UC-NRLF B M ATORY ASTRONOMY WILLSON GIFT OF R. 0/racy ^rav/ford UEPT. V^tf ^ -*p~^ J^""" *YT~ *# ^' LABORATORY ASTRONOMY BY ROBERT WHEELER WLLLSON, PH.D. PROFESSOR OF ASTRONOMY IN HARVARD UNIVERSITY GINN & COMPANY BOSTON NEW YORK CHICAGO LONDON COPYRIGHT, 1900, 1905 BY ROBERT W. WILLSON ALL RIGHTS RESERVED 66.1 ASTROHOHT DEPT. gtftenaeum GINN & COMPANY PRO- PRIETORS BOSTON U.S.A. %f- PREFACE THE subjects treated in elementary text-books of astronomy which are most difficult and discouraging to the beginner are those which deal with the diurnal motion of the heavens and the apparent motions of the sun, moon, and planets among the stars. A clear conception of these fundamental facts is, however, necessary to a proper understanding of many of the striking phenomena to which the study of astronomy owes its hold upon the intellect and the imagination. No adequate notion of those subjects which involve the ideas of force and mass can be given to the average student who has not mastered the elements of mechanics ; but to explain the motions of the heavenly bodies, the knowledge of a few principles of solid geometry and of the properties of the ellipse will suffice, no more, indeed, than may "be easily explained- in the pages of the text-book itself. Most of the difficulties which arise at the outset of the study may be satisfactorily met by methods which require the student to make and discuss simple observations and to solve simple problems. This necessity is recognized in many recent text-book's which introduce such methods to a greater or less extent, in all cases to great advantage and in some with marked success. I have gathered in this book some of those which I have found practicable, intending that they should explain in natural sequence those phenomena which depend on the diurnal motion, the moon's motion in her orbit and the change in position of that orbit, the motion of the sun in the ecliptic, and the geocentric motions of the planets. The methods chosen may be carried out with fair-sized classes and do not require a place of observation favored with an extensive view of the heavens. The gnomon-pin, the hemisphere, the cross- staff, a simple apparatus for measuring altitude and azimuth which iii iv PREFACE may be converted into an equatorial by inclining it at the proper angle, together with a few maps and diagrams, form an outfit so inexpensive that it may be supplied to each pupil, and much work may be done at home. It is obvious that the possibility thus offered of utilizing favorable opportunities for observation is especially valuable in a study which is so much dependent on the weather. All members of the class, too, will be doing the same or similar work at the same time, a principle of cardinal importance in elementary laboratory work with large classes. The meridian work of Chapter VI is added for the sake of logical completeness, to explain the determination of the zero of right ascensions, a subject which is usually neglected in the text-books and would not be included in an ordinary course. Nothing has been directly planned for teaching the names of the constellations and the use of star maps. The work of Chapters II, III, and IV, covering a period of some months, results in a very good acquaintance with the principal stars and asterisms. It may be assumed, too, that the teacher is familiar with the heavens and will gather the class as early as possible to introduce them at least to the polar constellations. The book is intended primarily for teachers, but much of it is suitable for use as a text-book, in spite of its rather condensed form. It is meant to be used in connection with one of the many admirable text-books on descriptive astronomy adapted to high- school pupils. The first six chapters were printed in 1900, and various changes and additions might now be made, notably an improvement in the protractor for laying off altitudes on the hemisphere, which is now so constructed that it may be used as a ruler for the accurate draw- ing of great circles. This permits a much simpler determination of the pole of a small circle than that described in the first chapter. ROBERT W. WILLSON HARVARD UNIVERSITY STUDENTS' ASTRONOMICAL LABORATORY December, 1905 TABLE OF CONTENTS CHAPTER I THE SUN'S DIURNAL MOTION PAGE Path of the Shadow of a Pin-head cast by the Sun upon a Horizontal Plane 1 Altitude and Bearing .......... 4 Representation of the Celestial Sphere upon a Spherical Surface. . . 6 The Sun's Diurnal Path upon the Hemisphere is a Circle a Small Circle except about March 20 and September 21 8 Determination of the Pole of the Circle ....... 9 Bearing of the Points of Sunrise and Sunset 11 The Meridian the Cardinal Points 11 Magnetic Decimation 12 Azimuth .............. 12 The Equinoctial ............ 14 Position of the Pole as seen from Different Places of Observation . . 15 Latitude equals Elevation of Pole 16 Hour-angle of the Sun .......... 17 Uniform Increase of the Sun's Hour-angle Apparent Solar Time . . 18 Declination of the Sun its Daily Change . . . . . .20 CHAPTER II THE MOON'S PATH AMONG THE STARS Position of the Moon by its Configuration with Neighboring Stars . .21 Plotting the Position of the Moon upon a Star Map .... 24 Position of the Moon by Measures of Distance from Neighboring Stars . 25 The Cross-staff 25 Length of the Month ' 29 Node of the Moon's Orbit 30 Errors of the Cross-staff . 31 VI TABLE OF CONTENTS CHAPTER III THE DIURNAL MOTION OF THE STARS PAGE Instrument for measuring Altitude and Azimuth 34 Adjustment of the Altazimuth .35 Determination of Meridian by Observations of the Sun .... 37 Determination of Apparent Noon by Equal Altitudes of the Sun . . 39 Meridian Mark 40 Selection of Stars Magnitudes . . . . . . . . .41 Plotting Diurnal Paths of Stars on the Hemisphere .... 42 Paths of Stars compared with that of the Sun .42 Drawing of Hemisphere with its Circles ....... 42 Rotation of the Sphere as a Whole 43 Declinations of Stars do not change like that of the Sun ... 43 Equable Description of Hour-angle by Stars ...... 43 Hour-angle and Declination fix the Position of a Heavenly Body as well as Altitude and Azimuth Comparison of the Two Systems of Coordinates ........... 44 Equatorial Instrument for measuring Hour-angle and Declination . . 45 Universal Equatorial Advantages of the Equatorial Mounting . . 45 CHAPTER IV THE COMPLETE SPHERE OF THE HEAVENS Rotation of the Heavens about an Axis passing through the Pole explains Diurnal Motions of Sun, Moon, and Stars . . . . .47 Relative Position of Two Stars determined by their Declinations and the Difference of their Hour-angles 48 Use of Equatorial to determine Positions of Stars . . . . .49 Use of a Timepiece to improve the Foregoing Method .... 50 Map of Stars by Comparison with a Fundamental Star . . . .53 Extension of Use of Timepiece to reduce Labor of Observation . . 54 The Vernal Equinox to replace the Fundamental Star Right Ascension 56 Sidereal Time Sidereal Clock 57 Right Ascension of a Star is the Sidereal Time of its Passage across the Meridian 58 Right Ascension of any Body plus its Hour-angle at any Instant is Side- real Time at that Instant ........ 58 Finding Stars by the Use of a Sidereal Clock and the Circles of the Equa- torial Instrument .......... 59 The Clock Correction 60 List of Stars for determining Clock Error 61 TABLE OF CONTENTS vii CHAPTER V MOTION OF THE MOON AND SUN AMONG THE STARS PAGE Plotting Stars upon a Globe in their Proper Relative Positions . . 63 Plotting Positions of the Moon upon Map and Globe by Observations of Declination, and Difference of Right Ascension, from Neighboring Stars ' . . . . .64 Variable Rate of Motion of the Moon . . . . ' . . . 65 Variable Semi-diameter of the Moon ........ 65 Position of Greatest Semi-diameter and of Greatest Angular Motion . ( 65 Plotting Moon's Path on an Ecliptic Map 65 Observations of Sun's Place in Reference to a Fundamental Star by Equa- torial and Sidereal Clock ........ 66 Sun's Place referred to Stars by Comparison with the Moon or Venus . 68 Plotting the Sun's Path upon the Globe the Ecliptic .... 70 CHAPTER VI MERIDIAN OBSERVATIONS Use of the Altazimuth or Equatorial in the Meridian 72 The Meridian Circle 73 Adjustments of the Meridian Circle ........ 74 Level ............. 74 Collimation ............. 78 Azimuth . ... 78 Determination of Declinations . . 80 Determination of the Polar Point . . . . . . . . 81 Absolute Determination of Declination ....... 81 Determination of the Equinox 83 Absolute Right Ascensions . ... . . . . . . .84 Autumnal Equinox of 1899 85 Autumnal Equinox of 1900 * . . .87 Length of the Year . 88 CHAPTER VII THE NAUTICAL ALMANAC Mean Time 91 The Equation of Time 92 Standard Time . 93 viii TABLE OF CONTENTS PAGE The Calendar Pages 94 Examination of the Several Columns ....... 99 Data for the Planets and Stars ........ 102 Comparison of Observations with the Ephemeris 102 Observations of the Moon with the Cross-staff ; Length of the Month . 103 Observation at Apparent Noon 104 Observations of the Planets. Observations of the Moon with Equatorial 105 Observations of the Sun's Place ........ 106 Determination of the Equinox ........ 107 CHAPTER VIII THE CELESTIAL, GLOBE Description of the Globe . . . , " . . . . . .108 Rectifying the Globe for a Given Place and Time . . . . . Ill The Sun's Place on the Globe . . . : ,. . ... . .112 The Altitude Arc 113 Problems which do not require Rectification of the Globe . . . 114 Problems which require Rectification of the Globe for a Given Time . 117 Finding an Hour-angle by the Globe . . . . . . .119 Reduction to the Equator . . . . . . . . 121 CHAPTER IX EXAMPLES OF THE USE OF THE GLOBE Problems which require Rectification of the Globe for a Given Place . 122 Rising and Setting of Stars 122 Sunrise . . . . 124 Altitude and Azimuth ; Hour-angle ........ 125 Finding the Time from the Sun's Altitude 126 Identifying a Heavenly Body by its Altitude and Azimuth at a Given Time 129 Aspect of the Planets at a Given Time ....... 130 Rising and Setting of the Moon 131 Twilight 133 Orientation of Building by Sun Observation 134 Latitudes in which Southern Cross is Visible . . . ... 135 The Midnight Sun ; the Harvest Moon 136 Change of Azimuth at Rising and Setting 137 Graduating a Horizontal Sundial . . 137 Graduating a Vertical Sundial 138 Determining Path of Shadow by Globe 139 The Hour-index 141 TABLE OF CONTENTS IX CHAPTER X THE MOTION OF THE PLANETS PAGE Elliptic Orbits a Result of the Law of Gravitation ..... 143 Properties of the Ellipse 144 To draw an Ellipse from Given Data . . . ' ' '. . . .145 Mean and True Place of a Planet ; Equable Description of Areas . 146 The Equation of Center . . .148 Measurement of Angles in Radians . 149 The Diagram of Curtate Orbits .151 To find the Elements of an Orbit from the Diagram . . . . 154 Place of the Planet in its Orbit - . .156 To find the True Heliocentric Longitude of a Planet . . . . 157 To find the Heliocentric Latitude ........ 161 Geocentric Longitude of a Planet . . . . ... . . 161 The Sun's Longitude and the Equation of Time 162 Geocentric Latitude .......... 163 Perturbations; Precession 166 The Julian Day 167 Right Ascensions and Declinations of the Planets 167 Configurations of the Planets 168 The Path of Mars among the Stars in 1907 169 LABORATORY ASTRONOMY PART I CHAPTER I THE DIURNAL MOTION OF THE SUN THE most obvious and important astronomical phenomenon that men observe is the succession o.f day and night, and the motion of the sun which causes this succession is naturally the first object of astronomical study. Every one knows that the sun rises in the east and sets in the west, but very many educated people know little more of the course of the sun than this. The first task of the beginner in astronomy should be to observe, as carefully as possible, the motion of the sun for a day. What is to be observed then? A little thought shows that it can only be the direction in which we have to look to see it at different times ; that is, toward what point of the compass how far above the ground. All astronom- / ical observation, indeed, comes down ultimately to this the direc- / tion in which we see things. The strong light of the sun enables us to make use of a very simple method depending on the principl that the shadow of a body lies in the same straight line with body and the source of light. Path of the Shadow of a Pin-head. If we place a pin upright on a horizontal plane in the sunlight and mark the position of the shadow of its head at any time, we thus fix the position of the sun at that time, since it is in the prolongation of the line drawn from the shadow to the pin-head. In order to carry out systematic 1 enables *^ ?inciple / 0# ith the/ ^ 2 LABORATORY ASTRONOMY observations by this method in such a form that the results may be easily discussed, it will be convenient to have the following appa- ratus : (1) A firm table in such a position as to receive sunlight for as long a period as possible. It is better that it should be in the open air, in which case it may be made by driving small posts into FIG. i the ground and securely fastening a stout plank about 18 inches square as a top. (2) A board, 18 inches long and 8 inches broad, furnished with leveling screws and smoothly covered with white paper fastened down by (3) thumb tacks. (4) A level for leveling the board. (5) A compass. (6) A glass plate, 6 inches long and 2 inches broad, along the median line of which a straight black line is drawn. (7) A pin, 5 cm. long, with a spherical head and an accurately turned base for setting it vertical. (8) A timepiece. Draw a straight pencil line across the center of the paper as FIG. 2 nearly as possible perpendicular to the length of the board. Place the board upon the table and level approximately. Put the com- pass on the middle of the pencil line and put the glass plate on the compass with its central line over the center of the needle ; turn the plate till its median line is parallel to the pencil line (Fig. 2), THE DIURNAL MOTION OF THE SUN 3 and swing the whole board horizontally, till the needle is parallel to the two lines, which are then said to be in the magnetic meridian. Press the leveling screws firmly into the table, and thus make dents by which the board may at any future time be placed in the same position without the renewed use of the compass. Level the board carefully, placing the level first east and west, then north and south. Place the pin in the pencil line, in the center if the observation is made between March 20 and September 20, but near the south- ern edge of the board at any other time of the year, pressing it firmly down till the base is close to the paper, so that the pin is perpendicular to the paper. Mark with a hard pencil the estimated center of the shadow of the pin-head, A (Fig. 3), noting the time by the watch to the nearest minute, affix a number or letter, and affix the same number to the recorded time of the observation in the note-book. It is a good plan to use pencil for notes made while FIG. 4 observing, and ink for computations or notes added afterward in discussing them. Kepeat at hourly, or better half -hourly, intervals, thus fixing a set of points (Fig. 4), through which a continuous curve may be drawn showing the path of the shadow for several hours. The same observation should be repeated two weeks later. LABORATORY ASTRONOMY ALTITUDE AND BEARING By the foregoing process we obtain a diagram on which is shown the position of the pin point, a magnetic meridian line through this point, and a series of numbered points showing the position of the shadow of the pin-head at different times ; the height of the pin is known and also the fact that its head was in the same vertical line with its point. In the discussion of these results, it will be convenient to proceed as follows : Eemove the pin and draw with a hard pencil a fine line, AB (Fig. 5), through the pinhole and the point marked at the first obser- vation. This line is called a line of bearing, and the angle which FIG. 5 it makes with the magnetic meridian is called the magnetic bearing of the line. This angle, which may be directly measured on the diagram by a protractor, fixes the position of the vertical plane which contains the observed point and passes also through the center of the pin-head and the sun. If this point bears KW. from the pin, the sun evidently bears S.E. Imagine a line, AC (Fig. 3), connecting the observed point with the sun's center and passing also through the center of the pin-head. The position of the sun in the vertical plane is evidently fixed by this line. The angle between the line of bearing and this line, BA C, is called the altitude of the sun ; it measures, by the ordinary con- vention of solid geometry, the angle between the sun's direction and the plane of the horizon. THE DIURNAL MOTION OF THE SUN o To determine this angle, lay off the line B'C' (Fig. 6), equal in length to the pin, 5 cm., draw a perpendicular through E\ and by means of a pair of compasses or scale laid between the two points A and B (Fig. 5), lay off the line A'B' on the perpendicular, draw A'C'f and measure the angle B'A'C' by a protractor. We now have the bearing and altitude of the sun at the time of the first observation, the bearing of the sun from the pin being opposite to that of the point from the pin. In like manner the altitude and bearing are determined for each observed point upon the path of the shadow, and noted against the correspond- ing time, in the note-book (to avoid con- fusion, it is convenient to make a separate figure for the morning and afternoon observations, as shown in Fig. 6). We have thus obtained a series of values which will enable us to study more easily the path of the sun upon the concave of the sky. Plotting the Sun's Path on a Spherical Surface. Probably the most evident method of accomplishing this object would be to construct a small concave portion of a sphere, as in the accom- panying figure, which suggests how the position of the sun might be referred to the inside of a glass shell. 6 LABORATORY ASTRONOMY But the hollow surface offers difficulty in construction and manipulation, and it requires but little stretch of the imagination to pass to the convex surface as follows. The glass shell, as seen from the other side, would appear thus : FIG. and we can more readily get at it to measure it, and moreover can more easily recognize the properties of the lines which we shall come to draw upon it, since we are used to looking upon spheres from the outside rather than from the inside, except in the case of the celestial sphere. On both Figs. 7 and 8 is shown a group of dots which have nearly the configuration of a group of stars conspicuous in the southern heavens in midsummer and called the constellation of Scorpio. It is evident that the constellation has the same shape in both cases, except that in Fig. 8 it is turned right for left or semi- inverted, as is the image of an object seen in a mirror. This prop- erty obviously belongs to all figures drawn on the concave surface as seen from the center, when they are looked at from the outside directly toward the center. So also the diurnal motion of the sun, which as we see it from the center is from left to right, would be from right to left as viewed from the outside of such a surface. This latter is so slight an inconvenience that it is customary to represent the motions of the heavenly bodies in the sky upon an opaque globe, and to determine the angles which these bodies describe about the center, by measuring the corresponding arcs upon the convex surface. THE DIURNAL MOTION OF THE SUN 7 Plotting on a Hemisphere. The apparatus required for plotting the sun's path consists of : a hemisphere,, a, 4{- inches in diameter ; a circular protractor, b, a quadrantal protractor, c, of 2^- inches FIG. 9 radius, and a pair of compasses, d, whose legs may be bent and one of which carries a hard pencil point. Determine by trial with the compasses the center of the base of the hemisphere, and mark two diameters by drawing straight lines upon the base at right angles through the center. Prolong these by marks about -J inch in length upon the convex surface. Place the Fio. 10 hemisphere exactly central upon the circular protractor, by bring- ing the marked ends of one of the diameters upon those divisions of the protractor which are numbered and 180, and the other on the divisions numbered 90 and 270. Determine and mark the 8 LABORATORY ASTRONOMY highest point of the hemisphere by placing the quadrant with its base upon the circular protractor, and its arc closely against the sphere, and marking the end of the scale (Fig. 10). Eepeat this with the arc in four positions, 90 apart on the base. The points thus determined should coincide ; if they do not, estimate and mark the center of the four points thus obtained. This point represents the highest point of the dome of the heavens the point directly overhead, called the zenith, and the zero and 180 points on the base protractor may be taken as representing the south and north points respectively of the magnetic meridian. The Sun's Path a Circle. To plot the altitude and bearing of the first observation, place the foot of the quadrant or altitude arc close against the sphere, the foot of its graduated face on the degree of the protractor which corresponds to the bearing. Mark a fine point on the sphere at that degree of the altitude arc corre- sponding to the altitude at the first observation. This point fixes the direction in which the sun would have been seen from the center of the hemisphere at the time of observation if the zero line had been truly in the magnetic meridian. Proceed in the same manner with the other observations of bearing and altitude, and thus obtain FIG. 11 a series of points (Fig. 11), through which may be drawn a con- tinuous line representing the sun's path upon that day. It will appear at once that the arcs between the successive points are of nearly equal length if the times of observation were equi- distant, and otherwise are proportional to the intervals of time THE DIURNAL MOTION OF THE SUN between the corresponding observations a property which does not at all belong to the shadow curve from which the points are derived. We thus have a noteworthy simplification in referring our observations to the sphere. It will also appear that a sheet of FIG. 12 stiff paper or cardboard may be held edgewise between the hemi- sphere and the eye, so as to cover all the points ; that is, they all lie in the same plane. This fact shows that the sun's path is a circle on the sphere. It is shown by the principles of solid geometry that all sections of the sphere by a plane are circles. If the plane of the circle passes through the center, it is the largest possible, its radius being equal to that of the sphere ; it is then called a great circle. Near the 20th of March and 22d of September it will be found that the path of the shadow is nearly a straight line on the diagram, and that the path of the sun is nearly a great circle ; that is, the plane of this circle passes nearly through the center of the sphere. In general, the shadow path is a curve, with its concave side toward the pin in summer and its convex side toward it in winter, while the path on the sphere is a small circle, that is, its plane does not pass through the center of the sphere. Determining the Pole of the Circle. It is proved by solid geometry that all points of any circle on the sphere are equidistant from two 10 LABORATORY ASTRONOMY points on the sphere, called the poles of the circle. It is important to determine the pole of the sun's diurnal path. Estimate as closely as possible the position on the sphere of a point which is at the same distance from all the observed points of the sun's path and open the compasses to nearly this distance. For a closer approximation to the position of the pole, place the steel point of the compasses at the point on the hemisphere correspond- ing to the first observation, a, and with the other (pencil) point draw a short arc, m (Fig. 12), near the estimated pole. Draw the arc n from the point of the last observation, G.M.T. Jan. 10 -21 59' 4".0 + 22". 