UC-NRLF OUTLINES OF NATURAL PHILOSOPHY, BEING HEADS OF LECTURES DELIVERED IN THE UNIVERSITY OF EDINBURGH, BY JOHN PLAYFAIR, I'KOFESSOR OF NATURAL PHILOSOPHY IN THE UNIVERSITY OF EDINBURGH, FELLOW OF THE ROYAL SOCIETY OF LONDON, SECRETARY OF THE ROYAL SOCIETY OF EDINBURGH, AND MEMBER OF THE ROYAL MEDICAL AND PHYSICAL SOCIETIES OF EDINBURGH. VOL. II. SECOND EDITION. EDINBURGH : Printed by Abernethy $ Walker, POR ARCHIBALD CONSTABLE AND COMPANY, EDINBURGH, AND LONGMAN, HURST, REES, ORME & BROWN, AND CADELL & DAVIES, LONDON. 1816. JIB1! - n/fu; Y a ,^--N7X'JJ "iO XT hours ; and as the whole circumference of the equator passes in that time over the meridian of any place, an arch of 15° passes in an hour ; of 1 °, in four minutes ; of 1' in 4 seconds of time; and so in proportion. ,18. To compare the period of a clock with the length of the siderial day. Observe the hour at which a star comes to the meri- dian on any two successive days ; the interval ought to be 24? hours ; if it is more, the clock goes too fast ; if less, too slow. By repeating these ob- servations day after day, the rate of the clock, or the quantity by which it daily advances on true time, or falls short of it, will be found. By short- ening the pendulum if the clock goes too slow, or lengthening it if the clock goes too fast, the mo- tion of the clock may be brought nearly to agree with that of the stars. 19- The period of the clock being thus adjust- ed, it is required to find what hour the clock marks, when the point of the vernal equinox is on the meridian of a given place. Observe OUTLINES OF NATURAL PHILOSOPHY, Observe the hour by the clock when the sun's centre is on the meridian ; and observe also the declina- tion of the sun at the same instant. Let the angle at which the sun's path cuts the equator, or what is called the Obliquity of the Ecliptic, = 23°.27'.30"; find an arch p, such that sin f = ta" Pe^ . ; tan23°.27/.W

f if it be after the sol- stice. The error thus found may either be cor- rected or allowed for. The clock is afterwards to be regularly compared with the southing of the stars. •K> . io a:?«'i ink <7K& 'n/la yt. i^ 'jtrii «o . yfinb Ji rfoid*, v.-j . on) SO. If a meridian circle pass through any star, the arch of the equator intercepted between that circle and the point of the vernal equinox is called the Right Ascension of the star ; and if a clock be regulated as above described, the time of a star's passage over the meridian, when turned into degrees, will be equal to its right ascension, _• » >>. ij>i . a. Suppose ASTRONOMY. 13 Suppose the hour according to siderial time, when a star passes the meridian, to be 4h. 24?i». 5Qsec. 4h. = 60° 24,m. = 6° 50sec. = 12'. 30" Right Ascen. = 66°. 12'. 30". h. The preceding observations, for determining the places of the stars, are all supposed to be made on the meridian ; and such, when they can be ob- tained, on account of their simplicity, are prefer- able to all others. It often happens, however, that the stars must be observed when they are not on the meridian, and their positions, with respect to the immoveable circles of the sphere, must then be derived from spherical trigonometry. c. The angle which the meridian of a star makes with the meridian of the place of observation, is called the star's Horary Angte, as it is the angle which measures the time between the instant of observa- tion and the star's passage over the meridian. 21. Of these five quantities, the Declination, the Altitude, the Azimuth, the Horary Angle of a star, and the Latitude of the place of ob- servation, if any three be given, the other two may be found from the resolution of the spheri- cal 14* OUTLINES OF NATURAL PHILOSOPHY. cal triangle PZS, fig. 1. contained by the arches joining the pole, the zenith, and the star. This general problem contains twenty cases, of which those that follow are the most useful in astrono- my. In all of these, we suppose d to be the declination, a the altitude, h the horary angle, z the azimuth of a star, and / the latitude of the place of obser- vation. 22. Suppose the altitude and azimuth of a star to be observed, and the latitude to be also known ; it is required to find the declination, and the horary angle of the star. Here #, «, I are given, to find d and h, or, in the spherical triangle PZS, the sides ZP, ZS are gi- ven, (the complements of I and d] and the angle PZS, between them to find the side PS, and the angle P. By letting fall a perpendicular from S on the meri- dian, we may obtain a solution by case 1st and 2d, of oblique-angled spherical triangles. The solution may also be expressed analytically thus : Find ASTRONOMY. Find an angle x, so that tan x = cos z X cot a, then 7 tan z X sin x tan // = — ; cos (£— x) COS X In applying these formulas it must be observed, that when z, or the azimuth counted from the north, is greater than 90°, cos z9 and therefore also tan x9 and x itself, are negative. The latter formula is useful, when the declination of a star is to be determined, and the former when the time is to be determined from the observations of a star not in the meridian. It is more usual, however, when the time is required, to have the declination of the star given, as in the following problem. 23. The declination and the altitude of a star being given, as also the latitude, to find the horary angle and the azimuth. Here the three sides of the triangle ZPS are given, to find the angles at P and Z. If df = complement of Dec. a' = complement of Alt. I' = complement of Lat. sin 16 OUTLINES OF NATURAL PHILOSOPHY. and sin /. 1 _ / 2 ~~ > sin /' X sin a' By the first formula, the hour can always be deter- mined from astronomical observations, if the time of the star's passing the meridian be known. The second formula serves to find the meridian, from the observation of the altitude of a known star. The meridian, however, is better found by obser- ving a star when it has the same altitude on the east and west sides of the meridian, and bisecting the difference of the azimuths, as in § 15. 24. Let a, h and / be given, to find d ; thai is, the altitude, the horary angle, and the latitude* to find the declination. a. Here two sides of the triangle ZPS are given, and the angle opposite to one of them, to find the third side. This falls under Case 7. of oblique-angled spherical triangles. b. Find ASTRONOMY. 17 b. Find an arc x, such that tan x = cos h X cot /. , cos x X sin a Next find an arc ?/, so that cos y = r— j . sin i- the declination is the complement of = y + x. This solution is from its nature ambiguous ; the sum of y and x must be taken, when the perpendicular from the zenith, on the circle of declination, falls within the triangle; their difference, when it falls without. 25. Let a, d, z be given to find /, or to find the latitude from observing the altitude and azimuth of a star, and knowing also its declina- tion. Here, again, in the triangle ZPS, two sides are gi- ven, and the angle opposite to one of them, to find the third side, the complement of latitude. The perpendicular must be let fall from the star on the meridian ; and the distance of this perpendicular, first from the zenith, and then from the pole, is found as in the last case; the sum or difference is the complement of the latitude. 26. Let a, d and h, be given to find / ; that is, the altitude and horary angle being observ- ed of a star, of which the declination is known, to find the latitude. VOL. II. B The 18 OUTLINES OF NATURAL PHILOSOPHY. The perpendicular is to be let fall, as in the last case, from the star on the meridian ; and there being two sides of the spherical triangle given, and the angle opposite to one of them, the calculation is as before. This problem is useful for finding the latitude, when two equal altitudes of a star are observed, and the interval of time between the observations. The half of the interval gives the horary angle, and so the latitude may be found as above. 27. In the above formulas it may happen, that a = 0, or that the star is in the horizon, or 90° from the zenith. The horary angle is then found, if the latitude and declination are given, from a right angled triangle, of which one of the sides, containing the right angle, is the ele- vation of the pole, the other, the arch between the star, when rising or setting, and the meri- dian ; and the hypothenuse is the distance of the star from the pole. In this case, the horary angle, (converted into time), is the time of half the stay of the star above the horizon, (or under it),, and if it be called H, cos H = tan / X tan d. The other side of the triangle, or the azimuth of the rising or setting star, is also called the Amplitude, and ASTRONOMY* 19 and is found from this formula, ,.^ sin d cos amplit. = - . cos/ This is much used in navigation, for ascertaining the variation of the compass. 28. In those stars which are less distant from the pole than the complement of latitude, the triangle formed by the star, the zenith and the pole, becomes right angled at the star, when the circle described by the star, in its diurnal motion, touches the vertical ; the azimuth is then a maximum, and is found from the solu- tion of a right-angled triangle, in which the hy* pothenuse, the complement of latitude, and one side, the complement of declination, are given, to find the angles, one of which is the azimuth, and the other the horary angle of the star. TT . cos Dec. Hence sin z = - cos Lat. , cot Dec. and cos h — = = cot Lat. cot Dec. X tan Lat. The first of these gives the azimuth when it is great- est, the second the hour when that happens. They are much used for finding the meridian line. 29. Another 20 OUTLINES OF NATURAL PHILOSOPHY. 29. Another purpose for which spherical tri- gonometry is employed, is, when the position of a star relatively to one circle of the sphere is given, to find its position relatively to ano- ther which is given in position with respect to the first. The position of the fixed stars, by the foregoing ob- servations, are found relatively to the equator; that is, the perpendicular arc from the star to the equator, is given, and also the distance of that perpendicular from a given point in the circum- ference of the equator ; but there is another cir- cle, called the Ecliptic, to which it becomes neces- sary to refer the stars. This circle cuts the equa- tor at the points of the vernal and autumnal equi- noxes, and is inclined to it at an angle known from observations which will be afterwards ex- plained, to be nearly 23°. 27'. 30". The distance of a star from this circle is called the Latitude of the star, and is reckoned north or south according as the star is on the same side of the ecliptic with the North Pole, or the South. The arc of the ecliptic intercepted between the per- pendicular from the star, and the point of the ver- nal equinox, is called the Longitude of the star. Let the angle formed by the equator and ecliptic, or the Obliquity of the Ecliptic be =0. 30* Suppose ASTRONOMY. 21 SO. Suppose than the right ascension and de- clination of a star are given, to find its longi- tude and latitude. Find an arc #, such that, cot x = sin z x.cot Dec. The declination, if north, is reckoned positive, if south, negative, and x has the same sign with it. Let y = x — seems to owe its white appearance to an incredible multitude of stars, which the eye can- not distinguish. Dr HERSCHEL has seen 1 16,000 stars pass through the field of his telescope in a quarter of an hour ; though the field was not more than 15 in diameter. 32. The most obvious distinction among the stars, is founded on their different magnitudes. Those of the first magnitude, are distinguished by particular names ; there are only ten visi- ble in Europe, which all astronomers have a- greed to belong to that class. a. The stars visible to the naked eye, are divided, in all, into six classes, not very accurately separa- ted from one another. The ancients counted 15 of the first, 45 of the second, 208 of the third. LA LANDE, § 557. H ALLEY, ASTRONOMY. 25 HALLEY, Phil. Trans. N° 364. Also VINCE, vol. r. p. 505. b. If c be the class reckoned from the first, 13 Xc2is nearly the number of stars in that class. 33. The fixed stars are not scattered over the heavens indiscriminately, but are disposed in groups ; to which, from the most remote anti- quity, names have been given from certain fi- gures of animals, conceived to be connected with them, which are called Constellations. a. In the beginning of astronomical science, it was only by such a device as this that men could speak of the stars, or describe them to one another. It is a remain of the ancient picture writing, that preceded alphabetical language. b. The number of the ancient constellations was 4-8 ; 24 have been added by the moderns, 14 in the southern hemisphere, and 10 composed ont of groups, not included in the ancient arrangement. The stars of each constellation are distinguished by the Greek letters, disposed in reference to their magnitude and position. For the method of distinguishing the constellations, see LA LANDE, torn. i. § 735., &c. 34. Many of the stars, which, to the naked eye, or through telescopes of small power, ap- pear §6 'OUTLINES OF NATURAL PHILOSOPHY. pear single, are found, with higher magnifiers, to consist of two, sometimes three or more stars, extremely near to one another. a. Dr HERSCHEL has observed no less than 700 of these multiple stars, of which only 42 were known before. In some of them the small stars are dif- ferent in brightness, and in the colour of their light. Thus y Andromedae is double ; the stars very unequal, the largest a reddish-white, the smallest a sky-blue, inclining to green. See VINCE, §669. 35. The fixed stars are not entirely exempt from change ; several stars which are mention- ed by the ancient astronomers having now ceas- ed to be visible, and some being now visible to the naked eye, which are not in the ancient ca- talogues. a. It was the appearance of a new star that induced HIPPARCHUS to begin his catalogue. A new star which appeared in Cassiopeia's Chair, in 1572, exceeded Sirius in brightness, and was seen at noon-day. It did not change its place, but gradu- ally decayed, and in about sixteen months disap- peared entirely. It is supposed to have appeared before in 945, and 1264. \7iNCE, § 704. b. A star of the same kind was seem by KEPLER in 1 604 ; and several similar facts are recorded. ct Algol, ASTRONOMY. %tf c. Algol, or /3 Persei, has been observed to have pe- riodical changes of brightness, that return at the distance of about 2 days and 21 hours. At its greatest brightness, it is of the second, at its least, of the fourth magnitude ; the change from the first state to the second is made in about three hours and a half, and the change back again in the same time ; during all the rest of the period it pre- serves its greatest brightness, Phil. Trans. 1773. Also VINCE, § 713. Some other stars are subject to similar variations. 36. Though the fixed stars are without sensi- ble motion relatively to one another, yet many of them, when observed very accurately, are found to change their places slowly. This was suspected by MAYER, but was first proved by Dr MASKELYNE. Thus the right ascension of Sirius diminishes annu- ally by two thirds of a second, and his declination, (which is south), increases by 1".2. Something of the same kind is observed of several others. VINCE, § 724-. Whether this motion is always in the same direction, or always at the same rate, is not yet known. 37. In many places of the heavens, spaces faintly luminous, or shining with a pale white light, §8 OUTLINES OF NATURAL PHILOSOPHY. light, and of an irregular shape, are discovered ; and on applying to them telescopes of great power, they are resolved into a multitude of small stars, distinctly separate, but extremely near to one another. These are called Nebulae. a. The Milky Way is a space of this kind, visible to the naked eye, and encompassing the whole hea- vens. DR HERSCHEL'S telescope discovers it to consist of a vast multitude of stars, (§ 31. d}. Other two nebula near the South Pole, distinguish- able by the naked eye, are called by sailors the Magellanic Clouds. The telescope shews them also to be composed of stars. b. The other nebula are not visible but with tele- scopes. HUYGENS discovered one in Orion's Sword, that appeared a bright spot on a dark ground, and seemed like an opening into some brighter region. c. Several more nebula had been observed before Dr HERSCHEL, by HALLEY, CASSINI, LA CAILLE, &c. and a catalogue of 103 was published by the French astronomers, in the Connaissance des Terns for 1783. Dr HERSCHEL has given a catalogue of more than 2000 discovered by himself. Phil. Trans. 1786, 1789, &c. 38. Dr HERSCHEL has also discovered nebu- lous stars, that is, single stars, surrounded by a faint ASTRONOMY. gp faint luminous light. This light has not been resolved into small stars, and Dr HERSCHEL be- lieves it to be the effect of a luminous fluid, Phil Trans. 1791. 39. As the apparent motion of a body, in any direction, may arise either from the real motion of the body in that direction, or from the motion of the spectator in the opposite j so the apparence of the diurnal motion of the heavenly bodies, round the earth, may either be produced by the real revolution of the heavens from east to west, or by the rotation of the earth, on its axis, from west to east. This principle is conformable to the experience of every day, and as there are, therefore, two ways of accounting for the phenomena of the diurnal mo- tion, we must choose that which is least liable to objection from other quarters. 40. It is no objection to the supposition, that the diurnal motion of the heavens arises from the motion of the spectator, that he himself is not sensible of his motion. The motion which any body has in common with other bodies, does not affect its state with respect of those bodies, and all the relative motions take place just as if the motion common to them all had no existence. This 30 OUTLINES OF NATURAL PHILOSOPHY. This has been explained under the heads of the first law of motion, and of the distinction between ab- solute and relative motion. 41. The reality of the diurnal revolution of the heavens, is liable to a great objection, as it supposes that a circular motion, in the same direction, is common to an immense number of bodies, far distant, and entirely detached from one another ; and that this motion is so regula- ted, that their revolutions are all performed in the same time, and in planes parallel to one another. The revolution of a detached body about a centre, or about an axis, cannot take place without a force constantly acting, to draw it out of the straight line in which it has at every instant a tendency to continue its motion. The revolution of a solid body, like the earth, on its axis, may arise from one original impulse ; its continuance requires no new action, but is a consequence of the inertia of matter. It was to obviate the difficulty arising from the de- tached and distant situation of the bodies, to which motions so perfectly coincident were ascribed, that the hypothesis of crystalline orbs was in- vented. To those who do not admit the exis- tence of these orbs, the diurnal revolution of the heavens can have no probability. 42. The ASTRONOMY. 31 42. The physical and mechanical objections to the rotation of the earth being entirely ob- viated, while many press so hard upon the op- posite hypothesis, the simplicity of the ex- planations afforded by the former, justifies us in admitting it as the cause of the apparent diurnal revolution of the heavens, at least till some fact, or some principle inconsistent with it shall be discovered. SECT. II. OF THE ATMOSPHERICAL REFRACTION. 43. A RAY of light, in passing through the at- mosphere, is bent into a curve, in the same ver- tical plane with the original ray, and concave toward the surface of the earth. But the ob- ject from which the ray comes, is seen in the direction which the ray has when it enters the eye, and therefore it appears elevated above its true place. This is called the Atmospherical, and sometimes the Astronomical Refraction. a. The effect of the atmospheric refraction alters the place of an object only in a vertical plane ; it in- creases 3% OUTLINES OF NATURAL PHILOSOPHY. creases the altitude, but does not affect the azi- muth. b. Hence all the altitudes, measured as in the prece- ding sections, require corrections to be applied to them before the true altitudes are obtained. The method of making these corrections is now to be explained. 44. From the principles of optics, it is known, that the rays which pass through the strata of the atmosphere at right angles, or which come from stars in the zenith, suffer no refraction, and that, at all other elevations, the quantity of the refraction is nearly as the tangent of the zenith distance. a. If x be the true distance of a star from the zenith, and y the refraction, so that the apparent distance is x — y, then, by the nature of refraction, sin # is to sin (x — y] in a constant ratio, suppose that of m to n ; and since sin (x — y) = sin x . cos y —cos x . sin j/, !! sin ,r=sin x . cos^ — cos x . siny, m or — = cos y — ^!_f. sin y. Now if y be very m sin x small, cosy = 1 nearly; and therefore sinj/ = (1 _ * ) ELf = (i _ n-} tan x. As sin y = y ml cos x ml nearly ASTRONOMY* 33 hearty, j/=(l — — "\ tan x 5 where the constant co- efficient 1 — —is to be determined by observa- m tion. If. It is evident that this approximation is imperfect, because it gives y infinite when ^=90°, or when the star is in the horizon ; and in all cases when the altitude is small, it gives the refraction too great. Dr BRADLEY found that the refraction was more nearly as the tangent of the apparent zenith distance, diminished by a certain multiple of the fefraction, or that y=. A .tan (x — b y}. He also found A = 57" and b = 3 nearly, so that y = 57" tan (x — 3 y). To use this formula, we may suppose, for a first ap- proximation, 3/ = 57" tan #, when x is not very great. If 07=90°, we must assume y^ and correct the value by trials* The manner in which the constant quantities A and b have been determined, remains to be explained. 45. As the determination of the latitude of a place, and of the declination of a star, involve the effect of refraction, they are not to be em- ployed for finding the refraction, except when VOL. II. C a 34f OUTLINES OF NATURAL PHILOSOPHY. a considerable error in them will not produce any sensible error in the refraction to be found. In conformity with this rule, the following method of finding the horizontal refraction may be em- ployed. 46. A star which rises and sets due east and west, would be as long under the horizon, as it is above it, if there were no refraction. It will be found, however, by observation, to be lon- ger above than below the horizon : take half the difference, and reduce it to degrees, or parts of a degree ; then multiply this last by the cosine of the latitude of the place ; the product is the horizontal refraction. a. In this way, the mean horizontal refraction will be found to be 33' 5 and here it would require a great error in the latitude to produce an error at all sensible in the refraction. b. This method requires a situation, where the stars to be observed rise and set in the sea. It may, however, be extended, though with less simplicity, to all the stars, and to their descending, not be- low the horizon, but below a given- altitude, or a. given parallel to the horizon. 47. Observe the altitude and azimuth of a star of a known declination at the same instant j from ASTRONOMY. 35 from the azimuth, the polar distance and the complement of latitude, compute the altitude ; the difference between this and the observed altitude is the refraction. a. This may be easily done with the Astronomical Circle, which measures altitudes and azimuths at the same time. The circumpolar stars are well suited to this observation. The triangle may be resolved by NAPIER'S rules, Mauduit, p. 76. anot. 2. See other methods, VINCE, § 176., &c. WOODHOUSE, p. 85., &c. LA LANDE, 2170., &c. I). If the time from the altitude of a known star, on the one side of the meridian, to the same altitude on the other side, be observed, the apparent alti- tude may be calculated from thence ; and if a cir- cumpolar star is used, great accuracy may be ob- tained in this way, as a small change in the alti- tude produces a great one in the horary angle ; so that to observe this latter angle, is the most ex- act way of coming to the knowledge of the appa- rent altitude. 48. The refraction varies with the state of the barometer and thermometer, and, for the same altitude, is nearly proportional to the density of the air at the earth's surface. a. Dr 36 OUTLINES OF NATURAL PHILOSOPHY. a. Dr BRADLEV has given a formula, expressing this variation of the refraction. Xet b = height of the mercury in the barometer in inches, k = height of FAHRENHEIT'S thermome- ter, z = the zenith distance, r = the mean refrac- tion computed by the rule, (§ 44. 6.), the correct br 400 refraction r = ——-X 29.6 350+7* This formula would be improved, by introducing the expression for the density or specific gravity of the air from vol. i. } 333. It will then be i> - _^L x (1—0.00222 (A — 50))- A«7«O 49. Not only are the stars elevated by refrac- tion, but all terrestrial objects are elevated from the same cause, by an angle equal to that which the straight line drawn from the eye to the ob- ject makes with the tangent to the path of the ray at the point where it enters the eye. The refraction of the heavenly and terrestrial bodies is differently estimated. That of the celestial bo- dy is the angle contained between the tangent to the curvilineal path of the ray where it is first ac- ted on by the atmosphere, and the tangent to the same curve when it enters the eye. The refrac- tion of a terrestrial body is the angle contained between the tangent at the eye and the chord of the arch intercepted between the object and the eye. Near ASTRONOMY. 3? Near the earth's surface, the curvilineal path of the ray of light may be supposed to coincide with the circle of equal curvature. 50. If the elevation of the top of a mountain from a point in the plane below, and the de- pression of that point from the top of the moun- tain, be both observed at the same time, the angle subtended at the earth's centre by the distance between these points, added to the ob- served elevation, and the sum diminished by the depression, is double of the refraction. This supposes the path of a ray of light, for a small part, to coincide with a circle. If, in fig 4., the arch from B to A be the path of a ray, and if AH and BF be perpendicular to AC, BC, in A and B ; the tangents EA, EB being drawn to the path of the ray, HAE is the apparent elevation of B from A, and FBE the apparent depression of A from B •, the true elevation being HAB, and the true depression FBA. It is evident that FBA = BAH+ACB, or true Dep. = true Elev.-f Hor. Ang. ; that is, true Elev. = true Dep. — Hor. Ang. But true Elev. = app. Elev. — Ref. $ and true Dep. = app. Dep. + Ref. ; therefore 2 Ref. = Hor. Ang. + app. Elev. — app. Dep. 51. The 3$ OUTLINES OF NATURAL PHILOSOPHY. 51. The terrestrial refraction found by means of the preceding theorem, when the elevation is not very great, varies from - to ~- of the angle subtended by the horizontal distance of the ob- jects ; and the radius of curvature of the ray, therefore, varies from twice to twelve times the radius o£the earth. In the mean state of the atmosphere, the refraction is about -7 of the horizontal angle, and the radius of purvature of the ray seven times the radius of the earth. The terrestrial refraction must vary with the density of the air, or with the barometer and thermometer. The great differences, however, remarked above, must be owing to some other cause. Jn the measurement of heights, the angle of elevation should be diminished by one-fourteenth of the angle corresponding to the horizontal distance, suppo- sing the refraction to be of the mean quantity. 52. The effect of refraction may also be allow- ed for, by computing the correction of curva- ture, as in § 248, vol. i., and taking one-seventh of it, for the number of feet, by which the ob- ject is rendered by the refraction higher than it ought to be. If ASTRONOMY. 39 If L be the length of the horizontal line in English 2L* miles, the correction for curvature in feet is — — » o 2L* and for refraction TTT-» and this last is to be subtracted from the former, which gives — - or — for the quantity to be added to the height as calculated trigonometrically, including both the corrections for curvature, and, in the mean state of the atmosphere, for refraction. When a height is calculated trigonometrically from the angle of apparent elevation observed at a gi- ven distance, both the preceding corrections maybe made on the angle itself, which on some occasions is more convenient. Divide the horizontal dis- tance in feet by 6075, the feet in a geographical mile or a minute of a great circle of the earth. Let the quotient = H, and let the observed angle of elevation = E, then the true angle corrected both for curvature and refraction is E -f- 0.428 H. SECT. III. FIGURE OF THE EARTH. 53. THE figure of the earth is understood to be determined by a surface at every point perpen- dicular 40 OUTLINES OF NATURAL PHILOSOPHY. dicular to the direction of gravity, or to the di- rection of the plumb-line, (§ 24. £.). This surface is the same that the sea would have if it were continued all round the earth ; or, if we were to trace curve lines, by levelling from a given point round the earth, in every direction, till they re- turned into themselves, the superficies in which all these lines would lie, is that which we consider as the superficies of the earth. The given point may be supposed any one, on the level of the sea. The figure bounded by this superficies, is that which is really measured by the combined methods of astronomy and practical geometry, and is to be carefully distinguished from the actual figure of the earth, including all its inequalities ; or from an average figure that should leave out as much of solid matter above it, as is included of empty space under it. 54. The length of an arch of the meridian, traced on the superficies above defined, may be measured by observing the latitude of the two extremities of the arc, and then measuring the distance between these points in fathoms, toises, pr any other known measure. The distance, as measured on the surface, divided by $he degrees, and parts of a degree contained in the difference ASTRONOMY. 41 difference of the latitudes, will give the length of a degree. ERATOSTHENES was the first who applied this method to the estimation of the earth's circumference. By measuring, or rather estimating, the difference of latitude between Alexandria and Syene, and also their distance, he concluded the circumference of ( the earth to be 250000 stadia. MONTUCLA, His- toire des Math. torn. I. p. 242. 2d edit. The length of a degree, or of any arch of the meri- dian, is determined by taking two points nearly at the distance of the arch required, and nearly north and south of one another. A series of triangles is then to be carried from the one point to the other by means of stations taken on the tops of hills or other elevated grounds. The angles of these tri- angles are to be measured, as also the azimuths of the sides, at the points where the series of triangles begins and ends. Thus, there is a series of triangles all given in species which connects the two extreme points, and the bearings of the sides of these triangles, in respect of the meridian of the first station, are also given. The lengths of the sides of the triangles in known measures, toises or fathoms, is next found by mea- suring a base on a level ground, and connecting it by angles with the sides of one of the triangles. Jn all this we proceed as if the triangles were plane, whereas they are in fact spherical, and the three angles 42 OUTLINES OF NATURAL PHILOSOPHY. angles of each exceed two right angles, by a quan- tity called the Spherical Excess ; which is to four right angles, as the area of the triangle to half the superficies of the globe. By knowing the spherical excess, the errors in the measures of the angles are discovered ; and it is also known, that if from each of the angles of any of the triangles, be subtracted one-third of the spherical excess, the sines of the angles so corrected are proportional to the lengths of the sides ; so that all the triangles, though strict- ly speaking spherical, may be resolved by the rules of plane trigonometry. IQ this way, the distance of the two points at first as- sumed is computed, as also their distance reduced to the meridian of either of them. , 55. When, by such accurate observations, the lengths of degrees were determined in different latitudes, they were found to increase gradual- ly from the Equator to the Pole. The radius of curvature of the meridian, therefore, increases as we go toward the Pole, and the curva- ture itself diminishes. The earth, therefore, is not a sphere, but is flattened at the Poles, so that the axis from Pole to Pole is less than the diameter of the Equator. Though it is only by experiment that the true figure of the meridian can be disco- vered, it has been found necessary to assume hypo- thetically, for its figure, the curve which is next in simplicity ASTRONOMY. 43 simplicity to the circle, viz. the Ellipsis, and also to suppose the superficies of the earth to be that of a spheroid generated by the revolution of this ellipsis, about its shorter axis. In many complex cases, this mode of approximating to the truth, by probable assumptions, is the sim- plest that can be pursued. The hypothesis thus assumed, must be rigorously submitted to the test of experience. 56. The solid contained by the radius of cur- vature, at any point in an ellipsis, and the square of the semiparameter of the greater axis, is equal to the cube of the normal at the same point. That is, if a and b are the semiaxes, r the radius of curvature, n the normal at any point, ri* = — x r. See FBISIUS de Anatysi Sect. Con. Opera, torn. i. p. 96. prob. 32. NEWTON'S Conic Sections, prop. 78. cor. 2. 