# Full text of "Elements of astronomy : designed for the use of students in the University"

## See other formats

Digitized by tine Internet Arciiive in 2007 witii funding from IVIicrosoft Corporation littp://www.arcliive.org/details/elementsofastronOOvincuoft THE ELEMENTS OF ASTRONOMY : DESIGNED FOR THE USE OF STUDENTS IN THE UNIVERSITY. BY THE Rev. S. VINCE, A.M. F.R.S. PLUMIAN PROFESSOR of ASTRONOMY and EXPERIMENTAL PHILOSOPHV in the UNIVERSITY qf CAMBRIDGE. FOURTH EDITION. CAMBRIDGE : printed hy J. Smith, Printer to the University ; AND SOLD BY DEIGHTON & SONS, NICHOLSON & SON, CAMBRIDGE; AND J.MAWMAN, 39, LUDGATE-STREET, LONDON. 1816. CONTENTS, Page Chap. I. Definitions 1 II. On the Doctrine of the Sphere ... 13 III. To determine the right Ascension, DecU- nation, Latitude and Longitude of the Heavenly Bodies 41 IV. On the Equation of Time 47 V. On the Length of the Year, the Precession of the Equinoxes from Observation and Obliquity of the Ecliptic ... 58 VI. On Parallax 65 VII. On Refraction 78 VIII. On the System of the World .... 83 IX. On Kepler's Discoveries 90 X. On the Motion of a Body in an Ellipse about the Focus 94 XI. On the Opposition and Conjunction of the Planets 106 XII. On the mean Motion of the Planets . . 110 XIII. On the greatest Equation, Excentricity, and Place of the Aphelia of the Orbits of the Planets 114 XIV. On the Nodes and IncUnations of the Orbits of the Planets 118 XV. On the apparent Motions and Phases of the Planets 124 XVI. On the Moon's Motion from Observation, and it's Pliaenomena 134 XVII. On the Rotation of the Sun and Planets 137 XVIII. On the Satellites iQy XIX. On the Ring of Saturn 184 XX. On the Aberration of Light .... 190 XXI. On the Eclipses of the Sun and Moon . 208 XXII. On the Transits of Mercury and Venus . 234 XXIII. On Comets 244 XXIV. On the Fixed Stars 257 XXV. On the Longitude of Places upon the Earth 283 8YSTE OF ASTRONOMY, CHAPTER L DEFINITIONS, Art. 1. Astronomy is that branch of Natural Philosophy which treats of the heavenly bodies. The determination of their magnitudes, distances^ and the orbits which they describe, is called plane or pure Astronomy ; and the investigatioti of the causes of their motions is called physical Astronomy. The former discoveries are made from observations on their apparent magnitudes and motions ; and the latter from analogy, by applying those principles and laws of motion by which bodies on and near the earth are governed, to the other bodies in the system. The principles of plane Astronomy only are what we here propose to treat of, and we shall begin with the explanation of such terms as are the foundation of the science. (2.) A great circle QRST of a sphere is ouej whose plane passes through it's center C; and w small circle BDHK is that whose plane does nuv" pass through it's center. A 2 DEFINITIONS. (3.) A diameter PCE of a sphere, perpendicular to any great circle QRST, is called the a .vis of that great circle ; and the extremities P_, E, of the axis, are called it's Poles. (4.) Hence, the pole of a great circle is 90° from every point of it upon the sphere ; because every angle PCR being a right angle, the arc P R is every where 90°* And as the axis P E is perpendicular to the circle QRST when it is perpendicular to any two radii CQ, CR, a point on the surface of the sphere 90^ distant from two points of a great circle wherever taken, will be the pole. (5.) All angular distances on the surface of a sphere, to an eye at the center, are measured by the arcs of great circles. (6.) Hence, all the triangles formed on the surface 6f a sphere, for the solution of spherical problems, must be formed by the arcs o^ great circles. (7.) Any two great circles bisect each other ; for both passing through the center of the sphere, their common section must be a diameter of each, and every diameter bisects a circle. (8.) Secondaries to a great circle, are great circles which pass through it's poles ; thus, P R E is a secondary to QRST. (9.) Hence, secondaries must be perpendicular to their great circles; for if one line be perpendicular to a plane, any plane passing through that line will DEFINITIONS, 3 also be perpendicular to it ; therefore as the axis PE of the great circle QRST is perpendicular to it, and is the common diameter of all the secondaries, they must all be perpendicular to the great circle. Hence also, every secondary, bisecting it's great circle (*7)j must bisect every small circle B DH K parallel to it ; for the plane of the secondary passes notonly through the center C of the great circle, but also the center A of the small circle jjarallel to it. (10.) Hence, a great circle passing through the poles of two great circles, must be perpendicular to each ; and vice versa, a great circle perpendicular to two other great circles must pass through their poles. (11.) If an eye be in the plane of a circle, that circle appears a straight line ; hence, in the repre- sentation of the surface of a sphere upon a plane, those circles whose planes pass through the eye, are represented by straight lines. (12.) The angle formed by the circumferences of two great circles on the surface of a sphere, is equal to the angle formed by the planes of those circles; and is measured by the arc of a great circle inter- cepted between them, and which is a secondary to each. For let C be the center of the sphere, P QE, PRE two great circles ; then as the circumferences of these circles at Pare perpendicular to the common intersection P CE, the angle at P between them is equal to the angle between the planes, by Euc. B. XI. Def 6. Now draw CQ, CR perpendicular to P CE, then as these lines are respectively parallel to the directions of the circumferences P Q, PR, at the point P, the angle QCR is equal to the angle at P formed by the two circles, Euc. B. XI. Prop. 10. ; and the angle QCR is measured by the arc QR of Si great circle whose pole is P, because PQ, PR are each 90^ ^ The figures in the parentheses refer to the Articles. A 2 DEFINITIONS. (13.) If at the intersection P of two great circles as a pole, a great circle QRST be described, and also a small circle BDHK parallel to it, the arcs QR, B D of the great and small circles intercepted between the two o^reat circles, contain the same number of degrees. For C and A are the centers of the respective circles, and QC is parallel to BA, and RC is parallel to DA; therefore by Euc. B. XI. Prop. 10. the angle BAD is equal to the angle QCR, consequently the arcs B D, QR contain the same number of degrees. Hence, the arc BD oi such a small circle measures the angle at the pole between the two great circles. Also, QR.BD:. QC: B^ :: radius : cos. BQ. (14.) The ajcis of the earth pep'q is that diameter pOp' about which it performs it's diurnal rotation ; and the extremities p, p', of this diameter, are called it's Poles. (15.) The terrestrial Equator is a great circle erqs of the earth perpendicular to it's axis. Hence, the axis and poles of the earth are the axis and poles of it*s equator. That half of the earth (suppose epq) which lies on the side of the equator which we inhabit, is called the iwrthern hemisphere, and the other ep'q the southern ; and the poles are respectiv^ely called the tiorth and south poles. (1 6.) The Latitude of a place on the earth's surface, is it's angular distance from the equator, measured upon a secondary to it ; thus, the arc e h measures the latitude of h. The secondaries to the equator are called Meridians. (17.) The Longitude of a place on the earth's surface, is an arc of the equator intercepted between the meridian passing through the place, and another, called the Jirst meridian, passing through that place from which you begin to measure ; thus, the longitude of the place z? on the meridian prp' measured from the first meridian pep\ is er. DEFINITIONS. (18.) If the plane of the terrestrial equator erqs be produced to the sphere of the fixed stars, it marks out a circle E RQS called the celestial equator; and if the axis of the earth pOp' be produced in like manner, the points P, P' , in the Heavens to which it is produced, are called Poles, being the poles of the celestial equator. The star nearest to each pole is called the Pole star. (19.) Secondaries, as PxP', to the celestial equa- tor are called Circles of Declination ; of these, 24 which divide the equator into equal parts, each con- taining 15°, are called hour circles. (20.) Small circles, as dfgh, parallel to the ce- lestial equator, are called Parallels of Declination. (21.) The sensible horizon is that circle ahc in the heavens whose plane touches the earth at the spectator h. The rational horizon is a great circle HO R in the heavens, passing through the earth's center^ parallel to the sensible horizon. (22.) Almacanter is a small circle parallel to the horizon. (23.) If the radius O 6 of the earth at the place h where the spectator stands, be produced both ways to the heavens, that point Z vertical to him is called the Zenith, and the opposite point A^ the Nadir. Hence, the zenith and nadir arc the poles of the 6 DEFINITIONS. rational horizon (3) ; for the radius produced being perpendicular to the sensible, must also be perpen- dicular to the rational horizon. (24.) Secondaries to the horizon are called veHical circles ; and being (9) perpendicular to the horizon, the altitude of an heavenly body is measured upon them. (25.) A secondary P E H P' common to the ce- lestial equator and the horizon of any place b, and which therefore (lo) passes through the poles P, Z, of each, is the celestial Meridian of that place. Hence, the plane of the celestial meridian of any place, conicides with the plane of the terrestrial meridian of the same place. (26.) The meridian ZP R H cuts the horizon in the point R, called the north point, and in the point H, called the south point ; P being the north pole. (27.) The meridian of any place divides the heavens into two hemispheres lying to the east and west; that lying to the east is called the eastern hemisphere, and the other, lying to the west, is called the tvestern hemisphere. (28.) The vertical circle which cuts the meridian of any place at right angles, is called the prime vertical ; and the points where it cuts the horizon are called the east and west points. Hence, the east and west points are 90^ distant from the north and south points. These four are called the cardinal points. (2^.) If a body be referred to the horizon by a secondary to it, the distance of that point of the horizon from the north or south points, is called it's Azimuth. The Amplitude is the distance from the east or west point. (30.) The Ecliptic is that great circle in the heavens which the sun appears to describe in the course of a year. (31.) The ecliptic and equator being great circles Biiist (7) bisect each other, and their inclination is DEFINITIONS. 7 called the obliquity of the eliptic ; also, the points where they intersect are called the equinoctial points. The times when the sun comes to these points are called the Equinoxes. (32.) The ecliptic is divided into 12 equal parts, called Sigfis: Aries t , Taurus b, Gemini n, Cancer <b , Leo SI , Virgo iik , Libra =■& , Scorpio ttj. , Sagittarius ^ , Capricornus vj , Aquarius ^ , Pisces X . The order of these is according to the motion of the sun. The first point of Aries coincides with one of the equinoctial points, and the first point of Libra with the other. The first six signs are called northern, lying on the north side of the equator ; and the last six are called southern, lying on the south side. The signs v^ , ^, X , t? b, n, are called ascending, the sun approaching our (or the north) pole whilst it passes through them; and ?s, SI, 11K, ===, TU, f , are called descending, the sun receding from our pole as it moves through them. (33.) When the motion of the heavenly bodies is according to the order of the signs, it is called dii^ect, or in consequentia ; and when the motion is in the contrary direction, it is called retrograde, or in ante- cedentia. The real motion of all the planets fs according to the order of the signs, but their apparent motion is sometimes in an opposite direction. (34.) The Zodiac is a space extending on each side of the ecliptic, within which the motions of all planets are performed, (35.) The right ascension of a body is an arc of the equator intercepted between the first point of Aries and a declination circle passing through the body, measured according to the order of the signs. (36.) The oblique ascension is an arc of the equator intercepted between the first point of Aries and that point of the equator which rises with any body, measured according to the order of the signs. (37.) The ascensional difference is the difterence between the right and oblique ascension. 8 DEFINITIONS. (38.) The Declination of a body is it's angular distance from the equator, measured upon a secondary to the equator drawn through the body. (39.) The Longitude of a star is an arc of the ecliptic intercepted between the first point of Aries and a secondary to the ecliptic passing through the star, measured according to the order of the signs. If the body be in our system, and seen from the sun, it is called the heliocentric longitude; but if seen from the earthy it is called the geocentric longitude ; the body in each case being referred perpendicularly to the ecliptic in a plane passing through the eye. (40.) The Latitude of a star is it's angular distance from the ecliptic, measured upon a secondary to the ecliptic drawn through the star. If the body be in our system, it's angular distance from the ecliptic seen from the earth is called the geocentric latitude ; but if seen from the sun, it is called the heliocentric latitude. (41.) Thus, if cy, Q be the equator, ^C the ecliptic, V the first point of Aries, s a star, and the <v^ p great circles sr, sp be drawn perpendicularly to f C and tQ; then ^p is it's right ascension, sp it's declination, s r it's latitude, and <^ r its longitude. The circle 5 r is called a Circle of Latitude. Hence, if we know the right ascension <^p, and declination ps of a body s, we know it's place ; for, take T/? = to the given right ascension, draw the meridian ps, and take ps = io the given declination, and s is the place of the body. In like manner, if we know the longitude t ^ of a body s, and latitude rs, we know the place of the body. (42.) The Tropics aje two parallels of declination DEFISITIOMS. 9 touching the ecliptic. One, touching it at the be- ginning of Cancer, is called the Tropic of Cancer ; and the other_, touching it at the beginning of Capricorn, is called the Tropic of Capricorn. The two points vvliere the tropics touch the ecliptic, are called the Solstitial points. (43.) The Colures are two secondaries to the ce- lestial equator; one, passing through the equinoctial points, is called the equinoctial colure ; and the other, passing through the solstitial points, is called the solstitial colure. The times when the sun comes to ihe solstitial points are called the Solstices. (44.) The Arctic and Antarctic circles are two parallels of declination, the former about the north and the latter about the south pole, the distances of which from the two poles are equal to the distances of the tropics from the equator. These are also called polar circles. (45.) The two tropics, and two polar circles, when referred to the earth, divide it into five parts, called Zones; the two parts within the polar circles are called the frigid zones ; the two parts between the polar circles and the tropics are called the temperate zones ; and the part between the tropics, is called the torrid zone. Small circles in the heavens are referred to the earth, and the contrary, by lines drawn to the earth's center. (46.) A body is in Conjunction with the sun, when it has the same longitude ; in Opposition, when the difference of their longitudes is 180°; and in Quad- 7'ature, when the difference of their longitudes is 90°. The conjunction is marked thus c5 , the oppo- sition thus 8 , and quadrature thus d . (47.) Syzygy is either conjunction or opposition. (48.) The Elongation of a body from the sun, is it's angular distance from the sun when seen from the earth. (49.) The diurnal parallax is the difference be- tween the apparent places of a body in our system lO DEFINITIONS. when referred to the fixed stars, if seen from the center and surface of the earth. Tlie annual parallax is the difference between the apparent places of a body in the heavens, when seen from the opposite points of the earth's orbit. (50.) The Argument is a term used to denote any quantity by which another required quantity may be found. For example, the argument of a planet's latitude is it's distance from it's node, because upon that the latitude depends. (51.) The Nodes are the points where the orbits of the primary planets cut the ecliptic, and where the orbits of the secondaries cut the orbits of their pri- maries. That node is called ascending where tlie planet passes from the south to the north side of the ecliptic ; and the other is called the descending node. The ascending node is marked thus Sh , and the de- scending node thus y . The line which joins the nodes is called the line of the nodes. (52.) If a perpendicular be drawn from a planet to the ecliptic, the angle at the sun between two lines, one drawn from it to that point where the per- pendicular falls, and the other to the earth, is called the angle of Commutation. (53.) The angle of Position is the angle at an heavenly body formed by two great circles, one passing through the pole of the equator, and the other through the pole of the ecliptic. (54.) Apparent noon is the time when the sun comes to the meridian. (55.) True or mean noon is 12 o'clock, by a clock' adjusted to go 24 hours in a mean solar day. (56.) The Equation of' time at noon, is the interval between true and apparent noon. (57.) A star is said to rise or set Cosmically, when it rises or sets at sun-rising; and when it rises or sets at ^un-setting, it is said to rise or set Achronically. (58.) A star rises Heliacally, when, after having been so near to the sun as not to be visible, it emerges DEFINITIONS. 1 1 out of the sun's rays, and just appears in the morning; and it sets Heliacallif, when the sun approaches so near to it, that it is about to immerge into the sun's rays and to become invisible in the evening- (59.) The curtate distance of a planet from the sun or earth, is the distance of the sun or earth from that point of the ecliptic where a perpendicular to it passes through the planet. (60) Aphelion is that point in the orbit of a planet which is furthest from the sun. (61.) Perihelion is that point in the orbit of a planet which is nearest the sun. (62.) Apogee is that point of the earth's orbit which is furthest from the sun, or that point of the moon's orbit which is furthest from the earth. (63.) Perigee is that point of the earth's orbit which is nearest the sun, or that point of the moon's orbit which is nearest the earth. The terms aphelion and perihelion are also applied to the earth's orbit. (64.) Apsis of an orbit, is either it's aphelion or perihelion, apogee or perigee ; and the line which joins the apsides is called the line of tlie apsides. (6*5.) Anomaly {true) of a planet, is it's angular distance at any time from it*s aphelion, or apogee — {mean) is the angular distance from the same point at the same time, if it had moved uniformly with it's mean angular velocity. {QQ.) Equation of the center is the difference between the true and meaw anomaly; this is some- times called the prosthapheresis. {67.) Nonagesimal degree of the ecliptic, is that point which is highest above the horizon. (68.) The mean place of a body, is the place where a body (not moving with an uniformly angular velo- city about the central body) would have been, if it had moved with it's mean angular velocity. The true place of a body, is the place where the body actually is at any time. 12 DEFINITIONS. ■^1^' (6^.) Equations are corrections which are appHed to the mean place of a body, in order to get it's true place. (70.) A Digit is a twelfth part of the diameter of the sun or moon. (71.) Those bodies which revolve about the sun in orbits nearly circular, are called Planets, or pyi- mart/ planets for the sake of distinction ; and those bodies which revolve about the pr^imary planets are called secondary planets, or Satellites. (72.) Those bodies which revolve about the sun in very elliptic orbits are called Comets. The sun, planets, and comets, comprehend all the bodies in what is called the Solar System. (73.) All the other heavenly bodies are called Fixed Stars, or simply Stars. (74.) Constellation is a collection of stars contained within some assumed figure, as a ram, a dragon, an hercules, 8^c. the whole heaven is thus divided into constellations. A division of this kind is necessary^ in order to direct a person to any part of the heavens which we want to point out. Characters used for the Sun, Moon and Planets. D $ e The Sun. The Moon. Mercury. Venus. The Earth. Mars. Ceres. Pallas. Juno. Vesta. Jupiter. Saturn. Georgian. Characters used for the Days of the PFeeh. © Sunday. D Monday. {? Tuesday. ^ Wednesday. h Thursday. Friday. Saturday. Chap. IL ON THE DOCTRINE OF THE SPHERE. (75.) A SPECTATOR upon the earth's surface conceives himself to be placed in the center of a concave sphere in which all the heavenly bodies are situated ; and by constantly observing them, he perceives that by far the greater number never change their relative situations, each rising and setting at the same interval of time, and at the same points of the horizon, and are therefore csiWed Jiied stars ; but that a few others, called planets, together with the sun and moon, are constantly changing their situations, each continually rising and setting at different points of the horizon, and at different intervals of time. Now the determination of the times of the rising and setting of all the heavenly bodies ; the finding of their position at any given time in respect to the horizon or meridian, or the time from their position ; the causes of the different lengths of days and nights, and the changes of seasons; the principles of dialling, and the like, constitute the doctrine of the sphere. And as the apparent diurnal motion of all the bodies has no reference to any particular system, or dispo- sition of the planets, but may be solved, either by supposing them actually to perform those motions every day, or by supposing the earth to revolve about an axis, we will suppose this latter to be the case, the truth of which will afterwards appear. {'jQ.) hetpep'q represent the earth, O it's center, b the place of a spectator, HZ R Nthe sphere of the fixed stars ; and although the fixed stars do not lie in the concave surface of a sphere of which the center 14 DOCTRINE OF THE SPHERE. of the earth is the center, yet, on account of the immense distance even of the nearest of them, their ^ N relative situations from the motion of the earth, and consequently the place of a body in our system referred to them, will not be affected by this suppo- sition. The plane ahc touching the earth in the place of the spectator, is called (21) the sensible horizon, as it divides the visible from the invisible part of the heavens ; and a plane HOR parallel to abc^ passing through the centre of the earth, is called the rational horizon ; but in respect to the sphere of the fixed stars, these may be considered as coin- ciding, the angle which the arc Ha subtends at the earth becoming then insensible, from the immense distance of the fixed stars. Now if we suppose the earth to revolve daily about an axis, all the heavenly bodies must successively rise and set in that time, and appear to describe circles whose planes are per- pendicular to the earth's axis, and therefore parallel to each other, because each body continues at the same distance from the equator, during the revolution of the earth about it's axis. Thus, all the stars will appear to revolve daily about the earth's axis, as if they were placed in the concave surface of a sphere having the earth in the center. Let therefore pp be that diameter of the earth about which it must DOCTRINE OF THE SPHERE. 15 revolve in order to give the apparent diurnal motion to the heavenly bodies, then />, />', are called it's poles; and if pp' be produced both ways to P, P\ in the heavens^, these points are called (18) the poles of the heavens, and the star nearest to each of these is called the pole star. Now, although the earth, from it's motion in it's orbit, continually changes it's place, yet as the axis always continues parallel to itself, the points P, P', will not, from the immense distance of the fixed stars, be sensibly altered ; we may therefore suppose these to be fixed points*. Produce O h both ways to Z and iV, and Z is the zenith, and xYthe nadir (23). Draw the great circle PZ HNR, and it will be the celestial meridian (25), the plane of which coincides with the terrestrial meridian pbp' passing through the place b of the spectator. Let erqs represent a great circle of the earth perpendicular to it's axis pp', and it will be the equator (15); and if the plane of this circle be ex- tended to the heavens, it marks out a great circle ERQS called the celestial equator (18). Hence, for the same reason that we may consider the points jP, P', as fixed, we may consider the circle ERQS as fixed. Now as the latitude of any place b on the earth's surface is measured by the degrees of the arc he (l6), it may be measured by the degrees of the arc Z E ', hence, as the equator, zenith, and poles in the heaven, correspond to the equator, place of the spectator, and poles of the earth, we may leave out the consideration of the earth in our further enquiries upon this subject, and only consider the equator, zenith, and poles in the heavens, and H R the rational horizon to the spectator. {77') Let the annexed figure represent the position * This is not accurately true, the earth's axis varying a little from its parallelism from the action of the moon. This is called the Nutation of the earth's axis, and was discorered by Dr. Bradley. 16 DOCTRINE OF THE SPHERE. of the Iieavens to Z the zenith of a spectator in north latitude, EQ, the equator, P, P, it's poles, HOR the rational horizon, PZHP' R the meridian of the spectator, and draw the great circle ZON perpen- dicular to ZP RH, and it is the prime vertical (28) ; R will he the north point of the horizon, and H the south (26), and O will be the east or west point, (28) according as this figure represents the eastern or western hemisphere. Draw also a great circle POP' perpendicular to the meridian. We must therefore conceive this figure to represent half a globe, and all the lines upon it to represent circles ; and if we conceive the eye to be vertical to the middle point O of the figure, all the circles which pass through that point will appear right lines; therefore the right lines ZON, POP, EOQ, HOR, must be considered as semicircles. Now as each circle HR, EQ, ZN, PP' is perpendicular to the meridian, it's pole must be in each (8, 9), therefore their common intersection O is the pole of the meridian. Draw also the small circles wH, nit, ae, Rv, yx\ parallel to the equator; and as the great circle POP' bisects EQ. in O, it must also bisect the small circles mt, a e, in r and c; for as £0 = 90°, tr and ec are each =90° (13); and as QO = 90°, m r and a c are each = 90" ; hence, ac-=-C€, and m r — r t. DOCTRINE OF THE SPHERE. I7 (78.) As all the heavenly bodies, in their diurnal motion, describe either the equator, or small circles parallel to the equator, according as the body is in or out of the equator ; if we conceive this figure to repre- sent the eastern hemisphere, QE, ae, mt, may repre- sent their apparent paths from the meridian under the horizon to the meridian above, and the points b, 0,s, are the points of the horizon where they rise. And as ae, QE, mt, are bisected in c, O, r, eb must be greater than ba, QO equal to OE, and ts less than s m. Hence, a body on the same side of the equator with the spectator, will be longer above the horizon than below, because eb is greater than ba; a. body in the equator will be as long above as below, because QO = O E ; and a body on the cow ^rary side will be longer below than above, because m s is greater than s t. iVnd the further ae, or m t^ are from the equator, the greater will be the difference of ab, be, and rns, sf, or of the times of continuing above and below the horizon; and the further they will rise from O. The bodies de- scribing ae, mt, rise at b and s ; and as O is the east point of the horizon, and R and /Tare the north and south points, a body, on the same side of the equator with the spectator, rises between the east and the north, and a body on the contran/ side rises between the east and the south, the spectator being supposed to be in the north latitude; and a body in the equator rises in the east at O. When the bodies come to d or n, they are in tlie prime vertical, or in the east ; hence, a body on the same side of the equator with the spectator comes to the east after it is risen, and a body on the contrary side, before it rises. The body which describes the circle Rv, or any circle nearer to P', never sets ; and such circles are called circles of per- petual apparition ; and the stars which describe tliem are called circumpolar stars. The body which de- scribes the circle wH, just becomes visible at H, and then it instantly descends below the horizon ; but tiie bodies which describe the circles nearer to P' are never 18 DOCTRINE OF THE SPHERE. visible. Such is the apparent diurnal motion of the heavenly bodies, when the spectator is situated any where between the equator and poles; and this is called an oblique sphere, because all the bodies rise and set obliquely to the horizon. As this figure may also represent the western hemisphere, the same circles ea, tm will represent the motions of the heavenly bodies as they descend from the meridian above the horizon to the meridian under. Hence, a body is at the greatest altitude above the horizon, when on the meridian, and at equal altitudes when equidistant on each side, from it, if the body have not changed it's declination. {7^') If the spectator be at the equator, then E coincides with Z, and consequently EQ with ZiV, and E2 therefore P P' with H R. Hence, as the equator EQ is perpendicular to the horizon, the circles ace, mrt, parallel to EQ must also be perpendicular to it; and as these circles are always bisected by PP\ they must now be bisected by H R. Hence, all the heavenly bodies are as long above the horizon as below, and rise and set at right angles to it, on which account this is called a i^ight sphere. (80.) If the spectator be at the pole, then P coin- cides with Z, and consequently P P' with Z N, and therefore EQ with HR. Hence, the circles mt, ae, parallel to the equator, are also parallel to the horizon ; therefore as a body in it's diurnal motion describes a circle parallel to the horizon, those fixed bodies in the heavens, which are above the horizon, must always DOCTRINE OF THE SPHERE. I9 continue above, and those which are below must always continue below. Hence, none of the bodies, ZP p^ by their diurnal motion, can either rise or set. This is called a joara//e/ sphere, because the diurnal motion of all the bodies is parallel to the horizon. These apparent diurnal motions of the fixed stars remain constant, that is, each always describes the same parallel of declination. (81.) The ecliptic, or that great circle in the heavens which the sun appears to describe in the course of a year, does not coincide with the equator, for during that time it is found to be only twice in the equator ; let therefore COL represent half the ecliptic, or half the sun's apparent annual motion ; C the first point of Capricorn, and L the first point of Cancer ; and this being a great circle, must cut the equator into two equal parts (7). Hence, as the apparent motion of the sun is nearly uniform, the sun is nearly as long on one side of the equator as on the other. (See Fig. in p. l6.) When therefore the sun is at q, on the same side of the equator with the spectator, describing the parallel of declination ae by it's apparent diurnal motion, the days are longer than the nights, and it rises at h to the north of the east point ; but when it is on the contrary side, at />, de- scribing mt, the days are shorter than the nights, and it rises at s to the south of the east point, the spectator being on the north side of the equator ; but when the j*un is in the equator, at O, describing QE, the days B 2 20 DOCTRINE OF THE SPHERE. and nights are equal, and it rises in the east, at O *. If ae, mt be equidistant from EQy then will he = 7ns, and ab = st; hence, when the sun is in these opposite parallels, the length of the day in one is equal to the length of the night in the other; therefore the mean length of a day at every place is 12 hours. Hence, at every place, the sun, in the course of a year, is half a year above, and half a year below the horizon f. It is manifest, also, that the days increase from the time the sun leaves C the beginning of Capricorn, till he comes to L the beginning of Cancer ; and that they decrease from the time the sun leaves the beginning of Cancer till he comes to the beginning of Capricorn. When the spectator is at the Equator, the sun at p or q describing the circles m t, ae, by it's apparent diurnal EZ motion, and these being bisected by the horizon, the sun will be always as long above as below the horizon. * The different degrees of heat in summer and winter, do not altogether arise from the different times which the sun is above the horizon, but partly from the different altitudes of the sun above the horizon ; the higher the sun is above the horizon, the greater is the number of rays which fall on any given space, and the greater also is the force of the rays. From all these circumstances arise the different degrees of heat in summer and winter. The increase of heat also as you approach the equator, arises from the two latter circumstances. t This is not accurately true, because the sun's motion in the ecliptic is not quite uniform, on which account it is not exactly as long on one side of the equator as on the other. If the major axis of the earth's orbit coincided with the line joining the equinoctial points, the times would be equal. This happened at the Creatioij. DOCTRINE OF THE SPHERE. 21 and consequently the days and nights will be always 12 hours long. There will however be some variety of seasons, as the sun will recede 23° . 28' on each side from the spectator. In this situation of the spectator, the sun will be vertical to him at noon when it is in the equator. And when the spectator is any where between the tropics, the sun will be vertical to him at noon, when it's declination is equal to the latitude of the place, and of the same kind, that is, when they are both north, or both south. When the spectator is at the Pole, the sun at j) or q is carried, by it's apparent <] ^^ r.f^ a V d V.^ "i h^ P'N diurnal moWon, in the circles mpt, aqe, parallel to the horizon ; hence, it never sets when it is in that part OL of the ecliptic which is above the horizon, nor rises when in that part OC which is below ; con- sequently there is half a year day, and half a year night. As the sun illuminates one half of the earth or 90° all round about that place to which he is ver- 22 DOCTRINE OF THE SPHERE. tical *, when he is i?i the equator, he will illuminate as far as each pole ; when he is on the north side of the equator, the north pole will be within the illumi- nated part, and the south pole will be in the dark part ; and when the sun is on the south side of the equator, the south pole will be within the illuminated part, and the north pole in the dark part. And when the sun is at the tropic, he illuminates 23° . 28' beyond one pole, and the other pole is 23^ . 28' within the dark part. Hence, the variety of seasons arises from the axis of the earth, which coincides with PP\ not being perpendicular to the plane of the ecliptic LOC, for if it were, the ecliptic and equator would coincide, and the sun would then be always in the equator, and consequently it would never change it's position in respect to the surface of the earth. If QR=.EH=z 23° . 28', the sun's greatest declination, then on the longest day the sun describes the parallel Rv, which just touching the horizon at R, shows that the sun does not descend on that day below the horizon, and therefore that day is 24 hours long. But when the sun comes to its greatest declination on the other side of EQj it describes ivH, and consequently does not ascend above the horizon for 24 hours, and therefore that night is 24 hours long. This therefore happens when EH, the complement of EZ the latitude (l6), is 23*^ . 28', or in latitude 66"^ . 32'. If EH, the com- plement of latitude, be less than 23° . 28', the sun will be above the horizon in summer, and below in winter, for more than 24 hours, and the longer above or below, as you approach the pole, where, as was before observed, it will be six months above, and as long below the horizon. The orbits of all the planets, and of the moon, are also inclined to the equator, as ap- pears by tracing their motions amongst the fixed * This is not accurately true, because, as the sun is greater than the earth, he will illuminate beyond 90°, by a quantity which is nearly equal, in minutes of a degree, to his apparent semidiameter. DOCTRINE OF THE SPHERE. 23 stars ; therefore, in the time in which each makes one revolution in its orbit, the same appearances will take place, as in the sun. All these different appearances in the motion of the moon, must therefoie happen in every month. It is also evident, that these variations of rising and setting must be greater or less, as the orbits are more or less inclined to the equator, as ap- pears by Art. 78. Hence, they must be greater in the moon than in the sun*. The apparent annual motion of the sun, and the real motion of the moon and planets, is from west to east, and therefore contrary to their apparent diurnal motion. (82.) Hitherto we have considered the motion of the heavenly bodies in the eastern hemisphere ; but if the figure represent the western hemisphere, all the reasoning will equally apply. The bodies will be just as long in descending from the meridian to the horizon, as in ascending from the horizon to the meridian ; the paths described will be similar ; and they will set in the same situation in respect to the west point of the horizon, as they rise in respect to the east ; that is, if a body rise to the north or south of the east, it will set at the same distance from the west towards the north or south. (83.) Having thus explained all the apparent diurnal motions of the heavenly bodies, with the cause of the variety of seasons, we shall proceed in the next place to show the method of determining the positions of the different circles, and the situation of the bodies in respect to the horizon, meridian, or any other circles, at any given time ; and having given their situation, to find the time ; for the understanding of * On account of the continual change of declination of the sun, moon, and planets, their apparent diurnal motions will not be ac- curately parallel to the equator ; in those cases therefore, where the declination alters sensibly in the course of a day, and where great accuracy is required, we must in our computations, take into consideration, the change of declination. 24 DOCTRINE OF THE SPHERE. which, a knowledge of" plane and spherical trigonome- try is all that is requisite. (84.) The altitude PR ofthe }wU above the horizon, is equal to the latitude of the place. For the arc ZE is (l6) the measure of the latitude ; but PE = ZR, each being— 90"; take away ZP which is common to both, and EZ=PR*. (85.) To find the latitude of a place, observe the greatest and least altitude of a circumpolar star, and apply the correction for refraction , in order to get tlie true altitudes, and half the sum icill he the alti- tude of the pole. For if yx be the true circle described by the star, then, as Px = Py, PR = ^x Ry + Rx. See the last figure. The latitude may also be thus found. Let eOt be the ecliptic; then when the sun comes to e, it's declination is the greatest, and eH is the z greatest meridian altitude; when the sun comes to the ecliptic at t, let ts be the parallel described on that * Fro-Ti hence arises the method of measuring the circumference ofthe earth ; for if a man travel upon a meridian till the height of the pole has altered one degree, he must then have travelled one de^rree; hence, by measuring that distance and multiplying it by 360 we get the circumterence ofthe earth. This was undertaken bv our countryman Mr. Norwood, who measured the distance between London and York, and observed the different altitudes of the pole at those places. Afterwards, the French mathematicians ^ measured DOCTRINE OF THE SPHERE. a5 dav, and then s H is t!ie least meridian altitude; and as Ee = Es, \x He \ Hs=HE the complement of the latitude. (86.) Half the difference of the suns greatest and least meridian altitudes. Is equal to the inclination of the ecliptic to the equator. For half He- Hs, or half Ae, is eqiial to Ee which (12) measures the angle EOe, the inclination of the ecliptic to the equator. (87.) The angle which the equator makes with the horizon, or the altitude of that point of the equator which is on the meridian, is equal to the complement of the latitude. For ZH is 90*^, and therefore EH is the comple- ment of EZ; and as OE=OH=Q0\ ^//measures (12) the angle EOH*. (88.) Let abcdxe be a parallel of declination de- scribed by an heavenly body in the eastern hemisphere, and draw the circles of declination Ph, Pc, Pd, Px, and the circles of altitude Zh, Zc, Zd, Zx. Now, as has been already explained, when the body comes to Z>, it rises ; at c it is at the middle point between a and e ; and at d it is due east ; and let x be it's place at any other time. Let us suppose this body to be the sun, and not to change it's declination in it's passage from a to e, and let us suppose a clock to be adjusted to go 24 hours in one apparent diurnal revolution of the sun, or from the time it leaves any meridian till it returns to it again, then the sun will always approach the meridian, or any other circle of declination, at the measured a degree. Cassini measured one in France. After that, Clairaut, 3Taupertuis, and several other mathemati- cians went to Lapland, and measured a degree, the length of which appears to be 60,2 English miles in the latitude of 45°; for the earth being a spheroid, the degrees in different latitudes are different. * See my Treatise on Plmie and Spherical Trigonometry, Art. 173. This is the Trigonometry referred to in the future part of this Work. 2& DOCTRINE or THE SPHERE. rate of 15*^ in an hour ; also, the angle which the sun describes about the pole will vary at the same rate, because (13) an arc Joe, which the sun at x has to de- scribe before it comes to the meridian, measures the angle xPe, called the hour angle. If therefore we .suppose the clock to show 12 when the sun is on the meridian at a or e, it will be 6 o'clock when he is at c. And as the sun describes angles about the pole P at the rate of 15" in an hour, the angle between any circle, Pt, of declination passing through the sun at x, and the meridian PE, converted into time at the rate of 15° for an hour, will give the time from apparent noon, or when the sun comes to the meridian. (89.) Given the sun's declination, and latitude of the place, tojind the time of' his rising, and azimuth at that time. The sun rises at h-, and in the triangle ^Z/?, hZ — 90°, hP = co-dec. P^ = co-lat. Now when one side of a triangle = 90°, it may be solved by the circular parts, taking the angles adjacent to the side = 90", and the complements of the other three parts, for the circular parts. Hence, (Trig. Art. 215.) rad. X cos. ZPb=^cot. bP X cot. ZP, or, rad. x cos. hour angle= tan. dec. X tan. lat. therefore (Trig. Art. 213). Log. tan. dcc.-{-log. tan. lat. — 10,=: log. cos. hour ang.fro??iapp,noo?i; which converted into time, at the rate of 15° for an hour, (see Table I. at the end), and subtracted from 12 o'clock, gives the apparent time of rising. Also, (Trig. Art. 215.) rad. x cos. bP = s'in. ZPx cos. PZb, or, rad. x sin. dec. = cos. lat. x cos. azi. therefore JOj-^-Iog. sin. dec. —log. cos. lat. = log. cos. azi. from North, Ex. Given the latitude of Cambridge 52^ . 12' . 35", to find the time of the sun's rising on the longest day, and azimuth at that time, assuming the greatest decli- nation of the sun 23" . 28'. DOCTRINE OF THE SPHERE. 27 Dec. 23° . 28' . 0". - tan. 9,6376106 Lat. 32° . 12'. 35. - tan. 10,1104699 - COS. 9,7480805 Convert this into time (Tab. I.) and it gives Sh. 19'.6", which subtracted from 1 2, gives 3h. 40' . 54'', the time when the sun's center is upon the rational horizon on the longest day ; Also, Dec. 23°. 28'. O". - 10, + sin. 19,6001 181 Lat. 52. 12. 35. - - cos. 9.7872996 Azi. 49. 28. 9. - - cos. 9,8128185 Hence, on the longest day, the sun rises 40°. 3 1'. 5 1" from the east towards the north. (90.) Tojind the sidHs altitude at six d clock. The sun is at c at 6 o'clock, and the angle ZPc is a right one ; hence, (Trig. Art. 212.) rad. x cos. Zc = cos. ZP X COS. Pc, or rad. x sin. alt. = sin. lat. x sin. dec. therefore Log. sir}, lat. + log. sin. dec. — 10, = /oo-. sin. alt. Ex. Taking the data of the last example, we have, Lat. 52«. 12'. 35" - - sin. 9,8977695 Dec. 23. 28. O - - sin. 9,6001181 Alt. 18. 20.32. - - sin. 9,4978876 (91.) To find the time when the sun comes to d the prime vertical^ and it^s altitude at that time. In this case, the angle dZP — (^0^; hence, (Trig. Art. 112.) rad. x cos. dP = cos. ZP x cos. Zd, or, rad. X sin. dec. = sin. lat. x sin. alt. therefore * This log. 9.7480805 is found in the tables to be the log. cosine of 55°. 57'. 13", but as the angle is manifestly greater than 90°, we must take its supplement. In the solution of spherical triangles, ambiguous cases will frequently arise ; for the determination of which, where the case is not evident, the reader is referred to my Treatise on Trigo?iometrj/. 28 DOCTRINE OF THE SPHERE. 10, -\-log. sin. dec- log. sin. lat. = log. sin. alt. Also, (Trig. Art. 212)rad. x cos.ZPd=cot. Pdx tan. PZ, or, rad. x cos. hour angle=tan. dec. x cot. lat. therefore log. tan. dec. + log. cot. lat. — 1 0, = log. cos. hour angle ; which, converted into time (Tab. I.), gives the time from apparent noon. Ex. Taking the data of the last example, we have, Dec. 23^28'. 0" - 10 + ,sin. 19,6*001 181 Lat.52.12.35 - - sin. 9,8977695 Alt. 30.15.31 - - sin. 9,7023486 Dec. 23.28. o - - tan. 9,6376106 Lat.52.12.35 - - cot. 9,8895301 Hour z 70. 19. 44. - - COS. 9,5271407 This angle 'JO'^. I9'. 44", converted into time, gives 4h. 41'. 19" the time from apparent noon. (22.) Given the latitude of the place, the suit's decUnation, and altitude, to jind the hour, and his azimuth. Let X be the sun's place ; then, (Trig. Art. 239) sin. Px X sin. PZ : rad.^ :: sin, j- X Px j-PZ+Zxx sin. I X Px + PZ - Zx : cos. I ZPx^ ; hence, ZPx is known, which converted into time (Tab. L) gives the time from apparent noon. Also, (Trig. Art. 239) sin. Zx X sin. ZP : rad.'* :: sin. | x Zx + ZP+Px x sin.^xZx + ZP — Px : cos. § xZP' ; hence, tlie azimuth xZP from the north is known. Ex. Given the lat. 34°. 55' N, sun's declination 23^ 22'. 57" N, and true altitude SS"". 59'. 39", to find the apparent time. Here, ZP = 55°. 5', Zx = 53°. O'. 21", Px = 67°. 37'. 3" : hence (Trig. Art. 239) DOCTRINE OF THE SPHERE. 29 P.r=67''. 37'. 3" - ar. CO. sin. 0,034019 ZP = 55. 5. - ar. CO. sin. 0,086193 Zx=53. O. 21 Sum 175. 42. 24 i Sum 87. 51.12 - - - sin. 9,999694 Z,v = 53. O. 21 DifF. 34. 50. 51 - - r. sin. 9,756932 2)19,876838 9,938419 the cosine of 29^ 47' . 44", half the angle ZPjj, .-. ZPjc = 59°. 35'. 28", which reduced into time gives 3 h. 58'. 22", the time from apparent noon. By the very same process, the angle JcZP is found. (93.) Given the error in altitude, tojind the error in time. Let m n be parallel to the horizon, and nx represent the error in altitude ; then, as the calculation of the time is made upon supposition that there is no error in the declination, we must suppose the body to be at m instead of x, and consequently the angle fnPx, or the arc q r, measures the error in time. Now nx : xm :: sifi. nmx : rad. (Trig. Art. 125) xm '. qr '.: cos.rx : rad. (Art. 13). hence, nx : qr :: sin. nmx x cos. rx : rad.^ .-. qr=nx x rad.^ — '■ ; but ZxP =znmx, nxm beintr the sm. nmx x cos. rx complement of both ; also, (Trig. Art. 221 .) sin. ZxP, or nmx, : sin. ZP :: sin. xZP : sin. xP, or cos. rx, .'. sin. n7nx x cos. rx = sin. ZP x sin.xZP ; hence, qr = rad.' rad.- nx X ^-f5 : 7yn - ^^^ ^ TT '■ '■ — • sm. ZP X sm. xZP cos. lat. x sm. azmi. Hence, the error is least on the prime vertical. All altitudes therefore, for the purpose of deducing the 30 DOCTRINE OF THE SPHERE. time, ought to be taken on, or as near to, the prime vertical as possible. Ex. In lat. 50*^. 12', if the error in alt. at an azim. 44°. 22' be l', then qr= l' x -^ ^— = 2',334 of ' ^ ,012 X, 690 a degree=9",336 in time. Hence, the perpendicular ascent of a body is quickest when it is on the prime vertical ; for nx varies as sin. azim. when qr and the lat. are given, (94.) Given the lat. of the place, and the suns declination, to find the time when twilight begins. Twilight is here supposed to begin when the sun is 18° below the horizon ; draw therefore the circle hyk parallel to the horizon, and 18" below it, and twilight , will begin when the sun comes to y, and Zy — 108° ; hence, (Trig. Art. 239) sin. Py x sin. PZ : rad.* :: sin. I X PZ + Py + 108^ x sin. § x PZ-i-Py— 108° : cos. I yPZ'^ ; therefore yPZ is known, which con- verted into time (Tab. I.), gives the time from appa- rent noon. The operation is the same as that in Art. 92. (95 .) To find the time when the apparent diurnal motion of a fixed star, is perpendicular to the horizon. Let yx be the parallel described by the star ; draw the vertical circle Zh, touching it at o, and when the star comes to o, it's motion is perpendicular to the horizon ; and as ZoP is a right angle, we have (Trig. DOCTRINE OF THE SPHERE. 31 Art. 212) rad. x cos. ZPo — tan. Po X cot. PZ, or rad. X cos, hour angle = cot. dec. x tan. lat. therefore, log. cot. dec. + log. tan. lat. — lO, = log. cos. hour angle; which converted into time (Tab. I.), gives the time from the star's being on the meridian. Hence, the time of the star's coming to the meridian being known, the time required will be known. (96.) To find the time of the shortest tiviUght. Let a 6 be the parallel of the sun's declination at the time required, draw cd indefinitely near and parallel to it, and TW 2i parallel to the horizon 18° below it; then vPw., sPt measure the duration of twilight on each parallel of declination, and when the twilight is shortest, the increment of the duration is=0, and these must be equal; hence, vPr = wPz, therefore vr =w z ; and as rs = tz, and r and z are right angles, rvs=zivt; but Pvr = dO° = Zvs, take Zvr from both, and PvZ=-rvs', for the same reason PwZ = zwt ; hence, PvZ=PwZ. Take re = wZ = 90°, and join Pe; and as Pv=:Piv, ve=ivZ, and Pve — PwZ, we have Pe — PZ; and if Pi/ be perpendicular to eZ, then will Zy—ye. Now, (Trig. Art. 224) cos. Pv : cos. Pe, or PZ, :: cos. vy : cos. ey, that is, sin. dec. ; sin. lat. :: sin. ey : cos. ey :: tan. ey = g° : rad. or, rad. : sin. lat. :: tan. 9° : sin. of the suns declination at the time of the shortest twilight ; and the logarithmic operation is, log. sin. lat. -^ log. tan. 9^^— 10, = log. sin. dec. Because PZ is never greater than 90^^, and Zy = 9", 32 DOCTRINE OF THE SPHERE. therefore Py is never greater than 90\ and it's cosine is positive; also, vif is ahvays greater than 90*^, there- fore it's cosine is negative; hence, (Trig. Art. 212. rad. being unity) cos. Pv (= cos. Pyx cos. vy) is ne- gative ; consequently Pv is greater than 90^^ ; there- fore the sun's declination is south. If, instead of taking RIV = 18^, we take it = the sun's diameter (2.v), we shall get the time of the year when the body of the sun is the least time in ascend- ing above the horizon ; hence, log. sin. lat. + log. tan. s — 10, = log. sin. dec. Thus we get the declination when the sun is the least time in rising ; and as the declination must be always very small, this event must happen when the sun is very near the equinox. (97 ■) Tojindthe duration of the shortest twilight. As zi?Psr=i;Pe, therefore ZPe = vPw, which mea- sures the shortest time. Now (Trig. Art. 212) rad. X sin. Zy=zii\n. PZ x sin. ZPy, or, rad. x sin. 9'^ = cos. lat. X sin. ZPy, therefore, 10, + log. sin. 9° — log. cos. lat. = log. siti. ZPy, which doubled gives ZPe, or vPw, which, converted into time (Tab. I.), gives the duration of the shortest twilight. Ex. To find the time of the year at Cambridge, when the twilight is shortest ; and the length of that twilight. Lat. 52°. 12'. 35" - - - - sin. 9,8977695 9' ----- tan. 9,1997125 Dec. 7^ 11'. 25" - - - - sin. 9,0974820 This declination of the sun gives the time about March 2, and October 11. 9°. 0'. O" - - 10,+sin. 19,1943324 Lat. 52. 12.35 - - - cos. 9,787299^ ZPy 14. 47. 27 - - - sin. 9,4070328 DOCTRINE OF THE SPHERE. 33 The double of this gives 29°. 34'. 54", which, con- verted into time, gives \h. 58'. 20'' for the duration of the shortest twilight, it being supposed to end when the sun is 18° below the horizon. (98.) To Jind the sun's declination, ivhen it is just twilight all night. Here the sun at a (Fig. p. 25.) must be 18° below the horizon ; therefore 1 8*^ + dec. Qa = RQ = EH— comp. of lat. of place ; hence, the sun's dec. = comp, lat. — 18° ; look therefore into the Nautical Alnianachy and see on what days the sun has this declination, and you have the time required. The sun's greatest declination being 23°. 28' it follows, that if the com- plement of latitude be greater than 41°. 28', or if the latitude be less than 48°. 32', there can never be twi- light all night. If the sun be on the other side of the equator, then it's dec.= 18°, -comp. lat. {99-) If the sun's declination Ee be greater than EZ, then the sun comes to the meridian at e to the north of the zenith Z of the spectator ; and if we draw the secondary Zqm touching the parallel ae of declina- tion described by the sun, then Rm is the greatest azimuth from the north which the sun has that day, the 1L_Z azimuth increasing till the sun comes to q, and then decreasing ; for a circle from Z to any other point of € a will cut RO nearer to R, and it will also cut ea in two points which have the same azimuth, they being in the same vertical circle; in this case, therefore, the 34 DOCTRINE or THE SPHERE. sun has the same azimuth twice in the morning. If, therefore, we draw the straight Hne Zu perpendicular to the horizon, the shadow of this Hne, being always opposite to the sun, will, in the morning, first recede from the south point //, and then approach it, and therefore will go backwards upon the horizon. But if we consider PP' as a straight line, or the earth's axis produced, the shadow of that line will not go backv^ards upon the horizon, because the sun always revolves about that line, whereas it does not revolve about the perpendicular Zv, it never getting to the south of it. Hence it appears, that the shadow of the sun upon a dial can never go backwards, because the gnomon of a dial is parallel to PP', and therefore the sun must always revolve about the gnomon. The time when the azimuth is greatest is found from the right angled triangle PqZ; for (Trig. Art. 212) rad. X COS. ZPq = tan. qP x cot. PZ, or^ rad. x cos. hour angle = cot. dec. x tan. lat. ; therefore, log. cot. dec. + log. tan. lat. — \0,r:zlog. cos. hour angle from apparent noon. (100.) It has hitherto been supposed, that it is 12 o'clock when the sun comes to the meridian ZHN (Fig. p. 20) and that the clock goes just 24 hours in the interval of the sun's passage from any meridian till it returns to it again. But if a clock be thus adjusted for one day, it will not continue to show 1 2 o'clock every day when the sun comes to the meridian, be- cause it is found by observation, that the intervals of time from the sun's leaving any meridian till it returns to it again, are not always equal ; this difference be- tween the sun and the clock is called the Equation of Time, as will be explained in Chap. IV. Hence, when the clock does not agree with the sun, and the sun is at ^, any arc xe is not the measure of the time from 12 o'clock, but from the time when the sun comes to the meridian, or from apparent noon, as it is called. (101.) The method of finding the hour angle for the time at which a body rises, has been upon the sup- DOCTRINE OF THE SPHERE. 35 position that the body is upon the rational horizon at the instant it appears, or 90° from the zenith ; but all bodies in the horizon are elevated by refraction 33' above their true places ; this therefore would make them appear when they are 33' below the rational horizon, or 90*^ + 33' from the zenith; also, all the bodies in our system are depressed below their true places by parallax, as will be afterwards explained ; therefore from this cause they would not appear till they were elevated above the rational horizon by a quantity equal to their horizontal parallax, or when distant from the zenith 90° — hor. par. Hence, from both causes together, a body becomes visible when it's distance ZFhom the zenith = 90° -}- 33' — hor. parallax, f^ being the place of the body when it becomes visible, Z the zenith, and P the polej hence, knowing ZF^, also ZP the complement of latitude, and Pf^the com- plement of declination, we can find the hour angle ZPf^. A fixed star has no parallax, therefore in this case ZF=90^ 33'. Tojindthe Time in which the Sun passes the 3Ieridian, or the horizontal Wire of a Telescope. (102.) Let moj be the diameter of ^" of the sun, esti- mated in seconds of a great circle ; then, as the se- conds in mx, considered as a small circle, must be increased in proportion as the radius is diminished, because (Trig. Art. 75) when the arc is given, the angle is inversely as the radius, we have, sin. Px, or c 2 36 DOCTRINE OF THE SPHERE. COS. dec. rjc, : rad. :: seconds d' in ma: of a great circle : the seconds in w^r ofthe small circle ea, which (13) is equal to the seconds in qr, or, in the angle rPq, and therefore the angle rPq — d" divided by cos. dec. (rad. being unity) = c?" x sec. dec, which measures the time in which the sun passes over a space equal to it's diameter, and consequently the time the diameter will be in passing over the meridian ; hence, 15" in space (corresponding to l" in time) : d" x sec. dec. in space :: l" in time : the time in seconds of passing the ... d'y. sec. dec. meridian = —-r, ■ • 15 (103.) Hence, qr, the sun's diameter in right ascension, is equal to d" y. sec. dec. If, therefore, the sun's diameter=32'=1920", and it's dec. = 20°, it's diameterinrightascension=1920"x l,o64=34'.2,"88. The same is true for the moon, if ^" = it's diameter. XT, * . ^ rad." (104.) By Art. 93. qr=znx x ^ : : — ^ ^ •' ^ COS. lat. X sin. azim. ,. . ,., rad.'- = (if«a?=r?",thesmi sdiameterja x r— : ; — ^ ' ' COS. lat. X Sin. azim. hence, as before, the time of describing qr, or the time in which the sun ascends perpendicularly through a space equal to if s diameter, or the time of passing , . , . . , d" rad.^ an horizontal wire, is equal to ■— 77X ^ . ^ ■ = — ' ' 15' COS. lat. X sin. azim. DOCTRINE OF THE SPHERE. 37 The same expression must also give the time which the body of the sun is in ascending above the horizon. If ^"=1980" the horizontal refraction, then d" di- vided by 15" = 132"; hence, refraction accelerates the rid * rising of the sun by 132" x ; ^—^ : — . ° "^ cos. lat. X sm. azmi. On the Principles of DiaUing. (105.) As the apparent motion of the sun about the axis of the earth, is at the rate of 1 5° in an hour, very nearly, let us suppose the axis of the earth to project it's shadow into the meridian opposite to that in which the sun is, and then this meridian will move at the rate of 1 b"* in an hour. Hence, let ^P/?/?^^ represent a meridian on the earth's surface, POp the earth's axis, z the place of the spectator, HKRf^B. great circle, of which z is the pole ; draw the meridians Pip, P2p, &c. making angles with PRpoflb^, 30°, &c. respectively, then, sup- posing PR to be the meridian into which the shadow of OP is projected at 12 o'clock, Pi, P2, &c. are the meridians into which it is projected at 1,2, &c. o'clock, and the shadow will be projected on the plane HKRF^ into the lines OR, 0\, 02, &c. and the angles ROl, RO2, &c. will be the angles between the 12 o'clock line and the 1,2, &c. o'clock lines. Now in the right 38 DOCTRINE OF THE SPHERE. angled triangle PRl, we have (84) PR the latitude of the place, and the angle i?Pl = 15*; hence, (Trig. Art. 210) rad. : tan. 15° :: sin. PR : ta?i. Rl ; in the same manner we may calculate the arcs H2, RS, &c. In this case, we make the earth's axis the gnomon, and the shadow is projected upon the plane HKRV. Take a plane ahcd at %, parallel to HKRV, and it is the sensible horizon (2l), and draw zr parallel to JPOp ; then, on account of the great distance of the sun, we may conceive it to revolve about zr in the same manner as about PO, and consequently the shadow will be projected upon the plane ahcd^ in the same manner as the shadow of PO is projected upon the j^hine HKRV^ and therefore the hour angles are calculated by the same proportion. This is an hori- zontal dial. ( 106.) Now let NLzK be a great circle perpendicu- lar to PRpH, and consequently perpendicular to the horizon at Zy and the side next to H is full south. Then, for the same reason as before, if the angles Npl, Np2j &c. be 15°, vSO", &c. the shadow of ^O will be DOCTRINE OF THE SPHERE. 