on the earth's surface. Of the remaining two, the last (which is recommended by Mr Baily) seems entitled to the preference, on account of its being independent of great accuracy in the rate of the clock or the position of the transit instrument.

The difference of the longitudes of the two stations is supposed to be nearly known from the chronometer, or by other means; and the object is to correct or determine the error of the first approximation.

Let A and B be the two stations, of which A is supposed to be the most westerly, and put

- \( r \): the difference (in sidereal time) of the transit of the moon's limb and the star previously agreed on at A; - \( r' \): the same difference at B; - \( \tau \): the apparent Greenwich time of the culmination of the moon at A; - \( \tau' \): the apparent Greenwich time of the culmination of the moon at B; - \( d \): the true declination computed for the time \( \tau \); - \( r \): the true radius \( \tau \); - \( d' \): the same quantities computed for the time \( \tau' \); - \( s \): the length of the true solar day in seconds; - \( m \): \( \tau \)'s motion in AR in half that interval, expressed in seconds of space;

\[ \begin{align*} d &= \text{true declination computed for the time } \tau \\ r &= \text{true radius } \tau \\ d' &= \text{the same quantities computed for the time } \tau' \\ s &= \text{length of the true solar day in seconds} \\ m &= \tau \text{'s motion in AR in half that interval, expressed in seconds of space; \end{align*} \] \( \chi = \) the assumed difference of longitude, in time; \( (\chi + e) = \) the correct difference of longitude.

The angle comprised between the two horary circles which pass respectively through the centre and limb of the moon, at the station A, is \( \frac{r}{\cos d} \), which being reduced to time, becomes \( \frac{r}{15 \cos d} \), and expresses the sidereal time in which the moon's semi-diameter passes the meridian.

Hence \( r = \frac{r}{15 \cos d} \) is the observed difference of the \( AR \) of the star and moon's centre at A, the upper sign being taken when the first or western limb of the moon is observed, and the under when it is the eastern limb, the star being supposed to precede the moon. In the same manner \( \frac{r'}{15 \cos d} \) is the sidereal time in which the semi-diameter of the moon passes the meridian at B, and \( r' = \frac{r'}{15 \cos d} \) the observed difference of the \( AR \) of the star and moon's centre. We have, therefore, by subtracting these two expressions,

\[ (r - r') = \frac{r}{15 \cos d} - \frac{r'}{15 \cos d} \]

for the observed difference of the \( AR \) of the moon's centre, during the time elapsed between the two observations. Put this difference equal to \( \Delta \); then \( \chi + \Delta \) is the difference between \( t \) and \( t' \) in sidereal time, which becomes

\[ (\chi + \Delta) = \frac{8640}{s} \]

when expressed in mean solar time.

Hence we have

\[ t = t' + (\chi + \Delta) \frac{8640}{s}; \]

consequently, when the apparent Greenwich time at one of the observatories is known, it is also known at the other observatory. Now, let \( a \) and \( a' \) be the moon's \( AR \) in space, computed for the times \( t \) and \( t' \) respectively (taken from the ephemeris); then the formula for the correction of the assumed difference of longitudes will evidently be

\[ \epsilon = \left( \frac{15 \Delta}{a - a'} \right) \frac{s}{2m}; \]

and this added to \( \chi \) gives \( \chi + \epsilon \), the corrected difference of longitudes.

**Problem XL.—To find the meridional zenith distance of the sun (or a star), from observations made near the meridian.**

Let P (fig. 124) be the pole, Z the zenith, and S the sun. Make

- \( l = 90^\circ - PZ = \) the latitude of the place; - \( D = 90^\circ - PS = \) the declination; - \( P = ZPS \) the hour angle; - \( Z = l - D = \) the true zenith distance; - \( Z' = ZS \) the observed zenith distance; - \( x = Z' - Z \) the correction.

The triangle \( PZS \) gives

\[ \cos ZS = \cos ZP \cos PS + \sin ZP \sin PS \cos ZPS, \]

or \( \cos Z = \sin l \sin D + \cos l \cos D \cos P \);

but \( \cos (l - D) = \cos l \cos D + \sin l \sin D \),

and \( \cos P = 1 - 2 \sin^2 \frac{1}{2} P \),

therefore,

\[ \cos Z' = \cos (l - D) - 2 \cos l \cos D \sin^2 \frac{1}{2} P, \]

or \( \cos Z' = \cos Z - 2 \cos l \cos D \sin^2 \frac{1}{2} P \).

Now, by the trigonometrical formula (Algebra, p. 240),

\[ \cos Z = \cos Z' = 2 \sin \frac{1}{2} (Z' - Z) \sin \frac{1}{2} (Z' + Z), \]

therefore,

\[ 2 \sin \frac{1}{2} (Z' - Z) \sin \frac{1}{2} (Z' + Z) = 2 \cos l \cos D \sin^2 \frac{1}{2} P, \]

But \( Z' - Z = x \), and \( Z' + Z = 2Z + x \),

therefore,

\[ 2 \sin \frac{1}{2} x \sin \frac{1}{2} (Z' + Z) = 2 \cos l \cos D \sin^2 \frac{1}{2} P, \]

that is,

\[ 2 \sin \frac{1}{2} x \cos \frac{1}{2} x \sin Z + 2 \sin^2 \frac{1}{2} x \cos Z. \]

or \( 2 \sin \frac{1}{2} x \cos \frac{1}{2} x + 2 \sin^2 \frac{1}{2} x \cot Z \)

\[ = \frac{2 \cos l \cos D}{\sin Z} \sin^2 \frac{1}{2} P. \]

In order to resolve this equation, make \( \cot Z = a \),

\[ \frac{\cos l \cos D}{\sin Z} \sin^2 \frac{1}{2} P = b, \]

and divide both sides by \( 2 \cos^2 \frac{1}{2} x \); it then becomes

\[ \tan \frac{1}{2} x + a \tan^2 \frac{1}{2} x = \frac{b}{\cos^2 \frac{1}{2} x} = b (1 + \tan^2 \frac{1}{2} x). \]

Whence

\[ \tan^2 \frac{1}{2} x + \frac{1}{a - b} \tan \frac{1}{2} x = \frac{b}{a - b}; \]

and, consequently,

\[ \tan \frac{1}{2} x = -\frac{1}{2(a - b)} = \frac{1}{2(a - b)} \sqrt{1 + 4b(a - b)}. \]

Developing this expression, and rejecting all the powers of \( b \) higher than the cube, we have

\[ \tan \frac{1}{2} x = b - ab^2 + (1 + 2a^2)b^3 - \ldots \]

Now the series which expresses the arc in terms of its tangent (see art. Algebra, p. 270), gives

\[ \frac{1}{2}x = \tan \frac{1}{2}x - \frac{1}{3} \tan^3 \frac{1}{2}x + \frac{1}{5} \tan^5 \frac{1}{2}x - \ldots \]

Therefore, by substituting, and rejecting the terms containing higher powers of \( b \) than the cube,

\[ \frac{1}{2}x = b - ab^2 + (1 + 2a^2)b^3 - \ldots \]

that is,

\[ x = 2b - 2ab^2 + (1 + 2a^2)b^3 - \ldots \]

and therefore, on restoring the values of \( a \) and \( b \), and dividing by \( \sin l \),

\[ x = 2 \left( \frac{\cos l \cos D}{\sin Z} \right) \cdot \frac{\sin^2 \frac{1}{2} P}{\sin l} - 2 \cot Z \left( \frac{\cos l \cos D}{\sin Z} \right)^2 \frac{\sin^2 \frac{1}{2} P}{\sin l} + \ldots \]

The last term of this series is scarcely sensible in any case; it is therefore only necessary to compute the first two.

If, instead of eliminating \( Z' \) by means of the equation \( Z' - Z = x \), we had eliminated \( Z \), the resulting expression would have been

\[ x = -2 \left( \frac{\cos l \cos D}{\sin Z} \right) \cdot \frac{\sin^2 \frac{1}{2} P}{\sin l} + 2 \cot Z \left( \frac{\cos l \cos D}{\sin Z} \right)^2 \frac{\sin^2 \frac{1}{2} P}{\sin l} - \ldots \]

**Problem XII.—Given the times of two observed equal altitudes of the sun, to find the true time of his meridional passage.**

Let P (fig. 126) be the pole, Z the zenith, ZPM the meridian, and A and C the places of the sun, before and after his meridional passage, when his zenith distances \( ZA \) and \( ZC \) are observed to be equal. In consequence of the variation of the sun's declination while he passes from A to C, the hour angles \( APM \) and \( MPC \) are unequal. Make \( BPM = APM = P \), and let \( BPC \) be denoted by \( \phi \). Now, \( APC = 2P + \phi \); half of which is \( P + \frac{1}{2} \phi \); therefore the true time of the meridional passage will be found by subtracting \( \frac{1}{2} \phi \) from the mean of the times of observed equal altitudes.

To find the value of \( \phi \), we have the equation

\[ \cos ZA = \cos ZP \cos PA + \sin ZP \sin PA \cos APM, \] Practical Astronomy.

cos. Z = sin. l sin. D + cos. l cos. D cos. P.....(1)

Now, let Δ = Bb be the small change of declination corresponding to the variation of the hour-angle from P to P + ϕ, and let D + Δ be substituted for D, and P + ϕ for P in this equation. It then becomes

\[ \cos Z = \sin l \sin (D + \Delta) + \cos l \cos (D + \Delta) \cos (P + \phi). \]

But as Δ and ϕ are very small arcs, their cosines may be made equal to the radius, and their sines equal to the arcs themselves; therefore, on expanding the above equation, it becomes

\[ \cos Z = \sin l \sin D + \cos l \cos D \cos P. \]

\[ + (\sin l \cos D - \cos l \sin D \cos P) \Delta \]

\[ - \cos l \cos D \sin P \phi + \cos l \sin D \sin P \Delta \phi; \]

whence, in consequence of equation (1), and rejecting the term multiplied by Δ ϕ,

\[ (\sin l \cos D - \cos l \sin D \cos P) \Delta = \cos l \cos D \sin P \phi; \]

therefore,

\[ \phi = \frac{\tan l}{\sin P} - \tan D \cot P. \]

Let t be the sidereal time of the first observation, t that of the second, and \( t' = t + 24 \). Also let the diurnal variation of D be denoted by v, expressed like the arcs ϕ and Δ in seconds; we have then \( 24^h : 24 : : v : \Delta \); whence \( \Delta = \frac{1}{12} \phi v \), and consequently

\[ \phi = \frac{1}{12} \phi v \left( \frac{\tan l}{\sin P} - \tan D \cot P \right). \]

Now, to convert ϕ into seconds of time, it is only necessary to divide the number to which it is equal by 15; therefore, expressed in time, ϕ becomes

\[ \phi = \frac{1}{12} \phi v \left( \frac{\tan l}{\sin P} - \tan D \cot P \right). \]

For the angle P we may substitute \( \theta \) (half the time elapsed between the two observations) converted into degrees, and suppose D to be the value of the declination at the instant of the meridional passage; the value of ϕ being so small as not to be affected by these substitutions. Therefore, ultimately,

\[ \frac{1}{2} \phi = \frac{1}{360} \phi \left( \frac{\tan l}{\sin \theta} - \tan D \cot \theta \right), \]

which, subtracted from \( \theta \), gives the true time of the meridional passage.

Problem XIII.—To compute the angle of the vertical, or the difference between the apparent and geocentric latitude arising from the spheroidal figure of the earth (fig. 127).

Let AMP be a quadrant of the elliptic meridian, C being the centre, and P the pole. The straight line VME, perpendicular to the ecliptic in M, determines the apparent zenith V of the place M, while ZMC drawn through C determines the true zenith Z. Hence the angle ADM is the apparent latitude of M, and ACM its geocentric latitude; and the angle ZMV between the true and apparent zenith is the angle of the vertical.

Let x and y be the rectangular co-ordinates of the point M, the origin being at C, and make AC = m, CP = n, ADM = l, and ZMV = v.

The equation of the ellipse gives

\[ y^2 = n^2 - \frac{n^2}{m^2} x^2, \]

whence

\[ ydy = - \frac{n^2}{m^2} xdx; \]

therefore,

\[ \frac{y}{x} = - \frac{n^2}{m^2} \frac{dx}{dy}. \]

But \( \frac{y}{x} = \tan ACM \), and \( \frac{dx}{dy} = \tan ADM = \tan l \);

therefore tan. ACM = \( \frac{n^2}{m^2} \tan l \). Now ZMV = DMC = ADM — ACM, therefore tan. ZMV = tan. (ADM — ACM) = \( \frac{\tan ADM - \tan ACM}{1 + \tan ADM \tan ACM} \);

that is,

\[ \tan v = \frac{(1 - \frac{n^2}{m^2}) \tan l}{1 + \frac{n^2}{m^2} \tan^2 l} = \frac{(m^2 - n^2) \tan l}{m^2 + n^2 \tan^2 l} \]

\[ = \frac{(m^2 - n^2) \sin l \cos l}{m^2 + n^2 \sin^2 l} = \frac{(m^2 - n^2) \sin l \cos l}{m^2 - (m^2 - n^2) \sin^2 l}. \]

Now, let \( m - n = 1 \); or \( m = n + 1 \); then, by substituting,

\[ \tan v = \frac{(2m - 1) \sin l \cos l}{m^2 - (2m - 1) \sin^2 l}, \]

or, neglecting the terms multiplied by \( \frac{1}{m^2} \), and recollecting that \( \sin l \cos l = \frac{1}{2} \sin 2l \),

\[ \tan v = \frac{\sin 2l}{2m} \left( 2 - \frac{1}{m} \right) \frac{1}{1 - \frac{2}{m} \sin^2 l}, \]

whence

\[ \tan v = \frac{\sin 2l}{2m} \left( 2 - \frac{1}{m} \right) \left( 1 + \frac{2}{m} \sin^2 l \right); \]

that is,

\[ \tan v = \frac{\sin 2l}{m} \left( 1 - \frac{1}{2m} + \frac{2}{m} \sin^2 l \right) \]

\[ = \frac{\sin 2l}{m} \left( 1 - \frac{1}{2m} + \frac{1}{m} \cos^2 l + \frac{1}{m} \sin^2 l \right), \]

but \( \cos^2 l - \sin^2 l = \cos 2l \); therefore,

\[ \tan v = \frac{\sin 2l}{m} \left( 1 + \frac{1}{2m} - 2 \cos 2l \right); \]

and \( 2 \sin 2l \cos 2l = 4 \sin 4l \); therefore,

\[ \tan v = \frac{\sin 2l}{m} + \frac{\sin 2l - \sin 4l}{2m^2}. \]

The last term of this expression cannot in any case amount to \( 2l \), so that the angle of the vertical is very nearly proportional to the sine of twice the latitude.

Problem XIV.—To compute the parallax of the moon, or a planet, in altitude.

Let P be the horizontal parallax,

p be the parallax of altitude,

Z be the apparent zenith distance.

It was shown in Chap. I. Sect. 2 of Theoretical Astronomy that sin. p = sin. P sin. Z; now let Z' be the true zenith distance, then Z = Z' + p, and consequently

\[ \sin p = \sin P \sin (Z' + p) \]

\[ = \sin P (\sin Z' \cos p + \cos Z' \sin p), \]

therefore

\[ \tan p = \sin P (\sin Z' + \cos Z' \tan p) \]

\[ \tan p (1 - \sin P \cos Z') = \sin P \sin Z', \]

whence

\[ \tan p = \frac{\sin P \sin Z'}{1 - \sin P \cos Z'}. \]

But by a well-known series,

\[ p = \tan p - \frac{1}{3} \tan^3 p + \frac{1}{5} \tan^5 p - \ldots, \text{&c.} \]

Substituting therefore the above expression for tan. p, and reducing the powers to series, we find

\[ p = \frac{\sin P \sin Z'}{\sin 1'} + \frac{\sin^2 P \sin 2Z'}{\sin 2'}, \frac{\sin^3 P \sin 3Z'}{\sin 3'}, \text{&c.} \] The first two terms of this series are in every case sufficient for the computation of \( p \).

**Problem XV.—To compute the parallax in right ascension.**

In fig. 128 let \( P \) be the pole, \( Z \) the zenith, \( A \) the true place of the moon or a planet, depressed on the vertical circle through the effects of parallax to \( B \). Having joined \( PA \) and \( PB \) by arcs of great circles, the corresponding variation in right ascension will be represented by the angle \( APB \). Make

- \( P = \) the horizontal parallax; - \( p = AB \), the parallax of altitude; - \( H = APB \), the parallax in right ascension; - \( l = 90^\circ - ZP \), latitude of place; - \( \Delta = PA \), the polar distance; - \( N = ZA \), the true zenith distance; - \( N + p = ZB \), the apparent zenith distance; - \( H = ZPA \), the hour angle.

In the parallactic triangle \( APB \) we have

\[ \sin PA : \sin AB :: \sin AB : \sin APB ; \]

that is,

\[ \sin \Delta : \sin p :: \sin AB : \sin H , \]

but \( \sin p = \sin P \sin (N + p) \). See Theoretical Astronomy, Chap. I. Sect. 2.

Therefore,

\[ \sin \Pi = \frac{\sin P \sin (N + p) \sin ABP}{\sin \Delta} \]

Now, in the triangle \( ZBP \), we have

\[ \sin ZB : \sin ZP :: \sin ZPB : \sin ZBP (ABP); \]

that is,

\[ \sin (N + p) : \cos l :: \sin (H + \Pi) : \sin ABP . \]

Therefore,

\[ \sin ABP = \frac{\cos l \sin (H + \Pi)}{\sin (N + p)}, \]

whence

\[ \sin \Pi = \frac{\sin P \cos l}{\sin \Delta} \sin (H + \Pi). \]

Make \( \frac{\sin P \cos l}{\sin \Delta} = A \); then

\[ \sin \Pi = A \sin (H + \Pi) = A (\sin H \cos \Pi + \cos H \sin \Pi); \]

whence

\[ \tan \Pi = A (\sin H + \cos H \tan \Pi), \]

and

\[ \tan \Pi (1 - A \cos H) = A \sin H; \]

that is,

\[ \tan \Pi = \frac{A \sin H}{1 - A \cos H}; \]

whence we have the following series,

\[ \Pi = A \frac{\sin H}{\sin 1^\circ} + A^2 \frac{\sin 2H}{\sin 2^\circ} + \text{etc.} \]

**Problem XVI.—To compute the parallax in declination.**

Make \( \sigma = PB - PA \) (fig. 129) = the parallax in declination.

From the triangle \( ZPA \) we get

\[ \cos PZA = \frac{\cos AP - \cos PZ \cos AZ}{\sin PZ \sin AZ}, \]

and from \( ZPB \)

\[ \cos PZB = \frac{\cos BP - \cos PZ \cos BZ}{\sin PZ \sin BZ}; \]

therefore

\[ \frac{\cos AP - \cos PZ \cos AZ}{\sin AZ} = \frac{\cos BP - \cos PZ \cos BZ}{\sin BZ}; \]

that is (retaining the notation of last problem),

\[ \frac{\cos \Delta - \sin l \cos N}{\sin N} = \frac{\cos (\Delta + \sigma) - \sin l \cos (N + p)}{\sin (N + p)}; \]

whence

\[ \cos (\Delta + \sigma) = \frac{\cos \Delta \sin (N + p)}{\sin N} \]

\[ = \frac{\sin l [\cos N \sin (N + p) - \sin N \cos (N + p)]}{\sin N} \]

\[ = \frac{\cos \Delta \sin (N + p)}{\sin N} - \frac{\sin l \sin p}{\sin N} \]

\[ = \frac{\cos \Delta \sin (N + p)}{\sin N} - \frac{\sin l \sin P \sin (N + p)}{\sin N}; \]

therefore

\[ \cos (\Delta + \sigma) = \frac{\sin (N + p)}{\sin N} [\cos \Delta - \sin l \sin P] \ldots (1) \]

Now, from the property of spherical triangles,

\[ \sin ZA : \sin PA :: \sin ZPA : \sin AZP, \]

therefore

\[ \sin AZP = \frac{\sin PA \sin ZPA}{\sin ZA}. \]

For the same reason

\[ \sin BZP = \frac{\sin PB \sin ZPB}{\sin ZB}; \]

therefore

\[ \frac{\sin PA \sin ZPA}{\sin ZA} = \frac{\sin PB \sin ZPB}{\sin ZB}, \]

that is,

\[ \frac{\sin \Delta \sin H}{\sin N} = \frac{\sin (\Delta + \sigma) \sin (H + \Pi)}{\sin (N + p)}, \]

whence

\[ \frac{\sin (N + p)}{\sin N} = \frac{\sin (\Delta + \sigma) \sin (H + \Pi)}{\sin \Delta \sin H}. \]

By substituting this in equation (1) there results

\[ \cos (\Delta + \sigma) = \frac{\sin (\Delta + \sigma) \sin (H + \Pi)}{\sin \Delta \sin H} \]

\[ [\cos \Delta - \sin l \sin P], \]

whence

\[ \cot (\Delta + \sigma) = \frac{\sin (H + \Pi)}{\sin H} \left( \cot \Delta - \frac{\sin l \sin P}{\sin \Delta} \right). \]

Make \( \frac{\sin l \sin P}{\sin \Delta} = \cot \alpha \), then

\[ \cot (\Delta + \sigma) = \frac{\sin (H + \Pi)}{\sin H} (\cot \Delta - \cot \alpha), \]

that is,

\[ \cot (\Delta + \sigma) = \frac{\sin (H + \Pi) \sin (\alpha - \Delta)}{\sin H \sin \alpha \sin \Delta}, \]

from which, as \( \Delta \) is known, \( \cot (\Delta + \sigma) \) may be computed, and thence \( \sigma \). A more convenient formula, however, may be obtained by proceeding as follows. (See Delambre, Abrégé d'Astronomie, p. 154).

From the equation

\[ \cot (\Delta + \sigma) = \frac{\sin (H + \Pi)}{\sin H} \left( \cot \Delta - \frac{\sin l \sin P}{\sin \Delta} \right) \]

we have

\[ \cot \Delta = \frac{\sin H \cot (\Delta + \sigma)}{\sin (H + \Pi)} + \frac{\sin l \sin P}{\sin \Delta}; \]

whence

\[ \cot \Delta - \cot (\Delta + \sigma) = \frac{\sin H \cot (\Delta + \sigma)}{\sin (H + \Pi)} - \cot (\Delta + \sigma) \]

\[ + \frac{\sin l \sin P}{\sin \Delta}; \text{ but } \cot \Delta - \cot (\Delta + \sigma) = Astronomy.

Practical sin. (Δ + π) cos. Δ - cos. (Δ + π) sin. Δ = sin. π sin. (Δ + π) sin. Δ = sin. (Δ + π) sin. Δ;

therefore

\[ \frac{\sin. \sigma}{\sin. (\Delta + \pi)} = \frac{\sin. l \sin. P}{\sin. \Delta} \]

\[ - \cot. (\Delta + \pi) [\sin. (H + \Pi) - \sin. H] \]

\[ \frac{\sin. (H + \Pi)}{\sin. (\Delta + \pi)} \]

Now, by the trigonometrical formulæ, the difference of the sines of two arcs is equal to twice the sine of half their difference multiplied by the cosine of half their sum, therefore

\[ \sin. (H + \Pi) - \sin. H = 2 \sin. \frac{1}{2} \Pi \cos. (H + \frac{1}{2} \Pi), \]

whence

\[ \frac{\sin. \sigma}{\sin. (\Delta + \pi)} = \frac{\sin. l \sin. P}{\sin. \Delta} \]

\[ - 2 \sin. \frac{1}{2} \Pi \cos. (H + \frac{1}{2} \Pi) \cot. (\Delta + \pi); \]

\[ \frac{\sin. (H + \Pi)}{\sin. (\Delta + \pi)}; \]

and sin \( \sigma = \sin. l \sin. P \cos. (\Delta + \pi) \)

\[ \frac{2 \sin. \frac{1}{2} \Pi \sin. \Delta \cos. (H + \frac{1}{2} \Pi) \cos. (\Delta + \pi)}{\cos. \frac{1}{2} \Pi \sin. (H + \Pi)} \]

or, since \( 2 \sin. \frac{1}{2} \Pi = \frac{\sin. \Pi}{\cos. \frac{1}{2} \Pi}, \)

\[ \sin. \sigma = \sin. l \sin. P \sin. (\Delta + \pi) \]

\[ \frac{\sin. \Pi \sin. \Delta \cos. (H + \frac{1}{2} \Pi) \cos. (\Delta + \pi)}{\cos. \frac{1}{2} \Pi \sin. (H + \Pi)} \]

But it was shown in the last problem that

\[ \frac{\sin. \Pi}{\sin. (\Delta + \pi)} = \frac{\sin. P \cos. l}{\sin. \Delta \sin. (H + \Pi)}, \]

whence

\[ \frac{\sin. \Pi \sin. \Delta}{\sin. (H + \Pi)} = \sin. P \cos. l; \]

consequently,

\[ \sin. \sigma = \sin. l \sin. P \sin. (\Delta + \pi) \]

\[ \frac{\sin. P \cos. l \cos. (H + \frac{1}{2} \Pi) \cos. (\Delta + \pi)}{\cos. \frac{1}{2} \Pi \sin. (H + \Pi)} \]

Make tan. \( x = \frac{\cos. (H + \frac{1}{2} \Pi) \cot. l}{\cos. \frac{1}{2} \Pi}; \) then, by substituting,

\[ \sin. \sigma = \sin. l \sin. P [\sin. (\Delta + \pi) - \tan. x \cos. (\Delta + \pi)], \]

or sin. \( \sigma = \frac{\sin. l \sin. P}{\cos. x} [\sin. (\Delta + \pi) \cos. x - \cos. (\Delta + \pi) \sin. x], \)

\[ \sin. \sigma = \frac{\sin. l \sin. P}{\cos. x} \sin. (\Delta + \pi - x), \]

\[ \sin. \sigma = \frac{\sin. l \sin. P}{\cos. x} [\sin. (\Delta - x) \cos. \pi + \cos. (\Delta - x) \sin. \pi]; \]

whence,

\[ \tan. \sigma = \frac{\sin. l \sin. P}{\cos. x} [\sin. (\Delta - x) + \cos. (\Delta - x) \tan. \pi]; \]

therefore (making \( \frac{\sin. l \sin. P}{\cos. x} = B \),

\[ \tan. \sigma = \frac{B \sin. (\Delta - x)}{1 - B \cos. (\Delta - x)} \]

consequently

\[ \sigma = B \frac{\sin. (\Delta - x)}{\sin. 1^\circ} + B^2 \frac{\sin. 2 (\Delta - x)}{\sin. 2^\circ} \]

\[ + B^3 \frac{\sin. 3 (\Delta - x)}{\sin. 3^\circ} + \ldots \]

Problem XVII.—To compute the altitude and longitude of the nonagesimal.

