Ǻ-º-º-º-º-º: - . . . . yºrº." tucKER.M.M.W. & Scott, Book-sellers and Stationers, jº ain Street, West of the Court House Cºrlisle. 3. who have for sale, a large collection of books, in the various departments of Literature. frees: |-•…“- -*.&- ~~ ~~~~）-- __ _- ----- - - -→ * - *p - ș· * {-, 4 *•* →---- ;· |-----�~~ ~ ~ |-● - AN & ELEMENTARY TREATISE # ON S T R O N O MT Y. IN TWO PARTS. THE FIRST, CONTAINING A CLEAR AND COMPENDIOUS VIEW OF THE THEORY. THE SEcoMD, A NUMBER OF PRACTICAL PROBLEMS. To WHICH ARE ADDED, Solar, Lumar, and some other ASTRONOMICAL TABLES. -** BY JOHN GUMMERE, EELEow of THE AMERICAN PHILosophic AI, soci ETY, AND comfºrsponDING MEMBER OF THE ACADEMY OF NATURAL SCIENCES, PHILADELPHIA. PHILADELPHIA: RUBLISHED BY KIMBER & SHARPLEss, No. 93, MARKET STREET. J. CRISSY AND G. GOODMAN, PRINTERs. 1822. EASTEuN DISTRICT of FENNsylvaSIA, To wit: is BE IT REMEMBERED, that on the second day of January, *śckkkºkkick -> LaSi in the forty-sixth year of the Independence of the United - # States of America, A. D. 1822, Kimber & Sharpless, of the said Žbººk-k-k-k-k-k-k-ki district, have deposited in this office, the title of a book, the right whereof they claim as proprietors, in the words following, to wit: “An Elementary Treatise on Astronomy. In two parts. The first, con- taining a clear and compendious view of the Theory. The second, a number of Practical Problems. To which are added, Solar, Lunar, and some other Astronomical Tables. By John Gummere, Fellow of • the American Philosophical Society, and Corresponding Member of the Academy of Natural Sciences, Philadelphia.” In conformity to the Act of Congress of the United States, intituled “An act for the encouragement of learning, by securing the copies of maps, charts, and books, to the authors and proprietors of such copies, during the times therein mentioned.” And also to the act, entitled “An act, Sup- plementary to an act, entitled, ‘An act for the encouragement of learning, by securing the copies of maps, charts, and books, to the authors and pro- prietors of such copies, during the times therein mentioned,’ and extending the benefits thereof to the arts of designing, engraving, and etching histori. cal and other prints.” 4. D. CALDWELL, Clerk of the Eastern District of Pennsylvania. (2.Élſ hºp &_*.* ſkø6 t {{A_2 . 12 3--4; 31 PREFACE. THE object in writing the present Treatise, has been to give, in a moderate compass, a methodical and scientific exhibition of the elementary principles of Astronomy, and to furnish the student with Rules and Tables for making some of the more useful and more interesting calculations. The work is divided into two Parts; the First containing the Theory, and the Se- cond, Practical Problems. - Particular attention has been given to the arrange- ment of the First Part. The different subjects are in- troduced in such order, as to make it unnecessary for the student to anticipate propositions in advance of those which he is studying. The Definitions are given as they are wanted in the course of the work, and after previous investigations have served to render them easily understood. - Astronomy, when taken in its whole extent, and with all its different methods, necessarily forms a large treatise. It is not therefore practicable to give those various methods, in a work of the size to which it has been thought proper to limit this. Neither are they important, except to those who devote very decided attention to this interesting science; and they must have recourse to more extended works. Most students are satisfied with obtaining a correct gene- ral knowledge of the subject, and of the means by which the principal facts have been discovered or can be established, without entering into all the investiga- g 1y PREFACE. tions necessary to render those means the most effica- cious in giving precision to the results. In conformity with these views, I have seldom given more than one method of determining any particular fact, and have avoided entering into minute details that did not ap- pear necessary to a proper comprehension of the subject. In the demonstrations, the student is supposed to be acquainted with Algebra, Geometry, Plane and Spheri- cal Trigonometry, and Conic Sections, or at least the properties of the Ellipse. As many persons study Astronomy who have no knowledge of the Differential Calculus, it has not been used, though in a few cases it might have been introduced with advantage. The Problems in the Second Part are principally for making calculations relative to the Sun, Moon and Fixed Stars. The Tables of the Sun and Moon, which are used in these, have been abridged from the Tables of Delambre and Burckhardt, and reduced to the meridian of Greenwich. Although the quantities are only given to whole seconds, and several small equations have been omitted, the places of the sun and moon, obtained from these tables will be very nearly correct, for any time within the period to which the tables of Epochs extend.* Tules are also given for obtaining the places and motions of the sun and moon for a given time from the Nautical Almanac. Each of the problems is illustrated by one wrought example; * The small table for reducing the moon’s parallax is adapted to the mean value of the parallax, and has a degree of accuracy corresponding to the other tables. As the same table is used for reducing the parallax when ob- tained more accurately from the Nautical Almanac, it would have been bet- ter to have adapted it to different values of the parallax; but this circum- stance was not thought of, in time to alter it. And as the change in the reduction, for a given latitude, is small, it is not perhaps important, PR EFACE. W and to most of them are added one or two unwrought questions, with the answers annexed, to serve as exer- cises for the student. While writing the present treatise, I have had re- eourse to several of the best modern publications on the subject; among which may be particularly mention- ed those of Vince, Woodhouse, Playfair, Delambre, Biot, and Laplace. From these I have adopted the methods which best suited my purpose, making, when it appeared necessary, such modifications in them as the plan of the work required. In the Projection of Eclipses and Occultations, a method is given which is believed to be new, and which renders the operation more sim- ple, without materially affecting the accuracy of the results. An easy method, derived from the former, is also given for tracing the central path of an eclipse of the sun. - In a work of this description, particularly when printed from manuscript, errors must be expected to occur; some proceeding from inadvertencies on the part of the author, and others occurring in the press. | Such as have been discovered, which, it is believed, include all that are important, are enumerated at the end of the volume. - - John GUMMERE. Burlington, JW. J. * 12mo. 22d, 1821. CONTENTS. PART I. CHAP. I. General Phenomena of the Heavens, - º 1 CHAP. II. Definitions of Terms.-Astronomical Instruments, 4 CHAP. III. Meridian Line.—Sidereal Day.—Diurnal Mo- tion.—Refraction, - ſº º tº 7 CHAP. IV. Latitude of a Place.—Figure and Extent of the Earth.—Longitude, º --> - - CHAP. W. On Parallax, - - -> tº a 22 CHAP. VI. Apparent Path of the Sun.—Fixed Stars, - 29 CHAP. VII. Sun's Apparent Orbit.—Kepler's Laws.--Kep- ler’s Problem, - - º sº CHAP. VIII. Equation of Time.—Right Ascension of Mid- Heaven, º * tº- - - 71 CHAP. IX. Circumstances of the Diurnal Motion.—Sun’s Spots, and Rotation on its Axis.-Zodiacal Light, 77 CHAP. X. Of the Moon, - *- - gº CHAP. XI. Eclipses of the Sun and Moon.—Occultations, 115 CHAP. XII. Of the Planets, tº gº tº- 155 16 46 CHAP. XIII. On Comets, - - tº- - 190 CHAP. XIV. Aberration of Light.—Nutation of the Earth's Axis.--Annual Parallax of the Fixed Stars, º 193 CHAP. XV. Nautical Astronomy, &_- sº - 198 CHAP. XVI. Of the Calendar, - tº tºº 2O7 CHAP. XVII. Universal Gravitation, and some of its Effects, 213 PART II. Catalogue of the Tables, º - tº- - 245 Observations and Rules relative to Quantities with different signs, tº- - º wº sº 250 PROB. I. To work a proportion by logistical logarithms, 254 PROB. II. From a table in which quantities are given, for each Sign and Degree of the Circle, to find the quan- tity corresponding to Signs, Degrees, Minutes and Seconds, - º ſº º sº 255 PROB. III. To convert Degrees, Minutes and Seconds of the Equator into Time, - sº &_ - 257 viii CONTENTS, PAGLe PROB. IV. To convert Time into Degrees, Miuutes and Seconds, - sº º º tº 257 PROB. W. The longitudes of two Places, and the Time at one of them being given, to find the corresponding Time * at the other, - tº a tº tºº - 258 PROB. VI To convert apparent time into mean, and the con- trary, º fººt 259 PROB. VII. To find the Sun's Longitude, Semidiameter and Hourly Motion, and the apparent Obliquity of the Ecliptic, for a given time, from the Tables, - 262 PROB. VIII. The Obliquity of the Ecliptic, and the Sun's Longitude being given, to find the Right Ascension and Declination, - tº sº tºº PROB. IX. Given the Obliquity of the Ecliptic and the Sun’s Right Ascension, to find the Longitude and Declination, 265 PROB. X. The Obliquity of the Ecliptic and the Sun's Longi- tude being given, to find the angle of Position, - 266 PROB. XI. To find from the Tables, the Moon's Longitude, Latitude, &c. º º tº gº PROB. XII. To find the Moon's Longitude, &c. from the Nautical Almanac, tºº tº gº - 274 PROB. XIII. To find the Reductions of Parallax and Latitude, 278 Prob. XIV. To find the Mean Right Ascension and Decli- nation, or Longitude and Latitude of a Star, from the 264 267 Tables, - ſº * - tº º 279 PROB. XV. To find the Aberrations of a Star in Right Ascem- sion and Đeclination, - tº gº - 280. PROB. XVI. To find the Nutations of a Star in Right Ascen- sion and Declination, dº gº sº 282 PROB. XVII. To find the Aberrations of a Star in Longitude and Latitude, - - 284 Prob. XVIII. To find the Nutation of a Body in Longitude, 285 PROB. XIX. The Obliquity of the Ecliptic and the Right Ascension and Declination of a Body being given, to , find the Longitude and Latitude, - gº PROB. XX. The Obliquity of the Ecliptic and the Longitude and Latitude of a Body being given, to find the Righ Ascension and Declination, - - 287 PROB. XXI. The Obliquity of the Ecliptic and the Longi- tude and Declimation of a Body being given, to find 285 the Angle of Position, º gºe 289 Prop. XXII. To find the Time of a Star's Passage over the Meridian, Eº 290, PROB. XXIII. To find the Time of the Moon’s Passage over the Meridian, gº tº êº º 291 / &ONTENTS, ix PROB. PROB. PROB. PROB. PROB. PROB. PROB. PROB. PROB. PROB. PROB. PROB. PROB. PROB. PROB. PROB. PROB. IPAGE, XXIV. To find the Time of the Sun's Rising and Setting, - - --> - XXV. To find the Time of the Moon's Passage over the Meridian, from the Nautical Almanac, -> 293 XXVI. To find the Moon’s Declimation from the Nautical Almanac, - - º - XXVII. To find the Time of the Moon's Rising or Setting, sº - - º - 296 . XXVIII. To find the Longitude and Altitude of the Nomagesimal, - º º - XXIX. To find the Moon's Parallax in Longitude and Latitude, * - -º sº tº XXX. To find the Time of New or Full Moon by the Tables, - - º .* XXXI. To find the Time of New or Full Moon by the Nautical Almanac, - tº- tºº XXXII. To determine what Eclipses may be expected to occur in a given year, and the Times nearly at 292 * 2.94. 295 301 309 which they will take place, º - 315 XXXIII. To Calculate an Eclipse of the Moon, 319 XXXIV. To Project an Eclipse of the Moon, - 324 XXXV To Project an Eclipse of the Sun, tº 328 XXXVI. To Calculate an Eclipse of the Sun, 337 XXXVII. To find by Projection the 1.atitudes and Longitudes of the Places at which an Eclipse of the Sun is Central, for different times during the continu- ance of the Central Eclipse, tº- 345 XXXVIII. To Project an Occultation of a Fixed Star by the Moon, - - - - - 348 XXXIX. Given the Moon's true Longitude, to find the corresponding time at Greenwich by the Nautical Almanac, - sº tºº - XL. To find the Longitude of a Place from the ob. served time of beginning or end of an Occultation of a Fixed Star by the Moon, - tºº Eºs 355 354 The following fllphabet is given in order to facilitate, to the stu- dent who is unacquainted with it, the reading of those parts in which the Greek letters are used: Letters. A cº B 3 & T y ſ A 3. E a Z č & H iſ © 9 % I K #. A. A M p. N y 5 : O 0 II 2 7: X, or g T. r1 T . q q. X 2. Y J, Śl & f Names. Alpha Béta Gamma Delta. Epsilon Zeta Eta Theta Iöta Rappa Lambda Mu Nu Xi Omicron Pi Rho Sigma Tau Upsilon Phi Chi Psi Omega AN TELEMENTARY TREATISE ON ASTRONOMY. PART I, CHAPTER. T. General Phenomena of the Heavens. 4. AstroNoMy is the science which treats of the ap- pearances, motions, distances, and magnitudes of the heavenly bodies. That part of the science in which the causes of their motions are considered, is called Phy- sical J1stronomy. • 2. If, in a clear night, we fix our attention on the heavens, and make continued or repeated observations on the stars, we shall find that they retain the same situations with respect to each other, but that with re- spect to the earth they undergo a continual change. Those to the eastward will be seen to ascend, and others will come into view or rise; those to the west- ward will be seen to descend and will go out of view, Or Set, 3. If we direct our attention to the north, different phenomena will present themselves. Many stars will 2 ' 2 ASTRONOMY. be seen that do not set.* They appear to revolve or describe circles about a certain star, that seems to re- main stationary: this stationary star is called the Polar Star. Those stars that do not set, are called Circum- polar Stars. 4. When the polar star is accurately observed, it ceases to appear stationary, and is found to have an apparent motion in a small circle, about a point from which the different parts of the circumference are equally distant. This point is called the JWorth Pole. It is in reality about this point, and not the polar star, that the apparent revolutions of the stars are performed. 5. The stars appear to move, from east to west, ex- actly as if attached to the concave surface of a hollow sphere which revolves on its axis in a space of time, nearly equal to 24 hours. This motion, which is com- mon to all the heavenly bodies, is called the Diurnal JMotion. 6. If we examine the situations of the Moon, on suc- cessive nights, we shall find that she changes her posi- tion among the stars, and advances from west to east. 7. The Sun also appears to partake of this motion, relative to the stars. This may be inferred from ob- serving the stars in the west after the sun has set. If our observations be continued for a number of succes- sive evenings, we shall find that the sun continually approaches to the stars, situated to the eastward of him. 8. Besides the sun and moon, there are ten stars which change their situations, with respect to the other stars, and have a motion among them. These are called Planets. Five of them, named, JMercury, Venus, * Here, and in other parts of the work, unless the contrary is mentioned, the Observer is supposed to be in the United States, or in the southern or middle parts of Europe. ASTRONOMY. 3 JMars, Jupiter, and Saturn, are visible to the naked eye, and were known to the ancients. The other five, named, Vesta, Juno, Ceres, Pallas, and Uranus,” can not be seen without the aid of a telescope, and have not been long known. f The stars which do not sensibly change their situa- tions, with respect to one another, are called Fiaced Stars. 9. There are some stars, that occasionally appear in the heavens, which have a motion among the fixed stars, and only continue visible for a few weeks or months. They are commonly accompanied by a faint brush of light, called a tail. These are named Comets. 10. If a person, placed on the margin of the sea, ob- serve a vessel receding from the land, he will first lose sight of the hull, then of the lower parts of the sails, and lastly of the topsails. This will be the case, in whatever direction the vessel pursues her course, or in whatever part of the earth the observation is made. We hence conclude that the surface of the sea is convex. It is also well known, that vessels have sailed entirely round the earth, in different directions. From these circumstances, it is inferred that the form of the earth is globular. 11. In astronomical investigations, except when great accuracy is required, it is usual to consider the earth as a sphere. f 12. The angular distance between any two of the fixed stars, is found to be the same, in whatever part of the earth’s surface the observation is made. It fol- lows, therefore, that the distance of the stars from the * The planet Uranus, which was discovered by Dr. Herschel, was by him hamed Georgian Sidus, in honour of his patron, King George III. By the French it was called Herschel. It is now generally known by the name given in the text. 4. - ASTRONOMY. earth is so great, that the earth’s diameter, compared with it, is insensible. * , 43. It is not supposed that the fixed stars are all at the same distance from the earth. But since their dis- tances are so immensely great that the most accurate observations do not indicate a difference, they are con- sidered as placed in the concave surface of a sphere, having the same centre with the earth. CHAPTER II. Definitions of Terms.—Astronomical Instruments. 4. THE straight line which passes through the North Pole, and through the centre of the earth, is called the .ſlavis of the Heavens. It is the line about which the heavens appear to revolve. 2 The point in which the axis of the heavens meets the southern part of the celestial sphere, is called the South Pole. | - 3. The points in which the axis of the heavens in- tersects the surface of the earth, are called the Worth and South Poles of the Earth. - 4. A plane which passes through the centre of the earth, and is perpendicular to the axis of the heavens, intersects the celestial sphere in a circle, which is called the Celestial Equator, or simply the Equator. The circle in which this plane cuts the surface of the earth, is called the Terrestrial Equator. 5. If at any place on the earth’s surface a straight line in the direction of gravity, that is in the direction of the plumb line, when a plummet is freely suspended and is at rest, be produced upward, the point in which it cuts the celestial sphere, is called the Zenith of the ASTRONOMY. 5 place. If the same line be produced downward, the point in which iſ cuts the opposite part of the sphere, is called the JW'adir. 6. A plane which passes through any place, and is perpendicular to the straight line joining the zenith and nadir, cuts the celestial sphere in a circle, which is called the Horizon, or, sometimes, the Sensible Hori- 2,070. . The circle in which a plane, passing through the centre of the earth, and parallel to the horizon, cuts the celestial sphere, is called the Rational Horizon. 7. A great circle which passes through the poles of the heavens and through the zenith of a place, is called the JMeridian of that place. The meridian cuts the horizon at right angles, in two points, called the JWorth and South Points of the Ho- Tizon. 8. The intersection of the plane of the meridian with the earth’s surface, is called the JMeridian Line, or Terrestrial JMeridian. 9. The arc of the meridian intercepted between the zenith and equator, is called the Latitude of the place. 10. Circles, which pass through the zenith and na- dir of any place, are called Vertical Circles, and are perpendicular to the horizon of the place. 14. The vertical circle which is at right angles to the meridian, is called the Prime Vertical. The prime vertical intersects the horizon in two points, called the East and West Points of the Horizon. 42. The are of a vertical circle, intercepted between a star and the horizon, is called the Altitude of the star; and the arc of the horizon, intercepted between the said vertical and the meridian, is called the flzimuth of the star. 6 ASTRONOMY. The definitions of other astronomical terms will be found in succeeding parts of the work, when such know- ledge of the subject shall have been obtained, as will render them easily understood. - - 43. For making astronomical observations various instruments are used, some of which, with the purposes to which they are applied, it will be proper briefly to Imention. - - • 44. The Astronomical Quadrant is an instrument used to take the altitude of a heavenly body. It is made of different sizes, but generally of two, three, or more feet radius. The quadrantal arc or limb is divided into 90 equal parts or degrees, and these degrees are sub- divided into smaller parts, according to the size of the instrument. To the quadrant a telescope is attached, having a motion about the centre of the quadrant and carrying with it a vernier index that moves along the graduated limb and increases its subdivisions, which, by this means, is generally extended to seconds. In the eye tube of the telescope a ring is placed, having two very fine wires attached to it, crossing each other at right angles in the centre. The intersection of these wires is made to coincide accurately with the focus of the eye glass, and serves to determine the line of sight, or, as it is technically called, the Line of Collimation of the telescope. - - 15. The Astronomical Circle is an instrument by which an observer may, at the same time, obtain the altitude and azimuth of a heavenly body. It has two graduated circles, one horizontal for the azimuth, and the other vertical for the altitude. A telescope is fixed to the vertical circle, and moves with it. Most astro- nomical observations may be accurately made with an instrument of this kind. - ASTRONOMY. 7 46. A Transit Instrument is a telescope fitted up in such a manner, that its line of collimation may be made to move accurately in the plane of the ineridian. It is used for observing the passage of a heavenly body over the meridian. 17. A JMicrometer is an instrument attached to tele- scopes, by means of which small angles may be mea- sured with an extreme degree of precision. 48. The Astronomical Clock is not very different from the common clock. Its pendulum rod is so con- structed, that its length is not sensibly affected by changes in the temperature of the air. The hours on the face are marked from 1 to 24. The student who wishes to see particular descrip- tions of astronomical instruments, accompanied by en- gravings, may consult Vince's Practical Astronomy, Traité D'Astronomie par Delambre, or Rees’s Cyclo- pedia. CHAPTER III. JMeridian Line.—Sidereal Day.—Diurnal JMotion.— Refraction. 1. LET Z R, Fig. 1. represent the northern part of the meridian of a place, Z, the zenith, P the pole, H. R0 the horizon, SS/G the circle which one of the fixed stars appears to describe in its diurnal motion, and S and S' different situations of the star, the former in the eastern and the latter in the western part of the heavens; also let PS and PS' be arcs of declination circles, ZSA and ZSB arcs of vertical circles, and let the situations S and S' of the star in its apparent circle be such that the altitudes AS and BS’ are equal. 8 ASTRONOMY. Then if, as it appears to do, the star continues at the same distance from the pole P, the arc PS = PS'; also because ZA = ZB, being each quadrants, and AS = BS', we have ZS = ZS', and PZ is common to the two triangles PZS and PZS'; therefore the angle TZS = PZS', and the arc AR = BR, these arcs be- ing the measures of the angles PZS and PZS'. Now, RA and RB are the azimuths of the star when in the situations S and S' (2.12.)* If therefore the altitude and bearing of a star be observed when in the eastern part of the heavens, and if its bearing be again observ- ed, when it arrives at the same altitude in the Western part of the heavens, the line bisecting the angle made by these bearings will be a meridian line, (2.8.) 2. In conformity with appearance, we have, in the preceding article, made the assumption, that the appa- rent diurnal motion of a star is in a circle. The proba- bility of this assumption being true, is increased by the fact that repeated accurate observations on the same star with different altitudes, or similar observations on any other star, give the same situation for the meridian line. 3. When an accurate meridian line has been thus obtained, an astronomical circle, or a transit instrument, may be so adjusted, that the line of collimation of its telescope, shall move in the plane of the meridian. 4. When by a good clock the exact time is observed from the time of a fixed star passing the meridian on any evening to the time of its passage on the next eve- ning, and this observation is repeated on several suc- cessive evenings, it is found that the interval of time * The first number refers to the chapter, and the second to the article. When a reference is made to an article in the same chapter, the number of the article only is given. GHAPTER III. • between its passages on any two succeeding evenings is the same. Similar observations on different stars give the same interval of time. This is true, not only for the time between two successive passages of a star over the meridian, but also for the time from a star being at any altitude to its return to the same altitude on the suc- ceeding evening. It appears therefore very probable that the diurnal motion of a star is uniform. 5. The time between two successive passages of a star over the meridian is called a Sidereal Ilay. And a clock that is so regulated as to move through 24 hours in the course of a sidereal day is said to be regu. lated to Sidereal Time. 6. We have inferred as probable, that the diurnal motion of a star is performed in a circle about the pole of the heavens, and that its motion in that circle is uni- form. If this is the case it is evident that the distance PS' is constant, and that the angle ZPS' must in- crease uniformly with the time. As the star moves through the whole circle or 360° in 24 sidereal hours, it must, if its motion is uniform, move through 15° in each hour, and consequently the angle ZPS' must in- crease at the rate of 45° per hour. Now, if PZ, the distance of the pole from the zenith, and PS', the dis- tance of a star from the pole, are known, and if the al- titude BS", or its complement, the zenith distance ZS of the star, be observed, the angle ZPS may be calcu- lated. Observations and calculations thus made on a star at different times during the same night, prove that the angle ZPS' varies as the time; and, therefore, that the diurnal motion of a star is uniform. 7. In the preceding article the distances PZ and PS' are supposed to be known. A method of obtaining them is now to be explained. Let F and G be the situ- 3 40 ASTRONOMY. ations of the star on the meridian above and below the pole. Then we have ZF = PZ — PF ZG = PZ + PG = PZ + PF therefore ZG – ZF = 2 PF, or PF = }(ZG – ZF) also ZG + ZF = 2 PZ, or PZ = }(ZG + ZF) Let K and L be the situations on the meridian of another star at a different distance from the pole. Then we have in like manner PZ = } (ZK + ZL.) But observations made on different stars at different dis- tances from the pole do not give the same result for PZ. It is found that the value of PZ thus obtained is less, as the distance of the star from the pole is greater; that is, as it is nearer to the horizon when on the meri- dian below the pole. When the altitude RP of the pole is 40 or 50°, and one of the stars observed is the polar star, and the other is one that at its passage of the meridian below the pole, is very near the horizon, the difference between the values of PZ obtained from them, amounts to about half a degree. This effect is produced by the action of the earth’s atmosphere on the rays of light from the stars, and is called Jltmospherical Refraction, and sometimes ſls- tronomical Refraction. OF REFRACTION. 8. It is known that when a ray of light passes ob- liquely from one medium to another of different density, its direction is changed. Let FHGB Fig 2, be a vertical section of a vessel whose sides are opaque. An object placed on the bottom at E could not, when the vessel is empty, be seen by an eye placed at O. But if the CHAPTER III. 14 vessel be filled with water, the object will become visible in the direction OB, and will appear as though it were really at D. A ray of light, therefore, which passes obliquely from water to air is refracted so as to make a greater angle with the perpendicular to the common surface, than if it passed on, without suffering a change in its direction. Again, a ray of light passing from an object at O, in the direction OB will, when the vessel is empty, meet the bottom in D. But if the vessel be filled with water, the ray of light will be refracted on entering the water, and will take the direction BE, so that to an eye at E, it will appear to come from the point A, and therefore the object will appear to be more elevated than it really is. The same effects, though different in degree, take place when a ray of light passes from air into a va- cuum, or the contrary. 9. The angle contained between the directions of the direct and refracted rays, is called the flngle of Re- fraction, or simply the Refraction. 10. It is found by experiment, that for the same two mediums, except when the ray of light passes very ob- liquely from one to the other, the sine of the angle con- tained between the direct ray and the perpendicular to the common surface, is equal to the sine of the angle contained between the refracted ray and the same per- pendicular, multiplied by some constant quantity. If ZB be perpendicular to the common surface FB, of the two mediums, and OB be the direct ray and AB the direction of the refracted ray, we have m sin ZBA = sin ZBO = sin (ZBA + ABO.) The value of m is constant for the same two mediums, but is greater or less, according as the difference of density of the mediums, is greater or less. 44. The almosphere extends to the height of some miles, and its surface is supposed to be nearly concen- tric with the surface of the earth. It has been found by experiments, in ascending high mountains, that its ^, 42 ASTRONOMY. density gradually decreases, with an increase of dis- tance from the general surface of the earth. Hence, a ray of light which enters it obliquely, passing continu- ally from a rarer to a denser medium, has its direction continually changed, and its path will therefore be a curve, concave towards the earth. This curve coincides with a vertical plane, because as the density of the at- mosphere, on each side of such plane, is the same, there is no cause for its deviating either way. Refraction, therefore, makes the apparent altitude of a star, great- er than the true; but it does not change its azimuth. The curvilinear path of a ray of light, passing through the atmosphere, differs but little from a right line, except near the horizon, where, on account of its greater obliquity, its direction undergoes a greater change, 42. The refraction, except near the horizon, varies, nearly as the tangent of the apparent zenith distance. Let N represent the apparent zenith distance of a star, and r, the refraction corresponding. Then N + r = the true zenith distance, and, therefore, (10.) #1 sin N = sin (N + r) = sin N cos r + cos N sim r (App.” 13.) cos N sin N But as r is small, cos r = 1, and sin r = r, nearly. Therefore º = (m — 1) tan N. Or m = COs r + sin r = cos r + cot N sin r. ſm = 1 + r cotN, or r = 43. It is evident that the refraction can not vary ac- curately as the tangent of the apparent Zenith distance, because in that case, when the zenith distance is 90°, it would be infinite. And in all cases, when the alti- gº tude is small, and consequently when the rays of light * Appendix at the end of part 1st. CHAPTER III. 43 enter the atmosphere very obliquely, it would be too great. T}r. Bradley found that the refraction is more nearly equal to the product of the tangent of the difference between the apparent zenith distance and three times the refraction, multiplied by a constant quantity. His formula is, r = 57" tan (N–3r); in which 57" is the refraction at 45° apparent zenith distance. 44. From the observed altitudes or zenith distances of two circumpolar stars, when on the meridian, both above and below the pole, the true distance of the pole from the zenith, the refraction for each meri- dian altitude of the stars, and the true polar distances of the stars, may be obtained. In Fig. 1. let P be the true place of the pole, G and F the true situations of a star on the meridian, below and above the pole, and L and K, those of another star. Also let N and n re- present the apparent zenith distances of the star at G and F, R. and r the corresponding refractions, N' and n' the apparent ze- mith distances of the other star at L and K, and R' and r", the corresponding refractions. Then, ZG = N +R, ZF = n+ r, ZL = N + R', and ZK = n + 1"; therefore (7), PZ = 4 (N+R+n+r) = 3 (N'--R + n + r",) and N-H R+n+ r = N + R' + n’ + r", or R+ r—R'—r' = N + n’—N—n. Now supposing the refraction to vary as the tangent of the ap- parent zenith distance, (12) we have, * R. * R.' *" ??? - 1 = — = — = = —-; whence tan N T tann T tan N T tann’ 2 R tan n R. tam Nº R tan m' T = ' = and r = ; therefore tam N' tan N tan NT’ 2 R tan n_R tam N' R tan n' = N'-H n' — N– n R+ *==== *T* *-as-s-s-s- tan N tan N tan N 14 ASTRONOMY. (N + n’—N—n) tan N tan N-i-tan n–tan N.—tan my Whence r, R and r become known, and consequently PZ = } (N+n+R+r); also the polar distance PF = Z.P –ZF = ZP— (n+r), and PK = ZP-ZK = ZP-(n + r.). This method of finding the refraction is by Boscovich. When neither of the zenith distances exceeds 70° or 75°, it gives it with considerable accuracy. From which we obtain R = 15. When the true distance of the pole from the zenith, and the apparent Zenith distance and corres- ponding refraction, of a star on the meridian, are known, the true polar distance of any other star may be determined from its observed meridian altitude or zenith distance. If M be the true situation of the star on the meridian, and we put n” for the observed apparent zenith distance, and r" for the corresponding refraction, we have, tan m tan m'' or r" = " tam n". 5 7" ** tam m consequently PM = PZ +ZM = Z.P.--n” + r", becomes known. 16. The true zenith distance of the pole and polar distance of a star being known, the refraction may be found for any observed altitude, by knowing also the interval between the times of observation and of the star’s passage over the meridian. Let Sº be the place of the star. Then in the triangle ZPS'', the sides ZP and PS" are given, and also from the observed in- terval of time, the angle ZPS". Consequently the side ZS" may be calculated The difference between ZS" thus obtained, and the observed zenith distance, is the corresponding refraction. In this way the refraction may be found for different apparent alti- tudes, from the horizon to the zenith. 47. The refraction at a given altitude is subject to CHAPTER III. 15 some change depending on the variation in the state of the air as indicated by the barometer and thermometer. The refractions which have place when the barometer stands at 29.6 inches and the thermometer at 50° are called mean refractions. Table II. contains the mean refractions for different altitudes from the horizon to the zenith. Above the altitude of 15 and 20°, these are at all times sufficiently accurate, except in cases when the greatest precision is required. Formulea have been investigated for obtaining the refraction with reference to the state of the barometer and thermometer; but these investigations do not be- long to an elementary treatise. OTHER EFFECTS OF REFRACTION. 18. As refraction elevates the heavenly bodies in verticle circles, and as these circles continually ap- proach each other from the horizon till they meet in the zenith, it is evident that the apparent distance of any two of those bodies must be less than the true dis- tance. 19. The refraction increases with an increase of zenith distance. The lower part of the sun or moon is therefore more refracted than the upper part, so that the vertical diameter is shortened and the body ap- pears of an elliptical form. This effect is most ob- servable near the horizon, where, on account of the more rapid increase of the refraction, the difference be- tween the vertical and horizontal diameters may amount to # part of the whole diameter. The hori- zontal diameter also suffers a slight diminution. (18) 20. At the true horizon the refraction is about 28%. Hence it follows that when any of the heavenly bodies are really in the horizon, they appear to be 28% above 46 ASTRONOMY. it, and that therefore refraction retards their setting, and accelerates their rising. 24. When the sun descends below the horizon of any place, its rays continue for some time to reach the upper parts of the atmosphere, and are refracted and reflected so as to occasion considerable light, which gradually diminishes as the sun descends farther be- low the horizon, and prevents an immediate transition from the light of day to the darkness of night. The same effect, though in a reverse order, takes place in the morning previous to the sun’s rising. The light thus produced is called the Crepusculum or Twilight. CHAPTER IV. Latitude of a Place.—Figure and Eatent of the Earth.-Longitude. 4. Let HZRN, Fig. 9, represent the meridian, Pp. the axis of the heavens, Z, the zenith, HOR, the hori- zon, and E00 the equator, the latter two seen edge- wise. Then ZQ is the latitude of the place(2.9). But ZQ = PQ–PZ = ZH-PZ = PH; therefore the la- titude of a place is equal to the altitude of the pole at that place. A method of obtaining the altitude of the pole has been shown (3.44). 2. Because ZR = 90°, RQ = 90°– ZQ; therefore the altitude of the point of the equator which is on the meridian, at any time is equal to the complement of the latitude of the place. FIGURE AND EXTENT OF THE EARTH, 3. By the figure of the earth is meant the general form of its surface, supposing it to be smooth, or that cHAPTER Iv. 47 the surface of the land corresponded with the surface of the ocean. This excludes the consideration of the irregularities in its surface, occasioned by mountains and vallies, which indeed are very minute when com- pared with the whole extent of the earth. 4. Experiment proves that the direction of gravity, at any place on the earth, is perpendicular to the free surface of still water: Hence it is perpendicular to the general surface of the earth at that place. The straight line which represents the direction of gravity, at any place, is called the Vertical. 5. Let EPQ p, Fig. 3, be a meridian of the earth, Pp the axis, EQ a diameter of the equator, and A and B two places on the meridian. If the earth be a sphere, the direction of gravity at each of the places will pass through the centre of the earth, and therefore the angle FCA will be the latitude of A, and ECB the latitude of B (2.9). Hence if the latitudes of A and B be de- termined (4), the angle ACB = ECB — ECA, be- comes known. The distance AB may be obtained by actual measurement. Then as the angle ACB : 360° ; : distance AB : to the circumference of the earth. 6. As the angle ACB : 19 : : distance AB : the length of a degree of latitude. Now, if the earth be a sphere, the length of a degree of latitude must be the same on any part of the meridian. But it has been found by observation and measurement, at different places, that the length of a degree increases in going from the equator towards the pole. At the equator the length of a degree is 68m. 1280yds, at latitude 45° it is 69m. 79yds, and at latitude 66°; it is 69m. 465 yds. 7. The increase in the length of a degree of latitude, as the latitude itself increases, proves that the meridian 4. #8 ASTRONOMY. is not a circle, and leads to the supposition that it is an ellipse, having the axis of the earth for its shorter axis, Let the ellipse EPQ p, Fig. 4, represent the meri- ilian, Pp the axis of the earth and EQ a diameter of the equator, Also let AD be perpendicular to the curve at A, and let the situation of the point B be such, that BF, perpendicular to the curve at B, may make the angle BFP = EDA. Then the angle EDA is the difference of latitude between the places E and A, and the angle BFP is the difference of latitude between the places B and P. Now it is evident from inspec- tion of the figure or from the consideration of the de- crease of curvature from E to P, that the distance BP, corresponding to a given difference of latitude near the pole, is greater than the distance EA, corresponding to the same difference of latitude, near the equator. 8. Analytical investigations founded on the measure of a degree in different latitudes and on different meri- dians, prove that a meridian is nearly an ellipse, and that the figure of the earth is nearly an oblate spheroid, having the polar diameter, to the equatorial in the ratio of 320 to 321, 9. Calculations, made from the most accurate mea- surements, give the mean diameter of the earth 7920 miles, the circumference 24880 miles, and the length of a degree of a great circle 69 ºr miles.* The differ- ence between the equatorial and polar diameters, is about 25 miles. 40. The angle contained in the plane of the meri- dian, between the radius of the equator and a straight line from any place to the centre of the earth, is called * These are the numbers, nearly, that are given in an ingenious essay by R. Adrain, Prof. of Math. in Columbia College, and published in the Trans- actions of the Amer. Philos. Society, Vol. 1. New Series, 4CHAPTER IV. 49 the Reduced Latitude of the place. And the angle contained between a vertical line at any place and the straight line to the centre of the earth, is called the Reduction of Latitude. Thus the angle ECA is the reduced latitude of the place A, and CAD is the re- duction of latitude. Since ECA = EDA – CAD, it is plain that the Reduced latitude is equal to the true latitude, dimin- ished by the reduction of latitude. 44. The true latitude of a place being given, the reduced latitude may be found by the following pro- portion. The square of the equatorial diameter, is to the square of the polar diameter, as the tangent of the true latitude, is to the tangent of the reduced latitude. Let the circle EBQ, Fig. 5, be described on the equatorial diameter EQ, and let AF be perpendicular to the ellipse at A, and BAG perpendicular to EQ. Put EQ = a, Pp = b, the true latitude of A = the angle EDA = L, and the reduced latitude ECA = R. Then, CG tan R = AG = DG tan L; hence Dg tan R. But (Comic Sections), GG tan L DG b° CG T a 2 #. tam R. b? Therefore = — tan L gº’ or a” : b% :: tan L : tari R. 42. The equatorial and polar diameters of the eartli, and the latitude of a place being given, the radius from the place to the centre of the earth may be found. Put the angle BCG = M; then to obtain the radius AG, we have BG = CG tan M, and AG = GG tan R; 20 ASTRONOMY. therefore BG º CG tan M -> tan M. tam R. AG T CG tan R. but (Comic Sections,) ; : ; tan M. {!, {!, a b? := - º - *- t - - s tºº t => Hence tan R ºf 5’ or tan M. b &ll b a 2 an L = b tan L. & And AC = BC sim CBG -: # a cos M sin CAG cos R. LONGITUDE, 13. We have shown (4) how to obtain the latitude of a place. But it is evident that the latitude is not of itself sufficient to designate the situation of a place, because all the points in a circle on the earth’s sur- face, parallel to the equator, have the same latitude. Something more then is necessary to designate, with precision, the situations of places. As the meridians cut the equator at right angles, they are conveniently made use of for this purpose. 14. A meridian which passes through some par- ticular place is called the First Meridian. The angle contained between the first meridian and a meridian through any place, is called the Longitude of that place. Longitude is measured by the arc of the equator, intercepted between the first meridian and the meridian passing through the place, and is called east or west according as the latter meridian is to the east or west of the first meridian. 45. Different nations have adopted different first meridians. The English reckon longitude from the meridian which passes through their Observatory, at Greenwich, near London; and the French from the meridian of their Observatory at Paris. As there is CHAPTER IV. 24 no public Observatory in the United States, there is not a uniformity with respect to a first meridian. Some reckon the longitude from the meridian of Washington, some from that of Philadelphia, and others from the meridians of other principal cities. But for astronomical purposes we reckon our longi- tude from the meridian of Greenwich or of Paris. 16. Since the diurnal motion of the stars is from east to west (1.5), any particular star must come to a given first meridian, sooner than to the meridian of a place which has west longitude, and later than to the meridian of a place which has east longitude (14); and the difference of times will be found by allowing one sidereal hour for each 15° of longitude, and in the same proportion for odd degrees, minutes and seconds (3.5). It follows therefore that, if the time at which some star passes the first meridian, be observed by an accurate watch or portable chronometer, regulated to keep sidereal time; and if it be then taken to a place to the east or west of the first meridian, and the time, at which the same star passes the meridian of that place, be observed by it, the difference of times, converted into degrees, by allowing 15° to the hour, will express the longitude of the place. There are various other methods of determining the longitudes of places, some of which will be noticed in succeeding parts of the work. Table 1. contains the latitudes of a number of the principal cities, and their longitudes from the meridian of Greenwich, expressed both in degrees and in time, 22 ASTRONOMY. CHAPTER W. Om Parallaa. 4. The directions in which a body is seen at the same instant, from different places on the earth’s sur- face, must in reality be different; but the distances of the fixed stars are so immensely great, (1.12), that for any one of them the difference is perfectly insensible. This is not the case with the sun, moon, and planets. They are sufficiently near to the earth, to have the di- rections in which they are seen, sensibly influenced by the situation of the observer. Astronomers, therefore, in order to render their observations easily comparable, and for convenience in calculation, reduce the situation of a body, as observed at any place on the earth’s sur- face, to the situation in which it would appear from the centre. The observed situation of the body is called its fly- parent place, and the situation in which it would ap- pear from the earth’s centre is called its True place. 2. The angle contained between two right lines, conceived to be drawn from a body, one to the centre of the earth and the other to the place of the observer, is called Parallaa. It is also sometimes called Paral- laac in Altitude. 3. Let ADE, Fig. 6, represent the earth, considered as a sphere, C its centre, A a place on its surface, Z. the zenith of the place A, and B the situation of a body; then will ZAB be the apparent zenith distance of the body, ZCB its true zenith distance, and ABC its parallax in altitude. The parallax ABC = ZAB — ZCB = apparent zenith distance — true zenith distance = 90° — appa- CIIAPTER V. 23 rent altitude — (90° — true altitude) = true altitude — apparent altitude. 4. The parallax in altitude of a body, when its ap- parent zenith distance is 90°, is called the Horizontal Parallaac. 5. Supposing a body to continue at the same dis- tance from the earth, the sine of the parallax in alti- tude is equal to the sine of the horizontal parallax, multiplied by the sine of the apparent zenith distance. Put R = AC = \mean radius of the earth, D = CB = distance of the body from the earth’s centre, N = ZCB = true zenith distance, = ABC = parallax in altitude, and a = the horizontal parallax; then N + p = ZAB = apparent zenith distance, And sin p = ; sin CAB = # sim ZAB = jºin (N + p.) º But when N + p = 20°, p becomes ar, Hence sin a = . sin 90° = ; Consequently sin p = sin a sin (N + p.) 6. The distance of a body from the centre of the earth is equal to the radius of the earth, divided by the sine of the horizontal parallax. o e R. Since sin a = D (5), We have D = R_. SII] ar Hence, as the radius of the earth has been deter- mined (4.9), when the horizontal parallax of a body is 24 ASTRONOMY. known, its distance from the centre of the earth is easily found. 7. The distances of the heavenly bodies are so great - that p, the parallax in altitude, and ar, the hori- Zontal parallax, are always very small angles; even for the moon which is much the nearest, the value of a does not at any time exceed 62'. We may there- fore, without sensible error, use p and w themselves, instead of their sines. If this be done, the last formu- lap, in the two preceding articles, become, p = w sin (N + p,) - * p 1 and D - R - R. 8. In the fraction ; of the last formula, 1 represents the radius and a the measure of the horizontal paral- lax. Hence, in order to render the numerator and de- nominator of the fraction homogeneous, if a be ex- pressed in seconds, we must also express the radius in seconds. Because 6.2831853 is the length of the circumference when the radius is 1, and 1296000 is the number of seconds in the circumference; we have 6.2831853 : A :: 4296000" : 206264".8 = the length of the radius expressed in seconds. Hence if the value of w is expressed in seconds, 206264. 1.8 zy" D = R. 9 If the meridian zenith distances of a body be ob- served on the same day, by two observers, remote from each other on the same meridian, and at places, whose CHAPTER V. 25 latitudes are known, its horizontal parallax: may, from thence be determined. Let AEA'Q, Fig. 7, represent a meridian of the earth, con- sidered as a sphere, C its centre, EQ a diameter of the equator, A and Aſ the situations of two observers, Z and Z'their zeniths, and B the situation of a body on the meridian. Put L = ECZ = latitude of the place A, L' = ECZ' = do. A', d = ZAB = apparent zenith distance at A, and d = Z'A'B = do. A'. Then, \ ACA' = ECZ + ECZ' = L + L', BAC = 180° — ZAB = 180° — d, BA'C = 180° — Z'A'B = 180° — d", \ and ABA' = 360° — (ACA' + BAC + BA/C) = d -- d. — (L + L'). Again (7) ABC = a sin d, A'BC = a sin d', and ABA' = ABC + A'BC = a sin d -- ar sin d'. ABA' _ d -- d' — (L + L') H := –– = CIAC6 Zºr sin d -- sin d' sin d -- sin d' 40. If the meridian zenith distances of a fixed star, which passes the meridian nearly at the same time with the body, be observed, as well as those of the body, the horizontal parallax may be obtained, independent of the latitudes of the places. For if S be the situation of the star when on the meridian, we then know, BAS = ZAS — ZAB and BA'S = Z'A'B — Z'A'S. But ABA' + BAS = BLS = ASA' + BA/S, or ABA' = BA'S – BAS + ASA'. Or since the angle ASA' is insensible (1), we have, ABA' = BA'S – BAS, BA'S – BAS sin d -- sin d" and zr = 5 26 ASTRONOMY. 41. It is not necessary that the two observers should be on precisely the same meridian; for if the meridian zenith distances of the body be observed on several successive days, its change of meridian Zenith distance in a given time will become known. Then if the dif. ference of the longitudes of the places is known (4.16), the zenith distance of the body as observed on one of the meridians, may be reduced to what it would be, if the observation had been made, in the same latitude on the other meridian, In the year 1751, Wargentin, at Stockholm, and La- caille, at the Cape of Good Hope, made the requisite observations on the planet JMars, and determined its parallax at the time of observation to be 34".64. Hence, (8), l 206264".8 D = R.º.º.º. 24".64 == R × 8374. The distance of Mars from the earth’s centré was, therefore, at the time of observation, 8371 times the mean radius of the earth. 12. For the moon, whose parallax is much greater than that of any other of the heavenly bodies, it is ne- cessary to take into view the spheroidical figure of the earth. Let the ellipse PE p 3, Fig. 8, represent a meridian of the earth, C its centre, EQ a diameter of the equator, and Z and Z' the true zeniths of the places A and A'. The angle 2AZ = CAd = reduction of latitude for the place A, and 2'A'Z' = CA'd' – reduction of latitude for the place A', may be found (4.11), and thence the angle zAB = ZAB — 2AZ and z'A'B = Z'A'B — 2'A'Z', are known. Now if w and a' be the horizontal parallaxes of the moon at CHAPTER W. 27 A and A', and R and R stand for the radii CA and CA, we have (7), * - R R! R! * = D = +; whence ar' = 2J'. Zºr =; whence s = Px Let d and dº stand for the reduced zenith distances 2AB and z'A'B; then (7), ABC = a sin d, ABC = a sin d = # wr sin d, and ABA = ABC + ABC = a sin d -- # w sin d', or R or sin d -- Rº a sin d = R × ABA'. Hence a = R _X ABA. © R sin d -- R' sin dº 43. The horizontal parallax of the moon, to an observer at the equator, is called the Equatorial Pa- 'pallaa. If a " = the equatorial parallax, and R' = CE = the ra- dius of the equator, then, R" 2J" — R" × ABA' R. R. sin d -- R sin dº f / 2J" " - 44. From observations made in the year 1751, by La- caille, at the Cape, and Lalande, at Berlin, and from other methods, which have been used for the same pur- pose, the moon’s equatorial parallax is found to vary from 53' 52" to 61' 32". Hence, D R, 206264".8 R" × 64, nearly = its greatest distance, 3212, 1 and D = R". ** = R' x 56, nearly – its least dis- tance. Consequently the moon’s mean distance is about 60 times the equatorial radius of the earth. 28 t ASTRONOMY. 45. The mean equatorial parallax of the moon is ; (53' 52" + 64’32") = 57 42". But the parallax at the mean distance is only 57' 22". Let D, D', and D", be the least, greatest, and mean distances of the moon from the earth, and w, w", a ", the corresponding pa- rallaxes. Then (4), R” R” D = + , D' = — and D" = –tº––. Sln 20" Sln zº' Sin ar'' R” R!! * (p + D) = 3 (+++) # ( ) * \sin z; Sin zº' / / 1. W Hence R” 1. ( R” + R!' ) sinzº T * \sin ar sin zº' 2 _ sin a + sin ar' = 2 sin # (a + ar) cos # (or — ar') sin zº’’’ sin a sin ar' sin a sin ar' (Ap. 20.) ſº sin zy sim zy' Sin ar' = (A) d / sin # (a + w() cos # (a — ar') As the arcs are small, we may, without material error, con- sider cos # (a — ar') = 1, and for the other terms take the arcs instead of their sines. We shall then have, zr” 2". zr' * 2 ZJ". zy' (B) ~ I (Tº) tº z. Tº From either of the expressions, A and B, the value of a " is found equal 57' 22". 16. The sun’s distance is so great, that its parallax can not be determined with precision by the preceding method. It may, however, be shown to be about 8" or 9". By a method that will be noticed hereafter, the sun’s mean horizontal parallax is found to be 8".7. Hence its distance is, D R, 306864".8 = 23708 x R' 8".7 CHAPTER VI. 29 CHAPTER VI. JApparent Path of the Sun-Fived Stars. 4. The sun partakes with the stars in the apparent diurnal motion; but the time between his passing the meridian on any day, and his passing it the next, is found to be greater than a sidereal day. The sun, therefore, appears to have a motion eastward among the fixed stars. The altitude of the sun, when on the meridian, is not the same on two successive days. On the 20th of March and 22d of September, it is about the same as the meridian altitude of the equator; from the former time to the latter, it is greater; and during the other part of the year, it is less. On the 21st of June the sun’s meridian altitude is greatest, and it then exceeds the meridian altitude of the equator about 23° 28'; on the 21st of December it is least, and is then less than that of the equator by the same quantity 23° 28'. The sun’s motion appears, therefore, to be in a. plane, cutting the ecliptic in two opposite points. 2. Let HZRN, Fig. 9, represent the meridian, HOR, the horizon, Pp the axis of the heavens, EQA the equator, ASFG the apparent path of the sun, P the north pole, and Z the zenith. Also, let S be the situa- tion of some bright star, which, in the latter part of March or in April, passes the meridian a short time before the sun.* Let the time at which the star passes the meridian be observed by a clock, accurately regu- lated to sidereal time. If then the altitude SR of the Sun’s centre, when on the meridian, be observed and corrected for refraction and parallax, and also the time * The brighter stars may be distinctly seen in the day time, with an as, tronomical telescope, 30 l ASTRONOMY. be observed, we have the polar distance PS = 180°– (PH + RS) = 180° — (latitude of the place + alti- tude of the sun’s centre), and the angle BPQ = the difference of times, converted into degrees (3.6). If similar observations be made on the same star and the sun, a few weeks after, when the sun has moved in its apparent path to S', we shall have PS' and the an- gle BPD. Consequently the angle SPS' = BPD — BPQ, becomes known. If the sun’s apparent path ASF be a great circle, SPS' will be a spherical triangle, in which we know the two sides PS, PS' and the contained angle SPS'; whence the angles PSS" and PS'S may be found. Then in the right angled triangle AQS, we have ASQ. = PSS! and QS = 90° — PS, with which the angle A may be found. We may also find the angle A from the triangle ADS', in which are given ASD = 180° — PS'S and DS' = 90° – PS'. The value of the angle A, thus determined from the two triangles AQS and ADS', is found to be the same. Hence the appa- rent path ASG of the sun, is a great circle. It there- fore cuts the equator in two points, A and F, at the distance of 180°. But little more than half of the circle is shown in the figure, as the whole, if accurately represented, would occupy too much room. 3. The great circle which the sun appears to de- scribe, is called the Ecliptic. 4. The points in which the ecliptic cuts the equator, are called the Equinoctial Points. The time when the sun is at the equinoctial point, in his passage from the south to the north side of the equator, is called the Vernal Equinoa; and the time, when he is at the other equinoctial point, is called the flutumnal Equinoa. --- CIIAPTER WI. t 31 The terms Vernal Equinoa and Autumnal Equinoa: are frequently applied to the equinoctial points them- selves. 5. The two points in the ecliptic, which are at 90° distance from the equinoctial points, are called the Sol- stitial Points. The point T represents the situation of the solstitial point on the north side of the equator; the other is on the part of the ecliptic left out of the figure. The time, when the sun is at the northern solstitial point is called the Summer Solstice, and the time, when he is at the southern solstitial point, is called the Winter Solstice. 6. A great circle, passing through the equinoctial points and the poles of the heavens, is called the Equi- noctial Colure. Another great circle, passing through the solstitial points, is called the Solstitial Colure. 7. The angle which the ecliptic makes with the equator, is called the Obliquity of the Ecliptic. The obliquity of the ecliptic is found to be 23° 28 near- ly (2). 8. Two small circles parallel to the equator and touching the ecliptic at the solstitial points, are called the Tropics. That, which is on the north side of the equator is called the Tropic of Cancer, and the other, the Tropic of Capricorn. Thus aſſb is the tropic of Cancer, and cde, the tropic of Capricorn. 9. Two small circles parallel to the equator and at a distance from the poles equal to the obliquity of the ecliptic, are called Polar Circles. The one, about the north pole, is called the flrctic Circle; the other, about the south pole, is called the flntarctic Circle. Thus fºh is the Arctic Circle, and kmn, the Antarctic. 10. Circles, corresponding to the tropics and polar circles, conceived to be drawn on the earth, divide its 32 ASTRONOMY. surface into five parts, called Zones. The part con- tained between the tropics, is called the Torrid Zone, the two parts between the tropics and polar circles, are called the Temperate Zones, and the other two parts within the polar circles, are called the Frigid Zones. 41. The ecliptic is supposed to be divided into twelve equal parts, which are called Signs. Each sign, therefore, contains 30 degrees. The division of the signs commences at the vernal equinox, and they are numbered in the direction of the sun’s apparent motion in the ecliptic. The signs of the ecliptic are; sometimes, designated by names or characters, instead of numbers. The names of the twelve signs with their correspond- ing numbers, and the characters by which they are usu- ally denoted, are, S. S. 0. Aries ‘P. 6. Libra = • 1. Taurus 8. 7. Scorpio 1ſt, 2. Gemini II. 8. Sagittarius 1. 3. Cancer ga. 9. Capricornus V3. 4. Leo Sl. 10. Aquarius &. 5. Virgo 102. 11. Pisces 3{. Aries, Taurus, Gemini, Cancer, Leo, and Virgo lie on the north side of the equator and are called JWorth- ern Signs. The others lie on the south side, and are called Southern Signs. Capricornus, Aquarius, Pisces, Aries, Taurus and Gemini are called ºffscending Signs, because while the sun is in them, his meridian altitude continually in- creases. Cancer, Leo, Virgo, Libra, Scorpio and Sagittarius are called Descending Signs, because the CHAPTER VI. 33 sun’s meridian altitude continually decreases, while he is in them. f 42. A zone of the heavens extending in breadth to 8 or 9° on each side of the ecliptic, is called the Zo- diac. Within the zodiac, all the planets perform their motions, except three of those recently discovered. 43. Any great circle, which passes through the poles of the ecliptic, is called a Circle of Latitude. 14. The arc of the ecliptic, intercepted in the order of the signs, between the vernal equinox and a circle of latitude, which passes through a star, is called the Longitude of the star. And the arc of the circle of latitude, intercepted between the star and the ecliptic, is called the Latitude of the star. Latitude is said to be north or south, according as the body is on the north , or south side of the ecliptic. 15. Any great circle, which passes through the poles of the equator, is called a Circle of Declination. 16. The arc of the equator, intercepted between the vernal equinox and a declination circle, which passes through a star, is called the Right flscension of the star. And the arc of the circle of declination, inter- cepted between the star and the equator, is called the Declination of the star. Declination is said to be north or south, according as the body is on the north or south side of the equator. Longitude and Right Ascension are both reckon- ed from the vernal equinox, round to it again, in the order of the signs. 47. The situations of the fixed stars, are generally expressed by right ascension and declination, and those of the sun, moon and planets, by longitude and latitude. With the obliquity of the ecliptic known, the longitude and latitude of a body may be obtained from the right 6 34 ASTRONOMY. ascension and declination, by means of spherical Tri- gonometry. On the contrary from the longitude and latitude, the right ascension and declination may be found. 18. Let EQ, Fig. 10, represent the equator, EC the ecliptic, P and P' their poles, E the vernal equinox, PSR a circle of de- climation and PSL a circle of latitude, both passing through a body at S. Then will ER be the right ascension, RS the decli- nation, EL the longitude and LS the latitude of the body. Put R = ER = right ascension, D = RS = declination, L = EL = longitude, A = LS = latitude, • = REL = obliquity of the ecliptic, a = RES, * and y = LES. 19. When the right ascension and declination are given to find the longitude and latitude, we have, tang RS tang ER, t Or == S 5 t S S- tang rur 5 ang RES sin ER, ang E cos RES - t t _ 90s LES tang ER. ang EL cos LES tang ES cos REST? and tang LS = tang LES sin EL. Or tang a = tang p. tang L = * (2–0) tang R. Sl]] R. COS (ß and tang x = tang (2–2) sin L. If attention be given to the rules for trigonometrical signs, and tang D be considered negative when the declination is south, these for- mulae will apply, whatever be the situation of the body; observing that the longitude and right ascension are always, either, both be- tween 90° and 270° or both between 270° and 90°, and that, when the tang a comes out negative, the latitude is south. Let S be the sun's place in the ecliptic; then ES' = sum's longitude, Hence CHAPTER VI. 35 * tan ER, 2 tude = tam ES' = -ºttº:- = tan. Sun's longitu cos RES' tan Sun's right ascension . . cos Obliquity of the ecliptic 20. When the longitude and latitude are given to find the right ascension and declination, we have, tang LES = tang LS. tang ES = tang EL. sin EL’ cos LES’ ... I's cos RES tang EL. tang ER = cos RES tang ES = . cos LEST? and tang RS = tang RES sin ER. tang X cos (y-º-o) tang L. O t - g A. == y r tang y in L’ tang R cos y and tang D = tang (y-º-w) sin R. * When the latitude is south, tang a must be considered nega- tive; and if tang D come out negative, the declination will be South. ^ * For the sum we have, tan Sun’s right ascension = tan ER = cos RES' tan ES' = cos Obliquity of the ecliptic X tan Sun’s longitude; and sim Sun's decl. = sin RS' = sin RES' sin ES' = sin Obliquity of the ecliptic × sin Sun's long. 21. The angle contained between a circle of latitude and circle of declination, both passing through the sun or a star, is called the flngle of Position of the star. If PP be produced to meet the equator in N, then in the tri- angle PSP, PS = complement of latitude, PS = complement of declination, PP = obliquity of the ecliptic, PPS = 180° — NPR = 180°–NR = 180° – (EN–ER) = 180° — (90° — ER) = 90° -- ER = 90° -- right ascension, and PP'S = EM — EL = 90° — longitude. With any three of these five parts given, the angle of Position PSP may be found. When the longitude, latitude, and obliquity of the ecliptic are given, we have, putting S = PSP the angle of position, (App. 37). 36 ASTRONOMY. cot S = cot PP" sin P'S — cos P'S cos PP'S sin PPS _ cot a cos ? – sin a cos (90°-L) sin (90° — L) - cot a cos A — sin A sin L c - cos L c sin L (; : cos x – sin a cos L \sin L' ) COt aſ ve = tan L. • COS X — Sin X | . SII] Make tan z = sin L tan a. cot OS 2. Then # = − = + = * * = COS 2. sin L sin L tan a T tanz sin ºf COS 2. º Hence cot S = tan L. ( § f. cos x – sin •) SII]. 2. tan L º º = *-tt. (cos 2 cos A — sim 2 sin x). S1 In 2 But (App. 14) cos 2 cos x — sin 2 sin x = cos (2 + x). tan L Therefore cot S = *-*. cos (2 + 2). (A) SIRT 2. "When the longitude, declination, and obliquity of the ecliptic, are given, we have Sin PP' Sin PP'S cos L sin a - { - —--—- (B) sin PS cos D For the sum, A = 0, and the formula cot a cos x — sin à sin L sin S = 'cot S = becom C cos L 2 eS, t t & cot S - cot & cos L’ or tans –“ H = Lian. f (C) cot w It is easy to see that the northern part P'S of the circle of lati- tude is to the west of the northern part PS of the circle of decli- nation, when the longitude is less than 90° or more than 270°, and to the east when it is between 90° and 270°. a CHAPTER VI. - 37 , SITUATIONS OF THE FIXED STARS. 22. In order to distinguish the fixed stars from each other, the ancients supposed the figures of men, ani- mals or other objects to be drawn on the concave sur- face of the heavens. This mode of distinction is still used. The group of stars contained within the con- tour of any such figure, is called a Constellation. The following tables exhibit the names of the prin- cipal constellations. 1. ANCIENT CONSTELLATIONS. JWorthern. 4. Ursa Minor, The Little Bear. 2. Ursa Major, The Great Bear. 3. Draco, The Dragon. 4. Caepheus, Caepheus. 5. Bootes, Bootes. 6. Corono Borealis, The Northern Crown. 7. Hercules, Hercules kneeling. 8. Tyra, The Lyre. 9. Cygnus, The Swan. 40. Cassiopea, The Lady in her Chair. 11. Perseus, Perseus. 42. Auriga, The Wagoner. 13. Serpentarius, Serpentarius. 14. Serpens, The Serpent. 15. Sagitta, The Arrow. 16. Aquila, The Eagle. 17. Delphinus, The Dolphin. 18. Equulus, The Horse’s Head. 19. Pegasus, The Flying Horse. 20. Andromeda, Andromeda. 21. Triangulum, The Triangle. 38 ASTRONOMY. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 45. . Ara, . Corona Australis, . Piscis Australis, : Constellations of the Zodiac. Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpio, Sagittarius, Capricornus, Aquarius, Pisces, Cetus, Orion, Eridanus, Lepus, Canis Major, Canis Minor, Argo, Hydra, Crater, Corvus, Centaurus, Tupus, The Ram. The Bull. The Twins. The Crab. The Lion. The Virgin. The Scales. The Scorpion. The Archer. The Goat. , The Water-bearer. The Fishes, Southern. 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 Altar. The Southern Crown. The Southern Fish. 2. NEW SOUTHERN CONSTELLATIONS. . Columba Noachi, Noah’s Dove. CŞHAPTER VI. * 39 44. 42. 43. 14. ! 4.* 4. The Royal Oak. Robur Carolinum, Grus, Phoenix, . Indus, Pavo, . Apus, ºlvis Indica, Apis, Musca, Chamelion, The Crane. The Phoenix. The Indian. The Peacock. The Bird of Paradise The Bee or Fly. The Chamelion. Triangulum Australe, The Southern Triangle. Piscis volans, Passer, Dorado, Xiphias, Toucan, Hydrus, The Flying Fish. The Sword Fish. The American Goose. The Water Snake. • 3. HEVELIUS’ CoNSTELLATIONS. JMade out of the Unformed Stars. Lynx, Leo Minor, Coma Berenices, Asteron and Chara, Cerberus, Vulpecula and Anser, Antinous, Scutum Sobieski, Lacerta, Camelopardalis, . Monoceros, Sextans, The Lynx. The Little Tion. Berenice’s Hair. The Greyhounds. Cerberus. The Fox and Goose. Antinous. Sobieski’s Shield. The Lizard. The Camelopard. The Unicorn. The Sextant. 23. The stars of a constellation are distinguished by the letters of the Greek alphabet, which are ap- plied to them according to their apparent, relative size. The principal star in the constellation is named 2, the | 40 ASTRONOMY. second in order 6, the third y, and so on. When the number of stars in a constellation, exceeds the letters in the Greek alphabet, the letters of the Roman alpha- bet are applied to the remainder in the same manner; and when these are not sufficient, the numbers 1, 2, 3, &c. are used to designate those that are left. Some of the fixed stars have particular names, as Sirius, Aldebaran, Arcturus, Regulus, &c. 24. The stars are also divided into classes, depend- ing on their apparent magnitudes. Those of the first class, are called stars of the first magnitude, those of the second, stars of the second magnitude, and so on, to stars of the siath magnitude, which includes all those that are just visible to the naked eye. Those stars which are not visible without the aid of a tele- scope, are called telescopic stars. 25. In the triangle ASQ, Fig. 9. having QS and the angle ASQ (2), AQ may be found; then AQ — QB = AB = the right ascension of the stars. When the right ascension of one star is obtained, the right ascension of any other may be found by observing the difference between the time of its passing the meridian and the time of the known star doing the same. Iets' be the situation of the stars when on the me- ridian, then its declination Bs = Qs' = Rs' — RQ = Rs' — (90° – PH) = Rs' + PH — 90° = the cor- rect meridian altitude of the star + latitude of the place — 90°. When the right ascensions and declinations of the stars have been obtained, their longitudes and lati- tudes may be calculated (19). 26. A table containing a list of fixed stars, desig- nated by their proper characters, and giving their right CHAPTER VI. 44 ascensions and declinations or their longitudes and latitudes, is called a Catalogue of those stars. 27. Hipparchus began the first catalogue of the fixed stars 120 years before the Christian era. This cata- logue, with some additions, was afterwards published by Ptolemy, and contained the situations of 1022 stars. The Britannic Catalogue, published by Flam- stead, in 1689, contained the situations of 3000 stars. Since that period various other catalogues have been published, some of which are very extensive. Bode's Atlas and Catalogue contain the situations of A7,000 stars. The Catalogues of Lacaille, Bradley, Mayer, and Maskelyne are not extensive, but they are valued for their accuracy. All the fixed stars, visible to the naked eye, with some others, are represented on celestial globes of 12 or 18 inches diameter. * 28. The number of stars visible with the best tele- scopes, amounts to several millions: but the number vi- sible to the naked eye, is much less than is generally supposed by those who only judge from the impressions made, when noticing them on a fine evening. The number thus visible at any one time above the horizon does not much exceed 1000. * 29. Many of the stars, which to the naked eye, or through telescopes of small power, appear single, are found with high magnifiers to consist of two, three, or more stars, extremely near to each other. 30. The fixed slaps are not entirely exempt from change. Several stars, which are mentioned by the an- cient astronomers, have now ceased to be visible, and some are now visible to the naked eye, which are not in the ancient catalogues. 31. Many spaces are discovered in the heavens, 7 42 ASTRONOMY. which are faintly luminous, and shine with a pale white light. On applying to them telescopes of great power, they are found to consist of a multitude of small stars, distinctly separate, but very near to each other. These are called JWebulae. The JMilky Way is a space of this kind, and is visible to the naked eye. PRECESSION OF THE EQUINOXES. 32. By comparing catalogues of the same fixed stars, formed at different periods, it is found that their lati- tudes continue very nearly the same, but that all their longitudes increase at the rate of 50".ſ in a year. As the latitude of a star is its distance from the ecliptic (14), it follows from the first mentioned circumstance, that the plane of the ecliptic remains fixed, or very nearly so, with respect to the situations of the fixed Stars. 33. The longitude of a star being the arc of the ecliptic, intercepted in the order of the signs, between the vernal equinox, and a circle of latitude passing through the star (44), it follows from the circumstance of all the stars having the same increase of longitude (32), that the vernal equinox must have a motion along the ecliptic in a direction contrary to the order of the signs, amounting to 50%.1 in a year. As the autumnal equinox is always at the distance of 180° from the ver- nal equinox (2 and 4), it must have the same motion. This retrograde motion of the equinoctial points, is called the Precession of the Equinoaces. 34. As the ecliptic remains fixed (32), its pole must also continue in the same place; but the equator and its pole must change their situations, otherwise there could not be a motion in the equinoctial points. Let E'C', Fig. 11, be the ecliptic, P its pole, pºpD a circle about CHAPTER WI. 43 the pole P, at a distance equal to the obliquity of the ecliptic, EQC the equator, and PpI.G the solstitial colure. Then because E is the pole of PLQ, the pole of the equator EQC is in PLQ; it is also in the small circle p"pſ); it is therefore at p. Now, if the vernal equinox E, move by a retrograde motion to E', the solstitial point L will move a like distance to L'; there- fore E'Q'C' will then be the equator, Pp'L'Q' the sol- stitial colure, and p' the pole of the equator. Hence the pole of the equator has a retrograde motion, in a small circle about the pole of the ecliptic, and at a dis- tance from the latter pole, equal to the obliquity of the ecliptic. The precession of the equinoxes being only 50". 1 in a year, it must require 25869 years for them to move through the whole of the ecliptic; and the pole of the equator will evidently require the same time to make its retrograde revolution about the pole of the ecliptic. 35. The change in the situation of the equator, which produces the precession of the equinoxes, must also produce changes in the right ascensions and declina- tions of the stars. These changes are different, accord- ing to the situations of the stars with respect to the equator and equinoctial points. The change which takes place in the right ascension or declination of a star in the course of a year, is called the film.nual Varia- tion in right ascension or declination. Let s be the situation of a star, psab a declination circle, when EQC is the situation of the equator, EE/ the annual precession in longitude = 50".1 (32), and Em and p'sa' declination circles, when the situation of the equator is E'Q'C'. Then the difference between a's and as, is the annual variation in declination of the star 8, and the difference between E'a' and Ea, is its an- 44 ASTRONOMY. nual variation in right ascension. The annual varia- tions in right ascension and declination, may be ex- pressed in formulae, involving only the right ascension and declination of the star, and the obliquity of the ecliptic. ANNUAL VARIATION IN DECLINATION. 36. Let Sn be equal to sp' and np' be joined by the arc of a grèat circle; then as the arcs pp' and p'n are evidently very small, we may, without sensible error, consider pp' as the arc of a great circle, and the angle pnp/ as a right angle, excepting, with respect to the latter, the case in which the star is very near the pole, We may also consider any very small arc as equal to its sine or tangent, * Put a = pp = E'Em = obliquity of the ecliptic, R = Ea = right ascension, and v = variation in declimation. Then pp' = pºp sin pº := 50.1 sin & 50'’. 1 sin a sin pp'P sin 90° pn = pp' cos p'ps = 50", 1 sin a cos p'ps. . But p'ps = 90° — apQ = EQ — aQ = Ea = R, and pn = sp — Sn = sp — sp' = 90° — as — (90 — a's) = d.'s — aS = v. Therefore v = 50". 1 sin a cos R. When the declimation is north, as in the figure, the sign of v is the same as the sign of cos R; but when the declimation is south, the sign of v must be contrary to that of cos R. ANNUAL VARIATION IN RIGHT ASCENSION. 37. Let D = as - declination of the star, and W = its an- nual variation in right ascension. Then W = E'a' — Ea = E'a' — mb = E'm + a'b,; E'm = E'E cos EE'm = 50" l cos of p'n = pp' sin p'pm = 50". 1 sin a sin R; pºp' = p'n - 50”.1 sin a sin R. and sin p's COS d'S CHAPTER VI. +5 50”.1 sin a sin R, sim a's a'b = d'sb sin a's = p&p' sin a's = COS d’s = 50". 1 sin a sin R tan a's. As the quantity 50". 1 sin a sin R, which is multiplied by tan a's, is very small, and as the difference between a's and as, can only be a few seconds, we may, without sensible error, consider tan a's = tan as – tan D. Therefore a'b = 50". 1 sin a sin R tan D, and W = E'm -- a'b = 50". 1 cos a -- 50". 1 sin a sin R tan D. When the declination is south, tan D must be taken negative. The second term of the value of V is negative when the right as- cension is less than 180° and declination south, or when the right ascension is more than 180° and declimation north. In either of these cases, when sin R tan D is so great as to make the second term exceed the first, the value of V is negative. E’m = 50”.1 cos w = 46", is the annual, retrograde motion of the equinoctial points along the equator. 38. In catalogues of the fixed stars, which express their situations by right ascensions and declinations, the annual variations in these are also stated, with their proper signs. In some catalogues the north po- lar distances of the stars are given, instead of the de- clinations. The variations will be the same, except that the sign will be different when the north polar dis- tance is less than 90°. 39. With the right ascension and declination of a star for a given time and their annual variations, its right ascension and declination may be found, with considerable accuracy, for a time a few years later or earlier. Put t = the number of years, then t. W = its change in right ascension and t. v = its change in declination, nearly. If the time for which the right ascension and declination are required, is after the given time, t. V and t. v must be applied, with their signs as determined by the preceding formulae; but, if 46 ASTRONOMY > it is before the given time, t must be considered nega- tive, which will change the signs of t. V and t. v. When, from the right ascension and declination of a star for a given time, its right ascension and declina- tion are required with accuracy, for a time several years earlier or later, they can be found' by rigorous formu- laº, which have been investigated for the purpose; or with nearly the same facility, by calculating the lon- gitude and latitude of the star for the given time (19), adding to the longitude the precession in longitude, which will be the product of 50". A by the interval of time expressed in years and parts, and then with the longitude thus obtained and the latitude, calculating the right ascension and declination. 40. In consequence of the precession of the equi- noxes, the twelve signs of the ecliptic, which about 2000 years ago, respectively corresponded with the twelve constellations of the zodiac, bearing the same names, have receded so far that the sign Taurus, now corresponds nearly with the constellation Aries. 41. In the preceding investigations, we have con- sidered the plane of the ecliptic as fixed and the obli- quity of the ecliptic as continuing always the same. This, though very nearly, is not strictly true. A com- parison of accurate observations, made at long intervals of time, proves that each is subject to a slight change. These changes will be noticed in a succeeding chapter. CHAPTER VII. Sun's flpparent Orbit—Repler’s Laws—Idepler’s Problem. 1. It has been shown (6.37) that the vernal equinox has a retrograde motion along the equator of 46" a CHAPTER WII. 47 year. This is its mean motion. It has been found from accurate observations that this motion is not uni- form. The place at which the vernal equinox would, at any time, be, if its motion was uniform, is called its JMean place, or the JMean Equinoa. 2. The motion of the mean equinox along the equa- tor, being 46" a year, its motion in one day must be # of a second nearly, which corresponds to +}w of a second in time (3.6). If therefore on any particular day the mean equinox be on the meridian at precisely the same instant with some fixed star, it would, in con- sequence of its retrograde motion, come, to the meridian on the succeeding day #g of a second earlier than the star. 3. Some astronomers, with a view to convenience in observing the right ascensions of the heavenly bo- dies, regulate their clocks so as to mark 0 h. 0 m 0 sec. when the mean equinox is on the meridian, and they call the interval between two of its consecutive passages over the meridian, a Sidereal Ilay. The term sidereal day is now generally used as here defined, and is to be thus understood in the fol. lowing parts of the work. But on account of the very small difference between its length and the length of the sidereal day as defined in a preceding chapter (8.5), we may consider them as equal in all cases that regard observations made during a single day or a Small number of days. 4. The time between two consecutive passages of the sun’s centre, over the meridian, is called a True Solar Day. In consequence of the sun’s motion east- Ward among the fixed stars (6.1), the length of a solar day is greater than that of a sidereal day. 48 ASTRONOMY. It is ascertained by observations that the length of a solar day is different at different times in the year, but that at the same time in different years it is very nearly the same. By comparing the number of solar days that elapse from the time that the sun passes the meridian on a given day in any year, to the time of its passage on the same day in some succeeding year, with the number of sidereal days and parts of a day, that elapse during the same time, it is found that the mean length of a solar day, called a JMean Solar Day, is equal to 24 h. 3 m. 56.555 sec. of sidereal time. Af 5. The ratio of 24 h : 24 h. 3 m. 56.555 sec. is the same as 1 : 1.0027.379, which is therefore the ratio of a mean solar day, to a sidereal day. Hence to reduce a given portion of mean solar time to the corresponding sidereal time, we must multiply by 1.0027379; and on the contrary to reduce sidereal, to mean solar time, we must divide by the same number. The excess of a mean solar day above a sidereal day is 3 m. 56.555 sec. in sidereal time; and in mean solar time it is 3 m. 55.94 sec. 6. By observing the meridian altitude of the sun’s centre, and correcting it for refraction and parallax, its north polar distance may be determined (6.2). If this be done on several successive days, about the 20th of March, it will be found, either that on some one of these days, the north polar distance is exactly 90°, and consequently that the sun is then at the equinox, or which is much more probable, that on the first of some two consecutive days the north polar dis- tance is greater, and on the second less, than 90°. From these observations, the time that the sun is at the vernal equinox may be determined. CHAPTER VII. 49 Let A, Fig. 12, be the sun's place on the first of these two days, B, its place on the second, CD a portion of the equator, P its north pole, and AB a portion of the ecliptic. Then will E be the place of the vernal equinox. The arcs PA and PB are known; and from the interval in si- dereal time between the two observations, the angle APB or its measure the arc CD, is likewise known, it being evidently equal to the excess of the interval above 24 hours, converted into de- grees or parts of a degree. But, tan AC = tan E sin EC, tan BD = tan E sin ED. therefore tan AC : tan BD : : sin EC : sin ED, or tan AC + tan BD : tan AC — tan BD : : sin EC -- sin ED : sin EC — sin ED. From whence we have (App. 32), sin (AC + BD) : sin (AC — BD) : : tan ; (EC + ED); tan ; (EC — ED), or tan ; (EC — ED) = tan 4 (EC = ED) sin (AC — BD) sin (AC - BD) * _ tan : CD sin (PA — 90 – 90 - PB) sº sin (PA — 90° -- 90° — PB) tan 3 CD sin (PA ; PB — 180°) * = sin (PA — PB) Now knowing CD and 4 (EC — ED), we know EC and ED, which are the sum’s distances from the equinox at the times of observation. The sum’s motion in right ascension during a day may be considered uniform, particularly near the equinox, as may be determined by observing it for several days about that time. If, therefore, CE converted into time (3.6), be added to the side- real time of the first observation, we shall have the sidereal time of the sun being at the equinox. 7. By similar observations made the ensuing year, the time of the sun's return to the vernal equinox, will be known. & te 8. The interval of time between two consecutive re- • 8 50 ASTRONOMY, turns of the sun to the vernal equinox is called a Tropical Fear. The ancient astronomers determined the length of the year from the sun’s return to the same tropic and thence applied to it the term tropical year, which is still retained. * * 9. The length of the tropical year is subject to a slight variation. By observations made at intervals of 50, 60 or 400 years, its mean length, expressed in mean solar time is found to be 365 d. 5 h. 48 m. 54.6 sec. Hence 365 d. 5 h. 48 m. 54.6 sec. : A day :: 360° : 59' 8.” 33 = sun’s mean motion in longitude during a mean solar day. 10. On account of the annual precession of the equinox in longitude, which is 50."4 (6,33), the sun only passes through an are of the ecliptic equal to 359° 59' 9".9, during a tropical year. 14. The time in which the sun passes through the whole 360° of the ecliptic, or which is the same thing, the interval of time between two consecutive returns of the sun to the same fixed star, is called a Sidereal Fear. Hence 359° 59' 9".9 : 360°: : 365 d. 5 h. 48 m. 54.6 sec. : 365 d. 6 h. 9 m. 14.5 sec. = the length of a sidereal year, expressed in mean solar time. The sidereal year therefore exceeds the solar, by 20 m. 49.9 sec. 42. When the sun's apparent diameter is accurately observed, at different seasons in the year, it is found to vary. It is greatest about the first of January and con- tinually decreases till about the first of July, when it is least. It then increases till the first of January. When greatest it is 32' 35".6, and when least, it GHAPTER VII, 54 is 34' 31". Consequently the mean diameter is 32! 3".3. As there is no reason to suppose that this change in the apparent diameter, is caused by a change in its real diameter, it is inferred that the sun’s distance from the earth is variable. 13. From a comparison of the sun’s apparent diame- ter, as observed at any two different times, we may ob- tain the ratio of its distances from the earth at those times. Let AB and A'B', Fig. 13, be the sun in two different situations, and E the place of the earth. Put 3 = apparent diameter AEB 3' = apparent diameter A'EB' D = ES and D' = ES’. Then D sin; 3 = AS = A'S' = D sin , 4', T) sin # 3' # 3' ..)" OT = = —- = − = −. D' sin # 3 # & 14. The apparent diameter of the sun, when at his mean distance from the earth, is 32' 2".7. In the same manner as for the moon's parallax (5.15), if 2, 3, and 3", be the sun's apparent diameter, at its greatest, least, and mean distances, we shall have, * = ** – 82 2.7. 