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Full text of "Practical Astronomy"

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Observation on Polaris for Azimuth 



Frontispiece 



PRACTICAL ASTRONOMY 

A TEXTBOOK FOR ENGINEERING SCHOOLS 

AND 

A MANUAL OF FIELD METHODS 



BY 



GEORGE L. HQSMER 

Associate Professor of Geodesy, Massachusetts Institute of Technology 



THIRD EDITION 



NEW YORK * 

JOHN WILEY & SONS, INC. 
LONDON: CHAPMAN & HALL, LIMITED 

TA-5C 1 



CCflPYRIGHT, 1910, 1917 AND 1925 
BY 

GEORGE L. HOSMER 



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 t instruments, the methods applicable to 
astronomical and geodetic instruments being treated b$t briefly. 
It has been the author's intention to produce a book%hich 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 refiner 
ments. Much space has been devoted to the Measurement of 
Time because this subject seems to cause the student more 
difficulty thar \y other branch of Practical Astronomy. The 
attempt has I v { J 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 

ill 



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. 



PREFACE TO THE THIRD EDITION 



THE adoption of Civil Time in the American Ephemeris and 
Nautical Almanac in place of Astronomical Time (in effect in 
1925) necessitated a complete revision of this book. Advantage 
has been taken of this opportunity to introduce several improve- 
ments, among which may be mentioned: the change of the no- 
tation to agree with that now in use in the principal textbooks 
and government publications, a revision of the chapter on the 
different kinds of time, simpler proofs of the refraction and 
parallax formulae, the extension of the article on interpolation 
to include two and three variables, the discussion of errors by 
means of differentiation of the trigonometric formulae, the in- 
troduction of valuable material from Serial 166, U. S. Coast 
and Geodetic Survey, a table of convergence of the meridians, 
and several new illustrations. In the chapter on Nautical As- 
tronomy, which has been re-written, tfee method bf Marcq Saint- 
Hilaire and the new tables (H. O. 201 and 203) for laying down 
Sumner lines are briefly explained. An appendix on Spherical 
Trigonometry is added for convenience of reference. The size 



PREFACE V 

of the book has been reduced to make it convenient for field use. 
This has been done without reducing the size of the type. 

In this book an attempt has been made to emphasize the 
great importance to the engineer of using the true meridian and 
true azimuth as the basis for all kinds of surveys; the chapter 
on Observations for Azimuth is therefore the most important 
one from the engineering standpoint. In this new edition the 
chapter has been enlarged by the addition of tables, illustrative 
examples and methods of observing. 

Thanks are due to Messrs. C. L. Berger & Sons for the use 
of electrotypes, and to Professor Owen B. French of George 
Washington University (formerly of the U. S. Coast and Geo- 
detic Survey) for valuable suggestions and criticisms. The 
author desires to thank those who have sent notices of errors 
discovered in the book and asks their continued cooperation. 

G. L. H. 

CAMBRIDGE, MASS., June, 1924. 



CONTENTS 



CHAPTER I 
THE CELESTIAL SPHERE REAL AND APPARENT MOTIONS 



i Practical Astronomy 


PAGE 

i 


2 The Celestial Sphere . 


i 


3. Apparent Motion of the Sphere. . . . 


3 


4 The Motions of the Planets 


2 


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 




8 Precession and Nutation 


. . 10 


o. Aberration of Lieht 


12 



CHAPTER II 
DEFINITIONS POINTS AND CIRCLES OP 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. 

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 

1 6. Relation between the Two Systems of Coordinates ................ 23 

CHAPTER IV 

RELATION BETWEEN COORDINATES 

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

18. Relation between Latitude of Observer and the Declination and Alti- 

tude of a Point on the Meridian ............................... 30 

vii 



Viii CONTENTS 

ART. PAGE 

19. The Astronomical Triangle ..................................... 31 

20. Relation between Right Ascension and Hour Angle ................ 36 

CHAPTER V 
MEASUREMENT OF TIME 

21. The Earth's Rotation .......................................... 4< 



22. Transit or Culmination ........................................ 4c 

23. Sidereal Day .................................................. 40 

24. Sidereal Time ................................................. 41 

25. Solar Day .................................................... 41 

26. Solar Time ................................................ 41 

27. Equation of Time ............................................ 42 

28. Conversion of Mean Time into Apparent Time and vice versa ........ 45 

29. Astronomical Time Civil Time ................................ 46 

30. Relation between Longitude and Time .......................... 46 

31. Relation between Hours and Degrees ............................ 49 

32. Standard Time ................................................ 50 

33. Relation between Sidereal Time, Right Ascension and Hour Angle of 

any Point at a Given Instant ................................. 52 

34. Star on the Meridian .......................................... 53 

35. Mean Solar and Sidereal Intervals of Time ....................... 54 

36. Approximate Corrections ...................................... 56 

37. Relation between Sidereal and Mean Time at any Instant. ......... 57 

38. The Date Line ................... ........................... 61 

39. The Calendar ................................................ 62 

CHAPTER VI 

THE AMERICAN EPHEMERIS AND NAUTICAL ALMANAC STAR 
CATALOGUES INTERPOLATION 

40. The Ephemeris ................................................ 6^ 

41. Star Catalogues ............................................... 6* 

42. Interpolation ................................................. 7, 

43. Double Interpolation .......................................... 7; 

CHAPTER VII 
THE EARTH'S^ FIGURE CORRECTIONS TO OBSERVED ALTITUDES 

44. The Earth's Figure. . .......................................... 7 

45. The Parallax Correction ........................................ 8. 

46. The Refraction Correction ............ " .......................... 84 

47. Semidiameters ............................................... 87 

48. Dip of the Sea Horizon ........................................ 88 

49. Sequence of Corrections. ... .................................... 89 



CONTENTS IX 

CHAPTER VIII 

DESCRIPTION OF INSTRUMENTS OBSERVING 

RT. PAGE 

50. The Engineer's Transit 91 

51. Elimination of Errors 92 

52. Attachments to the Engineer's Transit Reflector 95 

-53. Prismatic Eyepiece 96 

54. Sun Glass ' 96 

55. The Portable Astronomical Transit 96 

56. The Sextant 100 

57. Artificial Horizon 103 

58. Chronometer 104 

/ 59. Chronograph 105 

60. The Zenith Telescope 105 

61. Suggestions about Observing 107 

62. Errors in Horizontal Angles 108 

CHAPTER IX 

i THE CONSTELLATIONS 

63. The Constellations no 

64. Method of Naming Stars no 

65. Magnitudes 111 

66. Constellations near the Pole nr 

67. Constellations near the Equator 112 

68. The Planets 114 

CHAPTER X 

OBSERVATIONS FOR LATITUDE 

"69. Latitude by a Circumpolar Star at Culmination 115 

70. Latitude by Altitude of the Sun at Noon 117 

71. Latitude by the Meridian Altitude of a Southern Star 119 

72. Altitudes Near the Meridian 120 

73. Latitude by Altitude of Polaris when the Time is Known 122 

74. Precise Latitudes Harrebow-Talcott Method 125 

CHAPTER XI 
OBSERVATIONS FOR DETERMINING THE TIME 

75. Observations for Local Time 127 

76. Time by Transit of a Star 127 

77. Observations with Astronomical Transit 130 



X . CONTENTS 

ART. PAGE 

78. Selecting Stars for Transit Observations 131 

79. Time by Transit of the Sun * 134 

80. Time by Altitude of the Sun 134 

81. Time by Altitude of a Star 137 

82. Effect of Errors in Altitude and Latitude 139 

83. Time by Transit of Star over Vertical Circle through Polaris 140 

84. Time by Equal Altitudes of a Star v 143 

85. Time by Two Stars at Equal Altitudes 143 

88. Rating a Watch by Transit of a Star over a Range 151 

89. Time Service 151 



CHAPTER XII 
OBSERVATIONS FOR LONGITUDE 

90. Methods of Measuring Longitude 154 

91. Longitude by Transportation of Timepiece 154 

92. Longitude by the Electric Telegraph 155 

93. Longitude by Time Signals 156 

94. Longitude by Transit of the Moon 157 

CHAPTER XIII 
OBSERVATIONS FOR AZIMUTH 

95. Determination of Azimuth 161 

96. Azimuth Mark : 161 

97. Azimuth by Polaris at Elongation .* 162 

98. Observations near Elongation 166 

99. Azimuth by Elongations in the Southern Hemisphere 166 

100. Azimuth by an Altitude of the Sun 167 

101. Observations in the Southern Hemisphere 176 

102. Most Favorable Conditions for Accuracy 178 

103. Azimuth by an Altitude of a Star near the Prime Vertical 181 

104. Azimuth Observation on a Circumpolar Star at any Hour Angle 181 

105. The Curvature Correction 184 

106. The Level Correction 184 

107. Diurnal Aberration 185 

108. Observations and Computations 185 

109. Meridian by Polaris at Culmination 188 

ito. Azimuth by Equal Altitudes of a Star 195 

ill* Observation for Meridian by Equal Altitudes of the Sun in the Fore- 
noon and in the Afternoon 196 

112. Azimuth of Sun near -Noon 197 

113, Meridian by the Sun at the Instant of Noon 199 



CONTENTS xi 

ART. PAGE 

114. Approximate Azimuth of Polaris when the Time is Known 200 

115. Azimuth from Horizontal Angle between Polaris and ft Ursw Minoris . . 205 

116. Convergence of the Meridians 206 

CHAPTER XIV 

NAUTICAL ASTRONOMY 

117. Observations at Sea 211 

Determination of Latitude at Sea: 

> 118. Latitude by Noon Altitude of the Sun 211 

119. Latitude by Ex-Meridian Altitudes 212 

Determination of Longitude at Sea: 

1 20. By the Greenwich Time and the Sun's Altitude 213 

Determination of Azimuth at Sea: 

121. Azimuth of the Sun at a Given Time 214 

Determination of Position by Means of Sumner Lines: 

122. Sumner's Method of Determining a Ship's Position 215 

123. Position by Computation 219 

124. Method of Marcq St. Hilaire 222 

125. Altitude and Azimuth Tables Plotting Charts 223 

TABLES 

I. MEAN REFRACTION 226 

II. CONVERSION OF SIDEREAL INTO SOLAR TIME 227 

III. CONVERSION OF SOLAR INTO SIDEREAL TIME 228 

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

HORIZON 229 

V. TIMES OF CULMINATION AND ELONGATION OF POLARIS 230 

VI. FOR REDUCING TO ELONGATION OBSERVATIONS MADE NEAR ELON- 

GATION 231 

VII. CONVERGENCE IN SECONDS FOR EACH 1000 FEET ON THE PARALLEL. . 232 
VIII. CORRECTION FOR PARALLAX AND REFRACTION TO BE SUBTRACTED 

FROM OBSERVED ALTITUDE OF THE SUN 233 

IX. LATITUDE FROM CIRCUM-MERIDIAN ALTITUDES OF THE SUN 234 

., 2 sin 2 k T 

X. VALUES OF m = : rr 2 35 

sin i 

FORMS FOR RECORD Observations on Sun for Azimuth 240 

GREEK ALPHABET 242 

ABBREVIATIONS USED IN THIS BOOK 243 

APPENDIX A The Tides 244 

APPENDIX B Spherical Trigonometry 255 



PRACTICAL ASTRONOMY 



CHAPTER I 

THE CELESTIAL SPHERE REAL AND APPARENT 

MOTIONS 

i* 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 
82 is S%, 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. 




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, the ratio being 
about 8000 miles to 5,600,000,000 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. 



4 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 game (left-handed) direction, The 




FIG. 2. DIAGRAM OF THE SOLAR SYSTEM WITHIN THE ORBIT OP 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 tie stars are called fixed stars to distinguish them 
from the planetsj^t^ 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 




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- 



6 



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 




JX1U XII 

FIG. 4. APPARENT PATH or 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 ^erms " east " and " west " cannot be taken 
tiO mean the saixie^tjjey 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 
plane. 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 
f 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 




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 speed 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', W, cc f 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 66f , 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 



Summer Solstice 
(June ) 




Autumnal Equinox 
(Sept. **) 

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 (D, 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, 




E 
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 in 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. iif $y$ fince the earth is rotating with a high velocity and 



THE CELESTIAL SPHERE II 

resists this attraction, the actual effect is not to change per- 
manently 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 drbit, 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 vernal equinox, F, 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-EariWTrOrbit- 



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 
occupy points i, 2 and.?. 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 tfte 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 fn 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 that 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 J5, 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, 
.CJ3, Fig. n, and while it is fal^ng 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 t;ube is to ,be held in such a way that the raindrop shall pass 
it without touching the sides, it must be held at the 



THE CELESTIAL SPHERE IJ 

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 

^ 
tan <XQ = ~ , 

where is the angular displacement and is called the "constant 
of aberration." Its value is about 20.^5 . If CB is not per- 
pendicular to AB) then 

v . A 
sin a. = ~ sin A 



or approximately 

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



tan a = sin a. = sin B, 



CHAPTER II 

DEFINITIONS POINTS AND CIRCLES OF REFERENCE 

10. The following astronomical terms are in common use and 
are necessary in defining 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, arid is shown by the plumb lin 
or indirectly by means of the spirit level (OZ, Fig. 12). 

Zenith Nadir, 

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



14 



POINTS AND CIRCLES OF REFERENCE IS 

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 
Ut 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. 



jtf PRACTICAL ASTRONOMY 

Hour Circles. 

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

The 6-hour circle is the hour circle whose plane is perpendicu- 
lar to that of the meridian. 

Parallels of Declination. 

Small circles parallel to the plane of the equator are called 
parallels of declination (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 




FIG. 12. THE CELESTIAL SPHERE 



vertical circle, It is evident that different observers will in 
general Jjave different meridians. The meridian cuts the horizon 
in the norland south points (N, 5, Fig. 12). The intersection 



POINTS AND CIRCLES OF REFERENCE if 

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 

v '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 (F, 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 ecliptic midway between the equinoxes are 
called the winter and summer solstices. 



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 

n. Spherical Coordinates. 

The direction of a point in space may be defined 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 




FIG. 13. SpflERiCAL COORDINATES 

OA, being the origin of coordinates. Pass a plane OBC 
through OC perpendicular to OA B; 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 
codrdinates 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 
rf secondaries, each passing through the two poles of the primary, 
Fhe coordinate measured from the primary is an arc of a 

T* 



SYSTEMS OF COORDINATES ON THE SPHERE K) 

secoftdary 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 




W- 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 declination of the star S is -45; 
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 



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. 16. HOUR ANGLE AND DECLINATION 

MA. For the measurement of time the hour angle may be 
counted from the upper or the lower branch of the meridian. 
These three systems are shown in the following table. 



Name. 


Primary. 


Secondaries. 


Origin of 
Coordinates. 


ist coord. 


2nd coord. 


Horizon System 


Horizon 


Vert. Circles 


South point. 


Altitude 


Azimuth 




Equator 


Hour Circles 


Vernal Equi- 


Declin. 


Rt. Ascen. 








nox. 






Equator Systems 





it 


Intersection 





Hour Angle 








of Meridian 












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, 




FIG. 17. THE OBSERVER'S LATITUDE 

EQ being the equator and 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- 
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. 1 8. 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 fcenith, 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 
apparent motion, the outer sphere revolves once*per dayman 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 co5rdinate 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 




FIG. 19. PORTION OF 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 
cbmpute the coordinates of one system from those of another. 
The mathematical relations between the spherical coordinates 
are discussed in Cha IV. * ; 



SYSTEMS OF COORDINATES ON THE SPHERE 2$ 

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 S 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 




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 
decimation is ~io. Locate a star whose right ascension is g h 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? 

4. What changes take place in the azimuth and altitude of a star during 
twenty-four hours ? 

5. What changes take place in the right ascension and declination of the ob- 
server's zenith during a day ? 

6. A person in latitude 40 N. observes a star, in the west, whose declination is 
5 N. In what order will the star pass the following three circles; (a) the 6^ circle, 
(b) the horizon, (c) the prime vertical ? 



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 equa- 
tor, and N and S the north 
and south points of the ho- 
rizon. By the definitions, CZ 
(vertical) is perpendicular to 
SN (horizon) and CP (axis) 
is perpendicular to EC (equator). Therefore the arc PN = 

arc EZ. By the defini- 
tions EZ is the declina- 
tion of the zenith, or 
the latitude, and PN is 
the altitude of the ce- 
lestial pole. Hence the 
altitude of the pole is 
always equal to the lati- 
tude of the observer. The 
same relation may be 
seen from Fig. 22, in 
which NP is the north 
pole of the earth, OH is 
the plane of the hori- 
zon, the observer being 
FIG. 22 at O, EQ is the equator, 

and OP' is a line parallel 

to C-NP and consequently points to the celestial pole. It may 
readily be shown that ECO, the observer's latitude, equals 




2$ PRACTICAL ASTRONOMY 

HOP', the altitude of the celestial pole. A person at the equator 
would 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, ^nd all stars would 
be above the horizon i2 h and below i2 h during o* - revolution 



(S.Pole) S 




N (N.Pole) 



PIG. 2$. THE RIGHT SPHERE 
Appearance of Sphere to Observer at Earth's Equator. 

of the sphere. All stars in both hemispheres would be above 
the horizon at some time every day. (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 29 

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. (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 
Appearance of Sphere to Observer at Earth's Pole 

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 tfc an 



30 PRACTICAL ASTRONOMY 

observer in the northern hemisphere. If the observer travels 
jiorth 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." 




Circumpolars 
(Never Rise) 



FIG. 25. CtRcuMPOtAR STARS 

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

The relation between the latitude of the observer and the 
declination and altitude of a point on the observer's meridian 
may be seen by referring to Fig. 26. Let A be any point on the 
meridian, such as a star or the centre of the sun, moon, or a 
planet, located south of the zenith but north of the equator; then 

EZ = 4>, the latitude* 

EA = 5, the declination 

SA = A, the meridian altitude 

ZA = f , the meridian zenith distance. 

* The Greek alphabet is given oft p. 242. 



RELATION BETWEEN COORDINATES 31 

From the figure it is evident that 

If A is south of the equator 6 becomes negative, but the same 
equation applies in this case provided the quantities are given 




FIG. 26. STAR ON THE MERIDIAN 

their proper signs. If A is north of the zenith we should have 
= 8 f [2]; but if we regard f as negative when north of 
the zenith and positive when south of the zenith, then equation 
[i] covers all cases. When the point is below the pole the same 
formula might be employed by counting the declination beyond 
90. In such cases it is usually simpler to employ the polar 
distance, p, instead of the declination. 

If the star is north of the zenith but above the pole, as at B, 
then since p = 90 6, 

<t> = h-p. [3] 

If B were below the pole we should have 

t = h + p. [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 relation 
existing among the spherical coordinates may be obtained. This 
triangle is so frequently employed in astronomy and navigation 
that it is called the " astronomical triangle " or the " PZS 
triangle." In Fig. 27 the arc PZ is the complement of the 



32 PRACTICAL ASTRONOMY 

latitude, or co-latitude; arc ZS is the zenith distance or com- 
plement of the altitude; arc PS is the polar distance or com- 
plement of the declination; the angle at 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 5 (from 
the north point), or 360 minus the azimuth, according as S is 
west or east of the meridian. The angle at S is called the paral- 
lactic angle. If any three parts of this triangle are known the 
other three may be calculated. The fundamental formulae of, 
spherical trigonometry are (see p, 257) 

cos a = cos b cos c + sin b sin c cos A, [5] 

sin a cos B = cos b sin c sin b cos c cos A, [6] 

sin a sin B = sin b sin A. [7] 

[f we put A = /, B = S, C = Z, a = 90 - h, b = 90 - *, 
r = Q 5, then these three equations become 

sin h = sin < sin 5 + cos <t> cos 5 cos / [8] 

cos h cos 5 = sin < cos 8 cos < sin 5 cos / [9] 

cos h sin 5 = cos </> sin /. [10] 

[f A = /, S = Z, C = 5, a = 90 - A, b = 90 - 5, c = 90 - 0, 
then the [6] and [7] become 

cos h cos Z = sin 6 cos cos 5 sin </> cos / [n] 

cos h sin Z = cos 6 sin /. [12] 

[f A = Z, J3 = 5, C = /, a = 90 - 5, b = 90 - 0, c = 90 - A, 

then 

sin 5 = sin < sin A + cos < cos A cos Z [13] 

cos 5 cos 5 = sin < cos A cos < sin A cos Z [14] 

cos 5 sin S = cos < sin Z. [15] 

[f ,4 = Z, B = J, C = 5, a = 90 - 5, b = 90 - A, c = 90 - 0, 
then 

cos 5 cos / = sin h cos < cos h sin cos Z. [16] 



RELATION BETWEEN COORDINATES 



33 



Other forms may be derived, but those given above will suf- 
fice for all cases occurring in the following chapters. 

The problems arising most commonly in the practice of sur- 
veying and navigation 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. 




FIG. 27. THE ASTRONOMICAL TRIANGLE 

In following formulae let 

/ = hour angle 
Z = azimuth* 
h = altitude 

* The trigonometric formulae give the interior angle of the triangle, and con- 
sequently the azimuth from the north point, unless the form of the equation is 
changed so as to give the exterior angle. 



34 PRACTICAL ASTRONOMY 

f = zenith distance 
d declination 
p = polar distance 
<t> = latitude 
and 5 = H0 + h + p). 

For computing / any of the following formulae may be used. 



sin ^ = V "* * Sm ~ } [i7l- 

w cos sin p 



cos cos (' - sn (' 



tan 



* cos </> sin p 

__ * / cos ^ sin (s 

i 



cos / = 



- 7 - ^ / ^\ 
cos (5 - p) sm (5 - <t>) 

sin A sin sin 5 

- - - - 

COS <t> COS d 



r T 
[2oJ . 



cos / -- tan <t> tan d [200] 

cos cos 5 

cos (0 5) sin A r -, 

vers / = - ~ -~ - - [21] 

cos <t> cos 5 

For computing the azimuth, Z, froni the north point either 
toward the east or the west, we have 



s . n - *) sn (j 



COS COS 



1 ^ A /COS ^ COS (3 p) r ! 

COS * Z= V cosos/ [23] 



tan * Z = i . - sn ,- { 

cos s cos (s />) 

sin 6 sin <t> sin h r , 

COS Z = - - 21.. - [25] 

COS < COS A 



RELATION BETWEEN COORDINATES 35 

cos Z = - r tan 6 tan h [2 za] 

cos <j> cos h 

cos (<}> h) sin 5 r rl 

vers Z = - - - : - [26] 

cos <t> cos /& 

Only slight changes are necessary to adapt these to the direct 
computation of Z s from the south point of the horizon. For 
example, formulae [24], [25] and [26] would take the forms 



cot t z, = I - f j 

v cos s cos (s - p) ' 

~ sin sin h sin 5 r Ol 

cos Z, = - - - - [28] 

cos cos h 

cos (0 + h) + sin 5 r 1 

vers Z, = - ^ ! ^ - -- 29 

cos cos h 

While any of these formulae may be used to determine the 
angle sought, the choice of formulae should depend somewhat 
upon the precision with which the angle is defined by the func- 
tion. 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 true. The tangent, however, on account of its 
rapid variation, always gives the angle more precisely than either 
the sine or the cosine. It will be observed that some of the for- 
mulae require the use of both logarithmic and natural functions. 
This causes no particular inconvenience in ordinary 5-place 
computations because engineer's field and office tables almost 
invariably contain both logarithmic and natural functions. If 
7-place logarithmic tables are being used the other formulae 
will be preferred. 

The altitude of an object may be found from the formulae 

sin h = cos (<t> 5) 2 cos <t> cos 5 sin 2 1 / [30] 

or sin h = cos (<t> 5) cos <t> cos 5 vers /, [300] 

which may be derived from Equa. [8]. 



36 PRACTICAL ASTRONOMY 

If the declination, hour angle, and altitude ajre given, the 
azimuth is found by 

sin Z = sin t cos 5 sec h. [31] 

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

r, sin t r -, 

tan Z = : [32! 

cos 4> tan 8 sin cos t 

This equation may be derived by dividing [12] by [n] and then" 
dividing by cos 8. 

Body on the Horizon. 

Given the latitude and declination, find the hour angle and 
azimuth when the object is on the horizon. If in Equa. [8] 
and [13] we put h = o, we have 

cos / = tan d tan < [33] 

and cosZ = sin 5 sec <. [34] 

These formulae may be used to compute the time of sunrise or 
sunset, and the sun's bearing at these times. 

Greatest Elongation. 

A special case of the PZS triangle which is of great practical 
importance occurs when a star which culminates north of the 
zenith is at its greatest elongation. When in this position the 
azimuth of the star is a maximum and its diurnal circle is tan- 
gent to the vertical circle through the star; the triangle is there- 
fore right-angled at the point S (Fig. 28). The formulae for the 
hour angle? and azimuth are 

cos / = tan < cot 5 [35] 

and sinZ = sin p sec <, [36] 

from which the time of elongation and the bearing of the star 
may be found. (See Art. 97.) 

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 equa- 



RELATION BETWEEN COORDINATES 



37 




E 



FIG. 28. STAR AT GREATEST ELONGATION (EAST). 




FIG. 29. RIGHT ASCENSION AND HOUR ANGUS 



38 PRACTICAL ASTRONOMY 

tor on the outer sphere as graduated into hour% 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 ob- 
server'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 




FIG. 30 

shows how far 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 jbold 



RELATION BETWEEN COORDINATES 39 

true for all positions of S. The general relation existing between 
these coordinates is, then, 

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

Questions and Problems 

1. What is the greatest north declination a star may have and pass the meridian 
to the 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 hori- 
zon at Boston? 

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

7. Make a sketch of the celestial sphere like Fig. 12 corresponding to a lati- 
tude of 20 south and the instant when the vernal equinox is on the eastern horizon. 

8. Derive formula [36]. 

9. Compute the hour angle of Vega when it is rising in latitude 40 North. 
10. Compute the time of sunrise on June 22, in latitude 40 N. 



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 the 
period of rotation is not absolutely invariable, yet the variation? 
are exceedingly small, and the rotation is assumed to be uniform. 
The most natural unit of time for ordinary purposes is the solar 
day, or the time corresponding to one rotation of the earth with 
respect to the sun's direction. On*account of the motion of 
the earth around the sun once a year the direction of this refer- 
ence line is continually changing with reference to the direc- 
tions of fixed stars, and the length of the solar day is not the 
true time of one rotation of the earth. In some kinds of as- 
tronomical work it is more convenient to employ a unit of time 
based upon this true time of one rotation, namely, sidereal time 
(or star time). 

22. Transit or Culmination. 

Every point on the celestial sphere crosses the plane of 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 time of transit, or 
culmination, of that point over that meridian. When it is on 
that half of the meridian containing 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 mentioned the upper transit will be under- 
stood unless the contrary is stated. 

23. Sidereal Day. 

The sidereal day is the interval of time between two suc- 
cessive upper transits of the vernal equinox over the same 

40 



MEASUREMENT OF TIME 41 

meridian. If the equinox were fixed in position the sidereal day 
as thus defined would be the true rotation period with reference 
to the fixed stars, but since the equinox has a slow (and variable) 
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 of time from the true time of one 
rotation. The sidereal day actually used in practice, however, 
is the one previously defined and not the true rotation period. 
sflThis causes no inconvenience because sidereal days are not used 
for reckoning long periods of time, dates always being givfcn in 
solar days, so this error never becomes large. The sidereal day 
is divided into 24 hours and each hour is subdivided into 60 
minutes, and each minute into 60 seconds. When the vernal 
equinox is at upper transit it is o ft , or the beginning of the side- 
real day. This may be called " sidereal noon/' 

24. Sidereal Time. 

The sidereal time at a given meridian at any specified instant 
is equal to the hour angle of the vernal equinox measured from the 
upper half of that meridian. It is therefore a measure of the 
angle through which the earth has rotated since the equinox 
was on the meridian, and shows at once the position of the sphere 
at this instant with respect to the observer's meridian. 

25. Solar Day. 

A solar day is the interval of time between two successive 
lower transits of the sun's centre over the same meridian. The 
lower transit is chosen in order that the date may change at 
midnight. The solar day is divided into 24 hours, and each hour 
is divided into 60 minutes, and each minute into 60 seconds. 
When the centre of the sun is on the upper side of the meridian 
(uppey transit) it is noon. When it is on the lower side it is 
midnight. The instant of midnight is taken as o*, or the begin- 
ning of the civil day. 

26. Solar Time. 

The solar time at any instant is equal to the hour angle of the 
sun's centre plus 180 or 12 hours; in other words it is the hour 



42 PRACTICAL ASTRONOMY 

angle counted from the lower transit. It is the.angle through 
which the earth has rotated, with respect to the sun's direc- 
tion, since midnight, and measures the time interval that has 
elapsed. 

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 
different length at different seasons. In former times when sun 
dials were considered sufficiently accurate for measuring time, 
this lack of uniformity was unimportant. Under modern con- 
ditions, which demand accurate measurement of time by the 
use of clocks and chronometers, an invariable unit of time is 
essential. The time ordinarily employed is that kept by a 
fictitious point called the " mean sun/' which is imagined to 
move at a uniform rate along the equator,* its rate of motion 
being such that it makes one apparent revolution around the 
earth in the same time as the actual sun, that is, in 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. The. time 
indicated by the position of the real sun is called apparent solar 
time and is the time shown by a sun dial, or the time obtained 
by direct instrumental observation of the sun's position. Mean 
time cannot, of course, be observed directly, but must be derived 
by computation. 

27. Equation of Time. 

The difference between mean time and apparent time at any 
instant is called the equation of time and depends upon how much 
the real sun is ahead of or behind its average position. It is given 
in ordinary almanacs as " sun fast " or " sun slow." The 
amount of this difference varies from about i^m to +i6m. 

* 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 
ecliotic) plus small periodic terms. 



MEASUREMENT OF TIME 



43 



The exact interval is "given in the American Ephemeris and in 
the (small) Nautical Almanac for specified times each day. 

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 its orbit, and (2) the fact that the real sun 
moves in the plane of the ecliptic and the mean sun in the plane 
of the equator, and equal arcs on the ecliptic do not correspond 
to equal arcs in the equator, or equal angles at the pole. 
fa In the winter, when the earth is nearest the sun, the rate of 
angular motion about the sun is greater than in the summer 
(see Art. 6). The sun will then appear to move eastward in the 
sky at a faster rate than in summer, and its daily revolution about 




the earth will therefore be slower. This delays the instant of 
apparent noon, making the solar day longer than the average, 
and therefore a sun dial will " lose time." About April i the 
sun is moving at its average rate 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 due to this cause alone is 
* * 

about 8 minutes, either + or . 

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



44 



PRACTICAL ASTRONOMY 



sun ") moves uniformly along the ecliptic at the % average rate of 
the actual 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 F, the 
vernal equinox, at the same instant that 5" starts, then the arcs 

TABLE A. EQUATION OF TIME FOR 1910. 





ist. 


xoth. 


20th. 


30th. 


January 


- 2 m 26* 


- 7 27* 


~ II m 02* 


13 22* 


February 


- 12 A.I 


TA 24. 


12 CO 




March 


12 38 


10 36 


7 48 


4 45 


April 


4 08 


I 31 


4- o 58 


*r to 

+ 2 47 


May 


+ 2 


+ 2 4.2 


+ 3 42 


4- 2 48 


June 


+ 2 31 


+ O <C7 


i 08 


3 '15 


July 


2 27 


< OI 


- 6 06 


- 6 16 


J "v 
August 


- 6 ii 


5 19 


- 3 26 


o 46 


September 


o oo 


H- 2 48 


+ 6 20 


4- 9 46 


October 


-j- 10 05 


4~ 12 45 


H" IS oi 


4- 16 13 


November 


4- 16 18 


4~ 16 02 


4" 14 26 


4- ii 28 


December 


4- ii 06 


4-72*. 


4- 2 36 


2 21 













fr6n*ara! Mnrch 






\ 






\ 






K 



FIG. 32, CORRECTION TO MEAN TIME (TO GET APPARENT TIME) 

VS and VS f are equal, since both points are moving at the same 
rate. By drawing hour circles through these two points it will 
be seen that these^ hour circles do not coincide unless the points 
S and S' happeri to be at the equinoxes or at the solstices. Since 
S and 5' are not, in general, on the same hour circle they will not 
cross the meridian at the same instant, the difference in time 



MEASUREMENT OF TIME 4$ 

being represented by the arc aS. The maximum length of aS 
is about 10 minutes of time, and may be either + or . The 
combined effect of these two causes, or the equation of time, 
is shown in Table A and (graphically) in Fig. 32. 

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

Mean time may be converted into apparent time by adding 
algebraically the equation of time for the instant. The value 
i the equation of time is given in the American Ephemeris for 
o* civil time (midnight) at Greenwich each day, together with 
the proper algebraic sign. For any other time it must be found 
by adding or subtracting the amount by which the equation has 
increased or diminished since midnight. This correction is 
obtained by multiplying the hours of the Greenwich Civil Time 
by the variation per hour. 

Example. Find the apparent time at Greenwich when the mean time (Civil) 
is 14^ 30 on Oct. 28, 1925. The equation of time at o& Greenwich Civil Time is 
-fi6 m 053.00; the variation per hour is -fo.2i8. (The values are numerically in- 
creasing.) The corrected equation of time at 14^ 30 is therefore +i6 w 05*^00 
+ i4*.S X o*.2i8 = +i6 o^.oo + 3*.i6 = i6 o8*.i6. The Greenwich Appar- 
ent Time is 14* 30 -f- i6 o8*.i6 = 14* 46"* o8*.i6. 

When converting apparent time into mean time we may pro- 
ceed in either of two ways. Since apparent time is given and 
the equation is tabulated for mean time it is first necessary to 
find the mean time with sufficient accuracy to enable us to take 
out the correct equation of time. 

Example. The Gr. Apparent Time is 14* 46^ 08*. 16 on Oct. 28, 1925; find the 
Gr. Civil Time. Subtracting the approximate equation (-|-i6 TO 05^.00) we obtain 
14* 30 03* for the approximate Gr. Civil Time. The corrected equation is there- 
fore -fi6 053.00 -f o*.2i8X 14^.5 - -fi6 o8*.i6 and the Gr. Civil Time is 
14* 3o> oo.oo. 

If preferred the Ephemeris of the sun for the meridian of 
Washington (following the star lists) may be used. The equation 
! for Washington Apparent noon Oct. 28, 1925, is i6 m 08*49; 
varia. per hour = ~o*.i96. Since the longitude of Washing- 
ton is 5* o8 w 1 5*. 78 west, the Washington Apparent Time corre- 



46 PRACTICAL ASTRONOMY 

spending to Greenwich Apparent Time 14^46^08^16 is 9* 38* 
52*.38. The equation for this instant is i6 m 08*49 + 0-196 X 
2^.35 = _ ^ o8*.o3. This fails to check the equation derived 
above (+i6 w o8 s .i6) because the method of interpolation is im- 
perfect. If a more accurate interpolation formula is used the 
results check to hundredths. 

29. Astronomical Time Civil Time. 

Previous to 1925 the time used in the Ephemens was As- 
tronomical Time, in which o h occurred at the instant of noon,~ 
the hours being counted continuously up to 24*. In this system 
the date changed at noon, so that in the afternoon the Astro- 
nomical and Civil dates agreed but in the forenoon they differed 
one day. For example: f P.M. of Jan. 3 would be 7* Jan. 3 
in astronomical time; but 3* A.M. of May n would be 15*, May 
10, when expressed in astronomical time. 

Beginning with the issue for 1925 the time used in the Ephem- 
eris is designated as Civil Time, the hours being counted from" 
midnight to midnight. The dates therefore change at mid- 
night, as in ordinary civil time, the only difference being that 
in the 24-hour system the afternoon hours are greater than 12. 

For ordinary purposes we prefer to divide the day into halves 
and to count from two zero points; from midnight to 'noon is 
called A.M. (ante meridiem), and from noon to midnight is called 
P.M. (post meridiem). When consulting the Ephemeris or the 
Nautical Almanac it isjiecessary to add i2 h to the P.M. hours 
before looking up corresponding quantities. The data found 
opposite 3* are for 3* A.M.; those opposite 15* are for 3* P.M. 

30. Relation between Longitude and Time. 

The hour angle of the sun, counted from the lower meridian 
of any place, is the solar time at that meridian, and will be 
apparent or mean according to which sun is being considered. 
The hour angle of the sun from the (lower) meridian of Green- 
. wich is the corresponding Greenwich solar time. The difference 
between the two times, or hour angles, is the longitude of the 
place east or west of Greenwich, and expressed either in degrees 



MEASUREMENT OF TIME 



47 



or in hours according as the hour angles are in degrees or in hours. 
Similarly, the difference between the local solar times of any 
two places at a given instant is their difference in longitude in 
hours, minutes, and seconds. In Fig. 33, A' AC is the Green- 
wich solar time or the hour angle of the sun from A'\ B'BC is 
the time at P or the hour angle of the sun from B f . The differ- 
ence A 'J3', or AB, is the longitude of P west of Greenwich. 



Pole 



*,wich 




FIG. 33 

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 be found if C were to represent the vernal equinox. 
[n this case the arc AC would be the hour angle of the equinox, 
3r the Greenwich Sidereal Time. BC would be the Local Sidereal 
Time at P and AB would be the difference in longitude. The 
measurement of longitude differences is therefore independent of 
the kind of time used, provided the times compared are of the 
mme kind. 

The truth of the preceding may be more readily seen by no- 
ticing that the difference in the two sidereal times, at meridian 
A and meridian B, is the interval of sidereal time during which 



48 PRACTICAL ASTRONOMY 

a star would appear to travel from A to B. Since the star re- 
quires 24 sidereal hours to travel from A to A again, the time 
interval AB bears the same relation to 24 sidereal hours that the 
longitude difference bears to 360. The difference in the mean 
solar times at A and B is the number of solar hours that the 
mean sun would require to travel from A to 5; but since the 
sun requires 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 that 
the longitude difference bears to 360. The difference in longi-, 
tude is correctly given when either time is used, provided the 
same kind of time is used for both places. 

To Change from Greenwich Time to Local Time or from Local 
Time to Greenwich Time. 

The method of changing from Greenwich to local time (and 
the reverse) is illustrated by the following examples. Remem- 
ber that the more easterly place will have the later time. 



Example i. The Greenwich Civil Time is 19* 40*" lo^.o. Required the civil 
time at a meridian 4^ 50 2i s .o West. 

Gr. Civ. T. = igft 40 io*.o 
Long. West = 4 ft 50*** 21^.0 
Loc. Civ. T. = 14^ 49 W 49^.0 

P.M. 



Example 2. The Greenwich Civil Time is 3^ oo m . Required the local civil 
time at a place whose longitude is 8* 00 West. In this instance the time at the 
place is 8& earlier than 3^, that is it is 5^ before midnight of the preceding day, or 
19^. This may also be obtained by adding 24* to the given 3* before subtracting 
the longitude difference. 

Gr. Civ. T. = 27* oo 
Long. West = Sfi _ 
Loc. Civ. T. = 19* oom 

= fit oo m P.M. 

Example 3. The Greenwich Civil Time is 20^ oo. What is the time at a place 
3 ft east of Greenwich? 

Gr. Civ. T. = so* oo> 

Long. East = 3^ oo"* 

Loc. Civ. T. 



MEASUREMENT OF TIME 49 

31. Relation between Hours and Degrees. 

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

Since 24* = 360, 

we have i h = 15, 

i m = IS 7 , 
i s = is". 

(Dividing the second equation by 15 we have 

A m _ ,o. 

4 i , 
also 4* = i'. 

By means of these two sets of equivalents, hours may be con- 
verted into degrees, and degrees into hours without writing 
down the intermediate steps. If it is desired to state the process 
as a rule it may be done as follows: To convert degrees into hours, 
divide the degrees by 15 and call the result hours; multiply the 
remainder by 4 and call the result minutes; divide the minutes 
(of an angle) by 15 and call the result minutes (of time); mul- 
tiply the remainder by 4 and call it seconds; divide the seconds 
(of angle) by 15 and call the result seconds (of time). 

Example. Convert 47 if 35" into hours, minutes and seconds. 

47 =45+ 2= 3 *o8 
if = i$ f -f 2 ' = oi^oS* 
35" - 30" + 5" - Q2*.33 

Result = 3* og m 10^.33 

To convert hours into degrees, reverse this process. 

Example. Convert 6* 35 51* into degrees, minutes, and seconds. 

6* - 90 

35 m = 32" + 3 ro - 8 45' 
51* = 48* + 3* - 1 2' 45" 

Result = 98 57' 45" 

One should be careful to use m and 5 for the minutes and sec- 
onds corresponding to hours, and ', " for the minutes and sec- 
onds corresponding to degrees. 



$0 PRACTICAL ASTRONOMY 

It should be observed that the relation 15 == i h is quite in- 
dependent of the length of time which has elapsed. A star 
requires one sidereal hour to increase its hour angle 15; the 
sun requires one solar hour to increase its hour angle 15. In 
the sense in which the term is used here i fl means primarily an 
angle, not an absolute interval of time. It becomes an absolute 
interval of time only when a particular kind of time is specified. 

32. 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 by an 
amount equal to their difference in longitude expressed in hours, 
minutes, and seconds. Before 1883 it was customary in this 
country for each large city or town to use the mean solar time of 
a meridian passing through that place, and for the smaller towns 
in that vicinity to adopt the same time. Before railroad travel 
became extensive this change of time from one place to another 
caused no great difficulty, but with the increased amount of 
railroad and telegraph business these frequent and irregular 
changes of time became so inconvenient and confusing that in 
1883 a uniform system of time was adopted. The country is 
divided into time belts, each one theoretically 15 wide. These 
are known as the Eastern, Central, Mountain, and Pacific time 
belts. All places within these belts use the mean local time of 
the 75, 90, 105, and 120 meridians respectively. The time 
of the 60 meridian is called Atlantic time and is used in the 
Eastern part of Canada. The actual positions of the dividing 
lines between these time belts depend partly upon the location 
of the large cities and the points at which the railway companies 
change their time. The lines shown in Fig. 34 are in accordance 
with the decisions of the Interstate Commerce Commission in 
1918. Wherever the change of time occurs the amount of the 
change is always exactly one hour. The minutes and seconds 
of all clocks are the same as those of the Greenwich clock. When 
it is noon at Greenwich it is f A.M. Eastern time, 6* A.M. Central 
time, $ h A.M. Mountain time, and 4* A.M. Pacific time. 



MEASUREMENT OF TIME 




2 PRACTICAL ASTRONOMY 

Standard time is now in use in the principal countries of the 
world; in most cases the systems of standard time are based on 
the meridian of Greenwich. 

Daylight Saving time for any locality is the time of a belt that 
lies one hour to the east of the place in question. If, for ex- 
ample, in the Eastern States the clocks are set to agree with those 
of the Atlantic time belt (60 meridian west) this is designated 
as daylight saving time in the Eastern time belt. 

To Change from Local to Standard Time or the Contrary. 

The change from local to standard time, or the contrary, is 
made by expressing the difference in longitude between the given 
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 the given instant of 
time. 

Example i. Find the standard time at a place 71 West of Greenwich when the 
local time is 4* 20 oo* 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 there is i6 TO earlier than the local time. 
The standard time is therefore 4^ 04 oo* P.M. 

Example 2. Find the local time at a place 91 West of Greenwich when the 
Central Standard time is p ft oo w oo A.M. The difference in longitude is i = 4. 
Since the place is west of the 90 meridian the local time is earlier. The local time 
is therefore 8* 56 oos A.M. 

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

In Fig. 35 the hour angle of the equinox, or local sidereal time, 
at the meridian of P, is the arc A V. The hour angle of the 
star S at the meridian of 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 5 = a + t [37] 

where 5 = the sidereal time at P, a = the right ascension and' 
/ = the hour angle of the star. This relation is a general one 
will be founcl to hold true for all positions, except that it 



MEASUREMENT OF TIME 



S3 



will be necessary to add 24* to the actual sidereal time when the 
sum of a and t exceed 24*. For instance, if the hour angle is 
10* and the right ascension is 20* the sum is 30*, so that the 
actual sidereal time is 6*. When the sidereal time and the right 
ascension are given and the hour angle is required we must first 
add 24* (if necessary) to the sidereal time (24* + 6* = 30*) be- 
fore subtracting the 2o h right ascension, to obtain the hour angle 
10* . If, however, it is preferred to compute the hour angle in a 
direct manner the result is the same. When the right ascension 



Pole 




FIG. 35 

is 20* the angle from V westward to the point must be 24* 20 = 
4*. This 4* added to the 6* sidereal time gives 10* for the hour 
angle as before. 

34. Star on the Meridian. 

When a star is on any meridian the hour angle of the star at 
that meridian becomes o*. The sidereal time at the place then 
becomes numerically equal to the right ascension of the star. 
This is of great practical importance because one of the best 
methods of determining the time is by observing transits of 
stars over the plane of the meridian. The sidereal time thus 



54 PRACTICAL ASTRONOMY 

becomes known at once when a star of known, right ascension 
is on the meridian. 

35. 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 motion of the sun 
makes the intervals between the sun's transits greater by nearly 




FIG. 36 

4 m than the interval between the transits of the equinox, that is, 
the solar day is nearly 4 longer than the sidereal day. In 
Fig. 36, let C and C" represent the positions of the earth on two 
consecutive days. When the observer is at it is noon at his 
meridian. After the earth makes one complete rotation (with 
reference to a fixed star) the observer will be at 0', and the side- 
real time will be exactly the same as it was the day before when 
he was at O. But the sun's direction is How CO", so the earth 
must turn through an additional degree (nearly) until the sun is 
again on this observer's meridian. This will require nearly 4* 



MEASUREMENT OF TIME 55 

additional time. Since each kind of day is subdivided into 
hours, minutes and seconds, all of these units in solar time will 
be proportionally larger than the corresponding units of si- 
dereal 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 A , the sidereal clock would immediately begin to 
gain on the solar clock, the gain being exactly proportional to 
the time elapsed, that is, about io 5 per hour, or more nearly 
3 m 56^ per day. 

In Jig. 36 C and C may be taken to represent the earth's 
position at the date of the equinox and any subsequent date. 
The angle CSC will then represent that angle through which 
the earth has revolved in the interval since March 22, and the 
angle SC'X (always equal to CSC') represents the accumulated 
difference between solar and sidereal time since March 22. 
This angle is, of course, equal to the sun's right ascension. 
The angle SC'X becomes 24* or 360 when the angle CSC 
becomes 360; in other words, at the end of one year the sidereal 
clock has gained exactly one day. 

This fact enables us to establish the exact relation between 
the two time units. It is known that the tropical year (equi- 
nox to equinox) contains 365.2422 mean solar days. Since the 
number of sidereal days is one greater we have 

366.2422 sidereal days == 365.2422 solar days, 
or i sidereal day = 0.99726957 solar days, [38] 

and i solar day = 1.00273791 sidereal days. [39]! 

Equations [38] and [39] may be written - 

24* sidereal time = (24* 3 SS'.gog) mean solar time. [40) 
24* medn solar time = (24* + 3 56^.555) sidereal time. [4] 

These equations may be put into more convenient form for 
putation by expressing the difference in time as a correction 
be applied to any interval of time to change it from one unit 



$6 PRACTICAL ASTRONOMY 

the other. If I m is a mean solar interval and /,4:he correspond- 
ing number of sidereal units, then 

I s = I m + 0.00273791 X I m [42] 

and l m = I s - 0.00273043 X I s , [43] 

These give +9 s .8s65 and g s .&2<)6 as the corresponding cor- 
rections for one hour of solar and sidereal time respectively. 
Tables II and III (pp. 227-8) were constructed by multiplying 
different values of I m and I s by the constants in Equa. [42] and * 
[43]. More extended tables (II and III) will be found in the 1 
Ephemeris. 

Example i. Assuming that a sidereal chronometer and a solar clock start 
together at a zero reading, what will be the reading of the solar clock when the 
sidereal chronometer reads 9^ 23 5i 5 .o? From Table II, opposite 9^, is the cor- 
rection i m 28*466; opposite 23 and in the 4th column is 3*. 768, and opposite 
51* and in the last column is 0^139. The sum of these three partial corrections is 
xw 32^373; 9 ft 23"* 51*. 1 32^.373 = 9^ 22 185.627, the reading of the solar 
clock. 

Example 2. Reduce 7^ io m in solar time units to the corresponding interval in 
sidereal time units. In Table III the correction for 7** is +i w 085.995; for lo" 1 it 
is -l-i s -643. The sum, i"> 103.638, added to 7** io m gives 7* n 108.638 of sidereal 
time. 

It should be remembered that the conversion of time dis- 
cussed above concerns the change of a short interval of time from 
one kind of unit to another, and is like changing a distance from 
yards to metres. When changing a long interval of time such, 
for example, as finding the local sidereal time on Aug. i, when the 
local solar time is io ft A.M., we make use of the total accumulated 
difference between the two times since March 22, which is the 
same thing as the right ascension of the mean sun. 

36. Approximate Corrections. 

Since both corrections are nearly equal to 10* per hpur, or 4 
per day, we may use these as rough approximations. For a still 
closer correction we may allow 10* per hour and then deduct 
i* for each 6* in the interval. The correction for 6* would then 
be 6 X 10* i* = 59*. The error of this correction is but 



MEASUREMENT OF TIME 57 

0^.023 per hour for solar time and 0^.004 per hour for sidereal 
time. 

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

If in Fig. 35, Art. 37, the point B is taken to represent the 
mean sun, then equation [37] becomes 

S = <x s + t s [44] 

in which a s and t s are the right ascension and hour angle of the 
mean sun at the instant considered. If the civil time is repre- 
sented by T then t s = T + 12", and 

S = a s + T + i2 ft [45] 

which enables us to find sidereal time when civil time is given 
and vice versa. If the equation is written 

5 - T = a s + I2 h , [46] 

then, since the value of a s does not depend upon the time at 
any place but only upon the absolute instant of time considered, 
it is evident that the difference between sidereal time and civil 
time at any instant is the same for all places on the earth. The 
values of S and T will be different at different meridians, but 
the difference, S T, is the same for all places at the given 
instant. 

In order that Equa. [45] shall hold true it is essential that a s 
and T shall refer to the same position of the sun, that is, to the 
same absolute instant of time. The right ascension of the mean 
sun obtained from the Ephemeris is its value of o* of Greenwich 
Civil Time. To reduce this right ascension to its value at the 
desired instant it is necessary to increase it by a correction equal 
to the product of the hourly increase in the right ascension 
times the number of hours elapsed since midnight, that is, by 
the number of hours in the Greenwich Civil Time (T). The 
hourly increase in the right ascension of the mean sun is constant 
and equal to +9^.8565 per solar hour. This is the same quantity 
that was tabulated as the " reduction from solar to sidereal 



58 PRACTICAL ASTRONOMY 

units of time " and is given in Table III. The difference be- 
tween solar and sidereal time is due to the fact that the sun's 
right ascension increases, hence the two are numerically the 
same. It is not necessary in practice to multiply the above 
constant by the hours of the civil time, but the correction may 
be looked up at once in Table III. Similarly, Table II furnishes 
at once the correction to the right ascension for any number of 
sidereal hours. Equation [45] will not hold true, therefore, 
until the above correction to a s has been made, and this cor- 
rection may be regarded either as the increase in the right ascen- 
sion or as the change from solar to sidereal time, or the contrary. 




Suppose that the sun S (Fig. 37) and a star 5' passed the 
meridian opposite M at the same instant, and that at the civil 
time T it is desired to compute the corresponding sidereal time. 
Since the sun is apparently moving at a slower rate than the 
star, it will describe the arc M'MS ( = T) while the star describes 
the arc M'MS'. The arc SS' represents the gain of sidereal 
time on mean time during the mean time interval T. , But S' 



MEASUREMENT OF TIME 59 

is the position of the sun at o h and FS" is the right ascension 
(a s ) at o". The required, right ascension is VS (or a s at time J) 
so a s at o h must be increased by the amount SS', or the correction 
from Table III corresponding to T hours. 

The right ascension of the mean sun is given in the Ephemeris 
as " sidereal time of o* G. C. T." or " right ascension of the 
mean sun + 12*." For convenience we may write Equa. [45] 
in the form 

S = (a s + i2) + T [ 4S a] 

in which it is understood that a correction (Table III) is to be 
added to reduce the interval T to sidereal units. 

If the student has difficulty in understanding the process 
indicated by Equa. [450] it may be helpful to remember that 
all the quantities represented are really angles, and may be ex- 
pressed in degrees, minutes, and seconds. If all three parts, 
the sun's right ascension + 12", the hour angle of the mean sun 
from the lower meridian (T), and the increase in the sun's right 
ascension since midnight (Table III), are expressed as angles 
then it is not difficult to see that the hour angle of the equinox 
is the sum of these three parts. 

Another view of it is that the actual sidereal time interval 
from the transit of the equinox over the upper meridian to the 
transit of the " mean sun " over the lower meridian (midnight) 
is a s + i2 h ; to obtain the sidereal time (since the upper transit 
of the equinox) we must add to this the sidereal time interval 
since midnight, which is the mean time interval since mid- 
night plus the correction in Table III. 

