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SOLAR AND LUNAR 
 
 ECLIPSES 
 
 FAMILIARLY ILLUSTRATED AND EXPLAINED, 
 
 WITH THE 
 
 METHOD OF CALCULATING THEM 
 
 ACC0RDI5G TO THE 
 
 THEORY OF ASTRONOMY, 
 
 AS TAUGHT IH 
 
 NEW ENGLAND COLLEGES 
 
 BY JAMES H. COFFIN, A.M. 
 
 NEW YORK: 
 
 COLLINS, BROTHER & CO 
 1845. 
 
Entered according to Act of Congress, in the year 1845, 
 
 BY JAMES H. COFFIN, 
 In the Clerk's Office of the District Court of Connecticut. 
 
541 
 CC4 
 
 PREFACE 
 
 The design of this treatise, is to explain the " rationale" of some 
 of the most interesting astronomical calculations, in such a way 
 that the student may clearly see the reason of every step, and its 
 connection with the theory. In this respect it differs from many 
 others, which give the rules for calculating merely, without any 
 explanation of the reason of them. Being partly designed as a text 
 hook for colleges, the author has endeavoured to adapt it to the 
 design of college education, which is not so much to make adept 
 practitioners in any particular science, as to give broad and com- 
 prehensive views of the whole field. Hence, the principles of the 
 several sciences should be thoroughly understood by the student ; 
 but the application of them to practice by mere rules is foreign to 
 the design of a collegiate course of study. If, therefore, the calcu- 
 lations of astronomy are attended to at all in college, it should be 
 in such a way, that the connection with the theory may be appa- 
 rent, and that the two may mutually illustrate each other. In 
 many of treatises for colleges, this point seems to be overlooked. 
 Some of them contain tables for astronomical calculations which 
 are very minute and accurate, and at the same time, so constructed 
 and arranged as to reduce the labour of calculation as much as 
 possible ; but the student can see no connection between them and 
 the motions and perturbations which occupy his attention in the 
 study of the theory. In fact, one who has studied the theory with 
 ever so much thoroughness, has here very little advantage over 
 one who is entirely ignorant of it ; each being guided wholly by 
 rules that must appear entirely arbitrary. 
 
 Such tables not being adapted to the design of this treatise, the 
 author found it necessary to prepare a set differing somewhat in 
 
 111505 
 
IV PREFACE. 
 
 their construction from those in ordinary use. They are based, 
 for the most part, on those of Delambre and Burg, with the more 
 recent improvements of Airy and Bessel ; but varied in the plan of 
 construction so as to adapt them to this work. Calculations may 
 be made from them sufficiently accurate for the ordinary purposes 
 of an almanac ; yet, as they are not designed especially for that 
 purpose, the aim has been not so much to secure extreme accuracy 
 in the results, as to render them easily understood. 
 
 The quantities in the tables are given, for the most part, in 
 degrees and decimals, instead of signs, degrees, minutes, and sec- 
 onds, with a view to facilitate the labour of calculation, and to 
 secure the same degree of accuracy with a less number of figures. 
 
 The work is intended to be complete in itself, on the subject on 
 which it treats, for those who have a general knowledge of the 
 motions of the heavenly bodies ; yet, it would, doubtless, contribute 
 to a better understanding of it, to read some such work as Olm- 
 sted's or Herschell's Astronomy previously. 
 
 Williams College, October, 1843. 
 
CONTENTS. 
 
 PART I. 
 
 CHAPTER I. 
 
 GENERAL CONDITIONS OF AN ECLIPSE. 
 1, Causes of eclipses.— 2. Must occur near the Moon's nodes — 3. Moon's latitude in solar 
 eclipses. — 4. Do. in lunar eclipses. — 5. Ecliptic limits.— 6. Solar. — 7. Lunar. — 8. Eclipses 
 must occur six months apart. 
 
 CHAPTER II. 
 
 MEAN TIME OF AN ECLIPSE. 
 9. Time when the sun passes the moon's nodes, how found. — 10. Motions not uniform. — 
 11. Mean time of an eclipse, how found. — 12. Annual equations of moon's perigee and 
 node explained. — 13. The same applied. 
 
 CHAPTER ILL 
 
 EQUATION OF THE CENTRE. 
 14. Velocity must be variable in eliptical orbits.— 15. The same shown by the principle of 
 equal areas in equal times. — 15. Equation of the centre explained. — 17. The same applied. 
 
 CHAPTER IV, 
 
 TIONS OF LONGITUDE. 
 18. Moon's motion disturbed by sun's attraction. — 19. Annual equation of the moon's longi" 
 tude explained and applied. — 20. Acceleration of the moon's mean motion. — 21. Its discov- 
 ery. — 22. Theories to account for it. — 23. Earth's orbit becoming less eliptical. — 24. This 
 shown to be the cause of the acceleration. — 25. Secular equation applied. 
 
 CHAPTER V. 
 
 PERTURBATIONS IN THE MOON's MOTION, CONTINUED. VARIATION. 
 
 26. Introduction to the subject of the chapter. — 27. Variation. — 28. Sun's attraction sometimes 
 increases and sometimes diminishes moon's gravity. — 29. Moon's motion alternately ac- 
 celerated and retarded. — 30. Effect upon the shape of the orbit. — 31. Combined effects of 
 perturbation in velocity and shape. — 32. The Table for Variation explained.— 33. The 
 equation applied. — 34. Annual equation of variation explained and applied. 
 
 CHAPTER VI. 
 
 PERTURBATION IN THE MOOn's MOTION, CONTINUED. EVECTION. 
 
 35. Difference between Variation and Evection. — 36. Evection caused, in part, by the une- 
 quable motion of the apsides. — 37. Caused, in part, by a variation in the eccentricity of 
 the orbit. — 38. Eccentricity depends on relative attraction. — 39. The principle applied to 
 the moon's orbit. — 40. Apsides must progress on the whole. — 41. Their place, how deter- 
 mined, physically. — 42. Their motion, how affected by an increase of gravity. — 43. How 
 
Vi CONTENTS. 
 
 by a diminution of the same. — 44. How by a perturbation in velocity. — 45. Neutral points. 
 — 46. Addititious force. — 47. Ablatitious force. — 48. The two combined. — 47. Neutral 
 points determined. — 50. Increase of gravity in quadrature only equal to half the diminution 
 in syzygy. — 51. Tangential force. — 52. Apsides progress when within 45° of syzygy. — 
 53. Remain nearly stationary when between 45° and 54° 43' 56". — 54. Regress when 
 more distant. — 55. Progress exceeds the regress. — 56. Variation in eccentricity and motion 
 of apsides represented by a figure. — 57. Evection represented by the same figure. — 58. 
 Argument of evection explained. — 59. The same computed. — 60. Evectiun applied. — 61. 
 Annual equation of evection. 
 
 CHAPTER VII. 
 
 PERTURBATIONS IN THE MOON'S MOTION, CONCLUDED. NODAL EQUATION 
 
 OF THE MOON'S LONGITUDE. CORRECTION OF EQUATION OF THE CEN- 
 TRE. REDUCTION TO THE ECLIPTIC. 
 
 62. Disturbing force acts out of the plane of the moon's orbit.— 63. Theory of nodal equation 
 explained.— 64. Equation applied. — 65. Equation of the centre corrected.— 66. Reduction 
 explained and applied. 
 
 CHAPTER VIII. 
 
 LUNAR OR MENSTRUAL EQUATION OF THE SUN*S LONGITUDE. 
 67. Theory of lunar equation.— 68. The same applied.— 69. Effect of inequalities inappre- 
 ciable. 
 
 CHAPTER IX. 
 
 NUTATION IN LONGITUDE. 
 70. Longitude affected by precession of the equinoxes. — 71. Precession explained. — 72. 
 Caused, in part, by the moon's attraction.— 73. When most rapid.— 74. Lunar nutation, 
 how caused.— 75. Equation for lunar nutation explained.— 76. The same applied.— 77. 
 Solar nutation. 
 
 CHAPTER X. 
 
 TRUE TIME LONGITUDES AND ANOMALIES. 
 78. Correction in the time of new moon. — 79. Greenwich time. — 80. Correction of the lon- 
 gitudes.— 81. Synopsis.— 82. Preliminary equations explained.— 83. Application of the 
 same.— 84. Synopsis. 
 
 CHAPTER XL 
 
 ELEMENTS OF AN ECLIPSE. 
 85. List of elements.— 86. Time and longitudes.— 87. Obliquity of ecliptic to equator.— 88. 
 Moon's latitude.— 89. Hourly motions of sun and moon.— 90. Sun's apparent semidiame- 
 ter.— 91. Moon's semidiameter and parallax.— 92. Semidiameter of earth's shadow.— 93. 
 Angle of moon's path. — 94. Sun's declination. — 95. Elements collected. 
 
 CHAPTER XII. 
 
 DELINEATION OF A SOLAR ECLIPSE. 
 96. Appearance of the earth as viewed from the sun, while the latter advances from the 
 vernal equinox to the summer solstice. — 97. The same, while it advances to the autumnal 
 equinox. — 98. The same, while it advances through the winter solstice to the vernal equi- 
 nox. — 99. Subject illustrated by a terrestrial globe. — 100. Relative size of objects, how 
 measured in astronomy. — 101. To draw the northern half of the earth's disc, as seen from 
 
CONTENTS. VII 
 
 the sun. 102. To find the position of the north pole on the disc. — 103. To find the position 
 of any given place at noon on the disc. — 104. To find the same at midnight. — 105. To find 
 the same at six o'clock, morning and evening. — 106. To find the same at any other hour, 
 and to draw the apparent diurnal path of the place. — 107. The construction shows the 
 time of sunrise or sunset. — 108. To find the position of the moon when new, as projected 
 upon the earth's disc. — 109. To draw its path across the disc. — 110. To find its position at 
 any time. — 111. To find how many digits will be eclipsed. — 112. To find when the eclipse 
 will begin and end. — 113. To find the above by calculation. 
 
 CHAPTER XIII. 
 
 CENTRAL TRACK OF A SOLAR ECLIPSE. 
 Hi. Track may be found by the drawing in the last chapter. — 115. A better method by a 
 terrestrial globe. — 116. How to find where the centre of the moon's shadow first strikes 
 the earth. — 117. How to find where it leaves it. — 118. How to find where the eclipse is 
 central at any given time. — 119. Track may be found more accurately still by calcula- 
 tion. — 120. How to find, by calculation, where the centre of the moon's shadow strikes or 
 leaves the earth. — 121. How to find, by calculation, where the eclipse will be central at 
 any given time. — 132. The same, more fully explained. — 123. Results of the calculation 
 in the solar eclipse of May, 1854.— 124. General description of the track of that eclipse. 
 
 CHAPTER XIV. 
 
 DELINEATION OF A LUNAR ECLIPSE. 
 125. To draw the earth's shadow.— 126. To find the position of the moon when full, in res- 
 pect to the shadow.— 127. To draw its path.— 128. To find the position of the moon at any 
 lime during the eclipse.— 129. The same obtained by calculation.— 130. Results of the 
 calculation in the lunar eclipse of November, 1844. 
 
10 
 
 earth. The angle MES, is not difficult of cpmputation, being 
 equal to SEA+MEC+AEC, the latter of which is equal (Euc. 
 1, 32) to ECB— EAB. Hence, MES=SEA+MEC+ECB— EAB, 
 all of which are easily found. The two former are the apparent 
 semidiameters of the sun and moon, as viewed from the earth, and 
 the two latter the semidiameter of the earth, as viewed from the 
 moon and sun, or, respectively, the moon's and sun's horizontal 
 parallax. On account of the distance of the sun and moon from 
 the earth not being constant, the angle MES is subject to a varia- 
 tion in size, being sometimes 1° 38', and at other times not more 
 than 1° 14'. The reader will perceive that, when the sun and 
 moon are in conjunction, the angle MES is the moon's latitude ; 
 and the conclusion to which we have just arrived, may be express- 
 ed thus : If the latitude of the moon, when new, is less than 1° 38' 
 there may be an eclipse of the sun, and if it is less than 1° 14' there 
 must be one. 
 
 Fig. 2. 
 
 4. Again, let S (Fig. 2) represent the centre of the sun, E that 
 of the earth, and M that of the moon, just impinging upon the 
 earth's shadow. The angle MET, as represented in the figure, is 
 plainly the least possible, without producing a partial or total 
 eclipse of the moon. The angle MET=MEF+FET, both of 
 which can be easily computed. The former is the moon's appa- 
 rent simidiameter, as seen from the earth, and FET is that of the 
 section of the earth's shadow that eclipses the moon. Now, (Euc. 
 1, 32,) FET=BFE— FHE and FHE=AES— BAE ; therefore, 
 FET=BFE+BAE — AES, the two former of which are the lunar 
 and solar parallaxes, and the latter is the sun's apparent semidiam- 
 eter, as seen from the earth. That is, the apparent semidiameter of 
 the section of the earth's shadow thai eclipses the moon, is equal to 
 the sum of the parallaxes of the sun and moon, diminished by the 
 sun's apparent semidiameter. And if this angle be increased by 
 the moon's apparent semidiameter, MEF, we shall have the whole 
 angle MET, which is the least latitude the moon can have in oppo- 
 
11 
 
 sition, without being eclipsed. This angle varies in size, like the 
 analagous one in solar eclipses, just described, and for the same 
 reason. Its maximum value is 1° 4', and its minimum 50'. 
 
 5. The centre of ihe section of the earth's shadow that eclipses 
 the moon, must be situated in the plane of the ecliptic, directly op- 
 posite to the sun, and must, therefore, be at the same distance from 
 one of the moon's nodes that the sun is from the other. Now, we 
 wish to know how near the sun, in its annual course, may approach 
 to one of the moon's nodes, without occasioning eclipses ; or. in 
 other words, at what distance from the node the moon's track and 
 the ecliptic will have diverged, so as to be from 1° 14' to 1° 38' 
 apart, if our inquiry relates to solar eclipses — or, from 50' to 1° 4'» 
 if it relates to lunar. 
 
 6. Let AN (Fig. 3) represent a 
 portion of the ecliptic, BN a por- 
 tion of the moon's orbit, SM a por- 
 tion of a secondary to the ecliptic, 
 and N one of the moon's nodes. Then, in the right angled spheri- 
 cal triangle, SMN, we have the angle SNM=5° 7' 47".9,* and for 
 solar eclipses, the arc SM=1° 14' to 1° 38'. With these data, we 
 find NS to be from 13° 14' to 19° 42', according to the value we 
 give to SM. Hence, if the sun is within 19° 42' of the moon's 
 node, on either side, at the time of new moon, it may be eclipsed ; 
 and if it is within 13° 24', it must be. These distances are called 
 the solar ecliptic limits. 
 
 Since it takes the sun more than a lunar month, usually, to pass 
 over one of these arcs, it follows, that it must be eclipsed at every 
 passage, and, consequently, twice a year, at least. It may be 
 eclipsed twice during one passage ; once, just after it enters the 
 ecliptic limits, and again, just before it leaves them : but, if so, both 
 of the eclipses will be small, and not central upon any part of the 
 earth. 
 
 7. For lunar eclipses, the arc SM is 50' to 1° 4', and, by the 
 same process as above, we find the lunar ecliptic limits to be, irom 
 7° 47" to 13° 21' on each side of the node.f They embrace an 
 
 * This angle is subject to a slight variation, amounting, at it3 maximum, to 6' 47".lo. 
 t Baiiy. 
 
12 
 
 arc considerably less than is passed over by the sun in one luna- 
 tion, so that it often happens, that the sun passes the node without 
 there being any lunar eclipse. 
 
 The lunar ecliptic limits being so much less than the solar, 
 eclipses of the moon must be proportionably less frequent; yet, 
 since a lunar eclipse is always visible over half the earth's surface, 
 while one of the sun can be seen only over a very much smaller 
 section, there will, on an average, be a greater number of visible 
 eclipses of the moon, at any given place, than of the sun, 
 
 .8. As the moon's nodes are 180° apart, or, in opposite points of 
 the ecliptic, the interval between eclipses occurring at one node, 
 and those occurring at the other, must be about six months. Also, 
 since the nodes move backward about 19° in each year, eclipses 
 must happen, on an average, nearly three weeks earlier every year 
 than they did on the year preceding. The reader will see that 
 these conclusions are verified by past experience, if he will take the 
 trouble to examine the almanacs of former years. 
 
 CHAPTER II. 
 
 MEAN TIME OF AN ECLIPSE, AND THE MEAN LONGITUDES AND ANOMA- 
 LIES OF THE SUN AND MOON. 
 
 9. The sun, in its apparent annual course, leaves the vernal equi- 
 nox where its longitude is 0°, about the 21st of March, and moves 
 eastward, towards the moon's nodes, about 1° each day. Conse- 
 quently, it must arrive at either node, in about as many days after 
 the 21st of March, in any given year, as the longitude of the node, 
 in that year, contains degrees. At the next new moon before or 
 after the date thus found, (more frequently the former.) there will 
 be an eclipse of the sun. It is probable, though not certain, that 
 there will also be a lunar eclipse at the nearest full moon. It will 
 occur then if at all at that passage of the node. 
 
 10. The velocity of the motions of sun and moon in their respect- 
 ive orbits, is variable ; but it is more convenient in astronomical 
 calculations to regard it as uniform, and to make the necessary cor- 
 rections for the inequalities afterward. 
 
13 
 
 11. To explain the method of calculating the time of an eclipse, 
 we will take, as an example, the solar eclipse that will occur when 
 the sun passes the moon's ascending node, in the year 1854, Ta- 
 ble 2d, at the end of this volume, contains the time of new moon 
 in March of that year, as well as of every other during the present 
 century, and the longitude of the sun, moon, and moon's ascending 
 node, all calculated on the supposition of a uniform rate of motion. 
 It also gives the .mean anomolies of the sun and moon, i. e., the dis- 
 tance of each from its perigee. The longitude of the descending 
 node may be found by adding, or subtracting, 180° to, or from, 
 that of the ascending node. We might compute all these quanti- 
 ties from the data given in table 1st, but table 2d supercedes the 
 necessity. Although some of them will not be used in this chapter, 
 it is most convenient to take them all out together, and write them 
 as below. The longitude of the node on that year (see right hand 
 column of the table) is 64°.2158 ; consequently, the sun will arrive 
 at it about 64 days after the 21st of March, which carries the time 
 to May 24th. The eclipse in question will occur at the new moon 
 nearest that time. Entering table 3d,*' we next take out such a 
 number of lunations (in this ease, two) as, when added to the time 
 of new moon in March, will bring it near to the time when the sun 
 reaches the moon's node, and write it down, with the longitudes 
 and anomalies, under the corresponding quantities already taken 
 from table 2d. These must be added together, (with the exception 
 of the right hand column, where the lower number must be sub- 
 tracted from the upper, because the motion of the node is retro- 
 grade,) and we thus obtain the time of mean new moon in May. 
 The following shows the operation : — 
 
 Mean new moon in March,... 
 
 Add two lunations, 
 
 Mean new moon in May, 
 
 Time. 
 
 h. ?n. s, 
 28 10 42 50 
 59 1 28 6 
 
 26 12 10 5t 
 
 Sun's 
 Anom- 
 aly. 
 
 85.586 
 58.21 1 
 
 143.797 
 
 Sun's 
 Longi- 
 tude. 
 
 6°0194 
 58.2135 
 
 Moon's 
 Anom- 
 aly. 
 
 93.665 
 51.634 
 
 64.2329|145.299 
 
 Moon's I Longi- 
 Longi- tude of 
 tude. Node. 
 
 6.0194 64°2158 
 58.2135l-3.1275 
 
 64.2329 i 61.0883 
 
 Table 5th shows the month and day to which any number of 
 days found by the foregoing addition corresponds. 
 
 At this stage of the calculation, it is well to compare the longi- 
 tude of the sun and moon with that of the node, and if they do not 
 
 * Table 3d shows the length of any number of mean lunations, from one to thirteen, 
 with the mean motions of the sun and moon, both in longitude and anomaly, during the 
 same ; also, the mean motion of the moon's nodes. 
 
14 
 
 differ more than 20°, there may be an eclipse, though there will not, 
 probably, be one, if the difference is over I62 . If the difference is 
 too great, it shows that too many lunations were added, or too few, 
 and a correction must be made accordingly. A difference of over 
 11° shows that another eclipse, at that passage of the node, is pos- 
 sible, but not probable, unless it is as much as 14°. 
 
 12. At the time of new moon, the longitudes of the sun and moon 
 must be equal, and, according to our calculations, they are so at 
 the time of the*mean new moon in May just found. But this is on 
 the supposition, that their motions were uniform. To find whether 
 or not their longitudes are truly equal, we shall proceed, in the fol- 
 lowing chapters, to compute them, taking into account all the chief 
 inequalities in their motions ; and, if they come out alike, the time 
 of new moon is correctly found ; otherwise, we shall have to add 
 or subtract such an amount of time, as, with the relative velocities 
 of the sun and moon, at the time, will render them equal. As a 
 preparatory step, it is necessary to know, more accurately, the 
 the moon's anomaly, and the longitude of its node. 
 
 The progressive motion of the moon's perigee, and the retro- 
 grade motion of its nodes, being both caused by the sun's attrac- 
 tion, are most rapid when the sun is in its perigee, and constantly 
 grow slower and slower, till the sun reaches its apogee, where the 
 motion becomes the slowest. Consequently, as the sun leaves its 
 perigee, the moon's perigee immediately gets before, and its nodes 
 behind their mean place, and continue so till the sun reaches its 
 apogee, when, owing to the diminished rate of motion, their mean 
 and true places again coincide. The contrary takes place when 
 the sun is in the other half of its orbit. Hence it is apparent, that 
 the moon's anomaly, being reckoned from its perigee, must be less 
 than the mean when the sun's anomoly is less than 180°, and great- 
 er when greater ; showing that something must be subtracted from 
 the moon's anomaly in the former case, and added in the latter. 
 The same must also be true of the longitude of the moon's nodes. 
 These facts are indicated in Tables 6th and 7th, by the signs — 
 and + placed at the head of the column containing the argument. 
 By the term argument, is meant that quantity on which others 
 depend, and which determines their value. Thus, in this case, the 
 sun's anomaly determines what correction must be applied to the 
 moon's anomaly, or to the longitude of its node, and is, therefore, 
 the argument. 
 
15 
 
 13. Entering tables 6th and 7th, with the sun's anomaly as an 
 argument, we will take out the corrections which the foregoing 
 considerations show to be necessary, and which are denominated 
 Annual Equations of the moon's perigee and node, (taking care to 
 make a proper allowance for the odd degree and decimals of the 
 anomaly, as the tables give the equation only for every two de- 
 grees,) and apply the former to the moon's anomaly, and the latter 
 to the longitude of the node, according to the sign -f- or — at the 
 head of the column of the argument. Observe in these, and most 
 of the other tables, that the unit figure of the argument is placed 
 at the top or bottom of the table, and the other figures at the right 
 or left. When the latter is found at the left, we must look for the 
 former at the top, but when the latter is at the right, the former must 
 be sought for at the bottom. Opposite the latter, and in the same 
 column with the former, the equation is found. Thus, in our 
 eclipse, the sun's anomaly being 143°. 797, we look for the number 
 14 in the left hand column, and for 3 at the top. But, since 
 the latter number is not found, the equation being given in the ta- 
 ble only for 142° and 144°, we must note the difference between 
 these equations, and take a proper proportion of it for the excess of 
 the argument over 142°, viz. 1°.797. By this process, we find the 
 annual equation of the perigee to be — 0°.221 ; and of the node 
 — 0°.875. These, applied to the moon's anomaly, and the longitude 
 of the node, make the former 145°.078, and the latter 61°.0008. 
 
 CHAPTER III. 
 
 EQUATION OP THE CENTRE. 
 
 14. The first inequality in the apparent motions of the sun and 
 moon that claims attention, results from the eliptical form of their 
 orbits, in consequence of which their motion is accelerated while 
 passing from apogee to perigee, and retarded in the other half of 
 their orbits, moving quickest in perigee, and slowest in apogee. 
 The reason of this it is not difficult to discover. 
 
16 
 
 Let AGH, &c. (Fig. 4) represent the 
 moon's eliptical orbit, A and B the apsides, 
 and E the earth. Let the moon start from 
 perigee, at A, with its swiftest motion, and 
 consequently with its greatest centrifugal 
 force. The attraction of the earth, at E, 
 not being able to retain it at that distance, 
 G ' H it immediately begins to recede along the 
 
 curve AGH. Being constantly pulled back by the earth's attrac- 
 tion, since the angle EGH is obtuse, its motion is retarded, and 
 when it approaches the apogee, at B, its velocity has become so 
 much diminished, that the attractive force of the earth prevents it 
 from receding further. It thus arrives at apogee with its slowest 
 motion. Leaving B with a weak centrifugal force, the superior 
 attractive power of the earth at E, immediately begins to draw 
 the moon toward itself along the curve BOD, constantly hurrying 
 it onward at every point, as O, the angle EOD, contained between 
 the direction toward which it is drawn, and that toward which it 
 moves being now acute. By the time it reaches A, its velocity 
 becomes so much increased, that it is prepared again to leave peri- 
 gee in the same condition as at first, to pursue another similar 
 round. 
 
 15. We shall arrive at the same conclusion, if we apply the prin- 
 ciple, that when a body is retained in its orbit by a force directed 
 toward a fixed point, as the moon is toward the earth in this case, 
 the radius vector must describe equal areas in equal times. Hence 
 the moon must move slower when it is near B, than when it is 
 near A, in about the same ratio that its distance from the earth is 
 greater. Not only is the absolute velocity greatest in perigee, but 
 the angular velocity, with which only we are now concerned, is 
 rendered still greater, by reason of the diminished distance. 
 
 The same reasoning will apply, in every respect, to the earth 
 revolving in an eliptical orbit round the sun, and hence to the ap- 
 parent motion of the sun round the earth. 
 
 16. If we consider the mean place of the sun and moon to be 
 their true one, when in perigee, it will be, also, when in apogee ; 
 because each half of the orbit is described in the same time ; but, 
 as they start from perigee with their swiftest motion, they thus get 
 ahead of their mean place, nor do they, though constantly retarded, 
 
17 
 
 lose what they had gained, till the moment they arrive at apogee. 
 Also, as they pass the apogee with their slowest motion, they di- 
 rectly get behind their mean place, and it is not till the moment 
 they reach the perigee, that the continued acceleration of their 
 motion, in this half of their orbits, enables them to gain up what 
 they had thus lost. Consequently, they must always be ahead of 
 their mean place, when in the former part of their orbits, and behind . 
 it in the latter. This difference between the mean and true place 
 of the sun or moon, is termed the equation of its centre. 
 
 17. It is necessary to know in which half of their respective 
 orbits the sun and moon are found at the time of our predicted 
 eclipse : for, if either is moving from perigee to apogee, i. e., if its 
 anomaly (4) is less than 180°, we must add something to its longi- 
 tude already found, (11,) but subtract if in the other half of its orbit, 
 i.e., if its anomaly is over 180°. The manner of computing the 
 precise amount, will occupy our attention in another chapter. It 
 is sufficient, here, to remark, that if, at any time, a line, OC, be 
 drawn from the mean place of the moon to the centre of the elipse, 
 the angle EOC, which this line makes with the radius vector, is 
 very nearly equal to one-half the angular distance by which the 
 moon is before or behind its mean place : so nearly that some au- 
 thors have given this as a method of computing the equation of the 
 centre. We shall use it hereafter as a convenient approxima- 
 tion. For the present, we will dispense with the labour of compu- 
 tation, and take the equation directly from tables already prepared. 
 
 By the calculations in the last chapter, (11 and 13,) we found 
 that the sun's anomaly was 143°.797, and the moon's, as corrected, 
 145°.078. It appears, then, that each is moving from perigee to 
 apogee, and is, therefore, ahead of its mean place, so that we must 
 add something to their respective longitudes ; with these anoma- 
 lies, respectively, as arguments, we may now enter tables 8th and 
 9th, in the same manner as we did tables 6th and 7th, and take out 
 the equation of the centre of the sun, and of the moon, applying the 
 former to the the sun's longitude, and the latter to the moon's. The 
 equations we find to be + 1M165 and +3°.4154, and the resulting 
 longitudes 65°.3494 and 67°.6483. 
 
 2 
 
18 
 
 CHAPTER IV. 
 
 PERTURBATIONS IN THE MOON's MOTION. ANNUAL AND SECULAR EQUA- 
 TIONS OF LONGITUDE. 
 
 18. In the preceding calculations, we first regarded the sun and 
 .moon as revolving in circular orbits, with uniform angular velocity ; 
 then, in elipses, describing equal areas by the radius vector, in 
 equal times : but neither of these suppositions is strictly true. The 
 sun's attraction disturbs the motion of the moon round the earth, 
 producing numerous inequalities. 
 
 The method of calculating these, will be treated of in another 
 place. It will be sufficient for our purpose, here, simply to give 
 the theory of them, and then take the corresponding corrections 
 from the tables. 
 
 19. The most obvious effect of the sun's attraction, is to draw 
 the moon away from the earth, and thus enlarge its orbit. If this 
 influence were always the same, it would occasion no inequality : 
 but when the sun is in perigee, it is nearer to the earth, and conse- 
 quently to the moon, than at other times ; and the moon, therefore, 
 will be more attracted by it. The moon being thus drawn farther 
 away from the earth, when the sun is in this situation, and its peri- 
 odic time consequently increased, it must fall behind its mean place. 
 And although the attractive force of the sun diminishes as it leaves 
 perigee, allowing the moon to contract its orbit and lessen its peri- 
 odic time, it will not gain up what it had lost till the moment the 
 sun reaches apogee. By similar reasoning, we may see that the 
 moon must always be in advance of its mean place, so far as this 
 cause is concerned, when the sun is in the other half of its orbit. 
 There must then be applied to the moon's longitude a correction 
 depending on the sun's anomaly ; .being additive when the anomaly 
 is less than 180°, and subtractive when it is more. Such a correc- 
 tion is supplied in table 10th, entering which, with the degrees of 
 the sun's mean anomaly, in the manner described in article 13th, 
 we find the equation to be -.1118, which is to be applied both to 
 the moon's longitude and anomaly,* for the cause we have been 
 considering obviously affects both alike. The same is true of most 
 
 * The anomaly contains but three decimal places ; hence, in applying the corrections 
 to it, the 3d figure is given according to its nearest value. 
 
d9 
 
 of the other corrections that remain to be applied. The longitude, 
 as already obtained, (17,) is 67°.6483, and the anomaly (13) 
 145°.078. Applying trie above equation, they become 67°.5365, 
 and 144°.9G6 respectively. 
 
 20. Connected with the foregoing inequality in the moon's mo- 
 tion, there is another of great historical interest, from the theories 
 to which it formerly gave rise, viz., the acceleration of the moon's 
 mean motion. It is too small to be discovered by direct observa- 
 tion, but becomes quite sensible in the lapse of ages. 
 
 21. Dr. Haliey, wishing to know the precise length of a lunation, 
 went back to the ancient Chaldean observations, intending to ascer- 
 tain how many new moons had occurred between that time and 
 his own, and then to divide the time by this number, which would 
 give the average length of each. But he was surprised to find 
 that a lunation in those days was considerably longer than now. 
 By comparing the Chaldean, Alexandrian, Arabian, and the pre- 
 sent observations, he found that the lunar period grew successively 
 shorter. 
 
 22. Astronomers doubted the fact when it was first announced ; 
 but when they became satisfied of its truth, they set themselves to 
 work to account for it. The most probable theory was, that the 
 moon revolved in a resisting medium, which would cause it gradu- 
 ally to fall toward the earth, and thus, by reducing the size of the 
 orbit, make the periodic time less. It must seem paradoxical to 
 those who have not thought upon the subject, that such a cause 
 could produce the effect in question ; and that the retarding of the 
 motion could make it revolve in less time. But it should be con- 
 sidered, that by diminishing the moon's velocity, its centrifugal 
 force is diminished in a more rapid ratio, which would allow the 
 earth to draw it nearer to itself, and reduce the size of the orbit. 
 And it is demonstrable, that the gain in time from the latter cir- 
 cumstance, would more than counterbalance the loss from the for- 
 mer ; so that on the whole, the moon's period would be shortened. 
 
 The objection to this theory is, that comets, which are proved 
 to be extremely light bodies, pass through this medium w^ith little 
 or no resistance. Hence it was inferred that the cause, if it exist- 
 
20 
 
 ed at all, was not sufficient to produce the effect of which we arc 
 speaking. 
 
 Other theories were advanced, but none were satisfactory ; and 
 it was reserved for La Place to explain the true reason of this ac- 
 celeration of the moon's period, about sixty years ago. 
 
 23. Owing to the attraction of the other planets, the earth's orbit 
 is gradually becoming less and less eliptical, or, nearer and nearer 
 to a circle ; so that the sun is every year about 39| miles nearer 
 the centre of the elipse than it was on the year before. At this 
 rate, the earth's orbit would become a circle in 40,315 years ; an 
 event, however, that can never take place, for long before such a 
 period shall elapse, the change of which we are speaking, and 
 which is only an inequality of long period, will have reached its 
 limit, when the eccentricity of the orbit would again increase. 
 
 24. If it can be shown that the sun's attraction diminishes the 
 moon's gravity toward the earth, and thus increases the periodic 
 time, more than it would do if the earth revolved in a circle at the 
 same mean distance, it is manifest that so long as the change in the 
 shape of the earth's orbit, of which we have just spoken, goes on, 
 the moon's periodic time must grow less and less. 
 
 Let ADBE (Fig. 5) represent an elipti- 
 cal orbit, S the attracting body, placed in 
 one of the foci, C the centre of the elipse, 
 and F the other focus. It can be demon- 
 strated, that the mean distance of S from all 
 points in the orbit, is equal to AC or CB. 
 Take any two points in the orbit G and H, 
 equidistant from B and A. We propose to prove that the average 
 attraction of S upon the moving body, when at these points, is 
 greater than it is when the body is at its mean distance. And 
 since these are any points in the arcs AD and DB, if we prove it 
 for them, we prove it for the whole orbit. 
 