9 + 3 15. From the Sun’s apparent diameter and horizontal parallax, its real diameter may be determined. It is about 140 times the earth’s diameter. If d = 2 AS = sun's real diameter, we have, === e d = 2 D sin; 3 = (5.6) 2 R** º SIRT 2.3" SII] …" 3) = Rºº?= R.? 25° = R 1922".7 87.7 = R x 221 = 1103 times the earth's diameter. 52 ASTRONOMY. 46. The sun’s right ascension may be obtained by observing the sidereal time of its passage over the me- ridian. From the right ascension, the longitude may be calculated (6.49). The sun’s longitude thus obtain- ed at different times in the year, does not increase uni- formly with the time. Its greatest motion in longitude during a mean solar day, is 61' 10" and takes place at the time its apparent diameter is greatest. Its least mo- tion is 57' 11", and takes place when the apparent diameter is least. 47. The curve which the sun’s centre seems to de- scribe in the plane of the ecliptic, during a year, is called the flpparent Orbit of the sun. 18. A right line, conceived to be drawn from the centre of the sun to the centre of the earth, or to the centre of a planet, is called the Radius Vector of the earth or planet. A right line joining the centres of the earth and moon, is called the radius vector of the moon. 49. It appears from the change in distance between the sun and earth (12), that the sun’s apparent orbit is not a circle; or at least that the earth does not occupy the centre. Let ADBF, Fig. 44, be the apparent orbit of the sun, E the earth’s place, A the sun’s place when the apparent diameter is least, B its place when the appa- rent diameter is greatest, and D its place at some other time. The difference between the sun’s longitudes at A and D, that is when the apparent diameters are least and greatest, is found to be 480°. It follows therefore that EA and EB must form one straight line AB. If AB be bisected in C, then AC = # AB = } (AE + FB). -- The angle AED = sun’s long, at D — sun's long. at A. CHAPTER VII. 53 20. The apparent orbit of the sun, is an ellipse, having the earth in one focus. This fact was discover- ed by Kepler, and it is called Jºepler’s first Law. It is deduced from investigations, founded on the observ- ed apparent diameter of the sun at different longitudes. Put 3 = Sun's apparent diameter at A, 9' = do. B, 3tt de. D, m = # (3' + 3) = Sun’s mean apparent diameter, and n = } (9' – 3). - From the sum's longitude and apparent diameter, as obtained at different times in the year, it is found that whatever be the situation of D, we have d" = m — n cos AED. But (13) AE : EB : : 3': 3, AE : AE + EB :: * : * + 3, . AE : # (AE + EB) :: * : 4 (3 + 2), AC AE : AC : : * : m = fºº. 3". * = AE Again, AE : EB : : * : 3, - AE + EB : AE — EB : : 3' + 3 : 3' — 3, 2 AC : 2 EC :: ; (3' + 3) : ; (3 – 3), EC EC AC EC : EC : : m : n = + m = ±. : 3'- tº 3/ or AC * : * = AG” – AD RE * = AE *. Hence substituting the values of m and n, in m — n cos AED, we have, _AC y_EC 3" = *E*-*. * cos AED. t But (13) 3" = ; 9'; therefore, #. 9" — ; * cos AED = #”. #-ºws AED - # 54 ASTRONOMY. ED. (AE – EC cos AED) = AE. EB = (AC + EC). (AC — EC) = AC” — EC*, - AC2 — EC2 or PP = AC EOT. RED" But (Comic Sections”) the last equation expresses a property of an ellipse, of which AB is the transverse axis, C the centre, and E one of the foci. 24. The point which is the sun’s place, when most distant from the earth, is called the ºlpogee; and the point which is its place when nearest the earth, is call- ed the Perigee. Those points are also called Apsides; the most distant being called the Higher Apsis, and the nearest, the Lower flysis. The transverse axis, which joins the apsides, is called the Line of the Apsides. 22. The distance between the earth and the centre of the sun’s apparent orbit, is called the Eccentricity of the orbit. 23. Astronomers usually call the mean distance of the earth from the sun, a unit or 1, and express other distances in conformity with this assumption. 24. Considering the earth’s mean distance from the sun, equal 1, the eccentricity of the sun's apparent orbit is .0168 nearly. Put AC = 1, and e = EC = the eccentricity; then (20), -*m- m = em * = Röm = 1 m = em. 2't. or e = * = 32".3 3 = .0168 nearly. 7??, 1923".3 T 25. Let ADBP, Fig. 15, be the sun’s apparent orbit, IE the place of the earth, D the sun’s place at any time, * The proposition here referred to, though an important one, is omitted in several of our treatises on Comic Sections. It is therefore demonstrated in the appendix, article 51. - CHAPTER VII. 55 and A and B the apogee and perigee. Then by the apparent motion of the sun, the radius vector ED, moves about E, in such manner that the area of the sector AED, increases uniformly with the time. This fact, discovered by Kepler, is usually expressed by saying, the Radius Vector describes equal flreas in equal Times. It is called Repler’s Second Law. If the circle AGB be described on the transverse axis AB, and HDG be drawn perpendicular to AB, them (Comic Sections*), AC2 — EC2 = ED = AC — EC cos BCG = AC Taºpai. REB = BD = AC COS A -- EC cos ACG; AC2 — EC2 hence, EC cos ACG = ACTEC cos AED T AC = AC. EC cos AED – EC* AC – EC cos AED AC cos AED — EC cos AED — e or cos ACG = ±5 = H.REP (A). Now the area of ECG = # EC. CG sin ACG = } e sin ACG, area of ACG = AC. § arc AG = } arc AG, area of AEG = } (e sin ACG + arc AG) (B). T * –- = a-º-º-º-º-º-º-º-º-º-º-º-º-º: Hence the area AEDA AC" AEG = • 2 ACG + arc AG) (C). The eccentricity e being known (24), and also the angle AED, from the sun's longitudes at A and D (19), the angle ACG or arc AG becomes known (A); and thence (C) the area of the ellipti- cal sector AEDA. The area of the sector AEDA, thus obtained at different times in the year, is found to increase uniformly with the time. } . (e sin 26. If we suppose the sun to be situated at E, and the earth to revolve round it in the orbit ADBP, and * See Appendix, article 52, 56 AsTRONOMY. if we admit the distances of the fixed stars to be so great, that straight lines conceived to be drawn from the points E and A to meet at any one of them, will not contain an appreciable angle, or which amounts to the same thing, that these lines may be considered as sensibly parallel; then the sun will appear to move ex- actly in the same manner as on the supposition that it is really in motion about the earth at rest at E. Let us suppose some one of the fixed stars to coin- cide with the vernal equinox, and let EQ and TAQ', be two straight lines which being produced would meet at this star. Then on the supposition that the earth is at rest at E, QEA is the sun’s longitude at A, and QED is its longitude at D. Therefore AED = QED — QEA, is the difference of their longitudes at A and D. On the supposition that the sun is at rest at E, and considering EQ and TQ' as parallel, its longitude when the earth is at A, is 180° 4- EAT = 180° -- QEA. In like manner its longitude, when the earth is at D, is 180° -- QED. But 480° 4- QED — (180° 4- QEA) = QED — QEA = AED. The difference of its longitudes, and consequently its appa- rent motion, is therefore the same on the two suppo- sitions. g But when we consider that the sun’s diameter is more than a hundred times the diameter of the earth (15), and consequently its magnitude, more than a mil- lion times the magnitude of the earth, it seems more reasonable to suppose that the sun’s apparent motion is produced by a real motion of the earth, than to adopt the contrary supposition. 27. The apparent diurnal motion of the heavenly bodies from east to west, may likewise be accounted CHAPTER VII. 57 for, by admitting the earth to have a diurnal motion on its axis in a contrary direction, that is from west to east. And it is certainly more reasonable to suppose this diurnal motion of the earth, than to suppose that all the heavenly bodies, situated at various , and im- . mensely great distances, should have motions so ad- justed as to revolve round the earth in the same length of time. Various astronomical phenomena serve to prove that the earth really has these two motions; that is, an an- nual motion round the sun, and a diurnal motion on its axis. { During the annual motion the earth’s axis continues parallel to itself; or in other words, if we suppose a right line to remain fixed in the position which the earth’s axis has in one part of the orbit, the axis con- tinues during the whole annual revolution, nearly pa- . rallel to that line. As the earth’s axis is perpendicular to the plane of the equator, the angle contained between it, and a right line passing through the centre of the earth perpendicu- lar to the ecliptic, must be equal to the angle contained by these planes, that is to the obliquity of the ecliptic. The line perpendicular to the plane of the ecliptic is called the flavis of the Ecliptic. 28. Although the particular consideration of the planets, is referred to a succeeding part of the work, it may be here observed that all of them, including the earth as one, revolve round the sun, from west to east, at different distances and in different times; and that the moon revolves round the earth, and with it round the sun. This system of the sun and planets is called the Copernican System, from its inventor Copernicus. The order of the planets with respect to their dis- 9 58 ASTRONOMY. tances from the sun, is JMercury, Venus, the Earth, JMars, Vesta, Juno, Ceres, Pallas, Jupiter, Saturn and Uranus. 29. The same reasoning that has been used to prove that the sun’s apparent orbit is an ellipse and that the radius vector describes equal areas in equal times, ap- . plies, when we suppose the sun at rest, to prove that the earth’s orbit is an ellipse, having the sun in one focus, and that its radius vector describes equal areas in equal times. Kepler, extending his researches, found that the or- bits of the planets are ellipses and that the radius vec. tor of each describes equal areas in equal times. 30. Another important law, discovered by Kepler, is, that the square of the time in which any planet re- volves round the sun, is to the square of the time in which another planet does the same, as the cube of the mean distance of the former, from the sun, is to the cube of the mean distance of the latter. This relation, which is usually expressed by saying the Squares of the times of revolution of the Planets are as the Cubes of their mean distances from the Sun, is called Rep- ter’s Third Law. 31. Although astronomers have completely estab- lished the fact of the earth’s annual and diurnal mo- tions, yet with a view to convenience of expression, they still frequently speak of the sun’s orbit and of the motion of the sun. 32. The point in the earth’s or a planet's orbit, which is the most distant from the sun, is called the •Aphelion; and the nearest point is called the Perihe- lion. The terms apogee and perigee are only used to express the greatest and least distances from the earth. 33. Let D and F, Fig. 14, be two situations of the CIIAPTER VII. 59 sun in its apparent orbit, at which its apparent diame. ter is the same. Then ED = EF, and consequently the angle AED = AEF. Hence from the longitudes of the sun at F and D, the longitude of A, the apogee, becomes known. The following is however a more ac- curate method of determining the place of the apogee. The transverse axis AB divides the ellipse into two equal parts. The sun will therefore be as long in moving from A to B as from B to A (25). Hence it will be half a year in moving from A to B, and in this time it will change its longitude 180°. This will not be the case for any other points in the orbit, because no other right line, passing through E, divides the ellipse into equal parts. If therefore two longitudes of the sun be observed at the interval of half a year, and such that their difference is 480°, the position of the line of the apsides becomes known. We have supposed the position of the line of the apsides to remain fixed. But observations made aſ long intervals of time, prove that it has a slow motion in the order of the signs. Astronomers therefore mo- dify the preceding methods of determining the position of the apsides, so as to take notice of this motion, which however only amounts to a few seconds in a year. 34. Delambre determined the longitude of the apogee, for the beginning of the year 1800 to be 3 9° 29' 3" and its mean yearly increase in longitude to be 61".9. 35. If we divide 3 9° 29' 3" by 61".9, the quotient is 5786. Hence it appears that about 5786 years an- terior to the year 1800, the apogee coincided with the vernal equinox. It is worthy of remark, that it is about that period, at which chronologists generally fix the creation of the world. 36. If, from 64".9 the annual motion of the apogee 60 § ASTRONOMY. from the equinox, we subfract 50".1, the annual preces. sion of the equinoxes, the remainder which is 4.1".8, is the mean annual motion of the apogee. 37. The time between two consecutive returns of the sun to the apogee, is called an ºdnomalistie Fear. As 359° 59' 9".9 : 360° 0' + 1".8 : : 365d. 5h. 48m. 54.6 sec. : 365 d. 6 h. 13 m. 58.8 sec. = the length of the anomalistic year, expressed in mean solar time. The anomalistic year, therefore, exceeds the tropical, by 25 m. Z.2 sec. As 365 d. 6 h. 43 m. 58.8 sec. : 4 day: : 360° : 59" 8".46 = Sun’s mean motion from the apogee during a mean solar day. 38. At the time that the true place of the sun is at D, Fig. 15, let F be the place at which it would have been, if it had moved from the apogee with its mean angular motion of 59'8".16 a day. 39. The angle, contained between the line of the apsides and the radius vector, is called the True Jłnomaly. Thus AED is the true anomaly. 40. The angle, contained between the line of the apsides and the straight line from the earth to the mean place of the sun, is called the Mean Anomaly. Thus AEF is the mean anomaly. The angle ACG is called the Eccentric Anomaly. 41. The angle, which is the difference between the Mean and True anomalies, is called the Equation of the Centre. 42. The equation of the centre DEF, expresses the difference between the mean longitude QEF and the true longitude QED. The mean longitude and mean anomaly increase uniformly with the time. They may, therefore, be easily determined for any particular point of time. Then, if the equation of the centre, corres. CHAPTER VII. | 6 || ponding to the mean anomaly be known, the true lon- gitude becomes also known. 43. The problem for determining the true anomaly from the mean, and which therefore determines the equation of the centre, is called Jºepler’s Problem. It is a problem of great importance and has been solved in various ways. The following method combines simplicity with a requisite degree of accuracy. Let CL be drawn parallel to EF, LO parallel to GC, and EM perpendicular to GC produced. Put, T = time of describing the whole ellipse, t = time of describing AD, -3. e = EC, AC being = 1, ang. AED = true anomaly, = arc AG = ang. ACG = eccentric anomaly, = arc AL = ang. AEL = ang. AEF = mean anomaly, area of the circle AGB, N = area of the ellipse ADBP, p p = 6.28318 &c. = circumference of the circle. Then M = AC X arc AGB = 1 × 3. p = 4 p. 7. !, : == Now (Conic Sections) M : N :: AC : CR :: AEGA : AEDA, M : AEGA :: N: AEDA: ; T : t: : 360 : ang. AEF::p: z; T. hence AEGA = *, * = # P. z = # 2; p p But (25.B) AEGA = } e sin a -- a). Therefore z = a + e sin a & (D}. Again (25.A) cos w = < * * * , — 6 COS Ql cos 2 – e cos w cos w = cos w — e, cos u + ecos w cos w = e -- cos w, . e -- COS 3. COS M = ~o 1 + e cosa; (E). The last formula may be converted into another that will be more convenient for logarithmic computation. We have (App.12.), 62 ASTRONOMY. e -- C0S 4; tan * 4 1 — cos u 1 + ecosa. an # u = T-º-º- = * 4- cos tº 6 -– COS 3. 1 +'e cora, 1 + ecos a — e — cos 3: 1 + 6 cos a -- 6 -- COs a _ (1 — eX – (1 − e) cos w 1 – e 1 — cost Q + e) + (1 + 6) cos aſ tº- IT." TIE cos & — 0. 1 + e **- 2 1 tam ” # 3. hence tan ; u = tan # 3. V H. (F). We have now an expression for the true anomaly in terms of . the eccentric, and (D) an equation showing a relation between the mean and eccentric anomalies. But in consequence of the latter containing both w and sin ar, we can not, in a direct manner, obtain the eccentric from the mean anomaly. There are how- ever various methods of approximation which give its value to any required degree of accuracy. 44. We have by trigonometry, EM = EC sin ECM = EC sin ACG = e sin w; but (43.D) e sin a = z – a = arc AL – arc AG = arc LG. Therefore EM = arc LG. Because OL is parahlel to MG, and EM is perpendicular to it, OM = sin LG. The difference between EM and OM is there- fore equal to the difference between the arch LG and its sine; and in all cases for the sun and planets, this difference is small, as may be thus shown. Since LG = e sin w, it is evident that LG will be greatest when a = 90°, and then we have LG = c sin 90° = e. But for the sum’s apparent orbit e = .0168 nearly (24); therefore LG = .0168 = 0°57'3. The value of e for the most eccentric orbit of the planets is about .254, the mean dis- tance from the sum being expressed by 1. Consequently LG when greatest is about 14°4. The difference between an arc of 14°4 and its sine, is but lit- tle, and for an arc of 573 it is much less. It follows therefore that in all cases, OM is nearly equal to the arc LG or its equal the right line EM. Consequently, because OL is parallel to CG, EL is nearly parallel to it, and the angle AEL is nearly equal to Al-- ~~~~~~ +--- ~ ~~~~~~~!-- A ſºl ſ{ CHAPTER VI. 63 Put 9 = CEL — CLE, 70 = CEL, g = ACG — CEL = a – w. Then a = w -- y, By trigonometry, CL - EC : CL – EC : : tan ; (CEL + CLE) : tan # (CEL — CLE), or 1 + e : 1 —e : : tan # 2 : tan # 9. Therefore tan # 9 = *=* tan # 2. 1 + e w = CEL = # 2 + # 9. Now (43. D) and App. 13), 2 = a + e sin a = w -- y + e sin (w -- y) = w -- y + e sin w cos y + ecos w sin y. W But because y is very small we may take cos y = 1 and sin y = y. Then z = w -- y + e sin w -- e y cos w, J -- e y cos w = z — (w -- 6 sin w), 3) = 2 — w -– e Sin w); 1 + e cos w and a = w -- y, becomes then known. * If greater accuracy is required, put w -- y = 2, and y' = a — a '. Then reasoning as above we have; J’ = 2 — 'a' -– 6 sin º'. 1 + 6 coS a." and a = 4' + y'. The last formula is not necessary except very great accuracy is required. The value of a = w -- y, is not liable to a greater error than fºr of a second for the most eccentric orbit of the pla- nets. For the Sun, the error can not amount to more than about # of a second if we take a = w. When the value of a is found, we obtain the true anomaly from the expression tan # w = tan ; a v =: 1 + e 45. Some astronomers have proposed that the anomaly should always be reckoned from the perigee because it is necessary to de S0 for the orbits of comets, 64 ASTRONOMY. If 180° be added to the mean anomaly, reckoned from the perigee, rejecting 360° when the sum exceeds it, the result will be the mean anomaly reckoned from the apogee. anomaly from the apogee may be found by the preceding formulae. The difference between these will be the equation of the centre. Given the sun’s mean anomaly from the perigee 8' 25" or from the apogee 2° 25°, and the excentricity of the orbit.016774; EXAMPLE I. required the equation of the centre. Here z = 2* 25° = 85°, and 62 - .016774. Log. (1 — e) .983226 log. (1 + e) 1.016774 - Og. * 1 + e tam #2 tam # 6 - 10 = lº- 42° 30' & 41 32 38/.6 - 84 2 38.6 1 — e Log. TTe 1 — e l *-* * sº- tº 0g vº + 6 tam # 3. 42° 1' 19'.3 tan ; w 41° 32' 40" 70 - 2 83 5 20 2 = 85 0 O Then the true 9.9926533 0.0072244 ººmºsº ºmºmºmºmºse 9.985.4289 9.96205.25 tºmºmºsºms ºf mºmº 9.9474814 9.985.4289 ammammºmºmºmºsº 9.9927.144 9.95.47732 gºmºsºme ºf smºsºmeºmºmº 9.94.74876 equation of Sun's centre. CHAPTER VII. 65 EXAMPLE II. Given Mercury's mean anomaly from the aphelion 1' 28°, and the eccentricity of its orbit.205513; required the equation of the Gentre. ^. -- 2 = 58 and e = .205513. Log. (1 — e) - .794487 - * log, (1 + e) - 1.205513 tº 1 — e log. 1 + e - - º tº tan + 2 - - 29° 0' 0" tan # 9 - - 20 4 4.7 - Q!) = * * -º 49 4 4.7 Log. € - .205,513 sº tº gº º log. - 206264".8 - - - - sim w 49° 4' 4".7 dº tºº Eº ſº e sin w = 32025".2 = 11) = º gº dº 49 4 4.7 w -- e sin w = - 57 57 49.9 2 = tº tºg 58 O 0 2 — (w -- e sin w) 0 .205513 - sº sº lóg. 6 = COS 70 49° 4' 4".7 - dº tº Eº © COS 7.0 gº . 1346 dº sº gº 1 + ecos w 1.1346 180".1 9 - TI316 * = w -- y = 49°4'4”.7 + 1' 54".7 = 49° 5'59". 4 8° 53' 45/.2 - dº 2 10.1 = 130”.1 9.9000868 0.08] 1719 *=se tº wrºs-ºsmº 9.8.189149 9.7437.520 9,5626669 - 9.3128393 5 3144251 - 9.87822.72 tº ºn 4.5054916 9.3128393 - 9.8163495 * * *ºss = 114/7.7 =1/ 54”.7 9.129.1888 40 66 ASTRONOMY. leg. =: - - - - - - 9.8189149 1 + e I — e Ion. V * * * tº * wº tºº 9,90945'74 1 + e tan ; a - 24° 32'59".7 - wº - - 9 6591633 tan ; u - 20 20 44.9 'lſ, --> - 40 41 29.8 2 = - 58 0 0 u — z =– 17 18 30.2 = equation of the centre, 46. When the eccentricity of the orbit is known, the true radius vector, corresponding to any given mean anomaly, may thence be determined. Put r = ED = the radius vector; then because (25), ED = AC + EC cos ACG, we have, r = 1 + 6 cos ar. Hence, having computed the eccentric anomaly, corresponding to the given mean anomaly, we easily obtain the value of the ra- dius vector. 47. If we divide 360° by the number of hours in the time of revolution, of the sun or a planet, in its orbit, the result will be the mean hourly motion. If H be the mean hourly motion, H' the true hourly motion, e the eccentricity of the orbit, and r the radius vector, then we have for the true hourly motion, the following ex- pression. H = H. Y(*= *). 7.2 Let a and b be the places of the body, half an hour before and after it is at P. Them will the angle abb, be its true hourly mo- tion in this part of its orbit. With the centre E and radius EP describe the arc cd. Then since the angle aeb is very small, we CHAPTER VII. 67 may consider the elliptical sector a Flba, as equal to the circular sector cRdc. t i Now the circumference of the circle of which EP is radius, is equal EP. p = rp; and consequently the area of the same circle is = r. #rp = # r*p. Then, & 360° : H! :: ; rºp : cºde, H' cEdc = 㺠#" p. Again (43) 360° : H :: T : 1 :: N : abFa :: N : cFác, H H CR H Edc = –tº–. N = . *t, M = –tº– 1 — e”). 4 p. * = sā; 360°, AC # v ( – ’). #p H' H H .# r" p = — 1 — e”) # p, or H = H. Y (1 — e”). 2 •e 48. Because H and v (1 — e”) are constant quan- tities, it follows that the true motion of a body in an elliptical orbit, varies inversely as the square of the radius vector. Hence it continually increases from the apogee, where it is least, to the perigee, where it is greatest; and thence continually decreases to the apogee. 49. As the mean and true places of the body coin- cide at the apogee and perigee, and as near the apogee the true motion is less than the mean motion, the mean place will be in advance of the true place, from the apogee to the perigee. From the perigee to the apogee, the true place will be in advance of the mean place. It is therefore evident; that the equation of the centre, which is the difference between the mean and true places, and which at the apogee is nothing, must con- tinually increase, till the true motion becomes equal to the mean, when the equation will be greatest; and thence it will decrease to the perigee, where it again becomes nothing. In like manner, the equation of the 68 ASTRONOMY. centre increases from the perigee, till the true motion, which is then diminishing, becomes equal to the mean. It is them greatest; and afterwards decreases till it be- comes nothing, at the apogee. The parts of the orbit on each side of the line of the apsides, being symmetrical, the greatest equation on one side will be equal to the greatest equation on the other side. º The sun’s true longitude may be obtained each day from its observed right ascension (6.49). The differ- ence between its longitudes on any two consecutive days will be its true diurnal motion at that time. Hence we may, by repealed observations, find the time when the true diurnal motion is equal to the mean. Knowing then the mean and true longitudes, when this takes place in opposite parts of the orbit, we may obtain from thence the greatest equation of the centre. Let A = the mean longitude, and B = the true lon- gitude, at the time the motions are equal between the apogee and perigee; A = the mean longitude, and B' = the true longitude, when the motions are equal be- tween the perigee and apogee; and V = the greatest equation of the centre. Then, * B = A — V, IB = A' + W. Hence B" – B = A' — A + 2 W, 2 V = (B' – B) — (A’ — A), W = , (B' – B) — ; (A' — A). At the time of the greatest equation, the sun’s true motion continues very nearly the same for two or three days. Consequently the equation of the centre will remain very nearly the same during this time. The CIIAPTER VII. f 69 value of the greatest equation may therefore be deter- mined with great accuracy by this method, without the necessity of knowing very precisely the time, at which the true motion is equal to the mean. 50. In a preceding article, a method has been given of obtaining the eccentricity from the greatest and least apparent diameters. It may however be obtained much more accurately from the greatest equation. Put K number of seconds in V 206264".8 Then, it is found by means of an analytical investi- igation, which we shall omit, that e = K– 4 K –- ºr K – &c. 768 . 98.3040 51. By a comparison of observations made at distant periods, it has been discovered that the equation of the sun’s centre, and consequently the eccentricity of the orbit, are at the present period continually diminishing. The rate of diminution in the greatest equation is about 18".79 in a century. It follows, therefore, that the equation of the centre, as computed for a given time, will not be accurately true for a different time. It will, however, err but little for a few years, before and after the time, for which it is computed. A complete table of the equation of the sun’s centre, has a column con- taining the variation of the equation in a century, called the Secular variation, by means of which the correct equation may be obtained for different periods. 52. The force which causes heavy bodies, when left at liberty near the surface of the earth, to fall to it, is called the flttraction of Gravitation. Newton was led 70 ASTRONOMY. - 5 by reasoning which appertains to Physical Astrono- my, to adopt as a principle, that this attraction, de- creasing inversely as the square of the distance from the earth’s centre, extends to the moon and retains it, in an elliptical orbit about the earth; that the sun, moon and planets are endued with like attractive forces, which vary according to the same law; and that it is the sun’s attraction, which retains the earth and planets in their orbits. This general principle of attraction is called JWewton’s Theory of Universal Gravitation. A combination of various discoveries which have been made in astronomy since the time of Newton, has served completely to establish the truth of his theory. 53. If the earth was acted on by no other force than the attraction of the sun, its orbit would be accurately an ellipse, and the areas described by its radius vector in equal times, would be precisely equal. Its true lon- gitude would therefore be accurately expressed by its mean longitude, corrected by the equation of the cen- tre. But the attractions of the moon and planets extend to the earth, and some of them produce sensible, though slight effects on its motion. By the aid of very refined analytical investigations, the means have been obtained of calculating these effects, which are called Perturba- tions. Their whole amount may produce a change in the sun’s longitude of about 45"; but in general it is considerably less. Our best solar tables, which are those calculated by Delambre, contain the equations due to the attractions of the moon and planets. The equation of the centre and the amount of the perturbations, applied to the mean longitude of the sun, give its true longitude from the mean equinox (4). 54. The difference between the mean place of the equinox in the ecliptic, and its true place, is called the CHAPTER VIII. 74 Equation of the Equinoaces in Longitude, or sometimes, the Lunar JWutation. The greatest value of this equa- tion is about 18". A table, from which its value may be obtained for any particular time, forms a part of a complete set of solar tables. To obtain the sun’s true longitude from the true equinox, we must correct the mean longitude by the equation of the centre, the amount of the perturbations, and the equation of the equinoxes in longitude. 55. From the sun’s true longitude, we obtain its true right ascension, by the formula in the last chapter (6.20). Another method is by means of a table calcu- lated for the purpose. The difference between the longitude and right as- cension, is called the Reduction of the Ecliptic to the Equator. Tables have been calculated which, for a given obliquity of the ecliptic, give the reduction cor- responding to each degree or half degree of longitude, and also the variation in the reduction for a change of 4' in the obliquity. With such a table the reduction corresponding to a given longitude is easily obtained, and being applied to the longitude, it gives the right ascension. CHAPTER VIII. Equation of Time—Right ºffscension of JMid-Heaven. 1. Solar days, being determined by the apparent diurnal motion of the sun (7.4), are used for all the common purposes of life. Astronomers also generally use solar time except in determining the right ascen- sions of bodies. 2. In common reckoning the day begins at midnight, and is divided into two portions of 12 hours each. The 72 . ASTRONOMY., first 12 are from midnight to noon, and are usually de- signated by the letters A. M.” annexed to the number of the hour. The latter 12, from noon to midnight, are designated by the letters P. M.* The astronomical day begins at noon of the com- mon day, and the hours are reckoned on to 24. Hence, any given time from noon to midnight is expressed by the same day, and hour in astronomical and in common reckoning. But to express astronomically a given time from midnight to noon, we must diminish the number of the common day by a unit, and increase the number of the hour by 12. 3. The angle contained between the meridian and a declination circle passing through the sun or any one of the heavenly bodies is called the Distance of the body from the JMeridian, or the Hour angle of the body. The intercepted arc of the equator is the measure of this an- gle, and therefore designates the distance of the body from the meridian. 4. The point of the equator which is on the meridian at the same time with the sun, will, by the diurnal mo- tion, be 45° to the west at the end of a sidereal hour. But on account of the sun’s increase in right ascension, during the time, its distance from the meridian must be less than 15°. The sun does not therefore move from the meridian at the rate of 45° in a sidereal hour. But as the interval from the time the sun is on the meridian to its return to it again is divided into 24 so- lar hours, and as the distance is 360°, if we suppose the right ascension to increase uniformly during the day, the Sun’s diurnal motion from the meridian must be at the rate of 45° in a solar hour. * A. M. are the initials of Ante JMeridian, forenoon, and P. M. of Post JMeridian, afternoon. CHAPTER VIII. 73 5. Since 15° of the sun's distance from the meridian corresponds to 1 solar hour, 4° must correspond to 4 minutes in time, 1' to 4 seconds, and 1" to 4 thirds. . Hence to convert the sun’s distance from the meridian into time, if we multiply the distance in degrees, mi- nutes and seconds, by 4, the product of the seconds will be thirds of time, the product of the minutes will be seconds, and the product of the degrees will be mi- nutes. As an example let 17° 21' 36” be converted into time. 47° 24′ 36” 4. ** 4 h. 9nſ. 26 sec. 24 thirds. The sun’s distance from the meridian is sometimes called the time from noon in degrees. 6. If the apparent annual motion of the sun was in the equator and was uniform at the rate of 59'8".33 from the mean equinox, in the interval between two of its consecutive passages over the meridian, it is evident that the intervals would be mean solar days (7.9 and 4). But the motion of the sun is not in the equator, and in the ecliptic it is not uniform. There are, therefore, two causes of inequality in mean solar days. 7. Supposing, as in the last article, the sun to move uniformly in the equator with its mean daily motion in longitude from the mean equinox, the time when it would, in that case, be on the meridian, is called Mean JWoom. And time reckoned from mean noon is called JMean Time. * The time when the sun is really on the meridian is called Apparent JVoon. And time reckoned from ap- parent noon is called Apparent Time. 4 A. 74 ASTRONOMY. The difference between the apparent and mean time is called the Equation of Time. 8. If the sun’s mean longitude be corrected by the equation of the equinoxes in right ascension, The differ- ence between the true right ascension and the corrected mean longitude will be the equation of time in degrees. When the true right ascension is greater than the corrected mean longitude, the equation of time must be added to apparent time to obtain mean time, and when it is less, the equation must be subtracted. ^- But on the contrary, when mean time is to be re- duced to apparent time, the equation must be subtract- ed, if the true right ascension is greater than the cor- rected mean longitude, and must be added, if it is less. Let VQ Fig. 16, be the equator, VC the ecliptic, V the ver- mal equinox, A its mean place in the ecliptic, S the true place of the sun, and AB and SD, declination circles. Then will B be the mean place of the vermal equinox in the equator, WA be the equation of the equinoxes in longitude, VB, in right ascension, WS the sun’s true longitude, and VD its true right ascension. Also let BL be equal to the Sun's mean longitude from the mean equinox, at the time the true longitude is VS or true right ascen- sion WD, and M be the point of the equator which is on the me- ridian at that time. Then (5 and 7), MD = apparent time in degrees, ML = mean time, - and DL = WD — WL = equation of time. Now (App. 49) tan WB = tan VA cos V; or since WA and consequently VB, only amounts to a few seconds (7.54), WB = WA cos W = WA cos 23° 28′ = .917 WA. Hence DL = WD — WL = WD — (LB + BW) = WD — (LB + .917 WA). 9. If the equation of the equinoxes in right ascen- CHAPTER VIII. 75 sion be omitted, the error in the equation of time will seldom exceed 4 second. Since WA when greatest is only 18" (7.54), the value of .917 VA can not exceed 16".5, or 1 second 6 thirds, in time. 40. The equation of time may be further resolved into its component parts. If E be the equation of the centre, P the amount of the perturbations, q the equation of the equinoxes in longitude, and R the reduction to the equator, then, Equation of time in degrees = E + R + P + .083.g. Put ſ = sum’s mean longitude, and a = obliquity of the ecliptic; then (11) q cos w = equation of equinoxes in right ascension. Hence, VL = RL + BW = M + q cos o, WS = M + E + P + q (7.54), and WD = M + E + P + q + R (7.55). Therefore DL = WD — WL = E + P -- R + q – q cos a = E + P + R + q (1 — cos &) E + P + R × 2 q sin” # a (App. 8) > = E + R + P + .083 q. - 44. The term E depends on the sun’s mean anomaly, and the eccentricity of the orbit (7.30); and the sun’s mean anomaly depends on its mean longitude and the longitude of the apogee. Therefore the term E de- pends on the mean longitude of the sun, longitude of the apogee, and the eccentricity of the orbit. The lat- ter two change but little in a year. Hence for a given year, a change in E, depends pricipally on the mean longitude. With a given obliquity of the ecliptic, the term R. depends on the sun's true longitude (7.55), and there- 76 t ASTRONOMY. fore principally on the mean longitude and term E. A table may therefore be formed containing for each degree of the sun’s mean longitude, the sum of E and R., expressed in time. Such a table will only be true for the time for which it is calculated. But the greatest error only amounts to about 15 seconds in 100 years. Sometimes a column is annexed, containing the secular variation in this part of the equation of time. ** 42. The term P. must consist of several parts, each depending on the body which produces the perturba- tion. The greatest value of P is about 45" (7.53), which is 3 seconds, in time. The greatest value of the term .834, is about 4% of a second in time. In calcu- lations where great accuracy is not required, these terms may be omitted. 43. When E and R are equal and have contrary signs, their sum is nothing. Nearly at the same time, as the other terms only amount to a few seconds, the equation of time must also be nothing, This circum- stance takes place four times in the year. These times are about the 15th of April, 15th of June, 1st of September and 24th of December. 44. As the motions of clocks or watches, that are well made, are uniform, or nearly so, they can not cor- respond with the unequal motion of the sun. They should therefore, for the common purposes of life, be regulated to mean solar time. This is easily done by applying the equation of time, to the observed time of the sun’s passage over the meridian.* * A simple method of drawing a meridian line that will be sufficiently ac. curate, to regulate a clock for the common purposes of society, will be given in the next chapter. CHAPTER IX. — 7 7 RIGHT ASCENSION OF THE MID- HEAVEN. *. 45. The arc, contained in the order of the signs, be- tween the vernal equinox and the point of the equator, which is on the meridian at any time, is called the Right Jiscension of the JMid-heaven at that time. Thus WM is the right ascension of the mid-heaven, when the point M is on the meridian. - 46. The right ascension of the mid-heaven, VM, is equal to the sum of VD and DM; that is, to the sum of the true right ascension of the sun and the apparant time eacpressed in degrees. - It is also equal to the sum of VB, BL, and LM; that is, to the suin of the equation of the equinoaces in . right ascension, mean longitude of the sum, and the mean time ea pressed in degrees. * In either case, the time is to be reckoned from noon to noon; and if the sum exceeds 360°, its excess above 360°, is the right ascension of the mid-heaven. CHAPTER HX. Circumstances of the diurnal motion.—Sun's Spots, and rotation on its aais.-Zodiacal Light. 1. Let HZNR, Fig. 47, represent the meridian of a place, Z the zenith, P the north pole, P’ the south pole, HR the horizon, EQ the equator, S the situation of a body in the eastern part of the horizon, and PSD a de- clination circle passing through the body. . * 2. If PS, the polar distance, does not change, the body rising” at S will describe the arc SB, parallel to * It has been noticed that the rising and setting of the heavenly bodies are affected by refraction (3.20). It is also evident that for the sun, moon, and planets, parallax will produce some effect. But in this and the following ;:S ASTRONOMY, the equator, and come to the meridian at B. In descend. ing from the meridian to the horizon, it will describe a similar and equal arc in the western hemisphere. It will therefore be the same length of time in descending from the meridian to the horizon, as in ascending from the horizon to the meridian. 3. The angle SPZ, or its measure DE, converted into time, expresses the interval of time between the rising of the body and its passage over the meridian. This interval is called the Semi-diurnal flrc. As the body is 42 hours in passing from A to B, the differ- ence between the semi-diurnal arc and 12 hours, ex- presses the time in which the body ascends to the hori- zon from the meridian below, and is called the Semi- nocturnal flºc. 4. When PS = 90°, S coincides with 0, and the body is in the equator. As OPE is 90°, the semi-diur- nal arc will then be 6 hours. When PS is less than 90°, it is evident the angle SPZ will be more than 90°, and consequently the semi-diurnal arc, more than 6 hours. When PS is more than 90°, as PS', the angle S'PZ is less than 90°, and the semi-diurnal arc, less than 6 hours. 5. When PS, the distance of the body from the ele- vated pole, is less than PH, the latitude of the place, the body continues above the horizon and does not set. When the distance of a body from P', the depressed pole, is less than PR, which is equal to PH, the lati- tude of the place, the body continues below the hori- zon and does not rise. 6. The sun’s polar distance, when least, is about 66°4. Therefore, at a place whose latitude is greater articles, unless when the contrary is mentioned, these effects are not con- sidered. CHAPTER IX. 79 than 66°4, the sun, when nearest the elevated pole, will revolve above the horizon without setting, and will continue to do so, as long as its distance from the ele- vated pole is less than the latitude. At the opposite season of the year, when the sun is nearest the depress- ed pole, it will continue a like period of time, below the horizon without rising. 7. At either of the poles of the earth, the latitude is 90°; and, therefore, the sun will continue above the hori- zon all the time it is on the same side of the equator with the elevated pole, which is about half the year. During the other half of the year it will be below the horizon. 8. At the equator, the latitude is nothing, the axis PP' coincides with HR, and the angle SPZ becomes SHZ = 90°. The semi-diurnal arc is therefore 6 hours, whatever be the distance of the body from the pole. ! 9. It follows from the preceding articles, that at the equator, the days” are always 42 hours long; from the equator to 669; latitude, the longest day varies from 42 to 24 hours; and from thence to the pole it varies from 24 hours to 6 months. 40. The difference in the warmth diffused by the sun to different parts of the earth, or to the same part at different seasons of the year, depends principally on its continuance above the horizon, and on its meridian altitude. The change in the sun’s distance from the earth must produce some effect; but on account of the small de- gree of eccentricity of the earth’s orbit, the change in distance is only a small part of the whole distance, and consequently the difference in warmth depending on * The term day, here, implies the time of the sun's continuance above the horizon, 80 ASTRONOMY. this cause, will be inconsiderable. Indeed the sun is in perigee, or nearest the earth, about the first of January; which, in northern latitudes, is the time of our coldest weather. \ 41. Within the limits of the torrid zone (6.40), when the sun is on the meridian, the direction of its rays is always nearly, and sometimes quite, perpen- dicular to the surface of the earth. The heat is there- fore very great. On the contrary, within the frigid zones, the sun never rising far above the horizon, the direction of its rays is very oblique, and consequently it produces but little warmth. Within the temperate zones, the obliquity in the di- rection of the sun’s rays, when it is on the meridian, be- ing never, either very great or very small, a medium temperature is produced. J 42. At any place in the temperate zones, the differ- ence between the greatest and least meridian altitudes of the sun, is evidently equal to twice the greatest de- clination, that is, to nearly 47°. And as the days are longest when the meridian altitude is greatest, and shortest when it is least, the difference in temperature, between the opposite seasons of the year, in which these circumstances have place, must be considerable. 43. In the north temperate zone, Spring, Summer, JAutumn, and Winter, the four seasons, into which the year is divided, are considered as respectively commencing at the times of the Vernal Equinoa, Summer Solstice, Autumnal Equinoa, and Winter Solstice. 44. Since at the terrestrial equator, the poles are in the horizon, the celestial equator, and circles parallel to it, must cut the horizon at right angles. This posi- CHAPTER IX. g 81 tion of the sphere is called a Right Sphere. At the poles, the equator coincides with the horizon; and hence, circles parallel to it, are parallel to the horizon. This is called a Parallel Sphere. At any other place on the earth, the equator and its parallels cut the hori- zon obliquely; and such a position of the sphere is call- ed an Oblique Sphere. *- 15. A circle parallel to the horizon is called an ill- macantar. 16. The arc of the equator, contained in the order of the signs, between the vernal equinox and the point of the equator, which is in the horizon at the same time with any body, is called the Oblique .ſlscension of the body. Thus, if V be the place of the vernal equinox, WO is the oblique ascension of the body at S. 47. The arc, which is the difference between the right ascension of a body and the oblique ascension, is called the Ascensional Difference. Thus, OD is the ascensional difference of the body at S. 18. Given the latitude of the place and the sun's declination, to find the time of its rising or setting. In the triangle ZPS we have (App. 34), cos ZS = cos PZ cos PS + sin PZ sin PS cos SPZ, or because ZS = 90°, 0 = cos PZ cos PS -- sin PZ sim PS cos SPZ. & cos PZ cos PS - Hence cos SPZ = — sin PZ sin PS T T cot PZ cot PS = — tam PH cot PS, or cos semi-diurnal arc = — tan latitude x cot polar dis- łółnce. Another formula which is frequently used, may be derived from the preceding, Because, cos SPZ = cos DE = cos (90° 4- OD) = —sin OD, f 42 -- 82 ASTRONoMy. we have, sin OD = tan PH cot PS = tan PH tan DS, or sin flècensional diff. = tan latitude x tan declination. Now the angle OPE or its measure OE, being equal to 90° or 6 hours, it is evident that when the latitude and declination are both of the same name, that is, both, north or both south, the as- censional difference converted into time and added to 6 hours will give the semi-diurnal arc. When the latitude and declination are of different names, the ascensional difference in time, subtracted from 6 hours, gives the semi-diurnal arc. e The semi-diurnal arc expresses the time of sunset, and sub- tracted from 12 hours, gives the time of sunrise. 19. Given the latitude of the place and the sun's declination to find its azimuth at rising or setting. In the triangle ZPS we have (App. 34) cos PS = cos PZ cos ZS + sin PZ sin ZS cos PZS, or because ZS = 90°, cos PS = sin PZ cos PZS. cos PS cos PS sin PZ T cos PH' cos polar distance cos latitude Hence cos PZS = or cos azimuth = 20. Given the latitude of the place, the sun's altitude” and de- clination, to find the time of day. - Let S, Fig. 18, be the situation of the sun; and put, L = HP = the latitude, D = PS = 90° tsun's declination, H = SE = the sun's altitude, P = ZPS = the hour angle. Then in the triangle ZPS, we have (App. 34), cos ZS — cos PS cos ZP cos ZPS = sin PS sin ZFT’ *The sun's altitude may be obtained with considerable accuracy by means of a Sextant of Reflection and an Artificial Horizon. For a description of these and the manner of using them, the student is referred to Bowditch's JVavi. gation. g CHAPTER IX. t 83 cos (90° — H) — cos D cos (90° — L) sin D sin (90°– L) * _ sin H — cos D sin L * sin D cos L But (App. 8), cos P = 1 — 2 sin * } P, * and (App. 13), cos D sin L = sin (D + L)—sin D cos L. Hence 1 — 2 sin” . P = or cos P = sin H — (sin D + L) + sin D cos L sin D cos L sin H – sin (D + L) sº g + 1, - sin D cos L 2 sing 1 P - sin (D + D) – sin H sm 3 * sin D cos L g 2 cos; (D + L + H) sing (D + L-H) App. 21 – sin D cos L *2 (App. 21) v 2 cos (P + .+ # *) sin (p + 1L + HH_ H) 2 2 * *º sim D cos L J cos (p + ++ L + h) sin (ºr ºth+ 1 + H H ) sin #P = V/ 2 - 2 2 - — +– sin D cos L To determine the time accurately, the observation of the sun's altitude should not be made when the sun is near the meridian, as its altitude then changes but slowly. Neither should it be made when the sun is very near the horizon, as the correction for re- fraction cam not then be depended on with certainty. In general the best time is three or four hours before or after moon. It is the true altitude of the sun's centre that is to be used in the calculation. Hence the altitude of the lower or upper limb, as obtained by observation must be corrected for refraction, pa- rallax, and semi-diameter. 24. If the sun’s declination did not change, it is evi- dent that it would have equal altitudes at equal times before and after apparent noon. Hence if an observa- tion of the sun’s altitude was taken in the forenoon, and the time observed by a good clock or watch, and if in 84 ASTRONOMY. the afternoon the time was also observed when the sun had obtained the same altitude; half the interval added to the time of the first observation, would give the time shown by the clock or watch when the sun was on the meridian. The deviation from 12 o'clock would be the error of the clock with respect to apparent time. But in consequence of the sun’s change in declina- tion during the interval between the observations, it is necessary, in order to render this method accurate, to apply a correction to the time thus obtained. This cor- rection is called the Equation of Equal ſlltitudes. Tables have been calculated, from which the equation is easily obtained. With these tables, the method of obtaining the error of the clock by equal altitudes of the sun, is simple, and it is also very accurate. 22. Given the latitude of the place, the Sun's altitude and decli- malion, to find ils azimull. Let Z = PZS == the sun’s azimuth; them (App. 34), *=== S PZS * PS cos PZ cos ZS cos PZS sin PZ sin ZS 2 cos D — sin L sin II cos L cos H or cos Z = But (App. 9), cos Z = 2 cos”; Z — 1, and (App. 14), sin L sin H = cos L cos H – cos (L + H}. Hence, 2 cos”; Z — 1 = cos D + cos (L + H) — cos L cos H cos L cos Hl cos D + cos (L + H) 1 * cos L cos H 2 cos (L + H) + cos D cos L cos H _ 2 cos; (L + H + D) cos; (L + H- tº gº cos L cos H 2 cos”; Z = D) G 4 (App. 22) CHAPTER IX. f 85 2 cos ( or ºr " ) COS (ºr ºr " — D ) sº-c cos L cos H COS (p+,+h) COS (****-0) º COS $ Z = v/ cos L cos H 23. If the sun's declination did not change, it is evi- dent that equal azimuths would correspond with equal altitudes. As the change in declination is but little, for a few hours, particularly near the time of the sol- stices, the azimuths corresponding to equal altitudes, must be nearly equal. This circumstance furnishes a simple method of drawing a meridian line that will an- swer for determining the time of apparent noon, when great accuracy is not required. To do this, describe a number of concentric circles or arcs of circles on a smooth board. At the common centre of these arcs, fix a piece of thick, straight wire, and make it exactly perpendicular to the surface of the board. By the aid of a spirit level or even of a common plumb line level, fix the board so that its upper surface may be horizon. tal. On a clear day, observe, during the forenoon, when the extremity of the shadow cast by the wire, exactly coincides with one of the arcs, and mark the place. In the afternoon, observe when the extremity of the sha- dow coincides with the same arc. A straight line drawn from the place of the wire, through the middle point of the arc contained between the marks will be a meridian line. When the shadow of the wire coincides with this line, it is apparent noon. Greater accuracy will be obtained by extending the observation to several of the concentric arcs, and if 86 ASTRONOMY. they do not give the same line, taking for the meridian line, a mean between them. * 24. To find the time of the sun's apparent rising or Setting. At the time of the apparent rising or setting of the sum, the zenith distance ZS = 90° -- refraction — parallaw. Let R = reſraction — parallax; then ZS = 90° -- R, and by an investi- gation nearly similar to that in article 20th, we have, sin. (p + 1 + h) COS (p + 1 + 5 *s R) sin; P = v 2 * 2 w sin D cos L As it is not important to know the precise time of the rising or setting of the heavenly bodies, it is usual to omit the effects of refraction and parallax and to consider the bodies as rising or setting when they are really in the horizon. g 25. To find the time of the beginning or end of twilight. Twilight commences or ends when the sun is about 18° below the horizon. Therefore the zenith distance ZS = 90° -- 18°; and by substituting 18° instead of R in the formula in the last ar- ticle, we have, sin (p + L + ls) COS (p+, + 18° 18) sin # P = V/ to 2 sin D cos L If the time of the commencement of twilight be subtracted from the time of sunrise, the remainder will be the duration of twilight. 26. The duration of twilight at a given place, changes with the declination of the sun. In northern latitudes, it is longest when the sun has its greatest north declination; and shortest when the declination is a few degrees south. It is not designed to enter into an explanation of the different circumstances, relative to the duration of twilight, as they are of but little prac- tical utility. But the determination of the time of CHAPTER IX. 87 shortest twilight, being a problem that has claimed con- siderable attention, may be introduced. 27. Given the latitude of the place, to determine the duration of the shortest twilight and the Sun's declination at the time. In the solution of this problem, twilight is supposed to com- mence when the sum is at a given distance below the horizon, the sun to rise when it is really in the horizon, and its distance from the pole to remain constant, during the continuance of twilight. The sun’s distance below the horizon when twilight commences is generally assumed to be 18°. Put 20 = 18°, and let S Fig. 19, be the situation of the sun when twilight commences, and Sº its situation in the horizon. Then PS = PS', ZS' = 90°, ZS = ZD + DS = 90° -- 2 a. and the angle SPS', converted into time, expresses the duration of twilight. Let PCS be a spherical triangle having the sides respectively equal to the sides of the triangle ZPS', that is, PS = PS', PC = PZ and CS = ZS' = 90°. Also let ZC be the arc of a great circle joining Z and C. Then the angle CPS is equal to ZPS', and consequently ZPC = SPS'. Hence when the angle ZPC is the least possible, the twilight will be shortest. Now in the triangle ZPC, the two sides ZP and PC are con- stant; and therefore the angle ZPC will be least when the side ZC is least. But as the two sides CS and ZS of the triangle ZSC are constant, the side ZC will be least when the angle ZSC = 0; that is when the sides ZC and CS coincide with ZS. Hence when the twilight is shortest, the angle PS/Z = PSC = PSZ. - We have, Fig. 20, ZC = ZD — CD = CS – CD = DS = 2 a. And because PZ = PC, if PE bisect the angle ZPC, it will also bisect the base ZC, and be perpendicular to it. Hence, *—sºm, “-- a--a -- ~~~~ sin ZPE = sim ZE sin a sin 9° * sin P cos PH cos latitude Twice the angle ZPE converted into time, gives the duration of shortest twilight. 88 ASTRONOMY. Since the two right angled triangles ZPE and SPE have the same perpendicular PE, we have (App. 45.), cos ZE : cos ES :: cos PZ : cos PS, cos a : cos (90° 4- a) : : sin latitude : sin declination, or cos a : — sin a : sin latitude : sin declination. sin a COS (!, sin latitude = — tan a sin Hence sin declination = — latitude. The value of sin declin. being negative, shows that the decli- mation is of a different name from the latitude of the place. Hence in northern latitudes the declination is south. The times of the year, at which the shortest twilight has place may be ascertained by observing in a Nautical Almanacº, the days on which the sun has the declination, found by the above formula. 28. The ancients gave considerable attention to the rising or setting of a star or planet under the circum- stances noticed in the following definitions. But it is not now considered of much importance. 29. The Cosmical rising or setting of a star, is when it rises or sets at sunrise. The flehronical rising or setting of a star, is when it rises or sets at Sunset. The Heliacal rising of a star, is when it rises so long before sunrise as just to become visible above the horizon and then immediately to disappear in conse- quence of the increasing light from the sun; and its Heliacal setting, is when it sets so long after the sun as just to become visible before it descends below the horizon. 3. * The JVautical Almanac is a work published annually in England, and at present republished in New York, and may generally be obtained one or two years previous to the year for which it is calculated, CHAPTER IX. - 89 sun's spots, AND RotATION ON ITS Axis. 30. The sun presents to us the appearance of a lu- minous, circular disc. But it does not from thence fol- low that its surface is really flat; for all globular bo- dies when viewed at a great distance have this ap- pearance. Observations with a telescope show that the sun has a rotatory motion. And it is only a globu- lar body, that in presenting all its sides to us, would always appear under the form of a circle. 34. When the sun is viewed with a telescope, dark Spots of an irregular form are often seen on its disc; and continued or repeated observations show that they have a motion from east to west. Their number, po- sitions and magnitudes are extremely variable. Fre- quently, several may be seen at once; and at some pe- riods, for a year or more there are none visible. Their magnitude is sometimes such as to render them visible to the naked eye, when in consequence of a smoky or thick atmosphere, the sun can be thus viewed without injury. Each spot is usually surrounded with a pe- numbra, beyond which is a border of light more bril- liant than the rest of the sun’s disc. Sometimes a spot becomes visible on the eastern limb of the sun, tra- verses the disc in about 14 days, disappears in the west, and after a like interval reappears in the east. But it is not often that this happens, as the spots fre- quently dissolve and perish before they arrive at the western side; or having disappeared on that limb, do not reappear. The nature of the solar spots and the causes of these changes are to us unknown. 32. When a spot remains so long permanent, as to be seen twice in the same position on the sun’s disc, the interval is found to be about 27 days. But this 43 90 A$"I'l-ONOMY, interval does not express the real period of the sun’s rotation on its axis. For during this time the sun, by its apparent annual motion, has advanced nearly a sign forward in the ecliptic. The spot has, therefore, made that much more than a complete revolution, before it appears to a spectator on the earth, to be in the same position. *- 33. The apparent position of a spot with respect to the sun's centre may be obtained by observing, when the sun is on the meridian, the right ascensions and de- clinations, both of the spot and centre. From three or more observations of this kind, the time of the sun’s rotation and the situation of its equator with respect to the ecliptic, may be ascertained. The student who wishes a complete investigation of these, may be re- fered to the astronomy of Delambre or Biot. The fol- lowing results have been obtained by Delambre. The time of the sun’s rotation on its axis 25 d. 0 h. 47 m. Inclination of the sun’s equator to the ecliptic 7° 19'. The north pole of the sun is directed towards a point in the ecliptic, the longitude of which is 14*. 20°. Some astronomers make the time of the sun’s ro- tation on its axis 25 d. 12 h. 34. It is also found that the moon and such of the planets as admit of sufficiently nice observations to de- termine the fact, have motions on their axes. This forms a strong analogical proof in favour of the earth’s diurnal motion. ZODIA CA. L LIGHT. 35. A luminous appearance is sometimes seen after sunset or before sunrise, in the form of a cone or pyra- mid, with its base at that part of the horizon which the sun has just left or at which it is about to appear, and CHAPTER X. 91. having its axis in the same direction as the plane of the sun’s equator. This phenomenon is called the Zodiacal Light. From the circumstance of its direc- tion always corresponding with the sun’s equator, it seems to have some connection with the sun’s rotation. 36. The angle which the plane of the stin’s equator makes with the horizon of a given place, at the time of sunset or sunrise, is different for different positions of the sun in the ecliptic. In our northern climates the greatest inclination, at the time of sunset, has place about the 1st of March; and at this season of the year, the zodiacal light is generally most distinct. At other seasons when the inclination is less, the vapours near the horizon conceal it from our view. On account of the different states of the air, at the season most fa- vourable to its appearance, it is much more distinct in some years, than in others. The extent of the zodiacal light is various, being sometimes more than 400°, and sometimes not more than 40° or 50°. OHAPTER X. of the moon. 1. THE moon, next to the sun, is the most conspicu- ous of the heavenly bodies, and is particularly dis- tinguished by the periodical changes, to which its figure and light are subject. The different appearances which it presents, are called the Phases of the moon. 2. By repeatedly observing the moon, when on the meridian, it is found that it has a motion among the fixed stars, from west to east, and that it comes to the meridian about 50 minutes later on each succeeding day. This motion is not uniform, but at a meån, it is 13° 30' 92 ASTRONOMY. 35" in 24 mean solar hours. It is also found, that the moon is sometimes on the north, and sometimes on the south side of the ecliptic, continuing about as long on one side as on the other; and that its orbit nearly coin- cides with the plane of a great circle, which intersects: the ecliptic in an angle of about 5°. 3. The points in which the moon’s orbit cuts the plane of the ecliptic, are called the moon’s Modes. That node in which the moon is, when passing from the south to the north side, is called the dscending JWode. The other, in which it is, when passing from the north to the south side, is called the Descending JWode. The nodes are distinguished by the following characters. Ascending node, §2. Descending node, 25. 4. At periods of about a month each, the moon en- tirely disappears, and continues invisible during two or three days. About the middle of this time, the longi- tudes of the sun and moon are equal. It is then said to be ...Wew JMoon. 5. When the moon again becomes visible, it is seen soon after sunset, a little above the western part of the horizon, under the form of a circular segment, the ex- terior boundary being a semicircle, and the interior, a semi-ellipse, having for its greater axis, the diameter of the semicircle. This phase of the moon is called a Crescent, and is represented in Fig. 21. The points A and B are called the Cusps, or Horns. 6. The convex part of the crescent is turned towards the sun; and if a great circle bisect, at right angles, the line AB, which joins the cusps, it will pass through the sun. CHAP"I"]. R. X. 93 7. From day to day the luminous segment increases in breadth, the interior boundary becomes less concave, and the moon advances to the east of the sun, till in a little more than seven days from the time of new moon, the difference of their longitudes% is 90°. This situa- tion of the moon with respect to the sun, is called the First Quarter. Nearly at the same time the moon ap- pears as a semicircle, the right line, joining the cusps, becoming the boundary on the side opposite the sun. The moon is then said to be Dichotonized, or to be a Half JMoon. t 8. After this, the side opposite the sun becomes con- vex, and the convexity, as well as the breadth of the segment, increases till the longitudes of the sun and moon differ 180°, which is in about fifteen days from new moon. At this time the moon appears nearly as a complete circle. It is then said to be Full JMoon. A phase of the moon, between the first quarter and full moon, is represented in Fig. 22. When the moon appears under this shape, it is said to be Gib- bows. .º. N - 9. After full moon, the western side of the moon be- comes elliptical, and the convexity and breadth de- crease. In about twenty two days from the time of new moon, the longitudes of the sun and moon differ 270°. It is then said to be Last Quarter. About this time the moon is again dichotomized. After this, the western side of the moon becomes concave, and the breadth of the segment continues to decrease till the moon again becomes invisible, a day or two before new moon. - * By the difference of their longitudes, is meant the excess of the moon's longitude above that of the sun, the former being increased by 360°, when it is less than the latter. 94 ASTRONOMY., 40. The interval from new moon to new moon, or from full moon to full moon, is called a Lunation, or Lunar JMonth. Its mean length is 29 d. 12 h. 44 m. 3 sec. e 11. Any two of the heavenly bodies are said to be in Conjunction, when their longitudes are the same; and to be in Opposition, when their longitudes differ 180°. The points in the orbit, at which the moon is, when in conjunction or opposition, are called the Syzigies; those at which it is, when its longitude ex- ceeds the sun’s by 90° or 270°, are called the Quadra- tures; and the middle points, between the syzigies and quadratures, are called the Octants. Some of these are designated by characters, as follows: Conjunction, 6, Opposition, & , Quadrature, [I. , r 12. The time between two consecutive conjunctions or oppositions, of a body with the sun, is called a Sy- modic Revolution of that body.* i 43. It follows from the preceding articles, that the new moon has place, when the moon is in conjunction with the sun; the full moon, when it is in opposition; and that the first and last quarters have place when it is in quadratures. Also, that the synodic revolution of the moon is the same with a lunation or lunar month. 14. For a few days before and after new moon, we * As the distances of the planets, Mercury and Venus, from the sun, are each less than that of the earth (7.28), they can never be in opposition to the sun. But they may be in conjunction, either by being between the sun and earth, or by being on the opposite side of the sun. The former is called the Inferior, and the latter the Superior conjunction. For either of these pla. nets, a Synodic Revolution is the interval between two consecutive conjunc- tions of the same kind. a CHAPTER Xe 95 can discern the entire disc, the obscure part appearing with a faint dusky light. Near the time of full moon, this part is entirely invisible. 15. When the moon is viewed with a telescope, the line separating the light part from the dark, is seen to be very irregular and serrated; and its form varies even during the time of observation. Bright spots are fre- quently seen on the dark part, near the line of separa- tion. The light of these spreads, till it becomes united to the rest. The whole enlightened surface also ap- pears diversified, with spots of different shapes and dif- ferent degrees of brightness. These spots always re- tain the same position with respect to each other, and occupy nearly the same place on the disc. It therefore follows, that nearly the same face of the moon is always turned towards the earth. 16. A consideration of the preceding circumstances leads to the conclusion, that the moon is an opaque globular body, having its surface diversified with mountains and vallies; and that it shines by reflecting the light of some other body. And from the relative situations of the Sun, moon, and earth, at the times of the different phases, this body appears to be the sun. At the time of new moon, the sun and moon are in nearly the same direction from the earth, and as the moon’s distance is less than the sun’s (5.44 and 46), the enlightened side of the moon is turned from the earth, and it is therefore invisible. At the first or last quarter, the difference of longitudes of the sun and moon being 90° or 270°, it is plain that about one half of the enlightened side is turned towards the earth. At the full, the moon being in opposition to the sun, near- ly its whole enlightened side is turned towards the earth. - 96 ASTRONOMY. 17. If, at the time of new moon, the moon is at or near to one of its nodes, as it is an opaque body, and is then directly between the sun and earth, or very. nearly so, it must prevent, at least, some of the sun’s light from arriving at the earth. This is sometimes the case, and occasions what is called an Eclipse of the Sun. When at the time of full moon, the moon is at, or near the node, it is in the earth’s shadow, and an Eclipse of the JMoon is produced. When the moon passes between the earth and a star or planet, the cir- cumstance is called an Occultation of the star or pla- net. These phenomena will be considered in the next chapter. 18. The Elongation of a body is its angular dis- tance from the sun, as seen from the earth. 19. The apparent breadth of the visible, enlightened part of the moon, is nearly equal to the apparent di- ameter, multiplied by the square of the sine of the elongation. - Let E, Fig. 23, be the centre of the earth, MANB the moon, C its centre, and let the sun be in the direction CS. Also let ACB be perpendicular to CS, AD to EC, and EM, EN, tam- gents to the moon at M and N. As this is not a problem in which it is important to obtain great accuracy, we may suppose the moon to be in the plane of the ecliptic, and the rays of light from the sum, that arrive at the moon, to be parallel. Then AMB is the enlightened half of the moon, of which only the part AM is visi- ble from the earth. It is seen under the angle AEM. Put C = angle ECS, * * * = angle CEM = app. semidiam. of moon. Then CD = AC cos ACE - AC sin SCE = AC sin C, ED = EC — AC sin C, f g AD = AC sin ACE = — AC cos SCE = — AC cos C, tan AEC = *P AC cos C ED T T ECTAC sin C' CHAPTER X. 97 But AC = EC sin 3 (7.18). EC sin * cos C . Hence tan AEC = — EE — EC sin \sin CT _ _ sin 3 cos C Tsinº sin C" Or since AEC and 3 are both small, 3 cos C ADC = — T-jºin G, AEM – CEM – AEC = 2 + → *% 1 —— sin 3 sin C 3 cos C = 3 + 3 cos C – 3 cos C + 1 — sim 3 sin C 3 sim \cos C sin C = 3 + 3 cos C º + + 1 — sin 3 sin C As the last term is very small it may be rejected, and we shall ...have, * t AEM = 3 + 3 cos C = 3 (1 + cos C) = 2 3 cos” # C (App. 9) - - gºe - tºº tº- wº * - - A. On account of the small distance of the moon from the earth, compared with the sun's distance from both of them (5.14 and 16), we may, without material error in the present case, consider the line ES', drawn from the earth in the direction of the sun, as parallel to CS. Then, AEM = 2 + 3 cos C = 2 + 3 cos (180° – CES) = 9– 3 cos CES' = 3 (1 — cos CES) = 2 3 sin” # CES', (App. 8) B. & 20. In like manner as the moon reflects light to the earth, the earth reflects light to the moon, with this difference, that when the moon gives least light to the earth, the earth gives the most to the moon, and the contrary. It is the light reflected from the earth to the moon, and from it back to the earth, that renders the obscure part visible. 44 98 ASTRONOMY. \ Moon’s MEAN MOTION IN LONGITUDE. 21. If the time at which the moon’s centre passes the meridian be observed, its right ascension at that times becomes known. And from the observed alti- tude, corrected for refraction and parallax, the north polar distance and consequently the declination is ea- sily obtained (6.2). With the right ascension and de- climation, the longitude may be calculated (6.49). By repeated observations and calculations of this kind the interval from the time at which the moon has any given. longitude, till it again arrives at the same longitude, may be ascertained. This interval, which is the Tro- pical Revolution of the moon, is found to vary. From observations, made at distant periods, its mean length is determined to be 27.321582 mean solar days.” Hence 27.324582 d. : A d. : : 360° : 13° 40' 35" = moon’s daily mean motion in longitude. The mean motion of the moon, here given, corres- ponds to the commencement of the present century. A comparison of modern observation, with those of re- mote periods, proves that the moon’s mean motion is subject to a small, but thus far a continual Acceleration. Investigations in Physical Astronomy, have made known the cause of this acceleration, and have shown that it is really a periodical inequality in the moon’s motion, which requires a great length of time to go through its different values. 22. The moon's mean longitude, the 1st of January 1801 at mean noon, on the meridian of Greenwich, was 3, 28°46'56". 1. With the Epoch of the mean longi. , tude, and the moon's mean motion, the mean longitude may be determined for any other given time. * It may be observed that astronomers generally express such periods in mean solar time; unless the contrary is specified, CHAIPTER X. 99 Moon’s NODES AND INCLINATION OF THE ORBIT. 23. With different longitudes and latitudes of the moon, deduced from the observed right ascensions and declinations, the situations of the nodes and the incli- nation of the orbit may be found, by methods nearly similar to those employed for determining the place of the vernal equinox, and the obliquity of the ecliptic (7.6 and 6.2). 24. The moon’s nodes are not fixed; they have a retrograde motion, which is ascertained, by determin- ing their situations at different periods. This mo- tion is subject to several inequalities, and diminishes from century to century. At the commencement of the present century the mean length of a tropical revolution of the nodes, was 6798.54019 days or 18 years and 224,47944 days; and at that time the mean longitude of the ascending node was 0° 13° 52' 47". The annual mean motion of the nodes in longitude is 19°20' 26". 25. The inclination of the moon’s orbit to the plane of the ecliptic, varies from about 5° to 5° 18'. It is greatest when the moon is in quadratures and least when it is in syzigies. The mean inclination, which is always the same, is 5°8' 38". - Moon’s ORBIT. 26. Observations and investigations, similar to those made in the case of the sun, prove that the moon’s orbit is nearly an ellipse, having the centre of the earth in one focus, about which the radius vector describes areas nearly proportional to the times. 27. Considering the mean distance of the moon from the earth, a unit, or 1, the eccentricity of the orbit is .0548553, which gives the greatest equation of the cen- tre 69 47' 54".5. 400 ASTRONOMY. 28. The apsides of the moon’s orbit, have a direct motion, by which they perform a mean tropical revo- lution in 3234.4751 days, or 8 years and 309.537 days. At the commencement of the present century, the mean longitude of the perigee was 8' 26° 10'. 29. Besides the equation of the centre, the moon’s motion is subject to numerous inequalities, of which the three following are the principal. The most considerable is called the Evection, and was discovered by Ptolemy. It depends on the angu- lar distance of the moon from the sun, and of the moon from the perigee, and amounts, when greatest, to 1% 20' 30". The second is called the Variation, and was disco- vered by Tycho Brahe. It disappears when the moon is in the syzigies and quadratures, and is greatest when it is in the octants. It then amounts to 35'42". 30. The third is called the flnnual Equation, and depends on the mean anomaly of the sun. When greatest, it amounts to 11’ 12". 30. The motion of the moon's nodes, the change in the inclination of the orbit, the motion of the apsides, and the preceding inequalities in the moon’s motion, are caused by the sun’s attraction, and are completely explained by investigations in physical astronomy. These investigations have also led to the discovery of other minute inequalities in the moon’s motion, and thence conduced to the accuracy of tables for computing its place at any given time. 34. The most accurate tables of the moon are those by Burg, and by Burckhardt. The former employs 28 equations for the moon’s longitude, and the latter 36. The moon’s place calculated by either of these sets of f CIIAPTER X. 404 tables, always agrees, within a few seconds, with its place as determined by observation. DIFFERENT REVOLUTIONS OF THE MOON. 32. From the tropical revolution of the moon (21), by taking into view the motion of the equinoxes, the sun’s apparent motion, and the motions of the apsides, and nodes of the moon’s orbit, the other revolutions of the moon are easily determined. The following are the lengths of these revolutions, as given by Delambre. .* DAYS. 'Tropical revolution, - º - 27.32 15255 Sidereal revolution, - “º gºe 27.32 15830 Synodic revolution, tº gº - 29.5305885 Anomalistic revolution, tºs tº 27.5545704 Revolution from one node to the same, 27.2122222 Moon’s REvolution ON ITS Axis. 33. It has been stated (15), that the moon presents nearly the same face to the earth. It must therefore revolve on its axis in the same direction and same time that it revolves in its orbit. But accurate and continued observations, show that the moon’s spots do not pre- serve exactly the same situations on the disc. They are seen alternately to approach and recede from the edge. Those that are very near the edge, successively disappear and again become visible, making periodical oscillations, which are called the Librations of the II10011. 34. The librations of the moon are not occasioned by an unequal motion on its axis. For, admitting this motion to be uniform, and the axis to be nearly per- pendicular to the plane of the orbit, small portions on {02. AS'I'RONOMY. the east and west sides of the moon ought alternately to come into sight and to disappear, in consequence of its irregular motion in its orbit. This is conformable to observation, and is called the Libration in Longi- tude. 35. Besides the motion of the spots in an easterly and westerly direction, they are observed to have a small alternate motion from north to south. This is called the Libration in Latitude, and shows that the moon’s axis, though nearly, is not exactly, perpendicu- lar to the plane of its orbit. 36. In consequence of the earth’s diurnal motion, a spectator at the surface sees portions of the moon a little different, according to its different positions above the horizon. This is called the Diurnal or Parallactic Libration. 37. From calculations founded on accurate observa- tions of the lunar spots, it has been found, that the equator of the moon is inclined to the plane of the ecliptic in an angle of 4° 30'; and that the line, in which the plane of the equator cuts the plane of the ecliptic, is always parallel to the line of the nodes. 38. Iſ three planes be supposed to pass through the centre of the moon, one representing the equator of the moon, another the plane of its orbit, and the third be- ing parallel to the ecliptic; then the last will lie between the two others, and will intersect them in the same line, in which they intersect each other. It will make with the first, an angle of 4° 30'; and with the second, an angle of 5°9'. This curious fact was discovered by Cassini; and it has been explained by Lagrange from the theory of gravity. CHAPTER X. {03 \ Moon’s DIAMETER AND MOUNTAINS. 39. The greatest and least horizontal parallaxes of the moon, for a place on the equator are 58' 52" and 64' 32" (5.44). The corresponding apparent diame- ters are found to be 29' 22". and 33' 31". If d = real diameter of the moon and R = equatorial radius of the earth, we have (7.15), | 29' 39" R. 1708 – 2B. 170° – d = R. f* f. 53' 52" 3230 * 6460 3 O * very nearly. Hence the diameter of the moon is about fºr of the equatorial diameter of the earth, and consequently its surface is about +'s of the earth’s surface, and its vol- ume, about 4, of the earth’s volume. The moon’s di- ameter in English miles is 2160. 40. It has been observed (16) that the moon’s sur- face is diversified with mountains and vallies. The heights of some of these mountains, in comparison with the diameter of the moon, are found to exceed those of the earth. Though not a subject of much importance, it may be interesting to the student to know a method of ascertaining the heights of the lunar mountains. 41. Let ABO, Fig. 24, be the enlightened hemisphere of the moon, E the situation of the earth, ES' the direction of the sun from the earth, and SM a solar ray, touching the moon in O, which will be one of the points in the curve, separating the en- lightened, from the dark part of the moon. Also let M be the summit of a mountain, situated near to 0, and sufficiently eleva- ted to receive the sun’s light. To a spectator at E, the summit M of the mountain, will appear as a bright spot on the dark part of the moon. * The angle MEO may be measured by means of a micrometer, ſ04. ASTRONOMY. | attached to the telescope. In this case as in a former (19), we may consider ES’ as parallel to MS. We may also without ma-. terial error consider the angle MES’ as equal to the elongation CES'. Let OD be perpendicular on ME, and put, * = apparent diameter of the moon, d = AO = real diameter of the moon. OD. EO sin MEO We have, CO tan MCO = MO = - \{* = ** * **** e nave, * sin OME sim OME 'EO sin MEO =; EC cin MEO sin MEST sin CES' EC sin MEO 1 sin MEO H tan MCO = tº. * * = o ence, tam CO sin CES' sin # 3 sin CES' (7.13). nearly. _ 2 sin MEO T sin 3" sin CES OC COS wido-09 Height of the mountain = a M = MC – OC = 1 — cos MCO OC 1— cos MCO sin MCO = OC. sin MCOT T Tcos MCOT' sin MCO 1 — cosMCO 1 . : :... tan MCO = OC. —t C sin MEO all O = 0 cot # MCO tan MCO (App. 11). == = OC tan MCO tan # MCO = 3 OC tam” MCO = # OC. 4 o Sin? MEO = d. ( sin MEO ) sim? 3 sin? CES' sin 3 sin CES' ang. MEO ) - UWe (; sin CES/ " Dr. Herschel has made observations on a number of the lunar mountains. For one of these the data are, the angle MEO = 40".625, apparent diameter of the moon = 32° 5''.2, and the elongation = 125°8'. Hence taking the moon’s diameter 2160 miles, we easily obtain from the preceding formula, the height of the mountain = 1.45 miles. 42. Luminous spots, which are entirely unconnected with the phases, or in other words, are not the reflec- CHAPTER X. 405 tion of the sun’s light, are sometimes seen on the moon’s disc. These are supposed to be volcanoes. 43. If the moon was surrounded with an atmosphere, such as appertains to the earth, it would, by its action in changing the rays of light, produce a very sensible effect in the duration of an eclipse of the sun, or an oc- cultation of a star or planet (17). But various accu- rate observations prove that if any effect of this kind has place, it is extremely small. It therefore follows, that if the moon has any atmosphere, it must be, either very limited, or very rare. Moon’s PASSAGE over THE MERIDIAN. 44. In consequence of the moon’s daily increase in right ascension, it passes the meridian later on each -day than on the preceding (2). The daily retardation varies from about 38, to 66 minutes. To obtain the time of the moon’s passage on any particular day, let R = the excess of the moon’s right ascension above the sum's, at noon of the given day,” S = daily motion of the sum in right ascension, M = that of the moon, both being considered as uniform during the day, T = the required time of the moon’s passage, and A = the arc of the equator, which passes the meri- dian, between noon and the moon’s passage. - Then 24h : T : : M : º = moon’s motion in right ascension during the time T. Hence A – R + 4*. 24h - Also 24 : T : ; 360° 4 S : A – T. º -- S) * When the moon's ascension is less than the sun's, it must be increased by 360° or 24 h. 45 406 ASTRONOMY. Therefore T. (360° 4. S = R + T. M. 24h 24h T. (360° -- S) = 24). R + T. M., 24h. R. 24h. R. 350. TSTM - 24TSITM' 45. If M = the daily motion of a planet in right ascension, the preceding formula will give the time of its passage over the me- ridian, observing that when the motion of the planet is retrograde, the sign of M must be changed and the formula will them become, 24h R. 360° E S + M. For a fixed star, M = 0, and the formula becomes, 24h R. ~ 350Ts. Moon’s RISING AND SETTING. 46 On account of the moon's change in declimation, the semi- diurnal arc, found with its declination at the time of its passing the meridian (9.18), is not correct. If however the semi- diurnal arc, thus obtained, be applied to the time of the moon’s passage, by subtracting for the rising or adding for the setting, it will give the approximate time of rising or setting. To obtain the time more correctly, find the declination for the approximate time and again calculate the semi-diurnal arc, which must be corrected on account of the moon’s change in right as- cension. Thus, 24" + S – M ; 24" : : semi-diurnal arc : cor- rected semi-diurnal arc, which applied as before to the time of the moon’s passage, gives the time of the moon’s rising or setting, very nearly. If D = the difference between the times of the moon’s passage on two consecutive days, one of which precedes and the other fol- lows the required time of the moon’s rising or setting, the last correction may be made thus : As 24" : 24" + D : : semi- diurnal arc : corrected semi-diurnal arc. As the mean length of the semi-diurnal arc is about 6 hours; it is better, in the operation for obtaining the approximate time of CHAPTER X. $ 107 rising or setting, to make use of the moon’s declination 6 hours before or after its passage over the meridian, according as it is the rising or setting that is required. f Moon’s PARALLAX IN LONGITUDE AND LATITUDE. 47. The effect of parallax in changing the altitudes of the heavenly bodies has been shown in a preceding chapter and a method given for determining it. But this is not the only effect of parallax. It also changes the right ascension, declination, longitude, and lati- tude of a body. It may be proper here to investigate formulae for calculating the effect of parallax on the moon’s longitude and latitude, as they will be useful in our chapter on eclipses of the sun. * 48. Let HZR, Fig. 25, be the meridian, HR, the horizon, Z the zenith, EQ the equator, ECO the eclip- tic, P and p their poles, E the vernal equinox, A the true place of the moon, B its apparent place, as de- pressed by parallax, in the verticle circle ZABK, and pAa and pPl, circles of latitude, passing through the true and apparent places. By the effect of parallax, the true longitude Ea is changed to the ap- parent longitude Eb, and the true latitude Aa, to the apparent latitude Bb. d 49. The difference between the true and apparent longitude of a body, produced by parallax, is called the Parallaa in Longitude; and the difference between the true and apparent latitude, is called the Parallaa. in Latitude. 50. The point in the ecliptic, which is above the ho- rizon, and at 90° distance from the intersection of the ecliptic and horizon, is called the JWomagesimal De- gree of the Ecliptic. 51. The data usually given to calculate the moon’s 408 ASTRONOMY. parallax in longitude and latitude, are the moon’s true longitude, and its latitude, or distance from the north pole of the ecliptic, its horizontal parallax, the obliqui- ty of the ecliptic, the latitude of the place, and the right ascension of the mid-heaven. On account of the spheroidical figure of the earth, the horizontal parallax, at any given time, is different at different places (5.12). The parallax for a given place, is called the Reduced Parallaw. If the earth were a sphere, having a radius equal to the straight line joining the given place and the centre, it is evident that the parallax for all parts of it, would be the same as the reduced parallax for the given place. It is also plain, from the definition (4.10), that the reduced lati- tude of any place, is its latitude on the supposition of the earth being a sphere. If, therefore, the reduced latitude and parallax be used, the earth may be con- sidered as a sphere, which will simplify the investiga- tions for finding the parallax in longitude and latitude. 52. Let pPCD be a great circle, passing through p and P, and pZn] another, passing through p and Z. Because P is the pole of EQ, the pole of pſ’D is in EQ; and because p is the pole of EC, the pole of ppD is in EC; the point E, in which EQ and EC intersect each other, is, therefore, the pole of ppD; and con- sequently, ED and EC are each 90°. In like manner, because Z and p are the poles of HR and EO, the point O, is the pole of pZn], and On and OI are each 90°. Consequently, n is the nona- gesimal degree of the ecliptic. Also, En is the longitude of the monagesimal degree, and ni is its altitude. These quantities are used in finding the parallax in longitude and latitude, and must first be found. 53. Put w = Pp = obliquity of the ecliptic, H = Z.P = complement of the reduced latitude, CHAPTER X. 4.09 M = EM = right ascension of the mid-heaven, m = En = longitude of the monagesimal, h = p2 = 90°–Zn = m1 = altitude of the nomagesimal, L = Ea = moon’s true longititude, * : A = pa = moon's true distance from north pole of the eclipſie, or = moon’s horizontal parallax, reduced, s " ," II = moon’s parallax in longitude, * = moon’s parallax in latitude. In the triangle pp2, we have given Pp, PZ, and pPZ = 180° – ZPD = 180°– DM = 180° — (90° — EM) = 90° + EM - 90° 4- M, to find p2 = the altitude of the nonagesi- mal, and Pp.Z = Cn = 90° – En = the complement of the longitude of the monagesimal. & LONGITUDE OF THE NONAGESIMAL. Lets = Pp.Z + PZp, D = Pp.Z — PZp, E = 180°– ; S, and F = 180° — # D. Then (App. 41), 1 _ cos # (H — º l o tan # s=:: (HTaj' cot # (90° -- M) _ cos 4 (H – 2) T cos. (HT2) . tam # (90° — M) y 1. ( H – a \ . tan (180°–4 S)=. i # + % tan (180°–45. 