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Full text of "Textbook on practical astronomy"

TEXT-BOOK 

ON 



PRACTICAL ASTRONOMY 



BY 



GEORGE L. HOSMER 

Assist ant Professor of Civil Engineering, Massachusetts 
Institute of Technology 



FIRST EDITION 

FIRST THOUSAND 



NEW YORK 

JOHN WILEY & SONS 

LONDON: CHAPMAN & HALL, LIMITED 

1910 




COPYRIGHT, 1910 

BY 
GEORGE L. HOSMER 



Stanhope Jpresa 

F. H. CILSON COMPANY 
BOSTON. U.S.A. 



PREFACE 



THE purpose of this volume is to furnish a text in Practical 
Astronomy especially adapted to the needs of civil-engineering 
students who can devote but little time to the subject, and who 
are not likely to take up advanced study of Astronomy. The 
text deals chiefly with the class of observations which can be 
made with surveying instruments, the methods applicable to 
astronomical and geodetic instruments being treated but briefly. 
It has been the author's intention to produce a book which is 
intermediate between the text-book written for the student of 
Astronomy or Geodesy and the short chapter on the subject 
generally given in text-books on Surveying. The subject has 
therefore been treated from the standpoint of the engineer, who 
is interested chiefly in obtaining results, and those refinements 
have been omitted which are beyond the requirements of the 
work which can be performed with the engineer's transit. This 
has led to the introduction of some rather crude mathematical 
processes, but it is hoped that these are presented in such a way 
as to aid the student in gaining a clearer conception of the prob- 
lem without conveying wrong notions as to when such short-cut 
methods can properly be applied. The elementary principles 
have been treated rather elaborately but with a view to making 
these principles clear rather than to the introduction of refine- 
ments. Much space has been devoted to the Measurement of 
Time because this subject seems to cause the student more 
difficulty than any other branch of Practical Astronomy. The 
attempt has been made to arrange the text so that it will be a 
convenient reference book for the engineer who is doing field 
work. 

For convenience in arranging a shorter course those subjects 



iv PREFACE 

which are most elementary are printed in large type. The mat- 
ter printed in smaller type may be included in a longer course 
and will be found convenient for reference in field practice, par- 
ticularly that contained in Chapters X to XIII. 

The author desires to acknowledge his indebtedness to those 
who have assisted in the preparation of this book, especially to 
Professor A. G. Robbins and Mr. J. W. Howard of the Massa- 
chusetts Institute of Technology and to Mr. F. C. Starr of the 
George Washington University for valuable suggestions and crit- 
icisms of the manuscript. 

G. L. H. 

BOSTON, June, 1910. 



TABLE OF CONTENTS 



CHAPTER I 

THE CELESTIAL SPHERE REAL AND APPARENT MOTIONS 

ART. PAGE 

1. Practical Astronomy i 

2. The Celestial Sphere i 

3. Apparent Motion of the Sphere 3 

4. The Motions of the Planets 3 

5. Meaning of Terms East and West 6 

6. The Earth's Orbital Motion The Seasons 7 

7. The Sun's Apparent Position at Different Seasons 9 

8. Precession and Nutation 10 

9. Aberration of Light 12 



CHAPTER II 

DEFINITIONS POINTS AND CIRCLES OF REFERENCE 

10. Definitions 14 

Vertical Line Zenith Nadir Horizon Vertical Circles 
Almucantars Poles Equator Hour Circles Par- 
allels of Declination Meridian Prime Vertical Eclip- 
tic Equinoxes Solstices Colures. 



CHAPTER III 

SYSTEMS OF COORDINATES ON THE SPHERE 

11. Spherical Coordinates 18 

12. The Horizon System 19 

13. The Equator Systems 19 

15. Coordinates of the Observer 22 

16. Relation between the Two Systems of Coordinates 23 



VI TABLE OF CONTENTS 

CHAPTER IV 

RELATION BETWEEN COORDINATES 
ART. PAGE 

17. Relation between Altitude of Pole and Latitude of Observer. ... 27 

1 8. Relation between Latitude of Observer and the Declination and 

Altitude of a Star on the Meridian 30 

19. The Astronomical Triangle 31 

20. Relation between Right Ascension and Hour Angle 36 



CHAPTER V 
MEASUREMENT OF TIME 

21. The Earth's Rotation 39 

22. Transit or Culmination 39 

23. Sidereal Day 39 

24. Sidereal Time 40 

25. Solar Day 40 

26. Solar Time ' 40 

27. Equation of Time 41 

28. Conversion of Apparent Time into Mean Time and vice versa ... 43 

29. Astronomical and Civil Time 44 

30. Relation between Longitude and Time 45 

31. Relation between Sidereal Tune, Right Ascension and Hour Angle 

of any Point at a. Given Instant 48 

32. Star on the Meridian 49 

33. Relation between Mean Solar and Sidereal Intervals of Time. ... 49 

34. Relation between Sidereal and Mean Time at any Instant 52 

35. Standard Time 56 

36. The Date Line 58 

37. The Calendar 59 



CHAPTER VI 

THE AMERICAN EPHEMERIS AND NAUTICAL ALMANAC STAR 
CATALOGUES INTERPOLATION 

38. The Ephemeris 62 

39. Star Catalogues 69 

40. Interpolation 69 



TABLE OF CONTENTS Vll 

/ 
CHAPTER VII 

THE EARTH'S FIGURE CORRECTIONS TO OBSERVED ALTITUDES 

ART. PAGE 

41. The Earth's Figure 72 

42. Parallax 73 

43. Refraction 76 

44. Semidiameters 78 

45- Dip 79 

46. Sequence of Corrections 80 

CHAPTER VIII 

DESCRIPTION OF INSTRUMENTS OBSERVING 

47. The Engineer's Transit 82 

48. Elimination of Errors 83 

49. Attachments to the Engineer's Transit Reflector 86 

50. Prismatic Eyepiece 87 

51. Sun Glass 87 

52. The Portable Astronomical Transit 87 

53. The Sextant 88 

54. Artificial Horizon 91 

55. Chronometer 92 

56. Chronograph. 93 

57. The Zenith Telescope 94 

58. Suggestions about Observing 95 

CHAPTER IX 

THE CONSTELLATIONS 

59. The Constellations 98 

60. Method of Naming Stars 98 

61. Magnitudes 99 

62. Constellations near the Pole 99 

63. Constellations near the Equator 100 

64. The Planets 102 

CHAPTER X 

OBSERVATIONS FOR LATITUDE 

65. Latitude by a Circumpolar Star at Culmination 103 

66. Latitude by Altitude of the Sun at Noon 105 

67. Latitude by the Meridian Altitude of a Southern Star 107 



Vlll TABLE OF CONTENTS 

ART. PAGE 

68. Latitude by Altitudes Near the Meridian 108 

69. Latitude by Polaris when the Time is Known no 

70. Precise Latitude Determinations Talcott's Method 112 

CHAPTER XI 

OBSERVATIONS FOR DETERMINING THE TIME 

71. Observations for Local Time 114 

72. Time by Transit of a Star 114 

73. Observations with Astronomical Transit 117 

74. Selecting Stars for Transit Observations 117 

75. Time by Transit of the Sun 119 

76. Time by Altitude of the Sun 120 

77. Time by Altitude of a Star 123 

78. Tune by Transit of Star over Vertical Circle through Polaris. ... 124 

79. Time by Equal Altitudes of a Star 127 

80. Time by Two Stars at Equal Altitudes 128 

83. Rating a Watch by Transit of a Star over a Range 135 

84. Time Service 136 

CHAPTER XII 

OBSERVATIONS FOR LONGITUDE 

85. Methods of Measuring Longitude 139 

86. Longitude by Transportation of Timepiece 139 

87. Longitude by the Electric Telegraph 140 

88. Longitude by Transit of the Moon 141 

CHAPTER XIII 

OBSERVATIONS FOR AZIMUTH 

89. Determination of Azimuth 146 

90. Azimuth Mark 146 

91. Azimuth by Polaris at Elongation 147 

92. Observations near Elongation 149 

93. Azimuth by an Altitude of the Sun 151 

94. Azimuth by an Altitude of a Star !55 

95. Azimuth Observation on a Circumpolar Star at any Hour Angle . 155 

96. The Curvature Correction 158 




TABLE OF CONTENTS ix 

ART. PAGE 

97. The Level Correction 158 

98. Diurnal Aberration 158 

99. Meridian by Polaris at Culmination 161 

100. Azimuth by Equal Altitudes of a Star 164 

101. Observation for Meridian by Equal Altitudes of the Sun 165 

102. Observation of the Sun near Noon 166 

103. Combining Observations 167 

CHAPTER XIV 

NAUTICAL ASTRONOMY 

104. Observations at Sea 170 

Determination of Latitude at Sea: 

105. Latitude by Noon Altitude of the Sun 170 

106. Latitude by Ex-Meridian Altitudes 171 

Determination of Longitude at Sea: 

107. Longitude by the Greenwich Time and the Sun's Altitude. ... 172 

108. Longitude by the Lunar Distance 172 

109. Azimuth of the Sun at a Given Time 174 

no. Azimuth of the Sun by Altitude and Time 175 

in. Sumner's Method of Determining a Ship's Position 175 

112. Position by Computation 178 



TABLES 

I. MEAN REFRACTION 184 

II. CONVERSION OF SIDEREAL TO SOLAR TIME 185 

III. CONVERSION OF SOLAR TO SIDEREAL TIME 186 

IV. (A) SUN'S PARALLAX (B) SUN'S SEMIDIAMETER (C) DIP OF 

HORIZON 187 

V. TIMES OF CULMINATION AND ELONGATION OF POLARIS 188 

VI. CORRECTION TO OBSERVE ALTITUDE OF POLARIS 189 

VII. VALUES OF FACTOR 112.5 X 3600 X sin i" X tan Z e 190 

GREEK ALPHABET 190 

LIST OF ABBREVIATIONS 191 



PRACTICAL ASTRONOMY 



CHAPTER I 

THE CELESTIAL SPHERE REAL AND APPARENT 

MOTIONS 

1. Practical Astronomy. 

Practical Astronomy treats of the theory and use of astro- 
nomical instruments and the methods of computing the results 
obtained by observation. The part of the subject which is of 
especial importance to the surveyor is that which deals with the 
methods of locating points on the earth's surface and of ori- 
enting the lines of a survey, and includes the determination of 
(i) latitude, (2) time, (3) longitude, and (4) azimuth. In solving 
these problems the observer makes measurements of the direc- 
tions of the sun, moon, stars, and other heavenly bodies; he is 
not concerned with the distances of these objects, with their 
actual motions in space, nor with their physical characteristics, 
but simply regards them as a number of visible objects of known 
positions from which he can make his measurements. 

2. The Celestial Sphere. 

Since it is only the directions of these objects that are required 
in practical astronomy, it is found convenient to regard all 
heavenly bodies as being situated on the surface of a sphere 
whose radius is infinite and whose centre is at the .eye of the 
observer. The apparent position of any object on the sphere is 
found by imagining a line drawn from the eye to the object, and 
prolonging it until it pierces the sphere. For example, the 
apparent position of Si on the sphere (Fig. i) is at Si, which is 
supposed to be at an infinite distance from C; the position of 
S z is Sz, etc. By means of this imaginary sphere all problems 



2 PRACTICAL ASTRONOMY 

involving the angular distances between points, and angles 
between planes through the centre of the sphere, may readily be 
solved by applying the formulae of spherical trigonometry. 
This device is not only convenient for mathematical purposes, 
but it is perfectly consistent with what we see, because all celestial 
objects are so far away that they appear to the eye to be at the 
same distance, and consequently on the surface of a great sphere. 



sr 




FIG. i. APPARENT POSITIONS ON THE SPHERE 

From the definition it will be apparent that each observer sees 
a different celestial sphere, but this causes no actual inconve- 
nience, for distances between points on the earth's surface are so 
short when compared with astronomical distances that they are 
practically zero except for the nearer bodies in the solar system. 
This may be better understood from the statement that if the 
entire solar system be represented as occupying a field one mile 
in diameter the nearest star would be about 5000 miles away on 
the same scale; furthermore the earth's diameter is but a minute 
fraction of the distance across the solar system, thematic being 
about 8000 miles to 5,60x3,000,000 miles,* or one 7oo,oooth part 
of this distance. 

* The diameter of Neptune's orbit. 



THE CELESTIAL SPHERE 3 

Since the radius of the celestial sphere is infinite, all of the 
lines in a system of parallels will pierce the sphere in the same 
point, and parallel planes at any finite distance apart will cut 
the sphere in the same great circle. This must be kept constantly 
in mind when representing the sphere by means of a sketch, in 
which minute errors will necessarily appear to be very large. 
The student should become accustomed to thinking of the 
appearance of the sphere both from the inside and from an out- 
side point of view. It is usually easier to understand the spheri- 
cal problems by studying a small globe, but when celestial 
objects are actually observed they are necessarily seen from a 
point inside the sphere. 

3. Apparent Motion of the Celestial Sphere. 

If a person watches the stars for several hours he will see that 
they appear to rise in the east and to set in the west, and that 
their paths are arcs of circles. By facing to the north (in the 
northern hemisphere) it will be found that the circles are smaller 
and all appear to be concentric about a certain point in the sky 
called the pole ; if a star were exactly at this point it would have 
no apparent motion. In other words, the whole celestial sphere 
appears to be rotating about an axis. This apparent rotation 
is found to be due simply to the actual rotation of the earth 
about its axis (from west to east) in the opposite direction to 
that in which the stars appear to move.* 

4. Motions of the Planets. 

If an observer were to view the solar system from a point far 
outside, looking from the north toward the south, he would see 
that all of the planets (including the earth) revolve about the 
sun in elliptical orbits which are nearly circular, the direction 
of the motion being counter-clockwise or left-handed rotation. 

* This apparent rotation may be easily demonstrated by taking a photo- 
graph of the stars near the pole, exposing the plate for several hours. The 
result is a series of concentric arcs all subtending the same angle. If the 
camera is pointed southward and high enough to photograph stars near the 
equator the star trails appear as straight lines. 



PRACTICAL ASTRONOMY 



He would also see that the earth rotates on its axis, once 
per day, in a counter-clockwise direction. The moon revolves 
around the earth in an orbit which is not so nearly circular, 
but the motion is in the same (left-handed) direction. The 




FIG. 2. DIAGRAM OF THE SOLAR SYSTEM WITHIN THE ORBIT OF SATURN 

apparent motions resulting from these actual motions are as 
follows: The whole celestial sphere, carrying with it all the 
stars, sun, moon, and planets, appears to rotate about the earth's 
axis once per day in a clockwise (right-handed) direction. The 
stars change their positions so slowly that they appear to be fixed 
in position on the sphere, whereas all objects within the solar 
system rapidly change their apparent positions among the stars. 
For this reason the stars are called fixed stars to distinguish them 
from the planets; the latter, while closely resembling the stars 



THE CELESTIAL SPHERE 5 

in appearance, are really of an entirely different character. The 
sun appears to move slowly eastward among the stars at the rate 
of about i per day, and to make one revolution around the earth 

30" 




FIG. 3a. SUN'S APPARENT POSITION AT GREENWICH NOON ON MAY 22, 23, 

AND 24, 1910 




10 
Y IV III 

FIG. 3b. MOON'S APPARENT POSITION AT 14^ ON FEB. 15, 16, AND 17, 1910 

in just one year. The moon also travels eastward among the 
stars, but at a much faster rate; it moves an amount equal to 
its own diameter in about an hour, and completes one revolu- 



PRACTICAL ASTRONOMY 



tion in a lunar month. Figs. 3a and 3b show the daily motions 
of the sun and moon respectively, as indicated by their plotted 
positions when passing through the constellation Taurus. It 
should be observed that the motion of the moon eastward among 
the stars is an actual motion, not merely an apparent one like 
that of the sun. The planets all move eastward among the 
stars, but since we ourselves are on a moving object the motion 
we see is a combination of the real motions of the planets around 




VIRGO 





-10 



Spica 



-15 



XIII XII 

FIG. 4. APPARENT PATH OF JUPITER FROM OCT., 1909 TO OCT., 1910. 

the sun and an apparent motion caused by the earth's revolution 
around the sun; the planets consequently appear at certain 
times to move westward (i.e., backward), or to retrograde. 
Fig. 4 shows the loop in the apparent path of the planet Jupiter 
caused by the earth's motion around the sun. It will be seen 
that the apparent motion of the planet was direct except from 
January to June, 1910, when it had a retrograde motion. 

5. Meaning of Terms East and West. 

In astronomy the terms " east " and " west " cannot be taken 
to mean the same as they do when dealing with directions in one 



THE CELESTIAL SPHERE 



plane. In plane surveying " east " and " west " may be con- 
sidered to mean the directions perpendicular to the meridian 
line. If a person at Greenwich 
(England) and another person at 
the 1 80 meridian should both 
point due east, they would actu- 
ally be pointing to opposite points 
of the sky. In Fig. 5 all four of 
the arrows are pointing east at the 
places shown. It will be seen from 
this figure that the terms " east " 
and " west " must therefore be 
taken to mean directions of ro- 
tation. 

6. The Earth's Orbital Motion. The Seasons. 

The earth moves eastward around the sun once a year in an 
orbit which lies (very nearly) in one plane and whose form is that 




FIG. 5. ARROWS ALL POINT 
EASTWARD 




b V 
FIG. 6. THE EARTH'S ORBITAL MOTION 

of an ellipse, the sun being at one of the foci. Since the earth is 
maintained in its position by the force of gravitation, it moves, as 
a consequence, at such a speecl in each part of its path that the 



8 



PRACTICAL ASTRONOMY 



, 



line joining the earth and sun moves over equal areas in equal 
times. In Fig. 6 all of the shaded areas are equal and the arcs 
aa'j bb', cc' represent the distances passed over in the same num- 
ber of days.* 

The axis of rotation of the earth is inclined to the plane of the 
orbit at an angle of about 66^, that is, the plane of the earth's 
equator is inclined at an angle of about 23^ to the plane of the 
orbit. This latter angle is known as the obliquity of the ecliptic. 
(See Chapter II.) The direction of the earth's axis of rotation 
is nearly constant and it therefore points nearly to the same 
place in the sky year after year. 

The changes in the seasons are a direct result of the inclination 
of the axis and of the fact that the axis remains nearly parallel 

Vernal Equinox 
(March 21) 



Summer Solstice 
(June 21) 

Aphell 




Perihelion (Dec. 31) 



Winter Solstice 
(Dec. 21) 



Autumnal Equinox 
(Sept. 22) 

FIG. 7. THE SEASONS 

to itself. When the earth is in that part of the orbit where the 
northern end of the axis is pointed away from the sun (Fig. 7) 
it is winter in the northern hemisphere. The sun appears to be 

* The eccentricity of the ellipse shown in Fig. 6 is exaggerated for the sake 
of clearness ; the earth's orbit is in reality much more nearly circular, the 
variation in the earth's distance from the sun being only about three per cent. 



THE CELESTIAL SPHERE 9 

farthest south about Dec. 21, and at this time the days are 
shortest and the nights are longest. When the earth is in this 
position, a plane through the axis and perpendicular to the plane 
of the orbit will pass through the sun. About ten days later the 
earth passes the end of the major axis of the ellipse and is at its 
point of nearest approach to the sun, or perihelion. Although 
the earth is really nearer to the sun in winter than in summer, 
this has but a small effect upon the seasons; the chief reasons 
why it is colder in winter are that the day is shorter and the 
rays of sunlight strike the surface of the ground more obliquely. 
The sun appears to be farthest north about June 22, at which 
time summer begins in the northern hemisphere and the days are 
longest and the nights shortest. When the earth passes the 
other end of the major axis of the ellipse it is farthest from the 
sun, or at aphelion. On March 21 the sun is in the plane of 
the earth's equator and day and night are of equal length at all 
places on the earth (Fig. 7). On Sept. 22 the sun is again in 
the plane of the equator and day and night are everywhere 
equal. These two times are called the equinoxes (vernal and 
autumnal), and the points in the sky where the sun's centre ap- 
pears to be at these two dates are called the equinoctial points, 
or more commonly the equinoxes. 

7. The Sun's Apparent Position at Different Seasons. 

The apparent positions of the sun on the celestial sphere 
corresponding to these different positions of the earth .are shown 
in Fig. 8. As a result of the sun's apparent eastward motion 
from day to day along a path which is inclined to the equator, 
the angular distance of the sun from the equator is continually 
changing. Half of the year it is north of the equator and half of 
the year it is south. On June 22 the sun is in its most northerly 
position and is visible more than half the day to a person in the 
northern hemisphere (/, Fig. 8). On Dec. 21 it is farthest south 
of the equator and is visible less than half the day (Z>, Fig. 8). 
In between these two extremes it moves back and forth across 
the equator, passing it about March 21 and Sept. 22 each year. 



10 



PRACTICAL ASTRONOMY 



The apparent motion of the sun is therefore a helical motion 
about the axis, that is, the sun, instead of following the path 
which would be followed by a fixed star, gradually increases or 
decreases its angular distance from the pole at the same time 
that it revolves once a day around the earth. The sun's motion 
eastward on the celestial sphere, due to the earth's orbital motion, 




FIG. 8. SUN'S APPARENT POSITION AT DIFFERENT SEASONS 

is not noticed until the sun's position is carefully observed with 
reference to the stars. If a record is kept for a year showing 
which constellations are visible in the east soon after sunset, 
it will be found that these change from month to month, and at 
the end of a year the one first seen will again appear in the east, 
showing that the sun has apparently made the circuit of the 
heavens in an eastward direction 

8. Precession and Nutation. 

While the direction of the earth's rotation axis is so nearly 
constant that no change is observed during short periods of 
time, there is hi reality a very slow progressive change in its 
direction. This change is due to the fact that the earth is not 
quite spherical in form but is spheroidal, and there is in conse- 
quence a ring of matter around the equator upon which the 
sun and the moon exert a force of attraction which tends to pull 
the plane of the equator into coincidence with the plane of the 
orbit. But since the earth is rotating with a high velocity and 



THE CELESTIAL SPHERE 



II 



resists this attraction, the actual effect is not to permanently 
change the inclination of the equator to the orbit, but first to 
cause the earth's axis to describe a cone about an axis per- 
pendicular to the orbit, and second to cause the inclination of 
the axis to go through certain periodic changes (see Fig. 9). The 
movement of the axis in a conical surface causes the line of 
intersection of the equator and the plane of the orbit to revolve 
slowly westward, the pole itself always moving directly toward 
the vernal equinox. This causes the equinoctial points to move 
westward in the sky, and hence the sun crosses the equator each 
spring earlier than it would otherwise; this is known as the 




-Plcme-of-Eartlfs-Orbit- 



FIG. 9. PRECESSION OF THE EQUINOXES 

precession of the equinoxes. In Fig. 9 the pole occupies suc- 
cessively the positions /, 2 and J, which causes the point V to 
move to points i, 2 and 3- This motion is but 50". 2 per year, 
and it therefore requires about 25,800 years for the pole to make 
one complete revolution. The force causing the precession is 
not quite constant, and the motion of the equinoctial points is 
therefore not perfectly uniform but has a small periodic varia- 
tion. In addition to this periodic change in the rate of the 
precession there is also a slight periodic change in the obliquity, 



12 



PRACTICAL ASTRONOMY 



called Nutation. The maximum value of the nutation is about 
9"; the period is about 19 years. The phenomenon of preces- 
sion is clearly illustrated by means of the apparatus called the 
gyroscope. As a result of the precessional movement of the 
axis all of the stars gradually change their positions with refer- 
ence to the plane of the equator and the position of the equinox. 
The stars themselves have but a very slight angular motion, 
this apparent change in position being due almost entirely to the 
change in the positions of the circles of reference. 

9. Aberration of Light. 

Another apparent displacement of the stars due to the earth's 
motion is what is known as aberration. On account of the 
rapid motion of the earth through space, the direction in which 
a star is seen by an observer is a result of the combined velocities 
of the observer and of light from the star. The star always 
appears to be slightly displaced in the direction in which the 
observer is actually moving. In Fig. 10, if light moves from C 
to B in the same length of time that the observer moves from 
A to B, then C would appear to be in the direction AC. This 





FIG. 10 



FIG. ii 



may be more clearly understood by using the familiar illustra- 
tion of the falling raindrop. If a raindrop is falling vertically, 
CB, Fig. n, and while it is falling a person moves from A to B, 
then, considering only the two motions, it appears to the person' 
that the raindrop has moved toward him in the direction CA. 
If a tube is to be held in such a way that the raindrop shall pass 
through it without touching the sides, it must be held at the 



THE CELESTIAL SPHERE 13 

inclination of AC. The apparent displacement of a star due 
to the observer's motion is similar to the change in the apparent 
direction of the raindrop. 

There are two kinds of aberration, annual and diurnal. 
Annual aberration is that produced by the earth's motion in its 
orbit and is the same for all observers. Diurnal aberration is 
due to the earth's daily rotation about its axis, and is different 
in different latitudes, because the speed of a point on the earth's 
surface is greatest at the equator and diminishes toward the pole. 

If v represents the velocity of the earth in its orbit and V the 
velocity of light, then when CB is at right angles to AB the 
displacement is a maximum and 

v 
tan a = > 

where a is the angular displacement and is called the "constant 
of aberration." Its value is about 20."$. If CB is not per- 
pendicular to AB, then 

v 
tan a = sin B, 

where a is the angular displacement and B is the angle ABC. 

Problem 

Referring to Fig. 2, make a sketch showing the path which Jupiter appears 
to describe, in the plane of its motion, but considering the earth as a fixed 
point on the diagram. 



CHAPTER II 

DEFINITIONS POINTS AND CIRCLES OF REFERENCE 

10. The following astronomical terms are in common use and 
are necessary in denning the positions of celestial objects on the 
sphere by means of spherical coordinates. 

Vertical Line. 

A vertical line at any point on the earth's surface is the direc- 
tion of gravity at that point, and is shown by the plumb line 
or indirectly by means of the spirit level (OZ, Fig. 12). 

Zenith Nadir. 

If the vertical at any point be prolonged upward it will pierce 
the sphere at a point called the Zenith (Z, Fig. 12). This point 
is of great importance because it is the point on the sphere which 
indicates the position of the observer on the earth's surface. 
The point where the vertical prolonged downward pierces the 
sphere is called the Nadir (N', Fig. 12). 

Horizon. 

The horizon is the great circle on the celestial sphere cut by 
a plane through the centre of the earth perpendicular to the 
vertical (NESW, Fig. 12). The horizon is everywhere 90 from 
the zenith and the nadir. It is evident that a plane through the 
observer perpendicular to the vertical cuts the sphere in this 
same great circle. The visible horizon is the circle where the 
sea and sky seem to meet. Projected onto the sphere it is a 
small circle below the true horizon and parallel to it. Its dis- 
tance below the true horizon depends upon the height of the 
observer's eye above the surface of the water. 

Vertical Circles. 

Vertical Circles are great circles passing through the zenith 
and nadir. They all cut the horizon at right angles (HZJ, 
Fig. 12). 

14 



POINTS AND CIRCLES OF REFERENCE 1 5 

Almucantars. 

Parallels of altitude, or almucantars, are small circles parallel 
to the horizon (DFG, Fig. 12). 

Poles. 

If the earth's axis of rotation be produced indefinitely it will 
pierce the sphere in two points called the celestial poles (PP' 
Fig. 12). 

Equator. 

The celestial equator is a great circle of the celestial sphere 
cut by a plane through the centre of the earth perpendicular to 




FIG. 12. THE CELESTIAL SPHERE 



the axis of rotation (QWRE, Fig. 12). It is everywhere 90 
from the poles. A parallel plane through the observer cuts the 
sphere in the same circle. 



1 6 PRACTICAL ASTRONOMY 

Hour Circles. 

Hour Circles are great circles passing through the north and 
south celestial poles (PVP f , Fig. 12). 

Parallels of Declination. 

Small circles parallel to the plane of the equator are called 
parallels of decimation (BKC, Fig. 12). 

Meridian. 

The meridian is the great circle passing through the zenith and 
the poles (SZPL, Fig. 12). It is at once an hour circle and a 
vertical circle. It is evident that different observers will in 
general have different meridians. The meridian cuts the horizon 
in the north and south points (N,S, Fig. 12). The intersection 
of the plane of the meridian with the horizontal plane through 
the observer is the meridian line used in plane surveying. 

Prime Vertical. 

The prime vertical is the vertical circle whose plane is per- 
pendicular to the plane of the meridian (EZW, Fig. 12). It 
cuts the horizon in the east and west points (E, W, Fig. 12). 

Ecliptic. 

The ecliptic is the great circle on the celestial sphere which 
the sun's centre appears to describe during one year (AMVL, 
Fig. 12). Its plane is the plane of the earth's orbit; it is inclined 
to the plane of the equator at an angle of about 23 27', called the 
obliquity of the ecliptic. 

Equinoxes. 

The points of intersection of the ecliptic and the equator are 
called the equinoctial points or simply the equinoxes. That 
intersection at which the sun appears to cross the equator when 
going from the south side to the north side is called the Vernal 
Equinox, or sometimes the First Point of Aries (V, Fig. 12). 
The sun reaches this point about March 21. The other inter- 
section is called the Autumnal Equinox (A, Fig. 12). 

Solstices. 

The points on the equator midway between the equinoxes are 
called the winter and summer solstices. 



POINTS AND CIRCLES OF REFERENCE \J 

Colures. 

The great circle through the poles and the equinoxes is called 




FIG. 12. THE CELESTIAL SPHERE 

the equinoctial colure (PVP', Fig. 12). The great circle through 

the poles and the solstices is called the solstitial colure. 

i 

Questions 

1. What imaginary circles on the earth's surface correspond to hour circles? 
To parallels of declination? To vertical circles? 

2. What are the widths of the torrid, temperate and arctic zones and how are 
they determined? 



CHAPTER III 

SYSTEMS OF COORDINATES ON THE SPHERE 

ii. Spherical Coordinates. 

The direction of a point in space may be denned by means 
of two spherical coordinates, that is, by two angular distances, 
measured on a sphere along arcs of two great circles which 
cut each other at right angles. Suppose that it is desired to 
locate C (Fig. 13) with reference to the plane OAB and the line 




B -^^A. PRIMARY 

FIG. 13. SPHERICAL COORDINATES 

OA, O being the origin of coordinates. Pass a plane OBC 
through C and perpendicular to OAB: these planes will intersect 
in the line OB. The two angles which fix the position of C, or 
the spherical coordinates, are BOC and AOB. These may be 
regarded as the angles at the centre of the sphere or as the arcs 
BC and AB. In every system of spherical coordinates the two 
coordinates are measured, one on a great circle called the primary, 
and the other on one of a system of great circles at right angles 
to the primary called secondaries. There are an infinite number 
of secondaries, each passing through the two poles of the primary. 
The coordinate measured from the primary is. an arc of a 

18 



SYSTEMS OF COORDINATES ON THE SPHERE 19 

secondary circle; the coordinate measured between the secondary 
circles is an arc of the primary. 

12. Horizon System. 

In this system the primary circle is the horizon and the sec- 
ondaries are vertical circles, or circles passing through the zenith 
and nadir. The first coordinate of a point is its angular distance 
above the horizon, measured on a vertical circle; this is called 
the Altitude. The complement of the altitude is called the 
Zenith distance. The second coordinate is the angular distance 
on the horizon between the meridian and the vertical circle 
through the point; this is called the Azimuth. Azimuth may be 
reckoned either from the north or the south point and in either 
direction, like bearings in surveying, but the custom is to reckon 
it from the south point right-handed from o to 360 except for 
stars near the pole, in which case it is more convenient to reckon 




Azimuth 
FIG. 14. THE HORIZON SYSTEM 

from the north, and either to the east or to the west. In Fig. 14 
the altitude of the star A is BA ; its azimuth is SB. 

13. The Equator Systems. 

The circles of reference in this system are the equator and 
great circles through the poles, or hour circles. The first coor- 
dinate of a point is its angular distance north or south of the 



20 



PRACTICAL ASTRONOMY 



equator, measured on an hour circle; it is called the Declination. 
Declinations are considered positive when north of the equator, 
negative when south. The complement of the declination is 
called the Polar Distance. The second coordinate of the point 
is the arc of the equator between the vernal equinox and the foot 
of the hour circle through the point; it is called Right Ascension. 
Right ascension is measured from the equinox eastward to the 
hour circle through the point in question ; it may be measured in 
degrees, minutes, and seconds of arc, or in hours, minutes, and 




FIG. 15. THE EQUATOR SYSTEM 

seconds of time. In Fig. 15 the decimation of the star S is AS; 
the right ascension is VA. 

Instead of locating a point by means of declination and right 
ascension it is sometimes more convenient to use declination 
and Hour Angle. The hour angle of a point is the arc of the 



SYSTEMS OF COORDINATES ON THE SPHERE 



21 



equator between the observer's meridian and the hour circle 
through the point. It is measured from the meridian westward 
(clockwise) from o h to 24* or from o to 360. In Fig. 16 the 
declination of the star S is AS (negative); the hour angle is 




FIG. 1 6. HOUR ANGLE AND DECLINATION 

MA. It is evident that the hour angles of all points on the 
celestial sphere are always increasing. 

These three systems are shown in the following table. 



Name. 


Primary. 


Secondaries. 


Origin of 
Coordinates. 


ist coord. 


and coord. 


Horizon System 


Horizon 
Equator 


Vert. Circles 
Hour Circles 


South point. 
Vernal Equi- 


Altitude 
Declin. 


Azimuth 
Rt. Ascen. 








nox. 






Equator Systems 


M 


u n 


Intersection 
of Meridian 


" 


Hour Angle 








and Equator. 







22 



PRACTICAL ASTRONOMY 



14. There is another system which is employed in some 
branches of astronomy but will not be used in this book. The 
coordinates are called celestial latitude and celestial longitude; 
the primary circle is the ecliptic. Celestial latitude is measured 
from the ecliptic just as declination is measured from the equator. 
Celestial longitude is measured eastward along the ecliptic from 
the equinox, just as right ascension is measured eastward along 
the equator. The student should be careful not to confuse celes- 
tial latitude and longitude with terrestrial latitude and longitude. 
The latter are the ones used in the problems discussed in this book. 

15. Coordinates of the Observer. 

The observer's position is located by means of his latitude and 
longitude. The latitude, which on the earth's surface is the 
angular distance of the observer north or south of the equator, 
may be defined astronomically as the declination of the ob- 
server's zenith. In Fig. 17, the terrestrial latitude is the arc EO, 




r' 



FIG. 17. THE OBSERVER'S LATITUDE 

EQ being the equator and O the observer. The point Z is the 
observer's zenith, so that the latitude on the sphere is the arc 
E'Z, which evidently will contain the same number of degrees 
as EO. The complement of the latitude is called the Co-latitude. 



SYSTEMS OF COORDINATES ON THE SPHERE 23 

The terrestrial longitude of the observer is the arc of the equator 
between the primary meridian (usually that of Greenwich) and 
the meridian of the observer. On the celestial sphere the longi- "1 
tude would be the arc of the celestial equator contained between ( 
two hour circles whose planes are the planes of the two terrestrial^ 
meridians. 

1 6. Relation between the Two Systems of Coordinates. 

In studying the relation between different points and circles 
on the sphere it may be convenient to imagine that the celestial 
sphere consists of two spherical shells, one within the other. 




FIG. 18. THE SPHERE SEEN FROM THE OUTSIDE 

The outer one carries upon its surface the ecliptic, equinoxes, 
poles, equator, hour circles and all of the stars, the sun, the moon 
and the planets. On the inner sphere are the zenith, horizon, 
vertical circles, poles, equator, hour circles, and the meridian. 
The earth's daily rotation causes the inner sphere to revolve, 



24 PRACTICAL ASTRONOMY 

while the outer sphere is motionless, or, regarding only the 
apparent motion, the outer sphere revolves once per day on its 
axis, while the inner sphere appears to be motionless. It is 
evident that the coordinates of a fixed star in the first equatorial 
system (Declination and Right Ascension) are practically always 
the same, whereas the coordinates in the horizon system are 
continually changing. It will also be seen that in the first 
equatorial system the coordinates are independent of the ob- 
server's position, but in the horizon system they are entirely 
dependent upon his position. In the second equatorial system 
one coordinate is independent of the observer, while the other 
(hour angle) is not. In making up catalogues of the positions 
of the stars it is necessary to use right ascensions and declina- 
tions in defining these positions. When making observations 




E R w 

FIG. 19. THE SPHERE SEEN FROM THE EARTH (LOOKING SOUTH) 



with instruments it is usually simpler to measure coordinates 
in the horizon system. Therefore it is necessary to be able to 
compute the coordinates of one system from those of another. 
The mathematical relations between the spherical coordinates 
are discussed in Chapter IV. 



SYSTEMS OF COORDINATES ON THE SPHERE 25 

Figs. 18, 19, and 20 show three different views of the celestial 
sphere with which the student should be familiar. Fig.. 18 is 
the sphere as seen from the outside and is the view best adapted 
to showing problems in spherical trigonometry. The star 5 has 
the altitude RS, azimuth S'R, hour angle Mm, right ascension 
Vm, and declination mS; the meridian is ZMS' . Fig. 19 shows 
a portion of the sphere as seen by an observer looking southward; 
the points are indicated by the same letters as in Fig. 18. Fig. 20 




w 



FIG. 20. THE SPHERE PROJECTED ONTO THE PLANE OF THE EQUATOR 



shows the same points projected on the plane of the equator. 
In this view of the sphere the angles at the pole (i.e., the 
angles between hour circles) are shown their true size, and 
it is therefore a convenient diagram to use when dealing with 
right ascension and hour angles. 



26 



PRACTICAL ASTRONOMY 



Questions and Problems 

1 . What coordinates on the sphere correspond to latitude and longitude on the 
earth's surface? 

2. Make a sketch of the sphere and plot the position of a star having an altitude 
of 20 and an azimuth of 250. Locate a star whose hour angle is i6 h and whose 
declination is 10. Locate a star whose right ascension is 9** and whose declina- 
tion is N. 30. 

3. If a star is on the equator and also on the horizon, what is its azimuth? Its 
altitude? Its hour angle? Its declination? 



CHAPTER IV 



RELATION BETWEEN COORDINATES 

17. Relation between Altitude of Pole and Latitude of Ob- 
server. 

In Fig. 21, SZN represents the observer's meridian; let P be 
the celestial pole, Z the zenith, E the point of intersection of the 
meridian and the equator, and 
N and S the north and south 
points of the horizon. By the 
definitions, OZ (vertical) is 
perpendicular to SN (horizon) 
and OP (axis) is perpendicular 
to EO (equator). Therefore 
the arc PN = arc EZ. By the 



E 





FIG. 22 

definitions, EZ is the declination of the -zenith, or the latitude, 
and PN is the altitude of the celestial pole. Hence the altitude 
of the pole is always equal to the latitude of the observer. The same 
relation maybe seen from Fig. 22, in which P is the north pole 
of the earth, OH is the plane of the horizon, the observer being 
at O, EQ is the earth's equator, and OP' is a line parallel to CP 
and consequently points to the celestial pole. It may readily 
be shown that ECO, the observer's latitude, equals HOP', the 
altitude of the celestial pole. A person at the equator would 

27 



28 



PRACTICAL ASTRONOMY 



see the north celestial pole in the north point of his horizon and 
the south celestial pole in the south point of his horizon. If he 
travelled northward the north pole would appear to rise, its 
altitude being always equal to his latitude, while the south pole 
would immediately go below his horizon. When the traveller 
reached the north pole of the earth the north celestial pole 
would be vertically over his head. 

To a person at the equator all stars would appear to move 
vertically at the times of rising and setting, and all stars would 
be above the horizon i2 h and below i2 h during one revolution 



(S.Pole) S 




N (N.Pole) 



FIG. 23. THE RIGHT SPHERE 

of the sphere. All stars in both hemispheres would be above 
the horizon at some time every day. This is called the " right 
sphere" (Fig. 23). 

If a person were at the earth's pole the celestial equator would 
coincide with his horizon, and all stars in the northern hemi- 
sphere would appear to travel around in circles parallel to the 
horizon; they would be visible for 24* a day, and their altitudes 
would not change. The stars in the southern hemisphere would 
never be visible. The word north would cease to have its usual 



RELATION BETWEEN COORDINATES 



2 9 



meaning, and south might mean any horizontal direction. The 
longitude of a point on the earth and its azimuth from the 
Greenwich meridian would then be the same. This is called 
the "parallel sphere" (Fig. 24). 

At all points between these two extreme latitudes the equator 
cuts the horizon obliquely. A star on the equator will be above 




FIG. 24. THE PARALLEL SPHERE 

the horizon half the time and below half the time. A star north 
of the equator will (to a person in the northern hemisphere) be 
above the horizon more than half of the day; a star south of the 
equator will be above the horizon less than half of the day. If 
the north polar distance of a star is less than the observer's north 
latitude, the whole of the star's diurnal circle is above the hori- 
zon, and the star will therefore remain above the horizon all 
of the time. It is called in this case a circumpolar star (Fig. 
25). The south circumpolar stars are those whose south polar 
distances are less than the latitude; they are never visible to an 
observer in the northern hemisphere. If the observer travels 



PRACTICAL ASTRONOMY 



north until he is beyond the arctic circle, latitude 66 33' north, 
then the sun becomes a circumpolar at the time of the summer 
solstice. At noon the sun would be at its maximum altitude; 
at midnight it would be at its minimum altitude but would still 
be above "the horizon. This is called the " midnight sun." 

z 




Circumpolars 
(Never Rise) 



FIG. 25. CIRCUMPOLAR STARS 

18. Relation between Latitude of Observer and the Declina- 
tion and Altitude of a Star on the Meridian. 

The relation between the latitude, altitude, and declination 
at the instant when a star is crossing the observer's meridian may 
be seen from Fig. 26. Let A be a star on the meridian, south of 
the zenith and north of the equator; then 

EZ = L, the latitude, 

EA = D, the declination, 

SA = h, the altitude, 

ZA = z, the zenith distance. 



From the figure 

or 

and 

also 



ZA = EZ - EA 
z = L - D 

h = 90 - (L - D}; 
L = 90 - (h - D). 



[i] 

[2] 



RELATION BETWEEN COORDINATES 3 1 

If A is south of the equator the declination is considered 
negative, so the same equation will hold true for this case. 




FIG. 26. STAR ON THE MERIDIAN 



If the star is north of the zenith, as at B, it wil^be more con- 
venient to use the polar distance,* = 90 D. 



In this case NP = NB - PB 

or L h p. 

If B is below the pole the equation is 

L = h + p. 



[3] 
[4] 



19. The Astronomical Triangle. 

By joining the pole, zenith, and any star 5 on the sphere 
by arcs of great circles we obtain a triangle from which the rela- 
tion existing among the spherical coordinates may be obtained. 
This triangle is so frequently employed in astronomy and navi- 
gation that is it called the "astronomical triangle" or the "PZS 
triangle." In Fig. 27 the arc PZ is the complement of the 
latitude, or co-latitude; arc ZS is the zenith distance or comple- 
ment of the altitude; arc PS is the polar distance or complement 
of the declination; the angle P is the hour angle of the star if 
.S is west of the meridian, or 360 minus the hour angle if 
S is east of the meridian; and Z is the azimuth of S, or 360 
minus the azimuth, according as S is west or east of the meridian. 
The angle at S is called the parallactic angle ; it is little used in 
practical astronomy. If any three parts of this triangle are 



PRACTICAL ASTRONOMY 



known the other three may be calculated. The fundamental 
formulae of spherical trigonometry are 



cos a = cos b cos c + sin b sin c cos A , 
sin a cos B = cos b sin c sin b cos c cos ^4 , 
sin a sin 5 = sin b sin ^4 . 



L5J 
[6] 
[7] 



If we put A = P, B = S, C = Z, a = 90 - h, b = 90 - L, 
c = 90 D, then these become 

(sin h = sin L sin D + cos L cos D cos P, [8] 

cos /* cos S = sin L cos Z) cos L sin Z) cos P, [9] 

cos A sin 5 1 = cos L sin P. [10] 

If A = P, B = Z, C = S, a = 90 - h, b = 90 - D, c = 90 - "L, 
then 

cos h cos Z = sin D cos Z, cos D sin L cos P, [n] 

cos A sin Z = cos Z) sin P. [12] 

If ^ = Z, B = S, C = P, a = 90 - D, b = 90 - L, c = 90 - h, 
then 

sin D = sin L sin h + cos Z, cos h cos Z, [13] 

cos D cos 5 = sin L cos ^ cos Lsin k cos Z, [14] 

cos D sin 5 = cos L sin Z. [15] 

Other forms may be derived by assigning different values to 
the parts of the triangle ABC. The formulae given in the 
following chapters may in nearly all cases be derived from 
equations [5] to [15]. 