25 + 4 x 22". 25 = + 89".0 - 21 57' 35".0 April 10 + 7 53 3 .7 + 55 .48 + 4 x 55 .48 = + 221 .9 + 7 56 45 .6 Aug. 10 + 15 38 18 .2 -43 .73 -4 x43 .73 = -174 .9 + 15 35 23 .3 Nov. 10 - 17 6 18 .2 -42 .31 -4 x42 .31 =-169 .2 - 17 9.7 .4 The error in a declination determined by a simple interpolation from the preceding mean noon can never exceed 12". By the more accurate method given in the Explanation, it is always less than 0".l. To' make sure that the correction has been applied with the proper sign, it is sufficient to notice that the computed value must lie between the values for the including dates. Columns 7 and 8 contain the equation of time and its hourly difference. The correction to be applied is obtained, as in the pre- ceding examples, by multiplying the hourly difference by the num- ber of hours elapsed since Greenwich mean noon, and must either be added or subtracted so as to give a value between the values of the including dates. The heading of the column indicates whether the equation of time is to be added to or subtracted from mean time to give apparent time. Of course when it is additive to mean time it must be sub- tracted from apparent time to give mean time, as will appear on comparing the corresponding column of page I. Example. What is the equation of time January 10, 1900, at 3 h 45 m , Central standard time ? The corresponding G.M.T. is 9 h 45 m = 9 h .75 Eq. of T. at Gr. Mean Noon . + 7 m 39 s . 87 H. Diff. = K014 Change in 9 h .75 = 9.75 x 1 8 .014 9 .88 x 9.75 Eq. of T. at 3 h 45 m , Cent. T. . +7 49.75 Corr. =9 8 .88 The correction 9 s . 88 is added because the value of the equation January 11 is seen to be 8 m 3 8 .90, and the correction must be applied so as to increase numerically the value on January 10. 96 LABORATORY ASTRONOMY The ninth column contains the right ascension of the mean sun. Since at mean noon the mean sun is on the meridian and since (p. 59) the right ascension of a body which is on the meridian at a given instant equals the sidereal time at that instant, the right ascension of the mean sun at Greenwich mean noon equals the Greenwich sidereal time at Greenwich mean noon, and this explains the alternative heading which appears at the top of the column. Since the right ascension of the mean sun increases uniformly, the constant hourly difference requires no special column, but is given at the foot of the page. For interpolation it is most con- venient to use Table III, which occupies three of the last pages of the Almanac, and gives directly the multiples of 9 8 .8565 by each hour and minute up to 24 hours, thus saving the reduction of minutes to decimals of an hour. Example. Right ascension of mean sun, January 15, 1900, at 4 h 44 m 30 s . R.A. mean sun, Gr. Mean Noon . ... . . . . . 19 h 37 m 55 s . 26 Add 4M4 m 30 s x 9 s . 8505 (Table III) . . . . . 46.74 E.A. meansunat4h44 m 30 8 19 38 42 .00 This is obviously the sidereal time of mean noon at a place in longitude 4 h 44 m 30 8 west, and if desired a table of this quantity may be computed for such a place by adding 46 8 .74 to the values given each day in the Almanac for Greenwich. Page I. The quantities on page I are only used for reducing meridian observations of the sun, which are made, of course, at local apparent noon. This page is convenient when the Greenwich mean time has not been noted, for the time elapsed since the preceding Greenwich apparent noon is equal to the west longitude of the place of observation. This is the quantity, therefore, by which the hourly difference must be multiplied to give the correction. An example of the use of this page is given on page 104. All the quantities given on page I may be found more easily from page II if we know the G.M.T. for which they are required. The only quantity for which we are obliged to consult page I is the semi-diameter, and this never differs by so much as 0".01 from its value at mean noon. THE NAUTICAL ALMANAC 97 Page III. Column 2 gives the day of the year corresponding to the given date, and is convenient for finding the number of days intervening between dates. Thus, January 15, 1900, is the 15th day of the year and September 25 is the 268th ; hence from noon, January 15, to noon, September 25, is 268 15, or 253 days. Column 3 contains the sun's longitude measured from the vernal equinox of the given date. For some purposes it is more convenient to measure from the mean equinox of the beginning of the fictitious year, an epoch much used in astronomical calculations but of no intrinsic interest. The minutes and seconds of the longitude as thus measured are found in column 4. The longitude of column 3 is measured from the actual place of the equinox at the given date as affected by precession and nutation. Column 6 gives the sun's latitude, which is always nearly but not exactly zero, as will be explained further on in this chapter. Column 7 gives the logarithm of the earth's distance from the sun in astronomical units. An astronomical unit is equal to the semi-axis major of the earth's orbit, about 93,000,000 miles. For those unacquainted with logarithms the following table will make it easy to find by interpolation the approximate distance cor- responding to a given logarithm. Logarithm 9.9925000 corresponds to 0.9829 astronomical units. " 9.9950000 " " 0.9886 " " " 9.9975000 " " 0.9943 " " " 0.0000000 " ' 1.0000 " " " 0.0025000 " " 1.0058 " " " 0.0050000 " " 1.0116 " " " 0.0075000 " " 1.0174 Example. January 19, 1900, log radius vector = 9.99299, which is very nearly ^ of the way from 9.9925 to 9.9950 ; hence on that date the distance of the earth from the sun is of the way between 0.9829 and 0.9886, or 0.9840 astronomical units. The value can be obtained within less than ^1^ of its amount without interpolation by taking the nearest value of the logarithm given in the table. Column 9 gives the mean time at which the vernal equinox is on the meridian of Greenwich (when the number of hours is greater than 12 the time is after midnight, and therefore during the morning 98 LABORATORY ASTRONOMY hours of the next civil date). This quantity is sometimes used in converting sidereal to mean time, but its use may be easily avoided and is sufficiently treated in the Explanation. Page IV. The quantities on page IV relate to the moon. They are given for each 12 hours of Greenwich mean time, and seem to call for no explanation, except perhaps the symbol 6, signifying conjunction, which occurs once (and occasionally twice) upon each page, on the day before or after that of new moon. Since successive transits follow each other nearly 25 hours apart, in general one date in each month would be left blank, the moon crossing the meridian during the hour preceding noon of one date, and during the hour following noon of the succeeding date. The symbol 6 occupies the vacant space and marks the date of new moon. Pages V to XII contain the right ascension and declination of the moon for every hour of G.M.T., together with their differences for each minute of time. The rapid motion of the moon makes it necessary to give these quantities at shorter intervals than suffice for the sun, in order that an equal accuracy may be attained in interpolation. These are of course places as seen from the earth's center, and it is to be remembered that at any point on the earth's surface the moon may be displaced by parallax a little more than 1. On page XII are given the exact dates to the nearest hour of G.M.T. of the moon's phases and the times of perigee and apogee. Pages XIII to XVIII contain tables of " lunar distances," that is, distances for each three hours of Greenwich mean time between the moon's center and certain bright stars and planets not far from the plane of its motion ; the sun is included in the list, as the moon is often visible in full daylight, so that its distance from the sun may be easily measured. This table is used in determining longitude ; the local time being known, the G.M.T. may be found by the method of lunar distances, as follows : The distance from moon to star or sun being measured is found to lie between two distances given in the table ; the G.M.T. of the observation then lies between the hours corresponding to the two tabular distances, and its exact value may be determined by interpolation. The difference between this time and the known local time of the observation is the longitude. THE NAUTICAL ALMANAC 99 The method requires accurate observations, and troublesome com- putations are necessary to correct the measured distance for the effects of refraction and parallax so as to find the distance from moon to star as seen from the earth's center. Data for the Planets, Eclipses. Following the calendar pages of the Nautical Almanac are thirty pages giving the right ascension and declination and the time of meridian passage of the five planets which are visible to the naked eye, and three pages containing the right ascensions and declinations of 150 of the brighter fixed stars. A few pages are devoted to the eclipses of the year, with maps from which may be obtained the approximate times of the successive phases of the solar eclipses as seen from any given point of obser- vation on the earth. EXAMINATION OF THE SEVERAL COLUMNS Having given this general summary of the contents of the tables, we will now call attention to some of the interesting facts and rela- tions that appear on running through the various columns throughout the whole year. The date of the solstices may be determined as the days on which the sun's declination has its maximum northern and southern values. The date of the equinoxes may be found, from either the right ascension or declination columns, as the date on which the decli- nation changes sign, and the right ascension is either O h or 12 h ; the exact time may be found by interpolation. (See page 107.) The number of days between the equinoxes may be determined by using the column of days, page III. It will be found that the sun is for some days more than half the year in that part of its orbit which lies in the northern hemisphere. The column of hourly difference shows that the declination is changing slowly at the solstices and most rapidly at the equinoxes ; moreover, the change at the latter dates is nearly uniform both in right ascension and declination, as stated on page 85. If a right triangle 'be drawn with the difference in right ascension for the date of the equinox as base and difference in declination as alti- tude, the angle between the base and the hypotenuse measured by 100 LABORATORY ASTRONOMY a protractor will be found to be 23^. It obviously equals the angle between the equator and the ecliptic. Notice that the equation of time is the difference between right ascension of mean and true sun, as stated on page 92, thus : From the Almanac for 1900 (p. II), we have the following values : January 21, Sun's B,. A. = 20 h 13 m 2 8 .79 ; K.A. Mean Sun 20 h l m 34 s . 61. Subtracting the latter from the former, we have for the equation of time + ll m 28 8 .18. This is the value given on page II ; the positive sign indicates that it is to be added to apparent time to find mean time, or subtracted from mean time to find apparent time. The dates on which the equation of time is and dates and values of greatest and least equations should be noticed ; also that on those dates for which the equation is the values of the sun's right ascen- sion and declination, etc., on pages I and II, are the same, since apparent noon and mean noon coincide. For 1900 the civil dates are as follows : EQ. OF T. February 11 + 14 m 27 8 .28 April 15 May 15 - 3 m 49 8 .40 June 14 July 27 r . . . . ... .". + 6 m 17 8 .22 September 1 November 3 - 16 m 20 s . 40 December 25 The hourly difference of the right ascension of the mean sun has the same integers as the mean daily motion of 'the sun in longitude, 0.98565 0.98565 X 3600" 0.98565 ; for .98565 per day = , or - , per hour, and reducing this to seconds of time by dividing by 15, we find the motion of the mean sun to be 9 8 .8565 per hour. This illus- trates the fact that the mean motion of the sun in longitude (0. 98565 per day) is the same as that of the mean sun in right ascension (9 8 .8565 per hour), page 92. The column which gives the sun's latitude will repay an investi- gation. It appears at a glance that there is a small but regular change,, from south to north and return, with a period of about 27 or 28 days. THE NAUTICAL ALMANAC 101 The principal cause of this is that it is not the earth, but the center of gravity of the earth and moon, which describes an orbit in the plane of the ecliptic ; and by the known properties of the center of gravity, when the moon is above the ecliptic the earth must be below. It is not very difficult to show that from this cause the latitude may be 0".67 greater or less than when both bodies are in the ecliptic, that is, when the moon is at one of her nodes. The attractions of Venus and Jupiter also draw the earth out of the ecliptic by an amount which may reach 0".5. In January, 1900, this " planetary perturbation " was about + 0".13. The total range of lat- itude during the month (see page 178) was from -f 0".68 to 0".48. The moon was at her nodes January 12.33 and January 26.85. From the radius vector column (p. Ill) we may find the sun's distance at any date by the table on page 97. By comparing this with the semi-diameter column (p. I), it is shown that the sun's distance is inversely proportional to its angular semi-diameter. Thus, January 1, 1904 : Log r = 9.9926540, Dist. = 0.9832, Senii-diam. = 16' 17".90 and July 1, 1904 : Log r = 0.0072095, Dist. = 1.0167, Semi-diam. = 15'45".67 and 0.9832 : 1.0167 = 945".67 : 977".90, as appears on multiplying the means and extremes and comparing the products. The dates of the moon's perigee and apogee may be determined from the greatest and least semi-diameter, page IV, column 2, or from the greatest and least parallax in column 4. Since both semi- diameter and parallax are inversely proportional to the moon's distance from the earth, the latter may be determined by multiplying the former by a constant quantity. This constant is 3.6625, and it is not difficult to show that it is the ratio of the earth's equatorial radius to that of the moon. Compare the last two columns, noting that at new rnoon the moon comes to the meridian with the sun at noon and that at full moon (age 15 days) it comes to the meridian at midnight. 102 LABORATORY ASTRONOMY TABLES OF THE PLANETS AND STARS The data for the planets which follow the calendar pages illus- trate many facts which are explained in the text-books on descriptive astronomy. Ketrograde motion, for example, is shown by negative hourly differences in right ascension ; the stationary points occur on those dates on which the hourly difference changes sign ; opposition takes place when the time of transit is 12 h ; conjunction, when it is O h ; the retrograde motion is a maximum at opposition. By means of the right ascensions and declinations the path for the year may be plotted on a star map, for which purpose an ecliptic map (see page 65) is especially adapted. The time of passing the node may be found from the point where the path cuts the ecliptic, and the sidereal period from the interval between two passages of the same node. A series of Almanacs covering some years is useful in following the outer planets as well as for comparison of the calendar pages to show the repetition of the solar data after four years. The table of star places contains columns of annual variation, that is, the sum of the precession and proper motion (the latter always a very small quantity), which are useful in showing the effects of precession on the right ascensions and declinations of stars- in different parts of the heavens. Compare in this respect 8 Draconis, (3 Ursae Minoris, Polaris, y Pegasi, y Geminorum, and A. Sagitarii. COMPARISONS OF OBSERVATIONS WITH THE EPHEMERIS Many of the facts which we have obtained by observation in former chapters may be found in the columns of the Almanac, and after a thorough comprehension of the methods has been acquired much time may be saved by employing these data ; but it is to be remembered that facts thus obtained are not so thoroughly grasped or so easily retained. With this caution, we may compare some of the results of our previous work with the tables, to give an idea of the methods of using the latter. Following are comparisons of a THE NAUTICAL ALMANAC 103 few of the observations of the preceding chapters with the values given by the Ephemeris : Observations of the Moon. From careful measurement of the map on page 29, the moon's declination on January 9, 1900, at 10 P.M., was + 19.3, and its right ascension was 2 h 38 m . The place of observation was 4 h 44 m .5 west of Greenwich, and the time used was Eastern standard time, which is 5 hours slow of Greenwich ; the G.M.T. was therefore 15 h O m , at which time. the moon's declination and right ascension are given in the Ephemeris (p. 180) as + 18 48' and 2 h 39 m . The difference between the observed and calculated places is about ^ -in declination and l m in right ascension, mainly due to error of observation with the cross-staff. Length of the Month. We may use the Ephemeris to find the length of the month by seeking the next date at which the moon's right ascension and declination are the same, which is February 5, at about 21 hours, G.M.T., as will be seen from page VI for February. This gives 27 d 6 h as the period of the moon's revolution among the stars. Passing to page V for December, we find that the right ascension was again 2 h 39 m on December 3 at 19 hours, at which time the declination was 17 19'. This shows that the moon's orbit had shifted during this time so that it did not pass through exactly the same points of the heavens in these two months, its December path in the neighborhood of right ascension 2 h 39 m being l south of the corresponding point of its path in January. By column 2 of page III, January 9 is the 9th day of the year and December 3 is the 337th ; hence the moon completed an integral number of revolutions in 337 d 19 h - 9 d 15 h , or 328 d 4 h . The -period having been determined as 27 days approximately and 328 -s- 271 being nearly 12, it is evident that the number of complete revolutions between these dates is 12. Dividing 328 d 4 h by 12, we have 27 d 8 h as a closer approximation to the sidereal month. Taking the length of the successive months during the year, it is interesting to note how very considerable is the difference in length of the successive sidereal months due to the " perturbations " of the moon's motion. 104 LABORATORY ASTRONOMY Observations at Apparent Noon. The observations recorded on page 39 were made at Cambridge, in longitude 4 h 44 m .5 west of Greenwich, and the watch time of apparent noon was ll h 56 m 2 8 .9. By the use of the Almanac, we find the correction of the watch to standard time as follows : Since the observation was made at local apparent noon, it will be better to use page I of the Almanac, which gives for March 8, at Greenwich apparent noon, equation of time ll m 1 8 .46, to be added to apparent time, and hourly difference s . 61 9. The time of observation was 4 h 44 m .5, or nearly 4 h .75 later, and the change of the equation of time in this interval was 4.75 X s . 619 = 2 8 .93. As the equation of time was decreasing, its value at the time of observation was 10 m 5S S .53. Since no sign is appended to the hourly difference, we check this result by noting that it falls between the values tabulated for March 8 and 9. Hence : Camb. App. T. . . . 12 h O m s Eq. of T. (add) 10 58 .53 Camb. M.T. 12 10 58.53 Corr. for Long 4 44 30 G. M.T. of observation 16 55 28.53 Subtracting 500 Eastern Std. T. of observation 11 55 28.53 Observed watch time . . . . ... . ... 11 56 2.9 Corr. of watch to Std. T. . . . . . .... . . . . - 34 .37 The correction for longitude to give G.M.T. is added, because at any given instant the local time of any place is greater than that of a place to the westward, since the sun passes its meridian earlier and always has a greater hour-angle than at the western place. Kemembering that Cambridge is 15 m 30 s east of the meridian from which Eastern standard time is reckoned, we may find the watch correction more simply, thus : Camb. M.T 12 h 10 m 58 S .53 Reduction for Long, (subtract) - 15 30 Eastern Std. T 11 55 28 .53 Watch time 11 56 2 .9 At - 34 .37 Observations of the Planets. The data on page 52 show that on February 5, 1900, at 7 h 12 m (the watch keeping Eastern standard THE NAUTICAL ALMANAC 105 time), the right ascension of Venus was 9.64 = 38 m 33 8 .6 less than that of y Pegasi, which from the Ephemeris was O h 8 m 5 S .69 ; hence from this differential observation the right ascension of Venus was 23 h 29 m 32 s .09. The G.M.T. of the observation was 12 h 12 m = 12 h .2. The tables for Venus (p. 224) give : H. DlFFS. + 11 s . 106 +77". 31 x 12.2 x 12 .2 135.49 943 .2 2 m 15 8 .49 15' 43". 2 The observation differs from the Ephemeris by l m 38 s in right ascension and 8' in declination, although the method should give angles within 0.2. The discrepancy is much greater than usually occurs, and this observation of Venus is affected by some unexplained error ; it depends on a single reading of the hour-angle. To exhibit the usual accuracy, we may compare with the following observa- tions, made February 6 : FEBRUARY 5 R.A. OF VENUS At Gr. M. noon . . 23 h 25 m 38 8 .14 Diff. for 12 h .2 . . +2 15.49 DECL. -4 55' 46" + 15 43 At 12M2 m , G.M.T. . 2327 53.63 Observed values (p. 53) 23 29 32 .1 -4 40 3 -4 32 WATCH TIME H.A. DECL. 7 Pegasi . . . . . Venus 7h 3m IQs 5 10 64. 6 74 .05 + 15. 45 - 3.4 7 Pegasi ... . . . 7 10 65 .7 + 15 .5 Hence Venus preceded y Pegasi 8.90 = 35 m 36 s , Decl. = 3.4 0.53 = 3.93 ; and since the right ascension of y Pegasi was O h 8 m 6 s , our observation gives for the place of Venus at 12 h 5 m G.M.T., E. A. = 23 h 32 m 30 s , and 8 = - 3 56'. The Ephemeris gives E.A. = 23 h 32 m 17 S .2, and 8 = 4 9' 14".5. Observations of the Moon's Place. The data given on page 55 show that on February 6, 1900, the moon followed y Pegasi 46.7 = 3 h 6 m 48 s . The right ascension of y Pegasi was O h 8 m 6 s ; hence the moon's right ascension was 3 h 14 m 54 s , while its declination, given directly by the circle, was + 20. 4. The Eastern standard time was 7 h 42 m , corresponding to 12 h 42 m G.M.T. 106 LABORATORY ASTRONOMY The Ephemeris gives : MOON'S K.A. DECL. DIFFS. FOB l m At 12 h G.M.T 3 h 14 m 34 8 +20 25' + 2 s . 33 + 6". 2 Diff. for 42 m + 1 38 + 4 s x 42 x 42 At time of observation . . 3 16 12 + 20 29 97 .9 260.4 Observed values 3 14 54 + 20 24 I m 37 8 .9 + 4' 20" The agreement here is satisfactory considering that the moon is more than 45 from the star with which it is compared. Part of the difference is due to parallax. Observations of the Sun's Place. By the observation treated on page 67, the sun's right ascension and declination at 5 h 36 m 26 s , Cambridge sidereal time, March 29, 1899, by comparison with a Ceti, were found to be O h 33 m 19 s and + 3.6. To compare this with the Ephemeris of the sun, we must first find the Greenwich mean time corresponding to 5 h 36 m 26 8 , Cambridge sidereal time. Heretofore we have had given either local apparent time or standard time of observations, and the Greenwich mean time has been found by adding the equation of time and longitude in one case or an integral number of hours in the other. In this case we have given the local sidereal time, to find the corresponding Greenwich mean time. The first step is to find the Greenwich sidereal time by adding the longitude west of Greenwich, after which G.M.T. isfoundas follows : Gr. Sid. T. = 5 h 36 m 26 s + 4 h 44 m 30 s . = 10 h 20 m 56 8 March 29, Gr. Sid. T. of Gr. M. noon ..... 