 Join GS, GF and HS, and let SG bear any ratio, other than that 
 of equality, to GF; say 6 : 4. Then, since by the properties of the 
 elipse SG-f GF=AB=2CB, it follows that the ratio of SG to CB 
 is 6 : 5, and of GF, or its equal HS, to CB, 4 : 5. Therefore, since 
 the force of gravity is inversely proportioned to the square of the 
 distance, the attraction at the former point will be §£, and at the 
 
21 
 
 latter f- f of what it is at the mean distance. The average between 
 them is fff of the attraction at the mean distance, — exceeding it 
 
 Now the earth's orbit is much nearer circular than we have 
 supposed this to be, and the excess of attraction must be propor- 
 tionably less : but still there must be an excess, so long as it is 
 eliptical at all. Hence, as the earth's orbit becomes nearer circular, 
 the sun's attraction upon it, and consequently upon the moon, must 
 continually grow less, allowing the orbit of the latter to contract. 
 This would diminish the periodic time, and produce the very effect 
 that excited so much wonder in the mind of Dr. Halley, and the 
 astronomers of his time. 
 
 25. The tables for this work are based on the moon's motion, as 
 it existed in the year 1800, and we must, therefore, add to its lon- 
 gitude and anomaly the amount gained since that time, from the 
 cause just explained. Table 11th contains the required correction, 
 calculated at intervals of five years, during the present century. 
 Look for the year in the left hand column of the table, except the 
 unit figure, which is placed at the top, and opposite to the former, 
 and under the latter will be found the correction required, express- 
 ed in decimals of a degree, the first two places, which are ciphers, 
 being omitted. 
 
 The correction for the year 1854 is .0009, which, added to the 
 longitude and anomaly already found, (19,) makes the former 
 67°.5374, and the latter 144°.967. 
 
 CHAPTER V. 
 
 PERTURBATIONS IN THE MOOn's MOTION, CONTINUED. VARIATION. 
 
 26. The moon's motions grow more complicated the farther we 
 proceed. To investigate them thoroughly, is nothing less than a 
 solution of the famed Problem of the Three Bodies. The moon's 
 orbit, which we first regarded as a circle, and then an elipse, we 
 shall now find to be neither a circle nor an elipse,* but an irregular 
 
 * This statement seems to conflict with former ones, where the eliptical form of the 
 moon's orbit was asserted; but its mean shape was then intended, without taking into 
 account the irregularities. 
 
22 
 
 oval shaped figure, which is constantly changing its form. The 
 prospect before us, in trying to reduce such irregularities to order, 
 so as to see their precise influence on the moon's longitude, is suffi- 
 ciently appalling, but nevertheless, let us not be deterred from the 
 attempt. 
 
 To avoid misapprehension, it ought perhaps here to be remark- 
 ed, that these irregularities are not of such a nature as to set aside 
 our previous work, but only show that, under some circumstances, 
 they may occasion necessary corrections. 
 
 •27. And first let us, in this chapter, see what the shape of the 
 orbit would be, and how the moon would revolve in it, on the sup- 
 position that it was originally a circle round the earth, but drawn 
 out of shape by the sun's attraction. We shall in this way disco- 
 ver the cause of an observed inequality in the moon's motion, de- 
 nominated variation, and discovered by Tycho Brahe, A. D. 1590. 
 The modifications that the orbit would undergo, by supposing the 
 original figure an elipse instead of a circle, will occupy our atten- 
 tion in the next chapter. 
 
 28. Let S (Fig. 6) represent the sun, E the earth, and ADCB 
 the moon's orbit ; and let us suppose, for a moment, that the moon, 
 retains a circular orbit. Let D represent the place of the moon at 
 conjunction, C at apposition, and A and B when it is at the same 
 distance from the sun that the earth is, or very nearly in quadra- 
 ture. 
 
 First, let the moon be at A or B, in which case the moon and 
 earth being equally distant from the sun, must be equally attracted 
 by it, and consequently there would be no tendency to change 
 their direction from each other, but only to draw them nearer to- 
 gether, which would be precisely equivalent to increasing the earth's 
 power of gravity. 
 
 Next let it be in conjunction at D. Now the earth and moon 
 are both in the same direction from the sun ; but the moon being 
 nearest is more attracted, in the inverse ratio of the square of the 
 distance, i. e., SE 2 : SD 2 - The only effect, therefore, is to draw 
 the moon directly away from the earth, by virtue of the difference 
 in the attractive forces, which would be equivalent to diminishing 
 the earth's attraction. 
 
29. Again, — suppose the moon at any point M in the quadrant 
 AD. Being nearer to the sun than the earth is, it is more attract- 
 ed by it, and the effect is nearly* the same as though the earth was 
 not attracted at all, but the moon drawn along the line MS by a 
 force equal to the difference of the attractions. The direction of 
 this force, making an acute angle with that in which the moon 
 moves, must accellerate the motion in its orbit; and the same 
 would be true of every point in the quadrant AD. If the moon 
 were at M' any point in the quadrant DB, the difference of attrac- 
 tions acting along the line M'S, would tend to retard its motion. 
 
 Once more : let the moon be at any point M" in the quadrant 
 BC. Being further from the sun than the earth is, it is less attract- 
 ed by it, which is nearly as though it were drawn in the opposite 
 direction, along the line M"L. The effect would be to accelerate 
 the motion, in nearly the same manner as in the quadrant AD. If 
 the moon was at any point M'" in the quadrant CA, the difference 
 of attractions acting, as it were, along the line M'"N, would retard 
 its motion. 
 
 Thus the moon is alternately accelerated and retarded in the 
 differant quadrants ; moving swiftest in syzygy and slowest in 
 quadrature. Hence, from this cause alone, the moon would be in 
 advance of its mean place while passing from syzygy to quadrature, 
 and behind it while passing from quadrature to syzygy. 
 
 80. But the above is not the only reason. We have thus far, in 
 the present investigation, supposed the moon's orbit to retain its 
 circular form, notwithstanding the disturbing influence of the sun : 
 but this is not possible. To retain a body in a circular orbit, the 
 centripetal and centrifugal forces must be equal. But we have 
 just seen that the velocity in syzygy, as at C and D, (Fig. 7,) is 
 
 * Sufficiently near for our present purpose. The subject will be investigated more 
 critically hereafter. 
 
24 
 
 greater than at A and B : and as the centrifugal force is propor- 
 tioned to the square of the velocity, it must be greater. On the other 
 hand, it was shown (28) that the sun's attraction diminished the 
 moon's gravity toward the earth in syzygy, as at C and D, and in- 
 creased it in quadrature, as at A and B. Taking both these facts 
 into consideration, it is man. Fig. 7. 
 
 ifest that at A and B, the 
 centripetal force must con- 
 siderably exceed the centri- 
 fugal, while at C and D, the 
 centrifugal will be the great- 
 est, which would cause the 
 moon's track to fall within the circle at the former points, as to L 
 and N, and without it in the latter, as to F and G. 
 
 The effect would be to Fig. 8. 
 
 throw the orbit into some- 
 thing such a shape as is 
 represented in Fig. 8, viz: 
 a kind of oval, with its 
 longest diameter, AB, at 
 right angles to line ES, 
 drawn from the earth to 
 the sun. 
 
 Will this alteration in the the shape of the moon's orbit affect its 
 longitude ? To aid us in this investigation, we will circumscribe 
 the oval by a circle ; and to make the illustration more striking, 
 we will suppose the oval very much flattened, so as nearly to coin- 
 cide with AB, as in Fig. 9. Now, if the arc AF be divided in any 
 given ratio at the point L, and LE be Fi £- 9 - 
 
 drawn, it will cut the arc AD by no means 
 in the same ratio. AM will bear a much 
 greater ratio to MD than AL to LF. 
 Hence, if two bodies, whose periodic 
 times were equal, should start from A atf| 
 the same time, and move with uniform 
 velocity, one in the circle ALF, and the 
 other in the oval AMD, the former would 
 arrive at L before the latter would at M, 
 leaving it behind perhaps at N. The same reasoning may be ap- 
 plied to the other quadrants, though with the opposite effect in DB 
 
 
25 
 
 and CA, when it will show that the moon must be in advance of 
 its mean place. 
 
 31. There are two reasons, then, why the moon will be behind 
 its mean place when passing from quadrature to syzygy, but in 
 advance of it while passing from syzygy to quadrature ; 1st, from 
 its unequal motion, (29,) and 2d, from the shape of its orbit. The 
 maximum effect of the former to change the moon's place, is from 
 9' 17" to 10' 15", and the latter from 23' 56" to 20' 52", according 
 to the distance of the earth from the sun. When at its mean dis- 
 tance, the maximum effects are 9' 46" for the former, and 25' 24" 
 for the latter, amounting to 35' 10" for both united. * 
 
 32. If the four quadrants were perfectly symmetrical, a table 
 showing the correction required for each degree in one quadrant, 
 would answer for all the rest ; only the equation would be additive 
 when the moon is passing from syzygy to quadrature, i. e., in 
 the arcs DB or CA, and subtractive when it is passing from quad- 
 rature to syzygy, i. e., in the arcs AD and BC. But there is a 
 slight difference ; for, 1st, the disturbing influence is a little less in 
 the half of the orbit nearest the sun than in the other half, the dif- 
 ference of the squares of the distance of the earth and moon from 
 the sun, being a trifle less ; and 2d, the quadrants (so termed for 
 the sake of conciseness) nearest the sun contain a little less than 
 90° each, and the other two quadrants, each a little more than 90°, 
 for AB is not strictly a straight line, but an arc of the earth's or- 
 bit. A table for two quadrants would, however, be sufficient — one 
 in the half of the orbit next to the sun, and the other in the half 
 most remote from it, as, for example, DB and BC. Table 12th is 
 constructed in this way, where it will be seen that the equations 
 are additive for a little less than 90° after the moon leaves D, and 
 then subtractive to the end of the next quadrant. If the moon's 
 angular distance from the sun exceeds 180°, which would carry it 
 into the quadrants CA or AD, the degrees are found at the right 
 hand and bottom of the table, and direction is given to " reverse 
 the signs," so that the equations which were additive in DB become 
 subtractive in AD, and those which were subtractive in BC become 
 additive in CA. • 
 
26 
 
 33. The angular distance of the moon from the sun, (found by- 
 subtracting the longitude of the latter from that of the former, as 
 thus far corrected, borrowing 360° if necessary,) shows in which 
 quadrant the moon is. When the difference is from 0° to 90°, or 
 from 180° to 270°, the moon is passing from syzygy to quadrature, 
 but when it is from 90° to 180°, or from 270° to 360°, the moon is 
 passing from quadrature to syzygy. In the present case, the lon- 
 gitude of the sun (17) is 64°.3494, that of the moon (25) 67°.5374, 
 and the excess of the latter 2°. 1880. Entering table 12th with 
 this argument, in the same manner as directed in article 13th, the 
 equation is found to be +.0445. This is to be applied to the moon's 
 longitude and anomaly according to its sign. If the argument had 
 been over 180°, the sign of the equation would have to be changed 
 to — . After this equation is applied, the moon's longitude becomes 
 67°.5819, and the anomaly 145°.012. 
 
 34. The inequality to which this chapter is devoted, being occa- 
 sioned by the disturbing influence of the sun, must be more or less 
 according as the distance of that luminary varies, as we have al- 
 ready observed, (31.) In table 12th, and, consequently, in the 
 equation that was just applied, the sun is supposed to be at its mean 
 distance. Hence another correction becomes necessary, which 
 must evidently depend on the same circumstances as the last, to- 
 gether with another, viz., the distance of the sun from the earth, 
 which is determined by its anomaly. Accordingly in table 13th 
 two arguments are employed ; viz., 1st, the argument just used for 
 variation, which is to be sought for at the top or bottom of the 
 table, and 2d, the sun's anomaly at the right or left. If the former 
 is found at the top, we look for the latter at the left; but if at the 
 bottom, at the right. The equation is found opposite the latter, 
 and in the same column with the former. Since one argument is 
 given only for every 5° and the other for 10°, it is necessary to 
 institute a kind of double proportion for the units and decimals. It 
 is further to be noticed, that if both arguments are to be found in 
 the same gnomon, enclosed by the heavy lines about the table, the 
 equation is to be applied with its proper sign, as found in the table ; 
 but if one is found in the inner and the other in the outer gnomon, 
 the sign before the equation is to be changed from + to — , or 
 
27 
 
 from — to +. In the present case, the former argument (33) is 
 2°.1880, which being between 0° and 5°, is to be considered as 
 found in the inner gnomon at the top; and the latter (11) is 
 143°.797, which is found in the outer gnomon, at the left. Making 
 a proper allowance for the units and decimals, the equation is 
 +.0053 ; but the arguments being found, one in the inner and the 
 other in the outer gnomon, the sign must be changed, and the equa- 
 tion becomes — .0053. This applied to the moon's longitude and 
 anomaly, found in the last article, makes the former C7°.57G6, and 
 the latter 145°.007. 
 
 CHAPTER VI. 
 
 PERTURBATIONS IN THE MOON's MOTION, CONTINUED. EVECTION. 
 
 35. The inequality which is to occupy our attention in this chap- 
 ter was discovered by Ptolemy, A. D. 110, and is denominated 
 Evection. 
 
 For distinctness of conception, it is necessary to bear in mind 
 the precise difference between this correction and that treated of 
 in the last chapter, for there is danger of confounding them, since 
 both are caused by the disturbing force of the sun in the plane of 
 the ecliptic. That supposed the original form of the moon's orbit 
 a circle, this an elipse, and wholly dependent on its eccentricity ; 
 so that if the elipse had no eccentricity, there would be no correc- 
 tion for evection. That always elongated the orbit in the direction 
 of the quadratures ; this, we shall see, elongates it in the direction 
 of the syzygyes. That regarded the shape of the orbit as constant ; 
 this, as ever changing. An important element in this correction 
 is the irregular motion of the line of apsides ; that had no such line 
 to take into account. 
 
 36. It will be shown, that the progressive linjjpf the moon's ap- 
 sides is quite irregular ; that it sometimes progreSes more and some- 
 
 * 
 
times less rapidly ; sometimes remains stationary, and sometimes 
 even goes backward. Now in determining the moon's mean ano- 
 maly, (11,) all the motions were supposed uniform, and no correc- 
 tion has been made for any irregularity in the motion of the moon's 
 perigee, except that which resulted from the unequal distance of 
 the sun, (13.) But since the anomaly is reckoned from the peri- 
 gee, it must be subject to all the irregularities that the perigee itself 
 is. Hence, in applying the equation of the centre, (17,) we used 
 data that were erroneous, and the error that was introduced needs 
 to be corrected. 
 
 27. But this is not all. The greater the eccentricity of an orbit 
 is, the greater is the equation of the centre. Thus the equation of 
 the moon's centre is much greater than that of the sun with the 
 same anomaly, (compare tables 8th and 9th.) because the orbit of 
 the former is much the most eccentric. Now it will be shown 
 presently, that the eccentricity of the moon's orbit is ever varying, 
 and the equation of the centre, which depends upon it, must vary 
 likewise ; whereas table 9th is computed on the supposition of a 
 constant mean eccentricity. So that we not only made use of a 
 wrong anomaly in applying the equation of the centre, but also a 
 wrong eccentricity in the moon's orbit. The correction for the 
 combined effect of these two errors constitutes evection. 
 
 38. It will be demonstrated in its proper place, that if a revolving 
 body be retained in an eliptical orbit, by a force directed toward one 
 of the foci, the square of the distance of the body from that focus, 
 at any point in its orbit, must always be inversely proportioned to 
 the intensity of the attractive force at that point. Hence, if an in- 
 increase or diminution of attraction were to take place throughout 
 the orbit, proportional to the existing attractions at each point, 
 the size, but not the form of the orbit would be changed. The ec- 
 centricity would remain the same as before. But if the alteration 
 in the attractive force were in any other ratio, it obviously would 
 affect the shape of the orbit. If the perigeal gravity was made to 
 bear too great a ratio to the apogeal, it would be drawn in too 
 much at the former point, or too little at the latter, and the orbit 
 would become rrqgfc eccentric. Or if the apogeal gravity became 
 
29 
 
 too great in proportion to the perigeal, the orbit would be rendered 
 less eccentric, or more nearly circular. 
 
 39. To apply this to our subject, let E (Fig. 10) represent the 
 earth, ADBC the moon's orbit, A being the perigee and B the apo- 
 gee, and FGHI the sun's apparent orbit. First, let the sun be at S, 
 so that the line of apsides, AB, of the moon's orbit is directed to- 
 
 Fg. 10. 
 
 wards it, or lies in syzygy. The moon is more attracted by the 
 earth at A than it is at B, in the inverse ratio of AE 2 to EB* ; and 
 in order that the disturbing influence of the sun, which tends (28) 
 to diminish the earth's attraction at these points, should effect no 
 change in the shape of the moon's orbit, it must also (38) be more 
 at A than at B, in the same ratio. But instead of that, the sun's 
 disturbing influence is greater at B than tit A, for the difference 
 between SE 2 and SB 2 is greater than between SA 2 and SE 2 ; con- 
 sequently the relative difference in the attractive forces at A and 
 B toward E is increased, and the orbit must become more eccen- 
 
30 
 
 trie. In the same manner it may be shown that the eccentricity 
 of the moon's orbit must be increased when the sun is at S". 
 
 But if the sun were at S', and the moon at A or B, the latter 
 would be drawn toward the earth by the sun's disturbing influence, 
 and its gravity increased, (28 ;) but more at B than A in the ratio 
 of EB to E A, as will appear if we resolve the force in the direction 
 S'A into two others in the directions AE and ES', and that in the 
 direction S'B into two in the directions BE and ES'. In this case 
 the greatest addition is made to the least force, whereas to preserve 
 the shape of the orbit unchanged, the additional gravities should 
 be in proportion to the previously existing ones. The apogeai 
 gravity thus becomes too great in proportion to the perigeal, and 
 the eccentricity of the orbit is diminished, (38.) We shall arrive 
 at the same conclusion if the sun be supposed to be at S'". 
 
 Thus the eccentricity of the moon's orbit is greatest when the 
 line of its ipsides lies in syzygy, and least when it lies in quadrature. 
 It is plain that these changes in eccentricity occur, not instantane- 
 ously, but gradually, as the sun progresses in its orbit. The ec- 
 centricity must diminish while the sun is passing from S to S', or 
 from S" to S'", and increase while it is passing from S' to S", or 
 from S'" to S, being at its mean state when the sun is about half 
 way between these points, as at L, M, N and K. The eccentricity 
 must exceed the mean when it is in the quadrants KL, or MN, and 
 be less than the mean when it is in the quadrants LM and NK. 
 
 40. The investigation of the irregular motion of the line of the 
 apsides of the moon's orbit, on which the evection in part depends, 
 is considerably more difficult than any of the preceding, and will 
 require the reader's close attention. That it must, on the whole, 
 progress, will appear, when we consider that the average effect of 
 the sun's attraction is to draw the moon away from the earth, and 
 thus to render its orbit less curved than it would otherwise be. 
 Consequently, after the moon leaves its perigee, or apogee, where 
 its motion is at right angles with the radius vector, its angular mo- 
 tion round the earth must amount to more than 180° before its 
 path will have been deflected enough to intersect the radius vector 
 at right angles again. That is, it must be more than 180° from 
 
31 • 
 
 perigee to apogee, or from apogee to perigee.* And, further ; 
 since the attraction of the sun sometimes increases, and sometimes 
 diminishes the moon's gravity toward the earth, we should con- 
 clude the line of its apsides must sometimes regress and sometimes 
 progress. We must, however, go into a more minute investiga- 
 tion of this motion, to account for all the phenomena to which it 
 gives rise.f 
 
 41. If the moon, or any other body revolving in an eliptical orbit, 
 should be deflected from its natural course at any point by some dis- 
 turbing influence, so as to move at right angles to the radius vector, 
 the point where such deflection occurred would thenceforward become 
 one of the ' apsides of the orbit, provided it were not further dis- 
 turbed. 
 
 This is evident, from the fact that the apsides are the only points 
 in an eliptical orbit, where the curve is at right angles to the radi- 
 us vector. Also the body must still revolve in an elipse, or some 
 other conic section, for we shall demonstrate hereafter that simple 
 gravitation toward a fixed point can retain it in no other curve. 
 Whether the point of deflection will be the perigee or the apogee, 
 will depend on the velocity of the time ; if it be greater than the 
 mean, the point will be the perigee, but if less, the apogee. 
 
 42. To apply this principle, let us inquire what alteration must 
 be made in the attractive force of the earth, E, (Fig. 11.) to bring 
 the motion of the moon at right angles to the radius vector at the 
 points of its orbit C, D, F, and G, the two former being near the 
 perigee, A, where the velocity exceeds the mean, and the two lat- 
 ter near apogee, B, where it is less. While the moon is moving 
 from A, through S to B, the direction of its motion constantly makes 
 an obtuse angle with the radius vector, (as EFM ;) it would be 
 
 * Playfair. 
 
 t The articles between this and the 56th may be omitted, if the instructer should deem 
 it expedient. 
 
32 
 
 Fig. 11. 
 
 necessary, therefore, at the 
 point F, that the attractive 
 force of E should be in- 
 creased, to curve its mo- 
 tion to K, and thus bring 
 it at right angles with EF. 
 And, if it were so increas- 
 ed, the point F would (44) 
 henceforth become the ap- 
 ogee, instead of B. In 
 other words, the apogee 
 would have moved back- 
 ward from B to F ; and consequently the perigee from A to T, 
 for they must always be opposite each other. 
 
 The same reasoning will apply to the point D ; yet, if the deflect- 
 ing force should occur there, D would become the perigee instead 
 of the apogee, on account of the moon's greater velocity at that 
 point, (44,) and the apogee would be found in the direction of the 
 line DO ; so that the apsides would have moved forward from A 
 to D, and from B to O. In the other half of the moon's orbit, 
 where the direction of the motion continually makes an acute angle 
 (as EGR) with the radius vector EG or EC, it is manifest, that 
 the attraction of E must be diminished, in order to bring the mo- 
 tion at right angles, as GL and CH ; or, rather, a repulsive force 
 must be given to it. Such a deflection occurring at G, would 
 change the place of the apogee from B to G, or, would make it 
 move forward. If occurring at C, it would change the place of 
 perigee from A to C, or, would make it move backward. 
 
 43. If instead of increasing the gravity at F, it were diminished, 
 it is pretty clear that the apogee would move forward instead of 
 backward. To illustrate it, let us suppose the gravity to be great- 
 ly diminished, so much so as to be nearly destroyed. The moon, 
 being scarcely attracted at all toward E, will fly off nearly in a 
 tangent to the elipse at F, and the apogee will be found in that 
 direction, but infinitely distant. If then, we draw NP parallel to 
 the tangent FM, it will point to the place of the apogee, which has, 
 therefore, moved forward, equivalent to the arc from B to V. If 
 the attraction were less diminished, it would not move forward so 
 far, but the reasoning would hold good. 
 
33 
 
 Similar reasoning applied to the points D, C, and G, will show 
 that, by reversing the supposed alteration in the force of gravity 
 at those points, we shall reverse also the motion of the line of the 
 apsides. 
 
 Summing up our conclusions, we find that an increase of the 
 moon's natural gravity toward the earth, near apogee, on either 
 side, would cause the line of apsides to regress ; while a diminution 
 would cause it to progress ; and that the reverse takes place by an 
 alteration in the natural force of gravity when the moon is near its 
 perigee. I employ the terms natural gravity, and natural velocity, 
 to signify the gravity and velocity that the moon would have if it 
 revolved regularly in its eliptic orbit, undisturbed by the attraction 
 of any foreign body. 
 
 44. It is evident that an increase in the moon's velocity, and 
 consequently of its centrifugal force, must have nearly the same 
 effect as a diminution of the earth's attraction ; and vice versa. 
 Hence, if its velocity near apogee is, from any cause, rendered 
 greater than its natural velocity in that part of its orbit, the line of 
 apsides must move forward, or progress ; and the reverse, if such 
 an increase of velocity occurs near perigee. If both these causes 
 conspire, (and we will proceed to show that they do,) the progress 
 or regress of the apsides must be still more rapid. 
 
 45. It has been shown, (28,) that the sun's disturbing influence 
 increased the moon's gravity toward the earth in quadrature, but 
 diminished it in syzygy ; and we should suppose that there must 
 be intervening points, where it exerted no influence either way. 
 These points it is important for us to find, for when the moon is 
 at these, the apsides of its orbit must be at rest, so far as their mo- 
 tion is caused by a variation in gravity. 
 
 46. Let the moon be at any point M (Fig. 12) of its orbit, and 
 let the sun's attraction on it at that point be represented by m. 
 Resolving this force into two others in the directions ME and ES, 
 the proportion for the former (called the addititious force, because 
 it increases the moon's gravity toward the earth) will read SM : 
 ME ::m: the addititious force =gM m= slfcfME m - 
 
 3 
 
47. The proportion for the latter force, in the direction ES, will 
 read SM : ES : : ?n : the force required =g^-?w. But the earth is at- 
 tracted by the sun in the same direction, ES, and it is the differ- 
 ence of the attractions only that exerts any disturbing influence on 
 the moon in this direction. We will therefore find how much the 
 
 attraction of the sun on 
 the earth is, and subtract 
 it from that just found, 
 viz., — m. By the laws 
 of gravity ES 2 : SM 2 : : 
 m : the sun's attraction at 
 the distance ES. Hence 
 the earth is attracted 
 with a force equal to 
 ^m, which is to be sub- 
 tracted from ^77i. Re- 
 ducing the fractions to a 
 common denominator, 
 and subtracting, we have 
 Now SM= 
 very nearly; 
 
 ES 3 — SM 3 
 
 -m. 
 
 ES3+SM 
 
 ES— EF, 
 
 therefore, by involving 
 both sides, and rejecting 
 the 3d and 4th terms in 
 the right hand member 
 on account of their small- 
 ness, we have SM 3 = 
 ES 3 — 3ES 2 xEF. Sub- 
 stituting this value in the 
 place of SM 3 ' the above 
 fraction, which expresses 
 the disturbing influence 
 of the sun in the direction 
 ES, becomes *™l$m m = 
 fgm. Resolving this 
 force into two others, 
 one in the direction EM, 
 and the other at right 
 angtes to it ; i. e., in the 
 
 directions EG and SG, the proportion for the former (called the 
 ablatitious force, because it diminishes the moon's gravity toward 
 
the earth) will read SE : EG, or (since the triangles EGS and EFM 
 are similar) ME : EF : : ^m : the ablatitious force=^~^m. 
 
 48. The addititious and ablatitious forces acting in direct oppo- 
 sition, must neutralize each other at the points in the orbit where 
 they are equal, showing that, at such points, the sun's attraction 
 produces no effect on the gravity of the moon toward the e«th. 
 In that part of the orbit that lies between these points and quadra- 
 ture,.there wilL.be an increase of gravity, and between those points 
 and syzygy, at Cor D, (for the demonstration will apply to. either 
 half of the orbit ACB or ADB,) a diminution. 
 
 49. But since the fractions representing these forces have a com- 
 mon denominator, they will evidently be equal when their numera- 
 tors are equal, i. e., when 3EP=ME 2 ; or (extracting the square 
 root) when v/3xEF=ME; or (converting the equation into a 
 proportion) when ME : EF : : V3 : 1. But ME : EF : : 1 : cos. 
 MES ; therefore, by equality of ratios, v/3 : 1 : : 1 : cos. MES= 
 .5773672, which is the cosine of 54° 43' 56". Hence the gravity 
 of the moon toward the earth is diminished when it is within 54° 
 43' 56" of syzygy, and increased when it is within 35° 16' 4" -of 
 quadrature. 
 
 50. It is worthy of notice here, that the diminution of gravity in 
 syzygy is about double the increase in quadrature. The above 
 reasoning shows, that the ablatitious force is to the addititious as 
 3EF 2 : ME 2 - But in syzygy EF=ME, and 3EF=3ME 2 ; so that 
 the difference between them is 2ME 2 ; while, in quadrature, EF 
 becomes 0, and the ablatitious force disappears, leaving the additi- 
 tious force proportional to ME 2, which is half of 2ME 2 
 
 51. There remains, not yet investigated, the tangential force in 
 the direction GS, or MH, one of the parts into which we resolved 
 (47) that in the direction ES. Its precise amount it is not now ma- 
 terial for us to know : but it is to be observed, that its only influ- 
 ence is to retard the moon's motion from D to B ; since, being at 
 right angles to EM, it neither increases nor diminishes the moon's 
 gravity toward the earth. If the moon was supposed to be in any 
 of the other quadrants, and figures constructed on the same princi- 
 ple as this, we should see that the moon must be retarded in pass- 
 
36 
 
 ing from D to B, and from C to A ; but accelerated from A to D, 
 and from B to C. Hence its motion must be swiftest in syzygy at 
 C and D, slowest in quadrature at A and B, and a mean half way 
 between syzygy and quadrature. This is the same conclusion to 
 which we arrived by a less rigid process, in article 29th. 
 
 5% To show how the motion of the line of apsides is affected by 
 this perturbation in the moon's gravity and velocity, let us recur 
 again to Fig. 10th. Let the sun be at S, or S", so that the line of 
 apsides, AB, lies in syzygy. In this case, both the moon's velocity 
 in its orbit will be increased, and its gravity towards the earth di- 
 minished (48) at A and B. Consequently, (43 and 44,) the line of 
 apsides must move forward when the moon is near apogee, at B ; 
 but backward, when it is near perigee, at A ; and if the regress 
 near perigee is equal to the progress near apogee, they will balance 
 each other, so as, on the whole, to produce no change in the posi- 
 tion of the line of apsides. But they are not equal, for several rea- 
 sons. 
 
 1st. The diminution of the moon's gravity at these points, and 
 the increase of velocity, are both caused by the force in the direc- 
 tion ES, (Fig. 12,) which was obtained in article 47, by taking the 
 the difference of the attractions of the sun upon the earth and 
 moon, in that direction. Consequently, they must depend on the 
 difference between the distances of the moon and earth from the 
 sun ; and this difference is greater when the moon is in apogee, than 
 when it is in perigee. 
 
 2d. The forces causing the line of apsides to progress, act for a 
 longer time than those causing it to regress, because the moon is 
 longer in describing the apogeal than the perigeal half of its orbit. 
 
 3d. A given force would produce more effect on the moon when 
 it is in apogee than when it is in perigee, on account of its natural 
 velocity being less at the former point, and therefore more easily 
 deflected from its orbit. If a cannon ball were moving but one 
 foot in a second, it would not be very difficult to turn it out of its 
 course, but not so if it were moving 1000 feet per second. 
 
 From all these circumstances combined, the line of apsides pro- 
 gresses quite rapidly when the sun is at S, or S". And, since the 
 same causes operate in the same way, one (51) while the sun is 
 passing through the arcs KL and MN, extending 45° each way 
 from S and S", the direction of the line of apsides of the moon's 
 
57 
 
 orbit, and the other (49) through FG and HI, extending 54° 43' 56" 
 from the same points, it follows that the line of apsides must pro- 
 gress, at least, while the sun is in the quadrants KL and MN. 
 
 53. When the sun is in either of the small arcs FK, LG, HM or 
 NI, containing 9° 43' 56" each, the line of apsides is nearly station- 
 ary : for the perturbations, both in gravity and velocity, being 
 near their limits are very weak, and what small force they do ex- 
 ert is in opposition to each other, the former tending to make the 
 apsides progress, and the latter, regress. 
 
 54. Now let us suppose the sun at S' or S'", so that the line of 
 apsides shall be at right angles to ES', or shall be in quadrature. 
 The moon's gravity toward the earth, when at A and B, will now 
 be increased, (48,) and its velocity diminished, (51.) Consequently, 
 (43 and 44,) the line of apsides must progress when the moon is 
 near perigee, but regress when it is near apogee. And, by nearly 
 the same reasoning as employed above, (52,) it may be shown, that 
 the regress exceeds the progress ; so that, on the whole, the line 
 of apsides regresses. In like manner, it may be shown, that it re- 
 gresses, though less rapidly, when the sun is any where in the arcs 
 IF or GH. 
 
 55. The regress here will, however, be less rapid than the pro- 
 gress when the sun is in the arcs KL and MN, for the perturba- 
 tion in the moon's gravity is (50) but half as great. It will also 
 be of shorter duration, for the arcs IF and GH contain but 70° 32' 
 8" each, and the sun moving forward in its orbit about 1° per day, 
 while the line of apsides moves backward, on an average, about 
 1-6 of a degree per day, each arc will be passed over in about 61 
 days; when 1 he line will become nearly stationary for about 10 
 days, (the time occupied in passing one of the small arcs, as FK,) 
 and then begin to advance.* 
 
 * It may seem to the reader erronious, to ascribe any part of the progress of the moon's 
 apsides to the perturbation in velocity ; for, since that is equal in quadrature and syzygy, 
 and extends the same distance, 45°, from each, it would seem that, so far as this cause 
 is concerned, the regress, when the apsides are near quadrature, should be just equal to 
 the progress when they are near syzygy. Sir Isaac Newton took this view of the subject, 
 and was greatly perplexed at finding that he could account for but about half the motion 
 of the line of the apsides. The explanation here given, is, in substance, that of Clairaut, 
 who showed that, when the apsides regressed, they approached to meet the sun, thus 
 shortening the regressive arc, and, consequently, diminishing the perturbation in velocity; 
 but, when they progressed, they receded from the sun, lengthening the progressive arc, 
 and thereby increasing the perturbation in velocity. So that the perturbation would not 
 only be greater in the latter case than in the former, but extend through a greater arc. 
 
38 
 
 Since the line of apsides remains nearly stationary about 40 
 days in the year, moves backward about 122, and forward during 
 the remainder, amounting to about 203 days ; and since its forward 
 motion is more rapid than its backward, it is evident that, in the 
 'course of a year, it must, on the whole, advance. The rate of ad- 
 vance is found by observation, to be such as to carry it entirely 
 round the orbit in 3232 days, IS hours, |P minutes, and 29.4 se- 
 conds, or about nine years. 
 
 The foregoing explanations have, I trust, made the theory of the 
 variation in the eccentricity of the moon's orbit, and the irregular 
 motion of the line of its apsides tolerably clear to the reader. It 
 remains to make a practical application of it. 
 
 56. We have seen (39, 52 and 54) that the eccentricity of the 
 moon's orbit exceeds the mean, when the sun is in those quadrants 
 where it causes the line of the moon's apsides to advance, and is 
 less than the mean when the sun is in the other parts of its orbit : 
 also, that these changes occurred alternately, and nearly in alter- 
 nate quadrants, the lines of division being not far from half way 
 between syzygy and quadrature. We will eneavour to represent 
 these changes by a figure, the construction of which was devised 
 by Sir Isaac Newton for the purpose, and which observation shows 
 to be very nearly correct. 
 