The most common cases arising in the practice of surveying 
are: 

1. Given the declination, latitude, and altitude, to find the 
azimuth and the hour angle. 

2. Given the declination, latitude, and hour angle, to find the 
azimuth and the altitude. 



RELATION BETWEEN COORDINATES 



33 



In the following formulae 



let 



and also let 



P = the hour angle, 

'Z = the azimuth,* 

^h = the altitude, 

z = the zenith distance, 

D = the declination, 

p = the polar distance, 

L = the latitude, 




FIG. 27. . THE ASTRONOMICAL TRIANGLE 

For computing P any of the following formulae may be used, 
sin \ P = y/V sin H* + (L - D)} sin * [z-(L=M\ , [l6] 

V I T r\ 

, ? \ cos L cos Z> / 

* In the formulae which follow Z is reckoned from the north (interior angle) 
unless otherwise designated. 



34 



PRACTICAL ASTRONOMY 



sin * P = t/fcosjsinfr-j 
cos L cos Z) 



cos * P = 



CQS 



- />) sin (5 - 



cos D cos 



vers P = 



cos 5 sin (s h) 
,sin (s L) cos (5 

cos (L D) sin h 
cos Z cos 'D 



[18] 
[19] 

[201 



For computing the angle Z (measured from the north point) we 
have 



n 



_ ___ 
i 7 _ //sin* (* + L - D) cosH* + L + Z?)\ 



cos L sn 2 



in Z = 4 /sin (5 - A) sin (5 - L)\ 
* / 



sn 



cos L cos 



\ Z = J(cs S co S (s-p)\ 
y V cos L cos h I 



cos 



tan i Z = 



^ ~ sl 5 ~ 
cos ^ cos (s p) 



^ cos (L + h) + sin Z) 
,versZ g * = - r- - - 

cos L cos h 



[22] 

[23] 
[24] 
[25] 



While any of these formulae may be used to determine the angle 
sought, the choice of formula should depend somewhat upon 
the precision with which the angle is denned by the function. 
If the angle is quite small it is more accurately found through its 
sine than through its cosine; for an angle near 90 the reverse 
is the case. On account of the rapid variation of the tangent 
an angle is always more precisely determined by this function 
than by either the sine or the cosine. The versed sine formulae 
require the use of both natural and logarithmic functions, but 
are sometimes convenient. 



* In this case Z is reckoned from the south. 



RELATION BETWEEN COORDINATES 35 

For computing the altitude and azimuth the following for- 
mulae may be used: 

cos M tan P ... 

tan Z 8 = -T 77 - . * 26 

sin (L M) 

, cos Z g 

tan k = tan (L-lff 

where M is an auxiliary angle such that tan M = - ; Z s is 

cos P 

measured from the south point. 

The altitude may also be found from the formula 

sin h = cos (L D) 2 cos L cos D sin 2 \ P [28] 

or sin h = cos (L D) cos L cos D vers P, [29] 

which may be derived from Equa. [8]. 

If the declination, hour angle, and altitude are given, the 
azimuth is found from 

. ,, . ^cos D 
sin Z = sin P - 
cos h 

= sin P cos D sec h. [30] 

For computing the azimuth of a star near the pole when the 
hour angle is known the following formula is frequently used: 

sin P 



,, 
tan Z = 



cos L tan D sin L cos P 



This equation may be derived by dividing [12] by [n] and then 
simplifying the result by dividing by cos D. 

Given the latitude and declination, find the hour angle and 
azimuth of a star on the horizon. Putting h = o in Equa. [8] 
and [13] the results are 

cos P = tan D tan L [32] 

~ sin D T -, 

and cos Z = -- - L33 

cos L 

* For the derivation of this formula see Chauvenet's Spherical and Practical 
Astronomy, Vol. I, Art. 14. 



PRACTICAL ASTRONOMY 



A special case of the PZS triangle occurs when a star near the 
pole (circumpolar) is at its greatest east or west position, known 
as its greatest elongation. At this time the star's bearing or 
azimuth is a maximum and its diurnal circle is tangent to the 




E HORIZON A N 

FIG. 28. STAR AT GREATEST ELONGATON (EAST) 

vertical circle through the star (Fig. 28) ; the triangle is conse- 
quently right-angled at S. 
The formulae for this case are 



D 

cos p = 



tan L 



and \^ (sinZ = sin p sec L. j> [35] 

20. Relation between Right Ascension and Hour Angle. 

In order to understand the relation between the right ascen- 
sion and the hour angle of a point, we may think of the equator 
on the outer sphere as graduated into hours, minutes, and seconds 
of right ascension, zero being at the equinox and the numbers 
increasing toward the east. The equator on the inner sphere is 
graduated for hour angles, the zero being at the observer's 
meridian and the numbers increasing toward the west. (See 
Fig. 29.) As the outer sphere turns, the hour marks on the right 
ascension scale will pass the meridian in the order of the numbers. 
The number opposite the meridian at any instant shows how far 




37 



FIG. 29. RIGHT ASCENSION AND HOUR ANGLE 




FIG. 30 



38 PRACTICAL ASTRONOMY 

the sphere has turned since the equinox was on the meridian. 
If we read the hour angle scale opposite the equinox, we obtain 
exactly the same number of hours. This number of hours (or 
angle) may be considered as either the right ascension of the 
meridian or the hour angle of the equinox. In Fig. 30 the star 
S has an hour angle equal to AB and a right ascension CB. The 
sum of these two angles is AC, or the hour angle of the equinox. 
The same relation will be found to hold true for all positions of 
5. The general relation existing between these coordinates is, 
then, 

Hour angle of Equinox = Hour angle of Star + Right Ascen- 
sion of Star. [36] 

Questions and Problems 

1. What is the greatest declination a star may have and culminate south of the 
zenith? 

2. What angle does the plane of the equator make with the horizon? 

3. In what latitudes can the sun be overhead? 

4. What is the altitude of the sun at noon in Boston (42 21' N.) on 
December 22? 

5. What are the greatest and least angles made by the ecliptic with the horizon 
at Boston? 

6. In what latitudes is Vega (Decl. = 38 42' N.) a circumpolar star? 

7. Make a sketch of the celestial sphere as it appears to an observer in latitude 
20 South at the instant the vernal equinox is on the eastern horizon. 

8. Derive formula [35]. 






CHAPTER V 
MEASUREMENT OF TIME 

21. The Earth's Rotation. 

The measurement of intervals of time is made to depend upon 
the period of the earth's rotation on its axis. Although it is 
probable that this period is not absolutely invariable, yet the 
variations are too small to be measured, and the rotation is 
assumed to be uniform. The most natural unit of time for 
ordinary purposes is the solar day, or the time of one rotation 
of the earth with respect to the sun's direction. On account of 
the earth's annual motion around the sun the direction of the 
reference line is continually changing, and the length of the 
solar day is not the true time of one rotation of the earth on its ) 
axis. For this reason it is necessary in astronomical work to 
make use of another kind of time, based upon the actual period of 
rotation, called sidereal time (star time). 

22. Transit or Culmination. 

Every point on the celestial sphere crosses the meridian of an 
observer twice during one revolution of the sphere. The instant 
when any point on the celestial sphere is on the meridian of an 
observer is called the transit, or culmination, of that point over 
that meridian. When it is on that half of the meridian contain- 
ing the zenith, it is called the upper transit; when it is on the 
other half it is called the lower transit. Except in the case of 
stars near the elevated pole the upper transit is the only one 
visible to the observer; hence when the transit of a star is men- 
tioned the upper transit will be understood unless the contrary 
is stated. 

23. Sidereal Day. 

The sidereal day is the interval of time between two successive 
upper transits of the vernal equinox over the same meridian. 

39 



40 PRACTICAL ASTRONOMY 

If the equinox were absolutely fixed in position, the sidereal day 
as thus denned would be the true period of the earth's rotation; 
but since the equinox has a slow westward motion caused by the 
precessional movement of the axis (see Art. 8), the actual 
interval between two transits of the equinox differs about 
o s .oi from the true time of one rotation. The sidereal day actu- 
ally used in practice, however, is the one denned above and not 
the true rotation period. Sidereal days are not used for reckon- 
ing long periods of time, dates always being in solar days, so this 
error never becomes appreciable. The sidereal day is divided 
into 24 hours and each hour is subdivided into minutes and 
seconds. When the equinox is at upper transit it is O A , or the 
beginning of the sidereal day (sidereal " noon "). 

24. Sidereal Time. 

The sidereal time at a given meridian at any instant is the 
hour angle of the vernal equinox. It is therefore a measure of 
the angle through which the earth has turned since the equinox 
was on the meridian, and shows the position of the sphere at 
the given instant with respect to the observer's meridian. 

25. Solar Day. 

A solar day is the interval of time between two successive upper 
transits of the sun's centre over the same meridian. It is divided 
into 24 hours, each hour being divided into minutes and seconds. 
When the sun is on the upper side of the meridian (upper 
transit) it is noon, or o h solar time. When it is on the lower side 
of the meridian it is midnight. 

26. Solar Time. 

The solar time at a given meridian at any instant is the hour 
angle of the sun's centre at that instant. This hour angle is a 
measure of the angle through which the earth has turned with 
respect to the sun's direction, and consequently is a measure of 
the time elapsed since the sun was on the meridian. 

Since the earth revolves around the sun in an elliptical orbit 
in accordance with the law of gravitation, the apparent angular 
motion of the sun is not uniform, and the days are therefore of 



MEASUREMENT OF TIME 41 

unequal length at different seasons. In former times, when sun 
dials were considered sufficiently accurate for measuring time, 
this lack of uniformity was not important. Under modern 
conditions, which demand accurate measurement of time by the 
use of clocks, an invariable unit of time is essential. As a con- 
sequence, the time adopted for common use is that kept by a 
fictitious sun, or mean sun, which is conceived to move at a 
uniform rate along the equator,* its speed being such that it 
makes one apparent revolution around the earth in the same time 
as the true sun (i.e., one year). The fictitious sun is so placed 
that on the whole it precedes the true sun as much as it follows 
it. The time indicated by the position of the mean sun is called 
mean solar time, or simply mean time. The time indicated by 
the position of the real sun is called apparent solar time and is 
the time shown by a sun dial. 

27. Equation of Time. 

Since observations made on the sun for the purpose of deter- 
mining the time can give apparent time only, it is necessary to be 
able to find at any instant the exact relation between apparent 
and mean time. The difference between the two, which varies 
from +i6 m to i6 m (nearly), is called the equation of time. 
This quantity may be found in the Nautical Almanac for each 
day of the year. 

This difference between the two kinds of time is due to several 
causes, the chief of which are (i) the inequality of the earth's 
angular motion in the orbit, and (2) the fact that the true sun 
is on the ecliptic while the mean sun is on the equator. In the 
winter, when the earth is nearest the sun, the rate of angular 
motion about the sun must be greater than in summer in order 
that the radius vector shall describe equal areas in equal inter- 
vals of time. (See Fig. 6 and Art. 6.) The sun will then appear 

* This statement is true in a general way, but the motion is not strictly uniform 
because the motion of the equinox itself is variable. The angle from the equinox 
to the " mean sun " at any instant is the sun's " mean longitude " (along the 
ecliptic) plus periodic terms. 



42 PRACTICAL ASTRONOMY 

to move eastward in the sky at a faster rate than in summer, 
and its daily revolution about the earth will be slower. This 
delays the instant of apparent noon, making the apparent solar 
days longer than their average, and therefore a sun dial will 
" lose time." About April i the sun is moving at its average 
speed and the sun dial ceases to lose time; from this date until 
about July i the sun dial gains on mean time, making up what 
it lost between Jan. i and April i. During the other half of the 
year the process is reversed; the sun dial gains from July i to 
Oct. i and loses from Oct. i to Jan. i. The maximum difference 
in time due to this cause is about 8 minutes, either + or . 

The second cause of the equation of time is illustrated by 
Fig. 31. Assume that point S' (sometimes called the " first 




FIG. 31 

mean sun") moves uniformly along the ecliptic at the average 
rate of the true sun; the time as indicated by this point will 
evidently not be affected by the eccentricity of the orbit. If 
the mean sun S (also called " the second mean- sun") starts at 
V, the equinox, at the same instant that S' starts, then the arcs 
VS and VS' are equal, since both points are moving with the 
same speed. By drawing hour circles through these two points 
it will be seen that these hour circles do not coincide except 
when the points are at the equinoxes or at the solstices. Since 
the points are not on the same hour circle they will not cross the 
meridian at the same time, the difference in time being repre- 






MEASUREMENT OF TIME 



43 



sented by the arc aS. The maximum length of aS is about 
10 minutes of time, which may be either + or . The com- 
bined effect of these two causes, or the equation of time, is shown 
in the following table. 

TABLE A. EQUATION OF TIME FOR 1910. 





ISt. 


ioth. 


20th. 


30th. 


January 


+ yn 26s 


+ ym 2 ?s 


+ II* 02 


+ I3 TO 22* 


February 


+ 13 4-i 


+ 14 24 


+ IT. S.Q 




March 


+ 12 ^8 


+ 10 36 


+ 7 48 


+ 4 4^ 


April 


+ 4 08 


+ 1 31 


- o <S 


2 47 


May 


2 CS 


3 42 


3 42 


- 2 48 


June 


2 31 


- o ?7 


+ i 08 


+ * i; 


July . 


+ 3 27 


-'- ^ 01 


+ 6 06 


+ 6 16 


August 


+ 6 n 


+ 5 IQ 


+ * 26 


+ o 46 


September 


+ o oq 


- 2 48 


6 20 


o 46 


October 


10 o^ 


12 4? 


i? 01 


16 13 


November 


-16 18 


16 02 


14 26 


ii 28 


December 


ii 06 


7 2T. 


2 36 


+ 2 21 













28. Conversion of Apparent Time into Mean Time and vice 
versa. 

Apparent time may be converted into mean time by adding 

or subtracting the equation of time at the instant. Since the 

equation of time is given in the Nautical Almanac for Greenwich 

noon its value at the desired instant must be found by adding 

\ or subtracting the increase or decrease since Greenwich noon. 

Example i. Find the mean time of the sun's transit over the meridian of Boston 
on June 30, 1910. The apparent time at Boston is 12^ oo m oo s M. at the instant 
of the transit of the sun's centre, and this corresponds to 4 h 44 iS s apparent 
time at Greenwich, since the longitude of Boston is 4^ 44 i8 s west of Greenwich. 
The equation of time at Greenwich Apparent Noon is 3"* i4 s -92 (to be added to 
apparent time); the hourly change is o s .5oo (increasing). The correction to be 
applied to the equation of time is 4^.74 X o s .5oo = 2 s -37, making the equation of 
time at Boston noon 3"* 17*. 29. 

L. A. T.* = 12^00 oo s .oo 
Equa. of. T = 



3 17 -29 



L. M. T. = 12* 03 1 7 s . 29 



* A list of abbreviations will be found on p. 191. 




44 PRACTICAL ASTRONOMY 

Example 2. Find the local apparent time at Boston at 2 P.M. (local mean time) 
Oct. 28, 1910. The Greenwich Mean Time corresponding to 2 P.M. local mean 
time is 6 h 44"* i8 s P.M. The equation of time at G. M. N. Oct. 28, 1910, is 
i6 m 04 s .29 (to be added to mean time); the hourly increase is o s .2o8. The correc- 
tion to the equation of time is 6^.7 4 X o*.2o8 = i s .4o. The equation of time at 
2 P.M. is therefore i6 TO c>5 s .69. 

L. M. T. = 2 h oo m oo s .oo 

Equa. of T. = 16 05 .69 

L. A. T. = 2 ft i6 m o5 s .69 

29. Astronomical and Civil Time. 

For ordinary purposes it is found convenient to divide the 
solar day into two parts of i2 h each; from midnight to noon is 
called A.M. (ante meridiem), and from noon to midnight is 
called P.M. (post meridiem). The date changes at the instant 
of midnight. This mode of reckoning time is called Civil Reck- 
oning. In astronomical work this subdivision of the day is not 
convenient. For simplicity in calculation the day is divided 
into 24^, numbered consecutively from o h to 24^. As it is not 
convenient to have the date change during the night, the astro- 
nomical date begins at noon or o h . This is called Astronomical 
Time. In using the Nautical Almanac it should be remembered ^ 
that it is necessary to change the date and hours to astronomical y 
time before taking out the desired data. In order to change * 
from one kind of time to the other it is only necessary to remem- 
ber that the astronomical slay begins at noon of the civil day of 
the same date; that is, in the afternoon the dates and the hours 
will be the same, but in the forenoon the astronomical date is 
one day less and the hours are 1 2 greater than in the civil time. 

Examples. 

Astr. Time May 10, 15* = Civil Time May u, 3* A.M. 
" Jan. 3, 7* = " Jan. 3, 7 h P.M. 

From these examples the following rules may be derived : 
To change Civil Time to Astronomical Time, 
If A.M., add i2 h and drop i day from date, and drop the A.M. 
If P.M., drop the P.M. 



MEASUREMENT OF TIME 45 

To change Astronomical Time to Civil Time* 

If less than i2 h , mark it P.M. 

If greater than i2 h , subtract i2 h , add i day to date, and mark 
it A.M. 

30. Relation between Longitude and Time. 

The hour angle of the sun at any given meridian at a given 
instant is the local solar time at that meridian, and will be 
apparent or mean time according as the true sun or the mean 
sun is considered. The hour angle of the sun at Greenwich at 
the same instant is the corresponding Greenwich solar time. 
The difference between the two hour angles is the longitude 
of the place from Greenwich, expressed either in degrees or in 
hours according as the hour angles themselves are expressed 
in degrees or in hours. Similarly the difference in local solar 
time of any two places at a given instant is their difference in 
longitude in hours, minutes, and seconds. In Fig. 32, AC is 
the hour angle of the sun at Greenwich (G), or the Greenwich 
solar time. EC is the hour angle of the sun at the meridian 
through P, or the local solar time at P. The difference, AB, 
is the longitude of P west of Greenwich. It should be observed 
that the reasoning is exactly the same whether C represents the 
true sun or the fictitious sun. The same result would also be 
found if the point C were to represent the vernal equinox. The 
arc AC would then be the hour angle of the equinox, i.e., the 
Greenwich Sidereal Time. BC would be the Local Sidereal 
Time, and AB the difference in longitude. The measurement 
of longitude is therefore independent of the kind of time used, 
because in each case the angular distances to A and B are meas- 
ured from the same point C on the equator, and the difference in 
these angles does not depend upon the position of this point 
nor upon the speed with which this point has moved up to the 
position at C. 



* The student may find it helpful to plot the time along a straight line, and to 
write two sets of numbers, one for Civil Dates and the other for Astronomical Dates. 



4 6 



PRACTICAL ASTRONOMY 



The difference in the sidereal times at meridian A and meridian 
B (Fig. 32) is the interval of sidereal time during which a star 
would go from A to B. Since the star requires 24 sidereal hours 
to travel from meridian A to meridian A again, the time interval 
from A to B bears the same relation to 24* that the longitude 



Pole 




FIG. 32 

difference bears to 360. The difference in the mean solar times 
at A and B is the number of mean solar hours that the^sun 
would take to go from A to B, and since the sun takes 24 solar 
hours to go from A to A again, the time interval from A to B 
bears the same ratio to 24 solar hours as when sidereal time was 
used. The difference in longitude is therefore correctly given 
when either sidereal or solar times are compared. 

The method of changing from Greenwich to local time and the 
reverse is illustrated by the following examples. 

Example i. The Greenwich astronomical time is 7*40"* io s .o. Required the 
local time at a meridian 4 h $o m 2i s .o West. 

G. M. T. = 7 h 40 io s .o 
Long. West = 4 5 21 -Q 

L. M. T. = 2 h 49 49 s .o (P.M.) 



MEASUREMENT OF TIME 47 

Example 2. The Greenwich mean time is 3^ 20 i6 s .5. Required the local 
mean time at a place whose longitude is 120 10' West. 

G. M. T. + 24 h = 2j h 20 i6 s .5 

Long. West = 8 h oo m 40 s .o 

L. M. T. = 19* IQ W 36 s .s 

= 7 A i9 TO 36 s .5A.M. 

Example 3. The mean time at a place 3^ East longitude is io h A.M. Required 
the Greenwich mean time. 

L. M. T. = 22 h oo m oo s . o 

Long. East = 3^ oo m oo s .o 

G. M. T. = ig h oo m oo s . o 

= J h 00 00 S . O A.M. 

Since a circle may be divided either into 24^ or into 360, the 
relation between these two units is constant. From the fact that 

2 4 h = 360 
we have also i h = 15, 

i w = 15', 

i s - IS"- 
The following equivalents are also convenient: 



By means of these two sets of equivalents time may be con- 
verted into degrees, or the contrary, without writing down the 
intermediate steps. In the following examples the intermediate 
steps are written down in order to show the process followed. 

Example i. Convert 6 h 35 51* into degrees. 

6 h = 90 

35 W = 32 m + 3 m = 8 45' 
5 i s = 4 8 s + 3 s = __ .12' 45" 

Total = 98 57' 45" 

Example 2. Convert 47 17' 35" into hours. 

47 = 45 -(- 2 = 3 A o8 m 
17' = 15' + 2 ' = oi m o8 8 
35" = 30" + 5" = 02 .33 



Total = 3" 09 w i o s . 33 



4 8 



PRACTICAL ASTRONOMY 



It should be observed that the relation 15 = i h is quite 
independent of the length of time that has elapsed. A star 
takes one sidereal hour to move over 15 of hour angle; the sun 
takes one solar hour to move over 15 of hour angle. In the 
sense in which it is used here, i h means an angle, and not an 
absolute interval of time. 

31. Relation between Sidereal Time, Right Ascension, and 
Hour Angle of any Point at a Given Instant. 

In Fig. 33 the hour angle of the equinox, or local sidereal time 
at the meridian through P, is the arc A V. The hour angle of 



Pole 




FIG. 33 

the star 5 at the meridian through P is the arc AB. The right 
ascension of the star 5 is the arc VB. It is evident from the 
figure that 

AV =VB + AB, 
or (S.R.+ Pj [37] 

where R = the right ascension and P = the hour angle of the 
point S, and 5 = the sidereal time; or, in words, 

Sidereal Time = Right Ascension -\-Hour Angle. [38] 



MEASUREMENT OF TIME 49 

This relation is a perfectly general one and will be found to hold 
true for all points on the sphere, provided it is agreed to reckon 
the sidereal time beyond 24* when necessary. For example, if 
the hour angle is io h and the right ascension is 2o h , the resulting 
sidereal time is 30*. This means that the equinox has made a 
complete revolution and has gone 6 h , or 90, on the next revolu- 
tion; the actual reading of the sidereal clock would be 6 h . In 
the reverse case, when it is necessary to subtract 2O A from 6* 
to obtain the hour angle, the 6 h must first be increased by 24* 
and the right ascension subtracted from the sum to obtain the 
hour angle, io h . 

32. Star on the Meridian. 

At the instant when the star is on the meridian its hour angle 
is O A and the equation becomes 

Sidereal Time = Right Ascension; [39] 

that is, the right ascension of a star equals the local sidereal time 
at which that star crosses the meridian. (See Art. 20, p. 36.) 

33. Relation between Mean Solar and Sidereal Intervals of 
Time. 

It has already been stated that on account of the earth's 
orbital motion the sun has an apparent eastward motion among 
the stars of nearly i per day. This eastward movement of the 
sun makes the intervals between the sun's transits greater by 
nearly 4"* than the intervals between the transits of the equinox, 
that is, the solar day is nearly 4 longer than the sidereal day. 
In Fig. 34 let C and C' be the positions of the earth on two 
consecutive days. When the observer is at O it is local noon. 
After the earth makes one complete rotation, the observer will 
be at O', and the sidereal time will be exactly the same as it was 
the day before when he was at O. But the sun's direction is 
now C'O", so the earth must turn through the angle O'C'O" 
in order to bring the sun again on the observer's meridian. 
Since this angle is about i it takes about 4 longer to complete 
the solar day than it does to complete the sidereal day. Since 



5 PRACTICAL ASTRONOMY 

each kind of day is subdivided into hours, minutes, and seconds, 
all of these units in solar time will be proportionally longer than 
the corresponding units of sidereal time. If two clocks, one 
regulated to mean solar time and the other to sidereal time, were 
started at the same instant, both reading o h , the sidereal clock 
would immediately begin to gain on the solar clock, the gain 




FIG. 34 

being exactly proportional to the time interval, that is, about io s 
per hour, or more nearly 3"* 56* per day. 

In order to find the exact relation between the two kinds of 
time it should be observed that the number of sidereal days in 
the year is exactly one greater than the number of solar days, 
because the sun comes back to the equinox at the end of one 
year. The length of the tropical* year is found to be 365.2422 

* The tropical year is the interval of time between two successive passages of 
the sun over the vernal equinox. The sidereal year is the interval between two 
passages of the sun across the hour circle through a fixed star on the equator. On 
account of the movement of the equinox caused by precession, the tropical year is 
about 2O TO shorter than the sidereal year. 



MEASUREMENT OF TIME 51 

mean solar days. The relation between the two kinds of day is 
therefore 

366.2422 sidereal days = 365.2422 solar days, [40] 

or i sidereal day = 0.99726957 solar day, [41] 

and i solar day = 1.00273791 sidereal days. [42] 

Equations [41] and [42] may be written 

24 h sidereal time = (24* 3 55 S .9O9) mean solar time, 
24* mean solar time = (24* + 3 m 56^.555) sidereal time. 

These equations may be put in more convenient form for com- 
putation by expressing the difference in time as a correction 
to be applied to any interval of time to change it from one kind 
of unit to the other. If Im is a mean solar interval and 7 S the 
corresponding number of sidereal units, then 

I s = I m + .00273791 X I m [43] 

and Im = I s - .00273043 X / s . [44] 

Tables II and III are constructed by multiplying different values 
of I m and I 8 by these constants. More extended tables may be 
found in the Nautical Almanac. The use of Tables II and III 
is illustrated by the following examples. 

Examples. 

Reduce 9^ 23'" 5i s .oof sidereal time to the equivalent number 
of solar units. From Table II, opposite 9^ is the correction 
i m 28 S .466; opposite 23 in the 4th column is 3". 768; and 
opposite 5i s in the last column is o s .i39. The sum of these 
three partial corrections is i m 32 S .373, which is the amount to 
be subtracted from g h 23"* 5i s .o to reduce it to the equivalent 
solar interval, g h 22 1 8 s . 627. 

Reduce j h io m solar time to sidereal time. The correction for 
7 , Table III, is + i m o8 s .995, and for io m is i s .643. The sum, 
i m 1 0^.638, added to 7* io m gives 7* u w io s .638 of sidereal time. 

This reduction may be made approximately by the following 
rule: the correction equals io 8 per hour diminished by i 8 for 



PRACTICAL ASTRONOMY 






every 6* in the interval. The correction for 6 h would be 
6 X io 8 i s = 59 s . This rule is based on a change of 3"* 
56* per day. For changing solar into sidereal the error is 
o s .o23 per hour; for sidereal into solar the error is o s .oo4 per 
hour. 

It should be kept in mind that the .conversion of time discussed 
in this article concerns the change from one kind of unit to 
another, like changing from yards to metres, and is not the same 
as changing from the local sidereal time to the local solar time 
at a particular instant. 

34. Relation between Sidereal Time and Mean Solar Time 
at any Instant. 

If in Fig. 33, Art. 31, the point B is taken to represent the 
mean sun, then equation [37] becomes 

S = R s + P s , [45] 

where R s and P s are the right ascension and the hour angle of 
the mean sun at the instant considered. P s is the local mean 
time by the definition given in Art. 26. If the equation is 
written 

S - P s = R a , [46] 

then, since the value of the right ascension R s does not depen 
upon the time at any particular meridian, but only upon the 
absolute instant of time considered, it is evident that the differ- 
ence between sidereal time and mean time at any instant is the 
same for all places on the earth. The actual values of S and 
P a will of course be different at different meridians, but the 
difference between the two is a constant for all places for the 
given instant. In order that Equa. [45] shall hold true it is 
essential that R s and P s shall refer to the same position of the 
sun, that is, to the same absolute instant of time. The right 
ascension of the sun obtained from the Nautical Almanac is its 
value at the instant of the Greenwich Mean Noon preceding, 
that is, at the beginning of the astronomical day at Greenwich.* 



* The dates are always in mean solar days, not in sidereal days. 



MEASUREMENT OF TIME 



53 



To reduce this right ascension to its value at the desired instant 
it is necessary to multiply the hourly increase in the right ascen- 
sion of the mean sun by the number of solar hours elapsed since 
the instant of Greenwich Mean Noon. The hourly increase in 
the right ascension of the mean sun is constant and is evidently 
equal to the correction in Table III, for the difference between 
sidereal and solar time is caused by the sun's motion, and the 
amount of the difference for any number of hours is exactly 
equal to the increase in the right ascension. If it is desired to 
find the increase for any number of solar hours, Table III should 
be used; for sidereal hours use Table II. Equation [45] may 
be written 

5 = R s + P s + C, [47] 

where R s refers to the instant of the preceding local mean noon, 
and C is the correction (Table III) to reduce P s to a sidereal 
interval, or to reduce R s to its value at the time P s . 

In Fig. 35 suppose that the sun S and a star 5" passed the 
meridian M at the same instant, and at the mean time P s it is 

M 




FIG. 35 

desired to compute the sidereal time. Since the sun is moving 
at a slower rate than the star, it will describe the arc MS ( = P a ) 



54 PRACTICAL ASTRONOMY 

while the star moves from M to S'. The arc SS', or C, repre- 
sents the gain of sidereal on mean time in the mean time interval 
MS or P s . But S' is the position of the sun at noon, so that 
VS f is the sun's right ascension at the preceding mean noon, or 
R s . The right ascension desired is VS, so R s must evidently 
be increased by the arc SS', or C. 

If it is desired to find the mean solar time corresponding to 
a given instant of local sidereal time,, the equation is 

Sidereal internal from noon = S R s , [48] 

or Mean time = P s = S R s C', [49] 

where C' is the correction from Table II to reduce 5" Rs to a 
solar interval, and represents the increase in the sun's right 
ascension in 5 R s sidereal hours. 
Examples. 

To find the Greenwich Sidereal Time corresponding to Greenwich Mean Time 
-9* 22 m i8 s .6o on Jan. 7, 1907. The right ascension of the mean sun at Greenwich 
Mean Noon is found from the Nautical Almanac to be 19^ 03"* 36 s -38. The cor- 
rection to reduce 9* 22 i8 8 .6o to sidereal time (Table III) is +i m 32 8 -37. Then, 
applying Equa. [47], 

R s = i9 A 03"36 8 .38 

P s = 9 22 18 .60 

C = _ i 32 -37 

S = 2S h 2 7 m 27 S . 3S 

Sidereal Time = 4* 27"* 27^.35 

To find the Greenwich Mean Solar Time when the Greenwich Sidereal Time is 
4 h 27 27*.3S on Jan. 7, 1907. 

S = 2& h 2 7 m 2 7 S .35 

R s = 19 03 36 .38 

5 - R 9 = 9 23 50 .97 

C' (Table H) = -i 32.37 

Mean Time = 9* 22 TO i8*.6o 

If the change from sidereal to solar time (or vice versa) is to be 
made at any meridian other than Greenwich, the right ascension 
of the sun for local noon must be found by multiplying the 
increase per solar hour by the number of solar hours since Green- 
wich noon, that is, by the number of hours in the longitude, and 



MEASUREMENT OF TIME 55 

adding this to the value of R s from the Almanac if the place is 
west of Greenwich, subtracting if east.* The correction may 
be taken from Table III. If the sidereal time in the above 
example is assumed to be the time at a meridian 5 A (75) west of 
Greenwich, the computation would be modified as follows: 

R, f = ig h 03 3 6 8 . 3 8 

Correction for 5^ longitude = 49 .28 

R s = 19 04 25 .66 

S = 28 27 27 .35 



5 R s = 9 23 01 .69 
C - i 32 .24 



Local Mean Time = g h 2i m 29^45 

It is evident that at the instant of mean noon P s = o and 
R s = S. At mean noon, therefore, the sidereal time equals 
the right ascension of the mean sun. This quantity will be 
found in the Almanac under both headings, " Sidereal Time of 
Mean Noon" and " Right Ascension of the Mean Sun." (See 

P- 65.) 

The reduction of mean solar time to sidereal time, or the re- 
verse, may be made also by first changing the given local time 
to the corresponding instant of Greenwich time, then making 
the transformation as before, and finally changing back to the 
meridian of the place. Take, for example, the case given on 
page 54. 

Local Sidereal Time = 28^ 27 m 27 s .35 

Longitude = 5 oo oo 

Greenwich Sidereal Time = 33 27 27 .35 

R s at Gr. M. Noon = 19 03 36 .38 

Sidereal Interval from Noon = 14 23 50 .97 

C' = 2 21 .52 

Greenwich Mean Time = 14 21 29 .45 

Longitude = 5 oo oo 
Local Mean Time = g h 2i OT 29 s .45 

The result agrees with that obtained by the former method. 
This method is quite as simple as the preceding, especially when 

* It should be remembered that the sun's R. A. is always increasing. 



56 PRACTICAL ASTRONOMY 

Standard Time is to be computed, for the final correction will 
always be a whole number of hours. Care should be taken 
always to use the right ascension of the sun at the noon preced- 
ing the given time. Suppose that the instant of io h A.M. May 5 
is to be converted into sidereal time, the longitude of the place 
being 4 h 4.4 i8 s west. Civil time io h A.M. May 5 = Astr. 
time 22 h May 4. If the first method is followed, the right 
ascension of the sun employed should be that of noon May 4. 
If the reduction is made by first changing to Greenwich time, 
then 22* + 4 h 44 m i8 s = 26* 44 i8 s May 4 = 2 h 44 i8 s 
May 5. The right ascension for the latter case would be that 
for noon of May 5. 

35. Standard Time. 

From the definition of mean solar time it will be seen that at 
any given instant the solar times at two places will differ from 
each other by an amount depending upon the difference in the 
longitudes. All places will have different local times except 
where they happen to be on the same meridian. Previous to the 
year 1883 it was customary in this country for each large city 
or town to use the mean time at its own meridian, and for all 
other places in the vicinity to adopt the same time. Before 
railroad travel became extensive this change of time from one 
point to another caused no great difficulty, but with the in- 
creased amount of railroad and telegraph business these frequent 
and irregular changes in time became so inconvenient that in 
1883 a uniform system of time was adopted in the United States. 
The country is divided into time belts each theoretically 15 in 
width; these are known as the Eastern, Central, Mountain and 
Pacific time belts, and places in these belts use the mean local 
time of the 75, 90, 105 and 120 meridians respectively. The 
time at the 60 meridian is called Atlantic time and is used in 
the Eastern Provinces of Canada. The actual positions of the 
dividing lines between these belts depend upon the positions of 
the principal cities and the railroads (see Fig. 36), but the change 
of time from one belt to another is always exactly one hour. The 



MEASUREMENT OF TIME 



57 




58 PRACTICAL ASTRONOMY 

minutes and seconds of all clocks are the same as the minutes 
and seconds of the Greenwich clock. When it is noon at Green- 
wich it is 8 A.M. Atlantic time, 7 A.M. Eastern time, 6 A.M. 
Central time, 5 A.M. Mountain time, and 4 A.M. Pacific time. 

The change from local to standard time, or the contrary, con- 
sists in expressing the difference in longitude between the local 
meridian and the standard meridian in units of time, and adding 
or subtracting this correction, remembering that the farther 
west a place is, the earlier it is in the day at any given instant 
of time. 

Examples. 

Find the standard time at a place 71 west of Greenwich when 
the local time is 4 h 20 oo s P.M. In longitude 71 the standard 
time would be that of the 75 meridian. The difference in 
longitude is 4 = i6 m . Since the standard meridian is west of 
the 71 meridian, the time is i6 m earlier than the local time. The 
standard time is therefore 4* 04"* oo 8 P.M. 

Find the local time at a place 91 west of Greenwich when the 
Central time is 9* OO TO oo s A.M. The difference in longitude is 
i = 4 m . Since the place is west of the standard meridian, the 
time is earlier. The local time is therefore S h 56 oo s A.M. 

Standard time is used not only in the United States but in a 
majority of the countries of the world; in nearly all cases these 
systems of standard time are based on the meridian of Green- 
wich as the prime meridian. Germany, for example, uses the 
local mean time at the meridian i h east of Greenwich; Japan 
uses that of the meridian 9* east of Greenwich ; Turkey, 2 h east 
cf Greenwich, etc. 

36. The Date Line. 

If a person were to start at Greenwich at the instant of noon 
and travel westward rapidly enough to keep the sun always on 
his meridian he would get back to Greenwich 24* later, but his 
own (local) time would not have changed but would have 
remained noon all the time. In travelling westward at a slower 
rate the same thing occurs, only in a longer interval of time. 



MEASUREMENT OF TIME 59 

The traveller has to set his watch back every day in order to 
keep it regulated to the meridian at which his noon occurs. As 
a consequence, his watch has recorded one day less than it has 
actually run, and his calendar is one day behind that of a person 
who remains at Greenwich. If the traveller goes east he has to 
set his watch ahead every day, and after circumnavigating the 
globe his calendar is one day ahead of what it should be. In 
order that the calendar may be everywhere uniform, it is agreed 
to change the date at the meridian 180 from Greenwich. When- 
ever a ship crosses the 180 meridian going westward, a day is 
omitted from the calendar, and when going eastward a day is 
repeated. In practice the change is made at midnight near the 
180 meridian, not at the instant of crossing. The date line 
actually used does not follow the 180 meridian in all places, but 
is deflected so as not to separate the Aleutian islands, and in 
the South Pacific ocean it passes east of several groups of islands 
so as not to change the date formerly used in these islands. 

37. The Calendar. 

Previous to the time of Julius Caesar the calendar was based 
upon the lunar month, and, as this resulted in a continual change 
in the date at which the seasons occurred, the calendar was fre- 
quently changed in an arbitrary manner in order to keep the 
seasons in their places, the result being extreme confusion in the 
dates. In the year 45 B.C. Julius Caesar reformed the calendar 
and introduced one based upon a year of 365! days, since called 
the Julian calendar. The \ day was taken care of by making 
the year contain 365 days, except every 4th year, called leap 
year, which contained 366; the extra day was added to February 
in such years as were divisible by 4. The year was begun on 
Jan. i ; previously it had begun in March. Since the year con- 
tains actually 365^ 5^ 48 m 46*, this difference of n m 14* caused 
a gradual change in the dates at which the seasons occurred. 
After many centuries the difference had accumulated to about 
10 days, so in 1582 Pope Gregory XIII ordered that the calendar 
should be corrected by dropping ten days and that future dates 



6o 



PRACTICAL ASTRONOMY 



should be computed by omitting the 366th day in those leap years 
which occurred in century years not divisible by 400; that is, 
such years as 1700, 1800 and 1900 should not be counted as leap 
years. This is the calendar used at the present time. 

Questions and Problems 

1. (a) Prove by direct computation of sidereal time from Fig. 37 that 

R + P = 24 h + S, 

in which R and P are the right ascension and hour angle of the star S, and S is the 
sidereal time, or hour angle of V. 

(b) Prove the same relation when V is at the point V. (See Art. 31, 
p. 48.) 

2. Prove that the difference in longitude of two points is independent of the 
kind of time used, by selecting two points at which the solar time differs by say 
3 A , and then converting the solar time at each place into sidereal time. 




FIG. 37 



FIG. 38 



3. Make a design for a horizontal sun dial for a place whose latitude is 42 21' N. 
The gnomon ad (Fig. 38), or line which casts the shadow on the horizontal plane, 
must be parallel to the earth's rotation axis; the angle which the gnomon makes 
with the horizontal plane therefore equals the latitude. The shadow lines for the 
hours (X, XI, XII, I, II, etc.) are found by passing planes through the gnomon 
and finding where they cut the horizontal plane of the dial. The vertical plane 
adb coincides with the meridian and therefore is the noon (XII*) line. The other 
planes make, with the vertical plane, angles equal to some multiple of 15. In 
finding the trace dc of one of these planes on the dial it should be observed that the 
foot of the gnomon, d , is a point common to all such traces. In order to find another 
point c on any trace, or shadow line, pass a plane abc through some point a on the 
gnomon and perpendicular to it. This plane (the plane of the equator) will cut 
an east and west line cc on the dial. If a line be drawn in this plane making an 



MEASUREMENT OF TIME 6 1 

angle of n X 15 with the meridian plane, it will cut ce at a point c which is on the 
shadow line. Joining c with the foot of the gnomon gives the required line. 

In making a design for a sun dial it must be remembered that the west edge of 
the gnomon casts the shadow in the forenoon and the east edge in the afternoon; 
there will be of course two noon lines, and the two halves of the diagram will be 
symmetrical and separated from each other by the thickness of the gnomon. The 
dial may be placed in position by levelling the horizontal surface and then com- 
puting the watch time of apparent noon and turning the dial so that the shadow is 
on the XII A line at the calculated time. 

Prove that the horizontal angle bdc is given by the relation 

tan bdc = tan P sin L, 
in which P is the sun's hour angle and L is the latitude. 

4. Why are the sun's and moon's right ascension always increasing? 

5. The local apparent time at a point A is io h 30"* A.M. If the equation of 
time is + 3"* 25^.8, what is the local mean time? What is the astronomical mean 
time at the given instant? Assuming the longitude of A to be 95 West, what is 
the Greenwich Mean Time? What is the Central Standard Time? What is the 
local mean time at the same instant at a point B in longitude 110 W.? If the 
right ascension of the mean sun at G. M. N. is i8 A 4i TO oi s .6, what is the local 
sidereal time? What is the Greenwich Sidereal Time? 



CHAPTER VI 

THE AMERICAN EPHEMERIS AND NAUTICAL 
ALMANAC STAR CATALOGUES INTERPOLATION 

38. The Ephemeris. 

In the problems previously discussed it has been assumed that 
the coordinates of celestial objects and various other data men- 
tioned are known to the computer. These data consist of 
results calculated from observations made with large instruments 
at the principal observatories; these results are published by the 
government several years in advance in the American Ephemeris 
and Nautical Almanac.* The Almanac contains the declinations 
and the right ascensions of the sun, moon, planets and stars, as 
well as the angular semidiameters, horizontal parallaxes, the 
equation of time, and other data required in astronomical cal- 
culations. Since all of these quantities vary with the time, 
their values are usually given for equidistant intervals of Green- 
wich time or of Washington time. 

The Almanac is divided into three parts. Part I is computed 
for the meridian of Greenwich, and is arranged especially for 
the convenience of navigators. Part II is computed for the 
meridian of Washington, and is arranged chiefly for the con- 
venience of astronomers. Part III contains the data for pre- 
dicting phenomena, such as eclipses, occultations, etc. At the 
end of the book are certain tables computed especially for the 
use of the navigator and the surveyor. 

The first page of Part I is headed "At Greenwich Apparent 

* Similar publications by other governments are: the Nautical Almanac 
(Great Britain), the Berliner Astronomisches Jahrbuch (Germany), the Con- 
naissance des Temps (France), and the Almanaque Nautico (Spain). 

The word " ephemeris " means a table of coordinates of a celestial body given 
for equidistant intervals of time. 

62 



THE AMERICAN EPHEMERIS AND NAUTICAL ALMANAC 63 

Noon," and contains the following data for that instant for each 
day in the month: right ascension and declination of the sun 
and their hourly changes, sun's semidiameter, time of semi- 
diameter passing the meridian, and the equation of time with 
its hourly change. The second page is headed "At Greenwich 
Mean Noon," and contains the right ascension and declination 
of the sun with their hourly changes, the equation of time with 
its hourly change, and the right ascension of the mean sun or 
sidereal time at mean noon. These pages are the ones to be', 
used by the navigator or the surveyor when making observa-( 
tions on the sun. Whether Table I or Table II shall be used in 
any given case depends upon whether apparent time or mean 
time is the more convenient. If the declination, right ascension, 
or equation of time is required for the instant of the sun's transit 
over any meridian, then the local apparent time is noon and the 
Greenwich Apparent Time is equal to the west longitude of the 
place. The desired quantity is found by taking out its value 
for the instant of Greenwich Apparent Noon and increasing or 
decreasing it by the hourly change. multiplied by the number of 
hours in the longitude. If the quantity is to be found for some 
instant of local mean time or Standard Time, then the Greenwich 
Mean Time may be readily found, and it is therefore more con- 
venient to compute the required value from that given for Green- 
wich Mean Noon. If Local Time is used, the Greenwich Time 
is found by adding the longitude; if Standard Time is used, the 
Greenwich Time is found by adding 5^, 6 h , etc., according to 
the time belt indicated. The tabular quantity is then corrected 
for the time elapsed since Greenwich Mean Noon. Tables I and 
II for the month of January, 1912, are shown on pages 64 and 65. 
The third page of the Almanac contains data not usually required 
by the surveyor. On the fourth page are the semidiameter, 
horizontal parallax, and time of transit of the moon, with their 
hourly changes. On account of the rapidity with which the 
semidiameter and parallax vary, they are given for both Green- 
wich noon and midnight. The next eight pages contain the 



6 4 



PRACTICAL ASTRONOMY 



JANUARY, 1912 
AT GREENWICH APPARENT NOON 





. 


THE SUN'S 


-j 








g 




W.S 


Equa- 




Day 

of the, 
week. 