3.9 jDrojected into the lines Ol, 02, &c. at 1, 2, &c. o'clock, and the angles A^Ol, N^02, &c. will be mea- sured by the arcs A'"!, A 2, &c. Hence, in the right angled triangle 7?A^l, p^V=the complement of the latitude, and the angle Npl = l5^; therefore (Trig. Art. 210) rad. : tan. 15° :: ym. pN : tan. Nl ; in the same manner we find A^2, A^3, &c. Hence, for the same reason as for the horizontal dial, if zabc be a plane coinciding with NLzK, and sthe parallel to Op, .st will project it's shadow in the same manner on the plane zabc, as Op does on the plane NLzK, and therefore the hour angles from the 12 o'clock line are computed by the same proportion. This is a vertical south dial. In the same manner the shadow may be projected upon a plane in any position, and the hour angles calculated. (107.) In order to fix an horizontal dial, we must be able to tell the exact time of the sun's coming to the meridian ; for which purpose, find the time (92) by the sun's altitude when it is at the solstices, that being the best time of the year for the purpose, be- cause then the declination does not vary, and set a well-regulated watch to that time ; then, when the watch shows 12 o'clock, the sun is on the meridian ; at that instant, therefore, set the dial, so that the shadow of the gnomon may coincide with the 12 o'clock line, and it stands right. (108.) Hence, we may easily draw a meridian line upon an horizontal plane. Suspend a plumb line so that the shadow of it may fall upon the plane, and when the watch shows 12, the shadow of the plumb line is the true meridian. The common way is to describe several concentric circles upon an horizontal plane, and in the center to erect a gnomon perpendi- cularly to it, with a small round well defined head, like the head of a pin ; make a point upon any one ©f the circles where the shadow of the head falls upon 40 DOCTRINE OF THE SPHERE. it in the morning, and again where it falls upon the same circle in the afternoon ; draw two radii from these two points, and bisect the angle between them^ and the bisecting line will be a meridian line. This should be done when the sun is at the tropic, v/hen it does not sensibly change it's declination in the interval of the observations ; for if it do, the sun will not be equidistant from the meridian at equal altitudes. But this method is not capable of very great accuracy ; for the shadow not being very accurately defined, it is not easy to say at what instant of time the shadow of the head of the gnomon is bisected by the circle. If, how- ever, several circles be made use of, and the mean of the whole number of meridians so taken, be drawn, the meridian may be found with sufficient accuracy for all common purposes. (109.) To find whether a wall be full south for a vertical south dial, erect a gnomon perpendicularly to it, and hang a plumb line from it ; then when the watch, as above adjusted, shows 12, if the shadow of the gnomon coincide with the plumb line, the wall is full south. C't Chap. III. TO DETERMINE THE RIGHT ASCENSION, DECLINATION, LATITUDE AND LONGITUDE OF THE HEAVENLY BODIES. (110.) The foundation of all Astronomy is to deter- mine the situation of the fixed stars, in order to find, by a reference to such fixed objects, the places of the other bodies at any given time, and thence to deduce their proper motions. The positions of the fixed stars are found from observation, by knowing their right ascensions and declinations (4l); and these are found by means of the transit telescope and astronomical quadrant, as explained in my Treatise on Practical Astronomy ; and then, by computation, their latitudes and longitudes may be found. (ill.) As the earth revolves uniformly about it's axis, the apparent motion of all the heavenly bodies, arising from this motion of the earth, must be uni- form ; and as this motion is parallel to the equator {j^^^ the intervals of the times, in which any two stars pass over the meridian, must be in proportion to the arc of the equator intercepted between the two secondaries passing through them, because (13) this arc of the equator contains the same number of degrees as the arc of any small circle parallel to it, and com- prehended between the same secondaries ; and there- fore, if one increase uniformly, the other must. Hence, the right ascension of stai-s passing the meridian at different times, will differ in proportion to the difference of the times of their passing, that is, if one star pass the meridian 1 hour before another, the difference of their right ascensions is 15°. Hence, if the clock be supposed to go uniformly, we have the 42 RIGHT ASCENSION following rule : As the interval of the times of the succeeding passages of any one jixed star over the meridian : the interval of' the passages of any two stars :: 360° : their difference of 7'ight ascensio?is'^. By the same method we may find the difference of right ascensions of the sun or moon, when they pass the meridian, and a star, and therefore if that of the star be known, that of the sun or moon will j which conclusion will be more exact, if we compare them with several stars, and take the mean. (112.) Now to determine the right ascension of a fixed star, Mr. Flamstead proposed a method, by com- paring the right ascension of the star with that of the sun when near the equinoxes, the sun having the same declination each time ; and as this method has not been noticed by any writers, we shall give an ex- planation. Let AGCKE be the equator, ABCWE the ecliptic, -iS" the place of the star, Sm a secondary to the equator, and let the sun be at P, near to A, when it is on the meridian, and take CT-=PA, and diaw PL, TZ, perpendicular to AGC, and ZL is parallel to AC, and the sun's declination is the same at T as at P. Observe the meridian altitude of the sun when at P, and also the time of the passage of it's center over the meridian ; observe also at what time the star passes over the meridian, and then (111) find * A small correction must here be applied for the aberration of the star, in order to get the true difference of right ascensions, as will be explained ; because there is a small djfierence between the true and apparent places. OF THE HEAVENLY BODIES. 43 the apparent difference Lm of their right ascensions. When the sun approaches near to T, observe it's me- ridian altitude for several days, so that on one of them, at t, it may be greater, and on the next day, at e, it may be less than the meridian altitude at jP, so that in the intermediate time it must have passed through T; and drawing thy es, perpendicular to AGCE, observe, on these two days, the difference hrn, sm of the sun's right ascension and that of the star ; draw also sv parallel to Zo. Then, to find Zb, we may consider the variation both of the right ascension and declina- tion, to be uniform for a small time, and consequently to be proportional to each other; hence, vb (the change of meridian altitudes in one day) : oh (the dif- ference of the meridian altitudes at t and T, or the difference of declination) :: sh (the difference oi sm, bm found by observation) : Zh, which added to b m, or subtracted from it, according to the situation of m, gives Zm, to which add Lm, or take their difference, according to circumstances, and we get ZL, which subtracted from AGC, or 180", half the remainder will be ALj the sun's right ascension at the first ob- servation, to which add Z/m, and we get the star's right ascension at the same time. Instead of finding hZ, we might have found sZ, by taking TZ — es for the second term, and thence we should have got Zm. Thus we should get the right ascension of a star, upon supposition that the position of the equator had re- mained the same, and the apparent place of the star had not varied in the interval of the observations. But the intersection of the equator with the ecliptic has a retrograde motion, called the Precession of the Equinoxes ; also, the inclination of the equator to the ecliptic is subject to a variation, called the Nutation ; and from the aberration of the star, it's apparent place is continually changing; these must therefore be al- lowed for, by considering how much they have varied in the interval of the observations ; but these are not subjects to be treated of in an elementary treatise. 44 RIGHT ASCENSION AND Having thus determined the right ascension of one star, that of the rest may be found from it (ill). (113.) The practical method of finding the right ascension of a body from that of a fixed star, by a clock adjusted to sidereal time*, is this: Let the clock begin it's motion from O^ O'. o" at the instant the first point of Aries is on the meridian ; then, when any star comes to the meridian, the clock will show the apparent right ascension of the star, the right ascension being estimated in the time, at the rate of 15° for an hour, provided the clock is subject to no error, because it will then show, at any time, how far the first point of Aries is from the meridian. But as the clock is necessarily liable to err, we must be able, at any time, to ascertain what it's error is, that is, what is the difference between the right ascension shown by the clock, and the right ascension of that point of the equator which is at that time on the meridian. To do this, we must, when a star, whose apparent right ascen- sion is known, passes the meridian, compare it's appa- rent right ascension with the right ascension shown by the clock, and the difference will show the error of the clock. For instance, let the apparent right ascension of Aldeharan be 4h. 23'. 50" at the time when it's transit over the meridian is observed by the clock, and suppose the time shown by the clock to be Ah. 23'. 52", then there is an error of 2" in the clock, it giving the right ascension of the star 2" more than it ought. If the clock be compared with several stars, and the mean error taken, we shall have, more accurately, the error at the mean time of all the observations. These observations being repeated every day, we shall get the rate of the clock's going, that is, how fast it gains or loses. The error of the clock, and the rate of it's * A clock is said to be adjusted to sidereal time, when it is ad- justed to go 24 hours from the time a fixed star leaves the meridian till it returns to it, or it is the time of a revolution of the earth about it's axis. DECLINATION OF THE HEAVENLY BODIES. 45 coing, being thus ascertained, if the time of the true transit of any body be observed, and the error of the clock at the time be appUed, we shall have the right ascension of the body. This is the method by which the right ascension of the sun, moon, and planets are regularly found in observatories. (114.) The right ascension of the heavenly bodies being thus ascertained, the next thing to be explained is, the method of finding their declinations. Take the apparent altitude of the body, when it passes the meridian, by an astronomical quadrant, as explained in mv Treatise on Practical Astronomy ; correct it for parallax and refraction (Chap. VI. and VII.) and you get the true meridian altitude, Ht, or He, (Fig. page 25), the difference between which and the alti- tude HE of the equator (which, by Art. 87, is equal to the complement of the latitude previously deter- mined) is the declination Et, or Ee, required. Ex. On April 27, 1774, the zenith distance of the moon's lower limb, when it passed the meridian at Greenwich, was 68°. 19'. 37",3 ; it's parallax in alti- tude was 56'. 19",2, allowing for the spheroidical figure of the earth ; the barometer stood at 29, 58, and the thermometer at 49 ; to find the declination. Observed zenith distance of L.L. 68°. I9'. 37''3 Refr. cor. for bar. and ther. - + 2. 23 68. 22.00,3 Parallax - - 56. 19,2 True zenith distance of L.L. - 67. 25. 41,1 Semidiametcr ------ — 16. 35 Xrue zenith distance of the center Q7 . 9. 6,1 Latitude -51.28.40 Declination 50Mf/i - - - - ~ 15. 40. 26,1 The horizontal parallax and semidiameter may be taken from the Nautical Almanack; and the parallax 46 LATITUDE AND LONGITUDE. in altitude may be found, as will be explained when we come to treat of the parallax. Given the Right Ascension and Declination of an Heavenlif Body, and the OhUquiti/ of the Ecliptic, to find the Latitude and Longitude. (115.) Let ^> be the body, t C the ecliptic, ^ Q the equator, sr, sp perpendicular to r C, t Q- Then (Trig. Art. 212) tan. sp : rad. :: sin ^p : cot. s^^p. Hence, srp-hQTC=srr. Also, COS. s^p : rad. :: tan. p^ ; tan. s^ (Trig. Art. 21 9) rad. : cos.^v r :: tan. s t : tan. r^p (Trig. Art. 21 9) ,*. cos. s^p : COS. 5T r :: tan. p^ : tan. r ^ = COS. s rrx tdin. pr ., . . r .^ i -,7 1 i— the tangent of the lonmtude ; and COS. s ^p ° O ' the logarithmic operation is, ar. CO. log. COS. s Tp-\-^og. cos. s Tr-\-log. (mi. pY—lO,=: log. tan. r T. Also, (Trig Art. 210) rad. : sin. rr :: tan. rys : tan. sr the tangent oi latitude y and the logarithmic opera- tion is, log. sin. r T-\-log. tan. rxs— 10, = log. tan. sr. In this manner, the right ascensions and declinations of the fixed stars being found from observation, their latitudes and longitudes may be computed, and their places become determined (41) ; hence, a catalogue of the fixed stars may be made for any time. If the latitude and longitude be given, the right ascension and declination may be found in the same manner; considering y^C the equator, and y Q thq ecliptic. Chap. IV. ON THE EaUATION OF TIME. (116.) Having explained, in the last Chapter, the practical methods of determining the place of any body in the heavens, we come next to the considera- tion of another circumstrtiice not less important, that is, the irregula rit y of time as mea s ured by the s un. The best measure of time which we have, is a clock regulated by the vibration of a pendulum. But with whatever accuracy a clock may be made, it must be subject to go irregularly, partly from the imperfection of the workmansliip, and partly from the expansion and contraction of the materials by heat and cold, by which the length of the pendulum, and consequently the time of vibration, will vary. As no clock, there- fore, can be depended upon for keeping time accu- rately, it is necessary that we should be able to ascer- tain, at any time, how much it is too fast or too slow, and at what rate it gains or loses. For this purpose, it must be compared with some motion whicli is uni- form, or of which, if it be not uniform, you can ascer- tain the variation. The motions of the heavenly bodies have therefore been considered as most proper for this purpose. Now the earth revolving uniformly about it's axis, the apparent diurnal motion of the fixed stars about the axis must be uniform. If a clock, therefore, be adjusted to go 24 hours from the passage of any fixed star over the meridian till it returns to it again, it's rate of going may be at any time determined by comparing it with the transit of any fixed star, and observing whether the interval con- tinues to be 24 hours; if not, the difference shows 48 EOUATION OP TIME. how much it gains or loses in that time. A clock ad- justed to go 24 hours in this interval, is said to be adjusted to sidereal time. But if we compare a clock with the sun, and adjust it to go 24 hours from the time the sun leaves the meridian on any day, till he returns to it the next day, which is a true solar day, the clock will not, even if it go uniformly, continue to agree with the sun, that is, it will not show 12 when the sun comes to the meridian. (117.) For let P be the pole of the earth, vivyz it's equator, and suppose the earth to revolve about it's axis in the order of the letters v wy z;\et<^ DLE be the celestial equator, and ^ CL the ecliptic, in which D E the sun moves according to that direction. Let the sun be at a when it is upon the meridian of any place on any one day, and m the place when it is on the meridian the next day, and draw Pvae, Prmp, se- condaries to the equator, and let the spectator be at s on the meridian Pv, with the sun at a on his meridian. Then when the earth has made one revolution about it*s axis, P sv is come again into the same position ; but the sun having moved forward, the earth must continue to revolve, in order to bring the meridian Psv into the position Prni, so that the sun at m may be again in the spectator's meridian. Now the angle vPr js measured by the arc ep, which is the increase of the sun's right ascension in a true solar day, the right ascension being measured upon the equator EaUATION OF TIME. 49 T DLE (41) ; hence, the length of a true solar dai/t is equal to the time of the earth's rotation about it's axis + the time of it's describing an angle equal to the increase of the sun's right ascension on a true solar day . Now if the sun moved uniforrnlif^ and in the equator t DLE, this increase, ep, would be al- ways the same in the same time, and therefore the solar days would be always equal ; but the sun moves in the ecliptic t CL^ and therefore, if it's motion were uniforin, equal arcs (a m) upon the ecliptic would not give equal arcs {ep) upon the equator *. But the motion of the sun is not uniform, and there- fore am, described in any given time, is subject to a variation, which must, on this account also, necessarily make ep variable. Hence, the increase, ep, of the sun's right ascension in a day, varies from two causes, that is, from the ecliptic not coinciding with the equator, and from the unequal motion of the sun in the ecliptic ; therefore the length of a true solar day is subject to a continual variation ; consequently a clock, adjusted to go 24 hours for any one true solar day, will not continue to shew 12 when the suncomes to the meridian, because the intervals bythe clock will continue equal (the clock being supposed neither to gain nor lose), but the interval of the sun's passage over the meridian will vary. (118.) As the sun moves through 36o° of right ascension in 365^ days very nearly, therefore 36,5 1 days : 1 day :: 36o°. : 59'. 2>",2 the increase of right * For tjraw int parallel to ep, and suppose ?na to be indefinitely small ; then by plain trigonometry, ma : mt :: rad. : sin. mat, or Y" «e (Trig. Art. 125.) mf : ep :: cos. ae : rad. (Art. 13) ,'.ma : ep :: cq5. ae : sin. 7" ae :: (because sin. f a e =z COS. a Tex rad. _, • a ^ ^^^\ a «<> j- l Trig. Art. 212) cos. ae : cos. a Y^ex radius : hence, COS. ae the ratio of ma to ep is variable ; if therefore the sun's motion ma were uniform the corresponding increase ep of right ascension would not be uniform. D 50 EaUATION OF TIME. ascension in one day, if the increase were uniform, in which case the solar days would be equal, and these days are called mean solar days. If therefore a clock be adjusted to go 24 hours in a meffw solar day, it can- not continue to coincide with the sun, that is, to show 12 when the sun is on the meridian ; but the sun will pass the meridian, sometimes before 1 2 and some- times after. This difference is called the Equation of Time. A clock thus adjusted, is said to be adjusted to 7}iean solar time*. The time shown by the clock is called tj^ie or mea)i tiriie, and that shown by the sun is called apparent time. (119.) A clock adjusted to go 24 hours in a 7nean solar day, would coincide with an imagi nary star moy- ing uniformly in the equator wit h the sun's _rnean niotion^ 5,9'. 8'', 2 in rig h t ascens ion, if the star werelo set off f rom a ny given meridian when th e clo ck shows 12 ; that is, the clock would always show 12 when the star came to the meridian, because the interval of the passages of this star over the meridian would be a mean solar day. This star, therefore, if we reckon it's mo- tion from the meridian, in time, at the rate of 1 hour for 15^, would always coincide with the clock; that is, \vheii the clock shows 1 hour, the star's motion would be 1 hour in right ascension, reckoned in time at the rate of 1 5*^ for an hour ; when the clock shows 2 hours, the star's motion would be 2 hours ; and so on. . Hence, this star may be substituted instead of the clock; therefore, when the sun passes the given meridian, the difference between it's right ascension and that of the star, converted into time, is the differ- ence between the time when the sun is on the me- * As the earth describes an angle of 360°. 59'. 8",2 about it's axis in a mean solar clay of 24 hours, and an angle of 360o in a sidereal day, therefore 300°. 59'. 8",2 : 360°. ;: 2ik. : 23k. 56'. 4,"098 the length of a sidereal day in mean solar time, or the time from the passage of a fixed star over the meridian, till it returns to it again. EQUATION OF TIME. 51 ridian and 12 o'clock, or the equation of time; because the given meridian passes through the star at 12 o'clock, and it's motion, in respect to the star, is at the rate of 16° in an hour (121). (120.) Now, to compute the equation of time, let APLS be the ecliptic, ALv the equator, A the first A* point of aries, P the sun's apogee, S any place of the . sun ; draw Sv perpendicular to the equator, and take An = AP. When the sun departs from P, let the imaginary star set out from n with the sun's mean motion in right ascension, or in longitude, or at the rate of 59'. 8", 2 in a day, and when ii passes the meridian, let the clock be adjusted to 12, as described in the last article : these are the corresponding posi- tions of the clock and sun, as assumed by astronomers. Take nm=Ps, and when the star comes to m, the place of the sun, if it moved uniformly with it's mean motion, would be at s, but at that time let S be the place of the sun, and let the sun at S, and consequently V, be on the meridian ; then as ?n is the place of the imaginary star at that instant, 7nv is the equation of time. Let a be the mean equinox*, and draw az per- pendicular to A L ; then z on the equator would have coincided with a, if the equinox had moved uniformly; therefore we must reckon the mean right ascension from z. Now mv = Av—Am; but Am = Az-{-zm = Aax cos. aAz-\-zm= (because cos. aAz (23°. 28') =4-^ very nearly) \±-Aa + zm; hencejmv=Av — Z7n * The equinox has a retrograde motion, and that motion is not uniforni ; we here therefore suppose a to be the point where the equinox wouhl have been, if it moved uniformly with it's fitean velocity. D 2 52 EaUATION OF TIME. — -14-^^ ; but ^i; is the sun's true right ascension, z 7ti is the mean right ascension, or mean longitude, and i-i-^a jAz ^ is the equation of the eq uinoxes in right ascension ;| hence, the equation of time is equal toTIie difference of the suri^s true right ascension, and it's mean longitude corrected hi/ the equation of the equi- noxes in right ascension. When Am is less than Av, mean time precedes apparent^ and when greater, apparent time precedes mean ; for as the earth turns about it's axis in the direction Av, or in the order of right ascension, that body whose right ascension is least must come upon the meridian first ;fthat is, when the sun's true right ascension is ^rea/er than it's mean longitude corrected as above, we must add the equa- tion of time to the apparent, to get the mean time; and when it is less, we must subtract. To convert mean time into apparent, we must subtract in the former case, and add in the latter. This rule for computing the equation of time, was first given by Dr. Mashelyne in the Phil. Trans. 1764. (121.) As a meridian of the earth, when it leaves m, returns to it again in 24 hours, it may be con- sidered, when it leaves that point, as approaching a point at that time 360° from it, and at which g:i^^^'') it arrives in 24 hours. Hence, the relative velocity with which a meridian accedes to or recedes from m, is at the rate of 15° in an hour. Therefore, when the meridian passes through v, the arc v m, reduced into time at the rate of 15° in an hour, gives the equation of time at that instant. Hence, the equation of time is computed for the instant of apparent noon, or when the sun is on the meridian. Now the time of apparent noon in mean solar time, for which we com- pute.can only be known by knowingthe equation of time. To compute, therefore, the equation on any day, you must assume the equation the same as on that day four years before, from which it will differ but very little, and it will give the time of apparent noon, suffi- ciently accurate for the purpose of computing the EaUATION OF TIME. 63 equation. If you do not know the equation four years before, compute the equation for noon mean tirae^ and that will give apparent noon accurately enough. Ex. To find the equation of time on July 1, 1792, for the meridian of Greenwich, by Mayer'' s Tables. The equation on July 1, 1 788, was, by the Nautical Almanac, 3'. 28", to be added to apparent noon, to give the corresponding mean time; hence, for July 1, 1792, at 0//. 3'. 2", compute the true longitude*. * The reason of this operation will appear to those who under- stand the method of computing the place of the sun from the solar tables. The explanation of such matters comes not within the plan of this work. See tny Complete System of Astronomy. 54 EaUATION OF TIME. o ' CO o ' CO C^ r-< CO in CO s o * CO CO 1— 1 o * 12; -1 CO TP CO CI r-i r o < © bJD o •0 CO tP CO "co * as CO •^ CO C< F-H ft CO CO O bJD C o S CT3 1^ C< '^ ^ tfs «> «\ «^ CO 0^ ^ c^ a^ ^ "^ -^ lO J^ vC ^ C^ CO CO CO rH l-H 2 1 + 1 + 1 co' 1—1 d I— 1 CO Epoch for 1792. Mean Mot. July 1, 3' 28" Mean Longitude Equat. of Center Equat. ]) I. 2^ II. 2 in. S3 IV. True Longitude With this true longitude, and obhquity 23°. 27'. 48"j4 of the ecliptic, the true right ascension of the EaUATION OF TIME. 55 sun is found to he 3\ 11°. 5'. 4r',25 ; also, the equa- tion of the equinoxes in longitude = -0",6; hence, The mean longitude - - 3*. 10°. 13'. 25",4 -|-4-of-0",6 ----- - 0,55 Mean longitude corrected 3. 10. 13. 24,85 True right ascension - - 3. 11. 5. 41,25 Equation - _ - . _ 52. l6,4 Which converted into time, gives 3'. 29", 1 for the true equation of time; which must be added to apparent, to give the true time, because the true right ascension is greater than the mean longitude. (122.) The sun's apogee, P, has a progressive motion, and the equinoctial points, A, L, have a regressive motion ; the inclination also of the equator to the ecliptic is subject to a constant variation. Hence, the same Table of the equation of time cannot continue to serve for the same degree of the sun's longitude. Also, the sun's longitude at noon at the same place is different for the same days on different years, and it is for apparent noon that the equation is computed. For these reasons, the equation of time must be computed a-new for every year. (123.) The two inequalities are sometimes sepa- rately considered, thus: First, that arising from the obliquity of the ecliptic. Let the sun and the imagi- nary star set off together from L, and let us now assume LS= Lm ; and let each move uniformly with the mean velocity, and then they will come to S and m together. Now when L S h greater than 90*^, the hypothenuse, LS, is less than the base, Lv (Trig, Art. 197) ; there- fore Lm is less than Lv (the case represented by the Figure); the star therefore, being left behind, comes upon the meridian first, and consequently true time precedes apparent. But when LS is less than 90°, the hypothenuse, LS, is greater than the base, Lv\ there- fore Lm is greater than Lv, and m lies on the other side of V ; therefore the sun comes upon the meridian 56 EaUATION OF TIME. first ; consequently apparent time precedes true. Hence, from equinox to tropic, apparent time pre- cedes true ; and from tropic to equinox, true time precedes apparent. Secondly^ that arising from the unequal motion of the earth in it's orbit. Let us suppose the sun to move about the earth, instead of the earth about the sun, the effect here being just the same, and this supposition will render the explanation easier. Let the sun depart from the apogee, and let the imaginary star set off from thence at the same time, with the mean angular velocity of the sun. Now when the sun is at it's greatest distance, it's angular velocity it less than it's mean angular velocity. (Note to Art. 165), and consequently less than the velocity of the star; the star therefore getting forwarder than the sun, the sun comes upon the meridian first, as shown in Art, 120, and therefore apparent time precedes true ; and this will continue till the sun comes to it's least distance, where, having performed half it's re- volution, and the star also having performed half it's revolution, the sun and star will coincide, (see Art 168). Hence, from apogee to perigee, apparent time precedes true. Now the sun and star departing together from perigee, the sun's velocity is greater than that of the star ; the star therefore being left behind, comes upon the meridian first, and true time precedes appa- rent ; and this will continue till the sun comes to the apogee, where they again coincide. Hence, from perigee to apogee, true time precedes apparent. (124.) Whenever the time is computed from the sun's aUitude, that time must be apparent time, be- cause we compute it from the time when the sua comes to the meridian, which is noon, or 12 o'clock apparent time, and will differ from the time shown by a well-regulated watch or clock, by the equation of time. A clock or watch may therefore be regulated by a good dial ; for if you apply the equation, as be- fore directed, to the apparent time shown by the dial. EQUATION OF TIME. 5/ it will give the mean time, or that which the clock or watch ought to show. (125.) The equation of time was known to, and made use of by, Ptolemy. Tycfio employed only one part, that which arises from tlie unequal motion of the sun in the ecliptic; but Kepler made use of both parts. He further suspected, that there was a third cause of the inequality of solar days, arising from the unequal motion of the earth about it's axis. But the equation of time, as now computed, was not generally adopted till 1672, when Flamstead published a dis- sertation upon it, at the end of the works of Horrox. 58 Chap. V. ON THE LENGTH OF THE YEAR, THE PRECESSION OF THE EaUlNOXES FROM OBSERVATION, AND OBLiaUITY OF THE ECLIPTIC. (126.) From comparing the sun's right ascension every day with that of the fixed stars lying to the east and west, the sun is found constantly to recede from those on the west, and approach to those on the east ; hence, it's apparent annual motion is found to be from west to east; and the interval of time from it's leaving any fixed star till it returns to it again, is called a sidereal year, being the time in which the sun com- pletes it's revolution amongst the fixed stars, or in the ecliptic. But the sun, after it leaves either of the equinoctial points, returns to it again in a less time than it returns to the same fixed star, and this interval is called a solar or tropical year, because the time from it's leaving one equinox till it returns to it, is the same as from one tropic till it comes to the same again. This is the year on which the return of the seasonsdepends. On the Sidereal JTear. (127.) To find the length of a sidereal year' On any day when the sun is at Z on the meridian, (Fig. page 42), take the difference, Zm, between the sun's right ascension when it passes the meridian, and that of a fixed star, S •, and when the sun returns to the same part of the heavens the ne^tt year, compare it's right ascension with that of the same star for two days, one when their difference, b m, of right ascensions is LENGTH OF THE YEAR. 69 Jess, and the other when the difference, sm, is greater than the difference, Zm, before observed ; then bs is the increase of the sun's right ascension in the time, t ; and as the increase of right ascension may be con- sidered as uniform for a small time, we have bs : bZ :: t : the time, T, in which the right ascension is in- creased from b to Z ; this time, T, therefore, added to the time of the observed right ascension at b, gives the time when the sun is at the same distance, Zm, in right ascension from the star, which it was when ob- served at ^the year before ; the interval of these times is therefore the length of a sidereal year. The best time for these observations is about March 25, June 20, September 17, December 20, the sun's motion in right ascension being then uniform. Instead of ob- serving the difference of the right ascensions, you may observe that of their longitudes. If, instead of repeating the second observations the year after, there be an interval of several years, and you divide the observed interval of time when the difference of their right ascensions was found to be equal, by the number of years, you will have the length of a sidereal year more exactly. Ex. On April 1, 1669, at oh. 0.47", mean solar time, M, Picard observed the difference between the sun's longitude and that of Procyon to be 3\ 8°. 59'. 36", which is the most ancient observation of this kind, the accuracy of which can be depended upon; see Hist. Celeste, par 31. le Monnier, page 37. And on April 2, 1745, M. dela Caille foum], by taking their difference of longitudes on the 2d and 3d, that at llh. 10'. 45", mean solar time, the difference of their longitudes was the same as at the first observation. Now as the sun's revolution was known to be nearly 365 days, it is manifest that it had made seventy-six complete revolutions, in respect to the same fixed star, in the space of 76 years 1^/. llh. 6'. 58". Now in these 76 years, there were 58 of 365 days and 18 bis- sextiles of 366 days ; that interval therefore contains ^O LENGTH OF THE YEAR. 27759d. 1 1 A. 6'. 58"; which being divided by 76, the quotient is 365c?. 6h. 8'. 47". the length of a siderea! year. From the most accurate observations, the length of a sidereal year is found to be 3656?. 6h. g\ 11",5. On the Tropical Year. (128.) Observe the meridian altitude, «, of the sun on the day nearest to the equinox ; then the next year take it's meridian altitude on two following days, one when it's altitude, m, is less than a, and the next when it's altitude, ??, is greater than a, then n - m is the in- crease of the sun's declination in 24 hours; also, when the declination has increased by the quantity a — m, from the time when the meridian altitude m, was ob- served, the declination will then become a; and as we may consider the increase of declination to be uniform for a day, we have n~m : a — m v. 24 hours : the in- terval from the time when the sun was on the meridian on the first of the two days, till the sun has the same declination a, as at the observation the year before ; and thistinle, added to the time when the sun's altitude m was observed, gives the time when the sun's place in the ecliptic had the same situation in respect to the equinoctial points, which it had at the time of the ob- servation the preceding year; and the interval of these times is the length of a tropical year. If instead of repeating the second observation the next year, there be an interval of several years, and vou divide the interval between the times when the decli- nation was found to be the same, by the number of years, you will get the tropical year more exactly. Ex. M. Cassini informsus, that on March 20, 1672, his father observed the meridian altitude of the sun's upper limb at the Royal Obprvatory at Paris, to be 41°. 43' ; and on March 20, 1716, he himself observed the meridian altitude of the upper limb to be 41°. 27'. 10"; and on the 21st to be 41°. 51': therefore the difference of the two latter altitudes was 23'. 50", and PRECESSION OF THE EGLUINOXES. 6l of the two former 15', 50"; hence, 23'. 50'.: 15'. 50" :: 24 hours : I5h. 56'. 39"; therefore, on March 20, 1716, at Ibh. 56'. 39", the sun's declination was the same as on March 20, 1672. Now the interval between these two observations was 44 years, of which 34 consisted of 365 days each, and 10 of 366 ; there- fore the interval in days was 16070 ; hence, the whole interval between the equal declinations was 16070 days 15/?-. 56'. 39", which divided by 44, gives 365d. 5h. 49'. 0". 53'", the length of a tropical year from these observations. From the best observations, the length of a tropical year is found to be 365 c?. 5h, 48'. 48". To Jind the Precession of the Equinoxes from Observation. (129.) The sun returning to the equinox every year before it returns to the same poin t in th e heaven s, shows that the equinoctial points have a reLrograde^ motion, and this arises from the moti on of the e quator, which is caused by the attraction of the sun and moon upon the earth, in consequence of it's spheroidical figure. The eiFect of this is, that the longitude of the stars must constantly increase ; and by comparing the longitude of the same stars at dif- ferent times, the motion of the equinoctial points, or the precession of the equinoxes, may be found. (130.) Hipparchus was the first person who ob- served this motion, by comparing his own observations with those which Timocharis made 155 years before. From this he judged the motion to be one degree in about 100 years; but he doubted whether the observa- tions of Timocharis were accurate enough to deduce any conclusion to be depended upon. In the year 128 before J. C. he found the longitude of Firgin's Spike to be 5*. 24° ; and in the year 1750 its longi- tude was found to be 6\ 20**. l', the difference of 62 THE ANOMALISTIC YEAR. which is 26". 21'. In the same year he found the longitude of the Lyons Heart to be 3'. 29°. 50' ; and in 1750 it was 4'. 26°. 21', the difference of which is 26°. 31'. The mean of these two gives 26"^. 26' for the increase of longitude in I878 years, or 50". 40'" in a year for the precession. By comparing the observa- tions of Albategnius, in the year 878, with those made in 1738, the precession appears to be 51". 9'". From a comparison of 15 observations of Tychoj with as many made by M. de la Caille, the precession is found to be 50". 20"'. But M. de la Lande, from the observations of M. de la Caille, compared with those in Flatiistead's Catalogue, determines the secular pre- cession to be 1°. 23'. 45", or 50", 25 in a year. (131.) The precession being given, and also the length of a tropical year, the length of a sidereal year may be found by this proportion ; 36o° — 50",25 : 360°. :: 365d. bh. 48'. 48" : 365c?. 6h. 9'. 11"^ the length of the 'sidereal ye^x. 0n the Anomalistic Year. m (132.) The year, called the anomalistic year, is sometimes used by astronomers, and is the time from the sun's leaving its apogee, till it returns to it. Now the progressive motion of theapogeein a year is 1 1"5 75, and hence, the anomalistic must be longer than the sidereal 3^ear, by the time the sun takes in moving over ll",75 of longitude at it's apogee; but when the sun is in it's apogee, it's motion in longitude is 58'. 13" in 24 hours; hence, 58'. 13" : ll",75 :: 24 .hours : 4'. 50"-^-, which added to 365c?. 6h. 9'. 11|", gives liGbd. 6d. 14'. 2"^ the length of the anomalistic y ear. M. de la Lande determined this motion of the apogee, from the observations of M. de la Hire and those of Dr. Mashelyne. Cassini made it the same. OBLiaUITY OF THE ECLIPTIC. 63 On the OhUqidtij of the Ecliptic. (133.) The method used by astronomers to deter- mine the obliquity of the ecliptic, is that explained in Art. 86, by taking half the difference of the greatest and least meridian altitudes of the sun. The follow- ing is the obliquity, as determined by the different astronomers. »* Eratosthenes 230 years before «/. C. 23°. 51'. 20" Hipparchus 140 years before J. C. Ptolemy 140 years after J. C. Pappus in the year 390 - - - Alhategnius in 880 - _ - - Arzachel in 1070 - - . - , Prophatius inl3(50 - - - - Reg iomojit anus in 1460 - - - IValtherus in 1490 - - _ _ Copernicus in 1500 - - - - Tycho in 1587 ----- 23. Cassini (the Father) in 1656 Cassini (the Son) in 1672 - Flamstead in 1690 - - - De la Caille in 1/50 . . - Dr* Bradley in 1/50 - - Mayer in 1750 . _ . . Dr. Mashelyne in 1769 - - M. de la Landem 1768 - - The observations of Alhategnius. an Arabian, are here corrected for refraction. Those of Walt her us ^ M. de Caille computed. The obliquity by Tycho is here put down as correctly computed from his obser- vations. Also, the obliquity, as determined by Flam- stead, is corrected for the nutation of the earth's axis. These corrections M. de la Lande applied. (134.) It is manifest, from the above observations, that the obliquity of the ecliptic continually decreases ; 23. 51. 20 23. 51. 10 23. 30. 23. 35. 40 23. 34. 23. 32. 23. 30. 23. ^9- 47 23. 28. 24 23. ^9- 30 23. 29- 2 23. 28. 54 23. 28. 48 23. 28. 19 23. 28. 18 23. 28. 18 23. 28. 8,5 23. 28. 64 OBLiaUITY OF THE ECLIPTIC. and the irregularity which here appears in the diminu- tion, we may ascribe to the inaccuracy of the observa- tions, as we know that they are subject to greater errors than the irregularity of this variation. If we compare the first and last observations, they give a diminution of 70" in IGO years. If we compare the last with that of Tycho, it gives 45". The last, com- pared with that of Flamstead, gives 50". If we com- pare that of Dr. Mashelyne with Dr. Bradley* s and Mayer's, it gives 50". The comparison of Dr. Mashelyne's determination, with that of M. de la Lande, which he took as the mean of several results, gives 50". We may therefore state the secular diminu- tion of the obliquity of the ecliptic, at this time, to be 50", as determined from the most accurate observations. This result agre^es very well with that deduced from theory. 65 Chap. VI. ON PARALLAX. (135.) The center of the earth describes that circle in the heavens which is called the ecliptic ; but as the same object would appear in different positions in respect to this circle, when seen from the center and surface, astronomers always reduce their observations to what they would have been, if they had been made at the center of the earth, in consequence of which, the places of the heavenly bodies are computed as seen from the ecliptic, and it becomes a fixed plane for that purpose^ on whatever part of the earth's surface the observations are made. (136.) Let C be the center of the earth, ^ the place of the spectator on it's surface, S any object, Zjfl the sphere of the fixed stars, to which the places of all the bodies in our system are referred ; Zthe zenith, i/the horizon ; draw CSm, ASn, and m is the place of S seen from the center, and ?i from the surface. Now the plane S^^^C passing through the centerof the earth, must be perpendicular to it's surface, and consequently it will pass through the zenith Z ; and the points, m, n, lying in the same plane, the arc of parallax, m/z, must he in a circle perpendicular to the horizon ; and hence, the azimuth is not affected, if the earth be a sphere. Now the parallax, ?)in, is measured by the angle mSn, ov ASC; and (Trig. Art. 128) CS : CA :: sin. SAC, or sin. SAZ, : sin. ^^-S'Cthe parallax=: CA X sin. SAZ \ r^ i • + ^ • ^i — -; As CA IS constant, supposmg the earth to be a sphere, the sine of the parallax varies as the sine of the apparent zenith distance directly, and 66 ON PARALLAX. the distance of the body from the center of the earth inversely. Hence, a body in the zenith has no pa- rallax, and at a^ in the horizon it is the greatest. And zr if the object be at an indefinitely great distance, it has no parallax ; hence, the apparent places of the fixed stars are not altered by it. As ii is the apparent place, and m is called the true place, the parallax depresses an object in a vertical circle. For diflTerent altitudes of the same body, the parallax varies as the sine, .9, of the apparent zenith distance; therefore, if/>=:the hori- zontal parallax, and radius be unity, we have 1 :s :: p : ps, the sine of the parallax. To ascertain, therefore, the parallax at all altitudes, we must first find it at some given altitude. (137.) First method, for the siin. Aristarchus proposed to find the sun's parallax, by observing it's elongation from the moon at the instant it is dichoto- mized^ at which time the angle at the moon is a right angle; therefore we may find the angle which the distance of the moon subtends at the sun ; which, di- minished in the ratio of the moon's distance from the earth's center to the radius of the earth, would give the sun's horizontal parallax. But a very small error in the time when the moon is dichotomized (and it is impossible to be very accurate in this), will make so very great an error in the sun's parallax, that nothing ON PARALLAX. 6f can he deduced from it to be depended upon. Ven- (lelinus determined the angle of elongation when the moon was dichotomized, to be 89°. 45', from which the sun's parallax was found to be 15". But P. Riccioli found it to be 28" or 30" from like observations. (153.) Second method. Hipparchus proposed to find the sun's parallax from a lunar eclipse, by the following method. Let S be the sun, E the earth, Ev the length of it's shadow, 7nr the path of the moon in a central eclipse. Observe the length of this eclipse, and then, from knowing the periodic time of the moon, the angle mEr, and consequently the half of that angle, or nEr, will be known. Now the horizontal parallax, ErB, of the moon being known, we have the angle Evr = ErB — nEr; hence, we know EAB* =AES- Evr=JES- ErB + nEr; that is, the sun's horizontal parallax = the apparent semidiameter of the * Supply the line AE, £ 2 68 ON PARALLAX. sun - the horizontal parallax of the moon + the semi- diaYneter of the earth's shadow where the moon passes through it. The objection to this method, is, the great difficulty of determining the angle nEr, with suffi- cient accuracy; for any error in that angle will make the same error in the sun's parallax, the other quanti- ties remaining the same. By this method, Vtohmij made the sun's horizontal parallax 2'. 50". Tycho made it 3'. (139.) Third method, for the moon. Take the meridian altitudes of the moon, when it is at it's greatest north and south latitudes, and correct them for refraction; then the difference of the altitudes, thus corrected, would be equal to the sum of the two latitudes of the moon, if there were no parallax ; con- sequently the difference between the sum of the two latitudes and the difference of the altitudes, will be the difference between the parallaxes at the two alti- tudes. Now, to find from thence the parallax itself, let S^ s, be the sines of the greatest and least apparent zenith distances, P, p, the sines of the corresponding parallaxes; then as, when the distance is given, the parallax varies (136) as the sine of the zenith dis tance, ^ ^ , ^ ^ s X P — p S : s :: P : p; hence, S—s : s :: P - p : /»= — ^— — the parallax at the greatest altitude. This supposes that the moon is at the same distance in both cases ; but as this will not necessarily happen, we must cor- rect one of the observations, in order to reduce it to what it would have been, had the distance been the same, the parallax at the same altitude being inversely as the distance (136). If the observations be made in those places where the moon passes through the zenith of one of the observers, the difference between the sum of the two latitudes and the zenith distance at the other observation, will be the parallax at that altitude. (140.) Fourth method. Let a body, P, be observed from two places, A, B, in the same meridian, then ON PARALLAX. 69 the whole angle APB, is the sum of the two parallaxes of the two places. The parallax (136) APC=hor. par. X sin. PJL, taking A PC for si«. A PC, P and the parallax BPC = hor. par. x sin PB3I; hence, hor. par. x sin PAL -{-sm.PB3I=APB, .*. hor. par. = APB divided by the sum of these two sines. If the two places be not in the same meridian, it does not signify, provided we know how much the altitude varies from the change of declination of the body in the interval of the passages over the meridians of the two observers. Ex. On Oct. 5, 1751, M. de la Ca'ille, at the Cape of Good Hope, observed Mars to be l'. 25",8 below the parallel of X in Aquarius, and at 25° distance from the zenith. On the same day, at Stockholm, Mars was observed to be T. 57",/ below the parallel of \, and at 68°. 14' zenith distance. Hence, the angle APB is 3l",9, and the sines of the zenith dis- tances beingO,4226and 0,9287, the horizontal parallax was 23",6. Hence, if the ratio of the distance of the earth from Mars to it's distance from the sun be found, we shall have the sun's horizontal parallax, the hori- zontal parallaxes being inversely as their distances from the earth (136). (141.) Fifth method. Let EQ be the equator, P it's pole, Z the zenith, i; the true place of the 70 ON PARALLAX. hoAy, and r the apparent place, as depressed by paral- lax in the vertical circle ZK, and draw the secondaries. Pva, Prh ; then ab is the parallax in right ascension, and r 5 in declination. Now, m' I vs :: 1 (rad,) : sin. vrs^ or ZvP (Trig.Art.125) vs : ab :: cos. va : 1 (rad.) Art. 13. /. V7' : ab :: cos. va : sin. ZvP ; .^ r ■, vrx sin, ZvP , ^ , therefore ao =- .; but t;r = ^or. par. X COS. va sin.vZ (136), and (Trig. Art. 22l)sin.i;Z : sin. ZP :: sin. ZPv : sin. ZvP^'^h^ijlJlIhlE^ , there- sin. i^Z r V 1 i^i. i- I. hor.par. x sin..^P x sin.Z^P?' iore,by substitution, ab = L . COS. va Hence, for the same star, where the hor. par. is given, the parallax in right ascension varies as the sine of the T 1 A 1 i.1 z. ^b X COS. va hour angle. Also, the hor.par. = — — — — _ -.-. sm. ZP X sin. ZPv For the eastern hemisphere, the apparent place b, lies pn the equator to the east of a, the true place, and therefore the right ascension is increased by parallax ; but in the western hemisphere, b lies to the west of a, and therefore therightascension isdiminished. Hence, if the right ascension be taken before and after the meridian, the whole change of parallax in right ascen- sion between the two observations, is the sum (s) of the two parts before and after the meridian, and the ON PARALLAX. ^1 s X COS. vet hor, par.='-. — t^jt- — o» where 8= sum of the sines of the two hour angles. On the meridian there is 11 • ■ 1 4. • r 1 vr X sin. ZvP no parallax m right ascension^ tor flo= cos. va ' where the angle ZvP, and consequently it's sine, vanishes. (142.) To apply this rule, observe the right ascen- sion of the planet when it passes the meridian, compar- ed with that of a fixed star, at which time tliere is no ])arallax in right ascension ; about 6 hours after, take the difference of their right ascensions again, and ob- serve how much the difference, dy between the appa- rent right ascensions of the planet and fixed star has changed in that time. Next, observe the right ascen- sion of the planet for 3 or 4 days when it passes the meridian, in order to get it's true motion in right ascension; then, if it's motion in right ascension in the above interval of time, between the taking of the right ascensions of the fixed star and planet on and off tlie meridian, be equal to d, the planet has no parallax in right ascension; but if it be not equal to d, the difference is the parallax in right ascension ; and hence, by the last article tiie horizontal parallax will be known. Or one observation may be made before the planet comes to the meridian, and one after, by which a greater differ- ence will be obtained. Ex. On Aug. 15, 1/19? Mars was very near a star of the 5th magnitude in the eastern shoulder of Aqua- rius, and SitQh. 18' in the evening. Mars followed the star in lO'. 17", and on the iGth, at 4h. 21' in the morning, it followed it in 10'. l", therefore, in that interval, the apparent right ascension of Mars had increased 16" in time. But, according to observations made in the meridian for several days after, it appeared that Mars approached the star only 14" in that time, from it's motion; therefore 2" in time, or 30" in motion, is the effect of parallax in the interval of the ob- servations. Now the declination of Mars was 15°, the co-latitude 41°. 10', and the two hour angles 72 ON PARALLAX. 49°. 15', and 56°. 3g'; therefore the hor. par.=^ 30" X COS. 15° ■ - -^27^" But sin. 41°. 10' X sin. 49°. 15' + sm.56°.39' at that time, the distance of the earth from Mars was to it's distance from the sun, as 37 to 100, and there- fore the sun's horizontal parallax comes out 10",17. (143.) But, besides the effect of parallax in right ascension and declination, it is manifest that the latitude and longitude of the moon and planets must also be affected by it; and as the determination of this, in respect to the moon, is in many cases, particularly in solar eclipses, of great importance, we shall proceed to show how to compute it, supposing that we have given the latitude of the place, the time, and consequently the sun's right ascension, the moon's true latitude and lon- gitude, with her horizontal parallax. (144.) Let HZR be the meridian, t EQ the equator, p it's pole; y C the ecliptic, P it's pole j t the first point of Aries, JYQi? the horizon, Zthe zenith, Z^Z/ a secondary to the horizon passing through the true place r, and aparent place t, of the moon ; draw Pt,Pr^ which produce to a, drawing the small circle ts, parallel to ov ; then rs is the parallax in latitude, and ov the parallax in longitude. Draw the great circles, f P, PZAB, Ppde, and ZW, perpendicular to pe ; then, as X Pz=^o°, and t 7^=90°, Y must (4) be the pole oi Ppde, and therefore 6^7" =9^°; consequently d is one of the solstitial points,® orv;? ; also, draw Zxper:- ON PARALLAX. J6 peiidicular to Pr, and join Z'r ,p<v • Now ^ E, or the angle c^ pE, or Zp ^ , is the right ascension of the mid-heaven, which is known; PZ~AB (because AZ is the complement of both) the altitude of the highest point, A^ of the ecliptic above the horizon, called the nonagesimal degree, and ^y' Ay or the angle <Y' PA, is it's longitude; also, Zp is the co-latitude of the place, and the angle Zp W is the difference be- tween the right ascension of the mid-heaven ^cpEy and ^ e. Now, in the right-angled triangle Zp JVy (Trig. Art. 212) rad. x cos. /; = tan. p W y, cot. pZ; therefore, log. tan. pW=\0,-\-log. COS. p — log. cot. pZ, (Trig. Art. 213); hence, P PV—pTV-^pP, where the upper sign takes place when the right ascension of the mid-heaven is less than 180°, and the lower sign, when greater. Also, in the triangles WZp, IVZ P, we have (Trig. Art. 231) sin. Wp : sin. JVP :: tan. WP Z : tan. IVpz :: (Trig. Art. 82) cot. WpZ : cot. WPZ, or, tan. APcv ; therefore (Trig. Art. 49), Log.tan. APT = ar. co. lo. sin. Wp-\-log. sin.WP-\-lo. cot. WpZ—lO; and as we know ^p o,ov <^ Po, the true longitude of the moon, we know APo, or ZPcc. Also, in the triangle IVPZ, we have (Trig, Art. 21 9) cos. WP Z, or sin, AP^f : I'ad. :: tan. WP : tan. ZP ; therefore, Log. tan. ZP = 10, -|- log. tan. WP - log. sin. AP^ • Again, in the triangle ZPr, we know ZP, Pr, and the angle, P, from which the angle ZrP, or t rs, may be thus found. In the right-angled triangle ZPx, we know -ZP, and the angle P; hence, (Trig. Art. 212) rad. X cos. ZPx — cot. PZ x tan. P.r; therefore, Log. tan. Px= 10, -\-cos. ZPx — log. cot. PZ ; therefore we know rx-, hence, (Trig. Art. 231) sin. rx : sin. Px :: tan. ZPx : tan. Zrx, or trs; there- fore. Log. tan. Z)\v=ar. co, log. sin. rx-\-log.si7i. Px-\-log.tan.ZPx— 10; also, in the right-angled triangle Zrx, we have (Trig. 74 ON PARALLAX. Art. 212) rad. x cos. Z?'x=cot. Zrx tan. rx; there- fore. Log. cot. Zr = 10, + log. COS. Z roc — log. tan. ?'.r. With this true zenith distance Z?', find (136) the parallax, as if it were the apparent zenith distance, and it will give you the true parallax nearly ; add this therefore to the true zenith distance, and you will get nearly the apparent zenith distance, to which com- pute again (136) the parallax, and thus you will get the true parallax, rt, extremely nearly; then, in the right-angled triangle rst, which may be considered as plane, we have (Trig. Art. 125) rad. : cos. r ;: rt : rs, the parallax in latitude ; therefore, log. r.s=log. rt +log. cos. r— 10, = log. par. latitude. Also, rad. : sin. r :: rt : ts; therefore, log. ts. = log. rt + log. sin. 7' — 1 ; hence, (13) cos. tv : rad. :: ts : ov, the parallax in longitude ; therefore, log. ov = 10, + log. ts—log. COS. tv = log. par. longitude. Ex. On January I, 177^5 at 9^. apparent time, in lat. 53° N. the moon's true longitude was 3^ 18'^. 27'. 35", and latitude 4°. 5'. 30" S. and it's horizontal parallax 61'. 9"; to find its parallax in latitude and longitude. The sun's right ascension was 282°. 22'. 2" by the Tables, and its distance from the meridian 135^; also, the right ascension op E, of the mid-heaven, was 57°. 22'. 2" ; hence, the whole operation for the solu- tion of the triangles will stand thus : ^ CZpff^ =32°. 37'. 58" - 10, + cos. 19.9253864 ^jZp =37. 0. 0. - - cot. 10.1228856 ^ (p^ =32. 23. 57 - - tan. 9.8025008 Pp' =23. 28. O PfV =55. 51. 67 ON PARALLAX. 75 ^ ^pJV =32°. 23'. 57" - ar. CO. sin. O.2709805 fe ipfV =55. 51. 57 - - - sin. 9-9178865 N^<Z;?^=32. 37. 58 - - - cot. 10.1935941 l^J I^^JPr =^7- 29. 8 - - - tan. 10.3824661 H oPt =108. 27. 35 oPJ = 40. 58. 27 N rwp =55. 51. 57 - 10,+tan. 20.1688210 ^ )APr =67. 29. 8 - - sin. 9.9655700 :^1 ^ IZP =57. 56. 36 - - - tan. 10.2032510 (;g (ZPv =40. 58. 27 - 10,+cos. 19,8779500 S3 )ZP =57. 56. 36 - - - cot. 9.7967445 'x =50. 19. 33 - - -tan. 10.0812055 *r =94. 5. 30 = 43. 45. 47 - ar. co. sin. O.1600743 = 50. 19. 33 - - - sin. 9.8863144 U^lZPx =40.58.27 - - -tan. 9.9^87676 "^z ^^Zrx =44. 1. 16 - - -tan. 9.9851563 'Zrx =44. 1. 16 - 10,-hcos. 19.8567795 = 43. 45. 57 - - - tan. 9.98 12846 r =53. 6. 10 - - -cot. 98754949 Zr =53. 6. 10 - - -sin 9.9029362 Hor. par. = 6l'. 9" = 3669" - log. 3.5645477 rt uncorrected = 2934" = 48'.54"log. 3.4674839 App.zen.clist.Z#=53''.55'.4"nearly,sin.9,9075042 Hot. par. =6l'. 9"=3669" - log. 3.5645477 76 ON PARALLLAX. ^ rPar. rt cor. = 2965" = 49'. 25" log. 3.4720519 ^ J/r5 = 44°. 1'. 16" - - - - COS. 9.8567795 ^ U^^par. in Zaf. = 2132"=35'. 32" log. 3.3288314- >^cor. = 2965" log. 3.4720519 ^ ')trs = A4°.l'. iQ" . - - - 6111.9.8419369 .J^5 = 206l"=34'. 21" - - - log. 3.3139888 True lat. ?-o = 4°. 5'. 30" App. Iat.^i> = 7'0 + r5=:4°. 41'. 2" cos. 9.9985472 #^=2061" ----- 10, + log. 13.3139888 o?;par.in/o««f. = 2067"=34'.27" log. 3.3154416 The value of /y is 7'o — or + r*, according as the moon has N. or S. latitude. The order of the signs being from West to East, from A towards C is eastward, and from A towards ^ is westward ; now the parallax depressing the body from r to t, increases the longitude from o to ?; ; but if the point had been on the other side of ^, ov would have lain the contrary way; hence, when the body is to the East of the nonagesimal degree, the parallax increases the longitude ; and when it is to the West, it diminishes the longitude. (145.) According to the Tables of Mayer, the greatest parallax of the moon (or when she is in her perigee and in opposition) is 61'. 32"; the least parallax (or when in her apogee and conjunction) is 53'. 52", in the latitude of Paris; the arithmetical mean of these is 57'. 42"; but this is not the parallax at the mean distance, because the parallax varies inversely as the distance, and therefore the parallax at the mean distance is 57'. 24", an harmonic mean between the two. M. de Lamhre re-calculated the parallax from the same observations from which Mayer calculated it, and found it did not exactly agree with. Mayer s. He made the equatorial parallax 57'. H",4. M. de la Lande makes it57'. 3" at the equator, 56'. 53",2 at the ON PARALLAX. 77 pole, and 57'. l" for the mean radius of the earth, sup- posing the difference of the equatorial and polar diameters to be of the whole. From the formula 300 of Mm/er, the equatorial parallax is 57'. 11", 4. (146.) To find the mean distance, Cs,oi the moon, we have AC, the mean radius (?•) of the earth, : Cs, the mean distance (Z)) of the moon from the earth, :: sin. 57'. l"=JsC {145) : radius :: 1 : 6o,3 ; conse- quently Z>=6o,3r; but r = 3964 miles; hence, Z) = 239029 miles. (147.) According to M. de la Lande, the horizontal semidiameter of the moon : it's horizontal parallax for the mean radius (r) of the earth -.: 15' : 54'. 57",4, or very nearly as 3 : 11; hence, the semidiameter of the moon is -iV ^ = tV X 3964=1081 miles ; and as the magnitude of spherical bodies are as the cubes of their radii, we have the magnitudes of the moon and earth as 3^ : 1 1^, or as 1 : 49 nearly. 78 Chap. VI. ON REFRACTION. (148.) VVhen a ray of light passes out of a vacuum into any medium, or out of any medium into one of greater density, it is found to deviate from it's rectilinear course towards a perpendicular to the surface of the medium into which it enters. Hence, light passing out of a vacuum into the atmosphere will, where it enters, be bent towards a radius drawn to the earth's center, the top of the atmosphere being supposed to be spherical and concentric with the center of the earth ; and as, in approaching the earth's surface, the density of the atmosphere continually increases, the rays of light, as they descend, are constantly entering into a denser medium, and therefore the course of the rays will continually deviate from a right line, and describe a curve ; hence, at the surface of the earth, the rays of light enter the eye of the spectator in a different direc- tion from that in which they would have entered, if there had been no atmosphere; consequently the ap- parent place of the body from which the light comes, must be different from the true place. Also, the re- fracted ray must move in a plane perpendicular to the surface of the earth ; for, conceiving a ray to come in that plane before it is refracted, then the refraction being always in that plane, the ray must continue to move in that plane. Hence, the refraction is always in a vertical circle. The ancients were not unac- quainted with this effect. Ptolemy mentions a dif- ference in the rising and setting of the stars in different states of the atmosphere; he makes, however, no allowance for it in his computations from his observa- ON REFRACTION. 