57. Hence the radius of curvature, at any point of the meridian, and consequently the length of a degree at that point, may be express- ed in terms of the latitude : if r be the radius of curvature at a point, of which the latitude is A, a and b denoting as before, a* I* ( a* cos A* + b* sin A*)^ Let OUTLINES OF NATURAL PHILOSOPHY. Let ADB be one half of the meridian, A and B points in the Equator, C the centre of the earth, D the Pole, EA a perpendicular to the meridian at E, a point of which the latitude is A = EGA, H the centre of curvature* F a perpendicular on the axis ; then EG is the normal, or n, GF the sub- normal = 5 ; let CF = x, and FE = y ; then bz if— — (a* — x31}. But, by the property of the subtangents of the ellipsis, x = ^ s ; also s = n cos A, and y = n sin A ; therefore, by substi- tution, n* (b* sin A* + a* cos A*) = b*9 and n = ^ (a* cos ** + bz sin Az)4 Now, by § 56. r == --; n3 -9 therefore, If D be the length of a degree in lat. A, and m r= 57°.2957795, the number of degrees in an arc equal to the radius, then r = m D, and D = m (a* cos A* + 58. In ASTRONOMY. 45 58. In an ellipse where the eccentricity is small, or where a and b differ but by a small quantity c, this general formula may be reduced to more simplicity, by extracting the root of the denominator, and rejecting the powers of b greater than the first ; we have then mD = a(l — ?-?+!? sin A*). a. This value of m D may be changed into another, more convenient in calculation, by substituting for sin A* its value, 1 cos 2 A, from which is obtained b. At the Equator, A = 0, and cos 2 A = 1 ; so that mD = a (1 -- -)=0 — 2c. c. At the Pole, A = 90°, 2 A= 180? ; and since cos 180°= — 1, wD = 0+c. The degree of the meridian at the equator, is there- fore to the degree at the pole as a — 2 c to «-f c. H. In the parallel of 45°, 2 A ,-= 90°, and cos 2 A = 0 ; therefore m D = a — -. The 46 OUT1NES OF NATURAL PHILOSOPHY. The radius of curvature at the parallel of 45°, or m D, is nearly an arithmetical mean between the radius of the equator and half the earth's axis. The degree in the parallel of 4? 5 is also an arithmeti- cal mean between any two degrees equally distant from it on the north and on the south. c. The degree in any latitude is understood to be that of which the middle point is in that latitude. f. If, therefore, D be found by actual measurement in any two known latitudes, we shall have two equa- tions, in which a and c are the only unknown quantities, and from which, therefore, they may be determined. 59. The lengths of two degrees, of which the middle points are in given latitudes, being known ; it is required from thence to determine the diameter of the equator, and the axis of the earth, that is, the longer and the shorter axis of the elliptic meridian. a. Let D and D' be the given degrees, (the least, or that nearest the equator, being D), A and >f the lati- tudes of their middle points, a the semitransverse axis of the meridian, c the difference of the semi- axis ; we have the equations, ml) == ASTRONOMY. 47 C 3c '=tf— — COS 2 A'. ~ D) Hence c = 3 (cos 2 A -co* 2 A')' ,c 2(D'~ D) a" 3D (cos 2 A —cos 2 A')" Also, if — = w, a = These formulas will be reduced to others, mote con- venient for logarithmical calculation, by substitu- ting for cos 2 A — cos 2 A', its value, viz. 2 sin (A+A') X sin (A' — A); m (D' — D) . = 3 sin (V + A) X sm(A'-A)9 ^ , c_ D' — D and a~ 3 D sin (A' + A) X sin (A' — A)' When A is nothing, or when one of the degrees is at the equator, A + A' and A' — A, are each equal to ^ so c = ^(p/.— D). ; therefore the excess of the 3 sin* A degree. 48 OUTLINES OF NATURAL PHILOSOPHY. degree, in any latitude, above the degree at the equator, when divided by the square of the sine of the latitude, should always give the same quotient; or the excess of the degrees of the meridian above the degree at the equator, should be as the squares of the sines of the latitudes. o- m[\y — JJ) = 3 sin (/ + ,OX sin (/-*)' D' — D = — sin (A' -f A) X sin (A'— A). m c. If, then, D' and D are two contiguous degrees, so thatA'=A+l°,D'— D = — sin(2A+l°)xsin 1° ; and since sin 1° = .01745, D' — . D = ScX .01745 . - - - sin (2 A + 1°). The contiguous degrees, therefore, differ by a quan- tity proportional to the sine of twice the middle latitude. The difference is a maximum, when 2 A + 1 = 90°, or when the middle latitude is 45°. The ASTRONOMY. 4*9 The quantity — is called the Compression, and de- termines the species of the ellipsis. d. We shall now take for the determination of the fi- gure and magnitude of the earth, the five arches subjoined, as those that have been measured with the greatest care, and the best instruments ; as being the largest also, and the most distant from one another. Lat. Deg. in 1 Deg. in toises. ' fathoms. Country. 1. II. III. IV. V. 0° .O'.O" 1 1 .0.0 4-5 .0.0 52 .2.2 66 .20.10 56749 60480.2 56755 604-86.6 57011 ! 60759.4? 57074 ! 60826.6 57192 ' 60952.4- Peru. India. France. England. Lapland, j As five quantities may be combined, two and two, in ten different ways, ten results may be deduced from the comparison of these degrees ; and if the meridian were truly elliptical, and if there were no error in the observations, all these results would coincide. As the latter supposition cannot be ex- pected to hold, we must look for some difference in the results, and must choose only those combi- nations, in which the degrees are considerably dis- tant from one another, because in that way the er- rors of observation will least affect the conclusions. Such are the 1st and 3d, 1st and 4th, 1st and 5th j 2d and 3d, 2d and 4th, 2d and 5th ; 3d and 5th. VOL. II. D e. The 50 OUTLINES OF NATURAL PHILOSOPHY. e. The 3d degree, or that which is bisected by the pa- rallel of 4-5°, is to be reckoned the most accurate of all, as being deduced from the actual measure- ment of an arch of more than 12 degrees. This arch belongs to the meridian of Paris, which has been continued by the French mathematicians north to Dunkirk, lat. 51° 2' 9" N., and south to Formentera, the southernmost of the Balearic isles, in lat. 38° 38' 56", the distance being 705188.77 toises, (Base Metrique, torn. in. p. 298.) The arch of the same meridian from Dunkirk, till it is intersected by the parallel of Greenwich Observa- tory, lat. 51° 28' 39"-|, was also measured by Ge- neral ROY, and found to be 25238.5, or, more exactly, by DE LAMB RE, on applying certain cor- rections, 25241.9 toises. Thus the amplitude of the arch between the parallels of Greenwich and Formentera, is 12° 48' 43".5, and the distance 730430.67. This arch, however, is not exactly bi- sected by the parallel of 45°, Greenwich be- ing 6° 28' 39".5 distant from that parallel, and Formentera only 6° 20' 3".99. To have the y m length of an arch beginning at Formentera, and ter- minating just as far tothenorth,as Formentera is to the south of 45°, we must subtract from the dis- tance of Greenwich the number of toises corres- ponding to 8'35".51, which, as may be inferred from the measurement itself, is 8165.88 toises -, and thus we have 722264.79 for the length of an arch extending 6° 20' 3".99 on each side of the paral- lel of 45°. Now, ASTRONOMY. 51 Now, it was observed (§ 58. d.) that the degree in lat. 45° is an arithmetical mean between any two degrees equally distant from it no the north and south ; and therefore if an arch of any number of degrees, for example 6, be measured on the south of 4-5°, and an arch of as many degrees on the north, the two together will be 12 times the degree in the parallel of 45°. If, therefore, the above ex- tent, 722264.79, be divided by 12.6689, the ampli- tude of the arch, the quotient 5701 It is the.degree of the meridian bisected by the parallel of 45°. f. The degree at the Equator is deduced from an arch of 3° 7' measured in Peru by the French and Spa- nish academicians. Jt is stated by BOUGUER at 56753*; by CONDAMINE, at 56749 ; and we pre- fer the latter number, as there is reason to think, from comparing that degree with others, that it is too great. Figure dc la Tcrre, par M. BOUGUER, Sect. v. p. 272. g. The degree lat. H°, is a mean of 6 degrees, mea- sured by Major LAMBTON in Indostan. Asiatic Researches, vol. xn. p. 94. h. The degree in lat. 52° 2' 20" is from an arch of the meridian measured by Colonel MUDGE in the south of England. The mean is given here from a paper, Phil. Trans, for 1812, p. 332. r t f i. The degree at the Polar Circle, or in lat. 66° 20' 10", is from an arch of the meridian lately measured by SWANBERG and other Swedish academicians. Ex- position 52 OUTLINES OF NATURAL PHILOSOPHY. position des Operations faites en Lapponie, fyc. par SWANBERG. Stockholm, 1805. k. Of all these degrees, it may be said, that an error of 30 toises in the length, or 2" in the amplitude of the arch, is more than can be reasonably sup- posed. If the hypothesis of an elliptic meridian agree with them nearer than this quantity, it must be considered as having the support of observa- tion. But if it is found that these arches cannot be reconciled with the elliptic hypothesis, without supposing greater errors than those just mention- ed, that hypothesis must be either rejected, or re- garded as doubtful. 60. The five degrees in the last article, coin- cide in giving very nearly the same compression to the earth at the poles, and may all be repre- sented by the same equation, to an exactness much within the limits that have been assigned (§ 59. A.) a. By combining the degrees in the seven ways men- tioned, it will be found, that — is between .003 1 5 'Jo>: a aril r; and .00325. b. The mean between these, or .0032, it that which, on the whole, seems the nearest to the truth. It makes the sum of all the errors in the five degrees amountonly to21 toises, taking them with the same sign. Taking them with their proper signs, they nearly ASTRONOMY. nearly destroy one another. The compression may therefore be stated at .0032 = » and the equation which does most nearly represent the de- grees of the meridian, will from thence come out, D = 5701 1* — 272'.65 cos 2 A. In fathoms, D = 60759.472 — - 290.576 cos 2 A. In miles, D = 69.044 3299 cos 2 A. c. Hence, by the formulas (§ 59. £.), Toiscs. Fathoms. Miles, c = 10469.58 = 11158.8 = 12.680 a = 3271743.00 = 3486858.8 = 3962.349 I = 3261273.42 = 3475700. = 3949.669 Radius of curvature for the parallel of 45° = a — C- = 3266508t.21 =3481279^.4 = 3956 009 miles. The miles meant here are English miles. d. The circumference of the Elliptic Meridian may be found nearly by multiplying the mean degree, or that in the parallel of 45°. by 360. The result is 24855.84 miles. The 54 OUTLINES OF NATURAL PHILOSOPHY. The circumference of the Equator is 24-896. 16 miles, a little more than 40 miles greater than the prece- ding. The circumference of the Meridian may be found more accurately by the theorem for the rectifica- tion of the ellipsis. See LA CAILLE, Lemons Ele- mentaires de Math. § 954-. Also Base Metrique, torn. ii. p. 676. The French, from their late measurement, compared with that in Peru, make the compression .00324, and the quadrant of the meridian 5131111 toises ; which gives for theentirecircumference5468481.54 fathoms, or 24856.72 miles ; about 1 mile great- er than the result obtained above. e. The Geographical Mile, or that of which there are 60 in the length of the mean degree, is 1012.6 fa- thoms, = 6075.6 feet. 61. The semidiameter of the earth belonging x» to any latitude \, is nearly equal to a (1 sin* ^. a. This formula is found, by expressing the serai- diameter of the elliptic meridian, in terms of the latitude, in a manner similar to that employed $57, ASTRONOMY. 55 § 57, 58., then extracting the square root, and rejecting the powers of the compression greater than the first. See CAGNOLI Trigonometric, § 1558. b. The tangent of the angle ACE (fig. 3.), which the diameter answering to the latitude A, makes with the greater axis of the meridian, or with the plane of the equator, is — x tan A. CAGNOLI, § 1339. c. The angle at the centre being thus found, the angle which the semidiameter CE makes with the verti- cal EG is also found, being the difference between the former angle and the latitude of the place. The angle which the diameter makes with the ver- tical is greatest at 45°, where it amounts to 11' 9" when the compression is .00324. Base Metriquet torn. in. p. 292. If the compression is greater, this angle also becomes greater. d. The figure of the earth may not only be determi- ned by comparing two degrees of the meridian with one another, but it may also be found by com- paring a degree of the meridian, with the degree of a great circle perpendicular to the meridian, in the same latitude. 62. If we suppose the earth to be cut at any point by a plane perpendicular to the meridian in that 56 OUTLINES OF NATURAL PHILOSOPHY. that point, the centre of curvature of this sec* tion, at the point where it cuts the meridian, is the point in which the direction of gravity, or of the plumb-line, intersects the axis of the earth. «. The direction of gravity, if the earth be a solid of revolution, passes always through the axis of the earth. If, therefore, we conceive the plumb-line to be carried over an indefinitely small arch of the perpendicular to the meridian, either to the east or west, its direction will intersect the axis at the same point where it intersected it before, which point, therefore, is the centre of curvature of the arch, or the same with K (fig. 3.). EK is great- er than EH, and the degree of the perpendicular arch is greater than the degree of the meridian in the same ratio. b. The radius of curvature of the arch perpendicular to the meridian, is therefore the normal of the meridian, relatively to its shorter axis, or it is V a* cos* A + £* sin* p. 14. Edin. Trans, vol. v. 63. If D be the degree of the meridian, at a point of which the latitude is X, and A the de- gree of the curve perpendicular to the meridian at the same point, c = ASTRONOMY. e = s (A — D) X— -; 2 v cos* A \ Ml a = w A — - (A — D) tan* A ; 2 c A — D r nearly. * a 2 A cos Thus the figure of the earth is determined by the de->- gree of the meridian in any latitude, compared with the degree perpendicular to it. The degree of the section perpendicular to the meri- dian, is to the degree of the circle parallel to the equator, that is, to the length of the degree of longitude, as 1 to cos A. The degree of longitude is therefore A cos A. Edin. Trans, vol. v. p. 26. The manner in which the amplitude of the celestial arch is measured in the case of a perpendicular to the meridian, is not so direct as that which is fol- lowed in the case of an arch of the meridian itself. It is best done by azimuths or by determining the convergency of the meridians. If P be thepole, (fig. 5.), AP an arch of the meridian, AD an arch at right angles to it, or nearly so : Let the latitude of A, and also the angle PAD be found, and again at D the an- gle PDA ; then in the spherical triangle PAD, the angles at A and D are given, and also the side AP, from which the arch AD may be computed in de- grees and minutes j and its length also having been measured, OS OUTLINES OF NATURAL PHILOSOPHY. measured, the length of a degree becomes known. It is not necessary that AD should be accurately at right angles to either of the meridians AP, DP : If it is nearly so, it* length can easily be reduced to that of the perpendicular. This method of finding the amplitude of an arch per- pendicular to the meridian, is only applicable in high latitudes, where the convergency of the meri- dians is considerable. Near the equator, where the meridians became almost parallel, a small error in determining the azimuths will produce a very great one in the amplitude of the arch, so that this me- thod cannot be safely employed. The manner of finding the amplitude of the perpendicular arch ac- curately in such cases, depends on the methods of finding the difference of longitude, which are to be explained in the next section. It appears from the preceding investigations, that the earth is an oblate spheroid, generated by the revo- lution of an ellipsis about its shorter axis, that axis being to the longer axis as .9968 to 1. All the arches, however, that have been measured, do not agree equally in bringing out this result. In general, though the conclusions from arches, which are large, and at a considerable distance, are con- sistent with one another, the contrary holds where arches very near to one another, and more especi- ally, contiguous portions of the same arch, are com- pared together. It has been shewn, that, according to the elliptic hypothesis, the differences of the contiguous ASTRONOMY. 59 contiguous degrees ought to be proportional to the sines of twice the middle latitude, (§ 59. c.), and therefore ought to follow a very regular progres- sion. This, however, has not been found to take place in any instance of actual measurement. In the great arch measured in France, in those of the Peninsula of India, and in the south of England, the contiguous degrees have differed very irregu- larly, and by such considerable quantities, as could not be explained by any probable error of obser- vation. Local irregularities in the figure of the earth, manifesting themselves by the deflection of the plumb-line, seem to give rise to these anoma- lies. The deflections are but small, and disappear altogether when arches of great extent, and differ- ing by large quantities, are compared with one another. This subject will be more fully considered under the head of Physical Astronomy. SECT. IV. GEOGRAPHICAL PROBLEMS. 64f. THE situation of a point in a given superfi- cies is determined, when its distances are known from 60 OUTLINES OP NATURAL PHILOSOPHY, from two planes, which are given in position with respect to that superficies. a. That the determinations thus afforded may be the simplest possible, the two planes ought to be at right angles to one another ; and if the superficies is one having a centre, the planes should pass through that centre. b. In the case of the earth, the plane of the Equator having its position fixed by the diurnal motion, is naturally pointed out as one of the fixed planes, to which the positions of places in the earth's sur- face are to be referred. The position of every place, relatively to the equa- tor, is determined by finding its latitude as above defined. c. The other circle to which the position of places on the earth's surface is to be referred, must necessa- rily be a Meridian (a) ; but as none of the meri- dians is distinguished from another, by any cir- cumstance in the diurnal motion, of which they all partake alike, the particular meridian that is to be fixed on for the determination of geographical po- sitions, is a matter of arbitrary arrangement. d. When a meridian is chosen for a first meridian, or that to which all positions are to be referred, it is not by directly measuring the distance from it that such a reference is made, but by measuring the angle which the plane of the meridian passing through ASTRONOMY. Cl through any given place, makes with the plane of the first meridian. This angle is called the Ion* gitude of the place, and the diurnal motion fur- nishes us with the means of determining it. It is measured by the arch of the equator, inter- cepted between the first meridian and the me- ridian of the place, and is reckoned east or west, according as the place is east or west of the first meridian* - The ancients took for their first meridian, or that from which their longitude was counted, the meri- dian of the Fortunate Isles, a line passing, as they conceived, through the western extremity of the habitable earth. Many of the moderns have em- ployed the same meridian, or rather that of the Island of Ferro, one of the most westerly of the Canaries. In general, however, nations employ the meridian of their own metropolis, or of their principal observatory ; as we do that of Greenwich, the French that of Paris, &c. It has been propo- sed to take the meridian of Mont Blanc as the first meridian, being that of a point very remarkable in the natural history of the globe. It would be inconvenient, however, to take for a first meridian any point where astronomical observations are not constantly made. 65. The hour, as reckoned under any two me- ridians, is different, and the difference is proper- tional to the difference of longitude, or the angle which 62 OUTLINES OF NATURAL PHILOSOPHY. which the planes of these meridians make with one another. The hour at any place, as we have already seen, is determined by the passage of a certain star, or a certain point in the heavens, over the meridian of that place. But the star comes to any meridian sooner than to another farther to the west, by a space of time which is to 24? hours as the angle made by the two meridians to 360 degrees. For every 15°, therefore, contained in the angle which the meridians make, or in the difference of longi- tude, one hour is to be reckoned, by which the ac- count of time at the more westerly place is later, or the clock slower than at the other. For all other angles, the proportion is the same ; for one degree, four minutes of time ; for one minute of a degree, four seconds of time, &c. 66. If, therefore, we could find under one me- ridian the time which they reckon at the same instant under another, we should have the dif- ference of longitude, by converting the differ- ence of time into degrees at the rate just men- tioned. «. An obvious way, therefore, of discovering the dif- ference of longitude of any two places, is to have a watch, or portable chronometer, well regulated, ac- cording to the time at one of the places, and then to carry it to the other ; where, on being compared with ASTRONOMY. 63 with the time, as reckoned there, it will give the difference of longitude. b. Suppose at the first place, by repeated observa- tions, the star Sirius was found to pass the meri- dian at 10h 24m 35sec by the watch, and at the other at 8h 16m 22sec by t]ie same watch ; the dif- ference, 2h 8m 1 3s«c5 converted into degrees, gives 32° 3' 15" for the difference of longitude, by which the first of the two places is east of the se- cond. 67. In like manner, any phenomenon, the be- ginning or end of which is seen at the same in- stant by observers under different meridians, af- fords the means of determining the difference of longitude. G For, by this means, the difference of the reckoning at the two places is ascertained just as by the chro- nometer. Many of the phenomena of the heavens, as will be afterwards explained, serve as signals of this kind. Such signals may also be given on the earth, by the sudden kindling or extinguishing of lights on some elevated station, from which they may be seen at considerable distances. The observation of the convergency of the meridians, is also another way of determining the longitude, as already observed. To it, and the two now men- tioned, 6i OUTLINES OF NATURAL PHILOSOPHY. tioned, all the methods of finding the longitude may be reduced. 68. If the latitudes of any two places are gi- ven, and also their difference of longitude, their distance may be found by spherical trigonome- try. a. If the earth is considered as a sphere, then, in the spherical triangle contained by the arches joining the two places with one another, and with the pole, two sides are given, viz. the distances from the pole, or the complements of latitude, and the angle at the pole, or the difference of longitude ; and there- fore the 3d side may be found by the 2d case of oblique-angled spherical triangles. This side is the distance of the places expressed in degrees, &c. ; and may be turned into miles, by multiplying by 69.04-4, the mean length of a degree, (§ 60. b.) If the angles at the base or the azimuths are also re- quired, it will be best to resolve the triangle by NAPIER'S Formula. See Elem. of Geomet. Edin. 1810, p. 378. See also WOODHOUSE'S Trigono- metry 9 p. 126* • • b. But if the spheroidal figure is to be taken into ac- count, the calculation becomes more complex. For as, on this supposition, the directions of the plum- mets AD, BF, (fig. 6.) at the two places, if their latitudes are different, do not meet the axis in the same point $ these three lines do not contain a solid 3 angle, ASTRONOMY. 65 angle, and therefore the rules of trigonometry can- not be directly applied to them. If, however, C be the centre of the spheroid, and if AC and BC be joined, the angles PC A, PCB, are deduced from the latitudes, § 61. b. Then, in the solid angle at C, are given the two plane angles PCA, PCB, and the inclination of their planes, viz. the differ- ence of longitude, or the angle at P ; therefore the angle ACB may be found by the same case of spherical triangles as before. Hence the straight line AB is also found, the radii CA, CB being gi- ven, § 61. c. In this way also, are found the angles at the base of the triangle PAB, or those which the plane ACB makes with the planes ACP, BCP. These, however, are not the true azimuths, which are the angles that the plane ADB makes with ADP, and that ABE makes with PEB. To find these last ; if DB be drawn, then in the tri- angle BCD, BC, CD, and the angle BCD are gi- ven, whence DB is found. Then in the triangle ADB, all the three sides are given ; wherefore the angle ADB may be found. Next, in the triangle BED, the sides BE, ED, DB are given ; there- fore the angle EDB, and its supplement PDB are found. Therefore the three plane angles ADP, ADB, PDB, which contain the solid angle at D, are given ; whence the inclination of the planes may be found, and therefore the angle which the plane PAD makes with the plane ADB, that is, VOL. II. E the 66 OUTLINES OF NATURAL PHILOSOPHY. the angle PAB, such as it would be measured at A. In the same way the azimuth at B may be found. d. When these calculations are applied in small tri- angles, they naturally become much more simple. The process now described) contains a general so- lution of spheroidal triangles, which have one angle at the pole, whatever be the oblateness of the sphe- roid, and whatever be the magnitude of the tri- angles. 69. The Artificial Globe is a delineation of the surface of the earth, and the circles belong- ing to it, on the surface of a sphere, moveable about an axis ; it serves to give a correct notion of the figure and proportion of the parts into which the earth's surface is either naturally or artificially divided, as well as to resolve many of the problems of geography, when great accura- cy is not required. A contrivance of the same kind is applied to the hea- vens. The uses of the celestial and terrestrial globes are fully explained in most of the treatises on Astronomy and Geography, JO. A Map is |a representation of the whole, or of a part of the earth's surface on a plane ; and ASTRONOMY. $7 and though less correct, by being less expensive, and more portable than the preceding, it is of much more general use. It is impossible to delineate on a plane any figure that can accurately resemble one which is extend- ed in three dimensions. A certain degree of re- semblance may, however, be obtained, and, in the construction of maps, this has been sought for in two ways ; by the projection of the spherical sur- face on a plane, such as it would be seen to the eye situated in a particular point; or by the dffoelope- ment, that is, the spreading out of a spherical on a plane surface. 71- The Stereographic Projection, is a repre- sentation of the surface of a sphere on the plane of a great circle, such as it would appear to an eye situated in the pole of that circle, or in a point 90° distant from every part of its cirum- ference. a. It is usual to suppose the eye placed in the equa- tor, 90° distant from the equinoctial points, so that the plane of projection is the equinoctial colure. If the eye is placed in either of the equinoctial points, the plane of projection is a circle at right angles to the former, called (for a reason that will afterwards appear) the Solstitial Colure. The hemisphere concave to the eye, or on the side of the plane opposite to the eye, is first delineated. The OUTLINES OF NATURAL PHILOSOPHY. The eye is then supposed placed in the opposite pole, and the other hemisphere is in like manner represented. It is in this way that the Maps of the World are usually* constructed. b. The stereographic projection has these two very remarkable properties. 1. All the circles of the sphere, both great and small, are represented by circles in this pro- jection, 2. Any two circles cut one another in the projec- tion, at the same angle in which they cut one another on the surface of the sphere. Ac- cordingly, the parallels of latitude in this pro- jection cut the meridians at right angles. These properties contribute much to the simplicity and beauty of the construction, which, however, has this disadvantage, that the same area on the earth's surface, is represented by a much larger area near the equator, and especially towards the edges of the projection, than at a greater distance. Notwithstanding of this, the stereographic projec- tion is well adapted to Maps of the World, or pf Jarge portions of the globe. 72. The construction called FLAMSTEED'S Projection, (though it is rather a Developement than a Projection),, is very well contrived for the representation ASTRONOMY. 69 representation of smaller portions of the earth's surface. a. In this construction, a straight line is drawn for the meridian of the middle of the map, on which are marked off equal distances, to denote degrees of latitude. From a point in this line, as a centre, and with a radius that is to the length of the de- gree of latitude as the cotangent of the middle lati- tude to an arch of i degree, an arch of a circle is described, to represent the middle parallel of lati- tude. From the same centre are described other arches, through the different points marked off on the meridian of the middle of the map, which re- present the different parallels of latitude. On any one of these parallels, equal distances are set off on each side of the middle point, which are to the as- sumed degree of latitude, as the cosine of the lati- tude of that parallel to the radius. The degrees of longitude are thus marked on each parallel, and the curves which pass through the corresponding points in the different parallels are meridians. These are curved more and more on retiring from the middle of the map ; but unless the extent is very great, they afford a very good representa- tion of the convex surface. &. This construction has a very remarkable property, viz. that the quadrilaterals in the map, included be- tween the meridians and parallels of latitude, have the same ratio to one another nearly, with the qua- drilaterals which they represent on the surface of the 70 OUTLINES OF NATURAL PHILOSOPHY. the globe. See Memoirs sur la Projection des Cartes Geographiques, par M. HENRY, 4to, Paris, 1810, chap. 8me, p. 55., &c. 73. The construction which is called MERCA- TOR'S Projection, is chiefly used for nautical charts. In it the meridians are parallel lines j the degrees of longitude are all equal ; the pa- rallels of latitude are also parallel lines ; and the degrees of latitude increase on the chart in the same ratio that the degrees of longitude dimi- nish in the sphere, or in the spheroid. a. The consequence of making the degrees of latitude increase, in the manner described, is, that the de- grees of latitude and longitude bear to one another the same ratio in this chart, that they actually do on the surface of the earth : and as the meridians are all parallel, the rhumb-lines, or the lines of azi- muth, are straight lines. Hence the great use of this construction in navigation. b. This very ingenious contrivance is alluded to, though obscurely, by PTOLEMY. It was first used among the moderns by MERCATOR, whose name it bears ; but the principle of it was first explained by ED. WRIGHT in 1599, who shewed that the parts into which the meridian is divided, must be in- versely as the cosines, or directly as the secants of the latitude ; and he taught how it might be con- structed ASTRONOMY. 71 s.tructed by the addition of the secants of a series of arches taken in arithmetical progression. It was afterwards observed, that the meridian line thus divided, was analogous to a scale of loga- rithmic tangents of the half- complements of the latitudes ; this was at first only known as a fact, but was afterwards demonstrated by JAMES GRE- GORY, in his Exercitationes Geometric*?, 1668* Dr HALLEY proved the same in a more concise man- ner, Phil. Trans. N° 219., and greatly improved the construction of the chart. For other methods of constructing maps and charts, see VARENIUS, Sect. iv. chap. 32. Encyclopedic Methodique, art. Cartes. LORGNA, Principii di Geogrqfia, 4-to, Verona, 1789. Traite de Topo- graphic, &c. par L. PUISSANT, Liv. n. LA GRANGE, Mem. de Berlin, 1779. SECT. 72 OUTLINES OF NATURAL PHILOSOPHY. SECT. V. Of PARALLAXES. HAVING obtained an accurate notion of the figure of the earth, we are enabled to measure the lines either on its surface or in its interior, which must serve as the bases from which, by the rules of tri- gonometry, we are to deduce the distances of ob- jects observed in the heavens. Though the fixed stars are too far off, to have their distances thus ascertained, there maybe others, of which the dis- tances admit of being compared with the diameter of the earth. All distances that are not ascertained by the direct application of a measuring line, are determined on the same general principle; that is, from the change in their angular position, which is made by a known change in the position of the observer. This leads us to consider what is called the Parallax of an Object. 74. The parallax of any object in the heavens, is the difference of its angular position, as it would ASTRONOMY* 7$ would be seen from the centre of the earth, and as it is seen from a point on the surface. The parallax of an object, is therefore the same with the angle which the distance between the centre and a given point on the surface subtends at the object. Though an object to have no parallax, ought, strict- ly speaking, to be at an infinite distance, yet it will have no sensible parallax, if its distance is very great compared with the diameter of the earth. An angle of one-fourth of a second may be considered as insensible ; so that if the radius of the earth subtend an angle, at the distance of any object, less than one-fourth of a second, that object will be seen, from all points of the earth's surface, in the same position. Now, an arch of I" is .000004-&4S of the radius ; and the fourth of a second is therefore •°00001212 = and therefore, if a body is distant from the earth by 825082 of its semidiameters, it can have no sen- sible parallax. Though the centre of the earth is a point from which no observations can be made, yet as it is equally related to all the points on the surface, the posi- tions of the heavenly bodies may be most conveni- ently 74 OUTLINES OF NATURAL PHILOSOPHY. ently referred to it. When a star is seen in the zenith of any place, it is seen in the same position as if it were viewed from the centre. 75. The parallax of a body at a given dis- tance from the centre of the earth, is greatest when the body is seen in the horizon. This is called the Horizontal Parallax ; and the paral- lax at any given altitude, or the quantity by which the true altitude is diminished, is to the horizontal parallax as the cosine of the altitude to the radius. If P be the horizontal parallax, p the parallax at the altitude «, p == P X cos a. If r be the radius of the earth, supposing it spherical, r and d the distance of the body, - = sin P. ¥ When P is very small, P = -%, P being expressed? not in degrees, but in parts of the radius ; to have it in degrees, -j must be multiplied by m, the number of degrees in an arch equal to the radius. If the horizontal parallax is known, the distance d is known ; for d = - — =:. sin P The ASTRONOMY. J5 The distances of bodies having different horizontal parallaxes, are therefore inversely as the sines of those parallaxes, or, when the parallaxes are small, inversely as the parallaxes themselves. 76. If two observers, under the same meridian, but at a great distance from one another, ob- serve the zenith distances of the same star, when it passes the meridian on the same day, they can from thence determine the horizontal parallax. If the amplitude of the arch of the meridian inter- cepted between the zeniths of the observers, be called a, and if