"o 


Apparent 
right ascen- 


- 1 

Diff. 
for i 


Apparent 
declination. 


Diff. for 
i hour. 


Semi- 
diam- 


;al time ol 
meter pas 
meridian 


tion of 
time, to 
be add- 
ed to ap- 


Diff. 
for i 
hour. 







sion. 


hour. 






eter. 


"'I 


parent 
time. 






Q 












K 










h m s 


$ 


' II 


" 


' " 


i 


m s 


s 


Mon. 


i 


18 42 21.37 


11-053 


S. 23 5 43.7 


+ 11.19 


16 17.89 


71.09 


3 13-30 


i -193 


Tues. 


2 


1 8 46 46.49 


i i . 039 


23 i i-5 


12-34 


16 17.90 


71.04 


3 41-79 


1.179 


Wed. 


3 


18 51 11.26 


11.025 


22 55 51.6 


13-49 


16 17.90 


71.00 


4 9-03 


1.165 


Thur. 


4 


18 55 35-68 


11.009 


22 50 14-4 


+ 14-63 


16 17.91 


70.95 


4 37-71 


1.149 


Frid. 


5 


18 59 59-71 


10.992 


22 44 IO.O 


15-75 


16 17.91 


70.90 


5 5-ii 


I-I33 


Sat 


6 


19 4 23-33 


10-975 


22 37 38.5 


16.87 


16 17.90 


70.84 


5 32.li 


1. 115 


Sun. 


7 


19 8 46.53 


10-957 


22 30 40-3 


+ 17-98 


16 17.87 


70.78 


5 58.67 


1-097 


Mon. 


8 


19 13 9-28 


10-939 


22 23 15-3 


19-09 


16 17.84 


70.71 


6 24.78 


1.077 


Tues. 


9 


19 17 31-55 


10.919 


22 IS 23.9 


20.19 


16 17.80 


70-64 


6 50.41 


1-057 


Wed. 


10 


19 21 53-31 


10.896 


22 7 6.3 


+ 21.28 


16 17.76 


70.57 


7 15-54 


1.036 


Thur. 


ii 


19 26 14-54 


10.873 


21 58 22.7 


22.36 


16 17.72 


70.50 


7 40.15 


1.014 


Frid. 


12 


19 30 35-23 


10.850 


21 49 13.3 


23-43 


16 17.67 


70.42 


8 4.21 


0.990 


Sat 


13 


19 34 55-33 


10.826 


21 39 38.4 


+ 24-49 


16 17.61 


70.34 


8 27.70 


0.966 


Sun. 


14 


19 39 14-83 


10.800 


21 29 38.3 


25-53 


16 17-55 


70.26 


8 50.59 


0.941 


Mon. 


15 


19 43 33-71 


10-773 


21 19 I3.I 


26.57 


16 17.48 


70.17 


9 12.86 


0.914 


Tues. 


16 


19 47 51-96 


10.746 


21 8 23.3 


+ 27-59 


16 17.41 


70.08 


9 34-48 


0.887 


Wed. 


*7 


19 52 9-54 


10.717 


20 57 9.2 


28.59 


16 17-34 


69.98 


9 55-42 


0.859 


Thur. 


18 


19 56 26.42 


10.688 


20 45 31.0 


29-58 


16 17.26 


69.88 


10 15.68 


0.830 


Frid. 


19 


20 o 42.58 


10.658 


20 33 29.1 


+ 30-56 


16 17.18 


69.78 


10 35-24 


0.800 


Sat. 


20 


20 4 58.02 


10.628 


20 21 3-9 


31-53 


16 17.10 


69.68 


10 54.08 


0.769 


Sun. 


21 


20 9 12.71 


10-597 


20 8 15.7 


32.48 


16 17.01 


69-57 


ii 12.17 


0-738 


Mon. 


22 


20 13 26.64 


10.565 


19 55 4-8 


+33-41 


16 16.92 


69.47 


ii 29.49 


0.706 


Tu s. 


23 


20 17 39-8o 


10-533 


19 41 31.7 


34-33 


16 16.82 


69.36 


ii 46.04 


0.673 


Wed. 


24 


20 21 52.16 


10.500 


19 27 36.7 


35-24 


16 16.72 


69.26 


12 1. 80 


0.640 


Thur. 


25 


2O 26 3.72 


10.465 


19 13 20.0 


+ 36.13 


16 16.62 


69.15 


12 16.76 


0.607 


Frid. 


26 


2O 30 14.46 


10.430 


18 58 42.3 


37-oi 


16 16.52 


69.04 


12 30.91 


0-573 


Sat 


27 


20 34 24.38 


10.396 


18 43 43-9 


37-86 


16 16.41 


68.93 


12 44.24 


0-538 


Sun. 


28 


20 38 33-48 


10.361 


18 28 25.0 


+ 38-70 


16 16.29 


68.82 


12 56.74 


0.504 


Mon. 


29 


20 42 41-74 


10.326 


18 12 46.2 


39-52 


16 16. 16 


68.71 


13 l8.4I 


0.469 


Tues. 


30 


2O 46 49.16 


10.291 


17 56 48.0 


40.32 


16 16.03 


68.60 


13 19-25 


0-434 


Wed. 


31 


20 50 55.74 


10.256 


17 40 30.7 


41.11 


16 15.90 


68.48 


13 29.25 


0.400 


Thur. 


32 


20 S5 1-49 


10.222 


S. 17 23 54.5 


+41.89 


16 15.76 


68.37 


13 38.41 


0.365 



Note. The mean time of semidiameter passing may be found by subtracting o*.ig from the 
sidereal time. The sign + prefixed to the hourly change of declination indicates that south decli- 
nations are decreasing. 



JANUARY, 1912 
AT GREEN'WICH MEAN NOON 



Day 

of the 
week. 


Day of the month. 


THE SUN'S 


Equation 
of time, to 
be sub- 
tracted 
from mean 
time. 


Diff. 
for i 

hour. 


Sidereal 
time, or right 
ascension of 
mean sun. 


Apparent 
right ascen- 
sion. 


Diff. 
for i 
hour. 


Apparent 
declination. 


Diff. for 
i hour 


Mon. 
Tues. 
Wed. 


i 

2 

3 


h m s 
18 42 20.78 
18 46 45-81 
18 51 10.50 


^ 
i i . 049 
11-035 

II.O2I 


S. 23 5 44.3 

23 I 2.2 

22 55 52-5 


+ 11. 18 
12.33 
13-47 


m s 
3 13-24 
3 41-72 
4 9-85 


s 
i-i93 
1.179 
1.165 


h m s 
18 39 7-54 
18 43 4. 10 
18 47 0.66 


Thur. 
Frid. 
Sat. 


4 
6 


18 55 34.84 
18 59 58.79 
19 4 22.33 


II. 006 
10.990 
10.972 


22 50 15.5 
22 44 11.3 
22 37 40.1 


+ 14.61 
15-74 
16.86 


4 37-62 
5 S-oi 
5 32-00 


1.149 
I-I33 
1-115 


18 50 57.22 
18 54 53-78 
18 58 50.33 


Sun. 
Mon. 
Tues. 


7 
8 
9 


19 8 45-45 
19 13 8.12 
19 17 30.30 


10.953 
10.934 
10.914 


22 30 42.0 
22 23 17.3 
22 15 26.2 


+ 17-97 
19.08 
20.17 


5 58.56 
6 24.66 
6 50.29 


1.097 
1.077 
1-057 


19 2 46.89 
19 6 43.45 
19 10 40.01 


Wed. 
Thur. 
Frid. 


10 
ii 

12 


19 21 51.99 
19 26 13.15 
19 3 33-77 


10.893 
10.870 
10.847 


22 7 8.8 
21 58 25.5 
21 49 16.4 


+ 21.26 
22.34 
23-41 


7 15-42 
7 40.02 
8 4.08 


1.036 
1.014 
0.990 


19 14 36.57 
19 18 33-13 

19 22 29.68 


Sat 
Sun. 
Mon. 


13 

14 
IS 


19 34 53- 81 
19 39 13-25 
19 43 32.07 


10.822 
10.797 
10.770 


21 39 41.8 
21 29 42.0 

21 19 17.2 


+ 24-47 
25-51 
26.55 


8 27-57 
8 50-45 
9 12-71 


0.966 
0.940 
0.914 


19 26 26.24 
19 30 22.8o 
19 34 19-36 


Tues. 
Wed. 
Thur. 


16 
i? 
18 


19 47 50-25 
19 52 7-76 
19 56 24.58 


10-743 
10.715 
10.686 


21 8 27.7 
20 57 13-9 
20 45 36.1 


+ 27-57 
28.58 
29-57 


9 34-33 
9 55-28 
10 15.54 


0.887 
0.859 
0-830 


19 38 15-92 
19 42 12.48 
19 46 9.03 


Frid. 
Sat. 
Sun. 


19 

20 
21 


20 o 40.70 
20 4 56.09 
20 9 10.73 


10.656 
10.626 
10.595 


20 33 34.6 

2O 21 9.7 
20 8 21.8 


+ 30-55 
31-52 
32-47 


10 35-10 
10 53-94 
n 12.03 


0.800 
0.769 
0.738 


19 50 5-59 
19 54 2.15 
19 57 58-71 


Mon. 
Tues. 
Wed. 


22 
23 
24 


20 13 24.62 

20 I? 37.74 
20 21 50.06 


10-563 
10.530 
10.497 


19 55 n-2 
19 41 38.4 
19 27 43.7 


+ 33-40 
34-32 
35-23 


II 29.36 
ii 45-91 
12 1.68 


0.706 
0-673 
0.640 


20 i 55.27 
20 5 51.82 
20 9 48.38 


Thur. 
Frid. 
Sat. 


25 
26 

27 


20 26 1.58 
2O 3O 12.29 

20 34 22.18 


10.463 
10.429 
10-395 


19 13 27.4 
18 58 50.0 
18 43 51.9 


+ 36.12 
37-00 
37-85 


12 16.65 
12 30.80 
12 44-13 


0.607 
0-573 
0-538 


20 13 44-94 
20 17 41.50 
20 21 38.05 


Sun. 
Mon. 
Tues. 
Wed. 


28 
2Q 

30 
31 


20 38 31.25 
20 42 39.48 
20 46 46.87 

2 So 53-44 


10.360 
10.326 
10.291 
10.256 


18 28 33.4 
18 12 55.0 
i? 56 57-o 
17 40 39.9 


+ 38-69 
39-Si 
40.31 
41.10 


12 56.64 

13 8.31 
13 19-15 
13 29.16 


0.504 
0.469 
0-434 
0.400 


20 25 34 -61 

20 29 31.17 

20 33 27.72 
20 3? 24.28 


Thur. 


32 


20 54 59-17 


IO.22I 


S. 17 24 4.0 


+ 41-88 


13 38.33 


0-365 


20 41 20.84 


Note. The semidiameter for mean noon may be assumed the same as that for ap- 
parent noon. The sign 4- prefixed to the hourly change of declination indicates that 
>outh declinations are decreasing. 


Diff. for i 
Hour, 
+9 8 -8s6s. 
(Table III.) 



66 



PRACTICAL ASTRONOMY 



MEAN PLACES OF STARS, 1912. 

WASHINGTON, JANUARY I<*.OO6. 



Name of star. 


Magni- 
tude. 


Right 
Ascension. 


Annual 
Variation. 


Declination 


Annual 
Variation. 


v/ 33 Piscium 


4-7 


h m s 
o o 40.808 


i 
+ 3.o7l< 


o ' // 
6 II en. 4? 


rt 
-f- 20 176 


. a Andromedae (Alphe- 
ratz) 


2 2 


o 3 co 162 


2 OO5? 


4-28 3.6 16 t;o 


19 880 


v ft Cassiopeiae 


2 4 


O A 28 CO4. 


2 l834 


4- ?8 3O "CT O7 


19 862 


v Phoenicis 


2.O 


O 4 ^6.83.2 


^.O?2O 


46 i3 ?8.o6 


19 848 


2 2 Andromedae 


e i 


O C 44. C7I 


3 I088 


4-4C 34 C7 27 




y Pegasi 


2.Q 


o 8 42 161 


+ "3 0861 


4- IA 4.1 3O 7? 


* O *J 

4~ 20 02 1 


<r Andromedae 


4 1 


Oil 4? 6l 1 


2 I2OO 


4- -26 17 CO C2 


10 063 


t Ceti 


1 8 


O 14. ?6 674 


3O?7O 


o i 8 42 04 


IO O7A 


f Tucanae 


45 


O 1C 2O 74? 


1 lAO 1 ? 


6c 23 2O 80 




44 Piscium 


6 o 


O 2O ? ? 4.64 


2 O742 


+1 27 8 CO 




ft Hydri 


2.Q 


o 21 8 620 


+ 5 2O4.3 


77 A A CO 48 


4- 2O 27O 


at Phoenicis 


2 4. 


O 21 ?6 2 ?4 


2 O73I 


42 47 I OC 


TQ CCI 


1 2 Ceti 


6 


o 25 32 885 


-2 0621 


4. 26 36 2C 




I? Ceti t 


52 




2 0877 






f Cassiopeiae 


2 7 


O 3.2 3 748 


3 3.272 


4- C3 24 J.C 87 


10 843 


TT Andromedae. . . 


4 4. 




4- 3 TO7O 






e Andromedae 


4. e 


O ?? 1 \4.I'?I 


3..I63.8 


4-28 50 2 68 


IO C73 


5 Andromedae 


2. C 


O ^4 'I?. I 2.8 


7 2OI4. 


4- 3O 22 4.C 08 


TO 721 


a Cassiop. (Schedir}.-\ 
ft Phoenicis. 


var. 

A 6 


o 35 30-339 


3-3852 


+ 56 3 17-55 


19-774 


ft Ceti 


2 . 2 


O 3Q IO. ^8l 


+ 3 0126 


18 28 9 82 


4- 10 7oc 


o Cassiopeiae .... 


4 7 








TO 738 


21 Cassiopeise . 


c 6 








TO 7l8 


Andromedae. 


A 2 










17 Cassiopeise "\ 


* 6 


O J."? d.6. 12^ 


3, 6116 


-J- C7 2O CO C4 


IO 2OC 


8 Piscium 


4 6 








4- TO 63T 


X Hydri . 


So 










20 Ceti 


4 O 


o J.8 "?o cr^A 


3 064.1 


T 77 18 A7 


IO COC 


y Cassiopeiae 


2 2 


O Cl 27 2J.7 


3. tlQ^S 


4- OO T/L 2C C6 


IO C3Q 


fji Andromedae. 




o r T c T Rrc 






TO c6c 


a Sculptoris 


-1 A 


O CJ. 21 Ol8 


+ 2 8908 




4- IO.4.72 


43 H. Cephei 


4 C 


O ^6 71 2J.4 


7 C72C 


4- 8 c 4.7 81? 


IO.43C 


e P'scium 












ft Phcenicis .... f 










19 288 


H Cassiopeiae. . . 


r 7 








17 7C2 


17 Ceti 


3. 6 


T J. O *77C 








ft Andr~ xlae 


2 4. 










T Pisciui. 


4. 7 


i 6 48 60*5 








f Piscium f 


c 6 


T O 7 O?8 








K Tucanae f 


50 










f Piscium 


5^ 


T T3 T C C2C 








v Piscium 


4. 7 








18 o8c 


B Ceti 


5 8 


T TO 77 ACC 




8 78 17 87 


18 633 


5 Cassiopeiae 


2 8 




3 8083 






y Phcenicis 


3 A 




2 6o8l 


j.7 46 8 c8 




38 Cassiopeiae 


6 o 


I 2J. 7O 74O 




+ 69 48 43 86 


4-18 622 


17 Piscium 


37 


I 26 J.6 7O7 






T 8 623 


a Ursae Min.(Po/am)f 


2.1 


i 27 51.07* 


+ 27.8225 


+ 88 50 10.77 


+ 18.594 



13 Ceti. dup. 5 w .s. 6*.2, o*.3. 
a Cassiop.. var. irreg 2"*.2,2".8 
17 Cassiop.. comp.7"*.6, 5" s. pr. 



/3 Phoenicis, dup. 4 m .i,4 m .i,i*. 
Piscium, star 6 m .s, 24" n.f. 



K Tucanae, comp. 7 m , 6" n. 

a TJrsse Min., star g m , 18" s. pr. 



THE AMERICAN EPHEMERIS AND NAUTICAL ALMANAC 67 



APPARENT PLACES OF STARS, 1912. 

FOR THE UPPER TRANSIT AT WASHINGTON. 





33 Piscium. 


a Andromedae 


/tf Cassiopeia 


6 Phoenicis 




Mag. 4.7 


Mag. 2.2 


Mag. 2.4 


Mag. 3.9 


Mean so- 
lar date. 


Right 


Declina- 


Right 


Declina- 


Right 


Declina- 


Right 


Declina- 




Ascension 


tion S. 


Ascension 


tion N. 


Ascension 


tion N. 


Ascension 


tions 




h m 


/ 


h m 


o / 


/; m 


o ' 


h m 


/ 




O O 


- 6 ii 


3 


+ 28 36 


o 4 


+ 58 39 


4 


-46 13 




s 


" 


i 


" 


f 


" 


s 


" 


Jan. 0.2 


49-31 I0 


65-8 6 


49-23 x 


22-3 


26.88 


65-8 


56.6 7 


77-3 


10.2 


49-21 


66.4 


49-io ; 


21-5 j 


26-57 3 ' 


65-1 t l 


56.48 9 , 


76.9 4 


20. 2 


49-12 g 


66.8 


48.97 3 


2O.4 


26.28 29 


63-9 I6 


56.31 


76.0 9 


30.1 


49-04 '. 


67.1 3 


48.86 


19.0 


26.01 


62 '3 o 


56.16 ; 


74-7 I3 


Feb. 9 . i 


48.98 


67.2 


48.76 " 


17-5 5 


25-77 24 
' ' 19 


to-*:: 


56.04 9 


73-0 H 


19.1 


48.94 


67-1 2 


48.69 


16.0 


25.58 z 


57-9 a . 


55-95 


70.9 


29.1 


48.92 


66.9 


48.65 4 


14.4 


25.45 I 


55-4 2 J 


55-90 , 


68.6 23 


Mar. 10. o 


48.93 


66.4 


48.65 


12.9 


25-39 




55-89 


6 5-9 27 


20.0 


48.98 J 


6 5-7 I 


48.69 J 


ii. 5 4 


25-41 


5- 1 2 


55-92 


63.0 2 ' 


30.0 


49.06 




48.77 I3 


10.4 


2 5-5 c 


47-6 H 


56.01 x 9 


60.0 3 


Apr. 9 . o 


49-i8 H 


63.6 


48.90 


9 

9-5 


25.68 \ 


45-3 2 


56.14 |' 


56.8 3 


18.9 


49-34 20 


62.2 4 


49.08 Ic 


9-0 f 


25-93 . 


43-3 .. 


56.33 : 


53-6 32 


28.9 


49-54 , 


60.6 ; 


49-30 11 


8.8 


26.25 * 


41.8 J 


56.57 n 


50-4 32 


May 8.9 


49.78 4 


58.8 


49-56 


9- 2 f. 


26.63 


40-7 " 


56.85 2 


47-4 3 


18.8 


50.05 27 
J > 29 


i 


49.85 29 

3 32 


9.6 

IO 


27-07 4 g 


40. i 




44-5 29 


28.8 


50.34 ,_ 


54.7 




10.6 


27-55 , 


40. i 


57-54 3 


41.8 27 


June 7.8 


50.65 3 


52.6 


50.51 34 


11.9 13 


28.05 


40-5 . 


57-93 3 


39-4 24 


17.8 


50.97 : 


50.5 


50.86 3S 


13.6 1? 


28.57 


41.5 ! 


o 4 ' 


37-3 2I 


27.7 


51.30 33 


48.4 


51.21 3 * 


15-6 2 ' 


29.08 s 


43- 2 


58 -7 6 !! 


35-6 I7 


July 7.7 


51.62 3 


46.4 ' 


51.56 3 


J 7' 8 ! 


29.58 s 


45-0 


59.18 4 


34-4 I2 




3 1 


9 


33 


2 4 


^ 48 






. 8 


17.7 


51,93 2 g 


44-5 6 


51.89 


20.2 


30.06 


47-3 27 


59.58 


33-6 


27.7 




42.9 


52.19 3 ! 


22.7 2 


30.50 4 




59-96 3 


33-3 3 


Aug. 6 . 6 


5^47 \\ 


41-4 


52.47 


25.2 


30.89 39 


53-0 3 


60.31 3 


33-5 


16.6 


52.70 ; 


40.2 


52.71 , 4 


27- 8 2 < 


-j 2 , 34 


5 6 ' 2 L 


60. 61 , 


34-1 6 


26.6 


s 2 -^ ; 


39-3 I 


52.91 i6 


30-3 


31-55 22 


59-6 34 

34 


60.87 2 

' 20 


35-2 " 

I ? 


Sept. 5.5 


53.05 


38.7 


53.07 


32-8 


3L72 


63-0 


6l.07 


36.7 




53.16 'J 


38.3 


53-19 " 


35-' 


31-87 


66.5 3S 


61.21 J 


38.5 l8 


2 5-5 


53.24 


38-2 


53.27 




31.96 


69.9 34 


61.30 ? 


40.5 20 


Oct. 5.5 


53.28 ' 


38-3 


53-31 


39-2 2 


31-99 , 


73-2 33 


61.33 I 


42-7 2 


15-4 


53.28 


38.6 3 


53.32 I 




31-96 3 


76.3 29 


7 


45-o 23 

2 2 


25-4 


53.26 


39- 1 


53.29 


42.3 ' 


31-88 


79-2 26 


61.24 XI 


47.2 


Nov. 4 . 4 


53.21 I 


39-7 - 


53.24 : 


43-5 " 


1 4 

^ i 74 






49-3 2I 


14.4 


53.15 8 


40.4 ' 


53.17 


44-4 


31-55 2 


84.0 


5o . QO> "i! 


51.2 J 9 


24.3 


53.07 


41- 1 


53-07 


44-9 


2 2 

31-33 26 


85.7 ;. 7 


60. . 7 


52.8 l6 


Dec. 4.3 


52.98 * o 


41.8 7 


52.96 J 


45-2 I 




87-0 I3 


60. >J 2 9 


54-1 is 


14-3 


52.88 


42-5 6 


52.84 


45-i 


30.79 


87.8 


60.43 


54-9 . 


24.2 


52.78 j; 




52.71 


44-8 3 


30-49 , 


88.0 


60.23 

IQ 


55-2 3 


34-2 


52.68 " 


'.7 6 


52.58 3 


44.1 


3o.i9 3 


87.6 4 


60.04 


55-i 


Sec 8, Tan 5 


i . 006 o 109 


i. 139 +o. 545 


1.923 +1.643 


I . 446 I. 044 


Mean Place 


49 s - 898 59". 45 


SO 8 . 162 16". 59 


288.504 51". 97 


5 6 8 .832 58". 96 


DVa,Dcoa 


0.00 +0.01 


O . OO . 04 


. 00 . 1 1 


o.oo +0.07 


D\f*8, Da)5 


+ 0.4 o.o 


+ 0.4 o.o 


+ 0.4 o.o 


+ 0.4 o.o 



68 PRACTICAL ASTRONOMY 

moon's right ascension and declination for every hour of Green- 
wich Mean Time, together with the changes per minute of time. 
Following the ephemeris of the sun and moon for the twelve 
months are the ephemerides of the planets. The values given 
for Greenwich Mean Noon are for o h of the astronomical date 
or 12 M of the civil date. The word " apparent " used in these 
tables indicates that the correction for aberration has been 
applied to the coordinates, giving the position of the object as 
actually seen by the observer except for the effect of parallax 
and refraction. The " differences for i hour " are in all cases 
the rates of change at the instant, that is, the differential coeffi- 
cients, not the actual differences between the consecutive tabular 
values. 

Part II contains the following three lists of stars, the first 
headed " Mean places of Stars, 19 "; the second, "Apparent 
Places of Circumpolar Stars, 19 "; and the third, "Apparent 
Places of Stars, 19 "; all of these are computed for the instant 
of transit over the meridian of Washington (5* o8 TO 15*. 78 West 
of Greenwich). A list of south circumpolar stars is also given. 
The tables given on pages 66 and 67 of this volume are extracts 
from the first and third of these tables in the Almanac. The 
first table contains the coordinates of about 800 stars referred to 
the "mean equinox " at the beginning of the year, that is, to 
the position that the equinox would occupy at the beginning 
of the solar * year if it were not affected by small periodic terms 
of the precession. The second table gives the coordinates of 
about 1 5 north circumpolar stars. Precession causes the coordi- 
nates of circumpolar stars to vary more rapidly than those of 
equatorial stars; the coordinates are therefore given for every 
day in the year. The hours and minutes of right ascension and 
the degrees and minutes of declination are at the head of the 
column ; the column contains only the seconds. In the third table 

* The year here referred to, called also the Besselian fictitious year, is one used 
in computing star places; it begins when the sun's mean longitude is 280, that is, 
when the R. A. of mean sun is i8 h 40"*, which occurs about Jan. i. 



THE AMERICAN EPHEMERIS AND NAUTICAL ALMANAC 69 

are about 800 stars, the coordinates for which are given for every 
ten days. The only other table in Part II of particular interest to 
the surveyor is that headed " Moon Culminations." This table 
contains the data needed when determining longitude by observ- 
ing transits of the moon. (See Art. 88, p. 141.) 

In the latter part of the Almanac will be found the following 
useful tables: I, Times of Culmination and Elongation of Polaris; 
II, Conversion of Sidereal Time into Mean Time; III, Conver- 
sion of Mean Time into Sidereal Time; IV, Latitude by an 
Altitude of Polaris; V and VI, Azimuths of Polaris; VII, Intervals 
for 5 Cassiopeia and f Ursa Majoris. (See Art. 99, p. 161.) 

39. Star Catalogues. 

When it is necessary to make observations on stars not given 
in the list in the Ephemeris, their positions must be taken from 
one or more of the star catalogues. These give the mean place 
of the star at some definite epoch, such as the beginning of the 
year 1890, or 1900, together with the necessary data for reducing 
to the mean place of any other year. This data is usually 
obtained by combining the observations made at different obser- 
vatories and at different times, so that changes in the star's 
coordinates are accurately determined. After the position in 
the catalogue has been brought up to the mean place for the 
desired year, the apparent place of the star for the exact date of 
the observation is computed by means of the formulae and tables 
given in Part II of the Ephemeris. For ordinary observations 
made by the surveyor the list of stars given in the Ephemeris is 
always sufficient, but in special kinds of work, such as finding 
latitude by Talcott's Method, many other stars must be used. 

40. Interpolation. 

When taking data from the Ephemeris it is general necessary, 
in order to obtain the value for a particular instant, to interpolate 
between values of the function for stated times. In some cases 
this may be simple interpolation, in which the function is assumed 
to vary uniformly between the two values given and the desired 
value found by direct proportion. When the difference for one 



70 PRACTICAL ASTRONOMY 

hour is given, this rate of change at the given instant may be 
assumed to hold good between the given value and the following 
one. Since this is not usually quite true, it will be more accurate 
to interpolate from the nearest given value in the table. The 
change for one hour is to be multiplied by the number of hours 
between the given time and the tabular time. This correction 
is either added or subtracted, according to whether the function 
is increasing or decreasing and whether the preceding or follow- 
ing tabular value is used. 

Example. 

At Greenwich Mean Noon. 
Feb. Sun's declination Diff. i h 

1 Si7 2 4 'o 4 ".o + 4 i".88 

2 17 07 09 .8 42 .64 

It is desired to find the declination at the instant 22* G. M. T. 
Feb. i. Since this is much nearer to the moon of Feb. 2 than 
of Feb. i, it will be more accurate to multiply 42". 64 by 2 h and 
add this to 17 07' 09". 8 (since the declination is decreasing). 
The result is S 17 08' 35".!. By working forward from the 
value on Feb. i the result is S 17 08' 3 2" . 6. By using a more 
exact formula the result is found to be S 17 08' 35".o. 

If the successive values of the " diff. for i h " or " diff. for 
i m " have large differences, and if a precise value of the function 
is desired, it will be necessary to interpolate between the given 
values of the differential coefficients to obtain the rate of change 
at the middle of the interval over which we are interpolating, 
and to use this interpolated rate of change in computing the 
correction. 

Example. 

Time R. A. of the Moon . Diff. for I OT 

o* 4 ft 46 m n a .49 2 s . 54 2 1 

i 4 48 44.06 2.5436 

If it is desired to find the right ascension for o h 40, the " diff. 
for i m " to be used is that for the instant o h 20, the middle 



THE AMERICAN EPHEMERIS AND NAUTICAL ALMANAC 71 

of the interval from o h to o h 40. This value lies one third of 
the way from 2 s . 5421 to 2 S .5436, or 2 S .5426. The correction to 
the R. A. at o h is 2 S .5426 X 4O TO = 101.70* = i m 4i s .7o, the 
required R. A. being 4* 47"* 53 s . 19. 

For general interpolation formulae the student is referred to 
Chauvenet's Spherical and Practical Astronomy, Vol. I, to Doo- 
little's Practical Astronomy or to Hayford's Geodetic Astronomy. 

Questions and Problems 

1. Compute the sun's apparent declination when the M. L. T. is 8 h 30 A.M., 
Jan. 16, 1912, at a place 85 west of Greenwich (see p. 65). 

2. Compute the right ascension of the mean sun at local mean noon Jan. 10, 
1912, at a place 96 10' west of Greenwich. 

3. Compute the equation of time for local apparent noon Jan. 30, 1912, at a 
place 20 east of Greenwich. 

4. Explain the relation between the sun's angular semidiameter and the time 
of the semidiameter passing the meridian. 

5. What is the relation between the " right ascension of the mean sun " and 
the " apparent right ascension of the sun " on Jan. i, 1912? 



CHAPTER VII 

THE EARTH'S FIGURE CORRECTIONS TO OBSERVED 

ALTITUDES 

41. The Earth's Figure. 

The earth's form is approximately that of an oblate spheroid 
whose shortest axis is the axis of rotation. The actual figure 
deviates slightly from that of a perfect spheroid, but for most 
astronomical purposes these deviations may be disregarded. 
Each meridian may therefore be considered as an ellipse, and 
the equator and all parallels of latitude as circles. The semi- 
major axis of the meridian ellipse is about 3962.80 miles, and the 
semi-minor axis is 3949.56 miles in length. The length of i of 
latitude at the equator is 68.704 miles; at the pole it is 69.407 
miles. 

In locating points on the earth's surface by means of coordi- 
nates there are three kinds of latitude to be considered. The 
latitude as found by astronomical observation is dependent upon 
the direction of gravity as indicated by the spirit levels of the 
instrument, and is affected by any abnormal deviations of the 
plumb line* at this point; the latitude as found directly by ob- 
servations is called the astronomical latitude. The geodetic 
latitude is the latitude that would be found by observation if 
the plumb line were normal to the surface of the spheroid taken 
to represent the earth's figure, that is, if all of the irregularities 
of the surface were smoothed out. Evidently the geodetic 
latitude cannot be directly observed but must be found by com- 
putation. The geocentric latitude is the angle between the plane 
of the equator and a line drawn from the centre of the earth to 
the point on the surface. In Fig. 39 the line AD is normal to 

* These deviations are small, averaging about 3" or 4", but in some cases 
deviations of nearly 30" are found. 

72 



THE EARTH'S FIGURE 



73 



the earth's surface at A, and the angle ABE is the geodetic 
latitude of A. If the plumb line coincides with AD, this is 
also the astronomical latitude. The angle ACE is the geocen- 
tric latitude. The difference between the two, or angle BAC, 




is called the angle of the vertical, or the reduction of latitude. 

The geocentric latitude is always less than the observed lati- 
tude by an angle which varies from about o n' 30" in latitude 
45 to zero at the equator and the poles. Whenever observations 
are made at any point on the earth's surface it is necessary to 
reduce the observed values to their values at the earth's centre 
before they can be combined with other data referred to the 
centre. In making this reduction the geocentric latitude must 
be used if the exact position of the observer with reference to the 
centre is to be computed. For most of the observations treated 
in the following chapters it will not be necessary to consider the 
spheroidal shape of the earth; it will be sufficiently exact to regard 
it as a sphere. 

42. Parallax. 

The coordinates of a celestial object as given in the Ephemeris 
are referred to the centre of the earth, while the coordinates 



74 



PRACTICAL ASTRONOMY 



obtained by observation are necessarily measured from a point 
on the surface, and must be reduced to the centre. The case 
most frequently occurring in practice is that in which the altitude 
of an object is observed and the geocentric altitude is desired. 
For all objects except the moon the distance of the body is so 
great that it is sufficiently accurate to regard the earth as a 




FIG. 40 

sphere. In Fig. 40, the angle ZOS is the observed zenith dis- 
tance, or the complement of the observed altitude, and ZCS is 
the true zenith distance. This apparent displacement of the 
object on the celestial sphere is called parallax. The effect of 
parallax is simply to decrease the altitude without altering the 
azimuth of the body, provided the spheroidal form of the earth 
be disregarded. The difference in direction between the lines 
OS and CS, or the angle OSC, is the parallax correction. In the 
triangle OSC, angle COS may be considered as known, since the 
altitude or complement of ZOS is observed. The distance OC 
is the semidiameter of the earth (3956.1 miles), and CS is the 



THE EARTH'S FIGURE' 75 

distance from the earth's centre to the centre of the body ob- 
served. Solving this triangle, 

or 

sinS = smZOSX~ [50] 

It is evident that the parallax correction will be zero at the 
zenith and a maximum at the horizon. For the maximum, 
when ZOS = 90, 

. e OC r , 

sin 5 = > [51] 

which is the same for all places on the earth's surface if the earth 
is regarded as a sphere. If P h represents the maximum or 
horizontal parallax, then equation [50] may be written 

sin S = sin Ph sin z 

= sin P h cos //, [52] 

where h is the apparent altitude of the object. But S and Ph 
are usually very small angles, and the error is negligible if the 
sines are replaced by their arcs.* Equation [52] then becomes 

S" = P h " cos h, [53] 

where S" and P h " are both in seconds of arc. 

For the moon the mean value of the horizontal parallax is 
about o 57' 02" f; for the sun it is 8". 8; for the fixed stars it is 

* The sine may be expressed as a series as follows: 

x 3 x 5 
sin x = x Y + T~ ~ ' ' ' [54] 

Replacing sin x by x amounts to neglecting all terms after the first. Whether the 
error will be appreciable in any given case may be determined by computing the 
value of the first of the neglected terms. If x = i the neglected terms are less 
than .005 of i% of x. The error in an angle of i would be less than o".2. The 
moon is the only object whose parallax is nearly as large as i, so that for all other 
objects this approximation is usually allowable. Similarly for cos x = i, the terms 
neglected are those of the series 



' I [55] 

L 11 

t The moon's mean distance is 238,800 miles; the sun's mean distance is 
92,900,000 miles. 



7 6 



PRACTICAL ASTRONOMY 



too small to be detected. The horizontal parallaxes of objects 
in the solar system are given in the Nautical Almanac.* For the 
parallax of the sun for different altitudes see Table IV (A). 

43. Refraction. 

Refraction is the term applied to the bending of a ray of light 
by the atmosphere as it passes from a celestial object to the 
observer's eye. On account of the increasing density of the 
layers of air the rays of light coming from any object are bent 
downward into a curve, and consequently when the rays enter 
the eye they have a greater inclination to the horizon than they 
did before entering the atmosphere. For this reason all objects 
appear higher above the horizon than they actually are. In 




FIG. 41 

Fig. 41, S is the true position of a star and 5" its apparent position. 
The light from 5 is bent into a curve aO, and the star is seen in 
the direction of the tangent ObS'. The angle which must be 
subtracted from the altitude of S f to obtain the altitude of 6 1 is 
called the refraction correction. This angle is really the angle 
SOS', but on account of the great distance of celestial objects 

* On account of the spheroidal form of the earth the equatorial diameter is the 
greatest and the parallax at the equator is a maximum; the parallaxes are therefore 
given in the Ephemeris under the heading " Equatorial Horizontal Parallax." 



THE EARTH'S FIGURE 77 

and the small angle of refraction the correction may be con- 
sidered as the angle SbS'. From the figure it is evident that 

ZcS = ZOS' + S'bS, 

or z' = z + r, [56] 

where z' = the true and z = the apparent zenith distance and 
r = the refraction correction. The approximate law of astro- 
nomical refraction may be deduced by assuming that the bend- 
ing all occurs at point b. The general law of refraction, when a 
ray enters a refracting medium, is expressed by the equation 

sinz' = n sin z, [57] 

where n is the index of refraction of the given medium; for air 
its value is roughly about 1.0003. 

Substituting from Equa. [56], 

sin (z + r) = n sin z, [58] 

Expanding, sin z cos r + cos z sin r = n sin z. [59] 

Since r is a small angle (never greater than 40') it is allowable 
to put cos r = i and sin r = r; then 

sin z + r cos z = n sin z, 

and r cos z = (n i) sin z, 

or r = (n i) tan z. [60] 

Replacing n by 1.0003 an d dividing by sin i" to reduce r from 
circular measure to seconds of arc, 

f.oooO 

r" = - ^- tan 2 

(.000,005) 

= 60" tan z 

= 60" cot A, [61] 

where /t is the apparent altitude.* 

The value of n varies considerably with the temperature and 
the pressure of the air, so that equation [61] must be considered 
as giving only a rough approximation to the true refraction. 

* " Apparent" is used here simply to distinguish between the direction of the star 
as actually seen and the direction unaffected by refraction. In speaking of parallax, 
the word " apparent " has a different meaning, and in case of aberration, still 
another meaning. 



7 PRACTICAL ASTRONOMY 

For high altitudes this formula is nearly correct, but for altitudes 
under 10 it is not sufficiently exact. If both sides of the equa- 
tion are divided by 60 so that r is reduced to minutes, we have 
the extremely simple relation that the refraction in minutes equals 
the natural cotangent of the altitude. For altitudes measured 
with an engineer's transit this formula is close enough for alti- 
tudes greater than about 10. For more accurate values of the 
refraction Table I may be used. From the table it will be seen 
that the refraction correction is zero at the zenith, about i' at 
an altitude of 45, and about 6 34' at the horizon.* 

The following formula, due to Professor George C. Comstock, 
gives very accurate values of the refraction for altitudes greater 
than 20, and is sufficiently accurate for all field observations 
made with surveyors' instruments. 

983 b , r , -, 

r = I cot h, [62] 

460 + t 

in which b is the barometer reading in inches, and / is the tem- 
perature in Fahrenheit degrees. 
Example. 

Altitude 30, barometer 29. i m ', thermometer 81 F. 
log. 983 = 2.9926 
log. 29.1 = 1.4639 

460 colog. 541 = 7.2668 

81 cot h = 0.2386 

541 1.9619 

r = 9 i".6 
= i' 3 i".6 

44. Semidiameters. 

The discs of the sun and moon are circular, and their angular 
semidiameters are given for each day in the Ephemeris. Since 
measurements can only be taken to the edge, or limb, the altitude 
of the centre of the object is obtained by making a correction 

* The sun's diameter is about 3 2', slightly less than the refraction on the horizon; 
when the sun has actually gone below the horizon at sunset the entire disc is still 
visible on account of the 34' increase in its apparent altitude due to atmospheric 
refraction. 



THE EARTH'S FIGURE 



79 



equal to the semidiameter. The apparent angular semidiameters 
given in the Ephemeris may be affected in two ways, one by the 
change in the observer's distance because he is on the earth's 
surface, the other by the difference in the amount of refraction 
correction on the upper and lower edges of the disc. 

The semidiameter given in the Ephemeris is that as seen from 
the centre of the earth. When the object is in the zenith the 
observer is nearly 4000 miles nearer than when it is in the hori- 
zon. The moon is about 240,000 miles distant from the earth, 
so that the semidiameter is increased by about ^V part, or 
about 16". 

The vertical diameter of an object appears to be less than its 
horizontal diameter because the refraction lifts the lower edge 
more than it does the upper edge. The disc then presents the 
appearance of an ellipse. When the sun is rising or setting, the 
contraction is most noticeable. This contraction of the semi- 
diameter does not affect the correction to an observed altitude, 
but must be taken into account when the distance is measured 
between the moon's limb and a star or a planet. (See Art. 108.) 

For the angular semidiameter of 
the sun on the first day of each 
month see Table IV (B). 

45- Dip- 

If altitudes are taken from the 
sea horizon, as when observing 
on board ship with the sextant, 
the measured altitude must be 
diminished by the angular dip 
of the sea horizon below the true 
horizon. In Fig. 42 suppose the 
observer to be at 0; the true 

horizon is OB and the sea horizon 

FIG. 42 

OH. Let OP = h, the height in 

feet above the surface; PC = R, the radius of the earth; and 

D, the angle of dip. 



B 




80 PRACTICAL ASTRONOMY 



Then cos Z> = -^-j-^ [63] 

D 2 

Putting cos D = i , neglecting other terms in the series, 

D 2 h h , , v 

= (nearly). 



R + h R 



Replacing R by its value in feet, 20,884,000, and dividing by 
sin i' to reduce D to minutes, 



V 

V 



- X sin i' 

2 



[64] 

This shows the amount of dip unaffected by refraction. The 
effect of refraction is to apparently lift the horizon, and the dip 
affecting the observed altitude is therefore less than that given 
by the formula. If the coefficient 1.064 is taken as unity, the 
formula is nearer the truth and is simpler, although still some- 
what too large. Table IV (C) , based on a more exact formula, will 
be seen to give smaller values. For ordinary sextant observations 
made at sea, where the greatest precision is not required it is 
sufficient to take the dip in minutes equal to the square root of 
the height of the eye in feet, that is, 



D' = Vh ft. [65] 

46. Sequence of Corrections. 

Strictly speaking, the corrections to the latitude should be 
made in the following order : 

(i) Instrumental corrections; (2) dip (if at sea); (3) refraction; 
(4) semidiameter; (5) parallax. In practice, however, it is not 
always necessary to follow this order exactly. At sea the cor- 
rections are often taken together as a single " correction to the 
altitude." Care should be taken to use the refraction correction 



THE EARTH'S FIGURE 8 1 

for the limb observed, not for the centre, for if the altitude is 
small the two will differ appreciably. 

Problems 

1. Compute the sun's mean horizontal parallax. The sun's mean distance is 
92,900,000 miles; for the earth's radius see Art. 41. Compute the sun's parallax 
at an altitude of 60. 

2. Compute the moon's mean horizontal parallax. The moon's mean distance 
is 238,800 miles; for the earth's radius see Art. 41. Compute the moon's parallax 
at an altitude of 45, 



CHAPTER VIII 
DESCRIPTION OF INSTRUMENTS 

47. The Engineer's Transit. 

The engineer's transit is an instrument for measuring hori- 
zontal and vertical angles. For the purpose of discussing the 
theory of the instrument it may be regarded as a telescopic line 
of sight having motion about two axes at right angles to each 
other, one vertical, the other horizontal. The line of sight is 
determined by the optical centre of the object glass and the 
intersection of two cross hairs* placed in its principal focus. 
The vertical axis of the instrument coincides with the axes of 
two spindles, one inside the other, each of which is attached to a 
horizontal circular plate. The lower plate carries a graduated 
circle for measuring horizontal angles; the upper plate has two 
verniers, on opposite sides, for reading angles on the circle. 
On the top of the upper plate are two uprights, or standards, 
supporting the horizontal axis to which the telescope is attached 
and about which it rotates. At one end of the horizontal axis 
is a vertical arc, or a circle, and on the standard is a vernier, in 
contact with the circle, for reading the angles. The plates and 
the horizontal axis are provided with clamps and slow-motion 
screws to control the motion. On the upper plate are two spirit 
levels for levelling the instrument, or, in other words, for making 
the vertical axis coincide with the direction of gravity. 

The whole instrument may be made to turn in a horizontal 
plane by a motion about the vertical axis, and the telescope may 
be made to move in a vertical plane by a motion about the 
horizontal axis. By means of a combination of these two 

* Also called wires or threads; they are either made of spider threads or are 
lines ruled upon glass. 

82 






DESCRIPTION OF INSTRUMENTS 83 

motions, vertical and horizontal, the line of sight may be made 
to point in any desired direction. The motion of the line of 
sight in a horizontal plane is measured by the angle passed over 
by the index of the vernier along the graduated horizontal 
circle. The angular motion in a vertical plane is measured by 
the angle on the vertical arc indicated by the vernier attached 
to the standard. The direction of the horizon is denned by 
means of a long spirit level attached to the telescope. When 
the bubble is central the line of sight should lie in the plane of 
the horizon. To be in perfect adjustment, (i) the axis of each 
spirit level * should be in a plane at right angles to the vertical 
axis; (2) the horizontal axis should be at right angles to the 
vertical axis; (3) the line of sight should be at right angles to the 
horizontal axis; (4) the axis of the telescope level should be 
parallel to the line of sight, and (5) the vernier of the vertical 
arc should read zero when the bubble is in the centre of the level 
tube attached to the telescope. When the plate levels are 
brought to the centres of their tubes, and the lower plate is so 
turned that the vernier reads o when the telescope points south, 
then the vernier readings of the horizontal plate and the vertical 
arc for- any position of the telescope are coordinates of the 
horizon system (Art. 12). If the horizontal circles are clamped 
in any position and the telescope is moved through a complete 
revolution, the line of sight describes a vertical circle on the 
celestial sphere. If the telescope is clamped at any altitude and 
the instrument turned about the vertical axis, the line of sight 
describes a cone and traces out on the sphere a circle of equal 
altitudes, or an almucantar. 