79 tlonsj this correction^ therefore, must be appUed, where great accuracy is required. Archimedes ob- served the same in water, and thought the quantity of refraction was in proportion to the angle of incidence. Alhazen, an Arabian optician, in the eleventh century, by observing the distance of a circumpolar star from the pole, both above and below, found them to be different, and such as ought to arise from refraction. Snellkis, who first observed the relation between the angles of incidence and refraction, says, that PFalthei^us, in his computation, allowed for refraction; but Tyclm was the first person who constructed a table for that purpose, which, however, was very incorrect, as he supposed the refraction at 45° to be nothing. About the year l()6o, Cassini published a new table of refrac- tions, much more correct than that of Tycho ; and, since his time, astronomers have employed much at- tention in constructing more correct tables, the niceties of modern astronomy requiring their utmost accuracy. To find the quantity of refraction. (149.) First method. Take the altitude of the sun, or a star whose right ascension and declination are known, and note the time by the clock ; observe also the times of their transits over the meridian, and that interval gives the hour angle. Now, in the triangle z PZx, we know PZ and Px, the complements of lati- tude and declination, and the angle xPZ, to find the 80 ON REFRACTION. side Zx (92), the complement of whicli is the true altitude, the difference between which and the oh- served altitude, is the refraction at that altitude. Ex. On May 1, 1738. at bh. 20' in the morning, Cassini observed the altitude of the sun's center at Paris to be 5°. O'. 14'', and the sun passed the meridian at I2h. 0'. O", to find the refraction, the latitude bein^y 48°. 50'. 10", and the declination was 15°. 0'. 25". The sun's distance from the meridian was 6h. 40', which gives 100° for the hour angle xPZ ; also, PZ = 41°. 9'. 50", and Px=74\ 59'. 35"; hence, (Trig. Art. 233) Zx=85°. lO'. 8", consequently the true alti- tude was 4°. 49'. 52". Now to 5°. O'. 14", the appa- rent altitude, add 9" for the parallax, and we have 5°. 0'. 23" the apparent altitude corrected for parallax ; hence, 5°. O'. 23" -4°. 49'. 52"=10'.3l", the refrac- tion at the apparent altitude 5°. O'. 14". (150.) Second method. Take the greatest and least altitudes of a circumpolar star which passes through, or very near, the zenith, when it passes the meridian above the pole ; then the refraction being nothing in the zenith, we shall have the true distance of the star from the pole at that observation, the altitude of the pole above the horizon, being previously determined ; but when the star passes the meridian under the pole, we shall have it's distance affected by refraction, and the difference of the two observed distances, above and below the pole, gives the refraction at the apparent altitude below the pole. Ex. M. de la Caille, at Paris, observed a star to pass the meridian within & of the zenith, and conse- quently, at the distance of 41°. 4' from the pole ; hence, it must pass the meridian under the pole at the same distance, or at the altitude 7°- 46'; but the ob- served altitude at that time was 7°- 52'. 25" ; hence, the refraction was 6'. 25" at that apparent altitude. (151.) Let CAn be the angle of incidence, CAm the angle of refraction, and consequently 7nAn the quan- tity of refraction ; let TC be the tangent of C?n, mv ON REFRACTION. 81 its sine, nw the sine of Cn, and draw rm parallel to vw ; then, as the refraction of air is very small, we W V may consider m rn as a rectilinear triangle ; and hence, by si-milar triangles, Av : Am :: rn : mn = Am X rn , ^ . , , • r -J — ; but Am is constant, and as the ratio ot mv to nw is constant by the laws of refraction, their dif- ference, rn, must vary as nw ; hence, m?z varies as '"^ , ^rr^ AmXniv , . , . mv ^ ■j^; but CT=: — j^ , which varies as ^, be- cause Am is constant; hence, the refraction, mn, varies as CT, the tangent of the apparent zenith distance of the star, because the angle of refraction, CAm, is the angle between the refracted ray and the perpendicular to the surface of the medium, which perpendicular is directed to the zenith. Whilst, therefore, the refrac- tion is very small, so that rm« may be considered as a rectilinear triangle, this rule will be sufficiently accurate *. (152.) The twilight in the morning and evening, arises both from the refraction and reflection of the sun's rays by the atmosphere. It is probable that the reflection arises principally from the exhalations of various kinds with which the lower parts of the atmosphere are charged ; for the twilight lasts till the sun is further below the horizoa in the evening, than it is in tlie morning when it * For further information on this subject, see ray Complete Si/stem of Astronomj/. F 82 ON REFRACTION. begins ; and it is longer in summer than in winter. Now, in the former case, the heat of the day has raised the vapours and exhalations ; and in the latter, they will be more elevated from the heat of the season ; therefore, upon supposition that the reflection is made by them, the twilight ought to be longer in the evening than in the morning, and longer in summer than in winter. (153.) Another effect of refraction is, that of giving the sun and moon an oval appearance, by the refrac- tion of the lower limb being greater than that of the upper, whereby the vertical diameter is diminished. For suppose the diameter of the sun to be 32', and the lower limb to touch the horizon, then the mean refrac- tion at that limb is 33", but the altitude of the upper limb being then 32', it's refraction is only 28'. 6", the difference of which is 4', 54", the quantity by which the vertical diameter appears shorter than that parallel to the horizon. When the body is not very near the horizon, the refraction diminishing nearly uniformly, the figure of the body is very nearly that of an ellipse. S3 Chap. VIII. ON THE SYSTEM OF THE WORLD. (154.) When any effect or phaenomenon is discovered by experiment or observation, it is tbe business of Philosophy to investigate it's cause. But there are very few, if any, enquries of this kind, where we can be led from the effect to the cause by a train of mathe- matical reasoning, so as to pronounce with certainty upon the cause. Sir /. Newton, therefore, in his Principia, before he treats on the System of the World, has laid down the following Rules to direct us in our researches into the constitution of the universe. Rule I. No more causes are to be admitted, than what are sufficient to explain the phaenomenon. Rule II. Of effects of the same kind, the same causes are to be assigned, as far as it can be done. Rule III. Those qualities which are found in all bodies upon which experiments can be made, and which can neither be increased nor diminished, may be looked upon as belonging to all bodies. Rule IV. In Experimental Philosophy, proposi- tions collected from phaenomena by induction, are to be admitted as accurately, or nearly true, until some reason appears to the contrary. The principles, upon which the application of these Rules is admitted, are, the supposition that the opera- tions of nature are performed in the most simple manner, and regulated by general laws. And al- though their application may, in many cases, be very unsatisfactory, yet, in the instances to which we shall here want to apply them, their force is little inferior to F 2 84 SYSTEM OF THE WORLD. that of direct demonstration, and the mind rests equally satisfied as if the matter were strictly proved. (155.) The diurnal motion of the heavenly bodies may be accounted for, either by supposing the earth to be at rest, and all the bodies daily to perform their revolutions in circles parallel to each other ; or by sup- posing the earth to revolve about one of it's diameters as an axis, and the bodies themselves to be fixed, in which case their apparent diurnal motions would be the same. If we suppose the earth to be at rest, all the fixed stars must make a complete revolution, in parallel circles, every day. Now the nearest of the fixed stars cannot be less than 400000 times further from us than the sun is, and the sun's distance from the earth is not less than 93 millions of miles. Also, from the discoveries which are every day making by the improvement of telescopes, it appears that the heavens are filled with almost an infinity of stars, to which the number visible to the naked eye bears no proportion; and whose distancesare, probably, incom- parably greater than what we have stated above. But that an almost infinite number of bodies, most of them invisible, except by the best telescopes, at almost infi- nite distances from us and from each other, should have their motions so exactly adjusted, as to revolve in the same time, and in parallel circles, and all this without their having any central body, which is a physical impossibility, is an hypothesis, which, by the Rules we have here laid down, is not to be admitted, when we consider, that all the phaenomena may be solved, simply by the rotation of the earth about one of it's diameters. If therefore we had no other reason, we might rest satisfied that the apparent diurnal mo- tions of the heavenly bodies are produced bj^ the earth's rotation. But we have other reasons for this supposi- tion. Experiments prove that all the parts of the earth have a gravitation towards each other. Such a body, therefore, the greatest part of whose surface is a fluid, if it remain at rest, must, from the equal gra^ SYSTEM OF THE WORLD. 85 vitatlon of it's parts, form itself into a perfect sphere. But if we suppose the earth to revolve, the parts most distant from the axis must, from their greater velocity, have a greater tendency to fly off, and therefore that diameter which is perpendicular to the axis must be increased. That this must be the consequence appears from taking an iron hoop, and making it revolve swiftly about one of it's diameters, and that diameter will be diminished, and the diameter perpendicular to it will be increased. Now it appears from mensura- tion, that the earth is not a perfect sphere, but a spheroid, having the equatorial longer than it's polar diameter. That diameter therefore, about which the earth must revolve, in order to solve all the phaenomena of the apparent revolution of the heavenly bodies, is the shortest ; and as it necessarily must be the shortest, if the earth be supposed to revolve, this agreement affords a very satisfactory proof of the earth's rotation. Another reason for the earth's rotation is from analogy. The planets are opaque and spherical bodies, like to our earth ; now all the planets, on which sufficient ob- servations have been made to determine the matter, are found to revolve about an axis, and the equatorial diameters of some of them are visibly greater than their polar. When these reasons, all upon different prin- ciples, are considered, they amount to a proof of the earth's rotation about it's axis, which is as satisfactory to the mind, as the most direct demonstration could be. These, however, are not all the arguments which might be offered ; the situations and motions of the bodies in our system necessarily require this motion of the earth. (156.) Besides this apparent diurnal motion, the sun, moon, and planets, have another motion ; for they are observed to make a complete revolution amongst the fixed stars, in different periods; and whilst they are performing these motions in respect to the fixed stars, they do not always appear to move in the same direction, or in that direction in which their 86 SYSTEM OF THE WORLD. complete revolutions are made, but sometimes appear stationary, and sometimes to move in a contrary direc- tion. We will here briefly describe and consider the different systems which have been invented, in order to solve these appearances. Ptolemy supposed the earth to be perfectly at rest, and all the other bodies, that is, the sun, moon, planets, comets and fixed stars, to re- volve about it every day; but that, besides this diurnal motion, the sun, moon, planets and comets had a mo- tion in respect to the fixed stars, and were situated, in respect to the earth, in the following order ; the Moon, Mercury, Venus, the Sun, Mars, Jupiter, Saturn. These revolutions he first supposed to be made in circles about the earth, placed a little out of the center, in order to account for some irregularities of their motions; but as their retrograde motions and stationary appearances could not thus be solved, he supposed them to revolve in epicycloids, in the following manner. Let ABC be a circle, S the center, E the earth, ah c d another circle, whose center v is in the circumference of the circle ABC. Conceive the circumference of the circle ABC to be carried round the earth every twenty-four hours, according to the order of the letters, h and at the same time let the center v of the circle ahcd have a slow motion in the opposite direction, and let SYSTEM OF THE WORLD. 8/ a body revolve in this circle in the direction abed; then it is manifest, that by the motion of the body in this circle, and the motion of the circle itself, the body will describe such a curve as is represented bv Mrnhnop-, and if we draw tlie tangents El, Em, the body woald appear stationary at tlie points / and m, and it's mo- tion would be retrograde through /m, and then direct again. Now to make Venus and Mercurv always accompany the sun, the center v of the circle «^c</ was supposed to be always very nearly in a right line be- tween the earth and the sun, but more nearly so for Venus than for Mercury, in order to give each it's proper elongation. This system, althougli it will ac- count for all the motions of the bodies, yet it will not solve the phases of V^enus and Mercury ; for in this case in both conjunctions with the sun, they ought to appear dark bodies, and to lose their lights both ways from their greatest elongations ; wdiereas it appears from observation, that in one of their conjunctions they shine with full faces. This system therefore cannot be true. (157-) The system received by the Egyptians was this: That the earth is immoveable in the center, about which revolve in order, the Moon, Sun, Mars, Jupiter, and Saturn ; and about the Sun revolve Mer- cury and Venus. This disposition will account for the phases of Mercury and Venus, but not for the apparent motions of Mars, Jupiter, and Saturn. (158.) The next system which we shall mention, though posterior in time to the true, or Copernican System, as it is usually called, is that of Tycho Brake, a Polish nobleman. He was pleased with the Coper- nican System, as solving all the appearances in the most simple manner; but conceiving, from takiugtheliteral meaning of somepassages in scripture, that it was ne- cessary to suppose the earth to be absolutely at rest, he altered the system, but kept as near to it as possible. And he further objected to the earth's motion, because it did not, as he conceived^ affect the motions of 88 SYSTEM OF THE WORLD. comets observed in opposition, as it ought ; whereas, if he had made observations on some of them, he would have found that their motions could not otherwise have been accounted for. In his system the earth is supposed to be immoveable in the center of the orbits of the sun and moon, without any rotation about an axis ; but he made the sun the center of the orbits of the other planets, which therefore revolved with the sun about the earth. By this system, the different motions and phases of the planets may be solved, the latter of which could not be by the Ptolemaic System ; and he was not obliged to retain the epicyclods, in order to account for their retrograde motions and stationary appearances. One obvious objection to this system is, the want of that simplicity by which all the apparent motions maybe solved, and the necessity that all the heavenly bodies should revolve about the earth every day; also, it is physically impossible that a large body, as the sun. should revolve in a circle about a small body, as the earth, at rest in its center ; if one body be much larger than another, the center about which they revolve must be very near the large body; an argument which holds also against the Ptolemaic System. It appears also from observation, that the plane in which the sun must, upon this supposition, diurnally move, passes through the earth only twicein a year. It cannot therefore be any force in the earth which can retain the sun in it's orbit; for it would move in a spiral, continually changing it's plane. In short, the complex manner in which all the motions are accounted for, and the physical impossibility of such motions being performed, is a sufficient reason for rejecting this system ; especially when we consider, in how simple a manner all these motions may be ac- counted for, and demonstrated from the common prin- ciples of motion. Some of Tycho's followers, seeing the absurdity of supposing all the heavenly bodies daily to revolve about the earth, allowed a rotatory motion to the earth, in order to account for their SYSTEM OF THE WORLD. 89 diurnal motion ; and this was called the Semi- Tycho7iic System ; but the objections to this system are in other respects, the same. (159.) The system which is now universally re- ceived is called the Copcrnican. It was formerly taught by Pythagoras, who lived about 500 years before J. C. and Pkilolaus, his disciple, maintained the same ; but it was afterwards rejected, till revived by Copernicus. Here the Sun is placed in the center of the system, about which the other bodies revolve in the following order ; Mercury, Venus, the Earth, Mars, Jupiter, Saturn, and the Georgian Planet, which was lately discovered by Dr. Herschel ; beyond which, at immense distances, are placed the fixed stars ; the moon revolves about the earth, and the earth revolves about an axis. This disposition of the planets solves all the phaenomena, and in the most simple manner. For from inferior to superior conjunction, Venus and Mercury appear, first horned, then dichotomized, and next gibbous; and the contrary, from superior to infe- rior conjunction ; they are always retrograde in the inferior, and direct in their superior conjunction. Mars and Jupiter appear gibbous about their quadra- tures ', but in Saturn and the Georgian this is not sensible, on account of their great distances. The motions of the superior planets are observed to be direct in their conjunction, and retrograde in their opposition. All these circumstances are such as ought to take place in the Copernican System. The motions also of the planets are such as should take place upon physical principles. We may also further observe, that the supposition of the earth's motion is necessary, in order to account for a small apparent motion which every fixed star is found to have, and which cannot otherwise be accounted for. I'he harmony of the whole is as satisfactory a proof of the truth of this system, as the most direct demonstration could be ; we shall therefore assume this system. 90 Chap. IX. ON Kepler's discoveries. (l6o.) Kepler was the first who discovered the figures of the orbits of the planets to be elhpses, having the sun in one of the foci; this he determined in the following manner. (l6l.) Let S be the sun, 71/ Mars, I>, E, two places of the earth when Mars is in the same point M of it's orbit. When the earth was at D, he observed the difference between the longitudes of the sun and Mars, or the angle MDS; in like manner, he ob- served the angle MES. Now the places D, E, of the earth in it's orbit being known, the distances DS, ES, and the angle DSE, will be known ; hence, in the triangle DSE, we know DS, SE, and the angle DSE, to find DE, and the angles SDE, SED ; hence, we know the angles MDE, MED, and DE, to find MD; and lastly, in the triangle MDS, we know MD, DS, and the angle MDS, to find MS, the distance of Mars from the sun. He also found the angle 3ISD, the difference of the heliocentric longitudes of Mars and the earth. By this method, Kepler, from observa- tions made on Mars when in aphelion and perihelion KEPLER'S DISCOVERIES. 9I (for he had determined the position of the hne of the apsides, by a method which we sj^all afterwards_ex- plain^ independent of the form of the orbit, deter- mined the former distance from the sun to be 166/80, and the latter 138500, the mean distance of the earth from the sun being 100000; hence, the mean distance of Mars was 15 264 0_, and the excentricity of it's orbit 14140. He then determined, in hke manner, three other distances, and found them to be 14/7^0, 163100, 166255. Fie next calculated the same three distances, upon supposition that the orbit was a circle, and found them to be 14853.9, 16*3883, l66605 ; the errors therefore of the circular hypothesis were '^8Q, 7^^i 350. But he had too good an opinion of Ti/cIio*s observations (upon which he founded all these calcu- lations) to suppose that these differences might arise from their inaccuracy; and as the distance between the aphelion and perihelion was too great, upon suppo- sition that the orbit was a circle, he knew that the form of the orbit must be an oval ; Itaque plane hoc est : Orhita planet ce non est cir cuius, sed ingrediens ad latera utraque paulatim, iterumque ad circuli amplitudinem in perigceo ejciens, cujusmodi Jiguram itinens ovalem nppellitant, pag. 213*. And as, of all ovals, the elhpse appeared to be the most simple, he first supposed the orbit to be an ellipse, and placed the sun in one of the foci ; and upon calculating the above observed distances, he found they agreed toge- ther. He did the same for other points of the orbit, and found that they all agreed ; and thus he pro- nounced the orbit of Mars to be an ellipse, having the sun in one of it's foci. Having determined this for the orbit of Mars, he conjectured the same to be true for all the other planets, and upon trial he found it to be so. Hence, he concluded, That the six primary planets revolve about the sun in ellipses, having the sun in one of the foci. * See his Work, De Motihus StellcB Martis. 92 Kepler's discoveries. The relative mean distances of the planets from the sun are as follows: Mercury, 38^10; Venus, 72333 ; Earth, 100000; Mars, 15236,9; Jupiter, 320279; Saturn, 954072; Georgian, 1918352. (162.) Having thus discovered the relative mean distances of the planets from the sun, and knowing their periodic times, he next endeavoured to find if there was any relation between them, having had a strong passion for finding analogies in nature. On March 8. 16I8, he began to compare the powers of these quantities, and at that time he took the squares of the periodic times, and compared them with the cubes of the mean distances, but, from some error in the calculation, they did not agree. But on May 15, having made the last calculations ag^ain, he discovered his error, and found an exact agreement between them. Thus he discovered that famous law. That the squares of the periodic times of all the planets are as the cubes of their mean distances J roni the sun. Sir /. Newton afterwards proved that this is a necessary consequence of the motion of a body in an ellipse, re- volving about the focus. Prin. Phil. Lib. I. Sec. \. Pr. 15. (163.) Kepler also discovered, from observation, that the velocities of the planets, when in their apsides, are inversely as their distances from the sun ; whence it followed, that they describe, in these points, equal areas about the sun in equal times. And although he could not prove, from observation, that the same was true in every point of the orbit, yet he had no doubt but that it was so. He therefore applied this principle to find the equation of the orbit (as will be explained in the next Chapter), and finding that his calculations agreed v»'ith observations, he concluded that it was true in general, That the playlets describe about the sun. equal areas in equal times. This discoveiy was, perhaps, the foundation of the Priticipia, as it might probably suggest to Sir /. Newton the idea, that tlofi KEPLER S DISCOVERIES. 93 proposition was true in general, which he afterwards proved it to be. These important discoveries are the foundation of all Astronomy. (164.) Kepler also speaks of Gravity as a power which is mutual between all bodies ; and tells us, that the earth and moon would move towards each other, and meet at a point as much nearer to the earth than the moon, as the earth is greater than the moon, if their motions did not hinder it. He further adds, that the tides arise from the gravity of the waters towards the moon. 94 Chap. X. ON THE MOTION OF A BODY IN AN ELLIPSE ABOUT THE FOCUS. (165.) As the orbits which are described by the primary planets revolving about the sun, are ellipses having the sun in one of the foci, and each describes about the sun equal areas in equal times, we next pro- ceed to deduce, from these principles, such conse- quences as will be found necessary in our enquiries respecting their motions. From the equal description of areas about the sun in equal times, it appears * that * For if APSi be an ellipse described by a planet about the sun at S in the focus, the indefinitely small area, PSpy described in a given time, will be constant; draw P?- perpendicular to Sp ; and as the area SPp is constant for the same time, Pr varies as tt: Pr . 1 but the angle pSP varies as -— >, and therefore it varies as-^-^^; that is, in the same orbit, the angular velocity of a planet varies in- versely as the square of it's distance from the sun. For different planets, the areas described in the same time are not equal, and ^ • area SPp ^1 ^, , r>^ therefore Pr varies as — ^ — i., consequently the angle p.5P vanes area SPp as a; 1 > ^h^t is, the angular velocities of different planets, are as the areas described in the same time directly, and the squares of their distances from the sua inversely. MOTION OF A BODY, &C. 95 the planets rnave with unequal angular velocities about the sun. The proposition therefore, which we here propose to solve, is, given the periodic time of a planet, the time of it's motion from it's aphelion, and the excentricity of it's orbit, to find it's angular dis- tance from the aphelion, or it's true anomaly, and it's distance from the sun. This was first proposed by Kepler, and therefore goes by the name of Kepler's Problem. He knew no direct method of solving it, and therefore did it by very long and tedious tentative operations. (li6.) Let AGQ be the ellipse described by the body about the sun at S in one of it's foci, AQ the major, GC the semi-minor axis, ^ the aphelion, Q the perihelion, P the place of the body, AP^QE a circle, C it's center ; draw NPI perpendicular to AQ^ join PS, NS, and NC, on which produced Jet fall the perpendicular ST. Let a body move uniformly in the circle from A to D with the me«w angniar velocity of the body in the ellipse, whilst the body moves in the ellipse from ^ to P; then the angle ACD is the mean, and the angle ASP tlie true anomaly .: and the difference of these two angles is called the equation of 96 MOTION OF A BODY IN the planefs center, or prosthapheresis. Let p s= the periodic time in the ellipse or circle (the periodic times being equal by supposition), and ^=:the time of describing AP or AD ; then, as the bodies in the ellipse and circle describe equal areas in equal times about *S' and C respectively, we have area ADC : area of the circle :: t : p, area of the ellipse : area ASP :: p : t ; also, area of circ. : area of ellip. :: area A S ]V:sir en ASP;* therefore, area ADC : area ASP :: area ASN: area ASP ; hence, ADC = ASN ; take away the area ACN, which is common to both, and the area DCN = SNC, but DCN=l DNx CN, and SNC=i ST X CN ; therefore ST= DN. Now if t be given, the arc AD will be given ; for as the body in the circle moves uniformly, we have p \ t w 36o° : AD. Thus we may find the mean anomaly at any given time, knowing the time when the body was in the aphelion ; hence, if we can find ST, or ND, we shall know the angle NCA, called the excentric anomaly, from whence, by one proportion (167) we shall be able to find the angle ASP the true anomaly. The problem is therefore reduced to this ; to find a triangle CST, such, that the angle C+ the degrees of an arc equal to ST^ may be equal to the given angle ACD. This may be expeditiously done by trial in the following manner, given by M. de la Caille in his Astronomy. Find what arc of the circumference of the circle ADQE is equal to C^, by saying, 355 : 113 :: 180°. : 57°. 17'-44",8, the number of degrees of an arc equal in length to the radius CA ; hence, CA : CS :: 57°. 17'- 44",8 : the degrees of an arc equal to CS. Assume therefore the angle SCT, multiply it's sine into the * See my Treatise on the Conic Sections, second edit. prop. 7. of the Ellipse, cor. 3 and 4. And this is the Treatise referred to in the future part of this Work. AN ELLIPSE ABOUT THE FOCUS. ^1 degrees in CS, and add it to the angle SCT, and if it be equal to the given angle ACD, the supposition was right ; if not, add or subtract the difference to or from the first supposition, according as the result is less or greater than ACD, and repeat the operation, and in a very few trials you will get the accurate value of the angle SCT. The degrees in ST" may be most readily obtained, by adding the logarithm of CS to the loga- rithm of the sine of the angle SCT, and subtracting 10 from the index, and the remainder will be the lo- garithm of the degrees in ST. Having found the value of AN, or the angle ACN, we proceed next to find the angle ASP. (167.) Let V be the other focus, and put AC=l ; then by Eucl. B. II. P. 12. SP ' - Pv' = vS'-{-2vSx vl=.vS+2vIxvS=2Cv-{-2vIx2SC = 2CTx2SC; hence, SP-^ Pv:2C I:: 2SC : SP - Pv, or 2 : 2CI :: 2SC: SP-2-SP,OYl : CI .: SC : SP-l;hence, SP = 1 + C.V X CI= 1 + CSx COS. z ACN. But . I —COS. ASP — =-„ (Trig. Art. 94), ^ _^^^^^^^p = tan. ^ ASP-, also, (Trig. Art. 1 25) SP, or 1 + C^x cos..^ CiV ; rad. = 1 :: SI, or CS + CI, or CS + cos. ANC, : cos. ASP = CS + cos.ACN , ■ ' r'L' i^,\- . Hence, tan. i ASP^ ; ( = 1 + Cc) X cos.^CiV ' ^ ' ^ 1- COS. ASP \ _\ +CSx COS. ACN- CS- cos. ACN I 4- COS. A SP^ ~ 1 + CS X cos.^6'iV-|- C6'+ cos.^C'iV _ 1 - CS+cos.ACNx C8^ _ 8a-cos.^CiYx8Q ~ 1 + CS-^cos.ACN-\-CS+l~ SA + COS. ACNxSA 1 - COS.ACN SQ ^rj. . A • , X = l+cos..^CW ^ SA= (Trigonometry, Article 95) tan. I ACN'^ X tti ; therefore sJ SA : yj SO, :: tan. SA lACNi tan. | ASP, consequently we get ASP the true anomaly. 98 MOTION OF A BODY IN' Ex. Required the true place of Mercury on Aug. 526, 174O5 at noon, the equation of the center, and it's distance from the sun. By M. de la Caille's Astronomy, Mercury was in it's aphehon on Aug. 9, at 6h. Sf. Hence, on Aug. 26, it had passed it's apheHon l6d. l^h. 23' ; there- fore 82 d. 23h. 15'. 23" (the time of one revolution) : l6d. I7h. 23' :: 36o° : 68^ 26'. 28". the arc JD, or mean anomaly. Now (according to this author) CA : CS :: 1011276 : 211l65 (166) :: 57°. 7l'.44",8 : 1 1°. 57'. 50"= 43070", the value of CS reduced totlie arc of a circle, the log. of which is 4,6341749. Also, 68**, 26'. 28" = 246388". Assume the angle ^C7' to be 60°. =216000", and the operation (166) to find the- angle ACN, will stand thus. AN ELLIPSE ABOUT THE FOCUS. 99 4,6341749 9,9375306 log. of. 2i6'ooo=a 4,5717055 37300 253300 246388 6912 = 6 4,6341749 9,9287987 2O9088=a- 6 = 58^4'. 48" — C 4,5629736 36557 245645 246388 743-d 4,6341749 9,9297694 ...... 209831 =c + f/ = 58°.l7'.l I " = r 4,5639443 36639 246470 246388 82=/ 4,6341749 9,9296626 209749=^e-/ = 58^1o'.49'' = o- 4,5638375 36630 246379 246388 9 = //; hence, as the dif- ference between the value deduced from the assump- tion and the true value, is now diminished about nine times every operation, the next difference would be l" ; G 2 100 MOTION OF A BObY IN if therefore we add h to g, and then subtract 1", we get 58°. 15'. 57", for the true value of the angle ACN, the excentric anomaly. Hence, (167) find the true anomaly ASP, from the proportion there given, by logarithms, thus. Log. tang. 29°. 7'. 58"! . . 9,7461246 I Log. *S'Q=800lll . . . 2,9515751 126976997 J Log. 5^=1222441 . . . 3,0436141 Log. tang. 24M6M5" . . . 9,6540856 Hence, the true anomaly is 48°. 32'. 30". Now the aphelion A was in 8\ 13°. 54'. 30" ; therefore the true place of Mercury was \0\ 2°. 2/'. Hence, (166), 68°. 26'. 28" -48°. 32'. 30"= 19°. 53'. 58", the equation of the center. Also, 8P=1 + CS y. cos. z ACN= 1310983 the distance of Mercury from the sun, the radius of the circle, or the mean distance of the planet, being unity. Thus we are able to compute,at any time, the place of a planet in it's orbit, and it's distance from the sun ; and this method of computing the excentric anomaly appears to be the most simple and easy of application of all others, and capable of any degree of accuracy. (168.) As the bodies D and P departed from A at the same time, and will coincide again at Q, ADQ^ ^PQ being described in half the time of a revolution ; and as at A the planet moves with it's least angular velocity (by the Note to Art. l65), therefore from A to Q, or '\x\ the first ^va signs of anomaly, the angle ^CZ> will be greater than ASP, or the mea?i will be greater than the true anomaly ; but from Q to A, or in the last six signs, as the planet at Q moves with it's greatest angular velocity, the true will be greater than the 7nean anomaly. When the equation is greatest in going from A to Q, the mean place is before the true place, by the equation, and from Q to A, the true AN ELLIPSE ABOUT THE FOCUS. 101 place is before the mean place, by the equation; therefore, from the time the equation is greatest till it becomes greatest again, the difference between the true and mean motions, is twice the equation. From apogee to perigee, the true and mean motions are the same. (169.) The method ascribed by some writers to Seth IVard, Professor of Astronomy at Oxford, and published in l654, although, as M. de la Lande ob- serves, it is given both by JVard and Mercator to BuUialdiis, is less accurate than these we have already given; yet as it may, in many cases, serve as a useful approximation, it deserves to be mentioned. He assumed the angular velocity about the other focus v to be uniform*, and therefore made it represent the mean * That this is not true may be thus shown. With the center 5* and radius SlVzzV'^^x C£ describe the circle zH'^, then the area of this lA circle=area of the ellipse [Conic Sections, Ellipse, Prop. 7. Cor. 5): l«t 102 MOTION OF A BODY IK anomaly. Produce v P io r, and take Pr=PS; then in the triangle Svr, 7'v-\-vS -. rv-vS :: tan. i . A vS r-\-vrS: tan. J. z vSr — vrS (Trig. Ar t. 135); now ^ . rv+vS=^ AQ + l vS-JS, and j. ri ; - t?^ = I ^Q - I Z78= SQ ; also, tan. | . z t?«Sr + vrS - tan. J. z AvPi and tan. |. z vSr-vrS=: (as Pr = PS) tan. |. zy-Sr — P8r = tan. I . a ASP-, hence, ^Ae aphelion distance : perihelion distance :: fi/7?. o/' | the mean anomaly : ^«». ^ ^rwe anomaly. This is called the simple elliptic hypotlvesis. In the orbit of the earth, the error is never greater than I7"; in the orbit of the moon, it may be l'. 35". By this hypothesis, for 90° from aphelion and perihelion, the computed place is hackivarder than the true ; and for the other part it is foi uoarder. (170.) The greatest equation of the center may be easily found from the Note to Article 169, giving the dimensions of the orbit. For as long as the angular Jet a body, moving uniformly in it, make one revolution in the same time liie body does in the ellipse ; and let the bodies set otF at the same time from A and z, and describe AP, ?p, in tlie same time ; then the angle zSv is the mean, and ASP the true anomaly. Draw pS indefinitely near to PS, and Pr, po, perpendicular to po Sp, FP ; then Pr—po. Now the angle PFp A^nes as -rrp^ — — r; ^ut, in a given time, the area PSp is given, therefore Pr P-t' varies as -r-rr ; hence, the angle PF/), described in a given time, \, which is not a constant quantity. Also, Z PFp PF X PS : Z PSp :: PS : PF :: ^^^p^ • 7^- And by the Note to Art. 165, as equal areas are described in equal times in the circle and ellipse about S, the angular velocity about S in the circle, be- comes ;:• Hence, the angular velocity about S is greater or less than the mean angular velocity, according as PFx PS is less or greater than Sl'V-, or than ACk CE. Also, the angular velocity about .S is the same in similar points of the ellipse ia respect to the center, or at equal distances from the center. AN ELLIPSE ABOUT THE FOCUS. 103 velocity of the body in the circle is greater than that in the ellipse about S, the equation will increase, the A bodies setting out from A and z j and when they be- come equal, the equation must be the greatest ; this therefore happens when-^, ='~sW' " AC x CE ' '''' when ^Cx C£ = SP^ hence, SP is known. Let SfV represent this value of SP ; then as we know A Sfr, FIV{-2AC-SIV) will be known, and as SF is known, we can find the angle FSIV the true anomaly. Hence (167) yj SQ, : sj SA :' tan. ^ true anom. : tan ^ excen. anom. ACN, or tan. J SCT ; hence, as we know SC^ we can find SJ\ or 104 MOTION OF A BODY IN ND ; and to convert that into degrees, say, Rad. = 1 : ND :: 57°. 17'. 44",8 : the degrees in ND\ which added to or subtracted from the angle ACN, gives ACD the mean anomaly, the difference between which and the true anomaly is the «Tea^e*f equation- Thus we may find the equation at any other time, having given SP. (171.) The excentricity, and consequently the di- mensions of the orbit, may be known from knowing the greatest equation. For (170) the equation is greatest when the distance is a mean between the semi- axes major and minor, and therefore in orbits nearly circular, the body must be nearly at the extremity of the minor axis, and consequently the angle NCA, or SCT (Fig. p. 103) will be nearly a right angle, there- fore ST\s nearlv equal to SC ; also N'SJ will be very nearly equal to' PSA. Now the angle NCA-NSA (PS'A)=SNC, and DCA- NCA=nCN, add these together, and DCA - PSA = DCN+ SNC,wh\ch (as iVC is nearly parallel to DS) is nearly equal to 2 DCN\ that is, the difference between the true and mean anomaly, or the equation of the center, is nearly equal to twice the arc DN, or twice ST, or very nearly twice SC. Hence, 57°. if. 44", 8 : half the greatest equation :: rad. = l : SC the excentricity. But if the orbit be considerably excentric, compute the greatest equation to this excentricity ; and then, as the equa- tion varies very nearly as SC, say, as the computed equation : excentricity found :: given greatest equa- tion : true excentricity. Ex. If we suppose, with M.de la Caille, that Mer- cury's greatest equation is 24*^. 3'. 5"; then 57°. 17'. 44",8 : 12°. r. 32",5 :: 1 : ,209888 the excentricity very nearly. Now the greatest equation, computed from this excentricity, is 23°. 54'. 28",5 ; hence, 23°. 54'. 28",5 : 24°. 3'. 5" :: ,209888 : ,211165 the true excentricity. M. de la Lande makes the greatest equation 23°. 40', and the excentricity ,207745. (172.) The converse of this problem^that is, given AN ELLIPSE ABOUT THE FOCUS. 105 the excentricity and true anomaly, to find the mean, may be very readily and directly solved. The excen- tricity being given, the ratio of the major to the minor axis is known*, vvhich is the ratio of AY to PI\ hence, the angle ASP being given, we have PI : NI :.- tan. ASP : tan. ASN; therefore, in the triangle NCS, we know NC, CS, and the angle CSN, to find the angle SCN' (Trig.Art.128), the supplement of which is the angle ACN, or SCT; hence, in the right- angled triangle STC, we know SC and the angle SCT, to find 87' (Trig Art 128), which is equal to ND, the arc measuring the equation, which may be found by saying, radius : 87' :: 57°- I?'- 44",8 : the degrees in ND, which added to ACN, gives ACD the mean anomaly. To Jind the hourly Motion of a Planet in ifs Orbit, having give?! the mean hourly Motion. (173.) The hourly motion of a planet in it's orbit is found immediately from the Note to Art. 169; for it appears from thence, that the angles PSp, ff'Sw, de- scribed by the body at P in the ellipse, and the body W'\n the circle in the same time, are as SIV^'^ : SP^^ or as (see Fig. p. 103) ACx CE : SP- ; hence, SPp=: /4C V CP Pf^Sw X — rrjji — the hourly motion of a planet in it's orbitf the angle flTSw being the mean motion of the planet in an hour. For greater accuracy, SP must be taken at the middle of the hour. Thus we may easily compute a table of the hourly motions of the planets in their orbits. * For as ^C, CS ar e known, we have GC=\/(jfS~— ^C'=: ^AC'—SC''=s/AC-\-SCx AL—SC, for thecoraputation of vvhich by logarithms, see Trig. Art. 52. io6 Chap. XI. ON THE OPPOSITIONS AND CONJUNCTIONS OF THE PLANETS. (174.) The place and time of the opposition of a su- perior, or conjunction of an inferior planet, are the most important observations for determining tlie ele- ments of the orbit, because at that time the observed is the same as the true longitude, or that seen from the sun ; whereas, if observations be made at any other time, we must reduce the observed to the true longi- tude, which requires the knowledge of their relative distances, and which, at that time, are supposed not to be known. They also furnish the best means of examining and correcting the tables of the planets' motions, by comparing the computed with the ob- served places. (175.) To determine the time of opposition, observe, when the planet comes very near to that situation, the time at which it passed the meridian, and also it's right ascension (ill or 113) ; take also it's meridian alti- tude ; do the same for the sun, and repeat the obser- vations for several days. From the observed meridian altitudes find the declinations (114), and from the right ascensions and dechnations compute (115) the latitudes and longitudes of the planet, and the longi- tudes of the sun. Then take a day when the differ- ence of their longitudes is nearly 180% and on that day reduce the sun's longitude, found from observation when it passed the meridian, to the longitude found at the time (t) the planet passed, by finding from obser- vation, or computation, at what rate the longitude then increases. Now in opposition the planet is retrograde, and therefore the difference between the OPPOSITIONS OF THE PLANETSI. 107 longitudes of the planet and sun increases by the sum of their motions. Hence, the following rule : As the sum (S) of their daily motions in longitude : the difference (D) between 180° and the difference of their longitudes reduced to the same time [t)* (sub- tracting the sun's longitude from that of the planet to get the difference reckoned from the sun according to the order of the signs) :: 2Ah. : interval between that time {t) and the time of opposition. This interval added to or subtracted from the time it), according as the difference of their longitudes at that time was greater or less than 180°^ gives the time of opposition. \i this be repeated for several days, and the mean of the whole taken, the timew^ill be had more accurately. And if the time of opposition found from observation, be compared with the time by computation from the Tables, the difference will be the error of the Tables, which may serve as means of correcting them. Ex. On October 24, 17b'3, M. de la Lawf/e observed the difference between the right ascensions of ^ Aries, and Saturn, which passed the meridian at 12/t. 1/'. 17" apparent time, to be 8^. 5'. 7", the star passing first. Now the apparent right ascension of the star at that time was 25°. 24'. 33",6; hence, the apparent right ascension of Saturn was 1'. 3*^. 29'. 40",6' at 12 /z. 17'. 17" apparent time, or \2h. T. 37" mean time. On the same day he found, from observation of the me- ridian altitude of Saturn, that it's declination was 10°. 35'. 20" N. Hence, from the right ascension and de- clination of Saturn, it's longitude is found to be V. 4°. 50'. 56", and latitude 2^ 43'. 25" S. At the same time the sun's longitude was found by calculation to be T. l^ 19'. 22", which subtracted from P. 4°. 50'. 56", gives 6'. 3 \ 31'. 34"; hence, Saturn was 3°. 31'. 34'' * For this diflirencc shows how far the planet is from opposi- tion ; and the proposition is founded on this principle, that the sun approaches the star by spaces in proportion to the times; the spaces S and D must therefore be as the time 24//, and the time {t) to f'Pfiosilion. 108 CONJUNCTIONS OF THE PLANETS. beyond opposition, but being retrograde will after- wards come into opposition. Now, from the observa- tions made on several days at that time, Saturn's longi- tude was found to decrease 4'. 50" in 24 hours, and by computation, the sun's longitude increased 59'. 59" in the same time, the sum of which is 64'. 49"; hence, 64\ 49" : 3". 31'. 34" :: 24h. : 78/?. 20'. 20", which added to October 24, I2h. l'. 37", gives 27c?. I8h. 21'. 57" for the time of opposition. Hence, we may find the longitude of Saturn at the time of opposition, by saying, 24A. : 78^. 20'. 20" :: 4'. 50" : 15'. 47" the re- trograde motion of Saturn in 78A. 20'. 20", which sub- tracted from 1*. 4°. 50'. 56", leaves 1*. 4°. 35'. 9" the longitude of Saturn at the time of opposition. In like manner we may find the sun's longitude at the same time, in order to prove the opposition; for 24/z. : 78^. 20'. 20" :: 59'. 59" : 3°. 15'. 47", which added to 7». 1°. 19'. 22", the sun's longitude at the time of observa- tion, gives 7*. 4°. 35^ 9" for the sun's longitude at the time of opposition, which is exactly opposite to that of Saturn. Hence, we may also find the latitude of Saturn at the same time, by observing, in like manner, the daily variation, or by computation from the Tables after the elements of it's motions are known, and the Tables constructed ; by which it appears, that in the interval between the times of observation and opposi- tion, the latitude had increased 6", and consequently the latitude was 2**. 43'. 31". Thus we find the times of opposition of all the superior planets. (176.) The place and time of conjunction of an in- ferior planet may be found in like manner, when the elongation of the planet from the sun, near the time of conjunction, is sufficient to render it visible ; the most favourable time therefore must necessarily be when the geocentric latitude of the planet at the time of conjunction is the greatest. In the year 1689, Venus was in it's inferior conjunction on June 25, and it was observed on 21, 22, and 28 ; from which observa- tions it's conjunction was found to be at 1 3 A. 46' appa- CONJUNCTIONS OF THE PLANETS. 10^ rent time at Paris, in longitude as 4°. 53'. 40", and latitude 3°. l'. 40"N. The state of the air must be very favourable, that the time and place of the superior conjunction may be thus observed; for as Venus is then about six times as far from the earth as at it's in- ferior conjunction^ it's apparent diameter and the quantity of light which we receive from it, are so small, as to render it difficult to be perceived. But the most accurate method of observing the time of an inferior conjunction both of Venus and Mercury, is from ob- servations made upon them in their transits over the sun's disc. 110 Chap. XII. ON THE MEAN MOTIONS OF THE PLANETS. (177.) The determination of the mean motions of the planets, from their conjunctions and oppositions, would very readily follow, if we knew the place of the aphelia and excentricities of their orbits ; for then we could (166) find the equation of the orbit, and reduce the true to the mean place ; and the mean places being determined at two points of time, give the mean motion corresponding to the interval between the times. But the place of the aphelion is best found from the mean motion. To determine therefore the mean motion, independent of the place of the aphelion, we must seek for such oppositions or conjunctions, as fall very nearly in the same point of the heavens; for then the planet being very nearly in the same point of it's orbit, the equation will be very nearly the same at each ob- servation, and therefore the comparison between the true places will be nearly a comparison of their mean places. If the equation should differ much in the two observations, it must be considered. Now, by compar- ing the modern observations, we shall be able to get nearly the time of a revolution ; and then, by com- paring the modern with the ancient observations, the mean motion may be very accurately determined ; for any error, by dividing it amongst a great number of revolutions, will become very small in respect to one revolution. As this will be best explained by an ex- ample, we shall give one from M. Cassini {Elem. (iAstron. p. 362), with the proper explanations as we proceed. MEAN MOTIONS OF THE PLANETS. Ill Ex. On September l6, 1701, Saturn w&s in oppo- sition at 2h. when the place of the sun was nj^ 23°. 21'. 16", and consequently Saturn in K 23°. 21'. 16", with 2°. 27'. 45" south latitude. On September 10, 1730, the opposition was at 12^. 27', and Saturn in k 17°« 53'. 57", with 2°. 19'. 6" south latitude. On Sep- tember 23, 1731, the opposition was at I5h. 51' in T 0°.30'. 50", with 2°. 36'. 55" south latitude. Now the interval of the two first observations was 29 years (of which seven were bissextiles) wanting 5<^. l3/i.33'; and the interval of the two last was I3/. 136?. 3h. 24'. Also, the difference of the places of Saturn in the two first observations was 5°. 27'. I9", and in the two last it was 12°. 36'. 53". Hence, in ly. 13d. 3h. 24', Saturn had moved over 12°. 06'. 53"; therefore 12°. 36'. 53" : 5°. 27'. 19" :: ly. 136?. 3h, 24' : l63d. l2/i. 41', the time of moving over 5°. 27'- 19" very nearly, because Saturn, being nearly in the same part of it's orbit, will move nearly with the same velocity ; this therefore, added to the interval between the two first observations (because at the second observation Saturn wanted 5°. 27'. 19" of being up to the place at the first observa- tion), gives 29 common years l64c?. 23/i. 8', for the time of one revolution. Hence say 29?/. l64c?. 23h. 8' : 365^. :: 36o°. : 12°. 13'. 23". 50"' the mea?z an- nual motion of Saturn in a common year of 365 days, that is, the motion in a year if it had moved uni- formly. If we divide this by 365, we shall get 2'. O". 28'" for the mean daily motion of Saturn. The mean motion thus determined will be sufficiently accurate to determine the number of revolutions which the planet must have made, when we compare the modern with the ancient observations, in order to determine the mean motion more accurately. The most ancient observation which we have of the opposition of Saturn was on March 2, in the year 228, before J. C. at one o'clock in the afternoon, in thie meridian of Paris, Saturn bejng then in m^ S°. 23', 112 MEAN MOTIONS OF THE PLANETS. with 2°. 50' north lat. On February 26, 1714, at 8h. 15', Saturn was found in opposition in ijp 7'^. 56'. 46", with 2°. 3' north lat. From this time we must sub- tract 1 1 days, in order to reduce it to the same style as at the first observation, and consequently this oppo- sition happened on February 15, at 8^. 15'. Hence, the difference between these two places was only 26'. 14". Also, the opposition in 1/15 was on March 11, at l6^. 55', Saturn being then in nj 21°. 3'. 14", with 2°. 25' north lat. Now between the two first opposi- tions there were 1 942 years (of which 485 were bissex- tiles) wanting 14^. l6/i. 45', that is, 1943 common years, and I05d. 7/1. 15' over. Also, the interval be- tween the times of the two last oppositions was S^Sd 8h. 40', during which time Saturn had moved over 13°. 6'. 28"; hence,i3°. 6'.28" : 26'. 14" :: 378d. 8h. 40': 13d. \Ah. which added to the time of the opposition in 1714, gives the time when the planet had the same longitude as at the opposition in 228 before J. C. This quantity added to 1943 common years 105^. 7/?. 15', gives 1943?/. 118c?. 2\h. 15', in which interval of time Saturn must have made a certain complete num- ber of revolutions. Now having found, from the modern observations, that the time of one revolution must be nearly 29 common years l6Ad. 23h. 8', it fol- lows that the number of revolutions in the above interval was QQ\ dividing therefore that interval by 66, we get 292/. 1626?. Ah. 27' for the time of one re- volution. From comparing the oppositions in the years 1714 and 1715, the true movement of Saturn appears to be very nearly equal to the mean move- ment, which shows that the oppositions have been ob- served very near the mean distance ; consequently the motion of the aphelion cannot have caused any con- siderable error in the determination of the mean motion. Hence the mean annual motion is 12°. 13'. 45". 14'", and the mean daily motion 2'. O". 35'". Dr. Halley makes the annual motion to be 12°. 13'. 21". M. de Place makes it 12°. 13'. 36",8. As the revolu- MEAN MOTIONS OF THE PLANETS. 113 tion here determined is in that respect to the longitude of the planet, it must be a tropical revohition. Hence, to get tlie sidereal revolution, we must say, 2'. 0". 35'" : 24'. 42". 20'" (the precession in the time of a tropical revolution, Art. 130) :: 1 day : 12fi?. "jh. l'.57", which added to l^y. l62d. All. 2f, gives 29;/. 174^. \\h. 28'. 57" the length of a sidereal year of Saturn. Thus we find the periodic times of all the superior planets. The periodic times of the inferior are found from their conjunctions. The periodic times of the planets are as follows ; Mercury, 87^/. 23/i. 15'. 43",6 ; Venus, 224^. l67i. 49'. 10",6; Mars, \y. 32ld. 23h. 30'. 35",6 ; Jupiter, lly.3l5d. I4h. 27'. 10",8 ; Saturn, 29,y. 174^. 1/l 31'. ir',2 ; the Georgian, 83/y. 150^. 18//. 114 Chap. XIII. ON THE GREATEST EQUATION, EXCENTRICITY, AND PLACE OF THE APHELIA, OF THE ORBITS OF THE PLANETS. (178.) Having determined the mean motions of the planets, we proceed next to show the method of find- ing the greatest equation of their orbits, the excen- tricity, and place of their aphelia. For although, in order to determine the mean motions very accurately, these tilings were supposed to be known, yet, without them, the mean motions may be so nearly ascertained, that these elements may from thence be very accu- rately settled. By Art. 161, we may find the distance of a planet from the sun in any point of it's orbit. The problem therefore is, having given in length and posi- tion, three lines drawn from the focus of an ellipse, to determine the ellipse. (179.) Let SB, SC, SD, be the three lines ; pro- duce CB, CD, and take SB : SC :: EB : EC, and EKHI G SC : SD :: CF : DF, then SC- SB , SC :. BC : EC=^^y^, and SC'-SD : SC v. DC: CF= GREATEST EftUATIONS, &C. 113 —p^ rvT^. Join FE, and draw DK, CI, BH, per- pendicularto it. Now, by similar triangles, IC : IIB :: EC: EB :: (by const.) SC : SB; also, IC : KD :: CF: DF:: SC : SD. Hence, the proportion of /C, HB, KD, is the same as SC, SB, SD, conse- quently EF is the directrix of the ellipse passing through B, C, D, (Con. Sect. p. 31). Through .S' draw y^.^QG perpendicular to FE ; take GA : AS :: CI: CS, and GQ ; SQ :: CI: CS ; then Cl+CS : CS :: GS : -SQ = ~jr^ — ^rij '■> in like manner we find Kyi -j- Oo AS=j=;y — y^^ and y/, Q, will be the vertices of the conic section. (180.) Calculation. In the triangles *S5C, 5(71), we know two sides and the included angles, they be- ing the distances of the observed })laces in the orbit; hence, (Trig. Art. 135) we can find BC, CD, and the angles BCS, SCD, and consequently BCD. Hence (179) we know CE and CF, and the angle ECF be- ing also known, the angle CEF can be found. There- fore in the right-angled triangle CIE, CE and the angle E are given; hence, (Trig. Art. 128) C7 is known. Join SI; then in the triangle SIC we know CI, CSand the angle SCI [=BCS - ECI), hence, we know SI, and the angles CIS, CSI, and hence the angle SIG is known ; therefore, in the right-angled triangle SIG, we know SI and the angle SIG, from whence SG is found. Hence, (179) we know SA, SQ, half the difference of which is the excentricitv, and their sum = ^Q. Lastly, in the triangle BSO (O being the other focus) we know all the sides, to find the angle BSA (Trig. Art. 133), the distance of the aphelion from the observed place B. In the year 1/40, on July 17, August 26, Septem- ber 6, M. de la Caille found three distances of Mer- cury (the mean distance being lOOOO) as follows .- SB 1 lu GREATEST EftUATlON, &C. = 10351,5, lC==ll325,5,SD = 9672,l66,the angle BSC = 3\27°.0'.35", and CSD = 4r. 40'. 4". Hence, BSC=29\ 55'. 5", BC= 18941, 5CZ)=56°. 49', Ci> = 8124,5, i?CF=86^ 44'. 5", C£ = 215004, C^ = 55647, C£F=14°.41'.44", C/=54543, C^/--=124\ 47'. 45", C7^=9*'- 49'. 4", 6T=47281, -S/G=:8b<'. 10'. 56", ^'G= 465 89, ^P = 8010,5, 5^=12209, SO = 4198,5 ; hence, the excentricity = 2099,75, BSA = 7V. 37'. 23, or 2^ 1 r. 37'. 23", which ifdded to 6^ 2^ 13'.'51", the position of SB, gives 8\ 13^ 51', 14" for the place of the aphelion. Hence, the greatest equation is 24". 3'. 5". (181.) Or from the same data, the place of the aphelion and excentricity may be thus found. Put the semi-axis major = 1, SB = a, SD = b, SC=c, the angle BSD = v. BSC=u, BSQ=^x, OS=e, half the parameter = r. Then, by a well-known property of the ellipse (Conic Sect. EUipse, Prop. 16), (^ = YTec^x' b^ ^^ , c= ===; hence, r=« 1 -{-e.cos. v+.r I -\-e. COS. u + jc -]-ae.cos.x = h + be.cos.v7^=:c-\- ce.cos.u-h x ; there- b-a c-a fore, — T= = e = — , a.cos.x — b.cos.v + x a.cos.x- c .cos.u + x now for cos-iJ + o^, and cos. u + x, substitute cos. ?;.cos. X-- sin. t;sin..r, and cos. u. cos. cT — sin. m. sin. x (Trig. Art. 102,) and we shall have b — a a . cos. x — b . COS. v . cos. x -\- b . sin. v sm. x c — a a . cos. x — c . COS. u . cos, x-\-c. sin. z<. sin, x ' each denominator by cos. x, and we have it — a c — a divide a — b.cos.v-\-b.sin:i\tain.x a— c. cos. u-\-c.sm.uXa.n.x ' , b.c — a. COS. V — c. b- a. cos. u - a.c — b hence, tan. x = == — ; , b . c — t7.sin.t; — c .b — a . sin. u which gives the place of the perihelion. Hence^ vvc OF THE ORBITS OF THE PLANETS. 117 c—a , . . know e = — ===theexcentncity; con- a . COS. X — c. COS. II + X sequently I - e and 1 +e, the perihelion and aphehon distances, are known. The species of the elhpsc being determined, it's major axis may be thus found : Com- pute the mean anomaly corresponding to the angle CSBy then say, as that mean anomaly : 36o° :: the time of describing the angle CSB : the periodic time. The periodic time being known, the major axis is found (162) by Kepler* s Rule. For other practical methods, see my Complete System of Astronomy. (182.) All the epochs in our Astronomical Tables are reckoned from noon on December 31, in the com- mon years, and from January 1, in the bissextiles. The places of the aphelia for che beginning of 1/50, are. Mercury, S\ 13°. 33'. 58"; Venus, 10^ 7°. 46'. 42"; the Earth, 3^ 8^ 37'. 16"; Mars^ 5^ r. 28'. 14"; Jupiter, 6M0°. 21'. 4"; Saturn, 8\ 28°. 9'. 7", the Georgian, 11\ l6°. I9'. 30". Tlie excentricities of the orbits, the mean distance of the earth from the sun being 1 00000, are, M-^rcury, 7955,4; Venus, 498; the Earth, l681,395; Mars, 14183,7 ; Jupiter, 25013,3 ; Saturn, 53640,42 ; the Georgian, 908O4. The greatest equations are. Mercury, 23°. 40'. o" ; Venus, 0°. 47'. 20"; the Earth, V. 55'.36",5 ; Mars, 10°. 40'. 40" ; Jupiter, 5\ 30'. 38 ',3 ; Saturn, 6°. 26'. 42"; the Georgian, 5°. 27'. 16". The aphelia of the orbits of the planets have a mo- tion, which may be found, from finding tlie places of the aphelia of each at two different times. Those motions in longitude in 100 years are. Mercury, 1°. 33'. 45"; Venus, 1°. 21'. O"; the Earth, 1°. 43'."35"; Mars, 1°. 51'. 40"; Jupiter, 1°. 34'. 33"; Saturn, 1°. 50'. 7". According to the calculation of M. de la Grange, the aphelion of the Georgian Planet is progressive t5",17 in the year, from the action of Jupiter and Saturn; consequendy it's motion in longitude is 50",25+3",17=53",42. 118 Chap. XIV. ON THE NODES AND INCLINATIONS OF THE ORBITS OF THE PLANETS TO THE ECLIPTIC. (183.) From observing the course of the planets for one revolution, their orbits are found to be incHned to the ecliptic, for they appear only twice in a revolution to be in the ecliptic ; and as it is frequently requisite to reduce their places in the ecliptic, ascertained from observation, to the corresponding places in their orbits, it is necessary to know the inclinations of their orbits to the ecliptic, and the points of the ecliptic where their orbits intersect it, called the Nodes. But, pre- vious to this, we must show the method of reducing the places of the planets seen from the earth to the places seen from the sun, and how to compute the heliocentric latitudes. (184.) Let E be the place of the earth, P the planet, S the sun, v the first point of aries ; draw Pv perpendicular to the ecliptic, and produce Es to a. Compute, at the time of observation, the longitude of the sun seen at a (115), and you have the longitude of the earth at E, or the angle r 8E ; compute also . the longitude of the planet, or the angle y* Sv (115), NODES AND INCLINATIONS, &C. 119 and the difFerence of these two angles is the angle ESv of convmitation. Observe the place of the planet in the ecliptic, and the place of the sun being known, we have the angle vES of elongation in respect to longitude; hence, we know the angle SvE, which measures the difference of the places of the planets seen from the earth and the sun ; therefore, the place of the planet seen from the earth being known, the place seen from the sun will be known. Also, tan. PEv : rad. :: vP : Ev (Trig. Art. 123) rad. : tan. PSv :: vS : vP .