 Let AaBb (Fig. 13) represent the moon's eliptical orbit, in its 
 mean state, M the moon, E the earth, placed in one of the foci, and 
 EC the mean eccentricity. Now, as the eccentricity varies, the 
 centre C will sometimes approach toward E, as far as K, and 
 sometimes recede from it to I ; the distances CK and CI being the 
 greatest variation of the eccentricity from the mean. Describe 
 the circle IDKH, and join MC and ME. Let ES represent the 
 direction of the sun, and B the moon's mean perigee, i.e., the place 
 of the perigee if it progressed uniformly. Now, since it has been 
 shown (39) that when ES is at right angles to AB, the eccentricity 
 is least, and when it coincides with it, the greatest, it is plain that 
 it must be represented by EK in the former case, and EI in the 
 latter. And when ES is in any other position, the eccentricity 
 must be represented by a line longer than EK, and shorter than 
 EI. We shall effect this for every possible position of ES, by al- 
 ways making the angular distance of H from I, in the direction 
 IDKH, equal to twice the angle BES, The length of EH will be 
 
39 
 
 equal to the eccentricity at the time, very nearly. For example, 
 when BES=45°, H will coincide with D, making EH very little 
 longer than EC, and thus showing that the eccentricity exceeds 
 the mean in the same small ratio. Again, if BES=90°, H would 
 coincide with K, showing that the eccentricity was now a mini- 
 mum, which agrees with what we have already seen to be true. 
 In the same manner it may be shown, that at any other point, this 
 construction brings out very nearly the result it should do accord- 
 ing to our previous reasoning. 
 
 Fig. 13. 
 
 Not only will the length of EH represent the eccentricity of the 
 moon's orbit at all times, but its position will represent that of the 
 line of apsides very nearly, never varying from it more than 3'. 
 Hence, the point H may always be considered as the centre of the 
 eclipse ; thus changing the whole eclipse from the mean position 
 AaBb, to that represented by the dotted curve WXYZ. 
 
 57. It was remarked, (17.) that if two lines were drawn from the 
 mean place of the moon, one to the centre of the elipse, and the 
 other to the focus, round which it revolved, the angle at the moon, 
 
40 , 
 
 contained by these lines, would be equal to half the equation of the 
 centre, very nearly. If the eccentricity and position of the elipse 
 remained unchanged, the angle in question would be EMC ; but 
 when, in consequence of the change, H becomes the centre of the 
 elipse, the angle becomes EMH. Therefore HMC, which is the 
 difference between these two angles, must be half the effect of the 
 disturbing force of the sun ; or, in other words, it is half the evec- 
 tion we have been so long in quest of. Hence, if we can find out 
 a method of determining the size of this angle, or the conditions on 
 which its size depends, our task is over, for by doubling it we shall 
 have the correction required. 
 
 58. Draw Hs at right angles to MC. Then, since small angles 
 are nearly proportional to their sines, the line H? must always be 
 nearly proportional to the angle HMS, and consequently to the 
 evection. But Hs is also the sine of HCs, or its supplement HCR ; 
 therefore the evection is always proportional to the sine of HCR. 
 This angle we will proceed to find. The angle SEB=MEB— 
 MES ; therefore the angular distance of H from I, in the direction 
 IDK, (being, by construction, double of SEB,) =2MEB— 2MES. 
 The angles MEB and MCB are nearly equal,* the eccentricity of 
 the orbit being small ; therefore, subtracting MCB, or its equal 
 ICR, from the first member of our equation, and MEB from the 
 last, we have HCR=MEB— 2MES. Now MEB is the moon's 
 mean anomaly, and MES is the angular distance between the sun 
 and moon, or the excess of the mean longitude of the moon over 
 the true longitude of the sun. Hence the evection, which has 
 been shown to be proportional to the sine of HCR, is proportional 
 to the sine of the moon's mean anomaly diminished by twice the 
 excess of its mean longitude over the true longitude of the sun.f 
 
 * There is danger that the proportions of the different lines, as they appear in the figure, 
 may mislead the reader, and it is well to remember, that EC is but about 1-20, and KC 
 about 1-100 of CB. 
 
 t In this demonstration, the moon's longitude is supposed to exceed that of the sun. 
 But we shall arrive at the same conclusion if we suppose the sun's longitude the greatest ; 
 as, for example, if it be in the direction ES'. For, now, S'EB=MEB-f MES' ; and, 
 consequently, by the same construction and reasoning as in the other case, HCR=MEB 
 -f-2MES\ But the result is obviously the same, whether we add the angle MES', or 
 subtract its supplement ; that is, the angular distance of M from S', reckoned in the other 
 direction, S'AaBM. Now, this supplementary distance is the excess of the moon's lon- 
 gitude over that of the sun, borrowing 360°, or one revolution ; therefore, the angle HCR 
 is still equal to the moon's mean anomaly, diminished by twice the excess of its mean 
 longitude over the true longitude of the sun ; and the principle becomes general in its 
 application. 
 
41 
 
 If the reader will now turn to table 14th, he will notice that this is 
 the argument by which the evection is taken out in that table. 
 
 59. At the time of our predicted eclipse, the quantities are as 
 follows : — 
 
 The moon's mean anomaly is, (13,) - - ■ 145°.0780* 
 The moon's mean longitude is, (11,) 64°.2329 
 
 Subtract sun's corrected longitude, (17,) 65 .3494 
 
 358 .8835x2=357 .7070 
 
 Argument of evection, 147.3110 
 
 60. Entering table 14th with this number as an argument, in the 
 same manner as heretofore, the required correction is found to be 
 .7156, which the sign — , placed at the head of the left hand col- 
 umn, in which the argument is in this case found, shows to be 
 subtractive. It is plain also, from the figure, that it should be sub- 
 tractive; for, in adding the equation of the centre, (17,) we added 
 twice the whole angle EMC, which was too much by twice the 
 angle HMC. We now correct the error, by subtracting the equa- 
 tion just found from the longitude and anomaly previously obtained, 
 (34,) which leaves for the former 66°.8610, and for the latter, 
 144°.291. 
 
 61. All the remarks that were made in article 34 on the subject 
 of variation, will apply also to evection, since both are caused by 
 the sun's disturbing influence. The method of taking the requisite 
 correction from the table (table 15) is also the same, only that in 
 this case, the argument for evection, viz., the moon's mean anomaly 
 diminished by twice the excess of the moon's longitude over that 
 of the sun, is to be sought for at the top or bottom of the table, in- 
 stead of the argument for variation. 
 
 The correction, as found in the table, is — .0100, but since one 
 argument is found in the inner gnomon, at the bottom, and the 
 other in the outer one, at the right, the sign is to be changed, (34,) 
 and the correction becomes -f .0100, which, added to the longitude 
 and anomaly last found, makes them respectively 67°.8710, and 
 144°.301. 
 
 * This is the moon's mean anomaly, corrected by the annual equation of its perigee, 
 which it is proper to do, because that inequality affects only the average progressive mo- 
 tion of the perigee at different seasons of the year, and is in no way connected with that 
 which we are now considering, or any other which has reference to the position of the 
 moon in its orbit 
 
42 
 
 CHAPTER VIL 
 
 NODAL EQUATION OF THE MOON's LONGITUDE, AND REDUCTION TO THE 
 
 ECLIPTIC. 
 
 62. In the numerous corrections that we have had occasion to 
 apply to the moon's longitude and anomaly, growing out of the dis- 
 turbing influence of the sun, the orbits of both have been supposed 
 to lie in the same plane ; or the latter to lie in the plane of the or- 
 bit of the former. The first of these suppositions is never true, 
 and the latter only twice in a year ; viz., when the sun passes the 
 moon's nodes. At all other times, it is either on one side of the 
 plane of the moon's orbit or the other. Now it is evident that the 
 sun's disturbing influence, in the various ways we have been speak- 
 ing of, must be less than if it lay in the plane of the moon's orbit ; 
 for* in order to make our reasoning good, its attraction must be 
 resolved into two forces, one lying in the plane of the orbit, and 
 the other at right angles to it, which necessarily creates a loss- of 
 force. If it were always at a fixed mean distance from the plane, 
 a proper allowance might be made in computing the inequalities, 
 and the work would thus be accurate without further correction. 
 In fact, the quantities in the tables which we have been using, were 
 calculated on that supposition. But since the distance is variable, 
 additional corrections are necessary for all that we have applied 
 in the three preceding chapters. We will select, as an example, 
 the annual equation of the moon's longitude, discussed in chapter 
 4th, remarking, as we pass, that if we were to attempt to apply all 
 the corrections resulting from causes like that under consideration, 
 and from the effect of one correction in altering the argument from 
 which others had been obtained, our task would be endless. The 
 business is, at best, only a series of approximations. 
 
 63. When the sun is passing one of the moon's nodes, being in 
 the plane of the orbit, its attractive force exceeds the mean, so far 
 as the circumstance now under consideration is concerned, dilating 
 the moon's orbit (19) and increasing the periodic time more than 
 usual. The moon must therefore fall behind its mean place, and 
 continue to do so more and more, till the sun reaches its point of 
 mean distance, about 45° from the node. As the sun continues to 
 recede from the plane of the moon's orbit, its disturbing influence 
 
43 
 
 must grow less* allowing the moon to contract its orbit and shorten 
 its periodic time, till finally, when the former is 90°* from the node, 
 the latter will have gained up what it had lost, so that its mean and 
 true place will again coincide. 
 
 The reverse of all this will take place when the sun is in the 
 next quadrant. Its disturbing influence being a minimum at the 
 outset, the moon must get ahead of its mean place ; and -it wtfl not 
 lose what it thus gains till the sun reaches the next node. Hence, 
 if the sun's longitude exceeds that of one of the moon's nodes by- 
 less than 90°, something must be subtracted from the longitude and 
 anomaly, as already obtained ; but added, if the excess is greater 
 than 90°. Or, reckoning from the ascending node, a subtractive 
 equation must be applied in the 1st and 3d quadrants, and an addi- 
 tive one in the 2d and 4th. Such an equation is termed the Nodal 
 Equation of the moon's longitude.* 
 
 64. To find how far the sun is from the node, the longitude of 
 the latter (13) must be subtracted from that of the former, (17.) 
 Entering table 16th with the argument thus found, viz., 4°.3486, in 
 the same manner as we did table 6th and others, we find the equa- 
 tion to be .0026, which the sign — at the head of the column con- 
 taining the argument, as well as our previous reasoning, shows 
 must be subtractive. The resulting longitude of the moon becomes 
 66°.8684, and the anomaly 144°.298. , 
 
 65. The moon's anomaly is now altered considerably, by reason 
 of the various equations that have been applied to it, from what it 
 was when we used it to take out the equation of the centre, in arti- 
 cle 17th ; and since this equation is a very important one, our work 
 will be more accurate if we now take it out again, and by what- 
 ever amount it differs from what it was as first taken out, correct 
 the moon's longitude. The anomaly, as used in article 17th, was 
 145°.078, which gave as an equation +3°.4154, while now it is 
 but 144°.298, which gives for the equation -f-3°.4834, so that we 
 did not add enough to the moon's longitade by 0°.0680. Adding 
 this now, we have 66°.0364, which may be regarded as the true 
 longitude of the moon, reckoned on its orbit, or, as it is usually 
 termed, the true Orbit Longitude. 
 
 * I find no name for this equation in any treatise on astronomy that I have met with, 
 and have given it one that seems to be indicative of its character. 
 
44 
 
 66. The plane of the lunar orbit being inclined to that of the 
 ecliptic, causes longitudes reckoned on it to be different from what 
 they would be if reckoned on the ecliptic. And since the longitudes 
 of the heavenly bodies are referred to the latter, the orbit longitude 
 just found needs one more correction to reduce it to the ecliptic. The 
 argument, found by subtracting the longitude of the moon's node 
 from that of the moon itself, is 5°. 9356, and the corresponding 
 equation, obtained from table 17th, is — .0233. This applied, 
 leaves 66°.9131 for the moon's true longitude from the mean vernal 
 equinox. 
 
 CHAPTER VIII. 
 
 LUNAR, OR MENSTRUAL EQUATION OF THE SUn's LONGITUDE AND NUTA- 
 TION. 
 
 67. It was observed in article 2d, that any motion or change of 
 motion in the earth, produced apparently a precisely similar one 
 in the sun. Now, the earth, like the moon, revolves round the 
 common centre of gravity of the two, and is, therefore, subject to 
 inequalities in this motion, the same in kind as those we have been 
 considering in that of the moon, though far less in degree, owing 
 to the earth's greater weight, and consequently close proximity to 
 the centre of gravity. These inequalities, small in themselves, are 
 rendered vastly smaller in their effect upon the sun's apparent mo- 
 tion, by reason of the great distance of the latter. 
 
 Fig. 14. 
 
 Let S (Fig. 14) represent the sun, ABF the earth, E its centre, 
 M the moon, and C the common centre of gravity between the 
 earth and moon, about which both revolve. The distance from E 
 
45 
 
 to C is not far from 2970 miles, or about three-fourths of the earth's 
 radius. 
 
 It is manifest that the longitude of the sun, as seen from E, will 
 differ from its longitude as seen from C, by the angle CSE. When 
 the angle MES is either 0° or 180°, the angle CSE will disappear, 
 and when it is of any other size, the latter angle can be calculated ; 
 for, in the triangle CES, the two sides, CS and CE, and the angle 
 CES are known. We assume here, that E revolves in a circle 
 round C, keeping CE of uniform length. It is plain from the dia- 
 gram, that if the longitude of the moon exceeds that of the sun, the 
 latter will be increased by the angle CSE ; but the contrary, if 
 the longitude of the sun is greatest. In other words, if the longi- 
 tude of the moon, diminished by that of the sun, is less than 180°, 
 the equation will be additive, but if greater, subtractive. 
 
 » 
 
 68. The longitude of the sun, as found in chapter 3d, is 65°.3494, 
 and that of the moon, as finally corrected, (67,) 66°.9131. The 
 excess of the latter above the former is 1°.5637. Entering table 
 18th with this number, in the usual way, we find that, in the pres- 
 ent case, the correction is inappreciable, unless we extend our de- 
 cimals further ; so that the longitude obtained in chapter 3d is to 
 be considered correct. 
 
 69. The error occasioned by regarding the orbit of the earth's 
 centre as a circle instead x>f an elipse, as it in fact is, might be cor- 
 rected by introducing an equation corresponding to the equation of 
 the centre of the sun or moon. But it would never be necessary, 
 unless extreme accuracy were required ; for the whole correction 
 which we just undertook to apply, when a maximum, is but the 
 decimals of a degree, .0021 ; and since the eccentricity of the or- 
 bit amounts to hardly more than 1-20 of the radius, the error can 
 never be more than about .0001, which is less than half a second. 
 Much less then must the inequalities in the motion be appreciable. 
 
46 
 CHAPTER IX. ■ 
 
 NUTATION IN LONGITUDE. 
 
 .70. The equinoxes are not stationary, but move slowly, west- 
 ward, which necessarily affects the longitudes of all the heavenly 
 bodies, since they are reckoned from the vernal 'equinox. If the 
 rate of motion were uniform, the longitudes of the sun, moon, and 
 moon's nodes, which we have obtained in the preceding chapters, 
 would nevertheless be correct ; for the tables from which we ob- 
 tained the mean longitudes in article 11th, are based upon the sup- 
 position of a uniform rate of precession of the equinoxes, and allow- 
 ance is consequently made for it. But it is not uniform, and we 
 are now to look into the causes of the inequality, and make the 
 requisite correction in the longitudes on account of it. 
 
 71. If. a body were to revolve round the earth, in an orbit not 
 coinciding with the ecliptic, so that it would be sometimes. north and 
 sometimes south of the plane of the latter, we can see that whenever it 
 were thus situated, the sun's attraction must tend to draw it back into 
 the aforesaid plane. To illustrate by a diagram, let SD (Fig. 1 5) rep- 
 resent the plane of the ecliptic, and MM' that of the revolving 
 body, both seen edgewise,* S the sun, and E the earth. Fig. 15. 
 When the body is at M the sun's attration on it, in the di- 
 rection SM, may be resolved into t^o other forces, in the 
 directions SC and CM, the latter of which tends to draw 
 the body directly into the plane SD. In like manner, when 
 the body is at M', it is drawn toward the plane SD, by a 
 force represented by M'D ; and so of any other point out 
 of the plane of the ecliptic. The consequence is, that the« 
 body, as it revolves round its orbit, which we will suppose 
 it to do in an easterly direction, is drawn into the plane of 
 the ecliptic, and made to cross it sooner, that is, farther 
 westward, every succeeding revolution.* The same would 
 be true of any number of bodies similarly situated, so that 
 our reasoning will be good, even if they were multiplied to such a 
 
 * The reader will perceive, that the condition of our supposed body corresponds, in 
 every respect, with that of the moon revolving round the earth, and hence will see the 
 cause of the retrograde motion of the moon's nodes. 
 
47 
 
 degree as to form a continuous ring entirely round the orbit. Nor 
 will it alter the principle, if we suppose the bodies, or ring, very 
 near to the earth, or even attached to it, only that, in < the latter 
 case, they would communicate their * motion to the earth, and so 
 could not move without dragging the earth with them, which 
 would greatly deaden any motion, or .change of motion they would 
 otherwise have. 
 
 Now, from the spheroidal form of the earth, there is a protruber- 
 ant mass of matter about eighteen miles thick girdling its equator, 
 every particle of which is situated precisely as we supposed our 
 imaginary bodies or ring to be. The effect we have described 
 must therefore follow, and every place — as Quito, for example — 
 must every day, by the diurnal motion of the earth, cross the eclip- 
 tic at a point farther west than on the day previous. Or,, to illus- 
 trate farther ; suppose the plane of the ecliptic to be a vast sheet 
 of some material substance, with an orifice of sufficient size to ad- 
 mit the earth, and to allow it to revolve freely on its axis. Now, 
 if a man were stationed on Mount Chimborazo, (which we will 
 suppose to be on the equator, though it is not precisely so,) and 
 every time the earth rolled round, so as to carry him under the 
 plane, which would be every twelve hours, should mark on the 
 edge the place under which he passed it, these marks would be 
 continually farther and farther west by about fifteen feet. 
 
 72. The attraction of the moon also conspires with that of the 
 sun in causing a precession of the equinoxes ; for the plane of its 
 orbit being much more nearly coincident with the ecliptic than that 
 of the equator is, it may be regarded as another body lying in the 
 plane of the ecliptic, and conspiring with the sun in its influence 
 upon the earth. 
 
 73. It is evident from Fig. 15th, that the greater the inclination 
 of the planes SD and MM', the greater must be the forces repre- 
 sented by CM and DM', and consequently the more rapid must be 
 the retrograde motion of the points of intersection. So far, there- 
 fore, as the moon's influence is concerned, the greater the obliquity 
 of its orbit to the equator, the more rapid must be the precession 
 of the equinoxes. 
 
48 
 
 Let CD (Fig. 16) represent 
 a portion of the ecliptic, seen 
 edgewise, V the vernal equi- 
 nox, and EF and GH portions 
 of the moon's orbit, making an ^_ 
 angle of about 5° with the eclip- 
 tic ; GH representing it when 
 the ascending node is at V, and 
 EF when the ascending node 
 is there. Also, let AB represent a portion of the equator, making 
 an angle of about 23*° with the ecliptic. Then HVB, the inclina- 
 tion of the moon's orbit to the equator, when the ascending node is 
 at V, equals about 28|° ; and FVB, the inclination when the as- 
 cending node is there, equals about 18£°. 
 
 74. When, therefore, the ascending node, in its retrograde course, 
 passes the vernal equinox, which it does once in about nineteen 
 years, the rate of precession must considerably exceed the mean, 
 and the equinoxes must immediately get too far west, which would 
 increase the longitude af all the heavenly bodies. The same would 
 be true all the while that the node was slowly working its way 
 backward round to the autumnal equinox ; for though the rate of 
 precession would continually diminish, and become a mean when 
 the node was 90° back, or west of the vernal equinox, yet it would 
 take the whole of the next quadrant for it to lose what it had gain- 
 ed in the first. Thus, when the ascending node gets round to the 
 autumnal equinox, which would bring the descending node to the 
 vernal, all the longitudes would become right, or in their mean 
 state again. But the rate of precession is now a minimum, and on 
 principles similar to those we have been discussing, it is apparent, 
 that, while the node is passing through the other half of its orbit, 
 the longitude of all the heavenly bodies must be less than the mean. 
 Thus, for a period of about nine and a half years, all longitudes are 
 greater than the mean, and then, for the same period, less ; and so 
 on, alternately. This is called Lunar Nutation in Longitude. 
 
 75. It is plain, that when the ascending node is passing from the 
 vernal back to the autumnal equinox, its longitude must exceed 
 180°, and be less than 180° when it is in the other half of its orbit; 
 so that we can know by the longitude of the node, whether to add 
 to or subtract from our mean longitudes. 
 
49 
 
 76. At the time for which we are calculating, the mean longi- 
 tude of the ascending node (11) is 61°0883 ; and entering table 
 19th with this argument, we find the amount to be subtracted from 
 the longitudes of all the heavenly bodies at that time, is .0042, 
 which will leave us for the moon's longitude (G6) 6G°.9089 ; for the 
 sun's (18) G5°.3452, and for the moon's node (13) 60°.9966. 
 
 77. There is another inequality in the rate of the Precession of 
 the Equinoxes, called Solar Nutation, and occasioned by the vari- 
 able distance of the sun from the plane of the equator in the course 
 of a year. But it is so small (never amounting to much over one 
 second) that it may be disregarded without material error. 
 
 CHAPTER X. 
 
 TRUE TIME LONGITUDES AXD ANOMALIES. 
 
 78. By comparing the true longitudes of the sun and moon, found 
 in the last chapter, we find that the latter is greatest by 1°.5637, 
 which shows that the moon has passed by the sun, and that the 
 eclipse is over. It remains (12) for us to subtract such an amount 
 from the mean time of new moon (11) as it must have taken the 
 moon to gain this difference, and in order to do so, we must know 
 the relative velocities of the sun and moon in their orbits at the 
 time. 
 
 Their motions are swiftest in perigee, and grow slower as they 
 recede from it ; hence their anomalies are the proper arguments for 
 determining their motions. We may therefore enter tables 20th 
 and 21st, with the anomalies of the sun and moon respectively as 
 arguments, and take out their hourly motions. The former we 
 find to be .0401, and the latter .5018. 
 
 All the other inequalities treated of in the preceding chapters, 
 must likewise affect the moon's hourly motion, of which Variation 
 and Evection are the most important. The effect of Variation, as 
 is plain from the theory, is to increase the moon's velocity in syzy- 
 gy and diminish it in quadrature. Now, in an eclipse, the moon is 
 always in syzygy, and hence we must add to its hourly motion 
 the quantity given in the margin of table 21st. To correct the 
 
 4 
 
50 
 
 moon's hourly motion for Evection, we must enter table 22d with 
 the same argument that was used for that inequality in article GOth, 
 and in the middle column we find the equation, which is to be ap- 
 plied to the hourly motion according to its sign. 
 
 The following is the operation in the case before us : — 
 Moon's hourly motion, by table 21st, - - .5018 
 Add for Variation, .0115 
 
 .5133 
 
 Subtract for Evection, by table 22d, - - .0092 
 
 .5041 
 
 Subtract sun's hourly motion, - .0401 
 
 Hourly gain of the moon upon the sun, - - .4640 
 
 Now, by simple proportion, we can find how long it must have 
 taken the moon to gain the difference in the longitudes of the sun 
 and moon, viz. 1°.5637. Thus, 
 
 .4640 : 1 hour : : 1°.5637 : the time required, which is thus found 
 to be 3 hours, 22 minutes, and 12 seconds. This subtracted from 
 the time of mean new moon, found in article 11th, leaves for the 
 true time of new moon in May, 26d. 8h. 48m. 44sec. 
 
 79. The time thus found is Greenwich time, and to reduce it to 
 that of any other place, allowance must be made for the difference 
 of longitude, viz., 4 minutes of time for each degree of longitude. 
 It is also to be observed, that the astronomical day begins at noon, 
 and counts the 24 hours round to the next noon. 
 
 80. The longitudes and anomalies of the sun and moon must 
 now be corrected, by subtracting their motions during the correc- 
 tion just applied to the time. If that correction had been additive, 
 this would be so also. The amount can be easily found from their 
 hourly motions, thus — 
 
 One hour : the moon's hourly motion, viz., .5041 : : 3h. 22m. 12 
 sec. : the correction required in its longitude and anomaly, which 
 is thus found to be 1°.6988. 
 
 One hour : the sun's hourly motion, viz., .0401 : : 3h. 22m. 12 
 sec. : the correction required in its longitude and anomaly, which 
 is thus found to be 0M351. 
 
51 
 
 At the same time the longitude of the node must be corrected, 
 by taking from table 4th its motion during the same time, and ap- 
 plying it, with the contrary sign from that of the other motions, 
 because it moves in the opposite direction. 
 
 81. The reader will get a clearer idea of the process of calcula- 
 ting the time of an eclipse, if we now give a synopsis of the 
 work that we have been through in the foregoing chapters. 
 
 EXAMPLE. 
 Showing the method of calculating the time of a Solar Eclipse. 
 
 
 Time. 
 
 Sun's 
 
 Anom-» 
 
 aly. 
 
 Sun's 
 Longi- 
 tude. 
 
 Moon't 
 Anom- 
 aly. 
 
 Moon's 
 Longi- 
 tude. 
 
 Longi- 
 tude of 
 Node. 
 
 Mean new moon,'March, 1854 
 
 d. h. m. s. 
 28 10 42 50 
 59 1 28 6 
 
 85.586 
 58.211 
 
 6.0194 
 58.2135 
 
 93.665 
 51.634 
 
 6.0194 
 58.2135 
 
 64.2158 
 -3.1275 
 
 
 Mean new moon in May, 
 
 An. equa'n moon's per. & node 
 
 26 12 10 56 
 
 143.797 
 
 64.2329 
 
 145.299 
 —.221 
 
 64.2329 
 
 61.0883 
 —.0875 
 
 
 
 + 1.1165 
 
 +3.4154 
 
 Equation of the centre, 
 
 145.078 
 
 61.0008 
 —.0042 
 
 
 
 
 
 An'l equation of moon's long. 
 
 65.3494 
 
 145.078 
 —.112 
 
 67.6483 
 —.1118 
 
 
 
 
 Secular equa. of moon's long., 
 
 144.966 
 +.001 
 
 67.53615 
 +.0009 
 
 
 
 
 
 Variation, v 
 
 144.967 
 +.045 
 
 67.5374 
 + .0445 
 
 
 
 
 
 Annual equation of variation, . 
 
 145.012 
 —.005 
 
 67.5819 
 —.0053 
 
 
 
 
 
 Evection, 
 
 145.007 
 —.716 
 
 67.5766 
 —.7156 
 
 
 
 
 
 Annual equation of evection, . 
 
 144.291 
 +.010 
 
 66.8610 
 +.0100 
 
 
 
 
 
 Nodal equation of the ) 
 
 144.301 
 —.003 
 
 66.8710 
 —.0026 
 
 moon's longitude, $ 
 
 
 
 
 Corrrection of the equa- ) 
 
 144.298 
 
 66.8684 
 +.0680 
 
 tion of the centre, £ * * 
 
 
 
 
 
 Reduction to the ecliptic, 
 
 66.9364 
 —.0233 
 
 
 
 
 —.0042 
 
 —1.699 
 
 Lunar nutation, 
 
 66.9131 
 —.0042 
 
 
 3 22 12 
 
 —.135 
 
 Correct for difference in ) 
 long, of sun & moon, $ 
 
 65.3452 
 —.1351 
 
 66.9089 
 —1.6988 
 
 60.9966 
 +.0074 
 
 True time, long's & anomalies, 
 
 26 8 48 44 
 
 143.662 
 
 65.2101jl42.599 
 
 65.2101 
 
 61.0040 
 
52 
 
 82. It can hardly fail to suggest itself to the attentive reader, 
 that so great an alteration in the time, as that which was made in 
 article 78th, must in some degree vitiate the result of our work. 
 The anomalies, and nearly all the arguments for the inequalities 
 would vary in the interim. It would hence seem desirable to have 
 obtained, if possible, some nearer approximation to the true time at 
 the outset. The " preliminary equations"* in tables 27 and 28 are 
 designed for this purpose, but the theory of them could not be well 
 explained at that stage of our work, where it was necessary to in- 
 troduce them, if at all. 
 
 The construction of these tables is as follows. Since at the time 
 of new or full moon, the argument for evection is the same as that 
 for the equation of the moon's centre, viz., the moon's mean anom- 
 aly, the separate effects of the two on the time are united in the 
 first preliminary equation. Also, since both the equation of the 
 sun's centre, and the annual equation of the moon's longitude, de- 
 pend on the sun's anomaly, they are united in like manner in the 
 second preliminary equation. 
 
 After having found the time of mean new or full moon, as des- 
 cribed in article 11th, we apply to it these equations, and then take 
 from table 4th the mean motions in longitude and anomaly during 
 the time so applied. These motions applied to the mean longitudes 
 and anomalies at the mean time, give the mean longitudes and an- 
 omalies at the corrected time ; and w r e then proceed to calculate 
 the true longitudes at the corrected time, in the same manner as 
 we have done for the mean time in the foregoing chapters. We 
 will illustrate, by example, the method of using these tables, and at 
 the same time show how to calculate a lunar eclipse. 
 
 83. It will not be necessary to give much more than a synopsis 
 of the operation, as the time of a lunar eclipse is calculated precise- 
 ly in the same manner as one of the sun, only that the half lunation 
 in table 3d, is used in order to give the time of mean full moon, 
 and the longitudes of the sun and moon, instead of being made to 
 agree, are made to differ just 180°. 
 
 Let us inquire whether there will be an eclipse of the moon 
 when the sun passes its ascending node in the year 1844. 
 
 Turning to table 2d, we find that the longitude of the ascending 
 
 * These are the same as those usually termed 1st and 2d equations of the mean to 
 ihe true syzygy. 
 
53 
 
 node in March of that year is 258°, and we therefore (11) take 
 from table 3d such a number of lunations as, added to the half lu- 
 nation at the foot of the table, will contain 258 days, or thereabouts. 
 In 8£ lunations there are 251 days, which is the nearest to 258 
 that we can get from the table. This will carry the time forward 
 to November, and will give us for the longitude of the sun 244°, 
 and of the node about 245°. The sun will, therefore, be but 1° 
 from the node, and so far within the lunar ecliptic limit (7) that 
 there cannot fail to be an eclipse of considerable size. We will 
 proceed to calculate it. From tables 2d, 3d and 5th, we obtain 
 the following : — 
 
 
 Time. 
 
 Sun's 
 Anom- 
 aly. 
 
 Sun's 
 Longi- 
 tude. 
 
 Moon's 
 Anom- 
 aly. 
 
 Moon's 
 Longi- 
 tude. 
 
 Longi- 
 tude of 
 Node. 
 
 Mean new moon in March, 1844, 
 Add eight lunations, 
 
 d. h. m. s. 
 
 18 15 40 54 
 
 246 5 52 23 
 
 14 18 22 1 
 
 76.522 
 
 232.843 
 
 14.553 
 
 356.7831 
 
 232.8530 
 
 14 5534 
 
 132.366 
 206 535 
 192.908 
 
 356.7831 
 232.8530 
 194.5534 
 
 25°. 1245 
 —12.5102 
 — .7819 
 
 
 
 Mean full moon in November,... 
 
 24 15 55 18 
 
 323.9181 244.1895 
 
 171809 
 
 64.1895 
 
 244.8324 
 
 Entering tables 27 and 28, with the anomalies of the moon and 
 sun respectively as arguments, and making the proper proportions, 
 we obtian the " preliminary equations," the first of which is lh. 
 30m. lsec, subtractive, and the second 2h. 30m. J9sec, also sub- 
 tractive. The sum of the two is 4h. 0m. 20sec, which must be 
 subtracted from the mean time found above. Also the motions of 
 the sun and moon during this time, both in longitude and anomaly, 
 must be subtracted, and that of the node added, because it moves 
 the other way. The quantities in table 4th, from which we obtain 
 these motions, are given only for the units of the arguments, but 
 will answer just as well for tens, by removing the decimal point 
 one place to the right. Thus the sun's motion in anomaly, accord- 
 ing to the table, is, for 2 days, 1°.9712, and for 20 days 19°.712. 
 When used for units, the right hand figure may be omitted, and the 
 next reckoned according to its nearest value. The following is 
 the operation for finding the motions in the case before us : — 
 
 Motion in 4 hours, 
 
 Do. in 20 seconds, 
 
 Total motion in 4h. 0m. 20sec, 
 
 Sun's 
 Anom- 
 aly. 
 
 0°.164 
 0°.000 
 
 0°.164 
 
 Sun's 
 Longi- 
 tude. 
 
 0°.1643 
 0°.0002 
 
 0°.1645 
 
 MoonV 
 Anom- 
 aly. 
 
 Moon's 
 Longi- 
 tude. 
 
 2°.177 2°.1961 
 0°.003i 0°.O030 
 
 2°.180,2°.1991 
 
 Longi- 
 tude of 
 Node. 
 
 0°.00P l 
 0°.000( ! 
 
 ! 
 
 0°.008^ 
 
54 
 
 84. After applying these corrections to the time, longitudes and 
 anomalies, the process of calculation is, throughout, the same as 
 for a solar eclipse, with the exception already mentioned* and it is 
 unnecessary to go through it in detail. The following is a synop- 
 sis of the calculation : — 
 
 EXAMPLE. 
 
 Showing the method of calculating the time of a Lunar Eclipse. 
 
 
 Time. 
 
 Sun's 
 Anom- 
 aly. 
 
 Sun's 
 Longi- 
 tude. 
 
 Moon's 
 Anom- 
 aly. 
 
 Moen's 
 Longi- 
 tude. 
 
 Longi- 
 tude of 
 Node. 
 