Example i. To find the Greenwich Sidereal Time corresponding to the Green- 
wich Civil Time 9* oo" oo* on Jan. 7, 1925. The " right ascension of the mean sun 
-f 12* " for Jan. 7, 1925, is 7^ 04 090.74. The correction in Table III for o* is 
4-i" 28^.71. The sidereal time is then found as follows: 

(a s 4- 12*) at o* = 7* 04 m 09*. 74 

T = 9 oo oo 
Table III = i 28 .71 
5 = 16* os m 



60 PRACTICAL ASTRONOMY 

If it is desired to find the civil time T when the sidereal time 
S is given, the equation is 

T = S - (a s + 12 ,. [456] 

In this instance it is not possible to correct the right ascension 
at once for the change since o h , for that is not yet known. It is 
possible, however, to find the number of sidereal hours since 
midnight, for this results directly from the subtraction of the 
tabulated value of (a s + i2 h ) from S. T is therefore found by 
subtracting from this last result the corresponding correction 
in Table II. 

Example 2. If the Greenwich sidereal time i6> 05"* 38*45 had been given, to 
find the civil time, we should first subtract from S the tabulated value of <x s + 12^, 
obtaining the sidereal interval of time since midnight. This interval less the 
correction in Table II is the civil time, T. 

S = i6^os> 38*45 

(as + i2) at o = 7 04 09 .74 

Sidereal interval = 9*01 28^71 

From Table II we find 

for 9* - i m 28*466 

for Iaa .164 

for 28*. 7 1 = .078 

total corr. = i m 28^.708 

Subtracting this from the above sidereal interval we have 
T = 9> oo> oo*. 

Example 3. If the time given is that for a meridian other than that of Green- 
wich the corresponding Greenwich time may be found at once (Art. 30) and the 
problem solved as before. Suppose that the civil time is n ft at a place 60 (- 4*) 
west of Greenwich and the date is May i, 1925. The right ascension 
33" 36X86. Then, 

Local Civil Time = u^ oo oo* 
Add Longitude W. = 4 oo oo 
Gr. Civil Time = 15^ oo oo* 
(as H- i2) at o = 14 33 36 .86 
Table III = 2 27 .85 
Gr. Sid. Time - 29^ 36 04*. 71 

Subtract Long. W. = j 

Loc. Sid. Time 2536o4*.7i 
.71 



MEASUREMENT OF TIME 6l 

Example 4. If the local sidereal time had been given, to find the local civil 
time the computation would be as follows: 

Local Sidereal Time == i h 36 04*. 71 

Add Longitude W. = 4. 

Greenwich Sidereal Time = 5* 36 043.71 (add 24^) 

(a s + I2>) at o> = 14 33 36 .86 
. Sidereal Interval = i$h 02 27^.85 
Table II = 2 27.85 

Greenwich Civil Time = 1$** oo m oo* 

Subtract Longitude W. = _4 

Local Civil Time = n^ oo m 00* 

Example 5. Alternative method. The same result may be obtained by apply- 
ing to the tabulated a s + 12^ a correction to reduce it to its value at o^ of local 
civil time. The time interval between o& at Greenwich and o ft at the given place 
is equal to the number of hours in the longitude, in this case 4 solar hours. In 
Table III we find for 4 h the correction +393.426. The value of (a s + 12^) at o& 
local time is 14** 33 36*. 86 + 39M3 = 14^ 34 W 163.29. (If the longitude is east this 
correction is sub tractive). The remainder of the computation is as follows: 

Local Civil Time = 1 1* oo m oo 3 
(as -f- i z h ) at o^ (local) = 14 34 16 .29 
Table III == i 48 .42 
Local Sidereal Time = 25^ 36 043.71 
= i& 36"* 043.71 
Conversely, 

Local Sidereal Time = i h 36*** 043.71 (add 24*) 

(as + 1 2*) at o 7 * (local) = 14 34 16 .29 

Sidereal interval = n 7 * oi m 483.42 

Table II = 01 48 .42 

Local Civil Time = nft oo m oo*.oo 

38. The Date Line. 

If a person were to start at Greenwich at the instant of noon 
and travel westward at the rate of about 600 miles per hour, i.e., 
rapidly enough to keep the sun always on his own meridian, he 
would arrive at Greenwich 24 hours later, but his own (local) 
time would not have changed at all; it would have remained 
noon all the time. His date would therefore not agree with that 
kept at Greenwich but would be a day behind it. When travel- 
ling westward at a slower rate the same thing happens except 
that it takes place in a longer interval of time. The traveller 



62 PRACTICAL ASTRONOMY 

has to set his watch back a little every day in prder to keep it 
regulated to the meridian at which his noon occurs. As a con- 
sequence, after he has circumnavigated the globe, his watch has 
recorded one day less than it has actually run, and his calendar 
is one day behind that of a person who remained 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 to avoid these discrepancies 
in dates it has been agreed to change the date when crossing the 
1 80 meridian from Greenwich. Whenever a ship crosses the 
1 80 meridian, going westward, a day is omitted from the cal- 
endar; when going eastward, a day is repeated. As a matter 
of practice the change is made at the midnight occurring 
nearest the 180 meridian. For example, a steamer leaving 
Yokohama July i6th at noon passed the 180 meridian about 
4 P.M. of the 22d. At midnight, when the date was to be 
changed, the calendar was set back one day. Her log there- 
fore shows two days dated Monday, July 22. She arrived 
at San Francisco on Aug. i at noon, having taken 17 days for 
the trip. 

The international date line actually used does not follow the 
1 80 meridian in all places, but deviates so as to avoid separating 
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. 

39. 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 dates at which the seasons occurred, the calendar was 
frequently changed in an arbitrary manner in order to keep the 
seasons in their places. This resulted in extreme confusion in 
the dates. In the year 45 B.C., Julius Caesar reformed the 
calendar and introduced one based on a year of 365! days, since 
called the Julian Calendar. The J day was provided for by 
making the ordinary year contain 365" days, but every fourth 



MEASUREMENT OF TIME 63 

year, called leap year, was given 366 days. The extra day was 
added to February in such years as were divisible by 4. 

Since the year actually contains 365^ 5* 48 46*, this differ- 
ence 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. In order to rectify this error 
Pope Gregory XIII, in 1582, ordered that the calendar should 
be corrected by dropping ten days and that future dates 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, 1900 should not be counted as leap 
years. 

This change was at once adopted by the Catholic nations. 
In England it was not adopted until 1752, at which time the 
error had accumulated to n days. Up to that time the legal 
year had begun on March 25, and the dates were reckoned ac- 
cording to the Julian Calendar. When consulting records re- 
ferring to dates previous to 1752 it is necessary to determine 
whether they are dated according to " Old Style " or " New 
Style." The date March 5, 1740, would now be written March 
16, 1741. "Double dating/ 7 such as 1740-1, is frequently 
used to avoid ambiguity. 

Questions and Problems 

1. If a sun dial shows the time to be g 7 * A.M. on May i, 1025, at a place in longi- 
tude 71 West what is the corresponding Eastern Standard Time? The corrected 
equation of time is + 2 56*. 

2. When it is apparent noon on Oct. i, 1925, at a place in longitude 76 West 
what is the Eastern Standard Time? The corrected equation of time is -f 10"* 17*. 

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 



64 PRACTICAL ASTRONOMY 

foot of the gnomon, d } is a point common to all such traces. In order to find an- 
other 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 ce on the dial. If a line be drawn in this plane making an 
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 




d 

FIG. 38 

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* line at the calculated time. 
Prove that the horizontal angle bdc is given by the relation 

tan bdc = tan t sin <, 
in which / is the sun's hour angle and < is the latitude. 

4. 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*, and then converting the solar time at each place into sidereal time. 

5. On Jan. 20, 1925, the Eastern Standard Time at a certain instant is 7^30"* 
P.M. (Civ. T. 19* 3o). What is the local sidereal time at this instant at a place 
in longitude 72 10' West? (Right ascension of Mean Sun + i2 at o* G. C. T. = 
7 h 55 m 25 s -o.) 

6. At a place in longitude 87 30' West the local sidereal time is found to be 
19* 13** ios.5 on Sept. 30, 1925. What is the Central Standard Time at this in- 
stant? (The right ascension of Mean Sun -f 12* at o* G. C. T. = 0*32^ S3*.2.) 

7. If a vessel leaves San Francisco on July 16 and makes the trip in 17 days, 
on what date will she arrive at Yokohama? 



CHAPTER VI 

THE AMERICAN EPHEMERIS AND NAUTICAL 
ALMANAC STAR CATALOGUES INTERPOLATION 

40. The Ephemeris. 

In discussing the problems of the previous chapters it has 
been assumed that the right ascensions and declinations of the 
celestial objects and the various other data mentioned are known 
to the computer. These data consist of results calculated from 
observations made with large instruments at the astronomical 
observatories, and are published by the Government (Navy 
Dept.) in the American Ephemeris and Nautical Almanac. 
This may be obtained a year or two in advance from the Super- 
intendent of Documents, Washington, D.C., price one dollar, 
It contains the coordinates of the sun, moon, planets, and stars, 
as well as the semidiameters, parallaxes, the equation of time, 
and other necessary data. 

It should be observed that the quantities given in the Almanac 
vary with the time and are therefore computed for equidistant 
intervals of solar time at some assumed meridian, usually that 
of the Greenwich (England) Observatory. 

The Ephemeris is divided into three principal parts. Part 
I contains the data for the sun, moon, and planets, at stated 
hours of Greenwich Civil Time, usually at o* (midnight), or the 
beginning of the Civil Day. (Previous to 1925 such data were 
given for Greenwich Mean Noon.) Part II contains the lists 
of star places, the data being referred to the meridian of the 
U. S. Naval Observatory at Washington (5*08 1 5^.78 west of 
Greenwich); the instant being that of transit. Part III con- 

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

65 



66 PRACTICAL ASTRONOMY 

tains data needed for the prediction of eclipses, occupations, 
etc. At the end of the volume will be found a "series of tables 
of particular value to the surveyor. 

There is also published a smaller volume entitled The American 
Nautical Almanac* which contains data for the sun, moon, 
and stars referred to the meridian of Greenwich. The arrange- 
ment of the tables is somewhat different from that given in the 
Ephemeris. This almanac is intended primarily for the use of 
navigators. 

Whenever the value of a coordinate, or other quantity, is 
given in the Ephemeris, it is stated for a particular instant of 
Greenwich (or Washington) time, and the rate of change, or 
variation per hour, of the quantity is given for the same instant. 
These rates of change are the differential coefficients of the 
tabulated functions. If the value of the quantity is desired for 
any other instant it is essential that the Greenwich time for that 
instant be known. The accuracy with which this time must 
be known will depend upon how rapidly the coordinate is vary- 
ing. If the time given is local time it must be converted into 
Greenwich time as explained in Chapter V. 

On p. 67 is a sample page taken from the Ephemeris for 
1925. On p. 69 are given portions of the table of " mean 
places " of stars, both circumpolars and others. On pp. 70 
and 71 are extracts from the tables of " apparent places " in 
which the coordinates are given for every day for close circum- 
polars and for every 10 days for other stars. The precession of 
the equinoxes causes the right ascension of close circumpolar 
stars to vary much more rapidly and more irregularly than for 
stars nearer the equator; the coordinates are therefore given 
at more frequent intervals. On p. 72 are extracts from the 
Nautical Almanac and The Washington Tables of the Ephemeris 
for 1925. 

In ?art II of the Ephemeris will be found a table entitled 

* Sold %y the Superintendent of Documents, Washington, D. C., for 15 
cents. 



THE AMERICAN EPHEMERIS AND NAUTICAL ALMANAC 67 



SUN, 1925 
FOR o& GREENWICH CIVIL TIME 



Date 


s 

? 1 

r 


Apparent 
Right 
Ascension 


Var. 
Hour 


Apparent 
Declina- 
tion 


Var. 
Hour 


Semi- 
diam- 
eter 


Hor. 
Par. 


Equation 
of Time. 
App. - 
Mean 


Var. 
Hour 


Sidereal 
Time. 
Right As- 
cension of 
Mean Sun, 
+12* 






km s 


5 


'0 / II 


// 


// 





m s 


5 


h m s 


Jan. i 


Th 


18 43 si 25 


n 049 


-23 3 51.8 


+11 57 


16 17.80 


8.95 


- 3 20 85 


I 19} 


6 40 30.40 


2 


Fr 


18 48 16 27 


ii 035 


22 59 04 


12 72 


16 17 91 


8 95 


3 49 32 


1. 179 


6 44 26.95 


3 


Sa 


18 52 40 94 


11.020 


22 53 41 4 


13 86 


16 17 91 


8 95 


4 17-43 


1.163 


6 48 23.51 


4 


Su 


18 57 5 22 


II 003 


22 47 55 o 


IS oo 


16 17 91 


8 95 


4 45 15 


1.147 


6 52 20.07 


5 


Mo 


19 i 29 09 


10 986 


22 41 41 5 


16 13 


16 17 91 


8 95 


5 12 47 


1.129 


6 56 16.62 


6 


Tu 


19 S 52 S3 


10 967 


-22 35 i i 


+I7-25 


16 17.90 


8-95 


- 5 39 35 


I. IIO 


7 013.18 


7 


We 


19 10 15 50 


10 947 


22 27 53 9 


18.35 


16 17 88 


8.95 


6 5 77 


1.091 


7 4 9-74 


8 


Th 


19 14 37 99 


10 927 


22 20 2O I 


19 46 


16 17.86 


8.95 


6 31 70 


1.070 


7 8 6.30 


9 


Fr 


19 18 59 97 


10.905 


22 12 19 9 


20 55 


16 17.83 


8.95 


6 57-12 


1.048 


7 12 2.85 


10 


Sa 


19 23 21 41 


10.882 


22 3 53 6 


21.63 


16 17-79 


8-95 


7 22 OI 


1.025 


7 IS 59-41 


ii 


Su 


19 27 42 31 


10.859 


-21 55 i 5 


+22 71 


16 17-75 


8.95 


7 46 34 


1.002 


7 19 55-97 


12 


Mo 


19 S2 2 63 


10.834 


21 45 43 7 


23 77 


16 17 71 


8 95 


8 10 10 


o 978 


7 23 52.52 


13 


Tu 


19 36 22 35 


10.809 


21 36 o 6 


24 82 


16 17 65 


8.95 


8 33-27 


o 952 


7 27 49-08 


14 


We 


19 40 41 46 


10.783 


21 25 52 4 


25.86 


16 17 59 


895 


8 55 82 


0.927 


7 31 45.64 


IS 


Th 


19 44 59-94 


10.757 


21 IS 19 4 


26 89 


16 17 53 


8.95 


9 17 75 


o 900 


7 35 42.20 


16 


Fr 


19 49 17 77 


10.729 


21 4 21 9 


+27 90 


16 17 45 


8.95 


- 9 39 03 


-o 873 


7 39 38.75 


17 


Sa 


19 53 34 94 


10.702 


20 53 02 


28 90 


16 17 38 


8 95 


9 59.65 


o 845 


7 43 35-31 


18 


Su 


19 57 51 45 


10.673 


20 41 14 6 


29 9c 


16 17 29 


8 94 


10 19 59 


o 817 


7 47 31.87 


19 


Mo 


20 2 7 . 26 


10 644 


20 29 5 4 


3087 


16 17 21 


8 94 


10 38 83 


0.787 


7 51 28.42 


20 


Tu 


20 6 22 35 


10 614 


20 i 6 33 o 


31 83 


16 17 12 


8.94 


10 57 37 


0.758 


7 55 24.98 


21 


We 


20 10 36.73 


10 584 


-20 3 37 7 


+32 77 


16 17 02 


8.94 


ii 15 20 


-o 727 


7 59 21-53 


22 


Th 


20 14 50.37 


10 553 


19 50 19 9 


33 70 


16 16 92 


8-94 


II 32 28 


o 696 


8 318.09 


23 


Fr 


20 19 3.26 


10 521 


19 36 39 9 


34 62 


16 16.82 


8.94 


II 48.61 


o 665 


8 714.65 


24 


Sa 


20 23 15-37 


10 488 


19 22 38 


35 53 


16 16 71 


8.94 


12 4 17 


0.632 


8 II 11.20 


25 


Su 


20 27 26.70 


10 456 


19 8 14 7 


36 41 


16 16 60 


8-94 


12 18 95 


0.599 


815 7.76 


26 


Mo 


20 31 37 24 


10 422 


-18 S3 30 3 


+37 28 


16 16 49 


8 94 


-12 32 93 


-o 565 


8 19 4-31 


27 


Tu 


20 35 46 96 


10.388 


18 38 25.2 


38 13 


16 16 37 


8 94 


12 46 10 


o 532 


823 0.87 


28 


We 


20 39 55.87 


10.354 


18 22 59.9 


38.97 


16 16 25 


8.93 


12 58.45 


0.497 


8 26 57-42 


29 


Th 


20 44 3 95 


10.319 


18 7 14.7 


39-79 


16 16 13 


8.93 


13 9-97 


0.463 


8 30 53-98 


30 


Fr 


20 48 II 19 


10.284 


17 51 10. o 


40 60 


16 16 oo 


8.93 


13 20 66 


o 428 


8 34 50.54 


31 


Sa 


20 52 17.60 


10.249 


17 34 46.3 


+41 38 


16 15.87 


8.93 


-13 30.51 


-0.392 


8 38 47-09 


Feb. i 


Su 


20 56 23 16 


10.214 


17 18 3-9 


42.15 


16 15.74 


8-93 


13 39 51 


0.358 


8 42 43.65 


2 


Mo 


21 O 27.89 


10.179 


17 I 3-2 


42 90 


16 15.60 


8.93 


13 47-68 


0.323 


8 46 40.20 


3 


Tu 


21 4 31 78 


10.145 


16 43 44-6 


43 64 


16 15-45 


8.93 


13 55 oi 


o 288 


8 50 36.76 


4 


We 


21 8 34 83 


10. IIO 


1626 8.5 


44.36 


16 15.31 


8.93 


14 LSI 


0.253 


8 54 33-31 


5 


Th 


21 12 37 04 


10.075 


16 8 15.5 


+45 06 


16 15.15 


8.92 


-14 7-18 


0.219 


8 58 29.87 


6 


Fr 


21 16 38 43 


10.041 


15 50 5-8 


45 75 


16 14.99 


8.92 


14 12 02 


0.185 


9 2 26.42 


. 7 


Sa 


21 20 39.01 


10.007 


15 31 39-8 


46-41 


16 14.83 


8.92 


'14 16 04 


0.150 


9 622.98 


8 


Su 


21 24 38 77 


9-974 


IS 12 58-1 


47.06 


16 14.66 


8.92 


14 19 25 


0.117 


9 10 19-53 


9 


Mo 


21 28 37 74 


9-940 


14 54 0.9 


47.70 


16 14,49 


8.92 


14 21 66 


0.084 


9 14 16.08 


10 


Tu 


21 32 35-92 


9-907 


-14 34 48.7 


+48.32 


16 14.31 


8.92 


-14 23 28 


0.051 


9 18 12.64 


ii 


We 


21 36 33-31 


9 875 


14 IS 21.9 


48.91 


16 14.12 


8.92 


14 24 12 


0.019 


9 22 9.19 


12 


Th 


21 40 29.93 


9.844 


13 55 41.0 


49-50 


16 13.93 


8.91 


14 24 19 


+0.013 


926 5.75 


13 


Fr 


21 44 25 .So 


9 813 


13 35 46.2 


50.07 


16 13.74 


8.91 


14 23.51 


0.044 


930 2.30 


14 


Sa 


21 48 2O.94 


9.782 


13 IS 38 o 


50.6l 


16 13.54 


8.91 


14 22.09 


0.075 


9 33 58.86 


15 


Su 


21 52 15 34 


9-752 


12 55 16.8 


+51.15 


16 13.34 


8.91 


-14 19 94 


+0.105 


9 37 5541 


16 


Mo 


21 56 9 02 


9-722 


-12 34 43-0 


+51-66 


16 13.13 


8.91 


-14 17.07 


+0,134 


94151.96 



NOTE. o* Greenwich Civil Time is twelve hours before Greenwich Mean Noooof the same 
date. 



68 PRACTICAL ASTRONOMY 

" Moon Culminations." This table gives the data required 
in determining longitude by observing meridian* transits of the 
moon. (See Art. 94.) 

The tables at the end of the Ephemeris, already referred to, 
include : 
Table I. For finding the Latitude by an observed Altitude of 

Polaris. 

Table II. Sidereal into Mean Solar Time. 
Table III. Mean Solar into Sidereal Time. 
Table IV. Azimuth of Polaris at All Hour Angles. 
Table V. Azimuth of Polaris at Elongation. 
Table Va. For reducing to Elongation observations made near 

Elongation. 
Table VI. For finding, by observation, when Polaris passes the 

Meridian. 
Table VII. Time of Upper Culmination, Elongation, etc., and 

other tables. 
41. Star Catalogues. 

Whenever it becomes necessary to observe stars which are not 
included in the list given in the Ephemeris, their positions must 
be taken from one of the star catalogues. These catalogues 
give the mean place of each star at some epoch, such as the be- 
ginning of the year 1890, or 1900, together with the necessary 
data for reducing it to the mean place for any other year. The 
mean place of a star is that obtained by referring it to the mean 
equinox at the beginning of the year, that is, the position it 
would occupy if its place were not affected by the small periodic 
terms of the precession. 

The year employed in such reductions is that known as the 
Besselian fictitious year. It begins when the sun's mean longi- 
tude (arc of the ecliptic) is 280, that is when the right ascension 
of the mean sun is iS h 40**, which occurs about January i. After 
the catalogued position of the star has been brought up to the 
mean place at the beginning of the. given ye'ar, it must still be 
reduced to its " apparent place/' for the exact date of the ob- 



THE AMERICAN EPHEMERIS AND NAUTICAL ALMANAC 69 



MEAN PLACES OF TEN-DAY STARS, 1925 
FOR JANUARY 0^.654, WASHINGTON CIVIL TIME 



Name of Star 


Mag- 
ni- 
tude 


Spec- 
trum 


Right 
Ascension 


Annual 
Varia- 
tion 


An- 
nual 
P.M. 


Declination 


Annual 
Varia- 
tion 


Annual 
P.M. 


33 Piscium 


4-7 

2 2 

2 4 
3-9 
5-1 

2.9 
4-5 
3-8 
4 3 
6.0 

2.9 
2.4 
6 o 
5-2 
3-7 

4-4 
4-5 
3-5 

mr. 
4.6 


Ko 
A op 



Fo 

B2 

A2 

Ko 
F8 
Gs 

Go 
Ko 
K S 
Go 

B2 

B 3 
Gs 

Ko 

Ko 
Ko 


h m s 

o i 29.826 

o 4 30 419 
o 5 9 934 
o 5 36.485 
o 6 25 013 

o 9 22.290 
o 14 24.284 
o 15 36.413 
o 16 10.633 

21 33.433 

21 50.154 
22 34.886 

o 26 12.693 
o 31 23.214 

o 32 47 044 

o 32 52.213 
o 34 35 279 
o 35 18 777 

36 14-395 
o 37 46.988 


s 

+3.0713 

3.0979 
3.1906 
3.0484 
3.1131 

+3-0875 
3-1302 
3.0567 
3 I4H 
3.0747 

+3-1856 
2.9702 
3-0623 
3 0873 
3-3338 

+3.2002 
3.1665 
3 2044 

3.3926 
2 8371 


s 

.0006 

+ 0107 
+ .0681 
+ .0096 

+ .0021 

+.0003 
-.0044 

0013 
+ .2734 
.0014 

+ .6947 
+ .0187 

+ .OOH 

+.0273 
+.0036 

+.0019 
.0172 
-[-. ono 

+ .0063 
. 0046 


/ 

-67 37.68 

+28 40 35.02 
+58 44 10 16 
46 9 40.94 
+45 39 17-71 

+14 46 o.oo 

+36 22 10.01 

- 9 14 22.41 
65 18 54,60 
4- i 31 27.67 

-77 40 35-90 
-42 42 47-83 
- 4 22 17.32 
4 o 19 60 
+53 29 3-76 

+33 18 24.20 
+28 54 17.07 
4-30 27 2.27 

+56 7 34-54 
-46 29 49-23 


+20.135 

19.878 
19-859 
19 846 
20.032 

+20.018 

19 969 
21.167 
19-933 

+20.272 
19-544 
19.913 
19.840 
19.833 

+I9-839 
19-563 
19-710 

19.763 
19.741 



+0.091 

0.163 
0.180 
-0.193 
0.004 

O.OIO 

o 017 
0.030 
+1.172 
0.023 

+0.319 
-0.403 

000 

o 017 
0.007 

o.ooo 

-0.254 

-0.097 

0,032 
0.032 


a Andromedae 
(Alpheralz) 


Cassiopeiae 


Phcenicis 


22 Andromedae.. . 

7 Pegasi 
a Andromedae. .. . 
i Ceti 


f Tucanae . 


44 Piscium 


ft Hydri 


ct Phoenicis .... 


12 Ceti . 


13 Ceti t 


i" Cassiopeia*. ..... 


v Andromedae 
Andromedae 
6 Andromedae 
a Cassiopeise 
(Schedir) f 


/u Phoenicis. 





13 Ceti, dup., 



, o".3- 



a Cassiop., var. irreg., 2 W .2, 2 m ,8. 



MEAN PLACES OF CIRCUMPOLAR STARS, 1925 
FOR JANUARY 0^.654, WASHINGTON CIVIL TIME 



43 H. Cephei 

a Ursae Mm. 

(Polaris) t 

4 G. Octantis 

Groom bridge 750 
Groombridge 944 



31 G. Mensae 

f Mensae 

51 H. Cephei 

7 G. Octantis 

25 H. Camelopar- 
dalis 



Groombridge 1119 

f Octantis 

i H. Draconis 

t Chamaeleontis. . 
30 H. Camelopar- 
dalis 



77 Octantis 

Bradley 1672 

t Octantis 

32 H. Camelop. segj 
K Octantis 



4-5 

2.1 

5.6 
6-7 
6.4 

6.2 
5.6 
5-3 

6.4 



7-0 
5-4 
4-6 
5.2 



6.3 
6-3 
5-4 
5 3 

5-6 



Ko 

F8 
Ko 
F8 
Ko 

Ao 
Aa 
Ma 



Mb 
Ao 

K 3 o 
B 3 



Ao 
Fo 
Ko 

A2 

A2 



o 58 11.014 + 7.7785 +-0737 +85 51 20.58 +19.398 -0.004 



I 34 13 . 
I 41 32.636 

4 12 24 

5 37 43-075 



588 +31.1184 
V - 3.6690 . 

065 +17.7589 + 
+18.8025 ' 



5 44 41.365 

6 46 18 940 

7 5 56.472 
7 13 37-284 



8 23 37.298 +57 

9 7 52.155 
9 26 31.760 
9 36 8.988 



10 22 5.096 + 7-4985 



10 59 52.260 
12 14 31.648 + 

12 46 55-237 
2 48 34-024 

13 28 28.100 



II. 6 
- 4-9. . 
+28.9317 
-20.4769 



7 15 24.635 +12.7709 +.0131 



. 3310 
8.2911 
8.7249 
1.6820 



0.4296 
+ 6.0474 
+ 0.4586 
+ 9-2533 



+ .1528 
+ .0086 
.0132 
+ .0130 



.0118 

.0035 

.0582 
-.0145 



-.0376 
-.1153 
-.0059 

.0121 

.0460 

-.0578 

.0702 
+ .0368 
- 0183 
-.0770 



+88 54 u. 12 
-85 8 56.44 
+85 21 23.96 
+85 9 46.65 

-84 49 36.02 
-80 44 9-91 
+87 10 10.26 
-86 54 58-20 

+82 33 38.50 

+88 51 28.64 
-85 21 54-58 
+8l 39 35-81 
80 36 16.52 

+82 56 28.35 

-84 II 25.52 
+88 6 56.51 
-84 42 59-21 
+83 49 13-82 
-85 24 II. 14 



+18.376 
+18 137 
+ 9-III 
+ 1-942 

+ 1-425 

- 3-941 

- 5-722 
6.323 

- 6.524 

-11.738 
-14.608 
-15-743 
-16.205 

-18.234 

-19.363 
-19.946 
-19.602 
-19.580 
-18.594 



+0.001 

+0.028 
+0.042 

0.004 

+0.087 
+0.082 
-0.034 
+0.000 

- 0.047 

+0.018 
+0.044 
0.027 
+0.019 

+0.009 

0.005 
+0.058 
+0.024 

+0.016 
0.024 



a Ursae Min., star 9 m , 18" s. pr, 



32 H. Camelop., star 5 W .8, 21", 6 n. pr, 



PRACTICAL ASTRONOMY 



APPARENT PLACES OP STARS, 

CIRCUMPOLAR STARS 
For the Upper Transit at Washington 



43 H. Cephei. 
Mag. 4-5 


a Ursae Minoris 
(Polaris) 
Mag. 2.1 


4 G. Octantis 
Mag. 5.6 


Groombridge 750 
Mag. 6.7 


Groombridge 944 
Mag. 6.4 


h 


w G 


( 


H 


w 


( 


H 


Jj n 


, 


_; 


<J> r, 


, 


H 


0) 


j. 


G 


<l 


i a 


G 


-I 


1 n 


G 


< 

4J>'S 


S PJ 


G 


<J 


3 j-c 


G 


*^.g 


IB 


1 


5P u 


Q 4 " 


1 


*o 

2 


|J5 


4 

h cij 




I- 3 


4 

_rt 


5 


US 

CO+J 

Q 


1 


." 


1* 




A w 


/ 




h m 


/ 




h m 


/ 




h m 


/ 




h m 


1 


Jan. 


058 


+8551 


Jan. 


I 34 


+8854 


Jan. 


i 41 


-85 9 


Jan. 


4 12 


+8521 


Jan. 


5 37 


+859 




5 







s 


" 




5 


" 




s 


' 




5 


' 


08 


17.25 


34 50 


o 8 


46 55 


23 79 


o 8 


32 96 


23 31 


0-9 


36.14 


27 50 


o 9 


54 86 


44 32 


1.8 


i<5.93 


34-55 


i 8 


45 42 


23-91 


i 8 


32.71 


23 36 


i 9 


36 oo 


27 78 


1.9 


54 82 


44 64 


2.8 


16 65 


34 59 


2 8 


44 34 


24 oi 


2 8 


32 44 


23 41 


2 9 


3586 


28 04 


2 


54-78 


44-93 


3-8 


16.36 


34 62 


3 8 


43 30 


24 10 


38 


32 17 


23 46 


3 9 


35 72 


28 29 


3 


54 73 


45 23 


4 8 


16 09 


34 65 


4 8 


42 28 


24 20 


4 8 


31 89 


23-50 


4 9 


35 58 


2853 


4 


54 69 


45 53 


5 7 


IS 82 


34 70 


5 8 


41 26 


24 30 


5 8 


31 59 


23 50 


5 9 


35 46 


28.78 


5- 


54-66 


45 83 


67 


IS 55 


34 74 


68 


40 25 


24 41 


6 8 


31 31 


23 So 


69 


35-34 


29 05 


6 


54 63 


46 14 


7 7 


IS 27 


34 79 


7-8 


39 21 


24 51 


7 8 


31 04 


23-49 


7-9 


35 22 


29 31 


7 


54-60 


46.45 


8.7 


14.99 


34-83 


8 8 


38 14 


24 61 


8 8 


30 77 


23 45 


8 9 


35 10 


29 57 


8 


54 57 


46 76 


9 7 


14 69 


34-88 


98 


37 03 


24 7i 


98 


30.50 


23 4C 


9 9 


34 97 


29 86 


9 


54 53 


M OS 


10.7 


14 38 


34 93 


TO 8 


35 87 


24 81 


10.8 


30 25 


23-34 


10 9 


34 82 


30.15 


10 


54 48 


47 41 


n. 7 


14.06 


3496 


n. 8 


3466 


24 90 


ii 8 


30 oi 


23.27 


n. 9 


34.65 


30 43 


ii 


54 43 


47 75 


12.7 


13 73 


34 97 


12 8 


33 41 


24 98 


12 8 


29 76 


23.22 


12.9 


34-48 


30 70 


12,9 


54 35 


48 09 


13-7 


13 39 


34 97 


138 


32 14 


25 04 


13 8 


29 53 


23 17 


139 


34 28 


30 97 


13 9 


54 25 


48 43 


14 7 


13 06 


34 93 


14 7 


3086 


25 09 


14.8 


29 29 


23 12 


14 9 


34 08 


31 22 


14 9 


54 14 


48 76 


IS 7 


12 73 


34 89 


IS 7 


29 60 


25 11 


15 8 


29.04 


23 08 


15 9 


3385 


31 46 


IS 9 


54 03 


49.06 


16 7 


12 42 


34 84 


167 


28 40 


25 12 


167 


28 78 


23.05 


16.9 


33 65 


3167 


16 9 


53 90 


49-35 


17 7 


12 13 


34 79 


17 7 


27 27 


25 12 


17.7 


28 51 


23 oi 


17 8 


33 45 


31 87 


179 


53-77 


49 62 


18.7 


11.85 


34 72 


187 


26 21 


25 13 


18 7 


28 23 


22 96 


18 8 


33 26 


32 05 


189 


53 67 


49-89 


19-7 


11-59 


34 67 


19 7 


25 19 


25 13 


19 7 


27-93 


22 87 


19 8 


33 09 


32 23 


199 


53-58 


50.13 


20.7 


n 33 


34.64 


20 7 


24 21 


25 14 


20 7 


27 64 


22 78 


20 8 


32 92 


32 43 


20,9 


53.49 


50.37 


21 7 


II 07 


34.63 


21.7 


23 19 


25 18 


21 7 


27 35 


22 65 


21.8 


32.77 


32 64 


21 9 


53-41 


50 65 


22 7 


10.81 


34 61 


22.7 


22 II 


25 21 


22 7 


27.08 


22 52 


22 8 


32 61 


32 86 


22 9 


53 34 


50-93 


23-7 


10 53 


34-58 


23.7 


20 96 


25 22 


23 7 


26.83 


22 37 


23 8 


32 44 


33-10 


23 9 


53 26 


51-24 


24 7 


10 21 


34 55 


24.7 


19 74 


25 24 


24 7 


26.58 


22 21 


248 


32 25 


33 33 


24-9 


53 16 


51-55 


25 7 


989 


34 50 


25 7 


18.47 


25 25 


25 7 


26.36 


22.06 


258 


32.02 


33 56 


25 9 


53 03 


51.86 


267 


9 56 


34 41 


26 7 


17 19 


25 24 


26.7 


26.13 


21.91 


26 8 


31-79 


33 76 


26 9 


52 89 


52.16 


27-7 


9 23 


34 31 


27-7 


15 91 


25 20 


27 7 


25 89 


21 78 


278 


31-54 


33 95 


27-9 


52.71 


52 44 


28.7 


891 


34 19 


287 


14-68 


.25.14 


287 


25.65 


21 67 


28 8 


31 27 


34-11 


28.9 


52 53 


52 69 


29-7 


8 63 


34 05 


29.7 


13 52 


25.06 


29-7 


25.40 


21.56 


29.8 


31-02 


34 26 


29-9 


52 35 


52 94 


30.7 


8.35 


33 90 


30 7 


12 42 


24 98 


30 7 


25 13 


21 43 


30 8 


30-78 


34-39 


30-9 


52.17 


53 I? 


31-7 


8.10 


33 76 


31-7 


n 37 


24 89 


31 7 


24.86 


21 30 


31-8 


30.55 


34 49 


31-9 


52.00 


53.37 


13-85 +13.81 


52.42 +52.41 


11.84 11.80 


12.36 +12.31 


11.86 +11.82 


oh s&m 1 15.014 


ih 34 m 135.588 


ih 4im 325.636 


4/ I2W 245.065 


5ft 37W 43*.075 


+85 Si' 2o".s8 


+88 54' H".I2 


-85 8' 56".44 


+85 21' 23' '.96 


+85 9' 46".6S 



NOTE. o^ Washington Civil Time is twelve hours before Washington Mean Noon of the same 
date. 



THE AMERICAN EPHEMERIS AND NAUTICAL ALMANAC 71 

APPARENT PLACES OF STARS, 1925 
FOR THE UPPER TRANSIT AT WASHINGTON 



Washington 
Civil Time 



33 Piscium 
Mag. 4-7 



Right 
Ascension 



Declina- 
tion 



a Andromeda; 
(Alpheratz) 
Mag. 2 2 



Right 
Ascension 



Declina- 
tion 



j8 Cassiopeiae 
Mag. 2.4 



Right 
Ascension 



Declina- 
tion 



e Phoanicis 
Mag. 3-9 



Right 
Ascension 



Declina- 
tion 



.Jan. 

Feb. 
Mar. 

Apr. 
May 

June 
July 

Aug. 

Sept. 
Oct. 

Nov. 
Dec. 



7 
10.7 
20 7 
30.6 

96 

19.6 

1 6 
ii 5 
21.5 
31 5 

10.5 
20 4 
30 
10 4 
20.3 

30 3 
9 3 
19 3 
29 2 
9 2 

19 2 

29.2 

8.1 
18 i 
28.1 

7 o 
17.0 
27 o 

7 
16 9 

269 
5-9 
15-9 
25 8 
5.8 

15.8 
25 7 
35 7 



h m 
o i 



28.841 



28 532 

28 454 

28.399 

28 367 T 
28 368 *| 
28 402 J4 
28 473 jjjl 

28.583 ' 
28 733 *j> 

28 921 188 

29 145 
29 398 



- 6 7 

45 99 6 

46 bo ; 

47 13 ,j 
47-49 2' 

47 71 2 ; 



3247 



73 
47 59 
47 20 
46 58 
45 75 ] 

44 64 IV, 
43-31 ^ 
41 79 



o 4 

29-734 
29 586 
29 444 
29 314 

29 204 

29.118 
29 063 
29 046- 
29 070 
29 139 !^ 

% 2 M * 

29 625 20 ; 

29 871 2 4 

30.151 ts 



29 678 
-29 977 

30 285 
30 596 
30 901 



31 192 , 

31.463 ; 
31 707 : 

31 919 : 
32.093 : 

32.231 

32 327 

32 388 

32 410- 
32 402 

32.362 
32.302 

32 220 
32 127 
32 021 i 

31.908 . 
31 795 : 
31-684 ' 



sl= 

30 12 

" 24 174 
26.51 



s 



i'326 



21.21 

20 96 
20.94- 

21 16 

21.59 
22.17 

22 83 

23 60 

24 40 

25.22 

26 01 

74 
27-39 



+28 40 
38.63 . 

2iS 



36 42 



86 33 

55 
J7 
24 



34-99 
41 






24.81 

20 

2395 
24 09 



$3*3 



30 

30 78S 

31-120 

31 457 

31 75 



-54 

26.82 

28 43 

3 ' 31 






32.096 288 34-72 

" ' K * !"! 37 I3 

32 802 
33-044 : 



241 
2A7 

^4 



46 87 



33348 
33 371- 
33 358 

33 315 
33 244 
33.150 . 

33-036 ; 

32.908 J- 

32.770 
32.625 JJ 



43 

7156 
9457 
[I4 58 






5835-^8 

58 17 
57 69 
56.90 



h m 
o 5 



9-620 



8 436 g 

8$ '42 

8 015 - 
8 017 2 



10.056 

543 

1 O4I 

1 S$ 



370 

421 

Ufia; 

487 
498 



3 827 ; 

: 4 OI 9 

4 149 
4 215 

'4 l62 112 



4.050 
3 888 

3 68 
3 " 432 
3-152 



248 



12.846 
2 525 
2.198 



+5843 
8185 8 



77 96 
79 



.55 



ss 



70.24 ". 
73-53 : 

76.89 8 
80.27 338 
83 61 
86 .-82 

89.83 : 



-91 



10055 
ioo 6s 

100 20 



^ 



h m 

o 5 

s 

35 oio 
34-8II * 
34.628 3 
34 467 1Ui 
34 333 

34-232 
34.169 

34 147" 
34-173 
34 248 . 



34 374 



I?8 



7^ 3l8 
J ' 4 354 

35 728 } 

ii 



39 124 I31 
39 255 r \ 
39.328 Jg 

39 346 
3U g 

39 228 
39-I06J 22 
38 950 J| 

S:SS? 
S.ig~ 

37-958 



-46 9 

61-39 
61.07 3- 
60 28 , 7S 
5906 J 2 ^ 
57 43 ^ 

55 43 23 , 

53- 12 -_/j 



44-79 

41 72 

38. '_ 
35-58 



295 
307 



257 



21 43 
20.32 

19 67 
19 80 



i 194 
155 






190 
216 



35 



42.38 

42.80 
42-74 



Mean Place 
Sec a, Tan d 



29.826 
i. 006 



37-68 
0.107 



30.419 
1.140 



35-02 

+0.547 



9 934 
I 927 



70.16 
+1-647 



36 485 40-94 
I 444 -1.041 



+0.061 
+0.40 



+0.007 

+0.01 



+0.061 0.036 

+0.40 +0.02 



+0.062 o.no 

+0.40 +0.02 



+0.060 

+0.40 



+0.069 
+0.03 



NOTE, o" Washington Civil Time is twelve hours before Washington. Mean Noon of the same 
date. 



PRACTICAL ASTRONOMY 



SUN, JANUARY 1925 



G. C. T. 


Stm's 
Decl. 


Equation 
of Time 


Sun's 
Decl. 


Equation 
of Time 


Sun's 1 Equation 
Decl. of Time 


Sun's 
* Decl. 


Equation 
of Time 




Thursday I 


Monday 5 


Friday 9 


Tuesday 13 


h 


' 


m s 


/ 


m s 


/ 


m s 


o / 


m s 


o 


-23 3-9 


-3 20.9 


-22 41 7 


5 12.5 


22 12 3 


-6 57-1 


21 36.0 


-8 33-3 


2 


23 3-5 


3 23.2 


22 41.2 


5 14-7 


22 II. 6 


6 S9-2 


21 35.2 


8 35.2 


4 


23 31 


3 25.6 


22 4O.6 


5 17-0 


22 II. 


7 1.3 


21 34-4 


8 37-1 


6 


23 2.7 


3 28.0 


22 40.Z 


S 19-2 


22 10.3 


7 3-4 


21 33-5 


839-0 


8 


23 2 3 


330.4 


22 39-5 


5 21.5 


22 9 6 


7 5-5 


21 32 7 


8 40.9 


10 


23 1.9 


3 32-8 


22 39 


5 23.7 


22 8.9 


7 7-6 


21 31.8 


8 42.7 


12 


23 1.5 


3 35 I 


22 38 4 


5 26.0 


22 8.2 


7 9-6 


21 3I.O 


844.6 


14 


23 i.i 


3 37-5 


22 37-8 


5 28 2 


22 7-5 


7 H. 7 


21 30 I 


8 46.5 


16 


23 o 7 


3 39 9 


22 37-3 


5 30-4 


22 6 8 


7 13-8 


21 29 3 


848.4 


18 


23 0.3 


3 42 2 


22 36 7 


5 32.7 


22 6 


7 IS.8 


21 28.4 


8 50.2 


20 


22 59.8 


3 44-6 


22 36 2 


5 34-9 


22 53 


7 17-9 


21 27 6 


8 52.1 


22 


22 59 4 


3 47-0 


22 35-6 


5 37 I 


22 4 6 


7 20.0 


21 26 7 


8 54-0 


H. D. 


O.2 


1.2 


o 3 


i i 


o 4 


I 


o 4 


0-9 




Friday 2 


Tuesday 6 


Saturday 10 


Wednesday 14 


o 


-22 S9-o 


-3 49-3 


-22 35 o 


-5 39 4 


-22 3.9 


-7 22 


-21 25 9 


-8 55. 8 


2 


22 58.6 


3 51 7 


22 34.4 


5 41 6 


22 3 2 


7 24 I 


21 25.0 


8 57 7 


4 


22 58.1 


3 54-0 


22 33 9 


5 43 8 


22 2 4 


7 26.1 


21 24.1 


8 59-5 


6 


22 57-7 


3 56.4 


22 33 3 


5 46 


22 1-7 


7 28.2 


21 23-3 


9 1-4 


8 


22 57-3 


3 58 7 


22 32 7 


5 48.2 


22 1.0 


7 30 2 


21 22.4 


9 3-2 


10 


22 56.8 


4 I I 


22 32 I 


5 So 4 


22 0.3 


7 32 2 


21 21 5 


9 5-0 


12 


22 56.4 


4 3-4 


22 31 5 


5 52 6 


21 59 5 


7 34 2 


21 20 7 


9 6.9 


14 


22 56.0 


4 5-8 


22 30 9 


5 54-8 


21 58.8 


7 36.3 


21 19 8 


9 8.7 


16 


22 55 5 


4 8 I 


22 30 3 


5 57 o 


21 58.0 


7 38 3 


21 18 9 


9 io. S 


18 


22 55 I 


4 10 4 


22 29 7 


5 59 2 


21 57 3 


7 40 3 


21 18.0 


9 12.3 


20 


22 54 6 


4 12.8 


22 29-1 


6 1.4 


21 56 5 


7 42.3 


21 17 I 


9 14-1 


22 


22 54-2 


4 I5-I 


22 28 5 


6 36 


21 55-8 


7 44 3 


21 16.2 


9 16.0 


H. D. 


0.2 


1.2 


3 


i.i 


0.4 


I 


0.4 


0.9 



NOTE. The Equation of Time is to be applied to the G. C. T. in accordance with the sign as 
given, 
oft Greenwich Civil Time is twelve hours before Greenwich Mean Noon of the same date. 

SUN, 1925 
FOR WASHINGTON APPAPENT NOON 



Date 


Apparent 
Right 
Ascension 


V 9 r. 

per 
Hr. 


Apparent 
Declina- 
nation 


Var 
per 
Hr. 


Equation 
of Time. 
Mean 
App. 


Var. 
per 
fir. 


Semi- 
diam- 
eter 


S. T. of 
Sem. 
Pass. 
Merid. 


Sidereal 
Time of 
o h Civil 
Time 




km s 


5 


, 


" 


m 5 


s 


/ 


m s 


h m s 


Jan. I 


18 47 1.20 


11.043 


-23 o 25.6 


+12 40 


+ 3 41-29 


+1.183 


16 17.90 


i 11.04 


6 41 21.04 


2 


18 51 26.06 


II.O27 


22 55 14 3 


13 54 


4 9-51 


i 168 


16 17.91 


11.00 


6 45 17-59 


3 


18 55 50 54 


II. Oil 


22 49 35-7 


14.68 


4 37.36 


1. 152 


16 17.91 


10.95 


6 49 14.15 


4 


19 o 14.62 


10.994 


22 43 29.8 


I5.8i 


5 4-8o 


I 135 


16 17.91 


10.90 


6 53 10.71 


5 


19 4 38.26 


10.976 


22 36 56.8 


16.94 


5 3I.8I 


1.116 


16 17.90 


10.84 


6 57 7-26 


6 


19 9 1-45 


10.957 


-22 29 57-0 


+18.05 


+ 5 58 37 


+1.097 


16 17.89 


10.78 


7 I 3 82 


7 


19 13 24.16 


10.936 


22 22 30.5 


19.16 


6 24.45 


1.076 


16 17.87 


10.71 


7 S 38 


8 


19 17 46.36 


10.914 


22 14 37.6 


20.25 


6 50.02 


1.054 


16 17.83 


0.64 


7 8 56.94 


9 


19 22 8.03 


10.892 


22 6 18.5 


21-34 


7 15.07 


1.032 


16 17.80 


0.57 


7 12 53-49 


10 


19 26 29.15 


10.868 


21 57 33-5 


22.41 


7 39-56 


1.009 


16 17.76 


0.49 


7 16 50.05 


ii 


19 30 49-70 


10.844 


21 48 22.7 


+2 3 .48 


+ 8 3-49 


+0.985 


16 17.72 


0.41 


7 20 46.61 


12 


19 35 9.67 


10.819 


21 38 46.4 


24 54 


8 26.83 


0.960 


16 17.67 


0.33 


7 24 43 16 


13 
14 


19 39.29-02 
19 43 47-74 


10.793 
10.767 


21 28 45 o 
21 18 18 8 


25.58 
26.61 


8 49-57 
9 ii 68 


0-934 


16 17.61 
16 17.55 


0.24 
0.15 


7 28 39 72 
7 32 36.28 


15 


19 48 5-82 


10.740 


21 7 28 O 


27.63 


9 33-15 


o.88l 


16 17.48 


0.06 


7 36 32.84 


16 


19 52 23.25 


10.712 


20 56 12.9 


+28.63 


+ 9 53.96 


+0.853 


16 17.40 


9-97 


7 40 29.39 


17 


19 56 40.00 


10.684 


20 44 33-8 


29.62 


io 14.09 




16 17-32 


9.88 


7 44 25.95 


18 


20 56.06 


10.655 


20 32 31-0 


30.60 


io 33-54 


0.796 


16 17-23 


9.78 


7 48 22.51 


19 


20 5 11.41 


10.625 


20 20 4.9 


3L56 


io 52.29 


0.766 


16 17.14 


9.68 


7 52 19.06 


20 


20 9 26.04 


10.594 


20 7 15 8 


32 52 


ii 10.32 


0.736 


16 17.05 


I 9-57 


7 56 15.62 



NOTE. For mean time interval of semidiameter passing meridian, subtract o s .i9 from the 
Sidereal interval. 
c^ Washington CM Time is twelve hours before Washington Mean Noon of the same date. 



THE AMERICAN EPHEMERIS AND NAUTICAL ALMANAC 73 

servation, by employing formulae and tables given for the pur- 
pose in Part II of the Ephemeris. 

There are many star catalogues, some containing the positions 
of a very large number of stars, but determined with rather in- 
ferior accuracy; others contain a relatively small number of 
stars, but whose places are determined with the greatest accu- 
racy. Among the best of these latter may be mentioned the 
Greenwich ten-year (and other) catalogues, and Boss' Catalogue 
. of 6188 stars for the epoch 1900. (Washington, 1910,) 

For time and longitude observations, the list given in the 
Ephemeris is sufficient, but for special kinds of work where the 
observer has but a limited choice of positions, such as finding 
latitude by Talcott's method, many other stars must be ob- 
served. 

42. Interpolation. 

When taking data from the Ephemeris corresponding to any 
given instant of Greenwich Civil Time, it will generally be neces- 
sary to interpolate between the tabulated values of the function. 
The usual method of interpolating, in trigonometric tables, for 
instance, consists in assuming that the function varies uniformly 
between two successive values in the table, and, if applied to the 
Ephemeris, consists in giving the next preceding tabulated value 
an increase (or decrease) directly proportional to the time elapsed 
since the tabulated Greenwich time. If the function is repre- 
sented graphically, it will be seen that this process places the 
computed point on a chord of the function curve. 

Since, however, the " variation per hour/' or differential co- 
efficient of the function, is given opposite each value of the func- 
tion it is simpler to employ this quantity as the rate of change of 
the function and to multiply it by the time elapsed. An in- 
spection of the diagram (Fig. 39) will show that this is also a 
more accurate method than the former, provided we always 
work from the nearer tabulated value; when the differential 
coefficient is used the computed point lies on the tangent line, 
and the curve is nearer to the tangent than to the chord for any 



74 



PRACTICAL ASTRONOMY 



distance which is less than half the interval between tabulated 
values. 

To illustrate these methods of interpolating let it be assumed 
that it is required to compute the sun's declination at 2i h Green- 
wich Civil Time, Feb. i, 1925. The tabulated values for o* 
(midnight) on Feb. i, and Feb. 2, are as follows: 

Sun's declination Variation per hour 
Feb. i, o* - 17 18' o 3 ". 9 + 42"-i5 

Feb. 2, o* 17 01 03 .2 + 42 .90 



17 18*03'.'9 




Greenwich Civil Time 

FIG. 39 



Feb,2 



The given time, 21*, is nearer to midnight of Feb. 2 than it is 
to midnight of Feb. i, so we must correct the value 17 01' 
03". 2 by subtracting (algebraically) a correction equal to +42".9o 
multiplied by 3", giving 17 03' ii ff .g. If we work from 
the value for o*, Feb. i, we obtain 17 18' 03".9 42". 15 X 
21 = 17 03' i8".7. For the sake of comparison let us inter- 
polate directly between the two tabulated values. This gives 
-I7oi'o3".2 + A X i7'oo".7 = -i7o3'io".8. These 
three values are shown on the tangents and chord respectively 
in Fig. 39. It is clear that the first method gives a point nearer 
to the function curve than either of the others'. 