' — a ; therefore P = Hence, also, sn

Libra, =& Taurus, tf Scorpio, TCI Gemini, n Sagittarius, / Cancer, 23 Capricornus, >? Leo, SI Aquarius, zz Virgo, TO Pisces, H The 88 OUTLINES OF NATURAL PHILOSOPHY. The manner of referring anys point to the ecliptic, of which the position is known with respect to the equator, or of finding its longitude and latitude from its right ascension and declination, has been already explained, § 30. y F 9 y If the point is in the ecliptic ; if, for example, it is the sun itself of which the longitude is required, from knowing the right ascension or declination ; we have tan. #1 0 tan. long. © = CQg Qbl ^c > or sin long. © = /Ki> § 80- The character JR denotes sin Ubl. .be. J the Right Ascension. 8.9. The point in which the equator and ec« jUptic intersect, is not iinmoveable, but appears, in respect of the fixed stars, to recede towards the west, at the rate of 50"^ nearly per annum, or about 1° in 72 years. This motion is called the Precession of the Equinoxes ; and by means of it, the equinoctial points describe an entire circle in 25867 years, which istheannus magnus of the ancients. In consequence of this, the longitude of the stars continually increases, at the rate of a degree in 72 years nearly. When the zodiac was first delineated by the ancient astrono- mers, the middle of the constellation Aries was at the vernal equinox, from which it is now distant more ASTRONOMY. 8D more than 58° towards the east. HIPPARCHUS discovered the precession of the equinoxes, by a comparison of his own with more ancient obser- vations. 90. On account of the precession of the equi- noxes, the tropical year, or the time in which the sun moves from the vernal equinox to the vernal equinox again, is less than the time in which he moves from one star to the same star again ; the vernal equinox having gone west- ward so as to meet the sun. The time in which the sun goes from one fixed star to the same fixed star again, is called a Siderial Year, and is longer than the tropical year, by the time that the sun takes to move over 50"^ of his orbit. This amounts to .014119 of a day, or 20' 19".9 ; so that the siderial year is 365<1 6** 9«* 1s. 5. 91. The Obliquity of the Ecliptic is also sub- ject to change, and appears to have been con- stantly diminishing from the remotest date of astronomical observation : its present rate of diminution is nearly 50" in a century. A Chinese observation of the sun's altitude at the solstices, as ancient as the year 1 1 00 before Christ, has 90 OUTLINES OF NATURAL PHILOSOPHY. has been preserved ; and from it LA' PLACE dedu- ces the obliquity at that time, = 23° 54«' 2". Con- naissance des Terns, 181 1, p. 432. A series of observations, from the age of PYTHEAS, down to the present time, confirms the same ge- neral result. LA PLACE, ibid. LA LANDE, Astron. § 2738. ; VINCE, Astron. i. § 151. The mean obliquity for 1 750, was determined very exactly by LA CAILLE and BRADLEY, to be 23° 28' 19". The obliquity, beside this progressive diminution, is subject to slight periodic irregularities, which are not here considered. The diminution itself, though apparently progressive, will be found after- wards to be really periodical, and a part of a slow vibration, by which the obliquity of the ecliptic alternately increases and diminishes within very narrow limits. Apparent ASTRONOMY* 91 Apparent Orbit of the Sun. 92. If the sun's motion in the ecliptic be de- termined by observation from day to day, (§ 81. #.), it will be found, that it is not uni- form, but is swiftest about the beginning of Ja- nuary, and slowest about the beginning of July ^ being continually retarded from January to July, and accelerated from July to January. 93. If the apparent diameter of the sun be al- so observed every day by a micrometer, or any instrument that measures small angles with great exactness, it is found to vary, so as to be greatest when the angular motion is greatest, and least, when it is least. The angular velo- city, and the diameter of the sun, do not, how- ever, vary in the same ratio ; but the angular velocities, at any two points of time, are as the squares of the apparent diameters. Thus, if v and t/ are the angular velocities of the sun, or his diurnal advances in the ecliptic, at any two seasons- 92 OUTLINES OF NATURAL PHILOSOPHY. seasons of the year, d and d' his apparent diameters !! - dl v' ~~ d' v d* at the same times, v i i/ : i d* : d'z9 or ~, = — • 94. As the apparent diameters of the sun must be inversely as his rectilineal distances from the earth ; therefore the apparent velocities of the sun are inversely as the squares of his distances from the earth, -sft If D and D' be the real distances of the sun from the earth, at the two instants, when the angular velo- cities are v and r/, v : v' : : D'3 : Da ; so that v D* is a constant quantity ; that is, the product of the diurnal motion into the sun's distance from the earth remains always the same. 95. As it is easy to prove that v D* is double of the sector which the line drawn from the sun to the earth describes, while it moves through the angle v ; therefore the areas described by the line drawn from the sun to the earth, in equal increments of time, are equal, and, in any times whatever, are proportional to the times. This is the First Law of KEPLER, in as far as relates to the motion of the sun. . 96. On ASTRONOMY. 96. On comparing the sun's diameter, mea- sured from time to time, with his place obser- ved in the ecliptic, it is found, that if the sun's mean apparent diameter be m, his least diame- ter m — ??, and z9 his angular distance* at any time from the point in the ecliptic where his diameter is least, his apparent diameter at that time is m — n. cos z. " w = 32' 06".2, and m — n = 31 ' 32".8, so that n = 32".4, and mini: 19262 : 324, or as 59.45 to 1. The sun's apparent semidiameter is therefore always expressed by the formula (32' 6".2) — 32".4 X cos z = 32' 6".2 (1 — — J— • cos a). Off f «d Because the distance of the sun and earth must be in- versely as the apparent diameter of the sun, there- T> fore, if that distance be called y9y~ m(l — - cos z) m where B is a constant but indeterminate quantity. Now, if a be assumed for the mean distance of the n2" \ sun, and if B = m a (1 ), j/= - I — cos z m an 94 OUTLINES OF NATURAL PHILOSOPHY. an equation to an ellipsis expressing the relatioa between y or F&, (fig. 12.) the radius vector drawn to the focus, and the angle z or AFG which that line makes witli the transverse axis on the side of the centre. $f cmilv/ rvunh)') oi< in tujofj sit nK, 97* The orbit of the sun is therefore an el- lipsis of which the semi-transverse axis is a, /yj the eccentricity — a, and the semi-conjugate axs a\~ Thus the species of the sun's orbit is determined. For the property of the ellipsis quoted above, see Geometry of Curve Lines, (Prof. LESLIE,) Prop. 22. FRISIUS de Anal. Sect. Con. Prob. 29. It may also be easily derived from SIMSON'S Con. Sect. Prop. 4. Book 2. 98. The position of the transverse axis, or of what is called in astronomy, the Line of the Ap- sides, in respect of the fixed stars, is found, by taking the middle point between two positions of the sun, in which the diurnal motion is the same. The ASTRONOMY. 95 The place of the apsides thus found, may be correct- ed, on the principle, that the axis is the only line drawn through the focus of an ellipse, which bi- sects its area. The time, therefore, from the sun's being in one apsis, to his being 180° farther ad- vanced, is precisely half a year. When, therefore, two points are found in the sun's orbit, 180° or 6 signs distant from one another, and such, that the sun takes precisely half a year to pass from the one to the other ; they are the Higher and the Lower Apsis. LA CAILLE, Mem. de I'Acad. des Sciences, 1742. Also his Astronomic, p. 78. WOODHOUSE, Astron. p. 210. Thus the longitude of the aphelion, for the year 1 780, was determined by DE LAMBRE, from Dr MASKE- LYNE'S observations, to be 99° 8' 19".9, or 9° 8' 19".9 advanced into the sign Cancer. The sun passed through this point 4m past noon on the 3 1st of June, according to the time at Greenwich. BIOT, Astron. vol. I. p. 220. The eccentricity of the sun's orbit was found, in (5 97.) = - x a. A method of finding - more ac- m m curately than can be done merely by the observa- tions of the sun's diameter will be immediately explained. When so determined, it is found = .016814, the mean distance being 1. In the following propositions, the mean distance = a, and the eccentricity == e. 99. The 96 OUTLINES OF NATURAL PHILOSOPHY. 99. The position and the species of the orbit, being thus determined, the calculation of the sun's place for any given time, is reduced to the geometric problem, of drawing a line through the focus of an ellipsis, so as to cut off a sector between it and the transverse axis, having a gi- ven ratio to the whole elliptic area. If AGPH (fig. 13.) be the orbit of the sun ; F the focus in which the earth is placed ; and G the place of the sun at a given time ; then ihe time of the sun's describing the arch PG, or of the radius vector describing the sector PFG, is given ; and the time of an entire revolution, or of the radius vector describing the whole elliptic area, being al- so given, the ratio of the sector PFG, to the whole ellipsis AGPH, is given. If, from this, the posi- tion of the straight line FG can be determined, it is evident, that the angle PFG, and the position of G, are found. This is known by the name of KEPLER'S Problem ,- . it can only be resolved by approximation. a. The angle PFG, which measures the angular dis- tance of the sun from the Perigee, or lower apsis, is called the true anomaly. b. If a circle be described from the centre F, with a radius = y a (a* — . €*)*' its radius will be equal to i that ASTRONOMY. 97 that of the ellipsis •, and if a sector a FG' be taken in the circle, equal to AFG in the ellipsis, the angle AFG' is called the mean anomaly. This angle is proportional to the time, and in- creases at the rate of 59' 8". 3 daily. c. The angle GFG', or the difference between the mean and the true anomaly, is called the Equation of the Sun's Centre. d. If the true sun G, and the imaginary sun G', set out together from A and a9 the sectors AFG and a FG', increasing at the same rate, the angles AFG, and#FG', must increase unequally, and the point G' must get before G, or the mean anomaly must exceed the true. But when FG becomes less than FG', the angle AFG will come to increase faster than a FG'. At the Perigee they will coincide ; and at a certain point, between A and P, their difference, or the equation of the centre, will be the greatest possible. This will happen, when the increments of the true and of the mean anomaly are equal to one another, or when the sun's real angular velocity is equal to his mean, or to 59' 8".3. Among the great number of solutions which have been given of this celebrate;! problem, one dis- tinguished for the simplicity of the result, pro- ceeds on the supposition, that the angles at the superior focus are proportional to the time, or are the same with the mean anomaly ; which is not VOL. II, G far 93 OUTLINES OF NATURAL PHILOSOPHY. far from the truth, in elliptical orbits of small eccentricity. This solution was first proposed by BULLIALDUS, a French astronomer, and was after- wards adopted and improved by Dr SETH WARD, and is known in this country by the name of WARD'S Hypothesis. Another solution is distin- guished, for the simplicity of the principles, and the elementary nature of the reasoning employed, viz. that given by the late Dr MATHEW STEWART, in the Edinburgh Physical and Literary Essays^ vol. u. (1755) p. 105. ; and again republished in his Physical and Mathematical Tracts,}*. 4-04. Among the other solutions, those of NEWTON, Prin. Math. lib. i. prop. 30. Schol.; of SIMPSON, Essays, 4to, (1740) p. 47. ; of EULER, Comment. Petrop. torn, vu. ; and of IVORY, Edinburgh Transactions) vol. v., (the latter extending to the most difficult case of the problem, when the eccentricity is great), are particularly commendable. Of all these, however, it may be said, that though excellent when a numerical calculation only is re- quired, yet when the solution is to be a step in the investigation of other properties of the elliptic mo- tion, they cease to be of use, so that recourse must be had to such general theorems, as express the true anomaly in terms of the eccentricity, and of the mean anomaly. The first solution of this kind was given by CLAIRAULT, Theorie de la Lune% § 31. lemma 3d, &c. It was afterwards improved and extended by other mathematicians, particular- ASTRONOMY. 99 ty BOSSUT and JEAURAT. See also LA LANDE, Astronomic, torn, in, § 348 1. ; and CAGNOLI, 7H- gonometrie, § 1488. The series was continued by JEAURAT, as far as the 9th power of the eccentri- city. 100. It is proved in that solution of KEPLER'S Problem, that if 1 be the semitransverse, and e the eccentricity of an elliptic orbit, x the mean anomaly, reckoned from the Perigee, and y the equation of the centre ; 1 5 y- + (2^ — - e* -f — e*)smz . 960 960 This value of y, applied to the mean anomaly accord- ing to its sign, will give the true anomaly. a. When the equation of the centre is found from this formula, the constant coefficients must be reduced into degrees and minutes, by multiplying each of them 100 OUT1NES OP NATURAL PHILOSOPHY. them by 57°.29578, &c. the number of degrees in an arch equal to the radius, b. Because, in the case of the sun, c is small, (viz, = .0168 14), its powers above the third may be re- jected ; and so the equation reduced is, y = (1° 55' 26".35) sin x + (!' 12".68) sin 2 x. + (I".05)sin3,r. 101. The Radius Fee tor r, or the line drawn from the sun to the earth, may also be express- ed in terms of the mean anomaly, supposing the mean distance = 1. 2" 1 3' 16 3 45 3 5 lj) cos 5 must be changed. A small part of this series will give the radius vector in the case of the sun, with suf- ficient accuracy ; we may suppose r = 1 +r-£a — £ cos x — — e* cos 2 x. . Though the eccentricity is supposed to be found, from § QJ.9 it must be corrected by means of the greatest equation of the centre, determined by observation ; this equation being called g, if g „ = 57.29578 2 768 983040 As h is very small in the sun's orbit, being r= .038629, e = - h nearly; that is, e = .016814. Syst. du Monde, p. 116. 5 BIOT, Astron. Phys. torn. ii. p. 228. 2d edit. a. The greatest equation itself is found by observing the sun's place in the ecliptic, day after day, and comparing it with the mean place, calculated as in § 99. b. 103. Astronomical 102 OUTLINES OF NATURAL PHILOSOPHY. 103. Astronomical tables, constructed from the data, and on the principles now explained, serve to determine the sun's place in the eclip- tic, for any instant of time, either past or fu- ture. a. From the time of the sun's passing through the pe- rigee, when the true and mean place coincide, the mean place for any other time may he computed, by allowing for the interval an increase of longi- tude, at the rate of 59' 8".3per diem, and thus the mean anomaly is computed. From the mean ano- maly, is found the equation of the centre, con- tained in a table, which gives the quantity of that equation for every degree of mean anomaly; thence is given the true anomaly, and of course the true longitude. The tables are so constructed, as to give the mean place of the sun for the beginning of every year ; hence the mean place for any time of any year is easily found. When equations are thus ranged in tables, the quantity by which they are found out in the table, and on which their magnitude depends, are called the Arguments of the Equations. The tables of the sun's motion, are in reality nothing else than the expansion of the general formula contained in § 100. b. Thus, from the fact of the variation of the sun's ap- parent diameter, compared with the variation of the angular velocity of the same body, it is demon- strated, ASTRONOMY; 103 stratad, that, whether it be the earth or the sun which really moves* the one describes an ellipsis round the other, placed in the focus of that curve ; and that the line drawn from the moveable to the immoveable body, (the radius vector), describes areas round the latter proportional to the times. From these two general laws, astronomers have been able to determine the proportional distances of the sun and earth, and their relative positions, at all seasons of the year. e. The use of facts, in scientific investigation, could not be better illustrated than by this example. The simple factj that the sun on a particular day had a certain apparent diameter, and a certain rate of angular motion, is nothing in itself, and cannot, taken alone, lead to any conclusion : But a multitude of such facts, compared together, by the assistance of geometry, leads to the knowledge of the general law to which they are all subject ; this law leads, conversely, to a more precise and com- prehensive knowledge of facts, embracing at once the past, the present and the future, and reducing them all into one theorem. We are left, however, in the dark as to one question, whether it be the sun or the earth which describes the elliptic orbit. The appearance that the earth is at rest, and that the sun is in motion, is no ar- gument against the fact being entirely the reverse, as has already been made evident in the case of the OUTLINES OF NATURAL PHILOSOPHY. the earth's rotation. The decisive facts have not yet occurred, which are to determine, whether or not a motion of translation belongs also to that body. In order that the theory of the elliptical motion may have its conclusions extended either to the future or the past, a very accurate measure and reckon- ing of time are necessary, and we are now in pos- session of the data by which these may with cer- tainty be obtained. Of the Equation of time 9 and of the Kalendar. The arrival of the Sun in the meridian, being a more conspicuous phenomenon than that of a Star, has been taken to mark the beginning and end of the day, used for the purposes of civil life. Solar time, consists of days measured in this manner, and is used by astronomers, as well as by the people at large. Astronomers begin the day at noon, and reckon 24< hours round to noon again : in the com- mon reckoning, the day begins at midnight, and is divided into 24 hours, which are counted by 12 and 12. LA PLACE has proposed, that in this the astronomers should follow the people, as, by beginning the day at midnight, the whole of the sun's stay above the horizon falls in the same day. 104. The ASTRONOMY. 105 104. The difference between the solar and si- derial day, has already been shewn to consist of 3m 56s 33* of siderial, or 3m 55s 49* of solar time ; but the solar day thus defined, is only the average, or the mean 5 for, the time actual- ly employed by the sun to return from the me- ridian to the meridian again, is different at dif- ferent seasons of the year, § 92. 105. If the sun moved regularly forward in the equator, at the rate of 59' 8".3, every day, the solar days would be all equal, and their lengths as above determined ; but as ^the sun neither moves in the equator, nor moves in the eclip- tic, at a uniform rate, there are two causes that affect the length of the solar day. The time, therefore, which is reckoned by the arrival at the meridian of an imaginary sun, or one which is supposed to move in the equator, is calledmean solar time. When it is reckoned by the arrival the real sun on the meridian, it is called apparent time. 106. The difference between the right ascen- sion of the sun, and his mean longitude, con- verted into mean solar time, is the difference between 106 OUTLINES OF NATURAL PHILOSOPHY. tween the mean and the apparent time, and is called the Equation of Time. a. The conversion of degrees and minutes, into mean solar time, is performed by a different rule from their conversion into siderial time. For an hour of mean solar time, there must be reckon- ed 15° 2' 27".84-7 of the equator ; and so on, in the same proportion. I. There are four times in the year, when the mean longitude of the sun, and his true right ascension, are equal to one another j and, at these times, the apparent and the mean time coincide. These times happen, at present, about the 15th of April, the 15th of June, the 1st of September, and the 24th of December. From the first of the above periods to the second, the apparent time is before the mean, and the equation of time is subtractive, or must be taken from the apparent time, to give the mean. It is greatest at the 15th of May, when it amounts to 3' 58" to be subtracted. From the se- cond to the third interval, the equation is additive, the mean time being before the apparent ; and it becomes a maximum about the 25th of July, when it amounts to 6' 6". It becomes negative between the third and fourth interval, and reaches its ma- ximum on the 2d or 3d of November, when it amounts to 16' 15" subtractive. Dr HALLEV has given a geometrical construction, for determining the time when the apparent days are longest or shortest, and also when they are equal to the mean. ASTRONOMY. mean. See KEIL'S Astronomy ', Sect. 25. p. 322., &c. f. When the equation of time is a maximum, the length of the mean and apparent day is the same. When the equation of time is nothing, the lengths of the apparent and the mean day differ most from one another. On the 16th of June, for example, when the equation of time vanishes, the day is 12".5 shorter than the mean solar day. The dif- ference becomes less and less till the 26th or 27th of July, when the equation of time is stationary or a maximum^ being then 6ra 5s nearly; and the apparent day is then in length equal to the mean. The difference of the days increases from thence ; and on the beginning of September, the apparent day is 1 8s longer than the mean, Sac. d. Clocks ought to be regulated by the mean solar time; and when they are adjusted by the sun's passing the meridian, the equation of time must be applied. e. The mean length of the day is thus accurately de- termined. But in the reckoning of time for the purposes of astronomy, and of civil life, we must not count by days only, but by that assemblage of days which constitutes a year, and which is natu- rally pointed out as a division of time by the re- turn of the seasons. The sort of incommensura- bility that exists between the lengths of the day and of the sun's revolution, renders it somewhat difficult 108 OUTLINES OF NATURAL PHILOSOPHY. difficult to adjust the reckoning of both, when it is to be done in whole numbers. 107. If the revolution of the sun consisted of an entire number of days, for instance 365, the year would naturally be made to do the same, and there would be no difficulty in the forma- tion of the Kalendar, or in adjusting the reckon- ing in years and in days to one another. All the years would then be precisely of the same number of days, and would all begin and end with the sun in the same point of the ecliptic. If the time of the sun's revolution includes a fraction of a day, the case is different ; a year, and a revolu- tion of the sun, cannot then be precisely of the same duration. 108. As it would be inconvenient to begin the year at any point of time, but the begin- ning of a day, a year must consist of an entire number of days. If, therefore, the revolution of the sun include a fraction of a day, for ex- ample one-fourth, then one year cannot be made equal to one revolution of the sun ; but four years may be made equal to four revolutions of the sun. Suppose ASTRONOMY. 109 Suppose, for example, that 365. The reformation of the kalender did not take place in England till the year 1752. The secular years on which the intercalary days are omitted, are 1700, 1800, and 1900. c. Though this degree of accuracy is sufficient for the purposes of history, and even of astronomy, and may easily be carried farther, by similar mean* adopted in future ages, yet the problem of interca- lating, so that the difference between the computa- tions OUTLINES OF NATUEAL PHILOSOPHY. tions in the kalender, and the real motions of the sun, should always be the least possible, is not thereby completely resolved. The modes of in- tercalation best suited to that object, require all the integer numbers to be found which most near- ly express the ratio of the fraction .212264- to 1, See EULER, Siemens d'Algebre, torn. n. ; Addi- tions by LA GRANGE, § 20. Secular Variations in the apparent Motion of the Sun. The variations in the sun's motion which have now been described, are confined within short periods, during which they alternately increase and dimi- nish. There are others, which go on from one age to another, and are either continually pro- gressive, or circumscribed by periods of very long duration. These are so slow, that they are only perceived by comparing together observations made at a great distance of time. They are call- ed Seculur Inequalities. 111. By comparing very distant observations, it is found that the line of the apsides, or the longer axis of the sun's orbit, has a progressive motion, or a motion eastward : so that the apsis recedes 3 ASTRONOMY. 113 recedes from the vernal equinox 62", or by DE LAMBRE'S Tables 6l".9 annually. a. This motion includes the precession of the equi- noctial points, which is in the opposite direction, and amounts to £0".25 ; so that the real motion of the apsides eastward, in respect of the fixed stars, is 1 1".65 a-year, or 19' 4/'.^ in a century. b. Hence there is a difference between the Tropical Year, or the time of the sun's revolution from e- quinox to equinox, and what is called the Anoma- listic Year, or the time of the sun's revolution from either apsis to the same apsis again. As the apsis has gone in the same direction with the sun over 62" in a year, the sun must come to the place where the apsis was at the beginning of the year, and must move over 6C2" more before the anoma- listic year is.completed. The time required to this is.0l74?8 of a day, which, added to the tropical year, gives 365d.2:>9744, or 3654 6h 14«ni2s for the anomalistic. BIOT, Astron. torn, n § 91. c. The line of the apsides thus continually moving round, must at one period have coincided with the line of the equinoxes. The lower apsis or perigee in 1750, was 278°. 6211 from the vernal equinox, according to LA CAILLE ; and the higher apsis was therefore at the distance of 98°.6211. The time required to move over this arch, at the rate of C2" annually, is about 57?2 years, which goes back nearly 4000 before our era, — a period remarkable VOL. II. H for OUTLINES OF NATURAL PHILOSOPHY. for being that to which chronologists refer the creation of the world. At that period, the length of time during which the sun was in the northern signs, that is, on the north side of the equator, was precisely the same with that on which he was on the south, each being exactly half a year. At present, the apogee, where the sun's motion is slowest* being in the ninth degree of Cancer, more time by 7d 16b 30m 8s is consumed in the northern than the southern signs ; so great is the change which the motion of the apsides has pro- duced. About 464 years ago, the apogee was in the beginning of Cancer. d. When we would calculate the place of the sun for any time, (especially if very distant), we must be- gin with determining the position of the line of the apsides ; as it is from the lower apsis that the mean and true anomaly are both computed. e. The motion of the sun's apsides being 19' 4" in a century, with respect to the fixed stars, it requires a period of more than 108000 years to complete their siderial revolution. Their tropical revolu-* tion is 20903 years. 112* By a comparison of distant observations, the equation of the centre, and of consequence the eccentricity of the sun's orbit, are discover- ed to be continually diminishing. The ASTRONOMY. 115 The rate of diminution in the maximum of the equa- tion, is about 18".79 in a century. If the rate of this diminution were to continue uni- form, (which, however, we have not a right to sup- pose), the earth's orbit wouid become a circle in about 36300 years. 113. From the changes in the place of the apsides, in the equation of the centre of the sun's orbit, and in the obliquity of the ecliptic, the equation of time undergoes a change from one age to another. a. The equation of time is equal to the mean longi- tude) minus the true right ascension of the sun. The former of these is not affected by the place of the apogee or perigee, and depends only on the distance of time from the equinox. The latter, the true right ascension, is dependent on ail the three elements just mentioned. The true longi- tude of the sun, from which the right ascension is deduced, depends on the place of the perigee, and on the equation to the centre, both of which vary ; and it involves also the reduction from the eclip- tic to the equator, which depends on the angle these circles make with one another. The equa- tion of time, therefore, answering to the same day of the year, varies in different ages ; but for any age, past or future, maybe computed on the prin- ciples already explained. *. The 116 OUTLINES OF NATURAL FHILOSOPHY. #. The secular variation of the equation of time, is different for every different state of that equation : 'it is greatest about the time when the sun is in the perigee; and it is then 15".6. The Sun's Rotation on his Axis. 114. The face of the sun, when viewed with a telescope, though of a bright and intense light, far above that of any other object, is of- ten marked with dark Spots, which, when exa- mined from day to day, are found to traverse the whole surface from east to west, in the space nearly of fourteen days. a. These spots, though only visible with the telescope, are sometimes so large, as to subtend an angle nearly of one minute. Their number, position, and magnitude, are extremely variable. Each of them is usually surrounded with a Penumbra, be- yond which is a border of light, more brilliant than the rest of the sun's disk. When a spot is first discovered on the eastern limb, it appears like a fine line ; its breadth augments, as it approaches the middle of the disk, from which it diminishes as it goes over to the western limb, where, at last, it entirely disappears. The same spot, after four- teen days, is sometimes discovered again on the east ASTRONOMY. 117 east side. It is not often, however, that this hap- pens, as the spots dissolve and perish, from causes that are to us quite unknown. b. The spots of the sun were first discovered by GA- LILEO at Florence in 1611. Istoria e Demonstra- zione intomo alle Macchie Solari. Opere di GA- LILEO, torn, ii. p. 85., &c. Edit, di Padova, 1744. 115. The position of a spot, relatively to the sun's centre, may be determined by observing the difference between the time when the cen- tre of the sun passes the vertical wire of a tran- sit telescope, and when the spot does the same. This gives the difference in right ascension ; and the difference in declination may be deter- mined at the same time, by means of the move- able wire in the telescope. In this way, by ob- servations made day' after day, the path of the spot on the sun's disk maybe traced with great exactness. a. When the spots are followed in this way, many of them are found to change, and to disappear al- together in the course of a few days. Sometimes a number of small spots unite into one large spot ; at other times, a large spot separates in a number of small ones, which soon disappear entirely. A few of them, more permanent than the rest, may be followed by the observation here described, as long as two or three revolutions. 116. The 118 OUTLINES OF NATURAL PHILOSOPHY. 116. The Paths of the spots thus traced, are observed to be rectilineal at two opposite sea- sons of the year, the beginning of June and the beginning of December, and to cut the ecliptic nearly at an angle of 7° 20'. Between the first and the second of these seasons, the paths of the spots are convex upward, or to the north, and acquire their greatest curvature about the mid- dle of that period. In the following six months, the paths of the spots are convex toward the south, and go through the same series of changes. They appear to be elliptic arches. In the beginning of March and September, when the opening of the elliptic paths is at its maximum, • the smaller axis is to the greater as 13 to 100. LA LANDE, Astronomie> 3233. 