 Mean new moon in March, 1844, 
 
 d. h. m. s. 
 18 15 40 54 
 136 5 52 23 
 14 18 22 1 
 
 7°6.522 
 
 356.7831 
 
 132.366 
 
 356.7831 
 
 25°3.1245 
 
 
 14.553 14.5534 
 
 192 908 lOAjaan 
 
 -12.5102 
 — .7819 
 244.8324 
 +.0088 
 
 
 
 64.1895 
 
 
 24 15 55 18 
 — 4 20 
 
 323.918 244.1895 
 — .164: —.1645 
 
 171.809 
 
 1st cor. in time, with cor'pond'g motions, 
 
 —2.180—2.1991 
 
 Annual equation of moon's per. and node, 
 
 24 11 54 58 
 
 323.754 244.0250 
 
 169.629 
 + .217 
 
 61.9904 
 +1.0412 
 
 244.8412 
 +.0897 
 
 
 
 
 
 —1.1560 
 
 169.846 
 
 + .109 
 
 169.955 
 + .001 
 
 169.956 
 + .006 
 
 244.9309 
 +.0044 
 
 
 
 
 
 
 242.8690 
 
 63.0316 
 +.1087 
 
 63.1403 
 +.0006 
 
 63.1409 
 
 +.0058 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 169.902 
 + .003 
 
 169.965 
 — .193 
 
 63.1467 
 +.0034 
 
 63.1501 
 —.1930 
 
 
 
 
 
 
 
 
 
 
 169.772 
 + .007 
 
 62.9571 
 +.0068 
 
 
 
 
 
 
 169.779 
 + .001 
 
 169.780 
 
 62.9639 
 +.0013 
 
 
 
 
 
 
 62.9652 
 
 +.0063 
 
 62.9720 
 +.0077 
 
 
 
 
 
 
 
 
 
 + .0044 
 
 242.8734 
 — .0103 
 
 — .121 
 
 
 62.9797 
 
 +.0044 
 
 
 — 14 41 
 
 — ".010 
 
 Cor. for difference* in Ion. of sun & moon, 
 
 02.984 1 
 —.1210 
 
 244.9353 
 +.0004 
 
 7-010 time, longitudes and anomalies, 
 
 24 11 40 17 
 
 323.744 242.8631 
 
 169.659 
 
 62.8631 
 
 244.9357 
 
 * In lunar eclipses it is this difference, + or —180°. 
 
55 
 CHAPTER XL 
 
 ELEMENTS OF AN ECLIPSE. 
 
 85. The following elements or data, are all that are needed for 
 making any calculation that we may desire in regard to an eclipse, 
 either solar or lunar ; such as the place on the earth's surface, 
 where the sun will be centrally eclipsed at any given time, while 
 the eclipse lasts ; the portions of the earth where an eclipse will 
 be visible, and the time when it will commence, become a maxi- 
 mum, and terminate ; or the size of an eclipse, at any given place 
 and time. The 10th is not needed in solar eclipses, nor the 2d, 3d, 
 and 12th in lunar. 
 
 1st. The time of new or full moon. 
 
 2d. The longitudes of the sun and moon. 
 
 3d. The obliquity of the ecliptic to the equator. 
 
 4th. The moon's latitude. 
 
 5th. The sun's hourly motion. 
 
 6th. The moon's relative hourly motion, or the excess of its 
 hourly motion over that of the sun. 
 
 7th. The sun's apparent semidiameter as seen from the earth. 
 
 8th. The moon's do. 
 
 9th. The apparent semidiameter of the earth as seen from the 
 moon, which is the same as the moon's horizontal parallax. 
 
 10th. The apparent semidiameter of the earth's shadow where 
 it eclipses the moon, as seen from the earth. 
 
 11th. The angle of the moon's visible path with the ecliptic. 
 
 12th. The sun's declination. 
 
 86. The method of obtaining the first two, was explained in the 
 last chapter. 
 
 87. The mean obliquity of the ecliptic to the equator in the year 
 1840, was 24° 27' 62".52, but it decreases at the rate of about half 
 a second a year, owing to the attraction of the planets. Il is also 
 subject to an inequality, whose period is about 19 years, depending 
 on the longitude of the moon's node : for it is evident from the 
 theory of lunar nutation, that the moon's influence must affect the 
 
56 
 
 obliquity of the equator to the ecliptic, as well as the place of their 
 points of intersection. Table 23d gives the obliquity on the 1st of 
 January in each year of the present century after 1840, taking both 
 these causes into account, from which it can be readily found for 
 any time in the year by inspection. In January, 1854, it is 23° 27' 
 33".6, and in January, 1855, it is 23° 27' 35".7. Hence, at the 
 time of our solar eclipse in May, 1854, it is 23° 27' 34".5. 
 
 88. The moon's latitude depends on its distance from the node, 
 and the inclination of the plane of its orbit ; but the inclination, and 
 also the place of the node, varies according to the situation of the 
 sun in respect to the node. Hence there are two principal equa- 
 tions of the moon's latitude, one depending on the distance of the 
 moon from the node, and the other on that of the sun. Now in 
 eclipses, the distance of both luminaries from the node is the same, 
 so that the two equations may be combined into one, which is done 
 in table 24th. The argument is found by subtracting the longitude 
 of the node from that of the sun and moon, which leaves, in our 
 solar eclipse, 4°.2061, and in the lunar, 177°.9274. The moon's 
 latitude, as determined by these arguments, is, in the former case, 
 the decimals of a degree .3664, or a little over one-third of a de- 
 gree, and in the latter, .1810. It is to be noticed, that in table 24th 
 the figures of the argument at the right and left are whole degrees, 
 and those at the top and bottom the first place of decimals. 
 
 It must be readily seen, that the moon's latitude must be north 
 for the first 180° after it leaves the ascending node ; and that it 
 moves northerly, or ascends, through the first quadrant, and south- 
 erly, or descends, through the second : also, that in the other half 
 of the orbit, its latitude must be south, being descending in the first 
 quadrant, and ascending in the second. These facts are indicated 
 in the table by capital letters at the head of the columns containing 
 the argument. 
 
 89. The method of finding the 5th and 6th elements was explain- 
 ed in the last chapter, (78 :) but if much accuracy were required, 
 they w r ould have to be now computed over again for our solar 
 eclipse, because the anomalies have been changed. Calculated 
 from the anomalies as finally corrected, in the same manner as 
 was done in article 78, the hourly motion of the sun in our solar 
 
57 
 
 eclipse we find to be .0401, and in the lunar .0421 ; and the rela- 
 tive hourly motion of the moon is .4649 in the solar eclipse, and 
 .4523 in the lunar. 
 
 90. The 7th element is obtained from the 1st column of table 
 26th, where it is sufficiently explained. 
 
 91. Table 21st, columns 1st and 3d, give the 8th and 9th ele- 
 ments, so far as they depend on the elliptical form of the moon's 
 orbit. But the effect of Variation is to throw the orbit into a kind 
 of oval, with its shortest diameter lying in syzygy. From this 
 cause, the distance of the moon from the earth is less when it is 
 new or full than at other times, which must increase their apparent 
 size as viewed from each other. Hence the corrections in the 
 margin of table 21st. Also Evection, by altering the shape of the 
 moon's orbit, affects its apparent size and parallax, so that a far- 
 ther correction becomes necessary from table 22d. The following 
 shows the method of obtaining these elements for the eclipse of 
 May, 1854 :— 
 
 
 Semidiarneter. 
 
 Parallax. 
 
 Values taken from table 21st, 
 
 .2482 
 -f .0020 
 
 .9097 
 + .0073 
 
 "Variation, 
 
 
 Evection, 
 
 .2502 
 — .0023 
 
 .9170 
 — .0080 
 
 
 True semidiameter and parallax, 
 
 .2479 
 
 .9090 
 
 
 
 
 The semidiameter and parallax at the time of the lunar eclipse 
 in November, 1844, obtained in the same way, are .2453 and 
 .8989 respectively. 
 
 92. It was shown in article 4th, that our 10th element, viz., the 
 apparent semidiameter of the section of the earth's shadow that 
 eclipses the moon, is equal to the sum of the parallaxes of the sun 
 and moon, diminished by the sun's apparent semidiameter. The 
 sun's parallax may always be put down at .0024, and the methods 
 of obtaining the other data for finding this element have been al- 
 ready explained. Thus in the lunar eclipse which we have taken 
 as an example, 
 
58 
 
 The sun's parallax is .0024 
 
 The moon's do., as just found, is - - - - .8989 
 
 .9013 
 
 The sun's semidiameter (see 7th element) is - - .2707 
 
 The semidiameter of earth's shadow is - - - .6306 
 
 93. The angle which the moon's path makes with the eclip- 
 tic varies according to the moon's distance from the node. When 
 at the node, it makes the same angle as the planes of the two or- 
 bits; but when it is 90° from it, its motion becomes parallel to the 
 ecliptic. But the inclination of the planes also varies, depending, 
 as was remarked above, on the distance of the sun from the node. 
 The two influences may, however, be combined into one at the 
 time of an eclipse, in the same manner as in table 24th. And not 
 only does the real angle vary from both these causes, but the ap- 
 parent angle is increased by the earth's motion in the same direc- 
 tion ; and since the rate of the latter, as compared with the moon's 
 motion, is quite variable, the apparent angle must vary also from 
 this cause. All these causes are taken into account in table 25th. 
 This table has two arguments, viz., 1st, the difference between the 
 hourly motions of the sun and moon, (for the motion of the earth is 
 measured by the apparent motion of the sun,) the first two decimal 
 places of which are placed at the top ; and 2d, the distance of the 
 sun or moon from the node, which is placed at the right and left. 
 In the solar eclipse we are calculating, the former (89) is .4649, 
 and the latter (found by subtracting the longitude of the node from 
 that of the moon) 4°.2061. These give the angle 5° 44' 33", 
 ascending. In the same way the angle at the time of our lunar 
 eclipse is found to be 5° 45' 41" descending. 
 
 94. The sun's declination, which is our 12th element, can easily 
 be computed from its longitude by spherical trigonometry, since 
 the obliquity of the ecliptic is known, (87 ;) for its longitude, right 
 ascension, and declination form a right-angled spherical triangle, 
 of which an angle and one side is known. Table 26th gives the 
 declination calculated from the obliquity in the year 1840, which 
 is sufficiently exact for our purpose, though, after a lapse of years, 
 it must evidently need correction. Entering this table, with the 
 sun's longitude at the time of our solar eclipse as an argument, we 
 obtain the declination 21°.1871. 
 
59 
 
 Since the sun starts northerly from the vernal equinox, its decli- 
 nation must be north for the first 180°, and south through the rest 
 of the orbit. This fact is indicated by the capital letters at the 
 head of the columns of the argument. 
 
 95. The elements collected are as follows : — 
 
 1. True time of the eclipse, 
 
 2. Longitude of the sun and moon,.... 
 
 3. Obliquity of ecliptic to equator, 
 
 4. Moon's latitude, 
 
 5. Sun's hourly motion, 
 
 6. Moon's relative do 
 
 7. Sun's apparent semidiameter, 
 
 8. Moon's do 
 
 9. Moon's horizontal parallax,, 
 
 10. Semidiameter of earth's shadow, .... 
 
 1 1. Angle of moon's visible path with eclip. 
 
 12. Sun's declination, 
 
 Solar Eclipse. 
 
 d. k. m. 8. 
 May 26 8 48 44 , 
 65°.2101 
 23° 27' 34".5 
 
 0°.3664 (north) 
 
 0°.0401 
 
 0°.4649 
 
 0°.2635 
 
 0°.2479 
 
 0°.9090 
 
 5° 44' 33" (ascend.) 
 2l°.187l (north) 
 
 Lunar Eclipse. 
 
 d. h. m. s. 
 N"ov. 24 11 40 17 
 
 0°.1810 (north) 
 
 ()°.0421 
 
 0°.4523 
 
 0°.2707 
 
 Q°.2453 
 
 3°.8989 
 
 (J°.6306 
 
 5° 45' 4i" (descend.) 
 
 CHAPTER XII. 
 
 DELINEATION OF A SOLAR. ECLIPSE. 
 
 9G. To find whether a solar eclipse will be visible at a particu- 
 lar place, and if so, its size and general appearance there, it is more 
 convenient to first reverse the order of viewing the phenomenon, 
 and to suppose the spectator placed at the centre of the sun, to 
 look down upon the earth, and see the moon passing across its disc. 
 From so vast a distance, the earth wofla 1 appear to him, as the sun 
 does to us, like a flat circular disc. The circle of illumination 
 would be to our observer the circle of the disc, and all circles 
 whose planes were perpendicular to this would be seen edgewise, 
 and would appear to him like straight lines. Their arcs would seem 
 to be only of the length of the straight lines that they would sub- 
 tend, as viewed by him. Such circles as were seen obliquely, 
 would appear elliptical in their shape. Let us suppose him to take 
 
60 
 
 his station when the sun is in the vernal equinox, about the 21st of 
 March, and to retain it for a year, accompanying the sun in its ap- 
 parent annual round. Being always in the plane of the ecliptic, it 
 would appear to him like a straight line dividing the earth into two 
 equal parts, one half lying north and the other south of it. At first 
 also, being in the plane of the equator, it too would be seen as a 
 straight line, as well as all the parallels of latitude ; yet not paral- 
 lel with the ecliptic. The west end of the 
 equator would be north of the ecliptic 
 and the east end south, crossing it in the 
 centre at an angle of about 23^°, as in 
 Fig. 17, where AB represents the eclip- 
 tic, CD the equator, PP' the earth's axis, 
 P and P' its poles, GH the axis of the 
 ecliptic, and 1 1, 2 2, 3 3, &c, parallels 
 of latitude. 
 
 As the sun advances, it gets north of the plane of the equator and 
 of the parallels of latitude, and they will no longer appear as 
 straight lines, but will seem bent downward toward the south. 
 The earth's axis will approach to parallelism with that of the eclip- 
 tic ; and the poles revolving in circles whose planes are parallel to 
 that of the ecliptic, will seem to move in straight lines toward n 
 and s till on the 20th of June, or thereabouts, when the sun reaches 
 the summer solstice, the two axes will coincide, and the north pole 
 will be seen at n. The south pole will be invisible, being hid be- 
 hind a segment of the earth ; but if the earth were transparent it 
 would appear at s. 
 
 97. The sun still advancing, the earth's axis will appear again 
 on the other side of GH, the poles will approach toward N and S, 
 and the parallels of latitude will become less curved. And when 
 the sun reaches the auturmtal equinox, in September, the latter will 
 again become straight lines, but lying the opposite way from what 
 they did in March, and the poles will appear at N and S. 
 
 08. As soon as the sun has passed the autumnal equinox, the 
 parallels of latitude will appear curved again, but upward toward 
 the north, instead of downward, because the sun will now be on 
 the south side of their planes. The poles will recede along the 
 
61, 
 
 iines NP and SP', passing n and s when the sun is at the winter 
 solstice, in December, and finally arriving at P and P' about the 
 '21st of March, when everything assumes the same aspect»as when 
 he started. 
 
 99. A common terrestrial globe will serve to present these vari- 
 ous appearances to the reader's view, in a much clearer light than 
 can be done by any verbal description. Let the north pole be 
 elevated about 662 above the point marked " North," on the wood- 
 en horizon, and then the latter will represent the ecliptic. If now 
 the globe be placed upon a table in the centre of the room, so that 
 the wooden horizon may be on a level with the eye, and the read- 
 er, after having found the 21st of March on the horizon, should 
 retire across the room in that direction, he will see the globe pre- 
 cisely as represented in Fig. 17. Let him now pass slowly round 
 the room in the order of the months on the horizon, and all the ap- 
 pearances we have described will be presented to his view. 
 
 100. When he is in the direction marked May 26, he will have 
 a true representation of the earth, as it would appear to our obser- 
 ver at the sun,, at the time of the solar eclipse we have been calcu- 
 lating. We will endeavor to represent the same by a figure, and 
 also the appearance of the moon passing over the earth's disc. In 
 order to give proper proportions to the several parts of our figure, 
 we must be able to mark down their relative dimensions. In com- 
 mon plans and drawings these are given in miles, feet, inches, or 
 some other direct measure of length ; but in astronomy it is found 
 more convenient to determine them by the angle which they would 
 subtend when viewed from a given distance, as we did in the last 
 chapter. And this answers the purpose just as well, provided the 
 distance be sufficiently great, for the apparent would be very near- 
 ly proportional to the real size of the object. The dimensions 
 which respect the moon and earth, given in the last chapter, are 
 the angles that the objects would subtend at a distance of about 
 237,000 miles, or the distance between the earth and moon. The 
 reader must not here fall into the error of supposing that our obser- 
 ver has changed his position. He is still at the sun, and these an- 
 gles are given merely for the purpose of determining the relative 
 sizes of the objects that we wish to draw. 
 
62 
 
 *%; 
 
 *; 
 
 fj, t) 89 ?t> u so so 
 
 4-0 SO GO 70 80 so ic a 
 
 3 
 
 ^yrft T ri^fi^.l'j^iTufir'T tirrrl^TTl iTr j'i^^rr nn]i j Yrlrn-<TnT}7TnTiTi nfimJu iT 
 
G3 
 
 101. Our first business is to make a scale SS (Fig. 18) of any 
 convenient length, and divide it into 100 equal parts. This scale 
 may be considered to be of such length, that if seen perpendicularly 
 at the distance of the moon from the earth, it would subtend an 
 angle of one degree, and of course each of the parts would subtend 
 .01 of a degree. The earth's semidiameter (95) seen at that dis- 
 tance, subtends an angle of 0°.O090. If therefore, with this dis- 
 tance, taken from our scale, as radius, (counting the two first deci- 
 mals places, with a proper proportion for the two last,) we describe 
 the semicircle AGB, the line AB may represent that portion of the 
 plane of the ecliptic which intersects the earth, and the whole se- 
 micircle AGB, the half of the earth's disc that is seen north of it. 
 Or, which is the same thing, AB may represent the wooden hori- 
 zon of the globe adjusted as above described, and AGB the half of 
 the globe that is above it. 
 
 102. We will next find the position of the north pole of the 
 earth, i. e., the point on the disc where it would be seen. A glance 
 at the globe will show that it is not at the top, nor anywhere in 
 the circumference of the disc. In fact, we have shown (96) that 
 it must appear to move in the right line PN, leaving P about the 
 21st of March, and arriving at n about the 20th of June. Hence, 
 at the time of our eclipse, it must be between P and n. Its precise 
 position we are able to determine; for its motion in the small circle, 
 which, seen edgewise, is represented by the straight line PN, is 
 just equal to that of the sun in longitude. Consequently the num- 
 ber of degrees that it has moved from P must be equal to the sun's 
 longitude. The plane of this circle is evidently perpendicular to 
 the surface of the paper, yet for the purpose of bringing the gradu- 
 ation in sight, we will suppose it to turn on the diameter PN, till 
 it lies flat down, as PHN. The reader must not suppose that any 
 circle will be seen in this position by our observer ; it is merely 
 drawn so temporarily, for the purposes of measurement. The 
 sun's longitude being 65°.2101, the north pole must be that number 
 of degrees from P, which would bring it to T ; and this point, 
 when the circle is turned up again into its place, edgewise upon 
 the paper, would be seen at C, which is hence the position that the 
 north pole must occupy on the disc ; and the line CE must repre- 
 sent the northern half of the earth's axis. We shall have no fur- 
 ther use for the temporary circle PHN, as drawn, and may, if we 
 
64 
 
 choose, erase it, since it forms no part of the representation that 
 we wished to draw. It merely served to enable us to find the po- 
 sition of the north pole of the earth, which is now effected. 
 
 103. The delineation of a parallel of latitude on the disc will 
 show how the diurnal path of a place would appear to our obser- 
 ver. We will select for the purpose that of the Astronomical Ob- 
 servatory of Williams' College, lat 42° 42' 51", Ion. 73° 12' 33" 
 west from Greenwich ; the former of which, converted into de- 
 grees and decimals by the aid of tables 30 and 31, is 42°.7142. If 
 the latitude were just equal to the sun's declination, the sun would 
 be vertical at noon, and the Observatory would be seen precisely 
 in the centre of the disc at E ; but since it exceeds it by 21°.5271, 
 the Observatory must be seen that distance north of the point 
 where the sun is vertical, which when projected on the disc, would 
 become the sine of the arc, measured from E, on the axis EC. To 
 find the length of the sine, we may count the degrees upward from 
 A, and draw the sine DD, the length of which, when applied from 
 E toward C, will reach to the point 12. This point must therefore 
 be the apparent position of the Observatory at noon. 
 
 104. If the earth were transparent, it would be seen at midnight 
 considerably farther north, as is evident from an inspection of the 
 globe. The point antipodal to that at* which the sun is vertical, and 
 which also would be seen at E, is as many degrees south of the 
 equator as the sun's declination is north. Hence the distance of 
 the Observatory from this point at midnight, must be equal to the 
 latitude of the former added to the sun's declination, which amounts 
 to 63°.9013. This arc. like the former, when projected on the disc, 
 will be seen on the axis EC, equal only to the length of its sine, 
 which we may find in the same way, by counting the number of 
 degrees upward from A, drawing the sine, LL, and laying it off on 
 the axis from E to K. The point K will then represent the appa- 
 rent place of the Observatory at midnight. 
 
 105. The line K-12 will be the shortest diameter of the ellipse, 
 into which the parallel of latitude appears to be thrown by being 
 seen obliquely ; the point O, midway between K and 12, its centre, 
 and the line 606, drawn through O at right angles to EC, its long- 
 
65 
 
 est diameter. The lines 06. not being foreshortened by being seen 
 obliquely, will appear of the full length of the radius of the parallel, 
 which, we know, is the cosine of the latitude. The complement of 
 the latitude of the Observatory is 47°.2858, and we may find RJl, 
 its sine, in the same way as we did the others. Setting off the 
 distance RR each way from O to 6 and 6, we find the extremities 
 of the longest diameter, which must be the points on the disc where 
 the Observatory will be seen at 6 o'clock in the morning, and. at 
 the same hour in the evening. 
 
 106. Its position at any other hour in the day may be found by 
 the following process. Draw two circles, 6M6 and F'KV, one 
 on the longest and the other on the shortest diameter of the ellipse, 
 and divide each into 24 parts, in the points 7, 8, 9, 10, &c. corres- 
 ponding to the hours of the day. Through the division points of 
 the former circle, draw straight lines parallel to EG, the earth's 
 axis ; and through those of the latter, at right angles to it. Note 
 the points where the lines that pass through the same hour on both 
 the circles intersect each other, and through them draw the ellipti- 
 cal curve seen in the figure. This curve will represent the paral- 
 lel of latitude that we wished to delineate, or the path of the Ob- 
 servatory over the disc* The several points of intersection mark 
 its position at the different hours. The two last circles, with the 
 lines connected with them, except the path of the place, may be 
 drawn in pencil mark, that they may be erased after the latter is 
 drawn, since they are of no further use. 
 
 107. The construction we have just completed will show us, if 
 we wish, the time of sunrise or sunset, by noticing at what hour 
 the path of the place cuts the circle of the disc. In this case, it is 
 a little before 5 o'clock in the morning, and a little after 7 in the 
 evening. 
 
 108. The moon's latitude (95) is 0°.3664 north. We will there- 
 fore take this distance from the scale SS, and measure it from E 
 toward G, which gives X as the place where the centre of the moon 
 
 * The reader will readily see that if the sun's, declination had been as far south as it is 
 north in this case, the points K and 12 must exchange places, and the curve representing 
 the path of the Observatory must lie on the upper side of 606. 
 
6G 
 
 will appear to our observer to cross the line EC The propriety 
 of this step will appear from the following considerations : — 
 
 1st. The longitude of the moon, when new, is the same as that 
 of Jhe sun viewed from the earth, or of the earth viewed from the 
 sun ; it must therefore be seen in the line EG. 
 
 2d. Its latitude is north, therefore it must be seen above the line 
 AB, and not below it, as it would be if its latitude were south. 
 
 3d. Its latitude, as given in article 95, is supposed to be measur- 
 ed at the same distance as the other angles for which our scale 
 was made, so that the scale furnishes the proper measure. 
 
 The only error that is to be noticed, is that the moon's centre, 
 as seen by our observer projected on the earth's disc, would be a 
 little farther from the ecliptic than its real distance, owing to the 
 divergence of the visual line between the moon and earth. Since 
 however the distance of the sun is so great, the lines drawn from 
 the observer's eye to the centres of the moon and earth must be 
 very nearly parallel, and the divergence just named so small, that it 
 may be disregarded. 
 
 109. At the time of our eclipse, the moon's path makes an angle 
 of 5° 44' 33" with the ecliptic, tending north. If, therefore, we 
 draw Ey, making an angle of that size with EB, and YZ par- 
 allel to it, the latter line will represent the apparent track of the 
 moon's centre across the earth's disc. It passes X at 48 minutes 
 and 44 seconds after 8 in the evening, by Greenwich time, (95,) 
 which, by Williamstown time, is 4 minutes and 6 seconds before 4 
 in the afternoon. The precise point where it will be at 4 o'clock, 
 or any other instant we may choose to name, may be found from 
 its relative hourly motion, viz., 0°.4649. Taking this distance from 
 our scale, SS, dividing it into 12 equal parts, and thus making the 
 smaller scale, ss, we have its motion in 5 minutes. With the aid 
 of this we can judge by the eye how far it would move in 4 min- 
 utes and 6 seconds, and setting off this distance from X toward Z, 
 we find its position at 4 o'clock. Its position at the hours 2, 3, 5, 
 6, &c, may now be found by measuring off its hourly motion each 
 way from 4. The hourly divisions may now be subdivided at 
 pleasure. In the plate they are divided into quarter hours. 
 
 110. The appearance of the moon as projected upon the earth's 
 disc, may be shown by taking its semidiameter, 0°.2479, from the 
 
67 
 
 scale SS, and with it describing a cirele, as d b c ; selecting for a 
 centre, the point where the moon's centre will be at the time for 
 which we wish to represent its appearance. The plate shows how 
 it will appear at 32 minutes past 5, the time at which its centre 
 erosses the diurnal path of the Observatory, according to our draw- 
 ing. At that time the Observatory will be at W, a little more 
 than half way from 5 to G on its diurnal path, and very nearly co- 
 inciding with the moon's centre. It must therefore be invisible to 
 our observer, being hid behind the moon ; and the same must be 
 true of a large tract of country about it ; for although part of the 
 moon has passed off from the earth's disc, the remaining part cov- 
 ers the parallel of latitude between the hours 8 and 7, and some- 
 what more. This will amount to over 60° of longitude, reckoning 
 15° for each hour, which would extend from the Rocky Mountains 
 on the west, about one-third of the way across the Atlantic on the 
 east. Of course the inhabitants of this entire tract must be unable 
 to see the centre of the sun at the time of which we speak, and at 
 Williams' College almost its whole disc must be hid ; for it has just 
 been shown that the locality is almost in .perfect range with the 
 centres of the sun and moon. 
 
 111. It is not difficult to determine precisely What part of the 
 disc will be concealed from view, or eclipsed. Suppose a line to 
 be drawn from the Observatory to the centre of the sun, and a 
 point to be taken in it at the distance of the moon. Let another line 
 be drawn through this point, from one edge of the sun, and contin- 
 ued so as to meet the earth's disc, at a considerable distance, evi- 
 dently, from the Observatory. If now the upper end of this line 
 be carried round the circumference of the sun, the lower end would 
 describe a circle on the earth's disc, having the Observatory for 
 its centre. And the line itself would describe two similar cones, 
 having a common vertex near the moon, and their bases, one upon 
 the sun, and the other upon the abovementioned circle on the earth's 
 disc. As seen from the centre of the sun, this circle would have 
 precisely the same situation in respect to the moon, that the sun 
 would have as seen from the Observatory ; so that if a portion of 
 this circle be hid from our observer at the sun, by some intervening 
 object, a like portion of the sun would be hid when viewed from 
 the Observatory. We wish then, to determine the size and posi- 
 tion of this circle at 32 minutes past 5, that we may see how large 
 
68 
 
 a part of it is covered or concealed by the moon. As viewed from 
 the fixed point spoken of, near the moon, it and the sun would evi- 
 dently subtend the same angle. But the sun would subtend very 
 nearly the same angle, whether seen from that point or from the 
 earth, for the relative distances are nearly the same. Now the 
 sun's semidiameter, seen from the earth, at that time subtends an 
 angle of 0°.2635 ; therefore this circle must, at the distance of the 
 moon, subtend the same angle. Hence we may take this distance 
 from the scale SS, and with it discribe the circle a b c, from the 
 centre W, the position of the Observatory at the time, and we have 
 the circle in question ; showing that a very slender crescent of 
 light will be seen on the south side of the moon. 
 
 The width of this crescent is usually described, by dividing the 
 diameter of the circle into 12 equal parts, called digits, and seeing 
 how many of these it contains. In this case it contains about one- 
 fourth of one of these divisions, leaving 11 3-4 digits eclipsed. 
 
 112. The eclipse must evidently commence at Williams' College, 
 as soon as the moon and the circle that we last drew begin to in- 
 terfere, which must be as soon as the distance between their centres 
 becomes less than the sum of their semidiameters. The two semi- 
 diameters added together make 0°.5114, and we may take this dis- 
 tance from the scale SS, and setting one foot of our compasses on the 
 moon's path, some distance to the left of W, and the other on that 
 of the Observatory, move them backward or forward till both feet 
 stand on the same hour and minute, which must be the time when 
 the eclipse commences. By a similar operation at the right hand 
 of W, the time of the end may be found. 
 
 The results, according to our drawing, are as follows : — 
 
 h. in. 
 
 Beginning, - - - - 4 15 
 
 Greatest obscuration, 5 32 
 
 End, 6 38 
 
 Duration, 2 23 
 
 Digits eclipsed, 11 3-4 
 
 113. We may derive from Fig. 18th a method, by which the 
 size and appearance of a solar eclipse, at any given time, and place 
 may be calculated mathematically. The tabular latitudes and lon- 
 gitudes of the sun and moon are calculated for the centre of the 
 
69 
 
 earth, and are consequently correct only where the sun and moon 
 are vertical, or in the zenith. At all other places they would be 
 affected by parallax. If the place were situated so as to appear on 
 the disc above or below AB, the parallax would affect the latitude, 
 and if on the right or left of GE, the longitude would be affected. 
 Thus V is the place of the Observatory at 3 o'clock P. M. on the 
 day of our eclipse, and E that at which the sun is vertical and the 
 moon nearly so, not differing from it more than about J 9 . Hence 
 VE is the sine of the zenith distance, to which the total effect of 
 parallax is always proportional. Draw VX at right angles to AB, 
 and it will represent the effect of parallax upon the latitude of the 
 sun and moon, and EX upon the longitude. 
 
 Having computed the elements (95) for the time at which we 
 wish to represent the appearance of the eclipse, the arc GP, which 
 is equal to the obliquity of the ecliptic to the equator, will be known, 
 and calling the radius of the semicircle AGB unity, Vn and E?i can 
 be found. Again, since P/i, the radius of the circle NHP, is now- 
 known, and also the arc PT, being equal to the sun's longitude, Cn 
 can be found. Then in the right angled triangle EwC, the two 
 sides E;i and Cn are known, and we can find the angle GEC. 
 
 If we let s represent the sun's longitude, and m the obliquity of 
 the ecliptic to the equator, then tan. GEC=tan. mxcos. s. 
 
 The lines EK and El2 are the sines of the sum and difference of 
 the latitude of the place and the sun's declination, and are there- 
 fore known : hence EO, which is equal to half the sum, can be 
 found. 
 
 V 
 
 The line OO is known, being the cosine of the latitude, and also 
 the arc MF, being the hour arc from noon, or the interval between 
 noon and the time to which the calculations refer converted into 
 degrees ; hence 01, which is the sine of this arc, or its equal UV, 
 and also IF its cosine may be found. IV is equal to IF foreshort- 
 ened in the ratio of radius to the sine of the sun's declination, and 
 is therefore known ; and subtracting* it or its equal OU from EO, 
 we shall have EU. Now in the right angled triangle EUV, the 
 sides EU and UV are known, and we can find the side EV and 
 the angle UEV. 
 
 * If the sun's declination and the latitude of the place are both north or both south, 
 subtract ; hut if one is north and the other south, add. 
 
70 
 
 If we let / represent the latitude of the place, d the sun's declina- 
 tion, and t the hour arc from- noon, then 
 UV— sin. Zxcos. I 
 
 EU=sin. Zxcos. d±*cos. Zxsin. d cos-, t 
 sin. t 
 
 tan. UEV= 
 
 cotan. /xcos. c?±*sin. flfxcos. t 
 
 We now have the angles GEO and UEV, which, added togeth- 
 er and subtracted from 90°, leave the angle VEX. We have also 
 the line EV. Hence, in the right angled triangle VEX, we are 
 enabled to find EX and VX. Since the parallax of a heavenly 
 body at any altitude is equal to the horizontal parallax mutiplied 
 by the sine of the zenith distance.f which in this case is EV, it fol- 
 lows that if we multiply it by EX, we shall get the effect of paral- 
 lax on the longitude, and if by VX, on the latitude. 
 
 The value of EV and the angle UEV may be obtained, if pre- 
 ferred, by another process. The co-latitude of the place of obser- 
 vation, the co-declination of a heavenly body and its zenith distance 
 form a spherical triangle, in which the two former parts are in 
 this case known, and also the included angle, being the hour angle 
 from noon. Hence the third side can be found, the sine of which 
 is EV, and the remaining angles. Now, by the principles on which 
 Fig. 18 is constructed, the angle UEV is the same as that opposite 
 the co-latitude in the sperical triangle, and is therefore known. 
 
 Having corrected the latitudes and longitudes of the sun and 
 moon for the effect of parallax, their differences will form two sides 
 of a right angled triangle, and the distance between their apparent 
 centres will be the hypothenuse. By comparing the latter with 
 the apparent semidiameters of the sun and moon, the size of the 
 eclipse can be readily determined. 
 
 Our 8th element is the moon's apparent semidiameter, as seen 
 from the centre of the earth ; but the distance of the moon from 
 any place on the earth's surface at which it is visible (save when 
 it is in the horizon) is less than from the centre, which must cause 
 it to subtend a greater angle. The augmentation is a maximum 
 when the moon is in the zenith, and grows less when it recedes 
 from it ; hence the sine of the zenith distance, EV, is the proper 
 
 * See note on preceding page. 
 
 f Olmsted's Astronomy, Art. 82.— Herscheirs do., Art. 303.— Norton's do., Art. 98. 
 Gunmearc'a do., Chap, v., Art. 5. 
 
71 
 
 argument for determining the amount of the augmentation, and is 
 so used in table 29th. 
 
 We will show the results of such calculations as we have been 
 describing, by applying them to our eclipse at 32 minutes past 5, 
 the time at which.it is represented in Fig. 18th. After computing 
 the necessary elements for the time and the parallax, as above des- 
 cribed, we have the following : — 
 
 
 Sun. 
 
 Moon. 
 
 Latitude, 
 
 Sun. 
 
 Moon. 
 
 Longitude, 
 
 65°.2744 
 .0020 
 
 66°.0189 
 .7644 
 
 0°.0000 
 .0011 
 
 0°.4369 north 
 .4154 
 
 Parallax, 
 
 Parallax, 
 
 
 Difference, 
 
 
 65° .2724 
 65°.2545 
 
 0°.0179 
 
 65°.2545 
 
 U°0011 south 
 0°.0215 north 
 
 0°.0215 north 
 
 Difference, 
 
 0°.0226 
 
 
 
 Making these differences the legs of a right angled triangle, and 
 regarding them as straight lines, which we very safely do, since 
 they are so small, we find the hypothenuse to be 0°.0288, which is 
 the apparent distance between; the centres of the sun and moon. 
 
