THE UNIVERSITY OF ILLINOIS LIBRARY , 33-0 vdD I O ' G,S5 e IWWEMATieSUBMm Return this book on or before the Latest Date stamped below. /' 1 ELEMENTARY TREATISE 0N iU'V\ x \; r r.f i. IN TWO PARTS. THE FIRST, CONTAINING A CLEAR AND COMPENDIOUS VIEW OF THE THEORY\ THE SECOND, ' • \ t •* 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, PHILADELPHIA: PUBLISHED BY KIMBER & SIIARPLESS, NO. 93 , MARKET STREET. .T. PRISSY AND G. GOODMAN, PRINTERS. Eastern District of Pennsylvania, to wit: BE IT REMEMBERED, that on the second day of January, ln t ^ ie forty-sixt4 year of the Independence of the United States of America, A. D. 1822, Kimber & Sharpless, of the said 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.” D. CALDWELL, Clerk of the Eastern District of Penmylvama. 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- IV 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 propercomprehension of the subject. In the demonstrations, the student is supposed to he 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 suu and moon, obtained from these tables will be very nearly correct, for any time within the period to which the tables of Epochs extend.* Rules 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. PREFACE. V 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¬ course to several of the best modern publications on the subject; among which may be particularly mention¬ ed those of Vince , Wuodhonse, 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 , JV*. J. 1 2mo. 22d , 1821. •I ' t ! > 'Il l f* | ’ . ' . , r ' I 3 . ■ „ , CONTENTS PART I. PAGE. Chap. I. General Phenomena of the Heavens, 1 Chap. II. Definitions of Terms.—Astronomical Instruments, 4 Chap. III. Meridian Line.—Sidereal Day.—Diurnal Mo¬ tion.—Refraction, - - - - 7 Chap. IV. Latitude of a Place.—Figure and Extent of the Earth.—Longitude, - - - - 16 Chap. V. On Parallax, 22 Chap. VI. Apparent Path of the Sun.—Fixed Stars, - 29 Chap. VII. Sun’s Apparent Orbit.—Kepler’s Laws.—Kep¬ ler’s Problem, 46 Chap. VIII. Equation of Time.—Right Ascension of Mid- Heaven, - - - - 71 Chap. IX. Circumstances of the Diurnal Motion.—Sun’s Spots, and Rotation on its Axis.—Zodiacal Light, 77 Chap. X. Of the Moon, - - - - 91 Chap. XI. Eclipses of the Sun and Moon.—Occultations, 115 Chap. XII. Of the Planets, - - - 155 Chap. XIII. On Comets, - - - 190 Chap. XIV. Aberration of Light.—Nutation of the Earth’s Axis.—Annual Parallax of the Fixed Stars, - 193 Chap. XV. Nautical Astronomy, - - 19S Chap. XVI. Of the Calendar, - 207 Chap. XVII. Universal Gravitation, and some of its Effects, 213 PART II. Catalogue of the Tables, - 245 Observations and Rules relative to Quantities with different signs, - - - - 250 PitOB. 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, ----- 255 Prob. III. To convert Degrees, Minutes and Seconds of the Equator into Time, - 257 via CONTENTS. PA OF. Prob. TV. To convert Time into Degrees, Miuutes and Seconds, - - 257 Prob. V. The longitudes of two Places, and the Time at one of them being given, to find the corresponding Time at the other, ----- 258 Prob. VI To convert apparent time into mean, and the con¬ trary, - - - - - 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, - 264 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. - 267 Prob. XII. To find the Moon’s Longitude, &c. from the Nautical Almanac, - 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 Tables, - 279 Prob. XV. To find the Aberrations of a Star in Right Ascen¬ sion and Declination, - - - - 280 Prob. XVI. To find the Nutations of a Star in Right Ascen¬ sion and Declination, - 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, - - 285 Prob. XX. The Obliquity of the Ecliptic and the Longitude and Latitude of a Body being given, to find the Right Ascension and Declination, - - 287 Prob. XXI. The Obliquity of the Ecliptic and the Longi¬ tude and Declination of a Body being given, to find the Angle of Position, - - - 289 Prob. XXII. To find the Time of a Star’s Passage over the Meridian, - - - - 290 Prob. XXIII. To find the Time of the Moon’s Passage over the Meridian, - - - - 391 CONTENTS. IX PAGE. Prob. XXIV. To find the Time of the Sun’s Rising and Setting, - - * - - 292 Prob. XXV. To find the Time of the Moon’s Passage over the Meridian, from the Nautical Almanac, - 293 Prob. XXVI. To find the Moon’s Declination from the Nautical Almanac, - - - 294 Pr 3B. XXVII. To find the Time of the Moon’s Rising or Setting, - - - - 296 Prob. XXVIII. To find the Longitude and Altitude of the Nonagesirnal, - 29S Prob. XXIX. To find the Moon’s Parallax in Longitude and Latitude, - - - - -' 301 Prob. XXX. To find the Time of New or Full Moon by the Tables, - 309 Prob. XXXI. To find the Time of New or Full Moon by the Nautical Almanac, ... 313 Prob. XXXII. To determine what Eclipses maybe expected to occur in a given year, and the Times nearly at which they will take place, - - 315 Prob XXXIII. To Calculate an Eclipse of the Moon, 319 Prob. XXXIV. To Project an Eclipse of the Moon, - 324 Prob. XXXV To Project an Eclipse of the Sun, - 328 Prob. XXXVI. To Calculate an Eclipse of the Sun, 337 Prob. 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 continu¬ ance of the Central Eclipse, - 345 pRon XXXVIII. To Project an Occultation of a Fixed Star by the Moon, ----- 348 Prob. XXXIX. Given the Moon’s true Longitude, to find the corresponding time at Greenwich by the Nautical Almanac, - 354 Prob. 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, - 355 1 The following Alphabet 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. Names. A CL Alpha B /3£ Bela r y f Gamma A J Delta E t Epsilon z U Zeta H v, Eta © 3-0 Theta I i Iota K K Kappa A A Lambda M /LL Mu N v Nu s 1 Xi O o Omicron n zttt Pi p f f liho 2 Phi x* Chi ^ 4 , Psi fl 0> Omega AN ELEMENTARY TREATISE ON ASTRONOMY. PART I. CHAPTER I. General Phenomena of the Heavens. 1. 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 P/iy- sical Astronomy. 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 Volar 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 North 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 Motion. 6. If w r e 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 aud 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, Mercury , 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 Mars, 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 w ithout the aid of a telescope, and have not been long known. The stars which do not sensibly change their situa¬ tions, with respect to one another, are called Fixed 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 lienee 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. 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 named 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 ASTRONOMY. 4 earth is so great, that the earth’s diameter, compared with it, is insensible. 13. 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 . 1. The straight line which passes through the North Pole, and through the centre of the earth, is called the Axis 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 J\Torth 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 tbe same line be produced downward, the point in which it cuts the opposite part of the sphere, is called the Nadir. 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¬ zon. 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 Meridian of that place. The meridian cuts the horizon at right angles, in two points, called the North and South Points of the Ho¬ rizon. 8. The intersection of the plane of the meridian with the earth’s surface, is called the Meridian Line , or Terrestrial Meridian. 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. 11. 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. 12. The arc 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 Azimuth 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. 13. 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 mention. 14. 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. 1 5. 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 16. 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 meridian. It is used for observing the passage of a heavenly body over the meridian. 17- A Micrometer is an instrument attached to tele¬ scopes, by means of which small angles may be mea¬ sured with an extreme degree of precision. 18. 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 21. The student who wishes to see particular descrip¬ tions of astronomical instruments, accompanied by en¬ gravings, may consult Vince’s Practical Astronomy , Traite I)’Astronomie'par Delambre , or Rees’s Cyclo¬ pedia. CHAPTER III. Meridian Line.—Sidereal Bay.—Biurnal Motion .— Refraction. 1 . Let Z R, Fig. 1 . represent the northern part of the meridian of a place, Z the zenith, P the pole, II RO the horizon, SS'Gthe 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 ZS'B 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 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 PZS = 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 mad%to an article in the same chapter, the number of the article only i9 given. CHAPTER III, 9 between its passages on any two succeedirig 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 Day. 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 3ff0° 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 15° 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 10 ASTRONOMY. ations of tlie star on the meridian above and below the pole. Then we have ZF = PZ — PF ZGr = PZ + PGr = PZ + PF therefore ZB — ZF = % PF, or PF = \ (ZGr— ZF) also ZGr + ZF = 2 PZ, or PZ = f (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 Atmospherical Refraction , and sometimes As¬ 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 0. But if the CHAPTER III. 11 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 0, 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 Jingle 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. 11. The atmosphere 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 12 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. 12. 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 4 r = the true zenith distance, and, therefore, ( 10 .) m sin N == sin (N 4 r) = sin N cos r 4 . cos N sin r (App.* 13.) cos iN or m = cos r 4 - ——^ sin r = cos r 4 - cot N sin r. T sinN But as r is small, cos r — 1, and sin r = r, nearly. Therefore m = 1 -f r cot N, or r == cot N = (m r— 1) tan N. 13. 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 w ould be infinite. And in all cases, w hen the alti¬ tude is small, and consequently when the rays of light * Appendix at the end of part 1st, CHAPTER III. 13 enter tlie atmosphere very obliquely, it would be too great. Dr. 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, 14. 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¬ nith distances of the other star at L and K, and R' and /, the corresponding refractions. Then, ZG = N-pR, ZF = tt-f r, ZL = N'-j-R', andZK = n'-p r'; therefore (7), PZ == ~ (N-pR+n-pr) = -J (N'-f R'-pn'+r',) and N-pR-pn-pr = N'-pR'-pn'-pr', or R-pr—R'—r' = N'-pn'—N— n. Now supposing the refraction to vary as the tangent of the ap¬ parent zenith distance, (12) we have, r == m — 1 = R tan n R R r' tan N tan n R tan N' R+ tan N R tan n R' = tan N tan N R tan N' tan N tan N' and r' -= tan n' R tan n' tan N whence, therefore, — an - = N'+n'— N tan N n. 14 } ASTRONOMY. From which we obtain R = — W+n'—X-n) tan N tan N-ftan n —tan N —tan n Whence r, R and r become knoAvn, and consequently PZ = \ (N-j-w-f R-frj; also the polar distance PF = ZP— ZF = ZP — (»+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. Id. 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 he 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 n tan n" _ r tan n ". _ =-or r = -; r r" tan n consequently PM = PZ-f ZM = ZP + n" 4 - 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 S" 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. 17 . 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. For mule 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 I part of the whole diameter. The hori¬ zontal diameter also suffers a slight diminution. (18) 20. At the true horizon the refraction is about 38'f. Hence it follows that when any of the heavenly bodies are really in the horizon, they appear to be 38'| above 16 ASTRONOMY. it, and that therefore refraction retards their setting, and accelerates their rising. 21. 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 Crejiusculum or Twilight. CHAPTER IV. Latitude of a Place.—Figure and Extent of the Earth .— Longitude. 1. Let HZRN, Fig. 9, represent the meridian, Pp the axis of the heavens, Z the zenith, HOR the hori¬ zon, and EOQ 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 (3.14). 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. 17 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, v 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, Vp 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 EGA 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 (1), the angle ACB = ECB — EGA, 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 : 1° : : 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 1,8 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 EPQp, Fig. 4, represent the meri¬ dian, P/? the axis of the earth aud 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 BKP 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 pf 320 to 321. 0, 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 ot miles.* The differ¬ ence between the equatorial and polar diameters, is about 25 miles. 10. 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 Tran&, Actions of the Araer. Philos. Society, Vol. 1. New Series. CHAPTER IV. 19 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 . 11. 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, P p = 6, 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 (Conic Sections), CG tan L DG _ b 2 CG ~ a 2 Therefore tan R b 2 tan L ~ a 2 ’ or a 2 : b 2 : : tan L : tan R, 12. The equatorial and polar diameters of the earth, 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 AC, we have BG = CG tan M, and AG =* CG tan R; 20 ASTRONOMY. therefore CGtanM = tanM. AG CG tan R tan R but (Conic Sections,) — — V AG 6 Hence —or tan M = ^ tan R = — tan L = tan R 6 6 b a 2 b , r w tan L. a And AC — s ‘ n ^ a cos M sin CAG ~ cos R LONGITUDE. 13. We have shown (1) 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. 15. 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. 21 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 w ork. 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- ASTRONOMY. 22 CHAPTER Y. On Parallax . 1. 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, (1.12J, 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 Ap¬ 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 Parallax . It is also sometimes called Paral¬ lax 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- CHAPTER 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 Parallax . 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, p — ABC = parallax in altitude, and w = the horizontal parallax; then N + p — ZAB = apparent zenith distance, And sin p = ^ sin CAB == ^ sin ZAB = 5 sin BC BC D (N + p.) But when N -f p = 20\ p becomes w, Hence sin «■ = 5 sin 90° = D D Consequently sin p = sin sin (N -f 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. Since sin We have D Hence, as the radius of the earth has been deter¬ mined (4.9), when the horizontal parallax of a body is = 5(5), R sin w* ASTRONOMY. ■JM* 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 w, the hori¬ zontal parallax, are always very small angles; even for the moon which is much the nearest, the value of ar does not at any time exceed 62'. We may there¬ fore, without sensible error, use p and ar themselves, instead of their sines. If this be done, the last formu¬ lae, in the two preceding articles, become, p = w sin (N 4 - p ,) and D = — = R —. R w 8. In the fraction £ of the last formula, 1 represents the radius and ar the measure of the horizontal paral¬ lax. Hence, in order to render the numerator and de¬ nominator of the fraction homogeneous, if sr 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 6.2S31853 : i :: 1296000" : 206264".8 = the length of the radius expressed in seconds. Hence if the value of ar is expressed in seconds, D - R 206264 - 8 z .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 A' 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° — ZAB = 180° — d', and ABA' = 360° — (ACA' + BAC + BA'C) = d + d — (L -f L'). Again (7) ABC == & sin d, A'BC = -sr sin d', and ABA' = ABC -f A'BC = w sin d -j- sin d'. Hence zr =- - - = - + - ~ ( L + L 'l sin d + sin d' sin d -f sin d' 10. 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 t)f 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, and » = B _ A ' S ~ BAS sin d -f sin d' 26 ASTRONOMY. 11. 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 (1.16), 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 1J5 1 , Wargentin, at Stockholm, and La- eaille, at the Cape of Good Hope, made the requisite observations on the planet Mars , and determined its parallax at the time of observation to be 24".64. Hence, (8), 206264".8 24".64 = H x 8371. The distance of Mars from the earth's centre 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 2, 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 zAZ = CA d = reduction of latitude for the place A, and z'AZ' — CA 'd' = reduction oflatitude for the place A', may be found (4.11), and thence the angle zAB = ZAB — zAZ and s'A'B == Z'A'B — js'A'Z', are known. Now if w and be the horizontal parallaxes of the moon at CHAPTER V. 27 A and A', and R and R' stand for the radii CA and CA', we have (7), J? = D = —; whence «•' == «■. Let d and d' stand for the reduced zenith, distances rAB and s'A'B; then (7), ii' ABC = -or sin d, A BC = «■' sin d' = — sin d', ind ABA' = ABC + A BC = sin d + 5' R R w sin d 4 . R' «■ sin d' = R x ABA'. R x ABA' sin d\ Hence 13. The horizontal parallax of the moon, to ah observer at the equator, is called the Equatorial Pa¬ rallax . Ifw" = the equatorial parallax, and R" = CE = the ra¬ dius of the equator, then, ,, R" R" x ABA' R R sin d + R sin d' 14. 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 6 t' 32". Hence^ D = R" = R" x 64, nearly = its greatest distance, 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. ASTRONOMY. 15. The mean equatorial parallax of the moon is £ (53' 52" + 61' 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 thfe earth, and «•, ®-",the corresponding pa- rallaxes. Then (4), D = R ,D R' and D" = R' sin w i D + D') = i (-j sin w' R" Sin vr ‘ sin w sm R'' \ in w'/ But D" = -J D 4- D). Hence -3_ = 1 (-51 + -31) sin w" Vsin «• sm «■'/ 2 sin sr 4- sin wr' _ 2 sin ^ 4- 73 -) cos -J (w—w') sm sin sin w' sm w sm ar (Ap. 20.) smr = sin «• sin «■' _ sin i (ar 4 «•' COS (ar — -sr') As the arcs are small, we may, without material error, con¬ sider cos i (®- — «■') — 1, and for the other terms take the arcs instead of their sines. We shall then have, *•» = __ = 2g - g ' fB) ^ (w 4 - *■') ar 4 From either of the expressions, A and B, the value of «•" 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, Tl _ 20626F.8 8".7 = 23708 x R CHAPTER VI. 29 CHAPTER VI. Apparent Path of the Sun.—Fixed Stars. 1 . 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, P p 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 ASTRONOMY. be observed, we have tbe polar distance PS = 180° — (PH 4- HS) = 180° — (latitude of the place 4- 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 EPD. Consequently the angle BPS' = 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 AS'D = 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 ill 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 Eclijptic. 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 Equinox ; and the time, when he is at the other equinoctial point, is called the Autumnal Equinox . CHAPTER VI. 31 Tlie terms Vernal Equinox and Autumnal Equinox are frequently applied to the equinoctial points them¬ selves. 5. The two points in the ecliptic, which are at 90 c 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 (S). 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 aTb is the tropic of Cancer, and ede , 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 Arctic Circle; the other, about the south pole, is called the Antarctic Circle. Thus fgh is the Arctic Circle, and Jcmn , 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. 11. 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. 0. Aries r. s. 6. Libra 1. Taurus 8. 7. Scorpio *1 2. Gemini n. 8. Sagittarius t 3. Cancer 23. 9. Capricornus V? 4. Leo si. 10. Aquarius AW' (VW 5. Virgo 11. Pisces X Aries, Taurus, Gemini, Cancer, Leo, and Virgo lie on the north side of the equator and are called North¬ ern Signs. The others lie on the south side, and are called Southern Signs. Capricoruus, Aquarius, Pisces, Aries, Taurus and Gemini are called Ascending Signs , because w hile 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. IS. 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. 13. Any great circle, which passes through the poles of the ecliptic, is called a Circle of Latitude. 14k 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 1Right Ascension 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. 17. 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. W ith the obliquity of the ecliptic known, the longitude and latitude of a body may be obtained from the right 6 ASTRONOMY. 34 ascension and declination, by means of spherical Tri¬ gonometry. On the contrary from tlie 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¬ clination and PS L 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, of = REL = obliquity of the ecliptic, x = RES, and y = LES. 19. When the right ascension and declination are given to find the longitude and latitude , we have, tang RES = tang ES = sin ER cos RES tang EL = cos LES tang ES = cos LES ** n f ER ; cos RES and tang LS = tang LES sin EL. Or tang * = tang L = sin R cos x and tang a = tang (a;—*>) 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' = sun’s longitude. Hence CHAPTER VI. 35 . . , x Tio/ tan ER tan. Sun’s longitude = tan Eb = —— 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 , ™ C0S EES tan § EE - tang ER = cos RES tang ES =-— > 6 & cos LES and tang RS = tang RES sin ER. /a x tang a . 0 cos («.+ •} tang L. Or tang y = —; tang R = — sin L b cos y and tang D = tang (y+*>) 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 sun we have, tan Sun's right ascension == tan ER = cos RES' tan ES' = cos Obliquity of the ecliptic x tan Sun's longitude; and sin Sun's decl. = sin RS' = sin RES' sin ES' = sin Obliquity of the ecliptic x sin Sun's long . SI. The angle contained between a circle of latitude and circle of declination, both passing through the sun or a star, is called the tingle of Position of the star. If P'P be produced to meet the equator in N, then in the tri¬ angle P'SP, P'S = complement of latitude, PS = complement of declination, P'P = obliquity of the ecliptic, P'PS == 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 PP S cot a cos a — sin a cos (90 — L) sin (90 — L) cot a cos a — sin a sin L sin L ( c - \si cos L cot a COS A sin a , T /cot = tan L. [ -—cos A cos L \sin L - — sinAV \sin L / Make tan z = sin L tan a. _ 1 1 sin L Then C0 ~ = sin L tan — cot z — tan z sin cos z Hence cot S = tan L. f cos z . cos a — sin a^ Vsin z } tan L / . —-. (cos z cos a — sin z sm a). sin 2 But App. 14) cos z cos a — sin z sin a — cos (z -}- a). (A) Therefore cot S = . cos (z + a). sin z When the longitude, declination, and obliquity of the ecliptic, are given, we have sin PP' sin PP'S cos L sin sin PS cos D For the sun, a = o, and the formula (B) cot S = cot S = cot eo cos a — sin a sin L cos L , becomes, cot # cos L’ Or tan S = cos ~ _ £ tan a cot u (C) 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°. CHAPTER VI. 37 SITUATIONS OF THE FIXED STARS. 22. Iu 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. Northern . 1. 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. Lyra, The Lyre. 9. Cygnus, The Sw T an. 10. Cassiopea, The Lady in her Chair. 11. Perseus, Perseus. 12. 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, Constellations of the Zodiac. 22. Aries, The Ram. 23. Taurus, The Bull. 24). Gemini, The Twins. 25. Cancer, The Crab. 26. Leo, The Lion. 27. Virgo, The Virgin. 28. Libra, The Scales. 29. Scorpio, The Scorpion. 30. Sagittarius, The Archer. 31. Capricornus, The Goat. 32. Aquarius, The Water-bearer. 33. Pisces, The Fishes. Southern . 34. Cetus, The Whale. 35. Orion, Orion. 36. Eridanus, Eridanus. 37 . Lepus, The Hare. 38. Canis Major, The Great Dog. 39. Canis Minor, The Little Dog. 40. Argo, The Ship. 41. Hydra, Tlie Hydra. 42. Crater, The Cup. 43. Corvus, The Crow. 44. Centaurus, The Centaur. 45. Lupus, The Wolf. 46. Ara, The Altar. 47. Corona Australis. , The Southern Crown. 48. Piscis Australis, The Southern Fish. 2. NEW SOUTHERN CONSTELLATIONS. 4. Columba Noachi, Noah’s Dove. CHAPTER VI. 39 2. Robur Carolipum, 3. Gras, 4. Phoenix, 5. Indus, 6. Pavo, 7. Apus, Ms Indicay 8. Apis, Musca, 9. Chamelion, 10. Triangulum Australe, 11. Piscis volans, Passer , 12. Dorado, Xiphias , 13. Toucan, 14. Hydrus, The Royal Oak. The Crane. The Phoenix. The Indian. The Peacock. The Bird of Paradise The Bee or Fly. The Chamelion. The Southern Triangle. The Flying Fish. The Sword Fish. The American Goose. The Water Snake. 3. HEVELIUS’ CONSTELLATIONS. Made out of the Unformed Stars . The Lynx. 1. Lynx, 2. Leo Minor, 3. Coma Berenices, 4. Asteron and Chara, 5. Cerberus, 6. Yulpecula and Anser, 7. Antinous, 8. Scutum Sobieski, 9. Lacerta, 10. Camelopardalis, 11. Monoceros, 12. Sextans, The Little Lion. 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 *, the 40 ASTRONOMY. second in order 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 sixth 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 star s. 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. Let s' be the situation of the star s 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 cliaracters, and giving their right CHAPTER VI. 41 ascensions and declinations or tlieir 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 17^000 stars. The Catalogues ofLacaille, 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 stars are not entirely exempt from change. Several stars, which are mentioned bv the an¬ cient astronomers, have uow 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, ry 4 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 Nebidce. The Milky 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". 1 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 (14), 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".l 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 Equinoxes . 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, jp'pT) a circle about CHAPTER YI. 43 the pole P, at a distance equal to tlie obliquity of the ecliptic^ EQG the equator, and P/;LQ 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'pT); 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 j 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 Annual 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".l (32), and E m 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 a, 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 he 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 great 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 ea — pP = E'Ehi = obliquity of the ecliptic, R = Ea = right ascension, and v = variation in declination. Then , pYp' sin wP 50". 1 sin , . 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 pi = sp — sn = sp — sp* = 90 3 — as — (90 — a's) =. a's — as = v. Therefore v = 50". 1 sin cos R. When the declination is north, as in the figure, the sign of v is the same as the sign of cos R; but when the declination 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 V = its an¬ nual variation in right ascension. Then V = E 'a' —Ea = E'a' — mb = E m -p a'b.; E 'm =. E'E cos EE'm --= 50" 1 cos p'n = pp' sin p'pn = 50". 1 sin ^ sin R; p'n 50". 1 sin a sin R psp' = sin p'i ■ ; and cos a CHAPTER VI. 45 a’b = a'sb gin a’s = psp' sin a's = 50".1 sin a sin R sin a' cos a's = 50". 1 sin -f 50". 1 sin « 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 declination 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 a = 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. V = 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 formulse; 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¬ lae, 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". 1 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 S000 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 Apparent Orbit — Kepler’s Laws — Kepler’s Problem . 1. It has been shown (6.37) that the vernal equinox has a retrograde motion along the equator of 46" a CHAPTER VII. 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 Mean place, or the Mean Equinox, 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 T |o 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 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 Day. 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 (3.5), 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, imt 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 Mean Solar Day, is equal to 24 h. 3 m. 56.555 sec. of sidereal time. 5. The ratio of 24 h : 24 h. 3 m. 56.555 sec. is the same as 1 : 1.0027379, 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; aud 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. 55.91 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 he 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 -f tan BD : tan AC — tan BD : : sin EC -f sin ED : sin EC — sin ED. From whence we have (App. 32), sin (AC + BD) : sin (AC — BD) : : tan (EC + ED) : tan A (EC — ED), or tan A (EC — ED) = tan ?fEC 4- ED) sin (AC — BD) 2 v ' sin (AC + BD) tan l CD sin (PA — 90° — 90° -j- PB) sin (PA — 90° + 90° — PB) tan ■} CD sin (PA -f PB — 180°) sin (PA — PB) Now knowing CD and ^ (EC — ED), we know EC and ED, which are the sun’s distances from the equinox at the times of observation. The sun’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. 8. The interval of time between two consecutive re- 8 50 ASTRONOMY. turns of the sun to the vernal equinox is called a Tropical Year. 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 100 years, its mean length, expressed in mean solar time is found to be 365 d. 5 h. 48 m. 51.6 sec. Hence 365 d. 5 h. 48 m. 51.6 sec.: 1 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 iu longitude, which is 50."1 (6.33), the sun only passes through an arc of the ecliptic equal to 359° 59' 9".9, during a tropical year. 11. 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 Year . Hence 359° 59' 9".9 : 360° : : 365 d. 5 h. 48 in. 51.6 sec. : 365 d. 6 h. 9 m. 11.5 sec. = the length of a sidereal year, expressed in mean solar time. The sidereal year therefore exceeds the solar, by 20 m. 19.9 sec. 12. When the sun’s apparent diameter is accurately observed, at different seasons in the year, it is fouud 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 CHAPTER VII. 51 is 31' 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 $ = apparent diameter AEB = apparent diameter A'EB' D = ES and D' = ES'. Then D sin * * = AS = A'S' = D sin a and takes place when the apparent diameter is least. 17- The curve which the sun’s centre seems to de, scribe in the plane of the ecliptic, during a year, is called the Apparent Orbit of the sun. 18. A right line, conceived to he 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. 19 . It appears from the change in distance between the sun and earth (IS), that the sun’s apparent orbit is not a circle; or at least that the earth does not occupy the centre. Let ADBF, Fig. 14, be the apparent orbit of the sun, E the earth’s place, A the sun’s place wiien the apparent diameter is least, B its place w hen the appa¬ rent diameter is greatest, and I) 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 180°. It follows therefore that EA and EB must form one straight line AB. If AB he bisected in C, then AC = § AB = | (AE 4 . EB). 