-. tan. PEv : tan. PSv :: vS: Ev :: sin. SEv : sin. ESu; that is, the sine of elongation in longitude : sin. of the difference of the longitudes of the earth and planet :: tan. of the geocentric latitude ; tan. of the heliocentric latitude. When the latitude is small, Sv : Ev is very nearly as PS : PE, which, in opposition, is very nearly as PS : PS— SE. Or we may com- pute (167) the values of PS and SE, which we can do with more accuracy than we can compute the angfles SEv and ESv. The curtate distance Sv of the planet from the sun may be found, by saying, rad. : cos. PSv :: PS : Sv. (185.) Now to determine the place of the node, find the planet's heliocentric latitudes just before and after it has passed the node, and let a and b be the places in the orbit, 7n and n the places reduced to the ecliptic; then the triangles arnN, bnN (which we may 'consider as rectilinear) being similar, we have am : hn :: Nm : N^n; therefore, am-\-bn : am w Nm + Nn (mn) : Nm, or «m-F bn : mn :: am : Nm, that is, the sum of the two latitudes : the difference of the longitudes :: either latitude: the distance of the node from the longitude corresponding to that latitude. Or 120 NODES AND INCLINATIONS OF THE if we take the two latitudes from the earth, it will be very nearly as accurate when the observations are made in opposition. If the distance of the observa- tions should exceed a degree, this rule will not be suffi- ciently accurate, in which case we must 'make our computations for spherical triangles thus. Put m n = a, am = (i, bn=zh, Nm = x\ then (Trig. Art. 212) sin. a- X , ,., sin. cr ,. , . ^- ; — = cot. A = -, radms being unity; but tan. b tan. /^ o j ' (Trig. Art. lOl) sin. «— a: = sin. ax cos. <i' — sin..rx , sin. a X cos. x - sin. x x cos. a sin. x cos. a\ hence, therefore. tan. b sin. a X tan /3 tan.fi sin. X = tan.a?. tan. b + COS. a x tan./3 cos. x Ex. Mr. Bugge observed the right ascension and declination of Saturn, and thence deduced (1 14,184) the following heliocentric longitudes and latitudes. 1784. Apparent Time. Heliocentric Lon. Heliocentric Lat\ Julyl2,atl2\ 3'. 1" 9s. 20°. 3 7'. 29'' 0°. 3'. 13" N. 20,-11.29. 9 9. 20. 51. 53 0. 2. 41 Aug. 1,-10.38.25 9- 21. 13. 17 0. 1. 34 8,-10. 9. 9. 21. 26. 2 0. 0. 56. 21,- 9. 14.59 g. 21 49. 27 0. 0. 2 27,- 8. 50. 19 9. 22. 0. 12 0. 0. 27 S. 31,- 8. 33.47 9. 22. 7. 32 0. 0. 50 Sept. 5,- 8. 13. 45 9. 22. 16. 28 0. 1. 21 15,- 7. 33. 45 9. 22. 34. 32 0. 1. 59 Oct. 8- 6. 4. 23 — — ' 9. 23. 16. 15 0. 3. 35 From the observations on August 21 and 27, by considering the triangles as plane, cr = 44",5; from those on 21 and 31, :r=44",5; and from those on August 21, and September 5, ^ = 40"; the mean of these is x = 42" ; Mr. Bugge makes ^ = 41", probably by taking the mean of a greater number, or computing from considering them as spherical triangles; hence, the heliocentric place of the descending node was 9*. ORBITS OF THE PLANETS TO THE ECLIPTIC. 121 21°. 50'. 8",5. Now on August 21, at gh. 12'. 26" true time, Safurns heliocentric longitude was 9'. 21°. 49'. 27", and on 27, at 8/^ 49'. 23" true time, it was 9. 22°. 0'. 12"; therefore, in 5d, 23h. 36'. 57" Saturn moved lO'. 45" in longitude; hence, lO'. 45" : 41" :: 5d. 23h. 36'. 57" : 9/?. 7'. 44" the time of describing 41" in longitude, which added to August 21, 9^. 12'. 26", gives August 21, 18/^ 20'. 10", the time when Saturn was in it's node. The longitudes of the nodes of the planets for the beginning of 17^0^ are, Mercury, V. 15°. 20'. 43" Venus, 2^ 14°. 26'. 18"; Mars, 1^ 17°. 38'. 38" Jupiter, 3^ 7°. 55'. 32"j Saturn, 3^. 21°. 32'. 22" Georgian, 2^ 12°. 47'. (186.) To determine the inclination of the orbit, we have am the latitude of the planet, and m N it's dis- tance upon the ecliptic from the node ; hence, Trig. Art. 210) sin. j}iN : tan. am :: rad. : tan. of the ano-Ie iV. But the observations which are near the node must not be used to determine the inclination, as a very small error in the latitude will make a consider- able error in the angle. If we take the observation on July 20, it gives the angle 2^. 38'. 15": if we take that on October 8, it gives the angle 2°. 22'. 13"; the mean of these is 2°. 30'. 14", the inclination of the orbit to the ecliptic, from these observations. Or the inclina- tion may be found thus, (I87.) Find the angle PSv (184), then the place of the planet and that of it's node being given^ we know vN; hence (Trig. Art. 210), sin. v N : tan Pv :: rad. : tan. PNv the inclination of the orbit. 122 NODES AND INCLINATIONS OF THE On March 27, 1694, at 7/?. 4'. 40", at Greenwich, Mr. Flamstead determined the right ascension of Mars to be 115°. 48'. 55'', and it's decHnation 24\ lo'. 50" north; hence, (184) the geocentric longitude was 2B 23°. 26'. 1 2", and latitude 2°. 46'. 38". Let 8 be the sun, E the Earth, P Mars, v it's place reduced to the ecliptic. Now the true place of Mars (by calculation) seen from the sun was Si 28°. 44'. 14", and the place of the sun was t 7^- 34. 25"; hence, subtracting the place of the sun from the place of Mars seen from the earth, we have the angle i'>ES between the sun and Mars 105°. 51'. 4'/"; and the place of the earth being ==b 7°. 34'. 25", take from it the place of Mars, and we have the angle ^^V = 38°. 50'. 11"; hence, (I87) sin. 105°. 51'. 47" : sin. 38°. 50'. 11" :: tan. PEv = 2''.46'. 38" : tan. PSv- 1°. 48'. 36". Now the place of the node was in « 17". 15', which subtracted from SI 28°. 44'. 14", gives 101°. 29'. 14" for the distance vN of Mars from it's node; hence, sin. vN=: 101°. 29'. 14" : tan. Pv=l^. 48'. 36" :: rad. : tan. PNv=zl^. 50'. 50", the inclination of the orbit. Mr. Bugge makes the inclination to be 1°.50'. 56",56, for Mart 1 788. M. de la Lande makes it 1°. 51' for 178O. The inclination of the orbits of the planets are, Mercury, 7°. O'. o"; Venus, 3°. 23'. 35"; Mars, 1°. 51'. 0";' Jupiter, 1°. 18'. 56"; Saturn, 2^ 29'. 50"; Georgian, 46'. 20". (188.) The motion of the nodes is found, by com- paring their places at two different times ; from whence, that of Mercury in 100 years is found to be 1°. 12'. 10"; Venus, 0°. 51'* 40"; Mars, 0°. 46'. 40"; Jupiter, 0°. !^^' . 30"; Saturn, 0°. 55'. 30". This mo- tion is in respect to the equinox. The Georgian planet has not been discovered long enough to determine the motion of it's nodes from observation. M. de la Grange has found the annual motion to be 1 2",5 by theory. But if we take the density of Venus according to M. de la Lande, it will be 20". 40"', which he uses in his tables. ORBITS OF THE PLANETS TO THE ECLIPTIC. 123 Thus we have determined all the elements necessary for computing the place of a planet in it's orbit at any time ; but to facilitale the operatioD, which would be extremely tedious if we had only the elements thus irivcn, astronomers have constructed tables of their motions, by which their places at any tune may be very readily computed. Since the discovery of the Georgium Sldus, four other primary planets have been discovered : The first called Ceres, was discovered by 31. Piazzi at Palermo, Jan. 1, 1801 ; the second, called Pallas, was discovered by Dr. Others at Bretnen, March 28, 1802 ; the third, called Juno, was discovered by M. Harding at LUienthal, September 1, 1804; and the fourth, called /"e.vfa, was discovered by Dr. Olbers, March 29, I807. The following Table contains the elements of the orbits of the three first ; the orbit of the fourth is not yet computed. Elements. Epocfi of Mean Long. 1 803 for Merid. of Seeberg Long, of Aphelion - - Long, of ascending Node inclination of the Orbit - Excentricity - _ - - Log. of Mean Distance - iVIean Din in. Trop. Mot. Ceres. 30». 1 2'. 1",1 326. 2S. 4,4 81. 0.41 10. 37. 86 0,07 84699 0,442004 77l'/,0521< Pallas. 18° 13' I",] 301. 3.24,3 J72. 30.47 31-. 37.43 0,2461007 0,4417647 771 ",6802 Juno. 42.36.36 233. 11.40 171. 4.15 13. 3.38 0,254236 0,4256078 8I5",959 Dr. Herschel makes the diameter of Pallas 147 miles: and that of Ceres l6l,6 miles. 124 Chap. XV. ON THE APPARENT MOTIONS AND PHASES OF THE PLANETS. (I89.) As all the planets describe orbits about the sun as their center, it is manifest, that to a spectator at the sun they would appear to move in the direction in which they really do move, and shine with full faces. But to a spectator on the earth, which is in motion, tliey will sometimes appear to move in a direction con- trary to their real motion, and sometimes appear sta- tionary ; and as the same face which is turned towards the earth, is not turned towards the sun, except in con- junction and opposition, some part of the disc which is towards the earth will not be illuminated. These, with some other appearances and circumstances which are observed to take place among the planets, we shall next proceed to explain ; and as they are matters in which great accuracy i.s never requisite, being of no great practical use, but rather subjects of curiosity, we shall consider the motions of all the planets as per- formed in circles about the sun in the center, and lying in the place, of the ecliptic. (190.) To find the position of a planet when sta- tionary. Let S be the sun, E the earth, P the co- temporary position of the planet, Xy the sphere of the fixed stars, to which we refer the motions of the planets ; let EF, PQ, be two indefinitely small arcs described in the same time, and let EPy FQ pro- duced, meet at L ; then it is manifest, that whilst the earth moves from E to F, the planet appears stationary at L ; and on account of the immense distance of the fixed stars, EPL, FQL may be considered as parallel. Draw SE, SFw, SvP, and SQ ; then, as EP and APPARENT MOTIONS, &C. 125 Fa are parallel, the angle QFS--PES=PwS^ PES = ESF, and SPw - SQF= SvF- SaF= PSQ ; that is, the cotemporary variations of theangles E and P are as ESF : PSQ, the cotemporary variations of the angular velocities of the earth and planet, or (be- cause the angular velocities are inversely as the periodic times, or inversely in the sesquiplicate ratio of the distances) as SP^ : SE^, or, (if SP : .S'^ :: « 1) as ffli : 1^. But sin. SEP : sin. SPE being as SP :SE, or, a : 1, the cotemporary variations of these angles will be as their tangents*. Hence, if x and i/ be the sines of the angles SEP and SPE. we have x -. y ■.-. a : 1, and ?J Jl-x'' s/^ y' av : 1, whence 0?^ = a' a' and a:r=: . — the sine of the a'-fa + r Va^ + u-\-\ planet's elongation from tlie sun, when stationary. Ex. If P be the earth, and E Venus ; and we take the distances of the earth and Venus to be 100000 and 72333, we find :r = 0,4826*4 the sine of 28". 51'. 5", See the Optics, Art. 421. 126 APPARENT MOTIONS the elongation of Venus when stationary, upon the supposition of circular orbits. For excentric orbits, the points will depend upon the position of the apsides and places of the bodies at the time. We ma}-, however, get a very near approxi- mation thus. Find the time when the planet would be stationary if the orbits were circular, and compute for several days, about that time, the geocentric place of the planet, so that you get two days, on one of which the planet was direct, and on the other retro- grade, in which interval it must have been stationary, and the point of time when this happened may be de- termined by inter])olation. (191.) To find the time when a planet is stationary, we must know the time of it's opposition, or inferior conjunction. Let m and n be the daily angular ve- locities of the earth and planet about the sun, and v the angle PSE when the planet is stationary ; then m — ti, or n — m, is the daily variation of the angle at the sun between the earth and planet, according as it is ^ superior or inferior planet ; hence, m — ?i, or 'V V n—rn, -. v '.: \ day : , or , the time from •^ 711 — 71 n — 771 opposition or conjunction to the stationary points both before and after. Hence, the planet must be stationary twice every synodic* revolution. Ex. Let P be the earth, E Venus ; then by the Example to Art. I90, the angle SPE = 28°. 51',5 ; therefore, PSE =13° ; also, n — T}i = 3f; hence, 3/' : 13^ :: 1 day : 21 days the time between the inferior conj miction and stationary positions. (192.) If the elongation be observed when sta- tionary, we may find the distance of the planet from the sun, compared with the earth's distance, supposed * A Synodic revolution is the time betweea two conjunctions of the same sort, or two oppositions of a planet. AND PHASES OF THE PLANETS. 127 to be unity. For (190) a.' = hencOj «- + X' x'-l X a= — — = (if ^ = the tangent of the angle whose sine is x) a* - 1' a=t' ; consequently a = ~t^ + t \/ 1 H — J upon the supposition of circular orbits. (193.) A superior planet is retrograde in opposition, and an inferior planet is retrograde in it's inferior conjunction ; for let E be the earth, P a superior planet in opposition ; then, as the velocities are as the inverse square roots of the radii of the orbits, the su- perior planet moves slowest ; hence, if EF, PQ, be two indefinitely small cotemporary arcs, PQ is less than E F, and on account of the immense distance of the sphere yZ of the fixed stars, FQ must cut EP in some point x between P and ?«, consequently, the planet appears retrograde from m to ??, If P be the earth, and E an inferior planet in inferior conjunction, it will appear retrograde from v to w. These retro- grade motions must necessarily continue till the' 128 APPARENT MOTIONS planets become stationary. Hence, from this and the last Artide, a superior planet appears retrograde from it's stationary point before opposition to it's stationary point after ; and an inferior planet, from it's stationary point before inferior conjunction to it's stationary point after. (194.) If S be the sun, E the earth, F Venus or Mercury, and EFsl tangent to the orbit of the planet. then will the angle SEf^ be the greatest elongation of the planet from the sun ; which angle, if the orbits were circles having the sun in their center, would be found by saying, ES : *S'^ :: rad, : sin. SEV. But the orbits are not circular, inconsequence of which the angle E V S will not be a right angle, unless the greatest elongation happens vrhen the planet is at one of it's apsides. The angle SEP^ is also subject to an alteration from the variation oi S E and S f^. The greatest angle SJ5/^ happens, when the planet is in it's aphelion and the earth in it's perigee ; and the least angle SEV, when the planet is in it's perihelion and the earth in it's apogee. M. de la Lande has calcu- lated these greatest elongations, and finds them 47°. 48'. and 44°. 57' for Venm, and 28^. 20' and 17°. 3& for Mercury. If we take the mean of the greatest elongations of Venus, which is 46°. 22',5, it gives the angle FSE=43°. 37',5 : and as the difi'erence of the daily mean motions of Venus and the earth about the sun is 37', we have 37' : 43°. 37',5 :: 1 day : 70,7 AND PHASES OF THE PLANETS. 129 i^ays, the time that would elapse between the greatest elongations and the inferior conjunction, if the mo- tions had been uniform, which will not vary much from the true time. (195.) To delineate the appearance of a planet at any time. Let S be the sun, JE the earth, /"^ Venus, for example; aPl) the plane of illumination perpen- dicular to SFy cFd, the plane of vision perpendicular to E Vy and draw av perpendicular to cf/; then ca is the breadth of the visible illuminated part, which is projected by the eye into cv, the versed sine of CVa, or SVZ, for S Vc is the complement of each. Now the circle terminating the illuminated part of the planet, being seen obliquely, appears to be an ellipse (Con. Sect. p. 37); therefore, if cmdn represent the projected hemisphere of Venus next to the earth, mw, cd, two diameters perpendicular to each other, and we take ci;=the versed sine of SVZ, and describe the ellipse mvn, then cv is the axis minor, and mcnvm will represent the visible enlightened part, as it appears at the earth ; and from the property of the ellipse (Con. Sect. Ell. Prop. 7. Cor. 6.), this area varies as cv. Hence, the visible enlightened part : the whole disc ;: the versed sine of SFZ : diameter. Hence, Mercury and Fenus will have the same phases from their inferior to their superior conjunc- tion, as the moon has from the new to the full : and I 130 APPARENT MOTIONS the same from the superior to the inferior conjunction, as the moon has from the full to the new. Mars will appear gibbous in quadratures, as the angle SFZ will then differ considerably from two right angles, and consequently the versed sine will sensibly differ from the diameter. For Jiipiter\ Saturn, and the Georgian, the angle SVZ never differs enough from two right angles to make those planets appear gibbous, so that they always appear full-orbed. (196.) Let /^ he the moon; then as EV'\s\evy small compared with P^ S, ES, these lines will be very nearly parallel, and the angle SVZ very nearly equal to SEV\ hence, the visible enlightened part of the moon varies very nearly as the versed sine of ifs elongation. (197-) Dr. Halley proposed the following problem : To find the position of Venus when brightest, suppos- ing it's orbit, and that of the earth, to be circles, having the sun in their center. Draw Sr perpendicular to EVZ, and put a - SE, b=SV, x = EV, y - Vr ; then h—y is the versed sine of the angle SVZ, which versed sine varies as the illuminated part; and as the intensity of light varies inversely as the square of it's distance, the quantity of light received at the earth varies as ~^= A- *^; but by Euclid, B. II. P. 12. X X' X- '' rt* _ fc2 _ J^.1 a'^=l/--\-x^ + 2xy ; hence, y= — ^ ; substitute Ji X this for y, and we get the quantity of light to be as b a^-b^-x^ 2bx-a' + b--\-x^ *-T - • 5 = ^— " = a maximum ; put X 2x^ 2x^ ^ the fluxion=0, and we get x = ^3a^ -{- //' —2b. Now, if «=1, Z>=,72333, as in Dr. ^ff/Ze/s Tables, then .r = , 43036; hence, the angle ES V=22\ 21', but the angle ESV, at the time of the planet's greatest elonga- tion, is 43°. 40'; hence, Venus is brightest between it's inferior conjunction and it's greatest elongation ; also, the angle SEV=3()°. 44' y th§ elongation of Venus AND PHASES OF THE PLANETS. 13'! from tbt3 sun at the same time, and z SVZ=- VSE'\- VES=62°. 5', the versed sine of which is 0,53, radius being unity ; hence (195), the visible enhghtened part : whole disc. :: 0,53 : 2 ; Venus therefore appears a little more than one fourth illuminated, and answers to the appearance of the moon when five days old. Her diameter here is about 39 ", and therefore the en- lightened part is about 10",25. At this time, Venus is bright enough to cast a shadow at night. This situation happens about 36 days before and after it's inferior conjunction ; for, supposing Venus to be in conjunction with the sun, and when seen from the sun to depart from the earth at the rate of 3/' in 1 day, we have 3f : 22°. 21' :: 1 day : 36 days nearly, the lime from conjugation till Venus is brightest. (198.) If we apply this to Mercury, b=,3l7li and jc= 1,00058 ; hence, the angle ESF=78°. 55'l; but the same angle, at the time of the planet's greatest elongation, is 67°. 13'|. Hence, Mercury is brightest between it's greatest elongation and superior conjunc- tion. Also, the angle SEF=22°. 18'4, the elonga- tion of Mercury at that time. (199.) When Venus is brightest, and at the same time is at it's greatest north latitude, it can then be seen with the naked eye at any time of the day, when it is above the horizon; for when it's north latitude is the greatest, it rises highest above the horizon, and therefore is more easily seen, the rays of light having to come through a less part of the atmosphere, the higher the body is. This happens once in about eight years, Venus and the earth returning to the same parts of their orbits after that interval of time. (200.) Venus is a morning, star from inferior to su- perior conjunction, and an evening star from superior to inferior conjunction. For let S be the sun, E the earth, ACBD the orbit of Venus, arm, csn, two tan- gents to the earth, representing the horizon at </and c. Then the earth, revolving about it's axis according to the order abcj when a spectator is at a, the part rCm I 2 132 APPARENT MOTIONS of the orbit of Venus is above the horizon, but the sun is not yet risen ; therefore Venus, in going from r through C to m, appears in the morning before sun- rise. When the spectator is carried by the earth's ro- tation to c, the sun is then set, but the part nDs of Venus' orbit is still above the horizon ; therefore. Venus, in going from n through D to s, appears in the evening after sun-set. (201.) If two planets revolve in circular orbits, to find the time from conjunction to conjunction. Let P:=the periodic time of a superior planet, /? = that of an inferior, ^=the time required. Then P : 1 day :; 360° 360° : — p-the angle described by the superior planet 360° in 1 day ; for the same reason, is the angle de- scribed by the inferior planet in 1 day; hence, — p- is the daily angular velocity of the inferior planet AND PHASES OF THE PLANETS. 133 from the superior. Now if they set out from con- junction, they will return into conjunction again after the inferior planet has gained 3oO° ; hence, — 360° : 360" 1 day : t- jr^. J P—p This will also give the time between two oppositions, or between any two similar situations. 134 Chap. XVI. ON THE MOON'S MOTION FROM OBSERVATION AND ITS PHiilNOMENA. (202.) The moon being the nearest, and after the sun, the most remarkable body in our system, and also useful for the division of time, it is no wonder that the ancient astronomers were attentive to discover it's motions ; and it is a very fortunate circumstance that their observations have come down to us, as from thence it's mean motion can be more accurately settled, than it could have been by modern observations only ; and it moreover gav-e occasion to Dr. Halley, from the observations of- some ancient eclipses, to dis- cover an acceleration in it's mean motion. The proper motion of the moon, in it's orbit about the earth, is from west to east ; and from comparing it's place with the fixed stars in one revolution, it is found to describe an orbit inclined to the ecliptic ; it's motion also ap- pears not to be uniform ; and the position of the orbit, and the line of it's apsides are observed to be subject to a continual change. These circumstances, as they are established by observation, we come now to explain. To determine the Place of the Moon^s Nodes. (203.) The place of the moon's nodes may be de- termined as in Art. 185, or by the following method. In a central eclipse of the moon, the moon's place at the middle of the eclipse is directly opposite to the sun, and the moon must also then be in the node; calcu- late therefore the true place of the sun, or, which is more exact, find it's place by observation, and the MOONS MOTION, &C. 135 opposite point will be the true place of the moon, and consequently the place of it's node. Ex. M. Cassini, in his Astronomy, p. 281, informs us, that on April l6, 1707, a central eclipse was ob- served at Paris, the middle of which was determined to be at I3h. 48' apparent time. Now the true place of the sun, calculated for that time, was 0^ 26**. 19' 17" ; hence, the place of the moon's node was 6^. 26". 19'. 17". The moon passed from north to south latitude, and therefore this was the descending node. (204.) To determine the mean motion of the nodes, find (203) the place of the nodes at different times, and it will give their motion in the interval ; and the greater the interval, the more accurately you will get the mean motion. Mayer makes the mean annual mo- tion of the nodes to be 12^ 19'. 43", 1. On the Inclination of' the Orbit of the Moon to the Ecliptic. (205.) To determine the inclination of the orbit, observe the moon's right ascension and declination when it is 90° from it's nodes, and thence compute it's latitude (114), which will be the inclination at that time. Repeat the observation for every distance of the sun from the earth, and for every position of the sun in respect to the moon's nodes, and you will get the inclination at those times. From these observa- tions it appears, that the inclination of the orbit to the ecliptic is variable, and that the least inclination is about b", which is found to happen when the nodes are in quadratures; and the greatest is about S*". 18', which is observed to happen when the nodes are in syzygies. The inclination is also found to depend upon the sun*s distance from the earth. On the mean Motion of the Moon. (206.) The mean motion of the moon is found from observing it's place at two different times, and 136 MOON S MOTIOI* you get the mean motion in that interval, suppositTg the moon to have had the same situation in respect to it's apsides at each observation ; and if not, if there be a very great interval of the times, it will be sufficiently exact. To determine this, we must compare together the moon's places, first at a small interval of time from each other, in order to get nearly the mean time of a revolution; and then at a greater interval, in order to g.et it more accurately. The moon's place may be determined directly from observation, or deduced from an eclipse, (207.) M. Cassini, in his Astronomy, p. 394, ob- serves, that on September 9, 17 18, the moon was eclipsed, the middle of which eclipse happened at 8/^. 4', when the sun's true place was 5^. 16". 40'. This he compared with another eclipse, the middle of which was observed at 8 A. 32'. on August 29, 17^9, when the sun's place was 5*. 5°. 47'- In this interval of 35 id^ 28' the moon made 12 revolutions and 349". 7' over; divide therefore 354^. 28' by 12 revolutions + 349°. 7'. part of a revolution, and it gives 27^i 'jh. 6' for the time of one revolution. From two eclipses in 1699, 1717, the time was found to be 22d. *jh. 43'. 6". (208.) The moon was observed at Paris to be eclipsed on Sept. 20, 1717, the middle of which eclipse was at 6A. 2'. Now Ptolcmif mentions, that a total eclipse of the moon was observed at Babylon on March 19, 720 years before J. C. the middle of which happened at ^h. 30', at that place, which gives 6/l 48' at Paris. The interval of these times was 2437 years (of which 609 were bissextiles) ^]r4f days wanting 46'; divide this by 27^. lh. 43'. 6", and it gives 32585 re- volutions and a little above |. Now the difference of the two places of the sun, and consequently of the moon, at the times of observations, was 6^ 6°. 12'^. Therefore, in the interval of 24373/. 17 4d. wanting 46', the moon had made 32585 revolutions 6\6°. 12', which gives 27c?. 7h. 43'. 5" for the mean time of a revolution. This determination is very exact, as the FROM OBSERVATION. 137 moon was at each time very nearly at the same dis- tance from it's apside. Hence, the mean diurnal mo- tion is 13°. 10'. 35", and the mean hourly motion 32'. 56". 27'"|. M. de la Lande makes the mean diurtial motion 13". lO'. 35",02784394. This is the mean time of a revolution in respect to the equinoxes. The place of the moon at the middle of the eclipse has here been taken the same as that of the sun, whicli is not accurate, except for a central eclipse ; it is sufficiently accurate, however, for this long interval. (209.) As the precession of the equinoxes is 50",25 in a year, or about 4" in a month, the mean revolution of the moon in respect to the fixed stars must be greater than that in respect to the equinox, by the time the moon is describing 4" with it's mean motion, which is about 7". Hence, the time of a sidereal re- volution of the moon is 27<^. "^h. 43'. 12". (210.) Observe accurately the place of the moon for a whole revolution as often as it can be done, and by comparing the true and mean motions, the greatest difference will be double the equation. If two ob- servations be found, where the difference of the true and mean motions is nothing, the moon must then have been in it's apogee and perigee (168). Mayer makes the mean excentricity 0,05503568, and the corresponding greatest equation 6°. 18'. 3l"6. It is 6°. 18'. 32" in his last Tables, published by Mr. Mason, under the direction of Dr. Maskelyne. (211.) To determine the place of the apogee, from M. Cass'ml's observations, we have the greatest equa- tion = 5\ 1'. 44"5; therefore (171), 57°. 17'. 48"8 : 2°. 30'. 52"25 :: ^C= 100000 : CS=4'688 for the moon's excentricity at that time*. Now (Fig. p. 101.) let V be the focus in which the earth is situated ; then * The excentricity of the moon's orbit is subject to a variation, it being greatest wh«n the apsides lie in syzygies, and least when in quadratures. 138 moon's motion (169) supposing QSP to be the mean anomaly, as QvP is the true anomaly, their difference SPv is the equation of the orbit, which equation is here 37'. 50",5 ; and as PS=Pr, the angle t^r 6'= 18'. 55",25 ; hence, (Trigonometry, Art. 128) vS=z817^ • ^^ = 200000 :: sin. vrS= 18'. 5 5", 25 : sin. vSr, or QSr,=: 7°. 12'. 20", from which take t;r^=18'. 55",25, and we have QvP = 6°. 53'. 25" the distance of the moon from it's apogee; add this to 2^ 19". 40', the true place of the moon, and it gives 2^ 26'\ 33'. 25" for the place of the apogee on December 10, l685, at loh. 38'. 10" mean time at Paris. This therefore may be considered as an epoch of the place of the apogee. To determine the mean Motion of the Apogee. (212.) Find it's place at different times, and com- pare the difference of the places with the interval of the time between. To do this, we must first compare observations at a small distance from each other, lest we should be deceived in a whole revolution; and then we can compare those at a greater distance. The mean annual motion of the apogee in a year of 365 days is thus found to be 40°. 39'. 50", according to Mayer. Horrox, from observing the diameter of the moon, found the apogee subject to an annual equation of 12°,5. (213.) The motion of the moon having been ex- amined for one month, it was immediately discovered that it was subject to an irregularity, which sometimes amounted to 5° or 6°, but that this irregularity disap- peared about every 14 days. And by continuing the observations for different months, it also appeared, that the points where the inequalities were the greatest, were not fixed, but that they moved forwards in the heavens about 3° in a month, so that the motion of the moon, in respect to it's apogee, was about less than it's absolute motion ; thus it appeared that the apogee FROM OBSERVATION. 139 had a progressive motion. Ptolemy determined this first inequality, or equation of the orbit, from three lunar eclipses observed in the years 7^9 ^"d 720, before J. C. at Babylon by the Chaldeans; from which he found it amounted to 5°. l', when at it's greatest. But he soon discovered that this inequality would not account for all the irregularities of the moon. The distance of the moon from the sun, ob- served both by Hipparchus and himself, sometimes agreed with this inequality, and sometimes it did not. He found that when the apsides of the moon's orbit were in quadratures, this^r*^ inequality would give the moon's place very well ; but that when the apsides w^ere in syzygies, he discovered that there was a further inequality of about 2°~, which made the whole ine- quality about 7°-T. This second inequality is called the Evection, and arises from a change of excentricity of the moon's orbit. The inequality of the moon was therefore found, by Ptolemi/, to vary from about 5° to 7°!^, and hence the mean quantity was 6". 20'. Mayer makes it 6°. 18'. 3l",6. It is very extraordinary, that Ptolemy should have determined this to so great a degree of accuracy. We cannot here enter any further into the inequalities of the motion of the moon. They who wish to see more on this subject, may consult my Complete System of Astronomy. 140 MOONS DIAMETER. (214.) Times of the Revolutions of the Moon, ofifs Apogee and Nodes, as determined hy M. de la Lande. 27''.7\43'. 4",6r95 27. 7. 43. 11,5259 44. 2,8283 18. 33,9499 29.12. 27.13. 27. 5. 5. 35,603 8". 311. 8. 34. 57,6177 8. 312. 11. 11. 39,4089 18. 228. 4. 52. 52,0296 18. 223. 7. 13. 17,744 - - 13°. 10'.35"02784394 Tropical revolution Sidereal revolution Synodic revolution Anomalistic revolution Revolution in respect to the node - - Tropical revolution of the apogee Sidereal revolution of the apogee - - Tropical revolution of the node - - Sidereal revolution of the node - - • Diurnal motion of- the moon in respect | to the equinox - Diurnal motion of the apogee 0. 6.41,069815195 Diurnal motion of the node - O. 3. 10,638603696 The years here taken are the common years of 365 days. On the Diameter of the Moon. (215.) The diameter of the moon may be measured, at the time of it's full, by a micrometer; or it may be measured by the time of it's passing over the vertical wire of a transit telescope ; but this must be when the moon passes within an hour or two of the time of the full, before the visible disc is sensibly changed from a circle. To find the diameter by the time of it's passage over the meridian, let <f' = the horizontal di- ameter of the moon, c=sec. of it's declination, and m =the length of a lunar day, or the time from the moon's diameter. I4i passage of the moon over the meridian on the day we calculate, to the passage over the meridian the next day. Then (102) cd" is the moon's diameter in right ascension ; hence, 360'' : cd' :: m : the time {t) of passing the meridian ; therefore c?" = 360** x • If we '^ ° cm observe when the limb of the moon comes to the meridian, we can find the time when the center comes to it, by adding to, or subtracting from, the time when. the first or second limb comes to the meridian, half the time of the passage of the moon over the meridian. The time in which the semidiameter of the moon passes the meridian, may be found by two Tables, in the Tables of the moon's motion. (21 6.) Alhategnius made the diameter of the moon to vary from 29'. 30" to 35'. 20", and hence the mean is 32'. 25". Copernicus found it from 2f. 34". to 35'. 38", and therefore the mean 31'. 3&'. Kepler made the mean diameter 31'. 22". M. de la Hire made it 31'. 30". M. Ca^^im made the diameter from 29'. 30" to 33'. 38". M. de la Lande, from his own ob- servations, found the mean diameter to be 31'. 26"; the extremes from 29'. 22" when the moon is in apogee and conjunction, and 33'. 31" when in perigee and opposition. The mean diameter here taken, is the arithmetic mean between the greatest and least diameters ; the diameter at the mean distance is 31'. 7". (217.) When the moon is at different altitudes above the horizon, it is at different distances from the J42 moon's phases. spectator^ and therefore there is a change of the appa- rent diameter. Let C be the center of the earth, A the place of a spectator on it's surface, Z his zenith, M the moon; then (Trig. Art. 128) sin. CAM, or ZAM : «in. ZCM'.CM : AM^^^L^L^^:^^ ; but the sni. ZAM apparent diameter is inversely as it's distance ; hence, the apparent diameter varies as ". ' ^^,,^ , CM being * * sm. ZCM *' supposed constant. Now, in the horizon, -— -v^Wrv sm. ZCM may be considered as equal to unity ; hence, 1 : —. — TyTTnfi or sm. ZLM : sm. ZAM, or cos. true alt. sm. ZCM ' (a) : cos. apparent alt. (A) :: the horizontal diameter : the diameter at the apparent altitude (A). Hence, the horizontal diameter : it's increase :: cos.« : cos. A —cos. a = (Trig. Art. Ill) 2 sin. ^a + i A x sin. I a-j;A; therefore the increase of the semidiameter , • 1 •, sin ^ « -{- ^A X sm.^a-^A = nor. semidiameter x = = 1 £ — • cos, a from this we may easily construct a table of the increase of the semidiameter for any horizontal semidiameter, and then for any other horizontal semidiameter, the increase will vary in the same proportion. On the Phases of the Moon. (218.) By Art. I96, the greatest breadth of the visible illuminated part of the moon's surface, varies as the vei*sed sine of the moon's elongation from the sun, very nearly; and the circle terminating the light and dark part, being seen obliquely, appears an elhpse ; hence, the following delineation of the phases. Let E be the earth, 8 the sun, Mthe moon ; describe the circle a 6crf, representing the hemisphere of the moon uhich is towards the earth, projected upon the plane LIBRATION OF THE MOON. 143 of vision; ac, db, two diameters perpendicular to each ether ; take dv=zthe versed sine of elongation SEM, and describe the ellipse avc, and (195) adcva will re- present the visible enhghtened part; which will be horned between conjunction and quadratures ; a semi- circle at quadratures ; and gibbous between quadra- tures and opposition ; the versed sine being less than radius in the first case, equal to it in the second, and greater in the third. The visible enlightened part varying as dv, vie have, the visible enlightened pai^t ' whole : : versed sine of' elongation ; diameter. On the Libration of the Moon. (219,) Many Astronomers have given maps of the face of the moon ; but the most celebrated are those of Hevelius in his Selenographia, in which he has re- presented the appearance of the moon in it's different states from the new to the full, and from the full to the new; these figures Mayer prefers. Langrenus and Ricciolus denoted the spots upon the surface by the names of Philosophers, Mathematicians, and other celebrated men, giving the names of the most cele- brated characters to the largest spots; Hevelius marked them with the geographical names of places upon the earth. The former distinction is now generally fol- lowed. The spots upon the moon are caused by the moun- tains and vallies upon it's surface; for certain parts are found to project shadows opposite to the sun ; and 144 tlBRATION Ot THE MOOJ^. when the sun becomes vertical to any of them, they are observed to have no shadows ; these therefore are mountains; other parts are always dark on that side next to the sun, and illuminated on the opposite side; these therefore are cavities. Hence, the appearance of the face of the moon continually varies, from it's altering it's situation in respect to the sun. The tops of the mountains, on the dark part of the moon, are frequently seen enlightened at a distance from the cosines of the illuminated part. The dark parts have, by some, been thought to be seas, and by others, to be only a great number of caverns and pits, the dark sides of which, next to the sun, would cause those places to appear darker than others. The great irregularity of the line bounding the light and dark part, on every part of the surface, proves that there can be no very large tracts of water, as such a regular surface would necessarily produce a line, terminating the bright part, perfectly free from all irregularity. If there was much water upon it's surface, and an atmosphere, as con- jectured by some Astronomers, the clouds and vapours might easily be discovered by the telescopes which we have now in use ; but no such phsenomena have ever been observed. (220.) Very nearly the same face of the moon is always turned tovvards the earth, it being subject only to a small change within certain limits, those spots which lie near to the edge appearing and disappearing by turns ; this is called it's Libratlon, and arises from four causes. 1. Galileo, who first observed the spots of the moon after the invention of telescopes, discovered this circumstance ; he perceived a small daily variation '^ arising from the motion of the spectator about the center of the earth, which, from the rising to the setting of the moon, would cause a little of the western limb of the moon te disappear, and bring into view a little of the eastern limb. 2. He observed likewise, that the north and south poles of the moon appeared and disappeared by turns; this arises from the axis of the LIBRATION OF THE MOON. 145 itioon not being ; perpendicular to the plane of it*s orbit, and is ^called a libration in latitude. 3. From the unequal angular motion of the moon about the earth, and the uniform motion of the moon about it's axis, a little of the eastern and western parts must gradually appear and disappear by turns, the period of which is a month, and this is called a libration in longitude; the cause of this libration was first assigned by Ricciolus, but he afterwards gave it up, as he made many observations which this supposition would not satisfy. Hevelius, however, found that it would solve all the phaenomena of this libration. 4. Another cause of libration arises from the attraction of the earth upon the moon, in consequence of it's spheroidical figure. (221.) If the angular velocity of the moon about it*s axis were equal to it's angular motion about the earth, the libration in longitude would not take place. For if E be the earth, abed the moon at v and w, and avc be perpendicular to Ebvd ; then aba is that hemi- sphere of the moon at v next to the earth. When the moon comes to w, if it did not revolve about it's axis bwd would be parallel to bvd, and the same face would not lie towards the earth. But if the moon, by re- volving about it's axis in the direction abed, had brought b into the line Ew, the same face would have been turned towards the earth ; and the moon would have revolved about it's axis through the angle bwE, K 14S LIBRATION OF THfi MOON. which is equal to the alternate angle wEv, the angle \vhich the moon has described about the earth. (222.) When the moon returns to the same point of it's orbit, the same face is observed t6 he towards the earth, and therefore (221) the time of the revolu- tion in it's orbit is equal to the time about it's axis. But in the intermediate points it varies, sometimes a little more to the east, and sometimes to the west, be- comes visible ; and this arises from it's unequal angular motion wEv about the earth, whilst the angular mo- tion about it's axis is equal, in consequence of which these two Angles cannot continue equal, and therefore, by the last article, the same face cannot continue to- wards the earth. Hence, the greatest libration in longitude is nearly equal to the equation of the orbit, or about 7^2 at it's maximum, and would be accu^ rately so, if the axis of the moon were perpendicular to it's orbit ; for the difference of the moon's mean motion and true motion, or the equation of the orbit, is the same as the difference of the moon's motion about it's axis and it's true motion, which is the libration. The same face will be towards the earth in apogee and perigee, for at those points there is no equation of the orbit. If E be the earth, M the moon, pq it's axis, J" P M^r-7 E TV" yM not perpendicular to the plane of the orbit ah ; then at a the pole/? will be visible to the earth, and at b the pole q will be visible ; as the moon therefore revolves about the earth, the poles must appear and disappear by turns, causing the libration in latitude. This is exactly similar to the cause of the variety of our sea- sons, from the earth's axis not being perpendicular to the plane of it*s orbit. Hence, nearly one half of the moon is never visible at the earth. Also, the time of it's rotation about it's axis being a month, the length of the lunar days and nights will be about a fortnight LIBRATION OF THE MOON. 14/ each, they being subject but to a very small change, on account of the axis of the moon being nearly per- pendicular to the ecliptic. (223.) Hevelms (Selenographia, p. 245.) observed, that when the moon was at it's greatest north latitude, the libration in latitude was the greatest, the spots which are situated near the northern limb being then nearest to it; and as the moon departed from thence, the spots receded from that limb, and when the moon came to it's greatest south latitude, the spots situated near the southern limb were then nearest to it. This variation he found to be about l'. 45", the diameter of the moon being 30'. Hence it follows, that when the moon is at it's greatest latitude, a plane drawn through the earth and moon perpendicular to the plane of the moon's orbit, passes through the axis of the moon ; consequently the equator of the moon must intersect the ecliptic in a line parallel to the line of the nodes of the moon's orbit, and therefore, in the heavens, the nodes of the moon's orbit and of it's equator coincide. (224.) It is a very extraordinary circumstance, that the time of the moon's revolution about it's axis should be equal to that in it's orbit. Sir /. Newton, from the altitude of the tides on the earth, has computed that the altitude of the tides on the moon's surface must be 23 feet, and therefore the diameter of the moon per- pendicular to a line drawn from the earth to the moon, ought to be less than the diameter directed to the earth, by 186 feet ; hence, says he, the same face must al- ways be towards the earth, except a small oscillation; for if the longest diameter should get a little out of that direction, it would be brought into it again by the attraction of the earth. The supposition of D. de Mairan is, that that hemisphere of the moon next the earth is more dense than the opposite one, and hence, the same face would be kept towards the earth, upon the same principle as above. (225.) When the moon is about three days from the new, the dark part is very visible, by the light re- K2 148 ALTITUDJE OF fleeted from the earth, which is moon-light to the Lunarians, considering our earth as a moon to them; and in the most favourable state, some of the principal spots may then be seen. But when the moon gets into quadratures, it's great light prevents the dark part from being visible. According to Dr. Smith, the strength of moon-light, at the full moon, is ninety thousand times less than the light of the sun; but, from some experiments of M. Bouguer, he concluded it to be three hundred thousand times less. The light of the moon, condensed by the best mirrors, produces no sensible effect upon the thermometer. Our earth, in the course of a month, shows the same phases to the Lunarians, as the moon does to us ; the earth is at the full at the time of the new moon, and at the new at the time of the full moon. The surface of the earth being about 13 times greater than that of the moon, it affords 13 times more light to the moon than the moon does to the earth. On the Altitude of the Luna?' Mountains. (226.) The method used by Hevelius, and others since his time, to determine the height of a lunar mountain is this. Let SLM be a ray of light from L M tlie sun, passing the moon at L, and touching the top of the mountain at M ; then the space between L and M appears dark. With a micrometer, measure LM^ and compare it with LC \ then, knowing LC, we kn ow LM, a nd by Eucl. B. L p. 4/. CM = ^ CL- + L3I~ is known ; from which subtract Cpy THE LUNAR MOUNTAINS. 149 and we get the height jo 71/ of the mountain. But as Dr. Herschel ohserves, in the Phil. Trans. 178I, this method is only applicable when the moon is in quadra- tures ; he has therefore given the following general method. Let E be the earth ; draw EMn and Lo perpendicular to the moon's radius RC, and Lr parallel to on, also ME' perpendicular to SM. Now ML would measure it's full length when seen from the earth in quadratures at E\ but seen from E, it only measures the length of Lr. As the plane passing through SM, EM, is perpendicular to a line joining the cusps, the circle RLp may be conceived to be a section of the moon perpendicular to that line. Now it is manifest, that the angle SLo or LCR, is very nearly equal to the elongation of the moon from the sun ; and the triangles LrM, LCo, being similar, Lo : LC :: Lr : Z.M= ""r,^' = __, ^,„,,^ , ra- Lo sine of elongation' dius being unity. Hence, we find Mp as before. Ex. On June, 178O, at seven o'clock, Dr. Herschel found the angle under which LM, or Lr appeared, to be 40", 625, for a mountain in the south-east quadrant; and the sun's distance from the moon was 125°. 8', whose sine is ,8104 ; hence, 40",625 divided by ,8104, gives 50", 13, the angle under which LM would ap- pear, if seen directly. Now the semidiameter of the moon was 16'. 2'\6, and taking its length to be 10^ 150 ALTITUDE OF miles, wehavel6'.2",6 : 50",13 :: IO9O: LiI/=56,73 miles ; hence, 3Ip~l,4'^ miles. (227.) Dr. Herschel found the height of a great many more mountains, and thmks he has good reason to believe, that their altitudes are greatly over-rated ; and that, a few excepted, they generally do not exceed half a mile. He observes, that it should be examined whether the mountain stands upon level ground, which is necessary, that the measurement may be exact. A low tract of ground between the mountain and the sun will give it higher, and elevated places between will make it lower, than it's true height above the Common surface of the moon. (228.) On April 19, 1787, Dr. ^(?7'*c/ie/ discovered three volcanos in the dark part of the moon; two of^ them seemed to be almost extinct^ but the third showed an actual eruption of fire, or luminous matter, resembling a small piece of burning charcoal covered by a very thin coat of white ashes ; it had a degree of brightness about as strong as that with which such a coal would be seen to glow in faint day-hght. The adjacent parts of the volcanic mountain seemed faintly illuminated by the eruption. A similar eruption ap- peared on May 4, 1783. Phil. Trans. 1787. On March 7j 1794, a few minutes before eight o'clock in the evening, Mr. Wilk'ms^ of Norwich, an eminent architect, observed, with the naked eye, a very bright spot upon the dark part of the moon ; it was there when he first looked at the moon ; the whole time he saw it, it was a fixed, steady light, except the moment before it disappeared, when it's brightness increased ; he conjectures that he saw it about five minutes. The same phaenomenon was observed by Mr. T. Stretton, in St. John's-square, Clerkenwell, London. Phil. Trans. 1794. On April 13, 1793, and on February 5, 1794, Mr. Piazzi, Astronomer Royal at Palermo, observed a bright spot on the dark part of the moon, near Aristarchus. Several other Astronomers have THE LUNAR MOUNTAINS. 151 observed the same phaenomenon. See the Memoirs de Berlm, for 1788. (229.) It has been a doubt amongst Astronomers, whether the moon has any atmosphere ; some suspect- ing that at an occultation of a fixed star by the moon, the star did not vanish instantly, but lost it's light gradually ; whilst- others could never observe any such appearance. M. Schroetar of Lilianthan, in the duchy of Bremen, has endeavoured to establish the existence of an atmosphere, from the following observations. 1. He observed the moon when two days and an half old, in the evening soon after sun-set, before the dark part was visible, and continued to observe it till it became visible. The two cusps appeared tapering in a very sharp, faint prolongation, each exhibiting it's farthest extremity faintly illuminated by the solar rays, before any part of the dark hemisphere was visible. Soon after, the whole dark limb appeared illuminated. This prolongation of the cusps beyond the simicircle, he thinks, must arise from the refraction of the sun's rays by the moon's atmosphere. He computes also the height of the atmosphere, which refracts light enough into it's dark hemisphere to produce a twilight, more luminous than the light reflected from the earth when the moon is about 32° from the new, to be 1356 Paris feet ; and that the greatest height capable of refracting the solar rays is 53/6 feet, 2. At an occul- tation of Jupiter's satellites, the third disappeared, after having been about l" or 2" of time indistinct; the fourth became indiscernible near the limb ; this was not observed of the other two. Phil. Trans. 1792. If there be no atmosphere of the moon, the heavens, to a Lunarian, must always appear dark like night, and the stars be constantly visible ; for it is owing to the reflection and refraction of the sun's light by tlie at- mosphere, that the heavens^ in every part, appear bright in the day. 1^2 PHiENOMENON OF On the Phcenomenon of the Harvest Moojt. (230.) The full moon which happens at, or nearest to, the autumnal equinox, is called the Harvest moon ; and at that time there is a less difference between the times of it's rising on two successive nights, than at any other full moon in the year; and what we here propose, is to account for this phaenomenon. (231.) Let P be the north pole of the equator Q^U, HAO the horizon, EAC the ecliptic, A the yniiU first point of Aries ; then, in north latitudes, A is the ascending node of the ecliptic upon the equator, AC being the order of the sines, and AQ that of the appa- rent diurnal motion of the heavenly bodies. When Aries rises in north latitudes, the ecliptic makes the least angle with the horizon ; and as the moon's orbit makes but a small angle with the ecliptic, let us first suppose EAC to represent the moon's orbit. Let A be the place of the moon at it's rising on one night; now, in mean solar time, the earth makes one revolu- tion in 23A. 56'. 4", and brings the same point ^ of the equator to the horizon again ; but, in that time, let the moon have moved in it's orbit from A to c, and draw the parallel of declination tcm, then it is mani- fest that 3'. 56" before the same hour the next night, the moon, in it's diurnal motion^ has to describe c n THE HARVEST MOON. 153 before it rises. Now en is manifestly the least possible, when the angle CAn is the least, ^£^ being given. Hence, it rises more nearly at the same hour, wHen it's orbit makes the least ano;le with the horizon. Now at the autumnal equinox, when the sun is in the first point of Libra, the moon, at that time at it's full, will be at the first point of Aries, and therefore it rises with the least diiference of times, on two succeslsive nights; and it being at the time of it's full, it is more taken notice of; for the same thing happens every month when the moon comes to Aries. (232.) Hitherto we have supposed the ecliptic to represent the moon's orbit, but as the orbit is inclined to it at an angle of 5°. 9' at a mean, let xAsz represent the moon's orbit when the ascending node is at A, and As the arc described in a day; then the moon's orbit making the least possible angle with the horizon in that position of the nodes, thearcs'w, and consequently the difference of the times of rising, will be the least possible. As the moon's nodes make a revolution in about 19 years, the least possible diflference can only happen once in that time. In the latitude of London the least difi^erence is about 17'. (233.) The ecliptic makes the greatest angle with the horizon when the first point of Libra rises, conse- quently, when the moon is in that part of it's orbit, the diflference of the times of it's rising will be the greatest; and if the descending node of it's orbit be there at the same time, it will make the diflTerence the greatest possible ; and this diflference is about \h. 17' in the latitude of London. This is the case with the vernal full moons. Those signs which make the least angle with the horizon when they rise make the greatest angle when they set, and vice versa; hence, when the diflference of the times of rising is the least, the diflference of the times of setting is the greatest, and the contrary. (234.) By increasing the latitude, the angle zAuj and consequently sn is diminished; and when the 154 HORIZONTAL MOON. time of describing sn, by the diurnal motion, is 3'. 56"' the moon will then rise at the same solar hour. Let us suppose the latitude to be increased until the angle sAn vanishes, then the moon's orbit becomes coinci- dent with the horizon every day, for a moment of time, and consequently the moon rises at the same sidereal hour, or 3'. .56" sooner, by solar time. Now take a globe, and elevate the north pole to this latitude, and, marking the moon's orbit in this position upon it, turn the globe about, and it will appear, that at the instant after the above coincidence, one half of the moon's orbit, corresponding to Capricorn, Aquarius, Pisces, Aries, Taurus, Gemini, will rise; hence, when the moon is going through that part of it's orbit, or for 13 or 14 days, it rises at the same sidereal hour. Nowj taking tlie angle ocA E^b". 9', and the angle EAQ= 23". 28', the angle QAjo, or QA H, when the moon's orbit coincides with the horizon, is 28°. 37' ; hence, (87) the latitude is 6v. 23' where these circumstances take place. If the descending node be at A, then x lying above JE;, QAt, or QAH=lS^. I9', and the latitude is 71°- 41'. In any other situation of the orbit, the latitude will be between these limits. When the angle QA,v is greater than the complement of latitude, the moon will rise sooner the next day. As there is a complete revolution of the nodes in about 18 years 8 months, all the varieties of the intervals of the rising and setting of the moon will happen within that time. On the Horizontal Moon. (235.) The phaenomenon of the horizontal moon is this, that it appears larger in the horizon than in the meridian; whereas, from it's being nearer to us in the latter than in the former case, it subtends a greater angle. Gassendus thought that, as the moon was less bright in the horizon, we looked at it there with a greater pupil of the eye, and therefore it appeared HORIZONTAL MOQN. 165 larger. But this is contrary to the prhiciples of Optics, since the magnitude of the image upon the retina does not depend upon the pupil. This opinion was supported by a French Ahhe, who supposed that the opening of the pupil made the chrystalline humour flatter, and the eye longer, and thereby increased the image. But there is no connection between the muscles of the iris and the other parts of the eye, to produce these effects. Des Cartes thought that the moon appeared largest in the horizon, because, when comparing it's distance with the intermediate objects, it appeared then furthest off; and as we judge it's distance greatest in that situation, we of course think it larger, supposing that it subtends the same angle. This opinion was supported hy Tiw Wallis, in the Phil. Trans. N°. I87. Dr. Berkley accounts for it thus. Faintness suggests the idea of greater distance; the moon appearing most faint in the horizon, sug- gests the idea of greater distance, arid, supposing the visual angle the same, that must suggest the idea of a greater tangible object. He does not suppose the visible extension to be greater^ but that the idea of a greater tangible extension is suggested, by the altera^ tion of the appearance of the visible extenision. He says, l.That which suggests the idea of greater mag- nitude, must be something perceived ; for what is not perceived can produce no? visible effect. 2. It must be something which is variable, because the nioon does not always appear, of the same magnitude in the horizon. 3. It cannot lie in the intermediate objects, they remaining the same ; also, when these objects are excluded from sight, it makes no alteration. 4. It cannot be the visible magnitude, because that is least in the horizon ; the cause, therefore, must lie in the visible appearance, which proceeds from the greater paucity of rays coming to the eye, producing ^am^- ness. Mr. Rouoning supposes that the moon appears furthest from us in the horizon, because the portion of the sky which we see, appears not an entire hemisphere, 156 HORIZONTAL MOON. but only a portion of one; and in consequence of this, we judge the moon to be furthest from us in the horizon, and therefore to be then largest. Dr. Smithy in his Optics^ gives the same reason. He makes the apparent distance in the horizon to be to that in the zenith as 10 to 3, and therefore the apparent diameters in that ratio. The methods by which he estimated the apparent distances, may be seen in Vol. I. page 65. The same circumstance also takes place in the sun, which appears much larger in the horizon than in the zenith. Also, if we take two stars near each other in the horizon, and two other stars near the zenith at the same angular distance from each other, the two former will appear at a much greater distance from each other, than the two latter. Upon this ac- count, people are, in general, very much deceived in estimating the altitudes of the heavenly bodies above the horizon, judging them to be much greater than they are. Dr. Smith found, that when a body was about 23° above the horizon, it appeared to be half way between the zenith and horizon, and therefore at that real altitude it would be estimated to be 45° high. The lower part of a rainbow also appears broader than the upper part. And this may be considered as an ar- gument that the phaenomenon cannot depend entirely upon the greater degree of faintness in the object when in the horizon, because the lower part of the bow fre- quently appears brighter than the upper part, at the same time that it appears broader. Also, this cause could have no effect upon the distance of the stars ; and as the difference of the apparent distance of the two stars, whose angular distance is the same, in the horizon and zenith, seems to be fully sufficient to ac- count for the apparent variation of the moon's diameter in these situations, it may be doubtful, whether the faintness of the object enters into any part of the cause. 157 Chap. XVII. ON THE ROTATION OF THE SUN AND PLANETS, (236.) The times of rotation of the sun, and planets, and the position of their axes, are determined from the spots which are observed upon their surfaces. The position of the same spot, observed at three different times, will give the position of the axis ; for tliree points of any small circle will determine it's situation, and hence we know the axis of the sphere which is perpendicular to it. The time of rotation may be found, either from observing the arc of the small circle described by a spot in any time, or by observing the return of a spot to the same position in respect to the earth. On the Rotation of the Sun. (237.) It is doubtful by whom the spots on the sun were first discovered. Scheiner, Professor of Mathematics in Ingolstadt, observed them in May, 1611, and published an account of them in 1612, in a work entitled, Rosa Urs'ina. Galileo^ in the Pre- face to a work entitled, Tsforia, D'lmostrazioni^ intorno alle Macchte Solari, Roma, l6'l3, says, that being at Rome in 161I, he then showed the spots of the sun to several persons, and that he had spf)ken of them, some months before, to his friends at Florence. He imagmed them to adhere to the sun. Kepler, m his Ephemeris, says, that they were observed by the son of David Fabricius, who published an account of them in 1611. In the papers of Harriot, not yet printed, it is said, that spots upon the sun were ob- served on December 8, 1610. As telescopes were in 158 ROTATION OF THE SUN. use at that time, it is probable that each might make the discovery. Admitting these spots to adhere to the sun's body, the reasons for which we shall afterwards give, we proceed to show how the time of it's rotation may be found. (238.) M. CassiJii determined the time of rotation, from observing the time in which a spot returns to the same situation upon the disc, or to the circle of lati- tude passing through the earth. Let t be that interval of time, and let m be equal to the true motion of the earth in that time, and n equal to it's mean motion ; then 360° +m : 36o°+n :: t : the time of return if the motion had been uniform, and this, from a great number of observations, he determines to be 2'jd. I2h, 20' ; now the mean motion of the earth in that time is 27°. ;'. 