The nonagesimal is the point of the ecliptic where that circle intersects the vertical plane passing through its pole. It is consequently the highest point of the ecliptic above the horizon, or 90° from the horizon measured on the ecliptic.

Let HH' (fig. 129) be the horizon, EO the ecliptic, EQ the equator, Z the zenith, P the pole of the equator, and P' the pole of the ecliptic. The great circle passing through P' and P intersects EO and EQ at right angles in C and D, whence EC = ED = 90° = MQ, and therefore EM = DQ. In like manner PD = 90° = PC, consequently PP' = CD the obliquity of the ecliptic.

The great circle PZN which passes through the pole of the ecliptic and the zenith is a circle of latitude and also a vertical circle; hence the angles at N and I are right angles, and O is the pole of PZN; consequently ON = ON = 90°. Now since ON = 90°, the point N is the nonagesimal, and its altitude IN = 90° - ZN = ZP' = the complement of the altitude of the pole of the ecliptic, or the co-latitude of the zenith. In like manner its longitude EN = 90° - NC = 90° - ZPP'. Now in order to compute ZP' and the angle ZPP', we have given, in the triangle ZPP', the side ZP' = the co-latitude, PP' = the obliquity, and also the angle ZPP', for ZPP' = 180° - ZPD = 180° - MD = 180° - (ED - EM) = 180° - 90° + EM = 90° + EM, and EM (which is the right ascension of the zenith, or as it is technically called, the right ascension of the mid-heaven) is given, being equal to the sidereal time of observation converted into degrees.

Let R = EM the right ascension of the zenith, \( \lambda = PZ \) the co-latitude of the place (reduced by problem XIII.), \( \omega = PP' \) the obliquity of the ecliptic, K = PZ the co-latitude of the zenith or altitude of the nonagesimal, N = 90° - ZPP' the longitude of the nonagesimal.

By the trigonometrical formulæ,

\[ \cos. K = \cos. \lambda \cos. \omega + \sin. \lambda \sin. \omega \cos. (90° + R) \]

\[ = \cos. \lambda \cos. \omega + \sin. \lambda \sin. \omega \sin. R. \]

Assume tan. \( \phi = \tan. \lambda \sin. R \), then, by substituting and reducing, we obtain

\[ \cos. K = \frac{\cos. \lambda}{\cos. \phi} \cos. (\omega + \phi). \]

To find the longitude \( L \), we have

\[ \tan. N = \frac{\cot. \lambda \sin. \omega}{\sin. (90° + R)} - \cot. (90° + R) \cos. \omega \]

\[ = \cot. \lambda \sin. \omega \frac{\cos. R}{\cos. \phi} + \tan. R \cos. \omega. \]

But tan. \( \lambda = \frac{\tan. \phi}{\sin. R} \), whence cot. \( \lambda = \frac{\sin. R}{\tan. \phi} \),

whence, by substituting,

\[ \tan. N = \tan. R \left( \frac{\sin. \omega}{\tan. \phi} + \cos. \omega \right), \]

and on reducing,

\[ \tan. N = \frac{\tan. R}{\sin. \phi} \sin. (\omega + \phi). \]

Problem XVIII.—To compute the parallax in longitude and latitude.

Let A (fig. 129) be the place of the star situated on the vertical ZAR; Ea is its longitude, and PA the complement of its latitude, or its distance from the pole of the ecliptic. Suppose that, through the effects of parallax, it is depressed from A to B; its longitude then becomes Eb, and its distance from the pole of the ecliptic PB, so that ab is the variation in longitude, and PB - PA the variation in latitude, which it is required to compute. Now it is evident that these quantities will be given in terms of the different parts of the triangle APB, exactly in the same manner as the parallax in right ascension and declination has been found from the triangle APB. The angle ZPA, which was before denoted by Astronomy H, now becomes \( \angle ZPA = Na - Ea - EN \) — longitude of the star — longitude of the nonagesimal, and ZPB becomes \( \angle ZPB = Nb = Na + ab \). Retaining, therefore, the notation employed in the three last problems, and making \( L = Ea \) the longitude of the star, \( \Delta' = PA \) its distance from the pole of the ecliptic, \( k = ZP = 90^\circ - K \), \( \Pi = ab \) its parallax in longitude, \( \sigma' = PB - PA \) its parallax in latitude, we shall have \( \Pi \) and \( \sigma' \) from the same formula as \( \Pi \) and \( \sigma \) in Problems XV. and XVI., by changing \( I \) into \( k \), \( H \) into \( L - N \), and \( \Delta \) into \( \Delta' \). Hence (Problem XV.)

\[ \sin \Pi = \frac{\sin P \cos k}{\sin \Delta'}, \quad \sin (L - N + \Pi), \]

and by putting \( C = \frac{\sin P \cos k}{\sin \Delta'} \), we deduce

\[ \tan \Pi = \frac{C \sin (L - N)}{1 - C \cos (L - N)} \]

\[ \Pi = \sin C \frac{\sin (L - N)}{\sin \Pi} + C \frac{\sin 2(L - N)}{\sin 2\sigma'} + \ldots \]

In like manner the formula for the parallax in declination given in Problem XVI., viz.

\[ \cot (\Delta + \sigma) = \frac{\sin (H + \Pi)}{\sin \Pi} \left[ \cot \Delta - \frac{\sin I \sin P}{\sin \Delta} \right] \]

becomes

\[ \cot (\Delta' + \sigma') = \frac{\sin (L - N + \Pi)}{\sin (L - N)} \left[ \cot \Delta' - \frac{\sin k \sin P}{\sin \Delta'} \right], \]

from which, by proceeding in the same manner as in Problem XVI., and making

\[ \tan \sigma' = \frac{\cos (L - N + \Pi) \cot k}{\cos \frac{1}{2} \Pi} \]

we deduce \( \tan \sigma' \)

\[ = \frac{\sin k \sin P}{\cos \sigma'} \left[ \sin (\Delta' - \sigma') + \cos (\Delta' - \sigma') \tan \sigma' \right] \]

whence, on making \( \frac{\sin k \sin P}{\cos \sigma'} = D \),

\[ \tan \sigma' = \frac{D \sin (\Delta' - \sigma')}{1 - D \cos (\Delta' - \sigma')} \]

consequently

\[ \sigma' = D \frac{\sin (\Delta' - \sigma')}{\sin \Pi} + D \frac{\sin 2(\Delta' - \sigma')}{\sin 2\sigma'} + \ldots \]

**Problem XIX.** — Given the apparent altitudes of the moon and the sun, or a star, and the apparent distance between them, to find the true distance.

This is a problem of great importance in Practical Astronomy, because the observed distance between the moon and the sun or a star is the surest means the navigator has to determine his longitude.

To obtain the apparent distance and altitudes, it is convenient to have three observers: one, the most expert, takes the apparent distance of the limb of the moon from that of the sun, or from the star; another observes the moon's altitude at the moment of the observation of the distance; and a third takes the star's altitude. For greater accuracy, these simultaneous operations ought to be repeated several times, and a fourth assistant may, by a good watch, note the intervals of time between them. A mean of the whole will then be obtained, and the corresponding time by the watch, by which the true time at that place will be nearly known. If there be only one observer, he must take the altitudes immediately before and after the distance, and endeavour to allow for the change of altitude during the time between the observations.

The observed distance between the limbs of the moon and sun must be increased by their semidiameters to obtain the distance of their centres. If the distance from a star is taken, because its diameter is insensible, the distance is to be increased by the moon's semidiameter only.

The observed altitudes thus found are affected by parallax and refraction. The moon's horizontal parallax is given in the *Nautical Almanac*; from this the parallax at the time of observation may be found by Prob. XIV., but the navigator avails himself of aid from *The Requisite Tables*, which shorten the process of calculation. The sun's parallax is almost insensible, but it may be taken into account; that of a star is accounted nothing. The altitudes are diminished by the parallaxes, but they are increased by refraction. The sun's altitude is more increased by the latter than diminished by the former; but the reverse happens with the moon. At sea the altitudes must be also corrected for the dip of the horizon (that is, for the height of the observer above the surface of the sea), and the refraction for the height of the thermometer.

Supposing all this done, let S and M be the true places of the sun or star and moon (fig. 130), Z the zenith; because parallax and refraction take place only in vertical circles, the apparent place of the sun, \( s \), will be in the vertical \( ZS \) above \( S \), and that of the moon, \( m \), in \( ZM \) below \( M \). The apparent distance will be \( sm \).

Let \( A = 90^\circ - ZM \) be the moon's true altitude; \( a = 90^\circ - ZS \) the star's true altitude; \( H = 90^\circ - Zm \) the moon's apparent altitude; \( h = 90^\circ - Zs \) the star's apparent altitude; \( D = SM \) the true, and \( d = sm \) the apparent distance.

By spherical trigonometry, in the triangles \( SZM \), \( sZm \),

\[ \begin{align*} \cos D &= \cos A \cos a \\ \cos d &= \cos A \cos a \\ \cos H &= \cos a \\ \cos h &= \cos a \\ \end{align*} \]

From these equal values of \( \cos \), \( SZM \) we find

\[ \cos D = \cos A \cos a + \sin A \sin a \]

\[ = [\cos d + \cos (H + h) - \cos H \cos h] \cos A \cos a \]

\[ + \sin A \sin a \]

\[ = 2 \cos \frac{1}{2} (H + h + d) \cos \frac{1}{2} (H + h - d) \cos A \cos a \]

\[ - (\cos A \cos a - \sin A \sin a) \]

But the last term \( = \cos (A + a) \); subtract now both sides from 1; then, observing that by the calculus of sines (see Algebra)

\[ 1 - \cos D = 2 \sin^2 \frac{1}{2} D, \]

\[ 1 + \cos (A + a) = 2 \cos^2 \frac{1}{2} (A + a), \]

we have, after dividing by 2, and putting

\[ F = \frac{\cos A \cos a}{\cos H \cos h} \]

\[ \sin^2 \frac{1}{2} D = \cos^2 \frac{1}{2} (A + a) \]

\[ - \cos \frac{1}{2} (H + h + d) \cos \frac{1}{2} (H + h - d) \times F \]

\[ = \cos^2 \frac{1}{2} (A + a) \left[ 1 - \frac{\cos \frac{1}{2} (H + h + d) \cos \frac{1}{2} (H + h - d)}{\cos^2 \frac{1}{2} (A + a)} \times F \right]. \]

Let \( \theta \) be such an angle that

\[ \sin^2 \frac{1}{2} D = \cos^2 \frac{1}{2} (A + a) \cos^2 \theta; \]

and \( \sin \frac{1}{2} D = \cos \frac{1}{2} (A + a) \cos \theta \).

The value of \( \cos D \) in the first formula may serve to determine \( D \), but not conveniently; by the various steps of the analytic process it is transformed into another, viz. Astronomy.

The last, which gives sin. \( \frac{1}{2} D \), and consequently \( D \) readily by logarithmic calculation.

The final result is the rule found by Borda: as a practical rule it is very convenient, because no attention is necessary to the signs of the trigonometrical quantities. On this account it is well adapted to seamen.

**Problem XX.—To find the longitude of a ship at sea by an observation of the distance of the moon from the sun or a star.**

From the observed distance, the true distance as it would appear if it could be seen from the earth's centre, may be found by the last problem: Now by the Nautical Almanac this is given for every third hour, Greenwich time, therefore, by an easy calculation, the time at which the observation was made, as it would be given by a watch showing Greenwich time, may be obtained. But the time of observation as reckoned at the ship may be found from the ship's latitude, the moon's or star's zenith distances (found by the observation), and their polar distances as given by the Almanac. Therefore the difference between the time of the observation as estimated at the ship and the corresponding Greenwich time becomes known: This difference is the ship's longitude.

**Example.** At sea, June 5, 1793, about an hour and a half after noon, in \( 10^\circ 46' 50'' \) south latitude, and \( 149^\circ \) longitude, by account, by means of a set of lunar observations made at a height of about 20 feet above the surface of the sea, it was found that

Distance of nearest limbs of \( \odot \) and \( \oplus \) \( = 83^\circ 26' 46'' \) Altitude of lowest limb of \( \odot \) \( = 48^\circ 16' 10'' \) Altitude of upper limb of \( \odot \) \( = 27^\circ 53' 30'' \)

Hence the longitude of the ship is required.

**Reduction of the apparent to the true altitude.**

| Dist. of nearest limbs of \( \odot \) and \( \oplus \) | \( = 83^\circ 26' 46'' \) | | Semidiam. of \( \odot \) from Naut. Almanac | \( = 0^\circ 15' 46'' \) | | Augmentation of the latter, prop. to altitude | \( = 0^\circ 0' 7'' \) | | Apparent distance (\( d' \)) of centres | \( = 83^\circ 57' 33'' \) | | Altitude of sun's lower limb | \( = 48^\circ 16' 10'' \) | | Subtract for dip of horizon | \( = 0^\circ 4' 24'' \) | | Semidiameter | \( = 0^\circ 15' 46'' \) | | Apparent altitude of sun's centre (\( h \)) | \( = 48^\circ 27' 32'' \) | | Refr. — par. — correct for therm. | \( = 0^\circ 0' 43'' \) | | True altitude of sun's centre (\( a \)) | \( = 48^\circ 26' 49'' \) | | Altitude of moon's upper limb | \( = 27^\circ 53' 30'' \) | | Correct for dip | \( = 0^\circ 4' 24'' \) | | Semidiameter | \( = 0^\circ 15' 1'' \) | | Apparent altitude of moon's centre (\( H \)) | \( = 27^\circ 34' 5'' \) | | Par. — refr. + corr. for therm. | \( = 0^\circ 46' 43'' \) | | True alt. of moon's centre (\( A \)) | \( = 28^\circ 20' 48'' \) |

**Reduction of the apparent to the true distance by Prob. XIX.**

\[ \begin{align*} d &= 83^\circ 57' 33'' \\ h &= 48^\circ 27' 32'' \text{ ar. com. cos. } -1783835 \\ H &= 27^\circ 34' 5'' \text{ ar. com. cos. } -0523390 \\ \sum 159^\circ 59' 10'' \\ \frac{1}{2} \sum 79^\circ 59' 35'' \text{ cos. } 9-2396866 \\ d - \frac{1}{2} \sum 3^\circ 57' 58'' \text{ cos. } 9-9985857 \\ a &= 48^\circ 26' 49'' \text{ cos. } 9-8217187 \\ A &= 28^\circ 20' 48'' \text{ cos. } 9-9445275 \\ A + a &= 76^\circ 47' 37'' \text{ 39-2358960 } \end{align*} \]

Computation of time at Greenwich.

By Nautical Almanac for 1793

| Dist. \( p \) from \( \odot \) at 15 hours | \( = 83^\circ 6' 1'' \) | | Dist. \( p \) from \( \odot \) at 18 hours | \( = 84^\circ 28' 26'' \) |

Increase of dist. in 3 hours \( = 1^\circ 22' 25'' \)

\( D = 83^\circ 20' 55'' \)

Dist. \( p \) from \( \odot \) at 15 hours \( = 83^\circ 6' 1'' \)

\( = 0^\circ 14' 34'' \)

Hence it appears that in 3 hours the distance of the moon from the sun was increased by \( 1^\circ 22' 25'' = 4595'' \); and that between the time of making the observation, and the 15th hour, Greenwich time, the increase of distance was \( 0^\circ 14' 54'' = 894'' \). Now for small intervals the distances will increase nearly as the times, therefore we have

\( 4945'' : 894'' :: 3h. : 0h. \cdot 5425 = 32m. 33s. \)

Thus it appears that the observation must have been made 32 minutes 33 seconds after the 15th hour, Greenwich time.

We have yet to find the correct time of the observation as estimated at the ship. To determine this, we know the sun's altitude \( 48^\circ 26' 49'' \), and the sun's declination, which is given in the Almanac for Greenwich, noon; and from this the declination at the time of the observation is found by making allowance for the difference of longitude and the hour from noon at the ship, both known nearly. The declination thus found is \( 28^\circ 22' 48'' \) north. The ship's latitude, \( 10^\circ 16' 40'' \), is also known. Hence to find the hour angle by Problem IV, we have

\( \lambda \), the colat. \( = 79^\circ 43' 20'' \) \( \Delta \), the pol. dist. \( = 113^\circ 22' 48'' \) \( z \), the zenith dist. \( = 41^\circ 33' 11'' \)

To find \( P \), the hour angle, the computation (see formula of Prob. IV.) may stand thus:

\[ \begin{align*} \sin. \frac{1}{2} (z + \lambda - \Delta) &= 3^\circ 56' 51'' \quad 8-8978559 \\ \sin. \frac{1}{2} (z - \lambda + \Delta) &= 37^\circ 36' 19'' \quad 9-7854851 \\ \sin. \frac{1}{2} (\lambda + \Delta + z) &= 117^\circ 19' 39'' \text{ ar. comp. } 0-0513929 \\ \sin. \frac{1}{2} (\lambda + \Delta - z) &= 75^\circ 46' 28'' \text{ ar. comp. } 0-0135256 \\ \end{align*} \]

tan. \( \frac{1}{2} P = 12^\circ 27' 17'' \) \( = 9-3441297 \)

\( P = 24^\circ 54' 35'' \) \( = 1^\circ 39' 38'' \)

time at Greenwich \( = 15^\circ 32' 33'' \)

Long. from Greenwich westward \( = 13^\circ 52' 54'' \)

Therefore, longitude east of Greenwich is \( 10^\circ 7' 4'' \).

We have put down the result to seconds, as it comes out by the calculation; but such accuracy is not attainable in nautical practice. Delambre says, "even with the best instruments, the most skilful navigators find anomalies for which they cannot account. The lunar tables are in a state of continual improvement, yet we cannot be sure that there is not an error of \( 20'' \) in a distance, which will produce an error of \( 40'' \) of time. To this possible error of theory must be joined that of the observation, which may be still greater. The error of the time may therefore amount to \( 80'' \), which is equivalent to \( 20'' \) or \( 4'' \) of a degree. This should be the maximum of the error; and, in general, this degree of accuracy is sufficient. Indeed a greater accuracy than \( \frac{1}{4} \) or \( \frac{1}{4} \) of a degree may sometimes be obtained, but still it is uncertain." ### Table I

The Mean Time of New Moon in January, New Style, with the Mean Anomalies of the Sun and Moon, and the Moon's Mean Distance from the Ascending Node from 1801 to 1900 inclusive.

| Years | Mean New Moon in January | Solar Equation | Moon's Mean Anomaly | Sun's Mean Anomaly | Moon's Mean Distance from Ascending Node | |-------|--------------------------|---------------|---------------------|-------------------|----------------------------------------| | | D. H. M. S. | S. D. M. S. | S. D. M. S. | S. D. M. S. | S. D. M. S. | | 1801 | 14 7 39 9 | 0 14 16 42 | 9 10 35 50 | 1851 | 2 5 20 10 | | 1802 | 3 16 27 43 | 0 11 55 49 | 9 18 36 37 | 1852 | 2 5 17 17 | | 1803 | 2 14 0 21 | 0 10 11 33 | 9 18 36 37 | 1853 | 2 5 17 17 | | 1804B.| 11 22 48 55 | 0 8 21 21 | 9 11 10 33 | 1854 | 2 5 17 17 | | 1805 | 29 21 32 | 0 7 25 58 | 9 11 10 33 | 1855 | 2 5 17 17 | | 1806 | 19 5 10 7 | 0 6 6 46 | 9 18 48 36 | 1856 | 2 5 17 17 | | 1807 | 8 13 58 41 | 0 4 16 34 | 9 18 48 36 | 1857 | 2 5 17 17 | | 1808B.| 27 11 31 16 | 0 3 22 11 | 9 26 26 36 | 1858 | 2 5 17 17 | | 1809 | 15 20 19 52 | 0 2 2 0 | 9 15 42 29 | 1859 | 2 5 17 17 | | 1810 | 5 5 8 27 | 0 0 11 48 | 9 4 58 20 | 1860 | 2 5 17 17 | | 1811 | 24 2 41 4 | 0 11 17 25 | 9 23 20 31 | 1861 | 2 5 17 17 | | 1812B.| 13 11 29 38 | 0 9 27 13 | 9 12 56 22 | 1862 | 2 5 17 17 | | 1813 | 1 20 18 13 | 0 8 7 1 46 | 9 1 52 13 | 1863 | 2 5 17 17 | | 1814 | 20 17 50 50 | 0 7 12 38 | 9 20 14 24 | 1864 | 2 5 17 17 | | 1815 | 10 23 29 25 | 0 5 22 27 | 9 9 30 15 | 1865 | 2 5 17 17 | | 1816B.| 29 0 12 2 | 1 4 28 4 18 | 9 1 27 52 | 1866 | 2 5 17 17 | | 1817 | 17 9 0 36 | 1 3 7 52 | 9 17 8 17 | 1867 | 2 5 17 17 | | 1818 | 16 7 49 11 | 1 1 17 40 | 9 6 24 8 | 1868 | 2 5 17 17 | | 1819 | 25 15 21 46 | 0 9 23 17 | 9 24 46 19 | 1869 | 2 5 17 17 | | 1820B.| 15 0 10 23 | 1 11 3 6 | 9 14 2 10 | 1870 | 2 5 17 17 | | 1821 | 29 1 58 57 | 1 9 12 54 | 9 2 3 16 | 1871 | 2 5 17 17 | | 1822 | 22 6 31 34 | 1 8 18 31 | 9 21 40 12 | 1872 | 2 5 17 17 | | 1823 | 11 15 20 5 | 1 6 20 19 | 9 10 56 3 | 1873 | 2 5 17 17 | | 1824B.| 1 0 8 43 | 1 5 6 7 42 | 9 0 11 54 | 1874 | 2 5 17 17 | | 1825 | 21 21 41 29 | 1 4 13 44 | 9 0 15 34 | 1875 | 2 5 17 17 | | 1826 | 2 6 29 54 | 1 2 23 33 | 9 3 7 49 | 1876 | 2 5 17 17 | | 1827 | 27 4 2 3 | 1 2 19 10 | 9 2 26 12 | 1877 | 2 5 17 17 | | 1828B.| 16 12 51 5 | 0 8 58 25 | 9 15 27 67 | 1878 | 2 5 17 17 | | 1829 | 4 21 39 40 | 0 10 18 46 | 9 3 4 43 | 1879 | 2 5 17 17 | | 1830 | 23 19 12 17 | 0 9 24 23 | 9 2 3 59 | 1880 | 2 5 17 17 | | 1831 | 13 4 0 51 | 2 8 41 11 | 9 12 21 49 | 1881 | 2 5 17 17 | | 1832B.| 2 2 49 25 | 2 6 14 0 | 9 1 37 42 | 1882 | 2 5 17 17 | | 1833 | 10 20 22 3 | 2 5 19 37 | 9 19 29 52 | 1883 | 2 5 17 17 | | 1834 | 9 19 10 37 | 2 3 29 25 | 9 9 15 43 | 1884 | 2 5 17 17 | | 1835 | 26 16 43 15 | 3 3 5 2 30 | 9 5 27 37 | 1885 | 2 5 17 17 | | 1836B.| 18 1 31 49 | 3 1 14 50 | 9 5 16 53 | 1886 | 2 5 17 17 | | 1837 | 6 10 20 23 | 3 11 24 39 | 9 0 6 9 36 | 1887 | 2 5 17 17 | | 1838 | 25 7 53 1 | 3 11 0 16 | 9 0 24 31 | 1888 | 2 5 17 17 | | 1839 | 14 16 41 35 | 3 9 10 4 21 | 9 0 13 47 | 1889 | 2 5 17 17 | | 1840B.| 4 1 30 10 | 3 7 19 52 | 9 0 3 39 | 1890 | 2 5 17 17 | | 1841 | 21 2 2 47 | 4 6 25 29 | 9 0 21 25 | 1891 | 2 5 17 17 | | 1842 | 11 7 51 21 | 4 5 17 53 | 9 7 10 41 | 1892 | 2 5 17 17 | | 1843 | 30 23 50 | 4 10 55 | 9 2 29 3 41 | 1893 | 2 5 17 17 | | 1844B.| 19 14 12 33 | 4 2 20 43 | 9 7 16 19 | 1894 | 2 5 17 17 | | 1845 | 7 23 1 6 | 4 1 0 31 | 9 8 7 5 25 | 1895 | 2 5 17 17 | | 1846 | 26 20 33 45 | 5 0 6 8 36 | 9 8 25 57 | 1896 | 2 5 17 17 | | 1847 | 10 15 56 47 | 5 0 5 17 | 9 7 10 41 | 1897 | 2 5 17 17 | | 1848B.| 5 14 10 54 | 5 8 25 44 | 9 0 4 29 17 | 1898 | 2 5 17 17 | | 1849 | 23 11 43 31 | 5 8 1 22 | 9 0 22 51 | 1899 | 2 5 17 17 | | 1850 | 12 20 32 6 | 5 6 11 10 | 9 0 12 7 19 | 1900 | 2 5 17 17 | ### TABLE II.