30 37.57 Hence the sidereal interval elapsed since Gr. M. noon is 9 50 18 .43 And, by Table II, the quantity to be subtracted from this to give the equivalent mean interval is ... 1 36 .71 Hence the corresponding mean time interval is . . . 9 48 41 .72 This is the mean time interval since Greenwich mean noon, which of course is the required G.M.T. We may now determine the sun's place at 9 h 48 m , or 9 h .8, G.M.T., by means of page II of the Ephemeris, as follows : SUN'S R.A. DECL. H. DIFFS. At Gr. M. noon . . . ,0 h 31 m 33* + 3 24'.4 -f 9 s .l + 58" Diff. for 9 h .8 .... + 1 29 + 9 .5 x 9.8 x 9 .8 At time of observation . 33 2 -f 3 34 .9 89 568 Observed values (p. 67) . 33 19 + 3 36 l m 29 s 9'. 5 THE NAUTICAL ALMANAC 107 Determination of the Equinox. The following data from the Alma- nacs of 1899 and 1900 may be compared with the results of page 89 : AT GK. APP. NOON SUN'S DECL. DJFF. DATE OF EQUINOX BY INTERPOLATION 1899. 1900. Sept. 22.0 23.0 Sept. 23.0 24.0 + 018 / 8". 7 -0 5 13 .9 + 26 .4 -0 22 57 .5 23'22".6 23 23 .9 23' 22".6 0'26".4 , = Sept. 23.01880 23' 23". 9 The longitude of the place of observation was 4 h 48 m 40 8 W. 4 h 48 m 40 s 4.811H 24 h 24 days = O d .20046. Hence the local dates of the equinoxes were September 22.57574, 1899, and September 22.81834, 1900, and the length of the tropical year was 365.24260 days, as compared with the observed values September 22.714, 1899, September 22.856, 1900. 365.14 days. Observations of Star Places. The right ascensions and declinations of the stars given on pages 86 and 89 may be compared with the mean places given in the Nautical Almanac for 1899 and 1900, or, better, with the apparent places given in Part II of the American Ephemeris. From the latter we find for September 22, 1900 : E.A. 77 Serpentis 18 h 16 m 11 8 .4 X Sagittarii 18 21 51 .9 Vega "... 18 33 35.5 Altair . 19 45 57.9 DECL. - 2 55'. 3 - 25 28 .5 + 38 41 .8 + 8 36.6 which are in close agreement with the results of observation. CHAPTER VIII THE CELESTIAL GLOBE WHEN a globe such as that described on page 63 has had a num- ber of constellations plotted on it in their proper positions, and the sun's path added, showing the positions occupied by the sun at dif- ferent times of the year, it becomes a very useful apparatus for many purposes. If, for instance, it is so placed that its axis points to the pole, and is turned about the axis until the place of the sun as marked on the globe for a certain date is on the under side and in a vertical plane through the center, the sphere will represent the heavens as seen at midnight on the given date. When the globe has been so adjusted, if a straight line is drawn from the center to any star on the surface of the globe, the prolon- gation of this line will lead to the real star at the point which it occupies on the sphere of the heavens. Thus used, such a globe is helpful to a beginner in identifying the constellations. Obviously the plane of the sun's path on the globe, if extended to the heavens, will mark out the ecliptic, and all the hour-circles and parallels of declination will mark the corresponding circles in the sky. If the globe is turned slowly about its axis so that a point on the equator moves from east to west through 15 per hour, we have a sort of working model of the moving sphere of the heavens on which we may measure off arcs and angles and thus solve approxi- mately many problems suggesting themselves to one beginning to study the apparent motions of the heavens. Such an apparatus has from very early times been an important aid to astronomers and students of astronomy, and no aid is so useful in arriving easily at correct ideas on the subject. Especially was it useful and appropri- ate in those days when the mechanism of the heavens was believed to correspond closely to that of the model and the globe was regarded as being a fair representation of their actual construction, in fact, 108 THE CELESTIAL GLOBE 109 a representation of the eighth or outer sphere which carried the fixed stars, turning about a material axis somehow fixed in the " Primum Mobile." The planets moving inside, each in its crystal sphere, were treated by projecting them each on to its proper place on the outside sphere for any particular time to solve a given prob- lem. For the beginner, who stands to a certain extent in the place of the early astronomers, it is still most important in studying many problems. Usually the diagrams by which we illustrate our state- ments of astronomical problems are drawn as if the celestial sphere were seen from the outside as we see the globe. This is because it is impossible to represent on a plane any large part of a spherical surface as seen from the inside. As usually constructed for demonstration and the solution of problems, the celestial globe is made by building up layers of strong paper laid in glue upon a solid wooden sphere so as to cover it with a light but stiff shell, which is then cut through along a great circle, so that the core may be taken out. The two halves of the shell are fastened together by gluing on a strip of thin, strong cloth, and after passing an axis of stiff wire through the center, several layers of a mixture of glue and whiting are applied to the surface, each being smoothed before drying. The whole is then turned so as to form a very light and accurate spherical shell. Upon the surface are pasted gores of paper, on which the circles and principal stars are printed in such a manner as to lie in their proper places on the globe. The outlines of the constellations are shown on the plates, and the conventional figures which have been ascribed to them. A small circular piece centered on the pole completes the map. The figures are colored by hand, and the whole is then cov- ered with a hard, transparent varnish. Both equinoctial and ecliptic are graduated to degrees, and the hours of right ascension on the former are marked by Koman numerals, ' The places of the sun are usually indicated on the ecliptic at dates five days apart. Since the circuit of the sun is completed in 365J days, while the length of the year is sometimes 365 and sometimes 366 days, an average position of the sun must be chosen, which is done with sufficient accuracy by plotting its place for the second year after leap year. 110 LABORATORY ASTRONOMY The axis of the globe is supported by a stiff brass circle, so that the center of the sphere lies exactly in the plane of one of its faces, and this face is graduated into degrees, one semicircle near the outer edge from at either pole to 90 at the equator, and the other semicircle near the inner edge from at the equator to 90 at either pole. The inner graduation is used for measuring the angular distance from the equator to any point on the globe, that is, the declination of any point. The graduation on the outer edge is used for placing the axis at the proper angle to the horizon in rectifying the globe, as explained on page 111. This graduated circle which supports the axis is called the " brass meridian." It is mounted in two slots in a somewhat larger wooden circle called the "horizon," in such a manner that it is perpendicular to the latter and that its center lies in the plane of the upper surface of the wooden circle. The horizon is graduated on its inner edge, and each quadrant has two sets of numbers, one of which reads from at the prime vertical to 90 at the meridian, and the other from at the meridian to 90 at the prime vertical. These numbers serve for the direct reading of amplitude and bearing respectively, which are easily translated into azimuth, remembering that W. is 90, N. 180, and E. 270, if azimuth is measured from the south point toward the west from to 360. The brass meridian may be turned in its own plane, sliding easily in the slots so that the axis of the globe shall make any desired angle with the horizon. If the globe is accurately made and mounted, its center will coin- cide with the common center of the graduated face of the brass meridian and the upper surface of the horizon, whatever may be the inclination of the axis. No irregularities should appear in the small space between these circles and the surface of the globe when the latter is whirled rapidly on its axis. Some idea of the^correct placing of the circles on the globe may be obtained by noting whether all points of the equator and parallels come under the proper divisions of the brass meridian, whether all points of the equator pass through the east and west points of the horizon 90 from the graduated face of the brass meridian, and whether the points of the equator which lie in the east and west points of THE CELESTIAL GLOBE HI the horizon are twelve hours apart whatever the inclination of the axis. It is desirable to have a means of fixing a point on the globe by some mark that may be afterward removed without injuring the sur- face. Gummed paper should not be used : small pieces of unglazed paper when well moistened will adhere long enough for ordinary purposes. A good mark may be made with water-color paint mixed with glycerine so as to be very thick and applied with a rubber point or soft pen point. Such a mark may easily be removed with a moist- ened finger even after several weeks. Ink suitable for fountain pens is usually safe if removed within an hour or two. TO RECTIFY THE GLOBE In order that the globe shall represent the heavens at any partic- ular place, the axis must be inclined to the horizon by an angle equal to the latitude. This may be accomplished by rotating the brass meridian in its plane and measuring the angle of elevation of the pole by the outside graduation, which reads from at the pole to 90 at the equator. This process is called " rectifying " the globe for a given place. Having been rectified for a given place, the globe may be rectified for a given time by bringing it to such a position that a line drawn from its center to any star is parallel to the line drawn from the given place to the actual place of the star in the heavens at the given time. For this purpose, the pole being elevated to the proper inclination, that is, the latitude, the whole apparatus is turned on its base until the brass meridian is in the meridian of the place, and the globe is turned on the polar axis until some one point is known to be in the proper position ; then all points of the globe will be in their proper positions. The point chosen for this purpose will vary with circumstances. If the local sidereal time is given, it is only necessary to place the globe so that the hour-angle of the vernal equinox equals the given sidereal time. (See page 57.) This is easily done by the graduation 112 LABORATORY ASTRONOMY of the equator on the globe. When the hour-angle of the vernal equinox is l h , 2 h , 3 h , the reading of the equinoctial under the brass meridian is l h , 2 h , 3 h , etc., and the globe is therefore rectified to a given sidereal time by turning it about the polar axis until the given sidereal time is brought to the graduated face of the brass meridian. The vernal equinox will then be at the proper hour-angle and all points on the globe will be properly related to the corresponding points on the sky. If the apparent time is given, the globe may be rectified by the following process. Mark the place of the sun in the ecliptic for the given day. Bring this point to the meridian, which rectifies the globe for apparent noon ; then, to rectify it for the given ap- parent time, it is necessary to turn the globe until the hour-angle of the sun is equal to the given apparent time. This may be done by using the graduations of the equator as follows. Rectify for apparent noon and read the hours and minutes of the graduation on the equinoctial which comes under the brass meridian (this is the sidereal time of apparent noon). Add to this reading the given apparent time, and the sum will be the hours and minutes of the equatorial graduation that must be brought to the meridian to place the sun at the proper hour-angle. If local mean time is given, the apparent time may be obtained by applying the correction for the equation of time for the given date, and the globe may then be rectified for apparent time, as described in the last paragraph. If, as will generally be the case, standard time is given, this may be reduced to local mean time by applying the correction for longi- tude, and we may then proceed as before. We may here remark that in rectifying the globe for solar time we make use of the sun's place as marked on the ecliptic for the given date ; and that this place may be inaccurate by as much as half a degree is obvious from the following consideration. Suppose the place of the sun on the globe to be exact for any one year on February 28. It will be exact on March 1 or about 1 in error, according as the year has not or has the date February 29. The following table of the sun's longitude shows more clearly the nature of the facts. THE CELESTIAL GLOBE 113 YEAR FEBRUARY 20 MARCH 2 SEPTEMBER 23 1901 331. 2 34P.2 179. 8 1902 330 .9 341 .0 179 .5 1903 . . -V . . . 330 7 340 7 179 3 1904 330 .5 341 .5 180 .0 Average 330 .8 341 .1 179 .7 The values nearly repeat themselves after four years. It is obvious that by assuming an average value of the longitude for February 20, March 2, and September 23, we should sometimes be in error by about in. the sun's place, though never more, and by some such compromise the places must be selected for the posi- tion of the sun upon a globe for general use. The error that thus arises may amount to 2 m in the determination of the sun's right ascension from the globe. An indispensable attachment for the celestial globe is a thin flexible strip of brass graduated to degrees and so constructed that it may be attached to the brass meridian at its highest point by a pivot, about which it can be turned so as to be brought to coincide with any vertical circle ; its graduated edge may then be brought over any point 'on the globe and the azimuth of the point fixed by noting the place where the arc meets the graduations on the horizon. The altitude of the point may be directly read on the flexible arc, which is graduated from at the horizon to 90 at the place where it is fixed to the brass meridian. The graduations are continued below the horizon from to 18 for the purpose of determining the end of twilight (page 133). The flexible arc is usually called the "altitude arc." The globe thus equipped may be used for the approximate solu- tion of all problems which arise from the diurnal motion, some of which we will now discuss. These approximate solutions are not only sufficient for many purposes, but always indicate the proper statement of the problem for purposes of computation, and serve to detect gross errors in the numerical results. 114 LABORATORY ASTRONOMY PROBLEMS WHICH DO NOT REQUIRE RECTIFICATION OF THE GLOBE Many problems are independent of the position of the observer on the earth's surface, and for their solution it is immaterial at what angle the polar axis is inclined. By bringing the axis to the plane of the horizon, any star may be brought to view above the horizon, but unless it is convenient to stand so that one can look down upon the globe from above, it is often better to take a sitting position and place the polar axis nearly vertical. In following the solutions of the examples below, the accompanying figures serve to show whether the globe has been brought to the proper position. Problem 1. To find the right ascension and declination of a star. Kotate the globe until the. star is in the plane of the brass meridian ; note the hours, minutes, and seconds of that graduation of the equinoctial which falls under the brass meridian. This is the right ascension of the star. This value we may call the "meridian reading" of the equator and in future abbreviate to R.A.M. (right ascension of the meridian). The declination of the star equals that degree of the graduation of the meridian under which the star lies. Example 1. The star rj Ursse Majoris in the end of the Dipper handle is brought to the brass meridian (Fig. 42) and is found to lie halfway between the divisions 49 and 50 north of the equator ; the declination is therefore + 49.5. The meridian reading is 13 h 44 m , which is the star's right ascen- sion. (For reading the declination the graduations on the inner edge of the brass meridian must be used.) Problem 2. Given the right ascension and declination of a star, to find the star. Rotate the globe until the meridian reading (R.A.M.) is equal to the given right ascension, and under the brass meridian at the given declination will be found the star. FIG. 42. R.A.M. 13" 44' Decl. THE CELESTIAL GLOBE 115 FIG. 43. K.A.M. 19^ 46> ; Decl. + 8 Example 2. The right ascension of a certain star is 19 h 46 m and its declination + 8jL. What is the star ? The division on the equator marked 19 h 46 m is brought to the brass meridian (Fig. 43), and halfway between the graduations 8 and 9 on the meridian is found Altair, which is the star sought. Problem 3. To find the angular distance between two stars. Place the flexible quadrant along the sur- face of the globe so that its graduated edge passes through both stars, and read the graduation at the points where it touches each star ; the difference of the readings is the angular distance between the stars. The graduated edge should lie along the great circle; as this is not always easy to adjust, it is well to repeat the measure with the quadrant in different adjustments and take the smallest value obtained. An alternative method free from this source of error is to adjust the points of a pair of compasses so that they may just span the distance between the two stars. The com- passes may then be applied to the globe with one leg at the vernal equinox (0); the other leg being brought to the equinoctial its reading will give the angular distance between the stars. To guard against defects in the globe, the second point may be brought to the ecliptic, and the reading should be the same as on the equinoctial ; if the readings differ, the mean of the values should be taken. In the use of the compasses care must be taken not to scratch the surface of the FIG. 44. Length of Dipper 26 g^^ Example 3. The following measures were made to determine the distance between allrsse Majoris and TjUrsse Majoris. With the flexible quadrant applied to the globe (Fig. 44) so as to lie as nearly 116 LABORATORY ASTRONOMY as possible along the great circle between the stars, the readings were: ij URS.E MAJOKIS a UKS^E MAJOBIS DISTANCE 0.0 26.0 26.0 0.0 26.1 26.1 20.0 46.1 26.1 40.0 66.1 26.1 Here no difficulty was found in laying the arc along the great circle, as the distance is not great, and the value is taken to be 26.l. Adjusting the points of a pair of compasses to the stars and then placing the compasses with one point at the vernal equi- nox, the other point was found to reach to 25. 6 of right ascension on the equinoctial and to 25.6 of longitude on the ecliptic, which gives the distance between the stars as 25.6. Problem 4. To find the sun's longitude, rig/it ascension, and .declination at a given date. If the sun's place at different dates is marked on the ecliptic, its longitude may be read off directly on the graduations of the ecliptic. In all old globes, however, and in many modern ones the ecliptic is not thus marked, and the place of the sun must be found by determining the longitude by a table such as that given on page 173, which is nearly correct for the first half of the present century. A substitute for this table is generally to be found in the form of two contiguous concentric circles on the horizon circle, one graduated into degrees of longitude and the other into months and days, so that the line for a given date in the outer circle is found opposite the corresponding degree of the sun's longitude in the inner circle. Commonly also the divisions both of this circle and of the ecliptic are divided into groups of 30, each correspond- ing roughly to one month of time. The 30 of Aries reach from the first of Aries on March 20 to the first of Taurus on April 20, and so on in the order of the signs. Thus, opposite May 6 is the fifteenth degree of Taurus, corresponding to longitude 45 in the usual way of reckoning ; opposite January 1 is the tenth degree of Capricor- nus, nine complete signs and 10, or longitude 280. In the table on page 173 the equivalents of the degrees of longitude are given in signs and degrees. THE CELESTIAL GLOBE 117 By whatever method the sun's place in the ecliptic is fixed, its right ascension and declination are found by the method of Problem 1. Example 4- What are the sun's right ascension and declination on April 20 ? The longitude is found by the table to be 29.5, and on bringing this point of the ecliptic to the meridian (Fig. 45) it is found to be in declination -f-11^- , while the reading of the meridian is l h 50 m . The sun's right ascension is therefore l h 50 m and its decli- nation is 11- north. PROBLEMS WHICH REQUIRE RECTI- FICATION OF THE GLOBE FOR A GIVEN TIME Such are problems which require a deter- mination of the angle between the meridian and some one of the hour-circles of the Fl( >- 45. Sun's E.A. i* 53 m ; globe. They are independent of the latitude of the place of observation, but depend upon the position of the heavenly bodies with respect to the meridian. The brass meridian being taken as the meridian of the place of observation, the only quantities involved are differences of hour-angle and of right ascension, and it will be advisable here to collect the following rela- tions, which have already been explained. All time is measured by the continually increasing hour-angle of some point of the celestial sphere. Local sidereal time (Camb. Sid. T.) is the hour-angle of the vernal equinox. Local apparent (solar) time (Camb. App. T.) is the hour-angle of the sun. Local mean (solar) time (Camb. M. T.) is the hour-angle of the mean sun. For example, at 21 h 20 m , Camb. Sid. T., the hour-angle of the vernal equinox at Cambridge is 21 h 20 m ; at 10 h 30 m , Chicago apparent 118 LABORATORY ASTRONOMY time, the hour-angle of the sun at Chicago is 10 h 30 m ; at 5 h 10 m , New York mean time, the hour-angle of the mean sun at New York is 5 h 10 m . The hour-angle is in all cases measured westward from the observer's meridian up to 24 h . Greenwich mean time (G.M.T.) is the hour-angle of the mean sun measured from the meridian of Greenwich. When we say that a place is a certain number of hours and minutes of longitude west of Greenwich, we mean that the rotation of the earth brings the sun to the meridian of the place just so many hours and minutes after its arrival at the meridian of Greenwich. At local noon, then, its hour-angle, reckoned from the Greenwich meridian, is equal to the difference of longitude between the two meridians. As the sun thereafter moves westward equally from the two meridians, Green- wich time is always greater than that of any place west of it by exactly the difference of their longitudes. Therefore, to find the G.M.T. corresponding to a given local mean time, we add to the latter the longitude (west) from Greenwich. Standard time is directly obtained from G.M.T. by subtracting 4, 5, 6, 7, 8 hours, respectively, for Colonial, Eastern, Central, Moun- tain, and Pacific time. Thus, the "reduction for longitude," so called, from Cambridge mean time is + 4 h 44 m .5 to G.M.T. and -f- 4 h 44 m .5 5 h to Eastern standard time ; or, by a single operation, 15 m .5 directly to Eastern time. The " reduction for longi- tude" for San Francisco is -f 8 h 9 m .7 to Greenwich and + 8 h 9 m .7 - 8 h = + 9 m .7 to Pacific time. Problems, therefore, which involve standard time require a knowledge of the observer's longitude. Problem 5. To rectify the globe for a given sidereal time. Eotate the globe till the E.A.M. equals the given sidereal time. This brings the FIG. 46. Sid. T. 7" 50- vernal equinox to an hour-angle equal to the given sidereal time, and all points of the sphere into their proper relation to the meridian. THE CELESTIAL GLOBE 119 Example 5. To rectify the globe for 7 h 50 m sidereal time, rotate the globe until E.A.M. is 7 h 50 m (Fig. 46). Problem 6. The globe being rectified for a given sidereal time, to determine the hour angle of a body. Note the E.A.M. when the globe is in the given position; then bring the body to the meridian and read its right ascension. Sub- tract the latter reading from the former and the result is the hour- angle of the body. Since the reading of the meridian is always the sidereal time (page 59), this process exemplifies the equation H.A. = Sid. T. E.A. It is of course understood . that if in adding two times or hour-angles the result is greater than twenty-four hours, that amount is to be subtracted ; thus, an hour-angle of 35 h 25 m 10 8 corresponds to the same position as an hour-angle of ll h 25 m 10 s . Also, if it is required to subtract a larger froni a smaller hour- angle, the latter should be increased by twenty-four hours before performing the subtraction : thus, 6 h 41 m - ll h 17 m = 30 h 41 m - ll h 17 m = 19 h 24 m . Example 6. What is the hour-angle of Sirius at (a) 7 h 50 ra , sidereal time, and at (b) 4 h 20 m , sidereal time ? (a) Rectifying the globe, as in Problem 5, to 7 h 50 m Sid. T., the E.A.M. = 7 h 50 m . Bringing Sirius to the meridian (Fig. 47), E.A.M. = 6 h 41 m = E.A. of Sirius, as in Problem 1. Hence H.A. of Sirius at 7 h 50 m Sid. T. = 7 h 50 m - 6 h 41 m = l h 9 m (Fig. 46). (b) Eectifying to 4 h 20 m Sid. T., E.A.M. FlG " 47 ' KA ' of Siriu8 ' 6 " 41m = 4 h 20 m , and, as before, H.A. = 4 h 20 m - 6 h 41 m = 28 h 20 m - 6 h 41 m = 21 h 39 m . Problem 7. The globe being rectified for a given apparent time, to determine the hour-angle of a body. Bring the sun's place to the meridian and take the E.A.M. (this is the sun's right ascension, Problem 4). Eotate the globe through an hour-angle equal to the given apparent time, and the sun is brought to the required hour-angle ; the E.A.M. thus becomes H.A. 120 LABORATORY ASTRONOMY of the sun -f- R.A. of the sun, and the globe is properly rectified when this reading of the equator is brought under the meridian. Since H.A. + E.A. = Sid. T., the rule may be given as follows : Determine the sun's right ascension by the globe (Problem 4). Add the given apparent time. The sum is the sidereal time. For this sidereal time rectify the globe by Problem 5, and find the hour-angle by Problem 6. Example 7. What is the hour-angle of Sirius at 10 P.M., apparent time, February 13 ? Sun's R. A. by globe 21 h 50 m App. T 10 0_ Sid. T 7 50 R.A. of Sirius by globe ' 6 41 H.A. of Sirius 1 9 (Fig. 46) Problem 8. The globe being rectified for a given mean time, to determine the hour-angle of a body. Apply the equation of time (with the proper sign) to the given mean time to find the corresponding apparent time, and with this value rectify as in Problem 7. Example 8. What is the hour-angle of Sirius at 5 A.M., local mean time, July 10 ? Equation of time + 5 m (add to App. T.) July 10, 5 A.M = July 9 d 17 h O m Eq. of T. (subtract) 5_ App. T. . . *** 16 55 Sun's R.A. by globe (add) . . . 