ENCYCLOPÆDIA BRITANNICA.

THEORETICAL ASTRONOMY.

OF THE MOON.

Theoretical Astronomy. WHEN we reflect on the immense addition that has been made to our knowledge of the constitution of the universe by means of the telescope, we may reasonably indulge an expectation that the further advances yet to be made will be closely connected with the extension of the power of that invaluable instrument.

Besides the power which the telescope gives of penetrating into remote space, it enables us to discover the nature and constitution of the bodies which are nearer to us. The admirable discoveries of Newton, as will be explained in the sequel, have taught us that the planets are not all composed of matter of the same density. But besides the property of weight, which belongs to all matter whatever, it has others which are peculiar to certain bodies, such as solidity, fluidity, &c. The Newtonian law of gravity reveals nothing of these; and it is only by the aid of the telescope that we can ever hope to acquire the slightest knowledge of the constitution of the planetary matter.

The structure of the more remote planets is probably beyond our knowledge, even when aided by the most improved telescopes; but it seems not unreasonable to indulge a hope of going some way in ascertaining the nature of the bodies nearer to us. Jupiter's belts have with great probability been supposed to be masses of clouds in the region of his equator; and Herschel discovered, to the great delight of astronomers, that the polar regions of Mars assumed a whiteness of appearance when obliquely exposed to the light of the sun, just as if, like those of the earth, they were covered with snow. Above all, the moon presents to the astronomer an extensive field, not barely for conjecture, but for instructive observation. The solar spots have given occasion to much ingenious speculation; but the distance of the moon is only one four hundredth part of the distance of the sun from the earth, and in the same proportion our power of becoming acquainted with the constitution of the latter exceeds that of ever knowing intimately the constitution of the former. It is no wonder, then, that the attention of astronomers in all ages has been directed to the companion of our earth, whose structure may with great probability be supposed to resemble considerably the body round which it revolves, and of which it is not improbable that it may have once been a part.

CONTINUATION OF CHAP. III.

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

Theoretical Astronomy. The surface of the moon is diversified by dark patches, the number of which appears prodigiously increased in the telescope. These patches or dark spots 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 respective 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 Astronomy. 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 be 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 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. Theoretical Astronomy.

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 $1^\{\backslash circ\}\; 29\text{'}$, and Lalande $1^\{\backslash circ\}\; 43\text{'}$. According to the latest computations made from the observations of Bouvard, the mean inclination of the lunar equator to the ecliptic is $1^\{\backslash circ\}\; 28\text{'}\; 42\text{'}\text{'}$. Mr Baily makes it $1^\{\backslash circ\}\; 30\text{'}\; 10\text{'}\text{'}\; 8$. 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 $1^\{\backslash circ\}\; 30\text{'}$ with the first, and of $3^\{\backslash circ\}\; 39\text{'}$ 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 $3^\{\backslash circ\}\; 39\text{'}$ and $1^\{\backslash circ\}\; 30\text{'}$. 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 $1^\{\backslash circ\}\; 30\text{'}$ and $5^\{\backslash circ\}\; 9\text{'}$. Hence the difference between the longitudes of these two poles is constantly $180^\{\backslash circ\}$, 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 $180^\backslash circ$ 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 incinerosum. 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 Catoptrics, 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 remits 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 enlightened 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 tinging 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.)

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.

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 if they 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 illumine 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.

Fig. 48. 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 OEm is measured by the micrometer; therefore Om, which is the projected distance of OM, is a given quantity. Now $OM\; =\; \backslash frac\{Om\}\{\backslash cos.\; MOm\}$; and since OEm is a very small angle, OEm may be considered a right angle, consequently $MOm\; =\; MOE\; -\; 90^\backslash circ$; therefore $OM\; =\; \backslash frac\{Om\}\{\backslash cos.\; (MOE\; -\; 90^\backslash circ)\}\; =\; \backslash frac\{Om\}\{\backslash sin.\; MOE\}\; =\; \backslash frac\{Om\}\{\backslash sin.\; EOS\}$; 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 $OM\; =\; n\; \backslash cdot\; CO$; we shall then have $CM\; =\; CO\; \backslash sqrt\{1\; +\; n^2\}$, and the height of the mountain

$aM\; =\; CO(\backslash sqrt\{1\; +\; n^2\}\; -\; 1)\; =\; \backslash frac\{n^2\}\{2\}\; CO$ (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 $\backslash frac\{1\}\{2\}$, which gives the height of the mountain equal to $\backslash frac\{1\}\{8\}$ 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 $\backslash frac\{1\}\{2\}$. The highest mountains on the earth do not reach an elevation greater than $\backslash frac\{1\}\{1000\}$ 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 1980th 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 Séjour, 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 Séjour 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' 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'.5. The mean horizontal refraction observed at the surface of the earth is 0'.585, 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, ex-

tending from the cusps into the dark body of the moon. Theoretical ^Astronomy Its greatest breadth was 2'; and it extended 1' 20" 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'.94; 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.¹ Every thing 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, Selenotopographische Fragmente; Hooke's Micrographia, 1665; Lemonnier, Selenographie; Mayer, Cosmographische Nachrichten, 1748; Boscovich, De Lunæ Atmosphæra, 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.

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:

ἡ σελῆν ἐστὶν ἕνα, σελῆν ἄστρα, σελῆν πόλεις.

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.

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', the sun will be entirely invisible. 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 $\backslash sin.\; ECB\; =\; \backslash sin.\; SEF\; =\; \backslash frac\{SF\}\{SE\}\; =\; \backslash frac\{SA\; -\; EB\}\{SE\}\; =\; \backslash sin.\; R\; -\; \backslash sin.\; p$, 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 $CE\; =\; \backslash frac\{EB\}\{\backslash sin.\; ECB\}$, therefore making $EB$, the radius of the earth, $=\; a$, $CE\; =\; \backslash frac\{a\}\{\backslash sin.\; (R\; -\; p)\}$. 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 perigean, mean, and apogean distances, the following results will be obtained:—

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 $IMT$ (fig. 50) be a part of the moon's orbit, intersecting the lines AC and A'C in $m$ and $m\text{'}$; then $mm\text{'}$ is the geocentric diameter of the shadow, the half of which is measured by the angle $mEC$. 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.

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 $IEC$; but $IEC\; =\; EIC\; +\; EC\text{'}I$, and $EC\text{'}I\; =\; EC\text{'}I\; =\; CEA\; +\; CAE$; therefore $IEC\; =\; EIC\; +\; CEA\; +\; CAE$; or, retaining the same denominations as above, $IEC\; =\; P\; +\; p\; +\; R$. 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 $\backslash frac\{1\}\{2\}\; (1^\{\backslash circ\}\; 15\text{'}\; 24\text{'}\text{'}\; 30\text{'}\text{'})\; =\; 37^\{\backslash circ\}\; 42\text{'}\; 15\text{'}\text{'}$, and $\backslash frac\{1\}\{2\}\; (1^\{\backslash circ\}\; 31\text{'}\; 44\text{'}\text{'}\; 30\text{'}\text{'})\; =\; 45^\{\backslash circ\}\; 52\text{'}\; 15\text{'}\text{'}$; and the least and greatest values of $r$, the apparent semidiameter of the moon, are respectively $14\text{'}\; 45\text{'}\text{'}$ and $16\text{'}\; 45\text{'}\text{'}$ (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 $37^\{\backslash circ\}\; 42\text{'}\; 15\text{'}\text{'}\; +\; 14\text{'}\; 45\text{'}\text{'}\; =\; 52^\{\backslash circ\}\; 27\text{'}\; 15\text{'}\text{'}$, and the greatest $45^\{\backslash circ\}\; 52\text{'}\; 15\text{'}\text{'}\; +\; 16\text{'}\; 45\text{'}\text{'}\; =\; 62^\{\backslash circ\}\; 37\text{'}\; 15\text{'}\text{'}$. 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 $62^\{\backslash circ\}\; 37\text{'}\; 15\text{'}\text{'}$, from which the corresponding ecliptic limit $NC$ is easily computed by means of the formula

Supposing the angle $CNM$, that is, the inclination of the moon's orbit to the ecliptic, to be $5^\{\backslash circ\}$, which is its minimum value, the value of $NC$ is found from the logarithmic tables to be $12^\{\backslash circ\}\; 2\text{'}$ nearly. Hence an eclipse of the moon can only happen when she is within about $12^\{\backslash circ\}\; 2\text{'}$ of her node. Under the most favourable circumstances, however, the limits may extend to $13^\{\backslash circ\}\; 21\text{'}$. 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 $7^\{\backslash circ\}\; 4\text{'}$.

When the moon's disk only comes into contact with the shadow, as in fig. 51, the phenomenon is called an appulse; 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 in-

stant 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 $fgh$ (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 $Wfx$, $Vix$, proceed in their direct course without suffering any refraction; but those which enter the atmosphere between $f$ and $k$, 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 $mopqm$, where none of the sun's rays can enter; all the rest, $R,\; R$, 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 $R,\; R$.

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, $NO$ Plate the orbit of the moon, $C$ the centre of the terrestrial shadow, and $M$ the centre of the moon at the instant of 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
Astro-nomy, 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 P M' are given, and consequently C' M' the distance of the centres. Let us assume

• $\backslash lambda\; =\; CM$ or $\backslash gamma$'s latitude when in opposition,
• $s\; =\; \backslash odot$'s motion in longitude,
• $m\; =\; \backslash gamma$'s horary motion in longitude,
• $n\; =\; \backslash gamma$'s motion in latitude,
• $t\; =$ time from M to M',
• $c\; =\; CM\text{'}$ the distance of the centres.

Now, since we suppose that CP and QM' are the $\backslash gamma$'s motion in longitude and latitude respectively in the time $t$, it is evident that $CP\; =\; mt$, and $QM\text{'}\; =\; nt$. But CC' is the sun's motion, or the motion of the terrestrial shadow in longitude during the same time; therefore $CC\text{'}\; =\; st$. We have then $CP\; =\; mt\; -\; st$, and $PM\text{'}\; =\; \backslash lambda\; +\; nt$; consequently

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

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

which gives the two following values of $t$:

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 $c\; =\; P\; +\; p\; -\; R\; +\; r$ (neglecting the small augmentation of the shadow occasioned by the refraction of the atmosphere). In the case of the penumbra we must take $c\; =\; P\; +\; p\; +\; R\; +\; r$; and it is evident that, if in either case $\backslash lambda$ is of such a magnitude that $c$ is less than $\backslash lambda\; \backslash cos\; \backslash theta$, the value of $t$ will be impossible; in other words, no eclipse can take place.

VOL. IV.

If we suppose $P\; +\; p\; -\; R\; +\; r\; =\; \backslash lambda\; \backslash cos\; \backslash theta$, 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 appulse, 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 $P\; +\; p\; -\; R\; +\; r\; -\; \backslash lambda\; \backslash cos\; \backslash theta$; 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 $\backslash lambda\; \backslash cos\; \backslash theta\; -\; P\; -\; p\; +\; R\; +\; r$. 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 $c\; =$

$\backslash lambda\; \backslash cos\; \backslash theta$. In this case $t\; =\; -\backslash frac\{1\}\{n\}\; \backslash lambda\; \backslash sin^2\; \backslash theta$, 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 $\backslash frac\{12\}\{2r\}\; (P\; +\; p\; -\; R\; +\; r\; -\; \backslash lambda\; \backslash cos\; \backslash theta)$. Thus, taking the moon's apparent diameter at $33^\backslash circ\; 18\text{'}$, and supposing the eclipsed part to be $24\text{'}\; 52\text{'}\text{'}$, this part expressed in digits will be $\backslash frac\{24\text{'}\; 52\text{'}\text{'}\}\{33^\backslash circ\; 18\text{'}\}\; \backslash times\; 12\; =\; 8.96$ 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 $2^\backslash circ$. 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.

Theoretical Astronomy. 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 formulæ 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 $R\; \backslash cdot\; \backslash frac\{D\}\{d\}$ ($R$ being his apparent semidiameter as seen from the earth), and his horizontal parallax will be $p\; \backslash cdot\; \backslash frac\{D\}\{d\}\; \backslash cdot\; \backslash frac\{m\}\{a\}$. The formula then which expresses the length of the terrestrial shadow $CE$ (fig. 50), namely,

tween 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.

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 formulæ 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 $r\; -\; R\; \backslash cdot\; \backslash frac\{D\}\{d\}$; or,

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

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

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 $19^\{\backslash circ\}\; 44\text{'}$. If her distance from the node is less than $13^\{\backslash circ\}\; 33\text{'}$, 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 appulse; 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 whereon 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 appears largest, and the sun's least; and the total darkness shadow and 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 Astronomy. 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 $We$ 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 $Vd$ 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 $VXcg$, 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 $VXcg$ 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 $aba$, 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 twelfth 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 57 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 formulae 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 $17^\{\backslash circ\}\; 21\text{'}$, while those of the moon are only $11^\{\backslash circ\}\; 26\text{'}$, 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 eclipses in 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 $19\backslash frac\{1\}\{2\}$ 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 Astronomy 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 $19^\{\backslash circ\}3286$ 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 $360^\{\backslash circ\}\; -\; 19^\{\backslash circ\}3286$, or, as is found by a simple proportion, 346.619851 days. On comparing this with 29.5305887 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 Theoretical obtained in exactly the same manner as in solar eclipses; Astronomy, 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 hypothenuse 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 hypothenuse, 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.)

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 $47^\{\backslash circ\}$; 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 Esperus, 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 $45^\{\backslash circ\}$ and $47^\{\backslash circ\}$ 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 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 $47^\{\backslash circ\}$, 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. 1.) 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 $28^\{\backslash circ\}\; 20\text{'}$; 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.

Theoretical Astronomy. 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 1737 Mercury was observed to be eclipsed 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, 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 cornicular, 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 con-

venient to have recourse to observations of their apparent Theoretical Astronomy. 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°28 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 the supposition that the earth is the centre of the celestial
astronomy. motions. 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 abedef, &c. When the planet arrives at b, be-
fore it comes into opposition with the sun at A it becomes
stationary. From b to c it retrogrades, and is again sta-
tionary at c. Its motion then becomes direct, and its dis-
tance from the earth continues to increase while it runs
through the arc defgh, 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 hmn 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 hmn. 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 mo-
tions of the planets. We must next endeavour to deter-
mine 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 astro-
nomy, viz. to express the position of the heavenly bodies
in terms of the time reckoned from a given instant.