Quantities to be added to the Epochs in Table I. for the Nineteenth Century, in order to obtain the Epochs of the corresponding Years in other Centuries. The sign — indicates the past Centuries, and + the future, in respect to the Nineteenth Century.

| Years | Mean New Moon | Moon's Mean Anomaly | Sun's Mean Anomaly | Moon's Mean Distance from Ascending Node | |-------|---------------|---------------------|--------------------|----------------------------------------| | Old Style | | | | | | — 2600 | 22 20 36 13 | 10 25 47 43 | 1 29 10 40 | 4 8 0 22 | | — 2500 | 27 4 43 18 | 7 11 18 23 | 2 2 29 56 | 8 27 26 33 | | — 2400 | 2 0 6 19 | 3 1 0 2 | 1 6 42 53 | 0 16 12 30 | | — 2300 | 6 8 13 24 | 11 16 30 42 | 1 10 2 9 | 5 5 38 42 | | — 2200 | 10 16 20 29 | 8 2 1 22 | 1 13 21 25 | 9 25 4 53 | | — 2100 | 15 0 27 33 | 4 17 32 2 | 1 16 40 41 | 2 14 31 4 | | — 2000 | 19 8 34 38 | 1 3 2 42 | 1 19 59 37 | 7 3 57 15 | | — 1900 | 23 16 41 42 | 9 18 33 22 | 1 23 19 13 | 11 23 23 26 | | — 1800 | 28 0 48 47 | 6 4 4 2 | 1 26 38 29 | 4 12 49 37 | | — 1700 | 2 20 11 49 | 1 23 45 42 | 1 0 51 26 | 8 1 35 34 | | — 1600 | 7 4 18 54 | 10 9 16 22 | 1 4 10 42 | 0 21 1 45 | | — 1500 | 11 12 25 59 | 6 24 47 2 | 1 7 29 58 | 5 10 27 57 | | — 1400 | 15 20 33 4 | 3 10 17 42 | 1 11 49 14 | 9 29 54 8 | | — 1300 | 20 4 40 8 | 11 25 48 22 | 1 14 8 30 | 2 19 20 19 | | — 1200 | 24 12 47 13 | 8 11 19 3 | 1 17 27 46 | 7 8 46 30 | | — 1100 | 28 20 54 18 | 4 26 49 43 | 1 20 47 2 | 11 28 12 41 | | — 1000 | 3 16 17 20 | 0 16 31 22 | 0 24 59 59 | 3 16 58 38 | | — 900 | 8 0 24 25 | 9 2 2 2 | 0 28 19 16 | 8 6 24 49 | | — 800 | 12 8 31 29 | 5 17 32 42 | 1 1 38 31 | 0 25 51 0 | | — 700 | 16 16 38 34 | 2 3 3 22 | 1 4 57 47 | 5 15 17 11 | | — 600 | 21 0 45 39 | 10 18 34 2 | 1 8 17 3 | 10 4 43 22 | | — 500 | 25 8 52 43 | 7 4 4 42 | 1 11 36 18 | 2 24 9 33 | | — 400 | 0 4 15 45 | 2 23 46 21 | 0 15 49 15 | 6 12 55 30 | | — 300 | 4 12 22 50 | 11 9 17 1 | 0 19 8 31 | 11 2 21 41 | | — 200 | 8 20 29 54 | 7 24 47 41 | 0 22 27 47 | 3 21 47 52 | | — 100 | 13 4 36 58 | 4 10 18 21 | 0 25 47 3 | 8 11 14 3 | | New Style | | | | | | — 300 | 14 12 22 50 | 11 9 17 1 | 0 19 8 31 | 11 2 21 41 | | — 200 | 18 20 29 54 | 7 24 47 41 | 0 22 27 47 | 3 21 47 52 | | — 100 | 24 4 36 58 | 4 10 18 21 | 0 25 47 3 | 8 11 14 3 | | + 100 | 5 8 7 5 | 8 15 30 40 | 0 3 19 16 | 4 19 26 11 | | + 200 | 9 16 14 9 | 5 1 1 20 | 0 6 38 32 | 9 8 52 22 |

### TABLE III.

Secular Equations.

| Years | Time of Mean New Moon | Moon's Mean Anomaly | Sun's Mean Anomaly | Moon's Mean Distance from Ascending Node | |-------|-----------------------|---------------------|--------------------|----------------------------------------| | B.C. | H. M. s. | D. M. s. | M. s. | | | 800 | 3 52 5 | +6 56 7 | —18 44 | —1 22 19 | | 700 | 3 34 54 | 6 25 19 | 17 20 | 1 16 13 | | 600 | 3 18 21 | 5 55 39 | 15 59 | 1 10 21 | | 500 | 3 2 27 | 5 27 7 | 14 41 | 1 4 42 | | 400 | 2 47 11 | 4 59 45 | 13 17 | 0 59 18 | | 300 | 2 32 33 | 4 33 32 | 12 16 | 0 54 6 | | 200 | 2 18 35 | 4 8 29 | 11 8 | 0 49 9 | | 100 | 2 5 16 | 3 44 36 | 10 3 | 0 44 26 | | A.C. | H. M. s. | D. M. s. | M. s. | | | 1 | 1 52 86 | 3 21 53 | 9 2 | 0 39 56 | | 101 | 1 40 35 | 3 0 21 | 8 4 | 0 35 40 | | 201 | 1 29 14 | 2 39 59 | 7 9 | 0 31 39 | | 301 | 1 18 33 | 2 20 50 | 6 17 | 0 27 51 | | 401 | 1 8 31 | 2 2 52 | 5 29 | 0 24 18 | | 501 | 0 59 10 | 1 46 6 | 4 44 | 0 20 59 | | 601 | 0 50 30 | 1 30 32 | 4 2 | 0 17 54 |

VOL. IV. ### TABLE IV.

Mean Anomalies of the Sun and Moon, and Moon's Mean Distance from Ascending Node for Mean Lunations.

| No. | Mean Lunations | Moon's Mean Anomaly | Sun's Mean Anomaly | Moon's Mean Distance from Ascending Node | |-----|----------------|---------------------|--------------------|------------------------------------------| | | | s. d. m. s. | s. d. m. s. | s. d. m. s. | | 1 | January | 29 12 44 3 | 0 25 49 1 | 0 29 6 19 | | 2 | February | 28 1 28 6 | 1 21 38 2 | 1 28 12 39 | | 3 | March | 29 14 12 9 | 2 17 27 3 | 2 27 18 58 | | 4 | April | 28 2 56 11 | 3 13 16 3 | 3 26 25 17 | | 5 | May | 27 15 40 14 | 4 9 5 4 | 4 25 31 37 | | 6 | June | 26 4 24 17 | 5 4 54 5 | 5 24 37 56 | | 7 | July | 25 17 8 20 | 6 0 43 6 | 6 23 44 16 | | 8 | August | 24 5 52 23 | 6 26 32 7 | 7 22 50 35 | | 9 | September | 22 18 36 26 | 7 22 21 8 | 8 21 56 54 | | 10 | October | 22 7 20 29 | 8 18 10 9 | 9 21 3 14 | | 11 | November | 20 20 4 31 | 9 13 59 9 | 10 20 9 33 | | 12 | December | 20 8 48 34 | 10 9 48 10 | 11 19 15 52 | | 1/2 | | 14 18 22 1 | 6 12 54 30 | 0 14 33 10 |

### TABLE V.

First Equation for the Times of New and Full Moon.

| Argument | Moon's Mean Anomaly | |----------|---------------------| | O.s. | L.s. | | II.s. | III.s. | | IV.s. | V.s. | | 0° | 0 0 0 | | 1 | 0 9 28 | | 2 | 18 56 | | 3 | 28 24 | | 4 | 37 52 | | 5 | 47 19 | | 6 | 56 45 | | 7 | 6 10 | | 8 | 15 35 | | 9 | 24 59 | | 10 | 34 21 | | 11 | 43 42 | | 12 | 53 2 | | 13 | 2 20 | | 14 | 11 37 | | 15 | 20 51 | | 16 | 30 4 | | 17 | 39 15 | | 18 | 48 23 | | 19 | 57 29 | | 20 | 6 33 | | 21 | 15 34 | | 22 | 24 33 | | 23 | 33 28 | | 24 | 42 21 | | 25 | 51 10 | | 26 | 59 56 | | 27 | 8 39 | | 28 | 17 18 | | 29 | 25 54 | | 30 | 34 26 |

| XI.s. | X.s. | | IX.s. | VIII.s. | | VII.s. | VI.s. | ### TABLE VI.

Second Equation for the Times of New and Full Moon for the year 1801, with the Secular Variation.

| Argument | Sun's Mean Ascension | |----------|----------------------| | | |

| I. Secular Variation | II. Secular Variation | III. Secular Variation | IV. Secular Variation | V. Secular Variation | |----------------------|-----------------------|------------------------|----------------------|---------------------| | + | + | + | + | + | | 0° | | | | | | 1° | | | | | | 2° | | | | | | 3° | | | | | | 4° | | | | | | 5° | | | | | | 6° | | | | | | 7° | | | | | | 8° | | | | | | 9° | | | | | | 10° | | | | | | 11° | | | | | | 12° | | | | | | 13° | | | | | | 14° | | | | | | 15° | | | | | | 16° | | | | | | 17° | | | | | | 18° | | | | | | 19° | | | | | | 20° | | | | | | 21° | | | | | | 22° | | | | | | 23° | | | | | | 24° | | | | | | 25° | | | | | | 26° | | | | | | 27° | | | | | | 28° | | | | | | 29° | | | | | | 30° | | | | |

Multiply the Secular Variation by the number of years between the given time and the year 1801, and divide the product by 100. If the given time be before 1801, change the sign of the Secular Variation. ### TABLE VII.

Third Equation for the Times of New and Full Moon.

| Argument | Moon's Mean Anomaly plus Sun's Mean Anomaly | |----------|---------------------------------------------| | | O₆ + V₁₀ + I₄ + V₇ + II₈ + V₁₁ + III₁₂ + IV₁₃ + V₁₄ + VI₁₅ + VII₁₆ + VIII₁₇ + IX₁₈ + X₁₉ + XI₂₀ + XII₂₁ + XIII₂₂ + XIV₂₃ + XV₂₄ + XVI₂₅ + XVII₂₆ + XVIII₂₇ + XIX₂₈ + XX₂₉ + XXI₃₀ + XXII₃₁ + XXIII₃₂ + XXIV₃₃ + XXV₃₄ + XXVI₃₅ + XXVII₃₆ + XXVIII₃₇ + XXIX₃₈ + XXX₃₉ + XXXI₄₀ + XXXII₄₁ + XXXIII₄₂ + XXXIV₄₃ + XXXV₄₄ + XXXVI₄₅ + XXXVII₄₆ + XXXVIII₄₇ + XXXIX₄₈ + XL₄₉ + XLI₅₀ + XLII₅₁ + XLIII₅₂ + XLIV₅₃ + XLV₅₄ + XLVI₅₅ + XLVII₅₆ + XLVIII₅₇ + XLI₉₉ + XLII₁₀₀ + XLIII₁₀₁ + XLIV₁₀₂ + XLV₁₀₃ + XLVI₁₀₄ + XLVII₁₀₅ + XLVIII₁₀₆ + XLI₉₉ + XLII₁₀₀ + XLIII₁₀₁ + XLIV₁₀₂ + XLV₁₀₃ + XLVI₁₀₄ + XLVII₁₀₅ + XLVIII₁₀₆ |

### TABLE VIII.

Fourth Equation for the Times of New and Full Moon.

| Argument | Moon's Mean Anomaly minus Sun's Mean Anomaly | |----------|----------------------------------------------| | | O₆ + V₁₀ + I₄ + V₇ + II₈ + V₁₁ + III₁₂ + IV₁₃ + V₁₄ + VI₁₅ + VII₁₆ + VIII₁₇ + IX₁₈ + X₁₉ + XI₂₀ + XII₂₁ + XIII₂₂ + XIV₂₃ + XV₂₄ + XVI₂₅ + XVII₂₆ + XVIII₂₇ + XIX₂₈ + XX₂₉ + XXI₃₀ + XXII₃₁ + XXIII₃₂ + XXIV₃₃ + XXV₃₄ + XXVI₃₅ + XXVII₃₆ + XXVIII₃₇ + XXIX₃₈ + XXX₃₉ + XXXI₄₀ + XXXII₄₁ + XXXIII₄₂ + XXXIV₄₃ + XXXV₄₄ + XXXVI₄₅ + XXXVII₄₆ + XXXVIII₄₇ + XXXIX₄₈ + XL₄₉ + XLI₅₀ + XLII₅₁ + XLIII₅₂ + XLIV₅₃ + XLV₅₄ + XLVI₅₅ + XLVII₅₆ + XLVIII₅₇ + XLI₉₉ + XLII₁₀₀ + XLIII₁₀₁ + XLIV₁₀₂ + XLV₁₀₃ + XLVI₁₀₄ + XLVII₁₀₅ + XLVIII₁₀₆ | ### Table IX

Fifth Equation for the Times of New and Full Moon.

| Argument | Twice Moon's Mean Distance from Ascending Node minus Moon's Mean Anomaly | |----------|---------------------------------------------------------------| | | O.s. + VI.s. | I.s. + VII.s. | II.s. + VIII.s. | | 0° | 0 0 18 | 2 16 | 30° | | 1 | 0 3 21 | 2 17 | | | 2 | 0 5 23 | 2 19 | | | 3 | 0 8 25 | 2 20 | | | 4 | 0 11 28 | 2 21 | | | 5 | 0 14 30 | 2 22 | | | 6 | 0 16 32 | 2 23 | | | 7 | 0 19 34 | 2 24 | | | 8 | 0 22 37 | 2 26 | | | 9 | 0 25 39 | 2 27 | | | 10 | 0 27 41 | 2 28 | | | 11 | 0 30 43 | 2 28 | | | 12 | 0 33 45 | 2 29 | | | 13 | 0 35 47 | 2 30 | | | 14 | 0 38 49 | 2 31 | | | 15 | 0 41 51 | 2 32 | | | 16 | 0 43 53 | 2 32 | | | 17 | 0 46 55 | 2 33 | | | 18 | 0 48 57 | 2 34 | | | 19 | 0 51 58 | 2 34 | | | 20 | 0 54 0 | 2 35 | | | 21 | 0 56 2 | 2 35 | | | 22 | 0 59 4 | 2 35 | | | 23 | 1 1 5 | 2 36 | | | 24 | 1 4 7 | 2 36 | | | 25 | 1 6 9 | 2 36 | | | 26 | 1 9 10 | 2 37 | | | 27 | 1 11 12 | 2 37 | | | 28 | 1 14 13 | 2 37 | | | 29 | 1 16 15 | 2 37 | | | 30 | 1 18 16 | 2 37 | |

### Table X

Sixth Equation for the Times of New and Full Moon.

| Argument | Moon's Mean Distance from Ascending Node | |----------|------------------------------------------| | | O.s. + VI.s. | I.s. + VII.s. | II.s. + VIII.s. | | 0° | 0 0 40 | 1 40 | 30° | | 1 | 0 4 42 | 1 38 | | | 2 | 0 8 44 | 1 36 | | | 3 | 0 12 46 | 1 34 | | | 4 | 0 16 48 | 1 31 | | | 5 | 0 20 49 | 1 29 | | | 6 | 0 24 50 | 1 26 | | | 7 | 0 28 51 | 1 23 | | | 8 | 0 32 53 | 1 21 | | | 9 | 0 36 53 | 1 18 | | | 10 | 0 40 54 | 1 15 | | | 11 | 0 43 55 | 1 11 | | | 12 | 0 47 55 | 1 8 | | | 13 | 0 51 56 | 1 5 | | | 14 | 0 54 56 | 1 1 | | | 15 | 0 58 56 | 0 58 | | | 16 | 1 1 56 | 0 54 | | | 17 | 1 5 56 | 0 51 | | | 18 | 1 8 55 | 0 47 | | | 19 | 1 11 55 | 0 43 | | | 20 | 1 15 54 | 0 40 | | | 21 | 1 18 53 | 0 36 | | | 22 | 1 21 53 | 0 32 | | | 23 | 1 23 51 | 0 28 | | | 24 | 1 26 50 | 0 24 | | | 25 | 1 29 49 | 0 20 | | | 26 | 1 31 48 | 0 16 | | | 27 | 1 34 46 | 0 12 | | | 28 | 1 36 44 | 0 8 | | | 29 | 1 38 42 | 0 4 | | | 30 | 1 40 40 | 0 0 | | ### TABLE XI.

Seventh Equation for the Times of New and Full Moon.

| Argument | Twice the Moon's Mean Anomaly plus the Sun's Mean Anomaly | |----------|----------------------------------------------------------| | | O₆ + VI₆ + VII₆ + VIII₆ + IX₆ + X₆ + XI₆ + V₆ + IV₆ + III₆ | | 0° | 0 18 31 30° | | 5 | 3 21 33 25 | | 10 | 6 23 34 20 | | 15 | 9 25 35 15 | | 20 | 12 28 35 10 | | 25 | 15 29 36 5 | | 30 | 18 31 36 0 |

### TABLE XII.

Eighth Equation for the Times of New and Full Moon.

| Argument | Twice the Moon's Mean Anomaly minus the Sun's Mean Anomaly | |----------|-----------------------------------------------------------| | | O₆ - VI₆ - VII₆ - VIII₆ - IX₆ - X₆ - XI₆ - V₆ - IV₆ - III₆ | | 0° | 0 5 9 30° | | 5 | 1 6 9 25 | | 10 | 2 6 9 20 | | 15 | 3 7 10 15 | | 20 | 3 8 10 10 | | 25 | 4 8 10 5 | | 30 | 5 9 10 0 |

### TABLE XIII.

Ninth Equation for the Time of Full Moon only.

| Argument | Moon's Mean Anomaly | |----------|---------------------| | | O₆ + VI₆ + VII₆ + VIII₆ + IX₆ + X₆ + XI₆ + V₆ + IV₆ + III₆ | | 0° | 0 0 0 43 1 15 30° | | 1 | 0 1 0 45 1 16 29 | | 2 | 0 3 0 46 1 17 28 | | 3 | 0 5 0 47 1 17 27 | | 4 | 0 6 0 49 1 18 26 | | 5 | 0 8 0 50 1 19 25 | | 6 | 0 9 0 51 1 19 24 | | 7 | 0 11 0 52 1 20 23 | | 8 | 0 12 0 54 1 21 22 | | 9 | 0 14 0 55 1 21 21 | | 10 | 0 15 0 56 1 22 20 | | 11 | 0 17 0 57 1 22 19 | | 12 | 0 18 0 58 1 23 18 | | 13 | 0 20 0 59 1 23 17 | | 14 | 0 21 1 0 1 24 16 | | 15 | 0 22 1 1 1 24 15 | | 16 | 0 24 1 3 1 24 14 | | 17 | 0 25 1 4 1 25 13 | | 18 | 0 27 1 5 1 25 12 | | 19 | 0 28 1 6 1 25 11 | | 20 | 0 30 1 7 1 26 10 | | 21 | 0 31 1 8 1 26 9 | | 22 | 0 33 1 9 1 26 8 | | 23 | 0 34 1 9 1 26 7 | | 24 | 0 35 1 10 1 26 6 | | 25 | 0 37 1 11 1 27 5 | | 26 | 0 38 1 12 1 27 4 | | 27 | 0 39 1 13 1 27 3 | | 28 | 0 41 1 14 1 27 2 | | 29 | 0 42 1 15 1 27 1 | | 30 | 0 43 1 15 1 27 0 |

### TABLE XIV.

Tenth Equation for the Time of Full Moon only.

| Argument | Sun's Mean Anomaly | |----------|--------------------| | | O₆ + VI₆ + VII₆ + VIII₆ + IX₆ + X₆ + XI₆ + V₆ + IV₆ + III₆ | | 0° | 0 16 29 30° | | 5 | 3 19 30 25 | | 10 | 6 21 31 20 | | 15 | 9 23 32 15 | | 20 | 11 25 32 10 | | 25 | 14 27 33 5 | | 30 | 16 29 33 0 |

| XI₆ + X₆ + IX₆ + V₆ + IV₆ + III₆ | ### TABLE XV

Sun's Mean Motion from the Moon's Perigee and Ascending Node, and Variation of the Sun's Mean Anomaly, for Hours, Minutes, and Seconds.

#### For Hours

| Sun's Mean Motion from Moon's Perigee | Sun's Mean Anomaly | Sun's Mean Motion from Ascending Node | |--------------------------------------|--------------------|-------------------------------------| | H. M. S. | M. S. | M. S. | | 1 2 11 | 2 28 | 0 2 36 | | 2 4 22 | 4 56 | 0 5 12 | | 3 6 39 | 7 23 | 0 7 47 | | 4 8 44 | 9 51 | 0 10 23 | | 5 10 55 | 12 19 | 0 12 59 | | 6 13 7 | 14 47 | 0 15 35 | | 7 15 18 | 17 15 | 0 18 10 | | 8 17 29 | 19 43 | 0 20 46 | | 9 19 40 | 22 11 | 0 23 22 | | 10 21 51 | 24 38 | 0 25 58 | | 11 24 2 | 27 6 | 0 28 34 | | 12 26 13 | 29 34 | 0 31 9 |

#### For Minutes and Seconds

| Sun's Mean Motion from Moon's Perigee | Sun's Mean Anomaly | Sun's Mean Distance from Ascending Node | |--------------------------------------|--------------------|----------------------------------------| | M. S. | M. S. | M. S. | | 1 0 2 | 0 2 | 0 3 | | 2 0 4 | 0 5 | 0 5 | | 3 0 7 | 0 7 | 0 8 | | 4 0 9 | 0 10 | 0 10 | | 5 0 11 | 0 12 | 0 13 | | 6 0 13 | 0 15 | 0 16 | | 7 0 15 | 0 17 | 0 18 | | 8 0 17 | 0 20 | 0 21 | | 9 0 20 | 0 22 | 0 23 | | 10 0 22 | 0 25 | 0 26 | | 11 0 24 | 0 27 | 0 29 | | 12 0 26 | 0 30 | 0 31 | | 13 0 28 | 0 32 | 0 34 | | 14 0 31 | 0 34 | 0 36 | | 15 0 33 | 0 37 | 0 39 | | 16 0 35 | 0 39 | 0 41 | | 17 0 37 | 0 42 | 0 44 | | 18 0 39 | 0 44 | 0 47 | | 19 0 42 | 0 47 | 0 49 | | 20 0 44 | 0 49 | 0 52 | | 21 0 46 | 0 52 | 0 54 | | 22 0 48 | 0 54 | 0 57 | | 23 0 50 | 0 57 | 1 0 | | 24 0 52 | 0 59 | 1 2 | | 25 0 55 | 1 2 | 1 5 | | 26 0 57 | 1 4 | 1 7 | | 27 0 59 | 1 6 | 1 10 | | 28 1 1 | 1 9 | 1 13 | | 29 1 3 | 1 11 | 1 15 | | 30 1 6 | 1 14 | 1 18 | ### TABLE XVI

Equation of the Sun's Centre for 1801, with the Secular Variation.

| Argument | Sun's Mean Anomaly | |----------|--------------------| | O.s | Secular Variation | | I.s | Secular Variation | | II.s | Secular Variation | | III.s | Secular Variation | | IV.s | Secular Variation | | V.s | Secular Variation |

Multiply the secular variations by the number of years between the given time and the year 1801, and divide the product by 100. If the time is before 1801, change the sign of the secular variation. ### TABLE XVII. For the Moon's Latitude and Inclination of her relative Orbit to the Ecliptic in Eclipses.

| Argument | Moon's True distance from Ascending Node | |----------|----------------------------------------| | Latitude | Inclination | | O₆ | + | | + | VI₆ | | increasing | left. VI₆ | | + | VI₆ | | increasing | right. VI₆ |

| 0° | 0 0 0 0 30° | | 1 | 0 5 17 29 5 44 | | 2 | 0 10 33 28 5 44 | | 3 | 0 15 49 27 5 43 | | 4 | 0 21 5 26 5 43 | | 5 | 0 26 21 25 5 43 | | 6 | 0 31 36 24 5 42 | | 7 | 0 36 50 23 5 41 | | 8 | 0 42 4 22 5 41 | | 9 | 0 47 17 21 5 40 | | 10 | 0 52 30 20 5 39 | | 11 | 0 57 41 19 5 38 | | 12 | 1 2 51 18 5 37 | | 13 | 1 8 0 17 5 35 | | 14 | 1 13 8 16 5 34 | | 15 | 1 18 14 15 5 32 | | 16 | 1 23 19 14 5 31 | | 17 | 1 28 23 13 5 29 | | 18 | 1 33 24 12 5 27 | | 19 | 1 38 24 11 5 25 | | 20 | 1 43 23 10 5 23 |

+ decreasing. V₆. right. V₆. - decreasing. XI₆. left. XI₆.