8" ; hence, 360° + 27''. -7'. 8'' : 36o° :: 27d. I2h 29' : 25d. 14h. 8', the time of rotation. Elem. d'Astron. p. 104. (239.) When the earth is in the nodes of the sun's equator, and consequently in it's plane, the spots ap- pear to describe straight lines : this happens about the beginning of June and December. As the earth re- cedes from the nodes, the path of a spot grows more and more elliptical, till the earth gets 90° from the nodes, which happens about the beginning of Sep- tember and March, at which time the ellipse has it's minor axis the greatest, and is then to the major axis, as the sine of the inclination of the ?olar equator to radius. (240.) There has been a great difference of opinions respecting the nature of the solar spots. Scheiner sup- posed them to be solid bodies revolving about the sun, very near to it ; but as they are as long visible as they are invisible, this cannot be the case. Moreover, we have a physical argument against this hypothesis, which is, that most of them do not revolve about tlie sun in a plane passing through it's center, which they necessarily must, if they revolved, like the planets, about the sun. Galileo confuted Schelne7''s opinion, ROTATION OF THE SUN. 159 by observing that the spots were not permanent ; that they varied their figure; that they increased, and sometimes disappeared. He compared them to smoak and clouds. Hevelius appears to have been of the same opinion ; for in his Cometographla, p. 36o, speaking of the solar spots, he says, Hoec materia nunc ea ipsa est evaporatio et exhalatio [quia aliunde minime oriri potest) quae ex ipso corpore solis, ut supra ostensum est, expirafur et exhalatur. But the permanency of most of the spots is an argument against this hypothesis. M. de la Hire supposed them to be solid, opaque bodies, which swim upon the hquid matter of the sun, and which are sometimes en- tirely immersed. M. de la Lande supposes that the sun is an opaque body^ covered with a liquid fire, and that the spots arise from the opaque parts, like rocks, which, by the alternate flux and reflux of the liquid igneous matter of the sun, are sometimes raised above the surface. The spots are frequently dark in the middle, with an umbra about them ; and M. de la Lande supposes that the part of the rock which stands above the surface, forms the dark part in the center, and those parts which are but just covered by the ig- neous matter, form the umbra. Dr. f^il.son, Professor of Astronomy at Glasgow, opposes this hypothesis of M. c?e la Lande, by this argument. Generally speak- ing, the umbra immediately contiguous to the dark central part, or nucleous, instead of" being very dark, as it ought to be, from our seeing the immersed parts of the opaque rock through a thin stratum of the igneous matter, is, on the contrary, very nearly of the same splendour as the external surface, and the umbra grows darker the further it recedes from the nucleus; this, it must be acknowledged, is a strong argument against the hypothesis of M. de la Lande. Dr. fVilson further observes, that M. de la Z/a/7rtf<? produces no op- tical arguments in support of the rock standing above the surface of the sun. The opinion of Dr. If^ilson is, that the spots are excavations in the luminous matter i6g rotation of the sun. of the sun, the bottom of which forms the umbra. They who wish to see the arguments by which this is supported, must consult the Phil, Trans. 1/74 and 1783. Dr. Halley conjectured that the spots are formed in the atmosphere of the sun. Dr. Herschel supposes the sun to be an opaque body, and that it has an atmosphere; and if some of the fluids which enter into it's composition should be of a shining brilhancy, whilst others are merely transparent, any temporary cause which may remove the lucid fluid will permit us to see the body of the sun through the transparent ones. Seethe Phil. Trans. 1795. Dr. Herschel, on April 19, 1779? saw a spot which mea- sured 1'. 8",o6in diameter, which is equal in length to more than 31000 miles ; this was visible to the naked eye. Besides the dark spots upon the sun, there are also parts of the sun, called Faculce, Lucili, &c. which are brighter than the general surface; these always abound most in the neighbourhood of the spots them- selves, or where spots recently have been. Most of the spots appear within the compass of a zone lying 30° on each side of the equator; but on July 5, 178O, M. de la Lande observed a spot 40° from the equator. Spots which have disappeared, have been observed to break out again. The spots appear so frequently, that Astronomers very seldom examine the sun with their telescopes, but they see some ; Scheiner saw fifty at once. The following phaenomena of the spots are described by Scheiner and Hevelius. I. Every spot which has a nucleus, has also an umbra surrounding it. II. The boundary between the nucleus and umbra is always well defined. III. The increase of a spot is gradual, the breadth of the nucleus and umbra dilating at the same time. IV. The decrease of a spot is gradual, the breadth of the nucleus and umbra contracting at the same time. ROTATION OF THE PLANETS. l6l V. The exterior boundary of the umbra never con- sists of sharp angles, but is always curvilinear, how- ever irregular the outline of the nucleus may be. VI. The nucleus, when on the decrease, in many cases changes it's figure, by the umbra encroaching irregularly upon it. VII. It often happens, by these encroachments, that the nucleus is divided into two or more nuclei. VIII. The nucleus vanishes sooner than the umbra. IX. Small umbrae are often seen without nuclei. X. An umbra of any considerable size is seldom seen without a nucleus. XI. When a spot, consisting of a nucleus and umbra, is about to disappear, if it be not succeeded by a facula, or more fulgid appearance, the place it occupied is, soon after, not distinguishable from any other part of the sun's surface. On the Rotation of the Planets. (24 1 .) The Georgian is at so great a distance, that Astronomers, with their best telescopes, have not been able to discover whether it has any revolution about it's axis. (242.) Saturn was suspected by Cassini and Fato, in 16*83, to have a revolution about it's axis; for they one day saw a bright streak, which disappeared the next, when another came into view near the edge of it's disc; these streaks are cdWed Belts. In 1719» when the ring disappeared, Cassini saw it's shadow upon the body of the planet, and a belt on each side parallel to the shadow. When the ring was visible, he perceived the curvature of the belts was such as agreed with the elevation of the eye above the plane of the ring. He considered them as similar to our clouds floating in the air ; and having a curvature similar to the exterior circumference of the ring, he concluded that they ought to be nearly at the same L l63 ROTATION OF THE PLANETS. distance from the planet, and that consequently the atmos}3here of Saturn extended to the ring. Dr. Hcrschel found that the arrangement of the belts aluays followed the direction of the ring ; thus, as the ring opened, the belts began to show an incurvature answering to it. And during his observations on June 19, 20, and 21, 1780, he saw the same spot in three different situations. He conjectured, therefore, that Saturn revolved about an axis perpendicular to the plane of it's ring. Another argument in support of this, is, that the planet is an oblate spheriod, having the diameter in the dn-ection of the ring to the dia- meter perpendicular to it, as about 11 : 10, according to Dr. Herschel; the measures were taken vvith a wire micrometer prefixed to his 20 feet reflector. The truth of his conjecture he has now verified, having determined thai Saturn revolves about it's axis in \0h. 16', 0",4. Phil. Trans. 1794. The rotation is accord- ing to the order of the signs. (243.) Jupiter is observed to have belts, and also spots, by wftich the time of it's rotation can be very accurately ascertained. M. Cassim found the time of rotation to be 9^'- 5^', from a remarkable spot which he observed m l6(j-5. In October 1 69 1, he observed two bright spots almost as broad as the belts ; and at the end of the month he saw tvvo more, and found them to revolve in 9//. 51'; he also observed some other spots near Jupiter's equator, which revolved in ^h. 50'; and, in general, he found that the nearer the spots were to the equator, the quicker they revolved. It is probable, therefore, that t'le spots are not upon Jupiter's surface, but in it's atmosphere; and for this reasoi) also, that several spots whicli apj)eared round at first, grew obl(^n<^ by degrees in a direction parallel to the belts, and divided themselves into two or tliree spots. M. Moraldi, from a great many observations of the spot observed by Cassini in 1665, found tlie time of rotation to be 9/i, 56'; and concluded that the spots had a dependence upon the contiguous belt, as ROTATION OF THE PLANETS. l63 the spot had never appeared without tlie belt, though the belt had without the spot. It continued to appear and disappear till l6^4, and was not seen any more till 17O8; hence, he concluded, that the spot was some effusion from the belt upon a fixed place of Jupiter's body, for it always appeared in the same place. Dr. Herschel found the time of rotation of different spots to vary ; and that the time of rotation of the same spot diminished ; for the spot observed in 1788 revolved as follows. From February 25 to March 2, in ^h. 55'. 20" j from March 2 to the 14th, in ^h. 54'. 58" ; from April 7 to the 12th, in ^h. 51'. 35". Also, from a spot observed in 1 799, it's rotation was, from April 14 to the 19th, in 9/^. 51'. 45"; from April 19 to the 23d, in Qh. 50'. 48". This, he ob- serves, is agreeable to the theory of equinoctial winds, as it may be some time before the spot can acquire the velocity of the wind ; and if Jupiter's spots should be observed in different parts of it's revolution to be ac- celerated and retarded, it would amount almost to a demonstration of it's monsoons, and their periodical changes. M. Schroeter makes the time of rotation gh. 55'. 36",6 ; he observed the same variations as Dr. Herschel. The rotation is according to the order of the signs. This planet is observed to be flat at it's poles. Dr. Pound measured the polar and equatorial diameters, and found them as 12 : 13. Mr. Short made them as 13 : 14. Dr. Bradley made them as 12,5 ; 13,5. Sir /. Newton makes the ratio ^^ : 10^ by theory. The belts of Jupiter are generally parallel to it's equator, which is very nearly parallel to the ecliptic; they are subject to very great variations, both in respect to their number and figure ; some- times eight have been seen at once, and at other times only one ; sometimes they continue for three months without any variation, and sometimes a new belt has been formed in an hour or two. From their being subject to such changes, it is very probable that they L 2 l64 ROTATION OT THE PLANETS. do not adhere to the body of Jupiter, but exist in it's atmospliere. (244.) Galileo discovered the phases of Mars; after which, some Itahans in l636, had an imperfect view of a spot. But in 1666, Dr. Hook and M. Cassini discovered some well-defined spots; and the latter determined the time of the rotation to be 24h. 40'. Soon after, M. Maruldi observed some spots, and determined the time of rotation to be 2Ah 39'. He also observed a very bright part near the southern pole, appearing like a polar zone; this, he says, has been observed for 60 years-, it is not of equal bright- ness, more than half of it being brighter than the rest; and that part which is least bright, is subject to great changes, and sometimes disappears. Something like this has been seen about the north pole. The rotation is according to the order of the signs. Dr. Herschel makes the time of a sidereal rotation to be 24//. 39'. 2l"67, without the probability of a greater error than 2"34. He proposes to find the time of a sidereal ro- tation, in order to discover, by future observations, whether there is any alteration in the time of the revo- lution of the earth, or of the p!anets, about tiieiraxes; for a cha> ge of either would thus be discovered. He chose Mars, because it's spots are permanent. See the Phil. T'rans. 178I. From further observations upon Mars, which he published in Phil. Trans. 1/84, he makes it's axis to be inclined to the ecliptic 59°. 42', and 61". 18' to it's orbit; and the north pole to be directed to 17°. 47' of Pisces upon the ei liptic, and 19°. 28' on it's orbit. He makes the ratio of the di- ameters of Mars to be as 16 : 15. Dr. Maskelyne has carefully observed Mars at the timeof opposi ion, but could not perceive any difference in it's diameters. Dr. Hei^schel observes, that Mars has a considerable atmosphere. (245.) Galileo first discovered the phases of Verms in 161 1, and sent the discovery to William de" Medici ^ ROTATION OF THE PLANETS. l65 to communicate it to Kepler. He sent it in this cypher, //cec immaturce a me frustra leguntur, o, 1/ ; which put in order, is, Cyntklce figuras cemulatur jiiater amorian, that is, Venus emulates the phases of the moon. He afterwards wrote a letter to him, "giv- ing an account of the discovery, and explaining the cypher. In 1666, M. Casshii, at a time when Venus was dichotomized, discovered a bright spot upon it at the straight edge, like some of the bright spots upon the moon's surface ; and by observing it's motion, which was upon the edge, he found the sidereal time of rotation to be 23/^. 16'. In the year 1726, Bianchini made some observations upon the spots of Venus, and asserted the time of rotation to be 24^ days; that the north pole answered to the 10^^ degree of Aquarius, and was elevated from 15° to 20° above it's orbit; and that the axis continued parallel to itself. The small angle which the axis of Venus makes with it's orbit is a singular circumstance, and must cause a very great variety in the seasons. M. Cassini, the Son, has vin- dicated his Father, and shown, from Blanchims ob- servations being interrupted, that he might easily mistake different spots, for the same: and he con- cludes, that if we suppose the periodic time to be 23/?. 20', it agrees equally with their observations; but if we take it 24^ days, it will not at all agree with his Father's observations. M. Schroeter has endeavoured to show that Venus has an atmosphere, from observing that the illuminated limb, when horned, exceeds a semicircle ; this he supposes to arise from the refrac- tion of the sun's rays through the atmosphere of Venus at the cusps, by which they appear prolonged. The cusps appeared sometimes to run 15°. 19' into the dark hemisphere; from which he computes, that the height of the atmosphere, to refract such a quantity of light, must be 15156 Paris ieei. But this must depend on the nature and density of the atmosphere, of which we are ignorant. Phil. Trans. 1792. He makes the time of rotation to be 23A, 21', and concludes, from his ob. l66 ROTATION OF THE PLANETS. servationSj tliat there are considerable mountains upon this planet, Phil. Trans. 1795- Dr. Herschel agrees with M. Schroeter, that Venus has a considerable atmosphere; but he has not made any observations, by which he can determine, either the time of rotation, or the position of the axis. Phil. Trans. 1793. (246.) The phases of Meixury are easily distin- guished to be like those of Venus ; but no spots have yet been discovered, by which we can ascertain whe- ther it has any rotation. (247.) The fifth satellite of Saturn was observed by M. Cassini for several years, as it went through the eastern part of it's orbit, to appear less and less, till it became invisible ; and in the western part to increase again. These phaenomena can hardly be accounted for, but by supposing some parts of the surface to be incapable of reflecting light, and therefore, when such parts are turned towards the earth, they appear to grow less, or to disappear. As the same appearances returned again when the satellite came to the same part of it's orbit, it aiFords an argument that the time of the rotation about it's axis is equal to the time of it's revolution about it's primary, a circumstance similar to the case of the moon and earth. See Dr. Herschets account of this in the Phil. Trans. 1792. The appearance of this satellite of Saturn is not always the same, and therefore it is probable that the dark parts are not permanent. Dr. Herschel has discovered that all the satellites of Jupiter have a rotatory motion about their axes, of the same duration with their respective periodic times about their primaries. Phil. Tram. 1797. l67 Chap. XVIII. ON THE SATELLITES. (248.) On January 8, 1610, Galileo discovered the four satellites of Jupiter, and called them Medicea Sidera, or Medicean Stars, in honour of the family of the Medici, his patrons. This was a discovery, very important in it's consequences, as it furnished a ready method of finding the longitudes of places, by means of their eclipses; the eclipses led M. Roemer to the discovery of the progressive motion of light; and hence Dr. Bradley was enabled to solve an apparent motion in the fixed stars, which could not otherwise have been accounted for, (249.) The satellites of Jupiter, in going from the west to the east, are eclipsed by the shadow of Jupiter, and as they go from east to west, they are observed to pass over it's disc; hence, they revolve about Jupiter, and in the same direction as Jupiter revolves about the sun. The three first satellites are always eclipsed, when they are in opposition to the sun, and the lengths of the eclipses are found to be different at dif- ferent times : but sometimes the fourth satellite passes through opposition without being eclipsed. Hence it appears, that the planes of the orbits do not coincide with the plane of Jupiter's orbit, for, in that case they would always pass through the center of Jupiter's shadow, and there would always be an eclipse, and of the same, or very nearly the same, duration, at every opposition to the sun. As the planes of the orbits which they describe sometimes pass through the eye, they will then appear to describe straight lines passing l68 PERIODIC TIMES, AND through the center of Jupiter ; but at all other times they will appear to describe ellipses, of which Jupiter is the center. On the Periodic Times, and distances of Jupiter's Satellites. (250.) To get the mean times of their synodic re- volutions, or of their revolutions in respect to the sun, observe, when Jupiter is in opposition, the passage of a satellite over the body of Jupiter, and note the time when it appears to be exactly in conjunction with the center of Jupiter, and that will be the time of con- jmiction with the sun. After a considerable interval of time, repeat the same observation, Jupiter being in opposition, and divide the interval of time by the number of conjunctions with the sun in that interval, and you get the time of a synodic revolution of the satellite. This is the revolution which we have occa- sion principally to consider, it being that on which the eclipses depend. But, owing to the equation of Jupi- ter's orbit, this will not give the mean time of a synodic revolution, unless Jupiter was at the same point of it's orbit at both observations ; otherwise, we must pro- ceed thus. (251.) Let J IP R be the orbit of Jupiter, aS* the sun in one focus, and F the other focus ; and as the excentricity of the orbit is small, the motion about F may be considered (169) as uniform. Let Jupiter be in it's aphelion at A in opposition to the earth at T; and L a satellite in conjunction ; and let / be the place of Jupiter at it's next opposition with the earth at D, and the satellite in conjunction at G. Then, if the satellite had been at O, it would have been in conjunction with F, or in mean conjunction ; therefore it must describe the angle FIS before it comes to the mean conjunction, which angle is (169) the equa- tion of the orbit, according to the simple elliptic DISTANCES OF JUPITER S SATELLITES. 169 hypothesis, which may be here used, as the excentricity of the orbit is but small ; the angle FIS therefore mea- sures the difference between the mean synodic revolu- tions in respect to F, and the synodic revolutions in respect to the sun S. If, therefore, n be the number of revolutions which the satellite has made in respect to the sun, n x 36o° — *S7/^=the revolutions in respect to F; hence, n x 36o°—SIF: 36o° :: the time between the two oppositions : the tmie of a mean synodic revo- lution about the sun. (252.) As the satelHte is at O at the mean conjunc- tion, and at G when in conjunction with the sun, it is manifest, that if the angle FJS continued the same, the time of a revolution in respect to S would be equal to the time in respect to F, or to the time of a mean synodic revolution ; hence, the difference between the times of any two successive revolutions in respect to S and irrespectively, is as the variation of the angle FIS^ or variation of the equation of the orbit. When Jupiter is at j4, the equation vanishes, and the times of the two conjunctions at F and 8 coincide. When Jupiter comes to /, the mean conjunction at O hap]}eij 3 170 PERIODIC TIMES, AND after the true conjunction at G, by the time of de- scribing the angle SIF, the equation of Jupiter's orbit. This is what Astronomers call the^r^Y mequalily ; and by this inequality of the intervals of the times of the true conjunctions, the times of the eclipses of the satellites are affected. (253.) But as a conjunction of the satellite may not often happen exactly at the time when Jupiter is in opposition, the time of a mean revolution may be found, when he is out of opposition, thus. Let H be the earth when the satellite is at Z in conjunction with Jupiter at R; and let ^' be another position of the earth when the satellite is at C in conjunction with Jupiter at /; and produce RH, IF, to meet in M; then the motion of Jupiter about the earth, in this interval, is the same as if the earth had been fixed at M. Now the difference between the true and mean motions of Jupiter is RFI- RMI= FIM+FRM, which shows how much the number of mean revolu- tions, in respect to jP, exceeds the same number of apparent revolution*; in respect to the earth j hence, nX36o° - FIM- FRM : 3b"0° :: the time between tiie observations : the time of a mean synodic revolu- tion of thv" satellite. If C and Z lie on the other side of O and J', the angles FJM, FRM, must be added to 77 y 360° ; and if one lie on one side, and the other on tlie Dther, one must be added and the other subtracted, according to the circumstances. (234.) As it is difficult, from the great brightness of .fupi'er, to determine accurately the time when the saUlhle is in conjunction with the center of Jupiter as iv passes over it's di;^c, the time of conjunction is deter- mined by obs'-rving it's enlrance uptm the disc, and it's going off: but as tl is cannot be detertnined with so.u'uch accuracy as. the tunes of i?imicrsion into the shadow of Jupiter, and emersion from it, the time of r ' -M-'tion Can be most accurati ly determined from ;)ses. ..; jL,^t / be the center of Jupiter's shadow FG, DISTANCES OF JUPITER's SATELLITES. I7I Nmt the orbit of a satellite, N the node of the satel- lite's orbit upon the orbit of Jupiter; draw Iv perpen- dicular to 77V, and Ic to Nt ; and when the satellite comes to i;, it is in conjunction* with the sun. Now both the immersion at m and emersion at t of the second, third, and fourth satellites may sometimes be observed, the middle point of time between which, gives the time of the middle of the eclipse at c ; and by calculating cv, from knowing the angle A^and NI, we get the time of conjunction at v. If both the im- mersion and emersion cannot be observed, take the time of either, and after a very long interval of time, when an eclipse happens as nearly as possible in the same situation in respect to the node, take the time of the same phaenomenon, and from the interval of these times you will get the time of a revolution. By these different methods, M. Cassini found the times of the mean synodic revolutions of the four satellites to be as follows : First. Second. Third. Fourth. 1^18\28'.a6" 3\r3\l7'.54'Y.3\59'.36" l6'^l8^5'.7'' (256.) Hence it appears, that 247 revolutions of the first satellite are performed in 437^. 3^. 44' ; 123 re- * A satellite is said to be in conjunction, both when it is bet'veen the Sun and Jupiter, and when it is opposite to the Sun ; the latter may be called superior, and the former inferior conjunction. 172 PERIODIC TIMES, AND volutions of the second, in 437^. Sh. 41'; 6l revolu- tions of the third, in 43^d. 3h. 35', and 26 revolutions of the fourth, in 435^. 14^. 13'. Therefore, after an interval of 437 ^^y^) the three first satellites return to their relative situations within nine minutes. (257.) In the return of the satellites to their mean conjunction, they describe a revolution in their orbits, together with the mean angle a° described by Jupiter in that time; therefore, to get the periodic time of each, we must say, 360° 4- «° : 36o° :: time of a synodic revolution : the time of a periodic revolution; hence, the periodic times of each are ; First. Second. Third. Fourth. 1*^.1 8\27'.33" >^13^13'.42" 7'*.3\42'.33" l6^1(:)^32'.8" (258.) The distances of the satellites from the center of Jupiter may be found at the time of their greatest elongations, by measuring with a micrometer, at that time, their distances from the center of Jupiter, and also the diameter of Jupiter, by which you get their distances in terms of the diameter. Or it may be done thus. When a satellite passes over the middle of the disc of Jupiter, observe the whole time of it's passage, and then, the time of a revolution : the time of it's passage over the disc :: 36o° : the arc of it's orbit corresponding to the time of it's passage over the disc; hence, the sine of half that arc : radius :: the semidiameter of Jupiter : the distance of the satellite. Thus M. Cassini determined their distances in terms of the semidiameter of Jupiter to be, of thejirsty 5,67 ; of the *ecowc?, 9 ; of the third, 14,38; and of the fourth, 25,3. (259.) Or, having determined the periodic times, and the distance of one satellite, the distances of the others may be found from the proportion of the squares of the periodic times being as the cubes of their dis- tances. Mr. Pound, with a telescope 15 feet long, DISTANCES OF JUPITER's SATELLITES. I75 found at the mean distance of Jupiter from the earth, the greatest distance of the fourth satelHte to be 8'. 16"; and by a telescope 123 feet long;, he found the greatest distance of the third to be 4'. 42" ; hence, the greatest distance of the second appears to be 2'. 56'". 47'", and of the first, l'. 5 1". 6'". Now the diameter of Jupiter, at it's mean distance, was determined, by- Sir /. NeivtoUy to be 3fl ; hence the distances of the sateUites, in terms of the semidiameter of Jupiter, come out 6,966; 9,494; 15,141, and 26,63 re- spectively. Prln. Math. Lib. ter. Phcen. (260.) Hence, by knowing the greatest elongations of the satellites in minutes and seconds, we get their distances from the center of Jupiter, compared with the mean distance of Jupiter from the earth, by say- ing, the sine of the greatest elongation of the satelhte : radius :: the distance of the satellite from Jupiter • the mean distance of Jupiter from the earth. On the Eclipses of Jupiter's Satellites. (261.) Let 8 be the sun, ^Fthe orbit of the earth, /Jupiter, ate the orbit of one of it's satellites. When the earth is at E before the opposition of Jupiter, the spectator will see the immersion at a\ but if it be the first satellite, upon account of it's nearness to Jupiter, the immersion is never visible, the satellite being then always behind the body of Jupiter; the other three satellites may have both tiieir immersions and emer- sions visible ; but this will depend upon the position of the earth. When the earth comes to i^ after oppo- sition, we shall then see the emersion of the first, but can tiien never see the immersion ; but we may see both the emersion and immersion of the other three. Draw EIr ; then at, the distance of the center of the shadow from the center of Jupiter, referred to the orbit of the satellite, is measured at Jupiter by sr, or the angle sir, or the angle ELS. The satellite may be hidden behind the body at r without being eclipsed. 174 ECLIPSES OF JUPITER*S SATELLITES. which is called an Occultation. When the earth is at E, the conjunction of the satellite happens later at the earth than at the sun ; but when the earth is at F, it happens sooner. (262.) The diameter of the shadow of Jupiter, at the distance of any of the satellites, is best found by observing the time of an eclipse when it happens at the node, at which time the satelhte passes through the center of the shadow ; for tli^ time of a synodic revolution : the time the satellite is passing through the center of the shadow :: 36o° : the diameter of the shadow in degrees. But when the first and second ^atelhtes are in the nodes, the immersion and emersion cannot both be seen. Astronomers, therefore, compare the immersions some days before the opposition of Jupiter with the emersions some days after, and then, knowing how many synodic revolutions have been made, they get the time of the transit through the shadow, and thence the corresponding degrees. But on account of the excentricity of some of the orbits, ROTATION OF JUPITER S SATELLITES. 175 the times of the cen'^ral transit must vary ; forexample, the second satelHte is sometimes found to be ih. 50' in passing through the center of tlie shadow, and, sometimes 2/?. 54'; this indicates an excentricity. (263.) The duration of the echpses being very une- qual, shows tliat the orbits are inclined to the orbit of Jupiter ; sometimes the fourth satellite passes through opposition without suffering an eclipse. The duration of the eclipses must depend upon the situation of the nodes in respect to the sun, just the same as in a lunar eclipse ; when the line of the nodes passes through the sun, the satellite will pass through the center of the shadow ; but as Jupiter revolves about the sun, the line of the nodes will be carried out of conjunction with the sun, and the time of the eclipse will be shortened, as the satellite will then describe only a chord of a section of the shadow instead of the diameter. On the Rotation of the Satellites of Jupiter, (264.) M. Cassini suspected that the satellites had a rotation about their axes, as sometimes in their pas- sage over Jupiter's disc tliey were visible, and at other times not ; he conjectm'ed, therefore, that they had spots upon one side and not on the other, and that they were rendered visible in their passage when the spots were next to the earth. At different times also they appear of different magnitudes and of different brightness. The fourth appears generally the smallest, but sometimes the greatest ; and the diameter of it's shadow on Jupiter appears sometimes greater than the satellite. The third also appears of a variable magni- tude, and the like happens to the other two. Mr. Pound also observed, that they appear more luminous at one time than another, and therefore he concluded that they revolve about their axes. Dr. Herschel has discovered that they all revolve about their axes, in the 17^ Saturn's satellites. times in which they respectively revolve about Jupiter. On the Satellites of Saturn. (265.) In the year \6bb, Huygens discovered the fourth satelHte of Saturn ; and pubHshed a table of it's mean motion in 1659. In 1671, M. Cassini dis- covered the fifth, and the third in 1672 ; and in l684, the first and second ; and afterwards pubhshed Tables of their motions. He called them Sidera Lodoicea, in honour of Louis le Grand, in whose reign, and ob- servatory, they were discovered. Dr. Halley found, by his own observations in l682, that Huygens's Tables had considerably run out, they being about 15" in 20 years too forward, and therefore he composed new Tables from more correct elements. He also re- formed M. CassinVs Tables of the mean motions ; and about the year 1720, published them a second time, corrected from Mr. Pound's observations. He observes, that the four innermost satellites describe orbits very nearly in the plane of the ring, which, he says, is, as to the sense, parallel to the equator ; and that the orbit of the fifth is a little inclined to them. The following Table contains the periodic times of the five satellites, and their distances in semidiameters of the ring, as determined by Mr. Pound, with a micro- meter fitted to the telescope given by Huygens to the Royal Society. Mr. Pound first measured the distance of the fourth, and then deduced the rest from the proportion between the squares of the periodic times and cubes of their distances, and these are found to agree with observations. SATURN S SATELLITES. 177 Satel- lites. PeriodicTimes by Pound. Dist. in semid. oj Ring, by Pound. Dist. in semid. of Saturn by Pound. Dist. in semid. of Ring by Cassini. Dist. at the mean dist. of Saturn. I 1^21\18'.27" 2,097 4,893 1 ' * 0'.43",5 II 2. 17. 41. 22 2,686 6,286 01 0. 56 III 4. 12. 25. 12 3,752 8,754 H 1. 18 IV 15.22.41.12 8,698 20,295 8 3. V 79. 7.49. 25,348 59,154 23 8. 42,5 The last column is from Cassini; but Dr. Het^schel makes the distance of the fifth to be 8'. 3l",97, which is probably more exact. In this and the two next Tables, the satelhtes are numbered from Saturn as they were before the discovery of the other two. On June 9, 1749, at loh. Mr. Pound found the distance of the fourth satellite to be 3'. 7" with a telescope of 123 feet, and an excellent micrometer fixed to it ; and the satellite was at that time very near it's greatest eastern digression. Hence, at the mean distance of the earth from Saturn, that distance be- comes 2'. 58",21 ; Sir /. Newtoti makes it 3'. 4". (266.) The periodic times are found as for the satellites of Jupiter (251). To determine these, M. Cassini chose the time when the semi-minor axes of the eclipses which they describe, were the greatest, as Saturn was then 90° from their node, because the place of the satellite in it's orbit is then the same as upon the orbit of Saturn ; whereas in every other case, it would be necessary to apply the reduction in order to get the place in it's orbit. (267,) As it is difficult to see Saturn and tiie satel- lites at the same time in the field of view of a telescope, their distances have sometimes been measured, by ob- serving the time of the passage of the body of Saturn M 1/8 SATLTTIM S SATELLITES. over a wire adjusted as an hour circle in the field of the telescope, and the interval between the times when Saturn and the satellite passed. From comparing the periodic times and distances, M. Cassini observed that Kepler^ Rule (162) agreed very well with observa- tions. (26s.) By comparing the places of the satellites with the ring in difierent points of their orbits, and the greatest minor axes of the ellipses which they appear to describe, compared with the major axes, the planes of the orbits of the first four are found to be very nearly in the plane of the ring, and therefore are inclined to the orbit of Saturn about 30" ; but the orbit of the fifth, according to M. Cassini the Son, makes an angle with the ring of about 1 5 degrees. (269.) M. Cassini places the node of the ring, and consequently the nodes of the four first satellites, in 5^ 22'^ upon the orbit of Saturn, and 5^ 21° upon the ecliptic. M. Hay gens had determined it to be in 5^ 20\ 30'. M. MirakU, in 17 16, determined the longi- tude of the node of the ring upon the orbit of Saturn to be b\ 19°. 48'. 30"; and upon the ecliptic to be 5^ 16°. 20'. The node of the fifth satellite is placed by M. Cassini in 5^ 5" upon the orbit of Saturn. M. de la Lande makes it 5*. O". 2/'. From the observa- tion of M. Bernard, at Marseilles, in 1787, it appears that the node of this satellite is retrograde. (270.) Dr. Halley discovered that the orbit of the fourth satellite was excentric ; for, having found it's mean motion, he discovered that it's place by observa- tion was at one time 3" forwarder than by his calcula- tions, and at other observations it was 2°. 30' behind ; this indicated an excentricity ; and he placed the line of the apsides in lO^. 22% Phil. Trans. N°. 145. Saturn's satellites. 179 Tables of their Revolutions and Mean Motions, according to M. de la Lande. Scud. Diurnal Motion. Motion in 365 Days. I 6^ lo^4l^53" 4\ 4°. 44'. 42" II 4. 11. 32. 6 4. 10. 15. 19 III 2. 19. 41. 25 9- 16. 57. 5 IV 0. 22. 34. 38 10. 20. 39. 37 V 0. 4. 32. 17 7. 6. 23. 37- S'^^e/. Periodic Revolution. Sj/7iodic Revolution. I 1''.21\18'.26",222 1^2l'M8'.54",778 II 2. 17. 44. 51,177 2. 17. 45. 51,013 III 4. 12. 25. 11,100 4. 12. 27. 55,239 IV 15. 22. 41. 16,022 15. 23. 15. 23,153 V 79. 7. 53. 42,772 79. 22. 3. 12,883 (271.) M. Cassini observed, that the fifth sateUite disappeared regularly for about half it's revolution, when it was to the east of Saturn ; from which he concluded, that it revolved about it's axis ; he after- wards, however, doubted of this. But Sir /. Newton^ in his Pri?icipia, Lib. III. Prop. 17, concludes from hence, that it revolves about it's axis, and in the same time that it revolves about Saturn ; and that the varia- ble appearance arises from some parts of the satellite not reflecting so much light as others. Dr. Herschel has confirmed this, by tracing regularly the periodical ciiange of light througli more than ten revolutions, M 2 180 Saturn's satellites. which he found, in all appearances, to be cotemporary with the return of the satelHte to the same situation in it's orbit. This is further confirmed by some observa- tions of M. Bernard, at Marseilles, in 1787 ; and is a remarkable instance of analogy among the secondary planets. (272.) These are all the satellites which were known to revolve about Saturn, till the year 1789. when Dr. //er^cAe/, ina Paper in the PJiil. Trans, for that year, announced the discovery of a sixth satellite, interior to all the others^ and promised a further ac- count in another Paper. But in the intermediate time he discovered a seventh satellite, interior to the sixth ; and in a Paper upon Saturn audit's ring, in the PhiL Trans. 1790? he has given an account of the discovery; with some of the elements of their motions. He afterwards added Tables of their motions. (273.) After his observations upon the ring, he says, he cannot quit the subject without mentioning his own surmises, and that of several other Astrono- mers, of a supposed roughness of the ring, or inequality in the planes and inclinations of it's flat sides. This supposition arose from seeing luminous points on it's boundaries^ projecting like the moon's mountains ; or from seeing one arm brighter or longer than the other ; or even from seeing one arm when the other was invisible. Dr. Herschel was of this opinion, till he ^Tiw one of these points move off the edge of the ring in the form of a satellite. With his 20 feet telescope he suspected that he saw a sixth satellite ; and on August 19, 1787; marked it down as probably being one ; and having finished his telescope of 40 feet focal length, he saw six of it's satellites the moment he di- rected his telescope to the planet. This happened on August 28, 1789. The retrograde motion of Saturn was then nearly 4'. 30" in a day, which made it very easy to ascertain whether the stars he took to be satel- lites, were really so ; and in about two hours and an half after, he found that the planet had visibly carried SATELLITES OF THE GEORGIAN. 181 them all away from their places. He continued his observations, and on September 17, he discovered the seventh satellite. These two satellites lie within the orbits of the other five. Their distances from the center of Saturn are 36",7889, and 28",6689 ; and their periodic times are id. 8h. 33'. 8",9, and22/i. 37'. 22",9. The planes of the orbits of these satellites lie so near to the plane of the ring, that the diiFerence cannot be perceived. On the Satellites of the Georgian, (274.) On January 11, I787, as Dr. Herschel was observing the Georgian, he perceived, near it's disc, some very small stars, whose places he noted. The next evening, upon examining them, he found that two of them were missing. Suspecting, therefore, that they might be satellites which had disappeared in con- sequence of having changed their situation, he con- tinued his observations, and in the course of a month discovered them to be satellites, as he had first con- jectured. Of this discovery he gave an account in the Phil. Trans. 1787. (275.) In the Phil. Trans. 1788, he published a further account of this discovery, containing their pe- riodic times, distances, and positions of their orbits, so far as he was then able to ascertain them. The most convenient method of determining the periodic time of a satellite, is, either from it's eclipses, or from taking it's position in several successive oppositions of the planet ; but no eclipses have yet happened since the discovery of these satellites, and it would be waiting a long time to put in practice the other method. Dr. Herschel, therefore, took their situations whenever he could ascertain them with some degree of precision, and then reduced them, by computation, to such situa- tions as were necessary for his purpose. In comput- ing the periodic times, he has taken the synodic revolutions, as the positions of their orbits, at the 182 SATELLITES OF THE GEORGIAN. times when their situations were taken, were not suffi- ciently known to get very accurate sidereal revokitions. The mean of several results gave the synodic revolu- tion of the first satellite 8^/. l^jh- i'. 19 ",3, and of the second \3d. llh. b'. l",5. The results, he observes, of which these are a mean, do not much differ among themselves, and therefore the mean is probably toler- abl}'' accurate. The epochs from which their situations may, at any time^ be computed are, for the^r*^, Oct. 19j "^l^l, at 19/i. 1 1'. 28"; and for the second, at l/A. 22'. 40"; at which times they were 'jQ''. 43' north, following the planet. (276.) The next thing to be determined, in the elements of the satellites, was their distances from the planet; to obtain which, he found one distance by observation, and then the other from the periodic times (Article 162). Now, in attempting to discover the distance of the second, the orbit was seemingly ellip- tical. On March 18, 1787, at 8A. 2'. 50", he found the elongation to be 46",46, this being the greatest of all the measures he had taken. Hence, at the mean distance of the Georgian from the earth, this elonga- tion will be 44",23. Admitting, therefore, for the present, says Dr. Herschel, that the satellites move in circular orbits, we may take 44",23 for the true distance, without much error; hence, as the squares of the periodic times are as the cubes of the distances, the distance of the first satellite comes out 33",09, The synodic revolutions were here used instead of the sidereal, which will make but a small error. (277') The last thing to be done, was to determine the inclinations of the orbits, and places of their nodes. And here a difficulty presented itself which could not be got over at the time of his first observation ; for it could not then be determined which part of the orbit was inclined to the earth, and which /rom it On the two different suppositions, therefore, Dr. Herschel has computed the inclinations of the orbits, and the places of their nodes, and found them as follows. The orbit SATELLITES OF THE GEORGIAN. 183 of the second satellite is inclined to the ecliptic 99''. 43'. 53",3 or 81°. 6'. 44",4 ; it's ascending node upon the ecliptic is in 5^ 18% or 8^. 6° ; and when the planet comes to the ascending node of this satellite, which will happen about the year 1818, the northern half of the orbit will be turned towards the east or west, at the time of it's meridian passage. M. de la Lambre makes the ascending node in 5^ 21°, or 8*. 9°, from Dr. Herschel's observations. The situation of the orbit of the first satellite does not materially differ from that of the second. The light of the satellites is extremely faint ; the second is the brightest, but the difference is small. The satellites are probably not less than those of Jupiter. There will be eclipses of these sateUites about the year 1818, when they will appear to ascend through the shadow of the planet, in a direction almost perpendicular to the ecliptic. Since these discoveries were made, Dr. Herschel has discovered four more satellites of the Georgian, and found that their motions are all retrograde. Phil. Trans. 1798. 84 Chap. XIX. ON THE RING OF SATURN. (278.) Galileo was the first person who observed any thing extraordhiary in Saturn. The planet ap- peared to him like a large globe between two small ones. In the year 1610 he announced this discovery. He continued his observations till l6l2, when he was surprised to find only the middle globe ; but some time after he again discovered the globes on each side, which, in process of time, appeared to change their form ; sometimes appearing round, sometimes oblong like an acorn, sometimes semicircular, then with horns towards the globe in the middle, and grow- ing, by degrees, so long and wide as to encompass it, as it were, with an oval ring. Upon this, Huygens set about improving the art of grinding object glasses ; and made telescopes which magnified two or three times more than any which had been before made, with which he discovered very clearly the ring of Saturn ; and having observed it for some time, he published the discovery in 1656. He made the space between the globe and the ring equal to, or rather bigger than the breadth of the ring ; and the greater diameter of the ring to that of the globe as 9 to 4. But Mr. Pound, with a micrometer applied to Huygen''?, telescope of 123 feet long, determined the ratio to be as 7 to 3. Mr. JVIiiston in his Memoirs of the Life of Dr. Clark, relates, that the Doctor's Father once saw a fixed star between the ring and the body of Saturn. In the year 1675, M. Cassini saw the ring, and observed upon it a dark elliptical line, divid- ing it, as it were, into two rings, the inner of which RING OF SATURN. 185 appeared brighter than the outer. He also observed a dark belt upon the planet, parallel to the major axis of the ring. Mr. Hadley observed that the outer part of the ring seemed narrower than the inner part, and that the dark line was fainter towards it's upper edge ; he also saw two belts, and observed the shadow of the ring upon Saturn. In October, 1714, when the plane of the ring very nearly passed through the earth, and was approaching it, M. Maraldi observed, that while the arms were decreasing both in length and breadth, the eastern arm appeared a little larger than the other for three or four nights, and yet it vanished first, for, after two nights interruption by clouds, he saw the western arm alone. This inequality of the ring made him suspect that it was not bounded by exactly parallel planes, and that it turned about it's axis. But the best description of this singular phaenomenon is that given by Dr. Herschel, in the Phil. Trans. 1790> who, by his extraordinary telescopes, has discovered many circumstances which had escaped all other ob- servers. We shall here give the substance of his account. (279-) The black disc, or belt upon the ring of Saturn, is not in the middle of it's breadth ; nor is the ring subdivided by many such lines, as has been re- presented by some Astronomers ; but there is one ^ single, dark, considerable broad line, belt, or zone, which he has constantly found on the north side of the ring. As this dark belt is subject to no change whatever, it is probably owing to some permanent construction of the surface of the ring. This con- struction cannot be owing to the shadow of a chain of mountains, since it is visible all round on the ring; * In a Paper in the Phil. Trans. 1792, Dr. Hcrschel observes, that, " since the year 1774 to the present time, I can find only- four observations where any other black division of" the ring is njientioned, than the one which 1 have constantly observed ; these were all in June, 1780.'' ISG RING OF SATURN. for at the ends of the ring there could be no shade ; and the same argument will hold against any supposed caverns. It is moreover pretty evident, that this dark zone is contained between two concentric circles, as all the phaenomena answer to the projection of such a zone. The matter of the ring is undoubtedly no less solid than the planet itself; and it is observed to cast a strong shadow upon the planet. The light of the ring is also generally brighter than that of the planet; for the ring appears sufficiently bright, when the telescope affords scarcely light enough for Saturn. Dr. Herschel next takes notice of the extreme thinness of the ring. He frequently saw the first, second, third, fourth, and fifth satellites pass before and behind the ring, in such a manner that they served as an excel- lent micrometer to measure it's thickness by. It may be proper to mention a few instances, as they serve also to solve some phaenomena observed by other Astronomers, without having been accounted for in any manner that could be admitted consistently with other known facts. July 18, 1789, at \6h. 41'. 9" sidereal time, the third satellite seemed to hang upon the following arm, declining a little towards the north, and was seen gradually to advance upon it towards the body of Saturn ; but the ring was not so thick as the lucid point. July 23, at \^h. 41'. 8", the fourth satellite was a very little preceding the ring, but the ring appeared to be less than half the thickness of the satellite. July 27, at 20h. 15'. 12", the fourth satellite was about the middle, upon the following arm of the ring, and towards the south ; and the second at the farther end, towards the north ; but the arm was thinner than either. August 29, at 22/z. 12'. 25", the fifth satellite was upon the ring, near the end of the preceding arm, and the thickness of the arm seemed to be about ^ or J of the diameter of the satellite, which, in the situation it then was, he took to be less than one-second in diameter. At the same time the first appeared at a little distance following the fifth, RING OF SATURN. 1 87 in the shape of a bead upon a thread, projecting on both sides of the same arm ; hence, the arm is thinner than the first, which is considerably smaller than the second, and a little less than the third. October l6, he followed the first and second satellites up to the very disc of the planet ; and the ring, which was ex- tremely faint, did not obstruct his seeing them gra- dually approach the disc. These observations are sufficient to show the extreme thinness of the ring. But Dr. Herschel further observes, that there may be a refraction through an atmosphere of the ring, by which the satellites may be lifted up and depressed, so as to become visible on both sides of the ring, even though the ring should be equal in thickness to the smallest satellite, which may amount to 1000 miles. From a series of observations upon luminous points of the ring, he has discovered that it has a rotation about it's axis, the time of which is \0h. 32'. 13l'4. (280.) The ring is invisible* when it's plane passes through the sun, or the earth, or between them ; in the first case, the sun shines only upon it's edge, which is too thin to reflect sufficient light to render it visible ; in the second case, the edge only being op- posed to us, it is not visible, for the same reason ; in the third case, the dark side of the ring is exposed to us, and therefore the edge being the only luminous part which is towards the earth, it is invisible on the same account as before. Observers have difl^ered 10 or 12 days in the time of it's becoming invisible, owing to the dift'erence of the telescopes, and of the state of the atmosphere. Dr. Herschel observes that the ring was seen in his telescope, when we were turned to- wards the unenlightened side; so that he either saw the light reflected from the edge, or else the reflection * The disappearance of the ring is only with the telescopes in common use among Astronomers; for Dr. Herschel, with his large telescopes, has been able to see it in every situation. He thinks the edge of the ring is not flat, but spherical, or spheroidical. 188 RING OF SATURK of the light of Saturn upon the dark side of the ring, as we sometimes see the dark part of the moon. He cannot, however, say which of the two might be the case; especially as there are very strong reasons to think, that the edge of the ring is of such a nature as not to reflect much light. M. de la Lande thinks that the ring is just visible with the best telescopes in com- mon use, when the sun is elevated 3' above it's plane, or three days before it's plane passes through the sun; and when the earth is elevated 2'. 20" above the plane, or one day from the earth's passing it. (281.) In a paper in the Phil. Trans. 1790, Dr. //er^c^^/ ventured to hint at a suspicion that the ring was divided; this conjecture was strengthened by fu- ture observations, after he had an opportunity of seeing both sides of the ring. His reasons are these : l . The black division, upon the southern side of the ring, is in the same place, of the same breadth, and at the same distance from the outer edge, that it always appeared upon the northern side. 2. With his seven feet re- flector and an excellent speculum, he saw the division on the ring, and the open space between the ring and the body, equally dark, and of the same colour with the heavens about the planet. 3. The black division is equally broad on each side of the ring. From these observations, Dr. Herschel thinks himself authorised to say, that Saturn has two concentric rings, situated in one plane, which is probably not much inclined to the equator of the planet. The dimensions of the rings are in the following proportions, as nearly as they could be ascertained. Parts. Inside diameter of the smaller ring - 5900 Outside diameter ------ 7510 Inside diameter of the larger ring - 7740 Outside diameter ------ 8300 Breadth of the inner ring - - . - §05 Breadth of the outer ring - - - - 280 Breadth of the space between the rings 115 RING OF SATURN. 18^ In the Mem. de VAcad. at Paris, 1787:. M. de la Place supposes that the ring may have many divi- sions ; but Dr. Herschel remarks, that no observations will justify this supposition. (282). From the mean of a great many measures of the diameter of the larger ring. Dr. Herschel m?ikes it 46'",677 at the mean distance of Saturn. Hence, it's diameter : the diameter of the earth :: 25,8914 : 1. From the above proportion, therefore, the diameter of the ring must be 204883 miles ; and the distance of the two rings 2839 miles. (283.) The ring being a circle, appears elliptical from it's oblique position ; and it appears most open when Saturn is 90° from the nodes of the ring upon the orbit of Saturn, or when Saturn's longitude is about 2'. 17°, and 8'. 17°. In such a situation, the minor axis is extremely nearly equal to half the major, when the observations are reduced to the sun ; consequently the plane of the ring makes an angle of about 30° with the orbit of Saturn. 190 Chap. XX. ON THE ABERRATION OF LIGHT. (284.) In the year 1725, Mr. Molyneux^ assisted by l)r. Bradley, fitted up a zenith sector at Kew, in order to discover whether the fixed stars had any sensible parallax *, that is, whether a star would appear to have changed it's place whilst the earth moved from one extremity of the diameter of it's orbit to the other; or, which is the same, to determine whether the diameter of the earth's orbit subtends any sensible angle at the star. The very important discovery arising from their observations is so clearly and fully related by Dr. Bradley, in a letter to Dr. Halley, that I cannot do better than give it to the reader in his own words. Phil. Trans. N°. 406. (285.) "^ Mr. Molyneuxs apparatus was completed and fitted for observing, about the end of November, 1725, and on the third day of December following, the bright star in the head of Draco, marked t by Bayer, was for the first time observed as it passed near the zenith, and it's situation carefully taken with the instrument. The like observations were made on the 5th, nth, and 12th of the same month; and * Dr. Hook was the first inventor of this mfthod, and in the year 1669 he put it in practice at Gresham College, with a telescope 36 feet long. His first observation was July 6, at which time he found the bright star in the head of Draco, marked T by Bayer, passed about 2'. 12" northward from the zenith ; on July 9, it passed at the same distance ; on August 6, it passed 2'. 6" north- ward from the zenith; on October 2!, it passed 1'. 48" or 50" north from the zenith, according to his observations. See his Cutlcrian Lectures. ABERRATION OF LIGHT. I9I there appearing no material difference in the place of the star, a farther repetition of them at this season seemed needless, it being a part of the year wherein no sensible alteration of parallax in this star could soon be ex]3ected. It was chiefly, therefore, curiosity that tempted me (being then at Kew, where the instru- ment was fixed) to prepare for observing the star on December 17, when, having adjusted the instrument as usual, 1 perceived that it passed a little more south- wardly this day than when it was observed before. Not suspecting any other cause of this appearance, we first concluded that it was owing to the uncertainty of the observations, and that either this or the foregoing were not so exact as we had supposed ; for which rea- son we purposed to repeat the observation again, in order to determine from whence this difference pro- ceeded; and upon doing it on December 20, I found that the star passed still more southwardly than in the former observations. This sensible alteration the more surprised us, in that it was the contrary way from what it would have been, had it proceeded from an annual parallax of the star : but being now pretty well satisfied that it could not be entirely owing to the want of exactness in the observations, and having no notion of any thing else that could cause such an ap- parent motion as this in the star, we began to think that some chansre of the materials, &c. of the instru- ment itself might have occasioned it. Under these apprehensions we remained some tmie ; but bemg at length fully convinced, by several trials, of the great exactness of the instrument, and finding, by the gra- dual increase of the star's distance from the pole, that there must be some regular cause that produced it, we took care to examine nicely, at the time of each ob- servation, how much it was ; and about the beginning of March, 1726, the star was found to be 20" more southwardly than at the time of the first observation. It now indeed, seemed to have arrived at it's utmost jijnit southward, because in several trials made about !92 ABERRATION OF LIGHT. this time, no sensible difference was observed in it's si- tuation. By the middle of April it appeared to be re- turning back again towards the north ; and about the beginning of June it passed at the same distance from the zenith as it had done in December when it was first observed. From the quick alteration of the star's declination about this time (it increasing a second in three days) it was concluded that it would now proceed northward, as it before had gone southward of its present situa- tion ; and it happened as was conjectured, for the star continued to move northward till September following, when it again became stationary, being then near 20" more northwardly than in June, and no less than 39 more northwardly than it was in March. From Sep- tember the star returned towards the south, till it arrived in December to the same situation it was in at that time twelve months, allowing for the difference of declination on account of the precession of the equinox. This was a sufficient proof that the instrument had not been the cause of this apparent motion of the star, and to find one adequate to such an eft'ect, seemed a difficulty. A nutation of the earth's axis was one of the first things that offered itself upon this occasion, but it was soon found to be insufficient ; for though it might have accounted for the change of declination in 7 Draconis, yet it would not at the same time agree with the phaenomena in other stars, particularly in a small one almost opposite in right ascension to 7 Draconis, at about the same distance from the north pole of the equator J for though this star seemed to move the same way as a nutation of the earth's axis would have made it, yet it changing it's declination but about half as much as 7 Draconis in the same time, (as appeared upon comparing the observations of both made upon the same days at diflferent seasons of the year,) this plainly proved that the apparent motion of the stars was not occasioned by a real nutation, since, if that had been the cause, the alteration in both stars would have been nearly equal. ABERRATION OF LIGHT. I93 The great regularity of the observations left no room to doubt but that there was some regular cause that pioduced this unexpected motion, which did not de- pend on the uncertainty or variety of the seasons of the year. Upon comparing the observations with each other, it was discovered, that in both the fore-men- tioned stars, the apparent difference of declination from the maxima was always nearly proportional to the versed sine of the sun's distance from the equi- noctial points. This was an inducement to think that the cause, whatever it was^ had some relation to the sun's situation with respect- to those points. But not being able to frame any hypothesis at that time, suffi- cient to solve all the phaenomena, and being very desirous to search a little farther into this matter, I began to think of erecting an instrument for myself at Wansted ; that, having it always at hand, I might with the more ease and certainty enquire into the laws of this new motion. The consideration, likewise, of being able, by another instrument, to conlirm the truth of the observations hitherto made with Mr. Moll/Helix's, was no small inducement to me ; but the chief of all was, the opportunity 1 should thereby have of trying in what manner other stars were affected by the same cause, whatever it was. For Mr. Moly- tieuxs instrument being originally designed for ob- serving 7 Draconis (in order, as I said before, to try whether it had any sensible parallax) was so contrived as to be capable of but little alteration in it's direction, jiot above seven or eight minutes of a degree; and there being few stars within half that distance from the zenith of Kew, bright enough to be well observed, he could not with his instrument thoroughly examine how this cause affected stars differently situated with respect to the equinoctial and solstitial points of the ecliptic. These considerations determined me; and by the contrivance and direction of the very ingenious person Mr. Graham, my instrument was fixed up August N 194 ABERRATION OF LIGHT. 