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 Kepler’s first Law. It is deduced from investigations, founded on the observ¬ ed apparent diameter of the sun at different longitudes. Put ^ = sun’s apparent diameter at A, V = do. B, = do. D, m — J- (P + f) = sun’s mean apparent diameter, and n = \ ($' — ^). From the sun’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 , or H' = H. 48. Because H and v/ (i — e 2 ) 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 then 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. The sun’s true longitude may be obtained each day from its observed right ascension (6.19). The differ¬ ence between its longitudes on any two consecutive days will be its true diurnal motion at that time. Hence we may, by repeated observations, find the time when the true diurnal motion is equal to the mean. Know ing then the meau 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, B' = A' + y. Hence B' — B = A' — A + 2 V, 2V = (B' — B) — (A' — A), y = § (B' — B) — i (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. CHAPTER VII. 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. 00 . 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 lz _ number of seconds in V S06264".8 Then, it is found by means of an analytical investi- igation, which we shall omit, that e \ K — K s 768 087 983040 01 . 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 ce ntre, 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 Attraction of Gravitation . Newton was led 70 ASTRONOMY. 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 Newton’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 aud 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 (1). 54. The difference betw een the mean place of the equinox in the ecliptic, and its true place, is called the CHAPTER VIII. 71 Equation of the Equinoxes in Longitude, or sometimes, the Lunar Nutation. 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 1Reduction 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 T 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 Ascension of Mid-Heaven. 1 . Solar days, being determined by the apparent diurnal motion of the sun (7-4), are used for all the eommou 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 mill night to noon, anil 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 IS. 3. The angle contained between the meridian and a declination circle passing through the sun or any one of the heavenly bodies is called th v Distance of the body from the Meridian , 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 15° 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 15° 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 15° in a solar hour. * A. M. are the initials of Ante Meridian , forenoon, and P.M. of Post Meridian, afternoon. CHAPTER VIII. 73 5. Since 15° of the sun’s distance from the meridian corresponds to 1 solar hour, 1° 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. 17° SI' 38" _4__ lh. 9m. 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 5 9' 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 eeliptic 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 Noon. And time reckoned from mean noon is called Mean Time. The time when the sun is really on the meridian is called Apparent Noon. And time reckoned from ap¬ parent noon is called Apparent Time. 11 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¬ nal 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 vernal equinox in the equator, YA be the equation of the equinoxes in longitude, YB, in right ascension, VS 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 VD, 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 = VD — VL == equation of time. Now (App. 49) tan VB = tan VA cos V; or since VA and consequently VB, only amounts to a few seconds (7.54), VB = VA cos V = VA cos 23° 28' = .917 VA. Hence DL = VD — VL = VD — (LB + BV) = VD — (LB + .917 VA). 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 1 second. Since VA when greatest is only 18" (7.54), the value of .917 VA can not exceed 16",5, or 1 second 6 thirds, in time. 10 . 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 4- R + P 4- .083.^. Put M = sun’s mean longitude, and a = obliquity of the ecliptic; then ( 11 ) q cos = equation of equinoxes in right ascension. Hence, VL = RL + BV = M 4 - q cos sin D cos L , p sin (D -f L) — sin “a 1 = ------- H sin D cos L 2 cos \ (D + L + H)sinl(D + L —H) sin D cos L (App. 21) 2 cos V 2 ) Si “( D + L + H -H) sin D cos L cos D + L 4 . H\ sin D + L + H H sin i P = v/ 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 can not then be depended on with certainty. In general the best time is three or four hours before or after noon. 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. 21. 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 Altitudes, 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¬ nation^ to find its azimuth. Let Z = PZS == the sun’s azimuth; then (App. 34), cos PS — cos PZ cos ZS cos PZS == or cos Z == sin PZ sin ZS cos D — sin L sin H cos L cos H But (App. 9), cos Z 2 cos 2 |- Z L and (App. 14), sin L sin II = cos L cos H — cos (L -f- H) , Hence, 2 C0g 21 ^_i _ cos D 4 - cos (L -f H) — cos L cos II cos L cos H _ cos D -f cos (L 4 - H)_j cos L cos H 2 cos 2 -- Z = cos ( L + H ) + cos D cos L cos H _ 2 cos -J (L + H 4 - D) cos-i (L + H — D) cos L cos H (App. 22) CHAPTER IX. 2 COS COS cos \ Z = s/ — 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 So P + _L_+_ H ) cos ( D + L + H 2 °) cos L cos H JD±L±H) cqs |DrL + H_ D | cos L cos H ASTRONOMY. 8(5 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 sun, the zenith distance ZS = 90° -f- refraction — parallax. Let R = refraction —parallax; then ZS = 90° -f R, and by an investi¬ gation nearly similar to that in article 20th, we have, D + L + _Rj cos + L. + . R _ 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. 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 Rin the formula in the last ar¬ ticle, we have, sm siniP= v/. >i» (D4-_L±J£) co ,(D+_L + jy_ l8 .) 2 P = -;-fr- j - 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 tioilight and the sun's declination at the time . In the solution of this problem, twilight is supposed to com¬ mence when the sun 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 2a = 18°, and let S Fig. 19, be the situation of the sun when twilight commences, and S' its situation in the horizon. Then PS = PS', ZS' = 90°, ZS = ZD -f 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, sin ZPE — sin — s ‘ n a _ sin 9° sin PZ 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° -f a) : : sin latitude : sin declination , or cos a : — sin a : : sin latitude : sin declination. Hence sin declination = — — n —? sin latitude = — tan a sin cos a latitude. The value of sin declin. being negative, shows that the decli¬ nation 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 Almanac*, 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 Jlchronical 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 belo^v the horizon. * The Nautical 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. 31. 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. 3&. 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 13 ga ASTRONOMY. 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 rc- 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. 17 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 11 s . £0°. 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. ZODIACAL LIGHT. 3 5. 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 sun’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 100°, and sometimes not more than 40° or 50°. CHAPTER X. Of the Moon . 1. T. re 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 oneach succeeding day. This motion is not uniform, hut at a mean, it is 13° 10' 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 Nodes. That node in which the moon is, when passing from the south to the north side, is called the Ascending Node . The other, in which it is, when passing from the north to the south side, is called the Descending Node . The nodes are distinguished by the following characters. Ascending node, &. Descending node, £ 5 . 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 * ew Moon. 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. CHAPTER X. 93 7 . From clay 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 Dichotomized , or to be a Half Moon. 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 Moon. A phase of the moon, between the first quarter and fall moon, is represented in Fig. 22. When the moon appears under this shape, it is said to be Gib¬ bous. 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 7 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 7 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. ASTRONOMY. 10 . The interval from new moon to new moon, or from full moon to full moon, is called a Lunation , or Lunar Month . Its mean length is 29 d. 12 h. 44 m. 3 sec. 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 9 , and the middle points, between the syzigies and quadratures, are called the Octants . Some of these are designated by characters, as follows: Conjunction, <5, Opposition, 29' 22 " _ R 1762 53' 52" * 3230 2U. 1762 6460 2R. — very nearly. 11 Hence the diameter of the moon is about T 3 T of the equatorial diameter of the earth, and consequently its surface is about T ? of the earth’s surface, and its vol¬ ume, about - 1 - 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 0, 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, 104 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 z— real diameter of the moon. OD EO sin MEO We have, CO tan MCO = MO = sin OME EO sin MEO sin MES' Hence, tan MCO EC cin MEO sin CES' EC sin MEO CO* sin CES' 2 sin MEO nearly. sin sin OME sin MEO sin CES' (7.13). sin^ sin CES' Height of the mountain — a M = MC — OC OC = OC. = OC. 1 cos MCO sin MCO 1 —- cos MCO OC. cos MCO 1 — cos MCO sin MCO OC cos MCO sin MEO tan MCO = OC. 1 cot h MCO sin MCO tan MCO (App. 11). — OC tan MCO tan £ MCO = h OC tan 2 MCO _ A 0G 4 sin 2 MEO = d / sin MEO \ 2 — " 2 * sin 2 / sin 2 CES' * \sin ^ sin CES'/ . / ang. MEO \ 2 = * U 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. 403 lion 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 sun’s, at noon of the given day,* S = daily motion of the sun 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. T.M Then24 h : T :: M : = moon’s motion in right ascension during the time T. Hence A = R -f — 24“ Also 24" : T :: 360’ + S : A = T - ( 360 ° + S) 24" * When the moon’s ascension is less than the sun’s, it must be increased by 360° or 24 h. 10G ASTRONOMY. Therefore = R + ™ 24‘* 24 h T. (36(b 4 - S) = 24 h . R + T. M, 24 h . R 24 h . R “ 360° + S — M “ 24 h + S — M* 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 then become, T = 24 h R 360° + S + M’ For a fixed star, M = 0 , and the formula becomes, 24 u R 360° + moon’s rising and setting. 46 On account of the moon’s change in declination, 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 h -f S — M : 24 h :: semr-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 h : 24 h + 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. 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 pftb, 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 E b, and the true latitude A a, to the apparent latitude B b. 49. The difference between the true and apparent longitude of a body, produced by parallax, is called the Parallax in Longitude; and the difference between the true and apparent latitude, is called the Parallax 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 JSTonagesimal De¬ gree of the Ecliptic . 51. The data usually given to calculate the moon’s 108 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 Parallax . 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 I another, passing through p and Z. Because P is the pole of EQ, the pole of pPD is in EQ; and because p is the pole of EC, the pole ofpPD is in EC; the point E, in which EQ and EC intersect each other, is, therefore, the pole ofpPD; 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 0, is the pole of pZnl, and On and 01 are each 90°. Consequently, n is the nona- gesimal degree of the ecliptic. Also, E n is the longitude of the nonagesimal degree, and nl is its altitude. These quantities are used in finding the parallax in longitude and latitude, and must first be found. 53. Put = P p — obliquity of the ecliptic, H = ZP = complement of the reduced latitude, CHAPTER X. 109 M= EM == right ascension of the mid-heaven, n .== E n = longitude of the nonagesimal, h — pZ — 90° — Zn — nl = altitude of the nonagesimal, L = Ea = moon’s true longititude, A = pA = moon’s true distance from north pole of the ecliptic, 33 - — moon’s horizontal parallax, reduced, n = moon’s parallax in longitude, 7 T = moon’s parallax in latitude. In the triangle pPZ, we have given Pp, PZ, and pPZ = 180° ZPD = 180° — DM = 180° — (90° — EM) = 90° + EM = 90° -f M, to find pZ — the altitude of the nonagesi¬ mal, and PpZ = Cn = 90° — E n = the complement of the longitude of the nonagesimal. LONGITUDE OF THE NONAGESIMAL. Let S = PpZ + PZp, D = PpZ — PZp, E = 180° — * S, and F = 180° — | D. Then (App. 41), tan | S == cos -i (H — *) cos -J cos (H (H + *) — a) tan(180° |S)= cos cos (H + ") — *) cos cos (H (H + cos 4 (H + ») __ cos (H — *>) COS (H 4- a) or tan E = cos 1 (H H i! COS 4 (H + a) . cot ‘ (90° + M) . tan 4 (90° — M), . tan (180° — 45° + \ M) , tan 4 (270° + M) . tan 4 (M — 90°), . tan 4 (M — 90°). Again (App. 42), tan 4 D = cot ' (90° + M), sin 4 (H + *) Whence, by transforming as above, we have, tan F = sin % (H ~ tan t (M — 90°). sin | (H 4 - *) 110 ASTRONOMY. Now IS + |D = VpZ = 90° — n. Hence, n = 90° — (4 S -f 4 D), or w = 450° — (i S 4- 4 D) = 180° — 4 S + 180° — i D + 90°. Consequently, n = E -f F -f 90°, rejecting 360', when the sum exceeds that number. ALTITUDE OF THE NONAGESIMAL. We have (App. 44), tan ih = s -j n . 1 S . tan i (H — ») sin 3 D sin 3 E sin 4 F* tan 4 (H — *). (5.5). PARALLAX IN LONGITUDE. 54 . The triangle ApB, gives, . „ • » t> sin AB sin ZBp sin n == sin ApB =- L sin A p _ sin w sia (N 4. p) sin ZB p sin A In the triangle ZpB, the angle ZpB=nb=na 4- ab= Ea—En + cib—h—n -f n, and sin ZB» — sin P Z sin Z P B _ sin/i sin (L — n + n) sin ZB sin (N 4- p) tt ♦ rr sin®-sin h . , T , _v /ri . Hence sin n — —.—.— sm (L — n 4- n) (C). sin A sin A sin n = sin (L — n + n) = sin (L — n) cos n — n) sin n = sin (L — n) cot n 4 . cos (L — n) sin w sin h 4- cos (L — n) sin n sin A sin w sin h cot n sin A cos (L — n) sin w sin h sin (L — n) sin (L — n ) Make tan « = si n > sin h sin (L - ») = fe sin ( L - n) . sin A sm A CHAPTER X. lit Then, - sm A sin »■ sin h sin (L — ri) . cos it — cot it = --, sin it and cot n = cos w cos(L— ri) sin it sin (L — ri) sin (L — ri) cos it — cos(L — ri) sin it __ sin(L — n — u ) sin (L — n) sin it sin (L — ri) sin u Hence, tan n = — sin w. sm(L— n — u) PARALLAX IN LATITUDE. 65 . The triangles pZA and pZB give (App. 34), cos ZB — cos ^ — cos cos ^ — cos — cos cos ^ sin Zp sin ZA sin Zp sin ZB or, cos A —cos h cos N cos (A 4 . *■) — cos h cos (N 4 - p ) sin N sin (N 4 - p) cos A sin (N + p) — cos h cos N sin (N -f p) = cos (A -f ri) sin N — cos h sin N cos (N 4 p) cos (A 4 . «r) sin N = cos A sin (N 4 - p) — cos h [sin (N 4 - p) cos N — cos (N 4 - p) sin N] = cos A sin 4 - p) — cos h sin p (App. 13) — cos A sin (N 4 . p') — cos h sin zr sin (N 4 - p) (5.5) = sin (N 4 - p). (cos A — sin w cos h.) Therefore, cos (A 4 - n) = j Z.P), (cos A — sin cos h). sin N But, sin A P sin Z P A = s ; n p zb = sinB P 8inZ P B sin ZA ^ sin ZB Qr sin A sin (L — n) __ sin A 4 sin (L — n - f- n) sin N sin (N+p; sin (N 4 p) __ sin (A 4 gr> sin (L —n 4 - n) sin N sin A sin (L — n) Hence,cos(A + *) = >in . ( A .+ y)"° ( L ~ n + ?> sin w cos h) sin A sin (L — ri) m . (cos A — j s a 1 x sin (L — n 4 - n) , * sin^-cos/tx /t ,x cot (A + ri) = —4---(cot A---), (E) sin (L — ri) sin A 112 Make tan x = Then, cot (A 4 - *) ASTRONOMY. sin w co? /t sin A sin (L— n -f- n) sin (L —w) sin (L —n 4- n) ^cosA (cot A — tan x ) sin x cos ;) sin (L — n) ' sin A _sin (L — n 4 n) cos A cos x — sin A sin z sin (L — n) sin A cos x __ sin (L — n 4- n) cos (A 4- x ) sin (L — n) sin A cos x (.n The apparent latitude (A 4 - *■) being calculated by either of the formulae (E) and (F), we have ^ = (A 4- w) — 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 4 - tt) _ ^_ sin a* cos h sin (L — n - j- n) sin A co t a — s * n eos ^ cot (A 4- ,r ) sin A sin (L — n 4 - rr) 1 f a , x sin a-cos 1l . x cot A — cot (A 4 - yr) — - — cot (A 4 . *) _l ' sin A ^ sin (L — ri) cot (A 4 . «■) sin (L — n 4 - n) Hence (App. 27), sin tt sin a- cos h sin A sin ( A 4- x) sin A cot (A 4 - t) [sin (L— n 4 . n) — sin(L— n)J sin (L — w 4 - n) __ sin -zr cos h _ 2 sin § n cos (L — n 4 - h n) cot (A 4 - *•) sin A sin — n 4 - n j (App. 21) sin 5r — sin a* cos h sin (A 4 ») — CHAPTER X. 113 2 sin 2 II sin A cos (L — »i + |n) cos (A 4 . w) sin (L — n 4 - n) =s sin w cos h sin (A 4 . «■) sin n sin A cos (L — n-|-|n) cos (A 4 - «•) cos 2 n sin (L — n + n) (A PP . 7) But ( 54 . C), — s l n _ A — — sin ®- sin h. Therefore, ^ sin (L — n 4 - n) 5 sin «r = sin ar cos h sin (A f w) sin 73- sin h cos (L —n + in) cos (A 4- *■) cos 2 n tan li cos (L — n 4 - h n) cos 2 n ( 6 ) = sin®- cos h j^sin (A + «r) (A + cos Make tan y = _ tan /i cos (L — n 4 - $ n) cos 2 n . Then, sin 7r = sin ®* cos /i sin (A 4 - — _ in cos (A + I L cos y J 30S hr-, - , _ . "I —— |^sin (A + *■) cos y — cos (a 4- *) sm y J sin (a -f » — y) (H) sin -sr cos h cos sin ®* cos h cos y sin w cos h . , . -v sm (A — 1 /) cos *■ cos y cos (A — y) sin 1 = cos y sin w cos h . . . , sin®-cos/i , . sm (A — y) cot 5 r H-cos (A— y ) cos y COt IF = cos y sin ®- cos h sin (A — y cos y cot (A — y ) Make tan v = sin ®- cos h sin (A — y) cos y a) Then cot «• = cot v — cot (A — y) = -^-— sin v sin (A — y) (App. 21) TT . sin (A — y) sin v sin (A — y — v) 16 114 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 S' = moon’s horizontal semidiameter, S'' — moon’s apparent semidiameter at a given situation above the horizon. Then (7.13), sin S' __ D .p." gx BC _ sinBAZ __ sin (N + p) „ ■sinT = D 7 = C lg ' j AB “ sTnBCZ ii^N C ’ '' _ sin (A + g-) sin f — it + n) sin A sin ( L — n) sin ^ sin (A 4- *■) sin (L — n 4 - n) Hence sin S' = sin A sin (L — n ) (D We may obtain other formulae for expressing the relation be¬ tween S and which will be more convenient when the paral¬ lax in latitude is found by the second method. Let Ps r, Fig. 25, be the arc of a great circle, bisecting the angle ApB, and ZGsL, another arc of a great circle, perpen¬ dicular to pr. Then pG — pL, the angle pGL == pLG, Gps == § n, and Zps = npr = npa + apr = L — n 4 - |n. Hence tan ps == tan pZ cos Zps = tan h cos (L — n + 5 n) ~ tan ps tan h cos (L — n + h n) tan pG =--— = - v — --= tan y cos Gps cos 5 n (56) Therefore pG = p, AG = Ap — Gp = A — y and BL = Bp — Lp = A -f *• — y. Now Hence sin BL sinBZL sin AZG sin AG sin BZ sin L sin BL sin BZ sin AG sin AZ sin G sin AZ* or sin (A + ?r — y sin (N + p) sin S' sin (A — y) sin N sin S CHAPTER XI. 115 and sin (M) sin £ sin (A -f 57 — y ) sin (A y) The last formula may be reduced to another still more simple For we have (56 H and I), • *. > sin w cos w . . , . . tan v cos y sm (A -f n — in = —-f.and sin (A — y) — — -p V 1 civ. nr\c 7i* ^ ^ ' Sin ZT COS ll sinar cos h TT sin (A + 7 r — ?/) sin «■ Hence,-- - = ——, sin (A — y) tan v sin ^sin *■ and sin = tan v (N). CHAPTER XI. Eclipses of the Sun and Moon. — Occultations . 1. 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 abg , Fig. 26, be sections of the sun and earth, by a plane, passing through their centres S and E, and AaC and B6C, tangents to the circles ABG and abg. Disregarding, at present, the action of 116 ASTRONOMY. the earth’s atmosphere, in changing the direction of those rays of the sun which pass through it, the tri¬ angular space aCb will be a section of the earth’s shadow. The line EC is called the Axis of the earth’s shadow. 4. With a view to conciseness of expression in some of the succeeding articles, we shall put R = E b = ra¬ dius of the earth, P = moon’s horizontal parallax, J) = sun’s horizontal parallax, d = moon’s apparent semidiameter, and <1 = 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 = EC6 + EB6, or * = EC b + p. Therefore, EC5 = S — p, , T1 ~ R R.ft R.n and EC = . _^ f = = --. sinEC6 EC b S—p R< )i Now (5.8), the moon’s distance from the earth = 1-. Hence, as S — p is less than half P, the distance EC, is more than double the moon’s distance from the earth. 6. Let hMh' 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 A dh and B ch' be tangents to the sections of the sun and earth, crossing each other between them. When any part of the moon enters the space be¬ tween the lines dh and 5C, that part will, evidently, be deprived of a portion of the sun’s light, and will therefore appear less bright. As the moon approaches CHAPTER XI. 117 the line bG, its light continues to be diminished; and and when the edge comes in contact with bC, 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 cdlih' 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 MEZi' is the semi-diameter of the penumbra. 10. The semi-diameter of the earth’s shadow is ecpial to the sum of the moon and sun’s horizontal pa¬ rallaxes, less the apparent semi-diameter of the sun. Tn the triangle EmC, we have, the angle mEM = E mb — ECm. 118 ASTRONOMY. Now, the angle mEM is the semi-diameter of the earth’s sha¬ dow, E mb is the moon’s horizontal parallax = P, and ECm = P c 5 )- Hence, Semi-diameter of the Earth’s Shadow = P — (<^ — p) = P 4 - p-*- 11. The diameter of the earth’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 ^ = 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 4 . y -f d — • p-h From the triangle mEC, we have, mEM = E mb — ECm = d p.d P ^ — (19), d -j- P P -P —p') = d +p' — P —P S' P P — P 21. The greatest breadth of the moon’s shadow, at the earth, is about T X T . part of the earth’s diameter. The. expression for the breadth is, d—J V—p , 2U. To obtain the breadth, we have Mm *= an g* e -MEm __ l . R.n (d—• p4? 124 ASTRONOMY. 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 + cT P — p 2R. To obtain the breadth, we have, ,, __ EM. ang. ME/i' _ 1 R.?z (d -f F). P ~ n n T"* and hh' = 2 MV = d ~ t-. 2 R. F—p d+t P ~P R. Taking d = 881", ? = 977."8, P = 3232" and p = 8".7, we obtain she greatest value of, hh' = 2 R = 1—. P — p 3223 2 R = 2 R nearly. 40 J 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 + 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 conical surface, which circumscribes the sun and earth. Then the angle, M'ES = EM 6 + EC6 = P — p + * 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 4* d -f CHAPTER^!. 125 28. From the expression P — p 4- d + ^ 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 sun: but when this difference is less than 13° 14/, 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 sun. NUMBER 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 KINDS 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 Appulse . When a part, but not 126 ASTRONOMY. tlie whole of the moon, enters the earth’s shadow, the 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, if 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 oh- scures the whole of the sun, the eclipse is said to be Total. When the moon’s disc is entirely interposed between the spectator 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 An¬ nular. 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. ECLIPSES OF THE MOON. 32. The apparent distance of the centre of the moon from the axis of the earth’s shadow, and the arcs of the moon’s orbit and of the ecliptic passed through by these, during an eclipse of the moon, being necessarily small, may without 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. 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. Let t be some short interval of time expressed 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 CM' will be the distance of the centres at that time. Draw M'F perpendicular, and AMB parallel to NF. Then CC' is the motion of the centre of the shadow in the time f, 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 differ by 180°, from the longitude of the sun, the apparent motion of the sun is the same as that of the centre of the shadow. Therefore CC' expresses the sun’s motion in longitude in the time t . And consequently C'F = CF — CC' = the difference of the moon’s and sun’s motions in longitude, in the time t. 34. Make CG equal 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 f, from opposition. We therefore obtain the distance of the centres of the moon, and shadow, the same, if instead of allowing to each 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 difference of the motions of the moon and sun, in longitude. 35. From Astronomical Tables we can get the hourly motions of the sun 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 Mto M', m — moon’s hourly motion in longitude, n = moon’s hourly motion in latitude, r sun’s hourly motion in longitude, * = moon’s latitude at opposition, 128 ASTRONOMY. I *= angle M"MR, s = P -f ;> — ^ + -tv (P 4- p — ^) = semidiam. of earth’s shadow (10 and 15). Then CF = m.f, CC' = r.t and RM" = HM' = n.t, MR = CG = C F = CF— CC' = m.t — r.t — t. (m — r) y tan. I = tan M "MR = RM'' __ t.n RM t. (m —r) As the expression for the tangent of the angle M"MR does not involve f, it is evident the angle itself will continue the same, whatever be the value of t. Hence the point M" moves in the line PMQ, which is therefore called the moon's Relative Orbit . 36. In the triangle M'MR, we have, MM" = MR cos M'MR t.(m — r) cos I The distance MM" is the moon’s motion on the relative orbit, in the time t. If we take t — 1 hour, we have, 7H T The moon’s hourly motion on relative orbit = --. cos I 37. Let AB, Fig. 28, be the ecliptic, C the centre of the earth’s shadow at the time of opposition, and CK perpendicular to AB, a circle of latitude. Make CM = a, 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 -f d, describe arcs cutting DH in D and H; and with a radius — s — 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 DII, it is evident that D is the place of the moon’s centre at the beginning of the eclipse; E, its place at the beginning 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 — cl is less than CF, the eclipse can not be total. 38. Because CD = CH, and CF is perpendicular to DH, we CHAPTER XI. 129 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 be is parallel to CM, the angle FMC = Mc&. Therefore, MCF =*= bMc = I (35). MIDDLE OF THE ECLIPSE. 39. In the triangle MCF, MF = CM sin MCF = a sin I. But, taking x = interval of time between the middle $f the eclipse, and the time of opposition, we have (36), MF = *•<”— r ) . COS I Hence, x. (m — r) a sin I, or x = a sin I cos I cos I m — t Now if M = the time of the middle, we obtain, a cos I sin I M = T ? i = T ? m 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. BEGINNING AND END OF THE ECLIPSE. 40. Let B = the time of beginning, E = the time of the end, and x = the interval between the middle and either of these. Then, = DF = FC cos = >/ (s -f- d ) 2 — a 2 cos 2 I =* */ (s 4- d — a cos I). (s + d -f a cos I) qt x — cos I >/ (g 4- d — a cos I . (s 4. d + a cos 1) m — r Hence, B = M —- rr, and E *= M -f a, become known, 48 130 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 x' = the interval between each of these and the middle. Then, __ cos I V (s — d — A cos I). (5 — d 4 - cos I) m — r B' = M — x\ and E' = M 4 - x'. 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, 0 _ NS _ CS —CN_ CS —(CF —FN) ' TfNV t VNV “ t \NV CS + FN— CF __ 12 (CS + FN — CF) “ ^NV NV. 12 (P 4 -p P+d A cos I) ~2d _ (P 4- p 4- d — ^ —acos I). 6 _ CONSTRUCTION OF AN ECLIPSE OF THE MOON* 43. The times of the different circumstances of an eclipse of the moon, may easily be determined by a geometrical construction. CHAPTER IX. 131 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 II. ECLIPSES OF THE SUN. 44. It lias been shown in a preceding article (&7), that when the angular distance of the centres of the sun and moon is equal to P — jp + ^ 4- 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 — jp + ^ 4 - 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 moon. 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 determine the circumstances of the eclipse for any particular place. 132 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¬ ginning 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 sphere, 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 sun 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 effect of the earth’s atmos¬ phere in refractiong the rays of light. 48. A plane between the earth and sun, perpendicular to the straight line 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 CHAPTER XI. 133 ■of the earth, and which consequently passes through the earth’s centre, is called the Universal Meridian, When by the diurnal rotation of the earth on its axis, any place on its surface is brought to coincide with this plane, the sun 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 D, 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 LMbe 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, CE$ = EQU — ESU or Apparent semidiarneter 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 = FG ~ EC tan(P— p) EU Therefore,tanCEM= SIM — FG tanfP_ p) EC EU V ' Fh or,CEM = §r FD In like manner, MEL = —. (P— p). EU If X = MEL == the apparent distance of the projection of the sun’s centre from the universal meridian, Y = CEM = the ap¬ parent distance of the projection of the centre, from the plane 131 ASTRONOMY. passing through the centres of the earth and sun, perpendicular to the universal meridian, and R = EU == radius of the earth, we have, Y FD R * (P — p), and Y = FG R * (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 sun, 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 sun’s decli¬ nation. Put, D = TP as the sun’s declination, U = DLH = the hour angle from noon, II = CH = the reduced latitude of the place. Then IIL = EH sin HEP = R cos H, and DF == DL sin DLH = PIL sin U = R cos H sin U. Hence (54), X = (P— p) = cos H sin U. (P— p) A. R EL = EH cos HEP = R sin H, EM = EL cos TEP = R sin H cos D sin (II 4 - D) + sin(H — D) (App. 16) ML = EL sin TEP = R sin II sin D, LF = DL cos DLH = IIL cos U = R cos II 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 = RcosU. sm ( H + P )~ sl li H -~. D )(App. 17). £ When the latitude of the place and declination of the sun ate both north, as represented in the figure, ER = EM — MR. It will be the same, when the latitude and declination are both south. But when one is north and the other south, ER = EM + MR. Therefore, FG = ER = EM t MR = R. sin ( H + D ) + si » ( H — D A T R cos U. sin (H + D) — Sin (H — D) 2 Consequently (54), y _ sin (H + D) + sin (H — D) ( P __^ 2 ^ sin (H + D) - sin (H - D) cos v (p _ g 2 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. 136 ASTRONOMY, latitude of the place, and make EG, El, FH, and FK, each equal to D, the sun’s declination. Join GH, EF, and IK, and bisect Rv in N. Through N, draw LNM parallel to EF. Make NO equal to wE, 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 WY in U. In this construction, the sun’s declination is supposed to be north, and the time in the afternoon. When the declination is south, the semicircle ZRQ must be on the upper side of PO, and WN must be produced to meet it in T. When the time is before noon, the arc YW must be laid off from \ r to the right hand. Now, NO = 10 E = w¥ = CF sin FCw = cos II. (P — p) 7 SU = NV = NW cos WNL = NO sin U = cos H sin U.(P — p). Hence, (55. A), SU = X. Also, Cu = CH cos v CH = sin IICB. (P—p) = sin (H -f D). (P — p),- CR = CK cos RCK = sin KCB. (P —p) = sin (H—D). (P-rt, fixr Cv -f CR sin (H D) -|- sin (H D) /p v 2 2 * v Fh RN ~~ s * n _ p ^ 2 2 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. CHAPTER XI* 437 • NS == NT cos YNW = NR cos U = sjn(H + D)-sin(H-D), cqsn(p _ p)> CS == CN t NS= sin(H i- D) i- sin (H D) 2 t — iS.:~ P)~g'° ( H ~ D ) cosU.(P— p). 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 earth and sun, is equal to Y, the point U is the projection of the sun’s centre, for the spectator at the given time, and place (54). 57. It is evident from the construction, that the semicircles PYO and fRQ, depend only on P — p, the difference of the ho¬ rizontal parallaxes of the moon and sun, D the sun’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 fRQ, and draw Ts parallel to CD. The distance Ts, applied on YW, from V to U, will give the point U the same as before. 