In Lunar Eclipses change the designation "right" or "left" of the Inclination.

### TABLE XIX. For the Moon's Latitude in Eclipses.

| Argument | Preceding argument, plus Moon's True Anomaly | |----------|---------------------------------------------| | O₆ | + | | + | VII₆ | | + | VIII₆ |

| 0° | 0 13 22 30° | | 5 | 2 15 23 25 | | 10 | 4 17 24 20 | | 15 | 7 18 25 15 | | 20 | 9 20 25 10 | | 25 | 11 21 26 5 | | 30 | 13 22 26 0 |

- XIA. + X₆. IXA. + IIIA.

### TABLE XX. For the Moon's Latitude in Eclipses.

| Argument | Moon's True Anomaly plus True Distance from Ascending Node | |----------|-----------------------------------------------------------| | O₆ | + | | + | VI₆ | | + | VII₆ | | + | VIII₆ |

| 0° | 0 8 15 30° | | 5 | 1 10 15 25 | | 10 | 3 11 16 20 | | 15 | 4 12 16 15 | | 20 | 6 13 17 10 | | 25 | 7 14 17 5 | | 30 | 8 15 17 0 |

- XIA. + X₆. IXA. + IIIA.

### TABLE XXI. For the Moon's Latitude in Eclipses.

| Argument | Sun's True Anomaly plus Moon's True Distance from Ascending Node | |----------|------------------------------------------------------------------| | O₆ | + | | + | VI₆ | | + | VII₆ | | + | VIII₆ |

| 0° | 0 10 17 30° | | 5 | 2 11 18 25 | | 10 | 3 13 18 20 | | 15 | 5 14 19 15 | | 20 | 7 15 19 10 | | 25 | 8 16 20 5 | | 30 | 10 17 20 0 |

- XIA. + X₆. IXA. + IIIA. ### TABLE XXII.

For the Moon's Latitude in Eclipses.

| Argument | Sun's True Anomaly minus Moon's True Distance from Ascending Node | |----------|---------------------------------------------------------------| | | O. + VI. + VII. + VIII. - | | 0° | 0 16 28 30 | | 5 | 3 19 30 25 | | 10 | 6 21 31 20 | | 15 | 8 23 31 15 | | 20 | 11 25 32 10 | | 25 | 14 27 32 5 | | 30 | 16 28 33 0 |

### TABLE XXIII.

The Moon's Equatorial Horizontal Parallax, Semidiameter, and Horary Motion in her Orbit, at New and Full Moon; and the Semidiameter and Horary Motion of the Sun.

| Argument | Moon's True Anomaly, or Sun's Mean Anomaly | |----------|--------------------------------------------| | | Moon's Equatorial Horizontal Parallax. Moon's Semidiameter. Moon's Horary Motion. Sun's Semidiameter. Sun's Horary Motion. Argument. | | O. | 61 23 16 44 38 13 16 18 2 33 XII. 0 | | 6 | 61 22 16 43 38 11 16 18 2 33 24 | | 12 | 61 18 16 42 38 7 16 17 2 33 18 | | 18 | 61 12 16 41 37 59 16 17 2 33 12 | | 24 | 51 4 16 38 37 49 16 16 2 32 6 | | I. | 60 53 16 35 37 36 16 15 2 32 XI. 0 | | 6 | 60 41 16 32 37 21 16 15 2 32 24 | | 12 | 60 26 16 28 37 3 16 13 2 32 18 | | 18 | 60 9 16 23 36 43 16 12 2 31 12 | | 24 | 59 51 16 19 36 21 16 11 2 31 6 | | II. | 59 31 16 13 35 58 16 9 2 30 X. 0 | | 6 | 59 10 16 7 35 33 16 8 2 30 24 | | 12 | 58 49 16 2 35 7 16 6 2 29 18 | | 18 | 58 26 15 55 34 40 16 4 2 29 12 | | 24 | 58 3 15 49 34 14 16 3 2 28 6 | | III. | 57 39 15 43 33 46 16 1 2 28 IX. 0 | | 6 | 57 16 15 36 33 19 15 59 2 27 24 | | 12 | 56 53 15 30 32 53 15 58 2 27 18 | | 18 | 56 30 15 24 32 27 15 56 2 26 12 | | 24 | 56 8 15 18 32 2 15 55 2 26 6 | | IV. | 55 47 15 12 31 39 15 53 2 25 VIII. 0 | | 6 | 55 28 15 7 31 17 15 52 2 25 24 | | 12 | 55 10 15 2 30 56 15 51 2 25 18 | | 18 | 54 53 14 57 30 38 15 49 2 24 12 | | 24 | 54 38 14 53 30 22 15 48 2 24 6 | | V. | 54 25 14 50 30 8 15 48 2 24 VII. 0 | | 6 | 54 15 14 47 29 56 15 47 2 23 24 | | 12 | 54 6 14 45 29 47 15 46 2 23 18 | | 18 | 54 0 14 43 29 40 15 46 2 23 12 | | 24 | 53 57 14 42 29 36 15 46 2 23 6 | | VI. | 53 55 14 42 29 35 15 46 2 23 VI. 0 |

### TABLE XXIV.

Diminution of the Moon's Equatorial Horizontal Parallax, and of the Latitude of a Place, on account of the Spheroidal Figure of the Earth.

| Arguments | Latitude at the Side, and Moon's Horizontal Parallax at the Top. | |-----------|------------------------------------------------------------------| | Diminution of Parallax. | Diminution of Latitude. | | m. m. m. | m. m. m. | | 53 57 61 | 0 0 0 0-0 | | 0° | 0 0 0 2-0 | | 5 | 0 0 0 4-0 | | 10 | 1 1 1 5-8 | | 20 | 1 1 1 7-5 | | 25 | 2 2 2 8-9 | | 30 | 3 3 3 10-1 | | 35 | 3 4 4 11-0 | | 40 | 4 5 5 11-5 | | 45 | 5 6 6 11-7 | | 50 | 6 7 7 11-5 | | 55 | 7 8 8 11-0 | | 60 | 8 9 9 10-1 | | 65 | 9 10 10 9-0 | | 70 | 10 10 11 7-5 | | 75 | 10 11 12 5-9 | | 80 | 10 11 12 4-0 | | 85 | 11 12 12 2-0 | | 90 | 11 12 12 0-0 | ### TABLE XXV.

Epochs of the Mean Longitude of the Sun's Perigee, including the Secular Variation of the Precession of the Equinoxes.

| Years | Longitude of Sun's Perigee | Secular Variation of Precession of Equinoxes | |-------|----------------------------|---------------------------------------------| | B.C. | | | | Old Style | s. d. m. s. | s. | | 800 | 7 25 1 10 | —60 | | 700 | 7 26 43 20 | 57 | | 600 | 7 28 25 33 | 56 | | 500 | 8 0 7 47 | 53 | | | 400 | 8 1 50 4 | —50 | | | 300 | 8 3 32 24 | 49 | | | 200 | 8 5 14 45 | 46 | | | 100 | 8 6 57 9 | 44 | | A.D. | | | | Old Style | s. d. m. s. | s. | | 1 | 8 8 39 35 | —42 | | 101 | 8 10 22 3 | 39 | | 201 | 8 12 4 34 | 37 | | 301 | 8 13 47 7 | 34 | | | 401 | 8 15 29 43 | —33 | | | 501 | 8 17 12 20 | 30 | | | 601 | 8 18 35 0 | 27 | | | 701 | 8 20 37 43 | 25 | | | 801 | 8 22 20 28 | —23 | | | 901 | 8 24 3 15 | 21 | | | 1001 | 8 25 46 4 | 18 | | | 1101 | 8 27 28 56 | 15 | | | 1201 | 8 29 11 51 | —14 | | | 1301 | 9 0 54 47 | 11 | | | 1401 | 9 2 37 46 | 8 | | | 1501 | 9 4 20 48 | 6 | | | 1601 | 9 6 3 52 | —4 | | | 1701 | 9 7 46 58 | —1 | | | New Style | s. d. m. s. | s. | | 1501 | 9 4 20 46 | —6 | | 1601 | 9 6 3 50 | 4 | | 1701 | 9 7 46 56 | —1 | | 1801 | 9 9 30 5 | +1 | | 1901 | 9 11 13 16 | 4 | | 2001 | 9 12 56 30 | |

### TABLE XXVI.

Mean Motion of the Sun's Perigee in Years, Months, and Days.

| Years | Motion of Sun's Perigee | Months | Motion of Sun's Perigee | Months | Motion of Sun's Perigee | |-------|-------------------------|--------|-------------------------|--------|-------------------------| | | d. m. s. | January | m. s. | January | m. s. | | 1 | 0 1 2 | 0 0 7 | 1 | 0 0 7 | 1 | | 2 | 0 2 4 | 0 5 13 | 2 | 0 5 13 | 2 | | 3 | 0 3 6 | 0 10 19 | 3 | 0 10 19 | 3 | | 4 | 0 4 8 | 0 15 25 | 4 | 0 15 25 | 4 | | 5 | 0 5 9 | 0 20 31 | 5 | 0 20 31 | 5 | | | 6 0 6 11 | 0 26 | | 0 26 | | | | 7 0 7 13 | 0 31 | | 0 31 | | | | 8 0 8 15 | 0 36 | | 0 36 | | | | 9 0 9 17 | 0 41 | | 0 41 | | | | 10 0 10 19 | 0 46 | | 0 46 | | | | 20 0 20 38 | 0 52 | | 0 52 | | | | 30 0 30 57 | 0 57 | | 0 57 | | | | 40 0 41 16 | | | | | | | 50 0 51 35 | | | | | | | 60 1 1 54 | | | | | | | 70 1 12 13 | | | | | | | 80 1 22 32 | | | | | | | 90 1 32 51 | | | | | | | 100 1 43 10 | | | | | ### TABLE XXVII.

The Sun's Declination for the Year 1801, with the Secular Variation.

| Argument | Sun's True Longitude | |----------|----------------------| | | O. + VI. | Secular Variation | I. + VII. | Secular Variation | II. + VIII. | Secular Variation | | D. | M. | M. | D. | M. | D. | M. | | 0° | 0 | 0-0 | 11 | 29-1 | 20 | 10-4 | | 1 | 0 | 23-9 | 0-0 | 50-1 | 0-4 | 22-9 | | 2 | 0 | 47-8 | 0-0 | 12-9 | 0-4 | 35-1 | | 3 | 1 | 11-6 | 0-0 | 31-5 | 0-4 | 46-9 | | 4 | 1 | 35-5 | 0-1 | 52-0 | 0-4 | 58-3 | | 5 | 1 | 59-3 | 0-1 | 13-2 | 0-4 | 9-3 | | 6 | 2 | 23-1 | 0-1 | 32-2 | 0-4 | 19-9 | | 7 | 2 | 46-9 | 0-1 | 51-9 | 0-5 | 30-2 | | 8 | 3 | 10-6 | 0-1 | 11-5 | 0-5 | 40-0 | | 9 | 3 | 34-3 | 0-1 | 30-8 | 0-5 | 49-4 | | 10 | 3 | 57-9 | 0-1 | 49-8 | 0-5 | 58-4 | | 11 | 4 | 21-5 | 0-1 | 8-6 | 0-5 | 7-1 | | 12 | 4 | 44-9 | 0-2 | 27-2 | 0-5 | 15-2 | | 13 | 5 | 8-4 | 0-2 | 45-5 | 0-5 | 23-0 | | 14 | 5 | 31-7 | 0-2 | 3-5 | 0-5 | 30-3 | | 15 | 5 | 54-9 | 0-2 | 21-2 | 0-5 | 37-3 | | 16 | 6 | 18-1 | 0-2 | 38-7 | 0-6 | 43-7 | | 17 | 6 | 41-1 | 0-2 | 55-9 | 0-6 | 49-8 | | 18 | 7 | 4-1 | 0-2 | 12-8 | 0-6 | 55-4 | | 19 | 7 | 26-9 | 0-2 | 29-4 | 0-6 | 6-6 | | 20 | 7 | 49-7 | 0-3 | 45-6 | 0-6 | 5-3 | | 21 | 8 | 12-3 | 0-3 | 1-6 | 0-6 | 9-6 | | 22 | 8 | 34-7 | 0-3 | 17-3 | 0-6 | 13-4 | | 23 | 8 | 57-1 | 0-3 | 32-6 | 0-6 | 16-8 | | 24 | 9 | 19-3 | 0-3 | 47-6 | 0-6 | 19-8 | | 25 | 9 | 41-3 | 0-3 | 2-3 | 0-6 | 22-3 | | 26 | 10 | 3-2 | 0-3 | 16-6 | 0-6 | 24-3 | | 27 | 10 | 24-9 | 0-3 | 30-6 | 0-7 | 25-9 | | 28 | 10 | 46-5 | 0-4 | 44-2 | 0-7 | 27-0 | | 29 | 11 | 7-9 | 0-4 | 57-5 | 0-7 | 27-7 | | 30 | 11 | 29-1 | 0-4 | 10-4 | 0-7 | 27-9 |

Multiply the secular variation by the number of years between the given time and the year 1801, and divide the product by 100. If the given time be before 1801, change the sign of the secular variation. ### Table XXVIII

**Equation of Time for converting Mean Time into Apparent Time for the Year 1891, with the Secular Variation.**

| Argument | Secular Variation | Secular Variation | Secular Variation | Secular Variation | Secular Variation | Secular Variation | Secular Variation | Secular Variation | |----------|-------------------|-------------------|-------------------|-------------------|-------------------|-------------------|-------------------|-------------------| | | | | | | | | | | | 0° | -7°36' + 4°11'9" | | | | | | | | | 1 | -7°17' + 3°12'2" | | | | | | | | | 2 | -6°58' + 3°13'5" | | | | | | | | | 3 | -6°40' + 3°14'8" | | | | | | | | | 4 | -6°21' + 3°15'1" | | | | | | | | | 5 | -6°02' + 3°15'4" | | | | | | | | | 6 | -5°43' + 3°15'7" | | | | | | | | | 7 | -5°24' + 3°16'0" | | | | | | | | | 8 | -5°05' + 3°16'3" | | | | | | | | | 9 | -4°46' + 3°16'6" | | | | | | | | | 10 | -4°27' + 3°16'9" | | | | | | | | | 11 | -4°8' + 3°17'2" | | | | | | | | | 12 | -3°30' + 3°17'5" | | | | | | | | | 13 | -2°13' + 3°17'8" | | | | | | | | | 14 | -1°54' + 3°18'1" | | | | | | | | | 15 | -1°35' + 3°18'4" | | | | | | | | | 16 | -1°16' + 3°18'7" | | | | | | | | | 17 | -1°97' + 3°19'0" | | | | | | | | | 18 | -1°78' + 3°19'3" | | | | | | | | | 19 | -1°59' + 3°19'6" | | | | | | | | | 20 | -1°40' + 3°19'9" | | | | | | | | | 21 | -1°21' + 3°20'2" | | | | | | | | | 22 | -1°02' + 3°20'5" | | | | | | | | | 23 | -0°43' + 3°20'8" | | | | | | | | | 24 | -0°24' + 3°21'1" | | | | | | | | | 25 | -0°6' + 3°21'4" | | | | | | | | | 26 | +0°19' + 3°21'7" | | | | | | | | | 27 | +0°38' + 3°22'0" | | | | | | | | | 28 | +0°57' + 3°22'3" | | | | | | | | | 29 | +1°16' + 3°22'6" | | | | | | | | | 30 | +1°35' + 3°22'9" | | | | | | | |

Multiply the secular variation by the number of years between the given time and the year 1891, and divide the product by 100. If the given time be before 1891, change the sign of the secular variation. Sect. II.—Application of the Tables, and Projection of Eclipses.

Problem I.—To calculate the time of true new or full moon for any period within the limits of the nineteenth century.

Precept 1. Write out the time of mean new moon in January for the proposed year from Table I., together with the mean anomalies of the moon and sun, and the moon's mean distance from her ascending node, applying to each of these quantities the secular equation found by its side by addition or subtraction, according as it has the sign + or —. If you want the time of full moon in January, add the half-lunation at the foot of Table IV., with its anomalies, &c., to the former numbers if the new moon falls before the 15th of January; but if it falls after, subtract the half-lunation, with the anomalies, &c., belonging to it, from the former numbers, and write down the respective sums or remainders.

2. In these additions or subtractions, observe that 60 seconds make a minute, 60 minutes make a degree, 30 degrees make a sign, and 12 signs make a circle. When you exceed 12 signs in addition, reject 12, and set down the remainder. When the number of signs to be subtracted is greater than the number you subtract from, add 12 signs to the lesser number, and then you will have a remainder to set down. In the tables, signs are marked S, degrees D, minutes M, and seconds S.

3. When the required new or full moon is in any given month after January, write out from Table IV. such one of the mean lunations, with the anomalies, &c., us, added to the time of mean new or full moon in January, will make the mean new or full moon to fall within the given month, setting them below the number taken out for January.

4. Add all these together, and in leap-years (which in Table I. have the letter B annexed to them) subtract one day from the time of mean new or full moon when it happens after 28th February. You will then have the time of the required mean new or full moon, with the mean anomalies, and the moon's mean distance from the ascending node, which are the arguments for finding the proper equations.

5. With the signs and degrees of the moon's mean anomaly enter Table V., and therewith take out the first equation for reducing the mean syzygy to the true; taking care to make proportions in the table for the odd minutes and seconds of anomaly, as the table gives the equation only to whole degrees.

Observe in this and every other case of finding equations, that, if the signs are at the head of the table, their degrees are at the left hand, and are reckoned downwards; the equation being in the body of the table under or over the signs in a collateral line with the degrees. The signs + and — at the head or foot of the tables where the signs are found, show whether the equation is to be added to the time of mean new or full moon, or to be subtracted from it.

6. With the signs and degrees of the sun's mean anomaly enter Table VI. and take out the second equation for reducing the time of mean to that of new or full moon, with a proportional part of its secular variation in the column adjoining, corresponding to the number of years elapsed since 1801, the whole variation being adapted for a period of 100 years.

7. Add together the mean anomalies of the sun and moon, and with the sum enter Table VII. and take out the third equation. For this and the following equations it will be sufficient to compute the arguments to minutes, neglecting the seconds.

8. Subtract the sun's mean anomaly from the moon's mean anomaly, and with the remainder enter Table VIII. and take out the fourth equation.

9. Subtract the moon's mean anomaly from twice the moon's distance from the ascending node, and with the remainder enter Table IX. and take out the fifth equation.

10. The moon's mean distance from the ascending node is the argument of Table X., with which take out the sixth equation.

11. To twice the moon's mean anomaly add the sun's mean anomaly, and with the sum enter Table XI. and take out the seventh equation.

12. From twice the moon's mean anomaly subtract the sun's mean anomaly, and with the remainder enter Table XII. and take out the eighth equation.

13. These are all the equations for reducing the time of mean new moon to the time of true new moon; but for full moon other two equations are required, the argument for equation ninth being the moon's mean anomaly, which equation is exhibited in Table XIII.; and the argument of the tenth equation being the sun's mean anomaly, the equation being exhibited in Table XIV.

14. Add together the equations which have the sign of addition, and also those which have the sign of subtraction, and subtract the lesser sum from the greater, giving to the remainder the sign of the greater; and add or subtract the remainder, according as its sign denotes, to or from the time of mean new or full moon, and you have the time of true new or full moon required.

These tables are adapted to the meridian of Greenwich observatory; and for any other place, its longitude in time is to be added to or subtracted from the time given by the tables, according as it is to the east or west of Greenwich, and the time as reckoned at the given place is obtained. The tables begin the day at noon, and reckon forward from thence to the noon following. Thus January the 31st, at 22 hours 30 minutes 25 seconds of tabular time, is February 1st (in common reckoning), at 30 minutes 25 seconds after 10 o'clock in the morning. It is to be further observed, that the time obtained from the tables is mean time, or that shown by a well-regulated clock or watch. But to make it agree with solar or apparent time, or that given by a sun-dial, which is necessary in the computation of solar eclipses, you must apply the equation of time contained in Table XXVIII. as afterwards directed.

The method of calculating the time of any new or full moon, without the limits of the nineteenth century, will be shown farther on; and a few examples compared with the precepts will make the whole work plain. **Example I.**

Required the Time of True New Moon in May 1836 at Edinburgh, long. 0 hours 12 minutes 44 seconds west of Greenwich.

| Time of New Moon | Moon's Mean Anomaly | Sun's Mean Anomaly | Moon's Mean Distance from Ascending Node | |------------------|---------------------|--------------------|----------------------------------------| | 1836. B. | | | | | Secular equations | | | | | April | | | | | Four lunations | | | | | Sum | | | | | Subtract 1 day for leap-year | | | | | Sum of equations | | | | | Time of true new moon at Greenwich | | | | | Subtract for Edinburgh | | | | | True time of new moon at Edinburgh | | | |

**Equations.**

| Argument I. p mean anomaly | s. n. m. s. | h. m. s. | h. m. s. | |----------------------------|-------------|----------|----------| | II. ⊕ mean anomaly | | | | | Secular variation | | | | | III. p mean anomaly + ⊕ mean anomaly | 9 11 26 | 6 58 | | | IV. p mean anomaly - ⊕ mean anomaly | 0 14 48 | 2 42 | | | Twice p mean distance from ascending node | 0 7 14 | | | | V. Do. — p mean anomaly | 7 9 7 | | | | VI. p mean distance from node | 0 3 7 | | | | Twice p mean anomaly | 9 26 14 | | | | VII. Do. + ⊕ mean anomaly | 2 9 33 | | | | VIII. Do. — ⊕ mean anomaly | 5 12 35 | | | | Sum of equations | + 3 10 7 | | | | | — 5 34 9 | | | | | + 3 10 7 | | | | | — 2 24 2 | | |

**Example II.**

Required the Time of True Full Moon in September 1830 at Greenwich.

| Time of Full Moon | Moon's Mean Anomaly | Sun's Mean Anomaly | Moon's Mean Distance from Ascending Node | |-------------------|---------------------|--------------------|----------------------------------------| | 1830 | | | | | Secular equations | | | | | Subtract a half-lunation | | | | | Full moon | | | | | Eight lunations | | | | | Sum of equations | | | | | True full moon | | | | **Astronomy**

**Equations**

| Argument I | s. d. m. s. | n. m. s. | |------------|-------------|----------| | I. 10 | 8 1 20 | 7 21 32 | | II. 8 | 1 23 24 | | | III. 6 | 9 24 | | | IV. 2 | 6 38 | | | V. 1 | 24 40 | | | VI. 0 | 1 21 | | | VII. 4 | 17 25 | | | VIII. 0 | 14 39 | | | IX. 10 | 8 1 | | | X. 8 | 1 23 | |

Secular variation........... 10

| Secular variation........... | 10 | |-----------------------------|----|

| Sum of equations............. | + 3 54 10 | |-------------------------------|-----------|

**Problem II.** — To calculate the time of new and full moon in a given year and month of any particular century between the Christian era and the nineteenth century.