117. The preceding appearances may be ex- plained, by supposing the spots to be opaque bodies, attached to the luminous surface of the sun ; the sun having a revolution on an axis inclined at an angle of 7U 20' to the axis of the ecliptic. The apparent revolution of a spot is performed in 27 days ; but in this time the spot has done more than complete an entire revolution, having, in addition to it, gone over an arch equal to that which the sun has described in the same time in his orbit. This' reduces ASTRONOMY. 119 reduces the time of the sun's rotation on his axis to 25d 9h 36«>, All the observations do not agree exactly in bringing out this conclusion. LA LANDE, J 3276., &c. The problem, From three positions of a spot on the sun's disk, in respect of the ecliptic, to find the position of his equator, and of his axis of rotation, is of considerable difficulty. It was first directly resolved by BOSCOVICH in 1737. See LA LANDE, f 3257, who has also given a solution. Dr HER- SCHEL has made many important observations on the solar spots. Phil. Trans. 1794, 1801. 118. A phenomenon, from which ho informa- tion concerning the motion of the sun has yet been obtained, though, from some circum* stances, it appears to be connected with his ro* tation, is the Zodiacal Light ; a luminous ap- pearance, seen after sun-set, or before sun-rise, somewhat similar to the milky way, but of a fainter light, in the figure of an inverted cone or pyramid, with its base toward the sun ; ha* ving its axis inclined to the horizon, and nearly in the plane of the ecliptic* The Zodiacal Light was discovered by CASSINI in 1683 ; but there is reason to think, that it had been remarked before that period. The 120 OUTLINES OF NATURAL PHILOSOPHY. The length of the Zodiacal Light, taken from the sun upwards to its vertex, is various, from 45° to 100°, and even 120°. The season most favourable for observing this phe- nomenon, is about the beginning of March, after sun-set : the axis extends toward Aldebaran, and makes, with the horizon, an angle nearly of 64? degrees. The aspect of the Zodiacal Light is by no means uniform ; it is much brighter in some years than others. A remarkably brilliant appearance of it was observed at Paris, 16th February 1769. The Zodiacal Light appears to be inclined to the ecliptic at an angle of 7°, and to cut it in the 18th of Gemini; so that it is in the plane of the sun's equator, or perpendicular to his axis of rotation. LA LANDE, Astron. § 84-7. As the decisive facts have not yet occurred, which are to determine whether or not the apparent motion of the sun is to be explained by the real motion of the earth, the language hitherto employed is entirely that suggested by the appearances. The lower apsis of the elliptic orbit, or that nearest to the focus in which ei- ther ASTRONOMY. ther the sun or the earth is placed, has been called the Perigee ; supposing that it is the point where the sun in motion approaches nearest to the earth at rest. On the contrary supposition, it would be called the Pe- rihelion. The same applies to the terms Apogee and Aphelion. It must be observed, that the distances of the Sun and Earth, now treated of, are only the relative distances, referred to an imaginary standard, and not compared with any known magnitude, such as the diameter of the earth. The observations on which such a comparison as this must be founded, and by which the sun's parallax is ascertained, cannot be explained till the planetary system is understood. In the mean time, it may be re- marked, that even by help of the observations above described, it may be inferred, that the sun's parallax is less than 10". From observations of the sun's solstitial altitudes, we saw, § 87., how the latitude may be deduced. In order to make the deduction with accuracy, the sun's altitudes must be corrected for the parallax; and it may be easily shewn, that if the horizontal parallax of the sun is so great as 10", the lat. of Greenwich, for example, determined without ma- king the due allowance for it, would be erroneous, by 1". The latitude determined by observations of the stars, is free from this error, and therefore ought not to agree with the former by 7". Now, the correction necessary to be made, in order that the deterniinatipn deituced from the altitude of the sun* OUTLINES OF NATURAL PHILOSOPHY* sun, should agree with that from altitudes of the stars, is not easily fixed with perfect precision, but is certainly not so great as 1". The sun's horizon- tal parallax is therefore less than 10". If the correction above mentioned could be found with perfect precision, this method of determining the sun's parallax, on account of its simplicity, would be preferable to all others. Though it does not aiford a precise measure, it assigns a limit that may be used till a more exact result can be ob- tained. The parallax of the sun being less than 10", his dis- tance must exceed 20626 semidiameters of the earth; and his diameter must be to that of the earth in a greater ratio than that of 15',5 to 10", or of 93 to 1. $75. SECT. ASTRONOMY. SECT. VII. MOTION OF THE MOON. 119. THE Moon, next to the sun, is the most remarkable of all the heavenly bodies, and is particularly distinguished by the periodical changes to which its figure and light are sub- ject. These changes are called the PHASES of the Moon. Modb curvata in comua, moda &qua port zone divisa, modb sinuata in orbem, maculosa, eadtmque subito pr&nitenS) immensa orbepleno, et repente nutta% &c. PLIN. Hist. Nat. lib. 11. cap. 9. 120. At the time when the moon is due south, about midnight, her disk is an entire circle, sub- tending an angle not much different from half a degree. On the next night, she comes later to the meridian by about 4.8 minutes, and the west- ern part of her disk is no longer bounded by a circle, but by an elliptic line ; and this line, .every subsequent night, is seen to encroach more on the luminous pait, UU» on the seventh night, OUTLINES OF NATURAL PHILOSOPHY. night, it is nearly a straight line, and the disk a semicircle. The diminution continues, the disk becoming more and more concave to the west, till, about the end of another seven days, it dis- appears altogether. After a few days, the moon again appears, like a fine crescent, to the eastward of the sun, with its concavity turned toward the east, and continues to increase on that side till it become entirely full orbed, about -29 days from the time when it was last full. The line separating the light and the dark part of the moon is irregular and serrated, and its form va- ries while one is looking at it through the teles- cope : the light, as it advances, touches some points, while they are yet at a distance from the illuminated surface, and while all round them is dark. The light on them spreads, till it be uni- ted to the rest. The Moon, when full, is opposite to the Sun : when she disappears, or when it is New Moon, she is in conjunction with the sun : these two aspects of the moon are called the Syzygies. At the time when the moon appears as a semicircle, she is 90° dis- tant from the sun on either side : she is then said to be in the quadratures. The moon, during all these changes, advances among the fixed stars, at the rate of 13° 10'4 at an ave- rage ASTRONOMY. rage in 24? hours, and comes later to the meridian by about 4-8 minutes every dayr The motion of the moon is therefore in the same direction with that of the sun, or according to the order of the signs. ' . These Phases may all be explained oil the supposition, that the moon is an opaque body, spherical, or nearly so, which moves in a circular orbit round the earth, while it receives its light from the sun, The opacity of the moon is proved not only from her phases, but from the occultation of stars by the dark or invisible part of her orb. 122. When the place of the moon is observed every night, it is found that her motion is in the plane of a great circle, inclined at an angle (subject to a small variation) of nearly 5° 9' to the ecliptic. The line in which the plane of the moon's orbit cuts the ecliptic, is called the Line of the Nodes. r ' J The Ascending Node, is the point where the moon is in the ecliptic, ascending to the north : the Des- cending Node9 is that where she is in the ecliptic, descending to the south, . The OUTLINES OF NATURAL PHILOSOPHY. 123. The position of the nodes is found, by observing the longitude of the moon, when she has no latitude ; and it appears, by a compari- son of such observations, that the line of the nodes is not fixed, but has a slow retrogade mo- tion, at the rate of 3' 10",64 in a day ; so as to complete a revolution relatively to the fixed stars in 6793.421 £ days. BIOT, Astron. Phys. vol. n. p. 351. 2d edit. As the moment of the moon being in the ecliptic may not be actually observed, yeU from several observations, taken before and after, the exact time, by the method of interpolations, may be found. 124. By comparing places of the moon, ob- served at very remote periods, it is found that the secular mean motion of the moon, relative- ly to the fixed stars, is 1336 circumferences plus 306°. 48763, which gives for the time of a side- rial revolution 27d,32l66l. In the same Way, the tropical revolution, (taking in the precession of the equinoxes), is found to be 2^d.321582 ; and the diurnal motion relatively to the tropics 13°. 18636. It also has been found, on making these comparisons for very remote ages, that the secular mean motion was ASfRONOMY. 127 was less formerly than it is at present. This in- crease of the mean motion is called the Moon's Acceleration. It amounts to 9" in the present age; and the effect of it on the moon's place increases as the squares of the time. 125. The period of the phases, or the time from full moon to full moon, is greater than the tropical revolution, as being the time required to describe 360°, with an angular velocity that is equal to the difference between the angular velocity of the moon, and the angular velocity of the sun : this gives the period of the phases, or what is called the Lunar Month, equal to 29^.530588 = 29d 12 See BIOT, torn. n. p. 8^8. Also WOODHOUSE, p. 305, The above is LA PLACE'S determination, Moon's Orbit. 126. It may be found, from observations of the Moon's apparent diameter, as was done in the case of the Sun, that the orbit in which she moves is nearly an ellipsis, having the earth in its focus, about which the radius vector drawn to 128 OUTLINES OF NATURAL PHILOSOPHY. to the moon describes areas proportional to the time. a» The same may be also proved, from observations of the Moon's parallax ; the Moon being near enough to the earth to admit of her parallax being accurately determined. b. The Moon's parallax, when least, is 53' 50".99, and when greatest, 1° 1' 22".99. Hence the Moon's greatest distance is 6*'. 84 19 semidiameters of the earth, and her least 55-9164. The mean distance of the moon from the earth is therefore 59.8791, which is also the semi- transverse axis of her orbit. c. A more accurate determination of these elements, has fixed the mean equatorial parallax at 57' 11". 4; the greatest distance of the moon at 1.05518, the least at .94482, the mean distance being 1. Hence the eccentricity is .05518. In round numbers, the moon's mean distance is 60 semidiameters of the earth ; and her diameter is to the diameter of the earth as 15' to 57' 11". 4, or as 3 to 11 nearly. VINCE, Astron. vol. i. 171. . From the eccentricity of the moon's or bit, the equation of the centre is found, ASTRONOMY. 129 = (6° IS' 17".S) sin x + (12' -49"-7) sin 2 * + (42".3) sin 3 #, ,r being put for the mean ano- maly, reckoned from the perigee. This is the equation of the centre, as deduced from DE LAMBRE'S Tables, published in ViNC&'aAstron. vol. in. see p. 133., &c. The maximum of the equation is 6° 17' 28", and takes place when the mean anomaly = 86° 5' 0". . The axis of the lunar orbit is not at rest, but has a progressive motion, like that of the sun's orbit. This motion is 0° & 41" in a day, or 40° 41' 33" in a year ; so that it makes an en- tire revolution, relatively to the fixed stars, in 3232d.5807, or in a little more than nine years. The Tropical revolution of the perigee is shorter by K1056. The motions hitherto enumerated, are similar in the Moon and in the Sun. There are other inequa- lities peculiar to the moon. 129. The moon's longitude, calculated ac- cording to the laws of the elliptic motion, does not agree exactly with her true place, but re- quires to be corrected by an arch proportional VOL, JL I to 130 OUTLINES OF NATURAL PHILOSOPHY. to the sine of double the angular distance of the moon from the sun, minus her mean anomaly. a. If c be the mean longitude of the moon, and 0 that of the sun ; the mean anomaly of the moon being $, this inequality is (1° 21'5".5) X sin(2(«—0) — #). b. This inequality is called the Evection : it was dis- covered by PTOLEMY, and is, after the equation of the centre, the first of the lunar irregularities that was observed, c. The argument of it, or 2 ( d — ©)^-#, increases at the rate of 11° 18' 59", or ll°.3166^er day; so that its period is _ r,— ^r^> or31d.8119; that is, 1 1 t in the space of 31 days J9 hours 28 minutes near- ly, the evection runs through all its changes, and is beginning to renew them in the same order. At the new and full moon, or at the Syzygies> when d — 0 is either nothing or 180°, the ar- gument is — Xy which gives the evection negative, if X be less than 180°, and positive, if it be greater, contrary to what happens to the equation of the centre, which is therefore diminished in both cases. At the quadratures, the equation to the centre is increased by the evection. The evection appears, therefore, as an inequality in the equation of the centre, by which it increases when the moon is in the ASTRONOMY. 131 the quadratures, and diminishes when the moon is in the syzygies. 130. Another irregularity in the moon^s mo- tion, which vanishes at the syzygies and the -quadratures, and is greatest in the middle be- tween them, or in the octants, is called the Moon's Variation. It depends on the differ- ence of longitude of the sun and moon, or on their angular distance, and may foe represented by the formula (3,5' 42") sin 2 (<[ — 0). This inequality was discovered by TYCHO BRAKE'. Its period is 14-d.76.53, or half a lunar month. 131. The angular motion of the moon is also subject to a third irregularity, by which it is di- minished when the sun approaches the perigee, and increased when he approaches the apogee. This is called the Annual Equation : it depends entirely on the time of the year, or mean ano- maly of the sun, and is equal to — (1 1' 11". or r r a -f- V s* — a* sin /s* r These two values of t give the time of the beginning and end of the eclipse. The least distance of the centres, is found by substi- tuting - for t in the formula S'M' = V r*t* — 2 a r t X cos * + a*, S'M' L4& OUTLINES OF NATURAL PHILOSOPHY.. S'M' is then = Vaz — 2 az cos ft + a2 = 1 — 2 cos /3 -f i =5 2 a sin - 0 = 2 a multi- plied into the sine of half the apparent inclination of the moon's orbit. When s* = a* sin £% or 5 = a sin £» there can be no more than a contact of the moon and shade, which will happen at the instant expressed by When s = a sin £, sin /s = S-9 and if sin £-^^, there a a can be no eclipse, as sin ft = ; therefore, unless - be less than _-, there can be no eclipse. Now, r a A* r = m + n + _ . _ nearly ; so that 2 (m+w) A n + 2 (W + n) must be less than — , otherwise there can be no eclipse of the moon. It will serve as a limit, to consider whether — - — is m -f n less 143 less than - or whether the horizontal motion «? of the moon in latitude, have to the sum of the ho- rary motions of the moon and sun in longitude, a less ratio than the sum of the semidiameters of the moon and shadow has to the latitude of the moon at the time of the conjunction. When the moon merely touches the shadow, that is, when the nearest approach of the moon to the shadow is just equal to the sum of the semidiame- ters of the moon and of the section of the shadow, we have what is called an Appulse. To derive as much advantage from the knowledge of the ecliptic limits as possible, it is necessary to observe, that when the mean opposition is 12Q 36' distant from the node, there can be no eclipse 5 and that when it is less than 9° distant from it, there must be an eclipse. Between these limits 12° 36' and 9°, the matter is uncertain, and must be decided by the calculation of the true place of the moon, &c. These are the limits given by DE LAMBRE 5 WOODHOUSE'S Astron. p. 34?2, The limits are differently assigned by other writers, VINCE, Astron. vol. i. § 545. 145. The moon seldom disappears entirely in lunar eclipses ; even the spots may be distin- guished 144 OUTLINES OF NATURAL PHILOSOPHY. guished through the shade ; the nearer the moon happens to be to the earth, or the farther she is from the apex of the shadow, the dark- ness is the greater. The light that, by the refraction of the earth's atmo- sphere, is made to enter within the limits of ihe conical shadow, is no doubt the cause of this phe* nomenon. In some instances the moon has disappeared entirely, as in that mentioned by KEPLER in June 1620. HEVELIUS has taken notice of another, where the moon could not be seen even with a telescope, though the night was remarkably clear. 146. As an eclipse of the moon happens at the same instant of absolute time to all obser- vers, it is one of the phenomena, from the ob- servation of which the longitudes of places may be most directly inferred, § 67* On account of the ill-defined boundary of the sha- dow, this method of ascertaining the longitude does not admit of great precision. It is difficult to determine the beginning or end to less than a /* . oo minute of time. The arrival of the boundary of the shadow, at the dif- ferent spots, can be more accurately ascertained than the beginning or end ; and, therefore, as many 5 observations ASTRONOMY. 145 observations of that kind should be made as pos- sible. When several such observations, made under two different meridians, are compared, the mean may furnish a tolerably exact determination of the dif- ference of longitude. The comparison of the beginning or end, with cal- culations previously made, may also serve for find- ing the longitude, and may be useful for that pur- pose at sea. Solar Eclipses. 147. The length of the moon's shadow is less than that of the earth, in the same ratio that the diameter of the moon is less than the dia- meter of the earth ; that is, in the ratio of 1 to 3.562. «. Hence, when the Earth is in the aphelion, the length of the Moon's shadow is 59.730 ; and if the moon is in the perigee, its distance from the earth is only 55.902 ; so. that the shadow may reach the earth, and a total eclipse may take place. But if the moon were in her apogee, when her distance VOL. II. K is 146 OUTLINES OF NATURAL PHILOSOPHY. is 63.862, the shadow could not reach the earth, and the eclipse could not any where be total. b. When the earth is in the perihelion, the length of the moon's shadow is 57.76 ; and if at the same time the moon be in the perigee, or indeed nearer than her mean distance, a total eclipse may happen. c. The moon's mean motion about the centre of the earth is 33' in an hour ; and the shadow of the moon, therefore, traverses the surface of the earth when it falls on the surface perpendicularly, with a velocity of about 380 miles in a minute. When the shadow falls obliquely, its velocity appears greater in the inverse ratio of the sine of the obli- quity. Relatively to a point on the earth's surface, the sha- dow may go much faster than this, as its motion may be in an opposite direction to the diurnal ro- tation. The duration of a total eclipse, in any given place, cannot exceed 7' 58". LA LANDE, § 1777. An Annular Eclipse, or one where the sun's disk appears like a ring all round the moon, may last 12m 24s. To have a partial eclipse of the sun, it is not requi- site that the shadow should reach the earth ; it is sufficient that the distance of the centres of the sun and moon be less than the sum of their appa- rent semidiameters. 148. When ASTRONOMY. 147 148. When an opaque body is opposed to one which is luminous, there is a certain space be- hind the former body, from which the latter is only partially visible. This space is called the Penumbra of the opaque body. a. If the bodies are spherical, the penumbra is a cone, having for its angle the line joining the centres of the two bodies ; and for its vertex, the point in that line where tangents to the opposite sides of the two bodies intersect one another. b. Half the angle of the conical penumbra of the moon, is equal to the apparent semidiameter of the sun, plus the angle which the moon's semidiame- ter subtends at the sun. 149. To conceive the phenomena of a solar eclipse in general, we may consider the section of the moon's penumbra as advancing on the earth from the west, and as being viewed by an observer in the moon, in the same manner that an eclipse of the moon is viewed by an obser- ver on the earth. The observer might be placed any where, providing he saw the whole of that side of the earth which is turned to the sun 5 but it is most convenient to suppose him in the moon, that the arches, compa- red 148 OUTLINES OF NATURAL PHILOSOPHY. red with one another, may belong to circles of the same radius. r>mr 150. The penumbra in a solar eclipse, if view- ed from the moon, would subtend an angle e- qual to the sum of the diameters of the sun and moon, as seen from the earth. When the penumbra just touches the disk of the earth, the distance of their centres is equal to half the angle subtended by the section of the penum- bra, plus the moon's horizontal parallax ; that is, = semidiam. D -f- semid. Q + Hor. Par. D . If the least distance of the centre of the sun and moon, (the same with the least distance of the centres of the earth and of the penumbra), is less than this, there can be no eclipse. The greatest value of the above angle is about !• 34?' 27". Supposing this to be the distance at the time of the ecliptic conjunction, we get the dis- tance from the node 17° 21' 27". If the con- junction happens nearer to the node than this, there may be an eclipse. If it be more distant there can be none. The limits of solar eclipses being greater than of lu- nar, there are more eclipses of the sun than of the • moon, ASTRONOMY. moon, and that nearly in the ratio of 3 to 2 ; but fewer eclipses of the sun are observed in any given place than of the moon, as a lunar eclipse is visi- ble to a whole hemisphere ; but a solar only to a part. 151. The general phenomena of the eclipse being calculated to the time of a given meridi- an, the phenomena, as they will be observed at any particular place, may also be determined, by calculating the altitudes of the sun and moon, and the effects of their parallaxes for different instants of time, and then employing the me- thod of interpolation, to determine the time of the beginning and end, and the quantity of the greatest obscuration. a. Let the places of the sun and moon be found for an instant, far from the beginning of the eclipse, and from thence let their altitudes for the given place be computed, as also the effects of parallax in longitude and latitude. Let the difference of the apparent longitude of the two bodies thus found be called , and the latitude of the moon A. If the sun's parallax is included, let his parallax in latitude be applied to the moon, taking notice, whether it increases or diminishes the difference of latitude. Then ACB, fig. 15. be- ing an arch of the ecliptic, A the place of the sun, D of the moon, as just computed, P the pole of the ecliptic, and PDC a circle of longitude, AC **, 150 OUTLINES OF NATURAL PHILOSOPHY. == 3, and CD = A, and AD the distance of the centres, which we may call?/ = V 3* + A* because the triangle ACD may be regarded as rectilineal. In practice, y may be found by a construction ; or, if great accuracy is required, we may compute y from the trigonometrical formula, cos y = cos 1 X cos A* b. Let similar calculations be made for other two instants, separated by equal intervals of time m \ so that one may be near the middle, and another near the end of the eclipse. Let the distances of the centres found for these times be A, A7, A"; let ''>Tf the differences of these distances be DandD'; and let the second difference, or D — D' = A. Then if y be the distance of the centres for any time t9 reckoned from the instant for which the first computation is made, y = -•* + m The distance of the centres is thus expressed in terms of the time, and from this equation the time of the beginning and end of the eclipse, and the quanti- ty of greatest obscuration, may be determined. c. The time of the greatest obscuration is = w(D — \ A) -D— and this being substituted for t, the value of y will give the nearest approach of the centres. d. This ASTRONOMY. 151 d. This is only one of many methods that have been contrived for the calculation of eclipses. LA LANDE, Astron. torn. u. § 1860, &c. e. The geometrical construction, by means of a pro- jection, is sufficiently accurate for the prediction of eclipses. The most simple is that which supposes the observer to be placed in the sun, and to see the path which any place on the earth's surface de- scribes in consequence of the diurnal motion, pro- jected into an ellipsis on the plane of the earth's disk, while the path of the moon's shadow is pro- jected into a straight line on the same disk. LA CAILLE, Astron. VINCE, J 575. -y f. A very great addition to these methods of calcu- lation was made by LA CAILLE, viz., that by which a geographical representation is given of the path of the shadow, the quantities and the times of ob- scuration, &c. for all places of the earth. LA CAILLE, Astron. 1 166. LA LANDE, Astron. § 191 1. The construction of such a map may be made with sufficient accuracy by means of a celestial globe. Number . OUTLINES OF NATURAL PHILOSOPHY. Number of Eclipses. a. In the space of eighteen years, there are usually about 70 eclipses, 29 of the moon, and 41 of the sun. These numbers are nearly in the proportion of 2 to 3. b. Seven is the greatest number of eclipses that can happen in a year, and two the least. c. If there are seven, five must be of the sun, and two of the moon. If there are only two, they must be both of the sun ; for in every year there are at least two eclipses of the sun. d. There can never be more than three eclipses of the moon in a year ; and in some years there are none at all. c. Though the number of solar eclipses is greater than of lunar in the ratio of 3 to 2, yet more lunar than solar eclipses are visible in any particular place, because a lunar eclipse is visible to an entire hemisphere, and a solar is only visible to a part. VINCE'S Astron. § 588. f* Central Eclipses are comparatively rare phenome- na •, for though there are about 28 such eclipses in every cycle of eighteen years, yet the space over which ASTRONOMY. 153 which every one of them appears to be central, is a narrow belt, perhaps a mere mathematical line, traced across the enlightened hemisphere of the earth. g. A central eclipse is annular, when the angle sub- tended by the sun's diameter is greater than that subtended by the moon's 5 it is total for an instant, or sine mord, when these angles are equal ; and it is total for a portion of time that can never exceed eight minutes, when the angle subtended by the moon's diameter is greater than that subtended by the sun's. h. A central eclipse observed at London in April 1715, is described by Dr HALLEY. The darkness for a few minutes was so entire, that the stars be- came visible. Though the disk of the sun was wholly covered by the moon, a luminous ring of a faint pearly light surrounded the body of the moon the whole time. Its breadth was about a tenth of the moon's diameter. The longest time that the obscuration lasted any where in Britain, was about 3m 57s. Phil. Trans, vol. 29. p. 295., &c. ViNCE, Astron. vol. 1. § 585. Occupation 154 OUTLINES OF NATURAL PHILOSOPHY. • Occupation of Stars. 152. The same method used in calculating eclipses, may be applied to compute the occul- tation of a fixed star by the moon ; only, when the moon is distant from the ecliptic, the base of the right-angled triangle in the former con- struction, must not be supposed equal to the difference of longitude, but to that difference multiplied into the sine of the star's distance, from the pole of the ecliptic, or into the cosine of its latitude. If S (fig. 16.) be the star, D the moon, DE the diffe- rence of latitude, SE is not to be taken as equal to AC, the difference of longitude, but as equal to AC X sin SP = dif. Ion. X cos A. The distance SD = V SE3 -f DE% being thus found, the rest of the computation is as before. If at the time of the mean conjunction of the moon and a star, that is, when the moon's mean longi- tude is the same with the longitude of the star, their difference of latitude exceed 1° 37', there can be no occultation, but if the difference be less than 51', there must be an occultation somewhere on the face of the earth. Between these limits there is a doubt, which can only be removed by the cal- culation of the moon's true place. VINCE, vol. i. § 591: All ASTRONOMY. 155 All the stars of which the latitude is less than 4° 32' may suffer occultations by the moon, in any part of the earth. Dr BREWSTER has given a catalogue of the most remarkable stars subject to occulta- tions of the moon. Edinburgh Encyclopedia, art. Astronomy, vol. n. p. 668. SECT. IX. OF THE PLANETS. 153. IT was said, (§3. #.) that beside the sun and moon, ten of the stars have motions east- ward, peculiar to themselves. They are called Planets, and are distinguished by particular names, which, taken in the order of the celeri- ty of their motions, are Mercury, Venus, Mars, Vesta, Juno, Ceres, Pallas, Jupiter, Saturn, Uranus. a. The first two perform their revolutions in the heavens in less than a year, and are called inferior Planets ; the rest have their period greater than a year, and are called superior. b. Five of the Planets, Mercury, Venus, Mars, Jupi- ter, and Saturn, are very conspicuous, and have been known from immemorial time. PLINY says of them : " Suus quidem cuique color est ; Saturno candidus, Jovi clarus, Marti igneus, Lucifero can- dens, Vesperi refulgens, Mercurio radians j Soli cum • OUTLINES OF NATURAL PHILOSOPHY. cum oritur ardens, postea radians." Hist. Nat. lib. ii. cap. 16. By Lucifer is understood Venus, when seen in the morning before sun-rise. By Vesper, the same planet seen in the evening after sun-set. c. The other five planets are visible only through the telescope, and have been lately discovered ; Ura- nus by HERSCHEL, in 1781 ; Ceres by PIAZZI, in 1801 5 Pallas by OLBERS, in 1802 ; Juno by HARDING, in 1803; Vesta by OLBERS, in 1807. d. The planets have also particular characters, by which they are distinguisded ; these, in the order in which they have been enumerated, are, It is best to begin with the inferior planets, and with Venus, as that of which the phenomena are most easily observed. 154}. Venus, the most brilliant of the planets, always accompanies the sun, never receding from him more than 45°, and becoming, as she is on the east or west side, alternately the Evening or the Morning Star. a. Venus is the only planet mentioned in the Sacred Writings, and in the most ancient poets, such as HESIOD and HOMER. b. The ASTRONOMY. 157 b. The Evening and Morning Star, or the Hesperus and Phosphorus of the Greeks, were at first sup- posed to be different. The discovery that they are the same is ascribed to PYTHAGORAS. 155. This planet, when an evening star, and at her greatest distance from the sun, or at what is called her Greatest Elongation, appears, through the telescope, to have a semicircular disk, like the moon in the last quarter, with its convexity turned to the west. From that time, during her approach to the sun, her splendour increases for a while, though the quantity of the illuminated disk diminishes, like the moon in the wane ; and at the same time, her dia- meter, measured by the distance of the horns, increases. a. At the time of her greatest elongation, Venus is stationary with respect to the sun, or has the same motion in longitude. After that, her motion in longitude becomes slower than the sun's, and she comes nearer to the sun, as just remarked. At a certain point she becomes stationary with respect to the fixed stars, having no motion in longitude. After that, her motion becomes retrogade in re- spect of the fixed stars, and is directed west- ward. b. Venus at last approaches the sun, so as to be lost in his light ; and after some time, appears on the west 158 OUTLINES OF NATURAL PHILOSOPHY. west side, and is seen in the morning, before the sun rises. c; Though Venus in general is not visible at the time of her conjunction with the sun, she has sometimes been seen as a dark spot passing over the body of the sun. This is the phenomenon called the Tran- sit of Venus. Her diameter is then greatest, and measures nearly one minute. 156. As Venus proceeds to the westward, her disk is seen as a crescent continually increasing, at the same time that the diameter is diminish- ing. At the elongation of 45°, the disk is again a semicircle ; and from thence it increases, while the distance from the sun diminishes, till the planet is lost in the sun's rays ; her orb being almost a circle^ but its diameter not more than one-sixth of what it was at the form- er conjunction. a. The conjunction, which is preceded by the ap- proach to a full orb, or that which follows the wes- tern elongation of Venus, is called the superior conjunction, as she is then farthest frpm the earth. The other the inferior. b. The time of the greatest elongation of Venus is about sixty-nine days before or after the inferior conjunction, when she is between 39° and 40° dis- tant from the sun, and comes to the meridian 1*» 38«* either before or after noon ; her disk is then like- ASTRONOMY. 15$ like that of the moon five days before or after the conjunction. LA LANDE, § 1147. 157. After the superior conjunction, the orb of Venus increases in magnitude ; but the en- lightened part diminishes, just reversing the; former order, till she arrive at her greatest eastern elongation ; after which the phenome- na are repeated, as already mentioned ; and the period which circumscribes all those changes, or the time from one conjunction to the next conjunction of the same sort, is, in its mean quantity, 584 days. a. This is called the Synodical Revolution of Venus. She is retrograde with respect to the fixed stars during forty-two days of that period. 