THE AMERICAN EPHEMERIS AND NAUTICAL ALMANAC 75 



Whenever these methods are insufficient, as might be the 
case when the tabular intervals are long, or the variations in 
the " varia. per hour " are rapid, it is possible to make a closer 
approximation by interpolating between the given values of the 
differential coefficients to obtain a more accurate value of the 
rate of change for the particular interval employed. If we 
imagine a parabola (Fig. 40) with its axis vertical and so placed 
that it passes through the two given points, C and C', of the 
^function curve and has the same slope at these points, then it is 




FIG. 40. PARABOLA 



ky 



obvious that this parabola must lie very close to the true curve 
at all points between the tabulated values. By the following 
process we may find a point 'exactly on the parabola and conse- 
quently close to the true value. The second differential coeffi- 
cient of the equation of the parabola is constant, and the slope 
(dy/dx) may therefore be found for any desired point by simple 
interpolation between the given values of dy/dx. If we deter- 
mine the value of dy/dx for a point whose abscissa is half way 
between the tabulated time and the required time we obtain the 
slope of a tangent line, rj 1 ', which is also the slope of a chord of 



76 PRACTICAL ASTRONOMY 

the parabola extending from the point representing the tabulated 
value to the point representing the desired value ; % for it may be 
proved that for this particular parabola such a chord is exactly 
parallel to the tangent (slope) so found. By finding the value 
of the " varia. per hour " corresponding to the middle of the 
time interval over which we are interpolating, and employing 
this in place of the given " varia. per hour " we place our point 
exactly on the parabola, which must therefore be close to the 
true point on the function curve. In the preceding example this 
interpolation would be carried out as follows: 

From o* Feb. 2 back to 21* Feb. i is 3* and the time at the 
middle of this interval is 22* 30. Interpolating between 
+42 /; .90 and +42". 15 we find that the rate of change for 22* 

30 ra is +42".9o - X o".75 = + 4 2".86. 
The declination is therefore 

-I7oi'o3".2 - 3 X 42".86 = 17 03' n".8. 

This is the most accurate of the four values obtained. 

As another example let it be required to find the right ascen- 
sion of the moon at 9^ 40"* on May 18, 1925. The Ephemeris 
gives the following data. 

Green. Civ. Time Rt. Asc. Var. per Min. 

9* o 2 9 m 59^.56 2.0548 

10* O 32 02 .80 2.0531 

The Gr. Civ. Time at the middle of the interval from 9* to 
9* 40 is 9* 20 W , or one-third the way from the first to the sec- 
ond tabulated value. The interpolated " variation per minute " 
for this instant is 2.0542, one-third the way from 2.0548 to 2.0531. 
The correction to the right ascension at 9* is 40 X 2^.0542 = 
82*. 168 and the corrected right ascension is therefore 0^31 
2i*.73. If we interpolate from the right ascension at io h using 
a " var. per min." which is one-sixth the way from 2.0531 
to 2.054$ we obtain the same result. 



THE AMERICAN EPHEMERIS AND NAUTICAL ALMANAC 77 



43. Double Interpolation. 

When the tabulated quantity is a function of two or more 
variables the interpolation presents greater difficulties. If 
the tabular intervals are not large, and they never are in a well- 
planned table, the interpolation may be carried out as follows. 
Starting from the nearest tabulated value, determine the change 
in the function produced by each variable separately and apply 
these corrections to the tabulated value. For example in Table 
F, p. 203, we find the following: 

p sin t 



H. A. 


1925 


1930 


T h $2 


3o'-9 


30'.2 


j_h $6m 


3i -9 


3 I .2 



Suppose that we require the value of p sin t for the year 1927 
and for the hour angle i h 53^.5. We may consider that the 
value 30'. 9 is increased because the hour angle increases and is 
decreased by the change of 2 years in the date, and that these 
two changes are independent. The increase due to the i m .5 

increase in hour angle is X i'.o = o f .^S. The decrease due 

4.0 

2 
to the change in date is - X o'.y = o'.28. The corrected value 

is 30'. 9 + o'.38 o'.28 = 3i'.o. 

In a similar manner the tabulated quantity may be corrected 
for three variations. 

Example. Suppose that it is desired to take from the tables of the sun's azi- 
muth (H. O. No. 71) the azimuth corresponding to declination 4-n30 / and hour 
angle (apparent time from noon) 3^ 02^, the latitude being 42 20' N. From the 
page for latitude 42 we find 

Declination 





11 


12 


3* io m 


1 1 2 39' 


m 45' 


^oo m 


114 56' 


114 01' 



7 8 



PRACTICAL ASTRONOMY 



and from page for latitude 43 we find 

Decimation 





11 


12 


3 fc I0 m 


113 22' 


112 29' 


3/ oo m 


HS 4i' 


1 14 48' 



Selecting 114 56' as the value from which to start, we correct for the three varia- 
tions as follows: 

For latitude 42, decrease in io m time = 2 17'; decrease for 2 time = 27'.4. 
For latitude 42, decrease for 1 of decimation = 55'; decrease for 30' of decli- 
nation = 27 '.5. For 3^00"* increase for i of latitude = 45'; increase for 20' of 
latitude =15'. 

The corrected value is 

114 56' 27'.4 27'. 5 + 15' = 114 i6'.i 

For more general interpolation formulae the student is re- 
ferred to Chauvenet's Spherical and Practical Astronomy, Doo- 
little's Practical Astronomy, Hayford's Geodetic Astronomy, 
and Rice's Theory and Practice of Interpolation. 

Questions and Problems 

1. Compute the sun's apparent declination when the local civil time is 15'* 
(3* P.M.) Jan. 15, 1925, at a place 82 10' West of Greenwich (see p. 67). 

2. Compute the right ascension of the mean sun +12^ at local o^ Jan. 10, 1925, 
at a place 96 10' West of Greenwich (see p. 67). 

3. Compute the equation of time for local apparent noon Jan. 30, 1925, at a 
place 71 06' West of Greenwich. 

4. Compute the apparent right ascensioji of the sun at G. C. T. i6> on Jan. 
10, 1925, by the four different methods explained in Art. 42. 

5. In Table F, p. 203, find by double interpolation the value of p f sin t for / = 
8* 42^.5 and for 1926. 



CHAPTER VII 

THE EARTH'S FIGURE CORRECTIONS TO 
OBSERVED ALTITUDES 

44. The Earth's Figure. 

The form of the earth's surface is approximately that of an 
ellipsoid of revolution whose shortest axis is the axis of revolu- 
tion. The actual figure departs slightly from that of the ellip- 
soid but this difference is relatively small and may be neglected 
in astronomical observations of the character considered in this 
book. Each meridian may therefore be regarded as an ellipse, 
and the equator and the parallels of latitude as perfect circles. 
En fact the earth may, without appreciable error, be regarded 
is a sphere in such problems as arise in navigation and in field 
astronomy with small instruments. The semi-major axis of 
the meridian ellipse, or radius of the equator on the Clarke 
(1866) Spheroid, used as the datum for Geodetic Surveys in 
the United States, is 3963.27 statute miles, and the semi-minor 
(polar) axis is 3949.83 miles in length. This difference of about 
13 miles, or about one three-hundredth part, would only be 
noticeable in precise work. The length of i of latitude at the 
equator is 68.703 miles; at the pole it is 69.407 miles. The 
length of i of the equator is 69.172 miles. The radius of a 
sphere having the same volume as the ellipsoid is about 3958.9 
miles. On the Hay ford (1909) spheroid the semi-major axis 
is 3963.34 miles and the semi-minor is 3949.99 miles. 

In locating points on the earth's surface by means of spherical 
coordinates there are three kinds of latitude to be considered. 
The latitude as found by direct astronomical observation is 
dependent upon the direction of gravity as indicated by the 
spirit levels of the instrument; this is distinguished as the 
astronomical latitude. It is the angle which the vertical or 

70 



8o 



PRACTICAL ASTRONOMY 



plumb line makes with the plane of the equator. The geodetic 
latitude is that shown by the direction of the normal to the sur- 
face of the spheroid, or ellipsoid. It differs at each place from 
the astronomical latitude by a small amount which, on the aver- 
age, is about 3", but occasionally is as great as 30". This 
discrepancy is known as the " local deflection of the plumb line/' 
or the " station error "; it is a direct measure of the departure 
of the actual surface from that of an ellipsoid. Evidently the 
geodetic latitude cannot be observed directly but must be de- 




rived by calculation. If a line is drawn from any point on the 
surface to the center of the earth the angle which this line makes 
with the plane of the equator is called the geocentric latitude. 
In Fig. 41 AD is normal to the surface of the spheroid, and the 
angle ABE is the geodetic latitude. The plumb line, or line 
of gravity, at this place would coincide closely with AB, say 
AB', and the angle it makes (AB'E) with the equator is the 
astronomical latitude of A. The angle ACE is the geocentric 
latitude. The difference between the geocentric and geodetic 
latitudes is the angle BA C, called the angle of the vertical, or the 



CORRECTIONS TO ALTITUDES 8l 

reduction of latitude. The geocentric latitude is always less than 
the geodetic by an amount which varies from o n' 30" in 
latitude 45 to o at the equator and at the poles. Whenever 
observations are made at any point on the earth's surface it 
becomes necessary to reduce' the measured values to the corre- 
sponding values at the earth's centre before they can be com- 
bined with other data referred to the centre. In making this 
reduction the geocentric latitude must be employed if great 
exactness in the results is demanded. For the observations of 
the character treated in the following chapters it will be suffi- 
ciently accurate to regard the earth as a sphere when making 
such reductions. 

45. The Parallax Correction. 

The coordinates of celestial objects as given in the Ephemeris 
are referred to the centre of the earth, whereas the coordinates 
obtained by direct observation are necessarily measured from a 
point on the surface and hence must be reduced to the centre. 
The case of most frequent occurrence in practice is that in which 
the altitude (or the zenith distance) of an object is observed 
and the geocentric altitude (or zenith distance) is desired. For 
all objects except the moon the distance from the earth is so 
great that it is sufficiently accurate to regard the earth as a 
sphere, and even for the moon the error involved is not large 
when compared with the errors of measurement with small 
instruments. 

In Fig. 42 the angle ZOS is the observed zenith distance, and 
SiOS is the observed altitude; ZCS is the true (geocentric) 
zenith distance, and ECS is the true altitude. The object 
therefore appears to be lower in the sky when seen from O than 
it does when seen from C. This apparent displacement of the 
object on the celestial sphere is called parallax. The effect of 
parallax is to decrease the altitude of the object. If the effect 
of the spheroidal form of the earth is considered it is seen that 
the azimuth of the body is also affected, but this small error 
will not be considered here. In the figure it is seen that the 



82 



PRACTICAL ASTRONOMY 



difference in the directions of the lines OS and CS is equal to 
the small angle OSC, the parallax correction. When the object is 
vertically overhead points C, and S are in a straight line and 
the angle is zero; when S is on the horizon (at Si) the angle OSiC 
has its maximum value, and is known as the horizontal parallax. 




FIG. 42 

In the triangle OCS, the angle at may be considered j 
known, since either the altitude or the zenith distance has been 
observed. The distance OC is the semidiameter of the earth 
(about 3959 statute miles), and CS is the distance from the 
centre of the earth to the centre of the object and is known for 
bodies in the solar system. To obtain S we solve this triangle 
by the law of sines, obtaining 

OC 



sin S = sin ZOS X 



CS 



From the right triangle OSiC we see that 

OC 
= C5i' 



sn 



[48] 



CORRECTIONS TO ALTITUDES 83 

The angle Si, or horizontal parallax, is given in the Ephemeris 
for each object; we may therefore write, 

sin S = sin Si sin ZOS [49] 

or sin S = sin Si cos h. [50] 

At this point it is to be observed that S and Si are very small 
angles, about 9" for the sun and only i for the moon. We 
may therefore make a substitution of the angles themselves 
(in radians) for their sines, since these are very nearly the same.* 
This gives 

S (rad) = Si (rad) X cos h. [51] 

To convert these angles expressed in radians into angles ex- 
pressed in seconds f we substitute 

S (rad) = S" X .000004848 . . . 
and Si (rad) = S/' X .000004848 . . . , 

the result being S" = S/' cos h, [52] 

that is 

parallax correction = horizontal parallax X cos h. [53] 

* To show the error involved in this assumption express the sine as a series, 

x 3 . x* 

sm x x . . . 

3 5 

Since we have assumed that sin x = x the error is approximately equal to the next 
term, For x = i the series is 

sin i = 0.0174533 0.0000009 + o.ooooooo. 

The error is therefore 9 in the seventh place of decimals and corresponds to about 
o".i8. For angles less than i the error would be much smaller than this since 
the term varies as the cube of the angle. 

If, as is frequently done, the cosine of a small angle is replaced by i, the error 
is that of the small terms of the series 

* 2 , x* 

COS X = I 1 . . . . 

2 4 
For i this series becomes 

cos i = i 0.00015234-. 

The error therefore corresponds to an angle of 31 ".42, much larger than for the 
first series. 

t To reduce radians to seconds we may divide by arc i" ( = 0.000004848137) 
pr multiply by 206264.8, 



84 PRACTICAL ASTRONOMY 

As an example of the application of Equa. [53] let us compute 
the parallax correction of the sun on May i, 1925, when at an 
apparent altitude of 50. From the Ephemeris the horizontal 
parallax is found to be 8". 73. The correction is therefore 

8".73 X cos 50 = s".6i 

and the true altitude is 50 oo' O5".6i. 

Table IV (A) gives approximate values of this correction for 
the sun. 

46. The Refraction Correction. 

Astronomical refraction is the apparent displacement of a 
celestial object due to the bending of the rays of light from the 
object as they pass through the atmosphere. The angular 
amount of this displacement is the refraction correction. On 
account of the greater density of the atmosphere in the lower 
portion the ray is bent into a curve, which is convex upward, and 




FIG. 43 

more sharply curved in the lower portion. In Fig. 43 the light 
from the star S is curved from a down to 0, and the observer at 
O sees the light apparently coming from S', along the line bO. 
The star seems to him to be higher in the sky than it really is. 
The difference between the direction of S and the direction of 




CORRECTIONS TO ALTITUDES 85 

S' is the correction which must be applied either to the apparent 
zenith distance or to the apparent altitude to obtain the true 
zenith distance or the true altitude. A complete formula for 
the refraction correction for any altitude, any temperature, and 
any pressure, is rather complicated. For observations with a 
small transit a simple formula will 
answer provided its limitations are 
understood. The simplest method 
of deriving such a formula is to con- 
sider that the refraction takes place 
at the upper limit of the atmosphere 
just as it would at the upper surface 
of a plate of glass. This does not 
represent the facts but its use may 
be justified on the ground that the 
total amount of refraction is the FIG. 44 

same as though it did happen this 

way. In Fig. 44 light from the star S is bent at 0' so that it 
assumes the direction O'O and the observer sees the star appar- 
ently at 5". ZO'S (= f') is the true zenith distance, ZO'S' 
(= f) is the apparent zenith distance, and SO f S r (= r) is the 
refraction correction; then, from the figure, 

r ' = r + r. [54] 

Whenever a ray of light passes from a rare to a dense medium (in 
this case from vacuum into air) the bending takes place according 

to the law 

sin f ' = n sin f, [55] 

where n is the index of refraction. For air this may be taken 
as 1.00029. Substituting [54] in [55] 

sin (f + r) = n sin f . [56] 

Expanding the first member, 

sin f cos r + cos f sin r = n sin f . [57] 



86 PRACTICAL ASTRONOMY 

Since r is a small angle, never greater than about o 34', we may 
write with small error (see note, p. 83). 

sin r = r 
and cos r = i 

whence 

sin f + r cos f = n sin f [58] 

from which r = (n i) tan f [59] 

being in radians. 

To reduce r to minutes we divide by arc i'( = 0.0002909 . . .). 
The final value of r is therefore approximately 

r-ZSS&tot [6o] 

(min) .00029 

= tan [61] 

= cot h. [62] 

This formula is simple and convenient but must not be regarded 
as showing the true law of refraction. The correction varies 
nearly as the tangent of f from the zenith down to about f = 80 
(h = 10), beyond which the formula is quite inaccurate. The 
extent to which the formula departs from the true refraction 
may be judged by a comparison with Table I, which gives the 
values as calculated by a more accurate formula for a tempera- 
ture of 50 F. and pressure 29.5 inches. 

As an example of the use of this formula [62] and Table I 
suppose that the lower edge of the sun has a (measured) altitude 
of 31 30'. By formula [62] the value of r is i'.63, or i' 38". 
The corrected altitude is therefore 31 28' 22". By Table I 
the correction is i' 33", and the true altitude is 31 28' 27". 
This difference of 5" is not very important in observations made 
with an engineer's transit. Table I, or any good refraction 
table, should be used when possible; the formula may be used 
when a table of tangents is available and no refraction table is 
at hand. For altitudes lower than 10 the formula should not 
be considered reliable. More accurate refraction tables may be 



CORRECTIONS TO ALTITUDES 87 

found in any of the text books on Astronomy to which reference 
has been made (p. 78). Table VIII, p. 233, gives the refraction 
and parallax corrections for the sun. 

As an aid in remembering the approximate amount of the 
refraction it may be noted that at the zenith the refraction is 
o; at 45 it is i'; at the horizon it is about o 34', or a little 
larger than the sun's angular diameter. As a consequence 
of the fact that the horizontal refraction is 34' while the sun's 
diameter is 32', the entire disc of the sun is still visible (apparently 
above the horizon) after it has actually set. 

47. Semidiameters. 

The discs of the sun and the moon are sensibly circular, and 
their angular semidiameters are given for each day in the Ephem- 
eris. Since a measurement may be taken more accurately 
to the edge, or limb, of the disc than to the centre, the altitude 
of the centre is usually obtained by measuring the altitude of 
the upper or lower edge and applying a correction equal to the 
angular semidiameter. The angular semidiameter as seen by 
the observer may differ from the tabulated value for two reasons. 
When the object is above the horizon it is nearer to the observer 
than it is to the centre of the earth, and the angular semidiameter 
is therefore larger than that stated in the Ephemeris. When 
the object is in the zenith it is about 4000 miles nearer the 
observer than when it is in the horizon. The moon is about 
240,000 miles distant from the earth, so that its apparent semi- 
diameter is increased by about one sixtieth part or about 16". 

Refraction is greater for a lower altitude than for a higher 
altitude; the lower edge of the sun (or the moon) is always 
apparently lifted more than the upper edge. This causes an 
apparent contraction of the vertical diameter. This is most 
noticeable when the sun or the moon is on the horizon, at which 
time it appears elliptical in form. This contraction of the ver- 
tical diameter has no effect on an observed altitude, however, 
because the refraction correction applied is that corresponding 
to the altitude of the edge observed; but the contraction must be 



88 



PRACTICAL ASTRONOMY 



allowed for when the angular distance is measured (with the 
sextant) between the moon's limb and the sun, a star, or a planet. 
The approximate angular semidiameter of the sun on the first 

day of each month is given in 
Table IV (B). 

48. Dip of the Sea Horizon. 
If altitudes are measured above 
the sea horizon, as when observing 
on board ship with a sextant, the 
measured altitude must be dimin- 
ished by the angular dip of the sea 
horizon below the true horizon. 
In Fig. 45 suppose the observer to 
be 4 at ; the true horizon is OB 
and the sea horizon is OH. Let 
OP = h, the height of the ob- 
server's eye above the water sur- 
face, expressed in feet; PC = R, the radius of the earth, regarded 
as a sphere; and D, the angle of dip. Then from the triangle 
OCH, 

n 

D = ' 163] 




FIG. 45 



and neglecting 



Replacing cos D by its series i \- . 

terms in powers higher than the second, we have, 

D 2 h 

7" = R + h' 
Since h is small compared with R this may be written 

^! = A 

2 R _ 
D-\/p. 

(rad) T /C 

Replacing U by its value in feet (20,884,000) and dividing by 
arc i' ( = .0002000), to reduce D to minutes, 



CORRECTIONS TO ALTITUDES 89 

x VI 





4 R 

V- 

2 



R , 

- X arc i' 

2 



= 1.064 Vh. [64] 

This shows the amount of the dip with no allowance for refrac- 
tion. But the horizon itself is apparently lifted by refraction 
and the dip which affects an observed altitude is therefore less 
*than that given by [64]. If the coefficient 1.064 is arbitrarily 
taken as unity the formula is much nearer the truth and is very 
simple, although the dip is still somewhat too large. It then 

becomes 

D f = VhJt. [65] 

that is, tlie dip in minutes equals the square root of the height 
in feet. Table IV (C), based upon a more accurate formula, 

> will be seen to give smaller values. 
49. Sequence of Corrections. 

Strictly speaking, the corrections to the observed altitude 
must be made in the following order: (i) Instrumental cor- 
rections; (2) dip (if made at sea); (3) refraction; (4) semidi- 
ameter; (5) parallax. In practice, however, it is seldom neces- 
sary to follow this order exactly. The parallax correction for 
the sun will not be appreciably different for the altitude of the 
centre than it will for the altitude of the upper or lower edge; 
if the altitude is low, however, it is important to employ the 

'refraction correction corresponding to the edge observed, be- 
cause .this may be sensibly different from that for the centre. 
In navigation it is customary to combine all the corrections, 
except the first, into a single correction given in a table whose 
arguments are the " height of eye," and " observed altitude." 
(See Bowditch, American Practical Navigator, Table 46.) 

Problems 

i. Compute the sun's mean horizontal parallax. The sun's mean distance is 
92,900,000 miles; for the earth's radius see Art. 44. Compute the sun's parallax 
at an altitude of 60 



90 PRACTICAL ASTRONOMY 

2. Compute the moon's mean horizontal parallax. The moon's mean distance 
is 238,800 miles; for the earth's radius see Art. 44. Compute the moon's parallax 
at an altitude of 45. 

3. If the altitude of the sun's centre is 21 10' what is the parallax correction? 
the corrected altitude? 

4. If the observed altitude of a star is 15 30' what is the refraction correction? 
the corrected altitude? 

5. If the observed altitude of the lower edge of the sun is 27 41' on May ist 
what is the true central altitude, corrected for refraction, parallax, and semidiameter? 

6. The altitude of the sun's lower limb is observed at sea, Dec. i, and found 
to be 18 24' 20". The index correction of the sextant is -f-i' 20". The height, 
of eye is 30 feet. Compute the true altitude of the centre. 



CHAPTER VIII 
DESCRIPTION OF INSTRUMENTS 

50, 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 
tiorizontal axis. By means of a combination of these two 

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



92 PRACTICAL ASTRONOMY 

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 defined 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 parallel of alti-* 
tude. 

51. 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. 



DESCRIPTION OF INSTRUMENTS 93 

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 



94 PRACTICAL ASTRONOMY 

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 
turned 180 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. 



DESCRIPTION OF INSTRUMENTS 95 

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. 

52. 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 or a flash light 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 mat- 
ter to make a substitute, either from a piece of bright tin or by 
fastening^, piece of tracing cloth or oiled paper over the objec- 
tive. A hole about f 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 flash light must be held so that the light is 



96 PRACTICAL ASTRONOMY 

diffused in such a way as to make the cross hairs visible, but so 
as not to shine into the observer's eyes. 

53. 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. 

54. 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. 

55. The Portable Astronomical Transit. 

The astronomical transit differs from the surveyor's transit chiefly in size and 
the manner of support. The diameter of the object glass may be from 2 to 4 inches, fc 
and the focal length from 24 to 48 inches. The instrument is mounted on a brick 
or a concrete pier and may be approximately levelled by means of foot screws. 
The older instruments were provided with several vertical threads (usually 5 or n) 
in order to increase the number of observations that could be made on one star. 
These were spaced about J' to i' apart, so that an equatorial star would require 
from 2* to 4 s to move from one thread to the next. The more recent transits are 
provided with the " transit micrometer "; in this pattern there is but a single 
vertical thread, which the observer sets on the moving star as it enters the field of 
view, and keeps it on the star continuously by turning the micrometer screw, 
until it has passed beyond the range of observation. The passage of the thread 
across certain fixed points in the field is recorded electrically. This is equivalent 



DESCRIPTION OF INSTRUMENTS 




FIG. 46. PORTABLE ASTRONOMICAL TRANSIT 
From the catalogue of C. L. Berger & Sons 



98 PRACTICAL ASTRONOMY 

to observations on 20 vertical threads. The field is illuminated by electric lights 
which are placed near the ends of the axis. The axis is perforated and a mirror 
placed at the centre to reflect the light toward the eyepiece. *The motion of the 
telescope in altitude is controlled by means of a cJamp and a tangent screw. The 
azimuth motion is usually very small, just sufficient to permit of adjustment into 
the plane of the meridian. The axis is levelled or its inclination is measured by 
means of a sensitive striding level applied to the pivots. The larger transits are 
provided with a reversing apparatus. 

The transit is used chiefly in the plane of the meridian for the direct determina- 
tion of sidereal time by star transits. It may, however, be used in any vertical 
plane, and for either time or latitude observations. The principal part of the 
work consists in the determination of the instrumental errors and in calculating* 
the corrections. The transit is in adjustment when the central thread is in a plane 
through the optical centre perpendicular to the horizontal axis, and the vertical 
threads are parallel to this plane. For observations of meridian transits this 
plane must coincide with the plane of the meridian and the horizontal axis must 
be truly horizontal. 

The chief errors to be determined and allowed for are (i) the azimuth, or devia- 
tion of the plane of collimation from the true meridian plane; (2) the inclination 
of the horizontal axis to the horizon; and (3) collimation error, or error in the sight 
line. Corrections are also applied for diurnal aberration of light, for the rate of the 
timepiece, and for the inequality of the pivots. The corrections to reduce an ob- 
served time to the true time of transit over the meridian are given by formulae 
[66], [67], and [68]. These corrections would apply equally well to observations 
made with an engineer's transit, and are of value to the surveyor chiefly in showing 
him the relative magnitudes of the errors in different positions of the objects 
observed. This may aid him in selecting stars even though no corrections are 
actually applied for these errors. 

The expressions for the corrections to any star are 

Azimuth correction = a cos h sec 5 [66] 

Level correction = b sin h sec 5 [67] 

Collimation correction = c sec 5 [68] 

in which a, b, and c are the constant errors in azimuth, level, and collimation, ^ 
respectively, expressed in seconds of time, and h is the altitude and B the declination 
of the star. If the zenith distance is used instead of the altitude cos h and sin h 
should be replaced by sin f and cos f respectively. These formulae may be easily 
derived from spherical triangles. Formula [66] shows that for a star near the 
zenith the azimuth correction will be small, even if a is large, because cos h is nearly 
zero. Formula [67] shows that the level correction for a zenith star will be larger 
than for a low star because sin h for the former is nearly unity. The azimuth error 
a is found by comparing the results obtained from stars which culminate north of 
the zenith with those obtained from south stars; if the plane of the instrument 
lies to the east of south; stars south of the zenith will transit too early and those 
north of the zenith will transit too late. From the observed times the angle may 
be computed. The level error b is measured directly with the striding level, making 



DESCRIPTION OF INSTRUMENTS 



99 



readings of both ends of the bubble, first in the direct, then in the reversed positions, 
the angular value of one level division being known. The collimation error, c, 
is found by comparing the results obtained with the axis in the direct position with 
the results obtained with the axis in the reversed (end-for-end) position. 

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





Declinations. 






h 


o 


10 


20 


30 


40 


50 


60 


70 


80 


h ' 


^ 


























O 


w 





s . 


o*.o 


o*.o 


o*.o 


o*.o 


o t<f .o 


o s .o 


o s .o 


o*.o 


C)0 


fc 


1 


ro 


0.7 


0.7 


0.8 


0,8 


0.9 


i. I 


1.4 


2.0 


4.0 


80 


W 


1 


20 


1.4 


1.4 


1.4 


1.6 


1.8 


2. L 


2.7 


4.0 


7-9 


70 


i 


Ij 
fl 


30 


2.0 


2.0 


2.1 


2 -3 


2.6 


3- 1 


4.0 


5-* 


it- S 


60 


i 


M 


40 


2.6 


2.6 


2.7 


3-o 


3-4 


4.0 


5- 2 


7-5 


14.8 


50 


j> 


^ 


5 


3- 1 


3- 1 


3-3 


3-6 


4.0 


4.8 


6.1 


9.0 


17.6 


40 





1 


60 


3-5 


3-5 


3-7 


4.0 


4-5 


5-4 


6.9 


10. I 


19-9 


30 


~o 


< 


70 


3-8 


3-3 


4.0 


4.4 


4.9 


5-8 


7-5 


TI .O 


21 .6 


20 


1 




80 


3-9 


4.0 


4.2 


4.6 


5-2 


6.1 


7-9 


II.5 


22 . 7 


10 






90 


4.0 


4.1 


4.2 


4.6 


5- 2 


6.2 


S.o 


IT. 7 


23.0 


O 





Note. Use the bottom line for the collimation error. 



From the preceding equations Table B has been computed. It is assumed 
that the collimation plane is r', or 4*, out of the meridian (a = 4*); that the axis 
is inclined i', or 4*, to the horizon (b 4*); and that the sight line is i', or 4*, to 
the right or left of its true position (c = 4*). An examination of the table will 
show that for low stars the azimuth corrections are large and the level corrections 
are small, while for high stars the reverse is true. As an illustration of the use of 
this table, suppose that the latitude is 42, and the star's declination is +30; 
and that a = i' (4*) and b = 2' (8*). The altitude of the star = 90 - (42 30) 
78. The azimuth correction is therefore i s .o and the level correction is 2 X 
4*.6 = 9 s . 2. If the line of sight were }' (or i*) in error the (collimation) correction 
would be i*. 2. This shows that with a transit set closely in the meridian but with 
a large possible error in the inclination of the axis, low stars will give better results 
than high stars. This is likely to be the case with a surveyor's transit. If, how- 
ever, the inclination of the axis can be accurately measured but the adjustment 
into the plane of the meridian is difficult, then the high stars will be preferable. 
This is the condition more likely to prevail with the larger astronomical transits. 
For the complete theory of the transit see Chauvenet's Spherical and Practical 
Astronomy, Vol. II; the methods employed by the U. S. Coast and Geodetic 
Survey are given in Special Publication; No. 14. 



100 



PRACTICAL ASTRONOMY 



56. The Sextant. 

The sextant is an instrument for measuring the angular dis- 
tance between two objects, the angle always lying in the plane 
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. 47, about 60 long, and two mirrors 7 
and H, the first one movable, the second one fixed. At the 
centre of the arc, /, 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 7 to the mirror H and thence to 



DESCRIPTION OF INSTRUMENTS 



IOI 



point 0. At a point near (on the line 110) is a telescope of | 
low power for viewing the objects. Between the two mirrors 
and also to the left of // 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 I 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 







FIG. 48. SEXTANT 

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. j8. Prolonging CI and D H to meet at 0, it is seen that the 
angle between the two objects is 2 a 2 0. 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 
is variable, but since all objects observed are at great distances 



102 PRACTICAL ASTRONOMY 

the errors caused by changes in the position of O 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 double 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 op- 
posite side of the direct image. If the shade glasses are of 
different colors the contacts can be more precisely made. Half 
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. 



DESCRIPTION OF INSTRUMENTS 103 

57. 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 
t 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 




-Sextant 



FIG. 49. ARTIFICIAL HORIZON 

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 so accurate as the mercury 
surface but is often more convenient. The principle of the 
artificial horizon may be seen from Fig. 49. Since the image 
seen in the horizon is as far below the true horizon as the sun is 
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 



104 PRACTICAL ASTRONOMY 

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 
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. 

58. Chronometer. 

The chronometer is simply an accurately constructed watch 
with a special form of escapement. Chronometers may be 
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 03*. 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 



DESCRIPTION OF INSTRUMENTS 105 

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 hundred ths 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. 

59. 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. The paper 
on which the record is made is wrapped around a cylinder which is revolved by a 
clock mechanism at a uniform rate, usually once per minute. The pen which makes 
the record is placed on the armature of an electromagnet which is mounted on a 
carriage drawn horizontally by a long screw turned by the same mechanism. The 
mark made by the pen runs spirally around the drum, which results in a series of 
straight parallel lines when the paper is laid flat. The chronometer is connected 
electrically with the electromagnet and records the seconds by making notches in 
the line corresponding to the breaks in the circuit, one centimeter being equivalent 
, to one second. Observations are recorded by the observer by pressing a telegraph 
key, which also breaks (or makes) the chronograph circuit and makes a mark on 
the record sheet. If the transit micrometer is used the passage of the vertical 
thread over fixed points in the field is automatically recorded on the chronograph. 
The circuit with which the transit micrometer is connected operates on the " make " 
instead of the " break " circuit. When the paper is laid flat the time appears as 
a linear distance on the sheet and may be scaled off directly with a scale graduated 
to fit the spacing of the minutes and seconds of the chronograph. 

60. The Zenith Telescope. 

The Zenith Telescope is an instrument designed for making observations for 
latitude by a special method known as the Harrebow-Taicott method. The in- 
strument consists of a telescope attached to one end of a short horizontal axis 
which is mounted on the top of a vertical axis. About these two axes the telescope 
has motions in the vertical and horizontal planes like a transit. A counterpoise 
is placed at the other end of the horizontal axis to balance the instrument. The 
essential parts of the instrument are (i) a micrometer placed in the focal plane of 
the eyepiece for measuring small angles in the vertical plane, and (2) a sensitive 
spirit level attached to the vernier arm of a small vertical circle on the telescope 
tube for measuring small. changes in the inclination of the telescope. The tele- 
scope is ordinarily used in the plane of the meridian, but may be used in any ver- 
tical plane. 

The zenith telescope is put in adjustment by placing the line of sight in a plane 
perpendicular to the horizontal axis, the micrometer thread horizontal, the hori- 
zontal axis perpendicular to the vertical axis, and the base levels in planes per- 
pendicular to the vertical axis. For placing the line of sight in the plane of the 
meridian there are two adjustable stops which must be so placed and clamped 



io6 



PRACTICAL ASTRONOMY 



that the telescope may be quickly turned about the vertical axis from the north 
to the south meridian, or vice versa, and clamped in the plane of the meridian. 

The observations consist in measuring with 
the micrometer the difference in zenith dis- 
tance of two stars, one north of the zenith, 
one south of it, which culminate within a few 
minutes of each other, and in reading the scale 
readings of the ends of the bubble on the lati- 
tude level. The two stars must have zenith 
distances such that they pass the meridian 
within the range of the micrometer. 

A diagram of the instrument in the two* 
positions is shown in Fig. 50. The inclination 
of the telescope to the latitude level is not 
changed during the observation. Any change 
in the inclination of the telescope to the ver- 
tical is measured by the latitude level and 
may be allowed for in the calculation. The 
principle involved in this method may be seen 
from Fig. 51. From the zenith distance of 
the star Ss the latitude would be 




FIG. 50. THE ZENITH TELESCOPE 



and from the star S* 



The mean of the two gives 



4- 



[69] 



showing that the latitude is the mean of the two decimations corrected by a small 
angle equal to half the difference of the zenith distances. The declinations are 
furnished by the star catalogues, 
and the difference of zenith dis- 
tance is measured accurately 
with the micrometer. The com- 
plete formula would include a 
term for the level correction and 
one for the small differential re- 
fraction. This method gives 
the most precise latitudes that 
can be determined with a field 
instrument. 

It is possible for the surveyor 
to employ this same principle if his transit is provided with a gradienter screw and 
an accurate level. The gradienter screw takes the place of the micrometer. A 
level may be attached to the end of the horizontal axis and made to do the work 
of a latitude level. 




FIG. 51 



DESCRIPTION OF INSTRUMENTS 107 

ui. Suggestions about Observing with Small Instruments. 

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. If the ground is shaky three pegs 
may be driven and the points of the tripods set in depressions 
in the top of the pegs. 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 flash light 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. If a lantern is used it should be held so as to heat the 
instrument 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 steps. The observations should be arranged so as 
to eliminate instrumental errors by reversing the instrument; 
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 ob- 
serving. 

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 
same time increase the accuracy. In making observations by 



io8 



PRACTICAL ASTRONOMY 



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 
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. 

62. Errors in Horizontal Angles. 

When measuring horizontal angles with a transit, such, for 
example, as in determining the azimuth of a line from the pole- 
star, any error in the po- 
sition of the sight line, 
or any inclination of the 
horizontal axis will be 
found to produce a large 
error in the result, on 
account of the high alti- 
tude of the star. In 
ordinary surveying these 
errors are so small that 
they are neglected, but 
in astronomical work 
they must either be eliminated or determined and allowed for 
in the calculations. 

In Fig. 52 ZH is the circle traced out by the true collimation 
axis, and the dotted circle is that traced by the actual line of 




H 

FIG. 52. LINE OF SIGHT IN ERROR 
(CROSS-HAIR OUT) 



DESCRIPTION OF INSTRUMENTS 



tan 4 




FIG, 53. PLATE LEVELS ADJUSTED 
BUBBLES NOT CENTRED 



sight, which is in error by the small angle c. The effect of this 
on the direction of a star S is the angle SZH. 

In -Fig. 53 the vertical 
axis is not truly vertical, 
but is inclined by the 
angle i owing to poor 
levelling. This produces 
an error in the direction 
of point P which is equal 
to the angle HZ' P. If 
the vertical axis is truly 
vertical but the horizon- 
tal axis is inclined to the 
horizon by the angle i, 

owing to lack of adjustment, the error in the direction of the 
point (S) is the same in amount and equal to the angle 

H iZ// 2 in Fig. 54. 

Problems 

1. Show that if the sight 
line makes an angle c with the 
perpendicular to the horizontal 
axis (Fig. 52) the horizontal 
angle between two points is in 
error by the angle 

c sec h f c sec /?", 
where h r and h" are the alti- 
tudes of the two points. 

2. Show that if the hori- 
zontal axis is inclined to the 
horizon by the angle i (Figs. 53 

and 54) the effect upon the azimuth of the sight line is i tan h y and that an 
angle is in error by 

i (tan V - tan h"), 
where h' and h" are the altitudes of the points. 




FIG. 54. PLATE LEVELS CORRECT HORIZONTAL 
Axis OUT OF ADJUSTMENT 



CHAPTER IX 
THE CONSTELLATIONS 

63. 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. 

64. 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, ft 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. % 

no 



THE CONSTELLATIONS III 

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, 
o? Capricorni, meaning that the star passes the meridian after 
a 1 Capricorni. 

65. 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. 

66. 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 06' distant from the pole 
at the present time (1925). 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. 55). The 
lower left-hand star of this constellation, the one at the bottom 
of the first stroke of the W, is called b, 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 8 Cassiopeia. If a line be drawn on the sphere 



112 PRACTICAL ASTRONOMY 

between d Cassiopeia and f Ursa Majoris, it wilj 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 ft Cassiopeia, the one at the upper right-hand* 
corner of the W. Its right ascension is very nearly o h and 
therefore the hour circle through it passes nearly through the 
equinox. It is possible then, by simply glancing at ft Cassiopeia 
and the polestar, to estimate approximately the local sidereal 
time. When ft Cassiopeia is vertically above the polestar it 
is nearly o h sidereal time; when the star is below the polestar 
it is i2 h sidereal time; half way between these positions, left and 
right, it is 6 h and i8 h , respectively. In intermediate positions 1 
the hour angle of the star ( = sidereal time) may be roughly 
estimated. 

67. Constellations Near the Equator. 

The principal constellations within 45 of the equator are 
shown in Figs. 56 to 58. 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/ 7 are Aries, Taurus, Gemini, Cancer, Leo, Virgo, 
Libra, Scorpio, Sagittarius, Capricornus, Aquarius, and Pisces. 
These constellations were named many centuries ago, and the 



THE CONSTELLATIONS 113 

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. 56 to 58 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 co-latitude, 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. + 12* to the local civil 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 m = 13*08. The R. A. of sun + i2Ms 25* o8 m , or i* o8 w . 
At 9* P.M. the local civil time is 21*. i h oS m + 21* = 22* 08**. 
A star having a R. A. of 22* o8 m would therefore be close to the 
meridian at 9 P.M. 

Fig. 59 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. 



114 PRACTICAL ASTRONOMY 

68. 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 east or west of the sun; it will therefore be 
seen a little before sunrise or a little after sunset. Mars, Jupi- 
ter, and Saturn move in orbits which are outside of that of the 
earth and therefore appear to us to make a complete circuit of 
the heavens. Mars makes one revolution around the sun in 
i year 10 months, Jupiter in about 12 years, and Saturn in 
29! years. Jupiter is the brightest, and when looked at through 
a small telescope shows a disc like that of the full moon; four 
satellites can usually be seen 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; in a low-power telescope 
the planet appears to be elliptical in form. 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. 

69. 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 Art. 34 and Equa. [45]. 
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 
& 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- 

"5 



n6 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 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], p. 31. The 
true altitude h is derived from the reading of the vertical circle 
by applying the index correction with proper sign and then 
subtracting 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 would be practically a point of light in a perfect telescope, 
but, owing to the imperfections in the corrections for spherical and chromatic 
aberration, the image is irregular in shape and has an appreciable width. The 
image of the star should be bisected with the horizontal cross hair, 



OBSERVATIONS FOR LATITUDE 117 

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 oo 

True altitude = 43 36 30 

Polar distance = __ i 15 _2 ^ 

Latitude =~42~2i'~os" 

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' 30"; index correction = o"; declination = + 87 n' 25". 

Observed altitude = 39 33' 30" 
Refraction correction = i 09 



True altitude 39 3 2 2I 

Polar distance == 2^48 35^ 

Latitude == 4~2~2o' 56" 

70. 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 local 
mean time (whichever is used) as shown in Arts. 28 and 32. 
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, 



Il8 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 the sun's declination may be corrected as 
follows: If the Greenwich Time or the Standard Time is noted 
at the instant of the observation the number of hours since 
o* Gr. Civ. Time is known at once. If the time has not been 
observed it may be derived from the known longitude of the 
place. Since the sun is on the meridian the local apparent time 
is 12*. Adding the longitude we obtain the Gr. App. Time. 
This is converted into Gr. Civil Time by subtracting the equa- 
tion of time. The declination is then corrected by an amount 
equal to the " variation per hour " multiplied by the hours of 
the Gr. Civ. Time. The time need not be computed with great 
accuracy since an error of i m will never cause an error greater 
than i" in the computed declination. The latitude is com- 
puted by applying equation [i] or its equivalent. 

Example i. Observed maximum altitude of the sun's lower limb, Jan. 15, 1925, 
= 26 15' (sun south of zenith); index correction, -f-i'; longitude 7io6'W.; 
sun's declination Jan. 15 at o^ Greenwich Civil Time = 21 15' ig /f .4, variation 
per hour, -|~26".89; Jan. 16, 21 04' 2i".g; variation per hour, + 27".9o; equa- 
tion of time, ~g m 17 s ; semidiameter, 16' if '.53. 

Observed altitude = 26 15' Loc. App. Time = 12* 

Index correction -f-i' Longitude = 4 h 44 m 24 s 

26 1 6' Gr. App. Time = 16^44 24* 

Refraction ~i-9 Equa. Time = 9 17 

7 Gr. Civ. Time = 16" 53 41* 
Semidiameter 




Parallax -f-.i Decl. at o = -21 04' 21^.9 

263o'.s -f-27". 9 o X 7 h -i = 3 18 * 

Declination 21 07.7 Corrected Decl. = 21 of 4o".o 

Co-latitude 47 3 8'.2 

Latitude 42 ai',8 

Example 2. Observed maximum altitude of sun's lower limb June i, 1925 = 
44 48' 30" bearing north; index correction = o"; Gr. Civil Time = 14* 50"* 12*, 



OBSERVATIONS FOR LATITUDE 

declination of sun at o>, G. C. T., = +21 57' 13". 7; variation per hour, +21". n; 
semidiameter, 15' 48^.05 . 



Observed altitude 44 48' 30" 
Refraction 57 



Semidiameter 
h 
f = 9 o - h 


44 4/33" 
15 48 


45 03' 21" 
44 56 39 

+ 22 02 27 


2254 / i2 // South 



Decl. at o* = +21 57' 13".? 

+ 2i".u X i4*-84 = +5 13 -3 
Corrected Decl. +22 02' 27^.0 




71. 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 
* The observer is assumed to be in the northern hemisphere. 



120 PRACTICAL ASTRONOMY 

semidiameter corrections become zero, and that it is not neces- 
sary 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. 76. 

Example. Observed meridian altitude of 9 Serpentis = 51 45'; index cor- 
rection = o; decimation of star = +4 05' n". 

Observed altitude of Serpentis = 51 45' oo" 

Refraction correction = 45 

51 44' 15" 

Declination of star = + 4 05 n 

Co-latitude = 47 39' 04" 

Latitude = 42 20 56 

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. 

72. 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 with suffi- 
cient accuracy. To derive the formula for making the reduction to the meridian 
we employ Equa. [8], p. 32. 

sin h = sin < sin 8 + cos <f> cos 5 cos /. [8] 

This is equivalent to 

sin h = cos (< 6) cos <f> cos 6 vers t [70] 

or 

sin h cos (< 5) cos 4> cos 5 2 sin 2 - [71] 

Denoting by km the meridian altitude, 90 (<f> 6), the equations become 

sin hm sin h -f cos # cos 5 vers t [72] 

sin hm = sin h -j- cos </> cos 8 2 sin 2 - . [73] 

If the time is noted when the altitude is measured the value of t may be computed, 
provided the error of the timepiece is known. With an approximate value of <j> 
the second term may be computed and the meridian altitude hm found through its 
sine. If the latitude computed from hm differs much from the preliminary value 
a second computation should be made, using the new value for the latitude. These 
equations are exact in form and may be used even when t is large. The method 
may be employed when the meridian observation cannot be obtained. 

Example. Observed double altitude of sun's lower limb Jan, 28, 1910, with 
sextant and artificial horizon. 



OBSERVATIONS FOR LATITUDE 



121 



Mean 
I.C. 



Double Altitude 

56 44' 40" 
49 oo 
52 4Q 



Watch 
25* 



16 
17 



22 
10 



Ref r. and par. 



56 48' 47" 

+3Q 

2) 5 6 49 ' 17" 
28 2 4 ' 38" 
i 38 



Semidiameter 



28 23' oo" 
+ 16 16 



Watch corr. 

E. S. T. of observ. 

E. S. T. of app. noon 

Hour angle = 

/ = 



19* 
19 



n h 1 7 
ii 57 



39 W 43 s 
9 55' 45" 



log cos <f> 
log cos 5 
log vers t 
log corr. 
corr. 

nat. sin h 
nat. sin km 



h = 28 39' 16" 

= 9.86763 

= 9-97745 

= 8.17546 

= 8.02054 

= .01048 

= -47953 

= .49001 

hm = 29 20' 29" 

^ = 60 39 31 

5 = 18 18 20 



Assumed latitude 
Declination 

Loc. app. noon 
Equa. time 

Long. diff. 
E. S. T. of noon 


42 30' 
-i8i8' 20" 

1 2 h 00 m 00 s 
-13 03 


I2 h I Tm 035 
15 42 


n'57 m 21* 



<t> = 4 22l'll"N. 

Note: A recomputation of the latitude, using this value, changes the result to 

42 21' 04" N. 

When the observations are taken within a few minutes of meridian passage the 
following method, taken from Serial No. 166, U. S. Coast and Geodetic Survey, 
may be employed for reducing the observations to the meridian. This method 
makes it possible to utilize all of the observations taken during a period of 20 min- 
utes and gives a more accurate result than would be obtained from a single meridian 
altitude. 

From Equa. [73] 

sin hm sin h = 2 cos <j> cos 5 sin 2 - [74] 

By trigonometry, 

sin hm ~ sin h = 2 cos ^ (hm + h) sin \(hm h) 
therefore 



(hm ti) cos cos d sin 2 - sec \ (hm + h) 
2 



(75} 



since hm h is small, we may replace sin J (hm h) by \ (hm h} sin i"; and also 
replace i (h m + h) by h = 90 $* . 
Then [75] becomes 



hm h = cos cos 



" cosec f 



or 



h + cos ^ cos $ cosec f 



2 sin 2 - 
2 

sin j" 



[76] 



122 PRACTICAL ASTRONOMY 

Placing A = cos <j> cos 5 cosec f 

and m 

Then km 

The latitude is then found by 




[771 



Values of m will be found in Table X and values of A in Table IX. The errors 
involved in this method become appreciable when the value of / is more than 10 
minutes of time. 

The observations should be begun about 10 minutes before local apparent 
noon (or meridian passage, if a star is being observed) and continued until about 
10 minutes after noon. The chronometer time or watch time of noon should be 
computed beforehand by the methods as explained in Chapter V. In the example 
given on p. 123 the chronometer was known to be 27^ 2o m slow of local civil time, 
and the equation of time was $ m 53*. The chronometer time of noon was there- 
fore 12** H- 5 m 53* 27 20* = ii* 38"* 33*. 

The values of / are found by subtracting the chronometer time of noon from the 
observed times. The value of m is taken from Table X for each value of t. A is 
taken from Table IX for approximate values of <f>, d and |". The values of the 
correction Am are added to the corresponding observed altitudes. The mean of 
all of the reduced altitudes, corrected for refraction and parallax, is the true meri- 
dian altitude of the centre. 

73. Latitude by Altitude of Polaris when the Time is Known. 
The latitude may be found conveniently from an observed altitude of Polaris 
taken at any time provided the error of the timepiece is approximately known. 
Polaris is but a little more than a degree from the pole and small errors in the time 

have a relatively small effect upon the 
result. It is advisable to take several 
altitudes in quick succession and note 
the time at each pointing on the star. 
Unless the observations extend over a 
long period, say more than 10 minutes 
of time, it will be sufficiently accurate 
to take the mean of the altitudes and 
the mean of the times and treat this as 
the result of a single observation. If 
the transit has a complete vertical cir- 
cle, half the altitudes may be taken 
with the telescope in the direct posi- 
tion, half in the reversed position. 
The index correction should be care- 
fully determined. 

The hour angle (/) of the star must 
be computed for the instant of the 
FIG. 60 observation. This is done according 

to the methods given in Chapter V. 

In the following example the watch is set to Eastern Standard Time. This is first 
converted into local civil time (from the known longitude) and then into local 



A 




OBSERVATIONS FOR LATITUDE 



123 



Example. 



OBSERVATIONS OP SUN FOR LATITUDE 



Station, Smyrna Mills, Me. 
Theodolite of mag'r No. 20. 
Chronometer No. 245. 



Date, Friday, August 5, 1910 

Observer, H. E. McComb 

Temperature, 24 C. 









Vertical circle 


Sun's 


VP 






limb. 


. \s. 




A. 


B. 


Mean. 


u 


R 


n* 30 O4 S 


61 14/00" 


13 '3" 


6ii3'45" 


L 


L 


ii 31 16 


119 23 00 


20 00 


60 38 30 


L 


L 


ii 33 14 


119 22 30 


19 30 


60 39 oo 


U 


R 


ii 34 38 


61 16 30 


15 30 


61 16 oo 


U 


R 


ii 3 6 3 6 


61 17 oo 


15 30 


61 16 15 


L 


L 


ii 37 34 


119 21 30 


19 


60 39 45 


L 


L 


ii 39 32 


119 21 30 


19 oo 


60 39 45 


U 


R 


ii 40 33 


61 17 30 


16 oo 


61 16 45 


U 


R 


ii 42 46 


61 16 30 


15 oo 


61 15 45 


L 


L 


ii 43 30 


119 22 30 


20 00 


60 38 45 




Obs'd. max. alt. 


60 58 15 




R&P 


-27 




h 


60 57 48 




f 


29 02 12 




8 


17 06 18 




<i> 


46 08 30 



COMPUTATION OF LATITUDE FROM CIRCUMMERIDIAN 

ALTITUDES OF SUN 

Station, Smyrna Mills, Me. Date, August 5, 1910. 

Chron. correction on L. M. T. H-27 m 20* 
Local mean time of app. noon 12 05 53 
Chron. time of apparent noon ii 38 33 



/ 


m 


A 


Am 


Reduced h. of 
sun's limb. 


Reduced h. of sun. 


-gwi 29* 


i 4 i" 


1.36 


I 9 2" 


61 i6'57" 




-7 17 


104 




141 


60 40 51 


60 58' 54" 


-5 19 


56 




76 


60 40 16 




-3 55 


30 




41 


61 16 41 


60 58 28 


-i 57 


8 




II 


61 16 26 




-o 59 


2 




3 


60 39 48 


60 58 07 


+o 59 


2 




3 


60 39 48 




+ 2 OO 


8 




ii 


61 16 56 


60 58 22 


+4 13 


35 




48 


61 16 33 




+4 57 


48 




65 


60 39 50 


60 58 12 




Mean 


60 58 25 




R. &P. 


-27 




h 


60 57 58 




r 


29 02 02 




5 


17 06 19 







46 08 21 



124 PRACTICAL ASTRONOMY 

sidereal time (see Art. 37). The hour angle, /, is the difference between the sidereal 
time and the star's right ascension. 
The latitude is computed by the formula 

<i> = h - p cos / 4- 2- p* sin 2 / tan h sin i" [78] 

the polar distance, p, being in seconds. For the derivation of this formula see 
Chauvenet, Spherical and Practical Astronomy, Vol. I, p. 253. 

In Fig. 60 P is the pole, 5 the star, MS the hour angle, and PDA the parallel 
of altitude through the pole. The point D is therefore at the same altitude as 
the pole. The term p cos t is approximately the distance from S to E, a point on 
the 6-hour circle PB. The distance desired is SD, the diffeernce between the alti- 
tude of S and the altitude of the pole. The last term of the formula represents 
very nearly this distance DR. When 5 is above the pole DE diminishes SE; 
when S is below the pole it increases it. 

Example i. 

Observed altitudes of Polaris, Jan. 9, 1907 

Watch Altitudes 

649" 26* 43 28' 30" 

5 1 45 28 30 

54 14 28 oo 

56 45 28 oo 

Mean 6 h 53 02^.5 Mean 43 28' 15" 

Index correction, i' oo"; p = i n' 09" = 4269"; t is found from the observed 
watch times to be 13 50'. 7.* 

log p = 3-63033 log const. = 4-3845 

log cos t = 9.98719 log p 2 = 7.2607 

i / L j\ ~~~s loe sin 2 / 

ft?, 008 ' } : 



0.3798 

last term = H-2".4 

Observed alt. = 43 28' 15" 
Index corr. = i oo 
Refraction = i oo 

43 26' 15" 
ist and 2nd terms i 09 03 

Latitude 42 if 12" N. 