**Note.** — Prior to the sixteenth century the times are supposed to be reckoned according to the Julian calendar, practical or old style. Between it and the nineteenth century they may be reckoned according to either the Julian or Gregorian calendar, or old or new style.

**Precept 1.** Find a year of the same number in the nineteenth century, with that of the year in the century proposed, and take out the time of mean new moon in January for that year, with the mean anomalies, and the moon's mean distance from the node at that time, as already taught, neglecting the secular equations contained in Table I.

**Precept 2.** Take from Table II. as many complete centuries of years having the sign — prefixed, and titled either old style or new style, according to the given date, as, when subtracted from the above said year in the nineteenth century, will answer to the given year, and take out the time of mean new moon and its anomalies, &c., belonging to the said centuries, and add them to those for the year in the nineteenth century, and the sums, after applying to them the secular equations taken from Table III., making proportions for the odd years, will be the times and anomalies, &c., of mean new moon in January or February, according as the time is less or more than 31 days, in the given year of the century proposed. Then work in all respects for the time of true new or full moon, as shown in the above precepts and examples.

**Example III.**

Required the True Time of New Moon in July 1339, Old Style.

| Time of New Moon. | Moon's Mean Anomaly. | Sun's Mean Anomaly. | Moon's Mean Distance from Ascending Node. | |------------------|----------------------|---------------------|----------------------------------------| | 1839 | | | | | 500 | | | | | 1339 | | | | | Secular equations | | | | | Five lunations | May | | | | Equations | July | | | | Time of true new moon | July | | |

**Equations**

| Argument I | s. d. m. s. | n. m. s. | |------------|-------------|----------| | I. 8 | 23 27 51 | 9 50 21 | | II. 6 | 20 54 57 | | | Secular variation........... | 1 27 56 | | III. 3 | 14 23 | | | IV. 2 | 2 33 | | | V. 2 | 22 57 | | | VI. 5 | 23 13 | | | VII. 0 | 7 51 | | | VIII. 10 | 26 1 | |

Secular variation........... 10

| Secular variation........... | 10 | |-----------------------------|----|

| Sum of equations............. | + 8 26 6 | |-------------------------------|-----------|

**Problem III.** — To calculate the true time of new or full moon in any given year and month before the Christian era.

**Precept 1.** Find a year in the nineteenth century which, being added to the given number of years before Christ diminished by one, shall make a number of complete centuries.

2. Find this number of centuries in Table II. and add the time and anomalies belonging to it to those of the above-found year of the nineteenth century, applying the secular equations for the given year in Table III., and the sums will denote the time and anomalies, &c., of mean new moon in January or February of the given year before Christ. Then for the true time of new or full moon proceed as above taught, for any year between the Christian era and the nineteenth century, observing that the given year before Christ is or is not leap-year, according as the above-mentioned year in the nineteenth century is leap-year or not. ### Example IV.

Required the True Time of New Moon in September 610 before Christ.

| Time of New Moon | Moon's Mean Anomaly | Sun's Mean Anomaly | Moon's Mean Distance from Ascending Node | |------------------|---------------------|--------------------|------------------------------------------| | 1891 | | | | | 2500 | | | | | 610 B.C. | | | | | Secular equations| | | | | Eight lunations | August | | | | Equations | Sept | | | | Time of true new moon | Sept | | |

### Equations.

**Problem IV.**—To calculate the time of true new or full moon, according to the Gregorian calendar or new style, in any given year or month of the 20th or 21st century.

**Precept I.** Find a year of the same number in the nineteenth century with that of the year proposed, and take out the mean time and anomalies, &c., of new moon for that year in Table I., omitting the secular equations.

2. Take so many years from Table II. having the sign + prefixed, as, when added to the above-mentioned year in the nineteenth century, will answer to the given year in which the new or full moon is required; and take out the time of new moon, with its anomalies, for these complete centuries.

3. Add these together, and to the sum apply the secular equations for the given year found in Table III., then work in all respects as above shown.

### Example V.

Required the Time of True New Moon in August 1999.

| Time of New Moon | Moon's Mean Anomaly | Sun's Mean Anomaly | Moon's Mean Distance from Ascending Node | |------------------|---------------------|--------------------|------------------------------------------| | 1899 | | | | | + 100 | | | | | 1999 | | | | | Secular equations| | | | | Seven lunations | July | | | | Equations | August | | | | Time of true new moon | Aug | | | **Equations**

| Argument I | 2 12 29 16 | |------------|-------------| | II | 7 7 1 21 | | Secular Variation | 44 | | III | 9 19 31 | | IV | 7 5 28 | | V | 10 2 18 | | VI | 0 7 23 | | VII | 0 2 0 | | VIII | 9 17 57 |

| Sum of equations | -11 45 33 | |------------------|-----------|

**Problem V.** To find the true anomalies of the sun and moon, and the moon's true distance from the ascending node at the true time of new or full moon.

With the sum of equations already found for reducing the time of mean new or full moon to that of the true, enter Table XV., and take therefrom the sun's mean motion from the moon's perigee, the change of the sun's mean anomaly, and the sun's mean motion from the ascending node, and apply these quantities to the mean anomalies and mean distance from the node at the time of mean new or full moon by addition or subtraction, according as the sum of the equations has the sign + or -. Then with the sun's corrected mean anomaly as argument, take from Table XVI. the equation of the sun's centre, correcting it for the secular variation as directed at the bottom of the table, and add or subtract the same to or from the corrected mean anomalies and mean distance from the node, and there will be obtained the true anomalies of the sun and moon, and the moon's true distance from the ascending node at the time of true new moon or full moon.

**Example VI.**

Required the True Anomalies of the Sun and Moon, and the Moon's True Distance from the Ascending Node at the time of True New Moon in 1836.

| The Sum of Equations is — 2h. 24m. 2s. | |----------------------------------------| | Sun's Mean Motion from Moon's Perigee. | | Sun's Mean Anomaly. | | Sun's Mean Motion from Ascending Node. |

| 2 Hours | 0 0 4 22 | |---------|----------| | 24 Minutes | 52 | | 2 Seconds | 0 |

Mean anomalies, &c.

| 4 28 6 57 | | 4 13 19 2 | | 0 3 36 53 |

Equation of sun's centre

| 4 28 1 43 | | 4 13 13 7 | | 0 3 30 39 |

Secular variation

| 1 22 57 | | 1 22 57 | | 1 22 57 |

Moon's True Anomaly

| 4 29 24 36 | | 4 14 36 0 | | 0 4 53 32 |

Sun's True Anomaly

Moon's True Distance from Ascending Node

At True New Moon.

**Elements for the Projection of Solar Eclipses.**

When at the time of true new moon the moon's true distance from the ascending node is between 11° 11' 50" and 0° 18' 10", or 5° 11' 50" and 6° 18' 10", there may at that time be an eclipse of the sun to some place on the earth's surface; but if it is beyond those limits there can be no eclipse. At the new moon in May 1836, the moon's distance from the node being within the limits, there may be an eclipse of the sun at that time.

It being ascertained that there may be an eclipse, the elements for predicting it are to be obtained as explained in the following example of the solar eclipse in May 1836, which is to be predicted, as it will happen at Edinburgh, in latitude 55° 57' N.

1. The moon's latitude at the true time of new moon. The moon's true distance from the ascending node at the time of true new moon is the argument of Table XVII.; the moon's true anomaly minus her true distance from the ascending node is the argument of Table XVIII.; this argument plus the moon's true anomaly is the argument of Table XIX.; the moon's true anomaly plus her true distance from the ascending node is the argument of Table XX.; the sun's true anomaly plus the moon's true distance from the ascending node is the argument of Table XXI.; and the sun's true anomaly minus the moon's true distance from the ascending node is the argument of Table XXII. With these arguments enter the respective tables, and take out the proper quantities, and add together such as have the sign +, and also such as have the sign —, and the difference of the two sums is the moon's latitude required, north if the greater sum has the sign +, south if it has —. Example.

The Moon's Latitude at the time of true New Moon in May 1836.

| Arg. Table XVII | s. d. m. s. | + | s. d. m. s. | |-----------------|-------------|---|-------------| | XVIII | 4 24 31 | | 0 25 47 | | XIX | 9 23 56 | | 7 | | XX | 5 4 18 | | 7 | | XXI | 4 19 30 | | 13 | | XXII | 4 9 42 | | 25 |

Moon's latitude north..............+ 0 26 2

II. The inclination of the moon's relative orbit to the ecliptic is found in Table XVII, with the moon's true distance from the ascending node for argument. In the present instance the inclination is 5° 43', left, signifying that the axis of the moon's orbit is to the left hand of the northern axis of the ecliptic.

III. The semidiameter of the earth's disk is equal to the difference of the moon's horizontal parallax corrected for the latitude of the given place, and the sun's horizontal parallax, which may always be assumed equal to nine seconds. The moon's equatorial horizontal parallax is found in Table XXIII, with her true anomaly for argument; and the correction to be subtracted therefrom is obtained in Table XXIV, with the latitude of the place and equatorial parallax as joint arguments. Thus, in the present instance, the moon's equatorial horizontal parallax is...54° 26' The correction to be subtracted..................8'

Leaving for reduced horizontal parallax............54° 18' which being again diminished by the sun's horizontal parallax, there remains 54° 9' for the semidiameter of the earth's disk.

IV. The sun's semidiameter is also obtained in Table XXII, with his mean anomaly as argument. We have, therefore, sun's semidiameter 15° 51'.

V. The moon's semidiameter is likewise obtained in Table XXII, with her true anomaly as argument. Hence, moon's semidiameter = 14° 50'.

VI. The moon's horary motion from the sun is equal to the difference of the sun's and moon's horary motions; both of which are found in Table XXII, with the sun's mean anomaly and the moon's true anomaly as arguments. Hence

Sun's horary motion...............................2° 25' Moon's horary motion.............................30° 9' Moon's horary motion from the sun................27° 44'

VII. The sun's true longitude at the true time of new moon is equal to the longitude of the sun's perigee added to his true anomaly. The longitude of the perigee is obtained by taking from Table XXV, the epoch of the first year of the century to which the given year belongs, and adding thereto the motion in Table XXVI, answering to the remaining number of years, months, and days, but subtracting therefrom a proportional part of the secular variation of the precession of the equinoxes found in Table XXV, opposite the first year of the given century, corresponding to the remaining number of years.

Example.

Longitude of Perigee.

| Year | s. d. m. s. | |------|-------------| | 1801 | 9 9 30 5 | | 30 years | 30 57 | | 5 do. | 5 9 | | May | 20 | | 15 days | 2 |

Longitude of perigee...9 10 6 33 Sun's true anomaly....4 14 96 0 Sun's true longitude...1 24 42 33

VIII. The sun's declination is found in Table XXVII, with the sun's true longitude for argument, and is to be corrected by the secular variation given in the same table, as directed at the bottom. The declination is north or south according as it bears the sign + or —, and, in the table, is given to tenths of a minute.

Example.

Mean time of true new moon at Edinburgh on May 15, 1836..................1 51 11

Sun's true longitude 1 24 43 Equation of time, +3 59 Secular variation......—10×35 = —3

Apparent time of true new moon........................................1 55 7

X. The reduced latitude of the given place is obtained by subtracting from the true latitude the correction in the last column of Table XXIV, answering to the latitude. In this particular case, Edinburgh being in north latitude..............55° 57' The correction is........................................10° 8 Reduced latitude........................................55° 46° 2 To Project an Eclipse of the Sun geometrically.

Take from a scale of any convenient length as many equal parts as the semidiameter of the earth's disk contains minutes of a degree, which for Edinburgh at the time of the eclipse in May 1836 is $5^\circ 10'$, or $54^\circ$. Then with this quantity as a radius describe the semicircle AHB upon the centre C (Plate XCI, fig. 136); which semicircle shall represent the northern half of the earth's enlightened disk as seen from the sun. If the given place were in south latitude, the southern half of the earth's disk must be represented.

Upon the centre C raise the straight line CH perpendicular to the diameter ACB; so ACB shall be a part of the ecliptic, and CH its axis.

Being provided with a sector, open it to the radius CA in the line of chords; and taking from thence the chord of the sun's greatest declination or the obliquity of the ecliptic for the given time (in the present instance $23^\circ 28'$) in your compasses, set it off both ways from H to g, and to h in the periphery of the semi-disk; and draw the straight line gVh, in which the north pole of the disk will be always found.

When the sun's longitude is between O' and VI', the north pole of the earth is enlightened by the sun; but whilst the sun is in the other six signs, the south pole is enlightened and the north pole is in the dark.

And when the sun's longitude is between IX' and III', the northern half of the earth's axis C XII. P lies to the right hand of the axis of the ecliptic as seen from the sun; and to the left hand while the sun is in the other six signs. It is evident that a contrary rule prevails with regard to the southern half of the earth's axis.

Open the sector till the radius of the sines be equal to the length of VB, and take the sine of the difference of the sun's longitude from III' or IX', whichever it is nearest (in the present instance $35^\circ 17'$), in your compasses from the line of the sines, and set off that distance from V to P in the line gVh, because the northern half of the earth's axis lies to the right hand of the axis of the ecliptic in this case; and draw the straight line C XII. P for the earth's axis, of which P is the north pole. If the earth's axis had lain to the left hand of the axis of the ecliptic, the distance VP would have been set off from V towards g.

To draw the parallel of latitude of the given place, Edinburgh, or the path of that place, on the earth's enlightened disk, as seen from the sun from sunrise till sunset, take the following method:

Find the sum and difference of the reduced latitude $55^\circ 46'$ and the sun's declination $18^\circ 58'$, which are $74^\circ 44'$ and $36^\circ 48'$. Take these arcs from the line of sines on the sector, CA being radius, and set them off from C to the two points, each marked XII. in the line of the earth's axis.

Bisect XII.—XII., and through the point K draw the line VI. K VI. perpendicular to the axis. Then, making CA or CB the radius of a line of sines on the sector, take the co-latitude of Edinburgh, $34^\circ 14'$, from the sines in your compasses, and set it both ways from K to VI. and VI. These hours will be just in the edge of the disk at the equinoxes; but at no other time in the whole year.

With the extent K VI. taken into your compasses, set one foot in K as a centre, and with the other foot describe the semicircle VI., 7, 8, 9, 10, &c., and divide it into 12 equal parts. Then from these points of division draw lines parallel to the earth's axis C XII. P.

With the extent K XII. as a radius, describe the quadrant arc XII. f, and divide it into six equal parts, as XII. a, ab, bc, cd, de, ef; and through the division points a, b, c, d, e, draw the lines VII. e V., VIII. d IV., IX. e III., X. b II., and XI. a I., all parallel to VI. K VI., and meeting the former lines in the points VII., VIII., IX., X., XI., V., IV., III., II., and I.; which points shall mark the several situations of Edinburgh on the earth's disk, at these hours respectively as seen from the sun; and the elliptic curve VI., VII., VIII., &c. being drawn through these points, shall represent the parallel of latitude, or path of Edinburgh, as seen from the sun from six in the morning to six in the afternoon. On continuing the lines VII. p, VIII. o, &c. IV. u, V. x, &c. on the other side of VI. K VI., and setting off p V. equal to VII. p, o IV. equal to VIII. o, &c., and continuing the elliptic curve through the points V. IV. &c., VII. VIII. &c., the path of Edinburgh, as seen from the sun before six in the morning and after six in the afternoon, will be had; but it is needless to draw the curve farther than the points where it meets the periphery of the earth's disk, which represent the times of sunrise and sunsetting.

N.B. If the sun's declination had been south, the diurnal path of Edinburgh would have been on the upper side of the line VI. K VI. If the latitude of the given place were south, in which case the southern half of the earth's disk would be represented, the diurnal path between six in the morning and six in the afternoon would be between the line VI. K VI. and the centre of the disk, when the sun's declination was south, and the contrary when north. It is requisite to divide the horary spaces into quarters as in the figure, and if possible into minutes also.

In the present case the northern half of the axis of the moon's relative orbit lies to the left hand of the axis of the ecliptic. Make CB the radius of a line of chords on the sector, and taking therefrom the chord of $5^\circ 43'$, the inclination of the moon's relative orbit to the ecliptic, set it off from H to M on the left hand of CH, the axis of the ecliptic; then draw CM for the axis of the moon's orbit, and take the moon's latitude $26^\circ 2'$ from the scale CA in your compasses, and set it from C to y in the line CH, and through y draw the straight line NyS at right angles to the axis of the moon's orbit CM for the path of the penumbra's centre over the earth's disk.

Take the moon's horary motion from the sun, $27^\circ 44'$, in your compasses, from the scale CA (every division of which is a minute of a degree), and with that extent make marks along the path of the penumbra's centre, and divide each space from mark to mark into 60 equal parts or horary minutes by dots; and set the hours to every 60th minute in such a manner that the dot signifying the instant of new moon by the tables may fall into the point Z, half-way between the axis of the moon's orbit and the axis of the ecliptic; and then the rest of the dots will show the points of the earth's disk, where the penumbra's centre is at the instants denoted by them in its transit over the earth.

Apply one side of a square to the line of the penumbra's path, and move the square backwards and forwards, until the other side of it cuts the same hour and minute (as at 3 hours and 21 minutes) both in the path of Edinburgh and in the path of the penumbra's centre; and the particular minute or instant so pointed out is the instant of the greatest obscuration of the sun, at the place for which the construction is made, namely, Edinburgh in the present example.

---

1 Although a sector is a convenient instrument in these projections, yet it is not absolutely necessary. The intelligent student will be able to lay off an arc of any number of degrees, also to make an angle of a given number of degrees, in various ways. Take the sun's semidiameter, $15^\circ 51'$, in your compasses, from the scale CA, and setting one foot on the path of Edinburgh, at the point answering to the instant of the greatest obscuration, namely, at $2\frac{1}{2}$ minutes past three, with the other foot describe the circle UY, which represents the sun's disk as seen from Edinburgh at the greatest obscuration. Then take the moon's semidiameter, $14^\circ 50'$, in your compasses, from the same scale, and setting one foot on the path of the penumbra's centre at the point $2\frac{1}{2}$ minutes past three, describe the circle TX for the moon's disk as seen from Edinburgh at the time when the eclipse is at the greatest, and the portion of the sun's disk which is hid or cut off by the moon's will show the quantity of the eclipse at that time; which quantity may be measured on a line equal to the sun's diameter, and divided into 12 equal parts for digits. As the moon's disk is entirely contained within the sun's, the eclipse as seen from Edinburgh will be annular.

Lastly, take the sum of the semidiameters of the sun and moon, $30^\circ 41'$, from the scale CA, in your compasses; and setting one foot in the line of the penumbra's centre path, on the left hand from the axis of the ecliptic, direct the other foot toward the path of Edinburgh, and carry that extent backwards and forwards till both the points of the compasses fall into the same instants in both the paths, and these instants will denote the time when the eclipse begins at Edinburgh. Then do the like on the right hand of the axis of the ecliptic; and where the points of the compasses fall into the same instants in both the paths, they will show at what time the eclipse ends at Edinburgh.

These trials give $3\frac{1}{2}$ minutes after one in the afternoon for the beginning of the eclipse at Edinburgh at the points N and O, $2\frac{1}{2}$ minutes after three for the time of greatest obscuration, and $2\frac{1}{2}$ minutes after four at R and S for the time when the eclipse ends, according to apparent time. To have the mean time, or that shown by well-regulated clocks and watches, apply the equation of time in the contrary manner to that used for converting the mean time of new moon into apparent time. Therefore, in the present instance, subtract 3 minutes 56 seconds, or 4 minutes approximately, from the apparent times, and we have

- Beginning of eclipse at Edinburgh, $13\frac{1}{2}$ p.m. - Greatest obscuration ........................................... $2\frac{1}{2}$ - End of eclipse .................................................. $4\frac{1}{2}$

all according to mean time.

The Projection of Lunar Eclipses.

When the moon's mean distance from either of her nodes at the time of mean full moon is less than $13^\circ 21'$ there may be an eclipse of the moon; but if greater, there cannot be an eclipse.

We find by Example II. that at the time of mean full moon in September 1830, the moon's mean distance from the descending node is only $1^\circ 20' 38''$, which being so much less than the limit, there will then be an eclipse.

By Problem V. find the true anomalies of the sun and moon, and the moon's true distance from the ascending node, at the true time of full moon.

| Sum of Equations, + 3h. 54m. 10s. | |----------------------------------| | Sun's Mean Motion from Moon's Perigee. | Sun's Mean Anomaly. | Sun's Mean Motion from Ascending Node. | |----------------------------------|-------------------|----------------------------------| | 3 Hours ........................................ | 0 0 6 33 | 0 0 7 23 | 0 0 7 47 | | 54 Minutes .................................... | 1 58 | 2 13 | 2 20 | | 10 Seconds ..................................... | 0 | 0 | 0 | | Mean anomalies, &c. ....................... | + 8 31 | + 9 36 | + 10 7 | | Equation of sun's centre ................. | 10 8 9 51 | 8 1 33 0 | 6 1 30 40 | | Secular variation ......................... | -1 40 30 | -1 40 30 | -1 40 30 | | Moon's True Anomaly, ..................... | + 4 | + 4 | + 4 | | Moon's True Distance from Ascending Node. | 10 6 29 25 | 7 29 52 34 | 5 29 50 14 |

At True Full Moon.

The elements for constructing an eclipse of the moon are eight in number, as follows:

1. The true time of full moon, and at that time; 2. the moon's horizontal parallax; 3. the sun's semidiameter; 4. the moon's; 5. the semidiameter of the earth's shadow at the moon; 6. the moon's latitude; 7. the angle of the moon's visible path with the ecliptic; 8. the moon's true hourly motion from the sun.

Therefore,

1. To find the true time of full moon. Work as already taught in the precepts. Thus we have the true time of full moon August 1830 (see Example II.) to be the 2d day at 10h. 36m. 47s. mean time at Greenwich.

2. To find the moon's horizontal parallax. Enter Table XXIII. with the moon's true anomaly $10^\circ 6^\circ 29' 25''$, and thereby take out her horizontal parallax; which, by making the requisite proportions, will be found to be $59' 52''$.

3. 4. To find the semidiameters of the sun and moon. Enter Table XXIII. with the sun's mean anomaly and moon's true anomaly ($8^\circ 1^\circ 33' 0''$ and $10^\circ 6^\circ 29' 25''$), and thereby take out their respective semidiameters, the sun's $15^\circ 54'$, and the moon's $16' 19''$.

5. To find the semidiameter of the earth's shadow at the moon. Add the sun's horizontal parallax, which is always $9'$, to the moon's, which in the present case is $59' 52''$; the sum is $60' 1''$, from which subtract the sun's semidiameter, $15' 54''$, and the remainder, $44' 7''$, being increased by $50''$ for the effect of the earth's atmosphere, we have $44' 57''$ for the semidiameter of the earth's shadow, which the moon then passes through. 6. To find the moon's latitude. Proceed as already directed under the prediction of solar eclipses. Thus,

| Argument, Table XVII. | 5 29 50 14 | + 52 | |-----------------------|------------|------| | XVIII. | 4 6 39 | | | XIX. | 2 13 9 | | | XX. | 4 6 20 | | | XXI. | 1 29 43 | | | XXII. | 2 0 2 | |

+ 2 16 — 10

Moon's latitude north + 2° 6'

7. To find the angle of the moon's visible path with the ecliptic. Enter Table XVII. with the moon's true distance from the ascending node for argument, and the angle is found to be 5° 44', left, signifying that the axis of the moon's orbit is to the left hand of the northern axis of the ecliptic.

8. To find the moon's true horary motion from the sun. With the true anomaly of the moon and the mean anomaly of the sun, take out their horary motions from Table XXIII., and the sun's horary motion subtracted from the moon's leaves remaining the moon's true horary motion from the sun; in the present case 33° 58'.

These elements being found for the construction of the moon's eclipse in September 1830, proceed as follows:

Draw the line ACB for part of the ecliptic, and CD perpendicular thereto for the northern part of its axis, the moon having north latitude. (Plate XCI. fig. 137.)

Add the semidiameters of the moon's and earth's shadow together, which in the eclipse make 61' 16"; and take this in your compasses, from a scale of equal parts, and setting one foot on the point C as a centre, with the other foot describe the arch ADB, in one point of which the moon's centre will be at the beginning of the eclipse, and in another at the end thereof.

Take the semidiameter of the earth's shadow, 44' 57", in your compasses, from the scale, and setting one foot in the centre C, with the other foot describe the semicircle KLM for the northern half of the earth's shadow; because the moon's latitude is north in this eclipse.

Subtract the semidiameter of the moon from the semidiameter of the earth's shadow, and the remainder is 28' 38", which take in your compasses from the scale, and setting one foot on the point C as a centre, with the other foot describe the arch OPQ; in one point of which the moon's centre will be at the beginning of total darkness, and in another at the end thereof.

Draw the line CE on the left hand of the northern axis of the ecliptic, and making an angle of 5° 44' therewith, which line represents the northern part of the axis of the moon's orbit, the moon's latitude being north.