158. Hence it is evident, that the orbit of Venus surrounds the Sun, but excludes the Earth ; and that her motion, with respect to the sun, may be equable, notwithstanding the va- riety of appearances it puts on to a spectator in the earth. a. If a be the distance of the earth from the sun, and x that of. Venus from the sun, a -f x will be its distance from the earth, at the superior conjunc- tion, and a — x at the inferior; and by § 156. a -f- x : a — x : : 6 : 1 ; so that 2 a : 2 x : : 7 : 5, or as 10 to 7. If the earth's distance from the sun be called 10, that of Venus is 7 ; and this serves as a 160 OUTLINES OF NATURAL PHILOSOPHY. a first approximation to the distance of Venus, supposing her orbit a circle, with its centre in the sun. b. From the synodical revolution of 584 days, the periodic time of Venus is found nearly equal to 224 days. For if v be the angular velocity of the planet in her orbit, or her diurnal motion, that of the earth being 59' 8". 3 ; then the arch described by the sun, during a synodical revolution of Ve- nus is (59' 8".3) 584; and that described by Venus in the same time is 360°-f-(59' 8".3) 584 ; there- fore the diurnal motion of Venus in her orbit 360°-f(59'8".3) 584 __ , , 360° 584 ~ "~ . T ' ~ 1° 35' 56" ; and the periodic time = 360° 35'. 56 =r 224 days, nearly. 159. Mercury, like Venus, accompanies the sun, and never recedes from him farther than 28°. The greatest apparent diameter is 11".275, the least 5 a. Hence Mercury describes an orbit round the sun, at the mean distance, (computed as in the case of Venus), of 3.7 of the parts whereof the Earth's distance from the sun is 10. &. The synodic revolution of Mercury is 115^.877 nearly. i The ASTRONOMY. l6l The diurnal motion in his orbit 4° 5' 22". The periodic time 87d.97 nearly. 160. The points in which a planet has no la- titude, are called, as in the case of the Moon^ the Nodes qf the Planet. The Planet is then in the ecliptic, and one-half of its orbit lies on the north, the other on the south side of that plane. a . The line of the nodes of every planet, or the com- mon section of the plane of its orbit, with the plane of the ecliptic, passes through the Sun. This was discovered by KEPLER. See Dr SMALL'S Ac- count of KEPLER'S Discoveries, p. 154. b. The node through which the planet passes into the northern signs, is called its Ascending Node ; that through which it passes into the southern, is call- ed the Descending Node. c. The Heliocentric place of a planet, is its place as it would be seen from the Sun : the Geocentric, as it is seen from the Earth, 161. When the Earth is in the line of a pla- net's nodes, or, which is the same, when the Sun is seen from the Earth in that line, if the pla- net's elongation from the Sun, and its geocen- tric latitude be observed, the inclination of the orbit may be found ; for, the sine of the elon-* VOL. II. L gatiow 162 OUTLINES OF NATURAL PHILOSOPHY. gation is to the radius, as the tangent of the geocentric latitude to the tangent of the incli- nation. LA LANDE, § 1358. GREGORY'S Astron. Book in. prop. 20. If the planet be 90° distant from the Sun, the lati- tude observed is just equal to the inclination. KEPLER made use of this last method for deter- mining the inclination of the orbit of Mars. 162. If an inferior planet, at the inferior and superior conjunctions, or a superior planet at the opposition and conjunction, be also 90° dis- tant from the node ; from the observation of Its geocentric place, the inclination of the or- bit being known, the ratio of the planet's dis- tance from the Sun, to the Earth's distance from the Sun, may be found. . In the annexed figure, (fig. 17,), let S be the Sun, W the orbit of Venus, E the Earth ; then joining EV, EV, and drawing the perpendiculars VD, V'D', if ES = a, SV=&, and the angle ESV=I, SD = b cos I = SD'; so that ED = a — 6cosl, and ED' = a + b cos I. If the angle VES = A, and V'ES = *', then because VD = V'D', tan ,vhen sta- tionary. Arch of Re trogradation. Time of Re- trogradation. Synodic Re- volution Mercury, Venus, M*rv Jupiter, Saturn, Uranus, 18° 00' 28 48 136 48 U5 12 108 54 103 30 13° 30' 16 12 16 12 9 54 6 18 3 36 23 days. 42 73 121 139 151 118 days. 584 780 399 ,378 ,370 165. The apparent motion of an object is af- fected by the motion of the spectator ; and if there is not a certainty that he is at rest, when the motion of the former appears extremely ir- regular, it is natural to inquire, whether any mo- tion ASTRONOMY. 167 tion that can reasonably be ascribed to the lat- ter will explain the irregularities observed* In the present case, there is not a certainty that the spectator, (or the Earth), is at rest : the want of any information of his motion from terrestrial ob- jects affords none, as, according to a principle al- ready explained, the motions of bodies among one another, are nowise affected by any motion which is common to them all. 166. When a spectator moves, without being sensible of it, he necessarily transfers his own motion to the objects around him, estimating it in a direction opposite to that in which he has actually moved. Suppose an object to move from A to B, (fig. 18.), while an observer, unconscious of his own motion, is carried from C to D. From B draw BF equal and parallel to CD, but extending from B the op- posite way that CD does from C, and join AF ; AF will be the apparent path of the body, and F its apparent place, at the time when the spectator is really in D, and the body in B. If the velocity of the observer had been the same, but in an opposite direction, CD', BF' being drawn == CD', but in the opposite direction, the apparent motion of A would have been in the line AF' This 168 OUTLINES OF NATURAL PHILOSOPHY. This is evident, because CF' is equal to D'B, and makes the same angle with CD. Hence it is evident, that an object, without being really at rest, may be apparently so, if the observer is in motlori, and may even acquire an apparent motion, in a direction contrary to its real. Thus also the observer and the object both moving with perfect regularity, both describing concentric circles, for example, with uniform velocities, and directed the same way, the pne may become sta- tionary in respect of the other, and even acquire a motion in an opposite direction. 167* Suppose f? to be the distance of a planet (imagining it to describe a circle round the sun) from the sun, or the radius of its orbit, and t the elongation from the sun, at which it appears stationary, the radius of the circle in which the observer must move, in order to see the planet stationary at that elongation, being called *r, is found from the equation, a? + d x = d* cot* e ; «. This follows from what KEIL has demonstrated in his Astronomy, Sect. 27. ; LEMONIER, Institutions Astronomiques, ASTRONOMY. 169 Astronomiques, p. 585. LA LANDE has simplified the demonstration, J 1188. See also BJOT, vol. HI. p. 181., Note. b. If, by help of the above formula, we inquire, sup- posing the Earth to revolve in a year round the Sun, and Mars in 686.98 days, what must be the ratio of the distance of the Earth to the distance of Mars from the Sun, in order that the latter may be stationary at the elongation of 136Q? we shall find it to be that of 1 to 1.52, which is the ratio deduced from other phenomena. c. Not only are the stations thus explained, but the extent of the arches of progression and retrogra- dation also. This coincidence affords a strong presumption in favour of the system of the Earth's motion, or that which, from the name of its dis- coverer, is called the Copernican System. d. The same holds of Jupiter, Saturn, Uranus. The same motion of the Earth, and the same distance from the Sun, will account for the phenomena in all the cases ; so that whatever probability there is, from the phenomena of one planet, in favour of the Earth's motion, the same is increased in a quadru- plicate ratio, from considering the phenomena of all these four superior planets. e. The two inferior planets, give a similar increase of evidence. The four new planets are not taken into 170 OUTLINES OF NATURAL PHILOSOPHY. into account, as the times of their stations, &c. may not yet have been sufficiently determined by observation. f. On the strength of this evidence, we shall assume the motion of the Earth as a fact, and try whether it is consistent with the other phenomena of the planetary motions. Orbits of the Planets. 168. If a planet be observed twice in the same node, the node in the interval being supposed to remain fixed, the position of the line of the nodes may be determined, and also the distance of the planet from the sun at the times of ob- servation, a. Let a superior planet be observed in its node N, (fig. 19.), from the Earth at E, and after the planet has made an entire revolution, and return- ed to the point N, let the Earth be at E'. Then, from the time, and the construction of the Earth's orbit, EE' is given, and the angles SEE', SE'E. But the angles SEN, SE'N, are known by obser- vation 5 therefore the angles EE'N, E'EN, as also the ASTRONOMY. 1?1 the base EE' are given, and therefore the sides EN, E'N, are also given. Hence, from either of the triangles ESN, E'SN, SN is found, and also the angle ESN, or the heliocentric place of the node. GREGORY'S Astronomy, Book in. prop. 19. WOLFII, Elem. Astron. § 777. Thus the periodic time of the planet is determined, and also its mean motion. b. When observations of this kind are made at a con- siderable distance of time from one another, it is fo^nd that the nodes of every planet have a slow motion retrograde, or in a direction contrary to the order of the signs. 169. The distance of a planet from the Sun, and its heliocentric place, or its longitude as seen from the Sun, may be determined by ob- servations made at the time of its opposition to the Sun. If E (fig. 20.) be the Earth, S the Sun, P a planet, O its place reduced to the ecliptic, SN the line of the nodes ; the points S, E, and O are in the same straight line, because of the planet's opposition to the Sun, and the angle ESN, that is, OSN, is known from the last problem ; therefore the an- gle PSO or PSE, the heliocentric latitude, is also given. But the angle PEO, the geocentric latitude, is given by observation, and the base ES, from the theory of the Earth's motion ; therefore SP, or 172 OUTLINES OF NATURAL PHILOSOPHY. or the planet's distance from the Sun, is found. The position of PS, relatively to NS, is also thus determined ; for, in the right-angled spherical tri- angle, of which the base is the arch that mea- sures the angle OSN, and the perpendicular the arch which measures the angle PSO, the hypothe- nuse is the measure of the angle PSN, which the radius vector makes with the given line SN. b. Thus also, EP, the planet's distance from the Earth, is found. If, then, by observations made a little before and after the opposition, the diur- nal motion of the planet in its own orbit, relative- ly to E, be determined, the same may be found relatively to S, being to the other in the inverse ratio of SP to EP, When many oppositions of a planet are thus obser- ved, many different radii of the planetary orbits are determined, as well as the angular motions corresponding to them. 170. It appears, on laying down the radii determined as above, that the orbits of the pla- nets are ellipses, having the Sun in their com- mon focus ; and that the angular motions of a planet round the Sun, are inversely as the squares of its distances from the Sun ; so that the sectors described by the radius vector, are proportional to the time. These ASTRONOMY. 173 These two propositions, which have already been shewn to hold of the Earth's motion, are there- fore common to the motions of all the planets. They were discovered by KEPLER, and were first found out by him, with infinite ingenuity and la- bour, when he was endeavouring to determine the orbit of Mars. 171. When the focus of an ellipse, and three points in its circumference, are given, the el- lipse may be described ; and hence the plane- tary orbits may be determined, that is, the axis, the eccentricities, and thence the equations to the centres, &c. The application of this to find the three Elements of an orbit, the Eccentricity, the place of the Aphelion, and the Epoch, or radical mean place, for a given time, is in LA LANDE, § 1288, &c. Also VIUCE'S Astronomy > vol. i. § 257. See also NEU- TONII Prin. Math. lib. Inius^ prop. 21. Schol. 172. When the mean distances of the planets are compared, and also their periodical times, it is found that the squares of the periodical times are as the cubes of the distances. This great general fact was also discovered by KEP- LER, and is the third of the laws that bear the name of that astronomer. 173. When 174 OUTLINES OF NATURAL PHILOSOPHY. 173. When the elements of the orbit are found from observation, at periods considerably dis- tant from one another, the line of the apsidefc of each of the planets is discovered to have a slow motion forward. LA LANDE, § 1309, &c. The Elements of the Orbits of the different Planets are given in the annexed tables. I. Inclination of Long, ascend- Secular motion the Orbit to the ing Node for of the Node. Ecliptic for 1801. 1801. Mercury, 7° V 1" 46° 57' 31" — 13' 2" Venus, 3 23 32 74 52 52 — 31 10 Mars, 1 11 6 48 1 30 — 38 48 Vesta, 13 3 29 '171 6 38 Juno, 7 8 46 103 1 0 Ceres, 10 37 34, 80 1 3 Pallas, 34? 37 8 172 32 36 Jupiter, 1 18 43 98 25 34 — 26 17 Saturn, 2 29 38 111 55 27 — 37 54 Uranus, 0 46 26 72 51 14 — 59 57 II. ASTRONOMY. 175 II. Siderial Revo- Mean Dis- Eccentricity, lutions. tances. the mean Dis- tance being 1. Mercury, 87d.9692 0.38709 0.20551 Venus, 224- .7008 0.72333 0.0068S The Earth, 365 .2564 1. 0.01685 Mars, 686 .9796 1.52369 0.09313 Vesta, 1590 .998 2.66716 0.25494 Juno, 1335 .205 2.373 0.09322 Ceres, 1681 .539 2.76740 0.07835 Pallas, 1681 .709 2.76759 0.24538 Jupiter, 4332 .5963 5.20279 0.04817 Saturn, 10758 .9698 9.53877 0.05617 Uranus, 30688 .7127 19.18330 0.04667 III. 176 OUTLINES OF NATURAL PHILOSOPHY. Hi. Mean Longi- Mean Longitude Secular Mot. tude of the of the Perihe- of the Pe- Planets, lion. rihelion. 1st Jan. 1801. Mercury, 166° 0' 48" 74° 21' 46" 9' 43" Venus, 11 33 16 •28 37 1 4 28 Earth, 100 39 10 99 30 5 19 39 Mars, 64 22 57 332 24 24 26 22 Vesta, 267 31 4-9 249 43 0 Juno, 290,37 16 53 IS 41 Ceres, 264- 51 34- 146 39 39 Pallas, 252 43 32 121 14 1 Jupiter, 112 15 7 11 8 35 11 4 Saturn, 135 21 32 89 8 58 32 17 Uranus, 177 47 38 167 21 42 40 The first of the above Tables gives the inclination of the orbits, the position of the line of the nodes, and the secular motion of the nodes for all the planets. The sign minus, prefixed to the motions of the nodes, signifies that they are retrograde* The in- clinations of the orbits of Vesta, Ceres and Pallas, are greater than those of the other planets ; and the orbit of the last goes far beyond the zodiac. i The ASTRONOMY. 177 The secular variations of the nodes for these three planets, and for Juno, have not yet been determi- ned. The Second Table gives the times of the siderial re- volutions in days; the mean distances from the Sun, or the semi-transverse axes of the orbits, on the supposition that the mean distance of the Earth is 1 •, also the eccentricity of each orbit, supposing the semi- trans verse of that orbit to bel. The Third Table gives the mean longitude of each planet for the 1st of January 1801, to the meri- dian of Greenwich ; also the mean longitude of the perihelion, or lower apsis of the orbit, with the secular variation. See Exposition du Systeme du Monde, 3me ed. BIOT, torn. HI. § 29. WOOD- HOUSE, 'Ast. p. 286. VOL. IL M notation 178 OUTLINES OF NATURAL PHILOSOPHY. Rotation of the Planets. 176. Four of the planets, Venus, Mars, Ju- piter and Saturn, when examined with the te- lescope, appear to revolve on axes, in the same direction in which they revolve in their orbits ; the axis of each remaining always parallel*, or nearly parallel to itself. a. This conclusion is derived from the motion of cer- tain spots, which are distinguished, by the colour or intensity of their light, from the other parts of the planetary disk. In this way the time of rotation is also determined, as in the case of the Sun, (§ b. It is thus found, that Venus revolves in 23** 21m 9s, on an axis which makes a very small angle with the plane of the ecliptic. This was first observed by the elder CASSINI. LA LANDE, J 3341. c. Mars revolves in 1 day 39 minutes, on an axis in- clined at an angle of 59° 42' to the ecliptic. d. Jupiter revolves in 9^ 56m, on an axis nearly per- pendicular to the ecliptic. . The plane of the ring is in the plane of the equator of Saturn 5 it is inclined to the orbit of the pla- net, nearly at an angle of 30°, and remains always parallel to itself. 196. When Saturn is in the longitude of 5s €0°, or 11s 20°, the plane of the ring passes through the Sun, and the light then falling up- on it edgewise, it is no longer visible to us. a. This disappearance of the ring has been often ob- served. LA LANDE, § 3354. The disappearance of the ring from this cause, lasts only a few days ; for when Saturn has passed the node of the ring three or four minutes, or the Sun has risen above its plane by that quantity, the ring becomes visible, ~ . 197« The ring also disappears, when its plane passes through the Earth ; for its edge, or its thickness, being then directed to the eye, and being too fine to be seen, the planet appears quite round. a. In this case, the Earth requires to be elevated above the plane of the ring, at least half a degree, before the ring is seen, which makes it continue invisible sevejr ASTRONOMY. 191 seven or eight days before and after the passage of the Earth through its plane. I. This is what happens with ordinary telescopes ; but with the forty foot reflector of Dr HERSCHEL, the ring does not cease to be visible. The Doc- tor saw the satellites along the margin of the ring, like bright beads threaded on a string. 198. There is a third cause of the disappear- ance of the ring, viz. the Earth being placed on the side of the ring that is turned from the Sun. The ring may on these accounts disappear twice in the same year. LA LANDE, § 3357. Opuscula of BOSCOVICH, torn. v. Also SEJOUR, Anneau de Saturne. 199- The ring is divided into two by a dark line going all round concentric with the outer and inner circles. a. This was first observed by CASSINI, Mem. Acad. des Sciences, 1715. SHORT, with his twelve foot reflector, observed the same ; as Dr HERSCHEL has also done. b. To Dr HERSCHEL, and some other observers, it has appeared that there are more of these lines than one. It OUTLINES OF NATURAL PHILOSOPHY. It appears, therefore, that there are really two con- centric rings, or perhaps more. " .&' *' 200. The ring revolves on an axis at right angles to its own plane nearly, in the same time with the planet itself, or in a little more than ten hours. This observation we also owe to Dr HERSCHEL. «. The time of the revolution of the ring is .437 of a day, or IQh 29™ 17s. b.It is remarkable, that if a satellite at the mean dis- tance of the middle of the ring, revolved round Sa- turn, and obeyed the law of KEPLER in respect of the other satellites, it would revolve exactly in 17s. BIOT, Astron. Phys. torn. in. p. 96. Jiifiu c ertl .V'Ul , l^tfyd silt ftoiv/'o" -j Jbiuicn iTc gnioB onif SECT. *3*:&\ . '.IT'-ia 'ri: 5o< ASTRONOMY. 193 SECT. XI. OF COMETS. 201. A COMET is a luminous body, which ap- pears in the heavens only for a limited time, seldom exceeding a few months ; during which, beside the diurnal motion, of which it partakes in common with the other heavenly bodies, it has always a motion peculiar to itself, by which it changes its place among the fixed stars. Its appearance is usually that of a collection of va- pour, in the centre of which is a nucleus, for the most part, but indistinctly defined. In some Comets, the peculiar motion is progressive, in others retrograde. In the same comet, the motion is all nearly in one plane ; but in different Comets, these planes make all different angles with the ecliptic. At the beginning and end of the ap- pearance of a Cometj it deviates from the plane in which the middle part of its course lies. NEWTON de Systemate Mundi, § 59. The Comets have no parallax, and are therefore certainly beyond the limits of our atmosphere. VOL. II. N 202. A OUTLINES OF NATURAL PHILOSOPHY. . A Comet, when it first appears, is usual- ly surrounded by a faintly luminous vapour, to which the name of Coma has been given. As the Comet approaches the Sun, the coma be- comes more bright, and at length shoots out into a long train of luminous transparent va- pour, very much resembling a streamer, and in a direction opposite to the Sun. This forms the Tail of the Comet. As the Comet retires from the Sun, the tail grows less, and resumes nearly its first appearance. Those Comets which never come very near the Sun, have nothing but a coma or nebulosity round them during the whole time of their continuance. The tail is always transparent, so that the stars are distinctly seen through it, as they are even said to have been in some instances through the central part, or what was supposed the nucleus of the Co- met. The length and form of the tail are very various. Sometimes it is only a few degrees, ait others it is more than a quadrant. In the great Comet which appeared in 1680, the tail subtended an angle of 70° ; in that of 1618, an angle of 104°. The tail sometimes consists of diverging streams of light; that of 1744 consisted of six, all proceeding from the head, and all a little bent in the same di- rection. The ASTRONOMY. 195 The Comet of 1811 was remarkable for its beauty. The tail was composed of two diverging beams of faint light, slightly coloured, which made an angle from 15° to 20°, and sometimes much more, and were bent outward. The space between was com- paratively obscure. Abrege d'Ast. par DE LAM- BRE, lefon 21. § 57* 203. A Comet remains so short a time in Sight, and describes so small a part of its course within our view, that, from observation alone, without the assistance of hypothesis, we should not be able to determine the nature of its path. The hypothesis most conformable to analogy is, that the Comet moves in an ellipsis round the Sun placed in one of the foci, and that the radius vector from the Sun to the Comet de- scribes areas proportional to the times* As the ellipse in which a Comet moves is evidently very ecceatric, it will coincide very nearly with a parabola, at its vertex, or for all the time that a Comet remains in sight. If it be supposed that the Comet de- scribes an ellipsis or a parabola, in conformity to the laws of KEPLER, then from three geocen- tric places, known by observation, the orbit may be determined. The first solution of this problem was given by NEW- TON, Princip. lib. HI. prop. 41. He calls it Proble- ma 196 OUTLINES OF NATURAL PHILOSOPHY. ma longe difficillimum, and it may, therefore, be- readily believed, that the solution does not fall within the limits of an elementary treatise. At the same time, a very simple geometrical problem is the foundation of it. Arith. Universalis. The determination of the orbit implies that of the five quantities, which are its elements : 1* The inclination of the orbit. 2. The position of the line of the nodes. 3. The longitude of the pe- rihelion. 4. The perihelion distance from the Sun. 5. The time when the Comet is in the pe- rihelion. NEWTON'S solution being a laborious and indirect ap- proximation, the problem has been attempted by many others. -LA CAILLE, Astron. § 775, &c. BOSCOVICH, Opera, torn. in. p. 14. &c. The so- > lutions of BOSCOVICH and LA PLACE are illustra- ted by Sir H. ENGLEFIELD, in his Determination of the Orbits of Comets, Lond. 1793. LA LANDE has given a mechanical construction, that serves for finding the orbit nearly, § 3127. ; it is followed by VINCE, § 653. LAMBERT has demon- strated some remarkable properties of the orbits of Comets, in a work entitled Insigniores Orbitte Co- metarum proprietates, Aug. Vind. 1761, 8vo. See also LA GRANGE, Mem. de Berlin, 1783. ; The most perfect solution of all is supposed to be that of LA PLACE, Mechanique Celeste, The ASTRONOMY. 197 The latest, and in practice one of the best, is that of DE LAMBRE, Abrege d'Astron. legon 21. § 24. 205. The only Comet which is known with absolute certainty to have returned, is that of 1682, which, conformably to the prediction of Dr HALLEY, appeared in 1759- Dr HALLEY was led to this prediction by observing, that a Comet had appeared in 1 607, and another in 1531, and that the elements of their orbits, when calculated from the observations made on them, agreed nearly with those of the Comet of 1682, the period being between seventy-five and seventy-six years. Though there can be little doubt that it was the same Comet which was seen at all the four dates just mentioned, the appearances were considerably different. In 1531 the Comet was of a bright gold colour; in 1607 it was dark and livid; it was bright again in 1682, and obscure" in 1759. PJNGRE', Cometographiey torn. n. p. 189. The return of some of the other Comets is probable, though not certain. The great Comet of 1680, was supposed by Dr HAL- LEY to have a period of 575 years, and to be the same which had appeared a little before the death of JULIUS G/ESAR, in the year 44 A. C. j again, in reign of JUSTINIAN, in the year 531 P. C. ; and 198 OUTLINES OF NATURAL PHILOSOPHY. and in 1 106, in the reign of HENRY I. At all these periods, appearances of a great and terrible Co- met are recorded, but no such observations was made as can ascertain their identity completely. Synopsis Astronomic Cometicce^ subjoined to HAL- ILEY'S Astronomical Tables. 206. The Comet of 1680, mentioned above, is remarkable for having approached nearer to the Sun than any other that is known. At its. perihelion, its distance from the Sun was only -i-th part of the Earth's. It descended to the loo A Sun with great velocity, and almost perpendi- cularly, and ascended in the same manner, re- maining in sight for four months. When this Comet was in the perihelion, the diame- ter of the Sun must have subtended an angle of more than 100 degrees. See many interesting particulars with respect to it, Princip. torn. uir prop. 41. at the end. The phenomena of the tails of Comets, shew the ce- lestial spaces to be void of resistance. Some Comets have come very near the Earth. A Comet in 14?2 is said by REGIOMONTANUS to have moved over an arch of 120 degrees in one day •, and another in 1 760 moved over an arch of 4?1 degrees in the same space of time. As neither pf them could probably have described in its orbit more ASTRONOMY. 199 more than an arch of a few degrees, these extraor- dinary apparent changes of place can only have proceeded from vicinity to the Earth. £07. It appears that Comets contain very lit- tle matter, and have but a very feeble action on other bodies. In the year 1454, a Comet is said to have eclipsed the Moon $ so that it must have been very near to the Earth ; yet it had no sensible effects. The Comets just mentioned produced none. A Comet in 1770 came very near to the satellites of Jupiter, but caused no derangement in the sys- tem. 208. The number of Comets observed and re- corded, with more or less accuracy, exceeds 350. PINGRE', who wrote in 1783, enumerates 324 ; and 32 are now to be added. Of these, not so many as a third have been observed with such accuracy, as to allow the elements of their orbits to be asceratined. The elements of 98 have been computed, going back as far as the year 837, and coming down to the Comet of 1807. See FERGUSSON'S Astronomy, (BREWSTER'S edit.) vol. II. p. 360. SECT. #00 OUTLINES OF NATURAL PHILOSOPHY. SECT. XII. OF THE ABERRATION OF LIGHT, ANP 'THE NUTA- TION OF THE EARTH'S AXIS. Aberration of Light. £09- IF a ray of Light, coming in a straight line, and with a given velocity, pass through a tube, also moving in a straight line, with a gi- ven velocity, and remaining parallel to itself; the path of the ray, relatively to the tube, is the diagonal of a parallelogram, the sides of which are proportional to the velocities "of the Light, and of the tube, and in the same straight lines, one of the velocities only, being estimated in a direction opposite to its own. This is evident from the composition of motion. We may conceive what is here supposed to be ac- tually the case, with a ray of light coming from a fixed star, and passing through a telescope, or a tube, furnished with plain sights ; the tube having the same motion with the earth in its orbit, which, for a short time, may be regarded as rectilineal and uniform. The path of the ray in the tube, is therefore ASTRONOMY. 201 therefore not directed to the point from which the ray comes, and the line of collimation (the line in the tube along which the object is viewed) must be inclined, so as to have the direction of the path of the ray ; or of the diagonal just mentioned. The angle thus contained between the line drawn to the object, and the line of the tube in which the object is seen, is called the Aberration. 210. When the motion of the tube, or of the earth, is perpendicular to the motion of the raj, the aberration is a maximum, and is measured by the arch which the earth describes in its or- bit, in the time which light takes to come to it from the sun. In 8' 13", light moves over the radius of the earth's prbit : and, in the same time, the earth moves over an arch of 20".23'2, which is therefore the maxi- mum of the aberration ; and this is found by ob- servation to be accurately true. 211. Every star appears to describe an ellipsis in the heavens, of which the true place of the star is the centre ; the semi-transverse axis, 20V.232, in the direction of a tangent to the pa- rallel of latitude of the star ; and the semi-con- jugate axis, 20". 232 x sin Lat. of the star. SIMPSON'S Essays, 1740, p. 1. &c. If OUTLINES OF NATURAL PHILOSOPHY. If the star is in the pole of the ecliptic, it describes a circle, with the radius 20'.232, having the pole for its centre. If the star is in the ecliptic, it describes a straight line in that plane, and extending 20".232 on each side of the star. The purposes of practical astronomy require, that the change made by the aberration on the longi- tude and latitude, and on the right ascension and declination of a star, should be computed. . If L be the longitude of the sun at any time, and L' the longitude of a star, the aberra- tion of the star in longitude is — Q0".23<2 x cos (I/— L) m cos Lat. and the aberration in latitude is x sin (L' — L) sin Lat. These formulas were first given by CLAIRAUT. Mem. Acad. de Scien. 1737. See also CAGNOLI, Trig. § 1529. LA LANDE, § 2823. and DE LAMBRE, Ast. legon 19. § 20, 21. &c. 213. If A be the right ascension, and D the declination of a star, L being the sun's longi* tude, ASTRONOMY. tude, as before, the aberration of the star in right ascension is 19".17 X cos (A — L) — 0".83 X cos (A + L) m cosD and the aberration in declination is, sin D(19//.l7sin(A-L)-0//.83sin(A+L)-8//cosL XcosD). CAGNOLI, ibid. DE LA MERE, ibid. From these four formulas, all the effects of aberra- tion may be computed. Tables have been con- structed for facilitating the calculations. VINCE, Ast. vol. i. § 513. & 522. The discovery of the aberration was made by Dr BRADLEY, in 1725, from observation alone. He has given an account of it, in a letter to Dr HAL- LEY, Phil. Trans. N? 406. A very full extract of this letter is in VINCE'S Astron. vol. i. $ 503. Af- ter observation had satisfied him of the changes above stated, in the latitudes of different stars ; he discovered the cause to be the motion of light, combined with the motion of the earth, as above explained. . From the phenomena of the aberration, the motion of the earth in its orbit is more di- rectly proved than from any other fact in astro- »omy, Though OUTLINES OP NATURAL PHILOSOPHY. Though it is proved to demonstration, from facts above enumerated, that the Earth is far from being the centre of the planetary motions, yet all the ap- pearances hitherto mentioned, are consistent with what is calledjthe TYCHONIC System of the heavens, (from its inventor TYCHO BRAKE'), in which the Sun, accompanied by the planets, revolves in an or- bit round the Earth. This system, which its want of simplicity renders suspected, is entirely over- turned by the fact of the aberration, and the mo- tion of the Earth completely established. When the aberration was first discovered, it was thought that the velocity of light, as inferred from the eclipses of the satellites of Jupiter, did not per- fectly agree with it. It has, however, been found, from more accurate comparisons, that they per- fectly coincide. 215. It appears, that the light of the heaven- ly bodies- traverses the spaces between them and the earth with the same uniform velocity. The reflected light from the satellites, travels with the same velocity with the direct light of the fixed stars ; and the velocity of this last is the same from whatever distance it comes. There is reason to think, that light is accelerated by the action of transparent bodies. If so, these bodies also impart to light a velocity in the direc- tion of the earth's motion, and proportional to the increase ASTRONOMY. increase of velocity which they produce, because their action does not change the quantity of the aberration. 216. The aberration considered above, is that of a fixed star, or of an immoveable body. In the planets, there is another source of aberra- tion, in the motion of the planet itself, equal to the variation of its geocentric place, in the time that light takes to move from the planet to the earth. The simplest way of making allowance for this varia- tion, is to compute the place of the planet for an instant preceding the given instant, by the time that light takes to move from the planet to the earth. This w£$ the method used by Dr MASKE- LYNE. VlNCE, Ast. vol. I. § 527. No attention has been paid here to the velocity which places on the Earth's surface derive from the mo- tion of the Earth on its axis. This is too small to produce a sensible effect, as is shewn by DE LAMBRE, Astron. legon 19. J 23. Nutation of the Earth9 s Axis. 217. A small inequality, which has been ob- served in the precession of the equinoxes, and in the mean obliquity of the ecliptic, is known by the name of the Nutation. This $66 OUTLINES OF NATURAL PHILOSOPHY. This inequality, which affects all the heavenly bodies^ was discovered by Dr BRADLEY, while employed in verifying his theory of the aberration. The period of the changes of this inequality was ob- served to be eighteen years nearly, the same with the period of the revolution of the Moon's nodes ; and it was soon found, that the quantity of the in- equality depended on the place of the node. Cer- tain theoretical considerations thus led to the dis- covery of the precise form which this inequality assumes, when analytically expressed, and reduced into a formula. It is not certain that observation alone would have led to this conclusion ; but there is no doubt that the places of the heavenly bodies, when corrected by means of it, agree much better with observation than when the correction is not applied. 