1.9; 1727' As I had no convenient place where I conld make use of so long a telescope as Mr. Moly- neux's, I contented myself with one of but little more than half the length of his (viz. of about 12^ feet, his being 24 1) judging, from the experience which I had already had, that this radius would be long enough to adjust the instrument to a sufficient degree of exact- ness, and I have had no reason since to change my opinion ; for, from all the trials I have yet made, 1 am well satisfied, that when it is carefully rectified, it's situation may be securely depended upon to half a se- cond. As the place where my instrument was to be hung, in some measure determined it's radius, so did it also the length of the arch or limb on which the divisions were made to adjust it ; for the arch could not conveniently be extended farther than to reach to about 6°. 15' on each side my zenith. This indeed was sufficient, since it gave an opportunity of making choice of several stars very different both in magnitude and situation, there being more than two hundred in- serted in the British Catalogue, that may be observed with it. I needed not to have extended the limb so far, but that I was willing to take in Capella, the only star of the first magnitude that comes so near to my zenith. My instrument being fixed, I immediately began to observe such stars as I judged most proper to give me light into the cause of the motion already mentioned. There was variety enough of small ones, and not less than twelve that I could observe through all the sea- sons of the year, they being bright enough to be seen in the day-time, when nearest the sun. 1 had not been long observing, before I perceived that the notion we had before entertained, of the stars being farthest north and south when the sun was about the equinoxes, was only true of those that were near the solstitial colure ; and after I had continued my observations a few months, I discovered, what I then apprehended to be a general law, observed by all the stars^ viz. that ABERRATION OF LIGHT. 195 each of them became stationary, or was farthest north or soutli when they passed over my zenith at six o'clock either in the morning or evening. I perceived hke- wise, that whatever situation the stars were in, with respect to the cardinal points of the ecliptic, the appa- rent motion of every one tended the same way when they passed my instrument about the same hour of the day or night; for they all moved southward while they passed in the day, and northward in the night ; so that each was farthest north when it came about six o'clock in the evening, and farthest south when it came about six in the morning. Though 1 have since discovered that the maxima in most of these stars do not happen exactly when they come to my instrument at those hours, yet, not being able at that time to prove the contrary, and supposing that they did, I endeavoured to find out what propor- tion the greatest alterations of declination in different stars bore to each other ; it being very evident that they did not all change their declinations equally. I have before taken notice that it appeared from Mr. Molyneux's observations that 7 IJraconis altered it's declination about twice as much as the fore-mentioned small star almost opposite to it; but examining the matter more particularly, I found that the greatest alteration of declination in these stars was as the sine of the latitude of each respectively. This made me suspect, that there might he the like proportion be- tween the maxima of other stars; but finding that the observations of some of them would not perfectly cor- respond with such an hypothesis, and not knowing whether the small difterence 1 met with might not be owing to the uncertainty and error of the observations, I deferred the farther examination into the truth of this hypothesis, till I should be furnished with a series of observations made in all parts of the year; which might enable me, not only to determine what errors the observations are liable to, or Iiow far they may N 2 \q6 aberration of light. safely be depended upon ; but to judge whether there had been any sensible change in the parts of the in- strument itself. Upon these considerations I laid aside all thoughts at that time about the cause of the fore-mentioned phaenomena, hoping that I should the easier discover it, when I was better provided with proper means to determine more precisely what they were. When the yetir was completed, 1 began to examine and compare my observations, and having pretty well satisfied myself as to the general laws of the phse- nomena, I then endeavoured to find out the cause of them. I was already convinced, that the apparent motion of the stars was not owing to a nutation of the earth's axis. The next thing that oflfered itself, was an alteration in the direction of the plumb-line with which the instrument was constantly rectified; but this, upon trial, proved insufficient. Then I con- sidered what refraction might do; but here also nothing satisfactory occurred. At last I conjectured, that all the phaenomena hitherto mentioned, proceeded from the progressive motion of light and the earth's annual motion in it's orbit. For I perceived, if light was propagated in time, the apparent place of a fixed object would not be the same when the eye is at rest, as when it is moving in any other direction than that of the line passing through the eye and object; and that when the eye is moving in different directions, the apparent place of the object would be diflferent." This is Dr. Bradley's account of this very important discovery ; we shall therefore proceed to show that his principle will solve all the phaenomena. (28G.) The situation of any object in the heavens is determined by the position of the axis of the tele- scope annexed to the instrument with which we mea- sure ; for such a position is given to the telescope, that the rays of light from the object may descend down ABERRATION OF LIGHT. 197 the axis, and in that situation the index shows the angular distance required. Now if hght be progres- sive, when a ray from any object descends down the axis, the position of the telescope must be different from what it would have been, if light had been instan- taneous, and therefore the place to which the telescope is directed, will be different from the true place of the object. For let S' be a fixed star, FF the direction of the earth's motion, S'F the direction of a particle of light, entering the axis ac of a telescope at a, and moving through aF while the earth moves from c to F; then, if the telescope continue parallel to itself, the light will descend in the axis. For let the axis, wm, Fw, continue parallel to « c ; then, considering each motion * as uniform, the spaces described in the same time will continue in the same proportion ; but cF: aF:: en : av, and by supposition cF, aF, are described * The motion of the spectator arising from the rotation of the earth about it's axis is not here taken into consideration, it being so small as not to produce any sensible effect. 198 ABERRATION OF LIGHT. in the same time, therefore en, av, are described in the same time; hence, when the telescope comes into the situation «m, the particle of light will be in the axis at v ; and this being true for every instant, in this position of the telescope the ray will descend down the axis, and consequently the place of the star, deter- mined by the telescope at F, is *', and the angle S' Fs is the abcrratioiiy or tfie difference between the true place of the star and the place determined by the in- strument. Hence, if we take any line FS : Ft :: ve- locity of light : the velocity of the earth, and join St, and complete the parallelogram FtSs, the aberration will be equal to the angle FSt. Also S represents the true place of the star, and s the place determined by the instrument. (287.) As the place measured by the instrument differs from the true place, let us next consider how the progressive motion of light may effect the place of the star seen by the naked eye. If a ray of light fall upon the eye in motion, it's relative motion, in respect to the eye, will be the same as if equal motions were impressed in the same direction upon each, at the in- stant of contact ; for equal motions in the same direc- tion, impressed upon two bodies, will not affect their relative motions, and therefore the effect of one upon the other will not be altered. Let l^F be a tangent to the earth's orbit at F, which will represent the di- rection of the earth's motion at F, S' the star, join S'F, and produce it to G, and take FG : Fn :: the velocity of light : the velocity of the earth's in it's orbit, and complete the parallelogram nFGH, and draw the diagonal FH. Now as FG, nF, represent the mo- tions of light and of the earth in it's orbit, conceive a motion F?i equal, and opposite to JiFto be impressed upon the eye at F, and upon the ray of light, then the eye will be at rest, and the ray of light, by the two motions FG, F?i, will describe the diagonal FH -, this, therefore, is the relative motion of the ray of light in respect to the eye itself. Hence, the object appears ABERRATION OF LIGHT. 199 in the direction HF, and consequently it's apparent ])lace differs from it's true place by the angle GFH= FSt. It appears, therefore, that the apparent place of the object to the naked eye, is the same as the place determined by the instrument. We may therefore call the place, measured by the instrument, the appa- rent place. Many writers have given the explanation in this article, as the proof of the aberration, not con- sidering that the aberration is the difference between the true place and that determined by the instrument, or the instrumental error; indeed, in this case, the apparent place to the naked eye, coincides with the place determined by the instrument, and therefore no error has been produced by considering it in that point of view; but it introduces a vvrong idea of the subject ; the correction which we apply, or the aberration, is the correction of the place determined by the instru- ment, and therefore the investigation ought to proceed upon this principle; how much does the place, deter- mined by the instrument, differ fiom the true place.'' (288.) By Trigonometry, Art. 128, sin. FSt : sin. FtS :: Ft : FS :: velocity of the earth : velocity of light; hence, sine of aberration = sin. FtS k — r — r 1 • ■ ; therefore, if we consider the velocity of vel. of light ^ the earth and of light to be constant, the sine of abcr- ration, or the aberration itself, as it never exceeds 20", varies as sin. FtS^ and therefore is greatest when that angle is a right angle ; if, therefore, s be put for the sine of FtS, we have 1 (rad.) : s :: 20" : s x 20" the aberration. Hence, when Ft coincides with FS, or the earth is moving directly to or from a star^ there is no aberration. (289.) As (by observation) the angle FSt = 20", when FtS~()0°, we have, the velocity of the earth ; velocity of light :; sin. 20" : radius :: 1 : 10314. (290.) The aberration S's' lies, from the true place of the star, in a direction parallel to the direction of the earth's motion, and towards the same part. 200 ABERRATIOxNf OF LIGHT. (291.) Whilst the earth makes one revolution in it's orbit, the curve, parallel to the echptic, described by the apparent place of a fixed star, is a circle. For let AFBQ, be the earth's orbit, K the focus in which the sun is, H the other focus ; on the major axis AB describe a circle in the same plane ; draw a tangent yFZ to the point F, and Ky,HZ, perpendicular to it; then (Conic Sect. Ellipse, prop. 5), the points y and Z will be always in the circumference of the circle. Let S' be the true place of the star, any where out of the plane of the ecliptic, which therefore must be conceived as elevated above the plane AFBQ, and take tF : FS as the velocity of the earth to the velocity of light, and complete the parallelogram FtSs, and s will (286) be the apparent place of the star. Draw FL perpendi- cular to AB, and let WsVx be the curve described by the point s, and JVSV be parallel to Fh. Now (from physical principles) the velocity of the earth varies as rrr-, or as HZ (Con. Sec. El. p. 6) ; but tF, or Ss represents the velocity of the earth ; hence, Ss varies as HZ, Also, as Ss^ SF, are parallel to Fy, FL, the ABERRATION OF LIGHT. 201 angle sSF'=the angle z/FL, which is = theangle Z/fL, because the angieLFZ added to each makes two right angles, for in the quadrilateral figure LFZH, the angles L and Z are right ones. Hence, as Ss varies as HZ, and the angle iiSP=ZHA, the figures de- scribed by the points s and Z must be similar; but Z describes a circle in the time of one revolution of the earth in it's orbit ; hence, a" must describe a circle about S in the same time. And as Ss is always parallel to tF which lies in the plane of the ecliptic, the circle lVsf^x'\s parallel to the ecliptic. Also, as S and H are two points similarly situated in WP^ and AB, it appears that the true place of the star divides that diameter, which, although in a different plane, we may consider as perpendicular to the major axis of the earth's orbit, in the same ratio as the focus divides the major axis. But as the earth's orbit is very nearly a circle, we may consider S in the centre of the circle, without any sensible error. (292.) As we may, for the purposes which we shall here want to consider, conceive the earth's orbit AFBQ. to be a circle, and therefore to coincide with AyZB, if from the center C we draw Cs parallel to Ss, or yF, s will be the point in that circle corresponding to s iu the circle WsVx-, and as F/=90°, the apparent place of the star in the circle of aberration is always 90*^ before the place of the earth in it's orbit, and conse- quently the apparent angular velocity of the star and earth about their respective centers are always equa). It is further supposed, that the star S' is at an indefi- nitely great distance ; for the true place of the star is supposed not to be altered from the motion of the earth, and considering FH as always parallel to itself, it will always be directed to S' as a fixed point in the heavens. Hence also, as t!;e apparent place of the sun is opposite to that of the earth, the apparent place of the star, in the circle of aberration, is 90" hehiml that of the sun. (293). As a small part of the heavens may be con- 202 ABERRATION OF LIGHT. ceivcd as a plane perpendicular to a line joining the star and eye, it follows, from the principles of ortho- graphic projection, that the circle amhn parallel to the ecliptic described by the apparent place of the star, pro- jected upon that plane, will be an ellipse ; the apparent path of the star in the heavens will therefore be an ellipse, and the major axis will be to the minor, as radius to the sine of the star's latitude. For let CE be the plane of the ecliptic, P it's pole, PE a secondary to it, PC perpendicular to EC, C the place of the eye, and let ab be parallel to CE, then it will be that diameter of the circle anhm of aberration which is seen most obliquely, and consequently that diameter which is projected into t!ie minor axis of the ellipse; let Dui be perpendicular to ah, and it will be seen di- rectly, being perpendicular to a line drawn from it to the eye, and therefore will be the m*.ijor axis ; draw Ca, Cbd, and ad is the projection oi ab ; and as ad may be considered as a straight line, we have (Trig. Art. 128) mn or ah, the major axis, : ad the minor :: rad. : sin. ahd, or ECd the star's latitude. As the angle had is the complement of ahd, or of the star's latitude, the circle is projected upon a plane making an angle with it equal to the complement of the star's latitude. (294.) As the minor axis da coincides with a se- condary to the ecliptic, it must be perpendicular to it, and the major axis myi is parallel to it, it's position not being altered by projection. Hence, in the pole of ABERRATION OF LIGHT. 203 the ecliptic^, the sine of the star's latitude being radius, the clHpse becomes a circle; and in the plane of the ecliptic, the sine of the star's latitude being = 0, the minor axis vanishes, and the ellipse becomes a straight line, or rather a very small part of a circular arc. (295.) To find the aberration in latitude and lotigi- tude. Let ABCD be the earth's orbit, supposed to be a circle with the sun in the center at x, and con- ceive P to be in a line drawn from x perpendicular to ABCD, and to represent the pole of the ecliptic ; let 8 be the true place of the star, and conceive ape q to be the circle of aberration ])arallel to the ecliptic, and abed the ellipse into which it is projected ; let t 2^ be an arc of the ecliptic, and draw the secondary PSG to it, and (293) it will coincide with the minor axis bd into which the diameter j»</ is projected ; draw GCxA^ and it is parallel to pq, and Bx D perpendicular to AG must be parallel to the major axis a e ; then, when the earth is at A, the star is in conjunction, and in opposition when the earth is at C. Now, as the place of the star in the circle of aberration (292) is always 90" before the earth in it's orbit, when the earth is at A, B, C, D, the apparent places of the star in the circle will be at r/,7;, c, q, and in the ellipse at a, b, c, d; and to find the place of the star in the circle, when the earth is at any point E, take the angle pSs = ExBj and s will be the corresponding place of the star in the circle; and to find the projected place in the ellipse, draw 51; perpendicular to Se, and vt perjjendicular to Sc in the plane of the ellipse, and t will be the appa- rent place of the star in the ellipse; join st, and it will be perpendicular to v t, because the projection of the circle into the ellipse is in lines perpendicular to the ellipse; draw the secondary PvtK, which will, as to sense, coincide with vt, unless the star be very near to the pole of the ecliptic 3 therefore the rules here given will be sufficiently accurate, except in that case. Now as cvS is parallel to the ecliptic, S and v must have the same latitude ; hence, vt is the abenation in 204 ABERRATION OF LIGHT. latitude ; and as G is the true, and K the apparent place of the star in the echptic, GK is the aberration 7P J^ <r HS D G^ K> in longitude. To find these quantities, put m and n for the sine and cosine of the angle sSc, or CxE. the earth's distance from syzygies, radius being unity ; and as (293) the angle 5?;^= the complement of the star's latitude, the angle vst = t\ie star's latitude, for the sine and cosine of which put v and w, and put r = Sa, or S s ; then in the right-angled triangle Ssv (Trig. Art. 128) 1 : m :: r : sv = rm; hence, in the triangle i^ts, \ : V :: rm : tv = rvm the aberration in latitude. Also, in the triangle Ssv, 1 : n :: r : vS=:rn; hence, w (13) ; I :: 7m : GK= — the aberration in lonsitude ABERRATION OF LIGHT. 205 When the earth is in syzygies, 7«=0, therefore there is no aberration in latitude ; and, as n is then greatest, tliere is the greatest aberration in longitude ; if the earth be at A, or the star in conjunction, the apparent place of the star is at a, and reduced to the ecliptic at //; therefore GH'is the aberration, which diminishes the longitude of the star, the order of the signs being <~c GT', but when the earth is at C, or the star in opposition, the apparent place c reduced to the ecliptic is at F, and the aberration GFincreases the longitude; hence, the longitude is the greatest when the star is in opposition, and least when in conjunction. When the earth is in quadratures at Dor B, thenw = 0, and m is greatest ; therefore there is no aberration in longi- tude, but the greatest in latitude ; when the earth is at D, the apparent place of the star is at d, and the latitude is there increased; but when the earth is at B, the apparent place of the star is at h, and the lati- tude is diminished ; hence, the latitude is least in .quadrature before opposition, and greatest in quadra- ture after. From the mean of a great number of ob- servations. Dr. Bradley determined the value of /• to be 20". Ex. 1. What is the greatest aberration in latitude and longitude of 7 Ursce minoris, whose latitude is 73^ 13' ? First, 7«= 1, 'v =,9669 the sine of 75°. 13'; hence, 20" x 9669=19",34 the greatest aberration in latitude. For the greatest aberration in longitude, 20" n — \, iv = ,2bb\ ; hence, — — — = 7^"i4 the greatest ,2551 aberration in longitude. Ex. 2. What is the aberration in latitude and longi- tude of the same star, when the earth is 30" from syzygies ? Here m = ,5 ; hence, 19",34 x ,5 =9",G7 the aberration in latitude. If the earth be 30" beyond conjunction or before opposition, the latitude is di- minished; but if it be 30° after opposition or before conjunction, the latitude is increased. Also, w- = ,866 ; hence, 78",4 x ,866'=:67",89 the aberration in longi- 206 ABERRATION OF LIGHT. tude. If the earth be 30° from conjunction, the longitude is diminished; but if it be 30° from opposi- tion, it is increased. Ex. 3. For the Sun, ni=0 and ?i = 1, w=l ; hence, it has no aberration in latitude, and the aberration in longitude =r = 20" constantly. This quantity 20" of aberration of the sun, answers to it's mean motion in 8'. 7". 30'", which is therefore the time in which the light moves from the sun to the earth at it's mean dis- tance. Hence, the sun always appears 20" backwarder than it's true place. (296.) To find the aberration in j-ight ascension and declination. Let JEL be the equator, p it's pole ; ^4CL the ecliptic, P it's pole ; S the true place of the star, s the apjmrent place in the ellipse; draw the great circles, Psa, Psb, pSw, pSv, and Sv, st perpendicular to P h, p V. Now s V = rv m (2^5 ) ; also, Sv = rn\ hence, (Trig. Art. 123) rym {vs) : rn [Sxj) :: rad. : tan. .S^y = — = cotan. earth's dist. syzv. —. , — j = cosec. star's vm sni. star s lat. lat. X cotan. earth's distance from syzygies. Thus we immediately compute the angle Ssv\ compute also the angle of position Psp from the three sides of that tri- angle being given (Trig. Art. 239), ^"^ "^ get the ABERRATION OF LIGHT. 20/ anf>le Ssp, it being the sum or difference of Ssv and Psp. Put a and b for the sine and (josinc of Ssv, c and d for the sine and cosine of Ssp, ;2;=: cosine of the star's dechnation ; then (as sv, st, are the cosines of Ssv, Sst, to radius sS) b : d :: sv ( =:rvm) : st = rvm x ^= 20" x vm X r the aberration in declination ; and (as Sv, St, b are the sines of Ssv, Sst, to radius 5.9) a :c :: Sv { =rn) .8jf = ''— ; hence {I3),vw(= —^^) = 20" X ^ a V ' \ COS. doc. ^ «.!£ the aberration in right ascension. 208 Chap. XXI. ON THE ECLIPSES OF THE SUN AND MOON. (297.) An eclipse of the Moon is caused by it's en- tering into the earth's shadow, and consequently it must happen when the moon is in opposition to the sun, or at the full moon. An eclipse of the Sun is caused by the interposition of the moon between the earth and sun, and therefore it must happen when the moon is in conjunction with the sun, or at the new moon. If the plane of the moon's orbit coincided with the plane of the ecliptic, there would be an eclipse at every opposition and conjunction ; but the plane of the moon's orbit being inclined to the ecliptic, there can be no eclipse at opposition or conjunction, unless at that time the moon be at, or near to the node. For suppose MMinm' be the orbit of the moon, and let the other circle represent the plane of the earth's orbit, or that plane in which the sun S, appears as seen from the earth E, and let these two planes be inclined to each other, so that we may conceive the part MM'm to lie above, and the part mm'Mhe\ow the plane of the earth's orbit; and M, m, are the nodes. Now if the moon be at M, in conjunction, the three bodies are then in the same plane, and therefore the moon is interposed between the earth and sun, and causes an eclipse of the sun. But if the moon be at M when the sun comes into conjunction at S\ M is iiow elevated above the line joining E and S', and M may be so far from My that the elevation of M' above the line ES' may be so much, that the moon may not be interposed between E and S', in which case there ECLIPSES OF THE SUN AND MOON. 209 will be no eclipse of the sun. Whether, therefore, there will be an eclipse of the sun at the conjunction, or not, depends upon the distance of the moon from the node at that time. If the moon be at m at the time of opposition, then the three bodies being in the same right line, the shadow EV oi the earth must fall upon the moon, and the moon must suffer an eclipse. But if the moon be at tri at the time of opposition, rd may be so far below the shadow Ev of the earth, that the moon may not pass through it, in which case there will be no eclipse. Whether, therefore, there will be a lunar eclipse at the time of opposition, or not, depends upon the distance of the moon from the node at that time. If the two planes coincided, there would evidently be a central interposition every con- junction and opposition, and consequently a total 210 TO CALCULATE AN eclipse. Meton, who lived about 430 years before Christ, observed, that after 19 years, the new and full moons returned again on the same day of the month. The ancient Astronomers also observed, that at the end of 18 years 10 days, a period of 223 lunations, there was a return of the same eclipses ; and hence, they were enabled to foretel when they would happen. This is mentioned by Pliny the Naturalist, Lib. II. Ch. 13. and by Ptolemy, Lib. IV. Ch. 2. This restitution of eclipses depends upon the return of the following elements to the same state. — 1. The sun's place. 2. The moon's place. 3. The place of the moon's apogee. 4. The place of the ascending node of the moon. The exact recurrence of these can never take place ; but it so nearly happens in the above time, as to produce eclipses remarkably corre- sponding. In this manner Dr. Halley predicted and published a return of eclipses from 1700 to 17I8, many of them corrected from observations ; together with the following elements. — 1 . The apparent time of the middle. 2. The sun's anomaly. 3. The an- nual argument. 4. The moon's latitude. He says, that in this period of 223 lunations, there are 18 years 10 or 11 days (according as there are five or four leap-years) 'jh. 43'| ; that if we add this time to the middle of any eclipse observed, we shall have the re- turn of a corresponding one, certainly within ih. 30'; and that, by the help of a few equations, we may find the like series for several periods. To explain the Principles of the Calculation of an Eclipse of the Moon. (298.) The first thing to be done, is to find the time of the ?wea«* opposition. To get which, from * The time of the mean opposition is the time when the opposi- tion would have taken place, if the motions of the bodies had been uniform. ECLIPSE OF THE MOON. 211 the Tables of Epacts*, amongst the Tables of the moon's motion, take out the epact for the year and month, and subtract the sum from 29c?. \2h. 44'. 3", one synodic revolution of the moon, or two if neces- sary, so that the remainder may be less than a revolu- tion, and that remainder gives the time of the mean conjunction. If to this we add 14^:?. 18A. 22'. \'\b, half a revolution, it gives the time of the next mean opposition; or if we subtract, it gives the time of the preceding mean opposition. If it be leap-year, in January and February subtract a day from the sum of the epacts, before you make the subtraction. When the day of the mean conjunction is O, it denotes the last day of the preceding month. Ex. To find the times of the mean new and full moons in February, 1795' Epact 1795 - - 9^ 11^ 6'. 17" February - --1. 11. 15. 57 10. 22. 22. 24 29. 12. 44. 3 Mean new moon - 18. 14. 21. 49 14. 18. 22. 1,5 Mean full moon - 3. 19. 59.-47,5 (299«) To determine whether an eclipse may happen at opposition, find the mean longitude of the earth at the time of mean opposition, and also the longitude of the moon's nodej then, according to M. Cassini, if * The epact for any year is the ag€ of the moon at the beginning of the year from the last mean conjunction, that is, from the time when the mean longitudes of the sun and moon were last equal. The epact for any month is the age which the moon would have had at the beginning of the month, if it's age had been nothing at the beginning of the year ; therefore, if to the epact for the year, the epact for the month be added, the sum taken from 29 d. \2h_. 44-'. 3". or from twice that quantity if the sum exceed it, must give the time oimcan conjunction. O 2 212 TO CALCULATE AN the difference between the mean longitudes of the earth and the moon's node be less than 7°- 30', there must be an eclipse; if it be greater than 14*^. 30', there cannot be an eclipse; but between 7°« 30', and 14°. 30', there may, or may not, be an eclipse. M. de Lambre makes these limits 7°. 47', and 13°. 21'*. Ex, To find whether there will be an eclipse at the full moon on Februarys, 1795- Sun's mean long. at3^ 19'. 59'. 47",6.10M3°.27'.'20"8 Mean long, of the earth - - - 4.13.27.20,8 Long, of the moon's node - - - 4. 8. 1. 48,5 Difference - - 0. 5.25.32,3 Hence, there must be an eclipse. Examine thus all the new and full moons for a month before and a month after the time at which the sun comes to the place of the nodes of the lunar orbit, and you will be sure not to miss any eclipses. Or, having the eclipses for the last 18 years, if you add to the times of the middle of these eclipses, 18?/. lO^Z. 7A. 43' I, or I83/. lid. 7h. 43' |, (297) it will give the times when you may expect the eclipses will return. (300.) To the time of mean opposition, compute the true longitudes of the sun and moon, and the moon's true latitude; and find, from the Tables of their motions, the horary motions of the sun and moon in longitude, and the difference (d) of their horary motions is the relative horary motion of the moon in respect to the sun, or the motion with which the moon approaches to, or recedes from, the sun ; find also the moon's horary motion in latitude ; and suppose, at the time (t) of mean opposition, the moon is at the distance [m] from opposition ; then, as we * This may be found from Art. 306. by finding the true hmit, and then applying the greatest diflercnce of the true and mean places. ECLIFSE OF THE MOON. 213 may suppose the moon to approach the sun, or recede from it, uniformly, d : m :: 1 hour : the time {w) be- tween t and the opposition, which added to, or sub- tracted from, the time f, according as the moon is not yet got into opposition, or is beyond it, gives the time of the echptic opposition. (301.) To find the place of the moon in opposition, let 71 be the moon's horary motion in longitude; then, 1 hour '. w :: n : the increase of the moon's longitude in the time w, which applied to the moon's longitude at the time of the mean opposition, gives the true longitude of the moon at the time of the ecliptic opposition. The opposite point to that must be the true longitude of the sun. Find also the moon's true latitude at the time of opposition, by saying, 1 hour : IV :: the horary motion in latitude : the motion in lati- tude in the time iVy which applied to the moon's lati- tude at the time of the mean opposition, gives the true latitude at the time of the true opposition*. In like manner you may compute the true time of the ecliptic conjunction, and the places of the sun and moon for that time, when you calculate a solar eclipse. (302.) With the sun's horary motion in longitude, and the moon's in longitude and latitude, find the in- clination of the relative orbit, and the horary motion upon it. To do this, let L3Ibe the horary motion of the moon in longitude, S Al that of the sun ; draw Ma perpendicular to LM, and equal to the moon's horary motion in latitude ; take Sb = Ma, and parallel * For greater certainty, you may compute again, from the Tables^ the places of the sun and moon, and if they be not exactly in op- position, which probably may not be the case, as the moon's longi- tude does not increase uniformly, repeat the operation. This accuracy, however, in eclipses is generally unnecessary ; for the best lunar Tables cannot be depended upon to give the moon's longitude nearer than 10"; therefore the probable error from the Tables is vastly greater than that which arises from the motion in longitude not being uniform. Unless, therefore, very great accu- racy be required, this operation is unnecessary. 214 TO CALCULATE AN to it, and join La, Lb ; then La is the moon's true orbit, and Lb it's relative orbit in respect to the sun. Hence, LS (the difference of the horary motions in longitude) : Sb the moon's horary motion in latitude :: radius : tan. bLS, the inch nation of the relative orbit ; and cos. bLS : radius :: LS : Lb, the horary motion in the relative orbit. (303.) At the time of opposition, find, from the Tables, the moon's horizontal parallax, it's semi- diameter, and the semidiameter of the sun, the hori- zontal parallax of which we may here take = 9". (304.) To find the semidiameter of the earth's shadow at the moon, seen from the earth. Let AB be the diameter of the sun, TR the diameter of the earth, O and C their centers ; produce AT, BR, to meet at /, and draw OCT; let FGH be the diameter of the earth's shadow at the distance of the moon, and join OT, CF. Now the angle FCG= CFA- CIA, but CIA=OTA^TOC, therefore FCG = CFA- OTA + TOC, that is, the angle under which the se- midiameter of the earth's shadow, at the moon, ap- jyears, is equal to the sum of the horizontal parallaxes of the sun and moon diminished by the apparent semi- diameter of the sun. In eclipses of the moon, the shadow is found to be a little greater than this Rule ECLIPSE OF THE MOON. 215 gives it, owing to the atmosphere of the earth. This augmentation of the semidiameter is, according to M, Cassini, 20" \ according to M, Mowmer, 30" ; and according to M. de la Htre^ 6o". Mayer thinks the correction is about 77- of the semidiameter of the sha- 60 dow, or that you may add as many seconds as the semidiameter contains minutes. Some computers always add 50" ; but this must be subject to some un- certainty, (305.) As the angle CIT ( = OTA-TOC) is known, we have sin. TIC-, cos. TIC .: TC . C/the length of the earth's shadow. If we take the angle ATO=z\&. 3" the mean semidiameter of the sun, TOC=i^" the horizontal parallax of the sun, we have CIT=15\ 54"; hence, sin. 15'. 54" : cos. 15'. 54", or 1 : 216,2 :: TC: Cr=2l6,2 TC. (306.) The different eclipses which may happen of the moon, may be thus explained. Let CL represent the plane of the ecliptic, OR the moon's orbit, cutting O the ecliptic in the node A^; ^nd let SH represent a section of the earth's shadow at the distance of the moon from the earth, and M the moon at the time when she passes nearest to the center of the earth's shadow. Hence, if the opposition happen as in posi- tion I, it is manifest that the moon will just pass by the shadow of the earth without entering it, and there will be no eclipse. In position II, part of the moon will pass through the earth's shadow, and there will be a ^ar^k^ eclipse. In position III, the whole of the. 316 TO CALCULATE AN moon passes through the earth's shadow, and there is a total echpse. In position IV, the center of the moon passes through the center of the earth's shadow, and there is a total and central ecHpse. It is plain, therefore, that whether there will, or will not, be an eclipse at the time of opposition, depends upon the distance of the moon from the node at that time ; or the distance of the earth's shadow, or of the earth, from the node. Now in lunar eclipses we may take the angle at N=b°. 17', and in position I, the value of Ev is about l**. 3'. 30"; hence (Trig. Art. 221), sin. 5". 17' : rad. :: sin. 1°. 3'. 30" : sin. EN:=^\\\ 34'-, when, therefore, EN is greater than that quantity at the time of opposition, there can be no eclipse. This quantity 11**. 34' is called the ecliptic limit. (307.) Let Arh be that half of the earth's shadow which the moon passes through, NL the relative orbit of the moon ; draw Cmr perpendicular to NL, and let z be the center of the moon at the beginning of the eclipse, m at the middle, x at the end ; also, let AB be the ecliptic, and Cn perpendicular to it. Now in the right-angled triangle Cnm, we know Cn the latitude of the moon at the time of the ecliptic con- junction, and (302) the angle Cnm* the complement of the angle which the relative orbit of the moon makes with the ecliptic; hence (Trig. Art. 125) rad. : cos. * If the moon at n have north or south latitude increasing, the angle Oim is to be set off to the right j otherwise, to the left of Cn, ECLIPSE OF THE MOON. 217 Cnm :: Cn : nni, which is called the Reduction ; and rad. : sin. Cnm :: C?i : Cm. The horary motion [h) of the moon upon it's relative orbit being known, we know the time of describing mti, by saying, h : mn :: 1 hour : the time of describing mn. Hence, knowing the time of the ecliptic conjunction at n, we know the time of the middle of the eclipse at m. Next, in the right-angled triangle Cmz, we know Cm, and Cz the sum of the semidiameters of the earth's shadow and the moon, to find mz, which is done thus by loga- rithms; 2ismz=sJ Cz' - Cm^ = \/ Cz + Cmx Cz - Cm, the log. of mz-=ix log. Cz-{- Cm + log. Cz - Cm, (Trig. Art. 52). Hence, the horary motion of the moon being known, we know the time of describing zm, which subtracted from the time at m gives the time of the beginning, and added, gives the time of the end. The magnitude of the eclipse at the middle is represented by tr, which is the greatest distance of the moon within the earth's shadow, and this is mea- sured in terms of the diameter of the moon, conceived to be divided into 12 equal parts, called Digits, or Parts deficient ; to find which, we know Cm, the difference between which and Cr gives mr, which added to mt, or if m fall out of the shadow, take the difference between mr and 7nt, and we get tr ; hence, to find the number of digits eclipsed, say, mt : tr :: 6 digits, or 36o', (it being usual to divide a digit into 60 equal parts, and call them minutes,) : the digits eclipsed. If the latitude of the moon be north, we use the upper semicircle ; if south, we take the lower. (308.) If the earth had no atmosphere, when the moon was totally eclipsed, it would be invisible ; but, by the refraction of the atmosphere, some rays will be brought to fall on the moon's surface, upon which ac- count the moon will be visible at that time, and appear of a dusky red colour. M. Maraldi [Mem. de I Acad. 1723) has observed, that, in general, the earth's umbra, at a certain distance, is divided by a kind of penumbra. 518 ECLIPSES OF THE SUN. from the refraction of the atmosphere. This will ac- count for the circumstance of the moon being more visible- in some total eclipses than in others. It is said, that the moon, in the total eclipses in 1601, 1620, and 1642, entirely disappeared. (309.) An eclipse of the moon, arising from it's real deprivation of light, must appear to begin at the same instant of time to every place on that hemisphere of the earth Vk^hich is turned towards the moon. Hence, it affords a very ready method of finding the difference of longitudes of places upon the earth, as will be after- wards explained. The moon enters the penumbra of the earth before it comes to the umbra, and therefore it gradually loses it's light; and the penumbra is so dark just at the umbra, that it is difficult to ascertain the exact time when the moon's limb touches the umbra, or when the eclipse begins. When the moon has entered into the umbra, the shadow upon it's disc is tolerably well defined, and you may determine, to a considerable degree of accuracy, the time when any spot enters into the umbra. Hence, the beginning and end of a lunar eclipse are not so proper to deter- mine the longitude from, as the times at which the umbra touches any of the spots. On Eclipses of the Sun. (310.) An eclipse of the sun is caused by the inter- position of the moon between the sun and spectator, or by the shadow of the moon falling on the earth at the place of the observer. The different kinds of eclipses will be best explained by a figure. Let S be the sun, M the moon, JB or Jt'B' the surface of the earth; draw tangents pxvs, qzvr, from the sun to the same side of the moon, and ocvz will be the moon's umbra, in which no part of the sun can be seen ; if tangents ptbd, qwac, be drawn from the sun to the opposite sides of the moon, the space comprehended between the umbra and waCy tbd, is called the ECIPSES OF THE SUN. 219 penumbra, in which part of the sun only is seen. Now it is manifest, that if AB be the surface of the earth, the space mn, where the umbra falls, will suffer a total eclipse; the part am, bn, between the bounda- ries of the umbra and penumbra, will suffer a partial eclipse ; but to all the other parts of the earth there will be no eclipse. Now let A' B' be the surface of the earth, the earth being, at different times, at dif- ferent distances from the moon; then the space within rs will suffer an annular eclipse; for if tangents be drawn from any point o within rs to the moon, they must evidently fall within the sun, therefore the sun will appear all round about the moon in the form of a ring; the parts cr, sd, will suffer a partial eclipse; and the other parts of the earth will suffer no eclipse. In this case, there can be no total eclipse any where, as the moon's umbra does not reach the earth. Ac- 220 ECLIPSES OF THE SUN. cording to M. du Sejour, an eclipse can never be an- nular longer than 12'. 24", nor total longer than 7'. 58". (311.) The umbra xvz is a cone, and the penumbra wcdf the frustrum of a cone whose vertex is f^. Hence, if these be both cut through their common axis perpendicular to it, the section of each will be a circle, having a common center in the line joining the centers of the sun aud moon, and the penumbra in- cludes the umbra. (312.) The moon's mean motion about the center of the earth is at the rate of about 33' in an hour ; but 33' of the moon's orbit is about 2280 miles, which, therefore, we may consider as the velocity with which the moon's shadow passes over the earth ; but this is the velocity upon the surface of the earth where the shadow falls perpendicularly upon it, it being the velocity perpendicular to Mv; in every other place, the velocity over the surface will be increased in the proportion of the sine of the angle which Mv makes with the surface, in the direction of it's motion, to radius. But the earth having a rotation about it's axis, the relative velocity of the moon's shadow over any given point of the surface will be different from this ; if the point be moving in the direction of the shadow, the velocity of the shadow, in respect to that point, will be diminished, and consequently the time in which the shadow passes over it will be increased ; but if the point be moving in a direction contrary to that of the shadow, as is the case when the shadow falls on the other side of the pole, the time will be dimini^ed. The length of a solar eclipse is therefore affected by the earth's rotation about it's axis. (313.) The different eclipses of the sun may Jbe thus explained. Let CL represent the orbit of the earth, OR the line described by the centers of the moon's umbra and penumbra at the earth ; N the moon's node; SF the earth, E it's center; pn the ECLIPSES OF THE SUN. 221 moon*s penumbra, u the umbra. Then, in position I, the penumbra p n just passes by the earth, without falHng upon it, and therefore there will be no echpse. In position II, the penumbra pn falls upon the earth, but the umbra u does not; therefore there will be a partial eclipse where the penumbra falls, but no total eclipse. In position III, both the penumbra /?/?, and umbra u fall upon the earth; therefore, where the penumbra falls, there will be a partial eclipse, and where the umbra falls there will be a total eclipsp ; and to the other parts of the earth there will be no eclipse. Now the ecliptic litnit, may be thus found. The angle A^ may be taken 5°. 17'j and in position I, the value of Eu {u being the center of the umbra) is about 1°. 34'. 27"; hence (Trig. Art. 221) sin. 5°. if : rad. :: sin. V. 34'. 27" : sin. EN=17\2l\ 2f' the ecliptic limit; if therefore, at the time of conjunction, the earth be within this distance of the node, there will be an eclipse. (314.) An eclipse of the sun, or rather of the earth, without respect to any particular place, may be calcu- lated exactly in the same manner as an eclipse of the moon, that is, the times when the moon's umbra or penumbra first touches and leaves the earth ; hut to find the times of the beginning, middle, and end, at any particular place, the apparent place of the moon, as seen from thence, must be determined, and conse- 222 TO CALCULATE AN quently it's parallax in latitude and longitude must be computedj which renders the calculation of a solar echpse extremely long and tedious. To explain the Principles of the Calculation of an Eclipse of the Sun for any particular Place. (315.) Having determined {X\S) that there will be an eclipse somewhere upon the earth, compute, by the Astronomical Tables, the true longitudes of the sun and moon, and the moon's true latitude, at the time of mean conjunction (301) ; find also the horary mo- tions of the sun and moon in longitude, and the moon's horary motion in latitude ; and compute the time of the ecliptic conjunction of the sun and moon, in the same manner (300) as the time of the ecliptic opposition was computed. At the time of the ecliptic conjunction, compute (301) the sun's and moon's longitude, and the moon's latitude ; find also the hori- zontal parallax of the moon from the Tables of the moon's motion, from which subtract the sun's hori- zontal parallax, and you get the horizontal parallax of the moon from the sun. (31 6.) To the latitude of the given place, and the horizontal parallax of the moon from the sun (which we here use instead of the horizontal parallax of the moon, as we want to find what effect the parallax has in altering their apparent relative situations,) at_tjie_ time of the ecli ptic conjunct ion^ compute (144) the moon's parallax in latitude and longitude from the sun ; the parallax in latitude applied to the true lati- tude gives the apparent latitude [L) of the moon from the sun ; and the parallax in longitude shows the appare nt difference {D) of the longitudes of the sun and moon. (317.) Let S be the sun, CE the ecliptic, according ECLIPSE OF THE SUN. 223 to the order of the signs; take SM=D, draw MIV perpendicular to MSy and take it = L, then N is the apparent place of the moon, and SN= ^ D' + U is the apparent distance of the moon from the sun. (318.) If the moon be to the ea^"^ of the nonagesimal degree, tlie parallax increases the longitude ; if to the tvest, it diminishes it (Art. 144) ; hence, if the trme longitudes of the sun and moon be equal, in the former case the apparent place will be from aS" towards E, and in the latter, towards C. To some time, as an hour, after the true conjunction, if the moon be to the west of the nonagesimal degree ; or before the true conjunction, if the moon be to the east of the nonagesimal degree, find the sun's and the moon's true longitude, and the moon's true latitude, from their horary motions ; and to the same time compute the moon's parallax in latitude and longitude from the sun ; apply the parallax in latitude to the true lati- tude, and it gives the apparent latitude (/) of the moon from the sun; take the difference of the sun's and moon's true longitude, and apply the parallax in longitude, and it gives the apparent distance {d) of the moon from the sun in longitude. From S set off SP = d, and on EC erect the perpendicular PQ equal to I, and Q is the apparent place of the moon at one hour from the true conjunction ; and SQ ( = ^ d^ -\- 1*) is the apparent distance of the moon from the sun ; draw the straight line NQ, and it will represent the relative apparent path of the moon, considered as a straight line, in general it being very nearly so j it's value also 2!^4 TO CALCULATE AN represents the relative horary motion of the moon in the apparent orbit, the relative horary motion in longitude being MP. (319,) The difference between the moon's apparent distance in longitude from the sun at the time of the true ecliptic conjunction, and at the interval of an hour, gives the apparent horary motion (r) in longi- tude of the moon from the sun ; the difference (D) between the true longitude at the ecliptic conjunction, and the moon's apparent longitude, is the apparent distance of the moon from the sun in longitude at the true time of the ecliptic conjunction; hence, r : D :: 1 hour : the time from the true to the apparent conjunction, consequently we know the time of the apparent conjunction. To find whether this time is accurate, we may compute (from the horary motions of the sun and moon) their true longitudes, and the moon's parallax in longitude from the sun, and apply it to the true longitude, and it gives the apparent longitude, and if this be the same as the sun's longitude, the time of the apparent conjunction is truly found; if they be no t the same, find jrom thence the true time, as b efore. iTo the true time of the apparent conjunctTonTHfind the moon's true latitude from it's horary motion, and compute the parallax in latitude, and you get the apparent latitude at the time of the apparent conjunction. Draw SA perpendicular to CE, and equal to this apparent latitude ; then the point A will not probably fall in NQ ; but suppose it to fall in QiV, to which draw SB perpendicular, and NR parallel to PM. Then knowing NR ( = P3I), and QR { = QP^MN) we have NR : RQ :: rad. : tan. QNR, or ASB {Tng. Art. 123) Sin. QNR : rad. :: QR : QiV (Trig. Art. 128) The time of describing NQ in the apparent orbit being equal to the time from M to P in longitude, NQ is the horary motion in the apparent orbit. ECLIPSE OF THE SUN. 225 Rad. : sin. ASB :: AS : AB (Trig. Art. 125) Rad. : COS. ASB :: AS : SB. (320.) At the apparent conjunction the moon apr- pears at A, which time (319) is known; when the inoon appears at B, it is at it's nearest distance from the sun", and consequently the time is that of the greatest obscuration, (usually called the time of the middle,) provided there is an eclipse, which will always be the case, when SB is less than the sum of the apparent semidiatneters of the sun and moon. If, therefore, it appear that there will be an eclipse, we proceed thus to find it's quantity, and the beginning and end. As we may suppose the motion to be uni- form, QiV: AB :: the time of describing NQ : the time of describing AB, which added to or subtracted from the time at A, (according as the apparent latitude is decreasing or increasing), gives the time of the greatest obscuration. (321.) From the sum of the apparent semidiameters of the sun and moon, subtract BS, and the remainder shows how much of the sun is covered by the moon, or the parts deficient; hence, semid. o : parts defi- cient :: 6 digits : the digits eclipsed. If SB be less than the difference of the semidiameters of the sun and moon, and the moon's semidiameter be Xhe greater, the eclipse will be total; but if it be the less, the eclipse will be annular, the sun appearing all round the moon ; if B and S coincide, the eclipse will bQ central. (322.) Produce, if necessary, QN, and take SV,SIV, equal to the sum of the apparent semidiameters of the sun and moon, at the beg inning and end respectively ; then BF=JSF'^SB\ax^d BIF=.JSI^F'-.SB^-, and to find the times of describing these, say, as the hourly motion of the moon in the apparent orbit, or NQ, : BF :: 1 hour : the time of describing FB ^ and NQ : BfF.. 1 hour : the time of describing J5^A^ p 226 TO CALCULATE Aft which times, respectively subtracted from and added to the time of the greatest obscuration, give nearly the times of the beginning and end. But if accuracy be required, a different method must be adopted ; for we suppose yfV to be a straight line, which supposition will, in general, cause errors, too considerable to be neglected. It may, however, always serve as a rule to assume the time of the beginning and end. Hence it follows, that the time of the greatest obscuration at B, is not necessarily equidistant from the beginning and end. (323.) If the eclipse be total, take ST, SIV, equal to the difference of the semidiameters of the sun and moon, and then BF= Bir= J'SW^ - SB\ from whence we may find the times of describing J5^, BfV, as before, which we may consider as equal, and which applied to the time of the greatest obscuration at B, give the time of the beginning and end of the total darkness. (324.) To find more accurately the time of the beginning and end of the eclipse, we must proceed thus. At the estimated time of the beginning, find, from the horary motions, and the computed parallaxes, the apparent latitude f^D of the moon, and it's appa- rent longitude DS from the sun, and we have S f^= aJ SW^-i-Df'^y and if this be equal to the apparent semid. D 4- semid. © (which sum call S), the estimated time is the time of the beginning; but if Sl^he not equal to S, assume (as the error directs) another time at a small interval from it, before, if SFhe less than S, but after, if it be greater], to that time compute again the moon's apparent latitude mv, and apparent longitude Sm from the sun, and find Sv=^ Sm'^ + mv^; and if this be not equal to S, proceed thus ; as the difference of Sv and SF : the difference of Sv and SL {=^S) :: the above-assumed interval of time, or time of the motion through Vv, • the time through vLy ECLIPSE OF THE SUN. 22/ which added to or subtracted from the time at v, ac- cording as Sv is greater or less than SL, gives the time of the beginning. The reason of this operation is, that as F^v, vL, are very small;, they will be very nearly proportional to the differences of SF, Sv, and Sv, SL. But as the variation of the apparent distance of the sun from the moon is not exactly in proportion to the variation of the differences of the apparent longitudes and latitudes, in cases where the utmost accuracy is required, the time of the beginning thus found (if it appear to be not correct) maybe corrected_, by assuming it for a third time, and proceeding as before. This correction, however, will never be ne- cessary, except where extreme accuracy is required in order to deduce some consequences from it. But the time thus found is to be considered as accurate, only so far as the Tables of the sun and moon can be de- pended upon for their accuracy; and the best lunar Tables are subject to an error of 10" in latitude. Hence, accurate observations of an eclipse, compared with the computed time, furnish the means of correct- ing the lunar Tables. In the same manner, the end of the eclipse may be computed. (325.) As there are not many persons who have an opportunity of seeing a total eclipse of the sun, we shall here give the phaenomena which attended that on April 22, 1715. Capt. Stannyayi, at Bern in Switzer- land, says, *' the sun was totally dark for four minutes and a half; that a fixed star and planet appeared very bright; and that it's getting out of the eclipse was preceded by a blood-red streak of light, from it's left limb, which continued not longer than six or seven seconds of time ; then part of the sun's disc appeared, all on a sudden, as bright as Venus was ever seen in the night ; nay, brighter, and in that very instant gave a light and shadow to things, as strong as moon-light used to do.'