System of epicycles. 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
Plate LXXXV. is called the deferent, carries on its circumference the
Fig. 59. 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 mo-
tions 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 ve-
locity is the greatest possible, being the sum of the ve-
locities in the two circles. It is evident also that the ap-
parent 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 accele-
rated 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 ad-
vances according to the order of the signs, the planet
will continue to move directly till it arrives at the point s,
where its retrograde motion in the epicycle, in the direc-
tion sP, 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 s' 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.
station 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 ve-
locity 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 phenome-
na. Double and triple systems were therefore introduced,
in which the first epicycle was regarded as a second de-
ferent carrying its own epicycle, the second epicycle as
a third deferent, and so on, till every irregularity of mo-
tion was explained. It is easy to conceive all this me-
chanism, and even to reduce it to general formulæ; 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 epi-
cycloids, 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 em-
ploying other curves, to represent the planetary motions;
and the system of epicycles which resulted from this in-
quiry 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

Theoretical were never seen on the solar disk. Ptolemy, however, Astronomy 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, inasmuch 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 any thing 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 semicircle, 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; $\backslash odot$ 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 $\backslash gamma$, $\backslash delta$, $\backslash pi$, &c, in the centre 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 and show as many several stations from whence Mercury may be viewed from the earth. Suppose then the planet to be at 1 and the earth at $a$; draw a line from $a$ to 1, 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 through 4, 5, 6, 7, &c. we shall find his geocentric places successively at the points D, E, F, G, &c. 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 numeral 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, 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 $90^\backslash circ$ 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 Theoretical direct. 2. While the earth is in its nearest semicircle, the planet will be retrograde. 3. While the earth is near the planet 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, LXXXVI. 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 cde, 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, and 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 $c$, Jupiter would appear stationary, because the visual ray drawn through both planets does not sensibly differ from the tangent $Fa$ 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, in fig. 66, let $\backslash odot$ Plate be the sun, the little circle round it the orbit of the LXXXV. earth, of which abedefg 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 $\backslash odot$, fig. 67, be the sun, the little circle round it the orbit of the Plate earth, whereof ghiklmn is the nearest semicircle, OPQLXXXVI. 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 $Pg$ 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

Theoretician MN. Since the direct motion is swiftest when the Astronomy. 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 $\backslash odot$ 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 consequently his geocentric place will be at $C$; and in like manner his place may be found for the other months at 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 $SEV\; =\; \backslash phi$, and the angle $SVE\; =\; \backslash psi$. During a short time, at the station, the lines $EV$ and $EV$ ($Ee$ and $Vv$ being very small arcs) may be considered as parallel; therefore $\backslash phi\; =\; SEv\; =\; Sev\; +\; ESe\; =\; \backslash phi\; +\; \backslash delta\backslash phi\; +\; ESe$, whence $\backslash delta\backslash phi\; =\; -ESe$. In like manner $\backslash psi\; =\; SSe\; =\; Sev\; -\; VSe\; =\; \backslash psi\; +\; \backslash delta\backslash psi\; -\; VSe$; whence $\backslash delta\backslash psi\; =\; VSe$. But as the orbits are described uniformly, $T\; :\; t\; ::\; VSe\; :\; ESe$, and consequently $T\; :\; t\; ::\; \backslash delta\backslash psi\; :\; \backslash delta\backslash phi$. Now, the triangle $SEV$ gives the equation $\backslash sin.\; \backslash psi\; =\; \backslash frac\{R\}\{r\}\; \backslash sin.\; \backslash phi$, from which, by differentiating, we get

$R\; \backslash cos.\; \backslash phi\; :\; r\; \backslash cos.\; \backslash psi$. Comparing this with the above proportion, we have $T\; :\; t\; ::\; R\; \backslash cos.\; \backslash phi\; :\; r\; \backslash cos.\; \backslash psi$, whence $r\; \backslash cos.\; \backslash psi$

whence we derive

or, if we suppose $\backslash frac\{r\}\{R\}\; =\; r\text{'}$, and $\backslash frac\{t\}\{T\}\; =\; \backslash ell$,

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 $r\text{'}$, 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

0.48208; and $\backslash phi\; =\; 28^\backslash circ\; 49\text{'}$. The angle $\backslash psi$ is obtained from the equation $\backslash sin.\; \backslash psi\; =\; \backslash frac\{\backslash sin.\; \backslash phi\}\{r\text{'}\}$, which gives $\backslash psi\; =\; 138^\backslash circ\; 11\text{'}$.

The angle $VSE$ at the sun, or the heliocentric distance of Venus and the Earth, which we shall designate by $\backslash chi$, is $180^\backslash circ\; -\; (\backslash phi\; +\; \backslash psi)\; =\; 13^\backslash circ$. From this angle we likewise obtain the time from the station to the conjunction; for since the mean daily motion of Venus is $1^\backslash circ\; 36\text{'}\; 7\text{'}\text{'}.8$, and that of the earth $59\text{'}\; 8\text{'}\text{'}.3$, the relative daily motion is $36\text{'}\; 59\text{'}\text{'}.5$,

therefore the time from conjunction will be $\backslash frac\{13^\backslash circ\}\{36\text{'}\; 59\text{'}\text{'}.5\}$ or about 21 days, and consequently the retrograde motion of Venus will continue about 42 days.

If in the equation $\backslash sin.\; \backslash phi\; =\; \backslash frac\{1\}\{R\}\; \backslash left(\; \backslash frac\{r^2\; T^2\; -\; R^2\; t^2\}\{T^2\; -\; t^2\}\; \backslash right)^\{\backslash frac\{1\}\{2\}\}$ we suppose $rT\; =\; Rt$, the angle $\backslash phi$ 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 $rT$, $\backslash phi$ 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.

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 is, 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 S$\backslash phi$, E$\backslash phi$, be drawn from S and E to the first point of Aries; and because the parallax of the fixed stars is insensible, their distance being infinitely great in comparison of the distance of the earth from the sun, S$\backslash phi$ and E$\backslash phi$ are considered as parallel to one another. 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 formulae of spherical trigonometry. In fact, PL being a circle of latitude, the arc $\backslash phi$NL is the Heliocentric Longitude of the planet, $\backslash phi$N 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 $\backslash phi$: so that the longitude of the planet is obtained by adding NP to the longitude of the node. 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 $\backslash phi$EL and $\backslash phi$SL 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 $\backslash phi$EL and $\backslash phi$ES, 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 Prostaphaeresis 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

Theoretical earth is a tangent to the earth's orbit. In any other si-
Astronomy. tuation of the planet the sine of the annual parallax will
be to the sine of the elongation as SE to SL.

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 place 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 $+\; 180^\backslash circ$, 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 circum-
stances 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 circum-
stance 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 in-
stant 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 $\backslash angle\; EN$
is obtained from observation, being the geocentric longi-
tude of the node; and $\backslash angle\; ES$, the longitude of the sun, is
known from the solar tables. Hence, in the triangle
ENS the angle $\backslash angle\; SEN\; =\; \backslash angle\; EN\; -\; \backslash angle\; ES$ 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 de-
termined, and we shall then have the heliocentric longi-
tude $\backslash angle\; SN$ (equal to $180^\backslash circ\; -\; \backslash angle\; SN\; -\; \backslash angle\; ES$), 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, $\backslash frac\{SN\}\{SE\}\; =\; \backslash frac\{\backslash sin.\; SEN\}\{\backslash sin.\; ENS\}$

$=\; \backslash frac\{\backslash sin.\; (\backslash angle\; EN\; -\; \backslash angle\; ES)\}\{\backslash sin.\; (\backslash angle\; SN\; -\; \backslash angle\; EN)\}$; but from this equation nothing
can be deduced, inasmuch as it contains two unknown
quantities, SN and $\backslash angle\; SN$. 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. $\backslash frac\{SN\}\{Se\}\; =\; \backslash frac\{\backslash sin.\; SeN\}\{\backslash sin.\; eNS\}\; =\; \backslash frac\{\backslash sin.\; (\backslash angle\; eN\; -\; \backslash angle\; eS)\}\{\backslash sin.\; (\backslash angle\; SN\; -\; \backslash angle\; eN)\}$, exact-

ly 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 observa-
tions, this last equation will furnish a second relation be-
tween the two unknown quantities SN and $\backslash angle\; SN$, by com-

bining 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 ob-
servations 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 ob-
served 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
of the planet, which is then PEN (fig. 71), is given, as LXXXVI.

also its geocentric latitude PEL. Now, by reason of the
right-angled triangles PEL and PSL, we have $PL\; =\; EL$
$\backslash tan.\; PEL\; =\; SL\; \backslash tan.\; PSL$, consequently $\backslash tan.\; PSL$
$=\; \backslash frac\{EL\}\{SL\}\; \backslash tan.\; PEL$; but $EL\; :\; SL\; ::\; \backslash sin.\; LSN\; :\; \backslash sin.\; LES$,

therefore $\backslash tan.\; PSL\; =\; \backslash frac\{\backslash sin.\; LSN\}\{\backslash sin.\; LES\}\; \backslash tan.\; PEL$. Now, by Na-
pier's rules for circular parts, the right-angled spherical
triangle PNL gives also $\backslash sin.\; NL\; =\; \backslash cot.\; PNL\; \backslash tan.\; PL$;
whence $\backslash tan.\; PL\; =\; \backslash sin.\; NL\; \backslash tan.\; PNL$; or, since $\backslash tan.\; PL$
$=\; \backslash tan.\; PSL$, and $\backslash sin.\; NL\; =\; \backslash sin.\; LSN$, therefore $\backslash tan.\; PSL$
$=\; \backslash sin.\; LSN\; \backslash tan.\; PNL$. By equating these two values of
$\backslash tan.\; PSL$, we have

therefore, ultimately,

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 ac-
curacy by a simple proportion. It requires also to be re-
marked that the above method of determining the incli-
nation 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 er-
ror, especially if at the time of the observation the planet
is not very distant from its quadratures. In fact, if we
make $PNL\; =\; I$, $PEL\; =\; \backslash lambda$, and $LES\; =\; \backslash phi$, the above

equation becomes $\backslash tan.\; I\; =\; \backslash frac\{\backslash tan.\; \backslash lambda\}\{\backslash sin.\; \backslash phi\}$ which being differentiated

with respect to $I$ and $\backslash phi$, gives $d\; \backslash tan.\; I\; =\; -\backslash frac\{d\backslash phi\; \backslash cos.\; \backslash phi\; \backslash tan.\; \backslash lambda\}\{\backslash sin^2\; \backslash phi\}$.

But $d\; \backslash tan.\; I\; =\; \backslash frac\{dI\}\{\backslash cos^2\; I\}$ and from the equation itself we

derive $\backslash cos^2\; I\; =\; \backslash frac\{\backslash sin^2\; \backslash phi\}\{\backslash sin^2\; \backslash phi\; +\; \backslash tan^2\; \backslash lambda\}$; therefore, by substituting

From this it is evident that the error of inclination re-
sulting from an erroneous position of the node will be
smaller in proportion as $\backslash phi$ is greater, and will disappear
altogether when $\backslash phi$ (that is, LES or the geocentric dis-
tance of the planet from its node) is $90^\backslash circ$, that is to say, at
the quadratures.

The process now explained for determining the posi-
tion of the nodes gives at the same time the length of

theoretical 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. Theoretical 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 $50\backslash cdot1$, from the numbers given in the table.

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 elided. 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 the plane of the ecliptic in the same straight line $ESL$; LXXXVII. 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,

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 Astronomy, vector; for $\backslash tan.\; PL\; =\; \backslash tan.\; PNL\; \backslash sin.\; NL$; 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 $PS\; :\; SE\; ::\; \backslash sin.\; PES\; :\; \backslash sin.\; EPS$, whence $PS\; =\; SE\; \backslash frac\{\backslash sin.\; PES\}\{\backslash sin.\; EPS\}$ 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 four 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 those which express the mean distances of the planets from the sun.

These three general laws, which are necessary consequences of the law of gravitation, 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 classed 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, inconsiderable, 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 $\backslash mathcal\{O\}$ to $\backslash mathcal\{O\}\text{'}$, will be different from that in which it passes from $\backslash mathcal\{O\}\text{'}$ to $\backslash mathcal\{O\}$; 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. Having already

Theoretical explained in last section the manner in which the geocentric are converted into heliocentric longitudes, and reduced to the plane of the orbit, let us assume $v,\; v\text{'},\; v\text{'}\text{'}$, to represent three heliocentric longitudes so reduced, and $\backslash sigma$ the longitude of the perihelion, also reduced to the plane of the orbit. The true anomalies corresponding to the three observations will consequently be $v\; -\; \backslash sigma,\; v\text{'}\; -\; \backslash sigma,\; v\text{'}\text{'}\; -\; \backslash sigma$. Now, if, as in the series given in chap. ii. sect. 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 $nt,\; nt\text{'},\; nt\text{'}\text{'}$; and it will be observed, that although $t$ is unknown, yet as $t\text{'}\; -\; t$ and $t\text{'}\text{'}\; -\; t$ (the intervals between the first and the other two observations) are known, $t$ and $t\text{'}$ 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,

from which we must deduce the three unknown quantities $t,\; e,\; \backslash sigma$, 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

and if we assume

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

whence, by division,

Now, the numerator of this fraction

and, similarly, the denominator

an equation which gives $\backslash sigma$, the longitude of the perihelion reduced to the orbit, the angles $v,\; v\text{'},\; v\text{'}\text{'}$, being given by observation.

Having thus found $\backslash sigma$, and consequently $v\text{'}\; -\; \backslash sigma,\; v\text{'}\text{'}\; -\; \backslash sigma$, it is easy to determine the eccentricity $e$. From the equation

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

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

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

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 causes 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

Theoretical known the existence of inequalities, can neither detect Astronomy. 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-

ing. The perihelia are also gradually shifting their places Theoretical on the planes of the orbits. These motions are direct in Astronomy. the case of all the planets excepting Venus, the perihelion of whose orbit, when referred to the fixed stars, moves in a direction contrary to that of the signs.

The following table exhibits the mean motions of the planets, and the elements and positions of their ellipses on the planes of their orbits, at the commencement of the year 1801. For the four new planets the epoch is 1820.

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 his opposition, was observed at the same instant of time by Lacaille 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".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

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 $\backslash frac\{1\; \backslash times\; 8381\}\{0.435\}\; =\; 19226$ 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 approximative, 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 Astronomy rection 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°5 and 8°7. The mean 8°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. 505, 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.

Here the mean of the first 5 results is nearly... 8°59
of the next 5..... 8°41
of the next 4..... 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 Theoretical Astronomy we have

$\backslash sin.\; 8^\{\backslash circ\}6\; :\; 1\; ::\; \backslash text\{radius\; of\; earth\}\; :\; \backslash text\{sun\text{'}s\; distance\};$
that is, on reducing the radius of a circle to seconds,
sun's distance = $\backslash frac\{360^\{\backslash circ\}\; \backslash times\; 60\; \backslash times\; 60\}\{8^\{\backslash circ\}6\; \backslash times\; 2\; \backslash times\; 3\; \backslash cdot\; 14159\}\; =\; 23984\; \backslash text\{\; terrestrial\; radii.\}$

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 $23984\; \backslash times\; 4000\; =\; 95936000$ miles.

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.

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 $T\; :\; t\; ::\; 1\; :\; \backslash frac\{t\}\{T\}$

= part of a circumference described by A in the time t. But during the same time t, B describes a whole circumference; therefore $1\; -\; \backslash frac\{t\}\{T\}$ 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 $1\; -\; \backslash frac\{t\}\{T\}\; :\; 1\; ::\; t\; :\; S$, whence $S\; =\; \backslash frac\{Tt\}\{T-t\}$.