48. Elimination of Errors. 

It is usually more difficult to measure an altitude accurately 
with the transit than to measure a horizontal angle. While the 
precision of horizontal angles may be increased by means of 
repetitions, in measuring altitudes the precision cannot be 

* The axis of a level may be defined as a line tangent to the curve of the glass 
tube at the point on the scale taken as the zero point, or at the centre of the tube. 



84 PRACTICAL ASTRONOMY 

increased by repeating the angles, owing to the construction of 
the instrument. The vertical arc usually has but one vernier, 
so that the eccentricity cannot be eliminated, and this vernier 
often does not read as closely as the horizontal vernier. One 
of the errors, which is likely to be large, but which may be elimi- 
nated readily, is that known as the index error. The measured 
altitude of an object may differ from the true reading for two 
reasons: first, the zero of the vernier may not coincide with the 
zero of the circle when the telescope bubble is in the centre of 
its tube; second, the line of sight may not be horizontal when 
the bubble is in the centre of the tube. The first part of this 
error can be corrected by simply noting the vernier reading when 
the bubble is central, and applying this as a correction to the 
measured altitude. To eliminate the second part of the error 
the altitude may be measured twice, once from the point on the 
horizon directly beneath the object observed, and again from 
the opposite point of the horizon. In other words, the instru- 
ment may be reversed (180) about its vertical axis and the 
vertical circle read in 'each position while the horizontal cross 
hair of the telescope is sighting the object. The mean of the 
two readings is free from the error in the sight line. Evidently 
this method is practicable only with an instrument having a 
complete vertical circle. If the reversal is made in this manner 
the error due to non-adjustment of the vernier is eliminated at 
the same time, so that it is unnecessary to make a special deter- 
mination of it as described above. If the circle is graduated 
in one direction, it will be necessary to subtract the second 
reading from 180 and then take the mean between this result 
and the first altitude. In the preceding description it is assumed 
that the plate levels remain central during the reversal of the 
instrument, indicating that the vertical axis is truly vertical. 
If this is not the case, the instrument should be relevelled before 
the second altitude is measured, the difference in the two altitude 
readings in this case including all three errors. If it is not de- 
sirable to relevel, the error of inclination of the vertical axis may 



DESCRIPTION OF INSTRUMENTS 85 

still be eliminated by reading the vernier of the vertical circle 
in each of the two positions when the telescope bubble is central, 
and applying these corrections separately. With an instru- 
ment provided with a vertical arc only it is essential that the axis 
of the telescope bubble be made parallel to the line of sight, and 
that the vertical axis be made truly vertical. To make the axis 
vertical without adjusting the levels themselves, bring both 
bubbles to the centres of their tubes, turn the instrument 180 
in azimuth, and then bring each bubble half way back to the 
centre by means of the levelling screws. When the axis is truly 
vertical, each bubble should remain in the same part of its tube 
in all azimuths. The axis may always be made vertical by 
means of the long bubble on the telescope; this is done by set- 
ting it over one pair of levelling screws and centring it by means 
of the tangent screw on the standard; the telescope is then 
revolved about the vertical axis, and if the bubble moves from 
the centre of its tube it is brought half way back by means of 
the tangent screw, and then centred by means of the levelling 
screws. This process should be repeated to test the accuracy 
of the levelling; the telescope is then turned at right angles 
to the first position and the whole process repeated. This 
method should always be used when the greatest precision is 
desired, because the telescope bubble is much more sensitive 
than the plate bubbles. 

If the line of sight is not at right angles to the horizontal axis, 
or if the horizontal axis is not perpendicular to the vertical axis, 
the errors due to these two causes may be eliminated by com- 
bining two sets of measurements, one in each position of the 
instrument. If a horizontal angle is measured with the vertical 
circle on the observer's right, and the same angle again observed 
with the circle on his left, the mean of these two angles is free 
from both these errors, because the two positions of the horizontal 
axis are placed symmetrically about a true horizontal line,* and 

* Strictly speaking, they are placed symmetrically about a perpendicular to 
the vertical axis. 



86 PRACTICAL ASTRONOMY 

the two directions of the sight line are situated symmetrically 
about a true perpendicular to the rotation axis of the telescope. 
If the horizontal axis is not perpendicular to the vertical axis the 
line of sight describes a plane which is inclined to the true vertical 
plane. In this case the sight line will not pass through the zenith, 
and both horizontal and vertical angles will be in error. In 
instruments intended for precise work a striding level is provided, 
which may be set on the pivots of the horizontal axis. This 
enables the observer to level the axis or to measure its inclina- 
tion without reference to the plate bubbles. The striding level 
should be used in both the direct and the reversed position and 
the mean of the two results used in order to eliminate the errors 
of adjustment of the striding level itself. If the line of sight is 
not perpendicular to the horizontal axis it will describe a cone 
whose axis is the horizontal axis of the instrument. The line 
of sight will in general not pass through the zenith, even though 
the. horizontal axis be in perfect adjustment. The instrument 
must either be used in two positions, or else the cross hairs must 
be adjusted. Except in large transits it is not usually practicable 
to determine the amount of the error and allow for it. 

49. Attachments to the Engineer's Transit. Reflector. 

When making star observations with the transit it is necessary 
to make some arrangement for illuminating the field of view. 
Some transits are provided with a special shade tube into which 
is fitted a mirror set at an angle of 45 and with the central 
portion removed. By means of a lantern held at one side of 
the telescope light is reflected down the tube. The cross hairs 
appear as dark lines against the bright field. The stars can be 
seen through the opening in the centre of the mirror. If no 
special shade tube is provided, it is a simple matter to make a 
substitute, either from a piece of bright tin or by fastening a 
piece of tracing cloth or oiled paper over the objective. A hole 
about | inch in diameter should be cut out, so that the light from 
the star may enter the lens. If cloth or paper is used, the lan- 
tern must be held so that the light is diffused in such a way as 






DESCRIPTION OF INSTRUMENTS 87 

tQ render the cross hairs visible. The light should be held so as 
not to shine into the observer's eyes. 

50. Prismatic Eyepiece. 

When altitudes greater than about 55 to 60 are to be meas- 
ured, it is necessary to attach to the eyepiece a totally reflecting 
prism which reflects the rays at right angles to the sight line. 
By means of this attachment altitudes as great as 75 can be 
measured. In making observations on the sun it must be 
remembered that the prism inverts the image, so that with a 
transit having an erecting eyepiece with the prism attached the 
apparent lower limb is the true upper limb; the positions of the 
right and left limbs are not affected by the prism. 

51. Sun Glass. 

In making observations on the sun it is necessary to cover the 
eyepiece with a piece of dark glass to protect the eye from the 
sunlight while observing. The sun glass should not be placed 
in front of the objective. If no shade is provided with the 
instrument, sun observations may be made by holding a piece 
of paper behind the eyepiece so that the sun's image is thrown 
upon it. By drawing out the eyepiece tube and varying the 
distance at which the paper is held, the images of the sun and 
the cross hairs may be sharply focussed. By means of this 
device an observation may be quite accurately made after a 
little practice. 

52. The Portable Astronomical Transit. 

The astronomical transit differs from the surveyor's transit chiefly in size and 
in the manner of support. The diameter of the object glass may be anywhere 
from 2 to 4 inches, and the focal length from 24 to 48 inches. The instrument is 
set upon a stone or brick pier. The cross hairs usually consist of several vertical 
hairs (say n or more) instead of a single one as in the surveyor's transit. The 
motion in altitude is controlled by means of a clamp and a tangent screw. The 
azimuth motion is usually very small, simply enough to allow adjustments to be 
made, as the transit is not used for measuring horizontal angles. The axis is 
levelled or its inclination measured by means of a sensitive striding level. 

On account of the high precision of the work done with the astronomical transit 
the various errors have to be determined with great accuracy, and corresponding 
corrections applied to the observed results. The transit is chiefly used in the plane 



88 



PRACTICAL ASTRONOMY 



of the meridian for determining the times of transit of stars. The principal errors 
determined and allowed for are (i) azimuth, or deviation from the true meridian; 
(2) inclination of the horizontal axis; (3) collimation, or deviation of the sight line 
from the true perpendicular to the rotation axis. The corrections to reduce an 
observed time to the true time of transit across the meridian are given by formulae 
[66] to [68]. These corrections would apply equally well to observations with the 
engineer's transit, and serve to show the relative magnitudes of the errors for 
different positions of the objects observed. 

Azimuth correction = a cos h sec D, [66] 

Level correction = b sin h sec D, [67] 

Collimation correction = c sec D, [68] 

where a, b and c are the errors in azimuth, inclination and collimation respectively 
(expressed in seconds of time), and h is the altitude and D the declination of the 
star observed. From these formulae Table B has been computed. It is assumed 
that the instrument is i', or 4", out of the meridian (a = 4 s ); that the axis is 
inclined i', or 4 s , to the horizon (b = 4*); and that the sight line denned by the 
middle (or the mean) wire is i', or 4 s , to the right or left of its true position (c=4 8 ). 
The numbers in the table show the effect of these errors at different altitudes and 
declinations. 

TABLE B. ERROR IN OBSERVED TIME OF TRANSIT (IN 
SECONDS OF TIME) WHERE a, b OR c = i'. 





Declinations. 




2 


h 





10 


20 


30 


40 


50 


60 


70 


80 


h 


<2 


K 

w 





o*.o 


o s .o 


O S .O 


s . 


o*.o 


o s .o 


o s .o 


s . 


o s .o 


9 





























W 


c 


10 


0.7 


0.7 


0.8 


0.8 


0.9 


I . I 


1.4 


2 .O 


4.0 


80 




1 


20 


1.4 


1.4 


1.4 


1.6 


1.8 


2. I 


2-7 


4.0 


7-9 


70 


| 


"o 


3 


2.0 


2.O 


2. I 


2-3 


2.6 


3-i 


4.0 


5-8 


, "-5 


60 


3 


g 


40 


2.6 


2.6 


2.7 


3-o 


3-4 


4-0 


S- 2 


7-5 


14.8 


5 


j> 


^ 


5 


3.1 


3.1 


3-3 


3-6 


4-0 


4-8 


6.1 


9.0 


17.6 


40 


V 


a 


60 


3-5 


3-5 


3-7 


4-0 


4-5 


5-4 


6.9 


10. I 


19.9 


3 


3 


| 


7 


3-8 


3-8 


4.0 


4-4 


4-9 


5-8 


7-5 


II. 


21.6 


20 


S3 




80 


3-9 


4.0 


4.2 


4.6 


5-2 


6.1 


7-9 


"5 


22.7 


10 






90 


4-o 


4-1 


4.2 


4.6 


S- 2 


6.2 


8.0 


11.7 


23.0 


o 





Note. Use the bottom line for the collimation error. 

53. The Sextant. 

The sextant is an instrument for measuring the angular dis- 
tance between two objects, the angle always lying in the plane 



DESCRIPTION OF INSTRUMENTS 



8 9 



through the two objects and the eye of the observer. It is 
particularly useful at sea because it does not require a steady 
support like the transit. It consists of a frame carrying a 
graduated arc, AB, Fig. 43, about 60 long, and two mirrors / 
and H, the first one movable, the second one fixed. At the 
center of the arc, 7, is a pivot on which swings an arm IV, 6 to 
8 inches long. This arm carries a vernier V for reading the 




angles on the arc AB. Upon this arm is placed the index glass 
/. At H is the horizon glass. Both of these mirrors are set 
so that their planes are perpendicular to the plane of the arc 
AB, and so that when the vernier reads o the mirrors are parallel. 
The half of the mirror H which is farthest from the frame is 
unsilvered, so that objects may be viewed directly through the 
glass. In the silvered portion other objects may be seen by 
reflection from the mirror I to the mirror H and thence to 
point O. At a point near (on the line HO) is a telescope of 
low power for viewing the objects. Between the two mirrors 



90 PRACTICAL ASTRONOMY 

and also to the left of H are colored shade glasses to be used when 
making observations on the sun. The principle of the instru- 
ment is as follows : A ray of light coming from an object at 
C is reflected by the mirror / to H, where it is again reflected 
to O. The observer sees the image of C in apparent coincidence 
with the object at D. The arc is so graduated that the reading 
of the vernier gives directly the angle between OC and OD. 
Drawing the perpendiculars FE and HE to the planes of the 
two mirrors, it is seen that the angle between the mirrors is 
a )8. Prolonging CI and DH to meet at O, it is seen that the 
angle between the two objects is 2 a 2 /3. The angle between 
the mirrors is therefore half the angle between the objects that 
appear to coincide. In order that the true angle may be read 
directly from the arc each half degree is numbered as though it 
were a degree. It will be seen that the position of the vertex O 
is variable, but since all objects observed are at great distances 
the errors caused by changes in the position of are always 
negligible in astronomical observations. 

The sextant is in adjustment when, (i) both mirrors are per- 
pendicular to the plane of the arc; (2) the line of sight of the 
telescope is parallel to the plane of the arc; and (3) the vernier 
reads o when the mirrors are parallel to each other. If the 
vernier does not read o when the doubly reflected image of a 
point coincides with the object as seen directly, the index cor- 
rection may be determined and applied as follows. Set the 
vernier to read about 30' and place the shades in position for 
sun observations. When the sun is sighted through the tele- 
scope two images will be seen with their edges nearly in contact. 
This contact should be made as nearly perfect as possible and 
the vernier reading recorded. This should be repeated several 
times to increase the accuracy. Then set the vernier about 30' 
on the opposite side of the zero point and repeat the whole 
operation, the reflected image of the sun now being on the 
opposite side of the direct image. If the shade glasses are of 
different colors the contacts can be more precisely made. Half 



DESCRIPTION OF INSTRUMENTS 91 

the difference of the two (average) readings is the index correc- 
tion. If the reading off the arc was the greater, the correction 
is to be added to all readings of the vernier; if the greater reading 
was on the arc, the correction must be subtracted, 

In measuring an altitude of the sun above the sea horizon the 
observer directs the telescope to the point on the horizon ver- 
tically under the sun and then moves the index arm until the 
reflected .image of the sun comes into view. The sea horizon 
can be seen through the plain glass and the sun is seen in the 
mirror. The sun's lower limb is then set in contact with the 
horizon line. In order to be certain that the angle is measured 
to the point vertically beneath the sun, the instrument is tipped 
slowly right and left, causing the sun's image to describe an arc. 
This arc should be just tangent to the horizon. If at any point 
the sun's limb goes below the horizon the altitude measured is 
too great. The vernier reading corrected for index error and 
dip is the apparent altitude of the lower limb above the true 
horizon. 

54. Artificial Horizon. 

When altitudes are to be measured on land the visible horizon 
cannot be used, and the artificial horizon must be used instead. 
The surface of any heavy liquid, like mercury, molasses, or 
heavy oil, may be used for this purpose. When the liquid is 
placed in a basin and allowed to come to rest, the surface is 
perfectly level, and in this surface the reflected image of the sun 
may be seen, the image appearing as far below the horizon as 
the sun is above it. Another convenient form of horizon con- 
sists of a piece of black glass, with plane surfaces, mounted on a 
frame supported by levelling screws. This horizon is brought 
into position by placing a spirit level on the glass surface and 
levelling alternately in two positions at right angles to each 
other. This form of horizon is not as accurate as the mercury 
surface but is often more convenient. The principle of the 
artificial horizon may be seen from Fig. 44. Since the image 
seen in the horizon is as far below the true horizon as the sun is 



92 PRACTICAL ASTRONOMY 

above it, the angle between the two is 2 h. In measuring this 
angle the observer points his telescope toward the artificial 
horizon and then brings the reflected sun down into the field of 
view by means of the index arm. By placing the apparent 
lower limb of the reflected sun in contact with the apparent 
upper limb of the image seen in the mercury surface, the angle 
measured is twice the altitude of the sun's lower limb. The two 
points in contact are really images of the same point. If the 
telescope inverts the image, this statement applies to the upper 
limb. The index correction must be applied before the angle is 




Sextant 



FlG. 44 

divided by 2 to obtain the altitude. In using the mercury hori- 
zon care must be taken to protect it from the wind, otherwise 
small waves on the mercury surface will blur and distort the 
image. The horizon is usually provided with a roof -shaped 
cover having glass windows, but unless the glass has parallel 
faces this introduces an error into the result. A good substitute 
for the glass cover is one made of fine mosquito netting. This 
will break the force of the wind if it is not blowing hard, and 
does not introduce errors into the measurement. 

55. Chronometer. 

The chronometer is simply an accurately constructed watch 
with a special form of escapement. Chronometers may be 



DESCRIPTION OF INSTRUMENTS 93 

regulated for either sidereal or mean time. The beat is usually 
a half second. Those designed to register the time on chrono- 
graphs are arranged to break an electric circuit at the end of 
every second or every two seconds. The 6oth second is dis- 
tinguished either by the omission of the break at the previous 
second, or by an extra break, according to the construction of the 
chronometer. Chronometers are usually hung in gimbals to 
keep them level at all times; this is invariably done when they 
are taken to sea. It is important that the temperature of the 
chronometer should be kept as nearly uniform as possible, be- 
cause fluctuation in temperature is the greatest source of error. 
Two chronometers of the same kind cannot be directly com- 
pared with great accuracy, o s .i or o s .2 being about as close as 
the difference can be estimated. But a sidereal and a solar chro- 
nometer can easily be compared within a few hundredths of a 
second. On account of the gain of the sidereal on the solar 
chronometer, the beats of the two will coincide once in about 
every 3 m 05*. If the two are compared at the instant when the 
beats are apparently coincident, then it is only necessary to 
note the seconds and half seconds, as there are no fractions to 
be estimated. By making several comparisons and reducing 
them to some common instant of time it is readily seen that 
the comparison is correct within a few hundredths of a second. 
The accuracy of the comparison depends upon the fact that the 
ear can detect a much smaller interval between the two beats 
than can possibly be estimated when comparing two chronome- 
ters whose beats do not coincide. 

56. Chronograph. 

The chronograph is an instrument for recording the time kept by a chronometer 
and also any observations the times of which it is desired to determine. A piece 
of paper is wrapped about a cylinder, which is revolved by a mechanism at a uniform 
rate. A pen in contact with the paper is held on an arm, connected with the arma- 
ture of an electro-magnet, in such a way that the pen draws a continuous line which 
has notches in it corresponding to the breaks in the circuit made by the chro- 
nometer. By means of this instrument the time is represented accurately on the 
sheet as a linear distance. If it is desired to record the instant when any event 



94 



PRACTICAL ASTRONOMY 



^ 



occurs, such as the passage of a star over a cross hair, the observer presses a tele- 
graph key which breaks the same circuit, and a mark is made on the chronograph 
sheet. The instant of the observation may be scaled from the record sheet with 
great precision. 

57. The Zenith Telescope. 

The zenith telescope is an instrument designed for making observations for 
latitude by a special method devised by Capt. Andrew Talcott, and which bears 
his name. The instrument consists of a telescope having a vertical and a horizontal 
axis like the transit; the telescope is attached to one end of the horizontal axis in- 
stead of at the centre. The essential features of the instrument are (i) a microm- 
eter, placed in the focus of the eyepiece, for 

^^ \ r^*+^ measuring small differences in zenith distance, 

and (2) a sensitive spirit level, attached to a 
small vertical circle on the telescope tube, for 
measuring small deflections of the vertical axis. 
The telescope is used in the plane of the 
meridian. There are two stops whose positions 
can be regulated so that the telescope may be 
quickly shifted, by a rotation about the ver- 
tical axis, from the north meridian to the 
south meridian. The observation consists in 
measuring with the micrometer the difference 
in zenith distance of two stars, one north of 
the zenith and one south, which culminate 
- within a few minutes of each other, and in 
taking readings of the spirit level at the same 
time the micrometer settings are made. A 
FIG. 45. THE ZENITH TELESCOPE diagram of the instrument in the two posi- 
tions is given in Fig. 45. The inclination of 

the telescope to the vertical is not changed between the two observations, so it is 
essential that the zenith distances of the two stars should be so nearly equal that 
both will come within the range of the micrometer screw, usually 30' or less. 
The principle involved in this method may be seen from Fig. 46. From the 
observed zenith distance of the star Ss the latitude is 




and from the star 
Taking the mean, 



L = D n z n . 
L = HA, + /?) + i (*. - z n ). 



[69] 



The latitude is therefore the mean of the declinations corrected by half the djffer- 
ence of the zenith distances. The declination may be computed from the star 
catalogues, and the difference in zenith distance may be very accurately measured 
with the micrometer screw. It is evidently essential that the telescope should 



DESCRIPTION OF INSTRUMENTS 



95 



have the same inclination to the vertical in each case. If the inclination changes, 
however, the amount of this change is accurately determined from the level readings 
already mentioned (see Art. 70). 




FIG. 46 

58. Suggestions about Observing. 

The instrument used for making such observations as are 
described in this book will usually be either the engineer's transit 
or the sextant. In using the transit care must be taken to give 
the tripod a firm support. It is well to set the transit in position 
some time before the observations are to be begun; this allows 
the instrument to assume the temperature of the air and the 
tripod legs to come to a firm bearing on the ground. The 
observer should handle the instrument with great care, par- 
ticularly during night observations, when the instrument is 
likely to be accidentally disturbed. In reading angles at night 
it is important to hold the light in such a position that the 
graduations on the circle are plainly visible and may be viewed 
along the lines of graduation, not obliquely. By changing the 
position of the lantern and the position of the eye it will be 
found that the reading varies by larger amounts than would be 
expected when reading in the daylight. Care should be taken 
not to touch the graduated silver circles, as they soon become 
tarnished. The lantern should be held so as to heat the instru- 
ment as little as possible, and so as not to shine into the observer's 
eyes. Time may be saved and mistakes avoided if the program 
of observations is laid out beforehand, so that the observer knows 
just what is to be done and the proper order of the different 



9 6 



PRACTICAL ASTRONOMY 



steps. The observations should be arranged so as to eliminate 
instrumental errors, usually by means of reversals; but if this 
is not practicable, then the instrument must be put in good 
adjustment. The index correction should be determined and 
applied, unless it can be eliminated by the method of observing. 
In observations for time it will often be necessary to use an 
ordinary watch. If there are two observers, one can read the 
time while the other makes the observations. If a chronometer 
is used, one observer may easily do the work of both, and at the 



c sec. h 




FIG. 47 

same time increase the accuracy. In making observations by 
this method (called the " eye and ear method ") the observer 
looks at the chronometer, notes the reading at some instant, say 
at the beginning of some minute, and, listening to the half-second 
beats, carries along the count mentally and without looking at 
the chronometer. In this way he can note the second and 
estimate the fraction without taking his attention from the star 
and cross hair. After making his observation he may check his 
count by again looking at the chronometer to see if the two 
agree. After a little practice this method can be used easily 
and accurately. In using a watch it is possible for one observer 
to make the observations and also note the time, but it cannot 
be done with any such precision as with the chronometer, be- 
cause on account of the rapidity of the ticks (5 per second), 
the observer cannot count the seconds mentally. The observer 



DESCRIPTION OF INSTRUMENTS 



97 



must in this case look quickly at his watch and make an allow- 
ance, if it appears necessary, for the time lost in looking up and 

taking the reading. 

Problems 

i. Show that if the sight line makes an angle c with the perpendicular to the 
horizontal axis (Fig. 47) the horizontal angle between two points is in error by 
the angle 

c sec h' c sec h", 

where h' and h" are the altitudes of the two points. 



i tan h 




FIG. 48 

2. Show that if the horizontal axis is inclined to the horizon by the angle * 
(Fig. 48) the effect upon the azimuth of the sight line is i tan h, and that an angle 
is in error by 

i (tan h' - tan h"), 

where &' and h" are the altitudes of the points. 



CHAPTER IX 
THE CONSTELLATIONS 

59. The Constellations. 

A study of the constellations is not really a part of the subject 
of Practical Astronomy, and in much of the routine work of 
observing it would be of comparatively little value, since the 
stars used can be identified by means of their coordinates and a 
knowledge of their positions in the constellations is not essential. 
If an observer has placed his transit in the meridian and knows 
approximately his latitude and the local time, he can identify 
stars crossing the meridian by means of the times and the alti- 
tudes at which they culminate. But in making occasional 
observations with small instruments, and where much of the 
astronomical data is not known to the observer at the time, some 
knowledge of the stars is necessary. When a surveyor is be- 
ginning a series of observations in a new place and has no accu- 
rate knowledge of his position nor the position of the celestial 
sphere at the moment, he must be able to identify certain stars 
in order to make approximate determinations of the quantities 
sought. 

60. Method of Naming Stars. 

The whole sky is divided in an arbitrary manner into irregular 
areas, all of the stars in any one area being called a constellation 
and given a special name. The individual stars in any constel- 
lation are usually distinguished by a name, a Greek letter,* or 
a number. The letters are usually assigned in the order of 
brightness of the stars, a being the brightest, /3 the next, and so 
on. A star is named by stating first its letter and then the name 
of the constellation in the (Latin) genitive form. For instance, 

* The Greek alphabet is given on p. 190. 



THE CONSTELLATIONS 99 

in the constellation Ursa Minor the star a is called a Ursa 
Minoris; the star Vega in the constellation Lyra is called 
a Lyra. When two stars are very close together and have 
been given the same letter, they are often distinguished by the 
numbers i, 2, etc., written above the letter, as, for example, 
a 2 Capricorni, meaning that the star passes the meridian after 
a 1 Capricorni. 
* 61. Magnitudes. 

The brightness of stars is shown on a numerical scale by their 
magnitudes. A star having a magnitude i is brighter than one 
having a magnitude 2. On the scale of magnitudes in use a few 
of the brightest stars have fractional or negative magnitudes. 
Stars of the fifth magnitude are visible to the naked eye only 
under favorable conditions. Below the fifth magnitude a tele- 
scope is usually necessary to render the star visible. 

62. Constellations Near the Pole. 

The stars of the greatest importance to the surveyor are those 
near the pole. In the northern hemisphere the pole is marked 
by a second-magnitude star, called the polestar, Polaris, or 
a Ursa Minoris, which is about i 10' distant from the pole 
at the present time (1910). This distance is now decreasing 
at the rate of about one-third of a minute per year, so that for 
several centuries this star will be close to the celestial north pole. 
On the same side of the pole as Polaris, but much farther from 
it, is a constellation called Cassiopeia, the five brightest stars 
of which form a rather unsymmetrical letter W (Fig. 49). The 
lower left-hand star of this constellation, the one at the bottom 
of the first stroke of the W, is called 5, and is of importance to 
the surveyor because it is very nearly on the hour circle passing 
through Polaris and the pole; in other words its right ascension 
is nearly the same as that of Polaris. On the opposite side of 
the pole from Cassiopeia is Ursa Major, or the great dipper, a 
rather conspicuous constellation. The star f , which is at the 
bend in the dipper handle, is also nearly on the same hour circle 
as Polaris and 5 Cassiopeia. If a line be drawn on the sphere 






100 PRACTICAL ASTRONOMY 

between 8 Cassiopeia and f Ursa Majoris, it will pass nearly 
through Polaris and the pole, and will show at once the position 
of Polaris in its diurnal circle. The two stars in the bowl of 
the great dipper on the side farthest from the handle are in a 
line which, if prolonged, would pass near to Polaris. These 
stars are therefore called the pointers and may be used to find 
the polestar. There is no other star near Polaris which is 
likely to be confused with it. Another star which should be 
remembered is /3 Cassiopeia, the one at the upper right-hand 
corner of the W. Its right ascension is very nearly O A and 
therefore the hour circle through it passes nearly through the 
equinox. It is possible then, by simply glancing at Cassiopeia 
and the polestar, to estimate approximately the local sidereal 
time. When /3 Cassiopeia is vertically above the polestar it 
is nearly O A sidereal time; when the star is below the polestar 
it is 1 2 h sidereal time ; half way between these positions, left and 
right, it is 6 h and iS h , respectively. In intermediate positions 
the hour angle of the star ( = sidereal time) may be roughly 
estimated. 

63. Constellations Near the Equator. 

The principal constellations within 45 of the equator are 
shown in Figs. 50 to 52. Hour circles are drawn for each hour 
of R. A. and parallels for each 10 of declination. The approxi- 
mate declination and right ascension of a star may be obtained 
by scaling the coordinates from the chart. The position of the 
ecliptic, or sun's path in the sky, is shown as a curved line. The 
moon and the planets are always found near this circle because 
the planes of their orbits have only a small inclination to the 
earth's orbit. A belt extending about 8 each side of the ecliptic 
is called the Zodiac, and all the members of the solar system 
will always be found within this belt. The constellations along 
this belt, and which have given the names to the twelve " signs 
of the Zodiac," are Aries, Taurus, Gemini, Cancer, Leo, Virgo, 
Libra, Scorpio, Sagittarius, Capricornus, Aquarius, and Pisces. 
These constellations were named many centuries ago, and the 




FlG. 49. CONSTELLATIOJ 



MAPI 




BOUT THE NORTH POLE 




FIG. 50. PRINCIPAL FIXED STARS BETWEE: 






MAP II 




:CLINATIONS 45 NORTH AND 45 SOUTH 



*. 



2* *' 





JULY 



JUNE 



1C 



3d 



CANES VENATI 



- t 



CORONA 



-fa 



HERCUES 



20 






BOREALIS 



/ BOOTES 



P- 



.- 

8 V 



COMA BEF 






XVII 



SERPENS 

*a 
e-t- 

XVI 



I XV 



XIV 



OPHIUCHUS 



V. * 



VIRGO 



so 




Li. 



L.y 



SCORPIO 



FIG. 51. PRINCIPAL FIXED STARS BETWEI 




DECLINATIONS 45 NORTH AND 45 SOUTH 



NOVEMBER 



OCTOBER 



SEF 



40- 



LACERTA 



30 



ANDROMEDA 



10 



PEGASUS 



8 PISCES 




XXIII 



XXII 



XXI 



10 



-ft 



/ AQUARIUS 




CETUS 



CARRICORNU 



30 



PISCIS AUSTRALIS 



FIG. 52. PRINCIPAL FIXED STARS BETWEI 



MAP IV 



AUGUST 



JULY 




)ECLINATIONS 45 NORTH AND 45 SOUTH 




FlG. 53. CONSTELLATIO 




ABOUT THE SOUTH POLE 



THE CONSTELLATIONS IOI 

names have been retained, both for the constellations themselves 
and also for the positions in the ecliptic which they occupied at 
that time. But on account of the continuous westward motion 
of the equinox, the " signs " no longer correspond to the con- 
stellations of the same name. For example, the sign of Aries 
extends from the equinoctial point to a point on the ecliptic 
30 eastward, but the constellation actually occupying this 
space at present is Pisces. In Figs. 50 to 52 the constellations 
are shown as seen by an observer on the earth, not as they would 
appear on a celestial globe. On account of the form of pro- 
jection used in these maps there is some distortion, but if the 
observer faces south and holds the page up at an altitude equal 
to his colatitude, the map represents the constellations very 
nearly as they will appear to him. The portion of the map to be 
used in any month is that marked with the name of the month 
at the top; for example, the stars under the word " February " 
are those passing the meridian in the middle of February at 
about 9 P.M. For other hours in the evening the stars on the 
meridian will be those at a corresponding distance right or left, 
according as the time is earlier or later than 9 P.M. The approxi- 
mate right ascension of a point on the meridian may be found at 
any time as follows: First compute the R. A. of the sun by 
allowing 2 h per month, or more nearly 4 per day for every 
day since March 23, remembering that the R. A. of the sun is 
always increasing. Add this R. A. to the local mean time and 
the result is the sidereal time or right ascension of a star on the 
meridian. 

Example. On October 10 the R. A. of the sun is 6 X 2 h + 17 X 
4 TO = 13* o8 m . At g h P.M. (local mean time) the sidereal time 
is 13* o8 w + g h oo m = 22 h oS m . A star having a R. A. of 22* o8 m 
wgould therefore be close to the meridian at 9 P.M. 

Fig. 53 shows the stars about the south celestial pole. There 
is no bright star near the south pole, so that the convenient 
methods of determining the meridian by observations on the 
polestar are not practicable in the southern hemisphere. 



102 PRACTICAL ASTRONOMY 

64. The Planets. 

In using the star maps, the student should be on the lookout 
for planets. These cannot be placed on the maps because their 
positions are rapidly changing. If a bright star is seen near the 
ecliptic, and its position does not correspond to that of a star 
on the map, it is a planet. The planet Venus is very bright and 
is never very far from the sun; it will therefore be seen a little 
before sunrise or a little after sunset. Mars, Jupiter, and Saturn 
are outside the earth's orbit and therefore revolve around the 
earth. Jupiter is the brightest, and when looked at through a 
small telescope shows a disc like that of the full moon, and four 
satellites can usually be seen all lying nearly in a straight line. 
Saturn is not as large as Jupiter, but in a telescope of moderate 
power its rings can be distinguished, or at least the planet looks 
elongated. Mars is reddish in color and shows a disc. 



CHAPTER X 
OBSERVATIONS FOR LATITUDE 

IN this chapter and the three immediately following are given 
the more common methods of determining latitude, time, longi- 
tude, and azimuth with small instruments. Those which are 
simple and direct are printed in large type, and may be used for 
a short course in the subject. Following these are given, in 
smaller type, several methods which, although less simple, are very 
useful to the engineer; these methods require a knowledge of 
other data which the engineer must obtain by observation, and 
are therefore better adapted to a more extended course of study. 

65. Latitude by a Circumpolar Star at Culmination. 

This method may be used with any circumpolar star, but 
Polaris is the best one to use, when it is practicable to do so, 
because it is of the second magnitude, while all of the other 
close circumpolars are quite faint. The observation consists 
in measuring the altitude of the star when it is a maximum or a 
minimum, or, in other words, when it is on the observer's me- 
ridian. This altitude may be obtained by trial, and it is not 
necessary to know the exact instant when the star is on the 
meridian. The approximate time when the star is at culmina- 
tion may be obtained from Table V or by formulae [39] and [49]. 
It is not necessary to know the time with accuracy, but it will 
save unnecessary waiting if the time is known approximately. 
In the absence of any definite knowledge of the time of culmina- 
tion, the position of the pole star with respect to the meridian may 
be estimated by noting the positions of the constellations. When 
S Cassiopeia is directly above or below Polaris the latter is at 
upper or lower culmination. The observation should be begun 
some time before one of these positions is reached. The hori- 

103 



104 PRACTICAL ASTRONOMY 

zontal cross hair of the transit should be set on the star* and the 
motion of the star followed by means of the tangent screw of the 
horizontal axis. When the desired maximum or minimum, is 
reached the vertical arc is read. The index correction should 
then be determined. If the instrument has a complete vertical 
circle and the time of culmination is known approximately, it 
will be well to eliminate instrumental errors by taking a second 
altitude with the instrument reversed, provided that neither 
observation is made more than 4 m or 5 m from the time of culmi- 
nation. If the star is a faint one, and therefore difficult to find, 
it may be necessary to compute its approximate altitude (using 
the best known value for the latitude) and set off this altitude 
on the vertical arc. The star may be found by moving the 
telescope slowly right and left until the star comes into the field 
of view. Polaris can usually be found in this manner some time 
before dark, when it cannot be seen with the unaided eye. It 
is especially important to focus the telescope carefully before 
attempting to find the star, for the slightest error of focus may 
render the star invisible. The focus may be adjusted by look- 
ing at a distant terrestrial object or, better still, by sighting at 
the moon or at a planet if one is visible. If observations are to 
be made frequently with a surveyor's transit, it is well to have 
a reference mark scratched on the telescope tube, so that the 
objective may be set at once at the proper focus. 

The latitude is computed from Equa. [3] or [4]. The true 
altitude h is derived from the reading of the vertical circle by 
applying the index correction with proper sign and then subtract- 
ing the refraction correction (Table I). The polar distance is 
found by taking from the Ephemeris (Table of Circumpolar 
Stars) the apparent declination of the star and subtracting this 
from 90. 

* The image of a star is practically a point of light; if the telescope were perfect 
it would be actually a point, but, owing to the imperfections in the corrections 
for aberration, the image, even though perfectly distinct, has an appreciable 
width. The image of the star should be bisected with the horizontal cross hair. 



OBSERVATIONS FOR LATITUDE 105 

Example i. 

Observed altitude of Polaris at upper culmination = 43 37'; 
index correction = +30"; declination = +88 44' 35". 

Vertical circle = 43 37' oo" 

Index correction = +30 

Observed altitude =43 37 30 

Refraction correction = i 02 



True altitude = 43 36 28 

Polar distance = i 15 25 

Latitude = 42 21' 03" 

Since the vertical circle reads only to i' the resulting value for the 
latitude must be considered as reliable only to the nearest i'. 

Example 2. 

Observed altitude of 51 Cephei at lower culmination = 39 
33' 3"j index correction = o"; declination = -f- 87 n' 25". 

Observed altitude = 39 33' 30" 
Refraction correction = i 10 



True altitude =39 32 20 

Polar distance = 2 48 35 



Latitude = 42 20' 55" 

\A 66. Latitude by Altitude of Sun at Noon. 

The altitude of the sun at noon (meridian passage) may be 
determined by placing the line of sight of the transit in the plane 
of the meridian and observing the altitude of the upper or lower 
limb of the sun when it is on the vertical cross hair. The watch 
time at which the sun will pass the meridian may be computed 
by converting i2 h local apparent time into Standard or locaJ 
mean time (whichever is used) as shown in Arts. 28 and 35. 
Usually the direction of the meridian is not known, so the maxi- 
mum altitude of the sun is observed and assumed to be the same 
as the meridian altitude. On account of the sun's changing 
declination the maximum altitude is not quite the same as the 
meridian altitude; the difference is quite small, however, usually 
a fraction of a second, and may be entirely neglected for obser- 
vations made with the engineer's transit or the sextant. The 
maximum altitude of the upper or lower limb is found by trial, 



106 PRACTICAL ASTRONOMY 

the horizontal cross hair being kept tangent to the limb as long 
as it continues to rise. When the observed limb begins to drop 
below the cross hair the altitude is read from the vertical arc 
and the index correction is determined. The true altitude of 
the centre of the sun is then found by applying the corrections for 
index error, refraction, semidiameter, and parallax. In order 
to compute the latitude it is necessary to know the sun's declina- 
tion at the instant the altitude was taken. If the longitude of 
the place is known approximately (say within half a degree) the 
declination may be taken from the Nautical Almanac for the 
instant of Greenwich Apparent Noon and increased or decreased 
by the hourly change multiplied by the number of hours in the 
longitude. If the place is west of Greenwich the correction is to 
be added algebraically; if the place is east, it is to be subtracted. 
If the longitude is not known, but the Greenwich mean time is 
known, as would be the case if the timepiece kept either Green- 
wich time or Standard time, the declination may be computed 
by noting the watch time of the observation as nearly as possible 
and correcting the declination as follows: take out the declina- 
tion at Greenwich Mean Noon, and increase it by the hourly 
change multiplied by the number of hours since Greenwich Mean 
Noon. The latitude is then found from Equa. [2]. 

Example i. 

Observed maximum altitude of sun's lower limb, Jan. 8, 1906, = 25 06'; index 
correction = +i'; the longitude is 4>44*i8 s (= 71 04' 5) west; the declina- 
tion of the sun at Greenwich Apparent Noon = S 22 19' 33"; hourly change = 
+ i9".59; the semidiameter = 16' 17". 

Observed altitude = 25 06'. o Decl. at G. A. N. = - 22 19' 33" 
Index correction + i .o 19". 59 X 4 A - 74 = +i 33 

25 07 . o Decl. at L. A. N. = 22 18' oo" 

Refraction correction = 2.0 

25 05 . o 
Semidiameter = + 16 .3 

25 



Parallax 
Altitude of centre 
Declination 
Colatitude 
Latitude 


= 


+ .1 


= _ 


25 21.4 

22 l8 . O 


= 


47 39'- 4 
42 20'. 6 



OBSERVATIONS FOR LATITUDE 107 

Example 2. 

Observed double altitude of sun's upper limb at noon on Jan. 28, 1910 (with 
artificial horizon), = 59 17' 40"; Eastern Standard time = 11^57' ; index cor- 
rection = +30"; declination at Greenwich Mean Noon = S 18 21' 08"; hourly 
change = +39". 07; semidiameter = 16' 16". 

Decl. at G. M. N. = - 18 21' 08" Double alt. = 59 17' 40" 

39". 07 X 4^.95 +3' 13" I- C. __ 3 

Decl. at n* s? m = - 18 17' 55" 2)59 18 10 

29 39' 05" 
Refraction = i 43 



29 37 22 
Semidiameter = 16 16 



29 21 06 
Parallax = ' +8 



29 21 14 

Declination 18 17 55 



Colatitude = 47 39' 09" 

Latitude = 42 20' 51" 

67. By the Meridian Altitude of a Southern* Star. 

The latitude may be found from the observed maximum alti- 
tude of a star which culminates south of the zenith, by the 
method of the preceding article, except that the parallax and 
semidiameter corrections become zero, and that it is not necessary 
to note the time of the observation, since the declination of the 
star changes so slowly. In measuring the altitude the star's 
image is bisected with the horizontal cross hair, and the maxi- 
mum found by trial as when observing on the sun. For the 
method of finding the time at which a star will pass the me- 
ridian see Art. 72. 

Example. 

Observed meridian altitude of Serpentis = 51 45'; index correction = o; 
declination of star = +4 05' n". 

Observed altitude of Serpentis = 51 45' oo 

Refraction correction = 46 

Si 44 14 
Declination of star = + 4 05 1 1 



Colatitude = 47 39' 03" 

Latitude = 42 20' 57" 

* The observer is assumed to be in the northern hemisphere. 



108 PRACTICAL ASTRONOMY 

Constant errors in the measured altitudes may be eliminated 
by combining the results obtained from circumpolar stars with 
those from southern stars. An error which makes the latitude 
too great in one case will make it too small by the same amount 
in the other case. 

68. Altitudes Near the Meridian. 

If altitudes of the sun or a star are taken near the meridian they may be reduced 
to the meridian altitude provided the latitude and the times are known approxi- 
mately. To derive the formula for making the reduction take the fundamental 
formula given in Equa. [8] 

sin h = sin L sin D + cos L cos D cos P. 

This may be transformed into 

sin h = cos (L D) cos L cos D vers P, [70] 

or 

p 
sin h = cos (L D) cos L cos D X 2 sin 2 [71] 

2 

Transposing and denoting by h m the meridian altitude L D, the equation 
becomes 

sin h m = sin h + cos L cos D vers P, [72] 

p 

or sin h m = sin h + cos L cos D X 2 sin 2 [73] 

If the altitude h be measured and the corresponding time be noted, then the value 
of P becomes known. If L is known approximately, then the second term may be 
computed and h m , or L D, found through its sine. If the value of L derived 
from the first computation does not agree closely with the assumed value, a second 
computation should be made using the new value of L. When observations are 
taken within a few minutes of the meridian (say 15) the computation may be 
shortened by the use of the approximate formula 

C" = 112.5 X P 2 X cos L cos D sec h sin i", [74] 

in which C" is the correction in seconds of arc and P is the time from the meridian 
expressed in seconds of time. (Log 112.5 X sin i" = 6.7367). If Pis expressed in 
minutes the formula is 

C" = i ".9635 X P 2 X cos L cos D sec h. [75] 

This formula may be derived as follows: Transposing Equa. [73] 



p 

sin h m sin h = 2 cos L cos D sin 2 [76] 



By trigonometry 



p 

2 cos | (h m + K) sin (h m - h) = 2 cos L cos D sin 2 [77] 



OBSERVATIONS FOR LATITUDE 



109 



Since h is nearly equal to h m , cos 5 (h m + h) may be put equal to cos h; placing 

C = h m h, the equation becomes 

p 
sin 5 C = cos L cos .D sin 2 sec k. [78] 

C and P are both small angles and may be put in place of their sines, hence 

C = i P 2 X cos L cos D sec h. [79] 

To reduce C to seconds of arc and P to seconds of time the left member must be 
multiplied by sin i" and the right by (15 sin i") 2 , giving 

C" = cos L cos D sec h X P 2 X 112.5 sin i". [74] 

In using this formula it will be necessary to use an approximate value of L; a 
second approximation may be made if necessary. 

The method of " reduction to the meridian " given above should not be applied 
when the object observed is far from the meridian. 

Example i. 

Observed double altitude sun's lower limb Jan. 28, 1910. 

Double Alt. Q 
56 44' 40" 
49 oo 
52 4 



I.C. 