 The sine of the zenith distance is found by the calculations to be 
 .95678, and the consequent augmentation of the moon's semidiame- 
 ter, taken from table 29th, 0°.0007. The moon's apparent semidi- 
 ameter would otherwise be at this time 0°.2478, and applying this 
 correction, it becomes 0°.2485. The sun's remains the same as 
 was found in article 95th, viz., 0°.2G35, and the difference between 
 the two semidiameters is 0°.0150. By comparing this difference 
 with the distance between the centres, viz., 0°.0288, we see that 
 the sun's disc must extend beyond the moon's on one side by 
 0°.0438, or about 2' 38", while on the other it would fall short by 
 0°.0138 ; thus forming a slender crescent of light about the moon, 
 as we have before remarked. 
 
72 
 
 CHAPTER XIII. 
 
 CENTRAL TRACK OF A SOLAR ECLIPSE. 
 
 114. If we examine the plate principally referred to in the last 
 chapter, (Fig. 18,) we notice that the moon's track crosses the 
 path of the Astronomical Observatory of Williams' College twice ; 
 once about 20 minutes before 4, and again about 32 minutes past 
 5. In the first instance, it crosses far west of where the Observa- 
 tory will be at the time, and in the second, a little east. Counting 
 every hour as 15° of longitude, the point where it first crosses is 
 about 70° west of the Observatory, which carries it into the Paci- 
 fic ocean, not far from Astoria. It crosses the second time at a 
 point about 3° east of the Observatory, which is in the Atlantic 
 ocean, off Cape Ann. If we had numerous elliptical curves drawn 
 to represent the different parallels of latitude, we might, by a pro- 
 cess analagous to the foregoing, determine over what countries of 
 the earth the moon's centre, or more strictly, the centre of its 
 shadow would pass, from the time it first struck the disc on the 
 west side, till it passed off on the east. There is however an ea- 
 sier way of effecting this, by means of a figure of different con- 
 struction, used in connection with a terrestrial globe. 
 
 115. Let ANB (Fig. 10) represent the northern half of the 
 earth's disc, as seen from the sun, YZ the moon's track, with the 
 hours of Greenwich time marked on it, AB a portion of the eclip- 
 tic, as in Fig. 18, in the last chapter, and the curved lines seconda- 
 ries to it, orthographically projected at intervals of 15°. 
 
 Now to adjust the globe so as to correspond with this figure, 
 elevate the north pole, as directed in the last chapter, (99,) and at 
 the point that answers to N in the figure. 90° above the wooden 
 horizon, which now represents the ecliptic, or 23^° from the north 
 pole of the earth, screw on the graduated quadrant of altitude. , 
 By swinging the other end round, between the globe and the in- 
 side of the wooden horizon, it may be made to represent any of 
 the lines NA, N«, N6, &c. Or we may screw it on at S, and thus 
 represent the other half of the curves, which becomes necessary 
 when the moon's latitude is south. 
 
73 
 
 Fig. 19. 
 
 The point E in the figure corresponds, in the present case, to 
 that marked May 26th on the wooden horizon, and consequently 
 the points A and B must be found 90° from it on each side. If the 
 lower end of the quadrant of altitude is brought to one of these 
 points, we shall have a representation of the arc NA ; and if to the 
 other, of NB. Our figure shows that the moon's track crosses the 
 former about 16° 40' above A, and the latter about 29° above B. 
 |The graduation of the quadrant of altitude would readily show 
 where these places are on the globe, were it not for its diurnal 
 revolution, which we must next take into account. 
 
 116. It is evident that the sun must always be just rising at all 
 places situated on the line NA, and just setting at all on the line 
 
74 
 
 NB. Now, at the equator, the sun rises invariably at 6 o'clock in 
 the morning, and sets at the same hour in the evening. Therefore, 
 at 8 minutes before 7 in the evening, by Greenwich time, when, 
 according to our drawing, the moon's centre first strikes the earth's 
 disc, it must be just 6 o'clock in the morning at the place where 
 the equator cuts the arc NA, (compare Fig. 17.) The question is 
 then reduced to this, viz., at what place on the equator is it 6 
 o'clock in the morning, at the same time that it is 8 minutes before 
 7 in the evening by Greenwich time ? Converting the time into 
 longitude, the point in question is found to be in the Pacific ocean, 
 a little southwest of Mulgrave's island, in longitude 167° east from 
 Greenwich. 
 
 Turn the globe on its axis till this point is brought under, the 
 quadrant of altitude, (the latter being adjusted so as to represent NA,) 
 and count 16° 40' upward from the wooden horizon, and we shall 
 discover the place where the centre of the eclipse first strikes the 
 earth. The experiment shows it to be near the Caroline Islands, 
 lat. 7° north, and Ion. 164° east. 
 
 117. According to our drawing, the centre of the eclipse leaves 
 the earth at 29 minutes past 10 in the evening, by Greenwich time, 
 but at the point where the equator cuts the arc NB it is but 6 
 o'clock in the evening. The longitude corresponding to this differ- 
 ence in time is 67{° west, which is in the southern part of Vene- 
 zuela, in South America. Now turning the globe on its axis till 
 this point is brought under the graduated quadrant, (adjusted so as 
 to represent NB,) and counting upward 29° from the wooden hori- 
 zon, we find that the eclipse leaves the earth in the Atlantic ocean, 
 about 800 miles easterly from Bermuda. 
 
 118. Let it now be required to find where the eclipse is central 
 at any time during its passage across the disc; suppose at 10 
 o'clock P. M. 
 
 Through the point of division representing 10 o'clock on the 
 moon's track, draw sjc parallel to AB. The arc Bx contains about 
 27£°, which must also be the number of degrees in the arc m 10. 
 Bring the foot of the quadrant of altitude to m, which is 45° from 
 E. Turn the globe backward on its axis 7i° from its last posi- 
 tion, such being the amount of its motion between 10 o'clock and 
 29 minutes past 10, the time for which we last calculated. The 
 
75 
 
 27ith degree on the quadrant of altitude, reckoned upward from 
 the wooden horizon, will mark where the eclipse must be central 
 at 10 o'clock. We find, on trial, that iris near the west end of 
 Lake Superior, in North America. 
 i 
 
 119. These mechanical methods give a tolerable approximation 
 to the point where a solar eclipse will be central at any time ; and 
 by taking a sufficient number of points, we may delineate its gene- 
 ral track. But where much accuracy is required, recourse must 
 be had to calculation. The theory of the following process is the 
 same as that of the mechanical method just employed. 
 
 120. The first step is to find the time when the centre of the 
 moon's shadow first strikes or leaves the earth,* (I will here adopt 
 the latter,) and the sun's longitude at the time. The calculation is 
 a very easy one, if the moon's track across the disc is considered 
 as a straight line, and the earth as a perfect sphere. 
 
 F^ 20. Let ASBK (Fig. 20) repre- 
 
 sent the earth's disc, and YZ 
 the moon's track across it. 
 In the triangle ELM, the two 
 sides, EL and EM are known, 
 the former being the moon's 
 latitude at the time of new 
 moon, and the latter its equa- 
 torial parallax. Also, the an- 
 gle ELM is known, being 
 equal to that of the moon's 
 path with the ecliptic, increased by 90°. Hence we can find the 
 side LM, and the angle LEM, or its complement MEB. Knowing 
 the moon's hourly motion, we can easily tell how long it would 
 take it to pass from L to M, which, added to the time of new moon, 
 will give the time of its leaving the earth. 
 
 The sun's motion in longitude during this interval can be calcu- 
 lated from its hourly motion, and thus its longitude at the time the 
 eclipse leaves the earth will be known. 
 
 If still greater accuracy be required, the moon's latitude, and 
 
 * It is immaterial whether we take the time when the centre of the shadow strikes or 
 leaves the earth, provided wc make the other parts of the process to correspond. 
 
76 
 
 the difference in the longitudes of the sun and moon, at the time 
 last found, can be computed from the tables ; the former of which 
 is represented by the line MP, and the latter by EP. Also we 
 may compute again the angle of the moon's path with the ecliptic, 
 and the horary motions. With these data we could easily calcu- 
 late the distance of the point M from the circumference, reckoned 
 on the line YZ, and how long it would take the moon to pass 
 over it 
 
 121. Next, find the longitude of the sun, and the latitude and 
 longitude of the moon at the time for which we wish to calculate 
 the position of the centre of the eclipse. This may be done either 
 from their hourly motions and the angle of the moon's path with 
 1he ecliptic, or, if greater accuracy be required, directly from the 
 tables. 
 
 Let the sun's longitude thus found = s. 
 
 Let the moon's latitude do. = I. 
 
 Let the difference in the longitudes of the sun and moon = d. 
 
 Let the sun's longitude at the time the eclipse leaves the earth 
 
 Convert the time to which the calculations refer into degree?, 
 minutes and seconds, reckoning 15° for an hour; subtract there- 
 from 90°, (borrowing 360° if necessary,) and let the remainder 
 
 Let the moon's equatorial parallax, which may be regarded as 
 constant during the eclipse, =p. 
 
 Let the obliquity of the ecliptic to the equator =m. 
 
 Let P, Pi, Ps, &c. = sundry arcs and angles obtained during 
 the process of computation. 
 
 Let the required latitude of the centre of the shadow = x. 
 
 Let the required longitude, reckoned westerly from the meridian 
 of Greenwich, or of that place for which the time is given, = y. 
 
77 
 
 Then -=sin. P, 
 P 
 
 d • -r> 
 = T -:=Sin. Pi, 
 
 f p X COS. P 
 
 *S±Pl=P2, 
 
 sin. mxcos. P2=sin. P3, 
 
 tan. ???xsin. p2=tan. P4, 
 
 fsin. (P±P4)xcos. P3=sin. a?=the latitude, 
 
 cot. s' 
 
 cos. m 
 tan. P2 
 
 =tan. Ps, 
 Pe, 
 
 cos. m 
 
 sin. P3 
 
 cos. x 
 
 sin. #xtan. P7=Ps, 
 
 jp 5 — P 6 +^±Ps=y=the longitude. 
 
 The chief difficulty in applying these equations consists in know- 
 ing which of the four possible values to give to P, Pi, P2, &c. 
 The following statements will remove all doubt. 
 
 P, Pi, P3 and P4 are each always less than 90°. 
 
 Ps is always of the same affection as s' increased by 90°. 
 
 P6 is always of the same affection as P2. 
 
 P7 is less or greater than 90°, according as P is less or greater 
 than the complement of P4; it never exceeds 180°. 
 
 Ps is always of the same affection as P7. 
 
 122. It is impossible, by a mere description, to convey to the 
 reader a clear idea of the reason of the several steps of this pro- 
 cess ; but if he will take his globe, and adjust it in the same manner 
 as he would do to find the position of the centre of the eclipse by 
 the previous mechanical process, he may be able to discover suc- 
 
 * If after the new moon -j- ; if before it — . 
 
 t The sign before P4 is + if P2 is less than 180° ; but — if it is greater. And in the 
 latter case if P4 is greater than P, the latitude of the place will be opposite in character 
 to that of the moon ; i. e. if the moon's latitude is north that of the place will be south, 
 and the contrary. 
 
 X The sign before Ps is -f- if Ps is between 0° and 90°, or between 270° and 360° ; 
 but — if P2 is between 90° and 270°. 
 
78 
 
 cessively the arcs and angles expressed byP, Pi, &c. ; and hence 
 to understand the method by which they are obtained. 
 
 P is the distance of the centre of -the eclipse from the ecliptic, 
 measured on a secondary to it drawn upon the earth's surface, as m 
 10, (Fig. 19.) Or, it is the latitude of that point in the heavens, on 
 which an observer, placed at the centre of the earth, would see the 
 centre of the shadow at the earth's surface projected, if the earth 
 were transparent. 
 
 Pi is an arc of the ecliptic, intercepted between the aforesaid 
 secondary and the point where the sun is vertical. Or, it is the 
 difference between the sun's longitude and that of the aforesaid 
 point in the heavens. 
 
 Pa is the same arc increased by the sun's longitude. Or, it is the 
 longitude of the aforesaid point in the heavens. 
 
 P3 is an arc of &. great circle, drawn from the north pole of the 
 equator, perpendicular to the aforesaid secondary. 
 
 P4 is the arc of the secondary, intercepted between this perpen- 
 dicular and the north pole of the ecliptic. 
 
 Ps is the right ascension of that point in the equator where it is 
 cut by a secondary to the ecliptic passing through the centre of 
 the shadow on the earth's surface, at the time that, it leaves the 
 earth ; or, it is the right ascension of that point in the heavens on 
 which the centre of the shadow would be seen projected at that 
 time, by an observer at the centre of the earth. 
 
 P6 is the right ascension of that point in the equator, where it is 
 cut by the first mentioned secondary. 
 
 P7 is the angle at the centre of the shadow, or at the first men- 
 tioned point in the heavens, contained between secondaries to the 
 ecliptic and equator passing through that point. 
 
 Ps is the arc of the equator intercepted between these two secon- 
 daries. 
 
 133. To apply the process to a particular case, let it be required 
 to find the place where the solar eclipse which we have taken as 
 an example, will be central at 20 minutes and 51 seconds past 10, 
 
79 
 
 by Greenwich time. After the preparatory steps described in 
 articles 120 and 121, we have the following data and results : — 
 
 DATA. 
 
 
 RESULTS. 
 
 Time=10£. 20m. 51sec. 
 
 P = 28° 
 
 58' 
 
 22" 
 
 s'= 65° 16' 50" 
 
 Pi = 66 
 
 29 
 
 27 
 
 5=65° 16' 34" 
 
 P 2 =131 
 
 46 
 
 1 
 
 /= 0°.4394 
 
 P 3 = 15 
 
 22 
 
 40 
 
 d= 0°.7278 
 
 P 4 = 17 
 
 56 
 
 12 
 
 Z=65° 12' 45" 
 
 a;= 44 
 
 45 
 
 28=the latitude. 
 
 p= 0°.9072 
 
 Ps=153 
 
 21 
 
 5 
 
 ?rc=23° 27' 34".5 
 
 Pg=129 
 
 19 
 
 34 
 
 
 Pt= 21 
 
 55 
 
 45 
 
 
 P 8 == 15 
 
 49 
 
 38 
 
 
 y= 73 
 
 24 
 
 38— the longitude 
 
 Thus we find that the centre of the shadow, at the time just 
 mentioned, is in lat. 44° 45' 28", and Ion. 73° 24' 38", which is on 
 the west shore of Lake Champlain, about four miles north of the 
 village of Piattsburg. 
 
 J24. In bringing to a close our examination of the solar eclipse 
 of 1854, it may not be uninteresting to give a general description 
 of it. 
 
 The centre of the eclipse first strikes the earth at 54 minutes 
 and 27 seconds past 6 P. M., (Greenwich time,) in the Pacific 
 ocean, not far from the Caroline Islands, and travelling northeast- 
 wardly, nearly over the Sandwich Islands, strikes the American 
 coast a little north of Astoria, in Oregon Territory, about 24 min- 
 utes past 9. Crossing the Oregon Territory, it enters the British 
 possessions, and turning easterly, and then southeasterly, re-enters 
 the United States territories west of Lake Superior. At 13 min- 
 utes past 10 it crosses the outlet of Lake Superior, about 100 miles 
 N. W. from Michilimackinack. After reaching the settled parts 
 of Canada, it passes a little south of Bytown, travelling at a rate of 
 70 miles per minute, and reaches the St. Lawrence about 20 miles 
 below St. Regis, at 20 minutes past 10. Again entering the United 
 States, on the north line of New York, it arrives at the west shore 
 of Lake Champlain, about four miles north of Pittsburgh, at 20 
 minutes and 51 seconds past 10. It strikes the opposite shore in 
 
80 
 
 the south part of the town of Georgia, and from thence passes 
 through the following towns in Vermont, viz., Fairfax, Fletcher, 
 Cambridge, Stirling, Morriston, Elmore, Woodbury, Cabot, Dan- 
 ville, Barnet, Waterford, and reaches the Connecticut river at 21 
 minutes and 45 seconds past 10. Travelling now about 100 miles 
 per minute, it passes through the towns of Littleton and Bethlehem, 
 in New Hampshire, and from thence directly over the Notch in 
 the White Mountains, and through Adams and Chatham into 
 Maine. After passing through Fryeburg, Denmark, Bridgetown, 
 Sebago, and across the pond into Windham, Gray, Cumberland, 
 and Yarmouth, it strikes the Atlantic in the latter town, about ten 
 miles, in a direct line, from Portland. It leaves the earth at 28 
 minutes and 55 seconds past 10, in lat. 32° 36' 6", Ion. 49° 44' 12", 
 which is in the Atlantic ocean, about 800 miles east of Bermuda.* 
 
 Since the apparent size of the moon at the time of the eclipse is 
 less than that of the sun, (95,) the eclipse cannot be total at any 
 place ; but along the line we have described, the visible part of the 
 sun will appear as a very slender bright ring, encircling the moon. 
 This appearance will extend about fifty miles on each side, taking 
 in Burlington, Middlebury, Dartmouth, Bowdoin, and Waterville 
 colleges ; the ring will appear of uniform width only along the 
 central line. Such eclipses as this are called annular. 
 
 CHAPTER XIV. 
 
 DELINEATION OF A LUNAR ECLIPSE. 
 
 125. The delineation of a lunar eclipse is extremely simple, 
 since it consists merely in representing the passage of the moon 
 across the earth's shadow. To show the method of effecting it, 
 we will proceed to delineate the lunar eclipse of November, 1844, 
 from the elements obtained in chapter 11th. The angular dhnen- 
 
 * By taking into account several minute circumstances that we have disregarded, the 
 central track of the eclipse may vary slightly from this description, probably passing ten 
 or fifteen miles further north, and nearly over Bowdoin college. 
 
81 
 
 sions given in the 4th, 6th, 8th and 10th elements being supposed 
 to be all taken at the same distance, viz., the distance from the earth 
 to the moon, will serve as a measure for their absolute dimen- 
 sions, in the same manner as they did in the solar eclipse. 
 
 The first step is to make the larger and smaller scales SS and 
 ss (Fig. 21) just as was done in the solar eclipse, (101 and 109.) 
 Take from the longer one the semidiameter of the earth's shadow, 
 viz., 0°.6306, and with it describe the graduated circle BDAE, 
 which will represent the shadow. It is evident that the 
 plane of the ecliptic must bisect the shadow, apd we therefore 
 draw the two diameters AB and DE at right angles to each other, 
 the former to represent a section of the plane of the ecliptic, and 
 the latter its axis. 
 
 120. The moon's latitude is 0°.1810 north. We therefore take 
 this distance from the scale SS, and measure it upward from C 
 toward D, which gives M as the place of the moon's centre at the 
 time of full moon. If the latitude were south, the centre would be 
 found in the line CE. 
 
 127. Its path makes an angle of 5° 45' 41" with the ecliptic, 
 tending south. If, therefore, we draw CF, making an angle of that 
 size with CB, and YMZ parallel to it, the latter line will represent 
 the track of the moon's centre across the shadow. It passes M at 
 40 minutes and 14 seconds past 11 in the evening, by Greenwich 
 time, anp! its position at any other hour and minute may be found 
 by graduating the line YZ, as directed in article 109th. 
 
 128. By taking the moon's semidiameter, 0°.2453, from the scale 
 SS, and with it describing a circle from any point in its path as a 
 centre, the position of the entire disc will be shown, as it must exist 
 at the time indicated at its centre on the path. It is drawn in the 
 plate in five different positions : 1st, when it begins to impinge on 
 the shadow at G, which must be the commencement of the eclipse : 
 2d, when it just falls wholly within the shadow at H, at which 
 time the eclipse must begin to be total : 3d, when its centre is at 
 N, found by drawing CN perpendicular to YZ, and thus (Euc. 3, 
 2) bisecting the chord, which must be the middle of the eclipse : 
 4th, when it begins to leave the shadow at L, at which time it 
 
 6 
 
82 
 
 must cease to be total, and 5th, when it entirely leaves the shadow 
 at P, which must be the end of the eclipse. The respective cen- 
 tres are at R, S, N, T and V, and the time may be determined 
 very nearly by the drawing. 
 
 129. More accurate results may, however, be obtained by cal- 
 culating the length of MR, MS, MN, MT and MY, and then find- 
 ing by the relative hourly motion of the moon, how long it must 
 take it to pass over them. In the right angled triapgle CNM, the 
 side CM and the angle MCN are known, being our 4th and lltfr 
 elements, and we can find CN and MN. The sirfe CN is common 
 to the two right angled triangles SNC and RNC, and the sides CS 
 and CR are also known, the former being the difference, and the 
 lattjr the sum of our 8th and 10th elements. Jlpnce NS anc( NR 
 can be found, and likewise their equals NT and NV. Now, by 
 adding and subtracting MN, which is known, we shall have the 
 lines required. 
 
 130. The times obtained by this process are as follows :— ~. 
 
 Commencement of the eclipse, - ? 
 
 Begins to be total, - 
 
 Middle, - 
 
 Ceases to be total, 
 
 End of the eclipse, 
 
 Duration of total obscuration, - 
 Duration of the eclipse, - 
 
 The foregoing is Greenwich time, but can readily be converted 
 into that of any other place, by allowing for the difference of lon- 
 gitude. 
 
 If strict accuracy were aimed at, the elements should be calcu- 
 lated at several intervals during the eclipse, as they are liable to 
 vary considerably. 
 
 h. 
 
 m. 
 
 aec. 
 
 9 
 
 48 
 
 59 
 
 10 
 
 57 
 
 31 
 
 11 
 
 42 
 
 42 
 
 12 
 
 27 
 
 53 
 
 1 
 
 36 
 
 24 
 
 1 
 
 30 
 
 22 
 
 3 47 26 
 
N 
 
 2 
 
 S 
 
 O 
 
 a 
 
 02 
 
 H 
 
 H 
 
 « 
 M 
 
EXPLANATION OF TERMS, 
 
 AS USED IN THIS WORK 
 
 Ecliptic. The apparent annual path of the sun through the heavens. 
 
 Nodes. The points where the orbit of a planet intersects the plane of the ecliptic. 
 
 Ascending Node. That through which the planet passes from the south side of the 
 ecliptic to the north side. 
 
 Descending Node. That through which it returns 'to the south side. 
 
 Line of the Nodes. A straight line connecting the two nodes. 
 
 Equinoctial Points, or Equinoxes. The point where the ecliptic intersects the plane 
 of the equator. 
 
 Vernal Equinox. That through which the sun apparently passes from the south side of 
 the equator to the north side. 
 
 Autumnal Equinox. That through which it returns to the south side. 
 
 Solstitial Points, or Solstices. Points in the ecliptic midway between the equi- 
 noxes. 
 
 Perigee. The point where the sun or moon approaches nearest the earth. 
 
 Apogee. The point where they are most distant from the earth. 
 
 Apsis. The common name for apogee or perigee. 
 
 Apsides. The plural of apsis. 
 
 Line of the Apsides. A straight line joining the two apsides. 
 
 Conjunction. In the same direction as the sun. 
 
 Opposition. In an opposite direction from the sun. 
 
 Stzygy. The common name for conjunction and opposition. 
 
 Quadrature. Points in the moon's orbit midway between the syzygies. 
 
 Radius Vector. A straight line drawn from a revolving body to the centre about which 
 it revolves. 
 
 Latitude of a heavenly body. Its distance north or south of the ecliptic. 
 
 Longitude of a heavenly body. Its distance eastwardly from the vernal equinox, 
 measured on the ecliptic. 
 
 Right Ascension. The same, measured on the equator. 
 
 Declination. The distance of a heavenly body, north or south, from the equator. 
 
 Lunation. The time from one new or full moon to another. 
 
 Parallax. The apparent change in the place of a heavenly body, when viewed from 
 different points. It is always equal to the angle which a line connecting the 
 points of observation would subtend, when viewed from the body. 
 
Note. — It is thought best to omit, in the present edition of this 
 work, a sequel, or second part, now in manuscript, and to which 
 there have been several references in the foregoing pages, ex- 
 plaining the method of calculating most of the lunar motions and 
 inequalities directly from the laws of elliptical motion and the 
 principles of gravitation, without the aid of tables. It may ap- 
 pear hereafter. 
 
 Errata. — Page 15th, &c. for elipse read ellipse; page 27, near 
 the bottom, for syzygyes read syzygies. 
 
 Page 32, Fig. 11, the lines NE and EP should be in the same 
 straight line. 
 
 I 
 
ASTRONOMICAL TABLES 
 
LIST OF TABLES 
 
 1. Elements of Orbits of Sun and Moon. 
 
 2. Mean New Moons in March, &c. 
 
 3. Mean Lunations. 
 
 4. Mean Motions in hours, minutes and seconds. 
 5 Days of the Year reckoned from March. 
 
 6. Annual Equation of the Moon's Perigee. 
 
 7. Annual Equation of the Moon's Node. 
 
 8. Equation of the Sun's Centre. 
 
 9. Equation of the Moon's Centre. 
 
 10. Annual Equation of the Moon's Longitude. 
 
 11. Secular Equation of the Moon's Longitude. 
 12 Variation. 
 
 13. Evection. 
 
 14. Annual Equation of Variation. 
 Annual Equation of Evection. 
 Nodal Equation of Moon's Longitude. 
 Reduction to the Ecliptic. 
 
 18. Lunar or Menstrual Equation of the Sun's Longitude. 
 
 19. Lunar Nutation in Longitude. 
 Sun's Semidiameter and Hourly Motion. 
 
 Moon's Semidiameter, Hourly Motion and Equatorial Parallax. 
 Do. as affected by Evection. 
 Obliquity of the Ecliptic to the Equator. 
 Moon's Latitude in Eclipses. 
 
 Angle of the visible path of the Moon with the Ecliptic in Eclij 
 Sun's Declination. 
 
 27. First Preliminary Equation. 
 
 28. Second Preliminary Equation. 
 
 29. Augmentation of the Moon's Semidiameter. 
 
 30. To convert minutes into decimals of a degree. 
 
 31. To convert seconds into decimals of a degree. 
 
 15. 
 16. 
 17. 
 
 TABLE I. 
 
 Elements of Orbits of Sun and Moon. 
 
 Mean longitude, Jan. 1, 1801, .... 
 
 Sun. 
 
 Moon. 
 
 O / // 
 
 280 39 13.17 
 
 118 17' 8.3 
 
 Motion in 100 years or 36525 days, - - - - 
 
 36000 46 0.77 
 
 481267 52 41.6 
 
 Mean longitude of perigee, Jan. 1, 1801, 
 
 279 31 9.71 
 
 266 10 7.5 
 
 Motion of do. eastward in 100 vears, ... 
 
 1 42 56.0 
 
 4069 2 46.6 
 
 Longitude of Moon's JNode, Jan. 1, 1801, 
 
 
 13 53 17.7 
 
 Motion of do. westward, in 100 years, ... 
 
 
 1934 9 57.5 
 
TABLE II. 
 
 Mean New Moon, SfC. in March. 
 
 
 Mean New Moon in 
 
 Sun's Mean 
 
 Moon's Mean 
 
 Sun and Moon's 
 
 Longitude of 
 
 Node. 
 
 Year. 
 
 
 March. 
 
 
 Anomaly. 
 
 Anomaly. 
 
 Mean Longitude. 
 
 1800 
 
 d. 
 25 
 
 h. 
 
 
 m. 
 
 18 
 
 sec. 
 49 
 
 83°208 
 
 127.960 
 
 2.7° 39 
 
 28.8207 
 
 1810 
 
 5 
 
 6 
 
 36 
 
 43 
 
 63.167 
 
 63.441 
 
 342.8436 
 
 196.4758 
 
 1820 
 
 14 
 
 1 
 
 38 
 
 40 
 
 72.232 
 
 24.740 
 
 352.0799 
 
 2.5671 
 
 1830 
 
 23 
 
 20 
 
 40 
 
 37 
 
 81.295 
 
 346.038 
 
 1.3163 
 
 168.6535 
 
 1840 
 
 3 
 
 2 
 
 58 
 
 31 
 
 61.254 
 
 281.520 
 
 341.4458 
 
 336.3135 
 
 1841 
 
 22 
 
 
 
 31 
 
 8 
 
 79.624 
 
 257.140 
 
 359.8335 
 
 315.9844 
 
 1842 
 
 11 
 
 9 
 
 19 
 
 42 
 
 68.888 
 
 206.943 
 
 349.1144 
 
 297.2190 
 
 1843 
 
 
 
 18 
 
 8 
 
 17 
 
 53.152 
 
 156.746 
 
 333.3954 
 
 278.4536 
 
 1844 
 
 18 
 
 15 
 
 40 
 
 54 
 
 76.522 
 
 132.366 
 
 356.7831 
 
 253.1245 
 
 1845 
 
 8 
 
 
 
 29 
 
 28 
 
 65.786 
 
 82.169 
 
 346.0640 
 
 23^.3591 
 
 1846 
 
 26 
 
 22 
 
 2 
 
 6 
 
 84.156 
 
 57.789 
 
 4.4517 
 
 219.0300 
 
 1847 
 
 16 
 
 6 
 
 50 
 
 40 
 
 73.420 
 
 7.592 
 
 353.7326 
 
 200.2646 
 
 1848 
 
 4 
 
 15 
 
 39 
 
 15 
 
 62.684 
 
 317.395 
 
 343.0135 
 
 181.4991 
 
 1849 
 
 23 
 
 13 
 
 11 
 
 52 
 
 81.054 
 
 293.015 
 
 1.4012 
 
 161.1702 
 
 1850 
 
 12 
 
 22 
 
 
 
 27 
 
 70 318 
 
 242.819 
 
 350.6822 
 
 142.4048 
 
 1851 
 
 2 
 
 6 
 
 49 
 
 1 
 
 59.583 
 
 192.622 
 
 339.9631 
 
 123.6394 
 
 1852 
 
 20 
 
 4 
 
 21 
 
 38 
 
 77.952 
 
 168.242 
 
 353.3508 
 
 103.3103 
 
 1853 
 
 9 
 
 13 
 
 10 
 
 13 
 
 67.217 
 
 118-045 
 
 347.6317 
 
 84.5449 
 
 1854 
 
 28 
 
 10 
 
 42 
 
 50 
 
 85.586 
 
 93.665 
 
 6.0194 
 
 64.2158 
 
 1855 
 
 17 
 
 19 
 
 31 
 
 24 
 
 74.851 
 
 43.468 
 
 355.3003 
 
 45.4504 
 
 1856 
 
 7 
 
 4 
 
 19 
 
 59 
 
 64.115 
 
 353.271 
 
 344.5813 
 
 26.6851 
 
 1857 
 
 25 
 
 1 
 
 52 
 
 36 
 
 82.485 
 
 328.891 
 
 2.9689 
 
 6.3560 
 
 1858 
 
 14 
 
 10 
 
 41 
 
 11 
 
 71.749 
 
 278.694 
 
 352.2499 
 
 347.5906 
 
 1859 
 
 3 
 
 19 
 
 29 
 
 46 
 
 61.013 
 
 228.497 
 
 341.5308 
 
 328.8252 
 
 1860 
 
 21 
 
 17 
 
 2 
 
 24 
 
 79.383 
 
 204.117 
 
 359.9185 
 
 308.4961 
 
 1861 
 
 11 
 
 1 
 
 50 
 
 58 
 
 68.647 
 
 153920 
 
 349.1994 
 
 289.7307 
 
 1862 
 
 
 
 10 
 
 39 
 
 33 
 
 57.911 
 
 103 723 
 
 338.4803 
 
 270.9653 
 
 1863 
 
 19 
 
 8 
 
 12 
 
 10 
 
 76.281 
 
 79.343 
 
 356.8680 
 
 250.6362 
 
 1864 
 
 7 
 
 17 
 
 
 
 45 
 
 65.545 
 
 29.146 
 
 346.1490 
 
 231.8708 
 
 1865 
 
 26 
 
 14 
 
 33 
 
 22 
 
 83.915 
 
 4.766 
 
 4.5366 
 
 211.5417 
 
 1866 
 
 15 
 
 23 
 
 21 
 
 57 
 
 73.179 
 
 314.569 
 
 353.8176 
 
 192.7763 
 
 1867 
 
 5 
 
 8 
 
 10 
 
 31 
 
 62.443 
 
 264.372 
 
 343.0985 
 
 174.0110 
 
 1868 
 
 23 
 
 5 
 
 43 
 
 9 
 
 80.813 
 
 239.992 
 
 1.4862 
 
 153 6819 
 
 1869 
 
 12 
 
 14 
 
 31 
 
 43 
 
 70.077 
 
 189.795 
 
 350.7671 
 
 134.9165 
 
 1870 
 
 1 
 
 23 
 
 20 
 
 18 
 
 59.34i 
 
 139.599 
 
 340.0481 
 
 116.1511 
 
 1871 
 
 20 
 
 20 
 
 52 
 
 55 
 
 77.711 
 
 115.219 
 
 358.4357 
 
 95 8220 
 
 1872 
 
 9 
 
 5 
 
 41 
 
 29 
 
 66.976 
 
 65.022 
 
 347.7167 
 
 77 0567 
 
 1873 
 
 28 
 
 3 
 
 14 
 
 6 
 
 85.346 
 
 40.642 
 
 6.1043 
 
 56'7275 
 
 1874 
 
 17 
 
 12 
 
 2 
 
 41 
 
 74.610 
 
 350.445 
 
 355.3853 
 
 37*9622 
 
 1875 
 
 6 
 
 20 
 
 51 
 
 15 
 
 63.874 
 
 300.248 
 
 344.6662 
 
 19*1968 
 
 1876 
 
 24 
 
 18 
 
 23 
 
 53 
 
 82.244 
 
 275.868 
 
 3.0539 
 
 358*8677 
 
 1877 
 
 14 
 
 3 
 
 12 
 
 28 
 
 71.508 
 
 225.671 
 
 352.3348 
 
 340 1023 
 
 1878 
 
 3 
 
 12 
 
 1 
 
 3 
 
 60.772 
 
 175.474 
 
 341.6158 
 
 3213370 
 
 1879 
 
 22 
 
 9 
 
 33 
 
 40 
 
 79.142 
 
 151.094 
 
 0.0034 
 
 301 0079 
 
 1880 
 
 10 
 
 18 
 
 22 
 
 15 
 
 68.406 
 
 100.897 
 
 349.2844 
 
 2822425 
 
 1881 
 
 
 
 3 
 
 10 
 
 50 
 
 57.670 
 
 50.700 
 
 338.5653 
 
 2634771 
 
 1882 
 
 19 
 
 
 
 43 
 
 27 
 
 76.040 
 
 26.320 
 
 356.9530 
 
 243 1480 
 
 1883 
 
 8 
 
 9 
 
 32 
 
 2 
 
 65.304 
 
 336.123 
 
 346.2339 
 
 2243826 
 
 1884 
 
 26 
 
 7 
 
 4 
 
 39 
 
 83.674 
 
 311.743 
 
 4.6216 
 
 2040535 
 
 1885 
 
 15 
 
 15 
 
 53 
 
 14 
 
 72.938 
 
 261.546 
 
 353.9025 
 
 185'2881 
 
 1886 
 
 5 
 
 
 
 41 
 
 49 
 
 62.202 
 
 211.349 
 
 343.1835 
 
 1665228 
 
 1887 
 
 23 
 
 22 
 
 14 
 
 26 
 
 80.572 
 
 186.969 
 
 1.5712 
 
 1461937 
 
 1888 
 1889 
 
 12 
 
 7 
 
 3 
 
 
 
 69.836 
 
 136.772 
 
 350.8521 
 
 127*4283 
 
 1 
 
 15 
 
 51 
 
 35 
 
 59.100 
 
 86.575 
 
 340.1330 
 
 108'6629 
 
 1890 
 
 20 
 
 13 
 
 24 
 
 12 
 
 77.470 
 
 62.196 
 
 353.5207 
 
 88*3337 
 
 1891 
 
 9 
 
 22 
 
 12 
 
 47 
 
 66.734 
 
 11.999 
 
 347.8016 
 
 69*5683 
 
 1892 
 
 27 
 
 19 
 
 45 
 
 24 
 
 85.104 
 
 347.619 
 
 6 1893 
 
 492392 
 
 1893 
 
 17 
 
 4 
 
 33 
 
 59 
 
 74.368 
 
 297 422 
 
 3554703 
 
 30*473;) 
 
 1894 
 
 6 
 
 13 
 
 22 
 
 33 
 
 63.632 
 
 247 225 
 
 344.7512 
 
 11*7085 
 
 1895 
 1896 
 
 25 
 
 10 
 
 55 
 
 10 
 
 82.002 
 
 222 845 
 
 3 1389 
 
 351*3794 
 
 13 
 
 19 
 
 43 
 
 45 
 
 71.266 
 
 172 648 
 
 352*4 198 
 
 332.6140 
 
 1897 
 
 3 
 
 4 
 
 32 
 
 19 
 
 60.530 
 
 122 451 
 
 341.7007 
 
 3 13 8486 
 
 1898 
 
 22 
 
 2 
 
 4 
 
 56 
 
 78.900 
 
 98 071 
 
 0.0884 
 
 2^.5195 
 
 1899 
 
 11 
 
 10 
 
 53 
 
 31 
 
 68.164 
 
 47.874 
 
 349.3694 
 
 274.7542 
 
 1900 
 
 
 
 19 
 
 42 
 
 6 
 
 57.428 
 
 357.677 
 
 338.6503 
 
 255.9888 
 
 This Table shows the time of New Moon in March, of each year, with the longitudes, anom- 
 alies, &c. of the Sun and Moon at that time, on the supposition that all the motions are per- 
 torrned with uniform angular velocity. r 
 
TABLE III. 
 