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 $T\; =\; 365\; \backslash cdot\; 256$ days, and $t\; =\; 87\; \backslash cdot\; 969$ days; consequently $T\; -\; t\; =\; 277\; \backslash cdot\; 287$, and $S\; =\; \backslash frac\{365\; \backslash cdot\; 256\; \backslash times\; 87\; \backslash cdot\; 969\}\{277\; \backslash cdot\; 287\}\; =\; 115\; \backslash cdot\; 877$ days, which, therefore, is the time of a synodic revolution of Mercury.

In the case of Venus we have $t\; =\; 224\; \backslash cdot\; 700$, whence $T\; -\; t\; =\; 365\; \backslash cdot\; 256\; -\; 224\; \backslash cdot\; 700\; =\; 140\; \backslash cdot\; 556$, and consequently $S\; =\; \backslash frac\{365\; \backslash cdot\; 256\; \backslash times\; 224\; \backslash cdot\; 700\}\{140\; \backslash cdot\; 556\}\; =\; 583\; \backslash cdot\; 92$ days; which is the period of her synodic revolution.

Theoretical Astronomy. 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 $mT\; =\; \backslash frac\{nT\}\{T-t\}$, whence

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:—

Here the first approximating fraction is $\backslash frac\{1\}\{3\}$, 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 $3\; +\; \backslash frac\{1\}\{6\}\; =\; \backslash frac\{6\}\{19\}$, 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 $\backslash frac\{87969\}\{277287\}$, namely,

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 $\backslash frac\{t\}\{T\}$; but it is unnecessary to have recourse again to division, inasmuch as they are easily obtained from the above. For example, since the fraction $\backslash frac\{1\}\{3\}$ 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

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 $\backslash frac\{m\}\{n\}\; =\; \backslash frac\{t\}\{T-t\}$, we shall have $t\; =\; 224.7008$, $T-t\; =$

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

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 $20^\backslash circ$ or $24^\backslash circ$; in sixteen years the change of latitude increases to $40^\backslash circ$ or $48^\backslash circ$, 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, $113\; \backslash pm\; 8$ 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 $\backslash pm\; 8$ 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.)

Theoretical SECT. III.—Of the Physical Constitution of the Planets, their Magnitude, Rotation, and other remarkable objects.

Mercury.

Mercury is a small star, 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 $6^\{\backslash circ\}02\text{'}$; 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 $\backslash frac\{1\}\{117\}$ of the radius of the earth; one of the mountains in the moon has been estimated at $\backslash frac\{1\}\{218\}$ of her radius; the highest in Venus at $\backslash frac\{1\}\{125\}$; and one in Mercury at $\backslash frac\{1\}\{128\}$. 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 star 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 $75^\{\backslash circ\}$. 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 (perpendicular to CD, the axis of the ecliptic); 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 21.005 revolutions. That intervals of 121 days 14 hours 25 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, 165.09 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 165 revolutions exactly, supposing the observations to have been perfectly accurate. On dividing the intervals by 21, 125, 146, and 165 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 bable result at least will be obtained. In this manner Astronomy. 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 $\backslash frac\{1\}\{500\}$, 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' 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 fifth 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 1665; and after having continued his observations for a month, he found they re-

turned to the same situation in 24 hours and 40 min. The Theoretical planet was observed by some astronomers at Rome with Astronomy. 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 $2\backslash frac\{1\}\{2\}^\backslash circ$ 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, astro-Bright nomers had noticed that a segment of his globe about spots about the poles of Mars. the south pole exceeded the rest of his disk so much in brightness, that it appeared to project as if it were the 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 observa- Herschel's tions on this planet is given in the 74th volume of the account of Philosophical Transactions. Some of the remarkable ap- these spots pearances there described are represented in fig. 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 25° 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 Martian year appear very different from what we enjoy, when compared to the surprising duration of the years of Jupiter, Saturn, and the Georgium 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 Theoretical so faint before its occultation, that it could not be seen by Astronomy. 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 treme smallness, the difficulty of finding it after its emer-
Astronomy. gence 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 tutelar
goddess of that country, Ceres; and her emblem,
the sickle, ♁, 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 3.2 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 nebulousity which
surrounds the planet renders it almost impossible to dis-
tinguish the true disk; and hence arises the great dis-
crepancy 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 con-
ceives that there is little chance of discovering the period
of its rotation.

Discovery of Pallas.
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 tri-
angle with the stars 20 and 191 of Virgo in Bode's cata-
logue. 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 bril-
liancy; but after examining it for two hours, he remark-
ed 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 as-
cension had diminished 10', while its northern declination
had increased 20'. From observations continued during
a month, M. Gauss calculated an elliptic orbit, the eccen-
tricity of which amounted to .24764, much greater than
that of any of the other planets. He also found its in-
clination to be 34° 39', exceeding the aggregate in-
clinations of all the other planetary orbits; and its mean
distance 2.770552, 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 dis-
covered. 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 ex-
tended even to the poles. Dr Olbers gave the new planet
the name of Pallas, choosing for its symbol the lance, ♃,
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 subject of an ingenious memoir, in which he deter-
Astronomy. mined the explosive force necessary to detach a fragment
from a planet with a velocity that would cause it to de-
scribe 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 $\backslash frac\{121\; \backslash text\{\; or\; \}\; 156\}\{\backslash sqrt\{\backslash text\{mean\; distance\}\}\}$, 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 con-
sidering, 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 si-
milar to their actual orbits. This hypothesis served also
to explain the great eccentricities and inclinations by
which these planets are distinguished from the others be-
longing 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 proba-
bly 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 com-
mon nodes, in which the several fragments would neces-
sarily be found at each revolution. Dr Olbers therefore
proposed to examine carefully every month the two op-
posite parts of the heavens in which the orbits of Ceres
and Pallas intersect each other, with a view to the disco-
very 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 demonstrated the soundness of his conjecture,
and gave a degree of probability to his hypothesis.

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 consider-
able variations, the cause of which is uncertain.

While M. Harding, of the observatory of Lilienthal,
near Bremen, was engaged in forming a complete zodiac
of the small telescopic stars near the orbits of Ceres and
Pallas, with which these planets were likely to be con-
founded, he determined, on the 22d 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 defilé where, according
to Dr Olbers, an observer would be certain of detecting
in their passage the other fragments of the original plan-
et 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. Hard-
ing, 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 inter-
val 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 with-
out any nebulousity, and of a whitish colour. A few days

theoretical afterwards the elements of its orbit were computed by Gauss. This planet has received the name of Juno, and for its symbol ♃, the starry sceptre of the queen of Olympus.

Junio 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 nebulousity, 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 $5\backslash frac\{1\}\{2\}$ 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 $36^\{\backslash circ\}7\text{'}$, and varies between $45^\{\backslash circ\}8\text{'}$ and $30^\{\backslash circ\}$, would subtend an angle of $3^\{\backslash circ\}17\text{'}$ if seen at the same distance as the sun. From Jupiter the sun will appear under an angle of $6\text{'}$ 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 2444 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. 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 $52\backslash frac\{1\}\{2\}$ 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 $3^\{\backslash circ\}5\text{'}30\text{'}\text{'}$, 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 2.4 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 the compression of Jupiter is about $\backslash frac\{1\}\{13\}$th of his radius, the diameter of his equator being to that of his poles as 15 to 14 nearly, while that of the earth is only $\backslash frac\{1\}\{37\}$th. According to Struve, the equatorial diameter at the mean distance subtends an angle of $38^\{\backslash circ\}32\text{'}$, the polar $35^\{\backslash circ\}53\text{'}$; and the

ellipticity is $0.0728\; =\; \backslash frac\{1\}\{13.71\}$.

The annual parallax of Jupiter is less than $12\text{'}\text{'}$, consequently the earth, as seen from Jupiter, will never appear at a greater distance than $11^\{\backslash circ\}$ or $12^\{\backslash circ\}$ from the sun. The digressions of Mars would be $17^\{\backslash circ\}2\text{'}$, those of Venus $8^\{\backslash circ\}$, and those of Mercury only $4^\{\backslash circ\}16\text{'}$. 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 splen-
dour, 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.

Satellites of Jupiter. 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 fol-
lows 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 nar-
row 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 elon-
gation 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 minia-
ture all the phenomena of the planetary system.

Occulta-
tions and
eclipses of
the satel-
lites.
Since the satellites revolve in orbits about the huge orb
of Jupiter, it is evident that occultations of them must fre-
quently 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 re-
volution; the third very rarely escapes an occultation; but
the fourth more frequently, by reason of its greater dis-
tance. It is seldom that a satellite can be discovered
upon the disk of Jupiter, even by the best telescopes, ex-
cepting 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. Some-
times, however, a satellite in passing over the disk ap-
pears 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 sat-
ellite 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 sub-
ject to change, or, if they be permanent, like those of our
moon, that the different portions of the surfaces of the satel-
lites are not equally luminous, and that at different times
they turn different parts of their globes towards us. Possi-
bly both these causes may contribute to produce the phe-
nomena just mentioned. By reason of the spots, also, both
the light and apparent magnitude of the satellites are vari-
able; 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 Theoreti-
the distance of that planet from the sun, the satellites Astrono-
are illuminated by the sun very nearly in the same man-
ner with the primary itself; hence they appear to us al-
ways 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, con-
stantly turn the same face towards their primary, or com-
plete 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 semi-
circles, 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 suffi-
cient 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 vi-
sible 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 ad-
vances 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 phe-
nomena 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 pass-
ing from A to C, or when Jupiter advances from his op-
position 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-

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 $=\; \backslash frac\{1\}\{2\}$, of the second $=\; \backslash frac\{1\}\{2\}$, of the third $=\; \backslash frac\{1\}\{2\}$, of the fourth $=\; \backslash frac\{1\}\{2\}$. 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:—

These dimensions are adapted to the mean distance of Jupiter, namely 5.20279. Compared with the earth, the diameters of the satellites are approximately as follows:—I. $=\; \backslash frac\{1\}\{2\}$, II. $=\; \backslash frac\{1\}\{2\}$, III. $=\; \backslash frac\{1\}\{2\}$, IV. $=\; \backslash frac\{1\}\{2\}$. 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 9½ times the semidiameter of the terrestrial orbit, or nearly 900 millions of miles. His apparent diameter at his mean distance is only about 16.2, 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 $\backslash frac\{1\}\{50\}$ 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 .55, or about $\backslash frac\{1\}\{2\}$ 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 anse. Sometimes the anse 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 9½ 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 his 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. At present (1830) the south side is presented to us; it will be invisible in 1833; its northern side will become visible in 1838; it will again disappear in 1847, and 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:—

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.)

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. But it is very remarkable that the globe of Saturn appears to be flattened at the equator as well as at the poles. The polar compression extends to a great distance over the surface of the planet, and the greatest diameter is that of the parallel of 43° of latitude, where, consequently, the curvature of the meridians is also the greatest. The disk of Saturn, therefore, resembles a square of which the four corners have been rounded off. According to the latest observations of Herschel, the axis of rotation, the diameter of the

equator, and the greatest diameter under the parallel of Theorekal 43°, are to one another as the numbers 32, 35, 36. (See Astronomy, the Philosophical Transactions for 1806, Part II.)

Saturn is attended by seven satellites, but so small Satellites that they can only be seen by the help of powerful tele- of Saturn. scopes. Huygens first discovered one of these satellites in 1655. It is the sixth in the order of distance, and is the largest of them all. Four others were discovered about twenty years afterwards by Dominic Cassini; and, lastly, 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. All the satellites appear to revolve in the plane of the ring, with the exception of the two last. The inclinations of their orbits are, however, not known with much certainty.

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 radius of the ecliptic, or about 1900 millions of miles; and his sidereal revolution is performed in 83 years 150 days and 18 hours. 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 incon siderable. 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 appeared only as a star of the seventh magnitude. 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 recognised 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, so that Uranus is accompanied by six satellites; and the whole number of satellites now known to belong to the system is consequently eighteen.

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:—

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 $4\; +\; 3\; \backslash cdot\; 2^\{n-2\}$, 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 revolvibilis Observationes Bononienses, Bonon. 1666; Idem, Disceptatio Apologetica de Maculis Jovis et Martis, Bonon. 1667; Idem, Nouvelles Découvertes dans le Globe de Jupiter, Paris, 1690; Bianchini, Hesperi et Phosphori Nova Phenomena, 1728; Cassini, Éléments d'Astronomie, 1740; Schroeter, Aphroditeographische Fragmente, Helmstadt, 1796; Idem, Lilien-thalische Beobachtungen der neu entdeckten planeten, Göttingen, 1805; Gauss, Theoria Motus Corporum Celestium, Hamburg, 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 eclipsed except in passing from the west to the east of the planet. When an eclipse takes place 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 $3^\backslash circ\; 5\text{'}\; 30\text{'}\text{'}$. The inclination of the orbit of the second to the plane of the planet's equator is $27^\backslash circ\; 49\text{'}\; 2\text{'}\text{'}$. 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 $12^\backslash circ\; 20\text{'}$; 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 $3^\backslash circ\; 19\text{'}\; 24\text{'}\text{'}\; 6$ and $2^\backslash circ\; 51\text{'}\; 35\text{'}\text{'}\; 4$, and that of the third between $3^\backslash circ\; 17\text{'}\; 50\text{'}\text{'}$ and $2^\backslash circ\; 53\text{'}\; 10\text{'}\text{'}$. 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 $13\text{'}\; 16\text{'}\text{'}\; 4$, and at its minimum in 1777, only to $5\text{'}\; 7\text{'}\text{'}\; 5$. 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 $42^\backslash circ\; 58\text{'}\; 7\text{'}\text{'}$ 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 also participates in 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 $m\text{'}$, $m\text{'}\text{'}$, $m\text{'}\text{'}\text{'}$, the mean motions of the three satellites respectively, and their mean longitudes by $l\text{'}$, $l\text{'}\text{'}$, $l\text{'}\text{'}\text{'}$, these laws are expressed by the formulas,

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 $l\text{'}\; +\; 2l\text{'}\text{'}\; -\; 3l\text{'}\text{'}\text{'}\; =\; 0$. 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, there-

fore, 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 $30^\backslash circ$, 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, Lalande makes the inclination of this satellite $22^\backslash circ\; 42\text{'}$ to the orbit of Saturn, or $24^\backslash circ\; 45\text{'}$ 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 must maintain the ring, and the orbits of the interior satellites, in the plane of his equator. But the amount of the compression is unknown: 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 $67^\backslash circ\; 25\text{'}\; 47\text{'}\text{'}$; that of its inferior apside $203^\backslash circ\; 35\text{'}\; 7\text{'}\text{'}$. Its mean motion in 36525 days is 2290 revolutions $+\; 202^\backslash circ\; 12\text{'}$; in one day

coreal 22° 34' 37.186. The eccentricity is .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° 55'; 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.

The comets form a class of bodies belonging to the solar system, distinguished from the planets by their physical appearances and the great eccentricity of their orbits. By reason of the smallness of their diameters, and a nebulousness 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.