Take the moon's latitude, 2° 6', from the scale with your compasses, and set it from C to G in the axis of the ecliptic, and through the point G draw the straight line RSGTU, at right angles to the axis of the moon's orbit, for the path of the moon's centre. Then F, in the line CE, is the point in the earth's shadow where the moon's centre is at the middle of the eclipse; G, the point where her centre is at the instant of her ecliptical conjunction; and the middle between them, the point where her centre is at the time of true full moon by the tables.

Take the moon's horary motion from the sun, 33° 58', in your compasses, from the scale, and with that extent make marks along the line of the moon's path; then divide each space from mark to mark into 60 equal parts or horary minutes, and set the hours to the proper dots, in such a manner, that the dot signifying the instant of full moon (36 minutes and 47 seconds after ten) may be midway between the points F and G.

The point U, where the moon's path intersects on the right hand the arch described with the sum of the semidiameters of the moon and earth's shadow, denotes the instant when the eclipse begins, namely, at 52 minutes after eight; the point T, where the moon's path intersects on the right hand the arch described, with the difference of the semidiameters, denotes the instant when the moon's total darkness begins, namely, at 48 minutes after nine; the point F denotes the middle of the eclipse at 39 minutes after ten; the point S, where the moon's path intersects on the left hand the arch described, with the difference of the semidiameters, denotes the end of the moon's total darkness at 30 minutes after eleven; and the point R, where the moon's path intersects the arch described, with the sum of the semidiameters, denotes the end of the eclipse at 26 minutes after 12, all mean time according to the meridian of Greenwich. If the times reckoned by any other meridian are required, apply the longitude from Greenwich to the Greenwich times, by addition or subtraction, according as the place is east or west of Greenwich.

On F as a centre, with a radius equal to the moon's semidiameter, describe a circle which represents the moon's disk at the middle of the eclipse.

The line VX denotes the quantity eclipsed at the middle of the eclipse, which may be measured on a line equal to the moon's diameter, and divided into equal parts for digits. In the present case, the eclipse being total, the quantity eclipsed is said to be greater than the moon's diameter, and is found to be 21½ digits. ### Chapter III.

**Catalogue of Fixed Stars, and Table of Refractions.**

#### Sect. I.—Right Ascensions and North Polar Distances of 652 Stars, extracted from the Greenwich Twelve-Year Catalogue of 2156 Stars, and reduced to the epoch 1850, January 1; with the Annual Variations, including the Geometrical Precessions given in the British Association Catalogue, and the Proper Motions given in Mr. Mad's Paper in vol. xix. of the Memoirs of the Royal Astronomical Society.

| No. | Star's Name | Mag. | Mean R. A. 1850, Jan. 1 | Annual Variation in R. A. | Mean N. P. D. 1850, Jan. 1 | Annual Variation in N. P. D. | |-----|-------------|------|-------------------------|--------------------------|---------------------------|-----------------------------| | 1 | Andromeda, α | 2-3 | 0 3 38 57 | + 3 08 2 | 4 64 16 31 | -19 01 | | 2 | Cassiopeia, β | 2-3 | 0 1 12 32 | + 3 14 5 | 3 40 49 17 | -19 87 | | 3 | Ceti, α | 4-5 | 0 11 46 65 | + 3 05 9 | 5 39 22 67 | -19 97 | | 4 | Cassiopeia, γ | 4-5 | 0 24 39 69 | + 3 33 0 | 2 53 48 17 | -19 95 | | 5 | Cassiopeia, δ | 4-5 | 0 28 38 43 | + 3 29 5 | 5 55 46 16 | -19 89 | | 6 | Andromeda, ε | 4 | 0 30 38 27 | + 3 13 9 | 6 30 13 14 | -19 64 | | 7 | Cassiopeia, ε | 4 | 0 32 15 57 | + 3 35 4 | 1 17 10 47 | -19 82 | | 8 | Ceti, ζ | 4 | 0 36 34 42 | + 3 01 2 | 10 48 38 93 | -19 83 | | 9 | Cassiopeia, η | 4 | 0 39 12 15 | + 3 11 6 | 7 50 42 28 | -19 76 | | 10 | Andromeda, θ | 4 | 0 39 23 77 | + 3 16 3 | 6 32 59 82 | -19 65 | | 11 | Cassiopeia, ι | 4 | 0 40 54 20 | + 3 56 1 | 5 38 48 61 | -19 26 | | 12 | Pisces, α | 4 | 0 40 54 20 | + 3 56 1 | 5 38 48 61 | -19 26 | | 13 | Andromeda, ρ | 4 | 0 41 33 34 | + 3 27 3 | 4 44 24 34 | -19 72 | | 14 | Cassiopeia, ς | 4 | 0 47 41 79 | + 3 53 9 | 3 0 47 78 | -19 64 | | 15 | Andromeda, σ | 4 | 0 48 20 26 | + 3 29 2 | 5 18 54 19 | -19 66 | | 16 | Ursus Majoris | 4 | 0 49 9 48 | + 3 43 2 | 3 24 4 | -19 59 | | 17 | Sculptoris, α | 5-4 | 0 51 22 61 | + 2 89 9 | 12 10 8 62 | -19 52 | | 18 | Pisces, β | 4 | 0 55 9 68 | + 3 10 8 | 8 52 5 73 | -19 48 | | 19 | Andromeda, τ | 4-5 | 1 0 49 09 | + 4 33 3 | 3 33 34 38 | -19 34 | | 20 | Cassiopeia, υ | 4-5 | 1 1 20 96 | + 4 34 2 | 5 10 33 72 | -19 25 | | 21 | Pisces, γ | 4-5 | 1 1 55 55 | + 3 58 2 | 3 38 58 53 | -19 31 | | 22 | Pisces, δ | 4-5 | 1 5 53 5 | + 1 57 62 | 1 29 24 55 | -19 25 | | 23 | Pisces, ε | 4-5 | 1 5 53 5 | + 1 57 62 | 1 29 24 55 | -19 25 | | 24 | Cassiopeia, χ | 4-5 | 1 10 41 47 | + 3 70 4 | 1 3 9 0 0 | -19 16 | | 25 | Cassiopeia, ψ | 4-5 | 1 15 24 12 | + 4 11 1 | 3 29 17 23 | -19 00 | | 26 | Cassiopeia, ξ | 4-5 | 1 16 28 7 | + 3 50 1 | 3 29 47 50 | -18 91 | | 27 | Cassiopeia, ο | 4-5 | 1 16 31 58 | + 2 99 8 | 5 27 32 58 | -18 73 | | 28 | Ceti, η | 4-5 | 1 18 42 27 | + 3 54 0 | 4 22 10 70 | -18 77 | | 29 | Pisces, η | 4-5 | 1 22 19 71 | + 3 13 4 | 8 47 53 77 | -18 60 | | 30 | Pisces, ζ | 4-5 | 1 23 27 76 | + 3 75 4 | 7 25 45 41 | -18 74 | | 31 | Andromeda, η | 4-5 | 1 28 0 80 | + 4 38 5 | 4 29 48 42 | -18 20 | | 32 | Andromeda, θ | 4-5 | 1 28 48 67 | + 3 63 5 | 4 2 8 0 5 | -18 16 | | 33 | Pisces, η | 4-5 | 1 33 37 68 | + 3 11 1 | 6 15 26 28 | -18 36 | | 34 | Andromeda, ι | 4-5 | 1 41 17 23 | + 3 70 5 | 4 4 6 1 | -18 33 | | 35 | Pisces, η | 4-5 | 1 42 18 11 | + 2 03 7 | 10 25 50 03 | -17 97 | | 36 | Ceti, ξ | 5-4 | 1 43 32 68 | + 4 24 0 | 1 27 25 1 | -18 02 | | 37 | Cassiopeia, η | 5-4 | 1 44 23 83 | + 4 22 3 | 2 3 17 77 | -17 98 | | 38 | Cassiopeia, ζ | 5-4 | 1 44 23 83 | + 4 22 3 | 2 3 17 77 | -17 98 | | 39 | Trianguli, α | 4-3 | 1 44 32 54 | + 3 39 5 | 6 1 9 0 7 | -17 78 | | 40 | Pisces, η | 4-5 | 1 45 47 52 | + 3 29 4 | 8 37 33 98 | -17 83 | | 41 | Arietis, α | 3-2 | 1 46 21 73 | + 3 29 1 | 6 55 38 45 | -17 82 | | 42 | Cassiopeia, η | 5-4 | 1 49 43 95 | + 4 77 6 | 19 49 26 45 | -17 80 | | 43 | Cassiopeia, ζ | 5-4 | 1 50 44 28 | + 4 59 0 | 18 18 28 81 | -17 78 | | 44 | Cassiopeia, η | 5-4 | 1 54 42 62 | + 4 64 2 | 28 23 36 62 | -17 49 | | 45 | Trianguli, β | 4-5 | 2 0 41 7 | + 4 41 1 | 6 43 33 35 | -17 29 | | 46 | Trianguli, γ | 4-5 | 2 0 41 7 | + 4 41 1 | 6 43 33 35 | -17 29 | | 47 | Ceti, η | 4-5 | 2 5 3 17 | + 4 18 1 | 8 11 4 0 | -17 10 | | 48 | Andromeda, η | 5 | 2 15 38 72 | + 3 94 8 | 40 24 12 93 | -16 67 | | 49 | Bradley 332 | 4 | 2 16 49 70 | + 4 30 8 | 23 16 34 30 | -16 67 | | 50 | Ceti, η | 4 | 2 20 11 32 | + 3 17 7 | 8 12 54 06 | -16 40 | | 51 | Ceti, η | 5 | 2 28 0 41 | + 3 13 2 | 8 5 50 94 | -15 99 | | 52 | Perseus, α | 5 | 2 32 47 98 | + 3 75 2 | 50 26 37 1 | -15 58 | | 53 | Perseus, η | 4 | 2 33 58 79 | + 4 04 7 | 4 24 35 47 | -15 56 | | 54 | Perseus, η | 4 | 2 35 31 22 | + 3 06 8 | 8 23 58 02 | -15 42 | | 55 | Arietis, α | 5 | 2 36 47 61 | + 2 55 8 | 7 11 19 13 | -15 44 | | 56 | Arietis, α | 5 | 2 36 50 36 | + 3 22 9 | 8 31 20 65 | -15 47 | | 57 | Arietis, α | 5 | 2 36 50 36 | + 3 22 9 | 8 31 20 65 | -15 47 | | 58 | Arietis, α | 5 | 2 36 50 36 | + 3 22 9 | 8 31 20 65 | -15 47 | | 59 | Arietis, α | 5 | 2 36 50 36 | + 3 22 9 | 8 31 20 65 | -15 47 | | 60 | Arietis, α | 5 | 2 36 50 36 | + 3 22 9 | 8 31 20 65 | -15 47 | | 61 | Arietis, α | 5 | 2 36 50 36 | + 3 22 9 | 8 31 20 65 | -15 47 | | 62 | Arietis, α | 5 | 2 36 50 36 | + 3 22 9 | 8 31 20 65 | -15 47 |

No. 63, Perseus. The proper motion of this star in R. A., given in the British Association Catalogue, is + 0°129; this probably results from some error, and is not included in the annual variation. | No. | Star's Name | Mag. | Mean R.A. 1850, Jan. L. | Annual Variation in R.A. | |-----|-------------|------|------------------------|-------------------------| | | | | | | | 125 | Auriga (α) | 1 | 5 36 06 | +4 417 | | | | | | | | 126 | Leporis (μ) | 3 1/4| 6 11 36 | +2 026 | | | | | | | | 127 | Orionis (β) | 5 | 7 19 31 | +2 878 | | | | | | | | 128 | Auriga (γ) | 5 | 9 35 39 | +2 059 | | | | | | | | 129 | Orionis (δ) | 5 | 14 6 25 | +3 059 | | | | | | | | 130 | Auriga (δ) | 5 | 16 48 79 | +3 788 | | | | | | | | 131 | Orionis (ε) | 5 | 17 5 29 | +2 816 | | | | | | | | 132 | Auriga (ε) | 5 | 22 57 50 | +3 901 | | | | | | | | 133 | Orionis (η) | 5 | 24 20 70 | +3 002 | | | | | | | | 134 | Leporis (η) | 5 | 25 6 96 | +2 644 | | | | | | | | 135 | Orionis (η) | 5 | 27 54 38 | +2 943 | | | | | | | | 136 | Orionis (η) | 5 | 28 36 22 | +3 039 | | | | | | | | 137 | Orionis (η) | 5 | 28 40 14 | +2 322 | | | | | | | | 138 | Tauri (ζ) | 3 1/4| 28 40 14 | +2 327 | | | | | | | | 139 | Orionis (η) | 5 | 31 1 30 | +2 899 | | | | | | | | 140 | Orionis (η) | 5 | 33 11 47 | +2 026 | | | | | | | | 141 | Columba (α) | 2 | 34 13 14 | +2 177 | | | | | | | | 142 | Auriga (ζ) | 5 | 33 12 59 | +2 408 | | | | | | | | 143 | Tauri (ξ) | 5 | 43 53 99 | +3 766 | | | | | | | | 144 | Orionis (η) | 5 | 45 20 01 | +2 547 | | | | | | | | 145 | Columba (β) | 5 | 45 40 02 | +2 109 | | | | | | | | 146 | Orionis (η) | 5 | 47 3 12 | +2 324 | | | | | | | | 147 | Auriga (γ) | 4 1/2| 47 10 41 | +4 930 | | | | | | | | 148 | Auriga (δ) | 5 | 48 31 46 | +4 404 | | | | | | | | 149 | Auriga (ε) | 5 | 49 29 57 | +4 088 | | | | | | | | 150 | Leporis (η) | 4 1/2| 49 34 46 | +2 741 | | | | | | | | 151 | Columba (γ) | 5 | 52 1 27 | +2 127 | | | | | | | | 152 | Geminorum (α) | 5 | 53 13 53 | +3 631 | | | | | | | | 153 | Orionis (η) | 5 | 55 0 10 | +3 643 | | | | | | | | 154 | Orionis (η) | 5 | 55 0 57 | +3 561 | | | | | | | | 155 | Orionis (η) | 5 | 59 0 47 | +3 424 | | | | | | | | 156 | Piazzi (v) | 5 | 6 2 18 53 | +6 023 | | | | | | | | 157 | Auriga (γ) | 5 | 6 49 09 | +3 823 | | | | | | | | 158 | Geminorum (γ) | 5 | 6 49 43 | +3 619 | | | | | | | | 159 | Leporis (η) | 5 | 6 23 18 | +3 503 | | | | | | | | 160 | Geminorum (α) | 5 | 6 13 20 | +4 025 | | | | | | | | 161 | Geminorum (α) | 5 | 6 16 57 | +2 041 | | | | | | | | 162 | Geminorum (α) | 5 | 6 30 31 | +3 503 | | | | | | | | 163 | Geminorum (α) | 5 | 6 32 24 | +2 292 | | | | | | | | 164 | Geminorum (α) | 5 | 6 35 26 | +3 455 | | | | | | | | 165 | Geminorum (α) | 5 | 6 32 65 | +3 473 | | | | | | | | 166 | Geminorum (α) | 5 | 6 34 42 | +3 695 | | | | | | | | 167 | Geminorum (α) | 5 | 6 36 52 | +3 370 | | | | | | | | 168 | Camelopardalis (α) | 5 | 6 37 30 | +6 622 | | | | | | | | 169 | Canis Major (α) | 1 | 6 38 32 | +2 646 | | | | | | | | 170 | Auriga (γ) | 5 | 6 40 9 28 | +4 249 | | | | | | | | 171 | Geminorum (α) | 5 | 6 42 58 | +3 559 | | | | | | | | 172 | Canis Major (α) | 4 | 6 44 14 | +2 238 | | | | | | | | 173 | Lynx (α) | 4 | 6 44 15 | +5 225 | | | | | | | | 174 | Canis Major (α) | 2 | 6 52 43 | +3 562 | | | | | | | | 175 | Geminorum (α) | 4 | 6 55 39 | +3 502 | | | | | | | | 176 | Auriga (γ) | 5 | 6 59 11 | +3 146 | | | | | | | | 177 | Auriga (γ) | 5 | 1 15 57 | +4 141 | | | | | | | | 178 | Geminorum (α) | 5 | 1 35 15 | +3 826 | | | | | | | | 179 | Canis Major (α) | 2 | 7 2 17 59 | +2 438 | | | | | | | | 180 | Geminorum (α) | 4 | 7 29 20 | +3 454 | | | | | | | | 181 | Geminorum (α) | 3 | 7 11 9 66 | +3 592 | | | | | | | | 182 | Geminorum (α) | 5 | 7 16 24 | +3 737 | | | | | | | | 183 | Canis Major (α) | 3 | 7 18 9 70 | +2 368 | | | | | | | | 184 | Geminorum (α) | 5 | 7 19 27 | +3 806 | | | | | | | | 185 | Geminorum (α) | 5 | 7 23 12 | +3 843 | | | | | | | | 186 | Geminorum (α) | 5 | 7 26 40 | +3 708 | | | | | | | | 187 | Lynx (α) | 5 | 7 31 27 | +3 444 | | | | | | | | 188 | Canis Minor (α) | 1 | 7 31 27 | +3 444 | | | | | | | | 189 | Geminorum (α) | 4 | 7 35 23 | +3 626 | | | | | | | | 190 | Piazzi (v) | 1 | 7 36 7 2 | +3 681 | | | | | | | | 191 | Auriga (γ) | 4 | 7 37 45 | +2 405 | | | | | | | | 192 | Navis (α) | 4 | 7 42 59 | +2 523 | | | | | | | | 193 | Geminorum (α) | 5 | 7 44 18 | +3 682 | | | | | | | | 194 | Navis (α) | 5 | 7 50 24 | +2 576 | | | | | | | | 195 | Canis Major (α) | 5 | 7 54 17 | +3 695 | | | | | | | | 196 | Lynx (α) | 5 | 7 57 9 4 | +4 551 | | | | | | | | 197 | Navis (α) | 5 | 8 1 9 41 | +2 553 | | | | | | |

No. 220, Piazzi, lx. 37. The proper motion in R.A., included in the annual variation, is -0°-063; this is very questionable. ### Astronomy

#### Right Ascensions and North Polar Distances of 652 Stars

| No. | Star's Name | Mag | Mean N.P.D., Jan. 1 | Annual Variation in N.P.D. | |-----|------------------------------|-----|----------------------|----------------------------| | | | | | | | | | | | |

This table provides data on the right ascensions and north polar distances of 652 stars, listed by their names and magnitudes, along with their mean positions and annual variations.

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**Note:** The data includes stars such as **Sirius**, **Vega**, **Deneb**, **Altair**, and **Betelgeuse**, among others, indicating their relative positions and magnitudes in the night sky. ### Table: Right Ascensions and North Polar Distances of 652 Stars

| No. | Star's Name | Mag. | Mean R.A. 1859, Jan. L. | Annual Variation in R.A. | Mean N.P.D. 1859, Jan. L. | Annual Variation in N.P.D. | |-----|-------------|------|------------------------|--------------------------|---------------------------|---------------------------| | | | | | | | |

- **No. 468**, Groomebridge 2555. The annual variations do not include proper motions. - **No. 470**, Bradley 2313. The annual variation in R.A. does not include proper motion. - **No. 557**, Bradley 2727. The annual variation in R.A. does not include proper motion.

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**Notes:** - The table lists stars with their right ascensions (R.A.) and north polar distances (N.P.D.) from the year 1859, along with the annual variations in these values. - Proper motions are not included in the annual variations provided. | No. | Star's Name | Mag | Mean R.A. 1850, Jan. L. | Annual Variation in R.A. | Mean N.P.D. 1850, Jan. L. | Annual Variation in N.P.D. | |-----|-------------|-----|------------------------|-------------------------|--------------------------|---------------------------| | 567 | Capricorni, ζ | 4 | 21 18 5.57 | +3 438 113 3 24.46 | -15 23 | 22 32 20.99 | +3 333 117 49 29.04 | | 568 | Aquarii, γ | 3 | 21 23 39.59 | +3 162 96 13 41.74 | -15 57 | 22 33 58.88 | +2 865 79 57 0.63 | | 569 | Cygni, δ | 5 | 21 23 54.76 | +2 201 44 7 9.69 | -15 47 | 22 34 43.24 | +2 807 61 28 25.93 | | 570 | Cephei, β | 3 | 21 26 42.34 | +6 895 20 5 49.54 | -15 69 | 22 35 58.65 | +2 800 60 33 43.10 | | 571 | Cygni, ε | 4.5 | 21 28 20.59 | +2 248 45 4 19.18 | -15 73 | 22 41 38.64 | +3 182 104 22 58.54 | | 572 | Capricorni, λ | 5.4 | 21 28 40.34 | +3 359 110 8 6.69 | -15 84 | 22 42 40.09 | +2 885 66 11 21.44 | | 573 | Aquarii, ξ | 4 | 21 29 45.89 | +3 197 98 31 27.87 | -15 86 | 22 44 21.26 | +2 112 24 35 15.18 | | 574 | Capricorni, γ | 4.5 | 21 31 46.89 | +3 336 107 20 13.82 | -15 98 | 22 44 47.02 | +3 128 98 22 35.79 | | 575 | Capricorni, α | 5 | 21 31 16.16 | +3 058 100 32 49.84 | -16 13 | 22 46 48.80 | +3 188 106 37 1.68 | | 576 | Piscis Austr., ι | 5 | 21 36 0.69 | +3 595 123 42 25.82 | -16 12 | 22 47 55.55 | -0 006 7 38 32.07 | | 577 | Capricorni, μ | 5 | 21 36 46.17 | +2 117 98 31 27.87 | -16 12 | 22 49 21.07 | +3 331 120 24 57.91 | | 578 | Pegasi, α | 2.3 | 21 36 49.10 | +2 947 80 48 37.09 | -16 27 | 22 55 1.78 | +2 742 48 28 44.70 | | 579 | Cygni, α | 4.5 | 21 37 26.16 | +2 688 61 55 55.86 | -16 04 | 22 56 14.66 | +3 052 86 50 12.12 | | 580 | Cygni, β | 4.5 | 21 37 29.57 | +2 681 61 56 0.84 | -16 04 | 22 56 30.52 | +2 896 62 43 46.87 | | 581 | Capricorni, η | 5 | 21 38 45.22 | +3 318 106 48 18.83 | -16 09 | 22 57 17.31 | +2 281 75 36 3.27 | | 582 | Cephei, δ | 5 | 21 39 42.90 | +6 914 19 22 42.77 | -16 19 | 22 57 19.17 | +2 257 74 4 38.33 | | 583 | Cephei, ε | 5 | 21 41 7.52 | +1 729 29 34 12.21 | -16 45 | 22 57 19.17 | +2 257 74 4 38.33 | | 584 | Cygni, ε | 4.5 | 21 41 15.37 | +2 207 41 22 58.42 | -16 47 | 22 58 1.78 | +2 742 48 28 44.70 | | 585 | Groomb, 3599 | 5 | 21 44 21.69 | +1 098 30 32 17.12 | -16 65 | 22 58 1.78 | +2 742 48 28 44.70 | | 586 | Capricorni, μ | 5 | 21 40 16.05 | +3 258 105 15 18.52 | -16 70 | 22 58 1.78 | +2 742 48 28 44.70 | | 587 | Aquarii, α | 3 | 21 58 19.69 | +3 246 104 41 29.41 | -17 20 | 22 59 1.78 | +2 742 48 28 44.70 | | 588 | Aquarii, β | 3 | 21 59 16.04 | +1 738 29 6 9.28 | -17 24 | 22 59 1.78 | +2 742 48 28 44.70 | | 589 | Cephei, δ | 5.4 | 21 59 20.94 | +1 701 26 6 6.00 | -17 34 | 22 59 1.78 | +2 742 48 28 44.70 | | 590 | Pegasi, α | 5 | 22 23 35.10 | +2 655 57 33 31.10 | -17 43 | 22 59 1.78 | +2 742 48 28 44.70 | | 591 | Pegasi, β | 5 | 22 23 37.93 | +3 029 84 32 17.34 | -17 52 | 22 59 1.78 | +2 742 48 28 44.70 | | 592 | Pegasi, γ | 4 | 22 3 10.75 | +2 655 57 33 22.10 | -17 51 | 22 59 1.78 | +2 742 48 28 44.70 | | 593 | Pegasi, δ | 4 | 22 3 17.47 | +2 346 38 31 16.87 | -17 58 | 22 59 1.78 | +2 742 48 28 44.70 | | 594 | Cephei, η | 4.5 | 22 5 39.53 | +2 071 32 32 18.08 | -17 59 | 22 59 1.78 | +2 742 48 28 44.70 | | 595 | Aquarii, ξ | 4.5 | 22 8 54.77 | +3 170 98 31 42.32 | -17 72 | 22 59 1.78 | +2 742 48 28 44.70 | | 596 | Cephei, ε | 5.4 | 22 9 31.12 | +2 196 33 42 11.28 | -17 79 | 22 59 1.78 | +2 742 48 28 44.70 | | 597 | Aquarii, γ | 4 | 22 13 54.42 | +3 100 98 8 28.37 | -17 97 | 22 59 1.78 | +2 742 48 28 44.70 | | 598 | Pegasi, α | 5 | 22 14 24.94 | +2 759 62 25 24.31 | -17 97 | 22 59 1.78 | +2 742 48 28 44.70 | | 599 | Lacertae | 5 | 22 16 50.22 | +2 040 33 13 4.06 | -17 99 | 22 59 1.78 | +2 742 48 28 44.70 | | 600 | Aquarii, ε | 5 | 22 17 36.97 | +3 064 80 39 55.63 | -18 08 | 22 59 1.78 | +2 742 48 28 44.70 | | 601 | Lacertae | 5 | 22 21 21.65 | +3 087 90 47 8.23 | -18 25 | 22 59 1.78 | +2 742 48 28 44.70 | | 602 | Aquarii, η | 4 | 22 21 17.37 | +2 346 38 31 16.87 | -17 88 | 22 59 1.78 | +2 742 48 28 44.70 | | 603 | Aquarii, θ | 5.4 | 22 22 42.11 | +3 173 101 26 37.21 | -18 33 | 22 59 1.78 | +2 742 48 28 44.70 | | 604 | Piscis Austr., ι | 4 | 22 22 57.89 | +3 431 123 6 51.25 | -18 21 | 22 59 1.78 | +2 742 48 28 44.70 | | 605 | Cephei, Φ | var. | 22 23 38.64 | +2 211 32 21 54.4 | -19 29 | 22 59 1.78 | +2 742 48 28 44.70 | | 606 | Lacertae | 4 | 22 25 7.35 | +2 456 40 29 14.37 | -18 36 | 22 59 1.78 | +2 742 48 28 44.70 | | 607 | Aquarii, η | 4.3 | 22 27 38.70 | +3 082 90 33 20.61 | -18 39 | 22 59 1.78 | +2 742 48 28 44.70 | | 608 | Aquarii, θ | 5 | 22 29 59.02 | +3 108 90 3 0.84 | -18 42 | 22 59 1.78 | +2 742 48 28 44.70 | | 609 | Lacertae | 5 | 22 31 13.20 | +2 432 39 13 41.39 | -18 46 | 22 59 1.78 | +2 742 48 28 44.70 | | 610 | Cephei, η | 5 | 22 32 3.92 | +1 490 17 8 4.94 | -18 64 | 22 59 1.78 | +2 742 48 28 44.70 |

No. 590, Φ Cephei. The annual variations do not include proper motion.