59. From the similar triangles WNV and TNS, we hafe WN : TN : : WV : NS. But WN = NO, TN = NR, and NS = UV. Therefore, NO : NR : : WV : UV. Hence (Conic Sections , the point U is in an ellipse, of which PO is the trans¬ verse axis, and Rv 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 axes PQ and Rv. 49 138 ASTRONOMY. POSITION OF THE MOON ? S RELATIVE ORBIT ON THE CIRCLE OF PROJECTION. 60. Make Da and Db each equal 23° 28', the obliquity of the ecliptic; join ab and on it describe the semicircle adb. From b lay off the arc bet 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 a&, and through m, draw CmZ. 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 declination circle. In like manner, the line in which a circle of latitude passing through the centre of the sun, 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 = bd = sun’s longitude, a = Db = obliquity of the ecliptic, S = the angle of position. Then bn = C b sin b CD = Cb sin mn = nd cos bnd = bn cos L = C b sin » cos L. Cn = Cb cos b CD = Cb cos == c + x tan I, SF 3 + FG 2 == SG 2 , or (a — x) 2 + (c + x tan I) 2 = s 2 a 2 — 2ax + x 2 + c 2 -f 2cx tan I -f x 2 tan 2 I = s 2 (B) (1 4 . tan 2 I) .a 2 — 2 (a — c tan I) x =s 2 — a 2 —c 2 , 2 2. (a — ctan I) s 2 — a 2 —c 2 * 1 + tan 2 1 1 -f tan 2 1 5 2 _ 2. (ft — c tan I) , (a — ctanl ) 2 = s 2 — a 2 — c 2 1 -f tan 2 1 (1 + tan 2 1) 2 1 4 -tan 2 1 (a — ctanI) 2 (1 4 . tan 2 1 ) z CHAPTER XI. 143 __ (s 2 — a* — c 2 ). (1 + tan 2 1 ) + (a — ctan I) g (1 -f tan 2 1 ) 2 $ 2 4 - s 2 tan 2 1 — (c 2 4 - 2 a c tan I 4 - a 2 tan 2 1 ) ; (1 4 - tan 2 1) 2 s 2 . (1 4 - tan 2 1 ) — (c 4 - a tan I ) 2 = '(1 + tan 2 1 ) 2 “ s 2 _ (c 4 - a tan I) 2 \ ~ 1 4 - tan 2 I.' V s 2 . (1 4 - tan 2 I)/ =s 2 . cos 2 1 . (i - ( c+at y - c -°ii- 1 ) „,_ g-ctan l — 4 -cos I,/ (, _(i±ftanI)W_r\ 1 + tan 2 1 v s 2 / / , t\ o t ft .i ,(\ (c 4 - a tan I ) 2 cos 2 1\ x = (a — c tan I) cos 2 1 ± s. cos I v/ ^1 — ^- —J. - J i . . (c 4 - a tan I) .cos I Make sin 0 =^T- L - s Then, sin 2 < = £ + «tanI) 2 .cos 2 I ’ s 2 a j , /i (c + «tanD 2 cos 2 I\ / t . 2 \ . And -v/ — Z-- j = ^1 — sm 2 0 j=cos*. Hence, x = (a — c tan I) cos 2 1 ± 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 x in the for¬ mula, correspond to these two points. Hence the value of a; cor¬ responding to G, is, x = (a — c tan I) cos 3 I — s. cos I cos 0. When the moon’s apparent latitude at the time B 4 - t is less than at the time B — f, we must make, . . (c — a tan I) .cos I . , „ , sin 0 = A- 1 -, and we shall have s x = (a 4 - c tan I) .cos* I — s . cos I cos 0. 2tx Now, as m : a : :2t :-= interval from B — t to the true m time of the beginning. Hence if B' = the true time of beginning, we have, 144 ASTRONOMY. tan B = B — t l—l sin 6 — ( c±a tan I) cos 1 m s 4 - ((flTctan I) .cos 8 I —s cos I cos 6 The upper signs must be used when the apparent latitude is increasing , and themder, 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 4 . f, as for B — t and B 4 t. Then, if a, c, m,n,s, I and 0 , 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 1 = 2 , si ne= ^ otanl ) cosl , m s E' = E —t -f ( 5 . cos I cos 0 — (a± c tan I) cos 2 i) * ^ \ 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 -ft, the apparent distances of the moon from the sun, in longitude, and the moon’s apparent latitudes. Let SK be the ap¬ parent distance of the moon from the sun in longitude at the time G — t, SN, at the time G + £, and KI, NP perpendiculars to EC, the corresponding latitudes. Then if S q be drawn perpendicular to IP, q will be the place of the moon, when the apparent distance of the centres is least. Let x— KV, and a, c, m/n, s and I, de¬ signate the same quantit ies as before, but having the values apper¬ taining to the middle. From the similar triangles PIQ and SgV, we have, IQ : PQ : : Vq : SV. But IQ = m, PQ = w, \q = c 4 x tan I, and SV — a — x. Hence, m : n : : c + x tan I : : a — x , ?—* = 2 = tan I, c -f x tan I m x— ~ C - tan * = (a — c tan I) .cos* I 1 4 - tan 2 I v ' CHAPTER XI. 145 When the apparent latitude is increasing, x = (a 4 - c tan I) cos 2 I. Hence, if G' = the true time of greatest obscuration, we have, tanl=^, G=G— t 4 - (a=F c tan I) .cos 2 I —. m m 73. To find Sq, the nearest distance of the centres, we have (72), Yq = c ± x tan I = c ± tan I. (a^fc tan I) cos 2 I — c ± sin I cos I. (a c tan I). Hence, S q Yq cos I cos I ± sin I. (a qF c tan I) Now to find the quantity of the eclipse, if <^= the sun’s appa¬ rent semidiameter, and d — the moon’s apparent semidiameter at the time of the greatest obscuration, we have, Fig. 33. 1 ' > mn — Sq -f qn 4 - S m = S 7 + ^ — Sq + d—Sq = f + d — Sq = s—Sq = Digits Eclipsed cos I =F sin I.(atc tan I). Hence, * ( s —-Tasini.(aq=ctanI)\ .mn _ 6. \ cos I ' J ) 74. When Sq, the nearest apparent distance of the centres, is less than the difference, between $ and d, the eclipse will be either annular or total. It will be annular if l be greater than d, and total if d be greater than £ 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 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 S b, Fig. 34, be a circle of latitude, Sd a declination circle, and Sh a vertical circle, all passing through the sun’s centre. Then bSd will be the angle of position, and dSh the angle con- 20 146 ASTRONOMY. tained by the declination circle and vertical circle, passing through the sun. 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, II = the latitude of the place, a — the obliquity of the ecliptic, and U = the hour angle from noon. Then ( 6 . 20 and 21 ) cos A = sin L sin */, and tan bSd — cos L tan a (C) In the triangle PSZ, Fig . 18, we have (App. 87), crq £ _ co * s * n PS — cos ZPS cos PS * snTzps tan H sin A — cos U cos A sin U Hence, Fig. 34, tan dSk = sin a - cos Ucos^ sinU Also tan FS6 = 55 = £ - . (E) IS d The angle bSd will be to the left of S 6 , 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 S/i will be to the right of S d 7 in the forenoon, and to the left in the afternoon. By attending to these circumstances, and adding or subtracting accordingly, the an¬ gle b$h, and consequently ES h, becomes known. Thence, by ap¬ plying the angle FSG, we have the angle /iSG, 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 formulas, 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. CHAPTER XT. 147 78. 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 — l and E — £, 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— £, 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 — f), or which is the same G — B, and n = moon’s apparent motion in latitude during the same time. For the end we may give to m and w, 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 w, we obtain very nearly the inclinations of the apparent, relative orbit at the times of beginning, end, and greatest obscuration. As the value of x, in the equation B (70), must be small, its square may be omitted with but little error. We shall then have, a 2 — 2ax 4. c 2 4 2cx tan I = s 2 , 2. (ft — c tan I) .x — a 2 4 c 2 — s 2 , ct 2 -f c 2 — s 2 2. (a — ctan I)’ Hence B =B — 1+ ( - ?—+. c * ~~ ffj• .. G ~ B 2 2m ( a ^ c tan I) E' = E — t 4 ( ft2 4- c 2 » (E G ) 2m (a ± c tan I) q, _ q _ t [ft T c tan I) cos 2 I] . E — B) In each formulas the upper sign is to be used wheAthe 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 Jip- 148 ASTRONOMY. parent Ecliptic Conjunction , 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 regard 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 end of a lunar eclipse. PATH OF A CENTRAL ECLIPSE OF THE SUN. 80. The latitude and longitude of the place, to which the sun is centrally eclipsed, at a given time during the continuance of the central eclipse, may be easily 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 = (ED 2 — FD 2 ) = v/ [(P — p) 2 —X 2 ]. 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 FIFL, 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 noon, 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 pq 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 w ould have the projection of the sun’s centre also at D, would then have a central eclipse. Make TP = the sun’s de- CHAPTER XI. 149 dination, laying it to the left of T, when the declination is north , but to the right , when the declination is south , and draw PEP'. Through D, draw MDG parallel to TU, and DF parallel to AB. Then KM = y/ (EM 2 — EK 2 ) = ^ [ P _p)» —**]. With the centre E, and a radius equal to KM, describe an arc, cutting DF, in F, and through F, draw LFH, perpendicular to PP. Then PEH is the complement of the latitude of the place (80.). Make KG = FL, and through G, draw EGI. Then because NG = EK = X, and EN = G£ == FL, UEI is the hour angle, from the universal meridian at the required place (SO). 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 cen¬ 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 are on the same, or dif¬ 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 eclipse 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 Eclipse. 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 anti a star, that is, when the moon’s mean longitude is 150 ASTRONOMY. the same with the longitude of the star, their difference of latitude exceed 1° 37' there can not be an occultation; but if 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 El must be equal to the de¬ clination, and bd 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. To get the position of the circle of latitude, lay off the declina¬ tion of the star or planet, from D to &', and draw b'c parallel to ah. With the centre C and radius Cc, describe the arc cm', cut¬ ting dm, 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 O' 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 S b 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. 151 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, BC : EF : : AB : DE = BC But, EF = C a — BC cos BCE = BC cos BE. Hence, DE = AB - BCcos BE = AB cos BE ’ BC 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. 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 moon. 88 . It is evident from the formulae for computing the parallaxes in longitude and latitude, that those parallaxes are not sensibly ef- 152 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 x = the error in the moon’s longitude, and y = the error in the latitude. Then a + x 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 + x 2 . cos 2 Z + (c + y) 2 = s 2 , or, a 2 cos 2 Z + 2 ax cos 2 1 -f- x 2 . cos 2 1 -f c 2 2 cy -\-y 2 = s 2 . Now as x and y are small quantities, the terms involving their squares, may be omitted. Hence, if § (s 2 — a 2 cos 2 1 — c 2 ) = e, we have, a cos 2 1. x cy — e. As the errors x 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 2 1. x + c'y — e'. From these two equations the values of x and y are easily found. 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 x 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. 153 clination, the moon’s longitude and latitude may be calculated (0.19), 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 arc constructed, we most 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 21 154 ASTRONOMY. and latitude respectively, and x = the error in the difference of meridians. Then mx and nx will be the errors in the moon’* 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¬ midiameter, at the observed time of the beginning or end, we have, (a -f mx) 2 . cos 2 1 4 - (c 4 - nx) 2 = s 2 , or, a 2 cos 2 1 + 2 mx cos 2 1 + m 2 x 2 cos 2 1 4 - c 2 4 - 2 cnx -f n 2 x 2 = s 2 Now as the longitude of the place is supposed to be nearly known, x must be a small quantity, and the terms involving its square may be neglected. Hence we obtain, s 2 — c 2 — a 2 cos 2 1 jjj , - _ # 2 m cos 2 1 + 2 cn In a similar manner we may determine the longitude of a place, from an observation of an eclipse of the sun. 90. Some astronomers think the apparent diameter of the sun, obtained from observation and given in the solar tables, is t©o 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. Dusejour 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"£. He also supposes the moon's atmosphere inflects the rays of light, so as to produce an eflecton the beginning or end of a solar eclipse, or of an occultation, equal to CHAPTER XII* 155 a diminution of 2" in the semidiameter of the moon. This is called the Inflexion of the moon. Delambre 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. 1. Hitherto our attention has been chiefly directed to the Earth, and to those two conspicuous luminaries, the Hun 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 other. Their paths, therefore, cross the ecliptic. 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 motiou for several days. 2. The points in which the path of a planet cuts the plane of the ecliptic, are called the Nodes . The node through which the planet passes from the south to the north side of the ecliptic, is called the Ascending 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 sun. 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* 150 ASTRONOMY. tate Distance of the planet. The point is called the Deduced 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 Parallax. 6, The sun, earth, moon, and planets are frequently designated by characters, as follows. Sun O Juno TT 4 - Mercury - $ Ceres $ Venus $ Pallas - $ Earth - 0 Jupiter X Moon 3> Saturn - h Mars Vesta - -«») . . 1 + e. COS. X «•(!—e 2 ) . 1 -f- e. cos (x 4 - 6) »"= «• Q —«») 1 4 - e. cos (a; + , ??i 2 4- r 2 _ 2. (En 4- Gx 4. I n 2 4- £ ,r 2 ) m 2 4- r 2 _ Em 4- win 4- Gr 4 - r* m 2 4- r 2 and 6 = 2.(En + , G^ + in- + ^) we h m 2 + r 2 t 2 4. 2at = — b t 2 4- 2at 4- a 2 = a 2 — b ' + “ = ^ ^~ 6 > = a -Ta~8ir> _&C - i = ~A — & c 2a 8a 3 7 2 But _, and all the following terms of the value of t , are ex- 8 a 3 tremely small and may be neglected. Hence, t _b _En 4 GT.^n J 4§ 5r 2 2 a Em 4 - mn 4 . Gr 4- r*' _ En 4 - Gw 4 . ;§ n 2 _j_ § ' ’ * p_ q , i ~-g _ Etc -j- Gv 4~ 2 air 4 - i ^ 7r p (E 4 - n). m 4 (G 4 - *). r 46. Let T' be the time of the 2d internal contact for the earth’s centre, and 0', for the place A. And instead of E, G, m, r, n, *■, u and v, let E', G', m', r', rr, u' 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. 175 former. Attending to these changes, we have in like manner as before, T , = 6 , EV + GV + In'ii' +ji>V p = (E' 4- n'). m' — [G' + ** )• r ' Put a - E..+ G.4 inn,- jv* and (E + n). m + (G + sr). r p, _ E 'u' 4 - G v’ 4 - i if n' 4 - l v'v'_ ~~ (E' 4 ir;. »f — 4 - *■'). / Then T = 0 — /3P, and T' = *' 4 . / 8 'P, T' — T = 6’ — 6 4 - (/3 4 - /3'). P. If now d — V — 0 , and s = /3 4 - /3', we hare, T' — T = d 4 - sP. 47. If, for some other place in a latitude considerably different from that of A, d 1 and s' be the values of the expressions desig¬ nated by d and s, we shall have in like manner, T' — T = d' 4 - s'P. Hence, d 4 - sP = d! 4 - s'P, sP — s'P = d 1 — 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 ja' for the two different places of ob¬ servation. In the expression represented by fl, the values of E, G, m and r, are known. To obtain u 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 n = P sin, ft sin (L n 4 - n > sin. A sin sr = sin P cos h sin (A 4 - *-) ___ sinP sin h cos (L — n \ n) cos (A 4 - «•) cos \ n Or because n and are very small, 176 ASTRONOMY. sm n = sin P sin h sin L — n) sin A (Q) in % = sin P ^cos h sin A — sin h cos A cos L — n ^ (R) sirfl sin h sin L — n) Hence 45 , u = sin P sin A % sm 7r i • * • / . n v— — = - = cos h sin A — sin/i cos A cos L — nj. P sin P ^ J The other two quantities n and tt, contained in the expression represented by /3, depend on P, the quantity sought. But as n and 9 r 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, E u Gv Em - Gr and /3' = EV 4- GV E 'm' — G'r'’ we obtain a very near approximate value of P. Then, taking n 3 = wP, v vP, IT u' P', and «•' = v'P, and again calculating the values of /3 and /3', we obtain the correct value of P. 49. If sr' = the sun’s horozontal parallax at the time of the transit, sr, being the mean parallax, and V the radius vector, we have *r' = —. 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), sine a = ^_ - Sin ‘ --, and (37% V — v sin A _ zr v. sin l V — v Therefore P — «_«r' = *L_ _— ... ^ ineieiore r _ p * y y - y2 _ yj) V 2 — V» Hence, w = P. -— v 50. The transit of 1769 was observed at Ward bus, a small island on tlie north coast of Europe; and at CHAPTER XII. Otaheite in the South Sea, and the duration was found to he 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 mean 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¬ clined 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, 24 178 ASTRONOMY. in about 88 clays, 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 hut 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 he visi¬ ble in the United States. The calculation of a transit of Mercury is altogether similar to one of Venus. 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 beiug seen in opposition as Well as in conjunction. The disc of Mars does not present the phases of the two inferior planets, hut 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 17"> and when least, 3".5. The sidereal revolution of Mars is nearly 687 days, and its mean distance from the sun is 143 millions of tniles. Its diameter is about 4000 miles. CHAPTER XII. 179 Mars revolves on its axis from west to east in 24 h. 39 m., and 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 167 to 177. 58. W hen 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 180 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, ist 63. Careful and repeated observations, show that the motions of the satellites, 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 Kepler’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. 6-1. The crbits of the third and fourth satellites are elliptical. Those of the other two, have not been as¬ certained to differ sensibly from circles. 6J. The mutual attractions of the satellites on one another, produce inequalities in their motions, which must be taken into view, wlfm 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 exactly 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 ASTRONOMY". 182 that of the third , must always he 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 he 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 ilcemer a Danish astronomer, in the year I 67 J. 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. 183 Its velocity is therefore 180,000 miles per second, which is greater than any other, with which we are acquainted. 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 Be Terns , 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 another*. 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 snn 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. 484 ASTRONOMY. (lays, 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".1 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 40 to 44. 71 , 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° 21'. 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 . Saturn 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. URANUS 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".l 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. VESTA, 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 Yesta, by Olbers 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 186 ASTRONOMY, the angle, it makes with the ecliptic. This angle is nearly 35°. 75 . The following tallies contain the elements of the orbits of the planets and the periodic revolutions of the satellites. The elements of the four new planets are to be considered only as approximations. Sidereal Mevolutions of the Planets, Days. Mercury - 87.969258 Venus - 224.700t87 The Earth - 365.256384 Mars - 686.979646 Vesta - 1326.930 Juno - - 1594.023 Ceres - 1681.370 Pallas - 1685.619 Jupiter - 4332.5S5117 Saturn - - 10758.322161 Uranus - 30688.712687 Mean distances from the Sun, or Semi-axes of Orbits , the Earth's mean distance being = 1 . Mercury - 0.387098 Venus - - 0.723332 The Earth - 1.000000 Mars - - 1.523692 Vesta - - 2.36319 Juno - - 2.67035 Ceres - - 2.76722 Pallas - - 2.7718S Jupiter - 5.202776 Saturn - - 9.53S770 Uranus - - 19.183305 CHAPTER XII, isp jRatio of the Eccentricity to the Semi-transverse axis, at the beginning of 1801, with the Secular Varia¬ tion. The sign — indicates a diminution . Ratio of the Eccentricity. Secular Variation! Mercury - 0.205515 - 0.00000387 Venus 0.006861 - — 0.00006275 The Earth - - 0.016853 — 0.00004181 Mars 0.093307 - 0.00009019 Vesta - - 0.089128 Juno 0.254311 Ceres - - 0.078502 Pallas 0.241600 Jupiter - 0.048164 0.00016036 Saturn 0.056132 - -0.00031240 Uranus - 0.046670 — 0.00002521 Mean Longitudes, reckoned from the Mean Equinox, at the Epoch of Mean Noon, at Greenwich, January 1, 1801. Mercury 166° 0' 49 Venus - 11 33 3 The Earth 100 39 10 Mars - 64 22 56 Vesta - 293 32 34 Juno 72 55 28 Ceres - : 77 25 23 Pallas - - 65 22 5 Jupiter 112 15 23 Saturn - 135 20 18 Uranus r 177 47 39 188 ASTRONOMY, Mean Longitudes of the Perihelia, for the same Epochs as the preceding, with the Sidereal, Secular Variations. Long. Perihel. Sec :. Var. Mercury 74° 21' 47" - 9' 44" Venus 128 43 53 . — 4 28 The Earth - 99 30 5 - - 19 41 Mars - *> 332 23 57 26 22 Vesta 250 18 26 Juno 53 13 22 Ceres 146 46 32 Pallas 120 54 48 Jupiter ■» 11 8 36 11 5 Saturn 89 8 58 - - 32 17 Uranus 167 21 42 4 0 Inclinations of the Orbits to the Ecliptic, at the be¬ ginning of 1801, with the Secular Variations of the Inclinations to the true Ecliptic. Inclination. Sec. Var. Mercury 7° 0' 1" 18".l Venus - - 3 23 29 - — 4.5 The Earth 0 0 0 - 0 Mars - 1 51 6 — 0.3 Vesta 7 7 50 Juno - - 13 4 16 Ceres 10 37 31 Pallas - - 34 35 14 Jupiter 1 18 52 — 22.6 Saturn - 2 29 38 - - —15.5 Uranus 0 46 26 - 3.1 CHAPTER XII, 189 Longitudes of the Ascending Nodes at the beginning of 1801 j with the Sidereal , Secular Motions. Long, of Sec. Mot. Mercury - 45° 57' 31" — 13' 2" Venus - 74 52 39 - -31 11 The Earth 0 0 0 - 0 0 Mars - 48 0 3 - - — 38 49 Vesta 103 9 51 Juno 171 8 29 Ceres 78 53 31 Pallas - 172 32 31 Jupiter - 98 26 18 — 26 21 Saturn - 111 55 46 - - 32 22 Uranus - 72 51 14 — 59 59 Sidereal Revolutions of the Satellites , , and their Mean Distances from the Planets about which they re - valve. The distances are expressed in terms of the Equatorial Radius of the Planet. JUPITER. - Mean Dist. Sider. Revol. 1st Satellite 6.04853 1.7691378 Days. 2d - - 9.62347 - 3.5511810 3d 15.35024 7.154552S 4th - - 26.99835 - 16.6887697 SATURN. Mean Dist. Sider. Revol. 1st Satellite 3.351 0.94271 Days 2d - - 4.300 - 1.37024 3d - 5.284 - 1.88780 4th - 6.819 - - 2.73948 5th 9.524 4.51749 6th - - 22.081 - - 15.94530 7th 64.359 79.32960 490 ASTRONOMY. URANUS. Mean Dist. Sider. Revol* 1st Satellite 13.120 5.8926 Days 2d - - 17.022 - 8.7068 3d 19.845 10.9611 4th - - 22.752 - 13.4559 5th 45.507 38.0750 6th - - 91.008 - 107.6944 CHAPTER XIII. On Comets. 1 . It lias 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 along train of luminous trans- CHAPTER XIII. 191 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 1618, 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 (12.31), 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 Jlstronomie , Chajp. 33, art. 59, &c. 10 . The only comet which is known with certainty to have returned, is that of 1682, which conformably to the prediction of Dr. Halley, appeared in 1759. Halley was led to this prediction by observing that a comet had appeared in 1531 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. A*e thence inferred that instead CHAPTER XIV. 193 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. 11. The number of comets is not known, but it amounts to several hundred. CHAPTER XIV. • I Aberration of Light, Nutation of the Earth’s Axis , and the Annual Parallax of the fixed Stars. \ 1 . 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 aE 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 36 194 ASTRONOMY. that the earth is at A, will arrive at E, at the same time that the earth does. Let A A "a" and ES' be each parallel to A a. Then ao! is to ? AA' and a'a" is to A'A" in the ratio of Ea to EA. 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 A a 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, ,, . r —20". 253 cos (L —L') Jibtr. m Long. =-^- cos a Aber. in Lat. — 20",253 sin (L' — L) sin a. 6 . Let A be the right ascension and D the declina- CHAPTER XIV. 195 tion of the star, L being the sun’s longitude as before. Then, Aber. in Right Jlscen. _ 0".837 cos (A + L) — 19".416 cos (A — L) cos. D. Aber. in Decl. — sin D [19".416 sin (A— L) — 0".837 sin (A 4- L) — 8".0G6 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 &0".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. 10 . 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. 11. 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 196 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, w hich have been observed in the precession of the equinoxes, and in the mean obliquity of the ecliptic; are called Nutation. 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 w as 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. 15. 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". 1 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. 197 very nearly agree with the observed inequalities. The agreement becomes more exact, if 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. 16 . If N be the longitude of the moou’s ascending node, the variation in the obliquity of the ecliptic, pro¬ duced by the mutation is -f 9".6 cos N; the inequality in the motion of the equinoxes in longitude, sometimes called the Equation of the Equinoxes in Longitude , is 17"-9I6 sin N; and the inequality in tlieir motion in right ascension, called the Equation of the Equinoxes in Eight Ascension , is — 16".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, Nutation in Right Ascen. — — 16".462 sin N — 8".373 cos (A — N) tan D — 1'227 cos (A 4 - N) tan D. Nutation in Decl. = + 8 '.373 sin (A — N) + l". 227 sin (A + N), 19. The nutation does not affect the positions of the stars relative to oue another, nor to the plane or pole of the ecliptic; it only affects tlieir positions relative to the plane of the equator, or to the position of the earth’s axis. ANNUAL PARALLAX OF THE FIXED STARS. 20. The angle contained between two straight lines, conceived to be drawn from the sun and earth, and 19 $ ASTRONOMY. meeting at a fixed star is called the Annual Parallax of the star. SI. 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. 13 sec. would require more than 3 years to come from the star to the earth. CHAPTER XV. Nautical Astronomy, 1. Some 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 Sextant and Circle of Reflection, By either of these the altitude of any of the heavenly bodies, and the an- CHAPTER XV. 199 gulUr 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 Navigator. 2. The Mile 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 Mariner’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 Knots , 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, j Fig. 46, be the track of the ship, NS and N'S', meridians passing through A and B, and AO and BD parallels of latitude, Then in the triangle DAB, we have, ) 200 ASTRONOMY. AD — AB cos DAB, Or, Diff. of Lat. = Dist. x 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 earth is considered as a sphere or spheroid. But in consequence of currents and other causes, the distance and course can never he ob¬ tained with great accuracy, and consequently the dif¬ ference of latitude thus found, must he considered only as an approximation. 5. The triangle ADB, also gives, BD = Dist. x sin. Course. As the meridians are not parallel, but really 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 Middle Latitude , and is equal to half the sura 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 PDA, 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 BE* Hence (5), Fig. 46, CHAPTER XV. SOI Biff. of Long, of A and B BD _ Dist. x sin. Course cos Mid Lat. cos Mid. Lat. 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¬ ceding manner can not he 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 (42). 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 27 203 ASTRONOMY. cases the latitude may he 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. 10 . 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 clone 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 - Ween 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 S' 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 = PS = 90° ± sun’s declination, A = SPS' = interval of time, expressed in degrees, L = HP = latitude, where the greater altitude is taken, U = PSZ, V = ZSS', W = PSS', and X = SS'. Then from the right angled triangle PSG, we have, sin i X == sin h 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, /H + X + H'\ . /H + X + H' „\ sin i V = v/ __ ' 1 2 _ cos H sin X Then U = Wq:V 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 -f cos U sin D cos H = cos D sin H -f- (2 cos 2 5 U — I sin D cos H (App. 9), = 2 cos 2 h U sin D cos H — (sin D cos H — cos D sin H), = 2 cos 2 h U sin D cos H — sin (D — H). But, sin L = — cos (90° + L) = 1 — 2 cos 2 \ (90 4 - L) (App. 9) and, sin (D — H) = cos [90° _ (D — H)] = 2 cos 2 h [90° — (D —H)] —1. By substituting these values we have, cos 2 \ (90° 4 - L) = cos 2 5 [90° — (D — H)] — cos 2 5 U sin D cos H = cos 2 i [90° — (D — H)]. (1 cos 2 5 U sin D cos H ^ cos 2 h [90° — (D — H)] / Make sin M = cos \ U sin D cos H cosi [90° — (D — H)] 204 ASTRONOMY. Then cos 2 \ (90° + L) cos 2 \ [90 — (D — H ]. (1 — sin 2 M) =* cos 2 \ [90° — D — H)] cos 2 M. Or, cos i (90° + Lj = cos \ [90° — (D — IT] cos. M. 12. 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 u ncertainty. 13. 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. 14. 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 ZO, 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 sun or star. Put, IJ = 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. Then in the triangle ZMS, we have, (App. 34), 206 ASTRONOMY. cos Z = cos D —sin H sin A (App. 14) cos H cos A __ cos D — cos H cos A -f cos HI 4 - A) cos H cos A __ cos D 4- cos H 4 - A) _j cos H cos A In like manner from the triangle ZM'S' we have, rr cosD' 4- COS (H' + A') . cos Z*---—-— 1 . cos hi cos A Hence cos ^ + cos (H 4 - A) _ cos D' 4 - cos (H' 4 - A') cos H cos A cos H' cos A' , COS H COS A r ipv , /TT I A \1 Or, cos D =-—-—. [cos D 4 - cos (H 4 - A)] — cos H cos A cos (H' 4- A'). But, cos D 4- cos (H 4 - A) = 2 cos \ (H 4 - A 4 D) cos \ (H 4 - A — D), (App. 22) cos (IT 4 A') = 2 cos 2 \ (H' 4- A') — 1 (App. 9) cos D == 1 — 2 sin 2 \ D' (App. 8 ). Substituting these values^ and reducing, we have, sin 2 h D' == cos 2 |fH' 4 A') — cos \ (H A 4 - D) cos \ (H 4 - A — D) cos H' cos A' cos H cos A =e cos 2 \ (H' 4 - A'). cos 5 (H 4 A 4 - D) cos \ (H 4- A — D) cos H' cos A' cos H cos A cos 2 h (H' -f A') Make sin M = cos 5 (H 4- A 4- D) cos-2 (H 4- A — D) cos H' cos A.^ n/ cos H cos A cos I (H' 4- A') Then, sin 2 \ D' = cos 2 (H 4- A'). (1 — sin 2 M; = cos 2 \ (H' + A') cos 2 M, Or, sin \ D = cos (H' 4- A') cos M. CHAPTER XVI. SO? CHAPTER XVI. Of the Calendar . 1. The Calendar is a distribution of time into periods of different lengths, as years, months, weeks, and days. S. It has been shown that the tropical year contains 36 5 d. 5 h 48 ni. 51.6 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 Bissextile year. It is also frequently called Leaj) 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. 4. It is evident that the reckoning by the Julian calendar, supposes the length of the year to be 365J days. A year of this length is called a Julian Year . 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 Nice, which was held in the year 325, the Vernal Equinox fell on the 2 (st 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 ASTRONOMY. ten days earlier* that is, on the 11th of March. It was observed that by continuing to reckbti 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 10 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 21 st 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 1700 , 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¬ rian 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 t7^2. It is now used in all Christian countries, except Russia. CHAPTER XVJ. 209 8. The Julian and Gregorian calendars are also de¬ signated by the terms Old Style and JSF ew Style. In consequence of the intercalary days, omitted in the years 1700 and 1800, there is now 12 days difference between them. 9. The year is divided into twelve portions, called calendar months. Each of these contain, either 30 or 31 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. 