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 Dörffel 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 hyperbolas, 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 Burekhardt, 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 its 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 nebulousity. 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 first of the three periodic comets with which we are yet 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 1006; 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 ob-

ject so striking and so terrific could not fail, in a superstitious 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 will return to its perihelion in 1835; but whether on this occasion it will present any resemblance to its former appearances, or whether it will even be visible in Europe, cannot be certainly determined. The following table of its elements in 1835 is given by Pontécoulant (Théorie Analytique du Système du Monde, tome ii. p. 147):

The two other 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 Pontécoulant, are its elements for 1829–30, computed from the observations at Paramatta:

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.

The third periodic comet with which our knowledge of the solar system has recently been enriched receives its name from Biela, by whom it was first perceived in Bohemia, on the 25th of February 1825. The parabolic elements computed from the first observations presented a striking resemblance with those of two comets

Theoretical which had been observed in 1772 and 1806, which induced MM. Clausen and Gambart, the first 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 will return to its perihelion in November 1832, about the same time with Encke's. The following table of its elements has been computed from the observations of 1826, and the theory of the perturbations (Pontécoulant, tome ii. p. 158):

Such is the present state of astronomy with respect to periodic comets. There are two others besides, 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 1848.

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

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 130 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 Freret, 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 Gaius, 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 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 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

Theoretical branches, and surrounding the nucleus, was a space com-
Astronomy, paratively obscure.

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 nebulousity. 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 Lahire. But the nebulousity 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 163d 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 Theoretical curvature which is generally observed may be accounted Astronomy, 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

oretical 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 two 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 upwards of 150 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, Quæst. Natural. vii.; Hevelii Cometographia; Newton, De Mundi Systemate, et Princip. lib. iii. prop. 42; Halley, Synopsis Astronomiae Cometice, et Phil. Trans. vol. xxiv.; Euler, Mém. Acad. Berlin, 1756; Dionis du Séjour, Essai sur les Comètes, Paris, 1775; Lexell, Phil. Trans. 1779; Clairaut, Mém. Acad. Paris, 1760; Lambert, Lettres Cosmologiques, et Mém. Acad. Berlin, 1771; Bode, ibid. 1786, 1787; Sir H. Enfield on the Orbits of Comets, 4to, London; Pingré, Cometographie, 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.

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 cele-

tial 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 $\backslash alpha$, the next most conspicuous is called $\backslash beta$, the third $\backslash gamma$, 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 $\backslash alpha$ Aquarii is a star of the same order of brightness as $\backslash gamma$ 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; Rigel, 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; Fomalhaut; 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 (αὐγέωτες, 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 CONSTITUTIONS.

North of the Zodiac.

The constellations added by Hevelius are the following:—

The constellations added by Halley in the southern hemisphere are,—

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 Catalogue of stars. 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 Astronomia in 1690.

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

Catalogues of Lacaille.—The first of these, published in his Astronomia Fundamenta, contains 397 stars; the second, which is given in his Cælum 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 Astronomia, 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.

Theoretical Astronomy. Catalogue of the Astronomical Society of London, containing 2581 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.

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 it is even doubtful, after all the attempts that have been made to detect it, whether it is at all sensible to the best instruments. 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 immense line is 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 circle of the Greenwich Observatory, the chef d'œuvre of Troughton. Astronomers are, however, not entirely agreed as to the fact that the parallax is altogether insensible; and Dr Brinkley of the Dublin Observatory has perceived indications of its existence with regard to several stars. Dr Brinkley could dis-

cover no parallax in his observations on the circumpolar stars, with regard to which the changes of declination resulting from the precession are most appreciable. He did not even observe it in the case of the stars of the constellation 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 (α Lyre) he found a parallax of 1".13, and one of 1".42 in the star Athair, in the Eagle. Bradley supposed the parallax of Sirius to amount to 1". These facts are, however, disputed 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, thinks the probable value of the parallax cannot exceed 0".018, 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 $\backslash frac\{1\}\{\backslash sin.\; 1\text{'}\text{'}\}\; \backslash times\; 95,000,000\; =\; 206,264\; \backslash times\; 95,000,000\; =\; 19,595,080,000,000$, or about 20 billions 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 $206,264\; \backslash times\; 8$ 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 Magni-stars appear to be very different,—a circumstance which may be attributed either to a real diversity of magnitude, 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 $\backslash frac\{1\}\{2\}$, that of Aldebaran $1\backslash frac\{1\}\{5\}$, and that of Capella $2\backslash frac\{1\}\{5\}$. Supposing the measurement accurate, and the annual parallax of this last not to exceed $1\text{'}$, its volume would be equal to 20 million times that of the sun.

Number of the stars. 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 $15^\backslash circ$ by $2^\backslash circ$, 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 po-

sition 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 $\backslash lambda$ in the constellation Hercules. This conjecture has not, however, been confirmed, or rather it has been entirely overthrown, by subsequent observations. 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. The proper motions of the stars which have hitherto been remarked are evidently subject to no one assignable law, and are directed to many different points in space. It is therefore infinitely probable that they are due, in part at least, to a real dis-

Theoretical placement of the stars, and not to a general translation of Astronomy, the solar system.

Some recent observers have supposed they have detected proper motions in a great number of stars, the rates of which they have inserted in their catalogues along with those of the precession; but there is great discordancy in the determinations of different astronomers. 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 $\backslash gamma$ Pegasi, $\backslash alpha$ Ceti, Rigel, Sirius, Spica, $\backslash gamma$ and $\backslash beta$ Aquila, $\backslash alpha$ Cygni, $\backslash alpha$ Aquarii, and $\backslash alpha$ Pegasi, are all positive according to Zach, while Maskelyne considers them as being all negative. Such is the uncertainty respecting the proper motions even of the Greenwich stars, which, by reason of the frequent observations they have undergone, and the rigorous scrutinies to which they have been subjected, are probably those whose places are 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 Theoretical the errors of observation. The double star 61 Cygni is indeed supposed by Bessel to have an annual proper motion of $+5^\{\backslash circ\}06$ in right ascension, and $3^\{\backslash circ\}34$ 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 $0^\{\backslash circ\}2$, in the arc of a great circle. The following table, published by Mr Baily 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 $0^\{\backslash circ\}5$ 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.

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 $M\text{'}$ 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

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 $M\text{'}$, 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 develope their rates and directions, and assign the distant centres round which they are performed.

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 $\backslash delta$ in the Great Bear; others, like $\backslash beta$ 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, $\backslash delta$ Cephei one of $5\backslash frac\{1\}\{2\}$ days, $\backslash beta$ Lyrae one of $6\backslash frac\{1\}\{2\}$ days, $\backslash mu$ Antinoi 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 phenomena. 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 obscurations 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 Messrs Herschel and South have placed this singular 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 feeble 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 $\backslash xi$ Ursa Majoris, in which the angular velocity is $6^\{\backslash circ\}4$ 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. $\backslash alpha$ Castoris, $\backslash gamma$ Virginis, $\backslash zeta$ Canceri, $\backslash xi$ Bootis, $\backslash delta$ Serpentis, and that remarkable double star $\backslash beta$ 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 2100 of them. Messrs South and Herschel have given a catalogue of 380 in the Transactions of the

The Astronomical Royal Society for 1824, and South added 458 to the list 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, $\backslash xi$ of Cancer, and $\backslash xi$ of the Balanee, 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, $\backslash mathcal\{A\}$. 19 hours 8 minutes 56 seconds, P. D. $95^\{\backslash circ\}\; 45\text{'}\; 33\text{'}\text{'}$, 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), $\backslash mathcal\{A\}$. 6 hours 12 minutes, P. D. $84^\{\backslash circ\}\; 12\text{'}$, 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 Nebula, 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 atmospheric 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. Mr 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 Huygenian 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 cirrus 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 Mr 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 ($10\text{'}\text{'}$ or $12\text{'}\text{'}$), but totally devoid of any distinct outline, so that it is impossible to say precisely where the nucleus ends and the nebula begins. Its nebosity is of the most perfect milky, absolutely irresolvable kind, without the slightest tendency to that separation

Physical Astronomy. into flocculi above described in the nebula of Orion; nor is there any sort of appearance of the smallest star in the centre of the nipple. This nebula is oval, very bright, and of great magnitude, and altogether a most magnificent object."

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 for 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 as 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 any thing more than mere fancies and speculations. Even if every link in the chain were per-

fect, 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, Historia miræ Stellæ in Collo Ceti, anno 1660; Maupertuis, Figure des Astres, Œuvres, tome i. Lyon, 1756; Michel, Phil. Trans. 1784; Pigott, Phil. Trans. 1785, 1797; Lemonnier, Mém. Acad. Par. 1789; Lalande, Astronomie (885); Herschel, Phil. Trans. passim. (s.)

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

¹ This 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.

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,

I. 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 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 DF. 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, SF, 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 infinitely small, and their number infinitely great, that is, when the action of the centripetal force is continued.

But when the intervals of time become infinitely small, the rectilinear 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 rectilinear 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.

Physical Astronomy. 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 $ACA\text{'}$, fig. 110, be the curve which a body describes about the centre $S$. Let $Aa$ and $A\text{'}a\text{'}$ be two arcs of the curve, described in the same indefinitely small portion of time. Join $Sa$, $Sa\text{'}$, then the areas of the triangles $ASa$, $ASa\text{'}$ are equal by this proposition. At $A$ and $A\text{'}$, draw the tangents $AB\text{'}$, $A\text{'}B\text{'}$, and from $S$ let fall on them the perpendiculars $SB$ and $SB\text{'}$. Because the areas of the triangles $ASa$, $ASa\text{'}$ are equal, $Aa\; \backslash times\; SB\; =\; A\text{'}a\text{'}\; \backslash times\; SB\text{'}$, or $Aa\; :\; A\text{'}a\text{'}\; ::\; SB\text{'}\; :\; SB$; but $Aa$ is to $A\text{'}a\text{'}$ as the velocity of the body describing the curve at $A$ to its velocity at $A\text{'}$, therefore these velocities are inversely as the perpendiculars $SB$, $SB\text{'}$.

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 transverse, and $DE$ the conjugate axis, $C$ 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 $P\; =\; v$. 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 $v\; =\; c\; \backslash cdot\; \backslash frac\{DC\}\{SG\}$.

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 $SP\; :\; PF\; ::\; SG^2\; :\; SG\; \backslash times\; FH$. But $SG\; \backslash times\; FH\; =\; CD^2$, therefore $SP\; :\; PF\; ::\; SG^2\; :\; CD^2$, and $\backslash frac\{CD^2\}\{SG^2\}\; =\; \backslash frac\{PF\}\{SP\}$. Now $v\; =\; c\; \backslash cdot\; \backslash frac\{DC\}\{SG\}$, or $v^2\; =\; c^2\; \backslash cdot\; \backslash frac\{CD^2\}\{SG^2\}$, and therefore $v^2\; =\; c^2\; \backslash cdot\; \backslash frac\{PF\}\{SP\}$.

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 $SN$ = the transverse axis $AB$, $v^2\; =\; c^2\; \backslash cdot\; \backslash frac\{NL\}\{LS\}\; =\; c^2\; \backslash cdot\; \backslash frac\{SN\; -\; SP\}\{SP\}$. 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 $Bb$, and which is employed in accelerating the body moving in that line, is $G\; \backslash times\; \backslash frac\{Bf\}\{BE\}$; and the increment of the space

being $Bb$, therefore $2G\; \backslash times\; \backslash frac\{Bf\; \backslash times\; Bb\}\{BE\}$ is the momentary increment of the square of the velocity of the body at $B$. But $Bf\; \backslash times\; Bb\; =\; BE^2$, because $BEb$ is a right-angled triangle, and $Ef$ the perpendicular on the hypothenuse. Therefore $2G\; \backslash times\; \backslash frac\{Bf\; \backslash times\; Bb\}\{BE\}\; =\; 2G\; \backslash times\; \backslash frac\{BE^2\}\{BE\}\; =\; 2G\; \backslash times\; BE\; =\; 2G\; \backslash times\; Dd$. But $2G\; \backslash times\; Dd$ 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 $c\; \backslash times\; \backslash sqrt\{\backslash frac\{AF\}\{AS\}\}$, or (if $AN$ in the axis produced be taken equal to $AF$) $c\; \backslash times\; \backslash sqrt\{\backslash frac\{AN\}\{AS\}\}$. 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 $c\; \backslash times\; \backslash sqrt\{\backslash frac\{PF\}\{PS\}\}\; =\; c\; \backslash times\; \backslash sqrt\{\backslash frac\{NL\}\{SL\}\}$, 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 $c\; \backslash times\; \backslash sqrt\{\backslash frac\{LN\}\{LS\}\}$. 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 is it 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 $\backslash sqrt\{\backslash frac\{AB\}\{BC\}\}$ to 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 $v$, $v\; =\; c\; \backslash sqrt\{\backslash frac\{AB\}\{BC\}\}$, by hypothesis, and

therefore $2ABED\; =\; c^2\; \backslash cdot\; \backslash frac\{AB\}\{BC\}$; and since $AB\; =\; AC\; -\; BC$,

$2ABED\; =\; c^2\; \backslash cdot\; \backslash frac\{AC\; -\; BC\}\{BC\}\; =\; c^2\; \backslash left(\; \backslash frac\{AC\}\{BC\}\; -\; 1\; \backslash right)$. For the

same reason $2ABED\; =\; c^2\; \backslash left(\; \backslash frac\{AC\}\{BC\}\; -\; 1\; \backslash right)$, and therefore the

difference of these areas, or $2BbE$, that is, $2EB\; \backslash times\; Bb\; =\; c^2\; \backslash left(\; \backslash frac\{AC\}\{BC\}\; -\; \backslash frac\{AC\}\{BC\}\; \backslash right)\; =\; c^2\; \backslash cdot\; \backslash frac\{AC\; \backslash cdot\; Bb\}\{BC^2\}$. Wherefore, dividing

by $Bb$, $2EB\; =\; c^2\; \backslash cdot\; \backslash frac\{AC\}\{BC^2\}$; or $EB\; =\; c^2\; \backslash cdot\; \backslash frac\{AC\}\{BC^2\}$; now $c^2$ and

AG are given, therefore $EB$ is inversely as $BC^2$, 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 transverse axis, and if the gravitation at that distance

$=\; F$, and the mean distance itself $=\; a$, $F\; =\; c^2\; \backslash cdot\; \backslash frac\{a\}\{a^2\}\; =\; \backslash frac\{c^2\}\{a\}$, 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 semitransverse axis, $b$ the semiconjugate, then $\backslash pi\; ab\; =$ 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 conjugate axis, therefore $\backslash frac\{1\}\{2\}\; \backslash pi\; ab$ 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 $\backslash frac\{1\}\{2\}\; \backslash pi\; ab$ will give the number of seconds in which the revolution is completed, which is therefore $\backslash frac\{\backslash pi\; ab\}\{\backslash frac\{1\}\{2\}\; \backslash pi\; ab\}\; =\; \backslash frac\{2\backslash pi\; a\}\{c\}$, or

since $c^2\; =\; aF$, the time of a revolution $=\; \backslash frac\{2\backslash pi\; a\}\{\backslash sqrt\{aF\}\}\; =\; 2\backslash pi\; \backslash sqrt\{\backslash frac\{a\}\{F\}\}$.