This computation may be greatly shortened by the use of Table I of the Epheme- 
ris, or Table I of the Nautical Almanac. In the Ephemeris the total correction to 
the altitude is tabulated for every 3 of hour angle and for every 10" of declination. 
In the Almanac the correction is given for every io m of local sidereal time. 

Example 2. 

The observed altitude of Polaris on March 10, 1925 = 42 20'; Watch time = 
8* 49 30* P.M.; watch 30* slow of E. S. T. Long. 71 10' W. Index correction, 
-fi r . Declination of Polaris, +88 54' 1 8"; right ascension, i* 33 35*.6 

* If the error of the watch is known the sidereal time may be found by the 
methods explained in Chap, V. For methods of finding the sidereal time by direct 
observation see Chap. XI. 



OBSERVATIONS FOR LATITUDES 125 

Watch 8* 49 30* Observed alt. 42 20' 

Error 30 I. C. +i 

E. S. T. 8* 50"* oo P.M. R efr. -i .0 

Civ. Time = 20 50 oo h 42 20'. o 

Dif. Long. 15 20 Corr., Table I +13 .0 

Loc. Civ. Time = 21*05 2oS Latitude 42 33^.0 

Table III = 3 27 .9 

Sun's R. A. -f- 1 2 ft = ii 08 36.1 Note: From the Almanac the correc- 

Table III (Long.) 46_.8 tion for Loc. Sid. Time 8^ i8 io.8 

32^ i S m 10 s . 8 i s H-IS'.O. From the Ephemeris 

24 the correction for hour angle 6* 

Loc. Sid. Time = "8* 18" io*.8 "? ;.? and Declination +88 

Rt.Asc.Star = 133 35.6 & l8 * +13' ". The latter 

T , f . ^ 2 is more accurate. 

Hour Angle of Polaris 6 h 44 35^.2 

Example 3. Observed altitude of Polaris May 5, 1925 = 41 10' at Gr. Civ. 
Time 23^ 50"*. Longitude 5^ West. 

Gr. Civ. Time 23^ 50 oo*. 

Table III 3 54.91 

R. A. -J-i2* 14 49 23 .08 

38^43^17^.99 

24 

Greenwich Sidereal Time 14^43^* I 7 s -99 

_5 

Loc. Sid. Time 9^43 i7*.99 

R. A. Polaris i 33 30 .68 

Hour Angle, t, 8 h og m 47^.31 

Observed Altitude 41 10' oo" 

Refraction - i 05 

41 08' 55" 
Correction, Table I (Eph.) +35 58 

Correction, Table la (Eph.) 5 

Latitude 4i44 / 48"N. 

74. Precise Latitudes Harrebow-Talcott Method. 

The most precise method of determining latitude is that known as the " Har- 
rebow-Talcott " Method, in which the zenith telescope is employed. Two stars 
are selected, one of which will culminate north of the observer's zenith, the other 
pouth, and whose zenith distances differ by only a few minutes of angle. For 
convenience the right ascensions should differ by only a few minutes of time, 
say 5 to io m . The approximate latitude must be known in advance, that is, 
within i' or 2', in order that the stars may be selected. This may be determined 
with the zenith telescope, using the method of Art. 71. It will usually be neces- 
sary to consult the star catalogues in order to find a sufficient number of pairs 
which fulfill the necessary conditions as to difference of zenith distance and differ- 
ence of right ascension. 

If the first star is to culminate south of the zenith the telescope is turned until 
the stop indicates that it is in the plane of the meridian, on the south side, and then 
clamped in this position. The mean of the two zenith distances is then set on the 
finder circle and the telescope tipped until the bubble of the latitude level is in the 
centre of its tube. When the star appears in the field it is bisected with the mi- 
crometer wire; at the instant of passing the vertical wire, that is, at culmination, 
the bisection is perfected. The scale readings of the ends of the bubble of the lati- 



126 PRACTICAL ASTRONOMY 

tude level are read immediately, then the micrometer is read and the readir 
recorded. The chronometer should also be read at the instant of culminatior 
order to verify the setting of the instrument in the meridian. 

The telescope is then turned to the north side of the meridian (as indicated by 
the stop) and the observations repeated on the other star. Great care should be 
taken not to disturb the relation between the telescope and the latitude level. 
The tangent screw should not be touched during observation on a pair. 

When the observations have been completed the latitude may be computed by the 
formula 

4> = J(fc + 8 n ) + %(m s - mn) X R + Hfe + + l(r, - r n ) [79] 

in which ms t mn, are the micrometer readings, R the value of i division of the 
micrometer, l s , /, the level corrections, positive when the north reading is tl 
larger, and r$ , rn, the refraction corrections. Another correction would be require 
in case the observation is taken when the star is not exactly on the meridian. 

In order to determine the latitude with the precision required IP geodetic opt 
ations it is necessary to observe as many pairs as is possible during one night (say 
10 to 20 pairs). In some cases observations are made on more than one night in 
order to secure the necessary accuracy. By this method the latitude may readily 
be determined within o".io (or less) of the true latitude, that is, with an error 
of 10 feet or less on the earth's surface. 

Questions and Problems 

1. Observed maximum altitude of the sun's lower limb, April 27, 1925 = 61 28', 
bearing South. Index correction = 4-30". The Eastern Standard Time is 
ii* 42 A.M. The sun's declination April 27 at o>* Gr. Civ. T. = +13 35' 51". 3; 
the varia. per hour is -|-48".i9; April 28, +13 55' oi".o; varia. per hour, 4-47^.62; 
the semidiameter is 15' 55^.03 . Compute the latitude. 

2. Observed maximum altitude of the sun's lower limb Dec. 5, 1925 = 30 10'. 
bearing South. Longitude = 73 W. Equation of time = 4-9 22*. Sun's decli- 
nation Dec. 5 at cfi Gr. Civ. T. = 22 16' 54". o; varia. per hour 19^.85; 
Dec. 6, 22 24'37".s; varia. per hour, 18".77; semidiameter, 16' is // .84. 
Compute the latitude. 

3. The noon altitude of the sun's lower limb, observed at sea Oct. i, 1925 = 
40 30' 20", bearing South. Height of eye, 30 feet. The longitude is 35 10' W. 
Equation of time = -f-io 03*.56. Sun's declination Oct. i at o ft Gr. Civ. T. = 
2 53' 38". 2; varia. per hour = -58". 28; on Oct. 2, 3 16' 56".!; varia. per 
hour, 58". 20; semidiameter = i6'oo".57. Compute the latitude. rj 4 

4. The observed meridian altitude of 8 Crateris = 33 24', bearing South; 
index correction, +30"; declination of star = 14 17' 37". Compute the lati- 
tude. 

5. Observed (ex. meridian) altitude of a Celt at 3^ o8" 49$ local sidereal tim 
SB 51 21'; index correction = i'; the right ascension of a. Ceti = 2^ 57 24^.0 
declination = +3 43' 22". Compute the latitude. 

6. Observed altitude of Polaris, 41 41' 30"; chronometer time, o* 44 385.5 
(local sidereal); chronometer correction =* 34*. The right ascension of Polaris 
i 25 42*; the declination == -{-88 49' 29". Compute the latitude. 

7. Show by a sketch the following three points: i. Polaris at greatest elonga- 
tion; 2. Polaris on the 6-hour circle; 3. Polaris at the same altitude as the pole. 
(See Art. 73, p. 122, and Fig. 28, p. 37.) 

8. Draw a sketch (like Fig. 19) showing why the sun's maximum altitude is 
not the same as the meridian altitude. 



CHAPTER XI 
OBSERVATIONS FOR DETERMINING THE TIME 

75. Observation for Local Time. 

Observations for determining the local time at any place at 
y instant usually consist in finding the error of a timepiece 
the kind of time which it is supposed to keep. To find the 
Jiar time it is necessary to determine the hour angle of the sun's 
entre. To find the sidereal time the hour angle of the vernal 
^quinox 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. 

76. 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 
t^v* Ephemeris for the date. The difference between the ob- 
. irved chronometer time T and the right ascension a is the 
chronometer correction AT, 

>r AT - a - T. [So] 

If the chronometer keeps mean solar time it is only necessary 
to convert the true sidereal time a into mean solar time by 

127 



128 PRACTICAL ASTRONOMY 

Equa. [45], 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 bejl 
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. 
[45]) and the appearance of the star in the field looked for a 
little in advance of this time. If the data from the Ephemeris t 
are not at hand the computation may be made, with sufficient* 
accuracy for finding the star, by the following method: Com- 
pute 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. 
+ 12*) will be the mean local time within perhaps 2 m or 
3**. This may be reduced to Standard Time by the method 
explained in Art. 32. In the surveyor's transit the field of view 
is usually about i, so the star will be seen about 2 m before it 
reaches the vertical cross hair. Near culmination the star's 



OBSERVATIONS FOR DETERMINING THE TIME I2Q 

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 observations with the surveyor's 
transit. When a chronometer is used the " eye and ear method " 
is the best. (See Art. 61.) 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 altitude at culmination. (See Art. 71.) 

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 error may be found at once since the star's 
right ascension is the local sidereal time. If civil time is desired, 
the true sidereal time must be converted into local civil time, or 
into Standard Time, whichever is desired. 

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. 

Example. 

Observed the transit of a Hydra on April 5, 1925, in longitude 
5 71 2o m west. Observed watch time (approx. Eastern Standard 
Time) = 8* 48 24* P.M. or 20* 48 24* Civil Time. The right 
ascension of a Hydra for this date is g h 23 54^.84; the R. A. 
of the mean sun +12* is 12* 5 i m 06*48 at o h Gr. Civ! T. From 
Table III the correction for 5* 20 is + 52^.57. 

Rt. Asc. of Hydra +24* = 33** 23 54^.84 
Corrected R. A. sun -f- 1 2* = 12 51 59 .05 

Sid. int. since MVt. = 26^ 31 55^.79 

Table II = -3 21 .82 

Local Civil Time = 20^ 28"* 33^.97 

Red. to 75 merid. = 20 oo .00 

Eastern (Civ.) Time = 20* 48 33^.97 

Watch = 20 48 24 

Watch correction = +9^.97 (slow) 



130 PRACTICAL ASTRONOMY 

77. Observations with Astronomical Transit. 

The method previously described for the small transit is the same in principle 
as that used with the larger astronomical transits for determining sidereal time- 
The chief difference is in the precision with which the observations are made and 
the corrections which have to be applied to allow for instrumental errors. The 
number of observations on each star is increased by using several vertical threads 
or by employing the transit micrometer. These are recorded on the electric chrono- 
graph and the times may be scaled off with great accuracy. 

When the transit is to be used for time determination it is set on a concrete or 
brick pier, levelled approximately, and turned into the plane of the meridian as 
nearly as this is known. The collimation is tested by sighting the middle thread 
at a fixed point, then reversing the axis, end for end, and noting whether the thread - 
is still on the point. The diaphragm should be moved until the object is sighted 
in both positions. The threads may be made vertical by moving the telescope 
slowly up and down and noting whether a fixed point remains on the middle thread. 
The adjustment is made by rotating the diaphragm. To adjust the line of sight 
(middle thread) into the meridian plane the axis is first levelled by means of the 
striding level, and an observation taken on a star crossing the meridian near the 
zenith. This star will cross the middle thread at nearly the correct time even if 
the instrument is not closely in the meridian. From this observation the error of 
the chronometer may be obtained within perhaps 2 or 3 seconds. The chronometer 
time of transit of a circumpolar star is then computed. When this time is indicated 
by the chronometer the instrument is turned (by the azimuth adjustment screw) 
until the middle thread is on the circumpolar star. To test the adjustment this 
process is repeated, the result being a closer value of the chronometer error and a 
closer setting of the transit into the plane of the meridian. Before observations 
are begun the axis is re-levelled carefully. 

The usual list of stars for time observations of great accuracy would include 
twelve stars, preferably near the zenith, six to be observed with the " Clamp east," 
six with " Clamp west." This division into two groups is for the purpose of de- 
termining the collimation constant, c. In each group of six stars, three should be 
north of the zenith and three south. From the discrepancies between the results 
of these two groups the constant a may be found for each half set. Sometimes a 
is found by including one slow (circumpolar) star in each half set, its observed time 
being compared with that of the " time stars," that is those near the zenith. The 
inclination of the horizontal axis, b, is found by means of the striding level. The 
observed times are scaled from the chronograph sheet for all observations, and the 
mean of all threads taken for each star. This mean is then corrected for azimuth 
error by adding the quantity 

a sin f sec 5. [66] 

The error resulting from the inclination of the axis to the horizon is corrected by 
adding 

b cos sec 8 [67] 

and finally the collimation error is allowed for by adding 

c sec 5. [68] 



OBSERVATIONS FOR DETERMINING THE TIME 131 

Small corrections for the changing error of the chronometer and for the effect of 
diurnal aberration of light are also added. The final corrected time of transit is 
subtracted from the right ascension of the star, the result being the chronometer 
correction on local sidereal time. The mean of all of the results will usually give 
the time within a few hundredths of a second. 

78. 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, 
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. 99, 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 
>ight the telescope at much greater altitudes than about 70 



132 



PRACTICAL ASTRONOMY 



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 42 22' N., longitude, 7io6'W., 
date, May 5, 1925; the hours, between 8* and g h P.M., Eastern 
Standard Time. The limiting altitudes chosen are 10 and 65. 
The " sidereal time of o* Greenwich Civil Time/ 7 or " Right 
ascension of the mean sun +i2 h ," is 14* 49 23^.08. The local 
civil time corresponding to 2o h E. S. T. is 20^15^36*. The 
local sidereal time is therefore 20* 15** 36* + 14^ 49 23* + a cor- 
rection from Table III (which may be neglected for the present 
purpose) giving about 11^05^ for the right ascension of a star 
on the meridian at 8^ P.M. Eastern time. 

The co-latitude is 47 38', the meridian altitude of a star on 
the equator. For altitudes of 10 and 65 this gives for the 
limiting declinations +17 22' and 37 38' respectively. 

In the table of " Mean places " of Ten-Day Stars (1925) the 
following stars will be found. The complete list contains some 
800 stars. In the following list many stars between those given 
have been intentionally omitted, as indicated by the dotted lines. 



Star 


Mag. 


Rt. Asc. 


Decl. 










a Crateris . ... 


4 . 2 


loft ^6 m O7 5 .io5 


17 53' 57"-53 


d Leonis 


S-o 


10 56 41 .272 


+ 4 01 13 .67 


/3 Cratcris 


4-5 


ii 07 58 .012 


22 24 58 .47 


8 Leonis . . . 


2.6 


ii 10 07 .379 


-j-2O 56 05 .33 










TT Centauri 


4-3 


ii 17 34 .823 


-54 04 47 .36 


X Draconis 


4.1 


ii 26 58 .349 


-[-69 44 42 .71 


Hydros 


3-7 


ii 29 18 .582 


~3i 26 33 .36 


IT Chameleontis . , , . 
3 DTaconis 


5-7 
5.5 


ii 34 09 .389 
ii 38 18 .337 


-75 28 52 .97 
4-67 09 36 .24 


f Cr&tcris .... 


4.9 


ii 40 57 .535 


17 56 01 .41 










y Cofvi 


2.8 


12 ii 56 .763 


17 07 31 .85 











OBSERVATIONS FOR DETERMINING THE TIME 



133 



In this list there are three stars, ft Crateris, Hydra, and 
f Crateris, whose decimations and right ascensions fall within the 
required limits. There are 13 others which could be observed 
but were omitted in the above list to save space. After select- 
ing the stars to be observed the approximate watch time of 
transit of the first star should be computed. The times of the 
other stars may be estimated with sufficient accuracy by means 
of the differences in the right ascension. The watch times 
will differ by almost exactly the difference of right ascension. 
The altitudes (or the zenith distances) should be computed 
to the nearest minute. This partial list would then appear as 
follows: 



Star 


Mag. 


Approx. E. S. T. 


Approx. Alt. 


Crateris 
Hydros 


4-5 
3 -7 


19* 58 49* 
20 20 09 


25 13' 

16 ii 


Crateris 


4-9 


20 31 48 


29 42 











In searching for stars the right ascension should be examined 
first. As the stars are arranged in the list in the order of in- 
creasing right ascension it is only necessary to find the right 
ascension for the time of beginning the observations and then 
follow down the list. Next check off those stars whose decli- 
nations fall within the limits that have been fixed. Finally 
, note the magnitudes and see if any are so small as to make the 
star an undesirable one to observe. 

When the stars have been selected, look in the table of " Ap- 
parent Places of stars " to obtain the right ascension and decli- 
nation for the date. These may be obtained by simple inter- 
polation between the values given for every 10 days. The 
mean places given in the preceding table may be in error for any 
particular date by several seconds. With the correct right 
ascensions the exact time of transit may be calculated as pre- 
viously explained. 



134 PRACTICAL ASTRONOMY 

79. Time by Transit of Sun, 

The apparent solar time may be determined directly by ob- 
serving the watch times when the west and the east limbs of the 
sun cross the meridian. The mean of the two readings is the 
watch time of the instant of Local Apparent Noon, or 12* ap- 
parent time. This i2 h is to be converted into Local Civil Time 
and then into Standard Time. If only one edge of the sun's 
disc can be observed the time of transit of the centre may be 
found by adding or subtracting the " time of semidiameter 
passing the meridian." This is given in the Ephemeris for 
Washington Apparent Noon. The tabulated values are in 
sidereal time, but may be reduced to mean time by subtracting 
o*.i8 or o*.i9 as indicated in a footnote. 

Example. 

The time of transit of the sun over the meridian 71 06' W. is to be observed 
March 2, 1925. 

Local Apparent Time =12'* oo m oo 8 G. C. T. = 16^.95 
Equation of Time = 12 19 .93 

Local Civil Time = 12'* 12 ig s . 93 Equa. of Time at o^ = i2 m i6*.29 

Longitude diff.> 15 36 . 0^.517 X 7^.05 = 3 .64 

Eastern Standard Time = n h 56 43^93 Corrected Equa. of T. = i2 m 10^.93 

The observed time of the west and east limbs are n^ 55 47* and n h 57 56* re- 
spectively. The mean of these is n^ 56^ 51^.5; the watch is therefore 7*. 6 fast. 
The time of the semidiameter passing the meridian is i w o5*.i6. If the second 
observation had been lost the watch time of transit of the centre would be n* 55"* 
47* H- io5*.i6 = n h 56 5 2 s . 1 6, and the resulting watch correction would be 

-8.2. 

80. 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 



OBSERVATIONS FOR DETERMINING THE TIME 135 

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 
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 at o h Gr. Civ. 
Time is to be corrected by adding the " variation per hour " 
multiplied by the number of hours in the Greenwich civil time. 
If the watch used is keeping Standard time the Greenwich time 
is found at once (see Art. 32). If the watch is not more than 
2 m or 3 m in error the resulting error in the declination will not 
exceed 2" or 3", which is usually negligible in observations with 
small instruments. If the Standard time is not known but the 
longitude is known then the Greenwich time could be com- 
puted if the local time were known. Since the local time is the 
quantity sought the only way of obtaining it is first to compute 
the hour angle (/) using an approximate value of the declination. 
From this result an approximate value of the Greenwich civil 
time may be computed. The declination may now be computed 
more accurately. A re-computation of the hour angle (/), 
using this new value of the declination, may be considered 
final unless the declination used the first time was very much in 
s error. 

In order to compute the hour angle the latitude of the place 
must be known. This may be obtained from a reliable map or 
may be observed by the methods of Chapter X. The precision 
with which the latitude must be known depends upon how pre- 
cisely the altitudes are to be read and also upon the time at 



136 



PRACTICAL ASTRONOMY 



which the observation is made. When the sun is near the prime 
vertical the effect of an error in the latitude is small. 

The value of the hour angle is computed by applying any of 
the formulae for / in Art. 19. This hour angle is converted into 
hours, minutes and seconds; if the sun is west of the meridian 
this is the local apparent time P.M., but if the sun is east of the 
meridian this time interval is to be subtracted from i2 h to obtain 
the local apparent time. This apparent time is then converted 
into mean (civil) time by subtracting the (corrected) equation 
of time. The local time is then converted into Standard time 
by means of the longitude difference. The difference between 
the computed time and the time read on the watch is the watch 
correction. This observation is often combined with the ob- 
servation on the sun for azimuth, the watch readings and alti- 
tude readings being common to both. 



Example. 

Nov. 28, 1925. 

Lower limb 
(Tel. dir.) 

Upper limb 

(Tel. rev;) 

Mean 

Refraction and parallax 



is 

15 55 

16 08 

15 26'.o 

3 -3 

iS22 / . 7 



Lat. 42 22'; Long. 71 06'. 
Watch (E. S. T.) 
S h 39 W 42* A.M. 
8 42 19 

8 45 34 
8 47 34 



8" 43 47*. 2 E. S. T. (Approx.) 

5 



4> = 42 22' 

h - 15 22 .7 

p = III 17 .2 

25= 169 Ol'.9 

s 84 30^.9 log cos 8.98039 
s h = 69 08 .2 log sin 9.97055 
^ < SB 42 08 .9 log csc 0.17324 
s _ p = _. 26 46 .3 log sec 0.04924 

2)0.17342 



13* 43 m 47 s - 2 Gr. Civ. T. (Approx.) 

Decl. at <# = - 21 10' 58".8 

-27".i4 X i3*-7 = -6 ii .8 

5= -2ii7 / io".6 

p = in, 17 10 .6 



Eq. of t. at 
o*.828 X i3* 
Eq. of t. 



14*. 56 
ii .34 



-fi2o3*.22 



log tan - 



' 9 58671 



o ^A> 



= 21 06' 43 

42 13' 26" 
: 2 4 8* 53^.7 



OBSERVATIONS FOR DETERMINING THE TIME 137 

L. A. T. = 9*11 06X3 
Eq. of t. = +12 03 .2 

Loc. Civ. T. = 8* 59 03*. i 
Long, cliff. == 15 36 . 

Eastern Standard Time = 8 h 43"* 275.1 
Watch = 8 43 47 .2 

Watch fast 20*. i 

The most favorable conditions for an accurate determination 
of time by this method are when the sun is on the prime vertical 
land the observer is on the equator. When the sun is east or 
west it is rising or falling at its most rapid rate and an error in the 
altitude produces less error in the calculated houf angle than the 
same error would produce if the sun were near the meridian. The 
nearer the observer is to the equator the greater is the inclina- 
tion 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 preceding example the rise is i' in 
about & 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 correc- 
tion due to variations in temperature and pressure of the air 
are likely to be large. Observations should not be made when 
the altitude is less than 10 if it can be avoided. 

81. 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 me- 
ridian 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. 37. Since it is easy to select 



138 PRACTICAL ASTRONOMY 

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 Gr. Civ. Time with sufficient accuracy for correcting 
the right ascension and declination. 

Example. 

Observed altitude of a Bo'otis (Arcturus) on Apr. 15, 1925 = 40 10' (east). 
Watch, 8* 54 20* P.M. Latitude = 42 18' N., Long. = 71 18' W. Rt. Asc. a 
Bodtis, 14* i2 m 15^.6; decl. -f *934' 14"- Rt. Asc. Mean Sun +12* = i3^3O> 

32*.OI. 

Obs. alt. 40 10' 
Refr. -i .1 

h 40 o8'.9 

= 42 iS'.o log sec 0.13098 

h = 40 08 .9 

p = 70 25 .8 log esc 0.02584 

2)152 52 .7 

s = 76 26'.3 log cos 9-37 OI 3 
s h ~ 36 17 .4 log sin 9.77223 



log sin - = 9.64959 

-'= 26 30' 15" 

/ = 53 oo' 30" (east) 
= 3/ 32^02* (east) 

Rt. Asc. of star = 14* 12^* 153.6 

Loc. Sid. T. = io^4oi3*.6 

Long. W. = 4 45 12 . 

Gr. Sid. T. * 15* 25"* 25^.6 
R. A. Sun 4-i2 ft = 13 30 32 .o 

i* 54 m 53*-6 
Table II 18 .8 

Gr. Civil T. = i* 54^ 34^.8 

J _ 

Eastern. Stand. Time 20* 54"* 34*.8 

= 8 54 34.8P.M. 
Watch 8 54 20 P.M. 

Watch i4*.8 slow 



OBSERVATIONS FOR DETERMINING THE TIME 139 

82. Effect of Errors in Altitude and Latitude. 

In order to determine the exact effect upon i of any error in 
the altitude h let us differentiate equation [8] with respect 
to h, the quantities <t> and 5 being regarded as constant. 

sin h sin <t> sin d + cos < cos 8 cos t. [8] 

Differentiating, 

dt 

cos h = o cos 6 cos 5 sm / 

ah 

dt cos h 



dh cos < cos 6 sin 

= V- ^ by Equa. [12]. [81] 

cos sin Z J M 

An inspection of this equation shows that when Z = 90 or 270 
sin Z is a maximum and a minimum for any given value of <. 

(trl 

It also shows that the smaller the latitude, the greater is its 

cosine and consequently the smaller the value of . The most 

an 

favorable position of the body is therefore on the prime vertical. 
The negative sign shows that the hour angle decreases as the 
altitude increases. When Z is zero (body on meridian) the value 

of is infinite and t cannot be found from the observed altitude. 
dh 

The effect of an error in the latitude may be found by differ- 
entiating [8] with respect to <. The result is 

o = cos </> sin d + cos 6 ( cos sin t~ cos / sin <) 

a</> 

. dt 

cos <t> cos d sin / cos sm 6 sm <l> cos d cos / 
a<t> 

= cos/? cos Z by [n] 
. dt^ _ cos h cos Z 
d(l> cos <t> cos d sin t 
cosZ 

_ . by I I2 | 

sin Z cos ^ J l J 



140 PRACTICAL ASTRONOMY 

= !___. i 82 i 

cos <t> tan Z ' 

This shows that when Z = 90 or 270 an error in < has no effect 
on /, since = o. In other words, the most favorable position 

of the object is on the prime vertical. It also shows that the 
method is most accurate when the observer is on the equator. 

83. 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, 
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 m or $ m later; 
note the instant when this star passes the vertical cross hair. It will be of as- 
sistance in making the calculations if the altitude of each star is measured im- 
mediately 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 1925 is about i 26'; a star on the equator would then 
pass the vertical cross hair nearly 4 later than the computed time if Polaris is 
at eastern elongation (see Table B, p. 99). 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 in- 
verted positions of the telescope and the two results combined. A new setting 
should be made on Polaris just before each observation on a time-star. 

* For a complete discussion of this method see a paper by Professor George 
O. James, in the Jour. Assoc. Eng. Soc., Vol. XXXVII, No. 2; also Popular As- 
tronomy, No. 1 7 2. A method applicable to larger instruments is given by Professor 
Frederick H." Seats, in Bulletin No. 5, Laws Observatory, University of Missouri. 



OBSERVATIONS FOR DETERMINING THE TIME 



141 



In order to deduce an expression for the difference in time between the meridian 
transit and the observed transit let a and <x be the right ascensions of the stars, 
S and So the sidereal times of transit over the cross hair, i and to the hour angles 
of the stars, the subscripts referring to Polaris. Then by Equa. 37, p. 52, 

t = S - a. 

and to = So <XQ. 

Subtracting, h - / = ( ~ o) - (S - So). [83] 

The quantity 5 So is the observed interval of time between the two observa- 
tions expressed in sidereal units. If a mean time chronometer or watch is used the 
interval must be increased by the amount of the correction in Table III. Equa. 
[83] may then be written 

j - t = (a - ) - (T - To) - C [84] 

where T and To are the actual 
watch readings and C is the 
correction from Table III to 
convert this interval into side- 
real time. 

In Fig. 61 let P Q be the 
position of Polaris when it 
is observed; P, the celestial 
north pole; Z, the zenith of 
the observer; and S, the time 
star in the position in which 
it is observed. It should be 
noticed that when S is passing 
the cross hair, Polaris is not 
in the position Po, but has 
moved westward (about P) by 
an angle equal to the (sidereal) 
interval between the two ob- 
servations. Let PQ be the polar 
distance of Polaris ; and , 
v the zenith distances of the 
two stars; and h and ho their altitudes. 

Then in the triangle PoPS, 

sin S _ sin po 




sinS 



In triangle PZS y 



or 



sin P PS sin PoS 

sin P PS sin po cosec ( + f ) 

= sin (to t) sin po cosec (h -f- ho). 

sin (/) _ sin f 
sin S ~~ cos <f> 
sin ( /) = sin S cos h sec <. 



[85] 



[86] 



142 PRACTICAL ASTRONOMY 

Substituting the value of sin S in Equa. [85], 

sin ( /) = sin po sin (/ t) cosec (h + ho) cos h sec <. [87] 

Since / and po are small the angles may be substituted for their sines, and 

/ = po sin (to t) cosec (h -f ho) cos h sec <. [88] 

If the altitudes h and ho have not been measured the factor cos h may be replaced 
by sin (< -r- 6) and cosec (/5 + //o) may be replaced by sec (5 c) with an error 
of only a few hundredths of a second, 5 being the declination of the time star and c 
the correction in Table I in the Ephemeris or the Almanac. 

In this method the latitude <f> is supposed to be known. If it is not known, then 
the altitudes of the stars must be measured and $ computed. It will usually be 
accurate enough to assume that the observed altitude of the time star is the same 
as the meridian altitude, and apply Equa. [i]; otherwise a correction may be 
made by formula [77]. The latitude may also be found from the altitude of the 
polestar, using the method of Art. 73. 

After the value of / (in seconds of time*) has been computed it is 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 converted into local civil time and 
then into standard time and the watch correction obtained. 

If it is desired to find the azimuth of the line of sight this may be done by com- 
puting a in the formula 

a t sec h cos 6. [89] 

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

Example. 

Observation of o Virginia over Vertical Circle through Polaris; latitude, 42 21' 
N.; longitude 4^ 44 18^.3 W.; date, May 8, 1906. 

Observed time on Polaris & h 35 58* 

Observed time on o Virginis 8 39 43 

a I2i*oo m 26 s .3 Diff. 3 m 45* 

o I 24 35 .4 _ o / 



Table III 0.6 <t> ~ * = 33 06' 



05^.3 p = yi.gs 

= i58oi'.3 lo go =1.8564 5 = +9 15' 

log sin (to t) = 9.5732 c = +i 06.5 

log sec (5 - c) = 0.0044 D - c = 8 oS'.s 
log sin (0 5) = 9.7373 
log sec < = 0.1313 

log 4 SB 0.6021 

log / = 1.9047 

/ = -808.30 
= 1 20^.3 

* The factor 4 has been introduced in the following example in order to reduce 
minutes of angle to seconds of time. 



OBSERVATIONS FOR DETERMINING THE TIME 143 

The true sidereal time may now be found by subtracting i m 2o*.3 from the right 
ascension of o Virginis, the result being as follows: 

a = 12" oo^ 26^.3 

/ = I 20 .3 

S as Hft 59n 06X0 

The local civil time corresponding to this instant of sidereal time for the date is 
20& 55* i4*.5. t The corresponding Eastern Standard time is 20* 39 32^.8, or 8* 39 OT 
32^.8 P.M. The difference between this and the watch time, 8* 39 43*, shows that 
the watch was lo*^ fast. 

84. 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 
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. 

85. 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 observe first 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 approxi- 
mate 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. 



144 



PRACTICAL ASTRONOMY 



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 
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 east star is observed first. In this case the cross hair is set a little 
below the star. 

In Fig. 62 let nesw represent the horizon, Z the zenith, P the pole, S e the easterly 

star, and Sw the westerly star. 
Let t e and t w be the hour 
angle of Se and Sw, and let 
HSeSw be an almucantar, or 
circle of equal altitudes. 

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




t e * 
Taking the mean value of 

s, 



, 



M 



from which it is seen that 
the true sidereal time equals 
the mean right ascension cor- 
rected by half the difference 
FIG. 62 in the hour angles. To de- 

rive the equation for correct- 
ing the mean right ascension so as to obtain the true sidereal time let the funda- 
mental equation 

sin h = sin 5 sin <j> -f- cos 5 cos <f> cos / [8] 



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



OBSERVATIONS FOR DETERMINING THE TIME 



be differentiated regarding d and t as the only variables, then there results 

o sin < cos 8 cos 5 cos <f> sin t j- cos 4> cos t sin 6, 
from which may be obtained 

dt == tan<fr __ tang 
dS ~ sin* 



If the difference in the decimation is small, dd may be replaced by J (dw 
in which case <ft will be the resulting change in the hour angle, or i (t w fc). 
The equation for the sidereal time then becomes 



c _ 

~~ 



4- 



_ tan 61 
tan / J 



[92] 



Lsin/ 

in which (8 W 8 e } must be expressed in seconds of time. 5 may be taken as the 
mean of 8 e and fa- The value of t would be the mean of te and tw if the two stars 
were observed at the same instant, but since there is an appreciable interval be- 
tween the two times / must be found by 



, , 
[93] 



and T e being the actual watch readings. 

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 Boreal is . . 


2 3 








ft Tauri 


1.8 


IO A 28 m 


7 * 3 8 




a Bootis 


O.2 








f Getninorum 


A 


10 37 


7 47 




a Bo'dtis 


O 2 








5 Geminorum 
p Bootis 


3-5 
7 6 


10 48 


7 5 8 




a* Geminorum 


I .0 


11 OO 


8 10 




TT HydrcR . . . 


3. cr 








p Argus 


2 .9 


11 10 


8 20 






2.8 








11 Geminorum 


T. . C 


TI 19 


8 29 




a Serpentis 


2.7 








ot Canis Minoris 


0.5 


ii 35 


8 45 




ft Herculis 


2.8 








8 Geminorum 


3. "? 


ji 51 


9 ot 




oc. Serpentis 


2 . 7 








ft Cancri 


3.8 


12 02 


9 12 






2.7 








UydrcR 


3-5 


12 II 


9 21 




ft Libras 


2.9 








ce HydrcR 


2.1 


12 2O 


9 30 




ft Hercuhs 


2.8 , 










4-0 


12 32 


9 42 















146 



PRACTICAL ASTRONOMY 



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 decimations 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 American Ephemeris it is not difficult to select 
a sufficient number of pairs at any time for making an accurate determination 
of the local time. On page 141 is a short list taken from the American Ephemeris 
and arranged for making an observation on April 30, 1912. 

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 W. Date, Dec. 14, 1905. 



Star. 

a Ceti (E) 
8 A quite (W) 

Mean 
Diff. 

t 


Rt. Asc. 
2/i 57>n 22*. i 
19 20 43 .6 


23/1 ogm 02*. 8 
7 36 38.5 
4 13.7 


2) 7^40^52*.! 


= 3*5o26Xi 



57 36' 3i"-5 



Decl. 

+3 43' 69".! 
+ 2 55 44 .o 

+3 19' 56".6 
2) o 48 25 .1 



= 24' i2".6 

= -"96X84 



T e 



Watch. 

i8 m oo* 

22 13 



^ 20"* 06X5 

04 13 



Mean R. A. = 23^ 09*" 02*.8 
Corn = 01 41 .o 

Sid. Time = 23^ 07"* 2i*.8 

The local civil time corre- 
sponding to this is 

17 35 m 43 s A 
Long. diff. 15 42 .o 

Eastern time 17* 20 01*4 

= 5 20 01 .4 P.M. 
Watch reading 5 20 06 .5 Corr. 

Watch fast 5*.i 



2 

log tan <j> 
log esc t 


= 9.9598 log tan 5 = 
= o-c>735 log cot t = 


1.9001 ^n; 

8.7650 
9.8024 


= 2.0194 (n) 
104*. 6 _ 
~ 3-6 


0-5535 (n) 



IQI'.O = i 



OBSERVATIONS FOR DETERMINING THE TIME 



147 



86. Formula [91] 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 (ft tanAS tan 5 tan A5 tan d tan A6 



sin AJ ; 



sin/ 



tan/ 



tan/ 



. , 
vers A/ } 



TABLE C. CORRECTIONS TO BE ADDED TO A3 AND A/. 

(Equa. [94], Art. 86.) 



Arc or sine. 


Correction to 
A5, 


Correction to 
A. 


Arc or sine. 


Correction to 
AS. 


Correction to 
A*. 


8 
100 


8 
O.OO 


s 

O.OO 


6 

800 


8 
O.9O 


8 
0.45 


2OO 


O.OI 


O.OI 


850 


I. 08 


0-54 


300 


0.05 


O.O2 


900 


1.2 9 


0.64 


400 


o.u 


0.06 


95 


'-SI 


0.76 


500 


O.22 


O.II 


IOOO 


1.77 


0.88 


600 


0.38 


0.19 


1050 


2.05 


i. 02 


650 


0.48 


0.24 


IIOO 


2 -35 


1.17 


700 


O.6O 


0.30 


1150 


2.69 


1-34 


750 


0.74 


-37 


I20O 


3.06 


i-5 2 



TABLE D. CORRECTION TO BE ADDED TO A/t 

(Equa. [94], Art. 86) 





At (in seconds of time). 


ad 
term. 


IOO* 


200* 


300* 


400* 


500* 


600* 


700* 


8 oo 


goo 8 


IOOO* 


8 


8 


8 


8 


8 


8 


8 


a 


8 


9 


a 


100 


O.OO 


O.OI 


O.O2 


0.04 


O.O7 


O.IO 


0.13 


0.17 


O.2I 


0.26 


200 


O.OI 


0.02 


0.05 


0.08 


0.13 


O.I9 


0.26 


0-34 


0.43 


-53 


300 


O.OI 


0.03 


O.O7 


0.13 


O.2O 


O.29 


-39 


0.51 


0.64 


0.79 


400 


O.OI 


0.04 


O.IO 


0.17 


O.26 


0.38 


0.52 


0.68 


0.86 


i. 06 


500 


O.OI 


0.05 


0.12 


0.21 


-33 


0.48 


0.65 


0.85 


1.07 


1.32 


000 


O.O2 


O.O6 


0.14 


0.2S 


0.40 


o-57 


0.78 


1.02 


1.28 


i-59 


7OO 
800 


0.02 
O.O2 


0.07 
0.08 


0.17 
0.19 


0.30 
0.34 


0.46 
0-53 


0.67 
0.76 


0.91 
1.04 


1.18 
1-35 


1.50 
1.71 


1.85 

2. II 


900 


O.02 


0.10 


0.21 


0.38 


o-59 


0.86 


1.17 


1-52 


i-93 


2.38 


IOOO 


0.03 


O.II 


O.24 


O.42 


0.66 


-95 


1.30 


1.69 


2.14 


2.64 


IIOO 


0.03 


0.12 


O.26 


0.47 


o.73 


1.05 


1.42 


1.86 


2.36 


2.91 


1200 


0.03 


0.13 


0.29 


0.51 


0.79 


1. 14 


1-55 


2.03 


2-57 


3-17 



f The algebraic sign of this term is always opposite to that of the second term. 
Chauvenet, Spherical and Practical Astronomy, Vol. I, p. 199. 



148 



PRACTICAL ASTRONOMY 



where A 5 is half the difference in the declinations and AJ is the correction to the 
mean right ascension. If sin A/ and tan A 5 are replaced by their arcs and the 
third term dropped, this reduces to Equa. [91], except that A3 and At are finite 
differences instead of infinitesimals. In order to compensate for the errors thus 
produced let A 6 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 A/ (Table C). 
With the approximate value of A/ thus obtained the third 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 A 5 may therefore be extended without increasing the 
errors arising from the approximations. 

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' i5".2. 
R. A. i Geminorum = 7" 20 i6.8s; decl. = +27 58' 3o".8. 



14* n 37^.98 
7 20 16 .85 

2) 6* 51"* 213.13 

3/ 25"* 403.56 

t = 51 25' o8". 4 



log AS = 3.000993 

log tan <j> - 9.959769 

log esc/ 0.106945 

3.067707 

ist term = n68*.7i 
2d term = 352 .76 

A/ (approx.) 
Corr., Table C = + 
Corr., Table D = 

A/ = + 8i7*.o6 



27 58' 3o".8 
19 38 15 .2 

2) 8 20' i5".6 

A5 = 4 10' 07".8 

= 1000^.52 
Corr., Table C i .77 

A5 = ioo2*.29 

log A5 = 3.00099 
log tan 5 = 9.64462 
log cot/ = 9.90187 

2.54748 
2d term = 352.76 




Mean R. A. = 10 45 57 .42 
Sid. Time of Equal Alt. = 10* 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 



i = 



- E) - (V - ')) X ~> 



OBSERVATIONS FOR DETERMINING THE TIME 149 

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



Corr. 



30 sin S cos 5 30 cos < sin Z 



in which S 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. 

87. 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, S in the PZS triangle, can be avoided by the use of tables. Publica- 
tion No. 120 of the U. S. Hydrographic Office gives values 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 decli- 
nation 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 
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 S sought. 

Suppose that two stars have equal declinations and that at a certain instant 
thek 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. 63 BC is the increase in declination; BD is the almucantar through A, B 
and D- t 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 great 
circles, and the triangle BCD is not strictly a spherical triangle, but it may 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 5 and DBC is 90 - 5. The length of the arc CD 
is then BC cot S, or (fa - $<?) cot 5. The angle at f is the same as the arc CD' 
and equals CD sec 5. If (dw 8e) is expressed in minutes of arc and the cor- 

* See Arts, 87 and 122 for the method of using these tables. 



ISO PRACTICAL ASTRONOMY 

rection is to be in seconds of time, then, remembering that the correction is half 

the angle x y 

Correction = 2 (dw d e ) cotS sec 5. % [95] 

5 should be taken as the mean of the two declinations, and the hour angle, used 
in finding 5, 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 i (Si + S z ) sec i (to + to). [96] 

2 2 

By substituting arcs for the sine and tangent this reduces to the equation given 
above, except that the mean of Si and 52 is not exactly the same as the value of S 
obtained by using the mean of the hour angles. 

Z 




FIG. 63 



The example on p. 146 worked by this method is as follows. From the azimuth 
tables, using a declination of 42, latitude 3, and hour angle 3 h 50"*, 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(5 W 8e) = 96'.84log - 1.9861 (n) 
log cot S 0.0172 
log sec 5 0.0007 
log corr. 2.0040 (n) 
log corr. = 1 008.9 

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. 



OBSERVATIONS FOR DETERMINING THE TIME 151 

88. 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 
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^.91 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^.91 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. 

89. Time Service. 

The Standard Time used in the United States is determined 
by means of star transits at the U. S. Naval Observatory (George- 
town Heights) 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 telegraph companies and is relayed from 
the Arlington and Annapolis radio stations. For the territory 
west of the Rocky Mountains the time is determined at the 
Mare Island Navy Yard. 

The error of the standard (sidereal) clock is determined about 
15 times per month by transits of 6 to 12 stars over the meridian. 
Two instruments are employed as a check, one a large (6 inch) 
* A planet should not be used for this observation. 



PRACTICAL ASTRONOMY 

transit, the other a small one which may be reversed in the 
middle of a set of observations on a star. 

When signals are to be sent out the sending clocks are com- 
pared with the sidereal clock by means of a chronograph, the 
two clocks recording simultaneously. After the error of the 
sending clock is determined the clock is " set " correct by means 
of an automatic device which accelerates or retards its rate for 
a short time until a chronographic comparison shows that it is 
correct. The sending clock makes the signals through a relay 
directly onto the wires both for the wire and the wireless signals 
from Arlington and Annapolis. 

In order to test and keep record of the errors in these signals 
they are received and recorded on a chronograph at the ob- 
servatory. Thus the error of the sending clock and the error 
of the signal are on record and may be obtained for use in ac- 
curate work. The error of the time signal is rarely as much as 
a tenth of a second. 

The " noon " signal is sent out each day at 12* Eastern Stand- 
ard Time, the series of signals beginning at n* 55^ and ending 
at 12*. This signal may be heard on the sounder at any tele- 
graph office or railroad station. The sounder gives a click once 
per second. The end of each minute is shown by the omission 
of the 55th to 5Qth seconds inclusive, except for the noon signal, 
which is preceded by a silent interval of 10 seconds. A similar 
signal is sent out at io h P.M. Eastern Standard Time and is usu- 
ally relayed by radio stations so that it may be heard on an 
ordinary receiving set. 

Questions and Problems 

1. Compute the Eastern Standard time of the transit of Regulus (a Leonis) 
over the meridian 7io6'.o west of Greenwich on March 3, 1925. The right 
ascension of the star is 10* 04 235.8. The right ascension of the mean sun + 12* = 
10* 41 00^.26 at o of March 3, G. C. T. 

2. At what time (E. S. T.) will the centre of the sun be on the meridian on 
Apr. i, 1925 in longitude 7io6'.oW.? Equa. of time at o* G. C. T. Apr. i = 
4 i2*.47; varia. per hour = -f 0^.755 . 

3. Compute the error of the watch from the data given in prob. 5, p. 207. 



OBSERVATIONS FOR DETERMINING THE TIME 153 

4. Compute the error of the watch from the data given in prob. 6, p. 208. 

5. Compute the sidereal time of transit of d Capricorni over the vertical circle 
through Polaris on Oct. 26, 1906. Latitude = 42 i8'-5; longitude = 4^ 45"* 07*. 

'Observed watch time of transit of Polaris = j h 10 20 s ; of 5 Capricorni j h i^ m 
28 s , Eastern Time. The declination of Polaris = +88 48' 3i".s; right ascen- 
sion = i h 26 37 s . 9; declination of 5 Capricorni = i633'o2".8; right ascension 
== 21^41^*53^.3. c = 39'. 3. [Right ascension of mean sun -\-i2 h at time of 
observation 2 h 17 48*.3.] Compute the error of the watch. 

6. Time observation on May 3, 1907, in latitude 42 21'. o N., longitude 4 h 44 
i8 s .o W. Observed transit of Polaris at 7* i6 w 17^.0; of AI Hydras at 7** i8 m 50*. 5. 

^Declination of Polaris +88 48' 28". 3; right ascension = i^ 24^ 50^.2. Decli- 
nation of fj.Hydr& = 16 21' 53". 2; right ascension = io& 21 36*. i. c = -f SO'.L 
[Right ascension of mean sun + 12* at time of observation = 14^ 42"* 58 5 .O3.] 
Compute the sidereal time of transit of ^ Hydros over the vertical circle through 
Polaris and also the error of the watch in Standard time. 

7. Observation for time by equal altitudes, Dec. 8, 1904. 

Right Ascension Declination Watch 

a Tauri (E) 4^30^ 293.01 +16 18' 59^.9 7*34^56* 

aPegasi(W) 22 59 61.12 +14 4i 43 -7 7 39 45 

Lat. = 42 28'.o N.; long. = 4 h 44 i$ s .o. [Right ascension of mean sun -f-i2& 
at instant of observation = 5^48 w 44 s .4i.] Compute the sidereal time and the 
error of the watch. 

8. Observation for time by equal altitudes, Oct. 13, 1906. 

Right Ascension Declination Watch 

v Ophinchi (W) i7^53 w 52 s .i5 ~945 / 34"-6 7^i34o 

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

Lat. = 42 iS'.o; long. = 4^45 06 s . 8 W. [Right ascension of mean sun -fi2 ft 
at instant of observation = i a 26 m 34.29.] Compute the sidereal time and the 
error of the watch. 



CHAPTER XII 

OBSERVATIONS FOR LONGITUDE 

90. 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. 

91. 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 f the watch correction at the 
west station, d the number of days between the observations, 

154 



OBSERVATIONS FOR LONGITUDE 155 

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 
Diff. in time = Diff. in Long. = c + dr c' . [97] 

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 

iby 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 local sidereal 

time 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 
Jits 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** 10* slow on local mean time. The 
watch is known to be gaining 8* per day. The second obser- 
vation is made 48 hours after the first. The difference in longi- 
:ude is therefore 

+ iS m 40* - 2 X 8* - i4 m 10* = i m 14*. 
The meridian B is therefore i m 14* or 18' 30" west of meridian A. 

92. 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 
with break-circuit chronometers. The stars observed are selected in such a man- 
ner as to permit determining the instrumental errors so that the effect of these 
errors may be eliminated from the results. The stars are divided into two groups. 



I$6 PRACTICAL ASTRONOMY 

Half the number are observed with the axis in one position and the other half with 
the axis reversed. This determines the error of the sight lii\e. In each half set 
some of the stars are north of the zenith and some south. The differences in times 
of transit of these two groups measures the azimuth error. The inclination error 
is measured with the striding level. (See Arts. 55 and 77.) 

After the corrections to the two chronometers have been accurately determined 
the two chronographs are switched into the main-line circuit and signals sent either 
by making or breaking the circuit a number of times by the use of a telegraph key. 
These signals are recorded on both chronographs. In order to eliminate the error 
due to the time of transmission of the signal,* the signals are sent first in the di- 
rection E to W and then in the direction W to E. The mean of the two results is! 
free from the error provided it is constant during the interval. The personal errors 
of the observers are now nearly eliminated by the use of the impersonal transit 
micrometer, instead of by exchange of observers, as was formerly done. After all 
of the observations have been corrected for azimuth, collimation and level, 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. The diff- 
erence between the two is the difference in longitude expressed in time units. 

In the more recent work (since 1922) the longitude of a station is determined with 
reference to Washington by receiving the time signal by radio. This cuts the cost 
of the work in half since there is but one station to be occupied. 

By the telegraph method a longitude difference may be determined with an error 
of about QS.OI or about 10 feet on the earth's surface. 

93. Longitude by Time Signals. 

If it is desired to obtain an approximate longitude for any purpose this may be 
done in a simple manner provided the observer is able to obtain the standard time 
at some telegraph office or railroad station, or by radio, as given by the noon signal 
or the 10^ P.M. signal. He may determine his local mean time by any of the pre- 
ceding methods (Chapter XI). The difference between the local time and the 
standard time by telegraph or radio is the correction to be applied to the longitude 
of the standard meridian to obtain the longitude of the observer. 

Example. 

Altitude of sun, 27 44' 35"; latitude, 42 22' N.; declination, ^oc/oo," N., 
equation of time, -f-3 m 48 s . 8; watch reading, 4^ i8 m 13*. 8. From these data the 
local mean time is found to be 4* 33 43^.9, making the watch 15*" 30*. i slow. By 
comparison with the telegraph signal at noon the watch is found to be 6* fast of 
Eastern Standard Time. The longitude is then computed as follows: 

Correction to L. M. T. = +i5 w 30*.! 

Correction to E. S. T. = -oo 06.0 

Difference in Longitude = i$ m 36*. i 

= 354'oi". 5 

Longitude = 75 - 3 54'. = 71 o6'.o West 

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



OBSERVATIONS FOR LONGITUDE 157 

94. Longitude by Transit of the Moon. 

A method which is adapted to use with the surveyor's transit and which, although 
not precise, may be useful on 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 Ephemeris for every hour of Greenwich 
Civil Time; hence if the right ascension can be determined, the Greenwich Civil 
Time becomes known. 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 moon's bright limb and also of several stars 
whose declinations are nearly the same as that of the moon. The table of " Moon 

i Culminations " in the Ephemeris shows which limb (I or II) may be observed. 

k (See note on p. 159.) The observed interval of time between the moon's transit 
and a star's transit (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 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 edge a correction, taken from the Ephemeris, called 
" sidereal time of semidiameter passing meridian." In computing this correction 
the increase in right ascension during this short interval has been allowed for, so 
the result is not the right ascension of the centre at the instant of transit of the 
limb, but at the instant of transit of the centre. If the west limb was observed 
this correction must be added; if the east limb, 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 Civil Time corresponding to 
this instant is found by interpolation in the table giving the moon's right ascension 
for every hour. To obtain the Greenwich Civil Time by simple interpolation find 
the next less right ascension in the table and the " varia. per min." on the same 
line; subtract the tabular right ascension from the given right ascension (obtained 
from the observation) and divide this difference by the " varia. per min." The 
result is the number of minutes (and decimals of minutes) to be added to the tab- 
ulated hour of Greenwich Civil Time. If the " varia. per min." is varying rapidly 
it will be more accurate to interpolate as follows: Interpolate between the two 
values of the " varia. per min." to obtain a "varia. per min." which corresponds 

>to the middle of the interval over which the interpolation is carried. In observa- 
tions made with a surveyor's transit this refinement is seldom necessary. 

In order to compare the Greenwich time with the local time it is necessary to 
convert the Greenwich Civil Time into the corresponding instant of Greenwich 
Sidereal Time. The difference between this and the local sidereal time is the 
longitude from Greenwich. 

In preparing for observations of the moon's transit the Ephemeris should be 
consulted (Table of Moon Culminations) to see whether an observation can be 
made and to find the approximate time of transit. The time of transit may be 
obtained either from the Washington civil time of transit or from the Greenwich 
civil time in the first part of the Ephemeris. The tabular time must be corrected 
for longitude. The apparent altitude of the moon should be computed and allow- 



158 PRACTICAL ASTRONOMY 

ance made for parallax. The moon's parallax is so large that the moon would not 
be in the field of view if this correction were neglected. The horizontal parallax 
multiplied by the cosine of the altitude is the required correction. The moon will 
appear lower than it would if seen from the centre of the earth. The correction is 
therefore subtractive. 