218. The phenomena of the, nutation maybe represented by supposing, that while a point, which may be considered as defining the mean place of the pole of the equator, describes a cir- cle in the heavens, round the pole of the eclip- tic, at a distance from it equal to the mean ob- liquity of the ecliptic, and with a retrograde mo- tion of 50" annually ; another point, represent- ing the actual pole of the equator, moves round the former at the distance of 9", so as to be al- ways 90° more easterly than the Moon's ascend- ing ASTRONOMY. 20? ing node. The inequality thus produced in the precession of the equinoxes, and in the obliqui- ty of the ecliptic, will exactly agree with the appearances as observed. It was by this construction, that Dr BRADLEY repre- sented the irregularities he had observed. From the oscillatory motion thus ascribed to the pole of the equator, or to the axis of the Earth, the name of nutation is derived. 219. If N be the longitude of the moon's as- cending node, + 9". 6 cos N is the variation in the obliquity of the ecliptic produced by the nu- tation, and — IT' .94*6 sin N is the variation in the precession, or in the equinoctial points. The last of these equations affects the longitude of all the heavenly bodies equally. DE LAMBRE, Abrege d'Ast. legon 20. § 16. 220. If A be the right ascension of a star, and D its declination, the nutation in right as- cension is tan (D — 8".373 cos (A — N) — 1".227 cos (A + N)) ; and the nutation in declination is + 8".373 sin (A — N) + 1".227 sin (A + N). The 208 OUTLINES OF NATURAL PHILOSOPHY. The effect on the declination, therefore, does not in- volve the declination itself, and is the same for all stars having the same right ascension. The inequalities here considered, do not affect the position of the stars relatively to one another, nor to the plane or pole of the eclip- tic ; they affect thgir position only relatively to the plane of the equator, or to the position of the Earth's axis. It was from this general fact> combined with the re- lation observed between these inequalities, and the motion of the moon's nodes, that BRADLEY was led to the construction given above. In separating between the inequalities belonging to the nutation, and those depending on the aberration, (which ob- servation always presented as combined together), he has displayed great acuteness and sagacity* See WOODHOUSE, Astron. chap. 17. SECT. ASTRONOMY. 209 SECT. XIIL DIMENSIONS OF THE SOLAR SYSTEM. • HITHERTO, the distance of the Sun from the Earth has served as the unit, by which we have measured all other distances in the planetary sys- tem. It now remains, (in order to have a precise idea of those distances), to compare this unit with the diameter of the Earth, and of consequence with the known measures in which that diameter has already been expressed. This depends on the parallax of the Sun, which has been shewn to be less than 10", and, on account of its small- ness, difficult to be ascertained. The method which first presents itself, does not lead to any thing more precise than the limit just mentioned. Since the ratios of the distances of the planets from the Sun, to the distance of the Earth from the Sun, are known, if the parallax of any of the planets were discovered, that of the Sun would, of consequence, become known. This follows, readily, from the laws of the planetary motions. VOL. II. O Mars, OUTLINES OF NATURAL PHILOSOPHY. Mars, when in opposition to the Sun, is nearer the Earth than any other of the superior planets, and his parallax of consequence is the greatest. From the opposition of October 1751, above referred to, and observed by LA CAILLE, at the Cape, the pa- rallax of Mars was determined to be 24/'.6 ; and his distance from the Earth 3371 semidiame- ters of the latter. But, from the place of Mars in his orbit, his distance from the Earth at that time was A354* of the parts of which the Sun's distance from the Earth is 1 ; so that the parallax of the Sun is equal to that of Mars, multiplied by the above decimal, and is therefore 10".69; and therefore the distance of the Sun is 19226 semi- diameters of the Earth. This, however, cannot, any more than the former determination, be considered in any other light than as a limit, which the Sun's distance probably exceeds. The planet Venus, at the inferior con- junction, approaches nearer to the Earth than Mars does in opposition ; and, therefore, if Ve- nus can then be observed, which she may be when she passes over the disk of the Sun, that observation will afford the best means of ascer? taining the gjun's parallax. '^M";: '"' ; ""o- 'li"This ASTRONOMY. This was first remarked by Dr H ALLEY, and since that time two transits of Venus over the Sun have happened, which astronomers have taken the greatest pains to observe with accuracy. The general principle which connects this method of finding the parallax with the more elementary me- thods already explained, is all that can be given here. The details of the calculations must be left to the Treatises and M&noires which treat of them particularly. . Let S be the Sun (fig, 21,) and E the Earth, both supposed at rest, while Venus at V moves westward in her orbit, with the sum of her own angular velocity and that of the Sun. Let O and O' be the stations of two observers on the surface of the Earth, who see the transit begin when Venus is at the points V and V of her orbit. If the difference of the longitude of the two observers be known, the time that Ve- nus has taken to move over the arch V V is also known, and therefore the arch VV is given, or the angle which the line OO' subtends at the distance of the Sun. But OO' itself is given, and its ratio to the radius of the Earth ; there- fore the angle which that radius subtends at the Sun, or the horizontal parallax of the Sun, is given. This OUTLINES OF NATURAL PHILOSOPHY. This construction supposes the observers Q and O' to be either exactly, or nearly in the plane of the or- .;- bit of Venus, but it may be extended to cases in which that condition does not take place. Jt re- quires, too, that the longitude of the places of ob- servation should be accurately known. To avoid the necessity of this determination, the durations of the transit, as seen from different stations, have been preferred, for ascertaining the parallax. If we suppose observers, situated in respect of one another, so that the line which Venus is seen to de- scribe on the Sun's disk, is longer at the one sta- tion than the other-, the duration of the transit will be proportionally greater, and the difference will evidently depend on the distance of the obser- vers from one another, estimated in the direction perpendicular to the lines which Venus traces out on the surface of the Sun. The differences of du- ration, therefore, depend on the parallax of the 3un, or on a. function of it; and therefore when that function is known, the parallax may be infer- red, from the comparison of one value of it with another. See WOODHOUSE, Astron. p. 378, &c. The transit of 1769 was observed at Wardhus or the North Cape, and also at Otaheite in the South Sea, and was found to be longer at the former than at the latter by 23'. 10". This difference, suppo- sing the parallax to have been 8".83, should have amounted to 23'.26/;.95 ; and hence the parallax is deduced = 8".72. VINCE, Ast. i. § 622. Dr MAS- KELYNE'g ASTRONOMY. 213 KELYNE'S method of calculating the parallax from the duration of the transit, is there given. See also on this subject, LA LANDE, Astron. vol. n. liv. xr. 225. From a mean of the observations of the transit of 1769, taking it as calculated by SHORT, EULER, &c. the parallax of the Sun, for his mean distance, has been found to be 8". 7 3 ; and the mean distance, therefore, -- 23659 semidia- meters of the Earth, or 93595000 miles. VINCE, Astron. vol. I. § 629. Thus the scale is determined according to which the distances of the planets have been already set down, * - SECT. OUTLINES OF NATURAL PHILOSOPHY* SECT. XIV. OF THE ANNUAL PARALLAX AND DISTANCE OF THE FIXED STARS. • 226. THE fixed stars, as has been already 6hewn, have no parallax with respect to this Earth, or any line that can be measured on its surface ; and their distance is so great, that it is yet doubtful whether they have any parallax, even with respect to the orbit of the Earth round the Sun. A fixed star not only occupies exactly the same place in the heavens, from whatever point of the Earth's surface it is observed, bat it does so within a quan- tity so small as to be hardly measurable, even when viewed from opposite extremities of a dia- meter of the Earth's orbit, On the supposition that the star does change its si- tuation, when so viewed, the angle which measures that change is called the Annual Parallax of the Star. 227. If ASTRONOMY. If a fixed star had an annual parallax that was sensible, it would appear to describe an ellipsis, of which the greater semi-axis was equal to that parallax, and the semi-conjugate equal to the same, multiplied into the sine of the latitude of the star. The centre of this ellipse would be the place of the star, seen from the Sun ; the conjugate would have the direction of a circle of latitude, passing through the star and the pole of the ecliptic. . IfL be the longitude, and X the latitude of a star, of which the annual parallax, when a ma- ximum, is p j then, if the longitude of the Sun is S, the parallax of the star, in latitude, will be — p sin X cos (L — S) j and its parallax in longitude _£-sin(L — S). COSX DE LAMBRE, Abrege, legon 19. § 29. The effects of aberration and of parallax, are both to make the fixed stars describe ellipses, but quite different, in magnitude and position. BOSCOVICH, however, has demonstrated, that a fixed star, under the influence of both these causes, will still appear to describe an ellipsis about its true place. Di$$ertati& OUTLINES OF NATURAL PHILOSOPHY. Dissertatio de cfanuisjixarum aberrationibus. Ro- nwe, 1742. LA LANDE, Astron* § 2857. . - , . ,TT. , 229. When two stars appear very near to one another, or when their distance subtends a very small angle, the variations in that angle, at op- posite seasons of the year, may serve to deter- mine the parallax of the fixed stars. 1 This method of ascertaining the distance of the fixed stars, seems to have been first thought of by GALI- LEO, Syst. Cos. Diak 3. Pr HERSCHEL has also recommended it ; and the double stars which he has discovered, as well as the Lamp Micrometer which he has invented, give hopes that it may prove successful. 230. From the consideration of the quantity of the light of the fixed stars, compared with the quantity of the light of the Sim, it has been concluded, that the parallax of a star of the se- cond magnitude, is not more than f of a se- cond ; and of a star of the sixth magnitude, not more than TO- or TV of that quantity. The very ingenious paper in which these conclusions are dedbced, is by the Reverend Mr MITCHELL. Phil Trins. vol. Ivii. p. 234, &c. • ) : >qq« 281. The ASTRONOMY. . The attempts of astronomers to discover the annual parallax of the fixed stars by direct observation, have not yet been perfectly suc- cessful ; and it is therefore probable, that the quantity sought for does not exceed one se- cond. Dr BRINKLEY is of opinion, that he has ascertained the annual parallax of * Lyres to be 2".52. PhiL Trans. 1810, p. 204-. The attempts to discover the parallax, have chiefly been made on smaller stars, which, being proba- bly more distant than * Lyree, their parallax has been too small for observation. The observation of Dr BRINKLEY is of great authority, but, while single, cannot be considered as perfectly decisive. If we suppose the annual parallax not to exceed 1", the distance of the fixed stars cannot be less than 206265 times the radius of the Earth's orbit. As light traverses the latter in 8'. 13", it will require 3 years and 79 days to come from a fixed star to the Earth. or : This may be supposed true for stars of the second magnitude, even if those of the first have a paral- lax of two seconds. Though this distance is immense, it is probably small compared to that of the most remote of the bodies which 218 OUTLINES OF NATURAL PHILOSOPHY. which we see in the heavens. As it cannot be doubted, that the fixed stars are luminous bodies like the Sun, it is probable that they are not near- er to one another than the Sun is to the nearest of them. When, therefore, two stars appear like a double star, or very near to one anbther, the one must be placed far behind the other, but nearly in the same straight line, when seen from the Earth. The same must hold, at least in a certain de- gree, wherever a great number of stars are seen concentrated in a small spot. In the starry nebu- lae, therefore, such as the Milky Way, which de- rive their light from the number of small stars, ap- pearing as if in contact with one another, it is plain, that the most distant of these must be many thousand times farther off than the nearest, and light must, of course, require many thousand years to come from them to the Earth. The poet, per- haps, has been taxed with exaggeration, who spoke of " Fields of radiance, whose unfading light " Has travelled the profound six thousand years, " Nor yet arrived in sight of mortal things." Yet the fields which he describes, are far within the circle to which the observations of the astrono- mer extend ! AP- ASTRONOMY APPENDIX. ON THE METHOD OF DETERMINING BY OBSERVA- TION, THE CONSTANT CO-EFFICIENTS IN AN AS- SUMED OR GIVEN FUNCTION OF A VARIABLE QUANTITY. Observation makes known the places of the heavenly bodies only for instants, separated from one ano- ther by certain intervals of time ; but the purposes of science require, that they should be determined for any time whatever, or for every instant. This is rendered practicable by the consideration, that the magnitude which determines the place of any body, its longitude or latitude for example, is a variable quantity, between which and the time (a variable quantity also) a certain relation continues always the same. This relation may therefore be expressed by an equation, which, for every value of one of the variable quantities, will give the cor- responding value of the other; so that the one of them may be assigned in terms of the other, or in what is called a Function of it. The form of this function is sometimes known from theory or ana- logy, and at other times it is wholly unknown. In the 220 OUTLINES OF NATURAL PHILOSOPHY. the latter case, we must assume the simplest func- tion that can represent the observations, and this naturally consists of a series of terms, proceeding according to the powers of one of the variable quantities, with co-efficients which areconstant, but unknown quantities, to be found from the observa- tions. This is called the method of Interpolation, because it inserts a term in the midst of a number of others. 232. If oc and y are two variable quantities, of which several values have been determined from observation ; if y be assumed equal to a series of the powers of x, beginning from 0, and going on to as many terms as there are obser- vations, viz. y = A + B x + C a? -f D #*, &c. ; then, if for y and x, be put their corresponding values, as determined by observation, as many equations will arise as there are unknown co-ef- ficients, A, B, C and D to be found, from which they will become known. The most useful interpolations are, when the time is one of the unknown quantities, and when the in- tervals between the observations are equal, as is supposed in what follows : rr 233. Let a, a, a', a", a", be any number of quantities determined from observations made ; Dflfi c;»15 , dl ASTRONOMY. at the times 0, m9 %m, 3m, 4 we have .500 a + .866 b = + .318 .707 a + 1.000 b = — .334 .866 a + .866 b = — .083 .966 a + .500 b = + .044 1.000 a + 0 = + .05 .966 > — .5006 ;=— .021 .866 a — .866 b = — .084. By adding all these equations together, 5.971 a + 1.0766 = —.299. Then, ordering all the seven original equations, so that the terms involving b shall be affirmative, we obtain 2.207 a -f 4.598 b = + .1. From these, a = — .0527, and b = + .047^ This ASTRONOMY. This method of determining the co-efficients of a gi- ven function, or correcting them from observation, by means of what are called Equations of Condi- tion, is said to have been invented by TOBIAS MAYER of Gb'ttingen, and employed in the con- struction of his Lunar Tables. He has, how- ever, given no account of it in any of his works ; and DE LAMBRE says, that his own Astronomical Tables are the first in which it is certainly known to have been used. Astronomic, torn. n. § 85. It is now generally adopted by astronomers, and may be introduced with great advantage into many experimental investigations *. It has also been improved on by LE GENDRE, who determines the quantities sought, so that the sums of the squares of the errors shall be a minimum. The investigation is however too difficult to be in- troduced into an elementary treatise. OUT- • Any equation, expressing the relation that obtains among the co-effi- cients of another equation, is called an Equation of Condition, ; the phrase, however, is usually confined to differential equations. OUTLINES OF NATURAL PHILOSOPHY, ASTRONOMY. PART II. PHYSICAL ASTRONOMY. SECTION L OF THE FORCES WHICH RETAIN THE PLANETS IN THEIR ORBITS* £36. IF a body gravitating to a fixed centre, have a projectile motion impressed on it, in a line 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. Princip. Math. lib. i. prop. 1. «. Conversely, 230 OUTLINES OF NATURAL PHILOSOPHY. a. 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. b. 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 at those points. 9. By comparing this proposition with the first of KEPLER'S laws, it follows, that the primary planets all gravitate to the Sun, and that the secondary planets gravitate, each to its primary ; and thus the laws of Dynamics are immediately extended to the motions of the heavenly bodies. d. If the curve in which the body moves returns into itself, it is called an Orbit, as in the case of the pla- nets ; if it does not, or if it is not known that it does it, is called a Trajectory. 237. If of two bodies gravitating to the same centre, one descend in a straight line, and the other revolve in a curve ; then, if the velocities of these bodies are equal in any one case, where they are equally distant from the centre, they will always be equal where they are equally distant from it. Princip. lib. i. prop. 40. *. If, PHYSICAL ASTRONOMY. 231 a. If, instead of descending, the body that has the rectilineal motion should ascend from the centre, while the revolving body retires from it, the same thing holds of their velocities. b. When in these propositions a body is said to gra- vitate to a centre, or to be urged toward it by a centripetal force, no supposition is implied con- cerning the nature of that force, or the cause from which it proceeds. Nothing more is understood, than the mere existence of a fact, like the tenden- cy of bodies to fall at the surface of the Earth. 238. If a body descending from rest at a gi- ven point in a straight line, toward another gi- ven point to which it gravitates, have its velo- city always proportional to the square root of its distance from the first of these points, divi- ded by the square root of its distance from the second, it is urged toward this last point, by a force which is inversely as the square of the dis- tance from it. a. Let A, fig. 22. be the point from which the body falls, C the centre to which it tends ; let AD, per- pendicular to AC, be equal to a given line ; and as the centripetal force at A to the centripetal force at any point B in the descent of the body, so let AD be to the perpendicular BE, and let DEF be the curve which passes through the ex- tremities of all the perpendiculars so drawn ; then twice 232 OUTLINES OF NATURAL PHILOSOPHY. twice the area ABED is equal to the square of the velocity at B, (§ 100. vol. i.). Let this velo- city = 0; having bisected AC in G, let the velo- city in G be == c. Draw the perpendicular b e indefinitely near to BE. Then t>* : c* : : 2 ABED, therefore 2 ABED = c* X ~, or since AB=AC— CB, 2ABED=c*X .dnjjva i>flj = c*r^P— i\ VBQ j xAC \ For the, same reason, 2 A £ e D=e* X ( ™ — 1 ), and therefore the area 2 B & * E=c* ( — - wherefore, 2 B 6 X BE = c» , and dividing by B *, 9 BE = c» X =c» X . AG therefore, BE sr c* X . But BE represents the centripetal force at B any point in the line AC, and both c and AG are given, therefore the centripetal force at B is inversely as the square of BC, the distance from the centre of force. Therefore, PHYSICAL ASTRONOMY. 333 Therefore, if the velocity of a body attracted to a centre, be as the square root of the distance it has fallen through, divided by the square root of the distance between it and the centre, the centripe- tal force by which it is impelled is inversely as the square of its distance from that centre. Hence also, if the centripetal force at G be = F, and Afr r* if CG = «, F = e X TJ s= 1, and *F = c\ £39. If to a tangent of an ellipsis, a perpendi- cular be drawn from either focus, the distance of that focu3 from the point of contact will be to the distance, of the, other focus from the same point, as the square of the perpendicular drawn to the tangent, to the square of the seini-con- jugate axis. «. Let P (fig. 23.) be a point in the ellipsis ADBE, AB the transverse, DE tfie conjugate axis, C the centre, 8 and F the foci ; GPH a tangent to the ellipsis in P, SG the perpendicular on it from S. Draw SP, PF, and make FH perpendicular to GP. Because the angles SPG, FPH,are equal, from the nature of the ellipsis, the triangles SPG FPH are equiangular, and therefore SP : PF : : SG : FH, and SP : PJF : SG* : FH x SG. But OUTLINES OF NATURAL PHILOSOPHY. But FH X SG = CDZ, therefore SP : PF : : SG2 : CD*. 7 „ CD* PF 1. Hence . If a body urged by a centripetal force, directed to a fixed point, describe an ellipsis of which that point is a focus, the centripetal force must be inversely as the square of the dis- tance. Suppose a body falling in a straight line to the focus S, (fig. 24.) to have at A the same velocity that the body revolving in the ellipsis has at A. Let P be any other position of the revolving body, and with the radius SP let an arch of a circle be de- scribed, meeting AB in L ; the velocity of the fall- ing body at L, and of the revolving body at P, will be equal. If, then, the velocity of the revolving body at the mean distance, or at the point D, be called c, and its velocity at P be v, v : c : : CD : SG, (§ 236, £.), or * = , and *» = PF PTT X ^p, (j 239, a.) Therefore also » =c JEL . Now PHYSICAL ASTRONOMY. Now v is the velocity of the falling body at L, as well as of the revolving body at P ; therefore if LN be taken equal to FP, the velocity of the fall- ing body atLis = c i/ But because SL is equaltoSP,andLNtoPF,SN=SP+PF=AB, so that the point N is given. The centripetal force at L is therefore as ^Y~' $ 238., that is, as oiu — ^ , or inversely as the square of the distance. Q. E. D. The forces, therefore, which make the planets describe ellipses having the Sun in their common focus, are inversely as the squares of the distances from the centre of the Sun. a. As the same reasoning that is here applied to the ellipse, might be applied^to the hyperbola or the parabola, therefore the force tending to one of the foci, which is requisite to make a body describe a conic section, is inversely as the square of the dis- tance. b. The converse of this is true, viz. that if the cen- tripetal force be inversely as the square of the dis- tance, the body will describe a conic section. c. The 236 OUTLINES OF NATURAL PHILOSOPHY. c. The line NS, (fig. 24.) through which a body fall- ing toward S will have, at any distance from S, the same velocity that the revolving body has at that same distance, is equal to the transverse axis of the ellipsis ; and the velocity of the revolving body at D, its mean distance, is equal to that which is ac- quired by falling from N to K, the middle point between N and S. If d be a given distance from S9 and/ the centripetal force at that distance, be- eause - But a F = c% therefore d*f= a c* and c = ^L-, a* 242. If bodies describe different ellipses about a centre to which they are urged by centripetal forces in the inverse ratio of the squares of the distances, the squares of the times of re- volution will be as the cubes of the mean dis- tances* dfk The velocity c, at the mean distance a, is J . . , and a * therefore the sector described in a second by the radius vector, when the body is at its mean dis- tance, is — = — ~— y and as oft as this is con- 2a* tained PHYSICAL ASTRONOMY. tained in the whole elliptic area, of so many seconds does the time of the revolution consist. Now the area of the ellipsis of which a is the semi- transverse, and b the semi-conjugate, is «• a b ; therefore the time of revolution =r *• a b X ~ _ = ^ • b df* v * r, the value of a is affirmative, and the conic section is an ellipsis ; and this ellipsis has d* f its higher apsis at A, if v* ^^ -4- ; but when w* d* f 2 d* f is between the limits of — and - ^-, the lower r r apsis is at A. d. When PHYSICAL ASTRONOMY. 239 d. When v goes beyond this latter limit, or when tr* r -^ 2 d^f, the value of a is negative, and the trajectory becomes a hyperbola. 244. From the fact that action is always ac- companied by re-action, we conclude that gra- vitation among terrestrial bodies is the mutual tendency of the particles of matter to one ano- ther. It is therefore reasonable from analogy to suppose, that this is true in all cases, and that the force of gravitation toward different bodies, the distances being the same, is propor- tional to the quantities of matter, or the masses of the bodies. If the mass of a body be call- ed m, the gravitation to it at any distance x will be — . x* . Hence, in the formulas above, m may be inserted instead of d*f. , The quantities of matter in any two prL mary planets, are directly as the cubes of the mean distances at which their satellites revolve, and inversely as the squares of the periodic times of those satellites. a. This is proved by substituting m a for df*9 in the formula t = — — — ; we have thence OUTLINES OF NATURAL PHILOSOPHY. from which the proposition fallows readily. 1. By means of this theorem, the masses of the and of the plnnets Tvhich have satellites, may be compared with one another. c. In the Meccmiqttg Celeste^ they are calcalated front the most exact data, as below t Quantity of matter in the Sun being = I That in the Earth, = As the ratios of the Diameters of the planets are known from observation, the ratios of their Bwlks» being the same with those of the cubes of their diameters, are also known ; and hence the Densi- ties, which are proportional to the quantities of matter, divided by the bulks, are found. Density PHYSICAL ASTRONOMYc Density of the Sun, = 1 - of the Earth, = 3.9393 - of Jupiter, = 0.8601 - of Saturn, = 0.4951 — - of Uranus, = 1.1376 246. The immoveable point to which the pla- nets gravitate, is not the centre of the Sun, but the centre of gravity of the solar system. From the equality of action and re-action, the gravi- tation of the planets to the Sun must be accompa- nied by the gravitation of the Sun to the planetSj, so that the quantity of the motion of the former, estimated in any direction, must be equal to that of all the latter estimated in the opposite. The Sun, therefore, moves in an orbit, about the only point of which the condition cannot be disturbed by the mutual action of the surrounding bodies, viz. the centre of gravity of the whole. If there were only one planet, the Sun and that pla- net would describe similar conic sections, of which their common centre of gravity would be one of the foci ; their distances from that point being al- ways inversely as their masses. If there be a num- ber of planets, the path of the Sun will become a more complicated curve, but will be such as to furnish a centrifugal force in respect of each pla- net, just able to counteract the gravitation toward it. VOL. II. Q 24?. The OUTLINES OF NATURAL PHILOSOPHY. . The centre of the Sun is never distant by so much as his own diameter from the centre of gravity of the system. The diameter of the Sun is equal nearly to .009 of the radius of the Earth's orbit. Now, if we sup- pose the Sun, and all the great planets of the sys- tem, Jupiter, Saturn and Uranus, to be in a straight line, and the planets all on one side of the Sun, the centre of the Sun will be nearly the far- thest possible from the centre of gravity of the whole; yet we shall find on computation, that the distance is not greater than .0085 of the radius of the Earth's orbit. NEWTONI Princip. lib. in. prop. 12. . In the preceding propositions the ex- istence of the principle of gravitation, has been established by induction from the laws of KEP- LER ; and from that principle, by reasoning downward, conclusions have been obtained con- cerning the quantity of matter in the planets, to which observation, without the assistance of theory, never could have reached. It yet remains to be shewn, that the same force which occasions the descent of heavy bodies on the Earth's surface, at the rate of 16.09 feet per second, when diminished in the inverse ratio of the square of the distance, is just sufficient to re- tain the Moon in her orbit. The PHYSICAL ASTRONOMY. %4tS The siderial revolution of the Moon is 27a.32166, § 124. and her mean distance 59.879 semidiame- ters of the Earth. From this, on the supposition that the squares of the periodic times are as the cubes of the distances, the period of a body pro- jected, so as to describe a circle round the Earth, will be found lh.41516. From this the arch which the body describes in one second, is found to be 4/.13".48'"; and the deflection from the tangent in one second, if reduced to feet, comes out 16 and a small fraction, the same with the descent of a heavy body in one second, at the surface of the Earth. NEWTONI Princip. lib. in. prop. 4. On the subject of this section, see NEWTONI Priricip» Math. lib. ImuSj sectio 2da, 3tia, 7ma, 8va, lima. JOH. BERNOUILLI, Opera> torn. ImuSj NO. 86. EULERI Mechanica, torn. i. cap. 5tum. Particularly prop. 80, 81. FRISII Opera, torn. Stius, Cosmographia, lib. Imiw. Dr MATHEW STEWART, Tracts Phys. fy Math. Tract 1. The subject is there treated in a man* ner strictly geometrical. LA BANDE, liv. 2de, torn. m. VINCE, Astron. chap. 31. vol. u. Theorie des Mouvemens des Planetcs, par M. LA PLACE, lrepartie, Paris, 1784. SECT. 244 OUTLINES OF NATURAL PHILOSOPHY. SECT. II. OF THE FORCES WHICH DISTURB THE ELLIPTICAL MOTION OF THE PLANETS. VV HEN there are only two bodies that gravitate to one another, with forces inversely as the squares of their distances, it appears from the last section that they move in conic sections, and describe, about their common centre of gravity, equal areas in equal times, that centre either remaining at rest, or moving uniformly in a straight line. But if there are three bodies, the action of any one on the other two, changes the nature of their orbits, so that the determination of their motions be- comes a problem of great difficulty, distinguish- ed by the name of THE PROBLEM OF THE THREE BODIES. The solution of this problem, in its utmost generali- ty, is not within the power of the mathematical sciences, as they now exist. Under certain limita- tions, however, and such as are quite consistent with the condition of the heavenly bodies, it ad- mits of being resolved. These limitations are, that the force which one of the bodies exerts on the other two, is, either from the smallness of that body, or its great distance, very inconsiderable, in respect PHYSICAL ASTRONOMY. respect of the forces which these two exert on one another. The force of this third body is called a disturbing force, and its effects in changing the places of the other two bodies are called the disturbances of the system. Though the small disturbing forces may be more than one, or though there be a great number of remote disturbing bodies, the computation of their combined effect arises readily from knowing the effect of one ; and therefore the problem of Three bodies, under the conditions just stated, may be extended to any number. Two very different methods have been applied to the solution of this problem. The most perfect is that which embraces all the effects of the disturbances at once, and by reducing the momentary changes into fluxionary or differential equations, proceeds, by the integration of these, to determine the whole change produced in any finite time, whe- ther on the angular or rectilineal distance of the bodies. This method gives all the inequalities at once, ancl as they mutually affect one another. The other method of solution is easier, and more elementary, but much less accurate. It supposes the orbit disturbed to be nearly known, and pro- ceeds to calculate each inequality by itself, inde- pendently of the rest. It cannot, therefore, be exact,, OUTLINES OF NATURAL PHILOSOPHY. exact, and gives only a first approximation to the quantities sought ; but being far simpler than the other, it is much better suited to the elements of science. It is also the original method, and that which was first applied by Sir ISAAC NEWTON to , explain the irregularities of the Moon's motion. The same has been followed and improved by CA- LENDRINI, in his Commentary on the third Book of the Principia ; by FRISI in his Cosmographia ; and by VINCE, in the second volume of his Astronomy. The other method was not invented till several years later, when it occurred nearly about the same time to the three first geometers of the age, CLAIRAUT, EULER, and D'ALEMBERT. It was followed also by MAYER, and several others, but particularly by LA PLACE, who, in the Mecanique Celeste, has gi- ven a complete investigation of the inequalities both of the primary and secondary planets. I shall explain the resolution of the forces that is in some measure common to both methods ; and shall shew how their effects are to be estimated in some simple instances, going from thence to the enumeration of the results. I begin with the Moon's irregularities, as the easiest case of the problem. 