4. & M) cos 3 (H – «) = ** 3 \* T *l, tan 4 (270° g cos # (H + æ) an # (270° -- M) cos 3 (H – "), t 1. tººl ov T così (HT2) an # (M — 90°), or an E = ** (*= *), t I (M. – 90° I cos # (H + æ) an # (M – 90°). Again (App. 42), - tan : D = * * (H– ø). { } (90° * 2 sin # (H + w) CO 2. ( + M). Whence, by transforming as above, we have, tan F-ºn 3 (H-2). – 90° sin # (H + w} an 3 (M – 90°). 440 ASTRONOMY, Now 3 S + 3 D = Po2 = 90° — n. Hence, n = 90° – (# S + 3 D), or n = 450° — (3 S + 3 D) = 180° — 4 S + 180° — 3 D + 90°. * , Consequently, n = E + F + 90°, rejecting 360°, when the sum exceeds that number. ALTITUDE OF THE NONAGESIMAL. We have (App. 44), tan : h = **ś tan 4 (H — a) e J. e Il # _ sin # E wºº ... tan # (H — al. sim # F # ( ) PARALLAX IN LONGITUDE. 54. The triangle ApH, gives, * sin AB sin ZBp sin Ap sin a sin (N + p) sin ZBp (5.5). sin A sin II = sin ApH = In the triangle ZpB, the angle Zpb=nb-na + ab= Ea—En -- ab=L–n -- II, d sim ZB _sin p2 sin Zp.B. sin hsin (L– n + m). and Sim ZBP sin ZB sin (N + p) . -- sin a sin h . / II = − º - Hence sim sin A sin (L — n + II) (C), sin A sin a sin h + cos (L – m) sin II sin II = sin (L — n + II) = sim (L — n) cos II sim A ë * −r = sin (L — m) cot II + cos (L — n sin a sin h sin ( J + ( ) in A gº cot II = SII). cos (L–n) sin a sin hsin (L-7) Tsin (L–7) in a sin h si gººs sin arsin h . Make tan w = ** *** (D - º – w sin (L–n). sin A sim A CHAPTER X. 444 sin A COS M, T £n, º e e * tº º \sin a sin h sin (L – m) S10 'lſ, and cot II = #-Hº _ sin (L–n) cosw-cos (L-m) sinu sin (L–n—w). *- sin (L–n) sin w sin (L–n) sin w sin (L — m) - sin (L–n—w) sin w. Hence, tan II = PARALLAX IN LATITUDE. 55. The triangles p2A and p2B give (App. 34), cos Ap-cos Zp cos ZA, cos Bp-cos Zp cos ZB cos p2B = sin Zp sin ZA Tsim Zºsimº ET’ or 99% A — cosh cos N cos (A + +) — cosh cos (N + p.) 2 sim N -* Tsin (N + p.) Tº cos A sin (N + p.)— cosh cos N sin (N + p) = cos (A + wº sin N — cosh sin N cos (N + p.) cos (A + r.) sin N = cos A sin (N + p) — 90sh Isin (N + p.) cos N.— cos (N + p) sin N] = cos A sin (N + p) — cos h sin p - (App. 13) = cos A sin (N + p.) — cos h sin a sin (N + p) (5.5) = sin (N + p.). (cos A – sin ºr cos h.) Therefore, cos (A + x) = sin(N+ p) . (co8 A — sin a cosh). S11) n ** But, sin Apsin ZPA si _ sin Bosin Zp.B ul, sin ZA sin p2B sin ZB sin A sin (L–n) _ sin (A + ºr sin (L– n + II) Or, º -: —— * sin N sin (N + p.) sin (N + p). sin (A + +" sin (L – n + H) (D) sin N sin A sin (L — n) ~ 2. sin (A + x) sim (L – m + º º Iſ). (cos A — sin A sin (L — n) Hence, cos (A + z) = sin a cos h) sin (L — n + II) A — in-coºl), (E) A = cot (A + x) sin (L – m) ' (cot sin A 412 ASTRONOMY. sin a cosh -* Make tan a = sin A Then, cot (A + n) = *##". (cot A — taña) sin (L-n + n) (#. _ sin :) sin (L — n) sin A cosa. _ sin (L — n + II) cos A cosa — sin A sin & sin (L — n) sin A cos a sin (L– n + n) cos (A + 2) sin (L — n) sin A cos a (F). The apparent latitude (A + x) being calculated by either of the formulae (E) and (F), we have r = (A + ºr) — A. 56. We may obtain formulae that will give the parallax in la- titude, without first finding the apparent latitude, and which in some cases may be more convenient than those in the last article, We have (55.E) sin (L — n) cot (A + x) * tota tºº sim, a cos h sin (L – m + II) sim A^T cot A = sin a eos h S11] (L — n) cot (A + r.) Slm A sim (L — n + II) sin a cosh -— ” A sin A cot (A + x) + sin (L — n) cot (A + r.) sin (L– n + II) Hence (App. 27), sin ºr _ sin a cos h sin Asin (AIE *) T Tsin 2– cot (A + 1) [sin (L – m + II) — sin (L– n)] sin (L — n + II) cot A — cot (A + 1) = _ sin a cosh 2 sin & II cos (L — n + 3 II) cot (A + ºr) T - sin. A sin (L — n + II) (App. 21) sin ºr = sin a cos h sin (A + x) — €HAPTER X. | 413 2 sin 3 II sin A cos (L – n + 3 II) cos (A "+ a) sim (L – n + II) = sin ar cos h sin (A + a) sin m sin A cos (L – n 4-3 n) cos (A + a) a … cos à II sin (L – n + II) (App. 7) But (54.C), sin # !"# II) = sin a sin h. Therefore, sin zr = sin a cos h sin (A -H- cr) sin zr sin h cos (L – n -H- 3 II) cos (A -|- zr) (G) t~ ~ * •£ cos # II = sinar cosh [in (A -- zr) _ tan h cos (L – n + 3 II) cos # II cos (A + -)] Make tan y -º "ººº (L – m + 3 m). Then, cos # II te, e «. o sin y sin r = sin ar cos h | sin (A + a) — °“ 2. cos (A + a) cos y - sin ar cos h [sin (A + x) cos y -- cos (A +.a) sin y] c0s y sin ar cos h =---------- sin (A p=…*s H ¿î– sin (a + a -9) (H) _ Sin * cos h sin (A — y) cos * cos y sin º cos | cos (A — y) sin a cos y « h . • sin ar cos h 1 _ Sin º cos n (A ' — c0t ar -i- ----------- COS (A - cos y sin ( j) -l- cos y ( y) s… cos y — cot (A - cot ar = sin ar cos h sin (A — y ( J) Make tan o - *--* h sin (A —g) (I) cos y sin (A -- y – v) sin v sin (A —y) (App. 37) Them cot ar = cot v — cot (A — y) = sin (A — y) sin v sin (A -- y - v)' Hence tan * —, \ 46 1ſº ASTRONOMY. EFFECT OF PARALLAX ON THE APPARENT DIAMETER OF THE MOON. 57. The moon is nearer to any place on the earth’s surface when it is elevated above the horizon, than when it is in the horizon. The angle under which its diameter is seen, will therefore be greater in the for- mer case, than in the latter. * Let } = moon’s horizontal semidiameter, * * = moon’s apparent semidiameter at a given situation above the horizon. Then (7.13), sim \' D e BC sin BAZ sin (N + p) - - - Q'. 6 - = -- = —- sin .) D' (Fig. 6) AB sin BCZ sin N sin (A + ºr) sin ( — n + II) 55. D sin A sin ( L — n) (55. D), sin 3 sin (A + x) sin (L — n + II) sin A sin (L — n) * (5.5) Hence sin 3" = (L) We may obtain other ſormulae for expressing the relation be- tween 3 and 9', which will be more convenient when the paral- lax in latitude is found by the second method. Let Psºr, Fig. 25, be the arc of a great circle, bisecting the angle ApH, and ZGSL, another arc of a great circle, perpen- dicular to pr. Then pC = pl., the angle pGL = plG, Gps = } II, and Zps = mpr = mpa + apr = L — n + 3 II. Hence tan ps = tan p2 cos Zps = tan h cos (L – m + 3 II) tan ps tan h cos (L — n + 3 II) cos Gps T COS # II tan pG = = tam !/ (56) Therefore pG = y, AG = Ap — Gp = A — iſ and BL = Bp — Lp = A + ºr — y. Now sin BL sinBZL sin AZG sin AG. sin BZ sin L sin G sin AZ Hence sin BL sin BZ sin AG sin AZ sin (A + ºr — y sin (N + p.) sin º' r º = —x+- = —R- sin (A — y) sin N sin 3 CIIAPTER XI. 145 sin 3 sin (A + ºr — y) sin (A — y) and sin 9' = (M) The last formula may be reduced to another still more simple For we have (56 H and I), tºº, e sin ºr cos y e tan v cos y Sin (A + II — TI) = — and sin (A — u) = −-#. in (A + y) sin ºr cosh' ( y) sin a cos h sin (A — Q sin ºr |He C, ( + 7ſ. y) - l, 7 sin (A — y) lan () tº sin \sim and sin 3" = a. (N). tan v CHAPTER, XI. Eclipses of the Sun and JMoon.—0ccultations. 4. As the moon is an opaque body, and shines, only, by reflecting the sun’s light, when, at the time of full moon, it enters the earth’s shadow, it must become eclipsed. When, at the time of new moon, the moon passes between the sun and a spectator on the earth, it must occasion, to him, an eclipse of the sun. 2. If we suppose the sun and earth to be spheres, as they are, very nearly, the sun being much larger than the earth, the shadow of the earth must have the form of a cone, the length of which depends on the magnitudes of the bodies, and on their distance from each other. The moon’s shadow is also conical, but of less extent than that of the earth. EARTH’s SHADow. 3. Let ABG and alg, Fig. 26, be sections of the sun and earth, by a plane, passing through their centres S and E, and AaC and BbC, tangents to the circles ABG and abs. Tisregarding, at present, the action of 4:16 z ASTRONOMY. the earth's atmosphere, in changing the direction of those rays of the sun which pass through it, the tri- angular space a Cb will be a section of the earth’s shadow. The line EC is called the ſlacis of the earth’s shadow. 4. With a view to conciseness of expression in some of the succeeding articles, we shall put R = Eb = ra- dius of the earth, P = moon’s horizontal parallax, p = Sun's horizontal parallax, d = moon’s apparent semidiameter, and 3 = sun’s apparent semidiameter. 5. The earth's shadow extends to more than twice the distance of the moon. Put n = 206264".8 (5.8). From the triangle EBC, we have, SEB = ECb + EBb, or 3 = ECb + p. Therefore, ECb = 3–p, R _ R.n R.n and BC – i.e.- : - ; ºr Now (5.8), the moon’s distance from the earth = º ſº Hence, as 3–p is less than half P, the distance EC, is more than double the moon’s distance from the earth. 6. Let hyſh' be a circular arc, described with the centre E, and a radius equal to the distance between the centres of the earth and moon; and let Adh and Bch' be tangents to the sections of the sun and earth, crossing each other between them. º When any part of the moon enters the space be- tween the lines dh and bo, that part will, evidently, be deprived of a portion of the sun’s light, and will therefore appear less bright. As the moon approaches JG HAPTER XI. 447 the line bC, its light continues to be diminished; and and when the edge comes in contact with b0, the eclipse commences. Hence there is a gradual diminu- tion of the moon’s light, previous to the commencement of an eclipse of the moon. There is also a gradual in- crease in the light, after the eclipse has ended. This is conformable to observation. 7. If we suppose the line dh to revolve about EC, and form the surface of the frustum of a cone, of which cdhh' is a section, the space included within that sur- face is called the Penumbra. The earth’s shadow is sometimes called the Umbra. 8. The moon sometimes enters the penumbra, and again passes out, without any part entering the umbra or real shadow. In such cases, it sustains a diminu- tion of its light, but is not said to be eclipsed. 9. Any section of the earth's shadow or of the pe- numbra, by a plane perpendicular to the axis of the shadow, is a circle. If we suppose such a section of the shadow and penumbra, to be made at the distance of the moon, the apparent semi-diameter of the section of the earth’s shadow, as seen from the centre of the earth, is called the Semi-diameter of the Earth’s Sha- dow; and the apparent semi-diameter of the section of the penumbra, is called the Semi-diameter of the Pe- numbra. The angle MEm is the semi-diameter of the earth’s shadow, and the angle MEh! is the semi-diameter of the penumbra. 40. The semi-diameter of the earth’s shadow is equal to the sum of the moon and sun’s horizontal pa- rallaxes, less the apparent semi-diameter of the sun. In the triangle EmC, we have, the angle m|EM = Emb — ECm. 4 48 ASTRONOMY. Now, the angle mEM is the semi-diameter of the earth's sha- dow, Emb is the moon's horizontal parallax = P, and ECm = * — p (5). \ Hence, Semi-diameter of the Earth's Shadow = P — (3 — p) = P + p — 3. 44. The diameter of the carth’s shadow, at the moon, is more than double the apparent diameter of the moon, and consequently the moon may be entirely en- veloped in the shadow. If we take P = 57' 22", p = 8".7, and 3 = 16' 1".3, we obtain the mean semi-diameter of the earth’s shadow = 41’ 29”.4, and consequently the mean diameter = 82'58".8; which is more than twice the apparent diameter of the moon. 12. If, at the time of full moon, the apparent dis- tance of the moon’s centre, from the axis of the shadow, does not become less than P + p + d – 3, there can not be an eclipse. When the edge of the moon touches the earth's shadow, its centre is at a distance from it, equal to the moon’s semi-diameter. If, therefore, to the semi-diameter of the shadow, we add the moon’s semi-diameter, we have the distance of the moon’s centre from the axis of the shadow, at the beginning or end of an eclipse of the moon, equal P + p + d – 3. 13. The distance of the moon’s centre from the axis of the shadow, at the time of full moon, depends on the moon's distance from the node, and on the inclination of the orbit. We may, for a given inclination of the orbit, and given value of P + p + d – 3, determine within what distance from the node, the moon must be, in order that an eclipse may take place. ! CHAPTER XI. 149 : By taking the least and greatest inclinations of the orbit, the greatest and least values of P + p + d – 3, and also taking into view the inequalities in the mo. tions of the sun and moon, it has been found, accord- ing to Delambre, that when at the time of mean full moon, the difference of the mean longitudes of the moon and node, exceeds 12° 36', there can not be an eclipse; but when this difference is less than 9° there must be one. These numbers are called the Lunar Ecliptic Limits. 14. From tables of the mean motions of the sun, moon, and node, the mean time of any full moon, and the difference of the mean longitudes of the moon and node, at that time, are easily obtained. Then by the lunar ecliptic limits, we know whether or not these will be an ecliptic, except when the difference of longitudes of the moon and mode, is between 9° and 12° 36', in which case further calculation is necessary. 45. We have hitherto, considered the earth’s shadow as limited by those rays, which, passing from the edge of the sun, touch the corresponding side of the earth. But it is found that the observed duration of an eclipse of the moon, always exceeds the duration, computed on the supposition of the shadow being thus limited. This is accounted for, by supposing the most of those rays of light which pass near the surface of the earth to be absorbed by the earth’s atmosphere, as the effect of such an absorption would evidently be an increase in the extent of the earth’s shadow. In consequence of the gradual diminution of the moon’s light, previous to its entering the earth’s shadow, and gradual increase on leaving it (6), the time of the commencement or end of a lunar eclipse can not be observed with great accu- racy. Astronomers therefore differ with regard to the 420 ASTRONOMY. amount of the correction which should be made. It is however usual in computing an eclipse of the moon, to increase the semi-diameter of the earth’s shadow by a gº, parl; or which amounts to the same, to add as many seconds as the semi-diameter contains minutes. 46. There is another effect of the earth’s almosphere, which is perceptible in eclipses of the moon. Those rays of the sun which enter the earth’s atmosphere and are not absorbed, have their directions changed, so as to meet at a less distance than they otherwise would do. The rays through the atmosphere near the sur- face of the earth, are so much converged as to meet at a distance from the earth, less than that of the moon. In this way a sufficient quantity of light is thrown on the moon to render it visible, even when it is in the middle of the shadow. It then appears with a dull, red- dish light. 47. As an eclipse of the moon is occasioned by a real loss of light on the moon, and not by the interpo- sition of any body between the moon and the spectator on the earth, it must present the same appearance to all those who have the moon above their horizon during the eclipse, and observe it at the same time. It will be shown that the case is diſſerent with eclipses of the SUll]. A 18. The semi-diameter of the earth’s penumbra is equal to the sum of the moon’s horizontal parallax, sun’s horizontal parallax, and the apparent semidiameter of the sun. From the triangles ELh', and ELB, we have, MEh' = Eh'c + ELc = Eh'c + EBc + SEB. But Eh'c = P, EBC = p, and SEB = 3. Hence, Semi-diameter of the Penumbra = P + p + 3. CHAPTER XI, 42? MOON'S SHADOW. 49. The length of the moon’s shadow is about equal to the distance of the moon from the earth, being, some- times, a little greater, and, sometimes, a little less. Let now agb be considered as a section of the moon, and M the centre of the earth, then, the same figure will answer for de- termining the different circumstances relative to the moon’s sha- dow, that have been found for the earth's shadow. Put, R.' = Eb = moon’s radius, p' = EBb = sum’s horizontal parallax for the moon, * = SEB = Sun's apparent semi-diameter at the moon. We have, (5.8) SM = Rº, and EM = ** p PT' Now, Es – SM-EM = ** – º – RºſP-P). p P P.p R.n.) SM.) ..) Hence, (7.13) 3 = # = –1 – = - tº . , (7.18) ES R.m. (P — p) P –p Pp We have also (7.15) R' = # and SB = R.? hy w .R.'...)" 9 R.d P..) d [[ Pºº". P . Iºd. –tº– = –Pº- *r = -R-- it, ºr F-F = Fer R.d.ºn R’.m. R' n –E– ºC = ± = -: l -*. And, E ECB Ø\' — p! P.) * p.d P—p P — p R. d.n (P — p) R.m. P – p * 7T, T- 5- = -E-, -R---- (P.3 — p.d.).I P #P-p t 17 422, ASTRONOMY. In the last expression for the value of EC, the actor; €X- presses the moon's distance from the centre of the earth. And it s P–p is evident the other factor *P —p is greater than, equal to, or less than a unit, according as d is greater than, equal to, or less than 3. Hence the moon’s shadow extends beyond the centre of the earth, just to it, or not so far, according as the moon's apparent diameter is greater than, equal to, or less than the sun's apparent diameter. 20. The apparent semi-diameter, as seen from the moon, of a section of the moon’s shadow, at the earth, is equal to, P (2–2). Pi—F From the triangle mEC, we have, mEM = Emb — ECm = d – ()" — p’) = d -- p’— 3’ p.d P.9 P.d–P.9 (#9), d H- P — p P — p P —p (d— 3). & P 24. The greatest breadth of the moon’s shadow, at the earth, is about ºr part of the earth’s diameter. The expression for the breadth is, *-*. 2 R. P—p To obtain the breadth, we have / Mm = EM. angle.*MEm 1. R.n. (d–J). P. 77, m P P—p. CHAPTER XI, 423 or mm' = 2 Mm = d– 2. 2 R. P 3. if we take d = 1005".5, 3 = 945".5, P = 8692", and p = 8".7, which are, the greatest value of d, least of 3, and the corresponding values of P and p, we evidently obtain the greatest value of mm'. These numbers give, d — 3 60 $nm' = ū = --→ • P—p 2 R. 3684.3 i 2 R = 6T' 2 R nearly. 22. As the sun can only be entirely eclipsed at those parts of the earth’s surface on which the moon’s sha- dow falls, it is evident, from the preceding article, that in the most favourable cases, this phenomenon can only have place for a small portion of the earth. 23. When for a point on the earth, in the right line passing through the centres of the sun and moon, d = 3, the breadth of the shadow will be nothing. In this case, the sun will be entirely eclipsed, but it will not continue so, for any perceptible time. When d is less than 3, the expression for the breadth of the shadow becomes negative; the rays from the edge of the sun, which pass near the moon, crossing each other, in that case, before they arrive at the earth. In those parts of the earth where this has place, the edge of the sun will appear as a ring, surrounding the moon. 24. The apparent semi-diameter, as seen from the moon, of a section of the moon’s penumbra, at the earth, is equal to & P 3). —— (d. -- ). E From the triangles ELh' and ELB, we have, MEh! = Eh'c + ELc = Eh!c + EBc -- SEB = d -- p' + ) P.3 P.d + P.9 P £24 ASTRONOMY. 6 25. The greatest breadth of the moon’s penumbra, at the earth, is a little more than half the earth’s dia- meter. The expression for the breadth is, d + 3 To obtain the breadth, we have, EM. ang. MEl' 1, Rºn (d+ 2). P_ d 7, - a F TEL; P and hli' = 2 Mh' = d + 3. 2 R. P — p Taking d = 881", 2 = 977."8, P = 3232” and p = 8”.7, we obtain she greatest value of, d + 3 1859 23 hh' = ** T ... 2 R = ***. 2 R = ′. * = F == 3.223 40 Mh' = R. + 3 — p" 2 R nearly. 26. As no part of the sun can be hid by the moon, at those parts of the earth which are without the pe- numbra, the sun may be wholly visible for a large por- tion of the earth, while it is eclipsed either in part or entirely, in other parts. 27. If, at the time of new moon, the apparent dis- tance of the sun and moon does not become less than P — p + d -- 3, there can not be an eclipse of the sun to any part of the earth. Again, considering agb as a section of the earth, let M! be the place of the moon's centre, when in the comical surface, which circumscribes the sun and earth. Then the angle, MES = EM'b + ECb = P – p + 2. If to the value of M'ES, we add the apparent semi-diameter of the moon, we shall have, for the apparent distance of the cen- tres of the Sun and moon, at the beginning or end of an eclipse of the sun, the expression P-p + d -- 3, CHAPTER XI. 425 28. From the expression P – p + d -- 3, by taking into view the inclination of the orbit, and the inequali- ties in the motions of the sun and moon, it has been found, according to Delambre, that when at the time of mean new moon, the difference of the mean longitudes of the moon and node exceeds 19° 2', there can not be an eclipse of the sum; but when this difference is less than 13° 44', there must be one. These numbers are called the Solar Ecliptic Limits. 29. As the solar ecliptic limits exceed the lunar, eclipses of the sun must occur more frequently than those of the moon. But as the former are only visible to some parts of that portion of the earth, which has the sun above the horizon during the eclipse (26), and the latter to the whole of that portion which has the moon above the horizon (17), there are, for any given place, more visible eclipses of the moon than of the SUll. TNUMBER OF ECLIPSES IN A YEAR. 30. From the solar and lunar ecliptic limits, and the motions of the sun, moon, and node, it is found that the greatest number of eclipses that can take place in a year is seven; and that the least number is two. When there are seven eclipses in a year, five are of the sun, and two of the moon. When there are only two, they are both of the sun. In every year there are at least two eclipses of the Sun. DIFFERENT IXINDS OF ECLIPSES. 31. When the moon just touches the earth’s shadow, or approaches very near, without entering it, the cir- cumstance is called an .ſlppulse. When a part, but not #26 As TRONOMY. the whole of the moon, enters the earth’s shadow, (lity the phenomenon is called a Partial eclipse of the moon; when the moon enters wholly into the shadow, it is called a Total eclipse; and when the centre of the moon passes through the axis of the shadow, the eclipse is said to be Central. An exactly central eclipse of the moon seldom, iſ ever occurs. With regard to the sun, when the disc of the moon just touches, or approaches very near, to the disc of the sun, the circumstance is called an Appulse. When the moon obscures a part, and only a part, of the Sun, the eclipse is said to be Partial; and when the moon ob- scures the whole of the sun, the eclipse is said to be Total. When the moon’s disc is entirely interposed between the specialor and the sun, but in consequence of the apparent diameter of the moon, being less than that of the sun, the edge of the Sun is seen as a ring surrounding the moon (23), the eclipse is called ſln- mular. Lastly, when the straight line passing through the centres of the sun and moon, passes also through the place of the spectator, the eclipse is said to be Cen- tral. |ECLIPS]S O'I' TIIE MOON, 32. The apparent distance of the centre of the moon from the axis of the carth’s shadow, and the arcs of the moon’s orbil and of the ecliptic passed through by these, during an eclipse of the moon, being necessarily small, may williout material error, be considered as right lines. We may also consider the apparent motion of the sun in longitude and the motions of the moon, in longitude and latitude, as uniform, during the eclipse. These suppositions being made the calculation of the circum- stances of an eclipse of the moon, is very simple. chAPTER XI. 427 33. Let NF, Fig. 27, be a part of the ecliptic, NL a part of the moon’s orbit, C the centre of a section of the earth's shadow at the moon, CD perpendicular to NF, a circle of latitude, and M the centre of the moon at the instant of opposition. Then CM, which is latitude of the moon, in opposition, is the distance of the centres of the shadow and moon at that time. Lct t be some short interval of time cypressed in hours, and parts of an hour, and let C" and M' be the situation of centres of the shadow and moon at the time t before or after opposition. Then C'M' will be the distance of the centres at that time. Draw MF perpendicular, and AMB parallel to NF. Then CC' is the motion of the centre of the shadow in the time t, CF is the moon’s motion in longitude, and HM' its motion in latitude. Now as the longitude of the centre of the earth’s shadow, must always diſſer by 180°, from the longitude of the sun, the apparent motion of the sum is the same as that of the centre of the shadow. Therefore CC' expresses the Sum’s motion in longitude in the time t. And consequently C'F = CF – CC' = the differenee of the moon’s and sun’s motions in longitude, in the time t. 34. Make CG cqual to C'F, and GM" perpendicular to NF, and equal to FM'. Then CM" = C'M' = the distance of the centres of the moon, and earth's shadow at the time t, from opposition. We therefore obtain the distance of the centres of the moon, and shadow, the same, if instead of allowing to cach its proper motion we suppose the centre of the shadow to remain at rest at C, and the moon’s motion in longitude to be equal to the diſſerence of the motions of the moon and sum, in longitude. 35. From Astronomical Tables we can get the hourly motions of the sum and moon, in longitude, and the moon’s hourly motion in latitude. Then supposing the motions uniform, we easily obtain their values for any other short interval of time. Put T = time of opposition, t = time of moon’s centre passing from M to M', m = moon’s hourly motion in longitude, m = moon’s hourly motion in latitude, r = sun's hourly motion in longitude, ^ = moon's latitude at opposition, 428 ASTRONOMYe I = angle M”MR, s = P + p — 3 + ºr (P + p – 3) = semidiam. of earth's shadow (10 and 15). Then CF = m.t, CC' = r.t and RM" = HM’ = n.t, MR = CG = CF = CF – CC' = m.t –r.t= t. (m—r), an i = an MMR = ºr=#5===; As the expression for the tangent of the angle M'MR does not involve t, it is evident the angle itself will continue the same, whatever be the value of t. Hence the point M” moves in the ljne PMQ, which is therefore called the moon's Relative Orbit. 36. In the triangle M"MR, we have, f / MR. _ t.(m —r) MM" = cos MºMR T Tcos I gy The distance MM" is the moon’s motion on the relative orbit, in the time t. If we take t = 1 hour, we have, 77] — ?” cos F." 37. Let AB, Fig. 28, be the ecliptic, C the centre of the earth's shadow at the time of opposition, and CK perpendieular to AB, a circle of latitude. Make CM = x, Mb parallel to AB, and = m. —r, and be parallel to CK, and = n. Through M and c, draw DMcH, which will be the moon’s relative orbit. With the centre C and a radius = S, describe the circle KLPI, which will repre- sent the section of the earth’s shadow at the moon. With the same centre and a radius = S + d, describe arcs cutting DH in D and H; and with a radius = 8 – d, describe other arcs, cutting DH in E and G. From C, draw CF perpendicular to DH. Then supposing the moon to move in the direction DH, it is evident that D is the place of the moon’s centre at the beginning of the eclipse; E, its place at the begimming of the total eclipse; F, its place, when nearest the centre of the shadow; G, its place at the end of the to- tal eclipse; and H, its place at the end of the eclipse. When s — d is less than CF, the eclipse can not be total. 38. Because CD = CH, and CF is perpendicular to DH, we The moon’s hourly motion on relative orbit = CHAPTER XI. 429 have FD = FH. The point F, therefore designates the moon’s place at the middle of the eclipse. In the triangles MFC and Mbc, the angles F and b are right angles; and because bc is parallel to CM, the angle FMC = Mcb. Therefore, MCF = bMc = I (35). MIDDLE OF THE ECLIPSE. 39. In the triangle MCF, MF = CM sin MCF = 2 sin I. But, taking a = interval of time between the middle of the eclipse, and the time of opposition, we have (36), MF = *.(m-r). cos I a. (m—r) ſº 2 sin I cos I **-* = a sin I, or a = *****. os I ??? — T Now if M = the time of the middle, we obtain, y A cos I sim I 771 – ?" *. Hence, - The upper sign must be used when the latitude is increasing; and the lower, when it is decreasing. The nearest distance of the centre is CF = A cos I. Bíº GINNING AND END OF THE ECLIPSE. 40. Let B = the time of beginning, E = the time of the end, and a = the interval between the middle and either of these. Then, *(*-*) – D F = vTCFEFC: cos I = V (s -- d)” – 2° cos” I = M (ST à-xcos I). (sº d'I xcos I) _ cos I v (3 + d – a cos I). (s -- d -- a cos I) tºmºs 777 – ?” Hence, B = M — w, and E = M + æ, become known. OR 3. 48 430 ASTRONOMY. BEGINNING AND END OF THE TOTAL ECLIPSE. 41. Put B’ = the time of the beginning of the total eclipse, E = the time of the end, and w' - the interval between each of these and the middle. Then, cos I v (8– d – A cos I). (s— d -- cos I) 7?? — ?” y B' = M–a', and E' = M + æ'. a' = QUANTITY OF THE ECLIPSE. 42. In an eclipse of the moon, it is usual to suppose that dia- meter of the moon, which, produced if necessary, passes through the centre of the shadow, to be divided into twelve equal parts, called Digits, and to express the quantity of the eclipse by the number of those parts; that is contained within the shadow, at the time when the centres of the moon and shadow are nearest. When the moon is entirely within the shadow as in total eclipses, the quantity of the eclipse is still expressed by the number of digits of the moon’s diameter, which is contained in that part of a radius of the shadow, passing through the moon's centre, which is intercepted between the edge of the shadow and the inner edge of the moon. Thus, the number of digits contained in SN, ex- presses the quantity of the eclipse, represented in the figure. Hence, if Q = the quantity of the eclipse, we have, q__NS.–Cs-CN – CS-(CF–FN) # NV # NV # NW OS + FN—CF – 12 (CS + FN—CF) #, NV NW _ 12 (P + p – 3 + d – a cos I) 2 d (P + p + d – 3–2 cos I). 6. d sº * CONSTRUCTION OF AN ECLIPSE OF THE MOON, 48. The times of the different circumstances of an eclipse of the moon, may easily be determined by a geometrical construction, 4CHAPT EIR IX, § 431 within a minute or two of the truth. To render the construction explicit, suppose the time of opposition to be 8 h. 35 m. 20 sec. on some given day. Then, as 60 minutes : 35 m. 20 sec. : moon’s hourly motion on relative orbit : moon's distance from the point M at 8 o'clock. If this distance be taken in the dividers, and laid on the relative orbit, from M backwards to the point 8, it will give the moon’s place at that hour. Then taking in the di- viders, the moon's hourly motion on the relative orbit, and laying it on the orbit from 8 to 9, 9 to 10, 10 to 11, and backwards, from 8 to 7, and 7 to 6, we have the places of the moon’s centre at those hours respectively. By dividing the hour spaces into quarters, and subdividing these into 5 minute spaces or minute spaces, we easily perceive the times at which the moon’s centre is at the points D, E, F, G, and H. ECLIPSES OF THE SUN. 44. It has been shown in a preceding article (27), that when the angular distance of the centres of the sun and moon is equal to P — p + 3 + d, the edge of the moon just touches the luminous frustum of a cone, contained between the sun and earth. When the dis- tance of the centres becomes less than P — p + 3 + d, it is evident the moon must obscure a portion of the sun’s light to some part of the earth's surface. The instants at which the moon commences and ceases to prevent any part of the sun’s light from arriving at the earth, are called the Beginning and End of the General Eclipse of the sun. These times may be obtained either by calculation or construction, nearly in the same manner as the beginning or end of an eclipse of the II100Ile 45. Although the calculation of the general eclipse of the sun is equally simple with that of a lunar eclipse, it is very different when the object is to determiae the circumstances of the eclipse for any particular place. 4.32 ASTRONOMY. Then, it is necessary to take into view the situation of the place on the illuminated surface of the earth, or, which amounts to the same, to consider the effects of parallax. This circumstance renders the calculation tedious, at least, when it is desired to give to the re- sults, all the accuracy of which the problem is sus- ceptible. 46. We shall first give a method of obtaining results nearly true, by means of a geometrical construction. When the construction is carefully performed, on a large scale, the error in the time of be- gimming or end will not exceed one or two minutes. This method will therefore suffice, except considerable accuracy is required. We shall afterwards give a method of obtaining by calculation, from the results of the construction, others that will be more ac- curate. The earth will still be considered as a sphère, and accordingly the reduced latitude of the place, and the reduced parallax must be used (10.51). 47. A section of the earth made by a plane passing through its centre, perpendicular to the line joining the centres of the earth and sun, is called the Circle of Illumination. It forms nearly the boundary between the enlightened and dark parts of the earth's surface. As the sum is larger than the earth, it evidently enlightens a small portion more than one half of the earth. The enlightened part is still further increased by the cffect of the earth's atmos- phere in refractiong the rays of light. 48. A plane between the earth and sun, perpendicular to the straightline joining their centres, and at a distance from the earth's centre, equal to the distance of the centres of the earth and moon, is called the Plane of Projection. 49. If from the sun's centre to every point in the circumference of the circle of illumination, straight lines be conceived to be drawn, they will form the surface of a right cone, a section of which, by the plane of projection is a circle, and is called the Circle of Projection. 50. A plane passing through the centre of the sun and the poles | CHA PTER XI, 433 of the earth, and which consequently passes through the earth's centre, is called the Universal JMeridian, When by the diurnal rotation of the earth on its axis, any place on its surface is brought to coincide with this plane, the sum must be on the meridian of that place. 51. Let S, Fig. 29, be the centre of the sun, E, the centre of the earth, TSU the plane of the universal meridian, and let AUWT and PRQV, each conceived to be perpendicular to the plane of the paper, be, respectively, the circles of illumination and projection. If D be a place on the earth's surface, a spectator at B, will see the sun's centre in the direction of the line DS, which intersects the circle of projection in L. The point L is called the Projection of the Sun's Centre, for the spectator at D. 52. Let DF and LM be each perpendicular to TUS, the plane of the universal meridian, and FG and MC each perpendicular to ES, the line joining the centres of the earth and sun. Then DF and LM are the distances of the points D and Lfrom the universal me- ridian; and FG and MC are the distances of the same points from a plane perpendicular to the universal meridian, and passing through the centres of the sun and earth. 53. From the triangle EQS, we have, CES = EQU – ESU or flpparent semidiameter of the circle of Projection = P –p. 54. By similar triangles, EU : CQ :: ES: CS :: EN: CM. But CQ = EC tan CEQ = EC tan (P–p); and we may with- out sensible error consider EN = FG. Hence, EU: EC tan (P-p): ; FG: CM = ** **Cº-P). .# EU CM FG Therefore, tan CEM = ** = * *. amºmºmº erefore, tan CEM EC EU tan (P–p), FG CEM – - e. P– tº Or, M EU ( p) FD In like manner, MEL = *. (P–0). m like manner, EU (P–p) If X = MEL = the apparent distance of the projection of the Sum's centre from the universal meridian, Y = CEM = the ap- parent distance of the projection of the centre, from the plane 134 ASTRONOMY. passing through the centres of the earth and sun, perpendicular to the universal meridian, and R = EU = radius of the earth, we have, D F t X = º (P— p), and Y = º (P— p). 55. Let AUBT, Fig. 30, be a section of the earth by the plane of the universal meridian, PP' the earth's axis, P the north pole, P" the south pole, TU a diameter of the circle of illumination, CQ a diameter of the equator, ES the line joining the centres of the earth and sum, and D a given place on the earth. Also let DLH be a plane perpendicular to PP the earth's axis, DF a straight line perpendicular to LH, and consequently to the universal me- ridian, LM and FR, each parallel to AB, and LN and FG, each parallel to TU. Then, as in a preceding article (52), DF is the distance of the place from the universal meridian, and FG is its distance from the plane passing through the centres of the earth. and sun, perpendicular to the universal meridian. Also PH = PD = the complement of the latitude of the place, DLH = DPH = the hour angle from noon, and TP = AC = the Sum's decli- nation. Put, D = TP = the sun’s declination, U = DLH = the hour angle from noon, H = CH = the reduced latitude of the place. Then HL = EH sin HEP = R cos H, and DF = DL sin DLH = HL sin U = R cos H sin U. Hence (54), X = º P—p) = cos Hsin U. (P–p) A. Also, EL = EH cos HEP = R sin H, EM = EL cos TEP = R sin H cos D = R. sin (H + D) ; sin (H–D) (App. 16) ML = EL sin TEP = R sin Hsin D, LF = DL cos DLH = HL cos U = R cos H cos U. But from the similar triangles FNL and EML, we have, EL : LM :: FL ; LN, * CHAPTER IX, 135 or R sin H : R sin H sin D : : R cos H cos U : MR 1 : sin D : : R cos H cos U : MR. Hence, MR = R cos U sin D cos H = R cos U. sin (H + DH sin (H–D) (App. 17). When the latitude of the place and declimation of the sun are both north, as represented in the figure, ER = EM-MR. It will be the same, when the latitude and declimation are both south. But when one is north and the other south, ER = EM + MR. Therefore, w FG - ER– EMEMR – R.” (H+ P)+inº-P N º 2 + R cos U. sin (H + D)—sim (H–D) 2 Consequently (54), Y = sin (H + D) # sin (H–D). (P—p) + sin (H + D) Fº (H-D) cos U. (P— p). B PROJECTION of THE SUN’s CENTRE ON THE CIRCLE of PROJECTION. 56. Let AB, Fig. 31, be the line in which the plane, through the centres of the sun and earth, and perpendicular to the univer- sal meridian, intersects the circle of projection, and CD the inter- section of the universal meridian with the same circle. With the centre C, and a radius equal to P – p, describe the semicircle ADB, to represent the northern half of the circle of projection. With a sector,” make AE and BF, each equal to H, the reduced * The Sector is an instrument, generally made of ivory or box wood, about a foot in length, with a joint in the middle. There are several lines on each side of it. But the only one, we shall notice, is the line of chords, which is used to lay off a given number of degrees on the arc of a circle. This line is marked with the letter C. It consists of two lines, running each way from the centre of the joint to near the ends of the instrument; each line being divided into 60 parts or degrees, and each degree subdivided into halves. At the 60 on each line, there is a brass pin with a small puncture, 436 ASTRONOMYA - latitude of the place, and make EG, EI, FH, and FK, each equal to D, the sun's declination. Join GH, EF, and IK, and bisect Rw in N. Through N, draw LNM parallel to EF. Make NO equal to whº, and with the centre N, and radii NO and NR, describe the semicircles PYO and tRQ. With a sector, make YW equal to U, the hour angle from noon. Join WN, draw WV parallel to CD, and through T, draw STU parallel to PO, meet- ing WW in U. In this construction, the sun's declination is supposed to be north, and the time in the afternoon. When the declimation is south, the semicircle tRQ must be on the upper side of PO, and WN must be produced to meet it in T. When the time is before moon, the arc YW must be laid off from Y to the right hand. Now, NO = whº = whº = CF sin FCw = cos H. (P–p), SU = NW = NW cos WNL = NO sin U = cos H sin U.(P– p). Hence, (55. A), SU = X. Also, Cv = CH cos v CH = sin HCB. (P–p) =sin(H + D). (P — p), CR = CK cos RCK = sin KCB. (P–p) = sin (H–D). (P — p), CN=9. º CR. _sim (H + D) ; sin (H– D). (P--p), Co-CR _ sin (H + D)--sin (H D). (P—p), 2 2 RN = To lay off any number of degrees on the arc of a given circle, take the radius of the circle in the dividers, and setting one foot in the puncture at the end of one line, open the sector till the other foot of the dividers just reaches to the puncture at the end of the other line. Then setting one foot of the dividers to the given number of degrees on one line, open them till the other foot reaches to the same number of degrees on the other line. This distance, applied as a chord to the arc, will intercept the given number of degrees. To measure a given arc, open the sector as before. Then, taking the chord of the arc in the dividers, apply them to the line of chords, moving them without changing their opening, till each foot is at the same number of degrees on each line. This number of degrees will be the measure of the given arc. C] IAPTER XI. 437 NS = NT cos YNW = NR, cos U =" (H + D) º sin (H–D). cos U (P-p), CS = CN +: NS = sin (H + D) s sin (H–D). (P-p) +: sin (H + D)—sin (H–D) cos U. (P–p). 2 Hence (55. B), CS = Y. Since US, the distance of the point U, from the universal me- ridian, is equal to X, and CS, its distance from the plane, per- pendicular to the universal meridian, and passing through the centres of the carth and Sun, is equal to Y, the point U is the projection of the sun's centre, ſor the spectator at the given time, and place (54). 57. It is evident from the construction, that the semicircles PYO and tRQ, depend only on P — p, the difference of the ho- rizontal parallaxes of the moon and sun, D the sum’s declination, and H the latitude of the place. Therefore, since P – p and D may be considered as constant during the continuance of the eclipse, the same semicircles will answer for finding the point U at any other time, within that interval. 58. When it is required to find the projection of the sun's centre for several different times, on the same figure, it is better to omit drawing the lines WN and UTS, and instead, to lay the edge of a ruler from W to N, make a mark at T on the semicircle tRQ, and draw Ts parallel to CD. The distance Ts, applied on WW, from V to U, will give the point U the same as before. 59. From the similar triangles WNW and TNS, we have WN : TN : : WW : NS. But WN = NO, TN = NR, and NS = UV. Therefore, NO : NR :: WV : UV. Hence (Comic Sections), the point U is in an ellipse, of which P0 is the trans- verse axis, and Rw the conjugate. On account of the earth's diurnal motion, the position of U is continually changing. But it is evident, that as a long as P — p and D may be considered constant, its Path will be the ellipse described about the ºwes PO and Rw. 49 #38 ASTRONOMY. ! PoSITION OF THE Moon’s RELATIVE ORBIT ON THE CIRCLE OF PROJ ECTION. 60. Make Da and D5 each equal 23° 28, the obliquity of the ecliptic, join ab and on it describe the semicircle adb. From 6 lay off the arc bá equal to the sun's longitude. When the longi- tude exceeds 6 signs, its excess above 6 signs must be laid off from a to d. Draw Dm perpendicular to ab, and through m, draw Cm2. Since the universal meridian passes through the poles of the earth (50), it coincides with a circle of declination. Hence CD, its intersection with the circle of projection, may be considered as the arc of a declimation circle. In like manner, the line in which a circle of latitude passing through the centre of the sum, intersects the circle of projection, may be considered as the arc of a circle of latitude. The angle contained between these lines will be the angle of position for the sun (6.21). Put, L = ba = sum’s longitude, w = Db = obliquity of the ecliptic, S = the angle of position. Then br. = Cb sin b CD = Cb sin w, mn = nd cos brid = bncos L = Cb sin a cos L, Cn = Cb cos b CD = Cb cos o, tan DCZ = * = Cb sin a cos L T C. T Cb cos a But (6.21.C), tan S = tan a cos L. Hence, tam DCZ = tan S, or DCZ = S. Therefore CZ makes with CD, the angle DCZ equal to S, the angle of position. It is also evident, that by the construction, CZ will fall to the right hand, or west of CD, when the longitude is less than 90° or more than 270°, and to the left hand or east, when the longitude is more than 90° and less than 270°. Hence CZ is a circle of latitude, the plane of which passes through the sun’s centre. 61. Having CZ the circle of latitude, the moon’s relative orbit may be drawn, and the places of the moon's centre at different = tan a cos L. *. CHAPTER x1. 139 hours, be determined in the same manner as in an eclipse of the moon (37 and 43), only using the time of conjunction instead of the time of opposition. Let pil be the moon’s relative orbit; and let w be the place of the moon’s centre at the same time the projection of the sun's centre is at U. If the distance Uu is less than the apparent semi-diameters of the sum and moon, a part of the sun, at least, is then eclipsed. 62. Considering the earth a sphere, a vertical line (4.4) and eonsequently a verticle circle, at a given place, will pass through the centre. Hence, since C is in the straight line joining the centres of the sum and earth, CU is the intersection of the circle of projection with a verticle circle passing through the centre of the sun. The position of u, the moon’s centre, with regard to this circle, is therefore determined, ra CONSTRUCTION OF AN ECLIPSE OF THE SUN. t 63. The construction, Fig. 32, is for an eclipse of the sun on the 27th of August, 1821, in the morning, and it is adapted to the meridian and latitude of Philadelphia. The points 7, 8, 9, 10, and 11 on the line pſ, represent the situation of the moon’s cem- tre on the plane of projection, at those hours respectively. The other points 7, 8, 9, 10 and 11, represent the projections of the sum’s centre at the same times. The lines 7c, 8e, 9g, 10h, and 11k are drawn parallel to AB, meeting the lines VIIc, VIIIs, IXv, Xw, and XIa, in the points c, e, g, h, and k. Draw a right line MN, Fig. 33, and in it take any point S, to represent a fixed position of the sum’s centre. Then taking 7c, the distance from the moon’s centre at 7 o'clock to the line VIIc, lay it from S to the right hand to c. In like manner, make the distances Se, Sg, Sh, and Sk, in the line MN, respectively equal to the distances 8e, 9g, 10h, and 1 1k, observing that each distance is to belaid offto the right or left of the point S, according as the moon’s centre is to the right or left of the projection of the sun’s centre. Draw the lines ci, e8, g3, h!0 and k l 1, perpendicular to MN. Make ci equal to the distance ci on the line passing through the projection of the sum’s centre at 7 o'clock. Also make e8, g3, h10, and k l 1, respectively equal to the corresponding distances 140 ASTRONOMY. in Fig. 32, observing that each must be placed above or below the line MN, according as the moon’s centre is higher or lower than the projection of the sun’s centre. Then the points 7, 8, 9, 10, and 11, Fig. 33, will represent the positions of the moon’s centre, at those hours, with regard to S, the Sun’s centre. Join the points 7,8, 8,9, 9,10, and 10,11, and the broken line thus formed, which will deviate but little from one right line, will repre- sent very nearly the apparent relative orbit of the moon. With the centre S, and a radius equal to the sum of the apparent semi- diameters of the sum and moon, describe arcs cutting the moon's path in b and r, which will be the positions of the moon’s centre at the beginning and end of the eclipse. From S, draw Sq per- pendicular to the part 8,9, of the moon’s path; them q will be the position of the moon’s centre, when it is nearest to the centre of the sun, and consequently when the eclipse is greatest. If the hour spaces 7,8, 8,9, and 10,11, be each divided into quarters and these subdivided into three equal parts, or spaces of 5 mi- mutes, the times of beginning, greatest obscuration, and end, can be easily estimated. 64. The hour spaces on the moon’s apparent path are not equal, and therefore the times obtained from equal divisions of them, are not quite accurate. The error from this cause will not, how- ever, exceed one or two minutes; and if the construction be made for each half hour, it will be much less. - With the centre S, and a radius equal to the sun's apparent semi-diameter describe a circle to represent the sun’s disc; and with the centre q, and a radius equal to the moon’s apparent semi- diameter describe another circle to represent the moon’s disc. The part of the sun's disc, that is intercepted by the moon's, shows the part eclipsed. If Sq be produced to m and n, and mn be measured by the scale, used in the construction, the quantity of the eclipse will evidently be obtained by this proportion. ſts the Sun's apparent diameter: mn :: 12 digits : the digits eclipsed. Let u, Fig. 32, be the projection of the sun's centre at the time the eclipse commences. Then Cu will be a verticle circle passing through the sun's centre at that time (62). Draw SF, Fig. 38, making the angle FSM equal to the angle wob; then v is the CHAPTER XI, i44 sun's vertex. Now as the eclipse commences at the point a of the sun's disc, the angle vSa expresses the angular distance from the sum’s vertex, of the point at which the eclipse commences. The knowledge of this angle is important to the astronomer, who wishes to observe with accuracy the commencement of an eclipse of the sun. Without it he would not know at what part of the edge to fix his attention, while waiting to see the first impression. CALCULATION OF AN ECLIPSE OF THE SUN. 67, Let B designate the approximate time of the beginning of the eclipse, found by construction (63), and t some short interval of 4 or 5 minutes. Calculate for the time B —t, by means of as- tronomical tables, the Sun's longitude, hourly motion, and semi- diameter; also the moon’s longitude, latitude, horizontal parallax, semidiameter, and hourly motions in longitude and latitude. Then as our object is to obtain the difference of the apparent longitudes of the sun and moon, and the moon's apparent latitude, in order to obtain their apparent distance, subtract the sum’s horizontal paral- lax, from the reduced horizontal parallax of the moon, and con- sidering the remainder as the moon’s parallax, calculate the paral- lax in longitude and latitude (10.54 and 56), using the reduced latitude of the place*. To the moon’s true longitude and latitude, apply, respectively, the parallaxes in longitude and latitude ac- cording to their signs, and the apparent longitude and latitude are obtained. Take the difference between the true longitude of the sum and the apparent longitude of the moon, which will be the moon’s apparent distance from the sun, in longitude. 68. With the sun and moon's longitudes, the moon’s latitude, and their hourly motions at the time B —t, find the longitudes and, the moon’s latitude at the time B -- t. For this latter time, cal- culate the parallaxes in longitude and latitude, and thence deduce * This is equivalent to considering the apparent place of the sun, the same as the true, and referring the whole effect of parallax to the moon. It is not rigidly exact. For we virtually calculate the sun's parallax in longitude and latitude by making use of the moon’s longitude and latitude, instead of the sun's. But on account of the small quantity of the sun's parallax, and the little difference in the longitudes at the time of an eclipse, the error is quite insensible. 442 ASTRONOMY the apparent distance of the moon from the sun in longitude, and the moons's apparent latitude. 69. Let EC, Fig. 34, be a part of the ecliptic and S the place of the sum’s centre, which we shall consider fixed. Let SA be the apparent distance of the moon from the sum in longitude, and AD, perpendicular to EC, the apparent latitude, at the time B—t. Then D will be the moon’s apparent place. In like manner let L be the moon’s apparent place at the time B + t. The line DL, which does not sensibly differ from a straight line, represents the part of the moon’s apparent relative orbit, passed through during 2t, the interval between the times B — t and B -- t. 70. Let the point G in the line DL, be the place of the moon’s centre at the true time of beginning. Put, a = SA = appar. dist. of moon from sun in long. at the time B — t, c = AD = moon’s app. latitude at the time B —t, m = AH = diff of moon’s app. distances from the Sun in long. at B — t and B + t, * m = wi. = diff of moon’s apparent latitudes at B – t and B + t, s = SG = sum of app. semi-diameter of sun and moon, I = angle wiſ)L = inclimation of moon’s app. rel. orbit, a = AF, wl, n Then tan i = tan wbL = t = *, Dw T ºn Gv = Dv tan vLG = a tan I, SF = a – a., and FG = AD + Gv = c + a tam I, SF2 + FG% = SG”, or (a — «)* + (c + a tam I) *=s* a” — 2aa -- a.” + c2 + 2ca; tan I + æº tam” I = 8* (B) (1 + tan” I).a.” — 2 (a — c tam I) w = s” — a” – c’, 2. (a —ctan I), s”— a” – c’ —TTLET-" = -Tuniº 2. (a — c tan I) L (a – c tan I)” s”— a”—cº TTTT tanºi (TItanº I) 3 T TItanº IT (a – c tan I) * (1 + tan” I) * a;2 3.2 + CHAPTER x1. 443 X- (s”— a”— cº. (1 + tan” I) + (a – c tan I); (1 + tan” I)* = * * s” tan” I—(c’ + 2 a c tan I + a” tan” I) (1 + tan” I,” _ s”. (1 + tan” I)—(c + a tan I)* -- (1 + tan” I)* s? (-º) s”. (1 + tan”I) - TTºI.' = s”. cos” I (–6 rºpe) * & © S” a — citan I | ( erºtype) — “T * *** = + 8, cos I --> *TTT tani v/ I S” _ (c + d tan I) ºr) 2 a = (a-ctan I) cos” I-E s. cos I v(1 S Make sin 9 = (c -- a tan I).cos I S (c -- a tan I)”. cos” I s? & = v (l — sin” •)=cos. Then, sin”6 = And v (1–6– ºper) S2 Hence, a = (d−ctan I) cos” I: S. cos I cos 0. If the line DL were produced, it is evident there would be another point in it, such that its distance from S, would be the same with that of the point G. The two values of a in the for- mula, correspond to these two points. Hence the value of a cor- responding to G, is, a = (a-c tan I) cos” I — s. cos I cos 0. When the moon’s apparent latitude at the time B + t is less than at the time B–t, we must make, (c.— a tan I).cos I S w = (a + c tan I).cos” I — S. cos I cos 0. sin 6 = , and we shall have Now, as m :a:::2t : * = interval from B — t to the true 77? time of the beginning. Hence if B' = the true time of beginning, we have, £44 ASTRONOMY. tan I = *, sin = (c 4: a tan I) cos I, Tºº, S B' = B — t + (a+ctan I).cos” I–s cosicos ). 2. 7?? The upper signs must be used when the apparent latitude is increasing, and the under, when it is decreasing. The same is to be observed in the three following articles. 71. The end of the eclipse may be found in a similar manner. Let E be the approximate time of the end, and make the same calculations for E — t and E + t, as for B —t and B -- t. Then, if a, c, m, n, s, I and 9, designate the same quantities as in the last article, but having the values appertaining to the end of the eclipse, and E = the true time of end, we have, tan I = e tan I) cos *, sin – (**** ) co I. ??? S E' = E — t + (i. cos I cos 9 -- (a + c tan I) cos” 1). 2 : 72. To find the true time of greatest obscuration, let G be the approximate time, and calculate as before, for the times G —t and G + t, the apparent distances of the moon from the sun, in longitude, and the moon’s apparent latitudes. Let SK be the ap- N. parent distance of the moon from the sun in longitude at the time G—t, SN, at the time G + t, and KI, NP perpendiculars to EC, the corresponding latitudes. Then if Sq be drawn perpendicular to IP, q will be the place of the moon, when the apparent distance of the centres is least. Let w = KW, and a, c, m, n, s and I, de- signate the same quantities as before, but having the values apper- taining to the middle. From the similar triangles PIQ and SqW, we have, IQ: PQ: :Wq: SV. But IQ= m, PQ = n, W4- c + c tan I, and SV = a –a, Hence, m : n : ; c + a tan I : : a -a, —“Et—— “e tan I, c + a tan I m _ a - c tan I TTTTºi- (a – c tan I) cos” I CHAPTER XI. 445 When the apparent latitude is increasing, / a = (a + c tan I) cos” I. Hence, if G' = the true time of greatest obscuration, we have, tan I = *, G = G–t + (a + c tan I).cos” I. 2t. 7)? 777, 73. To find Sq, the nearest distance of the centres, we have (72), WQ = c + a tam I = c + tam I. (a + c tan I) cos” I = c + sin I cos I. (a + c tan I). Hence, Sq = Vq = -º- + sin I. (a + c tan I) cos I cos I Now to find the quantity of the eclipse, if 3– the sun's appa- rent semidiameter, and d = the moon’s apparent semidiameter at the time of the greatest obscuration, we have, Fig. 33. mn = Sq + qn + Sm =Sq + 3–Sq + d – Sq = 3 + d — Sq = s—Sq=s–––. F sin I. (a + c tan I). Hence, cos I C (s — `i +sin I. (a+ c tan D) Digils Edipºd – º – “..A. “Tº 74. When Sq, the nearest apparent distance of the centres, is less than the difference, between 3 and d, the eclipse will be either annular or total. It will be annular if 3 be greater than d, and total if d be greater than 3. The time when an eclipse commences or ceases to be total or annular may be found by the same for- mulae as the beginning or end of the eclipse, only making s = to the difference of 3 and d, and giving to a, c, m, &c. the values which they have in finding the time of greatest obscuration. 75. The greatest duration of an annular eclipse at any place is about 12 minutes, and of a total eclipse about 8 minutes. In a total eclipse the obscurity is such as to render the principal stars and most conspicuous planets distinctly visible. 76. Let Sb, Fig. 34, be a circle of latitude, Sd a declimation circle, and Sh a vertical circle, all passing through the sun's centre. Then b$d will be the angle of position, and dSh the angle con- 20 ,” 446 ASTRONOMY. \ tained by the declimation circle and vertical circle, passing through the sum. Put L = the sun’s longitude at the beginning of the eclipse, A = the sun's distance from the north pole of the equator, a = SF = the apparent difference of the sun and moon's longi- tudes, c = the moon's apparent latitude, H = the latitude of the place, a = the obliquity of the ecliptic, and U = the hour angle from moon. Then (6.20 and 21) cos A = sin L sin a, and tan b$d = cos L tan a (C) In the triangle PSZ, Fig. 18, we have (App. 37), l , cot Pz sin PS – cos ZPS cos PS t S = €0 Sin ZPS _ tan H sin A — cos U cos A esmºs sim U -- e. Hence, Fig. 34, tan dSh = tan Hsin A–. cos U cos A (D) sin U FG, C lso tan FSG = + · = −. sº - E Also tan FS d (E) The angle bSd will be to the leſt of Sb, when the sun's longi- tude is less than 90°, or more than 270°, and to the right when it is between 90° and 270°. The angle d Sh will be to the right of Sd, in the forenoon, and to the left in the afternoon. By attending to these circumstances, and adding or subtracting accordingly, the an- gle bSh, and consequently ESh, becomes known. Thence, by ap- plying the angle FSG, we have the angle hSG, contained between the vertical circle passing through the sun’s centre, and the line joining the centres of the Sun and moon, at the beginning of the eclipse. 77. Supposing the latitude of the place for which the calcula- tion is made, to be truly known, and also its longitude from the - place for which the solar and lunar tables are computed, the results obtained from the preceding formulae, when the calculations are carefully performed, will have an accuracy corresponding with that of the tables themselves, or very nearly so. But the best of these tables are liable to errors of a few seconds. Consequently the times obtained, will be liable to small errors, depending on the former. CIIAPTER XI. - 447 "i3. Except in cases, when the greatest precision is required in the results, it will not be necessary to calculate the longitudes of sun and moon, and the moon’s latitude, from the tables, only for the time G — t. The longitudes and latitude at the times B —t and E—t, may be found from the former by means of their hourly motions. It will also be sufficient to calculate the moon’s appa rent longitudes and latitudes for the times B — t, G – t and E—t. Then for the beginning of the eclipse we may take m = moon’s apparent relative motion in longitude during the time (G — t) — (B — t), or which is the same G — B, and n = moon’s apparent motion in latitude during the same time. For the - end wé may give to m and n, the values of the same quantities, for the time E — G. And for the greatest obscuration we may give to them the values of those quantities for the time E — B. With these values of m and n, we obtain very nearly the inclinations of the apparent, relative orbit at the times of beginning, end, and greatest obscuration. As the value of a, in the equation B (70), must be small, its square may be omitted with but little error. We shall then have, a” — 2aw -- cº -- 2ca; tan I = 8*, 2. (a — c tan I).a. = a” + c2 — s”, r a” + c2 — s” 2. (a – c tan I)” (a” + c2 — s”).(G–B) + 2m (a + c tan I) E’ = E — t + ["s” — (a’ + cºi. (E – G) 2m (a + c tan I) + [(a + c tan I) cos” I] . E-B) 770, * Hence B' = B — t 'G' = G — t In each formulae the upper sign is to be used when the apparent latitude is increasing, and the lower when it is decreasing. 79. Instead of finding the approximate times of beginning and end by construction, we may, though with more labour, perform the whole by calculation. Thus, let the sun's longitude and the moon’s apparent longitude and latitude be calculated for the time of new moon. From these longitudes we know, whether the ſp- 448 ASTRONOMi Y. parent Ecliptic Conjunclion, that is, the circumstance of the appa- rent longitude of the moon being the same with the Sun's longitude, has place before or after new moon. For a time an hour or two, earlier or later, than the new moon, according as the apparent con- junction is before or after, again calculate the sun's longitude and moon's apparent longitude and latitude. Then considering the apparent relative orbit of the moon as a straight line passing through the apparent positions of the moon with rogard to the Sun, at those times, we can obtain the approximate times of beginning, greatest obscuration and end, nearly in the same manner as the beginning, middle, and cnd of a lunar eclipse. IPATH OF A C EN 'I'lk AL FC LIPSE OF THE SUN. 80. The latitude and longitude of the place, to which the sun is centrally colipsed, at a given time during the continuance of the central eclipse, may be casily determined with considerable accuracy, by means of a geometrical construction. If we suppose the circle ATBU, Fig. 30, to be described with a radius equal to P-p, it is evident (54 and 55), that FD = X, and ER = FG = Y. Then in the right angled triangle EFD, we have EF = y (ED’ – FD*) = x/ [(P — p) * — X*]. Now if X and Y are known for any given time, EF and ER are known, and consequently the position of the point F. The po- sition of PE is determined by the sun's declination, and therefore that of the line HFL, which passes through F, perpendicular to PE. In the right angled triangle ELH, EL and EH being known, the angle PEH, the complement of the latitude of the place, is also known. In the right angled triangle, DLF, FL and FD being known, the angle DLH, which is the hour angle from moon, is known. 81. Let AUBT, Fig. 35, be the circle of projection, described with the radius P — p, TU the intersection of the universal me. ridian with this circle, AEB perpendicular to TU, and pſ, the moon’s relative orbit (60). If D be the place of the moon’s cen- tre at a given time, on a given meridian, it is manifest the place which would have the projection of the sum’s centre also at D, would then have a central eclipse. Make TP = the Sun's de- CIIA PTER XI. 449 climation, laying it to the left of T, when the declimation is north, but to the righl, when the declination is south, and draw PEP'. Through D, draw MDG parallel to TU, and DF parallel to AB. Then KM = y (EM” ––EK*) = x/ [P–p)*—w"). With the centre E, and a radius equal to KM, describe an arc, cutting DF, in F, and through F, draw I.F.H., perpendicular to PP. Then PEH is the complement of the latitude of the placc (80.). Make KG = FL, and through G, draw EGI. Then because NG = EK =X, and EN = GK = FL, UEI is the hour angle, from the universal meridian at the required place (80). The place will be to the west or east, of the universal meridian, according as the point D is to the right or left of TU. Now for the place, for which the construction is made, the time that the moon’s cem- tre is at D, and consequently the distance of the place from the universal meridian, is known. Hence for the same instant, wc know the distance, in time, of the given place, and required place, from the universal meridian; and by taking the difference, or sum of these distances, according as they arc on the same, or diſ- ferent sides of the universal meridian, the longitude in time, of the required place, from the given place, becomes known. By making the construction for every 15 or 20 minutes, during the continuance of the central eclipse, we shall have the latitudes and longitudes of a series of places, at which the colipse will be central. A curve line, drawn on a map or globe, through those places, will represent, what is called the Path of the Central JEclipse. 82. By a process but little different from the preceding, the longitudes and latitudes of those places that will have the eclipse of a given magnitude, for instance 6 or 9 digits, may be obtained. The places at which an eclipse will be central or of a given magnitude, may be determined more accurately by calculation. The methods of making these calculations, are given in our larger treatises on astronomy. OCCULTATIONS. 83. If at the time of mean conjunction of the moon and a star, that is, when the moon’s mean longitude is #50 ASTRONOMY, the same with the longitude of the star, their difference of latitude exceed 1°37' there can not be an occultation; but iſ the difference be less than 51", there must be an occultation somewhere on the earth. Between these limits there is a doubt, which can only be removed by the calculation of the moon’s true place. 84. The construction for an occultation, is nearly the same as for an eclipse of the sun. There is however a difference in some parts. The radius CB, Fig. 31, must be equal P, for a star, and P−p for a planet, p being the horizontal parallax of the planet. The arcs FH, FK, EG and EI must be equal to the de- climation, and bal to the longitude, of the star or planet. In de- termining the projected place of the star, for a given time, we must use the hour angle corresponding to the difference between the given time, and the time the star is on the meridian. Tô get the position of the circle of latitude, lay off the declina- tion of the star or planet, from D to b', and draw b'c parallel to ab. With the centre C and radius Cc, describe the arc cm', cut- ting dim, produced if necessary, in m'. Join Cm' which will be the circle of latitude, as is easily deduced from the expression for the angle of position (6.21. B.) The distance Cr' on the circle of latitude, must be equal to the difference or sum, of the latitude of the moon, and that of the star or planet, according as they are of the same, or of different names. It must be placed above C, when the moon is to the north of the other body, but below, when it is to the south. For a star, the moon’s motion in longitude, will be its rela- live motion in longitude. For a planet the moon’s relative mo- tion in longitude is obtained by subtracting the motion of the planet when direct, and adding it when retrograde.” In Fig. 33, it is evident, that for a star we must take Sb and Sr, each equal to the apparent semidiameter of the moon; and for a planet, equal to the sum of the apparent semidiameters of the moon and planet. * The apparent motion of a planet is sometimes retrograde. This cir- cumstance will be more particularly noticed in the next chapter. CHAPTER XI. 454. 85. When considerable accuracy is required, the moon's rela- tive motion on the ecliptic must be reduced to its motion on a pa- rallel to the ecliptic, passing through the star or planet. This is done by multiplying the relative motion in longitude by the cosine of the latitude of the star or planet. For if AB, Fig. 36, be an arc of the ecliptic, and DE the corresponding arc of a circle, parallel to it, we have, AB. EF BC : EF: ; AB : DE = *** ***. A E. BC But, EF = Ca = BC cos BCE = BC cos BE. Hence, DE = * * * = AB cos BE. 86. The difference between the calculation of an occultation and that of an eclipse of the sun, is easily deduced from what has been said in the two preceding articles. º, 87. Observations of an eclipse of the sun or of an occultation of a star, made at places, whose longitudes and latitudes are correctly known, furnish means of de- termining the errors of the tables at the time; and they are frequently used for that purpose, particularly those of an occultation. The positions of many of the stars are determined with great precision, and the moon’s parallax and apparent diameter are very accurately known. But the moon’s longitude and latitude, com- puted from the best lunar tables are liable to errors of several seconds. Hence if the observed time of be- ginning or end of an occultation, of a star whose posi- tion is well determined, does not agree with the time, obtained by calculation, the difference must depend on errors in the computed longitude and latitude of the Iſºl 0011. 88. It is evident from the formulae for computing the parallaxes in longitude and latitude, that those parallaxes are not sensibly ef- 452 ASTRONOMY. fected by small errors in the longitude and latitude. The errors in the apparent longitude and latitude, may therefore be considered the same as those in the true. 89. Let a and c be the apparent distances of the moon from the star, in longitude and latitude, respectively, as obtained by calcu- lation for the observed time of beginning, and l be the latitude of the star. Also let a = the error in the moon’s longitude, and y = the error in the latitude. Then a + æ and c + y will be the true, apparent distances of the moon from the star in longitude and latitude, at the observed time of beginning.” Consequently, if s = the moon’s apparent semidiameter, we have, (a + a *. cos’ l + (c -i- y) * = 8*, or, a” cosºl + 2 aa cos’ l + a”. cos” l + c + 2 cy +y” = 8*. Now as a and y are small quantities, the terms involving their squares, may be omitted. Hence, if # (s” — a” cos’ l– c’) = e, we have, a cos’ l. a. -- Cy = e. As the errors a, and y will not sensibly change during the con- tinuance of an occultation, another similar equation may be ob- tained from an observation of the end, and a calculation for that time. Thus, a'. Cos” l. ~ + c'y = €”. From these two equations the values of a and y are easily ſound. 90. The errors of the lunar tables, in longitude and latitude, may also be determined, by observing the moon’s right ascension and declination, either at the place for which the tables are constructed, or at any other whose longitude and latitude are accurately known. From the observed right ascension and de- * The signs of a and y, are both put affirmative. If either or both of them ought to be negative, it will be determined by the calculation. CHAPTER XI, 453 clination, the moon’s longitude and latitude may be calculated (6.49), and thence the errors ascertained. 91. In calculating the moon’s longitude and latitude for any instant of time, as reckoned at a given place on a meridian, different from that for which the tables are constructed, we must reduce the given time, to the time that is reckoned at the same instant at the latter place. This reduction depends on the difference of longitude of the two places, which for astronomical purposes is generally expressed in time; one mean so- lar hour corresponding to 15°. An error in the differ- ence of longitudes, will produce errors in the computed longitude and latitude of the moon. 92. An observation of an occultation, at a place whose longitude is not correctly known, furnishes one of the most accurate means of determining it. The ac- curacy will be increased, if on the day of occultation, the errors in the tables, have been ascertained by ob- servations at a known meridian. 93. Supposing the tables accurate, or that the errors have been ascertained and allowed, the difference be- tween the observed, and calculated time of the begin- ning or end of an occultation, or of an eclipse of the sun, at any place whose latitude is accurately known, must depend on errors in the computed longitude and latitude of the moon, produced by an error in the lon- gitude of the place. 94. A small error in the longitude of the place, or which is the same in the difference of time as reckoned at the two meridians, will very little affect the parallaxes in longitude and latitude, as is evident from the formulae for computing these quantities. Con- sequently the errors produced in the apparent longitude and lati- tude will be sensibly the same as in the true. 95. Let m and n be the moon’s hourly motions in longitude 24 454. ASTRONOMY. and latitude respectively, and a = the error in the diſſerence of meridians. Then ma; and na will be the errors in the moon's computed, apparent longitude and latitude. Hence, a and c, be- ing, as before, the computed, apparent distances of the moon from the star in longitude and latitude, and s the moon's apparent se- midiamcicr, at the observed time of the beginning or end, we have, (a + maj", cos’ l -- (c -- na)* = 8*, or, a” cos’ l 4-2 mºcosºl + m2 w" cos’ l-H cº-i- 2cna -- n°wº = s” Now as the longitude of the place is supposed to be nearly known, a must be a small quantity, and the terms involving its square may be neglected. Hence we obtain, s” — cº- a” cos” l 2 m cos’ l-- 2 cm {} = In a similar manner we may determine the longitude of a place, from an observation of an eclipse of the Sun. 96. Some astronomers think the apparent diameter of the sun, obtained from observation and given in the solar tables, is too great. They infer this from a com- parison of the observed time of the beginning or end of a solar eclipse, at a known meridian, with the time ob- tained by calculation, after making allowance for the errors in the tables in other respects. To account for it, they suppose the apparent diameter of the sun is amplified, by the very lively impression so luminous an object makes on the organ of sight. This amplification is called Irradiation. Duséjour thinks that in the cal- culation of solar eclipses, the semidiameter of the sun, as given by the tables, ought to be diminished by 3"g. He also supposes the moon’s atmosphere inſlects the rays of light, so as to produce an eſtect on the beginning or end of a solar eclipse, or of an occultation, equal to CIIAPTEIR XII, 455 a diminution of 2" in the semidiaumeter of the moon. This is called the Infleasion of the moon. T}clambre is of the opinion that the irradiation and inflexion are not well established, and that their ex- istence is very doubtful. CHAPTER XII. Of the Planets. 4. Hitherto our attention has been chiefly directed to the Earth, and to those two conspicuous luminaries, the Sun and Moon. We shall now take some notice of the bodies called Planets. These bodies, like the moon, are observed to be sometimes on one side of the ecliptic, and sometimes on the olher. Their paths, therefore, cross the ecliplic. Their apparent motions are very irregular; sometimes Direct, that is, from west to east, or according to the order of the signs; and some- times Retrograde, or from east to west. There are also times, at which a planet appears to be Stationary, or to have but very little motion for several days. 2. The points in which the path of a planet cuts the plane of the ecliptic, are called the JWodes. The node through which the planet passes from the south to the north side of the ecliptic, is called the ſlscending node. The other is called the Descending node. 3. The Geocentric place of a body, is its place as seen from the earth. The Heliocentric place, is its place as it would be seen from the sum. 4. If a straight line be conceived to be drawn from the centre of a planet, perpendicular to the plane of the ecliptic, the distance from the point, in which it meets the ecliptic, to the centre of the sun is called the Cur- 456 AST 18 ONOMY. tate Distance of the planet. The point is called the Reduced Place of the planet. 5. If the reduced place of a planet, the centre of the sun, and centre of the earth, be joined by three straight lines they will form a plane triangle, lying in the plane of the ecliptic. In this triangle, the angle at the centre of the earth is called the Elongation; the angle at the centre of the sun, the Commutation; and the angle at the reduced place of the planet, the Annual Parallaa. 6. The sun, earth, moon, and planels are frequently designated by characters, as follows. Sun º º G) Juno º - # Mercury - - ? Ceres - * % Venus - * Q Pallas - - ? JEarth * - © Jupiter º 2. Moon - * D Saturn - - 12 Mars * - 3 Uranus * H. Westa - * jt VIENUS. 7. We commence with Venus, il being the most bril- liant of the planets and one whose phenomena are easily observed. This planet always accompanies the sun, being seen alternately on the east and west side, and never receding from it more than about 45°. When it is to the east of the sum, it is seen in the evening and is called the Evening Star; and when to the west, it is seen in the morning and is called the Morning Star. The evening and morning star, called by the ancient Greeks, Hesperus, and Phosphorus, were at first thought to be diſferent stars. The discovery that they are the same, is ascribed to Pythagoras. 8. When Venus is the evening star, and is at its CIIAPTER XII. 457 greatest elongation from the sun, it appears through the telescope to have a semicircle disc, like the moon in quadratures, with its convexity turned to the west.* From that time as it approaches the sun, its splendour increases for a while, though the breadth of the illu- minated disc, diminishes, like the moon in the wane. At the same time, the diameter, measured by the dis- tance of the horns, increases. 9. Venus continues to approach the Sun, till at length it becomes invisible, in consequence of the Sun's supe- rior light. After some time it appears on the west side and is seen in the morning, before the sun rises. 40. Though Venus is not, in general, visible at the time of its conjunction with the sun, it has sometimes been seen as a dark spot, passing over the surf’s disc. This phenomenon is called a Transit of Venus. When Venus is thus seen on the disc of the sun, its ap-. parent diameter is easily measured, and is found to be ºnearly one minute. 44. As Venus proceeds to the westward of the sun, its disc is seen as a crescent, continually increasing at the same time the diameter is diminishing. The convexi- ty is then turned towards the east. When the planet is at its greatest western elongation, the disc is again a se- micircle. From that time, as it again approaches the sun, the visible disc, like the moon aſler the first quarter, approaches nearer to a circle, and just previous to its being lost in the sun’s rays, at the superior conjunction (10.42. Note), the disc, does not sensibly differ from a complete circle. Its diameter is then only about 40 seconds. 12. From the superior conjunction the diameter of Venus increases, but the apparent disc changes from a * In observing Venus with a telescope, it is better to have thc object chd, partly covered in order to diminish the light. 458 ASTRONOMY. full orb, till at the greatest eastern elongation, it again becomes a semicircle. The period, that circumscribes all these changes, is the same as the time from one conjunction to another of the same kind. This period, which is the synodic revolution (10.12) of Venus, is, at a mean, about 584 days. 43. The different phases of Venus are readily ac- eounted for, by supposing it to be an opaque, spherical body, revolving round the sun from west to east, at a distance less than that of the earth, and shining by re- flecting the sun’s light. 44 Venus never deviates more than a few degrees from the plane of the ecliptic. By supposing it to move in a circular orbit in the plane of the ecliptic, we can easily obtain an approximation to its distance from the sun, in parts of the earth’s distance. Calling the earth’s distance from the sun 1, let a, be the distance of Venus from the sun. Then its distance from the earth, at inferior conjunction will be 1 — as and at superior, A + ar. Hence, (7.43), 1 + a 1 — a : 60" : 10" :: 6: 1, and consequently,7a: = 5, or, a = .744. sº 45. Since the visible disc of Venus when at its great- est elongation, either eastern or western, is a semicircle, it is evident that, the annual parallax (5) of the planet is then a right angle. Hence taking 45° as the great- est elongation, the distance of Venus from the sun, found by trigonometry is .707. As the distance of Venus from the sun, in different positions is deter- mined to be nearly the same, it appears the orbit is nearly a circle. 46. From the synodic revolution (12), the periodic revolution may be determined. For, from the time Venus is in conjunction with the sun, till it is again in \ CHAPTER XII. 459 conjunction of the same kind, its angular motion about the sun, must exceed the earth’s by a complete revolution, or 360°. The periodic revolution of Ve- nus is 224 d. 17 h. nearly. If t = 584 = the synodic revolution in days, m = 59'8" = the earth's motion in one day, a = the diurnal motion of Venus, and p = the periodic time, we have, ta: = 360° + tim, or, a = m. + * - 1°. 36', 360° 360° x t d p = − = −. and p -- = Hºm = 224 d. 17 h. nearly. Another method of finding the periodic time will be given in a succeeding article. 47. A while after the greatest eastern elongation, Ve- nus comes nearly stationary with respect to the fixed stars, having for a short time, no sensible motion in lon- gitude. After that its motion becomes retrograde, and continues so till near the greatest western elongation, when being again a short time stationary, it afterwards becomes direct. The motion then continues direct through the remaining part of the synodic revolution. These are necessary consequences of the respective distances of the earth and Venus from the sun and of their respective motions in their orbits. As it will simplify the explanation and will produce no material error, we may still suppose the orbit of Venus to coineide with the plane of the ecliptic. Let S, Fig. 37, be the sun, BDPp the orbit of Venus, and Ee a part of the earth’s orbit. Also, let E and P, be the positions of the earth and Venus at the time of in- ferior conjunction, and e and p, their places one day af. 460 ASTRONOMY. ter. Draw eC parallel to ES. Now in the triangle eSp, we know eS = 1, Sv = .71 nearly, eSp = pSP — eSE = 1° 36' — 59' = 37. Whence the angle Sep is found = 4° 31'. But the angle SeC = ESe = 59'. Since, therefore, the angle Sep is greater than the angle Sec, the position of p is to the right hand or west of eC, and consequently the apparent motion of Venus is retrograde, in this part of the orbit. It is evident, without computation, that the motion of Venus must be direct, in the part of the orbit opposite to the earth. It is also plain that when the motion is changing from direct to retrograde or the contrary, the planet must appear stationary for a time. 48. Admitting the truth of Kepler’s third law (7.30), and supposing the planets and earth to move in circu- lar orbits in the plane of the ecliptic, it may be proved by analytical investigation, that the apparent motion of each planet must be retrograde in the part of the orbit next the earth, and direct in the opposite part. We may also, by such investigations, determine the Sta- tionary points, that is, the points at which the planet appears stationary; and likewise the times during which, the motions of each planet must appear retro- grade. Now the planets do not move in the plane of the ecliptic, nor in circular orbits; but none of them, or at least, none except Pallas, deviate so far in either respect, as much to affect the results obtained on those suppositions. The computed durations of the retrograde motions of the planets, are found to agree very nearly with the durations obtained by observation. This near agree- ment forms a strong proof in favour of the earth’s mo- tion and of the Copernican System. 19. When Venus is in the part of the orbit opposite CHAPTER XII, i 46ſ the earth, nearly the whole of its enlightened side is turned towards the earth. But on account of its greater distance, it does not then afford so much light, as when in a different parl of the orbit. It is found by calcu- lation that Venus gives the most light to the earth, when being in the inferior part of its orbit, its elonga- tion is 39° 43'. This takes place about 36 days be- fore and after inferior conjunction. Although at those times the chlightened disc is only about one fourth of the whole, the light is so great that Venus may be dis- tinctly seen, with the naked eye, in the day time, even when the sun is shining in its greatest splendour. This continues to be the case for several days at each time. POSITION OF THE NODES. 20. When a planet is at either of its nodes, it is in the plane of the ecliptic, and consequently its latitude. is nothing. Now from the observed right ascension and declination of a planet at any time, its geocentric longitude and latitude may be calculated (6.49). If several longitudes and latitudes be thus obtained, about the time the planet is passing from one side of the ecliptic to the other, the exact time at which its lati- tude is nothing, may be obtained by proportion, and also its longitude at that time. This longitude of the planet, will evidently be the geocentric longitude of the node. If similar observations and calculations be made when the planet returns to the same node, we shall again have the geocentric longitude of the node, which on account of the different position of the earth in its orbit, will be different from the former. From these two longitudes, supposing the node to have no motion, its heliocentric longitude may be determined. 22 462 ASTRONOMY, Let S, Fig. 38, be the sun, N, the mode, E, the place of the earth at the time the planet was found to be in the mode from the ſirst set of observations, and E' its place at the second time. Also let EQ, EQ and SQ, all parallel to each other, represent the di- rection of the vermal equinox. Pul V = SE = radius vector of the earth at the first time, the mean radius vector being = 1, S = QES = sum’s longitude, G = QEN = the geocentric longitude of the node, and V', S' and G' the same quantities for the second time. Then if v = SN = radius vector of the planet when at the mode, and N = QSN = the heliocentric longitude of the mode, we have, SEN = EQS — QEN = S – G, SNE = QAN – QSN = QEN — QSN = G —N, sin SNE: sin SEN: : SE: SN, sin (G —N) : sin (S — G) : : V : v, V. sin (S.–G) = v. sin (G-N). Inlike manner, W’. sin (S' — G') = v. Sim (G' — N). Therefore, V sin (S.–G) sin (G - N) sin G cos N– cos G sin N V. sin (SEG) T sin (GFEIN) T sin Gºcos N–cos Gºsin N _ sin G — cos Gitan N ,” T sin GF – cos Gºian N Hence, tan N = V. sin (S.–G) sin G' – V. sin (S' — G') sin G V sin (S.–G) cos G –W. sin (S-G) cos Gº V. sin (S.–G) sin (GEN) We have also v = 24. From observations made it distant periods, it is found that the nodes of the planels, have slow retro- grade unotions. 22. The heliocentric longitude of either node being determined from observations on the planet, when in that node, and the motion of the node being also as- certained, its heliocentric longitude may be found for any given time. When the heliocentric longitudes of CHAPTER X | [. {{53 the two nodes of a planet are thus determined for the same time, they are ſound to diſſer 180°. Hence it follows, that the line of the nodes, and consequently the plane of the orbil, passes through the centre of the Sll 11. IN CLINATION OF TIME Olkl} ["I'. 23. The place of the mode of a planel being known, the inclination of the orbit may be determined. To do this, find the geocentric longitude and latitude of the planet, at the time the longitude of the sun is the same with that of the mode. Let ENp, Fig. 39, be the plane of the ecliptic, E, the earth, S, the sun, N the mode, P, the planet, Pp, perpen- dicular to the ecliptic, and pſ), perpendicular to EN, the line of the modes. Then PDp is the inclination of the orbit. Put E = pDN = the difference between the geocentric longitude of the planet and the longitude of the sum, x = the geocentric latitude of the planet, and I = PDp = the inclination of the orbit. Then, Pp = Ep. tan A, and Dp = Ep. Sin E. Pp tan A Hence, tam I = ++ = * *. } Dp sin E B ERIODIC TIME. 24. The interval from the time the planet is in one of the nodes, till its return to the same, making allow- ance for the motion of the node, gives the sidercal revolution of the planet. The sidereal revolution of Venus is 224; d. 16 h. 49 m. 8 sec. IIELIOCENTRIC LONGITUDE AND LATITUDE, AND RA- 1) IUS VECTOR. 