Since the moon increases its right ascension about 2* 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. It has, however, one great advantage. If for any reason, 
such as an accident to the timepiece, a knowledge of the Greenwich time is com* 
pletely lost, it is still possible by this method to recover the Greenwich time with 
a fair degree of accuracy. 

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

Example. 

Observed transit of moon's west limb July 30, 1925, for longitude. Watch time 
of transit moon's west limb, 7^ 27** 14*; transit of <r Scorpii, 7* 29^ 20 s . 



a Scorpii 
Moon, I 

Interval 
Table III 

Sid. int. 
Rt. Asc. ff Scorpii 

Rt. Asc. Moon's limb 
Time of s. d. passing 

Rt. Asc. centre 


jh Qqin 20 s 
7 27 14 


2 06* 

0.3 


2 m 06^.3 

16 16 39 .50 


1 6* 14 33*. 20 

I 10 .36 


16* i$ m 43*- 56 



43^.56 = local sidereal time 

From Ephemeris 

July 31 Gr. Civ. T. Rt. Asc. Moon Varia. per Min. 

o* 16* i4 375.54 2.3986 

i 16 17 01 .64 2.4049 

2 m 245.10 63 



. 

i o6.02 = 66X02 log 1.81968 .33-0 x 63 = 14 

Interpolated varia. per min. = 2.4000 log 0.38021 144.1 



= 27 3o*.s 

G. C. T. = o 27 3o.so 
a s + i2& = 20 32 23 .49 
Table III 04.52 

Gr. Sid. T. 20* 59 s8*.5i 
Loc. Sid. T. 16 15 43-56 

Longitude 4* 44 W i4*-95 W. = 71 03' 45 7/ W: 



OBSERVATIONS FOR LONGITUDE 



IS9 



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' -f 5 08' to 23 27' - 5 08', that is, from 28 35' to 18 19', 





First Quarter 




Last Quarter 



c 



o 

Full Moon 

FIG. 64. THE MOON'S PHASES 

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. 64 shows the relative positions of the three bodies at several 
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- 



160 PRACTICAL ASTRONOMY 

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 toward the earth & 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. 

Questions and Problems 

1. Compute the longitude from the following observed transits: (0 Aquarii, 
5& i6 m 04 P.M.; TT Aquarii, 5* 24 40$; moon's west limb, 5^ 32 27*; X Aquarii, 
5* 5i m 47 s . Right ascensions; Aquarii, 22 h n m 27^.6; TT Aquarii, 22 h 2o m 04*. 6; 
X Aquarii, 22 h 47 i8.3. The sidereal time of semi-diameter passing meridian 
= 60*. 3. At Gr. Civ. T. 22 h the moon's right ascension was 22^ 27"* 53^.3. The 
varia. per min. = i s .98oo. The right ascension of the mean sun +12^ = 4^ 36 
29*. 7. 

2. Which limb of the moon can be observed for longitude by meridian transit 
if the observation is taken in the morning? 

3. At about what time (local civil) will the moon transit when it is at first 
quarter? 

4. The sun's corrected altitude is 57 15' 36"; latitude, 42 22' N.; corrected 
decimation, 18 58'.6 N,; corrected equation of time, 4-3 49 s .o; watch reading, 
i h 2Q m o8 s , P.M. Error of watch on Eastern Standard time by noon signal is lo 5 
(fast). Compute the'longitude. 

5. On April 2, 1925 the transit of Hydrce is observed; watch reading 7* 52 31^ 
P.M. At IO& P.M. E. S. T. the radio signal shows that the watch is 3* fast. The 
right ascension of the sun -f-i2* at cfl G. C. T. April 2, 19^5, is 12^39 i6 5 .82j 
on April 3, it is 12^ 43 i3 s ,38. Compute the longitude. 



CHAPTER XIII 
OBSERVATIONS FOR AZIMUTH 

95. Determination of Azimuth. 

^ The determination of the azimuth of a line or of the direction 
l of the true meridian is of frequent occurrence in the practice of 
the surveyor and is probably the most important to him of all 
the astronomical observations. In geodetic surveys, in which 
triangulation stations are located by means of their latitudes 
and longitudes, the precise determination of astronomical po- 
sition is of as great importance as the orientation; but in general 
engineering practice, in topographical work, etc., the azimuth 
observation is the one that is most frequently required. 

Too much stress cannot be laid on the desirability of employ- 
ing the true meridian and true azimuths for all kinds of surveys. 
The use of the magnetic meridian or of an arbitrary reference 
line may save a little trouble at the time but is likely to lay up 
trouble for the future. As surveys are extended and connected 
and as lines are re-surveyed the importance of using the true 
meridian becomes greater and greater. 

96. Azimuth Mark. 

When an observation is made at night it is frequently incon- 
venient or impossible 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 azimuth mark, which can be seen both 
at night and in the day, and then to measure the angle between 
this mark and the first object during the day. The azimuth 
mark usually consists of a lamp or a lantern placed inside a box 
having a small hole cut in the side through which the light may 
be seen. The size of the opening will vary with the distance, 
power of telescope, etc.; for accurate work it should subtend 
an angle not much greater than o".s to i".o. If possible the 

161 



162 PRACTICAL ASTRONOMY 

mark should be placed so far away that the focus of the tele- 
scope will not have to be altered when changing from the star 
to the mark. For a large telescope of high power the distance 
should be a mile or so, but for surveyor's transits it may be 
much less; and in fact the topography around the station may 
be such that it is impossible to place the mark as far away as is 
desirable. 

97. Azimuth of Polaris at Greatest Elongation. 

The 1 simplest method of determining the direction of the 
meridian with accuracy is by means of an observation of the 




FIG. 65. CONSTELLATIONS NEAR THE NORTH POLE. POLARIS AT 
WESTERN ELONGATION 



polestar, or any other close circumpolar, when it is at its greatest 
elongation. (See Art. 19, p. 36.) The appearance of the 
constellations at the time of this observation on Polaris may be 
seen by referring to the star map (Fig. 55) and Fig. 65. 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 computing the sidereal time when the star is at 
elongation and changing this into local civil time and then into 
standard time, by the methods of Arts. 37 and 32. 

To find the sidereal time of elongation first compute the 
hour angle (t e ) by Equa. [35] and express it in hours, minutes and 
seconds. If western elongation is desired, t e is the hour angle; 



OBSERVATIONS FOR AZIMUTH 



Zenith 



Eastern Elongation 



if eastern elongation is desired, 24* t e is the hour angle. The 
sidereal time is then found by adding the hour angle to the 
right ascension. An average 
value for t e for Polaris for 
latitudes between 30 and 
50 is about s* 56 of sidereal 
time, or 5" 55 of mean time; 
this is sufficiently accurate 
for a rough estimate of the 
time of elongation, and as 
it changes but little from 
year to year and in differ- We8tecnE i ongat io j 
ent latitudes, it may be used 
instead of the exact value for 
many purposes. Approxi- 
mate values of the times of 
elongation of Polaris may be 
taken from Table V. 

Example. Find the East- 
ern Standard time of western 
elongation of Polaris on April 
25, 1925, in latitude 42 22' 
N. , longitude 7 1 06' W. The 
right ascension of Polaris is 
i*33 m 27 s .i5; the declination is +88 54' 04".54. The right 
ascension of the mean sun +12" at o* G. C. T. = 14" 09 57*. 54; 
corrected for longitude it is 14* io w 44*. 26. 

log tan < = 9.96002 t e = 5* S5 m 57*-45 S = 7* 29** 22*.6o 

log tan 5 = 1.71717 a = i 33 27 .15 as + 12* = 14 10 44 .26 

log cos te = 8.24285 S 7 h 2g m 22 s . 60 Sid. int. =17* i8 w 38^.34 

t e = 88 59' 5i".7 Table II = 2 50.16 

= 5 55*55*45 Loc. Civ. T. = 17* 15* 4 8*.i8 

Long. diff. = 15 36 .00 

E. S. T. (civil) ==17* oo I2*.i8 

The transit should be set in position half an hour or so before 
elongation. The star should be bisected with the vertical cross 




FIG. 66 



164 



PRACTICAL ASTRONOMY 



hair and, as it moves out toward its greatest elongation, its 
motion followed by means of the tangent screw of the upper or 
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 elongation. About 
5 W before elongation, centre the plate levels, set the vertical hair 
carefully on the star, lower the telescope without disturbing its 
azimuth, and set a mark carefully in line at a distance of several 
hundred feet north of the transit. Reverse the telescope, re- 
centre the levels if necessary, bisect the star again, and set an- 
other point beside the first one. If there are errors of adjust- 
ment (line of collimation and horizontal axis) the two points 
will not coincide; the mean of the two results is the true point. 
The angle between the meridian and the line to the stake (star's 
azimuth) is found by the equation 

sin Z n sin p sec $ [36] 

where Z n is the azimuth from the north (toward the east or the 
west); p, the polar distance of the star; and 0, the latitude of 
the place. The polar distance may be obtained by taking the 
declination from the Ephemeris and subtracting it from 90; 
or, it may be taken from Table E if an error of 30" is permissible 
The latitude (0) may be taken from a map or found by obser- 
vation. (Chap. X.) The latitude does not have to be known 
with great precision; a differentiation of [36] will show that an 



error of i' in causes an error of only about 
for latitudes within the United States. 



" in Z n for Polaris 



TABLE E 
MEAN POLAR DISTANCE OF POLARIS 



Year 


Mean Polar Distance 


Year 


Mean Polar Distance 


1924 

1925 
1926 


Io6'o 7 ". 3 
05 48 .9 

05 30 -5 


1930 
1931 
1932 


I0 4 'I7".2 

i 03 59 .0 
i 03 40 .7 


1927 
1928 
1929 


05 12 .2 

04 53 -8 
04 35 -5 


1933 
1934 
1935 


I 03 22 .5 

I 03 04 .3 
I 02 46 .1 



OBSERVATIONS FOR AZIMUTH 165 

The above method is general and may be applied to any 
circumpolar star. For Polaris, whose polar distance in 1925 
is about io6', it is usually accurate enough to use the ap- 
proximate formula 

Z w " = p" sec * [98] 

in which Z w " and p" are expressed in seconds of arc. 

This computed angle (ZJ may be laid off in the proper direc- 
tion by means of a transit (preferably by daylight), using the 
method of repetitions, or with a tape, by measuring a perpen- 
dicular offset calculated from the measured distance to the stake 
and the star's azimuth. The result will be the true north- 
and-south line. 

It is often desirable to measure the horizontal angle between 
the star at elongation and some fixed point, instead of marking 
the meridian itself. On account of the slow change in the azi- 
muth there is ample time to measure several repetitions before 
the error in azimuth amounts to more than i" or 2". In lati- 
tude 40 the azimuth changes about i' in half an hour before 
or after elongation; the change in azimuth varies nearly as the 
square of the time interval from elongation. The errors of 
adjustment of the transit will be eliminated if half the angles 
are taken with the telescope erect, and half with the telescope 
inverted. The plate levels should be re-centred for each po- 
sition 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 25, 1925, in latitude 42 22' N. The declination of Polaris 
is +88 54' 04".54. Polar distance, p, is i 05' 55". 46 3955"-46. 

By formula [36] By formula [98] 

log sin p - 8.28272 log p" = 3.59720 

log sec <p = 0.13145 log sec < = 0.13145 

log sinZfi ~ 8.41417 logZ w " = 3.72865 

Zn - i 29' i3"-6 Z" = 5353"-6 

- x* 29' i3".6 



166 PRACTICAL ASTRONOMY 

If the mark set in line with the star is 630.0 feet away from the 
transit the perpendicular offset to the meridian is "calculated as 
follows: 

log 630.00 = 2.79934 
log tan Zn = 8.41432 

Jog offset = 1.21366 
offset = 16.355 ft- 

98. Observations Near Elongation. 

If observations are made on a close circumpolar star within 
a few minutes of elongation the azimuth of the star at the 
instant of pointing may be reduced to its value at elongation if 
the time of the observation is known. The formula for com- 
puting the correction is 

C = 112.5 X 3600 X sin i" X tanZ, X (T - T e y [99] 

in which Z e is the azimuth at greatest elongation, T is the ob- 
served time and T e the time of elongation. T T e must be 
expressed in sidereal minutes. The correction is in seconds of 
angle. Values of this correction are given in Table Va in the 
Ephemeris for each minute up to 25, or in Table VI of this book. 

Example. 

The horizontal angle from a mark to the right to Polaris is 2 37' 30", the watch 
reading 6^ 28 00 s when the star was sighted. The watch time of western elonga- 
tion is o 7 * 04 00 s . The azimuth of Polaris at elongation is i 37' 48". The cor- 
rection corresponding to a 24"* mean time interval, or a 24"* 04* sidereal interval, 
is 32". The horizontal angle from the mark to the elongation position of the star 
is 2 37' 30" 32" = 2 36' 58". The bearing of the mark is the sum of this and 
the azimuth at elongation, or 2 36' 58" + i 37' 48" = 4 14' 46". The bearing 
is therefore N. 4 14' 46" W. 

99. Azimuth by Elongations in the Southern Hemisphere. 

The method described in the preceding article may be applied 
to stars near the south pole, but since there are no bright stars 
within about 20 of the pole the observation is not quite so 
simple and the results are somewhat less accurate. As the polar 
distance increases the altitude of the star at elongation increases 
and the diurnal motion becomes more rapid. The increase in 
altitude causes greater inconvenience in making the pointings 
and also magnifies the effect of instrumental errors. On account 



OBSERVATIONS FOR AZIMUTH 167 

of the rapid motion of the star it is important to know before- 
hand both the time at which elongation will occur and the 
altitude of the star at this instant. 

The time of elongation is computed as explained for Polaris 
in Art. 91. The altitude may be found by the formula 

. , sin . , 
sm h = -72- = sm <j> sec p. 
sm 5 

There is usually time enough to reverse the transit and make 
one observation in each position of the axis, without serious 
error, if the first is taken when the star is 10' to 15' below eastern 
elongation, or the same amount above western elongation. 

Example. 

Mean observed horizontal angle between mark and a. Triang. Austr. at Eastern 
elongation May 31, 1920 = 35 10' 30". (Mark E. of star.) Decl. a Triang. 
Austr. = -6853'n"; right ascension = i6> 40 18^.5. Lat. = -3435 / (S.)j 
Long. = 58 25' W. 

The time and altitude are computed as follows: 

log tan <f> = 9.83849 log sin = 9.75405 

log tan 5 = 0.41326 log sin 5 = 9.96982 

log cos t e 9.42523 log sin h = 9.78423 

3 6o - te = 74 33'-6 h = 37 28'.; 

24^ te 4^ 58 14*. 4 

t e = 19 01 45 .6 

a 16 40 18 .5 

S = 1 1^42 04*. i 

The local civil time corresponding to n h 42 04*. i is 19^ 05 3 2 s . 5. 

For the azimuth of the star and the resulting bearing of the mark we have 

log cos 5 = 9-55657 
log cos <f> - 9.91556 

log sin Z = 9.64101 

Z * 25 56'.8 (East of South) 
Measured angle = 35 10 .5 

Bearing of mark S 61 O7'.3 E 

ioo. 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 



i68 



PRACTICAL ASTRONOMY 



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 

FIG. 67. POSITION OF cross hairs so that the vertical hair is 

SUN'S Disc A FEW SECONDS tangent to the right edge of the sun and 

BEFORE OBSERVATION , , , i i a n 

(A.M. observation in Northern the horizontal hair cuts off a small seg- 

Hemisphere.) 





(Fig. 67.)* 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 de- 

sired to determine the time accurately, 

so that the watch correction may be 

found from this same observation, it can FlG 68 p osmON OF 

be read more closely by a second ob- SUN'S Disc A FEW SECONDS 

_ . . , . - , . BEFORE OBSERVATION 

server. Both the horizontal and the (A . M . observation in Northern * 
vertical circles are read, and both angles Hemisphere.) 

and the time are recorded. The same observation may be re- 
peated three or four times to increase the accuracy. The instru- 
ment 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 vertical cross hair cuts a segment from 
the left edge (Fig. 68). The same number of pointings should 

* Tn the diagram only a portion of the sun's disc is visible; in a telescope of low 
power the entire disc can be seen. 



OBSERVATIONS FOR AZIMUTH 



169 



be taken in each position of the instrument. After the pointings 
on the sun are completed the telescope should be turned to the 





FIG. 69. POSITIONS OF SUN'S Disc A FEW SECONDS BEFORE OBSERVATION 

(P.M. Observation in Northern Hemisphere.) 




mark again and the vernier reading checked. If the transit 
has a vertical arc only, the telescope cannot be used in the re- 
versed position and the index correction must therefore be 



PRACTICAL ASTRONOMY 

determined. If the observation is to be made in the afternoon 
the positions will be those indicated in Fig. 69.** 

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 
the sun's centre. The azimuth is then computed by any one 
of the formulae [22] to [29]. 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 great a degree of precision as can 
be expected in this method. 

Example. 

Observation on Sun for Azimuth and Time. Lat., 42 29'^ N. ; long., 71 07'.$ W. 
Date, May 25, 1925. 



Hor. Circle 

(ver. A.) 
Mark o oo' 


Vert. Circle 


Watch 
(E. S. T.) 


L & L limbs 67 54 


43 35' 


2& 58"* 00 s P.M. 


68 ii 


43 20 


2 59 21 


68 26 


43 08 


3 oo 33 


(instrument reversed) 






R & U limbs 69 25 


43 25 


3 01 53 


69 39 


43 12 


3 03 05 


69 52 


43 oo 


3 04 10 


Mean 68 54^.5 


43 i6'-7 


3 oi io*.3 


Mark ooo' 


R&P -0.9 


12 


Hor. angle, 






mark to sun 68 54'. 5 


I. C. +i.o 


Civ. T. 15^ oi m 10^.3 




h - 43 16' 8 


5 






G. C. T. 20* oi io*.3 



* 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 



171 



Equa. [28] nat 
nat sin 8 = .357^6 
log sin <# = 
log sin h 

sin <j> sin h .46310 
numerator = .10524 

log " = 
log sec <J> = 
log sec h 
log cos Z s = 

z s 

Hor. angle = 
Azimuth of mark 


log 

9.82962 
9-83605 

9.66567 

9.02218 
0.13231 
0.13787 


9.29236 

7 8 4 i / .7 
68 54.5 


947'.2 



Sun's decl. at o* = +20 48' 55^,8 
-j-27".6o x 20&.02 = +9 12 .6 

6 = -f-2o58'o8". 4 
N 



If it is desired to compute the time 
from the same observation it may be 
found by formula [12], by [19], or by 
those derived on p. 174. The resulting 
Eastern standard time is 3^ 000*42*. 7, 
making the watch 27^.6 fast. (The equa- 
tion of time at 20^ Gr. Civ. T. is -j-3 1 * 




FIG. 70 



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 s sec h, where s is the semidiameter 
and h is the altitude of the centre. 

The following examples and explanations are taken from Serial 
166, U. S. Coast and Geodetic Survey, and illustrate the method 
of observing for azimuth and longitude with a small theod- 
olite as practised on magnetic surveys. 

Having leveled and adjusted the theodolite and selected a 
suitable azimuth mark, a well-defined object nearly in the 
horizon and more than 100 yards distant, the azimuth ob- 
servations are made in the following order, as shown in the 
sample set given on pages 172 and 173. 

Point on the mark with vertical circle to the right of the 
telescope (V.C.R.) and read the horizontal circle, verniers A 
and B. Reverse the circle, invert the telescope and point on 
the mark again, this time with vertical circle left (V.C.L.). 
Place the colored glass in position on the eyepiece and point on 
the sun with vertical circle left, bringing the horizontal and 
vertical cross wires tangent to the sun's disc. At the moment 
when both cross wires are tangent note the time by the chro- 



PRACTICAL ASTRONOMY 



nometer. If an appreciable interval is required to look from the 
eyepiece to the face of the chronometer, the observer should 
count the half-seconds which elapse and deduct the amount 
from the actual chronometer reading. The horizontal and 
vertical circles are then read and recorded. A second point- 
ing on the sun follows, using the" same limbs as before. The 
alidade is then turned 180 and the telescope inverted and 
two more pointings are made, but with the cross wires tangent 
to the limbs of the sun opposite to those used before reversal, \ 
This completes a set of observations. A second set usually fol- 
lows immediately, but with the order of the pointings reversed, 
ending up with two pointings on the mark. Between the two 
sets the instrument should be releveled if necessary. 

Form 266 

OBSERVATIONS OF SUN FOR AZIMUTH AND TIME* 



Station, Smyrna Mills, Me. 
Theodolite of mag'r No. 20. 
Mark, Flagpole on school building. 
Chronometer, 245. 



Date, Friday, August 5, 1910. 
Observer, H. E. McComb. 
Temperature, 20 C. 



Sun's 
limb 


v. c. 


Chronome- 
ter time 


Horizontal circle 


Vertical circle 


A 


B 


Mean 


A 


B 


Mean 




L 


Mark 


124 43 40 


43 50 


124 43 45 


/ // 


/ // 


/ // 




R 




304 43 40 


43 40 


304 43 40 


















124 43 42 












h m s 














53 


R 


8 25 54 


155 12 30 


12 50 


155 12 40 


41 10 30 


II 30 


41 ii oo 





R 


27 56 


155 40 40 


41 oo 


155 40 So 


41 31 oo 


31 30 


41 31 15 


EL 


L 


30 03 


337 oo 10 


oo 20 


337 oo 15 


138 47 oo 


45 30 


41 13 45 


Q 


L 


32 06 


337 28 20 


28 30 


337 28 25 


138 27 oo 


25 30 


41 33 45 






8 28 59-8 






336 20 32 






41 22 26 


















- 57 





L 


8 33 45 


337 51 oo 


51 20 


337 Si 10 


138 10 30 


09 oo 


41 so 15 


Q 


L 


35 59 


338 24 30 


24 50 


338 24 40 


137 48 30 


47 oo 


42 12 IS 


<S1 


R 


38 20 


158 10 oo 


10 20 


158 10 10 


43 10 30 


ii 30 


43 II oo 


S3 


R 


40 37 


158 41 50 


42 10 


158 42 oo 


43 31 30 


32 30 


43 32 00 






8 37 10 2 






338 17 oo 






42 41 23 


















-54 




R 


Mark 


304 43 40 


43 50 


304 43 45 










L 




124 43 20 


43 40 


124 43 30 


















124 43 38 









From U. S. Coast and Geodetic Survey Serial 166. 



OBSERVATIONS FOR AZIMUTH 



173 



The chronometer and circle readings for the four pointings of a 
set are combined to get mean values for the subsequent compu- 
tation. When the vertical circle is graduated from zero to 360, 
the readings with vertical circle right give the apparent altitude 
of one limb of the sun, while those with vertical circle left must 
be subtracted from 180 to get the apparent altitude of the other 
limb. The mean of the four pointings gives the apparent alti- 
tude of the sun's center. This must be corrected for refraction 
'and parallax to get the true altitude. 

Form 266 

OBSERVATIONS OF SUN FOR AZIMUTH AND TIME* 



Station, Smyrna Mills, Me. 
Theodolite of mag'r No. 20. 
Mark, Flagpole on school building. 
Chronometer, 245. 



Date, Friday, August 5, 1910. 
Observer, H. E. McComb. 
Temperature, 21 C. 



Sun's 
limb 


v. c. 


Chronome- 
ter time 


Horizontal circle 


Vertical circle 


A 


B 


Mean 


A 


B 


Mean 




R 


Mark 


280 45 oo 












45 20 


280 45 10 










L 




loo 45 20 


45 40 


100 45 30 


















280 45 20 












h m s 














1 


L 


3 12 38 


96 35 50 


36 20 


96 36 05 


143 03 oo 


00 OO 


36 58 30 


Si 


L 


14 38 


97 01 20 


oi 50 


97 oi 35 


143 23 oc 


20 OO 


36 38 30 


EJ 


R 


16 44 


278 04 10 


04 30 


278 04 20 


36 53 30 


53 oo 


36 53 IS 


F 


R 


18 46 


278 29 oo 


29 20 


278 29 10 


36 33 00 


32 00 


36 32 30 






3 IS 41 S 






277 32 48 






36 45 41 


















- I 08 


GT 


R 


3 20 12 


278 47 20 


47 40 


278 47 30 


36 18 oo 


17 30 


36 17 45 


ET 


R 


22 12 


279 12 OO 


12 20 


279 12 10 


35 58 oo 


57 30 


35 57 45 


i 


L 


23 SO 


98 57 40 


58 10 


98 57 55 


144 56 30 


53 30 


35 05 oo 


13 


L 


25 50 


99 22 40 


23 10 


99 22 55 


145 17 oo 


14 oo 


34 44 30 






3 23 01. o 






279 05 08 






35 31 IS 


















I 11 




L 


Mark 


100 45 20 


45 40 


100 45 30 










R 




280 45 20 


45 30 


280 45 25 


















280 45 28 









* From U. S. Coast and Geodetic Survey Serial 166. 

It is important to test the accuracy of the observations as 
soon as they have been completed, so that additional sets may 
be made if necessary. This may be done by comparing the 



174 PRACTICAL ASTRONOMY 

mean of the first and fourth pointings of a set with the mean of 
the second and third, or by comparing the rate of change in the 
altitude and azimuth of the sun between the first and second 
pointings, the third and fourth, fourth and fifth, fifth and sixth, 
and seventh and eighth. For the period of 15 or 20 minutes 
required for two sets of observations the rate of motion of the 
sun does not change much. 

COMPUTATION 
From formula [24] we have 

Z 

cot 2 = sec 5 sec (s p) sin (s </>) sin (s h) 
2 

and from [19], 

tan - = i/ ( cos 5 sin (s h) esc (s <t>) sec (s p) j 

// cos s cos (s p) sin 2 (s K)\ 
" V \sin (s - </>) sin (s - h) ' cos 2 (s - p)) 

Z 

= tan - sin (s h) sec (s p). 

2 

The angle Z s is the azimuth of the sun from the south point, 
east if in the morning, west if in the afternoon. 

The form of computation is shown in the following example, 
for the sets of observations at Smyrna Mills, Me., given above. 

The different steps of the computation are most conveniently 
made in the following order: . 

Enter the corrected altitude, mean readings of the horizontal 
circle for the pointings on the sun and on the mark, and the 
chronometer time for each set of observations in their proper 
places. Enter the value of latitude obtained from the latitude 
observations or other source. Compute the chronometer cor- 
rection on standard time for the time of each set of observations 
from the comparisons with telegraphic time signals. Unless 
the chronometer has a large rate its correction may be taken the 
same for two contiguous sets of observations. Compute the 
Greenwich time of observation for each set, and find from the 



OBSERVATIONS FOR AZIMUTH 



175 



Form 269 

COMPUTATION OF AZIMUTH AND LONGITUDE* 

Station, Smyrna Mills, Me. 



Date 


Aug. 


Aug. 5 


Aug. 5 


Aug. 5 


h 


41 21 29 


/ // 

42 40 29 


> II 

36 44 33 


35 30 04 


<t> 


46 08 21 


46 08 21 


46 08 21 


46 08 21 


P 


. 72 51 34 


72 51 39 


72 56 08 


72 56 12 


as 


160 21 24 


161 40 29 


155 49 02 


154 34 37 


s 


80 10 42 


80 50 14 


77 54 31 


77 17 18 


s- p 


7 19 08 


7 58 35 


4 58 23 


4 21 06 


s-h 


38 49 13 


38 09 45 


41 09 58 


41 47 14 


5-0 


34 02 21 


34 41 53 


31 46 10 


31 08 57 


log sec 5 


0.76807 


o 79795 


0.67887 


0.65749 


" sec (s - p) 


o 00355 


o 00422 


o 00164 


o 00125 


" sin (5 h) 


9 79718 


9 79091 


9.81839 


9 82371 


" sin (s </) 


9.74800 


9 75530 


9 72140 


9.71372 


" ctn i Zs 


o 31680 


o 34838 


0.22030 


0.19617 


" ctniZs 


o 15840 


0.17419 


o 11015 


0.09808 




o / // 


/ // 


/ II 


/ // 


Z from South 


69 33 04 


67 36 43 


75 37 17 


77 10 09 


Circle reads 


336 20 32 


338 17 oo 


277 32 48 


279 05 08 


S. Mer. " 


45 53 36 


45 53 43 


201 55 31 


201 54 59 


Mark " 


124 43 42 


124 43 38 


280 45 20 


280 45 28 


Azimuth of Mark 


78 50 06 


78 49 55 


78 49 49 


78 50 29 


Mean 


78 50 05 








log sec (s p) sin (s h) 


9 80073 


9 79513 


9.82003 


9.82496 


" tan i / 


9-64233 


9 62094 


9.70988 


9 72688 




o / n 


/ // 


1 It 


/ // 


t in arc 


47 23 24 


45 20 52 


54 17 25 


56 07 55 




k m s 


h m s 


h m s 


h m s 




-3 09 33 6 


-3 oi 23.5 


3 37 09 7 


3 44 31 7 


E 


4- 5 53 4 


4- 5 53 3 


+ 551 8 


4- 5 51 8 


Local time 


8 56 19 8 


9 04 29.8 


3 43 01.5 


3 50 23.5 


Chron. time 


8 28 59.8 


8 37**o.2 


3 IS 41-5 


3 23 oi. o 


A/ on local time 


+ 27 20.0 


4- 27 19 6 


4" 27 20.0 


4" 27 22.5 


A* on 75 merid. time 


5-8 


58 


58 


S 8 


AX 


-27 25 8 


-27 25.4 


-27 25.8 


-27 28.3 


Mean 


-27 26.3= - 6si'.6 


X 68 08'.4 



* From U. S. Coast and Geodetic Survey Serial 166, 



1 76 PRACTICAL ASTRONOMY 

American Ephemeris, or the Nautical Almanac, the sun's polar 
distance and the equation of time for that time" as previously 
explained. The succeeding steps require little explanation. 
As the horizontal circles of theodolites are with few exceptions 
graduated clockwise, and as the sun is east of south in the 
morning and west of south in the afternoon, it follows that in 
order to find the horizontal circle reading of the south point, 
the azimuth of the sun must be added to the circle reading of 
the sun for the morning observations and subtracted from it' 
for the afternoon observations. The horizontal circle reading 
of the south point subtracted from the mark reading gives the 
azimuth of the mark, counted from south around by west from 
o to 360. 

For the computation of /, the logarithms of sec (s p) and 
sin (s h) are found in the azimuth computation and their 
sum can be written down in its proper place. From that must 
be subtracted log ctn ^ Z s to find log tan \ /. The correspond- 
ing value of i is the time before or after apparent noon. If in 
the case of the morning observations ctn \ ti be substituted for 
tan 4 /, /i will be counted from midnight. The difference be- 
tween the chronometer correction on local mean time and the 
correction on standard time is the difference in longitude be- 
tween the standard meridian and the place of observation. 

101. Observations in the Southern Hemisphere. 

In making observations on the sun for azimuth in the southern 
hemisphere (latitude greater than declination) the pointings would 
be made on the left and lower limbs and on the right and upper 
limbs in the forenoon, and on the right and lower and on the 
left and upper limbs in the afternoon, as indicated in Fig. 71. 

If the instrument has no vertical and horizontal hairs but has 
cross hairs of the X pattern the sun's image may be placed in 
any two symmetrical positions instead of those indicated above. 

The same formulae used for the northern hemisphere may be 
adapted to the southern hemisphere either by considering the 
latitude < as negative and employing the regular forms, or by 



OBSERVATIONS FOR AZIMUTH 



177 



taking as positive, and using the south polar distance instead 
of the north polar distance when employing Equa. [24]; the 
resulting azimuth in this case will be that measured from the 
south point of the meridian. 

As an illustration of an observation made in the southern 
hemisphere the following observation is worked out by two 
methods. On April 24, 1901 (P.M.), the mean altitude of the 
sun is 22 12' 30"; the corrected declination is 12 40' 30" N.; 
[latitude, o 41' 52" S.; mean horizontal angle from mark, toward 





P.M. 



NORTH 



FIG. 71 



the left, to sun = 75 53' 30". 
computation is as follows: 



Employing formula [24] the 



-oVsa" 

22 12 30 
77 19 30 
9 8 5 0' 08" 

49 25 04 

50 06 56 
27 12 34 

s- p-2i 54 26 



2 S 
S 

S (f> 

s-h 



log sec o.i 8673 
log sin 9.88498 
log sin 9.66615 
log sec 0.05369 

2)9-78555 
log tan \ Z - 9.89278 

\Z= 37 59' 53" 

Z = N 75 59 46 W 
Hor. angle = 75 53 30 
True bearing of mark = N o 06' r6" W 

If in formula [24] we had used the south polar distance, 
102 40' 30", and considered the latitude as positive the result 
would have been the azimuth of the sun from the south point or 
Sio 4 oo'i4"W. 



178 PRACTICAL ASTRONOMY 

If formula [25] is employed we may take <t> positive, reverse 
the sign of 6 and obtain the bearing of the sun from the south. 

nat sin 5 == 0.21942 

log sin = 8.08558 

log sink = 9-57746 

sum = 7.66304 

nat sin <f> sin h 0.00460 

numerator = 0.22402 

log numerator = 9.35029 n 

log sec <f> = 0.00003 

log sec h = 0.03348 

log cos Z s ~ 9.38380 n 

Z s = S 104 oo' 15" W 
Measured angle] 75 53 30 

True Bearing of Mark = S 179 53' 45" W 
or N o 06' 15" W 

In this case it would have been quite as simple to solve [25] 
in its original form, obtaining the bearing from the north point. 
If the south latitude is greater than the sun's declination (say, 
lat. 40 S., decl. 20 S.) then the method used in the example 
would be preferable. 

102. Most Favorable Conditions for Accuracy. 

From an inspection of the spherical triangle Pole Zenith 
Sun, it may be inferred that the nearer the sun (or other observed 
body) is to the observer's meridian the less favorable are the 
conditions for an accurate determination of azimuth from a 
measured altitude. At the. instant of noon the azimuth becomes 
indeterminate. Also, as the observer approaches the pole the 
accuracy diminishes, and when he is at the pole the azimuth is 
indeterminate. 

To find from the equations the error in Z due to an error in h 
differentiate Equa. [13], regarding h as the independent variable; 
the result is 



(A7 \ 

cos h sin Z cosZ sin h ) 
dh / 

= sin < cos h cos c 
= cos 5 cos S by [14] 



or, cos <t> cos h sin Z = sin < cos h cos cos Z sin h 
tin 



OBSERVATIONS FOR AZIMUTH 179 

. dZ cos 8 cos S 



dh cos cos h sin Z 
cos S 



, which by [15] 



sin 5 cos h ' 

= I [102] 

cos h tan S 

If the declination of the body is greater than the latitude (and 
1 in the same hemisphere) there will be an elongation, and at this 
point the angle 5 (at the sun or star) will be 90; the error dZ 
will therefore be zero. For objects whose declinations are such 
that an elongation is possible it is clear that this is the most 
favorable position for an accurate determination of azimuth 
since an error in altitude has no effect upon Z. 

If the declination is less than the latitude, or is in the opposite 
hemisphere, the most favorable position will depend partly 
upon 5, partly upon h. From Equa. [15] it is seen that the maxi- 
mum value of 5 occurs simultaneously with the maximum value 
of Z, that is, when the body is on the prime vertical (Z = 90 
or 270). To determine the influence of h suppose that there 
are two positions of the object, one north of the prime vertical 
and one south of it, such that the angle S is the same for the two. 
The minimum error (dZ) will then occur where cos h is greatest; 
this corresponds to the value of h which is least, and therefore, 
on the side of the prime vertical toward the pole. The exact 
position of the body for greatest accuracy could be found for 
'any particular case by differentiating the above expression and 
placing it equal to zero. 

To find the error in the azimuth due to an error in latitude 
differentiate [13] with respect to d$. This gives 



(jrr V 

cos<sinZ- cosZsin<H 
d<t> ) 

= sin h cos < cos k c< 
= cos S cos t by [16] 



or cos h cos q> sin Z = sin h cos < cos h cos Z sin 
a<t> 



l8o PRACTICAL ASTRONOMY 

d? __ COS d COS / 

d<t> cos h cos 4> sin Z 

= sin Z cos t 
sin / cos < sin Z 



, r , 



tan 



From this equation it is evident that the least error in Z due ] 
to an error in <j> will occur when the object is on the 6-hour circle 
= 90).^ 

Combining the two results it is clear that observations on 
an object which is in the region between the 6-hour circle 
and the prime vertical will give results slightly better than 
elsewhere; observations on the body when on the other side 
of the prime vertical will, however, be almost as accurate. 
The most important matter so far as the spherical triangle 
is concerned is to avoid observations when the body is near 
the meridian. 

The above discussion refers to the trigonometric conditions 
only. Another condition of great importance is the atmospheric 
refraction near the horizon. An altitude observed when the 
body is within 10 of the horizon is subject to large uncertainties 
in the refraction correction, because this correction varies with 
temperature and pressure and the observer often does not know 
what the actual conditions are. This error may be greater 
than the error of the spherical triangle. When the two require- 
ments are in conflict it will often be better to observe the sun 
nearer to the meridian than would ordinarily be advisable, 
rather than to take the observation when the sun is too low for 
good observing. In winter in high latitudes the interval of 
time during which an observation may be made is rather limited 
so that it is not possible to observe very near the prime vertical. 
The only remedy is to obtain the altitude and the latitude with 
greater accuracy if this is possible. 



OBSERVATIONS FOR AZIMUTH l8l 

103. Azimuth by an Altitude of a Star near the Prime Vertical. 

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 semi- 
diameter corrections become zero. The declinatioh 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 par- 
tially eliminated by combining two observations, one on a star 
about due east and the other on one about due west. 

Example. 

Mean altitude of Regulus (bearing east) on Feb. n, 1908, is 17 36'. 8. Latitude^ 
42 21' N. The right ascension is ioft 03 293.1 and the declination is -j-i2 24' 57". 
Compute the azimuth and the hour angle. 

= 42 21' log sec = 0.13133 

^ = *7 33 -8 log sec = 0.02073 

cos .90788 <j> h 24 47'. 2 

sm - 2I 5 02 6 = 4-12 25 .o 

c s .69286 log = 9.84065 

log vers Z n = 9.99271 
Z n = 89 oa'.S 
The star's bearing is therefore N 89 02'.8 E. 

To obtain the time we may employ formula [12], 

log sin Z n = 9-99994 
log cos h = 9.97927 
log sec 5 = 0.01028 

log sin t = 9.98949 

t = -77 2 6'.7 

= 5* 09 46*. 7 (east) 

Right ascension 10 03 29 .1 

Sidereal time = 4^ 53 42*4 

104. 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 
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 



182 PRACTICAL ASTRONOMY 

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 en- 
gineer's transit) or with a direction instrument in which the circles are read with 
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 comparatively 
large effect upon the results. 1 

The star chosen for this observation should be one of the close circumpolar stars 
given in the circumpolar list in the Ephemeris. (See Fig. 72.) Polaris is the only 
bright star in this group and should be used in preference to the others when it is 
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 J 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 

* 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. 

t 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 sui'cient accuracy 
by holding Fig. 72 so that Polaris is in its true position with reference to the me- 
ridian (as indicated by the position of 8 Cassiopeice) and then estimating the dif- 
ference 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 elon- 
gation, i.e., i to sec #. 



OBSERVATIONS FOR AZIMUTH 



of the measured horizontal angles. The labor involved in this process is so great, 
however, that the common 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. 



XVIII* 




51 Cephei 



XII 



FIG. 72 

For a precise computation of the azimuth of the star formula [32] may be used, 

~ _ sin/ , , 

n cos </> tan d sin cos t 

the azimuth being counted from the north toward the east. 

A second form may be obtained by dividing the numerator and denominator 
by cos < tan 5, giving 

tanZ* = - 



cot 5 sec <f> sin / 



i cot 6 tan <f> cos t 
If cot 5 tan tf> cos / is denoted by a, then 



tan Z n = cot 5 sec <f> sin t - 



If values of log- 



are tabulated for different values of log a the use of this 



i a 

third form will be found more rapid than the others. Such tables will be found in 
Special Publication No. 14, U. S. Coast and Geodetic Survey.* 

For a less precise value of the azimuth the following formula will be found con- 
venient; 

Z = p sin / sec h [106] 



* Sold by the Superintendent of Documents, Washington, D. C., for 35 cents. 



1 84 PRACTICAL ASTRONOMY 

in which Z and p are both in seconds or both in minutes of angle. The error due 
to substituting the arcs for sines is very small. The precision of the computed 
azimuth depends largely upon the precision with which h can be measured. If the 
vertical arc of the transit cannot be relied upon it will be better to use formula 

[32]- 

105. The Curvature Correction. 

If the azimuth of the star corresponding to the mean of the observed times has 
been computed it is necessary to apply a curvature correction to this result to ob- 
tain the mean of all the azimuths corresponding to the separate hour angles. The 
curvature correction may be computed by the formula 



in which n is the number of pointings on the star in the set and r for each obser- 
vation is the difference (in sidereal time) between the observed time and the mean 
of the times for the set. The interval r is tabulated as a time interval for con- 
venience, but is taken as an angle when computing the tabular number. The 
sign of this correction always decreases the angle between the star and pole. 

. r 
2 sin 2 - 

Values of -: 77 are given in Table X. 
sin i ' 

The curvature correction may also be computed by the formula 

-tan Z [0.2930] S (T - To) 2 [1070] 



in which the quantity in brackets is a logarithm; 2(T To) 2 is the sum of the 
squares of the time intervals in (sidereal) minutes. This correction should be sub- 
tracted from the azimuth Z calculated for the mean of the observed times. If it 
is preferred to express the time interval in seconds the logarithm becomes [6.73672]. 
The curvature correction is very small when the star is near the meridian; near 
elongation it is a maximum. 

106. The Level Correction. 

The inclination of the horizontal axis should be measured by the striding level, 
wand e being the readings of the west and east ends of the bubble in one position 
of the level, and w' and e' the readings after reversal of the level. The level cor- 
rection is then 

= - | (w + w'} - (e + e'} \ tan h , [108] 

4 I J 

if the graduations are numbered in both directions from the middle, or 

= - ( (w - w') + (e + e'} \ tan h [109] 

4 I J 

if the graduations are numbered continuously in one direction. In this formula 
the primed letters refer to the readings taken when the level " zero " is west. In 



OBSERVATIONS FOR AZIMUTH 185 

both formulae d is the angular value of a level division and h is the altitude of the 
star. 

If the azimuth mark is not in the horizon a similar correction must be applied 
to the readings on the mark. Ordinarily this correction is negligible. 

When applying this correction it should be observed that when the west end of 
the axis is too high the instrument has to be turned too far west (left) when pointing 
at the star. The correction must therefore be added to the measured angle if the 
mark is west of the star; in other words the reading on a circle numbered clockwise 
must be increased. If the correction is applied to the computed azimuth of the 
mark the sign must be reversed. 

107. Diurnal Aberration. 

If a precise azimuth is required a correction should be applied for the effect of 
diurnal aberration, or the apparent displacement of the star due to the earth's 
rotation. The observer is being carried directly toward the east point of the hori- 
zon with a velocity depending upon his latitude. The displacement will therefore 
be in a plane through the observer, the east point, and the star. The amount of 
the correction is given by 

o".32 cos <f> cos Z n sec h [no] 

The product of cos < and sec h is always nearly unity for a close circumpolar and 
cos Z is also nearly unity. The correction therefore varies but little from ".32. 
Since the star is displaced toward the east the correction to the star's azimuth is 
positive. 

1 08. Observations and Computations. 

In the examples which follow, the first illustrates a method appropriate for 
small surveyor's transits. The time is determined by the altitude of a star near 
the prime vertical and the azimuth of Polaris is computed by formula [106]. Cor- 
rections for curvature, inclination and aberration are omitted. 

In the second example the time was determined somewhat more precisely and a 
larger number of repetitions was used. The instrument was an 8-inch repeater 
reading to 10". 

The third and fourth examples are taken from the U. S. Coast and Geodetic 
Survey Spec. Publ. No. 14, and illustrates the methods used by that Survey where 
the most precise results are required for geodetic purposes. 

Example i 
Observed altitudes of Regulus (east), Feb. n, 1908, in lat. 42 21'. 

Altitude Watch 

17 05' 7^ 12"* 1 6 s 

17 3i 14 3i 

17 49 16 07 

18 02 17 20 

The right ascension of Regulus is io&o3 TO 29*. i; the declination is 4-12 24' 57". 
From these data the sidereal time corresponding to the mean watch reading (7*15^ 
03^.5) is found to be 4* 53 423.7. 



186 PRACTICAL ASTRONOMY 

Observed horizontal angles from azimuth mark to Polaris. 
(Mark east of north.) 

Telescope Direct Time of pointing on Polaris 

Mark o oo' 7 20 38* 

23 oo 
Third repetition 201 48' 23 56 

Mean = 67 i6 / .o ? h 22 31*. 3 

Telescope Reversed 

Mark = o oo' 7* 27 09* 

28 17 
Third Repetition 201 54' 29 21 



Mean = 67 iS'.o 7* 28"* 

Altitude of Polaris at 7* 2O 38* = 43 03' 
Altitude of Polaris at 7 29 21 = 43 01 
Mean watch reading for Polaris 7 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 .1 
t = 54 38' 

P = 4251 
log/> = 3-62849 
log sin / = 9.91141 
log sec h 0.13611 

log azimuth = 3.67601 
azimuth = 4743" 

= i i9'.o 
Mean angle = 67 17 .o 

Mark East of North = 65 58' o 



Example 2. 
RECORD OF TIME OBSERVATIONS 

Polaris: Chronometer, i2 h 09 31^.5; alt., 41 15' 4" 
c Com: Chronometer, 12 13 37 . 5; alt, 25 34 oo 

Polaris: R.A. = i* 25 si.i; decl. = +88 49' 2 4"-8 , 
eCorvi: R.A. = 12* 5"* 30* -5J ded. = - 22 07' 21". o 

Chronometer R. A. Decl. 

a Serpentis (E) 12* 24 153.7 15* 39 W Si 8 - 6 +6 42' 20".; 

* Hydra: (W) 12 18 32 .o 8 42 00.5 + 6 44 5 -9 

(Lat. = 42 21' oo" N.; Long. & 44*1 X 8 . o W.) 

From these observations the chronometer is found to be io* 22*.! fast. 



OBSERVATIONS FOR AZIMUTH 



I8 7 



Example 2 (continued) 
RECORD OF AZIMUTH OBSERVATIONS 

Instrument (B. & B. No. 3441) at South Meridian Mark. Boston, May 16, 1910. 
(One division of level == i5 // .o.) 



Object. 


3 

i 


d 

M-4 
O 

^6 


Chronometer. 


Horizontal circle. 


Level readings 
and angles. 


Vernier A, 


B. 














W E 


Polaris . . 






ii* 2435*.o 


o oo' oo" 


oo" 


7-o 3-9 














5-8 5.1 








27 15.0 




















12.8 9.0 








28 31-5 






9.0 















3-8 




3 




30 oo.o 






Corr. = 1 2". 5 








31 20.5 






Alt. Polaris at 




























41 20' 30" 








32 27.0 






Alt. Polaris at 














II* 5T m O4 S .O 


Mark... 




6 




*39 33' 30" 


30" 


41 18' 40" 














Mean horizontal 














angle = 














66 35' 35"- o 


Polaris. . 












W E 








II 42 45.5 


39 33' 3o" 


30" 


5-i 5-8 














3-3 7-6 








09.0 






8.4 13-4 




4) 




45 I 5- 






8.4 








46 29.5 






__ 














Corr. 1 6". 5 








47 25.0 














48 54-5 








Mark... 




6 




*78 27' 30" 


20" 


Mean horizontal 














angle = 














66 28' 59". 2 














Alt. Polaris at 














12* o9 m 31*. 5 = 














41 15' 4o" 



* Passed 360. 



l88 PRACTICAL ASTRONOMY 

Example 2 (continued) 

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 .1 

Hour Angle of Polaris =1001 12. 4 

t =150 18' 06" 
log cos < = 9.868670 
log tan 5 = 1.687490 
log cos <f> tan 5 = 1.556160 
cos tan 8 = 35.9882 
log sin </> = 9.82844 
log cos t = 9.93884 
log sin cos t = 9.76728 

sin cos / = .5852 
denominator =36.5734 

log sin t = 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 . T 
Mark east of North = 65 45' 47", o 



109. 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 



OBSERVATIONS FOR AZIMUTH 



189 



Example 3 

RECORD AZIMUTH BY REPETITIONS. 

[Station, Kahatchee A. State, Alabama. Date, June 6, 1898. Observer, 

O. B. F. Instrument, lo-inch Gambey No. 63. Star, Polaris.] 

[One division striding level = 2. "67.] 



Objects. 


Chr. time 
on star. 


S 
"o 

1 


Repetitions. 


Level read- 
ings. 
W. E. 


Circle readings. 


Angle. 


o 


' 


A 


B 


d 

1 




h m s 


















/ // 


Mark 




D 


o 




178 


O3 


22 .5 


2O 


21 .2 




Star 


HAJQ 30 




J 


4c? TO 7 




**o 












t\j \j 






j *'"' / 

9-2 5-9 
















49 08 




2 


















52 Si 


D 


3 


9.6 5.6 






















5.2 17-0 
















56 10 


R 


4 


11.3 4.0 






















7-8 7,4 














Set No. 5.. 


14 59 12 




5 


















15 oi 55 


R 


6 


8.7 6.6 


IOO 


16 


2O 


2O 


2O 


72 57 50.2 










".9 3-4 
















14 54 17-7 






68.2 53.6 






















+ 14-6 














Star. . . 


i 5 04 44 


R 


i 


11.9 3.4 






















8.5 6.8 
















07 18 




2 


















09 54 


R 


3 


7-9 7-3 






















II .2 4-1 














Set No. 6. . 


14 15 


D 


4 


9.0 6.1 






















5-9 9-6 
















16 14 




5 


















15 18 24 




6 


5-9 9-6 






















9.1 6.2 














Mark 




D 






177 


27 


OO 


OO 


OO 


72 51 46.7 




15 ii 48.2 






69.4 53-i 






















+ 16.3 















PRACTICAL ASTRONOMY 



COMPUTATION AZIMUTH BY REPETITIONS 
[Kahatchee, Ala. = 33 13' 4o".33-] 



Date, 1898, set 


June 6 5 


June 6 6 


Chronometer reading .... 


14. Z4. 17 7 


15 ii 48 2 


Chronometer correction. 


ii i 


7T I 


Sidereal time . . .... 


14. <\3 4.6 6 


T r TT 17 I 


a. of Polaris 


I 21 2O 3 


I 21 2O 3 


t of Polaris (time) 


17 32 26 3 


1 2. 4.Q C 6 8 


t of Polaris (arc) 
5 of Polaris * 


203 06' 34". 5 

88 4.S 4.6 Q 


207 29' I2 /; .0 


log cot 6 


8 3343O 


8 374.30 


log tan < 
log cos t. . 


9.81629 

906367% 


9.81629 

904.708% 








log a (to five places) 
log cot- 5 


8.11426^ 

8 334.3O 1 ? 


8.09857W 
8 2.2.4.3OC; 


log sec 


O 077^3^ 


o 077^^5; 


log sin t 


9 "\Q383OW 


cc 

9 6042 i in 


, i 


9QQA387 


9QO/ie8/i 


10g i - a 






log ( tan A} (to 6 places) 


8 00005771 


8 07063 <;w 


A Azimuth of Polaris, from 
north * 


o 34.' 22" S 


o 40' 26" 8 


, 2 sin 2 Jr 
r and : n- 


tn s " 
7 47.7 119.3 

5 09-7 52-3 
i 26.7 4.1 


m s " 
7 04.2 98.1 
4 30.2 39.8 
i 54.2 7.1 


sin i 
Sum 


i 52.3 6.9 
4 54.3 47.2 
7 37.3 114.0 
742 g 


2 26.8 ii. 8 
4 25.8 38.5 
6 35-8 85.4 
280 7 


Mean. . 


\7 1 


46 8 


i^ 2sinHr 


I 7C8 


i 670 


log n^ sini" 
log (curvature corr.) 
Curvature correction 


9.758 

0.6 


9-741 
-0.6 


Altitude of Polaris = h. . 


12 07' 




- tan h = level factor .... 


O 4.IQ 


O 4.IQ 


Inclination 


+*, 6 


+4, I 


Level correction 


^ 

I s 


^ // 
I 7 


Angle, star mark 


72 <7 <?O.2 


72 <I 4.6.7 








Corrected angle 


72 <7 4.8 7 


72 "?I 4. 1 ? O 


Corrected azimuth of star * 


34 22.2 


o 40 26.2 


Azimuth of mark E of N 


77 -22 IO Q 


73 32 II 2 


Azimuth of mark 


180 co OQ.O 

2t{2 22 IO.Q 


180 oo oo.o 
2;^ ^2 ii .2 


(Clockwise from south) 







To the mean result from the above computation must be applied corrections for diurnal aberra- 
tion and eccentricity (if any) of Mark. Carry times and angles to tenths of seconds only. 
* Minus if west of north. 



OBSERVATIONS FOR AZIMUTH 



191 



Example 4 

HORIZONTAL DIRECTIONS 

[Station, Sears, Tex. (Triangulation Station). Observer, W. Bowie. In- 
strument, Theodolite 168. Date, Dec. 22, 1908.] 