56 48' 47" 
+3Q" 

2) 56 40' i7" 



28 24' 38' 
Refr. = - 1 46 



Watch, 
n* i5 m 25* 

l6 22 

17 10 


Watch corr. 

E. S. T. 
App. Noon 


n ft i6 m 19* 
+ i 19 


ii 57 21 



28 22 

s.d. = +16 



Hour angle = 39 43* 

P = 9 55' 45' 



52 
16 



28 39 08 
par. = +8 



h = 28 39' 16' 

log cos L = 9. 86763 

log cos D = 9. 97745 

log vers P = 8. 17546 



8. 02054 

. 01048 

nat sin h = . 47953 



Assumed lat.. = 
L. A. N. 
Eq. T 

L. M. T. 
Red. for long. = 

E. S. T. 
Sun's decl. at 
G. M. N. - - 
39".o7 X 4* 


42 30' 

12 OO OO 
+ 13 03 


12 13 03 
IS 42 


ii 57 21 

18 21' 08" 
.3 = 2' 48" 



nat sin h m = 49001 

h m = 29 20' 29" 
D = 18 18 20 



Cor'd. decl. = 18 18' 20" 



Co-lat. = 47 38' 49" 
Lat. = 42 21' n" 
A recomputation, using the corrected latitude, changes this result to 42 21' 04". 



no 



PRACTICAL ASTRONOMY 



Example 2. 

Observed altitude of y Ceti = 50 33'; index correction = i'; hour angle of 
y Ceti derived from observed time = 3 m 14^.2; declination = -f- 2 50' 30". 

log cos L = g. 8691 

log cos D = 9. 9995 

log sec h = o. 1967 

log const = 6. 7367 

2 log P = 4. 5765 



logC 



Observed altitude 
Index correction 



= 1-3785 
= 23 ". 9 

- 5o33'-o 
= i .o 



50 32-0 
Refraction correction = 0.8 



True altitude = 50 31'. 2 

Reduction to meridian = +0.6 



Meridian altitude 
Declination 

Co-latitude 
Latitude 



= 50 

= + 2 



31'- 8 
SO-5 



= 47" 4i'. 3 
= 42 i8'. 7 



The method of " ex-meridians altitudes," as it is sometimes called, may be used 
when the meridian observation is lost or when it is desired to increase the accuracy 
of the result by multiplying the number of observations. 

69. Latitude by Altitude of Polaris when the Time is Known. 

The altitude of Polaris varies slowly on account of its nearness to the pole, 
hence, if the sidereal time is known, the latitude may be found accurately by an 
altitude of this star taken at any time, because errors in the time have a rela- 
tively small effect upon the result. Several altitudes should be taken in succession, 
and the time noted at each pointing of the cross-hair on the star. For obser- 
vations made with the surveyor's transit and covering only a few minutes' time 
the mean of the altitudes may be taken as corresponding to the mean of the observed 
times. If the instrument has a complete vertical circle, half of the observations 
should be made with the instrument in the reversed position. The index correc- 
tion should be determined in each case. In order to compute the latitude it is 
necessary to know the hour angle of the star at the instant of observation. When 
a common watch is used for taking the time the star's hour angle is found by Equa. 
[47] and [37]. The latitude is then found by the formula 

L = h p cos P + | sin i' p 2 sin 2 P tan h [80] 

(log \ sin i' = 6. 1627 10) 

The derivation of the formula is rather complex and will not be given here. It 
is obtained by expanding the correction to h in a series in ascending powers of 



OBSERVATIONS FOR LATITUDE 



ill 



p, the small terms being neglected. The sum of all terms after that containing 
p 2 amounts to less than i" and these have therefore been omitted in Equa. [80]. 
In this equation p, the polar distance, is expressed in minutes of arc. Values of 
the last term may be taken with sufficient accuracy from Table VI. The alge- 
braic sign of the second term is deter- 
mined by the sign of cos P; the third 
term is always positive. In Fig. 54, 
P is the pole, S the star, MS the 
hour angle, and PDA the almucantar 
through P or circle of equal altitudes. 
It will be seen that the term p cos P 
is approximately the distance from 5 
to E, a point on the six-hour circle 
PB; the distance desired is SD, the 
angular distance of 5 above the al- 
mucantar through P. The last term 
in Equa. [80] is approximately equal 
to DE, each term in the series giving 
a closer approximation to the distance 
SD. 

Example. FIG. S4 

Observed Altitudes of Polaris, Jan. 9, 1907. 

Watch. Altitudes. 

6 h 49 26 s 43 28'. 5 

5i 45 28.5 

54 14 28 . o 

56 45 28.0 

Index correction = i'. o; p = 71'. 15; P is found from the observed watch times 
to be 13 50 . 7.* 

log p 1.8522 log constant =6.1627 

log cos P =9. 9822 log p 2 = 3. 7044 

log sin 2 P =8. 7578 

log p cos P = i. 8394 log tan h =9. 9762 




p cos P = 69'. 09 



Last term 

Observed alt. = 43 28'. 25 
I.C. - i. 



8.6011 
= + o'. 04 



Refraction 



i st. and and terms 
Latitude 



43 27'. 25 
i . 01 

43 26'. 24 
i 09 .05 

42 17'- 19 



* If the error of the watch is known the sidereal time may be found by Equa. [47]. 
For method of finding the sidereal time by observation see Chap. XI. The 
hour angle of the star is found by Equa. [38]. 



112 



PRACTICAL ASTRONOMY 



70. Precise Latitude Determinations. Talcott's Method. 

The most precise method of determining latitude is that known as " Talcott's 
Method," which requires the use of the zenith telescope. In making observations 
the observer selects two stars, one north of the zenith and one south of it, the two 
zenith distances differing by only a few minutes of angle, and the right ascensions 
differing by about 5 or 10 minutes of time. For the best results the zenith dis- 
tances should be small and nearly equal. If the first star culminates south of 
the zenith the telescope is turned about its vertical axis until the stop indicates 
that it is in the meridian and on the south side of the zenith. The telescope is 
tipped until the sight line has an inclination to the vertical equal to the mean of 
the two zenith distances.* It is clamped in this position and great care is taken 
not to alter its inclination until the observations on both stars are completed. 
When the star appears in the field the micrometer wire is set so as to bisect the 
star's image; at the instant of culmination the setting of the wire is perfected and 
the scale of the spirit level is read at the same time. The chronometer (regulated 
to local sidereal time) should be read when the bisection is made, so that the read- 
ing of the micrometer may be corrected if the star was not exactly on the meridian 
at that instant. The micrometer screw is then read. The telescope is then turned 
to the north side of the meridian, the inclination remaining unchanged, and a 
similar observation made on the other star. When both sets of micrometer read- 
ings and level readings have been obtained, the latitude is found by the formula 



L = 



+ D n ) + } (m, - ) X R + i (/. + W + i (r s -r n ), 



[81] 



in which m s , m n are the micrometer readings, R the value of i division of the mi- 
crometer, l s , l n the level corrections (positive when the north reading is the larger) 
and r s , r n the refraction corrections. Another correction must be added in case 
the observation is taken when the star is off the meridian. 

In order to determine latitude by this method with the precision required in 
geodetic operations, observations are made on several nights, and on each night 
a large number of pairs of stars is observed. By this method a latitude may be 
determined within about o". 05 which is equivalent to nearly 5 feet in distance 
on the earth's surface. 

Questions and Problems 

1. Observed maximum altitude sun's lower limb, April 27, 1910, = 61 05'. 
Index correction = + 30". The longitude is 4^ 44"* i8 s W. The sun's decl. 
at G. A. N. = N 13 38' 22". 3; diff. for i h = +48". 07; the semidiameter = 
15' 55". Compute the latitude. 

2. The observed meridian altitude of 8 Crateris = 33 24'; index correction = 
+ 30"; declination of star = 14 if 37". Compute the latitude. 

3. Observed altitude of a Ceti at 3* o8 m 49* L. S. T. = 51 21'; I. C. = i' 

* In order to compute these zenith distances it is necessary to know a rough 
value of the latitude, say within i' or 2'. This may be found by an observation 
with the zenith telescope using one of the preceding methods. 



OBSERVATIONS FOR LATITUDE 113 

the right ascension of a Ceti = 2 h 57"* 24*. 8; decimation = + 3 43' 22". Com- 
pute the latitude. 

4. Observed Altitude of Polaris, 41 41' 30"; chronometer time, g h 44 38*. 5 
(Loc. Sid. Time); chronometer correction, 34*. The R. A. of Polaris is i* 25 
42'; the declination is + 88 49' 29". Compute the latitude. 

5. Show by a sketch the positions of the following three points; i. Polaris 
at greatest elongation; 2. Polaris on the six-hour circle; 3. Polaris at the same 
altitude as the pole. 

6. What is the most favorable position of the sun for a latitude observation? 

7. What is the most favorable position of Polaris for a latitude observation? 

8. Draw a sketch showing why the sun's maximum altitude is not the same as 
the meridian altitude. 



CHAPTER XI 
OBSERVATIONS FOR DETERMINING THE TIME 

71. Observation for Local Time. 

Observations for determining the local time at any place at 
any instant usually consist in finding the error of a timepiece 
on the kind of time which it is supposed to keep. To find the 
solar time it is necessary to determine the hour angle of the sun's 
centre. To find the sidereal time the hour angle of the vernal 
equinox must be measured. In some cases these quantities 
cannot be measured directly, so it is often necessary to measure 
other coordinates and to calculate the desired hour angle from 
these measurements. The chronometer correction or watch 
correction is the amount to be added algebraically to the read- 
ing of the timepiece to give the true time at the instant. It is 
positive when the chronometer is slow, negative when it is fast. 
The rate is the amount the timepiece gains or loses per day; 
it is positive when it is losing, negative when it is gaining. 

72. Time by Transit of a Star. 

The most direct and simple means of determining time is by 
observing transits of stars across the meridian. If the line of 
sight of a transit be placed so as to revolve in the plane of the 
meridian, and the instant observed when some known star 
passes the vertical cross hair, then the local sidereal time at this 
instant is the same as the right ascension of the star given in 
the Nautical Almanac for the date. The difference between 
the observed chronometer time t and the right ascension a. 
is the chronometer correction T, 

or T = a-t. [82] 

If the chronometer keeps mean solar time it is only necessary 

to convert the true sidereal time a into mean solar time by 

114 



OBSERVATIONS FOR DETERMINING THE TIME 115 

Equa. [49], and the difference between the observed and com- 
puted times is the chronometer correction. 

The transit should be set up and the vertical cross hair sighted 
on a meridian mark previously established. If the instrument 
is in adjustment the sight line will then swing in the plane of 
the meridian. It is important that the horizontal axis should 
be accurately levelled; the plate level which is parallel to this 
axis should be adjusted and centred carefully, or else a striding 
level should be used. Any errors in the adjustment will be 
eliminated if the instrument is used in both the direct and re- 
versed positions, provided the altitudes of the stars observed 
in the two positions are equal. It is usually possible to select 
stars whose altitudes are so nearly equal that the elimination 
of errors will be nearly complete. 

In order to find the star which is to be observed, its approxi- 
mate altitude should be computed beforehand and set off on 
the vertical arc. (See Equa. [i].) In making this computation 
the refraction correction may be omitted, since it is not usually 
necessary to know the altitude closer than 5 or 10 minutes. 
It is also convenient to know beforehand the approximate time 
at which the star will culminate, in order to be prepared for the 
observation. If the approximate error of the watch is already 
known, then the watch time of transit may be computed (Equa. 
[49]) and the appearance of the star in the field looked for a 
little in advance of this time. If the data from the Nautical 
Almanac are not at hand the computation may be made, with 
sufficient accuracy for finding the star, by the following method : 
Compute the sun's R. A. by multiplying 4 by the number of 
days since March 22. Take the star's R. A. from any list of 
stars or a star map. The star's R. A. minus the sun's R. A. 
(Equa. [49]) will be the mean local time within perhaps 2"* or 
3 TO . This may be reduced to Standard Time by the method 
explained in Art. 35. In the surveyor's transit the field of view 
is usually about i, so the star will be seen about 2 before it 
reaches the vertical cross hair. Near culmination the star's 



Il6 PRACTICAL ASTRONOMY 

path is so nearly horizontal that it will appear to coincide with 
the horizontal cross hair from one side of the field to the other. 
When the star passes the vertical cross hair the time should be 
noted as accurately as possible. A stop watch will sometimes 
be found convenient in field obversations with the surveyor's 
transit. When a chronometer is used the " eye and ear method " 
is the best. (See Art. 58.) If it is desired to determine the 
latitude from this same star, the observer has only to set 
the horizontal cross hair on the star immediately after making 
the time observation, and the reading of the vertical arc will 
give the star's apparent altitide at culmination. (See Art. 67.) 

The computation of the watch correction consists in finding 
the true time at which the star should transit and comparing 
it with the observed watch time. If a sidereal watch or chro- 
nometer is used the star's right ascension is at once the local 
sidereal time. If mean time is desired, the true sidereal time 
must be converted into local mean solar time, or into Standard 
Time, whichever is desired. 

Example. 

Observed transit of a Hydra 8 h 48"* 58.*. 5, Eastern time, in 
longitude 5^ 20 west; date April 5, 1902. From the almanac, 
the star's R. A. =g h 22 48 S .4, and the sun's R. A. at G. M. N. = 
o h 51"* 24 S .6. To reduce this to the R. A. at local mean noon 
take from Table III the correction for 5^ 20"* which is +52 S .6. 
The corrected R. A. =o h 52 17*. 2. The local sidereal time, 
which is g h 22 48 .4, is then reduced to Standard Time as 
follows : 



R. 


A. Star 


= 9* 


22 m 


48 s . 


4 


R. 


A. Sun 


= 


52 


i? 


2 






8 


30 


3i 


2 




c 


= 


I 


23 


6 



Mean Local Time =8 29 07 .6 

Red. to 75 merid. = 20 oo . o 

Eastern Time =8 49 07.6 

Watch time =8 48 58 . 5 



Watch slow = 9 s . i 



OBSERVATIONS FOR DETERMINING THE TIME 117 

Transit observations for the determination of time can be 
much more accurately made in low than in high latitudes. 
Near the pole the conditions are very unfavorable. 

73. Observations with Astronomical Transit. 

The method just described is in principle the one in most common use for deter- 
mining sidereal time with the large astronomical transit. Since the precision at- 
tainable with the latter instrument is much greater than with the engineer's transit, 
the method must be correspondingly more refined. The number of observations 
on each star is increased by using a large number of vertical threads, commonly 
eleven. These times of transit are recorded by electric signals on the chrono- 
graph (see Art. 56, p. 93), and are scaled from the chronograph sheet to hundredths 
of a second. In this class of work many errors which have been assumed to be 
negligible in the preceding method are important and must be carefully determined 
and allowed for. The instrument has to be set into the plane of the meridian by 
means of repeated trials, and there is always a small remaining error in the azimuth 
of the sight line. This error in azimuth a is measured by comparing the observed 
times of rapidly moving (southern) stars and slowly moving (circumpolar) stars. 
The correction to any observed time for the effect of azimuth error is 

a cos h sec D. [66] 

The inclination of the axis to the horizon b is measured with the spirit level and 
the observed times are reduced to the meridian by adding the correction 

b sin h sec D. [67] 

The error in the sight line c is found by reversing the telescope in its supports 
and comparing observations made in the two positions. The correction to any 
observation is 

c sec D. [68] 

Corrections are also made for the effect of diurnal aberration and sometimes other 
minor corrections. 

74. Selecting Stars for Transit Observations. 

Before the observations are begun the observer should pre- 
pare a list of stars suitable for transit observations. This 
list should include the name or number of the star, its magni- 
tude, the approximate time of culmination, and its meridian 
altitude or its zenith distance. The right ascensions of consec- 
utive stars in the list should differ by sufficient intervals to give 
the observer time to make and record an observation and pre- 
pare for the next one. The stars used for determining time 
should be those which have a rapid diurnal motion, that is, 



n8 



PRACTICAL ASTRONOMY 



stars near the equator; slowly moving stars are not suitable 
for time determinations. Very faint stars should not be selected 
unless the telescope is of high power and good definition; those 
smaller than the fifth magnitude are rather difficult to observe 
with a small transit, especially as it is difficult to reduce the 
amount of light used for illuminating the field of view. The 
selection of stars will also be governed somewhat by a consider- 
ation of the effect of the different instrumental errors. An in- 
spection of Table B, p. 88, will show that for stars near the 
zenith the azimuth error is zero, while the inclination error is 
a maximum; for stars near the horizon the azimuth error is a 
maximum and the inclination error is zero. If the azimuth of 
the instrument is uncertain and the inclination can be accurately 
determined, then stars having high altitudes should be preferred. 
On the other hand, if the level parallel to the axis is not a sensi- 
tive one and is in poor adjustment, and if the sight line can be 
placed accurately in the meridian, which is usually the case 
with a surveyor's transit, then low stars will give the more accu- 
rate results. With the surveyor's transit the choice of stars is 
somewhat limited, however, because it is not practicable to 
sight the telescope at much greater altitudes than about 70 
with the use of the prismatic eyepiece and 55 or 60 without 
this attachment. 

Following is a sample list of stars selected for observations 
in a place whose latitude is 40 N., longitude 77 W., date May 5, 
1910, hours between S h and 9^ Eastern time; the limiting alti- 
tudes chosen are 10 and 65. The right ascension of the mean 
sun for the date is 2 h 50. Adding this to 8 A - o8 m = 7* 52, 
the local mean time, the resulting sidereal time is io h 42, 
which is approximately the right ascension of a point on the 
meridian at S h E. S. T. The limiting right ascensions are there- 
fore io h 42 and n h 42. The co-latitude is 50, which gives, 
for altitudes 10 and 65, the limiting declinations +15 and 
40. In the table of mean places for 1910 the following 
stars are given: 



OBSERVATIONS FOR DETERMINING THE TIME 
MEAN PLACES FOR 1910 



Star. 


Magn. 


Rt. Asc. 


Decl. 


1 Leonis 


c -I 


IcA 44 W 2,2 s 


+ 11 Ol' 


5 J Chameleontis 


A. 7 


TO 4.4 57 


80 04 


46 Leonis Minoris 


2 


TO 48 17 


4- 2.4 42 


Groombridge 1 706 


6 2 


IO C2 47 


+ 78 I? 


a Ursa Majoris 


2 .O 


10 58 ii 


+ 62 14 


rj Octantis 


6.1 


10 so s8 


84 07 


p 3 Leonis 


6.2 


II O2 IQ 


+ 227 


\f/ Urstz Majoris 


T.. 2 


II 04 2. 7 


+ 44 <O 


5 Leonis 


2 . 7 


1 1 OQ I Q 


+ 21 O I 


v Ursce Majoris 


}. 7 


II 1 3 2.7 


+ ?"? ^? 


5 Crateris 


2 .Q 


II 14 !JO 


14 17 


.T Leonis 


C I 


II 21 IO 


+ 7 21 


X Draconis 


4 


II 26 04 


+ 60 sO 


v Hydra ... 


2 8 


II 28 74 


2.1 2^ 


' v Leonis 


4 4 






X Ursa Majoris 


2 .Q 


ii 41 18 


+48 17 


/3 Leonis 


2 2 


TT 4/1 28 


-l-i c os 











From this list there are found seven stars whose declinations 
and right ascensions fall within or very close to the required 
limits. In the following list the times of transit and the alti- 
tudes have been computed roughly but with sufficient accuracy 
to identify the stars. 

OBSERVING LIST FOR TRANSIT OBSERVATIONS 



Star. 


Magn. 


Approx. E.S.T. 


Approx. Alt. 


/ Leonis 


C 2. 


8 h oo m 


61 01' 


fp Leonis 


6 2 


8 18 


s2 27 


5 Crateris 


3 O 


8 2,0 


2x 42. 


T Leonis 


c . i 


8 2,0 


2 2T 


Hydra 


2.8 


8 44 


18 38 


v Leonis 


J..4. 


8 48 


4O 4O 


ff Leonis 


2 2 


O OO 


6c o? 











75- Time by Transit of the Sun. 

The apparent solar time may be directly determined by 
observing the watch times when the west and east limbs of the 



120 PRACTICAL ASTRONOMY 

sun cross the meridian. The mean of the two readings is the 
watch time for the instant of Local Apparent Noon or i2 h M.. 
apparent time. This apparent time is to be converted into 
mean time and then into Standard Time. If only one limb 
of the sun can be observed the time of transit of the centre may 
be found by adding or subtracting the " time of semidiameter 
passing the meridian," which is given in the Nautical Almanac. 

Example. 

Observed transit of sun on Jan. 28, 1910, longitude 4* 44* i8 s W. Time of 
transit of W. limb = n h 54 53*; E. limb = n h 57"* n s ; mean of two limbs = 
ii* 56 m 02*. o. 
L. M. T. = i2 h oo m oo s Equa. of T. at G. A. N. =13"* oo s . 71 

Equa. T. = +13 03 . o o s . 485 X 4 A . 74 = 2 .30 

L. M. T. =12 13 03.0 Cor'd. Equa. T. = 13 03*. 01 

Red. to 75 = 15 42 .o 
E.S.T. =11 57 21.0 
Watch time = 1 1 56 02.0 



Watch slow = i m i9 s . o 

. 76. Time by an Altitude of the Sun. 

The apparent solar time may be determined by measuring 
the altitude of the sun when it is not near the meridian, and 
then solving the PZS triangle for the angle at the pole, which is 
the hour angle of the sun east or west of the meridian. The 
west hour angle of the sun is the local apparent time. The 
observation is made by measuring several altitudes in quick 
succession and noting the corresponding instants of time. The 
mean of the observed altitudes is assumed to correspond to the 
mean of the observed times, that is, the curvature of the path 
of the sun is neglected. The error caused by neglecting the 
correction for curvature is very small provided the sun is not 
near the meridian and the series of observations extends over 
but a few minutes' time, say io m . The measurement of alti- 
tude must of course be made to the upper or the lower limb 
and a correction applied for the semidiameter. The observa- 
tions may be made in two sets, half the altitudes being taken 
on the upper limb and half on the lower limb, in which case no 
semidiameter correction is required. The telescope should be 



OBSERVATIONS FOR DETERMINING THE TIME 121 

reversed between the two sets if the instrument has a complete 
vertical circle. The mean of the altitudes must be corrected 
for index error, refraction, and parallax, and for semidiameter 
if but one limb is observed. The declination must be corrected 
by adding to the declination at G. M. N. the hourly change mul- 
tiplied by the number of hours since G. M. N. It is necessary 
for this purpose to know the approximate Greenwich Mean 
Time. If the watch used is keeping Standard Time the G. M. T. 
is found at once. (Art. 35.) If the watch is not more than 
2 m or ^ m j n error the effect on the computed declination will be 
negligible for observations made with small instruments. If the 
longitude is known the declination may be corrected by first 
computing an approximate value of the local time and adding 
this to the longitude, obtaining the approximate G. M. T. 
With this approximate G. M. T. the declination may be cor- 
rected and the whole computation repeated. It will seldom be 
necessary to make a third computation. In order to compute 
the hour angle the latitude of the place must be known. The 
hour angle of the sun's centre P is then found by means of one 
of the formulae of Art. 19.* When the value of P is found it 
is converted into hours, minutes and seconds, and if the sun is 
east of the meridian it is subtracted from 1 2 h to obtain the local 
(civil) apparent time; if astronomical time is desired it should 
be subtracted from 24^. This apparent time is then converted 
into mean time by adding or subtracting the equation of time. 

* If tables of log versed sines, in addition to the usual tables, are available, 
then the following formula will be found convenient: 

_ cos (L D) sin k 

vers P = , ,. 83 

cos L cos D 

In case P is greater than 90 the formula below may be substituted: 

vers p. = sinA + co S (L + Z)) ) 
cos L cos D 

where P' = 180 - P. 

The sum or difference in the numerator must be computed with natural func- 
tions and the remainder of the computation performed by logarithms. 



122 



PRACTICAL ASTRONOMY 



The equation of time must be corrected for the time elapsed 
since G. M. N. The resulting mean time is to be converted 
into Standard Time, to which the watch is regulated. The 
difference between the computed result and the mean of the 
observed watch readings is the watch correction. 

The most favorable conditions for an accurate determination 
of time by this method are when the sun is on the prime vertical 
and when the observer is at the equator. When the sun is 
east or west it is rising or falling at its most rapid rate and an 
error of i' in the altitude produces less error in the calculated 
hour angle than does i' error when the sun is near the meridian. 
The nearer the observer is to the equator the greater is the in- 
clination of the sun's path to the horizon, and consequently the 
greater its rise or fall per second of time. If the observer were 
at the equator and the declination zero, the sun would rise or 
fall i' in 4 s of time. In the example given below the rise is 
i' in about 8 s of time. When the observer is near the pole 
the method is practically useless. Observations on the sun 
when it is very close to the horizon should be avoided, however, 
even when the sun is near the prime vertical, because the errors 
in the tabulated refraction correction due to variations in the 
temperature and pressure of the air are likely to be large. Ob- 
servations should not be made when the altitude is less than 
about 10 if this can readily be avoided. 



Example. 

Observation of Sun's Altitude for Time, Nov. 28, 1905. Lat. 42 21' N. Long. 
71 04. 5' W. 



Lower limb 
Tel. dir. 

Upper limb 
Tel. rev. 



Altitude 

14 4i' 
15 oo' 



j IS? 55' 



1 6 08' 



Mean = 

Refraction and parallax = 



26'. o 
3-3 



h = 15 22'. 7 



Watch (Eastern Time) 

& h 3Q m 42 s A.M. 

8 42 19 



8 45 
8 47 



34 
34 



Mean = 8 h 43 47*. 2 A.M. 

G. M. T. = i h 43 47. 2 (approx.) 



OBSERVATIONS FOR DETERMINING THE TIME 123 

L 42 21'. o sec o. 13133 Decl. at G. M. N. = 21 14' 54" 

h = 15 22.7 / -26".8iXA 73 = -46 
P = in 15.7 esc 0.03061 

Corrected decl. = 21 15' 40" 

25 = 178 59'. 4 p = 111 15' 40" 

s = 84 29'. 7 cos 8. 98196 
s h = 69 07'. o sin 9. 97049 

Eq. t. = i2 m 04 s . 29 

2)9.11439 .846X1.73 = I-46 



log sin P = 9. 55719 i2 m 02 s . 83 

i P = 21 08' 45" 
P = 42 17' 30" 
= 2 h 49 io s . o 



k, A , T - ' :r 5:i 

j"= j. //,+. -**>) 

Zl, t r * 



M.L.T. =8* 58 47 s - 2 

15 42 .o 



Eastern time = 8 h 43"* 05" . 2 
Watch time = 8 43 47 . 2 

Watch fast 42 s . o 



77. Time by the Altitude of a Star. 

The method of the preceding article may be applied equally 
well to an observation on a star. In this case the parallax and 
semidiameter corrections are zero. If the star is west of the meri- 
dian the computed hour angle is the star's true hour angle; 
if the star is east of the meridian the computed hour angle must 
be subtracted from 24^. The sidereal time is then found by 
adding the right ascension of the star to its hour angle. If 
mean time is desired the sidereal time thus found is to be con- 
verted into mean solar time by Art. 34. Since it is easy to select 
stars in almost any position it is desirable to eliminate errors in 
the measured altitudes by taking two observations, one on a 
star which is nearly due east, the other on one about due west. 
The mean of these two results will be nearly free from instru- 
mental errors, and also from errors in the assumed value of the 
observer's latitude. If a planet is used it will be necessary to 
know the G. M. T. with sufficient accuracy for correcting the 
right ascension and declination. 



124 



PRACTICAL ASTRONOMY 



Example. 
Observed altitude of 

Observed altitude 
Index correction 
Refraction correction = 



h = 



L = 42 iS'.o 

h = 44 53-o 
p = 66 41 . 4 



Jupiter (east), Jan. 9, 1907. Lat. = 42 18'. o; Long. = 

Eastern time = 7* 32 02 s 
44 55' 

- i Decl. at G. M. N. = + 23 18' 22". o 

i Hourly change = + i". oo 

G. M. T. = i2 h 32 02 s (approx.) 

44 53'- o* 12^.53 X i ".oo = +12". 5 

Corrected decl. = + 23 18' 34". 5 

p = 66 41' 25". 5 



s 
s 

* 17 s - 3 
- i 8 - 395 
-17 s . 5 

1 59 s - 8 


L 

h 

P 

s 


= 34 

= 32 

= 10 

= 76 


38'. 2 
03 . 2 
14.8 
56.2 


CSC 

sin 

sec 
cos 


o. 24537 
9. 72486 
o. 00698 

9-35416 






2)9^33137 




log 


tan 
* 

R. 


P = 
P = 
P = 

A. = 


9. 66568 
24 50' 

49 4* 

2o* 41 W 

6* i8 m 


57" 
54'; 

12 s . 

59 s - 


4 
8 



2 S = 152 112 . 4 
S 76 56 . 2 

R. A. at G. M. N. = 6* 19' 
Hourly change 

1 2 h . 53 X - i. 395 = 

Corrected R. A. = 6* 18' 



Sid. Time = 27* oo" 1 12 s . 2 

The local sidereal time is therefore 3^ oo w 1 2 s . 2 when the watch reading is 7 * 3 2 TO 02 s . 
The error of the watch may be found by reducing the sidereal time to Eastern Time. 
78. Time by Transit of Star over Vertical Circle through Polaris.| 
In making observations by this method the line of sight of the telescope is set 
in the vertical plane through Polaris at any (observed) instant of time, and the 
time of transit of some southern star across this plane is observed immediately 
afterward; the correction for reducing the star's right ascension to the true sidereal 
time of the observation is then computed and added to the right ascension. The 
advantages of the method are that the direction of the meridian does not have to 
be established before time observations can be begun, and that the interval which 
must elapse between the two observed times is so small that errors due to the 
instability of the instrument are reduced to a minimum. 

The method of making the observation is as follows: Set up the instrument and 
level carefully; sight the vertical cross hair on Polaris (and clamp) and note and 
record the watch reading; then revolve the telescope about the horizontal axis, 

* Parallax is negligible for this planet, as it is only about 2". 

f This method is given by Mr. George O. James in the Jour. Assoc. Eng. Soc. 
Vol. XXXVII, No. 2. In a later paper (Popular Astronomy No. 172) Mr. 
James gives the formula 

P = p sec L sin (L - D) sec (D - c) sin (P P), 

in which c is the correction from Table IV in the Nautical Almanac. This for- 
mula is preferable to that given in the text, provided the latitude is known, since 
it is not necessary to make a second approximation. A discussion of the method 
used with large instruments is given by Professor Frederick H. Sears in Bul- 
letin No. 5, Laws Observatory, University of Missouri. 



OBSERVATIONS FOR DETERMINING THE TIME 



I2 5 



being careful not to disturb its azimuth; set off on the vertical arc the altitude 
of some southern star (called the time-star) which will transit about 4 or $ m 
later; note the instant when this star passes the vertical cross hair. It will be of 
assistance in making the calculations if the altitude of each star is measured 
immediately after the time has been observed. The altitude of the time-star 
at the instant of observation will be so nearly equal to its meridian altitude that 
no special computation is necessary beyond what is required for ordinary transit 
observations. If the times of meridian transit are calculated beforehand the 
actual times of transit may be estimated with sufficient accuracy by noting the 
position of Polaris with respect to the meridian. If Polaris is near its elongation 
then the azimuth of the sight 
line will be a maximum. In 
latitude 40 the azimuth of 
Polaris for 1910 is about i 32'; 
a star on the equator would 
then pass the vertical cross hair 
nearly 4 later than the com- 
puted time if Polaris is at 
eastern elongation (see Table 
B). If Polaris is near western 
elongation the star will transit 
earlier by this amount. In 
order to eliminate errors in the 
adjustment of the instrument, 
observations should be made 
in the erect and inverted posi- 
tions of the telescope and the 
two results combined. A new 
setting should be made on 
Polaris just before each obser- 
vation on a time-star. 

In order to deduce an expression for the difference in time between the meridian 
transit and the observed transit let R and R be the right ascensions of the stars, 
5 and So the sidereal times of transit over the cross hair, P and P the hour angles 
of the stars, the subscripts referring to Polaris. Then by Equa. [37], 

P =5 - R 

and Po = So- RO] 

subtracting, P - P = (R - ) - (5 - So). [85] 

The quantity 5 S is the observed interval of time between the two observa- 
tions expressed in sidereal units. If an ordinary watch is used the interval must 
be reduced to sidereal units (Table III). Equa. [85] may then be written 

Po - P = (R - Ro) - (T - To) - C, [86] 

where T and T are the actual watch readings and C is the correction to reduce 
this interval to sidereal time^- 

In Fig. 55 let P be the position of Polaris; P, the celestial pole; Z, the zenith; 




126 



PRACTICAL ASTRONOMY 



and S, the time-star. Also let Z and Zo represent the azimuths of the two stars; 
p and PO, their polar distances; z and z , their zenith distances; and h and ha, 
their altitudes. In the triangle PPoZ, 



and in the triangle PZS 



sin P _ sin Zo 
sin zo sin /><>' 



sin P sin Z 



Since the azimuth of the sight line has not changed, 

smnl = 1 80 + sm 

or Z = 180 + Z 

and sin Z = sin Z . 

From Equa. [87] and [88], solving for sin P, there results 
sin P = sin p sin P sec Z) cos h sec 



[89] 



in which D is the declination of the time-star 5. Since the angles P and p Q are 
small they may be substituted for their sines, giving 

P = po sin PO sec D cos h sec &o. [90] 

In this equation the value of PO is unknown, and unless the local sidereal time is 
already known with accuracy it is necessary to determine PO by a series of approxi- 
mations. A rough value of PO ( = P'o) may be found from the equation 

P'o = (R- *) - (T - r ) - C. [91] 

Using this value of P' in Equa. [90] the result is P', an approximate value of P. 
A corrected value of PO is then obtained by the equation 

Po = P'oV P'- [92] 

With this new value of PO a new value of P may be computed. If the second 
value of P differs much (say 5') from the first value it will be necessary to make 
another computation of PO. It is usually possible to make a rough estimate of 
PO from the known value of the local time. The watch time of the observation 
on Polaris may be converted into local sidereal time and the hour angle P found 
by Equa. [47] and [37]. When a series of observations is made the hour angle 
of Polaris at all observations after the first may be closely estimated by adding 
to the value of PO at the first observation the time elapsed since the first obser- 
vation on Polaris. If the altitudes of the stars have not been measured it is 
usually accurate enough to take h = 90 L D (the meridian zenith distance) 
for the time-star, and for Polaris ho = L + po cos P , or better still h = L c, 
where c is the quantity given in Table IV at the end of the Nautical Almanac. 

The final value of P, the hour angle of the time-star at the instant it was observed, 
is the correction to be added to the right ascension of the time-star to obtain the 
local sidereal time of the observation on this star. This sidereal time may then be 
reduced to mean time or to standard time and the watch correction obtained. 



OBSERVATIONS FOR DETERMINING THE TIME 127 

The above method is applicable to transit observations made with a small 
instrument. For the large astronomical transit a more refined method of making 
the reductions must be used. 

Observation of o Virginis over Vertical Circle through Polaris; Lat.,42 21' N., 
Long., 4^ 44"* 18*. 3 W.; Date, May 8, 1906. 

Observed time on Polaris = 8 h 35"* 58 s 

Observed transit of o Virginis = 8 39 43 

Diff. = 3 45* 
R = i2 h oo m 26 s . 3 

Ro= i 24 35 .4 L = 42 21' 

D= + 9 15 



R R = io h 35 50*. 9 
T - T = 3 45 .o L-D= 33 06' 

C = .6 

#0 = ?i'. 85 

P'o = IOJ 3 2 OSM - log #o=I. 8564 

, Z 5 8 OI r -3 log sec D = 0.0057- 

P -19 -8 log sin (L- D) = 9. 7373 

D , , D , o , log sec (L - c) = o. 1238 

P'o+P = i57 41-5 log sin P' = 9.5732 

log P' = i. 2964 w 
P'= -19'. 79 

The log sin of (P' + P') = 9-5793; substituting this for log sin P' , the log P' 
is increased 61 units in the fourth place, giving 20'. 07 for P. Converting this 
into time it is 8o s . 3 or i m 20". 3, the desired correction. The true sidereal 
time may now be found by subtracting i m 2o s . 3 from the right ascension of 
o Virginis. The complete computation of the watch correction is as follows: 

R = T 2 h oo m 26 s . 3 
P = i 20 . 3 



5 = n h 59 TO o6 s .o 

R s = 3 p2 23 .6 

&h 5 6" 428.4 

C = i 27 .9 



M. L. T.= 8 h 55 14*. 5 
15 4i -7 



Eastern time = 8^ 39 32*. 8 
Watch time =8 39 43 

Watch fast = io s . 2 

79. Time by Equal Altitudes of a Star. 

If the altitude of a star is observed when it is east of the meridian at a certain 
altitude, and the same altitude of the same star again observed when the star is 
west of the meridian, then the mean of the two observed times is the watch reading 



128 



PRACTICAL ASTRONOMY 



for the instant of transit of the star. It is not necessary to know the actual 
value of the altitude employed, but it is essential that the two altitudes should be 
equal. The disadvantage of the method is that the interval between the two 
observations is inconveniently long. 

80. Time by Two Stars at Equal Altitudes. 

In this method the sidereal time is determined by observing when two stars 
have equal altitudes, one star being east of the meridian and the other west. If 
the two stars have the same declination then the mean of the two right ascensions 
is the sidereal time at the instant the two stars have the same altitude. As it is 
not practicable to find pairs of stars having exactly the same declination it is neces- 
sary to choose pairs whose declinations differ as little as possible and to introduce 
a correction for the effect of this difference upon the sidereal time. It is not 
possible to observe both stars directly with a transit at the instant when their 
altitudes are equal; it is necessary, therefore, to first observe one star at a certain 
altitude and to note the time, and then to observe the other star at the same alti- 
tude and again note the time. The advantage of this method is that the actual 
value of the altitude is not used in the computations; any errors in the altitude 
due either to lack of adjustment of the transit or to abnormal refraction are there- 
fore eliminated from the result, provided the two altitudes are made equal. In 
preparing to make the observations it is well to compute beforehand the approx- 
imate time of equal altitudes and to observe the first star two or three minutes 
before the computed time. In this way the interval between the observations 
may be kept conveniently small. It is immaterial whether the east star is observed 
first or the west star first, provided the proper change is made in the computation. 
If one star is faint it is well to observe the bright one first; the faint star may then 
be more easily found by knowing the time at which it should pass the horizontal 
cross hair. The interval by which the second observation follows the time of 
equal altitudes is nearly the same as the interval between the first observation 
and the time of equal altitudes. It is evident that in the application of this 
method the observer must be able to identify the stars he is to observe. A star 
map is of great assistance in making these observations. 

The observation is made by setting the horizontal cross hair a little above the 
easterly star 2 or 3"* before the time of equal altitudes, and noting the instant 
when the star passes the horizontal cross hair. Before the star crosses the hair 
the clamp to the horizontal axis should be set firmly, and the plate bubble which 
is perpendicular to the horizontal axis should be centred. When the first obser- 
vation has been made and recorded the telescope is then turned toward the westerly 
star, care being taken not to alter the inclination of the telescope, and the time 
when the star passes the horizontal cross hair is observed and recorded. It is 
well to note the altitude, but this is not ordinarily used in making the reduction. 
If the time of equal altitudes is not known, then both stars should be bright ones 
that are easily found in the telescope. The observer may measure an approxi- 
mate altitude of first one and then the other, until they are at so nearly the same 
altitude that both can be brought into the field without changing the inclination 
of the telescope. The altitude of the east star may then be observed at once and 



OBSERVATIONS FOR DETERMINING THE TIME 



129 



the observation on the west star will follow by only a few minutes. If it is desired 
to observe the west star first, it must be observed at an altitude which is greater 
than when the west star is observed first. In this case the cross hair is set a little 
below the star. 

In Fig. 56 let nesw represent the horizon, Z the zenith, P the pole, S e the easterly 
star, and S w the westerly star. 
Let P e and P w be the hour 
angle of S e and S w , and let 
HS e Sw be an almucantar, or 
circle of equal altitudes. 

From Equa. [37], for the 
two stars S e and S w , the 
sidereal time is 

5 = R w + Pw 
S = R e - Pe* 

Taking the mean value of 5, 
Rw~T~Re . Pw Pe r i 

s= - +-., [93] 

from which it is seen that 
the true sidereal time equals 
the mean right ascension 
corrected by half the differ- 
ence in the hour angles. To 
derive the equation for cor- 
recting the mean right ascension so as to obtain the true sidereal time let the 
fundamental equation 

sin h = sin D sin L + cos D cos L cos P [8] 

be differentiated regarding D and P as the only variables, then there results 

o = sin L cos D cos D cos L sin P -=- cos L cos P sin D, 

dD 

from which may be obtained 




FIG. 56 



dP_ 
dD 



tan L tan D 



[94] 



sin P tan P 

If the difference in the declination is small, dD may be replaced by (D w D e ), 
in which case dP will be the resulting change in the hour angle, or 5 (P w P e ). 
The equation for the sidereal time then becomes 

R w + Re . Dw - D e Ran L _ tan D~\ 



_ _ 



, 



_ 
sinP tanP ' 



[95] 



in which (D w D e ) must be expressed in seconds of time. D may be taken 
as the mean of D e and D w . The value of P would be the mean of P e and P w if 



P e is here taken as the actual value of the hour angle east of the meridian. 



130 



PRACTICAL ASTRONOMY 



the two stars were observed at the same instant, but since there is an appreciable 
interval between the two times P must be found by 

n ~n T* *T* 

p = R^_R + T^T. [9fi] 

If the west star is observed first, then the last term becomes a negative quantity. 
Strictly speaking this last term should be converted into sidereal units, but the 
effect upon the result is usually very small. In regard to the sign of the correction 
to the mean right ascension it should be observed that if the west star has the 
greater declination the time of equal altitudes is later than that indicated by the 
mean right ascension. In selecting stars for the observation the members of a 
pair should differ in right ascension by 6 to 8 hours, or more, according to the 
declinations. Stars above, the equator should have a longer interval between 
them than those below the equator. On account of the approximations made in 
deriving the formula the declinations should differ as little as possible. If the 
declinations do not differ by more than about 5, however, the result will usually 
be close enough for observations made with the engineer's transit. From the 
extensive star list now given in the Nautical Almanac it is not difficult to select 
a sufficient number of pairs at any time for making an accurate determination 
of the local time. Following is a short list taken from the American Ephemeris 
and arranged for making an observation on April 30, 1912. 

LIST FOR OBSERVING BY EQUAL ALTITUDES 
Lat., 42 21' N. Long., 4* 44 18* W. Date, Apr. 30, 1912. 



Stars. 


Magn. 


Sidereal time 
of equal alti- 
tudes. 


Eastern time 
of equal alti- 
tudes. 


Observed 
times. 


a Corona Borealis 


2 . T. 








ft Tauri . 


1.8 


10" 28 


7 h 3 gm 




a Bootis 


O.2 








f Geminorutn 


4 


io 37 


7 47 




a Bootis 


O.2 








8 Geminorum 


7. C 


10 48 


7 5 8 




p Bootis 


3.6 








of Geminorum 


i .9 


II OO 


8 10 




K Hydrce 


2. C 








o Areus. 


2 .O 


II IO 


8 20 




ft Herculis . . ... 


2.8 








rj Geminorum. ... ... 


7. e 


ii 19 


8 29 




a Serpentis 


2 . 7 








a Cnnis Minoris 


o. s 


ii 35 


8 45 




ft Herculis 


2.8 








S Geminorum 


7,. C 


ii 5i 


9 01 




a Serpentis 


2.7 








ft Cancri 


3.8 


12 02 


9 12 




a Serpenlis 


2.7 








e HydrcB 


7. e 


12 II 


9 21 




ft Libras 


2 .0 








a HydrcE 


2 . I 


12 2O 


9 3 




^3 Herculis 


2.8 








v Cancri 


4-9 


12 32 


9 42 















OBSERVATIONS FOR DETERMINING THE TIME 131 

Following is an example of an observation for time by the method of equal alti- 
tudes. 

Example. 

Lat, 42 21' N. Long., 4* 44 i8 s W. Date, Apr. 14, 1905. 

Star. Rt. Asc. Decl. Watch. 

a Ceti (E) 2* S7" 22 s . i + 3 43' 69". i 5* i8 oo 

8 Aquila (W) 19 20 43-6 +2 55 44 .o 5 22 13 



Mean 23* 09 02 s . 8 +3 19' 56". 6 $ h 20 06*. 5 

Diff. 7 36 38 . 5 2)-o 48' 25".! 04 13 

4 *3 7 D w -D f , 

- - e = 24 12 .6 

2) 7* 4Q 52*. i 2 _ _ Q6 . 8 . 