10. 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 . 11. As a common year consists of 365 days, or 52 weeks and 1 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 ou. It therefore follows, that whatever letter is the dominical letter, in any common 210 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. 12. 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 Gist day of the year; but i he 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. 13. From w hat 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. 14. 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, w hich 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 CHAPTER XVI. 211 number of years above 1S00. by 5, and divide the pro¬ duct by 4, neglecting the remain lee. 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 letter. 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 hy 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. 17* The Lunar Cycle , or as it is sometimes called, the Metonic Cycle, is a period of 19 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 Number. The first year of the Christian era was the 2d of the lunar 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 19. The remainder expresses the year of the cycle. If nothing remains, the year is the 19th of the cycle. 19. The Cycle of the Indiction is a period of 15 years. This period, which is not astronomical, was introduced at Home, 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. 19. The Julian Period is a period of 7980 years, obtained by taking the continued product of the num¬ bers 28 , 19 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 iu a given year of the period, as in the same year of the preceding period. The first year of the Christian era was the 4714th of the Julian period. Hence if 471*3 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 moou 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. 213 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 . 1. 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 214 ASTRONOMY. an Ellipse or some other of the curves, called Conic 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. 7. By taking into view Kepler’s third Iaw r , 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. 10. The existence of a similar force, in each of the planets that have no satellites, is inferred from the ef¬ fects which they are known to produce on one another and on the other planets. 41. 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. 12. A Projectile Force is the force by which a body is put in motion. 13. A Centripetal Force is the force by which a body revolving about another body is urged towards it. 14. A Centrifugal Force is the force by which a body revolving about another body, tends to recede from it. 15. Centripetal and Centrifugal forces are called Central Forces . Relative Masses of the Planets—Relative weight of a body at their surfaces. 16. 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. 17. The masses of those planets which have satel- 216 ASTRONOMY. lites may be found ina simpler manner and with greater accuracy. If 1 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 4 - m d 3 P 2 "l r M " D* ~f 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, d 3 P 2 M == -.. —very nearly. D 3 p 2 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 Monde. Sun Mercury Venus The Earth Mars Jupiter - Saturn Uranus 1 1 2025810 1 356632 1 337102 1 2546320 1 1066.09 1 3512.08 1 19504 CHAPTER XVII. 247 If the mass of the earth be denoted by 1, the mass of the moon, according to the most accurate deter¬ mination, is V £ T . 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 1. Sun 0.252 Mercury - 2.585 Venus ; - 1.024 The Earth - - 1.000 The Moon 0.615 Mars - - 0.656 Jupiter - 0.201 Saturn - 0.103 Uranus - 0218 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 , 27.70 Mercury - 1.70 Venus - 0.98 29 ASTRONOMY. SIS The Earth - 1 00 Mars 0.34 Jupiter - 2.33 Saturn - 1.02 Uranus - 0.93 THE CENTRE OF GRAVITY OF THE SOLAR SYSTEM. 21 . As all particles of matter attract each other, the sun 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 proportion ably 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 suits centre cau 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 b^ 7 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 35. From some minute changes in the situations of CHAPTER 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 Dusejour and Bnrckhardt 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 tbe actions of the others. 28. Tbe 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, p 2 : P 2 : 1 + m A 3 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 tlieir 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 tlieir 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. £21 33 . The problem of the three bodies has exercised the ingenuity of several of the most eminent mathema¬ ticians. But Laplace, in the Meeanique 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 ASTRONOMY. mass by the moon’s mean distance from the earfh, 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 ST 3 quadratures, at A and B*. Then, SM 2 : SE 2 : SE : = the 5i\r force with which the sun acts on the moon at M. In the line SE 3 MS, produced if necessary, take MD = _—; then MD repre- SM 2 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, mf> _ SE3 _ SN3 - (SM + MN') 3 SM 2 SM 2 SM 2 _ SM 3 -f 3 SM 2 x MN + 3 SM x MN 2 + MN 3 SM 2 But as MN is very small compared with SM, the two terms 3SM x MN 2 , and MN 5 may be omitted. Therefore, MD = SM 3 + 3 SM 2 x MN _ gM + 3 MN SM 2 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, 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 EK. 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- ASTRONOMY. 224 lion 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 x = the angle AEM. Then, EK = EM cos MEK r sin x EL — 3 EK = 3 r sin x , 3 r PM as LQ = EL cos x — 3 r sin x cos x = — sin 2 x, (App. 7). 2 Also, EQ = EL sin x — Sr sin 2 x, MQ = EQ — EM = 3 r sin 2 x — r = — r (1 — 3 sin 2 x)* Or using the affirmative sign to denote an increase in the moon’s gravity to the earth, MQ -f r 1 — 3 sin 2 x). Now if m — the mass of the sun, then the force which the sun TYl exerts on the earth may be expressed by —. Hence, ES : PM : : the force PM = —. a 2 — : the force PM. Therefore, a 2 PM ES m.3r sin 2x 2a 3 3mr Jo 3 " sin2x A. In like manner, the force MQ — —. MS — JHHL. (I —3 sin 2 x). B a 2 ES a 3 When the moon is in quadratures, x = 0 or 180°, and conse¬ quently. The force MQ = + —. a 3 When the moon is in syzigies, x = 90° or 270°, and, therefore, The foree MQ = — ^HOL. a 3 The force MQ is = 0, when 3 sin* x = 1, or sin x = y/ -J; that is, when x — 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 CHAPTER XVIT. 225 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 the mean diminution is ; r representing in this case the mean 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. 3/. The inequality in the moon’s motion, called the Annual Equation, (IO.&9), is the effect of the varia¬ tion in the distance of the earth from the sun. Since, in the expression JlilL., which designates the mean di- 2(i ^ 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 011 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 t\\e earth, SO 226 ASTRONOMY. in apogee and perigee, when the line of the apsides coincides with the line of the sjzigies, X and x, the distances at which the moon would be from the earth, in apogee and perigee, if she was not acted on by the sun, and G and g the perigean and apogean gra¬ vities in that case. Also put n = and supposing the earth’s 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 2 : x 2 : ’ G : g, and R 2 : r 2 : : G — 2rn : g — 2R n. Consequently, X 2 = G * 2 g' and 5! = r 2 g — 2Rn Now as G is greater than g> and 2rn, less than 2Rn, it is evi¬ dent that, is greater than —. g - 2Rn 8 g R2 # %2 Hence, -— is greater than —, r 2 x 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 sun. 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. I CHAPTER 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 2x. Hence, supposing the earth’s distance from the sun, and moon’s distance from the earth to remain constant, this force is proportional to sin 2x ; 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 least. 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 Acceleration of the Moon (10.21), by which her velocity appears subject to continual increase, and her period to continual diini- * The expressions, mean place, true place, mean motion and true motion, are hereto be understood, only in relation to the present inequality. 228 ASTRONOMY. notion, lias 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 ORBIT. 4fl. 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 sun’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 XVII. £29 MOTION OF THE MOON’S NODES, AND CHANGE IN THE INCLINATION 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, i9 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 nodes. These two lines lie in the plane of the ecliptic, which we may consider as desig¬ nated by the plane of the paper. Let EMHI conceived to be, from El, 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 nodes 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 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 node 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 nodes, 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 nodes, the motion of the nodes 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 nodes 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 node 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 node in EN' to the quad¬ rature in EA, the motion of the nodes will be again retrograde; and while she is moving from the quadrature in EA, to the node in El, 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 plain 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 nodes has the position nn' it is easy to determine from what has been said, that the motion of the nodes CHAPTER XVII. 231 will be direct while the moon is moving from the nodes 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 nodes. 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 x = the moon’s angular distance from the quadratures, S = IEL the an¬ gle contained between the line of the nodes 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 x. Hence, LI = EL sin LEI = 3r sin x sin S, MK = LH = LI sin LIH == 3r sin x sin S sin I, HI Or, (35), using — to denote the force exerted by the sun on the a 2 earth, m, r A/rTr m.MK 3mr . . a . , 1 he force MK =■ - == -sm x sin S sin I. a 3 a 3 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 x. And it is easy to perceive, by reference to the figure, that sin #, 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 nodes 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 nodes. astkoxomV. 232 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 the orbit will be di¬ minished; but if she is moving from the 90° to the node, 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 node 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 nn' 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 node in Ew, and will be increased from thence till she returns to the quadrature in EA. The 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 theses 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 4s 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 be 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 being about 7 it is evident, that if the earth is 31 234 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 w as to the mean density of the earth nearly as 5 to 9. 51. 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 eartli 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 Nutation in precession. The ine¬ quality in the sun’s action occasions a very small change in the obliquity of the ecliptic, called the Solar Nutation 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 Mean Precession of the Equinoxes. Another effect is an inequality in the precession called the Lunar JV* utation , and sometimes, the Equation of the Equinoxes. SECULAR VARIATION 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 23 6 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'. DIURNAL 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. 59. 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. - , \ s 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 50 in. 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 w ater 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. Fol- 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 JYeap Tides. 66. The highest of the spring tides or the lowest of the neap tides, is not the tide that has place nearest ASTRONOMY'. 338 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 9-1^1 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 somewdiat 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 w here the moon is, and in the opposite one. On the w r est 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- CHAPTER XVII. 289 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 sun, 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 snn 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 1. 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. »JI Oftf 14 i tut a r«M •• • ir APPENDIX TO PART I. Containing Trigonometrical Formulce; and Two Propositions in Conic Sections . Many of the Trigonometrical Formulas 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 Lacroix, 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 a, the radius being = 1. 1. sin 2 a . COS 2 a = 1 7. sin a r= 2 sin h a cos £ a 2 sin a = = tan a cos a 8. cos a ~ 1 — 2 sin 2 i a tan a 9. cos a = 2 cos \ 2 h a — 1 3. sin a = - v' 1 tan 2 a 10. tan 1 a . -- sin a 1 1 + cos a 4. cos a : sin a v 1 4- tan 2 a 11. cot 2 a = 1 - — cos a 5. tan a sin a 12. 1 — cos a cos a tan a 1 4 - cos a 6. cot a 1 cos a tan a sin a For two arcs a and b , of which a is supposed to be the greater. 32 212 APPENDIX TO PART I. 13. sin (a ± b) = sin a cos b ± cos a sin b 14. cos (a ± b) = cos a cos b sin a sin b 15. tan («± 6) = . t an a * tan b 1 -f tan a tan b 16. sin a cos b — ^ sin (a 4 - b) 4 . \ sin (a — b) IT. cos a sin 6 = § sin ( a 4 . 6 ) — £ sin (a — 6 ) 18. sin a sin 6 = § cos (a— b) — 5 cos ; a 4 - 6 ) 19. cos a cos b — h cos (a — b) + \ cos (a 4 . 6 ) 20. sin a 4 - sin b = 2 sin £ (a 4 - 6 ) cos § (a — b) 21 . stn a— sin 6 = 2 cos \ (a 4 - 6 ) sin h (a — b) 22 . cos b 4 - cos a = 2 cos h (a 4 6 ) cos \ (a — b) 23. cos b — cos a = 2 sin § (a 4 - b) sin \ (a — b) 24. tau a -f tan I 25. tan a — tan b — 26. cot b 4 - cot a = 27. cot b — cot a = 29. tan § (a— 6) = 30. cot 5 (« + ^) = 31. cot § (a — 5) = 32. tan 3 fa + b) sin (a 4 b) cos a cos b sin (a — -b ) cos a cos b sin (a 4 - b ) sin a sin b sin (a — ■5) sin a sin b sin a 4 . sin b cos a + cos b sin a — sin b cos a 4 - cos b sin a — sin b cos b — cos a sin a 4 sin b cos b — cos a sin a 4 sin sm a — sin b tan h (a — b) For a Spherical Triangle, in which A, B, and C are the ai gles, and a, b , and c, the opposite sides, as in Fig. 53. 33. sin A sin b = sin B sin a 34. cos a = cos A sin b sin c 4- cos b cos c 35. cos A = cos a sin B sin C — cos B cos C APPENDIX TO PART I, 243 36. cot a — 37. cot A = cot A sin B -f- cos B cos c sin c cot a sin & — cos C cos b 38. sin i A = y/ sin C sin \ ^ a -|- b — c) sin h (a c bf sin b sin c C0S i ( B — A ) 40. tan \ (b — a) = 41. tan § (B 4- A) cos i (B + A) sin £ i (B — A) sin 2 (B -f A) h C COS 2 (b — Z ^ cos h (b r sin 2 ( b — a) a) 43. 44. cot I C = tan h (B sin h (b a ) A s sin j (6 + a) sin 4 (6 •— a) cot ^ C = tan i (B + A) C 0 S JJ1 +tt) cos 5 (b — a) tan he — tan h (b — a) . g . J sin h (B — A) tan h c = tan h (b + a) C0S - cos 2 (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, AB4/ie transverse axis, E and F the foci, C the centre, and D a point in the curve, then , AC 2 — EC 2 ED = AC — EC cos AED* Let DH be perpendicular to AB. Then, ED 2 = DH 2 + EH 2 = DH 2 + (EC + CH) 2 APPENDIX TO PART I. 244 = DH 2 + EC 2 + 2 EC x CH + CH% and FD 2 = DH 2 + FH 2 =, DH 2 + (EC —CH) 2 DH 2 + EC 2 — 2EC x CH + CH 2 . Hence by subtraction, ED 2 —FD 2 4EC x CH. But ED 2 —FD 2 = (ED + FD) x (ED — FD) = 2AC x (ED — FD). Therefore, 2AC x (ED — FD) = 4 EC x CH, Or, ED — FD = 2 — CH . AC But ED f FD = 2AC. Hence by addition, 2 ED = 2AC + 2EC x CH AC ED AC EC x CH AC (A) ED x AC = AC 2 + EC x CH = AC 2 + EC x (EH — EC) = AC 2 + EC x (ED cos AED —EC) = AC 2 + EC x ED cos AED — EC 2 , ED x AC — ED x EC cos AED = AC 2 — EC 2 ED x (AC — EC cos AED) = AC 2 — EC 2 ED = AC 2 — EC 2 AC — EC cos AED 52. If the circle AGBM, Fig. 55, be described about AB, as diameter y and HG be produced to meet it in G, then , ED AC —EC cos BCG. From the preceding demonstration, we have (A), ED = AC + _L S 9.1* AC = AC + = AC + EC x CG cos ACG AC EC x AC cos ACG AC = AC + EC cos ACG = AC —EC cos BCG. AN ELEMENTARY TREATISE i'"'\ . r VA& , . i ASTRONOMY. . PART 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. Mean Astronomical Refractions. TABLE III. Mean Right Ascensions and Declinations of some of the Fixed Stars, for the beginning of 1820, with their Annual Variations. 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 VIH. 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 equations of the sun’s place. They are all given for mean noon at the meridian of Greenwich, 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 sun’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. Ml 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 Mean 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 noon 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 af¬ firmative. ASTRONOMY. 248 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 cgrresponds is - 3 ^-. 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. TABLES XLIV and XLV. Moon’s Semidiameter and the augmentation of the semidiame¬ ter 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 noon and midnight. TABLE LVII. Logistical Logarithms. This table is convenient in working ASTRONOMY. 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. SCHOLILTM. The tables of the Sun, which are those from IX to XIX, inclusive, are abridged from Delambre’s Solar Tables. And those of the Moon, 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 Affirmative sign f-, perfixed; and others by the Negative sign — . Those which have the affirmative sign prefixed, are called Positive or Affirmative quantities; and those with the negative sign, prefixed, are called Negative quantities. When a quantity is affirmative, the sign is frequently omitted; but when it is negative, the sign must always be used. I 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 negative 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" 7 2 3 4 Sum 12 17 Add+ 3' 15" — 8 12 — 5 1 + 2 17 Add —3' 51" — 4 10 — 1 15 Sum—9 16 Add—17' 10" — 4 3 + 12 4 + 18 59 Add —7' 14" 4-8 2 4 - 3 17 Sum 4-4 5 Add 4 - 3' 1" — 1 15 4 - 4 18 — 6 4 Sum — 7 11 Sum 4 - 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. From 5' 10" Sub. 3 21 Rem. 1 49 From — 8' 29" Sub. —3 2 Rem.—5 27 From 4' 11" Sub. 7 27 Rem. — 3 16 From — 2' 18" Sub. —7 11 Rem. + 4 53 From -f 2' 5" Sub. — 1 11 Rem. -f 3 16 From —4' 17" Sub. + 6 21 Rem. — 10 3S 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. JF/ien 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, Sine, 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 arc : : 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. TV hen 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. 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 Sine, Cosine, &c. of an arc, in either quadrant, with its appropriate sign, as exemplified in the following table. 252 ASTRONOMY. Arc 37 * 18/ 21" 114 35 10 247 12 3G 314 17 50 Sine + 9.78252 . 9.95872 — 9.96470 — 9.85475 Cosine 9.90060 — 9.61916 — 9.58811 4- 9.84409 Tangent 4- 9.88193 — 10.33956 4- 10.37659 — 10.01065 Cotang. 4- 10.1180 7 — 9.66044 4- 9.62341 — 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 arc 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 qu ntitv 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 found 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 — 9.85475 arc 225 42 10 or 314 17 50 Cosine + 9.90060 arc 37 18 18 or 322 41 42 Cosine — 9.61916 arc 114 35 11 or 245 24 59 Tangent + 9.8S1S3 arc 37 18 21 or 217 18 21 Tangent —1033956 arc 114 35 11 or 294 35 11 Cotangent + 9.62341 arc 67 12 36 or 247 12 36 Cotangent — 9.98935 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 necessary 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 Numbers . When the logarithm of a natural number 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, 251 * ASTRONOMY. The logarithm of .27 is 9.43136 of .027 is 8.43136 of—.027 is—8.43136 of .0027 is 7.43136 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 lo¬ 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. PROBLEMS FOR MAKING VARIOUS 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", what is its motion in 39 m. 22 sec.? Ans. 20' 58". As 60 m. - - - 0 : 39 m. 22 sec - 1830 : 31' 57" - - 2737 : 20' 58" 4567 2. If the moon’s declination change 2° 29' in 12 hours, what will be the change in 8 h. 21m. Ans. V 44'. As 12 h. - - - 6990 : 8h. 21m. - - 8565 :: 2° 29' - - 13831 22396 : 1° 44' - - 15406 3. When the sun’s hourly motion is 2' 31", what is its motion in 17 m. 18 sec. Am. 0' 44" 4. When the sun’s declination changes 22' 14" in 24 hours, what is its change in 19 h. 25 m.? Ans. 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 , Minutes 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 Argument. 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 I s 9° 31' 26", to find the corres¬ ponding quantity in table XXXII. Ans. 11° 13' 17". I s 9° gives 11° IT 15". The difference between IT IT 15" and the next following quantity in the table is 5' 9". As 60' : 3T 26" : : 5' 9" : 2'42".* To 11° IT 15" Add 2 42 11 13 57 2. Given the argument 10 s 13° 16' 54", to find the corres¬ ponding quantity in table XXXV. Ans. 93° 32' 37". 10 s 13° gives 93' 33' 40". The difference between 93° 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" From 93° 33' 40' Take 1 3 93 32 37 3. Given the argument 4 s 11° 57' 10", to find the corres¬ ponding quantity in table XV. Jins. 3° 24' 12". 4. Given the argument 3721, to find the corresponding quantity in table XXV. Ans. 4' 52" PROBLEM III. To convert Degrees , Minutes and Seconds of the Equator into Time. Multiply the quantiyby 4, and call the product of the seconds, thirds; of the minutes, seconds; and of the degrees, minutes. Exam. 1 . 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 . 7h. 48 m, 50 sec. 3. Convert 21° 52' 27", into time. Jins. 1 h. 27 m. 30 see. PROBLEM IV. To convert Time , into Degrees , Minutes 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 5h. 41 m. 10 sec. into Degrees, &c. h. m. sec. 5 41 10 60 4)341 10 34 85° 17' 30" 258 ASTRONOMY. 2 . Convert 7h. 48 m. 50 sec. into Degrees, &c. Ans. 117° 12' SO '. 3. Convert 11 h. 17 m. 21 sec. into Degrees, &c. Ans. 169° 20' 15". PROBLEM V. 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 ivest, 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. - - . 5 0 46 Time at Philadelphia, - 7 91131P.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. - - 4 56 4 Time at Greenwich, 11 8 11 24 P. M. 3. When it is Sept. 10th. 3h. what is the time as reckoned at ] Longitude of Paris, do. of New-Haven, Diff. of Long. Time at Paris, September, Diff. of Long. Time at New-Haven, Or September 10th, 10 20 m. 35 sec. P. M. at Paris, iw-Haven? h. m. sec. 0 9 21 E. 4 51 52 W. 5 1 13 d. h. m. sec. 10 3 20 35 5 1 13 9 22 19 22 i. 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? Ans. 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? Ans. Oct. 5th. 6 h. 40 m. 47 sec. A.M. 6. When it is noon, of the 10th of June at Greenwich, what is the time at Philadelphia? Ans. June 10th, 6h. 59 m. 14 sec. A. M. PROBLEM VI. 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. Then, 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 XIY, 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;'butthis 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 noon, on the me¬ ridian of Greenwich, and can easily be found 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 W'hen 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 isSh. 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, 14 20 15 12 Diff. of Long. - - - - 5 0 46 Time at Greenwich - - -15115 58 M. Long. 1821 i CD GO 48' 19' August 6 28 57 26 15d. 13 47 57 lh. - 2 28 16 m. . 39 Mean Long. 4 21 36 49 ASTRONOMY. 261 The equation of time in table XIX, corresponding to 4 s 21° 36' 49'' is + 4 m. 13sec. d. h. m. sec. Mean Time at Philadelphia, August 1821,14 20 15 12 Equation of time, sign changed - — 4 13 Apparant time - - - - 14 20 10 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. 21m. 17 see. 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. - - 5 0 46 Time at Greenwich - - 18 8 22 3 M. Long. 1821 - 9 s 8° 48'19" October 8 29 4 54 18 d. - - 16 45 22 8 h. 19 43 22 m. - 54 M. Long. 6 24 59 12Equat.oftime—14m.48sec. d. h. m. sec. Appar. Time at Greenwich, Oct. 1821, 18 8 22 3 Equation of time - - - — 14 48 Mean Time at Greenwich - - 188 715 3. On the 15th of May, 1821, when it is 7h. 12 m. P. M. mean time at Greenwich, what is the apparent time at Boston? Jins. 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? Ms. 3h. 16 m. 1 sec. P. M. 262 ASTRONOMY. PROBLEM VII. To find the Sun's Longitude , Semidiameter, and Hourly Mo¬ tion , and the apparent Obliquity of the Ecliptic , for a given time , from the Tables . jpor the Longitude. When the given time is not for the meridian of Greenwich, reduce it to that meridian by prob. V; 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 sun; 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 Sun’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, Stc. reject 12 or 24 signs, when the sum exceeds either of these quantities. In addin" any arguments, expressed in 100, or 1000, Stc. parts of the circle, when th are expressed by two figures, reject the hundreds from the sum; when three figures, the thousands; and when by four figures, the ten thousand. ASTRONOMY. 263 For the Semidiameter and Hourly Motion. 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. - - - - 5 0 46 Mean time at Greenwich - 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' 19" 9* 7° 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 0 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 i Eq. Sun’s cent. 8 2 7 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 - - + 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. Jins. Sun’s longitude 4 s 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. Jins. Sun’s longitude 11 s 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 Ascension. 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¬ sine, &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. 1 . Given the obliquity of the ecliptic 23° 27' 40", and the sun’s longitude 125° 31' 25", to find the right ascension and declination. cos. Obliquity tan. Long. tan. Right Ascen. sin. Obliquity sin. Long. sin. Decl. 23° 27' 40" 125 31 25 - 127 53 30 23° 27' 40" 125 31 25 - 18 54 23 N 9.96253 —10.14635 — 10.10888 9.60002 - 9.91055 9.51057 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"? Ans. Right ascension 33“ 1' 43", and declination 13° 18' 32" N. 3. Given the obliquity of the ecliptic 23° 27' 50", and the sun’s longitude 313° 36' 12"; what is the right ascension and declination? Ans. Right ascension 316° 4' 30", and declination 16° 45' 29" S. PROBLEM IX. Given the Obliquity of the Ecliptic and the Sun's Right Ascen - 5 ton, 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 Declination, 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 declination? 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 OS 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¬ clination? Ans. Longitude 55° 51' 16", and declination 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. Ans. 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 Cosine of the Lon- ASTRONOMY. S67 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. 1. Given the obliquity 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" - 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? Ans. 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? Ans. 16° 15' 7"; and the northern part of the circle of latitude lies to the east of the circle of declination. PROBLEM XI. To find, from the Tables , the Moon's Longitude , Latitude , Equa¬ torial Parallax , Semidiameter , and Hourly Motions , 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 nine, 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~ hie 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 Illrd 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 Vth, Vlth, Vllth, Vlllth, 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 the 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; whiclrwillbe North or South, according as the distance is less or greater than 90°. §70 ASTRONOMY. For the Equatorial Parallax. With the corrected arguments, Evection, Anomaly and Va¬ 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 Motion 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 Motion 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. 1. Required the moon’s longitude, latitude, equatorial parallax, semidiameter, and hourly motions in longitude and lati¬ tude, on the 6th of August 1821, at 8h. 46 m. 27 sec. A. M. mean time at Philadelphia. d. h. m. sec. Mean time at Philadelphia, August, 5 20 46 27 DifT. of Long. - 5 0 46 Mean time at Greenwich, August, 6 1 47 13 272 o CHHH O TJ< ^ © rH CM t—1 to c* CM 00 CM co CM rH CO to CO 00 Cl CO rH KOHOHH rH H Cl »0 h- Cl to rH © O O co H CO CM © rH CM N. to Tj« to oo co 00 © 00 r-t T-H rH © O H* CM 00 rH rH © K »o CO rf CM CM © CO hr 00 CM h}< lO rH rH OKOi hr h. 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CO CM CM CO CM to c0 CM CO to ‘to o <© c CM CO tO CO 00 CO ^ ‘ rH ■ rH & © CM hr , hr to hr CM © CO © to K rH to c 4> c o CM rH CM CO CO rj< © h~ © CO rH CO c o CO \S o c "O 3 ‘to c o J 4> rt 5 r* CM to CO CM Tjl © © CM CM O tl « Nutat. © CM CM CO JH 'rH *-> » hr CM © © to h- rH CM *j3 rH C0 CO ’tf CO O O © © CM to hr © I lo nH rH CO CM t}< rj* O e H © © tO rH hr CM 00 00 CM © h, © h- b- © © ^ © ^tO»OHH rH CO e o CO to CO CM © w 4 CO CM CM CO to 4> > © © © © w rH CM CM HhH CS 3 CT* W 1 . 6 o : ( q sh . . e % e So 5 ^ CO =3 H<©rH^H03 ASTRONOMY. if Sfl C* V «*§ o 2 CO hJ ^ < be a goo* o‘.2 ,2 03 -O 'O -g Cj- . W c* W £ 3 >» 03 « ctf E . o . c s << cr* W £ c ^§.2 *’■3 g S- 'C T3 rt rt « >|> « £ 3 Cfi cS ■ 1 KN.C^'^VS^COOOOCO VO o *Og cr> ^ VO rt 8? •< I II *s lo; , Ion V VI VII VIII IX X 3 C/1 *g; ^CN O O o s c « .0- b- O & C* a l—t CO C/l VO b ha . ■ c s- > c «« < a < > £ v o '-(0>O(NO00C0NN ^ b- i—i o *OC0CNTj*C0C0C0iH“r-(rH tN N H »-l rH 1-1 c* ^ VO b CO wi *0 H CN H rH VO T-t b^ CO iH CO iH Tjl 1-1 u* a o O O iH > iH ^ £ HWCO^*OVOKCOO)OH!MC'5^>0(OKCOO>0 rtHHHHHHHHHO) a Sum An. 273 30 An. 4 13 50 Moon’s Eq. Par. 54 31 -Moon’s Semidiameter 14' 51". As 32' 56" : 3/ 3" : : — O' 49" : —0' 45" = Moon’s hourly motion Sum 6 1 31 in latitude, tending South. Yar. 30 31 ASTRONOMY. 27 * 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 s 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 5h. 56 m. 14 sec. P. M. mean time at Philadelphia? Jins. Long. 6 s 3° 7' 12"; lat. 3° 54 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 Moon’s Longitude , Latitude , Hourly Motions , Equa¬ torial Parallax , and Semidiameter , for a given Time , from the Nautical Almanac. Reduce the given time to Apparent time at Greenwich. Then, For the longitude. Take from the Nautical Almanac, the two longitudes, for the noon and midnight, or midnight and noon, 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 ter)n , 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 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 Motion 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, \ 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 Declination may be found in the same manner. For the Hourly Motion 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 Parallax. 