C 

1 


Objects 
observed. 



H 


3* 


o 




Backward. 


4 


1 


4) d 


Direc- 
tion. 


Remarks. 


i 
I 


Morrison , . 


h m 
8 19 


D 


A 





f 




tf 

35 


tf 
35 






" 


i division of the 










B 






41 


41 








striding level 










C 






36 


34 


37.o 






= 4". 194 








R 


A 


180 


00 


36 


35 


















B 






32 


31 


















C 






35 


34 


338 


35.4 


00 






Buzzard.. 




D 


A 


53 


30 


43 


42 


















B 






41 


42 


















C 






34 


33 


39-2 














R 


A 


233 


30 


39 


37 


















B 






34 


32 


















C 






38 


38 


36 3 


378 


02 4 






Allen 




D 


A 


170 


14 


6l 


62 


















B 






57 


55 


















C 






61 


59 


59-2 














R 


A 


350 


14 


50 


49 


















B 






63 


60 


















C 






53 


53 


54-7 


57 o 


21.6 






Polaris 




D 


A 


252 


01 


54 


53 








W E 




h m s 






B 






54 


53 








9-3 28.0 


. 


i 48 35-5 






C 






51 


51 


52-7 






27-7 9-1 




i 51 06.0 






















- 


























18.4 0.5 18.9 




I 49 50.8 






























R 


A 


72 


01 


09 


09 








24-9 6-3 










B 






02 


01 








13.0 31-7 










C 






10 


08 


06.5 


29 6 































ii. 9 -I3-S 25.4 


























- 7.0 



192 



PRACTICAL ASTRONOMY 



COMPUTATION OF AZIMUTH, DIRECTION METHOD. 

[Station, Sears, Tex. Chronometer, sidereal 1769. - 32 33' 31' 

Instrument, theodolite 168. Observer, W. Bowie.] 



Date 1908, position 


Dec.. 22, i 

i 49 SO. 8 
- 4 37- S 
i 45 13 3 
I 26 41 9 
18 31.4 
4 37' 5i". o 
88 49 27.4 

8.31224 
9.80517 
9.99858 


2 

2 01 33.0 

- 4 37-5 
i 56 55-5 
r 26 41.9 
o 30 13.6 
733'2 4 ".o 

8.31224 
9-80517 
9-99621 


3 
2 16 31 o 

- 4 37 4 
2 II 53 6 
I 26 41 8 
o 45 ii 8 
ii 17' 57". o 

8.31224 
9-80517 
9.99150 


' L 

4 
2 43 28 8 
- 4 37 3 
2 38 Si 5 
i 26 41 8 
I 12 09.7 

I802'25".S 

8.31224^ 
9-80517 
9.978II 


Chronometer reading 


Chronometer correction 
Sidereal time 
a of Polaris 


t of Polaris (time) 
/ of Polaris (arc) 
d of Polaris 


log cot S . . 


log tan 


log cos / 




log o (to five places) 
log cot 8 


8.11599 

8.312243 

0.074254 
8.907064 

0.005710 


8.11362 

8 312243 
0.074254 
9.H8948 

0.005679 


8.10891 

8.312243 

0.074254 
9.292105 

0.005618 


8.09552 

8.312243 
0.074254 
9.490924 

0.005445 


log sec <f> 


log sin / 


io g -!- 


i a 


log ( tan A) (to 6 places) 


7.299271 
o 06 50.8 
m s 
2 30 
o 


7.511124 
o ii 09.2 
m s 

2 00 

o 


7.684220 
o 16 36.9 
m s 
3 18 

o 


7.882866 
o 26 15.0 
m s 
i 38 




A Azimuth of Polaris, from north* 
Difference in time between D. 
and R. . 


Curvature correction 


Altitude of Polaris = h 


33 46 
0.701 

-7.0 
-4-9 
252 01 29.6 


33 46 

0.701 

-7.2 
-S.o 
86 58 ii. 2 


33 46 
0.701 

-7.0 
~4-9 
281 54 27.0 


33 46 

0.70! 

-1.8 

-1-3 

116 45 48.6 


d 
~ tan h level factor 


4 
Inclination "f" 


Level correction 


Circle reads on Polaris 




Corrected reading on Polaris 


252 01 24.7 

170 14 57.0 


86 58 06.2 
5 IS S8.2 


281 54 22.1 

200 17 42.4 


116 45 47-3' 
35 18 45.4" 


Circle reads on mark 




Difference, mark Polaris 
Corrected azimuth of Polaris, from 
north* 


278 13 32 3 

o 06 50.8 
180 oo oo.o 


278 17 52.0 

o ii 09.2 
180 oo oo.o 


278 23 20.3 

o 16 36.9 
180 oo oo.o 


278 32 58.1 

o 26 15.0 
180 00 00.0 




Azimuth of Allen 


98 0641.5 


98 06 42.8 


98 06 43.4 


98 06 43-1 


(Clockwise from South) 



. To the mean result from the above computation must be applied corrections for diurnal aberra- 
tion and eccentricity 'if any) of Mark. 

Carry times and angles to tenths of seconds only. 

* Minus, if west of north. 

t The values shown in this line are actually lour times the inclination of the horizontal axis 
in terms of level divisions. 




OBSERVATIONS FOR AZIMUTH *93 

other star are in the same vertical plane, and then waiting a 
certain interval of time, depending upon the date and the star 
^served, 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 
the instant when the latter is in the meridian is 
computed as follows : In Fig. 73 P is the pole, P' is 
Polaris, S is the other star (6 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- 
a tracting this from 180 gives the angle ZP'P\ PP' 

IG ' 7 is known, and PZ is the colatitude of the observer. 
The triangle ZP f P may then be solved for ZPP', the desired 
togle. Subtracting ZPP f from 180 or i2 h gives the sidereal 
interval of time which must elapse between the two 
observations. The angle SPP r 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 & 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 



I 9 4 PRACTICAL ASTRONOMY 

required; but for many purposes it is sufficient to use a time 
interval calculated from the mean places of the stars and for a 
mean latitude of the United States. The time ititerval for the 
star 6 Cassiopeia for the year 1910 is 6 m .i and for 1920 it is about 
i2 m .3. For the star f Ursa Majoris the time interval for the 
year 1910 is approximately 6^.7, while for 1920 it is 11^.3. Be- 
ginning with the issue for 1910 the American Ephemeris and 
Nautical Almanac gives values of these intervals, at the end 
of the volume, for different latitudes and for different dates. 
Within the limits of the United States it will generally be nec- 
essary to observe on b Cassiopeia when Polaris is at lower 
culmination and on f Ursce 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 
8 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 
8 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 correctlv measured. 



OBSERVATIONS FOR AZIMUTH 



195 



no. 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 
*'n 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- 
, and the star should be one that can be identified with certainty 6^ or & later. 
* r^ould be taken to use a star which will reach the same altitude on the oppo- 
Jte side ot the meridian before daylight interferes with the observation. In the 




-RM. 



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. 74 . 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 
ferst observation. When trie 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 



IQ4 
196 



PRACTICAL ASTRONOMV 
PRACTICAL ASTRONOMY 



is turned to the same reference mark. The bisector of the angle between the two 
directions is the meridian line through the transit. It will usually be found more 
practicable to turn angles from a fixed mark to the star than to set stakes. The 
accuracy of the result may be increased by observing the star at several different 
altitudes and using the mean value of the horizontal angles. In this method ti t 
index correction (or that part of it due to non-adjustment) is eliminated, since it 
is the same for both observations. The refraction error is also eliminated, pro- 
vided it is the same at the two observations. Error in the adjustment of the hori- 
zontal axis and the line of sight will be eliminated if the first half of the set is taken 
with the telescope direct and the second half with the telescope reversed. With 
a transit provided with a vertical arc (180) this cannot be done. Care should b ej 
taken to re-level the plates just before the observation is begun; the levelling should)* 
not, of course, be done between the pointing on the mark and the pointing on the 
star, but may be done whenever the lower clamp is loose. 

in. 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 = 



cos <f> sin t 



in which d is the hourly change in declination multiplied by the number of hours 
elapsed between the two observations, < is the latitude, and / 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. 

VARIATION PER HOUR IN SUN'S DECLINATION 
(1925) 



Day of 
month 


Jan. 


Feb. 


Mar. 


Apr. 


May 


June 


July 


Aug. 


Sept. 


Oct. 


Nov. 


Dec. 


I 


+ 12" 


+42" 


+ 57" 


+ 58" 


+46" 


+ 21" 


- 9" 


~37" 


-54" 


-58" 


-49" 


-24' 


5 


16 


45 


58 


57 


43 


17 


13 


40 


55 


58 


46 


2O 


10 


22 


48 


5Q 


56 


40 


12 


18 


43 


57 


57 


43 


14 


15 


27 


5i 


59 


54 


36 


7 


23 


46 


58 


56 


39 


9 


20 


32 


54 


59 


52 


32 


+ 2 


27 


49 


5 


54 


35 


~~3 


25 


36 


+ 56 


59 


49 


28 


-3 


32 


5i 


59 


52 


30 


+3 


30 


+41 




+ 58 


+46 


+ 23 


~8 


-36 


-53 


-5 


-50 


-25 


+9 



OBSERVATIONS FOR AZIMUTH 197 

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 
iri position, the upper clamp loosened, and the telescope pointed at the sun. It 
P, 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 accu- 
rately as possible. The altitude and the horizontal angle are both read. In the 
afternoon 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 
altitude 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 
mark to the south point of the horizon. The algebraic sign of the correction is 
determined from the fact that if the un 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' f Alt., 24 58' 



U&L limbs 



Hor. Angle, 357 14' 15" U & R limbs Hor. Angle, 162 28' oo" 



Time, 7^ ig m 30* [ Time, 4** i2 15* 

\ elapsed time = 4 h 26- 22** 

t = 66 35' 30" Incr. in decl. = + 52" X 4^.44' 

log sin/ = 9.96270 = -f 230^.9 

log cos = 9.86902 

9.83172 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" 

112. 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 



198 PRACTICAL ASTRONOMY 

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 Timel 
can be obtained accurately by the noon signal and the longitude can be 
obtained from a map within about 500 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 observations 
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 there- 
fore be used in determining the time. The observed watch reading 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, t. The 
azimuth is then found by the formula 

sin Z = sin / sec h cos 5. [12] 

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. A 



Example. 

Observation on the sun for azimuth. 
Lat. 42 21'. Long. 4* 44 18* W. Date, Feb. 5, 1910. 



Hor. Circle 

Mark, o oo' 

app. L & L limbs, 29 or 

app. U & R limbs, 28 39 

Mean, 28 50' 



Vert. Circle. 



3i 49' 
31 16 



Refr., 
h = 



1.6 



Watch. 
(30* fast) 

- -V 22* 



i I 44 20 



5 = -l602 / 32.2 // 

Eq. t. 



Watch corr. 
E. S. T. 

L. M. T. 

Eq. t. 

L. A. T. 



log sin t 
log cos 5 
log sec h 

log sin Z 
Z 

Hor. Circle 

Azimuth 



-30 
= n* 43^ 21* 
15 42 
= 1 1* 59 03* 

14 09 .1 

15 06 .1 
" 346'.5 

= 8.81847 

: 9-98275 
= 0.06930 

8.87052 
s 4i5'-4 
: 2 8 50 



. 
326 S4 '.6 



OBSERVATIONS FOR AZIMUTH 199 

113. Meridian by the Sun at the Instant of Noon. 3 " 

If the error of the watch can be determined within about one second, and the 
sun's decimation is such that the noon altitude is not too high for convenient ob- 
serving and accurate results, the following method of determining the meridian 
will often prove useful. Before beginning the observation the watch time of local 
apparent noon should be computed and carefully checked. If the centre of the 
sun can be sighted accurately with the vertical hair at this time the line of sight 
will be in the meridian, pointing to the south if the observer is in the northern 
hemisphere. As this is not usually practicable the vernier reading for the south 
point of the horizon may be found as follows: Set the " A " vernier to read o, 
sight on a reference mark (such as a point on the line whose bearing is to be found) 

|and clamp the lower motion. Loosen the upper clamp and, about io m before noon, 
set the vernier so that the vertical hair is a little in advance of the west edge of 
the sun's disc. Read the watch as each limb passes the vertical hair and note the 
vernier reading. Then set the vernier so that the line of sight is nearly in the 
meridian and repeat the observation. It is best to make a third set as soon as 
possible after the second to be used as a check. 

The mean of the two watch readings in each observation is the reading for the 
centre of the sun. This may be checked roughly by reading also the time when 
the sun's disc appears to be bisected. From the first and second vernier readings 
dad the corresponding watch times compute the motion of the sun in azimuth per 
second of time. Then from the second watch reading and the watch time of ap- 
parent noon, the difference of which is the interval before or after noon, compute 
the correction to the second vernier reading to give the reading for the meridian. 
The third set of observations may be used in conjunction with the first to check the 
preceding computations, by computing the second vernier reading from the ob- 
served time and comparing with the actual reading. The reading for the south 
point may also be computed by using the first and third or the second and third 
observations. 

The accuracy of the results depends upon the accuracy with which the error of 
the watch may be obtained, upon the reliability of the watch, and the accuracy of 
the longitude, obtained from a map or by observation. When the sun is high the 

, sights are more difficult to take and the sun's motion in azimuth is rapid so that an 

terror in the watch time of noon produces a larger error in the result than when it 
has a low altitude at noon. The method is therefore more likely to prove satis- 
factory in winter than in summer. In winter the conditions for a determination 
of meridian from a morning or afternoon altitude of the sun are not favorable, so 
that this method may be used as a substitute. The method is more likely to give 
satisfactory results when the observer is able to obtain daily comparisons of 
his watch with the time signal so that the reliability of the watch time may be 
judged. 

? 

* The method described in this article was given by Mr. T. P. Perkins, 
Engineering Dept, Boston & Maine R. R., in the Engineering News, March 31, 
1904. 



200 PRACTICAL ASTRONOMY 

Example. 

On Jan. i, 1925, in latitude 42 22' N., longitude 71 os'.6 W., the " A " vernier 
is set at o and cross hair sighted at mark; the vernier is then set to read 42 40' 
(to the right). The observed times of transit of the west and east edges of the sun 
over the vertical hair are n h 36^ 39* and n* 39 oi s . The vernier is then set at 
45 04'.$; the times of transit are n* 460*10* and n* 48^31*. As a check the 
vernier is set on 45 35', the times of transit being n>48 m o8 s and n ft 50^31*. 
The watch is 13* slow of Eastern time. Find the true bearing of the mark from 
the transit. 

Local Apparent Time = 12* oo"* oo* 

Equation of Time = 3 40 .8 

Local Civil Time =12* 03 40^.8 

Longitude difference = 15 37 .6 



Eastern Standard Time n h 48 03^.2 

Watch slow 13 



Watch time of apparent noon = n* 47 50^.2 

Interval, ist obs. to 2nd obs. = g m 30^.5 = 570^.5 

Interval, 2nd obs. to noon = 295.7 

Diff. in vernier readings = 2 24^5 = 144'. 5 

The correction (x) to the 2nd reading (45 04'. 5) is found by the proportion 
x : 144-5 = 29.7 : 570.5 



The vernier reading for the meridian is therefore 45 04^.5 + 7'.$ = 45 12' making 
the bearing of the mark S 45 12' E. 

114. Approximate Azimuth of Polaris when the Time is Known. 

If the error of the watch is known within half a minute or so, the azimuth of 
Polaris may be computed to the nearest minute, that is, with sufficient precision 
for the purpose of checking the angles of a traverse. The horizontal angle between 
Polaris and a reference mark should be measured and the watch time of the point- 
ing on the star noted. It is desirable to measure also the altitude of Polaris at 
the instant, although this is not absolutely necessary. A convenient time to make^ 
this observation is just before dark, when both the star and the cross hairs can be 
seen without using artificial light. The program of observation would be: i. 
Set on o and sight the mark, clamping the lower clamp. 2. Set both the vertical 
and the horizontal cross hairs on the star and note the time. 3. Read the hori- 
zontal and the vertical angles. 4. Record all three readings. The method of 
repetition may be employed if desired. 

If the American Ephemeris is at hand the azimuth of Polaris may be taken out 
at once from Table IV when its hour angle and the latitude are known. The watch 
time of the observation should be converted into local sidereal time and the hour 
angle of the star computed. The azimuth may be found by double interpolation 

in TnKlA TV nTnliAm ^\ 



OBSERVATIONS FOR AZIMUTH 



2OI 



Example. 

On May 18, 1925, the angle from a reference mark (clockwise) to Polaris was 
36 10' 30"; watch time & h 10"* 20* P.M.; wat^h slow 10 s ; altitude 41 21'. 5; lat- 



itude 42' 
Fig. 75-) 



22'N.; longitude 7io6'W. Find the azimuth of the mark. (See 

First Solution 
Watch reading 8> iow 20* P.M. 



Watch correction 
Eastern Time 

Local Time 
Loc. Civ. T. 
Table III 
Table III (Long.) 



+ 10 



io m 30* P.M. 
15 36 



Loc. Sid. T. 
a, Polaris 

t, Polaris 

From Table IV, Ephemeris, azimuth 
Measured horizontal angle 

^ Bearing of mark 

North 
Polaris 



Mark 



8 h 26 06* P.M. 

20 26 06 

3 21 .4 
46.7 

15 40 38.3 
36^ io m 525.4 

12 10 52 .4 

i 33 38.6 



o 3 o'. 9 West 
36 10.5 




West 



West 



East 



South 
FIG. 75 




East 



Polaris 



If the Ephemeris is not at hand the azimuth may be found from the tables on 
pp. 203 and 204 of this volume. The watch time of the observation is corrected 
for the known error of the watch and then converted into local time. From Table 
V the local civil time of the upper culmination of Polaris may be found. The 
difference between the two is the star's hour angle in mean solar units. This 
should be converted into sidereal units by adding 10 s for each hour in the interval. 
(Fig. 76.) It should be observed that if the time of upper culmination is less than 



202 



PRACTICAL ASTRONOMY 



the observed time the difference is the hour angle measured toward the west, and 
the star is therefore west of the meridian if this hour angle, is less than 12*. If 
the time of upper culmination is greater than the observed time the difference 
is the hour angle measured toward the east (or 24^ the true hour angle) and the 
star is therefore east of the meridian if this angle is less than 12^. 
To obtain the azimuth from Tables F and G we use the formula 



Z' = p' sin t sec h. 



106] 



Table F gives values of p' sin t for the years 1925, 1930, and 1935, and for every 
4 m (or i) of hour angle. To multiply by sec h, enter Table G with the value of 
p' sin I at the top and the altitude (h) at the side. The number in the table is to be^ 
added to p' sin t, to obtain the azimuth Z'. 

It is evident that the result might be obtained conveniently by using the time 
of lower culmination when the hour angle (from U. C.) is nearer to 12** than 
too*. 

If the altitude of the star has not been measured it may be estimated from the 
known latitude by inspecting Figs. 65 and 72 and estimating how much Polaris is 
above or below the pole at the time of the observation. This correction to the 
altitude may also be obtained from Table I in the Ephemeris if the hour angle of 
the star is known. 

As an illustration of the use of these tables we will work out the example given 
on p. 201. 

Second Solution 



Observed time 
Watch correction 

Eastern time 



Local time 
Loc. Civ. T. 
Upper culmin. 

10* X i<A6 
Hour angle 



8 low 20* P.M. 

+10 

8* io m 30* P.M. 
15 36 

8 26 06* P.M. 

20 26 .1 
Q 50 .7 



i .8 



10^ 37^.2 



Table V, U. C., May 15, io O2.i 
Corr. for 3 days n .8 

May 18 ' 9^ 50^.3 

Corr. for 1925 -fo .2 

Corr. for long. -fo .2 

- 

Upper culmin. o& 50^.7 



From Table F, p' sin / = o 23'. 2 
From Table G^corr. 7 .7 

Azimuth = o 3o'.p 
Measured horizontal angle = 36 10 .5 

Bearing of mark N. 36 41'. 4 W. 

Since the observation just described does not have to be made at any particular 
time it is usually possible to arrange to sight Polaris during twilight when terres- 
trial objects may still be seen distinctly and no illumination of the field of the 
telescope is necessary. In order to find the star quickly before dark the telescope 
should be focussed upon a very distant object and then elevated at an angle equal 
to the star's altitude as nearly as this can be judged. It will be of assistance in 



OBSERVATIONS FOR AZIMUTH 



203 



TABLE F 
Values of p sin / for Polaris (in minutes) 



t 


* 
I92S 


1930 


1935 


/ 


/ 


1925 


1930 


1935 


/ 


h m 








h m 


h m 








h m 





0.0 





0.0 


12 00 


3 oo 


46.5 


45-4 


44 4 


9 oo 


4 


I.I 


i.i 


i i 


ii 56 


04 


47-3 


46.2 


45 2 


56 


8 


2 3 


2 2 


2 2 


52 


08 


48.1 


47-0 


45 9 


52 


12 


3 4 


3-4 


3 3 


48 


12 


48.8 


47 7 


466 


48 


o 16 


4-6 


4-5 


4-4 


II 44 


316 


49 6 


48.4 


47 4 


844 


20 


5-7 


5 6 


5 5 


40 


20 


50.4 


49 2 


48.1 


40 


24 


6 8 


6 7 


6.6 


36 


24 


51-2 


49 9 


48 8 


36 


28 


8.0 


78 


7-6 


32 


28 


51-9 


So 6 


49 5 


32 


o 32 


9 I 


89 


8.7 


II 28 


3 32 


52.6 


51 3 


50 i 


8 28 


36 


10.3 


10. 1 


98 


24 


36 


53-3 


52.0 


50.8 


24 


40 


II. 4 


II. 2 


10 9 


20 


40 


53 9 


52. 6 


51-4 


20 


44 


12.5 


12 3 


12.0 


16 


44 


54 6 


53 3 


52 o 


16 


o 48 


13 7 


13-4 


13.0 


II 12 


348 


55 2 


53 9 


52.6 


8 12 


52 


14.8 


14 4 


14 I 


08 


52 


55 8 


54 5 


53 2 


08 


56 


15 9 


15-5 


15.2 


04 


56 


56 4 


55-0 


53-8 


04 


I 00 


17.0 


16.6 


16.2 


II 00 


4 oo 


57-0 


556 


54 4 


8 oo 


I 04 


18 i 


17 7 


17.3 


10 56 


4 04 


57-6 


562 


54 9 


756 


08 


19-2 


18.8 


183 


52 


08 


58 i 


567 


55 4 


52 


12 


20 3 


19 9 


19 4 


48 


12 


58 6 


57-2 


559 


48 


16 


21.4 


20.9 


20 4 


44 


16 


59 2 


57-8 


56.4 


44 


I 20 


22.5 


22 


21.5 


10 40 


4 20 


59-6 


58.2 


56.9 


7 40 


24 


23 5 


23 o 


22.5 


36 


24 


60. i 


58-7 


57-3 


36 


28 


24 6 


24 I 


23.5 


32 


28 


60.6 


59-2 


57-8 


32 


32 


25 7 


25 I 


245 


28 


32 


61 o 


59-6 


582 


28 


136 


26.7 


26 1 


25 5 


10 24 


436 


61 4 


60 


58.6 


7 24 


40 


27,8 


27 2 


26.5 


20 


40 


61.8 


60.4 


59-0 


20 


44 


28 8 


28 2 


27.5 


16 


44 


62.2 


60.8 


59-3 


16 


48 


29-9 


29.2 


28.5 


12 


48 


62.6 


61 i 


59-7 


12 


I 52 


30.9 


30 2 


29.5 


10 08 


4 52 


63.0 


61.4 


60 o 


708 


56 


31 9 


31-2 


30 4 


04 


56 


63-3 


61.8 


60.3 


04 


2 00 


32.9 


32 I 


31-4 


10 00 


5 oo 


63.5 


62.0 


60.6 


7 oo 


04 


33-9 


33 I 


32.3 


9 56 


04 


638 


623 


60.9 


6 56 


2 08 


34-9 


34.1 


33-3 


9 52 


508 


64.1 


62.6 


61.2 


652 


12 


35-8 


35 


34 2 


48 


12 


64.3 


62.9 


61.4 


48 


16 


36.8 


35 9 




44 


16 


64.6 


63.1 


61.6 


44 


20 


37-7 


36 8 


36 o 


40 


20 


64.8 


63.3 


61.8 


40 


2 24 


38.7 


37-8 


36.9 


9 36 


5 24 


65.0 


63.4 


62 o 


636 


28 


39-6 


38-7 


37-8 


32 


28 


65.2 


63.6 


62.2 


32 


32 


40.5 


39-6 


38.6 


28 


32 


65.3 


63.8 


62.3 


28 


36 


41.4 


40.4 


39-5 


24 


36 


65.5 


63.9 


62.4 


24 


2 40 


42.3 


41-3 


40.4 


9 20 


5 40 


65.6 


64.0 


62.5 


6 20 


44 


43 2 


42.2 


41.2 


16 


44 


65.6 


64.1 


62.6 


16 


48 


44-0 


43-0 


42.0 


12 


48 


65.7 


64.2 


62.7 


12 


52 


44-9 


43-8 


42.8 


08 


52 


65.8 


64.2 


62.7 


08 


256 


45-7 


44-6 


43-6 


9 04 


S56 


65.8 


64.2 


62.8 


6 04 


3 oo 


46.5 


45-4 


44-4 


9 oo 


6 oo 


65.8 


64.3 


62 8 


6 oo 



204 



PRACTICAL ASTRONOMY 



TABLE G.- CORRECTION FOR ALTITUDE 



p sin t 


Proportional Parts 


Alt. 


10' 


20' 


30' 


40' 


50' 




















9' 


60' 


i' 


2' 


3 


4' 


5' 


6' 


7' 


8' 


IS 


o' 4 


o'.7 


i' i 


i' 4 


i'. 8 


2'.i 


o' o 


o' i 


o' i 


o' i 


0' 2 


0' 2 


o'.2 


o'-3 


o'-3 


18 


o .5 


I .0 


i 5 


2 .1 


2 .6 


3 -I 


I 


.1 


.2 


2 


o 3 


o 3 


o .4 


o .4 


o 5 


21 


o .7 


i .4 


2 .1 


2 .8 


3-6 


4 3 


.1 


.1 


.2 


o .3 


o 4 


o 4 


o -5 


o 6 


0.6 


24 


o .9 


i 9 


2 .8 


3 8 


4 -7 


5 -7 


I 


2 


o .3 


o 4 


o -5 


o .6 


o 7 


o 8 


o -9 


27 


I 2 


2 4 


3 7 


4 9 


6 i 


7 -3 


I 


2 


o 4 


o -5 


o 6 


o 7 


o ,9 


I .0 


i .1 


30 


I 5 


3 -I 


4-6 


6 .2 


7 7 


9 3 


2 


o 3 


o 5 


o 6 


o .8 


o 9 


i i 


I .2 


I -4' 


31 

32 


I 7 
I 8 


3 3 
3-6 


5 -o 
5 4 


6 7 
7 -2 


8 .3 
9 o 


10 O 

10 .8 


2 
2 


o .3 

o 4 


o 5 
o 5 


o 7 
o 7 


o .8 
o 9 


I .0 


I 2 


I 3 


I 5 


33 
34 


I 9 
2 I 


3 8 
4 .1 


5 8 
6 .2 


7 -7 
8 2 


9 6 
10 3 


ii 5 

12 4 


2 


o 4 


o 6 
o .6 


o 8 
o .8 


I 
I 


I 2 
I .2 


I 3 
I 4 


I -5 
I .6 


i -7 
I .9 


35 


2 .2 


4 .4 


6.6 


8 .8 


II 


13 -2 


2 


o .4 


o .7 


o 9 


I I 


I -3 


i 5 


i .8 


2 .0 


36 


2 4 


4 7 


7 -I 


9 -4 


ii 8 


14 2 


.2 


o 5 


o .7 


o 9 


I 2 


i .4 


I -7 


i 9 


2 .1 




2 5 


5 -o 


7 6 


10 I 


12 6 


15 I 


3 


o 5 


o 8 


I 


i 3 


I 5 


i 8 


2 


2 3 


38 


2. 7 


5 -4 


8 .1 


10 .8 


13 5 


16 I 


o .3 


o 5 


o .8 


i .1 


i 3 


I 6 


i -9 


2 I 


2 -4 


39 


2 .9 


5 -7 


8 .6 


n .5 


14 -3 


17 .2 


o -3 


o 6 


o 9 


i .1 


i 4 


I .7 


2 .0 


2 .3 


2 .6 


40 


3 I 


6 i 


9 2 


12 2 


15 3 


18.3 


o 3 


o .6 


o 9 


I .2 


i 5 


i 8 


2 I 


2 4 


2 -7 


40-30 


3 2 


6 3 


9 5 


12 6 


15-8 


18.9 


o 3 


o 6 


I 


i 3 


i .6 


i -9 


2 .2 


2 .5 


2 .8 


4i 


3 3 


6-5 


9.8 


13 -o 


16 3 


19 -5 


o 3 


o 7 


I 


I 3 


i .6 


2 


2 3 


2 .6 


2 .0- 


41-30 


3 -4 


6 7 


10 .1 


13 -4 


16 8 


20 I 


o 3 


o 7 


I .0 


I 4 


i 7 


2 


2 4 


2 7 


3 -0 


42 


3 -5 


6-9 


10 .4 


13 8 


17 3 


20 7 


o 3 


o .7 


I 


I .4 


i 7 


2 .1 


2 4 


2 .8 


3 .1 


42-30 


3-6 


7 -I 


10.7 


14 2 


17.8 


21 3 


o 4 


o 7 


I I 


I 4 


i 8 


2 .2 


2 .5 


2 9 


3 2 


43 


3 -7 


7 .3 


II .0 


14 7 


18 4 


22 .0 


o .4 


o 7 


I I 


I 5 


l .8 


2 2 


2 6 


2 .9 


3 -3 


43-30 


3 8 


7 6 


u 4 


15 -I 


18 9 


22 7 


o 4 


o 8 


I I 


I 5 


i 9 


2 -3 


2 .7 


3 -0 


3 -4 


44 


3 -9 


7-8 


II .1 


IS 6 


19 -5 


23 -4 


o 4 


o 8 


I 2 


I .6 


2 


2 -3 


2 .7 


3 i 


3 -5 


44-30 


4 o 


8 .0 


12 .1 


16 .1 


20 I 


24 I 


o 4 


o .8 


I .2 


I .6 


2 .0 


2 .4 


2 .8 


3.2 


3-6 


45 


4 -I 


8.3 


12 .4 


16 .6 


20 7 


24 -9 


o .4 


o .8 


I .2 


I -7 


2 .1 


2 S 


2 .9 


3 -3 


3 7 


45-30 


4 3 


8-5 


12 8 


17 .1 


21 3 


25 6 


o .4 


o .8 


I -3 


i 7 


2 I 


2 .6 


3 -o 


3 -4 


3-8 


46 


4 .4 


8 8 


13 -2 


17.6 


22 


26 4 


o 4 


o 9 


I 3 


I .8 


2 2 


2 .6 


3 -I 


3 5 


3 -9 


46-30 


4 -5 


9 o 


U 6 


18 .1 


22 6 


27 2 


o -5 


o .9 


I 4 


i 8 


2 3 


2 -7 


3 -2 


3 -6 


4 i 


47 


4-7 

A ft 


9 3 

Q fi 


14 o 


18.7 


23 3 


28 .0 

28 8 


o .5 


o .9 


I 4 


i -9 


2 -3 


2. 8 

2Q 


3 3 


3 -7 

i 8 


4 -2 


47~"30 
48 


4 - 

4-9 


y .v 
9 9 


14 .8 


19 .8 


24 -7 


29 -7 


o 5 
o 5 


I .0 


I 4 
I 5 


1 9 

2 


2 4 
2 5 


y 
3 o 


3 4 
3 -5 


6 - 

4 o 


4 -3 

4 4 


48-30 


5 -i 


10 .2 


IS -3 


20 .4 


25 -5 


30 .5 


o 5 


I 


I 5 


2 


2 .6 


3 i 


3-6 


4 -I 


4-6 


49 


5 2 


10.5 


IS .7 


21 .0 


26 .2 


31 -5 


o 5 


I .0 


I .6 


2 .1 


2 6 


3 -I 


3 -7 


4.1 


4 -7 


49-30 


5 -4 


10 .8 


16 2 


21 6 


27 o 


32 -4 


o 5 


I .0 


I .6 


2 .2 


2 .7 


3 2 


3-8 


4 -3 


4 -9 


50 


5-6 


II .1 


16.7 


22 2 


27.8 


33 -3 


o .6 


I .1 


I .7 


2 .2 


2 .8 


3 3 


3 9 


4 4 


5 -0 - 


50-30 


5 -7 


II .4 


17 .2 


22 .9 


28 .6 


34 3 


o .6 


I .1 


I .7 


2 .3 


2 -9 


3 4 


4 o 


4 6 


5 -2 


51 


5 -9 


II .8 


17 7 


23.6 


29 -5 


35 .3 


o .6 


I .2 


I .8 


2 .4 


2 -9 


3 -5 


4 i 


4 -7 


5 3 


51-30 


6 .1 


12 .1 


18 .2 


24 3 


30 .3 


36.4 


o .6 


I .2 


I .8 


2 .4 


3 .0 


3-6 


4 .3 


4-9 


5 -5 


52 


6 .2 


12 .5 


18 7 


25 .0 


31 -2 


37 -5 


o 6 


I .2 


i -9 


2 5 


3 .1 


3 7 


4 -4 


5 -o 


5-6 


52-30 


6.4 


12.8 


19 3 


25.7 


32 I 


38.6 


o .6 


I -3 


i 9 


2 .6 


3 -2 


3 -9 


4-5 


5 -i 


5-8 


53 


6 .6 


13 2 


19.8 


26.5 


33 -I 


39 -7 


0.7 


I -3 


2 .0 


2 .6 


3 -3 


4 .0 


4.6 


5 -3 


6 .0 


53-30 


6 .8 


13-6 


20 .4 


27 .2 


34 -I 


40 -9 


o .7 


I .4 


2 .0 


2 .7 


3 -4 


4 -I 


4-8 


5 -4 


6.1 


54 


7 -o 


14-0 


21 .0 


28 .1 


35 -I 


42 .1 


o .7 


I .4 


2 .1 


2 .8 


3 -5 


4 -2 


4 -9 


5-6 


6-3 


54-30 


7 -2 


14 .4 


21 .7 


28 .9 


36.1 


43 -3 


o .7 


I .4 


2 .2 


2 -9 


3-6 


4 -3 


5 .0 


5-8 


6.5 


55 


7 -4 


14 -9 


22 .3 


29-7 


37 -2 


44 -6 


o .7 


i -5 


2 .2 


3 .0 


3 -7 


4 .5 


5 .2 


5 -9 


6.7 



OBSERVATIONS FOR AZIMUTH 



205 



finding the horizontal direction of the star if its magnetic bearing is estimated 
and the telescope turned until the compass needle indicates this bearing. If 
there is so much light that the star proves difficult to find it is well to move 
k the telescope very slowly right and left. The star may often be seen when it 
is in apparent motion, while it might remain unnoticed if the telescope were 
motionless. 

115. Azimuth from Horizontal Angle between Polaris and /9 "Ursa Minoris.* 

In order to avoid the necessity for determining the time, which is often the chief 
difficulty with the preceding methods, the azimuth of Polaris may be derived from 
the measured horizontal angle between it and some other star, such as Ursa 
\Minoris. If the horizontal angle between the two stars is measured and the lati- 
Hude is known the azimuths of the stars may be calculated. 

The observation consists in sighting at a mark with the vernier reading o, 
then sighting Polaris and reading the vernier, and finally sighting at /3 Ursa Minoris 
and reading the vernier. The difference between the two vernier readings is the 
difference in azimuth of the stars (neglecting the slight change in the azimuth of 
Polaris during the interval). From an inspection of the table the correspond- 
ing azimuth of Polaris may be found. This azimuth combined with the vernier 
reading for Polaris is the azimuth of the mark. 

The following example is taken from the publication f referred to: 

FIELD RECORD 

Simultaneous Observations on a and /3 Ursa Minoris for Azimuth 
(a. observed first) 



Date: Friday, Nov. 9, 1923, P.M. 

Telescope Direct 



Latitude 37 57' 15" N. 



Point 
Sighted 


A Vernier 


Angle between 
a and/3 


Angle between 
a and mark 


Time 
Interval 


Polaris 
& Urs. Min. 
Mark 


00 00' 00" 
346 12 OO 

346 03 oo 


13 48' oo" 


i3S7'oo" 


24 s 




Telescope Inverted 




Polaris 
Urs. Min. 
Mark 


00 oo' oo" 

346 22 00 

346 04 oo 


13 38' oo" 

2)27 26' oo" 


I3 56' oo" 

2)27 "S3' oo" 


62* 
2)86 

43 s 


13 43' oo" 


13 56' 30" 



* This method and the necessary tables were published by C. E. Bardsley, 
Rolla, Mo., 1924. 

t School of Mines and Metallurgy, University of Missouri; Technical Bulletin, 
Vol. 7, No. 2. 



2O6 



PRACTICAL ASTRONOMY 



COMPUTATION RECORD 
Selected Values for Interpolation from Table I 



No. 


Latitude 37 


Latitude 37 57' 15" 


Latitude 38 


Az. at 


Angle 
between 
a and/3 


Az.a 


Angle 
between 
a andp 


Az.a 


Angle 
between 
a and/8 


133 

134 


o4i'.9 

o 38 .7 


13 46'.i 
13 9 -3 


o42'.38 

(o 41 .51) 
o 3p .18 


13 53' -06 
(13 43 .00} 
13 15 -88 


o 42'. 4 

o 39 -2 


i353'-4 
13 16 .2 



Interpolated Azimuth of Polaris Table I = o4i / .5i = 

Correction for declination = 

Correction for right ascension = 

Polaris East of North = 

Angle Polaris to Mark = 

Mark West of North =" 



o"4i 



+05 
+ 25 



13 



42 or 
56 30 



13 
1 80 



14 

00 



29" 

00 



Azimuth of Mark from South = 1 66 45' 31" 

The time interval should ordinarily be kept within one minute, so that the 
observations are as nearly simultaneous as possible. If, however, the order of the 
pointings is reversed in the second half set the error due to this time interval is 
nearly eliminated. Double pointings might be made on /3 Ursa Minoris, one 
before and one after that on Polaris, from which the simultaneous reading might 
be interpolated. 

116. Convergence of the Meridians. 

Whenever observations for azimuth are made at two different points of a survey 
for the purpose of verifying the angular measurements, the convergence of the 
meridians at the two places will be appreciable if the difference of their longitudes 
is large. At the equator the two meridians are parallel, regardless of their differ- 
ence of longitude; at the pole the convergence is the same as the difference ini 
longitude. It may easily be shown that the convergence always equals the differ- 
ence in longitude multiplied by the sine of the latitude. If the two places are in 
different latitudes the middle latitude should be used. Table VII was computed 
according to this formula, the angular convergence in seconds of angle being 
giyen for each degree of latitude and for each 1000 feet of distance along the parallel 
of latitude. 

Whenever it is desired to check the measured angles of a traverse between two 
stations at which azimuths have been observed the latitude differences and depar- 
ture differences should be computed for each line and the total difference in de- 
parture of the two azimuth stations obtained. Then in the column containing the 
number of thousands of feet in (his departure and on a line with the latitude will 
be found the angular convergence of the meridians. The convergence for num- 



OBSERVATIONS FOR AZIMUTH 207 

bers not in the table may be found by combining those that are given. For in- 
stance, that for 66,500 feet, in lat. 40, may be found by adding together 10 times 
the angle for 6000, the angle for 6000, and one-tenth the angle for 5000. The result 
is 549".3, the correction to be applied to the second observed azimuth to refer the 
line to the first meridian. 

Example. 

Assume that at Station i (lat. 40) the azimuth of the line i to 2 is found to be 

82 15' 20", and the survey proceeds in a general southwesterly direction to station 

21, at which point the azimuth of 21 to 20 is found by observation to be 269 10' 

,00". The calculation of the survey shows that 21 is 3100 feet south and 15,690 

^feet west of i. From the table the convergence (by parts) for 15,690 feet is 2' 

^09". 7. Therefore if the direction of 21 20 is to be referred to the meridian at i 

this correction should be added to the observed azimuth, giving 269 12' 09". 7. 

The difference between the observed azimuth at i and the corrected azimuth at 21 

(-180) is 6 56' 49". 7, the total deflection, or change in azimuth, that should be 

shown by the measured angles if there were no error in the field work. 

Problems 

1. Compute the approximate Eastern Standard Time of the eastern elongation 
of Polaris on Sept. 10. The right ascension of the star is i^ 35 32*4. For the 
approximate right ascension of the mean sun at any date and the hour angle of 
Polaris at elongation see Arts. 76 and 97. 

2. Compute the exact Eastern Standard Time of the eastern elongation of 
Polaris on March 7, 1925. The right ascension of the star is i^ 33"* 38^.00; the 
declination is +88 54' i9".i8; the latitude of the place is 42 2i'.5 N. The 
right ascension of the mean sun +12* on March 7, at & Gr. Civ. T. is io& 56 
46*47. 

3. Compute the azimuth of Polaris at elongation from the data of Prob- 
lem 2. 

4. Compute the local time of eastern elongation of a 2 Centauri on April i, 1925, 
in latitude 20 South. Compute the altitude and azimuth of the star at elongation. 
The right ascension of the star is 14* 34** 32^.65; its declination is 60 31' 26". 12. 
The right ascension of the mean sun + 12^ at o& G. C. T. is 12^ 35"* 20*. 27. 

5. Compute the azimuth of the mark from the following observations on the 
sun, May 25, 1925. 

Ver. A. Alt. Watch (E. S. T.) 

Mark o 

O 7i 01 ' 40 46' 3* i3n 33* P.M. 

7i 16 40 33 3 i4 So 

7i 28 40 22 3 15 50 

(telescope reversed) 

72 21' 40 42' 3*i65o* 

~ 72 32 40 32 3 i7 48 

U , , 72 41 40 23 3 18 36 

Mark o 



208 PRACTICAL ASTRONOMY 

Lat. = 42 29'.s N.; long. - 71 o;'.s W. I. C. = o". Declination at G. C. T. 
eft = +20 48' 55".8; varia. per hour = -f-27".6o. Equa. of time = +3 m 19*48; 
varia. per hour = 0^.230. 

6. Compute the azimuth of the mark from the following observations on the 
sun, May 25, 1925. 

Ver. A. Alt. Watch (E. S. T.) 
Mark o 

O 76 25' 35 28' 3*4237*p.M. 

76 39 35 13 3 43 57 

76 49 35 02 3 44 55 

(telescope reversed) 



O 



77 37 35 25' 3* 45 m 5& 

77 49 35 12 3 46 55 

78 oo 35 oo 3 47 58 



Lat. = 42 29'.s N.; long. = 71 07'.$ W. I. C. = o'. Declination at G. C. T. 
o = -j-2o48' 55". 8; varia. per hour = -f-27".6o. Equa. of time = +3 19*48; 
varia. per hour = 03.230. 

7. On March 2, 1925, in latitude 42 01' N., longitude 71 07' W., the hori- 
zontal angle is turned clockwise from a mark to the sun with the following results: 
Left and lower limbs; hor. circle, 53 56'; altitude, 40 31'; watch, n^ 58"* 50*. 
Right and upper limbs; hor. circle, 55 09'; altitude, 41 02'; watch i2 h 00** 20 s . 
Watch is 3* fast of E. S. T. The declination of the sun at o& G. C. T. = -7 28' 
4o".8; varia. per hour, +57^.06. Equa. of time, 12"* 28*45; varia. per hour 
4-0*496. Compute the azimuth of the mark. 

8. On March 2, 1925 vernier A is set at o and telescope pointed at a mark. 
Vernier A is then set to read (clockwise) 50 01'; west edge of sun passed at n* 
44 43* and east edge at 11^46 53*, by watch. Vernier is next set at 53 18'; 
west edge of sun passed at n h S4 m 44* and east edge at n h 56 55*. Watch is 3* 
fast of Eastern Standard Time. The latitude is 42 01' N., longitude is 71 07' 
west. The equation of time at o 7 * G. C. T, March 2, 1925 is i2 m 28*45; varia. 
per hour, +0*496. Compute the azimuth of the mark. 

9. The transit is at sta. B; vernier reads o when sighting on sta. A. At 
8& oo m P.M. (E. S. T.) Polaris is sighted; alt. = 41 25'. Horizontal angle 113 30' * 
(clockwise) to star. Date, May 8, 1926. Compute the bearing of B. A. 

10. Prove tnat the horizontal angle between the centre of the sun and the right 
or left limb is s sec k where s is the apparent angular semidiameter and h is the 
apparent altitude. 

11. Prove that the level correction (Art. 106, p. 184) is i tan h where i is the 
inclination as given by the level. 

12. Why could not Equa. [106], p. 183, be used in place of Equa. [31], p. 36, 
in the method of Art. 112? 

13. If there is an error of 4* in the assumed value of the watch correction and 
an azimuth is determined by the method of Art. 112, (near noon) what would be the 



OBSERVATIONS FOR AZIMUTH 



relative effect of this error when the sun is on the equator and when it is 23 south? 
Assume that the latitude is45N. (See Table B, p. 99.) 

14. At station A on Aug. 5, 1925, about 5* P.M. a sun observation is made to 
^obtain the bearing of AB. The corrected altitude is 22 29', the latitude is 

42 29' N., the corrected declination is 16 56' N., and the hori- 
zontal angle from J5, clockwise, to the sun is 102 42'. 

After running southward to station E an observation is made 
on Polaris, the watch time being 8>* 50"* P.M. (E. S. T.). The 
altitude is 42 05' and the horizontal angle from sta. D toward 
,the left to Polaris is 2 09'. The longitude is approximately 

Kso'w. 

Find the error in the angles of the survey. 

15. May 8, 1925, in lat. 42 22' N., long. 71 06' W, transit at 
sta. i, o on sta. 2. Horizontal angle clockwise to sun, L and 
L limbs, 183 52', alt. 45 21', watch 3* 35 oo 5 ; horizontal angle 
to sun, U and R limbs, 183 25', alt. 44 6 37', watch 3^ 36^ 15*. 
Index correction 0^.5. Corrected declination, +16 59^.0 (N). 

May 8, 1925, transit at sta. 2, o on sta. i. Horizontal angle 
(clockwise) to Polaris 113 30'; alt. 41 25'; watch 8> oo E. S. T. 
Compute the bearing of i 2 from each observation. 

16. With the transit at station 21, on June 7, 1924, in latitude 

42 29^.5 N., longitude, 71 07'. 5 W., the following sights are taken on the sun, the 
reference mark being station 22; 




Hor. Circle. 


Mark 


o 


Sun, L and L 


155 43' 


Sun, L and L 


155 55 


Sun, U and R 


156 50 


Sun, U and R 


157 00 



Vert. Arc. 

42 03' 

41 52 

42 ii 
42 02 



Watch (E. S. T.) 

3^ I4 m 20 s P.M. 

3 15 27 
3 16 30 
3 17 22 



Index correction -f-i'. Sun's declination (corrected) = +2247 / > 3. 

The deflection angle at sta. 22 is 5 26' Rj at 23 it is 7 36' L; at 24 it is 
t 2n'R. 

At sta. 24 an observation is taken on Polaris', o on sta. 25; first angle, at 7^ 02^ 
05* E. S. T., 252 44' (clockwise); second repetition, at 7^04^ 25*, 145 29'; third 
repetition, at 7" o6* 40*, 38 14'. The altitude is 41 27'. Station 24 is 2800 
feet east of station 21. 

Compute the error in the traverse angles between stations 21 and 25, assuming 
that there is no error in the observations on the sun and Polaris. 

17. Differentiate the formula 

sin Z = sin p sec </> 

it7 f!7 

to obtain -r- and 3- and from these compute the error hi Z produced by an error 
a<f> dp , 

of i' in or an error of i" in 5. 



210 PRACTICAL ASTRONOMY 

18. Observation on sun Oct. 21, 1925, for azimuth. 

Watch (E. S. T.) 

7* I0* 30* A.M. 
7 II 22' 

7 12 30 
7 13 32 

Index correction to altitude = + *' 

Decimation at o*, G. C. T., Oct. 21 = 10 26' 02^.5 ; varia. per hour = 
53"-78. Latitude = 42 29'.$; longitude = 71 07^.5 W. Equa. of time a 
o -fi5* ii*.2i; varia. per hour, +.417. Compute the azimuth of the marl 





Hor. Angle. 


Altitude 


Mark 







Sun, L and R 


iS 55' 


10 26' 


Sun, L and R 


16 06 


10 35 


Sun, U and L 


IS 46 


ii 18 


Sun, U and L 


IS 57 


ii 28 


Mark 








CHAPTER XIV 
NAUTICAL ASTRONOMY 

117. 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 

118. 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. 70. 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 



212 



PRACTICAL ASTRONOMY 



point of the arc. (Fig. 77.) 
following example. 



This method is illustrated by the 



Example. 

Observed altitude of sun's lower limb 69 21' 30", bearing north. Index cor- 
rection = i' 10"; height of eye = 18 feet; corrected sun's decimation = 
+9oo / 26" (N). The approximate latitude and longitude are n3o / S, 15 oo' 
W. The corrections for dip, refraction, parallax and semidiameter may be taken 
out separately; in practice the whole correction is taken from Bowditch, American 
Practical Navigation, Table 46. The latitude is computed by formula [i]. Ifi 
the sun is bearing N the zenith distance is marked S, and vice versa. The zenith 
distance and the declination are then added if both are N or both are S, but sub- 
tracted if one is N and one is S; the latitude will have the same name (N or S) as 
the greater of the two. 



Observed alt. 
Correction 



69 21' 30" 

+ 10 18 



Tab. 46 
I.C. 



True altitude 69 31' 48" 
Zenith distance 20 28 1 2 S 
Declination 9 oo 26 N 

Latitude 11 27' 46" S 



-t-ll' 28" 

+ 1 10 

+10' 18" 



o 



Sea 



FIG. 77 



Horizon 



119. Latitude by Ex-Meridian Altitude. 

If for any reason the altitude is not taken precisely at noon the latitude may be 
found from an altitude taken near noon provided the time is known. If the in- 
terval from noon is not over 25 minutes the correction may be taken from Tables 
26 and 27, Bowditch. For a longer interval of time formula [300] should be used. 
When using Table 26, look up the declination at the top of the page and the latitude 
at the side; the tabular number (a) is the variation of the altitude in one minute 
from meridian passage. To use Table 27, find this number (a) at the side and the 
number of minutes (/) before or after noon at the top; the tabular number is the 
required correction, at 2 . 

Example i. 

The observed altitude of the sun's lower limb Jan. i, 1925 is 26 10' 30" bearing 
south; chronometer time, 15^ 3o io; chronometer 15* fast. Height of eye 18 
feet; I. C. = o". The decimation is - 23 oo'.S; equa. of time is 339.3. Lat. 
by dead reckoning, 40 40' N; long, by dead reckoning, 50 02' 30". 



NAUTICAL ASTRONOMY 213 

Chron. 15* 3o* io Table 46, H-IO' 19" Obs. alt. = 26 10' 30" 

Corr. 15 I. C. oo Corr. = ~}-io ig 



G. C. T. 15* 29^ 55* Corr. -fio' 19" True Alt. = 26 20' 49" 

E( l ua - ~3 39 -3 aP = 54 

G. A. T. i$ 26"* i5.7 Table 26, h = 26 21' 43" 

Lon &- 3 20 10 Lat. 41 1 a j" 5 Zenith Dist. = 63 38 17 N 

L.A.T. 12^06^05^.7 Dec -- 2 3J ' Declination 23 oo 48 S 

Table 27 ^ Latitude = 40 37' 29" N 
54" 



i"5 
6w.! j 



Example 2. 

Observed altitude of sun's lower limb Jan. 20, 1910, = 20 05' (south); T. C. = 
o'; G. A. T. i* 3S 28*; lat. by D. R. = 49 20' N.; long, by D. R. 16 19' W.; 
height of eye, 16 feet; corrected decimation, 20 14' 27" S. Find the latitude. 
If this is solved by Equa. [300] the resulting latitude is 49 11' N. 

Determination of Longitude at Sea 

120. By the Greenwich Time and the Sun's Altitude. 

The longitude of the ship may be found by measuring the sun's 
altitude, calculating the local time, and comparing this with the 
Greenwich time as shown by the chronometer. The error of the 
chronometer on Greenwich Civil Time and its rate of gain or 
loss must be known. The error of the chronometer may be 
checked at sea by the radio time signals. In solving the triangle 
for the sun's hour angle the latitude of the ship and the declina- 
tion of the sun are required, as well as the observed altitude. 
The latitude used is that obtained from the last preceding ob- 
servation brought up to the time of the present observation by 
allowing for the run of the ship during the interval. This is the 
latitude " by dead reckoning." On account of the uncertainty 
of this (D. R.) latitude it is important to make the observation 
when the sun is near the prime vertical. The formula usually 
employed is a modified form of Equa. [17] (see also p. 259). 