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 $t\text{'}$ be the times of revolution for two different planets, of which the mean distances are $a$ and $a\text{'}$, and the gravitation at those distances $F$ and $F\text{'}$,

and, by what has just been shown, $t\; :\; t\text{'}\; ::\; \backslash frac\{a^\{\backslash frac\{1\}\{2\}\}\}\{F^\{\backslash frac\{1\}\{2\}\}\}\; :\; \backslash frac\{a\text{'}^\{\backslash frac\{1\}\{2\}\}\}\{F\text{'}^\{\backslash frac\{1\}\{2\}\}\}$, or

$t^2\; :\; t\text{'}^2\; ::\; \backslash frac\{a\}\{F\}\; :\; \backslash frac\{a\text{'}\}\{F\text{'}\}$. But $F\; :\; F\text{'}\; ::\; a^2\; :\; a\text{'}^2$, by what is already

shown (Art. 9), therefore $t^2\; :\; t\text{'}^2\; ::\; \backslash frac\{a\}\{a^2\}\; :\; \backslash frac\{a\text{'}\}\{a\text{'}^2\}$, or $t^2\; :\; t\text{'}^2\; ::\; a^3\; :\; a\text{'}^3$, 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 $\backslash frac\{c^2\; a\}\{x^2\}$ ($x$ being the distance from the centre of force), and the same is true of every individual planet; but whether $c^2\; a$ 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 $c^2\; a$ 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 $c^2\; a$ 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 $c^2\; a$ seems therefore to depend on the central body of each system of planets, and the precise nature of this connection requires to be further 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 semitransverse, or the mean distance to be found, $=\; a$, the semiconjugate $=\; b$, the velocity at the distance $a\; =\; c$, 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. $bc\; =\; rv$; and if $F$ denote the centripetal force at the distance $a$, $c^2\; =\; aF$, and $F\; =\; \backslash frac\{c^2\}\{a^2\}$. But $F\; =\; \backslash frac\{d^2f\}\{a^2\}$, therefore $\backslash frac\{c^2\}\{a\}\; =\; \backslash frac\{d^2f\}\{a^2\}$, and $c\; =\; d\backslash sqrt\{\backslash frac\{f\}\{a\}\}$. Hence, by substituting for $c$, $bd\backslash sqrt\{\backslash frac\{f\}\{a\}\}\; =\; rv$, and $b^2d^2f\; =\; ar^2v^2$. But $b^2\; =\; AS\; \backslash times\; SB\; =\; r(2a-r)$, wherefore $r(2a-r)d^2f\; =\; ar^2v^2$, and $a\; =\; \backslash frac\{rd^2f\}\{2d^2f\; -\; rv^2\}$. 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 $v^2\; =\; \backslash frac\{d^2f\}\{r\}$.

16. But if $2d^2f\; =\; rv^2$, or $v^2\; =\; \backslash frac\{2d^2f\}\{r\}$, 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 ; but when $v^2\; >\; \backslash frac\{d^2f\}\{r\}$, and less $\backslash frac\{2d^2f\}\{r\}$, $A$ is the lower apsis.

18. When $v^2$ goes beyond this last limit, or when $v^2\; >\; \backslash frac\{2d^2f\}\{r\}$, 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 transverse 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 transverse axis; and therefore the foci and the transverse 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 $c$ the velocity of her motion.

Then, by Art. 14, $ac^2\; =\; r^2g$, and therefore $c\; =\; r\backslash sqrt\{\backslash frac\{g\}\{a\}\}$.

Now, the circumference of the circle described by the moon is $2\backslash pi\; a$, and this, divided by $c$, gives the periodic

time of the moon in seconds, or $\backslash frac\{2\backslash pi\; a\}\{r\}\; \backslash times\; \backslash sqrt\{\backslash frac\{a\}\{g\}\}\; =\; t$, so that $t\; =\; \backslash frac\{4\backslash pi^2\; a^3\}\{r^2\; g\}$, and $a^3\; =\; \backslash frac\{r^2\; g\; t^2\}\{4\backslash pi^2\}$. Hence $\backslash frac\{a^3\}\{r^3\}\; =\; \backslash frac\{g\; t^2\}\{4\backslash pi^2\; r^2\}$, and $\backslash frac\{a\}\{r\}\; =\; \backslash left(\backslash frac\{g\; t^2\}\{4\backslash pi^2\; r^2\}\backslash right)^\{\backslash frac\{1\}\{3\}\}$. Hence, as $g$, $r$, and $t$ are known, we may find $\backslash frac\{a\}\{r\}$, 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 $\backslash frac\{a\}\{r\}\; =\; 60.218$.

Now the mean equatorial parallax of the moon is found by observation = $57^\backslash circ\; 0\text{'}\; 9\text{'}\text{'}$, 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 re-action 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 loadstone, 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 laminae 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 $\backslash frac\{1\}\{DE^2\}$, and

that same force, reduced to the direction EC, is as $\backslash frac\{1\}\{DE^2\}$

$\backslash times\; \backslash frac\{EC\}\{DE\}\; =\; \backslash frac\{EC\}\{DE^2\}$. From the centre C, with the radius DC,

let a circle DKH be described, and indefinitely near it the circle $d\; k\; h$; then, since every particle in the ring of matter contained between these circles has its attraction

proportional to $\backslash frac\{EC\}\{ED^2\}$, the attraction of the whole ring will

be as $\backslash frac\{EC\}\{ED^2\}$ multiplied into the number of particles, or into the solidity of the ring. But if $EC\; =\; a$, $ED\; =\; x$, and $AC\; =\; r$, $CD^2\; =\; x^2\; -\; a^2$, and the surface of the ring $=\; 2\backslash pi\; x\; x$. If then the thickness of the plate AFBG = $m$,

the solidity of the ring = $2\backslash pi\; m\; x\; x$, and its attraction in the direction EC is $=\; \backslash frac\{2\backslash pi\; m\; x\; x\; \backslash times\; a\}\{x^3\}\; =\; \backslash frac\{2\backslash pi\; m\; a\; x\}\{x^2\}$, the fluent of which taken so as to vanish when $DC\; =\; 0$, or when $x\; =\; a$,

is $2\backslash pi\; m\; -\; \backslash frac\{2\backslash pi\; m\; a\}\{x\}\; =\; 2\backslash pi\; m\; \backslash left(1\; -\; \backslash frac\{a\}\{x\}\backslash right)$ = 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 $2\backslash pi\; m\; \backslash left(1\; -\; \backslash frac\{EC\}\{EA\}\backslash right)$.

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 $2\backslash pi\; m\; \backslash left(1\; -\; \backslash frac\{EF\}\{ED\}\backslash right)$.

Let $CE\; =\; a$, $AC\; =\; r$, $ED\; =\; x$, $EF\; =\; y$, and the attraction above will be $2\backslash pi\; m\; \backslash left(1\; -\; \backslash frac\{y\}\{x\}\backslash right)$; and if $x$ and $y$ be variable, the quantity $m$ in this formula, or the thickness of the circular plate, will be $=\; y$, and therefore the attraction of the plate $=\; 2\backslash pi\; y\; \backslash left(1\; -\; \backslash frac\{y\}\{x\}\backslash right)$. 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 $AF\; \backslash times\; FB\; =\; (y\; -\; a\; +\; r)(a\; +\; r\; -\; y)\; =\; r^2\; -\; a^2\; +\; 2ay\; -\; y^2\; =\; DF^2\; =\; x^2\; -\; y^2$. Hence $r^2\; -\; a^2\; +\; 2ay\; =\; x^2$, or $y\; =\; \backslash frac\{a^2\; -\; r^2\; +\; x^2\}\{2a\}$, and therefore

$y\; =\; \backslash frac\{x^2\}\{a\}$. By substituting these values of $y$ and $\backslash dot\{y\}$ in the Astronomical expression for the attraction of the circular plate, that

But the attraction of this circular plate may be considered as the fluxion of the attraction of the spherical segment, generated by the revolution of the arch AD, and therefore the fluent of the above fluxionary quantity will give the attraction of that segment. Now, this fluent

determined that the fluent may be equal to 0, when the arch $AD\; =\; 0$, or when $x\; =\; y\; =\; a\; -\; r$. Therefore $C$

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 $=\; \backslash frac\{4\backslash pi\; r^3\}\{3a^2\}$. But $\backslash frac\{4\backslash pi\; r^3\}\{3\}$ 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, $\backslash frac\{m\}\{x^2\}$ 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 $a\text{'}$ be the mean distances at which satellites revolve about any two planets, $m$ and $m\text{'}$ the quantities of matter in those planets, $t$ and $t\text{'}$ their periods

of revolution; it has been shown that $t\; =\; \backslash frac\{2\backslash pi\; a^\{3/2\}\}\{df^\{1/2\}\}\; =\; \backslash frac\{2\backslash pi\; a^\{3/2\}\}\{m^\{1/2\}\}$,

and consequently $t\; :\; t\text{'}\; ::\; \backslash frac\{a^\{3/2\}\}\{m^\{1/2\}\}\; :\; \backslash frac\{a\text{'}^\{3/2\}\}\{m\text{'}^\{1/2\}\}$, and $m\; :\; m\text{'}\; ::\; \backslash frac\{a^3\}\{t^2\}\; :\; \backslash frac\{a\text{'}^3\}\{t\text{'}^2\}$.

The masses, therefore, of any two planets are as the cubes of the mean distances at which their satellites revolve, 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 most correct data, it is found that, making

Or if we make the mass of the Earth 1, that of the Sun = 329630, of Jupiter 330.6, 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 19.23°, that of the earth would subtend 17.4°, of Jupiter 186.8°, of Saturn 177.7°, and of Uranus 74. 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 4.713, which is its mean density, that of water being = 1, then

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 gra-

vity. 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 $\backslash frac\{S+E\}\{SP^2\}$.

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 force 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 $\backslash frac\{1\}\{SE^2\}\; :\; \backslash frac\{1\}\{SM^2\}\; ::\; SE\; :\; SM^2$ = the force by which the sun

draws the moon in the direction MS. Take $MG\; =\; \backslash frac\{SE^3\}\{SM^2\}$ 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 gra-

Physical the earth is composed of the two parts MF and MO, which
Astronomy 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 oppo-
site 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, = $a$; EM, the radius vector
of the moon's orbit, = $r$; the angle CEM = $\backslash phi$; the
mass of the sun = $m$. The force SE, then, which retains

the earth in its orbit, is $\backslash frac\{m\}\{a^2\}$, and the sun's force in the di-

rection SM, if ML be drawn perpendicular to ES, is $\backslash frac\{m\}\{SM^2\}$

The part of this force, which is in the direction ES or

MK, is therefore $\backslash frac\{ma\}\{(a^2\; -\; 2ar\; \backslash sin\; \backslash phi\; +\; r^2)^\{\backslash frac\{3\}\{2\}\}\}$. By raising the

denominator to the power $-\backslash frac\{3\}\{2\}$, rejecting the terms which

involve the higher powers of $r$, and multiplying $ma$ by those

that are left, the force MK comes out = $\backslash frac\{m\}\{a^2\}\; \backslash left(\; 1\; +\; \backslash frac\{3r\}\{a\}\; \backslash sin\; \backslash phi\; \backslash right)$

nearly. Taking away from this ES or MH = $\backslash frac\{m\}\{a^2\}$, there re-

mains the force MN = $\backslash frac\{m\}\{a^2\}\; \backslash times\; 3r\; \backslash sin\; \backslash phi$.

Hence the force MO = $\backslash frac\{m\}\{a^2\}\; \backslash cdot\; 3r\; \backslash sin^2\; \backslash phi$; and the force

NO at right angles to the radius vector = $\backslash frac\{m\}\{a^2\}\; \backslash cdot\; 3r\; \backslash sin\; \backslash phi$

$\backslash times\; \backslash cos\; \backslash phi\; =\; \backslash frac\{m\}\{a^2\}\; \backslash cdot\; \backslash frac\{3r\}\{2\}\; \backslash sin\; 2\backslash phi$; also the force MF = $\backslash frac\{mr\}\{a^2\}$,

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 $\backslash frac\{mr\}\{a^2\}\; (1\; -\; 3\; \backslash sin^2\; \backslash phi)$.

30. At the quadratures where $\backslash phi$ vanishes, this force is

$\backslash frac\{mr\}\{a^2\}$, and is affirmative, increasing the moon's gravity to the

earth. At a certain point, between the quadratures and
the syzygies, when $3\; \backslash sin^2\; \backslash phi\; =\; 1$, or $\backslash sin\; \backslash phi\; =\; \backslash frac\{1\}\{\backslash sqrt\{3\}\}$, that is,

when $\backslash phi\; =\; 35^\backslash circ\; 15\text{'}\; 5\text{'}\text{'}$, 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 $\backslash sin\; \backslash phi$ increases,
the quantity $1\; -\; 3\; \backslash sin^2\; \backslash phi$ is negative, and the moon's gra-

force is $-\backslash frac\{2mr\}\{a^2\}$, and by this quantity the moon's gravita-
tion is diminished. Physical Astronomy.

The mean quantity of the force which is thus contin-
ually 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 $\backslash phi$ with the

line of the quadratures, this force = $\backslash frac\{mr\}\{a^2\}\; (1\; -\; 3\; \backslash sin^2\; \backslash phi)$;

multiplying by $\backslash dot\{\backslash phi\}$, we have $\backslash frac\{mr\}\{a^2\}\; (\backslash dot\{\backslash phi\}\; -\; 3\backslash dot\{\backslash phi\}\; \backslash sin^2\; \backslash phi)$, the fluent

of which = $\backslash frac\{mr\}\{a^2\}\; (-\backslash frac\{1\}\{2\}\; \backslash phi\; +\; \backslash frac\{3\}\{2\}\; \backslash sin\; \backslash phi\; \backslash cos\; \backslash phi)$, and this,

when $\backslash phi$ is an entire circumference or four right angles,
is $\backslash frac\{mr\}\{a^2\}\; \backslash times\; -\backslash frac\{\backslash pi\}\{2\}$. This is the sum of the forces for an en-

tire revolution, and when divided by $\backslash pi$, gives the mean

force $-\backslash frac\{mr\}\{2a^2\}$ 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 or-
bit, and divided by the cube of the sun's distance from the
earth; at the syzygies it is diminished by twice this quan-
tity; and the effect on the whole is a diminution by one half
of the same quantity.

If $\backslash frac\{mr\}\{a^2\}$ be reduced to its numerical value, supposing the

moon's gravitation to the earth to be 1, it is found = $\backslash frac\{1\}\{174\}$

nearly. Hence the mean disturbing force of the sun is

nearly = $\backslash frac\{1\}\{583\}$ 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 plan-
et, the other immediately affects the motion of the plan-
et in the orbit. When any of these inequalities is ex-
pressed 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 $19^\backslash circ\; 19\text{'}\; 42\text{'}\text{'}\; 3$. 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 di-
rected 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-revolu-
tion, 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 $19^\backslash circ\; 19\text{'}\; 42\text{'}\text{'}\; 3$ in a year. The manner of de-

Physical Astronomy. ducing 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 $180^\backslash circ$, 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 incurved 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 $180^\backslash circ$.