25. The place of the node and the inclination of the orbit, being known, we may deduce the heliocentric 164: ASTRONOMY. longitude and latitude of a planet, from the geocentric longitude and latitude, obtained from observation. Let pFSN, Fig. 40, be the plane of the ecliptic, E, the earth, S, the sum, P, the planet, N the mode, Pp perpendicular to the ecliptic, pID perpendicular to SN, and EQ and SQ. the di- rection of the vermal equinox. p Put N = the heliocentric longitude of the node, S = the sun’s longitude, E = the earth's longitude = S +180°, G = the geo- centrical longitude of the planet, A = the geocentric latitude, L = the heliocentric longitude, l = heliocentric latitude, I = the inclination of the orbit, W = the radius vector of the earth, and v = the radius vector of the planet. Then, NSp = L – N, SEp = G—S, ESp = E—L, EpS = 180° —SEp — ESp = 180°– G + S- E + L = 180 — G + S — 180°– S + L = L — G. Now, Ep. tan x = PP = Sp. tam. l, tan. A Sp sin SEp sin (G – S) tan. Ep T sin Esp T sin (E = L)’ tan. A sin (E-L) = tan, l sin (G — S) (A). Also, Dp = Sp. sin (L–N), Sp. tan. l = Pp = Dp. tam. I = Sp. sin (L — N) tan. I, t tan. l = sin (L–N), tan I. * Hence, tan. A sin (E-L) = sin (L–N) sin (G-S). tan. I, or, tan. A sin [(E — N) — (L — N)] = sin (L –– N) sin (G — S) tam I, * tan a sin (E–N) cos (L–N)—tan Acos (E–N) sin (L–N) = sin (L — N) sin (G — S) tan. I, iam x sin (E — N) — tan a cos (E — N) tam (L — N) = tan (L–N) sin (G–S) tan. I, tan (L– N) = tan A sin (E–N) o tan a cos (E-N) + sín (G–S) tan I Hence, L = (L–N) + N, becomes known; We have also (A), CHAPTER XII, t 465 tan X sin (E — L) * sin (GIS) ' From the triangle EpS, we have, Sp = ES sin SEp V sin (G — S) tan, l = T Tsin Eps 7 sin (LEG) Sp V sin (G – S) Hence, v = cos PSp T cos.T. sin (L– G)' 26. The sum of the longitude of the node, and of the angle contained in the order of the signs, between the right line, joining the sun and node, and the radius vector of the planet, is called the Orbit Longitude of the planet. Thus the orbit longitude of the planet at P, Fig. 40, is QSN + NSP. If L' = the orbit longitude, them, Dp = SD. tan (L–N), and SD. tan (L'—N) = DP = . º COS _SD. tan (L–N) cos I Hence, tan (L-N)- ºngº). COS. - iONGITUDE OF THE PERIHELION, ECCENTRICITY, AND SEMI-TRANSWERSE AXIS. 27. Assuming the orbit of the planet to be an ellipse, we may, from the heliocentric, orbit longitude, and the radius vector, found for three different times, de- termine the longitude of the perihelion, the eccentricity, and the semi-transverse axis. Let APD, Fig. 41, be the orbit, A the perihelion, and P, P., and P'', the three positions of the planet. Then SP, SP, and SP", the three radius vectors are known, and also from the lon- gitudes, the angles PSP and PSP". - 4 Putv = SP, v' = SP', v' = SP", w = ASP, 0 = PSP', 2 = 466 ASTRONOMY." PSP", a = semi-transverse axis, and de = the eccentricity. Then, (Conic Sections*) w = – “. (1 — e”) tº- º º B 1 + e. Cos. a. º w = * (*=*) tº- º tº C 1 + e. COs (a + 6) wº- * (*-*) - D 1 + e. cos(x + 2) From B and C, v + v e. cos. a = v' + v'e. cos (a + 9) Or, e = v'— v . E. v. COs, a - v'. cos (a + 6) In like manner from B and D, 6 = v" — v F v. COs a — v". COs (a + 2) Put v' — v = m, and v" — v = n, then from E and F, we have, n v. cosa – v'. cos. (a + 6) n v. cos. º –V, cos. (a + 2) v. cos. a. - v'. COS. 9 cos. a - v' sin. 9 sin. a. v. Cos. a. — v". COS. 9 COS. a. — v". Sin p sin.a. v — v'. cos. 6 — v'. Sim. 6 tan. a. v — v". cos. © — v". Sin. © tan. a." ºn (v — v'. cos. 6) – m. (v — v". cos. 2) Tº v'. Sin. 9 — m v". Sin. © w ºmmº emº Hence, tam. a = The value of a being determined, if it be subtracted from the orbit longitude of the planet in the first position, the remainder will be the orbit longitude of the periheliom. If L be the ecliptic longitude, and L' the orbit longitude of the perihelion, we have, (26), - tan (L– N) = tan (L' — N) cos I. The value of e, the ratio of the eccentricity to the semi-trans- verse axis may be found from either of the expressions, E and F; and a, the semi-axis, from either B, C, or D. *. * See Appendix, article 51. CHAPTER XII. 467 28. When the longitude of the perihelion, the ec- centricity, and the semi-transverse axis, of the orbit of any planet, are determined from several sets of obser- vations, not very remote from each other, they are found, respectively, to be very nearly the same. Hence it appears, the assumption, that the orbit is an ellipse, is true, or at least nearly so. & 29. From observations made on the different planets, at remote periods, it is found that the perihelions have slow motions. The motion of the perihelion of Venus is retrograde. Those of the other planets are direct. The eccentricities of the orbits are also subject to continued, but very minute changes. Some of them are at present increasing; others diminishing. The semi-transverse axes of the orbits do not change. This fact was first discovered by La Grange, from in- vestigations in Physical Astronomy, and it is found to be conformable to observation. EPO CH OF A PLANET BEING AT THE PERIHELION OF ITS ORBIT. 30. From several observations of the planet about the time it has the same longitude as the perihelion, the correct time of its being at the perihelion, may be easily determined by proportion. ELEMENTS OF THE ORBIT OF A PLANET. 31. The longitude of the ascending node of the or- bit, the inclination of the plane of the orbit to the eclip- tic, the mean motion of the planet round the sun, the mean distance of the planet from the sun, or which is the same, the semi-transverse axis of its orbit, the ec- centricity, the longitude of the perihelion, and the time 468 ASTRONOMY. when the planet is in the perihelion, are called the Elements of the Orbit. 32. There are various other methods, for deter- mining the elements of the orbit, besides those which have been given in the preceding articles. Those which are founded on observations of the planet, when in conjunction, opposition, and in the nodes, are among the most accurate. The elements of the orbit, may also be determined with tolerable accuracy, by certain methods of estima- tion and computation, without extending the observa- tions to the time of the planet’s passage through the node. These methods were applied on the discovery of the new planets. * 33. When the elements of the planet’s orbit have been accurately determined, from a great number of observations, the equation of the centre may be calcu- lated, and tables may be formed, which will give the heliocentric longitude, latitude, and the radius vector, for any given time. But most of the planets are sen- sibly affected by the mutual attractions, among one another. The perturbations thus produced, have been calculated, for several of the planets, and form a part of complete tables of the planets. GEO CENTRIC LONGITUDE AND LATITUD E. 34. From the heliocentric longitude and latitude of a planet, as obtained from the tables, to find the geocen- tric longitude and latitude. Put p = EpS = annual parallax, Fig. 40, E = SEp = the elongation, S = ESp = the commutation, l = PSp = the helio- centric latitude, A = PEp = the geocentric latitude, W = ES = CHAPTER XII. 469 radius veetor of the earth, and v = SP = radius vector of the planet. Then Sp = v. cos. l. Now, by trigonometry, ES + Sp: ES — Sp:: tan 3 (EpS + SEp) : tan 3 (EpS — SEp), or, W -- v. cos. l ; W — v. cos. l :: tan 4 (p + E): tan * (p – E). W — v. cos. l e º ...t E V -- v. cos. l an # (p + E) Hence, tan A (p-E) = – ſº tº tan (180—s) V -- v. Cos. l v. cos. l = –– an (90° – 3 S). v. Cos. l 1 + 1 . –H W -* - * Puttan. - * * * Then (App. 15), tam 3 (p – E) = H. tan. (90 — 3 S) = tan (45°– 9). tan (90° — 3 S). And E = 3 (p + E) — 3 (p —E) = 90°– 3 S — 3 (p-E). If G = longitude of the sun, then Geocen, long. of the Planet = 0 + E = 9 + 90 — 3 S — * (p — E). For the geocentric latitude, we have (25. A), sim. E. tam. l tan. A = —- sin S 35. When a planet is in conjunction or opposition, the angles of elongation and commutation, and consequently their sines, are each, mothing. In these cases the geocentric latitude can not be found by the preceding formula. It may however be easily de- termined in a different manner. Let E, Fig. 42, be the earth, $ the sum, and P the planet, in inferior conjunction. Then, Sp = v. cos. l, and Pp = v. sin. l. Also PP = Ep. tan A = (V – v. cos. l). tan. A. Hence, (V – v. cos. l), tan A = v. sin, l, 23 470 ASTRON 6) MY. v. sin l or tam A = −. W –v. cos. l 36. From the triangles Spp and Epp, Fig. 40, we easily ob- tain the distance of the planet from the earth. Thus, EP. sin. A = Pp = v. sin. l. v. sin. 1 OP EP 2: * e \ Sin A 37. Let a = sun's mean horizontal parallax, that is the par- Tallax when the earth is at its mean distance from the Sun, desig- nated by a unit or 1, p = the horizontal parallax of the planet, in a given position, and R = the radius of the earth. Then (5.7), 1. z = R = EP. p, or p = ± = zº. sin. 3. EP v. sin. l & The parallax of Venus when in inferior conjunction is about 30”. ſº 38. It has been found from observations, that the apparent semidiameter of Venus, when at a distanee from the earth, equal to the earth’s mean distance from the sum, is 8".27. Hence if 3 = the apparent semidiameter of Venus, when at a given point P, we have (7.13), * = 8".27 8''.27 sin. A ºsmºsºs EP. T. p. sin. W. " TRANSIT OF VENUS. 39. When Venus is in inferior conjunction, if it is so near either node, that its geocentric latitude is less than the apparent semidiameter of the sun, it must pass between the earth and sun, and produce the pheno- menon, called a transit (40). This phenomenon is of great importance on account of its use in determining the parallax of the Sun, and thence the real distances and magnitudes of the bodies in the solar system. CHAPTER XII. * 47ſ Its use for this purpose was first made known by Dr. Halley. º 40. A synodic revolution of Venus, being about 584 days, 5 synodic revolutions will be 2020 days, or 8 years nearly. It follows therefore, that 8 years after Venus has been in inferior conjunction, it is again in inferior conjunction, nearly on the same day in the year, and consequently nearly in the same part of the heavens. Hence if it was near the node, at the former conjunction, it is also near the same node at the latter. If there was a transit in the first instance, the planet may perhaps be near enough to the node, in the second, to occasion another transit. A more particular calcu- lation shows that whenever there is one transit of Ve- nus, there must, generally, be another at the same node, 8 years before or after. At the end of a second period of 8 years, the planet is too far from the node, at conjunction, for a transit to take place. . When there have been two transits at one mode, the next two take place at the other node, but not till more than a century after the former. { The last transits of Venus were in the years 1764 and 4769. The next will be in 1874 and 1882. 41. The time of inferior conjunction of Venus, may be found from tables of the sum and Venus. From the geocentric longi- tude and latitude of Venus, calculated for the time of conjunction, and also for a time, an hour before or after conjunction, the geo- centric hourly motions of Venus in longitude and latitude will be known. The geocentric hourly motion in longitude, being re- trograde must be added to the sum’s hourly motion, to obtain the relative hourly motion, which will also be retrograde. From the hourly motion in latitude, and the relative hourly motion in longi- tude, the inclination of the relative orbit may be found, in the same manner, as the inclination of the moon’s relative orbit, in 472 ASTRONOMY. eclipses of the moon. The apparent semidiameter of Venus may be found by the formula in a preceding article (38). 42. Let LMN9, Fig. 43, represent the disc of the sun, C its centre, AB a part of the ecliptic, the order of the signs, being from A to B, CL a circle of latitude, and vie the relative orbit of Venus. The apparent motion of Venus being retrograde, will be in the direction, from v to w. The position of the relative orbit is adapted to the transit of 1769, at which time the latitude of Venus was north, decreasing. With the centre C and a radius equal to the sum of the semi- diameters of the sun and Venus, let arcs be described, cutting vu, in v and w, and with a radius equal to the difference of the semi- diameters, let other arcs be described, cutting vu) in a and e. The situations of Venus with respect to the sum, when at the points v, a, e, and w, are respectively called the First Eaternal Contact, First Internal Contact, Second Internal Contact, and Second Ex- ternal Contact. 43. The times of the different contacts as seen from the cen- tre of the earth, may be calculated in the same manner as the beginning and end of an eclipse of the moon, using the geocentric latitude of Venus, instead of the moon’s latitude, the sum or dif- ference of the semidiameters of the sum and Venus, instead of the sum of those of the earth’s shadow and moon, the geocentric hourly motion of Venus in latitude, instead of the moon’s hourly motion in latitude, and the relative hourly motion of Venus in longitude, instead of that of the moon. f PARALLAXFS OF THE SUN AND VENUS. 44. Let E, Fig. 44, be the earth, V, Venus, and aa'b'b, the disc of the sun. Also let AGB be the path of Venus over the sun’s disc as seen from E, the carth’s centre, the latitude of Venus being north. Then to a spectator at I towards the north pole, the path will be a Hb, which being nearer the centre is greater than the former. Consequently the duration of the transit as seen from T, is longer, than as seen frem the centre of CHAPTER XII. 173 the earth. To a spectator at K, towards the south pole, the path will be a'Fb', which is less than A.B. Hence the duration is shorter than as seen from the earth’s centre. The difference between the durations as seen from I and K, depends on the parallaxes of Venus and the sun, or rather on the difference between the parallax of Venus and that of the sun. This differ- ence is called the Relative Parallaac of Venus. From the observed durations of the transit at the two places, the relative parallax, and thence, the parallax of the sun, may be determined. * Let a' Fig 43, be the place of the centre of Venus, at the 1st internal contact as seen from a place A in north latitude, where the commencement is accelerated, and end retarded by the effects of parallax, and let as and a R be each perpendicular to AB. Put, T = time of 1st internal contact for the earth’s centre, 0 = the same for the place A, t = T – 6, E = CS = the elongation of Venus at the time T, G = aS = the geocentric latitude at the same time, m = relative hourly motion of Venus in longitude, * = hourly motion of Venus in latitude, P = relative parallax of Venus, II = relative parallax in longitude at the time 6, ! a = relative parallax in latitude at the same time. Then CR = E + mt + II*, and a'R = G + r + wº, CR” -- aſ R* = a C* = a(X = CS” + aSº, or, (E + mt + II)” + (G + r + +)* = E” + G*, * The unknown quantities II and z, are applied with affirmative signs, al- though, one of them at least, in the present case, must be negative. This produces no error in the investigation. In the numerical computation, these quantities must be used with the signs, which they will be determined to have in particular cases. 174: ASTRONOMY. 2Emt + 2EII + 2mtDI + mºt” + II* + 2Gri + 2Gº + 2rta + 42H2 + * = 0, * (m” + r"). t” + 2. (Em + m1 + Gr -- ra). t = —2. (EII + Gºr + # II* + # *) *. t” + 2. (Em -- mm + Gr-|- ra). i. m” + r.” _ _% (En -- G+ + # II* + 3 +”) m” -- rº tº If a = Em -- mTI + Gr-- ror J m? -- rº and 5 – 3. Cº-F 9: + 2* + **), we have, m” -- rº t” + 2at = — b * + 2at + a” = d”—b ł -- d = v (a”—b) F a—"—". — &c. 20, 80% O But º and all the following terms of the value of t, are ex- (l tremely small and may be neglected. Hence, b EII – Gr. 3 II* + 3 +” 2a. Em + m1I + Gr-H ra _ EII + Gr-- 3 II* + 3 +” (ETII).m. TOGT2). F Put II = up, and r = vP, then, _Eu + Gw -- * uſi + 3 vº. P d (EIT). mTOGT2).” ‘’ and, Eu + Gv -- # wri + 3 vor = 6 -– t = 0 — . P T + (E + II). m + (G + x). r g==º d = 46. Let T'be the time of the 2d internal contact for the earth's centre, and 9', for the place A. And instead of E, G, m, r, II, w, as and v, let E, G', m', 'r', II', 'r', w and v' be the values of the cor- responding quantities at the 2d internal contact. As the latitude is decreasing and the 2d internal contact for the place A, is later than for the earth’s centre, the value of r must be taken, negative, in obtaining the latitude at the latter time, from that, at the \, CHAPTER XII. 475 former. Attending to these changes, we have in like mammer as before, E'w' + G'v' + # w!'ſſ' + # wa'. P (E' + II'). m. — (G' + ºr'). r Ew -- Gv -- # w II -- # vºr and (E + II). m + (G + x). r Eu' + G'v' + 3 wºn' + 3 v'a', = (E -i- II'). m'— (G + ºr'). r" - Then T = 0 — 3P, and T′ = 9' + 3'P, T'— T = 6' — 8 + (3 + 6'). P. If now d = 9' – 9, and s = 8 + 9', we have, * T' —T = d -H, sp. T' = 0 + {Put {3 = g’ 47. If, for some other place in a latitude considerably different from that of A, d' and 8' be the values of the expressions desig- mated by d and 8, we shall have in like manner, T'— T = d' -- s'P. Hence, d -- sp = d' + s^P, sB — S'P = d’ – d, P = d' — d S — S' 48. In this expression for the value of P, the two quantities d and d' are known from observation. The quantities s and s' de- pend on the values of 3 and 3' for the two different places of ob- servation. In the expression represented by 3, the values of E, G, m and r, are known. To obtain w and v, let L = the geo- centric longitude of Venus, and A = its distance from the north pole of the ecliptic. Then (10,52.C. and 54.G), sin P sin. h sin (L — n + II) sin. A sin a = sin P cos h sin (A + x) _sin P sin h cos L – n + 3 II) cos (A + 2) COS # II G 9r because II and z, are very small, sim II = , and, 476 ASTRONOMiY. sin II = sin P sin h sim L — wº sin A j (Q) sin w = sin P (co. h sin A — sin h cos A cos L —w) (R) II sin II sin h sin L – m) P sin P --- sin A ’ Hence (45 , w = Zr sin ºr © & f w = 1 = +3 = cosh sin A — sinh cos A cos (L–n). P sin P \ The other two quantities II and ºr, contained in the expression represented by 3, depend on P, the quantity sought. But as II and or are very small, it is evident from examination of the expression, that we obtain a near value of 3, if the terms, in which they en- ter, are omitted. Hence by taking, Eu -- Gv E'w' /.../ T En TGF _ E'w' -- G'v — F--—-73 E'm' — G'r' and 3' we obtain a very near approximate value of P. Then, taking II = uP, w = vT, II* = wº", and ºr' = v'P, and again calculating the values of 3 and 3', we obtain the correct value of P. 49. If ar' = the sun’s horozontal parallax at the time of the transit, ar, being the mean parallax, and V the radius vector, we Zy" have Z3°/ -º - e. W At the time of a transit, the latitude of Venus is so small that we may consider its sine as equal to its tangent, and its cosine = 1. Hence (35), v. sin. i – v. ' and (37), sine A = a sin A gº 2J" v. Sin l W — v Therefore P = p — ar' = V *— -y == WHy; Hence, a = P. V* – Vo. Q) 50. The transit of 1769 was observed at Wardhus, a small island on the north coast of Europe, and at CHAPTER XII, 477 Otaheite in the South Sea, and the duration was found to be longer at the former place, than at the latter, by 23 m. 10 sec. The sun’s mean horizontal parallax, determined from the observations, made at those places, is 8".7. From observations made at other places, re- sults a little different were obtained. By taking the mean of the results deduced from the most accurate ob- servations, astronomers have fixed the parallax at 8".6 or 8”.7; some adopting one number, and some the other. 51. Taking the sun’s parallax 8".7, the earth’s imean distance from the sun is 23708 semidiameters of the earth (5.8), or 94,000,000 English miles, nearly. Thence from the sidereal revolutions of the earth and Venus, the mean distance of Venus from the sun, found by Kepler’s third law (7.30), is 68 millions of miles. From the observed diameter of Venus, when at a known distance from the earth, its real diameter is easily found. It is about 7600 miles. 62. It is found that Venus revolves on its axis, from west to east, in 23 h. 21 m., and that its axis is in- elined to the ecliptic in an angle of about 15°. MERCURY. 53. Mercury, like Venus, always accompanies the sun. Its greatest elongation is about 23°. The phe- nomena of Mercury correspond in almost every part, with those of Venus, only that it is farther from the earth and nearer to the sun, and consequently more dif- ficult to be observed. Its greatest and least apparent diameters are 11”.2 and 5". It can only be seen, by the naked eye, when in the most favourable positions. Mercury makes a sidereal revolution round the sun, 24 478 ASTRONOMY. in about 88 days, at a mean distance of 37 millions of miles. Its diameter is a little more than 3,000 miles. On account of the proximity of Mercury to the sun, it is difficult to determine whether it revolves on its axis. Shroeter thinks he has ascertained, that it makes a revolution, like the earth and Venus, from west to east, in 24 h. 5 m.; and that its axis makes but a small angle with the ecliptic. 54. When at the time of inferior conjunction, Mer- cury is in either node, or very near to it, a transit of Mercury takes place. Transits of Mercury occur much more frequently than those of Venus. The next five will take place in the years, 1822, 1832, 1835, 1845, and 1848. Of these, the last four will be visi- ble in the United States. The calculation of a transit of Mercury is altogether similar to one of Venus. f 55. Mercury and Venus are called Inferior planets, because their orbits are included within the earth's. The others are called Superior, because their orbits are without the earth’s. MARS, 56. Mars and all the other superior planets, differ from Mercury and Venus, in being seen in opposition as well as in conjunction. The disc of Mars does not present the phases of the two inferior planets, but it is observed, in particular situations, to deviate very sensibly from a circle. The apparent diameter of Mars, undergoes considerable change. When great- est it is 47", and when least, 3".5. The sidereal revolution of Mars is nearly 687 days, and its mean distance from the sun is 44.3 millions of miles. Its diameter is about 4000 miles. GHAPTER XII. 479 Mars revolves on its axis from west to east in 24 h. 39 m., afid its axis is inclined to the ecliptic in an angle of 59°42'. Its polar diameter is less than the equatorial. According to the measures of Arago, these diameters are to each other in the ratio of 189 to 194. JUPITER AND ITS SATELLITES. 57. Jupiter is the most brilliant of the planets, ex- cept Venus. Its apparent diameter when greatest is 44".5, and when least, 30". The sidereal revolution of Jupiter is about 4333 days, or nearly 12 years, and its mean distance from the sun is nearly 490 millions of miles. The diameter of Jupiter is 89,000 miles, which is more than 11 times the diameter of the earth. The magnitude of Jupiter is therefore more than 1300 times that of the earth. Jupiter revolves from west to east, on an axis nearly perpendicular to the ecliptic, in 9 h. 56 m. Its polar diameter is to its equatorial diameter, in the ratio of 467 to 177. 58. When Jupiter is examined with a good telescope, its disc is observed to be crossed near the centre by several obscure spaces which are nearly parallel to each other, and to the plane of the equator. These are called the Belts of Jupiter. 59. When Jupiter is viewed with a telescope, even of moderate power, it is seen accompanied by four small stars, nearly in a straight line parallel to the ecliptic. These always accompany the planet, and are called its Satellites. They are continually changing their positions with respect to one another, and to the planet, being sometimes all to the right, and some- times all to the left; but more frequently some on each side. The greatest distances to which they recede 480 ASTRONOMY. from the planet on each side, are different for the dif- ferent satellites, and they are thus distinguished; that being called the First satellite, which recedes to the least distance; that the Second, which recedes to the next greater distance, and so on. The satellites of Jupiter were discovered by Galileo in 1610. 60. Sometimes a satellite is observed to pass between the sun and Jupiter, and to cast a shadow which de." scribes a chord across the disc. This produces an eclipse of the sun, to Jupiter, analogous to those which the moon produces on the earth. It follows that Jupi- ter and its satellites are opaque bodies, which shine by reflecting the sun’s light. Jupiter being an opaque and nearly spherical body, must project a conical shadow in a direction opposite to the sun. When either of the satellites enters this shadow, it must suffer an eclipse and consequently be- come invisible. Observations show that this is the case. The satellites are frequently seen, even when con- siderably distant from the planet, to grow faint, and in a little time, entirely to disappear. The third and fourth satellites are sometimes observed, after having been eclipsed, again to become visible on the same side of the disc. These phenomena indicate that the satel- lites of Jupiter are little moons which revolve round that planet, in like manner as the moon does round the earth. 61. The satellites are sometimes on the opposite side of Jupiter, from the earth, and consequently become in- visible. Sometimes they are between the earth and Jupiter, in which case they are not easily distinguish- ed from the planet itself. - * When a satellite is invisible in consequence of en- tering into the shadow of Jupiter, the phenomenon is called an Eclipse of the satellite. CHAPTER XII. 481 f 62. Careful and repeated observations, show that the motions of the salellites, are from west to east, in orbits, nearly circular, and making small angles with the plane of Jupiter’s orbit. Observations on the eclipses of the satellites make known their synodic re- volutions, from which their sidereal revolutions are easily deduced. From measurements of the greatest ap- parent distances of the satellites from the planet, their real distances are determined. 63. A comparison of the mean distances of the sa- tellites, with their sidereal revolutions, proves that Repler’s third law with respect to the planets applies also to the satellites of Jupiter. The squares of their sidereal revolutions are as the cubes of their mean dis- tances from the planet. The planets Saturn and Uranus are also attended by satellites, and the same law has place with them. 64. The orbits of the third and fourth satellites are elliptical. Those of the other two, have not been as- certained to differ sensibly from circles. 65. The mutual attractions of the satellites on one another, produce inequalities in their motions, which must be taken into view, when it is designed to deter. mine from calculation, their positions, at any given time, with accuracy. In investigating this subject, La Place discovered two very remarkable conditions, that connect the mean motions of the first three satellites. He found, That the mean motion of the first satellite, added to twice the mean motion of the third, is eacactly equal to three times the mean motion of the second. He also found, That the mean longitude of the first satellite, less three times that of the second, more twice 482 ASTRONOMY. .* that of the third, must always be equal to 180°. It follows from this circumstance that the longitudes of these three satellites can never be the same at the same time, and consequently that they can never be all eclipsed at once. *. 66. The satellites of Jupiter undergo periodical changes in brightness. From very attentive observa- tions of these changes, Dr. Herschel infers that each satellite revolves on its own axis in the same time that it makes a sidereal revolution round the planet. 67. Observations on the eclipses of Jupiter’s satel- lites, have led to the discovery of a very important fact; which is, that the Transmission of light is Successive. When Jupiter is in opposition, the eclipses of the satellites, happen earlier, than they ought to do, ac- cording to the known durations of their revolutions, and on the supposition that the transmission of light is instantaneous. On the contrary, when Jupiter is near conjunction, they happen later than they ought to do on the preceding supposition. The variations are the same for all the satellites, and are found evidently to be connected with the distance of Jupiter from the earth; the eclipses happening later as the distance is greater. These circumstances are easily explained and the amount of the retardation, accurately account- ed for, by allowing light to occupy 16 m. 26 sec. in traversing with a uniform motion, a distance equal to the transverse axis of the earth's orbit. The discovery of the successive transmission of light was made by Roemer a Danish astronomer, in the year 1674. 68. Since light is 16 m. 26 sec. in passing over a distance equal to the diameter of the earth’s orbit, it must be 8 m. 13 sec. in passing from the sun to the earth, when these bodies are at their mean distance. .* CHAPTER xII. * 483 Its velocity is therefore 190,000 miles per second, which is greater than any other, with which we are acquainted. f 69. The eclipses of Jupiter’s satellites furnish a simple means of determining the longitudes of places on land, with considerable accuracy. The tables for calculating these eclipses, constructed by Delambre, and founded on the theory of La Place, give the times of beginning or end of the eclipses, with very little error. These times are calculated and in- serted in the Nautical Almanac, for the meridian of Greenwich, and in the Connaissance De Tems, for the meridian of Paris. As a satellite really loses its light by entering into the shadow of Jupiter, the commencement of an eclipse must be seen at the same instant by all ob- servers, however distant from one anotherº. If, there- fore an eclipse of one of the satellites, be observed at a place whose longitude is required, the difference be- tween the observed time, and the time computed for the meridian of Greenwich, will give the difference of meridians, supposing the tables to be accurate. This method of finding the longitude can not be em- ployed at sea, because the motion of the vessel, pre- vents the use of telescopes of sufficient power, for ob- serving the eclipses. * SATURN WITH ITS SATELLITES, AND RING. 70. Saturn revolves round the sun in about 10758 * This supposes the telescopes used by the observers to be of equal good- ness. For, since the diminution of light is gradual (60), two observers, by the side of each other, but using telescopes of differnt power, will not lose sight of the satellite at the same instant. The observation also depends on the state of the air, and in some measure on the eye of the observer. 184 ASTRONOMY. days, or nearly 29; years, at the distance of 900 mil- lions of miles. Its diameter is 79,000 miles. The greatest and least apparent diameters of Saturn, are 20'’. A and 16”.3. - Saturn revolves on its axis from west to east, in 40 h. 16 m. Its axis is inclined to the ecliptic in an angle of about 60°. The polar diameter is to the equatorial, in the ratio of 10 to 11. 74. The planet Saturn is distinguished from all the other planets, in being surrounded by a broad, thin ring, which is entirely detached from the body of the planet. It is ascertained to be opaque and to shine by reflecting the sun’s light. This ring was discovered by Huygens, and is discernible, when in favourable positions, with telescopes of small power. The plane of the ring is inclined to the ecliptic in an angle of 31°24'. Consequently, the face of the ring can never be turned directly towards the earth. It is generally seen under the form of an eccentric ellipse. The ring becomes invisible when the enlightened face is turned from the earth. On account of its little thickness, it is also invisible in two other cases. These are, when the plane of the ring, produced, passes through the centre of the earth, and when it passes through the centre of the sun. The ring revolves round an axis, perpendicular to its plane, and passing through the centre of Saturn, in 40 h. 29 m. Observations with telescopes of high power, show that the ring of Saturn, really consists of two concen- tric rings, entirely separate from each other. The breadth of the interior ring is 20,000 miles; of the ex- terior 7000, and of the space between them 2800. The \ CHAPTER XII. 185 distance from the centre of the planet to the inside of the interior ring is 73 000 miles. $ 72. Saturii is accompanied by seven satellites, which move round it from west to east in orbits that are nearly circular. The orbits of the first six, nearly coincide with the plane of the ring; that of the seventh, makes a less angle with the ecliptic. URANU'S AND ITS SATELLITES. 73. The planet Uranus was discovered by Dr. Herschel in the year 1781. It revolves round the sun in 30689 days, or a little more than 84 years, at the distance of 1800 millions of miles. Its diameter is 35000 miles. The greatest and least apparent diame- ters are 4". 1 and 3".7. The distance of Uranus is so great that its revolution on its axis has not been as- certained. According to the observations of Herschel, Uranus is accompanied by six satellites, which revolve in orbits nearly perpendicular to the plane of the ecliptic. WESTA, JUNO, CERES AND PALLAs. 74. These four planets, although less distant than several of the others, are so extremely small that they can only be seen with telescopes of considerable power. Ceres was discovered by Piazzi on the first day of the present century; Pallas, by Olbers in 1802; Juno, by Harding in 1803; and Vesta, by Gibers in 1807. They revolve from west to east, in orbits not very different in extent, and contained between those of Mars and Jupiter. The orbit of Pallas differs from those of all the other planets in the greatness of 25 £86 ASTRONOMY. the angle, it makes with the ecliptic. This angle is nearly 35°. 75. The following tables contain the elements of the orbits of the planets and the periodie revolutions of the satellites. The elements of the four new planets are to be considered only as approximations. Sidereal Revolutions of the Planets. $ Days. | Mercury tº a tº º sº tº 87.969.258 Venus gº º gº &º 224.70078? The Earth wºg tºg * 365.256384 Mars - * * wº tºº º 686.97.9646 Westa * tºg wº tºss 1326.930 Juno - tº tºº wº - 1594,023 Ceres gº º * gº 1681.370 Pallas gº {º º - 1685.619 Jupiter - ſº º º 4332.585117 Saturn tº * -º wº - 10758.322161 Uranus - agº. *sº dº 30688,712687 JMean distances from the Sun, or Seini-aaces of the Orbits, the Earth’s mean distance being = 1. Mercury tºº tºº º - , 0.387098 Venus tº º tºº sº - 0.723332 The Earth - º sº tºg 1.000000 Mars wº tº gº * - 1.523692 Westa - sº wº * > tº 2.363.19 Jumo sºg tº tºº. tº - 2.67035 Ceres - tº tºº wº º 2.76.722 Pallas are - - tº - 2.77.188 Jupiter tºº º wº - 5.202776 Saturn - - - tºg - 9,538770, Uranus tºº ſº tº • 19, 183305 CHAPTER XII. 487 Ratio of the Eccentricity to the Semi-transverse axis, at the beginning of 1801, with the Secular Varia- tion. Mercury Venus - The Earth - Mars ºn tº Vesta - * Juno - Ceres - - Pallas t- Jupiter Saturn - Uranus The sign — indicates a diminution. JMean Longitudes, reckoned from the JMean Equinoa, at the Epoch of JMean JYoon, at Greenwich, January 4, 1801. Mercury Venus The Earth Mars Westa º Juno Ceres * Pallas - Jupiter - Saturn TJramus Ratio of the Eccentricity. 0.205515 0.006861 s 0.016853 0.093307 0.08912S 0.254311 0.078502 0.241600 0.048164 0.056132 0.046670 Secular Variation, - 0.00000387 — 0.00006275 — 0.00004181 0.00009019 O.00016036 - — 0.00031:240 — 0.00002521 2* gº 166° 0' 49” 11 33 3 100 39 10 64 22 56 293 32 34 72 55 28 77 25 23' 65 22 5 112 15 23 135 20 18 177 47 39 488 ASTRONOMY. JMean Longitudes of the Perihelia, for the same Epochs as the preceding, with the Sidereal, Secular Variations. w Long. Perihel. Sec. War. Mercury - - 74° 21' 47" - 9' 44” Venus - tº- 128 43 53 - — 4 28 The Earth - - 99 30 5 - - 19 41 Mars - ſº 332 23 57 tºº 26 22 Vesta º - 250 18 26 Juno - sº 3 13 22 Ceres tº - 146 46 32 Pallas - º 120 54 48 Jupiter * h - 11 8 36 -> 11 5 Saturn - tº-º 89 8 58 - - 32 17 - 4 0 Uranus tº - 167 21 42 Inclinations of the Orbits to the Ecliptic, at the be- ginning of 1804, with the Secular Variations of the Inclinations to the true Ecliptic. Inclination. Sec. War. Mercury - *g 7° 0' 1" tº- 18”.1 Venus - sº - 3 23 29 º — 4.5 The Earth --> 0 0 0 - - O Mars - tº ºt - 1 51 6 º — 0.3 . Westa - * 7 7 50 Juno - sº - 13 4 16 Ceres sº º 10 37 31 Pallas - - - 34 35 14 Jupiter - - 1 18 52 - — 22.6 Saturn Eº - 2 29 38 - - — 15.5 Uranus - *- 0 46 26 - - 3.1 CHAPTER XII. 489 Longitudes of the ſlacending JWodes at the beginning of 480ſ, with the Sidereal, Secular Motions. Mercury - Venus - The Earth Mars - Westa - Juno - Ceres - Pallas - Jupiter - Saturn Uranus - Long. of $2. 45° 5' 31" 74 5 0 48 103 171 78 172 32 98 26 111 72 2 O 39 - Sidereal Revolutions of the Satellites, and their JMean Distances from the Planets about which they re- volve. The distances are ea pressed in terms of the Equatorial Radius of the Planet. 3 5] 29 31 31 18 46 51 14 Sec. TMot. - — 13' 2" - — 31 11 0 0 - — 38 49 was gº ſº – 26 21 - — 32 22 * * — 59 59 1st Satellite 2d - º 3d dº 4th - ſº - 1st Satellite 2d - 3d - gº 4th - 5th ſº 6th - 7th ** JUPIT ER. Mean Dist. 6.04853 – 9.62347 - 15.350.24 - 26.99835 - SATURN. Mean IDist. 3.351 - 4.300 – 5.284 - 6.819 - 9.524 - - 22.081 - 64.359 - Sider. Revol. 1.7691378 Days. – 3.551 1810 7.15455.28 - 16.6887697 Sider. Revol. - 0.94271 Days 1.37024 º 1,88780 - 2.73948 gº 4.51749 - 15.94530 - 79.32960 496) ASTRONOMY, URANUS. Mean Dist. Sider. Revol. 1st Satellite - 13,120 º - 5.8926 Days 2d - £º – 17.022 - tº 8.7068 3d tº tº a 19.845 wº - 10.961 i 4th - tºº - 22.752 - - . 13,4559 5th tº iº 45.50% tºº - 38.0750 6th - us - 91.008 - - 107.6944 CHAPTER XIII. On Comets. 4. It has already been said (1.9) that a Comet is a body which occasionally appears in the heavens, has a motion among the fixed stars, and only continues visi- ble for a short period. The appearance of a comet is usually that of a collection of vapour, in the centre of which is a nucleus, that is, in general, not very dis- tinctly defined. The motions of some comets are direct, and of others retrograde. In the same comet the motion continues nearly in one plane, passing through the sun’s centre; but for different comets the planes make very different angles with the ecliptic. It is found that when a comet first becomes visible, its distance from the sun is is diminishing; and that when it ceases to be visible, the distance is increasing. 3. When a comet first appears, its nucleus is usually surrounded by a faintly luminous vapour, to which the name of Coma has been given. As the comet ap- proaches the sun, the coma becomes more bright, and at length shoots out into a long train of luminous trans- ÇHAPTER XIII. 491 parent vapour, in a direction opposite to the sun. This forms the Tail of the comet. As the comet recedes from the sun, the tail precedes it, being still in a direction opposite to the sun, and grows less, till at length the comet resumes nearly its first appearance. In those comets which do not ap- proach very near to the sun, the coma does not extend into a tail. The tail is always transparent, so that the stars are distinctly seen through it. 4. The length and form of the tail are very various. In some, the length is only a few degrees, and in others it is more than a quadrant. In the great comet which appeared in 1680, the tail extended to a dis. tance of 70°; and in that of 1648, to the distance of 104°. 5. It is supposed that in the near approach of a comet to the sun, the heat becomes so intense as to melt and evaporate the exterior part, and thus to . form round the interior, an atmosphere of vapour which is the coma. And that the more volatile parts of this vapour being acted on by the impulsion of the sun’s rays, are moved in a direction opposite to the sun, and thus form the tail. 6. Comets have been sometimes observed to pass very near to some of the heavenly bodies, without pro- ducing any sensible effect on their motions. It is hence inferred that the quantities of matter which they con- tain is very small. 7. A comet remains so short a time in sight, and de- scribes so small a part of its course within our view, that, from observation alone, without the assistance of hypothesis, it would be impossible to determine the na- ture of its path. The hypothesis most conformable to analogy is, that the comet moves in an ellipse, having 192 ASTRONOMY. the sun in one of the foci, and that the radius vector describes areas, proportional to the times. As the ellipse, in which a comet moves, is evidently very eccentric, the part of it in the vicinity of the ver- tex or perihelion, and through which the comet passes while it continues visible, must coincide very nearly with a parabola. 8. The elements of a comet’s orbit are, the incli- nation of the orbit, the position of the line of the nodes, the longitude of the perihelion, the perihelion distance from the sun, and the time when the comet is in the perihelion. These are less in number than those of the orbit of a planet (42.34), because the observations that can be made during one appearance of a comet are not sufficient to determine with any degree of accuracy, the transverse axis of the orbit and the periodic time. 9. Assuming the orbit of a comet to be an ellipse or porabola and that the radius vector describes areas proportional to the times, the elements may be deter- mined from three observed, geocentric places of the comet. This, though a problem of considerable dif. ficulty, may be performed in a great variety of ways; almost every noted astronomer of latter time, having given a method of his own. One of the latest and in practice one of the best, is that given by Delambre in his flstronomie, Chap. 33, art. 59, &c. 10. The only comet which is known with certainly to have returned, is that of 1682, which conformably to the prediction of Dr. Halley, appeared in 4759. Halley was led to this prediction by observing that a comet had appeared in 1534 and another in 1607, and that the elements of their orbits, when calculated from the observations made on them, agreed nearly with those of the comet of 1682. i.e. thence inferred that instead CHAPTER XIV, 493 of three different comets, it was the same comet that had appeared at those times, and that its period was between 75 and 76 years. º 11. The number of comets, is not known, but it amounts to several hundred. CHAPTER XIV. Jlberration of Light, JWutation of the Earth’s ſlavis, and the ſlnnual Parallua of the fiaced Stars. 4. Dr. Bradley in the course of some accurate ob- servations on the fixed stars, found that their apparent places were subject to small changes, amounting when greatest to about 40". He also ascertained that those changes were annual, as their magnitudes were the same at the same time in each year. These observa- tions were commenced in the year 1725, and continued for several years. After several unsuccessful attempts, to explain the cause of these periodical changes, it occurred to him that in consequence of the progressive motion of light, and of the earth’s motion in its orbit, the apparent place of a star, ought generally to be different from the true place. 2. Let OB, Fig. 45, be a portion of the earth’s or- bit, so small that it may be considered as a right line, and the earth’s motion in it, uniform; and let ES be the direction of a fixed star, from the point E. Also let AE be the distance through which the earth moves in some short portion of time, and a B. the distance through which a particle of light moves in the same time. Then a particle of light, which, coming from the star in the direction SE, is at a, at the same time 26 194 ASTRONOMY. that the earth is at A, will arrive at E, at the same time that the earth does. Let A'a', A"a" and ES’ be each parallel to Aa. Then aa' is to AA’ and a'a" is to A'A" in the ratio of Ea to E.A. Consequently when the earth is at A', the particle of light is at a', and when the earth is at A", the particle of light is at a". The particle of light therefore, continues in the same direction from the earth, that is, in the direction Aa or ES.' Hence it meets the earth at E, in the di- rection S'E. To an eye at E, the particle of light en- tering it in the direction S'E, appears to come from a star in the direction ES’. What has been said of a single particle, will apply to all the particles coming from the star, and entering the eye. Consequently the star appears to be in the direction ES'. 3. The angle which expresses the change pro- duced in the apparent place of a body, by the mo- tion of light combined with the motion of the spectator, is called the Aberration. Thus S'ES is the aberration of the star S. 4. Various formulae have been investigated for com. puting the effect of aberration on the longitudes, lati- tudes, right ascensions and declinations of the heavenly bodies, and particularly of the fixed stars. Of those that apply to the fixed stars, the following are some of the most simple. 5. Let L be the longitude of the sun at the time for which the aberration is required, and L' and A, the longitude and latitude of the star. Then, — 20". 253 cos (L – L') COS X ./lber. in Lat. = 20".253 sin (L' — L) sin x. Jīber. in Long. = 6. Let A be the right ascension and D the declina- CIH.A.PTER XIV. 495 tion of the star, L being the sun’s longitude as before. Then, Jlber. in Right ſlscen. 0".837 cos (A + L) — 19".416 cos (A — L) --. cos. D. sº Jaber. in Decl. = sin D [19".416 sin (A–L) — 0".837 sin (A + L) —8”.066 cos L cos D] 7. In catalogues of the fixed stars, the mean places are given. By means of the preceding formula, or by small tables which have been calculated for the pur- pose, the aberration may be found, and thence the ap- parent place of the star. 8. In consequence of the aberration, each star ap- pears to describe an ellipse in the heavens, of which the true place is the centre; the semi-transverse axis is 20".253 and the semi-congugate is 20".253 sin A. 9. The supposition of the earth’s annual motion, serves fully to explain the phenomena of the aberra- tion. And the amounts of the aberrations for different stars, and at different times, computed on that suppo- sition, are found, exactly to agree with observation. These circumstances form the strongest proof of the reality of the earth’s annual motion. 40. The aberration of the sun, which has place only in longitude is — 20".253. Thus the sun’s ap- parent place is always about 20".25, behind its true place. Solar tables give the apparent place of the sun, as affected by aberration, and it is this which is generally wanted. 14. For a planet, the aberration is different from what it is for a fixed star; because the planet changes its place during the time that light is passing from it 496 ASTRONOMY. to the earth. The aberration is therefore increased or diminished by the geocentric motion of the planet during this time. For the moon, the aberration is always very small, only amounting to a fraction of a second. 12. Besides the aberration produced by the annual motion of the earth, there is another, called the Diurnal aberration, which is produced by the earth’s motion on its axis. This is however so small as to be nearly insensible. NUTATION. 13. Small inequalities, which have been observed in the precession of the equinoxes, and in the mean obliquity of the ecliptic; are called JWutation. These inequalities were discovered by Dr. Bradley while employed in verifying his theory of the aberration. 14. The period of the changes of these inequalities was observed to be about the same as the period of the revolution of the moon’s nodes; and it was found that the quantities of the inequalities depended on the place of the node. 45. The phenomena of the nutation may be repre- sented by supposing, that while a point, which may be considered as defining the mean place of the pole of the equator, describes a circle in the heavens, round the pole of the ecliptic, at a distance from it, equal to the mean obliquity of the ecliptic, and with a retro- grade motion of 50". A annually, another point repre- senting the true pole of the equator, moves round the former at the distance of 9", so as to be always 90° more easterly than the moon’s ascending node. The inequalities thus produced in the precession of the equinoxes, and in the obliquity of the ecliptic, will CHAPTER XIV. 497 very nearly agree with the observed inequalities. The agreement becomes more exact, iſ instead of supposing the true pole to describe a circle about its mean place, it be supposed to describe an ellipse, having its semi- transverse axis equal 9".6 and its semi-conjugate 7".5. 46. If N be the longitude of the moon's ascending mode, the variation in the obliquity of the ecliptic, pro- duced by the mutation is + 9".6 cos N; the inequality in the motion of the equinoxes in longitude, sometimes called the Equation of the Equinoaces in Longitude, is 47".946 sin N; and the inequality in their motion in right ascension, called the Equation of the Equinoaces in Right Ascension, is — 46".462 sin N. 17. The equation of the equinoxes in longitude, equally effects the longitudes of all the stars. The equation of the equinoxes in right ascension also affects the right ascensions of all the stars, but it only forms a part of the nutation in right ascension. 18. If A be the right ascension of a star, and D its declination, then, JYutalion in Right ſlscen. = — 16”.462 sin N–8".373 cos (A —N) tan D — 1"227 cos (A + N) tam D. JNutation in Decl. = *- + 8 .373 sin (A–N) + 1".227 sin (A + N), 49. The nutation does not affect the positions of the stars relative to one another, nor to the plane or pole of the ecliptic; it only affects their positions relative to the plane of the equator, or to the position of the earth’s 3, X1S. ANNUAL PARALLAX of THE FIXED STARs. 20. The angle contained between two straight lines, conceived to be drawn from the sun and earth, and 198 ASTRONOMY. meeting at a fixed star is called the flnnual Parallaa: of the star. 24. Observations have been made by several astro- nomers, on different stars, and at times when the earth was in opposite parts of its orbit, with the view of ascertaining whether they have any sensible parallax. Dr. Brincly deduced from his observations, that the parallax of & Lyrae, when greatest, that is, when the line joining the sun and earth is perpendicu- lar to the line joining the sun and star, is nearly 3”. But Pond, the present Astronomer Royal of England, from a series of very accurate observations on the same star, makes the greatest parallax only 0".26. 22. If we suppose the annual parallax of a star, when greatest to be 1", the distance of the star will be 206265 times the radius of the earth’s orbit. This distance is so immensely great, that light, which tra- verses the distance from the sun to the earth in 8 m. 43 sec. would require more than 3 years to come from the star to the earth. CHAPTER, XV. JW'autical flstronomy. 4. SoNE of the most useful practical applications of astronomy are those which serve to make known to the Navigator, his latitude and longitude, when at sea. The continual agitation in the motion of a ship does not permit the use of instruments, which are adjusted by a plumb line or spirit level. The astronomical in- struments used at sea, are the Hadley’s Quadrant, the Seactant and Circle of Iteflection. By either of these the altitude of any of the heavenly bodies, and the an- CHAPTER XV. 499 gular distance between them, within certain limits, may be obtained with considerable accuracy. The sextant and circle of reflection are made with greater accuracy than the quadrant, and are principally used for mea- suring the angular distance of the moon from the sun or a fixed star. For descriptions of these instruments and of the methods of adjusting and using them, the student is referred to Bowditch’s Practical JWavigator. 2. The JMile used in measuring distances at sea, is the 60th part of a degree. So that a mile just corres- ponds to a minute. 3. The course on which a ship sails is determined by an instrument called a JMariner’s Compass; and the rate at which she sails by an instrument called a Log. The latter is a piece of board in the form of a sector of a circle, the circular part of which is loaded with lead, so that when in the water it may keep a verticle position. To the log is attached a line of con- siderable length, divided into spaces called R'nots, and and wrapped round a light reel. The length of a knot is such, that when the log is thrown into the sea, and the line allowed to run freely off the real, the number of knots which pass off in a half minute, indicates the number of miles the ship is then sailing in an hour. 4. That portion of the surface of the ocean, which a ship traverses during a few hours, or even during a day, does not differ much from a plane. Supposing it to be a plane, and also that the meridians are paral- lel to one another, let AB, Fig. 46, be the track of the ship, NS and N'S', meridians passing through A and B, and AC and BD parallels of latitude, Then in the triangle DAB, we have, 200 ASTRONOMY. AD = AB cos DAB, Or, Diff. of Lat. = Dist. × cos. Course. It is proved by writers on the theory of navigation, that this expression for the difference of latitude is ri- gidly true, even when the earthis considered as a sphere or spheroid. But in consequence of currents and other causes, the distance and course can never be ob- tained with great accuracy, and consequently the dif. ference of latitude thus found, must be considered only as an approximation. 5. The triangle ADB, also gives, BD = Dist. x sim. Course. As the meridians are not parallel, but réally con- verge towards the poles, each way from the equator, it is evident that BD is greater than the distance be- tween the meridians on one of the parallals of lati- tude and less than that on the other, except when A. and B are on opposite sides of the equator and equally distant from it. It is in general nearly equal to the distance between the meridians on a parallel of lati- tude, midway between the parallels passing through A and B. The latitude of this parallel is called the JMiddle Latitude, and is equal to half the sum or half the difference of the latitudes of A and B, according as they are on the same, or on opposite sides of the equator. 6. If AB, Fig. 36, be considered as a part of the equator, P its pole, ED a parallel of latitude, and PHDA, PEB meridians, passing through any two places, then (11.85) AB, the difference of longitude of the places is equal to ED divided by the cosine of B.E. Hence (5), Fig. 46, - CHAPTER XV. t 20ſ. Diff. of Long, of fl and B Fºl BD == Dist. x sim. Course cos Mid Lat. cos Mid. Lät. 7. The computed differences of longitude and lati- tude, applied to the longitude and latitude of the place A, give nearly those of the place B, when the distance between the places is not great. The longitude and latitude thus found are called the Estimated longitude and latitude. 8. As the longitude and latitude, found in the pre- eeding manner can not be depended on, except for a short time, it is necessary that the navigator should be able to determine them by observation. When the weather is favourable the latitude is determined each day at noon, by observation of the sun's altitude about that time. Several altitudes being observed, it is easy to infer the greatest altitude that the sun acquires, which is the meridian altitude. But this altitude is the apparent altitude of the under or upper limb, usually the former, and must be corrected for refraction, paral- lax and semidiameter. It also requires another cor- rection. The observation gives the altitude above the visible horizon; and consequently, as the observer is on the deck of the vessel, several feet above the sur- face of the water, it is too great. A small table ac- companies every treatise on navigation, containing a correction, depending on the height of the eye, which is to be subtracted from the observed altitude. This correction is called the Dip of the Horizon. From the correct meridian altitude of the sun, the latitude is easily determined (4.2). 9. Sometimes the sun is hid by clouds, so as to prevent the observation of the meridian altitude, and yet it is visible at other times in the day. In such - 27 20? ASTRONOMY. cases the latitude may be found from two observed al- titudes with the interval of time between them. The interval ought if possible to be two or three hours; and one of the altitudes should be as near to noon as cir- cumstances will admit. 40. When the altitudes are taken at different places, as is generally the case at sea, the less one should be reduced to what it would have been, if it had been taken at the same instant, at the place where the other is taken. This may be done with sufficient accuracy in a very simple manner. Let A, Fig. 47, be the place where the less altitude is taken, B that where the greater is taken, and AS the line in which the horizon is intersected by a vertical circle passing through A and the sun, at the time of observing the less altitude. From the bearing of the sun at that time, and the course the ship is sailing, the angle BAS is known. On AS let fall the perpendicu- lar BD. Then it is evident that the altitudes of the sun, at B and D, at the same time, are the same. But the altitude at D is greater than the altitude at the same time at A, by the number of minutes contained in AD. Consequently AD is the correction of the less altitude. Hence the * Correction = AD = AB cos BAS. When the angle BAS is greater than 90°, the cosine is negative, and the correction must be subtracted from the altitude. 11. Given two altitudes of the sun, with the interval of time be- tween the observations, to find the latitude of the place. Let Z, Fig 48, be the zenith of the place at which the greater altitude is taken, P the pole, S the place of the sun at the time of the less altitude, and Sº its place at the time of the greater, As the Sun's declination changes but little in the course of a few hours, PS and PS' may be each considered as equal to the sun’s polar distance at the middle of the time between the observations; and consequently the triangle PSS' may be regarded as isosceles. If PG be perpendicular to SS', it will bisect it in G. Put, CHAPTER XV. - 203 H = ES = the less altitudes, reduced, * & H' = FS' = the greater altitude, ^, D -a, PS - 90° ºf sun’s declination, A = SPS' = interval of time, expressed in degrees, L == HP = latitude, where the greater altitude is taken, U = PSZ, W = ZSS', W = PSS', and X = SS.’ Then from the right angled triangle PSG, we have, sin # X = sin # A sin D, and cot. W = cos D tan & A. From the triangle ZSS', by an investigation exactly similar to that in article 20th, chap. 9th. we have, * (*****) in (ºr ºth - H) cos H sin X Then U = W = W The upper sign has place, when the sun passes the meridian on the opposite side of the zenith from the elevated pole, and the under, when it passes on the same side. Now from the triangle PSZ, cos ZP = cos PS cos ZS + cos PSZ sin PS sin ZS, Or, sin L = cos D sin H + cos U sin D cos H = cos D sin H + (2 cos” # U — I sin D cos H (App. 9), = 2 cos” # Usin D cos H –(sin D cos H – cos D sin H), = 2 cos” # Usin D cos H – sin (D–H). But, sin L = —-cos (90° -- L) = 1 — 2 cos” # (90 + L) (App. 9) and, sin (D–H) = cos [90° — (D–H)] = 2 cos” # [90° — (D–H)]— 1. * - By substituting these values we have, ef cos” # (90° -- L) = cos” # [90°—(D–H)] — cos” # Usin D cos H * = cos” # [90°– (D–H)]. (l !' __cos” # Usin D cos H ) cos” # [90°–(D–H)]/’ cos 3 U vsin D cos H Make sin M = * in M = #####, 204 AS'TRONOMY, Then cos” # (90° -- L) = cos” # [90—(D–Hºl. (1–sinº M) = cos” # [90° – D – H)] cos” M. - Or, cos 3 (90° + L) = cos # [90° — (D–H}] cos. M. 42. When the latitude is determined, the time may be obtained by an observation of the sun’s altitude, a few hours from noon (9.20). Supposing the watch or chronometer, used on board the vessel to have been well regulated and set, previously to leaving port, and that it keeps time accurately, the difference between the time obtained from observation, and that shown by the watch, gives the difference of longitude, in time. But the best time keeper can not be entirely depended on, and therefore the longitude, thus obtained, is liable to uncertainty. 43. ‘If the true angular distance between the centres of the moon and sun, or between, the centre of the moon and some star, near the ecliptic, be obtained from calculations, founded on, the observed angular distance; and the time when they are at that distance, be determined by calculation for the meridian of Green- wich; then the difference between the calculated time, and the time of observation as reckoned at the meri- dian of the ship, will give the longitude from Green- wich. l 44. The Nautical Almanac contains the distances of the moon from the sun, and from several stars that are best adapted to the purpose. The distances are given to every third hour. It is therefore easy to determine by proportion, the time when either distance is of a given magnitude. 15, The observed distance must be corrected for the semidiameter of the moon when the observation is of the moon and a star, and for the semidiameter of the Sun and moon, when it is of those bodies, so as to give CHAPTER XV, 205 the apparent distance of the centres. To obtain the true distance, the apparent distance must be corrected for the effects of refraction and parallax. This re- quires that the altitudes of the bodies should be known. 16. The altitudes may be taken by two assistant ob- servers, at the same time that the principal one ob- serves the angular distance. If there is but one ob- server, he can first take several altitudes of the bodies; then several distances; and afterwards several more altitudes, noting the times of all the observations. Thence it it easy to infer, with sufficient accuracy, the altitudes corresponding to the mean of the dis- tances. 17. Given the apparent distance of the moon and sun, or of the "moon and a fixed star, and the altitudes of the bodies, to determine the true distance. Let Z, Fig. 49, be the zenith, ZH the vertical passing through the moon, and Z0, that passing through the sun, or a star. Then as the moon is more depressed by parallax, than it is eleva- ted by refraction, the apparent place is below the true place. But for the sun or a star as the parallax is very little or insensible, the apparent place is above the true place. Let M be the apparent place and M' the true place of the moon; and S the apparent place and S' the true place of the sum or star. Put, H = HM = apparent altitude of the moon, H = HM! = true do. A = OS = apparent altitude of the sun or star, A' = OS' = true do. D = MS = apparent distance, D' = M'S' = true do. Them in the triangle ZMS, we have, (App. 34), 206 AstroNoMy. cos D–sin Hsin A cos H cos A * * cos D — cos H cos A + cos (H + A). A =" App. 14 -- & cos H cos A (App. 14) cos D + cos H + A) — 1 cos H cos A sº cos Z = In like manner from the triangle ZM'S' we have, cos D' + cos (H' + A') 1 cos Z = cos H' cos A' e \ Hence cos D + cos (H + A) _ 90s D' -- cos (H' + Aſ) & cos H cos A cos H’ cos A' 2 Or, cos D' = cos H' cos A'. [cos D + cos (H + A)]– "cos Hºcos Á cos (H' + A'). But, } cos D + cos (H + A) = 2 cos 3 (H + A + D) cos 3 (H + A— D), (App. 22) cos (H' + A') = 2 cos” # (H' + A') — 1 (App. 9) cos D' = 1 — 2 sin” # D' (App. 8). Substituting these values, and reducing, we have, sin” # D' = cos” # (H' + A') — cos 3 (H -- A + D) cos # (H + A — D) cos H' cos A' cos H cos A r = cos” # (H' + A'). cos 3 (H + A -- D) cos 3 (H + A — D) cos H'cos A. cos H cos A 1 — cos” # (H' + Aſ) Make sin M = * (e. # (H + A + D) cos 3 (H -- A —D) cos H' cos A.) v \– cos H cos A - cos # (H' + A') Then, sin” D = cos”; (H + Aſ). (1 — sin” M) = cos” # (H' + A!) cosº M, - Or, sin; D = cos; (H' + A') cos M. ===s CHAPTER XVI. * 207 CHAPTER XVI. Of the Calendar. 4. THE Calendar is a distribution of time into periods of different lengths, as years, months, weeks, and days. 2. It has been shown that the tropical year contains 365 d. 5 h. 48 m. 54.6 sec. (7.9). But in reckoning time for the common purposes of life, it is most con- venient to have the year to contain a certain number of whole days. In the calendar established by Julius Caesar, and thence called the Julian calendar, three successive years are made to consist of 365 days, each; and the fourth, of 366 days. The year, which con- tains 366 days, is called a Bisseatile year. It is also frequently called Leap year. The others are called Common years. The added day in a bissextile year is called the Intercalary day. 3. According to the Julian Calendar, and reckoning from the epoch of the christian era, every year, the number of which is exactly divisible by 4, is a bissex- tile; and the others are common years. 2^ 4. It is evident that the reckoning by the Julian calendar, supposes the length of the year to be 365; days. A year of this length is called a Julian Fear. A Julian year, therefore, exceeds the true astrono- mical year, by 11 m, 8.4 sec. This difference amounts to rather more than a day, in 130 years. 5. At the time of the Council of JW'ice, which was held in the year 325, the Vernal Equinox fell on the 21st of March, according to the Julian calendar. But by the lat- ter part of the 16th century, in consequence of the ex- Cess of the Julian year above the true solar year, it came 208 w ASTRONOMY. ten days earlier, that is, on the 11th of March. It was observed that by continuing to reckon according to the Julian calendar, the seasons would fall back, so that in process of time they would correspond to quite dif- ferent times of the year. This reckoning also led to irregularity in the times of holding certain festivals of the church. The subject, claiming the attention of Pope Gregory XIII. he, with the assistance of several astronomers, reformed the calendar. To allow for the 40 days, by which the vernal equinox had fallen back from the 21st of March, he ordered that the day following the 4th of October 1582, should be reckoned the 15th, instead of the 5th. And in order to keep the vernal equinox to the 21st of March, in future, it was concluded that three intercalary days should be omitted every four hundred years. It was also con- cluded that the omission of the intercalary days should take place in those centurial years, the numbers of which, were not divisible by 400. Thus the years A700, 1800, and 1900, which, according to the Julian calendar would be bissextiles, would, according to the reformed calendar, be common years. 6. The calendar, thus reformed, is called the Grego- vian Calendar. It is easy to perceive, by a short cal- culation, that time reckoned by the Gregorian calen- dar, agrees so nearly with that reckoned by true solar years,' that the difference does not amount to a day in 4000 years. 7. The Gregorian calendar was at once adopted in Catholic countries, but in those, where the Protestant Religion prevailed, it did not obtain a place, till some time after. In England and her colonies, it was not introduced till the year 1752. It is now used in all Christian countries, except Russia. * GHAPTER XVI, 209 8. The Julian and Gregorian calendars are also de- signated by the terms Old Style and JWew Style. In consequence of the intercalary days, omitted in the years 1700 and 1800, there is now 42 days difference between them. 9. The year is divided into twelve portions, called calendar months. Each of these contain, either 30 or 34 days, except the second month, February, which in a common year, contains 28 days, and in a bissex- tile, 29 days; the intercalary day being added at the last of this month. - 40. It was formerly customary to designate the days of the week in the calendar by the first seven letters of the alphabet, always placing them so, that A corres- ponded to the first day of the year, B to the second, C to the third, D to the fourth, E to the fifth, F to the sixth, G to the seventh, A to the eighth, B to the ninth, and so on. According to this arrangement, whatever letter designates any given day of the week in the first part of the year, continues to designate the same, throughout the year. The letter designating the first day of the week, or Sunday, is called the Dominical Letter. 14. As a common year consists of 365 days, or 52 weeks and 4 day, the last day of each year must fall on the same day of the week as the first, and the next year must commence one day later in the week. Con- sequently the day of the week which was the first day of the former year, and was designated by A, is the seventh day of the second, year, and is designated by G; that which was the second, and was designated by B, in the former year, is the first, and is designated by A in the second, and so on. It therefore follows, that whatever letter is the dominical letter, in any common 28 240 ASTRONOMY. year, the letter next preceding it in the alphabet, is the dominical letter in the following year, except the for- mer was A. in which case the second is G. - 42. In every common year, the first day of March, is the 60th day of the year, and consequently corres- ponded to the letter D. in bissextile years, on ac- count of the intercalation, the 1st of March is the 61st day of the year; but the letter D was still made to correspond to it, and the letters for the remaining part of the year were arranged accordingly. It therefore follows that, after the 29th of February, any given day of the week was designated by the letter in the alpha- bet, next preceding that, by which it was designated in the first two months. Consequently a bissextile had two dominical letters, one of which appertained to January and February, and the other, which was the next preceding letter in the alphabet, appertained to the other ten months. & 43. From what has been said, it follows that the do- minical letters succeed one another in a retrograde or- der, that is in the order G, F, E, D, C, B, A, G, F &c.; and that each bissextile has two, in the same order. - It is now usual to retain only the dominical letter in the calendar, and to designate the other days of the week by numbers, or by their names. 44. The year 1800, which was a common year, commenced on the fourth day of the week, and conse- quently the dominical letter was the 5th of the alpha- bet, which is E. From thence, taking into considera- tion, that every four years in which a bissextile is in- cluded, requires five dominical letters in a retrograde order, it is easy to find the dominical letter for any year in the present century. To do this, multiply the 4. CIIAPTER XVI. 244. number of years above 1800, by 5, and divide the pro- duct by 4, neglecting the remainder. Divide the quo. tient by 7, and subtract the remainder from 5; or from 12, when the remainder is equal to, or greater than 5. The last remainder is the number of the dominical Hetter. Delambre, in the 38th chapter of his Astronomy has given the investigation of a formula for finding the do- minical letter in any century, according to the Gre- gorian calendar. . . 15. There are some periods of time, which though they are not now much used, it may be proper briefly to notice. 16. The Solar Cycle is a period of 28 years, in which, according to the Julian calendar, the days of the week return to the same days of the month, and in the same order. The first year of the christian era was the 10th of this cycle. Consequently if 9 be ad- ded to the number of any year, and the sum be divided by 28, the remainder will be the number of the year of the solar cycle. When there is no remainder, the year is the 28th of the cycle. 47. The Lunar Cycle, or as it is sometimes called, the JMetonic Cycle, is a period of 49 years, in which the conjunctions, oppositions, and other aspects of the moon, return on the same days of the year. The sy- nodic revolution of the moon being 29.5305885 days, 235 revolutions are 6939.688 days; which differs only about an hour and a half from 19 Julian years. The number by which the year of the lunar cycle is desig- nated, is frequently called the Golden JWumber. The first year of the christian era was the 2d of the iunar cycle. Hence to find the year of the cycle, for any given year, add 1 to the number of the year, and 212 cHAPTER XVII. divide by 49. The remainder expresses the year of the cycle. If nothing remains, the year is the 19th of the cycle. - 49. The Cycle of the Indiction is a period of 15 years. This period, which is not astronomical, was introduced at Rome, under the emperors and had re- ference to certain judicial acts. To find the cycle of the indiction for a given year, add 3, and divide by 15. The remainder expresses the year of the cycle. * 49. The Julian Period is a period of 7980 years, obtained by taking the continued product of the num- bers 28, 49 and 15. After one Julian period the dif- ferent cycles of the sun, moon and indiction, return in the same order, so as to be just the same in a given year of the period, as in the same year of the preceding period. The first year of the christian era was the 4744th of the Julian period. Hence if 4713 be added to the number of a given year, the result will be the year of the Julian period. 20. The Epact as an astronomical term is the mean age of the moon at the commencement of a year, or in other words, it is the interval between the commence- ment of the year and the time of the last mean new moon; and is expressed in days, hours, minutes and seconds. 21. The Epact, as given in the calendar, is nearly the age of the moon at the commencement of the year, expressed in whole days, and was introduced for the purpose of finding the days of mean new and full moon throughout the year, and thence the times of cer- tain festivals. Without entering into any explanation of the reason of the rule, it must suffice here to observe, that the Epact for any year during the present century CHAPTER XVII. 2ſ 3 may be found by multiplying the golden number of the year by 11, adding 19 to the product and dividing the sum by 30. The remainder is the Epact for the year. CHAPTER xvii. Universal Gravitation and some of its effects. 4. It is designed to give in this chapter a general view of some of the effects of the attraction of gravita- tion, without entering into very minute investigations. The propositions, contained in the first four of the fol- lowing articles, are demonstrated in treatises on Me- chanics. 2. If a body put in motion, be urged towards a fixed point, not in the direction of its motion, by a force continually acting upon it, it will move in a curve; and the straight line drawn from the body to the point, will describe areas proportional to the times. 3. Conversely, if a body move in a curve, in such manner, that the straight line drawn from it, to some point, describes areas proportional to the times, the body is urged towards the point by a force continually acting on it. - By Kepler’s first and second laws, the planets re- volve in curves about the sun, and their radius vectors describe areas proportional to the times. Consequently the planets are urged towards the sun by forces con- tinually acting on them. - 4. If a body revolving about a point, be continually urged towards that point, by a force which varies in- versely as the square of the distance, it will move in 244 ASTRONOMY. an Ellipse or some other of the curves, called Comic Sections. 5. If a body continually urged by a force, directed to some point, describe an ellipse of which that point is a focus, the force must vary inversely as the square of the distance. It therefore follows, from Kepler’s second law, that each planet is continually urged towards the sun, by a force which varies inversely as the square of the dis- tance from the sun’s centre. 6. Since each planet is urged towards the sun by a force, varying inversely as the square of the distance, it is reasonable to suppose, instead of a distinct force for each planet, a single force residing in the sun, and varying from planet to planet according to the same law. f 7. By taking into view Kepler’s third law, for the motions of the planets, it is proved that the sun is the centre of a force, which, acting on the particles cf mat- ter in all the planets, and varying in intensity, inversely as the square of the distance from the sun’s centre, re- tains them in their orbits. 8. As the motions of the satellites of Jupiter, Saturn and Uranus are conformable to Kepler’s third law, it is proved in like manner that each of these planets is the centre of a force, which varying in intensity in- versely as the square of the distance from the centre of the planet, extends to the satellites and retains them in their orbits. 9. The earth has but one satellite, and therefore Kepler’s third law does not apply to it. But by in- vestigations, founded on the distance which a heavy body falls at the earth’s surface in one second of time, compared with the distance which the moon recedes CHAPTER XVII. 215 in the same time from a tangent to its orbit, towards the earth, it is proved that the force of gravity, va- rying inversely as the square of the distance, extends to the moon and retains it in its orbit. 40. The existence of a similar force, in each of the planets that have no satellites, is inferred from the ef. ſects which they are known to produce on one another and on the other planets. 44. The circumstances mentioned in the preceding articles serve to prove that all particles of matter are urged towards one another, with a force which varies inversely as the square of the distance. This force is called the Force of Gravitation. w 42. A Projectile Force is the force by which a body is put in motion. 43. A Centripetal Force is the force by which a body revolving about another body is urged towards it. 44. A Centrifugal Force is the force by which a body revolving about another body, tends to recede from it. g 45. Centripetal and Centrifugal forces are called Central Forces. Relative JMasses of the Planets—Relative weight of a body at their surfaces. 46. The relative quantities of matter or masses of the sun, planets, and satellites may be determined with considerable accuracy, from the effects which they produce in disturbing the motions of each other. For these effects depend on the quantities of matter of the disturbing bodies and on their distances; and the dis- tances are known from the methods of plane astro- nomy. - 47. The masses of those planets which have satel- 246 ASTRONOMY. lites may be foundina simpler manner and with greater accuracy. If A denote the mass of the sun, M the mass of a planet, m the mass of one of its satellites, D. the mean distance of the planet from the sun, d the mean distance of the satellite from the planet, and P and p the periodic times of the planet and satellite re- spectively; then it is proved that M -- m dº Pº ITM - Dº nº As the mass of the satellite is small compared with that of the planet, and the mass of the planet is small compared with that of the sun, we have, M = #. * nearly. 18. The following table exhibits the relative quan. tities of matter or masses of the sun and planets as given by Laplace in the fourth edition of his Systeme Du JMonde. * #º- Sun *. tºº º sº I I Mer sº sº ſº e *====== cury 2025810 Venus - : - tºº tºº I 356.632 The Earth - - - –– t 337.102 Mars - tº º —— *. - 2546320 Jupiter - - - is –– 1066.09 Saturn gº - - –– - 3512.08 Uranus * . ſº a 1 19504 CHAPTER XVII. 247 If the mass of the earth be denoted by 4, the mass of the moon, according to the most accurate deter- mination, is sºr. 19. The densities of bodies are proportional to their quantities of matter, divided by their bulks. The fol- lowing table contains the densities of the sun, moon and planets, the density of the earth being denoted by 4. Sun - tº tº wº 0.252 Mercury - - - - 2.585 Venus - e- 3 tº 1.024 The Earth - - - - 1.000 The Moon * sº iº- 0.615 Mars - - Gº- - - 0.656 Jupiter - * tº- ſº 0.201 Saturn - - - - 0.103 Uranus - º º º 0.218 20. Supposing the planets to be exactly spherical and not to revolve on their axes, the weight of the same body at their different surfaces would be pro- portional to their quantities of matter, divided by the squares of their diameters. But the centrifugal force, at the surface of a planet that revolves on its axis, di- minishes the weight of a body, placed on it, particu- larly near the equator. The diminution thus pro- duced, on any of the planets, is not however very con- siderable. The following table, taken from Vince's Astronomy, exhibits the relative weight, nearly, of a body at the surface of the sun and planets, its weight at the surface of the earth being denoted by 1. Sun - nº dº ... 27.70 Mercury - º º - 1.70 Venus - tº- sº ſº 0.98 29 248 ASTRONOMY. The Earth - tº tº - i.00 Mars - * tº tº 0.34 Jupiter - gº iº - 2.33 Saturn - dº gº sº 1.02 Uranus sº gº sº - 0.93 THE CENTRE OF GRAVITY OF THE SOLAR SYSTEM. 21. As all particles of matter attract each other, the sum must be attracted towards a planet, in like man- ner as the planet is, towards the sun. But as the quantity of matter in the sun is far greater than that in any of the planets, its attraction at a given distance must be proportionably greater. 22. If there were only one planet, the sun and that planet would describe similar ellipses, of which their common centre of gravity, would be one of the foci: their distances from that point, being always inversely as their quantities of matter. As there are several planets revolving round the sun, the path of the sun’s centre must be a more complicated curve. But the quantity of matter in all the planets, taken together, being very small, compared with that in the sun, the extent of the curve described by the sun’s centre can not be very great. 23. It is found by computation, that the distance be- tween the sun’s centre and the centre of gravity of the system, can never be equal to the sun’s diameter. 24. It is proved by writers on Mechanics that the centre of gravity of a system of bodies is not affected by the mutual actions of these bodies on one another; and that unless there are extraneous actions, the centre of gravity will either remain at rest or move uniformly in a right line. 25. From some minute changes in the situations of CHIAPTER XVII. 219 some of the fixed stars, called the Proper motions of those stars, Dr. Herschell has inferred that the centre of gravity, and consequently the whole system, of the sun and planets, is in motion towards the constellation Hercules. But the investigations of Duséjour and Tłurckhardt have shown that the observations, hitherto made, are not sufficient to prove the existence of any such motion. KEPLER’s LAws. 26. Kepler’s laws, with regard to the motions of the planets, have been thus far considered as rigor. ously true. It may now be proper to inform the stu- dent that the mutual actions of the heavenly bodies on each other, cause slight deviations from those laws, as they are stated in the preceding part of the work. 27. If the radius vector and mean distance of a pla- net be reckoned from the centre of gravity of the sys- tem, to the centre of the planet, or when the planet has satellites, to the centre of gravity of the planet and satellites, the first and second laws will hold true, ex- cepting so far as the motion of the planet is affected by the actions of the others. 28. The third law, as applied to any two of the planets, is affected not only by the actions of the other planets, but also by the quantities of matter in the two planets themselves. If p and P be the periodic revo- lutions of any two of the planets, a and A their mean distances from the centre of gravity of the system, and m and M their quantities of matter, that of the sun be. ing denoted by 1, then, disregarding the actions of the other planets, *: Pe. . . * : * p 1 + m, 1 + M. 220 ASTRONOMY. PROBLEM OF THE THREE BODIES. 29. If we suppose only two bodies to gravitate to- wards each other, with forces inversely as the squares of their distances, and to revolve about their common centre of gravity, they would move in conic sections, and the radius vectors would describe areas propor- tional to the times; the centre of gravity either remain- ing at rest or moving uniformly in a right line. But if there are three bodies, the action of any one on the other two, changes the nature of their orbits, so that the determination of their motions becomes a problem of the greatest difficulty, distinguished by the name of The Problem of the Three Bodies. --- 30. The solution of the problem of the three bodies, in its utmost generality, is not within the power of the mathematical sciences, as they now exist. Under cer- tain limitations, however, and such as are quite con- sistent with the condition of the heavenly bodies, it admits of being resolved. These limitations are, that the force which one of these bodies exerts on the other two, is, either from the smallness of that body, or its great distance, very inconsiderable, in respect of the forces which these two exert on one another. 31. The force of the third body is called a disturb. ing force, and its effects in changing the places of the other two bodies are called the disturbances of the System. 32. Though the small disturbing forces may be more than one, or though there be a great number of remote disturbing bodies, the computation of their combined effect arises readily from knowing the effect of one; and therefore the problem of three bodies, under the conditions just stated, may be extended to any number. CHAPTER XVII. 224 33. The problem of the three bodies has exercised the ingenuity of several of the most eminent mathema- ticians. But Laplace, in the JMecanique Celeste, has extended the solution farther than any other person. He has given a very complete investigation of the ine- qualities, both of the planets and satellites. INEQUALITIES OF THE MOON. 34. The moon is attracted at the same time by both the earth and sun; it is only, however, the difference between the gravitations of the earth and moon to- wards the sun that disturbs the motion of the moon about the earth. If the sun were at an infinite distance, they would be attracted equally, and in parallel straight lines; and in that case their relative motions would not be in the least disturbed. But his distance, although very great in respect of that of the moon, yet can not be supposed infinite; the moon is alternately nearer the sun, and farther from him than the earth, and the straight line which joins her centre and that of the sun, forms with the terrestrial radius vector an angle which is continually varying; thus the sun acts une- qually and in different directions on the earth and moon, and from this diversity of action there must re- sult inequalities in her motion which depend on her position in respect of the sun. 35. At the quadratures, the gravity of the moon to the earth is increased in consequence of the sun’s ac- tion, by a quantity equal to the product of the mass of the sun, by the distance of the moon from the earth, divided by the cube of the earth’s distance from the sun; at the syzigies it is diminished by twice this quantity; and the effect in the whole, is a diminution of the moon’s gravity, equal to the product of the sun’s 222 ASTRONOMY. mass by the moon’s mean distance from the earth, di- vided by twice the cube of the earth’s distance from the sun. And the value of the mean diminution is equal to a 358th part of the whole gravity of the moon to the earth. It is a well known proposition in Mechanics, that if AB and AD, Fig. 50, represent the quantities and directions of two forces acting on a point or body at A, and the parallelogram ABCD be completed, the diagonal AC will represent the quantity and di- rection of a single force which would produce the same effect as the two forces. The substitution of a single force as the equiva- lent of two others, is called the Composition of Forces. On the contrary if AC, represent the quantity and direction of a single force acting on a body at A, and any parallelogram ABCD is described about AC as a diagonal, the adjacent sides AB and AD will represent the quantities and directions of two forces that are just equivalent to the single force. The substitution of two forces, as the equivalent of a single force, is called the Resolution of Forces. Let ACBO, Fig. 51, represent the orbit of the moon which may in this investigation be considered, as coinciding with the plane of the ecliptic. Also let S be the sun, E the earth, M the place of the moon in her orbit, and AB, perpendicular to SE, the line of the quadratures. Let the line SE represent the force which the sun exerts on the earth at E or on the moon, when in quadratures, at A and Bº. Then, SM*: SEP: SE: †- the force with which the sun acts on the moon at M. In the line MS, produced if necessary, take MD = ; then MD repre- M2 sents the force which the sun exerts on the moon at M. Let the * Strictly speaking, as the quantity of matter in the earth is greater than that in the moon, the forces which the sun exerts on the earth and moon when at equal distances, are not equal. But the effects of those forces, in moving the bodies, are equal, and it is these effects, which is the subject under consideration. CHAPTER XVII. 223 force MD be resolved into the two MH and MG, one of which, MH is equal and parallel to ES. Then since the force MH is equal and parallel to ES, it will have no tendency to change the relative motions or positions of the earth and moon. The other force MG, will therefore represent, in quantity and direction, the whole effect of the sun’s action in disturbing the moon’s motion in her orbit. Let SM be produced to meet the diameter AB in N. Then because the angle ESN is very, small, being when greatest only about 7”, the line, SN may be considered equal to SE. Hence, MD SE* - SN’ (SM + MN)* SM2 SM2 SM2 _ SM* + 3SM × MN + 3 SM × MN* + MNº —- SM3 º But as MN is very small compared with SM, the two terms 3SM × MN”, and MN* may be omitted. Therefore, * +...