No. 649, Bradley, 3195. The annual variation in R.A. does not include proper motion. ### Sect. II.—Table of Atmospheric Refractions, with Corrections for the Height of the Barometer and Thermometer.

| App. Altitude | Refr. B. 30° | Diff. for Th. 50° | Diff. for +1 Alt. | Diff. for +1 B. | |---------------|--------------|------------------|-----------------|----------------| | 0 | 33 51 | 11 7 | 8 4 | 4 0 | | 5 | 32 11 | 11 3 | 7 6 | 10 | | 10 | 31 58 | 10 9 | 7 3 | 20 | | 15 | 31 5 | 10 5 | 7 0 | 30 | | 20 | 30 13 | 10 1 | 6 7 | 40 | | 25 | 29 24 | 9 7 | 6 4 | 50 | | 30 | 28 37 | 9 4 | 6 1 | 5 0 | | 35 | 27 51 | 9 0 | 5 9 | 10 | | 40 | 27 6 | 8 7 | 5 6 | 20 | | 45 | 26 24 | 8 4 | 5 4 | 30 | | 50 | 25 43 | 8 0 | 5 1 | 40 | | 55 | 25 3 | 7 7 | 5 0 | 50 | | 1 | 24 25 | 7 4 | 4 7 | 6 0 | | 5 | 23 48 | 7 1 | 4 6 | 10 | | 10 | 23 13 | 6 9 | 4 5 | 20 | | 15 | 22 40 | 6 6 | 4 4 | 30 | | 20 | 22 8 | 6 3 | 4 2 | 40 | | 25 | 21 37 | 6 1 | 4 0 | 50 | | 30 | 21 7 | 5 9 | 3 9 | 7 0 | | 35 | 20 38 | 5 7 | 3 6 | 10 | | 40 | 20 10 | 5 5 | 3 6 | 20 | | 45 | 19 43 | 5 3 | 3 5 | 30 | | 50 | 19 17 | 5 1 | 3 4 | 40 | | 55 | 18 52 | 4 9 | 3 3 | 50 | | 2 | 18 29 | 4 6 | 3 2 | 5 0 | | 5 | 18 5 | 4 6 | 3 1 | 10 | | 10 | 17 43 | 4 4 | 3 0 | 20 | | 15 | 17 21 | 4 3 | 2 9 | 30 | | 20 | 17 0 | 4 1 | 2 8 | 40 | | 25 | 16 40 | 4 0 | 2 6 | 50 | | 30 | 16 21 | 3 9 | 2 7 | 5 0 | | 35 | 15 62 | 3 7 | 2 7 | 10 | | 40 | 15 43 | 3 6 | 2 6 | 20 | | 45 | 15 23 | 3 5 | 2 5 | 30 | | 50 | 15 0 | 3 4 | 2 4 | 40 | | 55 | 14 51 | 3 3 | 2 3 | 50 | | 3 | 14 33 | 3 2 | 2 3 | 10 | | 5 | 14 19 | 3 1 | 2 2 | 20 | | 10 | 14 4 | 3 0 | 2 1 | 30 | | 15 | 13 59 | 2 9 | 2 0 | 40 | | 20 | 13 33 | 2 8 | 1 9 | 50 | | 25 | 13 21 | 2 7 | 1 8 | 60 | | 30 | 13 5 | 2 7 | 1 7 | 70 | | 35 | 12 53 | 2 6 | 1 6 | 80 | | 40 | 12 41 | 2 5 | 1 5 | 90 | | 45 | 12 23 | 2 4 | 1 4 | 100 | | 50 | 12 16 | 2 4 | 1 3 | 110 | | 55 | 12 3 | 2 3 | 1 2 | 120 |

**Explanation of the Table of Refractions.**

This table is computed upon principles explained by the late Dr Young in the Philosophical Transactions for 1819; and it appears to agree more perfectly with the latest observations than any other table before published.

The apparent altitude being found in the first column, the second shows the refraction when the barometer stands at 30 inches, which is its mean height above the level of the sea, and the thermometer at 60° of Fahrenheit. The third column contains the difference to be subtracted or added for every minute of altitude, reckoned from the nearest number in the first column. The fourth shows the number of seconds to be added for every inch that the height of the barometer exceeds 30, or to be subtracted for each inch that it wants of 30; and the last contains the number of seconds to be subtracted for each degree that the thermometer stands above 50°, or to be added for each degree that its height wants of 50°.

If great accuracy be required, we must also deduct from the observed height of the barometer 0.003 inch for each degree that the thermometer near it is above 50°, and add an equal quantity for an equal depression. In fact, however, the table is so nearly accurate as to make the temperature correction estimated from the height of the thermometer within; and if we employed the height of the thermometer without, which would be more consistent with the theory, it would probably be necessary to suppose the standard temperature of the table 48° only (or rather 47°), instead of 50°.

**Examples.**

1. At 7° 10' 13", barometer 29-87, thermometer 66°, the refraction is 6° 52' 26", from twenty-two observations of Bradley.

2. At 19° 18' 19", barometer 30-045, thermometer 34°, the refraction is 3° 51' 5", from three observations of Bradley.

3. At 13° 43', barometer 29-85, thermometer 45°, the refraction is 3° 55' 65", from 156 observations of Mr Pond.

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1. Alt. 7° 20' R. 7° 8" Diff. Alt. -9° B. 14° 3 Th. -93 + 1° 62 1° 47° = 1° 3 -13 -16

2. Alt. 19° R. 2° 47° 7" Diff. Alt. -16 B. 5° 61 Th. -34 + 2° 93 1° 19° = 1° 3 + 645 + 16

3. Alt. 13° 40' R. 3° 55' 5" Diff. Alt. -29 B. 7° 39 Th. -42 + 3° 55' 66 - 67 -11 -18 + 2 41

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**Error.**

- 0° 01 2° 05 + 3° 5 There are two principal objects to be accomplished by astronomical instruments; the one is the extension and improvement of the science; and the other its application to geography, navigation, and the ordinary wants of society. Here we give the name astronomical instruments to such as in their application are directed to the heavenly bodies, as a telescope is directed to a star, and the axis of a sun-dial to the pole; but we do not consider as astronomical instruments orreries, and machines composed of wheels and pinions, such as exhibit imperfectly representations of the celestial motions. Globes are indeed appropriate furniture in an observatory, because they truly exhibit the relative positions of the stars and the different countries; and they serve to resolve approximately the different problems of the sphere. But complex orreries and planetary clocks are mere playthings. They excite admiration by the ingenuity displayed in their construction, but they are of no practical use. Showmen pretend to teach astronomy by their assistance; but their inability in giving just notions of the dimensions and magnitudes of the bodies which form the solar system may be easily conceived by reflecting, that if in an orrery the earth be represented by a sphere one inch in diameter, the representation of the sun should be nine feet; also, that if the representation of Mercury describe a circle of about four inches radius, then the orbit of Uranus should have its radius sixteen feet.

Judicious teachers of astronomy may, however, employ with advantage simple contrivances to facilitate the acquiring of correct notions of the celestial motions; but they will direct the attention to a single object at a time, and not attempt to exhibit all the phenomena of the heavens at once. Wooden wheels and catgut bands are just as useful for the purpose in question as metallic wheels and pinions. A globe moved by the hand round a candle will serve to show the changes of the seasons; and in like manner the phases of the moon, her nodes and their motion, and the nature of eclipses, may be all explained by simple and easy contrivances.

1. Astronomical Telescope.

In a subsequent part of this work the theory of the telescope, and the various kinds of telescopes, will be fully explained. As, however, the telescope forms an essential part of almost all complex astronomical instruments, it will be proper to explain here in a general way its principles and use.

The astronomical telescope is composed of two principal parts, the object-glass, and the eye-glass or eye-piece. These are in opposite ends of a tube; and in its application the former is next the object, and the latter next the eye. In telescopes of the best construction the object-glass is composed of two and sometimes of three pieces. (See ACHROMATIC GLASSES.) We shall here, however, suppose it of the simplest form; that is, a very thin double convex glass, the opposite sides being portions of spherical surfaces.

Let ABDE (Plate XC. fig. 131) be a lens of this form, C being the centre of a sphere, of which ABE is a portion of the surface. Let L be any point in an object to be viewed with the telescope, and let a ray of light proceeding from L fall perpendicularly on the convex surface in B; it will pass through the glass without being turned out of its direction, and will proceed straight forward on the prolongation of the line LBD.

Let another ray LI fall obliquely on the convex surface at I; by the principles of optics this will be refracted, Practical that is, turned out of the direction LI, and take a new direction IPP. Draw CI to the centre; the new direction will be such that the sine of the angle of incidence MIL or CIL will be to the sine of the refracted angle CIP or CIP' in the constant ratio of a given number n to 1.

We have therefore

\[ n = \frac{\sin CIL}{\sin CIP} = \frac{\sin ICP}{\sin ICL} \cdot \frac{CL}{IP} \cdot \frac{IP}{CP} \]

Let us suppose the point I to be near B; then IP = BP nearly, and IL = BL nearly, and we have

\[ n = \frac{CL}{BP} \cdot \frac{BP}{CP} \text{ nearly} \]

When the ray arrives at the concave surface ADE, it suffers a second refraction in passing into the air, and changes its direction from IP into a new direction IL'. Let C' be the centre of a sphere, of which ADE is a part of the surface; draw the radius CP', and we shall have

\[ \sin C'TL' = \sin C'TP' :: n : 1; \text{ and therefore} \]

\[ n = \frac{\sin C'TL'}{\sin C'TP'} \cdot \frac{\sin C'TP'}{\sin C'TL'} \cdot \frac{C'L'}{IP} \cdot \frac{IP}{CP'} \]

We suppose the arcs IB, PD, to be small; therefore IP = PD nearly, and IL' = DL' nearly; hence

\[ n = \frac{C'L'}{DL'} \cdot \frac{PD}{CP'} \text{ nearly}. \]

Since the thickness of the lens is supposed to be inconsiderable, we may assume that PD = PB nearly, and that DL' = BL nearly, and then we have

\[ n = \frac{CL}{BP} \cdot \frac{PB}{CP} \text{ nearly} \]

Let \( r = CB, r' = CB, \Delta = BL, \Delta' = BL', v = PB, \) and equations (1) and (2) become

\[ n = \frac{v(r + \Delta)}{\Delta(v - r)}; \quad n = \frac{v(r' + \Delta')}{\Delta'(v + r')} \]

from these we obtain

\[ \frac{n}{v} = \frac{n-1}{r} - \frac{1}{\Delta}, \quad \frac{n}{v} = \frac{1}{\Delta} - \frac{n-1}{r'}; \]

and hence

\[ \frac{1}{\Delta} + \frac{1}{\Delta'} = (n-1)\left(\frac{1}{r} + \frac{1}{r'}\right). \]

When a ray of light passes out of air into glass, the sine of the angle of incidence is to the sine of the angle of refraction as 3 to 2 nearly, or as \( \frac{3}{2} \) to 1. In this case \( n = \frac{3}{2} \), and

\[ \frac{1}{\Delta} + \frac{1}{\Delta'} = \frac{1}{2}\left(\frac{1}{r} + \frac{1}{r'}\right). \]

This expression will be the very same if we put \( r \) instead of \( r' \), and \( r' \) instead of \( r \); also, if we put \( \Delta \) for \( \Delta' \), and \( \Delta' \) for \( \Delta \). Hence we may infer, that whichever of the two convex sides of the object-glass be turned towards the object L, the value of \( \Delta' \) will be the same for a given value of \( \Delta \); also, that if \( L' \) be the focus to which rays issuing from L converge after refraction, then L will be the focus to which rays issuing from \( L' \) would converge after passing through the object-glass.

If \( \Delta \), the distance of L from the object-glass, be very great, then \( \frac{1}{\Delta} \) will be very small in respect of \( \frac{1}{\Delta'} \); and in the case of the heavenly bodies \( \frac{1}{\Delta} \) vanishes, and we have simply

\[ \frac{1}{\Delta} = \frac{1}{2}\left(\frac{1}{r} + \frac{1}{r'}\right). \]

If we suppose both sides of the object-glass to be alike convex, that is, \( r = r' \), then \( \Delta = \Delta' \); hence we learn, 1st, that all rays which come from any point whatever of a very remote object, and which traverse a double convex glass of equal curvature on both sides, are united by refraction about its centre of sphericity, which is called its principal focus; 2d, that rays which proceed from a point at the centre of sphericity do, after refraction, proceed in lines which may be considered as parallel. All these conclusions are only true approximately. They suppose the convex arcs ABE ADE small, and the thickness of the glass very little; nevertheless they differ but little from truth; the focus L is not indeed a mathematical point, but has a certain magnitude, which varies with the distance of the object and the breadth of the lens.

If we now suppose that two lenses, BD, bd (fig. 132), are adapted to the extremity of a tube, so that their centres of sphericity coincide at the same point F, and that this point and the centres of the lenses are in the same straight line, then, from what has been explained, it follows that rays coming from a distant object L, after passing through the lens BD or object-glass, will be collected at F (which is therefore called the focus of parallel rays), and will there form an image of the object L; also, that all the rays, after crossing in the focus F, will proceed forward; but in passing through the second lens or eye-glass bd, they will be again refracted, and emerge on the other side in parallel lines; and if they enter an eye now situated at O, these parallel rays will produce distinct vision.

The eye does not see directly the remote object AC, but only its image formed at the focus F, and this respect to the object is inverted; for the rays proceeding from A and falling on the object-glass at B, are by refraction turned into the direction BF, and meet the glass at &. In like manner, the rays which are emitted from C, and pass through the object-glass at D, meet the eye-glass on the opposite side at B; thus the object AC and its image formed at F have opposite positions.

The astronomical telescope, then, differs from the common telescope for viewing objects at a distance, in reversing the position of objects seen through it, also the direction of their motions, the upper limb of the sun or moon appearing the lower, and all the heavenly bodies appearing to move from west to east. This, which would be an inconvenience with terrestrial objects, is of no consequence in viewing the stars.

The surface of the object-glass being always much greater than that of the eye-glass, which has a shorter focus, and all the rays which fall on the surface BD of the former being collected on the surface bd of the latter, they are there condensed, and the illumination is increased in the inverse proportion of the areas of the glasses; so that if the intensity of the light which falls on the object-glass be represented by I, that on the eye-glass will be \(\frac{(BD)^2}{(bd)^2}\); hence telescopes in general render objects more luminous and more easily distinguished. They also magnify objects; for let A be the centre and B the border of an object (fig. 133). The point A is visible to the eye O, by the ray ADaEO which traverses the object-glass D and eye-glass E, but suffers no refraction. (We here do not consider the oblique rays, which, proceeding from A, are collected at the principal focus.) The border B is visible by the ray BDb at the focus b of the object-glass. This ray meets the eye-glass at d, and is there turned by refraction into the direction de; and, in emerging from the glass at e, is again refracted to O, its focus, so that OE is parallel to EB. The image is seen under the angle \(eOE = bEa\); but \(ab = Da \tan D\), therefore

\[ \tan E = \frac{Da}{Ea} \tan D = \frac{R}{r} \tan D, \text{ or } E = \frac{R}{r} D; \]

\(R\) being the radius of sphericity of the object-glass, and \(r\) that of the eye-glass. The angle under which the object is seen is therefore increased in the proportion of the two radii, and the magnifying power is the greater, as the radius of the eye-glass is less than that of the object-glass.

Common astronomical telescopes generally magnify from 70 to 100 times; some even magnify 300 times. This, however, must not be understood in a rigorous sense; if, for example, we expect to see the moon 100 times larger with a telescope which is said to magnify 100 times, we may be disappointed. To produce this effect, the telescope ought to magnify more: it only represents the moon under an angle 100 times greater; but it is not by the visual angle alone that we judge of magnitude; our opinion is greatly influenced by the distance at which we suppose the object. When we see an object under the angle AKB (fig. 134), nothing determines whether this object is truly AB, or CD, or EFF; and according as we judge it to be in the first, or second, or third of these positions, or in one more remote, we assign to it magnitudes always increasing although the angle is still the same. But this judgment being uncertain, and such as cannot be subjected to calculation, the magnifying power of a telescope is in practice estimated by the angle of vision, which can always be exactly determined.

Let F be the principal focus of the object-glass C (fig. 135). If the angle HCG is such, that HG (= 2 HF) = 2 CF tan. HCF is equal to the diameter of the interior tube of the telescope, the angle HCG is called the field of view: every object whose focal image is greater than HG cannot be seen entirely in the telescope. This happens in the case of the sun and moon when viewed with telescopes of about 81 feet, such as are used in considerable observatories. In these, the sun's image will be about 9½ inches. This exceeds the diameter of the tube. But the opening is yet more contracted by a perforated diaphragm, which, besides other purposes, serves to cut off the rays irregularly reflected from the inside of the tube, also those which produce colour in the image. To determine the field in view as limited by the diaphragm, then, we have this equation: \(2 \tan HCF = \frac{HG}{CF} = \frac{2HF}{CF}\). If we put \(r\) for HF, the radius of the diaphragm, and \(R\) for CF, the radius of sphericity of the object-glass, the field of view in seconds is \(\frac{2r}{R \sin 1^\circ}\).

2. Dorpat Telescope.

The late Joseph Fraunhofer of Munich, a most skilful artist and experimenter in optics (whose demise in 1826, in the prime of life, was a great loss to science), constructed a magnificent refracting telescope for the observatory of the Imperial University at Dorpat. It was received by Professor Struve in the year 1825, and has since been found to fulfil most satisfactorily his expectation and the intentions of the maker. As this is one of the most magnificent instruments of the kind that has hitherto been constructed, and described by a figure, we have given an engraving of it, copied from the Memoirs of the Astronomical Society. (See Plate XII.)

The object-glass of this telescope is about 9½ inches in diameter, and its focal length about 14 English feet. The main tube is 138 feet; and, in addition, there is the small tube which holds the eye-pieces. Of these there are four; the least magnifying power is 175, and the greatest 700. After the telescope was received at Dorpat, a perfect micrometrical apparatus was ordered to be made for it. This was to consist of four annular micrometers, of which two were to be double; a lamp circular micrometer, with four The frame-work of the stand is made of oak, and the tube of deal, veneered with mahogany. The whole weight of the telescope and its counterpoises is supported at one point, namely, at the common centre of gravity of all the ponderous parts. These weigh 3000 Russian pounds, of which the frame-work contains 1000; the remaining 2000 are so balanced in every position, that the telescope may be turned, with ease and certainty, in every direction towards the heavens.

The basis of the frame is formed of two cross beams, each nine feet seven inches long. The ends of these are seen in the figure at A, B, C, D. They are braced by four smaller bars forming a square, one of which is seen at E. This braced cross is fastened to the floor by eight screws, six of which are seen in the figure. A perpendicular post, about six feet high and seven inches square, is fixed over the centre of the cross, and is propped at the north, east, and west sides by three curved stays, denoted by G, G', G'', which are fixed at their lower ends to the beams of the cross, and at the upper to the vertical post. An inclined beam H of the same thickness rests on the southern end of the meridian beam of the cross, and is attached to the vertical beam in a position parallel to the polar axis. This axis, shown in the figure at I, is a cylinder of steel 39 inches long, and proportionally thick. It turns in two collars, and its lower end, which is rounded and polished, rests on a steel plate attached to the bearing piece K, which is secured to the inclined beam H, and has therefore very little friction, the weight being supported by friction rollers near the common centre of gravity; and a counterpoise L is applied to support the axis in any position. There is a circle 13 inches in diameter, graduated to minutes of time, fixed to the lower end of the axis, and furnished with verniers. The axis of vertical motion of the telescope, which has nearly the dimensions of the polar axis, passes through a brass tube at right angles to the latter; the tube, which is seen at M, forms a part of the frame, and is fastened to the upper end of the polar axis by twelve screws. This axis carries the circle of declination, which is 19 inches in diameter, and is divided to every 10", with a vernier reading 10" or 5" by estimation. The tube of the telescope is fixed to the frame-work nearer to the eye end than the middle, and has two counterpoises attached to levers, which balance the two ends, and prevent the natural tendency of the longer end to bend. The brass frame holding the two axes appears on the figure clamped to the tube by two strong rings, one at each end of the centre of motion. A bent lever, carrying the weight O, embraces by a double ring the near end of the axis of the declination circle. The axis itself carries another weight; and by this and the weight O it is counterpoised. The slow motion in altitude is given to the telescope by a Hooke's joint applied to the screw of the clamp, which has a spring urging it against a strong iron bar P, attached to the end of the cylinder M, that forms a stop to the circle; and a slow equatorial motion is given by a second Hooke's joint taking hold of an endless screw, acting with the racked edge of the hour circle, while a spring presses it into action uniformly, and a lever is employed to raise it out of the rack when necessary. The handles taking hold of these screws extend to the reach of the observer, who can thus point his telescope in right ascension and declination with the same certainty as the best meridian instrument.

A regular sidereal motion is communicated to the instrument by clock-work, which keeps a star apparently at rest in the centre of the field of view; and there is a contrivance by which the sidereal can be changed into a practical solar, also to a lunar angular motion.

This almost invaluable instrument cost 10,500 florins (about 950 pounds sterling). The price, although it may appear considerable, yet barely covered the expense of the workmanship of its construction. This relinquishment of the profit of trade does great credit to the ingenious and liberal-minded artists, Fraunhofer, and Utzschneider, the chief of the optical establishment at Munich.

3. Lord Ross's Reflecting Telescope.

This celebrated telescope has two object-mirrors of 6 feet in diameter, and of 53 feet focal length. The preparations for the casting were commenced in 1842, and many experiments were necessary before the best proportions of tin and copper, of which the alloy for the mirrors was composed, were discovered. The respective weights of the specula are 83 and 4 tons, and the most refined precautions were necessary in the construction of the furnaces for fusing these immense masses of metal, in transferring it to the moulds prepared for the casting, and in cooling it with the requisite degree of slowness, to prevent flaws or cracks. The polishing process was performed by causing the polishers to make strokes backwards and forwards in imitation of the manner in which the hand would perform the same process by means of the action of a small steam-engine. This engine gave motion to a beam, which, by the intervention of a crank and connecting rods, gave motion to the polishers. A very ingenious system of levers was devised for the support of the mirror, so that it should remain in every required position without strain. Three similar systems of these levers rest on fulcrum immediately under the centres of gravity of the three equal sectors into which the surface of the mirror may be supposed to be divided. Each system consists of one triangle with its point of support directly under its centre of gravity, on which it freely oscillates, and each triangle carries at its angles other three triangles similarly supported; and finally, at each angle of these last-named triangles, are placed three balls rotating freely, on which the mirror ultimately rests.