 Mean Lunations. 
 
 No. 
 
 
 
 
 
 Sun's Mean 
 
 Moon's Mean 
 
 Sun and Moon's 
 
 Longitude of 
 
 Lun. 
 
 Mean Lunations. 
 
 Anomaly. 
 
 Anomaly. 
 
 Mean Longitude. 
 
 Node. 
 
 
 d. 
 
 A. 
 
 m. 
 
 sec. 
 
 
 
 
 
 1 
 
 29 
 
 12 
 
 44 
 
 3 
 
 29.105 
 
 2%.817 
 
 2§.1067 
 
 1.5%38 
 
 2 
 
 59 
 
 1 
 
 28 
 
 6 
 
 58.211 
 
 51634 
 
 58 2135 
 
 3.1275 
 
 3 
 
 88 
 
 14 
 
 12 
 
 9 
 
 87.316 
 
 77.451 
 
 87.3202 
 
 4.6914 
 
 4 
 
 118 
 
 2 
 
 56 
 
 12 
 
 116.421 
 
 103.268 
 
 116.4270 
 
 6.2551 
 
 5 
 
 147 
 
 15 
 
 40 
 
 14 
 
 145.527 
 
 129.085 
 
 145.5/337 
 174.6405 
 
 7.8189 
 
 5 
 
 177 
 
 4 
 
 24 
 
 17 
 
 174.632 
 
 154.906 
 
 9.3827 
 
 7 
 
 206 
 
 17 
 
 8 
 
 20 
 
 203.738 
 
 180.718 
 
 203.7472 
 
 109465 
 
 8 
 
 236 
 
 5 
 
 52 
 
 23 
 
 232.843 
 
 206.535 
 
 232.8530 
 
 12.5102 
 
 9 
 
 265 
 
 18 
 
 36 
 
 26 
 
 261.948 
 
 232.352 
 
 261.9607 
 
 14.0740 
 
 10 
 
 295 
 
 7 
 
 20 
 
 29 
 
 291.054 
 
 258.169 
 
 291.0674 
 
 15.6378 
 
 11 
 
 324 
 
 20 
 
 4 
 
 32 
 
 320.159 
 
 283.986 
 
 320.1742 
 
 17.2016 
 
 12 
 
 354 
 
 8 
 
 48 
 
 35 
 
 349.264 j 
 
 309.803 
 
 349.2809 
 
 18.7654 
 
 13 
 
 383 
 
 21 
 
 32 
 
 37 
 
 378.370 
 
 335.620 
 
 378.3877 
 
 20.3291 
 
 h 
 
 14 
 
 18 
 
 22 
 
 1 
 
 14.553 
 
 192.908 
 
 *14.5534 
 
 .7819 
 
 Note. — The true quantities for one lunation, from which these tables are calcu- 
 
 lated, are as follows, viz. 
 
 Length of a lunation, - 29d. 
 
 Sun's mean motion in Anomaly in one lunation, 
 Moon's do. ----- 
 Sun and Moon's mean motion in Longitude, do. 
 Mean motion of the Node, do. 
 
 12A. 
 
 44m. 2.88sec. 
 
 29°.10535764 
 
 25 .81692410 
 
 29 .10674457 
 
 1 .56377989 
 
 TABLE IV. 
 
 
 Mean Motions of the Sun and Moon. 
 
 
 
 Sun's 
 
 Sun's 
 
 Moon's 
 
 Moon's 
 
 Longitude of 
 
 Days. 
 
 Anomaly. 
 
 Longitude. 
 
 Anomaly. 
 
 Longitude. 
 
 Node. 
 
 1 
 
 8.9856 
 
 8.98565 
 
 1§.0650 
 
 1§.17640 
 
 .05295 
 
 2 
 
 1.9712 
 
 1.97129 
 
 26.1300 
 
 26 35279 
 
 .10591 
 
 3 
 
 2.9568 
 
 2.95694 
 
 39.1950 
 
 39.52919 
 
 .15886 
 
 4 
 
 3.9424 
 
 3.94259 
 
 52 2600 
 
 52 70559 
 
 .21182 
 
 5 
 
 4.9280 
 
 4.92824 
 
 65.3250 
 
 65.88198 
 
 .26477 
 
 6 
 
 5.9136 
 
 5.91388 
 
 78 3900 
 
 79.05838 
 
 .31773 
 
 7 
 
 6 8992 
 
 6.89953 
 
 91.4549 
 
 92.23477 
 
 .37068 
 
 8 
 
 7.8848 
 
 7.88518 
 
 104.5199 
 
 105.41117 
 
 .42364 
 
 9 
 
 8 8704 
 
 8.87083 
 
 117.5849 
 
 118.58757 
 
 .47659 
 
 Hours. 
 
 bun's 
 Anomaly. 
 
 .0411 
 .0821 
 .1232 
 .1643 
 .2053 
 .2464 
 .2875 
 .3285 
 .3696 
 
 feun's 
 Longitude. 
 
 .84107 
 .08214 
 .12321 
 .16427 
 .205:34 
 .24641 
 .28748 
 .32855 
 .36962 
 
 Moon's 
 Anomaly. 
 
 0.5444 
 1.0887 
 1.6331 
 2.1775 
 2.7219 
 3.2662 
 3.8106 
 4.3550 
 4.8994 
 
 Moon's 
 Longitude. 
 
 .5^902 
 1.09803 
 1.64705 
 2.19607 
 2.74508 
 3.29410 
 3.84312 
 4 39213 
 4.94115 
 
 Longitude of 
 Node. 
 
 .00221 
 .00441 
 .00662 
 .00883 
 .01104 
 .01324 
 .01545 
 .01765 
 .01986 
 
 For the moon increase this by 180°. 
 
TABLE NO. IV CONTINUED. 
 
 Min- 
 
 Sun's Lon. 
 
 Moon's 
 
 Moon's Lon- 
 
 Lon. of 
 
 Se- 
 
 Sun's Lon. 
 
 Moon's Lon. 
 
 utes. 
 
 and Anom. 
 
 Anom. 
 
 gitude. 
 
 Node. 
 
 conds. 
 
 and Anom. 
 
 and Anom. 
 
 1 
 
 .0<?068 
 
 .0891 
 
 .00°915 
 
 •06004 
 
 1 
 
 .00001 
 
 .0(?015 
 
 2 
 
 .00137 
 
 .0181 
 
 .01830 
 
 .00007 
 
 2 
 
 .00002 
 
 .00030 
 
 3 
 
 .00205 
 
 .0272 
 
 .02745 
 
 .00011 
 
 3 
 
 .00003 
 
 .00046 
 
 4 
 
 .00274 
 
 .0363 
 
 .03660 
 
 .00015 
 
 4 
 
 .00005 
 
 .00061 
 
 5 
 
 .00342 
 
 .0454 
 
 .04575 
 
 .00018 
 
 5 
 
 .00006 
 
 .00076 
 
 6 
 
 .00411 
 
 .0544 
 
 .05490 
 
 .00022 
 
 6 
 
 .00007 
 
 .00091 
 
 7 
 
 .00479 
 
 .0635 
 
 .06405 
 
 .00026 
 
 7 
 
 .00008 
 
 .00107 
 
 8 
 
 .00548 
 
 .0726 
 
 .07320 
 
 .00029 
 
 18 
 
 .00009 
 
 .00122 
 
 9 
 
 .00616 
 
 .0817 
 
 .08235 
 
 .00033 
 
 9 
 
 .00010 
 
 .00137 
 
 
 
 
 
 
 TABLE 
 
 V. 
 
 
 
 
 
 
 Bays of 
 
 the year reckoned from March. 
 
 
 
 Mar. 
 1 
 
 April. 
 
 32 
 
 May. 
 
 62 
 
 June. 
 
 July. 
 
 Aug. 
 154 
 
 Sept. 
 185 
 
 Oct. 
 
 Nov. 
 
 Dec. 
 276 
 
 Jan. 
 
 Feb. 
 338 
 
 93 
 
 123 
 
 215 
 
 246 
 
 307 
 
 2 
 
 33 
 
 63 
 
 94 
 
 124 
 
 155 
 
 186 
 
 216 
 
 247 
 
 277 
 
 308 
 
 339 
 
 3 
 
 34 
 
 64 
 
 95 
 
 125 
 
 156 
 
 187 
 
 217 
 
 248 
 
 278 
 
 309 
 
 340 
 
 4 
 
 35 
 
 65 
 
 96 
 
 126 
 
 157 
 
 188 
 
 218 
 
 249 
 
 279 
 
 310 
 
 341 
 
 5 
 
 36 
 
 66 
 
 97 
 
 127 
 
 158 
 
 189 
 
 219 
 
 250 
 
 280 
 
 311 
 
 342 
 
 6 
 
 37 
 
 67 
 
 98 
 
 128 
 
 159 
 
 190 
 
 220 
 
 251 
 
 281 
 
 312 
 
 343 
 
 7 
 
 38 
 
 68 
 
 99 
 
 129 
 
 160 
 
 191 
 
 221 
 
 252 
 
 282 
 
 313 
 
 344 
 
 8 
 
 39 
 
 69 
 
 100 
 
 130 
 
 161 
 
 192 
 
 222 
 
 253 
 
 283 
 
 314 
 
 345 
 
 9 
 
 40 
 
 70 
 
 101 
 
 131 
 
 162 
 
 193 
 
 223 
 
 254 
 
 284 
 
 315 
 
 346 
 
 10 
 
 41 
 
 71 
 
 102 
 
 132 
 
 163 
 
 194 
 
 224 
 
 255 
 
 285 
 
 316 
 
 347 
 
 11 
 
 42 
 
 72 
 
 103 
 
 133 
 
 164 
 
 195 
 
 225 
 
 256 
 
 286 
 
 317 
 
 348 
 
 12 
 
 43 
 
 73 
 
 104 
 
 134 
 
 165 
 
 196 
 
 226 
 
 257 
 
 287 
 
 318 
 
 349 
 
 13 
 
 44 
 
 74 
 
 105 
 
 135 
 
 166 
 
 197 
 
 227 
 
 258 
 
 288 
 
 319 
 
 350 
 
 14 
 
 45 
 
 75 
 
 106 
 
 136 
 
 167 
 
 198 
 
 228 
 
 259 
 
 289 
 
 320 
 
 351 
 
 15 
 
 46 
 
 76 
 
 107 
 
 137 
 
 168 
 
 199 
 
 229 
 
 260 
 
 290 
 
 321 
 
 352 
 
 16 
 
 47 
 
 77 
 
 108 
 
 138 
 
 169 
 
 200 
 
 230 
 
 261 
 
 291 
 
 322 
 
 353 
 
 17 
 
 48 
 
 78 
 
 109 
 
 139. 
 
 170 
 
 201 
 
 231 
 
 262 
 
 292 
 
 323 
 
 354 
 
 18 
 
 49 
 
 79 
 
 110 
 
 140 
 
 171 
 
 202 
 
 232 
 
 263 
 
 293 
 
 324 
 
 a55 
 
 19 
 
 50 
 
 80 
 
 111 
 
 141 
 
 172 
 
 203 
 
 233 
 
 264 
 
 294 
 
 325 
 
 356 
 
 20 
 
 51 
 
 81 
 
 112 
 
 142 
 
 173 
 
 204 
 
 234 
 
 265 
 
 295 
 
 326 
 
 357 
 
 21 
 
 52 
 
 82 
 
 113 
 
 143 
 
 174 
 
 205 
 
 235 
 
 266 
 
 296 
 
 327 
 
 358 
 
 22 
 
 53 
 
 83 
 
 114 
 
 144 
 
 175 
 
 206 
 
 236 
 
 267 
 
 297 
 
 328 
 
 359 
 
 23 
 
 54 
 
 84 
 
 115 
 
 145 
 
 176 
 
 207 
 
 237 
 
 268 
 
 298 
 
 329 
 
 360 
 
 24 
 
 55 
 
 85 
 
 116 
 
 146 
 
 177 
 
 208 
 
 238 
 
 269 
 
 299 
 
 330 
 
 361 
 
 25 
 
 56 
 
 86 
 
 117 
 
 147 
 
 178 
 
 209 
 
 239 
 
 270 
 
 300 
 
 331 
 
 362 
 
 26 
 
 57 
 
 87 
 
 118 
 
 148 
 
 179 
 
 210 
 
 240 
 
 271 
 
 301 
 
 332 
 
 363 
 
 27 
 
 58 
 
 88 
 
 119 
 
 149 
 
 180 
 
 211 
 
 241 
 
 272 
 
 302 
 
 333 
 
 364 
 
 28 
 
 59 
 
 89 
 
 120 
 
 150 
 
 181 
 
 212 
 
 242 
 
 273 
 
 303 
 
 334 
 
 365 
 
 29 
 
 60 
 
 90 
 
 121 
 
 151 
 
 182 
 
 213 
 
 243 
 
 274 
 
 304 
 
 335 
 
 366 
 
 30 
 
 61 
 
 91 
 
 122 
 
 152 
 
 183 
 
 214 
 
 244 
 
 275 
 
 305 
 
 336 
 
 
 31 
 
 
 92 
 
 
 153 
 
 184 
 
 
 245 
 
 
 306 
 
 337 
 
 
 Tl 
 
 lis Tal 
 
 jle shows the mc 
 
 nth and day to wh 
 
 ich any number 
 
 jf days i 
 
 n a year, 
 
 reck 
 
 aned f 
 
 rom t 
 
 he 1st 
 
 of Ma 
 
 rch, c< 
 
 jrresp< 
 
 )nds. 
 
 
 
 
 
TABLE VI. 
 
 Annual Equation of the Moon's Perigee, 
 Argument — Sun's Anomaly. 
 
 r Arg. 
 
 0° 
 
 2° 
 
 4° 
 
 6° 
 
 8° 
 
 10° 
 
 Arg. 
 
 ~0 
 
 .000 
 
 .013 
 
 .026 
 
 .038 
 
 .051 
 
 .063 
 
 + 
 35 
 
 1 
 
 .063 
 
 .076 
 
 .088 
 
 .101 
 
 .113 
 
 .126 
 
 34 
 
 2 
 
 .126 
 
 •138 
 
 .149 
 
 .161 
 
 .172 
 
 .183 
 
 33 
 
 3 
 
 .183 
 
 .194 % 
 
 .205 
 
 .216 
 
 .226 
 
 .236 
 
 32 
 
 4 
 
 .236 
 
 .245 
 
 .255 
 
 .264 
 
 .273 
 
 .282 
 
 31 
 
 5 
 
 .282 
 
 .290 
 
 .298 
 
 .305 
 
 .312 
 
 .319 
 
 30 
 
 6 
 
 .319 
 
 .325 
 
 .331 
 
 .337 
 
 .342 
 
 .347 
 
 29 
 
 7 
 
 .347 
 
 .351 
 
 .355 
 
 .359 
 
 .362 
 
 .365 
 
 28 
 
 8 
 
 .365 
 
 .367 
 
 .369 
 
 .370 
 
 .371 
 
 .371 
 
 27 
 
 9 
 
 .371 
 
 .371 
 
 .371 
 
 .370 
 
 .369 
 
 .367 
 
 26 
 
 10 
 
 .367 
 
 .365 
 
 .362 
 
 .359 
 
 .355 
 
 .351 
 
 25 
 
 11 
 
 .351 
 
 .347 
 
 .342 
 
 .337 
 
 .331 
 
 .324 
 
 24 
 
 12 
 
 .324 
 
 .318 
 
 .311 
 
 .304 
 
 .296 
 
 .288 
 
 23 
 
 13 
 
 .288 
 
 .280 
 
 .271 
 
 .261 
 
 .252 
 
 .242 
 
 22 
 
 14 
 
 .242 
 
 .231 
 
 .221 
 
 .210 
 
 .199 
 
 .188 
 
 21 
 
 15 
 
 .188 
 
 .177 
 
 .165 
 
 .153 
 
 .141 
 
 .129 
 
 20 
 
 16 
 
 .129 
 
 .117 
 
 .104 
 
 .092 
 
 .079 
 
 .066 
 
 19 
 
 17 
 
 .066 
 
 .052 
 
 .039 
 
 .026 
 
 .013 
 
 .000 
 
 18 
 
 Arg. 
 
 10° 
 
 8° 
 
 6° 
 
 4° 
 
 2° 
 
 0° 
 
 •Arg. 
 
 The sun's attraction causes a progressive motion in the line of the moon's apsides, which affects 
 the place of the perigee; and since the distance of the sun varies in different seasons of the 
 year, its attraction also varies. This causes the motion to be more rapid at some times than at 
 others, occasioning inequalities in the moon's anomaly, for which this Table furnishes the 
 correction. 
 
 TABLE VII. 
 
 Annual Equation of the Moon's Node. 
 Argument — Sun's Anomaly. 
 
 Arg. 
 
 0° 
 
 2° 
 
 4° 
 
 6° 
 
 8° 
 
 10° 
 
 Arg. 
 
 
 
 .0000 
 
 .0053 
 
 .0106 
 
 .0159 
 
 .0212 
 
 .0264 
 
 + 
 35 
 
 1 
 
 .0264 
 
 .0316 
 
 .0367 
 
 .0418 
 
 .0469 
 
 .0520 
 
 34 
 
 2 
 
 .0520 
 
 .0570 
 
 .0619 
 
 .0667 
 
 .0714 
 
 .0760 
 
 33 
 
 3 
 
 .0760 
 
 .0805 
 
 .0849 
 
 .0892 
 
 .0934 
 
 .0975 
 
 32 
 
 4 
 
 .0975 
 
 .1015 
 
 .1053 
 
 .1090 
 
 .1126 
 
 .1160 
 
 31 
 
 5 
 
 .1160 
 
 .1192 
 
 .1223 
 
 .1253 
 
 .1281 
 
 .1308 
 
 30 
 
 6 
 
 .1308 
 
 .1333 
 
 .1356 
 
 .1378 
 
 .1398 
 
 .1417 
 
 29 
 
 7 
 
 .1417 
 
 .1434 
 
 .1449 
 
 .1462 
 
 .1373 
 
 .1481 
 
 28 
 
 8 
 
 .1481 
 
 .1488 
 
 .1493 
 
 .1496 
 
 .1499 
 
 .1500 
 
 27 
 
 9 
 
 .1500 
 
 .1498 
 
 .1494 
 
 .1488 
 
 .1481 
 
 .1473 
 
 26 
 
 10 
 
 .1473 
 
 .1463 
 
 .1451 
 
 .1437 
 
 .1421 
 
 .1402 
 
 25 
 
 11 
 
 .1402 
 
 .1381 
 
 .1359 
 
 .1336 
 
 .1313 
 
 .1289 
 
 24 
 
 12 
 
 .1289 
 
 .1263 
 
 .1235 
 
 .1205 
 
 .1273 
 
 .1138 
 
 23 
 
 13 
 
 .1138 
 
 .1103 
 
 .1067 
 
 .1030 
 
 .0992 
 
 .0953 
 
 22 
 
 14 
 
 .0953 
 
 .0913 
 
 .0871 
 
 .0828 
 
 .0784 
 
 .0740 
 
 21 
 
 15 
 
 .0740 
 
 .0695 
 
 .0648 
 
 .0600 
 
 .0551 
 
 .0501 
 
 20 
 
 16 
 
 .0501 
 
 .0452 
 
 .0403 
 
 .0355 
 
 .0306 
 
 .0257 
 
 19 
 
 17 
 
 .0257 
 
 .0206 
 
 .0155 
 
 .0104 
 
 .0052 
 
 2° 
 
 .0000 
 
 18 
 
 Arg. 
 
 10° 
 
 8° 
 
 6° 
 
 4° 
 
 0° 
 
 Arg. 
 
 The retrograde motion of the moon's nodes is caused by the sun's attraction, and as the dis- 
 tance of that luminary varies in different seasons of the year, its attraction must also vary, pro- 
 ducing inequalities in the motion of the nodes, for which this Table supplies the correction. 
 
TABLE VIII. 
 
 Equation of the Sun's Centre. 
 Argument — Sun's Anomaly. 
 
 Arg. 
 
 0° 
 
 1° 
 
 2° 
 
 3o 
 
 40 
 
 50 
 
 6° 
 
 70 
 
 8° 
 
 9° 
 
 10° 
 
 Arg. 
 35 
 
 + 
 
 
 .0000 
 
 .0343 
 
 .0685 
 
 .1028 
 
 .1370 
 
 .1711 
 
 .2052 
 
 .2393 
 
 .2733 
 
 .3071 
 
 .3409 
 
 1 
 
 .3409 
 
 .3746 
 
 .4081 
 
 .4415 
 
 .4748 
 
 .5079 
 
 .5408 
 
 .5736 
 
 .6062 
 
 •6386 
 
 .6708 
 
 34 
 
 2 
 
 .6708 
 
 .7028 
 
 .7345 
 
 .7660 
 
 .7972 
 
 .8282 
 
 .8590 
 
 .8895 
 
 .9196 
 
 .9495 
 
 .9790 
 
 33 
 
 3 
 
 .9790 
 
 1.0083 
 
 1.0372 
 
 1.0658 
 
 1.0941 
 
 1.1220 
 
 1.1493 1.1767 
 
 1.2035 
 
 1.2299 
 
 1.2559 
 
 32 
 
 4 
 
 1.2559 
 
 1.2815 
 
 1.3067 
 
 1.3315 
 
 1.3559 
 
 1.3799 
 
 1.403411.4264 
 
 1.4491 
 
 1.4711 
 
 1.4928 
 
 31 
 
 5 
 
 1.4928 
 
 1.5140 
 
 1.5347 
 
 1.5549 
 
 1.5747 
 
 1.5939 
 
 1.6127 1.6309 
 
 1.6486 
 
 1.6658 
 
 1.6824 
 
 30 
 
 6 
 
 1.6824 
 
 1.6986 
 
 1.7142 
 
 1.7293 
 
 1.7439 
 
 1.7578 
 
 1.7712 1.7841 
 
 1.7965 
 
 1.8112 
 
 1.8194 
 
 29 
 
 7 
 
 1.8194 1.8301 
 
 1.8401 
 
 1.8496 
 
 1.8585 
 
 1.8669 
 
 1.8747 1.8819 
 
 1.8885 
 
 1.8945 
 
 1.9000 
 
 28 
 
 8 
 
 1.9000 1.9049 
 
 1.9091 
 
 1.9128 
 
 1.9159 
 
 1.9185 
 
 1.9204 1.9217 
 
 1.9225 
 
 1.9226 
 
 1.9223 
 
 27 
 
 9 
 
 1.9223 1.9213 
 
 1.9197 
 
 1.9175 
 
 1.9148 
 
 1.9115 
 
 1.9075 1.9030 
 
 1.8980 
 
 1.8924 
 
 1.8862 
 
 26 
 
 10 
 
 1.8862 
 
 1.8794 
 
 1.8721 
 
 1.8642 
 
 1.8557 
 
 1.8467 
 
 1.8372| 1.8270 
 
 1.8164 
 
 1.8052 
 
 1.7935 
 
 25 
 
 1] 
 
 1.7935 
 
 1.7812 
 
 1.7684 
 
 1.7551 
 
 1.7412 
 
 1.7269 
 
 1.7120; 1.6966 
 
 1.6807 
 
 1.6644 
 
 1.6475 
 
 24 
 
 12 
 
 1.6475 
 
 1.6301 
 
 1.6123 
 
 1.5940 
 
 1.5752 
 
 1.5560 
 
 1.5362J 1.5161 
 
 1.4955 
 
 1.4745 
 
 1.4530 
 
 23 
 
 13 
 
 1.4530 
 
 1.4311 
 
 1.4088 
 
 1.3861 
 
 1.3630 
 
 1.3395 
 
 1.3155 1.2915 
 
 1.2666 
 
 1.2415 
 
 1.2162 
 
 22 
 
 14 
 
 1.2162 
 
 1.1904 
 
 1.1643 
 
 1.1379 
 
 1.1111 
 
 1.0840 
 
 1.0566 
 
 1.0289 
 
 1.0009 
 
 .9726 
 
 .9441 
 
 21 
 
 15 
 
 .9441 
 
 .9152 
 
 .8861 
 
 .8567 
 
 .8271 
 
 .7973 
 
 .7672 
 
 .7369 
 
 .7064 
 
 .6757 
 
 .6448 
 
 20 
 
 16 
 
 .6448 
 
 .6137 
 
 .5825 
 
 .5510 
 
 .5194 
 
 .4877 
 
 .4558 
 
 .4238 
 
 .3917 
 
 .3594 
 
 .3271 
 
 19 
 
 17 
 
 .3271 
 
 .2946 
 
 .2621 
 
 .2296 
 
 .1969 
 
 .1641 
 
 .1314 
 
 .0985 
 
 .0657 
 
 .0329 
 
 .0000 
 
 18 
 Arg. 
 
 Arg. 
 
 10o 
 
 9° 
 
 8° 
 
 70 
 
 6° 
 
 50 | 40 
 
 3° 2° - 
 
 1° 
 
 Oo 
 
 Owing to the elliptical form of the sun's apparent orbit, it does not revolve with 
 uniform angular velocity, and this Table shows the correction in its longitude, 
 required to be made on that account. 
 
 The epoch of this Table is 1840. 
 
 TABLE IX. 
 
 Equation of the Moon's Centre. 
 Argument — Moon's Anomaly. 
 
 Arg. 
 
 + 
 
 
 0° 
 
 1° 
 
 2o 
 
 3° 
 
 40 
 
 5° 
 
 6° 
 
 70 
 
 8° 
 
 90 
 
 10O 
 
 Arg. 
 
 0.0000 
 
 0.1181 
 
 0.2362 
 
 0.3542 
 
 0.4721 
 
 0.5897 
 
 0.7072 
 
 0.8244 
 
 0.9412 
 
 1.0578 
 
 1.1738 
 
 35 
 
 1 
 
 1.1738 
 
 1.2896 
 
 1.4048 
 
 1.5223 
 
 1.6336 
 
 1.7471 
 
 1.8600 
 
 1.9721 
 
 2.0835 
 
 2.1944 
 
 2.3041 
 
 34 
 
 2 
 
 2.3041 
 
 2.4131 
 
 2.5212 
 
 2.6284 
 
 2.7346 
 
 2.8398 
 
 2.9439 
 
 3.0470 
 
 3.1490 
 
 3.2498 
 
 3.3494 
 
 33 
 
 3 
 
 3.3494 
 
 3.4478 
 
 3.5450 
 
 3.6409 
 
 3.7354 
 
 3.8287 
 
 3.9205 
 
 4.0109 
 
 4.0999 
 
 4.1874 
 
 4.2735 
 
 32 
 
 4 
 
 4.2735 
 
 4.3580 
 
 4.4410 
 
 4.5223 
 
 4.6022 
 
 4.6804 
 
 4.7569 
 
 4.8318 
 
 4.9050 
 
 4.9765 
 
 5.0463 
 
 31 
 
 5 
 
 5.0463 
 
 5.1143 
 
 5.1806 
 
 5.2451 
 
 5.3078 
 
 5.3687 
 
 5.4277 
 
 5.4849 
 
 5.5403 
 
 5.59385.6454 
 
 30 
 
 6 
 
 5.6454 
 
 5.6952 
 
 5.7480 
 
 5.7889 
 
 5.8330 
 
 5.8750 
 
 5.9152 
 
 5.9534 
 
 5.9897 
 
 6.02406.0564 
 
 29 
 
 7 
 
 6.0564 
 
 6.0868 
 
 6.1152 
 
 6.1417 
 
 6.1663 
 
 6.1889 
 
 6.2094 
 
 6.2281 
 
 6.2448 
 
 6.25956.2722 
 
 28 
 
 8 
 
 6.2722 
 
 6 2830 
 
 6.2919 
 
 6.2988 
 
 6.3033 
 
 6.3068 
 
 6.3079 
 
 6.3070 
 
 6.3043 
 
 6.2997 
 
 6.2931 
 
 27 
 
 9 
 
 6.2931 
 
 6.2847 
 
 6.2743 
 
 6.2621 
 
 6.2481 
 
 6.2322 
 
 6.2144 
 
 6.1948 
 
 6.1734 
 
 6.1502 
 
 6.1252 
 
 26 
 
 10 
 
 6.1252 
 
 6.0984 
 
 6 0699 
 
 6.0396 
 
 6.0076 
 
 5.9739 
 
 5.9384 
 
 5.9013 
 
 5.8624 
 
 5.8220 
 
 5.7799 
 
 25 
 
 11 
 
 5.7799 
 
 5.7362 
 
 5 6908 
 
 5.6439 
 
 5.5954 
 
 5.5453 
 
 5.4937 
 
 5.4406 
 
 5 3860 
 
 5.3300 
 
 5.2723 
 
 24 
 
 12 
 
 5.2723 
 
 5.2135 
 
 5 1531 
 
 5.0913 
 
 5.0281 
 
 4.9636 
 
 4.8978 
 
 4.8306 
 
 4.7622 
 
 4.6924 
 
 4.6215 
 
 23 
 
 13 
 
 4.6215 
 
 4.5493 
 
 4 4759 
 
 4.4013 
 
 4.3256 
 
 4.2487 
 
 4.1707 
 
 4 0915 
 
 4 0114 
 
 3 9302 
 
 3.8480 
 
 22 
 
 14 
 
 3.8480 
 
 3.7620 
 
 3 6806 
 
 3.5954 
 
 3.5093 
 
 3 4223 
 
 3.3344 
 
 3.2457 
 
 3.1562 
 
 3 0658 
 
 2.9747 
 
 21 
 
 15 
 
 2 9747 
 
 2.8828 
 
 2 7901 
 
 2.6968 
 
 2.6028 
 
 2 5081 
 
 2.4127 
 
 2.3168 
 
 2 2203 
 
 2 1232 
 
 2 0256 
 
 20 
 
 16 
 
 2.0256 
 
 1.9275 
 
 18289 
 
 1.7298 
 
 1.6303 
 
 15304 
 
 1.4301 
 
 1.3294 
 
 1.2285 
 
 I 1272 
 
 10256 
 
 19 
 
 17 
 
 Arg. 
 
 1.0256 
 ~K)o 
 
 0.9238 
 
 0.8217 
 
 0.7194 
 
 0.6170 
 6° 
 
 0.5144 
 
 0.4117 
 
 0.3088 
 
 0.2059 
 
 0.1030 
 lo 
 
 0.0000 
 
 18 
 
 9° 
 
 8° 
 
 70 
 
 5° 
 
 40 
 
 30 
 
 2o 
 
 00 
 
 Arg. 
 
 Owing to the elliptical form of the moon's orbit, it does not revolve with uni- 
 form angular velocity, and this Table shows the correction in its longitude re- 
 quired to be made on that account. 
 
 2 
 
TABLE X. 
 
 Annual Equation of the Moorts Longitude. 
 Argument — Sun's Mean Anomaly. 
 
 Arg. 
 
 
 0° 
 
 2° 
 
 4 o 
 
 6° 
 
 8° 
 
 10° 
 .0318 
 
 Arg. 
 