The same method may be applied to a star or a planet. In 
this case the longitude is obtained from the sidereal time. As 
the observation is ordinarily computed the Gr. Civ. T. is con- 
verted into Gr. Sid. T. and the hour angle of the star at Green- 
wich then computed. The solution of the pole zenith star 



214 PRACTICAL ASTRONOMY 

triangle gives directly the hour angle of the star at the ship's 
meridian. The difference between the two hour angles is the 
longitude. 

Example. 

Observed altitude of sun's lower limb on Aug. 8, 1925 (P.M.), = 32o6 / 3o // ; 
chronometer 20* 37^*40*; chronometer correction, i^^o*; index correction, 
-fi' oo". Height of eye 12 feet. Lat. by D. R., 44 47' N. Sun's decimation at 
20 s , G. C. T,, +16 07^.9; H. D., o'.7. Equa. of time at 20*, 5 w 33*.i; 
H. D. -hos.,3. 

Chron. 20* 37^ 40* Decl. 20* -fi6o7 / .9 

C - C -i 30 -0.7 X 0.6 -.4 

G. C. T. 20* 36^ io Decl. + 16 o/.$ 

Eq. 5 32.9 

G. A. T. 20& 30"* 37*.! Eq. t. 20 h $m 33*.! 

+0.3 X 0.6 +.2 



Lat. 44 47' log sec 0.14888 Eq. t. -< 

Alt. 32 18 30" log esc 0.01743 

pol. dist. 73 5 2 3 log cos 9.39909 Obs. alt. 3 2 06' 30" 

2 )iso_58__ log sin 9.83520 a 46 +"ii 

half sum 75 29 log hav 1 9.40060 ^ ^ 32 l8 ' 3O " 

half sum alt. 43 10 30 t 4* oo* 48*. 7 (Bowditch, Table 45) 

L. A. T. 16 oo 48 .7 
G. A. T. 20 30 37 .1 

Long. = 4^ 29 48^.4 
= 67 27^.1 W. 

Determination of Azimuth at Sea 

121. Azimuth of the Sun at a Given Time. 

For determining the error of the compass and for other pur- 
poses it is frequently necessary to know the sun's azimuth at an 
observed instant of time. The azimuth may be computed by 
any formula giving the value of Z when /, <f> and d are known. 
In practice it is not usually necessary to calculate Z, but its 
value may be taken from tables. Publication No. 71 of the U. S. 
Hydrographic Office gives azimuths of the sun for every i 
of latitude and of declination and every io m of hour angle. Burd- 
wood's and Davis's tables may be used for the same purpose. 
For finding the azimuth of a star or any object whose declination 
is greater than 23 Publication No. 120 may be used. 



NAUTICAL ASTRONOMY 215 

Example. 

As an illustration of the method of using No. 71 suppose that we require the sun's 
azimuth in latitude 42 01' N, declination 22 47' S, and hour angle, or local ap- 
parent time, ,9* 25** 20 s A.M. Under lat. 42 N and declination 22 S, hour angle 
tf* 2o> we find the azimuth N 141 40' E. The corresponding azimuth for lat. 
4.3 is 141 50', that is 10' greater. The azimuth for lat. 42, decl. 23 and hour 
ingle op 2ois 142 n', or 31' greater. For lat. 42, decl. 22, and hour angle 9* 30"* 
the azimuth is 143 47', or 2 07' greater. The first azimuth, 141 40' must be 
increased by a proportional part of each one of these variations. The desired 
izimuth is therefore 

141 40' + ~ X 10' + i* X 31' + ^ X 127 = 143 I2 '. 
Go oo 10 

The azimuth is N 143 12' E or S 36 48' E. 

If at the time stated (gh 25 i8) the compass bearing of the sun were S 17 E, 
the total error of the compass would be 19 48', the north end of the compass being 
west of true north. If the " variation of the compass " per chart is 24 W, the 
deviation of the compass is 24 19 48' = 4 12' E. 



Determination of Position by Means of Stunner Lines 

122. Stunner'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. 



2l6 



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 ceutreT- 
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 Une. 




FIG. 78 

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. 78, 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 J?, the coordinates being 4 A and (say) 
20 02' N, and the radius of which is 30. If the ship's position 



NAUTICAL ASTRONOMY 217 

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 
:ircle. 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. 
[n practice the circles would seldom have as short radii as those 
in Fig. 78; 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. 120), 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 itseli 



2l8 



PRACTICAL ASTRONOMY 



in the direction of the ship's course and a distance equal to the 
ship's run. In Fig. 79, 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. 79 

observation is A f 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 5 is the position of the ship 
at the time of the second observation. Since the bearing of 
the Sumner line is always at right angles to the bearing of the 
sun, it is evident that the line might be plotted from one latitude 
and one longitude instead of two. If the assumed latitude and 
the calculated longitude are plotted and a line drawn through 
the point at right angles to the direction of the sun (as shown by 

* The nautical mile (6080.20 feet) is assumed to be equal to an arc of i' of a 
Great Circle on anypart of the earth's surface. 



NAUTICAL ASTRONOMY 2IQ 

the azimuth tables) the result is the Sumner line; the ship is 
somewhere on this line. The two-point method of laying down 
.the line really gives a point on the chord and the one-point 
r Jnethod gives a point on the tangent to the circle of position. 
The second method is the one usually employed for the plotting 
the lines of position. 

One of the great advantages of this method is that even if 
one observation can be taken it may be utilized to locate 
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. 

123. Position by Computation. 

The latitude and longitude of the point of intersection of the position lines may 
be calculated more precisely than they can be taken from the chart. When the first 
altitude is taken a latitude is assumed which is near to the true latitude (usually 
the D. R. lat.), and a longitude is calculated. The azimuth of the sun is taken out 
of the table for the same lat. and hour angle. From the run of the ship between 
the first and second observations the change in lat. and change in long, are cal- 
culated, usually by the traverse tables. (Tables i and 2, Bowditch). These 
differences are applied as corrections to the assumed lat. and calculated long. This 
places the ship on the corrected Sumner line (corresponding to A'B', Fig. 79). 
When the second observation is made this corrected latitude is used in computing 
the new longitude. The result of two such observations is shown in Fig. 80. 
jk Point A is the first position; A' is the position of A after correcting for the run 
^'the ship; B is the position obtained from the second observation using the lati- 
tude of A '. The distance A'B is therefore the discrepancy in the longitudes, 
\ owing to the fact that a wrong latitude has been chosen, and is the base of a triangle 
the vertex of which is C, the true position of the ship. The base angles A' and B 
jLiQ the same as the azimuths of the sun at the times of the two observations. If 
we drop a perpendicular from C to A'B, forming two r|ght triangles, then 

Bd = OJcotZ 2 
A'd^CdcotZ,. 
>or 

A/>2 = A< cot Za 
Api A<cotZi 

where A0 is the error in latitude and Ap the difference in departure. In order to 



220 



PRACTICAL ASTRONOMY 



express Bd and A'd as differences in longitude (AX) it is necessary to introduce the 

factor sec <, giving, 

AX 2 = A< sec cot Z 2 1 , , 

AXi = A< sec <#> cot Z l \ U 7J 

These coefficients of A0 are called " longitude factors " and may fre taken' from 
Bowditch, Table 47. These formulae may also be obtained by differentiation. 

To find A<, the correction to the latitude, the distance A'B AXi + AX 2 is 
known, the factors sec <f> cot Z are calculated or taken from the table, and then A< 
is found by 

A< ** = sec <j> cot Zi -f- sec <f> cot Z 2 * \ 

Having found A</>, the corrections AXi, AX 2 , are cpmputed from [107]. 




FIG. 80 

If one of the observations is taken in the forenoon and one in the afternoon the 
denominator of [108] is the sum of the factors; if both are on the same side of the 
meridian the denominator is the difference between the factors. The difference 
between- the two azimuths should not be less than 30 for good results. When the 
angle is small the position will be more accurately found by computation than by 
plotting. If two objects can be observed at the same time and their bearings differ 
by 30 or more the position is found at once, because there is no run of the ship to 
be allowed for. This observation might be made on the sun and the moon, or on 
two bright stars or planets. It should be observed that the accuracy of the result- 
ing longitude depends entirely upon the accuracy of the chronometer, just as in 
the method of Art. 1 20. 



NAUTICAL ASTRONOMY 221 

Example. 

Location of Ship by Sumner's Method. 

On Jan. 4, 1910 at Greenwich Civil Time 13* 1 20*33* the observed altitude of the 
L is 15 53' 30"; index correction = o"; height of eye, 36 feet; lat. by D. R. 
[fc oo ; N. 

At Gr. Civ.*T. 18* 05** 31* the observed altitude of the sun is 17 33' 30"; index 
correction o"; height of eye, 36 feet. The run between the observations was N 89 
W, 45 miles. 

First Observation 

G. C. T. 13* i2 33* Observed alt. 15 53' 30" Declination 22 47' 04" 

112 47' 04" 

-4 W Si'-a 



iEqua. 4 51 Table 46 


+ 7 ii 


Polar dist. 


*$. A. T. 13* 07** 42* true alt. 


16 oo' 41" Equa. t. 


alt. 


16 


oo'. 7 






lat. 


42 


00 


sec 


0.12893 


p. d. 


112 


47.1 


CSC 


0.03528 




2)I70 


47'-8 






half sum 


85 


23'-9 


cos 


8.90433 


remainder 69 23 .2 


sin 


9-97I27 






log 


hav. / 


9.03981 


/ 2^ 34 W 40* 


Sun's Az. S 36 48' E 


L. A 


. T. = 


9 25 


20 


Az. factor 1.80 


G.A 


.T. = 


13 07 


42 



Long. 3* 42"* 22* 

- 55 35' 30" W. 

Lat. 42 oo' N Long. 55 35 '. S W. 
run o .8 N run i oo .7 W. 

Cor'd. Lat. 42 oo'.S N Cor'd. Long. s6 3 6'.2W. 

Second Observation 

G. C. T. 18* 05 31* Observed alt. 17 33' 30" Declination -22 45' 50" 
Equa. 4 56 .8 Table 46 +7 31 p. d. 112 45 50 

G. A. T. 18* oo 355.2 true alt. 17 41' 01" Equa. t. -4 56.8 

alt. i74i'.o 

lat. 42 oo .8 sec 0.12902 

p. d. 112 45 .8 esc 0.03522 

2)172 27 .6 

half sum 86 i3'.8 cos 8.81790 
remainder 6832'.8 sin 9.96882 

log hav. i . 8.95096 

Sun's Az. A 33 30' W t 2* 19** 07* 

Az. factor 2.03 L. A. T. 14 19 07 

G. A. T. 18 oo 35 

Long. 3* 41* 28* 



222 PRACTICAL ASTRONOMY 

Cor'dLong. 56 36'. 2 i9'.4 X 1.80 = 34'$,9, corr. to ist Long. 

2d Long. = 55 22 ig',4 X 2.03 * 39 . -^corr. to 2d Long. 

Diff. i i4'.2 = 74'.2 

ist Long. 56 36'.2 2d L O ng., 55 22' 

74'- 2 , A rr frtlaf Corr. 34-9 corr. * 39 -3 

; = 19 .4. COrr. to lat. -75 ; , rz : 

1.80+2.03 V ^' 56i.3 e 56 oi'.3 

. ' . Lat. = 42 2o'.2 N . ' . Long. = 56 01^,3 W. 

nc 

124. Method of Marcq St. Hilaire. 

Instead of solving the triangle for the angle at the pole, as explained in thc ^ pre- 
ceding article, we may assume a latitude and a longitude, near to the true position 
and calculate the altitude of the observed body. If the assumed position does note 
happen to lie on the Sumner line the computed altitude will not be the same as 
the observed altitude. The difference in minutes between the two altitudes is 
the distance in miles from the assumed position to the Sumner line. If the observed 
altitude is the greater then the assumed point should be moved toward the sun by 
the amount of the altitude difference. A line through this point perpendicular 
to the sun's direction is the true position line. It is now customary to work 
up all observation by this method except those taken when the sun is exactly on the 
meridian or close to the prime vertical. The former may be worked up for latitude 
as explained in Art. 118. The latter may be advantageously worked as a " time 
sight " or " chronometer sight " as in Art. 120. 

The formula for calculating the altitude is 

Hav. zen. dist. = hav. (Lat. ~ Decl.) -f cos Lat. cos Decl. hav. (hour angle) 

in which (Lat. Decl.) is the difference between Lat. and Decl. when they have 
the same sign, but their sum if they have different signs. The altitude is 90 
minus the zenith distance. To illustrate this method the first observation on 
p. 221 will be worked out. If we assume Lat. = 42oo'N, and Long. = 56 
30' W, the hour angle (t) is computed as follows: 



G. C. T. 

Long. 

L. C. T. 
Equa. t. 

L. A. T. 
Lat. 
Decl. 


13*12^33* 
3 46 oo 


log hav. 9.05922 
log cos 9.87107 
log cos 9.96471 


qh 2 fyn 335 
-4 51 .2 


9^ 2i m 41*. 8 
42 oo' 

-22 47.1 


Lat. Decl. 


64 47'- 1 


log 8.89500 
number .07852 
nat. hav. .28699 


Zen. dist. 


74 23.7 


nat. hav. .36551 


Calc. alt. 
Obs. alt. 

Alt. diff. 


15 36.3 

16 00.7 




24 .4 toward sun 


Sun's az. S 


36 48' E. 





NAUTICAL ASTRONOMY 223 

From the point in lat. 42 oo' N, long. 56 30' W, draw a line in direction S 36 48' E. 
On this line lay off 24.4 miles (i' of lat. i naut. mile) toward the sun. Through 
this last point draw a line in direction S 53 12' W. This is the required position 
line. 

125. Altitdde and Azimuth Tables Plotting Charts. 

To facilitate the graphical determination of position the Hydrographic Office 
publishes two sets of tables containing solutions of triangles, and a series of charts 
designed especially for rapid plotting of lines of position. 

The table designated as H. O. 201 gives simultaneous altitudes and azimuths of 
the sun (or any body whose declination is less than 24) for each whole degree of 
platitude and declination and each io w of hour angle. Since it is immaterial what 
fepoint is assumed for the purpose of calculation, provided it is not too far from the 
true position, interpolation for latitude and hour angle may be avoided by taking 
the nearest whole degree for the latitude, and a longitude which corresponds to an 
hour angle that is in the table, that is, some even io*. By interpolating for the 
minutes of the declination the altitude and azimuth are readily taken from the 
table. The difference between the altitude from the table (calculated h) and the 
observed altitude is the altitude difference to be laid off from the assumed position, 
toward the sun if the observed altitude is the greater. To work out the example 
of Art. 124 by this table we should enter with lat. = 42, hour angle 2^40"* and 
decl. 23. Interpolating for the 13' difference in declination, the corresponding 
altitude is 15 24'. 7 and the azimuth is N 142. i E. The longitude corresponding 
to an hour angle of 2 h 40 (g h 40 L. A. T.) is 3* 47 42* or 56 55^5 W. If we plot 
this point (42 N, 56 55'. 5 W) and then lay off 36'. o toward the sun (N 142.1 E) 
we should find a position on the same Sumner line as that obtained in Art. 124. 
Small variations in the azimuth will occur when the assumed position is changed. 
The different portions of the Sumner line will not coincide exactly in direction be- 
cause they are tangents to a circle. 

The table designated as H. O. 203 gives the hour angle and the azimuth for 
every whole degree of latitude, altitude, and declination. In this table the decli- 
nations extend to 27. When using this table we assume an altitude which is a 
whole degree but not far from the observed altitude. It is necessary to interpo- 
late, as before, for the minutes of declination; this is easily done by the use of the 
*utes of change per minute which are tabulated with the hour angle and the azi- 
muth. In working out the preceding example by the use of H. 0. 203 we might 
use lat. 42 N, alt. 16, decl. 22 47'.! S. The resulting hour angle is 2 h 34"* 50^.6 
(or L. A. T. 9^ 25^ 09^.4) and the azimuth is 143.!. The longitude corresponding 
to this hour angle is 55 38' W. Plotting this position (42N., 55 38' W) and 
laying off o'. 7 (the difference between the observed alt. i6oo'.7 and the tabular 
alt. 1 6) toward the sun we obtain another point on the same position line. 

The charts designed for plotting these lines show each whole degree of latitude 
and longitude.* The longitude degrees are 4 inches wide and the latitude degrees 

* No. 3000, sheet 7, extends from 35 N to 40 N; sheet 8 extends from 40 N 
t04sN; etc. 



224 PRACTICAL ASTRONOMY 

proportionally greater. On certain meridians and parallels are scales of minutes 
for each degree. The minutes on the latitude scale serve also*as a scale of nautical 
miles for laying off the altitude differences. A compass circle is provided for lay- 
ing off the azimuths. A pair of dividers, a parallel ruler and a pencil are all the 
instruments needed for plotting the lines. 



TABLES 



226 



PRACTICAL ASTRONOMY 



TABLE I. MEAN REFRACTION. 
Barometer, 29.5 inches. Thermometer, 50 F. 



App. Alt. 


Refr. 


App. Alt. 


Refr. 


App. Alt. 


Refr. 


App. Alt. 


Refr. 


ooo' 


33' 5i" 


10 00' 


5' 13" 


20 oo' 


2' 3' 


35 oo' 


i' 21" 


30 


28 ii 


30 


4 59 


30 


2 32 


36 oo 


I 18 


I 00 


23 Si 


II 00 


4 46 


21 OO 


2 28 


37 oo 


i 16 


30 


20 33 


30 


4 34 


30 


2 24 


38 oo 


I 13 


2 00 


i7 55 


12 OO 


4 22 


22 00 


2 20 


40 oo 


I 08 


30 


15 49 


30 


4 12 


30 


2 17 


42 oo 


i 03 


3 oo 


14 07 


13 oo 


4 02 


23 oo 


2 14 


44 oo 


o 59 


30 


12 42 


30 


3 54 


30 


211 


46 oo 


o 55 


4 oo 


II 31 


14 oo 


3 45 


24 oo 


2 08 


48 oo 


o 5i 


30 


10 32 


30 


3 37 


30 


2 5 ' 


50 oo 


o 48 


5 oo 


9 40 


15 oo 


3 30 


2 5 oo 


2 O2 


52 oo 


o 45 


30 


8 56 


30 


3 23 


26 oo 


i 57 


54 oo 


o 41 


6 oo 


8 IQ 


16 oo 


3 i7 


27 oo 


i 52 


56 oo 


o 38 


30 


7 45 


30 


3 10 


28 oo 


i 47 


58 oo 


o 36 


7 oo 


7 i5 


17 oo 


3 05 


29 oo 


i 43 


60 oo 


o 33 


30 


6 49 


30 


2 59 


30 oo 


1 39 


65 oo 


o 27 


8 oo 


6 26 


18 oo 


2 54 


31 oo 


i 35 


70 oo 


21 


30 


6 05 


30 


2 49 


32 oo 


i 31 


75 


o 15 


9 oo 


5 46 


19 oo 


2 44 


33 oo 


i 28 


80 oo 


O IO 


3 


5 29 


3 


2 40 


34 oo 


i 24 


85 oo 


o 05 


IO OO 


5 i3 


20 oo 


2 36 


35 oo 


I 21 


90 oo 


o oo 



TABLES 



227 



TABLE II. FOR CONVERTING SIDEREAL INTO MEAN SOLAR 

. TIME. 

* 

(Increase in Sun's Right Ascension for Sidereal h. m. s.) 
Mean Time == Sidereal Time C. 



Sid. 
Hrs. 


Corr. 


Sid. 
Min. 


Corr. 


Sid. 
Min. 


Corr. 


Sid. 
Sec. 


Corr. 


Sid. 
Sec. 


Corr. 


to "~~ 
I 


m s 

o 9.830 


I 


8 

0.164 


31 


8 

5-079 


I 


8 

0.003 


31 


8 

0.085 


2 


o 19.659 


2 


0.328 


32 


5.242 


2 


0.005 


32 


0.087 


3 


b 29.489 


3 


0.491 


33 


5.406 


3 


O.OO 


33 


0.090 


4 


o 39,318 


4 


0-655 


34 


5-570 


4 


o.on 


34 


0.093 


5 


o 49.148 


5 


0.819 


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 


1.147 


37 


6.062 


7 


0.019 


37 


O.IOI 


8 


I 18.636 


8 


1.311 


38 


6.225 


8 


0.022 


38 


0.104 


9 


I 28.466 


9 


1.474 


39 


6.389 


9 


0.025 


39 


0.106 


10 


I 38.296 


10 


1.638 


40 


6-553 


10 


0.027 


40 


o. 109 


ii 


I 48.125 


ii 


1.802 


4i 


6.717 


ii 


0.030 


41 


O.II2 


12 


i 57-955 


12 


1.966 


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 


15 


2-457 


45 


7-372 


15 


O.04I 


45 


0.123 


16 


2 37- 2 73 


16 


2.621 


46 


7-536 


16 


O.044 


46 


O.I26 


17 


2 47.102 


17 


2.785 


47 


7.700 


17 


0.046 


47 


O.I28 


18 


2 56.932 


18 


2.949 


48 


7.864 


18 


0.049 


48 


O.I3I 


iQ 


3 6.762 


19 


3-II3 


49 


8.027 


19 


O.052 


49 


0.134 


20 


3 16.591 


20 


3- 2 77 


50 


8.191 


20 


0-055 


50 


- I 37 


21 


3 26.421 


21 


3-440 


51 


8-355 


21 


0.057 


51 


0.139 


22 


3 36-250 


22 


3.604 


52 


8.519 


22 


O.O60 


52 


0.142 


23 


3 46.080 


23 


3.768 


53 


8.683 


23 


0.063 


53 


0.145 


24 


3 55-909 


24 


3-932 


54 


8.847 


24 


O.066 


54 


0.147 






25 


4.096 


55 


9.010 


25 


0.068 


55 


0.150 






26 


4-259 


56 


9.174 


26 


0.071 


56 


o^sa 






27 


4-423 


57 


9.338 


27 


0.074 


57 


0.156 






28 


4.5^7 


58 


9.502 


28 


0.076 


58 


0.158 







29 
30 


4.75i 
4.9i5 


I 9 
60 


9.666 
9.830 


29 
30 


0.079 
O.082 


S 


0.161 
0,164 



228 



PRACTICAL ASTRONOMY 



TABLE III. FOR CONVERTING MEAN SOLAR INTO SIDEREAL 

TIME. 

(Increase in Sun's Right Ascension for Solar h. m. s.) 
Sidereal Time = Mean Time + C. 



h 


Corr. 


j! 


Corr. 


ll 


Corr. 


| 


Corr. 





Corr. 


I 


m s 
o 9.856 


i 


8 


31 


8 

5-93 


i 


8 
0.003 


31 


8 

0.085 


2 


o 19.713 


2 


0.329 


32 


5- 2 57 


2 


0.005 


32 


0.088 


3 


o 29.569 


3 


0.493 


33 


5.421 


3 


O.OO8 


33 


0.09O 


4 


39.426 


4 


0.657 


34 


5.585 


4 


o.on 


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 


5-9I4 


6 


0.016 


36 


0.099 


7 


I 8.995 


7 


I.I50 


37 


6.078 


7 


0.019 


37 


O.IOI 


8 


I 18.852 


8 


I .314 


38 


6.242 


8 


0.022 


38 


0.104 


9 


I 28.708 


9 


1.478 


39 


6.407 


9 


0.025 


39 


0.107 , 


10 


I 38.565 


10 


1.643 


40 


6.57i 


10 


0.027 


40 


O.IIO 


ii 


I 48.421 


ii 


T.807 


41 


6-735 


ii 


0.030 


41 


O.II2 


12 


I 58.278 


12 


I.97I 


42 


6.900 


12 


0.033 


42 


O.II5 


13 


2 8.134 


13 


2.136 


43 


7.064 


13 


0.036 


43 


O.II8 


14 


2 17.991 


14 


2.300 


44 


7.228 


14 


0.038 


44 


O.I20 


15 


2 27.847 


15 


2.464 


45 


7-392 


15 


0.041 


45 


0.123 


16 


2 37.704 


16 


2.628 


46 


7-557 


16 


0.044 


46 


0.126 


*7 


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.I2I 


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 


3.450 


51 


8.378 


21 


0.057 


51 


0.140 


22 


3 36.842 


22 


3.6l4 


52 


8.542 


22 


0.060 


52 


0.142 


23 


3 46.699 


23 


3.778 


53 


8.707 


23 


0.063 


53 


0.145 * 


24 


3 56.555 


24 


3-943 


54 


8:871 


24 


0.066 


54 


O.I4 






25 


4.107 


55 


9.035 


25 


0.068 


55 


O.I5I 






26 


4-271 


56 


9.199 


26 


0.071 


56 


- 1 53 






27 


4-435 


57 


9.364 


27 


0.074 


57 


0.156 






28 


4.600 


58 


9.528 


28 


0.077 


58 


0.160 






29 


4.764 


59 


9.692 


29 


0.079 


59 


0.162 






30 


4*928 


60 


9.856 




0.082 


60 


0.164 



TABLES 

TABLE IV. 

PARALLAX SEMIDIAMETER DIP. 



229 



(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. 





9" 


I 


o' 59" 


10 


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 S^ 


90 


o 


10 


3 06 






ii 


3 15 






12 


2 24 


(B) Sun's semidiameter. 


13 


J *T- 

3 3 2 




14 


3 40 


Date. 


Semidiameter. 


15 

16 


3 4 
3 55 






X 7 


4 02 


Jan. i 


1 6' 1 8" 


18 


4 09 


Feb. i 


16 16 


19 


4 16 


Mar. i 


16 10 


20 


4 23 


Apr. i 
May i 


16 02 
T5 54 


21 
22 


4 29 
4 36 


June i 


15 48 


2 3 


4 42 


July i 


15 46 


24 


4 48 


Aug. r 


15 47 ' 


2 5 


4 54 


Sept. i 


i5 53 


26 


5 oo 


Oct. i 


16 oi 


27 


5 06 


Nov. i 


1 6 09 


28 


5 ii 


Dec. i 


16 15 


29 


5 i7 






30 


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 






100 


9 48 



230 



PRACTICAL ASTRONOMY 



TABLE V 



LOCAL CIVIL TIME OF THE CULMINATIONS AND ELONGATIONS OF POLARIS IN 

THE YEAR 1922 

(Latitude, 40 N.; longitude, 90 or 6* west of Greenwich) 



Civil date 


1922 


East 
elongation 


Upper 
culmination 


West 
elongation 


Lower 

culmination 


January i . . . . 




h m 
12 54.7 

ii 59-3 
10 52.2 

9 56.9 
9 01.7 
8 06.5 
6 59-6 

6 04.5 

5 01.6 
4 06.8 
3 oo . i 

2 05.3 

I 02 .7 
o 07.9 


h m 
18 50.0 

17 54-7 
16 47.5 
15 52.2 
14 57-o 
14 01.8 

12 54.9 

II 59.8 

10 56.9 

IO O2. I 

8 55-4 
8 00.6 
6 58.0 
6 03.2 

4 56.7 
4 01.9 
2 55-3 

2 00.4 

o 57.6 

f 02.8 

(23 58.7 

22 51.9 
21 56.8 

20 53-8 

19 58.5 


h m 

o 49 2 


h m 
6 51-9 
5 56.6 
4 49-4 
3 54-2 
2 58.9 

2 03.7 

o 56.8 
f o 01.7 
(23 57-8 

22 55.0 
22 OO.I 
20 '53.S 
19 58.6 

18 56.0 
18 01.3 

16 54-7 
iS 59-9 
14 53-3 
13 58.5 
12 55-7 

12 OO . 7 

10 53-8 
9 58.7 
8 55-7 
8 00.5 


January 15 




23 5o.o 

22 42.8 

21 47-5 

20 52.3 

19 57-1 
18 50.2 

i7 5S-i 
16 62.2 
15 57-4 
*4 50.7 
13 55-9 
12 53-3 
ii 58.5 
10. 52.0 
9 57-2 
8 50.6 
7 55-7 
6 52-9 

5 57-9 

4 5i-i 
3 56.o 
2 53-o 

i 57 7 


February i . 




February 15 . . 




March i 




March 15 




April i 




April 15 




May i 




May 15 




June i 




Tune i <j 




Tulv i 




July 15 




August i 




22 57-5 

22 O2.7 
20 56.1 
20 01.2 

18 58.4 
18 03.4 

16 56.6 
16 01.5 

14 58.5 
14 03 . 2 


August 15 . . . . 




September i . . 




September 15 
October i . . . . 






October 15. . . 




November i . . 




November 15 




December i . . 




December 15 . 








A. To refer 
For 


the tabular values to years other than 1922. 
year 1923 add 1^.4 
f add 2 . 8 up to Mar. i . 
( subtract i . i on and after Mar. i 
1925 add o .2 
1926 add i .5 
1927 add 2 .7 
f add 4 . i up to Mar. i 
1 add o . 2 on and after Mar. i 
1929 add i .6 
1930 add 3 .1 
1931 add 4-5 
jadd 5 .9 up to Mar. i 
1 add a .0 on and after Mar. i 



TABLES 



231 



B. To refer to any calendar day other than the first and fifteenth of each month 
SUBTRACT the quantities below from the tabular quantity for the PRECEDING DATE. 



, Day of 
1 month. 


Minutes. 


No. of days 
elapsed. 


Day of 
month. 


Minutes. 


No. of days 
elapsed. 


2 or 16 


3-9 


i 


10 or 24 


35-3 


9 


3 17 


7-8 


2 


ii 25 


39-2 


10 


4 18 


n.8 


3 


12 26 


43-1 


IX 


5 19 

6 20 


15-7 
19.6 


4 
5 


13 27 
14 28 


47-0 
Sl.o 


12 

13 


7 21 


23-5 


6 


29 


54-9 


14 


8 22 


27-4 


7 


30 


58.8 


IS 


9 23 


31-4 


8 


31 


62.7 


16 



C. To refer the table to Standard Time: 

* (a) ADD to the tabular quantities four minutes for every degree of longi- 
cude the place is west of the Standard meridian and SUBTRACT when the 
place is east of the Standard meridian. 

(b) Times given in the table are A.M., if less than 12^; those greater than 
12* are P.M. 

D. To refer to any other than the tabular latitude between the limits of 10 and 
50 north: ADD to the time of west elongation o^.io for every degree south 
of 40 and SUBTRACT from the time of west elongation o w .i6 for every degree 
north of 40. Reverse these operations for correcting times' of east elon- 
gation. 

- E. To refer to any other than the tabular longitude: ADD o"*.i6 for each 15 
east of the ninetieth meridian and SUBTRACT o*.i6 for each 15 west of the 
ninetieth meridian. 

TABLE VI 
FOR REDUCING TO ELONGATION OBSERVATIONS MADE NEAR ELONGATION 



X. Azimuth 
Nat Elon 

Time* N, 


i o' 


1 10' 


1 20 


i3o' 


i 40' 


1 50' 


2 0' 


2 10' 


Azimuth / 
at Elon/^ 

/ Time* 






















m 




O 


O 


O.O 





0.0 


0.0 








m 




i 





O 





4 o i 


-f O.I 


4- o i 


4-o i 


4- o.i 


i 


2 


-f O.I 


4-02 


+ 0.2 


2 


0.2 


0.3 


0.3 


o 3 


2 


3 


0.3 


0.4 


o 4 


o 5 


o.S 


0.6 


0.6 


o 7 


3 


4 


0.5 


0.6 


o 7 


o 8 


0.9 


I.O 


i.i 


I 2 


4 


s 


4- 0.9 


4- I 


4 i i 


4- i .1 


-f I 4 


4- i 6 


+ i 7 


4- i 9 


5 


6 


1.2 


I 4 


i 6 


I 8 


2 I 


2-3 


2.5 


2 7 


6 


7 


1.7 


2 


2 2 


2 5 


2.8 


31 


3-4 


3 7 


7 


8 


2 2 


2 6 


2 9 


3 3 


3 7 


4.0 


4.4 


48 


8 


9 


2.8 


3 2 


3-7 


4 2 


46 


S.i 


5-6 


6 o 


9 


10 


4- 3-4 


-f 4-0 


4- 4-6 


+ 5-1 


4- 5-7 


-f 6.3 


+ 6.9 


4- 7-4 


10 


ii 


4.1 


4-8 


5 5 


6.2 


6-9 


7.6 


8-3 


9 o 


ii 


12 


4-9 


5 8 


6 6 


7-4 


8.2 


9-0 


9 9 


10.7 


12 


13 


5 8 


6 8 


7 7 


8-7 


9-7 


10 6 


II. 6 


12.6 


13 


14 


6.7 


7 8 


9-o 


10. 1 


II. 2 


12.3 


13 4 


14.6 


14 


" 3 


4- 7-7 
8.8 


-f- 9 o 

10 2 


4io.3 
ii 7 


4-H.6 

13 2 


4-12.8 
14.6 


4-I4-I 
16.1 


+I5-4 
17-5 


+16.7 

19.0 


15 
16 


> 17 


9-9 


ii. 5 


13 2 


14-9 


16.5 


18.2 


19.8 


21.5 


17 


18 


ii. i 


12 9 


14.8 


16.7 


18.5 


20.4 


22.2 


24.1 


18 


19 


12 4 


14 4 


16.5 


18.6 


20.6 


22.7 


24-7 


26.8 


19 



Sidereal time from elongation. 



232 



PRACTICAL ASTRONOMY 



TABLE VII 

CONVERGENCE IN SECONDS FOR EACH 1000 FEET ON THi 

PARALLEL 



Lat. 


Distance (East or West) 





1000 


2000 


3000 


4000 


5000 


6000 


7000 


8000 


gooo 


o 
20 


u 
3-51 


// 
7,01 


// 
10.51 


a 
14.02 


H 
17.52 


H 
21.03 


// 
24-53 


// 
28.04 


// 
31.54 


21 


3-78 


7-57 


n-35 


15-13 


18.91 


22.69 


26.48 


30.26 


34.04 


22 


3.98 


7.96 


11.94 


15-92 


19-90 


23.88 


27.86 


31-84 


35.83 


23 


4.18 


8.36 


12-55 


16.73 


20.91 


25.09 


29-27 


3346 


37.64 


24 


4-39 


8-77 


13.16 


17-54 


21-93 


26.32 


30.70 


35.09 


3947 


25 


4-59 


9.19 


13.78 


18.37 


22.97 


27.56 


32 15 


36.75 


41-34 


26 


4.80 


9-6i 


14-42 


19.22 


24.02 


28.83 


33-63 


38.44 


43.24 


27 


5-02 


10.04 


15.06 


20.08 


25.10 


30.11 


35-13 


40.15 


45.17 


28 


5-24 


10.48 


15-71 


20.95 


26.19 


31-42 


36.66 


41.90 


47.13 


29 


546 


10.92 


16.38 


21.84 


27.30 


32.76 


38.22 


43-68 


49.14 


30 


5.69 


n-37 


17.06 


22.74 


28.43 


34.12 


39-80 


45-49 


5LI7 


31 


5-92 


11-83 


17.75 


23.67 


29-59 


35.5i 


41.42 


47-34 


53-26 l 


32 


6.16 


12.31 


18.46 


24.62 


30.77 


36.92 


43-08 


49-23 


55.38 


33 


6-39 


12.78 


19.17 


25.57 


31.96 


38-36 


44-75 


5i.i5 


57.54 


34 


6.64 


13.29 


19.92 


26.57 


33-21 


39.85 


46.49 


53.13 


59-77 


35 


6.89 


13-79 


20.68 


27-58 


34-47 


41.37 


48.26 


55.15 


62.05 


36 


7-15 


14-31 


21.46 


28.61 


35-77 


42.92 


50.07 


57.22 


64.38 


37 


7.42 


14.84 


22.26 


29.67 


37-09 


44.51 


5L93 


59-35 


66.77 


38 


7.69 


15-38 


23-08 


30.77 


38.46 


46.15 


53.84 


6i.53 


69.22 


39 


7.97 


15-95 


23.92 


31-89 


39-86 


47.83 


55.8o 


63.77 


71.74 


40 


8.26 


16.52 


24.78 


33.04 


41.30 


49.56 


57-82 


66.08 


74.34 


41 


8-55 


17.11 


25.67 


34-22 


42.78 


51.33 


59.89 


68.45 


77-00 


42 


8.86 


17.72 


26.58 


35-45 


44.31 


53-17 


62.03 


70.89 


79.76 


43' 


9.18 


18.36 


27-53 


36.71 


45-89 


55.o6 


64.24 


73.42 


82.60 


44 


9-50 


19.01 


28.51 


38.01 


47.52 


57*02 


66.52 


76.02 


85.53 


45 


9.84 


19.68 


29.52 


39.36 


49-20 


59-04 


-68.88 


78.72 


88.56 


46 


10.19 


20.38 


30-57 


40.76 


50.95 


61.13 


71.32 


81.51 


9i.7o. 


47 


iQ-55 


21.10 


31-65 


42.20 


52.76 


63-31 


73-86 


84.41 


94.96 


48 


10.93 


21.85 


32.78 


43-71 


54.63 


65-56 


76.49 


87-41 


98.34 


49 


11.32 


22.63 


33-95 


45.27 


56.59 


67.90 


79.22 


90.54 


101.85 


5<> 


11.72 


23-45 


35-17 


46.89 


58.62 


70.34 


82.06 


93.78 


105.51 



TABLES 



233 



TABLE VIII 

CORRECTION FOR PARALLAX AND REFRACTION TO BE SUBTRACTED FROM 
OBSERVED ALTITUDE OF THE SUN 



Apparent 
Taltittide 


Temperature, centigrade 




Apparent 
altitude 


-10 





+10 


+20 


+30 


+40 


+50 







3636 


35 IS 


34 oo 


32 50 


31 45 


30 45 


29 50 


o 
O 


I 


26 10 


25 12 


24 18 


23 28 


22 42 


22 00 


21 2O 


I 


2 


19 35 


18 51 


18 ii 


17 34 


16 59 


16 26 


15 57 


2 


3 


IS 20 


14 46 


14 15 


13 46 


13 18 


12 53 


12 30 


3 


4 


12 31 


12 03 


ii 37 


II 13 


10 50 


10 30 


10 II 


4 


5 


10 29 


10 05 


9 44 


9 24 


9 05 


8 48 


832 


5 


6 


8 59 


8 38 


8 20 


8 03 


7 47 


7 32 


7 18 


6 


7 


7 49 


7 31 


7 15 


7 oo 


6 46 


633 


6 21 


7 


8 


655 


639 


6 25 


6 12 


5 59 


5 48 


5 37 


8 


9 


6 ii 


5 57 


5 44 


5 32 


5 21 


5 ii 


5 oi 


9 


10 


5 34 


5 22 


5 10 


4 59 


4 49 


4 39 


4 30 


10 


ii 


5 04 


4 52 


4 42 


4 32 


4 23 


4 14 


406 


ii 


12 


4 39 


4 29 


4 19 


4 10 


4 01 


3 53 


346 


12 


13 


4 17 


4 07 


358 


3 50 


3 42 


3 35 


3 28 


13 


14 


3 58 


3 49 


3 41 


3 33 


326 


3 19 


3 13 


14 


15 


3 42 


3 34 


326 


3 19 


3 12 


306 


3 oo 


IS 


16 


3 27 


3 19 


3 12 


3 05 


2 59 


2 53 


2 47 


16 


17 


3 14 


3 07 


3 oo 


2 54 


2 48 


2 42 


2 37 


17 


18 


3 02 


2 55 


2 49 


2 43 


2 37 


2 32 


2 27 


18 


19 


2 52 


2 45 


2 39 


2 33 


2 28 


2 23 


2 19 


19 


20 


2 42 


236 


2 30 


2 25 


2 20 


2 15 


2 II 


20 


21 


2 33 


2 27 


2 22 


2 I? 


2 12 


2 08 


2 04 


21 


22 


2 26 


2 20 


2 15 


2 IO 


2 06 


2 02 


I 58 


22 


23 


2 18 


2 13 


2 08 


2 03 


I 59 


I 55 


i Si 


23 


24 


2 12 


2 07 


2 02 


i 58 


i 54 


I 50 


i 46 


24 


25 


2 OS 


2 OO 


I 56 


I 52 


i 48 


I 44 


I 41 


25 


26 


2 00 


i 55 


i Si 


i 47 


I 43 


I 39 


I 36 


26 


27 


I 55 


I 50 


1 46 


i 42 


i 38 


I 35 


I 32 


2? 


28 


I 49 


I 45 


I 41 


I 37 


I 34 


I 31 


I 28 


28 


29 


i 45 


I 41 


i 37 


I 33 


I 30 


i 27 


I 24 


29 


30 


I 41 


I 37 


I 33 


I 30 


r 26 


I 23 


I 21 


30 


32 


I 33 


I 29 


i 26 


I 23 


I 19 


I I? 


i 15 


32 


34 


I 26 


I 22 


I 19 


I 16 


I 13 


I II 


I 09 


34 


36 


I 19 


I 16 


I 13 


I 10 


i 08 


i 05 


I 03 


36 


38 


I 13 


I 10 


I 07 


I 04 


I 02 


I OO 


058 


38 


40 


I 08 


I 05 


I 02 


I OO 


5 8 


o 56 


o 54 


40 


42 


i 03 


I 00 


058 


o 56 


o 54 


o 52 


o 50 


42 


44 


o 59 


o 56 


o 54 


o 52 


o 50 


o 48 


o 47 


44 


46 


o 54 


o 52 


o 50 


o 48 


046 


o 45 


o 43 


46 


48 


o Si 


o 49 


o 47 


o 45 


o 44 


o 42 


41 


48 


So 


o 47 


o 45 


o 43 


o 41 


o 40 


038 


o 37 


SO 


55 


o 39 


o 37 


o 36 


o 35 


o 33 


o 32 


o 31 


55 


60 


o 32 


o 30 


o 29 


o 28 


o 27 


o 26 


- o 25 


60 


,65 


o 25 


o 24 


o 23 


O 22 


O 21 


20 


20 


65 


70 


o 20 


o 19 


o 18 


o 17 


o 17 


o 16 


o 15 


70 


75 


o 14 


o 14 


o 13 


O 12 


J2 


12 


II 


75 


80 


O IO 


o 09 


o 09 


o 09 


o 08 


o 08 


o 08 


So 


85 


o 04 


o 04 


o 04 


o 04 


o 04 


o 04 


o 03 


85 


90 


OO 


OO 


o oo 


O OO 


OO 


00 


00 


90 



234 



PRACTICAL ASTRONOMY 



TABLE IX 
LATITUDE FROM CIRCUM-MERIDIAN ALTITUDES OF THE SUN 

[A = cos 5 cos <f> cosec ] 



X 


6 


7 


8 


9 


10 


11 


12 


13 


14 


15 


16 


17 


18 


I 





95i 


8.14 


7.12 


6.31 


5-67 


5 15 


4.70 


4 33 


4.01 


3 74 


3 49 


3 27 


308 


o 


i 


9 53 


8,16 


7 13 


6.33 


569 


5-16 


4.72 


4-35 


4 03 


3-75 


3 50 


3-29 


3.09 


I 


2 


9-54 


8.17 


7-14 


634 


5 70 


5-17 


4.73 


436 


4 04 


3-76 


3 52 


330 


3 II 


2 


3 


9 54 


8 17 


7 15 


6.35 


5-71 


5 18 


4 74 


4 37 


4 05 


3-77 


3 53 


3 31 


3 12 


3 


4 


9-54 


8.17 


7 15 


6.35 


5 71 


5-19 


4-75 


4-38 


406 


3-78 


3-54 


3 32 


3-13 


4 


5 


9-53 


8.17 


7-15 


635 


5 7i 


5 19 


4 76 


4 39 


4.07 


3-79 


3-55 


3 33 


3 14 


5 


6 


9-Si 


8 16 


7 14 


635 


5 71 


5 19 


4 76 


4 39 


4 07 


3-80 


3 55 


3 34 


3 IS 


6 


7 


9 49 


8.14 


7.13 


634 


5 71 


5 19 


4 76 


4 39 


4 07 


38o 


3-56 


3 34 


3-15 


7 


8 


9-47 


8 12 


7 12 


633 


5-70 


5 18 


4 75 


4 39 


4 07 


38o 


3-56 


3 35 


3 16 


8 


9 


9 44 


8.10 


7.10 


631 


5 69 


5 17 


4 74 


4 38 


4 07 


380 


356 


3 35 


3 16 


9 


10 


9-41 


8.07 


7 07 


6 29 


5.67 


5 16 


4 73 


4 37 


406 


3 79 


3 55 


3 34 


3 16 


10 


ii 


9-36 


8 04 


7 04 


,6 27 


5 65 


5 14 


4-72 


4 36 


4 05 


378 


3 55 


3 34 


3 IS 


ii 


12 


9-31 


8 oo 


7 01 


6.24 


5 63 


5 13 


4 70 


4 35 


4 04 


3-77 


3 54 


3 33 


3 15 


12 


13 


9-25 


7 95 


6 97 


6 21 


56o 


5 10 


4 68 


4-33 


4 03 


3-76 


3 53 


3 32 


3 14 


13 


14 


9-19 


7-90 


6 93 


6.18 


5 57 


508 


4 66 


4 31 


4.01 


3-75 


3 52 


3-31 


3 13 


14 


IS 


9-13 


7-85 


6 89 


6.14 


5 54 


5 05 


4 64 


4 29 


399 


3-73 


3-SO 


3-30 


3 12 


IS 


16 


9.06 


7-79 


6 84 


6.10 


5 51 


5 02 


4 61 


4 2? 


3-97 


3 71 


3 49 


3-29 


3 10 


16 


i? 


8.98 


7 73 


6 79 


605 


5 47 


4 98 


4 58 


4 24 


3 95 


369 


3 47 


3 27 


3 09 


17 


18 


8 90 


766 


673 


6.00 


5 42 


4 95 


4 55 


4 21 


3 92 


36? 


3 45 


3-25 


308 


18 


19 


8.81 


7-59 


667 


5-95 


5 38 


4 91 


4 51 


4 18 


3-89 


3.64 


3-43 


3-23 


3-06 


19 


20 


8.72 


7 SI 


6.60 


5 90 


5 33 


4 86 


4 47 


4 IS 


386 


362 


3 40 


3 21 


3-04 


20 


21 


8.63 


7-43 


6.54 


5 84 


5 28 


482 


4 43 


4 ii 


3-83 


3 59 


3 37 


3 19 


3 02 


21 


22 


8.53 


7-35 


646 


5-78 


5 22 


4 77 


4-39 


4 07 


3.8o 


356 


3 34 


3-i6 


2 99 


22 


23 


8.42 


726 


6 39 


5 71 


5 16 


4 72 


4 35 


4 03 


376 


3 52 


3 31 


3-13 


2 97 


23 


24 


8-31 


7-17 


631 


5 64 


5 10 


4.66 


4 30 


3-99 


3 72 


3 49 


3-28 


3 10 


2.94 


24 


25 


8.20 


7 07 


623 


5 57 


5 04 


4 61 


4 25 


3 94 


3 68 


3 45 


3 25 


3 07 


2 91 


25 


26 


8.08 


697 


614 


5 49 


498 


4 55 


4 19 


389 


36 3 


3 41 


3 21 


3 04 


2.88 


26 


27 


7-96 


6.87 


6 05 


5.42 


4 91 


4 49 


4 14 


3 84 


3 59 


3 37 


3 17 


3 oo 


285 


27 


28 


783 


6.76 


596 


5-34 


4 84 


4 43 


4 08 


3 79 


3 54 


3 32 


3-13 


296 


2 81 


28 


29 


7 71 


6 65 


5 87 


5 25 


476 


4 36 


4 02 


3-74 


3 49 


328 


3-09 


2 93 


2.78 


29 


30 




6.54 


5 77 


5-17 


469 


4-29 


3-96 


3-68 


3 44 


3 23 


3 05 


2.89 


2.74 


3C 


31 






567 


5-08 


4.61 


4.22 


3 90 


362 


3 39 


3-18 


3.00 


2.85 


2.70 


31 


32 








4 99 


4- S3 


4 IS 


383 


3-56 


3 33 


3 13 


296 


2.80 


2.66 


32 


33 










4 45 


4 07 


3-77 


3-SO 


3-28 


308 


2.91 


2.76 


2.62 


33 


34 












4 oo 


3 70 


3 44 


3 22 


3 03 


2.86 


2 71 


2.58 


34 


35 














3.63 


338 


3 16 


2 97 


2.81 


2.66 


2.54 


35 


36 
















3 31 


3-10 


2 92 


276 


2 62 


2-49 


36 


37 


















3-04 


2.86 


2.70 


2.57 


2.44 


37 


38 




















2.8o 


2.65 


2.52 


2.40 


3 


39 






















2.60 


2.46 


2. 35 


39 


40 
























2.41 


2.30 


40 


41 


























2.25 


41 



TABLES 



23S 



TABLE IX (Continued) 
LATITUDE FROM CIRCUM-MERIDIAN ALTITUDES OF THE SUN 



\ r 

*~\ 

_\ 


19 


20 


21 


22 


23 


24 


25 


26 


27 


28 


29 


30 


31 


I 





2.90 


2.75 


2.6l 


2.48 


2.36 





















I 


2 92 


2. 7 6 


2 62 


2.49 


2-37 


2.26 
















i 


2 


2 94 


2. 7 8 


2.64 


2.51 


2 39 


2.28 


2.18 














2 


3 


2 95 


2-79 


2 65 


2.52 


2.40 


2 29 


2.19 


2 10 












3 


4 


2.96 


2.80 


2 66 


2 53 


2 41 


2.30 


2 20 


2. II 


2. 02 










4 


5 


2.97 


2.81 


2.67 


2 54 


2.42 


2 32 


2.21 


2.12 


2 03 


1-95 








5 


I 6 


2.98 


2.82 


2.68 


2 55 


2 43 


2-33 


2.22 


2 13 


2.04 


1.96 


i 89 






6 


7 


2.98 


2 82 


2.69 


2 56 


2 44 


2.33 


2.23 


2.14 


2 05 


1-97 


i 90 


1.83 




7 


8 


2 99 


2.83 


2.69 


2.56 


2 45 


2 34 


2.24 


2.15 


2 06 


198 


1.91 


1.84 


1.77 


8 


9 


2-99 


2.83 


2.70 


2 57 


2.45 


2 35 


2.25 


2.15 


2 06 


1-99 


i 92 


1.84 


1.78 


9 


10 


2-99 


28 4 


2 70 


2 57 


2 46 


2 35 


2.25 


2.16 


2.07 , 


2 00 


i 92 


I 85 


1-79 


10 


ii 


2-99 


283 


2 70 


2 57 


2.46 


2 35 


2.25 


2.16 


2 08 


2 00 


I 93 


I 86 


1-79 


ii 


12 


298 


2 8 3 


2 70 


2 57 


2.46 


2-35 


2.26 


2.17 


2 08 


2 00 


I 93 


1.86 


i. 80 


12 


13 


2 98 


2.82 


269 


2 57 


2 4 6 


2 35 


2.26 


2.17 


2.08 


2 00 


1-93 


i 86 


i. 80 


13 


14 


2-97 


2.81 


269 


2 56 


2.45 


2 35 


2.25 


2.17 


2.08 


2 01 


i 93 


I 87 


i. 80 


14 


15 


296 


2 81 


2 68 


2 56 


2-45 


2 35 


2.25 


2.16 


2 08 


2.OO 


i 93 


1.87 


1.80 


15 


16 


2 95 


2 80 


2 6 7 


2 55 


2-44 


2-34 


2.25 


2.16 


2 08 


2.00 


I 93 


1.87 


i. 80 


16 


27 


2.94 


2 79 


2 66 


2 54 


2-43 


2 33 


2.24 


2.15 


2 07 


2 OO 


I 93 


i 86 


i. 80 


17 


18 


2 92 


2 7 8 


2.65 


2 53 


2.42 


2-33 


2.23 


2 15 


2.06 


2 00 


1-93 


1.86 


i. 80 


18 


19 


2.90 


2.76 


2.64 


2 52 


2.41 


2.32 


2.22 


2.14 


2.06 


1-99 


1-92 


1.85 


i. 80 


19 


20 


2.89 


2 75 


2.62 


2 51 


2.40 


2 30 


2.21 


2.13 


2.05 


1.98 


I 92 


i 85 


1-79 


20 


21 


287 


2-73 


2 6l 


2 49 


2 39 


2 29 


2.20 


2.12 


2 04 


i 97 


I.QI 


I 84 


i 79 


21 


22 


2.84 


2.71 


2 59 


2 48 


2 37 


2 28 


2.19 


2. II 


2.03 


i 96 


1.90 


I 84 


1.78 


22 


23 


2 82 


2.69 


2 57 


2.46 


2.36 


2 26 


2.18 


2.10 


2 02 


i 95 


I 89 


I 83 


i 77 


23 


24 


2 80 


2 66 


2 55 


2 44 


2.34 


2 25 


2.16 


2 08 


2 01 


1.94 


1.88 


1.82 


1.76 


24 


25 


2 77 


2 64 


2.52 


2 42 


2.32 


2 23 


2.14 


2.07 


2.00 


1-93 


i 87 


i 81 


i 75 


25 


26 


2.74 


2 6l 


2.50 


2 39 


2 30 


2.21 


2.13 


2.05 


198 


1.91 


i 85 


I 79 


i 74 


26 


27 


2.71 


2 59 


2 47 


2.37 


2 27 


2 19 


2. II 


2.03 


196 


i 90 


i 84 


I 78 


i 73 


27 


28 


2.68 


2 5 6 


2.44 


2.34 


2 25 


2.17 


2.09 


2.01 


i 95 


1.88 


i 82 


I 77 


i 71 


28 


29 


2.65 


2 53 


2.42 


2 32 


2.23 


2 14 


2.06 


1-99 


1-93 


1.86 


I 80 


I 75 


i 70 


29 


30 


2.61 


2 49 


2-39 


2.29 


2 20 


2.12 


2.04 


1-97 


1.91 


1.84 


i 79 


I 73 


I 68 


30 


31 


2 58 


2.46 


2.36 


2.26 


2 I? 


2 09 


2.02 


i 95 


1.88 


I 82 


i 77 


1.71 


1.66 


31 


132 


2 54 


2-43 


2 32 


2 23 


2 14 


2.06 


1-99 


I 92 


1.86 


1. 80 


i 75 


169 


1.65 


32 


33 


2.50 


2.39 


2.29 


2.20 


2. II 


2.04 


1-97 


1.90 


1.84 


I 78 


I 73 


I 67 


163 


33 


34 


2.46 


2-35 


2.25 


2.17 


2 08 


2.01 


1-94 


1.87 


l.8l 


1.76 


1.70 


I 65 


1.61 


34 


35 


2.42 


2 31 


2 22 


2 13 


2 05 


1.98 


1.91 


1.85 


1.79 


I 73 


i 68 


1.63 


i 59 


35 


36 


2.38 


2.27 


2.18 


2 IO 


2 O2 


1-95 


1.88 


1.82 


1,76 


1.71 


I 66 


I 61 


i 56 


36 


37 


2.33 


2.23 


2 14 


2 06 


I 98 


1.91 


1.85 


1-79 


I 73 


1.68 


1.63 


1.58 


1-54 


37 


38 


2.29 


2.19 


2 10 


2 02 


1.95 


1.88 


1.82 


1.76 


I 70 


1.65 


i. 60 


1.56 


1.52 


38 


39 


2.24 


2.15 


2*06 


1.98 


I.9I 


i 85 


1.78 


1-73 


1.67 


1.63 


i 58 


1-54 


1-49 


39 


40 


2.20 


2.10 


2.02 


1-94 


1.88 


1.81 


1.75 


1.70 


1.64 


1. 60 


1-55 


LSI 


1-47 


40 


42 


2.10 


2.01 


1-94 


1.86 


1.80 


1.74 


1.68 


1.63 


1.58 


I 54 


1-49 


1-45 


1.42 


42 


'44 






1.85 


1.78 


1.72 


1.66 


l.6i 


1.56 


I. Si 


1-47 


1-43 


1.40 


1.36 


44 


46 










1.64 


1.58 


1-53 


1-49 


1-45 


1.41 


i 37 


1-34 


1.30 


46 


48 














1.46 


1.41 


1-38 


1-34 


I.3I 


1.27 


1.24 


48 


50 


















1.30 


1.27 


1.24 


1. 21 


1. 18 


SO 



236 



PRACTICAL ASTRONOMY 



TABLE CX (Continued) 
LATITUDE FROM CIRCUM-MERIDIAN ALTITUDES OF THE SUN 



v 






























,\ 


32 


33 


34 


35 


36 


37 


38 


39 


40 


41 


42 


,3 


44 




9 


1.72 


























9 


10 


1.72 


1.66 
























10 


ii 


1-73 


1.67 


1.62 






















II 


12 


1.73 


1.68 


1.62 


1-57 




















12 


13 


1.74 


1.68 


1.63 


158 


1-53 


















13 


14 


1-74 


1.68 


163 


1.58 


1-53 


1.48 
















14 


15 


1.74 


1.69 


1.63 


1.58 


i 53 


I 49 


1.44 














IS 


16 


1-74 


1.69 


1.63 


1.58 


I. S3 


i 49 


1. 45 


1.41 












16 


17 


1-74 


1.69 


1.64 


i. 59 


1.53 


i 49 


1.45 


1.41 


1.37 










17 


18 


1.74 


1.69 


1.63 


i 59 


I 53 


1.49 


I 45 


1. 41 


1.37 


1.33 








18 


19 


1.74 


1.68 


1.63 


1.58 


1-53 


1-49 


I 45 


I.4I 


1.37 


1.33 


1.30 






19 


20 


1.73 


1.68 


1.63 


1.58 


1-53 


1-49 


1-45 


I 41 


I 37 


1-33 


1-30 


1.2? 