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 noon 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- ASTRONOMY 276 tude, from the Nautical Almanac, on the 6th of August, 1821, at 8h. 40 m. 54 sec. apparent time at Philadelphia. d. h. m. sec. Appar. time at Philadelphia, August 5 20 40 54 Diff. of Long. - 5 0 46 Appar. time at Greenwich, August 6 1 41 40 For the Longitude and Hourly Motion in Longitude . Mean of Longitudes 1st Diff. 2d Diff. 2d Diff. 5th midn. 6th noon 6th midn. 7th noon 7 s 12° 8'55" 7 18 7 55 7 24 9 18 8 0 13 38 5° 59' 0" A. 6 1 23 6 4 20 2'23" 2 57 B. + 2'40" T h. b. m. 12:1 41 A sec. 40 6° 1' 23" : 51' l."7, fourth term. Second Longitude - Fourth term - Cor. 2d diff. from tab. LVI 7 s 18° 7' 55" 51 1.7 — 9.7 Moon’s true Longitude 7 18 58 47 1 nn 12 A B - - - 8 m. 28 sec. + 2 40 L. L. L. L. 8504 13522 C - - +0 23 22026 A 6° r 23" i B, sign changed, - — 1 20 C - - - + 0 23 12)6 0 26 Hor. mot. in long. 30' 2".2 ASTRONOMY. 277 For the Latitude axd Hourly Motion in Latitude. Mean of Latitudes 1st Diff. 2d Diff. 2d Diff. 5th midn. 6th noon 6th midn. 7 th noon — 4” 50' 53" — 5 1 55 — 5 9 43 — 5 14 18 — 11' 2" A. — 7 48 — 4 35 + 3' 14" + 3 13 B. + 3' 13".5 T A h. h. m. sec. 12 :1 41 40 : : — 7' 48" : — 1' 6".l, fourth term. Second Latitude - - —5° 1' 55" Fourth term - - - — 1 6.1 Cor. 2d diff. - - - — 11.7 Moon’s true Latitude T 8 m. 28 sec. B +3 13".5 5 3 13 S. L. L. 8504 L. L. 12696 C + 0 27 21200 A - - - - —7' 48” § B, sign changed - — 1 37 C +0 27 12)—8 58 Hor. mot. in lat. - —0' 44".8, tending south. Moon’s semidiam. from N. Aim. 14' 53" do. eq. parallax - - - 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 S78 ASTRONOMY. Greenwich. Jins. Long. 2 s 23° 7' 43"; lat. 5° O' 36" N; equat. par. 57' 57"; semidiam. 15'49"; hor. mot. in long. 34' 3".8; and hor. mot. in lat. O' 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? Jins. Long. 11 s 7° 43' 8"; lat. 0° 17' 6"N; equat. par. 59' 47"; semidiam. 16 19"; hor. mot. in long. 36' 4".2; and hor. mot. in lat. 3' 19".3, tending north. PROBLEM XIII. The Moon's Equatorial Parallax, and the Latitude of a Place being given , to find the Reduced Parallax 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 c 27' N, to find the reduced parallax and latitude. Equatorial parallax 54' 31" Reduction - 5 Reduced Parallax 54 26 Latitude of Philadelphia 29° 57' N. Reduction - 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. Jins. 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. Jins . Reduced par. 57' 18", and reduced lat. 32° 40'N. ASTRONOMY. 279 PROBLEM XIV. To find the Mean Right Ascension 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 multiply 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.* Then, 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" Var. for 1 yr. 166 d. - - - - -j- 1 10 Mean right ascen. required - - 149 42 49 Mean declin. begin, of 1820, Var. for 1 yr. 166 d. Mean declin. required 12 ° 50' 36" N — 25 52 10 11N. 2 . Required the mean longitude and latitude of Regulus on the 15th of June, 1821. Mean longitude, begin, of 1810, table VIII, 4 s 27° 11 ' 18" Var. for 11 y. 166 d. ... + 9 33 Mean long, required - - - - 4 27 20 " 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 Yl, the number of days corresponding to the month, and adding to it the odd days* 280 ASTRONOMY. Mean latitude, begin, of 1810 0 ° 27' 36" N Var. for 11 y. 1 G 6 d. - 4 . 2 Mean latitude, required - ~ - 0 27 38 N. 3. Required the mean right ascension and declination of £ Tauri , on the 6 th 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 /3 Tauri , on the 6 th of November 1822? Ans . Mean long. 2 s 20 ° 5 59", and mean Iat. 5° 22 ' 27" N. PROBLEM XV. To find the Aberration of a Star , in right Ascension and Decli¬ nation, for a given Day. Find the mean right ascension and declination 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 0 , the right ascension of the star by A, and the declination by D. With the argument 0 , take the quantity x from table LX; add it to ©j attending to the sign and from the sum, subtract A. Then, For the Aberration in Right Ascension. With argument 0 , take from table LX, the log. a , with its proper sign, and to it. add the Cosine of (0 4 - x — A), and the arithmetical complement of the Cosine of D, rejecting the tenp 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 Aberration in Declination. Add together the log. a , the Sine of (0 + x — A), and the Sine of D, and reject the tens in the index of the sum. Take the natural ASTRONOMY. 281 number, corresponding to the sum, and call it m. With the ar¬ guments © 4- D and e— D, or when the declination is south, with these arguments, each increased by VI 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 ihe mean declination. Exam. 1 . What are the aberrations in right ascension and de¬ clination, of Regains, on the 15th of June, 1821, the sun’s lon¬ gitude on that day, being 2 s 23° 58'? A = 149° 42' 49" = mean right ascen. of Regulus. D = 12 50 11 N = mean declin. do, © = 2 s 23° 58' = sun’s longitude. © - 2 s 23° 58' Xy from table LX, 4- 0 30 © + x - - 2 24 28 A - 4 29 43 © 4- x —A - 9 24 45 = 294° 45' log. a from tab. LX, _ . — 1.3061 cos. (e|i- -A) 294 c ‘ 45' 9.6.219 cos D 12 50 Ar. Co. 0.0110 Aber. in right ascen. - -8". 69 - -0.9390 log. a - . . . - — 1.3061 sin (© 4- x — A) - 294 :° 45' - —9.9581 sin D 12 50 N. - 9.3466 m - + 4".08 0.6106 Arg. (© -f D) = 3 s 6° 48', gives - 4- 0".4S Arg. v ®— D) = 2 11 8, gives - - — 1.30 m - - - - - 4- 4.08 Aber. in declination . _ - 4- 3.26 2. Required the aberrations in right ascension and declination, of Antares , on the 11 th of March, 1821, the sun’s longitude being 37 \ 383 ASTRONOMY. 11 s 20° 38'. Jins. Aber. in right ascen. -f- 5".43, in declin, — 0".74. 3. On the 6 th of November 1822, the sun’s longitude will be 7 s 13° 32'; what will be the aberrations in right ascension and de¬ clination of /3 Tamil Jins. Aber. in right ascension -f 18".54, and in declination 4 - 0".15. PROBLEM XVI. To find the Mutations of a Star in Right Ascension and Decli¬ nation, for a given Time. Find the Supplement of the Mooh’s Node, from tables XX, XXI and XXII, and subtract it from 12 s 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 Mutation in Right Ascension. With the argument N, take from table LXII, the log. 5, with its propersign, and to it, add the Cosine of (N 4 - 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 number corresponding to the sum, to a quantity, taken fromtable LXIII, with the argument N, and the result will be the nutation in right ascension, to be applied to the mean right ascension. \ For the Mutation in Declination. To the log. b , add the Sine of (N 4 - B — A), or when the de¬ clination is south, the Sine of (N 4 . B — A -f VI s ) rejecting the tens in the index, and the natural number corresponding to the sum, will be the nutation in declination, to be applied to the mean declination. Exam. 1 . Required the nutations in right ascension and de¬ clination, of Rigel, on the 19th of July 1825, ASTRONOMY. £88 By prob. XIV, A = 76° 33' 8" and D = 8° 24' 36" S. Supp. of Node 1825 3 s 0° 25' July - - - - 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 - 2 16 33 N + B —A - 5 29 2 = 179° 2 log. 6, from tab. LXII, m — 0.8623 cos. (N + B — A) 179° 2' — 9.9999 tan. D 8 25 - — 9.1702 nat. numb. — 1".0S — 0.0324 From tab. LXIII. + 16.25 Nut. in right ascen. + 15,17 log. b. _ — 0.8623 sin. (N + B — A + VI s ) 359' 3 2' — 8.2271 Nut in declin. + d".i2 + 9.0894 2. Required the nutations, in right ascension and declination, of Antares on the 11th of March, 1821, Jbis. Nut. in right ascen. + 3".70, and in declin, + 9".23. 3. What will be the nutations, in right ascension and declina¬ tion of a Tauri , on the 6th of November, 1822. Jins, Nut, in right ascen. + 14".61, and in declin. + 7".3. 284 ASTRONOMY. PROBLEM XVII. To find the Aberrations 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 a. Then, For the Aberration in Longitude. Add together the constant logarithm, — 1.30649, the Cosine of (L —-L'), and the arithmetical complement of the Cosine of a, 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 a, 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 3 s 27 21'. Byprob. XIV, L' = 3* 11° 37' 51" and a = 39° 22' 31" S. cos. (L — L') 15° 43' — 1.30649 9.98345 cos. a 39 23 Ar. Co. 0.11187 Aber, in long. — 25".22 — 1.40181 sin (L — L') 15° 43' - — 1.30649 9.43278 sin a 39 23 - - 9.80244 Aber. in lat. — 3".48 - — 0.54171 2. On the 25th of April, 1822, the sun’s longitude will be I s 4° 41'; what will be the aberrations in longitude and latitude ASTRONOMY. 28 5 of Regulus, at that time? Ans. Aber. in long. + 7".81, and in lat. + O'M 5. 3. Required the aberrations in longitude and latitude of Vir- ginis , on the 10th of August, 1821. Jins. Aber. in long.—• 16".14, and in lat. q- 0".15. PROBLEM XVIII. To find the Nutation of a Body in Longitude f Find the mean longitude of the moon’s ascending node, as in prob. XVI, and to its Sine, add the constant logarithm — 1.25396, rejecting the tens in the index. The natural number corresponding to the sum, will be the nutation in longitude, to be applied to the mean longitude. Exam. 1. 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 s 6° 29'. — 1.26396 sin. long, of node 336° 29' — 9.60099 Nut. in long, q- 7".16 0.85495 2. Required the nutation in longitude of p Virginis , on the 10th of August, 1821. Jins . q. 7".48. 3. What will be the nutation in longitude of Regulus , on the 25th of April, 1822? Jins, q- 11".12. PROBLEM XIX. The Obliquity of the Ecliptic and the Right Ascension and De¬ clination of a Body being given , to find the longitude and La¬ titude. 4 Designate the obliquity of the ecliptic by E. To the Tangent of the declination marked negative when the declination 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°. 4 For the Longitude. Add together the Cosine of the difference between B and E, the Tangent of the right ascension, and the arithmetical comple¬ ment of the Cosine 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 declination 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 nutations, 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 - 145 15 13 - — 9.84113 E 23 27 46 cos. (B~E) 121 47 27 — 9.72166 tan. Right Asc. 211 52 37 - 9.79371 cos. B - 145 15 13 Ar. Co. — 0.08529 tan. Long. 201 44 16 9.60066 tan. (B a? E) 141° 47'27" - -10.20774 sin. Long. 201 44 16 - 9.56863 tan. Lat. 30 51 37N 9.77637 2. Given the obliquity of the ecliptic 23° 27' 47", the right ascension of Rigel , 76° 28'21", and the declination 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? Ms. 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 Ascension 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 Ascension. Add together the Cosine 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 2SH ASTRONOMY. right 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- Wording as the given ones are mean or apparent. Exam. 1. Given the obliquity of the ecliptic 23° 27' 46", the longitude of Arcturus 20V 44' 16", and the latitude 30° 5T 37" N, to find the right ascension and declination. tan. Lat. sin. Long. - 30° 51' 37" N 201 44 16 Ar. Co. 9.77637 — 0.43137 tan. B - E - 121 47 28 23 27 46 -10.20774 cos. (B + E) tan. Long, cos. B 145 15 14 - 201 44 16 - 121 47 28 Ar. Co. — 9.91471 9.60066 — 0.27834 tan. Right Asc. 211 52 36 9.79371 tan. (B -f- E) sin. Right Asc. 145° 15' 14" - 211 52 36 — 9.84113 — 9.72271 tan. Declin. 1 £ o 9.56384 2. Given the obliquity of the ecliptic 23° 27 r 47", the longi¬ tude of Rigel IV 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? Ans. 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 Angle of Position. Add together the Cosine 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 occupa¬ 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 northern part of the circle of latitude lies to the west of the circle of declination; bnt when the longitude is between 90 c and 270% it lies to the east. Exam. 1. Given the obliquity of the ecliptic 23’ 27' 46", the longitude of Arcturus 201° 44' 16", and the declination 20° 7' 5" N. to find the angle of position. cos. Long. 20P 44' 16" 9,96797 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. Ans. 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? Ans. 9° 6' 43 '. 38 290 ASTRONOMY. PROBLEM XXII. The Sutfs Right Ascension on two consecutive days at noon, and the Right Ascension of a Star being given , to find the time of its Passage over the Meridian. Subtract the sun’s right ascension on the first of the two given ilays, 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 maybe found nearly, by subtracting the sun’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 21 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 right ascension 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 Antares at the same ASTRONOMY. 291 time 16 h. 18 m. 28 sec. to find the time of its passage over the meridian. Jins. 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 sun’s right ascension on the 15th and 16th, from the part of the Nautical Almanac, con¬ tained in table LV. Jins . 4 h. 27 m. 33 sec. PROBLEM XXIII. The Right Ascensions of the Sun and Moon , in time , being given , on two consecutive days at woon, to find the time of the Moon's Passage over the Meridian. Subtract the right ascension of the sun on the first day at noon, from that on the second. Subtract the right ascension of the moon on the first day at noon, 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 tbe 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, 23h. 30 m. 1 sec.; the moon’s right as¬ cension on the 11 th, 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 Take 23 26 21 From 7 2 47 Take 6 4 3$ 1st remt 3 40 2d rem. 58 12 292 h. m. sec. From 6 4 35 Take 23 26 21 ASTRONOMY* h. in, sec. From 24 3 40 Take 58 12 3drem. 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 noon, 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 llth, 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 declination 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 sun’s declination between noon 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 declination may be ASTRONOMY. 293 found for the lime 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° 57' - - 9.92304 tan. Decl. 18 52 S - - 9.53368 sin. Asc. Diff. 16 38 9.45672 4 1 h. 6 m. 32 sec. 6 0 Semi-diur. arc 4 h. 53 m. time of sunset, 7 h 7 m. time of sunrise. 2. Required the time of the sun’s rising and setting at St. Petersburg, when the declination is 23° 28' N. Jins. 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? Jins. Sunrise 5h. 19 m. and sunset 6 h. 41 m. PROBLEM XXV. To reduce the time of the Moon's Passage over the Meridian of Greenwich , as given in the Nautical Almanac y to the time of its Passage over the Meridian 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, 1 @21. li. m. Passage at Greenwich, on the 17th, - 15 44 Reduction from table LVIII. - - - 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? Jins. 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. Jins. On the 22d day at 7 h. 45 m. A. M. in common reckoning. PROBLEM XXVI. i . * , ... L.Jli * nail From the Moon's Declination , as given in the Nautical Almanac , 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, tens of minutes and minutes of the change in declination, at the top. The sum of these, added to the declination at the noon or midnight next preceding the time, when the declination is in- ASTRONOMY. £9 5 treasing, but subtracted when it is decreasing, will give the re¬ quired declination. Note 1. When the declinations at one noon or midnight, and at the following midnight or noon, 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. h. m. Time at Philadelphia, August, Diff. of long. - 15 10 25 5 1 Time at Greenwich, 15 15 26 Declination, the 15th at midnight, do. 16th at noon, 0° 14' S 3 14 N Change in 12 hours, - 3 28 Declination, the 15th, at midn. Sum of quantities from table LIX - 0° 14' S 0 59 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. Jins. 18° 15' N. 3. Required the moon’s declination, on the 2d of August, 1821, at 1 b. 28 m. A. M. apparent time at New York. Jins. 0° 59' S. 296 ASTRONOMY. PROBLEM XXVII. To find the time of the moon’s Rising or Setting at a given Place , on a given astronomical day , by the aid of the Nautical Almanac. 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 declination for the resulting time, reduced to the meridian of Greenwich. With the latitude of the place and the moon’s declination, 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 difference 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. 1. Required the time of the moon’s rising at Philadel* liaon the 18th of August, 1821. d. h. m. Passage over mer. of Greenwich, «• 18 16 37 Reduction, - 12 Passage over mer. of Philadelphia, . 18 16 49 Subtract - 6 0 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' . 9.92304 tan. Declin. - 19 40 N. - 9.55315 sin. Ascen. diff. 17 25 9.47619 4 lh. 10m. 6 0 Semi-diur. arc, 7h. 10m. d. h. m. Moon’s passage over mer. of Philadelphia, , 18 16 49 Semi-diur. arc. - 0 - 7 10 Approximate time of moon’s rising, • 18 9 39 Diff. of Long. - - 5 1 Time at Greenwich, - 18 14 40 Moon’s declin. on the 18th, at 14h. 40m. is 19° 26' N, 39 298 ASTRONOMY. 9.92304 9.54754 tan. Lat. - 39° 57' tan. Declin. - 19 26 N. sin. Ascen. diff. 17 11 4 9.47058 lh. 9 m, 6 0 Semi-diur. arc, 7 9 Correction, 15 Semi-d. arc cor. 7 24 Moon’s passage, Corrected Semi-diur. arc. d. h. m. 18 16 49 7 24 Time of moon’s rising, - - - IS 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 JVonagesimal 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 half sum, logarithm C. ASTRONOMY. 299 For the given time, reduced to mean time at Greenwich, find the sun’s mean longitude and the argument N, from tables IX, X, XIII, and XIV. To the sun’s mean longitude, increased bj 2°, apply, according to its sign, the nutation 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 Cosine 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 nonagesimal * 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 nonagesimal. 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 nonagesi¬ 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' 43" N, and this taken from 90% leaves the polar dis¬ tance 50 3 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". 4 diff. 13° 23' 11" cos. 9.98803 tan. + 10, 19.37654 # sum 36 51 6 cos. 9.90319 tan. C. 9.87478 A. 0.08484 B. 9.50176 The sun’s longitude taken from the tables, for the given time, and increased by 2°, is 5 s 5° 24' 38", and the argument N is 71. The nutation, taken from table XVIII, with argument N, is + 7". Hence, the sun s mean longitude from the true equinox is 5 s 5° 24'45", or 155° 24' 45". The given time of day expressed ASTRONOMY. 301 astronomically, is 19 h. 30 m. 21 sec.; which, in degrees, is 292° 35' 15". Given time, in degrees, - 292° 35' 15" Sun’s mean long. - - 155 24 45 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. — 8.24192 E 178° 47' 4" tan. —8.32676 B. 9.50176 F 179 36 50 tan. —7.82852 90 0 0 -- 3 alt. non. 88 23 54 long, nonages. cos. — 9,99990 C. 9.87478 Ar. Co. cos.—0.00001 36° 50'47" tan. 9.87469 73 41 34 alt. nonages. 2. Required the longitude and altitude of the nonagesimal degree of the ecliptic at Philadelphia, on the 27th of August, 1821, at 8 h. 53 m. 20 sec. A. M. mean time. Ans. 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. Ans. Long. 121° 21' 25", and alt. 69° 30' 44". PROBLEM XXIX. The Longitude and Altitude of the Nonagesimal Degree of the Ecliptic , and the Moon's True Longitude , Latitude , Equatorial Parallax , and Horizontal Semidiameter being given , to find the Apparent Longitude and Latitude as affected by Parallax , and the Augmented Semidiametcr of the Moon, 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 fixed star , the first remain¬ der is the reduced parallax. Take the difference between the moon’s longitude and the longitude of the nonagesimal degree, and call it D. When the moon’s latitude is north , subtract it from 90°, but when it is south , 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 nonagesimal h y 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 Sine of h, and the arith¬ metical complement of the Sine 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 u'. Add together the logarithm c, and the Sine of (D -f w'), and the result will be the logarithm ofp, the parallax in longitude. ASTRONOMY. 303 Except when great accuracy is required, the last operation need not be performed, and p may be placed instead of it'. Add p to the moon’s true longitude, when the latter is greater than the longitude of the nonagesimal, 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 Cosine of h, and the result will be the logarithm of an arc x. But in occulta- tions , add together the logarithm of P, the Cosine of /i, 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 + ®)j and the arithmetical comple¬ ment of the Sine of d, and the result will be the logarithm of the arc x. To d, add a?, 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 -f Ip), and the Cosine of (d 4- #), 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 x , will give the apparent polar dis¬ tance. And the difference between this and 90°, w ill 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 x and z, 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, Which can only be used when the Apparent Latitude is small , as in Eclipses of the Sun. Add together, the logarithm of P, the Cosine of and the •arithmetical complement of the Sine of d, and the result will be ASTRONOMY. 304 the logarithm of an arc x. Add together, the logarithm of x , 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 «, the Sine of (D -f 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 x, attending to the sign of the latter. Then, add together, the logarithm of p , the logarithm of the difference between (d -f x ) and 90°, the arithmetical comple¬ ment of the logarithm of w, 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 -f- x), 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 nonagesi- 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 + u ), 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 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 nonagesi- 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 semi diameter. * The rules in the first method, are deduced from formulae C, G, and Lj 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. Reduction, 55° 18' 5 Moon’s long. 152° 31' 1' 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 3304" log. 3.51904 h - 73° 41'34" - sin. 9.98217 d - 89 48 22 - Ar. Co. sin. 0.00000 c. 3.50121 D - 64 7 7 sin 9,95410 u * 2853" - log. 3.45531 c. 3.50121 D 4 -u 64 54 40 - sin. 9.956.96 u' 2872" log. 3.45817 c. 3.50121 D -f ti' 64 54 59 - sin. 9 95698 V True long. 47'52".l - log. 3.45819 152 31 1 App. long. 153 18 53.1 P - - log. 3.51904 h - 73° 41' 34" - - cos. 9.44838 a? 15 27.7 - log. 2.96742 P * - - log. —3.51904 k D + ip - d 4 - x 64 31 3 90 3 49.7 - sin. 9.98217 cos. 9.63371 cos. — 7.04434 z 1.5 - log. 0.17926 40 30(5 ASTRONOMY. Ap.pol.dist. 90° 3'51".2 90 0 0 Ap. lat. - 0 3 51.2 S. Hor. semidiam. - 903'' - - log. 2.95569 App. pol. dist. 90° 3'51" - - sin. 10.00000 D m - - - - - sin. 9.95696 d - - - - - Ar. Co. sin. 0.00000 D Ar. Co. sin. 0.04590 Augmented semidiam. 15'9" - log. 2.95855 BY THE SECOND METHOD. p 3304" - log. 3.51904 h - 73°41'34" cos. 9.44838 d - 89 48 22 - Ar. Co. sin. 0.00000 x - 15 27.7 - log. 2.96742 h - 73 41 34 - tan. 10.53379 D - 64 7 7 sin. 9.95410 u 47 33 log. 3.45531 D - - Ar. Co. sin. 0.04590 D + « 64 54 40 sin. 9.95696 P 47 52 - log. 3.45817 (d + x) — 90% 3 49.7 log. 2.36116 u - Ar. Co. log. 6.54469 d Ar. Co. sin. 0.00000 App. lat. 3 51.2 S. log. 2.36402 Moon’s true long. - - 152° 31 ' 1 " P ~ 47 52 App. long. 153 18 53 ASTRONOMY. 307 Hor. semidiam. - 903" - - log. 2.95569 D-f-tt - - - - - sin. 9.95696 d - - - *! Ar. Co. sin. 0.00000 D Ar. Co. sin. 0.04590 Augm. semid. - 15' 9" - - log. 2.95855 i , 2. Given the longitude of the nonagesimal 67° 29' 8", the alti¬ tude 57° 56' 36", the moon’s true longitude 3 s 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". p h - d 3661" - - 57°56'36" - 94 5 30 - Ar. Co. log. 3.56360 sin. 9.92815 sin. 0.00111 D 40 58 27 c. 3.49286 sin. 9.81672 u - 34 0 - log. 3.30958 D -j-it - 41 32 27 c. 3.49286 sin. 9.82161 v! 34 23 - log. 3.31447 D + ti' - 41 32 50 - c. 3.49286 sin. 9.82167 P 34 23.1 - log. 3.31453 True long. 3 8 18 27 35 App. long. 3 19 1 58.1 308 ASTRONOMY. p - log. 3.56360 h ■ - 57° 56' 36" - cos. 9.72490 d - - 94 5 30 - sin. 9.99889 v - 32 18 log. 3.28739 d + v - 94 37 48 - sin. 9.99858 d - Ar. Co. sin. 0.00111 % - 32 16.8 - log. 3.28708 P . log. — 3.56360 h - sin. 9.92815 D + ip - 41 15 39 cos. 9.87605 d + x - 94 37 46.8 - cos. — 8.90696 2 - - + 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 - u - sin. 9.82161 d - Ar. Co. sin. 0.00111 D - - Ar. Co. sin. 0.18328 Augm. semidiam. 16' 50".8 log. 3.00455 3. About the middle of the eclipse of the sun, on the 27th of August, 1821, the longitude of the nonagesimal, 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 nonagesimal, at Philadelphia, was 121° 21'25", altitude 69° 30' 44", moon’s true longitude 153 e ASTRONOMY. 309 56' 15", latitude 0° 3' 47" N, reduced parslllax 55' 10", and semidiameter 15' 3"; required the apparent longitude and lati¬ tude, and the augmented semidiameter. Jins. 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 Moon, for a given Year and Month. For Mew Moon . 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 bissextile 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 YI, 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 IY, take the corresponding equations from table VII, and add them to the time of mean new moon; the sum will be the Approximate 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 sun’s longitude is greater than the moon’s, but subtracted , when it is to, 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 V. For Full Moon. 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 subtracted , 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, 311 M. New Moon. I. II. III. IV. d. h. m. 1821, 2 17 59 0092 7859 80 78 8 lun. 236 5 52 6468 5737 22 93 238 23 51 6560 3596 02 71 Days, 212 August, 26 23 51 1 . 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 s 3° 57'12 Moon’s do. 5 3 56 43 Difference, - - 0 29 Moon’s hourly motion in long, is 30' 55" Sun’s do. - 2 25 Difference, ... 28 30 m. m. sec. As 28' 30" : 29" :: 60 : 1 1, the correction. d. h. m. sec. Approx, time of new moon, August, 27 3 17 0 Correction, - + 1 1 True time, in mean time at Greenwich, 27 3 18 1 Equation of time, - —- 1 19 Apparent time at Greenwich, 27 3 16 42 Diff. of Meridians, - 5 0 46 Apparent time at Philadelphia, 26 22 15 56 312 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 h lun. 14 18 22 404 5359 58 50 6 lun. 177 4 24 4851 4303 92 95 202 J3 6 5559 5449 11 0 Days, 181 July, 21 23 6 I. 2 55 II. 13 7 III. 5 IV. 20 f | Mltll v*1 July, 22 15 33 Approximate time. Moon’s true long, found for the approx, time, is 9 s 29° 24' 51" Sun’s do. - 3 29 25 23 R. 5 29 59 28 6 0 0 0 Diff. 0 32 Moon’s hourly motion in long, is Sun’s do. 29' 34' 2 23 Difference, - 27 11 m. m. sec. As 27'11" : 32" :: 60 : 1 11, the correction. ASTRONOMY. 313 Approximate time of full moon, July, Correction, - True time, in mean time at Greenwich, Equation, - Apparent time at Greenwich, Diff. of meridian, - Apparent time at Philadelphia, d. h. m. sec. 22 15 33 0 + 1 n 22 15 34 11 — 6 2 22 15 28 9 5 0 46 22 10 27 23 3. Required the time of New Moon in July, 1821, expressed in apparent time at Philadelphia. Ans. 28 d. 9 h. 9 m. 58 sec. P. M. 4. Required the time of Full Moon in July, 1821, expressed #n apparent time, at Philadelphia. Ans. 14 d. 11 h. 17 m. 47 sec. PROBLEM XXXI. To find the Time of New or Full Moon in a given Month , by the Nautical Almanac . 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 New Moon. Take the two longitudes of the moon, for the midnight and noon, or noon 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 noons. 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 44 314 ASTRONOMY. 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, and 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. V. For Full Moon. Proceed exactly as for the new moon, except that each of the sun’s longitudes must be increased by VI 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 VI. Exam. 1. Required the time of new moOn in August, 1821, by the Nautical Almanac. In this example T is 3 h. 17 m. 0’s Long. C >s Long. Distances. 1st Diff. 2d Diff. 26th midn. 27th noon 27th midn. 28th noon 4 s 26° 2' 0" 5 2 15 44 5 8 26 28 5 14 34 24 5 s 3° 2 O' 25" 5 3 49 25 5 4 18 25 5 4 47 26 — 7° 18'25" — 1 33 41 + 483 4- 9 46 58 4-5°44'44" A.4-5 41 44 -j-5 38 55 —3' 0" — 249 [B.—2 54 Second distance - 1° 33' 41" Equat. 2d dilf. - - — 17 D. 1 33 24 As 5° 41' 44" : 1° 33' 24" :: 12h. : 3h. 16m. 51 se*\, time of new moon, in apparent time at Greenwich. 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 expected to occur in any given i/ear, 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 new 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 37, 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, ofO, 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 bissextile , 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 new 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 circte. 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 Mam. 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; blit 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 sun, 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 night, it will be visible at that place. Exam. 1. Required the eclipses that may be expected in the year 1822, and the times nearly, at which they will take place. ASTRONOMY, SI For the Eclipses of the Sun. M. New Moon. I. II. HI. IV. N ‘ 1822, 1 lun. d. h. m. 21 15 32 29 12 44 0602 S08 7182 717 78 15 66 99 930 85 51 4 16 31 1410 7899 93 65 15 Feb. I. II. III. IV. 20 4 16 7 38 19 29 13 11 As the sum of the numbers N, comes within 37 of 0, there must be an eclipse. Feb. 21 7 47 Mean time at Greenwich. M. New Moon. I. II. III. IV. N. 1822, 7 lun. d. h. m. 21 15 32 206 17 8 0602 5659 7182 5020 78 7 66 94 930 596 228 8 40 212 6261 2202 85 60 526 August, I. II. III. IV. 16 8 40 1 24 0 40 16 14 As the sum of the numbers N, comes within 37 of 500, there must be an eclipse. August, 16 11 14 Mean time at Greenwich 318 ASTRONOMY 1822 h lun. Hun. M. New Moon. I. II. III. IV. N. d. h. m. 21 15 32 0602 7182 78 66 930 14 18 22 404 5359 58 50 43 6 21 10 0198 1823 20 16 887 29 12 44 808 717 15 99 85 36 9 54 1006 2540 35 15 972 31 Feb. 5 9 54 I. 6 52 II. 0 20 III. 4 IV. 29 Feb. 5 *17 39 As the sum of the numbers N, although it comes within 35 of 1000, does not come with¬ in 25, the eclipse may be considered doubtful. It may, however, be observed, that further calculation by the next problem would show that tl^ere will be a small eclipse. Mean time at Greenwich. M. Full Moon. I. II. III. IV. N. d. h. m. 1822 6 21 10 0198 1823 20 16 8S7 7 lun. 206 17 8 5659 5020 7 94 596 213 14 18 212 5857 6843 27 10 483 August, I. II. III. IV. 1 14 18 2 14 19 26 3 26 As the sum of the numbers N, comes within 25 of 500, there must be an eclipse. 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. Ans. One of the moon on the 26th of January, at 5 h. 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 6 h. 50 m. A. M.; and one of the moon on the 23d of July, at 3h. 33 m. A. M. ASTRONOMY. 319 PROBLEM XXXIII. To Calculate an Eclipse of the Moon. Find the approximate time of full moon, by prob. XXX, and for 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 ihe 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. For the Moon's Latitude at the True Time of Full Moon . 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 sun. Increase the result by a ^ part, and it will be the semi • diameter of the earth’s shadow, which call S. For the Inclination of the Moon'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 ASTRONOMY, Add tog-ether the constant logarithm 3.55630, the Cosine of I, and the arithmetical complement of the difference between the hourly motions of the moon and sun, in longitude, rejecting the fens in the index, and call the resulting logarithm R. For the Time of the Middle 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 J, 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 moon, add the Cosine 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 -f d 4- 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 x , 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. \ ^ * 5 ' 1 ‘ t • ♦ m "* For the Times of Beginning and End of the Total Eclipse. To, and from, the difference of S and d, add and subtract e. 