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 $r\; \backslash times\; \backslash frac\{F-3f\}\{F-5f\}$; 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 $360^\backslash circ\; \backslash times\; \backslash left(\backslash frac\{F-3f\}\{F-5f\}\backslash right)^\{\backslash frac\{3\}\{2\}\}$.

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 $=\; \backslash frac\{1\}\{357.7\}$, the motion of the apsides will come out $=\; 3^\backslash circ\; 1\text{'}\; 20\text{'}\text{'}$ 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 $6^\backslash circ\; 9\text{'}$, 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 evection, 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 evection is accurately deduced from the theory of gravitation, it appears

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 $\backslash Delta$ be the angular distance of the moon from the sun, are,

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, as the cube of his distance from the earth; 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 $(11\text{'}\; 12\text{'}\text{'})\; \backslash times\; \backslash sin.\; \backslash text\{\; sun\text{'}s\; mean\; anomaly\}$, 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 $10\text{'}\text{'}$ 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.

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

Physical Astronomy. 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 Bureckhardt 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 $\backslash frac\{1\}\{2025810\}$, of Venus $\backslash frac\{1\}\{405871\}$, and of Mars $\backslash frac\{1\}\{2546320\}$; 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 $A\; \backslash times\; \backslash frac\{PS^2\}\{PP\text{'}^2\}$. Let

$PR\; =\; A\; \backslash times\; \backslash frac\{PS^2\}\{PP\text{'}^2\}$; 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 5° 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 $\backslash frac\{1\}\{79\}$ 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°3 annually, and the line of the apsides moves forward 15°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, $5\; \backslash cdot\; 7\; \backslash sin.\; (\backslash text\{long.\; \}\; \backslash phi\; -\; 3\; \backslash text\{\; long.\; \}\; \backslash delta)$; of the earth, $7\; \backslash cdot\; 2\; \backslash sin.\; (\backslash text\{long.\; \}\; \backslash odot\; -\; \backslash text\{long.\; \}\; \backslash phi)$. Several inequalities are produced in Jupiter.

48. The inequalities of the small planets Vesta, Juno, Ceres, and Pallas, have not yet been computed. The disturbances which they must suffer from Mars and Jupiter are no doubt 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 two of these planets 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°8, and his aphelion forward 6°96. The secular change in the inclination of the orbit is 22°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°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°4 annually, and the aphelion forward at the rate of 19°4; the secular change of the inclination is — 15°5, and the secular diminution of the equation of the centre 2°1.

There is, besides these variations in the orbits, an inequality in the motion of each of these planets, which it has been found very difficult to explain, and has only lately been fully accounted for, according to the theory of gravity, by the profound investigations of Laplace. These inequalities are both of a long period, viz. 918.76 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

and that which must be applied to $S$ is

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 $5S\; -\; 2I\; -\; n\; \backslash times\; 58^\backslash circ\; 88\text{'}$ requires all that time to increase from 0 to 360°.

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° annually, and the aphelion forward at that of 2°55. The ec-

centricity is diminishing, and the secular variation of the greatest equation of the centre is 11°03. Physical Astronomy.

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 $2\; \backslash cdot\; 30^\backslash circ\; \backslash times\; \backslash sin.\; (S\; -\; 2U\; +\; A)$.

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 general remark is, that one element in every planetary orbit, viz. the mean distance, is exempted from all change; 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 ellipse, 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 may be expected again about the year 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

Physical have so altered the original orbit as to render the comet
Astrology. 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 for-
merly 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 unex-
pected, confirmation of the theory of gravity.

Though the comets are so much disturbed by the ac-
tion of the planets, yet it does not appear that their re-
action produces any sensible effect. The comet of 1770
came so near to the earth as to have its periodic time in-
creased by 2.246 days, according to Laplace's computa-
tion; 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 gra-
vity of the earth is insensible, and the mass of the comet,
therefore, less than $\backslash frac\{1\}\{500\}$th of the mass of the earth. The
same comet also passed through the system of the satel-
lites of Jupiter without causing any derangement of their
motions. Hence it is reasonable to conclude, that no ma-
terial 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 ob-
served 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 re-
lation which takes place between the mean motions of the
first three satellites; the mean motion of the first satel-
lite, 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 satellite moves nearly in the plane of Jupiter's
equator, and has no eccentricity, except what is commu-
nicated 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 orbit of the second satellite moves on a fixed
plane, to which it is inclined at an angle of $27^\{\backslash circ\}\; 13\text{'}$, and
on which its nodes have a retrograde motion, so that they
complete a revolution in 29.914 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 dis-
turbances. This satellite has no eccentricity but that
which it derives from the action of the third and fourth.
The third satellite moves on a fixed plane, to which it is
inclined at an angle of $12^\{\backslash circ\}\; 20\text{'}$, and its nodes make a tro-
pical revolution backwards in 141.739 years. The equator
of Jupiter is inclined to the plane of his orbit at an angle
of $3^\{\backslash circ\}\; 5\text{'}\; 30\text{'}\text{'}$. 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 emanation from the equation of the
centre of the fourth satellite. The first equation is refer-
able to an apsis which has an annual motion of $2^\{\backslash circ\}\; 36\text{'}\; 39\text{'}\text{'}$
forward in respect of the fixed stars; the second equa-
tion 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 $13^\{\backslash circ\}\; 16\text{'}$. In 1777 the
equations were opposed, and their difference was $5^\{\backslash circ\}\; 6\text{'}$.

The two last inequalities were perceived by Mr War-
gentin, by observation alone; but their exact amount, and
the law which they observe in their changes, he could not
discover. The orbit of the fourth satellite moves on a
fixed plane, to which it is inclined at an angle of $14^\{\backslash circ\}\; 58\text{'}$;
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 $24^\{\backslash circ\}\; 33\text{'}$ to the equator of Jupiter;
the orbit is sensibly elliptical, and its greater axis has an
annual motion of $42^\{\backslash circ\}\; 58\text{'}\; 7\text{'}\text{'}$. 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 = .027337; and as the mass
of the moon is $\backslash frac\{1\}\{79.89\}\; =\; .012517$, 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 some-
what 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 inequa-
lities 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 $A\; \backslash sin.\; nt$ or $A\; \backslash cos.\; nt$, 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
$A\; \backslash sin.\; nt$ 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 $360^\{\backslash circ\}$, 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 hap-
pens 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 destroy-
ing its own effect.

It would be far otherwise if into the value of any in-
equalities a term entered of the form $A\; \backslash times\; nt$, $A\; \backslash tan.\; nt$,

Practical Astronomy. A sin. ut 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 infinite to one, that, without a cause particularly directed to that object, such a conformity could not have arisen in the motions of thirty-one different 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. (N.)

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; also those which are continually making 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 result.

The division of labour in this, as in other subjects, has been attended with advantage. One class of astronomers now survey the heavens by telescopes of more or less power. These, within the last fifty years, have enriched the science by the discovery of five new planets in addition to the six known from the most remote ages. The late Sir William Herschel took the lead in this noble labour, and he has been followed by many others. By their exertions various comets have been observed; and the periodic times and orbits of three have now been determined with such certainty as to enable us to predict their return. A second class observe the heavenly bodies with instruments which combine increased powers of seeing, with the means of measuring minute angular distances or considerable angles with great accuracy. The established national observatories, and some private observatories, conducted with not less zeal and intelligence, are engaged in this important labour. These employ transit instruments, clocks, mural or meridian circles, quadrants, equatorial or azimuth and altitude instruments, of the best workmanship. Their object is less to add new facts than to improve and extend those which are already known.

By the observations made in observatories, the science is continually advancing towards perfection; but, like all knowledge founded on observation, there will ever be room for further improvement.

Although it is only a few individuals, comparatively speaking, that can possess telescopes of such power as to give reasonable hopes of extending astronomical knowledge by the discovery of new phenomena, yet, by the liberality of governments, the united efforts of learned bodies, and the fortunate possession of leisure, pecuniary means, and zeal for the improvement of the science evinced by some private gentlemen, telescopes of extraordinary dimensions and excellence have been constructed, and considerable additions have of late years been made to our knowledge of the heavens in regions almost beyond the flight of human imagination.

The great instruments, such as are placed in national observatories, are also in general beyond the reach of private astronomers; yet a few wealthy amateurs of the science have rivalled these establishments, both in the acquisition of instruments and in their application. With instruments of an inferior construction little can be done towards improving the primary elements of astronomy, because the results they give are less correct and less to be depended on than those obtained by superior instruments. When the measure of an angle is determined to the accuracy of a second, it is of no use to have another measure which may err as much as ten seconds. However, it cannot be doubted that the best instruments now employed will in time be superseded by others more perfect. After a certain degree of accuracy has been obtained, it is only by slow degrees and much labour that we can go beyond it; and then, like differentials of different orders, every

Practical addition to the accuracy of a result is small in comparison to that which went before it.

Another branch of practical astronomy, of no small importance, applies the knowledge which the labour of many years, or even ages, has accumulated, to the improvement of geography, and to navigation, and the wants of society. For this instruments are also required. These are the portable transit instrument and chronometer; portable azimuth and altitude circles; reflecting and repeating circles; sextants, quadrants, and other minor instruments, down to the convenient garden sun-dial. These instruments are within the reach of ordinary cultivators of astronomy, and they are the most useful, because the advantage they yield is immediate and obvious: for by their aid the exact position on the earth of every point visited by a traveller may be determined; ships may be conducted in safety and with certainty to remote regions; and true time may be ascertained in all places. A third department of practical astronomy may comprehend the calculations required for reducing the observations made in observatories; that is, in disengaging them from the effects of refraction, parallax, nutation, aberration, instrumental errors, &c. and fitting them for their place in the annals of the science. The time spent in making an observation is in fact often small in comparison to that required for its reduction. The observation is a process purely mechanical, requiring only steady attention, and an expertness in observing a moving body, and in estimating small portions of time by the beat of the clock, which may be acquired by practice. But the reduction requires the application of much theoretical knowledge of the science, and an acquaintance with some of the refinements of modern analysis; at least these must be possessed by the person who frames the rules and directs the operations, although a less degree of knowledge may suffice for the mere numerical calculation.

The observations of the traveller and navigator require also the application of mathematical science. Plane and spherical trigonometry are indispensable for the intelligent navigator; and the latter of these is required for the simplest observations, such, for example, as are made by a quadrant or sextant, to find true time. It frequently happens that the navigator is deficient in theory, contenting himself with the practice of the calculation of his lunars, as he calls observations made to ascertain the distance of the moon from the stars. This is an evil which, we fear, sometimes leads to the loss of many lives and much property.

Another kind of calculations is required in the construction of Almanacs and Ephemerides. These are made under the direction of men well versed in the theory, although the mere labour of numerical calculation can be executed by correct arithmeticians according to prescribed rules; and, to insure accuracy, each is commonly performed by two persons, and their results compared by a third.

The attentive reader of the preceding treatise will readily understand that much practical astronomy is involved in it; necessarily indeed, for the theory and the practice cannot be entirely disengaged from each other. Thus, it is shown (chap. II. sect. 1.) how the obliquity of the ecliptic may be derived from observations of the sun's declination on three successive days about the time of the summer solstice; and again, in section 2, a formula is given for finding the eccentricity of the earth's orbit from the greatest equation of the centre, as determined from observations; also another for determining the true anomaly from the mean: these and various others, in chap. III. and elsewhere, may be regarded as applications of practical astronomy. We shall therefore here only further give some examples of the more important problems in what may be called spherical astronomy.

The important and very general problem of determining what will be the position of any proposed body of the system at any given time, past or to come, belongs to the division of the subject which we are now treating. Its correct solution, however, requires an apparatus of tables far beyond what can be comprised in the space to which this article must be limited. Even the single case of the moon would require a very considerable number of tables and precepts. Such a body of tables is, however, not necessary even in works professing on astronomy. The British Nautical Almanac, the Connaissance des Temps, and the Berlin Ephemeris (Berliner Astronomisches Jahrbuch), are the sources from which the practical astronomer usually derives a knowledge of the phenomena he expects to happen, and for the observation of which he must prepare beforehand. The same observation applies to various other subsidiary tables, of which there are professed collections. One of the most copious is the first of two large volumes, forming a work on practical astronomy, by the Rev. Dr Pearson; this contains tables expressly intended for the reduction of astronomical observations, and many others of great importance to the professed astronomer. There is a smaller collection, entitled Tables to be used with the Nautical Almanac, by the Rev. W. Lax, professor of astronomy and geometry in the University of Cambridge. This is most essential to the navigator, for whose use also a much larger work was published, viz. A complete Collection of Tables for Navigation and Nautical Astronomy, by Joseph de Mendoza Rios. There is another and later work, published in 1829 in France, entitled Nouvelles Tables Astronomiques et Hydrographiques, par V. Bagay, professeur d'hydrographie. This must be valuable to the French navigator; but it is not less so to practical astronomers of all countries, on account of its portable size, considering that it contains a table of logarithmic sines and tangents to every second of the quadrant. We must also notice with much approbation a neat and comprehensive collection of astronomical tables and formulae by Mr Baily, late president of the Astronomical Society of London. This work is chiefly adapted to the practical astronomer on land. There is one part of practical astronomy which at all times has, in a particular manner, excited the attention of mankind: we mean the determination of the exact time of new and full moon, and the prediction of eclipses. To determine these with as much accuracy as is sufficient to gratify ordinary students of practical astronomy, tables have been purposely constructed, and are given in the sequel; with the addition of plain precepts, applicable without any considerable degree of mathematical knowledge. By means of them the young astronomer may find the time of an eclipse of the sun and moon, and exhibit its appearance by a geometrical construction. The places of upwards of 500 fixed stars, and some tables useful in practical astronomy, are also given; and the principal instruments of astronomy are described and exhibited by engravings in the superior style which distinguishes the present from all former editions of this work.

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 XCIII.) be the pole of the equator $O\backslash gamma\; Q$, $E$ the pole of the ecliptic $I\backslash gamma\; L$, and $S$ the place of the star. Let $PR$, the circle of latitude passing through

S, meet the equator in R; and EM, the circle of declination, meet the ecliptic in M; also let $\backslash gamma\; E$, $\backslash gamma\; P$, be arcs of great circles passing through the equinoctial point $\backslash gamma$ and the poles of the ecliptic and equator respectively. Let us now make

$L\; =\; \backslash gamma\; M$ the longitude of S,

$l\; =\; SM$ the latitude,

$AR\; =\; \backslash gamma\; R$ the right ascension,

$D\; =\; SR$ the declination,

$\backslash omega\; =\; L\backslash gamma\; Q$ the obliquity of the ecliptic.