7 20* Sid. T 15 R.A. of Sirius (Problem 1) (subtract) 6 41 H.A. of Sirius 17 34 Problem 9. The globe being rectified for a given standard time, to determine the hour-angle of a body. Apply the reduction for longitude to find the corresponding mean time and rectify as in Problem 8. * The sun's place is marked on the globe for noon of the indicated date. It is therefore more accurate in this problem to make use of the sun's place for July 10 and in general for the nearest noon, which is always that of the civil date. THE CELESTIAL GLOBE 121 Example 9. At Chicago (longitude -f 5 h 50 m ) what is the hour- angle of Sirius at 6.30 P.M., Central standard time, October 30? Red. for Long. Chicago T. to Central T. - 10 m Eq. of T. - 16 m (subtract from App. T.) Central standard time 6 h 30 m Red. of Long, to Chicago M. T. Chicago M. T Eq. of T. (add to M. T.) . . +10 6 40 +16 App. T 6 56 Sun's R.A. by globe (add) 14 23 Chicago Sid. T 21 19~ R.A. of Sirius (subtract) 6 41 H.A. of Sirius (by Problem 5) 14 38~ Reduction to the Equator. In the solution of Example 4, page 117, it was shown that when the sun's longitude is 29.5 its R.A. is l h 50 m , or 27.5. The quantity which must be added to the longitude of a point on the ecliptic to find its E.A. (in this case 2) is called the " reduction to the equator " and is used in finding the equation of time as explained in Chapter X. Its value for any given point of the ecliptic may be found by the globe as in Example 4. Following are the results : LONGITUDE and 180 10 190 20 200 30 210 40 220 50 60 70 80 90 230 240 250 260 270 RED. TO EQUATOR 0.0 -0.8 -1 .5 -2 .1 -2 .4 -2 .4 -2 .2 - 1 .6 -0 .9 0.0 LONGITUDE RED. TO EQUATOR 90 and 270 o.o 100 280 + 0.9 110 290 + 1.6 120 300 + 2 .2 130 310 + 2.4 140 320 + 2 .4 150 330 + 2.1 160 340 + 1.5 170 350 + 0.8 180 360 0.0 CHAPTER IX EXAMPLES OF THE USE OF THE GLOBE MOST of the problems with which we have to deal require that the observer's exact place on the earth shall be known, that is, his latitude as well as his longitude ; and in order that they may be solved it is necessary that the globe should be rectified to the latitude by inclining the axis to the horizon by an angle equal to the latitude. This chapter contains some typical examples and the methods by which they are solved, with references to the problems of the preceding chapter.* Attention should be paid to the arrangement of the solutions, and all numerical results should be fully labeled so that it may be seen how they are obtained and combined. In all the problems, unless otherwise stated, the globe must be rectified to the latitude of Cambridge, 42.4 N. The longitude may be assumed 411 4^m wes f; o f Greenwich. Example 10. At what sidereal time do the Pleiades rise at Cambridge ? Rectify the globe by raising the north pole to such an angle that the graduation 42.4 on the outside edge of the brass merid- ian coincides with the surface of the hori- zon. Rotate the globe about the polar axis until the Pleiades are in the plane of the eastern horizon (Fig. 48). The R.A.M. equals the sidereal time sought, 20 h 12 m . This result is independent of the longitude. The Pleiades rise at any place in latitude 42.4 K at 20 h 12 m of local sidereal time. FIG. 48. Rising of Pleiades : 20h 12 Camb. Sid. T. * These solutions were obtained with a not very accurate globe nine inches in diameter. Better results may be obtained with a larger globe in good condition. 122 EXAMPLES OF THE USE OF THE GLOBE 123 Example 11. At what apparent time do the Pleiades rise at Cam- bridge on October 30 ? Determine the sidereal time, as in the last example, 20 h 12 m . The sun's right ascension is determined to be 14 h 17 m by bringing it to the meridian (Fig. 49), as in Prob- lem 4, and the relation App. T. = Sid. T. Sun's E.A. gives 20 h 12 m - 14 h 17 m = 5 h 55 m Camb. App. T. Example 12. At what Cambridge mean time do the Pleiades rise October 30 ? Eq. of T. = - 16 m (subtract from App. T.). The apparent time being 5 h 55 m by the last example, the mean time is 5 h 55 m 16 m = 5 h 39 m . Example 13. At what Eastern standard time do the Pleiades rise at Cambridge ~ . , nr . n October 30? The arranement of the work is as follows : FIG. 49. October 30: Sun's Camb. Sid. T. by globe (Example 10) ... ..... 20 h 12 m Sun's R.A. by globe (Problem 4) ......... 14 17 Camb. App. T. (Example 11) ........ . '. . ~6 55~ Eq. of T. by table ....... - . , ...... - 16 Camb. M. T. (Example 12) 5 39 Red. to E. Std. T. - 16 E. Std. T. of rising of Pleiades 5 23 Example 14- At what standard time do the Pleiades set at Cambridge March 1 ? Bringing the Pleiades to the western horizon, we have, as in Example 13 : Camb. Sid. T. by globe (Fig. 50) ... 11* 15 m Sun's R.A. (Problem 4) 22 50 Camb. App. T :...... 12 25 Eq. of T. by table . . . . . . . . . + 13 Camb. M. T . ... . 12 38 Red. to E. Std. T -16 E. Std. T. of setting of Pleiades March 1 12 22 Example 15. What is the standard time FIG. 50. Pleiades setting : . . ~ , . , ,_- + ~ n Sid. T. iib i 5 m of sunrise at Cambridge on May 15 ? 124 LABORATORY ASTRONOMY Mark the place of the sun on the ecliptic for May 15 and bring this point to the plane of the eastern horizon (Fig. 51). The R.A.M. gives the Camb. Sid. T. by globe 20 h 28 m Sun's R.A. (Problem 4) by globe . . . 3 28 Camb. App. T T? 00~ Eq. of T. by table _ -4 Camb. M. T 16 56 Red. for Long 16 Std. T. of sunrise May 15 ~IQ 40~ Or, May 16, 4 h 40 m A.M. But see the note to Prob- lem 8. Since the place of the sun was taken for May 15, the solution gives the time of sunrise for that civil date. FIG. 51. Sunrise May 15: Sid. T. 20h 28"> Example 16. What is the azimuth of the sun at Cambridge at sunrise June 21 ? The sun's place for June 21, being brought to the horizon as in the preceding problem, was found to be on the division 59 of the graduation which reads from zero at the north point of the horizon to 90 at the east point (Fig. 52) ; its bearing, therefore, is N. 59 E., and its azimuth reckoned from the south point is 180 + 59, or 239. The graduation on the inner edge of the horizon has a second set of numbers begin- ning with at the east and west points and running to 90 at the north and south points. By means of this amplitudes may be directly measured. The amplitude of the sun in this case was E. 31 N. Example 17. At Cambridge, September 10, in the afternoon, the sun's altitude is 20. What is its azimuth? For the solution of this problem the alti- tude arc must be applied to the brass meridian, attaching the clamp so that the 90 mark of the gradua- tion is as exactly as possible under the graduation 42.4 on the inner edge of the brass meridian ; this is at the highest point FIG. 52. Sunrise June 21 : Sun's Bearing N. 59 E. ; Az. 239 EXAMPLES OF THE USE OF THE GLOBE 125 of the globe, corresponding to the zenith of the sphere in latitude 42.4 north. The longitude of the sun for September 10 being found, by the circles printed on the horizon for this purpose, to be 17.7 in Virgo, or five signs and 17.7 = 167.7, this point was brought into the southwest quadrant halfway from the south to the west point and the altitude arc made to pass through it ; the altitude was seen to be approximately 40. The foot of the arc was then moved about 20 toward the west point and the sun's place brought to it ; the altitude was now about 30. The foot of the arc was moved again about 20 farther toward the west point and the sun's place brought to it, the sun's altitude being about 15. The arc FIG. 53. September 15 : Sun's was now moved Alt. 20 ; Az. 77.5 , . . . back a few degrees toward the south and by a few trials a position found (Fig. 53) such that the sun's place coincided exactly with the division marking an altitude of 20 ; the zero of the graduated edge of the arc was then halfway between 77 and 78 of the graduation on the inner edge of the horizon circle. The bearing was then S. 77.5 W. and the azi- muth 77.5. Example 18. At Cambridge Altair is east of the meridian at an altitude of 30. Find its azimuth and hour-angle and the sidereal time. Bringing the place of Altair to 30 on the flexible arc, as described in the last problem, the bearing is found to be S. 73 E. Hence the azimuth is 287. With the same adjustment the E.A.M. is 15 h 56 m , which is the sidereal time. By bringing Altair to the meridian, its right ascension is found to be 19 h 43 m , and, by Problem 5, H.A. = 15 h 56 m - 19 b 43 m = 20 h 13 m . FIG. 54. Alt. of Altair 30 : H.A. 20h 13-" ; Az. 287; Sid. T. 15* 56* 126 LABORATORY ASTRONOMY Example 19. On September 10, at Cambridge, in the forenoon, the sun's altitude is 20. What is the local mean time ? The sun's longitude being 167.7, as in Example 17, its place is brought to 20 on the flexible arc in the southeast quadrant (at a bearing S. 78 E., with which compare the result of Problem 17) and the problem solved as follows : Sun's forenoon Alt. 20 R.A.M. 6 h 42 m Sun's R. A. (Problem 4) 11 13 App. T 19 29 Eq. of T. by table -3 Camb. M. T 19 26 FIG. 55. Sun's Alt. 20; Or - , 7 26A.M. K.A.M. G> 42> It would appear that our result means 7.26 A.M. of the following day. But it is to be remembered that we have used the sun's place for September 10 (the places are marked for noon), and our solution then applies more nearly to the morning of that date. Example 19 is perhaps the most important that we have solved, since it illustrates the method by which the longitude is determined at sea. The sun's altitude is measured by a sextant and its hour-angle computed. From the apparent time thus obtained the local mean time is found as above and compared with G-.M.T. kept by a chronometer. Example 20. On July 10, at Cambridge, what is the sun's hour-angle when it is in the prime vertical ? What is the local mean time ? In the summer half of the year the sun is in the prime vertical once in the fore- noon and once in the afternoon, so that there will be two solutions of the problem. The place of the sun July 10 is found by the table to be in longitude 107.7. The altitude arc being adjusted with its foot FIG. 5G. Sun in Prime Verti- cal : July 10, forenoon ; R.A.M. 3h 3"> EXAMPLES OF THE USE OF THE GLOBE 127 at the east point of the horizon, the sun's place is brought to the graduated edge of the are and R.A.M. noted. The altitude arc being brought in the same way to coincide with the west quadrant of the prime vertical, the sun's place is brought again to the gradu- ated edge and R.A.M. noted. Then the sun's right ascension is determined, and the results may be recorded and the computation made in the following form : Sun in prime vertical A.M. R.A.M 3 h 3 m Sun's R.A. by globe . '. .' . . . '. . . 7 20 App. T > . . . . . 19 43 Eq. of T. by table + 5 Local M. T. . 19 48 Ll h 36 m 7 20 4 16 + 5 4 21 Or 7 48A.M., 4 21p.M. Example 21. At Cambridge, at O h sidereal time, what bright stars are seen near the meridian ? What are their declinations ? Eectify the globe for latitude + 42.4. Eotate the globe until the R.A.M. is O h , and the following stars will be found near the meridian : y Pegasi, Decl. + 14.0 ; a Andromedse, Decl. + 27.5; ft Cassiopeise, Decl. + 58; Polaris, of course, but too near the pole to be seen on the globe ; y Ursse Majoris, Decl. 54 ; 8 Ursse Majoris, Decl. 58. The two latter are below the pole, and to determine their declinations the globe must be rotated 180 to bring them under the inner graduations of the meridian. Notice that the four first stars lie along the same hour-circle, which is 'the equinoc- tial colure, in R.A. O h , and that this circle is divided roughly by them into multiples of 15, thus : Polaris to (3 Cassiopeise, 30 ; ft Cassiopeise to a Andromedse, 30; a Andromedse to y Pegasi, 15. By continuing the line of stars about 15 we arrive at Decl. = 0, R.A. = 0, that is at the vernal equinox, which though marked by no conspicuous star is easily fixed by this alignment. FIG. 57. Stars on Meridian at Cambridge at V* Side- real Time 128 LABORATORY ASTRONOMY Example 22. What is the standard time corresponding to O h of sidereal time at Cambridge October 10 ? The sidereal time being given, this problem is similar to Exam- ples 13, 14, and 15, and illustrates the general process of passing from sidereal to mean or standard time by means of the globe, thus : Sid. T .............. ..... O h O m Sun's R. A. by globe ...... - ' ....... 13 _ 4 App. T ................... 10 56 Eq. of T ............... ... - 13 Camb. M. T ................. 10 43 Red. for Long, to Std. T ...... *. . . .--.'. . - 16 Eastern standard time 10 27 Example 23. Find the altitude and azimuth of Arcturus at 8 P.M., standard time, at Cambridge, September 10. This problem requires the globe to be rectified for both latitude and time. The latter adjustment is made as follows : Std. T 8 h O m Red. for Long + 16 Camb. M. T 8 16 Eq. of T. by table (add to M.T.) . . . +3 App. T 8 19 R.A. Sun by globe 11 15 Camb. Sid. T 19 34 Rectify for Cambridge, Lat. + 42.4. Rotate the globe till the E.A.M. is 19 h 34 m . FIG. 58. Arcturus: Septem- ber 10, 8 P.M., E. std. T. ; Apply the altitude quadrant so as to pass Ait. 20 ; AZ. 98 through Arcturus, and we find its altitude 19.5, and its bearing K 80.5 W.; hence its azimuth is 99.5. Example 24. What constellation is rising in the east at 9 P.M., Eastern standard time, at Cambridge, November 10? As in the preceding problem : Std. T 9 h O m Red. for Long, to Camb. M. T. . . . , + 16 Camb. M. T 9 16 Eq. of T. by table (subtract from App. T.) + 15 Camb. App. T. 9 31 R.A. of Sun by globe 16 3__ Sid. T. 34 EXAMPLES OF THE USE OF THE GLOBE 129 To rectify for time rotate the globe till the E.A.M. is O h 34 m . It will be found that the constellation of Orion has just risen above the eastern point of the horizon. Compare the form of this solution with that of Example 13, which is the inverse of this, the rising of a star being given and the standard time sought. PROBLEMS INVOLVING THE USE OF THE NAUTICAL ALMANAC Example 25. At Cambridge, November 30, 1904, at 5 h 15 m P.M., standard time, a bright star is seen due southwest about 10 above ,, , . , T ,. , . . ., , . FIG. 59. Orion rising: Cam- the horizon. .No other stars being visible in bridge, November 10, 9 P.M., the twilight, it is desired to identify the star. E. Std. T . Red. for Long. . Camb. M.T Eq. of T. (subtract from A pp. T.) .... App. T .- . R.A. of Sun Camb. Sid. T. . Std. T. 5 h I 5 + 16 5 31 + 11 5 16 42 20 22 8 Rectifying for Cambridge, Lat.-f- 42.4, and for 22 h 8 m Sid. T., it is found, by means of the altitude arc (Fig. 60), that there is no star upon the globe at the given altitude and azimuth, the nearest star being FIG. 60. Star 10 above South- cr Centauri, which FIG. 61. Star brought to Merid- re;^^: >uld *t t>e visi- 22" 7 ble at that altitude ian : R.A 18> 56 ; Decl. - 130 LABORATORY ASTRONOMY in twilight. The exact point being marked is brought to the meridian and found to be in R.A. 18 h 56 m and Decl. 23^ (Fig. 61). The fact that its position is very near the ecliptic suggests that it may be a planet, and on consulting the Almanac it is found that on November 30 the right ascension of Venus is 19 h 4 m and its declination 24.7, or within about 2 of the observed place. Example 26. Which of the planets that are visible to the naked eye are above the horizon at Cambridge at 8 P.M., standard time, October 1, 1904 ? From the Nautical Almanac are taken the following data for the given date : R.A. DECL. Mercury ............ ll h 25 m + 5.l Venus ............. 13 55 - 11 .4 Mars ............ . 10 10 + 12 .7 Jupiter ............ . 1 44 + 9 .1 Saturn ........... . . 21 10 - 17 .0 Marking these places upon the globe and rectifying for the given place and time, it is at once seen that the first three are below the western horizon, while Jupiter is 20 above the east point of the horizon and Saturn approaching the meridian at an altitude of about 30. Where only an approximate result is desired, it will often be sufficient to neglect the corrections for longitude and equation of time, the sum of which at Cambridge never amounts to much more than half an FIG. 62. pjaneta, October i, hour! Thig of CQurse assumeg standard time to equal apparent time. Thus, in this problem we may bring the sun to the meridian and, noting E.A.M. = 12 h 30 m and adding 8 h , we have 20 h 30 m ( 30 m ) as the E.A.M. corresponding to 8 h apparent time. The general terms in which the answer is given above will apply equally well, and some time is saved where only the general aspect of the heavens is required. EXAMPLES OF THE USE OF THE GLOBE 131 Example 27. At what standard time does Jupiter set at Cam- bridge December 25, 1904? By the tables in the Nautical Almanac, we find that on the given date the right ascension of Jupiter is l h 17 m and its declina- tion -f 6.8. Marking this place on the globe and bringing it to the western horizon, the E.A.M. is 7 h 38 m , which is the sidereal time. Converting to standard time : Sid. T 7 h 38 m Sun's R.A. by globe 18 12 App. T 13 26 Eq. of T . . Camb. M. T 13 26 Red. for Long - 16 Std. T 13 10 Or 1 10 A.M. Example 28. At what time does the moon rise at Cambridge December 25, 1904 ? If the moon's position were known directly from the Nautical Almanac, the solution of p IGi 63. j up iter setting : this problem would be similar to the last: R - A - M - 7h 38ffi J E - std - T. 13 h 10 m but the moon's right ascension and declina- tion are changing so rapidly that we must reach the result by approx- imation. We may first assume the moon's place at rising to be the same as at standard noon, December 25 (or 5 h , G.M.T.), and at that time the Almanac gives the moon's right ascension 8 h 54 m , Decl. H- 14.9. Marking this place on the globe and bringing it to the eastern horizon, we find E.A.M. = l h 56 m , and continue the computation as in the second column of the table below. (See Example 15.) G.M.T. 5 h 12 h 28 m 12 h 45 Moon's Place 8 h 54, + 14. 9 9 h ll m , + 13.9 9 h 13 m , + 13.9 R.A.M. . . . l h 56 m 2 h 13 2 h 18 m R.A. of Sun. . 18 12 18 12 18 12 App. T. ... 7 44 8 Eq. of T. . . . 0_ Camb. M. T. . 7 44 Red. for Long. . 16 E. Std. T. . 7 28 7 45 7 50 132 LABORATORY ASTRONOMY FIG. 64. Moonrise at Cam- bridge December 25, 1904 : K.A.M. 2* 18 This gives as the approximate time of moonrise 7 h 28 m , E. Std. T., or 12 h 28 m , G.M.T., and finding the moon's place for this time, R.A. 9 h ll m , Decl. + 13.9, we better our result by the computation shown in the third column, which gives 7 h 45 m , E. Std. T., or 12 h 45 m , G.M.T. With this value we find the moon's place 9 h 13 m , f 13.9, and compute as in the last column, finding E. Std. T. = 7 h 50 m . As this is within ten minutes of the time for which the data were assumed, and since in ten minutes the moon's right ascension, as shown by the difference column, changes by 24 s , a quantity too small to be surely measured on an ordinary 10-inch globe, we may regard the last solution as suffi- ciently accurate. It would appear that the two last results should be in closer agreement, since the difference in the assumed times is only seventeen minutes ; the two first measures, however, were not made with care, as only approxi- mate values were sought. It is obviously an advantage to estimate the approximate time of moonrise as closely as possible before beginning the solution : this may be done by noting the age of the moon (page IV of the month) and remem- bering that the moon rises and sets about 48 m , or O h .8, later each night than the night before, and that at new moon sun and moon rise and set together. Assuming that the sun rises at 6 A.M. and sets at 6 P.M., stand- ard time, we shall find an approximate value of the standard time of moonrise or moonset by adding to these times a number of hours equal to eight-tenths of the moon's age in days. Thus, in the pre- ceding problem, the moon's age being eighteen days on December 25, we add 0.8 x 18 h = 14 h .4 to 6 A.M. to find the time of moonrise ; FIG. 65. Moonset at Cam- bridge December 18, 1904 : R.A.M. 9* 41" EXAMPLES OF THE USE OF THE GLOBE 133 this gives 8 h .4 P.M. as the approximate time, which is within an hour of the final result. Example 29. Find the time at which the moon sets at Cam- bridge December 18, 1904. The moon's age is found by the Ephemeris to be eleven days ; hence we add 9 h to 6 h P.M., and have as the approximate time of moonset 15 h , corresponding to 20 h , G.M.T. We may record the successive approximations as follows : Assumed G.M.T. Moon's R.A. and Decl. FIRST APPROXIMATION 20 h SECOND APPROXIMATION 2 h R.A.M. at moonset Sun's R.A. . . App. T Eq. of T. . . . Red. for Long. Std. T. 17 39m 46 15 53 - 4 -16 15 33 20 h 33 m Qh 41m 17 46 15 55 -20 15 35 A single recomputation will always be sufficient if the moon's place is first determined by computing from its age. MISCELLANEOUS EXAMPLES Example 30. Find the duration of twilight at Cambridge March 1. Evening twilight ends when the sun has sunk so far below the horizon that his direct rays can no longer fall upon and be reflected by any particles in that portion of the atmos- phere which lies above the plane of the hori- zon. This is usually assumed to be the case when the sun is 18 below the horizon. Bringing the sun's place for March 1 to the horizon, and then, by means of the extension of the altitude arc, to point 18 below the horizon (Fig. 66), we have the following values R.A.M. at sunset . . 