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 x\ 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. 321 For the Quantity of the Eclipse. Add together the constant logarithm 0.77815, the logarithm of (S -f 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 nortl^md 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 moon’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 manner, 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. Do. hourly motion, Do. semidiameter, Moon’s longitude, Do. latitude, Do. equatorial parallax, Do. semidiameter, Do. hor. mot. in long. Do. do„ in lat. - 9 29 24 51 9 10 N. 54 1 d. 14 43 29 34 2 43, tending north. 3 s 29° 25' 23" 2 23 15 46 42 322 ASTRONOMY. d. h. m. sec. Approx, time of full moon, July, 22 15 33 0 Correct, found by prob. XXX. - -fill True time, in mean time at Greenwich, 22 15 34 11 Equat. of time, - - — 6 2 Apparent ti^| at Greenwich, - 22 15 28 9 DifF. of Long. - - - 5 0 46 Apparent time at Philadelphia, T. 22 10 27 23 m m. sec. As 60 : 1 11 :: 2' 43" : 3", the correct, of lat. Moon’s lat. at approx, time, - 9' 10"N. Correction, - - - -f 3 Moon’s lat. at true time, - - 9 13 N. Moon’s equatorial parallax, - 54' 1" Sun’s do. 9 Sum, - - - - 54 10 Sun’s semidiameter, « - 15 46 38 24 Add - - - - 0 38 Semidiam. of earth’s shadow, - S. 39 2 Moon’s hor. mot. less sun’s, 1631" Ar. Co. log. 6.78755 Moon’s hor. mot. in lat. 163 - log. 2.21219 I 5° 42' tan. 8.99974 ASTRONOMY, 323 3.55630 l ... 5 42 cos. 9.99785 Moon’s bor. mot. less sun’s, Ar. Co. log. 6.78755 R. 0.34170 Moon’s lat. - - 553 ft log. 2.74272 I - - - 5° 42' - sin. 8.99704 t 121 sec. = 2 m. 1 sec. T - 10 h. 27 m. 23 sec. log. 2.08146 Middle, 10 h. 25 m. 22 sec. Moon’s lat. log. 2.74272 I - - cos. 9.99785 c - 550" = 9' 10" - log. 2.74057 S + d + c - 3775' log. 3.57692 s + d —c - 2675 log. 3.42732 2)7.00424 3.50212 ; *. ’ R. 0.34170 sec. li. m. sec. s = 6980 = 1 56 20 log. 3.84382 Middle, h. 10 1 m. sec. 25 22 56 20 Beginning, 8 29 2 End, 12 21 42 A. M. of 23d day. S — d+c - 2009' S — d — c - 909 log. 3.30298 log. 2.95856 2)6.26154 3.13077 R. 0.34170 sec. m. sec. a;' = 2968 = 49 28 log. 3.47247 log. 3.47247 3i24 ASTRONOMY. Middle, of - h. m. sec. 10 25 22 49 28 Beginning of the total eclipse, 9 35 54 End do. - 11 14 50 S 4 - d — c d 883" Ar. Co. log. log. 0.77815 3.42732 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. Jins. Moon rises about sunset, 8 § digits eclipsed, Ecliptic opposition, - - 7h. lGm. Middle, - - - - 7 23 End, - - 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. Jins. Beginning, January 16th, at 2 17 A.M. Middle, - - - - 3 43 Ecliptic opposition, - - - 3 49 End,.5 8 Digits eclipsed 9.4, on moon’s northern limb. PROBLEM XXXIV. To Project an Eclipse of the Moon. 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 be 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. 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. 8£G 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 II, 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 the 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 manner, 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 II. describe circles to represent the moon at the begin¬ ning, 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. m Elements Collected. True time of full moon, July 22d, 10 h. Semidiam of earth’s shadow, Moon’s latitude, north , - Moon’s hor. mot. from sun, in long. Moon’s hor. mot. in lat. tending north , Moon’s hor. mot. from sun, on rel. orb. Moon’s semidiameter, - Sum of semidiam. of earth’s shadow and moon, Difference of * do. 27 m. 23 sec, 39' 2" = 39'.03 9 13= 9.22 27 11 = 27.18 2 43 = 2.72 27 20 = 27.33 14 43 = 14.72 53 45 = 53.75 24 19 = 24.32 Draw Al}, 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 M6, 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 sun 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. Through C, draw SCT perpendicular to PQ, intersecting it in P, 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 s 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 IL 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 RY, and on it describe the semicircle RTY. With the sector, opened to the radius OY or OR, make the arc VU, equal to the sun’s longitude. When the longitude exceeds VI signs, take YI signs from it, and set off the remainder from R, round towards V. 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 M b, 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 sun 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 noon. 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 sarhe radius, make the arcs EG, El, FH and FK, each equal to the sun^s declination, and join GH and IK. Bisect vw in N, and through N, draw 6 N 18, parallel to EF. Make N 6 and N 18, each equal to Er or rF, and on 6 N 18, describe the semicircle 6 Y 18. With the centre N and radius Nv or Nw, describe the circle avxw Take the intervals between noon 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 an¬ 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 uvxw; and when the sun’s declination is south , produce them to meet the same circle on the other side of N. From AST110N0MY. 331 the points in which these lines intersect the circle uvxw, when the sun’s declination is north, but from the points in which, being produced, they meet it, when the declination 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 point s. 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 lines through the others. Take from Fig. 57, the dis¬ tances from the sun’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 hours, denoted by their numbers. With the centre s, and a radius equal to the sum of the semi- 332 ASTRONOMY. 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 E and a radius greater than half the distance of these points, de¬ scribe two arcs, cutting each other in a. Lav the edge of a ruler from s to a, and draw the line DsGn, intersecting the apparent orbit in G, which will be the moon s place at the time ^)f 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 INI. 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 Be, G/i and E/, respectively parallel to the lines joining M and the hours next folloiving those points, and meeting the lines joining M and the hours next preceding the same points, in the points c, h and /. Draw c&, /ig, and/e, respectively parallel to lines joining L, and the hours next preceding the points B, G and E. Then the minutes corresponding to h, connected with the hour next pre¬ ceding B, those corresponding to g, connected 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 eclipse. 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 sun’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 s 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 DII : : 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, lav 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 C n. Make the angle CsV, Fig. 58, equal to the angle BO*, Fig . 57, and join sB. Then v will represent the sun’s vertex at the beginning of the eclipse, z the place at w hich the eclipse commences, and the angle VsB, 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. 22 h. 15 m. 56 sec. Semidiameter of the circle of projection, 55' 1 " = 55'.02 Sun’s longitude, - 153° 57 ' Sun’s declination, north , - - - 10 4 Moon’s latitude, north , - - - 341= 3. 6 S Moon’s hor. mot. from sun, in long. - 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,. 7 36 = 7.6 Sun’s semidiameter, - 15 52 = 15.87 Moon’s do..15 3 = 15.05 Sum of semidiameters of sun and moon, - 30 55 = 30.92 Latitude of Philadelphia, reduced, - 39 46 N. ASTRONOMY. 331? 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¬ dian TC Y, 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 RY, and on it describe the semicircle RTV. With the sector opened to the radius OR or OY, make the arc VTU equal to 153° 57', the sun’s longitude. Draw UW perpendicular to RY, 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 M5, below M6, 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, El, FII and FK, each equal to 10° 4', the sun’s declination, and join GH and IK, intersecting the universal meridian in io and v. Bisect vw in N; through N, draw 6 N 18, parallel to EF, and make N 6, and N 18, each equal to rE or rF. With the centre N and radius N 6 or N 18, describe the semicircle 6 Y 18, and with the same centre, and radius Nr or Nw, describe the circle uvxxe. 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 lines N 15, N 30, N 45, N 60, and N 75, intersect the circle uvxw , 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 SXIX, 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 ofs, 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 Be pa- 336 ASTRONOMY, rallel to M, XX; G h parallel to M, XXI; and E/parallel to ML. Draw cb 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 5, describe a circle to represent the sun, and with 15 .05, the moon’s semidiameter, 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 : 8f digits, the quantity of the eclipse. The interval between the time of beginning and noon is 4h. 29 m. which in degrees is 67° 15'. With the sector opened to the radius N 6 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 On, and make the angle CsV, Fig. 58, equal to BCn, and join sB The measure of the angle BsV 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 sun’s and moon’s discs, with the line sY 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, lid. 23 h. 57 m. 40 sec. Semidiam. of circle of projection, - 57'24" = 57'.4 Sun’s longitude, - 323° 18 Sun’s declination, south, - - 13 46 Moon’s latitude, north, - - - 42 10 = 42.17 Moon’s hor. mot. from sun, in long. - 31 4 = 31.07 Moon’s hor. mot. in lat. tending south , - 3 4= 3.07 Moon’s hor. mot. from sun, on rel. orb. - 31 13 = 31.22 Fourth term,. 30 0 = 30.00 ASTRONOMY. 337 Sun’s semidiameter, - l(y 14" *= 16'.23 Moon’s do. - - - - 15 42 = 15.7 Sum of semicfiameters, - - - - 31 56 = 31.93 Latitude of Philadelphia, reduced, - 39° 46 Result of Projection. d. h. m. Beginning, - - 12 11 7 A. M. Greatest obscuration, - 0 42 P. M. End, •« - - 2 11 Digits eclipsed Ilf, on sun’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 ASTRONOMY. 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¬ ning 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 ofN 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 V. 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 difference, 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 sup’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 i 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; arid 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 negative , when the apparent latitude at the approximate time of greatest obscuration is increasing , but affirmative , 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 arc V. Take the sum of G and V, 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 affirmative, 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. 341 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 Cosine 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 IT, and the arithmetical complement of the logarithm of the sun’s semi- diameter, and the result, rejecting the tens in the index, will bfr the logarithm of the digits eclipsed. Note 1. If Y be applied, with a contrary sign, to IT, it will give the apparent distance of the centres of the sun 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 sun’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 com¬ 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 eclipse of the sun, on the 27th of August, 1821, found by projection for the latitude and meridian of Philadelphia, is 7 h. 31m. 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¬ fies will be obtained: 342 ASTRONOMY, Approx, time of Approx, time of Approx, time Beginning. Greatest Obscur. of End. 7 b. 31m. 8h. 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. 1510 1513 1514 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 239” log. 2.37840 N - — 268 - log. —2.42813 M 1735 Ar. Co. log. 6.76070 V — 37 log. —1.56723 G 1844 W - 1807" S + H - . 2101" log. 3.32243 S —H 1623 log. 3.21032 . .. . .. > 2)6.53275 L 1847" log. 3.26637 T 4620 sec. log. 3.66464 G + L - 3691" log. 3.56714 L —G 3 log. 0.47712 2 M 3470 Ar. Co. log. 6.45967 W 1807 Ar. Co. log. 6.74304 Correction, 8 sec. log. 0.91161 Approx, time of Begi n. 7h. 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 940" log. 2.97313 N - 433 log. 2.63649 M . - 1740 Ar. Co. log. 6.75945 V + 234 log. 2.36907 G - 1631 W 1865 S + H 2806" log. 3.44809 S — H 926 log. 2.96661 2)6.41470 L 1612" log. 3.20735 T 5160 sec. log. 3.71265 G + L 3243" log. 3.51095 G — L 19 log. 1.27875 2 M 3480 Ar. Co. log. 6.45842 W 1865 Ar. Co. log. 6.72932 Correction, 49 sec. log. 1.69009 Approx, time of End, lOli. 14m. Osec. True time of End, 10b. 13m. 11 sec. ASTRONOMY 344 For Greatest Obscuration , and Digits Eclipsed . 0 == 4 - 109"; H = 507"; S » 1865"; T = 9780 sec. M = 3475"; and N = — 701". N M — 701' log.—2.84572 3475 Ar. Co. log. 6.45905 I H 16S°36' 507" - tan. — 9.30477 log. 2.70501 • ■ i i —102 - + 109 log.—2.00978 W I - 4- 7" - 168° 36' log. 0.84510 cos. 9.99135 i • . ' * X 10.83645 cos. 9.99135 9780 sec. log. 3.99034 3475" Ar. Co. log. 6.45905 Correction, Approx. time,G. Obscur. 4 - 19 sec. 8 b. 48m. 0 sec. log. 1.27719 True time, G. Obscur. 8h.48m. 19 sec. I - X. 10.83645 tan. — 9.30477 • i • • tx CO — 1 " - 1865 log. —0.14122 S' 1864 H 507" - log. 2.70501 - - Ar. Co. cos. 0.00865 H' - 517" - log. 2.71366 S' — H' Sun’s semidiameter, 0.77815 1347" - log. 3.12937 952 Ar. Co. log. 7.02136 Digits eclipsed, 8.5 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. 7m. A. M ; greatest obscuration, Oh. 42 m. P. M.; and end, 2 h. 11 m. P. M. Required the true times, and the quantity of the eclipse. Jins, Beginning, 11 h. 7 m. 12 sec. A. M.; greatest obscuration, Oh. 41 m. 29sec. P. M.; end, 2 h. 10m. 32 sec. P. M.; digits eclipsed, lli. 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 PC p, and EC perpendicular to it. For* the Place at which the Sun is Centrally Eclipsed, on the Me¬ ridian. From the point n, in which the relative orbit intersects the universal meridian, draw nq parallel to AB. Then E^, 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 noon and the time on the relative orbit, correspond- * See note, pages 135 and 13(5. 34G 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 Zessthan 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 af and et, parallel to AB, meeting the universal meridian in/and t; and from or through/ and Z, draw frs and utv, parallel to EC, cutting or meeting the line P p in r and u 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 Er 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 /is to the right or left of Pp. Take the distance ut, and lay it in like man¬ ner from k to w , above or below AB, according as Z is to the right or left of P p Through g and ic, draw the lines Cgh and Cwx. By means of the sector, measure the arc Y/i, and call it west or east , according as h is to the right or left of the universal meridian. Take the interval between noon 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 names, 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 a , 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 cen¬ tre is at e, find the longitude of the place which then has the cen¬ tral eclipse. |r . \ • 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 witlr the centre C, describe an arc, cutting TK in K, on the left of the universal meridian. Through K, draw pKz, parallel to EC. Then Ei/, 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 P p. Through Z, draw CZS. Then, with the arc YS 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 manner as directed above, for the arc Y h 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 sun 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. 348 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 plain without further explanation. The latitudes and longitudes obtained from it are as follows: Beginning centr. eclipse, 21 h. - 22 h. - On the Meridian, 23 h. - Oh. End centr. eclipse, ' Lat. 30° 45' N. 29 20 N. 17 15 N. 14 40 N. 3 15 N. 15 40 S. 23 0 S. Long, from Philad. 41° 45' W. 4 50 E. 24 0 E. 26 45 E. 36 15 E. 59 15 E. 81 10 E. PROBLEM XXXVIII. To Project an Occultation of a Fixed Star by the Moon^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 rr^, 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 sun’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 0 and radius OR, describe the circle RTV. Make the arc VU equal to the star’s longitude, setting it from V, in the direction VUT, and draw Urn parallel to CT. Make the arc Dp equal to the star’s declination, and draw pa parallel to RV. With the centre C and radius Ca, describe the arc aq , meeting Urn, produced if necessary, in 9 , and through < 7 , 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 M6, 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, El, FH and FK, each equal to the star’s declination, and join GH, EF and IK. Bisect vw in N, and through N, draw XNZ paral¬ lel to EF. With the centre N, and a radius equal to rE, describe the semicircle XYZ; and with the same centre and the radius Nv, describe the circle uvxw. 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 lines 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 uvxw on the opposite side of N. From the points in which the lines drawn to the centre N, intersect the circle itimr, 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 w hich they meet will be the star’s places on the circle of projec- ASTRONOMY. 3d 1 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/. From h, draw hi parallel to a line joining L and the moon’s place at the hour next preceding I; and from /, 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 nCB and tCB, Fig. 61. Through I, draw bv, parallel to sn, and through E, draw 353 ASTRONOMY. b'v' parallel to st; then bv and b'v' will represent vertical circles passing through the moon's centre at the times of immersion and emersion. The angles vis 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 Vd, Fig. 61, equal to the remainder, setting it round in the direction VTd, and join dO. Make the arc Yh equal to the moon’s latitude, above RV when the latitude is north , but below when it is south , and draw gh parallel to RV, intersecting dO, produced if necessary, in e, and CT in z. Make zs equal OR, and join se. Make the angle LC k equal to zse , and on the same side of LC that se is of CD. Make the angle ks C, Fig. 62, equal to &CB, Fig. 61, and through 1 and E, draw gd and g'd' parallel to ks; also, through the same points draw mr and mV perpendicu¬ lar to gd and g'd'. Then the points d and g will designate the positions of the moon’s cusps with respect to the vertical circle bv f at the immersion, and the points d' and g', the same with respect to b'v\ at the emersion. Make the arcs ru and rV, each equal to the remainder obtained above, by subtracting the sun’s longitude from that of the star, and draw uc and u'c' parallel to dg and d'g\ and meeting mr and mV 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'r' 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 6v, placed in a vertical position, and Fig . 64, does the same for the emer¬ sion. ASTRONOMY. 353 Note 1. No notice has been taken in the above rule, of the aberrations and nutations 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: The 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 nutation, 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 Cosine 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 ©f the sum of the semidiameters of the sun and moon. 3. The projection or calculation of an occupation 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 occupation 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 follows: 63 292 30 Conjunction, in appar. time at Washington, January, Star’s passage over meridian, - Semidiameter of circle of projection, Star’s longitude, * - Sun’s longitude, Star’s latitude, ... Moon’s latitude, - Moon’s dist. from star in latitude, north , Star’s declination, north , Moon’s hor. mot. in longitude, Moon’s hor. mot. in latitude, tending south , Moon’s semidiameter, - Latitude of Place, ... Fourth term, - d. h. m. 12 6 53 8 32 59' 29" = 59'.48 11 o 5 15 5 S. 55 S. 10 = 40.17 45 4 40 10 35 57 = 35.95 0 45 = 0.75 16 14 = 16.23 38 53 31 45 = 31.75 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 5h. 49 m.; and of the emersion, 6 h. 46 m. PROBLEM XXXIX. Given the Moon's true Longitude to find the corresponding time at Greenwich by the Nautical 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 seepnd differences from table LYI, 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. a 55 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. 5th midn. 6th noon 6th midn. 7th noon Longitudes. 1st Diff. 2d Diff. Mean of 2d Diff. 7* 12° 8' 55" 7 18 7 55 7 24 9 18 8 0 13 38 5°59' 0" A. 6 1 23 | 6 4 20 2 23 2 57 B. + 2' 40'' Given longitude, - - 7 s 18° 58' 47" 2d longitude, - - - 7187 55 Difference, - 50 52 Equat. 2d Diff. - - - -J- 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 fixed star by the moon y 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 nutation. When the moon’s apparent latitude and the latitude of the star are of the same name, take ASTRONOMY. S5C) their difference; but when they are of different names, take tlieir sum; the result will be the moon’s apparent distance from the star in latitude. Call this distance d, and the augmented semidiame¬ ter s. Add together the logarithms of (s 4 - d) and (s — d), and to half their sum, add the arithmetical complement of the Cosine of the stars 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 subtracting 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 observed 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 sun’s longitude must be used instead of the star’s; d must be taken equal to the moon’s apparent latitude, and s equal to the sum of the sun’s semidiame¬ ter and the augmented semidiameter of the moon. It may also be observed, that the sun’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 ASTRONOMY. 3 57 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, 5 h. 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, Do. latitude, south, Moon’s parallax in longitude, the moon being to the east of the nonagesimal, 63° 5 11 ' 45 18".2 6.1 24 59.8 Moon’s apparent latitude, south, - 5 37 37.1 Moon’s appar. dist. from star in latitude, d. 7 29 Moon’s augmented semidiameter, 16 23.6 s + d - - 1432". 6 log. 3.15613 s — d - - - 534.6 log. 2.72803 2)5.88416 2.94208 Star’s latitude, 5° 45' Ar. Co. cos. 0.00219 c - - 879".6 e= 14' 39".6 log. 2.94427 Star’s longitude, - 63° 11' 18".2 c . 14 39.6 Moon’s apparent longitude, 62 56 38.6 Parallax in longitude, 24 59.8 Moon’s true longitude, 62 31 38.8 Appar. time at Greenwich when the moon h. m. sec. had that longitude, 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. , ■ ' , , ■s 1 , ■ . • * , - i ■ '■ . ' ASTRONOMICAL ' • ' ■ • ' ' ■' ■ * ' ' i- .A W • *<. uV r t- , I w-Jt bt -)i ■*j39 . iWriH ' .i to/*’?*»< TABLE I 3 Latitudes , and Longitudes from the Meridian of Greenwich , of some Cities , and other conspicuous Places. Names of Places. Latitude. Longitude in Degrees. Longitude in Time. o in h. m. s. Amsterdam, Holland, 52 22 17N. 4 53 15E. 0 19 33 Athens, Greece, 37 58 IN. 23 46 14E. 1 35 5 Baltimore, U. States, 39 23 ON. 76 50 0W. 5 7 20 Bergen, Norway, 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 Hoi. 34 3 OS. 151 15 0E. 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, 36 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 0E. 1 15 36 Charleston. U. States, 32 50 ON. 79 48 0W. 5 19 12 Constantinople, Turkey, 41 1 27N. 28 55 15E. 1 55 41 Copenhagen, Denmark, 55 41 4N. 12 35 6E. 0 50 20 Dublin, Ireland, 53 21 11N. 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 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. 0 57 3 New-Haven, U. States, 41 18 ON. 72 58 0W. 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. 0 9 21 Pekin, China, 39 54 13N. 116 27 45E. 7 45 51 St. Petersburg, Russia, 59 56 23N. 30 18 45E. 2 1 15 Philadelphia, U. States, 39 56 55N. 75 11 30W. 5 0 46 Point Venus, Otaheite, 17 29 17S. 149 30 15W. 9 58 1 Quebec, Canada, 46 47 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 SOW. 1 45 46 Washington, U. States, 38 53 ON. 76 55 30W. 5 7 42 4 TABLE II Mean Astronomical Refractions, Ap. Alt. Refr. Ap. Alt. Refr. Ap. Alt. Refr. Ap. Alt. Refr. 1 0° 0' 33' 0" 4° 0' IT 51" 12° 20' 4' 16" 45° 0' O' 57" 0 5 32 10 4 10 11 29 12 40 4 9 46 0 0 55 0 10 31 22 4 20 11 8 13 0 4 3 47 0 0 53 0 15 30 35 4 30 10 48 13 20 3 57 48 0 0 51 0 20 29 50 4 40 10 29 13 40 3 51 49 0 0 49 0 25 29 6 4 50 10 11 14 0 3 45 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 35 52 0 0 44 0 40 27 0 5 20 9 23 15 0 3 30 53 0 0 43 0 45 26 20 5 30 9 8 15 30 3 24 54 0 0 41 0 50 25 42 5 40 8 54 16 0 3 17 55 0 0 40 0 55 25 5 5 50 8 41 16 30 3 10 56 0 0 38 1 0 24 29 6 0 8 28 17 0 3 4 57 0 0 37 1 5 23 54 6 10 8 15 17 30 2 59 58 0 0 35 1 10 23 20 6 20 8 3 18 0 2 54 59 0 0 34 1 15 22 47 6 30 7 51 18 30 2 49 60 0 0 33 1 20 22 15 6 40 7 40 19 0 2 45 61 0 0 31 1 25 21 44 6 50 7 30 19 30 2 39 62 0 0 30 1 30 21 15 7 0 7 20 20 0 2 35 63 0 0 29 1 35 20 46 7 10 7 11 20 30 2 31 64 0 0 28 1 40 20 18 7 20 7 2 21 0 2 27 65 0 0 26 1 45 19 51 7 30 6 53 21 30 2 24 66 0 0 25 1 50 19 25 7 40 6 45 22 0 2 20 67 0 0 24 1 55 19 0 7 50 6 37 23 0 2 14 68 0 0 23 2 0 18 35 8 0 6 29 24 0 2 7 69 0 0 22 2 5 18 11 8 10 6 22 25 0 2 2 70 0 0 21 2 10 17 48 8 20 6 15 26 0 1 56 71 0 0 19 2 15 17 26 8 30 6 8 27 0 1 51 72 0 0 18 2 20 17 4 8 40 6 1 28 0 1 47 73 0 0 17 2 25 16 44 8 50 5 55 29 0 1 42 74 0 0 16 2 30 16 24 9 0 5 48 30 0 1 38 75 0 0 15 2 35 16 4 9 10 5 42 31 0 1 35 76 0 0 14 2 40 15 45 9 20 5 36 32 0 1 31 77 0 0 13 2 45 15 27 9 30 5 31 33 0 1 28 78 0 0 12 2 50 15 9 9 40 5 25 34 0 1 24 79 0 0 11 2 55 14 52 9 50 5 20 35 0 1 21 80 0 0 10 3 0 14 36 10 0 5 15 36 0 1 18 81 0 0 9 3 5 14 20 10 15 5 7 37 0 1 16 82 0 0 8 3 10 14 4 10 30 5 0 38 0 1 13 83 0 0 7 3 15 13 49 10 45 4 53 39 0 1 10 84 0 0 6 3 20 13 34 11 0 4 47 40 0 1 8 85 0 0 5 3 25 13 20 11 15 4 40 41 0 1 5 86 0 0 4 3 30 13 6 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 12 0 4 23 44 0 0 59 89 0 0 1 TABLE III, 5 Mean Right Ascensions and Declinations of some of the Fixed Stars, for the beginning of 1820 , with their Annual Variations. Names and Magnitude. Right Asc. An. Var. Declination. An. Var. y Pegasi, - - Mag. o o , n 0 59 35 // + 46.1 o , n 14 10 56N. t + 20.0 « Polaris, - - 2.3 14 13 7 216.4 88 20 55N. + 19.4 a Arietis, - - - 3 29 15 38 50.3 22 36 23N. +17.3 a. Ceti, - - - 2 43 13 8 46.7 3 22 39N. -t-14.5 y 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 Tauri, - - - 2 78 43 47 56.7 28 26 42N. + 3.8 £ Tauri, - - - 3 81 43 14 53.6 21 1 23N. + 2.8 » Gemi norum, 2.3 91 0 6 54.3 22 32 56N. — 0.4 fx 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 Geminorum, - 3 107 20 25 53.8 22 18 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 7N. — 8.0 u Leonis, - - - O 149 22 33 49.2 17 38 10N. —17.3 Regulus, -* Virginia, - - - 1 149 41 39 48.1 12 50 36N. —17.3 - 3 175 19 45 46.9 2 46 44N. —20.3 y Virginia, - - 3 188 8 4 45.3 0 27 38S. + 20.0 et Virginis, - - - 1 198 55 30 47.2 10 13 5S. + 19.0 A returns, - - 1 211 51 45 40.9 20 7 28N. —19.0 tt 2 Librae, - - 2.3 220 14 2 49.5 15 17 13S. + 15.4 cT Scorpii, /3 Scorpii, - - - 3 237 25 37 52.9 22 5 59S. + 10.9 - 2 238 44 48 52.0 19 18 13S. + 10.5 Antares, - - 1 244 35 49 54.9 26 1 21S. + 8.7 ct Lyra;, - - - - 1 277 42 37 30.4 38 37 19N. 4- 2.9 00 10 ] L000 6* 42 TABLE XXXI Erection. Argument. Evection, corrected, 0* I s 1 I s lil S IV S 1. V« 1 0° 1°30 0" 2° 10'43" 2° 40'10" 2° 50'25" 2° 39' 8" 2° 9' 43" 1 1 31 25 2 11 57 2 40 51 2 50 23 2 38 25 2 8 29 2 1 32 51 2 13 9 2 41 30 2 50 20 2 37 40 2 7 16 3 1 34 16 2 14 21 2 42 8 2 50 15 2 36 55 2 6 2 4 1 35 42 2 15 31 2 42 45 2 50 9 2 36 8 2 4 47 5 1 37 7 2 16 41 2 43 21 2 50 1 2 35 19 2 3 32 6 1 38 32 2 17 50 2 43 55 2 49 52 2 34 30 2 2 16 7 1 39 57 2 18 58 2 44 27 2 49 41 2 33 40 2 1 0 8 1 41 21 2 20 5 2 44 59 2 49 29 2 32 48 1 59 43 9 1 42 46 2 21 11 2 45 29 2 49 15 2 31 55 1 58 26 10 1 44 10 2 22 17 2 45 57 2 49 0 2 31 2 1 57 8 11 1 45 34 2 23 21 2 46 24 2 48 43 2 30 7 1 55 49 12 1 46 58 2 24 24 2 46 50 2 48 26 2 29 11 1 54 30 13 1 48 21 2 25 26 2 47 14 2 48 6 2 28 14 1 53 11 14 1 49 44 2 26 28 2 47 37 2 47 45 2 27 16 1 51 51 15 1 51 7 2 27 28 2 47 59 2 47 23 2 26 17 1 50 31 16 1 52 29 2 28 27 2 48 19 2 47 0 2 25 17 1 49 11 17 1 53 51 2 29 25 2 48 37 2 46 35 2 24 16 1 47 50 18 1 55 12 2 30 21 2 48 54 2 46 8 2 23 14 1 46 29 19 1 56 33 2 31 17 2 49 10 2 45 41 2 22 11 1 45 7 20 1 57 53 2 32 11 2 49 24 2 45 12 2 21 7 1 43 46 21 1 59 13 2 33 5 2 49 37 2 44 41 2 20 2 1 42 24 22 2 0 32 2 33 57 2 49 48 2 44 9 2 18 56 1 41 2 23 2 1 51 2 34 48 2 49 58 2 43 36 2 17 50 1 39 39 24 2 3 9 2 35 38 2 50 6 2 43 2 2 16 43 1 38 17 25 *2 4 26 2 36 26 2 50 13. 2 42 26 : 2 15 34 1 36 54 26 2 5 43 2 37 13 2 50 19 2 41 49 2 14 25 1 35 32 27 2 6 59 2 37 59 2 50 23 2 41 11 2 13 16 1 34 9 28 2 8 15 2 38 44 2 50 25 2 40 31 2 12 5 1 32 46 29 2 9 30 2 39 28 2 50 26 2 39 50 2 10 54 1 31 23 30 2 10 43 2 40 10 2 50 25 2 39 8 2 9 42 1 30 0 TABLE XXXI, 43 Evection. Argument. Evection, corrected. Yl* VII s VIII* IX* X* XI i 0° 1‘ 30' 0" 0‘ >50' 18" 0° 20' 52" 0° 9'34" 0' >19'50" 0‘ >49'16" 1 1 28 37 0 49 6 0 20 10 0 9 34 0 20 32 0 50 30 2 1 27 14 0 47 55 0 19 29 0 9 35 0 21 16 0 51 45 3 1 25 51 0 46 44 0 18 49 0 9 37 0 22 1 0 53 1 4 1 24 28 0 45 34 0 18 11 0 9 41 0 22 47 0 54 17 5 1 23 6 0 44 26 0 17 34 0 9 47 0 23 34 0 55 33 6 1 21 43 0 43 17 0 16 58 0 9 54 0 24 22 0 56 51 7 1 20 20 0 42 10 0 16 24 0 10 2 0 25 12 0 58 9 8 1 18 58 0 41 4 0 15 50 0 10 12 0 26 3 0 59 28 9 1 17 36 0 39 58 0 15 19 0 10 23 0 26 55 1 0 47 10 1 16 14 0 38 53 0 14 48 0 10 36 0 27 48 1 2 7 11 1 14 52 0 37 49 0 14 19 0 10 50 0 28 43 ' 1 3 27 12 1 13 31 0 36 46 0 13 51 0 11 5 0 29 39 1 4 48 13 1 12 10 0 35 44 0 13 25 0 11 23 0 30 35 1 6 9 14 1 10 49 0 34 43 0 13 0 0 11 41 0 31 33 1 7 31 15 1 9 29 0 33 43 0 12 37 0 12 1 0 32 32 1 8 53 16 1 8 9 0 32 44 0 12 14 0 12 23 0 33 32 1 10 16 17 1 6 49 0 31 46 0 11 54 0 12 45 0 34 34 1 11 39 18 1 5 30 0 30 49 0 11 34 0 13 10 0 35 36 1 13 2 19 1 4 11 0 29 53 0 11 16 0 13 35 0 36 39 1 14 26 20 1 2 52 0 28 58 0 11 0 0 14 3 0 37 43 1 15 50 21 1 1 34 0 28 5 0 10 45 0 14 31 0 38 48 1 17 14 22 1 0 17 0 27 12 0 10 31 0 15 1 0 39 55 1 18 39 23 0 59 0 0 26 20 0 10 19 0 15 33 0 41 2 1 20 3 24 0 57 44 0 25 30 0 10 8 0 16 5 0 42 10 1 21 28 25 0 56 28 0 24 40 0 9 5§ 0 16 39 0 43 19 1 22 53 26 0 55 13 0 23 52 i 0 9 51 0 17 15 0 44 29 1 24 18 27 0 53 58 0 23 5 ! 0 9 45 0 17 52 0 45 39 1 25 44 28 0 52 44 0 22 20 | 0 9 40 0 18 30 0 46 51 1 27 9 29 0 51 31 0 21 35 0 9 36 0 19 9 0 48 3 1 28 34 30 0 50 18 0 20 52 0 9 34 0 19 50 0 49 16 1 30 0 TABLE XXXII 44 Equation of Moon's Centre . Argument. Anomaly, corrected. 0* 1 S 11* ill* IV* V* 0° 7° O' 0" 10 20' 58" 12° 38'44" 13° 17' 35" 12° 16' 21'' 9° 58' 29" 1 7 7 5 10 26 52 12 41 43 13 17 5 12 12 48 9 52 58 2 7 14 10 10 32 42 12 44 35 13 16 28 12 9 11 9 47 24 3 7 21 15 10 38 27 12 47 20 13 15 44 12 5 29 9 41 48 4 7 28 19 10 44 8 12 49 59 13 14 53 12 1 41 9 36 10 5 7 35 23 10 49 43 12 52 30 13 13 56 11 57 49 9 30 29 I 6 7 42 26 10 55 14 12 54 55 13 12 52 11 53 52 9 24 46 7 7 49 28 11 0 39 12 57 12 13 11 41 11 49 50 9 19 1 8 7 56 28 11 6 0 12 59 23 13 10 24 11 45 44 9 13 13 9 8 3 28 11 11 15 13 1 26 13 9 1 11 41 33 9 7 24 10 8 10 26 11 16 24 13 3 23 13 7 31 . 