P= 3 h 50^26^ 9& ' 84 

= 57 36' 31". 5 

Mean R. A. = 23^ og m 02*. 8 

Corr. = 01 41.0 

Sid. Time = 23^ o?" 1 21*. 8 r> n 

T> i ^-'u> J^e o^/\ o/-/\ 

^s = 17 30 43 . 2 log - = 1.9861 (n) 1.9861 (n) 

rfc ?6 W i8 s 6 lo S tan L = 9- 9598 log tan Z> = 8. 7650 
' 



55 2 csc P ~ - 735 log cot P ~ 9- 



MLT = c* 4V 4. =2. 0194 (n) 0.5535 (n) 

15 42 o -f 6 6 -3.6 

Eastern time = 5^ 2o m oi s . 4 
Watch time =5 20 06 . 5 Corr - 

Watch fast = 5 s . i 

8 1 . Formula [94] may be made practically exact by means of the following device. 
Applying Equa. [8] to each star separately and subtracting one result from the 
other we obtain the equation* 

tan L tan AZ? tan D tan AZ? . tan D tan AZ? 

sin AP = r~ s vers AP, k)?! 

sin P tan P tan P 

where AZ? is half the difference in the declinations and AP is the correction to 
the mean right ascension. If sin AP and tan AZ? are replaced by their arcs 
and the third term dropped, this reduces to Equa. [94], except that AZ? and AP 
are finite differences instead of infinitesimals. In order to compensate for the 
errors thus produced let AZ? be increased by a quantity equal to the difference 
between the arc and the tangent (Table C); and let a correction be added to the 
sum of the first two terms to allow for the difference between the arc and sine of 
AP (Table C). With the approximate value of AP thus obtained the third 

* Chauvenet, Spherical and Practical Astronomy, Vql. I, p. 199. 



132 



PRACTICAL ASTRONOMY 



term of the series may be taken from Table D. By this means the precision of 
the computed result may be increased, and the limits of AZ) may therefore be 
extended without increasing the errors arising from the approximations. 



TABLE C. CORRECTIONS TO BE ADDED TO AZ> AND AP 
(Equa. [99], Art. 81) 



Arc or sine. 


Correction to 
A>. 


Correction to 
AP. 


Arc or sine. 


Correction to 
AD. 


Correction to 
AP. 


1 
IOO 


8 

o.oo 



o.oo 


s 

800 


s 
0.90 


8 

0-45 


2OO 


O.OI 


O.OI 


850 


1. 08 


0-54 


3 00 


0.05 


O.O2 


900 


1.29 


0.64 


40O 


0. II 


O.O6 


95 


s 


0.76 


500 


0.22 


O.II 


1000 


1.77 


0.88 


OOO 


0.38 


0.19 


1050 


2.05 


I .02 


650 


0.48 


0.24 


IIOO 


2 -35 


I.I7 


7 00 


O.OO 


0.30 


1150 


2.69 


1-34 


75 


0.74 


-37 


I2OO 


3.06 


i-S 2 



TABLE D. CORRECTION TO BE ADDED TO AP * 
(Equa. [99], Art. 81) 





AP (in seconds of time). 


2d 

term. 


IOO S 


200 8 


3oo 8 


400 


5oo 8 


6oo s 


700* 


8oo 


900* 


IOOO S 


9 


8 


S 


s 


S 


8 


8 


s 


a 


8 


s 


IOO 


O.OO 


O.OI 


O.O2 


O.O4 


0.07 


O. IO 


0.13 


0.17 


O.2I 


0.26 


2OO 


O.OI 


O.O2 


0.05 


0.08 


0.13 


o. 19 


0.26 


0-34 


0-43 


o-S3 


300 


O.OI 


0.03 


O.O7 


0.13 


O.2O 


0.29 


o-39 


0.51 


0.64 


0.79 


4OO 


O.OI 


0.04 


0. 10 


0.17 


0.26 


0.38 


0.52 


0.68 


0.86 


i. 06 


500 


O.OI 


O.OS 


0.12 


0.21 


o-33 


0.48 


0.65 


0.85 


1.07 


1.32 


6OO 


0.02 


0.06 


0.14 


0.25 


0.40 


o-57 


0.78 


i .02 


!.28 


i-59 


7OO 


O.O2 


O.O7 


O.I 7 


0.30 


0.46 


0.67 


0.91 


1.18 


1.50 


1.85 


800 


O.O2 


0.08 


0.19 


0-34 


0-53 


0.76 


i .04 


i-35 


I.7I 


2. II 


90O 


-O.O2 


O.IO 


O.2I 


0.38 


o-S9 


0.86 


1.17 


1.52 


i-93 


2.38 


I OOO 


0.03 


O.II 


O.24 


O.42 


0.66 


o-9S 


1.30 


1.69 


2.14 


2.64 


IIOO 


0.03 


0.12 


0.2O 


0.47 


o-73 


1.05 


1.42 


1.86 


2.36 


2.91 


I2OO 


0.03 


0.13 


O.29 


0.51 


0.79 


1.14 


i-55 


2.03 


2-57 


3-17 



* The algebraic sign of this term is always opposite to that of the second term. 



OBSERVATIONS FOR DETERMINING THE TIME 133 

Example. 

Compute the time of equal altitudes of a Bootis and i Geminorum on Jan. i, 
1912, in latitude 42 21'. R. A. a Bootis = 14^ n m 37*. 98; decl. = + 19 
38' 15". 2. R. A. i Geminorum = ? h 20 i6 s . 85; decl. = + 27 58' 30". 8. 

I4 h Iim 37 s. 9 8 27 58' 30". 8 

7 20 16 .85 19 38 15 . 2 



2) 6 h 51 2i s . 13 2)8 20' 15". 6 

3^ 25"* 40*'. 56 AZ> = 4 10' 07". 8 

P = 51 25' 08". 4 = 1000 s . 52 

Corr., Table C = i . 77 

AZ? =1002*. 29 

log AZ> = 3.000993 log AD = 3.00099 

log tan L =9. 959769 log tan D = 9. 64462 

log esc P =o. 106945 log cot P =9. 901. 87 

3.067707 2.54748 

1168'. 71 352-76 

352 . 76 



8 1 5 s - 95 

Corr., Table C = + .48 
Corr., Table D = + . 63 



AP = + 8i7.o6 

= + 13"* 37 s - 06 
Mean R. A. = 10 45 57 . 42 

Sid. Time of Equal Alt. = io h 59 34*. 48 

For refined observations the inclination of the vertical axis should be measured 
with a spirit level and a correction applied to the observed time. With the engi- 
neer's transit the only practicable way of doing this is by means of the plate-level 
which is parallel to the plane of motion of the telescope. If both ends of this 
level are read at each observation, denoting the reading of the object end and E 
the eye end of the bubble, then the change in the inclination is expressed by 



where d is the angular value of one scale division in seconds of arc. The correction 
to the mean watch reading is 

_ _ i _ _ _ i _ 
30 sin 5 cos D 30 cos L sin Z 

in which 5 may be taken from the Azimuth* tables or Z may be found from the 
measured horizontal angle between the stars. If the west star is observed at a 
higher altitude than the east star (bubble nearer objective), the correction must 
be added to the mean watch reading. If it is applied to the mean of the right 
ascensions the algebraic sign must be reversed. 

* See Arts. 82 and 109 for the method of using these tables. 



134 



PRACTICAL ASTRONOMY 



82. The correction to the mean right ascension of the two stars may be con- 
veniently found by the following method, provided the calculation of the paral- 
lactic angle, 5 in the PZS triangle, can be avoided by the use of tables. Publica- 
tion No. 1 20 of the U. S. Hydrographic Office gives value of the azimuth angle 
for every whole degree of latitude and declination and for every io m of hour angle. 
The parallactic angle may be obtained from these tables (by interpolation) by 
interchanging the latitude and the declination, that is, by looking up the declin- 
ation at the head of the page and the latitude in the line marked " Declination." 
For latitudes under 23 it will be necessary to use Publication No. 71. 

In taking out the angle the table should be entered with the next less whole 
degree of latitude and of declination and the next less io m of hour angle, and the 
corresponding tabular angle written down; the proportional parts for minutes 
of latitude, of declination, and of hour angle are then taken out and added alge- 
braically to the first angle. The result may be made more accurate by working 




FIG. 57 

from the nearest tabular numbers instead of the next less. The instructions given 
in Pub. 1 20 for taking out the angle when the latitude and declination are of 
opposite sign should be modified as follows. Enter the table with the supplement 
of the hour angle, the latitude and declination being interchanged as before, and 
the tabular angle is the value of 5 sought. 

Suppose that two stars have equal declinations and that at a certain instant 
their altitudes are equal, A being east of the meridian and B west of the meridian. 
If the declination of B is increased so that the star occupies the position C, then 
the star must increase its hour angle by a certain amount x in order to be again 
on the almucantar through B. Half of the angle x is the desired correction. 
In Fig. 57 BC is the increase in declination; BD is the almucantar through 
A, B and D; and CD is the arc of the parallel of declination through which the 
star must move in order to reach BD. The arcs BD and CD are not arcs of 
grfet circles, and the triangle BCD is not strictly a spherical triangle, but it may 



OBSERVATIONS FOR DETERMINING THE TIME 135 

be shown that the error is usually negligible in observations made with the engi- 
neer's transit if BCD is computed as a spherical triangle or even as a plane triangle. 
The angle ZBP is the angle S and DEC is 90 - 5. The length of the arc CD 
is then BC cot S, or (D w D e ) cot S. The angle at P is the same as the arc 
CD' and equals CD sec D. If (D w D e ) is expressed in minutes of arc and the 
correction is to be in seconds of time, then, remembering that the correction is 
half the angle x, 

Correction = 2 (D w D e ) cot S sec D. [98] 

D should be taken as the mean of the two declinations, and the hour angle, used 
in finding S, is half the difference in right ascension corrected for half the watch 
interval 

The trigonometric formula for determining the correction for equal altitudes is 

tan = sin cot J (Si + S 2 ) sec j (D l + D 2 ). [99] 

2 2 

By substituting arcs for the sine and tangent this reduces to the equation given 
above, except that the mean of Si and S 2 is not exactly the same as the value of S 
obtained by using the mean of the hour angles. 

The example on p. 131 worked by this method is as follows. From the azimuth 
tables, using a declination of 42, latitude 3, and hour angle 3* SO TO , the approxi- 
mate value of S is 44 05'. Then from the tabular differences, 

Correction for 21' decl. = 22' 
Correction for 20' lat. = + 07 
Correction for 26* h. a. = + 02 
The corrected value of S is therefore 43 52' 

2 (D w - D e ) = - 96'. 84 log = i. 9861 (n) 
log cot S = o. 0172 
log sec D = o. 0007 



log corr. = 2. 0040 (n) 
log corr. = ioo s . 9 

This solution is sufficiently accurate for observations made with the engineer's 
transit, provided the difference in the declinations of the two stars is not greater 
than about 5 and the other conditions are favorable. For larger instruments 
and for refined work this formula is not sufficiently exact. 

The equal-altitude method, like all of the preceding methods, gives more precise 
results in low than in high latitudes. 

83. Rating a Watch by Transit of a Star over a Range. 

If the time of transit of a fixed* star across some well-defined 
range can be observed, the rate of a watch may be quite accu- 
rately determined without knowing its actual error. The 
disappearance of the star behind a building or other object 

* 
* A planet should not be used for this observation. 



136 



PRACTICAL ASTRONOMY 



when the eye is placed at some definite point will serve the pur- 
pose. The star will pass the range at the same instant of sidereal 
time every day. If the watch keeps sidereal time, then its 
reading should be the same each day at the time of the star's 
transit over the range. If the watch keeps mean time it will 
lose 3 m 55 s .Qi per sidereal day, so that the readings on successive 
days will be less by this amount. If, then, the passage of the 
star be observed on a certain night, the time of transit on any 
subsequent night is computed by multiplying 3 55 s -9i by the 
number of days intervening and subtracting this correction 
from the observed time. The difference between the observed 
and computed times divided by the number of days is the daily 
gain or loss. After a few weeks the star will cross the range in 
daylight, and it will be necessary before this occurs to transfer 
to another star which transits later in the same evening. In 
this way the observations may be carried on indefinitely. 

84. Time Service. 

The Standard Time used for general purposes in this country 
is determined by observations at Washington and is sent out 
to all parts of the country east of the Rocky Mountains by 
means of electric signals transmitted over the lines of the tele- 
graph companies. For the territory west of the Rocky Moun- 
tains the time is determined at the Mare Island Navy Yard 
and distributed by telegraphic signals. The error of the sidereal 
clock of the observatory is determined at frequent intervals 
by observing star transits. The sidereal clock is then compared 
with a mean-time clock, by means of a chronograph, and the 
error of this clock on mean time is computed. The mean-time 
clock is then compared with another mean-time clock especially 
designed for sending the automatic signals. When the error 
of this sending clock is found it is " set " (to Eastern Standard 
Time) by accelerating or retarding the motion of the pendulum 
until the error is reduced to a negligible quantity. The series 
of signals sent out each day begins at 11^55 ""A.M., Eastern 
time, and continues for five minutes. The clock mechanism 



OBSERVATIONS FOR DETERMINING THE TIME 137 

is arranged to break the circuit at the end of each second; 
this makes a click on every telegraph sounder on the line, or 
a notch on the sheet of a chronograph placed in the circuit. 
The end of each minute is shown by the omission of the 55th 
to 59th seconds inclusive, except for the noon signal, which 
is preceded by a ten-second interval. During this ten-second 
interval the local circuits controlling the time-balls,* which 
are dropped by this same signal, are thrown into the main 
circuit. The signals sent out in this way are seldom in error 
by an amount greater than one or two tenths of a second. The 
break in the circuit which occurs at the instant of noon, Eastern 
time, drops all the time-balls, corrects the clocks placed in the 
circuit, and gives a click on every telegraph sounder on the line. 
In many seaports the wireless telegraph lines are also thrown 
into the circuit and the signal thus made available at sea. 

Questions and Problems 

1. Compute the approximate Eastern time of transit of Regulus over the me- 
ridian 71 04/.5 West of Greenwich on March 21, 1908. The R. A. of Regulus 
is io h 03 m 29'.!; R s at G. M. N. = 23^ 54 23*.99. 

2. Compute the error of the watch from the data given in prob. 6, p. 169. 

3. Observed time of transit of 5 Capricorni over the vertical circle through 
Polaris, Oct. 26, 1906. Latitude = 42 iS'.s; longitude = 4 h 45 07*. Ob- 
served watch time of transit of Polaris = 7 h io m 20 s ; of d Capricorni = 7^ 13 28 s , 
Eastern Time. Declination of Polaris = + 88 48' 31 ".3; right ascension = 
i h 26 37 s -9- Declination of d Capricorni = 16 3.3' 02". 8; right ascension = 
2I h ^w 538.2. The right ascension of the Mean Sun at Local Mean 
Noon = i4 h i6 m 34 S .6. Compute the error of the watch on Eastern Time. 

4. Time observation on May 3, 1907, in latitude 42 2i'.o, longitude 4^ 44 
i8 s .o. Observed transit of Polaris = *] h i6 m 17 '.o; of M Hydra = -j h i& m 5o s -5. 
Decl. of Polaris = + 88 48' 28"-3; R. A. = i h 24 so s .2. Decl. of M Hydra = 

- i62i's3".2;R. A.= 10^21 36*.!. R. A. of Mean Sun at G. M. N. = 2^40 
563.63. Find the error of the watch. 

5. Observation for time by equal altitudes, Dec. 18, 1904. 

R. A. Decl. Watch. 

a Tauri (E) 4 h 3 m 2 9 s -i + l6 18' S9"-9 j h 34 56* 

a Pegasi (W) 22 59 61 .12 + 14 41 43 .7 7 39 45 

Lat. = 42 28'.o; long. = 4^44 15* .o. R. A. Mean Sun at G. M. N. = 17" 
46 40 S .38. 

* Time-balls are now in use in the principal ports on the Atlantic, Pacific, 
and Gulf coasts and on the Great Lakes. 



138 PRACTICAL ASTRONOMY 

6. Time by equal altitudes, Oct. 13, 1906. 

R. A. Decl. Watch 

vOphiuchi (W) ij h 53 52*.!$ -9 45' 34".6 7* 13"* 49* 

t Ceti (E) o 14 40 .99 9 20 25 .7 7 28 25 

Lat. = 42 18'; long. = 4^ 45 o6*.8. R. A. of Mean Sun at G. M. N.= 
13* 24 323.56. 

7. Show by differentiating Equa. [8] that the most favorable position of the 
sun for a time observation is on the prime vertical. The differential coefficients 

-jT- and -r= should be a minimum to give the greatest accuracy. The expres- 
sions obtained may be simplified by means of Equa. [12]. 

8. Compute the watch correction from the observation given on p. 124. The 
R. A. of the mean sun on Jan. 9, 1907, was 19^ n m 29 8 .49, 



CHAPTER XII 
OBSERVATIONS FOR LONGITUDE 

85. Method of Measuring Longitude. 

The measurement of the difference in longitude of two places 
depends upon a comparison of the local times of the places at 
the same absolute instant of time. One important method 
is that in which the timepiece is carried from one station to 
the other and its error on local time determined in each place. 
The most precise method, however, and the one chiefly used 
in geodetic work, is the telegraphic method, in which the local 
times are compared by means of electric signals sent through a 
telegraph line. Other methods, most of them of inferior accu- 
racy, are those which depend upon a determination of the moon's 
position (moon culminations, eclipses, occultations) and upon 
eclipses of Jupiter's satellites, and those in which terrestrial 
signals are employed. 

86. Longitude by Transportation of Timepiece. 

In this method the error of the watch or chronometer with 
reference to the first meridian is found by observing the local 
time at the first station. The rate of the timepiece should be 
determined by making another observation at the same place 
at a later date. The timepiece is then carried to the second 
station and its error determined with reference to this meridian. 
If the watch runs perfectly the two watch corrections will 
differ by just the difference in longitude. Assume that the first 
observation is made at the easterly station and the second at 
the westerly station. To correct for rate, let r be the daily 
rate in seconds, + when losing when gaining, c the watch 
correction at the east station, c' the watch correction at the 
west station, d the number of days between the observations, 

139 



140 



PRACTICAL ASTRONOMY 



and T the watch reading at the second observation. Then the 
difference in the longitude is found as follows : 

Local time at W. station = T + c' 

Local time at E. station = T + c + dr 

Diflf. in time = Diff. in Long. = c-\- dr c' . [i] 

The same result will be obtained if the stations are occupied 
in the reverse order. 

If the error of a mean-time chronometer or watch is found 
by star observations, it is necessary to know the longitudes 
accurately enough to correct the sun's right ascension. If a 
sidereal chronometer is used and its error found on L. S. T. this 
correction is rendered unnecessary. 

In order to obtain a check on the rate of the timepiece the 
observer should, if possible, return to the first station and again 
determine the local time. If the rate is uniform the error in 
its determination will be eliminated by taking the mean of the 
results. This method is not as accurate as the telegraphic 
method, but if several chronometers are used and several round 
trips between stations are made it will give good results. It is 
useful at sea and in exploration surveys. 

Example. 

Observations for local mean time at meridian A indicate 
that the watch is 15 40* slow. At a point B, west of A, the 
watch is found to be 14 io s slow on local mean time. The 
watch is known to be gaining 8 s per day. The second obser- 
vation is made 48 hours after the first. The difference in longi- 
tude is therefore 

+ I5 m 40 8 - 2 X 8 s - i4 m io s = i* i 4 s . 
The meridian B is therefore i m 14" or 18' 30" west of meridian A. 

87. Longitude by the Electric Telegraph. 

In the telegraphic method the local sidereal time is accurately determined by 
star transits observed at each of the stations. The observations are made 
with large portable transits and are recorded on chronographs which are connected 



OBSERVATIONS FOR LONGITUDE 141 

with break-circuit chronometers. The stars observed are chosen in such a manner 
as to determine the errors of the instruments so that these may be eliminated 
from the results as completely as possible. Some of the stars are slowly moving 
(circumpolar) stars and others are more rapidly moving stars near the zenith; 
a comparison of these two makes it possible to compute the azimuth of the line 
of collimation. Half of the stars are observed with the instrument in one position, 
half in the reversed position; this determines the error in the sight line. The 
inclination error is measured with the striding level. 

After the corrections to the two chronometers have been accurately determined 
the two chronographs are switched into the main-line circuit and signals are sent 
by breaking the circuit a number of times by pressing a telegraph key. These 
signals are recorded on both chronographs. In order to eliminate the error due 
to the time required in transmitting a signal,* these signals are sent first in one 
direction (E-W) and afterward in the opposite direction (W-E). In this 
manner the transmission time is eliminated, provided it is constant. The personal 
errors of the observers are eliminated by the observers exchanging places in the 
middle of the series; i.e., the above operation would be repeated for about five 
nights with the observers in one position and then for five nights after the observers 
have exchanged positions. After all of the observations have been corrected for 
instrumental errors, and the error of the chronometer on local sidereal time is 
known, each signal sent over the main line will be found to correspond to a certain 
instant of sidereal time at the east station and a different instant of sidereal time 
at the west station. This difference is the difference in longitude. The mean of 
all these values is the final difference free from errors in transmission time and per- 
sonal errors. By this method the difference in longitude may be determined with 
an error of perhaps 10 to 20 feet on the earth's surface. 

88. Longitude by Transit of the Moon. 

A method which is easily used with the surveyor's transit and which, although 
not precise, may be of use in exploration surveys, is that of determining the moon's 
right ascension by observing its transit over the meridian. The right ascension 
of the moon's centre is tabulated in the Nautical Almanac for every hour of 
Greenwich Mean Time; hence, if the right ascension can be determined, the 
Greenwich time can be computed. A comparison of this with local time gives 
the longitude. 

The observation consists in placing the instrument in the plane of the meridian 
and noting the time of transit of the bright limb f of the moon and also of several 
stars whose declinations are nearly the same as that of the moon. The observed 
time interval between the moon's transit and that of a star (reduced to sidereal 
time if necessary), added to or subtracted from the star's right ascension, gives 
the right ascension of the moon's limb. A value of the right ascension is obtained 

* In a test made in 1905 it was found that the time signal sent from Washington 
reached Lick Observatory, Mt. Hamilton, Cal., in 0^.05. 

t The table of moon culmination in the Ephemeris shows which limb (I or II) 
may be observed. See also note, p. 143. 



142 PRACTICAL ASTRONOMY 

from each star and the mean value used. To obtain the right ascension of the 
centre of the moon it is necessary to apply to the right ascension of the limb a 
correction taken from the Ephemeris called " sidereal time of semidiameter passing 
meridian." In computing this correction the increase in the right ascension 
during this short interval has been allowed for; so the result is not the right ascen- 
sion of the centre at the instant of the observation, but its right ascension at the 
instant of the transit of the centre over the meridian. If the west limb was 
observed this correction must be added; if the east limb was observed it must be 
subtracted. The result is the right ascension of the centre at the instant of 
transit, which is also the local sidereal time at that instant. Then the Greenwich 
Mean Time corresponding to this instant is found by interpolating in the table 
giving the moon's right ascension for every hour. To obtain the G. M. T. by 
simple interpolation find the next less right ascension in the table and the "diff. 
for i m " on the same line; subtract the tabular right ascension from the given 
right ascension (found from the observation) and divide this difference by the 
" diff. for I TO ." The result is the number of minutes and decimals of minutes 
to be added to the hour of G. M. T. opposite the tabular right ascension used. 
If the " diff. for i m " is varying rapidly it will be more accurate to interpolate as 
follows. Interpolate between the two values of the " diff. for i m " and obtain a 
" diff. for I TO " which corresponds to the middle of the interval over which the inter- 
polation is carried. In observations made with the surveyor's transit this more 
accurate interpolation is seldom necessary. 

In order to compare the Greenwich time with the local time it is necessary to 
convert the G. M. T. just obtained into the corresponding instant of Greenwich 
Sidereal Time. The difference between this and the local sidereal time is the longi- 
tude from Greenwich. 

In preparing for observations of the moon's transit the Nautical Almanac 
should be consulted (Table of Moon Culminations) to see whether an observation 
can be made and to find the approximate time of transit. The civil date should 
be converted into astronomical before entering the Almanac. The time of the 
moon's transit may be taken from the column headed " Mean time of transit " 
and corrected for longitude, or it may be computed from the approximate right 
ascension. The altitude of the moon should be computed as for a star, and 
in addition the parallax correction should be applied. The moon's parallax 
is so large that the moon probably would not be in the field of the telescope 
at all if this correction were neglected. The horizontal parallax multiplied 
by the cosine of the altitude is the correction to be applied; the moon will 
appear lower than it would if seen at the centre of the earth, so the correction is 
negative. 

Since the moon increases its right ascension about 2 s in every i m of time it is 
evident that any error in determining the right ascension will produce an error 
about thirty times as great in the longitude, so that this method cannot be made to 
give very precise results. 

Following is an example of an observation for longitude by the method of 
moon culminations made with an engineer's transit. 



OBSERVATIONS FOR LONGITUDE 



Example. 

Observed transit of Moon on Jan. 9, 1900, for longitude. Moon's west b'mb 
passed cross hair at 6 A 59"* 378.7; 5 Ceti passed at 7^ O2 m 578.0; and y Ceti passed 
at 7* 06"* 428.0. 



5 Ceti 
Moon's Limb 



Sid. int. 
R. A. 5 Ceti 

R. A. M.'s Limb 



-jh 02 m 578.0 y Ceti 

6 59 37-7 Moon's Limb 



03' 



= h 



Mean = 

Time of s. d. passing merid. = 
R. A. M.'s Centre 



55 



7* o6 m 42* . o 
6 59 37 -7 



I .16 



= 03 19 . 85 Sid. int. = 07"* 05*. 46 

= 2 34 23 . 02 R. A. 7 Ceti =2 38 08 . 77 



31 038.17 R.A.M.'s Limb = 2 h 31"* 03*. 31 
31 03 .31 



3i' 

I 



03 s . 24 
08 .86 



$2 12*. 10 = L. S. T. 



G. M. T. 



From the Nautical Almanac. 



R. A. Moon 

2 h 29 m 55*. 77 

2 32 12 .32 



Diff. for i" 

2. 2748 
2. 2767 



32"* 12 s . 10 

2 9 55 -77 

2 m 168.33 = 1368.33 log 2. 13459 G. M. T. = n h 59" 528.86 

Interpolated Diff . i m = 2.2767 logo. 35730 R.A.M.S. = 19 14 15 .92 

i 52 .86 
M. T. Interval = 59 TO .88i log 1. 77729 

G. S. T. = 7 h i6 m 018.64 

G. M. T. = i i h 59 m 528.86 L. S. T. =2 32 12.10 

Long. W. = 4 h 43 49 s -54 



NOTE. It has already been stated that the moon moves eastward on the celes- 
tial sphere at the rate of about 13 per day; as a result of this motion the time of 
meridian passage occurs about 51"* later (on the average) each day. On account 
of the eccentricity of its orbit, however, the actual retardation may vary consid- 
erably from the mean. The moon's orbit is inclined at an angle of about 5 08' 
to the plane of the earth's orbit. The line of intersection of these two planes ro- 
tates in a similar manner to that described under the precession of the equinoxes, 
except that its period is only 19 years. The moon's maximum declination, there- 
fore, varies from 23 27'+ 5 08' to 23 27' - 5 08', that is, from 28 35' to 18 19', 



144 



PRACTICAL ASTRONOMY 



according to the relative position of the plane of the moon's orbit and the plane of 
the equator. The rapid changes in the relative position of the sun, moon, and earth, 
and the consequent changes in the amount of the moon's surface that is visible 
from the earth, cause the moon to present the different aspects known as the 
moon's phases. Fig. 58 shows the relative positions of the three bodies at several 



First Quarter 




Last Quarter 



o 

Full Moon 

FIG. 58. THE MOON'S PHASES 



different times in the month. The appearance of the moon as seen from the earth 
is shown by the figures around the outside of the diagram. 

It may easily be seen from the diagram that at the time of first quarter the 
moon will cross the meridian at about 6 P.M. ; at full moon it will transit at mid- 
night; and at last quarter it will transit at about 6 A.M. Although the part of the 
illuminated hemisphere which can be seen from the earth is continually changing, 
the part of the moon's surface that is turned towa-d the earth is always the same, 
because the moon makes but one rotation on its axis in one lunar month. Nearly 
half of the moon's surface is never seen from the earth. 



OBSERVATIONS FOR LONGITUDE 145 

Questions and Problems. 

1. Compute the longitude from the following observed transits: 6 Aquarii, 
$h i6 TO O4 S ; TT Aquarii, 5^ 24 40"; moon's W. limb, $ h 32 27*5 X Aquarii, 
5 h 5i TO 47 s - R- A. Aquarii = 22* n m 2j s .6; R. A. ir Aquarii, = 22^ 2o m 04*. 6; 
R. A. \ Aquarii = 22^47 i8 s .3; sidereal time of semidiameter passing meri- 
dian = 6o*.3; at G. M. T. 10*, moon's R. A. = 22^ 27 53 s .s; diff. for i w = 
i s .g8oo; R. A. mean sum at G. M. N. = i6 h 38 28*.o. 

2. Can the longitude be computed by comparing G. M. T. and L. M. T.? 

3. Which limb can be observed in a P. M. observation of a moon culmination ? 

4. At about what time (mean local) will the moon transit at first quarter ? 



CHAPTER XIII 
OBSERVATIONS FOR AZIMUTH 

89. Determination of Azimuth. 

The determination of the azimuth of a line is of frequent 
occurrence in the practice of the surveyor, and is the most 
important to him of all the astronomical problems. On account 
of the high altitudes of the objects observed, as compared with 
those observed in surveying, the adjustments of the instru- 
ment and the elimination of errors are of unusual importance 
in these observations. All of the precautions mentioned in 
Chapter VIII in regard to stability of the instrument, etc., 
should be carefully observed : the instrument should be allowed 
to stand for some time before observations are begun; temper- 
ature changes from any source, such as heat from the lamp or 
from the hand, are to be avoided; the clamps and tangent screws 
should be used with the same care as in triangulation work if 
the greatest accuracy is desired in the results. 

90. Azimuth Mark. 

When the observation is made at night it is frequently incon- 
venient to sight directly at the object whose azimuth is to be 
determined ; it is necessary in such cases to determine the azimuth 
of a special mark called the azimuth mark, which can be seen 
both at night and in daylight, and then to measure the angle 
between this mark and the first object during the day. The 
azimuth mark usually consists of a lamp set inside of a box 
having a small hole cut in the side, through which the light 
may shine. The size of the opening should be determined by 
the distance of the mark; for accurate work it should subtend 
an angle not greater than about o".5 to i".o. If possible, 
the mark should be a mile or more distant, so that the focus 

of the telescope will not have to be altered when changing from 

146 



OBSERVATIONS FOR AZIMUTH 147 

the star to the mark. It is frequently necessary, however, 
to set the mark nearer on account of the topographic and other 
conditions. 

91. Azimuth of Polaris at Elongation. 

The simplest method of determining the direction of the 
meridian with accuracy is by means of an observation of the 
polestar, or any other close circumpolar, when it is at its greatest 
elongation. (See Art. 19, p. 31.) The appearance of the 
constellations at the time of this observation on Polaris may 
be seen by referring to Fig. 49. When the polestar is west of 
the pole the Great Dipper is on the right and Cassiopeia on the 
left. The exact time of elongation may be found by comput- 
ing the sidereal time when the star is at elongation, and convert- 
ing this into mean solar time (local or standard) by the methods 
of Arts. 34 and 35. To find the sidereal time of elongation first 
compute the hour angle P by Equa. [34] and then convert it 
into time. If western elongation is desired, then P is the hour 
angle; if eastern elongation is desired, then 24^ P is the true 
hour angle. The sidereal time is then found by Equa. [37]. 
An average value of P for Polaris for latitudes between 30 
50 is about 5^55 m ; this is sufficiently accurate for computing 
the time of elongation for many purposes. Approximate values 
of the times of elongation of Polaris may be taken from Table V. 

Example. ,-i 

Find the Eastern time of Elongation of Polaris on April 6, 
1904, in lat. 42 21'; long. 4 A 44 m i8 s W. The right ascension, 
is i h 23 48 S .3; the declination is + 88 47' 43".6; the sun's 
right ascension is o h 57 m 22^44. 

log tan I = 9. 95977 P = 5 h 55 36". S S = ? h ig m 24". 8 

log tan D = i. 67723 R = i 23 48.3 R s = o 58 09.1, 

log cos P =8.28254 S = 7 h 19 24*. 8 6 h 2i m 15 s . 7 

P = 88 54' 07" + C' = - i 02 . 5 

= $h ([JOT 36 s . 5 

M. L. T. = 6 h 20 i3 s . 2 
IS 42 




E. S.T. = 6> 04"* 3i.a 



148 PRACTICAL ASTRONOMY 

The transit should be set in position half an hour or so 
before elongation. The star is bisected by the vertical cross 
hair, and as it moves out toward its greatest elongation its 
motion is followed by means of the tangent screw of the upper 
or the lower plate. Near the time of elongation the star will 
appear to move almost vertically, so that no motion in azimuth 
can be detected for five minutes or so before or after elonga- 
tion. About 5 m before elongation, centre the plate levels, set 
the cross hair carefully on the star, lower the telescope with- 
out disturbing its azimuth, and set a stake or a mark carefully 
in line at a distance of several hundred feet north of the transit. 
Reverse the telescope, recentre the levels if necessary, bisect 
the star again, and set another point beside the first one. If 
there are errors of adjustment the two points will not coincide; 
the mean of the two is the true point. The angle between the 
meridian and the line to the stake (the star's azimuth) is found 
by the equation 

sin Z = sin p sec L [35] 

where Z is the azimuth from the north; p, the polar distance of 
the star; and L, the latitude of the place. L does not have to 
be known with great precision; an error of i' in L produces 
only about i" error in the azimuth of Polaris for latitudes within 
the United States. The above method may be applied to any 
close circumpolar star. For Polaris, whose polar distance is 
about i 10', it is usually accurate enough to use the formula 

Z" = p" sec L, [101] 

in which Z" and p" are expressed in seconds of arc. This 
computed angle may be laid off in the proper direction with a 
transit (by daylight), using the method of repetitions, or with 
a tape, by means of a perpendicular offset calculated from the 
measured distance to the stake and the calculated azimuth 
angle. (Fig. 59.) The result is the true north and south line. 
It is often desirable to measure the horizontal angle between 



OBSERVATIONS FOR AZIMUTH 



149 



Star 



the star at elongation and some fixed point instead of marking 
the meridian itself. On account of the slow change in azimuth 
there is ample time to measure several repetitions before the 
error in azimuth amounts to more than i" or 
2".* The errors of adjustment of the transit 
will be eliminated if half of the angles are 
taken with the telescope erect and half in- 
verted. The plate levels should be recentred 
for each position of the instrument before the 
measurements are begun and while the telescope 
is pointing toward the star. 

Example. 

Compute the azimuth of Polaris at greatest 
elongation on April 6, 1904, in latitude 42 
21' N. The declination of the star for the given 
date is + 88 47' 43 // .6. 

log sin p = 8. 32267 
log sec L = o. 13133 



log sin Z = 8. 45400 

Z =1 37' 47". 9 




FIG. 59 



By using the angles in place of the sines, neglecting fractions 
of a second, the following result is obtained : 



P = 4336" 

log p = 3. 6371 
log sec L = o. 1313 



logZ" =3-7684 
Z" = 5867 

= i 37' 47" 

92. Observations Near Elongation. 

If the observation is made on Polaris at any time within half 
an hour of elongation, the azimuth of the star at each pointing 



* In latitude 40 the azimuth changes about i' in half an hour before or after 
elongation; the change in azimuth varies approximately as the square of the time 
from elongation. 



PRACTICAL ASTRONOMY 



of the telescope may be reduced to its value at elongation, 
provided the time is known. The formula for this reduction is 

C = 112.5 X 3600 X sin i" X tanZ e X (T - T e ) 2 * [102] 

in which Z e is the azimuth at elongation; T, the observed time; 
T e , the time of elongation ; and C, the correction in seconds of 



* For the rigorous demonstration of this for- 
mula, which is rather complex, see Doolittle's. 
Practical Astronomy. The following proof, 
although inexact, gives substantially the same 
result. In Fig. 60 S is the position of Polaris 
and E its position when at greatest elongation, 
the angle SPE, or i, being not greater than 
about 8. In the triangle SPM , 

tan MP = tan PS cos SPM 
Since the arcs are small, we may put 




Vm 

FIG. 60 

i 2 
Replacing cos i by the series i [-> 



MP = SP cos 5PM, 
MP = p cos i. 
EM = EP - PM 

P ~~ P cos * 



EM = p 

2 



In the triangle ZME, Z.E = 90, and ZM = ZP (nearly), whence 



sin M ZE = 



cos L 



sin p i i* , . -. 

= T Xp (nearly), 

COS L 2 

i 2 
= sin Z e Xp-> 

in which Z e is the azimuth at elongation. Replacing sin MZE by its arc in seconds 
(C") and reducing i to seconds of time, 



C" = - X sin i" X (6o) 2 X (is) 2 X sinZ e . 



[103] 



Replacing sin Z e by tan Z e produces an error of only about o". 02 for Polaris in 
latitude 40 and reduces [103] to [102]. 



OBSERVATIONS FOR AZIMUTH 151 

arc. T T e must be in minutes of (sidereal) time. The 
factor 112.5 x 36 x sin I/f x tanZ e may be computed, and 
then all observations made at the same place at about the same 
date may be reduced by multiplying the square of the time in- 
tervals in minutes by the factor computed. Table VII gives 
values of the factor for values of Z e ranging from i to 2. 
These corrections will also be found in Table Via at the end 
of the Nautical Almanac. 

Example. 

Three repetitions of the angle between Polaris at western elongation and a 
mark supposed to be on the meridian, April 6, 1904. Lat. 42 21'; long. 71 O4'.5 W. 
The observed times are 6* 28 m 30*, 6^ 31 2o s and 6^ 34 2o s . First reading of 
vernier = o oo'; last reading of vernier = 4 51' oo". The R. A. of Polaris = 
i h 2$ 488.3; its declination = + 88 47' 43".6. R. A. Mean Sun at G. M. N. = 
o fc jj7 TO 22 S .44. 

From this data the Eastern time of elongation is found to be 6 h 04 3i s .2. 
The intervals (T - T e ) are 23 58^.8, 26 48 S .8 and 29 48*. 8. The azimuth 
of the star at elongation is i 37' 48". From Table VII the factor is found to 
be .0559. The resulting corrections are 32", 40" and 50". Adding these to the 
third reading, the sum is 4 53' 02". One third of this is i 37' 41", the measured 
angle between the mark and the star at elongation. The meridian mark is there- 
fore 7" west of north, according to this observation. 

93. Azimuth by an Altitude of the Sun. 

In order to determine the azimuth of a line by means of an 
observation on the sun the instrument should be set up over 
one of the points marking the line and carefully levelled. The 
plate vernier is first set at o and the vertical cross hair sighted 
on the other point marking the line. The colored shade glass 
is then screwed on to the eyepiece, the upper clamp loosened, 
and the telescope turned toward the sun. The sun's disc should 
be sharply focussed before beginning the observations. In 
making the pointings on the sun great care should be taken 
not to mistake one of the stadia hairs for the middle hair. If 
the observation is to be made, say, in the forenoon (in the 
northern hemisphere), first set the cross hairs so that the ver- 
tical hair is tangent to the right edge of the sun and the hori- 
zontal hair cuts off a small segment at the lower edge of the 



152 



PRACTICAL ASTRONOMY 




disc. (Fig. 6 1.)* The arrow in the figure shows the direction of 
the sun's apparent motion. Since the sun is now rising it will 
in a few seconds be tangent to the horizontal hair. It is only 
necessary to follow the right edge by means of the upper plate 
tangent screw until both cross hairs are 
tangent. At this instant, stop following 
the sun's motion and note the time. If 
it is desired to determine the time accu- 
rately, so that the watch correction may be 
found from this same observation, it can 
be read more closely by a second observer. 
FIG. 61. POSITION OF Both the horizontal and the vertical 
SUN'S Disc A FEW SECONDS circles are read, and both angles and the 
BEFORE OBSERVATION t j me are rec orded. The same observa- 

(A. M. Observation in Northern . . . , . 

Hemisphere.) tion may be repeated three or four 

times to increase the accuracy. The instrument should then 
be reversed and the set of observations repeated, except that 
the horizontal cross hair is set tangent to 
the upper edge of the sun and the ver- 
tical cross hair cuts a segment from the 
left edge (Fig. 62). The same number 
of pointings should be taken in each 
position of the instrument. After the 
pointings on the sun are completed the 
telescope should be turned to the mark FlG 62 p osmc >N OF 
again and the vernier reading checked. SUN'S Disc A FEW SECONDS 
If the transit has a vertical arc only, the BEFORE OBSERVATION 
telescope cannot be used in the reversed (A- M ' Hembphere a ) No 
position and the index correction must therefore be determined. 
If the observation is to be made in the afternoon the positions 
will be those indicated in Fig. 63.! 

* In the diagram only a portion of the sun's disc is visible; in a telescope of low 
power the entire disc can be seen. 

t It should be kept in mind that if the instrument has an inverting eyepiece 
the direction of the sun's apparent motion is reversed. If a prism is attached to 
the eyepiece, the upper and lower limbs of the sun are apparently interchanged, 
but the right and left limbs are not. 




OBSERVATIONS FOR AZIMUTH 



153 



In computing the azimuth it is customary to neglect the cur- 
vature of the sun's path during the short interval between the 
first and last pointings, unless the series extends over a longer 
period than is usually required to make such observations. 
If the observation is taken near noon the curvature is greater 
than when it is taken near the prime vertical. The mean of 
the altitudes and the mean of the horizontal angles are assumed 
to correspond to the position of the sun's centre at the instant 
shown by the mean watch reading. The mean altitude read- 
ing corrected for refraction and parallax is the true altitude of 





FIG. 63. POSITIONS OF SUN'S Disc A FEW SECONDS BEFORE OBSERVATION 

(P. M. Observation in Northern Hemisphere.) 

the sun's centre. The azimuth is then computed by any one 
of the formulae on page 34. The resulting azimuth combined 
with the mean horizontal circle reading gives the azimuth of 
the mark. Five-place logarithmic tables will give the azimuth 
within 5" to 10", which is as precise as the azimuth can be 
determined by this method. 

If for any reason only one limb of the sun has been observed, 
the azimuth observed may be reduced to the centre of the sun 
by applying the correction 5 sec h, where S is the semidiameter 
and h is the altitude of the centre. 

If one has at hand a set of tables containing log versed sines (such as are in- 
cluded in railroad engineering tables) the following formulae will sometimes be 
found useful. 

cos (L + h) + sin D [104] 



vers Z s = 



and 



cos L cos h 



cos (L h) sin D 
versZ n = - =-^ -. 
cos L cos h 



[105] 



154 



PRACTICAL ASTRONOMY 



The sum or difference in the numerator must be computed by natural functions 
and the remainder of the work performed by means of logarithms. 

Example. 

Observation on Sun for Azimuth. 

Lat. 42 21' N. Long. 4* 44"* 18* W. Date, Nov. 28, 1905. 

Hor. Circle. Vert. Circle. Watch. 

Ver. A. B. A. M. 

Mark 

R&Llimbs 311 48 48.5 14" 

R & L limbs 312 20 20 15 



238 14' 

311 48 

312 20 

(instrument reversed) 

L & U limbs 312 27 

L & U limbs 312 52 

Mark 238 14 



41 
oo 



8 42 19 



26. 5 



14 



15 55 

16 08 



45 
47 



34 
34 



Mean reading on 

mark 
Mean reading on 

sun 

Mark N. of sun 

L = 42 21'. o 
h = 15 22 .7 
p = in 15.7 



238 14'. o 

312 21.7 


Mean = 
R&P = 

h = 


15 26' 
3-3 


74 07'. 7 


15 22'. 7 



Mean = 8 h 43 47* 



G. M. T = 



43 



Sun's Decl. at G. M. N. =- 21 
- 2 6".8i X i m .73 



47" 

14' 54". 4 

- 46 . 4 



2 5 = 168 59'. 4 
s = 84 29'. 7 

s L = 42 08'. 7 

5 h = 69 07 . o 

s p = 26 46 . o 

s = 84 29.7 



Declination = 21 15' 40". 8 
N. Polar distance = m 15' 40". 8 

log sin 9. 82673 
log sin 9.97049 
log sec o. 04922 
log sec 1.01804 

2)0. 86448 



log tan \Z n = o. 43 22 4 

\ Z n = 69 42'. 9 

Zn = 139 25'. 8 

Mark N. of sun = 74 07 . 7 



Bearing of Mark= N 65 18'. i E 

By differentiating Equa. [13] it may be shown that when the latitude is greater 
than the sun's declination the greatest accuracy in the azimuth, so far as errors 
in altitude are concerned, is secured when the sun is somewhere between the prime 
vertical and the six-hour circle; the exact position for maximum accuracy depends 
upon the latitude and upon the parallactic angle. If an observer were on the 
equator and the sun's declination zero, the motion would be vertical and the 
change in azimuth would be zero. In the preceding example the azimuth increases 
about i' 50" for an increase of i' in the altitude. Errors in the azimuth due to 
errors in the assumed value of the latitude are a minimum when the sun is on the 
six-hour circle. Observations very near the horizon, however, are subject to errors 



OBSERVATIONS FOR AZIMUTH 155 

in the refraction, since the tabular values of the mean refraction may be largely 
in error for very low altitudes under the temperature and pressure conditions 
existing at the time of the observation. The general rule is therefore to avoid 
observations near the meridian and also those within 10 or less of the horizon. 