* The inference drawn from these phae- nomena is, that the moon has an atmosphere. J. C. FaciSf at Geneva, says, *' ther^ was seen, P 2 228 TO CALCULATE AN during the whole time of the total immersion, a white- ness, which seemed to break out from behind the moon, and to encompass it on all sides equally ; it's breadth was not the twelfth part of the moon's dia- meter. Venus, Saturn, and Mercury were seen by many ; and if the sky had been clear, many more stars might have been seen, and with them Jupiter and Mars. Some gentlewomen in the country saw more than 16 stars ; and many people on the mountains saw the sky starry, in some places where it was not overcast, as during the night at the time of the full moon. The duration of the total darkness was three minutes." Dr. J. J. Scheuchzer, at Zurich, says, *' that both planets and fixed stars were seen ; the birds went to roost; the bats came out of their holes ; and the fishes swam about j we experienced a manifest sense of cold; and the dew fell upon the grass. The total darkness lasted four minutes." (326.) Dr. //a//ei/*, who observed this eclipse at Lon- don, has thus given the phaenomena attending it. " It was unwersally observed, thatwhen the last part of the sun remained on it's east side, it grew very faint, and was easily supportable to the naked eye, even through the telescope, for above a minute of time before the total darkness ; wjiereas, on the contrary, my eye could not endure the splendour of the emerging beams in the telescope from the first moment. To this, per- * The Doctor begins his account thus. " Though it be certain, ■from the principles of Astronomy, that there happens necessarily a central eclipse of the sun, in some part or other of the terra- queous globe, about twenty»eight times in each period of eighteea years; and that of these, no less than eight do pass over the parallel of London, three of which eight are total with continuance ; yet, from the great variety of the elements, w^hereof the calculus of eclipses consists, it has so happened, that since March 20, 1140, 1 cannot find that there has been a total eclipse of the sun seen at London, though in the mean time the shade of the moon has often passed over other parts of Great Britain." ECLIPSE OF THE SUN. 229 haps, two causes concurred ; the one that the pupil of the eye did necessarily dilate itself during the dark- ness, which before had been much contracted by look- ing on the sun. The other, that the eastern parts of the moon, having been heated with a day near as long as thitty of our's, must of necessity have that part of it's atmosphere replete with vapours, raised by the long-continued action of the sun; and, by conse- quence, it was more dense near the moon's surface^ and more capable of obstructing the lustre of the sun's beams. Whereas at the same time the western edge of the moon had suffered as long a night, during which time there might fall in dews, all the vapours that were raised in the preceding long day ; and for this reason, that part of it's atmosphere might be seen much more pure and transparent. About two minutes before the total immersion, the remaining part of the sun was reduced to a very fine horn, whose extremities seemed to lose their accuteness, and to become round like stars. And for the space of about a quarter of a minute, a small piece of the southern horn of the eclipse seemed to be cut off from the rest by a good interval, and appeared like an oblong star round at both ends; which appearance could proceed from no other cause, but the inequali- ties of the moon's surface, there being some elevated parts thereof near the moon's southern pole, by which interposition, part of that exceedingly fine filament of light was intercepted. A few seconds before the sun was totally hid, there discovered itself round the moon a luminous ring, about a digit, or perhaps a tenth part, of the moon's diameter in breadth. It was of a pale whiteness, or rather pearl colour, seeming to me a little tinged with the colours of the iris, and to be concentric with the moon ; whence 1 concluded it was the moon's atmo- sphere. But the great height of it, far exceeding that of our earth's atmosphere ; and the observations of some one who found the breadth of the ring to increase 230 ECLIPSES or THE SUJT. on the west side of the moon, as the emersion ap- proached ; together with the contrary sentiments of those, whose judgement I shall always revere, make me less confident, especially as in a matter whereto I gave not all the attention requisite. Whatever it was, this ring appeared much brighter and whiter near the body of the moon, than at a dis- tance from it ; and it's outward circumference, w^hich was ill defined, seemed terminated only by the extreme rarity of the matter it was composed of; and in all respects resembled the appearance of an enlightened atmosphere viewed from far : but whether it belonged to the sun or the moon, I shall not at present under- take to decide. During the whole time of the total eclipse, I kept my telescope constantly fixed on the moon, in order to observe what might occur in this uncommon appear- ance, and I saw perpetual flashes or coruscations of light, which seemed for a moment to dart out from behind the moon, now here, now there, on all sides, but more especially on the western side, a little before the emersion ; and about two or three seconds before it, on the same western side, where the sun was just coming out, a long and very narrow streak of dusky, but strong red light, seemed to colour the dark edge of the moon, though nothing like it had been seen immediately after the immersion. But this instantly vanished upon the first appearance of the sun, as did also the aforesaid luminous ring;. As to the degree of darkness, it was such, that one might have ex[3€cted to have seen more stars than were seen in London ; the planets Jupiter, Mercury, and J'enuSy were all that were seen by the gentlemen of the Society from the top of their house, where they had a free horizon ; and 1 do not hear that any one in town saw more than Capella and Aldeharan of the fixed stars. Nor was the light of the ring round the moon capable of eflfacing the luster of the stars, for it was vastly inferior to that of the full moon, and so ECLIPSES OF THE SUN. 251 weak, that I did not observe it cast a shade. But the under-parts of the hemispherej particularly in the south-east under the sun, had a crepuscular bright- ness ; and all round us, so much of the segment of our atmosphere as was above the horizon, and was without the cone of the moon's shadow, was more or less enlightened by the sun's beams; and it's reflection gave a diffused light, which made the air seem hazy, and hindered the appearance of the stars. And that this was the real cause thereof, is manifest by the darkness being more perfect in those places near which the center of the shade passed, where many more stars were seen, and in some, not less than twenty, though the light of the ring was to all alike. I forbear to mention the chill and damp, with which the darkness of this eclipse was attended, of which most spectators were sensible, and equally judges; or the concern that appeared in all sorts of animals, birds, beasts, and fishes, upon the extinction of the sun, since ourselves could not behold it without some sense of horror." (327.) If a conjunction of the sun and moon happen at, or very near, the node, there will be a great solar eclipse; but, in this case, at the preceding opposition, the earth was not got into the lunar ecliptic limits, and at the next opposition it will be got beyond it ; hence, at each node there may happen only one solar eclipse, and therefore in a year there may happen only two solar eclipses. There must be one conjunction in the time in which the earth passes through the solar ecliptic limits, and consequently there must be one solar eclipse at each node ; hence, there must be two solar eclipses at least in a year. If an opposition happen ju«t before the earth gets into the lunar ecliptic limit, the next opposition may not happen till the earth is got beyond the limit on the other side of the node ; consequently there may not be a lunar eclipse at the node ; hence, there may 232 ON THE NUMBER OF not be an eclipse of the moon in the course of a year. When, therefore, there are only two eclipses in a year, they must be both of the sun. If there be an eclipse of the moon as soon as the sua gets within the lunar ecliptic limit, it will be got out of the limit before the next opposition ; consequently there can be only one lunar eclipse at the same node. But as the nodes of the moon's orbit move backwards about 19° iu a year, the earth may come within the lunar ecliptic limits, at the same node, a second time in the course of a year, and therefore there may be three lunar eclipses in a year; and there can be no more. If an eclipse of the moon happen at, or very near to, the node, a conjunction may happen before and after, whilst the earth is within the solar ecliptic limits; hence there may, at each node, happen two eclipses of the sun and one of the moon ; and in this case, the eclipses of the sun will be small, and that of the moon large. When, therefore, the eclipses do not happen a second time at either node, there may be six eclipses in a year, four of which will be of the sun, and two of the moon. But if, as in the last case, an eclipse should happen at the return of the earth within the lunar ecliptic limits at the same node a second time in the year, there may be six echpses, three of the sun and three of the moon. There may be seven eclipses in a year. For twelve lunations are performed in 354 days, or in 1 1 days less than a common year. If, therefore, an eclipse of the sun should happen before Jan uary 11, and the re be at; that, and at the next node, two solar and one lunar eclipse at each ; then the twelfth lunation from the first eclipse will give a new moon within the y. ar, and (on account of the retrograde motion of the moon's nodes) the earth may be got within the solar ecliptic limits, and there may be another solar eclipse. Hence, when here are seven eclipses in a year, five will be of the sun and two of the moon. This is upon sup- ECLIPSES IN A YEAR. 233 position that the first eclipse is of the sun ; but if the first eclipse should be of the moon, there may be three of the sun and four of the moon. As there are seven eclipses in the year but seldom, the mean number will be about four. The nodes of the moon move backwards about 19^^ in a year, which arc the eartli describes in about I9 days, consequently the middle of the seasons of the ecHpses happens every year about I9 days sooner than in the preceding year. The ecliptic limits of the sun (313) are greater than those of the moon (306), and hence, there will be more solar than lunar eclipses, in about the same pro- portion as the limit is greater, that is, as 3 : 2 nearly. But more lunar than solar eclipses are seen at any given place, because a lunar eclipse is visible to a whole hemisphere at once ; whereas a solar eclipse is visible only to a part, and therefore there is a greater proba- bility of seeing a lunar than a solar eclipse. Since the moon is as long above the horizon as below, every spectator may expect to see half the number of lunar eclipses which happen. For the calculation of eclipses, and all the circum- stances respecting them, see my Complete System of Astronomy, 234 Chap. XXII. ON THE TRANSITS OF MERCURY AND VENUS OVER THE SUN'S DISC. (328.) When Dr. Halley was at St. Helena, whither he went for the purpose of making a catalogue of the stars in the southern hemisphere, he ohserved a transit of Mercury over the sun's disc ; and, by means of a good telescope, it appeared to him that he could deter- mine the time of the ingress and egress, without it's being subject to an error of l"*; upon which he im- mediately concluded, that the sun's parallax might be determined by such observations, from the difference of the times of the transit over the sun, at different places upon the earth's surface. But this difference is so small in Mercury, that it would render the conclu- sion subject to a great degree of inaccuracy; in Venus, however, whose parallax is nearly four times as great as that of the sun, there will be a very considerable difference between the times of the transits seen from different parts of the earth, by which the accuracy of the conclusion will be proportionably increased. The Doctor, therefore, proposed to determine the sun's parallax from the transit of Venus over the sun's disc, observed at different places on the earth ; and as it was not probable that he himself should live to observe the * Hence, Dr. Ilallej/ concluded, that by a transit of Venus, the sun's distance might be determined with certainty to the 600th part of the whole ; but the observations upon the transits which happened in !761 and 1769, showed that the time of contact of the liaibs of the Sun and Venus could not be determined to that degree of certainty. ON THE TRANSITS OF MERCURY, &C. 235 next transits, which happened in 1761 and 1769, he very earnestly recommended the attention of thein to the Astronomers who should be alive at that time. Astronomers were therefore sent from England and France to the most proper parts of the earth to observe both those transits, from the result of which, the parallax has been determined to a veiy great degree of accuracy. (329.) Kepler was the first person who predicted the transits of Venus and Mercury over the sun's disc; he foretold the transit of Mercury in l631, and the transits of Venus in l631 and 176'!. The first time Venus was ever seen upon the sun, was in the year 1639, on November 24, at Hoole, near Liverpool, by our countryman Mr. Horrox, who was educated at Emanuel College in this University. He was em- ployed in calculating an Ephemerisfrom the Lausherg Tables, which gave, at the conjunction of Venus with the sun on that dav, it's apparent latitude less than the semidiameter of the sun. But as these Tables had so often deceived him, he consulted the Tables con- structed by Kepler, according to which, the conjunc- tion would be at 8//. l' a. m. at Manchester, and the planet's latitude 14'. 10" south; but, from his own corrections, he expected it to happen at 3h, 57' p.m. with 10' south latitude. He accordingly gave this in- formation to his friend Mr. Crahtree, at Manchester, desiring him to observe it; and he himself also pre- pared to make observations upon it, by transmitting the sun's image through a telescope into a dark chamber. He described a circle of about six inches diameter, and divided the circumference into 360°," and the diameter into 120 equal parts, and caused the sun's image to fill up the circle. He began to observe on the 23d, and repeated his observations on the 24th till one o'clock, when he was unfortunately called away by business; but, returning at 15' after three o'clock, he had the satisfaction of seeing Venus upon the sun's- disc, just wholly entered on the left side, so that the 236 ON THE TRANSITS OF MERCURY limbs perfectly coincided. At 35' after three, he found the distance of Venus from the sun's center to be 13', 30"; and at 45' after three, he found it to be 13'; and the sun setting at 50' after three o'clock, put an end to his observations. From these observations, Mr. Horrox endeavoured to correct some of the elements of the orbit of Venus. He found Venus had entered upon the disc at about 62°. 30' from the vertex towards the right on the image, which, by the telescope, was in- verted. He measured the diameter of Venus, and found it to be to that of the sun, as 1,12 : 30, as near as he could measure. Mr. Crabtree, on account of the clouds, got only one sight of Venus, which was at 2>h. Ab'. Mr. Horrox * wrote a Treatise, entitled Venus in Sole visa, but did not live to publish it ; it was, however, afterwards published by Hevelius. Gassendus observed the transit of Mercury which happened on November f, l631, and this was the first which had ever been observed; he made his observa- tions in the same manner that Horrox did after him. Since his time, several transits of Mercury have been observed, as they frequently happen ; whereas only two transits of Venus have happened since the time of Horrox. If we know the time of the transit at one node, we can determine, in the following manner, when they will probably happen again at the same node. (330.) The mean time from conjunction to conjunc- tion of Venus or Mercury being known (Art. 201), and the time of one mean conjunction, we shall know the time of ail the future mean conjunctions ; observe, therefore, those which happen near to the node, and compute the geocentric latitude of the planet at the time of conjunction, and if it be less than the appa- * The diflBculties which this very extraordinary person had to encounter with in his astronomical pursuits, he himself has de- scribed, in the Prolegomena prefixed to his Opera Postfmna^ published by Dr. fVallis, AND VENUS OVER THE SUN's DISC. 23^ rent semidiameter of the sun, there will be a transit of the planet over the sun's disc ; and we may determine the periods when such conjunctions happen, in the following manner. Let P = the periodic time of the earth, p that of Venus or Mercury. Now that a transit may happen again at the same node, the earth must perform a certain number of complete revolu- tions in the same time that the planet performs a cer- tain number, for then they must come into conjunction again at the same point of the eai'th's orbit, pr nearly in the same position in respect to the node. Let the earth jjerform x revolutions whilst the planet performs y revolutions; then will Px—j)y^ therefore -=■£. Now P = 365,256, and for Mercury, jO = 87,968; . P X p 87,968 ., , . -^ . , . , tnerelore- = ~ = -n -. = (by resolving; it into its y P 365,25b ^ ^ ^ continued fractions) -, — , — , — ,— ^, ■, &c. That ^ 4' 25' 29' 54'137'191' is, 1, 6, 7, 13, 33, 46, &c. revolutions of the earth are nearly equal to 4, 25, 29, 54, 13 7, I9I, &c. revolu- tions of Mercury, approaching nearer to a state of equality, the further you go. The first period, or that of one year, is not sufficiently exact ; the period of six years will sometimes bring on a return of the transit at the same node ; that of seven years more fre- quently ; that of 13 years still more frequently, and so on. Now there was a transit of Mercury at its de- scending node, in May, 17S6; hence, by continually adding 6, 7, 13, 33, 46, &c. to it you get all the years when a transit may be expected to happen at that node. In 1789, there was a transit at the ascending node, and therefore, by adding the same numbers to that year, you will get the years in which the transits may be expected to happen at that node. The next tran- sits at the descending node will happen in 1832, 1845, 1878, I891 ; and at the ascending node, in 1815, 1822, 1835, 1848, I861, 1868, 1881, I894. 238 ON THE TRANSITS OF MERCURV T^ T^ ^ y X p 224,7 ¥ov Jenus, p = 224,7', hence, ^ = -p=^^^-^ 8 235 713 o r.., ^ , =^"-r7 "TTTT-j -, ifcc. Inereiore the periods are 8, 13' 382' 1159 235, 713, &c. years. The transits at the same node will therefore, sometimes, return in eight years, but oftener in 235, and still oftener in 713, &c. Now, in 1769, a transit happened at the descending node in June, and the next transits at the same node will be in 2004, 2012, 2247, 2255, 249O, 2498, 2733, 2741, and 2984. In 1639, a transit happened at the ascend- ing node in November ; and the next transits at the same node will be in 18/4, 1882, 21 17, 2125, 236o, 2368, 2603, 2611, 2846, and 2854. These transits are found to happen, by continually adding the periods, and finding the years when they may be expected, and then computing, for each time, the shortest geocentric distance of Venus from the sun's center at the time of conjunction, and if it be less than the semidiameter of the sun, there will be a transit. A new Method of computing the Effect of Parallar, in accelerating or t^etarding the Time of the Be- ginning or End of a Transit of Venus or Mercury over the Sun's Disc. By Nevil Maskelyne, D. D. F. R. S. and Astronomer Royal. (331.) The scheme which is here given, relates particularly to the transit of J'^enu-s over the sun which happened in 1769. Let C represent the center of the sun LQ, P the celestial north pole of the equa- tor, PC a meridian passing through the sun, Z the zenith of the place, ADB ?S the relative path of Venus, ^ being the relative place of the descending node ; A the geocentric place of Venus at the ingress, B at the egress, and D at the nearest approach to the sun's center, as seen from the earth's center, and o the appa- rent place of Venus at the egress to an observer whose AND VENUS OV^ER THE SUn's DISC. 239 zenith is Z\ draw ouZ^ and u is the true place of Venus when the apparent place is at o, and>«^is the parallax in altitude of Venus from the suii; and the time of contact will be diminished by the time which Venus takes to describe u B ; draw 7i d honE parallel to A B, meeting Z B produced in E, and Bti, A ill, tangents to the circle, and let ChD be perpendicular to AB. Now the trapezium aoEB, on account of the small ness of it's sides, may be considered as rec- tilinear, and from the magnitude oi'ZB compared with Bu, BE may be considered as parallel to no, conse- quently no E B may be considered as a parallelogram, and therefore Eo may be taken equal to Bu. Now Eo = Enisino, according as E falls without or within iin 240 ON THE TRANSITS OF MERCURY the circle LQ of the sun's disc; and (Trig. Art. 128) En : EB :: sin. EBn = cos. CBZ : sm. BnE —sin, T>/-rrk ^I>T^ U r EBx COS.CBZ , BLD — Q.OS. CBD; hence, En = ^^m^ — ; ^"d ' cos. CBD (by Euclid) no = — , = -jj^ very nearly ; but Bn : BE :: sin. B En = s\n. Z B D : sin. BnE:^ cos. CBD; , ^ „ ^ BE'xsm.ZBD' , theretore Bn = ^. „ , ,> ; hence, 720 = COS. CBD' —TTi ' ^,r> i^n - Put h = horizontal parallax of AB X cos. CBD"^ ^ Venus from the sun ; then (136) BE = hx sin. -Zo = h X sin. Z B ', hence, 11 B = E = E ?i :h no = h X s'ln.Z B X COS. CBZ .j^ . „„j s'm. ZBD^ ^^^ (.jSjj Xfi xsm.Z/i X jji^^^^cBD = (Trig. Art. 80) // x sin. ZB x cos.CBZ x sec. CBD . /?^ xsin. Zl^^ X sin. ZBD^ x sec. CBD' ^, X ' rn • A he pa- A iJ rallax, therefore, consists of two parts ; one part varies as h, and the other as /i% the other quantities being the same. Put #" = the time which Venus takes, by it's geocentric relative motion, to describe the space // ; to find which, let ?« be the relative horary motion of ,r , 7 1 ^ „ .„ hxSGoo" Venus; then ?/i : h :: 1 hour = 3600 : t = . m Hence, to find the time of describing uB, we have, h : h X sin. ZB X cos. CBZ x sec. CB D ± h' X sin. ZB' X sin. ZBD' x sec. CBD' ^ ^ . — . r-n — ■: t : t x sm. AB ZB X cos. CBZx sec. CJ5/J + t xh X sm.ZB' X sin. ZBD' x sec. CBD' describing uB, or tlic effect of parallax in accelerating or retarding the time of contact; the upper sign is to be used when CBZ is acute, and the lower sign when it is obtuse. If CBZ be very nearly a right angle, but obtuse, it may happen that nE may be less than the time of AND VENUS OVER THE SUn's DISC. 241 no, in which case, nE is to be taken from no, according to the rule. The principal part 7iE of the effect of parallax will increase or diminish the planet's distance from the sun's center, according as the angle ZBC is acute or obtuse ; but the small part tio of parallax will always increase the planet's distance from the center; take, therefore^ the sum or difference of the effects, with the sign of the greater, as to increasing or de- creasing the planet's distance from the center of the sun. The second part of the correction will not ex- ceed 9" or lo" of time in the transits of Venus in 1761 and 1769, where the nearest apjiroach of Venus to the sun's center was about 10'. In the transit of 3Iercurij, the first part alone will be sufficient, except the nearest distance be much greater. If we suppose the 7nean horizontal parallax of the sun to be 8", 8 J, then, by calculation from * the above expression, it appears that the total durationat Wardhus was lengthened by parallax 1 1'. l6",88,and diminished at Otaheite by 12'. 10".07; hence, the computed dif- ference of the times is 23'. 26",95 3 but the observed difference was 23'. 10". (332.) Hence, the correct parallax may be accu- rately found as follows. Because the observed differ- ence of the total durations at Wardhus and Otaheite is 23'. lo", and the computed difference, from the assumed mean horizontal parallax of the sun 8",83, is 23'. 26",95, the true parallax of the sun is less than that assumed. Let the true parallax be to that as- sumed as 1 — e to 1, and (331) the first parts of the computed parallax will be lessened in the ratio of 1 — e : 1 ; and the second parts, in the ratio of 1 - ef to 1, or of 1 - 2e to 1 nearly. All the first parts, viz. 406",05 ; 287",05 ; 34l",48;382",47,inall = l417",05, combine the same way to make the total duration longer at Wardhus than at Otaheite. As to the * See my Co?>iplete Sj/stem of Astronomy, Chap. 25. 242 ON THE TRAJISITS OF MERCURY second parts, the effects at Wardhus are — 7">3l and -8",91, and at Otaheite are+l",63 and_-f4",4g, in all =--10", 10. Therefore 1417",05 xl—e- 10",10 X 1 - 2e= 1390" the excess of the observed total dura- tion at Wardhus abo ve that at Qtaheite ; or 1417",05 — 10",10 — 1390" = 1417",05-20"j20xe; and e = ^.„ =0,0121. Hence, the mean horizontal 1396",85 parallax of the sun^8",83x 1 -0,0121 =8",723l6; we assume, therefore, the mean horizontal parallax of the sun = 8" J. ♦ Hence, the radius of the earth : the distance of the sun from the earth ;: sin. 8"! : rad. :: 1 : 23575. (333.) The effect of the parallax being determined, the transit affords a very ready method of finding the difference of the longitudes of two places where the same observations were made. For, compute the effect of parallax in time, and reduce the observations at each place to the time, if seen from the center of the earth, and the difference of the times is the differ- ence of the longitudes. For example, the times at Wardhus and Otaheite, at which the first internal con- tact would take place at the earth's center^ are Qh. 40'. 44",6, and \2h. 38'. 25",07, the difference of which is 12h. 2'. 19",53 = 180°. 34'. 53", the difference of the meridians. From the mean of 63 results from the transits of Mercury, Mr. Short found the difference of the meridians of Greenwich and Paris to be 9'. 15"; and from the transit of Venus in 1761, to be 9'. lO" in time. (334.) The transit of Venus affords a very accurate method of finding the place of the node. For by the observations made by Mr. Rittenhouse^ at Norriton in the United States of America, the least distance CD was observed to be 10'. 10"; hence, drawing C^per- pendicular to C?5, cos.DCF=S°. 28'. 54" : rad. :: CD= 10'. 10" : CF= 10'. 17", the geocentric latitude AND VENUS OVER THE SUN's DISC. 243 of Venus at the time of conjunction; and * 0,72626 : 0,28895 :: 10'. 17" : 4'. 5", the hehocentric latitude CV of Venus; hence, considering C?s/^as a right Uned triangle, tan. ^^ C=3°. 23'. 35" : rad. :: the helio- centric latitude CF=4\ b" : C^ =1°. 8'. 52", which added to 2'. 13°. 26'. 34", the place of the sun, gives '2\ 14°. 35'. 26" for the place of the ascending node of the orbit of Venus. (335.) The time of the ecliptic conjunction may be thus found. Find, at any time (t), the difference (d) of longitudes of Venus and the sun's center; find also the apparent geocentric horary motion (jn) of Venus from the sun in longitude, and then say, m : d :: 1 hour : the interval between the time (t) and the con- junction, which interval is to be added to or subtracted from f, according as the observation was made before or after the conjunction. In the transit in 1761, at 6/1. 31'. 46", apparent time at Paris, M. de la Lands found d = 2'. o4:\4: and m = 3'. 57",4; hence, 3'. 57",4 : 2'. 34",4 :: 1 hour : 39'. l", which subtracted from 6h. 31'. 46", because at that time the conjunction was past, gives 5/?. 52'. 45" for the time of conjunction from this observation. We may also thus find the latitude at conjunction. The horary motion of Venus in lati- tude was 35",4; hence, 60' : 39',1 :: 35",4 : 23", the motion in latitude in 39',1, which subtracted from 10'. 1",2, the latitude observed at 6//. 31'. 46", gives 9'. 38",2 for the latitude at the time of conjunction. * 0,72626 is the distance of Venus from the sun, her distance from the earth being 0,28895 ; and the angle subtended by CF is inversely as the distance from CV. & 2 244 Chap. XXIII. ON THE NATURE AND MOTION OF COMETS. (336.) Comets are solid bodies, revolving in very ex- centric ellipses about the sun in one of the foci, and are therefore subject to the same laws as the planets, but differ in appearance from them ; for as they ap- proach the sun, a tail of light, in some of ibem, begins to appear, which increases till the comet comes to it's perihelion, and then it decreases again, and vanishes ; others have a light encompassing the nucleus, or body of the comet, without any tail. The most ancient philosophers supposed comets to be like planets, per- forming their revolutions in stated times. Aristotle^ in his first book of Meteors, speaking of comets, says, " But some of the Italians, called Pythagoreans, say, that a Comet is one of the Planets, but that they do not appear unless after a long time, and are seen but a small time, which happens also \.o Mercury T Seneca also, in Nat. Quest. Lib. vii. says, " Apollonius af- firmed, that the Comets were, by the Chaldeans, reckoned among the Planets, and had their periods like them." Seneca himself also, having considered the phaenomena of two remarkable comets, believed them to be stars of equal duration with the world, though he was ignorant of the laws that governed them ; and foretold, that after-ages would unfold all these mysteries. He recommended it to Astronomers to keep a catalogue of the comets, in order to be able to determine whether they returned at certain periods. Notwithstanding this, most Astronomers, from his time till Tycho Brake, considered them only as meteors, existing in our atmosphere. But that Astronomer, MOTION OF COMETS. 245 finding, from his own observations on a comet, that it had no diurnal parallax, placed them above the moon. Afterwards Kepler had an opportunity of observing two comets, one of which was very remarkable ; and from his observations, which aiforded sufficient indica- tions of an annual parallax, he concluded, " that comets moved freely through the planetary orbs, with a motion not much different from a rectilinear one; but of what kind he could not precisely determine." Hevelius embraced the hypothesis of^a rectilinear motion ; but, finding his calculations did not perfectly agree with his observations, he concluded, " that the path of a comet was bent in a curve line, concave to- wards the sun." He supposed a comet to be generated in the atmosphere of a planet, and to be discharged from it, partly by the rotation of the planet, and then to revolve about the sun in a parabola by the force of projection and it's tendency to the sun, in the same manner as a projectile upon the earth's surface describes a parabola. At length came the famous comet in 168O, which descending nearly in a right line towards the sun, arose again from it in like manner, which proved it's motion in a curve about the sun. G. S. Doerfell^ Minister at Plaven in Upper Saxony, made observations upon this comet, and found that it's mo- tion might be very well represented by a parabola, having the sun in it's focus. He was ignorant, how- ever, of all the laws by which the motion of a body in a parabola is regulated, and erred considerably in his parabola, making the perihelion distance about twelve times greater than it was. This was published five years before the Principia,\n which work Sir /. Newton having proved that Kepler's law, by which the motions of the planets are regulated, was a necessary conse- quence of his theory of gravity, it immediately foU lowed, that comets were governed by the same law ; and the observations upon them agreed so accurately with his theory, as to leave no doubt of it's truth. That comets describe ellipses^, and not parabolas oy 246 MOTIOxV OF COMETS. hyperbolas, Dr. Halley (see his Synopsis of the Astronomy of Comets) advances the following reasons. " Hitherto I have considered the orbits as exactly parabolic, upon which supposition it would follow, that comets, being impelled towards the sun by a cen- tripetal force, would descend as from spaces infinitely distant, and, by their so falhng, acquire such a ve- locity, as that they may again fly off into the remotest parts of the universe, moving upwards with a perpetual tendency, so as never to return again to the sun. But since they appear frequently enough, and since none of them can be found to move with an hyperbolic mo- tion, or a motion swifter than what a comet might acquire by it's gravity to the sun, it is highly probable they rather move in very excentric elliptic orbits, and make their returns after long periods of time ; for so their number will be determinate, and, perhaps, not so very great. Besides, the space between the sun and the fixed stars is so immense, that there is room enough for a comet to revolve, though the period of it's revolution be vastly long. Now, the latus rectum of an ellipsis is to the lotus rectum of a parabola, which has the same distance in it's perihelion, as the distance in the aphelion, in the eli psis, is to the whole axis of the ellipsis. And the velocities are in a sub- duplicate ratio of the same; wherefore, in very excen- tric orbits, the ratio comes very near to a ratio of equality ; and the very small difference which happens, on account of the greater velocity in the parabola, is easily compensated in determining the situation of the orbit. The principle use, therefore, of the Table of the elements of their motions, and that which indeed induced me to construct it, is, that whenever a new comet shall appear, we may be able to know, by com- paring together the elements, whether it be any of those which have appeared before, and consequently to determine it's period, and the axis of it's orbit, and to foretel it's return. And, indeed, there are many things which make m.e believe that the cornet^ which MOTION OF COMETS. 24/ Apian discovered in the year 1531, was the same with that which Kepler and Longomontcmus more accu- rately described in the year 1607 ; and which I myself have seen return, and observed in the year l682. All the elements agree, and nothing seems to contradict this my opinion, besides the inequality of the periodic revolutions; which inequality is not so great neither, as that it may not be owing to physical causes. For the motion of Saturn is so disturbed by the rest of the planets, especially Jupiter, that the periodic time of that planet is uncertain for some whole days together. How much more, therefore, will a comet be subject to such like errors, which rises almost four times higher than Saturn, and whose velocity, though increased but a very little, would be sutficient to change it's orbit, from an elliptic to a parabolical one. And I am the more confirmed in my opinion of it's being the same ; for in the year 1456,. in the summer-time, a comet was seen passing retrograde between the earth and the sun, much after the same manner ; which, though nobody made observations upon it, yet, from it's period, and the manner of it's transit, 1 cannot think difFei-ent from those I have just now mentioned. And since looking over the histories of comets, I find, at an equal interval of time, a comet to have been seen about Easter in the year 1305, which is another double period of 151 years before the former. Hence, I think, I may venture to foretel that it will return again in the year 1758." (337.) Dr. Halley computed the effect of Jupiter upon this comet in l682, and found that it would in- crease it's periodic time above a year, in consequence of which he predicted it's return at the end of the year 1758, or the beginning of 1759. He did not make his computations with the utmost accuracy, but, as he himself informs us, kvi calamo. M. Clairauf com- puted the eflfects both of Saturn and Jupiter, and found that the former would retard it's return in the last pe- riod 100 days, and the latter 511 days; and he deter- 248 MOTION OF COMETS. mined the time when the comet would come to it's perihelion to be on April 15, 1/^9^ observing that he might err a month, from neglecting small quantities in the computation. It passed the perihelion on March 13, within 33 days of the time computed. Now if we suppose the time stated by Dr. Hallei/ to mean the time of it's passing the perihelion, then if we add to that 100 days, arising from the action of Saturn, which he did not consider, it will bring it very near to the time in which it did pass the perihehon, and prove his computation of the effect of Jupiter to have been very accurate. If he meant the time when it would first appear, his prediction was very accurate, tor it was first seen on December 14, 17^8, and his computation of the effects of Jupiter will then be more accurate than could have been expected, considering that he made his calculations only by an indirect method, and in a manner confessedly not very accurate. Dr. Halley, therefore, had the glory, first to foretel the return ot a comet, and the event answered remarkably to his pre- diction. He further observed, that the action of Jupiter, in the descent of the comet towards it's perihelion in l682, would tend to increase the inclina- tion of it's orbit; and accordingly the inclinati(m in l682 was found to be 22' greater than in ib'O/. A learned Professor (Dr. Long's Astronomy, p. 562) in Italy to an English gentleman, writes thus : — *' Though M. de la Lande, and some other French gentlemen, have taken occasion to find fault with the inaccuracies of Halleys calculation, because he him- self had said, he only touched upon it slightly; never- theless they can never rob him of the honour, — First, of finding out that it was one and the same comet which appeared in l682, 1607, 1531, 1456, and 1305. — Secondly, of having observed, that the planet Jupiter would cause the inclination of the orbit of the comet to be greater, and the period longer. — Thirdly, of hav- ing foretold that the return thereof might be retarded till the end of 1/58, or the beginning of 1759." MOTION OF COMETS. 249 From the observations of M, Messier upon a comet in 1 770, Mr. Edric Prosperin, Member of the Royal Academies of Stockhohn and Upsal, showed, that a parabolic orbit would not answer to it's motions, and he recommended it to Astronomers to seek for the eUiptic orbit. This laborious task M. Lexell under- took, and has shown that an ellipse, in which the periodic time is about five years and seven months, as^rees very well with the observations. See the Phil. Trans. 17 79. As the ellipses which the comets de- scribe are very excentric, Astronomers, for the ease of calculation, suppose them to move in parabolic orbits, for that part which lies within the reach of observa- tion, by which they can very accurately find the place of the perihelion ; it's distance from the sun; the in- chnation of the plane of it's orbit to the ecliptic, and the place of the node. But it fails not within the plan of this work to enter into an investigation of these matters. For this, I refer the reader to my Complete System of Astronomy. (338.) It is extremely difficuk to determine, from computation, the elliptic orbit of a comet, to any degree of accuracy ; for when the orbit is very excen- tric, a very small error in the observation will change the computed orbit into a parabola, or hyperbola. Now from the thickness and inequality of the atmo- sphere with which the comet is surrounded, it is im- possible to determine with any great precision, when either the limb or center of the comet passes the wire at the time of observation. And this uncertainty in the observations will subject the computed orbit to a great t-rror. Hence, it happened, that M. Bouguer determined the orbit of the comet to be an liyperbola. M. Elder first determined the same for the comet in 17/4; but, having received more accurate observa- tions, he found it to be an ellipse. The period of the comet in l680 appears, from observation, to be 575 years, which Mr. Eider, by his computalion, deter- mined to be l66'| years. The only safe way to get 250 MOTION OF COMET?. the periods of comets, is to compare the elements of all those which have been computed, and where you find they agree very well, you may conclude that they are elements of the same comet, it being so extremely improbable that the orbits of two different comets should have the same inclination, the same perihelion distance, and the places of the perihelion and node the same. Thus, knowing the periodic time, we get the major axis of the ellipse ; and the perihelion distance being known, the minor axis will be known. When the elements of the orbits agree, the comets may be the same, although the periodic times should vary a little ; as that may arise from the attraction of the bodies in our system, and which may also alter all the other elements a little. We have already observed, that the comet which appeared in 1759, had it's periodic time increased considerably by the attraction oi Jupiter ^nA Saturn. This comet was seen in l682, 1607, and 1531, all the elements agreeing, except a little variation of the periodic time. Th. Halley sus- pected the comet in- 168O, to have been the same which appeared in II06, 531, and 44 years before Christ. He also conjectured, that the comet observed by /Jpian, in 1532, was the same as that observed by Hevelius, in 1661 ; if so, it ought to have returned in ^79^i but it has never been observed. But M. Mechain having collected all the observations in 1532, and calculated the orbit again, found it to be sensibly different from that determined by Dr. Halley^ w^hich renders it very doubtful whether this was the comet which appeared in 1661 ; and this doubt is increased, by it's not appearing in 1790- The comet in 177^. whose periodic time M. Lexell computed to be five years and seven months, has not been observed since. There can be no doubt but that the path of this comet, for the time it was observed, belonged to an orbit whose periodic time was that found by M. Lexell, as the computations for such an orbit agreed so very well with the observations. But the revolution was proba- NATURE AND TAILS OF COMETS. 251 biy longer before 1770; for as the comet passed very- near to Jupiter in 17^7? it's periodic time might be sensibly increased by the action of that planet ; and as it has not been observed since, we may conjecture, with M. Lexell, that having passed in 1772 again into the sphere of sensible attraction of Jupiter, a new disturbing force might probably take place and destroy the effect of the other. According to the above ele- ments, the comet would be in conjunction with Jupiter on August 23, 1779> and it's distance from Jupiter would be only-^i-g- of it's distance from the sun ; conse- quently the sun's action would be oniy-^J-^ part of that of Jupiter. What a change must this make in the orbit ! If the comet returned to it's perihelion in March, 177^) it would then not be visible. See M. Lexell's account in the Phil. Tram. 1779. The elementsof the orbits of the comets, in 1264 and 1556, were so nearly the same, that it is very probable it was the same comet; if so, it ought to appear again about the year 1848. On the Nature and Tails of Comets. (339.) Comets are not visible till they come into the planetary regions. They are surrounded with a very dense atmosphere, and from the side opposite to the sun they send forth a tail, which increases as the comet approaches it's perihelion, immediately after which it is longest and most luminous, and then it is generally a little bent and convex towards those parts to which the comet is moving; the tail then decreases, and at last it vanishes. Sometimes the tail is observed to put on this figure ^ towards it!s extremity, as that did in 1796- The smallest stars arc seen through the tail, notwithstanding it's immense thickness, which proves that it's matter rnust be extremely rare. The opinion of the ancient philosophers, and oi Aristotle himself, was, that the tail is a very thin fiery vapour 252 NATURE AND TAILS OF COMETS. arising from the comet. Apian, Cardan, Tycho, and others, believed that the sun's rays, being propagated through the transparent head of the comet, were re- fracted, as in a lens. But the figure of the tail does not answer to this ; and, moreover, there should be some reflecting substance to render the rays visible, in like manner as there must be dust or smoke flving about in a dark room, in order that a ray of light entering, it may be seen by a spectator standing side- ways from it. Kepler supposed, that the rays of the sun carry away some of the gross parts of the comet which reflect the sun's rays, and give the appearance of a tail. Hevel'ms thought that the thinnest parts of the atmosphere of a comet are rarified by the force of the heat, and driven from the fore part and each side of the comet towards the parts turned from the sun. Sir /. Newton thinks, that the tail of a comet is a very thin vapour, which the head, or nucleus of the comet, sends out by reason of it's heat. He supposes, that when a comet is descending to it's perihelion, the vapours behind the comet, in respect to the sun, being rarified by the sun's heat, ascend, and take up with them the reflecting particles with which the tail is composed, as air rarefied by heat carries up the parti- cles of smoke' in a chimney. But as, beyond the atmosphere of the comet, the aetherial air [aura cetherea) is extremely rare, he attributes something to the sun's rays carrying with them the particles of the atmosphere of the comet. And when the tail is thus formed, it, like the nucleus, gravitates towards the sun, and by the projectile force received from the comet, it describes an ellipse about the sun, and ac- companies the comet. It conduces also to the ascent of these vapour,*, that they revolve about the sun, and therefore endeavour to recede from it; whilst the at- mosphere of the sun is either at rest, or moves with su(^h a slow motion as it can acquire from the rotation of the sun about it's axis. These are the causes of the ascent of the tails in the neighbourhood of the sun. NATURE AND TAILS OF COMETS. 253 where the orbit has a greater curvature, and the comet moves in a denser atmosphere of the sun. The tail of the comet, therefore, being formed from the heat of the sun, will increase till it comes to it's perihelion, and decrease afterwards. The atmosphere of the comet is diminished as the tail increases, and is least immedi- ately after the comet has passed it's perihelion, where it sometimes appears covered with a thick black smoke. As the vapour receives two motions when it leaves the comet, it goes on with the compound motion, and therefore the tail will not be turned directly from the sun, but decline from it towards those parts which are left by the comet ; and meeting with a small resistance from the aether, will be a little curved. When the spectator, therefore, is in the plane of the comet's orbit, the curvature will not appear. The vapour, thus rarefied and dilated, may be at last scattered through the heavens, and be gathered up by the planets, to supply the place of those fluids which are spent in vegetation and converted into earth. This is the substance of Sir /. Newton's account of the tails of comets. Against this opinion, Dr. Hamilton, in his Philosophical Essays, observes, that we have no proof of the existence of a solar atmosphere ; and if we had, that when the comet is moving in it's perihelion in a direction at right angles to the direction of it's tail, the vapours which then arise, partaking of the great velocity of the comet, and being also specifically lighter than the medium in which they move, must suflfer a much greater resistance than the dense body of the comet does, and therefore ought to be left behind, and would not appear opposite to the sun ; and afterwards they ought to appear towards the sun. Also, if the splen- dour of the tails be owing to the reflection and refrac- tion of the sun's rays, it ought to diminish the lustre of the stars seen through it, which would have their light reflected and refracted in like manner, and conse- quently their brightness would be diminished. Dr. Halleyy in his description of the Aurora Borealis in 254 NATURE AND TAILS OF COMETS. 1716', says, " the streams of light so much resembled the long tails of comets, that at first sight they might well be taken for such." And afterwards, " this light seems to have a great affinity to that which the effluvia of electric bodies emit in the dark." Phil. Trans. N". 347. D. de Mairan also calls the tail of a comet, the aiiroi'a borealis of the comet. This opinion Dr, Hamilton supports by the following arguments. A spectator, at a distance from the earth, would see the aurora borealis in the form of a tail opposite to the sun, as the tail of a comet lies. The aurora borealis has no effect upon the stars seen through it, nor has the tail of a comet. The atmosphere is known to abound with electric matter, and the appearance of the electric matter in vacuo is exactly like the appearance of the aurora borealis, which, from it's great altitude, may be considered to be in as perfect a vacuum as we can make. The electric matter in vacuo suffers the rays of light to pass through, without being affected by them. The tail of a comet does not expand itself sideways, nor does the electric matter. Hence, he supposes the tails of comets, the aurora borealis, and the electric fluid, to be matter of the same kind. We may add, as a further confirmation of this opinion, that the comet in 1607 appeared to shoot out at the end of it's tail. Le P. Cysat remarked the undula- tions of the tail of the comet in 16I8. Hevelms ob- served the same in the tails of the comets in l652 and 1661. M. Pingre took notice of the same appear- ance in the comet of 1769' These are circumstances exactly similar to the aurora borealis. Dr. Hamilton conjectures, that the use of the comets may be to bring the electric matter, which continually escapes from the planets, back into the planetary regions. The arguments are certainly strongly in favour of this hypothesis ; and if this be true, we may further add, that the tails are hollow ; for if the electric fluid only proceed in it's first direction, and do not diverge side- ways, the parts directly behind the comet will not be NATURE AND TAILS OF COMETS. 255 filled with it; and this thinness of the tails will ac- count for the appearance of the stars through them. (340.) In respect to the nature of comets, Sir /. Newton observes, that they must be solid bodies like the planets; for if they were nothing but vapours, they must be dissipated when they come near the sun; for the comet in 168O, when it was in it's perihelion, was less distant from the sun than one-sixth of the sun's diameter, consequently the heat of the comet at that time was to the heat of the summer sun as 28000 to 1. But the heat of boiling water is about three times greater than the heat which dry earth acquires from the summer's sun ; and the heat of red-hot iron about three or four times greater than the heat of boiling water. Therefore the heat of dry earth at the comet, when in it's perihelion, was about 2000 times greater than red-hot iron. By such heat, all vapours would be immediately dissipated. (341.) This heat of the comet must be retained a very long time. For a red-hot globe of iron, of an inch diameter, exposed to the o})en air, scarce loses all it's heat in an hour; but a greater globe would retain it's heat longer, in proportion to it's diameter, because the surface, at which it grows cold, varies in that pro- portion less than the quantity of hot matter. There- fore a globe of red-hot iron, as large as our earth, would scarcely cool in 50000 years. (342.) The comet in 168O, coming so near to the sun, must have been considerably retarded by the sun's atmosphere, and therefore, being attracted nearer at every revolution, it will at last fall into the sun, and be a fresh supply of fuel for what the sun loses by it's constant emission of light. In like manner, fixed stars which have been gradually wasted, may be supplied with fresh fuel, and acquire new splendour, and pass for new stars. Of this kind are those fixed stars which appear on a sudden, and shine with a wonderful bright- ness at first, and afterwards vanish by degrees. Such is the conjecture of Sir /. Newton. 256 NUMBER OF COMETS. (343.) From the beginning of our sera to this timcj it is probable, according to the best accounts, that there have appeared about 500 comets. Before that time, about 100 others are recorded to have been seen, but it is probable that not above half of them were comets. And when we consider, that many others may not have been perceived, from being too near the sun — from appearing in moon-light — from being in the other hemisphere — from being too small to be per- ceived, or which may not have been recorded, we might imagine the whole number to be considerably greater; but it is likely, that of the comets which are recorded to have been seen, the same may have ap- peared several times, and therefore the number may be less than is here stated. The comet in 17^6, which first appeared on August 1, was discovered by Miss Caroline Herschel, a sister of Dr. Herschel ; since that time, she has discovered three others. As the plan of this work does not permit us to give the methods by which the orbits of comets may be com- puted, and all the opinions respecting them, if the reader wish to see any thing further on the subject, I refer him to my Complete System of Astronomy ; or to a Treatise, entitled Cometographie, oii Traite His- torhjne et Theorique des Cometes, par M. Pingre', //. Tom. quarto. Paris, 1784 ; or Sir H. Engle- field's Determination of the Orbits of Comets, a very valuable work, in which the ingenious Author has explained, with great clearness and accuracy, the manner of computing the orbits of comets, according to the methods of Boscovich andM. de la Place. 257 Chap. XXIV, ON THE FIXED STARS. (344.) All the heavenly bodies beyond our system, are called Fixed Stars, because (except some few) tliey do not appear to have any proper motion of their own. From their immense distance, they must be bodies of very great magnitude, otherwise they could not be visible ; and when we consider the weakness of re- flected light, there can be no doubt but that the}^ shine witli their own light. They are easily known from the planets, by their twinkling. The number of stars visible at once to the naked eye is about 1000 ; but Dr. Herschel, by his improvements of the reflect- ing telescope, has discovered that the whole number is great, beyond all conception. In that bright tract of the heavens, called the Milky IVay^ which, when ex- amined by good telescopes, appears to be an immense collection of stars which gives that whitish appearance to the naked eye, he has, in a quarter of an hour, seen 116000 stars pass through the field of view of a tele- scope of only 15' aperture. Every improvement of his telescopes has discovered stars not seen before, so that there appears to be no bounds to their number, or to the extent of the universe. These stars, which can be of no use to us, are probably suns to other systems of planets. (345.) From an attentive examination of the stars with good telescopes, many, which appear only single to the naked eye, are found to consist of two, three, or more stars. Dr. Maskelyne had observed a Herculis to be a double star : Dr. Hormhy had found tt Bootis R 258 ON THE FIXED STARS. to be double ; M. Cassini, Mr. Mayer, Mr. Plgott, and many other Astronomers, had made discoveries of the hke kind. But Dr. Herschel, by his improved telescopes, has found about 700, of which, not above 42 had been observed before. We shall here give an account of a few of the most remarkable. a Herculis, Flam. 64, a beautiful double star ; the two stars very unequal, the largest is red, and the smallest blue, inclining to green. h Lyrce, Flam. 12, double, very unequal, the largest red, and smallest dusky ; not easily to be seen with a magnifying power of 227. a Geminorum, Flam. 6Q, double, a little unequal, both white; with a power of 146 ; their distance ap- pears equal to the diameter of the smallest. 6 Lyrce, Flam. 4 and 5, a double-double star; at first sight it appears double at a considerable distance, and, by a little attention, each will appear double ; one set are equal, and both white ; the other unequal, the largest white, and the smallest inclined to red. The interval of the stars, of the unequal set, is one diameter of the largest, with a power of 227. 7 Andromedce, Flam. 57, double, very unequal, the largest reddish white, the smallest a fine bright sky-blue, inclining to green. A very beautiful object. a Ursce minoris. Flam. 1, double, very unequal, the largest white, the smallest red. /3 Lyrce, Flam. 10, quadruple, unequal, white, but three of them a little inclined to red. a Leonis, Flam. 32, double, very unequal, largest white, smallest dusky. e Bootis, Flam. 36, double, very unequal, largest reddish, smallest blue, or rather a faint lilac ; very beautiful. h Draconh, Flam. 39, a very small double star, very unequal, the largest white, smallest inclining to red. X Orionis, Flam. 39, quadruple, or rather a double star, and has two more at a small distance, the double ON THE FIXED STARS. 259 star considerably unequal, the largest white, smallest pale rose colour. ^ Librae y Flam, ultima, double-double, one set very unequal, the largest a very fine white. (U Cygni, Flam. 78, double, considerably unequal, the largest white, the smallest blueish. fjL Herculis, Flam. 86, double, very unequal ; the small star is not visible with a power of 2/8, but is seen very well with one of 46o ; the largest is inclined to a pale red, smallest duskish, a Capricorni, Flam. 5, double, very unequal, the largest white, smallest dusky. V Lyroe, Flam. 8, treble, very unequal, the largest white, smallest both dusky. a Lyroe, Flam. 3, double, very unequal, the largest a fine brilliant white, the smallest dusky ; it appears with a power of 227. Dr. Herschel measured the diameter of this fine star, and found it to be 0",3553. (346.) These are a few of the principal double, &o. stars mentioned by Dr. Herschel, in his catalogues, which he has given us in the Phil. Titans. 1782 and 178.5. The examination of double stars with a tele- scope, is a very excellent and ready method of proving it's powers. Dr. Herschel recommends the following method. The telescope and the observer having been some time in the open air, adjust the focus of the tele- scope to some single star of nearly the same magni- tude, altitude, and colour of the star to be examined; attend to all the phaenomena of the adjusting star as it passes through the field of view, whether it be per- fectly round and well defined, or affected with little appendages playing about the edge, or any other cir- cumstances of the like kind. Such deceptions may be detected by turning the object glass a little in it's cell, when these appendages will turn the same way. Thus you will detect the imperfections of the instrument, and therefore will not be deceived when you come to examine the double star. (347.) Several stars, mentioned by ancient Astrono- R 2 260 QN THE FIXED STARS, mers, are not now to be found, and several are now observed, which do not appear in their catalogues. The most ancient observation of a new star, is that by Hipparchus, about 120 years before J. C. which oc- casioned his making a catalogue of the fixed stars, in order that future Astronomers might see what altera- tions had taken place since his time. We have no account where this new star appeared. A new star is also said to have appeared in the year 130; another in 389; another in the ninth century, in 15° of Scorpio ; a fifth in 945 ; and a sixth in 1264 ; but the accounts we have of all these are very imperfect. (348.) The first new star we have any accurate ac- count of, is that which was discovered by Cornelius Gemma, on November 8, 1572, in the Chair of Cassiopea. It exceeded Sirius in brightness, and was seen at mid-day. It first appeared bigger than Jupiter f but it gradually decayed, and after sixteen months it entirely disappeared. It was observed by Ti/cho Brake, who found that it had no sensible parallax ; and he concluded that it was a fixed star. Some have supposed that this is the same which ap- peared in 945 and 1264, the situation of it's place favouring this opinion. (349.) On August 13, 1596, David Fahricius ob- served a new star in the Neck of the Whale, in 25°. 45' of Aries, with 15°. 54' south latitude. It disap- peared after October in the same year. Phocy Hides Holwarda discovered it again in 1637, not knowing that it had ever been seen before ; and after having disappeared for nine months, he saw it come into view again. Bullialdus determined the periodic time be- tween it's greatest brightness to be 333 days. It's greatest brightness is that of a star of the second magnitude, and it's least, that of a star of the sixth. It's greatest degree of brightness, however, is not always the same, nor are the same phases always at the same interval. (350.) In the year 160O; IVilliam, Janseniits dis- ON THE FIXED STARS. 26l covered a changeable star in the Neck of the Swan. It was seen by Kepler, who wrote a Treatise upon it, and determined it's place to be 16". 18'x:c^, with 55°. 30' or 32' north latitude. Ricciolus saw it in l6l6, 1621, 1624, and 1629. He is positive that it was invisible in the last years from l640 to l650. M. Cassini saw it again in l655 ; it increased till 1660, and then grew less, and at the end of 1 66 1, it disap- peared. In November 1665, it appeared again, and disappeared in 168I. In 1715 it appeared of the sixth magnitude, as it does at present, (351.) On June 20, 1670, another changeable star was discovered near the Swans Head, by P. Anthelme. It disappeared in October, and was seen again on March 17, 1671. On September 11, it disappeared. It appeared again in March 1672, and disappeared in the same month, and has never since been seen. It's longitude was r. 52'. 26" of ^ , and it's latitude 47". 25'. 22"N. The days are here put down for the new style. (352.) In 1686, Kirxhius observed ^ in the Swan to be a changeable star; and, from twenty years ob- servations, the period of the return of the same phases was found to be 405 days ; the variations of it's mag- nitude, however, were subject to some irregularity. (353.) In the year l604, at the beginning of October, Kepler discovered a new star near the heel of the right foot of Serpentarius, so very brilliant, that it exceeded every fixed star, and even Jupiter, in mag- nitifde. It was observed to be every moment changino- into some of the colours of the rainbow, except when it was near the horizon, when it was generally white. It gradually diminished, and disappeared about October 1605, when it came too near the sun to be visible, and was never seen after. It's longitude was 17°. 40'of ^ , with l"". 56' north latitude, and was found to have no parallax. (354.) Montanari discovered two stars in the con- stellation of the Ship J marked /3 and 7 by Bayer^ to b<5 262 ON THE FIXED STARS. wanting. He saw them in l664, but lost them in 1668. The star 6 in the tail of the Serpent, reckoned by Tycho of the third, was found, by him, of the fifth magnitude. The star |0 in Serpe.ntarius did not appear, from the time it was observed by him, till 1695. The star \// in the Lion, after disappearing, was seen by him in 1667. He observed also, that /3 in Medusas Head varied in it's magnitude. (355.) M. Cassini discovered owe new star of the fourth, and two of the fifth magnitude in Cassiopea; 2\&oJive new stars in the same constellation, of which three have disappeared ; two new ones in the beginning of the constellation Eridanus, of the fourth and fifth magnitude ; and four new ones of the fifth or sixth magnitude, near the north pole. He further observed, that the star, placed by Bayer near e of the Little Bear, is no longer visible ; that the star A of Andi^o- meda, which had disappeared, had come into view again in 1695 ; that in the same constellation, instead of one in the Knee^ marked v, there are two others come more northerly; and that ^ is diminished ; that the star placed by Tycho, at the end of the Chain of Andromeda, as of the fourth magnitude, could then scarcely be seen ; and that the star which, in Tycho's catalogue, is the twentieth of Pisces, was no longer visible. (356.) M. Maraldi observed, that the star k in the left leg of Sagittarius, marked by Bayer of the third magnitude, appeared of the sixth, in 1671 ; in 1676 it was found, by Dr. Halley, to be of the third; in 1693 it could hardly be perceived, but in 1693 and 1694 it was of the fourth magnitude. In 1704 he discovered a star in Hydra to be periodical; it's posi- tion is in a right line with those in the tail marked TT and 7. The time between it's greatest lustre, which is of the fourth magnitude, was about two years ; in the intermediate time it disappeared. In 1666, Hevelius says, he could not find a star of the fourth magnitude in the eastern scale of Libra, observed by ON THE FIXED STARS. 263 Tycho and Bayer; but Maraldi, in 1709, says, that it had then been seen for 15 years, smaller than one of the fourth. (357".) t/. Goodrlche, Esq. has determined the pe- riodic variation of Algol, or /3 Persei (observed by Montanari to be variable) to be about Q,d. 21 h. It's greatest brightness is of the second magnitude, and least of the fourth. It changes from the second to the fourth in about three hours and a half, and back again in the same time, and retains it's greatest bright- ness for the other part of the time. (358.) Mr. Gooflfr/c/ire also discovered, that j3 Lyras was subject to a periodic variation. The following is the result of his observations. It completes all it's phases in 12 days I9 hours, during vvhicli time, it undergoes the following changes: — 1. It is of the third magnitude for about two days. 2. It diminishes in about ].\ days. 3. It is between the fourth and fifth magnitude for less than a day. 4. It increases in about two days. 5. It is of the third magnitude for about three days. 6. It diminishes in about one day. 7- It is something larger than the fourth mag- nitude for a little less than a day. 8. It increases in about one day and three quarters to the first point, and so completes a whole period. See the Phil. Trans. 1785. He has also found, that ^ Cephei is subject to a periodic variation of 5d. 8/z. 37'j; during which time it undergoes the following changes : I. It is at it's greatest brightness about one day thirteen hours. 2. It's diminution is performed in about one day eighteen hours. 3. It is at it's greatest obscuration about one day twelve hours. 4. It increases in about thirteen hours. It's greatest and least brightness is that between the third and fourth, and between the fourth and fifth magnitudes. (359.) E. Pigott, Esq. has discovered ?