11 37 17 9 1 52 : 11 8 17 22 11 21 29 13 5 12 13 5 54 11 32 57 8 55 39 12 8 24 17 11 26 27 13 6 55 13 4 12 11 28 33 8 49 44 13 8 31 10 11 31 20 13 8 30 13 2 23 11 24 5 8 43 47 14 8 38 1 11 36 8 13 9 59 13 0 27 11 19 32 8 37 49 15 8 44 50 11 40 49 13 11 20 12 58 26 11 14 55 8 31 49 16 8 51 36 11 45 25 13 12 34 12 56 18 11 10 14 8 25 48 17 8 58 20 11 49 54 13 13 41 12 54 5 11 5 30 8 19 46 18 9 5 1 11 54 18 13 14 41 12 51 45 11 0 41 8 13 42' 19 9 11 39 11 58 35 13 15 34 12 49 19 10 55 49 8 7 38 20 9 18 15 12 2 47 13 16 20 12 46 47 10 50 53 8 1 32 21 9 24 47 12 6 52 13 16 59 12 44 10 10 45 53 7 55 26 22 9 31 16 i 12 10 50 13 17 31 12 41 27 10 40 50 7 49 18 23 9 57 42 12 14 42 13 17 56 12 38 38 10 35 43 7 43 10 24 9 44 4 !12 18 28 13 18 14 12 35 43 10 30 33 7 37 1 25 9 50 23 12 22 7 13 18 24 12 32 43 10 25 20 7 30 52 26 9 56 38 12 25 40 13 18 28 12 29 37 10 20 4 7 24 42 27 10 2 49 12 29 6 13 18 25 12 26 26 10 14 45 7 18 32 28 10 8 56 12 32 25 13 18 16 12 23 10 10 9 22 7 12 21 29 10 14 59 12 35 38 13 17 59 12 19 48 10 3 57 7 6 11 30 10 20 58 ; 12 38 44 13 17 35 12 16 21 9 58 29 7 0 0 TABLE XXXII, 45 Equat ion of Moon's Centre . Argument. Anomaly, corrected. --ft—— VI* VII* VIII s IX* X s XI* 0° 7° O' 0" 4° 1' 31" 1°43'39" 0°42'25" i°2r’i6" 3° 39' 2" 1 6 53 49 3 56 3 1 40 12 0 42 1 1 24 22 3 45 1 2 6 47 39 3 50 38 1 36 50 0 41 44 1 27 35 3 51 4 3 6 41 28 3 45 15 1 33 34 0 41 35 1 30 54 3 57 11 4 6 35 18 3 39 56 1 30 23 0 41 32 1 34 20 4 3 22 5 6 29 8 3 34 40 1 27 17 0 41 36 1 37 53 4 9 37 6 6 22 59 3 29 26 1 24 17 0 41 46 1 41 32 4 15 55 r 6 16 50 3 24 17 1 21 22 0 42 4 1 45 18 4 22 18 8 6 10 42 3 19 10 1 18 33 0 42 29 1 49 10 4 28 44 9 6 4 34 3 14 7 1 15 50 0 43 1 153 8 4 35 13 10 5 58 28 3 9 7 1 13 12 0 43 40 1 57 13 4 41 45 11 5 52 22 3 4 11 1 10 41 0 44 26 2 1 24 4 48 21 12 5 46 17 2 59 19 1 8 15 0 45 19 2 5 42 4 54 59 13 5 40 14 2 54 30 1 5 55 0 46 19 2 10 5 5 1 40 14 5 34 12 2 49 46 1 3 42 0 47 26 2 14 35 5 8 24 15 5 28 11 2 45 5 1 1 34 0 48 40 2 19 11 5 15 10 16 5 22 11 2 40 28 0 59 33 0 50 1 2 23 52 5 21 59 17 5 16 13 2 35 55 0 57 37 0 51 30 2 28 39 5 28 50 18 5 10 16 2 31 27 0 55 48 0 53 5 2 33 32 5 35 43 19 5 4 21 2 27 3 0 54 6 0 54 47 2 38 31 5 42 37 20 4 58 28 2 22 43 0 52 29 0 56 37 2 43 35 5 49 34 21 4 52 36 2 18 27 0 50 59 0 58 33 2 48 45 5 56 32 22 4 46 47 2 14 16 0 49 36 1 0 37 2 54 0 6 3 31 23 4 40 59 2 10 10 0 48 19 1 2 48 2 59 21 6 10 32 24 4 35 14 2 6 8 0 47 8 15 5 3 4 46 6 17 34 25 4 29 31 2 2 11 0 46 4 1 7 30 3 10 17 6 24 37 26 4 23 50 1 58 19 0 45 7 1 10 1 3 15 52 6 31 41 27 4 18 11 1 54 31 0 44 16 1 12 40 3 21 33 6 38 45 28 4 12 35 1 50 49 0 43 32 1 15 25 3 27 18 6 45 50 29 4 7 2 1 47 11 0 42 55 1 18 17 3 33 8 6 52 55 30 4 1 31 1 43 39 0 42 25 1 21 16 3 39 2 7 0 0 46 TABLE XXXIII \ Variation * Argument. Variation, corrected. 0* I s 11* 111* IV 1 i V* 0° 0°38 / 0" l c 8' 1" 1° 6- 58" 0°35 / 54" 0° ' 5' 29" 0‘ 3 6' 2" 1 0 39 13 1 8 35 1 6 18 0 34 40 0 4 54 0 6 42 2 0 40 26 1 9 7 1 5 36 0 33 27 0 4 21 0 7 24 3 0 41 39 1 9 36 1 4 52 0 32 13 0 3 51 0 8 8 4 0 42 52 1 10 3 1 4 5 0 31 0 0 3 22 0 8 55 5 0 44 4 1 10 28 1 3 17 0 29 47 0 2 56 0 9 44 6 0 45 16 1 10 50 1 2 27 0 28 34 0 2 33 0 10 34 7 0 46 28 1 11 9 1 1 35 0 27 22 0 2 12 0 11 27 8 0 47 38 1 11 26 1 0 42 0 26 11 0 1 54 0 12 22 9 0 48 48 1 11 41 0 59 4-6 0 25 1 0 1 38 0 13 19 ’ 10 0 49 57 1 11 53 0 58 49 0 23 51 0 1 24 0 14 17 11 0 51 6 1 12 2 0 57 50 0 22 42 0 1 14 0 15 17 12 0 52 13 1 12 9 0 56 50 0 21 34 0 1 5 0 16 19 13 0 53 19 1 12 13 0 55 48 0 20 28 0 1 0 0 17 22 14 0 54 24 1 12 15 0 54 45 0 19 22 0 0 57 0 18 27 15 0 55 27 1 12 14 0 53 41 0 18 18 0 0 57 0 19 33 16 0 56 30 1 12 10 0 52 35 0 17 15 0 0 59 0 20 41 17 0 57 31 1 12 4 0 51 28 0 16 13 0 1 4 0 21 50 18 0 58 30 1 11 55 0 50 21 0 15 13 0 1 11 0 23 0 19 0 59 28 1 11 44 0 49 12 0 14 15 0 1 22 0 24 11 20 1 0 24 1 11 30 0 48 2 0 13 17 ! 0 1 34 0 25 23 21 1 1 19 1 11 14 ! ! o 46 52 0 12 22 0 1 50 0 26 36 22 1 2 11 1 10 55 0 45 40 0 11 28 0, 2 8 0 27 50 23 1 3 2 1 10 34 0 44 29 0 10 37 0 2 28 0 29 4 24 1 3 51 1 10 10 0 43 16 0 9 47 0 2 51 0 30 20 25 1 4 38 1 9 44 0 42 3 0 8 59 0 3 17 0 31 36 26 1 5 23 ' i 1 9 15 0 40 50 0 8 13 0 3 45 0 32 52 .27 1 6 6 1 i 1 8 44 0 39 36 0 7 29 0 4 16 0 34 9 28 1 6 47 ! ! i 8 11 0 38 22 0 6 47 0 4 48 0 35 26 29 1 7 25 1 1 1 7 36 0 37 8 0 6 7 0 5 24 0 36 43 30 1 8 1 ; l 6 58 0 35 54 0 5 29 0 6 2 0 38 0 TABLE XXXIII, 47 Variation . Argument. Variation, corrected. VI* VII s VIII s IX* X s XI* 0° 0°38 0" l c 9' 58" 1° 10'30" 0° 40' 6" o c 1 9' ’ 2" 0° 7'58" 1 0 39 17 1 10 36 1 9 53 0 38 52 0 8 24 0 8 35 2 0 40 34 1 11 11 1 9 13 0 37 38 0 7 49 0 9 13 3 0 41 51 1 11 44 1 8 31 0 36 24 0 7 15 0 9 54 4 0 43 8 1 12 15 1 7 47 0 35 10 0 6 45 0 10 37 5 0 44 24 1 12 43 1 7 1 0 33 57 0 6 16 0 11 22 6 0 45 40 1 13 9 1 6 13 0 32 44 0 5 50 0 12 9 7 0 46 55 1 13 32 1 5 23 0 31 31 0 5 26 0 12 58 8 0 48 10 1 13 52 1 4 31 0 30 19 0 5 5 0 13 49 9 0 49 24 1 14 10 1 3 38 0 29 8 0 4 46 0 14 41 10 0 50 37 1 14 26 1 2 42 0 27 58 0 4 29 0 15 36 11 0 51 49 1 14 38 1 1 45 0 26 48 0 4 16 0 16 32 12 0 53 0 1 14 48 1 0 47 0 25 39 0 4 4 0 17 30 13 0 54 10 1 14 56 0 59 47 0 24 31 0 3 56 0 18 29 14 0 55 19 1 15 1 0 58 45 0 23 25 0 3 50 0 19 30 15 0 56 27 1 15 3 0 57 42 0 22 19 0 3 46 0 20 32 16 0 57 o o 1)0 1 15 3 0 56 38 0 21 15 0 3 45 0 21 36 17 0 58 38 1 15 0 0 55 32 0 20 12 0 3 47 0 22 41 18 0 59 41 1 14 54 0 54 25 0 19 10 0 3 51 0 23 47 19 1 0 43 1 14 46 0 53 18 0 18 10 0 3 58 0 24 54 20 1 1 43 1 14 35 0 52 9 0 17 11 0 4 7 0 26 3 21 1 2 41 1 14 22 0 50 59 0 16 14 0 4 19 0 27 12 22 1 3 38 1 14 6 0 ■ 49 49 0 15 18 0 4 34 0 28 22 23 1 4 33 1 13 48 0 . 48 38 0 14 25 0 4 51 0 29 32 24 1 5 25 1 13 27 0 47 26 0 13 33 0 5 10 0 30 44 25 1 6 16 1 13 3 0 46 13 0 12 43 0 5 32 0 31 55 26 1 7 5 1 12 38 0 45 0 0 11 54 0 5 57 0 33 8 27 1 7 52 1 12 9 0 43 47 0 11 8 0 6 23 0 34 20 28 1 8 36 1 11 39 0 42 33 0 10 24 0 6 53 0 35 33 29 1 9 18 1 11 6 0 41 20 0 9 42 0 7 24 0 36 47 30 1 9 58 1 10 30 0 40 6 0 9 2 0 7 58 0 38 0 48 TABLE XXXIV, Reduction. Argument. Suppl. of Node -f Moon’s Orbit Longitude. G* VI s 1* VIP IL* VIII* ill* IX* IV* X* V* XI* 0° 7 0' 1' 3" V 3" • 7 , o// ’ 12' 57" ' 12 '57" 1 6 46 0 56 1 10 7 14 13 4 12 50 2 6 31 0 49 1 18 7 29 13 10 12 42 3 6 17 0 43 1 26 7 43 13 17 12 33 4 6 3 0 38 1 35 7 57 13 22 12 25 5 5 48 0 33 1 44 8 12 13 27 12 16 6 5 34 0 28 1 54 8 26 13 32 12 6 7 5 20 0 24 2 3 8 40 13 36 11 56 8 5 6 0 20 2 14 8 54 13 40 11 46 9 4 53 0 17 2 24 9 7 13 43 11 36 10 4 39 0 14 2 35 9 21 13 46 11 25 11 4 26 0 12 2 46 9 34 13 48 11 14 12 4 12 0 10 2 58 9 48 13 50 11 2 13 3 59 0 9 3 9 10 1 13 51 10' 50 14 3 46 0 8 3 22 10 13 13 52 10 38 15 3 34 0 8 3 34 10 26 13 52 10 26 16 3 22 0 8 3 46 10 38 13 52 10 13 17 3 9 0 9 3 59 10 50 13 51 10 1 18 2 58 0 10 4 12 11 2 13 50 9 48 19 2 46 0 12 4 26 11 14 13 48 9 34 20 2 35 0 14 4 39 11 25 13 46 9 21 21 2 24 0 17 4 53 11 36 13 43 9 7 22 2 14 0 20 5 6 11 46 13 40 8 54 23 2 3 0 24 5 20 11 56 13 36 8 40 24 1 54 0 28 5 34 12 6 13 32 8 26 25 1 44 0 33 5 48 12 16 13 27 8 12 26 1 35 0 38 6 3 12 25 13 22 7 57 27 1 26 0 43 6 17 12 33 13 17 7 43 28 1 18 0 49 6 31 12 42 13 10 7 29 29 1 10 0 56 6 46 12 50 13 4 7 14 30 1 3 1 3 7 0 12 57 | 12 57 7 0 TABLE XXXV, 49 Moon’s Distance from the North Pole of the Ecliptic. Argument. Suppl. of Node -j- Moon’s Orbit Longitude. III* IV* v , , j VI* Ml S Vlli* 0° 84°39'16" 85° 20'43" 87°13'47" 89°48 / ' 0" 92° 22' ' 13" 94° 15' 17" 30° 1 84 39 19 85 23 27 87 18 28 89 53 23 92 26 52 94 17 57 29 2 84 39 27 85 26 16 87 23 12 89 58 46 92 31 27 94 20 31 28 3 84 39 41 85 29 10 87 27 58 90 4 8 92 36 0 94 23 1 27 4 84 40 1 85 32 9 87 32 48 90 9 31 92 40 30 94 25 25 26 5 84 40 27 85 35 12 87 37 39 90 14 52 92 44 56 94 27 45 25 6 84 40 58 85 38 20 87 42 33 90 20 14 92 49 19 94 29 59 24 7 84 41 34 85 41 33 87 47 30 90 25 35 92 53 39 94 32 8 23 8 84 42 17 85 44 50 87 52 28 90 30 55 92 57 56 94 34 12 22 9 84 43 5 85 48 11 87 57 29 90 36 14 93 2 9 94 36 11 21 10 84 43 58 85 51 37 88 2 31 90 41 33 93 6 18 94 38 4 20 11 84 44 57 85 55 7 88 7 36 90 46 50 93 10 24 94 39 52 19 12 84 46 2 85 58 42 88 12 42 90 52 7 93 14 27 94 41 35 18 13 84 47 12 86 2 20 88 17 50 90 57 22 93 18 25 94 43 13 17 14 84 48 27 86 6 3 88 23 0 91 2 36 93 22 20 94 44 45 16 15 84 49 49 86 9 50 88 28 11 91 7 49 93 26 10 94 46 11 15 16 84 51 15 86 13 40 88 33 24 91 13 0 93 29 57 94 47 32 14 17 84 52 47 86 17 35 88 38 38 91 18 10 93 33 40 94 48 48 13 18 84 54 25 86 21 33 88 43 53 91 23 18 93 37 18 94 49 58 12 19 84 56 7 86 25 36 88 49 10 91 28 24 93 40 53 94 51 3 11 20 84 57 56 86 29 42 88 54 27 91 33 29 93 44 23 94 52 2 10 21 84 59 49 i 86 33 51 88 59 46 91 38 31 93 47 49 94 52 55 9 22 85 1 48 |86 38 4 89 5 5 91 43 32 93 51 10 94 53 43 8 23 85 o O 52 86 42 21 89 10 25 91 48 30 93 54 27 94 54 26 7 24 85 6 1 86 46 41 89 15 46 I 91 53 27 93 57 40 94 55 2 6 25 85 8 15 86 51 4 89 21 7 91 58 21 94 0 48 94 55 33 5 26 85 10 35 86 55 30 89 26 29 92 3 12 94 3 51 94 55 59 4 27 85 12 59 87 0 0 89 31 52 92 8 1 94 6 50 94 56 18 O O 28 85 15 29 87 4 32 89 37 14 92 12 48 94 9 44 94 56 33 2 29 85 18 3 87 9 8 89 42 37 92 17 32 94 12 33 94 56 41 1 30 85 20 43 87 i 13 47 89 48 0 92 22 13 94 15 17 94 56 44 0 IP 1* 0* XI l X* IX* 30 TABLE XXXVI Equation II. of the Moon's Polar Distance . Argument II, corrected. 111> IV* V* VI* Vil* VIII* 0° O' 14" 1' 24" 4' 3 7 V 9' 0" 13' 23" 16' 36" 30° 1 0 14 1 29 4 45 9 9 13 31 16 40 29 2 0 14 1 34 4 53 9 18 13 39 16 45 28 3 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 O <6 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 11 7 15 5 17 26 16 15 0 32 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 42 3 15 7 20 11 51 15 37 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 , 15 49 17 40 9 22 0 52 3 36 7 47 12 17 ! 15 55 17 41 8 23 0 56 3 43 7 56 12 26 16 0 17 42 7 24 0 59 3 51 8 5 12 34 16 6 17 43 6 25 1 3 3 58 8 14 12 42 16 11 17 44 5 26 1 7 4 6 , 8 23 12 51 16 16 17 45 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 37 9 0 13 23 16 36 17 46 0 II* I s 0 s XI* X* IX* TABLE XXXVII. Equation III. of the Polar Distance. Argument Moon’s True Longitude. 111* IV* V* VI* VII* VIII* 0° 16" 15" 12" 8" 4" 1" 30° 6 16 14 11 7 3 1 24 12 16 14 10 6 3 0 18 18 16 13 10 5 2 0 12 24 15 13 9 5 1 0 6 30 15 12 8 4 1 0 0 II* I* 0* XI* X* IX* TABLE XXXVIII TABLE XXXIX 51 To Convert Degrees and Equations of Polar Distance. Minutes into Decimal Arguments. 20 of Long.; V to IX, cor- Parts. rected; and X, not corrected. Degrees Dec. Arg., 20 V. VI. VII. VIII IX. : X. Arg.j ami Min. parts. 250 0" 56" 6" 3" 25" 3" Tr 250 1° 5' 005 260 0 56 6 3 25 3 n 240 1 26 4 270 0 56 6 3 25 o O n 230 1 48 5 280 1 55 6 3 25 3 n 220 2 10 6 290 1 55 7 3 25 4 n 210 2 31 7 300 1 55 7 4 25 4 TT 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 12 350 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 17 100 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 22 50 8 17 23 460 11 36 24 19 17 23 23 40 8 38 24 470 12 35 25 21 17 25 24 30 9 0 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 31 960 11 31 32 550 19 22 37 32 12 38 32 950 11 53 33 560 20 20 39 33 11 40 33 940 12 14 34 570 21 19 40 34 11 41 34 930 12 36 35 580 22 17 41 36 10 43 35 920 12 58 36 590 23 16 43 37 10 44 36 910 13 19 37 600 24 15 44 38 9 46 37 900 13 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 29 5 53 47 5 56 43 790 17 38 49 720 29 5 53 47 5 56 43 780 18 0 50 730 30 4 54 47 5 57 43 770 18 22 51 740 30 4 54 47 5 57 43 760 18 43 19 5 52 53 750 i 30 4 54 47 5 57 43 •750 TABLE XL Morris Equatorial Parallax . Argument. Argument of the Evection. 1 1* 11* III* 1 n ‘ V* 0° 1 28" 1 ' 23" 1' 9" O' 50" O'32" O' 18" 30° 1 1 28 1 23 1 8 0 49 0 31 0 18 29 2 1 28 1 22 1 8 0 49 0 30 0 18 28 3 1 28 1 22 1 7 0 48 0 30 0 17 27 4. 1 28 1 22 1 7 0 47 0 29 0 17 26 5 1 28 1 21 1 6 0 47 0 29 0 17 25 6 1 28 1 21 1 5 0 46 0 28 0 17 24 7 1 28 1 20 1 5 0 46 0 28 0 16 23 8 1 28 1 20 1 4 0 45 0 27 0 16 22 9 1 28 1 20 1 4 0 44 0 27 0 16 21 10 1 28 1 19 1 3 0 44 0 26 0 16 20 11 1 28 1 19 1 2 0 43 0 26 0 15 19 12 1 27 1 18 1 2 0 42 0 25 0 15 18 13 1 27 1 18 1 1 0 42 0 25 0 15 17 14 1 27 1 17 1 0 0 41 0 24 0 15 16 15 .1 27 1 17 1 0 0 40 0 24 0 15 15 16 1 27 1 16 0 59 0 40 0 24 0 15 14 17 1 27 1 16 0 59 0 39 0 23 0 14 13 18 1 26 1 15 0 58 0 39 0 23 0 14 12 19 1 26 1 15 0 57 0 38 0 22 0 14 11 20 1 26 1 14 0 57 0 37 0 22 0 14 10 21 1 26 1 14 0 56 0 37 0 21 0 14 9 22 1 25 1 13 0 55 0 36 0 21 0 14 8 23 1 25 1 13 0 55 0 36 0 21 0 14 7 24 1 25 1 12 0 54 0 35 0 20 0 14 6 25 1 25 1 12 0 53 0 34 0 20 0 14 5 26 1 24 1 11 0 53 0 34 0 20 0 14 4 27 1 24 1 11 0 52 0 on JJ 0 19 0 14 3 28 1 24 1 10 0 51 0 33 0 19 0 13 2 29 1 23 1 10 0 51 0 32 0 19 0 13 1 30 1 23 1 9 0 50 0 32 0 18 0 13 0 XI s X* IX* VIII* VII* VI* TABLE XL! 53 Moon's Equatorial Parallax. Argument* Anomaly. 0* I s il* III* 1\ /« V i 0° 58' 58" 58' 27" 57' 8" 55' 30" 54' 2" 53' 3" 30° 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 14 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* X* IX* vm« VXI* VI* TABLE XLII TABLE XLIII 45 Moon's Equatorial Parallax . Argument. Argument of the Va¬ riation. Reduction of the Parallax , and also of the Latitude . Argument. Latitude. 0* 1* li* III* IV* V* 0° 56" 42" 16" 4" 18" 44" 30° 1 56 41 15 4 18 45 29 2 55 41 14 4 19 46 28 3 55 40 14 4 20 46 27 4 55 39 13 4 21 47 26 5 55 38 12 4 22 48 25 6 55 37 12 4 23 48 24 r 55 36 11 5 24 49 23 8 55 35 10 5 24 50 22 9 54 35 10 5 25 50 21 10 54 34 9 6 26 51 20 11 54 33 9 6 27 51 19 12 53 32 8 6 28 52 18 13 53 31 8 7 29 53 17 14 52 30 7 7 30 53 16 15 52 29 7 8 31 53 15 16 51 28 6 8 32 54 14 17 51 27 6 9 33 54 13 18 50 26 6 9 34 55 12 19 50 25 5 10 35 55 11 20 49 24 5 10 35 55 10 21 49 24 5 11 36 56 9 22 48 23 4 12 37 56 8 23 47 22 4 12 38 56 7 24 47 21 4 13 39 56 6 25 46 20 4 14 40 57 5 26 45 19 4 14 41 57 4 27 45 18 4 15 42 57 3 28 44 18 4 16 42 57 2 29 43 17 4 17 43 57 1 30 42 16 4 18 44 57 0 XI* X* IX s VIII* VII* i vp Lat. Red. of Paral. Red. of Lat. 0° 0' O' 0" 3 0 1 12 6 0 2 23 9 0 3 32 12 0 4 39 15 1 5 43 18 1 6 44 21 1 7 40 24 2 8 31 27 ' 2 9 16 30 3 9 55 33 3 10 28 36 4 10 54 39 5 11 13 42 5 11 25 45 6 11 29 48 6 11 25 51 7 11 14 54 8 10 56 57 8 10 30 60 9 9 57 63 9 9 18 66 10 8 33 69 10 7 42 72 10 6 46 75 11 5 45 78 11 4 41 81 11 3 33 84 11 2 24 87 11 1 12 90 11 , 0 0 TABLE XLIV. Moon's Semidiameter. Argument. Equatorial Parallax. Eq. Par. Semidi.tEq. Par. Semidi. Eq. Par. Semidi. 53 0" 14' 27" 56' 0" 15 16" 59' ' 0" 16' 5" 53 10 14 29 56 10 15 18 59 10 16 7 53 20 14 32 56 20 15 21 59 20 16 10 53 30 14 35 56 30 15 24 59 30 16 13 53 40 14 37 56 40 15 26 59 40 16 16 53 50 14 40 56 50 15 29 59 50 16 18 54 0 14 43 57 0 15 32 60 0 16 21 54 10 14 46 57 10 15 35 60 10 16 24 54 20 14 48 57 20 15 37 60 20 16 26 54 30 14 51 57 30 15 40 60 30 16 29 54 40 14 54 57 40 15 43 60 40 16 32 54 50 14 57 57 50 15 46 60 50 16 35 55 0 14 59 58 0 15 48 61 0 16 37 55 10 15 2 58 10 15 51 61 10 16 40 55 20 15 5 58 20 15 54 61 20 16 43 55 30 15 7 58 30 15 56 61 30 16 46 55 40 15 10 58 40 15 59 61 40 16 48 55 50 15 13 58 50 16 2 61 50 16 51 56 0 15 16 59 0 16 5 62 0 16 54 TABLE XLV. Augmentation of Moon's Semidiameter. Argum. Appar. Alt. TABLE XLVI. Moon's Horary Motion in Longitude . Arguments. 2, 3, 4 and 5 of Long. Arg. 2 3 4 5 Arg. 0 6" 1" 3" 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 Ap. Alt Augm. 6° 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 56 TABLE XLVII, Moon's Horary Motion in Longitude . Argument. Argument of the Evection. 0* I* II* III* IV* 1 0° V 20" V 15" V 0" O'39" O'20" O'6" 30° 1 1 20 1 14 0 59 0 39 0 19 0 6 29 2 1 20 1 14 0 58 0 38 0 19 0 5 28 3 1 20 1 14 0 58 0 37 0 18 0 5 27 4 1 20 1 13 0 57 0 37 0 18 0 5 26 5 1 20 1 13 0 56 0 36 0 17 0 4 25 6 1 20 1 12 0 56 0 35 0 16 0 4 24 7 1 20 1 12 0 55 0 35 0 16 0 4 23 8 1 20 1 11 0 54 0 34 0 15 0 4 22 9 1 20 1 11 0 54 0 33 0 15 0 3 21 10 1 20 1 11 0 53 0 33 0 14 0 3 20 11 1 20 1 10 0 52 0 32 0 14 0 3 19 12 1 19 1 10 0 52 0 31 0 13 0 3 18 13 1 19 1 9 0 51 0 31 0 13 0 3 17 14 1 19 1 9 0 50 0 30 0 12 0 2 16 15 1 19 1 8 0 50 0 29 0 12 0 2 15 16 1 19 1 8 0 49 0 29 0 11 0 2 14 17 1 18 1 7 0 48 0 28 0 11 0 2 13 18 1 18 1 7 0 48 0 27 0 11 0 2 12 19 1 18 1 6 0 47 0 27 0 10 0 2 11 20 1 18 1 5 0 46 0 26 0 10 0 1 10 21 1 18 1 5 0 46 0 25 0 9 0 1 9 22 1 17 1 4 0 45 0 25 0 9 0 1 8 23 1 17 1 4 0 44 0 24 0 8 0 1 7 24 1 17 1 3 0 44 0 23 0 8 0 1 6 25 1 16 1 3 0 43 0 23 0 8 0 1 • 5 26 1 16 1 2 0 42 0 22 0 7 0 1 4 27 1 16 1 1 0 41 0 22 0 7 0 1 3 28 1 15 1 1 0 41 0 21 0 7 0 1 2 29 1 15 1 0 0 40 0 20 0 6 0 1 1 30 1 15 1 0 0 39 0 20 0 6 0 1 0 XL* 1 X* IX* Mil* VII* VI* TABLE XLVIII, 57 Moon's Horary Motion in Longitude. Arguments. Sum of preceding equations, and Anomaly, cor¬ rected. 0" 10'' 20 " 30" 40" 50" 60" 70" 80" 90" 100" 0* 0° 4" 5" 6 " 8" 9" 10" 11" 12" 14" 15" 16" XII s 0° 5 4 5 6 8 9 10 il 12 14 15 16 25 10 4 5 7 8 9 10 11 12 13 15 16 20 15 4 5 7 8 9 10 11 12 13 15 16 15 20 5 6 7 8 9 10 11 12 13 14 15 10 25 5 6 7 8 9 10 11 12 13 14 15 5 I 0 5 6 7 8 9 10 11 12 13 14 15 XI 0 5 5 6 7 8 9 10 11 12 13 14 15 25 10 6 7 7 8 9 10 11 12 13 13 14 20 15 6 7 8 8 9 10 11 12 12 13 14 15 20 7 7 8 9 9 10 11 11 12 13 13 10 25 7 8 8 9 9 10 11 11 12 12 13 5 II 0 7 8 8 9 9 10 11 11 12 12 13 X 0 5 8 8 9 9 lo 10 10 11 11 12 12 25 10 8 9 9 9 10 10 10 11 11 11 12 20 15 9 9 9 1G 10 10 10 10 11 11 11 15 20 9 10 10 10 10 10 10 10 10 10 11 10 25 10 10 10 10 10 10 10 10 10 10 10 5 III 0 10 lu 10 lu 10 ; 10 10 10 i 10 10 10 IX 0 5 11 11 11 10 10 10 10 10 9 9 9 25 10 11 11 11 11 10 10 10 9 9 9 9 20 15 12 11 11 11 10 10 10 9 9 9 8 15 20 12 12 11 11 10 10 10 9 9 8 8 10 25 13 12 12 11 11 10 9 9 8 8 7 5 IV 0 13 12 12 11 11 10 9 9 8 8 7 VIM 0 5 13 13 12 11 11 10 9 9 8 7 7 25 10 14 13 12 11 11 10 9 9 8 7 6 20 15 14 13 12 12 11 10 9 8 8 7 6 15 20 14 13 12 12 11 10 9 8 8 7 6 10 25 14 13 13 12 11 10 9 8 7 7 6 5 V 0 15 14 13 12 11 10 9 8 7 6 5 VII 0 5 15 14 13 12 11 10 9 8 7 6 5 25 10 15 14 13 12 11 10 9 8 7 6 5 20 15 15 14 13 12 11 10 9 8 7 6 5 15 20 15 14 13 12 11 10 9 8 7 6 5 10 25 15 14 13 12 11 10 9 8 7 6 5 5 vr o 15 14 13 12 11 10 9 8 7 6 5 VI 0 1 0" 10" 20" 30" 40" 50" 60" 70" 80' 90" 10 0" 8* 58 TABLE XL1X Moon's Horary Motion in Longitude. Argument. Anomaly, corrected. 0’ 1* II s m. IV s V» 0° 34' 51" 34/ 14" 32'39" 30' 45" 29' 6" 28' 1" 30° 1 34 51 34 12 32 36 30 42 29 3 27 59 29 2 34 51 34 9 32 32 30 38 29 0 27 58 28 3 34 51 34 7 32 28 30 34 28 58 27 56 27 4 34 51 34 4 32 24 30 31 28 55 27 55 26 5 34 50 34 1 32 21 30 27 28 52 27 54 25 6 34 50 33 59 32 17 30 23 28 50 27 53 24 7 34 49 33 56 32 13 30 20 28 47 27 51 23 8 34 49 33 53 32 9 30 16 28 45 27 50 22 9 34 48 33 50 32 5 30 13 28 42 27 49 21 10 34 47 33 47 32 2 30 9 28 40 27 48 20 11 34 46 33 44 31 58 30 6 28 37 27 47 19 12 34 45 33 41 31 54 30 2 28 35, 27 46 18 13 34 44 33 38 31 50 29 59 28 33 27 45 17 14 34 43 33 35 31 46 29 56 28 30 27 45 16 15 34 42 33 32 31 42 29 52 28 28 27 44 15 16 34 41 33 28 31 38 29 49 28 26 27 43 14 17 34 39 33 25 31 35 29 46 28 24 27 42 13 18 34 38 33 22 31 31 29 42 28 22 27 42 12 19 34 36 33 18 31 27 29 39 28 20 27 41 11 20 34 34 33 15 31 23 29 36 28 18 27 41 10 21 34 33 33 12 31 19 29 33 28 16 27 40 9 22 34 31 33 8 31 15 29 30 28 14 27 40 8 23 34 29 33 5 31 12 29 26 28 12 27 39 7 24 34 27 33 1 31 8 29 23 28 10 27 39 6 25 34 25 32 58 31 4 29 20 28 9 27 39 5 26 34 23 32 54 31 0 29 17 28 7 27 39 4 27 34 21 32 50 30 57 29 14 28 5 27 38 3 28 34 19 32 47 30 53 29 12 28 4 27 38 2 29 34 16 32 43 30 49 29 9 28 2 27 38 1 30 34 14 32 39 30 45 29 6 28 1 27 38 0 XI s X s IX s VIII s VII s VI s TABLE l 59 Moon’s Horary Motion in Longitude. Arguments. Sum of preceding equations, and Arg. of Variation. to 28' 29' 30' 31' 32' 33' 34' 35' 36' 37' 0* 0° 0" 1" 2" 4" 5" 6" 7" 8" 10" 11" 12" XII* 0° 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 10 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 0 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 0 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 11 10 9 8 7 6 5 4 3 2 1 10 25 10 9 8 8 7 € 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 V 0 3 3 4 5 5 6 7 7 8 9 9 VII 0 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 10 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 0 0 1 2 3 5 6 7 9 10 11 12 VI 0 . 27' 28' 29' 30' 31' 32' 33' 34' 35' 36' 37' TABLE LI Moon's Horary Motion in Longitude . Argument. Argument of the Variation. 0 s I* II s m* IV* 1 \ ft 0° 1' 17" 0' 58": O' 20" O' 2" O' 22" 1' 0" 30° 1 1 17 0 57 0 19 0 2 0 23 1 1 29 2 1 17 0 55 0 18 0 3 0 24 1 2 28 3 1 17 0 54 0 17 0 3 0 25 1 3 27 4 1 17 0 53 0 16 0 3 0 26 1 4 26 5 1 17 0 52 0 15 0 3 0 27 1 5 25 6 1 16 0 51 0 14 0 3 0 29 1 6 24 7 1 16 0 49 0 13 0 4 i 0 30 1 7 23 8 1 16 0 48 0 12 0 4 : 0 31 1 8 22 9 1 15 0 47 0 11 0 4 0 33 1 9 21 10 1 15 0 45 0 11 0 5 0 34 1 10 20 11 1 14 0 44 0 10 0 5 0 35 1 11 19 12 1 14 0 43 0 9 0 6 0 37 1 12 18 13 1 13 0 41 0 8 0 6 0 38 1 13 17 14 1 13 0 40 0 8 0 7 0 39 1 13 16 15 1 12 0 39 0 7 0 8 0 40 1 14 15 16 1 11 0 38 0 6 0 8 0 42 1 1j 14 17 1 11 0 36 0 6 0 9 0 43 1 15 13 18 1 10 0 35 0 5 0 10 0 44 1 16 12 19 1 9 0 34 0 5 0 11 0 46 1 16 11 20 1 8 0 32 0 4 0 11 0 47 1 17 10 21 1 7 0 31 0 4 0 12 0 48 1 17 9 22 1 6 0 30 0 4 0 13 0 50 1 18 8 23 1 5 0 29 0 3 0 14 0 51 1 18 7 24 1 4 0 27 0 3 0 15 0 52 1 18 6 25 1 3 0 26 0 3 0 16 0 54 1 19 5 26 1 2 0 25 0 3 0 17 0 55 1 19 4 27 1 1 0 24 0 3 0 18 0 56 1 19 3 28 1 0 0 23 0 2 0 19 0 57 1 19 2 29 0 39 0 21 0 2 0 20 0 59 1 19 1 30 0 58 0 20 0 2 0 22 1 0 1 19 0 ' XI s 1 X* IX* VIII* VII* VI* TABLE LH 61 Moon's Horary Motion in Longitude . Argument. Argument of the Reduction. 0 * 1 * 11 * III* IV* V* 0 ° 2 " 6 " 14" 18" 14" 6 " 30° 1 2 6 14 18 14 6 29 2 2 7 14 18 13 6 28 3 2 7 15 18 13 5 27 4 2 7 15 18 13 5 26 5 2 7 15 18 13 5 25 6 2 8 15 18 12 5 24 7 2 8 16 18 12 4 23 8 2 8 16 18 12 4 22 9 2 8 16 18 12 4 21 10 3 9 16 17 11 4 20 11 3 9 16 17 11 4 19 12 3 9 16 17 11 4' 18 13 3 9 17 17 11 3 17 14 3 10 17 17 10 3 16 15 3 10 17 17 10 3 15 16 3 10 17 17 10 3 14 17 3 11 17 17 9 3 13 18 4 11 17 16 9 3 12 19 4 11 17 16 9 3 11 20 4 11 17 16 9 3 10 21 4 12 18 16 8 2 9 22 4 12 18 16 8 2 8 23 4 12 i 18 16 8 2 7 24 5 12 18 15 8 2 6 25 5 13 18 15 7 2 5 26 5 13 18 15 7 2 4 27 5 13 18 15 7 2 3 28 6 13 18 14 7 2 2 29 6 14 18 14 6 2 1 30 6 14 18 14 6 2 0 XI* X* IX* Vlil* VII* VI* / 62 TABLE L11I, Moon's Horary Motion in Latitude . Argument. Arg. I, of Latitude. 0 * + I' + 1 U + III* — IV*— V* — 0 ° 2' 58" 2 34" V 29" O' 0 " V 29" 2' 34" 30° 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 43 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 41 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 0 XI»+ x* + IX*+ VIII*— VII*— VI*- TABLE LIV. Moon's Horary Motion in Latitude . Argument. Arg. II, of Latitude. o + 1* + 1I*+ 111* — IV*— V* — 0° 4" 4" 2" 0" 2" 4" 30° 6 4 3 2 0 3 4 24 12 4 3 1 1 3 4 18 18 4 3 1 1 3 4 12 24 4 3 0 2 3 4 6 30 4 2 0 2 4 4 0 XI*+ x*+ IX*-f- vrn* — VII* VI*— TABLE LV, 63 Nautical Almanac. I. AUGUST 1821. r^ Phases of the Moon. 0 ) c £ _ > O Full Moon, 13. 2. 8 (£ Last Quarter, 19.18.49 Q Q • New Moon, 27. 3.17 w. Th. 1 2 Lammas-Day. Other Phenomena. F. 3 D. H. M. Sa. 4 3.17.47 J) a r% 10 - - lj Stationary. Sun. 5 7/A Sunday after Trinity. 11 - - $ Stationary. M. . 6 Transfig. of our Lord. 19 - - 2/ Stationary. Tu. 7 Name of Jesus. Princess 23. 1. 1 0 enters W. 8 [Amelia born. 25.17. 0 D $ Th. 9 26.14.30 D a SI F. 10 St. Lawrence. 27 - - O eclipsed in vis. Sa. 11 Prs. of JSninsTvick bom. 21.11.58 D /3 8 31. 1.32 D a 11 % Sun. 12 8th Sunday aft. Trinity. M. 13 Prince of IVales b. Tu, 14 W. 15 Assumption. Th. 16 Duke of York bom. F. 17 Sa. 18 Sun. 19 9th Sunday aft. Tnnity. M. 20 Tu. 21 - Duke of Clarence bom. W. 22 Th. 23 F. 24 St. Bartholomew. Sa. 25 Sun. 26 10/A Sunday aft. Trinity. M. 27 Tu. 28 St. Augustine. * W. 29 St. John Bapt.beheaded. Th. 30 F. 31 64 TABLE LV. Nautical Almanac. AUGUST 1821. II. Days of the Week. Days of the Month. THE SUN’S Equation ofTime, Add. I DiflT. Longitude. Right Ascen. in Time. jDeclination North. S. D. M. S. H. M S. D. M.S. M. S. S. W. Th. F. Sa. Sun. 1 2 3 4 5 4. 8.49.37 4. 9.47. 5 4.10.44.33 4.11.42- 2 4.12.39.32 8.45. 2,9 8.48.55.9 8.52.48 2 8.56.39.9 9. 0.31,0 18. 4.20 17.49. 6 17.33.35 17.17.46 17. 1.41 5.58.4 5.548 5.50.5 5.45 7 5.40,3 3.6 43 4.8 5,4 6,1 6.7 7.3 7.9 8.4 9,1 9.7 10,2 10.7 11.3 11.8 12.3 12,8 13.3 13.7 14.2 14.6 15.1 15,5 15.8 16.3 16.7 17,0 17.4 17.8 18.2 M. Tu. W. Th. F. 6 7 8 9 10 4.13.37. 2 4.14.34.34 4.15.32. 6 4.16.29.39 4.17.27.12 9. 4.21,5 9. 8.11,3 9.12. 0,6 9.15.49.2 9.19.37.3 16.45.19 16.28.41 16.11.47 15.54.37 15.37.13 5.34.2 5.27,5 5.20.2 5.12.3 5. 3,9 Sa. Sun. M. Tu. W. 11 12 13 14 15 4.18.24.47 4.19.22.23 4.20.20. 0 4.21.17.39 4.22.15.18 9.23.24,7 9.27.11.6 9.30.58,0 9.34.43.7 9.38.29,0 15.19.33 15. 1.39 14.43.30 14.25. 7 14. 6.30 4.54.8 4.45.1 4.34.9 4.24.2 4.12.9 Th. F. Sa. Sun. M. 16 17 18 19 20 4.23.12.59 4.24.10.42 4.25. 8.26 4.26. 6.12 4.27. 4. 0 9.42.13,7 9.45.57.9 9.49.41,6 9.53.24.9 9.57. 7,6 13.47.40 13.28.37 13. 9.21 12.49.51 12.30.10 4. 1,1 3.48,8 3.36,0 3.22,7 3. 9,0 Tu. W. Th. F. Sa. 21 22 23 24 25 4.28. 1.49 4.28.59.41 4.29.57.34 5. 0.55.29 5. 1.53.26 10. 0.50,0 10. 4.31,9 10. 8.13,3 10.11.54,4 10.15.35,0 12.10.17 11.50.11 11.29.55 11. 9.27 10.48.49 2.54,8 2.40,2 2.25,1 2. 9,6 1.53 8 Sun. M. Tu. W. Th. 26 27 28 29 30 5. 2.51.25 5. 3.49.25 5. 4.47.26 5. 5.45.30 5. 6.43.34 10.19.15.2 10.22.55,1 10.26.34.5 10.30.13.6 10.33.52.3 10.28. 0 10. 7. 2 9.45.53 9.24.36 9. 3. 9 1.37,5 1.20,8 1. 3,8 0.46 4 0.28,6 1 F. 31 5. 7.41.41 10.37.30,7 8.41.34 0.10,4 TABLE LV, 65 Nautical Almanac. III. AUGUST 1821. Days. Time of Sun’s Semidiameter passingMerid. THE SUN’S Place of the Moon’s Node . Semidi¬ ameter. H ourly Motion. Logar. Distance. M. S. M. S. M. S. S. D. M 1 7 13 19 25 1. 6,4 1. 5,9 1, 5,5 1. 5,0 1. 4,6 15.47,5 15.48.4 15.49.4 15.50.5 15.51,7 2.23,6 2.23.9 2.24,2 2.24,5 2.24.9 0.00620 0,00581 0 .00536 0.00485 0.00429 11. 5.43 11. 5.24 11. 5. 5 11. 4.46 11 . 4.27 ECLIPSES OF THE SATELLITES OF JUPITER. MEAN TIME. I. Satellite. II. Satellite. III. Satellite. Irmnei'sions. 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 13 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 21.18.58 1 1 IV. Satellite. *9 66 TABLE LV •Nautical Almanac . AUGUST 1821. IV. THE PLANETS. Heliocentric. Geocentric. Declin. Rt. asc. Passage t/5 a P Long. Lat. Long. Lat. in time. Merid. S. D..M. D. M. S. D. M. D. M. D. M. H. M- H. M. $ MERCURY Gr. Elong. 19d. Inf 6 Id. llh. 1 10. 7.44 6.55S 4. 9.36 4.54S 13. 8N 8.43 23.51 4 10.18.14 7. 0 4. 7.24 4.38 13.57 8.34 23.32 7 10.29.34 6.49 4. 5.37 4. 7 14.54 8.28 23.15 10 11.11.53 6.19 4. 4.34 3.25 15.49 8.24 23. 1 13 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. 7.41 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 2 VENUS. 1 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 2.37 5.20. 3 1.15 5. 5 11.25 1.32 25 7. 4.21 2.13 5.27.22 1. 4 2 . 2 ' 11.52 1.37 $ MARS. 1 1.24.16 0.12N 2.23.19 0. 9N 23.27N 5.31 20.45 7 1.27.32 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 0.30 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 % JUPITER. 1 0.17.59 1.18S 0.29.39 1.22S 10. 5N 1.52 17. 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. 0. 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 k SATURN. 1 0.20.34 2. SOS 0.26.40 2.34S 7.54N 1.43 16.55 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 ¥ GEORGIAN. 1 9. 1.33 0.15S 8.29.39 0.15S 23.43S 17 58 9.12 11 9. 1.40 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 Nautical Almanac, V. AUGUST 1821. 0 ) PU table lv, Nautical Almanac . IX. AUGUST 1821. CO O l O O rH oo rj< at co co to b- to to to io rH rtf CM to rtf to tO CM rtf - t Cj 1—< C5t rH rtf ot CONOO'^Nr}' tO CM CM X rtf V3 CM rH rH to rtf to rH rtf tO rH tO CM tO tH oo >o co O Oi oo to d tO CO to to H O 09 b- tO *0 tO rtf CO CO b- to o cn at rH co O co rH CO CO rtf at lo rH rtf CM to to rj< H H Tj< H rtf CM rH • to to d rtf 00 to to O N. K rH d rH rH co co rH tO CO CM CO rtf rH CM CO rtf X . rtf d 00 rtf co co o bl d cm o rH Ot bl p to rtf to to h o at b. to to rj« to rtf co CO to to Ot O O © rH at rtf cm at to rH CM CO to co to co *o co co rtf CM to •H CM Ot bl at Ot to ^ to H O CM to d d X CM rH to rtf CO rH CM to CO to to cm d to 00 to CM Ot to co rH CM rH Ot d to rtf to to rH-O Ot b- to to ^ rH rH CO to CO CO at co cm co to CO O to CO H O 00 CO 00 CM ■c rtf CM cm rH CM rtf to rtf CO CO .fcp M. 00 CO rtf 0 s } d d rtf CO H to CO CO to CO ^ CM to rtf rtf to CM to at oo CM GM CO § 00 rtf o rH bl d d co o k d co rtf CM O d to rtf co b- to cm o at oo to to rtf tO to rtf . CM rH O rtf rH CM at rtf co co • CM CO rtf rtf rH to CO CM CZ3 co . co l 1—( . to rH JC M. to CO ^ . d . to rH ot rH bl d , 00 bl X to rtf 1 CM 1 CO CM H CO to CO *— 1 • Ot d GM i d ■ oo »o cm at to rtf O at 00 to to rtf • co CM Q tO rtf CO to to rtf co’ CSOH rtf CM GM b- at 00 rH rtf to CM CM • to • CO rH rtf to rtf to ■ rH to s CM CO tO . *° . to o to to O b- , oo »o !> rtf CO rtf H K CO i rH 1 rH to co I O b- CO rH 00 to rH Ot 00 b- »0 rtf CM CO 1 to CO tO rtf • p CO rtf CO to co’ O O rtf b. r^ rtf 00 00 tO O CM rH CM CM >0 1 rtf • to to CM CO ' ^ ^ -G ItH Ot 00 CO , rH . to at co cm rtf o * to CO CO rtf rtf , bl rtf s CM rH CM * to co at d i CM . rH 00 d CM O bl 1 CO to d to rtf co O rH at 00 b~ to rtf rH to rtf CO to b, K CO b~ CO 00 CM Ot to CM tO rH rtf CM to • to • CM CM CO CM 1 rH CO rH 5 t—I to CO Ot . bl d , b~ 00 rH- 00 00 CM . bl CO rtf ' CM CO rtf '-2 rH cr t ^ CO TH rH rH co rH . to H K . d d 1 CO O bl rtf rH Ot , 00 CO rtf P to *o CO to to rH O 00 b- tO rtf to rtf CO c n ON00 oo at o K 00 Ot O H CM CO OlOHH < H H H rH rH CM rH rH rH CM CM CM CM CM CO CO p CO 05 S- rt to »CJ>Ori 5> O) H CO 00 N r? K VO CO CO O b- CM CO rH VO CM CO tf co CM CM CM CO CM CM CO tf CO CO tf rH 55 • lOKCOOOwO) CM tf CO Oo rH oi o' co o' ‘n p in (o k a o h CM co tf VO CO vo co CO oo < rH H —« CO cm tf © co hh cm cm N. CO O O vo O O OO CO CO tf rH vo vo vo >*C! tf vo CO vo CO tf VO rH tf —* COOOrtKOH W CO CM rH VO o CM CO VO K CO o •§ ^ ^ ’"J . . . ’"i VO VO CM rH CM VO tf VO CM p cocoKcoaoH © CM VO nI © hr. O tf 00 CO T rtf VO CO bn <30 O rH CM CO tf vo N. CO Vo co is. Oo :* rH y—4 P CO* (OOiKCTCOOOiiH KOV)0 O © VO © tf 55 CM CO CO V) rtf • tf CM CO CM # CO tf CO <5 JZ q CMCOCM^OOOOOb-CO , CM Oo rH CO CO © © © vo •> X CO lo ^ tf co CO H ' CM CM VO CO tf rH rH CO <5 • COtfV0©i^OO©rH 1 rH CO VO CO VO © CO h- rH CO tf CO is. Oo D p -tf COlOKOOOiOO) CO tf VO CO cc rH rH 1 CO OlKWHOKOlCO CM tf 00 is. , VO tf tf VO CM CO © CM CO o HC5H rH CM tf CM vo CM CM 2j r* fcpH CO CO rH rtf is. VO CO CO CM hi is. CO CO 00 vo CO CO in fH CM CO CO CM rH rH CM tf • VO V) rH Vo CM rH CM tf • CMCOtfCO© CO © tf © tf CO rH tO is. rH VO VO VO CM ' CM CM tf CM CM tf CO CO Vo CO H CM CO tf tf CO (O CO , CO O CM VO CM VO © CO p tf CO(OKOOOOOH CM tf VO CO co tf vo i^ oo o rH rH O § CO rH 00 CO CO CO O CO O 1 CO VO 00 H K CO rH © © © © rH CM CM VO CM VO CO rH rH co rH VO VO CM rH 3s< 8 © p OOOKOCOOoCOtf 1 CO VO rH VO O © VO © © © © tf tf tf tf O £ tf tf tf tf CONCOCO VO VO rH tf tf CO s p OoOHCMCOtfVOCO . CO 00 rH CO CO O tf K rH © © -u P > COV}(ONOOO)OH ' CM CO Vo CO K CO tf vo is. CO © rH FH oi rH > H(MC0tf CO d 5 £ G G ill el a CO a H '£. CO c < TABLE LV, 73 Nautical Almanac . XI. AUGUST 1821. aj T? O h. cn to to ■y yen cn to »0 Tf co co rH hP CN *0 CN CN rH cn co CN y CN d CN CN -H O CN X! IS H}< to cn y to y co * Tp tO CN »0 to cn d co to r-f p to hr to h. to h- to to hr ^ *0 cc to cO o >o rH hr CO 00 to co y JO • ^ y CN CN co CO CN CO cn y zz *o d o y CO CN rH co co oo d •—1 rO CO CO to rH y rH .X • CN y ss to oo 00 CN to O rH tO h- to h. Tp tO h- y »o v» co oo co 00 y o CN CN 00 tO to to CN CN CO to CN i CN — M. hi d rH to y y o co d cn V) rH CN CN CO . ^ tv 2 ' ^ X 1-1 CN cn cn CO d h. O CO d - 1 VO h- to h- to h- y to hr y 02 T? O o cn cn to rf rH y rH rH ■i* co co CN y CO CN »0 CN rH rH to 1~H O'* co >o oo to to cn -y h. 00 1 hr -§ *?; y CN CO ^ to y to h y »o to § • Cn rH hoco rH Tf tO CO rH Tp • 00 2 to h- to h. 00 to h- y to h. oo y CO 00 rH cn cn oo rH o >o cn y o to CN GN y • GN y rH co £Z „ to cn to to cn 00 y 00 rH X rH to rH . CO rH CN 1 2 co cn d cn CN • CO CONOCO y to h. oo • to to to to 00 hr y CO to co CO o to CN rH tO CN »0 o 1 . rH CO . T_ l 10 ' Tj| co >o ■ !~ rH cn hi hi d . 00 to co o y . d > **c to CN rH CN y CO »o cn y y y to 00 y hi d 1 rH rH to 00 rH • to 2 to to to to oo y to to co y ai cn iH y to h. 00 CO to H to hr CN rH CN to to i to to co h y ’ CN j= s hi © CN cn hi th CO y , d to y CJ rH tO ■S3 to h. cn to cn « cn O CO hr O p to to to to h. to y to to oo a5 CO 00 to hr y rH cn to to to 00 co . o y y rH y y to • y ' to CN 2* S «M y rH CN CN 00 CN , rH 00 co co fN cn , rH CO CO CO CN ’ CN CN CO CN CN tO Q y *o hr H rj* ^ O K h- co cn Cn O rH rH cn co y OHH Q H H H rH rH rH rH CN CN CN CN CN CN CO co * Jj x ? CO ctt cn V jy "5 3 2 '« c« bo eg V c o r Jl 5 C* 0.1 .0.1 o.s ! O.S 10.2 1*0.3 5 0.3 ! 0.3 1 10 10 50 0.4 0.9 1 .: 1.8 2.2 o.c >0.1 .0.1 o.s ! O.S 10.3 : 0.' > 0.4 ; 0.4 1 20 10 40 0.5 1.0 1.5 2.0 2.5 o.c 10.1 0.1 o.s ! O.S ! 0.3 ; O.c 3 0.4 < 0.4 1 30 10 30 0.5 1.1 1.6 2.2 2.7 0.1 0.1 0.2 0.2 !0.: l 0.3 0.4 L 0.4 • 0.5 1 40 10 20 0.6 1.2 1.8 2.4 3.0 0.1 0.1 0.2 0.2 ; 0.3 ; 0.4 0.4 L 0.5 ; 0.5 1 50 10 10 0.6 1.3 1.9 2.6 3.2 0.1 0.1 0.2 0.3 0.3 ; 0.4 0.5 > 0.5 0.6 2 0 10 0 0.7 1.4 2.1 2.8 3.5 0.1 0.1 0.2 0.3 0.3 0.4 0.5 0.6 0.6 2 10 9 50 0.7 1.5 2.2 3.0 3.7 0.1 0.1 0.2 0.3 0.4 0.4 0.5 0.6 0.7 2 20 9 40 0.8 1.6 2.3 3.1 3.9 0.1 0.2 0.2 0.3 0.4 0.5 0.5 0.6 0.7 2 30 9 30 0.8 1.6 2.5 3.3 4.1 0.1 0.2 0.2 0.3 0.4 0.5 0.6 0.7 0.7 2 40 9 20 0.9 1.7 2.6 3.5 4.3 0.1 0.2 0.3 0.3 0.4 0.5 0.6 0.7 0.8 2 50 9 10 0.9 1.8 2.7 3.6 4.5 0.1 0.2 0.3 0.4 0.5 0.5 0.6 0.7 0.8 3 0 9 0 0.9 1.9 2.8 3.8 4.7 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.7 0.8 3 10 8 50 1.0 1.9 2.9 3.9 4.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 3 20 8 40 1.0 2.0 3.0 4.0 5.