+ x *s-sm is MN Or, SD = 3 MN. As the angle ESM is very small, and SD is also small, the line DG must very nearly coincide with SE, and consequently the point G with the point L. We may therefore consider ML as the force by which the sun disturbs the motion of the moon. Now, MD = Y EL + LS = ES = HM = DG =SD +LS, very nearly, or, EL = SD, very nearly. Hence, if MK be perpendicular to SE, we have, EL = 3 MN = 3 ER. Let the force of ML be resolved into two others, one MQ, in the direction of the radius vector, and the other MP in the direc- tion of a tangent to the orbit at M. Then the force MQ increases or diminishes the gravity of the moon to the earth, according as the point Q falls between E and M, or in EM produced. The other force MP increases or diminishes the moon’s angular mo- 224 ASTRONOMY. tion about the earth. Since the moon's orbit does not differ much from a circle, the angle QMP may be considered as a right an- gle. Put a = SE, r = EM, and a = the angle AEM. Then, EK = EM cos MEK -- r sin a EL = 3 ER = 3 r sima, PM = LQ = EL cos w = 3 r sin a cosa. =} sin 2 a., (App. 7). ~. Also, EQ = EL sin a = 3 r sin” a, MQ = EQ—EM = 3 r sin” a - r = —r (1 — 3 sin” wy. Or using the affirmative sign to denote an increase in the moon’sgravity to the earth, MQ = + r(1–3 sin” w). Now if m = the mass of the sun, then the force which the sun exerts on the earth may be expressed by #. Hence, Q, ES : PM : : * : the force PM. Therefore, (l, m PM m.8r sin 2a: 3my the force PM = º' TS T T2 .3 ~ 2. Sin 2 a. A. In like manner, the force MQ = #: * - +. (1 — 3 sin” a). B When the moon is in quadratures, a = 0 or 180°, and conse- quently, The force MQ = + 1. (l, When the moon is in syzigies, a = 90° or 270°, and, therefore, The foree MQ =–3. (l, The force MQ is - 0, when 3 sin” a = 1, or sin a = v 4; that is, when a = 35° 15'52". The moon’s gravity to the earth is therefore increased while she is within about 35° of the quadratures, on either side, and is CIHAPTER XVIf, 225 s diminished in all the remaining part of the orbit; and the greatest diminution is double the greatest increase. It follows therefore that in the whole, the moon's gravity to the earth is diminished by the action of the sun. A short fluxional investigation proves that . . . . . 7)??" & & & ſº \ the mean diminution is glº 3" representing in this case the mean (l, distance of the moon from the earth. And it has been found that the value of this expression is equal to the 358th part of the whole gravity of the moon to the earth. 36. From the diminution of her gravity by a 358th part, the moon describes her orbit at a greater distance from the earth, with a less angular velocity, and in a longer time, than if she were acted on, only by the attraction of the earth. 37. The inequality in the moon’s motion, called the Jìnnual Equation, (10.29), is the effect of the varia- tion in the distance of the earth from the sun. º te wº 7) º gº e & Since, in the expression +. which designates the mean di- Cl minution in the moon's gravity to the earth, the quantities m and r, are constant, it follows, that the mean diminution is inversely pro- portional to the cube of the earth's distance from the sun. Hence as the earth approaches the perihelion, its distance diminishing, the mean diminution of the moon’s gravity to the earth must increase; the moon’s distance from the earth must become greater than it otherwise would be; and consequently its motion must be slower. The contrary takes place as the earth approaches the aphelion. 38. The Evection is produced by an inequality in the sun’s disturbing force, depending on the variation in the moon’s distance from the earth, and on the posi- tion of the moon with respect to the line of the Syzigies. Let R and r denote the distances of the moon from the earth, 30 226 ASTRONOMY. in apogee and perigee, when the line of the apsides coincides with the line of the syzigies, X and w, the distances at which the moon would be from the earth, in apogee and perigee, if she was not acted on by the sum, and G and g the perigean and apogean gra- vities in that case. Also put n = +: and supposing the earth's (l, distance from the sun to remain constant, n will be constant. Then (35), G–2rn and g – 2Rn, will be the perigean and apogean gravities of the moon, when the line of the apsides coincides with the line of the syzigies. Hence, X* : * : - G :g, and Rº: *:: G–2Fn : g – 2Rn. Consequently, ^- X* G -ā- - —? 3. 3. and Rí G– *. r” g — 2Rn 0. Now as G is greater than g, and 2rn, less than 2Rn, it is evi- dent that, ~, G–2rn g — 2Rn is greater than G. § 2 2 Hence, R* is greater than X. r2 3, 2 It therefore follows, that when the line of the apsides coincides with the line of the syzigies, the ratio of the apogean distance of the moon to the perigean distance, and consequently the eccen- tricity of the orbit, is increased by the action of the sum. In like manner it may be shown that when the line of the apsides coincides with the line of the quadratures, the sun’s action di- minishes the eccentricity of the orbit. The change in the ec- centricity of the orbit produces a change in the equation of the centre; which change is the evection. 39. The Variation is produced by a part of the sun’s disturbing force, which acts in the direction of a tan- gent to the moon’s orbit. CHAPTEIR XVII. 227 It has been shown (34.A), that MP, the part of the sun's force which acts in the direction of a tangent to the orbit, is equal to #. sin 2a. Hence, supposing the earth’s distance from the sum, and moon’s distance from the earth to remain constant, this force is proportional to sin 2w; that is, to the sine of twice the distance of the moon from the quadratures. It is therefore great- est in the octants; and nothing in the syzigies and quadratures. Supposing the moon to set out from the quadrature A, the tan- gential force MP continually accelerates her motion, till she ar- rives at the syzigy C; the force then changes its direction and retards her motion. Consequently at C the motion is greatest. As the moon advances from C, her motion is continually re- tarded till she arrives at B, where it is heast. It is then accele- rated till it becomes greatest at 0, and again retarded till it be- comes least at A. Hence, as the motion is greatest in the syzigies and least in the quadratures, and as the degree of retardation is the same as that of acceleration, we may infer that the mean motion* has place when the moon is in the octants. Now as the moon moves from the quadrature A with a motion less than her mean motion, her mean place will be in advance of her true place, and will become more and more so, till at the oc- tant, the true motion is equal to the mean. The difference be- tween the true and mean places is then the greatest. For after that, the true motion being greater than the mean, the true place will approach nearer to the mean, till at the syzigy C, they coincide. It is equally plain, that at the octant between C and B, the moon’s true place will be most in advance of the mean place, and that at B, they will again coincide. Corresponding effects take place in the two remaining quadrants, 40. The inequality called the Jicceleration of the JMoon (10.21), by which her velocity appears subject to continual increase, and her period to continual dimi- *The expressions, mean place, true place, mean motion and true motion, are here to be understood, only in relation to the present inequality. 228 ASTRONOMY, mution, has been found by Laplace to be a Secular equation, depending on a change in the eccentricity of the earth’s orbit, produced by the actions of the planets, and which requires several thousand years to go through its different values. MoTION OF THE APSIDEs of THE Moon’s or BIT. 41. The motion of the apsides is produced by the action of the sun, in diminishing the moon’s gravity to the earth. If the moon was only acted on by the earth’s attraction, it would describe an ellipse, and its angular motion would be just 180°, from one apsis to the other; or which is the same, from one place where the orbit cuts the radius vector at right angles, to the other. But in consequence of the change produced in the moon’s gravity to the earth, by the action of the sun, the moon’s path is not an ellipse. When the effect of the sun's action is a diminu- tion of the moon’s gravity, she will continually recede from the ellipse that would otherwise be described, her path will be less bent, and she must move through a greater distance before the radius vector intesects the path at right angles. She must there- fore move through a greater angular distance than 180°, in going from one apsis to the other, and consequently the apsides will ad- vance. On the contrary, when the gravity is increased by the sun's action, the moon’s path will fall within the ellipse which she would otherwise describe, its curvature will be increased, and the distance through which she must move before the radius vector in- tersects her path at right angles, will be less. The apsides will therefore move backwards. Now it has been shown (35) that the sum’s action, alternately diminishes and increases the moon’s gravity to the earth. The motion of the apsides will therefore be alternately direct and retrograde. But as the diminution has place during a much longer part of the moon’s revolution, and is besides greater than the increase, the direct motion will exceed the retrograde. Consequently in an entire revolution of the moon, the apsides have a progressive motion. CHAPTER XVIK. 229 MOTION OF THE Moon’s NoDES, AND CHANGE IN THE IN CLINATION OF THE ORBIT. 42. The direction, in which the sun’s disturbing force acts on the moon, does not, except in some par- ticular cases, coincide with the plane of the moon’s orbit; this force therefore produces a tendency in the moon to quit that plane, one of the effects of which, is a change in the position of the line of the nodes; and another, is a change in the inclination of the plane of the orbit to that of the ecliptic. Let OL, Fig. 52, be the line passing through the centres of the earth and sun, and IN’ the line of the modes. These two lines lie in the plane of the ecliptic, which we may consider as desig- mated by the plane of the paper. Let EMHI conceived to be, from EI, above the plane of the paper, be the plane of the moon’s orbit, NM a part of the northern half of the orbit, and AB a plane, seen edgewise, perpendicular to the line EL. When the moon is in this lattter place it is in quadrature. Let ML designate the quantity and direction of the sun's dis- turbing force when the moon is at M. Now when the line of the modes coincides with OL the line of the syzigies, ML will coincide with the plane of the moon’s orbit, and will therefore have no tendency to make the moon deviate from that plane. Also, since EL is equal to three times the distance of the moon from AB (34), when the moon is in the plane AB, that is when she is in quadrature, L will coincide with E, and consequently ML will be in the plane of the orbit, and will have no tendency to make the moon move from it. At all other times, the force ML, not acting in the plane of the orbit, will tend to make the moon quit that plane; or instead of supposing the moon continually to pass from one plane to another, we may conceive the plane it- self to change its position. Let LH be drawn perpendicular to IHME the plane of the moon's orbit. Then if MH be joined, and the parallelogram MHLK be completed, the lines MH and MK will represent, in t **, 230 ASTRONOMY. quantities and directions, two forces that are together equivalent to ML. The force MH acting in the plane of the orbit, has no tendency to change the position of that plane. The tendency of the other force MK, acting at right angles to the plane of the or- bit, will be to bend the moon's path towards the ecliptic, or from it. When the effect of the force, MK is to bend the moon’s path to- wards the ecliptic, the moon will meet the ecliptic sooner than it would otherwise do, and consequently the mode will move back- wards. On the contrary, when the force MK bends the moon’s path from the ecliptic, the moon will not meet the ecliptic so soon as it would otherwise do, and therefore the node will move for- ward. Now it is plain that when the points L and M are on the same side of the line of the modes, the force MK tends to make the moon’s path bend towards the ecliptic; and when they are on opposite sides, it tends to make the path bend from the ecliptic. Hence when the points L and M are on the same side of the line of the modes, the motion of the modes is retrograde; and when on opposite sides, it is direct. When the line of the nodes has the position NN’ the points L and M will be on the same side of it, while the moon is moving from the node N to the next quadrature in EB; and therefore the motion of the modes is retrograde. When the moon has passed the quadrature, the point L falls on the other side of E, in EO; and therefore while the moon is moving from the quadrature to the next mode in EN', the point, L and M will be on opposite sides of the line of the nodes, and the motion of the nodes will be direct. While the moon is moving from the mode in EN’ to the quad- rature in EA, the motion of the modes will be again retrograde; and while she is moving from the quadrature in EA, to the mode in EI, it will be direct. Hence, while the moon is moving from the nodes to the quadratures, the motion of the nodes is retro- grade; and while she is moving from the quadratures to the nodes, it is direct. It is therefore plaim that the retrograde motion has place during a longer portion of the moon's synodic revolution, than the direct motion. When the line of the modes has the position mn' it is easy to determine from what has been said, that the motion of the modes CHAPTER XVII. J 23ſ will be direct while the moon is moving from the modes to the quadratures; but retrograde while she is moving from the quad- ratures to the nodes; and therefore, that, in the whole synodic re- volution of the moon, the retrograde motion has place during a longer time than the direct motion. It appears then that in each synodic revolution of the moon, the nodes alternately retreat and advance, but that in all cases, except when the line of the nodes nearly coincides with the line of the syzigies, the motion is retrograde during a longer time than it is direct. Let the plane LIH be perpendicular to IN the line of the modes. Then the angle LIH is the inclination of the plane of the moon’s orbit to the ecliptic. As LH is perpendicular to the plane of the orbit, the angle IHL is a right angle. Put a = the moon’s angular distance from the quadratures, S = IEL the am- gle contained between the line of the modes and the line of the syzigies, I = LIH the inclination of the orbit, and r = EM = radius vector of the moon. Then (35), EL = 3r sin a. Hence, LI = EL sin LEI = 3r sin a sin S, MK = LH = LI sin LIH = 3r sin a sin S sin I, Or, (35), using # to denote the force exerted by the sun on the earth, m.MK 3mr The force MK = 3 — —- sin a sin S sin I. (0 (l, Now during any one revolution of the moon, none of the quan- tities which enter into the expression for the force MK, varies much, except sin a. And it is easy to perceive, by reference to the figure, that sin w, and consequently the force MK, always ac- quires its greatest value, during the time the motion of the nodes is retrograde. As in each synodic revolution of the moon, the modes retreat during a longer time than they advance, and as the force which causes the motion is greatest while they retreat, the retrograde motion must exceed the direct motion, and the result in the whole must be a retrograde movement of the modes. 232 ASTRONOMY. When the tendency of the force MK is to bend the moon's path towards the ecliptic, if the moon is then moving from the node to the 90° from it, the inclination of Nhe orbit will be di- minished; but if she is moving from the 90° to the mode, the in- clination will be increased. On the contrary, when the tendency of the disturbing force is to bend the path from the ecliptic, the inclination of the orbit will be increased when the moon is moving in the first 90° from the node, and will be diminished when she is moving from the 90° to the node. Hence when NN’ is...the line of the nodes, if the moon set out from the quadrature in EA, the inclination of the orbit will be continually diminished till she is 90° past the node N; and will then be increased till she arrives at the quadrature in EB; from thence to the 90° past the mode in EN', the inclination will be again diminished, and will then be increased till she again arrives at the quadrature in EA. The diminution will therefore be greater than the increase. But when mn" is the line of the nodes, if the moon set out from the quadra- ture in EA, the inclination will only be diminished till she ar- rives at the 90° from the node in En', and will be increased from thence to the quadrature in EB; it will then be diminished, till she is 90° from the mode in En, and will be increased from thence till she returns to the quadrature in EA. Thé increase will therefore exceed the diminution. Thus in some synodic revolu- tions of the moon the inclination of the orbit is diminished, and in others it is increased as much. The result is a mean inclina- tion which does not change. 43. Disturbances in the motions of the earth and planets are necessary effects of the actions of these bodies on one another; but it is not designed to take any other notice of them here, than to mention one impor- tant fact. 44. Lagrange and Laplace have proved that no terms only those which alternately increase and di- minish, can enter into the expressions for the disturb- ances of the planets. This proves that the system is CHAPTER XVII. 233 stable; that it does not involve any principle of de- struction in itself, but is calculated to endure for ever, unless the action of an external power is intro- duced. FIGURE OF THE EARTH. 45. It has already been inferred from observation (4.8) that the figure of the earth is an oblate spheroid, of which the greater axis, that is, the diameter of the equator is to the less, the axis of revolution as 321 to 320. 46. Since the earth revolves on its axis, it is evi- dent, that its parts are all under the influence of a cen. trifugal force, varying with their distances from that axis, and that if the whole were a fluid mass, the columns towards the equator, being composed of parts that, having a greater centrifugal force, tend more to recede from the axis, must extend in length, in order to balance the columns in the direction of the axis. By this means an oblateness or elevation at the equator would be produced, similar, in some degree at least, to that which the earth has been found to possess. 47. A homogeneous fluid of the same mean density with the earth, and revolving on its axis in the same time that the earth does, would be in equilibrium, if it had the figure of an oblate spheroid, of which the axis was to the equatorial diameter as 229 to 230. 48. If the fluid mass, supposed to revolve on its axis, be not homogeneous, but he composed of strata. that increase in density towards the centre; the solid of equilibrium will still be an elliptic spheroid, but of less oblateness than if it were homogeneous. 49 Hence as the ellipticity of the earth is less than 2}r, being about air, it is evident, that if the earth is 34 234 ASTRONOMY. a spheroid of equilibrium, it is denser toward the in- terior. 50. The greater density of the earth towards the centre has been proved by very accurate observations made on the sides of the mountain Schehallier, in Scotland, by Dr. Maskelyne. From the effect of the mountain in changing the direction of a plumb line suspended near it, and from the known figure and bulk of the mountain determined by a survey, it was found that the mean density of the mountain was to the mean density of the earth nearly as 5 to 9. 54. The inequalities on the surface of the earth, and the unequal distribution of the rocks which compose it, with respect to density, must produce great local ir- regularities in the direction of the plumb line, and are probably in part the causes of the inequalities observed in the measurement of contiguous arches of the meri- dian, even when the work has been conducted with the greatest skill and accuracy. These irregularities are so considerable that the ellipticity of the spheroid which agrees best with the measurement of some de- grees, is nearly double what may be accounted the mean ellipticity. 52. From accurate observations of the lengths of pendulums oscillating seconds at places in different latitudes, the relative force of gravity at the places may be determined and from thence the ellipticity of the earth. PRECESSION OF THE EQUINOXES AND NUTATION OF THE EARTH's AXIS. * 53. The physical investigation of the precession of the equinoxes is a subject of considerable difficulty. It CHAPTER XVII. 235 must suffice here, just to state that the precession is produced by the actions of the sun and moon on those parts of the earth which are on the outside of a sphere, conceived to be described about the earth’s axis. 54. The sun’s action produces a retrograde move- ment of the equinoctial points, which is nearly, but not quite uniform. This movement may be separated into two parts; one a continued mean precession of the equinoxes; the other an inequality in the precession called the Solar JWutation in precession. The ine- quality in the sun’s action occasions a very small change in the obliquity of the ecliptic, called the Solar JWutation in the obliquity. 55. The moon’s action produces effects similar to those produced by the sun, only greater in degree. One effect is a mean precession of the equinoxes, which combined with the mean precession produced by the sun, forms the whole JMean Precession of the Equinoaces. Another effect is an inequality in the precession called the Lunar JWutation, and sometimes, the Equation of the Equinoaces. SECULAR WARIATION OF THE OBLIQUITY OF THE ECLIPTIC. 56. The orbits of the planets not coinciding with the plane of the ecliptic, their actions on the earth tend to make it quit that plane. The effect is, a small va- riation in the position of the plane of the ecliptic. From these causes, the obliquity of the ecliptic has been, and still continues to be, diminished. The di- minution at the present period is about 52" in a cen- tury. In process of time the same causes must pro- duce an increase in the obliquity. 57. The secular variation of the obliquity of the 236 ASTRONOMY, ecliptic was less in former ages than it is at present. It has now acquired nearly its greatest value, and will begin to decrease about the 23d century of our era. Lagrange has shown that the total diminution in the obliquity, reckoning, from that in 1700, must be less than 5°23'. DIU RNAL ROTATION. 58. It is proved by minute investigation that the ac- tions of the sun and moon, combined with the change in the position of the ecliptic, must produce changes in the duration of a revolution of the earth on its axis, that is, in the length of the day. But the same inves- tigation also proves that those changes are so indefi- nitely minute, that, being periodical, they can never become sensible, even to the nicest observation. 69. When, from the washing of rains, or from other causes, any matter is made to descend from the higher parts of mountains to a position that is nearer the earth’s axis, its velocity will be diminished, and the velocity lost, being communicated to the mass, must tend to accelerate the diurnal motion. But no changes known to us, in the position of the matter of the earth, can ever produce any sensible alteration in the earth’s rotation on its axis. 60. The conclusion drawn from a full examination of the subject is, that the duration of the earth’s rota- tion may be regarded as perfectly unchangeable. OF THE TIDES. 61. The alternate rise and fall of the surface of the ocean, twice in the course of a lunar day, or of 24 h. .# CHAPTER XVII. 237 30 m. 48, sec. of mean solar time, is the phenomenon known by the name of the Tides. 62. The time from one high water to the next, is, at a mean, 12 h. 25 m. 24 sec. The instant of low water is nearly, but not exactly, in the middle of this interval; the tide, in general, taking nine or ten mi- nutes more in ebbing than in flowing. 63. The time of high water is principally regulated by the position of the moon, and in general, in the open sea, is from two to three hours after that body has passed the meridian, either above or below the horizon. But on the shores of the larger continents, and where there are shallows and obstructions to the motion of the water, the interval between the time of the moon’s passage of the meridian, and the time of high water, is very different at different places. The difference is so great, that at many places the time of . high water seems to precede the moon’s passage. For any given place, the time of high water is al- ways nearly at the same distance from that of the moon’s passage over the meridian. 64. Though the tides seem to be chiefly regulated by the moon, they appear also, in some degree to be under the influence of the sun. Thus, at the syzigies, when the sun and moon are on the meridian together, supposing other circumstances to be the same, the tides are the highest; at the quadratures, when the sun and moon are 90° distant, the tides are the least. 65. The tides about the time of the syzigies are called the Spring Tides; and those about the time of the quadratures, are called the JWeap Tides. 66. The highest of the spring tides or the lowest of the neap tides, is not the tide that has place nearest 238 ASTRONOMY. the syzigy or quadrature, but is in general the third, and in some cases, the fourth following tide. At Brest, in France, the tides of the syzigies rise to the height of 19.317 feet, and those of the quadratures only to 9.451 feet; which is not quite half the former quantity. In the Pacific Ocean, the rise in the first case is 5 feet, and in the second, between 2 and 2.5 feet. \ 67. The height of the tide changes with a change in the moon’s distance from the earth. Other circum- stances being the same, the tide is highest when the moon is in perigee, and the least when she is in apogee. The tides also depend on the sun's distance from the earth but in a less degree, than on that of the moon. In our winter the spring tides are greater than in the ... summer, and the neap tides smaller. 68. The tides depend, to some extent, on the posi- tions of the sun and moon with respect to the equator. When the moon is in the northern signs, the tide of the day, in all the northern latitudes, is somewhat greater than that of the night. The contrary has place when the moon is in the southern signs. 69. If the tides be considered relatively to the whole earth, and to the open sea, there is a meridian, about 30° eastward of the moon, where it is always high water, both in the hemisphere where the moon is, and in the opposite one. On the west side of this meri- dian the tide is flowing, on the east, it is ebbing; and on the meridian at right angles to the same, it is low water. In consequence of the earth's diurnal rota- tion, these meridians move westward; but they pre- serve nearly the same distance from the moon, only ap- CIIAPTER XVII. 239 proaching a little nearer to her at the syzigies, and going farther off at the quadratures. The great Wave which, in this manner, constitutes the tide, is an undulation in the waters of the ocean, in which there is very little progressive motion, except when it passes over shallows, or approaches the shores. 70. The facts, which have been enumerated, clearly indicate that the tides are produced by the actions of the sun and moon; but in a geater degree by that of the moon. It has been shown (35) that the sun's action, increases or di- minishes the moon’s gravity to the earth, according to her posi- tion with respect to the line of the syzigies, or of the quadra- tures. In like manner, the Sun's action increases or diminishes the gravity of a particle of matter at the earth’s surface, ac- cording to its position with respect to a plane passing through the centre of the earth, at right angles to the line joining the centres of the earth and sun. Within about 35 of this plane on each side, the gravity at the surface is increased; and at the remain- ing parts, that is for about 55° around the points in which the line of the centres intersects the surface, the gravity is di- minished. Now as the particles of water easily yield to any impression, the surface of the ocean will, in consequence of the change in the gravity of its different parts, assume a figure different from that which it would otherwise have. Around the points in which the line of the centres intersects the surface, the gravity being di- minished, the surface will be at a greater distance from the cen- tre; and in the middle parts between these points, the gravity being increased, the surface will be nearer the centre. In con- sequence of the earth's diurnal rotation, it will successively be different parts of the surface, that will thus have the distance from the centre increased and diminished. From what has been said it is easy to perceive that so far as it depends on the sun's 240 ASTRONOMY. action, it is high water at the same time in opposite parts of the earth; and that the consecutive high waters must follow each other at intervals of half a solar day. The moon produces effects exactly similar to those of the sum, but much greater in degree, and succeeding one another at in- tervals of half a lunar day. 71. At the time of the syzigies the actions of the sun and moon are combined in producing the tides; but at the quadratures they act in opposition to each other. The result is, much greater tides at the syzigies than at the quadratures. Observations have made known that the former are to the latter, nearly as 2 to 4. Consequently the effect of the moon’s action must be to that of the sun, nearly as 3 to 1. 72. The relative effects of the actions of the sun and moon in producing the tides, must depend on their dis- tances and masses; and as their distances and relative effects are known, their relative masses may from thence be determined. 73. Great extent is necessary, in order that the sea should be sensibly affected by the actions of the sun and moon; for it is only by the inequality of that action, on different parts of the mass of waters, that their equilibrium is disturbed; and this inequality can not sensibly have place, unless a great extent of water be included. 74. The tides which are experienced in narrow seas, and on shores far from the main body of the ocean, are not produced in those seas by the direct actions of the sun and moon, but are waves propagated by the great diurnal undulation. APPENDIX TO PART I. Containing Trigonometrical Formulae; and Two Propositions in Conic Sections. Many of the Trigonometrical Formulae included in the follow- ing collection, are used in the present work. They are intro- duced here and numbered in order to facilitate the references. Their demonstrations may be seen in any complete treatise on Trigonometry. Nearly all of them are contained and demon- strated in a good work on the subject by Laeroix, which has been translated and published at Cambridge, New England. Those which are not contained in that work, are easily de- duced from others that are. For a single arc or angle q, the radius being = 1. 1. sin” a + cos” a = 1 7. Sin a = 2 sin # a cos 3 a. 2. sin a = tan a cos a 8. CoS a = 1 — 2 sin” # a e tan a 9. COs a = 2 cos” # a - I 3. sin a = — * vº 1 + tan” a 10. tan ; a = | * * 4 1 1 + cos a e C0S 0 = — e ,--— SIFT. C. VTT tan a 11. cot 3 g = 4 tº * sin a 1 — cos a 5. tan a = 1 — cos & COS & 12. tan” # a = −: 1 COS (º, 1 + COs a 6. Cot a = - d tam d. sin a For two arcs a and b, of which a is supposed to be the greater. 32 242 A PPENDIX. TO PART ¿ 13 14. 15. y 16. 7. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30, 3l. 32. sin (a + b) = sin a cos b = cos a sin b cos (a + b) = cos a cos b =F sin a sin b tan a + tan b tam (a + b) = º- ( ) 1 - tan a tan b sin a cos b = 3 sin (a + b) + 3 sin (a —b) cos a sin b = 3 sin (a + b) — 3 sin (a —b) sin a sin b = 3 cos (a —b)— 3 cos (a + b) cos a cos b = 3 cos (a —b) + 3 cos (a + b) sin a -- sin b = 2 sin 3 (a + b) cos 3 (a — b) stn a — sin b = 2 cos 3 (a + b) sin 3 (a —b) cos b -- cos a = 2 cos 3 (a + b) cos 3 (a — b) cos b— cos a = 2 sin 3 (a + b) sin 3 (a — b) tau a + tan b = sin (a + b) cos a cos b tan a — tan b = sin (a = 0) cos a cos b cot b + cot a = sin (a + b) sin a sin b cot b — cot a = sin (a=b) sin a sin b tan 3 (a + b) = sin a + sin b cos a + cos b tan 3 (a— b) = º a — sin b cos a + cos b co (a + b) — º" º =º º cos b — cos a sin a + sin b cot 3 (a — b) = 3 ( ) cos b— cos a tan 3 (a + b) sin a l- sin b tan 3 (a —b) sin a — sin b For a Spherical Triangle, in which A, B, and Care the am- gles, and a, b, and c, the Opposite sides, as in Fig. 53. 33. 34. 35. sin A sin b = sin B sin a cos a = cos A sin b sin c + eos b cos c cos A = cos a sin B sin C— cos B cos C APPENDIX TO PART I. 243. cot A sin B -- cos B cos c 36. cot a = e Sl]] C 37. cot A = cot a sin b — cos C cos b f o - sin C 38. Sin A – vºn 3 (4 + º-º) in 3 (“H 6- 9) sin b sin c 39. tan # (b -- a) = tan # cos 3 (B — A) # (b -- a) ***T*ETA, sin 3 (B — A) sin à (B + A) 40. tan # (b — a) = tan # c 41. tan 3 (B + A) = cot 3 C cos # (b — a) cos . (b -- a) 42. tan 3 (B — A) = cot & c sin # (b — a) \ sin # (b -- a) cot & C = tan : (B – A "* (* + 2 . sin à (b — a) 43. cot # C = tan # (B A) cos 3 (b. 4-a) * 2 (B + ):#; tan 3 c = tan # (b — a) sin * (B -- A) 44. SIIl # º tam # c = tan b cos # + all # (b + a) cos # (B — A) For a right angled spherical triangle in which C is the right angle, and the opposite side c, the hypothenuse, as in Fig. 54. 45. cos c = cos a cos b 48. tan a = sin b tan A 46. cos c = cot A cot B 49. tan a = cos B tan c 47. Sin a = sin c sin A : 50. COs A = sin B cos a 51. If ADBL Fig. 55, be an Ellipse, AB the transverse aris, E and F the foci, C the centre, and D a point in the curve, then, AC2 — EC2 AC — EC cos AED" Let DH be perpendicular to AB. Then, ED* = DH2 + EH = DH2 + (EC + CH). ED = 244 APPENDIX TO PART I. = DH2 - EC2 -- 2 EC x CH + CH*, and FD = DH* - FH2 → DH* + (EC–CH)* BH2 + EC2 — 2EC x CH + CH*. Hence by subtraction, - ED2 — FD2 = 4EC x CH. But ED2–FD* = (ED + FD) × (ED —-FD) == 2AC × (ED — FD). Therefore, 2AC x (ED — FD) = 4 EC × CH, 2EC x CH — FD = −t. Or, ED D AC But ED -- FD = 2AC. Hence by addition, 2EC x CH ED = 2AC + “º EC x CH ED = A mºsºmsºmº-º-º-ºsmºsºmºsº wº sº gº A C + ºxº (A) ED X AC = AC2 + EC x CH = AC” + EC x (EH —EC) = AC” + EC x (ED cos AED —EC) = AC” -- EC x ED cos AED — EC*, ED X AC — ED x EC cos AED = AC” – EC” ED x (AC — EC cos AED) = AC” — EC” ED = AC” – EC*_. - AC — EC cos AED * 52. If the circle AGBM, Fig. 55, be described about AB, as a diameter, and HG be produced to meet it in G, then, l ED = AC — EC cos BCG, From the preceding demonstration, we have (A), ED = AC + EC_x CH AC = AC + EC × CG COS ACG AC = AC + EC x AC cos ACG. AC = AC + EC cos ÁCG = AC — EC cos BCG. AN ELEMENTARY TREATISE ON ASTRONOMY. BART II. CATALOGUE of the Tables with observations respecting some of them. TABLE I. Latitudes, and Longitudes from the Meridian of Greenwich, of some Cities and other conspicuous places. \, TABLE II. A Mean Astronomical Refractions. TABLE III. Mean Right Ascensions and Declinations of some of the Fixed Stars, for the beginning of 1820, with their Annual Wariations. t TABLE IV. Mean New Moons &c. in January. ' The time of mean new moon in January of each year has been diminished by 15 hours, which has been added to the equations in Table VII. Thus, 4 h. 20 m. has been added to the first equations; 10 h. 10 m. to the second; 10 minutes to the third; and 20 minutes to the fourth. By this means the equations for finding the approximate time of new or full moon, are all made additive. 246 ASTRONOMY. TABLES V, VI, and VII. These tables are used with the preceding one, in finding, nearly, the true time of new or full moon. TABLE VIII. Mean Longitudes and Latitudes of some of the Fixed Stars, for the beginning of 1810, with their Annual Variations. TABLE IX. Sun's Mean Longitude, the Longitude of the Perigee, and Arguments for finding some of the small cquations of the sun’s place. They are all given ſor mean noon at the meridian of Grecnwich, on the first of January for common years, and on the second of January for bissextiles. The sun’s longitudes and the longitudes of his perigee have, each, been diminished by 2” As each is diminished by the same quantity, the mean anomaly, which is obtained by subtracting the longitude of the perigee, from the sun's longitude, and which is the argument for the equa- tion of the centre, is not affected. The Argument I, is for the equation depending on the action of the moon; Argument II, is for that depending on the action of Jupiter; Argument III, is for that depending on the action of Venus; and Argument N, is for the Nutation, or equation of the equinoxes. Of the 2° which has been subtracted from the sum’s mean lon- gitudes, 1° 59' 30" is added to the equation of the centre, and 10" to each of the small equations due to the actions of the Moon, Jupiter and Venus. TABLE X. Motions of the Sun and Perigee and change in the arguments, for Months. TABLES XI and XII. Sun's Hourly Motion and Semidiameter. These two tables would, in order, come after table XVIII, but are put in the ASTRONOMY. 24/7 place which they occupy with a view to convenient arrangement on the pages. TABLES XIII and XIV. Sun's Motions for Days, Hours, Minutes and Seconds. TABLE XV. Equation of the Sun’s Centre. TABLE XVI. Small equations of Sun's Longitude. TABLE XVII. Mean Obliquity of the Ecliptic for the beginning of each year contained in the table. TABLE XVIII. Nutation in Longitude, Right Ascension and Obliquity of the Ecliptic. TABLE XIX. Equation of Time, to convert Apparent into JMean Time. TABLE XX. Epochs of the Moon's Mean Longitude and of the Arguments for finding the Equations which are necessary in determining the True Longitude and Latitude of the Moon. They are all given for mean moon at the meridian of Greenwich, on the first of January for common years, and on the second of January, for bissextiles. The Argument for the Evection is diminished by 29', the Anomaly by 1° 59', the Argument for the Variation by 8° 59', the Mean Longitude by 9° 44'; and the Supplement of the Node is increased by 7'. This is done to balance the quantities which are applied to the different equations to render them aſ- * firmative, 248 ASTRONOMY. TABLES XXI to XLII, inclusive. These tables together with table XX, are for finding the Moon's True Longitude, Latitude and Equatorial Parallax. / TABLE XLIII. Reductions of Parallax and of the Latitude of a Place. The reduction of parallax is for obtaining the parallax at any given place from the equatorial parallax. The reduction of latitude is for reducing the true latitude of a place as determined by obser- vation, to the corresponding latitude on the supposition of the earth being a sphere. The ellipticity to which the numbers in the table corresponds is ###. This differs a little from what is believed to be the most accurate determinations of the ellipticity, which make it from ### to ###. But the difference is too small to be regarded unless its value was known to a greater de- gree of precision. A TABLES XLIV and XLV. Moon's Semidiameter and the augmentation of the semidiame- ier depending on the altitude. TABLES XLVI to LIV, inclusive. Moon's Hourly Motions in Longitude and Latitude. TABLE LV. Contains 11 pages of the Nautical Almanac, taken from the month of August for that of 1821. TABLE LVI. Second differences. This table is useful for finding from the Nautical Almanac, the Moon's longitude or latitude for any time between moon and midnight. TABLE LVII. Logistical Logarithms. This table is convenient in working ASTRONOMY. ºf 249 proportions, when the terms are minutes and seconds, or degrees and minutes; or hours and minutes. TABLE LVIII. Change in Moon's Right Ascension from the Sun. This table serves to find the time of the moon's passage over the meri- dian of any given place, from the time of its passage, as given in the Nautical Almanac for the meridian of Greenwich. It is also convenient in a calculation for the rising or setting of the moon, to determine the correction of the semi-diurnal arc, which de- pends on the moon's change in right ascension from the sun. TABLE LIX. Change in Moon's Declination. This table is convenient in finding from the Nautical Almanac, the moon’s declination for any intermediate time between noon and midnight. TABLES LX, to LXIII, inclusive. These are tables, calculated by M. Gauss, for finding the Aber- ration and Nutation, of a Star, in Right Ascension and Declina- tion. TABLE LXIV. Semi-diurnal Arcs for the Latitude of 39°57' North. SCHOLIUM. The tables of the Sum, which are those from IX to XIX, inclusive, are abridged from Delambre's Solar Tables. And those of the Moom, which are from XX to LIV, inclusive, are abridged from Burckhardt's Lunar Tables. As some small equations, and also the tenths of seconds are omitted, all the quantities obtained from these tables will be liable to small errors, None of these errors will, however, exceed a few seconds. It may be proper here to inform the student, that when, in the following problems, he meets with the expressions, Sun's true longitude, Moon's true longitude, &c. he is to understand them as implying the true values of those quantities so far as they can be obtained from the tables used. 33 250 ASTRONOMY. Observations and Rules, relative to Quantities with different Signs. } IT is frequently convenient, in computations, to designate cer- tain quantities by the flífirmative sign + , perfixed; and others by 'the JNegative sign — . Those which have the affirmative sign prefixed, are called Positive or Affirmative quantities;" and those with the negative sign, prefixed, are called JNegative quantities. When a quantity is affirmative, the sign is frequently omitted; but when it is negative, the sign must always be used. To add quantities, having regard to their signs. When all the quantities have the same sign, add them as in common arithmetic, and prefix that sign to the sum. When the quantities have dif- ferent signs, add the affirmative quantities into one sum, and the megative into another. Then take the difference of these two sums and prefix the sign of the greater. These rules will be il- lustrated by the following examples. * Add 2' 11" Add – 3' 51" Add — 7' 14” ºf 2 — 4 10 + 8 2 3 4 — I 15 + 3 17 Sum 12 17 Sum — 9 16 Sum + 4 5 Add + 3' 15" Add-17' 10" Add + 3' 1" — 8 12 — 4 3 — 1 i5 — 5 1 + 12 4 + 4 18 + 2 17 + 18 59 — 6 4 Sum — 7 11 Sum + 9 50 Sum 0 0 To Subtract quantities, having regard to their signs. Suppose the sign of the quantity which is to be subtracted, to be changed; that is, if it is affirmative, suppose it to be negative; and if it is negative suppose it to be affirmative. Then proceed as in the above rule for adding quantities. Thus, *. ASTRONOMY. 251 From 5' 10" From 4' 11" From + 2' 5" Sub. 3 21 Sub. 7 27 Sub. — 1 11 Rem. 1 49 Rem. — 3 16 Rem. -- 3 16 Erom — 8' 29" From — 2' 18" From — 4' 17" Sub. — 3 2 Sub. — 7 11 . Sub. -- 6 21 Rem. — 5 27 Rem. -- 4 53 Rem. — 10 38 To find the Logarithmic Sine, Cosine, Tangent, or Cotangent of an arc, with its proper Sign, from Tables that extend only to each minute of the quadrant. 2. When the given arc does not exceed 180°. With the given arc, or when it exceeds 90°, with its supplement to 180°, take out from the table, the required, Sime, or Tangent, &c. When there are seconds, take out the quantity corresponding to the given de- grees and minutes; also take the difference between this quantity and the next following one, in the table. Then 60" : the odd seconds of the given are : ; the difference : a fourth term. This fourth term, added to the quantity taken out, when it is in- creasing, but subtracted, when it is decreasing, will give the re- quired quantity. When the given arc exceeds 180°. Subtract 180° from it, and proceed as before. When the arc exceeds 270°, it is more con- venient, and amounts to the same, to subtract it from 360°. To determine the Sign of the quantity. Call the arc from 0° to 90°, the first quadrant; from 90° to 180°, the second quadrant; from 180° to 270°, the third quadrant; and from 270° to 360°, the fourth quadrant. Then, The Sine of the arc is affirmative for the first and second quad- rants; and negative, for the third and fourth. The Cosine, is affirmative for the first and fourth quadrants; and negative, for the second and third. t The Tangent and Cotangent, are affirmative for the first and third quadrants; and negative, for the second and fourth. By attending to the preceding rules, the student will easily find the Sime, Cosine, &c. of an arc, in either quadrant, with its appropriate sign, as exemplified in the following table. 252 ASTRONOMY. Arc Sine Cosine Tangent otang. 37: 18, 21” -- 9.78252 + 9.90060 + 9.88193 + 10.11807 114 35 10 + 9.95872 – 9.6.1916 — 10.33956 – 9.66044 247 12 36 – 9.96470 — 9.58811 -- 10.37659 + 9.62341 314 1z 50 – 9.85475 + 9.84409 — 10.01065 — 9.98935 Note. The signs are seldom placed before affirmative loga- rithms; but they must not be omitted before negative ones. The Logarithmic Sine, Cosine, Tangent, or Cotangent of an are being given, to find the arc. When the given quantity can be found in the table, under or over its name, take out the corresponding arc. When the given quantity is not found exactly in the table, and the arc is required to seconds, take out the degrees and minutes correspºnding to the next less quantity, when that quantity is increasing; but to the next greater when it is decreasing. Take the difference between the quantity corresponding to the degrees taken out, and the next following one in the table; also take the difference between the same quantity and the given one. Then, the first difference: the second :: 60" : the number of seconds which is to be annexed to the degrees and minutes. Then, * For a Sine. When it is affirmative, the required arc will be, either the arc found in the table, or its supplement to 180°. When the sine is negative, the required arc will be, either the arc found in the table, increased by 180°, or its supplement to 360°. For a Cosine. When it is affirmative, the required arc will be, either the arc found in the table, or its supplement to 360°. When the cosine is negative, the required arc will be, either the supplement of the arc found in the table, to 180°, or that arc, in- creased by 180°. For a Tangent or Cotangent. When it is affirmative, the re- quired arc will be, either the arc found in the table, or that arc, increased by 180°. When the tangent or cotangent is negative, the required arc will be, either the supplement of the arc ſound in the table, to 180°, or its supplement to 360°. ASTRONOMY. 253 These rules are exemplified by the quantities in the following table. Sine + 9.78252, arc 37° 18' 21" or 142° 41' 39" Sine — 9.854.75 arc 225 42 10 or 314 17 50 Cosine + 9.90060 are 37 18 18 or 322 41 42 Cosine — 9.6.1916 arc 114 35 11 or 245 24 59 Tangent + 9.88193 arc 37 18 21 or 217 18 21 Tangent — 10,33956 arc 114 35 iſ or 294 35 11 Cotangent + 9.62341 arc 67 12 36 or 247 12 36 Cotangent — 9.98985 arc 134 17 51 or 314 17 51. Note. Tables which extend only to five decimals, will give the arc, for a tangent or cotangent, true to the nearest second, for a few degrees, near to 0°, 90°, 180°, or 270'; for a sine, near to 0° or 180°; and for a cosine, near to 90° or 270. In other cases they can not be depended on, to give the seconds accurately. They are, however, sufficient for many calculations; particularly, when the nature of the problem does not make it mecessary that the required arc or angle should be determined with great ac- curacy. As almost every mathematical student is furnished with a set of such tables, and as an example worked by them, will serve as well to illustrate a rule, as if worked by those which are more ex- tensive, they will be used, when necessary, in working the ex- amples and questions in the following problems. Observations, relative to the Signs of the Logarithms of Natural JNumbers. ** When the logarithm of a matural mumber is used in calcula- tion its sign is affirmative or negative, according to that of the number. When the natural number is a decimal, in order to avoid a dif- ficulty with respect to the sign, the arithmetical complement, of the index is used. Thus, when there is no cypher between the decimal point and first significant figure, the index is 9; when there is one cypher, the index is 8; when there are two cyphers, the index is 7; and so on. Thus, ! 254 ASTRONOMY. y The logarithm of .27 is 9,43186 of .027 is 8.431.36 of − .027 is — 8,431.36 of , .0027 is 7.4.3136 of— .0027 is — 7,43136 When, in order to get the product or quotient of quantities several logarithms, or logarithms and the arithmetical comple- ments of logarithms are added together, if they are all affirmative, or if there is an even number of negative ones, the resulting lot garithm will be affirmative; but if there is an odd number of ne- gative ones, the resulting logarithm will be negative. When the resulting logarithm of a calculation, is the logarithm of a natural number, the number will be affirmative or negative, according to the sign of the logarithm. | When in any of the calculations on the following problems, the resulting logarithm is the logarithm of a natural number, if the index is 9, or near to 9, as 8, 7, &c. the number will be a decimal. When the index is 9, there must be no cypher between the decimal point and first significant figure. When the index is 8, there must be one cypher; when the index is 7, there must be two cyphers; and so on. { *-ºs- IPROBLEMS FOR MARING WARIOUS AS., TRONOMICAL CALCULATIONS. PROBLEM I*. To work, by logistical logarithms, a proportion, the terms of which are minutes and seconds of a degree or of time, or hours and minutes. - With the minutes at the top and seconds at the side, or if a term consists of hours and minutes, with the hours at the top and * Perhaps in strict language, this and a few of the following problems are not properly called Astronomical. They are however for performing sub- sidiary operations, in astronomical calculations, \ ASTRONOMY, 255 minutes at the side, take from table LVII the logistical logarithms of the three given terms, and proceed in the usual manner of working a proportion by logarithms. The quantity, in the table, corresponding to the resulting logarithm will be the fourth term. Note 1. The logistical logarithm of 60' is 0. * 2. The student will easily perceive that proportions that are worked by logistical logarithms, may also be worked by the common rule in arithmetic. ExAM, 1. When the moon's hourly motion is 31' 57", wha is its motion in 39 m. 22 sec. 2 Ams. 20' 58". \ As 60 m. - tº gº 0 : 39 m. 22 sec - gº 1830 : : 31' 5’7” – – - 2737 : 20' 58” ' – & 4567 2. If the moon's declination change 2°29' in 12 hours, what will be the change in 8 h. 21 m. ſlns. 1° 44'. As 12 h. ſº sº – 6990 8 h. 21 m. - * * 85.65 ; : 2° 29' - tº - 13831 22396 1° 44' sº sºng 15406 3. When the sun’s hourly motion is 2' 31", what is its motion in 17 m. 18 sec. Jäms. 0' 44” 4. When the sun’s declimation changes 22' 14" in 24 hours, what is its change in 19 h. 25 m. ſins. 17' 59" PROBLEM II. / From a table in which quantities are given, for each Sign and Degree of the circle, to find the quantity corresponding to Signs, Degrees, JMinutes and Seconds. Take out, from the table, the quantity corresponding to the given signs and degrees; also take the difference between this 256 *. AstroNOMY. | quantity and the next following one. Then, 60' : odd minutes and seconds :: this difference : a fourth term. This fourth term added to the quantity taken out, when the quantities in the table are increasing; but subtracted, when they are decreasing, will give the required quantity. Note 1. When the quantities change but little from degree tº degree, the required quantity may frequently be estimated, with- out the trouble of making a proportion. Note 2. The given quantity with which a quantity is taken from a table, is called the flrgument. Note 3. In many tables, the argument is given in parts of the circle, supposed to be divided into a 100, a 1000, or 10000, &c. parts. The method of taking quantities from such tables is the same as is given in the above rule; except that when the argu- ment changes by 10, the first term of the proportion must be 10, and the second, the odd units; when the argument changes by 100, the first term must be 100, and the second, the odd parts between hundreds; and so on. ExAM. 1. Given the argument 1° 9° 31' 26", to find the corres- ponding quantity in table XXXII. ſins. 11° 13' 17". 1° 9° gives 11° 11' 15". The difference between 11° 11’ 15" and the next following quantity in the table is 5' 9". As 60' : 31' 26" : : 5' 9" : 2, 42".” To 11° 11’ 15” Add 2 42 11 13 57 2. Given the argument 10: 18° 16' 54", to find the corres. ponding quantity in table XXXV. ſins. 93° 32' 37". A 10° 13° gives 93° 33' 40". g w The difference between 98° 33' 40" and the next following quantity in the table, is 3'43". * The student can work the proportion, either by common arithmetic, or by logistical logarithms, as he may prefer. ASTRONOMY, 257 As 60' : 16' 54" : : 3' 43” : I’ 3”. From 93° 33' 40' Take 1 3 93 32 37 3. Given the argument 4, 11° 57' 10", to find the corres- ponding quantity in table XV. Ans, 3° 24′ 12". 4. Given the argument 3721, to find the corresponding quantity in table XXV. Ans. 4' 52" PROBLEM III. To convert Degrees, JMinutes and Seconds of the Equator into Time. - Multiply the quantiy by 4, and call the product of the seconds, thirds; of the minutes, seconds; and of the degrees, minutes. ExAM. I. Convert 72° 17' 42", into time. 72° 17' 42” 4 4 h. 49 m. 10 sec. 48". = 4 h. 49 m. 11 sec. nearly. 2. Convert 117° 12' 30", into time. Jins. 7 h. 48 m. 50 sec. 3. Convert 21° 52'27', into time. Ans. 1 h. 27 m. 30 sec. PROBLEM IV. / To convert Time, into Degrees, JMinutes ond Seconds. Reduce the time to minutes, or minutes and seconds; divide by 4, and call the quotient of the minutes, degrees; of the seconds, minutes; and multiply the remainder by 15, for the seconds. ExAM. 1. Convert 5 h. 41 m. 10 Sec. into Degrees, &c. h. m. Sec. 5 41 10 60 4)341 10 85° 17' 30" 34 258 ASTRONOMY. 2. Convert 'i h. 48 m. 50 sec. into Degrees, &c. ºffms. 11.7° 12' 30". 3. Convert 11 h. 17 m. 21 sec. into Degrees, &c. ſins. 169° 20' 15". PROBLEM W. The Longitude of two Places, and the Time at one of them being given, to find the corresponding Time at the other. Express the given time astronomically. Thus, when it is in the morning, add 12 hours, and diminish the number of the day, by a unit. When the given time is in the afternoon, it is already, in astronomical time. Find the difference of longitude of the two places, by sub- tracting the less longitude from the greater, when they are both of the same name, that is both east, or both west; but by adding the two longitudes together, when they are of different names. When one of the places is Greenwich, the longitude of the other, is the difference of longitude. Then, if the place, at which the time is required, is to the east of the other place, add the difference of longitude, in time, to the given time; but if it is to the west, subtract the difference of lon- gitude, from the given time. The sum or remainder is the re- quired time. Note. The longitudes of the places mentioned in the following examples, are given in table 1. ExAM. 1. When it is August 8th, 2 h. 12 m. 17 sec. A. M. at Greenwich, what is the time, as reckoned at Philadelphia? d. h. m. sec. Time at Grenwich, August, 7 14 12 17 Diff. of Long. wº tºº e 5 046 Time at Philadelphia, tº ºf 9 II 31 P. M. 2. When it is April 11th, 3 h. 15 m. 20 sec. P. M. at New York, what is the corresponding time at Greenwich? ASTRONOMY. 259 d. h.m. sec. Time at New York, April, 11 3 15 20 Differ. of Long. - tº ºs 4 56 4 *=s**** Time at Greenwich, 11 8 11 24 P. M. 3. When it is Sept. 10th. 3 h. 20 m. 35 sec. P. M. at Paris, what is the time as reckoned at New-Haven? h. m. Sec. Longitude of Paris, gº 0 9 21 E. do. of New-Haven, 451 52 W. Diff of Long. sº – 5 1 13 d. h. m. sec. Time at Paris, September, 10 3 20 35 Diff of Long. tºº * 5 1 13 Time at New-Haven, - 9 22 19 22 Or September 10th, 10 h. 19 m. 22 sec. A. M. 4. What it is January 15th, 9 h. 12 m. 10 sec. P. M. at Washington, what is the corresponding time at Berlin? ſins. Sept. 16th, 3 h. 13 m. 21 sec. A. M. 5. When it is Oct. 5th. 7 h. 8 m. A. M. at Quebec, what is the time at Richmond? fins. Oct. 5th. 6 h. 40 m. 47 sec. A. M. 6. When it is moon, of the 10th of June at Greenwich, what is the time at Philadelphia? Ans. June 10th. 6 h. 59 m. 14 sec. A. M. PROBLEM WI. • The Apparent Time being given, to find the corresponding Mean Time; or the Mean Time being given, to find the Ap- parent. When the given time is not for the meridian of Greenwich, re- duce it to that meridian by the last problem. Them, from the tables take out the sun’s Mean Longitude corresponding to this time, Thus, from table IX, take the longitude, corresponding to the given year; and from tables X, XIII, and XIV, take the 260 * ASTRONOMY. motions in longitude, for the months, days, &c. The sum, re- jecting 12 signs, when it exceeds that quantity, will be the Sun's Mean Longitude as given by the tables. With the Sun’s Mean Longitude, thus found, take the Equation of Time from table XIX. Then, when Apparent Time is given, apply the equation with the Sign it has in the table; but when Mean Time is given, apply it with a contrary Sign; the result will be the Mean or Apparent Time, required. Note 1. In taking the sun’s longitude from the tables, it is not necessary to regard the seconds in the given time. Note 2. The Sun's mean longitude, found from the tables in this work, is always two degrees less than its true value; but this difference is allowed for, in arranging the numbers in table XIX. Note 3. The Equation of Time is given in the Nautical Almanac for each day in the year, at apparent moon, on the me- ridian of Greenwich, and can casily be ſound for any intermediate time by proportion. When Apparent Time is given to find Mean, the equation is to be applied according to its Title; but when Mean, is given, to find Apparent, it must be applied, con- trary to its Title. The Equation is given on the second page of each month. See the second page of table LV. ExAM. 1. On the 15th of August, 1821, when it is 8 h. 15 m. 12 sec. A. M. mean time at Philadelphia, what is the ap- parent time at the same place? d. h. m. sec. Time at Philadelphia, August 1821, 1420 15 12 Diff of Long. : º gº ſº - 5 0 46 Time at Greenwich - gº - 15 1 15 58 M. Long. 1821 sº - 9s 8° 48' 19'' August º 6 28 57 26 15 d. dº * 13 47 57 1 h. - tº &\s 2 28 16 m. - gº 39 Mean-Long - 4 21 36 49 AsTRONOMY. 264 *~ The equation of time in table XIX, corresponding to 4s 21° 36'49" is + 4 m. 13 sec. - d. h. m. Sec. Mean Time at Philadelphia, August 1821, 14 2015 12 Equation of time, sign changed º — 4 13 Apparant time - - - - 14 2010 59 Or, August 1821, 15 d. 8 h. 10 m. 59 sec. A. M. 2. On the 18th of October, 1821, when it is 3h. 21 m. 17 sec. P. M. apparent time, at Philadelphia, what is the mean time at Greenwich? d. h. m. sec. Time at Philadelphia, October 1821, 18 3 21 17 Diff. of Long. gº tºmº º 5 0 46 Time at Greenwich tºº gº 18 8 22 3 M. Long. 1821 - 9s 8° 48' 1927 October 8 29 4 54 18 d. - - 16 45 22 8 h. ſº 19 43 22 m. gº 54 M. Long. 6 24 59 12 Equat, oftime—14m.48sec. d. h. m. sec. Appar. Time at Greenwich, Oct. 1821, 18 8 22 3 Equation of time tºg tº tº º — 14 48 Mean Time at Greenwich gº - 18 8 7 15 3. On the 15th of May, 1821, when it is 7 h. 12 m. P. M. mean time at Greenwich, what is the apparent time at Boston? Jłns. 2 h. 31 m. 41 sec. P. M. 4. On the 17th of September 1821, when it is 10 h. 25 m. 32 sec. A. M. apparent time at New York, what is the mean time at Greenwich? Ans, 3 h. 16 m. 1 sec. P. M. 262 ASTRONOMY. PROBLEM WII. To find the Sun's Longitude, Semidiameter, and Hourly JMo- tion, and the apparent Obliquity of the Ecliptic, for a given time, from the Tables. * For the Longitude. When the given time is not for the meridian of Greenwich, reduce it to that meridian by prob. W.; and if it is apparent time reduce it to mean time, by the last problem. With the mean time at Greenwich, take from tables IX, X, XIII, and XIV, the quantities corresponding to the year, month, day, hour, minute, and second, and find their sums.” The sum in the column of mean longitudes will be the tabular mean longi. tude of the sum; the sum in the column of perigee, will be the tabular longitude of the perigee; and the sums in the columns I, II, III, and N, will be the arguments for the small equations of the sun's longitude, and for the equation of the equinoxes, which forms one of them. Subtract the longitude of the perigee from the sun's mean lon- gitude, borrowing 12 signs when necessary; the remainder is the sun’s Mean Anomaly. With the mean anomaly take the equa- tion of the Sum’s centre from table XV; and with the arguments I, II, and III, take the corresponding equations from table XVI. The equation of the centre and the three other equations, added to the mean longitude, gives the Sun’s True Longitude, reckoned from the mean equinox. & With the argument N, take the equation of the equinoxes, or which is the same thing, the Nutation in Longitude, from table XVIII, and apply it, according to its sign, to the true longitude already found, and the result will be the true longitude, from the apparent equinox, | * In adding quantities that are expressed in signs, degrees, &c. reject 12 or 24 signs, when the sum exceeds either of these quantities. In adding any arguments, expressed in 100, or 1000, &c, parts of the circle, when they are expressed by two figures, reject the hundreds from the sum; when b three figures, the thousands; and when by four figures, the ten thousan' ASTRONOMY. 263 For the Semidiameter and Hourly JMotion. With the sun’s Mean Anomaly, take the Hourly Motion and Semidiameter, from tables XI and XII. For the Apparent Obliquity of the Ecliptic. To the Mean Obliquity, taken from table, XVII, apply, ac- cording to its sign, the Nutation in Obliquity, taken from table XVIII, with the argument N, and the result will be the Apparent Obliquity. Note. In the Nautical Almanac the Sun's Longitude is given for each day in the year at apparent noon; and the Semidiameter and Hourly Motion are given for several times in each month.* Either of these quantities may easily be found for any interme- diate times, by proportion. The Apparent Obliquity of the Ecliptic is given in the beginning of the Almanac, for each three months in the year, and is easily estimated for any intermediate time. ExAM. 1. Required the Sun's Longitude, Hourly Motion, and Semidiameter, and the Apparent Obliquity of the Ecliptic, on the 18th of October, 1821, at 3 h. 20 m, 18 sec. P. M. mean time at Philadelphia. --- d. h. m. sec. Mean time at Philadelphia Oct. 1821, 18 3 20 18 Diff of Long. - tºº sº ſº 5 0 46 Mean time at Greenwich gºt 18 8 21 4 * See the second and third pages of table LV. 264 ASTRONOMY. | M. Long. Long. Perigee. I II III | N 1821 9s 8° 48' 197 || 9s 70 50/ 43// | 920 || 782 260 | 036 Octob. 8 29 4, 54 46 || 250 | 684 || 468 || 40 18 d. 16 45 22 3 574 43 29 2 8 h. , 19 43 O 11 0|| 0 || 0 21 m. 52 4 sec. 0 | 9 7 51 32 || 755 509 || 757 78 * ————| 6 24 59 10 6 24, 59 10 Eq. Sun’s cent. 8 27 || 9 17 7 38 Mean Anomaly. I 4 Sun’s Hourly Motion 2, 29// II 10 | Sun’s Semidiameter - 16 5 III 6 6 25 7 57 || M. Obliq. Ecliptic 1821, 23° 27' 46” Nutation. + 9 Nutation - * º —H 8 Sun's true long. | 6 25 8 6 | Appar. Obliquity - 23 27 54 2. Required the Sun's longitude, hourly motion, and semi- diameter, and the obliquity of the ecliptic, on the 19th of August, 1821, at 7 h. 4 m. 51 sec. A. M. apparent time at Phila- delphia. Ans. Sun's longitude 4° 26° 6'43"; hourly motion 2' 25"; semidiameter 15' 51"; obliquity of the ecliptic 23° 27' 55". 3. Required the Sun's longitude, hourly motion, and semi- diameter, and the obliquity of the ecliptic, on the 21st of Feb- ruary, 1824, at 9 h. 6 m. 17 sec. P. M. mean time at Philadel- phia. ſins. Sun's longitude 11° 2° 27'46"; hourly motion 2' 31"; semidiameter 16' 12"; obliquity of the ecliptic 23° 27° 49''. PROBLEM VIII. The Obliquity of the Ecliptic and the Sun's longitude being given, to find the Right Ascension and Declination. For the Right flscension. To the Cosine” of the Obliquity, add the Tangent of the Lon- gitude, and reject 10 from the index; the resulting logarithm will be the the Tangent of the Right Ascension which must always be taken in the same quadrant as the longitude. *By the terms Sine, Cosine, &c. are here meant the logarithmic Sine, Co- sime, &c. The same is to be understood when the terms are used in the rules for working any of the following problems. ASTRONOMY. 265 For the Declination. To the Sine of the Obliquity, add the Sine of the Longitude, and reject 10 from the index; the resulting logarithm will be the Sine of the Declination, which must always be taken out less than 90°; and it will be North or South, according as the sign is af- firmative or negative. Note. The Sun's right ascension, and declination are given, in the Nautical Almanac, for each day in the year. See table LV. ExAM. I. Given the obliquity of the ecliptic 23° 27' 40", and the sun’s longitude 125° 31' 25", to find the right ascension and declination. 2 cos. Obliquity 23° 27' 40” º 9,96253 tan. Long. - 125 31 25 - — 10.14635 tan. Right Ascen. 127 53 30 — 10, 10888 sin. Obliquity 23° 27' 40" - 9.60002 sin. Long, - 125 31 25 - - 9.91055 sin. Decl. - 18 54 23 N - 9.5105.7 2. The obliquity of the ecliptic being 23° 27' 40", what is the sun's right ascension and declination, when his longitude is 35° 19' 30"? ºffms. Right ascension 33° 1'43", and declimation 13° 18! 32” N. º 3. Given the obliquity of the ecliptic 23° 27' 50", and the sun's longitude 313° 36' 12"; what is the right ascension and declimation? Ans. Right ascension 3.16° 4' 30", and declination 16° 45' 29" S, PROBLEM IX. Given the Obliquity of the Ecliptic and the Sun's Right ºffscen- sion, to find the Longitude and Declination. For the Longitude. To the arithmetical complement of the Cosine of the Obliquity, 35 266 ASTRONOMY. add the Tangent of the Right Ascension; and the resulting loga- rithm will be the Tangent of the Longitude, which must be taken in the same quadrant as the Right Ascension. For the Declination. To the Tangent of the Obliquity, add the Sine of the Right Ascension, and reject 10, from the index; the resulting logarithm will be the Tangent of the Declimation, which will be North or South, according as the sign is affirmative or negative. ExAM. 1. Given the Obliquity of the ecliptic 23° 27' 50", and the sun's right ascension 215° 12' 27"; what is the longitude and declimation? cos Obliquity 23° 27' 50" Ar. Co. 0.03748 tan. Right Asc. 215° 12 27 - - 9.84857 tan. Long. 217 34 5 - - 9.88605 tan. Obliquity 23° 27' 50" - 9.63755 sin. Right Asc. 215 12 27 - — 9.76083 tan. Declin. 14 3 0 S wº 9,39838 2. When the obliquity of the ecliptic is 23° 27' 50", and the sun's right ascension 53° 31' 20", what is the longitude and de- climation? /lns. Longitude 55° 51' 16", and declimation 19° 14" 24" N. #. 3. Given the obliquity of the ecliptic 23° 27' 40", and the sun's right ascension 187° 15' 21"; required the longitude and declination, ſins. Longitude 187°54' 6", and declination 3° 8' 15" S. PROBLEM, X The Obliquity of the Ecliptic and the Sun's Longitude being given, to find the angle of Position. To the Tangent of the Obliquity, add the Cosime of the Lon- ASTRONOMSY. 267 gitude, and reject 10, from the index; the resulting logarithm will be the Tangent of the angle of Position, which must always be taken less than 90°. - The northern part of the circle of latitude will be to the West or East of the Northern part of the circle of declination, accor- ding as the sign of the tangent of the Angle of Position is affirma- tive or negative. -- ExAM. I. Given the obliqnity of the ecliptic 23° 27' 50", and the sun's longitude 112° 19 17”, to find the angle of Position. tan. Obliquity 23° 27' 50" tº 9,63755 cos. Long. - 112 19 17 = — 9.57956 tan. Angle of Posit. 9° 21'41" - — 9.21711 The northern part of the circle of latitude lies to the east of the circle of declination. 2. Given the obliquity of the ecliptic 23° 27' 50", and the sun's longitude 77° 47' 30"; what is the angle of position? ſins. 5° 14' 40”; and the northern part of the circle of latitude lies to the west of the circle of declination. 3. When the obliquity of the ecliptic is 23° 27' 50", and the sun's longitude 225° 41' 12", what is the angle of position? ſins. 16° 15' 7"; and the northern part of the circle of latitude lies to the east of the circle of declimation. * PROBLEM XI. To find, from the Tables, the JMoon's Longitude, Latitude, Equa- torial Parallaw, Semidiameter, and Hourly JMotions, in Longitude and Latitude, for a given time. When the given time is not for the meridian of Greenwich, re- duce it to that meridian; and when it is apparent time, reduce it. to mean time. 268 ASTRONOMY., With the mean time at Greenwich, take out, from tables XX, XXI, XXII, XXIII, and XXIV, the arguments, numbered 1, 2, 3, &c. to 20, and find their sums, rejecting the ten thousands, in the first mine, and the thousands, in the others. The resulting quantities will be the arguments for the first twenty equations of Longitude. With the same time, and from the same tables, take out the remaining arguments and quantities, entitled Evection, Anomaly, Variation, Longitude, Supplement of the Node, II, V, VI, VII, VIII, IX, and X; and add the quantities in the column for the Supplement of the Node. For the Longitude. With the first twenty arguments of longitude, take, from tables XXV to XXX, the corresponding equations, and place their sum in the column of Evection. Then, the sum of the quantities in this column will be the corrected argument of Evection. With the corrected argument of Evection, take the Evection from table XXXI, and add it to the sum of the preceding equa- tions. Place the resulting sum, in the column of Anomaly. Then, the sum of the quantities in this column will be the cor- rected Anomaly. With the corrected Anomaly, take the Equation of the Centre from table XXXII, and add it to the sum of all the preceding equations. Place the resulting sum, in the column of Variation. Then, the sum of the quantities in this column will be the corrected argument of Variation. With the corrected argument of Variation, take the variation from table XXXIII, and add it to the sum of all the preceding equations; the result will be the sum of the first twenty three equations of the Longitude. Place this sum in the column of Longitude. Then, the sum of the quantities in this column will be the Orbit Longitude of the Moon, reckoned from the mean equinox. ASTRONOMY. 269 Add the Orbit Longitude, to the Supplement of the Node. The result will be the argument of the Reduction. It will also be the 1st argument of Latitude. With the argument of Reduction, take the Reduction from ta- ble XXXIV, and add it to the Orbit Longitude. Also, with the 19th argument, which is the same as argument N, for the Sun's Longitude, take the Nutation in Longitude, from table XVIII, and apply it, according to its sign, to the last sum. The result will be the Moon's true Longitude from the Apparent equinox. For the Latitude. Place the sum of the first twenty three equations of Longitude, taken to the nearest minute, in the column of Arg. II. Then the sum of the quantities in this column will be Arg. II of Latitude, corrected. The Moon's true Longitude is the IIIrd argument of Latitude. The 20th argument of Longitude is the IVth argu- ment of Latitude. Convert the degrees and minutes, in the sum of the first twenty three equations of Longitude, into thousandth parts of the circle, by taking from table XXXVIII, the number corresponding to them. Place this number in the columns V, VI, VII, VIII, and IX; But not in column X. Then the sums of the quantities in columns, V, VI, VII, VIII, IX, and X, rejecting the thousands, will be the Wth, WIth, VIIth, VIIIth, IXth, and Xth arguments of Latitude. With the sum of the Supplement of the Node, and the Moon's Orbit Longitude, which is Arg. I of Latitude, take the Moon's distance from the North Pole of the Ecliptic, from table XXXV; and with the remaining nine arguments, take thc corresponding equations from tables XXXVI, XXXVII, and XXXIX. The sum of these ten quantities will be the Moon's true distance from the North Pole of the Ecliptic. The difference between this distance and 90°, will be the Moon's true latitude; which will be JNorth or South, according as the distance is less or greater than 90°. *y 270 ASTRONOMYe For the Equatorial Parallax. With the corrected arguments, Evection, Anomaly and Wa- riation, take the corresponding quantities from tables XL, XLI, and XLII. Their sum will be the Equatorial Parallax. For the Semidiameter. With the Equatorial Parallax, take the Moon's Semidiameter from table XLIV. For the Hourly JMotion in Longitude. With the arguments, 2, 3, 4, and 5, of Longitude, rejecting the two right hand figures in each, take the corresponding equa- tions fromtable XLVI. Also with the correct argument of Evec- tion, take the equation from table XLVII. With the Sum of the preceding equations at top, and the cor- rect anomaly at the side, take the equation from table XLVIII. Also with the correct anomaly take the equation from table XLIX. With the Sum of all the preceding equations at the top, and the correct argument of Variation, at the side, take the equation from table L. With the correct argument of Variation, take the equation from table LI. And with the argument of Reduction, take the equation from table LII. These three equations added to the sum of all the preceding ones, will give the Moon's Hourly Motion in Longitude. - For the Hourly JMotion in Latitude. With the 1st and 2d arguments of Latitude, take the corres- ponding quantities from tables LIII and LIV, and find their sum, attending to the signs. Then 32° 56' : the moon's true hourly motion in Longitude :: this sum : the moon’s true hourly motion in Latitude. When the sign is affirmative the moon is tending North; and when it is negative, she is tending South, ASTRONOMY. 274 ExAM. I. Required the moon's longitude, latitude, equatorial parallax, semidiameter, and hourly motions in longitude and lati- tude, on the 6th of August 1821, at 8 h. 46 m. 27.sec. A. M. mean time at Philadelphia. . . d. h. m. see. Mean time at Philadelphia, August, 5 20 46 27 Diff. of Long. ems - - 5 0 46 Mean time at Greenwich, August, 6 1 47 13 272 ASTRONOMY. 88 89 8 I 2 ºpnļīšuo， I anū) suooj， 8 + - '3uoſ uſ 'qejn!!! ºpnºſťTJO I 3 IV , 92 ºg gr 9 | 69 ģç gȚ 2 ŞI Ç9 8I Z | 9I 9 - - - uoņonpºțI 672||387|| ŽIŤ|133||663||90|| † çg 6| iſ ſé řż 5 | Ģſ ģç gi 2 | csſ№gºſ“ g)38 9ý ZŐ 9 | 09 Iç 9I II 8I [8] [8] [8] [8] | ºg 9€ 39 9Ț8 I 9IŤ Ziff IŽI 99‘enbºſ go tums 0 |0 |0 |0 |0 |004449*oas gI 0 |0 |0 |0 | 0 | zg987 92,99 8398 930 I 32,‘UDI Z Ž 0 || I || I ||I lº | I || 32899 3863 080ý Ø92. 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O/ 5// 3 do. 2 4 do. - 1 5 do. 3 Evection 1 19 1 30 An. & Sum Eqs. 6 Anomaly 27 58 29 34 Var. & Sum Eqs. 9 Variation 3 Reduction 17 Hourly Motion in Long. 30 3 Moon’s Hourly Motion in Latitude, |Argequations | § ArgºFqu, ) 'slong. 1 Oo 18' 51// 2 1 39 3 20 4 5 42 5 1 36 6 2 39 7 1 39 8 17 9 1 4 10 16 11 14 12 6 13 27 14 20 15 19 16 10 17 8 18 3 19 7 20 17 O 36 14, EV. 1 11 27 Sum. 1 47 41 An. 4, 13 50 Sum 6 1 31 War. 30 31 Sum. 6 32 32 | Arg. [Eqs. D’s Lat. I 940 43/~57// II 16 57 D’s long. 2 20, long. 4. V 45 VI 7. VII 3 VIII 20 IX 18 X 43 95 3 16 90 0 O Moon's Lat. 5 3 16 S. Arg. |Y's Eq. Par. EV. 17 277 An. 53 0 War. 4. Moon’s Eq. Par. 54 31 Moon's Semidiameter 14' 51". As 32° 56'7: 30' 3// : I — 0’ 51/ II | +-- 2 Sum — 0 49 : — 0'49" : —0'45" = Moon's hourly motion in latitude, tending South. § 274 ASTRONOMY. 2. Required the Moon's longitude, latitude, equatorial parallax, semidiameter, and hourly motions in longitude and latitude, on the 27th of April, 1821, at 9 h. 43 m. 30 sec. P. M. mean time at Baltimore. Jins. Long. 11° 13° 32' 13"; lat. 6' 57" N, equat. par. 60' 0"; semidiam. 16' 21"; hor, mot. in long. 36"11"; and hor. mot. in lat. 3' 14", tending north. 3. What will be the Moon's longitude, latitude, equatorial pa- rallax, semidiameter, and hourly motions in longitude and latitude, on the 19th of August, 1822, at 5 h. 56 m. 14 sec. P. M. mean time at Philadelphia? ſins. Long. 6' 3" 7" 12"; lat. 3° 51 35" S : equat, par. 56' 19'; semidiam. 15' 21"; hor. mot. in long. 32’ 7"; and hor. mot. in lat. 2' 1", tending south. PROBLEM XII. To find the JMoon's Longitude, Latitude, Hourly JMotions, Equa- torial Parallaw, and Semidiameter, for a given Time, from the JWautical fillmanac. Reduce the given time to Apparent time at Greenwich. Then, For the longitude. Take from the Nautical Almanac, the two longitudes, for the moon and midnight, or midnight and moon, next preceding the time at Greenwich, and also the two immediately following these, and set them in succession, one under another. Then, having regard to the signs, subtract each longitude, from the next follow- ing one, and the three remainders will be the first differences. Call the middle one A. Subtract each first difference from the following, for the second differences. Take the half sum of the second differences and call it B. - Call the excess of the given time at Greenwich, above the time of the second longitude, T. Then 12 h : T :: A : fourth term, which must have the same sign as A. With the time T at the side, take from table LVI, the quantities corresponding to the minutes, tens of seconds, and seconds of B, at the top, the sum of these, with a contrary sign to that of B, will be the correction of second differences. ASTRONOMY. 275 A The sum of the second longitude, the fourth term, and the cor- rection of second differences, having regard to the signs, will be the required longitude. For the Hourly JMotion in Longitude. To the logistical logarithm of # of T, add the logistical loga- rithm of B, and find the quantity corresponding to the sum. Call this quantity C, and prefix to it the same sign as that of B. Or C may be found without logarithms; thus, 12 h : T : : B. : C. - Divide the sum of A, 3 B with its sign changed, and C, by 12, and the quotient will be the required hourly motion in longitude. For the Latitude. Prefix to north latitudes the affirmative sign, but to south lati- tudes the negative sign, and then proceed in the same manner as for the longitude. The resulting latitude will be north or south, according as its sign is affirmative or negative. Note. The Moon's Declimation may be found in the same }Y] all 116]'. For the Hourly JMotion in Latitude. With T, and the values that A and B have, in finding the lati- tude, find the hourly motion in latitude, in the same manner as directed for finding the hourly motion in longitude. When the resulting hourly motion in latitude is affirmative, the moon is tending north; and when it is negative, she is tending south. For the Semidiameter and Equatorial Parallaw. The Moon's semidiameter and equatorial, horizontal parallax may be taken from the Nautical Almanac with sufficient accuracy by simply proportioning for the odd time between moon and mid- night or midnight and noon. EXAM. 1. Required the Moon's longitude, latitude, equatorial parallax, semidiameter, and hourly motions in longitude and lati- 276 ASTRONOMY. tude, from the Nautical Almanac, on the 6th of August, 1821, at 8 h. 40 m. 54 sec. apparent time at Philadelphia. d. h. m. sec. Appar. time at Philadelphia, August 5 20 40 54 Diff. of Long. º º ſº 5 0 46 Appar. time at Greenwich, August 6 1 41 40 For the Longitude and Hourly JMotion in Longitude. Mean of Longitudes 1st Diff. 2d Diff. 2d Diff. 5th midn. 7s 12° 8' 55" 5° 59' O” Öth moon |7 18 7 55 2' 23” f / ſº f P §§§ 3 ; is A. : , ; ; ; * +** 7th noon 8 0 13 38 T A h. h. m. sec. 12 : 1 41 40 :: 6° 1' 23” : 51’ 1.”, fourth term. Second Longitude tº tº - 7s 18° 7' 55" | Fourth term - sº gºs -. tº 51 1.7 Cor. 2d diff. from tab. LV gº — 9.7 Moom's true Longitude ºf 18 58 47 # T gº tº º 8 m. 28 sec. L. L. 8504 B - tºº + 2 40 L. L. 13522 C dºg - + 0 23 22026 A tº º sº 6° 1' 23” # B, sign changed, - – 1 20 | C tºº gº - + 0 23 12)6 0 26 Hor. mot, in long. - 30' 2".2 ASTRONOMY. i 277 For the Lalitude and Hourly JMotion in Latitude. Mean of Latitudes 1st Diff. | 2d Diff. 2d Diff. 5th midn.-4 50 ºf nº sº —- 6th noon i- 5 1 55 * -- 3' 14" 6th midn. — 5 9 43 A. — 7 48 | 13 B. -- 3' 13”.5 7th moon | — 5 14 18 — 4 35 T A h. h. m. Sec. , 12 : 1 41 40 : : — 7' 48" : — 1' 6".1, fourth term. Second Latitude - - — 5° 1' 55" Fourth term - º - — 1 6.1 Cor. 2d diff - º - " - 11.7 Moon's true Latitude º 5 3 13 S. # T 8 m. 28 sec. L. L. S504 B + 3 13".5 L. L. 12696 C + 0 27 21200 A º- º tº *- — 7' 48’’ # B, sign changed -g — 1 37 C º º º - + 0 27 12) — 8 58 Hor. mot. in lat. - — 0' 44".8, tending south. Moon’s semidiam. from N. Alm. 14' 53” do. eq. parallax º tº - 54 32. Note. The quantities found in this example, from the Nautical Almanac, are for the same time as those found in example 1st of the last problem, from the tables in this work. It may be seen that there is not much difference in them. 2. Required the Moon's longitude, latitude, equatorial parallax, semidiameter, and hourly motions in longitude and latitude, on the 21st of August 1821, at 16 h. 20 m. 33 sec, apparent time at 273 ASTRONOMY. Greenwich. Ans. Long. 2; 23° 7' 48"; lat. 5° 0' 36' N; equat. par. 57' 57"; semidiam. 15'49"; hor. mot. in long, 34' 3".8; and hor. mot. in lat. 0'59".4, tending South. 3. What were the Moon's longitude, latitude, equatorial pa- rallax, semidiameter, and hourly motions in longitude and lati- tude, on the 14th of August 1821, at 2 h. 8 m. 2 sec. P. M. ap- parent time at Philadelphia? Ans. Long, 11° 7° 43' 8"; lat. 0° 17' 6' N; equat, par. 59' 47"; semidiam, 16' 197; hor. mot, in long. 36'4".2; and hor. mot. in lat. 3, 19".8, tending north. tº PROBLEM XIII. The JMoon's Equatorial Parallas, and the Latitude of a Place being given, to find the Reduced Parallaw and Latitude. With the Latitude of the place, take the Reductions from table XLIII, and subtract them from the Parallax and Latitude. ExAM. 1. Given the equatorial parallax 54' 31", and the latitude of Philadelphia 39° 27' N, to find the reduced parallax and latitude. Equatorial parallax tºp tºs 54' 31 '' Reduction gº tºº tºº gº 5 Reduced Parallax º gº 54 26 Latitude of Philadelphia st 29° 57° N. Reduction - sº º ſº 11 Reduced Lat. of Philadelphia 39 46 N. 2. Given the equatorial parallax 60° 0", and the latitude of Boston 42° 23' N. to find the reduced parallax and latitude. Ans. Reduced par. 59' 55", and reduced lat. 42° 12 N. 3. Given the equatorial parallax 57' 21", and the latitude of Charleston 32° 50' N. to find the reduced parallax and latitude. ..ſins. Reduced par. 57' 18", and reduced lat, 32° 40' N. ASTRONOMY. 279 PROBLEM XIV. To find the JMean Right flscension and Declination, or Lon- gitude and Latitude of a Star for a given Time, from the tables. Take the difference between the time for which the table is constructed and the given time, and ſultiply the annual variation, by the number of years in this difference; the product will be the number of years in this difference; the product will be the va- riation for the years. Reduce the odd time to days.” Them, 365 days : number of days :: annual variation : proportional part. This proportional part, added to the variation for the years, will be the whole variation, which applied to the quantity given in the table, with its proper sign, when the given time is after the time for which the table is constructed, but with a contrary sign when it is before, will give the required quantity. ExAM. 1. Required the mean right ascension, and declination of Regulus, on the 15th of June 1821. Mean right ascen. begin. of 1820, table III, 149° 41' 39" War. for 1 yr. 166 d. - * > sº - + 1 10 Mean right ascen. required tº-º; tºº 149 42 49 Mean declin. begin. of 1820, gº - 12° 50' 36” N War. for 1 yr. 166 d. º tº gº *ms 25 Mean declin. required - * = tº 52 to 11 N. 2. Required the mean longitude and latitude of Regulus on the 15th of June, 1821. Mean longitude, begin. of 1810, table VIII, 4, 27° 11, 18" War. for 11 y. 166 d. tº gº tº tº + 9 33 Mean long, required - Lº tº - 4 27 20'ſ 51 * This when the given time is after the time for which the table is con. structed, may be done very simply by taking from table VI, the number of days corresponding to the month, and adding to it the odd days. 280 ASTRONOMY. Mean latitude, begin. of 1810 - tº . O° 27' 36” N War. for 11 y. 166 d. - gº tºº - + 2 Mean latitude, required - tº sº 0 27 38 N. 3. Required the mean right ascension and declination of g Tauri, on the 6th of November 1822. Ans. Mean right ascen. 78° 46' 29” and mean declin. 28° 26' 53' N. 4. What will be the mean longitude and latitude of g Tauri, on the 6th of November 1822? Ans. Mean long. 2, 20° 5' 59", and mean lat. 5° 22' 27” N. PROBLEM XV. | To find the flberration of a Star, in right ºffscension and Decli- ^ſition, for a given Day. Find the mean right ascension and declimation of the star, for the given time by the last problem. Also find the sun’s true lon- gitude for noon of the given day by prob. VII, or take it from the Nautical Almanac. Designate the sun’s longitude by e, the right ascension of the star by A, and the declination by D. With the argument e, take the quantity a from table LX; add it to e, attending to the sign and from the sum, subtract A. Then, For the ſlberration in Right flscension. With argument e, take from table LX, the log. a, with its proper sign, and to it. add the Cosine of (e -- a - A), and the arithmetical complement of the Cosine of D, rejecting the tens in the index of the sum. The natural number, corresponding to the resulting logarithm will be the aberration in right ascension, to be applied to the mean right ascension. For the Jīberration in Declination. Add together the log. a, the Sine of (ê -- a - A), and the Sine of D, and reject the tens in the index of the sum. Take the natural ASTRONOMY. 284 number, corresponding to the sum, and call it m. With the ar- guments e -- D and e— D, or when the declimation is south, with these arguments, each increased by WI signs, take the cor- responding quantities from table LXI. The sum of these quan- tities, and m, giving attention to the signs, will be the aberration in declination, to be applied to the mean declimation. ExAM. 1. What are the aberrations in right ascension and de- clination, of Regulus, on the 15th of June, 1821, the sun’s lon- gitude on that day, being 2: 23° 58'? A = 149°42'49" = mean right ascen. of Regulus. D = 12 50 11 N = mean declin. do. e = 2* 23° 58' = sun's longitude. © tºº ſº *ge 2s 23° 58' a', from table LX, + 0 30 G + æ - tº - 2 24 28 A - sº gº 4 29 43 e -- a - A - 9 24 45 = 294° 45' log. q, from tab. LX, gº - — 1,3061 cos. (6 + æ- A 294° 45' . 9.6219 cos D ...tº tºº tº - 12 50 Ar. Co. 0.01.10 Aber. in right ascen. —8".69 - - — 0.9390 log, a - tº tºº * , tº * > — 1.3061 sin (e -- a - A) - 294° 45' - —9.9581 sin D tº e gº - 12 50 N. - 9,3466 ???, gº uº. sº + 4.08 tº 0.6106 Arg. (e -- D) = 3' 6° 48', gives - + 0”.48 Arg. (e–D) = 2 11 8, gives - - — 1.30 % - sº gº gº gº º gº + 4.08 Aber. in declination tº gº º - + 3.26 2. Required the aberrations in right ascension and declination, of flntares, on the 11th of March, 1821, the Sun's longitude being 37 282 ASTRONOMY. l 11' 20° 38'. ſhis. Aber. in right ascen. -- 5".43, in declin. — 0”.74. 3. On the 6th of November 1822, the sun's longitude will be 7° 13° 32'; what will be the aberrations in right ascension and de- climation of 3 Tauriº ſhis. Aber. in right ascension + 18".54, and in declination + 0". 