/ Antinoi to be a variable star, with a period of 'jd. 4h. 38'. The changes happen as follows : 1. It is at it's greatest brightness 44 + hours. 2. It decreases 62 + hours. 264 ON THE FIXED STARS. 3. It is at it's least brightness 30 ± hours. 4. It increases 36 + hours. When most bright, it is of the third or fourth magnitude, and when least, of the fourth or fifth. See the Phil. Trans. 1785. (36o.) In the Phil. Tram. 1796, Dr. Herschel has proposed a method of observing the changes that may happen to the fixed stars j with a catalogue of their comparative brightness, in order to ascertain the per- manency of their lustre. (361.) Dr. Herschel, in a Paper of the Phil. Timns. 1783, upon the proper motion of the solar system, has given a large collection of stars which were formerly seen, but are now lost; also a catalogue of variable stars, and of new stars ; and very justly observes, that it is not easy to prove that a star was never seen before ; for though it should not be contained in any catalogue whatever, yet the argument for it's former non- appearance, which is taken from it's not having been observed before, is only so far to be regarded, as it can be made probable, or almost certain, that a star would have been observed, had it been visible. (362.) There have been various conjectures to ac- count for the appearances of the changeable stars. M. Maupertuis supposes, that they may have so quick a motion about their axes, that the centrifugal force may reduce them to flat oblate spheroids, not much unlike a mill-stone; and it's plane may be inclined to the plane of the orbits of it's planets, by whose attrac- tion the position of the body may be altered, so that when it's plane passes through the earth, it may be almost or entirely invisible, and then become again visible as it's broadside is turned towards us. Others have conjectured, that considerable parts of their sur- faces are covered with dark spots, so that when, by the rotation of the star, these spots are presented to us, the stars become almost or entirely invisible. Others have supposed, that these stars have very large opaque bodies revolving about and very near to them, so as to obscure them when they come in conjunction with us. ON THE FIXED STARS. 2()5 The irregularity of the phases of some of them shows the cause to be variable, and therefore may, perhaps, be best accounted for by supposing that a great part of the body of the star is covered with spots, wliich appearand disappear hke those on the sun's surfoce. The total disappearance of a star may probably be the destruction of it's system ; and the appearance of a new star, the creation of a new system of planets. (363.) The fixed stars are not all evenly spread through the heavens, but the greater part of them arc collected into clusters, of which it requires a large magnifying power, with a great quantity of light, to be able to distinguish the stars separately. With a small magnifying power and quantity of light, they only appear small whitish spots, something like a small light cloud, and thence they were called Nehulce. There are some nehulce, however, which do not receive their light from stars. For in the year 1 656, Huygens discovered a nebula in the middle of Orion^s Sword; it contains only seven stars, and the other part is a bright spot upon a dark ground^ and appears like an opening into brighter regions beyond. In l6l2, Simon Marius discovered a nebula in the Girdle of Andromeda. Dr. Halleif, when he was observing the southern stars, discovered one in the Centaur, but this is never visible in England. In 1714, he found another in Hercules, nearly in a line with ^' and j; of Bayer. This shows itself to the naked eye, when the sky is clear and the moon absent. M. Cassini dis- covered one between the Great Dog and the Ship, which he describes as very full of stars, and very beautiful, when viewed with a good telescope. There are two whitish sjjots near the south pole, called, by the sailors, the Magellanic Clouds, which, to the naked eye, resemble the milky way, but, through telescopes, they appear to be composed of stars. M. de la Caille in his catalogue of fixed stars observed at the Cape of Good Hope, has remarked 42 nebulae which he observed, and which he divided into three 266 ON THE FIXED STARS. classes; 14, in which he could not discover the stars; 14, in which he could see a distinct mass of stars ; and 14, in which the stars appeared of the sixth magnitude, or below, accompanied with white spots, and nebulae of the first and third kind. In the Connolssance des Temps, for 1783 and 17^4, there is a catalogue of 103 tiebulae, observed by Messier and Mechain, some of which they could resolve, and others they could not. But Dr. Herschel has given us a catalogue of 2000 nebulae and clusters of stars, which he himself has discovered. Some of them form a round, compact system; others are more irregular, of various forms ; and some are long and narrow. The globular systems of stars appear thicker in the middle than they would do if the stars were all at equal distances from each other; they are, therefore, condensed towards the center. That the stars should be thus accidentally disposed, is too improbable a supposition to be ad- mitted; he supposes, therefore, that they are thus brought together by their mutual attractions, and that the gradual condensation towards the center is a proof of a central power of that kind. He further observes, that there are some ad(^itional circumstances in the appearance of extended clusters and nebulae, that very much favour the idea of a power lodged in the brightest part. For although the form of them be not globular, it is plainly to be seen that there is a tendency towards sphericity, by the swell of the dimensions as they draw near the most luminous place, denoting, as it were, a course, or tide of stars, setting towards a center. As the stars in the same nebula must be very nearly all at the same relative distance from us, and they appear nearly of the same size, their real magnitudes must be nearly equal. Granting, therefore, that these nebulae and clusters of stars are formed by their mutual attrac- tion. Dr. Herschel concludes that we may judge of their relative age by the disposition of their component parts, those b> ing the oldest which are most com- pressed. He supposes the milky way to be a nebula, ON THE FIXED STARS. 2d7 of which our sun is one of it's component parts. See the Phil Trans. 1786 and 1789. (364.) Dr. Herschel has discovered other phae- nomena in the heavens, which he calls Nebulous Stars; that is, stars surrounded with a faint luminous atmosphere, of a considerable extent. Cloudy or ne- bulous stars, he observes, have been mentioned by several Astronomers j but this name ought not to be applied to the objects which they have pointed out as such ; for, on examination, they prove to be either clusters of stars, or such appearances as may reason- ably be supposed to be occasioned by a multitude of stars at a vast distance. He has given an account of seventeen of these stars, one of which he has thus de- scribed. " November 13, 1790- A most singular phaenomenon ; a star of the eighth magnitude, with a faint luminous atmosphere, of a circular form, and of about 3' diameter. The star is perfectly in the center, and the atmosphere is so diluted, faint, and equal throughout, that there can be no surmise of it's con- sisting of stars ; nor can there be a doubt of the evident connexion between the atmosphere and the star. Another star not much less in brightness, and in the same field of view with the above, was perfectly free from any such appearance." Hence, he draws the following consequences. Granting the connexion be- tween the star and the surrounding nebulosity, if it consist of stars very remote which give the nebulous appearance, the central star, which is visible, must be immensely greater than the rest ; or if the central star be not larger than common, how extremely small and compressed must be those other luminous points which occasion the nebulosity ! As, by the former supposi- tion, the luminous central point must far exceed the standard of what we call a star, so, in the latter, the shining matter about the center will be much too small to come under the same denomination ; we therefore either have a central body which is not a star, or a star which is involved in a shining fluid, of a nature totally 268 ON THE FIXED STARS. unknown to us. This last opinion Dr. Herschel adopts. The existence of tliis shining matter, he says, does not seem to be so essentially connected with the central points, that it might not exist without them. The great resemblance there is between the chevelure of these stars, and the diffused nebulosity there is about the constellation Orion, which takes up a space of more than 6o square degrees, renders it highly proba- ble that they are of the same nature. If this be ad- mitted, the separate existence of the luminous matter is fully proved. Light reflected from the star could not be seen at this distance. And, besides, the out- ward parts are nearly as bright as those near the star. In further confirmation of this, he observes, that a cluster of stars will not so completely account for the milkiness, or soft tint of the light of these nebulae, as a self-luminous fluid. This luminous matter seems more fit to produce a star by it's condensation, than to depend on the star for it's existence. There is a tele- scopic milky way extending in right ascension from bh. 15'. 8" to 5^.3.9'. \" , and in polar distance from 87**. 46' to 98°. 10'. This, Dr. Herschel thinks, is better accounted for, by a luminous matter, than from a collection of stars. He observes, that perhaps some may account for these nebulous stars, by supposing that the nebulosity may be formed by a collection of stars at an immense distance, and that the central star may be a near star, accidently so placed ; the ap- pearance, however, of the luminous part does not, in his opinion, at all favour the supposition that it is pro- duced by a great number of stars; on the other hand, it must be granted that it is extremely difficult to admit the other supposition, when w^e know that nothing but a solid body is self-luminous, or, at least, that a fixed luminary must immediately depend upon such, as the flame of a candle upon the candle itself See Dr. HerscheVs Account, in the Phil Trans. 1791. ON THE CONSTELLATIONS. 26*9 On the Constellations. {SQb.) As soon as Astronomy began to be studied, it must have been found necessary to divide the heavens into separate parts, and to give some rejjre- sentations to them, in order that Astronomers might describe and speak of the stars, so as to be understood. Accordingly we find that these circumstances took place very early. The ancients divided the heavens into Constellations, or collections of stars, and repre- sented them by animals, and other figures, according to the ideas which the dispositions of the stars sug- gested. We find some of them mentioned by Job ; and although it has been disputed, u hether our trans- lation has sometimes given the true meaning to the Hebrew words, yet it is agreed, that they signify con- stellations. Some of them are mentioned by Homer and Hesiod, but Aratus professedly treats of all the ancient ones, except three which were invented after his time. The number of the ancient constellations was 48, but the present nuniber upon a globe is abmit 70 ; by rectifying which, and setting it to correspond with the stars in the heavens, you may, by comparing them, very easily get a knowledge of the different con- stellations and stars. Those stars which do not come into any of the constellations, are called Unformed Stars. The stars visible to the naked eye are divided into six classes, according to their magnitudes ; the largest are called of the first magnitude, the next of the second, and so on. Those which cannot be seen without telescopes, are called Telescopic Stars. The stars are now generally marked upon maps and globes with Bayer's letters ; the 1st letter in the Greek alphabet being put to the greatest star of each con- stellation ; the 2d letter to the next greatest, and so on ; and when any more letters are wanted, the Italic ifO ON THE CONSTELLATIONS. letters are generally used; this serves as a name to the star, by which it may be pointed out. Twelve of these constellations lie upon the ecliptic, including a space about 15'' broad, called the Zodiac, within which all the planets move. The constellation Aries, or the Ra?7i, about 2000 years since, lay in the^r,s^ sign of the ecliptic ; but, on account of the precession of the equinox, it now lies in the second. The follow- ing are the names of the constellations, and the number of the stars observed in them by different Astronomers. Antinoiis was made out of the unformed stars near Aquila; and Coma Berenices out of the unformed stars near the Lions Tail. They are both mentioned by Ptolemy, but as unformed stars. The constellations as far as the Triangle, with Coma Berenices, are northern ; those after Pisces, are southern. THE ANCIENT CONSTELLATIONS. c as e5^ ^ ^ ^ ^ Ursa Minor The Little Bear 8 7 12 24 Ursa Major The Great Bear 35 29 7^ 87 Draco The Dragon 31 32 40 80 Caepheus Caepheus 13 4 51 35 Bootes Bootes 23 18 52 54 Corona Boreal i s The Northern Crown 8 8 8 21 Hercules Hercules kneeling 29 28 45 113 Lyra The Harp 10 11 ir 21 Cygnus The Swan 19 18 47 81 Cassiopea The Lady in her Chair 1 3 26 37 55 Perseus Perseus 29 29 46 59 Auriga The Waggoner 14 9 40 66 Serpentarius Serpentarius 29 15 40 74 Serpens The Serpent 18 13 22 64 Sagitta The Arrow 5 5 5 18 Aquila The Eagle? 15 12 23 71 Autinous Antinous 5 3 19 Delphinus The Dolphin 10 10 14 18 ON THE CONSTELLATIONS. 2^1 THE ANCIENT CONSTELLATIONS CONTINUED. Equulus Pegasus Andromeda Triangulum Aries Taurus Gemini Cancer Leo Coma Berenices Virgo Libra Scorpius Sagittarius Capricornus Aquarius Pisces Cetus Orion Eridanus Lepus Canis Major Canis Mmor Argo Hydra Crater Corvus Centaurus Lupus Ara Corona Australis Pisces Australis The Horse's Head The Flying Horse Andromeda The Triangle The Ram The Bull The Twins The Crab The Lion ) Berenice's Hairj The Virgin The Scales The Scorpion The Archer The Goat The Water-bearer The Fishes The Whale Orion Eridanus The Hare The Great Dog The Little Dog The Ship The Hydra The Cup The Crow The Centaur The Wolf The AUar The Southern Crown The Southern Fish ft, 1 Ho 1 4 4 6 10 20 19 38 89 23 23 47 m 4 4 12 16 18 21 27 m 44 43 51 141 25 25 38 85 23 15 '^'^ 83 35 30 49 95 14 21 43 32 2>?> 50 110 17 10 20 51 24 10 20 44 31 14 22 69 28 28 '^^ 51 45 41 47 108 38 2.'o 39 113 22 21 45 ^1 38 42 62 78 34 10 27 84 12 13 16 19 29 13 21 31 2 2 13 14 45 3 4 64 27 19 31 6o 7 3 10 31 7 4 9 ^1 35 19 24 7 9 13 12 18 24 272 ON THE CONSTELLATIONS. THE NEW SOUTHERN CONSTELLATIONS. Columba Noachi Robur Carol inum Grus PhcEiiix Indus Pavo Apus, Avis Ind'ica Apis, Musca Chamaeleoii Triangulum Australe Piscis volans, Passer Dorado, Xiphias Toucan Hydrus Noah's Dove - - - - 10 Tlie Royal Oak - - - . 12 I'he Crane ----- 13 The Phoenix - - - - 13 The Indian - - - - 12 The Peacock - - - - 14 The Bird of Paradise - - 11 The Bee, or Fly - - - 4 The Chameleon - - - 10 The South Triangle - - 5 The FlyingFish - - - 8 The Sword Fish - - - 6 The American Goose - 9 The Water Snake - - 10 HEVMHUS's CONSTELLATIONS, MADE OUT OF THE UNFORMED STARS. Lynx Leo Minor Asteron and Chara Cerberus Vulpecula and Anser Scutum Sobieski Lacerta Camelopardalis Monoceros Sextans The Lynx The Little Lion The Greyhounds Cerberus The Fox and Goose Sobieski's Shield The Lizard The Camelopard The Unicorn The Sextant 19 23 4 27 7 32 19 11 44 53 25 35 16 58 31 41 Besides the letters which are prefixed to the stars, many of them have names, as Regulus, Sirins, Arcturus, &c. (366.) Kepler, who was afterwards, in this conjec- ture, followed by Dr. Halley, has made a very inge- nious observation upon the magnitudes and distances CATALOGUES OF THE FIXED STARS. 273 of the fixed stars. He observes, that there can be only 13 points upon the surface of a sphere as far distant from each other as from the center ; and supposing the nearest fixed stars to be as far from each other as from the sun, he concludes that there can be only thirteen stars of the first magnitude. Hence, at twice that distance from the sun, there may be placed four times as many, or 52; at three times that distance, nine times as many, or 1 1/; and so on. These num- bers will give, pretty nearly, the number of stars of the first, second, third, &c. magnitudes. Dr. Halley further remarks, that if the number of stars be finite, and occupy only a part of space, the outward stars would be continually attracted towards those which are within, and, in process of time, they would coalesce and unite into one. But if the number be infinite, and they occupy an infinite space, all the parts would be nearly in equilibrio, and consequently, each fixed star being drawn in opposite directions, would keep it's place, or move on till it had found an equilibrium. Pkil Tram. N°. 364. On the Catalogues of the Fixed Stars. (367.) At the time o^ Hipparchuso^Bhode^, about 120 years before J. C. a new star appeared, upon which he set about numbering the fixed stars, and re- ducing them to a Catalogue^ that posterity might know whether any changes had taken place in the heavens, Ptolemy^ however, mentions that Tymocharis and Arystillus left several observations made 180 years before. The catalogue of ///jo/>a re// w a contained 1022 stars, with' their latitudes and longitudes, which Ptolemy published, with the addition of four more. These Astronomers made their observations with an armillary sphere, placing the armilla, or hoop repre- senting the ecliptic, to coincide with the ecliptic in the heavens by means of the sun in the day-time, and 2J4 CATALOGUES OF THE FIXED STARS. then they determined the place of the moon in respect of the sun by a moveable circle of latitude. The next night by the help of the moon (whose place before found they corrected by allowing for it's motion in the interval of time) they placed the hoop in such a situa- tion as was agreeable to the present moment of time, and then compared, in like manner, the places of the stars with the moon. Thus they found the latitudes and longitudes of the stars ; it could not, however, be done with such an instrument to any very great degree of accuracy. Ptolemy adapted his catalogue to the year 137 ^ft^'' J.C.; but supposing, with Hipparchns, who made the discovery, the precession of the equi- noxes to be r in 100 years,instead of about 72 years, lie only added 2". 40'. to the numbers in Hipparchus for 26'5 years (the difference of the epochs) instead of 3". 42'. 22", according to Dr. Mciskelyne^ Tables. To compare his Tables, therefore, with the present, we must first increase his numbers by 1°. 2'. 22", and then allow for the precession from that time to this. The next Astronomer who observed the fixed stars a-new, was Uliigh Beigh, the grandson of Tamer- lane the Great; he made a catalogue of 1022 stars, reduced to the year 1437- WiU'iam, the most illus- trious Landgrave of Hesse, made a catalogue of 400 stars which he observed; he computed their latitudes and longitudes from their observed right ascensions and declinations. In the year 1610, Tycho Brake's cata- logue of '/J'H' stars was published from his own observa- tions, made with great care and diligence. It was afterwards, in 1627, copied into the Rudolplune Tables^ and increased by 223 stars, from other observa- tions of Tycho. Instead of a zodiacal armilla, Tycho substituted the equatorial arm ilia, by which he ob- served the difference of right ascensions, and the de- chnations, out of the meridian, the meridian altitude being always made use of to confirm the others. From thence he computed the latitudes and longitudes. Tycho compared Fe?ius with the sun, and then the CATALOGUES OF THE FIXED STARS 2'/5 Other stars with Venus, allowing for it's parallax and refraction ; and having thus ascertained the places of a few stars, he settled the rest from them ; and although his instrument was very large, and constructed with great accuracy, yet, not having pendulum clocks to measure his time, his observations cannot be very ac- curate. The next catalogue was that of R. P. Rlcciolus, which was taken from Ti/cho's, except 101 stars which he himself had observed. //eiT/m^- of Dantzick, in 1690, published a catalogue of 1930 stars, of which 950 were known to the ancients ; 603 he calls his own, because they had not been accurately observed by any one before himself; and 37/ of Dr. Hallei/y which were invisible to his hemisphere. Their places were fixed for the year 1660. The British Cata- logue, which was published by Mr. Flaimtead, con- tains 3000 stars, rectified for the year 1689. They are distinguished into seven degrees of magnitude (of which the seventh degree is telescopic) in their proper constellations. This catalogue is more correct than any of the others, the observations having been made with better instruments. He also published an Atlas Ccelestis, or maps of the stars, in which each star is laid down in it's true place, and delineated of it's own magnitude. Each star is marked with a letter, begin- ning with the first letter a of the Greek alphabet for the largest star of each constellation, and so on, ac- cording to their magnitudes, following, in this respect, the charts of the same kind which were published by J. Bayer, a German, l603. In the year 1757, M. de la Caille published his Fundamenta Asfronomice^ in which there is a catalogue of 397 stars ; and in 1763, he published a catalogue of 1942 southern stars, from the tropic of Capricorn to the south pole, with their right ascensions and declinations for 1750. He also published a catalogue of zodiacal stars in the Epheme- rides from I765 to 1774. Mr. Mayer also published a catalogue of 600 zodiacal stars. In the A'autical Almanac for 177^? there is published a catalogue of s 2 tifS PROPER MOTIONS OF 380 stars observed by Dr. Bi^adley, with their longi- tudes and latitudes. In the year 1782, J. E. Bode^ Astronomer at Berlin, published a set of Celestial Charts, containing a greater number of stars than in those of Mr. Flamstead, with many of the double stars and nebulae. He also published, in the same work, a catalogue of stars, that of Flamstead being the foundation, omitting some stars, whose positions were left incomplete, and altering the numbers; to which he has added stars from Hevelius, M. de la Caille, Mayer, and others. In the year 177^'j there was published at Berlin, a work entitled Recueil de Tables Astronomiques, in which is contained a very large catalogue of stars from Hevelius, Flamstead, M.. de la Caille, and Dr. Bradley, with their latitudes and longitudes for the beginning of 1800; with a catalogue of the southern stars of M. de la Caille \ of double stars ; of changeable stars, and of nebulous stars. This is a very useful work for the practical Astronomer. But the most complete catalogue is that published by the Rev. Mr. Wollastoyi, F. R. S. in 1789, entitled, A Specimen of a General Astronomical Catalogue, arranged in Zones of North Polar Distance, and adapted to January 1, 1790; containing a Compara- tive Fiew of the Mean Positions of Stars, as they come out upon Calculation from the Tables of several principal Observers. On the Proper Motions of the Fixed Stars. (368.) Dr. Mashelyne, in the explanation and use of his Tables, which he published with the first volume of his Observations, observes, that many, if not all the fixed stars, have small motions among themselves, which are called their Proper Motions ; the cause and laws of which are hid, for the present, in almost equal obscurity. From comparing his own observations at that time, with those of Dr. Bradley, Mr. Flamstead^ THE FIXED STARS. ^77 and Mr. Roemer, he then found the annual proper motion of the following stars, in right ascension, to be, ofSirius - 0",63, of Castor - o",28, of Procyon - 0",8, of Pollux — O" ,^3 , of Regulus — 0'\4l, of Arcturus — l",4, and o{ a Aquiloc -f o",57; and o^ Sirius in north polar distance r',20, and of Arcturus 2",01, both southwards. But since that time he had continued his observations, and from a catalogue of right ascen- sions of 36 principal stars (which he communicated to Mr. fVoUaston, and which is found in his work), it appears that 35 of them have z. proper motion m right ascension. (369.) In the year 1759, M. Mayer observed 80 stars, and compared them with the observations of Roemer in 1706. M. Mayer is of opinion, that (from the goodness of the instruments with which the ob- servations were made) where the disaojreement is at least 10" or 15", it is a very probable indication of a proper motion of such a star. He further adds, that when the disagreement is so great as he has found it in some of the stars, amongst which is Fomahandy where the difference was 21" in 50 years, he has no doubt of a proper motion. Dr. Herschel following Mayers judgement of his own and Roemer's observa- tions, has compared the observations, and leaving out of his account all those stars which did not show a dis- agreement amounting to lO", he found that 56 of them had a proper motion. From thence he attempts to deduce the motion of the solar system in the fol- lowing manner. (370.) If the sun be in motion as well as the stars, the effects will be altered according to their motion, compared with the motion of our sun. Some of them, therefore, from their own proper motions, might de- stroy, or more than counteract, the effects arising from the motion of the sun. In whatever direction our system should move, it would be very easy to find what effect in latitude and longitude would have taken place upon any star, by means of a celestial globe, by 2f8 PROPER MOTIONS OF conceiving the sun to move from the center upon any radius directed to the point to which the sun is mov- ing. Dr. Herschel describes the effect thus. Let an arc of 90° be apphed to the surface of a globe, and aUvays passing through that point to which the motion of the system is directed. Then whilst one end moves along the equator, the other will describe a curve passing through it's pole, and returning into itself; and the stars in the northern hemisphere, within this curve, will appear to move to the north ; and the rest will go to the south. A similar curve may be described in the southern hemisphere, and like appearances will take place. (371.) Now Dr. Herschel first takes the seven stars before mentioned, whose proper motions had been de- termined by Dr. Maskelyne ^ and he finds, that if a point be assumed about the 77** of ^'ig^^^ ascension, and the sun to move from it, it will account for all the motions in right ascension. And if, instead of sup- posing the sun to move in the plane of the equator, it should ascend to a point near to X HercuUs, it will ac- count for the observed change of declination of Sinus and Arctiirus. In respect to the quantity of motion, of each, that must depend upon their unknown rela- tive distances ; he only speaks here of the directions of the motions. (372.) He next takes twelve stars from the cata- logue of 56, whose proper motions have been deter- mined from a comparison of the observations oi Roemer and Mayer, and adds to them Regulus and Castor \ these have all a proper motion in right ascension and declination, except Regulus, which has none in decli- nation. Of these 27 motions, the above-supposed motion of the solar sytem will satisfy 22. There are also some remarkable circumstances in the quantities of these motions. Arcturus and Sirius being the largest, and therefore, probably, the nearest, ought to have the greatest apparent motion ; and so we find they have. Also, Arcturus is better situated to have a mo- THE FIXED STARS. 2/9 tion in right ascension, and it has the greatest motion. Several other facts of the same kind are found also to take place. But there is a very remarkable circum- stance in respect to Castor. Castor is a double star; now, how extraordinary must appear the concurrence, that two such stars should both have a proper motion so exactly alike, that they never have been found to vary a second ! This seems to point out the common cause, the motion of the solar system. (3'/ 3.) Dr. Hersdiel next takes 32 more of the same catalogue of 56 stars, and shows that their mo- tions agree very well with his supposed motion of the solar system. But the motions of the other 12 stars cannot be accounted for upon this hypothesis. \n these, therefore, he supposes the effect of the solar motion has been destroyed and counteracted by their own proper motions. The same may be said of I9 stars out of the 32, which only agree with the solar motion one way, and are, as to sense, at rest in the other. According to the rules of philosophising, therefore, which direct us to refer all phaenomena to as few and simple principles as are sufficient to explain them, Dr. Herschel thinks we ought to admit the motion of the solar system. Perhaps, however, this argument cannot be properly applied here, because there is no new cause or principle introduced, by sup- posing each star to have a proper motion. Admitting the doctrine of universal gravitation, the fixed stars oufifhtto move as well as the sun. But the sun's mo- tion, as here estimated, cannot be owing to the action of a body upon it which might give it a rotatory motion at the same time, as M. de la Lande conjectures; because a body acting on the sun, to give it it's rota- tion about it's axis, would not, at the same time, give it that progressive motion. See Dr. Herschel' ^ Acconwi in the Phil. Trans. 1/83. (374.) But it will be proper to consider how far this motion of the solar system agrees with the proper motion of the 35 stars determined by T)v,,Maskelyn^, 280 THE ZODIACAL LIGHT. Now, upon supposition that the sun moves, as conjec- tured by Dr. Herschel, that motion will account for the motion of 20 of them, so far as regards their di- rections ; but the motion of the other 15 is contrary to that which ought to arise from this supposition. As some of the stars must have a proper motion of their own, even upon the hypothesis of a solar motion, and which probably arises from their mutual attraction, it is very probable that they have all a proper motion from the same cause_, but most of them so very small, as not yet to have been discovered. And it might also happen, that such a motion might be the same as that which would arise from the motion of the solar system. Yet it must be confessed, that the circumstance of Castor, and the motions, both in right ascension and declination, of many of the stars being such as arise from this hypothesis, with the apparent motion of those stars being greatest which are probably nearest, form a strong argument in it's favour. •^ On the Zodiacal Light. (315.) The Zodiacal Light is a pyramid of light which sometimes appears in the morning before sun-rise, and in the evening after sun-set. It has the sun for it's basis, and in appearance resembles the Aurora Borealis. It's sides are not straiglit, but a little curved, it's figure resembling a lens edge- ways. It is generally seen here about October and March, that being the time of our shortest twilight; for it cannot be seen in the twilight ; and when the twilight lasts a considerable time, it is withdrawn before the twilight ends. It was observed by M. Cassini, in l683, a little before the vernal equinox, in the evening, extending along the ecHptic from the sun. He thinks, however, that it has appeared formerly, and afterwards disappeared, from an observation of Mr. J. Children/, in a book published in l66l, en- titled, Britannia Baconia. He says, that " in the THE ZODIACAL LIGHT. 281 month of February, for several years, about six o'clock in the evening, after twilight, he saw a path of light tending from the twilight towards the Pleiades^ as it were touching them. This is to be seen whenever the weather is clear, but best when the moon does not shine. I believe this phsenomenon has been formerly, and will hereafter appear always at the above-men- tioned time of the year. But the cause and nature of it I cannot guess at, and therefore leave it to the en- quiry of posterity." From this description, there can be no doubt but that this was the zodiacal light. He suspects also, that this is what the ancients called Trabes, which word they used for a meteor, or im- pression in the air like a beam. Pliny ^ lib. II. p. 26, says, Emicant Trabes, quos ducosvocaut. Des Cartes also speaks of a phsenomenon of the same kind. M. Fatio de Didlller observed it immediately after the discovery by M. Cassini, and suspected that it had always appeared. It was soon after observed by M. Kirch and Eimmart in Germany. In the 37^ear I/O/? on April 3, it was observed by IVIr. Derham, in Essex. It appeared in the western part of the heavens, about a quarter of an hour after sun-set, in the farm of a py- ramid, perpendicular to the horizon. The base of this pyramid he judged to be the sun. It's vertex reached 15° or 20° above the horizon. It was throughout of a dusky red colour, and at first appeared pretty vivid and strong, but faintest at the top. It grew fainter by degrees, and vanished about an hour after sun-set. This solar atmosphere has also been seen about the sun in a total solar eclipse, a luminous ring appearing about the moon at the time when the eclipse was total. (376.) M. Fatio conjectured, that this appearance arises from a collection of corpuscles encompassing the sun in the form of a lens, reflecting the light of the sun. M. Cassini supposed that it might arise from an infinite number of planets revolving about the sun ; so that this light might owe it's existence to these bodies. 282 THE ZODIACAL LIGHT. as the milky way does to an innumerable number of fixed stars. It is now, however, generally supposed, that it is matter detached from the sun by it's rotation about it's axis. The velocity of the equatorial parts of the sun being the greatest, would throw the matter to the greatest distance, and, on account of the dimi- nution of velocity towards it's poles, the height to which the matter would there rise would be diminish- ed j and as it would probably spread a little sideways, it would form an atmosphere about the sun something in the form of a lens, whose section perpendicular to it's axis would coincide with the sun's equator. And this agrees very well with observation. There is, however, a difficulty in thus accounting for this phae- nomenon. It is very well known, that the centrifugal force of a point of the sun's equator is a great many times less than it's gravity. It does not appear, there- fore, how the sun, from it's rotation, can detach any of its gross particles. If they be particles detached from the sun, they must be sent off by some other unknown force; and in that case they might be sent off equally in all directions, which would not agree with the observed figure. The cause is probably owing to the sun's rotation, although not immediately to the centrifugal force arising therefrom. 283 Chap. XXV. ON THE LONGITUDE OF PLACES UPON THE SURFACE OF THE EARTH. {3YJ.) The situation of any place upon the earth's surface is determined from it's latitude and longitude. The latitude maybe found from t'ne meridian altitude of the sun, or a known fixed star; from two altitudes of the sun, and the time between ; and by a variety of other methods. These operations are so easy in practice, and opportunities are so continually offering themselves, that the latitude of a place may generally be determined, even under the most unfavourable cir- cumstances, to a degree of accuracy sufficient for all nautical purposes. But the longitude cannot be so readily found. Philip III. King of Spain, was the first person who offered a reward for it's discovery ; and the States of Holland, soon after, followed his example, they being at that time rivals to Spain, as a maritime power. During the minority of Louis XV. of France, the regent power promised a great reward to any person who should discover the longitude at sea. In the time of Charles II. about 1675, the Sieur de St. Pierre, a Frenchman, proposed a method of finding the longitude by the moon. Upon this, a commission was granted to Lord Viscount Brounker, President of the Royal Society, Mr. Flamstead, and several others, to receive his proposals, and give opinion respecting it. Mr. Flamstead gave his opinion, that if we had Tables of the places of the fixed stars, and of the moon's motions, we might find the longitude, but not by the method proposed by the Sieur 284 THE LONGITUDE OF PLACES de St. Pm-re. Upon this, Mr. Flamsfead was ap- pointed Astronomer Royal, and an Observatory was built at Greenwich for him ; and the instructions to 'him and his successors were, " That they should apply themselves with the utmost care and diligence to rectify the Tables of the motions of the heavens, and the places of the fixed stars, in order to find out the so- much desired longitude at sea, for the perfecting of the art of navigation." (378.) In the year 1714, the British Parliament offered a reward for the discovery of the longitude ; the sum of lOOOO/. if the method determined the longitude to 1° of a great circle, or 60 geographical miles; of 15000/. if it determined it to 40 miles; and of 20000/. if it determined it to 30 miles; with this proviso, that if any such method extend no further than 80 miles adjoining to the coast, the proposer shall have no more than half such rewards*. The Act also appoints the First Lord of the Admiralty, the Speaker of the House of Commons, the First Commissioner of Trade, tlie Admirals of the Red, White, and Blue Squadrons, the JVIaster of Trinity-House, the President of the Royal Society, the Royal Astronomer at Green- wich, the two Savihan Professors at Oxford, and the Lucasian and Plumian Professors at Cambridge, with several other persons, as Commissioners for the Lono-i- tude at Sea. The Lowndian Professor at Cambridge was afterwards added. After this Act of Parliament, several other Acts passed, in the reigns of George l\. and \\l, for the encouragement of finding the longi- tude. At last, in the year I774, an Act passed, re- pealing all other Acts, and oflfering separate rewards to any person who shall discover the longitude, either by the lunar method, or by a watch keeping true time, within certain limits, or by any other method. The I * See Widston's account of ih? proceedings to obtain this Act, iri the Preface to his Longitude discovered hi/ Jupiter's Planets. UPON THE EARTH S SURFACE. 285 Act proposes, as a reward for a time-keeper, the sum of 5000/. if it determine the longitude to 1°, or 6o geo- graphical miles ; the sum of 'JbOOl. if it determine the same to 40 miles ; and the sum of lOOOO/. if it deter- mine the same to 30 miles, after proper trials specified in the act. If the method be by improved solar and lunar Tables, constructed upon Sir /. Newton\ theory of gravitation, the author shall be entitled to 5000/. if such Tables shall show the distance of the moon from the sun and stars within 15" of a degree, answering;- to about 7' of longitude, after making an allowance of half a degree for the errors of observation. And for any other method, the same rewards are offered as those for the time-keeper, provided it gives the longitude true within the same limits, and be practicable at sea. The commissioners have also a power of giving smaller rewards, as they shall judge proper, to any one who shall make any discovery for finding the longitude at sea, though not within the above limits. Provided, however, that if such person or persons shall afterwards make any further discovery so as to come within the above-mentioned limits, such sum or sums, as they may have received, shall be considered as part of such greater reward, and deducted therefrom accordingly. (379.) After the decease of Mr. Flatnstead, Dr. Halley, who was appointed to succeed him, made a series of observations on the moon's transit over the meridian, for a complete revolution of the moon's apogee, which observations being computed from the Tables then extant, he was enabled to correct the Tables of the moon's motion. And as Mr. Hadley had then invented an instrument by which altitudes could be taken at sea, and also the moon's distance from the sun or a fixed star, Dr. Halley strongly re- commended the method of finding the longitude from such observations, having found, from experience, the impracticability of all other methods, particularly at sea. 286 METHODS OF FINDING THE LONGITUDE Tojind the Longitude by the Moon^s Distance from the San^ or ajlxed Star. (380.) The steps by which the longitude is found by this method, are these : From the observed altitudes of the moon and the sun, or a fixed star, and their observed distance, com- pute the moon's true distance from the sun or star. From the Nautical Almanac^ find the time at Greenwich when the moon was at that distance. From the altitude of the sun or star, find the time at the place of observation. The diffcM'cnce of the times thus found, gives the difference of the longitudes. (381.) To find t!ie true distance of the moon from the sun or star by observation, let Z be the zenith, S the apparent place of the sun or a star, s the true place, z il/the apparent place of the moon, m it's true j)lace; then the altitudes of 71/ and s being known, by observa- tion, the refractions Ss, Mm are known; also MS is known by observation ; hence, in the triangle ZS3I, we know S M the apparent distance, SZ, Z M, the complements of the apparent altitudes, to find the angle Z (Trig. Art. 239) ; ^"d then in the triangle sZm, we know the angle Z, and sZ, mZ, the comple- UPON THE earth's SURFACE. 28/ ments of the true altitudes, to find sm the true distance* (Trig. Art. 233). Ex. Suppose on June 29, 1793, the sun's apparent zenith distance ZS was observed to be 70*^. 56'. 24", the moon's apparent zenith distance ZM to be 48°. 63'. 58", their apparent distance S3I to be 103". 29'. 27", and the moon's horizontal parallax to be 58'. 35"; to find their ttue distance sm. The true distance sm, computed by the above method, is 103^ 3'. 18". (382.) The^rwe distance of the moon from the sun or star being thus found, we are next to find the time at Greenwich. For this purpose, the sun or such fixed stars are chosen, as lie in or very near the moon's way, so that, looking upon the moon's motion to be uniform for a small time, the moon may be considered as approaching to, or receding from, the sun or star uniformly. To determine, therefore, the time at Greenwich corresponding to any given true distance of the moon from the sun or star, the true distance is computed in the Nautical Almanac for every three hours, for the meridian of Greenwich. Hence, con- sidering that distance as varying uniformly, the time corresponding to any other true distance may be thus computed : Look into the Nautical Almanac, and take out two distances, one next greater and the other next less than the true distance deduced from observa- tion, and the diflference D of these distances gives the access of the moon to, or recess from, the sun or star in three hours; then take the diflference d between the moon's distance at the beginning of that interval and the distance deduced from observation, and then say, D : d :: 3 hours : the time the moon is acceding to, or receding from, the sun or star by the quantity d; which * There are shorter methods than this direct one, of compu'ing the true distance, as the reader will see in my Complete Sj/stein of Astronomy ; but we here purpose only to explain ihe principles' by which the longitude qjay be thus found. 288 METHODS OF FINDING THE LONGITUDE added to the time at the beginning of the interval, gives the apparent time at Greenwich, corresponding to the given true distance of the moon from the sun or star. Ex. On June 29, 1 793, in latitude 52°. 12'. 35", the sun's altitude in the morning was, by observation, 19°. 3'. 36", the moon's altitude was observed to be 41°. 6'. 2", the sun's declination at that time was 23°. 14'. 4", and the moon's horizontal parallax 58'. 35"; to find the apparent time at Greenwich. True dist. of d from by Art. 381. - 103°. 3'. 18" Truedisthy Armt. Jim. on June'2Q,iit3h.l03. 4. 58 Truedist.byiVfl?«^.^/m.onJune29,at6/z.l01. 26'. 42 d= O. 1. 40 D= 1. 38. 16 Hence, 1°. 38'. 16". : O". 1'. 40" : 3h. :: 0\ 3'. 3", which added to 3h. gives 3^. 3'. 3^', the apparent time at Greenwich, when the ^r«e distance was 103°. 3'. 18". (383.) Find the apparent time at the place of ob- servation, by the altitude of the sun (12). Then the difference of the times at Greenwich, and at the place of observation, is the distance of the meridians in time. (384.) Now to find the apparent time at the place of observation, we have the sun's declination 23". 14'. 4", it's altitude 19°. 3'. 36", it's refraction 2'. 44", and parallax 8"; hence, it's true altitude was 19''. 1? and therefore it's true zenith distance was 70°. 59' ; also, the complement of declination was 66°. 45'. 36"; hence, by Art. 92 : 66°. 46'. 56' - - ar. CO. sin. 0,0367325 37. 47.25 - - ar. CO. sin. 0,2127004 70. 56. 24 175. 29. 45 87. 44. 52 - - - - sin. 9,9996644 16. 48. 28 . - - - sin. 9,46oi408 2)19,7092381 9,8546190 UPON THE EARTH S SURFACE. 289 the cosine of 44°. 18'. 52", which doubled gives 88°. 37'. 44", the hour-angle from apparent noon, which in time gives 5h. 54'. 31", the time before apparent noon, or 18h. 5'. 29", on June 28. Hence, Apparent time at place of observ. June 28, 18''. 5'. 29'' . at Greenwich, June 29, 3. 3. 3 Difference of meridians in time . . - 8. 57. 34 Which converted into degrees, gives 134°. 23'. 30", the longitude of the place of observation west of Greenwich. To find the Longitude hy a Time-heeper. (385.) Let the time-keeper be well regulated, and set to the meridian of Greenwich ; then if it neither gain nor lose, it will always show the time at Green- wich. Hence, to find the longitude of any other place, find the mean time from the sun's altitude by Art. 92 ; and observe, at the instant of taking the alti- tude, the time by the watch ; and the difference of these times, converted into degrees, at the rate of 15*^ for an hour, gives the longitude from Greenwich. If, for example, the time by the watch, when the altitude was taken was Qh. 19', and the mean time deduced from that altitude was 9/i. 23', the difference 3h. 4', converted into degrees, gives 46° the longitude of the place ea^t from Greenwich, because the time at the place of observation is for^warder than that at Green- wich. Thus the longitude could be very readily determined, if you could depend upon the watch. But as a watch will always gain or lose, before it is sent out, it's gaining or losing every day for some time, a month for instance, is observed ; this is called the rate of going of the watch, and from thence the mean rate ol" going is thus found. (386.) Suppose, for instance, I examine the rate of a watch for 30 days ; on some of those days I find it has X 290 METHODS OF FINDING THE LONGITUDE gained, and on some it has lost; add together all the quantities which it has gained, and suppose they amount to 17"; add together all the quantities which it has lost, and let the sum be 13"' ; then the difference 4" is the mean rate of gaining for 30 days, which divided by 30, gives 0",133 for a mean daily rate of gaining. Or you may get the mean daily rate thus. Take the difference between what the watch was too fast, or too slow, on the first and last days of observa- tion, if it be too fast or too slow on each day j but take the sum, if it be too fast on one day and too slow on the other, and divide by the number of days between the observations*. And to find the time at the place of trial at any future period by this watch, you must put down, at the end of the trial, how much the watch is too fast or too slow; then subtract from the time shown by the watch, 0"133 x number of days from the end of the trial, being the exact quantity which it has gained according to the above mean rate of gaining, and you are then supposed to get the true time affected with the error at the end of the trial. This would be all the error, if the watch had continued to gain ac- cording to the above rate; and although, from the different temperatures of the air to which the watch may be exposed, and from the imperfection of the workmanship, this cannot be expected, yet, by taking it into consideration, the probable error of the time will be diminished. In watches which are under trial at the Royal Observatory at Greenwich, as candidates for the rewards offered by Parliament for the discovery of the longitude, this allowance of a mean rate, to be applied in order to get the time, is not granted by the Act ofParliament, but it requires that the watch itself should go within the limits assigned; the Commission- ers, however, are so indulgent as to grant the applica- * For further information on this subject, see Mr. Wales's Mclhod of finding the Longitude at Sea. UPON THE earth's SURFACE. 2QI tion of a mean rate, which is undoubtedly favourable to the watches. (387.) As the rate of going of a watch is subject to vary from so many circumstances, the observer, when- ever he goes ashore and has sufficient time, should compare his watch, for several days, with the mean time deduced from the altitude of the sun or a star, by which he will be able to determine it's rate of going. And whenever he comes to a place whose longitude is known, he may correct his watch, and set it to Green- wich time. For instance, if he goto a place known to be 30". east longitude from Greenwich, his watch should be two hours slower than the time at that place. Find therefore, the time at that place by the altitude of the sun or a tixed star, and correct it by the equation of time, and compare the time so found with the time by the watch when the altitude was taken, and if the watch be two hours slower than the time deduced from observation, it is right ; if not, correct it by the dif- ference, and it again gives Greenwich time. (388.) In long voyages, unless you have sometimes the means of adjusting the watch to Greenwich time, it's error will probably be very considerable, and conse- quently the longitude deduced from it will be subject to a proportional error. In short voyages, a watch is undoubtedly very useful, and also in long ones, where you have the means of correcting it from time to time. It serves to carry on the longitude from one known place to another, supposing the interval of time not to be very long ; or to keep the longitude from that which is deduced from a lunar observation, till you can get another observation. Thus the watch may be rendered of great service in Navigation^ To find the Longitude hij an Eclipse of the Moon, and of Jupiter's Satellites. (389.) By an eclipse of the moon. This eclipse begins when the umbra of the earth first touches the 292 METHODS OF FINDING THE LONGITUDE moon, and it ends when it leaves the moon. Having the times calculated when the eclipse begins and ends at Greenwich, observe the times when it begins and ends at any other place; the difference of these times converted into degrees, gives the difference of longi- tudes. For as the phases of the moon in an eclipse happen at the same instant to every observer, the dif- ference of the times at different places, when any phase is observed, will give the difference of the longi- tudes. This would be a very ready and accurate method, if the time of the first and last contact could be accurately observed ; but the darkness of the pe- numbra continues to increase till it comes to the umbra, so that, until the umbra actually gets upon the moon, it is not discovered. The umbra itself is also very badly defined. The beginning and end of a lunar eclipse cannot, in general, be determined nearer than l' of time; and very often not nearer than 2' or 3'. Upon these accounts, the longitude, from the observed beginning and end of an eclipse, is subject to a con- siderable degree of uncertainty. Astronomers, there- fore, determine the difference of the longitudes of two places by corresponding observations of other phases, that is, when the umbra bisects any of the spots upon the moon's surface. And this can be determined with, a greater degree of accuracy than the beginning and end ; because, when the umbra is gotten upon the moon's surface, the observer has leisure to consider and fix upon the proper line of termination, in which he will be assisted by running his eye along the circum- ference of the umbra. Thus the coincidence of the umbra with the spots may be observed with tolerable accuracy. The observer, therefore, should have a good map of the moon at hand, that he may not mistake. The telescope, to observe a lunar eclipse, should have but a small magnifying povi^er with a great deal of light. The shadow comes upon the moon on the east side, and goes off on the west ; but if the telescope invert, the appearances will be contrary. UPON THE earth's SURFACE. 293 (390.) The eclipses o{ Jupiter''?, satellites afford the readiest method of determining the longitude of places at land. It was also hoped that some method might be invented to observe them at sea, and Mr. Irwin made a chair to swing for that purpose, for the observer to sit in ; but Dr. Maskeli/ne, in a voyage to Barba- does, under the direction of the Commissioners of Longitude, found it totally impracticable to derive any advantage from it ; and he observes, that, '^ consider- ing the great power requisite in a telescope for making these observations well, and the violence as well as irregularities of the motion of a ship, I am afraid the complete management of a telescope on ship-board will always remain among the desiderata. However, I would not be understood to mean to discourage any attempt, founded upon good principles, to get over this difficulty." The telescopes proper for making these observations, are common refracting ones from 15 to 20 feet, reflecting ones of 18 inches or 2 feet, or the 46-inch achromatic with three object glasses, which were first made by Mr. Dolland. On account of the uncertainty of the theory of the satellites, the observer must be settled at his telescope a few minutes before the expected time of an immersion ; and if the longitude of the place be also uncertain, he must look out proportionably sooner. Thus, if the longitude be uncertain to 2°, answering to eight minutes of time, he must begin to look out eight minutes sooner than is mentioned above. However, when he has observed one eclipse, and found the error of the Tables, he may allow the same correction to the calculations of the Ephemeris for several months, which will advertise him very nearly of the time of expecting the eclipses of the same sateHite, and dispense with his attending so long. Before the opposition of Jupiter to the sun, the immersions and emersions happen on ,the west side of Jupiter, and after opposition, on the east side ; but if the telescope invert, the appearance will be the con- trary. Before opposition, "the immersions only of the 294 METHODS OF FINDING THE LONGITUDE first satellite are visible; and after opposition, the emersions only. The same is generally the case with respect to the second satellite ; but both immersion and emersion are frequently observed in the two outer satellites. (391.) When the observer is waiting for an emer- sion, as soon as he susj)ects that he sees it, he should look at his watch, and note the second, or begin to count the beats of the clock, till he is sure that it is the satellite, and then look at the clock, and subtract the number of seconds which he has counted from the time then observed, and lie vvilHiave the time of emer- sion. If Jupiter be 8° above the horizon, and the sun as much below, an eclipse will be visible ; this may be determined near enough by a common globe. (392.) The immersion or emersion of a satellite being observed according to apparent time, the longi- tude of the place from Greenwich is found, by taking the difference between that time and the time set down in the Nautical Almanac, which is calculated for ap- parent time. Ex. Suppose the emersion of a satellite to have been observed at the Cape of Good Hope, May 9, IT^/? ^^ lOh. 4&. 45" apparent time; now the time in the Nautical Almanac is 9//. 33'. 12"; the difference of which time is \h. 13'. 33", the longitude of the Cape east o{ Greenwich in time, or 18°. 23'. 15". (393.) But to find the longitude of a place from an observation of an eclipse of a satellite, it is better to compare it with an observation made under some well- known meridian, than with the calculations in the Ephemeris, because of the imperfection of the theory : but where a corresponding observation cannot be ob- tained, find what correction the calculations of the Ephemeris require, by the nearest observations to the given time that can be obtained; and this correction, applied to the calculation of the given eclipse in the Ephemeris, renders it almost equivalent to an actual observation. The observer must be careful to regulate UPON THE earth's SURFACE. 295 liis clock or watch by apparent time, or at least to know the difference ; this may be done, either by equal altitudes of the sun, or of proper stars; or the latitude being known, from one altitude at a distance from the meridian, the time may be found by Art. 92. (3^)4.) In order the better to determine the dif- ference of longitudes of two places from corresponding observations, the observers should be furnished with tlie same kind of telescopes. For at an immersion, as the satellite enters the shadow, it grows fainter and fainter, till at last the quantity of light is so small that it becomes invisible, even before it is immersed in the shadow ; the instant, therefore, that it becomes invisi- ble will depend upon the quantity of light which the telescope receives, and it's magnifying power. The instant, therefore, of the disappearance of a satellite will be later the better the telescope is, and the sooner it will appear at it's emersion. Now the immersion is the instant the satellite is wholly gotten into the shadow, and the emersion is the instant before it begins to emerge from the shadow ; if, therefore, two telescopes show the disappearance or appeaiance of the satellite at the same distance of time from the immer- sion or emersion, the difference of the times will be the same as the difference of the true times of their immersions and emersions, and therefore will show the difference of longitudes accurately. But if the observed titne at one place be compared with the computed time at another, then we must allow for the difference between the apparent and true times of immersion or emersion, in order to get the true time where the observation was made, to compare with the true time from computation at the other place. This difference may be found, by observing an eclipse at any place whose longitude is known, and ci»mparing it with the time by computation. Observers, therefore, should settle the difference accurately by the mean of a great number of observations thus compared witfi the computation, by which means the longitude will i^S METHODS OF FINDING tHE LONGITUDE. be ascertained to a much greater accuracy and cer- tainty. After all this precaution, however, the different statesof the air at different times, and also the different states of the eye, will introduce a small degree of un- certainty ; the latter case may perhaps, in a great measure, be obviated, if the observer will be careful to remove himself from all warmth and light for a little time before he makes the observation, that the eye may be reduced to a proper state ; which precaution the observer should also attend to, when he settles the difference between the apparent and true times of immersion and emersion. Perhaps also the difference arising from the different states of the air might, by proper observations, be ascertained to a considerable degree of accuracy; and as this method of determining the longitude is of all others, the most ready, no means ought to be left untried to reduce it to the greatest certainty. ( 29r ) TABLE I. For converting Degrees, Minutes and Seconds, into Time, at the Rate of 36o Degrees for 24 Hours. Beg. Hou. Min. Deg. Hou. Min. i>ec. o/" Mm. Mm Sec. Min. Min. Sec. ^Sec. 1 4 30 o 1 ,o66 2 8 40 2 40 2 ,133 3 12 50 3 20 3 .2 4 16 6o 4 4 ,266 5 20 70 4 40 5 ,333 6' 24 80 5 20 6 ,4 7 28 90 6 7 ,466 8 32 100 6 40 8 ,533 9 36 200 13 20 9 ,6 10 40 300 20 10 ,666 20 1 20 TABLE IL For converting Time into Degrees, Minutes, and Seconds, at the Rate of 24 Hours for 36o Degrees. Hou . Deg. Min. See. Deg. Min. .Set-. Dec. oj Sec. Sec. 1 15 1 15 ,1 1,5 2 30 2 30 ,2 3,0 3 4 45 6o 3 4 1 45 ,3 ,4 4,5 6,0 5 6 7 8 75 90 105 120 5 6 7 8 1 1 1 2 15 30 45 ,5 ,7 ,8 7,5 9,0 10,5 12,0 9 10 135 150 9 10 2 2 15 30 ,9 13,5 11 165 20 5 12 180 30 7 30 16 240 40 10 20 300 50 12 30 Lately published hy the same Author , I. A CONFUTATION of ATHEISM, from the Laws and Constitution of the Heavenly Bodies ; in Four Discourses preached before the University of Cambridge. With an Introduction ex- plaining in a familiar manner thePrinciplesof Plane and Physical Astronomy: with Notes, and an Appendix, 1 Vol. Svo. Boards, 45, 6d. II. The CREDIBILITY of the SCRIPTURE MIRACLES Vindicated, in answer to Mr. Hume ; in Two Discourses preached before the University of Cambridge. — Second Edition, enlarged. To which are added, Notes and Remarks upon Mr. Hume's Principles and Reasonings. Price 2*. III. The ELEMENTS of the CONIC SECTIONS, adapted to the Use of Students in Philosophy. — Fourth Edition, improved and enlarged. Price 2s. 6d. IV. A Treatise on PLANE and SPHERICAL TRIGO- NOMETRY, with an Introduction, explaining the Nature and Use of Logarithms, adapted to the Use of Students in Philosophy. Third Edition, improved. Price 4s. 6d. V. A Course of Lectures in MATHEMATICS and PHILOSOPHY, by the same Author and Dr. Wood. PLEASE DO NOT REMOVE CARDS OR SLIPS FROM THIS POCKET UNIVERSITY OF TORONTO LIBRARY ...la'Uiirn/r.'ii'iitH