4 h 20 m R.A.M. at end of twilight 6 0_ Difference . 1 40 FIG. 66. End of Twilight at Cambridge March 1 134 LABORATORY ASTRONOMY which equals the change in the sun's hour-angle, or the time elapsed between sunset and the end of twilight. Example 31. At what hour, apparent time, does morning twi- light begin at Cambridge June 21? June 21. Sim's place 18 below E. horizon, R. A. M 20 h 8 m Sun's R.A. by globe . . . 6 App. T 14 . 8 Or 28 A.M. Example 32. At what point of the horizon does the first glim- mer of dawn appear in latitude 42.4 on June 21? Bringing the sun's place by trial to the altitude arc at a point 18 below the horizon (Fig. 67), the reading on the horizon at the graduated edge of the altitude arc is E. 57 N. = Az. 213 ; and as this is the nearest point of the horizon to the sun when it is 18 below the horizon, it is at this point or a little to the south that the first light will appear. Example S3. How many hours can the sun shine into north windows June 21 in latitude 41? By the method of Example 15, it is found that the apparent times of sunrise and sun- set on June 21 are 4 h 30 m A.M. and 7 h 30 m FIG. 67. Dawn at Cambridge P.M., and by the method of Example 20, june2i,at2fr8A.M.: Sun's that the sun is in the prime vertical at 7 h Az 213 56 m A.M. and 4 h 4 m P.M. Hence from 4 h 30 m to 7 h 56 m A.M. and from 4 h 4 m to 7 h 30 m P.M., a total of 6 h 52 m , the sun shines on the north face of an east and west wall. The length of the day is fifteen hours. Example 84. August 20, in latitude 42, longitude 4 h 48 m , at ten minutes past 10 A.M., Eastern standard time, the sun begins to shine upon the front wall of a building. How does the building face ? Since at the given time the sun is in the same vertical plane with the front wall of the building, the problem requires us to determine the direction of this plane by finding the sun's azimuth, which may be done as follows : EXAMPLES OF THE USE OF THE GLOBE 135 Rectifying for latitude 42^-, we have : Std. T. 10 h 10 m A.M = 22 h 10 m Red. for Long, (from E. Std. T.) + 12 Local M.T 22 22 Subtract Eq. of T. (additive to App. T.) . . . . ' . . - 3 App. T 22 19 Sun's R.A 10 1_ Sid. T 8 20 Eectifying for this time and bringing the altitude arc to the sun's place for August 20, we find the sun's azimuth to be 315. Hence the front wall is in a line from southeast to northwest, and the building fronts southwest. Example 35. What is the greatest north- ern latitude in which all of the four bright stars of the Southern Cross are visible? What must be the time of year ? Rectifying the globe for the equator, the Southern Cross (about R.A. 12 h , Decl. - 60) is brought to the meridian and the brass meridian is moved in its own plane until the lowest star is brought to the horizon at its south point. The elevation of the pole above the north horizon is then read on the brass meridian and found to be 28, which FIG. 68. August 20: std.T.io* is the required latitude. The star being still 10m ; R ; A - M - 8h ""J Sun ' 8 Az. 315 in the same position, the altitude arc is then used to mark the points of the ecliptic which are 18 below the horizon. These are found to be at points occupied by the sun January 2 and May 25, and between these dates, therefore, the whole cross may be above the horizon in latitude 28 in the full darkness of night, the sun being below the twilight limit. Example 36. What is the latest date at which we can see Sirius in the evening twilight in latitude 42? Sirius is visible when the sun is about 10 below the horizon, and cannot be seen later than the day on which he sets at the instant that the sun is 10 below the horizon. Rectifying for 42 and bringing Sirius to the western horizon, we find that the point of the ecliptic which is 10 below the horizon is 136 LABORATORY ASTRONOMY the place occupied by the sun on May 15, which is, therefore, the required date. Example 37. Between what dates is the sun visible at midnight at the North Cape, in latitude 70 north ? Eectifying the globe for 70 north and rotating the globe slowly, it is found that points on the ecliptic in longitudes 58 and 122 can be brought exactly to the north point of the horizon ; any point between these may be brought to the meridian below the pole and above the horizon. The dates at which the sun occupies these posi- tions are May 19 and July 25, and between these dates the sun will always come to the meridian at midnight above the horizon. Example 38. Illustrate the " harvest moon " by finding the time of moonrise at Edinburgh, latitude 56, on successive dates about the time of full moon, September 24, 1904. As only approximate results are desired, we may take from the Ephemeris the moon's place for 6 h P.M., G.M.T., and solve as follows: 1904 R.A. DECL. K.A.M. AT MOONRISE SUN'S ll.A. APPARENT TIME September 22 22h 36 m -8 17 h 22m 12 h O m 5 h 22 m 23 23 22 -4 17 42 12 4 5 38 24 7 - 1 18 9 12 7 2 25 52 + 3 18 28 12 10 6 18 26 1 38 + 7 18 51 12 14 6 37 And it appears that the moon rises about twenty minutes latter each night than it did on the previous night. Example 39. Find the time of moonrise at Edinburgh on succes- sive nights at full moon, March 31, 1904. We have, as in Example 38, the moon's place at 6 h P.M., G-.M.T. : 1904 R.A. DECL. R.A.M. AT MOONRISE SUN'S E.A. APPARENT TIME March 30 . . 11* 56 m + 1 Q 5h 5 7 m 0^ 38 5 h IQm 31 . . 12 52 -4 7 20 41 6 39 April 1 . . . 13 49 -8 8 42 44 7 58 EXAMPLES OF THE USE OF THE GLOBE 137 Therefore the full moon at the time of the vernal equinox rises about one hour and twenty minutes later each night. (Notice and explain the difference in the accuracy attained in these two examples.) Example 40- Find the rate at which 8 Orionis is changing its azimuth at rising and setting in latitude 42. Rectifying for 42 and bringing 8 Orionis to the eastern horizon, we find R.A.M. = 23 h 23 m ; Az. = 271. Increasing the hour-angle half an hour by making R.A.M. = 23 h 53 m , we find, by the alti- tude arc, Az. = 276. Bringing the star to the western horizon, we have K.A.M. = ll h 24 m ; Az. = 89^. Decreasing the hour-angle by making E.A.M. = 10 b 54 m , we find Az. = 84 half an hour before setting. In both cases the diurnal rotation causes the azimuth to increase at the rate of 5 in half an hour. By solving the same problem for stars in various parts of the heavens, as, for instance, Vega, y Pegasi, Airfares, and a Gruis, it appears that stars of whatever declinations, when near the horizon, are increasing their azimuths by about 10 per hour in latitude 42. (This is the rate at which the plane of the pendulum appears to revolve in Foucault's experiment.) Example 41. To mark the hour-lines on a horizontal sundial for use in latitude 42. The gnomon of an ordinary sundial (Fig. 69) is directed toward the pole, and its shadow at apparent noon falls upon the horizontal dial on the line of XII hours, which, when properly adjusted, lies in the direc- tion of the meridian. The shadow at that time is in a line drawn through the foot of the gnomon toward azi- muth 180. It always passes through the intersection of the gnomon with the dial and, continually shifting F- 69 - Horizontal sundial, / Latitude 42 toward the east, at any instant lies in the plane containing the sun and the gnomon. This plane cuts the celestial sphere in the sun's hour-circle. The shadow, therefore, is a line which passes through the foot of the gnomon and whose azimuth is that of the intersection of the sun's hour-circle with 138 LABORATORY ASTRONOMY the plane of the horizon. For a given hour-angle the position of this line will be the same whatever the position of the sun upon its circle, and is therefore the same for a given apparent time whatever the time of year. We may find the azimuth of the intersection of a given hour- circle with the horizon by means of the globe as follows. Rectify- ing the globe for 42, the vernal equinox is brought to the meridian, so that the equinoctial colure cuts the horizon at azimuth 180. In this position R.A.M. is O h , and the azimuth of the shadow is 180. Increasing the hour-angle of the colure by successive incre- ments of 15, we have the following values for the azimuths of the hour-lines : FOB THE P.M. HOURS: I II III IV V VI VII R.A.M. 1* 2 3 4 5 6 7 )URS: AND SIMILARLY FOR THE A.M. HOURS : Azimuth of Shadow R.A.M. Azimuth of Shadow 190 XI 23 170 201 X 22 159 214 IX 21 146 230 VIII 20 130 249 VII 19 111 270 VI 18 90 291 V 17 69 If the hour-circles are shown for each 15 as on most modern globes, it is sufficient to bring one hour-circle to the meridian and note the points where the other circles cut the horizontal plane ; Fig. 57 shows the globe rectified to 42 and O h Sid. T., and therefore in position for reading the azimuths of the successive hour-lines directly on the horizon. Example 42. To mark the hour-lines of a vertical sundial for use in latitude 42 N., the bearing of the plane being W. 24 S. Here the shadow of the gnomon falls upon a vertical plane, and the line for noon is a vertical line through the intersection of the gnomon with the plane. At any given hour after noon the shadow falls below the gnomon and to the east of the XII line (Fig. 70), since it marks the inter- section of the plane of the dial by the sun's hour-circle. It makes an angle with the XII line which may be defined as the " nadir EXAMPLES OF THE USE OF THE GLOBE 139 distance " of the line of intersection of the two planes, and this is equal to the zenith distance of that part of the same line which lies above the gnomon. This problem therefore requires us to find the zenith distance of the intersection of the sun's hour-circle with the vertical plane for a given hour-angle of the sun, and may be solved with the globe as follows : Rectify the globe for latitude 42, and adjust the altitude arc to the zenith with its foot at azimuth 66 on the horizon ; its plane then corresponds to that of the dial. Bringing the vernal equinox to the meridian, R.A.M. = O h , the equinoctial colure intersects the altitude arc at zenith distance 0. Increasing the hour-angle of the colure, as in Example 41, we have successively FIG. 70. Vertical Dial, Latitude 42 HOUK-LlNE I II III IV V R.A.M. lh 2 3 4 5 ZENITH DISTANCE OF INTERSECTION 13 30 49 70 90 which gives the angles of the afternoon lines from the noon line. Setting the arc at azimuth 246, we find in the same way HOUR-LINE XI X IX VIII VII VI V R.A.M. 23* 22 21 20 19 18 17 ZENITH DISTANCE OF INTERSECTION 11 22 33 44 56 70 90 which gives the morning lines. The A.M. and P.M. divisions will not be symmetrical about the XII line unless the vertical plane faces due south. Example 4&- Find the path of the shadow of a pin head on a horizontal plane at Cambridge March 21, from 8 A.M., apparent time, to 5 P.M., apparent time. 140 LABORATORY ASTRONOMY Rectifying the globe for latitude 42, bringing the sun's place to hour-angles which correspond to the successive hours from 8 A.M. to 5 P.M., and measuring its altitude and azimuth in each position by the altitude arc, we have the following results : APP. TIME RA.M. ALTITUDE AZIMUTH DISTANCE 8 h A.M. 20h 22 291 12.5cm. 9 21 32 303 8.1 10 22 40 318 6.0 11 23 46 337 4.9 Noon 48 4.5 l h P.M. 1 46 22 4.9 2 2 40 42 6.0 3 3 32 56 8.0 4 4 22 C8 12.6 5 5 11 80 26.5 To construct the curve we must know the length of the pin ; assuming this to be 5 cm. long, a point on the paper is chosen to represent the point vertically under the pin head, and through it is drawn a line to represent the meridian, and other lines are FIG. 71. Azimuth, of Shadow drawn at the azimuths differing by 180 from those given in the above table. (See Fig. 71.) The shadow path will cross these lines at the corresponding hours. To find the distance of any point of the shadow path from the foot of the pin, we may reverse the process explained on page 5. Drawing a line from C, the center of the base in Fig. 6, through the divisions of the protractor corresponding to any one of the alti- tudes of the above table and measuring the line A'B', we have the distance in centimeters from the foot of the pin to the point where the shadow falls on the corresponding azimuth line. The last column of the above table gives the distances measured in this manner. EXAMPLES OF THE USE OF THE GLOBE Fig. 72 shows the shadow path as thus constructed, and it is evidently a straight line. This will always be the case on the day of the equinox, when the sun is in the equator and its diurnal path is consequently a great circle. FIG. 72. Path of Shadow THE HOUR-INDEX The globe is usually provided with an arrangement by means of which approximate solutions may be made of problems involving time without the use of the graduations of the equinoctial. This process is so simple that its explanation might well have preceded that of the method of finding the sun's hour-angle given on page 112 and used in Problem 7. It is, however, very inaccurate, and should only be chosen where an error of several minutes is unimportant. The most convenient form given to the attachment is that of a small pointer fixed to the brass meridian in such a manner that it revolves about the same center as the polar axis, but with sufficient friction to keep it fixed in any position where it may be placed. This pointer, or " hour-index," lies close to the surface of the globe, which revolves freely under it. The end of the index lies over a small circle on the globe, about 15 from the pole ; and this circle is graduated into hours and quarters in two groups of 12 hours each, numbered in the same direction as the graduations of the equinoctial. The following example illustrates the use of the hour-index, which in this case gives sufficiently good results with less trouble than the method already explained. Example 44. Find the apparent times, October 1, 1904, of rising and setting of the planets whose places are given on page 130. 142 LABORATORY ASTRONOMY Mark the places of the planets and of the sun ; bring the latter to the meridian and set the hour-index to read XII noon. Eotate the globe through any angle, and the reading of the index will equal the hour-angle of the sun in its new position, and thus will give directly the corresponding apparent time. We may, therefore, rapidly determine the apparent time of rising and setting of all the planets by bringing each in turn to the eastern and western horizon and noting the reading of the hour-index. The hour-index may be adjusted to give local mean time or standard time directly by making it read the local mean time or standard time of apparent noon when the sun is brought to the meridian. Thus, for October 1, at Cambridge, longitude 4 h 44 m : App. T. of App. noon 12 h O m Eq. of T -10 Camb. M. T. of App. noon 11 50 Red. for Long 16 Std. T. of App. noon 11 34 And the index should be set to read ll h 34 m when the sun is on the meridian, in order to give Eastern standard time. CHAPTER X THE MOTIONS OF THE PLANETS IT has been the aim of the preceding chapters to show how the diurnal motion and the motion of the sun and moon among the stars may be studied in such a manner that the student shall acquire and fix his knowledge in large part by his own observations. There remains to be considered the motion of the planets, which cannot be studied in the same way because they move so slowly that a long time would be required to obtain a sumcient number of observations on which to base a satisfactory theory. It is of course desirable, however, during the continuance of the observa- tions on the moon and stars to include the planets in order to establish a few fundamental facts, such as that they never appear far from the ecliptic and that in general they move from west to east like the sun and moon, but that when opposite the sun, so that they come to the meridian at midnight, they are moving from east to west among the stars. Their places in the heavens should be occasionally observed, for comparison with the places derived from the theory which forms the subject of the present chapter. In treating of this theory we shall first assemble the few prin- ciples which have been shown to account for the observed motions, and shall then show how these principles may be applied to the graphical solution of problems involving the determination of the place in the heavens of a planet as seen from the earth at any given time. These problems serve to illustrate and explain the phenomena resulting from the planetary motions, as the globe problems of the preceding chapter serve for those resulting from the diurnal rotation of the earth. Results of the Law of Gravitation. In consequence of the attrac- tion of the sun, each planet describes an ellipse, having the sun in one focus ; this is "Kepler's first law." The mutual attractions of the planets produce " perturbations " of their motion, but in no case 143 144 LABORATORY ASTRONOMY are these perturbations sufficient to alter the place of the planet by so much as one degree from its place as determined by the sun's attraction. Jupiter may be displaced about 0.3 and Saturn nearly 0.8 ; but with this exception no displacement of a planet amounts to J. The asteroids are subject to much greater perturbations. The orbit of each planet is in a plane which remains nearly fixed, and the planes of all the orbits are so nearly coincident with the ecliptic that the projections of their paths on the ecliptic are no more distorted than the roads of a moderately rugged country are distorted in their representations on an ordinary plane map. This fact makes it as easy to determine their motions by an accurate map of their orbits on the plane of the ecliptic as to follow the motion of a traveler over a well-charted country, when his point of departure and rate of travel are known. PROPERTIES OF THE ELLIPSE An ellipse may be drawn by putting two pins upright in a board, as in Fig. 73, laying a knotted loop of thread on the board so as to include both pins, and then putting the point of a well-sharpened pencil on the surface inside the loop. Let the pencil be moved out FIG. 73. Drawing an Ellipse so as to form the loop into a triangle, and then drawn along the surface so as to pass successively through all the points which it can reach without allowing the thread to become slack. The curve which it follows will be an ellipse whose shape and size will depend only on the distance between the pins and the size of the loop. The form of the curve is shown in Fig. 74. F 1 and F 2 are the foci, AB the major axis, and C, which bisects both F^FZ and AB, is the center of the ellipse. PF-^ is the radius THE MOTIONS OF THE PLANETS 145 vector from any point P to F lf and PF 2 the radius vector to F 2 . They are usually represented by r and r 2 . r + r z is a constant for all points of the ellipse, being always equal to the length of the thread minus FiF z . For the point A and since from the sym- metry of the curve 2? AF l = BF 2 , A C is usually represented by a, and CF-, or CF 2 by c. J FIG. 74. Fundamental Points and Lines Since 2 c equals the distance between the foci, and 2 a + 2 c the length of the thread, the shape and size of the ellipse are completely fixed by the values of a and c. The ratio c/a is called the eccentricity and is repre- sented by e ; it is always less than unity. The line along which the major axis lies is called the line of apsides. To draw a Given Ellipse. Let it be required to draw an ellipse whose semi-major axis is one inch, and eccentricity i, with one focus at the point F 1 of Fig. 75, and with its major axis inclined 30 to the horizontal. Draw the line of apsides MN at the proper angle. Since e = J, we locate C one-fourth of an inch from F l on the line of apsides. Take F 2 at an equal distance beyond C, make the total length of the thread 2 inches = 2 a -f- 2 c, and draw the ellipse as shown in the figure. The dotted line surrounding the ellipse is a circle drawn about C as a center with a radius of one inch (equal to the semi major axis). It is worthy of notice that the ellipse differs but little from this circle, the greatest distance between the two being about y^ of an inch. With a less eccentricity the agreement of the two curves is closer. For e = 0.10 the difference is but .005 of the semi major axis, so that an ellipse of that eccentricity whose semi major axis is two inches differs at no point more than T ^ of 146 LABORATORY ASTRONOMY an inch from a circle struck about its center with a radius of two inches. If the orbits of the planets are drawn with their true eccentricities and with a line 0.01 inch in width, and in each case a circle is struck with radius a about the center of the ellipse, and having a width of .01 inch, no white space will be anywhere visi- ble between the two lines unless the diameter of the circle is greater Horizontal FIG. 75. Ellipse drawn with Given Constants than about 1 inch for Mercury, 4^ inches for Mars, 17 inches for Jupiter, and 12^ inches for Saturn. For Venus and the earth the circles may be several feet in diameter. The orbits may therefore be represented by such circles with a considerable degree of accuracy. MEAN AND TRUE PLACE OF A PLANET Having considered the geometrical properties of the planetary orbits, it is next in order to inquire as to the law which regulates the motions of the planets in their orbits. Since the sun is at one focus of the orbit, the planet's distance from the sun varies continually. It is nearest the sun at the peri- helion point, which is at one extremity of the major axis. Aphelion occurs at the opposite end of the major axis, and the planet is then at its greatest distance. THE MOTIONS OF THE PLANETS 147 Kepler's second law states that the planet moves in such a way that its radius vector sweeps over equal areas in equal times. The appli- cation of this principle will be evident from the following illustration. Fig. 76 represents the orbit of Mercury in its true proportions. The period of the revolution of the planet is eighty-eight days, in which time the radius vector sweeps over the whole area of the ellipse. To pass from perihelion to aphelion would require forty- four days, or one-half the period, since the area described is one- half the area of the whole ellipse. It is not difficult to fix very nearly the point reached by the planet twenty-two days after pass- ing through perihelion. It will then have accomplished a quarter of a revolution, and be at such a point P that the area ASP is one-quarter of the ellipse, or one-half of A PBS, so that APS equals BPS. It may be shown that this point must be very nearly in the line Pf drawn perpen- dicular to the major axis through f, the " empty " focus of the orbit, as it is sometimes called. Assuming P to be on this line, and drawing a perpen- dicular Sk through the focus occupied by the sun, and also the radius vector PS, we have from the symmetry of the ellipse, Area ASk equal Area BfP, and the triangle PkS evidently equals the triangle PfS. The difference of the two areas ASP and BSP is therefore the segment of the ellipse cut off by the chord Pk ; this segment is so very small that the area ASP is very nearly equal to BSP. The angle ASP through which the planet has moved about the sun since perihelion is called its " true anomaly." In this case it is about 110. We may now infer that the true anomalies of Mer- cury 22, 44, 66, and 88 days after perihelion would be about 110, 180, 250, and 0, respectively. FIG. 76. Equal Areas in the Ellipse 148 LABORATORY ASTRONOMY It is convenient to refer the motion of the planet to that of a hypothetical planet moving in the orbit in such a way as to be at perihelion with the real planet and describe equal angles in equal times ; thus the anomaly of the so-called " mean planet " after 22, 44, 66, and 88 days would be 90, 180, 270, and 360, respectively. The Equation of Center. The quantity to be added to the anomaly of the mean planet, or briefly, the " mean anomaly " of the planet, in order to find its true anomaly, is called the "equation of center"; in the cases above given it is for the four positions 0, + 20, 0, and 20. It is always positive for values of the mean anomaly between and 180, and negative for values between 180 and 360. It appears from Fig. 77, in which P and P' mark the true and mean places of the planet re- spectively, that at all points from perihelion A to aphelion B, the true anomaly ASP is greater than the mean anomaly A SP', while from aphelion to perihelion ASP is less than ASP'. The value of the mean anomaly being given for any time, its value for any other time is easily found, since it increases uniformly from to 360 in the time required for the planet to make one revolution. The mean anomaly being known, we may pass to the true anomaly by means of a table of the equation of center (page 174), in which the value of the latter is given for each degree or ten degrees of the planet's mean anomaly. The computation of these tables lies far beyond our scope, but it is worth while to note that approximate values of the equation of center may be found by a graphical method, which rests upon the principle that in describing equal areas about one focus of an FIG. 77 THE MOTIONS OF THE PLANETS 149 ellipse of small eccentricity, a planet describes very nearly equal angles about the other focus. If then the ellipse be carefully constructed on a large scale, say with a major axis of ten inches, and through the empty focus lines be drawn making angles of 10, 20, 30, etc., with the line of F 2 FIG. 78 apsides, these lines will cut the ellipse at the places occupied by the true planet when its mean anomalies are 10, 20, 30, etc. Fig. 78 shows one-half of the orbit of Mercury divided into six equal parts in this manner. The true places being thus fixed, and lines drawn from each to the sun, the true anomalies may be read off with a protractor ; and by comparison with the mean anomalies the equation of center for each ten degrees of mean anomaly may be determined. MEASUREMENT OF ANGLES IN RADIANS It has been assumed that the student is familiar with the ordi- nary method of measuring angles in degrees. For some purposes it is convenient to select a different unit, the u radian." One radian is the angle subtended by an arc whose length (measured by a flexible scale laid along the curve of the arc) is equal to that of the radius. This angle measured in the ordinary way is found to be 57.3 = 3438', or 206,265". If the length of an arc a is known, and also the radius of the circle r, the angle subtended by the arc is a/r (arc -^ radius) radians. Thus in a circle two feet in diameter, an arc of one inch subtends an angle of 1/12 radian, 6 inches of 0.5 radian, 1 foot of 1 radian, etc. Since 1 radian equals 57. 