Since $\backslash gamma\; M\; =\; \backslash gamma\; L\; -\; ML\; =\; 90^\backslash circ\; -\; ML$, and $ML$ is the measure of the angle SEP, we have $\backslash gamma\; M\; =\; L\; =\; 90^\backslash circ\; -\; SEP$, and, consequently, $\backslash sin\; L\; =\; \backslash cos\; SEP$. Again, since $\backslash gamma\; R\; =\; \backslash gamma\; Q\; -\; RQ\; =\; 90^\backslash circ\; -\; RQ$, and $RQ$ is the measure of the angle RPQ, we have $\backslash gamma\; R\; =\; AR\; =\; 90^\backslash circ\; -\; RPQ$; whence $\backslash sin\; AR\; =\; \backslash cos\; RPQ\; =\; -\backslash cos\; EPS$. In like manner we have $l\; =\; SM\; =\; 90^\backslash circ\; -\; ES$, whence $\backslash sin\; l\; =\; \backslash cos\; ES$; and also $D\; =\; SR\; =\; 90^\backslash circ\; -\; PS$, whence $\backslash sin\; D\; =\; \backslash cos\; PS$. Thus the problem depends on the solution of the oblique angled spherical triangle PES.

From the known properties of spherical triangles, we have

$\backslash cot\; PS\; \backslash sin\; EP\; =\; \backslash cot\; PES\; \backslash sin\; EPS\; +\; \backslash cos\; EP\; \backslash cos\; EPS$; therefore, by substituting the values just found, and observing that $EP\; =\; \backslash omega$, we have

Assume an angle $\backslash phi$ such that $\backslash tan\; \backslash phi\; =\; \backslash sin\; AR\; \backslash cot\; D$, then

whence, by substituting,

and by reducing,

a formula from which the longitude $L$ is easily computed by means of the logarithmic tables.

To find the latitude $l$, we have

$\backslash cos\; ES\; =\; \backslash cos\; EP\; \backslash cos\; PS\; +\; \backslash sin\; EP\; \backslash sin\; PS\; \backslash cos\; EPS$, which gives

Assume as before, $\backslash tan\; \backslash phi\; =\; \backslash cot\; D\; \backslash sin\; AR$, then, by substituting,

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

$\backslash cot\; ES\; \backslash sin\; EP\; =\; \backslash cot\; EPS\; \backslash sin\; PES\; +\; \backslash cos\; EP\; \backslash cos\; PES$, that is,

Assume $\backslash tan\; \backslash psi\; =\; \backslash sin\; L\; \backslash cot\; l$; then

and by substituting this value of $\backslash tan\; l$ in the above equation, it becomes

that is,

whence

To find the declination, we have

that is,

and by substituting $\backslash frac\{\backslash tan\; \backslash psi\}\{\backslash sin\; L\}$ for $\backslash cot\; l$,

therefore

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 $\backslash gamma\; TR$. The following are the formulae:

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 $\backslash lambda\; =\; AP$ the complement of the latitude,

$\backslash Delta\; =\; PV$ 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.

1. To find $Z$ we have

Assume $\backslash tan\; \backslash phi\; =\; \backslash tan\; \backslash Delta\; \backslash cos\; P$; then by substituting,

whence, by reducing,

2. To find $A$, we have from spherical trigonometry the formula

Assume $\backslash cot.\; \backslash psi\; =\; \backslash frac\{\backslash cot.\; \backslash Delta\}\{\backslash cos.\; P\}$, whence $\backslash cot.\; \backslash Delta\; =\; \backslash cot.\; \backslash psi\; \backslash cos.\; P$,

and, by substituting,

whence

3. To find $V$ we have

Assume $\backslash cot.\; \backslash chi\; =\; \backslash frac\{\backslash cot.\; \backslash lambda\}\{\backslash cos.\; P\}$, or $\backslash cot.\; \backslash lambda\; =\; \backslash cot.\; \backslash chi\; \backslash cos.\; P$, then,

by substituting and reducing as above, we have

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

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 $SS\text{'}$ be the part of the parallel described by the sun between his rising and passing the prime vertical, and $PS,\; PS\text{'}$ hour circles.

Let $L$ = the latitude of the place = $PB$,

$D$ = the sun's declination = $90^\backslash circ\; -\; PS$,

$v$ = the time from midnight to sunrise = 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 = hour-angle $SPB$,

$z$ = the sun's altitude when in the prime vertical = $OS\text{'}$.

In the spherical triangle $PBS$, right angled at $B$,

and in the spherical triangle $PZS$, right angled at $Z$,

Now $PB=L$, $PZ=90^\backslash circ-L$, $PS=PS\text{'}=90^\backslash circ-D$, $BS=90^\backslash circ-x$, $ZS\text{'}=90^\backslash circ-z$; hence we have these four formulae:

The hour-angles $v$ and $y$ must be converted into time by allowing $15^\backslash circ$ to one hour.

PROBLEM VI.—To find how much the rising of the sun or a star is advanced by refraction.

Let $ZPB$ be the meridian (fig. 123), $Z$ the zenith, $P$ the pole, and $OB$ the horizon. Let $SS\text{'}$, 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 $S\text{'}$ somewhere in the vertical circle $ZD$ produced downward, and the time of rising is accelerated by the small angle $SPS\text{'}$.

Let $L$ denote the latitude of the place = $90^\backslash circ\; -\; PZ$; $D$ the declination of the star = $90^\backslash circ\; -\; PS$; $H$ the hour-angle $BPS$, reckoned from midnight; $r$ the arc $DS$ of the vertical $ZS$, the effect of refraction; and $x$ the angle $SPS\text{'}$, the acceleration of time of rising.

In the spherical triangle $ZPS$,
$\backslash cos.\; ZS\text{'}\; =\; \backslash cos.\; ZP\; \backslash cos.\; PS\text{'}\; +\; \backslash sin.\; ZP\; \backslash sin.\; PS\text{'}\; \backslash cos.\; ZPS$,
Now, $\backslash cos.\; ZS\text{'}\; =\; \backslash cos.\; (90^\backslash circ\; +\; r)\; =\; -\backslash sin.\; r$,
and $\backslash cos.\; ZPS\text{'}\; =\; -\backslash cos.\; BPS\text{'}\; =\; -\backslash cos.\; (H\; -\; x)$;

therefore

In like manner, in the triangle $ZPS$, in which $\backslash cos.\; ZS\; =\; \backslash cos.\; 90^\backslash circ\; =\; 0$,

hence, by subtraction,

but $\backslash cos.\; (H\; -\; x)\; -\; \backslash cos.\; H\; =\; 2\; \backslash sin.\; \backslash frac\{1\}\{2\}\; x\; \backslash sin.\; \backslash left(\; H\; -\; \backslash frac\{x\}\{2\}\; \backslash right)$
(ALGEBRA, § 239, D);

Now, $x$ and $r$ being small angles, we may consider $2\; \backslash sin.\; \backslash frac\{1\}\{2\}\; x\; =\; x$, $\backslash sin.\; r\; =\; r$, and $\backslash sin.\; \backslash left(\; H\; -\; \backslash frac\{x\}\{2\}\; \backslash right)\; =\; \backslash sin.\; H$.

and here $H$ is determined by the formula

The value of $x$ just found is only a near approximation; let it be denoted by $x\text{'}$, and let $H\text{'}\; =\; H\; +\; x\text{'}$, and we shall have more nearly

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,

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 $18^\backslash circ$.

Let $a$ denote the arc, DS the depression, L the latitude of the place $=\; 90^\backslash circ\; -\; ZP$, D the sun's declination $=\; 90^\backslash circ\; -\; PS$, H the hour angle SPB (from midnight), $x$ the angle SPS.

Then, $\backslash cos.\; ZS\text{'}\; =\; \backslash cos.\; ZP\; \backslash cos.\; PS\text{'}\; +\; \backslash sin.\; ZP\; \backslash sin.\; PS\text{'}\; \backslash cos.\; ZPS$;

that is, because $\backslash cos.\; ZS\text{'}\; =\; -\backslash sin.\; a$, and $\backslash cos.\; ZPS\; =\; -\backslash cos.\; SPB$,

$-\backslash sin.\; a\; =\; -\backslash sin.\; L\; \backslash sin.\; D\; -\; \backslash cos.\; L\; \backslash cos.\; D\; \backslash cos.\; (H\; -\; x)$; hence we obtain

Now, by Problem V. $\backslash cos.\; H\; =\; \backslash tan.\; L\; \backslash tan.\; D\; \backslash dots\; \backslash dots\; \backslash dots\; (1)$ Let $\backslash phi$ be such an angle, that

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 $PS\; =\; \backslash Delta$, $PM\; =\; \backslash Delta\text{'}$, the angle $SPM\; =\; P$, and SM, the distance sought, $=\; d$. We have then

Assume $\backslash tan.\; \backslash phi\; =\; \backslash tan.\; \backslash Delta\; \backslash cos.\; P$; then, by substituting and reducing, as in Problem II., we obtain

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 $15^\backslash circ$ or $20^\backslash circ$ 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 $\backslash lambda$ the correct zenith distance of the pole, or the co-latitude; then $\backslash lambda\; =\; \backslash frac\{1\}\{2\}(Z\; +\; Z\text{'})\; +\; \backslash frac\{1\}\{2\}(R\; +\; R\text{'})$.

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,

according as the star is situated S or S', 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

and if below the pole, at $a\text{'}$, then $Ca\text{'}\; =\; 180^\backslash circ\; -\; D$, and

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

$\backslash tau$ = the difference (in sidereal time) of the transit of the moon's limb and the star previously agreed on at A;

$\backslash tau\text{'}$ = the same difference at B;

$t$ = the apparent Greenwich time of the culmination of the moon at A;

$t\text{'}$ = the apparent Greenwich time of the culmination of the moon at B;

$d$ = $\backslash gamma$'s true declination { computed for the time $\backslash tau$;

$d\text{'}$, $\backslash tau\text{'}$ = the same quantities computed for the time $t\text{'}$;

$s$ = the length of the true solar day in seconds;

$m$ = $\backslash gamma$'s motion in $\backslash text\{AR\}$ in half that interval, expressed in seconds of space;

$\backslash chi$ = the assumed difference of longitude, in time;
$(\backslash chi\; +\; \backslash epsilon)$ = 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 $\backslash frac\{r\}\{\backslash cos.\; d\}$, which being reduced

to time, becomes $\backslash frac\{r\}\{15\; \backslash cos.\; d\}$ and expresses the sidereal time in which the moon's semi-diameter passes the meridian.

Hence $\backslash tau\; \backslash pm\; \backslash frac\{r\}\{15\; \backslash cos.\; d\}$ is the observed difference of the $\backslash mathcal\{A\}$ 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 $\backslash frac\{r\text{'}\}\{15\; \backslash cos.\; d\}$ is the sidereal time in which the semi-diameter of the moon passes the meridian at B, and $\backslash tau\; \backslash pm\; \backslash frac\{r\text{'}\}\{15\; \backslash cos.\; d\}$ the observed difference of the $\backslash mathcal\{A\}$ of the star and moon's centre. We have, therefore, by subtracting these two expressions,

for the observed difference of the $\backslash mathcal\{A\}$ of the moon's centre, during the time elapsed between the two observations. Put this difference equal to $\backslash Delta$, then $\backslash chi\; +\; \backslash Delta$ is the difference between $t$ and $t\text{'}$ in sidereal time, which becomes $(\backslash chi\; +\; \backslash Delta)\; \backslash frac\{8640\}\{s\}$ when expressed in mean solar time.

Hence we have

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 $a\text{'}$ be the moon's $\backslash mathcal\{A\}$ in space, computed for the times $t$ and $t\text{'}$ respectively (taken from the ephemeris); then the formula for the correction of the assumed difference of longitudes will evidently be

and this added to $\backslash chi$ gives $\backslash chi\; +\; \backslash epsilon$, the corrected difference of longitudes.

PROBLEM XI.—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

$l\; =\; 90^\backslash circ\; -\; PZ$ = the latitude of the place;

$D\; =\; 90^\backslash circ\; -\; PS$ = the declination;

$P\; =\; ZPS$ the hour angle;

$Z\; =\; l\; -\; D$ = the true zenith distance;

$Z\text{'}\; =\; ZS$ the observed zenith distance;

$x\; =\; Z\text{'}\; -\; Z$ the correction.

The triangle $PZS$ gives

$\backslash cos.\; ZS\; =\; \backslash cos.\; ZP\; \backslash cos.\; PS\; +\; \backslash sin.\; ZP\; \backslash sin.\; PS\; \backslash cos.\; ZPS$,
or $\backslash cos.\; Z\text{'}\; =\; \backslash sin.\; l\; \backslash sin.\; D\; +\; \backslash cos.\; l\; \backslash cos.\; D\; \backslash cos.\; P$;

but $\backslash cos.\; (l\; -\; D)\; =\; \backslash cos.\; l\; \backslash cos.\; D\; +\; \backslash sin.\; l\; \backslash sin.\; D$,

and $\backslash cos.\; P\; =\; 1\; -\; 2\; \backslash sin^2\; \backslash frac\{1\}\{2\}\; P$,

therefore,

$\backslash cos.\; Z\text{'}\; =\; \backslash cos.\; (l\; -\; D)\; -\; 2\; \backslash cos.\; l\; \backslash cos.\; D\; \backslash sin^2\; \backslash frac\{1\}\{2\}\; P$,
or $\backslash cos.\; Z\text{'}\; =\; \backslash cos.\; Z\; -\; 2\; \backslash cos.\; l\; \backslash cos.\; D\; \backslash sin^2\; \backslash frac\{1\}\{2\}\; P$.

Now, by the trigonometrical formula (ALGEBRA, p. 240),

$\backslash cos.\; Z\; -\; \backslash cos.\; Z\text{'}\; =\; 2\; \backslash sin.\; \backslash frac\{1\}\{2\}\; (Z\text{'}\; -\; Z)\; \backslash sin.\; \backslash frac\{1\}\{2\}\; (Z\text{'}\; +\; Z)$,

therefore,

$2\; \backslash sin.\; \backslash frac\{1\}\{2\}\; (Z\text{'}\; -\; Z)\; \backslash sin.\; \backslash frac\{1\}\{2\}\; (Z\text{'}\; +\; Z)\; =\; 2\; l\; \backslash cos.\; D\; \backslash sin^2\; \backslash frac\{1\}\{2\}\; P$.
But $Z\text{'}\; -\; Z\; =\; x$, and $Z\text{'}\; +\; Z\; =\; 2Z\; +\; x$,

therefore,

$2\; \backslash sin.\; \backslash frac\{1\}\{2\}\; x\; \backslash sin.\; (Z\; +\; \backslash frac\{1\}\{2\}\; x)\; =\; 2\; \backslash cos.\; l\; \backslash cos.\; D\; \backslash sin^2\; \backslash frac\{1\}\{2\}\; P$,
that is,

In order to resolve this equation, make $\backslash cot.\; Z\; =\; a$, $\backslash frac\{\backslash cos.\; l\; \backslash cos.\; D\}\{\backslash sin.\; Z\}\; \backslash sin^2\; \backslash frac\{1\}\{2\}\; P\; =\; b$, and divide both sides by $2\; \backslash cos^2\; \backslash frac\{1\}\{2\}\; x$; it then becomes

Whence

and, consequently,

Developing this expression, and rejecting all the powers of $b$ higher than the cube, we have

Now the series which expresses the arc in terms of its tangent (see art. ALGEBRA, p. 270), gives

Therefore, by substituting, and rejecting the terms containing higher powers of $b$ than the cube,

that is,

and therefore, on restoring the values of $a$ and $b$, and dividing by $\backslash sin.\; l$,

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 $Z\text{'}$ by means of the equation $Z\text{'}\; -\; Z\; =\; x$, we had eliminated $Z$, the resulting expression would have been

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 $\backslash phi$. Now, $APC\; =\; 2P\; +\; \backslash phi$; half of which is $P\; +\; \backslash frac\{1\}\{2\}\backslash phi$; therefore the true time of the meridional passage will be found by subtracting $\backslash frac\{1\}\{2\}\backslash phi$ from the mean of the times of observed equal altitudes.