 + 
 35 
 
 .0000 
 
 .0064 
 
 .0128 
 
 .0192 
 
 .0255 
 
 1 
 
 .0318 
 
 .0381 
 
 .0443 
 
 .0505 
 
 .0567 
 
 .0627 
 
 34 
 
 2 
 
 .0627 
 
 .0687 
 
 .0747 
 
 .0805 
 
 .0863 
 
 .0920 
 
 33 
 
 3 
 
 .0920 
 
 .0974 
 
 .1028 
 
 .1081 
 
 .1132 
 
 .1183 
 
 32 
 
 4 
 
 .1183 
 
 .1232 
 
 .1280 
 
 .1326 
 
 .1370 
 
 .1413 
 
 31 
 
 5 
 
 .1413 
 
 .1454 
 
 .1494 
 
 .1532 
 
 .1568 
 
 .1602 
 
 30 
 
 6 
 
 .1602 
 
 .1634 
 
 .1664 
 
 .1692 
 
 .1719 
 
 .1743 
 
 29 
 
 7 
 
 .1743 
 
 .1764 
 
 .1785 
 
 .1803 
 
 .1818 
 
 .1832 
 
 28 
 
 8 
 
 .1832 
 
 .1843 
 
 .1852 
 
 .1859 
 
 .1864 
 
 .1866 
 
 27 
 
 9 
 
 .1866 
 
 .1865 
 
 .1864 
 
 .1860 
 
 .1853 
 
 .1843 
 
 26 
 
 10 
 
 .1843 
 
 .1832 
 
 .1819 
 
 .1802 
 
 .1784 
 
 .1764 
 
 25 
 
 Jl 
 
 .1764 
 
 .1742 
 
 .1717 
 
 .1690 
 
 .1662 
 
 .1630 
 
 24 
 
 12 
 
 .1630 
 
 .1597 
 
 .1562 
 
 .1525 
 
 .1486 
 
 .1446 
 
 23 
 
 13 
 
 .1446 
 
 .1404 
 
 .1359 
 
 .1313 
 
 .1265 
 
 .1216 
 
 22 
 
 14 
 
 .1216 
 
 .1165 
 
 .1113 
 
 .1059 
 
 .1004 
 
 .0947 • 
 
 21 
 
 15 
 
 .0947 
 
 .0890 
 
 .0831 
 
 .0771 
 
 .0711 
 
 .0649 
 
 20 
 
 16 
 
 .0649 
 
 .05S6 
 
 .0523 
 
 .0459 
 
 .0395 
 
 .0330 
 
 19 
 
 17 
 
 .0330 
 
 .0264 
 
 .0198 
 
 .0132 
 
 .0066 
 
 .0000 
 
 18 
 
 Arg. 
 
 10° 
 
 8° 
 
 6° 
 
 4° 
 
 2° 
 
 0° 
 
 Arg. 
 
 The attraction of the sun upon the moon tends to draw it away from the earth 
 and thus dilate its orbit ; and since the distance of the sun from the earth varies 
 in different seasons of the year, its attraction also varies. This causes a variation 
 in the size of the moon's orbit, as well as in its velocity, producing inequalities in 
 its longitude, for which this Table gives the required corrections. 
 
 TABLE XI. 
 
 Secular Equation, showing the Acceleration of the Moon's Mean 
 
 Motion. 
 
 Argument — The date. 
 
 Arg. 
 
 
 
 5 
 
 181 
 
 00 
 
 01 
 
 182 
 
 01 
 
 02 
 
 183 
 
 03 
 
 04 
 
 184 
 
 05 
 
 06 
 
 185 
 
 07 
 
 09 
 
 186 
 
 11 
 
 13 
 
 187 
 
 15 
 
 17 
 
 188 
 
 19 
 
 21 
 
 189 
 
 24 
 
 27 
 
 190 
 
 30 
 
 33 
 
 The eccentricity of the earth's orbit is, and has been for ages, slowly diminish- 
 ing, which renders the sun's disturbing influence on the moon less and less every 
 year, thus allowing the orbit of the latter to contract, diminishing its periodic time. 
 This Table contains the necessary corrections from this cause, at intervals of five 
 years during the present century. 
 
 Note. — The numbers in this Table are the 3d and 4th places of decimals of a 
 degree ; therefore in using them, two cyphers must be prefixed as decimals. 
 Thus for 33 read .0033. 
 
TABLE XII 
 Variation. 
 
 Argument- 
 
 —Moon's Longitude diminished 
 
 by that of the Sun. 
 
 
 Arg. 
 
 oo 
 
 lo 
 
 2o 
 
 3o 
 
 40 1 50 
 
 60 
 
 70 1 80 1 9° 
 
 lOo 
 
 Arg. 
 
 - 
 
 
 
 +.0000 
 
 +.0204 
 
 +.0407 
 
 +.0610 
 
 +.0812+.1013 
 
 +.1212 
 
 +.1410 +.1607 +.1801 
 
 +.1993 
 
 35^ 
 
 i 
 
 +.1993 
 
 +.2183 
 
 +.2370 
 
 +.2553 
 
 +.2734 +.2911 
 
 +.3084 
 
 +.3254 +.3419 +.3580 
 
 +.3736 
 
 34 
 
 
 2 
 
 +.3736 
 
 +.3887 
 
 +.4034 
 
 +.4175 
 
 +.4312 
 
 +.4443 
 
 +.4568 
 
 +.4687 +.4801 4-4908 
 
 +.5009 
 
 33 
 
 
 3 
 
 +.5009 
 
 +.5104 
 
 +.5192 
 
 +.5274 
 
 +.5349 
 
 +.5418 
 
 +.5479 
 
 +.5534 +.5581 +.5622 
 
 +.5656 
 
 32 
 
 
 4 
 
 +.5656 
 
 +.5682 
 
 +.5701 
 
 +.5713 
 
 +.5718 
 
 +.5716 
 
 +.5706 
 
 +.5639 +.5666 +.5634 
 
 +.5597 
 
 31 
 
 ■r. 
 
 5 
 
 +.5597 
 
 +.5551 
 
 +.5499 
 
 +.5440 +.5374 
 
 +.5302 +.5222 +.5137|+.5044'+.4946 
 
 +.4841 
 
 30 
 
 
 6 
 
 +.4841 
 
 +.4730 
 
 +.4613 
 
 +.4490 +.4361 
 
 +.4227 +.4088 +.3944 +.3794 +.3640 
 
 +.3481 
 
 29 
 
 DQ 
 
 7 
 
 +.3481 
 
 +.3317 
 
 +.314S 
 
 +.2978 +.2802 
 
 +.2623 +.2440 +.2254 +.2065 +.1874 
 
 +.1680 
 
 28 
 
 8 
 
 +.1680 
 
 +.1483 
 
 +.1284 
 
 +.1034 
 
 +.0882 
 
 +.0679 +.0475 +.0270 +.0054 
 
 -.0142 
 
 -.0348 
 
 27 
 
 
 9 
 
 -.0348 
 
 -.0554 
 
 -.0760 
 
 -.0965 
 
 -.1170 
 
 -.1373 
 
 -.1575 
 
 -.1775 
 
 -.1973 
 
 -.2170 
 
 -.2364 
 
 26 
 
 
 10 
 
 -.2364 
 
 -.2555 
 
 -.2744 
 
 -.2929 
 
 -.3112 
 
 -.3291 
 
 -.3466 
 
 -.3638 
 
 -.3805 
 
 -.3969 
 
 -.4127 
 
 25 
 
 8 
 
 11 
 
 -.4127 
 
 -.4281 
 
 -.4430 
 
 -.4575 
 
 -.4714 
 
 -.4847 
 
 -.4975 
 
 -.5097 
 
 -.5214 
 
 —5324 
 
 -.5429 
 
 24 
 
 - 
 
 12 
 
 -.5429 
 
 -.5526 
 
 -.5618 
 
 -.5703 
 
 -.5781 
 
 -.5853 
 
 -.5918 
 
 -.5976 
 
 -.6026 
 
 -.6070 
 
 -.6107 
 
 23 
 
 l* 
 
 13 
 
 -.6107 
 
 -.6137 
 
 -.6159 
 
 -.6174 
 
 -.6182 
 
 -.6183 
 
 -.6176 
 
 -.6162 
 
 -.6140 
 
 -.6111 
 
 -.6076 
 
 22 
 
 9 
 
 14 
 
 -.6076 
 
 -.6033 
 
 -.5982 
 
 -.5925 
 
 -.5860 
 
 -.5789 
 
 -.5711 
 
 -.5626 
 
 -.5534 
 
 -.5435 
 
 -.5330 
 
 21 
 
 15 
 
 -.5330 
 
 -.5220 
 
 -.5101 
 
 -.4977 
 
 -.4347 
 
 -.4712 
 
 -.4571 
 
 -.4424 
 
 -.4271 
 
 -.4114 
 
 -.3952 
 
 20 
 
 
 16 
 
 -.3952 
 
 -.3785 
 
 -.3614 
 
 -.3437 
 
 -.3257 
 
 -.3073 
 
 -.2885 -.2694 
 
 -.2500 
 
 -.2302 
 
 -.2102 
 
 19 
 
 
 17 
 
 -.2102 
 
 -.1899 
 
 -.1694 
 
 -.1487 
 
 -.1278 
 
 -.1067 
 
 -.0855-.0642 
 
 -.0429 
 
 -.0215 
 
 -.0000 
 ~0o 
 
 18 
 
 - 
 
 Arg. 
 
 10O 
 
 9° 
 
 8o 
 
 70 
 
 60 
 
 50 
 
 40 | 30 
 
 20 
 
 lo 
 
 Arg. 
 
 Reverse the Signs. 
 
 Owing to the disturbing influence of the sun, the moon is alternately accelera- 
 ted and retarded in the different quadrants, reckoning from syzygy. This Table 
 contains the corrections in its longitude resulting from this cause. 
 
 TABLE XIII. 
 Annual Equation of Variation. 
 
 Arguments — The argument of Variation at the top and bottom, and 
 the Sun's Mean Anomaly at the sides. 
 
 
 130 1 135 
 
 
 
 
 1 270 | 275 | 280 | 285 1 290 | 295 | 300 | 305 
 
 310 1 315 
 
 
 
 
 | 5 | 10 
 
 15 1 20 1 25 1 30 1 35 
 
 40 
 
 45 
 
 Arg. 
 
 
 Arg. 
 
 180 | 185 1 190 
 
 195 I 200 1 205 | 210 | 215 
 
 220 
 
 225 
 
 
 
 180 
 
 .00001 -.0064 
 
 -.0125 
 
 -.0LS3 
 
 -.0233 
 
 -.0281 
 
 -.0314 
 
 -.0342 
 
 -.0358 
 
 -.036-1 
 
 180 
 
 360 
 
 10 
 
 190 
 
 +.0008 -.0053 
 
 -.0114 
 
 -.0172 
 
 -.0225 
 
 -.0267 
 
 -.0306 
 
 -.0333 
 
 -.0353 
 
 -.0358 
 
 170 
 
 350 
 
 20 
 
 200 
 
 +.0019 -.0042 
 
 -.0100 
 
 -.0156 
 
 -.0211 
 
 _.0250 
 
 -.0286 
 
 -.0317 
 
 -.0333 
 
 -.0342 
 
 160 
 
 340 
 
 30 
 
 210 
 
 +.0028 -.0028 
 
 -.0083 
 
 -.0133 
 
 -.0181 
 
 -.0225 
 
 -.0261 
 
 -.0286 
 
 -.0306 
 
 -.0314 
 
 150 
 
 330 
 
 40 
 
 220 
 
 +.0033 -.0014 
 
 -.0061 
 
 -.0108 
 
 -.0153 
 
 -.0192 
 
 -.0225 
 
 -.0250 
 
 -.0267 
 
 -.0281 
 
 140 
 
 320 
 
 50 
 
 230 
 
 +.0042+.0000 
 
 -.0039 
 
 -.0081 
 
 -.0119 
 
 -.0153 
 
 -.0181 
 
 -.0206 
 
 -.0225 
 
 -.0233 
 
 130 
 
 310 
 
 60 
 
 240 
 
 +.0047+.0017 
 
 -.0019 
 
 _.0050 
 
 -.0081 
 
 -.0108 
 
 -.0133 
 
 -.0156 
 
 -.0172 
 
 -.0184 
 
 120 
 
 300 
 
 70 
 
 250 
 
 +.0050+.0028 
 
 +.0006 
 
 -.0019 
 
 -.0039 
 
 -.0061 
 
 -.0083 
 
 -.0100 
 
 -.0114 
 
 -.0125 
 
 110 
 
 290 
 
 80 
 
 260 
 
 +.0053 +.0042 
 
 +.0028 
 
 +.0017 
 
 +.0000 
 
 -.0014 
 
 -.0028 
 
 -.0042 
 
 -.0053 
 
 -.0064 
 
 100 
 
 280 
 
 90 
 
 270 
 
 +.0053+.0053 
 
 +.0053 
 
 +.0047 
 
 +.0042 
 
 +.0033 
 
 +.0028 
 
 +.0019 
 
 +.0008 
 
 .0000 
 
 90 
 
 270 
 
 100 
 
 280 
 
 +.0053+.0064 
 
 +.0072 
 
 4..0078 
 
 +.0081 
 
 +.0083 
 
 -I-.0081 
 
 +.0078 
 
 +.0072 
 
 +.0064 
 
 80 
 
 260 
 
 110 
 
 290 
 
 4-. 0050 +.0072 
 
 +.0089 
 
 + 0106 
 
 ,.0119 
 
 +.0128 
 
 +.0133 +.0133 
 
 +.0133 
 
 +.0125 
 
 70 
 
 250 
 
 120 
 
 300 
 
 +.0047+.0078 
 
 +.0108 
 
 T.0131 
 T.0153 
 +.0169 
 
 +.0153 
 
 +.0169 
 
 +.0181 
 
 +.0189 
 
 +.0186 
 
 +.0184 
 
 60 
 
 240 
 
 130 
 
 310 
 
 +.0042+.0031 
 
 +.0119 
 
 ■.0181 
 T.0206 
 T.0225 
 +.0233 
 
 +.0206 
 
 +.0206 
 
 +.0233 
 
 +.0239 
 
 +.0233 
 
 50 
 
 230 
 
 140 
 
 320 
 
 +.0033 +.0083 
 
 4-.0128 
 
 +.0236 
 
 +.0225 
 
 +.0275 
 
 +.0281 
 
 +.0281 
 
 40 
 
 220 
 
 150 
 
 330 
 
 +.0028+.0084 
 
 +.0133 
 
 +.0181 
 
 +.0258 
 
 +.0286 
 
 +.0306 
 
 +.0317 
 
 +.0314 
 
 30 
 
 210 
 
 160 
 
 340 
 
 +.0019 4-. 0078 
 
 4-.0133 
 
 +.0189 
 
 +.0275 
 
 +.0306 
 
 +.0328 
 
 +.0339 
 
 +.0342 
 
 20 
 
 200 
 
 170 
 
 350 
 
 +.0003|+.0072 
 
 4-.0133 
 
 +.0186 
 
 +.0239 
 
 +.0281 
 
 +.0317 
 
 +.0339 
 
 +.0353 
 
 +.0358 
 
 10 
 
 190 
 Arg. 
 
 
 90 1 85 
 
 80 1 75 
 
 70 
 
 65 1 60 1 55 1 50 1 45 
 
 Arg. 
 
 
 270 1 265 
 
 260 1 255 
 
 250 
 
 245 1 240 1 235 I 230 | 225 
 
 
 
 
 180 1 175 
 
 170 1 165 1 160 1 155 1 150 1 145 1 140 1 135 
 
 
 
 
 360 J 355 
 
 350 I 345 1 340 | 335 | 330 | 325 | 320 | 315 
 
 
 
 The inequality in the moon's motion, denominated Variation, and for which 
 Table 12th furnishes the correction, being occasioned by the disturbing influence 
 of the sun, must be greater or less according as the distance of the earth from that 
 luminary varies. In that Table the earth is supposed to be at its mean distance ; 
 hence another correction becomes necessary, which this Table furnishes. 
 
TABLE XIV. 
 
 Evection. 
 
 Argument — The Moon's Mean Anomaly diminished by twice the excess 
 
 of the Moon's Mean Longitude over the True Longitude of the Sun. 
 
 Arg. 
 
 0° 
 
 1° 
 
 2° 
 
 3° 
 
 4° 
 
 5° 
 
 6° 
 
 70 
 
 8° 
 
 9° 
 
 10° 
 
 Arg. 
 
 
 
 .0000 
 
 .0238 
 
 .0475 
 
 .0713 
 
 .0950 
 
 .1187 
 
 .1423 
 
 .1659 
 
 .1894 
 
 .2129 
 
 .2363 
 
 + 
 35 
 
 1 
 
 .2363 
 
 .2596 
 
 .2829 
 
 .3061 
 
 .3291 
 
 .3521 
 
 .3750 
 
 .3978 
 
 .4204 
 
 .4428 
 
 .4651 
 
 34 
 
 2 
 
 .4651 
 
 .4873 
 
 .5093 
 
 .5312 
 
 .5530 
 
 .5744 
 
 .5958 
 
 .6170 
 
 .6379 
 
 .6587 
 
 .6793 
 
 33 
 
 3 
 
 .6793 
 
 .6996 
 
 .7197 
 
 .7396 
 
 .7593 
 
 .7787 
 
 .7979 
 
 .8168 
 
 .8355 
 
 .8539 
 
 .8720 
 
 32 
 
 4 
 
 .8720 
 
 .8899 
 
 .9074 
 
 .9247 
 
 .9417 
 
 .9584 
 
 .9748 
 
 .9909 
 
 1.0067 
 
 1.0222 
 
 1.0374 
 
 31 
 
 5 
 
 1.0374 
 
 1.0522 
 
 1.0667 
 
 1.0808 
 
 1.0947 
 
 1.1082 
 
 1.1213 
 
 1.1341 
 
 1.1465 
 
 1.1586 
 
 1.1703 
 
 30 
 
 6 
 
 1.1703 
 
 1.1817 
 
 1.1926 
 
 1.2032 
 
 1.2135 
 
 1.2234 
 
 1.2323 
 
 1.2420 
 
 1.2507 
 
 1.2590 
 
 1.2670 
 
 29 
 
 7 
 
 1.2670 
 
 1.2745 
 
 1.2816 
 
 1.2884 
 
 1.2948 
 
 1.3007 
 
 1.3063 
 
 1.3114 
 
 1.3162 
 
 1.3206 
 
 1.3245 
 
 28 
 
 8 
 
 1.3215 
 
 1.3281 
 
 1.3312 
 
 1.3339 
 
 1.3362 
 
 1.3381 
 
 1.3396 
 
 1.3407 
 
 1.3414 
 
 1.3417 
 
 1.3415 
 
 27 
 
 9 
 
 1.3415 
 
 1.3410 
 
 1.3400 
 
 1.3386 
 
 1.3369 
 
 1.3347 
 
 1.3321 
 
 1.3291 
 
 1.3257 
 
 1.3220 
 
 1.3178 
 
 26 
 
 10 
 
 1.3178 
 
 13132 
 
 1.3082 
 
 1.3028 
 
 1.2971 
 
 1.2909 
 
 1.2844 
 
 1.2774 
 
 1.2701 
 
 1.2624 
 
 1.2543 
 
 25 
 
 11 
 
 1.2543 
 
 1.2459 
 
 1.2370 
 
 1.2278 
 
 1.2182 
 
 1.2083 
 
 1.1980 
 
 1.1872 
 
 1.1763 
 
 1.1650 
 
 1.1533 
 
 24 
 
 12 
 
 1.1533 
 
 1.1412 
 
 1.1288 
 
 1.1161 
 
 1.1031 
 
 1.0897 
 
 1.0760 
 
 1.0619 
 
 1.0476 
 
 1.0330 
 
 1.0180 
 
 23 
 
 13 
 
 1.0180 
 
 1.0027 
 
 .9872 
 
 .9713 
 
 .9552 
 
 .9388 
 
 .9221 
 
 .9051 
 
 .8879 
 
 .8704 
 
 .8526 
 
 22 
 
 14 
 
 .8526 
 
 .8346 
 
 .8164 
 
 .7979 
 
 .7792 
 
 .7603 
 
 .7411 
 
 .7217 
 
 .7021 
 
 .6822 
 
 .6622 
 
 21 
 
 15 
 
 .6622 
 
 .6420 
 
 .6217 
 
 .6011 
 
 .5803 
 
 .5594 
 
 .5383 
 
 .5171 
 
 .4957 
 
 .4742 
 
 .4525 
 
 20 
 
 16 
 
 .4525 
 
 .4307 
 
 .4087 
 
 .3867 
 
 .3646 
 
 .3422 
 
 .3199 
 
 .2975 
 
 .2749 
 
 .2523 
 
 .2296 
 
 19 
 
 17 
 
 .2296 
 10° 
 
 .2068 
 
 .1840 
 
 .1611 
 
 .1382 
 
 .1152 
 
 .0922 
 
 .0692 
 
 .0462 
 
 .0231 
 
 .0000 
 
 18 
 
 Arg. 
 
 9° 
 
 8° 
 
 7° 
 
 6° 
 
 5° 
 
 40 
 
 3° 
 
 2° 
 
 1<? 
 
 0° 
 
 Arg. 
 
 The disturbing influence of the sun causes a variation in the eccentricity of the 
 moon's orbit, and in the position of the major axis. This affects the moon's lon- 
 gitude, producing inequalities, for which this Table gives the required corrections. 
 
 TABLE XV. 
 
 Annual Equation of Evection. 
 
 Arguments — The argument for Evection at the top and bottom, 
 the Sun's Mean Anomaly at the sides. 
 
 and 
 
 Arg. 
 
 | 1 10 1 20 1 30 | 40 i 50 | 60 | 70 1 80 1 90 | 
 
 
 Arg. | 180 | 190 1 200 1 210 1 220 1 230 1 240 | 250 1 260 | 270 
 
 
 
 180 
 
 .0000i-.0081i-.0l61 
 
 -.0236 
 
 -.0303; -.0364, -.0408 
 
 -.0444, -.0467 
 
 -.0472 
 
 366 
 
 ran 
 
 10 
 
 190 
 
 +.0036 -.0047 -.0122 
 
 -.0200 
 
 -.0272 -.0331 
 
 -.0386 
 
 -.0425! -.0453 
 
 -.0467 
 
 350 
 
 170 
 
 20 
 
 200 
 
 +.0072-00051-0083 
 
 -.0158 
 
 -.0228 
 
 -.0294 
 
 -.0347 
 
 -.0394 
 
 -.0425 
 
 -.0444 
 
 340 
 
 160 
 
 30 
 
 210 
 
 +.0108 +.0033 
 
 -.0042 
 
 -.0111 
 
 -.0181 
 
 -.0244 
 
 -.0303 
 
 -.0347 
 
 -.0386 
 
 -.0408 
 
 330 
 
 150 
 
 40 
 
 220 
 
 +.0136 +.0072 
 
 +.0005 
 
 -.0964 
 
 -.0128 
 
 -.0189 
 
 -.0244 
 
 -.0294 
 
 -.0331 
 
 -.0364 
 
 320 
 
 140 
 
 50 
 
 230 
 
 +.0164+.0108 
 
 +.0050 
 
 -.0011 
 
 -.0072 
 
 -.0128 
 
 -.0181 
 
 -.0228 
 
 -.0272 
 
 -.0303 
 
 310 
 
 130 
 
 60 
 
 240 
 
 +.0186+.0142 
 
 +.0092 
 
 +.0042 
 
 -.0011 
 
 -.0064 
 
 -.0111 
 
 -.0158 
 
 -.0200 
 
 -.0236 
 
 300 
 
 120 
 
 70 
 
 250 
 
 +.0200+.0175+.0133 +.0092 
 
 +.0050 
 
 +.0005 
 
 -.0042 
 
 -.0083 
 
 —0122 
 
 -.0161 
 
 290 
 
 110 
 
 SO 
 
 260 
 
 +.0211 +.01921+.0175+.0142 
 
 +.0108 
 
 +.0072 
 
 +.0033 
 
 -.0005 
 
 -.0047 
 
 -.0081 
 
 280 
 
 100 
 
 90 
 
 270 
 
 +.0211 
 
 +.0211 +.0200 
 
 +.0186 
 
 +.0164 
 
 +.0136 +.0108 
 
 +.0072 
 
 + 0036 
 
 .0000 
 
 270 
 
 90 
 
 100 
 
 280 
 
 +.0211 
 
 +.0219+.0228 
 
 +.0222 
 
 +.0214 
 
 +.0200+.0175 
 
 +.0150 
 
 +.0117 
 
 +.0081 
 
 260 
 
 80 
 
 110 
 
 290 
 
 +.0200 
 
 +.0228+.0242 
 
 +.0256 
 
 +.0258 
 
 +.0253+.0241 
 
 +.0219 
 
 +.0195 +.0161 
 
 250 
 
 70 
 
 120 
 
 300 
 
 +.0186 
 
 +.0222+.0256 
 
 +.0278 
 
 +.0294 
 
 +.0300|+.0297 
 
 +.0286 
 
 +.0264 +.0236 
 
 240 
 
 60 
 
 130 
 
 310 
 
 +.0164 
 
 +.02)4+0258 
 
 +.0294 
 
 +.0320 
 
 +.0339+.0344 
 
 +.0342 
 
 +.0328 
 
 +.0303 
 
 230 
 
 50 
 
 140 
 
 320 
 
 +.0136 
 
 +.0200+.0253 
 
 +.0300 
 
 +.0339 
 
 +.0364+.038:j 
 
 +.0386 
 
 +.0381 
 
 +.0364 
 
 220 
 
 40 
 
 150 
 
 330 
 
 +.0108 
 
 +.0175+.0241 
 
 +.0297 
 
 +.0344 
 
 +.0383+.0405 
 
 +.0422 
 
 ,.0422 
 +.0450 
 
 +.0408 
 
 210 
 
 30 
 
 160 
 
 340 
 
 +.0072 
 
 +.0150+.0219 
 
 +.0286 
 
 +.0342 
 
 +.0386+.0422 
 
 +.0422 
 
 +.0444 
 
 200 
 
 20 
 
 170 
 
 350 
 
 +.0036 
 
 +.0117 +.0195 
 
 +.0264 
 
 +.0328 
 
 +.0381 +.0422 
 
 +.0450 
 
 +.0464 +.0467 
 
 190 
 
 10 
 
 180 
 
 360 
 
 +.0000 
 
 +.0081 +.0161 
 
 +.02361+.0303 
 
 +.03641+.0408 
 
 +.0444 
 
 +.0467+.0472 
 
 180 
 
 
 
 
 
 180 i 170 1 90 |Arg. 
 
 Arg. 
 
 
 360 ! 350 i 310 1 330 1 320 1 310 1 300 1 290 | 280 1 270 I 
 
 
 The inequality in the moon's motion denominated Evection, and for which Table 14th furnishes 
 the correction, being occasioned by the disturbing influence of the sun, must be greater or less 
 according as the distance of the earth from that luminary varies. In that Table the earth is 
 supposed to be at its mean distance ; hence another correction becomes necessary, which this 
 Table furnishes. 
 
 Note. — The foregoing explanation is true so far as it goes ; but the values given in the Table 
 are partially referrible to a variation in the argument for Evection occasioned by the Annual 
 Equation of the Moon's Longitude. And we might on this principle add several other Tables, 
 which should contain corrections required by the alteration of the arguments of previous ones; 
 but this would not accord with the simplicity of our design. 
 
TABLE XVI. 
 
 Nodal Equation of the Moon's Longitude. 
 Argument — The Sun's longitude diminished by that of the Moon's Node. 
 
 Arg. 
 
 0° 
 
 2° 
 
 4° 
 
 6" 
 .0036 
 
 8° 
 
 10° 
 
 Arg. 
 
 
 
 IS 
 
 .0000 
 
 .0012 
 
 .0024 
 
 .0048 
 
 .0059 
 
 + 
 17 
 
 & 
 
 1 
 
 19 
 
 .0059 
 
 .0071 
 
 .0082 
 
 .0092 
 
 .0102 
 
 .0112 
 
 16 
 
 34 
 
 2 
 
 20 
 
 .0112 
 
 .0121 
 
 .0129 
 
 .0137 
 
 .0144 
 
 .0150 
 
 15 
 
 33 
 
 3 
 
 21 
 
 .0150 
 
 .0156 
 
 .0161 
 
 .0165 
 
 .0169 
 
 .0171 
 
 14 
 
 32 
 
 4 
 
 22 
 
 .0171 
 
 .0173 
 
 .0174 
 
 .0174 
 
 .0173 
 
 .0171 
 
 13 
 
 31 
 
 5 
 
 23 
 
 .0171 
 
 .0169 
 
 .0165 
 
 .0161 
 
 .0156 
 
 .0150 
 
 12 
 
 30 
 
 6 
 
 24 
 
 .0150 
 
 .0144 
 
 .0137 
 
 .0129 
 
 .0121 
 
 .0112 
 
 11 
 
 29 
 
 7 
 
 25 
 
 .0112 
 
 .0102 
 
 .0092 
 
 .0082 
 
 .0071 
 
 .0059 
 
 10 
 
 28 
 
 8 
 
 26 
 
 .0059 
 
 .0048 
 
 .0036 
 
 .0024 
 
 4° 
 
 .0012 
 
 ~~ 2~ 
 
 .0000 
 
 9 
 
 27 
 
 Arg. 
 
 10° 
 
 8° 
 
 6° 
 
 0° 
 
 Arg. 
 
 The influence of the sun tending to dilate the moon's orbit is greatest when the 
 former is in the plane of the latter; i.e. when passing the nodes, and least when 
 farthest from them, occasioning an inequality in the moon's motion, for which 
 this Table furnishes the necessary correction. 
 
 TABLE XVII. 
 
 Reduction. 
 Argument — The Longitude of the Moon, diminished by that of its Node. 
 
 Arg. 
 
 o- 
 
 .0000 
 .0386 
 .0726 
 .0979 
 .1113 
 .1113 
 .0979 
 .0726 
 .0386 
 
 Arg. 
 
 10 
 
 1° I 
 
 0039 
 0423 
 0756 
 0998 
 1119 
 1105 
 0958 
 .0696 
 .0349 
 ~9° 
 
 2° 
 
 0079 
 0460 
 ,0785 
 .1016 
 1124 
 
 •A 
 
 0937 
 .0664 
 ,0311 
 
 0118 
 0496 
 .0813 
 1033 
 1127 
 10961.1087 
 
 .0914 
 .0632 
 0273 
 
 7° 
 
 0157 
 0531 
 0837 
 1048 
 1129 
 1075 
 ,0891 
 .0599 
 .0235 
 6° 
 
 J?!!- 6 .! 1 
 
 .0196.0235 
 
 .0565.0599 
 
 .0866 .0891 
 
 .1062 
 
 .1130 
 
 .1062 
 
 .0866 
 
 .0565 
 
 .0196 
 
 9° 10° 
 
 0386 
 0726 
 0979 
 1113 
 1113 
 0979 
 ,0726 
 .0386 
 0000 
 
 OP 
 
 Ar 
 
 *• 
 
 + 
 
 + 
 
 17 
 
 35 
 
 16 
 
 34 
 
 15 
 
 33 
 
 14 
 
 32 
 
 13 
 
 31 
 
 12 
 
 30 
 
 11 
 
 29 
 
 10 
 
 28 
 
 9 
 
 27 
 
 Arg. 
 
 The preceding Tables give the Moon's Longitude in its orbit: this reduces it to 
 the Ecliptic. 
 
 TABLE XVIII. 
 
 Lunar or Menstrual Equation of the Sun's Longitude. 
 Argument — The Longitude of the Moon diminished by that of the Sun. 
 
 Arg. 
 
 0° 
 
 5° 
 
 10° 
 
 Arg. 
 
 + 
 
 — 
 
 
 
 
 + 
 
 _ 
 
 
 
 18 
 
 .0000 
 
 .0002 
 
 .0004 
 
 17 
 
 35 
 
 1 
 
 19 
 
 .0004 
 
 .0005 
 
 .0007 
 
 16 
 
 34 
 
 2 
 
 20 
 
 .0007 
 
 .0009 
 
 .0010 
 
 15 
 
 33 
 
 3 
 
 21 
 
 .0010 
 
 .0012 
 
 .0013 
 
 14 
 
 32 
 
 4 
 
 22 
 
 .0013 
 
 .0015 
 
 .0016 
 
 13 
 
 31 
 
 5 
 
 23 
 
 .0016 
 
 .0017 
 
 .0018 
 
 12 
 
 30 
 
 6 
 
 24 
 
 .0018 
 
 .0019 
 
 .0020 
 
 11 
 
 29 
 
 7 
 
 25 
 
 .0020 
 
 .0020 
 
 .0021 
 
 10 
 
 28 
 
 8 
 
 26 
 
 .0021 
 
 .0021 
 
 .0021 
 
 9 
 
 27 
 
 Arg. 
 
 10° 
 
 5° 
 
 0° 
 
 Arg. 
 
 The revolution of the earth around the common centre of gravity of the moon 
 and earth, affects the sun's apparent place, causing a change in its longitude for 
 which this Table furnishes the correction. 
 
TABLE XIX. 
 
 Lunar Nutation in Longitude. 
 Argument — Longitude of the Moon's Ascending Node. 
 
 Arg. 
 
 1 Arg. 
 
 
 Arg. 
 
 Arg. 
 
 — 
 
 — 
 
 
 + 
 
 + 
 
 0° 
 
 180° 
 
 .0000 
 
 180° 
 
 360° 
 
 10 
 
 170 
 
 .0009 
 
 190 
 
 350 
 
 20 
 
 160 
 
 .0017 
 
 200 
 
 340 
 
 30 
 
 150 
 
 .0024 
 
 210 
 
 330 
 
 40 
 
 140 
 
 .0031 
 
 220 
 
 320 
 
 50 
 
 130 
 
 .0037 
 
 230 
 
 310 
 
 60 
 
 120 
 
 .0042 
 
 240 
 
 300 
 
 70 
 
 110 
 
 .0046 
 
 250 
 
 290 
 
 80 
 
 100 
 
 .0047 
 
 260 
 
 280 ' 
 
 90 
 
 90 
 
 .0048 
 
 270 
 
 270 
 
 The Precession of the Equinoxes, being occasioned in part by the attraction of 
 the moon lying out of the plane of the earth's equator, must be more or less rapid 
 according to the obliquity of the plane of its orbit to that of the equator, which 
 depends on the longitude of tbe moon's nodes. The longitudes of all the heav- 
 enly bodies, being reckoned from the Vernal Equinox, must be affected by any 
 change in the place of the point from which they are reckoned, and therefore need 
 a correction from this cause, which this Table supplies. 
 
 TABLE XX. 
 
 Sun's Semi-diameter and Horary Motion. 
 Argument — Sun's Anomaly. 
 
 Arg. 
 