3, an arc of one inch in the above circle 150 LABORATORY ASTRONOMY subtends 1/12 x 57.3; and, in general, radians are transformed to degrees, minutes, or seconds of arc by multiplying by 57.3, 3438, and 206,265, respectively; and degrees, minutes, or seconds to radians by dividing by 57.3, 3438, and 206,265, respectively. The use of the radian is especially convenient in problems in- volving an angle so small that the corresponding arc nearly equals its chord or the perpendicular drawn from one extremity of the arc to the radius drawn through its other extremity. The method is illustrated by the following instances : 1. The moon's distance is 240,000 miles, and its angular diameter is 31', or 31/3438 radian. Its diameter in miles is given by the equation 24^00 = 38' HenC6 D = 2164 miles ' approximately. 2. The height of a tree is 30 feet, and the length of its shadow is 150 feet. The altitude of the sun is a/r = 30/150 = 0.2 radian = 11.46. The true value obtained by trigonometrical computation is 11.54, differing by .08, and this approximate method will give results within 0.l so long as the angle does not exceed this value. 3. By means of a sextant the angle between the water line of a distant war ship (Fig. 79) and the top of its military mast is found .O-OiL- FIG. 79 to be 17' 10". The height of the mast is known to be 120 feet. Assuming this height to be equal to the arc subtended by the measured angle, we have 17' 10" = 0.005 radian = ? = beight of ">Bt r distance of ship and the distance of the ship is about 8000 yards. THE MOTIONS OF THE PLANETS 151 DIAGRAM OF CURTATE ORBITS Fig. 80 represents a diagram of the orbits of the five inner planets projected on the plane of the ecliptic, which serves to solve many problems regarding the planetary motions. The diagram is of convenient size for actual use, if its dimensions are such that one astronomical unit equals about f of an inch. In order to show how small is the distortion of the orbits as pro- jected, we may compare the length of the radius vector to any point in the orbit with that of its projection on the ecliptic, which is called the " curtate " distance from the sun. Even in the case of the orbit of Mercury, which has the greatest inclination, the curtate distance differs from the true distance at most by y^, in the case of Venus by less than ff J^, and in the case of all the other planets by less than y^ 1 ^- If the scale of the diagram is such that one astronomical unit equals 1| inches, no radius vector drawn in any one of the " curtate " orbits will differ from the corresponding radius vector drawn in the actual orbit by so much as ^1^ of an inch ; and by referring to the data given on page 146 it will be seen that on that scale the elliptic orbits may be represented with considerable accuracy as circles. The position of the line of apsides is fixed by the longitude of perihelion, page 174; the distance c of the center of the ellipse from the sun is found from the ratio c /a = e, and a circle struck about the center with a radius a very closely represents the curtate orbit ; the distances c and a are of course to be laid off from the scale of astronomical units. To draw such a diagram is a useful exercise, and by careful draw- ing and erasure a single diagram may serve for many problems, but it is convenient to have several printed copies when it is desired to preserve the solutions. It is also convenient to have diagrams on which an astronomical unit equals 2j, j, and f inches, respectively, the first extending to the orbit of Mars, the second to that of Jupiter, and the third to that of Saturn. The larger scale should be used for problems referring to Mercury and Venus, while the smaller scales are required for the major planets. 152 LABORATORY ASTRONOMY ELEMENTS OF THE SIX INNER PLANETS, JAN. I, 1900 Sj~M S | e t y h MUD DUUnco 0.887 0.72S 1.000 1.524 6.203 9.639 Eoetttridt, 0.2056 0.0068 0.0168 0.0933 0.0482 0.0561 Inclination 7*0 3*.4 1*9 1*3 2*5 Longitad. of A.- eeoduigNod. 17*1 76*.7 48*7 99*4 U2*7 Longitod. of Peri- helion 76* 1304 101*2 S34!2 12*7 9o!tt Meu Longitude, Gr. Meu Nooa 182?22 344*33 100*67 294*27 238*13 266*61 Sidereal Period, lieu Solar Dy> 87*9693 224-701 865*266 686*979 4332*58 10759*2 Mean dailj motion 4*09234 1*60213 0*98661 o!62403 0*08309 0?03346 EQUATION OF CENTER Mm, A "*" A*.* 0* 0.0 0.0 0.0 0.0 0.0 0.0 860 10 5.4 0.1 tj 2.1 1.0 1.2 350 10.5 0.3 0.7 4.1 2.0 2.4 340 14.9 0.4 1.0 6.9 2.9 8.4 830 18.6 0.5 1.2 7.5 3.7 4.4 820 21.1 0.0 .5 8.8 4.4 6.2 310 22.8 0.7 .7 9.8 4.9 5.8 300 23.6 0.7 .8 10.4 5.3 6. 290 23.6 0.8 .9 10.7 5.5 6. 280 22.9 M .9 10.6 u 6. 270 21.7 0.8 19 10.2 M 6. 260 19.9 0.7 .8 0.6 5.1 5. 250 15.3 o.n 1.4 7.6 4 4.7 230 12.5 0.5 1.2 6.8 S. 3.9 220 50 9.6 0.4 0.9 4.8 2 3.0 210 60 6.5 0.3 0.8 S.S 1. 2.1 200 70 3.3 0.1 0.3 1.7 0. 1.0 190 80 0.0 0.0 0.0 0.0 0. 0.0 180 FIG. 80. Diagram of Curtate Orbits THE MOTIONS OF THE PLANETS 153 On the plan of each orbit the symbol of the planet is placed at the perihelion point, whose position is thus approximately known at a glance. That part of the orbit which is above the plane of the ecliptic is marked with a full line, and the part below is marked by a broken line. The line of nodes is therefore determined as a line joining the two points where the character of the line changes. This line, of course, passes through the sun. The inclinations of the orbit planes are shown by the triangles which appear below the diagram, each marked by the symbol of the planet to whose orbit it pertains. A scale of astronomical units is printed at the bottom. The attached tables (see page 174) give the values of the elements of each orbit and certain other quantities which are required in finding the place of the planet in its orbit at a given time. Measurements may be made on the diagram between any two points by laying a strip of paper with its straight edge through the points, and marking the edge of the strip opposite each point. By laying the straight edge along the scale the distance in astro- nomical units is found. Instead of the paper strip a pair of com- passes may be used. The map shows the orbits as they would be seen from the north side of the ecliptic, and the motions of the planets as thus seen are always counter-clockwise about the sun. The plane of the map is that of the ecliptic, and it is so oriented on the paper that hori- zontal lines drawn from left to right would strike the celestial sphere at the vernal equinox. Therefore the direction which on an ordinary terrestrial map would be east on this map is toward longitude zero ; up is toward longitude 90, down toward longitude 270, and the direction of any other line on the map is fixed by determining the angle which it makes with the line drawn to the vernal equinox. Thus, the line in Fig. 81 from E to M makes an angle of 45 with the line SR, and is therefore directed ^toward longitude 45, and EJ is directed toward longitude 260. By draw- ing lines through the sun parallel to EM and EJ, respectively, the longitude may be read off directly on the circle which bounds the diagram. 154 LABORATORY ASTRONOMY 1 _L 1 Z 3 4 5 6 FIG. 81. Direction of a Line fixed by Longitude To find the Elements of an Orbit. The elements of the planetary orbits may be obtained from measurements on the diagram. These elements are as follows : a Semi-axis major of the ellipse or mean distance. e Eccentricity of the ellipse = c/a, where c is distance of focus from center. TT Heliocentric longitude of perihelion. Heliocentric longitude of node. i Inclination of plane of orbit to plane of the ecliptic. THE MOTIONS OF THE PLANETS 155 To find a draw a straight line from the perihelion point of the orbit through the sun to cut the orbit at the aphelion point. This is the line of apsides. Measure the distance from perihelion to aphelion along the line of apsides in astronomical units. This gives the major axis of the ellipse, one-half of which is the value of a. To find c, bisect the major axis and thus fix the center of the ellipse. The distance from focus to center may then be measured in astronomical units. This is the value of c; it is not regarded as one of the elements, since it is fixed by the values a and e. To find e, determine c/a from the above measurements. To find TT, prolong the line of apsides through the perihelion point ; the reading at the point where it cuts the graduated circle is the longitude of perihelion. To find Q, , prolong the line of nodes through the point where the planet moving counter-clockwise passes from the dotted por- tion of the orbit to the full line. The reading at the point where this line cuts the graduated circle is the longitude of the ascending node. To find the inclination i, measure the angle of the proper triangle by a protractor ; or, more accurately, measure the altitude h and the base b of the triangle ; h/b is equal to the inclination in radians. 57.3 h/b = i in degrees. The following measurements were made on the orbit of Jupiter : Sun to perihelion ... ............. 4.96 Sun to aphelion ..... .... ....... 5.42 Major axis ...... ........ ..... 10.38 a Semi-axis a ................ 5.19 Center to perihelion ............. 5.19 Focus to perihelion ............. 4.96 c Center to focus c .... ...... .... 0.23 . a 5.19 TT The line of apsides cuts the circle at 12. 7. Q The line to ascending node cuts the circle at 99.4. i The altitude of the triangle is 0.13 and the base 5.43; hence i = h/b = 0.13/5.43 = 0.024 radian = 57.3 x 0.024 = 1.37. 156 LABORATORY ASTRONOMY PLACE OF THE PLANET IN ITS ORBIT If the heliocentric longitude of a planet is known, it may be plotted at its proper place on the diagram by drawing a line from the sun to that division of the graduated circle which indicates the given longitude ; the intersection of this line with the orbit gives the required place. When, for instance, the heliocentric longitude of Jupiter is 280, the intersection falls very close to the descend- ing node. In this particular case the place of the planet is com- pletely known, since it is in the ecliptic. Usually the planet is FIG. 82. The Z Coordinate many millions of miles from the ecliptic, but its exact distance may be easily found by the use of its inclination triangle. This will appear by consideration of Fig. 82, which represents a diagram in which the orbit of Jupiter has been cut through along the heavy line, and the part of the orbit which is above the ecliptic turned up around the line of nodes so as to be at the proper incli- nation. The exact angle is insured by supporting it by wedges having the proper angle. THE MOTIONS OF THE PLANETS 157 The height of the planet at P above the plane of the ecliptic, which we shall call its " Z coordinate," or simply Z, is evidently the altitude of a right-angled triangle whose small angle is i (the inclination of the orbit), and whose base is the line drawn from the place of the planet on the diagram to the line of nodes. This line (which practically equals the hypotenuse) we will call U. To find Z, then, it is sufficient to measure U on the diagram and to lay off the same distance along the horizontal side of the incli- nation triangle. The vertical line drawn to the hypotenuse from the point thus fixed gives the length of Z in astronomical units. A far more accurate method is to make use of the obvious relation Z/U = i in radians, or 57.3 Z/U = i in degrees. Thus, for Jupiter Z = U x 1.3/57.3 = 0.023 U. TO FIND THE TRUE HELIOCENTRIC LONGITUDE OF A PLANET To find the true position of any planet at a given time we must first know its mean anomaly at that time, and then, by applying the equation of center, find the correspond- ing value of the true anomaly which enables us to place the planet Eo at the proper position in its orbit. Thus, if the earth's mean anomaly is 70, we find by the table, page 174, that the equa- tion of center is + 1.8, and hence the true anomaly is 71.8. Since the longitude of perihelion is 101.2, the true heliocentric longitude is 101.2 + 71.8, or 173.0, and this value enables us to plot the earth in its proper place on the diagram, Fig. 83. 158 LABORATORY ASTRONOMY We may find the mean anomaly if we know the number of days elapsed since perihelion, and the mean daily motion along the orbit. The fact that the planets move very nearly in the ecliptic, so that the motion in the real and curtate orbit is very nearly the same, makes it easier to proceed in a somewhat different manner, as follows : In the Table of Elements appended to the chart is given the "mean daily motion w (in heliocentric longitude), which is found by divid- ing 360 by the period in days. This quantity enables us by a simple multiplication to find the mean motion, or increase in heliocentric longitude of the mean planet in any given number of days. Knowing the mean (heliocentric) longitude at any given epoch, the mean longitude at any later date is found by addition of the mean motion in the elapsed time. The Table of Elements supplies the necessary " longitude at the epoch " for Greenwich mean noon, January 1, 1900. We may summarize the process of finding the planet's true helio- centric longitude as follows : Let E be the longitude at the epoch, t " " elapsed time in days, /u, " " mean daily motion, TT " " longitude of perihelion, M " " mean anomaly, v " " true anomaly, / " " true longitude (heliocentric). First find the mean anomaly at the time t, as follows : pi = Mean motion in elapsed time, E -f- fA.t = Mean longitude at given date, E + fjit TT = Mean anomaly. With this value of the mean anomaly find the equation of center by the table, and since True anomaly = Mean anomaly -f Equation of center, or v = E + pi TT + Equation of center, and True longitude = v -f- TT, we have directly True longitude = E + ^t + Equation of center. The form of the computation is shown in the solution of the following problem : THE MOTIONS OF THE PLANETS 159 Find the true place of Mars and the earth May 8, 1905, at Greenwich, midnight. The elapsed time may be found as follows : Gr. Mean Noon. Jan. 1, 1900, to Jan. 1, 1901 365 days 1902 365 1903 365 1904 365 1905 366 Jan. 1, 1905, to Feb. 1, 1905 31 Mar. 1, 1905 28 Apr. 1, 1905 31 May 1, 1905 30 Noon. May 1 to Midn., May 8, 1905 7.5 Elapsed time = 1953.5 days. For Mars td = 0. 52403 x 1953.5 = 1023. 69. For the earth & = 0.98561 x 1953.5 = 1925.39. MARS EARTH Mean longitude Jan. 1, 1900 = E 294. 27 100. 67 Mean motion 1953.5 days = nt 1023 .69 1925 .39 E + fit 1317 .96 2026 .06 Subtract complete revolutions ...... 1080 . 1800 . Mean longitude May 8.5, 1905 ...... 237 .96 226 .06 Subtract longitude of perihelion ir [~ 334.2 101.2 ~| Mean anomaly M |_ 263.8 124.9 J Equation of center . . . . . . . . . . - 10.36 + 1.50 True longitude / = E + id + Equation of center 227.60 227.56 It will be noted that in each case the value of E + ^t has been diminished by an integral number of revolutions : 3 x 360 for Mars and '5 x 360 for the earth. It appears, also, that the num- bers inclosed in brackets enter the computation only for the purpose of obtaining the equation of center which is then applied directly to the mean longitude following the equation I = E + fjit + equation of center. On plotting the planets it appears that Mars is exactly opposite the sun, as indeed is evident from the fact that the earth and Mars are in the same heliocentric longitude. The Ephemeris gives May 8, 8 P.M., G.M.T., as the time of opposition. The actual distance between Mars and the earth, as measured on the diagram, is 0.56 astronomical units, or fifty-two million miles. 160 LABORATORY ASTRONOMY The planet may be plotted with a very fine-pointed, hard pencil, against the edge of a ruler passing through the sun and the point of the graduated circle whose reading equals the planet's true helio- centric longitude. It is quite an advantage to have the ruler of a transparent substance in order that its edge may be correctly placed on the graduations. A better method, however, is to put a pin through the sun's place firmly into the drawing board or table, and pass around the pin a long loop of smooth black thread. The other end of the loop is FIG. 84. Plotting with a Loop held between the thumb and forefinger, with the threads slightly separated (about ^ of an inch). The loop is then drawn taut, and the middle of the white space between the threads may be bisected by the proper point on the graduation ; the place of the planet is then marked by putting the point of the pencil exactly midway between the threads where they intersect the orbit (Fig. 84). The planet having been placed in its true position on the orbit by plotting it as above, so that its curtate radius vector is drawn toward the true heliocentric longitude, its place is completely known if we measure U and find Z, as on page 157. The usual method of fixing the distance of the planet from the ecliptic is to give its heliocentric latitude b, or angular distance from the ecliptic, which may be found thus (Fig. 85) : THE MOTIONS OF THE PLANETS 161 b = Z/r = angular distance (radians) of planet above ecliptic as seen from the sun. Combining this with Z = U x i (radians), as explained on page 157, b (radians) = i (radians) ; and turning each side of the equation into degrees by multiplying by 57.3, we have (57.36)=- x (57.3 i), or The inclinations are so small that the latitude is always well determined by this method. FIG. 85. Heliocentric Latitude GEOCENTRIC POSITIONS When a planet has been placed on the diagram by its heliocentric coordinates, we may find its position as seen from the earth ; that is, we may find the longitude and latitude of that point of the celestial sphere upon which it is seen projected by an observer upon the earth. The line drawn from the earth to the planet is called the "line of sight," and its projection on the ecliptic is the line from the earth to the planet on the diagram. If this line is horizontal, it cuts the celestial sphere at the vernal equinox, and the planet's geocen- tric longitude is zero. Geocentric Longitude. The angle between the (projected) line of sight and the line drawn to the vernal equinox is the planet's geo- centric longitude. It is equal to the angle between the line of sight and the line drawn from the sun to the zero of the graduated circle. This angle may be measured in several ways : 1. By prolonging the line of sight, if necessary, till it cuts the line of equinoxes on the diagram, and measuring the angle with a protractor. 162 LABORATORY ASTRONOMY 2. By drawing a line through the sun parallel to the line of sight, and noting the point where it cuts the graduated circle. 3. The most accurate method is usually the following: Bring a straight edge to pass accurately through the places of earth and planet. Note the points of intersection with the graduated circle. FIG. 86. Geocentric Longitude Call the reading where the line of sight (from earth to planet) cuts the circle A, and the other (opposite) reading B. Then the geocentric longitude of the planet is A+B 90, if A is less than B ; and \- 90, if A is greater than B. This may be proved by 2 the theorem that the angle between two chords of a circle is meas- ured by the half sum or half difference of the included angles, according as they intersect inside or outside the circle. Better than a straight edge is a fine line on a transparent ruler (celluloid, glass, mica, tracing cloth), or a stretched thread laid over the two points. Fig. 86 illustrates the three methods, the heliocentric longitudes of the earth and Venus being 150 and 90, respectively. The angle at C measured by the protractor is 13, the line through S parallel to AB cuts the graduated circle at 13.0, while the readings at A and B are 20.0 and 186.0, so that - 90 = 13.0. The Sun's Longitude and the Equation of Time. It is an important fact that, since the line of sight to the sun is drawn to a point THE MOTIONS OF THE PLANETS 163 whose heliocentric longitude is opposite to that of the earth, the sun's geocentric longitude is always 180 + the earth's heliocentric longitude. The sun appears to move about the earth in an orbit whose ele- ments are the same as those of the earth about the sun, except that E and TT are each greater by 180. The sun's mean longitude is therefore 280.67 -f- fit and its mean anomaly is 280.67 -f /* 281.2, where t is. the number of days since January 1, 1900, and /A is the earth's mean daily motion. To find the sun's true longitude we add to the mean longitude the equation of center taken from the table for the earth, and from the true longitude we may find the R.A. by adding the reduction to the equator (page 121). We may therefore write : Sun's R.A. = Sun's mean longitude + Eq. center -f- Red. to equator. Sun's R.A. Sun's mean longitude = Eq. center + Red. to equator. And since the sun's mean longitude equals the R.A. of the mean sun (page 92), Sun's R.A. R.A. of mean sun = Eq. center -f- Red. to equator. The first member of the last equation is the equation of time , whose approximate value may thus be computed for any date : Jan. 31, 1900. ^t = 30 x 0.9S56 = 29.57 E + pt = 280.67 + 29.57 = 310 .24 E + iLt-Tr = 310.24 - 281.2 = 29 .04 Equation of center = + .97 + 0.97 True longitude 311.21 Red. to equator + 2 .4 Equation of time + 3.37 or 13.5 minutes to be added to apparent time. Geocentric Latitude. The geocentric latitude ft of the planet is the angular distance of the planet from the ecliptic as seen from the earth. It is found by the same method as that used for finding the heliocentric latitude b. (See Fig. 87.) Draw the line A from earth to planet on the diagram. Z/A equals the angle (3 in radians, and Z U x i (in radians). 