To find the value of $\backslash phi$, we have the equation $\backslash cos.\; ZA\; =\; \backslash cos.\; ZP\; \backslash cos.\; PA\; +\; \backslash sin.\; ZP\; \backslash sin.\; PA\; \backslash cos.\; APM$;

Practical Astronomy. that is (employing the denominations of last problem),
$\backslash cos.\; Z\; =\; \backslash sin.\; l\; \backslash sin.\; D\; +\; \backslash cos.\; l\; \backslash cos.\; D\; \backslash cos.\; P\; \backslash dots\; (1)$

Now, let $\backslash Delta\; =\; Bb$ be the small change of declination corresponding to the variation of the hour-angle from $P$ to $P\; +\; \backslash phi$, and let $D\; +\; \backslash Delta$ be substituted for $D$, and $P\; +\; \backslash phi$ for $P$ in this equation. It then becomes

$\backslash cos.\; Z\; =\; \backslash sin.\; l\; \backslash sin.\; (D\; +\; \backslash Delta)\; +\; \backslash cos.\; l\; \backslash cos.\; (D\; +\; \backslash Delta)\; \backslash cos.\; (P\; +\; \backslash phi)$.
But as $\backslash Delta$ and $\backslash phi$ 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

$\backslash cos.\; Z\; =\; \backslash sin.\; l\; \backslash sin.\; D\; +\; \backslash cos.\; l\; \backslash cos.\; D\; \backslash cos.\; P$
$+\; (\backslash sin.\; l\; \backslash cos.\; D\; -\; \backslash cos.\; l\; \backslash sin.\; D\; \backslash cos.\; P)\; \backslash Delta$
$-\; \backslash cos.\; l\; \backslash cos.\; D\; \backslash sin.\; P\; \backslash phi\; +\; \backslash cos.\; l\; \backslash sin.\; D\; \backslash sin.\; P\; \backslash Delta\; \backslash phi$;
whence, in consequence of equation (1), and rejecting the term multiplied by $\backslash Delta\; \backslash phi$,
$(\backslash sin.\; l\; \backslash cos.\; D\; -\; \backslash cos.\; l\; \backslash sin.\; D\; \backslash cos.\; P)\; \backslash Delta\; =\; \backslash cos.\; l\; \backslash cos.\; D\; \backslash sin.\; P\; \backslash phi$;
therefore,

Let $t$ be the sidereal time of the first observation, $t\text{'}$ that of the second, and $t\text{'}\; -\; t\; =\; 24$. Also let the diurnal variation of $D$ be denoted by $v$, expressed like the arcs $\backslash phi$ and $\backslash Delta$ in seconds; we have then $24^h\; :\; 24^s\; ::\; v\; :\; \backslash Delta$, whence $\backslash Delta\; =\; \backslash frac\{1\}\{24\}\; v$, and consequently

Now, to convert $\backslash phi$ into seconds of time, it is only necessary to divide the number to which it is equal by 15; therefore, expressed in time, $\backslash phi$ becomes

For the angle $P$ we may substitute $\backslash theta$ (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 $\backslash phi$ being so small as not to be affected by these substitutions. Therefore, ultimately,

which, subtracted from $\backslash theta$, 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

But $\backslash frac\{y\}\{x\}\; =\; \backslash tan.\; ACM$, and $-\backslash frac\{dx\}\{dy\}\; =\; \backslash tan.\; ADM\; =\; \backslash tan.\; l$; Practical Astronomy.

therefore $\backslash tan.\; ACM\; =\; \backslash frac\{n^2\}\{m^2\}\; \backslash tan.\; l$. Now $ZMV\; =\; DMC\; =\; ADM\; -\; ACM$, therefore $\backslash tan.\; ZMV\; =\; \backslash tan.\; (ADM\; -\; ACM)\; =\; \backslash frac\{\backslash tan.\; ADM\; -\; \backslash tan.\; ACM\}\{1\; +\; \backslash tan.\; ADM\; \backslash tan.\; ACM\}$;

that is,

Now, let $m\; -\; n\; =\; 1$, or $n\; =\; m\; -\; 1$; then, by substituting,

or, neglecting the terms multiplied by $\backslash frac\{1\}\{m^2\}$, and recollecting that $\backslash sin.\; l\; \backslash cos.\; l\; =\; \backslash frac\{1\}\{2\}\; \backslash sin.\; 2l$,

whence

that is,

but $\backslash cos^2\; l\; -\; \backslash sin^2\; l\; =\; \backslash cos.\; 2l$; therefore,

and $2\; \backslash sin.\; 2l\; \backslash cos.\; 2l\; =\; 4\; \backslash sin.\; 4l$, therefore,

The last term of this expression cannot in any case amount to $2\text{'}$, 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 $P$ = the horizontal parallax,

$p$ = the parallax of altitude,

$Z$ = the apparent zenith distance.

It was shown in Chap. I. Sect. 2 of THEORETICAL ASTRONOMY that $\backslash sin.\; p\; =\; \backslash sin.\; P\; \backslash sin.\; Z$; now let $Z\text{'}$ = the true zenith distance, then $Z\; =\; Z\text{'}\; +\; p$, and consequently

therefore

whence

But by a well-known series,

Substituting therefore the above expression for $\backslash tan.\; p$, and reducing the powers to series, we find

Practical Astronomy. 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;
• $\backslash Pi$ = $APB$, the parallax in right ascension;
• $l$ = $90^\backslash circ\; -\; ZP$, latitude of place;
• $\backslash Delta$ = $PA$, 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
$\backslash sin.\; PA\; :\; \backslash sin.\; AB\; ::\; \backslash sin.\; ABP\; :\; \backslash sin.\; APB$;

that is,
$\backslash sin.\; \backslash Delta\; :\; \backslash sin.\; p\; ::\; \backslash sin.\; ABP\; :\; \backslash sin.\; \backslash Pi$,
but $\backslash sin.\; p\; =\; \backslash sin.\; P\; \backslash sin.\; (N+p)$. See THEORETICAL ASTRONOMY, Chap. I. Sect. 2.
Therefore,

Now, in the triangle $ZBP$, we have
$\backslash sin.\; ZB\; :\; \backslash sin.\; ZP\; ::\; \backslash sin.\; ZPB\; :\; \backslash sin.\; ZBP\; (ABP)$;
that is,
$\backslash sin.\; (N+p)\; :\; \backslash cos.\; l\; ::\; \backslash sin.\; (H+\backslash Pi)\; :\; \backslash sin.\; ABP$.
Therefore,

whence

Make $\backslash frac\{\backslash sin.\; P\; \backslash cos.\; l\}\{\backslash sin.\; \backslash Delta\}\; =\; A$; then
$\backslash sin.\; \backslash Pi\; =\; A\; \backslash sin.\; (H+\backslash Pi)\; =\; A\; (\backslash sin.\; H\; \backslash cos.\; \backslash Pi\; +\; \backslash cos.\; H\; \backslash sin.\; \backslash Pi)$;
whence

and

that is,

whence we have the following series,

PROBLEM XVI.—To compute the parallax in declination.
Make $\backslash sigma\; =\; PB\; -\; PA$ (fig. 129) = the parallax in declination.

From the triangle $ZPA$ we get

and from $ZPB$

therefore

that is (retaining the notation of last problem),
$\backslash frac\{\backslash cos.\; \backslash Delta\; -\; \backslash sin.\; l\; \backslash cos.\; N\}\{\backslash sin.\; N\}\; =\; \backslash frac\{\backslash cos.\; (\backslash Delta\; +\; \backslash sigma)\; -\; \backslash sin.\; l\; \backslash cos.\; (N+p)\}\{\backslash sin.\; (N+p)\}$;
whence

therefore

Now, from the property of spherical triangles,
$\backslash sin.\; ZA\; :\; \backslash sin.\; PA\; ::\; \backslash sin.\; ZPA\; :\; \backslash sin.\; AZP$,
therefore

For the same reason

therefore

that is,

whence

By substituting this in equation (1) there results

whence

Make $\backslash frac\{\backslash sin.\; l\; \backslash sin.\; P\}\{\backslash sin.\; \backslash Delta\}\; =\; \backslash cot.\; \backslash alpha$, then

that is,

from which, as $\backslash Delta$ is known, $\backslash cot.\; (\backslash Delta\; +\; \backslash sigma)$ may be computed, and thence $\backslash sigma$. A more convenient formula, however, may be obtained by proceeding as follows. (See De-lambre, Abregé d'Astronomie, p. 154).

From the equation

we have

whence

therefore

Now, by the trigonometrical formulae, 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

whence

But it was shown in the last problem that

Make $\backslash tan.\; x\; =\; \backslash frac\{\backslash cos.\; (H\; +\; \backslash frac\{1\}\{2\}\; \backslash Pi)\; \backslash cot.\; l\}\{\backslash cos.\; \backslash frac\{1\}\{2\}\; \backslash Pi\}$; then, by substituting,

or $\backslash sin.\; \backslash sigma\; =$

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 $90^\backslash circ$ 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 = $90^\backslash circ$ = MQ, and therefore EM = DQ. In like manner PD = $90^\backslash circ$ = PC, consequently PP = CD the obliquity of the ecliptic.

The great circle PZNI 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 PZNI; consequently OI = ON = $90^\backslash circ$. Now since ON = $90^\backslash circ$, the point N is the nonagesimal, and its altitude IN = $90^\backslash circ$ - 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 = $90^\backslash circ$ - NC = $90^\backslash circ$ - 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' = $180^\backslash circ$ - ZPD = $180^\backslash circ$ - MD = $180^\backslash circ$ - (ED - EM) = $180^\backslash circ$ - $90^\backslash circ$ + EM = $90^\backslash circ$ + 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, $\backslash lambda$ = PZ the co-latitude of the place (reduced by problem XIII.), $\backslash omega$ = PP' the obliquity of the ecliptic, K = PZ the co-latitude of the zenith or altitude of the nonagesimal, N = $90^\backslash circ$ - ZPP the longitude of the nonagesimal.

By the trigonometrical formulae,

Assume $\backslash tan.\; \backslash phi\; =\; \backslash tan.\; \backslash lambda\; \backslash sin.\; R$, then, by substituting and reducing, we obtain

To find the longitude I, we have

whence, by substituting,

and on reducing,

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

Practical Astronomy. (fig. 128). The angle ZPA, which was before denoted by H, now becomes ZPA = Na = Ea = EN = longitude of the star = longitude of the nonagesimal, and ZPB becomes ZPB = Nb = Na + ab. Retaining, therefore, the notation employed in the three last problems, and making L = Ea the longitude of the star, Δ' = PA its distance from the pole of the ecliptic, k = ZI' = 90° - K, Π' = ab its parallax in longitude, σ' = PB - PA its parallax in latitude, we shall have Π' and σ' from the same formulae as Π and σ in Problems XV. and XVI., by changing l into k, H into L - N, and Δ into Δ'. Hence (Problem XV.)

and by putting C = $\backslash frac\{\backslash sin.\; P\; \backslash cos.\; k\}\{\backslash sin.\; \backslash Delta\text{'}\}$, we deduce

In like manner the formula for the parallax in declination given in Problem XVI., viz.

becomes

from which, by proceeding in the same manner as in Problem XVI., and making

we deduce $\backslash tan.\; \backslash sigma\text{'}$

whence, on making $\backslash frac\{\backslash sin.\; k\; \backslash sin.\; P\}\{\backslash cos.\; \backslash sigma\text{'}\}\; =\; D$,

consequently

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 ob-

tain 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° - ZM be the moon's true altitude; a = 90° - ZS the star's true altitude; H = 90° - Zm the moon's apparent altitude; h = 90° - Zs the star's apparent altitude; D = SM the true, and d = sm the apparent distance.

By spherical trigonometry, in the triangles SZM, sZm,

From these equal values of cos. SZM we find

But the last term = cos. (A + a); subtract now both sides from 1; then, observing that by the calculus of sines (see ALGEBRA)

we have, after dividing by 2, and putting

Let θ be such an angle that

and in this expression the value of F is put instead of it, then we have

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.

Practical the last, which gives $\backslash sin.\; \backslash frac\{1\}\{2\}\; D$, 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 $10^\{\backslash circ\}\; 46\text{'}\; 50\text{'}\text{'}$ south latitude, and $149^\{\backslash circ\}$ 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 $\backslash odot$ and $\backslash gamma$ ... $83^\{\backslash circ\}\; 26\text{'}\; 46\text{'}\text{'}$
Altitude of lowest limb of $\backslash odot$ ... $48^\{\backslash circ\}\; 16\text{'}\; 10\text{'}\text{'}$
Altitude of upper limb of $\backslash gamma$ ... $27^\{\backslash circ\}\; 53\text{'}\; 30\text{'}\text{'}$.
Hence the longitude of the ship is required.

Reduction of the apparent to the true altitude.

Reduction of the apparent to the true distance by Prob. XIX.

Computation of time at Greenwich.

By Nautical Almanac for 1793

Hence it appears that in 3 hours the distance of the moon from the sun was increased by $1^\{\backslash circ\}\; 22\text{'}\; 25\text{'}\text{'}\; =\; 4595\text{'}\text{'}$, and that between the time of making the observation, and the 15th hour, Greenwich time, the increase of distance was $0^\{\backslash circ\}\; 14\text{'}\; 54\text{'}\text{'}\; =\; 894\text{'}\text{'}$. Now for small intervals the distances will increase nearly as the times, therefore we have

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 $48^\{\backslash circ\}\; 26\text{'}\; 49\text{'}\text{'}$, 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 $23^\{\backslash circ\}\; 22\text{'}\; 48\text{'}\text{'}$ north. The ship's latitude, $10^\{\backslash circ\}\; 16\text{'}\; 40\text{'}\text{'}$, is also known. Hence to find the hour angle by Problem IV. we have

To find $P$ the hour angle, the computation (see formula of Prob. IV.) may stand thus:—

Long. from Greenwich westward $13^\{\backslash circ\}\; 52\text{'}\; 54\text{'}\text{'}\; 7$

Therefore, longitude east of Greenwich is $10^\{\backslash circ\}\; 7\text{'}\; 4\text{'}\text{'}\; 3$. 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 $20\text{'}\text{'}$ in a distance, which will produce an error of $40\text{'}\text{'}$ 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 $80\text{'}\text{'}$, which is equivalent to $20\text{'}\text{'}$, or $\backslash frac\{1\}\{2\}d$ 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 $\backslash frac\{1\}\{2\}d$ or $\backslash frac\{1\}\{4\}d$ of a degree may sometimes be obtained, but still it is uncertain.

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.