15. PROBLEM XVI. To find the JNutations of a Star in Right flscension and Decli- nation, for a given Time. Find the Supplement of the Moon's Node, from tables XX, XXI and XXII, and subtract it from 12° 0° 7'; the remainder will be the Mean Longitude of the Moon's Ascending Node. Designate the right ascension of the body by A, the declination by D, and the mean longitude of the moon’s node by N. With the argument N, take the quantity B, from table LXII; add it to N, attending to the sign, and from the sum, subtract A. For the JNutation in Right flscension. With the argument N, take from table LXII, the log. b, with its propersign, and to it, add the Cosime of (N + B–A), and the Tangent of D, marking it negative, when the declination is south, and reject the tens in the index of the sum. Apply the natural mumber corresponding to the sum, to a quantity, taken fromtable LXIII, with the argument N, and the result will be the mutation in right ascension, to be applied to the mean right ascension. For the JNutation in Declination. To the log, b, add the Sine of (N + B — A), or when the de- clination is south, the Sine of (N + B — A + WP) rejecting the tens in the index, and the natural number corresponding to the sum, will be the mutation in declination, to be applied to the mean declimation. £XAM. 1. Required the mutations in right ascension and de- clination, of Rigel, on the 19th of July 1825. ... ASTRONOMY. . * 283 By prob. XIV, A = 76° 33' 8" and D = 8° 24′ 36° S. Supp. of Node , 1825 tº - - 3s 0° 25' July - tºn sº g- 9 35 19 d. - - - 57 3 10 57 12 0 7 N - - - 8 19 10 B, from tab. LXII. — 3 35 N + B - - - 8 15 35 A gº º º - 2 16 33 N + B — A - - 5 29 2 = 179° 2' log, b, from tab. LXII, - - – 0.8623 cos. (N + B — A) 179° 2' - —9.9999 tan. D *- tº 8 25 - — 9.1702 mat. numb. - . — 1".08 — 0.0324 From tab. LXIII, + 16.25 Nut, in right ascen. + 15.17 log. b, - - - - - – 0.8623 sin. (N + B – A + VI) 359° 2' —8.2271 Nut in declin. - ' -- 0”.12 + 9,0894 sº 2. Required the mutations, in right ascension and declination, of flntares on the 11th of March, 1821. Jins, Nut. in right ascen. +3".70, and in declin, -ī- 9”.23. 3. What will be the mutations, in right ascension and declina- tion of 3 Tauri, on the 6th of November, 1822. Jins. Nut, in right ascem. -- 14".61, and in declin. + 7".3. 284 $ ASTRONOMY, t" PROBLEM XVII. To find the flberrations of a Star in Longitude and Latitude, for a given Time. Designate the sun’s longitude on the given day by L, the mean longitude of the star by L', and the mean latitude by 2. Then, For the Aberration in Longitude. Add together the constant logarithm, - 1.80649, the Cosine of (L–L'), and the arithmetical complement of the Cosine of 2, rejecting the tens in the index of the sum. The natural number, corresponding to this sum, will be the aberration in Longitude, to be applied to the mean longitude. For the Aberration in Latitude. Add together the constant logarithm, - 1.30649, the Sine of (L–L'), and the Sine of X, rejecting the tens in the index, and the natural number corresponding to the sum, will be the aberra- tion in latitude, to be applied to the mean latitude. ExAM. 1. Required the aberrations in longitude and latitude, of Sirius, on the 20th of July, 1821, the sun's longitude being 3s 27° 21'. Byprob. XIV, L' = 3' 11° 37' 51" and x = 39° 22' 31" S. f —1.80649 cos. (L – L') 15° 43' iº 9,98345 COS. A 39 23 Ar. Co. 0, 11187 Aber. in long. –25.22 — 1.40181 * — 1.30649 sin (L– L') 15° 43' - gº 9.48278 sin a 39 23 tºº - 9.80244 Aber. in lat. — 3”.48 tº - – 0.54171 2. On the 25th of April, 1822, the sun's longitude will be 1° 4° 41'; what will be the aberrations in longitude and latitude ASTRONOMY. 285 of Regulus, at that time? flns. Aber, in long. + 7".81, and in lat. + 0”.15. 3. Required the aberrations in longitude and latitude of 3 Wir- sinis, on the 10th of August, 1821. ſins. Aber, in long- 16”.14, and in lat. -- 0". 15. PROBLEM XVIII. To find the Nutation of a Body in Longitude. Find the mean longitude of the moon’s ascending mode, as in prob. XVI, and to its Sime, add the constant logarithm— 1.25396, rejecting the tens in the index. The natural number corresponding to the sum, will be the mutation in longitude, to be applied to the mean longitude. ExAM. I. Required the nutation in longitude of Sirius, on the 20th of July, 1821. The mean longitude of the moon’s ascending node at the given time is 11° 6° 29'. — 1.25396 sin. long. of mode 336°29′ – 9.60099 Nut. in long. + 7”. 16 0,85495 2. Required the mutation in longitude of 9 Virginis, on the 10th of August, 1821. Ans. -- 7"-48. 3. What will be the mutation in longitude of Regulus, on the 25th of April, 1822? Ans. -- 11", 12. PROBLEM XIX. The Obliquity of the Ecliptic and the Right flscension and De- clination of a Body being given, to find the longitude and La- titude. --- Designate the obliquity of the ecliptic by E. To the Tangent of the declination marked negative when the declimation is south, add the arithmetical complement of the Sine of the right ascen- 286 ASTRONOMY. sion; the result will be the Tangent of an arc, which, call B. The arc B must be taken acccording to the sign, but always less than 180°. * For the Longitude. Add together the Cosime of the difference between B and E, the Tangent of the right ascension, and the arithmetical comple- ment of the Cosime of B, rejecting 10 from the index; the result will be the tangent of the longitude, which must be taken ac- cording to the sign, observing also that the longitude and right as- cension are always, either both between 90° and 270°, or reckon- ing in the order of the signs, both between 270° and 90°. For the Latitude. To the Tangent of the difference between B and E, which must be marked negative, not only when the difference is greater than 90°, but also when E is greater than B, add the Sine of the longitude, rejecting 10 from the index; the result will be the Tangent of the latitude, which must always be taken less than 90°, and will be north or south, according as the sign is affirma- tive or negative. Note. When the mean obliquity of the ecliptic and the mean right ascension and declimation are used, the results will be the mean longitude and latitude. But when the apparent obliquity of the ecliptic, found by prob. VII, and the apparent right ascen- sion and declination, found by applying to the mean right ascen- sion and declination, the aberrations and mutations, obtained by problems XV and XVI, are given, the results will be the apparent longitude and latitude. ExAM. 1. On the 10th of April, 1821, the mean right ascen- sion of Arcturus was 211° 52' 37", the mean declination 20° 7' 4" N, and the mean obliquity of the ecliptic 23° 27'46". What were its longitude and latitude. ASTRONOMY, 287 tan. Declin. - 20 7' 4"N - 9.56384 sin. Right Asc. 211 52 37 Ar. Co.— 0.27729 tan. B ſº - 145 15 13 - — 9.841.13 E. sº ſº 23 27 46 º cos. (Bº E) 121 47 27 - — 9.72166 tan. Right Asc. 211 52 37 - 9.793.71 cos. B * - 145 15 13 Ar. Co. — 0.08529 tan. Long. 201 44 16 tºº 9.60066 tan. (B an E) 141° 47' 27’’ - – 10.20774 --- sin. Long. - 201 44 16 - — 9.56863 tam. Lat. - 30 51 37 N 9.77637 2. Given the obliquity of the ecliptic 23° 27' 47", the right ascension of Rigel, 76°28' 21", and the declimation 8° 25' 2"S, on the 1st of January, 1820, to find the longitude and latitude. Ans. Long. 74° 18' 51", and lat. 31°8'45" S. 3. On the first of January, 1821, the right ascension of Pro- cyon was 112°28'49", the declination 5° 40' 35" N, and the obliquity of the ecliptic 23° 27' 46". What were its longitude and latitude? Ans. 113° 18' 55", and 15° 59' 0" S. PROBLEM XX. The Obliquity of the Ecliptic, and the Longitude and Latitude of a Body being given, to find the Right ºffscension and Declination. Designate the obliquity of the ecliptic by E. To the Tangent of the latitude, marked negative when the latitude is south, add the arithmetical complement of the Sine of the longitude; the re- sult will be the tangent of an arc, which call B. The arc B must be taken, according to the sign, but always less than 180°. For the Right flscension. Add together the Cosime of the sum of B and E, the Tangent of the longitude, and the arithmetical complement of the Cosine of B, rejecting 10 from the index; the result will be the Tangent of the 288 ASTRONOMY, rift ascension, which must be taken according to the sign, observing also that the right ascension and longitude are always, either both between 90° and 270°, or reckoning in the order of the signs, both between 270° and 90°. For the Declination. To the Tangent of the sum of B and E, add the Sine of the right ascension, rejecting 10 from the index; the result will be the Tangent of the declination, which must always be taken less than 90°, and will be north or south, according as the sign is af- firmative or negative. Note. The quantities found will be mean or apparent, ac- cording as the given ones are mean or apparent. ExAM. 1. Given the obliquity of the ecliptic 23° 27' 46", the longitude of flrcturus 201° 44' 16", and the latitude 30° 51' 37" N, to find the right ascension and declimation. tan. Lat. tºg 30° 51' 37" N º 9.77637 sin. Long. - 201 44 16 Ar. Co. — 0.431.37 tam. B - - 121 47 28 sº — 10.20774 E - º 23 27 46 cos. (B -- E) 145 15 14 - - - 9.91471 tan. Long. - 201 44 16 tº - 9.60066 cos. B - - 121 47 28 Ar. Co. — 0.27834 tam. Right Asc. 211 52 36 º 9.79371 tan. (B + E) 145° 15' 14" - - — 9.841.13 sin. Right Asc. 211 52 36 sº — 9.72271 tam. Declin. 20 ºf 5 N - - 9.56384 2. Given the obliquity of the ecliptic 23° 27' 47", the longi- tude of Rigel 74° 18' 51", and the latitude 31°8'45" S, to find the right ascension and declination. Ans. Right Ascen. 76° 28′ 21", and declin. 8° 25' 1" S. 3. When the obliquity of the ecliptic was 23° 27'46", the ASTRONOMY. 289 longitude of Procyon 113° 18' 55", and the latitude 15° 59' 0" S, what were the right ascension and declination? ſins. Right ascen. 112°28' 48", and declin. 5° 40' 35" N. PROBLEM XXI. The Obliquity of the Ecliptic, and the Longitude and Declination of a Body being given, to find the flngle of Position. Add together the Cosime of the longitude, the Sine of the obli- quity, and the arithmetical complement of the Cosine of the de- clination, taking them all affirmative, and reject 10 from the in- dex; the result will be the Sine of the angle of position; which, in all cases where the problem is used in calculating an occulta- tion of a planet or star, by the moon, must be taken less than 90°. When the longitude is less than 90° or more than 270°, the morthern part of the circle of latitude lies to the west of the circle of declination; brit when the longitude is between 90° and 270°, it lies to the east. ExAM. I. Given the obliquity of the ecliptic 23° 27' 46", the longitude of flrcturus 201° 44' 16", and the declimation 20° 7' 5' N. to find the angle of position. cos. Long. 201° 44' 16" 9.96.797 sin. Obliq. 23 27 46 9,60005 cos. Declin. 20 7 5 Ar. Co. 0.02734 sin, Ang. Posit. 23 11 46 9.59536 The circle of latitude lies to the east of the circle of declination. 2. Given the obliquity of the ecliptic 23° 27' 47", the longi- tude of Rigel 74° 18' 51", and the declination 8° 25' 1" S; re- quired the angle of position. Jins. 6° 14' 50". 3. When the obliquity of the ecliptic was 23° 27'46", the longi- tude of Procyon 113° 18' 55", and the declination 5° 40' 35' N; what was the angle of position? ſins. 9° 6'43". 38 290 * ASTRONOMY. PROBLEM XXII. The Sun's Right.ſlscension on two consecutive days at noon, and the Right.ſlscension of a Star being given, to find the time of its Passage over the JMeridian. Subtract the sun's right ascension on the first of the two given days, from that on the second, and also from the right ascension of the star; increasing, when necessary, the latter quantities by 360°, or by 24 hours, according as the right ascensions are ex- pressed in degrees, or in time. Then, as the first remainder, in- creased by 360°, or by 24 hours : the second :: 24 hours: the time of the star's passage over the meridian. Note 1. The time of a star's passage may be found nearly, by subtracting the sum’s right ascension at noon, from the right as- cension of the star, and diminishing the remainder by 1, 2, or 3 minutes, according as the remainder is near to 6, 12 or 18 hours. 2. The sun's right ascension is given in the Nautical Almanac, for each day at apparent noon on the meridian of Greenwich, and may easily be found for any other meridian, by proportion. ExAM. 1. From the Nautical Almanac, the sun’s right as- cension on the 11th of March, 1821, at apparent noon, at Phila- delphia, was 23 h. 26 m. 21 sec. and on the 12th, it was 23 h. 30 m. 1 sec. Required the time at which Sirius passed the meridian, its right ascension being then 6 h. 37 m. 16 sec. h. m. Sec. h. m. sec. From 23 30 1 From 6 37 16 Take 23 26 2.1 Take 23 26 21 1st, rem. 3 40 2d. rem. 7 10 55 h. m. sec. h.m. sec. h. h.m. sec. 24 3 40 : 7 10 55: ; 24 : 7 9 49 time required. 2. Given the sun's rightascension on the 10th of April, 1821, at apparent noon, at Boston, 1 h. 15 m. 39 sec., on the 11th, 1 h. 19 m. 19 Sec., and the right ascension of flntares at the same ASTRONOMY. 29ſ sº time 16 h. 18 m. 28 sec. to find the time of its passage over the meridian. Ans. 15 h. 0 m. 31 sec. 3. Required the time of Arcturus’ passage over the meridian of Philadelphia, on the 15th of August, 1821, finding the right ascension of the star by Prob. XIV, and the sum’s right ascension on the 15th and 16th, from the part of the Nautical Almanac, con- tained in table LW. Jins. 4 h. 27 m. 33 sec. PROBLEM XXIII. The Right flscensions of the Sun and JMoon, in time, being given, on two consecutive days at noon, to find the time of the JMoon's Passage over the JMeridian. Subtract the right ascension of the sun on the first day at moon, from that on the second. Subtract the right ascension of the moon on the first day at moom, from that on the second. Subtract the right ascension of the sun on the first day at noon, from that of the moon, increasing the latter, when necessary, by 24 hours. Add the first remainder to 24 hours, and from the sum, subtract the second remainder. Then, as the result: third remainder : : 24 hours: time of the moon's passage over the meridian. Note. The time of the moon’s passage over the meridian of Greenwich is given for each day in the Nautical Almanac. The times of the passages of the planets are also given for several days in each month. See table LV. ExAM. 1. Given the sun's right ascension on the 11th of March, 1821, at apparent noon, at Philadelphia, 23 h. 26 m. 21 sec. and on the 12th, 23 h. 30 m. 1 sec.; the moon’s right as- cension on the 11th, 6 h. 4 m. 35 sec. and on the 12th, 7 h. 2 m. 47 sec. required the time of the moon's passage over the me- ridian. h. m. sec. h. m. Sec. From 23 30 1 From 7 2 47 Take 23 26 21 Take 6 4 35 1st remt 3 40 2d rem, 58 12 292 ASTRONOMY, h. m. Sec. h. m, sec. from 6 4 35 º From 24 3 40 Take 23 26 21 Take 58 12 3d rem. 6 38 14 23 5 28 h. m. sec h.m. Sec. h. h.m. sec. -- As 23 5 28: 6 38 14:: 24: 6 53 54, time required. 2. Given the sun's right ascension on the 10th of April, 1821, at apparent moon, at Boston, 1 h. 15 m. 39 sec. and on the 11th, 1 h. 19 m. 19 sec. the moon’s right ascension on the 10th, 8 h. 35 m. 43 sec. and on the 11th, 9 h. 24 m. 35 sec. required the time of the moon’s passage. Jins. 7 h. 34 m. 20 sec. 3. Given the sun's right ascension on the 13th of August, 1821, at apparent noon at Greenwich, 9 h. 30 m. 58 sec. and on the 14th, 9 h. 34 m. 44 sec. the moon’s right ascension on the 13th, 21 h. 28 m. 12 sec. and on the 14th, 22 h, 21 m. 36 sec. required the time of the moon’s passage. Ans. 12 h. 22 m. 50 sec. PROBLEM XXIV. The Latitude of a Place and the Sun's Declination at noon being given, to find the time of his Rising and Setting. To the Tangent of the latitude of the place, add the Tangent of the sun's declination, rejecting 10 from the index; the result will be the Sine of the ascensional difference, which must be taken less than 90°, and reduced to time. The ascensional difference, added to 6 hours, when the latitude and declimation are both of the same name, that is, both north or both south, but subtracted from 6 hours, when they are of differ- ent names, will give the semi-diurnal arc. The semi-diurnal arc expresses the time of sunset, and sub- tracted from 12 hours, gives the time of sunrise. Note. 1. In the above rule, no notice is taken of the change in the sum’s declination between moon and the time of his being in the horizon, nor of the effect of refraction in changing the time of his rising and setting. When the time of the sun's apparent rising or setting is required with precision, the declimation may be ASTRONOMY. 293 found for the time of rising or setting as given by the above rule, and then the calculation, performed by the formula in art. 24, chap. IX. But this is seldom necessary. 2. The rising or setting of a planet or star may be found by calculating the semi-diurnal arc as in the above rule, and sub- tracting it from the time of the body’s passage over the meridian for the rising, and adding it, for the setting. EXAM. 1. Required the time of the sun’s rising and setting at Philadelphia, on the 25th of January, 1821, the declination at noon of that day being 18° 52' S. tan. Lat. 39° 577 gº ſº 9.92304 tan. Decl. 18 52 S gº - 9.53368 sin. Asc. Diff. I6 38 9.45672 4 1 h. 6 m. 32 sec. 6 0 &===ºmºsºms Semi-diur. arc 4 h. 53 m. time of sunset, 7 h 7 m. time of sunrise. \ 2. Required the time of the sum’s rising and setting at St. Petersburg, when the declimation is 23° 28' N. Ans. Sun rises at 2 h. 46 m. and sets at 9 h. 14 m. 3. At what time did the sun rise and set at Philadelphia, on the 21st of August, 1821; ſins. Sunrise 5 h. 19 m. and sunset 6 h. 41 m. PROBLEM XXV. To reduce the time of the JMoon's Passage over the JMeridian of Greenwich, as given in the Nautical fllmanac, to the time of its Passage over the JMeridian of any other Place. Take from the Nautical Almanac, the difference between the time of the moon’s passage on the given day, and the next fol- lowing or next preceding day, according as the place is in west 294 ASTRONOMY, or east longitude. Then take from table LVIII, the quantity cor- responding to this difference at the top and the difference of lon- gitude, in time, at the side. This quantity will be the reduction and being added to the time of the moon's passage over the meri- dian of Greenwich on the given day, when the place is in west longitude, but subtracted, when it is in east longitude, will give the required time of passage, in the time, reckoned at the given place. $ ExAM. 1. Required the time of the moon’s passage over the me- ridian, at Philadelphia, on the 17th of August, 1821. h. m. Passage at Greenwich, on the 17th, - 15 44 Reduction from table LVIII. - tºº º 11 Passage at Philadelphia, 17th. - - 15 55 Or in common reckoning, on the 18th at 3 h. 55 m. A. M. 2. What was the time of the moon’s passage over the meri- dian of Boston, on the 10th of August, 1821 ° ſºns. 9 h. 50 m. P. M. 3. Reduce the time of the moon's passage over the meridian, as given in the Nautical Almanac for the 21st of August, 1821, to the time of passage at New York. Ans. On the 22d day at 7 h. 45 m. A. M. in common reckoning. PROBLEM XXVI. From the JMoon's Declination, as given in the JNaulical fllmanac, for each noon and midnight, to find the Declination, nearly, for a given Time and Place. Reduce the given time to apparent time at Greenwich. Then, taking the change in the moon’s declination, for the 12 hours within which the time at Greenwich falls, find in table LIX, the quan- tities corresponding to the time at the side, and to the degrees, tems of minutes and minutes of the change in declination, at the top. The sum of these, added to the declination at the moon or midnight next preceding the time, when the declimation is in- ASTRONOMY., 295 creasing, but subtracted when it is decreasing, will give the re- quired declination. * Note 1. When the declinations at one moon or midnight, and at the following midnight or moon, are of different names, their sum is the change in declination for 12 hours. 2. When the sum of the quantities taken from the table is to be subtracted from the declination and is greater than it, the latter must be subtracted from the former, and the name changed from North to South, or from South to North. * ExAM. 1. Required the moon’s declination on the 15th of August, 1821, at 10 h. 25 m. P. M. apparent time at Phila- delphia. d. i. Iſle Time at Philadelphia, August, tº 15 10 25 Diff of long. sº tº-e wº wº 5 1 Time at Greenwich, - tº - 15 15 26 Declination, the 15th at midnight, - 0° 14' S do. 16th at noon, sº 3 14 N Change in 12 hours, gº sº gº 3 28 Declimation, the 15th, at midn. & - 0° 14' S Sum of quantities from table LIX - 0 59 $ºmmºns Required declin. - - - - 0 45 N. 2. Required the moon's declination, on the 18th of August, 1821, at 4 h. 10 m. P. M. apparent time at Philadelphia. ..ſins. 18° 15' N. * 3. Required the moon's declination, on the 2d of August, 1821, at 1 h. 28 m. A. M. apparent time at New York. Ans. 0° 59' S. 296 *- ASTRONOMY. J PROBLEM XXVII. \ To find the time of the moon's Rising or Selling at a given Place, on a given astronomical day, by the aid of the JNautical s Jälmanac. Find the time of the moon’s passage over the meridian of the given place by Prob. XXV. To, or from the time of the passage, according as the moon’s setting or rising is required, add or subtract 6 hours, and find, by the last problem, the moon’s declimation for the resulting time, reduced to the meridian of Greenwich. With the latitude of the place and the moon’s declimation, find the semidiurnal arc, as in Prob. XXIV, and apply it to the time of the moon’s passage over the meridian, by subtracting for the ri- sing, or adding for the setting; the result will be the approximate time of rising or setting. Find the moon’s declination for the approximate time of rising or setting, reduced to the meridian of Greenwich, and with this declination, again calculate the semidiurnal arc. Take the diſſerence between the times of the moon’s passage over the meridian of Greenwich, on the given day and the next preceding, or next following one, according as the rising or setting is required. From table LVIII, take the quantity corresponding to this difference at the top, and the semi-diurnal arc, last found, at the side. This quantity will be a correction, which, added to the semi-diurnal, will give the corrected semi-diurnal arc. Apply the corrected semi-diurnal arc to the time of the passage over the meridian of the given place, by subtracting for the moon’s rising, or adding for the setting; the result will be the required time, sufficiently accurate for all common purposes. Note. When it is required to make many calculations of the moon's rising or setting, for any particular place, they may be much abbreviated by little expedients, which it would be trouble- some to specify. It may however be observed that the operation is considerably facilitated by having a table of semi-diurnal arcs, calculated for the latitude of the place, similar to table LXIV, which is adapted to the latitude of Philadelphia. ASTRONOMY. 297 ExAM. I. Required the time of the moon’s rising at Philadel- phia on the 18th of August, 1821. * d. h. m. Passage over mer. of Greenwich, tº 18 16 37 Reduction, tº tº wº ºn- tº 12 cºmmºmºmº Passage over mer. of Philadelphia, - 18 16 49 Subtract - ſº sºn ºf tºº ſº 6 0 * g 18 10 49 Diff of Long. - - - , - 5 1 Time at Greenwich, - - - - 18 15 50 Moon's declin. on the 18th, at 15h. 50m. is 19° 40' S. tan. Lat. - 39° 57. ſº tº 9.92304 tan. Declin. - 19 40 N. sº - 9.55315 šin. Ascen, diff. 17 25 tº: º 9.47619 4 1h. 10m. 6 0 Semi-diur. arc, 7h, 10m. d. h. m. Moon's passage over mer. of Philadelphia, 18 16 49 Semi-diur. arc. - * - - ºf 10 V Approximate time of moon's rising, - 18 9 89 Diff of Long. - gº ſº tº tº 5 1 Time at Greenwich, gº * gº 18 14 40 Moon's declin. on the 18th, at 14h. 40m, is 19° 26' N. 39 298 - ASTRONOMY. tam. Lat. - 39° 57? tºº eº 9.92304 tan. Declin. - 19 26 N. º - 9.54754 sin. Ascen. diff. 17 11 ºf ~s 9.47058 4 1 h. 9 m. 6 0 *-* Semi-diur, arc, 7 9 Correction, 1 5 Semi-d. arc cor. 7 24 - l d. h. m. Moon's passage, - º tºr º 18 16 49 Corrected Semi-diur. arc, - gº wº "7 24 Time of moon’s rising, - t- tº 18 9 25 2. Required the time of the moon's setting, at Philadelphia, on the 11th of August, 1821. Jins. 15 h. 38 m., or in common reck- oning, on the 12th, at 3 h. 38 m. A. M. 3. Required the time of the moon’s rising, at New York, on the 21st of August, 1821. Jins. 11 h. 36 m. P. M. PROBLEM XXVIII. To find the Longitude and Altitude of the Nonagesimal Degree of the Ecliptic, for a given time and place. Find the reduced latitude of the place by problem XIII: and when it is north, subtract it from 90°, but when it is south, add it to 90°, for the reduced distance of the place from the north pole. Take half the difference between this quantity and the obliquity of the ecliptic: also, half the sum of the same quantities. From the Cosine of the half difference, subtract the Cosine of the half sum, and call the result, logarithm A. From the Tangent of the half difference, with the index increased by 10, subtract the Tangent of the half sum, and call the result, logarithm B. Also, call the Tangent of the lialf sum, logarithm C. \ ASTRONOMY. 299 For the given time, reduced to mean time at Greenwich, find the sum’s mean longitude and the argument N, from tables IX, X, XIII, and XIV. To the sun's mean longitude, increased by 2”, apply, according to its sign, the mutation in right ascension, taken from table XVIII, with argument N, and it will give the Sun's mean longitude, reckoned from the true equinox. To the Sun's mean longitude from the true equinox, add the mean time of day, at the given place, expressed astronomically and reduced to degrees, and reject 360° from the sum, when it exceeds that quantity. The result will be the right ascension of the mid-heaven.* From the right ascension of the mid-heaven, subtract 90°, the former being first increased by 360°, when necessary, and call half the remainder R. To the logarithm A, add the Tangent of R, and the result will be the Tangent of an arc E, which must be taken according to the sign, but less than 180°. To the Tangent of E, add the loga- rithm of B, rejecting 10 from the index, and the result will be the Tangent of an arc F, which must also be taken according to the sign, and less than 180°. The sum of the arcs E and F, and 90°, rejecting 360°, when the sum exceeds that quantity, will be the longitude of the nonagesimal degree. Add together the logarithm C, the Cosime of E, and the arith- metical complement of the Cosine of F, and reject 10 from the index: the result will be the Tangent of half the altitude of the nonagesimal degree. Note 1. The above rule, which differs but little in substance from that given by Bowditch in his Practical Navigator, is gene- ral for all places, except within the North polar circle. And the only difference there, is, that for the longitude of the monagesimal * When the sun's true longitude has been previously calculated for the same time, for which the right ascension of the mid-heaven is wanted, it is evident the tabular mean longitude and the argument N, are already known. It may also be observed, that the right ascension of the mid-heaven is equal to the sum of the sun’s true right ascension, and the apparent time ex- pressed astronomically and reduced to degrees: 360° being rejected when the sum exceeds that quantity. - 300 ASTRONOMY. degree, 90° must be added to the arc E, and the arc F subtracted from the sum. .# 2. When the longitude and altitude of the nonagesimal degree are required, at any given place for several different times in the same day, which is generally the case, the same logarithms, A, B and C, when they have been once found, will answer for all the other operations. Indeed, the obliquity of the ecliptic changes so slowly, that except great accuracy is required, the same loga- rithms may be used in calculations, for a time several years dis- tant from the time for which they were obtained. 3. The last part of the above rule gives the distance of the zenith of the place from the north pole of the ecliptic, which is not always the real altitude of the monagesimal. Generally in the southern hemisphere, and frequently in the northern hemisphere, near the equator, it is the supplement of the altitude. But it sim- plifies the rule for the parallaxes, to which this problem is preli- minary, and produces no error, to use the same term in all cases. ExAM. 1. Required the longitude and altitude of the nomagesi- mal degree of the ecliptic, at Philadelphia, on the 27th of August, at 7 h. 30 m. 21 sec. A. M. mean time, the obliquity of the eclip- tic being then 23° 27' 55". The reduced latitude of Philadelphia, found by problem XIII, is 39° 45' 48" N, and this taken from 90°, leaves the polar dis- tance 50° 14' 17"; the difference and sum of this quantity and the obliquity of the ecliptic are 26° 46' 22" and 73° 42' 12"; half difference 13° 23' 11"; half sum 36° 51’ 6”. # diff. 13° 23' 11" COS. 9.98803 tan. + 10, 19.37654 # sum 36 51 6 cos. 9.903.19. tan. C. 9.87.478 sº A. 0.08484 B. 9.5oing The Sun's longitude taken from the tables, for the given time, and increased by 2”, is 5'5" 24' 38", and the argument N is 71. The mutation, taken from table XVIII, with argument N, is + 7”. Hence, the sum's mean longitude from the true equinox is 5' 5" 24'45", or 155° 24′ 45". The given time of day expressed ASTRONOMY. 30ſ. astronomically, is 19 h. 30 m. 21 sec.; which, in degrees, is 2.92* 35' 15". Given time, in degrees, & a 29.2° 35' 15" Sun's mean long. - tº - 155 24 45 e ———— Right ascen, mid-heaven, - 88 0 0 90 0 0 2)358 0 0 R. 179 0 0 A. 0.08484 R. 179° 0' 0" tan, – 8.24.192 E 178° 47' 4" tan. — 8.32676 cos. – 9.99990, B. 9.501.76 C. 9.87.478 F^179 36 50 tan. —7.82852 Ar. Co. cos. – 0.00001 90 0 0 emºsºm-ºsmºmºmºmº # alt. nom. 36° 50' 47" tan, 9.87469 88 23 54 long. nonages. games-m-m-m-m- 73 41 34 alt. nonages. 2. Required the longitude and altitude of the monagesimal degree of the ecliptic at Philadelphia, on the 27th of August, 1821, at 8 h. 53 m. 20 sec. A. M. mean time. Jins. Long. 105° 2' 18", and alt. 72° 43' 32". 3. Required the longitude and altitude of the nonagesimal de- gree, at Philadelphia, on the 27th of August, 1821, at 10 h. 14 m. A. M. apparent time. Jins. Long. 121° 21' 25", and alt. 69° 30° 44''. PROBLEM XXIX. The Longitude and flltitude of the JNonagesimal Degree of the Ecliptic, and the JMoon's True Longitude, Latitude, Equatorial Parallaw, and Horizontal Semidiameter being given, to find the Apparent Longitude and Latitude as affected by Parallaw, and the Jługmented Semidiameter of the JMoon, for a given place. 302 ASTRONOMY. Find the reduction of parallax, by problem XIII, and subtract it from the equatorial parallax; and in eclipses of the Sun, subtract from the remainder, the sun's parallax, which is 8".7, or 9” may be used without material error. Call the last remainder the Reduced parallax. In occultations of a fived star, the first remain- der is the reduced parallax. Take the difference between the moon's longitude and the longitude of the monagesimal degree, and call it D. When the moon’s latitude is north, subtract it from 90°, but when it is soulh, add it to 90°; the difference or sum will be the moon’s distance from the north pole of the ecliptic, which call d. Call the alti- tude of the monagesimal h, and the reduced parallax P. | Of the two following methods of finding the apparent longitude and latitude, it may be observed, that the first is general, and may be used either in eclipses or occultations. The second is applica- ble, only in eclipses of the Sun, or when it is known that the appa- rent latitude is small. It is more concise than the first, and though not quite so accurate, yet the errors will seldom exceed 2 or 3 tenths of a second. In working by either method, the student must observe, that when logarithms are directed to be added together, the tens in the resulting index are to be rejected. When the loga- rithm of an arc is to be taken, the arc must first be reduced to seconds; and when an arc is found, corresponding to a logarithm, it is seconds. FIRST METHOD, Which may be used, either in Eclipses of the Sun, or in Occulta- tions. Add together the logarithm of P, the Sime of h, and the arith- metical complement of the Sime of d, and call the resulting loga- rithm c. To the logarithm c, add the Sine of D, and the result will be the logarithm of an arc u. Add together the logarithm c, and the Sine of (D -- w), and the result will be the logarithm of an arc wº. Add together the logarithm c, and the Sine of (D + w), and the result will be the logarithm of p, the parallax in longitude.. * ASTRONOMY. 303 Except when great accuracy is required, the last operation meed not be performed, and p may be placed instead of w. Jidd p to the moon’s true longitude, when the latter is greater than the longitude of the monagesimal, but subtract, when it is less, and the result will be the apparent longitude. When the apparent latitude is necessarily small, as in eclipses of the sun, add together the logarithm of P, and the Cosime of h, and the result will be the logarithm of an arc al. But in occulta- tions, add together the logarithm of P, the Cosime of h, and the Sine of d, and the result will be the logarithm of an arc v. To d, add v, attending to the sign of the latter. Then add together the logarithm of v, the Sine of (d. 4- v), and the arithmetical comple- ment of the Sine of d, and the result will be the logarithm of the al'C Q}. To d, add aſ, attending to the sign of the latter. Then add to- gether, the logarithm of P, marked negative, the Sine of h, the Cosine of (D + 3 p.), and the Cosime of (d. 4- a), and the result will be the logarithm of an arc z. The arc z, applied according to its sign, to the sum of d and w, will give the apparent polar dis- tance. And the difference between this and 90°, will be the ap- parent latitude, which will be north or south, according as the polar distance is less or greater, than 90°. The sum of a and 2, regard being had to their signs, will be the parallax in latitude. Add together the logarithm of the moon’s horizontal semi- diameter, the Sine of the apparent polar distance, the Sine of (D + w), the arithmetical complement of the Sine of d, and the arithmetical complement of the Sine of D, and the result will be the logarithm of the augmented semidiameter. SECOND METHOD, A Which can only be used when the ſpparent Latitude is small, as in Eclipses of the Sun. Add together, the logarithm of P, the Cosine of h, and the arithmetical complement of the Sine of d, and the result will be 30+ ASTRONOMY. the logarithm of an arc v. Add together, the logarithm of a, the Tangent of h, and the Sine of D, and the result will be the loga- rithm of an arc u. Add together, the logarithm of w, the Sine of (D + w,) and the arithmetical complement of the Sine of D, and the result will be the logarithm of p, the parallax in longitude. Take the sum of d and a, attending to the sign of the latter. Then, add together, the logarithm of p, the logarithm of the difference between (d. -- a) and 90°, the arithmetical comple- ment of the logarithm of u, and the arithmetical complement of the Sine of d, and the result will be the logarithm of the apparent latitude, which will be north or south, according as (d. 4- a), is less or greater than 90°. - The parallax in longitude, p, added to the moon’s true longi- tude, when the latter is greater than the longitude of the monagesi- mal, but subtracted, when it is less, gives the apparent longitude. Add together, the logarithm of the moon’s horizontal semi- diameter, the Sine of (D + w), the arithmetical complement of the Sine of d, and the arithmetical complement of the Sime of D, and the result will be the logarithm of the augmented semi- diameter.* Note. In eclipses of the sun, it is not strictly the apparent longitude and latitude of the moon that are found by the preceding rules, but the values of those quantities, including the sun’s pa- rallax in longitude and latitude, which are the values wanted in the calculation. ExAM. 1. About the time of beginning of the eclipse of the sun, on the 27th of August, 1821, the longitude of the nomagesi- mal degree, at Philadelphia, was 88° 23'54", the altitude 73°41’ 34", moon’s true longitude 152° 31' 1", true latitude 0°11' 38" N, equatorial parallax 55' 18", and semidiameter 15' 3"; re- quired the apparent longitude and latitude, and the augmented semidiameter. * The rules in the first method, are deduced from formulae C, G, and L. of articles 54, 56, and 57, chap. X. Those in the second, from C, F, and L. of articles 54, 55, and 57. ASTRONOMY. 305 Equat, par. 55° 18' Moon's long. 152°31' 1" Reduction, 5 Long. nonag. 88 23 54 55 13 D = 64 7 7 Sun's paral. 9 h = 73 41 34 P = 55 4 d = 89 48 22 BY THE FIRST METHOD, P sº * 3304// - tº log. 3.51904 h - 78° 41' 34" - - sin. 9.982.17 d º - 89 48 22 - Ar. Co. sin. 0.00000 c. 3.50.121 D - - 64 7 7 wº iº sin 9,954.10 'll, tº º 2853” tº - log. 3.45531 c. 3,501.21 D + w - 64 54 40 tºº - sin. 9.95696 tº' tº º 2872” - tº log. 3.45817 c. 3,501.21 D + w( - 64 54 59 -. - sin. 9.95698 - 47' 52”.1 - log 3.45819 p - True long. 152 31 1 App. long. 153 18 53.1 {} P h 3C ; 2. tºº tºº gº º tº a gº - log. 3.51904 - 78° 41' 34" - gº cos. 9.44838 - - 15 27.7 - - log 2.96742 9. tº - " - - - log. — 3,51904 wº º - sin. 9.982 17 # p - 64 31 3 - ſº cos, 9,633.71 a - 90 3 49.7 - cos. – 7.04434 tº gº 1.5 iº - log, 0.17926 40 ASTIRONOMY. Ap. pol. dist. 90° 3' 51".2 90 0 0 Ap. lat. - 0 , 3 51.2 S. Hor. semidiam. - 903" * - log. App. pol. dist. 90° 3' 51" - ſº 2.95.569 D + w * tºº sº &º - sin. 9.95696 d - sº tºº º - Ar. Co. sin. 0.00000 D tº * , tºº sº Ar. Co. sin. 0.04590 Augmented semidiam. 15' 9" tº log. 2.95855 BY THE SECOND METHOD. P tº º * 3304" . tº log. 3.5.1904 h - - 73° 41' 34" gº cos. 9.44838 d tº - 89 48 22 - Ar. Co. sin. 0.00000 {} - tºº 15 27.7 - - log. 2.96742 h sº - 73 41 34 - tºº tan, 10.53379 D - - 64 "I '7 gº - sin. 9.95.410 Qſ, dº tºg 47 33 * - log. 3.45531 D - gº wº gº - Ar. Co. sin. 0.04590 D + w - 64 5440 * - - sin. 9.95696 p sº - 47 52 - * log. 3,45817 (d. 4- a)—90°, 3 49.7 ſº - log. 2,361.16 Q!, - º º - Ar. Co. log. 6.54469 d gº tº gº tº e Ar. Co. sin. 0.00000 App. lat. gº 3 51.2 S. - log. 2.36402 Moon's true long. - 152° 31' 1" p - gº; sº ſº tº 47 52 App. long. - sº sº 153 18 53 sin. 10.00000 ASTRONOMY. 307 Hor. semidiam. - 903" - tº -D + w - º - º tº d - tº- º - - Ar. Co. D - - - - Ar. Co. Augm. semid. - 15' 9" - - log. sin. sin. sin. log. 2.95.569 9.95696 0.00000 0.04590 2.95855 2. Given the longitude of the nomagesimal 67°29'8", the alti- tude 57° 56' 36", the moon’s true longitude 3, 18°27'35", lati- tude 4° 5' 30" S, reduced parallax 61' 1", and horizontal semi- diameter 16' 40”; to find the moon’s apparent longitude, latitude, and augmented semidiameter. / D = 40° 58' 27", h = 57° 56' 36", d = 94° 5' 30", P = 61' 1". * / Ar. Co. P *- tº 3661” - h - - 57° 56' 36" - " - d - - 94 530 - D tº 40 58 27 º, 70 - º 34 0 - D +w - 41 32 27 - w' tº- - 34 23 - D + w! - 41 32 50 - p -> º 34 23.1 -y True long. 3, 18 27 35 App. long. 3 19 1 58.1 log. sim. sin. sin. log. sin. log. sin. zº | log. 3.56360 9.92815 0.00111 3.49.286 9,81672 3.30958 3.49286 9,82161 3.31.447 3.49.286 9,8216? 3.31453 308 ASTRONOMY. P tº º tº- tºº log. 3.56360 h - - 57° 56' 36” g- - cos. 9.724.90 d - - 94 5 30 - Y- sin. 9.99889 Q) - tº 32 18 - - log. 3.28789 d -- v - 94 37 48 - *º sin. 9.99858 d tº º º Ar. Co. sin. 0.00111 a - - 32 16.8 - - log. 3.28708 P º º t- - log. — 3.56360 h - -. sº º sin. 9.92815 D + 3 p - 41 15 39 - - cos. 9.87605 d + a - 94 37 46.8 tº cos. – 8.90696 2: tº - + 3 8.3 - - log, 2.27476 Ap. pol. dist. 94.40 55.1 *-*. 90 0 0 App. lat. 4 40 55.1 Hor. semidiam. 1000" - - log. 3.00000 App. pol. dist. 94° 40' 55" - - sin. 9.99855 D + w - tº- ſº tº- º sin. 9.82161 d º tº º tº Ar. Co. sim. 0.00111 D - º - º - Ar. Co. sim. 0.18328 Augm. semidiam. 16' 50".8 - log. 3.00455 f 3. About the middle of the eclipse of the sun, on the 27th of August, 1821, the longitude of the monagesimal, at Philadelphia, was 105° 2' 18", the altitude 72° 43' 32', moon’s true longitude 153° 13' 52", latitude 0° 7' 42" N. reduced parallax 55' 12", and semidiameter 15' 3"; required the apparent longitude and latitude, and the augmented semidiameter. Ans. App. long. 153° 53'27"; app. lat. 0° 8'44" S.; augm. semidiam. 15' 12".2. 4. About the end of the eclipse of the sun, on the 27th of Au- gust, 1821, the longitude of the monagesimal, at Philadelphia, was 121° 21' 25", altitude 69° 30'44", moon’s true longitude 153° ASTRONOMY. 309 56' 15", latitude 0° 3' 47" N, reduced parallax 55' 10", and . semidiameter 15' 3"; required the apparent longitude and lati- tude, and the augmented semidiameter. Ans. App. long. 154°24' 21"; app. lat. 0° 15' 40" S; augm. semidiam, 15'14",4 PROBLEM XXX. To find from the Tables, the Time of New or Full JMoon, for a given Year and JMonth. For JNew JM00m. Take from table IV. the mean new moon in January, for the given year, and the arguments I, II, III, and IV. Take from table V, as many lunations, and the corresponding arguments I, II, III, and IV, as the given month is months past January, and add these quantities to the former, rejecting the ten thousands in the first two arguments, and the hundreds in the other two. Take the number of days corresponding to the given month, from the second or third column of table VI, according as the given year is a common or a bisseatile year, and subtract it from the sum, in the column of mean new moon; the remainder will be the tabular time of mean new moon, in the given month. If the number of days, taken from table VI, is greater than the sum of the days in the column of mean new moon, as will sometimes be the case, one lunation more than is directed above, with the corresponding ar- guments, must be added. With the arguments I, II, III, and IV, take the corresponding equations from table VII, and add them to the time of mean new moon; the sum will be the ſlpproximate time of new moon, ex- pressed in mean time at Greenwich. For the approximate time of new moon, calculate by problems VII and XI, the true longitudes and hourly motions in longitude of the sun and moon. Take the difference between the longitudes, and also between the hourly motions. Then, as the difference between the hourly motions : the difference between the longi- tudes :: 60 minutes : the correction. The correction, added to the approximate time of new moon, when the sum's longitude is greater than the moon's, but subtracted, when it is less, will give 310 * ASTRONOMY. the true time of new moon, expressed in mean time at Greenwich. This time may be changed to apparent time, at any given meri- dian, by problems VI and W. For Full JM00m. When the time of mean new moon in January of the given year is on, or after the 16th, subtract from it, and the arguments I, II, III and IV, a half lunation, with the corresponding argu- ments, taken from table V, increasing when necessary, either or both of the first two of the former by 10,000, and of the two latter by 100; but add them, when the time is before the 16th. The result will be the tabular time of mean full moon in January, and ...the corresponding arguments. Proceed to find the approximate time of full moon, in the same manner as directed for the new moon.* Calculate the true longitudes and hourly motions in longitude of the sun and moon, for the approximate time of full moon. Subtract the sun's longitude from the moon's, and call the re- mainder R. Also, subtract the hourly motion of the sun from that of the moon. Then, as the difference of the hourly motions : the difference between R and VI signs :: 60 minutes : the correction. The correction, added to the approximate time of full moon, when R is less than VI signs, but Sublracted, when it is greater, will give the true time of full moon. ExAM. 1. Required the time of New Moon in August, 1821, expressed in apparent time at Philadelphia. * When the half lunation and arguments are to be added, the addition may be left till the proper number of lunations, with their corresponding arguments, are placed under, and thus make one addition serve. ASTRONOMY. 31%. M. New Moon. I. II. III. IV. * d. h. m. g * t w 1821, 2 17 59 0092 7859 80' | 78 8 Jun. 236 5 52 6468 5737 22 93 238 23 51 6560 || 3596 || 02 || 71 Days, 212 August, 26 23 51 H. 0 54 II. 2 13 III. 9 IV. 10 August, 27 3 17 | Approximate time. Sun's true long, found for the approx. time, is 5' 3°57' 12 Moon’s do. 4. tº ſº 5 3 56 43 Difference, tºº tº *g tºº 0 29 Moon's hourly motion in long. is * 30 55'. Sun’s do. - gº - 2 25 Difference, tº ſº sº 28 30 Iſl. Iſl. SeC. As 28' 30" : 29" :: 60 : 1 1, the correction. d. h. m. sec. Approx. time of new moon, August, 27 3 17 O Correction, * * gºes tº tº- + 1 1 True time, in mean time at Greenwich, 27. 3 18 1 Equation of time, gº ſº # = ſº — 1 19 Apparent time at Greenwich, -, 27 3 16 42 Diff of Meridians, º tºº tº 5 0 46 Apparent time at Philadelphia, - 26 22 15 56 342 ASTRONOMY. 2. Required the time of Full Moon in July, 1823, expressed in apparent time at Philadelphia. M. New Moon.} I. II. , , III. IV. d. h. m. 1823, 11 0 20 0304 || 5787 61 55 # lum. || 14 18 22 404 || 5359 58 || 50 6 lum. 177 4 24 || 4851 || 4303 || 92 || 95 goº & Tºo Tsujii To Days, 181 July, 21 23 6 I. 2 55 II. 13 7 III. 5 IV. 20 | July, l 22 15 33 || Approximate time. Moon's true long, found for the approx. time, is 9°29°24' 51" Sun’s do. tº -> 3 29 25 23 R. 5 29 59 28 6 0 0 0 Diff. 0 32 Moon's hourly motion in long, is - 29' 34" Sum’s do. º tº - -, - 2 23 Difference, - & sº º sº 27 11 IY). Iſl. SęC. As 27' 11" : 32" :: 60 : 1 11, the correction. \ ASTRONOMY, 343 d. h. m. sec. Approximate time of full moon, July, 22 15 33 0 Correction, de wº º + 1 11 True time, in mean time at Greenwich, 22 15 34 11 Equation, - º º — 6 2 Apparent time at Greenwich, - 22 15 28 9 Diff. of meridian, - º * 5 0 46 Apparent time at Philadelphia, - 22 10 27 23 3. Required the time of New Moon in July, 1821, expressed in apparent time at Philadelphia. Jins. 28 d. 9 h. 9 m. 58 sec. P. M. 4. Required the time of Full Moon in July, 1821, expressed in apparent time, at Philadelphia. Jins. 14 d. 11 h. 17 m.47 sec. PROBLEM XXXI. To find the Time of New or Full JMoon in a given JMonth, by the JNautical fllmanac. The times of new and full moon are given to the nearest minute, on the first page of each month, in the Nautical Almanac. To find the time of either, to seconds, call the hours and minutes of the time given in the Almanac, or their excess above 12, hours, T. For JNew JMoon. Take the two longitudes of the moon, for the midnight and noon, or moon and midnight, next preceding the time given in the Al- manac, and also the two immediately following, and place them in order, one below another. Do the same with the sun's longi- tudes for the same times, observing that the sun’s longitude at midnight is half the sum of the longitudes, at the preceding and following moons. Subtract each longitude of the sun, from the corresponding longitude of the moon, noting the signs; the re- mainders will be the distances of the moon from the sun at those 44 3f4 ASTRONOMY. • t times. Subtract each of these distances from the one next fol- lowing, and the remainder will be the first differences. Call the middle one of these A. Subtract each first difference from the next following one, for the second differences. Take the mean, or which is the same thing, the half sum of the second differences, aſid call it B. With B at the top, and the time T, at the side, take from ta- ble LVI, the equation of second differences, and apply it with the same sign as B, to the second of the distances, taken affirmative, and call the result D. Then, A : D :: 12 hours : time of new moon. The time thus obtained will be apparent time at Green- wich, and it may be reduced to any other meridian by prob. W. For Full JMoon. Proceed exactly as for the new moon, except that each of the sun's longitudes must be increased by WI signs. Note. The times of the first and third quarters may be found, to seconds, in the same manner, except that the sun's longitudes must be increased by III or IX signs, instead of WI. ExAM. I. Required the time of new moon in August, 1821, by the Nautical Almanac. In this example T is 3h. 17 m. | ()'s Long. | (C’s Long. Distances. | 1st Diff. |2d IDiff. 26th midn.| 4s 26° 2' 0// 5s 3920/25/|— 7o 18/25// 5944/44// Q/ ()// Žinoon 5 #1; 44 |554; 25 H-1 3: Ai Aiºi." –º 37th midn.|5 & 26 38 || 3 4 1333 || 3 || 3 || 3 |AT; 4 G| —249 28th noon | 5 14 34 24 || 5 447 26 |+ 9 46 58 B.-254 . Second distance tº 1° 33'- 41’’ * Equat. 2d diff. - - - — 17 D. 1 33 24 As 5° 41' 44” : 19 33' 24" :: 12h. : 3 h. 16 m. 51 sec., , time of new moon, in apparent time at Greenwich, f - ASTRONOMY. 315 2. Required the time of full moon, in August, 1821, by the Nautical Almanac. Jins. 13th day, at 9 h. 7 m, 22 Sec. A. M. apparent time at Philadelphia. PROBLEM XXXII. \ To determine what Eclipses may be eaſpected to occur in any given year, and the Times nearly, at which they will take place. For the Eclipses of the Sun. : Take, for the given year, from table IV, the time of mean mew moon in January, the arguments and the number N.” If the number N differs less than 53, from 0, 500, or 1000, an eclipse of the sun may be expected at that new moon. If the difference is less than-87, there must be one. When the difference is between 37 and 53, there is a doubt, which can only be removed by cal- culation. If an eclipse may or must occur in January, calculate the ap- proximate time of new moon by problem XXX, and it will be the time nearly, at which the eclipse will take place, expressed in mean time at Greenwich. This time may be reduced to the me- ridian of any other place by problem V. Look in column N of table V, and, excluding the number be- longing to the half lunation, seek the first number that, added to the number N of the given year, will make the sum come within 53, of 0, 500, or 1000. Take the corresponding lunations and arguments, and this number N, and add them to the similar quan- tities for the given year. Take from the second or third column of table VI, according as the given year is common or bisseatile, the number of days next less than the sum of the days in the column of mean new moon, and subtract it from the time in that column; the remainder will be the tabular time of mean new moon in the month corresponding to the days, taken from table VI. At this mew moon an eclipse of the sun may be expected; and if the sum of the numbers N, differs less than 37 from the numbers mention- *The number N in this table, designates the sun’s mean distance from the moon's ascending node, expressed in thousandth parts of the circle. 316 ASTRONOMY, ed above, there must be one. Find the time nearly, of the eclipse, by calculating the approximate time of new moon as directed above. If there are any other numbers in the column N of table V, that when added to the number N of the given year, will make the sum come within the limit 53, proceed in a similar manner to find the times of the eclipses. Note. When the time at which an eclipse of the sun will take place is thus found, nearly, and reduced to the meridian of a given place in north latitude, if it comes during the day time, and if the sum of the numbers N, or the number N itself when the eclipse is in January, is a little above 0, or a little less than 500, there is a probability that the eclipse will be visible at the given place. When the number N in January, or the sum of the numbers N, in other months, is more than 500, the eclipse will seldom be visible in northern latitudes, except near the equator. For the Eclipses of the JMoon. When the time of new moon in January of the given year is on, or after the 16th, subtract from it, from the arguments, and the number N, a half lunation, the corresponding arguments, and the number N; but when it is before the 16th, add them. The results will be the time of mean full moon in January, and the corresponding arguments, and number N. Proceed to find the times at which, eclipses of the moon, may or must occur, exactly as directed for the sum, except that the limits 35 and 25, must be used instead of 53 and 37. Note. In an eclipse of the moon, when the time is found nearly, and reduced to the meridian of a given place, if it comes in the might, it will be visible at that place. ExAM. 1. Required the eclipses that may be expected in th: year 1822, and the times nearly, at which they will take place. ASTRONOMY. 317 1822, 1 lun. Feb. II. III. Feb. 1822, 7 lun. August, H. . II. Iii. IV. August, For the Eclipses of the Sun. M. New Moon.] I. II. III. IV. N | —r d. h. m. - | 21 15 32 || 0602 || 7182 | 78 66 |930 29 12 44 H S08 || 717 | 15 99 || 85 51 4 16 || 1410 || 7899 || 93 || 65 15|| 31 As th f the numbers N, 20 ; : cº, jū."; ºf §h...". be i 19 29 an eclipse. 13 ** 11 21 7 47 Mean time at Greenwich. M. New Moon. I. II. III. IV. N. d. h. Iſl. S 21 15 32 0602 || 7182 78 | 66 930 206 17 8 5659 5020 || 7 || 94 596 22s 840 6261 2202 || 85 |60 | 526 | 212 16 8 40 As the sum of the numbers N, comes 1 24 within 37 of 500, there must be an 0 40 eclipse, 16 14 16 11 14 || Mean time at Greenwich, 348 ASTRONOMY. M. New Moon. I. II. III. IV. N. \ d. h. m. 1822 21 15 32 0602 || 7182 78 66 || 930 ; lum. / 14 18 22 404 || 5359 58 || 50 43 6 21 10 || 0198 || 1823 20 | 16 || 887 'i lum. / 29 12 44 808. 717 15 99 85 36 9 54 || 1006 || 2540 || 35 | 151 972 31 As the s F t bers N, altl h it Feb. 5 9 54 come j."; jºid.". In Ot ...'. I. 6 52 in 25, the eclipse may be considered doubtful. II 0 20 It may, however, be observed, that further º calculation by the next problem would show | sº that there will be a small eclipse. Feb. 5 17 39 Mean time at Greenwich: \ M. Full Moom. I. II. III. IV. N. d. h. m. 1822 6 21 10 0198 1823 20 16 887 7 lum. 206 17 8 5659 5020 7 || 94 596 213 14 18 5857 | 6843 27 | 10 || 483 212 August, 1 14 18 As the sum of the numbers N, comes I 2 14 within 25 of 500, there must be an Ii. 19 26 eclipse. III. 3 IV. 26 August, 2 12 27 | Mean time at Greenwich. 2. Required the eclipses that may be expected in 1823, and the times nearly, at which they will take place, expressed in mean time at Greenwich. Jins. One of the moon on the 26th of January, at 5h, 24 m. P. M.; one of the sun on the 11th of February, at 3 h. 12 m. A. M.; one of the sun on the 8th of July, at 6h. 50 m. A. M.; and one of the moon on the 23d of July, at 3 h. 33 m. A. M. - ASTRONOMY. 319 PROBLEM XXXIII. To Calculate an Eclipse of the JMoon. Find the approximate time of full moon, by prob. XXX, and ſor this time, calculate the sun's longitude, semidiameter and hourly motion, and the moon's longitude, latitude, equatorial pa- rallax, semidiameter and hourly motions in longitude and latitude. Then find the true time of full moon as directed in prob. XXX, and reduce it to apparent time at the place for which the calcula- tion is to be made. Call the reduced time, T. W For the JMoon's Latitude at the True Time of Full JMoon. As 1 hour : correction for the time of full moon :: moon's hourly motion in latitude : correction of latitude. When the true time of full moon, expressed in mean time at Greenwich, is later than the approximate time, the correction of latitude must be added, if the latitude is increasing,” but subtracted, if it is de- creasing; but when the true time is earlier than the approximate time, the correction must be subtracted, if the latitude is increasing, but added, if it is decreasing. The result will be the moon's lati- tude at the true time of full moon. For the Semidiameter of the Earth's Shadow. To the moon’s equatorial parallax, add the sun's, which may be taken 9", and from the sum, subtract the semidiameter of the sum. Increase the result by a gº part, and it will be the semi- diameter of the earth’s shadow, which call S. For the Inclination of the JMoon's Relative Orbit. . To the arithmetical complement of the logarithm of the differ- ence between the hourly motions in longitude of the moon and sun, add the logarithm of the moon’s hourly motion in latitude, and the result will be the Tangent of the inclination, which call I. * When the moon’s latitude is north, tending north, or south, tending south, it is increasing; but when it is north, tending south, or south, tending north, it is decreasing. 320 ASTRONOMYe Add together the constant logarithm 3.55680, the Cosime of I, and the arithmetical complement of the difference between the hourly motions of the moon and sum, in longitude, rejecting the tens in the index, and call the resulting logarithm R. For the Time of the JMiddle of the Eclipse. Add together the logarithm R, the logarithm of the moon's lati- tude at the true time of full moon, and the Sine of I, rejecting the tens in the index, and the result will be the logarithm of an interval t, in seconds of time, which, added to T, when the lati- tude is decreasing, but subtracted, when it is increasing, will give the time of the middle of the eclipse. For the Times of Beginning and End. To the logarithm of the moon's latitude at the true time of full moom, add the Cosime of I, rejecting the tens in the index, and the result will be the logarithm of an arc, which call c. Call the moon's semidiameter, d. To, and from, the sum of S and d, add and subtract c. Then add together the logarithms of the results, S + d -- c and S + d — c, divide the sum by 2, and to the quotient add the logarithm R, and the result will be the logarithm of an interval a, in seconds of time, which subtracted from, and added to, the time of the middle, will give the times of the beginning and end. Note. If c is equal to, or greater than the sum of S and d. there can not be an eclipse. For the Times of Beginning and End of the Total Eclipse. To, and from, the difference of S and d, add and subtract c. Them add together the logarithms of the results, S – d -- c and S—d — c, divide the sum by 2, and to the quotient add the logarithm R, and the result will be the logarithm of an interval s', in seconds of time, which subtracted from, and added to, the time of the middle, will give the times of the beginning and end of the total eclipse. Note. When c is greater than the difference of S and d, the eclipse can not be total. Astronomy. $2ſi For the Quantity of the Eclipse. Add together the constant logarithm 0.77815, the logarithm of (S + d – c), and the arithmetical complement of the logarithm of d, rejecting the tens in the index, and the result will be the logarithm of the quantity of the eclipse, in digits. Note 1. In partial eclipses of the moon, the southern part of the moon is eclipsed when the latitude is north, and the northern part when the latitude is south, 2. When the eclipse commences before sunset, the moon rises about the same time the sun sets. To obtain the quantity of the eclipse nearly, at the time the moon rises, take the difference be- tween the time of sunset and the middle of the eclipse. Then, as 1 hour : this difference : ; difference between the hourly motions of the moon and sun, in longitude : a fourth term. Add together the squares of this fourth term and of the arc c, both in seconds, and extract the square root of the sum. Use this root instead of c, in the above rule, and it will give the quantity of the eclipse at the time of the moofi's rising, very nearly. When the eclipse ends after sunrise in the morning, the quantity at the time of the moon’s setting may be found in the same mammer, only using sunrise in- stead of sunset. ExAM. 1. Required to calculate, for the meridian of Philadel- phia, the eclipse of the moon, in July, 1823. The approximate time of full moon, is July 22, at 15h. 33 m. Sun' longitude at that time, 3, 29°25'23" Do. hourly motion, º º 2 23 Do. semidiameter, sº - 15 46 Moon's longitude, º 9 29 24 51 I}o. latitude, - de 9 10 N. Do, equatorial parallax, - 54 1 Do. semidiameter, - d. 14 43 Do. hor. mot, in long. - 29 34 Do. do. in lat. - - 2 43, tendingnorth, e 42 322 ASTRONOMY. d. h. m. sec. Approx. time of full moon, July, , 22 15 33 0 Correct, found by prob. XXX. -> + 1 11 True time, in mean time at Greenwich, 22 15 34 11 Equat. of time, - º º — 6 2 Apparent time at Greenwich, - 22 15 28 9 Diff of Long. dº -> * 5 0 46 Apparent time at Philadelphia, T. 22 10 27 23 II]. Iſl. S&C. * As 60 : 1 11 :: 2'43" : 3", the correct. of lat. Moon's lat. at approx. time, gº 9' 10"N, Correction, - º º + 3 f : Moon's lat. at true time, - º 9 13 N. Moon's equatorial parallax, cº 54! 1" Sun's do. º tº - 9 Sum, - wº wº-> - 54 10 Sun's semidiameter, - º 15 46 38 24 Add º ** ºg tº 0 38 Semidiam. of earth's shadow, - S. 39 2 Moon's hor. mot, less sun's, 1681". Ar. Co. log. 6.78755 Moon's hor. mot. in lat. 168 º log. 2.21219 1 - - - 548 - - tan, 8.89914 ASTRONOMY. 323 3.55630 I - - - 5 42 - cos. 9.99785. Moon's hor, mot, less sun's, Ar. Co. log. 6.78755 - • R. 0.341.70 Moon's lat., - º 553” - log. 2.74272 I tº - - 5°42' º cº sin. 8.997.04 t 121 sec. = 2 m. 1 see. - log. 2,08146 T - 10 h. 27 m: 23 sec. Middle, 10h. 25 m. 22 sec. *: Moon's lat. - - º * log 2.74272 I - dº * * tº - cos. 9.99785 e - 550 – 9 10 - - log 2.4057 S4 d 4 c - 3775" - log. 3.57692 S + d –c _ • 2675 . - ... log. 3.42732 * 2)7.00424 3.50212 R. 0.84170 sec. h.m. sec. ſº a = 6980 = 1 56 20 - - - log. 3.84.382 h. m. sec. i Middle, - - - 10 25 22 * - - - - - 1 56 20 Beginning, §º dº 8 29 2 , S End, " - - - 12 21 42 A.M. of 23d day. S— d -- c * . 2009” - log. 3,30298 S—d—c. - 909 gº log. 2.95856 ---. 2)6.26154 3.13077 R. 0.84170 g sec. In, S60, tº º a' = 2968 = 49 28 iſ ºs - log, 3,47247, 3% ASTRONOMY. h. m. sec. Middle, '• sº 10 25 22 a’ - - sº º 49 28. Beginning of the total eclipse, 9 35 54 End do. - - 11 14 50 * 0.77815 S + d – c sº • sº log, 3.42732 d - tºp 883'' Ar. Co. log. 7.05404 Digits eclipsed, - 18.2 - - log. 1.25951 2. Required to calculate for the meridian of Philadelphia, the eclipse of the moon, on the 2d of August, 1822. wins. Moon rises about sunset, 83 digits eclipsed, Echiptic opposition, - - 7h. 16m. A Middle, ſº wºe sºng - 7 23 End, - tº ſº ſº 8 55 Digits eclipsed 9, on moon's northern limb. 3. Required to calculate for the meridian of Philadelphia, the eclipse of the moon, in January, 1824. h. m. sºns. Beginning, January 16th, at 2 17 A. M. Middle, tº gº - - 3 43 Ecliptic opposition, - - - 3 49 End, - - tºº {º tº a 5 8 Digits eclipsed 9.4, on moon's northern limb. PROBLEM XXXIV. To Project an Eclipse of the JMoon. Find the true time of full moon, the moon’s latitude at that time, the semidiameter of the earth's shadow, the sun's hourly motion, and the moon's semidiameter and hourly motions in longi- tude and latitude, as directed in the last problem. | ASTRONOMY. 325 To the moon's hourly motion from the sun in longitude,” add 9", and it will give the moon's hourly motion from the sun, on the Relative Orbit, with sufficient accuracy for a construction. Draw any right line AB, Fig. 56, for a part of the ecliptic, and in it, take a point C. Take the semidiameter of the earth's shadow from a scalef of equal parts, and with the centre C, de- scribe a circle RST, to represent a section of the earth's shadow. Through C, draw KCL, perpendicular to AB. Take the moon's latitude from the scale, and set it on the line KL, from C to M, above AB, when the latitude is north, but below, when it is south. Then M will be the moon's place at the time of ecliptic opposition, or full moon. Draw Mb parallel to AB, and to the left of KL, and make it equal to the moon's hourly motion from the sun in longi- tude. Make bc perpendicular to Mb, and equal to the moon's hourly motion in latitude, drawing it above Mb, when the latitude is tending north, but below, when it is tending south. Through M and c, draw the indefinite right line PQ, and it will be the moon’s relative orbit. t Make the proportion, as 60 minutes : minutes and seconds of the true time of full moon :: moon’s hourly motion from the sun on the relative orbit : a fourth term. Take this fourth term from the scale, and lay it on the relative orbit from M to the right hand, and it will give the moon's place at the whole hour next preceding the time of full moon. Place the number of the hour to the point thus obtained. Then commencing at this point, and with the moon's hourly motion from the sun on the relative orbit, in the dividers, set off equal spaces on the orbit, on each side of the point, and thus obtain the moon’s places at some of the whole hours, preceding and following the hour, mentioned above. Put the numbers of the hours to these places. Divide each hour space into four equal parts, for quarters, and these into five minute or minute spaces. Through C, draw SCT perpendicular to PQ, and the inter- * Which is the difference of their hourly motions in longitude. f A suitable scale is one of 10 minutes to an inch. It may also be observed, that it is most convenient, to reduce the seconds in the quantities to be taken off, to decimals of a minute. 326 ASTRONOMY, section F, will be the moon's place at the middle of the eclipse, With the centre C, and a radius equal to the sum of the semi- diameters of the earth's shadow and moon, describe arcs cutting PQ in D and H, the moon’s places at the beginning and end of the eclipse. With the same centre, and a radius equal to the dif- ference of the semidiameters of the earth's shadow and moon, pro- vided this difference is greater than CF, describe other arcs, cut- ting PQ in E and G, the moon's places at the beginning and end of the total eclipse. If the difference of the semidiameters of the earth's shadow and moon is less than CF, the eclipse will not be total. From the divisions on thc relative orbit, the times at which the moon is at the points D, F and H, and consequently the times of the beginning, middle and end of the eclipse, are easily estimated. In like manmer, when the eclipse is total, the beginning and end of the total eclipse are determined from the points E and G. With the moon’s semidiameter for a radius, and the centres D, F and H, describe circles to represent the moon at the begin- ming, middle and end of the eclipse. Take the distance NS, when the latitude is north, but UT, when the latitude is south, and measure it on the scale. Then, as the moon’s semidiameter : this distance :: 6 digits : the digits eclipsed. Note. The quantities used in constructing an eclipse are fre- quently called the Elements. It is convenient to have them col- lected in order, before commencing the construction. The true time of full moon, which is one of the elements, may be expressed either astronomically or in common reckoning; the former is per- haps the most convenient. ExAM. 1. Required to construct the eclipse of the moon, in July, 1823, taking the time for the meridian of Philadelphia. The elements for this construction, the most of which have been found in the 1st example of the last problem, are as fol- lows: ! ASTRONOMY. 327 Elements Collected. True time of full moon, July 22d, 10 h. 27 m. 23.sec. Semidiam. of earth’s shadow, iºn g- 39'2" = 39.03 Moon's latitude, north, - - - - - 9 13 = 9.22 , Moon's hor. mot. from Sun, in long. iſ º 27 11 = 27.18 Moon's hor. mot. in lat, tending north, - 2 48 = 2.72 Moom’s hor, mot. from sun, on rel. Orb. - 27 20 – 27.33 Moon's semidiameter, - - - - 14 43 = 14.72 Sum of semidiam. of earth’s shadow and moon, 53 45 = 53.75 Difference of do. " - ſº- 24 19 = 24.32 Draw AB, Fig. 56, take the point C, and through it draw K CL, perpendicular to AB. Take the moon's latitude 9.22, from the scale, and lay it on KL, from C to M, above AB, because the latitude is north. Draw Mb parallel to AB, and make it equal to 27.18, the moon’s hourly motion from the sun in longitude. Draw be perpendicular to Mb, on the upper side, because the latitude is tending north, and make it equal to 2.72, the moon’s hourly mo- tion in latitude. Through M and c, draw the relative orbit, PQ. As 60 minutes : 27 m. 23 sec. the minutes and seconds of the true time of full moon :: 27.33, the moon’s hourly motion from the sum on the relative orbit : 12.47, the fourth term. Take this fourth term and lay it on the relative orbit, from M to 10, the moon's place at the 10th hour, in this example. Take 27.33, the moon’s hourly motion from the sun on the relative orbit, and lay it from 10 to 9, and 9 to 8, and on the other side, from 10 to 11, 11 to 12, and 12 to 13, for the moon’s places at those hours. Di- vide the hour spaces into quarters, and these into five minute or minute spaces. A * Through C, draw SCT perpendicular to PQ, intersecting it in F, which will be the moon’s place at the middle of the eclipse. With the radius 53.75, the sum of the semidiameters of the earth's shadow and moon, and the centre C, describe arcs, cut- ting the relative orbit in D and H, the moon’s places at the be- ginning and end of the eclipse. With the radius 24.32, the dif- ference of the semidiameters of the earth’s shadow and moon, describe arcs, cutting the relative orbit in E and G, the moon’s \ 328 ASTRONOMY. \ places at the beginning and end of the total eclipse. The times designated by the points D, F, H., E and G, agree nearly with the beginning, middle and end of the eclipse, and beginning and end of the total eclipse, found in the 1st example of the last problem. With 14.72, the moon's semidiameter, for a radius, describe the circles about the centres D, F and H. Take the distance NS, and measure it on the scale, and it will be found to be about 44.65. Then, 14.72:44.65 :: 6 digits: 18.2 digits, the quan- tity of the eclipse. * 2. Construct the eclipse of the moon, mentioned in the 2d ex- ample of the last problem, and the results will be found to agree nearly with the answer there given. 3. Construct the eclipse of the moon, mentioned in the 3d ex- ample of the last problem. PROBLEM XXXV. To Project an Eclipse of the Sun, for a given place. Calculate the approximate time of new moon by prob. XXX, and for that time, calculate the sun's longitude, semidiameter and hourly motion, and the moon's longitude, latitude, equatorial pa- rallax, semidiameter and hourly motions in longitude and latitude. Find the true time of new moon by prob. XXX, and reduce it to apparent time at the given place, expressing it astronomically. Also, find the moon’s latitude at the true time of new moon, from the hourly motion in latitude, in the same manner as directed in prob. XXXIII, for finding the latitude at the true time of full moon. With the sun's longitude at the approximate time of new moon, neglecting the seconds, and taking the obliquity of the ecliptic 23° 28, find the sun's declination by prob. VII. Find the moon's hourly motion from the sun on the relative orbit, by adding 9" to the difference of their hourly motions in longitude. Find the reduced latitude of the place and the reduced parallax, by prob. XIII. From the moon’s reduced parallax, subtract the sun's parallax, which may be taken 9", and the remainder will be the Semidiameter of the Circle of Projection. & ASTRONOMY. 329. * Draw a right line AB, Fig. 57, and in it take a point C. Take the semidiameter of the circle of projection from a scale of equal parts, and with the centre C, describe, on the upper side of AB, the semicircle ADB, to represent the northern half of the circle of projection. When the latitude of the place is south, the whole circle must be described. Through C, and perpendicular to AB, draw the line TCY, to represent the universal meridian. With a sector,” opened to the radius AC or CB, set off from D, the arcs DV, DR, each equal to the obliquity of the ecliptic, which may be taken 23° 28′; join RV, and on it describe the semicircle RTV. With the sector, opened to the radius OV or OR, make the arc VU, equal to the sun's longitude. When the longitude exceeds VI signs, take WI signs from it, and set off the remainder from R, round towards W. Draw UW perpendicular to RV, and through W, draw CWL, and it will be the projection of the circle of latitude, which passes through the moon at the time of new moon. Take the moon’s latitude from the scale, and lay it on CL, from C to M, above AB, when it is north, but on LC produced, below AB, when it is south. Then M will be the moon's place at the true time of ecliptic conjunction. From M, draw Mb perpen- dicular to CL, to the left hand, and make it equal to the moon’s hourly motion from the sun in longitude. Draw be perpendicular to Mb, above, when the moon’s latitude is tending north, but below, when it is tending south, and make it equal to the moon’s hourly motion in latitude. Through M and c, draw the moon’s relative orbit PQ. Make the proportion, as 60 minutes : minutes and * For the manner of using the sector, see the note at the bottoms of pages 135 and 136. To what is there said respecting the manner of using it, may be added, that when an arc greater than 60°, is to be laid off, it may be done by applying the radius of the circle as a chord to the arc, as many times successively as 60° is contained in the arc to be laid off, and then with the sector, laying off from the last point, an arc equal to the remainder. When a very small arc is to be laid off with a sector, it is better to add some constant arc to it, for instance 10°. Then taking the chord of the sum from the sector, lay it on the arc, from the given point to a second one, and taking the chord of the arc which was added, set it from the second point backwards, towards the first. The arc, intercepted between the last point and the given one, will be the arc which was to be laid off. 43 330 ASTRONOMY. seconds of the true time of new moon : : moon's hourly motion from the sun on the relative orbit : a fourth term. Take this fourth term from the scale, and lay it on the relative orbit, from M to the right hand, and it will give the moon’s place at the whole hour next preceding the time of new moon. Take the moon's hourly motion from the sum on the relative orbit, from the scale, and with it, lay off equal spaces on each side of the moon's place, just found, and thus obtain the moon's places for four or five other hours, contiguous to the time of new moon, some of them preced- ing and some following it. When the time of new moon is several hours before noon, there should be more places found for hours preceding the time of new moon, than for the hours following it, and the contrary, when the time of new moon is several hours past moon. To each of the moon s places, thus found, put the number of the hour. With a sector, opened to the radius AC or CB, set off arcs equal to the reduced latitude of the place, from A to E and B to F, on the semicircle above AB when the latitude is north, but below, when it is south, and join EF. With the sector, opened to the same radius, make the arcs EG, El, FH and FK, each equal to the sun's declimation, and join GH and IK. Bisect vu in N, and through N, draw 6 N 18, parallel to EF. Make N6 and N 18, each equal to Er or rR', and on 6 N 18, describe the semicircle 6 Y 18. With the centre N and radius Nv or Nw, describe the circle wºul. Take the intervals between moon and each of the hours marked on the relative orbit, and convert them into de- grees, allowing 15° to each hour, and they will be the hour angles from noon. With the sector, opened to the radius N 6 or N 18, lay off from Y, on the semicircle 6 Y 18, the arc being produced above 6 N 18, when necessary, arcs equal to each of the hour am- gles, laying them to the right, when the hours are in the forenoon, but to the left, when they are in the afternoon, and at the extremi- ty of each arc, place the number of degrees which it contains. From these points, draw lines parallel to the universal meridian DY. Also, from the same points, draw lines to the centre N, in- tersecting the circle wavo; and when the sum’s declimation is south, produce them to meet the same circle on the other side of N. From ASTRONOMY. 33 || \ the points in which these lines intersect the circle wºw, when the sun's declination is north, but from the points in which, being produced, they meet it, when the declimation is south, draw lines parallel to EF, to meet respectively, the corresponding lines, drawn parallel to the universal meridian; and the points in which they meet will be the sun's places, on the circle of projection, at the hours to which the lines correspond. At each of these points place the number of the hour to which it belongs. The points 6 and 18, are always the sun's places at those hours. When the declination is north, the point v is the sun's place at noon, desig- nated by 0; but when the declination is south, the point w is the sun’s place at noon. From the places of the moon at the hours, marked on the relative orbit, draw lines parallel to AB or EF, to meet the lines, produced, if necessary, which are parallel to the universal meridian, and pass through the Sun's places at the same hours, in the points S. Draw a right line AC, Fig. 58, and in it take a points. Take the distance from each of the points S, in Fig. 57, to the cor- responding place of the moon on the relative orbit, and lay it on AC, from s, to the right or left, according as the moon's place is to the right or left of the point S, and at the extremity of each distance, put the number of the hour, to which the distance cor- responds. Through each of these points, draw lines perpendicular to AC. This may be most conveniently done, by drawing through one of the points a perpendicular line, and then parallel to this, drawing limes through the others. Take from Fig. 57, the dis- tances from the sum’s place at each of the hours, marked on the relative orbit, to the corresponding point S, and place them on the perpendiculars, from the same numbers on the line AC in Fig. 58, above or below AC, according as the point S in Fig. 57, is above or below the sun's place. At the extremities of these dis- tances, place the same numbers that are on the line AC. Join each adjacent two of these extremities, and the broken line thus formed will be a near representation of the moon’s apparent, rela- tive orbit, and the points on it will be the moon’s places at the Thours, denoted by their numbers. With the centre s, and a radius equal to the sum of the semi- 332 ASTRONOMY. sº diameters of the sun and moon, describe arcs, cutting the appa- rent relative orbit in B and E, which will be the moon's places at the beginning and end of the eclipse. With the centres B and F and a radius greater than half the distance of these points, de- scribe two arcs, cutting each other in a. Lay the edge of a ruler from s to a, and draw the line DSGd, intersectiug the apparent orbit in G, which will be the moon s place at the time of greatest ob- scuration. From the moon's place on the apparent orbit at the whole hour next following the end of the eclipse, draw a right line LN in any convenient direction, and taking any short dis- tance in the dividers, lay it over 12 times, from L to the point M. Then LM is to be considered as representing an hour, divided into parts of 5 minutes each, which must be reckoned from L towards M. Join M and each of the hour points on the apparent orbit. From the points B, G and E, draw the lines Bc, Gh and Ef, respectively parallel to the lines joining M and the hours next following those points, and meeting the lines joining M and the hours next preceding the same points, in the points c, handf. Draw cb, hg, and fe, respectively parallel to Hines joining L, and the hours next preceding the points B, G and E. Then the minutes corresponding to b, connected with the hour next pre- ceding B, those corresponding to g, commected with the hour next preceding G, and those corresponding to e, connected with the hour next preceding E, will be the times of the beginning, great- est obscuration and end of the colipse. If a circle, described about the centre s, with a radius equal to the difference of the semidiameters of the sun and moon, cuts the apparent orbit, the eclipse will be annular or total; annular when the sum's semidiameter is greater than the moon’s; total when it is less. The beginning or end of the annular or total eclipse, when either has place, may be found in the same manner as the begin- ning or end of the eclipse, taking the points in which the circle cuts the apparent orbit. About the centres 8 and G, with radii respectively equal to the semidiameters of the sun and moon, describe circles to represent those bodies. Take the distance DH, and applying it to the scale, obtain its measure. Then, as the sun's semidiameter : measure of DH :: 6 digits : digits eclipsed. ASTRONOMY. 333 Take the interval between the beginning of the eclipse and noon, and convert it into degrees. With the sector opened to the radius N 6, or N 18, Fig. 57, lay off from Y, on the semicircle 6 Y 18, the arc being produced if necessary, an arc containing this number of degrees, laying it to the right or left, according as the time of beginning is before or after noon, and proceed to find the sun's place on the circle of projection for the time of begin- ning, in the same manner as directed above, for other times. Mark this place of the sun with the letter n, and join Cn. Make the angle CSV, Fig. 58, equal to the angle BCn, Fig. 57, and join sR. Then v will represent the sun's vertex at the beginning of the eclipse, z the place at which the eclipse commences, and the angle WSB, the angular distance of this point from the sun’s Vertex. Note. The times of beginning, &c. obtained by projection, are only approximate values. But when the construction is carefully made, they will seldom err more than one or two minutes. ExAM. 1. Required the times, &c. of the eclipse of the sun of August 27th, 1821, at Philadelphia. The different elements necessary for the construction are easily, found, and are as follows: Elements Collected. True time of new moon, August, 26 d. 22h. 15 m. 56 sec. Semidiameter of the circle of projection, 55' 1" = 55'.02 Sun's longitude, - tº {sº tºº 153°57' Sum’s declimation, north, - º sº 10 4 - Moon’s latitude, north, gº º iº 3 41 = 3.68 Moon’s hor. mot. from sum, in long. & e 28 30 = 28.5 Moon's hor. mot. in lat. tending south, - 2 51 = 2.85 Moon's hor. mot, from sun, on rel. orb. - 28 39 – 28.65 Fourth term, - &=º º tºº tº - 7 36 = 7.6 Sum's semidiameter, - $º aſº gº 15 52 = 15.87 Moon’s do. gº { } º tº - 15 3 = 15.05 Sum of semidiameters of sun and moon, 30 55 = 30.92 Latitude of Philadelphia, reduced, tºe 39 46 N. 334 ASTRONOMY. Draw AB, Fig. 57, and take the point C. Take 55.02, the semidiameter of the circle of projection, from a scale of equal parts, and with the centre C, describe the semicircle ADB. Through C, and perpendicular to AB, draw the universal meri- diam TCY, cutting ADB in D With a sector opened to the radius AC or CB, make the arcs DR, DV, each equal to 23°28', the obliquity of the ecliptic; join RV, and on it describe the semicircle RTV. With the sector opened to the radius OR or OV, make the arc WTU equal to 153°57', the sun's longitude. Draw UW perpendicular to RV, and through W, draw CWL. Take 3.68, the moon’s latitude, from the scale, and lay it on CL, from C to M, above AB, because the latitude is north. Draw Mb perpen- dicular to CL, and make it equal to 28.5, the moon’s hourly mo- tion from the sun in longitude. Draw be perpendicular to Mb, below Mb, because the latitude is tending south, and make it equal to 2.85, the moon’s hourly motion in latitude. Through M and c, draw the moon's relative orbit PQ. Take 7".6, the fourth term, from the scale, and lay it on the relative orbit, from M to XXII, the moon's place at that hour. Take 28,65, the moon's hourly motion from the sun on the relative orbit, from the scale, and set it over from XXII, backwards to XXI, XX, and XIX, and forwards to XXIII, for the moon s places at those hours. With the sector opened to the radius AC or CB, make the arcs AE and BF, each equal to 39° 46', the reduced latitude of Philadelphia. With the sector opened to the same radius, make the arcs EG, EI, FH and FK, each equal to 10° 4', the sun's declination, and join GH and IK, intersecting the universal meridian in w and v. Bisect vu in N; through N, draw 6 N 18, parallel to EF, and make N 6, and N 18, each equal to rB, or rR. With the centre N and radius N 6 or N 18, describe the semicircle 6 Y 18, and with the same centre, and radius Nv or Nw, describe the circle uva w. The in- tervals between noon and the hours, marked on the relative orbit, are 1, 2, 3, 4 and 5 hours, and these in degrees are 15°, 30°, 45°, 60° and 75°. With the sector opened to the radius N 6 or N 18, lay off these arcs on the semicircle 6 Y 18, all of them from Y to the right hand, because the hours are all in the forenoon. From the points' 15, 30, 45, 60 and 75, which are the extremities of the ASTRONOMY. 335 arcs, draw the lines 15, 23; 30, 22; 45, 21; 60,20; and 75, 19, parallel to the universal meridian TY; and from the same points, draw lines to the centre N, not producing them, because the sun’s declination is north. From the points in which the limes N 15, N 30, N 45, N 60, and N 75, intersect the circle wav, draw lines parallel to EF, respectively meeting the lines 15,23; 30,22; 45,21; 60,20; and 75, 19, in the points 23, 22, 21, 20, and 19, which are the sun’s places at those hours. From the points XIX, XX, XXI, XXII and XXIII, draw, parallel to AB or EF, the lines XIXS, XXS, XXIS, XXIIS, and XXIIIS, meeting the lines 75, 19; 60,20; 45,21; 30, 22; and 15, 23, in the points S. Draw AC, Fig. 58, and in it take the point S. Take the distances S.XIX, SXX, SXXI, SXXII, and SXXIII, Fig. 57, and set them on the line AC, Fig. 58, from s to 19, 20, 21, 22 and 23, placing the first three to the right of s, because the moon’s places at those hours are to the right of the corresponding points S, and the other two to the left, because the moon’s places are at those hours to the left of the corresponding points S. Draw 21, XXI, perpen- dicular to AC, and parallel to it, draw 19, XIX; 20, XX; 22, XXII; and 23, XXIII. Take the distances S19, S20, S21, S22, and S23, Fig. 57, and set them in Fig. 58, from 19 to XIX, 20 to XX, 21 to XXI, 22 to XXII, and 23 to XXIII, setting the first two above AC, because the points S are above the sun’s places, and the others below, because the points S are below the sun’s places. Join XIX, XX; XX, XXI; XXI, XXII; and XXII, XXIII, for the apparent relative orbit of the moon. Take 30.92, the sum of the semidiameters of the sun and moon, from the scale, and with the centre s, describe arcs cutting the apparent orbit in B and E, the moon’s places at the beginning and end. With the centres B and E, and a radius greater than half the distance be- tween them, describe arcs cutting each other in a ; and with the edge of a ruler, applied to s and a, draw the line DSGH, inter- secting the apparent orbit in G, the moon’s place at the greatest obscuration. From the point XXIII, in the apparent orbit, draw LN, and taking some short distance in the dividers, lay it over 12 times, from L to M, and number the divisions as in the figure. Join M, XIX; M, XX; M, XXI; and M, XXII, and draw Bc pa- 336 ASTRONOMY. rallel to M, XX; Gh parallel to M, XXI; and Ef parallel to ML. Draw ch parallel to L., XIX; hg parallel to L., XX; and fe parallel to L, XXII. Then, attending to the rule, it is easy to perceive that the beginning of the eclipse is at 19 h. 31 m.; the greatest obscuration at 20 h. 48 m ; and the end at 22 h. 14 m. Take 15'.87, the sun’s semidiameter, from the scale, and with the cen- tre s, describe a circle to represent the sun, and with 15'.05, the moon’s semidiarmeier, taken from the scale, and the centre G, describe another circle, to represent the moon. The distance DH, -- applied to the scale, will be found to measure 22.9. Then, 15'.87 : 22.9 :: 6 digits : 83 digits, the quantity of the eclipse. The interval between the time of beginning and moon is 4h. 29 m. which in degrees is 67° 15'. With the sector opened to the radius N6 or N 18, Fig. 57, lay off this arc on the semicircle 6 Y 18, from Y to the right hand, because the time is in the forenoon, and find n, the sun's place at that time, in the same manner as for other times. Join Cn, and make the angle CSW, Fig. 58, equal to BCn, and join sR, The measure of the angle Bs"W is 26°, which is the angular distance of the point at which the eclipse commences from the sun's vertex to the right hand. In Fig. 59, is a reduced representation of the sum’s and moon’s discs, with the line SW placed in a vertical position. 2. Required to calculate the elements, and project an eclipse of the sun, for the latitude and meridian of Philadelphia, that will occur in February, 1831. Elements. True time of new moon, February, 11 d. 23h. 57 m. 40 sec. Semidiam. of circle of projection, tº , 57'24" = 57.4 Sun’s longitude, - tºº gºt tº e 323° 18 Sun's declination, South, - - - 13 46 Moon's latitude, north, tº * º 42 10 = 42, 17 Moon's hor. mot, from sun, in long. sº 31 4 = 31.07 Moon's hor. mot, in lat, tending south, ſº 3 4 = 3.07 Moon's hor. mot. from sun, on rel. orb. - 31 13 = 31.22 Fourth term, gº gº tº; º - * 30 0 = 30,06 * ASTRONOMY. 