164 LABORATORY ASTRONOMY Hence, by reasoning applied on page 161, FlG. 87. Geocentric Latitude The whole process of finding geocentric latitude and longitude is illustrated in the following example : To find the positions of the five inner planets at Greenwich mean noon, July 6, 1907, the elapsed time from January 1, 1900, being 2742 days (see page 167). $ 9 d 1 3 E 182.22 344.33 100.67 294 27 238.13 Add fit ..'.... 11221 .20 4393 .04 2702 .54 1436 .89 227.83 E 4- id 11403 42 4737 37 2803 21 1731 16 465 96 Subtr. complete revolutions 11160 . 4680 . 2520 . 1440 . 360. E -f fit 243 .42 57 .37 283 .21 291 .16 105.96 Subtract TT 75 .9 130 .2 101 .2 334 .2 12.7 Mean anomaly .... Eq. of center by table Heliocentric longitude I . 167 .5 + 4 .10 247 .52 287 .2 -0 .73 56 .64 182 .0 -0 .06 283 .15 317 .0 -7 .89 283 .27 93.3 + 5.47 111.43 Plotting the planets on the diagram, we determine the geocentric places by finding the following values : THE MOTIONS OF THE PLANETS 165 9 * i From diagram, A . . . . 130. 4 80. 1 283. 2 111 2 u u B 302 3 266 9 103. 3 288 8 u U 0.14 0.22 1.16 1.09 u From u table A i 0.70 70 1.60 3. 4 0.40 1 9 6.26 1 3 A + J 3 90 126 35 83 5 283 25 110 2 U X I 8 . - 0.47 - 5. 51 + 0.21 A FIG. 88. Geocentric Places, July 6, 1907 166 LABORATORY ASTRONOMY The signs attached to the latitudes are fixed by the fact that Jupiter is in the full-line part of its orbit and therefore above the ecliptic, while all the other planets are in the dotted parts of their orbits and therefore in south latitudes. Since the full line extends from the longitude of the ascending node to that of the descending node, which is 180 greater, we may also fix the sign of /3 by the following rule : From the true heliocentric longitude subtract that of the ascend- ing node ; if I Q> < 180, the latitude is positive ; if I & > 180, the latitude is negative. Thus, in the above example : $ 9 cf y 1 247. 5 56. 6 283. 3 111.4 Q 47 .1 75.7 48.7 100.1 I _ o 200 .4 340 .9 234 .6 11 .3 B . nesr. near. nesr. pos. Perturbations. The longitudes above obtained are liable to an error of more than a tenth of a degree if the elapsed time exceeds a half century, and the perturbations which are neglected may add somewhat to the error. The effect of the mutual perturbations of Jupiter and Saturn may be approximately corrected by adding to the mean longitudes the following quantities : 3 h h 1800-1890 + 0.3 1800-1840 -0.8 1940-1960 -0.4 1890-1950 + .2 1840-1870 -0 .7 1960-1980 -0 .3 1950-1990 + .1 1870-1910 -0 .6 1980-1990 -0 .2 1990-2000 .0 1910-1940 -0.5 1990-2000 -0 .1 Effect of Precession. The true longitudes found by the method above described are referred to the equinox of 1900, the point from which the mean longitude of the table is measured. Since the vernal equinox moves along the ecliptic 50" per year toward the west, or nearly 6' in seven years, the longitudes meas- ured from the true equinox of 1907 will be about 0.l greater than THE MOTIONS OF THE PLANETS 167 if measured from the equinox of 1900. This "reduction to the equinox of date" is 50" x t, or 0.014 t, where t is the number of years elapsed since 1900. The Julian Day. The process of computing the elapsed time used on page 159 is tedious and liable to error where the elapsed time is considerable. Where the interval between distant dates is to be accurately determined astronomers find it convenient to make use of the number of each day in the Julian period. It is sufficient here to say that January 1, 4713 B.C., was the first day of this period, and the Ephemeris gives each year the number of the Julian day for January 1 ; thus, the 1st of January, 1900, was No. 2415021 in the cycle. To find the number for any given date, we turn to page III of the corresponding month, add the day of the year (taken from the second column), and subtract 1. The table on page 175 gives for each year from 1800 to 2000 a number one less than that of the Julian day corresponding to Jan- uary 1 of the given year. The subsidiary table for months gives for each month a number one less than the day of the year cor- responding to the first of the given month. It is easy to see that by adding together the year number, month number, and day of the month, we get the corresponding Julian day. Thus we compute the interval from January 1, 1900, to July 6, 1907, as follows : Year number for 1900, 2415020 1907 2417576 Month number for January, _ July 181 Day of month, 1 6 6 Julian day, 2415021 2417763 2415021 Elapsed time, 2742 Right Ascensions and Declinations of the Planets. By means of the geocentric latitudes and longitudes which we have thus determined the planets may be placed in their respective positions upon the globe. The proper longitude being found upon the ecliptic of the globe, the latitude is laid off on a strip of paper by placing it along the ecliptic and marking off the proper number of degrees along its edge. The paper is then applied to the globe so as to mark off this distance perpendicular to the ecliptic. The latitude is never so 168 LABORATORY ASTRONOMY great as 8, so that no serious error in the place will occur if the strip is not exactly perpendicular to the ecliptic. The place of the planet being thus marked on the globe, its right ascension and declination may be determined, and problems relat- ing to its diurnal motion, such as its times of rising and setting, may be solved by the methods of Chapters VIII and IX. CONFIGURATIONS OF THE PLANETS The elongation of a planet is its distance from the sun along the ecliptic as seen from the earth. It is therefore equal to the differ- ence of the geocentric longitudes of the sun and planet. The elonga- tion is measured either way from the sun up to 180, at which point the planet is at opposition, or opposite the sun. When the elongation is zero the sun and planet are in the same longitude, and the planet is in conjunction with the sun. The symbols and <$ are used for opposition and conjunction, respectively. When the longitude of the planet is greater than that of the sun it is east of the latter, and follows it in its diurnal revolution. It is therefore above the horizon at sunset and is said to be an "evening star,' 7 since it is visible in the twilight after sunset except when near conjunction. On the other hand, all planets whose longitudes are less than that of the sun precede it, and they will be above the horizon at sunrise and therefore visible at dawn, except when very near conjunction. They are then " morn- ing stars," just as stars in eastern elongation are evening stars. The geocentric longitude of the sun, July 6, 1907, is 103.2 (since the earth's heliocentric longitude is 283. 2, page 164). The longi- tude of Jupiter being 110. 2, its elongation is about 7 east, and it is an evening star, though too close to the sun to be visible ; it will become a morning star about July 14. The elongation of Mars is very nearly 180, and it is at opposi- tion and becoming an evening star. The longitude of Venus is 84.6 ; it is 18.6 west of the sun and is a morning star. On referring to the diagram (Fig. 88), and remembering that it moves more rapidly than the earth, it is evident that it is approaching conjunction beyond the sun (" superior" conjunction), after which it will pass THE MOTIONS OF THE PLANETS 169 to eastern elongations and be an evening star. Mercury's longitude is 126 ; it is 23 east of the sun, and referring to the diagram, we see that it is approaching conjunction between the earth and sun (" inferior " conjunction), after which it will be a morning star. The preceding principles enable us to find the place of a planet at any given date, and thus to answer many of the questions which continually suggest themselves to one interested in watching the courses of the planets in the sky. It is evident, for instance, from the problems solved on pages 159 and 164, that in 1907 the greater proximity of Mars to the earth offers conditions for the study of its surface which are much more favorable than those of the opposition of 1905. The oppositions of Mars recur at an average interval of about 780 days, which is the synodic period of the earth and Mars, as explained in the text-books of descriptive astronomy. We may fix the dates of other oppositions approximately, as in August, 1877, September, 1909, November, 1911, December, 1913, etc., and by computing for the first and last days of those months a closer approximation to the day of opposition may quickly be made, and finally a careful computation for the exact date will fix the time within a few hours. The geocentric place and the distance of the planet may then be found. It appears that favorable oppositions occur in the summer, and that the planet is then quite a distance south of the equator, so that it is far from the zenith of any northern observatory. The satellites of Mars were discovered in 1877, and in the same year an expedition was sent to the island of Ascension to observe Mars for a determination of the solar parallax. In conclusion we will consider the motion of Mars during the summer of 1907, to illustrate the form which, the computation takes when many places are to be found at comparatively short intervals. We first carefully determine the mean longitudes of Mars and the earth for March 22 to be 235.51 and 178.74, respectively, and then easily form the second column of the following schedule by successive additions of 10.48 and 19.71, the mean motions of the two planets in twenty days. 170 LABORATORY ASTRONOMY The third column is formed for Mars by writing the longitude of perihelion 334. 2 on the upper edge of a slip of paper and placing it under the numbers of the second column successively, subtracting from each to find the corresponding mean anomaly. The same result is more easily obtained by adding in the same way 25.8 (360-334.2) to each number in the second column. The third column is checked by noting that the differences of the successive values are 10.48, which insures the accuracy of both columns. The equation of center is taken from the table and entered in the fourth column, and the true heliocentric longitude found by adding corresponding numbers of the second and fourth columns. The same process gives the earth's true heliocentric longitude. The labor is by no means proportionate to that required in computing a single place, and the comparison of the successive numbers of each column is an important aid in detecting errors. MARS THE EARTH Date E + fj.t E + u.t-n Eq. of Center I E + nt E+H'-TT Eq. of Center I Mar. 22 235. 51 261.3 - 10.3 225. 2 178. 74 77.5 + 1.9 180. 6 April 11 245 .99 271 .8 -10 .6 235 .4 198 .45 97 .2 + 1 .9 200 .3 May 1 256 .47 282 .3 -10 .6 245 .9 218 .16 117 .0 + 1 -7 219 .9 21 266 .95 292 .8 -10 .3 256 .6 237 .87 136 .6 + 1.3 239 .2 June 10 277 .43 303 .2 -9.5 267 .9 257 .58 156 .4 + 0.7 258 .3 30 287 .91 313 .7 -8.3 279 .6 277 .29 176 .1 + 0.1 277 .4 July 20 298 .39 324 .2 - 6 .8 291 .6 297 .00 195 .8 - .5 296 .5 Aug. 9 308 .87 334 .7 - 5 .0 303 .9 316 .71 215 .5 -1 .1 315 .6 29 319 .35 345 .1 - 3 .1 316 .3 336 .42 235 .2 -1 .5 334 .9 Sept. 18 329 .83 355 .6 -0.9 328 .9 356 .13 254 .9 -1.8 354 .3 Oct. 7 340 .31 6 .1 + 1.3 311 .6 15 .84 274 .6 -1 .9 13 .9 The planets were plotted from the above data on a scale of 1.6 inches to the astronomical unit, the boundary circle being 9| inches in diameter. The values of A, B, U, and A were determined and the geocentric longitudes and latitudes A and ft found as in the following table : THE MOTIONS OF THE PLANETS 171 A B U A A. /3 March 22 . . 244. 6 103. 7 + 0.13 1.10 264. 1 + 0.2 April 11 . . 254 .8 112 .6 -0.16 0.91 273 .7 -0 .3 May 1 . . 263 .9 119 .2 0.44 0.75 281 .5 - 1 .2 21 . . 271 .4 120 .9 0.69 0.60 286 .1 -2 .2 June 10 . 277 .8 117 .1 0.90 0.50 287.4 -3 .4 30 . . 282 .2 108 .0 1.10 0.44 285 .0 -4 .8 July 20 . . 285 .7 94.3 1.25 0.42 280 .0 -5 .7 Aug. 9 . . 289 .7 84 .9 1.36 0.47 277 .3 -5 .5 29 . . 296 .8 84.1 1.39 0.54 280.4 -4 .9 Sept. 18 . . 305 .0 88.2 1.36 0.64 286 .6 -4 .0 Oct. 7 . . 316 .6 98 .4 1.27 0.76 297 .5 -3 .2 310 300 290 280 270 260 250 240 FIG. 89. Path of Mars in the Summer of 1907 172 LABORATORY ASTRONOMY In order to form an idea of the path described by the planet among the stars, the positions may be plotted on an ecliptic map, as in Fig. 89, which shows the form of the loop in the constellation of Sagittarius. During March the motion of the planet is eastward, or in the direction of increasing longitudes, and is said to be " direct." The rate of motion diminishes from one-half degree per day at the out- set to half that amount in May, and soon after the beginning of June the planet reaches its first " stationary point " and begins to move slowly in the opposite direction in longitude, or "retrograde." Its continuous motion in latitude toward the south prevents it from exactly retracing its path and causes it to describe a "loop." Its velocity in the retrograde arc increases to a maximum of a quarter of a degree per day at opposition early in July, and then decreases until the second stationary point is reached about August 9, when the planet resumes its direct motion. The exact dates of the stationary points may be found by com- puting a few places in the neighborhood of June 10 and August 9. The Ephemeris gives the dates as June 5 and August 8. THE MOTIONS OF THE PLANETS 173 TABLE III AVERAGE VALUES OF THE SUN'S LONGITUDE AND THE EQUATION OF TIME LONGITUDE LONGITUDE MEAN LONGITUDE EQ. OF TIME Jan. 1 . . . 280. 3 >J 10. 3 280. 1 + 3">. 5 11 . . . 290 .5 >? 20 .5 289 .9 + 7 .9 21 ... 300 .7 XX 30 .7 299 .8 + 11 .3 31 ... 310 .8 SZS 10.8 309 .6 + 13 .6 Feb. 10 ... 320 .9 SXH 20 .9 319.5 + 14 .4 20 ... 331 .1 X 1 .1 329 .4 + 14 .0 Mar. 2 ... 341 .3 X 11.3 339 .5 + 12 .4 12 ... 351 .4 X 21 .4 349 .3 + 10 .0 22 . 1 .3 f> 1 .3 359 .2 + 7 .1 April 1 ... 11 .2 | 11 .2 9 .0 + 4 .1 11 . 21 .0 P 21 .0 18.9 + 1 .2 21 . 30 .8 8 0.8 28 .7 - 1 .2 May 1 ... 40 .6 8 10.6 38 .6 - 2 .9 11 ... 50.3 8 20.3 48 .5 - 3 .7 21 ... 59 .9 8 29.9 58 .3 - 3 .6 31 . . . 69 .5 n 9 .5 68 .2 - 2 .6 June 10 ... 79 .0 n 19 .0 78 .0 - .9 20 ... 88 .6 n 28.6 87 .9 + 1 .2 30 ... 98 .1 EB 8 .1 97 .7 + 3 .3 July 10 . . . 107 .7 ZB 17 .7 107 .6 + 5 .0 20 ... 117 .2 5 27 .2 117 .5 + 6 .1 30 ... 126 .7 1 6.7 127 .3 + 6 .2 Aug. 9 ... 136 .3 1 16 .3 137 .2 + 5 .4 19 ... 145.9 1 25.9 147 .0 + 3 .6 29 ... 155 .6 TTJ? 5 .6 156 .9 + .9 Sept. 8 ... 165 .3 H 15 .3 166 .7 - 2 .3 18 . 175 .0 -n^ 25 .0 176 .6 - 5 .8 28 ... 184 .8 4 .8 186 .5 - 9 .2 Oct. 8 ... 194 .6 :: 14 .6 196 .3 -12 .3 18 . . 204 .5 24.5 206 .2 -14 .7 28 ... 214 .5 Til 4 .5 216 .0 -16 .1 Nov. 7 ... 224 .5 TH. 14 .5 225 .9 -16 .2 17 . . 234 .6 m. 24 .6 235 .7 - 15 .0 27 ... 244 .7 t 4.7 245 .6 -12 .4 Dec. 7 . . . 254 .8 t 14 .8 255 .4 - 8 .5 17 ... 265 .0 t 25 .0 265 .3 - 3 .9 27 ... 275 .2 >? 5.2 275 .2 + 1 .0 Jan. 6 . 285 .4 V? 15.4 285 .0 + 5 .7 174 LABORATORY ASTRONOMY IP CO >O CO CO co co O (M rH co co o o CO i - CO TH cc T-H O o o o o o o o o O O O o o o O TH iO O5 IO rH CO CO CO O5 t-~ O5 CO CO iO CO iO CO O x> S rH S > B Noon. Midnight. Noon. Diff. for i Hour Midnight. Diff. for i Hour. Meridian of Greenwich. Diff. for i Hour. Noon. i , ., . , b ra m d I 16 19.6 16 23.4 59 48-5 +1-33 60 2.8 + 1.05 6 29-5 2 16 26.4 16 28.3 60 13.6 0.75 60 2O.7 +0.44 o 57-9 2.43 O.g 3 16 29.2 16 29.2 60 24.1 +0.13 60 23.8 -0.17 i 55-o 2-33 1.9 4 16 28.1 16 26.2 60 20.0 -0.45 60 13.0 -0.70 2 49.6 2.22 2.9 5 16 23.6 16 20.2 60 3.2 0.92 59 50-9 I.IO 3 4*-9 2.14 3-9 6 16 16.3 16 12.0 59 36.7 1.25 59 20.9 1.36 4 32.8 2.IO 4-9 7 16 7.4 16 2.6 59 4-o -1.44 58 46-4 -1.48 5 23.1 2.10 5-9 8 *5 57-7 15 52-8 58 28.4 1.50 58 10.2 1.50 6 13.9 2.1 3 6.9 9 15 47-9 15 43-i 57 52-2 1.49 57 34-5 1.46 7 5-6 2.18 7-9 10 15 38.4 15 33-8 57 17-2 -1.43 57 0.3 -1.38 7 58-5 2.22 8.9 ii 15 2 9-3 15 25.0 56 44.0 J-34 56 28.2 1.29 8 52.2 2.24 9.9 12 15 20.9 15 16.9 56 13-0 1.24 55 58.4 1.19 9 45-8 2.22 10.9 13 15 i3-i 15 9-5 55 44-4 -1.14 55 3i-i -1.09 10 38.4 2.16 11.9 H 15 6.0 J5 2.7 55 18.3 1.03 55 6.2 0.98 ii 29.1 2.06 12.9 15 H 59-6 H 56.7 54 54-8 0.92 54 44- 2 0.85 12 17.2 1.95 13-9 16 H 54- 1 H 5i-7 54 34-5 -0.77 54 25-7 -0.68 13 2.7 1.8 5 14.9 *7 14 49.6 14 47.9 54 '8.1 -58 54 u-8 0.47 13 46.1 1.76 15-9 18 14 46.5 H 45-6 54 6.9 0.34 54 3-5 -O.2I 14 27.8 I.7I 16.9 19 14 45.2 14 45.2 54 i-9 0.06 54 2.1 +0.10 15 8. 5 1.69 17.9 20 H 45-9 14 47.1 54 4-4 +0.28 54 8.8 0.46 15 49-3 I. 7 I 18.9 21 14 48.9 H 5i-4 54 15-5 0.65 54 24.6 0.85 16 30.9 1.76 19.9 22 H 54-5 H 58.3 54 36-1 +1.06 54 50-1 + 1.27 17 14-3 1.86 20.9 23 15 2.8 15 7-9 55 6.5 1.48 55 25.4 1.6 7 1 8 0.4 1.99 21.9 24 15 13-7 15 20.0 55 46-5 1.85 56 9-9 2. 02 18 49.8 2.14 22.9 25 15 26.9 15 34-2 56 35-i +2.17 57 i-9 +2.29 19 43.0 2.29 23-9 26 15 4i-8 15 49-7 57 30-0 2.37 57 58.8 2.41 20 39-5 2.42 24.9 27 15 57-6 16 5.4 58 27.8 2.40 58 56-4 2.34 21 38.5 2.49 25.9 28 16 12.9 16 19.9 59 23-9 +2.22 59 49-6 +2.04 22 38.3 2.48 26.9 29 16 26.2 16 31.6 60 12.8 I.8l 60 32.8 1-52 23 37-3 2-43 27.9 30 16 36.0 16 39.3 60 49.0 I.lS 61 0.9 +0.80 6 28.9 31 16 41.3 16 41.9 61 8.2 +0.40 61 10.6 o.oo o 34.5 2.34 0.4 32 16 41-3 16 39-4 61 8.3 -0.39 61 1.3 -0.75 I 29.8 2.26 1.4 179 JANUARY, 1900. VII GREENWICH MEAN TIME. THE MOON'S RIGHT ASCENSION AND DECLINATION. Hour. Right Ascension. Diff for i Minute. Declination. Diff. for i Minute. Hour. Right Ascension. Diff. for i Minute. Declination. Diff. for I Minute. TUESDAY 9. THURSDAY n. h m 2 5 1.09 2.2544 N.l6 38 10.6 9.442 o h m 8 3 55 9-21 2.3237 N.22 4 39.6 3*944 i 2 7 16.41 8.2563 16 47 34.2 9-343 i 3 57 28.65 2.3242 22 8 32.5 3.818 2 2 9 3I-85 2.258? 16 56 51.8 9.243 2 3 59 48-12 2.3247 22 12 17.8 3*692 3 2 ii 47-39 2.2600 17 6 3.4 9.142 3 4 2 7-62 2.3252 22 15 55-5 3.566 4 2 14 3.05 2.2619 17 15 8.9 9.040 4 4 4 27.14 2.3254 22 19 25.8 3.440 5 2 16 18.82 2.2637 17 24 8.2 8.937 5 4. 6 46.67 2.3257 22 22 48.4 3.3I3 6 2 18 34.70 2.2656 17 33 1-4 8.835 6 9 6.22 2.3260 22 26 3.3 3.186 7 2 2O 50.69 2.2675 17 41 48.4 8.731 7 ii 25.79 2.3262 22 29 IO.7 3.059 8 2 23 6.8O 2.2693 17 50 29.1 8.627 8 13 45-36 2.3262 22 32 10.4 2.932 9 2 25 23.01 2.2712 17 59 3-6 8.521 9 16 4.93 2.3262 22 35 2.5 2.805 10 2 27 39.34 2.2730 18 7 31.6 8.414 10 18 24.50 2.3262 22 37 47.0 8.677 ii 2 29 55.77 2.2748 '8 15 53-3 8.307 ii 4 20 44.07 2.3261 22 40 23.8 2-549 12 2 32 12.32 2.2767 18 24 8.5 8.199 12 4 23 3-63 2.3259 22 42 52.9 2.422 *3 2 34 28.98 2.2785 18 32 17.2 8.091 13 4 25 23.18 2.3257 22 45 14.4 2.295 4 2 36 45-74 2.2802 18 40 19.4 7.982 M 4 27 42.71 2.3254 22 47 28.3 2.167 5 2 39 2.61 2.2820 18 48 15.1 7.873 15 4 30 2.23 2.3251 22 49 34-5 2.039 16 2 41 19.58 2.2837 18 56 4.2 7.76a 16 4 32 21.72 2.3246 22 51 33-o I.9 7 2 43 36.66 2.2856 19 3 46-6 7.651 17 4 34 41.18 2.3241 22 53 23.9 1.784 18 2 45 53.85 2.2873 19 ii 22.4 7-539 18 4 37 0.61 2.3235 22 55 7-1 1.657 9 2 48 11.14 2.2890 19 18 51-4 7-427 19 4 39 20.00 2.3229 22 56 42.7 1.529 20 2 5 28.53 2.2907 19 26 13.7 7.315 20 4 41 39.36 2.3222 22 58 10.6 1.401 21 2 52 46.02 2.2923 19 33 29.2 7.201 21 4 43 58.67 2.32:4 22 59 30.8 1.273 22 2 55 3-6: 2.2940 19 40 37.8 7.086 22 4 46 17-93 2.3207 23 43-4 1. 147 23 2 57 21.30 2.2956 N.I9 47 39-5 6.971 23 4 48 37-15 2.3197 N.23 i 48.4 1.019 WEDNESDAY 10. FRIDAY 12. 2 59 39.08 2.2972 N.I9 54 34-3 6.856 4 50 56-30 2.3187 N.23 2 45-7 0.892 i 3 i 56-96 2.2987 20 I 22.2 6.741 I 4 53 I5-40 2.3177 23 3 35-4 0.764 2 3 4 14-93 2.3003 20 8 3.2 6.624 2 4 55 34-43 2.3166 23 4 17-4 0.637 3 3 6 33.00 2,3018 20 I 4 37.1 6.507 3 4 57 53-39 s-3'54 23 4 51-9 0.511 4 3 8 5i-i5 2.3032 2O 21 4.0 6.389 4 5 o 12.28 -3Ma 23 5 18.7 0.384 5 3 'I 9-39 2.3047 2O 27 23.8 6.271 5 5 2 31.09 2.3128 23 5 38.o 0.258 6 3 13 27.72 2.3061 20 33 36-5 6.152 6 5 4 49-82 2.3115 23 5 49-7 0.132 7 3 15 46-12 2.3074 20 39 42.1 6.033 7 5 7 8.47 2.3102 23 5 53-8 + 0.006 8 3 18 4.61 2.3087 20 45 40.5 5.913 8 5 9 27.04 2.3087 23 5 50-4 - 0.120 9 3 20 23.17 2.3100 20 51 31.7 5-793 9 5 " 45-51 2.3070 23 5 39-4 0.246 10 3 22 41.81 2.3113 20 57 15.7 5.672 10 5 M 3-88 .3053 23 5 20.9 0.371 ii 3 25 0.53 2.3125 21 2 52.4 5-55* ii 5 16 22.15 2.3037 23 4 54-9 0.495 12 3 27 19.31 2.3136 21 8 21.9 5.43I 12 5 18 40.32 2.3019 23 4 21.5 0.619 13 3 29 38.16 2.3147 21 13 44.1 5.308 13 5 20 58.38 2.3001 23 3 40.6 0-744 14 3 3i 57.o8 2.3158 21 18 58.9 5.186 14 5 23 16.33 2.2982 23 2 52.2 0.868 15 3 34 16.06 2.3168 21 24 6.4 5.064 15 5 25 34.16 2.2962 23 * 56-4 0.992 16 3 36 35-io 2.3178 21 29 6.6 .941 16 5 27 51.87 2.2942 23 o 53.2 1.115 17 3 38 54-20 2.3187 21 33 59-3 .8.7 17 5 30 9-46 2.2921 22 59 42.6 1.237 18 3 4i 13-35 2.3196 21 38 44.6 .693 18 5 32 26.92 2.2899 22 58 24.7 1.360 19 3 43 32.55 2.3204 21 43 22.5 .569 19 5 34 44.25 8.2877 22 56 59-4 1.482 20 3 45 51.80 2.3212 21 .47 52.9 445 20 5 37 i-44 2.2853 22 55 26.8 1.604 21 3 48 ii. 10 2.3220 21 52 15.9 .320 21 5 39 18.49 2.2830 22 53 46-9 1.725 22 3 50 30-44 2.3226 21 56 31.3 .194 22 5 4i 35.40 2.2807 22 51 59.8 1.845 23 3 52 49.8i 2.3231 22 39.2 4-069 23 5 43 52-17 2.2782 22 50 5-5 1.965 24 3 55 9-21 2.3237 N.22 4 396 3-944 24 5 46 8.79 2.2757 N.22 48 4.0 2.085 180 VI. FEBRUARY, 1900. GREENWICH MEAN TIME. THE MOON'S RIGHT ASCENSION AND DECLINATION. Hour. Right Ascension. Diff.for t Minute. Declination. Diflf. for i Minute. Hour. Right Diff.for i Minute. Declination. Diff.for i Minute. MONDAY 5. WEDNESDAY 7. i m s s O t n h m s 8 O * M * 51 20.62 a. 295* N.is 26 59.8 10.261 o 3 42 30*05 8.3283 N.2I 30 28.1 4-685 i 53 38-36 2.2962 15 37 12.4 10. 159 i 3 44 49-75 2.3283 21 35 5-4 4.558 2 55 56-16 8.2971 15 47 18.9 10.057 2 3 47 9-45 2.3283 21 39 35-1 4*432 3 58 14.01 2.2981 15 57 19-3 9*954 3 3 49 29.15 2.3282 21 43 57-2 4.305 4 o 31-93 a. 2991 16 7 13.4 9.850 4 3 5i 48.84 2.3280 21 48 11.7 4*179 5 2 49.90 2.3001 16 17 1.3 9*745 5 3 54 8.51 3.3378 21 52 18.7 4.052 6 5 7-94 3.3012 16 26 42.8 9.639 6 3 56 28.17 2.3376 21 56 18.0 3.935 7 7 26.04 2.3021 16 36 18.0 9-533 7 3 58 47-82 3.3272 22 O 9.7 3.798 8 9 44.19 2.3030 16 45 46.8 9.427 8 4 * 7-44 2.3268 22 3 53.8 3.672 9 12 2.40 2.3040 16 55 9.2 9.319 9 4 3 27.04 2.3264 22 7 30.3 3.544 10 14 20.67 2.3050 17 4 25.1 9.210 10 4 5 46-61 3.3259 22 10 59.2 3.417 ii 16 39.00 2.3059 17 3 34-4 9.100 ii 4 8 6.15 2.3254 22 14 20.4 3.290 12 18 57.38 2.3069 17 22 37.1 8.990 12 4 10 25.66 2.3248 22 17 34.0 3.163 13 21 15.83 2.3080 17 3i 33-2 8.880 13 4 12 45.13 2.3242 22 20 39.9 3-035 14 23 34-34 3.3089 17 40 22.7 8.769 M 4 15 4-57 2.3236 22 23 38.2 8.908 15 25 52-90 2.3098 17 49 5-5 8.657 5 4 17 23.96 2. 3 228 22 26 28.9 .781 16 28 11.52 2.3108 17 57 4!