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 3 30 8 30 1.0 2.1 3.1 4.1 5.2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 3 40 8 20 1.1 2.1 3.2 4.2 5.3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1.0 3 50 8 10 1.1 2.2 3.3 4.3 5.4 0.1 0.2 0.3 0.4 0.5 0.7 0.8 0.9 1.0 4 0 8 0 1.1 2.2 3.3 4.4 5.6 0.1 0.2 0.3 0.4 0.6 0.7 0.8 0.9 1.0 4 10 7 50 1.1 2.3i 3.4 4.5. 5.7 0.1 0.2 0.3 0.5 0.6 0.7 0.8 0.9 1.0 4 20 7 40 : 1.2 2.3; 3.5 4.6; 5.8 0.1 0.2 0.3 0.5 0.6 0.7 0.8 0.9 1.0 4 30 7 30 : 1.2 2 3: 3.5 4.7: 5.9 0.1 3.2 0.4 < 3.5 0.6 0.7 0.8 0.9 1.1 4 40 7 20 : 1.2 2.4L 3.6 4.8: 5.9 0.1 0.21 0.4 ( 3.5 3.6i 0.7 0.8 1.0 1.1 4 50 7 io : 1.2: 2.4L 3.6 4.8 1 5.0 0.1 0.21 0.4 ( 3.5< 3.6 < 3.7 0.8 1.0 1.1 5 0 7 0 ] L.2 ‘ 2.4 C 3.6 ■ 4.9 ( 5.1 0.1 3.21 3.4 ( 3.5 ( 3.6 ( 3.7 0.9 1.(3 1.1 5 10 6 50 1 l.2i 2.5 C 3.7 ■ 4.9 ( 5.1 0.1 ( 3.2 ( 3.4 ( 3.5 ( 3.6 ( 3.7 0.9 1.0 1.1 5 20 6 40 ] 1.2 ‘ 2.5 c 3.7 • 4.9 ( 5.1 0.1 ( 3.2 ( 3.4 ( 3.5 ( 3.6 ( 3.7 0.9 1.0 1.1 5 30 6 30 1 L.2S 2.5 3 3.7 . 5.0 ( 5.2 0.1 ( ).2( 3.4 C 3.5 C 3.6 C 3.7 0.9 1.0 1.1 5 40 6 20 1 L.2 S 2.5: 3.7. 5.06 5.2 0.1 ( 3.2( 3.4 C 3.5 C 3.6 C 3.7 0.9 1.0 1.1 5 50 6 10 }1 .25 2.53 3.7 : 5.0 6 5.2 0.1 ( ).2( 3.4 C 3.5 C 3.6 C 3.7 0.9 1.0 1.1 6 0 6 0 (1 ..3]5 2.63 3.8 ; 5.06 5.3 0.1 ( ).2( 3.4 C 3.5 C 3.6 C 3.7 0.9 1.0 1.1 70 TABLE LVIT, Logistical Logarithms. 0 / 1 ' 2 ' 3 ' 4 ' 5 6 ' 7 ' 0 60 120 180 240 300 360 420 0 " 00000 17782 14771 13010 11761 10792 10000 9331 1 35563 17710 14735 12986 11743 10777 9988 9320 2 32553 1 7639 14699 12962 11725 1 0763 9976 9310 3 30792 17570 14664 12939 11707 10749 9964 9300 4 29542 17501 14629 12915 11689 10734 9952 9289 5 28573 17434 14594 12891 11671 10720 9940 9279 6 27782 17368 14559 12868 11654 10706 9928 9269 7 27112 17302 14525 12845 11636 10692 9916 9259 8 26532 17238 14491 12821 11619 10678 9905 9249 9 26021 17175 14457 12798 11601 10663 9893 9238 10 25563 17112 14424 12775 11584 10649 9881 9228 11 25149 17050 14390 12753 11566 10635 9869 9218 12 24771 16990 14357 12730 11549 10621 9858 9208 13 24424 16930 14325 12707 11532 10608 9846 9198 14 24102 16871 14292 12685 11515 10594 9834 9188 15 23802 16812 14260 12663 11498 10580 9823 9178 16 23522 16755 14228 12640 11481 10566 9811 9168 17 23259 16698 14196 12618 11464 10552 9800 9158 18 23010 16642 14165 12596 11447 10539 9788 9148 19 22775 16587 14133 12574 11430 10525 9777 9138 20 22553 16532 14102 12553 11413 10512 9765 9128 21 22341 16478 14071 12531 11397 10498 9754 9119 22 22139 16425 14040 12510 11380 10484 9742 9109 23 21946 16372 14010 12488 11363 10471 9731 9099 24 21761 16320 13979 12467 11347 10458 9720 9089 25 21584 16269 13949 12445 11331 10444 9708 9079 26 21413 16218 13919 12424 11314 10431 9697 9070 27 21249 16168 13890 12403 11298 10418 9686 9060 28 21091 16118 13860 12382 11282 10404 9675 9050 29 20939 16069 13831 12362 11266 10391 9664 9041 30 20792 16021 13802 12341 11249 10378 9652 9031 TABLE LVII // Logistical Logarithms. 0 ' 1 ' 2 ' 3 ' 1 4 ' 5 ' 6 ' 7 ' 0 60 120 180 240 300 360 420 30 " 20792 16021 13802 12341 i 11249 10378 9652 9031 31 20649 15973 13773 12320 11233 10365 9641 9021 32 20512 15925 13745 12300 11217 10352 9630 9012 33 20378 15878 13716 12279 11201 10339 9619 9002 34 20248 15832 13688 12259 11186 10326 9608 8992 35 20122 15786 13660 12239 11170 10313 9597 8983 36 20000 15740 13632 12218 11154 10300 9586 8973 37 19881 15695 13604 12198 ! 11138 10287 9575 8964 38 19765 15651 13576 12178 11123 10274 9564 8954 39 19652 15607 13549 12159 | 11107 10261 9553 8945 40 19542 15563 13522 12139 11091 10248 9542 8935 41 19435 15520 13495 12119 11076 10235 9532 8926 42 19331 15477 13468 12099 11061 10223 9521 8917 43 19228 15435 13441 12080 11045 10210 9510 8907 44 19128 15393 13415 12061 11030 10197 9499 8898 45 19031 15351 13388 12041 11015 10185 9488 8888 46 18935 15310 13362 12022 ' 10999 10172 9478 8879 47 18842 15269 13336 12003 10984 10160 . 9467 8870 48 18751 15229 13310 11984 1 10969 10147 9456 8861 49 18661 15189 13284 11965 10954 10135 9446 8851 50 18573 15149 13259 11946 10939 10122 9435 8842 51 18487 15110 13233 11927 ! 10924 10110 9425 8833 52 18403 15071 13208 11908 10909 10098 9414 8824 53 18320 15032 13183 11889 10894 10085 9404 8814 54 18239 14994 13158 11871 1 10880 10073 9393 8805 55 18159 14956 13133 11852 10865 10061 9383 8796 56 18081 14918 13108 11834 10850 10049 9372 8787 57 18004 14881 13083 11816 10835 10036 9362 8778 58 17929 14844 13059 11797 10821 10024 9351 8769 59 17855 14808 13034 11779 10806 10012 9341 8760 60 17782 14771 13010 11761 10792 10000 9331 8751 78 TABLE LV1I Logistical Logarithms. 8 ' 9 ' 10 ' 11 ' 12 ' 13 ' 14 ' 15 ' 1 I6 ' 1 480 540 600 660 720 780 840 900 960 0 " 8751 8239 7782 7368 6990 6642 6320 6021 5740 1 8742 8231 7774 7361 6984 6637 6315 6016 5736 2 8733 8223 7767 7354 6978 6631 6310 6011 5731 3 8724 8215 7760 7348 6972 6625 6305 6006 5727 4 8715 8207 7753 7341 6966 6620 6300 6001 5722 5 8706 8199 7745 7335 6960 6614 6294 5997 5718 6 8697 8191 7738 7328 6954 6609 6289 5992 5713 7 8688 8183 7731 7322 6948 6603 6284 5987 5709 8 8679 8175 7724 7315 6942 6598 6279 5982 5704 9 8670 8167 7717 7303 6936 6592 6274 5977 5700 10 8661 8159 7710 7302 6930 6587 6269 5973 5695 11 8652 8152 7703 7295 6924 6581 6264 5968 5691 12 8643 8144 7696 7289 6918 6576 6259 5963 5686 13 8635 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 8599 8104 7660 7257 6888 6548 6233 5939 5664 18 8591 8097 7653 7251 6882 6543 6228 5935 5660 19 8582 8089 76 46 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 5911 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 7193 6830 6494 6183 5892 5620 28 8504 8020 7583 7187 6824 6489 6178 5888 5615 29 8496 8012 7577 7181 6818 6484 6173 5883 5611 30 8487 8004 7570 7175 6812 6478 6168 5878 5607 1 TABLE LVII 79 Logistical Logarithms. 8' 9' 10' 11' 12' 13' 14' 15' 16' 480 540 600 660 720 780 840 900 960 30" 8487 8004 7570 7175 6812 6478 6168 5878 5607 31 8479 7997 7563 7168 6807 6473 6163 5874 5602 32 8470 7989 7556 7162 6801 6467 6158 5869 5598 33 8462 7981 7549 7156 6795 6462 6153 5864 5594 34 8453 7974 7542 7149 ! 6789 6457 6148 5860 5589 35 8445 7966 7535 7143 | 6784 6451 6143 5855 5585 36 8437 7959 7528 7137 i 6778 6446 6138 5850 5580 37 8428 7951 7522 7131 6772 6441 6133 5846 5576 38 8420 7944 7515 7124 | 6766 6435 6128 5841 5572 39 8411 7936 7508 7118 I 6761 6430 6123 5836 5567 40 8403 7929 7501 7112 ! i 6755 6425 6118 5832 5563 41 8395 7921 7494 7106 j 6749 6420 6113 5827 5559 42 8386 7914 7488 7100 I 6743 6414 6108 5823 5554 43 8378 7906 7481 7093 6738 6409 6103 5818 5550 , 44 8370 7899 7474 7087 6732 6404 6099 5813 5546 45 8361 7891 7467 7081 6726 6598 6094 5809 5541 46 8353 7884 7461 7075 ; 6721 6393 6089 5804 5537 4 7 8345 7877 7454 7069 6715 6388 6084 5800 5533 48 8337 7869 7447 7063 | 6709 6383 6079 5795 5528 49 8328 7862 7441 7057 6704 6377 6074 5790 5524 50 8320 7855 7434 7050 6698 6372 6069 5786 5520 51 8312 7847 7427 7044 6692 6367 6064 5781 5516 52 8304 7840 7421 7038 6687 6362 6059 5777 5511 53 : 8296 7832 7414 7032 6681 6357 6055 5772 5507 54 i 8288 7825 7407 7026 6676 6351 6050 5768 5503 55 8279 7818 7401 7020 6670 6346 6045 5763 5498 56 8271 7811 7394 7014 6664 6341 6040 5758 5494 57 8263 7803 7387 7008 6659 6336 6035 5754 5490 58 8255 7796 7381 7002 6653 6331 6030 5749 5486 59 8247 7789 7374 6996 6648 6325 6025 5745 5481 60 8239 7782 7368 6990 6642 6320 6021. 5740 5477 80 TABLE LVII Logistical Logarithms. 17 ' 18' 19' 20' 21' 22' 23' 24' | 25' 1020 1080 1140 1200 1260 1320 1380 1440 1 1500 0 " 5477 5229 4994 4771 4559 4357 4164 3979 1 3802 1 5473 5225 4990 4768 4556 4354 4161 3976 3799 . 2 5469 5221 4986 4764 4552 4351 4158 3973 3796 3 5464 5217 4983 4760 4549 4347 4155 3970 3793 4 5460 5213 4979 4757 4546 4344 4152 3967 3791 5 5456 5209 4975 4753 4542 4341 4149 3964 3788 6 5452 5205 4971 4750 4539 4338 : 4145 3961 3785 7 5447 5201 4967 4746 4535 4334 • 4142 3958 3782 8 5443 5197 4964 4742 4532 4331 ! 4139 3955 3779 9 5439 5193 4960 4739 4528 4328 4136 3952 3776 10 5435 5189 4956 4735 4525 4325 : 4133 3949 3773 11 5430 5185 4952 4732 4522 4321 4130 3946 3770 12 5426 5181 4949 4728 4518 4318 4127 3943 3768 13 5422 5177 4945 4724 4515 4315 4124 3940 3765 14 5418 5173 4941 4721 4511 4311 4120 3937 3762 : 15 5414 5169 4937 4717 4508 4308 4117 3934 3759 16 5409 5165 4933 4714 4505 4305 4114 , 3931 3756 17 5405 5161 4930 4710 4501 4302 4111 3928 3753 18 5401 5157 4926 4707 4498 4298 4108 3925 3750 19 5397 5153 4922 4703 4494 4295 ’ 4105 3922 3747 20 5393 5149 4918 4699 4491 4292 4102 3919 3745 21 5389 5145 4915 4696 4488 4289 4099 , .3917 3742 22 5384 5141 4911 4692 4484 4285 4096 ; 3914 3739 23 5380 5137 4907 4689 4481 4282 4092 | 3911 3736 24 5376 5133 4903 4685 : 4477 4279 1 4089 | 3908 3733 25 5372 5129 4900 4682 4474 4276 4086 ( 3905 3730 26 5368 5125 4896 4678 4471 4273 ! 4083 1 3902 3727 2 7 5364 5122 4892 4675 4467 4269 4080 ! 3899 3725 28 5359 5118 i 4889 4671 4464 4266 1 4077 i 3896 5722 , 29 5355 5114 4885 4668 4460 4263 4074 j 3893 3719 : 30 5351 5110 4881 4664 4457 4260 | 4071 i 3890 * 3716 TABLE LVII. 81 Logistical Logarithms . 17' 18' 19' 20' 21' 22' j 23' 24' 25' 1020 1080 1140 1200 1260 1320 j 1380 1440 1500 30" 5351 5110 4881 4664 4457 4260 4071 3890 3716 31 5347 5106 4877 4660 4454 4256 4068 3887 3713 32 5343 5102 4874 4657 4450 4253 4065 3884 3710 33 5339 5098 4870 4653 4447 4250 4062 3881 3708 34 5335 5094 4866 4650 4444 4247 4059 3878 3705 35 5331 5090 4863 4646 4440 4244 4055 3875 3702 36 5326 5086 4859 4643 4437 4240 4052 -3872 3699 37 5322 5082 4855 4639 4434 4237 4049 3869 3696 38 5318 5079 4852 4636 4430 4234 4046 3866 3693 39 5314 5075 4848 4632 4427 4231 4043 3863 3691 40 5310 5071 4844 4629 4424 4228 4040 3860 3688 41 5306 5067 4841 4625 4420 4224 4037 3857 3685 42 5302 5063 4837 4622 4417 4221 4034 3855 3682 43 5298 5059 4833 4618 4414 4218 4031 3852 3679 44 5294 5055 4830 4615 4410 4215 4028 3849 3677 45 5290 5051 4826 4611 4407 4212 4025 3846 3674 46 5285 5048 4822 4608 4404 4209 4022 3843 3671 47 5281 5044 4819 4604 4400 4205 4019 3840 3668 48 5277 5040 4815 4601 4397 4202 4016 3837 3665 49 5273 5036 4811 4597 4394 4199 4013 3834 3663 50 5269 5032 4808 4594 4390 4196 4010 3831 3660 51 5265 5028 4804 4590 4387 4193 4007 3828 3657 52 5261 5025 4800 4587 4384 4189 4004 3825 3654 53 5257 5021 4797 4584 4380 4186 4001 3822 3651 54 5253 5017 4793 4580 4377 4183 3998 3820 3649 55 5249 5013 4789 4577 4374 4180 3995 3817 3646 56 5245 5009 4786 4573 4370 4177 3991 3814 3643 57 5241 5005 4782 4570 4367 4174 3988 3811 3640 58 5237 5002 4778 4566 4364 4171 3985 3808 3637 59 5233 4998 4775 4563 4361 4167 3982 3805 3635 60 5229 | 4994 4771 4559 4357 4164 3979 3802 3632 *11 82 FABLE LVII Logistical Logarithms . 26' 27 ' 28' 29 ' 30' 31' 32' 33' 34' j 1560 1620 1680 1740 1800 1860 1920 1980 2040 0 " 3632 3468 3310 3158 3010 2868 2730 2596 2467 1 1 3629 3465 3307 3155 3008 2866 2728 2594 2465 2 3626 3463 3305 3153 3005 2863 2725 2592 2462 3 3623 3460 3302 3150 3003 2861 2723 2590 2460 4 3621 3457 3300 3148 3001 2859 2721 2588 2458 5 3618 3454 3297 3145 2998 2856 2719 2585 2456 6 3615 3452 3294 3143 2996 2854 2716 2583 2454 7 3612 3449 3292 3140 2993 2852 2714 2581 2452 8 3610 3446 3289 3138 2991 2849 2712 2579 2450 9 3607 3444 3287 3135 2989 2847 2710 2577 2448 10 3604 3441 3284 3133 2986 2845 2707 2574 2445 11 3601 3438 3282 3130 2984 2842 2705 2572 2443 12 3598 3436 3279 3128 2981 2840 2703 2570 2441 13 3596 3433 3276 3125 2979 2838 2701 2568 2439 14 3593 3431 3274 3123 2977 2835 2698 2566 2437 15 3590 3428 3271 3120 2974 2833 2696 2564 2435 16 3587 3425 3269 3118 2972 2831 2694 2561 2433 17 3585 3423 3266 3115 2969 2828 2692 2559 2431 18 3582 3420 3264 3113 2967 2826 2689 2557 2429 19 3579 3417 3261 3110 2965 2824 2687 2555 2426 20 3576 3415 3259 3108 2962 2821 2685 2553 2424 21 3574 3412 3256 3105 2960 2819 2683 2551 2422 22 3571 3409 3253 3103 2958 2817 2681 2548 2420 23 3568 3407 3251 3101 2955 2815 2678 2546 2418 24 3565 3404 3248 3098 2953 2812 2676 2544 2416 25 3563 3401 3246 3096 2950 2810 2674 2542 2414 26 3560 3399 3243 3093 2948 2808 2672 2540 2412 27 3557 3396 3241 3091 2946 2805 2669 2538 2410 1 28 3555 3393 3238 3088 2943 2803 2667 2535 2408 1 29 3552 3391 3236 3086 2941 2801 2665 2533 2405 j 80 3549 3388 3233 3083 2939 2798 2663 2531 2403 TABLE LVII 83 Logistical Logarithms . 26' 27 ' 28' 29' 30' 31' 32' 33' 34' 1560 1620 1680 1740 1800 1860 1920 1980 2040 30" 3549 3388 3233 3083 2939 2798 2663 2531 2403 31 3546 3386 3231 3081 2936 2796 2660 2529 2401 32 3544 3383 3228 3078 2934 2794 2658 2527 2399 33 3541 3380 3225 3076 2931 2792 2656 2525 2397 34 3538 3378 3223 3073 2929 2789 2654 2522 2395 ( 35 3535 3375 3220 3071 2927 2787 2652. 2520 2393 I 36 3533 3372 3218 3069 2924 2785 2649 2518 2391 37 3530 3370 3215 3066 2922 2782 2647 2516 2389 38 3527 3367 3213 3064 2920 2780 2645 2514 2387 39 3525 3365 3210 3061 2917 2778 2643 2512 2384 40 3522 3362 3208 3059 2915 2775 2640 2510 2382 41 3519 3359 3205 3056 2912 2773 2638 2507 2380 42 3516 3357 3203 3054 2910 2771 2636 2505 2378 43 3514 3354 3200 3052 2908 2769 2634 2503 2376 44 3511 3351 3198 3049 2905 2766 2632 2501 2374 45 3508 3349 3195 3047 2903 2764 2629 2499 2372 46 3506 3346 3193 3044 2901 2762 2627 2497 2370 47 3503 3344 3190 3042 2898 2760 2625 2494 2368 48 3500 3341 3188 3039 2896 2757 2623 2492 2366 49 3497 3338 3185 3037 2894 2755 2621 2490 2364 50 3495 3336 3183 3034 2891 2753 2618 2488 2362 51 3492 3333 3180 3032 2889 2750 2616 2486 2359 52 3489 3331 3178 3030 2887 2748 2614 2484 2357 53 3487 3328 3175 3027 2884 2746 2612 2482 2355 54 3484 3325 3173 3025 2882 2744 2610 2480 2353 55 3481 3323 3170 3022 2880 2741 2607 2477 2351 56 3479 3320 3168 3020 2877 2739 2605 2475 2349 57 3476 3318 3165 3018 2875 2737 2603 2473 2347 58 3473 3315 3163 3015 2873 2735 2601 2471 2345 59 3471 3313 3160 3013 2870 2732 2599 2469 2343 60 3468 3310 3158 3010 2868 2730 2596 2467 2341 ! 84 TABLE LVII Logistical Logarithms. 35' 36' 37 ' 38' 39' 40' 41' 42' 43' 2100 2160 2220 2280 2340 2400 2460 2520 2580 0" 2341 2218 2099 1984 1871 1761 1654 1549 1447 1 2339 2216 2098 1982 1869 1759 1652 1547 1445 2 2337 2214 2096 1980 1867 1757 1650 1546 1443 3 2335 2212 2094 1978 1865 1755 1648 1544 1442 4 2333 2210 2092 1976 1863 1754 1647 3542 1440 1 5 2331 2208 2090 1974 1862 1752 1645 1540 1438 | 6 2328 2206 2088 1972 1860 1750 1643 1539 1437 ; 7 2326 2204 2086 1970 1858 1748 1641 1537 1435 ; 8 2324 2202 2084 1968 3856 1746 1640 1535 1433 9 2322 2200 2082 1967 1854 1745 1638 1534 1432 10 2320 2198 2080 1965 1852 1743 1636 1532 1430 11 2318 2196 2078 1963 1850 1741 1634 1530 1428 12 2316 2194 2076 1961 1849 1739 1633 1528 1427 ' 13 2314 2192 2074 1959 1847 1737 1631 1527 1425 j 14 2312 2190 2072 1957 , 1845 1736 1629 1525 1423 ! 15 2310 2188 2070 1955 1843 1734 1627 1523 1422 i . I 16 2308 2186 2068 1953 1841 1732 1626 1522 1420 ' 17 2306 2184 2066 1951 1839 1730 1624 1520 1418 18 2304 2182 2064 1950 1838 1728 1622 1518 1417 19 2302 2180 2062 1948 1836 1727 1620 1516 1415 20 2300 2178 2061 1946 1834 1725 1619 1515 1413 i 21 2298 2176 2059 1944 1832 1723 1617 1513 1412 j 22 2296 2174 2057 1942 1830 1721 1615 1511 1410 1 23 2294 2172 2055 1940 1828 1719 1613 1510 1408 i 24 2291 2170 2053 1938 1827 1718 1612 1508 1407 ; 25 2289 2169 2051 1936 1825 1716 1610 1506 1405 1 26 2287 2167 2049 1934 1823 1714 1608 1504 1403 | 27 2285 2165 2047 1933 1821 1712 1606 1503 1402 i 28 2283 2163 2045 1931 1819 1711 1605 1501 1400 29 2281 2161 2043 1929 1817 1709 1603 1499 1398 30 2279 2159 2041 1927 1816 1707 1601 1498 1397 J TABLE LYII 85 Logistical Logarithms. 35' 36' 37' 38' 39' 40' 41 42' 43' 2100 2160 2220 2280 2340 2400 2460 2520 2580 30" 2279 2159 2041 1927 1816 1707 3601 1498 1397 31 2277 2157 2039 1925 1814 1705 1599 1496 1395 32 2275 2155 2037 1923 1812 1703 1598 1494 1393 33 2273 2153 2035 1921 1810 1702 1596 1493 1392 : 34 2271 2151 2033 1919 1808 1700 1594 1491 1390 1 35 2269 2149 2032 1918 1806 1698 1592 1489 1388 36 2267 2147 2030 1916 1805 ! 1696 1591 1487 1387 37 2265 2145 2028 1914 1803 1 i 1694 1589 1486 1385 38 2263 2143 2026 1912 1801 ! ! 1693 1587 1484 1383 , 39 2261 2141 2024 1910 1799 1691 1585 1482 1382 ’ 40 2259 2139 2022 1908 1797 1689 1584. 1481 1380 41 2257 2137 2020 1906 1795 1687 1582 1479 1378 42 2255 2135 2018 1904 1794 1686 1580 1477 1377 43 2253 2133 2016 1903 1792 1684 1578 1476 1375 44 2251 2131 2014 1901 1790 1682 1577 1474 1373 45 2249 2129 2012 1899 1788 1680 1575 1472 1372 46 2247 2127 2010 1897 1786 1678 1573 1470 1370 1 47 2245 2125 2009 1895 1785 1677 1571 1469 1368 48 2243 2123 2007 1893 1783 1675 1570 1467 1367 | 49 2241 2121 2005 1891 1781 1673 1568 1465 1365 50 2239 2119 2003 1889 1779 1671 1566 1464 1363 51 2237 2117 2001 1888 1777 1670 1565 1462 1362 52 2235 2115 1999 1886 1775 1668 1563 1460 1360 53 2233 2113 1997 1884 1774 1666 1561 1459 1359 54 2231 2111 1995 1882 1772 1664 1559 1457 1357 55 2229 2109 1993 1880 1770 1663 1558 1455 1355 '56 2227 2107 1991 1878 1768 1661 1556 1454 1354 57 2225 2105 1989 1876, 1766 1659 1554 1452 1352 58 2223 2103 1987 1875 1765 1657 1552 1450 1350 59 2220 2101 1986 1873 1763 1655 1551 1449 1349 60 2218 2099 1984 1871 1761 1654 1549 1447 1347 86 TABLE LVH Logistical Logarithms. 44' 45' 46' 47' 48' 49' 5 O' 51' 52~ 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 i 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 941 852 764 679 595 20 1314 1217 1123 1030 939 850 763 678 594 1 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 924 835 749 1 663 580 TABLE LVIL 87 Logistical Logarithms. 44' ' 45' 46' 47' 48' 49' 50' 51' 52' 2640 2700 2760 2820 2880 2940 3000 3060 3120 30" 1298 1201 1107 1015 924 835 749 663 580 31 1296 1200 1105 1013 923 834 747 662 579 32 1295 1198 1104 1012 921 833 746 661 577 33 1293 1197 1102 1010 920 831 744 659 576 34 1291 1195 1101 1008 918 830 743 658 574 35 1290 1193 1099 1007 917 828 741 656 573 36 1288 1192 1098 1005 915 827 740 655 572 37 1287 1190 1096 1004 914 825 739 654 570 38 1285 1189 1095 1002 912 824 737 652 569 39 1283 1187 1093 1001 911 822 736 651 568 40 1282 1186 1091 999 909 821 734 649 566 41 1280 1184 1090 998 908 819 733 648 565 42 1278 1182 1088 996 906 818 731 647 563 43 1277 1181 1087 995 905 816 730 645 562 44 1275 1179 1085 993 903 815 729 644 561 45 1274 1178 1084 992 902 814 727 642 559 46 1272 1176 1082 990 900 812 726 641 558 47 1270 1174 1081 989 899 811 724 640 557 48 1269 : 1173 1079 987 897 809 723 638 555 49 1267 1171 1078 986 896 808 721 637 554 50 1266 1170 1076 984 894 806 720 635 552 51 1264 1168 1074 983 893 805 719 634 551 52 1262 1167 1073 981 891 803 717 633 550 53 1261 1165 1071 980 890 802 716 631 548 54 1259 1163 1070 978 888 801 714 630 547 55 1257 1162 1068 977 887 799 713 628 546 56 1256 1160 1067 975 885 798 711 627 544 57 1254 1159 1065 974 884 796 710 626 543 58 1253 1157 1064 972 883 795 709 624 541 59 1251 1156 1062 971 881 793 707 623 540 60 1249 | 1154 1061 969 880 792 706 621 539 88 TABLE LVII Logistical Logarithms. 53' 54 ' 55 ' 1 56< 57 ' | 58' 59' 3180 3240 3300 3360 3420 j 5480 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 7 629 448 369 291 214 139 64 8 528 447 367 289 213 137 63 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 13 521 440 361 283 206 131 57 14 520 439 359 282 205 130 56 15 518 438 358 280 204 129 55 16 517 436 357 279 202 127 53 17 516 435 356 278 201 126 52 18 514 434 354 276 200 125 51 19 513 432 353 275 199 124 50 20 512 431 352 274 197 122 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 38 30 498 418 339 261 185 110 I 36 TABLE LV1I 89 Logistical Logarithms. 53' 54' 55' 56' 57 58' 59' 3180 3240 3300 3360 3420 3480 3540 30" 498 418 339 261 185 110 36 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 47 475 395 316 239 163 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 52 468 388 310 233 157 83 10 53 467 387 309 232 156 82 8 54 466 386 307 230 155 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 4 58 460 381 302 225 150 75 2 59 459 379 301 224 148 74 1 60 458 378 300 223 147 73 0 *12 90 TABLE LYIII Change of Moon’s liight Ascension from the Sun . Time. 38m 39m 40m 41m 42m 43 m 44m 45 m 46m 47 m 48m 49 m 50 m 51m 52in h. m. m. m. m. m. m. m. m. m. m. m. m. m. m. m. m. & 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 20 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 40 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 20 2 2 2 2 2 2 2 2 3 3 3 3 o 3 3 1 40 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 2 0 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 2 20 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 2 40 4 4 4 5 5 5 5 5 5 5 5 5 6 6 6 3 0 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 3 20 5 5 6 6 6 6 6 6 6 7 7 7 7 7 7 3 40 6 6 6 6 6 7 7 7 7 7 7 7 8 8 8 4 0 6 6 7 7 7 7 7 7 8 8 8 8 8 8 9 4 20 7 7 7 7 8 8 8 8 8 8 9 9 9 9 9 4 40 7 8 8 8 8 8 9 9 9 9 9 10 10 10 10 5 0 8 8 8 9 9 9 9 9 10 10 10 10 10 11 11 5 20 9 9 9 9 9 10 10 10 10 10 11 11 11 11 12 5 40 9 9 9 10 10 10 10 11 11 11 11 12 12 12 12 6 0 9 10 10 10 10 11 11 11 11 12 12 12 12 13 13 6 20 10 10 11 11 11 11 12 12 12 12 13 13 13 13 14 6 40 11 11 11 11 12 12 12 12 13 13 13 14 14 14 14 7 0 11 11 12 12 12 13 13 13 13 14 14 14 15 15 15 7 20 12 12 12 13 13 13 14 14 14 14 15 15 15 16 16 7 40 12 12 13 13 13 14 14 14 15 15 15 16 16 16 17 8 0 13 13 13 14 14 14 15 15 15 16 16 16 17 17 17 8 20 13 14 14 14 15 15 15 .16 16 16 17 17 17 18 18 8 40 14 14 14 15 15 16 16 16 17 17 17 18 18 18 19 9 0 14 15 . 15 15 16 16 16 17 17 18 18 18 19 19 19 9 20 15 15 16 16 16 17 17 18 18 18 19 19 19 20 20 9 40 15 16 16 17 17 17 18 18 19 19 19 20 20 21 21 10 0 16 16 17 17 17 18 18 19 19 20 20 20 21 21 22 10 20 16 17 17 18 18 19 19 19 20 20 21 21 22 22 22 10 40 17 17 18 18 19 19 19 20 20 21 21 22 22 23 23 11 0 17 18 18 19 19 20 20 21 21 22 22 22 23 23 24 11 20 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 11 40 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 12 0 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 TABLE LVIII 91 Change of Moon’s Right Ascension from the Sun. Time. 53m 54m 55 m 56m 57m 58m 59m 60m 61m 62 m 63m 64in 65 m 66 m h m. m. m. m. m. m. m. IT). m. m. m. m. m. m. m. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 20 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 40 1 1 2 2 2 2 2 2 2 2 2 2 o 2 1 0 2 2 2 2 2 2 2 2 o 6 3 3 3 3 3 1 20 3 3 3 3 3 3 3 3 3 3 3 4 4 4 1 40 4 4 4 4 4 4 4 4 4 4 4 4 5 5 2 0 4 4 5 5 5 5 5 5 5 5 5 5 5 5 2 20 5 5 5 5 6 6 6 6 6 6 6 6 6 6 2 40 6 6 6 6 6 6 7 7 7 7 7 7 7 7 3 0 7 7 7 7 7 7 7 7 8 8 8 8 8 8 3 20 7 7 8 8 8 8 8 8 8 9 9 9 9 9 3 40 8 8 8 9 9 9 9 9 9 9 10 10 10 10 4 0 9 9 9 9 9 10 10 10 10 10 10 11 11 11 4 20 10 10 10 10 10 10 11 11 11 11 11 12 12 12 4 40 10 10 11 11 11 11 11 12 12 12 12 12 13 13 5 0 11 11 11 12 12 12 12 12 13 13 13 13 14 14 5 20 12 12 12 13 13 13 13 13 14 14 14 14 14 15 5 40 13 13 13 13 13 14 14 14 14 15 15 15 15 15 6 0 13 13 14 14 14 14 15 15 15 15 16 16 16 16 6 20 14 14 15 15 15 15 16 16 16 16 17 17 17 17 6 40 15 15 15 16 16 16 16 17 17 17 17 18 18 18 7 0 15 16 16 16 17 17 17 17 18 18 18 19 19 19 7 20 16 16 17 17 17 18 18 18 19 19 19 20 20 20 7 40 17 17 18 18 18 19 19 19 19 20 20 20 21 21 8 0 18 18 18 19 19 19 20 20 20 21 21 21 22 22 8 20 18 19 19 19 20 20 20 21 21 22 22 22 23 23 8 40 19 19 20 20 21 21 21 22 22 22 23 23 23 24 9 0 20 20 21 21 21 22 22 22 23 23 24 24 24 25 9 20 21 21 21 22 22 23 23 23 24 24 24 25 25 26 9 40 21 22 22 23 23 23 24 24 25 25 25 26 26 27 10 0 22 22 23 23 24 24 25 25 25 26 26 27 27 27 10 20 23 23 24 24 25 25 25 26 26 27 27 28 28 28 10 40 24 24 24 25 25 26 26 27 27 28 28 28 29 29 11 0 24 25 25 26 26 27 27 28 28 28 29 29 30 30 11 20 25 25 26 26 27 27 28 28 29 29 30 30 31 31 11 40 26 26 27 27 28 28 29 29 30 30 31 31 32 32 12 0 26 27 27 28 28 29 29 30 30 31 31 32 32 33 92 TABLE L1X, Change in Moon’s Declination . lime. 1° c 1° 0 10 i | 10 / 20' 31/ W 50' Time. h. m. o / O r o r ' / r / ' h. m. 0 0 0 0 0 0 0 0 0 0 0 0 0 12 0 0 20 0 2 0 3 0 5 0 1 1 1 1 12 20 0 40 0 3 0 7 0 10 1 1 2 2 3 12 40 1 0 0 5 0 10 0 15 1 2 2 3 4 13 0 1 20 0 7 0 13 0 20 1 2 3 4 6 13 20 1 40 0 8 0 17 0 25 1 3 4 6 7 13 40 2 0 0 10 0 20 0 30 2 3 5 7 8 14 0 2 20 0 12 0 23 0 35 2 4 6 8 10 14 20 2 40 0 13 0 27 0 40 2 4 7 9 11 14 40 3 0 0 15 0 30 0 45 2 5 7 10 12 15 0 3 20 0 17 0 33 0 50 3 6 8 11 14 15 20 3 40 0 18 0 37 0 55 3 6 9 12 15 15 40 4 0 0 20 0 40 1 0 3 7 10 13 17 16 0 4 20 0 22 0 43 1 5 4 7 11 14 18 16 20 4 40 0 23 0 47 1 10 4 8 12 16 19 16 40 5 0 0 25 0 50 1 15 4 8 12 17 21 17 0 5 20 0 27 0 53 1 20 4 9 13 18 22 17 20 5 40 0 28 0 57 1 25 5 9 14 19 24 17 40 6 0 0 30 1 0 1 30 5 10 15 20 25 18 0 6 20 0 32 1 3 1 35 5 11 16 21 26 18 20 6 40 0 33 1 7 1 40 6 11 17 22 28 18 40 7 0 0 35 1 10 1 45 6 12 17 23 29 19 0 7 20 0 37 1 13 1 50 6 12 18 24 31 19 20 7 40 0 38 1 17 1 55 6 13 19 26 32 19 40 8 0 0 40 1 20 2 0 7 13 20 27 33 20 0 8 20 0 42 1 23 2 5 7 14 21 28 35 20 20 8 40 0 43 1 27 2 10 7 14 22 29 36 20 40 9 0 0 45 1 30 2 15 7 15 22 30 37 21 0 9 20 0 47 1 33 2 20 8 16 23 31 39 21 20 9 40 0 48 1 37 2 25 8 16 24 32 40 21 40 10 0 0 50 1 40 2 30 8 17 25 33 42 22 0 10 20 0 52 1 43 2 35 9 17 26 34 43 22 20 10 40 0 53 1 47 2 40 9 18 27 36 44 22 40 11 0 0 55 1 50 2 45 9 18 27 37 46 23 0 11 20 0 57 1 53 2 50 9 19 28 38 47 23 20 11 40 0 58 1 57 2 55 10 19 29 39 49 23 40 12 0 1 0 2 0 3 0 10 20 30 40 50 24 0 TABLE LIX, 93 Change of Moon’s Declination . Time. r | 2' 3' 4' 5' 6' 7' 8' 9' lime. h. m. t ! / / / / f h. in 0 0 0 0 0 0 0 0 0 0 0 12 0 0 20 0 0 0 0 0 0 0 0 0 12 20 0 40 0 0 0 0 0 0 0 0 0 12 40 1 0 0 0 0 0 0 0 1 1 1 13 0 1 20 0 0 0 0 1 1 1 1 1 13 20 1 40 0 0 0 1 1 1 1 1 1 13 40 i 2 0 0 0 0 1 1 1 1 1 1 14 0 2 20 0 0 1 1 1 1 1 2 2 14 20 2 40 0 0 1 1 1 1 2 2 2 14 40 3 0 0 0 1 1 1 1 2 2 2 15 0 3 20 0 1 1 1 1 2 2 2 2 15 20 3 40 0 1 1 1 2 2 2 2 3 15 40 4 0 0 1 1 1 2 2 2 3 3 16 0 4 20 0 1 1 1 2 2 3 3 3 16 20 4 40 0 1 1 2 2 2 3 3 3 16 40 5 0 0 1 1 2 2 2 3 3 4 17 0 5 20 0 1 1 2 2 3 3 4 4 17 20 5 40 0 1 1 2 2 3 3 4 4 17 40 6 0 1 1 1 2 2 3 3 4 4 18 0 6 20 1 1 2 2 3 3 4 4 5 18 20 6 40 1 1 2 2 3 3 4 4 5 18 40 7 0 1 1 2 2 3 3 4 5 5 19 0 7 20 1 1 2 2 3 4 4 5 5 19 20 7 40 1 1 2 3 3 4 4 5 6 19 40 8 0 1 1 2 3 3 4 5 5 6 20 0 8 20 1 1 2 3 3 4 5 6 6 20 20 8 40 1 1 2 3 4 4 5 6 6 20 40 9 0 1 1 2 3 4 4 5 6 7 21 0 9 20 1 2 2 3 4 5 5 6 7 21 20 9 40 1 2 2 3 4 5 6 6 7 21 40 10 0 1 2 2 3 4 5 6 7 7 22 0 10 20 1 2 3 3 4 5 6 7 8 22 20 10 40 1 2 3 4 4 5 6 7 8 22 40 11 0 1 2 3 4 5 5 6 7 8 23 0 11 20 1 2 3 4 5 6 7 8 8 23 20 11 40 1 2 3 4 5 6 7 8 9 23 40 12 0 1 2 3 4 5 6 7 8 9 24 00 94 TABLE LX For the Aberration of a Star in Right Ascension and Declination . Argument. Sun’s true Longitude. 0 VI s 1* Vll* II* VIII* Log. a X Log. a X Log. a X — + — 4 * — + 0° 1.2690 0° 0' 1.2790 2°1T 1.2977 2° 6' 30' 1 1.2690 0 5 1.2796 2 14 1.2983 2 3 29 2 1.2691 0 11 1.2802 2 16 1.2988 2 0 28 3 1.2692 0 16 1.2808 2 18 1.2993 1 57 27 4 1.2692 0 22 1.2815 2 20 1.2998 1 54 26 5 1.2693 0 27 1.2821 ! 2 21 1.3003 1 51 25 6 1.2695 0 32 1.2827 2 23 1.3008 1 47 24 7 1-2696 0 37 1.2834 2 24 1.3012 1 44 23 8 1.2698 0 43 1.2840 2 25 1.3017 1 40 22 9 1.2700 0 48 1.2847 2 26 1.3021 1 36 21 10 1.2703 0 53 1.2853 2 27 1.3025 1 32 20 11 1.2705 0 58 1.2860 2 28 1.3028 1 28 19 12 1.2708 1 3 1.1866 2 28 1.3032 1 24 18 13 1.2711 1 8 1.2873 2 28 1.3036 1 20 17 14 1.2714 1 12 1.2879 2 28 1.3039 l 16 16 15 1.2718 1 17 1.2886 2 28 1.3042 1 11 15 16 1.2721 1 22 1.2892 2 28 1.3045 1 7 14 17 1.2725 1 26 1.2899 2 27 1.3048 1 3 13 i 18 1.2729 1 30 1.2905 2 27 1.3050 0 58 12 i 19 1.2733 1 34 1.2912 2 26 1.3053 0 53 11 20 1.2738 1 39 1.2918 2 25 1.3055 0 49 10 21 1.2742 1 42 1.2924 2 24 1.3057 0 44 9 | 22 1.2747 1 46 1.2931 2 22 1.3059 0 39 8 23 1.2752 1 50 1.2938 2 21 1.3060 0 34 7 24 1.2757 1 53 1.2944 2 19 1.3061 0 30 6 25 1.2762 1 57 1.2949 2 17 1.3063 0 25 5 26 1.2768 2 0 1.2956 2 15 1.3064 0 20 4 - 27 1.2773 2 3 1.2961 2 13 1.3064 0 15 3 28 1.2779 2 6 1.2966 2 11 1,3065 0 10 2 29 1.2785 2 9 1.2972 2 8 1.3065 0 5 1 30 1.2790 2 11 1.2977 2 6 1.3065 0 0 0 Log 1 , a X Log. a X Log. a X V s XI s IV* X* III* IX s TABLE LXI, 95 For the Aberration of a Star in Right Ascension and Declination . Argument. Sun’s Longitude, more or less the Star’s Declination. 0 * I s 11* IIP IV s V s — — — -f + + 0 ° 4".03 3".49 2".02 0".00 2''.02 3".49 30° 1 4.03 3.46 1.95 0.07 2.08 3.53 29 2 4.03 3.42 1.89 0.14 2.14 3.56 28 3 4.03 3.38 1.83 0.21 2.20 3.59 27 4 4.02 3.34 1.77 0.28 2.26 3.63 26 5 4.02 3.30 1.70 0.35 2.31 3.66 25 6 4.01 3.26 1.64 0.42 2.37 3.68 24 7 4.00 3.22 1.58 0.49 2.43 3.71 23 8 3.99 3.18 1.51 0.56 2.48 3.74 22 9 3.98 3.13 1.45 0.63 2.54 3.77 21 10 3.97 3.09 1.38 0.70 2.59 3.79 20 11 3.96 3.04 1.31 0.77 2.65 3.81 19 12 3.95 3.00 1.25 0.84 2.70 3.84 18 13 3.93 2.95 1.18 0.91 2.75 3.86 17 14 3.91 2.90 1.11 0.98 2.80 3.88 16 15 3*90 2.85 1.04 1.04 2.85 3.90 15 16 3.88 2.80 0.98 1.11 2.90 3.91 14 17 3.86 2.75 0.91 1.18 2.95 3.93 13 18 3.84 2.70 0.84 1.25 3.00 3.95 12 19 3.81 2.65 0.77 1.31 3.04 3.96 11 20 3.79 2.59 0.70 1.38 3.09 3.97 10 21 3.77 2.54 0.63 1.45 3.13 3.98 9 22 3.74 2.48 0.56 1.51 3.18 3.99 8 23 3.71 2.43 0.49 1.58 3.22 4.00 7 24 3.68 2.37 0.42 1.64 3.26 4.01 6 25 3.66 2.31 0.35 1.70 3.30 4.02 5 26 3.63 2.26 0.28 1.77 3.34 4.02 4 27 3.59 2.20 0.21 1.83 3.38 : 4.03 3 28 3.56 2.14 0.14 1.89 3.42 4.03 2 29 3.53 2.08 0.07 1.95 3.46 4.03 1 30 3.49 2.02 0.00 2.02 3.49 4.03 0 — — — 4~ 4- ■+■ XI s X* IX* VIII s VII s VP 96 TABLE LX1I For the «5V *station in Right Jlscension and Declination. Argument. Mean Longitude of Moon’s Ascending Node. 0 * VI* % I s VII* II* Vlll* Log. b B j L og- B Log. b B 0 ° 0.9844 0 ° 0' 0.9588 6 ° 45' 0.8960 7° 48' 30° 1 0.9844 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 12 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 0.9825 2 2 0.9442 7 49 0.8795 6 29 22 9 0.9821 2 17 0.9422 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 0.9787 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 16 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 I 18 0.9750 4 24 0.9231 8 25 0.8637 3 56 12 19 0.9739 4 37 0.9208 8 25 0.8625 3 38 11 20 0.9728 4 50 0.9186 8 25 0.8615 3 20 10 21 0.9716 5 3 0.9163 8 24 0.8605 3 1 9 22 0.9704 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 1 0.8565 0 41 2 29 0.9604 6 35 0.8983 7 55 0.8563 0 21 1 ; 30 0.9588 6 45 0.8960 7 48 0.8563 0 0 0 _ 4* _ + _ 4_ Log. b B Log. b B Log. b B V* XI s IV* X* III* IX* TABLE LXIII. 97 For the Nutation in Right Ascension and Declination . Argument. Mean Longitude of Moon’s Ascending Node. 0 * 1 * ii s Hi* IV* V* 0 ° 0".00 8".2 7 14".33 16"54 14"33 8 ".27 30° 1 0.29 8.52 14.47 16.54 14.18 8.02 29 2 0.'58 8.77 14.61 16.53 14.03 7.77 28 3 0.87 9.01 14.74 16.52 13.83 7.51 27 4 1.15 9.25 14.87 16.50 13.72 7.25 26 5 1.44 9.49 14.99 16.48 13.55 6.99 25 6 1.75 9.72 15.11 16.45 13.38 6.73 24 7 2.02 9.96 15.23 16.42 13.21 6.46 23 8 2.30 10.19 15.34 16.38 13.04 6.20 22 9 2.59 10.41 15.45 16.34 12.86 5.93 21 10 2.87 10.63 15.55 16.29 12.67 5.66 20 11 3.16 10.85 15.64 16.24 12.49 5.39 19 12 3.44 11.07 15.73 16.18 12.30 5.11 18 13 3.72 11.28 15.82 16.12 12.10 4.84 17 14 4.00 11.49 15.90 16.05 11.90 4.56 16 15 4.28 11.70 15.98 15.98 11.70 4.28 15 16 4.56 11.90 16.05 15.90 11.49 4.00 14 17 4.84 12.10 16.12 15.82 11.28 3.72 13 18 5.11 12.30 16.18 15.73 11.07 3.44 12 19 5.39 12.49 16.24 15.64 10.85 3.16 11 20 5.66 12.67 16.29 15.55 10.63 2.87 10 21 5.93 12.86 16.34 15.45 10.41 2.59 9 22 6.20 13.04 16.38 15.34 10.19 2.30 8 23 6.46 13.21 16.42 15.23 9.96 2.02 7 24 6.73 13.38 16.45 15.11 9.72 1.75 6 25 6.99 13.55 16.48 14.99 9.49 1.44 5 26 7.25 13.72 16.50 14.87 9.25 1.15 4 27 7.51 13.83 16.52 14.74 9.01 0.87 3 28 7.77 14.03 16.53 14.61 8.77 0.58 2 29 8.02 14.18 16.54 14.47 8.52 0.29 1 30 8.27 14.33 16.54 14.33 8.27 0.00 0 4- ! 4- + + 4“ 4- XI* X* IX* VIII* VII* VI* 13 * 98 TABLE LXIV, Semidiurnal Arcs for 39° 57' North Latitude. North Declination. Sooth Declination. O' 20 ' 40' 0 ' 2 O' 40' h. m. h. m. h. m. h. m. h. m. h. m. 0 ° 6 0 6 1 6 2 6 0 5 59 5 58 1 6 3 6 4 6 6 5 57 5 56 5 54 2 6 7 6 8 6 9 5 53 5 52 5 51 cy O 6 10 6 11 6 12 5 50 5 49 5 48 4 6 13 6 15 6 16 5 47 5 45 5 44 5 6 17 6 18 6 19 5 43 5 42 5 41 6 6 20 6 21 6 22 5 40 5 39 5 38 7 6 24 6 25 6 26 5 36 5 35 5 34 8 6 27 6 28 6 29 5 33 5 32 5 31 9 6 30 6 32 6 33 5 30 5 28 5 27 10 6 34 6 35 6 36 5 26 5 25 5 24 11 6 37 6 39 6 40 5 23 5 21 5 20 12 6 41 6 42 6 43 5 19 5 18 5 17 13 6 45 6 46 6 47 5 15 5 14 5 13 14 6 48 6 49 6 51 5 12 5 11 5 9 15 6 52 6 53 6 54 5 8 5 7 5 6 16 6 56 6 57 6 58 5 4 5 3 5 2 17 6 59 7 1 7 2 5 1 4 59 4 58 18 7 3 7 4 7 6 4 57 4 56 4 54 19 7 7 7 8 7 10 4 53 4 52 4 50 20 7 11 7 12 7 14 4 49. 4 48 4 46 21 7 15 7 16 7 18 4 45 4 44 4 42 22 7 19 7 21 7 22 4 41 4 39 4 38 23 7 23 7 25 7 26 4 37 4 35 4 34 24 7 28 7 29 7 30 4 32 4 31 4 30 25 7 32 7 33 7 35 4 28 4 27 4 25 26 7 36 7 38 7 40 4 24 4 22 4 20 27 7 41 7 43 7 44 4 19 4 17 4 16 28 7 46 7 47 7 49 4 14 4 13 4 11 29 7 51 7 52 7 54 4 9 4 8 4 6 30 7 56 7 57 7 59 4 4 4 3 4 1 THE END ERRATA. Page Line 14 18 u 19 15 11 23 19 24 13 26 25 27 4 29 20 34 8 35 20 36 19 61 17 75 17 u 22 76 15 96 19 97 3 98 5 104 13 '. r/.it. Fig