If it is desired to compute the hour angle of the sun from the same observations 
used in determining the azimuth, it may be found by formula [19], in which case 
no new logarithms have to be taken from the tables; or it may be found by the 

equation 

sin P = sinZ cos h sec D. [12] 

The value of P and the error of the watch obtained by the use of this formula 
are given below.* 

log sinZ = 9. 81317 
log cos h = 9. 98416 
log sec D = o. 03061 



log sin P = 9. 82794 

P = 42 17' 26" 

= 2^ 49 m 09* . 7 
L. A. T. =9 10 50 .3 
Eq. T. = 12 02 .8 



L. M. T. = S h 58 47*. 5 
IS 42 



E. S.T. = 8* 43 TO 5 s -5 
Watch =8 43 47 . 2 



Watch fast = 4i s - 7 

94. Azimuth by an Altitude of a Star. 

The method described in the preceding article applies equally 
well to an observation on a star, except that the star's image is 
bisected with both cross hairs and the parallax and semidiameter 
corrections become zero. The declination of the star changes so 
little during one day that it may be regarded as constant, and 
consequently the time of the observation is not required. Errors 
in the altitude and the latitude may be partially eliminated by 
combining two observations, one on a star about due east and 
the other on one about due west. 

95. Azimuth Observation on a Circumpolar Star at any Hour Angle. 

The most precise determination of azimuth may be made by measuring the 
horizontal angle between a circumpolar star and an azimuth mark, the hour angle 

* See also Art. 102, p. 166, and Art. no, p. 175. 



156 



PRACTICAL ASTRONOMY 






of the star at each pointing being known. If the sidereal time is determined 
accurately, by any of the methods given in Chapter XI, the hour angle of the star 
may be found at once by Equa. [37] and the azimuth of the star at the instant 
may be computed. Since the close circumpolar stars move very slowly and 
errors in the observed times will have a small effect upon the computed azimuth, 
it is evident that only such stars should be used if precise results are sought. The 
advantage of observing the star at any hour angle, rather than at elongation, is 
that the number of observations may be increased indefinitely and greater accuracy 
thereby secured. 

The angles may be measured either with a repeating instrument (like the 
engineer's transit) or with a direction instrument in which the circles are read with 



XVIII 




if/51 Cephei 



XII 



FIG. 64 

great precision by means of micrometer microscopes. For refined work the instru- 
ment should be provided with a sensitive striding level. If there is no striding 
level provided with the instrument* the plate level which is parallel to the hori- 
zontal axis should be a sensitive one and should be kept well adjusted. At all 
places in the United States the celestial pole is at such high altitudes that errors 
in the adjustment of the horizontal axis and of the sight line have a compara- 
tively large effect upon the results. 

The star chosen for this observation should be one of .the close circumpolar stars 
given in the special list in the Nautical Almanac. (See Fig. 64.) Polaris is the only 
bright star in this group and should be used in preference to the others when it is 

* The error due to inclination of the axis may be eliminated by taking half 
of the observations direct and half on the image of the star reflected in a basin 
of mercury. 



OBSERVATIONS FOR AZIMUTH 157 

practicable to do so. If the time is uncertain and Polaris is near the meridian, 
in which case the computed azimuth would be uncertain, it is better to use 51 
Cephei,* because this star would then be near its elongation and comparatively 
large errors in the time would have but little effect upon the computed azimuth. 
If a repeating theodolite or an ordinary transit is used the observations consist 
in repeating the angle between the star and the mark a certain number of times 
and then reversing the instrument and making another set containing the same 
number of repetitions. Since the star is continually changing its azimuth it 
is necessary to read and record the time of each pointing on the star with the 
vertical cross hair. The altitude of the star should be measured just before and 
again just after each half-set so that its altitude for any desired instant may be 
obtained by simple interpolation. If the instrument has no striding level the 
cross-level on the plate should be recentred before the second half-set is begun. 
If a striding level is used the inclination of the axis may be measured, while the 
telescope is pointing toward the star, by reading both ends of the bubble, with the 
level first in the direct position and then in the reversed position. 

In computing the results the azimuth of the star might be computed for each 
of the observed times and the mean of these azimuths combined with the mean 
of the measured horizontal angles. The labor involved in this process is so great, 
however, that the practice is first to compute the azimuth corresponding to the 
mean of the observed times, and then to correct this result for the effect of the 
curvature of the star's path, i.e., by the difference between the mean azimuth 
and the azimuth at the mean of the times. The formula for the azimuth is 

z _ sinP 

cos L tan D sin L cos P 

The formula given below, although not exact, is sufficiently accurate for all work 
except refined geodetic observations. 

Z" = p" sin P sec h, [106] 

in which Z" and p" are in seconds of arc. In this formula the arcs have been 
substituted for their sines. The precision of the computed azimuth depends 
chiefly upon the precision with which h can be determined. If the vertical arc 
cannot be relied upon, and the latitude is known accurately, the first formula 
may be preferred. If desired, the altitude of Polaris may be computed by formula 
[80] and its value substituted in [106]. 

* 51 Cephei may be found by first pointing on Polaris and then changing the 
altitude and the azimuth by an amount which will bring 51 Cephei into the field. 
The difference in altitude and in azimuth may be obtained with sufficient accuracy 
by holding Fig. 64 so that Polaris is in its true position with reference to the 
meridian (as indicated by the position of S Cassiopeia) and then estimating the 
difference in altitude and the difference in azimuth. It should be remembered 
that the distance of 51 Cephei east or west of Polaris has nearly the same ratio 
to the difference in azimuth that the polar distance of Polaris has to its azimuth 
at elongation, i.e., i to sec L. 



158 PRACTICAL ASTRONOMY 

96. The Curvature Correction. 

If we let Ti, Tz, T 3> etc. = the observed times, T = the mean of these times, 
Zi, Zz, Z 3 , etc, = corresponding azimuths, and Z the azimuth at the instant T , 
then 

Zi + Z 2 + Z n =Zo _ ^^ [Q j I 2 (T _ n)2 * [iQ7] 

w n 

The quantity in brackets is the logarithm of a constant; 2 (T T ) 2 is the sum of 
the squares of the time-intervals (in minutes and decimals) reduced to sidereal 
intervals. The azimuth is therefore computed by first finding Z by Equa. [31] 
and then correcting it by means of the last term of Equa. [107]. 

If it is desired to express ( T T ) in seconds of time the constant log becomes 
[6.73672]. When the star is near culmination the curvature correction is very 
small; near elongation it is a maximum. 

97. The Level Correction. 

The inclination i of the axis as determined by the striding level is given in 
seconds of arc by 

i = [(w + a/) -._(.+ e')] - , [108] 

4 

where w and e are the readings of the west and east ends of the bubble for the 
direct position, and w' and e' are the same for the reversed positions, and d is the 
angular value of one division of the level scale. The correction to the measured 
horizontal angle is 

C = i tan h. [109] 

If the west end of the axis is too high (i positive) the telescope has to be turned 
too far west in pointing at the star; the correction must therefore be added to the 
measured angle if the mark is west of the star, subtracted if east. If the instru- 
ment has no striding level the error must be eliminated as completely as possible 
by relevelling between the half-sets. 

98. Diurnal Aberration. 

Strictly speaking, the computed azimuth of the star should be corrected for 
diurnal aberration, the effect of which is to make the star appear farther east 
than it actually is, because the observer is being carried due east by the diurnal 
motion of the earth. The correction is 

cos L cos Z 

" 3I9 X cos h [IIO] 

For all but the most precise observations it may be taken as o".32, since the factor 

cos L cos Z . , , 

= is never far from unity. 

cos h 

*For the derivation of the formula see Doolittle's Practical Astronomy and 
Hayford's Geodetic Astronomy. 



OBSERVATIONS FOR AZIMUTH 



159 



Example i. 
RECORD OF AZIMUTH OBSERVATIONS 

Instrument (B. & B. No. 3441) at South Meridian Mark. Boston, May 16, 1910. 
(One division of level = i5".o.) 







ci 




Horizontal circle. 




Object. 


"o 


i 

"o 


Chronometer. 




Level readings 
and angles. 








1 







Vernier A. 


B. 
















W E 


Polaris. . 






II A 2 4 "35.o 


o oo' oo" 


oo" 


7- 3-9 














5-8 5-i 








27 15.0 




















12.8 9.0 














9.0 








28 31.5 










4* 










3-8 




Q 




30 oo.o 






Corr. = 1 2". 5 














Alt. Polaris at 








31 20.5 






jjA ^m 2O s -5 = 














41 20' 30" 








32 27.0 






Alt. Polaris at 














n& 51"* 04 s . o = 


Mark. . . 




6 




*39 33' 3o" 


30" 


41 i 8' 40" 














Mean horizontal 














angle = 














66 35' 35"- o 


Polaris. . 












W E 








II 42 45.5 


39 33' 30" 


30" 


5-i 5-8 














3-3 7-6 




o 




44 09 . o 






8-4 13-4 








45 iS-o 






8-4 




g 




46 29.5 






5-o 














Corr. = 1 6". 5 








47 25.0 














48 54-5 








Mark. . . 




6 




* 7 8 27' 30" 


20" 


Mean horizontal 














angle = 














66 28' 59". 2 














Alt. Polaris at 














i2 h og m 31*. 5 = 














41 15' 4o" 



* Passed 360. 



l6o PRACTICAL ASTRONOMY 

RECORD OF TIME OBSERVATIONS 
Polaris: Chronometer, 12*09"* 31*. 5; alt., 41 15' 40" 
e Corvi: Chronometer, 12 13 37 .5; alt., 25 34 oo 

Polaris: R. A. = i h 25 si s .i; decl. = +88 49' 24". 8 
/Corvi: R. A. = i2 h 5"* 30*. 5; decl. = 22 07' 21". o 

Chronometer R. A. Decl. 

a Serpentis (E) i2 h 24 15*. 7 15*39"* Si 8 - 6 +6 42' 20". 7 

f Hydra: (W) 12 18 32 .o 8 42 oo . 5 + 6 44 58 . 9 

(Lat. = 42 21' oo" N.; Long. = 4* 44 i8. o W.) 

From these observations the chronometer is found to be io w 22*. i fast. 

COMPUTATION OF AZIMUTH 

Mean of Observed times = n* 37"* 25*. 6 

Chronometer correction = 10 22.1 

Sidereal time = n 27 03 . 5 

R. A. of Polaris = i 25 51 . i 

Hour Angle of Polaris = 10 01 12 .4 

P = 150 18' 06" 
log cos L = 9. 868670 
log tan D = i. 687490 
log cos L tan D = 1.556160 
cos L tan D = 35. 9882 
log sin L = 9. 82844 
log cos P = 9. 93884 
log sin L cos P = 9. 76728 
sin L cos P = . 5852 
denominator = 36. 5734 

log sin P = 9. 694985 

log denom. = 1.563165 

log tan Z = 8. 131820 

Z = o 4 6' 34". 2 

Curvature correction = 2. i 

Azimuth of star = o 46 32.1 

Measured angle, first half = 66 35' 35". o 
Level correction = 12.5 

Corrected angle = 66 35 22.5 

Measured angle, second half = 66 28 59 . 2 
Level correction = +16 .5 

Corrected angle = 66 29 15 . 7 

Mark east of star = 66 32 19 . i 
Mark east of North = 65 45' 47". o 



OBSERVATIONS FOR AZIMUTH 161 

Example 2. 

Observed altitudes of Regidus (east), Feb. n, 1908, in lat. 42 21'. 

Altitude Watch 

17 05' j h i2 m 16* 

17 3i J 4 3i 

17 49 l6 7 

18 02 17 20 

The right ascension of Regidus is 10* 03"* 29*. i; the declination is + 12 24' 57". 
From these data the sidereal time corresponding to the mean watch reading 
( 7 & I5 m 038.5) is found to be 4 h 53 42^.7. 

Observed horizontal angles from azimuth mark to Polaris. 

(Mark east of north.) 

Telescope Direct Time of pointing on Polaris 

Mark, o oo' 7 A 20 38* 

23 oo 
Third repetition 201 48' 23 56 



Mean= 67 16'. o -j h 22 318.3 

Telescope Reversed 

Mark= o oo' 7 27 09 

28 17 

Third Repetition 201 54' 29 21 

Mean = 67 18'. o 7 h 28 15 s . 7 

Altitude of Polaris at f h 2o m 38* = 43 03' 
Altitude of Polaris at 7 29 21 = 43 01 
Mean watch reading for Polaris = j h 25 23*. 5 
Corresponding sidereal time =5 04 04 . 4 
Right Ascension of Polaris = i 25 32 .3 
Hour-angle of Polaris =3 38 32 . i 
P = 54 38' 

P = 4251 
log p = 3. 62849 
log sin P = 9. 91141 
log sec h = o. 13611 



log azimuth = 3. 67601 
azimuth = 4743" 

= i 19'. o 
Mean angle = 67 17 .o 



Mark East of North = 65 58'. o 

99. Meridian by Polaris at Culmination. 

The following method is given in Lalande/s Astronomy and 
was practiced by Andrew Ellicott, in 1785, on the Ohio and 
Pennsylvania boundary survey. The direction of the meridian 
is determined by noting the instant when Polaris and some 




162 PRACTICAL ASTRONOMY 

other star are in the same vertical plane, and then waiting a 
certain interval of time, depending upon the date and the star 
observed, when Polaris will be in the meridian. At this instant 
Polaris is sighted and its direction then marked on the ground 
by means of stakes. The stars selected for this observation 
should be near the hour circle through the polestar; that is, 
their right ascensions should be nearly equal to that 
of the polestar, or else nearly i2 h greater. The stars 
best adapted for this purpose at the present time are 
d Cassiopeia and f Ursa Majoris. 

The interval of time between the instant when 
the star is vertically above or beneath Polaris and 
)P the instant when the latter is in the meridian is 
computed as follows : In Fig. 65 P is the pole, P' is 
Polaris, S is the other star (8 Cassiopeia) and Z is 
the zenith. At the time when S is vertically under 
P' ', ZP'S is a vertical circle. The angle desired is 
ZPP', the hour angle of Polaris. PP'*and PS, the 
polar distances of the stars, are known quantities; 
P' PS is the difference in right ascension, and may 
be obtained from the Ephemeris. The triangle P'PS 
may therefore be solved for the angle at P'. Sub- 
tracting this from 180 gives the angle ZP'P; PP f 
is known, and PZ is the colatitude of the observer. 
The triangle ZP'P may then be solved for ZPP', the desired 
angle. Subtracting ZPP' from 180 or i2 h gives the sidereal 
interval of time which must elapse between the two 
observations. The angle SPP' and the side PP' are so 
small that the usual formulae may be simplified, by replacing 
sines by arcs, without appreciably diminishing the accuracy 
of the result. A similar solution may be made for the upper 
culmination of 5 Cassiopeia or for the two positions of the 
star f Ursa Majoris, which is on the opposite side of the 
pole from Polaris. The above solution, using the right ascen- 
sions and declinations for the date, gives the exact interval 






OBSERVATIONS FOR AZIMUTH 163 

required; but for many purposes it is sufficient to use a time 
interval calculated from the mean places of the star and for a 
mean latitude of the United States. The interval for the star 
8 Cassiopeia for the year 1901 is 3 m .o; for 1910 it is 6 m .i, the 
annual increase being o m .35. For f Ursa Majoris the interval 
for 1901 is 3 m .7; for 1910 it is 6 m -7, the annual increase being 
o m .33. Beginning with the issue for 1910 the American 
Ephemeris and Nautical Almanac contains values of these 
intervals (Table VII) for different latitudes and for different 
dates. Within the limits of the United States it will generally 
be necessary to observe on d Cassiopeia when Polaris is at 
lower culmination and on f Ursa Majoris when Polaris is at 
upper culmination. 

The determination of the instant when the two stars are in 
the same vertical plane is necessarily approximate, since there is 
some delay in changing the telescope from one star to the other. 
The motion of Polaris is so slow, however, that a very fair 
degree of accuracy may be obtained by first sighting on Polaris, 
then pointing the telescope to the altitude of the other star (say 
d Cassiopeia} and waiting until it appears in the field; when 
d Cassiopeia is seen, sight again at Polaris to allow for its 
motion since the first pointing, turn the telescope again to 
5 Cassiopeia and observe the instant when it crosses the verti- 
cal hair. The motion of the polestar during this short interval 
may safely be neglected. The tabular interval of time corrected 
to date must be added to the watch reading. When this com- 
puted time arrives, the cross hair is to be set accurately on 
Polaris and then the telescope lowered in this vertical plane and 
a mark set in line with the cross hairs. The change in the 
azimuth of Polaris in i m of time is not far from half a minute 
of angle, so that an error of a few seconds in the time of sighting 
at Polaris has but little effect upon the result. It is evident that 
the actual error of the watch on local time has no effect what- 
ever upon the result, because the only requirement is that the 
interval should be correctly measured. 



164 



PRACTICAL ASTRONOMY 



100. Azimuth by Equal Altitudes of a Star. 

The meridian may be found in a very simple manner by means of two equal 
altitudes of a star, one east of the meridian and one west. This method has the 
advantage that the coordinates of the star are not required, so that the Almanac 
or other table is not necessary. The method is inconvenient because it requires 
two observations at night several hours apart. It is of special value to surveyors 
in the southern hemisphere, where there is no bright star near the pole. The star 
to be used should be approaching the meridian (in the evening) and about 3* or 
4^ from it. The altitude should be a convenient one for measuring with the tran- 
sit, and the star should be one that can be identified with certainty 6 h or S h later. 
Care should be taken to use a star which will reach the same altitude on the oppo- 
site side of the meridian before daylight interferes with the observation. In the 




P.M. 



A.M. 



northern hemisphere one of the stars in Cassiopeia might be used. The position 
at the first (evening) observation would then be at A in Fig. 66. The star should 
be bisected with both cross hairs and the altitude read and recorded. A note or 
a sketch should be made showing which star is used. The direction of the star 
should be marked on the ground, or else the horizontal angle measured from some 
reference mark to the position of the star at the time of the observation. When 
the star is approaching the same altitude on the opposite side of the meridian 
(at B) the telescope should be set at exactly the same altitude as was read at the 
first observation. When the star comes into the field it is bisected with the ver- 
tical cross hair and followed in azimuth until it reaches the horizontal hair. The 
motion in azimuth should be stopped at this instant. Another point is then set 
on the ground (at same distance from the transit as the first) or else another angle 



OBSERVATIONS FOR AZIMUTH 165 

is turned to the same reference mark. The bisector of the angle between the two 
directions is the meridian line through the transit point. It is evident that the 
index and refraction errors are eliminated, because they are alike for the two 
observations. If one observation is made with the telescope direct and the other 
with the telescope reversed, the other instrumental errors will be eliminated. Care 
should be taken to level the instrument just before the observations. The accu- 
racy of the final result may be increased by observing the star at several different 
altitudes and using the mean value of the different results. 

loz. Observation for Meridian by Equal Altitudes of the Sun in the Forenoon 
and in the Afternoon. 

This observation consists in measuring the horizontal angle between the mark 
and the sun when it has a certain altitude in the forenoon and measuring the 
angle again to the sun when it has an equal altitude in the afternoon. Since the 
sun's declination will change during the interval, the mean of the two angles will 
not be the true angle between the meridian and the mark, but will require a small 
correction. The angle between the south point of the meridian and the point 
midway between the two directions of the sun is given by the equation 

Correction = =. -> [m] 

cos L sin P 

in which d is the hourly change in declination multiplied by the number of hours 
elapsed between the two observations, L is the latitude, and P is the hour angle 
of the sun, or approximately half the elapsed interval of time. The correction 
depends upon the change in the declination, not upon its absolute value, so that 
the hourly change may be taken with sufficient accuracy from the Almanac for 
any year for the corresponding date. 

In making the observation the instrument is set up at one end of the line whose 
azimuth is to be determined, and the plate vernier set at o. The vertical cross 
hair is set on the mark and the lower clamp tightened. The sun glass is then put 
in position, the upper clamp loosened, and the telescope pointed at the sun. 
It is not necessary to observe on both edges of the sun, but is sufficient to sight, 
say, the lower limb at both observations, and to sight the vertical cross hair on 
the opposite limb in the afternoon from that used in the forenoon. The hori- 
zontal hair is therefore set on the lower limb and the vertical cross hair on the left 
limb. When the instrument is in this position the time should be noted as accurately 
as possible. The altitude and the horizontal angle are both read. In the after- 
noon the instrument is set up at the same point, and the same observation is made, 
except that the vertical hair is now sighted on the right limb; the horizontal hair 
is set on the lower limb as before. A few minutes before the sun reaches an alti- 
tude equal to that observed in the morning the vertical arc is set to read exactly 
the same altitude as was read at the first observation. As the sun's altitude de- 
creases the vertical hair is kept tangent to the right limb until the lower edge 
of the sun is in contact with the horizontal hair. At this instant the time is again 
noted accurately; the horizontal angle is then read. The mean of the two circle 
readings, corrected for the effect of change in declination, is the angle from the 



1 66 PRACTICAL ASTRONOMY 

mark to the south point of the horizon. The algebraic sign of the correction is 
determined from the fact that if the sun is going north the mean of the two ver- 
nier readings lies to the west of the south point, and vice versa. The precision 
of the result may be increased by taking several forenoon observations in suc- 
cession and corresponding observations in the afternoon. 
Example. 
Lat. 42 18' N. Apr. 19, 1906. 

A.M. Observations. P.M. Observations. 

Reading on Mark, o oo' oo" Reading on Mark, o oo' oo" 

( Alt., 24 58' ( Alt., 24 58' 

U & L limbs < Hor. Angle, 357 14' 15" U & R limbs ] Hor. angle, 162 28'oo" 

( Time, ; A 19"* 30* ( Time 4* i2 m 15* 

5 elapsed time = 4^ 26 22 s 

p = 66 35' 30" Incr. in decl. = + 52" X 4*. 44 

log sin P = 9.96270 = + 230". 9 

log cos L = 9.86902 



9.83152 Mean Circle Reading =79 51' 08" 

log 230". 9 = 2.36342 Correction = 5 40 



2.53170 True Angle = S 79 45' 28" E. 

Corr. = 340". 2 Azimuth = 280 14' 32" 

1 02. Azimuth of Sun near Noon. 

The azimuth of the sun near noon may be determined by means of Equa. [30], 
provided the local apparent time is known or can be computed. If the longitude 
and the watch correction on Standard Time are known within one or two seconds 
the local apparent time may be readily calculated. This method may be useful 
when it is desired to obtain a meridian during the middle of the day, because the 
other methods are not then applicable. 

If, for example, an observation has been made in the forenoon from which a 
reliable watch correction may be computed, then this correction may be used in 
the azimuth computation for the observation near noon; or if the Standard Time 
can be obtained accurately by a comparison at noon and the longitude can be 
obtained from a map within about 1000 feet, the local apparent time may be 
found with sufficient accuracy. This method is not usually convenient in mid- 
summer, on account of the high altitude of the sun, but if the altitude is not 
greater than about 50 the method may be used without difficulty. The obser- 
vations are made exactly as in Art. 93, except that the time of each pointing is 
determined more precisely; the accuracy of the result depends very largely upon 
the accuracy with which the hour angle of the sun can be computed, and great 
care must therefore be used in determining the time, The observed watch read- 
ing is corrected for the known error of the watch, and is then converted into local 
apparent time. The local apparent time converted into degrees is the angle at 
the pole, P. The azimuth is then found by the formula 

sin Z = sin P sec h cos D. [30] 

Errors in the time and the longitude produce large errors in Z, so this method 
should not be used unless both can be determined with certainty. 



OBSERVATIONS FOR AZIMUTH 

Example. 

Observation on the sun for azimuth. 

Lat. 42 21'. Long. 4* 44 i8 s W. Date, Feb. 5, 1910. 



I6 7 



Hor. Circle. Vert. Circle. Watch. 
Mark, o oo' (3O S fast) 
app. L & L limbs, 29 01 31 49' ii' l 43 wl 22 s 
app. U & R limbs, 28 39 31 16 n 44 20 


Mean, 28 50' 31 32'. 5 n>> 
Refr., i . 6 Watch corr. = 


43 5i s 
-30 


1 1 

9 
, X 


^ = 3 I ,3 / -9 E. S. T. =11* 
D = - 1 6 06' 04". 5 


43" 2i s 

15 42 


LA/T T TT^ 


S9 m 03 s 
14 09 


D = 16 02' 32". 2 Eq. t. = 


Eq. t. =i4o8 s . 05 L. A. T. = n' 
.217 X 4 h - 7 i .02 P = 


1 44 53. 
I ^m 6s 

46'. S 


3 



Eq. t. = 



09 s . 07 



log sin P = 8. 81847 
log cos D = 9. 98275 
log sec h = o. 06930 



log sinZ = 8. 87052 

Z = 4 15'. 4 
Hor. Circle = 28 50 



Azimuth = S 33 05'. 4 E 
= 326 54'. 6 

103. Combining Observations. 

From the foregoing descriptions of field methods of observing, it will be seen 
that but few of these methods are quite independent of the data obtained by other 
observations, and in the practice of the engineer it often happens that no one of 
the quantities which he desires can be completely determined until some or all 
of the others are known approximately. The latitude may be determined directly 
by means of a star at culmination, but it may be inconvenient or impracticable 
in many cases to wait until either Polaris or a southern star comes to the meridian. 
In all of the methods of determining time it is necessary to know either the latitude 
or the direction of the meridian before the time can be directly computed. In all 
of the methods of determining azimuth either the time, the latitude, or both must 
be known. Where all of these quantities are entangled it is usually necessary to 
obtain the true values by a series of approximations. In most cases, however, 
very few approximations are necessary to give the greatest accuracy afforded by 
the observations. 

If it is necessary to determine a precise azimuth, and nothing whatever is known 
in regard to the latitude of the point or the local time, then all three may be accu- 
rately determined by making observations of transits across the vertical circle 



1 68 PRACTICAL ASTRONOMY 

through Polaris, measuring the altitudes of all the stars, and then repeating the 
horizontal angle between Polaris and an azimuth mark. The measured altitudes 
of Polaris and the time-star make it possible to compute the sidereal time by two 
approximations (Art. 78) without knowing the latitude. When the time is known 
the latitude may be found by Arts. 68 and 69. If the instrument has only a 
vertical arc (180), then the altitudes of the southern stars may be measured and 
the first approximation to the latitude found from these observations. The alti- 
tudes of Polaris may then be calculated closely enough for computing Equa. [106]. 
After the time and the latitude are known the azimuth is found directly. By 
using the instrument in the two positions and increasing the number of obser- 
vations the precision of all of the results may be increased. 

The same results may be obtained by using the method of equal altitudes 
(Arts. 80-82), combined with measured altitudes of the polestar and observations 
for azimuth. By selecting a pair of stars having a large difference in right ascen- 
sion or a small difference in declination, the time may be fairly well determined 
by using an estimated latitude obtained by estimating a correction to the observed 
altitude of Polaris. When the time is known approximately, a new value of the 
latitude may be obtained, and with this new latitude the time may be recomputed. 
The azimuth may then be found as before. 

A very rapid but not very precise way of determining these three quantities 
and also checking the azimuth is to sight on the mark, then to sight on the pole- 
star, reading both the horizontal and vertical angles, and finally to sight on a prime 
vertical star, reading both angles. Using an estimated latitude the PZS triangle 
may be solved for P; with this value of P a close value of the latitude is found, 
and the hour angle is then recomputed. If the latitude and the time are known 
the azimuth may be determined from the polestar and checked by the azimuth 
from the star near the prime vertical. 

It is well when determining azimuths for surveying purposes to obtain checks 
by methods which are independent of one another. For example, if the azimuth 
is being found by angles measured to Polaris, a check may be obtained by turning 
an angle from some star near the prime vertical (Art. 77) and measuring its alti- 
tude simultaneously. Observations made on both east and west stars will increase 
the accuracy. The azimuth thus computed is inferior in accuracy to that found 
from Polaris, but the fact that it is independent makes it a valuable check against 
mistakes or large errors in the Polaris observations. A sun observation made late 
in the afternoon may be used in a similar way to check an evening observation 
on Polaris. 

Questions and Problems 

1. What error is caused by making the approximations in deriving formula 
[35]? 

2. Derive formula [106]. 

3. Show that if the declination is less than the latitude the most favorable 
conditions for determining azimuth by an altitude of the sun occur when the sun 



OBSERVATIONS FOR AZIMUTH 169 

is between the six-hour circle and the prime vertical. For greatest accuracy 

j 7 /77 

and TF should be a minimum. Differentiate Equa. [13] and simplify by means 

dli dL 

of [12] and [15]. 

4. Show that the factor cos L cos Z sec h (Equa. [no]) is always nearly equal 
to unity. 

5. Compute the approximate local mean time of eastern elongation of Polaris 
on Sept. 10. R. A. of Polaris, i h 25. See Art. 63, p. 101, for an approximate 
method of finding the R. A. of the mean sun. Use 5^ 55 for the hour angle of 
Polaris at elongation (see Art. 91, p. 147). 

6. Observation on sun May 15, 1906, for azimuth. Vernier A, on mark, read 
o oo'. On the sun, right and lower limbs, vertical circle read 43 36'; vernier A 
read 168 59' (right-handed); E. S. T., 2 h 52 45 s P.M. Upper and left limbs, 
vernier A read 169 52'; vertical circle read 42 33'; E. S. T., 2 h 55^ 37* P.M. Dec- 
lination at G. M. N. = + 18 42' 43". 6; diff. for i^ = + 3 5". 9 4. The latitude 
of the place is 42 21' N; longitude 71 05' W. Compute the azimuth of the mark. 

7. Compute the azimuth of Jupiter from the data given in Art. 77, p. 124. 

8. Prove that the horizontal angle between the centre of the sun and the 
right or left limb is 5 sec h where 5 is the apparent angular semidiameter and h is 
the apparent altitude. 

9. Prove that the level correction (Art. 97) is i tan h. 

10. Why could not Equa. [106] be used in place of Equa. [30] in the method of 
Art. 102 ? 

n. If there is an error of 4 s in the assumed value of the watch correction and 
an azimuth is determined by the method of Art. 102 (near noon), what would be 
the relative effect of this error when the sun is on the equator and when it is 23 
South? Assume the latitude to be 45 N. (See Table B.) 

12. Make a set of azimuth observations by the method of Art. 93 (three point- 
ings in each position of the instrument), and plot a curve using altitudes for ordinates 
and horizontal angles for abscissae; also plot a curve using altitudes and times for 
the two coordinates. 



CHAPTER XIV 
NAUTICAL ASTRONOMY 

104. Observations at Sea. 

The problems of determining a ship's position at sea and the 
bearing of a celestial object at any time are based upon exactly 
the same principles as the surveyor's problems of determining 
his position on land and the azimuth of a line of a survey. The 
method of making the observations, however, is different, 
since the use of instruments requiring a stable support, such as 
the transit and the artificial horizon, is not practicable at sea. 
The sextant does not require a stable support and is well adapted 
to making observations at sea. Since the sextant can be used 
only to measure the angle between two visible points, it is 
necessary to measure all altitudes from the sea-horizon and to 
make the proper correction for dip. 

Determination of Latitude at Sea 

105. Latitude by Noon Altitude of Sun. 

The determination of latitude by measuring the maximum 
altitude of the sun's lower limb at noon is made in exactly the 
same way as described in Art. 66. The observation should be 
begun a little before local apparent noon and altitudes measured 
in quick succession until the maximum is reached. In measur- 
ing an altitude above the sea-horizon the observer should bring 
the sun's image down 'until the lower limb appears to be in 
contact with the horizon line. The sextant should then be 
tipped by rotating right and left about the axis of the telescope 
so as to make the sun's image describe an arc; if the lower limb 
of the sun drops below the horizon at any point, the measured 
altitude is too great, and the index arm should be moved until 
the sun's image is just tangent to the horizon when at the lowest- 

170 



NAUTICAL ASTRONOMY 



171 



point of the arc. (Fig. 67.) This method is illustrated by the 
following example. 

Example. 

Observed altitude of sun's lower limb 69 21' 30", bearing north. Index cor- 
rection = i' 10"; height of eye = 18 feet; sun's declination at G. A. N. = 
N 8 59' 32"; diff. i h = + 54"-43- Approx. lat. = 11 30' S; approx. long. = 
i^ OO OT W. 



Obs'd alt. = 69 21' 30' 
Corr. = + 10 16 



Alt. centre = 69 31' 46" 
Declination = 9 oo 26 



Colatitude = 78 32' 12" 
Latitude = 11 27' 48" S 



Corrections 

I. C. = - i' 10" 

Dip = 4 12 

r & p = o 20 

S. D. = +i5 58 



Corr. = + 10' 1 6" 



Decl 



8 59' 32" 
+ 54 

9 oo' 26" 



O 



Sea 



Horizon 



FIG. 67 



106. Latitude by Ex-Meridian Altitudes. 

If for any reason the noon altitude has been lost, an altitude may be measured 
near noon and this altitude corrected to the corresponding noon altitude by 
Equa. [72]. In order to make this " reduction to the meridian " it is necessary 
to know the sun's hour angle. If the altitude is taken within a few minutes of 
noon the reduction may be made by the more convenient formula, [74]; in practice 
this is done by means of tables. 

Example. 

Observed altitude Jan. 20, 1910 = 20 05'; I. C. = o; G. M. T. = i* 35 
28 s ; lat. by dead reckoning = 49 20' N; longitude by dead reckoning = i h 05 
20*; height of eye = 16 feet; decl. at G. M. N. = 20 15' 02" S; diff. for i h 
+ 32 ".o; S. D. = i6'i7". 



G. M. T. = 
Long. W = 



35 m 2 gs 

05 20 



H. A. = cA 30* 

= 7 32' 



08* 



S. D. = + 16' 17' 

I. C. = oo 

Dip- = ~ 3 55 

r & p 2 30 



cos L = 9. 8140 
cos D = 9. 9723 
vers P = 7. 9361 



7.7224 

Corr. = .0053 
sin A = . 3461 



Decl. G. M. N. = 
Diff. for i h .i = 



IS' 02' 

35 



Decl. = 20 14' 27" S 



Corr. = + 9' 52' 
Obs. Alt. = 20 05 

sin h m = . 3514 

True Alt. = 20 14' 52" h m = 20 34' \ 

Decl. = 20 14 \ 
Colat. = 40 49' 
Lat. = 49 n' 



172 PRACTICAL ASTRONOMY 

Determination of Longitude at Sea 

107. By the Greenwich Time and the Sun's Altitude. 

The usual method of finding the longitude at sea is to determine 
the local mean time from an observed altitude of the sun (Art. 
76) and to compare this with the Greenwich Mean Time as 
shown by the chronometer. The error of the chronometer at 
some previous date and its daily gain or loss are supposed to be 
known. This is the same in principle as the method of Art. 86. 
The value of the latitude used in solving the PZS triangle must 
be that of the ship at the time the observation is made; this 
latitude must be found by correcting the latitude by observation 
at the previous noon for the run of the ship in the interval. 
This is called the latitude by " dead reckoning." On account 
of the large errors which may enter into this estimated latitude 
it is important that the observation (" time-sight "). should be 
made when the sun is near the prime vertical. 

Example. 

True alt. May 19, 1910 (P.M.) = 44 05'; G. M. T. = 6^ 55 io s . Lat. by 
dead reckoning = 42 oo' N; decl. at G. M. N. = 19 38' 20" N; diff. i h = 
-f- 32".?; equa. of time = 3 W 44*. i; deer, per i h = o s .i. 

L = 42 oo' sec = .1289 Decl. G. M. N. 19 38' 20" N 
D = 19 42 sec = . 0262 +32"_7 X 6^.9 = 3 46 

cos = .9252 L D = 22 18 Cor'd. decl. = 19 42' 06" N 

sin = . 6957 h - 44 05 



diff. = . 2295 log = 9. 3608 



log vers = 9.5159 Equa.G.M.N. = 3"* 44*. i 
H. A. = 3^ n m 07 s o s .i X 6^.9 = . 7 

Eq. t. = 3 43 

Cor'd Eq. t. = 3 43*. 4 



L. M. T. = 3 h 07 TO 24* 
G. M. T. = 6 55 10 



Long. W = 3^ 47 TO 46* 
= 56 56' i W. 

1 08. By a Lunar Distance. 

The accuracy of the preceding method is wholly dependent 
upon the accuracy of the chronometer giving the Greenwich 
time. With steam vessels making short trips and carrying 



NAUTICAL ASTRONOMY 173 

several chronometers this method gives the longitude with 
sufficient accuracy. In the days when commerce was carried 
on chiefly by means of sailing vessels the voyages were of long 
duration, and consequently the error of the chronometer could 
be verified only at long intervals; furthermore, the chronom- 
eters of that time were far less perfect than those of to-day, and 
their rates were subject to greater irregularities. Under these 
circumstances the method just described sometimes became 
wholly unreliable; in such cases the 
method of " lunar distance " was 
used. Although this method is 
necessarily of inferior accuracy it has 
the advantage of being entirely inde- 
pendent of the chronometer time. In 
the Nautical Almanac previous to the 
issue for 1912 there were given the 
geocentric distances of the moon 
from several bright stars, planets, and FlG 68 

the sun, for every 3^ of Greenwich 

Mean Time. If a lunar distance were measured at sea and this 
distance reduced to the centre of the earth, the corresponding 
instant of G. M. T. could be found by interpolation in these 
. tables. 

The observation requires that the altitudes of the moon and 
the sun or star should be measured simultaneously with the 
distance, and that the chronometer should be read at the same 
instant. In Fig. 68 let Z be the observer's zenith, M' the appar- 
ent and M the true position of the moon, and S f and S the appar- 
ent and true positions of the sun. The sun's apparent position 
is higher than its true position because its refraction is greater 
than its parallax. The moon's true position is higher than its 
apparent position because the parallax correction is the greater. 
The measured distance S'M' is to be reduced to the true dis- 
tance SM. In the triangle ZS'M' the three sides have been 
measured and the angle Z may be computed. Then in the 




174 PRACTICAL ASTRONOMY 

triangle ZSM the angle Z and the sides ZS and ZM are known, 
because the refraction and parallax corrections are known, and 
MS may be computed. By interpolating in the tables, the 
true G. M. T. corresponding to the instant of this observation 
may be obtained, the difference between this and the observed 
chronometer time being the error of the chronometer on G. M.T. 
The longitude may then be found by comparing the true G. M. T. 
with the local time computed from the sun's altitude. 

In the Ephemeris for 1912 the tables of lunar distances have 
been omitted, as lunar observations are no longer considered 
to be of practical value to the navigator. 

109. Azimuth of the Sun at a Given Time. 

For determining the error of the compass and for other pur- 
poses it is frequently necessary at sea to know the sun's azimuth 
at an observed instant of time. If the observed time be con- 
verted into local apparent time the azimuth Z may be computed 
by the following formulae.* 

tan \ (Z+S] = cot \ P sec | (p + co-L) cos | (p co-L) , [112] 
tan \ (Z S) = cot f P esc \ (p-\-co-L) sin | (p co-L). [113] 

In these formulae co-L is the co-latitude. In practice the azimuth 
is taken from tables computed by use of these formulae. 
Burdwood's and Davis's Azimuth Tables give the azimuth for 
each degree of P, L, and p, the former ranging from Lat. 30 to 
Lat. 60 and the latter from 30 N to 30 S. Publication 
No. 71 of the U. S. Hydrographic Office gives azimuths of the 
sun for latitudes up to 61. For finding the azimuth of an 
object having a declination greater than 24 publication No. 120 
of the Hydrographic Office may be used. 

Example. 

Find the sun's azimuth when L = 42 01' N, D = 22 47' S, P = g h 25 i8 s . 
From Publ. No. 71 for L = 42, D = 22, P = q h 20, the azimuth is N 141 40' E. 
The corresponding azimuth for L = 43 is 141 50', that is, an increase of 10' 
for i; the azimuth for L = 42 D = 23, and P = g h 20, is 142 n', or an 
increase of 31' for i of declination; for L = 42, D = 22, and P = g h 30 the 

* Napier's Analogies. 



NAUTICAL ASTRONOMY 



175 



azimuth is 143 47', or an increase of 2 07' for io m , or 12'. 7 for i m . The desired 
azimuth is therefore 141 40' + & X 10' + f X 31' + 5.3 X i2 ; .7 = 143 12'. 
The azimuth from the south point is therefore S 36 48' E. 

When the azimuth is determined for the purpose of finding the error of the com- 
pass the observation is usually taken near sunrise or sunset, which is not only a 
convenient time for making the pointings at the sun but is a favorable time for ac- 
curate determination of the azimuth. 

no. Azimuth of the Sun by Altitude and Time. 

When the altitude of the sun is observed for the purpose of 
finding the local time, the azimuth at the same instant may be 
computed by the formula 

sin Z = sin P cos D sec h. [12] 

Example. 

Find the sun's azimuth when P= 34 46'.4 (P.M.), D = 22 45' 50", h = 

17 4i' 

log sin P = 9. 75612 
log cos D = g. 96478 
log sec h = o. 02 102 



log sin Z = g. 74192 

Z =S33 3 o'.2W 

in. Sumner's Method of Determining a Ship's Position.* 

If the declination of the sun and the Greenwich Apparent 
Time are known at any instant, these two coordinates are the 
latitude and longitude respectively of a point on the earth's 
surface which is vertically under the sun's centre and which 
may be called the " sub-solar " point. If an observer were at 
the sub-solar point he would have the sun in his zenith. If 
he were located i from this point, in any direction, the sun's 
zenith distance would be i; if he were 2 away, the zenith 
distance would be 2. It is evident, then, that if an observer 
measures an altitude of the sun he locates himself on the cir- 
cumference of a circle whose centre is the sub-solar point and 
whose radius (in degrees) is the zenith distance of the sun. 
This circle could be drawn on a globe by first plotting the posi- 
tion of the sub-solar .point by means of its coordinates, and 

* This method was first described by Captain Sumner in 1843. 



176 



PRACTICAL ASTRONOMY 



then setting a pair of dividers to subtend an arc equal to the 
zenith distance (by means of a graduated circle on the globe) 
and describing a circle about the sub-solar point as a centre. 
The observer is somewhere on this circle because all positions 
on the earth where the sun has this measured altitude are located 
on this same circle. This is called a circle of position, and any 
portion of it a line of position or a Sumner line. 




FIG. 69 

Suppose that at Greenwich Apparent Time i h the sun is 
observed to have a zenith distance of 20, the declination being 
20 N. The sub-solar point is then at A . Fig. 69, and the observer 
is somewhere on the circle described about A with a radius 20. 
If he waits until the G. A. T. is 4^ and again observes the sun, 
obtaining 30 for his zenith distance, he locates himself on the 
circle whose centre is B, the coordinates being 4^ and (say) 
20 02' N, and the radius of which is 30. If the ship's position 



NAUTICAL ASTRONOMY 177 

has not changed between the observations it is either at S or 
at T; in practice there is no difficulty in deciding which is the 
correct point, on account of their great distance apart. A 
knowledge of the sun's bearing also shows which portion of the 
circle contains the point. If, however, the ship has changed its 
position since the first observation, it is necessary to allow for 
its " run " as follows. Assuming that the ship has sailed 
directly away from the sun, say a distance of 60 miles or i, 
then, if the first observation had been made while the ship was 
in the second position, the point A would be the same, but the 
radius of the circle would be 21, locating the ship on the dotted 
circle. The true position of the ship at the second observation 
is, therefore, at the intersection S'. If the vessel does not actu- 
ally sail directly away from or directly toward the sun it is a 
simple matter to calculate the increase or decrease in radius 
due to the change in the observer's zenith. 

This is in principle Sumner's method of locating a ship. 
In practice the circles would seldom have as short radii as those 
in Fig. 69; small circles were chosen only for convenience in 
illustrating the method. On account of the long radius of the 
circle of position only a small portion of this circle can be shown 
on an ordinary chart; in fact, the portion which it is necessary 
to use is generally so short that the curvature is negligible and 
the line of position appears on the chart as a straight line. In 
order to plot a Sumner line on the chart, two latitudes may be 
assumed between which the actual latitude is supposed to lie; 
and from these, the known declination, the observed altitude, 
and the chronometer reading, two longitudes may be computed 
(Art. 107), one for each of the assumed latitudes. This gives 
the coordinates of two points on the line of position by means 
of which it may be plotted on the chart. Another observation 
may be made a few hours later and the new line plotted in a 
similar manner. In order to allow for the change in the radius 
of the circle due to the ship's run between observations, it is 
only necessary to move the first position line parallel to itself 



i 7 8 



PRACTICAL ASTRONOMY 



in the direction of the ship's course and a distance equal to the 
ship's run. In Fig. 70, AB is a line obtained from a 9 A.M 
observation on the sun and by assuming the latitudes 42 and 
43. A second observation is made at 2 P.M.; between 9^ and 
2 h the ship has sailed S 75 W, 67'.* Plotting this run on the 
chart so as to move any point on AB, such as x, in the direction 
S 75 W and a distance of 67', the new position line for the first 




FIG. 70 

observation is A'B'. The P.M. line of position is located by 
assuming the same latitudes, 42 and 43, the result being the 
line CD. The point of intersection S is the position of the ship 
at the time of the second observation. Since the bearing of 
the sun is always at right angles to the bearing of the Sumner 
line, it is evident that one point and the bearing would be 
sufficient to locate the line on the chart. 