20 


21 


1.73 


1.68 


163 


158 


i 53 


1.49 


1-45 


I 41 


I 37 


1-33 


1-30 


1.27 


1.24 


21 


22 


1.72 


167 


1.62 


1.58 


i 53 


1.49 


i 45 


1.41 


i 37 


i 33 


1-30 


I 27 


1.24 


22 


23 


1.72 


1.66 


1.62 


i 57 


1-53 


1.48 


1.44 


1.41 


1.37 


1-33 


1.30 


1.27 


1.24 


23 


24 


I.7I 


i 66 


I.6i 


1.57 


1-52 


1.48 


1.44 


1.41 


1-37 


1-33 


1.30 


1.27 


1,24 


24 


25 


1.70 


165 


I 60 


1.56 


I. SI 


1.47 


1-43 


I 40 


1.36 


1-33 


1.30 


1.26 


1.23 


2 f 


26 


1.69 


1.64 


1.59 


1.55 


1.51 


1.47 


I 43 


1-39 


1.36 


1.32 


I 29 


1.26 


1.23 


26 


27 


1.68 


1.63 


1.58 


1.54 


I 50 


1.46 


I 42 


1.38 


1.35 


1.32 


1.29 


1.26 


1.23 


27 


28 


1.66 


I 62 


1-57 


1-53 


1-49 


1-45 


I 41 


1.38 


1-34 


1.32 


I 28 


1.25 


1.22 


28 


29 


1.65 


1. 60 


1.56 


1.52 


I 48 


1-44 


I 40 


I 37 


1-34 


1.31 


1.27 


1.24 


1.22 


29 


30 


1.63 


59 


I 55 


i 50 


146 


I 43 


i 39 


i 36 


1-33 


I 30 


1.27 


1.24 


1. 21 


30 


31 


1.62 


57 


1-53 


1.49 


1-45 


1.42 


i 38 


i 35 


1.32 


1.29 


1.26 


1.23 


I 20 


31 


32 


1.60 


.56 


I 52 


1.48, 


I 44 


I 40 


1.37 


1.34 


1.31 


1.28 


1,25 


1.22 


I 19 


32 


33 


1.58 


54 


I 50 


I 46 


I 42 


1-39 


1-36 


1-33 


1.30 


1.27 


1.24 


1. 21 


I.I8 


33 


34 


1.56 


.52 


1.48 


1-45 


1.41 


1.38 


1-34 


1.31 


1.28 


1.25 


1.23 


1.20 


1.18 


34 


35 


1.54 


50 


1-47 


1-43 


1-39 


1-36 


1-33 


1.30 


1.27 


1.24 


1. 21 


I- 19, 


1.16 


35 


36 


1.52 


.48 


1-45 


1.41 


1.38 


1-34 


I 31 


I 28 


1.26 


1.23 


1.20 


I.I8 


1. 15 


36 


37 


i So 


.46 


1-43 


1.39 


1.36 


1.33 


1-30 


1.27 


1.24 


1. 21 


1. 19 


1. 17 


1 14 


37 


38 


1.48 


44 


I 41 


1.37 


1-34 


I.3I 


1.28 


1.25 


1.23 


1.20 


1. 17 


I. IS 


1. 13 


38 


39 


1.46 


42 


1.38 


1-35 


1-32 


1.29 


1.26 


1.24 


1. 21 


1.18 


1.16 


I 14 


I. II 


39 


40 


1.43 


.40 


136 


I 33 


1.30 


I 27 


1.24 


1.22 


T.I9 


I I? 


1. 14 


1. 12 


1. 10 


40 


42 


1.38 


-35 


1.32 


1.29 


1.26 


I 23 


1.20 


1.18 


1.16 


1. 13 


i. II 


1.09 


1.07 


42. 


44 


1.33 


30 


I 27 


1.24 


1. 21 


1. 19 


1.16 


1.14 


1. 12 


1.09 


1.07 


i. 05 


1.04 


44 


46 


1.27 


.24 


I 22 


1.19 


1.16 


1. 14 


1. 12 


1. 10 


1.07 


1.05 


1.04 


1.02 


1. 00 


46 


48 


1. 21 


19 


1.16 


1. 14 


i. ii 


1.09 


1.07 


1.05 


1.03 


1. 01 


.99 


.98 


.96 


48 


SO 


I. IS 


1,13 


1. 10 


1. 08 


i. 06 


1.04 


1.02 


1. 00 


,98 


97 


.95 


94 


.92 


50 


55 


1. 00 


98 


96 


.94 


.92 


91 


89 


.88 


.86 


.85 


.84 


.82 


.81 


55 


60 












.76 


75 


.74 


73 


72 


71 


.70 


.69 


60 


65 






















.SB 


57 


57 


65 



TABLES 



237 



^ TABLE IX (Continued) 
LATITUDE FROM CIRCUM-MERIDIAN ALTITUDES OF THE SUN 



y 

A 


. 4S 


46 


47 


48 


49 


50 


5i 


52 


53 


54 


55 


56 


57 


I 


22 


1. 21 


























22 


23 


1. 21 


1.18 
























23 


24 


1. 21 


1.18 


i. IS 






















24 


25 


1. 2O 


1.18 


i. IS 


1. 12 




















25 


26 


1.20 


1. 17 


I. IS 


I 12 


.10 


















26 


27 


1.20 


I 17 


1.14 


1. 12 


.10 


1.07 
















27 


.28 


1. 19 


i 17 


1. 14 


I 12 


09 


1.07 


05 














28 


**> 


1. 19 


I.l6 


1. 14 


I. II 


.09 


I 07 


04 


1.02 












29 


30 


I 18 


i 16 


i 13 


I II 


08 


1. 06 


04 


1.02 


1. 00 










30 


31 


I.I8 


i 15 


1. 13 


1. 10 


.08 


I. Of) 


.04 


1.02 


1. 00 


0.98 








31 


32 


1. 17 


i 14 


1. 12 


I IO 


07 


1.05 


.03 


I.OI 


99 


97 


o.95 






32 


33 


i 16 


1.14 


I. II 


I 09 


07 


i 05 


.03 


I.OI 


99 


97 


95 


0.93 




33 


34 


I. IS 


1. 13 


1. 10 


I 08 


.06 


1.04 


.02 


1. 00 


98 


96 


94 


.93 


0.91 


34 


35 


1.14 


I 12 


I IO 


I 07 


05 


1.03 


.01 


99 


.98 


.96 


94 


92 


91 


35 


36 


1. 13 


I II 


I. 09 


I 0? 


.05 


1.03 


01 


99 


97 


95 


93 


92 


.90 


36 


37 


I 12 


1. 10 


i 08 


1. 06 


.04 


1.02 


.00 


.98 


-96 


94 


93 


91 


.90 


37 


38 


I. II 


I 08 


i. 06 


1.04 


.02 


I.OI 


99 


97 


95 


94 


.92 


.90 


89 


38 


^ 


1.09 


I O? 


1.05 


I 03 


.01 


I 00 


-98 


.96 


94 


93 


.91 


.90 


.88 


39 


%o 


I 08 


I 06 


1.04 


I 02 


I 00 


.98 


97 


95 


93 


.92 


90 


.89 


.8? 


40 


42 


1. 05 


I 03 


1. 01 


99 


98 


96 


94 


93 


91 


.90 


.88 


87 


.86 


42 


44 


I 02 


I 00 


.98 


97 


95 


93 


92 


90 


.89 


.88 


.86 


.85 


.84 


44 


46 


98 


97 


95 


93 


92 


.90 


.89 


.88 


.86 


.85 


.84 


.82 


.81 


46 


48 


94 


93 


.92 


.90 


.89 


.87 


.86 


.85 


.83 


.82 


.81 


.80 


79 


48 


50 


91 


.89 


.88 


.86 


.85 


.84 


83 


.82 


.80 


79 


78 


.77 


.76 


50 


55 


.80 


-79 


.78 


77, 


.76 


75 


74 


73 


.72 


71 


.70 


.69 


.68 


55 


60 


.68 


.67 


.67 


.66 


65 


.64 


64 


.63 


.62 


.61 


.61 


.60 


.60 


60 


fis 


.56 


.56 


55 


54 


54 


S3 


53 


52 


52 


51 


51 


50 


50 


65 


70 






43 


43 


42 


.42 


.42 


41 


.41 


.41 


.40 


40 


.40 


70 


\ r 




























f/ 


\ 


58 


59 


60 


61 


62 


63 


65 


67 


69 


71 


73 


' 78 


83 


/ 


\ 




























L 


,35 


0.89 


























35 


36 


.88 


0.87 
























36 


37 


.88 


.86 


0.85 






















37 


38 


87 


.86 


.84 


0.83 




















38 


39 


.87 


.85 


.84 


.82 


0.81 


















39 


40 


.86 


.84 


.83 


.82 


.80 


0.79 
















40 


42 


.84 


.83 


.82 


.80 


79 


78 


o 75 














42 


44 


.82 


.81 


.80 


79 


-78 


-76 


74 


0.72 












44 


46 


.80 


.79 


.78 


77 


76 


.75 


72 


.70 


0.69 










46 


48 


.78 


77 


76 


.75 


.74 


73 


71 


.69 


.6? 


0.65 








48 


*;5o 


75 


74 


73 


72 


.71 


.70 


.69 


.67 


.65 


.63 


0.62 






50 


1)55 


.68 


.67 


.66 


.65 


.64 


.64 


.62 


.61 


.60 


58 


0.57 


0.54 




55 


^60 


59 


.58 


58 


.57 


57 


.56 


55 


.54 


53 


52 


51 


.49 


0.46 


60 


65 


49 


49 


.48 


.48 


.48 


.47 


47 


,46 


45 


.44 


43 


.42 


.40 


65 


70 


39 


39 


39 


39 


38 


.38 


.38 


37 


37 


36 


36 


35 


34 


70 



PRACTICAL ASTRONOMY 



TABLE X 



Values of m 



r 


O m 


,m 


2 m 


3 W 


4 m 


5 


6 w 


7 W 


8 W 


s 


* 





* 


* 


* 


* 


* 


* 


* 





o.oo 


i 96 


7-85 


17-67 


31 42 


49-09 


70 68 


96 20 


125.65 


i 


o oo 


2 03 


798 


17-87 


31 68 


49-41 


71.07 


96 66 


126.17 


2 


o oo 


2.10 


8 12 


18 07 


31 94 


49 74 


71.47 


97.12 


126.70 


3 


o.oo 


2 16 


8.25 


18 27 


32 20 


50 07 


71 86 


97 58 


127.22 


4 


O.OI 


2 23 


8-39 


18 47 


32 47 


50.40 


72.26 


98.04 


127 75 


s 


O.OI 


2 31 


8 52 


18 67 


32 74 


50.73 


72 66 


98 50 


128.28 


6 


O.O2 


2 38 


8.66 


18.87 


33-01 


51.07 


73 06 


98.97 


I28.8I 


7 


O.O2 


2-45 


8 80 


19 07 


33 27 


51-40 


73.46 


99 43 


129-34 


8 


o 03 


2 52 


8 94 


19 28 


33 54 


Si 74 


73-86 


99 90 


129.87 


9 


o 04 


2 60 


9.08 


19 48 


33 81 


52 07 


74 26 


loo 37 


130.40 


10 


o 05 


2 67 


9-22 


19 69 


34-09 


52 41 


74-66 


100.84 


130 94 


ii 


o 06 


2 75 


9 36 


19 90 


34 36 


52 75 


75 06 


ioi 31 


131 47 


12 


0.08 


2 83 


9 so 


20 II 


M 64 


53 09 


75 47 


101 78 


132.01 


13 


o 09 


2 91 


9 64 


20.32 


34 91 


53 43 


75 88 


102.25 


132 55 


14 


O.II 


2 99 


9-79 


20 53 


35 19 


53 7,7 


76 29 


IO2 72 


133 09 


IS 


12 


3 07 


9 94 


20 74 


35.46 


54-H 


7669 


103 20 , 


133 63 


16 


0.14 


3 15 


o 09 


20 95 


35 74 


54 46 


77 10 


103 67 


134 I? 


17 


o 16 


3 23 


o 24 


21 16 


36 02 


54 80 


77.51 


104.15 


134 71 


18 


o 18 


3 32 


o 39 


21 38 


36 30 


55-15 


77 93 


104 63- 


135 25 


19 


21 


3 40 


0.54 


21.60 


36 58 . 


55 5b 


78.34 


105 10 


135 80 


20 


O.22 


3 49 


0.69 


21.82 


368 7 


55 84 


78 75 


105 58 


136 34 


21 


o 24 


358 


, o 84 


22.03 


37 IS 


56.19 


79 16 


I06.O6 


136 88 


22 


o 26 


3 67 


I.OO 


22.2$ 


37 44 


56 55 


79 58 


106.55 


I37-4C ., 


23 


28 


3 76 


1. 15 


22 47 


37 72 


56 90 


80 oo 


107.03 


137- 9& 


24 


0.32 


3-85 


I 31 


22.70 


38.01 


57 25 


80.42 


107.51 


138 53 


25 


0.34 


3 94 


ii 47 


22 92 


38 30 


57 60 


80.84 


107.99 


139-08 


26 


o 37 


4 03 


ii 63 


23 14 


38 59 


57.96 


81 26 


108 48 


139 63 


27 


0.40 


4 12 


ii 79 


23 37 


38 88 


58.32 


81 68 


108.97 


140.18 


28 


o 43 


4-22 


11-95 


23 60 


39 17 


58.68 


82 10 


109.46 


140.74 


29 


o 46 


4-32 


12 II 


23.82 


39.46 


59 03 


82 52 


109 95 


141.29 


30 


0.49 


4 42 


12.27 


24 05 


39 76 


59 40 


82 95 


no 44 


141.85 


31 


o 52 


4 52 


12 43 


24 28 


40 05 


59 75 


83 38 


no 93 


142 40 


32 


o 56 


462 


12.60 


24 Si 


40.35 


60. ii 


83 81 


"I 43 


142.96 


33 


o 59 


4.72 


12.76 


24.74 


40.65 


60 47 


84 23 


ni-92 


143 52 


34 


0.63 


4 82 


12.93 


24 98 


40 95 


60 84 


84.66 


112.41 


144 08 


35 


o 67 


492 


13-10 


25.21 


41 25 


61 20 


85-09 


112.90 


144-64 


36 


0.71 


5 03 


13 27 


25 45 


41-55 , 


61 S7 


85 52 


113-40 


145-20 


37 


o 75 


5 13 


13 44 


25 68 


41 85 


61 94 


85 95 


113 90 


145.76 


38 


o 79 


S 24 


13.62 


25 92 


42.15 


62 31 


86.39 


114 40 


146.33 


39 


083 


5-34 


13 79 


26.16 


42.45 


62.68 


86.82 


114 90 


14689 


40 


, o 87 


5-45 


13 96 


26.40 


42.76 


63 05 


87 26 


US 40 


147 46 


41 


0.91 




14 13 


26 64 


43-06 


63 42 


87.70 


115.90 


148.03 


42 


0.96 


5 67 


14 31 


26 88 


43-37 


63.79 


88 14 


116 40 


148 60 


43 


1. 01 


578 


14 49 


27 12 


43-68 


64.16 


88.57 


116.90 


149.17 


44 


i. 06 


5 90 


14 67 


27 37 


43-99 


64.54 


89.01 


II7-4I 


149-74 


45 


I IO 


6 01 


14 85 


27.61 


44-30 


64 91 


89 45 


117 92 


ISO 31 


46 


I 15 


6 13 


15 03 


27.86 


44 61 


65.29 


89.89 


118.43 


150.88 


47 


I 20 


6 24 


IS 21 


28.10 


44 92 


65-67 


90 33 


118.94 


151-45 


48 


1.26 


636 


15-39 


28.35 


45-24 


66 05 


90.78 


"9 45 


152 03 


49 


I.3I 


6.48 


15 57 


28.60 


45-55 


66.43 


91.23 


119.96 


I52.6I 


50 


1.36 


6.60 


15.76 


28.85 


45.87 


66.81 


91.68 


120.47 


153.19 , 


SI 


1.42 


6.72 


15 95 


29 10 


46.18 


67.19 


92.12 


120 98 


153-77 ; 


52 


I 48 


6.84 


16.14 


29 36 


46.50 


67.58 


92-57 


121.49 


154-35 


53 


1-53 


6.96 


16.32 


29.61 


46.82 


67-96 


93.02 


122.01 


154-93 


54 


1-59 


7-09 


16.51 


29.86 


47-14 


68.35 


93-47 


122 53 


155.51 


55 


1.65 


7.21 


16.70 


30.12 


47-46 


68.73 


93-92 


123 05 


156.09 


56 


1.71 , 


7-34 


16,89 


30.38 


47-79 


69.12 


94 38 


123-57 


156-67 


57 


1.77 


7-46 


17.08 


30.64 


48.11 


69 51 


94.83 


124.09 


157 25 


58 


1.83 


7.60 


17.28 


30.90 


48.43 


69-90 


95-29 


124.61 


157.84 


59 


1.89 


7-72 


17-47 


31 16 


48.76 


70.29 


95 74 


125.13 


158.43 



TABLES 
TABLE X (Continued) 



239 



2 sin* J T 
sin j." 


T 


9 m 


lo m 


n w 


I2 m 


I3 w 


14"* 


I5 W 


i6 m 


S 


%> 


tt 





" 


* 


* 


* 


f - - 





159-02 


196.32 


237-54 


282 68 


331-74 


384 74 


441-63 


502.46 


I 


159.61 


196.97 


238.26 


283 47 


332.59 


385 65 


442 62 


503-50 


2 


160.20 


197-63 


238 98 


284 26 


333-44 


386.56 


443-60 


504 55 


3 


160 80 


198.28 


239-70 


285 04 


334-29 


387 48 


444 58 


505 60 


4 


161.39 


198.94 


240.42 


285.83 


335-15 


388.40 


445-56 


506.65 


5 


161.98 


199 60 


241 14 


286.62 


336 oo 


389 32 


446 55 


507-70 


6 


^62.58 


200 26 


241 87 


287.41 


336-86 


390 24 


447-54 


508.76 


7 


163.17 


2OO 92 


242 60 


288.20 


337-72 


391 16 


448.53 


509.81 


8 


163 77 


201 59 


243-33 


289.00 


338 58 


392 09 


449-51 


510.86 


9 


164.37 


202 25 


244 06 


289.79 


339-44 


393-01 


450.50 


511 92 


10 


164 97 


202.92 


244-79 


290.58 


340.30 


393 94 


451 . 50 


512.98 


ii 


165.57 


203 58 


245-52 


291-38 


34I.I6 


394 86 


452 49 


514 03 


12 


166.17 


204 25 


245 25 


292.18 


342 02 


395 79 


453 48 


515.09 


13 


166 77 


204 92 


246.98 


292.98 


342.88 


396.72 


454 48 


516.15 


14 


167.37 


205.59 


247 72 


293.78 


343 75 


397 65 


455 47 


517.21 


IS 


167 97 


206.26 


248.45 


294.58 


344 62 


398.58 


456-47 


518.27 


16 


168 58 


206.93 


249-19 


295 38 


345-49 


399 52 


457 47 


519.34 


17 


169.19 


207 . 60 


249-93 


296.18 


346 36 


400.45 


458.47 


520.40 


18 


169.80 


208 . 27 


250.67 


296 99 


347 23 


401 38 


459 47 


521.47 


19 


170 41 


208 94 


251 41 


297 79 


348.10 


402 32 


460.47 


522.53 


20 


171 . 02 


209.62 


252.15 


298 60 


348 97 


403 26 


461.47 


523-60 


21 


171 63 


210 30 


252.89 


299.40 


349-84 


404.20 


462.48 


524.67 


32 


172 24 


210 98 


253 63 


300 21 


350 71 


405 14 


463.48 


525-74 


33 


172 85 


2ii 66 


254 37 


3OI 02 


351 58 


406 08 


464.48 


526.81 


24 


173-47 


212 34 


255 12 


301.83 


352 46 


407.02 


465.49 


527-89 


25 


174-08 


213.02 


255.87 


302.64 


353 34 


407.96 


466.50 


528.96 


26 


174 70 


213.70 


256 62 


303.46 


354 22 


408 90 


467.51 


530.03 


2? 


175 32 


214 38 


257-37 


304 27 


355-10 


409 84 


468.52 


531 II 


28 


175 94 


215 07 


258 12 


305 09 


355.98 


410 79 


469.53 


532.18 


29 


176.56 


215 75 


258.87 


30S-90 


356.86 


4ii 73 


470.54 


533.26 


30 


177.18 


216 44 


259.62 


306.72 


357 74 


412.68 


471-55 


534-33 


31 


177 80 


217.12 


260.37 


307.54 


358.62 


413 63 


472.57 


535-41 


32 


178 43 


217 81 


26l 12 


308.36 


359 51 


414 59 


473.58 


536 . 50 


33 


179 05 


218.50 


261 88 


309 18 


360.39 


415-54 


474-60 


537.58 


34 


179-68 


219.19 


262.64 


310.00 


361 . 28 


416.49 


475-62 


538.67 


35 


180.30 
180 93 


219 88 

220 58 


263.39 
264 15 


310 82 
311.65 


362.17 
363.07 


417.44 
418.40 


476.64 
477.65 


539-75 
540.83 


37 


181.56 


221.27 


264-91 


312.47 


363.96 


419 35 


478 67 


541.91 


38 


182.19 


221.97 


265.68 


313 30 


364-85 


420.31 


479-70 


543-00 


39 


182.82 


222.66 


266.44 


314.12 


365.75 


421.27 


480.7^ 


544.09 


40 


183 46 


223.36 


267.20 


314.95 


366.64 


422.23 


481.74 


545.18- 


41 


184.09 


224.O6 


267 96 


315.78 


367.53 


423.19 


482.77 


546.27 


42 


184.72 


224.76 


268.73 


316.61 


368.42 


424.15 


483.79 


547.36 


W 


185-35 


225.46 


269.49 


317.44 


369-31 


425.11 


484.82 


548.45 


14 


185.99 


226.16 


270 26 


318.27 


370.21 


426.07 


485.85 


549-55 


IS 


186.63 


226.86 


271.02 


319.10 


37I.II 


427.04 


486.88 


550.64 


16 


187.27 


227-57 


271.79 


319.94 


372.01 


428.01 


487.91 


SSi-73 


W 


187.91 


228.27 


272 56 


320 78 


372 92 


428.97 


488.94 


552.83 





188.55 


228.98 


273-34 


321 62 


373-82 


429-93 


489.97 


553.93 


19 


189.19 


229-68 


274.11 


322.45 


374-72 


430.90 


491.01 


555.03 


>o 


189.83 


230.39 


274-88 


323.29 


375.62 


431.87 


492.05 


556.13 


51 


190.47 


231 10 


275.65 


324.13 


376.52 


432.84 


493-08 


557-24 


>2 


191.12 


231.81 


276.43 


324-97 


377-43 


433 82 


494-12 


558.34 


>3 

>4 


191 76 
192.41 


232.52 
233.24 


277.20 
277-98 


325.81 
326.66 


378-34 
379-26 


434-79 
435.76 


495-15 
496.19 


559-44 
560.55 


;s 


193-06 


233.95 


278.76 


327 50 


380.17 


436.73 


497.23 


561.65 


57 


193.71 
194.36 


234.67 
235.38 


279-55 
280.33 


328.35 
329.19 


381.08 
381.99 


437-71 
438.69 


498-28 
409-32 


562.76 
563.87 


_.. 59 


195-01 
195-66 


236.10 
236.82 


28l . 12 
281.90 


330-04 
330 89 


382.90 
383 82 


439.67 
440.65 


500.37 
501 41 


%3 



240 



PRACTICAL ASTRONOMY 



Lat. 



OBSERVATION ON SUN FOR AZIMUTH 

* 

of line from to 
Long Date 



192 



Object 


Horizontal circle 


Vertical circle 


Watch 


Mark 






M. 









h m s 











(ft) 

















Mark 




Mean 


Mean 


Mean reading 
on Sun 

Hor. angle, 
Mark to Sun 




LC. 

refr. 


* 

sf> 






h 


G. C. T. 



nat sin 5 
log sin <j> 
log sin h 
sin <t> sin h 

sin 8 sin <f> sin h 

log numerator 
log sec <j> 
log sec h 
log cos Z 

7 

Hor. angle 
Azimuth of line 



logs. 



Decl. at o G. C. T. 



corr., 





COS Z 



sin 8 sin $ sin h 

- - - ^r - 

cos ^ cos h 



TABLES 



241 



OBSERVATION ON SUN FOR AZIMUTH 
of Line from to 



Date, 192 

P.M. 


Object 


Hor. Circle 


Vert. Circle 


Watch 


">Mark 


/ 


/ 


h m s 




















^l ^ 








1 










Mark 




Mean 


Mean 


Mean, I. C. 
Hor. Ang. Refr. 5& 
Mk. to 



Alt. 



G. C. T. 



Lat log sec . 

Alt log sec . 

Nat. Cos Sum 

Nat. Sin Decl 



Sum. 



log. 



log vers 7j S . 

Z, = 

Hor. Ang. = 



Azimuth to. 



242 



PRACTICAL ASTRONOMY 



GREEK ALPHABET 



Letters. 
A, a, 



A,*, 



5,?, 



*,*, 
A,X, 



Name. 

Alpha 

Beta 

Gamma 

Delta 

Epsilon 

Zeta 

Eta 

Theta 

Iota 

Kappa 

Lambda 

Mu 



Letters. 



2, o-, 5, 



X 



Name. 

Nu 

Xi 

Omicron 

Pi 

Rho 

Sigma 

Tau 

Upsilon 

Phi 

Chi 

Psi 

Omega 



ABBREVIATIONS USED IN THIS BOOK 243 



ABBREVIATIONS USED IN THIS BOOK 

T or V = vernal equinox 
R. A. or a = right ascension 
d = declination 
p = polar distance 
h = altitude 
f = zenith distance 
Az. or Z = azimuth 

/ = hour angle 
= latitude 
X = longitude 
Sid. or S = sidereal time 

Eq. T. = equation of time 
G. C. T. = Greenwich Civil Time 
U. C. = upper culmination 
L. C. = lower culmination 
/. C. = index correction 
ref . or r = refraction 
par. or p = parallax 
s. d. or s = semidiameter 



APPENDIX A 

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 rnoon were 
at rest the surface of the water beneath the moon would be 

244 



THE TIDES 



245 



elevated as shown* in Fig. 81 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 

*% and JB'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 



FlG. 81 

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 
thaii 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. 



246 



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 orlast quai 
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 Decimation. 

When the moon is on the equator the two successive high 
tides are of nearly the same height. When the moon is north 




FIG. 82 

or south of the equator the two differ in height, as is shown in 
Fig. 82. 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 aooeee. 



THE TIDES 247 

Priming and L'agging 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. 83, 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 quartets; at the 
>oints 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 (JP) 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 difier 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. 

Observatio'n 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 



248 



PRACTICAL ASTRONOMY 




THE TIDES 249 

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 macrk 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 
; en 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 wayes 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 



250 



PRACTICAL ASTRONOMY 




FIG. 84 



Fig. 84. If there are waves on the surface of J;he 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. 85), 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. 




FIG. 85 



Such a gauge may be read from a considerable distance by 
means of a transit telescope or field glasses. 



* Tubes of this sort are manufactured for use in water gauges of steam boilers. 



THE TIDES 251 

Location of Gauge. 

The spot chosen for setting up the gauge should be near the 
(pen 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, until a 
little while after the maximum or minimum is reached. The 
Height of the water surface sometimes fluctuates at the time 

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. 86a and 86b.) 
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. 87 it will be seen that the mean of a and 6, 



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 




HIGH WATER 

MACHIAS BAY, ME. 

JUNE 8, 1905. 



14.S "fete 



Eastern Time 
FIG. 80a 

Eastern Time 



LOW WATER 

MACHIAS BAY, ME. 

JUNE 10, 1905. 




1.80 



FIG. 86b 



one case too small and in the other case too great. The propei 
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 253 

and Geodetic Survey. By examining the predicted heights the 
exact relation may be found between mean sea level and the 
jnean 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, 6, c, d, e, and /is 0.2 ft. 




FIG. 87 

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 



254 PRACTICAL ASTRONOMY 

annually by the United States Coast ancl 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. 



APPENDIX B 
SPHERICAL TRIGONOMETRY 

The formulae derived in the following pages are those most 
frequently used in engineering field practice and in navigation. 
Many of the usual formulae of spherical trigonometry are pur- 
posely omitted. It is not intended that this appendix shall 
serve as a general text book on spherical trigonometry, but merely 
that it should supplement that part of the preceding text which 
deals with spherical astronomy. 

A spherical triangle is a triangle formed by arcs of great 
circles. If from the vertices of the triangle straight lines are 
drawn to the centre of the sphere there is formed at this point a 
triedral angle (solid angle) the three face angles of which are 
measured by the corresponding sides of the spherical triangle, 
and the three diedral angles (edge angles) of which are equal to 
the corresponding spherical angles. For any triedral angle a 
spherical triangle may be formed, by assuming that the center 
of the sphere is at the vertex of the angle, and assigning any 
arbitrary value to the radius. The three faces (planes) cut 
>out arcs of great circles which form the sides of the triangle. 
The solution of the spherical triangle is really at the same time 
the solution of the solid angle since the six parts of one equal 
the six corresponding parts of the other. Any three lines pass- 
ing through a common point define a triedral angle. For example, 
the earth's axis of rotation, the plumb line at any place on the 
surface of the earth, and a line in the direction of the sun's 
centre, may be conceived to intersect at the earth's centre. 
The relation among the three face angles and the three edge 
angles of this triedral angle may be calculated by the formulae 

255 



256 



PRACTICAL ASTRONOMY 



of spherical trigonometry. The sphere employed, however, is 
merely an imaginary one. 

The fundamental formulae mentioned on p. 32 may be derived 
by applying the principles of analytic geometry to the 1 spherical 
triangle. In Fig. 88 the radius of the sphere is assumed to be 
unity. If a perpendicular CP be dropped from C to the XY 
plane, and a line CP' be drawn from C perpendicular to OX 




FIG. 88 



then x and y may be expressed in terms of the parts of the 
spherical triangle as follows: 

#= cos a 

y = sin a cos B 

z = sin a sin B. 



APPENDIX B 



257 



P' 



If we change to a new axis OX', CM being drawn perpendicular 
to OX', then we have (/ being negative in this figure) 

x f = cos b 

y' = sin b cos A 

z' = sin b sin A . 

From Fig. 89, the formulae for trans- 
formation are 

x x' cos c y f sin c 
y = x' sin c + y' cos c 




z = z. 



FIG. 89 



By substitution, 

cos a = cos 6 cos c + sin b sin c cos A (i) 

sin a cos B = cos 6 sin c sin & cos c cos ,4 (2) 

sin asm B = sin & sin A (3) 

Corresponding formulae may be written for angles B and C. 

By employing the principle of the polar triangle, namely, that 
the angle of a triangle and the opposite side of its polar triangle 
are supplements, we may write three sets of formulae like (i), 
(2) and (3) in which each small letter is replaced by a large letter 
and each large letter replaced by a small letter. For example, the 
first two equations would be 

cos A = cos B cos C + sin B sin C cos a (a) 

sin A cos b = cos B sin C sin B cos C cos a. (b) 

^There will also be two other sets of equations for the angles B 
and C. 

Equation (i) may be regarded as the fundamental formula 
of spherical trigonometry because all of the others may be de- 
rived from it. All problems may be solved by means of (i), 
although not always so conveniently as with other special forms. 

Solving for A, Equa. (i) may be written 

cos a cos b cos c 



cos A 



sin 6 sin c 



(la) 



258 PRACTICAL ASTRONOMY 

in which form it may be used to find any angle A when the three 
sides are known. See Equa. [20] and [2oa],*and [25] and [250], 
pp. 34 and 35. If each side of the equation is subtracted from 
unity we have 

cos b cos c + sin b sin c cos a 



cos A = 



sin b sin c 



A cos (b c) cos a , , 

or vers A = - ^ -. - (2) 

sin b sin. c 



or 2 sn 



2 sn sn 

^4 

Dividing (3) by 2 and denoting sin 2 by " hav." the " haver- 

2 

sine," or half versed-sine, we may write 

hav. A = sin * ( g + b ~ C ^ sin * (a "" b + C ^ (4) 

sin 6 sin c 

From (2) we may derive [21] and [26] by substituting A = /, 
J = 90 5, c = 90 0, and a = 90 h\ or A = Z, b 90 
A, c = 90 </>, and a = 90 d. 
By putting 5' = | (a + 6 + c), (3) becomes 

.4 4 / sin (s f 6) sin (/ c) , , 

sm- = W - ^ - ' . - (5) 

2 v sin b sm c 

If we add each member of (ia) to unity we may derive by a 
similar process 

A 4 /sin s f sin ($' a) ^ 

cos- = U - r-rA - - (6) 

2 v sin b sin c 

Dividing (5) by (6) we have 



sm s f sm (/ - a ) 

Formulae [17], [22], [18], [23], [19] and [24] may be derived 
from (5), (6), (7) respectively or, more readily, from the inter- 
mediate forms, like (3), by putting s = | (<t> + h + p) and 



APPENDIX B 259 

A = i or Z. For example, if in (3) we put A = /, a = 90 A, 

b = 90 <, and = p, then 

-2! = ?in^ (go 6 - /? + 90 - <fr - ft) sin | (90 - /?- 90 
2 ~~ cos sin ft 



__ sin | (180 - Q + A + ft) sin H<fr - ^ + P) 
cos < sin ft 

Us = %(<!> + k + p) then 

2! = cos 5 sin (s /Q 
2 cos <t> sin 

from which we have [17]. 
Formula (4) or [17] may be written 

hav . , = cos * sin (s - h) 
cos </> sin p 

the usual form (in navigation) for the calculation of the hour 
angle from an observed altitude of an object. 

For the purpose of calculating the great-circle distance be- 
tween two points on the earth's surface formula (i) may be put 
in the form 

hav. (dist.) = hav. (j> A <fo) + cos <f> A cos <fo hav. AX (9) 

in which 0^, <f> B are the latitudes of two points on the earth's 
surface and AX their difference in longitude. <t> A ~ <t> B means the 
difference between the latitudes if they are both N or both S; 
the sum if they are in opposite hemispheres. 
Formula (9) is derived by substituting the co-latitudes for 

b and c and AX for A in (i) which gives 


cos (dist.) = cos (90 <f> A ) cos (90 <fo) 

+ sin (90 $ A ) sin (90 <t> B ) cos AX. 

If we add and subtract cos $ A cos <t> B in the right-hand member 
we obtain, 



260 PRACTICAL ASTRONOMY 

cos (dist.) = sin $ A sin <t> s + cos <f> A cos fa cos <f> A cos #5 

+ cos 4> A cos <fo cos AX 
= cos (<f> A ~ <B) cos #4 cos < 5 (i cos AX) 
= cos (04 0#) cos 04 cos 0B vers AX. 

Subtracting both members from unity 

i cos (dist.) == i cos (<t> A 5 ) + cos <$>A cos B vers AX 
or vers (dist.) = vers ($ A ~ <t> B ) + cos <t> A cos <t> B vers AX. 

Dividing by 2 

hav. (dist.) = hav. ($4 0#) + cos ^ cos <j> B hav. AX. (9) 

Note: A table of natural and logarithmic haversines may be 
found in Bowditch, American Practical Navigator. (Table 45.) 

The same formula may be applied to the calculation of the 
zenith distance of an object. In this case it is written 

Hav. f = hav. (< 8) + cos < cos <t> hav. t. ' (10) 

This is the formula usually employed in the method of Marcq 
Saint Hilaire. 

Right Triangles 

By writing formulae (i), (2), (3), (a) and (&) in terms of the 
three parts and placing C - 90, we may obtain the following 
ten right triangle formulae. 

cos c = cos a cos b 

. A sin a . D sini 

sin A = -: sin B = -; 

sin c sin c 

A tan b D tan a 

cos A = - cos B = - 

tan c tan c 

tan a . tan b (n) 

tan A = -sr tan B = -r-r- 

sin sin k a 

. cos B . cos ^4 

sin A = - r- sm B 



cos 6 cos 

cos c = cot A cot 5. 



APPENDIX B 26l 

These are readily remembered from their similarity to the 
corresponding formulae of plane triangles. 

To solve a right triangle select the three formulae which involve 
the two given parts and one of the three parts to be found. To 
check the results select the formula involving the three parts 
just computed. The computed values should satisfy this 
equation. 

Radians Degrees, Minutes, and Seconds. 

If the length of an arc is divided by the radius it expresses 
the central angle in radians. The number obtained is the cor- 
responding length of arc on a circle whose radius is unity. The 
unit of measurement of angles in this system is the radius of 
the circle, that is, an angle of i is an angle whose arc equals 
the radius, and therefore contains about 57-3. 

Since the ratio of the semi-circumference to the radius is ?r, 
there are TT radians in 180 of the circumference.* The conversion 
of angles from degrees into radians (or -w measure, or arc-measure) 
is effected by multiplying by the ratio of these two. 

no 

Angle in degrees = angle in radians X 

7T 

and ' angle in radians = angle in degrees X - 

i So 

To convert an angle in radians into minutes multiply by 

7T 

11 i ( T 

- 3437'-77; or divided by = .0002909. This latter 

IoO X OO 

number, the arc i', is nearly equal to sin i' or tan i'. 
To convert an angle expressed in radians into seconds multiply 

by = 206264.8; or, divide by the reciprocal, 

.00000,48481,36811, the arc i"; this number is identical with 
sin i" or tan i" for 16 decimal places. 



262 PRACTICAL ASTRONOMY 

Area of a Spherical Triangle. 

In text books on geometry it is shown tfiat " the area of a 
spherical triangle equals its spherical excess times the area of 
the tri-rectangular triangle/' the right angle being the unit of 
angles. If A represents the area of any spherical triangle, whose 
angles are A, B, and C, then 

A + B + C ~ 180 4 
X 



or A = 



9 o ' x 8 

(A + B + C - 1 80) 



180' 



180 
, if e is the spherical excess in degrees 



_ e"irR 2 if e" is the spherical excess in sec- 

"" 180 X 60' X 60" ? onds. (12) 

Spherical Excess. 

If Equa. (12) is solved for e" we have 

_ A 1 80 X 60 X 60 
e __ x - 

A v> , , the constant being the number of sec- 
= - X 206264.8, , . & .. 

R~ onds in one radian 

A , x 



R* arc i" 

Note: Arc i" = .00000,48481,36811; it is the reciprocal of 
the above constant, and is the length of the arc 'which subtends 
an angle of i" when the radius is unity. 

Solid Angles. 

Any solid angle* may be measured by an area on the surface 
of the sphere in the same manner that plane angles are measured 
by arcs on the circumference of a circle. The extent of the 
opening between the planes of a triedral angle is proportional to 

* A solid angle is one formed by the intersection of any number of planes in a 
common point. The triedral angle is a special case of the solid angle. 



APPENDIX B 263 

the area ot the corresponding spherical triangle, or in other words 
proportional to the spherical excess of that triangle. This is 
true not only of triangles, but also of spherical polygons and 
spherical* areas formed by circles (sectors). The unit of 
measurement of the solid angle is the steradian. A unit (plane) 
angle is one whose arc is equal in length to the radius of the circle; 
that is, it intercepts an arc whose length is R. The steradian is a 
solid angle which intercepts on the surface of the sphere an area 
equal to Jf? 2 ; or it intercepts on the sphere of unit radius an area 
equal to unity. Just as the plane angle (in radians) when mul- 
tiplied by the radius gives the length of arc, so the solid angle or 
the spherical excess (in radians) when multiplied by R 2 gives 
the area of the spherical triangle. To obtain a more definite 
idea of the size of this angle we may compute the length of arc 
from the centre to the circumference of a small circle having an 
area equal to R 2 (or i on a sphere of unit radius). This comes 
out about 32 46' + . The spherical area enclosed by the paral- 
lel of latitude 57 14' corresponds to one steradian in the angle 
of the cone whose apex is at the centre of the globe. 

Functions of Angles near o or 90. 

When obtaining from tables the values of sines or tangents of 
small angles (or angles near to 180) and cosines or cotangents of 
angles near to 90 there is some difficulty encountered in the 
interpolation, on account of the rapid rate of change of the 
logarithms. In practice these values are often found by ap- 
proximate methods which enable us to avoid the use of second 
differences in the interpolation. There are two assumptions 
which may be made, which result in two methods of obtaining 
the log functions. 

For sines/ we may assume that 

sin x = x" X sin i" 
or log sin x = log x" + log sin i". 

The log sin i" = 4.685 5749. 



264 PRACTICAL 

This method is accurate for very small angles; the limiting 
value of the angle for which the sine may be so computed 
depends entirely upon the accuracy demanded, that is, upon the 
number of places required. 

Example. Find the value of log sin o 10' 25" from a 5-place table. 

log sin i" = 4-68557 
log 625" = 2.79588 

log sin x = 7.48145 
The result is correct to five figures. 

A similar assumption may be made for tangents of small 
angles or for cosines and cotangents of angles near 90. 

For angles slightly larger than those for which the preceding 
method would be employed, we may assume that 

sin (A + a") sin A 



A + a" 



sin A 



The ratio of changes slowly and is therefore very nearly 

A. 

the same for both members of the -equation. We may there- 
fore compute the log sin (A + a") by the equation, 

log sin (A + a!'} = log (A" + a") + log 8 -^ 



in which A 11 signifies that the angle must be reduced to seconds. 
The latter logarithm is given in many tables in the margin 
of the page. It may be computed for any number which is 
stated in the table. This method is more accurate than the 
former. 

Example. 

Find the value of log sin 2 01' 30" in a five-place table. 

2 01' 30" = 7290". In the marginal table is found opposite " S " the log- 
arithm 4.68548 which is the difference between log sin A and log A. If this 
is not given it may be computed by taking from the table the nearest log sin, 
say log sin 2 01' = 8.54642 and subtracting from it the log of 7260" 3.86004* 
The result is 4.68548. 



APPENDIX B 265 

Then 

log ^- 4.68548 
log 7290" - 3-86273 
log sin x ~ 8.54821 
This result is correct to five figures. 



INDEX 



Aberration of light, 1 2 
Adjustment of transit, 92, 98, 130 
Almucantar^ 15 
Altitude, 19 

of pole, 27 

Angle of the vertical, 80 
Annual aberration, 13 
Aphelion, 9 
Apparent motion, 3, 28 

time, 42 
Arctic circle, 30 
Aries, first point of, 16, 113 
Astronomical time, 46 

latitude, 79 

transit, 96, 130 

triangle, 31, 134 
Atlantic time, 50 
Attachments to transit, 95 
Autumnal equinox, 16 
Axis, 3, 8 
Azimuth, 19, 161 

mark, 161 

tables, 77, 149, 214, 223 



B 



yBearings, 19 
Besselian year, 68 



Calendar, 62 

Celestial latitude and longitude, 22 

sphere, i 
Central time, 50 
I Charts, 223 

Chronograph, 105, 130, 152, 155 
Chronometer, 58, 104, 108, 130, 155 

correction, 127 

sight, 213 



267 



Circum-meridian altitude, 123, 212 

Circumpolar star, 29, 115, 181 

Civil time, 46 

Clarke spheroid, 79 

Coast and Geodetic Survey, 99, 121, 

171, 185, 247 
Co-latitude, 22 

Comparison of chronometer, 104 
Constant of aberration, 13 
Constellations, 10, no 
Convergence of meridians, 206 
Cross hairs, 91, 96 
Culmination, 40, 115, 128 
Curvature, 134, 183, 184 

D 

Date line, 61 
Daylight saving time, 52 
Dead reckoning, 213 
Decimation, 20 

parallels of, 16 
Dip, 88 
Diurnal aberration, 13, 185 

inequality, 246 



Eastern time, 50 

Ebb tide, 244 

Ecliptic, 16, 112, 114 

Ellipsoid, 79 

Elongation, 36, 162 

Ephemeris, 65 

Equal altitude method, 143, 195, 196 

Equation of time, 42 

Equator, 15 

systems, 19 
Equinoxes, 9, 16, 41 
Errors in horizontal angle, 108 

in spherical triangle, 139, 178 



268 



INDEX 



Errors in transit observations, 98, 130, 

132 

Eye and ear method, 108 
Eyepiece, prismatic, 96, 132 



Figure of the earth, 79 
Fixed stars, 2, 4, 68 
Flood tide, 244 
Focus, 116 



Geocentric latitude, 80 
Geodetic latitude, 80 
Gravity, 79, 91 
Gravitation, 7 

Greenwich, 23, 45, 50, 61, 65 
Gyroscope, 12 

H 

Harrebow-Talcott method, 73, *O5 
Hayford spheroid, 79 
Hemisphere, 9 
Horizon, 14 

artificial, 103 

glass, loo 

system, 19, 92 
Hour angle, 20, 38 

circle, 16 
Hydrographic office, 149, 214, 223 



Illumination, 95, 107 

Index error, 93, 102, 107, ir6 r 2i2 

Interpolation, 73 



Lagging, 247 
Latitude, 22, 27, 115 

astronomical, geocentric and geo- 
detic, 80 

at sea, 211 

reduction of, 81 
Leap-year, 63 
Level correction, 126, 184 
Local time, 47 



Longitude, 22, 66, 154 
at sea, 213 

M 

Magnitudes, in 
Marcq St. Hilaire, 222 
Mean sun, 42, 57 

time, 42 
Meridian, 16 
Micrometer, 96, 105, 182 
Midnight sun, 30 
Moon, apparent motion of, 5 

culminations, 66, 157 
Motion, apparent, 3, 28 
Mountain time, 50 

N 

Nadir, 14 
Nautical almanac, 43, 66 

mile, 218 

Naval observatory, 151 
Neap tide, 246 
Nutation, 10 

O 

Object glass, 91, 95, 96 
Obliquity of ecliptic, 8, n, 16 
Observations, 65 
Observer, coordinates of, 22 
Observing, 107 
Orbit, 3 
of earth, 7 



Pacific time, 50 
Parabola, 75 

Parallactic angle, 32, 149 
Parallax, 81, 158 

horizontal, 83 
Parallel of altitude, 15 

of declination, 16 

sphere, 29 
Perihelion, 9 

Phases of the moon, 159, 246 
Planets, 3, 114, 151, 220 
Plumb-line, 14, 80 



INDEX 



269 



Pointers, 112 
\>lar distance, 20 

ole, 3, ii, 15 

r star, in, 162 
recession, 10, 113 

-recision, 35 
Prediction of tides, 253 
Primary circle, 18 

Prime vertical, 16, 137, 139, 179, 181 
Priming, 247 
"Prismatic eyepiece, 96, 132, 170 

/ 

R 

dian, 83, 261 
inge, 151 
of tide, 244 
late, 127 

'eduction to elongation, 166 
of latitude, 81 

to the meridian, 120, 121, 126 
lector, 95 

raction, 84, 137, 143, 180 
orrection, 84 
ffect on dip, 89 
5ect on semidiameters, 87 
idex of, 85 

vetrograde motion, 6 
light ascension, 20, 36 

sphere, 28 
^ocation, 3, 40 
.un of ship, 217, 219 






Sea-horizon, 211 
Reasons, 7 

Secondary circles, 18 
)emidiameter, 82, 87 

contraction of, 87 
Jextant, 100, 211 

'ereal day, 40 
ime, 41, 52, 57 

gns of the Zodiac, 112 
Mar day, 41 

system, 2 

time. AI. <M. S7 



Solid angle, 255, 262 

Solstice, 1 6 

Spherical coordinates, 18, 31 

excess, 262 
Spheroid, 10, 79 
Spirit level, 14 
Spring tides, 246 
Stadia hairs, 168 
Standard time, 50 
Standards of transit, 91 
Star, catalogues, 68, 106, 125 

fixed, 4 

list, 45, 107, 130, 131, 145 

nearest, 2 
Steradian, 263 

Striding level, 95, 98, 128, 182 
Sub-solar point, 215 
Summer, 9 

Sumner's method, 215 
Sumner line, 216, 223 
Sun, altitude of, 117, 134, 167, 211, 213 

apparent motion of, 5 

dial, 42 

fictitious, 42 

glass, 96 



Talcott's method, 73, 105, 125 
Telegraph method, 155 

signals, 151 
Tide gauge, 249 

tables, 247, 253 
Tides, 244 
Time, service, 151 

sight, 213 
Transit, astronomical, 96 

engineer's, 91, 95 

time of, 40 

Transportation of timepiece, 154 
Tropical year, 55 

V 

Vernal equinox, 16 
Vernier of sextant, 100 
of transit. OT 



INDEX 



Vertical circle, 14, 140 

line, 14 
Visible horizon, 14 

W 

Washington, 45, 66 

Watch correction, 129, 151 

Winter, 8 

Wireless telegraph signals, 151, 156 



Year, 55, 68 



Zenith, 14 

distance, 19 

telescope, 105, 125 
Zodiac, 112