337 -Sun's semidiameter, tºº º tº . - 16'14" = 167.23 Moon's do. - dº tº . tº tº 15 42 = 15.7 Sum of semidiameters, - " - sº - 31 56 = 31.93 Latitude of Philadelphia, reduced, - 39° 46 . Result of Projection. d. h. m. Beginning, tº- º 12 11 7 A. M. Greatest obscuration, sº O 42 P. M. End, w †- tº - 2 11 Digits eclipsed 113, on sum’s south limb. Eclipse commences about 101*, from the sun's vertex to the right hand. PROBLEM xxxvi. To Calculate an Eclipse of the Sun, for a given Place, having given the Approximate Times, obtained by Projection. From the sun's longitude and hourly motion, previously found for the approximate time of new moon, find his longitude at the approximate times of beginning, greatest obscuration, and end of the eclipse. Also, find the sun's semidiameter and the apparent obliquity of the ecliptic for the approximate time of new moon. These change so slowly that they may be considered the same, during the continuance of the eclipse. Calculate the moon's lon- gitude, latitude, equatorial parallax and semidiameter for the ap- proximate times of beginning, greatest obscuration and end.* Calculate by problems XXVIII. and XXIX, the moon's appa- rent longitude, latitude, and augmented semidiameter, for the ap- proximate times of beginning, greatest obscuration and end, using * When great accuracy is not required, it will be sufficient to calculate the moon’s longitude, latitude, equatorial parallax, semidiameter, and hourly motions in longitude and latitude, for the approximate time of greatest ob- scuration, and by means of the hourly motions, find the longitude and lati- tude for the approximate times of beginning and end. The parallax and semidiameter may, without material error, be considered the same, during the eclipse. 44 338. \ AST'RONOMY. the reduced latitude of the place, and the difference between the reduced parallax of the moon and the sun's parallax. It is neces- sary to know for each of the apparent latitudes, whether it is in- creasing or decreasing. This may be determined by observing that, when at the beginning and end of any short interval of time, they are both of the same name, the apparent latitude is increasing or decreasing, according as it is greater or less at the end of the interval, than at the beginning. When they are of different names, it is decreasing at the beginning of the interval, and increasing at the end. For the Beginning. Subtract the moon’s apparent longitude at the approximate time of beginning from the sun's longitude at the same time, increasing the latter by 360°, when necessary, and call the remainder G. Call the moon's apparent latitude at the approximate time of be- ginning, H, the sum of the moon’s augmented semidiameter, at the same time, and the sun’s semidiameter, S, and the interval between the approximate times of beginning and greatest obscura- tion, T. Subtract the moon's apparent longitude at the approximate time of beginning, from its apparent longitude at the approximate time of greatest obscuration, increasing the latter by 360°, when neces- sary; and do the same with the sun’s longitudes at the same times. Take the difference between the remainders, and call it M. When the moon’s apparent latitudes at the approximate times of begin- ming and greatest obscuration, are of the same name, take their difference; but when they are of different names, take their sum; and call the difference or sum, N. The value of N must be mark- ed negative, when the apparent latitude at the approximate time * of beginning is increasing, but affirmative, when it is decreasing. Add together, the logarithm of H, the logarithm of N, and the arithmetical complement of the logarithm of M, and the result, rejecting the tens in the index, will be the logarithm of a small arc W. Apply V, according to its sign, to G, and call the result W. To the logarithm of the sum of S and H, add the logarithm ASTRONOMY. 339 of their diſſerence, and divide the sum by 2; the result will be the logarithm of an arc L. Add together, the logarithm of T, the logarithm of the sum of G and L, the logarithm of their difference, the arithmetical com- plement of the logarithm of 2 M, and the arithmetical complement of the logarithm of W, and the result, rejecting the tens in the index, will be the logarithm of a correction, which, added to the approximate time of beginning, when G is greater than L, but subtracted, when G is less than L, will give the true time of be- ginning very nearly. For the End. Subtract the sun's longitude at the approximate time of the end, from the moon’s apparent longitude at the same time, increasing the latter by 360°, when necessary, and call the remainder G. Call the moon’s apparent latitude at the approximate time of the end, H, the sum of the moon’s augmented semidiameter, at the same time, and the sun’s semidiameter, S, and the interval be- tween the approximate times of greatest obscuration and end, T. Subtract the moon’s apparent longitude at the approximate time of greatest obscuration, from its apparent longitude at the approximate time of the end, increasing the latter by 360°, when necessary; and do the same with the sum’s longitudes at the same times. Take the difference between the remainders, and call it M. When the moon’s apparent latitudes at the approximate times of greatest obscuration and end are of the same name, take their difference; but when they are of different names, take their sum; and call the difference or sum, N. The value of N must be mark- ed affirmative, when the apparent latitude at the approximate time of the end is increasing, but negative, when it is decreasing. Find the quantities V, W, L, and the correction, as directed for the beginning. The correction, added to the approximate time of the end, when G is less than L, but subtracted, when G is greater than L, will give the true time of the end, 340 ASTRONOMY. For the Greatest Obscuration, and Quantity of the Eclipse. Subtract the moon’s apparent longitude at the approximate time of greatest obscuration, from the sun's longitude at the same time, increasing the latter by 360°, when necessary; and when the remainder is a small arc, mark it affirmative, and call it G; but when it is near to 360°, subtract it from 360°, and marking the second remainder negative, call it G. Call the moon’s apparent latitude at the approximate time of greatest obscuration H, the sum of the moon's augmented semidiameter, at the same time, and the sun’s semidiameter, S, and the interval between the ap- proximate times of beginning and end, T. Subtract the moon's apparent longitude at the approximate time of beginning, from its apparent longitude at the approximate time of the end, increasing the latter by 360°, when necessary; and do the same with the sun’s longitudes at the same times. Take the difference between the remainders, and call it M. When the moon’s apparent latitudes, at the approximate times of begin- ning and end, are of the same name, take their difference; but when they are of different names, take their Sum; and call the dif- ference or sum, N. The value of N must be marked negalive, when the apparent latitude at the approximate time of greatest obscuration is increasing, but affirmalive, when it is decreasing. To the logarithm of N, add the arithmetical complement of the logarithm of M, and the result will be the Tangent of an arc I, which must be taken out according to the sign, but less than 180°. To the Tangent of I, add the logarithm of H, and the result, re- jecting the tens in the index, will be the logarithm of a small are W. Take the sum of G and W, attending to their signs, and call, it W. Add together, the logarithm of W and the Cosine of I, taken affirmative, and call the resulting logarithm X. Add to- gether, the logarithm X, the Cosine of I, taken aſſirmative, the logarithm of T, and the arithmetical complement of the logarithm f M, and the result, rejecting the tens in the index, will be the logarithm of a correction, which applied, according to its sign, to the approximate time of greatest obscuration, will give the true time, ASTRONOMY. 34f Add together, the logarithm X, and the Tangent of I, and the result, rejecting the tens in the index, will be the logarithm of a small arc Y. Apply Y to S, according to its sign, and call the result S'. To the logarithm of H, add the Cosime of I, taken af- firmative, and the result, rejecting the tens in the index, will be the logarithm of an arc H'. Add together, the constant logarithm 0.77815, the logarithm of the difference between S' and H', and the arithmetical complement of the logarithm of the sun’s semi- diameter, and the result, rejecting the tens in the index, will be the logarithm of the digits eclipsed. Noſe 1. If Y be applied, with a contrary sign, to H', it will give the apparent distance of the centres of the sum and moon, at the time of greatest obscuration. When this distance is less than the difference between the sun’s semidiameter and the augmented semidiameter of the moon, the eclipse is either annular or total; annular, when the sum’s semidiameter is the greater of the two; total, when it is the less. 2. When the point of the sun's disc, at which the eclipse CO HQ - mences, is required with greater accuracy than is given by the projection, it may be obtained by the formulae in chapter XI, art, 76. 3. Supposing the longitude, &c. to be accurate, the times ob- tained by this problem will be true, within a few seconds. When greater accuracy is required, the calculation may be made by the formulae in chap. XI, articles 67 to 76. ExAM. 1. The approximate time of the beginning of the eclipsc of the sun, on the 27th of August, 1821, ſound by projection for the latitude and meridian of Philadelphia, is 7 h. 31 m. A. M.; greatest obscuration, 8 h. 48 m. A. M.; and end, 10 h. 14 m. A. M. Required the true times and quantity of the eclipse. By reducing each of the given times to mean time at Green- wich, and calculating the sun’s and moon's longitudes, &c. for those times; and then calculating the parallaxes, the following quanti- ties will be obtained: 342 ASTRONOMY. Approx. time of Approx. time of Approx. time Beginning. Greatest Obscur. of End. 7 h. 31 m. 8 h. 48 m. 10 h. 14 m. Sun's true longitude, 153° 50' 36” 153°53'42" 153° 57' 10", Sun's semidiameter, 15 52 15 52 15 52 Moon's appar. long. 153 19 52 153 51 53 154 24 21 Moon's appar. lat. 3 59 S. 8 27 S. 15 40S Moon's augm. semid. 15 10 15 13 15 14 For the Beginning. G = 30'44" = 1844"; H = 3'59" = 239"; S = 31'2" = 1862"; T = 1 h. 17 m. = 4620 sec.; M = 28' 55" = 1735"; and N = — 4'28" = — 268". H - º 239" - log. 2.37840 N - º — 268 - log. — 2.42813 M ſº º 1735 Ar. Co. log. 6.76070 V - ſº- — 37 - log. — 1.56723 G Lºs sº 1844 W - - 1807” S + H - - 2101" - log. 3,32243 S – H t- 1623 - log. 3.21032 2)6.53275 L wº - 1847" - log. 3.26637 T - tº- 4620 sec. log. 3,66464 G + L - tº 3691'.' - log. 3.56714 L– G - - 3 - log, 0.47712 2 M wº sº 3470 Ar. Co. log. 6.45967 W - º 1807 Ar. Co. log. 6.74304 Correction, 8 sec. log, 0.91161 Approx. time of Begin. 7 h. 31 m.0 True time of Begin. 7h, 30m, 52.sec. ASTRONOMY. * 343 For the End. G = 1631"; H = 940”; S = 1866"; T = 5160 sec.; M = 1740”; and N = + 433”. *. H - * = tº 940” - log. 2.97.313 N - gº tº 433 - log. 2.63649 M º gº tº 1740 Ar. Co. log. 6.75945 V - sº + 234 - log. 2.36907 G wº tºº - 1631 --- W tº ºn 1865 S -- H - ſº 2806" - log, 8.44809 S – H tº. - 926 - log. 2.96661 2)6.41470 L dº gºe 1612" - log. 3.20735 T ey {º 5160 sec. log. 3.71265 G + L º - 324.3" - log. 3.51095 G— L tº tº 19 - log. 1.27875 2 M wº - 3480 Ar. Co. log. 6.45842 W tº 1865 Ar. Co. log. 6.72932 \ Correction, - , - 49 sec. log. 1.69009 Approx. time of End, 10h. 14 m. 0 sec. True time of End, 10b. 18m. 11 sec. 34% ASTRONOMY. Approxime G. Obscur. \ For Greatest Obscuration, and Digits Eclipsed. G = + 109"; H = 507’’; S = 1865"; T M = 3475"; and N = — 701". hº = 9780 Sec.; N > º — 701” log. — 2.84572 M tºo º tº- 3475 Ar. Co. log. 6.45905 I. > - 168° 36' - tan. – 9.30477 H tº- tºº - 507" - log. 2.70501 V tº- a- - — 102 - log. —2.00978 G º º + 109 * W . - tº tº- 7" - log. 0.84510 I tº- sº 168° 36' tº- cos. 9...991.35 X 10.83645 I tº º º gº º cos. 9.99 ſ 35 T ºp lº º 9780 sec. log. 3.99034 M . - tº- gº 3475" Ar. Co. log. 6.45905 Correction, - - + 19 sec. log. 1.27719 8h. 48m. 0 sec. True time, G. Obscur. 8h. 48m. 19 sec. X. I º - º tºº Y tº- º º — 1" S tº-> tº º 1865 S’ tº º - 1864 • H - tº- - 507’’ - I tº a wº º gº Ar. Co. H' * tº- - 517" - S' – H' - tº- 1347" - Sun’s semidiameter, 952 Ar. Co. 8.5 * Digits eclipsed, 10.83645 tan. — 9.30477 - log. — 0.14122 log. 2.70501 cos. 0.00865 log, 2.71366 0.77815 log. 3.12937 log. 7.02136 log, 0.92888 ASTRONOMY. 345 2. The approximate time of the beginning of the eclipse of the sun, that will occur on the 12th of February, 1831, found by projection, for the latitude and meridian of Philadelphia, is 11 h. 7 m. A. M.; greatest obscuration, 0 h. 42 m. P. M ; and end, 2 h, 11 m. P. M. Required the true times, and the quantity of the eclipse. Ans. Beginning, 11 h. 7 m. 12 sec. A. M.; greatest obscuration, 0 h. 41 m. 29 sec. P. M.; end, 2 h. 10 m. 32 sec. P. M.; digits celipsed, 113. PROBLEM XXXVII. To find by Projection, the Latitudes and Longitudes of the Places at which an Eclipse of the Sun is Central, for different times during the continuance of the Central Eclipse. Draw AB, Fig. 60, and perpendicular to it, draw the univer. sal meridian DCY. With the centre C, and a radius equal to the semidiameter of the circle of projection, describe the circle of projection ADBY; and proceed as, directed in prob. XXXV, to draw the moon’s relative orbit, and to find the moon’s places on it at such whole hours as will fall on the circle of projection, or near to it. Or when the eclipse has been previously projected for a particular place, this part may be obtained by pricking it off from that projection. Divide the hour spaces on the relative orbit into five minute or minute spaces. With a sector opened to the distance AC or CB, make the arc DP equal to the sun's declina- tion, laying it to the left, when the declination is north, but to the right, when it is south; draw PCp, and EC perpendicular to it. For the Place at which the Sun is Centrally Eclipsed, on the JMe- g ridian. From the point n, in which the relative orbit intersects the universal meridian, draw ng parallel to AB. Then Eq, measured with the sector,” will give the latitude of the place, which will be north or south, according as q is above or below E. The inter- val between moon and the time on the relative orbit, correspond- * See note, pages 135 and 136. 45 346 ASTRONOMY. ing to n, converted into degrees, will give the longitude of the place, reckoned from the meridian of the place for which the pro- jection is made; and it will be east or west, according as the time, reckoned astronomically, is more or less than 12 hours. For the Places at which the Eclipse Commences or Ceases to be Central. The central eclipse commences when the moon’s centre is at a, and ends when it is at e. From the points a and e, draw the lines qf and et, parallel to AB, meeting the universal meridian in fand t; and from or through f and t, draw frs and utv, parallel to EC, cutting or meeting the line Pp in r and w Then Es, measured with the sector, will give the latitude of the place at which the eclipse is central, when the moon’s centre is at a, and Ev will give the latitude of the place at which it is central when the moon’s centre is at e. These latitudes will be north or south, according as the points s and v are above or below E. - From a and e, draw ad and ek, parallel to the universal meri- dian. Take the distance fr, and lay it from d to g, on ad, pro- duced if necessary, above or below AB, according as fis to the Tight or left of Pp. Take the distance ut, and lay it in like man- mer from k to w, above or below AB, according as t is to the right or left of Pp. Through g and w, draw the lines Cgh and Cwa). By means of the sector, measure the arc Yh, and call it west or east, according as h is to the right or left of the universal meridian. Take the interval between moon and the time on the relative orbit, corresponding to a, and convert it into degrees, and it will give the hour angle, which must be marked west or east, according as the time is more or less than 12 hours. When the arc and hour angle are of the same name, take their difference; and if the arc is the greater of the two, mark the difference also with the same name; but if the arc is the less of the two, mark the difference with a contrary name. When the arc and hour angle are of different mames, take their sum, and mark it with the same name as the arc. The result in either case will be the longitude of the place at which the eclipse is central when the moon’s centre is at q, ASTRONOMY. 347 reckoned from the meridian of the place for which the construc- tion is made. In like manner, with the arc Ya, and the time the moon’s cem- tre is at e, find the longitude of the place which then has the cen- tral eclipse. For the Place at which the Eclipse is central at any other time during the continuance of the Central Eclipse. Let T be the moon’s place at the given time. Through T, draw ZMN parallel to the universal meridian, and TK parallel to AB. Take the distance MN, and with the centre C, describe an arc, cutting TK in K, on the left of the universal meridian. Through K, draw yRz, parallel to EC. Then Ey, measured with the sector, will give the latitude of the place which has a central eclipse when the moon’s centre is at T; the latitude being north or south, according as y is above or below E. Take the distance Kz, and lay it on NMZ, from M to Z, below AB, when K is to the left of Pp, as is generally the case; but above AB, if K is to the right of Pp. Through Z, draw CZS. Then, with the arc Y.S and the time the moon’s centre is at T, find the longitude of the place at which the eclipse is central in the same mammer as directed above, for the arc Yh and the time of the moon’s centre being at a. Note. From the latitudes and longitudes thus determined for a number of times during the eclipse, the path of the central eclipse may be drawn on a map. These latitudes and longitudes, determined by projection, can not be depended on as accurate. But when the construction is carefully performed, they will sel- dom err more than 15' or 20', and will therefore serve to ascer- tain nearly the places at which the eclipse will be central. ExAM. Required the latitudes and longitudes of the places at which the eclipse of the sum of August, 1821, will be central on the meridian, will commence and cease to be central, and at which it will be central at the whole hours during its continuance, reckoned on the meridian of Philadelphia. 343 ASTRONOMY. This is the eclipse, projected in the first example of prob. XXXV. Fig. 60, contains all the lines necessary for determining the latitudes and longitudes required in this example; and taken in connection with the above rule, it is sufficiently plaim without further explanation. The latitudes and longitudes obtained from it are as follows: Lat. Long. from Philad. Beginning centr. eclipse, 30° 45' N. 41° 45' W. 21 h. º gº 29 20 N. 4 50 E. 22 h. - tºº - 17 15 N. 24 0 E. On the Meridian, gº 14 40 N. 26 45 E. 23 h. - sº - 3 15 N. 36 15 E. 0 h. º º 15 40 S. 59 15 E. End centr, eclipse, - 23 0 S. 81 10 E, PROBLEM XXXVIII. To Project an Occultation of a Fixed Star by the JMoon, for a given Place. The times of the conjunctions of the moon with such stars as may suffer occultations, somewhere on the earth, are given in the Nautical Almanac, on the first page of each month. Thus, on the first page of table LV, the line, 3 d. 17 h. 47 m. D a ty, means that the moon is in conjunction with a Virginis, on the 3rd day of the month, at 17 h. 47 m. apparent time at Greenwich. When the occultation is visible at Greenwich, the times of beginning and end, or which is the same, of Immersion and Emersion, are given, instead of the time of conjunction. Find by prob. XIV, the star's mean longitude, latitude, right ascension and declination for the day of the occultation. For the time of conjunction, find by prob. XII, the moon’s latitude, equa- torial parallax, semidiameter, and hourly motions in longitude and latitude; also, find for the same time, the sun's longitude and right ascension. When the time of conjunction is not given, take from the Nautical Almanac, the two longitudes of the moon next less than the star's longitude, and the two next greater. Then, ASTRONOMY. 349 with them and the star's longitude, taken four times, instead of the sun's longitudes, the time of conjunction may be found in the same manner as is directed in prob. XXXI, for finding the time of new moon; except that T must be found by proportion; thus, the quantity A : the 2nd distance :: 12 h. : T. Reduce the time of conjunction to the meridian of the given place. When the latitudes of the moon and star are of the same name, take their difference, and it will be the moon's distance from the star in latitude; if the moon’s latitude is the greater of the two, this distance must be marked with the same name as the latitudes; but if the moon’s latitude is the less of the two, the dis- tance must be marked with the contrary name. When the lati- tudes are of different names, their sum will be the moon’s distance from the star in latitude, and it must be marked with the same name as the moon’s latitude. Subtract the sum’s right ascension from the right ascension of the star, expressed in time, increasing the latter by 24 hours when necessary, and the remainder will be the time of the star’s passage over the meridian.* Make the pro- portion, as 60 m. : the minutes and seconds of the time of con- junction of the moon and star :: the moon’s hourly motion in longitude : a fourth term. Draw AB, Fig. 61, and CT perpendicular to it. With the centre C and a radius equal to the moon’s parallax, describe the semicircle ADB, for the northern half of the circle of projection. When the latitude of the place is south, describe the whole circle. Make the arcs DR and DV, each equal to 23° 28, the obliquity of the ecliptic; join RV, and with the centre O and radius OR, describe the circle RTV. Make the arc WU equal to the star's longitude, setting it from W, in the direction WUT, and draw Um parallel to CT. Make the arc Dp equal to the star’s declination, and draw pa parallel to RW. With the centre C and radius Ca, describe the arc aq, meeting Um, produced if necessary, in q, and through q, draw the circle of latitude CL. Take the moon’s dis- tance from the star in latitude, from the scale, and when the dis- * This is not accurately the time of the star’s passage over the meridian, but it is nearly so. It is the star’s distance from the sun in right ascension at the time of conjunction, which is the quantity wanted in the projection. 350 ASTRONOMY. tance is north, set it on CL, from C to M, above AB; but when the distance is south, it must be set on LC produced, below AB. Draw Mb perpendicular to CL, and make it equal to the moon’s hourly motion in longitude. Draw be parallel to CL, above or below Mb, according as the moon's hourly motion in latitude is tending north or south, and make it equal to the hourly motion in latitude. Through M and c, draw the moon’s relative orbit PQ. Take the fourth term from the scale, and set it on the relative orbit, from M to the right hand, for the moon’s place at the whole hour, next preceding the time of conjunction. With the moon’s hourly motion in longitude, in the dividers, set off on the relative orbit, one or two equal spaces on each side of this point, to obtain the moon’s places at some of the contiguous hours, and mark each point with the proper number of the hour. Make the arcs AE and BF, each equal to the latitude of the place, and EG, EI, FH and FK, each equal to the star's declimation, and join GH, EF and IK. Bisect vu in N, and through N, draw XNZ paral- lel to EF. With the centre N, and a radius equal to rB, describe the semicircle XYZ; and with the same centre and the radius Nv, describe the circle www. Take the intervals between the time of the star's passage over the meridian and each of the hours mark- ed on the moon’s orbit, and the results converted into degrees, will give the hour angles for the star, at those hours respectively. Set off from Y, on the semicircle XYZ, produced if necessary, arcs equal to these hour angles, setting them to the right for the hours on the orbit, which are earlier than the time of the star's passage over the meridian, but to the left, for the hours which are later. From the extremities of the arcs, draw limes parallel to the meridian CD, and others to the centre N, producing the latter when the star’s declination is south, to meet the circle uva:w on the opposite side of N. From the points in which the lines drawn to the centre N, intersect the circle uvau, when the declination of the star is north, or from the points in which they meet it on the opposite side of N, when the declination is south, draw lines pa- rallel to XZ, to meet respectively the corresponding lines drawn from the extremities of the arcs, parallel to CD; and the points in which they meet will be the star's places on the circle of projec- ASTRONOMIY. 351 tion, at the hours to which the arcs appertain. Mark each of these points with the proper number of the hour. Through each of the moon's places at the hours marked on the relative orbit, draw lines parallel to EF, to meet respectively the lines which are parallel to CD, and pass through the star's places at the same hours, in the points S. Draw a right line AC, Fig. 62, and in it take a point s, to represent a fixed position of the star’s place. Transfer the distances between the points S and the moon and star's places in Fig. 61, to Fig. 62, and draw the moon’s appa- rent relative orbit, in the same manner as directed for the moon and sun in prob. XXXV. With the centre s and a radius equal to the moon's semidiameter, describe arcs cutting the apparent orbit in I and E, the moon's places at the immersion and emersion; and about the points I and E, with the same radius, describe cir- cles to represent the moon’s disc. From the moon’s place on the apparent orbit, at the hour next following the emersion, draw a right line LN in any convenient direction, and lay off from L to M, twelve equal spaces, to represent intervals of five minutes each, and number them as in the figure. Join M and the moon’s places at the hours marked on the apparent orbit. From the points I and E, draw lines respectively parallel to the lines join- ing M and the moon’s places at the hours next following the points I and E, and meeting the lines joining M and the moon's places at the next preceding hours, in the points h and f. From h, draw hi parallel to a line joining L and the moon's place at the hour next preceding I; and from f, draw fe parallel to the line joining L and the moon’s place at the hour next preceding E. Then the minutes corresponding to i, connected with the hour next preceding I, will be the time of the immersion; and the minutes corresponding to e, connected with the hour next preceding E, will be the time of emersion. Take the intervals between the time of the star's passage over the meridian and the time of im- mersion and emersion, and converting them into degrees, find in Fig. 61, the points n and t, the star's places at those times, in the same manner as directed for the whole hours. Make the angles nsC and tsC, Fig. 62, respectively equal to the angles nGB and tCB, Fig. 61. Through I, draw by, parallel to Sn, and through E, draw 352 ASTRONOMY. b'v' parallel to sl; then by and b'v' will represent vertical circles passing through the moon's centre at the times of immersion and emersion. The angles vſs and v'Es will be the angular distances from the moon’s vertex at which the immersion and emersion take place. If it is required to find the moon’s phase, and its position with regard to a vertical circle passing through the centre at the times of immersion and emersion, it may be done with sufficient accu- racy as follows: Subtract the sun's longitude from the longitude of the star, increasing the latter by 360°, when necessary, and make the arc Wil, Fig. 61, equal to the remainder, setting it round in the direction WTil, and join d0. Make the arc Wh equal to the moon's latitude, above RW when the latitude is north, but below when it is south, and draw gh parallel to RV, intersecting d0, produced if necessary, in e, and CT in 2. Make 2s equal OR, and join se. Make the angle LCk equal to zse, and on the same side of LC that se is of CD. Make the angle ksC, Fig. 62, equal to kCB, Fig. 61, and through I and E, draw ga and g’d parallel to ks; also, through the same points draw mr and m'r' perpendicu- lar to gil and g’d'. Then the points d and g will designate the positions of the moon’s cusps with respect to the vertical circle by, at the immersion, and the points d' and g', the same with respect to b'v', at the emersion. Make the arcs rw and rºw', each equal to the remainder obtained above, by subtracting the sum’s longitude from that of the star, and draw we and w'c' parallel to dg and d'g', and meeting mr and m'r' in the points c and c'. Describe the cir- cular arcs deg and d'c'g'. Then, when the above remainder is less than 180°, degr and d'c'g'' will be near representations of the enlightened disc of the moon; but when the remainder is greater than 180°, dmgc and d'm'g'c' will be representations of the en- lightened disc. For the example to which the figures are adapted, Fig. 63, represents the position of the moon’s phase and of the star at the emersion, with respect to the vertical circle by, placed in a vertical position, and Fig. 64, does the same for the emer- Sl()]], \ | ASTIRONOMY. 353 Note 1. No notice has been taken in the above rule, of the aberrations and mutations of the star, nor of some other small cor- rections of the elements, as they would produce but little effect on the results obtained by the projection. 2. The calculation of the true times of immersion and emersion, from the approximate times, may be made nearly in the same manner as for the beginning and end of an eclipse of the sun. There are, however, the following differences in the calculation: he approximate times of immersion and emersion must be used instead of the approximate times of beginning and greatest ob- scuration, or greatest obscuration and end of the eclipse. The star's longitude, corrected for aberration and mutation, must be used instead of the sun's longitudes. The apparent distances of the moon from the star in latitude, must be used instead of the moon’s latitudes. To the logarithm of G, obtained from the moon and star's longitudes, add the Cosine of the star's latitude, reject- ing the tens in the index, and use the natural number correspond- ing to the sum, instead of G. To the logarithm of M, add the Cosime of the star's latitude, rejecting the tens in the index, and use the natural number corresponding to the sum, instead of M. Lastly, the moon's augmented semidiameter must be used, instead of the sum of the semidiameters of the Sun and moon. 3. The projection or calculation of an occultation of a Planet by the moon, may be performed in nearly the same manner as for a fixed star. The planet's right ascension, declination, geo- centric longitude and latitude, may be obtained from the Nautical Almanac, and must be used instead of those of the star. The moon’s hourly motions from the planet in longitude and latitude, must be used instead of the hourly motions of the moon. When great accuracy is required, the parallax and semidiameter of the planet must be taken into view; but it is not thought necessary to notice here the manner of doing this. ExAM. Required to project, for the latitude and meridian of Washington, an occultation of y Tauri, by the moon, which took place in January, 1813, the elements obtained from the Nautical 46 3+5 * ASTRONOMY. Almanac for that year, and by the problems referred to in the rule, being as follws: d. h. m. Conjunction, in appar. time at Washington, January, 12 6 53 Star's passage over meridian, º sº tº 8 32 Semidiameter of circle of projection, tº 59' 29" = 59'.48 Star's longitude, - tº tº . 63° 11 Sun’s longitude, gº º 292 30 | Star's latitude, ſº º sº 5 45 5 S. Moon's latitude, tºº ſº - 5 4 55 S. Moon's dist. from star in latitude, north, 40 10 = 40.17 Star's declination, north, gº ſº 15 10 Moon's hor. mot. in longitude, 3 ſº 35 57 = 35.95 Moon's hor. mot, in latitude, tending south, 0 45 = 0.75 Moon's semidiameter, Gº tºº ſº 16 14 = 16.23 Latitude of Place, tº tºº ſº 38 53 Fourth term, - tº sº dº 31 45 = 31.75 tº The Figures 61, 62, 63 and 64, are adapted to this example, and need no further explanation. The time of immersion, ob- tained by the construction, is 5 h. 49 m.; and of the emersion, 6 h. 46 m. TROBLEM XXXIX. Given the JMoon's true Longitude to find the corresponding lime at Greenwich by the JNautical Almanac, the approximate time be- ing given. Call the hours and minutes, &c. of the approximate time, or their excess above 12 hours, T. Take, from the Almanac, the two longitudes of the moon next less, and two next greater than the given longitude, and find the first and second differences, and the arcs A and B, as directed in prob. XII. With the time T and the arc B, take the equation of second differences from table LVI, and apply it with the same sign as B, to the difference between the second longitude and the given one, and call the result D. Then A : D :: 12 hours : to ASTRONOMY. 355 the required time, reckoned from the noon or midnight corres- ponding to the second longitude. The time thus obtained will be apparent time at Greenwich. ExAM. Required the time at Greenwich when the moon’s longi- tude is 7, 18° 58' 47", the approximate time being August 6th, 1821, at 1 h. 41 m. Here, T = 1 h. 41 m. Longitudes. | 1st Diff. 2d Diff. Mean of * 2d Diff. 5th midn. 7, 12° g #" sº wº 6th noon | 7 18 55 * 2 23 g 6th midn. 7 24, 9 18 A. : : ; 2 57 B. -- 2'40' # j | 8 |o 13 $5 wº Given longitude, dº * 7s 18° 58' 47" 2d longitude, tº º ſº sº 7 18 7 55 Difference, - - wº º 50 52 Equat, 2d Diff. - &= dº + 9.7 D. 51 1.7 6° 1' 23" : 51' 1".7 :: 12 h. : 1 h. 41 m. 40 sec. the time re- quired. PROBLEM XL. Given the Latitude of a Place and the observed apparent Time of the Beginning or End of an Occultation of a fived star by the ºngon, to find the Longitude of the place, it being supposed to be nearly known by estimation. By means of the estimated longitude, reduce the observed time to the meridian of Greenwich, and for that time calculate the moon’s true longitude, latitude, and semidiameter; and then the parallax in longitude, the apparent latitude, and the augmented semidiameter. Also, find the star's longitude and latitude, cor- rected for aberration and mutation. When the moon's apparent latitude and the latitude of the star are of the same name, take \ 356 ASTRONOMY. their difference; but when they are of different names, take their sum; the result will be the moon’s apparent distance from the star in latitude. Call this distance d, and the augmented semidiame- ters. Add together the logarithms of (8 + d) and (s — d.), and to half their sum, add the arithmetical complement of the Cosime of the star's latitude, and the result will be the logarithm of a small arc c. When the calculation is for the beginning, subtract c from the star's longitude: but when it is for the end, add c to the star's longitude; and the result will be the moon's apparent longi- tude at the observed time of beginning or end. . To the moon’s apparent longitude, thus found, apply the parallax in longitude, by adding when the moon is to the west of the nonagesimal, but by subleacting when it is to the east; and the result will be the moon’s true longitude, as deduced from the observation. Find from the Nautical Almanac, by the last problem, the time at Greenwich when the moon has this longitude, the approximate time being the obscrved time of beginning or end reduced to the meridian of Greenwich. Then, on the supposition that the tables are accurate, the difference between the time found from the Nautical Almanac, and the observed time of beginning or end, will be the longitude of the place in time. If the longitude thus found differs considerably from the estimated longitude, the ope- ration should be repeated. Note 1. When the immersion and emersion are both observed, the longitude should be deduced from each, and the mean of the results taken as the longitude of the place. 2. The above rule with a little change, will serve to determine the longitude of a place from the observed time of beginning or end of an eclipse of the sun. To do this, the sum’s longitude must be used instead of the star's; d must be taken equal to the moon’s apparent latitude, and 8 equal to the sum of the sun’s semidiame- ter and the augmented semidiameter of the moon. It may also be observed, that the sum’s latitude being nothing, the arithmetical complement of its Cosine will be nothing. ExAM. The beginning of the occultation of y Tauri, mentioned in the example to prob. XXXVIII, was observed by Bradley and *s ASTRO-NOMY. 357 Pease, at a distance of nearly two miles from the Capitol in Washington. The apparent time of immersion, after allowance made for the error of the watch, was 5 h. 46 m. 49 sec.; the re- duced latitude of the place of observation, 38°42'59" N.; and its estimated longitude in time, 5h. 7 m. 50 sec. west. Required the longitude of the place of observation, making use of the fol- lowing elements, obtained from the Nautical Almanac for 1813, or calculated by preceding problems: | Star’s corrected longitude, * * wº 63° 11’ 187.2 Do. latitude, South, tº - 5 45 6.1 Moon's parallax in longitude, the moon being to the east of the nonagesimal, gº 24 59.8 Moon's apparent latitude, south, £º º 5 37 37.1 Moon's appar. dist. from star in latitude, d. 7 29 Moon’s augmented semidiameter, tº s, 16 23.6 8 + d {º - . 14327.6 gº Jog. 3.15613 8 — d - gº ** 534.6 & log. 2.72803 2)5.88416 2.94208 Star's latitude, 5° 45' Ar. Co. cos. 0.00219 C tº dº 879”.6 = 14' 39".6 log. 2.94427 Star's longitude, sº {º - 63° F 1' 18".2 C - - tº & º tº . 14 39.6 Moon's apparent longitude, - 62 56 38.6 Parallax in longitude, sº º 24 59.8 * s — Moon's true longitude, * . 62 31 38.8 Appar. time at Greenwich when the moon h. m. sec. had that longitude, gº tº-g tºº 10 54 39.4. Appar. time of immersion, observed, - 5 46 49 Longitude, in time, of the place of observation, 5 7 50.4 END OF PART II. ASTRONOMICAL TABILIES., TABLE I. 3 Latitudes, and Longitudes from the JMeridian of Greenwich, of some Cities, and other conspicuous Places. z w Names of Places. Latitude. º º O / // h. m. s. Amsterdam, Holland, 52 22 17N. 4 53 15E. 0 19 33 Athens, Greece, 37 58 1N. 23 46 14E. 1 35 5 Baltimore, U. States, 39 23 ON. 76 50 0W. 5 7. 20 Bergen, Morway, 60 24 ON. 5 20 40E. 0 21 23 Berlin, Germany, 52 31 45N. 13 22 15E. 0 53 29 Boston, U. States, 42 23 ON. 71 4 0W. 4 44 16 Botany Bay, New Hol. 34 3 OS. 151 15 OE. 10 5 0 Brest, France, 48 23 14N. 4 28 45W. 0 17 55 Bristol, England, 51 27 6N. 2 35 29W.] 0 10 22 Cadiz, Obs. Spain, S6 32 ON. 6 17 22W. 0 25 9 Cairo, Egypt, 30 2 21N, 31 18 45E, 2 5 15 Canton, China, 23 8 9N. |113 2 45E. 7 32 11 Cape G. Hope, Africa, 33 55 15S. 18 24 OE. 1 15 36 Charleston. U. States, 32 50 0N. 79 48 OW.] 5 19 12 Constantinople, Turkey, 41 1 27 N. 28 55 15E. 1 55 41 Copenhagen, Denmark, 55 41 4N. | 12 35 6E. 0 50 20 Dublin, Ireland, 53 21 11.N. 6 18 45W. 0 25 15 Edinburgh, Scotland, 55 57 57N. 3 10 15W. 0 12 41 Greenwich, Obs. England, 51 28 40N. 0 0' 0 () () () London, England, 51 30 49N. 0 5 30W. 0 0 22 Madrid, Spain, 40 24, 57N. | 3 42 15W. || 0 14, 49 Naples, Italy, 40 50 15N. 14, 15 45E. O 57 3 New-Haven, U. States, 41 18 ON. 72 58 OW. 4, 51 52 New-York, U. States, 40 42 40N. 74 1 OW. 4, 56 4 Paris, Obs. France, 48 50 14N. 2 20 15E. O 9 21 Pekin, China, 39 54, 13N. 116 27 45E. 7 45 51 St. Petersburg, lêussia, 59 56 23N. 30 18 45E. 2 1 15 Philadelphia, U. States, 39 56 55N. 75 11 30W. 5 O 46 Point Venus, Otaheite, 17 29 17S. 149 30 15W. 9 58 1 Quebec, Canada, 4647 30N. 71 9 45W, 4 44 39 Richmond, U. States, 37 30 ON. 77 58 OW.; 5 11 52 Rome, Italy, 41 53 54N. 12 28 15E. 0 49 53 Stockholm, Sweden, 59 20 31N. 18 3 30E. 1 12 14 Vienna, Germany, 48 12 40N. 16 22 45E. 1 5 31 Wardhus, Lapland, 70 22 36N. 26 26 30W.}* 1 45 46 Washington, U. States, 38 53 0N. 76 55 30W.] 5 7 42 .* TABLE II. \ JMean Astronomical Refraclions. Ap. Alt. Refr. Ap. Alt. Refr. 1Ap. Alt. Ap. Alt. Refr. " Oo. 07 || 33/ O'' 40 O' | 11/51// 120 20/ 450 07 || 0/57// O 5 32 10 4, 10 || 11 29 || 12 40 46 0 0 55 0 10 || 31 22 || 4, 20 | 11 8 || 13 0 || 4. 47 0 0 53 O 15 30 S5 4, 30 || 10 48 || 13 20 3 48 0 0 51 0 20 29 50 4, 40 || 10 29 || 13 40 || 3 49 0 0.49 0 25 | 29 6 : 4 50 | 10 11 || 14 0 || 3 50 0 0 48 0 30 28 22 : 5 0 || 9 54 14, 20 || 3 40 51 0 0 46 0 35 27 41 || 5 10 || 9 38 14, 40 || 3 52 0 0 44 O 40 || 27 0 || 5 20 || 9 23 15 0 || 3 5S O 0 43 0 45 26 20 # 5 30 || 9 8 15 30 || 3 54, O 0 41 || 0 50 || 25 42 || 5 40 || 8 54 16 0 || 3 55 0 0 40 0 55 || 25 5 || 5 50 || 8 4.1 : 16 30 3 56 () 0 38 | 1 0 || 24, 29 || 6 0 || 8 28 || 17 0 || 3 57 0 O 37 1 5 || 23 54 6 10 || 8 15 17 30 2 58 0 0 35 1 10 || 23 20 6 20 8 3 18 0 || 2 59 0 0 34 1 15 22 47 6 30 || 7 51 18 30 2 60 0 0 33 1 20 22 15 6 40 || 7 40 ; 19 0 || 2 61 () 0 31 1 25 21 44 6 50 || 7 30 19 30 2 62 0 0 30 1 30 21 15 # 7 O 7 20 20 0 || 2 3 0 0 29 1 35 | 20 46 7 10 || 7 11 20 3 2 64, 0 0 28 1 40 20 18 7 20 || 7 2 || 21 0 || 2 65 0 0 26 1 45 19 51 7 30 || 6 53 21 30 2 66 0 0 25 1 50 | 19 25 7 40 || 6 45 22 0 || 2 67 0 0 24 1 55 | 19 0 || 7 50 || 6 37 23 0 || 2 68 0 0 23 2 0 | 18 35 ; 8 0 || 6 29 24; 0 || 2 69 0 0 22 2 5 18 11 : 8 10 || 6 22 25 0 || 2 70 0 0 21 2 10 || 17 48 8 20 6 15 # 26 0 || 1 71 O 0 19 2 15 17 26 8 30 || 6 8 27 0 || 1 72 0 0 18 2 20 | 17 4 || 8 40 || 6 1 28 0 || 1 73 0 0 17 2 25 | 16 44 8 50 || 5 55 ; 29 0 || 1 74 0 0 16 2 30 | 16 24, 9 0 || 5 48 30 0 || 1 75 0 0 15 2 35 | 16 4 || 9 || 0 || 5 42 || 31 0 || 1 76 0 0 14 2 40 15 45 9 20 || 5 36 32 0 || 1 77 O 0 13 2 45 | 15 27 || 9 30 || 5 31 : 33 0 || 1 78.0 0 12 2 50 | 15 9 || 9 40 || 5 25 || 34 0 || 1 79 0 0 11 2 55 | 1.4 52 || 9 50 | 5 20 # 35 0 || 1 80 0 0 10 3 O || 14 S6 10 0 || 5 15 # 36 0 || 1 81 0 O 9 3 5 14, 20 * 10 15 5 7 || 37 0 || 1 82 0 O 8 3 10 || 14 4 ; 10 30 || 5 0 || 38, 0 || 1 83 0 O 7 3 15 || 13 49 : 10 45 4, 53 || 39 0 || 1 84 0 || 0 6 3 20 | 13 34 11 0 || 4 47 40 0 || 1 8 || 85 0 O 5 3 25 | 13 20 11 15 440 : 41 0 || 1 5 I 86 0 0 4 3 30 13 6 ſ 11 30 || 4 34 || 42 0 || 1 3 || 87 0 0 3 3 40 | 12 40 11 45 || 4, 29 : 43 0 || 1 1 88 0 0 2 3 50 | 12 15 A, 12 0 || 4 23 I 44 0 | 0 59 89 0 0 1 TABLE III. ! § JMean Ilight flscensions and Declinations of some of the Fived Stars, for the beginning of 1820, with their flnnual Variations. \Mames and Magnitude. Right Asc.] An. Var. Declination. An. Var. f Mag. " ' " // O / // p > y Pegasi, - - - , 3 || 0 59 35 | + 46.1 14 10 56N. +20.0 a Polaris, - - 2.3 || 14 13 7 || 216.4 || 88 20 55N. +19.4 & Arietis, - - - 3 29 15 38 50.3 22 36 23N. + 17.3 a Ceti, - - - 2 || 43 13 8 46.7 || 3 22 39N. +14.5 3, Tauri, - - - 2 62 23 23 51.0 | 15 11 3N. + 9.2 Aldebaran, - - 1 | 66 24 0 51.4 | 16 8 19N. + 7.8 Capella, - - 1 || 75 51 7 66.3 45 48 8N. + 4.5 Rigel, - - - 1 || 76 28 21 51.8 || 8 25 2S. — 4.7 g Tauri, - - - 2 78 43 47 56.7 28 26 42N. + 3.8 & Tauri, - - - 3 || 81 43 14 53.6 21 1 23N. -- 2.8 * Geminorum, 2.3 | 91 0 6 54.3 22 32 56N. — 0.4 & Geminorum, - 3 | 93 1 0 54.5 22 35 48N. — 1.1 y Geminorum, 2.3 96 49 37 52.0 | 16 32 36N. — 2.5 Sirius, - - - 1 || 99 18 18 39.8 | 16 28 33S. + 4.4 J. Geminorum, - 3 |107 20 25 53.8 || 22 1.8 15N. – 6.0 Procyon, - - 1.2 112 28 2 47. 1 || 5 40 46N. — 8.6 |Pollux, - - 2.3 |113 34 16 55.3 28 27 7.N. — 8.0 a Leonis, - - - 3 |149 22 33 49.2 || 17 38 10N. —17.3 Regulus, - - 1 ||149 41 39 48.1 | 12 50 36N. —17.3 g Virginis, - - - 3 |175 1945 46.9, 2 46 44N. —20.3 y Virginis, - - 3 |188 8 4 45.3 || 0 27 38S. +20.0 2 Virginis, - - - 1 198 55 30 47.2 10 13 5S. +19.0 Arcturus, - - 1 |211 51 45 40.9 || 20 7. 28.N. —19.0 a 2 Librae, - - 2.3 [220 14, 2 49.5 15 17 13S. +15.4 J Scorpii, - - 3 |237 25 37 52.9 22 5 59S. -- 10.9 g Scorpii, - - - 2 238 44 48 52.0 | 19 18 13s. |-|- 10.5 Antares, - - 1 |244, 35 49 54.9 26 1 21S. + 8.7 a Lyrae, - - - - 1 |277 42 37 30.4 || 38 37 19N. –– 2.9 a Sagittarii, - -2.3 (281. 1 28 55.8 26 30 34S. — 3.7 z Sagittarii, - - 3 |284 45 47 53.5 21 18 OS. — 5.0 a 1 Capricorni, 3.4 .301 54 56 50.0 || 13 3 22S. 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GN. ONGNQ GNQ CNR GN c^ *AX GITISIVJ, Gº, II on º co so - q} –， – – – – ºś • GUYQUr Caer Gas $ $ $ $ $ $ &9 CO CAO CAO CAO CAO co on v. CO NO Q * I 92, O 98 II I 9& I 22, O #& SI I | ?& & 82, O fºlſ GI I 83 & 68, 0 || 7 AI I | 33 g 08 0 | #9 8T I | [& 2.I & O 9% O2, T | 0& Ig 98 0 || 88 & I | 6T &g jºg 0 || OS 7& I | 8 I WI 93 0 || 73, 93, I 2I A3 23 0 || 8 I 83, I | 9 | 2, 63 () &I 08 I GI 82, Off O || 2 38 I | WI 99. If, O & #8 I | SI/ #2, 8; 0 | 89 98 I &I #g #7 0 | #9 A8 I • • • • → + QUY QUY QUY QUY · - （№ on ~i (o H） NQ N9NO º Ř – № $3， $ C_n }=-4. NO $ $ $ $3 $ <> <> <> <> o G3 *** }） }-， ſ-a ſ-a NO ºr oſ º co to Q }}> CO C0 C0 C0 C0 on to ſº NO → H< * CD C O O O }-} }， }） }） }） Caeſ)NO }=4. O QUr}→ CO QUY }}> GUY CO → NO *N OD ON ON QUY NO QUY NO}}> Q NO OD CO → <> <> <> <> <> º ſº ſº ſº ſº ſº. № }， & Co co o o º *ą. C o C> C O ©r çuş № №. ！!！ NO ！!！ ：） & 0 ON Õ Õo ND °N o o C> <> o Oº Ò9 CAO CAO CLO O QUY CAO NO → CO +）, ±> NO O He-4 Co *NĮ ~Q e w -k- - 39 p.2, O | 8%. 6 I | 93 3 IZ, 0 | [8 j, I 63 8 & 0 || 9T 9. I 8& ## 82, O & 8 I A& 8 O2, 0 || Alf & I 08 I 5. 9 3, 9 6 Q 0. O O O O }}> }}> Om On Ur }=4 }） }=4 ſ-a }） }}> <ſ> º \!> £> È È È È È }=4 }）; Q„O Q> *N O CAO }， }） }， }， }） NO NO NO NO NO Co *NI QUY QL9 }， GUY «O O NO QUY CO sIA g|IIA. sIIA. ‘LNGINſibaw *. ** s\{I "241"100 sans on ſº unimba sy. ‘Āſeuouv učaſ sºuns l sIX 46 TABLE XVI, TABLE XVII, Small Equations of Sun's Longitude. JMean Obliquity of the w Ecliptic. I I l Arg, J. II. III. Arg. I II. Years.TM. Obſiqu. 0 | 10%| 10’’ 10”; 500 | 10"| 10"| 1 1821 230 2746// g / 320 | 15 || 14 || 19 820 330 | 15 || 14 | 19 830 340 | 15 || 14 | 20 840 350 15 13 20 # 850 10 1853 23 27 29 10 1854 || 23 27 28 11 1855 23 27 28 1856 || 23 27 27 12 1857 23 27 27 12 1858 || 23 27 26 13 1859 || 23 27 26 1860 23 27 25 360 | 15 || 13 | 20 860 370 || 14 || 12 19 : 870 380 || 14 12 19 880 390 || 14 | 12 19 890 400 || 13 || 11 18 900 410 || 13 || 11 17 910 420 13 11 17 920 10 || 10 || 11 || 9 || 510 || 10 10 1822 || 23 27 46 20 | 11 11 || 9 : 520 || 9 || 10 || 8 1823 23 27 45 30 || 11 || 12 || 8 || 530 || 9 || 10 || 7 1824, 23 27 44, 40 || 11 || 13 || 8 || 540 || 9 || 10 || 7 1825 || 23 27 44 50 | 12 14 || 7 || 550 | 8 || 10 || 6 1826 || 23 27 43 _60 | 12 || 14 || 7 || 560 || 8 || 9 || 5 1827 || 23 27 43 70 12 15 || 7 : 570 || 8 || 9 || 4 1828 || 23 27 42 80 13, 15 || 7 || 580 || 7 || 9 || 3 1829 || 23 27 42 90 13 | 16 || 7 || 590 || 7 || 9 || 3 1830 || 23 27 41 , 100 | 13 | 16 || 7 || 600 || 7 || 9 || 2 -* 1831 || 23 27 41 110 || 14 || 17 || 7 || 610 || 6 || 8 || 1, 1832 || 23 27 40 120 || 14 || 17 || 7 || 620 || 6 || 8 || 1 1833 || 23 27 40 130 || 14, 18 || 8 || 630 || 6 || 8 || 1 1834 || 23 27 39 140 | 15 18 || 8 || 640 || 5 || 7 || 0 1835 || 23 27 39 150 | 15 18 || 9 || 650 || 5 || 7 || 0 t 1836 || 23 27 38 166) 15 18 || 9 || 660 || 5 || || 6 || 0 1837 || 23 27 38 170 | 15 18 || 10 || 670 || 5 || 6 || 1 1838 || 23 27 37 180 | 15 | 18 10 680 || 5 || 6 || 1 1839 || 23 27 37 190 | 16 18 || 11 || 690 || 4 || 5 || 2 1840 23 27 36 200 | 16 || 18 || 11 || 700 || 4 || 5 || 2 * 1841 || 23 27 36 210 | 16 | 18 || 12 || 710 || 4 || 4 || 3 1842 23 27 35 220 | 16 || 18 12 || 720 || 4 || 4 || 3 1843 || 23 27 35 230 | 16 || 18 || 13 730 || 4 || 4 |, 4 1844 || 23 27 34 240 | 16 || 17 | 14 740 || 4 || 3 || 5 1845 || 23 27 34 250 | 16 || 17 | 14 || 750 || 4 || 3 || 6 1846 || 23 27 33 260 | 16 || 17 | 15 # 7.60- 4 6 1847 23.27 33 270 | 16 || 16 16 || 770 || 4 7 1848 || 23 27 32 280 | 16 || 16 || 17 || 780 || 4. 8 1849 || 23 27 31 290 | 16 || 16 || 17 790 4. 8 1850 || 23 27 30 300 | 16 || 15 18 # 800 4. 9 1851 || 23 27 30 310 | 16 15 18 810 || 4. 9 1852 23 27 29 | 5 5 5 5 5 6 6 6 7 7 7 1. iiiii 1 3 430 | 12 || 11 | 16 || 930 | 8 440 | 12 || 11 || 15 - 940 || 8 13 450 | 12 || 10 || 14 950 | 8 13 460 | 11 10 13 || 960 9 12 470 | 11 || 10 || 13 970 9 12 480 | 11 || 10 | 12 || 980 9 11 490 || 10 || 10 || 11 990 10 11 500 || 10 || 10 || 10 1000 || 10 10 *IIIAX GITIH.W.I. &W #3 or T-Fººd T-ToT-Troof or T–To T-HTo T-HToog Oſ I I 066 . OI I I 067 OI & 3, OS6 OE 3, & 08% 6 : 8. S 0.26 || 6 8 S 02:7 6 7 # 096 ft 6 # ſº O9% 6 -- | g – || 9 - || 096 6 — | g + | 9 + | 09% 6 9 4. O#76 || 6 9 4. Oyj, 6 4. 9 086 || 6 4. 8 O37 | 8 8 6 036 ... 8 8 6 O 8 6 OI OI6 || 8 6 OI OI? 8 : + | OI — II — 006 || 8 — OI + II + | 00% 2. OI II 068 || A. OI II 068 2. II &I 088 || 2. IT ØI 088 4. 3I SI (928 A. &I SI 0.18 9 8 I #I 098 - 9 SI #I 098 9 + | 81 — | g I — Ogg 9 — SI -- | g I + | Ogg 9. WI GI O78 || 9 # I QI Ofg 9. #I 9I OS8 9 #I 9I OSS * j, GI 9I 0&S # GI 9I 038 | 8 GI 2I OI8.4 S GI AI OTS 8 + | 91 — 21 — | 008 | g – 91 + | 21 + | 008 & 9T AI O64 & 9I AI 06& & 9I 8I 08A & | 9 | 8I 083, I 9I 8T 0.22 I 9I 8L ,023 I + || 9 | 8I 092 || I 9I — 8 I '09& O – || 9T – SI — | 092 + 0 + | 9L -- 8L + || Ogg I 9I 8I 074. I 9'ſ 8T 0% I 9T 8I 08/. I 9I 8I 083, & 9I 8T 03/, & 9T 8T 033 | 6 9I AI OIA & 9I AI OI& 8 – || 9 || – || ZI — 002 || 8 + | 91 + | ZI + 002, 8 QI AL O69 || 8 GI AI 06I j, GI 9I 039 || ||7 GI 9I 08T 9. FI 9I 0.29 || 9 WI 9I 02I 9. FI GI 099 || G #I GI 09 I 9 — SI – gT – || 099 || 9 + | g + g|I + Og| 9 £I #I 079 9 SI #I 07I 2. ØI SI 0.89 °F. A. &I SI OSI 2. II &I 039 T. A. II &I 0&I 4. OI II OI9 . A Of II OLI 8 — OI - || II – 009 : 8 -H | OI + II —H | 00I 8 6 OI 069 || 8 | 6 OI 06 8 8 6 O89 : 8 8 6 O8 8 2. 8 0.29 ± 6 4. 8 02. 6 9 2. 099 || 6 9 4. 09 | 6 – | g – 9 - || Ogg | 6 + | g + | g + |0s 6 # # 079 || 6 º y Off 6 8 8. O89 || 6 | 8 S 08 6 & 3, 039 || OI 3, & 03, OI | I I OLG | OT I I OI ,01 – 26 -– 20 – dog ºr + ,6 + 6 + | 0 ‘buqo 'ºosV ºf 5uorſ 'N buqo osvºj '3tion 'N 'N IO ºpon op jo quauaiddns 'LNGINſibay - ‘uoy!ppm.N.' * - 48 TABLE XIX. * Equation of Time, to convert Apparent into Mean. Sun's Mean Longitude. ARGUMENT. ·* KO N）, CD Cò №ſ $ ſs - in co – co\^ N. GNR +→ QO OD OD <> o+--+ GNQ Crò <* AròCGOĆO ∞??+ ' ' <ń Ń – šeš | Gð ;）（ * & †y-+ c^) GN | <º ： Gº <říGº <* , co ſp$3 （$ $ $ $3 >goº oº — ~ ~ ~<> <> <> <> Q> | -! ;-+ +-+ GNR GN || CN， co co Gº <Ř | <ſº º Aro Aro Arp j co qo qo №, №. +-+ || $ ſo ~ ~ ~ ſo ſo | co o № co oCO ON? \^ N- || On Q > w-， ſ-， y-+ | Q CO CO co o || N. cro co crò gael?vo vrae <# | <ſº co co CN -Ar> <ſ， co | Gº Arp <† 6.5È ºgg}}> |×gºo ſo ſo ſo º so ſ to ſo ſo ſo ſo ſ to to to to so ſ to to < < <- [ <** • • • • • • • oº º +Aº+ și o co to o cº <ſ«O CO O x={ GNRc^ <* <řł co coGNQ O CO «O cròC GO CN. N. GNRvo on GN <> <> | +-+ +-+ +-+ +-+ +-+ -+-+ || | o o w-， CN co <ſ， un«O N， OO ON Oș→ CN^ Go <ř+ \,^)O！ Gº Go <† \pGO N） CO OD O ș-4 | ► | y=4 \，={ ſ=\ y={GN ſ GNR GNR GN GNR GNR ſ GNR GNR GN-GN， CO º XIX CITIgIV,L 6) s Ig 9 -H 0g gI +l 2 gI +| 22; 0 - 7g SI – gg gI – Og Of 9 89 SI 09 II A9 0 I? SI | A2, gli 63 69 9 g FI 99 II 82, I A9 8I 8I GI 83 AI A II FI 9I II 89 I 2I f I | 6 GI A & 98. A AI fI A9 OI 83 3 93 fI | 89 FI 9& fg A 22, 7I 89 OI 69 6 69 7I 97 p I 92, 9I 8 92, 7 I 8I OI 62, 8 39 f I 98 7I fra, I9 8 08 FI | A9 6 69 S 8 9 I f& 7I [ 82, 8 f 8 38 7I | 98 6 82, fº i7I g I II FI 32, 9 6 ipg 7I SI 6 89 fº f& GI 89 9I | Iº, 82 6 98 fI | 09 8 A3 9 89 GI i f SI 03 If 6 99 7I | A3 8 99 9 & SI | 6g 9 6I 89 6 99 7I 9 8 i73 9 67 GI 7I gI 8I fI OI gg 7I 88 A 29 9 99 9 I 69 & I AI 09 OI 99 FI SI A 0& A I 9I gf & I 9I 9f OI 09 f I Aif 9 Aif A 9 9I 93 & I 9 I & II 92, 7I Iº, 9 7I 8 6 9I 6 & I | 7 I AI II ga, FI f,9 9 07 8 &I 9I 39 II SI 39 II 9I f I 92, 9 9 6 fI 9I f, 9 II & I 9f II OI FI 69 7 I8 6 9I 9I 9I II II O & I 9 7I | 09 fº 99 6 9I 9I 39 OI OI SI & I gg 8I | 2, 7 03 OI 9 I 9I 68 OI 6 92, 3 I Alf, 9 I S8 8 8 f OI i7I 9I 03 0I | 8 68 3I A9 8I | f 9. 9 II 3 I 9 I 0 0 I | A Ig & I | A, 8I | f8 3 83 II 6 9I If 6 9 2, SI 9I i i 2, 67 II 9 9I I3 6 9 9I SI S I | G8 I OI & I I 9I I 6 f, S2, 9I 0g & I | f I 63 3I 99 GI 07-8 8 98 9I 98 2, I | 78 0 8f & I 67 GI 6I 8 2, &# 8I 33 3 I i 0 -- A SI 8f, 9 I 69 2 I 0g gt -- A gI + 28, 0 – fa, gli – gg gI - 88 2 - 0 202S º UDI ”39S ”UI “998 ”ULI QºS º UI “ O9S º UII “OoS ”UI O sIX sX sXI sIIIA sII A sIA “apn113uo I ueaIN s.unS LNGINnpaV rueaIN opul luoreddV 1.toſuoo oi “aul L ſououpnbar ara 20 TABLE XX. JMoon's Epochs. A Years. ITV 2 || 3 || 4 || 3 || 6 || 7 || 8 TV 1821 0027 | 8365 5389 1368 || 6970 7714 6319 7024 7800 1822 00:20 5573 || 5054, 6112 || 94.41 || 3512 || 7380 9481 | 6664. 1823 0012 2782 4720 || 0856 1913 || 9309 | 8440 1938 5528 1824 B. 0033 0640 5426 5887 || 4720 5478 9559 4787 44.17 1825 0026 784.9 5092 || 0631 || 7 192 | 1276 0619 7243 3280 1826 0018 5057 || 4758 || 5375 9663 || 7073 | 1680 | 97.01 2144 1827 O011 |,2265 4424 || 0119 2135 | 2871 2740 2158 1008 | 1828 B. 0032 0124 || 5129 || 5150 494.2 9040 || 3859 || 5007 || 98.96 1829 00:24 || 7332 4795 98.94 || 74.14 4837 || 4919 7463 87.60 1830 0017 | 4541 4461 || 4638 [9885 | 0635 | 5979, 9921 || 7623 1831 O010 || 1749 || 4127 | 9381 || 2357 6432 || 7040 2378 || 6487 1832 B, 0030 9607 4833 4412 5164. || 2601 || 81.58 5226 5376 1833 0023 | 6816 4499 || 9156 7636 8399 || 9219 7683 4239 1834 0016 || 4024 4164 || 3900 || 0107 || 4.196 || 0279 || 0140 || 3103 1835 - || 0009 || 1232 3830 8644 2579 || 9993 || 1340 2598 || 1967 1836 B. 00:29 9091 || 4536 || 3675 5386 6163 2458 || 5446 || 0856 1837 00:22 6299 || 4202 | 84.19 7858 1960 || 3518 7903 || 9719 1838 00.15 3508 || 3868 3163 0329 7757 || 4579 || 0360 | 8583 1839 0008 || 0716 3534 7907 || 2801 || 3555 5639 || 2818 7447 1840 B, 0028 8575 || 4239 2938 5608 9724 6758 || 5666 6335 1841 0021 || 57.83 3906 7682 8080 5522 7818 8123 || 51.99 1842 O014 || 2991 35.71 2425 || 0551 1319 8879 || 0580 4062 1843 0007 || 0200 3237 7169 || 3023 7116 |9939 3038 2926 1844.B. .0027 8058 3943 2200 5830 3286 1058 5886 | 1815 1845 0020 5266 3609 || 6944 8302 9083 2118 8343 || 0678 1846 0013 24.75 3275 1688 0773 || 4880 3179 0800 9542 1847 0006 9683 2941 6432 3245 0678 || 4239 || 3257 | 8406 1848 B. 0026 7542 3646 1463 || 6052 | 6847 || 5358 || 6106 || 7295 1849 0.019 || 4750 || 3312 6207 || 8524, 2644 6418 8563 6158 1850 0012 1958 2978 || 0951 || 0995 | 8442 7479 || 1020 5022 1851 0005 || 9167 2644 5695 3467 4239 || 8539 || 3477 3885 1852 B. 0025 7025 || 3350 0726 6274 || 0408 9658 || 6326 2774 1853 0018 4233 3016 || 54.69 || 87.46 6206 || 0718 8782 | 1637 1854, 0.011 || 1442 2681 || 0213 || 1217 | 2003 || 1778 1240 0501 1855 0004 || 8650 || 2347 || 4957 || 3689 7801 || 2839 3697 9365 1856 B. 00:24 6509 || 3053 9988 || 6496 || 3970 3957 || 6546 || 8254 1857 O017 3717 | 2719 || 4732 | 8968 97.67 5018 9002 || 7117 1858 0010 || 0925 || 2385 94.76 || 1439 5565 6078 || 1460 || 5981 1859 0003 || 8134 || 2051 || 4220 3911 1362 7139 3917 || 4845 1860 B. 0023 5992 || 2756 9251 / 6718 7531 | 8257 6765 |º TABLE XX. 24 JMoon's Epochs. HTTTTTTTTTTTTTTTTTTTTTT5-T-I5-55- 1821 || 620 917 | 842 || 142 979 || 067 923 || 331 134 || 036 || 036 1822 226 278 562 615 172 208 || 282 | 684 || 609 || 090 202 1823 833 639 281 || 088 || 366 || 348 || 641 || 036 || 084 || 143 || 369 1824 B. 509 || 030 || 070 595 || 659 || 519 || 037 431 585 197 537 1825 116 391 || 790 068 853 || 659 || 397 783 || 060 251 703 1826 | 722 || 752 || 510 || 54i 047 800 || 756 136 536 304 869 ; 1827 | 329 || 113 229 014 24.1 940 115 488 || 011 || 358 || 036 1828 B. 005 || 505 || 019 521 533 || 111 || 511 883 || 512 412 204 1829 || 612 866 | 738 || 994 | 727 251 871 235 987 466 370 1830 || 219 || 226 458 || 468 || 921 392 || 230 588 || 462 519 || 536 1831 825 587 177 940 | 115 532 589 940 | 937 || 573 | 703 1832 B. 502 || 979 || 967 || 447 | 408 || 704 || 985 335 || 438 627 871 1833 108 340 | 687 920 | 602 | 844 345 688 || 913 | 681 || 037 1834 || 715 701 || 406 || 393 || 796 || 984 || 704 || 040 || 388 || 734 203 1835 | 321 || 061 | 125 866 989 || 124 063 393 || 863 788 370 1836 B. 998 || 453 915 || 373 || 282 296 || 459 || 787 364 | 842 538 1837 605 || 814 || 635 | 846 || 476 436 819 || 140 840 | 895 704. 1838 211 || 175 354 || 319 670 576 178 492 315 949 870 1839 818 536 || 074 || 792 || 864 716 537 | 845 790 003 || 037 1840 B. 494 || 927 | 863 || 299 || 157 | 888 || 933 239 291 || 056 205 1841 || 101 288 583 || 772 || 351 || 028 293 || 592 || 766 110 || 371 1842 | 707 || 649 302 245 544 168 652 944 241 164 537 1843 314 || 010 || 022 || 718 || 738 308 || 012 || 297 716 218 704. 1844 B. 990 || 402 || 811 225 || 031 480 | 407 691 217 272 872 1845 597 763 531 698 |225 620 767 || 044 692 || 325 | 038 1846 203 123 250 | 171 || 419 760 126 396 167 379 204 1847 810 || 484 || 970 644 613 || 901 || 486 || 749 643 || 433 371 1848 B. 486 || 876 || 759 || 151 || 905 || 072 | 881 || 143 || 144 || 487 || 539 1849 || 093 || 237 479 624 || 099 212 241 496 || 619 540 705 1850 700 || 597 | 199 || 097 293 || 352 600 | 848 || 094 594 | 871 1851 || 306 || 958 918 570 || 487 493 960 201 569 648 || 038 || 1852 B. 983 || 350 | 707 || 077 | 780 | 664 355 595 || 070 701 || 206 1853 589 711 || 427 550 974 804 || 715 948 || 545 755 372 1854 196 || 072 147 || 023 168 944 || 074 || 300 020 | 809 || 539 1855 | 802 || 432 || 866 496 || 361 || 085 434 || 653 495 863 705 1856 B. 479 || 824 656 || 003 || 654 256 829 || 047 || 996 || 916 873 1857 || 086 185 375 || 476 | 848 396 189 400 471 970 03 1858 || 692 || 546 095 || 949 || 042 537 || 548 || 752 947, 024 206 1859 299 || 907 814 || 422 236 677 || 908 || 105 || 422 || 078 || 372 1860 B. l. 975 298 604 || 929 529 | 848 303 499 || 923 || 131 || 540 33. • *XX IIHVJ, 0 79 6T 0 | #7 A.I. 6 8 || 93, 87 0 8 || 8 & 8 II '81 098T 0& 08, A3, A. | 39 8& AI OT || 0T 99 8I j, | 88 IS T 9 698 I. 7I AS AI 8 || 93, IS / 9 || 09 &I 0& I & 0 II II | 898.I 6 #8 8 [I] I WI 8& I | 68, 6&. I& OI! 88 8& 0& 9 A98. I S II 63, 9 || 93 98 8T 6 || 6 97 && A. | S 29 6& III '91998I && ZS 9 & £ff A'ſ, 93 y | #9 89 OT # 88 9 8& 9 998.I. AL 7I A3, 6 || 6 I OI AY O 78 9T &I I | 7 98 Z 0 #98. II Ig ZI 9 || 89 &S Z 8 || SI 38 £I OI, jºg & AI 9 998.I 9 83, 8 I | 68, 99 AZ, 9 || 89 87 jz I Z || 9 &S 93 0 || "8 & 98.I 93, #9 GT 8 || 93 9 9 II. 89 I & F | 98 If ſº, 9 I98I 02, IG 9 ºff II 63, 93, 9 8T 8T # I # OT #7 I 098. I j/I 8 Jø II 97 Ig 9T & Zg ºff S 9 OI! S3 88 SI Z. 678 I 8 97 AI A I& #I Z OT || ZS [g 9 A, 9 A, 83, I "8: 878.I 82, II 92, & 63, g3, GT G | 88, p. 98, 8 98 9T. I& Z. Aft'8T 8& 87 ST OI iſ 87 g I & I& 93 0 || 9 g; 0 & 978T AI 93, 9 9 || 68 OI 9& 8 &# 28 A2, 6 98 SI OI 8 g?8I &I & A& I | #I 88 9T # 22, #79 83, 9 || A. &W 6L & ‘8 pººl IS 82, j. 6 || 2:2, 77 fº, II Z. A ZI 9 || AS IS ZI 8 gi/8I 93, 9 93, ſº | Ag 9 SI Z | Zī, 83, 8 I O | A 0& A3 & 378.I IZ, &# ST 0 || 38 62, 9 8 22, OW 6L 6 AS 87 9 6 Ij,8I gT 6T 9 8 || 8 & 9 gº, OI! / Ag O2, 9 || 8 AL 9'ſ 8 ‘8 O78.I g8 97 ST 8 GI & # 9 || 89 6 6 9 88 98, 7L 6 688T 63, 32, # IIſ O9 gº, ſº I 38 92, OI 0 || 6 99 8& 8 988.I 7& 69 ft2, 9 92, 8; 7ſ 6 || 2:I gº II 6 || 68 8& 8 OI. A.88.I 6I 98 gT & O II g g 39 69 &T 9 || 6 &9 &I '7 '81988.I 88 & 8& 6 || 6 && SI 0 | 88 &I L S 07 I II OI! 988.I 88 68 SI 9 || 7% ºf 8 8 8T 62, & O II 08 0& # #88T 8& 9T # I | 0& A. pg, 9 || 99 gift & 6 0% 89 63, OI. 888. I Sö 89 #3, 8 || 79 63 ºf II 88 & 9 9 II 23, 6 9 || '8, 388. I &# 6T & 7 || 8 [? &2, 9 #2, 9ſ 92, Z | If 98 Z II IS8I A8 99 && II 88 9 ST & # 39 jº. II & I 9 AI 9 O88I 38 £g £I Z | ST 93, 9 OI! ## 9% gº, 8 & SS 93, II 638 I 26 01 W S 67 87 S6 g | 73 g 26 g | SI & 9 9 |'81838: Aſ 98 II OI. 99 69 I I OI SI 91 & | 77 II 7 O A38 I & SI & 9 & 22, 2.2, 8 || 09 WS 9T III WI 07 SI 9 938 - A8 09 && I | A gy &T # IS [g AI 8 || 77 8 8& O 938. I IS AZ, 8 I 6 || 87 A. § 0 || II 8 6T G | GI AS & A. '91 W381 I9 89 0& 7 || IQ 8T II A A9 03 A 2, 97 97 0 I 838. I 97 OS II 0 || A3. If I 8 AS AS 8 II 9T gT OT / 338 I , I'7/2 o’, s? |/|& W of 3 solſ|/AI/79 s5 s8 |/27/87 oëIsſ I38I ‘apnºſition__"uonelle A • MIeuouv ‘tion oa Aq ‘S.189A ‘spoda suooſſ' TABLE XX. 23 JMoon's Epochs. Years. Supp. of Node Il. v. VI. VII. VIII.] IX. X. 1821 0s 139 3/29//; Os 270,417 | 706 || 711 || 074 || 079 637 596 1822 1 2 23 11 || 4, 18 13 || 120 124 382 386 717 536 1823 1 21 42 56 || 8 8 45 533 536 689 || 692 || 796 || 475 1824 B. 2 11 5 47 || 0 10 26 981 988 || 026 || 032 912 || 420 ; 1825 3 O 25 29 || 4 0 58 395 401 || 333 338 992 || 359 1826 3 19 45 11 || 7 21 30 809 || 813 || 641 645 072 299 1827 4, 9 4 53 || 11 12 2 223 225 949 951 | 151 238 1828 B. 4, 28 27 46 || 3 |13 43 670 677 285 291 || 267 182 1829 5 17 47 29 || 7 4, 15 084 090 592 || 597 || 34?" | 122 } 1830 6 7 7 11 || 10, 24, 47 || 498 || 502 || 900 904 || 427 | 062 1831 6 26 26 53 || 2 15 19 912 || 914 | 208 || 210 || 506 || 001 # 1832 B. 7 15 49 46 6 17 0 || 360 || 366 545 550 622 |945 1833 8 5 9 28 10 7 32 774 || 779 || 852 856 || 702 885 1834, 8 24, 29 11 1 28 4 || 187 191 || 159 163 782 825 1835 9 (3 48 53 5 18 36 601 || 603 || 467 469 || 861 || 764 1836 B. 10 3 11 46 9 20 18 048 || 055 804 809 977 || 708 1837 10 22 31 28 || 1 10 50 || 463 || 468 111 || 116 057 | 648 1838 11 11 51 10 || 5 1 22 || 876 880 || 419 || 423 | 137 588 1839 || 0 1 10 52 8 21 54 || 290 292 | 726 729 217 | 527 # 1840 B. || 0 20 33 45 || 0 23 35 | 738 744 || 063 || 069 || 332 || 471 1841 1 9 53 28 4 14, 7 || 152 157 || 370 375 412 411 1842 1 29 13 10 || 8 4 39 566 569 || 678 | 682 492 350 1843 2 18 32 52 | 11 25 11 # 980 || 980 986 || 988 572 290 1844 B. 3 7 55 45 3 26 52 427 433 322 328 687 || 23 1845 3 27 15 27 || 7 17 24 840 | 846 629 || 634 767 174. # 1846 4, 16 35 9 || 11 7 56 254 258 937 941 847 bl3 1847 5 5 54 52 || 2 28 38 || 668 || 670 245 247 || 927 || 053 1848 B. 2 25 17 45 7 0 9 || 116 122 || 582 587 042 997 1849 6 14 37 27 | 10 20 41 || 531 535 | 889 | 893 | 122 || 937 1850 7 3 57 9 || 2 11 13 944 || 947 196 || 200 202 : 876 ' 1851 | 7 28 16 51 | 6 1 45 | 358 || 359 504 || 506 282 | 816 1852 B. 8 12 39 44 || 10 3 27 806 || 811 | 841 | 846 398 || 760 1853 9 1 59 26 || 1 23 59 220 223 148 152 477 | 700 1854 9 21 19 9 || 5 14, 31 634 636 || 456 : 459 557 | 639 1855 10 10 38 51 || 9 5 3 || 047 || 048 || 763 765 637 579 1856 B. 11 0 1 44 || 1 6 44 || 495 500 | 100 } 105 753 523 1857 11 19 21 26 4, 27 16 || 909 || 912 407 || 411 832 463 1858 0 8 41 8 || 8 17 48 || 323 325 715 718 912 402 1859 0 28 O 51 || 0 8 20 | 736 737 023 O24 992 3:2 1860 B. l. 1 17 23.43. 4 10_1 | 184, 189 359 || 364 || 108_j 286 TABLE XXI. JMoon's JMotions for JMonths. Months. TT3T3TAT5 T5 TFT-3T9 Com. |000000000000|000000000000000000000000 Jan. 3 is 9973|93508960|97139664|9628|9942.9610|9976) Com. | 849) 146|2246|8896. 402|1533|17892099 753 Feb. 3." §§ 3; ; March, - 1615|8343||1371,6931|9797]1951|34043027|1433 April, - - 2464|8490.36.16|5827, 199|34845193.5.1262186 May, - - 3285,7986,4822,4436. 265/4646.69246835|2914 June, - - 41348.1337067|3332 66661798713|8934,3667 July, - - 4955,7629,8273/1942| 732.7341|443]. 643,4396 August, - - 58047776|| 518; 838||1134|88742233,2742,5148 September, 6653|7922|2764|9734|1536,408.40214842}5901 October, - |74747419,3969|8343|16021569||5752}5550,6630 November, 8325||7565|6215||7239|2004,3102.75:1|8649|7382 l)ecember, 1914417062|742015848(20704264,9272| 35818111 TABLE XXI, l JMoon's JMotions for JMonths. Months. Evection. Anomaly. Variation. M. Long. J Com. Os Oo Of Oſ/ Os Oo O/ 0// Os 09 0/ O// Os Oo 07 0// * } Bis. 11 1841 1 |11 1656 6 11 17 48 33 |11 16 49 25 reb. 3 Qom. || 23 4; 43 || 1; 9 33 Q 17. 54 48 || 1838.5 CD. 3. 11 9 29 43 || 1 1 56 59 0 5 43 21 || 1 5 17 31 March, - 10 7 40 26 || 1 20 50 4 |11 29 15 15 || 1 27 24 27 April, - - || 9 28 29 8 || 3 5 50 57 || 0 17 10 3 || 3 15 52 32 May, - - - || 9 || 7 58 51 4 747 56 || 0 22 53 24 || 4 21 10 3 June, - - || 8 28 47 33 5 22 4849 || 1 10 48 11 || 6 9 38 9 July, - - - || 8 8 17 16 || 6 24 45 48 || 1 16 31 32 || 7 14 55 40 August, - || 7 29 5 59 || 8 9 46.42 2 4 26 20 9 3 23 46 September, 7 19 54 41 || 9 24 47 35 | ? 22 21 7 |10 21 51 52 October, - || 6 29 24, 24 10 26 44, 34 || 2 28 4, 28 |11 27 9 22 November, 6 20 13 6 || 0 11 45 27 | 3 15 59 16 || 1 15 37 28 f l)ecember, 5 29 42 49 I 1342 26 || 3 21 42 3 2 20 54 59 TABLE XXI, 25 y JMoon's JMotions for JMonths. ‘MI-TTTTTTTTTTTTTTTTTTTTTTTTT20 J. com, looooooooooooooooooooo looo |000 000 000 * } Bis. 930 |969 |930 |966 |901 |969 |963 |958 |974 |000 |000 F Com. |175 |965 184 59 || 74 |946 |135 |304 |805 || 5 || 14 eb. 3. 105 |934, 114 25 |975 is 16 || 98 |262 |779 || 5 || 14 March, - 139 836 |157 | 16 |851 |801 j159 |482 |532 || 9 || 27 April, - - 314|801 |342 || 76 |925 |747 |294 |786 336 || 13 || 41 May, - - |419 |735 |456 101 |899 |663 |392 ||47 |115 18, 55 June, - - |593 j700 |640 160 |973 j609 |527 |351 |920 22 69 July, - - 698 |634 |754 185 |948 |525 |625 |613 |699 || 27 | 83 | August, - - 873 |599 |938 245 22 |471 759 |917 303 ||31 || 97 September, 48 |563 |123 |304 96 |417 |894, 221 |308 || 36 |111 October, - 152 |497 |237 |329 || 71 j$33 |992 |483 87 | 40 |125 November, 327 |462 |421 388 (145 |279 |127 |787 |892 || 45 139 December, 1432 396 |535 |414 (120 194 |225 || 49 |670 || 49 |153 TABLE XXI, JMoon's JMotions for JMonths. Months. TSUPEGFNOTETITTV TVſ. TVIITVIIITIXTX. s Com | 0° 0° 0' 0//| Cs 0° 0' 000 000 |000 000 000 |000 * * Bis. 11 29 56 49 | 11 18 51 |966 |961 |972 |966 |964 |995 reb. SQom. | 9 || 8 39 || || 15 43 ||34 |224|875 45 |1}} 163 CIO. 3. 0 1 35 19 || 11 4 34 || 20 |185 |847 || 11 || 75 159 March, - || 0 3 7 27 | 9 27 59 || 7 ||330 |666 |989 |114 313 April, - - || 0 4.45 57 || 9 13 42 61 |554 |542 34 |225 478 May, - - - || 0 6 21 16 || 8 18 15 81 |738 |389 46 |300 |638 June, - - || 0, 7 59 46 || 8 3 58 |136 |962 |264 91 |41.1 |802 July, - - - || 0 9 35 5 || 7 8 32 156 #147 112 |103 |486 |962 August, - || 0 11 13 35 | 6 24 15 210 371 |987 |147 |597 126 September, 0 12 52 5 || 6 9 58 |265 595 |862 |193 |708 |291 October, - || 0 14 27 24 5 14, 32 285 |780 |710 204 1783 |451 November, 0 16 5 53 5 () 15 |339 4, 585 |250 |894, j615 | December, 0 1741 13 ; 4 4 49_359 [188 |432 1261 1969 '775 4% TABLE XXII. JMoon's JMotions for Days. —i 1 2 3, 5 6 7 8 9 1 |0000 || 0000 || 0000 || 0000 || 0000 || 0000 || 0000 || 0000 0000 2 27 , 650 | 1040 287 336 || 372 58 || 390 24, 3 55 | 1300 || 2080 || 574 671 || 744 115 781 || 49 4 82 1950 || 3121 861 | 1007 || 1116 || 173 || 1171 3 5 109 2600 || 4161 || 1148 || 1342 1488 231 1561 97 6 || 137 3249 5201 || 1435 | 1678 1860 289 1952 121 7 | 164 3899 || 6241 1722 || 2013 2232 || 346 2342 146 8 || 192 || 4549 || 7281 | 2009 || 2349 2604 || 404 2732 170 9 219 || 5199 || 8321 2296 || 2684. 2976 462 3122 194, 10 246 5849 || 9362 2583 3020 || 3348 || 519 3513 219 11 274 6499 || 402 2870 3355 3720 577 3903 || 243 12 301 || 7149 1442 31.57 3691 4093 || 635 || 4293 267 13 || 328 || 7799 || 2482 || 3444 4026 4,465 692 || 4684, 291 14, 356 | 8449 || 3522 3731 4362 4837 750 5074, 316 15 383 || 9098 || 4563 | 4018" | 4698 || 5209 808 || 5404 || 340 | 16 || 411 || 97.48 5603 || 4305 || 5033 || 5581 || 866 5854 || 364 17 438 || 398 || 6643 || 4592 || 5369 || 5953 : 923 || 6245 || 389 18 465 || 1048 || 7683 || 4878 5704 || 6325 981 | 6635 || 413 19 493 1698 || 8723 || 5165 6040 | 6697 || 1039 || 7025 437 20 520 2348 9763 5452 6375 | 7069 || 1096 || 7416 || 461 21 548 || 2998 || 804 || 5739 || 6711 || 74.41 || 1154 7806 || 486 22 || 575 3648 || 1844 6026 || 7046 7813 | 1212 || 8196 || 510 23 602 || 4298 2884 || 6313 | 7382 8185 1269 8586 534 24 630 || 4947 3924 | 6600 77.17 | 8557 | 1327 | 8977 || 559 25 657 5597 4964. | 6887 || 8053 | 8929 || 1385 93.67 583 26 684, 6247 6005 || 7174 8389 9301 || 1443 97.57 607 27 712 | 6897 7045 7461 8724 9673 1500 148 631 28 739 7547 8085 7748 9060 45 | 1558 538 || 656 29 || 767 8.197 91.25 8035 | 9395 || 417 | 1616 928 680 30 794, 8847 165 8322 97.31 || 789 | 1673 1319 || 704 31 821 9497 1205 || 8609 66 ſ 1161 1731 1709 || 729 TABLE XXIIs JMoon's JMotions for Days. TDay ITOTITIATISTITISTISTIFTTSTIST55 1 000 000 |000 000 |000 000 |000 |000 |000 000 000 70 31 || 70 || 34 99 || 31 37 42 26 140 62 |141 | 68 [198 || 61 |73 | 84 || 52 210 || 93 |211 ||103 |297 92 110 126 || 78 : 351 |156 |352 |171 496 |153 |183 |210 130 421 |187 |423, 205 |595 |183 ||220 252 |156 491 |2|18 |493 (239 694 |214, 256 (294 |182 9 |561 |249 |564. 273 |793 |244, 293 ||336 [208 10 |631 |280 |634, 308 |892 |275 |329 |379 |234 2 3 4. 5 281 H125 |282 |137 397 122 ||146 |168 || 104. 6 7 8 11 702 |311 705 342 |992 |305 366 |421 260 | 12 (772 |342 (775 376 91 |336 403 463 |286 13 |842 |374 |845 |410 |190 |366 |439 |505 |312 14, 1912 |405 |916 |444, 289 |397 |476 |547 |337 15 1982 |436 |986 |478 |388 |427 |5|12 |589 |363 16 || 52 |467 || 57 |513 |487 |458 |549 |631 |389 * 17 |122 |498 |127 547 |587 |488 |586 (673 |415 18 193 529 |198 |581 |686 |519 |622 |715 441 | 19 |263 |560 |268 (615 (785 |549 |659 |757 |467 20 |333 |591 |339 (649 |884 |580 |695 799 |493 21 |403 |623 |409 |683 |983 ||611 /32 |841 |519 22 |473 |654, 480 (718 82 (641 769 |883 |545 3 (543 (685 (550 (752 |182 (672 |805 |925 1571 24, 1614, 716 |621 (786 |281 |702 |842 |967 597 25 |684 |747 |691 (820 |380 |733 |878 || 9 |623 26 (754 (778 |762 [854 (479 |763 |915 52 (649 27 |824 |809 832 |888 |578 794 |952 94 |675 28 |894, 840 |903 |923 |677 |824, 1988 |136 701 29 |964. |872 973 |957 |777 855 25 178 727 30 || 34, 1903 || 43 |991 |876 885 61 (220 753 31 105 1934 [114 || 25 1975 1916 198 (262. IZ79 1 1 TABLE XXIl. JMoon's JMotions for Days. Days Evection. Anomaly. Variation. M. Long. î 1 || 0s 00 0' 0//| Os Oo O/ O//| 0s 00 07 0// 0s 0° 0' 0// 2 : 0 11 18 59 || 0 13 3 54 || 0 12 11 27 || 0 13 10 35 3 () 22 37 59 || 0 26 7 48 || 0 24, 22 53 || 0 26 21 10 4 || 1 3 56 58 || 1 9 1:1 42 || 1 6 34, 20 || 1 9 31 45 5 1 15 15 58 || 1 22 15 36 || 1 18 45 47 || 1 22 4.2 20 6 1 26 34, 57 || 2 5 19 30 || 2 0 57 13 || 2 5 52 55 7 || 2 7 53 57 || 2 18 23 24 || 2 13 8 40 2 19 3 30 8 || 2 19 12 56 || 3 || 27 18 || 2 25 20 7 || 3 2 14, 5 9 || 3 0 31 55 || 3 14, 31 12 || 3 7 31 34 || 3 15 24 40 10 || 3 11 50 55 || 3 27 35 6 || 3 19 43 0 || 3 28 35 15 11 || 3 23 - 9 54 || 4 10 39 0 || 4 1 54, 27 || 4 11 45 50 12 || 4 4 28 54 || 4, 23 42 54 || 4, 14 § 54 || 4 24, 56 25 13 || 4, 15 47 53 5 6 46 48 4, 26 17 20 || 5 8 7 0 14 || 4 27 6 53 || 5 19 50 42 || 5 || 8 28 47 || 5 21 17 35 15 5 8 25 52 6 2 54 36 || 5 20 40 14 6 4 28 10 16 || 5 19 44, 51 6 15 58 29 || 6 2 51 40 6 17 38 45 17 | 6 1 3 51 6 29 2 23 || 6 15 3 7 || 7 O 49 20 18 || 6 12 22 50 7 12 6 17 | 6 27 14 34 || 7 13 59 55 19 6 23 41 50 || 7 25 10 11 || 7 9 26 1 || 7 27 10 30 20 || 7 5 0 49 || 8 8 14, 5 || 7 21 37 27 || 8 10 21 5 21 || 7 16 1.9 49 || 8 21 17 59 || 8 3 48 54 || 8 23 31 40 22 || 7 27 38 48 || 9 4, 21 53 || 8 16 0 21 || 9 6 42 16 23 || 8 8 57 47 || 9 17 25 47 8 28 11 47 9 19 52 51 24 8 20 16 47 10 0 29 41 || 9 |0 23 14 ||10 3 3 26 25 9 1 35 46 || 10 13 33 35 | 9 22 34, 41 ||10 16 14, 1 26 || 9 12 54 46 10 26 37 29 |10 446 7 |10 29 24 36 27 | 9 24, 13 45 |11 9 41 23 10 16 57 34 |11 12 35 11 28 10 5 32 45 11 22 45 17 10 29 9 1 |11 25 45 46 29 (10 16 51 44 || 0 5 49 11 11 11 20 28 || 0 8 56 21 30 10 28 10 43 || 0 18 53 5 11 23 31 54 0 22 6 56 31 |11 9 29, 43 || 1 1 56 59 || 0 5 43 21 1 5 17 31 TABLE XXII, 29 JMoon's JMotions for Days. Days Sup. of Node. II. V. VI. V [I. VIII.] IX. ( X. 1 Oo O/ 0// 0s 0° 0' | 000 || 000 || 000 || 000 || 000 || 000 2 || 0 3 11 11 9 || 34 || 39 28 || 34 || 36 5 . 3 || 0 6 21 22 18 68 79 56 | 67 || 72 11 4 || 0 9 32 1 3 27 | 102 || 118 85 || 101 || 108 || 16 5 || 0 12 42 1 14 37 || 136 || 158 || 113 || 135 | 1.43 21 6 || 0 15 53 1 25 46 || 170. 197 || 141 169 179 27 7 || 0 19 4 2 6 55 204 || 237 169 202 || 215 32 8 || 0 22 14 2 18 4 || 238 || 276 | 198 || 236 251 || 3 9 || 0 25 25 2 29 13 272 316 226 270 287 43 10 || 0 28 36 3 10 22 || 306 || 355 254 303 || 323 48 11 || 0 31 46 3 21 31 340 || 395 || 282 337 || 358 53 12 || 0 34, 57 4, 2 40 || 374 || 434 || 311 371 || 394 || 58 13 || 0 38 8 4, 13 50 | 408 || 474 339 405 || 430 64. 14 || 0 41 18 4, 24, 59 442 || 513 || 367 || 438 || 466 69 15 || 0 44, 29 5 6 8 || 476 553 || 395 || 472 502 || 74. 16 || 0 47 39 5 17 17 || 510 || 592 || 424 || 506 538 80 17 || 0 50 50 5 28, 26 || 544 632 452 539 573 || 85 18 || 0 54, 1 6 9 35 578 || 671 480 573 || 609 || 90 19 || 0 57 11 6 20 44, 612 || 7 || 1 || 508 || 607 || 645 96 | 20 1 0 22 7 1 53 | 646 || 750 || 537 641 681 | 101 21 || 1 3 33 7 13 3 | 680 || 790 565 674 || 717 | 106 22 || 1 6 43 7 24 12 || 714 || 829 || 593 || 708 || 753 112 23 1 9 54, 8 5 21 748 || 869 || 621 742 788 117 24 || 1 13 5 8 16 30 782 908 || 650 || 775 824 122 25 || 1 16 15 8 27 39 816 || 948 678 809 || 860 | 128 26 1 19 26 9 8 48 || 850 987 | 706 | 843 896 || 133 27 || 1 22 36 T9 19 57 | 884, 027 | 734, 877 932 138 28 || 1 25 47 10 1 6 918 066 762 910 || 968 143 29 || 1 28 58 10 12 16 || 952 | 106 || 791 || 944 || 003 || 149 30 || 1 32 8 10 23 25 || 986 || 145 || 819 978 || 039 || 154 3 1 35 19 11 4 34 || 020 185 | 847 i011 || 075 | 159 30 TABLE XXIII. JMoon's JMotions for Hours. Hours. 1 2 3 4. 5 6 || 7 8 9 1 || 1 || 27 || 43 | 12 || 14 | 16 || 2 | 16 || 1 2 || 2 || 54 87 24 || 28 || 31 || 5 || 33 || 2 3 || 3 || 81 || 130 || 36 || 42 47 || 7 || 49 3 4 T 3 T 103 || 173 || 48 || 36|| 62 ||10|| 65||7 5 || 6 || 135 | 217 | 60 || 70 || 78 12 || 81 5 6 || 7 || 162 260 | 72 | 84 || 93 14 || 98" | 6 7 || 8 || 190 303 | 84 || 98 || 109 || 17 | 114 || 7 8 || 9 || 217 | 347 || 96 || 112 || 124 | 19 || 130 || 8 9 || 10 || 244 || 390 108 || 126 || 140 22 || 146 || 9 10 || 11 271 || 433 | 120 || 140 || 155 24 | 163 || 10 11 || 12 298 || 477 131 || 154 171 26 || 179 11 12 || 14 || 325 520 || 143 168 || 186 || 29 || 195 || 12 13||13 || 352 563 | 155 | 182 | 202 |31 || 211 || 13 14 || 16 || 379 607 || 167 196 || 217 | 34 228 || 14. 15 || 17 | 406 || 650 || 179 || 210 || 233 || 36 || 244 15 16 || 18 || 433 || 693 191 224 248 || 38 || 260 | 16 17 | 19 || 460 | 737 203 || 238 264 41 || 276 17 18 20 || 487 || 780 215 252 279 || 43 293 18 19 || 22 || 515 823 227 266 || 295 || 46 || 309 || 19 20 || 23 || 542 | 867 || 239 || 280 || 310 48 325 20 21 24 || 569 910 || 251 294 326 50 341 21 22 || 25 || 596 || 953 263 || 308 || 341 53 358 22 23 26 || 623 || 997 || 275 322 || 357 55 || 374 23 24 27 || 650 |1040 | 287 336 || 372 || 58 390 24 *III XX GI I9IV,I, 3G 93 | 27 | 28 | IS | 66 | 78 | 02 | I8 | 0A | 78, 92 | 07 | 98 | 63 | 96 | 88 | A9 | 09 | A9 | 83 f3 | 68 | 79 | 83 | I6 | I9 | 79 | 88 | f9 | 32, 82, A9 | 38 | 93 | A8 | 08 | I9 | A3 | I9 | Iº, 32 | 98 | IS | 93 | 88 | 83 | 89 | 93 | 89 | 02, i 16 | 88 | 66 | V6 | 82 | 26 | 99 | 96 | 99 | 6I 6I | 39 -,88 | S3 | 7A | G8 | 89 | 93 | 89 | 8I 8I | 08 | 93 | I& | 02 | 73 | 09 | gg | 0g | lI | 2I | 83 | 93 | 06 | 99 | S& | 27 | Ig | 27 | 9I i 9I | 9g | gg | 6I | z9 | Ig | 77 | 6I | ti l gI gI | 93 | I& | 8I | 89 | 03 | IV | 8I | If I FI 7I | 83 | 06 , 9I | 79 | 8I | 88 | AI | 88 | 9 I 9I | Iº, 8I | GI | 6f | AI | 98 | 9I | G9 | & I gr . 61 | 2I | I | gi | 9I | gg | I | zg | it | II | 8I | 9 I | SI | I 7 | 7I | 6% | 9I | 63 | OI OI | 9I | 7I | II | 28 | SI | 92 | & I | 93 | 6 3-6 | 7I | & I | 0I | 98 | II | 82 | 0I | S3 | 8 8 | gI | II | 6 | 66 | 0I | 06 | 6 | 06 | 2 9 | II | 6 | 8 | G& | 6 | 8I | 8 | 8I | 9 9 | 6 | 8 | 9 | I2 | 2 | 9I | 9 | 9I | 9 7 | 2 | 9 | 9 | 9I | 9 | & I | g & I | - 7 3 | g | 9 | | | gI | | | 6 | 7 | 6 8 & | fr | 8 | 8 | 8 | 3 | 9 | 8 | 9 6 I | 3 | 3 | I | | | I | 8 | I | 8 | I 8I l 2I ' 9I l gr i pl I gr º gl º II | OI 'sino H. “sanoH toſ suonoArs.u00A' TABLE XXIII. JMoon's JMotions for Hours, Hours. Evection. Anomaly. Variation. Longitude. 1 09 28/ 17// Oo 32/40//| Oo 30/29// Oo 32/ 56// 2 0 56 35 1 5 19 1 O 57 1 5 53 3 1 24 52 1 37 59 1 31 26 1 38 49 - 4. 1 53 10 2 10 39 || 2 || 54 2 11 46 5 2 21 27 2.43 19 || 2 32 23 2 44 42 6 2 49 45 3"|15 58 3 2 52 3 17 3 7 3 18 2 3 48 38 3 33 20 3 50 35 8 3 46 20 4, 21 18 4, 3 49 4, 23 32 9 4, 14 37 4, 53 58 4 34 17 || 4 56 28 10 4 42 55 5 26 3 5 4, 46 5 29 25 11 5 11 12 5 59 17 5 35 15 6 2 21 12 5 39 30 6 31 57 || -6 5 43 6 35 17 13 || 6 7 47 || 7 4 37 || 6 36 12 || 7 8 14 14 6 36 5 7 37 16 7 6 40 7 41 10 15 || 7 4, 22 8, 9 56 || 7 37 9 8 14, 7 16 || 7 32.40 | 842 36 || 8 7 38 || 8 47 3 17 8 O 57 9 15 16 8 38 6 9 20 0 18 8 29 15 9 47 55 9 8 35 9 52 56 19 8 57 32 | 10 20 35 9 39 3 || 10 25 53 20 9 25 50 | 10 53 15 10 9 32 10 58 49 21 9 54, 7 || 11 25 55 10 40 1 || 11 31 46 22 10 22 24 || 11 58 34 || 11 10 29 || 12 4, 42 23 10 50 42 | 12 31 14 || 11 40 58 || 12 37 39 24 || 11 1859 13 3 54 | 12 11 27 | 13 10 35 £2 * ; : ['Sº Iº. 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Q> Q> OY . ~ Co ço co N N | N. ~ o) on on l & & & & \!> | ► ► ► 00 00 || &O NO NO NO NO ! ;-） --★ → （） o ł co |-|-/ <> <> <> | <> <> <> <> <> | <> Q> <> Q Q | <> C o C> <> i o o <> o. ſ. o o o o o ſº ſo — — + + – – + | ~ ~ ~ ~ ~ | ~ ~ ~ ~ ~ | ~ ~ ~ ~ ~ || o’s o o c | s o oſo e | ± + + + ~ ~ | ~ ~ ~ <> <> | <> <> <> • <> | e <> e <> | <> • • • • | s <> <> e <> | № ~–> + + + ~ ~ | ~ ~ ~ ~ ~ | ~ ~ ~ ~ ~ | ~ ~ ~ ~ ~ || …o s b s <> | <> <> • <> ~ | ¡ ¿ +-+ +-+ +→. +→. +→. || — — — -^ <> | <> o. o <> <> | <> <> <> • <> | <> <> o <> s { c c c c <> | 5: } → }}> QD CQ Q Q > C wº ‘spuoodS pup sojnunſ toſsuo!!oſſ 8,100Hſ * AIXX CITEIWAL (#8 | 9 || || 9 || || 9 I #I | 08 #I | f | 3& 9T WI 9T 03, 9 Ll 6 ft/I 08 9I 9T. 9{ } ºf 6& QI | f | 99 g|I} ºf 7L Zī, QI. Oft ST 63, 9I | VI g| || 8T | 8& QI | f | 33 g Li ST FI g|I g|I|| &T SI 83, SI I WI gL | 8 || || A8, QI # | 67 WI; &# SI &; pl. ºf &I) /ć, 7I SI | #I &I 92, 3. 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I2, i I | 27 8 | I | 6 | I | & | I | gI | 3 | gI | II | 6 | 88 | 16 | I | 9 - pº s 8 | I | 3 | I | 3 | I | 2 I | 2 | 2, I | OI | 6 | 38 | 03 | I | 97 8 | I | 2 | I l a | I | zI | g | II | 0I | 6 gg | 0g | I | 77 8 | I | 3 | I | & | I | 3I | 2 | II | 0I | 6 | I8 | 6I | I | Si | 8 | I | 3 | I | & | I | II | 3 | II | 0I | 8 | 08 | 6I | I | & 8 | I | 2 | I | 2 | I | II | 2 | II | OI | 8 | 08 | 6I | I I7 8 l I | 3 | I l & | I | II | 2 | OI | 6 | 8 | 63 | 8I | I | 07 8 | I | & | I | & | I | II | 3 | OI | 6 | 8 | 83 | 8I | I | 68 8 | I | & | I | & 1 I | 0T | & | OI | 6 | 8 | A3 | EI | I | 88 8 | I | & . I | & | I | 0I | I | 0I | 6 | 2 | 23 | 2 | | I | 28 8 | I | 3 | I | 6 | I | 0I | I | 6 | 8 | 2 | 93 | 9t | I | 98 3 l I & l I l & I I I OI | I | 6 | 8 | A | 93 | 9I | I | 98 & | I | 2 | I | 2 | I | 6 | I | 6 | 8 | 2 | 92 | 9I | I | 78 & | I | 3 | I | 2 | I | 6 | I | 6 | 8 | A | fa | GI | I | 89 & | I | 3 | I | 3 | I | 6 | I | 8 | 4 | 9 | 86 | VI | I | 38 6 | I | I | I | I | 0 | 8 | I | 8 | A | 9 | gg | VI | I | 18 I 7 I | SI & I l II ! OI I 6 l 8 | A | 9 l 9 l f I 9 l & I I I I IN A spu009S pup sºgnu Aſ toſ suongoA's.tlooIA' TABLE XXIV. 37 * JMoon's JMotions for JMinutes and Seconds. . ..] Evect. # Sup. Min Anom. Variat. Long. nod. II. Sec. Ev. An. Var'Lon. 31 14/37/16/ 52/15/45/177 1//| 4// 14/ 31 15// 17// 167; 171/H 32 |15 5 17 25 16 15 |17 34 || 4 || 15 32 15 || 17 | 16 18 33 15 34 17. 58 |16 46 |18 7 || 4 || 15 33 | 16 18 || 17 | 18 34, 16 2 |18 30 |17 16 18 40 || 4 || 16 34 16 18 || 17 | 19 35 |16 30 19 3 |17 47 |19 13 5 16 35 | 17 | 19 18 19 36 (16 58 |19 36 18 17 |19 46 || 5 || 17 36 17 | 20 18 20 37 17 27 20 8 |18 48 (20 19 || 5 || 17 37 18 20 19 20 38 17 55 |20 41 |19 18 20 52 || 5 || 18 38 18 || 21 19 21 39 18 23 21 14, 19 49 |21 25 || 5 | 18 39 18 || 21 20 21 40 |18 52 |21 46 |20 19 |21 58 || 5 || 19 40 | 19 22 || 20 22 41 19 20 22 19 20 50 |22 31 || 5 || 19 41 19 22 || 21 22 42 |19 48 ||22 52 |21. 20 |23 3 || 6 || 20 42 20 || 23 21 || 23 43 |20 16 |23 24, 21 51 |23 36 6"| 20 43 | 20 23 22 || 24 44 (20 45 |23 57 22 21 24, 9 || 6 || 21 44 21 | 24 22 || 24 45 21 13 24, 30 |22 52 |24 42 || 6 || 21 45 21 || 24 || 23 || 25 46 21 41 ||25 2 |23 22 ||25 15 || 6 || 21 46 22 J 25 || 23 || 25 47 (22 10 |25 35 |23 53 25 48 || 6 || 22 47 22 26 || 24 26 48 |22 38 |26 8 |24, 23 (26 21 6 22 48 || 23 26 || 24 26 49 |23 6 (26 40 24 54, 26 54 || 6 || 23 49 || 23 27 25 27 50 |23 34 27, 13 ||25 24, 27 27 || 7 || 23 50 24 27 || 25 27 51 24 3 27 46 ||25 55 28 0 || 7 || 24 51 || 24 28 26 || 28 52 |24, 31 |28 18 (26 25 |28 33 || 7 || 24 52 25 28 26 || 28 53 (24 59 |28 51 (26 56 |29 6 || 7 || 25 53 || 25 29 || 27 29 54 (25 28, 29 24 |27 26 |29 39 || 7 || 25 54 26 29 || 27 || 30 55 (25 56,129 56 (27 56 |30 12 || 7 || 26 55 26 30 28 30 56 (26 24, 30 29 (28 27 30 45 || 7 || 26 56 || 26 || 30 28 31 57 26 52 |31 2 28 57 |31 18 || 7 || 27 57 27 || 31 29 31 58 (27 21 |31 34, 29 28 31 51 || 8 || 27 58 27 32 29 || 32 59 |27 49 |32 7 (29 58 |32 23 8 || 28 59 || 28 32 || 30 || 32 60 (28 17 132 40 30 29 |32 56 || 8 || 28 60 28 33 30 33 38 TABLE XXV, First Equation of Moon's Longitude. ARGUMENT 1, Arg. 1. Diff. Alg. 1 | Diff. 0 12' 40" to 5000 | 12' 40” 100 || 11 58 ; 5100 | 13 20 º 200. 11 16 || 3 || 5299 || 14 1 || || 399 || 1Q 34 || || || 5399 || 14 41 || 33 499 || 9 53 || || || $400 | 15 20 ſº 500 9 12 || 0 || 5500 | 16 0 || 3: 600 8 33 ||38||5609 | 16 38 37 700 Z 54 || 3 || 3.99 || 17 13 || || 800 | 7 ||6 || 5 || 3399 || 17 52 §§ 900 || 6 49 || || 3999 || 1827 | . 1000 || 6 6 || 3 || 6000 19 ||3: 1100 5 33 ||31 || $199 || 1933 1200 5 2 || 6 || 6200 || 20 4 ; 1300 4 32 ... I 5300 20 33 9 1400 || 4 Å 33 $499 || 21 1 ; 1500 || 3 40 || 3 || 6500 21 27 | 3: 1600 || 3 17 | 21 | 6609 || 21 50 22 1700 || 2 #6 13 || 3.99 22 12 13 1800 238 || || || 3899 22 31 i. 1900 2 22 || 3 || 69% 22.48 is 2000 || 2 9 || || || 7000 || 23 3 i. 2100 || || 5 || 8 || 7199 || 23 15 | 10 2200 || || 59 || 3 || 7299 || 23 25 | } 2300 | 1.44 || 3 || 7399 || 23 32 3 2400 I 41 || 6 || 400 23 37 , 2500 | 1.41 || 3 || 7500 || 23 39 || 6 2600 | 1.43 || 5 || 7609 || 23 39 3 2700 || || 48 || 7 || 7.99 || 23 36 s 2800 || || 53 | 16 || Z399 || 23 30 2900 2 .5 || 3 || 7999 || 23 22 | 1. 3000 || 2 17 | 3 || 8000 || 23 11 i. 3190 2 32 17 | #199 || 32 53 16 3299 || 2 49 ið $299 || 22 43 i. 3300 3 8 || 3 || 3399 || 22 24 3. 3400 || 3 30 || 3 || $499 || 23 3 || 3: 3500 || 3 53 || 3 || 8500 21 40 || 3: 3600 419 8600 || 21 15 3700 || 4, 46 . 8700 20 48 § 3800 || 5 ||6 || || || 3:09 | 20 18 || || 3909 || 5 47 || 3 || 3999 || 1947 ||3: 4000 || 6 19 gº 9000 | 1914 | . 4199 || 5 § 35 || 3:99 || 13 49 36 4200 28 #| 939 18 4 ||3: 4399 || 3 ||37 || 3393 || 17 33 ||3: 4499 || 3 42 || 3 ||34.9 1643 4500 9 20 | 3 || 9500 | 16 8 || || 4609 || 9 || || 40 || 9599 || 1537 41 4799 || 1039 || 3 |9.99 || 1446 || 3: ;|##|*|#|#2;|3: 41 9 || 13 22 || 33 5000 || 12 40 10000 | 12 40 * TABLE XXVIe . 39 ARGUMENTS 2 to 7. º || C. 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Go Go Go <><> Q> <> <> | o oo <> o co | Q Q Q Q Q | Q Q Q Q Q !” Q Q Q Q Q>o <> C > #|$§§§§ | §§§§§ | §§§§§ | §§§§§ | §§§§§ |$§§§§ |§§§§Ē | §§§§§ [șĒĒĒĒ$ $3 $ <$ | № GŘ ČŇ ČR ÖN | ČŇ （* ří ;） （ | （ （ （-4. № 4-3 | \，«)© © oº on où I OY OY OY OY OY I OY OO OO OO OO || OO OO OO OO CON- \~ NN- ~ È+ +-+ ++ (N cº | <ſº 'n to oo o I Gº <# № on gº | ſº go ++ \^ oo | OY AQ op go ºc | <> <† № ++ ： | co QN AQ QQ QN | ſº co Q CQ (Q || op 2 § § ſºOĎ OD OS r-+ | ~ ~{ r-4 r-4 CN || CNR CN Co co co | <† <ř. <> <> <> <> | <> <> <> <> <> | <> <> <> <> o. | <> <> <> o co | c <> o. o c | +-+ +-+ +-+ +-+ +-+ | +-+ +-+ +-+ +-+ +-+ | +-+ +-+ +-+ +-+ +-+ | ~ ~ ~ ~ ~y={ x={ ſ={ 'ſ- • èo «o № co on l o Gº Ar» o «Q --+ | №o -- Go <> Ar) ſ os co № ++ \^ ł co r* cº tº og y={ r-4 r-{ ſ={ y-{CNR CNR CN CO CO<* <# un Aro Ar)y={ ſ={ CNRco cºſ ºſſ ºf Aro\^)+-， CNR CNR Cro C^66 <* <† <† <ſ è> <> o. o <> i <> <> | <> o. o o <> | <> <> <> <> i - - - - -| | +-+ +-+ +-+ +-+ +-+GN CNR CN GN GN || CN ON CNR GNR GN *… ò o co № to | <* Qº Q so GN | co <† o un <> | <ſº cº co ~ o | <† № I o co to o cº«o o Aro o «O | y={ CO <řł r-， OO C^ Go Go Cro C^)CO CO GN GNR GNſ-， p-{ w=-{\^ <* <ř， co coON ！={ r-{AroAro <ſ， co CN CN<† <* c^) Go GNUGNR ſ-º ſ-º ！={ № co cº co co | co co co co co | co co co co co | Gº Gº cº cº & | Gº! Gº Gº cº − | − −/ − − −o o co o C> I Q> <> O O Q> s─º è> <> o. № <* I - № Gº №. --. | <† № on o -ſ | cº dº cº − <> I on № to <ſ， cº I o co ſo <† cooO Oº o ș-{ c^b | QO OY GO CO Q^) CY) C^) CNR GNR GNR || CNR ſ-{ r-4\^ <> <> Q> ~ № Gº co co <* | ſo to co on -ſ | co ſo co o co | so gì cº ſo on ! Gº to o, ço ſo | c <† №. ！! <ſ<* N） ON ON <> o c <> | <> o. o o <> | <> o. o o co ſ o o c <> <> | <> <> <> <> <> I - +-+ +-+ +-+ +-++-+ +-+ +-+ +-+ +-+ | +-+ +-+ +-+ +-+ +-+ • *… NS- N- SO AO COO №. cº on <*ON CYD N）. Y-{N） OD +={ c^p <†\^» N - OO OO ONC) ſ-{ ſ={ CNR C^)Grò qo CD cº «oy-， GO r-， N） C^ \^ ^^^ A^ A^ A^\^ <† <ł co còg}{ſ} （!！! *\^ <† <† cô GNY！{\^ <# Gae)G^) GNR ſ-{\^)An <† <Ř Co -ſ co GNR GN ） +-+ § <* <* <† <> <> :： | <> C > Q> | <> o co oO CDo o C> Qo o I o C> o C>.<> | o o C> Q><> o. o o C> | <> <> <> O $|$$$$$ |$$$$$ | ĢĒĢ||ĶĒĒĒĒĒ |§§§§§ | §§§§§3ŞE Ş Ş Ş | Ģ Ģ Ģ Ģ Ģ C2 ºg Co | Q S2 +-+ +-+ +-+ | ~£ © gº N. «Q | <！ Gº oy № sſ | sº on to co o <Ř GNR H <ſº co | +--+ T <ři CN<ſſ Co - An có | +-ſ ſõ đN ~ ~ <ț¢ £ © ® ° C，5 （+ \ð | č， T ~ ~ă č3 ��• Õn CO N- So º CO | ON H © ºp №. !! !！ Sì， çº ſº go Í o) N： «Q \，) cº | Q ++ o) co go į Aro <ř. Gº – o >OAn aº što | Arſ A^. Arš ir ūō | <ř （* <ř <ň <ň | <Ë SË čº čõ đã į c5 cř, čº ćeš ČÁ GN GN ON ON GN CNGNR GNR ſ-{ y-\ +-+y={ x={ v={ ſ=\ r=|p={ ſ=\ y=\ y={ ſ=ły=\ y={ ſ=\ p={ ++y={ ſ=\ y={ ſ=\ p=-{ *）. • Cº Xº S2 $2 00 OA | 92 G og 30 GN || N. ~ <# to N. || N. Go <* + + №. ! CN ſo o co <} | \e» ſo ſes <！ Gº ∞ Gº <ť voy={co <# <ſ > I 9È CO № so ſo ſº | <ſº co CN + + +--+ | c on co N. so\^ <# Gºo CN +--+ | o CO N） CO \r)<ř， co GN o O) þ=ągº co ſo cº đã cóco co co co coGNQ GNR CNR CNR GN || GNR p={ ſ=-{ +++ +---+ | y={ y-\ y，\ y={ GR GNR GN GN GNR GN | CN Gº Gº Gº GºoGN GNR GN GNR GN || GN GN GN GN GN | ON GN GN GNR GN *）. № cº º ſo od -- I Gº – os ſo oC \r) CO +→ GN | +---+ o), «O GN GO | O ſ-+ +++ © OO ∞gº cº cº -\^ r： Oº Oº co ſ co <* * ſo ſoC9 CQ OQ OY OQ || OĎ OD O) ſo o } o © o C> o paeſș * * * * *<† •†• <ſ: <ſt <#1<ř. An - to Gº N | ON N- → to oCD y={ GNR Crò CO | C^ Gº ！! O， co ſ cry o A^ Go CN （r) ++ <ſco Aro Gº <# #--+$ $ $ $3 $3y'=+ C^D ^^>GNQĢ Ģ Ģ Ģ Ķ? ∞*_ © | $2 r.， GSI SŤ 30 № | Q o) ：-） Gº <ſGN. CY) \r) ko NJ | O Q> ！! C\^) GO CO OY O gº coco co <† ）. GNR CO <㺠Aro || «o N）, co o, oGO N， CO O Q | +---+ CN Go <ří Arò ſ to N）, Oo Oº o ！{r=-{ ſ={ w={ ſ={ GNGº Gº Gº! Gº GºGN GN GN GN co TABLE XXXI. 43 Evection. ARGUMENT. Evection, corrected. ！{ w=-1 Q：> O O Oo *ų. *ae © C \to ++ N）, co | H OD CO N， N= | N. CO on r-4 ço į so o CN vo o | <ſº o co co cCO tae ++ co <ſſ=+ COArbGN №ſ į è c – co <ſ ： 1 to co oſ o Gº { co <ſ, to № co | <> – co <ř ſo ' N， co s – cº•OO |× | <ń Ńń OES ŠĶ Ķ ĻŠ Į Š Š ŠĶŞE !！!?? №! ;º+-+ +-+ GNR GN GNR§ § § §§ €. © C O C O Oo o o y={ w=-{p={ ſ=\{ x={ ſ={ w=-{y={ ſ=\ y={ ſ=\ y={y={ w= +-+ +-+ +-+y=\ y=\ y={ ſ=\ y=| *）. èo cº co + + N）） <$ | ON GN co ſe) op | cº où ºn cºp GN | Gº <ť so on cº | og \^ Gº <> o) | op on ſ-º co zo $3|$3 $9<+ coGN p-{Arş <* <> C | <> <> <> <> | C. C <> <> C | <> <> C C C | <> <> C o C> | <> o. o o o *） *） # <ř. \r) №. --★ N. | <ſº GN CN Go to | © ® co ！! ！! !! GO AQ O AO GO | y=， z=+ CO \rb O) | \eo GN o os os Ş; Ş, 3 § § § | Ģº ?!$3$3 | $3 º $3 G§ § 2:3 º ¡ ¿ - g3 ºg} | ??? o ? Gae. ;：4 | №n on on on on on I on o c o o į C → -+ - GN | ON CN cº cº <ſ> | <ſ; A^ ^^ Go to į №, №, co on on p=-{O ſ-º ！={ ſ=\ y={y={ y-\ y，\ r={ x={y={ ſ=\ y=\ y={ w=-{y={ x={ x={ w={ ſ={！{ y-{ y-+ +-+ +-4 © o o o o <> | <> <> <> | <> O O O Q> | <> o. o o C> | C o o o o ! o O o o o • Čº o od o) – t） | <ř <† <ſ， to <> | Aſp – o co on l –， aº o co <ſ →\，) ș-， CN •ří p-{ c^)\to GN \r) +-+ <ſſ=+ \^ GNcroy'=+ \^) C^D p-{<ří Gro --+AròAro <* <ř， co cº =3 | à> <> o. co co nº ſ to to A^ v^ <> o c on >GNQ GNR r + H → + +y=-\ y={ ſ=\ y={ ſ=(y={ x={ ſ={ ſ=\ y={y={ ſ=\ y=， y={ r={y=\ y={ ſ-º ！！! © o o c <> <> | <> <> | <> <> | o c <> <> | <> <> o S CO (O Ap <Ř SH (o || N. o <ř. OO co | O CO <ř， co co«Q O， co OO | Arò GN o o o ſae+--+ Aro <# cro CN | +-+ +-+ Aro Arp | on N. to A^ | A^ <† <* <ř. <> <> <> | Q C <> <> Q> | <> Q> <> C C | C. C <> <> <> | <> C C Co o *… čo n- <;º º co too CO CO <ř. CO co P-， Aro GNGN. Ao co r* | Arp çr) y = <ři CN<ř， co ſ-º Arp | C^ p-{ <# GN º | > co N.+-+ o CO №. «O | O) N）, «O > [ đô ĞCNR GNR r，\ y={ ſ={ | y={ ſ={ ſ={ x={Aro Ar) Aro Q ！={ w=! y={ +--+ CN， co <Ë Arò GN º II XXX GITI CIV,L ºsse 0 0 2 68, 89 6 | Ig 9I gIl g8 2I SI FF 88 gli 89 02 OI II 9 2| 29 8 OT 87 6I gIl 6g 2I SI 88 gg &I 6g ti oi I2, 3I 2 33 6 OI OI S3 & I 9I 8I SI 98, g8 2, I 99 8 OI 38 8I A 97 7I OI 93 93 & I 92 8I SI; 9 62, gli 6t & OI & Vg 2 V 06 OI| 28 66 gli 8g 8I gI | 07 gg zI 8g 9s 6 29 08 A | 02 92 0I| Sf 38 3T 78, 8I SI A Z2, 3I | S2, 09 6 I A8 AI 88 08 OII Sf G8 & I 7I SI 9I 82, 8 I & I: 7 pt 6 QI SI | Si 98 QI 88 88 g 99 ZI SI & I gI gº 28 6 8I 67 2| 09 07 0Il 26 If gI Ig 2L SI| 0g OI gIi 9I I8 6 93 gg 2 gg gf OI OI FV & Il 6g 9I SI | gg 9 gIi 27 V8 6. 39 I 8| Sg 09 OI Al 7 97 3Il 02 9I SI | Alif & ZIl 9I 8I 6 88 2 8| 67 gg OI 6I 67 & I 78 gI SI | gg 8g II 68 II 6 | 2 f SI 8 If O II gi Ig & I ſi f I SI 3I 7g II| I g 6 97 6I 8| 08 G III 9 79 & I] [if gI SI 7g 67 II) 02. 89 8 87 gg 8| VI OI II SI 9g gi vs & I gIl sg gº II 9g Is 8 l 67 Is 8 gg VI II 93 8g gIl 06 II gri 6; 07 Il 09 º 8 s67 Z9 8| 39 6I III A 3 O SI 69 6 SI S 98 II| I 38 8 Alf 97 8 9 72, II SZ & GI| OS 9 SI 6& I8 II OI I8 8 '75 6 7 8 99 82, II & I 7 SI | gg 9 8I 22, 92, II 2 I F& 8 6S G9 8| A9 39 II f9 9 SI & I g 8Il 6% Iº, II & AI 8 I | 28 I 6| AI A9 II| IS A SI| 98, 9 SI | fº, 9I II| 98, 0I 8 f& A 6| SS Il II I 6 SI 9%, I SI GI II II 82, 9 8 SI SI 6| fif gif II fa, OI SI S2, 6g & I | O 9 II 82, 99 A | 8 I 6 I 6| 09 67 II I 7 II SI 2,I 2g & I 69 0 IT 86, 67 A | A. 97 73 6| 39 Sg II &g & I SI Gg 7g & Il 7 I G9 0I| 9% 37 A | 9 6g 08 6| 67 2g II 99 SI SI | 08 gg gI| Si 6 0I gg 98 2 | 9 OI 98 6| If I & II Sg FI SI 69 67 gT 8 F7 0I 6I 83 A | 7 97 If 6) 63 g & Il pif, GI SI | 02 Alf & I Z%, 88 OI 9I I& A | 8 i72, Aif 6| II 6 &I 82, 9I SI C8 77 & I 37 38 0I| OI 7I A | & 89 gg 6| 87 & I & II g AI SI SF IT &I &g 9& OI| 9 A. A | I a68, 89 o6.716,9I o&I ag8 AIoSI/i77/88 o&I//89 06 0Il 0 0 o2 lo0 sA s AI sIII sIl sI s0 penoa1.too “ApeuouV INGINnpav “ouluog scuoolſ ſo uompnbGI TABLE XXXII, 45 yf JMoon's Centre. walion 0 ** Eq ARGUMENT. Anomaly, corrected. ！ **） **） ©q ++ <ſı ;-） ON N. | \p CO <# GO AQ || №ſ pº Q <ț¢ © | po № co № <！ GN. H | \to \^ <ň do co | co co cº co co$ $ $ $3<> *） ！ | © \r, \ + N） co on 1 arb QN QO xo ++ [ op <# ！+ CO \r) { ++ CO \r) ON OS Į SO co o №. ！! !! ！! CO AQ GNR |×gº º to ſo+-+ GN GN Crò <ïſ | čo co co co <* <ſt | <† <† <† <† <# | <ſ: <ſt un ſo Arp | Ar) A^. Ao lo ſo | ln (o qo qp co | so «o co co N. æ------*ų. èo Gº Arp ºf Q go | gº do o co cº | <ſſ Gº Arp ſp +-+ | gº gº Gº r+ \ſq | Q < > ≡ ≈Q N） | gº gº co oo GN +---， CN Gro Aro GNR \r)C^) +-+ +-+！={GN, <ğıco r*Ao co cº co co<†GN. ON 30 CC rºi }）{ș * * * * *<# <# <# <ři ºři | <ſſ <# <# <†: <ř. | \to \^ \r) A^. Ao | Ar)+-+ +-+ +-+ +-+ GN? Ō o o o o o I o o o o o I o co o o o į o o o o co | c +→ -+ +-+ +-+ | +-+ +-+ +-+ +-+ +-+ *）. ！：- ò， cº o <ſ， cº N | № QN CQ 92 Gº! | -3 \r， \p OS SÊ | 22 №. 09 so gº | Q $2 o CO <ť || N- So Q （Q XQ a, į šo – to cº oº — | — cº co vo - | * – so sº co | co co º cº | so co –+--+ C^D ^^) GNR Fi į čº o to co o № | <ſ ： co ſº co ſ o co aº co - || @ № ſº Sſ Şş i © © og № © | ſq sſ; C2 QQ QQ |×ș + co co co osGN GNR p={ P--+ +-+y=-{Aro Aro Aro Aſo aro■ <ř: <# <ři ºří<† •†•<ř: <† ſ-º ș={ ſ=\ y={ r={ ſ=*！={ ſ={ ſ={ ſ=\ y={ſ={ r={ r={ ſ={ ſ={QD QD Q> O O© C O C QDO O CD O Q> **） `-º co co vo šo q | sq N， o № № ! ;-） on o ©. Aº | co ſº № co CQ || N. Go <> CO +-+ | oſ №t C}' + Q ��cro º crò r-， Aro <ſ> | CN -{ r-4r-， y-， co <ř.GN, ſeſ õN （ " <ří | CN +-+ +-+ +-+ | +--+ CO <# H co ț¢）{*…•→• - þæ{co o aero Oº <† || Oº <ſ. O <ři oſſ | <ſſ on <# O \to | Q Aº + + N） CN | OD ſ'iſſāGNQ GNR y={ r-{\p að co N） − Aro ob || CN co o <ſ， co | CN (Q <> <ſ， co | Q co o <† po į QN QQ Q \，) gº | gº Cae Gº! № ！！! Þ>ș3ș53%čº №, №, º ſē Ī ī£5 <ř. <ī£ cò GR | ČŇ ;-） -- T \o | ſr$ <ří º co GN | ON H H №, to co Go to lo | eo qo qo eo Aro | Aro ſe Aro Aro Aro | ſo ſº ſe aº O！{ v={ ſ={ w=( ）{+ { v={ ſ-{ y-\ y={GN GNQ GNR GNR GNGN GN CN Cºo CryC^) C^) C^ (！) C^ <> <> <> C c <> | C. C <> <> o | O Co o C> C | C o C> o c | <> o. o o o I o co o o o ~ ~ © Q> N – N –O <ři r-， CN <ſ© CO CO y-， №.Aro (o co <Ř CN vº GNQ \r) CNR \r). GN. Aro || Crý r-{ \f) co CN | +--+Ār) {r} | \r) · H GN CO | \r) CN Aro （-） | <# ！! <ſ: GN >ìn <ſ> <ſº cº co Oº į ON ON +-+ +-+ +-+ | +-+ +-+ ++ o o į o ++ +-+ +-+ +-+ | -! Gº Gº! Gº co | co <} <+ \^ \o O <> <> <> <> <> | c C o o o I o C> o o o I o C> <> <> <> | <> o. o o o I o o o o o ~ *）. SÌ， Q N. CO < > N - | <ſſ GN ！! ++ y + | ON <> <> | <> <> <> <> C | C. C. o o C | C C C C C | C C C o o į o o o o o *）. № co ſo gº ſo N | № to cº to o l o so op to ： | ſº oo — ON ON | Oº o od to co | <> to cº co <ſ Aro ſ-º co \n)！=-{CN co <# <} <ſA^ ^^ <# <† <ř.CO ON CN r-+\O <* CN +---+Arò co GºArb gael|- GO （n) <* <É CO | GNR -， o CD po | № $@ \ ^ Siſ go | ĢN ！! Go O) op { Q \，) <ſ， co CS ] © oº oo N. aro E?\^ \to | Är) A^. Aro Ar) Arý ſ \r) A^ ■^ <† <# | <ſſ <Ř <} <ř. | <> O O Q> O + C o C> <> O || C. o o o C> | o o “o o ~ ~ ș-{ \r) №. «O CO CO© OD CO + + CGN. O co Aro <ſo <† \^ <ſ， o<} \^ <łº o\ry± & O CO CO CO CN || Ap$ $ $3r-+ +-+ +-+ | +-{ \O o \ c – – – – i cº CN ON ON CN i GN (N - → -+ | ~ o c <> o. i on co co № to Or-º r={ | +-+ +-+ +-+ +-+ +-+ | +--+ +-+ +-+ +-+ +--+ | +-+ +-+ +-+ +-+ +-+ | +-+ +-+ +-+ +-+ ſ-º ！! ！! ！! ！! ;-+ | +-+ +-+ +-+ +-+ +-+ | +---+ +-+ +-+ +-+ +-+ | +-+ +-+ +-+ +-+ +-+ | +-+ +-+ +-+ +-+ +--+ | +-+ +-+ +-+ +={ w=-{ ~. <> Cº. № © ÇN | o C> O +-+ | +-+ +-+ +-+ +-+ +-+ | +-+ +-+ ++ r^-{ y-{ O•æ C r + GNR CO <ſº ºſ coCo co GN CO cro± ON CO <# Ato$ $3 È $2 | № oo on on <> r： || Oº Oº co <și ſo ſ to № co o oN- QQ Q Q> -4 || c^ <> <> <> <> <> | <> <> [ o o os o oC C C C Q> | Q Q Q> co o • № sť Q ‹› （Q so | > @ ſo to ob ſ to <ſ， to o toOn <ț¢ £ + C CN || N. co cº <;， oo GN. ~★ → <ři r-4\^ Gº<Ë GNRſg ºgę ſęr-+ CO \rò r-{ c^pAO CN Arſ CN Arò % | à co № № so so | wº so ſo <* | Aro Go «o N）, №, O <> <> <> <> <> <> | <> o. o o o I o o co o os© C O C o I o o o o o S GO GNY CO <# Go N）ř， ſ，+ O CO COCO O p-{ \r) ON<+ CO \r) co cry<ſº co <Ř ON? Cº. �� to co 6º – ſoC^ p-{\^><řł co Crò Gº rº-4r-+ +-+ CN Go <;º\^)GN， <ſ Þ4 | § © ® 2 ſº gº | Q - o gº N. I to ſo <> <> | <> <> [ o o co o oC C C C C | o C> <> o • È {2 c^ ， ^ — | co gº - co cº | q ~ ~ ſo ºCY OD CO SO CO | O N） C^D O GO 2, ſ go ſo H co sºy-， GNR co Cry <ſ， cº CN + +<ſſ Go GNR •ł F Í SÐ © on CO N. № ſ to A^ <ř, co GN į – o os co №.Ş> O) og N. GO ] © cºp GNR r， Q Þ-gº\r) \r) \ryAn <† <Ř <† C o C> Go oQ> O O CD O • • © SQ （→ <ț¢ ºn Cº) | O GN gº <> ſo | oo oo qo ） cGNQ GO CO N）, C/O | QO OY OY GO O ∞$ $3'）; $3 $CO \r) - GN | co <Ř AFS^ON <ři CNGO CO Grb •ł• →<> w-， y-， GNR GNC^) C^ <ř* <††: <řł Ary<# <Ř co co coCNR CNR p-" +=ł C |×? ===???????? È ÈĒ Ē Ē （G ſg+-+ +-+ +-+-+-+ +-+ | ~ ~ ~ ~ ~ ++ y + r， r+ r）{ r-+ | +-+ +-+ +-+ +-+ +-+ | +---+ +-+ +-+ +-+ +-4r=\ y={ y-\ y={' +-+ | y={ ſ=\ y={ ſ=\ y={ s— <> N, Ñ， º CO SŤ | Q \Q <> <ſ， N - || O) o o os R）»）, OO C^ ^^ GO | MºD GN GO CO CO ¿>>';'）;CNR<Ř Aro P-， GN Cry<ſr-+ +-+ CN<ř， co Gºo GN r +Arp C^ p-{ \r> §， | ? a Q + cq sſ | so to cº oſ o l – co <; io toGNQ Go <ſ\ \n \O || N. N. CO OY OY Þ-gº do * * * * | < < < <ă ſă | io iš ſă să ſă <> <> <> <> <> <> | <> <> <> <> o \! o o o o osș-{ y--{ ， } + y + | +-+ +++ ſ-º ！! ！={ O r-º QN CO <ři \^ | GO N） CO CO o ] -!<# \ºy-， GNQ C^ <} \r> | P-R ON<řip={|-; Ç Ç Ę ĘON ON CN CN GN | GNR GNR GNR GN co } º AIXXX GITICIVI, 8ip 0 A | A9 & i | A9 & I | 0 A | 8 I | 9 I | 08 fI A | 7 SI | 09 GI | 97 9 | 99 0 | OI I 62, 62, A | OI SI | 27 & I | I8 9 | 67 0 | 8I I | 8%, 87 A | AI 8 I | 88 3 I | AI 9 | Sf 0 | 92 I | A2, A9 A | 23 SI | 93 3 I | 8 9 9 0 | 98 I | 93 & I 8 | A2, 8I | 9I & I | 87 g | 89 0 | 77 I | G8, 93 8 | 38 SI | 9 & I | 78 9 | 82, 0 | 79 I | 72, 07 8 | 99 8I | 99 II | 03 g | 73 0 | 9 & | 88, º, '79 8 | 07 8I | 97 II | 9 g | 03 0 | 7I & | 3% A, 6 | Sf SI | 98 II | S9 7 | 2I O | 78, 3 | Iº, Iº, 6 | 97 SI | G8, II | 68 7 | 7I O | 98 % | 03 79 6 | 87 SI | 7I II | 92, 7 | 3I O | 97 & | 6 I 87 6 | 09 SI | & II | 3I 7 | 0I 0 | 89 % | 8I I OI | I9 8I | 09 OI | 69 9 | 6 0 | 6 9 | AI SI OI | 39 8I | 88 OI | 97 9 | 8 0 | 23 8 | 9I 9& OI | 29 SI | 92, OI | 7S 9 | 8 O | 78 S | GI 88 OI | 39 SI | SI OI | 33 9 | 8 0 | 97 9 | 7I 09 OI | I9 SI | I OI | 6 8 | 6 0 | 69 8 | 8I & II | 09 SI | 87 6 | 89 & | OI O | & I 7 | & I '7I I I | 87 SI | 79 6 | 9f & &I O | 9& 7 | II g2, II | 97 9I | Iº, 6 | G8 & | 7I O | 68 7 | OI 98 II | 87 SI | A 6 | 72, 3 | AI O | Sg 7 | 6 97 II | 07 SI | 79 9 | 7I & | 02, 0 | 9 9 | 8 99 II | 99 9I | 07 8 | S & | F& 0 | 03 9 | A, 9 & I | 38 SI | 93 8 | 79 I | S2, 0 | 78 9 | 9 9I & I | A2, 9I | & I 9 | 77 I | SS 0 | 87 9 | 9 9& & I | && SI | A9 2 | G8 I | 88 0 | S 9 | 7 88 C,I | AI SI | 97 A | 92, I | Sf 0 | AI 9 | 8 37 & I | OI SI | 6% A | 8I I | 67 0 | I9a 9 | & 09 & I | 7 SI | 7I A | OI I | 99 0 | 97 9 | I A9 AGI | 29 8I 40 A 48 I | 8 I | 0 A o0 sIX s A | sX s AI | sXI s IIIl s III A siIl s[lA sl) sſ A s0 opniguo I liquo suooIN + apoNgo Iddns LNINnpav - v, “u0110mpo?I TABLE XXXV. JMoon's Distance from the JNorth Pole of the Ecliptic. ARGUMENT. Suppl. of Node + Moon's Orbit Longitude. IIIs IV's Vs VIs Wils Viişs O 84939/16// 84; 39 19 84 39 27 84; 39 41 84 40 1 84, 40 27 850207437 85 23 27 85 26 16 85 29 10 85 32 9 85 35 12 87o 13/47// 87 18 28 87 23 12 87 27 58 87 32 48 87 37 39 890 48' 0// 89 53 23 89 58 46 90 4, 8 90 9 31 90 14, 52 92.922/13// 92 26 52 92 31 27 92 36 0 92 40 30 92 44 56 94.915/17// 94; 17 57 94, 20 31 94, 23 1 94, 25 25 94, 27 45 1 i| 84 40 58 84 41 34 84, 42 17 84 43 5 84. 43 58 85 38 20 85 4.1 33 85 44, 50 85 4.8 11 85 51 37 87 42 33 87 47 30 87 52 28 87 57 29 88 2 31 90 20 14. 90 25 35 90 30 55 90 36 14, 90 41 33 92 4.9 19 92 53 39 92 57 56 93 2 9 93 6 18 94, 29 59 94, 32 8 94 34 12 94 36 11 94 38 4 1. 1. i : 84 44, 57 84. 46 2 84 47 12 |84, 48 27 84 49 49 85 55 7 85 58 42 86 2 20 86 6 3 86 9 50 88 7 36 88 12 4.2 88 17 50 88 23 0 88 28 11 90 46 50 90 52 7 90 57 22 91 2 36 91 7 49 93 93 93 93 93 10 24. 14, 27 18 25 22 20 26 10 94 39 52 94 41 35 94 43 13 94 44, 45 94 46 11 . : 84. 51 15 84 52 47 84 54, 25 84. 56 7 84. 57 56 86 13 40 86 17 35 86 21 33 86 25 36 86 29 42 88 33 24 88 38 38 88 43 53 88 49 10 88 54, 27 91 91 91 91 13 0 18 10 23 18 28 24, 91 33 29 93 29 57 93 33 40 93 37 18 93 40 53 93 44, 23 94, 47 32 94 48 48 94; 49 58 94, 51 3 94; 52 2 O * 2 : 2 ; : : 84 59 49 85 1 48 85 3 52 85 6 1 85 8 15 86 33 51 86 38 4 86 42 21 86 46 41 '86 51 4. 88 59 46 89 5 5 89 10 25 89 15 46 89 21 7 91 91 91 91 91 38 31 43 32 48 30 53 27 58 21 47 49 51 10 54, 27 57 40 0 48 93 93 93 93 94 94; 52 55 94 53 43 94 54, 26 94, 55 2 94, 55 33 i : cº O J 85 10 35 85 12 59 85.15 29 85 18 3 85 20 43 86 55 30 |87 0 0 '87. 4 32 87 9 8 & 1347 89 26 29 89 31 52 89 37 14. 89 42 37 89 48 0 92 3 12 92 8 1 92 12 48 92 17 32 92 22 13 94 3 51 94 6 50 94, 9 44 94; 12 33 94, 15 17 | 94, 55 59 94, 56 18 94, 56 33 94 56 41 94 56 44 Ils | 1. Os 7% XI's Xs IX's TABLE XXXVI. Equalion II. of the Moon's Polar Distance. ARGUMENT II, corrected. * IIIs IV's Vs V]'s Wils VIIIs Oo. 0/14// 1 / 24//| 4/.377 9/ 0//| 13/23//| 16/36// 300 1 || 0 || 4 || 1 29 || 4 45 9 9 || 13 31 || 16 40 29 2 : 0 14 || 1 34 || 4, 53 9 18 || 13 39 || 16 4.5 28 S | 0 14 || 1 39 || 5 1 9 27 | 13 47 | 16 49 27 4 || 0 15 || 1 44 || 5 9 9 37 13 54 16 53 26 5 || 0 16 || 1 49 || 5 18 9 46 14, 2 | 16 57 25 6 || 0 17 | 1 54 || 5 26 9 55 14, 9 || 17 1 || 24, 7 || 0 |18 || 2 0 || 5 34 || 10 4 || 14, 17 || 17 4 || 23 8 || 0 19 || 2 5 || 5 43 10 13 14 24 || 17 8 22 9 || 0 20 2 11 || 5 51 10 22 || 14, 31 \| 17 11 || 21 10 || 0 22 || 2 17 | 6 0 || 10 31 14, 38 || 17 14, 20 11 || 0 23 2 23 || 6 9 || 10 40 || 14, 45 17 17 | 19 12 || 0 |25 || 2 29 || 6 17 | 10 49 14, 52 17 20 18 13 || 0 27 || 2 35 | 6 26 10 58 || 14, 59 || 17 23 17 14 || 0 29 || 2 41 || 6 35 | 1 || 7 || 15 5 || 17 26 16 15 || 0 3 2 48 || 6 44, 11 16 || 15 12 17 28 15 16 || 0 34 || 2 54, 6 53 || 11 25 | 15 18 || 17 31 14 17 | 0 37 || 3 1 || 7 2 | 11 34 15 25 | 17 33 || 13 18 || 0 40 || 3 8 || 7 11 || 11 43 15 31 || 17 35 | 12 19 || 0 4.2 || 3 15 || 7 20 | 11 51 15 3 17 36 | 11 20 || 0 45 3 22 || 7 29 12 0 | 15 43 || 17 38 10 21 || 0 49 || 3 29 || 7 38 | 12 9 5 15 49 || 17 40 9 22 || 0 52 || 3 3 7 47 | 12 17 | 15 55 17 41 8 23 || 0 56 || 3 4.3 || 7 56 | 12 26 16 0 || 17 42 7 24, O 59 || 3 51 8 5 || 12 34, 16 6 || 17 43 6 25 | f 3 || 3 58 8 14 | 12 42 16 11 || 17 44 5 26 || 1 7 || 4, 6 8 23 | 12 51 | 16 16 || 17 4.5 4. 27 | 1 11 || 4 13 || 8 32 | 12 59 | 16 21 || 17 45 3 28 || 1 15 4, 21 8 42 | 13 7 | 16 26 || 17 46 2 29 || 1 20 4, 29 8 51 | 13 15 | 16 31 17 46 1 30 ( 1 24 4 3 9 0 || 13 23 16 36 || 17 46 0 IIs Is Os YIS Xs IXs TABLE XXXVII. Equation III. of the Polar Distance. ARGUMENT, Moon's True Longitude. Y II is IV s Vs V is V [ls His Oo 16// 15// 12// 8// 4// 1// 300 6 16 14 11 7 3 1 24 12 16 14 10 s 6 3 O 18. 18 16 13 10 5 2 O 12 24 I5 13 9 5 I 0 6 30 15 12 8 4. l O 0 IIs Is 0s Xīs Xs [X's g TABLE XXXVIII. #! TABLE XXXIX. 51 To Convert Degrees and Equations of Polar Distance. JMinutes into Decimal ARGUMENTs. 20 of Long.; W to IX, cor- Parts. rected; and X, not corrected. Degrees] Dec. Arg. 20 ! V. V1. VII. VIII | | X. X. Arg. and Min. parts. 350|TO7-567|T67|T57|T357|T37|TTEUT 19 5/ | 003 260 0 || 56 || 6 || 3 || 25 || 3 || 11 |240 1 26 4. 270 0 || 56 || 6 || 3 || 25 || 3 || 11 |230 1 48 5 - || 280| 1 || 55 || 6 || 3 || 25 || 3 || 11 |220 2 10 6 290 1 || 55 || 7 || 3 || 25 || 4 || 11 |210 2 31 7 300|| 1 || 55 || 7 || 4 || 25 || 4 || 11 |200 2 53 8 310|| 1 || 54 || 8 || 4 || 24 || 5 | 12 190 3 14 9 320|| 2 || 53 || 8 || 5 || 24 || 6 |- 12 |180 3 36 10 330| 2 || 53 || 9 || 5 || 24 || 6 || 13 |170 3 58 11 340|| 3 || 52 || 10 || 6 || 23 || 7 || 13 |160 4, 19 – 12 35U' 3 || 51 | 11 || 7 || 23 || 8 || 14 |150 4 41 13 360 4, 50 | 12 || 8 || 23 || 9 || 14 ||140 5 2 14 370|| 4 || 49 || 13 || 9 || 22 || 10 || 15 |130 5 24 15 380 / 5 || 48 || 14 10 22 11 16 |120 5 46 16 390| 6 || 46 | 15 || 11 || 21 | 13 || 17 |110 6 7 17 400|| 6 || 45 16 | 12 || 21 || 14 || 17 |1UU 6 29 18 410|| 7 || 44, 17 | 13 20 15 18 90 6 50 19 420) 8 || 42 | 18 || 14 | 20 | 17 | 19 || 80 7 12 20 430| 9 || 41 || 20 | 15 19 - 18 20 | 70 7 34, 21 440|| 10 || 39 || 21 || 17 | 19 20 21 || 60 7 55 22 450| 10 || 38 || 23 | 18 || 18 || 22 || 2:2 || 50 8 17 23 460| 11 || 36 24, 19 || 17 || 23 23 40 8 3 24 470 12 || 35 || 25 | 21 17 || 25 || 24, 3 9 O 25 480; 13 || 33 27 22 | 16 27 25 | 20 9 22 26 490|| 14 32 28 24 | 16 || 28 || 26 || 10 9 43 27 500 15 30 30 || 25 15 || 30 27 |000 10 5 || 28 510 16 || 28 || 31 || 26 || 14 || 32 28 |990 10 26 29 520, 17 | 27 | 33 28 14, 33 29 |980 10 48 30 530 18 25 || 34 29 || 13 || 35 | 30 |970 11 10 || 31 540 19 24 || 36 || 31 | 12 || 37 || 31 1960 11 3 32 550 19 22 || 37 || 32 | 12 || 38 || 32 1950 11 53 3 560; 20 | 20 | 39 || 33 || 11 | 40 || 33 |940 12 14 34 570 21 | 19 | 40 34 || 11 || 41 || 34 1930 12 36 || 35 580. 22 || 17 || 41 || 36 | 10 || 43 35 |920 12 58 || 3 590] 23 16 || 43 37 || 10 || 44 || 36 |910 13 19 || 3 600 24 | 15 44 38 || 9 || 46 || 37 1900 3 41, 38 610| 24 || 13 || 45 39 || 9 || 47 37 890 14, 2 || 39 620 25 | 12 46 40 || 8 || 48 || 38 |880 14, 24 | 40 630 26 || 11 || 47 41 || 8 || 50 39 |870 14. 46 41 640; 26 || 10 || 48 || 42 || 7 || 51 40 860 15 7 || 42 650 27 | 9 || 49 || 43 || 7 || 52 40 |850 15 29 43 660] 27 | 8 || 50 44 || 6 || 53 || 41 |840 15 50 44 670 28 || 7 || 51 || 45 || 6 || 54 || 41 |830 16 12 45 680) 28 || 7 || 52 || 45 || 6 || 54 || 42 |820 16 34 || 46 690] 29 || 6 || 52 46 || 6 || 55 42 [810 16 55 || 47 700 29 || 5 || 53 46 || 5 || 56 42 |800 17 17 || 48 710| 2 5 || 53 || 4-7 || 5 || 56 || 43 Z90 | | 3 || 4. 720, 29 || 5 || 53 || 47 || 5 || 56 || 43 Z80 18 O 50 730 S 4, 54 47 || 5 || 57 || 43 |770 18 22 51 740. 30 || 4 || 54 47 || 5 || 57 43 |760 18 43 52 750) 3 4 || 54 47 | 5 || 57 ' 43 1750 TX GlrTgTV,L sIA sIIA | sIIIA | sXI sX ! sIX O SI 0 | 8I 0 | 38 0 | 09 0 | 6 I | 88 I | 08 I SI 0 | 6I 0 | 29 0 | Ig 0 | 0I I | 98, I | 68, 6 8I 0 | 6I O | S8 0 | Ig 0 | OI I | fº, I | 83 9 7I 0 | 6I 0 | 88 0 | 29 0 | II I | 7% I | A3 f, i7I O | 02 O | 79 0 | Sg O | II I | 72, I l 93 9 fI O | 03 0 | 79 0 | 9g 0 | 3I I | 92 I | 98, 9 i7I O | 08, 0 | 98 0 | 79 O | 3I I | 98, I | 73 A. i7I O | I& 0 | 98 0 | 99 0 | SI I | G& I | 9% . 8 i7I O | TZ 0 | 98 0 | 9g 0 | SI I | 98, I | C2, 6 fI O | Iº, 0 9 O | 99 O | 7I I | 93 I | I3 7I O | 38, 0 9 O | Ag 0 | 7I I | 93 I | 03 7I 0 | 32, 0 | 88 0 | Ag 0 | GI I | 93 I | 6 I i7I O | S3 0 | 68 0 | 89 O | GI I | 93 I | 8I 7I O | S2, 0 | 68 0 | 69 0 | 9I I | A3 I | AI GI O | 73 O | 07 0 | 69 O | 9I I | A3 I | 9I 9I O | 73 0 i 07 0 | O I | AI I | A2, I | 9I GI O | 72, O | If O | 0 I | AI I | A8, I | 7I 9I O | 96, 0 | 37 0 . I I | 8I I | 28, I | 8 9I 0 | G8, 0 | 37 0 | & I | 8I I | A8, I | & I GI O | 9%, 0 | Sf O | & I | 6I I | 83 I | II 9I O | 92, 0 | fif O | S I | 6L I | 82, I | 0I 9I 0 | A3 0 | fif 0 | 7 I | 08, I | 82, I | 6 9I O | A8, 0 | Gif O | 7 I | 03 I 3 I | 8 9I O | 83 0 | 97 0 | 9 I | 03 I | 83 I | A, AI O | 82, 0 | 97 0 | 9 I | I& I | 83 I | 9 AI 0 | 63 0 | A 7 0 | 9 I | I2, I | 83 I | 9 AI 0 | 62, 0 | Alf 0 | A. I | 3% I | 86 I | 7 AI 0 | 08 0 | 87 0 | A. I | 33 I | 8%, I | 8 8I 0 | 08 0 | 67 0 | 8 I | 3% I | 82, I | 2, 8I 0 | I9 0 | 67 0 | 8 I | 98, I | 8% I | I a/8I 0 a 68 0 09 A0 / 6 /I A/86 /I //86 /I lo0 sA sAI | sITT sII a I s0 “uomooAGI au, Jo uoun3 IV LNGIUmp IV appand pm.topmba s.uvoIA' TABLE XLI. 53 JMoon's Equatorial Parallaa. ARGUMENT. Anomaly. ) Os |s iIs II is IVs Vs 09| 58/58// 58/27//] 57 8// 55/30//| 54/ 2// 53/ 3// 300 1 : 58 58 58 25 || 57 5 || 55 27 | 53 59 53 2 || 29 2 || 58 58 || 58 23 57 2 || 55 23 53 57 53 0 || 28 3 58 57 || 58 21 || 56 58 55 20 || 53 54 || 52 59 || 27 4 || 58 57 || 58 19 56 55 55 17 | 53 52 52 58 26 5 : 58 57 58 16 56 52 55 14 53 50 52 57 || 25 6 58 56 58 14 || 56 49 || 55 11 || 53 47 52 56 || 24 7 58 56 || 58 12 56 45 55 7 || 53 45 || 52 55 23 | 8 || 58 55 58 10 || 56 42 || 55 4 || 53 43 52 54 || 22 9 58 55 58 7 || 56 39 55 1 || 53 41 || 52 53 21 10 || 58 54, 58 5 56 36 54 58 || 53 38 || 52 52 | 20 11 58 53 58 2 56 32 || 54, 55 53 36 || 52 51 | 19 12 || 58 53 58 0 || 56 29 54, 52 || 53 34 || 52 50 | 18 13 || 58 52 57 57 || 56 26 || 54 49 53 32 || 52 49 17 14 || 58 51 57 55 56 22 54, 46 53 30 || 52 49 16 15 58 50 57 52 56 19 54 43 53 28 52 48 15 16 58 49 || 57 49 || 56 16 || 54 40 53 26 || 52 47 | 1.4 17 58 48 57 46 56 13 54 37 || 53 24 52 47 || 13 18 || 58 46 || 57 44 || 56' 9 || 54 34 53 22 || 52 46 | 12 19 || 58 45 || 57 41 || 56 6 || 54, 31 || 53 21 || 52 45 || 11 20 '58 44, 57 38 56 3 54, 29 53 19 52 45 || 10 21 58 42 || 57 35 | 55 59 || 54, 26 || 53 17 52 45 9 22 || 58 41 || 57 32 || 55 56 54, 23 53 15 52 44 || 8 23 58 39 57 29 55 53 54, 20 || 53 14 || 52 44 || 7 24 58 38 || 57 26 55 49 || 54, 18 || 53 12 || 52 43 || 6 25 58 36 || 57 23 55 46 || 54, 15 || 53 10 || 52 43 || 5 26 58 34 57 20 55 43 || 54, 12 || 53 9 || 52 43 || 4 27 | 58 33 || 57 17 | 55 40 || 54, 10 || 53 7 || 52 43 || 3 28 58 31 57 14 55 36 54, 7 || 53 6 52 43 || 2 29 || 58 29 || 57 11 || 55 33 54, 4 || 53 4 || 52 43 || 1 30 || 58 27 | 57 8 || 55 30 54 2 | 53 3 || 52 43 || 0 XI? Xs IXs VIIIs VIIs VIs 45 TABLE XLII. TABLE XLIII, JMoon's Equatorial Parallaw. Reduction of the Parallas, ARGUMENT. Argument of the Wa- and also of the Latitude. riation. ARGUMENT. Latitude. ()s Is IIs IIIs IVs | Vs Lat Red. of ) Red. of - “| Paral. Lat. Oo 56// 42// 16// 4// 18// 44// 300 1 || 56 || 41 || 15 4. 18 45. 29 Oo O / 07 O'ſ 2 || 55 || 41 || 14 4. 19 || 46 28 3 0 1 12 3 55 | 40 || 14 4. 20 46 27 6 : , 0 2 23 4 || 55 || 39 3 4. 21 47 26 9 O S 32 5 || 55 || 38 12 4. 22 || 48 || 25 12 O 4, 39 15 1 5 43 6 55 37 12 4. 23 || 48 || 24 7 || 55 36 || 11 5 24 49 3 18 1 6 44 8 || 55 || 35 | 10 5 24, 50 | 22 21 1 7 40 9 || 54 || 3 10 5 25 || 50 || 21 24 2 8 31 10 || 54 || 34 9 6 26 || 51 | 20 27 2 9 16 3 3 9 55 11 54 33 || 9 || 6 || 27 | 51 | 19 12 || 53 32 || 8 6 28 52 18 33 3 10 28 13 53 || 31 8 7 29 53 || 17 36 4. 10 54, 14, 52 3 7 7 30 || 53 16 3 5 11 13 15 52 29 7 8 || 31 || 53 15 42 5 11 25 45 6 11 29 16 || 51 || 28 6 8 32 || 54 | 1.4 17 || 51 27 6 9 33 || 54 | 1.3 48 6 11 25 18 50 26 6 9 34 55 12 51 7 11 14 19 || 50 25 5 10 35 55 11 54 8 10 56 20 49 || 24, 5 || 10 35 55 || 10 57 8 10 30 60 9 9 57 21 || 49 || 24 5 11 3 56 9 22 || 48 3 4 12 37 56 8 63 9 9 18 23 47 22 4, 12 38 56 7 66 10 8 33 24 47 21 4 || 13 3 56 6 69 10 7 42 | - 25 || 46 20 4 14 40 || 57 5 72 10 | 6 46 75 11 5 45 26 45 19 4 14 41 || 57 4. * 27 || 45 18 4 || 15 42 || 57 3 78 11 4 41 28 I 44 18 4 | 16 42 57 2 81 11 3 33 29 || 4:3 17 4 || 17 43 || 57 1. 84 11 2 24 30 42 | 16 4 18 44 57 O 87 11 1 12 90 11 0 0 xis x Ixºlviii. vii. vi. TABLE XLIV. ARGUMENT. JMoon’s Semidiameter. Equatorial Parallax. Eq. Par. semidiºq. Par. Semidi. Eq. Par. Semidi. 53/ O// 53 10 53 20 53 30 53 40 14, 27; 56 on 14, 29 14, 32 14, 35 14, 37 56 10 56 20 56 30 56 40 18 21 24 26 16// 59/ Oſº 59 10 59 20 59 30 59 40 16' 5// 16 7 16 10 16 13 16 16 53 54 54 54 54 50 O 10 20 30 14, 40 14, 43 14. 46 14, 48 14, 51 56 50 57 0 57 10 57 20 57 30 29 32 35 15 37 15 40 59 50 60 0 60 10 60 20 60 30 18 21 24 26 29 54 54 55 55 55 40 50 O 10 20 14, 54 14, 57 14, 59 15 2 15 5 57 40 57 50 58 0 58 10 58 20 15 43 15 46 15 48 15 51 15 54 60 40 60 50 61 0 61 10 61 20 16 32 35 16 37 16 40 16 43 55 30 55 40 55 50 56 0 15 7 15 10 15 13 15 16 58 30 58 40 58 50 59 0 15 56 15 59 16 2 16 5 61 30 61 40 61 50 62 0 16 46 16 48 16 51 16 54 TABLE XLV. .Augmentation of JMoon's Semidiameter. ARGUM. Appar. Alt. Ap. Aft|Augm. 60 2/ 12 3 18 5 24 6 30 8 36 9 42 11 48 12 54 13 60 14 66 15 72 15 78 16 84. 16 90 16 TABLE XLVI. JMoon's Horary JMotion in Longitude. ARGUMENTs. 2, 3, 4 and 5 of Long. Arg. 2 || 3 || 4 || 5 |Arg. O 6//| 1// 37' 3//|100 5 || 5 || 2 || 3 || 3 || 95 10 || 5 || 2 || 3 || 3 || 90 15 || 4 || 2 || 3 || 3 || 85 20 || 4 || 3 || 2 || 2 || 80 25 || 3 || 3 || 2 || 2 || 75 30 || 2 || 3 || 2 || 2 || 70 - 35 || 2 || 4 || 1 || 1 || 65 40 || 1 || 4 || 1 || 1 || 60 45 || 1 || 4 || 1 || 1 || 55 50 || 0 || 5 || 1 || 1 || 50 9ſ; *IFAIX GI’IgVJ, sIA sIIA sIl? A s), I s\{ sIX O I 0 || 9 0 || 0& 0 | 68 0 || 0 || || 9 || I | 08 I I 0 || 9 0 || 0& 0 || 07 O || 0 || || 9 || I 63, & I O | A 0 || I& 0 | If 0 | I I | g I I | 8& 8 I O | A 0 || 3& 0 | If? O | T I 9I I Z3, j, I 0 || A 0 || 3& 0 || 37 0 || 3 || || 9 || | | 9& 9. I 0 || 8 || 0 || 8& 0 || 87 0 || S T | 9T I | 93, 9 I 0 || 8 0 || 8& 0 | #7 0 || 3 I ZI I | 73, A | [ 0 || 8 0 || 7& 0 | ## 0 || ||7 || || ZI [ ] § , 8 I 0 || 6 0 || 93 0 || 97 0 || 7 || || AT I &&. 6 I 0 || 6 0 || 93 0 || 9% 0 || 9 || || 8 I I I& OI | [ 0 || OI 0 || 93 0 || 97 0 || 9 || || 8L I 0& IT & 0 || OT 0 || A& 0 || Zī, 0 || 9 || || 8T I 6I &I & 0 || II O | A3 0 || 87 0 || Z. I | 8T I 8T 8I & 0 || II () || 8& 0 || 8; 0 || 2 I | 8T I | AI 7I & 0 || II 0 | 66, () || 67 0 || 8 || || 6’ſ I | 9 | 9I & 0 } &I 0 | 68, 0 || 09 0 || 8 I 6I I | 9 I '91 & O &I 0 || 08 0 || Og O || 6 || || 6 I I | 7'ſ AI 8 0 || 8 || 0 || I9 O Iç () || 6 || || 6T I | 8 8L | S 0 || 8T 0 || IS 0 || 39 0 || OI I 6I I &I 6T | 8 0 || 7 || 0 | &g 0 || &g 0 || OT I | 0& I | II 03, 8 O | #I 0 || 88 O | 89 0 || II I 0& I | OI I& 8 0 || 9 || 0 || 38 0 || 79 0 || II I | 0& I | 6 3& W 0 || gl 0 | #8 O || 79 0 || II I | 0& I | 8 8& # 0 || 9 || 0 || GS 0 || Qg 0 || &I I | (3 I | A. 7& # 0 || 9 || 0 || SS 0 || 99 () &I I | 0& I | 9 93 || 7 0 || AI 0 || 98 0 || 99 0 || 8T I | 0& I | 9 93 || 9 O | 8 || 0 || A.8 0 || 29 0 || 8 || || || 0& I 7 A3 || 9 0 || 8 || 0 || Z8 0 || 89 0 || 7T I | 03, I | 8 8& 9 () 6T 0 || 88 0 | 89 0 || 7 I I | 0& I & 63, 9 0 || 6 || 0 | 68 0 || 69 0 || 7T I | 0& T | I o08 /9/0 |//06/0 |/68 /0 |//0 /T |,9]. , I liſ0& / I loſ) sA sAI sl II sII s] s0 'uonoaaq alſ, Jo Juoun; IW ‘LNGIWnbūW ‘opng|3uo'I w! wouloſſ fiſp.toPE s.100W’ Ág *IIIATX KITEIWJL //00I), 106 || 108 || 102 || 109 | 1,09 || 107 /,08 || 103 (,,OI //0 O IA 9 9 A | 8 6 OI II | &I 8T | #I GI | 0 IA 9. g 9 A | 8 6 OI II &I SI #I | QI 9& OI G 9 A l 8 6 OI II &I | ST | FI 9T || 0& G I G 9 A | 8 6 OT | II &I 8I 7T | GI GI Oć, 9. 9 || A. | 8 6 OI II | &I | 8 || || 7 || || QI OI 9& G 9 2. 8 6 OI II &I SI WI 9T 9 0 IIA 9 9 || A. 8 6 Of II &I 8T | #I | 9 || || 0 A G 9 2 || 2 || 8 6 OI II &I SI 8I WI 93, OI 9 : A | 8 || 8 || 6 || OT | II &I &I SI WI 0& 9 I’ 9 4. 8 8 6 OT | II | &I &I SI | f | | 9T | 03 9 2 8 6 6 OT | II II &I SI 7I OI 93 2. A | 8 6 6 OT | II II &I | SI 8T 9 0 IIIA | A. 8 8 6 6 OT | II II &I &I 8T | 0 AI 9 2. 8 8 6 6 OT | II | II &I &I | 8 || || 93, OI 8 8 6 6 OT || OT || OT | II II &I &I | 03 | SI 8 6 6 6 OT || OI OT | II | II II | &I SI 03 6 6 6 6 OI OT || OT | II II | II II OI 93, 6 6 6 OI OI | OT || OT || OT | II II II | 9 O XI OI OT || OI OT || OT || OT || OT || OT || OT | QI OI O III 9. OT || OI OT || OT || OT || OI OT || OI | OT || OT || OI 9& OI II | OT || 0 || || OT || OI OT || OT || OT || OT || 0 || || 6 || 0& 9I II II || II | OL | OT || OL OT || OI | 6 6 6 GI 03, &I II | II II OT || OI OI | 6 6 6 8 OI 93, &I &T II II OT || OT || 0 || || 6 6 8 8 G O X | 8 I &I | &I II | II | OT || 6 6 8 8 Z || 0 II G &I &I &I II | II OT || 6 || 6 8 8 A 93 OI &L | 8'ſ &I II | II OI | 6 6 || 8 A | A. | 06 GI #I 8I &I &I | FI OT || 6 || 8 8 A | 9 || 9 I 03 7I | ST | SI &I II OT || 6 || 8 4. A | 9 || 0T 93, 9I 'WI SI &I II | OT || 6 8 4. 9 9 9 O IX GI VI 8I &I II OT || 6 8 4. 9 9. O I G 9I 'WI | 8I &I II | OT || 6 8 Z 9 9 || 93 OI 9I W L | 8 &T | II OT | 6 8 A | 9 || 9 || 06 GI 9I GI SI &I II || 0 || || 6 8 Z || 9 || 7 || 9T 0& 9I | 9 || || 8 || || &I II | OI | 6 || 8 A || 9 || 7 || OI 93, 9I | 9 || || 7 || || &I II OT || 6 || 8 || 9 9 || ||7 || 9 o0 sIIX |09T |09.I ſ/WI WGI Will |z|01 a6 /8 /9 |/9 ºf loſ) so //00I}//06 l/08 //02. ',09 //09 //07 //08 //03, l,0I //0 "paloa.I -100 ‘ĀſeuouW pue “suogenba 3upaoald Jo uns ‘SINGWnbūW ‘ophyāuoT un uoyoAſ flap.1011 s.100MP 58 TABLE – XLIX, 's JMoon's Horary JMotion in Longitude. ARGUMENT. Anomaly, corrected. 0: Is | |S III's IVs Vs O O 34' 51// 34, 51 34, 51 34, 51 34 51 34 50 34' 14’ſ 34 12 34, 9 34; 7 34 4. 34 1 32/.39// 32 36 32 32 32 28 32 24 32 21 30/.45// 30 42 30 38 30 34 30 31 30 27 29' 6"/ 29 3 29 O 28 58 28 55 28 52 28/ 1// 27 59 27 58 27 56 27 55 27 54 1 :i 34, 50 34, 49 34, 49 34, 48 34, 47 33 59 33 56 33 53 33 50 33 47 32 17 32 13 32 9 32 5 32 2 30 23 30 20 30 16 30 13 30 9 28 50 28 47 28 45 28 42 28 40 27 53 27 51 27 50 27 49 27 48 i i 34, 46 34 45 34 44 34 43 34 42 33 44 33 41 33 38 33 35 33 32 30 6 30 2 29 59 29 56 29 52 28 37 28 35 28 33 28 30 28 28 27 47 27 46 27 45 27 45 27 44 1 6 34 41 34, 39 34; 38 34, 36 34 34 33 28 33 25 33 22 33 18 33 15 29 49 29 46 29 42 29 39 29 36 28 26 28 24 28 22 28 20 28 18 27 43 27 42 27 42 27 41 27 41 34 33 34 31 , 34, 29 34, 27 34, 25 33 12 33 8 33 5 33 1 32 58 29 33 29 30 29 26 29 23 29 20 28 28 28 12 28 10 28 9 16 14 27 40 27 40 27 39 27 39 27 39 34, 23 34, 21 34, 19 34 16 34 14 32 54 32 50 32 47 32 43 32 39 29 17 29 14 29 12 29 9 29 6 28 28 28 28 28 : 27 39 27 38 27 38 27 38 27 38 ii XIS Xs VIIIs VIIs VIs TABLE L. 59 JMoon's Horary JMotion in Longitude. ARGUMENTs. Sum of preceding equations, and Arg. of Wariation. 277 (28/29/30/31/32/33/34/35 36 |37/ Os Oo O// 1// 2// 4// 5// 6// 7ſ, 8// 10// 11// 12// XIIs Oo 5 || 0 || 1 || 2 || 4 || 5 || 6 || 7 || 8 ||10 |11 12 25 10 || 0 || 1 || 3 || 4 || 5 || 6 || 7 || 8 || 9 |11 [12 20 15 || 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 |}0 |11 15 20 | 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 |10 |11 10 25 || 2 || 3 || 4 || 4 || 5 || 6 || 7 || 8 || 8 || 9 |10 5 I 0 || 3 || 4 || 4 || 5 || 5 || 6 || 7 || 7 || 8 || 8 || 9 || XI O 5 || 4 || 4 || 5 || 5 || 6 || 6 || 6 || 7 || 7 || 8 || 8 25 10 || 5 || 5 || 5 || 6 || 6 || 6 || 6 || 6 || 7 || 7 || 7 20 15 || 6 || 6 || 6 || 6 || 6 || 6 || 6 || 6 || 6 || 6 || 6 15 20 || 7 || 7 || 7 || 7 || 6 || 6 || 6 || 5 || 5 || 5 || 5 10 25 || 8 || 8 || 7 || 7 || 6 || 6 || 6 || 5 || 5 || 4 || 4 5 II 0 || 9 || 9 || 8 || 7 || 7 || 6 || 5 || 5 || 4 || 3 || 3 | X 0 5 10 || 9 || 8 || 8 || 7 || 6 || 5 || 4 || 4 || 3 || 2 25 10 |11 10 || 9 || 8 || 7 || 6 || 5 || 4 || 3 || 2 || 1 20 15 11 10 || 9 || 8 || 7 || 6 || 5 || 4 || 3 || 2 || 1 15 20 |12 |11 ||10 || 8 || 7 || 6 || 5 || 4 || 2 || 1 || 0 10 25 |12 |11 |10 || 8 || 7 || 6 || 5 || 4 || 2 || 1 || 0 5 III 0 |12 |11 ||10 || 8 || 7 || 6 || 5 || 4 || 2 || 1 || 0 || IX O w 5 |12 |11 ||10 || 8 || 7 || 6 || 5 || 4 || 2 || 1 || 0 25 10 |12 |11 ||10 || 8 || 7 || 6 || 5 || 4 || 2 || 1 || 0 20 15 11 10 || 9 || 8 || 7 || 6 || 5 || 4 || 3 || 2 || 1 15 20 [1] [10 || 9 || 8 || 7 || 6 || 5 || 4 || 3 || 2 || 1 10 25 10 || 9 || 8 || 8 || 7 || 6 || 5 || 4 || 4 || 3 || 2 5 IV 0 || 9 || 8 || 8 || 7 || 7 || 6 || 5 || 5 || 4 || 4 || 3 || VIII 0 5 || 8 || 8 || 7 || 7 || 6 || 6 || 6 || 5 || 5 || 4 || 4 25 10 ) 7 || 7 || 7 || 6 || 6 || 6 || 6 || 6 || 5 || 5 || 5 20 15 || 6 || 6 || 6 || 6 || 6 || 6 || 6 || 6 || 6 || 6 || 6 15 20 || 5 || 5 || 5 || 6 || 6 || 6 || 6 || 6 || 7 || 7 || 7 10 25 || 4 || 4 || 5 || 5 || 6 || 6 || 6 || 7 || 7 || 8 || 8 5 W 0 || 3 || 3 || 4 || 5 || 5 || 6 || 7 || 7 || 8 || 9 || 9 || VII O 5 || 2 || 3 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 9 ||10 25 10 || 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 ||10° 11 20 15 || 0 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 ||0 |12 15 20 | 0 || 1 || 2 || 4 || 5 || 6 || 7 || 8 ||10 (11 |12 10 25 || 0 || 1 || 2 || 3 || 5 || 6 || 7 || 9 ||10 11 12 5 VI O || 0 || 1 || 2 || 3 || 5 || 6 || 7 || 9 |10 |11 |12 || VI O 27, 287|29/ 130' 31' 132' 33' |34 |357 (36' 37/ ° IT CITFIW.L IA sIIA sIIIA s}{I s}{ sIX O || 6T I | 0 || || 33 0 || & 0 || 0& 0 | 89 0 || 09 I 6I I 69 0 || 0& 0 | & 0 || I& 0 || 69 0 || 62, & 6T I | A9 O || 6 || 0 | & 0 || 8& 0 || 0 || || 83, 8 || 6’ſ I 99 0 || 8I 0 || 8 0 || 7& 0 | I I A3, 7 || 6T I 99 0 || ZI 0 || 8 0 || 93 0 || 3 || || 93, | 9 || 6T I | 79 0 || 9 || 0 || 8 0 || 92, 0 || 8 || || 93, 9 || 8T I 39 0 || 9 || 0 || 8 0 || 23, 0 || 7 || || 72, A 8T I | [g 0 || 7 || 0 || 8 O | 68, 0 || 9 || || 8 8 8T I | 09 O || 8 || 0 || 7 0 || 08 0 || 9 || || 33 6 ZI I 87 O &I 0 | # 0 || IS O | A. I I& Oſ. ZI I 27 O II 0 || 7 0 || 38 0 || 8 || || 02, II | 9T I | 9% 0 || II 0 || 9 0 || 78 0 || 6 || || 6’ſ &T | 9T I | #7 O OI () | g O G8 0 || OT || || 8T 8I GI I 87 () 6 0 || 9 0 || 98 0 || IT I ZI 7I g|I I Čiž () 8 0 || 9 0 || 88 0 || IT I 9T g|I | #I [ . Off () 8 0 || Z 0 | 68 0 || &T I 9T 9I | ST I 68 0 || Z 0 || 8 0 || 07 0 | ST I | ?I Ží | ºf f | # 0 | g o | g o iſ o ºi i gi 8T | &l I | Z9 0 || 9 0 || 6 0 || 87 0 || 7T I &I 6T | II I gS O | g O OI O ºf 0 || 7T I I IT O2, OI. I #8 O | g 0 | II O | 97 () gT I OI I& 6 I 88 0 || 7 0 || II 0 || Ziff () 9T I | 6 28, 8 T | [8 O | # 0 &I 0 || 87 0 || 9T T | 8 8& A. I | 08 0 | # 0 || 8 || 0 || 67 0 || 9T I A. #2, 9 I | 68, 0 & 0 | #I 0 | [g Q 9T I 9 92, g I Z& 0 || 8 0 || 9 I 0 || 39 0 || AI T | 9 96 || 7 || || 93 0 || 8 0 || 91 0 | 89 0 || ZI T | 7 AZ || 8 || || 93 0 || 8 O j ZI 0 || 79 0 || AT I | 8 88, I & I | 73 0 || 8 0 || 8 || 0 || 99 0 || AI I & 6& | [ I | 8& 0 | & 0 || 6T () A9 0 || AI I I o08 |//0 / I |z,3& 10 /ć 10 l/0& 10 l/89 10 l/2,I 11 loſ) •A | AI III | II | | | 0 •uogenie A alſ, Jo JuaunāIV INCIWmbay ‘9pm.5u0T uy ºtoyoſ'ſ fi.lp.10H suoopſ' * [ITI GITISHVJ, W9 sIA. sIIA |s|IIA sXI six s]X O || 3 || 9 || WI | 8I WI 9 || 08 I & | 9 || 7 || || 8 || || 7 || || 9 || 63, & | 3 || 4 || 71 || 8 || 8 || 9 |8& 8 || 3 || A. GI 8T | 8T | 9 || Alſº, 7 || 3 || 2 || 9 || 8T | 8 || 9 |9& 9 & | 2. 9I 8T || 8 || || 9 || 93, 9 || 3 || 8 || 9 || || 8T &I | 9 || 73 A & | 8 || 9T 8I &I | 7 || 8& 8 & | 8 9I | 8T | &T | 7 || 33 6 & | 8 || 9T 8I &I | 7 || I& OI | 8 || 6 || 9T | ZI II | 7 || 03 II | 8 || 6 || 9T 2T | II || 7 || 6 I 3. I | 8 || 6 9I AI II | f | 8T SI 8 || 6 || AI ZI II | 8 || ZI 71 || 8 || OT | ZI ZI OL | 8 || 9T 9I | 8 || OT || AI ZI OT || 8 || 9T 9I | 8 || OT | ZI ZI OT || 8 || VI 2I | 8 || II | ZI | ZI | 6 || 8 || 8 8I 7 || II | ZI 9T || 6 || 8 || &I. 6T || 7 || II || AI 9T || 6 || 8 || II 03 || 7 || II AI | 9 || || 6 || 8 || OI I3 || 7 || &I 8T | 9T || 8 || 3 || 6 33, 7 || &I | 8 || || 9 || || 8 || 3 || 8 83 || 7 || &T || 8 || || 9 || || 8 || 3 || 2 % | 9 || &I 8I GI | 8 || 3 || 9 93, 9 || 8T 8I 91 | A & 9 92, 9 || 8T || 8 || || GI Z & 7 A3 || 9 || 8 || || 8 I | 9 || || 2 || 3 || 8 83 || 9 || 8 || || 8T || 7 || || 2 || 3 || 3 66, 9 || 7 I 8T | #I | 9 || 3 || I o08 || 09 |//7I /8T //7T |/9 |/3 loſ) ‘s A 's AI sl11 || sſD | sl s0 'uomompaºl aul Jo quaum3+W ‘INGIWnt5AV ‘opmåuoT up uoyoAſ ſup40H 8.100W. TABLE LIII. JMoon's Horary JMotion in Latitude. ARGUMENT. Arg. I, of Latitude. f 0.4-11. 4- |Il-4- III – Ivº-Iv.– 09 2/58// 2' 34"| 1/29/ O O" | 1/29//| 2/34// 300 1 || 2 58 || 2 33 || 1 27 || 0 3 1 32 || 2 36 29 2 || 2 58 || 2 31 || 1 24 || 0 6 1 35 || 2 37 28 3 || 2 58 2 29 || 1 21 || 0 9 1 37 || 2 39 27 4 2 58 || 2 28 1 18 || 0 12 1 40 || 2 40 26 5 2 57 2 26 || 1 15 || 0 16 1 42 || 2 41 || 25 6 || 2 57 2 24 || 1 13 || 0 19 1 45 || 2 43 24 7 || 2 57 || 2 22 || 1 10 || 0 22 1 47 || 2 44 || 23 8 || 2 56 || 2 20 || 1 7 || 0 25 1 50 || 2 45 22 9 || 2 56 2 19 || 1 4 || 0 28 1 52 || 2 46 || 21 10 || 2 55 || 2 17 || 1 1 || 0 31 1 55 || 2 47 20 11 || 2 55 || 2 15 || 0 58 || 0 34 1 57 || 2 48 19 12 || 2 54 || 2 12 || 0 55 || 0 37 1 59 || 2 49 18 13 || 2 53 || 2 10 || 0 52 || 0 40 2 2 2 50 | 17 14 || 2 53 || 2 8 || 0 49 || 0 43 2 4 || 2 51 | 16 15 2 52 || 2 6 || 0 46 || 0 46 2 6 || 2 52 | 15 16 || 2 51 || 2 4 || 0 4.3 || 0 49 2 8 || 2 53 14, 17 2 50 2 2 || 0 40 || 0 52 2 10 || 2 53 13 18 2 49 1 59 || 0 37 || 0 55 2 12 2 54 || 12 19 || 2 48 || 1 57 || 0 34 || 0 58 2 15 2 55 | 11 20 2 47 1 55 || 0 31 || 1 1 2 17 || 2 55 10 21 || 2 46 || 1 52 | 0 28 || 1 4 2 19 || 2 56 9 22 || 2 45 || 1 50 || 0 25 || 1 7 2 20 2 56 8 23 2 44 1 47 || 0 22 1 10 2 22 || 2 57 7 24 || 2 43 || 1 45 || 0 19 || 1 13 2 24 2 57 6 25 || 2 4.1 1 42 || 0 16 || 1 15 2 26 || 2 57 5 26 2 40 1 40 || 0 12 || 1 18 2 28 2 58 4 27 2 39 || 1 37 || 0 9 || 1 21 2 29 || 2 58 3 28 2 37 1, 35 | 0 6 || 1 24, 2 31 || 2 58 2 29 || 2 36 || 1 32 || 0 3 || 1 27 2 33 2 58 1 30 || 2 34 || 1 29 || 0 0 || 1 29 2 34 || 2 58 O | | |x| + |x| + |x|+|viſiº-vii. vis- TABLE LIV. * JMoon's Horary JMotion in Latitude. ARGUMENT. Arg. II, of Latitude. 0°-H | 1 + II*-H III" — IV*— Vs — 0° 4// 4// 2// 0// 2// 4// 300 6 4. 3 2 0 3 4. 24 12 4. 3 I 1 3 4. 18 18 4. 3 1 1 3 4. 12 24 4. 3 O 2 3 4. 6 30 4. 2 O 2 4. 4. O ſ XI.--|x|+|x|+|VIII —lvii. VI-I TABLE LY, 63 * JWautical fllmanac. - I. AUGUST 1821. | # # IPhases of the Moon. § 5 ^. ;: 2. f —— º º Sundays, Holidays, : : Terms, &c. ID. H. M. O O D) First Quarter, 5.14.10 §, £, O Full Moon, 13. 2. 8 & & (C Last Quarter, 19.18.49 © New Moon, 27. 3.17 ¥. : Lammas-Day. Other Phenomena. F. 3 D. H. M. . Sa. 4. 3.17.47 D) 0. ty 10 - - la Stationary. Sun. 5 || 7th Sunday after Trinity. || 11 - - š Stationary. MI. 6 Transfig. of our Lord. 19 - - 21 Stationary. Tu. 7 | Name of Jesus. Princess. 23. 1. 1 () enters in W. 8 [Amelia born. 25.17. 0 ) ; Th. 9 26. 14.30 D) a F. 10 | St. Lawrence. 27 - - G) eclipsed invis. Sa. | 11 | Prs. of Brunswick born. 21. 11.58 ) g 8 31. 1.32 ) a my Sun. 12 | 8th Sunday aft. Trinity. M. 13 [Prince of Wales b. Tu. | 1.4 W. 15 Assumption. Th. 16 | Duke of York born. F. 17 Sa. 18 Sun. 19 9th Sunday aft. Trinity. M." | 20 Tu. 21 | Duke of Clarence born. W. 22 t Th. 23 F. 24 || St. Bartholomew. Sa. 25 Sun. 26 10th Sunday aft. Trinity. M. 27 Tu. 28 St. Augustine. W. 29 St. John Bapt. beheaded. Th. 30 ** F. 31 64 TABLE LW. JWautical fllmanac. AUGUST 1821. II. ; : 5 $ 5 THE SUN’S > || > t 9 (l) E # —|Equation O 3 Longitude Right Ascen. Declination of Time.|Diff. É. É. § º £n Time. JVorth. Jidd. ſº | c. S. D. M. S. H. M. S. D. M. S. M. S. s. W. 1 || 4. 8.49.37 8.45. 2,9 | 18. 4.20 5.58,4 a Th. | 2 || 4. 9.47. 5 8.48.559 17.49. 6 || 5.548 3,5 F. 3 || 4.10.44.33 | 8.52.48.2 | 17.33.35 | 5.50,5 : 43 Sa. | 4 || 4.11.42. 2 | 8.56.39.9 || 17.17.46 || 5.457 || 48 Sun. | 5 || 4.12.39.32 9. 0.310 || 17. 1.41 5.40.3 || 3,4 6,1 M. 6 || 4.13.37. 2 9. 4.21,5 | 16.45.19 5.34,2 tu. || 7 || 4.14.34.34 || 9.8.11.3 | 16.28.41 || 5.27.5 | 6′. w. || 8 || 4.15.32.6 9.12. 0.6 | 16.11.47 5.202 || Z3 Th. | 9 || 4.16.29.39 || 9.15.49.2 | 15.54.37 5.12.3 | "...9 F. 10 || 4.17.27.12 || 9.19.373 || 15.37.13 || 5.3.9 | 84 9,1 Sa. 11 4.18.24.47 | 9.23.24,7 15.19.33 4.54,8 A. Sun. 12 4.19.22.23 9.27.11.6 | 15. 1.39 4.45.1 |..}} M. 13 || 4.20.20. Ó 9.30.580 14.43.30 4.34.9 ||9.2 Tu. | 1.4 || 4.21.17.39 9.34.437 14.25.7 || 4.242 ||9|| W. 15 4.22.15.18 9.38.29.0 | 1.4.6.30 4.12.9 |113 11.8 Th. 16 || 4.23.12.59 || 9.42. 13,7 13.47.40 || 4. 1,1 lºs F. 17 || 4.24.10.42 9.45.37.9 || 13.28.37 || 3.48.8 ||3: sa. 18 4.25. 3.26 9.49.416 || 13.9.21 || 3.360 |}}} Sun. 19 || 4.26. 6.12 9.53.24.9 | 12.49.51 3.22.7 ||3.3 M. 20 4.27. 4.0 9.57.76 | 12.30.10 || 3.9.0 |13,7 14,2 Tu. | 3 || 4-2. 1:49 ||9. 9.399 || 3:19-17 | 2:54:3 |14.6 W. 22 || 4-23:39-41 | 19. 4.34.9 || || 33-11 || 3:49,3 . Th: |33 || 4-29.57.34 19.3.13.3 || 14-23-3 || 3.23.1 |. F. 24 5. Q.53.39 ||9-14.544 || ||...}. 2.96 ||. Sa. 25 5. 1.53.26 10.15.35,0 10.48.49 | 1.53,8 |** -16,3. Sun. 36 5. 2.31-25 | 19.13-15.2 | 19.28. Q | 1.3%; 16.7 | M. 2. 5. 3.49.25 | 19.32-33,1 | 19.7.2 | 1.20.8 ||...} Tu. 28 5. 4.47.26, 10.26.34,5 9.45.53 | 1. 3,8 #. W. 29 |5. 5.43.39 19.39.13.6 2.24.33, 9.434 i. Th. 30 5. 6.43.34 10.33.52,3 9. 3. 9 || 0.28,6 “” 18,2 F. : 31 5. 7.41.41 10.37.30,7 |, 8.41.34 0.10,4 TABLE LV. 65 JVautical Almanac. III. AUGUST 1821. ~ * THE SUN'S 4 Time of Sum’s Place of the • ‘Semidiameter|| Semidi- Hourly Logar, Moon's # passing Merid.] ameter. Motion. |Distance. Node. C M. S. M. S. M. S. | S. D. M. l 1. 6,4 15.47,5 2.23,6 0.00620 ||11. 5.43 7. 1. 5,9 15.48,4 2.23,9 0.00581 ||11. 5.24 13 1. 5,5 15.49,4 || 2.24,2 0.00536 || 11. 5. 5 19 1. 5,0 15.50,5 2.24,5 0.00485 ||11. 4.46, 25 1. 4,6 15.51,7 2.24,9 0.00429 ||11. 4.27 ECLIPSES OF THE SASTELLITES OF JUPITER. ; * JME.A.JV" TIME. I. Satellite. II. Satellite. III. Satellite. 21. 18.58 Immersions. Immersions. Days. | H. M. S. || Days. | H. M. S. Days. H. M. S. *2 13.43.56 #4. 11. 3.24. #6 12. 7.12 Im. 4. 8.12.22 8 0.22.37 #6 14.31.28 E. 6 2.40.46 #11 13.40.50 13 16. 8.10 Im. 7 21. 9. 13 15 3. 0. 1 3 18. 31.27 E. 9 15.37.36 18 16, 18. 11 20 20. 9.30 Im. 11 10. 6. 4 22 5.37.21 20 22.31.51 E. 13 4.34.28 25 18.55.31 28 0.10. 7 Im. 14 23. 2.55 29 8. 14.40 28 2. 31.36 E. 16 17.31. 19 * 18 11, 59.48 20 6, 28.13 22 0.56.41 23 | 19.25. 7 * #25 13.53.36 27 8.22, 1 29 2.50. 32 30 IV. Satellite. #9 TABLE LV. JW autical fllmanac. AUGUST 1821. 1W. THE PLANETS. Heliocentric. GeoCentric. th Declin. Rt. asc. Passage ă Long. Lat. Long. | Lat. in time. | Merid. s. D. M. D.M. s. D. M. D. M. || D. M. Th.M. H. M. § MERCURY Gr. Elong. 19d. Inf 3 1d. 11h. 10. 7.44 W 6.55S || 4. 9.36 || 4.54S || 13. 8N | 8.43 23.51 , 10. 18.14 || 7. 0 || 4. 7.24 || 4.38 || 13.57 8.34 23.32 10.29.34 || 6.49 || 4. 5.37 4. 7 || 14.54 8.28 23.15 11, 11.53 || 6.19 || 4. 4.34 || 3.25 || 15.49 8.24 23. 1 11.25.22 || 5.26 || 4. 4.31 || 2.35 || 16.38 8.25 22. 52 16 0.10. 9 || 4. 8 || 4. 5.33 | 1.43 || 17.14 8.30 22.47 19 || 0.26. 18 2.24 || 4. 7.4.1 0.52 || 17.32 8.39 22.47 22 | 1.13.43 0.18S || 4.10. 51 0. 6S || 17.26 8.53 22.50 25 | 2. 2. 7 | 1.56N || 4. 14.55 || 0.34N || 16.55 9. 10 22.57 28 || 2.21. 2 4. 1 || 4.19.43 | 1. 5 || 15.57 9.30 23. 7 31 || 3. 9.47 ( 5.38 || 4.25. 2 | 1.28 || 14.34 9.51 23.17 T 9 VENUS. % 1 ſ 5.25.39 3.20N || 4.28. 1 | 1.30N || 13.35N 10. 3 1. 18 7 || 6. 5.22 || 3.11 || 5. 5.22 | 1.28 || 10.55 || 10.31 1.23 13 6.15. 3 || 2.56 || 5.12.42 | 1.22 8. 4 || 10.58 1.27 19 | 6.24.43 2.3 5.20. 3 | 1.15 5. 5 || 11.25 1.32 35 | 7.4.31 2.13 || 5.27.22 | 1.4 2. 2 | 11.52 | 1.3 & MARS. 1 | 1.24. 16 || 0.12N 2.23.19 || 0. 9N || 23.27N 5.31 20.45 7 ; 0.18 || 2.27. 19 |. 0.14 || 23 40 5.48 || 20.39 13 || 2. 0.45 || 0.24 || 3. 1.17 || 0.19 || 23.46 6. 6 || 20.34 19 2. 3. 56 0.3 3. 5.11 0.24 || 23.45 6. 23 || 20. 28 25 || 2. 7. 6 || 0.36 || 3. 9. 3 || 0.29 || 23.38 6.40 | 20.23 21 JUPITER. TOI7.59 TITEST0.35.55TIESTIO-5NTT 53 TIZ-5- 7 || 0.18.32 | 1.18 || 0.29.57 | 1.24 || 10. 10 1.53 | 16.43 13 || 0.19. 5 | 1.18 || 1. O. 9 || 1.25 || 10. 12 1.54 | 16.21 19 0.19.38 | 1.18 || 1. 0.13 | 1.27 || 10. 12 1.55 15.59 25 || 0.20.11 | 1.17 || 1. 0.10 || 1 28 || 10.10 1.54 | 15.36 b SATURN. TTO.30.54TE.50ST0.35.10 T5...IST7.5INTITTTT6.55 7 || 0.20.47 2.30 || 0.26.43 || 2.36 7.53 1.43 | 16.32 13 0.20.59 || 2.30 || 0.26.42 | 2.38 7.51 1.43 | 16. 9 19 || 0.21. 12 || 2.30 || 0.26.37 || 2.39 7. 48 1.43 | 15.47 25 0.21.24 2.30 || 0.26. 28 2.41 7.44 1.42 15.24 H GEORGIAN. ITS. I.33 TOISSTL3.35.35TO.15STE. 155TTF 53 9. F2 11 9 #| 0.15 || 8. 29.24 0.15 || 23.43 || 17. 57 8.33 21 | 9. 1.47 ( 0.15 || 8.29.14 || 0.15 || 23.43 || 17. 57 7.55 TABLE LV. 67 JWautical Almanac. W. AUGUST 1821. # # THE MOON'S (L) (1) -5 | E º & … . . Longitude. Latitude. O O Oſ) CD g ã Noon. JMidnight. JVoon. JMidnight. s, D.M. s. s. D. M. s.l. D. M. s. D. M. S. W. 1 || 5.18.28.49 || 5.24.32.24 fl 1. 14.51 S | 1.46.44 S Th 2 6. 0.33. 19 || 6. 6.32. 3 2. 17. 13 2.46. I F. 3 : 6.12.29. 7 || 6.18.25. 3 3.12.53 3.37.36 | S.A. 4 || 6.24.20.24 7, 0.15.46 3.59.56 4. 19.43 1 Sum. || 5 || 7. 6.11.44 || 7.12. 8.55 || 4.36.46 4.50.53 | M. 6 7.18. 7.55 || 7.24. 9. 18 || 5. 1.55 5. 9.43 Tu. || 7 || 8. (). 13.38 || 8. 6.21.29 || 5.14. 8 5. 15. 1 W. 8 || 8.12.33. 19 || 8.18.49.35 5.12.17 5. 5.48 Th. | 9 || 8.25. 10.40 || 9. 1.36.53 : 4. 55.30 4.41.22 F. 10 | 9. 8. 8.27 | 9. 14.45.29 || 4.23.25 4. 1.42 Sa. | 11 | 9.21.28. 0 || 9.28. 15.54 3.36.22 3. 7.39 Sun. 12 || 10. 5. 9. 1 ||10. 12. 7. 1 || 2.35.50 2. 1.18 M. 13 || 10. 19. 9.30 10.26. 15.57 || 1.24.33 0.46. 7 S Tu. || 14 | 11. 3.25.47 11.10.38.23 0. 6.38 S | 0.33. 14 N W. 15 11.17.53. 4 |11.25. 9. 6 || 1.12.45 N | 1.51.14 Th. J 6 O. 2.25. 49 O. 9.42.32 2.27.58 3. 2. 17 F. 17 | 0.16.58.37 || 0.24. 13.27 3.33.3 4. 1.26 Sa..., | 18 | 1. 1.26.31 | 1. 8.37.23 - 4.25.17 4.44.50 Sun. 19 | 1.15.45.41 1.22.51. 6 1 4.59.50 5.10.10 M. 20 | 1.29.53.21 ; 2. 6.52.15 ± 5.15.45 5. 16.3 '1'u. 21 2. 13.47.41 2.20.39.33 5.12, 51 5. 4.38 W. 22 || 2.27.27.49 || 3. 4.12, 27 4.52. 12 4.35.49 Th. 23 || 3.10.53.27 | 3.17.30.50 : 4.15.49 3.52.32 F 24 3.24. 4.38 || 4. 0.34, 54 - 3.26.23 2.57. 48 Sa. 25 4. 7. 1.42 4.13.25, 6 2.27.11 1.54.58 Sun. 26 || 4.19.45.10 || 4.26. 2, 0 1.21.36 0.47, 30 N Ml. 27 | 5. 2. 15.44 5. 8.26.28 0.13. 7 N | 0.21. 8 S Tu. 28 || 5.14.34.24 || 5.20.39.41 0.54.53 S | 1.27.46 W. 29 || 5.26.42.33 6. 2.43. 14 1.59.26 2.29.35 Th. 30 6. 8.42. 3 6. 14.39.19 2.57.55 3.24. 11 F. 31 || 6.20.35.23 6.26.30.38 3.48. 9 4. 9.37 68 TABLE LV. JWautical fllinamac. WI. AUGUST' 1821. . •º º; 5 THIE MOON’S. § 5 ;: | > # | # Right Ascension. Declination. ‘5 ‘8 |A Passage *-- §, £, 8°1 Merid. | Wºon. JMidnight, JYoon. JMidnight. cq cº * | F | D.T.M.T D. M. D. M. D. M. Tºp M. W. 1 || 5 || 2.35 | 168.55 174.17 ± 3.25 N | 0.32 N Th. 2 || 6 || 3.15 179.36 | 184.54 2.19 S 5. 8 S F. | 3 || 7 || 3.54 || 190. 13 195.35 7.54 10.35 Sa. || 4 || 8 || 4.35 201. 2 | 206.35 13.10 15.38 Sun, | 5 || 9 || 5.18 212, 16 || 218. 7 || 17. 57 20. 6 M. 6 || 10 || 6. 4 224. 9 230.23 22. 5 23.50 Tu. || 7 || 11 || 6.53 || 236.50 || 243.29 || 25.20 26.34 W. 8 || 12 || 7.46 250.20 257.22 27.29 28. 4 Th. | 9 || 13 || 8.42 264.33 271.50 28.18 28. 9 F. 10 || 14 || 9.39 || 279. 10 || 286.31 || 27.36 26. 39 Sa. | 11 || 15 || 10.36 293.50 301. 4 ; 25.19 23.36 Sun. 12 | 16 11.31 || 308.11 || 315. 11 21.31 19. 7 M. 13 || 17 | 12.23 || 322. 3 || 3:28.47 | 16.26 13.30 Tu. || 14 | 18 || 13.14 || 335.24 || 341.55 10.22 7. 5 W. 15 | 19 14. 3 || 348.23 || 354.49 : 3.41 S 0.14 S Th. | 16 || 20 || 14.53 1.15 || 7.43 3. 14 N | 6.39 N F. 17 | 21 15.44 14.15 20. 54 9.58 I3. 9 Sa. | 18 22 || 16.37 27.41 || 34.38 || 16. 8 18.53 Sun. 19 || 23 17.34 || 41.45 || 49. 1 || 21.21 23. 30 M. 20 | 24 18.33 56.27 | 64. 2 || 25.17 26.4() Tú. 21 || 25 || 19.33 71.43 79.26 27.39 28. 12 W. 22 || 26 20.32 87. 8 94.45 ° 28.19 28. () Th. 23 27 21.29 102.14 || 109.33 27. 16 26. 9 F. 24, 28 22.21 || 1 16.38 || 123.29 24.42 22.56 Ša, 25 29 23. 9 || 130. 6 136.28 20.54 18.39 Sun. 26 || 30 23.54 , 142.37 || 148.34 || 16. 12 i.S. 36 M. 27 | 1 d; 154.20 | 159.57 : 10.63 8. 5 Tu. 28 || 2 || 0.36 || 165.27 17().51 5. 14 N 2.22 N W. 29 || 3 || 1.17 176. 1 1 | 181.30 (). 31 S 3.22 S Th. 30 || 4 || 1.57 186.49 192. 9 -6.11 8.55 F. | 31|| 5 || 2.37 | 197.33 ſ 203.2 11.34 14.7 TABLE LV. 69 | JW'autical ſllmanac. VII. AUGUST 1821. 24 || 5 TIII. MOON'S * Q1) º: w O ;: 3. * | * | Semidiameter. IIor. I’arallax. Proportional i. &: Logarithm. O O § § JVoon. Midnight: JWoom, Midnight ſº | C. . . . . . M. S. M. S. M. S. M. S. JVoon. JMidnight W. 1 || 14.59 || 14.55 : 54. 53 || 54.41 .. 5158 5.174, Th. || 2 || 14.53 || 14.50 54.31 54.22 || 5187 5199 F. 3 || 14.49 14.47 54. 16 54.12 5207 5213 Sa. 4 14.47 | 1.4.47 54.11 || 54. 12 5214 5213 Sum. || 5 || 14.49 || 14.50 54. 16 || 54.22 5207 5 199 IM. 6 || 14.53 14.56 ± 54.31 54.42 || 5187 517.3 Tu. || 7 || 14.59 15. 4 || 54.55 || 55.11 : 51.56 5135 W. 8 15. 8 15.14 || 55.29 || 55.49 5111 5()85 Th. | 9 || 15.20 | 15.26 66. 10 || 56.33 : 5058 5028 F, 10 || 15.32 | 15.39 56.57 | 57.21 : 4998 4967 I 5.46 | 15.52 . 57.45 58. 8 - 4937 4908 15.58 || 16. 4 || 58.30 || 58.51 || 4881 4,855 M. 13 16. 9 || 16. 13 : 59.10 || 59.26 4832 4,812 Tu. || || 4 || 16.17 | 16.20 59.40 59.51 ± 4.795 4782 W, 15 | 16.22 | 16.23 59.59 || 60. 3 4772 4768 A. Sh { 7. ; ; Th. 16 || 16.24 | 16.23 : 60. 5 || 60. 3 4765 4768 I'. 17 | 16.22 || 16.20 59.58 || 59.51 - 4774, 4782 Sa. | 18 16. 17 | 16. 14 59.42 59.31 4793 4806 Sun. 19 | 16. 11 | 16. 7 : 59. 18 59. 4 || 4822 4,839 M. 20 | 16. 3 || 15.59 ± 58.50 58.35 - 4.856 4875 Tu, 21 15.55 15.50 58. 19 58. 3 || 4895 4.915 W. 22 15.46 15.42 57. 17 | 57.31 4935 4.955 Th. 23 15.37 15.33 - 57.15 56.59 ± 4975 4995 F. 24, 15.29 || 15.25 56.44 56.29 5014 50.34. Sa. 25 | 15.21 | 15.17 , 56.14 || 55.59 ± 5053 , 6072 Sun. 26 15.13 | 15. 9 || 55.45 55.31 5090 || 5108 M. 27 | 15, 5 15. 2 : 55.18 55. 5 5125 || 51.43 Tu. 28 || 14.59 || 14.56 54.53 54.42 : 51.58 || 517.3 W. 29 || 14.53 || 14.50 64.32 54.23 : 5.186 || 51.98 Th. 30 || 14.49 || 14.47 54.16 || 54.10 5207 || 52.15 F. 31 || 14.46 || 14.45 : 54. 6 54. 4 $221 5223 –mºmº 70 TABLE LV. ł JW'autical fllmanac. VIII. AUGUST 1821. Igº zgºgg9I 03°01’98 | 93'89-'99 | 88° 97'88 | 29 **$'OŤ || 6I º Sgºzº | zºº II · # # | 2 · 6 · 9# | ggº gły， Zſ,ÇI 29°98’ 67 | 8I’93’IŞ | 98° ŞI’89 | 09’ I ‘99 || 89°6'ſ '99 || 6ç， Zgº 3ç | gç， çgº 09 | 69. gſ. ºgg#Iºspaſy o 9 I“ I “†9 | Iž“8ý“ 99 | 99° 98° 29 || 0 ‘83” 69 || 19 · 6 · 12 | 63 · 9çº z2 | zgº ziņºſ, 2 | g • 63 · 92ŞIș 99 'ſ I'82 | 79° 0 ° 08 | 9çºçº º I8 | O ‘Igº g3 | - - -* = -<！-- - -->- - -Ź, I «2»ZI“ Zºº & || I ’33’97 || 6ç* 9çº 97 || OI º zgº 37 | zgº Z * OgŹI [ '87° 19 | 98°8’I ’89 || II **9°79 | 8Ť’63’99 || 61 · 9 · 89 || gſ， Oſ， 69 | 9 · 9I · Ig | GI · Ig - ggIIºſseſſºaſ p ØØ ‘93” （9 | ŞI’I ’99 || 99” Ç9' 29 | Igº OI º 69 || - - ---★ → •■ ■ ■- - -OI *3Ğ°03′87 | 08° 8′67 | Zſº 9I 'Ig | z I • ç#ºgg | gſ, º gI ， †çOI Aſ ī || Ž9°01' 29 | 18°68’89 | 9 · 8 · 09 || 99，99 · 19 || I · ç · 99 || īģ· ģg-ſg | ğę·ī7 · 99 || š'qneųIeuoĀ QŽI 63° 49 | 98° 29′89 | 33°93’ 02 | 0 ‘89” IZ || 23°03′92 || 97° 27' （2 | 35 · și · 92 | Ģ· ſ-228M. 9Ğ’8 º 62 | 29 ° 79°08 || ZI’I ’38 | †gº Zgº g3 || - - -… • -- - ，s• • •■4 IĞ‘83” I9 || 98° 98°39 | 9 ° 09’ 99 || Ziff * g * Ç9 || gſ; ， ZI · 99Z 09° 18′ 29 | 9_°9′′89 | 08'0 '02 | † ’9 I“ IZ || $7，6&， &2 | 3z"#ff， g2 | g[ · 59 ·#2 || Ōi-ſí-579·æ[ſnby o II ’6őº 24 || II****82 | ŽI’69’62 || 6 I’WI’ 18 | 93' 63，8 | zººřſ， g3 | 5g， 59 · †g || 5#-##-539 Iſ; * 9ç* 099 88°23' 38 | 33′99° 88 | 6 ‘93” 98 | 79'89-'99 || Zgº zgº 39 | Izº Iç• 69 | † • Oz • IĘ | ģ#·ğ#·ğř,† 88° ZITÝŽ | 03° 97'97 | 6 - 9 Iº 27 | I **** 8Ť || 9çº & I“ Og | 9ç• Iſº Ig || 0 · Iī · çç | ğ ·õ#·řç9søJēļuV 33:6 （99 || №’88° 29 | 8 · 8 · 69 | IŤ' 29' 09 || Igº z ºzg || 6 · Zgº g9 | 9 · 2 · çg | ÎI · Žç-ggØ� S㺠2. ’89 | 67’ 28’ 69 || Võ ’8 º 12 | 8 · 68° 32 || 9 ‘OI’ſ Z | 5 · Iň · çZ | ZgºgI ， ZZ | 2ș· §. 32I 'S "W 'CI | 'S "W 'CI | 'S ' IN ‘CI | -8 ° W ‘CI ! 'S ' IN ‘CI | 's . IN ‘CI | 's · IN ‘CI | · s · f） ºg*Sºude N ‘SÁøOISit}}S *([[XX"UȚIII AX'QAX*?\/?\upſ/A^*ųXI*UȚIA*UĻIII*wooAr "{{SIH ™IO LSVGI SWIV LS WOH， CINV ‘Nns WoºHJ HQJ NGO S. NOOIN JO SĢIONVLSIGI TABLE LV, ^~ JW'autical fllmanac. AUGUST 1821. IX. ? ţI * ## * #9I ”S 9ff º gI ， 99 || 6 Iº IŤ“ ZS | çç• 6 • 69 | 39 ’89.” Off || &T º 2 * & * | 9çº çSº Sy | IŤ ** *9* | IS ‘99”.9†[8 çº, ºg • 957 | †& ’ IS ‘67 | 62; • O • Ig || 88’ 63° 39 || 89 · 89°89 || VI º 8&º 99 || ° 29’99 | 9 Iº 23°8908ºsaugļu V Zç* 9ç* Sg | 97°93’ I9 | Øſ; * 9ç* 39 | 97°93’ț79 || ~ ~ ~• • •• ſ） →• © ®63 ȘI · I · Off | †gº zgº IŤ || 8 ** * $ſ; † 99” Ç9 ***ý | Içº Z * 97 || 0 ° 07′ ZŤ | zgºz I º 6783 ççºiſſ; * Og | IŤ“ ZI“ (39 | 69°09” Çç | 09’ S， ’99 | ŞI’ 29’99 | 87°09′89 | 99’’， ‘09 | 99° 89°I933 9 ſº z I º 99 | ŞI • Ziff ºff9 || Lç* Igº 99 || Tºyº 99° 29 || 77° 18′ 69 || 69 º 9 * IZ || 83° Øſ; * 32 || 6 • 8I?? ZIŻ,� g º ffyg º çZ | OI_° 09’ 22 | 09º 9 º 62 | 8 º 8，7° 08 || 67° 6'Iº 2,8 || Ziff º 99 · 88 || 89 · 88° 98 || 2:2, º II º 2803unS 9ųJL 6çº 957 º 88 | 87°93’ 06 | 09 *f， * 36 | 9 ’87 º 86 || 38° 13’ 96 || @I ‘O º 26 | † º 68° 86 || 8 * 8I° 00'I | 61 †gº Zç* IOL | Içº 98° SOȚI OS’9 Iº ȘOI| 03° 99° 9OI LØ ”99 ’80I| .39° 9I“ OII || 79 · 99 º III| 93° 29′ S II || 8I 3 º 31 ° çȚI| 69 º 89°9'II| 69 º 68° 8'[[| 8 * I3* 0& Iſi - - -• • •&= ±∞, ∞• • * *ZI Z * 98° 0903� IŤ * 6I ºgg || 6g'ſ ºff9 || IŞ° 6'ſ '99 | 97° 579 º 29 || VI° 03′ 69 | ſg º ç • Ig | Ziff º Iç* 39 | 9ç* 29 * #96Ț*Xn][OŁ OI ºțgº 99 || Oſſ; * OI º 89 | 03° 29' 69 || 8 Iºff º I2 ſ + - -ſ-º • №s* （ ） •• • •8{ 33° √3* 08 ſ 08° 57 ° 38 | Igº çiff º 99 | #çº 92, º 99 | Z * 6 * 298I ççº Içº 89 | 9 Iº çgº Oý | 9 °6'[' &ý | 83° 8 ºff || I '87° 957 || 0 - ggº Zºff | Oz, º 3 Iº 6Ť || Zç* & * IgZſºue.leqĐpĮV 0çº 67° 39 || 29 · 99° ºg | 9 IºØſ, ’99 | 67°8 '89 || 89'99 º 69 || Zgº ziſ; ' I9 || Oz, º 63° 99 | çº, º 9 Iº 999}[ S ‘IN ‘CI | S ‘IN ‘CI | ''S ‘IN ‘CI | ‘S “IN ‘CI I 'S 'IN ‘CI | 'S ' IN ‘CI | 'S "W ‘CI | 'S ‘IN ‘CI·skeq | ſººſ N ·ųIxx |‘QIII AX‘qAx | '7ņāņupīAr*UIXI* UȚIA*UĻIII‘uoq^^SJ（8ļS 72 TABLE LV. JWautical Almanac. AUGUST 1821. 9 I“ Off • OOI I ÇI 3 ' I9"86 || 89” I “26 | 09’ OEI ’96 | 99°93’ 96 || 7 *ggº I6 | çg’97“ 68 | Içº Zgº 28 | 62 · 5 · 99 { #t ZI’IĞŤ8 | 2，88° 38 | 63° 9′′08 | 99° 29’ 82 | 08’OI · ZZ || && && （ 92 | 63，99° 92 | Oç• 57， IZ | g[ | sºugļuç 23，8 ° 02 || IĞ” ZI’89 | IS’ IS ‘99 || 69'97"#9 || 97°0 · 99 | 0çº ç'I' I9 | ŞI • Igº 69 | 9ç， głº „g ſ ží 69'ſ '99 | 33°6'I ‘79 || 9 · 98°39 | OT '99-'09 || 99° OI º 67 | Çgºgg， ZŤ | Igº 97° çŤ | I · ç ·## | ſit 89°8õ’ŒŽ | 9 ºgſ， Oſſ | & & · 68 | 07' &&' 28 i 0 ºg ſºgg | zººg • ſg | 97， †gºzº | grºgººog | or Z8* 0ý ’92, i OI 9Ğ'$ （94 | 28’’žō’82 | 0Īº Zºº L2 | 9 ‘OI’02 | 03’-99° 89 || Zç* 9çº 99 || çç° 02' 99 || II · ç† • çg | š ȘŽIŞ 39 | 9ŻŻ8:09 | $_0_（6$ | OŽ（9329 | 98， IS ‘99 || 8Ť“ ZI’79 | 61'ſºzg || 8 · II · Ig || 3 | x, eoţds ŽIŤ 88:6Ť | 29°9 : 87 | ĢI’88-‘9Ť | 01’ I ‘97 || Ogº 64° 9′ | ±± 29. Iſ | zg· 9z · Off | ÇI” çç· 39 | 7 IĞIŽĞİ 28 | 68° 89'98 || 0 I’83， † 8 || 89°39′39 || Zſºzzº Ig | g9 º zgº 6g | Z - ggººg | gę， gg · 93 | 9 8 **õ’93 | 3纞çºgg | gººggºgg | ZŤ'99-'02 | - - - | - - - | - - - | - - - - || $ I “SI” IŻI| 93° 97°6II| .8 ° 03’8II| 0 - †g · 9II || 9 ŽI’83， ȘII| ŽŽ’ſ ‘’’II| 08’ 28’， II| 38 ºg Iº III || 67° 27′ 60|| 6 Iº gz '901|| ? * 6ç· 901] © · çç-çõt | 7 ȘI ȚIŽO Į | 68°27′30īļ ȘI **Ğ’IOT | & ’ I ‘00 || 0 · 89'96 || Z. "ÇI （ 26 | yzºzº º 96 || 0ç• 53， #5 | 9 Ķ4.3$ | 9. §Ž:ſſ | $9.33:06 | 1970 （68 || 89°38′28 || 0 ° 21'98 || «I’99**8 | 8żºğšº gē | g | runs otſi, 8ŻIII， 38 | 0I 09’ 08 | 98° 83' 62 | 8 · 2 · 82 || gs” gº '92 || I - †gº çZ | Oçº z * #2 || 59 · 55-37 | # 23:6 [[4 || 89° 29′ 69 | 2L '98-'89 | 68** I * 29 || Zçº zgº ç9 | z I - Igºț9 | zg· 6 · 99 | ğž· Ž#·īg | g $3|$3.209 | 83°S º 69 | & I“ IŤ， 29 | 79°8’I ’99 # 63，9çºğ9 | 29 · 99° 99 | 9 Iº II ºgg || 33 ºgſ- og | ğ I8°93’ 67 | 73°3 : 87 | 8 · 68° 9′ | Øſºg Lºgº || 9 *ggº gº | 6I-gż-żţ | Ogºſ ·ī£ | ff •õ#·šį | \ ’S 'IN ‘CI | ‘S ‘IN ‘CI | 'S "W 'CI | 's · IN ‘CI ! 'S ‘IN ‘CI | 's · IN ‘CI | 's · IN ‘CI | S -IN -qÁ'Saue N *SťCI> -*. ''TIXX ' 'TITIA X_I_°CIA X_I ºſºittp:ſw | ‘qXI‘UȚIA ! ' UNITI‘uoqĄº„SŁt}} S Hºffſ JO LSZIÁM SHVJ.S WOŁGI CINY. ‘Nºns INOH3 HAL NGO S. Noow sio sºon vision ;* TABLE, LV, 73 JWautical Almanac. AUGUST 1821. XI. †gº ºg º Sg | I - S 9 · 39°39 | Þý“OI’IÇ | 13’65’’ 67 | 29* 23°8′ | 08’9 · 257 || 0 ‘gº’ gº | Zgº czºł+ \ öğ- † .© | ° UIR) 6 ° Oºſ; * Ify | ޺, º 8I º Oț7→ =•* •«-» «） *）（.*• → *，* • •09|çº}}S 3 QJ）. 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I | OI/2T | 89998 || I I886 || 0000I | 3620I | [92II | OIOSI | IAAj/I 3822 I | 00000 / 10 O&# 098 || 008 || Of'& OSI 0&T | 09 O | 1.4 19 9. 157 18 1% * I /0 ‘suſly.up.50T (pops!.50T *IIAT GITQIV, L A4 I928 09.28 6928 82.28 A828 0000I &IOOI 78,00I 9900ſ 6700I 3620ſ 9080I Ić80I GS80T 0980I I92. II 622. II 262. II 9I8II #83 II OIOSI WSOSI 6908I 8808T 80IST IZZ7I 808?I ##9; I T887I 8I67I Č822. I ç982. I 6362 I #008T I808I 96.28 GO88 #I88 $73,88 8888 I900I 8200I 9800I 8600I OIIOI 998OI 0880I 768OI 60601 #360I 398 II IZ8II 688 II 806II 236TI 88TSI 89 ISI 88ISI 8038. I 8838. I 9967 I 766; I 3809 I TA.09.I OIIST 69 [8][ 6838I 03881 8078.I A878.I X I988 I988 # 0288 62.88 3,3][OI GSIOI 27IOI 09 IOI 32 IOI 6860ſ #960I 6960I #860I 6660I 976 II 996 II ‘786II 800&I 3303, I 6938. I #838T OISSI 9888.I 39SSI 67 ISI 68 ISI 63,39T 6939T OISGI 82.98.I I998.I I928 I XI 98.68|I 8888 8688 A.068 AI68 93.68 G8IOI 26TOI OI&OT 8330I 98&OI g|IOII OSOII g;/OII I90II 920 IT I?0&I I90&I O80&I 660&I 6II&I 8888.I GI?SI I?pg|I 8978.I 96% SI IGSGI 8689T 995/9I 2279 I 0399. I I806.I 8&I6I 8666. I I886.I 98.76 I g868 g?68 #968 j968 8268 87&OI I9&OI 723OT A830ſ 0080ſ I60II AOIII 8& III 88 III j/9III 68I&I 69 I&I 82. I&I 86 ICI 8IóðI 339.8T 6798 || 92.98.I #099 | 3898.I S999'ſ 2099 I I999 I G69SI 0.729 I `. I Ø996. I 99.261 I886.I 00003, 8868 3,668 3006 ł zºos I306 I806 SISOI 9380I 6880T Č980ſ G980ſ 828OI 0ZIII 98.III IO&II 2I&II 883, II 67&II 683&I 696&I 6233ſ 008&I O38& I I#8&I 0998.I 8899. I 9IZSI gizZSI 8.228 I 3089. I 98.29 I 6889 I 8/89 I G&69 I 82.69 I Ió09 I 3&IO& 87&0& 82.80% 3T903 67906, 36203, //,8 O3% 008 0% 08 I 0&I 09 0 19. riff 18 13, | I 10 ‘sutilitanºot ſpºusgå0T 78 TABLE LVII, Logistical Logarithms. Tº 8. 9/ 10/ 11’ 12/ 13/ 14/ 15/ 16/ 480 || 540 600 | 660 | 720 780 | 840 | 900 || '960 O// 8751 8239 7782 | 7368 6990 | 6642 6320 | 6021 5740 1 8742 8231 7774. 7361 6984 6637 || 6315 6016 || 5736 , 2 87.33 8223 7767 | 7354 6978 6631 || 6310 || 6011 || 5731 8724, 8215 7760 | 7348 6972 6625 6305 || 6006 || 57.27 87.15 8207 || 7753 | 7341 6966 | 6620 | 6300 || 6001 || 5722 8706 || 8199 || 7745 | 7335 | 6960 | 6614 || 6294 || 5997 || 5718 8688 || 8183 || 7731 || 7322 /6943 | 6603 6284 || 5987 5709 8679 || 8175 || 7724 | 7315 6942 6598 || 6279 5982 5704 8670 8167 || 7717 | 7309 || 6936 || 6592 || 6274 5977 5700 J 4. 5 6 8697 || 8191 || 7738 || 7328 |,6954 | 6609 || 6289 || 5992 || 5713 7 8 9 § 10 || 8661 | 8159 || 7710 | 7302 || 6930 6587 | 6269 || 5973 || 5695 11 86.52 || 8152 7703 || 7296 || 6924 6581 | 6264. || 5968 5691 12 || 8643 8144, 7696 || 7289 || 6918 6576 6259 || 5963 5686 13 86.35 | 8136 7688 || 7283 || 6912 || 6570 | 6254 5958 5682 14, 8626 8128 7681 | 7276 || 6906 || 6565 | 6248 || 5954 5677 15 8617 | 8120 || 7674 || 7270 6900 || 6559 || 6243 5949 5673 16 8608 8112 || 7667 | 7264. 6894 || 6554, 6238 || 5944 5669 ! 17 | 3599 || $104 || 7660 | 7257 | 6888 || 6343 | 6333 || 3339 || 3664 18 8591 8097 7653 || 7251 | 6882 6543 6228 5935 5660 19 8582 8089 || 7646 7244 | 6877 || 6538 6223 5930 5655 .20 8573 8081 7639 7238 | 6871 6532 6218 5925 5651 | 21 | 8565 8073 || 7632 | 7232 | 6865 6527 | 6213 || 5920 ! 5646 22 || 8556 8066 || 7625 | 7225 | 6859 6521 6208 5916 || 5642 23 8547 | 8058 || 7618 || 7219 | 6853 || 6516 6203 || 591.1 : 5637 24 || 8539 8050 || 7611 || 7212 | 6847 || 6510 || 6198 || 5906 || 5633 25 | 8530 8043 7604 || 7206 | 6841 || 6505 || 6193 5902 || 5629 26 || 8522 || 8035 7597 | 7200 6836 6500 6188 || 5897 5624 27 | 8513 || 8027 || 7590 7.193 || 6830 || 6494 6183 5892 || 5620 28 || 8504 || 8020 || 7583 || 7187 | 6824 || 6489 || 61.78 || 5888 5615 | 29 | 8496 || 8012 || 7577 | 7181 | 6818 6484 || 6173 || 5883 || 5611 30 | 8487 8004 || 7570 717.5 ! 68.12 64.78 || 6168 || 5878 || 5607 TABLE LVII. 79 Logistical Logarithms. { 8/ 9/ 10/ 11/ 12/ 14/ 15/ 16/ 480 540 600 660 720 840 900 960 8487 84.79 8470 84.62 8453 8445 8004, 7.997 7989 7981 7974. 7966 7570 7563 7.556 7549 7542 7535 7175 71.68 7162 71.56 7149 7143 6812 6807 6801 6.795 6789 6784. 6168 61.63 6,158 6153 6148 6143 5878 5874, 5869 5864, 5860 5855 5607 5602 5598 5594, 5589 5585 } 8437 84.28 84.20 84.11 8403 7959 7951 7944, 7936 7929 7528 7522 75.15 7508 7501 7137 7131 7124 7.118 7112 6778 6772 i. 6766 6761 6755 61.38 6133 6128 6123 61.18 5850 5846 584.1 5836 5832 5580 5576 5572 5567 5.563 8395 8386 8378 8370 8361 7921 7914. 7906 7899 7891 7494 7488 7481 7474. 74.67 7106 7100 7.093 7087 7081 6749 6743 6738 6732 6726 61.13 6108 6103 6099 6094 5827 5823 5818 5813 5809 5559 5554, 5550 5546 5541 8353 8345 833 8328 8320 7884, 7877 7869 7862 7855 7461 7454. 7.447 744], 7434 7075 7069 7063 7.057 7.050 6721 6715 6709 6704. 6698 6089 6084 6079 6074 6069 5804. 5800 5795 5790 5786 5537 5533 5528 5524, 55.20 8312 8304 8296 8288 8279 7847 7840 78.32 78.25 7818 7427 74.21 74.14. 7407 7401 7044 7038 7032 7026 7020 6692 6687 6681 6676 6670 6064. 60.59 6055 60.50 6045 5781 5777 5772 5768 5763 5516 5511 5507 5503 5498 8271 8263 8255 8247 8239 7811 7803 7796 77.89 77.82 7394 7387 7381 7374, 7368 7014 7008 7002 6996 6990 6664. 6659 6653 6648 6642 6040 6035 6030 6025 6021 5758 5754 5749 5745 5740 5494 5.490 5486 5481 5477 80 TABLE LVII. Logistical Logarithms. 17/ 18/ Ig/T T20, TT2. 22/ 1020 1080 1140 1200 1320 0// 5477 54.73 5469 54.64 5460 5456 5229 5225 5221 5217 52.13 5209 4994 4,990 4986 4,983 4979 4975 4771 4768 4764 4760 4757 475.3 4357 4354 4351 4347 4,344 4341 54.52 5447 : 5443 5439 5435 5205 5201 5197 5193 5189 4971 4967 4964 4,960 4.956 47.50 4746 4742 4,739 4735 4.338 433 4331 4328 4.325 23/ 1380 4164 4.161 4,158 4155 4152 4149 *** * 4.145 4142 4,139 4136 41.33 24/ ſ ! –- 25/ 1440 1300 **------ ! 3979 3976 3973 3970 3967 $964 J802 3799 3796 3793 37.91 S788 3961 3958 3955 3952 3949 3785 37.82 3779 3775 3773 54:30 5426 5422 54.18 5414 518.5 5181 5.177 5173 5169 4,952 4949 494.5 494.1 4937 4732 4728 4724 4721 4717 4.321 4318 4315 4311 4308 4130 4,127 4,124 4,120 4,117 3946 3943 3940 3937 3934 3770 3768 3765 3762 37.59 5409 5405 5401 5397 5393 5165, 516.1 5157 5153 5149 4933 4,930 4,926 4922 4,918 4,714 4,710 4707 4703 4699 4,305 4302 4298 4295 4.292 4114 4,111 4.108 4,105 4102 3931 3928 3925 3922 3919 3756 375.3 3750 3747 3745 5389 5384 5380 5376 5372 5145 514.1 51.37 5133 5129 4.915 4,911 4907 4903 4900 4,696 4692 4,689 4685 4,682 4289 4285 4282 4.279 4.276 4,099 4096 4,092 4089 4086 3917 3914. 3911 3908 3905 3742 3739 3736 3733 3730 5368 5364 5359 5355 5351 5125 5122 5118 5114 5110 4,896 4,892 4889 4.885 4.881 4678 4.675 4.671 4668 4664 4273 4.269 4.266 4.263 4260 4083 4080 4077 | 3902 3899 3896 4074 3893 4071 3890 37.27 3725 3722 3719 3716 TABLE 84 LVII. Logistical Logarithins. 17/ 18/ 19/ 20/ 21/ 22/ y/ J 24/ 1020 1080 1140 1200 1260 1320 1380 1440 5351 5347 5343 5339 5335 53.31 5110 5106 5102 5098 5094. 5090 4881 4877 4,874, 4870 4.866 4863 4664 4,660 4657 4653 4,650 4646 4457 4454, 4450 4447 4444 4440 4260 4.256 4.253 4.250 424.7 4244 4071 40.68 4065 4,062 4059 4055 3890 3887 3884, 3881 3878 3875 5326 5322 5318 5314. 5310 5086 5082 5079 5075 5071 4859 4.855 4852 4848 4844 4643 4,639 4636 4632 4,629 4437 4434 4430 44.27 44.24 4.240 4,237 4234 4231 4.228 4052 4049 4046 4043 4040 3872 3869 3866 3863 3860 5.306 5302 5298 5294, 5290 5067 5063 5059 5055 5051 484.1 4837 4,833 4,830 4826 4.625 4622 4618 4615 4611 4420 44.17 4414 44.10 4.407 4224. 4221 4218 4215 4212 4037 4034 4031 4028 4025 3857 3855 3852 3849 3846 5285 5281 5277 5273 5269 5048 5044 5040 5036 5032 4,822 4819 4815 481.1 4,808 4,608 4604, 4601 4597 4594 4.404 4400 4397 4394 4390 4209 4.205 4202 4,199 4,196 4,022 4019 4016 4013 4010 384.3 3840 3837 3834. 38.31 5265 5261 5257 5253 5249 5028 5025 5021 50.17 5013 4804. 4800 4797 4793 4,789 4,590 4587 4584, 4580 4577 4.387 4.384 4.380 4377 4,374 4193 4,189 4.186 4.183 4,180 4007 4004 4001 3998 $995 3828 38.25 3822 3820 3817 5.245 5241 5237 5233 5229 5009 5005 5002 4,998 4994 4.786 4782 4778 4775 4771 4,573 4,570 4,566 4.563 4.559 4,370 4367 4364 4.361 4.357 4.177 4174. 4,171 4,167 4164 3991 3988 3985 3982 3979 3814, 3811 3808 3805 3802 * { { 82 TABLE LVII. Logistical Logarithms. 26/ 37 28/ 29/ 30/ 31/ 32! 1560 1620 1680 1740 1800 1860 1920 */ 0 3632 3629 3626 3623 36.21 3618 34,68 3465 3463 3460 34,57 3454 3310 3307 3305 3302 3300 3.297 $158 3155 3153 3150 3148 314.5 3010 3008 3005 3003 3001 2998 2868 2866 2863 2861 2859 2856 27.30 2728 2725 2723 2721 2719 1 3615 361.2 3610 3607 3604 34.52 3449 3444 3441 3.294, 3292 3289 3287 $284 3143 3140 3138 3135 3.133 2996 2993 2991 2989 2986 2854 2852 2849 2847 284.5 2716 2714, 2712 2710 2707 ; : } : 3601 3598 3596 3593 3590 3438 34.36 34.33 3431 3428 3282 3279 3276 $274, 3271 3130 3128 3125 3123 3.120 2984, 2981 2979 2977 2974 2842 2840 2838 2835 2833 2705 2703 2701 2698 2696 i : 3587 3585 3582 3579 3576 3.425 3423 3420 34.17 34:15 3269 3.266 3264 3.261 3259 31.18 31.15 31.13 31.10 3.108 2972 2969 2967 2965 2962 2831 2828 2826 2824. 2821 2694 2692 2689 2687 2685 3574 3571 3568 3565 $563 34:12 3409 3407 3404 3401 3.256 3253 3251 3.248 3246 3105 3103 3101 3098 3096 2960 2958 2955 295.3 2950 2819 2817 2815 2812 28.10 2683 2681 2678 2676 2674, 3560 35.57 3555 35.52 3549 3.399 3396 3393 3.391 3388 3243 3241 3.238 3236 3233 3093 3091 3088 3086 3083 2948 2.946 2943 2941 2939 2808 2805 2803 2801 2798 2672 2669 2667 2665 2663 TABLE LVII. 83 Logistical Logarithms. 26/ 27/ 28/ 29/ 30/ 31/ 32/ 34/ 1560 1620 1680 1740 1800 1860 1920 2040 3549 3546 3544 3541 3.538 3535 3388 3386 3383 3380 3378 3375 3233 3231 3228 3225 3.223 3220 3083 3081 3078 3076 3073 3071 2939 2936 2934 2931 2929 2927 2798 2796 2794 2792 2789 2787 2663 2660 2658 2656 2654 2652 2403 2401 2399 2397 2395 2393 3533 35.30 3527 3525 3522 3372 3370 3367 3365 3362 3.218 3215 3.213 3.210 3208 3069 3066 3064. 3061 3059 2924, 2922 2920 29.17 2915 2785 2782 2780 2778 2775 2649 2647 2645 2643 2640 2391 2389 2387 2384 2382 3519 3516 3514 3511 3508 3359 3357 33.54 3351 3349 3205 3203 3200 31.98 3195 3056 3054 3052 3049 3047 2912 2910 2908 2905 2903 2773 2771 2769 2766 27.64 2638 2636 2634 2632 2629 2380 2378 2376 2374. 2372 3506 3503 3500 3497 34.95 3346 3344 3341 3.338 3336 31.93 3190 3.188 3.185 3183 3044 3042 3039 3037 3034 2901 2898 2896 2894. 2891 27.62 2760 2757 2755 2753 2627 2625 2623 2621 2618 2370 2368 2366 2364 2362 3492 3489 3487 3484 3481 3333 33.31 3328 3325 3323 3.180 3178 $175 3173 3170 3032 3030 3027 3025 3022 2889 2887 2884. 2882 2880 2750 2748 2746 2744. 2741 2616 2614, 2612 2610 2607 2359 2357 2355 2353 2351 34.79 34.76 3473 34.71 3468 3320 33.18 3.31.5 33.13 3310 3.168 3165 3.163 3160 3158 3020 3018 3015 3013 3010 2877 2875 2873 2870 2868 2739 2737 2735 2732 27.30 2605 2603 2601 2599 2596 2469 2467 2349 234.7 2345 2343 2341 *IIAI GIT IV.I. 86%I I09I ( 202I | 9TSI Z36ſ [#03, 69 I& 667 I | 809 I | 60ZI AISI 636|| || Sp0& [9][& I09I 909ſ II/I 6L8 I I86 I | g;0& | 89 I& 809 I 909 I &IZI I38T | SS6 I Aft|O& Q9I& #09 I 809 I | # IZI | 838 I | 786.I 670& Z913, 909 I | 0I9I 9TAI Gö8I 986.I | IQ0& 6913, 809 I &I9I | 8 IAI 2&SI | 886.I | 890& | 0ZI& 0ISI SI9I 6L2T 8&SI | 076I 99.03, &ZI& IIGI 9 [9ſ I&A.I OS8T & F6][ 2908, 72.I& SIGI A.I.9 I | 832. I ČSSI #76 I 690& 92.I& GIGI 6I9| || 932. I | WSSI 976T | IQ02, 8ZI& 9IGI | 039 || || Z32I 988.I 876 I &903, 08T & 8ISI 339|| || 832T | 888 [ 096 I 7903, &SI& 0&SI | 739T. 08/I | 688 I I96I 990& 78 [3 339 I | 9391 || 382. I | [78I | 896 I | 890& 98.I& S&SI | 239T | 782T | SW9I 996 I | 020& | 88I& 939 I | 6391 || 982I gp8 I 296 I 3203 || 06I& A39 I IS9I 2S2I Zī/SI 696 I | 7203 || 36|Ić 839 I 889T | 68ZI 6781 I96I 9203 || 76I& OSGI 789 I I?AI 093I | 896 || || 820& 961& 389 I 989 I SF/I & 98.I Q96'ſ 0803 || 8613 $789 I 889 I | 972.I 798 I 296 I 3803, 0033 GSGI 0.79ſ 972.I 998 I | 896 I | #803 || 3063 Z8GI | [j,9| || 87/I | 898 I 0.26T | 980& | 7063 689 I | SjøI | 09/I | 098 I | 326|| || 880& 9066 ORQI GP9 I 39.2I C98I | 726L | 0603 || 8033 379 I 279 I | #92. I | 898 I | 926. I | 3603 || 0I63 779 I 8F91 gg/L | 998I | 826T | #60& &I63 979 I 099 I | 29 ZI Z98I | 086T 9603 '7I63, 279 I &99T | 69ZI | 698 I ČS6I 8603 || 916& 67g I 799 I | [92I [28T || 786 I | 660& 8I& 0&gó, 097& 007& 0783, 083& | 0363 || 09 I& /&P /Ijz /07 /68 /88 /28 /99 'stuply.up#OT polls?50T TABLE 85 LVII, Logistical Logarithms. 38/ 39/ 40/ 41 4.2/ 43/ 2280 2340 2400 24.60 2520 2580 1927 1925 1923 1921 1919 1918 1816 1814 I812 1810 1808 1806 1707 1705 1703 1702 1700 1698 1601 I599 1598 I596 1594, 1592 1498 1496 1494 1493 1491 1489 1397 1895 1893 1392 1390 1388 1916 1914 1912 1910 1908 1805 1803 1801 1799 1797 1696 1694, 1693 1691 1689 1591 1589 1587 I585 1584, 1487 1486 1484 1482 1481 1387 1385 1383 1382 1380 1906 1904 1903 1901 1899 1795 1794 1792 1790 1788 1687 1686 1684 1682 1680 1582 1580 1578 1577 1575 1479 1477 1476 1474 1472 1378 1377 1375 1373 1372 1897 1895 1893 1891 1889 1786 1785 1783 1781 1779 1678 1677 1675 1673 1671 1573 1571 1570 1568 1566 1470 1469 1467 1465 1464 1370 1368 1367 1365 1363 1888 1886 1884 1882 1880 1777 1775 1774, 1772 1770 1670 1668 1666 1664 1663 1565 1563 1561 1559 1558 1462 1460 1459 1457 1455 1362 1360 1359 1357 1355 1878 1876 1875 1873 1871 1768 1766 1765 I763 1761 1661 1659 1657 1655 1654 1556 1554, 1552 1551 1549 1454. 1452 1450 1449 1447 1354 1352 1350 1349 1347 86 "l'AlèILE LW 11. Logistical Logarithms. TT 44/ 45' 4.67 47/ 48/ 49/ 50/ 51/ 537 2640 2700 2760 2820 2880 2940 || 3000 3060 3120 0// 1347 | 1249 1154, 1061 969 || 880 || 792 || 706 || 621 1 1345 1248 1152 1059 968 878 790 704 || 620 2 || 1344, 1246 1151 || 1057 || 966 || 877 || 789 || 703 || 619 3 || 1342 | 1245 1149 || 1056 || 965 : 875 || 787 702 || 617 4 || 1340 | 1243 | 1148 || 1054 || 963 874 786 700 616 5 § 1339 1241 1146 1053 962 872 785 699 || 615 6 || 1337 1240 | 1145 || 1051 960 871 783 || 697 || 613 7 || 1335 | 1238 || 1143 1050 959 869 || 782 696 || 612 8 || 1334 || 1237 || 1141 || 1048 || 957 || 868 780 694 | 610 9 || 1332 || 1235 | 1140 || 1047 956 || 866 || 779 || 693 609 10 || 1331 || 1233 || 1138 || 1045 954 || 865 777 | 692 608 11 || 1329 || 1232 || 1137 || 1044 953 || 863 || 776 690 || 606 12 || 1327 | 1230 | 1135 | 1042 | 951 862 || 774 | 689 605 13 || 1326 || 1229 || 1134 || 1041 || 950 860 | 773 | 687 | 603 | 14 || 1324 | 1227 | 1132 || 1039 || 948 # 859 || 772 | 686 || 602 15 || 1322 || 1225 | 1130 || 1037 / 947 | 857 || 770 | 685 | 601 16 || 1321 1224 1129 || 1036 945 856 || 769 | 683 || 599 17 | 1319 || 1222 || 1127 | 1034 944 855 || 767 | 682 598 18 || 1317 | 1221 1126 || 1033 942 | 853 || 766 | 680 || 596 19 || 1316 1219 || 1124 || 1031 94.1 | 852 || 764 || 679 || 595 20 1314 || 1217 | 1123 | 1030 | 939 850 763 678 594 21 || 1313 || 1216 || 1121 || 1028 938 | 849 || 762 | 676 || 592 22 || 1311 1214 || 1119 1027 936 847 || 760 675 || 591 23 || 1309 || 1213 1118 1025 | 935 | 846 || 759 673 590 24 || 1308 || 1211 1116 || 1024 || 933 844 || 757 | 672 588 25 || 1306 || 1209 || 1115 1022 || 932 843 || 756 || 670 587 26 || 1304 || 1208 || 1113 || 1021 || 930 | 841 754 | 669 585 27 | 1303 || 1206 || 1112 || 1019 929 | 840 || 753 | 668 || 584 28 || 1301 || 1205 || 1110 || 1018 927 | 838 || 751 | 666 || 583 29 || 1300 || 1203 1109 || 1016 || 926 837 750 | 665 581 30 1298 1201 | 1107 || 1015 I 924 || 835 | 749 | 663 580 GITIEIWAL *IIAT A8 689 || I39 || 902 362 088 696 || IQOI | #9 II 67&I 09 079 || 839 202 862. I88 || IA6 || 390.I 99 II | IGöI 69 I?9 || 739 60Z 9.6/ 888 & 26 790I Z9 II | 89&T | 89 879 || 939 OI/ 962 788 || 726 990I | 69 II || 79&I Zg 779 || Z39 II/ 862, 988 || 92.6 || A901 || 09 II | 99&I | 99 979 || 839 SIA 662. A.88 || A6 8901 || 39 II | 29&I 99 Aftg 089 || 7 [Z I08 888 8/6 || 020I 89(I 69&I #9 879 || I89 || 9T.A. 308 O68 086 || IAOI 99 II | [9&T | 89 099 || 889 || ZIZ, 808 I68 I86 820I, 29II | 39&I &g I99 || 789 || 6 IA 908 868 886 || 720I | 89 II 79&T | Ig 399 || 989 || 0&Z 908 || 768 || 786 || 920I | OZII 99&I 09 799 || 289 || I&A. 808 968 || 986 820I | IAII A9&I | 6′y 999 || 889 || 8&Z 608 A68 Z86 || 620L | 8/II | 69&I | 87 A99 || 0%) #32, II8 668 686 T80I | WA II | 0/3, I Zī, 899 || I?9 || 93/. 2, I8 006 || 066 380I 9/II &/3, I | 97 699 || 379 Z32, j/I8 306 &66 || 780I | 84II | 72&I 97 I99 || 779 6&Z 918 806 || 866 980ſ 62II | 92&I #7 399 979 OS 9I8 906 966 280T | [8][I A&I | 87 999 || 2:59 I& 818 906 || 966 || 880 I 38 II | 8/3, I &? 999 || 879 || 882. 618 806 || 866 || 060ſ 78.II | 08&I I? 999 || 679 || 78/. Ić8 606 | 666 [60T 98II 38&I Off 899 || IG9 || 982, 338 II6 IOOL | 860 I | Z8II | 88&I | 68 699 || 399 || 28 #68 &I6 || 300|| || 960T | 68 II | 98&I 88 0/9 || 799 || 68Z. 968 7I6 || 700I 9601 || 06LI Z8&I A3 &/9 || 999 || 07/ A38 9I6 || 900I 8601 || 36|II | 88&I 98 82.9 || 999 || IWA, 82.8 ZI6 | 2001 | 6601 || 86II | 06&I g8 ‘729 || 899 || 872. 088 8I6 || 800I [OII | 96II | I6&I | 78 92.9 699 || 772. I88 036 OLOI &OTI | A6TI / 86&I | 68 Al/9 | [99 || 972. 888 Ić6 &IOI | 70II 86 II | 96&I | 33 62.9 || 399 || AyA 788 836 || 8 IOI 90II | 00&I 96&I IQ 089 || 899 || 67/, 988 7&6 || 9 IOI A0II I03I 863.I ſ/09 0&18 0908 || 0008 || 0766 0886 || 0686 || 0946 || 0046 || 0796 , 139 ZI9 109 /6? /87 /27 | 197 | 197 /77__I__ 'supljamä07 ſpous?.50T TABLE LVII. Logistical Logarithms. 53/ 54/ 55/ 56/ | 57/ 58/ 59/ 3180 || 3240 || 3300 || 3360 || 3420 | 3480 || 3540 0// 539 458 378 300 223 147 73 1 537 456 377 298 221 146 72 2 536 455 375 297 220 145 71 3 535 454. 374, 296 219 143 69 4. 533 452 373 294, 218 142 68 5 532 451 371 293 216 141 67 6 531 450 370 292 215 140 66 7 529 448 369 291 214 139 64 8 528 447 367 289 213 137 3 9 526 446 366 288 211 136 62 10 525 444 365 287 210 135 61 11 524 443 363 285 209 134 60 12 522 442 362 284. 208 132 58 13 521 440 361 283 206 13 57 14 520 439 359 282 205 13 56 15 518 4.38 358 280 204 129 55 16 517 436 357 279 202 127 - || 53 17 516 435 356 278 201 I26 52 18 514 434 354 276 200 125 51 19 513 432 353 275 199 124 50 20 512 431 352 274, 197 I22 49 21 510 430 350 273 196 121 47 22 509 428 349 271 195 120 46 23 507 427 348 270 194 119 45 24 506 426 346 269 192 117 44 25 505 424 345 267 191 116 42 26 503 423 344 | 266 190 115 41 27 502 422 342 265 189 114 40 28 501 420 341 264 187 112 39 29 499 419 340 262 186 111 3 30 498 4.18 339 261 185 110 36 TABLE LVIIe Logistical Logarithms. 53/ 54/ 55/ 56/ 57 587 59/ 3180 || 3240 || 3300 || 3360 | 3420 | 3480 || 3540 30" || 498 4.18 339 261 185 110' 3 31 497 416 337 260 184 109' 35 32 495 415 336 258 182 107 34 33 494 414 335 257 181 106 33 34 493 412 333 256 180 105 31 35 491 411 332 255 179 104. 30 36 490 410 331 253 177 103 29 37 489 408 329 252 176 101 28 38 || 487 407 328 251 175 100 || 27 39 || 486 406 327 250 174 99 25 40 || 484 404 326 248 172 98 24 41 || 483 403 324 247 171 96 23 42 || 482 | 402 || 323 246 || 170 95 || 22 43 || 480 400 322 || 244 169 94. 21 44 479 399 320 243 167 93 19 45 || 478 398 319 242 166 91. 18 46 476 396 318 241 165 90 17 47 475 395 316 23 163 89 16 ; 48 474 394 315 . . .238 162 88 15 49 472 392 314 237 161 87 13 50 471 391 313 235 160 85 12 51 470 390 311 234. 158 84 11 52 468 388 310 23 157 83 10 53 467 387 309 23 156 82 8 : 54 466 386 307 23 155 80 7 55 464 384 306 229 153 79 6 56 463 383 305 || 228 152 78 5 57 462 382 304 227 151 77 | 58 460 381 302 225 150 75 2 , 59 459 379 301 224 148 74. 1 60 458 378 300 223 147 73 || 0 #1 •IIIA'I GITISIVAL 06 gTgzTgTzTzTFETETETETETETJETETETTTTTzi 96 | 96 | 76 | 72, | 82, | 82 | 2,3 | 2,2, | Ig | IZ, | 03 | 02 | 6I | 6I | 8 | | 07 II 96 | 76 | 76 | 86 | 82 | 2,3 | 2,2, | Iz, 1 I2 | 02 | 03 | 6I | 6I | 8 I | 8 I | 02, II 76 | 86 | 86 | 2,2, i za, | ZZ | IZ | I2 | 02 | 02, | 6I | 6 I | 8 I | 8I I ZI | 0 II 82, f 86 | 66 | 66 | I2, I IZ, | 02 | 02 | 61 | 6 I | 6I | 9I | 9I | AI | ZI | 07 01 && | 66 | 66 | Ia I IZ, 1 02, | 06 | 61 | 6I | 6I I 8I | 8I | ZI | AI I 9 I | 03 01 &, | 16 | I3 | 03 | 02 | 02 | 6 I | 6 L | 8I | 8 I | ZI | ZI | ZI | 9I | 9I | 0 0 I I2, | I3 | 03 | 03 | 6I | 6I | 6I | 8 I | 9I | AI I ZI I ZI | 9I | 9I | 9T | 07 6 02, | 06 | 6 I | 6 I | 6I I 8I | 8 I | 8 I | ZI | ZI | 9I | 9I | 9I | g I | 9 I | 02, 6 6I | 61 | 6 I | 8 I | 8I I 8I | AI I ZI | 9I | 9I | 9I | çi | ÇI I ÇI | 7 I | 0 6 6I I 8I | 8 I | 8 I | AI I ZI | AI I 9 I | 9I | 9I | çI | çI | 7 I | 7 I | 71 | 07 8 8I I 8I | AI I ZI | ZI | 9I | 9I | 9I | çI | çI | ÇI | VI | 7 I | 7 I | 8 I | 04 8 ZI | ZI | AI | 9I | 9T | 9I | ÇI I ÇI I ÇI | jzI | f7I | f7T | 9I | 9I | ST | 0 8 ZI | 9I | 9I | 9I | çI | g I | çI | VI | VI | VI | 9I | gI | SI I ZI | ZI | 07 A 9I | 9I | 9I | 9I | ÇI | f7I | 7 I | 7 I | VI | g I | 9I | SI I ZI I ZI | ZI | 03 Z 9I I 9 I | 9I | 7 I | f7I | 7 I | 9I | 9I | 3 I | 9I | gl | ZI ( ZI I II | II | 0 A fI | VI | VI | FI | g I | SI I SI I ZI | gI | 2,T | zI | II | II | II i II | 07 9 7 I | SI I 8I | SI | g I | ZI I ZI I ZI I ZI | II | II | II | II | OI | 0 I | 06 9 SI I 8I | ZI I ZI | 2,I I ZI | II | II | II i II | OI | OI | OI | OI | 6 | 0 9 ZI | 6I I ZI | ZI | II | II | II | II • | OI | OI | OI | OT | 6 | 6 | 6 | 07 9 ZI | II | II i II | II | OI | 01 | 01 | 01 | 01 | 6 | 6 | 6 | 6 | 6 | 06 9 II | II | 0 I | OI | OI | OI | OI | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 0 9 .| OI | 01 | 01 | 01 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 9 | 8 | 8 | 2 | 07 7 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 9 | 8 | 8 | Z | Z | Z | Z | 06 7 6 | 8 | 8 | 8 | 9 | 8 | 8 | 2 | Z | Z | Z | 2 | 2 | 9 | 9 | 0 7 8 | 8 | 8 | 2 | 2 | 2 | 2 | Z | Z | Z | 9 | 9 | 9 | 9 | 9 | 07 8 Z | Z | Z | Z | Z | Z | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 06 8 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 0 8 9 | 9 | 9 | 9 | Ç | Ç | ç | ç | ç | Ç | 9 | 9 | 7 | 7 | 7 | 07 % g | 9 | G | g | g | g | 7 | V | W | 7 | 7 | 7 | 7 | 7 | 7 | 06 Z 7 | 7 | 7 | 7 | 7 | 7 | 7 | | | | | 7 | 9 | 8 | 8 | 8 | 8 | 0 & 7 | 7 | 8 | 9 | 9 | 9 | 9 | G | g | 8 | 9 | 8 | 8 | 8 | 8 | 07 I 8 | 8 | 9 | 8 | 9 | 9 | 9 | Z | Z | Z | Z | Z | 6 | 6 | 3 | 06 I «, | 6 | 3 | 2 | 2 | 2 | Z | Z | Z | Z | Z | Z | 6 | 6 | 6 | 0 I I | I | I | I | I | I | I | I | I | I | I | I | I | I | I | 07 0 I | I | I | I | I | I | | | I | I | I | I | I | I | I | I | 08, 0 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 0 “Uli * UUU * ULI "UU! 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Change in JMoon's Declination. , q}: o c <> | <> <> <> | <> Q <> {, <> Q <> | <> Q <> | <> Q <> | <> Q Q | <> © <> | <> | <> © <> | <> <> c h c c c c \, # | E º $ $ | ° §§CN <ří ſi Gº <ſGN， <ſGº <#1GNQ <#GN <říGN <#GN， <ſ§ § || Sºș Ș ;-（ | ）， CN Gº Gº | cº cº cº | <ſſ <: <ť | vº vº vº | so ſo so | ► ► ► | co co co | on on off | Q Q Q | ±± − × | QN QQ QQ || cº co go <ſ `----p={ ſ=\ y={y={ ſ=\ y={►| y={ x={y=-{ ſ=\ y=}►| P={ y--+ſ=\ y，\ y={w=\ y=， y={y=， y=， y={GNQ GNR GNQGNQ GNQ GNRGNQ GNR GNQGNQ GNR GNR GN. *…o – co | w-， CN || co <;， «o || N. CO OD | <> r） CN || C^ <ří vo || N. CO OA | Q ± QN || c^ <ři ſo | № co gº | c - GN || co <ſ， co | N） co do o <#r-+ +-+ +-+ | +-+ +-+ +-+ | +-+ +-+ +-+ | CNR GN GN | ON ON CR | ÓN GN GN | Go Cry čº || c^5 cò co | © cºp 5，5 <ři às | < o H CN || CN co <ſ> | \^ ºo N. || N. CO OD || C. H. GN | GN co <ſſ | Ar) so №. ! № co oo | Q ± GN | QR go sſ | Arp ºp №. ! № go où o croy=\ y={ P--;+-+ +-+ +={y={ x={ y--+r={ y-\ y=-{GNQ GNR GNGN. ON ONGNQ GNR GNGNQ GNR GNQ Gro > | < c +-+ +-+ | ON GN co | co <† <† || An to co | ~ ~ CO | Oo o， o | c -+ +-+ | CN Gº co | c^ <† <† | Ao to co | №, №, oo | off o) on Q GNQ{r=\ y={ ſ=-{y={ ſ={ ş-，y=\ y={ ++{ș-4 y={ ſ={y={ ſ={ ſ=-{y=\ y={ y--; GNR > | < o c +-+ | +-+ +-+ +-+ | Gº Gº GN | Gº co co | co <ſ> <ſt | <ſº <* A^ | A^ ^^ ºo | so so to | ► ► ► | ► co co | co od opo o sº x={ ■■■■ *-a，-ae caeae：№ſ < o \r> | \r) ∞ Ao | <> AQ © | Ar? Q \to | <> Ar> <> | ln Q \，) | © ® Q | Aq Q \p | c \co O || An Q \，) | Q An Q | \Q Q \Q <> &3y=łr-+ QQ GNRco co <ří<* Aro Aroy=|y-， GN GNRco co <#<Ť， Aro Ar>！{+--+ GN GNRco co <ſ<† Aro Ar) o <> | <> <> | o co | <> <> o \ y-+ +-+ +-+ | +-+ +-+ +-+ | +-+ +-+ +-+ | +-+ +-+ +-+ | CN Gº CN || CN GNR GNR || CNR ON GN || CNR GNR GNR GO cº)O £^ NJ | O CO №. ! o co N - || C. CO N. | O CO N） I O CO №. ! O CO \N. I O CO N- || C. GO N- || Q CQ N - || CD GO ÈN）. O §< <>N-+--+ +-+ +-+ | GNR GNR GN || c^p c^p c^p | <ſº | O C O į o o co | C. O Q> | <> <> C | ► | +-+ +-+ | +-+ +-+ +-+ | +-+ +-+ +-+ | +-+ +-+ +-+ | ~{ y-\ x={ | y=\ y={ v={ CNR < o Gº co | \^ \~ OO || o CR co j \ep N） co I o GN GO | Aº № co I o GN co | Aº №. oOº I, Q GS， CO | Aſ? № op | © QN QO | \Q. № © <> ſ-r=\ y={ ș= {y=\ y={ ſ={GNQ GNR GNRGNQ GNR GNco co GoGo co co<ří ºſº <ří<# <† <#4Aro Arp Ap\rx \to \rb o o o C> | o o ] o (o o ] o o o I o o co I o C> o ] o C> o ſ o o co | C C C | C C C | C C C | C. C <> r： gö: <> <> o l o o o I o o co | c o o I o c o į o o cv | <> | o <> | <> <> <> | <> Q Q | <> Q Q | <> Q G C EEºŞ şGN， <ſGº <#GNQ <#GN， <ſGN， <;ºGNQ <*CN <ſCNQ <řiGNQ <ț¢GN, <ſGN | — — — || ON ON oº | co co co | <* * * | so ſo vo | so ſo so | ~ ~ ~ || co co co | om o ol | S. 22 | !！ ！! ！! ?? *XIT CITEIWJL 36 §::::::: | E E E | © o co | co co co | ~ ~ ~ || o. o. o.. | ∞ or vi | ► ► ► | cº cº cº | ſº ºº ºº | ** ** **• • • F | -} Įsg） | sg） | sg = | sg= | sºº | §§ • | §§ • | §§ • | §še | $še | $še | №šeĚ# , , , , , , , , } ） ） ） | — — — | — — — | — — — | — — — | <> º. º į º SP SP© c o į o <> <> | <> <> | <> <> <> < ! ！！ ！, \, \, \, | No No No | No N9 — | — — — | — — — | — — — | — — — | — — — | — ^-^ <> | <> º. º ! ººº<> < ! NQ …, …, …, … | ∞, ∞, √3 | No No No | No No No Í No No No | No No — | — — — | — — — | — — — | — -^ <> | <> º.º | ºſº º ~ | ° „ , ， ） | ≤ ∞ √3 | -5 & 5 & 0 | ， ، ، | co No No ! No No so | ſo so so | No H+ + | +-+ +-+ +-+ | — — — | — <> <> | <> ºº →• •,, ，,, ，,, ，,, ] ） ） ） | ► ► ► | ► co co | co co co | co co No | N9 NO NO | NO NO NO | NO + H+ | +-+ +-+ +-+ | +-* * * º | SP º º ~ | °} os os os e, | …, …, … | ∞, ∞, ∞ | ► ► ► | ► ► co | co co co | Co CO NO ¡ NO NO NO | NO NO − | − − − | − − º | SP º º ~ | °}} → → → o, | os os os ſ os os cae | & & & | ► ► ► | ► ► co | Co do co | &O CO NO | NO NO NO ! So Hº Hº | ** * * * * | SP º º ~~{ oo oo oo ~Q || ~ ~T （N.OD OD ODOY CÒ QUrQUY QUY QUY}}> \!> }}>}}> \!> CAOCAO CAQ CAONo so so | so so –ſ-º ſ-a }-，• • • • ¡ ºg co co º co | cae, co ~ | ~ ~ ~ | os os os || op v vs || ~ ~ ~ | ► ► ► | CO C0 C0 | C-9 NO NO | NO NO + | ** * * * * | SP º º ~«Q ĒĒĒĒĒĒĒĒĒĒĒĒĒĒTĘ TĘ TĘ TĘ Ę Ę | 555 | 5:5, 5. || $ $ $ | z） ≥ ± | §§§F || ? §§§） | §§3） | § （3） | §§3o | § 8o | §§ • | $š s | $šo | § 3 s | $še§ 8o | §§ • 3} 'uoupuyload suoolſ' ſo 25umlo *X"I GIT ºf W.J. #6 sX! s][[ l_s\{_s\I | IX A t a | p 3orſ | ac 1 p 30"I | 2C p :SorI | , 0 || 0 O gg.09" I 9 & 2262, ‘I II & | 0626.' I | 08 I 9 O || 9909" I 8 & &Z62,' I 6 & 98.26"I 66 2, OT 0 || 99.08.' I II & 9963,” I 9 & 62.2% I | 86 g | g I O | #909 ‘I SI & I962, “I S & 8/26"I | 26 # 06 0 | #90s. I # 91 & 996&’ I | 0 & | 8926. I 96. g gº, O | 8909" I ZI & | 6766, "I 29 I | &926. I | 96 9 || Og O | [909" I 6I & #762, ‘ I 89 I | Z926.' I 76 4 S 0 || 0908' I Ić, & | 896&’ I 09 I | 392&T | 86 9 68 0 || 6909 I &&. & IQ63,” I | 97 I | Z726' I | 66 6 ## 0 || Zgog" | | #2, &, #263,” I &# I &723° I I& OI | 67 0 | g g09" I 92, & 8I62,' I | 68 I | 8823." I | 06 II | 89 0 | 89.08.' I 92, & &I62,' I 78 I 8823" | | 6 || &I | 89 0 || 090S' I Z& & g062,' I | OS I | 6&2&" I 8T SI & I | 8708'I 28, & | 668&’ I 9& I | 9326’ I | 2 | }I | 2 I g?08 I 8& & &68& "I Č& I | I&A3' I 9T gT | II I ’ I | 82, & 9992, 'I ZI I | 8TA&’ I | 9 | 9I 9I I 6908' I 8& 2, 6292,” I &I I 7L/6' I 7I 2I | 0& I | 99.08.' I 8& & | 8282, “I 8 I | II/3' I | 8 || 8T | 73, I 3909" I 8& 2, 998I* I | S I | 802&’ I &I 6L | 83, I 83.08.' I 8& & ) 0.996," I 89 0 || 902&" T | II 0& &S I 93.09° I Z2, 3 || 9992,” [ . §§ 0 || 802&’ I | 0T Ić, 98 I IZOS' I 98, & Zł83, " I '87 0 || 0023° I | 6 & | Off I ZIOS' I gº, & Oż92,” I gº 0 || 869&’ I | 8 8& ºf I &IOS' I 7& 2, #888,” I 28 0 || 9693. I A #6 2; T | 800S I 86 & 2686 I | 68 0 gó93° I 9 96 || Ig I | COOS' I Iz, & Iz82, "I 28, 0 || 869&’ I 9 96 || 79 I | 866&’ I 02, & | g I92,” I && 0 || 3693. I 7 Z& 29 I | S66&’ I 8I & 8088,” I 9I 0 || 3693.” T | 8 8& O & 886&” I 9I & 2,088,” I II 0 || I69&’ I & 63 || 8 & 886&’ I WI & | 9622, 'I 9 0 || 069&’ I I /08 || 9 o', 226&’ I /IIoſº | 06Z&* I /0 oC) 069&" I loo + | – || + | – || + || – 3C p :3or QC p :3orſ 3C tº 3on sIIIA sli sIIA siſ sſ. A 0 –-º-º: ‘opng|3uo'I only sºuns 'LNGWnbºw ‘uo?!pu!!920ſ pup u01.swoost" 115??I up apps p ſo woup.t.19qū’ all to I TATBLE LXI, d © / y 2007 (º.ſ. Q070. if a Star in Right Jiscens Declimat ion o O w <> OD OOoo oo №. to Arp | <ſ， co CN -{ c | on oo N. GO A^ | <* c^ Gº r+ C cºp GNR GN+--+ +-+ +-+ +-+ +--+ | +→. +→. +={ y-\ r=-{ CY c^) GOy-， ~ř. GO CO o+--+ c^p \^ GO N-CO O C №ſ GNRCNR co coCo § +§ iſ ſăco cò Cò đã õ | oſ ºſ oſ oſ oſ ſ oſ oſ <> <> º&&&&&+ № cºc^) Grocº co co | co co co co co | co co co co co | co co <* <* <* | <* <* <* <* <ſ ©$ ºg( C) +++co oo <* oº | Ap o \^ Go Arp | © ® Q <† Q | CO CO QR SQ Q | Sił Q GX SG2 C} \ , v. > + | ޺ ºGN c^2<ğ ºğ và đã ¡ ¿ Ñ Ñ Õ off | āſ oſ ē Ģ Ģ Į ~; ~ Oſ dº cº | cº º 2 * * * ſ += •!Ģ Ģ Ģ Ģ Ģ GN | Gº! Gº GN ON CN || CNR CNR CNR CNR CN | ON ON CO COco co co co co | co co co C^ coį> C)„4 co us ! Cº dº co co o ſ №. ！! ！! Co sº | +\ CO \ſq ， Q ! № № © Şİ， SÐ | ► 92 9è （Q 9R�^ È +šºs Ē5;$3<Ř <ří Arò qo №. 1 N - OO OD O' C~; ~; oſ cº cº | <ť vº vº se ºs№ cº cº oſ º+E > S S S S S | s c c c c | o o co o -ſ | +-+ +-+ +-+ +-+ +-+ | ~ ~ ~ ~ ~ | ~ ~ ~ ~ Gºį> GN.• urs co co № co | <ſº go + Arp op | ++ y^ oo → <ſº | go º SĂ N- S | $2 $2 Q QS XQ || 9& № sº №. 92 ±§§§§§§či ſ; id <ń õ | čo dº H -i & | oſ oſ cò № № | ſq ： < < : cº | Gº! Gº r: | <> <> C <> <> | <> <> <> <> <>}， O º ſo CN CO <łº o | Q QNR OG<ř. o Aro o Ar> | <> \to co ſº p" | <ſt op cºp №. ！! | 92 Q <† 9º 9（! 24§ § ¶ ã § čňGNR GNR y +è ă ä ã có | āſ Ņ Ņ © ® 1 và <Ë SË čº cº | oſ oſ -; e º% Èo cô cô co co co | cCº cº CN Gº GN || CN CN Gº GN ON ! Gº GNR GNR GNR GNR cº)CNoo qo <# #-ſ on l №. <łº w + CO GO I CO ON SQ QO QQ 3ëſ&&&Ğ ğ Ģ Ģ Ķ Ļ Ņ Ņ Ņ © ® | © √∞ √° √≠ √ſº： ~* <† <ſco co co co co i co co co co co I co co CO CO GO× | O ko N. CO CO Q+--+ CN c^p <Ť， ArbGO N， OO Q！ ?） ：-（ -， CNR || CNR CN ON CNR CNR | GR GNR GN | . 30 ARGUMENT. Sun's Longitude, more or less the Star's Declination. For the ſlberrat VIs 96 TAl3LE LXII. For the JWutation in Right flscension and Declination. ARGUMENT. Mean Longitude of Moon's Ascending Node. 0s Vls , Is VII's IIs V111s Log. b B Log. b JB Log. b B 09 || 0.984.4 || 0° 0' + 0.9588 || 6°45' 0.8960 70 48' 300 1 0.984.4 || 0 15 0.9571 || 6 54 || 0.8939 || 7 40 29 2 || 0.9843 || 0 31 || 0.9554 || 7 ||3 || 0.8917 || 7 32 28 3 || 0.9842 || 0 46 || 0.9536 || 7 ſ? I 0.8896 || 7 23 || 27 4 || 0.9840 || 1 1 || 0.9518 || 7 20 0.8875 || 7 14 || 26 5 || 0.9837 || 1 16 || 0.9500 || 7 28 || 0.8854 || 7 4 || 25 6 0.9834 || 1 32 : 0.9481 || 7 36 || 0.8834 || 6 53 || 24 7 || 0.9830 || 1 47 0.9462 || 7 43 || 0.8814, 6 42 23, 8 || 0.98.25 || 2 2 || 0.94.42 || 7 49 || 0.8795 || 6 29 22 9 || 0.9821 || 2 17 || 0.94.22 || 7 55 0.8776 || 6 17 | 21 10 || 0.9815 2 31 0.9402 || 8 1 || 0.8758 6 3 20 11 || 0.9809 || 2 46 || 0.9382 8 6 0.8740 || 5 49 || 19 12 0.9802 || 3 |1 0.9361 || 8 10 || 0.8723 5 35 18 13 || 0.9795 || 3 15 0.9340 || 8 14 0.8707 || 5 20 17 14 0.97.87 || 3 29 || 0.9318 || 8 17 0.8692 || 5 4 || 16 15 0.9779 || 3 45 0.9297 || 8 20 0.8677 || 4 48 || 15 I6 0.9770 3 57 : 0.9275 || 8 23 0.8663 || 4 31 || 14, 17 || 0.9760 || 4, 11 0.9253 8 24 0.8650, 4, 14 || 13 18 || 0.9750 || 4 24 0.9231 || 8 25 || 0.8637 || 3 56 | 12 19 || 0.97.39 || 4 37 0.9208 || 8 25 0.8625 || 3 38 || 11 20 0.97.28 || 4 50 0.9.186 || 8 25 (). 8615 3 20 10 21 0.97.16 || 5 3 || 0.9163 || 8 24 (). 8605 || 3 1 || 9. 22 || 0.97.04 || 5 16 || 0.9140 || 8 23 0.8596 || 2 41 || 8 23 0.9691 || 5 28 0.9118 || 8 21 || 0.8588 || 2 22 || 7 24 0.9678 || 5 40 0.9095 || 8 |18 || 0.8582 || 2 2 || 6 25 0.9664. || 5 51 0.9072 || 8 15 0.8576 || 1 42 || 5 26 0.9650 || 6 3 0.9050 || 8 11 || 0.8571 || 1 22 || 4. 27 || 0.9635 | 6 14, 0.9027 | 8 6 0.8568 || 1 2 3 28 0.9620 || 6 24 0.9005 || 8 || || 0.8565 || 0 41 2 29 0.9604 || 6 35 fl. 0.8983 || 7 55 0.8563 || 0 21 | 1 30 0.9588 6 45 . 0.8960 7 48 || 0.8563 || 0 0 || 0 * –– *-4 + *- + Log. b IB Log. b JB Log. b T} Vs XIs TV K - IIIs IX's 26 *IIIX'I GITEIV.I. X:3'ſ } sl/A sIIA sIIIA a YI sX sIX + + | + | + | + O ()()"O | A3' 8 98’’;I #9 '91 SS “WI Č," 8 OS I 66° 0 || 39 °8 Aff "WI #9 ° 9 I 81°7'ſ 30° 9 62, 3 89 ° 0 || AA '8 I9'? I $g ‘9L | SO “WI ZZ * Z. 8& S 28° 0 || IO' 6 WA." FI 39 ° 9T | 88 ‘SI 19 ° 2 Z8, , # GI* I | 98, '6 A8 °ºf I | 09 '9'ſ &A' SI G& ‘A 98, 9 '97" I | 6p' 6 66" FI 87°9'I 99 “S 66° S) 93, 9 GA," I &A,” 6 II* GI 97°9 I | 89° SI 92" 9 || ||7& 2. 30.’ & 96’ 6 92, '91 || 37°9'I Ić, 8 97' 9 || 8 8 08’ & 6T'OI 78°g I S'9'ſ 70° SI 06' 9 || 32, 6 69 & I'ſ "OT g?" GI V8 °9'I 98" &I $6" Q I& OI 28° 2, 89° OT | g g "SI 63, '9'ſ 29°3, I } 99 ° 9 || 02, II | 9 I* 9 98 “OI | #9° GI W& ‘9T | 6'7" &I | 68° S 6I &I fr;7° S 20° II | 82° GI 8I* 9T || 08" &I II* 9 || 8 || QI &A,” S 88,” II 38° g| || &I'9T OI" &I W8° F | AI #I 00"? | 67° IT | 06” gi 90° 9T | 06’ II 99 °W 9T gT | 83°W OZ’ II | 86°g I 86°gſ' | OZ’ II | 8& ‘W | g I 9I 99 °W 06’ll 90'9| || 06’ gT | 67 II 00°W VI AI #8 "'7 OI" &I &I '91 38° GI 88,” II &Z" & SI 8T | II* G | 08" &I 8I '91 S.A." g| || A.O." IT | #7° S &I 6T | 68 ° 9 6'7" &I } j^* 9T | #9 gT 98* OT || 9 I* 9 || II 0& 99 ° 9 || A9" &I 6&’9| || 99 °g I | 89° 0'I | Z8° & OI I& 86° 9 || 98" &L | 78°9'I 97" GT | IV* OT || 69 °3, 6 ØØ 08, '9 | #0 "SI 88°9'I #8 "SI 6I* OT || 08' 3, 8 83, 97° 9 || IC, “SI 37°9'I S3" gl 96° 6 30° & A. #2, S2’ 9 || 88° SI 97°9'I II* GI 32°6 92,” I 9 £2, 66° 9 99 “S 87°9'I | 66° WI 6'ſ '6 77" I g 93, G& " A. 32°8'I | 09 ‘9T || A8 ' 'WI 92,” 6 SI " I y Z8, IQ "A 88° SIJ &S '91 || 72, "'WI | [O' 6 A8" () | 8 83, ZZ * Z. SO''WI 89 °9'I [9 ‘’VI ZZ” 8 89 () & 62, 30° 8 || 8 || "WI #9 ° 91 ZF'ſ I £9 8 63' 0 || I o08 || A&’/,8 88"/WT | 79 /9I | 88”//WI A.& /8 || 00"/0 ſo() s.A sAI sIII sl I sT s0 “apoM 3upuaoSW S.u.00IN Jo apmúšuo"I utoN INCIWnbºw 'uoy!pu!!200 pup worswoost" 11%I w? won|ppm.A alſ, 10:1 983 TABLE LXIV. SEMIDIURNAL ARCs for 39°57' JNorth Latitude. ~ | ~£ © ® № 99 sĒ rº | go * r+ № <ť | Q № co o ſo | Gº çp să º co | cº co <ă o ſp | c to – to - 5 | <>+ A^ (^p \p <ř: <ř. º Aro Ar} \r> <† <ſſ | co crò c^) Gºº GN || CN－ +-+ +-+ +-+ºr) \tº \^ <ř. | <řł co cº co © | CN +-+ +-+ º 1° | 2 to so in to so so | so so so 'n to ſ to so ſo so ſo | wo < < < < 1 < < < < < | < < < < <* -3& 23E <> ț> $2 $2 № co | S2 $2 CQ © $2 | gº oſ ºn GN CO | <ſº -- № go o) | \tº + N. GN go | <ſ， o <ſ> on <ſ § ļº！=）?Aro Arp Aro <ř. <+ <ſſ | <† <} <† <† <† ：: | | 'å Gº to o cº so oº | gº gº gł gº º I º cº º r* * | go Gº to e <;: | co gº ſo º ſo | e <;: a) sł gº 5 |> | =+-+ +-+ +-+ | GNR GN GN co co | <ſ： <ſt <;º Aro Aro | Ary+-+ +-+ | +---+ GNR GN co co | <ſ: <ſ， º Aro ºro § 1º || … ſo ſo ſo so so so | so to so so so | ſo to ſo to ſo ſ to № N N N | ~ ~ ~ ~ ~ | ~ ~ ~ ~ ~ £ ‘ō |\ | & - & Co — ſo co | ±± ſº gº gº ſº | gn og «Q op gº | №. ！! <ſº co cº | to º ſo gº cº | co vº № gº № $ ſº | 5+-+ +-+ +-+ | GN GN GNR co co | co <† <# <ř. Aro | \e)+--+ | +--+ ON ON CN co | co <ț¢ <ři Aro Aro º 1° 1 - to to so to so so | so ſo to so so ſ to so to to to ſ to N. N. ~ ~ | ~ ~ ~ ~ ~ | ~ ~ ~ ~ ~ -C || ~ •==* ~3 C. CO № o cºn №. ! º sť ſº o <ſ> | № ++ lº cº gº | so ob co N - || Aro o cº co cº | ſo ： «Q ± ſo Źļs€+-+ +-+ ++ ſ GNR GN Gº co coc^ <# <† •} \oAro Aròy=|;-+ +-+ CN Gº Cry | C^ <† •† \rò lo º so so ºc so «o so | so to to so so | so «o so «o ſo ſ to to № № №. ! № № № № №. ! № № № № № O C ≡ CNR CO <ř* Aro | so N → CO on o { ++ GNQ co <# Aro || ko ?） oo on o | +---+ CNR co <ſº ºro || ko N. CO Q Q> N-ŞE | | | Ş | $3 Р޺ | § § 3： $2， $ | 33 $ $3 （$ $ | $ $ $ $ $3 THE END, Page Line 14 18 G6 19 15 11 23 19 24 13 26 25 27 4 29 20 34 8 35 20 36 19 61 17 75 17 { % 22 76 H 5 96 19 97 3 98 5 104 13 CC 14 110 11 112 7 114 13 115 5 119 16 123 7 124 10 12"| 6 ERRATA. tan M." tan m” For , read q’ º For PM = PZ + ZM = PZ + n” + r", read PM it. ZM — PZ == n." + 7" — PZ. For Formule, redd Formulae. For 20°, read 90°. 2.T. R For R’ read zº" For PEp2, read PEp0. IFor R” Yead R’ For ecliptic, read equator. For PSL, read PSL. For of the star, read of the sum or star. For L tan w, read cos L tam w. For ang. AEL. read ang. ACL. I'or RL, read BL. For x , read +. For .834, read .0830. For sine of the clongation, read sine of half the elongation. For EE, read EC. For times, read time. , I — cos MCO 1 — cos MCO For * ~ * , read – tº: sin MCO cos MCO For sin MEO, read sin MCO. For ***, read sin E. sinº, Fº sin F For latitude, read polar distance. For sin (– n + II), read sin (L – m + II). For sin (A + II — y), read sin (A + ºr — y). For these will be an ecliptic, read there will be an eclipse. For 3684.3, read 3683.3. For she, read the. For is latitude, read is the latitude. ERRATA. Page Line 130 *C , 133 142 149 160 162 164 [75 192 197 204 234 299 302 303 25 26 23 4 304 314 14 11 12 (3 + d — a cos I) read = 2d read = (s + d — a cos I). 6 — — d º For CES, read CEQ For 3 (0-0 tan |), read & (a — c tan I). 3. 1 + tan” I 1 + tan” I For PP, read PP'. For Sv, read Sp. For EQS, read QES. . For geocentrical, read geocentric. For 52 and 54, read 54 and 56. For porabola, read parabola. For mutation, read mutation. For semidiameter, read semidiameters. For Schehallier, read Shehallien. For logarithm of B, read logarithm B. For Take the difference between the moon’s longi- tude and the longitude of the monagesimal degree, and call it D, read Subtract the longitude of the nonagesimal from the moon’s longitude, increasing the latter by 360°, if necessary, and when the re- mainder is less than 180°, it is the moon’s distance to the east of the nomagesimal, which call D; but when the remainder is greater than 180°, subtract it from 360°, and the second remainder will be D, the moon’s distance to the west of the monagesimal. For Add p to, &c. read Add p to the moon's true longitude, when the moon is to the east of the nonagesimal; but subtract, when it is to the west, and the result will be the apparent longitude. For The parallax in longitude, p, &c. substitute the same as directed the last above. For time of new moon, read time of new moon, reck- oned from the noon or midnight to which the second distance corresponds. //av/e Z. *. -- * Young & /)e//e/e7: Xú . * . . . . . ºr *. * , X, - - * - N. * *-*----' //av/e 2 Fº . Z.6. E" Xàtory &//e/Zeke" &c., 4, . . f : * PZaze 3. T Fºy. 30. Young ("Z)cłże/cy J', ', ... ---7 ...' Sº º |.. () * ** , , s \ . w -A zf - I- -N - Young & Zºe/Reker Jº ~~ 2. % AZazze 5. r 4. * - f & # tº: - 4. º Fig. 52. A. E S S Åg,” a I TH T) L Aºy, 55. Young & Ze/7 *º- Aºzze 6'. Q.. 123 R ºffl *= º C Al Fig. 59. Young & 7), 77,7,7- .V.". • * **** , ºr > y \* - O E A ºf, \ .* ZZaZe 7. 7) |- ZD, S. ('J, "ZO % }ining & De/leker o'e. RSITY OF MICHIGAN Ill. E O1 |\ ,- O700 * ... ." | O780 | H ; |E sº C---§- - # * :z.is:•,”..• wºº-":r Sº...SE%ºº Eiº#E. Sºl * - - • ‘s `` - ** º -- " . a, - Y - Rs - - - º 'AT’ACJ wº. Jºy sºlº". Ow-wºº Cº.º.º.º.º.º.º. ºOº. # ºº i-*-ºº --- -º * DUFFIE *. In ſº: LIRRARY || # -- - - . . . - sº a - . § wº-s - - * Yū; - Zºº & º . . . . \\ . . . . . ... • * - bºº- : . . . º ||||||||||III TTTTTTTºwſº |||||||||I||||IIIf: Fº wº-gº s sº e g is º e º º sº º sº, º ax*º z É C! | | | | | C £º º C C C ſº C ſº C C º C C | C | ſ | C C C ſ ! | | º [. ſº t | TI I IC ( : II.'T OF s 2’ TIII. T.V. I I .V. N. I* 13 ICS I R Y - "I" 12 1& I.M. N \ SSO (“I .\"I"I ON * DUPL A 426928 ·*、 ir •- ' , -' ），}';：ºx,"ſºſ '. ' ’ .：§§§§§ , , （**）（$*§§ *:: ** º: º is: -