The tube of the telescope provided for the use of these enormous mirrors is of wood, fixed to a cube of 10 feet, which has in one of its sides folding doors for the admission of the specula. The specula themselves are placed on frames which run on railroads leading into this cube, these frames being the same on which they were cast and polished.

The telescope is placed between two immense piers 70 feet long and nearly 50 feet high, which serve as supports for the apparatus necessary for giving the requisite elevations of the tube, and for directing it to a given object, and also for the galleries which carry the observer near its upper end. It carries near its upper extremity the apparatus for the Newtonian small mirror, though provision has been made in the construction of the observing galleries for its use as a Herschelian telescope, without a second reflection, for the purpose of avoiding loss of light.

The observations made with this wonderful instrument have been chiefly confined to nebulae, in many of which some remarkable discoveries have been made. In the Phil. Trans. for 1850, are some beautiful drawings of some of these mysterious structures, amongst which the spiral forms of Messier 51 and Messier 99 deserve particular attention.

4. Transit Instrument, Meridian Circle, and Astronomical Clock.

The primary problem in geography is to determine the exact position of any proposed point on the earth's surface in respect of the equator and some assumed meridian, as that of Greenwich or Paris; that is, to find its latitude and The corresponding problem in astronomy is to determine the position of every fixed star, and in general of any celestial object, in respect to the equinoctial circle in the heavens (or else the pole) and a circle passing through the pole and the intersection of the equator and ecliptic; that is, to find its declination or polar distance, and its right ascension (chap. i. sect. i.). The former of these is found in great observatories by the mural circle, and the latter by the transit instrument or transit circle, and the clock.

The first transit instrument of which we have any account was that of Roemer, which he described in 1700. (Miscel. Ber. tom. iii.) Dr Halley placed a transit instrument in the Greenwich observatory in 1721. The axis was iron, and the telescope about five feet in length. This has been long laid aside, but is still preserved as a relic. Transit instruments of the present day are of two forms; one, the most common, is adapted to the determination of the right ascension only, the other to the determination of both right ascension and declination, either at once or by separate observations.

We have selected for description the transit instrument now in Sir James South's observatory at Kensington, which was constructed for him in 1820, by Troughton, with all the care that artist could bestow on it; and which, as far as the just proportions of its parts are concerned, he regarded as his happiest production. The instrument, with its various parts, is represented in Plate XCIII. Figures 1 and 3 represent two views of it. EO is an achromatic telescope, of which E is the end next the eye; AA' is its axis of motion, with which the tubes are closely united at their junction, so as to form but one body. The extremities of the axis rest in notches formed on two cheeks of metal at AA', which are firmly attached to the inner faces of two stone pillars PP'; and B, B', B", B", are four braces connecting the tubes and axis. These are the parts of the instrument which, on inspection, immediately meet the eye. When adjusted, the axis of motion of the instrument is truly horizontal and perpendicular to the plane of the meridian, and the optical axis of the telescope is in the plane of the meridian; the object of the whole construction is to keep it precisely in that plane, whatever position be given to the telescope by turning on its axis.

The object-glass of the telescope has four inches of clear aperture, and its focal length is seven feet two inches. The body of the telescope and the axis are formed of conical tubes firmly united in a spherical centre-piece, on which their wider ends rest, and cover two-thirds of its surface; thus the tube of the telescope is formed of two pieces, which taper towards its extremities, where their diameters are nearly the same as that of the object-glass. The axis in like manner tapers equally towards its extremities.

The centre-piece is perforated in the direction of the telescope, and also in that of the axis; the width of the first opening being a little more than the radius of the object-glass, and that of the second just enough to allow the light of a lamp placed near the end of the axis to pass uninterruptedly to the centre illuminator. The ends of all the four cones, where they join the sphere, are strengthened by circular pieces of cast brass, which extend full four inches into the cones, and are fixed by solder and pins. They are turned concave in front, so as to fit the surface of the sphere into which they are rabatted, and serve to keep the opposite branches of the axis and telescope straight and perpendicular to one another; and to these pieces are attached rings for the reception of the screws which bind the whole together. The four branches of the axis and telescope are solely united by tension bars. These pass through the sphere, six in the direction of the axis and four in that of the telescope. They are arranged at equal distances between corresponding parts, but so as neither to obstruct the rays of the object-glass nor the light of the lamp that falls on the illuminator. They screw into the rings of the brass pieces which enter the cones. The tension bars serve a most important purpose in giving stiffness and permanence of form to the instrument; and he that would imitate it would do well to study Sir James South's description in the Phil. Trans. for 1826. Fig. 3 is a section through the axis, and exhibits the six bars which bind together the cones of the axis, and also the places of the four which are perpendicular to them, and which connect the tubes of the telescope. Fig. 4 is a section through the telescope; the bars of the telescope are shown lengthwise, whilst those of the axis are perpendicular. In both figures the illuminator within the telescope is shown, in one the polished surface, and in the other the back of the plate. The illuminator crosses the tube of the telescope at an angle of 45°. This position requires that the opening in it, through which the light coming through the object-glass passes, should be an ellipse. The braces B, B', B", B", extending from the cones of the axis to those of the telescope, are attached to the former about two inches from the pivots, and to the latter about ten inches from the centre-piece. They exert but a very slight pressure, and might have been omitted in this instrument. They were added in imitation of the Greenwich transit, to which they are essentially necessary.

The apparatus for giving the telescope any required altitude is shown at the eye end in figures 1 and 2, but on a larger scale in fig. 5. It consists of two complete circles, six inches in diameter, firmly attached to the eye end of the telescope; each is provided with two opposite verniers, subdividing its divisions into minutes of a degree. The indexes have clamps and slow moving screws, and microscopes are attached to the verniers; a spirit level is also attached to the index of each circle. The apparatus is adjusted by setting the index to the place of the star, and then, the telescope being moved round till the bubble of the level stands in the middle of its range, the star will traverse the field between the two horizontal wires. If two stars differ but little in right ascension, as Capella and Rigel, so as not to allow time for changing the index which was set to the altitude of that which came first, then the index of the other circle may be set to the altitude of the following star, and both observed. When the same object is to be observed by direct vision, and also by reflexion, then one of the indexes may be set to point the telescope to the direct place of the star, and the other to its reflected image.

Figures 7 and 8 exhibit the plates or side-pieces, and Ys in which the pivots of the axis rest. The plates, which are semicircular, are imbedded in the stone piers, and are firmly screwed to them. Figure 7 represents the eastern plate, in which the contrivance for placing the axis truly level is contained. This adjustment is made by a piece of which the upper end is formed into a Y, and which may be moved vertically, but not laterally. To raise or depress it gradually, there is a piece having a short cylindrical part in the middle; also a fine screw at its upper end, which works in the moveable piece, and a coarse screw at its lower, which works in the fixed plates. The cylinder has holes by which it can be turned round by a capstan pin. By the ingenious contrivance of the two screws, the sliding piece is moved vertically, but slowly; for the space gone through is only the difference of the spaces through which it would have been pushed or drawn by each screw acting by itself. Fig. 8 shows the western plate. In this the Y piece admits only of a horizontal motion for the purpose of placing the instrument in the meridian. The adjustment is effected by two screws, which work in the opposite sides of the piece, and whose heads abut against the fixed plate. To produce motion one of them must be screwed and the other unscrewed by equal Practical quantities while the observer's eye is at the telescope; and to effect this the screws are connected by pinion work put in motion by a handle hanging down close to the inside of the western pier. (See fig. 1.)

Fig. 9 is a bird's-eye view of the head of one of the piers. This is meant to show the apparatus for relieving the pivots of the axis and the Ys from a great part of the weight which would otherwise bear on them. Immediately behind the adjustable Y piece, but rather broader, is a plain piece of brass having a Y cut in its upper end: a lever also is seen, one extremity of which passes into a hole made in this Y piece, while the other end carries a weight. The bar of the lever is expanded into a circle whose centre is about one third of the lever's length distant from the pivot of the axis. The circle admits the illuminating lantern. Two steel screws, with blunted, hard, and polished points, are inserted in the diameter at right angles to the direction of the lever: these rest on hardened and polished planes, which are let into the stone pier, and together form the fulcrum, in the manner of a balance. The weight is a short cylinder hooked on the end of the lever: it is hollow to receive small shot, introduced, as a counterpoise, to relieve, more or less at pleasure, the instrumental portion of the pivot, and also the instrumental Y piece, of weight.

Fig. 6 is a perspective view of the eye end of the telescope. In it a micrometer is shown, which moves a plate contiguous to that in which the five transit wires are inserted: one wire is contained in the moveable plate, and is intended to facilitate the observation of the pole-star and others near it.

In fig. 1, on the eastern side of the telescope, a projecting finger-screw is seen. This gives motion to an apparatus within the tube of the telescope for regulating the quantity of light projected by the illuminator on the transit wires.

The Greenwich transit instrument, also the workmanship of Troughton, and one of his much admired productions, was placed in that observatory in 1816. In its construction it is nearly the same as Sir James South's, but of different dimensions. The object-glass of the telescope is 5 inches in clear aperture, and the focal length 10 feet: its horizontal axis, including the pivots, is 3 feet 10 inches. This instrument was dismounted in the year 1851, and its use was replaced by that of the great transit circle. For a more detailed account the reader may consult the volumes of the Greenwich Observations.

All transit instruments have a meridian mark, that is, a mark on some remote object, by which it may be ascertained at any time whether the instrument be truly in the meridian.

Transit or meridian circles are in their nature quite analogous to the instruments we have described; with, however, the important addition of a graduated circle of considerable dimensions as a principal part of their construction. The meridian circle which Troughton constructed for Mr Groombridge, but which afterwards passed into the hands of Sir James South, is a fine example of this kind of instrument. A figure and description of it may be seen in Dr Pearson's valuable work on Practical Astronomy. The largest and finest instrument at present is that which was erected at Greenwich in 1850.

A clock of the very best construction is an indispensable companion of the transit instrument. This may be regulated so as to show either mean solar or sidereal time, according to the principal objects in view, the one being always easily convertible into the other. The attention of ingenious men has been long directed to the construction of the astronomical clock; but on this subject we shall have occasion to treat fully elsewhere in our work.

The transit instruments here described are of the most expensive kind, and adapted to the higher efforts in the cultivation of astronomy; but there are also portable transits which travellers may use, and which may serve to determine true time. In all applications of the instrument, the axis must be placed truly horizontal by means of the level; the line of collimation, that is, a line between the centre of the object-glass and the centre of the cross wires, must move in a great circle, or, which is the same thing, it must be perpendicular to the axis; and the vertical circle which it describes must be the meridian; these are the three principal adjustments. The Nautical Almanac gives the exact time when the sun or certain considerable stars pass the meridian. The observer gives the telescope such a position that the star must appear in the field of view in its passage. When it is seen, he notes the exact time, by the beat of the clock, when it crosses each wire. Of these wires there are generally five or seven in the focus of the eye-piece, and all transits (whether perfect or imperfect) are always reduced to the time of passage over an imaginary line corresponding to the mean of these.

5. The Mural Circle.

The Royal Observatory at Greenwich has two mural quadrants, each about eight feet radius; these are fixed on a massive structure of hewn stone in the form of a parallelopiped, one on each side, and hence their name. That whose telescope is directed towards the north is chiefly of iron, and was erected by Graham in 1725, for the lunar observations of Halley: it was, however, redivided by Bird in 1753. The other, for the southern part of the meridian, is of brass, and was constructed by Bird, and placed in its position in 1750: with this Bradley and Maskelyne successively made their observations for forty-six years.

Experience had shown, early in the present century, that entire circles have a great advantage over quadrants; accordingly, the use of the Greenwich quadrants was discontinued, and in their stead a mural circle six feet in diameter, constructed by Troughton, was placed in the observatory on 12th June 1812. A second mural circle, nearly a copy of the other, constructed by Thomas Jones (another eminent artist), was, in 1825, placed in the observatory, fronting the former, and at a distance of seven feet to the east; and with these two instruments simultaneous observations were, during the administration of Mr Pond, daily made of the zenith distances of stars, the zenith point being obtained by a combination of observations made by direct vision and by reflection in a trough of mercury. Soon after the accession of Mr Airy, the use of Jones's circle was discontinued, as we have before had occasion to mention.

Figs. 1 and 2 of Plate XCIV, give two views of Troughton's mural circle. In fig. 1 the circle is seen obliquely on the front or eastern face of the wall, with the greater part of its apparatus. The breadth of the wall from north to south is seven feet, its thickness from east to west four feet, and its height ten feet. It is formed of four stones laid one on another. The third stone has in its under side a semicircular groove, cut in its middle from west to east, of six inches radius. The upper side of the second stone being worked level, forms the diameter of this semicircular arch-hole, and supports the axis-work of the instrument at about five feet above the floor. The real centre of the instrument is about five inches higher. The nucleus of the Greenwich circle is an octagon of eight inches diameter at the corners; its depth is three inches, and a circular perforation of six inches and a half is made through its whole depth. The outward faces of the octagon, which are each three inches square, support eight of the circle's conical radii, to which they are screwed and steadily-pinned. The other eight radii are fitted in closely, each one between two of the former, so that their lower ends come down on the corners of the octagon. The limb of the circle consists of two rings, the interior one having its plane parallel, and the exterior perpendicular, to the plane of the circle, so that, when united, their section will be represented by the letter T. The interior or flat ring has in the engraving the appearance of passing through clefts in the middle of the outer ends of the sixteen radii, which are there solid. The perpendicular ring is fitted close on the exterior edge of the other, to which it is screwed, and also to the ends of the conical radii. The cones are bound together at half the distance from the double ring to the centre by a circle of interposed bracing bars. The circular aperture of the octagon is shut up by plates before and behind, which are fastened to the octagon by strong steel screws. The posterior plate has a large circular hole, and the anterior a smaller one, both truly wrought. Into these the axis of motion is fitted, and united to the octagon and circle by means of screws. The axis is a cone of brass nearly seven inches in diameter in front, but behind only half as much, and nearly four feet long; this works in a socket, which at each end receives it, and in which it fits with the greatest possible exactness. The two parts which fit the axis are soldered into a strong brass tube, larger than the tube of the axis, but nearly of the same shape. On the tube of the sockets in front is soldered a strong perforated plate or upright bearing piece, at right angles to the axis, which nearly fills the semicircular aperture in the wall; and at the remote end is soldered a short cylinder, the use of which will be explained. It is there that the adjustments for placing the circle in the meridian, and for levelling the axis, are performed. Two strong horizontal plates are fastened on the lower surface (which is flat) of the perforation through the wall, one before and the other behind. The bearing piece of the socket in front only rests upon the plate, but behind the bearing cock and plate are screwed together. In front the plate and bearing piece are connected by a conical piece of hardened steel, which is fixed under the middle of this piece, and fits nicely into a hole in the plate, but so as to revolve. At this end of the axis these parts do not come quite in contact; for there are fixed under the bearing piece, at each extremity, about ten inches apart, two short props, like buttons of hardened steel, the spherical surfaces of which rest upon planes of hardened steel fixed in the plate. The central conical piece prevents the circle from sliding sideways when angular motion is given round this conical piece to bring the instrument into the plane of the meridian. It has been stated that a short cylinder was soldered on the remote end of the cone of the sockets. This passes into a perforation in the cock behind, which perforation is greater than that of the cylinder. Two fine-threaded screws at right angles to each other work in the cock, one vertically for levelling, and the other horizontally for meridian adjustment. The two screws only press with their points against the sides of the short cylinder; but opposite to them are the ends of two small cylinders standing in the same line, which are urged forward with spiral springs, and thus force the short cylinders into contact with the screws. The telescope is seen on the face of the instrument; its focal length is six feet two inches, which is the exact outer diameter of the circle; the aperture is four inches, and its common magnifying power about 150. The telescope is attached to the circle at the centre by a steel axis, which passes through the proper axis of motion from end to end, and was indeed the arbor on which the axis was turned. The weight of the telescope is supported on its own axis, and it may be fixed to the circle in any position by means of two clamps which keep hold of the border of the circle. The graduation of the instrument is on the convex cylindrical surface of the exterior ring, therefore the reading microscopes have their direction parallel to the plane of the instrument. The divisions on Troughton's circle are made on a narrow ring of white metal composed of four parts gold to one of palladium; and the figures which count the degrees are engraved on a like ring of platinum.

In Jones's circle the divisions are on gold. None of these Practical metals tarnishes in the least degree. The divisions are by Astronomy lines, and suited to wires which cross at an acute angle in the reading microscopes. The degrees are cut into 5 spaces, and are numbered from the pole southward to the same pole again, viz. from 0° to 360°. The 5 spaces are subdivided by the microscopes to single seconds; and a division representing this quantity on the micrometer head may be easily estimated to the tenth of a second. There are six reading microscopes; which, during Mr Airy's direction, were always used.

In order that the circle may move easily round on its axis, there is an apparatus for counterpoising it, or for lifting the whole weight, without which the load would press altogether on the lower side of the front socket. This is effected by means of two large rollers, shown below the axis in fig. 1. The rollers, set in a double frame, act on the edge of the centre flanch, nearly in contact with the radial cones. Two perpendicular bars of steel, at about the height of the centre, are connected with the frame of the rollers by hook and eye; and these bars are in a similar manner suspended by two beams, each resembling a common balance, at the top of the wall. The part which appears in front is shown in fig. 1, and one of the beams, its fulcrum and counterpoising weight, near the top of the wall, in fig. 2. This apparatus produces a simple lift, without any tendency to affect the due motion of the circle's axis.

There is another flat, circular ring, somewhat larger than the graduated one, fastened at several places to the wall, and nearly touching it. On this ring the clamp and screw for slow motion slide, and may be clamped to it at any part of the ring.

The plumb-line of the mural circle, seen in fig. 2, is for placing the axis truly horizontal. The apparatus by which the plummet is suspended was applied by dovetail fittings occasionally to the wall near its top, see fig. 1. The apparatus itself is shown in fig. 2. In fig. 1 fixed microscopes are seen on the telescope, near its end, for viewing the plumb-line.

The wires of the telescope are illuminated by a diagonal reflecting plate in the middle of the tube, which receives the light by a circular aperture, seen in fig. 1, in a line with the centre of the circle. A lantern, at four or five feet distance, placed in the line of the axis, throws light on the field of view.

The instrument seen in a vertical position on the back of the wall is a zenith tube. It was erected to discover, if possible, the parallax of a star that passes very near the zenith; but was never used.

The mural circle has as an accompaniment a clock; and since all observations made with it require to be corrected for refraction, which depends on the state of the atmosphere (see table in page 148), every recorded observation must have annexed to it the height of the barometer and thermometer.

We have now described the instruments which constitute the principal furniture of an observatory, and which are all that the present state of the science absolutely requires. We proceed next to describe some others which are extremely convenient, and almost equally necessary in the science.

6. Equatorial Instruments.

An equatorial instrument is of great value to a practical astronomer, for by means of it he can direct his telescope at once to any object, however minute, whose right ascension and declination are known; and conversely, he can determine the right ascension and declination of any object to which it is directed, although out of the meridian.

Plates XCV. and XCVI. give a representation of Sir James South's five-foot equatorial instrument. The greater part of this instrument is composed of tinned iron plate; and its Practical characteristics are lightness, steadiness, promptness in adjustment, and capability of retaining them.

Fig. 1 of Plate XCV represents the instrument as viewed at right angles to the declination circle. The polar axis is about 10 feet long; the lower end is a pivot attached to a cone, which, reckoning upwards, is about a fourth of the whole length. The higher side of the cone is cut in a sloping direction, as seen in the figure, for the purpose of more conveniently observing the vicinity of the pole. From the upper end of the cone the polar axis branches into two parts, between which is room for the declination circle and the head of the observer. These two branches are again united at the top by an open frame of bell metal, represented in fig. 2, to which the upper pivot is attached. This frame, as well as the iron-work which composes it, is so contrived as to present the least possible surface to obstruct the telescope; for the same reason the pivot at the top of the telescope is made as small as possible, whilst that at the lower end is considerably larger. Both ends of the axis are supported on stones, the northern end rising within about four inches of the level of the declination circle, the rest of the support being of wrought iron. At the southern end the stone rises very little above the floor, but a cast-iron frame supports the pivot at the height of about two feet. The Y, or angle which supports the lower pivot, is placed upon the frame, and provided with two screw adjustments, one for giving the axis its due elevation, and the other for bringing the instrument to the meridian. The form of the iron-work will be understood by consulting the different figures. The two branches of the polar axis on their upper sides are formed of broad planes, making one continued plane. On these the axis and reading microscopes of the declination circle are fixed. The instrument is self-balanced by the position and figure of its parts, and the addition of a weight fixed to the conical part of the polar axis. The diameter of the declination circle is four feet, the length of the telescope five feet, and of the axis about thirty-two inches.

In Plate XCV, fig. 1, the declination circle appears quite plain, like the head of a drum, with the telescope directed towards the equator. In Plate XCVI, fig. 1, the polar axis is considerably fore-shortened, from the position of the draughtsman in making the drawing of the instrument. In this figure the edge of the declination circle is shown as a short cylinder, with the telescope protruding beyond it. In this figure, also, the shape of the declination axis, and the two principal microscopes for reading the declination, are shown. There is a third microscope, which indicates zenith distances. This is seen in Plate XCV, fig. 1, between the eye end of the telescope and the instrument's elevated pole. In the same figure is shown a narrow brass ring, whereon the graduation is made.

The hour-circle, two feet in diameter, is fastened to the lower end of the polar axis; its edge is seen in Plate XCV, fig. 1, and its under side in Plate XCVI, fig. 1 and fig. 3. One of the reading microscopes is well seen in fig. 1 of Plate XCV, and both of them less perfectly in the other two figures. The circle is of brass, and the divisions (fine lines) are on an inlaid ring of platinum, corresponding to twenty seconds each; these are subdivided by the microscopes to tenths of seconds. The declination circle is divided to spaces of five minutes, which are subdivided by the micrometer screw of the microscopes to single seconds. The instrument is furnished with two ground levels. The divided side of the declination circle is quite flat; but the opposite face is articulated, showing how the parts are united. It is on this side that the levels would be seen; one of them is parallel to the telescope, and the other to the declination axis.

The clamps and screws for slow motion are unusual, but remarkably good. Instead of the common mode of clamping the circle, in this instrument the clamp is made to grasp the axis. There is soldered on each axis a ring of brass, the outer edge of which is broad and cylindrical. On this fixed ring a moveable one is fitted, and afterwards cut into three equal parts; these are again united in two of the three sections by joints, like those which bind the different parts of a watch chain together. At the third juncture the clamping takes place, a projecting part of the ring having been there cut through, leaving one half on each side of the section. Here the ring gapes, but a screw passes through the projecting pieces or ears on each side of the disjunction, and in bringing them together, grasps the axis with a firm embrace. To the middle of the tripartite rings are attached long arms of tinned iron plate, at the extremities of which the slow-moving screws have their places. The fixed stud is in the lower screw, placed in the iron support; that of the upper one is in the polar axis. The long screw for slow motion in right ascension is acted on by a contrate wheel and a pinion at right angles to the plane of the circle, as shown in fig. 1, Plate XCV.; a long handle is attached to it, and shown leaning against the northern pier. A similar screw for declination, but without the contrate part, is seen in Plate XCVI, fig. 1.

The illumination of the wires of the telescope is made by a small lantern placed at one end of the declination axis; and there is a contrivance between the nozzle of the lantern and the end of the axis, by which the light is adapted to different observations.

The eye-piece of the telescope is represented in Plate XCVI, fig. 2, in which there is seen the edge of a graduated circle, the front of a quadrant, and two small spirit-levels. There is likewise shown, but partially, a double parallel line micrometer, which also measures angles of position.

This instrument, when first constructed, was designed to be placed where a meridian mark could not be obtained. A mark, however, could be placed and seen to the westward; and to take advantage of this, the axis of the declination circle was converted into a telescope with two object-glasses of equal focus, two sets of cross wires, and an eye-glass that might be placed in either end; a mark was then built up to the level of the axis, and in a line at right angles with the meridian, and this formed a substitute for a meridian mark. The instrument bears no maker's name, but the scheme of its fabric was devised by the late Captain Huddart, F.R.S. The brass-work, &c., was made by J. and E. Troughton, under his direction; and the object-glass for the telescope of 3½ inches aperture by P. and J. Dollond. To preserve the tinned work from oxidation, it is well covered with white paint, and varnished; thus it has not only a neat appearance, but can be easily cleaned at any time.

With this instrument, and another equatorial of seven-feet focal length, the object-glass being made by Tulley, Sir James South and Sir John Herschel made observations on the apparent distances and positions of 380 double stars; a work of which we have had occasion previously to make mention.