-5 8.544 16 4 19 43-3- 2.3221 22 29 II.9 S.653 17 30 30.20 2.3117 18 6 10.8 8.431 17 4 22 2.6l 2.3212 22 31 47-3 8.537 18 32 48-93 3.3126 18 14 33.2 8.317 18 4 24 21.86 2.3204 22 34 15.1 8.400 19 35 7.7i 2.3135 18 22 48.8 8.202 19 4 26 41.06 2.3195 22 36 35.3 a.373 20 37 26.55 2.3144 18 30 57-5 8.087 20 4 29 0.20 8.3184 22 3 8 47 .8 8.145 21 39 45-44 2.3153 18 38 59-3 7.972 21 4 31 19.27 8.3173 22 40 52.7 3.018 22 42 4-39 2.3162 18 46 54.1 7.856 22 4 33 38-28 3.3163 22 42 50.O 1.892 2 3 44 23-39 2.3170 N.i8 54 42.1 3-739 23 4 35 57-23 2.3152 N.22 44 39.7 1.765 TUESDAY 6. THURSDAY 8. o 46 42.43 2.3177 tf.ig 2 22.9 7.622 o 4 38 16.10 2-3139 N.22 46 21.8 1.638 i 49 1-52 8.3186 19 9 56.7 7.504 i 4 40 34.90 2.3127 22 47 56.3 1.512 2 51 20.66 3.3194 19 17 23.4 7-386 2 4 42 53-62 2.31I3 22 49 23.2 1.386 3 53 39.85 2.3202 19 24 43.0 7-267 3 4 45 12.26 2.3100 22 50 42.6 1.260 4 55 59-oS 8.3208 19 3i 55-5 7.148 4 4 47 30-82 2.3086 22 51 54-4 1.134 5 58 18.35 2.3215 19 39 0.8 7.028 5 4 49 49-29 2.3071 22 52 58.7 1.008 6 3 o 37.66 2.3222 19 45 58.9 6.908 6 4 52 7-67 3.305J 22 53 55-4 o.88z 7 3 2 57.01 2.3228 19 52 49.8 6.787 7 4 54 25.95 3.3039 22 54 44.6 0-757 8 3 5 16.40 8.3234 *9 59 33-4 6.666 8 4 56 44-14 2.3023 22 55 26.3 0.632 9 3 7 35-82 2.3240 20 6 9.7 6.545 9 4 59 2.23 2.3007 22 56 0.5 0.508 10 3 9 55-28 2.3246 20 12 38.8 6.424 10 5 I 20.22 2.2989 22 56 27.3 0.384 ii 3 12 14.77 8.3251 20 19 0.6 6.303 ii 5 3 38-10 2.2971 22 56 46.6 0.260 12 3 14 34-29 2.3256 20 25 15.0 6.178 12 5 5 55-87 2.2932 22 56 58.5 0.136 13 3 16 53.84 2.3260 20 31 22.0 6.056 13 5 8 I3-52 2.2933 22 57 2.9 + 0.012 14 3 9 i3-4i 2.3264 20 37 21.7 3*933 14 5 10 31.06 3.2913 22 56 59-9 0.1(1 15 3 21 33.01 2.3268 20 43 14.0 5.8o8 15 5 12 48-48 2.2893 22 56 49.6 0.233 16 3 23 52-63 2.3271 20 48 58.8 5-685 16 5 15 5-78 2.2872 22 56 31.9 0.357 17 3 26 12.26 2.3273 20 54 36.2 S.56I J 7 5 17 22.95 2.2851 22 56 6.8 0*479 18 3 28 31.91 2.3277 21 O 6.1 3-436 18 5 19 39-99 2.2829 22 55 34-4 0.601 19 3 30 51-58 8.3279 21 5 28.5 5.3i* 19 5 21 56.90 2.2807 22 54 54-7 0.733 20 3 33 "-26 2.3281 21 10 43.5 3-187 20 5 24 13-67 8.2784 22 54 7-7 0.843 21 3 35 30-95 2.3282 21 15 50.9 5.061 21 5 26 30.31 2.2761 22 53 13-5 0.963 22 3 37 50-65 2.3283 21 20 50.8 4-936 22 5 28 46.80 2.2737 22 52 12. 1 1.084 23 24 3 40 10.35 3 42 30.05 2.3283 2.3283 21 25 43.2 N.2I 30 28.1 4.8u 4.685 23 24 5 3 3-i5 5 33 19.36 8.3713 8.2689 22 5 3-4 N.22 49 47.6 1.204 1.323 181 XVIII. FEBRUARY, 1900. GREENWICH MEAN TIME. LUNAR DISTANCES. ** P Name and Direction of Object Midnight P.L. of Diff. XVk P.L.. of Diff. XVIIIk P.L. of Diff. XXIk P.L, Diff. 18 Pollux W. 86 46 29 3081 88 15 2 3076 89 43 41 3069 91 12 28 3063 Regulus W. 49 49 o 3051 51 18 10 344 52 47 28 3037 54 16 55 3030 Antares E . 50 7 41 3038 48 38 15 3032 47 8 42 3026 45 39 2 3022 JUPITER E . 50 57 29 3062 49 28 33 3056 47 59 30 3050 46 30 19 3044 SATURN E . 74 45 5 1 3050 73 16 4 3044 71 47 22 3038 70 17 57 33i a Aquilae E . 102 48 44 3500 101 28 20 3489 ioo 7 43 3479 98 46 55 3468 19 Pollux W. 98 38 27 3027 100 8 6 3020 K 37 54 301 r I3 7 53 3003 Regulus W. 61 46 34 2989 63 17 i 2979 64 47 40 2969 66 18 31 2960 Antares E 38 8 59 2992 36 38 36 2985 35 8 4 2977 33 37 23 2971 JUPITER E . 39 2 19 3007 37 32 15 998 36 2 O 2989 34 3i 34 2980 SATURN E . 62. 48 39 2994 61 18 19 2985 59 47 48 2976 58 17 5 2967 a Aquilae E 92 o 7 3421 90 38 14 3412 89 16 ii 3403 87 53 58 3395 20 Regulus W. 73 55 57 2906 75 28 8 2894 77 o 34 2882 78 33 16 2870 JUPITER E . 26 56 24 2930 25 24 43 23 52 49 2909 22 2O 41 2898 SATURN E . 50 40 26 2914 49 8 25 2903 47 36 10 2891 4 6 3 39 2878 a Aquilae E . 81 o 35 3356 79 37 28 3350 78 14 14 3343 76 50 52 3337 SUN E. 109 33 56 3275 to8 9 15 3262 106 44 19 3249 105 19 8 3235 21 Regulus W. 86 20 53 2802 87 55 18 2788 89 30 i 2773 9i 5 4 2758 Spica W. 32 18 23 2793 33 53 o 2779 35 27 56 2763 37 3 12 2747 SATURN E . 38 16 56 36 42 42 2797 35 8 10 2782 33 33 18 2766 a Aquilae E . 69 52 25 3313 68 28 29 33" 67 4 30 3309 65 40 29 3307 SUN E. 98 9 o 3162 96 42 5 3U6 95 14 5 1 3130 93 47 18 3114 22 Spica W. 45 4 53 2666 46 42 19 2649 48 20 8 2631 49 5 8 21 2613 SATURN E . 25 33 48 2686 23 56 49 2669 22 19 28 2652 20 41 44 2634 Aquilae E . 58 40 32 3322 57 16 46 3330 55 53 9 334 54 29 44 3352 SUN E. 86 24 25 3028 84 54 47 3009 83 24 45 5989 81 54 19 2971 23 Spica W. 58 15 32 2522 59 56 H 2504 61 37 21 2485 63 18 55 2467 a Aquilae E . 47 37 27 3468 46 16 27 3505 44 56 8 3548 43 36 37 3598 SUN E. 74 16 17 2876 72 43 28 2856 71 10 13 2836 69 36 32 2817 24 Spica W. 7i 53 21 2373 73 37 34 2355 75 22 14 4335 77 7 22 2317 JUPITER W. 24 51 20 2407 26 34 45 2387 28 18 39 2367 30 3 i 2348 SUN E. 61 41 44 2719 60 5 29 2699 58 28 48 2680 56 5i 4i 2661 25 Spica W. 85 59 40 2228 87 47 26 2210 89 35 38 2194 91 24 15 2177 JUPITER W. 38 51 44 2256 40 38 49 2238 42 26 20 2220 44 *4 J 7 2203 SUN E . 48 39 50 2571 47 o 15 2554 45 20 17 2538 43 39 56 2522 26 Spica W. TOO 33 22 2101 102 24 19 2087 104 15 38 2074 106 7 17 2061 Antares W. 55 5 57 2113 56 36 37 2098 58 47 39 208 5 60 39 2 2071 JUPITER W. 53 20 ii 2126 55 10 31 2112 57 i 12 2093 58 52 14 2085 SATURN W. 29 40 6 2120 3i 30 35 2105 33 21 26 2092 35 12 37 2079 SUN E. 35 13 9 2457 33 30 55 8446 31 48 26 2438 30 5 46 2433 27 JUPITER W. 68 12 4 2030 70 4 51 2021 7i 57 52 2013 73 5i 6 2005 SATURN W. 44" 33 2025 46 26 7 2017 4 8 19 15 2009 50 12 36 2000 SUN E. 21 31 15 2439 19 4 8 3 6 2456 18 6 21 2 4 8l 16 24 41 2520 182 MARCH, 1900. AT GREENWICH APPARENT NOON. THE SUN'S j 8 | Sidereal Equation of i z. Semi- Time, to be 1 1 diameter Added to 3 "S Apparent Diff. for Apparent Diff. for Semi- Passing Apparent Diff. for I .1 Right Ascension. i Honr. Declination, diameter. Meridian, Time. i Hour. Thur I h m s 22 47 57.02 s 9-37 1 S. 7 38 24.5 +56.96 16 9.26 65-37 m 12 34.70 .485 Frid 2 22 51 41.63 9-350 7 15 34-2 57,22 16 9.02 65-30 12 22.8O .506 Sat. 3 22 55 25.75 9-330 6 52 37-9 57-47 16 8.77 65-23 12 10.41 .527 SUN. 4 22 59 9.38 9.310 6 29 35.9 +57-70 16 8.52 65.16 " 57-52 547 Mon. 5 23 2 52.53 9.291 6 6 28.7 57-91 16 8.27 65.09 II 44.16 .566 Tues 6 23 6 35.24 9.272 5 43 16.5 58.11 1 6 8.02 65.03 II 30.35 0.584 Wed. 7 23 10 17.52 9.254 5 *9 59-8 +58.29 16 7.77 64.97 II l6.I2 0.602 Thur. 8 2 3 13 59-37 9.237 4 56 39-0 58.46 16 7.52 64.91 II 1.46 0.619 Frid. 9 23 17 40.83 9.221 4 33 H-4 58.61 16 7.26 64.86 10 46.40 0.635 Sat. 10 23 21 21.91 9.206 4 9 46.5 +58.74 1 6 7.00 64.81 10 30.97 0.650 SUN. ii 23 25 2.6 4 9.192 3 46 15-5 58.86 16 6.74 64.76 10 15.19 0.665 Mon. 12 23 28 43.04 9.178 3 22 41.9 58.96 1 6 6.48 64.71 9 59-07 0.679 Tues. 13 23 32 23.11 9.165 2 59 6.1 +59-05 16 6.22 64.66 9 42-65 0.691 Wed. H 23 36 2.90 9.154 2 35 28.3 59.12 16 5.96 64.62 9 25.93 0.702 Thur 15 23 39 42.42 9.144 2 II 48.9 59.18 1 6 5.69 64.58 9 8.95 0.712 Frid 16 23 43 21.72 9-134 I 48 8.4 +59-23 16 5.42 64-55 8 51-73 0.721 Sat. i7 23 47 0.80 9.125 I 2 4 27.0 59-26 16 5-15 64.52 8 34-31 0.730 SUN. 18 23 50 39.68 9.118 i o 45.0 59.27 *6 4.88 64-49 8 16.68 0.738 Mon. 19 23 54 18.38 9.113 o 37 2.8 +59-27 16 4.61 64.47 7 58-87 0.744 Tues. 20 2 3 57 5 6 -94 9.106 S. o 13 20.8 59.26 16 4.34 64-45 7 4-93 0.750 Wed. 31 o i 35-37 9.101 N. o 10 20.7 59.23 1 6 4.07 ' 64.43 7 22.85 0-755 Thur. 22 o 5 I3-7I 9.098 o 34 i-3 +59-i8 16 3.79 6 4 . 4I 7 4-69 o-759 Frid. 23 o 8 51.97 9-095 o 57 40-8 59-12 16 3-51 64.40 6 46.44 0.762 Sat. 2 4 12 30.17 9.093 i 21 18.6 59-05 16 3.24 64'39 6 28.14 0.764 SUN. 25 o 16 8.33 9.092 i 44 54.6 +58.97 1 6 2.96 64.39 6 9.80 0.765 Mon. 26 o 19 46.47 3.092 2 8 28.1 58.86 16 2.69 64.38 5 5i-45 0.765 Tues. 27 o 23 24.61 9-093 2 3i 59-i 58.74 16 2.41 64.38 5 33-09 0.764 Wed. 28 o 27 2.78 9.094 2 55 27.0 +58.60 16 2.13 64.38 5 14-77 0.763 Thur. 29 o 30 41.00 9-096 3 18 51.5 58.44 16 1.86 64-39 4 56-48 0.761 Frid. 30 o 34 19.27 9-098 3 42 12.2 58.27 16 1.58 64.40 4 38.23 0-759 Sat. 31 o 37 57.60 9.101 4 5 28.7 58.09 16 1.31 64.41 4 20.05 0.756 SUN. 32 o 41 36.01 9.105 N. 4 28 40.7 +57.90 16 1.03 64.42 4 1^8 0.752 NOTE. The mean time of semidiameter passing may be found by subtracting o.i8 from the sidereal time. The sign + prefixed to the hourly change of declination indicates that south declinations are decreasing; nonb declinations, increasing. 183 II. MARCH, 1900. AT GREENWICH MEAN NOON. Day of the Week. Day of the Month. THE SUN'S Equation of Time, to be Subtracted from Mean Time. Diff. for i Hour. Sidereal Time, or Right Ascension Mean Sun. Apparent Right Ascensioa Diff. for i Hour. Apparent Declination. Diff. for i Hour. Thur. Frid. Sat. I 2 3 h m s 22 47 55.05 22 51 39.70 22 55 23.86 s 9-371 9-350 9-330 S. 7 38 36-4 7 15 46-0 6 52 49-6 57-22 57-47 12 34.80 12 22.90 12 10.51 0.485 0.506 0-527 22 35 20.24 22 39 16.80 22 43 13-35 SUN. Mon. Tues. 4 5 6 22 59 7-53 23 2 50.72 23 6 33.47 9.310 9.291 9.272 6 29 47.4 6 6 40.0 5 43 27.6 +57-70 57-91 58.11 II 44.27 1 1 30.46 0-547 0.566 0.584 22 47 9.90 22 5 I 6.45 22 55 3.01 Wed. Thur. Frid. 7 8 9 23 10 15.79 23 13 57-68 23 17 39-i8 9-254 9-237 9.221 5 20 10.7 4 56 49-7 4 33 24.9 +58.29 58.46 58.61 II 16.23 ii i-57 10 46.51 0.602 0.619 0.635 22 58 59-56 23 2 56.11 23 6 52.66 Sat. SUN. Mon. 10 ii 12 23 21 20.30 23 25 1.07 23 28 41.51 9.206 9.192 9.178 4 9 56.8 3 46 25.6 3 22 51.8 +58.74 58.86 58.96 10 31.08 10 15.30 9 59-i8 0.650 0.665 0.679 23 10 49.22 23 14 45-77 23 18 42.32 Tues. Wed. Thur. 13 H 23 32 21.63 23 36 1-47 23 39 41.04 9.165 9.154 9.144 2 59 15-7 2 35 37-7 2 II 58.0 +59-05 59.12 59.18 9 42.76 9 26.04 9 9.06 0.691 0.702 0.712 23 22 38.88 23 26 35-43 2 3 30 31-98 Frid. Sat. SUN. 16 17 18 23 43 20.38 23 46 59.50 23 50 38.42 9.134 9.125 9.118 I 48 17.2 1 24 35-5 i o 53.2 +59-23 59.26 59-27 8 51-84 8 34-4i 8 16.78 0.721 0.730 0.738 23 34 28.54 23 38 25.09 23 42 21.64 Mon. Tues. Wed. 19 20 21 23 54 17.17 23 57 55-77 o i 34.25 9.112 9.106 9.101 o 37 10.7 S. o 13 28.4 N. o 10 13.4 +59-27 59.26 59-23 7 58.98 7 22.95 0-744 0.750 0-755 23 46 18.19 23 50 14.74 23 54 "-So Thur. Frid. Sat. 22 23 2 4 o 5 12.63 o 8 50.93 O 12 29.18 9.098 9-095 9-093 33 54-3 o 57 34-i I 21 I2'.2 +59-I8 59.12 59.05 7 4-78 6 46-53 6 28.22 0-759 0.762 0.764 23 58 7-85 O 2 4.40 o 6 0.96 SUN. Mon. Tues, 25 26 27 o 16 7.39 o 19 45.58 o 23 23.77 9.092 9.092 9-093 i 44 48.5 2 8 22.3 2 31 53-6 +58.97 58.86 58.74 6 9.88 5 5i-5 2 5 33-i6 0.765 0.765 0.764 o 9 57-51 o 13 54.06 o 17 50.61 Wed. Thur. Frid. Sat. 28 29 3 31 o 27 1.99 o 30 40.25 o 34 18.56 o 37 56.94 9.094 9.096 9.098 9.101 2 55 21.8 3 1 8 46-7 3 42 7-7 4 5 24-5 +58.60 58.44 58.27 58.09 5 14-83 4 56.54 4 38-29 4 20. 1 2 0.763 0.761 0.759 0.756 O 21 47.16 o 25 43.72 o 29 40.27 o 33 36.82 SUN. 3 2 o 41 35.40 9.105 N. 4 28 36.8 +57-90 4 2.03 0.752 o 37 33-37 NOTE. The semidiameter for mean noon may be assumed the same as that for apparent noon. The sign + prefixed to the hourly change of declination indicates that south declinations are decreasing ; north declinations, increasing. Diff. for i Hour, (Table III.) 184 II. APRIL, 1900. AT GREENWICH MEAN NOON. Day of the Week. | i THE SUN'S Equation of Time, to be Subtracted Diff. for Sidereal Time, Right Ascension Mean Sun. Apparent Right Ascension. Diff. for i Hour. Apparent Declination. Diff. for j Hour. Added to Mean Time. Mon.' Tues. 2 3 h m s o 41 35.40 o 45 13.96 o 48 52.63 9.105 9.109 9.114 N. 4 28 36.8 4 51 44.2 5 14 46-3 +57.90 57.69 57-47 4 2.03 3 44-04 3 26.15 0.752 0.748 0-743 h m s o 37 33-37 o 41 29.93 o 45 26.48 Wed. Thur. Frid. 4 5 6 o 52 31.44 o 56 10.39 o 59 49.49 9.120 9.126 9-133 S 37 42-7 6 o 33.2 6 23 17.3 +57-23 56.97 56.70 3 8.40 2 50.8l 2 33.36 0-737 0.731 0.724 o 49 23.03 o 53 19-59 o 57 16.14 Sat. SUN. Mon. 7 8 9 i 3 28.79 i 7 8.27 i 10 47.97 9.141 9.150 9.160 6 45 54-7 7 8 25.1 7 30 48-2 +56.41 56.11 55.80 2 16.09 i 59-02 I 42.17 0.716 0.707 0.697 i i 12.69 i 5 9-24 i 9 5.8o Tues. Wed. Thur. 10 ii 12 i 14 27.91 i 18 8.ii I 21 48.58 9.170 9.181 9-193 7 53 3-7 8 15 ii. i 8 37 10.3 +55-48 55-15 54.80 I 25.57 I 9.21 53-13 0.687 0.676 0.664 1 *3 2.35 i 16 58.90 I 20 55.46 Frid. Sat. SUN. Mon. Tues. Wed. 13 H 16 18 i 25 29.35 i 29 10.43 i 32 51-84 i 36 33.60 i 40 15.72 i 43 58.24 9.206 9.219 9-233 9.248 9.264 9.280 8 59 0.8 9 20 42.4 9 42 14-7 10 3 37.4 10 24 50.1 10 45 52.7 +54-43 54.05 +53-24 52.82 52-39 o 37-34 o 21.86 o 6.72 0.651 0.638 0.624 0.609 0-593 0.576 I 24 52.01 I 28 48.56 i 32 45-12 i 36 41-67 i 40 38.22 i 44 34.78 o 8.07 o 22.50 o 36.54 Thur. Frid. Sat. 19 20 21 i 47 41.17 i 51 24.51 i 55 8.29 9.297 9-3I5 9-334 ii 6 44.7 ii 27 25.9 ii 47 55.8 +51.94 51.48 51.01 o 50.16 I 3-38 I 16.15 0.559 0.541 0.523 i 48 31.33 i 52 27.88 i 56 24.44 SUN. Mon. Tues. 22 2 3 2 4 i 58 52.53 2 2 37.24 2 6 22.42 9-353 9-373 9-393 12 8 14.2 12 28 20.7 12 48 15.0 +50.52 50.02 49.50 I 28.46 I 40.32 I 51.68 0.504 0.484 0.464 2 20.99 2 4 J 7-55 2 8 14.10 Wed. Thur. Frid. y 27 2 10 8.09 13 54.26 17 40.94 9-413 9-434 9-455 13 7 56.8 13 27 25.7 13 46 41.4 +48.97 48.43 47.87 2 2.56 2 12.95 2 22.82 0.443 0.422 0.401 2 12 10.65 2 16 7.21 2 20 3.76 Sat. SUN. Mon. 28 29 y- 21 28.13 25 15.84 29 4-07 9.476 9.498 9.520 14 5 43-5 14 24 31.7 14 43 5.6 +47.29 46.70 46.10 2 32.18 2 41.03 2 49.36 0.380 0.358 0.336 2 2 4 0.32 2 27 56.87 2 31 53-42 Tues. 3* 2 32 52.82 9-542 N.is i 25.0 +45-49 2 57.16 0.314 2 35 49-98 NOTE. The semidiameter for mean noon may be assumed the same as that for apparent noon. The sign + prefixed to the hourly Change of declination indicates that north declinations are increasing. Diff. for i Hoar, + 9 '.8 5 6 5 . (Table III.) 185 II. AUGUST, 1900. AT GREENWICH MEAN NOON. THE SUN'S Equation of j -a Time, Sidereal 1 S to be Subtracted Time, a 1 from Right Ascension g Apparent Diff. for Apparent Diff. for Added to Diff. for of >, Right Ascension. i Hour. Declination. i Hour. Mean Time. i Hour. Mean Sun. a Q Wed. I h m s, 8 44 41.30 s 9.716 N.i8 5 i.i -37.61 6 8.12 s 0.140 h m s 8 38 33.18 Thur. 2 8 48 34.18 9.690 17 49 49-7 38.34 6 4.44 o.i 66 8 42 29.74 Frid. 3 8 52 26.44 9.664 17 34 21.0 39-05 6 0.15 0.191 8 46 26.29 Sat. 4 8 56 1 8.08 9-638 17 18 35.2 -39-75 5 55-23 0.217 8 50 22.85 SUN. 5 9 o 9.10 9.613 17 2 32.6 4-45 5 49-70 0.243 8 54 19-4 Mon. 6 9 3 59-5i 9.587 16 46 13.5 4I.I3 5 43-55 0.269 8 58 15.96 Tues. 7 9 7 49-3 1 9.562 16 29 38.3 -41.80 5 36-80 0.294 9 2 12.51 Wed. 8 9 " 38-5 1 9-537 1 6 12 47.1 42.46 5 29.44 0.319 9 6 9.07 Thur. 9 9 15 27.11 9-5I3 15 55 4-3 43.10 5 21.48 0-344 9 10 5.62 Frid. 10 9 19 15-12 9.489 15 38 18.2 -43-73 5 12.94 0.368 9 14 2.18 Sat. 1 1 9 23 2.56 9.465 15 20 41.0 44-35 5 3-82 0.392 9 i7 58.73 SUN. 12 9 26 49.43 9.442 15 2 49.1 44.96 4 54-14 0.415 9 21 55-29 Mon. 13 9 30 35-75 9.419 14 44 42.7 -45.56 4 43-91 0.438 9 25 51-84 Tues. H 9 34 21.54 9-397 14 26 22. 46.15 4 33-14 0.460 9 29 48.40 Wed. 15 9 38 6.80 9-375 H 7 47-5 46.72 4 21.85 0.482 9 33 44-95 Thur. 16 9 4i 5*-55 9-354 13 48 59-4 -47-28 4 10.04 0.503 9 37 4*-50 Frid. 17 9 45 35-79 9-333 13 29 57.9 47.83 3 57-73 0.524 9 41 38.06 Sat. 18 9 49 *9-54 9.3I3 13 10 43.5 48.36 3 44-93 0-544 9 45 34-6i SUN. 19 9 53 2.81 9-293 12 51 16.4 -48.88 3 31-64 0.564 9 49 31.17 Mon. 20 9 56 45.61 9.273 12 31 37.0 49-38 3 17-89 0.583 9 53 27.72. Tues. 21 10 o 27.94 9.254 12 II 45.6 49.88 3 3-67 0.602 9 57 24.28 Wed. 22 10 4 9.83 9.235 II 51 42.6 -50.36 2 49.00 0.620 10 i 20.83 Thur. 23 10 7 51.27 9.217 II 31 28.2 50.83 2 33-89 0.638 10 5 17.38 Frid. 2 4 10 ii 32.28 9.199 11 II 2.8 51-28 2 18.34 0.656 10 9 13.94 Sat. 25 10 15 12.86 9.182 10 50 26.8 -51.72 2 2.37 0.674 10 13 10.49 SUN 26 10 18 53.03 9.165 10 29 40.5 52.14 i 45-99 0.691 10 17 7.04 Mon. 2 7 10 22 32.8l 9.149 10 8 44.3 52-55 i 29.21 0.707 10 21 3.60 Tues. 28 10 26 12.19 9-133 9 47 38-4 -52.94 i 12.04 0.723 10 25 0.15 Wed. 2 9 10 29 51.20 9.118 9 26 23.2 53-32 o 54.50 0.738 10 28 56.70 Thur. 30 10 33 29.86 9.104 9 4 59-i 53-69 o 36.60 0-753 10 32 53.26 Frid. 31 10 37 8.16 9.090 8 43 26.3 54.04 o 18.35 0.767 10 36 49.81 Sat. 32 10 40 46.13 9.076 N. 8 21 45.2 -54-38 o 0.23 0.781 10 40 46.36 NOTE. The semidiameter for mean noon may be assumed the same as that for apparent noon. Diff. for i Hour, The sign preBxed to the hourly change of declination indicates that north declinations are + 9'.8 5 6 5 . decreasing. (Table IIL) 186 SEPTEMBER, 1900 III. AT GREENWICH MEAN NOON. THE SUN'S | 1 i TRUE LONGITUDE. Logarithm of the Radius Vector Mean Time 5 i Diff. for LATITUDE. of the Diff. for of I s A y t Hour. Earth. I Hour. Sidereal Noon i 244 158 34 13-5 33 23.3 145-25 0.18 0.0038298 -44.4 h m s 13 17 2.70 2 245 159 32 20.2 31 29.9 145.31 0.06 0.0037217 44-9 13 13 6.80 3 246 160 30 28.3 29 37.9 M5-37 + 0.07 0.0036125 43-4 13 9 10.89 4 247 161 28 37.8 27 47-3 M5-43 -f- O.2O 0.0035024 -45-9 13 5 I4-98 5 162 26 48.7 25 58-2 145.49 0.28 0.0033914 46.3 13 i 19.07 6 249 163 25 1.2 24 10.6 M5-55 o-35 0.0032799 46.6 12 57 23.17 7 250 l6 4 23 15.3 22 24.5 145-62 + 0.40 0.0031678 -46.8 12 53 27.26 8 9 251 252 165 21 3I.O 1 66 19 48.4 2O 40.1 18 57-5 145.69 M5-77 0.42 0.40 0.0030554 0.0029427 46-9 47-o 12 49 31.35 12 45 35-45 10 253 167 18 7.8 17 16.7 145-85 + 0.35 0.0028297 -47-1 12 41 39.54 ii 254 1 68 1 6 29.0 15 37-9 M5-93 0.27 0.0027166 47-2 12 37 43.64 12 255 169 14 52.3 14 i.o 146.01 0.15 0.0026032 47-3 12 33 47-73 13 256 170 13 17.6 12 26.3 146.10 + 0.03 0.0024894 -47-5 12 29 51.82 H 257 171 ii 45.2 10 53-7 146.19 O.IO 0.0023751 47-7 12 25 55.91 15 258 172 10 14.9 9 23.4 146.28 0.23 O.OO226O3 48.0 12 22 O.OI 16 259 173 8 46.8 7 55-2 146-37 -0.35 0.0021448 -48-3 12 18 4.10 17 260 174 7 20.9 6 29.2 146.46 0.47 0.0020285 48.6 12 14 8.19 18 261 175 5 57-2 5 5-5 146.56 0.56 O.OOI9II4 49.0 12 10 12.29 19 262 176 4 35.8 v 3 43-9 146.65 0.63 0.0017934 -49-4 12 6 16.38 20 263 177 3 16.5 2 24.6 146.74 0.68 0.0016743 49.8 12 2 20.48 21 264 178 i 59-3 i 7-3 146.83 0.70 0.0015543 50.2 II 58 24.57 22 265 178 60 44.2 59 52-0 146.91 0.70 0.0014332 50.6 ii 54 28.66 23 266 179 59 31.0 58 38.8 147.00 0.66 O.OOI3II2 51.0 ii 50 32.76 24 267 1 80 58 20.0 57 27.7 147.08 0.61 O.OOII882 5'-4 ii 46 36.85 25 268 181 57 10.9 56 r8. 5 147.16 -0.53 0.0010642 -51.8 ii 42 40.94 26 269 182 56 3.7 55 "-3 147.24 0.44 0.0009394 52.1 ii 38 45.04 27 270 183 54 58-4 54 5-9 M7-32 0.33 0.0008137 52.4 ii 34 49-13 28 271 184 53 55-o 53 2.4 147-39 0.21 0.0006872 -52.7 ii 30 53.22 29 272 185 52 53-4 52 0.7 147-47 0.08 0.0005603 33-o ii 26 57.32 30 273 186 51 53.6 51 0.8 147-54 + 0.04 0.0004328 53-2 ii 23 1.41 31 274 187 50 55-5 50 2.6 147.62 4-0.17 0.0003049 -53-3 ii 19 5-5 NOTE. The numbers in column A correspond to the true equinox of the date ; in column A' to the Diff. for t Hour. OT829-6. mean equinox of January o d .o. (Table II.) 187 II. NOVEMBER, 1900. AT GREENWICH MEAN NOON. THE SUN'S Day of the Week. 8 ?. 1 I Equation of Time, to be Added to Mean Time. Diff. for Sidereal Time, Right Ascension Mean Sun. Apparent Right Ascensioa Diff. for i Hour. Apparent Declination. Diff. for i Hour. Thur Frid. Sat i 2 3 h m s 14 24 57-33 14 28 52.67 14 32 48.80 9.790 9.823 9-856 S. I 4 22 58.5 I 4 4 2 8.4 15 I 4.0 -48.20 47.61 47.01 16 18.76 16 19.97 1 6 20.40 0.067 0.034 0.00 1 14 41 16.09 14 45 12.64 14 49 9.20 SUN. Mon. Tues. 4 5 6 14 36 45.72 14 40 43.46 14 44 42.01 9.889 9.923 9.958 15 19 44.8 15 38 10-5 15 56 20.6 -4 6 -39 45-75 45.09 16 20.03 16 18.85 16 16.85 0.032 0.065 0.100 H 53 5-75 14 57 2.31 15 o 58.86 Wed. Thur. Frid. 7 8 9 14 48 41.40 14 52 41.62 14 56 42.70 9-993 10.028 10.063 16 14 14.9 16 31 52.8 16 49 14.1 -44.42 43-73 43-03 16 14.02 16 10.35 16 5-83 0-135 0.170 0.206 15 4 55-42 15 8 51.97 15 12 48.53 Sat. SUN. Mon. 10 1 1 12 15 o 44.63 15 4 47-43 15 8 51.09 10.099 10.135 10.171 17 6 18.2 17 23 4.8 17 39 33-6 -42.31 41-57 40.82 16 0.45 15 54,2 1 15 47.10 0.242 0.278 0.314 15 16 45.08 15 20 41.64 15 24 38.19 Tues. Wed. Thur. 13 14 J 5 15 12 55.63 15 17 1.03 15 21 7.30 10.207 10.243 10.279 17 55 44-i 18 ii 35.9 18 27 8.6 -40.05 39-26 38.46 15 39-12 15 30.27 15 20.56 0.351 0.387 0.423 15 28 34.75 15 32 31-30 15 36 27.86 Frid. Sat. SUN. 16 17 18 15 25 14.43 15 29 22.42 15 33 31.26 10.315 10.351 10.386 18 42 21.8 18 57 15-1 19 ii 48.1 -37- 6 4 36.80 35-95 15 9.98 H 58.55 14 46.27 0.459 0.494 0.529 15 40 24.42 15 44 20.97 15 48 17.53 Mon. Tues. Wed. 19 20 21 15 37 40.94 15 41- 51.46 15 46 2.79 10.421 10.455 10.489 19 26 0.5 19 39 5^-8 19 53 21.8 -35.08 34.19 33-29 H 33-14 14 19.18 14 4.40 0.564 0.598 0.632 15 52 14.08 15 56 10.64 16 o 7.20 Thur. Frid. Sat. 22 23 24 15 50 14.94 15 54 27.87 15 58 41.59 10.522 10.555 10.587 20 6 29.8 20 19 15.8 20 3 I 39.2 -32.37 31-44 30.50 13 48.82 13 32-43 13 15-27 0.666 0.699 0-731 16 4 3-75 16 8 0.31 16 ii 56.86 SUN Mon. Tues. 25 26 27 16 2 56.07 16 7 11.29 16 ii 27.24 10.619 10.649 10.678 20 43 39.8 20 55 17.2 21 6 31.0 -29-54 28.57 27-58 12 57-35 12 38.69 12 19.30 0.762 0.792 0.822 16 15 53-42 16 19 49.98 16 23 46.53 Wed. Thur. Frid. 28 2 9 30 16 15 43.88 l6 20 1.22 16 24 19.22 10.707 10.736 10.764 21 17 2O-9 21 27 46.7 21 37 48.0 -26.58 25.56 24.54 II 59-21 II 38.43 II 16.99 0.851 0.880 0,907 16 27 43.09 1 6 31 39.65 16 35 36.20 Sat 31 16 28 37.86 10.790 S.2i 47 24.5 -23.51 10 54.90 0.933 16 39 32.76 NOTE. The semidiaraeter for mean noon may be assumed the same as that for apparent noon. The sign prefixed to the hourly changs of declination indicates that soutn declinations are Diff. for i Hour, 4-9'.8s65. mcreasmg. (Table III.) 188 VENUS, 1900. GREENWICH MEAN TIME. JANUARY. FEBRUARY. | Apparent Right Ascension. . Var. of R.A. for i Hour. Apparent Declination. Var. of Decl. for i Hour. Meridian A 7. Apparent Right Var. of R.A. Hour. Apparent Declination. Var. of Decl. for i Meridian 5 Passage. Passage. 1 Noon. Noon. MM*. Noon. 1 Noon. Noon. Noon. Noon. t 20 39 23.93 +12.819 -20 9 51.7 +46.64 156-8 I 23 746.06 +H.S33 -6 58 20.4 +75-82 2 22.9 2 20 44 30.90 12.763 19 5 55-3 48.06 i 57-9 2 23 12 .5.26 11.199 6 27 55.5 76.24 2 23.4 3 204936.51 12.706 1931 25.0 49-45 i 59.1 3 23 16 43-65 11.167 5 57 20.9 76.63 2 23.9 4 20 54 40.75 12.648 19 ii 21.7 50.82 2 O.2 4 23 21 II. 26 11.136 5 26 37.4 76.99 224.5 5 20 59 43-61 12.5 y> 18 50 46.0 52.15 2 1.3 5 23 25 38.14 11.106 4 55 45-8 77-31 2 25.0 6 21 4 45.08 +12.532 -18 29 38.8 +53-45 2 2.4 6 23 30 4-31 +11.077 -4 24 46.7 +77-60 225.5 7 21 94,5-15 12.474 18 8 0.8 54-72 2 3-5 7 23 34 29.81 11.050 3534J.I 77-86 2 25.9 8 21 14 43.82 12.4 5 174552.7 55-95 2 4-5 8 23 38 54-68 11.024 3 22 29.6 78.09 2 26.4 9 21 19 41.08 12-357 17 23 15-4 57-15 2 5-5 9 23 43 18.95 11.000 2 51 13.0 78.19 2 26.9 10 21 24 36.95 12.299 17 o 9.6 58-32 2 6. 5 10 2347 42.66 10.978 2 19 52.O 78.46 227.3 ii 21 2931.42 +12.241 -16 36 36.1 +59-46 2 7-5 ii 2352 5-85 +10.957 -I 48 27.4 +78.60 227-7 12 21 34 24.51 I2.I 3 16 12 35.7 60.57 2 8.4 12 23 56 28.56 10.937 I l6 59.8 78.70 2 28.2 13 21 39 16.22 12.1 f 1548 9.2 61.64 2 9-3 13 o o 50.82 10.919 o 45 29.9 78.77 2 28.6 14 2144 6.57 12.0 15 23 17.3 62.68 2 1O.2 M o 5 12.68 10.903 -o 13 58.6 78.82 2 29.1 15 21 48 55.57 12.014 14 58 0.9 63.69 2 II. I '3 o 934.17 10.889 +o 17 33.5 78.84 2 29.5 16 21 5343-25 + 11.959 -14 32 20.7 +64-66 2 II.9 16 1355-34 +10.876 +049 5.7 +78.83 2 29.9 17 21 58 29.61 11.905 14 6 17.5 65.60 2 12.8 17 o 18 16.22 10.865 I 20 37.2 78.79 230.3 18 22 3 14.70 11.852 13 39 52.0 66.51 2 13.6 18 22 36.86 10.856 1 52 7.4 78.72 230.7 19 22 7 58.53 ii. 8 13 13 5-i 67-39 2 14.3 J9 o 26 57.29 10.848 2 23 35.6 78.62 2 3I.I 20 22 12 41.11 11.749 12 45 57.6 68.23 2 15.1 20 o 31 17.54 10.841 2 55 i.o 78.49 2 31-5 21 22 17 22.48 +11.699 -12 18 30.2 +69.04 2.5.8 21 o 35 37-66 +10.836 +3 26 23.0 +78.33 2 3'-9 22 22 22 2.67 11.650 ii 5043.7 69.83 2 16.6 22 o 39 57-69 10.833 3 57 4-8 78.14 2 32-3 23 22 26 41.70 11.602 II 22 38.8 70.58 2 17-3 23 o 44 17.66 10.832 4 28 53.8 77-92 2 32.6 24 22 31 19.60 11-556 10 54 16.6 71.29 2 l8.0 2 4 o 48 37.62 10.832 5 o 1.2 77-68 233-0 25 22 35 56-41 11.511 10 25 37.6 71-97 2 18.7 25 o 52 57-59 10.833 5 31 2.4 77.41 233-4 26 224032.15 +11.4 3 - 9 56 42-5 +72.62 2 19-3 26 o 57 17.62 +10.836 +6 i 56.6 +77." 233-8 27 22 45 6.86 11.426 927 32.1 73-24 2 19-9 2 7 i i 37-74 10.840 6 32 43.2 76.77 2 34-2 28 22 49 40.57 11.385 858 7-4 73-82 2 20.5 28 i 557-96 10.846 7 3 21.4 76.40 2 34-6 29 22 54 13-32 11.345 8 28 29.0 74-37 2 21. 1 29 i 10 18.33 10.853 7 33 50-5 76.01 2 35-o 3 225845.13 11.306 7 58 38-0 74.89 2 21-7 30 i 14 38.88 10.860 8 4 9.8 75-59 2 35-4 31 23 3 16.03 + 11.269 - 7 28 34.9 +75-37 222-3 31 i 18 59.62 +10.869 +8 34 18.5 +75-14 235-8 32 23 746.06 +11.233 - 6 58 20.4 +75-82 2 22.9 32 i 23 20.59 +10.879 +9 4 16.0 +74-65 2 36.2 Day of the Month. 1st. 6th. llth. 16th 21st. 26th. 31st. Day of the Month. 6th. 10th. 15th. 20th. 25th. Semidiameter Hor. Parallax LS 593 6.03 6.14 6.15 6.25 6.35 ' 6.24 6.37 6.50 6.47 6.59 6.73 Semidiameter . . 6.6 Hor. Parallax 6.8 3 6.78 6.95 7.15 1 703 7-19 7-37 7-31 7-57 i NOTE.- -The sign + indicates north declinations ; the sign indicates south declinations. 189 14 DAY USE RETURN TO DESK FROM WHICH BORROWED ASTRON-MATH-STAT. This book is due on the last date stamped below, or on the date to which renewed. Renewed books are subject to immediate recall. MAY 2 3 1994 LD 21-1001 LD 21-40m-10,'65 (F7763slO)476 General Library University of California Berkeley M298787 V/Vyy