 Semi-d ameter. 
 
 Horary motion. 
 
 Arg. 
 
 0° 
 
 .2717 
 
 .0425 
 
 360° 
 
 10 
 
 .2716 
 
 .0424 
 
 350 
 
 20 
 
 .2714 
 
 .0423 
 
 340 
 
 30 
 
 .2710 
 
 .0422 
 
 330 
 
 40 
 
 .2705 
 
 .0421 
 
 320 
 
 50 
 
 .2700 
 
 .0419 
 
 310 
 
 60 
 
 .2694 
 
 .0417 
 
 300 
 
 70 
 
 .2687 
 
 .0415 
 
 290 
 
 80 
 
 .2679 
 
 .0413 
 
 280 
 
 90 
 
 .2671 
 
 .0411 
 
 270 
 
 100 
 
 .2663 
 
 .0409 
 
 260 • 
 
 110 
 
 .2655 
 
 .0407 
 
 250 
 
 120 
 
 .2648 
 
 .0405 
 
 240 
 
 130 
 
 .2642 
 
 .0403 
 
 230 
 
 140 
 
 .2637 
 
 .0401 
 
 220 
 
 150 
 
 .2632 
 
 .0400 
 
 210 
 
 160 
 
 .2628 
 
 .0399 
 
 200 
 
 170 
 
 .2626 
 
 .0398 
 
 190 
 
 180 
 
 .2625 
 
 .0397 
 
 180 
 
 The elliptical shape of the sun's apparent orbit causes it to vary, both as to ap- 
 parent size and velocity. The values of these are given in this Table, at intervals 
 of 10° throughout the entire orbit. 
 
TABLE XXI. 
 
 Moon's Semi-diameter, Horary Motion and Equatorial Parallax. 
 Argument — Moon's corrected Anomaly. 
 
 Argument. 
 
 Semi-diam -ter. 
 
 Horary motion. 
 
 Equat. Parallax. 
 
 Argument. 
 
 0° 
 
 .2743 
 
 .6133 
 
 1°.0051 
 
 360 
 
 5 
 
 .2742 
 
 .6131 
 
 1°.0049 
 
 355 
 
 10 
 
 .2740 
 
 .6122 
 
 1°.0042 
 
 350 
 
 15 
 
 .2737 
 
 .6106 
 
 1°.0030 
 
 345 
 
 20 
 
 .2733 
 
 .6086 
 
 1°.0014 
 
 340 
 
 25 
 
 .2727 
 
 .6061 
 
 .9993 
 
 335 
 
 30 
 
 .2719 
 
 .6031 
 
 .9968 
 
 330 
 
 35 
 
 .2712 
 
 .5994 
 
 .9939 
 
 325 
 
 40 • 
 
 .2703 
 
 .5956 
 
 .9907 
 
 *>0 
 
 45 
 
 .2693 
 
 .5911 
 
 .9871 
 
 315 
 
 50 
 
 .2682 
 
 .5864 
 
 .9833 
 
 310 
 
 55 
 
 .2671 
 
 .5816 
 
 .9792 
 
 305 
 
 60 
 
 .2659 
 
 .5767 
 
 .9749 
 
 300 
 
 65 
 
 .2647 
 
 .5714 
 
 .9705 
 
 295 
 
 70 
 
 .2635 
 
 .5661 
 
 .9659 
 
 290 
 
 75 
 
 .2623 
 
 .5608 
 
 .9613 
 
 285 
 
 80 
 
 .2611 
 
 .5556 
 
 .9567 
 
 280 
 
 85 
 
 .2599 
 
 .5503 
 
 .9521 
 
 275 
 
 90 
 
 .2586 
 
 .5450 
 
 .9475 
 
 270 
 
 95 
 
 .2574 
 
 .5397 
 
 - .9429 
 
 265 
 
 100 
 
 •2562 
 
 .5350 
 
 .9384 
 
 260 
 
 105 
 
 .2550 
 
 .5303 
 
 .9342 
 
 255 
 
 110 
 
 .2539 
 
 .5256 
 
 .9302 
 
 250 
 
 115 
 
 .2528 
 
 .5214 
 
 .9263 
 
 245 
 
 320 
 
 .2518 
 
 .5173 
 
 .9227 
 
 240 
 
 125 
 
 .2509 
 
 .5136 
 
 .9194 
 
 235 
 
 130 
 
 .2500 
 
 .5100 
 
 .9162 
 
 230 
 
 135 
 
 .2492 
 
 .5069 
 
 .9133 
 
 225 
 
 140 
 
 .2485 
 
 .5039 
 
 .9108 
 
 220 
 
 145 
 
 .2479 
 
 .5014 
 
 .9086 
 
 215 
 
 150 
 
 .2474 
 
 .4992 
 
 .9066 
 
 210 
 
 155 
 
 .2469 
 
 .4972 
 
 .9048 
 
 205 
 
 160 
 
 .2465 
 
 .4958 
 
 .9034 
 
 200 
 
 165 
 
 .2462 
 
 .4944 
 
 .9023 
 
 195 
 
 370 
 
 .2460 
 
 .4936 
 
 .9016 
 
 190 
 
 175 
 
 .2459 
 
 .4932 
 
 .9011 
 
 185 
 
 180 
 
 .2458 
 
 .4930 
 
 .9009 
 
 180 
 
 The principle and construction of this Table is the same as that of Table 20th. 
 At the time of new or full moon the quantities in this Table must be increased 
 for the effect of Variation as follows, viz. 1st column, .0020; 2d do. .0115; 3d do. 
 .0073. 
 
TABLE XXII. 
 
 Moon's Semi-diameter, Hourly Motion, and Equatorial Parallax, as 
 affected by Evection. 
 
 Argument — The same as for Evection, Table 14th. 
 
 Arg. 
 
 Semi-diameter. 
 
 Hourly motion. 
 
 Equatorial Parallax. 
 
 Arg. 
 
 0° 
 
 + .0029 
 
 +.0112 
 
 +.0105 
 
 360° 
 
 10 
 
 -f.0028 
 
 +.0109 
 
 +.0103 
 
 350 
 
 20 
 
 +.0027 
 
 +.0103 
 
 +.0098 
 
 340 
 
 30 
 
 +.0025 
 
 +.0097 
 
 +.0091 
 
 330 
 
 40 
 
 +.0022 
 
 +.0086 
 
 +.0081 
 
 320 
 
 50 
 
 +.0019 
 
 +.0070 
 
 + .0068 
 
 310 
 
 60 
 
 +.0014 
 
 + .0055 
 
 +.0052 
 
 300 
 
 70 
 
 + .0009 
 
 + .0036 
 
 +.0035 
 
 • 290 
 
 80 
 
 +.0005 
 
 +.0019 
 
 +.0018 
 
 280 
 
 90 
 
 - .0000 
 
 - .0002 
 
 - .0001 
 
 270 
 
 100 
 
 - .0005 
 
 - .0020 
 
 - .0019 
 
 260 
 
 110 
 
 -.0010 
 
 - .0038 
 
 - .0037 
 
 250 
 
 120 
 
 - .0014 
 
 - .0055 
 
 - .0054 
 
 240 
 
 130 
 
 - .0019 
 
 - .0071 
 
 - .0068 
 
 230 
 
 140 
 
 -.0022 
 
 -.0083 
 
 - .0080 
 
 220 
 
 150 
 
 - .0024 
 
 -.0094 
 
 - .0089 
 
 210 
 
 160 
 
 - .0026 
 
 - .0103 
 
 - .0096 
 
 200 . 
 
 170 
 
 -.0027 
 
 -.0107 
 
 - .0100 
 
 190 
 
 180 
 
 -.0028 
 
 - .0109 
 
 - .0103 
 
 180 
 
 All the inequalities in the moon's longitude, for which the foregoing Tables 
 give the corrections, must likewise affect its apparent size, hourly motion, and 
 equatorial parallax. Variation and Evection are the only ones that it is important 
 to take into account, the former of which may be considered constant at the time 
 of new or full moon, and this Table gives the requisite correction for the latter. 
 
 TABLE XXIII. 
 
 Obliquity of the Ecliptic to the Equator. 
 Argument — The date. 
 
 Arg. 
 
 184 
 
 185 
 186 
 
 18? 
 
 188 
 189 
 
 23 27 15.1 
 23 27 24 3 
 
 23 27 43.2 
 23 27 26.0 
 
 23 27 33.6;23 27 30 7 
 23 27 18.4 23 27 20.1 
 23 27 20.8:23 2717.4 
 23 27 13.4i23 2716.0 
 
 3 
 
 23 27 40.423 27 37. 1 23 27 33.5 
 23 27 28.3 23 27 31.0 23 27 33.6 
 23 27 27.2 23 27 23.7 23 27 21.3 
 23 27 235 23 27 26 23 27 27.9 
 23 27 13.9 23 27 10.8 23 27 8.4 
 23 27 18.3 23 27 20.0 23 27 20.7 
 
 o / II 
 
 23 27 29.6 
 23 27 35.7 
 23 27 17.6 
 23 27 29.0 
 23 27 6.8 
 23 27 20 5 
 
 6 
 
 23 27 27.1 
 23 27 37.2 
 23 27 15.8 
 23 27 29. 
 23 27 6.3 
 23 27 19.4 
 
 23 2724.8 
 23 27 377 
 23 2714.9 
 27 28.5 
 23 27 7.0 
 23 2717.3 
 
 2 23 
 
 23 2723.6 
 23 27 37.3 
 23 27 15.1 
 23 2726 7 
 23 27 8.6 
 23 27 14.4 
 
 23 27 23.4 
 23 27 36.0 
 23 27 15.9 
 23 27 24.1 
 23 27 10.8 
 23 2711.0 
 
 The obliquity of the ecliptic to the equator is slowly diminishing, owing to the 
 attraction of the planets, and is also subject to an inequality whose period is about 
 nineteen years, caused by the attraction of the moon, and called Nutation. This 
 Table gives the obliquity on the 1st of January in each year, taking both these 
 causes into account. 
 
TABLE XXIV. 
 
 MoorCs Latitude in Eclipses. 
 Argument — Moon's Longitude diminished by that of its Node. 
 
 Arg. 
 
 S.D. 
 
 lsoio.oboo 
 
 181J0.0873 
 182 ! 0.1746 
 
 1880.6950 
 
 0.2617 
 0.3486 
 0.4353 
 0.5219 
 0.6085 
 
 0.7811 
 0.8672 
 0.9529 
 1.0382 
 1.1233 
 1.2082 
 1.2925 
 1.3765 
 Arg. 1°.0 
 
 0087 
 0960 
 1834 
 2704 
 3572 
 4440 
 5306 
 6173 
 7036 
 7897 
 8758 
 9614 
 0467 
 1318 
 2167 
 3009 
 3849 
 
 0.0175 
 
 .9 
 
 0.1048 
 0.1921 
 0.2791 
 0.3659 
 0.4526 
 0.5392 
 0.6258 
 0.7122 
 0.7983 
 0.8844 
 
 0.9700 
 
 1.0552 
 
 1.14031 
 1.2251 1 
 1.3093 1 
 1.3933 1 
 
 JB 
 
 0262 
 
 1135 
 
 2008 
 
 287810 
 
 37460 
 
 46130 
 
 54780 
 
 63430 
 
 72080 
 
 80690 
 
 8930;0 
 
 97850 
 
 0638 1 
 
 1488 
 
 2335 
 
 3177 
 
 4017 
 
 .7 
 
 0350 
 1222 
 2095 
 ,2965 
 3832 
 4699 
 5565 
 6431 
 7295 
 8155 
 9015 
 9870 
 0723 
 1573 
 2420 
 3261 
 4101 
 
 .6 
 
 0.0437 
 0.1310 
 0.2182 
 0.3052 
 0.3919 
 0.4786 
 0.5652 
 0.6517 
 0.7381 
 0.8241 
 0.9101 
 0.9955 
 1.0808 
 1.1658 
 1.2504 
 1.3345 
 1.4185 
 
 .5 
 
 0524 
 1397 
 2269 
 3139 
 4005 
 4873 
 5739 
 6604 
 7467 
 8327 
 9187 
 0041 
 0893 
 1743 
 2588 
 3428 
 4269 
 
 0.0611 
 0.1484 
 0.2356 
 0.3226 
 0.4092 
 0.4960 
 0.5826 
 0.6690 
 0.7553 
 0.8413 
 0.9273 
 1.0126 
 1.0978 
 1.1828 
 1.2672 
 1.3512 
 1.4353 
 
 4 1 3. 
 
 .8 
 
 0.0698 
 0.1572 
 0.2443 
 0.3313 
 0.4180 
 0.5046 
 0.5912 
 0.6778 
 0.7639 
 0.8499 
 0.9358 
 1.0211 
 1.1063 
 1.1913 
 1.2756 
 1.3596 
 1.4437 
 
 .2 
 
 .9 lo.O Arg. 
 
 0.07850 
 
 0.1659 
 
 0.2530 
 
 0.3399 
 
 0.4267 
 
 0.5133 
 
 0.5999 
 
 0.6864 
 
 0.7725 
 
 0.8585 
 
 0.9444 
 
 1.0296 
 
 1.1148 
 
 1 1998 I 
 
 l!2841 1 
 
 1 3680 1 
 
 1.4521 
 
 N.D. S-A 
 
 73 1791359 
 1746 178 358 
 2617,177 357 
 3486176356 
 4353 175,355 
 5219! 174 1 354 
 60851173 353 
 6950'172 352 
 7811J171 351 
 8672170i 350 
 9529169 349 
 0382|168 348 
 1233167,347 
 2082166 346 
 .2925165 345 
 .3765 164 344 
 
 1-4605 163] 343 
 
 Arg. 
 
 The moon has sometimes a north and sometimes a south latitude, owing to the 
 obliquity of the plane of its orbit to that of the ecliptic. This Table gives the 
 latitude for every tenth of a degree of longitude, reckoned 17° either way from 
 each node. The capital letters at the hea'd of the columns of the argument show 
 whether the latitude is north or south, and whether it is ascending or descending. 
 
 TABLE XXV. 
 
 Angle of the visible path of the Moon with the Ecliptic in Eclipses. 
 
 Arguments — Horary motion of the Moon from the Sun at the top, and 
 the Moon's distance from the Node at the right and left. 
 
 N. A. 
 
 ~6° 
 
 S. D. 
 
 180° 
 
 .44 
 
 .46 
 
 .48 
 
 .50 
 
 .52 
 
 .54 
 
 .56 
 
 .58 
 
 .60 
 
 N. D. 
 
 S. A. 
 
 5°47' 
 
 5°46' 
 
 5°45' 
 
 5044/ 
 
 5°43' 
 
 5°42' 
 
 5°41' 
 
 5°40' 
 
 5°39' 
 
 180° 
 
 360° 
 
 3 
 
 183 
 
 5 46 
 
 5 45 
 
 5 44 
 
 5 43 
 
 5 42 
 
 5 41 
 
 5 40 
 
 5 40 
 
 5 39 
 
 177 
 
 357 
 
 6 
 
 186 
 
 5 45 
 
 5 44 
 
 5 43 
 
 5 42 
 
 5 41 
 
 5 40 
 
 5 39 
 
 5 39 
 
 5 38 
 
 174 
 
 354 
 
 9 
 
 189 
 
 5 42 
 
 5 41 
 
 5 40 
 
 5 39 
 
 5 38 
 
 5 38 
 
 5 37 
 
 5 36 
 
 5 35 
 
 171 
 
 351 
 
 12 
 
 192 
 
 5 39 
 
 5 38 
 
 5 37 
 
 5 36 
 
 5 35 
 
 5 34 
 
 5 34 
 
 5 33 
 
 5 32 
 
 168 
 
 348 
 
 15 
 
 195 
 
 5 35 
 
 5 34 
 
 5 33 
 
 5 32 
 
 5 31 
 
 5 30 
 
 5 30 
 
 5 29 
 
 5 28 
 
 165 
 
 345 
 
 The angle of the moon's path with the ecliptic, which depends upon its dis- 
 tance from the node, is apparently increased by the earth's motion in the same 
 direction. This Table gives the apparent angle, taking both these facts into con- 
 sideration. 
 
TABLE XXVI. 
 
 The Sun's Declination. 
 Argument — Sun's Longitude. 
 
 / 
 
 g 
 
 6 2^/?>.? 
 
 Arg. 
 
 Oo 
 
 1° 
 
 2° j 3o j 40 
 
 50 
 
 60 
 
 7o 
 
 80 
 
 9° 
 
 lOo 
 
 Arg. 
 
 N. 
 
 
 
 s. 
 
 IS 
 
 0.0000 
 
 0°3980 
 
 0°7961 1°1939 
 
 1.5911 
 
 1°9S83 
 
 2°3S50 
 
 2°7809 
 
 3°1761 
 
 3°5703 
 
 3°9639 
 
 s. 
 35 
 
 N. 
 
 17 
 
 I 
 
 iy 
 
 3.9639 
 
 4.3564 
 
 4.7477 5.1380 
 
 5.5267 
 
 5.9142 
 
 6.3000 
 
 6.6839 
 
 7.0664 
 
 7.4472 
 
 7.8259 
 
 34 
 
 16 
 
 Si 
 
 20 
 
 7.8259 
 
 8.2025 
 
 8.5767 8.9489 
 
 9.3189 
 
 9.6858 
 
 10.0506 
 
 10.4128 
 
 10.7720 
 
 11.1284 
 
 11.4817 
 
 33 
 
 15 
 
 B 
 
 2 l 
 
 11.4817 
 
 11.8317 
 
 12.1789,12.5225 12.8628 
 
 13.1997 
 
 13.5328 
 
 13.8622 
 
 14.1877 
 
 14.5092 
 
 14.8270 
 
 32 
 
 14 
 
 4 
 
 22 
 
 14.8270 
 
 15.1402 
 
 I5.4494jl5.7539 16.0542 
 
 16.3500 
 
 16.6408 
 
 16.9273 
 
 17.2086 
 
 17.4850 
 
 17.7561 
 
 31 
 
 13 
 
 b 
 
 23 
 
 17.7561 
 
 18.0223 
 
 18.2833.18.5386 18.7886 
 
 19.0331 
 
 19.2717 
 
 19.5042 
 
 19.7314 
 
 19.9528 
 
 20.1681 
 
 30 
 
 12 
 
 6 
 
 24 
 
 20.1681 
 
 20.3770 
 
 20.579520.776120.9664 
 
 21.1500 
 
 21.3267 
 
 21.4975 
 
 21.6614 
 
 21.8186 
 
 21.9689 
 
 29 
 
 11 
 
 V 
 
 25 
 
 21.9689 
 
 221122 
 
 22.2486 22. 378 1I22.5003 
 
 22.6153 
 
 22.7234 
 
 22.8242 
 
 22.9178 
 
 23.0042 
 
 23.0831 
 
 28 
 
 10 
 
 8 
 
 26 
 
 23.0831 
 
 23.1544 
 
 23.2184 23.2750 
 
 23.3242 
 6° 
 
 23.3658 
 5° 
 
 23.3997 
 
 23.4261 
 
 23.4450 
 
 23.4564 
 
 23.4603 
 
 27 
 
 9 
 
 Arg. 
 
 10o 
 
 90 
 
 8° ) 7° 
 
 40 
 
 3o 
 
 2° 
 
 10 
 
 Oo 
 
 Arg. 
 
 The plane of the ecliptic not coinciding with that of the equator, the sun is 
 sometimes north of the equator and sometimes south. This is called its declina- 
 tion, and this Table shows its amount for every degree of longitude. The epoch 
 of the Table is 1840. 
 
 TABLE XXVII. 
 
 1st Preliminary Equation. 
 
 Argument — Moon's Anomaly. 
 
 
 0° 
 
 10 
 
 2o 
 
 30 
 
 40 
 
 50 
 
 6° 
 
 70 
 
 8° 
 
 9=> 
 
 10O 
 
 > 
 + 
 
 h. ra. s. 
 
 h. m. s. 
 
 h. 111. s. 
 
 h. m. s. 
 
 h. m. s. 
 
 h. m. s. 
 
 h. m. s. 
 
 h. m. s. 
 
 h. ra. s. 
 
 h. m. s. 
 
 h. m. s. 
 
 
 
 
 
 9 34 
 
 19 8 
 
 28 41 
 
 38 13 
 
 47 44 
 
 57 13 
 
 1 6 41 
 
 1 16 7 
 
 125 31 
 
 134 54 
 
 35 
 
 1 
 
 134 54 
 
 144 16 
 
 153 36 
 
 2 2 53 
 
 2 12 8 
 
 2 21 19 
 
 2 30 28 
 
 2 39 34 
 
 2 48 39 
 
 2 57 43 3 6 45 
 
 34 
 
 2 
 
 3 6 45 
 
 3 15 44 
 
 3 24 42 
 
 3 33 38 
 
 3 42 32 
 
 3 123 
 
 4 7 
 
 4 8 47 
 
 4 17 25 
 
 4 26 1 
 
 4 34 33 
 
 33 
 
 3 
 
 4 34 33 
 
 4 43 2 
 
 4 51 15 
 
 4 59 42 
 
 5 7 56 
 
 5 16 5 
 
 5 24 9 
 
 5 32 9 
 
 5 40 4 
 
 5 47 54 
 
 5 55 38 
 
 32 
 
 4 
 
 5 55 38 
 
 6 3 16 
 
 6 10 49 
 
 6 18 18 
 
 6 25 40 
 
 6 32 56 
 
 6 40 6 
 
 6 47 6 
 
 6 54 8 
 
 7 1 2 
 
 7 7 50 
 
 31 
 
 5 
 
 7 7 50 
 
 7 14 30 
 
 7 21 2 
 
 7 27 22 
 
 7 33 36 
 
 7 39 46 
 
 7 45 46 
 
 7 5133 
 
 7 57 23 
 
 8 3 12 
 
 8 8 59 
 
 30 
 
 6 
 
 8 8 59 
 
 8 14 33 
 
 8 20 18 
 
 8 25 44 
 
 8 31 
 
 8 36 6 
 
 8 41 2 
 
 8 45 48 
 
 8 50 24 
 
 8 54 50 
 
 8 58 6 
 
 29 
 
 7 
 
 8 58 6 
 
 9 3 13 
 
 9 7 9 
 
 9 10 54 
 
 9 14 28 
 
 9 17 51 
 
 9 21 3 
 
 9 24 4 
 
 9 26 54 
 
 9 29 33 
 
 9 32 1 
 
 28 
 
 A 
 
 9 32 1 
 
 9 34 18 
 
 9 36 24 
 
 9 38 19 
 
 9 40 3 
 
 9 4136 
 
 9 42 59 
 
 9 44 11 
 
 9 45 12 
 
 9 46 3 
 
 9 46 44 
 
 27 
 
 9 
 
 9 46 44 
 
 9 47 14 
 
 9 47 33 
 
 9 47 46 
 
 9 47 54 
 
 9 47 49 
 
 9 47 36 
 
 9 47 13 
 
 9 46 38 
 
 9 45 52 
 
 9 44 53 
 
 26 
 
 K) 
 
 9 44 53 
 
 9 43 42 
 
 9 42 21 
 
 9 40 51 
 
 9 39 8 
 
 9 37 14 
 
 9 35 12 
 
 9 32 58 
 
 9 30 32 
 
 9 27 58 9 25 12 
 
 25 
 
 11 
 
 9 25 12 
 
 9 22 14 
 
 9 19 5 
 
 9 15 43 
 
 9 12 9 
 
 9 8 25 
 
 9 4 31 
 
 9 25 
 
 8 56 10 
 
 8 5145 8 47 8 
 
 24 
 
 12 
 
 8 47 8 
 
 8 42 18 
 
 8 37 19 
 
 8 32 11 
 
 8 26 53 
 
 8 2124 
 
 8 15 46 
 
 8 9 57 
 
 8 3 56 
 
 7 57 45 
 
 7 5124 
 
 23 
 
 13 
 
 7 5124 
 
 7 44 51 
 
 7 38 9 
 
 7 31 18 
 
 7 24 10 
 
 7 17 9 
 
 7 9 52 
 
 7 2 24 
 
 6 54 46 
 
 6 47 
 
 6 39 4 
 
 22 
 
 14 
 
 6 39 4 
 
 6 30 57 
 
 6 22 41 
 
 6 14 19 
 
 6 5 51 
 
 5 57 17 
 
 5 48 37 
 
 5 39 51 
 
 5 30 57 
 
 5 2156 
 
 5 12 48 
 
 21 
 
 If) 
 
 5 12 48 
 
 5 3 33 
 
 4 54 11 
 
 4 44 42 
 
 4 35 6 
 
 4 25 20 
 
 4 15 26 
 
 4 5 26 
 
 3 55 21 
 
 3 45 11 
 
 3 34 58 
 
 20 
 
 16 
 
 3 34 58 
 
 3 24 42 
 
 3 14 24 
 
 3 4 3 
 
 2 53 38 
 
 2 43 9 
 
 2 32 34 
 
 2 21 54 
 
 2 11 10 
 
 2 23 
 
 149 33 
 
 19 
 
 17 
 
 < 
 
 149 33 
 
 138 40 
 
 127 44 
 
 1 16 46 
 
 1 5 48 
 
 54 50 
 
 43 52 
 
 32 54 
 
 2156 
 
 10 58 
 
 000 
 
 18 
 
 < 
 
 10° 
 
 90 
 
 8° 
 
 70 
 
 6« 
 
 50 
 
 40 
 
 3° 
 
 2o 
 
 1° 
 
 0o 
 
 When the moon's anomaly is less than 180°, it is in advance of its mean place at 
 time of new or full moon by reason of the Equation of the Centre, but behind 
 it by Evection, (Tables 9 and 14,) yet on the whole it is in advance; conse- 
 quently it will overtake the sun sooner than it would otherwise do, and something 
 must be subtracted from the mean time. The contrary takes place when the anom- 
 aly is more than 180°. This Table shows the amount of time to be added or sub- 
 tracted from these causes. 
 
TABLE XXVIII. 
 
 2d Preliminary Equation. 
 Argument — Sun's Anomaly. 
 
 OP 
 
 + 
 
 
 
 1 
 
 2 
 
 ■3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 10 
 II 
 12 
 13 
 11 
 
 152 
 16 
 1? 
 
 m. s. 
 
 
 44 28 
 27 31 
 
 7 45 
 43 57 
 14 49 
 39 30 
 57 27 
 
 7 59 
 10 53 
 
 6 10 
 54 4 
 35 3 
 
 9 36 
 38 44 
 
 3 12 
 24 10 
 42 39 
 
 10° 
 
 1Q 
 m. s 
 
 4 29 
 48 52 
 3141 
 1135 
 47 18 
 17 35 
 4140 
 53 52 
 
 8 37 
 10 45 
 
 5 18 
 52 29 
 32 45 
 
 6 45 
 35 22 
 59 26 
 20 6 
 38 26 
 
 9 
 
 go 
 
 m. s. 
 
 8 56 
 53 13 
 35 49 
 
 15 20 
 50 36 
 20 20 
 43 45 
 
 12 
 
 9 10 
 10 33 
 
 422 
 50 50 
 30 26 
 
 3 51 
 3157 
 55 37 
 
 16 
 34 11 
 
 HO 
 
 :p 
 
 13 23 
 
 57 36 
 
 39 56 
 
 19 5 
 
 53 49 
 
 23 
 
 45 44 
 
 126 
 
 9 39 
 
 10 16 
 
 3 23 
 
 49 7 
 
 28 3 
 
 40 
 
 ii. in. 
 
 17 
 
 1 1 
 141 
 
 2 22 
 
 2 57 
 
 3 25 
 
 3 47 
 
 4 2 
 4 10 
 4 9 
 4 2 
 3 47 
 3 25 
 
 50 
 
 54 2 57 
 
 28 29 
 5146 
 1153 
 28 55 
 
 7« 
 
 2 25 
 147 
 
 1 7 
 25 
 
 h. m. 
 500 22 
 56 
 
 1 
 47 
 
 
 35 
 38 
 35 
 
 4 
 
 55 
 18 
 1? 
 36 
 53 
 9 
 54 
 45 
 39 
 
 60 
 
 60 
 
 2 26 
 
 3 
 3 28 
 
 3 49 
 
 4 3 
 4 10 
 4 9 
 4 1 
 3 45 
 3 23 
 2 54 
 2 21 
 144 
 1 3 
 21 
 
 h. m. 
 
 26 
 
 1 10 
 52 
 
 2 30 
 
 3 3 
 3 30 
 
 3 51 
 
 4 4 
 4 10 
 4 8 
 3 59 
 3 43 
 3 20 
 2 51 
 2 17 
 1 40 
 59 
 17 
 
 5° 
 
 4° 
 
 70 
 
 h. m 
 
 31 10 
 
 1 14 49 
 156 5 
 
 2 33 35 
 
 3 6 10 
 3 32 50 
 
 3 52 49 
 
 4 5 37 
 4 10 49 
 4 8 21 
 3 58 27 
 3 4123 
 3 17 51 
 2 48 30 
 2 14 14 
 1 36 10 
 55 1 
 12 51 
 
 30 
 
 9° 
 
 HO 
 
 m. s. ft. in. 
 35 36 40 
 19 5 123 
 12 3 
 6 2 40 
 6311 
 3 37 
 
 
 
 37 
 9 
 
 35 6 
 54 26 
 
 6 29 
 10 54 
 
 7 41 
 57 2 
 39 18 
 15 9 
 45 18 
 10 36 
 32 12 
 51 4 
 
 8 35 
 
 2 rj 
 
 3 55 
 
 4 7 
 4 10 
 4 6 
 3 55 
 3 37 
 3 12 
 2 42 
 2 6 
 128 
 46 
 
 
 h. in. s. 
 44 28 35 
 
 10O 
 
 10 
 
 27 31 
 2 7 45 
 
 2 43 57 
 
 3 14 49 
 3 39 30 
 
 3 57 27 
 
 4 7 59 
 4 10 53 
 4 6 10 
 3 54 4 
 3 35 
 3 9 36 
 2 38 44 
 2 3 12 
 124 10 
 42 39 
 0_4_0 
 
 Oo 
 
 When the sun's anomaly is less than 180°, it is before and the moon behind the 
 mean place, by reason of the Equation of the Centre (Table 8) of the former, 
 and the Annual Equation of the Longitude (Table 10) of the latter. For both rea- 
 sons, then, the moon will not overtake the sun so soon as it would otherwise do, 
 and consequently something must be added to the mean time of New or Full Moon. 
 The contrary takes place when the anomaly is more than 180° ; and this Table 
 shows the amount of time to be added or subtracted from these causes. 
 
 TABLE XXIX. 
 
 Augmentation of the Moon's Semi- diameter. 
 
 Argument — Distance of the place (as projected on the disc) from the 
 
 earth's centre. 
 
 Tables 21 and 22 show us the apparent semi-diameter of the 
 moon as viewed from the centre of the earth; but the distance 
 of the moon from any place on the earth's surface at which it 
 is visible (save when it is in the horizon) is less than from the 
 centre, which must cause it to subtend a greater angle. This 
 Table shows the augmentation in the moon's apparent semi- 
 diameter from this cause. 
 
 Arg. 
 
 
 + 
 
 
 
 
 .0045 
 
 10 
 
 .0045 
 
 20 
 
 .0044 
 
 30 
 
 .0043 
 
 40 
 
 .0041 
 
 50 
 
 .0038 
 
 60 
 
 .0035- 
 
 70 
 
 .0031 
 
 80 
 
 .0024 
 
 90 
 
 .0015 
 
 100 
 
 .0000 
 
TABLE XXX. 
 
 To convert minutes into decimals of a degree. 
 Argument — The number of minutes. 
 
 Arg. 
 
 
 0' 
 
 I' 
 
 2' 
 
 3' 
 
 4' 
 
 5' 
 
 6' 
 
 7' 
 
 8' 
 .1333 
 
 9' 
 
 .0000 
 
 .0167 
 
 .0333 
 
 .0500 
 
 .0667 
 
 .0833 
 
 .1000 
 
 .1167 
 
 .1500 
 
 1 
 
 .1667 
 
 .1833 
 
 .2000 
 
 .2167 
 
 .2333 
 
 .2500 
 
 .2667 
 
 .2833 
 
 .3000 
 
 .3167 
 
 2 
 
 .3333 
 
 .3500 
 
 .3667 
 
 .3833 
 
 .4000 
 
 .4167 
 
 .4333 
 
 .4500 
 
 .4667 
 
 .4833 
 
 3 
 
 .5000 
 
 .5167 
 
 .5333 
 
 .5500 
 
 .5667 
 
 .5833 
 
 .6000 
 
 .6167 
 
 .6333 
 
 .6500 
 
 4 
 
 .6667 
 
 .6833 
 
 .7000 
 
 .7167 
 
 .7333 
 
 .7500 
 
 .7667 
 
 .7833 
 
 .8000 
 
 .8167 
 
 5 
 
 .8333 
 
 .8500 
 
 .8667 
 
 .8833 
 
 .9000 
 
 .9167 
 
 .9333 
 
 .9500 
 
 .9667 
 
 .9833 
 
 TABLE XXXI. 
 
 To convert seconds into decimals of a degree. 
 Argument — The number of seconds. 
 
 Arg. 
 
 0" 
 
 1" 
 
 2" 
 
 3" 
 
 4" 
 
 5" 
 
 6" 
 .0017 
 
 7" 
 
 8" 
 
 9" 
 
 
 
 .0000 
 
 .0003 
 
 .0006 
 
 .0008 
 
 .0011 
 
 .0014 
 
 .0019 
 
 .0022 
 
 .0025 
 
 1 
 
 .0028 
 
 .0031 
 
 .0033 
 
 .0036 
 
 .0039 
 
 .0042 
 
 .0044 
 
 .0047 
 
 .0050 
 
 .0053 
 
 2 
 
 .0056 
 
 .0058 
 
 .0061 
 
 .0064 
 
 .0067 
 
 .0069 
 
 .0072 
 
 .0075 
 
 .0078 
 
 .0081 
 
 o 
 
 .0083 
 
 .0086 
 
 .0089 
 
 .0092 
 
 .0094 
 
 .0097 
 
 .0100 
 
 .0103 
 
 .0106 
 
 .0108 
 
 4 
 
 .0111 
 
 .0114 
 
 .0117 
 
 .0119 
 
 .0122 
 
 .0125 
 
 .0128 
 
 .0131 
 
 .0133 
 
 .0136 
 
 5 
 
 .0139 
 
 .0142 
 
 .0144 
 
 .0147 
 
 .0150 
 
 .0153 
 
 .0156 
 
 .0158 
 
 .0161 
 
 .0164 
 
QB5HI 
 
 an