112. Position by Computation. 

The coordinates of the point of intersection of the lines of position may be 
calculated more precisely than they can be taken from the chart. When the first 

* The nautical mile (6080.27 f eet ) is assumed to be equal to an arc of i' on 
any part of the earth's surface. 



NAUTICAL ASTRONOMY 



179 



altitude is measured the navigator assumes a latitude which is near the true lati- 
tude, and from this calculates the corresponding longitude. The approximate 
azimuth of the sun is also calculated from the same data. (Equa. [30].) The 
run of the ship up to the time of the second observation is reduced to the difference 
in latitude and the difference in longitude from the known course and speed of 
the vessel. These two differences are applied as corrections to the assumed lati- 
tude and the calculated longitude. This places the ship on the new Sumner line 
(corresponding to A'B', Fig. 70). When the P.M. observation is made the corrected 
latitude is used in computing the new longitude. The result of these two obser- 
vations is shown in Fig. 71. Point A is the first position; A' is the position of A 

A 




FIG. 71 

corrected for the run of the ship; B is the position obtained by the P.M. observation 
using the latitude of A'. A'B is then the discrepancy in the longitudes, due to 
the fact that a wrong latitude has been chosen, and is the base of a triangle the 
vertex of which, C, is the true position of the ship. The base angles A' and B 
are the azimuths of the sun at the times of observation. In practice this triangle 
is often solved as follows:* Dropping a perpendicular from C to A'B, forming 
two right triangles, 

Bd = Cd cot Z 2 , 
and 

A'd = CdcotZi, 
or 

A/> 2 = AL cot 2, 

A/>i = AL cot Zi, 

* See A. C. Johnson's " On Finding the Latitude and Longitude in Cloudy 
Weather." 



180 PRACTICAL ASTRONOMY 

where AL = the error in latitude and Ap the difference in departure. In order 
to express Bd and A'd as differences in longitude (AM) it is necessary to introduce 
the factor sec L, giving 

AM 2 = AL sec L cotZ 2 , [114] 

AM 2 = AL sec L cot Z\. [115] 

To find AL, the correction to the latitude, the distance A'B = AM 2 + AMi 
is known, the factors sec L cot Z may be found from the approximate latitude 
and the sun's azimuths, therefore 

A'B 
sec L cot Zi + sec L cot Z 2 

Having found AL, the corrections AMi and AM 2 are found by [114] and [115]. 
Since the factors sec L cot Z are, in practice, taken from a table and the operations 
indicated in Equa. [114], [115], and [116] are easily performed with the slide rule 
the method is in reality a rapid one. 

In the above description the observations are taken one in the forenoon and one 
in the afternoon, but any two observations, provided the position lines intersect at an 
angle over 30, will give good results. If the observations are both on the same 
side of the meridian the denominator of [116] becomes the difference of the factors 
instead of the sum. If two objects can be observed at the same time, and their 
bearings differ by 30 or more, the position of the ship is obtained at once, since 
there is no run of the ship to be applied. This observation might be made upon 
two bright stars or planets at twilight. It should be observed that the accuracy 
of this method depends upon the accuracy of the chronometer, just as in the 
methods of Art. 107. 

One of the great advantages of this method is that even if only one observation 
can be taken it may be utilized to locate the ship along a (nearly) straight line; 
and this is often of great value. For example, if the first position line is found 
to pass directly through some point of danger, then the navigator knows the 
bearing of the point, although he does not know his distance from it; but with 
the single observation he is able to avoid the danger. In case it is a point which 
it is desired to reach, the true course which the ship should steer is at once known . 
The following example illustrates the method of computing the coordinates of the 
point of intersection. 



NAUTICAL ASTRONOMY l8l 

Example. 

Location of ship by Sumner's Method, Jan. 4, 1910. 

At chronometer time i h 12 48* the sun's lower limb is observed to be 15 53' 
30"; index corr. = o"; height of eye = 36 ft.; chronometer is 15* fast of G. M. T. 
Latitude by dead reckoning, 42 oo' N. At chronometer time 6 h 05"* 46 s the alti- 
tude of the sun's lower limb = 17 33' 30"; index corr. = o"; height of eye, 36 ft; 
chronometer, 15 s fast. The run between the observations was i' N and 60' W. 

First Observation 

Semidiam. = + 16' 17" Declination at G. M. N. = 22 47' 22". 3 

dip = - 5 53 + i5"-i5 X i h .2 + 18 . 2 

r&p = 3 14 

Decl. = 22 47' 04". i 

Corr. = + 07' 10" p = 112 47' 04". i 

Obs. Alt. = 15 53' 30" 



True Alt. = 16 oo' 40" 



L = 42 oo' seco. 12893 Equa. t. = 4 49*. 80 

p = 112 47 csco.03528 i s .i45 X i h .2 = i .37 
h = 16 oo . 7 

Cor'd Eq. t. = 4"* 51*. 17 



170 47 .7 

5 = 85 23'. 8 cos 8. 90448 Sun's Az.* = S 36 48' E 

s h = 69 23 . i sin 9. 97126 cot Az. X sec Lat. = i. 80 

2)9. 03995 



log sin P = g. 51998 
% P = 19 20'. 2 
P = 38 40'. 4 

= 2 h 34 m 41*. 6 

L. A. T. = g h 25 i8 s . 4 

Eq. t. = 451.2 



L. M. T. = g h 30 m 09*. 6 
G. M. T. = i 12 33 



Long. = 3^ 42 23". 4 Lat. = 42 oo' 

= 55 35'- 8 Run = i' 

Run = + 60' 

Cor'd Lat. = 42 01' 



Cor'd Long. = 56 35'. 8 



* By table or by Equa. [30]; see Art. 93, p. 155. 



l82 



semidiam. = 
dip = 

r&p = 

Con. = 
Obs. Alt. = 



PRACTICAL ASTRONOMY 
Second Observation 



+ i6' 17" Decl. at G. M. N. = 

- 5 53 + I5"i5 X 6*.i = 

- 2 54 

Decl. = 

+ /30" p = 

33 30 



- 22 47' 22". 3 

+ i 32 .4 

- 22 45' 49". 9 
112 45' 49". 9 



True Alt. = 17 41' <*>' 



L = 42 01' sec o. 1 2904 

p = 112 45-8 CSC O.O3522 

h= 17 41 



"Eq. t. = 4 498. 80 
. 145 X 6.i = 6 . 98 



Cor'd Eq. t. = 4 56*. 78 
172 27 . 8 

s = 86 13. 9 cos 8. 81771 Sun's Az.* = S 33 30' W 

s h = 68 32'. 9 sin 9. 96883 cot Az. X sec Lat. = 2. 03 



log sin P = 9. 47540 
I P = 17 23'. 2 

P=3446'. 4 
L. A. T. = 2* i9 os 8 . 6 
Eq. t. = 4 56 . 8 



L. M. T. = 2^ 24"* 02*. 4 
G. M.T. = 6 05 31 



Long. = 3* 4i m 28*. 6 
= 55 22'. i 



ist Long. = 56 35'. 8 19. 2 X i. 80 = 34'. 6 Corr. to ist Long. 

2d Long. = 55 22 . i 19. 2 X 2. 03 = 39 . o Corr. to 2d Long. 

Diff. = i 13'. 7 = 73 '. 7 ist Long. = 56 35'. 8 2d Long. = 55 22'.! 

Corr. = 34 . 6 Corr. = 39 .o 



- 7 



- 
i. 80 + 2 03 = IQ/ " 2 Corr> to ^ e Lat - Long- = 56 i'- 2 Long. = 56 oi'.i 

.'. Lat. = 42 20' N. /. Long. = 56 01' W. 



* By table or by Equa. [30]; see Art. 93, p. 155. 



TABLES 



184 



PRACTICAL ASTRONOMY 



TABLE I. MEAN REFRACTION. 



Barometer, 2 9. 5 inches. 



Thermometer, 50 P. 



App. Alt. 


Refr. 


App. Alt. 


Refr. 


App. Alt. 


Refr. 


App. Alt. 


Refr. 


ooo' 


33' 5i" 


10 oo' 


5' 13" 


20 OO' 


2' 36" 


35 oo' 


l' 2l" 


30 


28 ii 


3 


4 59 


3 


2 3 2 


36 oo 


I 18 


I OO 


23 Si 


II OO 


4 46 


21 OO 


2 28 


37 oo 


I 16 


30 


20 33 


30 


4 34 


30 


2 24 


38 oo 


I 13 


2 OO 


i7 55 


12 OO 


4 22 


22 00 


2 20 


40 oo 


I 08 


3 


15 49 


30 


4 12 


3 


2 I 7 


42 oo 


I 03 


3 


14 07 


13 oo 


4 02 


23 oo 


2 14 


44 oo 


o 59 


3 


12 42 


30 


3 54 


3 


2 II 


46 oo 


o 55 


4 oo 


II 31 


14 oo 


3 45 


24 oo 


2 08 


48 oo 


o 51 


3 


10 32 


3 


3 37 


3 


2 05 


5 


o 48 


5 


9 4o 


15 


3 30 


25 oo 


2 02 


5 2 oo 


o 45 


30 


8 56 


3 


3 23 


26 oo 


I 57 


54 oo 


o 41 


6 oo 


8 19 


16 oo 


3 i7 


27 oo 


I 52 


56 oo 


o 3 8 


30 


7 45 


3 


3 10 


28 oo 


i 47 


58 oo 


o 36 


7 PO 


7 i5 


17 oo 


3 05 


29 oo 


1 43 


60 oo 


o 33 


30 


6 49 


30 


2 59 


30 oo 


i 39 


65 oo 


o 27 


8 oo 


6 26 


18 oo 


2 54 


31 oo 


35 


70 oo 


O 21 


3 


6 05 


30 


2 49 


32 oo 


3i 


75 oo 


IS 


9 oo 


5 46 


19 oo 


2 44 


33 


28 


80 oo 


O IO 


30 


5 29 


30 


2 40 


34 oo 


24 


85 oo 


o 05 


10 00 


5 i3 


20 oo 


2 36 


35 o 


21 


90 oo 


o oo 



TABLES 



TABLE II. FOR CONVERTING SIDEREAL INTO MEAN SOLAR 

TIME. 

(Increase in Sun's Right Ascension in Sidereal h. m. s.) 

Mean Time = Sidereal Time C'. 



Sid. 
Hrs. 


Corr. 


Sid. 
Min. 


Corr. 


Sid. 

Min. 


Corr. 


Sid. 
Sec. 


Corr. 


Sid. 
Sec. 


Corr. 


I 


m s 
9.830 


I 


8 

o. 164 


31 


s 
5-079 


I 


8 
0.003 


31 




0.085 


2 


19.659 


2 


0.328 


32 


5.242 


2 


0.005 


32 


0.087 


3 


o 29.489 


3 


0.491 


33 


5.406 


3 


O.OO8 


33 


0.090 


4 


o 39.318 


4 


0.655 


34 


5-570 


4 


O.OII 


34 


0.093 


5 


o 49.148' 


5 


0.8l9 


35 


5-734 


5 


0.014 


35 


0.096 


6 


o 58.977 


6 


0.983 


36 


5.898 


6 


0.016 


36 


0.098 


7 


I 8.807 


7 


I.I47 


37 


6.062 


7 


0.019 


37 


O.IOI 


8 


I 18.636 


8 


I.3II 


38 


6. 225 


8 


0.022 


38 


o. 104 


9 


I 28.466 


9 


1.474 


39 


6.389 


9 


0.025 


39 


o. 106 


10 


I 38.296 


10 


1.638 


40 


6-553 


10 


0.027 


40 


o. 109 


ii 


I 48.125 


ii 


I. 802 


4i 


6.717 


ii 


0.030 


4i 


O. 112 


12 


i 57-955 


12 


I .966 


42 


6.881 


12 


0.033 


42 


0.115 


13 


2 7.784 


13 


2.130 


43 


7-045 


13 


0.035 


43 


0.117 


14 


2 17.614 


14 


2.294 


44 


7.208 


14 


0.038 


44 


O.I2O 


15 


2 27.443 


IS 


2-457 


45 


7-372 


15 


0.041 


45 


0.123 


16 


2 37-273 


16 


2.621 


46 


7-536 


16 


O.O44 


46 


o. 126 


17 


2 47.102 


17 


2.785 


47 


7.700 


17 


0.046 


47 


0.128 


18 


2 5 6 -93 2 


18 


2.949 


48 


7.864 


18 


0.049 


48 


0.131 


IQ 


3 6 -762 


19 


3-ii3 


49 


8.027 


19 


0.052 


49 


0.134 


20 


3 16.591 


20 


3-277 


50 


8.191 


20 


0-055 


50 


0-137 


21 


3 26.421 


21 


3-440 


Si 


8-355 


21 


0.057 


Si 


0.139 


22 


3 36.250 


22 


3.604 


52 


8.519 


22 


O.o6o 


52 


o. 142 


23 


3 46.080 


23 


3.768 


53 


8.683 


23 


0.063 


53 


0.145 


24 


3 55-909 


24 


3-93 2 


54 


8.847 


24 


O.O66 


54 


0.147 






25 


4.096 


55 


9.010 


25 


0.068 


55 


0.150 






26 


4-259 


56 


9.174 


26 


O.O7I 


56 


0.153 






27 


4-423 


57 


9.338 


27 


0.074 


57 


0.156 






28 


4-587 


58 


9.502 


28 


0.076 


58 


0.158 






29 


4-751 


59 


9.666 


29 


0.079 


59 


0.161 






30 


4.915 


60 


9.830 


30 


0.082 


60 


o. 164 



i86 



PRACTICAL ASTRONOMY 



TABLE III. FOR CONVERTING MEAN SOLAR INTO SIDEREAL 

TIME. 

(Increase in Sun's Right Ascension in Solar h. m. s.) 
Sidereal Time = Mean Time + C. 



0) v 

o 

%x 


Corr. 


B 

3* 


Corr. 


rt C 

Ii 


Corr. 


c . 

$ Z 
S w 


Corr. 


C . 
cd o 
<u v 
g W 


Corr. 


I 


m s 
o 9.856 


i 


s 
0. 164 


31 


s 
5-093 


i 


s 
0.003 


31 


s 

0.085 


2 


o 19.713 


2 


0.329 


32 


5.257 


2 


0.005 


32 


0.088 


3 


o 29.569 


3 


0-493 


33 


5-421 


3 


0.008 


33 


0.090 


4 


o 39.426 


4 


0.657 


34 


5-585 


4 


O.OII 


34 


0.093 


5 


o 49.282 


5 


0.821 


35 


5-75 


5 


0.014 


35 


0.096 


6 


o 59-139 


6 


0.986 


36 


5-914 


6 


0.016 


36 


0.099 


7 


I 8.995 


7 


1.150 


37 


6.078 


7 


0.019 


37 


O.IOI 


8 


I 18.852 


8 


1-3*4 


38 


6.242 


8 


O.O22 


38 


o. 104 


9 


I 28.708 


9 


1.478 


39 


6.407 


9 


0.025 


39 


o. 107 


10 


I 38-565 


10 


1.643 


40 


6-571 


10 


O.O27 


40 


O.IIO 


ii 


I 48 . 42 1 


ii 


1.807 


4i 


6-735 


ii 


0.030 


4i 


O. 112 


12 


I 58.278 


12 


1.971 


42 


6.900 


12 


0-033 


42 


O.II5 


13 


2 8.134 


13 


2.136 


43 


7.064 


13 


0.036 


43 


0.118 


14 


2 17.991 


14 


2.300 


44 


7.228 


14 


0.038 


44 


O. I2O 


15 


2 27.847 


IS 


2.464 


45 


7-39 2 


15 


0.041 


45 


0.123 


16 


2 37.704 


16 


2.628 


46 


7-557 


16 


O.O44 


46 


o. 126 


17 


2 47.560 


17 


2 -793 


47 


7.721 


17 


0.047 


47 


0.129 


18 


2 57-417 


18 


2 -957 


48 


7.885 


18 


0.049 


48 


0.131 


19 


3 7-273 


19 


3.121 


49 


8.049 


19 


0.052 


49 


0.134 


20 


3 17.129 


20 


3-285 


50 


8.214 


20 


0-055 


50 


0.137 


21 


3 26.986 


21 


3-45 


Si 


8.378 


21 


0.057 


51 


o. 140 


22 


3 36.842 


22 


3-614 


52 


8.542 


22 


O.o6o 


52 


o. 142 


23 


3 46.699 


23 


3-77 8 


53 


8.707 


23 


0.063 


53 


0.145 


24 


3 56.555 


24 


3-943 


54 


8.871 


24 


O.O66 


54 


o. 148 






25 


4.107 


55 


9-035 


25 


0.068 


55 


0.151 






26 


4.271 


56 


9.199 


26 


0.071 


56 


0.153 






27 


4-435 


57 


9-364 


27 


0.074 


57 


o. 156 






28 


4.600 


58 


9.528 


28 


0.077 


58 


o. 160 






29 


4.764 


59 


9.692 


29 


0.079 


59 


o. 162 






3 


4.928 


60 


9.856 


30 


0.082 


60 


o. 164 



TABLES 

TABLE IV. 

PARALLAX SEMIDIAMETER DIP. 



I8 7 



(A) Sun's parallax. 


(C) Dip of the sea horizon. 


Sun's altitude. 


Sun's parallax. 


Height of eye 
in feet. 


Dip of sea 
horizon. 


O 


9" 


I 


o' 59" 


IO 


9 


2 


i 23 


20 


8 


3 


i 42 


3 


8 


4 


i 58 


40 


7 


5 


2 II 


5 


6 


6 


2 24 


60 


4 


7 


2 36 


70 


3 


8 


2 46 


80 


2 


9 


2 56 


90 





IO 


3 06 






ii 

12 


3 15 

"? 24 


(B) Sun's semidiameter. 


13 


O *"T 

3 32 




14 


3 40 


Date. 


Semidiameter. 


15 

16 


3 48 
3 55 






I 7 


4 02 


Jan. i 


1 6' 18" 


18 


4 09 


Feb. i 


16 16 


19 


4 16 


Mar. i 


16 10 


20 


4 23 


Apr. i 


16 02 


21 


4 29 


May i 


i5 54 


22 


4 3 6 


June i 


15 48 


2 3 


4 42 


July ! 


15 46 


2 4 


4 48 


Aug. i 


I 5 47 


2 5 


4 54 


Sept. i 


15 53 


26 


5 


Oct. i 


16 01 


27 


5 6 


Nov. i 


1 6 09 


28 


5 ii 


Dec. i 


16 15 


2 9 


5 i7 






3 


5 22 






35 


5 48 






40 


6 12 






45 


6 36 






5 


6 56 






55 


7 16 






60 


7 35 






65 


7 54 






70 


8 12 






75 


8 29 






80 


8 46 






85 


9 02 






90 


9 18 






95 


9 33 






IOO 


9 48 



i88 



PRACTICAL ASTRONOMY 



TABLE V. 

Local Mean (Astronomical) Times of Culmination and Elongation of Polaris 
for 1910, computed for Longitude 90 West of Greenwich and for Latitude 40 N.* 



Date. 


Upper culmi- 
nation. 


Western elon- 
gation. 


Lower culmi- 
nation. 


Eastern elon- 
gation. 


Jan. i 


6* 44 


12* 3.9 OT 


l& h 42 


A 4Q m 


Tan. i5. . 


5 48 


II 44 


17 46 


27 40 


Feb. i 


4 41 


10 76 


1 6 7,9 


22 42 


Feb. i $ 


1 46 


4i 


15 44 


21 47 


Mar. i 


2 Cl 


8 46 


14 4Q 


2O C2 


Mar. 15 


i <;6 


7 61 


12 54 


10 ^7 


Apr. i 


O 40 


6 44 


12 47 


l8 50 


Apr. i 5 . . 


27 ^O 


6 40 


II 52 


17 ^^ 


May i 


22 47 


4 4^ 


10 40 


16 ^2 


May 15 


21 Z2 


3 <>I 


9 .4 


I 1 ? ^7 


June i 


20 4? 


2 44 


8 47 


14 =50 


Tune i 5 . . 


10 ^O 


i 49 


7 S2 


17 CC 


Tuly i . . 


18 48 


o 47 


6 qo 


12 C? 


Tuly i 5 . . 


17 5,4 


2? 4Q 


; s6 


II >Q 


Aug. i 


16 47 


22 42 


4 48 


IO ^1 


Aug. i 5 . . 


15 52 


21 47 


7 64 


Q ^7 


Sept. i. 


14. 4.5 


2O 4O 


2 47 


8 50 


Sept. i5.. 


11 5O 


10 4% 


I ^2 


7 ^^ 


Oct. i 


12 47 


18 42 


o 40 


6 ^2 


Oct. 15 


II 52 


17 47 


2? 5O 


c C7 


Nov. i 


10 46 


16 41 


22 44 


4 t;o 


Nov. 15 


5O 


15 46 


21 48 


7 CC 


Dec. i 


8 47 


14 42 


2O 45 


2 tT2 


Dec. i5. . 


7 52 


17 47 


19 5o 


I ^7 













* This table may be used to find the approximate times for any year. For dates 
falling between those given in the table the times may be found by interpolation, 
the daily difference being about 4. For the method of converting this local time 
into Standard time see Art. 35. 



TABLES 



189 



TABLE VI. CORRECTION TO THE ALTITUDE OF POLARIS* 

(Equa. [80], Art. 69.) 





Latitudes. 


H.A. 


10 


15 


20 


25 


3 


35 


40 


45 


50 


55 


o 


















// 


// 


IO 
20 


o 

i 


o" 

I 


2 


2 


3 


4 


4 


5 


2 

6 


2 
7 


3 


2 


3 


4 


5 


6 


8 


9 


II 


13 


16 


40 


3 


5 


7 


9 


n 


13 


15 


18 


22 


26 


5 


5 


7 


IO 


12 


15 


1 8 


22 


26 


31 


37 


60 


6 


9 


12 


IS 


19 


2 3 


2 7 


33 


39 


47 


70 


7 


IO 


14 


18 


22 


27 


3 2 


38 


46 


55 


80 


7 


n 


IS 


20 


24 


29 


35 


42 


49 


60 


90 


8 


n 


16 


20 


25 


3 


36 


43 


5i 


61 


IOO 


7 


n 


i5 


19 


24 


29 


35 


4i 


49 


59 


no 


6 


IO 


13 


i7 


21 


26 


3i 


37 


44 


53 


1 20 


5 


8 


n 


i5 


18 


22 


26 


3i 


37 


45 


130 


4 


6 


9 


ii 


14 


17 


20 


24 


29 


35 


140 


3 


4 


6 


8 


IO 


12 


14 


17 


20 


24 


150 


2 


3 


4 


5 


6 


7 


9 


10 


12 


iS 


1 60 


I 


i 


2 


2 


3 


3 


4 


5 


6 


7 


170 





o 





I 


i 


i 


i 


i 


2 


2 



* This table is calculated for a polar distance = i 10'. An increase of i' in the 
polar distance produces an increase of about 3% in the tabulated term. The hour 
angle in the table is measured from o at upper culmination either to the east or to 
the west. 



PRACTICAL ASTRONOMY 



TABLE VII. 
VALUES OF FACTOR 112.5 X 3600 X SIN i" TAN Z e . 



Z e 


Factor. 


Ze 


Factor. 


Z e 


Factor. 


ioo' 


343 


I20' 


0457 


i4o' 


OS? 1 


01 


.0348 


21 


.0463 


4i 


577 


02 


354 


22 


.0468 


42 


0583 


3 


.0360 


2 3 


.0474 


43 


.0589 


04 


.0366 


24 


.0480 


44 


594 


5 


.0371 


2 5 


.0486 


45 


.0600 


06 


0377 


26 


.0491 


46 


.0606 


07 


3 8 3 


27 


.0497 


47 


.0611 


08 


.0388 


28 


53 


48 


.0617 


OQ 


394 


29 


.0508 


49 


.0623 


IO 


.0400 


3 


.0514 


5 


.0629 


II 


.0406 


3 1 


.0520 


Si 


.0634 


12 


.0411 


3 2 


.0526 


52 


.0640 


13 


.0417 


33 


0531 


53 


.0646 


14 


.0423 


34 


0537 


54 


.0651 


IS 


.0428 


35 


543 


55 


.0657 


16 


0434 


36 


.0548 


56 


.0663 


i? 


. 0440 


37 


0554 


57 


.0669 


18 


.0446 


38 


.0560 


58 


.0674 


19 


.0451 


39 


.0566 


59 


.0680 



GREEK ALPHABET 



Letters. 


Name. Letters. Name. 


A, a, 


Alpha 


N, v, Nu 


B , 0, 


Beta 


H, f, Xi 


r , 7, 


Gamma 


O, o, Omicron 


A 8 

, 


Delta 


n, TT, pi 


E , e , 


Epsilon 


P, /, Rho 


z , 


Zeta 


2, cr, 5, Sigma 


H,<7, 


Eta 


Trr\ 
, r, Tau 


, 0,tf, 


Theta 


T, f, Upsilon 


I,*, 


Iota 


< I ) , </>, Phi 


K,, 


Kappa 


X, x , Chi 


A,X, 


Lambda 


, ^, Psi 


M, p, 


Mu 


fi, G>, Omega 



TABLES IQI 

ABBREVIATIONS USED IN THIS BOOK 

T or V = vernal equinox. 
R. A. or Rt. Asc. = right ascension. 
D or Decl. = declination. 

p = polar distance. 
h or Alt. = altitude. 

z = zenith distance. 
P or H. A. = hour angle. 
L or Lat. = latitude. 
Long. = longitude. 
Sid. = sidereal. 
Sol. = solar. 

G. M. N. = Greenwich Mean Noon. 
G. M. T. = Greenwich Mean Time. 
G. A. T. = Greenwich Apparent Time. 
G. S. T. = Greenwich Sidereal Time. 
L. M. N. = Local Mean Noon. 
L. M. T. = Local Mean Time. 
L. A. T. = Local Apparent Time. 
L. S. T. = Local Sidereal Time. 
Eq. T. = equation of time. 
Astr. = astronomical time. 
Civ. = civil. 

E. S. T. = Eastern Standard Time. 
U. C. = upper culmination. 
L. C. = lower culmination. 
or U. L. = upper limb. 
or L. L. = lower limb. 

RL ,LL = right limb, left limb. 

* = star. 
Corr. = correction. 
I. C. = index correction, 
r. or refr. = refraction correction. 

p = parallax correction. 
s. d. = semidiameter. 
Z or Az. = azimuth. 
N, E, S, W = north, east, south, west. 



APPENDIX 

THE TIDES 
The Tides. 

The engineer may occasionally be called upon to determine 
the height of mean sea level or of mean low water as a datum 
for levelling or for soundings. The exact determination of these 
heights requires a long series of observations, but an approxi- 
mate determination, sufficiently accurate for many purposes, 
may be made by means of a few observations. In order to 
make these observations in such a way as to secure the best 
results the engineer should understand the general theory of 
the tides. 

Definitions. 

The periodic rise and fall of the surface of the ocean, caused 
by the moon's and the sun's attraction, is called the tide. The 
word " tide " is sometimes applied to the horizontal movement 
of the water (tidal currents), but in the following discussion 
it will be used only to designate the vertical movement. When 
the water is rising it is called flood tide; when it is falling it is 
called ebb tide. The maximum height is called high water; the 
minimum is called low water. The difference between the two 
is called the range of tide. 

Cause of the Tides. 

The principal cause of the tide is the difference in the force 
of attraction exerted by the moon upon different parts of the 
earth. Since the force of attraction varies inversely as the 
square of the distance, the portion of the earth's surface nearest 
the moon is attracted with a greater force than the central 
portion, and the latter is attracted more powerfully than the 
portion farthest from the moon. If the earth and moon were 
at rest the surface of the water beneath the moon would be 

192 



THE TIDES 193 

elevated as shown in Fig. 72 at A. And since the attraction 
at B is the least, the water surface will also be elevated at this 
point. The same forces which tend to elevate the surface at 
A and B tend to depress it at C and D. If the earth were 
set rotating, an observer at any point O, Fig. 72, would be 
carried through two high and two low tides each day, the approx- 
imate interval between the high and the low tides being about 
6j hours. This explanation shows what would happen if 
the tide were developed while the two bodies were at rest; but, 
owing to the high velocity of the earth's rotation, the shallow- 
ness of the water, and the interference of continents, the actual 




>Moon 



tide is very complex. If the earth's surface were covered with 
water, and the earth were at rest, the water surface at high 
tide would be about two feet above the surface at low tide. 
The interference of continents, however, sometimes forces the 
tidal wave into a narrow, or shallow, channel, producing a 
range of tide of fifty feet or more, as in the Bay of Fundy. 

The sun's attraction also produces a tide like the moon's, 
but considerably smaller. The sun's mass is much greater 
than the moon's but on account of its greater distance the ratio 
of the tide-producing forces is only about 2 to 5. The tide 
actually observed, then, is a combination of the sun's and the 
moon's tides. 



194 



PRACTICAL ASTRONOMY 



Effect of the Moon's Phase. 

When the moon and the sun are acting along the same line, at 
new or full moon, the tides are higher than usual and are called 
spring tides. When the moon is at quadrature (first or last quar- 
ter), the sun's and the moon's tides partially neutralize each other 
and the range of tide is less than usual ; these are called neap tides. 

Effect of Change in Moon's Declination. 

When the moon is on the equator the two successive high 
tides are of nearly the same height. When the moon is north 




FIG. 73 

or south of the equator the two differ in height, as is shown in 
Fig. 73. At. point B under the moon it is high water, and the 
depth is greater than the average. At B', where it will again 
be high water about 12 later, the depth is less than the average. 
This is known as the diurnal inequality. At the points E and Q, 
on the equator, the two tides are equal. 

Effect of the Moon's Change in Distance. 

On account of the large eccentricity of the moon's orbit 
the tide-raising force varies considerably during the month. 
The actual distance of the moon varies about 13 per cent, and 
as a result the tides are about 20 per cent greater when the moon 
is nearest the earth, at perigee, than they are when the moon 
is farthest, at apogee. 



THE TIDES 195 

Priming and Lagging of the Tides. 

On the days of new and full moon the high tide at any place 
follows the moon's meridian passage by a certain interval of 
time, depending upon the place, which is called the establish- 
ment of the port. For a few days after new or full moon the 
crest of the combined tidal wave is west of the moon's tide and 
high water occurs earlier than usual. This is called the priming 
of the tide. For a few days before new or full moon the crest 
is east of the moon's tide and the time of high water is delayed. 
This is called lagging of the tide. 

All of these variations are shown in Fig. 74, which was con- 
. structed by plotting the predicted times and heights from the U. S. 
Coast Survey Tide Tables and joining these points by straight 
lines. It will be seen that at the time of new and full moon the 
range of tide is greater than at the first and last quarters; at the 
points where the moon is farthest north or south of the equator 
(shown by N, S,) the diurnal inequality is quite marked, 
whereas at the points where the moon is on the equator () 
there is no inequality; at perigee (P) the range is much greater 
than at apogee (A). 

Effect of Wind and Atmospheric Pressure. 

The actual height and time of a high tide may differ consider- 
ably from the normal values at any place, owing to the weather 
conditions. If the barometric pressure is great the surface is 
depressed, and vice versa. When the wind blows steadily into 
a bay or harbor the water is piled up and the height of the tide 
is increased. The time of high water is delayed because the 
water continues to flow in after the true time of high water has 
passed; the maximum does not occur until the ebb and the effect 
of wind are balanced. 

Observation of the Tides. 

In order to determine the elevation of mean sea level, or, 
more properly speaking, of mean half-tide, it is only necessary 
to observe, by means of a graduated staff, the height of high 
and low water for a number of days, the number depending upon 



196 



PRACTICAL ASTRONOMY 




THE TIDES 197 

the accuracy desired, and to take the mean of the gauge read- 
ings. If the height of the zero point of the scale is referred to 
some bench mark, by means of a line of levels, the height of the 
bench mark above mean sea level may be computed. In order 
to take into account all of the small variations in the tides 
it would be necessary to carry on the observations for a series 
of years; a very fair approximation may be obtained, however, 
in one lunar month, and a rough result, close enough for many 
purposes, may be obtained in a few days. 

Tide Gauges. 

If an elaborate series of observations is to be made, the self- 
registering tide gauge is the best one to use. This consists of 
a float, which is enclosed in a vertical wooden box and which 
rises and falls with the tide. A cord is attached to the float 
and is connected by means of a reducing mechanism with the 
pen of a recording apparatus. The record sheet is wrapped 
about a cylinder, which is revolved by means of clockwork. 
As the tide rises and falls the float rises and falls in the box 
and the pen traces out the tide curve on a reduced scale. The 
scale of heights is found by taking occasional readings on a 
staff gauge which is set up near the float box and referred to a 
permanent bench mark. The time scale is found by means of 
reference marks made on the sheet at known times. 

When only a few observations are to be made the staff gauge 
is the simplest to construct and to use. It consists of a vertical 
graduated staff fastened securely in place, and at such a height 
that the elevation of the water surface may be read on the 
graduated scale at any time. Where the water is compara- 
tively still the height may be read directly on the scale; but 
where there are currents or waves the construction must be 
modified. If a current is running rapidly by the gauge but 
the surface does not fluctuate rapidly, the ripple caused by the 
water striking the gauge may be avoided by fastening wooden 
strips on the sides so as to deflect the current at a slight 
angle. The horizontal cross section of such a gauge is shown in 



198 



PRACTICAL ASTRONOMY 




FIG. 75 



Fig. 75. If there are waves on the surface of the water the height 

will vary so rapidly that accurate readings cannot be made. In 

order to avoid this difficulty a 
glass tube about f inch in di- 
ameter is placed between two 
wooden strips (Fig. 76), one of 
which is used for the graduated 

scale. The water enters the glass tube and stands at the height 

of the water surface outside. In order to check sudden varia- 
tions in height the water is allowed to enter this tube only 

through a very small tube (i mm inside diameter) placed in a 

cork or rubber stopper at the lower end 

of the large tube. The water can rise 

in the tube rapidly enough to show the 

general level of the water surface, but 

small waves have practically no effect 

upon the reading. For convenience the 

gauge is made in sections about three 

feet long. These may be placed end to 

end and the large tubes connected by 

means of the smaller ones passing 

through the stoppers. In order to read 

the ' gauge at a distance it is convenient 

to have a narrow strip of red painted 

on the back of the tube or else blown 

into the glass.* Above the water surface 

this strip shows its true size, but below 

the surface, owing to the refraction of 

light by the water, the strip appears 

several times its true width, making 

it easy to distinguish the dividing line. 

Such a gauge may be read from a considerable distance by 

means of a transit telescope or field glasses. 




FIG. 76 



* Tubes of this sort are manufactured for use in water gauges of steam boilers. 



THE TIDES 199 

Location of Gauge. 

The spot chosen for setting up the gauge should be near the 
open sea, where the true range of tide will be obtained. It 
should be somewhat sheltered, if possible, against heavy seas. 
The depth of the water and the position of the gauge should be 
such that even at extremely low or extremely high tides the 
water will stand at some height on the scale. 

Making the Observations. 

The maximum and minimum scale readings at the times of 
high and low tides should be observed, together with the times 
at which they occur. The observations of scale readings should 
be begun some thirty minutes before the predicted time of high 
or low water, and continued, at intervals of about 5 m , until a 
little while after the maximum or minimum is reached. The 
height of the water surface sometimes fluctuates at the time 
of high or low tide, so that the first maximum or minimum 
reached may not be the true time of high or low water. In 
order to determine whether the tides are normal the force and 
direction of the wind and the barometric pressure may be 
noted. 

Reducing the Observations. 

If the gauge readings vary so that it is difficult to determine 
by inspection where the maximum or minimum occurred, the 
observations may be plotted, taking the times as abscissae and 
gauge readings as ordinates. A smooth curve drawn through 
the points so as to eliminate accidental errors will show the posi- 
tion of the maximum or minimum point. (Figs, yya and yyb.) 
When all of the observations have been worked up in this way 
the mean of all of the high-water and low-water readings may 
be taken as the scale reading for mean half-tide. There should 
of course be as many high-water readings as low- water readings. 
If the mean half- tide must be determined from a very limited 
number of observations, these should be combined in pairs 
in such a way that the diurnal inequality does not introduce 
an error. In Fig. 78 it will be seen that the mean of a and b, 



2OO 



PRACTICAL ASTRONOMY 



or the mean of c and d, or e and/, will give nearly the mean half- 
tide; but if b and c, or d and e, are combined, the mean is in 



14.9 



14.7 



14.6 



14.5 



HIGH WATER 

MACHIAS BAY, ME. 

JUNE 8, 1905. 



Eastern Time 
FIG. yya 

Eastern Time 





1.80 



FIG. 



one case too small and in the other case too great. The proper 
selection of tides may be made by examining the predicted 
heights and times given in the tables issued by the U. S. Coast 



THE TIDES 201 

and Geodetic Survey. By examining the predicted heights the 
exact relation may be found between mean sea level and the 
mean half-tide as computed from the predicted heights corre- 
sponding to those tides actually observed. The difference be- 
tween these two may be applied as a correction to the mean 
of the observed tides to obtain mean sea level. For example, 
suppose that the predicted heights at a port near the place of 
observation indicate that the mean of a, b, c, d, e, and/ is 0.2 ft. 




FIG. 78 

below mean sea level. Then if these six tides are observed and 
the results averaged, a correction of 0.2 ft. should be added to 
the mean of the six heights in order to obtain mean sea level. 

Prediction of Tides. 

Since the local conditions have such a great influence in 
determining the tides at any one place, the prediction of the 
times and heights of high and low water for that place must be 
based upon a long series of observations made at the same point. 
Tide Tables giving predicted tides for one year are published 



202 



PRACTICAL ASTRONOMY 



annually by the United States Coast and Geodetic Survey; 
these tables give the times and heights of high and low water 
for the principal ports of the United States, and also for many 
foreign ports. The method of using these tables is explained 
in a note at the foot of each page. A brief statement of the 
theory of tides is given in the Introduction. 
. The approximate time of high water at any place may be 
computed from the time of the moon's meridian passage, pro- 
vided we know the average interval between the moon's transit 
and the following high water, i.e., the " establishment of the 
port." The mean time of the moon's transit over the meridian 
of Greenwich is given in the Nautical Almanac for each day, 
together with the change per hour of longitude. The local 
time of transit is computed by adding to the tabular time the 
hourly change multiplied by the number of hours in the west 
longitude; this result, added to the establishment of the port, 
gives the approximate time of high water. The result is nearly 
correct at the times of new and full moon, but at other times 
is subject to a few minutes variation. 



INDEX 



Aberration of light, 12 
Adjustment of transit, 83, 87 
Almucantar, 15, 83, in 
Altitude, 19 

of pole, 27 

Angle of the vertical, 73 
Annual aberration, 13 
Aphelion, 9 
Apparent motion, 3, 28 

time, 41 
Arctic circle, 30 
Aries, first point of, 16 
Astronomical time, 44 

transit, 87, 117 

triangle, 31, 120 
Atlantic time, 56 
Attachments to transit, 86 
Autumnal equinox, 16 
Axis, 3, 8 
Azimuth, 19, 146 

mark, 146 

tables, 174 

Bearings, 19 
Besselian year, 68 

Calendar, 59 

Celestial latitude and longitude, 22 

sphere, i 
Central time, 56 
Chronograph, 93, 136, 141 
Chronometer, 92, 141, 173 

correction, 114 

Circumpolar star, 29, 103, 155 
Civil time, 44 
Co-latitude, 22 
Colure, 17 
Comparison of chronometer, 93 



203 



Constant of aberration, 13 
Constellations, 10, 98 
Cross hairs, 82, 87 
Culmination, 39, 61, 103 
Curvature, 120, 153, 158 

Date line, 58 

Dead reckoning, 172 

Declination, 20 

parallels of, 16 
Dip, 79 
Diurnal aberration, 13, 158 

inequality, 194 

Eastern time, 56 
Ebb tide, 192 
Ecliptic, 16, loo, 102 
Elongation, 36, 147 
Ephemeris, 62 

Equal altitude method, 128, 164 
Equation of time, 41 
Equator, 15 

systems, 19 
Equinoxes, 9, 16 
Errors in horizontal angle, 97 

in transit observations, 88, 118 
Eye and ear method, 96 
Eyepiece, prismatic, 87, 118 

Figure of the earth, 72 
Fixed stars, 2, 4, 68 
Flood tide, 192 
Focus, 104 

Gravity, 82 
Gravitation, 7 
Greenwich, 23, 45, 52, 172 
Gyroscope, 12 



204 



INDEX 



Hemisphere, 9 
Horizon, 14 

artificial, 91 

system, 19, 83 
Hour angle, 20, 36 

circle, 16 
Hydrographic office, 134, 174 

Index error, 84, 90, 106 
Interpolation, 69 

Lagging, 195 
Latitude, 22, 27, 103 

astronomical, geocentric and geodetic, 
72 

at sea, 170 

reduction of, 73 
Leap-year, 59 
Level correction, 158 
Local time, 45 
Longitude, 22, 45, 139 

at sea, 172 

Lunar distance, 172 

Magnitudes, 99 
Mean sun, 41, 55 

time, 41 
Meridian, 16 
Micrometer, 94, 112, 156 
Midnight sun, 30 
Moon, apparent motion of, 5 

culminations, 69, 141 
Motion, apparent, 3, 28 
Mountain time, 56 

Nadir, 14 

Nautical almanac, 43, 62 

mile, 178 
Neap tide, 194 
Nutation, 10 

Object glass, 82, 87 
Obliquity of ecliptic, 8, n, 16 



Observations, 62 
Observing, 95 

Observer, coordinates of, 22 
Orbit, 3 
of earth, 7 

Pacific time, 56 

Parallactic angle, 31, 134, 154 

Parallax, 63, 73 

correction, 74 

horizontal, 75 
Parallel of altitude, 15 

of declination, 16 

sphere, 29 
Perihelion, 9 

Phases of the moon, 144, 194 
Planets, 3, 102, 135 
Plumb-line, 14, 72 
Pointers, 100 
Pole, 3, ii, 15 

star, 99, 162 
Polar distance, 20 
Precession, 10, 101 
Prediction of tides, 201 
Primary circle, 18 
Prime vertical, 16, 122, 172 
Priming, 195 
Prismatic eyepiece, 87, 118, 152 

Radius vector, 41 
Range, 135 

of tide, 192 
Rate, 114 
Reduction to elongation, 150 

of latitude, 73 

to the meridian, 109 
Refraction, 76 

correction, 76 

effect on dip, 80 

index of, 77 
Retrograde motion, 6 
Right ascension, 20, 36 

sphere, 28 
Rotation, 3, 39 
Run of ship, 177 



INDEX 



205 



Sea-horizon, 170 
Seasons, 7 

Secondary circles, 18 
Semidiameter, 63, 78 

contraction of, 79 
Sextant, 80, 88, 170 
Sidereal day, 39 

time, 40, 49, 52 
Signs of the Zodiac, 100 
Solar day, 40 

time, 40, 49, 52 

system, 2 
Solstice, 1 6 

Spherical coordinates, 18, 31 
Spheroid, 10, 72 
Spirit level, 14 
Spring tides, 194 
Stadia hairs, 151 
Standard time, 56 
Standards of transit, 82 
Star catalogues, 69, 94 

fixed, 4 

list, 119, 130 

nearest, 2 

Striding level, 86, 87, 115, 156 
Sub-solar point, 175 
Summer, 9 

Sumner's method, 175 
Sumner line, 176, 179 
Sun, altitude of, 105 

apparent motion of, 5 

dial, 41 

fictitious, 41 

glass, 87 



Talcott's method, 69, 94, 112 
Telegraph method, 140 

signals, 136 
Tides, 192 
Tide gauge, 197 

tables, 201 
Time ball, 137 

service, 136 

sight, 172 

star, L25 
Transit, astronomical, 87 

engineer's, 78, 82 

time of, 39 

Transportation of timepiece, 139 
Tropical year, 50 

Vernal equinox, 16 
Vernier of sextant, 90 

of transit, 82 
Vertical circle, 14, 124 

line, 14 
Visible horizon, 14 

Washington, 62 

Watch correction, 114, 139 

Winter, 8 

Wireless telegraph signals, 137 

Year, 50, 68 

Zenith, 14 

distance, 19 

telescope, 94, 112 
Zodiac, 100 



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