249. The Moon, in her motion round the Earth, is disturbed by the action of the Sun ; her gravity to the Earth is increased near the quadratures, PHYSICAL ASTRONOMY. quadratures, and diminished both at the oppo- sition and the conjunction, and the areas de- scribed by the radius vector, except near the quadratures, are never exactly proportional to the times. a. Let ADBC (fig. 25.) be the orbit, nearly circu- lar, in which the Moon M revolves round the Earth at E. Let the Sun be at S, and let the line SE denote the force of the Sun on the Earth •, then SE3 •c-\ift *s tne attracti°n of the Sun on the Moon at M ; let MG be equal to this line, and joining ME, let the parallelogram KF be described, of which GM is the diagonal, and the sides MF, MK are in the directions of ME and ES; and let HS be drawn parallel to ME. The force MG may be resolved into the two MF, MK, of which MF, directed toward the centre E, 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 direc- tion of the line joining the centre of the Sun and Earth ; but it is only the excess of this force above that by which the Sun draws the Earth, that dis- turbs the relative position of the Moon and Earth. If, therefore, ES, which denotes the force of the Sun on the Earth be taken from MK, the remain- der KH will be the force by which the Sun draws OUTLINES OF NATURAL PHILOSOPHY. draws tne Moon from the Earth, in the direc- tion of the line passing through the centres of the Sun and Earth. This force at the conjunction, exceeding that part of the disturbing force which draws the Moon to the Earth, tends to diminish the Moon's gravity to the Earth. At the opposition B it does the same, by becoming negative ; for the Earth is then drawn more than the Moon, and the difference is nearly the same as at the conjunction. t>. If MN be taken equal to HK, and NO made per- pendicular to the radius vector, the force MN is resolved into two, one directed from M to O, les- sening the gravity of the Moon to the Earth, and the other directed from O to N, parallel to the tangent to the Moon's orbit at M, and therefore accelerating the Moon from C to A, retarding her from A to D, and so alternately in the other two quadrants. At the quadratures C and D, the force HK vanishes, and the only remaining force is directed to the centre of the Earth, so that the areas are there proportional to the times. c. The analytical values of these forces are next to be found. Draw CED the line of the quadratures, put SE = the investigation would lead to nothing, as a would disappear from the equation ; but if it be r* carried to the terms that involve — , an equation a may be found from which a can be determined. Dr STEWAE/?, therefore, having first compared the mean disturbing force with the gravity of the Moon to the Earth, from the motion of the ap- sides, proceeded to determine the ratio of the former force to the force that retains the Earth in its orbit, by carrying the approximation to the terms that involved the square of the Sun's dis- tance, and from that computed the Sun's parallax at 6".9. Sun's Distance, prop. 10. This value of the parallax is no doubt too small ; it is only an inconsiderable part of the disturbing force of the Sun, into which his distance enters as an element, and therefore any deduction founded on it must be liable to error. MAYER has also sought to determine the Sun's pa- rallax from one of the lunar equations, as deduced from the solution of the problem of the three bodies. The co-efficient of this equation involves the Sun's parallax, and gives it equal to 7 ".8, and MAYER thinks that the error cannot exceed a 24-th part. Theoria Lunae, § 51. 263. The OUTLINES OF NATURAL PHILOSOPHY. %63. The Tables of the Moon's motions in the state to which they are now brought, contain 28 equations for the longitude of the Moon j 12 for the latitude, and 13 for the horizontal parallax, or the distance of the Moon from the Earth ; and their greatest probable error does not exceed 12 seconds. It is very much to the introduction of the equation of the period of 85 years, which nobody before LA PLACE had discovered, that the superior accu- racy of the New Tables is to be ascribed. The observations at Greenwich have afforded the data from which the theory of gravity, and the integral calculus have extracted these memorable results. SECT, PHYSICAL ASTRONOMY, SECT. III. DISTURBANCES IN THE MOTIONS OF THE PRIMARY PLANETS PRODUCED BY THEIR ACTION' ON ONE ANOTHER, IT is necessary, in this inquiry, to know the quanti- ties of matter in the different planets ; and these have been already calculated for the planets which have satellites. The masses of Venus and Mars have been computed by M. LA PLACE from some disturbances which they appear to produce on the Earth's motion. The mass of Mercury has been estimated, from supposing his density, and that of the Earth, to be inversely as their mean distances from the Sun. This law holds with respect to the Earth, Jupiter and Saturn, and analogy au- thorises the extension of it to Mercury. From knowing the density and the bulk, the quantity of matter is inferred. The mass of the Sun being 1, that of Mercury is •, of Venus * 2025810 38313?' of Mars ] 84?60gg» those of the other being as already stated, § 24-5* The OUTLINES OF NATURAL PHILOSOPHY. The gravitation of one planet to another, is express- ed by the quantity of matter in each, divided by the square of the distance ••> and therefore the two bodies tend to come together with a force that is as the sum of their masses divided by the square of the distance ; so that when the motion of both is referred to one only, the force must be express- ed by the sum of the masses divided by the square of the distance* The disturbances produced by the action of the pri- mary planets on one another, are of more difficult investigation than those produced by the Sun on the motions of the Moon, because the disturbing body is not at an immense distance, as in the lat- ter case. The only sure way of subjecting them to calculation, is by a direct solution of the Pro- blem of the Three Bodies ; the part of which that may be accounted quite elementary is now to be considered. ;fo vlt 264. The forces which act upon a body, to however many centres they tend, and whatever law they may obey, may be resolved into the •directions of three lines or axes, given in po- sition, at right angles to one another. This is evident from Dynamics, vol. i. § 70. The advantage of this resolution of forces for deter- mining the motion of a body attracted to several centres. PHYSICAL ASTRONOMY. 267 centres, appears to have been first suggested by MACLAURIN, Fluxions, $470. It has been very useful in the solution of the problem of The three bodies. 265. Let S be the Sun, (fig. 26.), and P and P' two planets referred to the plane of the ecliptic, each by three rectangular co-ordi- nates, PQ, QR, RS, and PQ', Q'R', R'S, pa- rallel to the three axes SO, SN, NM : let SR = x, RQ = y, PQ = z. Let all the forces that draw P in the direction of SM, or in a line parallel to #, whether arising from the action of the Sun S, or the Planet P', be denoted by F j all those in the direction of SN Q"r# by F ; and those in the direction of SO or z, by F" ; and • let t be the momentary increment or fluxion of the time ; then # = — F> y = — Fx t*9 and For from the principles of Dynamics, (1. § 100.) • 7? — F x = v v ; and (1. J 56.) v = — £08 OUTLINES OF NATURAL PHILOSOPHY. = -r, because / is supposed constant 5 there- fore v v ' = Jl X J?L = ^LfL and hence that is # = — F **. In the same manner are found the values of y and of z. On these three equations is founded the solution of the problem of The three bodies, 266. The same things being supposed, let the co-ordinates by which the planet P' is referred to the ecliptic be #', yr, z' -9 let SP = +y* + * = r, SP' = and the distance of the planets, or PP' = Jx _ xj + (y _yj + (z — zj = y. Also let m be the mass of the planet P, and m* that of F, the mass of the Sun being 1, The whole force PHYSICAL ASTRONOMY. force parallel to SM or #, that affects the rela- tive motion of P to S, is (1 -f- m)x , jg^g' i m' q r^ r/J a* x' For the action of the Sun on P being resolved as above, the part of it that is in a direction parallel to x or to SM is — ; and the action of P' on P being resolved in like manner, the part of it paral- mf (x — - x*} lei to x is — - — '. Now, supposing x to vary by the momentary increment or fluxion x9 xf remaining constant, the increment of q, or q will, be such that -¥- = x x. , and therefore x f m (x—-x) = m q ^ ^jgjj^ therefore, is the force by which P' acts directly on P ; and if to this be added the Sun's force in the same direction, viz. — , the amount of the direct action of S and P' x -' on P, in a line parallel to SM, is - + * Now, S70 OUTLINES OF NATURAL PHILOSOPHY. . Now the Sun, by the united action of the planets P and P', is drawn in the direction opposite to this - m x mr a/ Jast by a force = -75- -f — 7— , and as the Sun is here considered as immoveable, we must conceive this force to be transferred to P in the opposite direction* Thus the whole action on P, or the force F = x m x , m' x' , m' q -3 H T> 7* f^7J __ (1 + m) x , m' x? m' q ~~ ar" By writing in this formula j/, yr andj/, instead of x> 0 + and in like manner x> andi, we have F = + ? m' zf m'q -- Ji — i -- «~ r J The substitution of these values of F, &c. in the three formulas of the last article, will give three fluxionary equations, on which the motion of P depends. The same being done for P', there will come out six fluxionary equations, from the integration of which PHYSICAL ASTRONOMY* which the motions both of P and P' may be de- termined. This is the problem of the Three Bodies, as far as the principles of Dynamics are concerned. The rest depends on the integral calculus, and involves a multitude of investigations which do not belong to this place. See Memoir of LA PLACE, sur tes Inegalites de Jupiter et Saturn. Acad. des Sciences, 1785. Also Mecanique Celeste, liv. vi. What is given above, is merely an elementary expo- sition of the dynamical principles employed in these investigations ; a sketch follows of some of the most important conclusions deduced from them. 267. The place of every planet in its orbit, is changed by the action of the other planets, and the orbit itself is changed in all its elements but two, the mean distance from the Sun and the mean motion of the Planet. a. In the orbit of a planet, the line of the nodes, the inclination to the plane of the ecliptic, the line of the apsides, and the eccentricity, all vary. The lines of the nodes and of the apsides revolve conti- nually; but the inclination of the orbit and its ec- centricity are only subject to small periodical va- riations on each side of a mean, front which they never depart far. *. The OUTLINES OF NATURAL PHILOSOPHY. b. The quantity and direction of these motions, as determined by the theory of gravity, agree entire- ly with the results of observation. 268. Mercury is so near the Sun, that his place in his orbit is not sensibly affected by the action of the other planets ; but his orbit is ne- vertheless disturbed, both in its form and posi- tion. a. The disturbances of the four elements of the orbit are as in the following Table, where, instead of the secular change of the eccentricity, that of the great- est equation of the centre is set down. The first column gives the planet, to which the disturbances opposite to it in the succeeding columns are to be ascribed ; the second is the annual motion of the nodes ; the third of the apsides, both in respect of the fixed stars ; the fourth contains the secular variations of the inclination ; the fifth those of the greatest equation of the centre. Action-of Node. Aphel. Inc. Orb. Eq. Cent. ?' — 5".57 — 0 .87 — 0 .14 — 2 .18 — 0 .12 4- 4".14 4- 0 .84 4- 0 .04 4- 1 .56 4- 0 .08 4- 3".Q4 4- 0 .58 — 0 .22 — 1 .26 4- 0 .02 4- 9".46 4- 0 .86 4- 9 .87 4- 1 .04- — 8 .88 4- 6 .66 4- 2 .16 4-20.4-3 The PHYSICAL ASTRONOMY. The sign — , when applied to the node, or the aphe- lion, implies that the motion is retrograde. 6.- The motions of the node and of the apogee, given in Astronomical Tables, are different from the above, as the annual procession of the equinoxes is included. Thus the annual motion of the nodeis — 8".98 + 50".25 = + 4-1 ".2 7, in respect of the equinoxes, though it is only — 8".98 in re- spect of the fixed stars. 269. In like manner, the place of Venus in her orbit is not sensibly disturbed, but the or- bit itself is subject to change. Node. Aphel. Inc. Orb. Eq. Cent By $ + 0".16 — ±".30 + 1".91 — 9".02 ? — 7 .46 © — 6 .69 — 5 .06 — 9 .02 * ~0 .29 •f 1 .18 — 0 .42 — 0 .64 V — 5 .13 + 6 .38 + 2 .60 — 6 .16 T? — 0 .09 + 0 .08 , 0 .35 — 0 .14 — 14". 70 + 18".40| + 4".47 — 24/r.98 270. In the orbit of the Earth, the apsides move forward annually at the rate of 11".807719 in respect of the fixed stars ; and the eccentri- city diminishes, so that the secular variation of the greatest equation of the centre is — 17".66. VOL. II. S a. Of OUTLINES OF NATURAL PHILOSOPHY. a. Of the motion of the apsides, 5" is due to Venus, 1".2 to Mars, and 5". ,5 to Jupiter nearly. b. In the secular diminution of the equation to the centre, + 4/'.l 8 is the effect of Venus, — - 4/'.94< of Mars, — 16".02 of Jupiter, and the rest is produ- ced by Mercury and Saturn. . As 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, the regularity of the Earth's motion is disturbed on that account, and the Earth is forced out of the plane of the ecliptic. a. The irregularities thus communicated to the Earth are, by observers on its surface, transferred to the Sun ; the Sun, therefore, has a motion in longi- tude, by which he alternately advances before the point that describes the elliptical orbit in the hea- vens, and falls behind it ; and in like manner al- ternately ascends above the plane of the ecliptic, and descends below it. b. These inequalities are small. The mass of the Moon is about ^th of that of the Earth ; the distance, therefore, 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 in the Sun's place must be less than his horizontal parallax. c. The PHYSICAL ASTRONOMY. 275 The greatest latitude which the Sun can have, is equal to the horizontal parallax, multiplied into the sine of the Moon's greatest latitude. This cart hardly amount to a second ; it is called the men- strual parallax, and was first mentioned by Mr SMEATON, Phil. Trans. 1768. See also Mecanique Celeste, torn. in. p. 106. . The place of Mars in his orbit is sensi- bly affected by the action of Venus, the Earth, and Jupiter. The principal inequality produced by Jupiter is — 25".6 sin (Long. $ — Long, if) + 16".8 sin (Long. $ —Long, if); By the Earth, 7 ".2 sin (Long. 0 — Long. $ ) ; and by Venus, 5'. 7 sin (Long. ? — 3 Long. $). There are, besides these, some other small equations. See DE LAMBRE'S Tables in VINCE'S Astronomy^ vol. in. p. 48., &c. In the orbit of Mars, the eccentricity is diminishing. The secular variation of the greatest equation of the centre is — 37". The annual and siderial motion of the aphelion is 16".75 forward, and of the node 23".25 backward. These motions are chiefly produced by the three planets just named. 273, The 276 OUTLINES OF NATURAL PHILOSOPHY. 273. 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 con- siderable, and, on account of their vicinity, though their masses are small, they may some- what disturb the motions of one another. Their action on the other bodies in the system is pro- bably insensible. As two of these planets have nearly the same perio- dic time, they must preserve nearly the same dis- tance, and the same aspect with regard to one another. This offers a new case in the computa- tion of disturbing forces, and may produce equa- tions of longer periods than are yet known in our system. . The action of Jupiter and Saturn on one another, produces an inequality in the motion of each, of considerable amount, and of a long period, viz. 918.76 years. a. tin express a number of years reckoned from the beginning of 1750, S the mean longitude of Sa- turn, and I of Jupiter, reckoned from the same time, then the great equation that must be applied to the mean longitude of Jupiter, or to I, is + (20W.5 — n X 0".042733) X sin (5 S — 2 I + 5°.34'.8" — « X 58".88) ; and PHYSICAL ASTRONOMY, 277 and that which must be applied to S is — .(48'. 44" — n X 0".l) X sin (5 S — 2 I + 5°.34/.8" — n X 58".88). These equations are to one another nearly in the ra- tio of 3 to 7. As the quantity 5S — 2 I — n X 58".S8, requires 918.76 years to increase from 0 to 360 degrees, therefore the above equations require that period to run through all their changes. See LA PLACE, Mem. Acad. des Sciences, 1785, 1786, Also LA LANDE, Astron. torn. in. § 3670. Besides these two great inequalities, there are ten others, arising from the action of Saturn, to which Jupiter is subject, and which may amount when greatest to 11 '5 6" ; there are also six to which Saturn is subject from the action of Jupiter, and these may amount to For the particular forms of these equations, see LA LANDE, ibid, and VINCE, Astron. vol. in. p. 94-. and 102. 276. The motion of the apsides, and the change of eccentricity in the orbits of Jupiter and Saturn, are chiefly produced by the ac^ tion of those planets on one another; but in the disturbance which the planes of their orbits suffer, the other planets have a sensible effect. JUPITER. I. OUTLINES OF NATURAL PHILOSOPHY. JUPITER. Node. Aphel. Inc. Eq. Cent 8 — 0".31 — 0".95 ? —17 .56 + 0".01 —IT .67 © — 0,.01 + 0 .01 * — 0 .39 — 1 .06 — 0".02 % — 6 .95 TI + 5 ,.88 + 6 .56 — 7 .51 + 56 .28 — 19".34 + 6".58 —27".ll + 56".26 SATURN. Node. Aphel. Inc. Eq. Cent. \ — 0".ll — 8 .66 — 1 .1 —26 .65 0 $ — 0 .14 — 1 .25 It —12 .28 + 15 .99 + 5 .88 -I'. 50". 6 J — 0 .14 — 20".93 + 15 .99 23". 11 -1'.50".6 e. The eccentricity and the greatest equation of Ju- piter, are increasing, from the action of Saturn ; and those of Saturn decreasing, from the re-action of Jupiter. Their secular variations are nearly in the proportion of 1 to 2. 277* Uranus PHYSICAL ASTRONOMY. #79 277- Uranusy on account of his great dis- tance, suffers no disturbance in his motion but from Saturn and Jupiter. The principal ine- quality depends on Saturn ; which, if S be the longitude of that planet, U of Uranus, and A of the aphelion of Saturn, is £.30" sin(S— 2U+ A). For the other equations, see LA LANDE, Astron. in. § 3671. Also the New Tables^ VINCE, Astron. in. p. 106. The orbit of Uranus is also disturbed. The node moves backward at the rate 34>",25 annually, and the aphelion forward at that of 2".55. The ec- centricity increases, and the secular variation of the greatest equation of the centre is + 11 ".03. When, in the preceding Tables, a planet is repre- sented as producing a change in the place of its node, it must be understood that it does not pro- duce this effect by its action on its own orbit, but by its action on the plane of the ecliptic. 278. Comets, in describing their elliptic orbits round the Sun, have been found to be disturb- ed by the action of the larger planets, Jupi- ter 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 £80 OUTLINES OF NATURAL PHILOSOPHY. for a limited portion of time, by considering the orbit as an ellipsis, the elements of which are continually changing. This is the method of LA GRANGE, and is followed in the Mecanigue Celeste, Part u. Chap. 9. ij&mfe.sdi Dr HALLEY, when he predicted the return of the comet of 1 682, 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-appear- ance to the end of the year 1758, or the beginning of 1 759. He professed, however, to have made this calculation hastily, or, as he expresses it, levi calamo. Synop- sis of the Astronomy of Comets.1 CLAIRAUT, on calculating with great care and labour the effects both of Jupiter and Saturn, found that the return of the Comet would be retarded 511 days by the former, and 100 by the latter ; in consequence of which he foretold that its return to its perihelion would be on the 15th of April 1759- He said 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 5 thus affording PHYSICAL ASTRONOMY. a very remarkable verification of the theory of Gravity, and the calculation of Disturbing Forces. This Comet may be expected again about the year 1835. The investigations of LA PLACE will ren- der it much easier to calculate the quantity by which its arrival may be anticipated or retarded by the action of the planets. A Comet, which was observed in 1770, had a mo- tion, when carefully examined, which could not be reconciled with a parabolic orbit, but which might be represented by an elliptic orbit of mo- derate eccentricity, in which it revolved in the space of five years and eight months. This Co- met, however, had never been seen in any former revolution, nor has it been seen in any subsequent one. 280. Mr BURKHARDT, on tracing the path of this Comet, found that between the years 1767 and 1770, it had been very near to Jupiter, and again had come very near to that planet in 1779 ; he therefore conjectured, that the dis- turbance of Jupiter might have so altered its original orbit, as to render the Comet for a time visible from the Earth ; and may have so changed it again, after one revolution, as to re- store the Comet to the same region in which it had formerly moved. This conjecture has been confirmed by a careful application of the for- mulas of the Mecanique Celeste. Mr 282 OUTLINES OF NATURAL PHILOSOPHY. i Mr BURKHARDT found that the Comet had come so near to Jupiter between 1767 and 1770, that it may have been brought from an orbit of which the semi-transverse was 13.293, (that of the Earth's or- bit being 1), and in which it revolved in a period of 48.466 years, to one in which the semi-trans- verse was 3.178, and in which it revolved in five years and eight months, as it was at that time ob- served to do. While revolving in this orbit, it came near to Jupiter again ; and its time of revo- lution and its distance were so changed, that the latter became 6.388, and the former 16 years. In this orbit it cannot, any more than in its first, come so near the Earth as to be visible. The preceding is the greatest instance of disturbance that has yet been discovered among the bodies of our system, and furnishes a very happy and unex- pected application of the theory of Gravitation. . 281. Though the Comets are disturbed in so great a degree by the action of the Planets, they do not appear by their re-action to pro- duce any sensible effects. This must no doubt arise from the small quantity of matter which a Comet contains. The Comet of 1770 came so near to the Earth, as to have its periodic time increased by 2.046 days, ac- cording to LA PLACE'S computation, and if it had been* equal in mass to the Earth, it would have made PHYSICAL ASTRONOMY. 283 made an augmentation of 2h 48m nearly in the length of the year. But it is certain, the same author adds, that if an augmentation so great as even two seconds in the length of the year, had taken place at that time, it would have been dis- covered from Dr MASKELYNE'S observations, when compared by DE LAMBRE for the construc- tion of the New Astronomical Tables. The mass of the Comet, therefore, cannot have been Ti^th of the mass of the Earth. The same Comet passed through the midst of the satellites of Jupiter without producing the smallest effect. Mecanique Celeste, vol. iv. p. 250. It is reasonable, therefore, to think, that no mate- rial, or even sensible alteration has ever been pro- duced in our system by the action of a Comet. % SECT. £84 OUTLINES OF NATURAL PHILOSOPHY. SECT. IV, OF THE DISTURBANCES WHICH THE SATELLITES OF JUPITER SUFFER FROM THE ACTION OF ONE ANOTHER. THE application of the same principles to the satellites of Jupiter, has fully explained all the irregularities which had been observed in their motions, and has reduced under known laws several others, of which the existence had been indistinctly perceived. A very remarkable relation takes place between the mean motions of the first three satellites, as re- marked § 185; the mean motion of the first satel- lite -f- twice that of the third, being equal to three times the mean motion of the second, reckoning from any instant of time. LA PLACE has shewn, Mecanique Celeste, liv. n. chap. 8. that if the pri- mitive mean motion of these satellites was near this proportion, their mutual action on one ano- ther must in time have brought about an accu- rate conformity to it. It PHYSICAL ASTRONOMY. 285 It follows, that Long. 1st Sat — , 3 Long. 2d Sat. + 2 Long. 3d Sat. = a constant quantity ; and it has been found, since ever the satellites were ob- served, that this constant quantity has been near- ly equal to 180°. This last must be the result of original constitution. 283. The first satellite moves nearly in the plane of Jupiter's equator, and has no eccentri- city except what is communicated to it from the third and fourth ; the irregularities of one of these small planets producing similar irregu- larities in the rest. It has beside an inequality chiefly produced by the action of the second, and circumscribed by the period of 437.659 days. The orbit of the second satellite moves on a fixed plane, to which it is inclined at an angle of 27'. 13", and on which its nodes have a retrograde motion, so that they complete a re- volution in 29.914 years. The motion of the nodes of this satellite is one of the principal data that have been used for determining the masses of the satellites, which are so necessary to be known in computing their disturbances. This satellite has no eccentricity but what it de- rives from the action of the third and fourth. 285. The OUTLINES OF NATURAL PHILOSOPHY. 285. The third satellite moves on a fixed plane that is between the equator and the orbit of Ju- piter, and is inclined to that plane at an angle of 12'.20", its nodes making a tropical revolu- tion, (retrograde), in 141.739 years. The equator of Jupiter is inclined to the plane of his orbit, at an angle of 3°.5/.27// ; the fixed planes on which the planes of the orbits move, are determi- ned by theory, and probably could never be dis- covered by observation alone. 286. The orbit of the third satellite is eccen- tric ; but appears to have two distinct equations of the centre ; one which really arises from its own eccentricity ; and another, which theory shews to be an emanation from the equation of the centre of the fourth satellite. The first equation is referable to an apsis, which has an annual motion of 2° 36' 39" forward in respect of the fixed stars ; the 2d equation is referable to the apsides of the 4th satellite. These two equations may be considered as forming one equation of the centre, referable to an apsis that has an irregular motion. The two equations coincided in 1682, and the sum of their maxima was 13' 16". In 1777, the equations were oppo- sed, and their difference was 5' 6". Observation PHYSICAL ASTRONOMY. 287 Observation alone led M. WARGENTIN to the know- ledge of these inequalities, but he could not dis- cover the law. 287. The orbit of the fourth satellite moves on a fixed plane, to which it is inclined at an angle of 14/ 58", and its nodes complete a side- rial revolution (backward) in 531 years- The fixed plane itself is inclined at an angle of 24' 33" to the equator of Jupiter ; the orbit is very sensibly elliptical, and the line of the ap- sides has an annual motion of 43'34".7« The motion of the apsides of this satellite is one- of the principal data from which the quantities of matter have been determined. 288. If the mass of Jupiter be supposed uni- ty, the mass of the 1st Sat. = .0000173281 of the 2d = .0000232355 of the 3d = .0000884972 of the 4th = .0000426591 If the mass of the Earth be put = 1, that of the third satellite will be found .027337. Now the mass of the Moon is — = .OU599, and therefore the DO. 5 quantity of matter in the third satellite is about twice as great as that in the Moon. The fourth satellite 288 OUTLINES OF NATURAL PHILOSOPHY. satellite is nearly equal to the Moon. Mecanique Celeste, 2 and the density of the matter contained in it = d, the quantity of matter, or m = — , and so the attraction When PHYSICAL ASTRONOMY. When x = r, this expression becomes - - - , o which is therefore the attraction at the surface of the sphere. The force of attraction or of gravity at the surfaces of different spheres, are therefore as their densities multiplied into their radii. The force with which a particle placed any where within a sphere, is urged toward the centre of the sphere, is proportional to its dis- tance from that centre. From what goes before, it is plain, that it is not at- tracted by any part of the sphere more distant from the centre than itself. It is therefore attract- ed only by the sphere at the surface of which it is placed. . If a particle be placed any where in the interior of an elliptic spheroid ; and if through it there be described a spheroidal surface, simi- lar and similarly situated to the surface of the spheroid, the particle will be urged only by the attraction of the spheroid contained within this surface, and the force of that attraction will be the force acting on a particle similarly situated on OUTLINES OF NATURAL PHILOSOPHY. on the surface of the exterior spheroid, as the transverse axis of the interior to the transverse axis of the exterior spheroid. . 296. If in an oblate spheroid differing little from a sphere, b be the polar semi-axis, b + c the radius of the equator, and

= 90°, this becomes c 3\ /T 5/ * {&••$**•. P ' wnere ^ = °> For a very elementary demonstration of this theorem, see the Notes on NEWTON'S Principia, by MADAME DU CHASTELLET, Principes, &c. torn. n. p. 237, > 297' If in an oblate spheroid, a be the semi- transverse axis, and e the eccentricity of the meridian j then, if gravity at the surface, in the plane PHYSICAL ASTRONOMY. plane of the equator, be called g9 the force of gravity at any distance «r, from the centre, in the same plane, will be MACLAURIN'S Fluxions, § 659. Supposing e9 as in the former articles, to be the dif- ference of the two semi-axes of the meridian, e% = 2 a c, and the above will become, It is evident from this, that the attraction of a sphe- roid, in the plane of its equator, does not decrease exactly in the inverse ratio of the square of the dis- tance ; on which account a small inequality is produced in the motion of the satellites belonging to planets which are not entirely spherical. The Moon is subject to an inequality arising from this cause, as are also the satellites of Jupiter. SECT. 296 OUTLINES O£ NATURAL PHILOSOPHY. SECT. VI. FIGURE OF THE EARTH. 298. FROM observation it has already been in- ferred that the Figure of the Earth is nearly that of an oblate spheroid, of which the great- er axis, the diameter of the equator, is to the less, the axis of revolution, as 312 to 311. The strict meaning of the phrase, the Figure of the Earth, has already been defined, and must be carefully kept in view, in searching into the causes which have determined it. Since the Earth revolves on its axis, it is evident, that its parts are all under the influence of a cen- trifugal force, proportional to their distances from that axis, and that if the mass were fluid, the » columns toward the equator, being composed of parts that are lighter, must extend Mn length, in order to balance the columns in the direction of the axis. By this means an oblateness or\elevation at the equator would be produced, similar, in some degree at least, to that which the Earth has been found to possess. Though it is not evident how the centrifugal force would produce such an effect on PHYSICAL ASTRONOMY. 297 on a solid body like the Earth, it may throw some light on the matter, to inquire into the figure which a homogeneous fluid would put on, its parts being all supposed to gravitate to one another, and at the same time to be under the influence of a centrifugal force, by which they all tended to re- cede from an axis given in position. To the equilibrium of a fluid mass, it is necessary that any two columns reaching to the surface from any point in the interior of the fluid, should ba- lance one another, or should press equally on that point. Conversely, when this is the case, however many the forces may be that act on the particles of the fluid, the mass must remain in equilibria. It follows from this, when a fluid is at rest its surface is at every point perpendicular to the direction of the diagonal resulting from all the forces which act at that point. 299* This equilibrium of the columns will take place in a mass of homogeneous fluid revolving on an axis, if it be formed into an oblate sphe- roid, such that the polar semi-axis is to the ra- dius of the equator, as the attraction at the equator diminished by the centrifugal force at the same place, to the attraction at the pole. This proposition was first demonstrated by MAC- LAURIN. Fluxions, torn, u, § 636 to 641. 300. Hence OUTLINES OF NATURAL PHILOSOPHY. 300. Hence the fraction which expresses the ellipticity of the meridian, or the excess of the equatorial above the polar radius, is to that which expresses the ratio of the centrifugal force at the equator, to gravity at the same place, as 5 to 4. This is also demonstrated by MACLAURIN, and fol- lows readily, from comparing the last theorem with those already stated concerning the attraction of spheroids. 301. The centrifugal force is to the gravity at the equator, nearly as 1 to 289. A body under the equator falls in a second through 15.0515 French feet, as is concluded from the length of the pendulum vibrating seconds. Now the centrifugal force is measured by the de- flection of a body from the tangent produced by the Earth's rotation in 1 second. But a point of the equator in a second of mean solar time, de- scribes an arch of 15".04*17 of a degree ; the versed sine of which, multiplied into the radius of the equator, in French feet, gives the deflec- tion, which is found to be — — of the direct de- scent. 302. A PHYSICAL ASTRONOMY. 299 302. A homogeneous fluid, therefore, of the same mean density with the Earth, and revol- ving on an axis in the space of %3h 56m 4sec of solar time, would be in equilibria if it had the figure of an oblate spheroid, of which the axis was to the equatorial diameter as 230 to This is accordingly the figure which NEWTON a- scribes to the Earth. Princip. lib. 3. prop. 19. In the spheroid thus constituted, the gravitation at the equator will be to that at the pole as 229 to 230 ; and if g be the gravitation at the equator, and gf that at any other point, of which the lati- tude is A, g' = g ( 1 + ^ 5 also if I be the length of the pendulum that vibrates seconds at the equator, and V the length of a pendulum that does the same in Latitude A, /' = I ( 1 -J — — -? ^ \ NEWTON'S investigation of the figure of the Earth, though very ingenious, involved some assumptions which prevented it from being quite satisfactory. A very accurate and elegant demonstration was af- terwards given by MACLAURIN ; and the investiga- tion was improved and rendered more analytical by CLAIRAUT, Fig. de la Terre. But 300 OUTLINES OF NATURAL But though it was thus demonstrated that the parts of a homogeneous fluid, on which the figure of the oblate spheroid just described was any how indu- ced, would be in equilibria, yet it was not shewn conversely, that, whenever an equilibrium takes place in such a fluid mass, the figure of the mass ftiust be the oblate spheroid in question. D'ALEM- BERT indeed shewed, that there are more sphe- roids than one in which the state of equilibrium may be maintained ; and this result, though it was not observed by MACLAURIN, might have been inferred from his solution. LE GENDRE af- terwards proved, that the solids of equilibrium. must always be elliptic spheroids, and that in ge- neral there are two spheroids which satisfy the conditions. In the case of a homogeneous mass of the mean den- sity of the Earth, revolving in the space of 23h 56' 4", one of the spheroids is that which has been mentioned ; the other, is one in which the equato- rial diameter is to the polar, as 681 to 1. Mem. Acad. des Sciences, 1784. LA PLACE has added the limitation which follows. 303. A fluid and homogeneous mass, of the mean density of the Earth, cannot be in equili- brium with an elliptic figure, if the time of its rotation be less than 2h 25m 17sec ; if the time of revolution is greater than this, there will al- ways V PHYSICAL ASTRONOMY. 301 ways be two elliptic spheroids, and not more, in which an equilibrium may be maintained. If the density of the fluid is greater than the mean density of the Earth, the time of rotation with which the equilibrium ceases to be possible, is had, by dividing 2h 25m I7sec by the square root pf the density of the fluid, that of the Earth being unity. LA PLACE, Theorie du Mouvement et dc la Figure des Planetes, Paris, 1784-, p. J26. 304. If the fluid mass, supposed to revolve on its axis, be not homogeneous, but be composed of strata that increase in density toward the centre ; the solid of equilibrium will still be an elliptic spheroid, but of less oblateness than if it were homogeneous. This was demonstrated by CLAIRAUT, Theorie> &c. NEWTON fell into the mistake of supposing the contrary to be the case, or that the greater densi- ty toward the centre would be accompanied with greater oblateness. If the density increase, so as at the centre to be infinite, the ellipticity is 2 x 289 = ~57lf » wnicn is the case of the least el- lipticity. — - is the case of the greatest. 305. Hence OUTLINES OF NATURAL PHILOSOPHY. 305. Hence, as the ellipticity of the Earth has been shewn to be less than Q' O^2- T near- ly), it is evident, that if the Earth is a spheroid of equilibrium, it is denser toward the interior. The greater density of the Earth toward the centre, is in itself probable, and has been put beyond all doubt by very accurate experiments, made on the sides of the mountain Schehallien in Perthshire, by the late Dr MASKELYNE. By observations of the zenith distances of stars, the difference of the latitude of two stations on the south and north sides of that mountain, was determined. A trigonometrical survey of the mountain also ascertained the distance between the same two points, and thence, from the known length of a degree of the meridian, under that pa- rallel, the difference of the latitude of the stations was inferred, and was found less by 11 ".6 than by the astronomical observations. The zeniths of the stations had therefore been sepa- rated from one another more than in the usual pro- portion of the meridian distance ; and this could only arise from the plummet on each side being attracted toward the body of the mountain. From PHYSICAL ASTRONOMY. 303 From the quantity of this change of direction, the ratio of the attraction of the mountain to the at- traction of the whole Earth, or to the force of gravity, was found to be that of 1 to 17804. The bulk and figure of the mountain being also given from the survey, the mean density of the moun- tain was found to be to the mean density of the Earth nearly as 5 to 9. Phil. Trans, vol. LXVIII. p. 781. Also HUTTON'S Tracts, vol. in. p. 62. The mean density of the Earth, is therefore nearly double the density of the rocks which compose Schehallien, which seem again to be considerably more dense than the mean of those which form the exterior crust of the Earth. Phil. Trans. 1811, p. 347. 306. If the density of the Earth increase toward the centre, the elliptic! tj of the sphe- roid, and the diminution of gravity from the pole to the equator will not both be expressed by the same fraction as in the case of the homo- geneous spheroid, but the sum of the two frac- tions will always be equal to the same quan- tity, viz. — - or .008695. 1 Lo This theorem was first given by CLAIRAUT, and is of great use, as serving to explain the connection be- tween the ellipticity, as ascertained from the mea- surement 304* OUTLINES OF NATURAL PHILOSOPHY. surement of degrees, and from experiments with the pendulum. As, in the actual figure of the Earth, the compression, or the ellipticity, is nearly .0032, § 60. if we take this from .008695, the remainder, .005495, or -— , is the diminution of gravity from the pole to the equator. And the gravitation at any other . point of the spheroid, is g ( 1 + .005495 sin* A), g being the gravity at the equator. The length of an isocronous pendulum is expressed by the same formula. This agrees nearly with the observa- tions on the length of the pendulum in different latitudes. See a Table of them, VINCE, Astron. vol. n. p. 105. LA LANDE, Astron. $ 2712. BIOT, Astron. torn. ur. p. 14-8.' The lengths of the pendulum in different latitudes, are less subject to irregularities than the lengths of degrees ; the intensity of gravity being, as might be expected, less affected by local variations than its direction. 307. The inequalities on the surface of the Earth, and the unequal distribution of the rocks which compose it, with respect to density, must produce great local irregularities in the direc- tion of the plumb-line, and are probably the causes of the inequalities observed in the mea- surement 3 PHYSICAL ASTRONOMY. 305 surement of contiguous arches of the meridian, even where the work has been conducted with the greatest skill and accuracy. This is exemplified in the great arch of the meridian measured across France ; and in those measured in England and Hindostan. The cause may be something concealed under the surface, which can at present only be a subject of hypothetical, or, at best, of analogical reasoning. These irregularities are so considerable, that the spheroid which agrees best with the degrees in France, is one having an ellipticity of - , near- ly double of what may be accounted the mean el- lipticity. 308. Notwithstanding these irregularities, the figure of the Earth has made such an approxi- mation to the spheroid of equilibrium, as indi- cates either the original fluidity of the entire mass, or the gradual acquisition of a spheroidal figure in consequence of the repeated waste and reconsolidation of the parts near the sur- face. VOL. II. U If 306 OUTLINES OF NATURAL PHILOSOPHY. If the whole mass of the Earth was ever in a fluid- state, it must have been so from the action of heat. The insolubility of the greatest part of rocks and minerals in water, and the immense bulk of wa- ter that would be required for dissolving even those that are soluble, are insuperable objections to the hypothesis of aqueous formation. The igneous formation is not subject to either of these difficulties. The spheroidal figure may also have been gradually acquired without supposing the original fluidity of the whole mass. 309. If in a terraqueous body, however irre- gular in its primitive form, the prominent parts are subject to be worn down, and the detritus io be carried to the lower parts, occupied by wa- ter, where they acquire a horizontal stratifica- tion, and are, by certain mineral operations, afterwards consolidated into stone ; such a body, in the course of ages, must acquire a sur- face every where at right angles to the direc- tion of gravity, and consequently more or less approximating to a spheroid of equilibrium. The natural history of the Earth gives great coun- tenance to the suppositions here introduced ; which therefore seem to furnish a very rational explanation of the ellipticity belonging to the Earth PHYSICAL ASTRONOMY. 307 Earth, and to the planets which are known to re- volve on their axes. The distribution of the solid materials in the interior of the Earth, will very much affect the nature of this solid, and the manner in which the figure is acquired must probably prevent the approxima- tion from ever being complete. The distribution of the materials at any distance un- der the surface is unknown, and we have no means of examining into it, but by the measurement of degrees, or by experiments on the pendulum, and on the plumb-line, like those at Schehallien. 310. On the whole, the facts known, from ob- servation, about the figure of the Earth, agree in general with theory ; but there are in the expressions of that theory, so many quantities which are yet indeterminate, that the perfect coincidence of the two cannot be affirmed. SECT. 308 OUTLINES OF NATURAL PHILOSOPHY. SECT. Vll. OF THE PRECESSION OF EQUINOXES. 311. THE precession of the equinoxes, is the slow angular motion by which the intersection of the equator and ecliptic goes backward, at the rate of 50r/| annually ; while the inclina- tion of these planes continues nearly the same ; so that the pole of the equator describes a cir- cle about the pole of the ecliptic in the space of 257^8 years nearly. In seeking for the cause of this phenomenon, it is natural to inquire how the gravitation toward dis- tant bodies, such as the Sun and Moon, may af- fect the Earth's rotation on its axis. 312. From what has been proved of the force with which the Sun disturbs the motion of the Moon, it is evident, that every particle in that hemisphere of the Earth which is turned to- ward the Sun, is drawn toward that body, while every particle in the other hemisphere is drawn in the opposite direction, the force that acts on PHYSICAL ASTRONOMY. 309 on any particle being as its distance from the plane that separates these hemispheres. From this it follows, that if the Earth were a perfect sphere, the solar forces acting on the opposite he- mispheres, would exactly balance one another, and could produce no motion in the Earth or its axis. 313. The Earth may be considered as a sphere circumscribed by a spheroidal shell or meniscus, thickest at the equator. The tendency of the Sun's action on this meniscus, except at the time of the equinoxes, is always to make it turn round the intersection of the equator with the ecliptic, towards the plane of this latter circle. For the matter of the meniscus may be regarded as forming a ring round the Earth, in the plane of the equator. Now, the solar force acting on the part of this ring that is above the ecliptic, may at every point be resolved into two 5 one of which is in the plane of the equator, and the other perpen- dicular to it. The result of all the latter, must be a force tending to impress on the ring a motion round its intersection with the ecliptic. The same holds of the half of the ring that is under the ecliptic. . Hence, if the equator had no other mo- tion, it would turn round its intersection with the 310 OUTLINES OF NATURAL PHILOSOPHY. the ecliptic, till it coincided with that plane ; the line of their intersection remaining all the while at rest. The mean quantity of the solar force which is thus decomposed, is — — r cos $, where m is the mass of the Sun, a the mean distance of the Sun from the Earth, r the radius of the equator, and S the declination of the Sun. The part of this force which is perpendicular to the plane of the equator, and which tends to make it move round the line of its nodes, is, ™ r cos 3 x sin 2. 315. As the ring which surrounds the equa- tor, at the same time that it has the tendency just described, revolves on an axis perpendicu- lar to its plane in twenty-four hours, it will not revolve on either of these axes, but on one in the same plane which divides the angle between them? so that the sine of its angular distance from each axis, is in the inverse ratio of the an-< gular velocity round that axis. If the arch, round the intersection of the equator and ecliptic, which the solar force acting upon the ring of the mcniscusy would make the Earth de- scribe PHYSICAL ASTRONOMY. 311 scribe in i an indefinitely small portion of time, be q> ; and if the angle described in the same time by the diurnal revolution be v, the axis of that revo- lution will be changed by an angle of which the . tangent is — The substitution of the tangent for the ratio of the two sines mentioned in the proposition, is made, because the angle which the axis'of the diurnal re- volution makes with the intersection of the equa- tor and ecliptic, is a right angle. 316. If the force which produces the motion chap. 13. p. 271. 2de edit. If the precession due to the Sun's force be 1 5". 3, that which is produced by the Moon is 35/; ; which is to the former nearly as 7 to 3. If the effect of the Sun were reduced to 12.5, that of the Moon would be triple of it, which is agreeable to the la- test results deduced from the theory of the tides, as will be seen in the next section. 320. The PHYSICAL ASTRONOMY. 31J 320. The action of the Moon produces also an inequality which diminishes the precession by a quantity proportional to the sign of the dis- tance of the Moon's ascending node from the vernal equinox ; and, besides this, a diminution in the obliquity of the ecliptic, proportional to the cosine of the said distance. These two inequalities constitute the Nutation j and the result here stated from the theory of gravita- tion, is conformable to that which was before gi- ven from observation, § 218. Conformably to what is said there, these two inequalities may be expressed by the revolution of the extremity of the Earth's axis produced to the Heavens, and describing an ellipse, as there represented ; of which the greater axis is to the less, as the cosine of the obliquity of the ecliptic to the cosine of twice that obliquity. . The precession, on account of variations in the solar action, as well as in the lunar, is subject to some inequalities, not included in the preceding theorems. The amount of the precession, including all these ine- qualities, may be calculated for any period of time, by a formula given in the Mecanique Celeste, which, reduced to the sexagesimal notation, is 50".412 X t — - 4626" sin. 1".394. X t + 12954" sin 1".4298 X t. t is the number of years reckoned from 1750. As SI 8 OUTLINES OF NATURAL PHILOSOPHY, As the annual precession is not always the same, the length of the tropical year, in remote ages, has been somewhat different from what it is at present. In the age of HIPPARCHLTS, it was about 10" long- er. The siderial year, as already observed, re- mains invariable. Diurnal Rotation. 32C2. The velocity of the Earth's rotation on its axis, or the length of the day, is not affect- ed by the action of the Sun or Moon, in such a degree as can ever become sensible, even to the nicest observation. Systeme du Monde, p. 271. The small inequalities so produced, do not accumulate by time, but quickly compensate one another. 323. The motions of bodies near the surface of the Earth, tend, in some cases, to alter the velocity of the diurnal rotation ; but if these motions are only such as we at present per- ceive, their effects, like the preceding, must for ever remain insensible. A body which, by descending from a height, or by moving from the equator toward the poles, comes nearer to the Earth's axis, tends to accelerate the motion PHYSICAL ASTRONOMY. 319 motion about that axis, because it brings with it, into its new situation, more velocity than it can retain, and must therefore communicate a part of it to the general mass. The contrary happens when a body recedes from the axis, and the quan- tity of the acceleration or retardation thus produ- ced, may be calculated, on the principle, that the total momentum of the Earth must remain the same, notwithstanding of any change that can arise from the action of its parts on one another. From this it follows, that if any body change its dis- tance from the axis, the momentum of rotation of the whole Earth, is to the change in the momen- tum of rotation of the body, as the velocity of the diurnal rotation to the variation in that velocity, arising from the motion of the body. As the first of these terms is incomparably greater than the second, so must the third be than the fourth. Hence, though the degradation of mountains, and the carrying of matter by the rains from a higher to a lower level, are effects that go on continually, the amount can never be so great as to be sen- sible. In some cases, even these small changes are imme- diately compensated. The constant current in the atmosphere, from the poles to the equator, is counteracted, in the retardation it tends to pro- duce, by the contrary current, which, in the supe- rior 320 OUTLINES OF NATURAL PHILOSOPHY. rior regions of the air, sets from the equator to- ward the poles. . In former ages, however, if the changes that have happened on the surface, or in the interior of the Earth, have been as great as some Geologists suppose, the change in the diurnal rotation may have been very consider- able. If the ocean once stood at the height of 1 5,000 feet above its present level, a quantity equal in bulk to a 440th part of the whole Earth, must have passed from being above the level of the present sea to be under it. If the mean density of water were the same with that of the Earth, it may be calculated easily, that the time of the diurnal revolution must, on this account, be shortened by 5^.682. As the Earth's mean density is to that of water nearly as 4?.71 to 1, this acceleration is reduced to 1m 325. The changes on the surface, or in the in- terior of the Earth, may have produced great variations in the position of the Earth's axis, relatively to the Earth itself, as well as in the lime of the diurnal rotation. If the Earth had originally a very irregular figure ; and if, as above suggested, it has acquired its pre- sent PHYSICAL ASTRONOMY. 321 sent form, by the wearing down of the more pro- minent parts, and their subsequent consolidation in the form of horizontal strata, the axis of the Earth's rotation may have been very different from what it now is ; it may have gone through a long series of changes, and may have carried the equator, and the accumulation of waters which accompanied it, over regions from which they are now far distant. Many facts in the natural history of the Earth, and of the mineral kingdom, give countenance to these suppositions ; and if it be true that the more an- cient strata have been set on edge, and that conti- nents have been raised up by the action of an ex- pansive force in the interior of the Earth, such ac- tion may have materially assisted in changing the position of the Earth's axis. Obliquity of the Ecliptic. 3%5. The position of the ecliptic is subject to change by the action of the planets ; each of them producing a retrograde motion in the in- tersection of the plane of the ecliptic with the plane of its own orbit. This does not affect the inclination of these two planes, nor does it af- fect the plane of the equator, but it neverthe- VOL. II. X less OUTLINES OF NATURAL PHILOSOPHY. less changes the inclination of the ecliptic to the equator, and also the line of their intersec- tion. This change in the inclination, and in the position of the line of the equinoxes, is easily deduced by spherical trigonometry from the retrogradation of the intersection of the two planes, and from the constancy of their inclination. See LA LANDE, Astron. § 2751, &c. 326. The variations in the obliquity of the ecliptic, thus produced, are among the number of the secular inequalities which have long pe- riods, and, after reaching a maximum, return in a contrary direction. As far back as observation goes, the obliquity of the ecliptic has been diminishing, and is doing so at present, by 52" in a century ; it will not, however, always continue to diminish, but in the course of ages will again increase, oscillating backwards and forwards on each side of a mean, from which it never can depart far. The secular variation of the obliquity was less in an- cient times than it is at present ; it is now near its maximumi, and will begin to decrease in the 22d century of our era. LA GRANGE has shewn, that the total change of the obliquity, reckoning from that in 1700, must be less than 5° 23'; Mem. Acad. de Berlin, 1782, p. 284. PHYSICAL ASTRONOMY. 323 p. 284?. Also, that the changes in the inclinations of the planetary orbits, are all periodical, and can- not carry the planes of those orbits .beyond the li- mits of the zodiac, or 8° on either side of the ecliptic. By the retrogradations of the nodes of the ecliptic and the planetary orbits, the preces- sion of the equinoxes is diminished by a small quantity, which is at present about 0".261 annual- ly. Ibid. p. 28 1 . All this is quite independent of the figure of the Earth, and would be the same though the Earth were truly spherical. THE first solution of the problem of the Precession was given by NEWTON, Princip. lib. in. prop. 39. It is not free from error; but it displays, in a strong light, the resources of genius contending with the imperfections of a science not sufficiently advanced for so arduous an investigation. One mistake has already been pointed out, $318; ano- ther consisted in supposing, that the motion of the nodes of the Ring surrounding the equator, must be the same with those of the Moon, allow- ing for the different time of rotation, and the dif- ferent inclination to the ecliptic. D'ALEMBERT corrected these mistakes, and gave an accurate, though prolix solution, which also compre- hended in it the theory of the Nutation. Precession des OUTLINES OF NATURAL PHILOSOPHY. des Equinoxes, Paris, 174-9. A solution equally correct and original, was given about the same time by EULER, Mem. Acad. de Berlin, 1749. Two solutions, in the Philosophical Transactions for 1754 ancj 1756, continued to follow the method of NEWTON. The first of these was by SYLVABELLE,' the second by WALMESLEY ; and this last, though it retained both the defects just mentioned, is re- markable for the elegance of the demonstrations. It extended the problem to the nutation of the Earth's axis, and it treated of the diminution of the obliquity of the ecliptic by the action of the planets. A memoir by LA GRANGE, on the Libration .of the Moon, which was crowned by the Academy of Sci- ences at Paris in 1769, contained an excellent so- lution of the problem of the Precession. SIMPSON, in his Miscellaneous Tracts, has given the solution already referred to, which is one of great simplicity and correctness. Its only defect is, that it does not clearly point out the means by which the uniform inclination of the Earth's axis is main- tained. Another very elegant solution, is that of FRISI above referred to ; Theoria Diurni Motus, Opera, torn. in. p. 292, LA LANDE has followed SIMPSON, as has also VINCE, in his Astronomy. The latest solution is that of Professor ROBERTSON of Oxford, Phil. Trans. 1807; PHYSICAL ASTRONOMY. r 325 1807 ; it is also after the method of SIMPSON, and the investigation is accurate and concise. The solution of LA PLACE, in the Mecanique Celeste, must be considered as the most perfect, and that which can most certainly be said to include, and to estimate with accuracy, all the causes which have a share in this phenomenon. There is, however, one defect it may be said to have, that as it pro- ceeds entirely by the calculus, it does not suffi- ciently carry the imagination along with it. SECT. OUTLINES OF NATURAL PHILOSOPHY. SECT. VIII. Of THE TIDES. 327. THE alternate rise and fall of the surface of the sea twice in the course of a lunar day, or of 24h 50m 48sec of mean solar time, is the phe- nomenon known by the name of the Tides. The time from one high- water to the next, is, at a mean, 12h 25m 24sec. The instant of low-water is nearly, but not exactly, in the middle of this inter- val ; the tide, in general, taking nine or ten mi- nutes more in ebbing than in flowing. At the time of new and full Moon, the tide is the highest, and the interval between the consecutive tides is the least, viz. 12h 19m 28sec. At the qua- dratures, or when the Sun and Moon are 90° dis- tant, the tides are the least, and the interval be- tween them is the greatest, viz. 12& 30m 7sec. f 328. The time of high- water is mostly regu- lated by the Moon, and in general, in the open sea, is from two to three hours after that planet has PHYSICAL ASTRONOMY. has been on the meridian, either above or un- der the horizon. On the shores of the larger continents, and where there are shallows and obstructions, there are great irregularities in this respect, and when these exceed six hours, it may seem as if the high- water preceded the Moon's transit over the meridian* For any given place, the hour of high- water is al- ways nearly at the same distance from that of the Moon's passage over the meridian. This constant dependence on the Moon, is the rea- son why the tides are considered as astronomical phenomena. 329. Though the tides seem to be chiefly re- gulated by the Moon, they appear also, in a certain degree, to be under the influence of the Sun. Thus, at the syzygies, when the Sun and Moon come to the meridian together, the tides, cceteris paribus, are the highest ; at the quadra- tures, or when the Sun and Moon are 90° dis- tant, the tides are least. The former are call- ed the Spring, the latter the Neap Tides. The highest of the Spring Tides is not the tide that immediately follows the syzygy, but is in general die third, and, in some cases, the fourth. At 328 OUTLINES OF NATURAL PHILOSOPHY. At Brest, the tides of the syzygies rise to 19.317 feet ; and those of the quadratures to 9.151, not quite the half of the former quantity. In the Pacific Ocean, the rise, in the first case, is 5 feet j in the second, 2 or 2.5. The greater the rise of high- water above the level of a fixed point, the greater the depression of the corresponding low- water relatively to the same point. To estimate the height of the tide, it seems best to take the excess of the mean of the two con- secutive high-water marks, above the intermediate low- water. This is the method of LA PLACE. 330. The height of the tide is affected by the vicinity of the Moon to the Earth, and increa- ses, cceteris paribus, when the parallax and ap- parent diameter of the Moon increase, but in a higher ratio. The greatest variation of the Moon's semidiameter above or below the mean is about ^th of the wholer and the corresponding variation of the tide at the syzygies is ^tn °f its mean quantity. Sys- teme du Monde, p. 77. 331. The rise of the tide is affected by the declination of the luminaries ; it is greatest, cceteris paribus, at the equinoxes, and least at the solstices. When PHYSICAL ASTRONOMY. 329 When the Moon is in the northern signs, the tide of the day, in all northern latitudes, is somewhat greater than the tide of the night. The contrary happens when the Moon is in the southern signs. 332. If the tides be considered relatively to the whole Earth, and to the open Sea, it is evi- dent, that there is a meridian, about 30° east- ward of the Moon, where it is always high- wa- ter, both in the hemisphere where the Moon is, and in the opposite ; on the west side of this circle, the tide is flowing ; on the east, it is ebbing ; and on the meridian, at right angles to the same, it is everywhere low-water. These meridians move westward, preserving nearly the same distance from the Moon, only approach- ing nearer to her at the syzygies, and going farther off at the quadratures. In high latitudes, whether south or north, the rise and fall of the tides are inconsiderable. It is pro- bable that at the poles there are no tides. The great Wave which, in this manner, constitutes the tide, is to be considered as an undulation, or reciprocation of the waters of the ocean, in which there is, except when it passes over shallows, or ap- proaches 330 OUTLINES OF NATURAL PHILOSOPHY. preaches the shores, little or no progressive mo- tion. In all this, no regard is had to the operation of local causes, winds, currents, &c. by which these gene- ral laws are modified, or overruled. 833. The dependence of the phenomena just enumerated on the motion of the Sun and Moon, naturally suggests an inquiry into the effects which the action of these bodies may produce on the waters which cover so large a proportion of the Earth's surface. 334". If m be the mass, and a the distance of the Sun, r the mean radius of the Earth, z the distance of the Sun from the zenith of any place (or the distance of that place from the point to which the Sun is vertical) a particle of matter at that place is drawn toward the Sun by a force equal to — cos 2; besides having 977 *y* its gravity increased by another force = . This is derived from the resolution of forces, in the same way as when the Sun's action on the Moon was investigated. When PHYSICAL ASTRONOMY. When the point is in the hemisphere opposite to the Sun, cos z becomes negative, so that the force draws the particle from the Earth, in a direction opposite to that which it has in the other hemi- sphere. If m' be the mass of the Moon, a' its distance from the Earth, 2' the distance of the Moon from the zenith, the forces by which the Moon affects the gravity of bodies on the Earth's surface^is express- ed by the same formulas. 335. The preceding force may again.be resol- ved into two, one of which, in the direction of the Earth's radius, and opposite to gravity, is _ — — cos z , and, with that already mentioned, as increasing gravity, viz. — p, makes the whole force by which the Sun's action diminishes gra- , Smrcosz* — mr vity, to be - ^ - . The other force, draws the particles everywhere ho- rizontally toward the point that has the Sun in its zenith, and is equal to °— — r sin 2 z * In the he- £ ct misphere OUTLINES OF NATURAL PHILOSOPHY. misphere opposite to the Sun, this attraction is to- ward the point which has the Sun in its nadir. 336. If the surface of the Earth be covered by a fluid, the parts will move in obedience to these forces ; those columns, where the Sun is vertical, will be rendered lighter, and will be lengthened, in order- to be in equilibria with those at the distance of 90°, which will of course be shortened. The same thing exactly will happen in the opposite hemisphere ; and if the waters on the Earth's sur- face were at rest, the Ocean would form itself into an oblong spheroid, with its longer axis passing through the attracting body. When there are two attracting bodies, their effects will most nearly co- incide, when they are nearest to one another ; and will be equal to .the sum of the two effects taken separately. When they are 90° distant, the effect produced will be the difference of the separate ef- fects. This is NEWTON'S explanation of the tides. He calculates the mean force by which the Sun diminishes the force of gravity at the surface, as equal to one 1 2868200 part of gravity. Hence he deduces the longer axis, and the greatest rise of the tide produced by the Sun at 9.2 inches. De Systemate Mundi. Opera, torn. in. p. 212. A different determination h given in the Principia, lib. in. prop. 36. The PHYSICAL ASTRONOMY. 333 The force drawing the water horizontally is not included here ; its tendency is to increase the ef- fect just calculated. The force — —, by which the solar force everywhere increases gravity, need not be taken into account, and does not affect the equilibrium of the water. The height to which the sea would thus rise at high- water, above the level which it would stand at if acted on by gravity alone, is twice as great as its depression under that level at low-water. This is easily demonstrated from the content of the sphe- roid remaining always the same. 337. The preceding is sufficient to shew, that the phenomena of the tides are effects that might be expected from the principle of gravitation. It is, however, an approximation from which ex- act results cannot be obtained, since a material element has been left out, namely, the motion of the water, on which the forces of the Sun and Moon are exerted. 338. The rapid motion of the waters, in sequence of the diurnal rotation, prevents them from assuming, at every instant, the figure which 334? OUTLINES OF NATURAL PHILOSOPHY. which the equilibrium of the forces acting on them would require ; so that they oscillate con- tinually, alternately approaching to that figure, and receding from it. To resolve the problem of the tides, including the condition of the diurnal motion, is a matter of great difficulty, and requires all the latest improve- ments of the Calculus. LA PLACE is the only one who has ventured to undertake it, and he calls it " le Probleme leplus epineux de toutc la Mecanique " Celeste" Exposition du Syst. du Monde, chap. JO. p. 248. edit. 2v USE ONLY ~~JUN 251985 — — _ ^CIRCULATION DEPT. rfafw DEPT. LD21A-40m-8,'7l (P6572slO)476-A-32 General Library University of California Berkeley ??KELEY LIBRARIES 8001023^58 .3